  The non-minimal coupling of a scalar field is considered in the framework of Ashtekar's new variables formulation of gravity. A first order action functional for this system is derived in which the field variables are a tetrad field, and an SL(2,C) connection, together with the scalar field. The tetrad field and the SL(2,C) connection are related to the Ashtekar variables for the vacuum case by a conformal transformation. A canonical analysis shows that for this coupling the equations of Ashtekar's formulation of canonical gravity are non-polynomial in the scalar field. (to be published in Phys. Rev. D) 
  In the derivation of a pure spin connection action functional for gravity two methods have been proposed. The first starts from a first order lagrangian formulation, the second from a hamiltonian formulation. In this note we show that they lead to identical results for the specific cases of pure gravity with or without a cosmological constant. 
  Motivated by the apparent dependence of string $\sigma$--models on the sum of spacetime metric and antisymmetric tensor fields, we reconsider gravity theories constructed from a nonsymmetric metric. We first show that all such "geometrical" theories homogeneous in second derivatives violate standard physical requirements: ghost-freedom, absence of algebraic inconsistencies or continuity of degree-of-freedom content. This no-go result applies in particular to the old unified theory of Einstein and its recent avatars. However, we find that the addition of nonderivative, ``cosmological'' terms formally restores consistency by giving a mass to the antisymmetric tensor field, thereby transmuting it into a fifth-force-like massive vector but with novel possible matter couplings. The resulting macroscopic models also exhibit ``van der Waals''-type gravitational effects, and may provide useful phenomenological foils to general relativity. 
  The two lineal gravities --- based on the de Sitter group or a central extension of the Poincar\'e group in 1+1 dimensions --- are shown to derive classically from a unique topological gauge theory. This one is obtained after a dimensional reduction of a Chern--Simons model, which describes pure gravity in 2+1 dimensions, the gauge symmetry being given by an extension of ISO(2,1). 
  General relativity has previously been extended to incorporate degenerate metrics using Ashtekar's hamiltonian formulation of the theory. In this letter, we show that a natural alternative choice for the form of the hamiltonian constraints leads to a theory which agrees with GR for non-degenerate metrics, but differs in the degenerate sector from Ashtekar's original degenerate extension. The Poisson bracket algebra of the alternative constraints closes in the non-degenerate sector, with structure functions that involve the {\it inverse} of the spatial triad. Thus, the algebra does {\it not} close in the degenerate sector. We find that it must be supplemented by an infinite number ofsecondary constraints, which are shown to be first class (although their explicit form is not worked out in detail). All of the constraints taken together are implied by, but do not imply, Ashtekar's original form of constraints. Thus, the alternative constraints give rise to a different degenerate extension of GR. In the corresponding quantum theory, the single loop and intersecting loop holonomy states found in the connection representation satisfy {\it all} of the constraints. These states are therefore exact (formal) solutions to this alternative degenerate extension of quantum gravity, even though they are {\it not} solutions to the usual vector constraint. 
  It has recently been shown by Goldberg et al that the holonomy group of the chiral spin-connection is preserved under time evolution in vacuum general relativity. Here, the underlying reason for the time-independence of the holonomy group is traced to the self-duality of the curvature 2-form for an Einstein space. This observation reveals that the holonomy group is time-independent not only in vacuum, but also in the presence of a cosmological constant. It also shows that once matter is coupled to gravity, the "conservation of holonomy" is lost. When the fundamental group of space is non-trivial, the holonomy group need not be connected. For each homotopy class of loops, the holonomies comprise a coset of the full holonomy group modulo its connected component. These cosets are also time-independent. All possible holonomy groups that can arise are classified, and examples are given of connections with these holonomy groups. The classification of local and global solutions with given holonomy groups is discussed. 
  Damour, Deser and McCarthy have claimed that the nonsymmetric gravitational theory (NGT) is untenable due to curvature coupled ghost modes and bad asymptotic behavior. This claim is false for it is based on a physically inaccurate treatment of wave propagation on a curved background and an incorrect method for extracting asymptotic behavior. We show that the flux of gravitational radiation in NGT is finite in magnitude and positive in sign. 
  We consider the general procedure for proving no-hair theorems for static, spherically symmetric black holes. We apply this method to the abelian Higgs model and find a proof of the no-hair conjecture that circumvents the objections raised against the original proof due to Adler and Pearson. 
  We examine gravitational waves in an isolated axi--symmetric reflexion symmetric NGT system. The structure of the vacuum field equations is analyzed and the exact solutions for the field variables in the metric tensor are found in the form of expansions in powers of a radial coordinate. We find that in the NGT axially symmetric case the mass of the system remains constant only if the system is static (as it necessarily is in the case of spherical symmetry). If the system radiates, then the mass decreases monotonically and the energy flux associated with waves is positive. 
  Using the tetrad formalism, we carry out the separation of variables for the massive complex Dirac equation in the gravitational and electromagnetic field of a four-parameter (mass, angular momentum, electric and magnetic charges) black hole. 
  We have obtained the correct expression for the centrifugal force acting on a particle at the equatorial circumference of a rotating body in the locally non-rotating frame of the Kerr geometry. Using this expression for the equilibrium of an element on the surface of a slowly rotating Maclaurin spheroid, we obtain the expression for the ellipticity (as discussed earlier by Abramowicz and Miller) and determine the radius at which the ellipticity is maximum. 
  We consider the problem of extracting physical predictions from the wave function of the universe in quantum cosmological models. We state the features of quantum cosmology an interpretational scheme should confront. We discuss the Everett interpretation, and extensions of it, and their application to quantum cosmology. We review the steps that are normally taken in the process of extracting predictions from solutions to the Wheeler-DeWitt equation for quantum cosmological models. Some difficulties and their possible resolution are discussed. We conclude that the usual wave function-based approach admits at best a rather heuristic interpretation, although it may in the future be justified by appeal to the decoherent histories approach. 
  We explore the thermodynamics of a general class of two dimensional dilatonic black-holes. A simple prescription is given that allows us to compute the mass, entropy and thermodynamic potentials, with results in agreement with those obtained by other methods, when available. 
  The origin of seed perturbations in the Universe is studied within the framework of a specific minisuperspace model. It is shown that the `creation' of the Universe as a result of a quantum transition from a flat empty spacetime would lead to a flat FLRW (Friedmann Lema\^\i tre Robertson-Walker) Universe with weak inhomogeneous perturbations at large wavelengths. The power spectrum of these perturbations is found to be scale invariant at horizon crossing (i.e., the Harrison-Zeldovich spectrum). It is also recognised that the seed perturbations generated in our model would be generically of the isocurvature kind. 
  A method for taking the $D\to 2$ limit of D-dimensional general relativity is constructed, yielding a two-dimensional theory which couples gravitation to conserved stress-energy. We show how this theory is related to those obtained via an alternative dimensional reduction approach. 
  This paper derives and analyzes exact, nonlocal Langevin equations appropriate in a cosmological setting to describe the interaction of some collective degree of freedom with a surrounding ``environment.'' Formally, these equations are much more general, involving as they do a more or less arbitrary ``system,'' characterized by some time-dependent potential, which is coupled via a nonlinear, time-dependent interaction to a ``bath'' of oscillators with time-dependent frequencies. The analysis reveals that, even in a Markov limit, which can often be justified, the time dependences and nonlinearities can induce new and potentially significant effects, such as systematic and stochastic mass renormalizations and state-dependent ``memory'' functions, aside from the standard ``friction'' of a heuristic Langevin description. One specific example is discussed in detail, namely the case of an inflaton field, characterized by a Landau-Ginsburg potential, that is coupled quadratically to a bath of scalar ``radiation.'' The principal conclusion derived from this example is that nonlinearities and time-dependent couplings do {\em not} preclude the possibility of deriving a fluctuation-dissipation theorem, and do {\em not} change the form of the late-time steady state solution for the system, but {\em can} significantly shorten the time scale for the approach towards the steady state. 
  In this paper a quantum mechanical phase space picture is constructed for coarse-grained free quantum fields in an inflationary Universe. The appropriate stochastic quantum Liouville equation is derived. Explicit solutions for the phase space quantum distribution function are found for the cases of power law and exponential expansions. The expectation values of dynamical variables with respect to these solutions are compared to the corresponding cutoff regularized field theoretic results (we do not restrict ourselves only to $\VEV{\F^2}$). Fair agreement is found provided the coarse-graining scale is kept within certain limits. By focusing on the full phase space distribution function rather than a reduced distribution it is shown that the thermodynamic interpretation of the stochastic formalism faces several difficulties (e.g., there is no fluctuation-dissipation theorem). The coarse-graining does not guarantee an automatic classical limit as quantum correlations turn out to be crucial in order to get results consistent with standard quantum field theory. Therefore, the method does {\em not} by itself constitute an explanation of the quantum to classical transition in the early Universe. In particular, we argue that the stochastic equations do not lead to decoherence. 
  Gravitational instantons, solutions to the euclidean Einstein equations, with topology $R^3 XS^1$ arise naturally in any discussion of finite temperature quantum gravity. This Letter shows that all such instantons (irrespective of their interior behaviour) must have the same asymptotic structure as the Schwarzschild instanton. Using this, it can be shown that if the Ricci tensor of the manifold is non-negative it must be flat. One special case is when the Ricci tensor vanishes; hence one can conclude that there is no nontrivial vacuum gravitational instanton. This result has uses both in quantum and classical gravity. It places a significant restriction on the instabilities of hot flat space. It also can be used to show that any static vacuum Lorentzian Kaluza-Klein solution is flat. 
  This article gives necessary and sufficient conditions for the formation of trapped surfaces in spherically symmetric initial data defined on a closed manifold. Such trapped surfaces surround a region in which there occurs an enhancement of matter over the average. The conditions are posed directly in terms of physical variables and show that what one needs is a relatively large amount of excess matter confined to a small volume. The expansion of the universe and an outward flow of matter oppose the formation of trapped surfaces; an inward flow of matter helps. The model can be regarded as a Friedmann-Lema\^\i tre-Walker cosmology with localized spherical inhomogeneities. We show that the total excess mass cannot be too large. 
  We derive a formula for the nonequilibrium entropy of a classical stochastic field in terms of correlation functions of this field. The formalism is then applied to define the entropy of gravitational perturbations (both gravitational waves and density fluctuations). We calculate this entropy in a specific cosmological model (the inflationary Universe) and find that on scales of interest in cosmology the entropy in both density perturbations and gravitational waves exceeds the entropy of statistical fluctuations of the microwave background. The nonequilibrium entropy discussed here is a measure of loss of information about the system. We discuss the origin of the entropy in our cosmological models and compare the definition of entropy in terms of correlation functions with the microcanonical definition in quantum statistical mechanics. 
  The action of Ashtekar gravity have been found by Cappovilla, Jacobson and Dell. It does not depend on the metric nor the signature of the space-time. The action has a similar structure as that of a massless relativistic particle. The former is naturally generalized by adding a term analogous to a mass term of the relativistic particle. The new action possesses a constant parameter regarded as a kind of a cosmological constant. It is interesting to find a covariant Einstein equation from the action. In order to do it we will examine how the geometrical quantities are determined from the non-metric action and how the Einstein equation follows from it. 
  A consequence of non-Gaussian perturbations on the Sachs-Wolfe effect is studied. For a particular power spectrum, predicted Sachs-Wolfe effects are calculated for two cases: Gaussian (random phase) configuration, and a specific kind of non-Gaussian configuration. We obtain a result that the Sachs-Wolfe effect for the latter case is smaller when each temperature fluctuation is properly normalized with respect to the corresponding mass fluctuation ${\delta M\over M}(R)$. The physical explanation and the generality of the result are discussed. 
  The local Lorentz and diffeomorphism symmetries of Einstein's gravitational theory are spontaneously broken by a Higgs mechanism by invoking a phase transition in the early Universe, at a critical temperature $T_c$ below which the symmetry is restored. The spontaneous breakdown of the vacuum state generates an external time and the wave function of the Universe satisfies a time dependent Schrodinger equation, which reduces to the Wheeler-deWitt equation in the classical regime for $T < T_c$, allowing a semi-classical WKB approximation to the wave function. The conservation of energy is spontaneously violated for $T > T_c$ and matter is created fractions of seconds after the big bang, generating the matter in the Universe. The time direction of the vacuum expectation value of the scalar Higgs field generates a time asymmetry, which defines the cosmological arrow of time and the direction of increasing entropy as the Lorentz symmetry is restored at low temperatures. 
  A family of solutions to low energy string theory is found. These solutions represent waves traveling along "extremal black strings" 
  Canonical transformations relating the variables of the ADM-, Ashtekar's and Witten's formulations of gravity are computed in 2+1~dimensions. Three different forms of the BRST-charge are given in the 2+1 dimensional Ashtekar formalism, two of them using Ashtekar's form of the constraints and one of them using the forms suggested by Witten. The BRST-charges are of different rank. 
  The Cosmological Principle states that the universe is both homogeneous and isotropic. This, alone, is not enough to specify the global geometry of the spacetime. If we were able to measure both the Hubble constant and the energy density we could determine whether the universe is open or closed. Unfortunately, while some agreement exists on the value of the Hubble constant, the question of the energy density seems quite intractable. This Letter describes a possible way of avoiding this difficulty and shows that if one could measure the rate at which light-rays emerging from a surface expand, one might well be able to deduce whether the universe is closed. 
  We construct the propagator of a non-relativistic non-interacting particle in a flat spacetime in which two regions have been identified. This corresponds to the simplest "time machine". We show that while completeness is lost in the vicinity of the time machine it holds before the time machine appears and it is recovered afterwards. Unitarity, however, is not satisfied anywhere. We discuss the implications of these results and their relationship to the loss of unitarity in black hole evaporation. 
  We present new evidence in support of the Penrose's strong cosmic censorship conjecture in the class of Gowdy spacetimes with $T^3$ spatial topology. Solving Einstein's equations perturbatively to all orders we show that asymptotically close to the boundary of the maximal Cauchy development the dominant term in the expansion gives rise to curvature singularity for almost all initial data. The dominant term, which we call the ``geodesic loop solution'', is a solution of the Einstein's equations with all space derivatives dropped. We also describe the extent to which our perturbative results can be rigorously justified. 
  We present a class of exact solutions to the constraint equations of General Relativity coupled to a Klein - Gordon field, these solutions being isotropic but not homogeneous. We analyze the subsequent evolution of the consistent Cauchy data represented by those solutions, showing that only certain special initial conditions eventually lead to successfull Inflationary cosmologies. We argue, however, that these initial conditions are precisely the likely outcomes of quantum events occurred before the inflationary era. 
  It is shown that for some particular value of the cosmological constant depending on the gauge coupling constant a continuous one-parameter family of Einstein-Yang-Mills wormholes exists which interpolates between the instanton and the gravitating meron solutions. In contradistinction with the previously known solutions the topological charge of these wormholes is not quantized. For all of them the contribution of gravity to the action exactly cancels that of the gauge field. 
  After reviewing the context in which Euclidean propagation is useful we compare and contrast Euclidean and Lorentzian Maxwell-Einstein theory and give some examples of Euclidean solutions. 
  Here I examine how to determine the sensitivity of the LIGO, VIRGO, and LAGOS gravitational wave detectors to sources of gravitational radiation by considering the process by which data are analyzed in a noisy detector. By constructing the probability that the detector output is consistent with the presence of a signal, I show how to (1) quantify the uncertainty that the output contains a signal and is not simply noise, and (2) construct the probability distribution that the signal parameterization has a certain value.   From the distribution and its mode I determine volumes $V(P)$ in parameter space such that actual signal parameters are in $V(P)$ with probability $P$. If we are {\em designing} a detector, or determining the suitability of an existing detector for observing a new source, then we don't have detector output to analyze but are interested in the ``most likely'' response of the detector to a signal. I exploit the techniques just described to determine the ``most likely'' volumes $V(P)$ for detector output corresponding to the source. Finally, as an example, I apply these techniques to anticipate the sensitivity of the LIGO and LAGOS detectors to the gravitational radiation from a perturbed Kerr black hole. 
  A choice of time-slicing in classical general relativity permits the construction of time-dependent wave functions in the ``frozen time'' Chern-Simons formulation of $(2+1)$-dimensional quantum gravity. Because of operator ordering ambiguities, however, these wave functions are not unique. It is shown that when space has the topology of a torus, suitable operator orderings give rise to wave functions that transform under the modular group as automorphic functions of arbitrary weights, with dynamics determined by the corresponding Maass Laplacians on moduli space. 
  The quasilocal energy of gravitational and matter fields in a spatially bounded region is obtained by employing a Hamilton-Jacobi analysis of the action functional. First, a surface stress-energy-momentum tensor is defined by the functional derivative of the action with respect to the three-metric on ${}^3B$, the history of the system's boundary. Energy density, momentum density, and spatial stress are defined by projecting the surface stress tensor normally and tangentially to a family of spacelike two-surfaces that foliate ${}^3B$. The integral of the energy density over such a two-surface $B$ is the quasilocal energy associated with a spacelike three-surface $\Sigma$ whose intersection with ${}^3B$ is the boundary $B$. The resulting expression for quasilocal energy is given in terms of the total mean curvature of the spatial boundary $B$ as a surface embedded in $\Sigma$. The quasilocal energy is also the value of the Hamiltonian that generates unit magnitude proper time translations on ${}^3B$ in the direction orthogonal to $B$. Conserved charges such as angular momentum are defined using the surface stress tensor and Killing vector fields on ${}^3B$. For spacetimes that are asymptotically flat in spacelike directions, the quasilocal energy and angular momentum defined here agree with the results of Arnowitt-Deser-Misner in the limit that the boundary tends to spatial infinity. For spherically symmetric spacetimes, it is shown that the quasilocal energy has the correct Newtonian limit, and includes a negative contribution due to gravitational binding. 
  We have shown that two of the most studied models of lineal gravities - Liouville gravity and a ``string-inspired'' model exhibiting the main characteristic features of a black-hole solution - can be formulated as gauge invariant theories of the Poincar\'e group. The gauge invariant couplings to matter (particles, scalar and spinor fields) and explicit solutions for some matter field configurations, are provided. It is shown that both the models, as well as the couplings to matter, can be obtained as suitable dimensional reductions of a 2+1-dimensional ISO(2,1) gauge invariant theory. 
  The gravitational field in a spatially finite region is described as a microcanonical system. The density of states $\nu$ is expressed formally as a functional integral over Lorentzian metrics and is a functional of the geometrical boundary data that are fixed in the corresponding action. These boundary data are the thermodynamical extensive variables, including the energy and angular momentum of the system. When the boundary data are chosen such that the system is described semiclassically by {\it any} real stationary axisymmetric black hole, then in this same approximation $\ln\nu$ is shown to equal 1/4 the area of the black hole event horizon. The canonical and grand canonical partition functions are obtained by integral transforms of $\nu$ that lead to "imaginary time" functional integrals. A general form of the first law of thermodynamics for stationary black holes is derived. For the simpler case of nonrelativistic mechanics, the density of states is expressed as a real-time functional integral and then used to deduce Feynman's imaginary-time functional integral for the canonical partition function. 
  We consider the quantum gravity and cosmology of a Jordan-Brans-Dicke theory, predicted by string effective actions. We study its canonical formalism and find that the constraint algebra is that of general relativity, as a consequence of the general covariance of scalar-tensor theories. We also analyze the problem of boundary conditions and propose that they must be imposed in the Jordan frame, in which particles satisfy the strong equivalence principle. Specifically, we discuss both Hartle-Hawking and wormhole boundary conditions in the context of quantum cosmology. We find quantum wormhole solutions for Jordan-Brans-Dicke gravity even in the absence of matter. Wormholes may affect the constants of nature and, in particular, the Brans-Dicke parameter. Following Coleman's mechanism, we find a probability distribution which is strongly peaked at zero cosmological constant and infinite Brans-Dicke parameter. That is, we recover general relativity as the effective low energy theory of gravity. 
  I review the equivalence between duality operators on two-forms and conformal structures in four dimensions, from a Clifford algebra point of view (due to Urbantke and Harnett). I also review an application, which leads to a set of "neighbours" of Einstein's equations. An attempt to formulate reality conditions for the "neighbours" is discussed. 
  Bounds to the Nordtvedt parameter are obtained from the motion of the first twelve Trojan asteroids in the period 1906-1990. From the analysis performed, we derive a value for the inverse of the Saturn mass 3497.80 \pm 0.81 and the Nordtvedt parameter -0.56 \pm 0.48, from a simultaneous solution for all asteroids. 
  The wave function of the universe is evaluated by using the Euclidean path integral approach. As is well known, the real Euclidean path integral diverges because the Einstein-Hilbert action is not positive definite. In order to obtain a finite wave function, we propose a new regularization method and calculate the wave function of the Friedmann- Robertson-Walker type minisuperspace model. We then consider a homogeneous but anisotropic type minisuperspace model, which is known as the Bianch type I model. The physical meaning of the wave function by this new regularization method is also examined. 
  Feynman's sum-over-histories formulation of quantum mechanics is reviewed as an independent statement of quantum theory in spacetime form. It is different from the usual Schr\"odinger-Heisenberg formulation that utilizes states on spacelike surfaces because it assigns probabilities to different sets of alternatives. Sum-over-histories quantum mechanics can be generalized to deal with spacetime alternatives that are not ``at definite moments of time''. An example in field theory is the set of alternative ranges of values of a field averaged over a spacetime region. An example in particle mechanics is the set of the alternatives defined by whether a particle never crosses a fixed spacetime region or crosses it at least once. The general notion of a set of spacetime alternatives is a partition (coarse-graining) of the histories into an exhaustive set of exclusive classes. With this generalization the sum-over-histories formulation can be said to be in fully spacetime form with dynamics represented by path integrals over spacetime histories and alternatives defined as spacetime partitions of these histories. When restricted to alternatives at definite moments of times this generalization is equivalent to Schr\"odinger-Heisenberg quantum mechanics. However, the quantum mechanics of more general spacetime alternatives does not have an equivalent Schr\"odinger-Heisenberg formulation. We suggest that, in the quantum theory of gravity, the general notion of ``observable'' is supplied by diffeomorphism invariant partitions of spacetime metrics and matter field configurations. By generalizing the usual alternatives so as to put quantum theory in fully spacetime form we may be led to a covariant generalized quantum mechanics of spacetime free from the problem of time. 
  We show how, by considering the cumulative effect of tiny quantum gravitational fluctuations over very large distances, it may be possible to: ($a$) reconcile nucleosynthesis bounds on the density parameter of the Universe with the predictions of inflationary cosmology, and ($b$) reproduce the inferred variation of the density parameter with distance. Our calculation can be interpreted as a computation of the contribution of quantum gravitational degrees of freedom to the (local) energy density of the Universe. 
  A pedagogical introduction is given to the quantum mechanics of closed systems, most generally the universe as a whole. Quantum mechanics aims at predicting the probabilities of alternative coarse-grained time histories of a closed system. Not every set of alternative coarse-grained histories that can be described may be consistently assigned probabilities because of quantum mechanical interference between individual histories of the set. In the quantum mechanics of closed systems, containing both observer and observed, probabilities are assigned to those sets of alternative histories for which there is negligible interference between individual histories as a consequence of the system's initial condition and dynamics. Such sets of histories are said to decohere. Typical mechanisms of decoherence that are widespread in our universe are illustrated. Copenhagen quantum mechanics is an approximation to the more general quantum framework of closed subsystems. It is appropriate when there is an approximately isolated subsystem that is a participant in a measurement situation in which (among other things) the decoherence of alternative registrations of the apparatus can be idealized as exact. Since the quantum mechanics of closed systems does not posit the existence of the quasiclassical domain of everyday experience, the domain of the approximate aplicability of classical physics must be explained. We describe how a quasiclassical domain described by averages of densities of approximately conserved quantities could be an emergent feature of an initial condition of the universe that implies the approximate classical behavior of spacetime on accessible scales. 
  Recent work on canonical transformations in quantum mechanics is applied to transform between the Moncrief metric formulation and the Witten-Carlip holonomy formulation of 2+1-dimensional quantum gravity on the torus. A non-polynomial factor ordering of the classical canonical transformation between the metric and holonomy variables is constructed which preserves their classical modular transformation properties. An extension of the definition of a unitary transformation is briefly discussed and is used to find the inner product in the holonomy variables which makes the canonical transformation unitary. This defines the Hilbert space in the Witten-Carlip formulation which is unitarily equivalent to the natural Hilbert space in the Moncrief formulation. In addition, gravitational theta-states arising from ``large'' diffeomorphisms are found in the theory. 
  It is suggested that not only the curvature, but also the signature of spacetime is subject to quantum fluctuations. A generalized D-dimensional spacetime metric of the form $g_{\mu \nu}=e^a_\mu \eta_{ab} e^b_\nu$ is introduced, where $\eta_{ab} = diag\{e^{i\theta},1,...,1\}$. The corresponding functional integral for quantized fields then interpolates from a Euclidean path integral in Euclidean space, at $\theta=0$, to a Feynman path integral in Minkowski space, at $\theta=\pi$. Treating the phase $e^{i\theta}$ as just another quantized field, the signature of spacetime is determined dynamically by its expectation value. The complex-valued effective potential $V(\theta)$ for the phase field, induced by massless fields at one-loop, is considered. It is argued that $Re[V(\theta)]$ is minimized and $Im[V(\theta)]$ is stationary, uniquely in D=4 dimensions, at $\theta=\pi$, which suggests a dynamical origin for the Lorentzian signature of spacetime. 
  In Minkowski spacetime it is well-known that the canonical energy-momentum current is involved in the construction of the globally conserved currents of energy-momentum and total angular momentum. For the construction of conserved currents corresponding to (approximate) scale and proper conformal symmetries, however, an improved energy-momentum current is needed. By extending the Minkowskian framework to a genuine metric-affine spacetime, we find that the affine Noether identities and the conformal Killing equations enforce this improvement in a rather natural way. So far, no gravitational dynamics is involved in our construction. The resulting dilation and proper conformal currents are conserved provided the trace of the energy-momentum current satisfies a (mild) scaling relation or even vanishes. 
  The origin of the phenomenological deterministic laws that approximately govern the quasiclassical domain of familiar experience is considered in the context of the quantum mechanics of closed systems such as the universe as a whole. We investigate the requirements for coarse grainings to yield decoherent sets of histories that are quasiclassical, i.e. such that the individual histories obey, with high probability, effective classical equations of motion interrupted continually by small fluctuations and occasionally by large ones. We discuss these requirements generally but study them specifically for coarse grainings of the type that follows a distinguished subset of a complete set of variables while ignoring the rest. More coarse graining is needed to achieve decoherence than would be suggested by naive arguments based on the uncertainty principle. Even coarser graining is required in the distinguished variables for them to have the necessary inertia to approach classical predictability in the presence of the noise consisting of the fluctuations that typical mech- anisms of decoherence produce. We describe the derivation of phenomenological equations of motion explicitly for a particular class of models. Probabilities of the correlations in time that define equations of motion are explicitly considered. Fully non-linear cases are studied. Methods are exhibited for finding the form of the phenomenological equations of motion even when these are only distantly related to those of the fundamental action. The demon- stration of the connection between quantum-mechanical causality and causalty in classical phenomenological equations of motion is generalized. The connec- tions among decoherence, noise, dissipation, and the amount of coarse graining necessary to achieve classical predictability are investigated quantitatively. 
  This is the write-up of my lectures at the NATO Summer School held in Salamanca in June 1992. The paper deals with the problem of time in quantum gravity. All the major schemes are reviewed. Please note that the paper is in two parts for ease of email transmission; this is part 1. The mailer from gr-qc may further subdivide these two sections. 
  We extend the argument that spacetimes generated by two timelike particles in D=3 gravity (or equivalently by parallel-moving cosmic strings in D=4) permit closed timelike curves (CTC) only at the price of Misner identifications that correspond to unphysical boundary conditions at spatial infinity and to a tachyonic center of mass. Here we analyze geometries one or both of whose sources are lightlike.   We make manifest both the presence of CTC at spatial infinity if they are present at all, and the tachyonic character of the system: As the total energy surpasses its tachyonic bound, CTC first begin to form at spatial infinity, then spread to the interior as the energy increases further. We then show that, in contrast, CTC are entirely forbidden in topologically massive gravity for geometries generated by lightlike sources. 
  The Weyl curvature inside a black hole formed in a generic collapse grows, classically without bound, near to the inner horizon, due to partial absorption and blueshifting of the radiative tail of the collapse. Using a spherical model, we examine how this growth is modified by quantum effects of conformally coupled massless fields. 
  We construct an effective action for gravity in which all homogeneous solutions are nonsingular. In particular, there is neither a big bang nor a big crunch. The action is a higher derivative modification of Einstein's theory constructed in analogy to how the action for point particle motion in special relativity is obtained from Newtonian mechanics. 
  Various physical properties of cosmological models in (1+1) dimensions are investigated. We demonstrate how a hot big bang and a hot big crunch can arise in some models. In particular, we examine why particle horizons do not occur in matter and radiation models. We also discuss under what circumstances exponential inflation and matter/radiation decoupling can happen. Finally, without assuming any particular equation of state, we show that physical singularities can occur in both untilted and tilted universe models if certain assumptions are satisfied, similar to the (3+1)-dimensional cases. 
  Progress in numerical relativity has been hindered for 30 years because of the difficulties of avoiding spacetime singularities in numerical evolution. We propose a scheme which excises a region inside an apparent horizon containing the singularity. Two major ingredients of the scheme are the use of a horizon-locking coordinate and a finite differencing which respects the causal structure of the spacetime. Encouraging results of the scheme in the spherical collapse case are given. 
  A very simple wormhole geometry is considered as a model of a mode of topological fluctutation in Planck-scale spacetime foam. Quantum dynamics of the hole reduces to quantum mechanics of one variable, throat radius, and admits a WKB analysis. The hole is quantum-mechanically unstable: It has no bound states. Wormhole wave functions must eventually leak to large radii. This suggests that stability considerations along these lines may place strong constraints on the nature and even the existence of spacetime foam. 
  Simon argued that the semi-classical theory of gravity, unless with some of its solutions excluded, is unacceptable for reasons of both self-consistency and experiment, and that it has to be replaced by a constrained semi-classical theory. We examined whether the evidence is conclusive. 
  The simple physics of a free particle reveals important features of the path-integral formulation of relativistic quantum theories. The exact quantum-mechanical propagator is calculated here for a particle described by the simple relativistic action proportional to its proper time. This propagator is nonvanishing outside the light cone, implying that spacelike trajectories must be included in the path integral. The propagator matches the WKB approximation to the corresponding configuration-space path integral far from the light cone; outside the light cone that approximation consists of the contribution from a single spacelike geodesic. This propagator also has the unusual property that its short-time limit does not coincide with the WKB approximation, making the construction of a concrete skeletonized version of the path integral more complicated than in nonrelativistic theory. 
  The phenomenon of linearisation instability is identified in models of quantum cosmology that are perturbations of mini-superspace models. In particular, constraints that are second order in the perturbations must be imposed on wave functions calculated in such models. It is shown explicitly that in the case of a model which is a perturbation of the mini-superspace which has $S^3$ spatial sections these constraints imply that any wave functions calculated in this model must be SO(4) invariant. (This replaces the previous corrupted version.) 
  In the context of a Poincar\'e gauge theoretical formulation, pure gravity in 3+1-dimensions is dimensionally reduced to gravity in 2+1-dimensions with or without cosmological constant $\Lambda$. The dimensional reductions are consistent with the gauge symmetries, mapping ISO(3, 1) gauge transformations into ISO(2,1) ones. One of the reductions leads to Chern-Simons-Witten gravity. The solutions of 2+1-gravity with $\Lambda\le 0$ (in particular the black-hole solution recently found by Banados, Teitelboim and Zanelli) and those of 1+1-dimensional Liouville gravity, are thus mapped into 3+1-dimensional vacuum solutions. 
  We consider the Einstein equation with first order (semiclassical) quantum corrections. Although the quantum corrections contain up to fourth order derivatives of the metric, the solutions which are physically relevant satisfy a reduced equations which contain derivatives no higher than second order. We obtain the reduced equations for a range of stress-energy tensors. These reduced equations are suitable for numerical solution, are expected to contain fewer numerical instabilities than the original fourth order equations, and yield only physically relevant solutions. We give analytic and numerical solutions or reduced equations for particular examples, including Friedmann-Lema\^\i tre universes with cosmological constant, a spherical body of constant density, and more general conformally flat metrics. 
  Reversing a slight detrimental effect of the mailer related to TeXability 
  The scope of the paper has been broadened to include a more complete discussion of the following topics: The derivation of composition laws in quantum cosmology. The connection between the existence of a composition law in the sum over histories approach to relativistic quantum mechanics and quantum cosmology, and the existence of a canonical formulation. 
  Accurate limits for the violation of the Principle of Equivalence have been found from the comparison of the redshifts of two identical nuclear species in different chemical environments. 
  We construct an explicit class of dynamic lorentzian wormholes connecting Friedmann-Robertson-Walker (FRW) spacetimes. These wormholes can allow two-way transmission of signals between spatially separated regions of spacetime and could permit such regions to come into thermal contact. The cosmology of a network of early Universe wormholes is discussed. 
  Family of exact spacetimes of D=3 Einstein gravity interacting with massless scalar field is obtained by suitable dimensional reduction of a class of D=4 plane-symmetric Einstein vacua. These D=3 spacetimes describe collisions of line-fronted asymptotically null excitations and are generically singular to the future. The solution for the scalar field can be decomposed into the Fourier-Bessel modes around the background solitons. The criteria of regularity of incoming waves are found. It is shown that the appearance of the scalar curvature singularities need not stem from singularities of incoming waves. Moreover, in distinction to D=4 case, for all solutions with regular incoming waves the final singularities are inevitable. 
  The order $\hbar$ fluctuations of gauge fields in the vicinity of a blackhole can create a repulsive antigravity region extending out beyond the renormalized Schwarzschild horizon. If the strength of this repulsive force increases as higher orders in the back-reaction are included, the formation of a wormhole-like object could occur. 
  Quantum stress-energy tensors of fields renormalized on a Schwarzschild background violate the classical energy conditions near the black hole. Nevertheless, the associated equilibrium thermodynamical entropy $\Delta S$ by which such fields augment the usual black hole entropy is found to be positive. More precisely, the derivative of $\Delta S$ with respect to radius, at fixed black hole mass, is found to vanish at the horizon for {\it all} regular renormalized stress-energy quantum tensors. For the cases of conformal scalar fields and U(1) gauge fields, the corresponding second derivative is positive, indicating that $\Delta S$ has a local minimum there. Explicit calculation shows that indeed $\Delta S$ increases monotonically for increasing radius and is positive. (The same conclusions hold for a massless spin 1/2 field, but the accuracy of the stress-energy tensor we employ has not been confirmed, in contrast to the scalar and vector cases). None of these results would hold if the back-reaction of the radiation on the spacetime geometry were ignored; consequently, one must regard $\Delta S$ as arising from both the radiation fields and their effects on the gravitational field. The back-reaction, no matter how "small", 
  We analyze the degree of equivalence between abelian topologically massive, gauge-invariant, vector or tensor parity doublets and their explicitly massive, non-gauge, counterparts. We establish equivalence of field equations by exploiting a generalized Stueckelberg invariance of the gauge systems. Although the respective excitation spectra and induced source-source interactions are essentially identical, there are also differences, most dramatic being those between the Einstein limits of the interactions in the tensor case: the doublets avoid the discontinuity (well-known from D=4) exhibited by Pauli-Fierz theory. 
  A weak-field solution of Einstein's equations is constructed. It is generated by a circular cosmic string externally supported against collapse. The solution exhibits a conical singularity, and the corresponding deficit angle is the same as for a straight string of the same linear energy density. This confirms the deficit-angle assumption made in the Frolov-Israel-Unruh derivation of the metric describing a string loop at a moment of time symmetry. 
  It has been speculated that Lorentzian wormholes of the Morris- Thorne type might be allowed by the laws of physics at submicroscopic, e.g. Planck, scales and that a sufficiently advanced civilization might be able to enlarge them to classical size. The purpose of this paper is to explore the possibility that inflation might provide a natural mechanism for the enlargement of such wormholes to macroscopic size. A new classical metric is presented for a Lorentzian wormhole which is imbedded in a flat deSitter space. It is shown that the throat and proper length of the wormhole inflate. The resulting properties and stress-energy tensor associated with this metric are discussed. 
  We compute the group element of SO(2,2) associated with the spinning black hole found by Ba\~nados, Teitelboim and Zanelli in (2+1)-dimensional anti-de Sitter space-time. We show that their metric is built with SO(2,2) gauge invariant quantities and satisfies Einstein's equations with negative cosmological constant everywhere except at $r=0$. Moreover, although the metric is singular on the horizons, the group element is continuous and possesses a kink there. 
  A gauge and diffeomorphism invariant theory in (2+1)-dimensions is presented in both first and second order Lagrangian form as well as in a Hamiltonian form. For gauge group $SO(1,2)$, the theory is shown to describe ordinary Einstein gravity with a cosmological constant. With gauge group $G^{tot}=SO(1,2)\otimes G^{YM}$, it is shown that the equations of motion for the $G^{YM}$ fields are the Yang-Mills equations. It is also shown that for weak $G^{YM}$ Yang-Mills fields, this theory agrees with the conventional Einstein-Yang-Mills theory to lowest order in Yang-Mills fields. Explicit static and rotation symmetric solutions to the Einstein-Maxwell theory are studied both for the conventional coupling and for this unified theory. In the electric solution to the unified theory, point charges are not allowed, the charges must have spatial extensions. 
  We describe the geomety of a set of scalar fields coupled to gravity. We consider the formalism of a differential Z_2-graded algebra of $2\times 2$ matrices whose elements are differential forms on space-time. The connection and the vierbeins are extended to incorporate additional scalar and vector fields. The resulting action describes two universes coupled in a non-minimal way to a set of scalar fields. This picture is slightly different from the description of general relativity in the framework of non-commutative geomety. 
  In order to investigate the effects of vacuum polarisation on mass inflation singularities, we study a simple toy model of a charged black hole with cross flowing radial null dust which is homogeneous in the black hole interior. In the region $r^2 \ll e^2$ we find an approximate analytic solution to the classical field equations. The renormalized stress-energy tensor is evaluated on this background and we find the vacuum polarisation backreaction corrections to the mass function $m(r)$. Asymptotic analysis of the semiclassical mass function shows that the mass inflation singularity is much stronger in the presence of vacuum polarisation than in the classical case. 
  The appearance of two geometries in one and the same gravitational theory is familiar. Usually, as in the Brans-Dicke theory or in string theory, these are conformally related Riemannian geometries. Is this the most general relation between the two geometries allowed by physics ? We study this question by supposing that the physical geometry on which matter dynamics take place could be Finslerian rather than just Riemannian. An appeal to the weak equivalence principle and causality then leads us the conclusion that the Finsler geometry has to reduce to a Riemann geometry whose metric - the physical metric - is related to the gravitational metric by a generalization of the conformal transformation. 
  The inverse problem, to reconstruct the general multivector wave function from the observable quadratic densities, is solved for 3D geometric algebra. It is found that operators which are applied to the right side of the wave function must be considered, and the standard Fierz identities do not necessarily hold except in restricted situations, corresponding to the spin-isospin superselection rule. The Greider idempotent and Hestenes quaterionic spinors are included as extreme cases of a single superselection parameter. 
  I attempt to answer the question of the title by giving an annotated list of the major results achieved, over the last six years, in the program to construct quantum general relativity using the Ashtekar variables and the loop representation. A summary of the key open problems is also included. Also included are expositions of several new results including the construction of spatially diffeomorphism invariant observables constructed by coupling general relativity to matter fields. 
  The spontaneous breaking of local Lorentz invariance in the early Universe, associated with a first order phase transition at a critical time $t_c$, generates a large increase in the speed of light and a superluminary communication of information occurs, allowing all regions in the Universe to be causally connected. This solves the horizon problem, leads to a mechanism of monopole suppression in cosmology and can resolve the flatness problem. After the critical time $t_c$, local Lorentz (and diffeomorphism) invariance is restored and light travels at its presently measured speed. The kinematical and dynamical aspects of the generation of quantum fluctuations in the superluminary Universe are investigated. A scale invariant prediction for the fluctuation density amplitude is obtained. 
  We consider the cosmological amplification of a metric perturbation propagating in a higher-dimensional Brans-Dicke background, including a non trivial dilaton evolution. We discuss the properties of the spectral energy density of the produced gravitons (as well as of the associated squeezing parameter), and we show that the present observational bounds on the graviton spectrum provide significant information on the dynamical evolution of the early universe. 
  The basic features of the complex canonical formulation of general relativity and the recent developments in the quantum gravity program based on it are reviewed. The exposition is intended to be complementary to the review articles available already and some original arguments are included. In particular the conventional treatment of the Hamiltonian constraint and quantum states in the canonical approach to quantum gravity is criticized and a new formulation is proposed. 
  We prove that the flux of gravitational radiation from an isolated source in the Nonsymmetric Gravitational Theory is identical to that found in Einstein's General Theory of Relativity. 
  Tunneling rate is investigated in homogenous and anisotropic cosmologies. The calculations is done by two methods: Euclidean and Hamiltonian approaches. It is found that the probability decreases exponentialy as anisotropy is increased. 
  We derive the contributions of spin-orbit and spin-spin coupling to the gravitational radiation from coalescing binary systems of spinning compact objects. We calculate spin effects in the symmetric, trace-free radiative multipoles that determine the gravitational waveform, and the rate of energy loss. Assuming a balance between energy radiated and orbital energy lost, we determine the spin effects in the evolution of the orbital frequency and orbital radius. Assuming that a laser interferometric gravitational observatory can track the gravitational-wave frequency (twice the orbital frequency) as it sweeps through its sensitive bandwidth between about 10 Hz and one kHz, we estimate the accuracy with which the spins of the component bodies can be determined from the gravitational-wave signal. 
  A perturbed Reissner-Nordstr\"om-de Sitter solution is used to emphasize the nature of the singularity along the Cauchy horizon of a charged spherically symmetric black hole. For these solutions, conditions may prevail under which the mass function is bounded and yet the curvature scalar $R_{\alpha\beta\gamma\delta} R^{\alpha\beta\gamma\delta}$ diverges. 
  We show that the dynamics of a spatially closed Friedmann - Robertson - Walker Universe conformally coupled to a real, free, massive scalar field, is chaotic, for large enough field amplitudes. We do so by proving that this system is integrable under the adiabatic approximation, but that the corresponding KAM tori break up when non adiabatic terms are considered. This finding is confirmed by numerical evaluation of the Lyapunov exponents associated with the system, among other criteria. Chaos sets strong limitations to our ability to predict the value of the field at the Big Crunch, from its given value at the Big Bang. (Figures available on request) 
  In a parametrized and constrained Hamiltonian system, an observable is an operator which commutes with all (first-class) constraints, including the super-Hamiltonian. The problem of the frozen formalism is to explain how dynamics is possible when all observables are constants of the motion. An explicit model of a measurement-interaction in a parametrized Hamiltonian system is used to elucidate the relationship between three definitions of observables---as something one observes, as self-adjoint operators, and as operators which commute with all of the constraints. There is no inconsistency in the frozen formalism when the measurement process is properly understood. The projection operator description of measurement is criticized as an over-idealization which treats measurement as instantaneous and non-destructive. A more careful description of measurement necessarily involves interactions of non-vanishing duration. This is a first step towards a more even-handed treatment of space and time in quantum mechanics. (This paper was written for the festschrift of Dieter Brill.) 
  The global structure of 2-dimensional dilaton gravity is studied, attending in particular to black holes and singularities. A gravitational energy is defined and shown to be positive at spatial singularities and negative at temporal singularities. Trapped points are defined, and it is shown that spatial singularities are trapped and temporal singularities are not. Thus a local form of cosmic censorship holds for positive energy. In an analogue of gravitational collapse to a black hole, matter falling into an initially flat space creates a spatial curvature singularity which is cloaked in a spatial or null apparent horizon with non-decreasing energy and area. 
  A generalization of Newtonian gravitation theory is obtained by a suitable limiting procedure from the ADM action of general relativity coupled to a mass-point. Three particular theories are discussed and it is found that two of them are invariant under an extended Galilei gauge group. 
  We review recent developments in differential topology with special concern for their possible significance to physical theories, especially general relativity. In particular we are concerned here with the discovery of the existence of non-standard (``fake'' or ``exotic'') differentiable structures on topologically simple manifolds such as $S^7$, \R and $S^3\times {\bf R^1}.$ Because of the technical difficulties involved in the smooth case, we begin with an easily understood toy example looking at the role which the choice of complex structures plays in the formulation of two-dimensional vacuum electrostatics. We then briefly review the mathematical formalisms involved with differentiable structures on topological manifolds, diffeomorphisms and their significance for physics. We summarize the important work of Milnor, Freedman, Donaldson, and others in developing exotic differentiable structures on well known topological manifolds. Finally, we discuss some of the geometric implications of these results and propose some conjectures on possible physical implications of these new manifolds which have never before been considered as physical models. 
  We discuss the Starobinskii-Unruh process for the Kerr black hole. We show how this effect is related to the theory of squeezed states. We then consider a simple model for a highly relativistic rotating star and show that the Starobinskii-Unruh effect is absent. 
  An exact solution of Einstein's field equations for a static spherically symmetric medium with a radially boost invariant energy-momentum tensor is presented. In the limit of an equation of state corresponding to a distribution of radially directed strings there is a $1/r$ correction to Newton's force law. At large distances and small accelerations this law coincides with the phenomenological force law invented by Milgrom in order to explain the flat rotation curves of galaxies without introducing dark matter. The present model explaines why the critical acceleration of Milgrom is of the same order of magnitude as the Hubble parameter. 
  The increasing entropy, large-scale isotropy and approximate flatness of the universe are considered in the context of signature change, which is a classical model of quantum tunnelling in quantum cosmology. The signature change hypothesis implies an initial inflationary epoch, the magnetic half of the Weyl curvature hypothesis, and a close analogue of the conformal singularity hypothesis. Adding the electric half of the Weyl curvature hypothesis yields, for a perfect fluid, only homogeneous and isotropic cosmologies. In the cosmological-constant case, the unique solution is the Vilenkin tunnelling solution, which gives a de Sitter cosmology. 
  A weak-field solution of Einstein's equations is constructed. It is generated by a circular cosmic string revolving in its plane about the centre of the circle. (The revolution is introduced to prevent the string from collapsing.) This solution exhibits a conical singularity, and the corresponding deficit angle is the same as for a straight string of the same linear energy density, irrespective of the angular velocity of the string. 
  Using the squeezed state formalism the coherent state representation of quantum fluctuations in an expanding universe is derived. It is shown that this provides a useful alternative to the Wigner function as a phase space representation of quantum fluctuations. The quantum to classical transition of fluctuations is naturally implemented by decohering the density matrix in this representation. The entropy of the decohered vacua is derived. It is shown that the decoherence process breaks the physical equivalence between vacua that differ by a coordinate dependent phase generated by a surface term in the Lagrangian. In particular, scale invariant power spectra are only obtained for a special choice of surface term. 
  We study magnetically charged black holes in the Einstein-Yang-Mills-Higgs theory in the limit of infinitely strong coupling of the Higgs field. Using mixed analytical and numerical methods we give a complete description of static spherically symmetric black hole solutions, both abelian and nonabelian. In particular, we find a new class of extremal nonabelian solutions. We show that all nonabelian solutions are stable against linear radial perturbations. The implications of our results for the semiclassical evolution of magnetically charged black holes are discussed. 
  The tunneling approach for entropy generation in quantum gravity is applied to black holes. The area entropy is recovered and shown to count only a tiny fraction of the black hole degeneracy. The latter stems from the extension of the wave function outside the barrier. In fact the semi-classical analysis leads to infinite degeneracy. Evaporating black holes leave then infinitely degenerate "planckons" remnants which can neither decay into, nor be formed from, ordinary matter in a finite time. Quantum gravity opens up at the Planck scale into an infinite Hilbert space which is expected to provide the ultraviolet cutoff required to render the theory finite in the sector of large scale physics. 
  We consider the quantum evolution of the space-independent mode of a $\lambda {\phi}^4$ theory as a minisuperspace in the space of all $\phi$. The motion of the wave packet in the minisuperspace is then compared to the motion of a wave packet in a larger minisuperspace consisting of the original minisuperspace plus one space-dependent mode. By comparing the motion of the two packets we develop criteria that tell us when the quantum evolution in the space-independent minisuperspace gives us useful information about the true evolution in the larger minisuperspace. These criteria serve as a toy model for similar(but much more complex) criteria that will tell us whether or when quantized gravitational minisuperspaces can possibly give any useful information about quantum gravity. 
  A large class of solutions of the Einstein-conformal scalar equations in D=2+1 and D=3+1 is identified. They describe the collisions of asymptotic conformal scalar waves and are generated from Einstein-minimally coupled scalar spacetimes via a (generalized) Bekenstein transformation. Particular emphasis is given to the study of the global properties and the singularity structure of the obtained solutions. It is shown, that in the case of the absence of pure gravitational radiation in the initial data, the formation of the final singularity is not only generic, but is even inevitable. 
  This is a review of cosmological models prepared for the Pont d'Oye workshop on the origin of structure in the universe. The classes of models are discussed in turn, and then some of their uses are considered. 
  This review was given at the 65th birthday meeting of D.W. Sciama, The Renaissance of General Relativity and Cosmology, to be published by Cambridge University Press. It presents progress in the understanding of non-standard relativistic cosmologies during Sciama's career, organized by the areas of application rather than the mathematical types of the models. 
  This is an article contributed to the Brill Festschrift, in honor of the 60th birthday of Prof. D.R. Brill, which will appear in the Vol.2 of the Proceedings of the International Symposia on Directions in General Relativity. In this article we present the (1+1)-dimensional method for studying general relativity of 4-dimensions. We first discuss the general formalism, and subsequently draw attention to the algebraically special class of space-times, following the Petrov classification. It is shown that this class of space-times can be described by the (1+1)-dimensional Yang-Mills action interacting with matter fields, with the spacial diffeomorphisms of the 2-surface as the gauge symmetry. The constraint appears polynomial in part, whereas the non-polynomial part is a non-linear sigma model type in (1+1)-dimensions. It is also shown that the representations of $w_{\infty}$-gravity appear naturally as special cases of this description, and we discuss briefly the $w_{\infty}$-geometry in term of the fibre bundle. 
  We study the emergence of string instabilities in $D$ - dimensional black hole spacetimes (Schwarzschild and Reissner - Nordstr\o m), and De Sitter space (in static coordinates to allow a better comparison with the black hole case). We solve the first order string fluctuations around the center of mass motion at spatial infinity, near the horizon and at the spacetime singularity. We find that the time components are always well behaved in the three regions and in the three backgrounds. The radial components are {\it unstable}: imaginary frequencies develop in the oscillatory modes near the horizon, and the evolution is like $(\tau-\tau_0)^{-P}$, $(P>0)$, near the spacetime singularity, $r\to0$, where the world - sheet time $(\tau-\tau_0)\to0$, and the proper string length grows infinitely. In the Schwarzschild black hole, the angular components are always well - behaved, while in the Reissner - Nordstr\o m case they develop instabilities inside the horizon, near $r\to0$ where the repulsive effects of the charge dominate over those of the mass. In general, whenever large enough repulsive effects in the gravitational background are present, string instabilities develop. In De Sitter space, all the spatial components exhibit instability. The infalling of the string to the black hole singularity is like the motion of a particle in a potential $\gamma(\tau-\tau_0)^{-2}$ where $\gamma$ depends on the $D$ spacetime dimensions and string angular momentum, with $\gamma>0$ for Schwarzschild and $\gamma<0$ for Reissner - Nordstr\o m black holes. For $(\tau-\tau_0)\to0$ the string ends trapped by the black hole singularity. 
  We analyze carefully the problem of gauge symmetries for Bianchi models, from both the geometrical and dynamical points of view. Some of the geometrical definitions of gauge symmetries (=``homogeneity preserving diffeomorphisms'') given in the literature do not incorporate the crucial feature that local gauge transformations should be independent at each point of the manifold of the independent variables ( = time for Bianchi models), i.e, should be arbitrarily localizable ( in time). We give a geometrical definition of homogeneity preserving diffeomorphisms that does not possess this shortcoming. The proposed definition has the futher advantage of coinciding with the dynamical definition based on the invariance of the action ( in Lagrangian or Hamiltonian form). We explicitly verify the equivalence of the Lagrangian covariant phase space with the Hamiltonian reduced phase space. Remarks on the use of the Ashtekar variables in Bianchi models are also given. 
  Spherical configurations that are very massive must be surrounded by apparent horizons. These in turn, when placed outside a collapsing body, must propagate outward with a velocity equal to the velocity of radially outgoing photons. That proves, within the framework of (1+3) formalism and without resorting to the Birkhoff theorem, that apparent horizons coincide with event horizons. 
  We investigate the sensitivity of individual LIGO/VIRGO-like interferometers and the precision with which they can determine the characteristics of an inspiralling binary system. Since the two interferometers of the LIGO detector share nearly the same orientation, their joint sensitivity is similar to that of a single, more sensitive interferometer. We express our results for a single interferometer of both initial and advanced LIGO design, and also for the LIGO detector in the limit that its two interferometers share exactly the same orientation. We approximate the evolution of a binary system as driven exclusively by leading order quadrupole gravitational radiation. To assess the sensitivity, we calculate the rate at which sources are expected to be observed, the range to which they are observable, and the precision with which characteristic quantities describing the observed binary system can be determined. Assuming a conservative rate density for coalescing neutron star binary systems we expect that the advanced LIGO detector will observe approximately 69~yr${}^{-1}$ with an amplitude SNR greater than 8. Of these, approximately 7~yr${}^{-1}$ will be from binaries at distances greater than 950~Mpc. We explore the sensitivity of these results to a tunable parameter in the interferometer design (the recycling frequency). The optimum choice of the parameter is dependent on the goal of the observations, e.g., maximizing the rate of detections or maximizing the precision of measurement. We determine the optimum parameter values for these two cases. 
  We examine the relationship between the decoherence of quantum-mechanical histories of a closed system (as discussed by Gell-Mann and Hartle) and environmentally-induced diagonalization of the density operator for an open system. We study a definition of decoherence which incorporates both of these ideas, and show that it leads to a consistent probabilistic interpretation of the reduced density operator. 
  We investigate the time neutral formulation of quantum cosmology of Gell-Mann and Hartle. In particular we study the proposal discussed by them that our Universe corresponds to the time symmetric decoherence functional with initial and final density matrix of low entropy. We show that our Universe does not correspond to this proposal by investigating the behaviour of small inhomogeneous perturbations around a Friedman-Robertson-Walker model. These perturbations cannot be time symmetric if they were small at the Big Bang. 
  A real tunneling solution is an instanton for the Hartle-Hawking path integral with vanishing extrinsic curvature (vanishing ``momentum'') at the boundary. Since the final momentum is fixed, its conjugate cannot be specified freely; consequently, such an instanton will contribute to the wave function at only one or a few isolated spatial geometries. I show that these geometries are the extrema of the Hartle-Hawking wave function in the semiclassical approximation, and provide some evidence that with a suitable choice of time parameter, these extrema are the maxima of the wave function at a fixed time. 
  We confirm that the diagonal elements of the Gell-Mann and Hartle's decoherence decoherence functional are equal to the relative frequencies of the results of many identical experiments, when a set of alternative histories decoheres. We consider both cases of the pure and mixed initial states. 
  If string theory describes nature, then charged black holes are not described by the Reissner-Nordstrom solution. This solution must be modified to include a massive dilaton. In the limit of vanishing dilaton mass, the new solution can be found by a generalization of the Harrison transformation for the Einstein-Maxwell equations. These two solution generating transformations and the resulting black holes are compared. It is shown that the extremal black hole with massless dilaton can be viewed as the ``square root" of the extremal Reissner-Nordstrom solution. When the dilaton mass is included, extremal black holes are repulsive, and it is energetically favorable for them to bifurcate into smaller holes. (To appear in the Festschrift for Dieter Brill) 
  According to recent reports there is an excess correlation and an apparent regularity in the galaxy one-dimensional polar distribution with a characteristic scale of 128 $h^{-1}$ Mpc. This aparent spatial periodicity can be naturally explained by a time oscillation of the gravitational constant $G$. On the other hand, periodic growth features of bivalve and coral fossiles appear to show a periodic component in the time dependence of the number of days per year. In this letter we show that a time oscillating gravitational constant with similar period and amplitude can explain such a feature. 
  We estimate the entropy associated to a background of squeezed cosmic gravitons, and we argue that the process of cosmological pair production from the vacuum may explain the large amount of entropy of our present universe. 
  Measurement is a fundamental notion in the usual approximate quantum mechanics of measured subsystems. Probabilities are predicted for the outcomes of measurements. State vectors evolve unitarily in between measurements and by reduction of the state vector at measurements. Probabilities are computed by summing the squares of amplitudes over alternatives which could have been measured but weren't. Measurements are limited by uncertainty principles and by other restrictions arising from the principles of quantum mechanics. This essay examines the extent to which those features of the quantum mechanics of measured subsystems that are explicitly tied to measurement situations are incorporated or modified in the more general quantum mechanics of closed systems in which measurement is not a fundamental notion. There, probabilities are predicted for decohering sets of alternative time histories of the closed system, whether or not they represent a measurement situation. Reduction of the state vector is a necessary part of the description of such histories. Uncertainty principles limit the possible alternatives at one time from which histories may be constructed. Models of measurement situations are exhibited within the quantum mechanics of the closed system containing both measured subsystem and measuring apparatus. Limitations are derived on the existence of records for the outcomes of measurements when the initial density matrix of the closed system is highly impure. (Festschrift for Dieter Brill). 
  I show that the gravitational scattering amplitudes of a spin-5/2 field with mass $m\ll M_{Pl}$ violate tree-level unitarity at energies $\sqrt{s}\approx\sqrt{mM_{Pl}}$ if the coupling to gravity is minimal. Unitarity up to energies $\sqrt{s}\approx M_{Pl}$ is restored by adding a suitable non-minimal term, which gives rise to interactions violating the (strong) equivalence principle. These interactions are only relevant at distances $d\lequiv 1/m$. 
  We study a three-parameters family of solutions of the Brans-Dicke field equations. They are static and spherically symmetric. We find the range of parameters for which this solution represents a black hole different from the Schwarzschild one. We find a subfamily of solutions which agrees with experiments and observations in the solar system. We discuss some astrophysical applications and the consequences on the "no hair" theorems for black holes. 
  (One typo corrected and one incorrect statement removed. Extra details on conserved quantities and symmetry algebras added). 
  A positive mass theorem for General Relativity Theory is proved. The proof is 4-dimensional in nature, and relies completely on arguments pertaining to causal structure, the basic idea being that positive energy-density focuses null geodesics, and correspondingly retards them, whereas a negative total mass would advance them. Because it is not concerned with what lies behind horizons, this new theorem applies in some situations not covered by previous positivity theorems. Also, because geodesic focusing is a global condition, the proof might allow a generalisation to semi-classical gravity, even though quantum violations of local energy conditions can occur there. 
  An error in the gauge fixed quantization of section 3 is corrected. The result is a much simpler treatment of the clock field, leading to a simplification of the gauge fixed quantum theory and the treatment of the semiclassical limit. 
  It is argued that the observed Thermodynamic Arrow of Time must arise from the boundary conditions of the universe. We analyse the consequences of the no boundary proposal, the only reasonably complete set of boundary conditions that has been put forward. We study perturbations of a Friedmann model containing a massive scalar field but our results should be independent of the details of the matter content. We find that gravitational wave perturbations have an amplitude that remains in the linear regime at all times and is roughly time symmetric about the time of maximum expansion. Thus gravitational wave perturbations do not give rise to an Arrow of Time. However density perturbations behave very differently. They are small at one end of the universe's history, but grow larger and become non linear as the universe gets larger. Contrary to an earlier claim, the density perturbations do not get small again at the other end of the universe's history. They therefore give rise to a Thermodynamic Arrow of Time that points in a constant direction while the universe expands and contracts again. The Arrow of Time does not reverse at the point of maximum expansion. One has to appeal to the Weak Anthropic Principle to explain why we observe the Thermodynamic Arrow to agree with the Cosmological Arrow, the direction of time in which the universe is expanding. 
  The authors have introduced recently a ``microcanonical functional integral" which yields directly the density of states as a function of energy. The phase of the functional integral is Jacobi's action, the extrema of which are classical solutions at a given energy. This approach is general but is especially well suited to gravitating systems because for them the total energy can be fixed simply as a boundary condition on the gravitational field. In this paper, however, we ignore gravity and illustrate the use of Jacobi's action by computing the density of states for a nonrelativistic harmonic oscillator. (Festschrift for Dieter Brill) 
  We show several kinematical properties that are intrinsic to the Bianchi models with compact spatial sections. Especially, with spacelike hypersurfaces being closed, (A) no anisotropic expansion is allowed for Bianchi type V and VII(A\not=0), and (B) type IV and VI(A\not=0,1) does not exist. In order to show them, we put into geometric terms what is meant by spatial homogeneity and employ a mathematical result on 3-manifolds. We make clear the relation between the Bianchi type symmetry of space-time and spatial compactness, some part of which seem to be unnoticed in the literature. Especially, it is shown under what conditions class B Bianchi models do not possess compact spatial sections. Finally we briefly describe how this study is useful in investigating global dynamics in (3+1)-dimensional gravity. 
  The configuration-space topology in canonical General Relativity depends on the choice of the initial data 3-manifold. If the latter is represented as a connected sum of prime 3-manifolds, the topology receives contributions from all configuration spaces associated to each individual prime factor. There are by now strong results available concerning the diffeomorphism group of prime 3-manifolds which are exploited to examine the topology of the configuration spaces in terms of their homotopy groups. We explicitly show how to obtain these for the class of homogeneous spherical primes, and communicate the results for all other known primes except the non-sufficiently large ones of infinite fundamental group. 
  Microscopic black holes are sensitive to higher dimension operators in the gravitational action. We compute the influence of these operators on the Schwarzschild solution using perturbation theory. All (time reversal invariant) operators of dimension six are included (dimension four operators don't alter the Schwarzschild solution). Corrections to the relation between the Hawking temperature and the black hole mass are found. The entropy is calculated using the Gibbons-Hawking prescription for the Euclidean path integral and using naive thermodynamic reasoning. These two methods agree, however, the entropy is not equal to 1/4 the area of the horizon. 
  We discuss the structure of a gravitational euclidean instanton obtained through coupling of gravity to electromagnetism. Its topology at fixed $t$ is $S^1\times S^2$. This euclidean solution can be interpreted as a tunnelling to a hyperbolic space (baby universe) at $t=0$ or alternatively as a static wormhole that joins the two asymptotically flat spaces of a Reissner--Nordstr\"om type solution with $M=0$. 
  In quantum models of gravity, it is surmized that configurations with degenerate coframes could occur during topology change of the underlying spacetime structure. However, the coframe is not the true Yang--Mills type gauge field of the translations, since it lacks the inhomogeneous gradient term in the gauge transformations. By explicitly restoring this ``hidden" piece within the framework of the affine gauge approach to gravity, one can avoid the metric or coframe degeneracy which would otherwise interfere with the integrations within the path integral. This is an important advantage for quantization. 
  The gravitational field of monopoles, cosmic strings and domain walls is studied in the quadratic gravitational theory $R+\alpha R^2$ with $\alpha |R|\ll 1$, and is compared with the result in Einstein's theory. The metric aquires modifications which correspond to a short range `Newtonian' potential for gauge cosmic strings, gauge monopoles and domain walls and to a long range one for global monopoles and global cosmic strings. In this theory the corrections turn out to be attractive for all the defects. We explain, however, that the sign of these corrections in general depends on the particular higher order derivative theory and topological defect under consideration. The possible relevance of our results to the study of the evolution of topological defects in the early universe is pointed out. 
  We present a singularity free class of inhomogeneous cylindrical universes filled with stiff perfect fluid $(\rho = p)$. Its matter free $ (\rho = 0)$ limit yield two distinct vacuum spacetimes which can be considered as analogues of Kasner solution for inhomogeneous singularity free spacetime. 
  A basic problem in quantizing a field in curved space is the decomposition of the classical modes in positive and negative frequency. The decomposition is equivalent to a choice of a complex structure in the space of classical solutions. In our construction the real tunneling geometries provide the link between the this complex structure and analytic properties of the classical solutions in a Riemannian section of space. This is related to the Osterwalder- Schrader approach to Euclidean field theory. 
  We analyse the dynamics of the collision of two spherical massive shells in a generally spherically symmetric background, obtaining an expression from the conservation law that imposes a constraint between the different parameters involved. We study the light-like limit and make some comparisons of the predictions of our master equation with the results obtained in the case of collision of light-like shells, like the short life of white holes or the mass inflation phenomena. We present some particular cases of the constraint equation. 
  The method of the calculation of effective potential (in linear curvature approximation and at any loop) in massless gauge theory in curved space- time by the direct solution of RG equation is given.The closed expression for two-loop effective potential is obtained.Two-loop effective potential in scalar self-interacting theory is written explicitly.Some comments about it as well as about two-loop effective potential in standard model are presented. 
  A coordinate-free approach to limits of spacetimes is developed. The limits of the Schwarzschild metric as the mass parameter tends to 0 or $\infty$ are studied, extending previous results. Besides the known Petrov type D and 0 limits, three vacuum plane-wave solutions of Petrov type N are found to be limits of the Schwarzschild spacetime. 
  A solvable 2-dimensional conformally invariant midi-superspace model for black holes is obtained by imposing spherical symmetry in 4-dimensional conformally invariant Einstein gravity. The Wheeler-DeWitt equation for the theory is solved exactly to obtain the unique quantum wave functional for an isolated black hole with fixed mass. By suitably relaxing the boundary conditions, a non-perturbative ansatz is obtained for the wave functional of a black hole interacting with its surroundings. 
  We develop a renormalization-group formalism for non-renormalizable theories and apply it to Einstein gravity theory coupled to a scalar field with the Lagrangian $L=\sqrt{g} [R U(\phi)-{1/2}  G(\phi) g^{\mu\nu} \partial_{\mu}\phi \partial_{\nu}\phi- V(\phi)]$, where $U(\phi), G(\phi)$ and $V(\phi)$ are arbitrary functions of the scalar field. We calculate the one-loop counterterms of this theory and obtain a system of renormalization-group equations in partial derivatives for the functions $U, G$ and $V$ playing the role of generalized charges which substitute for the usual charges in multicharge theories. In the limit of a large but slowly varying scalar field and small spacetime curvature this system gives the asymptotic behaviour of the generalized charges compatible with the conventional choice of these functions in quantum cosmological applications. It also demonstrates in the over-Planckian domain the existence of the Weyl-invariant phase of gravity theory asymptotically free in gravitational and cosmological constants. 
  Generalizing the results of Joshi and Dwivedi in Commun.Math.Phys. 146, p.333 (1992), it is pointed out that strong curvature naked singularities could occur in the self-similar gravitational collapse of any form of matter satisfying the weak energy condition for the positivity of mass-energy density. 
  This talk presents some progress achieved in collaboration with A.Linde and D.Linde towards understanding the true nature of the global spatial structure of the Universe as well as the most general stationary characteristics of its time-dependent state with eternally growing total volume. 
  We study a theory which generalizes the nonminimal coupling of matter to gravity by including derivative couplings. This leads to several interesting new dynamical phenomena in cosmology. In particular, the range of parameters in which inflationary attractors exist is greatly expanded. We also numerically integrate the field equations and draw the phase space of the model in second order approximation. The model introduced here may display different inflationary epochs, generating a non-scale-invariant fluctuation spectrum without the need of two or more fields. Finally, we comment on the bubble spectrum arising during a first-order phase transition occurring in our model. 
  Two sets of spatially diffeomorphism invariant operators are constructed in the loop representation formulation of quantum gravity. This is done by coupling general relativity to an anti- symmetric tensor gauge field and using that field to pick out sets of surfaces, with boundaries, in the spatial three manifold. The two sets of observables then measure the areas of these surfaces and the Wilson loops for the self-dual connection around their boundaries. The operators that represent these observables are finite and background independent when constructed through a proper regularization procedure. Furthermore, the spectra of the area operators are discrete so that the possible values that one can obtain by a measurement of the area of a physical surface in quantum gravity are valued in a discrete set that includes integral multiples of half the Planck area. These results make possible the construction of a correspondence between any three geometry whose curvature is small in Planck units and a diffeomorphism invariant state of the gravitational and matter fields. This correspondence relies on the approximation of the classical geometry by a piecewise flat Regge manifold, which is then put in correspondence with a diffeomorphism invariant state of the gravity-matter system in which the matter fields specify the faces of the triangulation and the gravitational field is in an eigenstate of the operators that measure their areas. 
  The geometry of the spinning black holes of standard Einstein theory in 2+1 dimensions, with a negative cosmological constant and without couplings to matter, is analyzed in detail. It is shown that the black hole arises from identifications of points of anti-de Sitter space by a discrete subgroup of $SO(2,2)$. The generic black hole is a smooth manifold in the metric sense. The surface $r=0$ is not a curvature singularity but, rather, a singularity in the causal structure. Continuing past it would introduce closed timelike lines. However, simple examples show the regularity of the metric at $r=0$ to be unstable: couplings to matter bring in a curvature singularity there. Kruskal coordinates and Penrose diagrams are exhibited. Special attention is given to the limiting cases of (i) the spinless hole of zero mass, which differs from anti-de Sitter space and plays the role of the vacuum, and (ii) the spinning hole of maximal angular momentum . A thorough classification of the elements of the Lie algebra of $SO(2,2)$ is given in an Appendix. 
  We use a $\lambda\Phi^4$ scalar quantum field theory to illustrate a new approach to the study of quantum to classical transition. In this approach, the decoherence functional is employed to assign probabilities to consistent histories defined in terms of correlations among the fields at separate points, rather than the field itself. We present expressions for the quantum amplitudes associated with such histories, as well as for the decoherence functional between two of them. The dynamics of an individual consistent history may be described by a Langevin-type equation, which we derive. \noindent {\it Dedicated to Professor Brill on the occasion of his sixtieth birthday, August 1993} 
  A unified theory of all forces should be nonsingular. In such a unified theory, Einstein's general relativity will be a very low curvature effective theory. At larger curvatures, new terms will become important. The question then arises as to whether it is possible to construct an effective action for gravity which leads to a nonsingular theory. In this work, we construct an effective action for gravity in which all homogenerous and isotropic solutions are nonsingular. In particular, there is neither a big bang nor a big crunch. Preliminary work indicates that our construction provides a theory in which anisotropies decrease at large curvatures, and in which the inside of a black hole is singularity free. 
  In this paper we solve for the quantum propagator of a general time dependent system quadratic in both position and momentum, linearly coupled to an infinite bath of harmonic oscillators. We work in the regime where the quantum optical master equation is valid. We map this master equation to a Schroedinger equation on Super-Hilbert space and utilize Lie Algebraic techniques to solve for the dynamics in this space. We then map back to the original Hilbert space to obtain the solution of the quantum dynamics. The Lie Algebraic techniques used are preferable to the standard Wei-Norman methods in that only coupled systems of first order ordinary differential equations and purely algebraic equations need only be solved. We look at two examples. 
  An elementary review of quantum cosmology. (Talk given at Texas/Pascos 1992 at Berkeley) 
  A simple direct explicit proof of the generalized second law of black hole thermodynamics is given for a quasistationary semiclassical black hole. 
  It is shown that the attempt to extend the notion of ideal measurement to quantum field theory leads to a conflict with locality, because (for most observables) the state vector reduction associated with an ideal measurement acts to transmit information faster than light. Two examples of such information-transfer are given, first in the quantum mechanics of a pair of coupled subsystems, and then for the free scalar field in flat spacetime. It is argued that this problem leaves the Hilbert space formulation of quantum field theory with no definite measurement theory, removing whatever advantages it may have seemed to possess vis a vis the sum-over-histories approach, and reinforcing the view that a sum-over-histories framework is the most promising one for quantum gravity. 
  Formulae are derived for the spectra of scalar curvature perturbations and gravitational waves produced during inflation, special cases of which include power law inflation, natural inflation in the small angle approximation and inflation in the slow roll approximation. 
  The form of the initial value constraints in Ashtekar's hamiltonian formulation of general relativity is recalled, and the problem of solving them is compared with that in the traditional metric variables. It is shown how the general solution of the four diffeomorphism constraints can be obtained algebraically provided the curvature is non-degenerate, and the form of the remaining (Gauss law) constraints is discussed. The method is extended to cover the case when matter is included, using an approach due to Thiemann. The application of the method to vacuum Bianchi models is given. The paper concludes with a brief discussion of alternative approaches to the initial value problem in the Ashtekar formulation. 
  We discuss the appearance of time-asymmetric behavior in physical processes in cosmology and in the dynamics of the Universe itself. We begin with an analysis of the nature and origin of irreversibility in well-known physical processes such as dispersion, diffusion, dissipation and mixing, and make the distinction between processes whose irreversibility arises from the stipulation of special initial conditions, and those arising from the system's interaction with a coarse-grained environment. We then study the irreversibility associated with quantum fluctuations in cosmological processes like particle creation and the `birth of the Universe'. We suggest that the backreaction effect of such quantum processes can be understood as the manifestation of a fluctuation-dissipation relation relating fluctuations of quantum fields to dissipations in the dynamics of spacetime. For the same reason it is shown that dissipation is bound to appear in the dynamics of minisuperspace cosmologies. This provides a natural course for the emergence of a cosmological and thermodynamic arrow of time and suggests a meaningful definition of gravitational entropy. We conclude with a discussion on the criteria for the choice of coarse-grainings and the stability of persistent physical structures. Invited Talk given at the Conference on The Physical Origin of Time-Asymmetry Huelva, Spain, Oct. 1991, Proceedings eds. J. J. Halliwell, J. Perez-Mercader and W. H. Zurek, Cambridge University Press, 1993 
  Symmetric gauge fields and invariant metrics in homogeneous spaces are found. Their use for finding exact solutions of the Einstein-Yang-Mills (EYM) equations is discussed. 
  Brill waves are the simplest (non-trivial) solutions to the vacuum constraints of general relativity. They are also rich enough in structure to allow us believe that they capture, at least in part, the generic properties of solutions of the Einstein equations. As such, they deserve the closest attention. This article illustrates this point by showing how Brill waves can be used to investigate the structure of conformal superspace. This article is written as a contribution to the Dieter Brill Festschrift. 
  The purpose of these lectures is to discuss in some detail a new, non-perturbative approach to quantum gravity. I would like to present the basic ideas, outline the key results that have been obtained so far and indicate where we are headed and what the hopes are. The audience at this summer school had a diverse background; many came from high energy physics, some from mathematical physics and the rest from general relativity. Therefore, I have tried to keep the technicalities --particularly proofs and even the number of equations-- to a minimum. My hope is that a research student in any of these three fields should be able to get a bird's eye view of the entire program. In particular, I have kept all three perspectives in mind while discussing the difficulties one encounters and strategies one adopts and in evaluating the successes and limitations of the program. 
  We trace the development of ideas on dissipative processes in chaotic cosmology and on minisuperspace quantum cosmology from the time Misner proposed them to current research. We show 1) how the effect of quantum processes like particle creation in the early universe can address the issues of the isotropy and homogeneity of the observed universe, 2) how viewing minisuperspace as a quantum open system can address the issue of the validity of such approximations customarily adopted in quantum cosmology, and 3) how invoking statistical processes like decoherence and correlation when considered together can help to establish a theory of quantum fields in curved spacetime as the semiclassical limit of quantum gravity. {\it Dedicated to Professor Misner on the occasion of his sixtieth birthday, June 1992.} To appear in the Proceedings of a Symposium on {\it Directions in General Relativity}, College Park, May 1993, Volume 1, edited by B. L. Hu, M. P. Ryan and C. V. Vishveshwara (Cambridge University Press 1993)~~~~umdpp 93-60 
  In several of the class A Bianchi models, minisuperspaces admit symmetries. It is pointed out that they can be used effectively to complete the Dirac quantization program. The resulting quantum theory provides a useful platform to investigate a number of conceptual and technical problems of quantum gravity. 
  A canonical transformation is performed on the phase space of a number of homogeneous cosmologies to simplify the form of the scalar (or, Hamiltonian) constraint. Using the new canonical coordinates, it is then easy to obtain explicit expressions of Dirac observables, i.e.\ phase space functions which commute weakly with the constraint. This, in turn, enables us to carry out a general quantization program to completion. We are also able to address the issue of time through ``deparametrization'' and discuss physical questions such as the fate of initial singularities in the quantum theory. We find that they persist in the quantum theory {\it inspite of the fact that the evolution is implemented by a 1-parameter family of unitary transformations}. Finally, certain of these models admit conditional symmetries which are explicit already prior to the canonical transformation. These can be used to pass to quantum theory following an independent avenue. The two quantum theories --based, respectively, on Dirac observables in the new canonical variables and conditional symmetries in the original ADM variables-- are compared and shown to be equivalent. 
  By simple arguments, we have shown that Karolyhazy's model overestimates the quantum uncertainty of the space-time geometry and leads to absurd physical consequences. The given model can thus not account for gradual violation of quantum coherence and can not predict tiny experimental effects either. 
  We scrutize the commonly used criteria for classicality and examine their underlying issues. The two major issues we address here are that of decoherence and fluctuations. We borrow the insights gained in the study of the semiclassical limit of quantum cosmology to discuss the three criteria of classicality for a quantum closed system: adiabaticity, correlation and decoherence. We then use the Brownian model as a paradigm of quantum open systems to discuss the relation of quantum and thermal fluctuations and their role in the transition from quantum to classical. We derive the uncertainty relation at finite temperature. We study how the fluctuations of a quantum system evolve after it is brought in contact with a heat bath and analyse the decoherence and relaxation processes. From the effect of fluctuations on decoherence we show the relation between these two sets of criteria of classicality. Finally, we briefly comment on the issue of nonintegrability in quantum open systems. 
  Methods and concepts for the study of phase transitions mediated by a time-dependent order-parameter field in curved spacetimes are discussed. A practical example is the derivation of an effective (quasi-)potential for the description of `slow-roll' inflation in the early universe. We first summarize our early results on viewing the symmetry behavior of constant background fields in curved but static spacetimes as finite size effect, and the use of derivative expansions for constructing effective actions for slowly-varying background fields. We then introduce the notion of dynamical finite size effect to explain how an exponential expansion of the scale factor imparts a finite size to the system and how the symmetry behavior in de Sitter space can be understood qualitatively in this light. We reason why the exponential inflation can be described equivalently by a scale transformation, thus rendering this special class of dynamics as effectively static. Finally we show how, in this view, one can treat the class of `slow-roll' inflation as a dynamic perturbation off the effectively static class of exponential inflation and understand it as a dynamical critical phenomenon in cosmology. 
  We show how the concept of quantum open system and the methods in non-equilibrium statistical mechanics can be usefully applied to studies of quantum statistical processes in the early universe. We first sketch how noise, fluctuation, dissipation and decoherence processes arise in a wide range of cosmological problems. We then focus on the origin and nature of noise in quantum fields and spacetime dynamics. We introduce the concept of geometrodynamic noise and suggest a statistical mechanical definition of gravitational entropy. We end with a brief discussion of the theoretical appropriateness to view the physical universe as an open system. 
  The problem of information loss in black hole formation and the associated violations of basic laws of physics, such as conservation of energy, causality and unitarity, are avoided in the nonsymmetric gravitational theory, if the NGT charge of a black hole and its mass satisfy an inequality that does not violate any known experimental data and allows the existence of white dwarfs and neutron stars. 
  Generalized symmetries of the Einstein equations are infinitesimal transformations of the spacetime metric that formally map solutions of the Einstein equations to other solutions. The infinitesimal generators of these symmetries are assumed to be local, \ie at a given spacetime point they are functions of the metric and an arbitrary but finite number of derivatives of the metric at the point. We classify all generalized symmetries of the vacuum Einstein equations in four spacetime dimensions and find that the only generalized symmetry transformations consist of: (i) constant scalings of the metric (ii) the infinitesimal action of generalized spacetime diffeomorphisms. Our results rule out a large class of possible ``observables'' for the gravitational field, and suggest that the vacuum Einstein equations are not integrable. 
  A relevant reference ([14]) has been added. 
  Four-dimensional Euclidean spaces that solve Einstein's equations are interpreted as WKB approximations to wavefunctionals of quantum geometry. These spaces are represented graphically by suppressing inessential dimensions and drawing the resulting figures in perspective representation of three-dimensional space, some of them stereoscopically. The figures are also related to the physical interpretation of the corresponding quantum processes. 
  Contents:  Introduction. The Present State of the Universe.   What Can We Expect From a Complete Cosmological Theory?   An Overview of Quantum Effects in Cosmology.   Parametric (Superadiabatic) Amplification of Classical Waves.  Graviton Creation in the Inflationary Universe.   Quantum States of a Harmonic Oscillator.   Squeezed Quantum States of Relic Gravitons and Primordial Density Perturbations.  Quantum Cosmology, Minisuperspace Models and Inflation.   From the Space of Classical Solutions to the Space of Wave Functions.  On the Probability of Quantum Tunneling From "Nothing".  Duration of Inflation 
  A model of two--dimensional gravity with an action depending only on a linear connection is considered. This model is a topological one, in the sense that the classical action does not contain a metric or zweibein at all. A metric and an additional vector field are instead introduced in the process of solving equations of motion for the connection. They satisfy the constant curvature equation. It is shown that the general solution of these equations of motion can be described by using the space of orbits under the action of the Weyl group in the functional space containing all pairs formed by a metric and a vectorfield. It is shown also that this model admits an equivalent description by using a family of actions depending on the metric and the connection as independent variables. 
  A theory of gravitation is constructed in which all homogeneous and isotropic solutions are nonsingular, and in which all curvature invariants are bounded. All solutions for which curvature invariants approach their limiting values approach de Sitter space. The action for this theory is obtained by a higher derivative modification of Einstein's theory. We expect that our model can easily be generalized to solve the singularity problem also for anisotropic cosmologies. 
  We consider the (massless) scalar field on a 2-dimensional manifold with metric that changes signature from Lorentzian to Euclidean. Requiring a conserved momentum in the spatially homogeneous case leads to a particular choice of propagation rule. The resulting mix of positive and negative frequencies depends only on the total (conformal) size of the spacelike regions and not on the detailed form of the metric. Reformulating the problem using junction conditions, we then show that the solutions obtained above are the unique ones which satisfy the natural distributional wave equation everywhere. We also give a variational approach, obtaining the same results from a natural Lagrangian.   (PACS numbers 04.20.Cv and 02.40.+m.) 
  The Kerr geometry is represented as being created by a source moving along an analytical complex world-line. The equivalence of this complex world-line and an Euclidean version of complex strings (hyperbolic strings) is discussed. It is shown that the complex Kerr source satisfies the corresponding string equations. The boundary conditions of the complex Euclidean strings require an orbifold-like structure of the world-sheet. The related orbifold-like structure of the Kerr geometry is discussed. 
  We study the quantization of the curved spacetime created by ultrarelativistic particles at Planckian energies. We consider a minisuperspace model based on the classical shock wave metric generated by these particles, and for which the Wheeler - De Witt equation is solved exactly. The wave function of the geometry is a Bessel function whose argument is the classical action. This allows us to describe not only the semiclassical regime $(S\to\infty),$ but also the strong quantum regime $(S\to0).$ We analyze the interaction with a scalar field $\phi$ and apply the third quantization formalism to it. The quantum gravity effects make the system to evolve from a highly curved semiclassical geometry (a gravitational wave metric) into a strongly quantum state represented by a weakly curved geometry (essentially flat spacetime). 
  The motion of relativistic particles around three dimensional black holes following the Hamilton-Jacobi formalism is studied. It follows that the Hamilton-Jacobi equation can be separated and reduced to quadratures in analogy with the four dimensional case. It is shown that: a) particles are trapped by the black hole independently of their energy and angular momentum, b) matter alway falls to the centre of the black hole and cannot understake a motion with stables orbits as in four dimensions. For the extreme values of the angular momentum of the black hole, we were able to find exact solutions of the equations of motion and trajectories of a test particle. 
  A general definition of a black hole is given, and general `laws of black-hole dynamics' derived. The definition involves something similar to an apparent horizon, a trapping horizon, defined as a hypersurface foliated by marginal surfaces of one of four non-degenerate types, described as future or past, and outer or inner. If the boundary of an inextendible trapped region is suitably regular, then it is a (possibly degenerate) trapping horizon. The future outer trapping horizon provides the definition of a black hole. Outer marginal surfaces have spherical or planar topology. Trapping horizons are null only in the instantaneously stationary case, and otherwise outer trapping horizons are spatial and inner trapping horizons are Lorentzian. Future outer trapping horizons have non-decreasing area form, constant only in the null case---the `second law'. A definition of the trapping gravity of an outer trapping horizon is given, generalizing surface gravity. The total trapping gravity of a compact outer marginal surface has an upper bound, attained if and only if the trapping gravity is constant---the `zeroth law'. The variation of the area form along an outer trapping horizon is determined by the trapping gravity and an energy flux---the `first law'. 
  It is shown that for a wide class of analytic Lagrangians which depend only on the scalar curvature of a metric and a connection, the application of the so--called ``Palatini formalism'', i.e., treating the metric and the connection as independent variables, leads to ``universal'' equations. If the dimension $n$ of space--time is greater than two these universal equations are Einstein equations for a generic Lagrangian and are suitably replaced by other universal equations at bifurcation points. We show that bifurcations take place in particular for conformally invariant Lagrangians $L=R^{n/2} \sqrt g$ and prove that their solutions are conformally equivalent to solutions of Einstein equations. For 2--dimensional space--time we find instead that the universal equation is always the equation of constant scalar curvature; the connection in this case is a Weyl connection, containing the Levi--Civita connection of the metric and an additional vectorfield ensuing from conformal invariance. As an example, we investigate in detail some polynomial Lagrangians and discuss their bifurcations. 
  The apparent superluminal growth in the size of a sufficiently large part of the universe can be ascribed to the special relativistic effect of time dilation. 
  We consider the consequences of describing the metric properties of space- time through a quartic line element $ds^4=G_{\mu\nu\lambda\rho}dx^\mu dx^\nu dx^\lambda dx^\rho$. The associated "metric" is a fourth-rank tensor $G_{\mu\nu\lambda\rho}$. We construct a theory for the gravitational field based on the fourth-rank metric $G_{\mu\nu\lambda\rho}$ which is conformally invariant in four dimensions. In the absence of matter the fourth-rank metric becomes of the form $G_{\mu\nu\lambda\rho}=g_{(\mu\nu}g_{\lambda\rho )}$ therefore we recover a Riemannian behaviour of the geometry; furthermore, the theory coincides with General Relativity. In the presence of matter we can keep Riemannianicity, but now gravitation couples in a different way to matter as compared to General Relativity. We develop a simple cosmological model based on a FRW metric with matter described by a perfect fluid. Our field equations predict that the entropy is an increasing function of time. For $k_{obs}=0$ the field equations predict $\Omega\approx 4y$, where $y={p\over\rho}$; for $\Omega_{small}=0.01$ we obtain $y_{pred}=2.5\times 10^{-3}$. $y$ can be estimated from the mean random velocity of typical galaxies to be $y_{random}=1\times10^{-5}$. For the early universe there is no violation of causality for $t>t_{class}\approx10^{19}t_{Planck}\approx 10^{-24}s$. 
  A set of coordinates in the non parametric loop-space is introduced. We show that these coordinates transform under infinite dimensional linear representations of the diffeomorphism group. An extension of the group of loops in terms of these objects is proposed. The enlarged group behaves locally as an infinite dimensional Lie group. Ordinary loops form a subgroup of this group. The algebraic properties of this new mathematical structure are analized in detail. Applications of the formalism to field theory, quantum gravity and knot theory are considered. 
  Some time ago, Sorkin (1975) reported investigations of the time evolution and initial value problems in Regge calculus, for one triangulation each of the manifolds $R*S^3$ and $R^4$. Here we display the simple, local characteristic of those triangulations which underlies the structure found by Sorkin, and emphasise its general applicability, and therefore the general validity of Sorkin's conclusions. We also make some elementary observations on the resulting structure of the time evolution and initial value problems in Regge calculus, and add some comments and speculations. 
  The authors have recently proposed a ``microcanonical functional integral" representation of the density of quantum states of the gravitational field. The phase of this real--time functional integral is determined by a ``microcanonical" or Jacobi action, the extrema of which are classical solutions at fixed total energy, not at fixed total time interval as in Hamilton's action. This approach is fully general but is especially well suited to gravitating systems because for them the total energy can be fixed simply as a boundary condition on the gravitational field. In this paper we describe how to obtain Jacobi's action for general relativity. We evaluate it for a certain complex metric associated with a rotating black hole and discuss the relation of the result to the density of states and to the entropy of the black hole. (Dedicated to Yvonne Choquet-Bruhat in honor of her retirement.) 
  We show that the gravitational field equations derived from an action composed of i) an arbitrary function of the scalar curvature and other scalar fields plus ii) connection-independent kinetic and source terms, are identical whether one chooses nonmetricity to vanish and have non-zero torsion or vice versa. 
  We consider source-free electromagnetic fields in spacetimes possessing a non-null Killing vector field, $\xi^a$. We assume further that the electromagnetic field tensor, $F_{ab}$, is invariant under the action of the isometry group induced by $\xi^a$. It is proved that whenever the two potentials associated with the electromagnetic field are functionally independent the entire content of Maxwell's equations is equivalent to the relation $\n^aT_{ab}=0$. Since this relation is implied by Einstein's equation we argue that it is enough to solve merely Einstein's equation for these electrovac spacetimes because the relevant equations of motion will be satisfied automatically. It is also shown that for the exceptional case of functionally related potentials $\n^aT_{ab}=0$ implies along with one of the relevant equations of motion that the complementary equation concerning the electromagnetic field is satisfied. 
  This bibliography attempts to give a comprehensive overview of all the literature related to the Ashtekar variables. The original version was compiled by Peter Huebner in 1989, and it has been subsequently updated by Gabriela Gonzalez and Bernd Bruegmann. Information about additional literature, new preprints, and especially corrections are always welcome. 
  This paper is an application of the ideas of the Born-Oppenheimer (or slow/fast) approximation in molecular physics and of the Isaacson (or short-wave) approximation in classical gravity to the canonical quantization of a perturbed minisuperspace model of the kind examined by Halliwell and Hawking. Its aim is the clarification of the role of the semiclassical approximation and the backreaction in such a model. Approximate solutions of the quantum model are constructed which are not semiclassical, and semiclassical solutions in which the quantum perturbations are highly excited. 
  The effective action of a (1+2)-dimensional defect is obtained as an expansion in powers of the thickness.Considering non-straight solutions as the zero order term, the corrections to the Nambu action are found to depend on the curvature scalar and on the gaussian curvature . 
  We consider the induced $2$d-gravity in the minisuperspace approach. The general solution to the Wheeler-DeWitt equation is given in terms of different kind of Bessel functions of purely real or imaginary orders. We study the properties of the corresponding probability distribution finding a kind of phase transition at the critical point $\nu=0$. 
  We study the loop representation of the quantum theory for 2+1 dimensional general relativity on a manifold, $M = {\cal T}^2 \times {\cal R}$, where ${\cal T}^2$ is the torus, and compare it with the connection representation for this system. In particular, we look at the loop transform in the part of the phase space where the holonomies are boosts and study its kernel. This kernel is dense in the connection representation and the transform is not continuous with respect to the natural topologies, even in its domain of definition. Nonetheless, loop representations isomorphic to the connection representation corresponding to this part of the phase space can still be constructed if due care is taken. We present this construction but note that certain ambiguities remain; in particular, functions of loops cannot be uniquely associated with functions of connections. 
  Testable conditional probabilities appear to be restricted to single hypersurfaces (marvelous moments) and depend only on stationary observables. Observable evolution, such as a change of entropy, should be expressed as a dependence upon clock time, not upon inaccessible coordinate time. 
  A program is outlined concerning the set of all solutions of the hyperbolic Ernst equation on a two-dimensional manifold whose underlying topological space is the same as the domain of all Ernst potentials for colliding plane gravitational wave pairs. The aim of the program is to construct and apply a non-trivial extension of the group of Kinnersley-Chitre transformations. This is to be done by employing the formalism of a homogeneous Hilbert problem. In this first paper of a series, the aforementioned program is completely carried out for the collinear polarization case. 
  We study the decoherence properties of a certain class of Markovian quantum open systems from both the Decohering Histories and Environment Induced Superselection paradigms. The class studied includes many familiar quantum optical cases. For this class, we show that there always exists a basis which leads to {\em exactly} consistent histories for any coarse graining {\em irrespective} of the initial conditions. The magnitude of the off--diagonal elements of the reduced density matrix $\rho$ in this basis however, depends on the initial conditions. Necessary requirements for classicality as advanced by the two paradigms are thus in direct conflict in these systems. 
  A generic outcome of theories with scalar-tensor coupling is the existence of inflationary attractors, either power-law or de Sitter. The fluctuations arising during this phase are Gaussian and their spectrum depends on the wavenumber $k$ according to the power-law $k^{1/(1-p)}$, where $p$ is the inflationary power-law exponent. We investigate to which extent these properties depend on the coupling function and on the potential. We find the class of models in which viable attractors exist. Within this class, we find that the cosmic expansion and the scaling of the fluctuation spectrum are independent of the coupling function. Further, the analytical solution of the Fokker-Planck equation shows that the deviations from Gaussianity are negligible. 
  We reformulate the symmetries of Gurses [Phys. Rev. Lett. 70, 367 (1993)] in a more abstract, more geometrical manner. The type (b) transformation of \gurses\ is related to a diffeomorphism of the differentiable manifold onto itself. The type (c) symmetry is replaced by a more general type (c-bar) symmetry that has the nice property that the commutator of a type (c-bar) generator with a type (a) generator is itself of type (c-bar). We identify a differential constraint that transformations of type (c) and (c-bar) must satisfy, and which, in our opinion, may severely limit the usefulness of these transformations. 
  We argue that purely local experiments can distinguish a stationary charged particle in a static gravitational field from an accelerated particle in (gravity-free) Minkowski space. Some common arguments to the contrary are analyzed and found to rest on a misidentification of ``energy''. 
  It is shown that if a representation of a *-algebra on a vector space $V$ is an irreducible *-representation with respect to some inner product on $V$ then under appropriate technical conditions this property determines the inner product uniquely up to a constant factor. Ashtekar has suggested using the condition that a given representation of the algebra of quantum observables is a *-representation to fix the inner product on the space of physical states. This idea is of particular interest for the quantisation of gravity where an obvious prescription for defining an inner product is lacking. The results of this paper show rigorously that Ashtekar's criterion does suffice to determine the inner product in very general circumstances. Two versions of the result are proved: a simpler one which only applies to representations by bounded operators and a more general one which allows for unbounded operators. Some concrete examples are worked out in order to illustrate the meaning and range of applicability of the general theorems. 
  It is shown that there exist families of asymptotically flat solutions of the Einstein equations coupled to the Vlasov equation describing a collisionless gas which have a Newtonian limit. These are sufficiently general to confirm that for this matter model as many families of this type exist as would be expected on the basis of physical intuition. A central role in the proof is played by energy estimates in unweighted Sobolev spaces for a wave equation satisfied by the second fundamental form of a maximal foliation. 
  We point out an incompleteness of formulations of gravitational and gauge theories that use traces of holonomies around closed curves as their basic variables. It is shown that in general such loop variables have to satisfy certain inequalities if they are to give a description equivalent to the usual one in terms of local gauge potentials. 
  We show how Gromov's spaces of bounded geometries provide a general mathematical framework for addressing and solving many of the issues of $3D$-simplicial quantum gravity. In particular, we establish entropy estimates characterizing the asymptotic distribution of combinatorially inequivalent triangulated $3$-manifolds, as the number of tetrahedra diverges. Moreover, we offer a rather detailed presentation of how spaces of three-dimensional riemannian manifolds with natural bounds on curvatures, diameter, and volume can be used to prove that three-dimensional simplicial quantum gravity is connected to a Gaussian model determined by the simple homotopy types of the underlying manifolds. This connection is determined by a Gaussian measure defined over the general linear group $GL({\bf R},\infty)$. It is shown that the partition function of three-dimensional simplicial quantum gravity is well-defined, in the thermodynamic limit, for a suitable range of values of the gravitational and cosmological coupling constants. Such values are determined by the Reidemeister-Franz torsion invariants associated with an orthogonal representation of the fundamental groups of the set of manifolds considered. The geometrical system considered shows also critical behavior, and in such a case the partition function is exactly evaluated and shown to be equal to the Reidemeister-Franz torsion. The phase structure in the thermodynamical limit is also discussed. In particular, there are either phase transitions describing the passage from a simple homotopy type to another, and (first order) phase transitions within a given simple homotopy type which seem to confirm, on an analytical ground, the picture suggested by numerical simulations. 
  A dynamically preferred quasi-local definition of gravitational energy is given in terms of the Hamiltonian of a `2+2' formulation of general relativity. The energy is well-defined for any compact orientable spatial 2-surface, and depends on the fundamental forms only. The energy is zero for any surface in flat spacetime, and reduces to the Hawking mass in the absence of shear and twist. For asymptotically flat spacetimes, the energy tends to the Bondi mass at null infinity and the \ADM mass at spatial infinity, taking the limit along a foliation parametrised by area radius. The energy is calculated for the Schwarzschild, Reissner-Nordstr\"om and Robertson-Walker solutions, and for plane waves and colliding plane waves. Energy inequalities are discussed, and for static black holes the irreducible mass is obtained on the horizon. Criteria for an adequate definition of quasi-local energy are discussed. 
  We use the quantum Brownian model to derive the uncertainty relation for a quantum open system. We examine how the fluctuations of a quantum system evolve after it is brought in contact with a heat bath at finite temperature. We study the decoherence and relaxation processes and use this example to examine 1) the relation between quantum and thermal fluctuations; and 2) the conditions when the two basic postulates of quantum statistical mechanics become valid. 
  The purpose of this review is to describe in some detail the mathematical relationship between geometrodynamics and connection dynamics in the context of the classical theories of 2+1 and 3+1 gravity. I analyze the standard Einstein-Hilbert theory (in any spacetime dimension), the Palatini and Chern-Simons theories in 2+1 dimensions, and the Palatini and self-dual theories in 3+1 dimensions. I also couple various matter fields to these theories and briefly describe a pure spin-connection formulation of 3+1 gravity. I derive the Euler-Lagrange equations of motion from an action principle and perform a Legendre transform to obtain a Hamiltonian formulation of each theory. Since constraints are present in all these theories, I construct constraint functions and analyze their Poisson bracket algebra. I demonstrate, whenever possible, equivalences between the theories. 
  The propagation of scalar and spinor fields in a spacetime whose metric changes signature is analyzed. Recent work of Dray et al. on particle production from signature change for a (massless) scalar field is reviewed, and an attempt is made to extend their analysis to the case of a (massless) spin-half field. In contrast to their results for a scalar field, it is shown here---for $SL(2,C)$ spinors---that although there are inequivalent forms of the Dirac equation that can be used to propagate a spinor in a signature changing spacetime, none of these forms gives rise to a conserved inner product on the space of solutions to the field equations. 
  The change of signature of a metric is explained using simple examples and methods. The Klein-Gordon field on a signature-changing background is discussed, and it is shown how the approach of Dray et al.\ can be corrected to ensure that the Klein-Gordon equation holds. Isotropic cosmologies are discussed, and it is shown how the approach of Ellis et al.\ can be corrected to ensure that the Einstein-Klein-Gordon equations hold. A straightforward calculation shows that a well defined Ricci tensor requires the standard junction condition, namely vanishing of the second fundamental form of the junction surface. 
  Exotic smooth manifolds, ${\bf R^2\times_\Theta S^2}$, are constructed and discussed as possible space-time models supporting the usual Kruskal presentation of the vacuum Schwarzschild metric locally, but {\em not globally}. While having the same topology as the standard Kruskal model, none of these manifolds is diffeomorphic to standard Kruskal, although under certain conditions some global smooth Lorentz-signature metric can be continued from the local Kruskal form. Consequently, it can be conjectured that such manifolds represent an infinity of physically inequivalent (non-diffeomorphic) space-time models for black holes. However, at present nothing definitive can be said about the continued satisfaction of the Einstein equations. This problem is also discussed in the original Schwarzschild $(t,r)$ coordinates for which the exotic region is contained in a world tube along the time-axis, so that the manifold is spatially, but not temporally, asymptotically standard. In this form, it is tempting to speculate that the confined exotic region might serve as a source for some exterior solution.  Certain aspects of the Cauchy problem are also discussed in terms of ${\bf R^4_\\Theta}$\ models which are ``half-standard'', say for all $t<0,$ but for which $t$ cannot be globally smooth. 
  We investigate the quantum to classical transition of small inhomogeneous fluctuations in the early Universe using the decoherence functional of Gell-Mann and Hartle. We study two types of coarse graining; one due to coarse graining the value of the scalar field and the other due to summing over an environment. We compare the results with a previous study using an environment and the off-diagonal rule proposed by Zurek. We show that the two methods give different results. 
  We investigate the occurrence and nature of naked singularity for the inhomogeneous gravitational collapse of Tolman-Bondi dust clouds.It is shown that the naked singularities form at the center of the collapsing cloud in a wide class of collapse models which includes the earlier cases considered by Eardley and Smarr and Christodoulou. This class also contains self-similar as well as non-self-similar models. The structure and strength of this singularity is examined and the question is investigated as to when a non-zero measure set of non-spacelike trajectories could be emitted from the singularity as opposed to isolated trajectories coming out. It is seen that the weak energy condition and positivity of energy density ensures that the families of non-spacelike trajectories come out of the singularity. The curvature strength of the naked singularity is examined which provides an important test for its physical significance and powerful curvature growth near the naked singularity is pointed out for several subclasses considered. The conditions are discussed for the naked singularity to be globally naked. Implications for the basic issue of the final fate of gravitational collapse are considered once the inhomogeneities in the matter distribution are taken into account. It is argued that a physical formulation for the cosmic censorship may be evolved which avoids the features above. Possibilities in this direction are discussed while indicating that the analysis presented here should be useful for any possible rigorous formulation of the cosmic censorship hypothesis. 
  Recent research has indicated that negative energy fluxes due to quantum coherence effects obey uncertainty principle-type inequalities of the form $|\Delta E|\,{\Delta \tau} \lprox 1\,$. Here $|\Delta E|$ is the magnitude of the negative energy which is transmitted on a timescale $\Delta \tau$. Our main focus in this paper is on negative energy fluxes which are produced by the motion of observers through static negative energy regions. We find that although a quantum inequality appears to be satisfied for radially moving geodesic observers in two and four-dimensional black hole spacetimes, an observer orbiting close to a black hole will see a constant negative energy flux. In addition, we show that inertial observers moving slowly through the Casimir vacuum can achieve arbitrarily large violations of the inequality. It seems likely that, in general, these types of negative energy fluxes are not constrained by inequalities on the magnitude and duration of the flux. We construct a model of a non-gravitational stress-energy detector, which is rapidly switched on and off, and discuss the strengths and weaknesses of such a detector. 
  Cosmological perturbations generated quantum-mechanically (as a particular case, during inflation) possess statistical properties of squeezed quantum states. The power spectra of the perturbations are modulated and the angular distribution of the produced temperature fluctuations of the CMBR is quite specific. An exact formula is derived for the angular correlation function of the temperature fluctuations caused by squeezed gravitational waves. The predicted angular pattern can, in principle, be revealed by the COBE-type observations. 
  Within the framework of the minimum quadratic Poincare gauge theory of gravity in the Riemann-Cartan spacetime the dynamics of homogeneous anisotropic Bianchi types I-IX spinning-fluid cosmological models is investigated. A basic equation set for these models is obtained and analyzed. In particular, exact solutions for the Bianchi type-I spinning-fluid and Bianchi type-V perfect-fluid models are found in analytic form. 
  It is suggested that gravity waves could, in several cases, be detected by means of already (or shortly to be) available technology, independently of current efforts of detection. The present is a follow-up on a recently suggested detection strategy based on gravity-wave-induced deviations of null geodesics. The new development is that a way was found to probe the waves close to the source, where they are several orders of magnitude larger than on the Earth. The effect translates into apparent shifts in stellar angular positions that could be as high as $10^{-7}$ arcsec, which is just about the present theoretical limit of detectability. (Calculation improved; results unchanged.) 
  We prove that the Thin Sandwich Conjecture in general relativity is valid, provided that the data $(g_{ab},\dot g_{ab})$ satisfy certain geometric conditions. These conditions define an open set in the class of possible data, but are not generically satisfied. The implications for the ``superspace'' picture of the Einstein evolution equations are discussed. 
  We propose a new discrete approximation to the Einstein equations, based on the Capovilla-Dell-Jacobson form of the action for the Ashtekar variables. This formulation is analogous to the Regge calculus in that the curvature has support on sets of measure zero. Both a Lagrangian and Hamiltonian formulation are proposed and we report partial results about the constraint algebra of the Hamiltonian formulation. We find that the discrete versions of the diffeomorphism constraints do not commute with each other or with the Hamiltonian constraint. 
  These are the author's lectures at the 1992 Les Houches Summer School, ``Gravitation and Quantizations''. They develop a generalized sum-over-histories quantum mechanics for quantum cosmology that does not require either a preferred notion of time or a definition of measurement. The ``post-Everett'' quantum mechanics of closed systems is reviewed. Generalized quantum theories are defined by three elements (1) the set of fine-grained histories of the closed system which are its most refined possible description, (2) the allowed coarse grainings which are partitions of the fine-grained histories into classes, and (3) a decoherence functional which measures interference between coarse grained histories. Probabilities are assigned to sets of alternative coarse-grained histories that decohere as a consequence of the closed system's dynamics and initial condition. Generalized sum-over histories quantum theories are constructed for non-relativistic quantum mechanics, abelian gauge theories, a single relativistic world line, and for general relativity. For relativity the fine-grained histories are four-metrics and matter fields. Coarse grainings are four-dimensional diffeomorphism invariant partitions of these. The decoherence function is expressed in sum-over-histories form. The quantum mechanics of spacetime is thus expressed in fully spacetime form. The coarse-grainings are most general notion of alternative for quantum theory expressible in spacetime terms. Hamiltonian quantum mechanics of matter fields with its notion of unitarily evolving state on a spacelike surface is recovered as an approximation to this generalized quantum mechanics appropriate for those initial conditions and coarse-grainings such that spacetime geometry 
  We analyze the global structure of a family of Einstein-Maxwell solutions parametrized by mass, charge and cosmological constant. In a qualitative classification there are: (i) generic black-hole solutions, describing a Wheeler wormhole in a closed cosmos of spatial topology $S^2\times S^1$; (ii) generic naked-singularity solutions, describing a pair of ``point" charges in a closed cosmos; (iii) extreme black-hole solutions, describing a pair of ``horned" particles in an otherwise closed cosmos; (iv) extreme naked-singularity solutions, in which a pair of point charges forms and then evaporates, in a way which is not even weakly censored; and (v) an ultra-extreme solution. We discuss the properties of the solutions and of various coordinate systems, and compare with the Kastor-Traschen multi-black-hole solutions. 
  We discuss the limits of validity of the semiclassical theory of gravity in which a classical metric is coupled to the expectation value of the stress tensor. It is argued that this theory is a good approximation only when the fluctuations in the stress tensor are small. We calculate a dimensionless measure of these fluctuations for a scalar field on a flat background in particular cases, including squeezed states and the Casimir vacuum state. It is found that the fluctuations are small for states which are close to a coherent state, which describes classical behavior, but tend to be large otherwise. We find in all cases studied that the energy density fluctuations are large whenever the local energy density is negative. This is taken to mean that the gravitational field of a system with negative energy density, such as the Casimir vacuum, is not described by a fixed classical metric but is undergoing large metric fluctuations. We propose an operational scheme by which one can describe a fluctuating gravitational field in terms of the statistical behavior of test particles. For this purpose we obtain an equation of the form of the Langevin equation used to describe Brownian motion. 
  Existence of maximal hypersurfaces and of foliations by maximal hypersurfaces is proven in two classes of asymptotically flat spacetimes which possess a one parameter group of isometries whose orbits are timelike ``near infinity''. The first class consists of strongly causal asymptotically flat spacetimes which contain no ``black hole or white hole" (but may contain ``ergoregions" where the Killing orbits fail to be timelike). The second class of spacetimes possess a black hole and a white hole, with the black and white hole horizons intersecting in a compact 2-surface $S$. 
  A four dimensional generally covariant field theory is presented which describes non-dynamical three geometries coupled to scalar fields. The theory has an infinite number of physical observables (or constants of the motion) which are constructed from loops made from scalar field configurations. The Poisson algebra of these observables is closed and is the same as that for the 3+1 gravity loop variables in the Ashtekar formalism. The theory also has observables that give the areas of open surfaces and the volumes of finite regions. Solutions to all the Hamilton-Jacobi equations for the theory and the Dirac quantization conditions in the coordinate representation are given. These solutions are holonomies based on matter loops. A brief discussion of the loop space representation for the quantum theory is also given together with some implications for the quantization of 3+1 gravity. 
  We find the quantum analogues of Carlini-Mijic wormholes and consider their application to the cosmological constant problem. In a simple model with $\Lambda$ only we differ with the results of Strominger. 
  This is a review of the aspirations and disappointments of the canonical quantization of geometry. I compare the two chief ways of looking at canonical gravity, geometrodynamics and connection dynamics. I capture as much of the classical theory as I can by pictorial visualization. Algebraic aspects dominate my description of the quantization program. I address the problem of observables. The reader is encouraged to follow the broad outlines and not worry about the technical details. 
  Following an idea close to one given by C. G. Torre (private communication), we prove that Riemannian spaces (M,g) and (M,h) that are related by a Gurses type (b) transformation [M. Gurses, Phys. Rev. Lett. 70, 367 (1993)] or, equivalently, by a Torre-Anderson generalized diffeomorphism [C. G. Torre and I. M. Anderson, Phys. Rev. Lett. xx, xxx (1993)] are neighborhood-isometric, i.e., every point x in M has a corresponding diffeomorphism phi of a neighborhood V of x onto a generally different neighborhood W of x such that phi*(h|W) = g|V. 
  By attaching basis vectors to the components of matter fields, one may render free action densities fully covariant. Both the connection and the tetrads are quadratic forms in these basis vectors. The metric of spacetime, which is quadratic in the tetrads, is then quartic in the basis vectors. 
  If the Universe contains at least one inflationary domain with a sufficiently large and homogeneous scalar field, then this domain permanently produces new inflationary domains of all possible types. We show that under certain conditions this process of the self-reproduction of the Universe can be described by a stationary distribution of probability, which means that the fraction of the physical volume of the Universe in a state with given properties (with given values of fields, with a given density of matter, etc.) does not depend on time. This represents a strong deviation of inflationary cosmology from the standard Big Bang paradigm. 
  Based on the standard statistical interpretation of mixed quantum states, a unique family of consistent histories has been constructed for the quantum Brownian motion in the Caldeira-Leggett reservoir. Analytic solutions have been shown in the Markovian regime: they are uniquely defined coherent wave packets travelling near classical trajectories. 
  We continue the investigation of formation of trapped surfaces in strongly curved , conformally flat geometries. Initial data in quasi-polar gauges rather then maximal ones are considered. This implies that apparent horizons coincide with minimal surfaces. Necessary and sufficient conditions for the formation of trapped surfaces are given. These results can be generalized to include the case with gravitational radiation. We found that mass of a body inside a fixed volume is bounded from above if geometry of a Cauchy slice is smooth. 
  It is shown that only a narrow class of inflationary models can possibly agree with the available observational data on the anisotropy of the cosmic microwave background radiation (CMBR). These models may be governed by ``matter'' with the effective equation of state $-1.2 <p/ \epsilon < -0.6$ which includes the De-Sitter case $p/ \epsilon = -1$. 
  Various works have suggested that the Bondi--Sachs--Penrose decay conditions on the gravitational field at null infinity are not generally representative of asymptotically flat space--times. We have made a detailed analysis of the constraint equations for ``asymptotically hyperboloidal'' initial data and find that log terms arise generically in asymptotic expansions. These terms are absent in the corresponding Bondi--Sachs--Penrose expansions, and can be related to explicit geometric quantities. We have nevertheless shown that there exists a large class of ``non--generic'' solutions of the constraint equations, the evolution of which leads to space--times satisfying the Bondi--Sachs--Penrose smoothness conditions. 
  We show how to measure cosmological parameters using observations of inspiraling binary neutron star or black hole systems in one or more gravitational wave detectors. To illustrate, we focus on the case of fixed mass binary systems observed in a single Laser Interferometer Gravitational-wave Observatory (LIGO)-like detector. Using realistic detector noise estimates, we characterize the rate of detections as a function of a threshold signal-to-noise ratio $\rho_0$, the Hubble constant $H_0$, and the binary ``chirp'' mass. For $\rho_0 = 8$, $H_0 = 100$ km/s/Mpc, and $1.4 \msun$ neutron star binaries, the sample has a median redshift of $0.22$. Under the same assumptions but independent of $H_0$, a conservative rate density of coalescing binaries ($8\times10^{-8}\,{\rm yr}^{-1}\,{\rm Mpc}^{-3}$) implies LIGO will observe $\sim 50\,{\rm yr}^{-1}$ binary inspiral events. The precision with which $H_0$ and the deceleration parameter $q_0$ may be determined depends on the number of observed inspirals. For fixed mass binary systems, $\sim 100$ observations with $\rho_0 = 10$ in the LIGO detector will give $H_0$ to 10\% in an Einstein-DeSitter cosmology, and 3000 will give $q_0$ to 20\%. For the conservative rate density of coalescing binaries, 100 detections with $\rho_0 = 10$ will require about 4~yrs. 
  During the evolution of density inhomogeneties in an $\Omega=1$, matter dominated universe, the typical density contrast changes from $\delta\simeq 10^{-4}$ to $\delta\simeq 10^2$. However, during the same time, the typical value of the gravitational potential generated by the perturbations changes only by a factor of order unity. This significant fact can be exploited to provide a new, powerful, approximation scheme for studying the formation of nonlinear structures in the universe. This scheme, discussed in this paper, evolves the initial perturbation using a Newtonian gravitational potential frozen in time. We carry out this procedure for different intial spectra and compare the results with the Zeldovich approximation and the frozen flow approximation (proposed by Mattarrese et al. recently). Our results are in far better agreement with the N-body simulations than the Zeldovich approximation. It also provides a dynamical explanation for the fact that pancakes remain thin during the evolution. While there is some superficial similarity between the frozen flow results and ours, they differ considerably in the velocity information. Actual shell crossing does occur in our approximation; also there is motion of particles along the pancakes leading to further clumping. These features are quite different from those in frozen flow model. We also discuss the evolution of the two-point correlation function in various approximations. 
  The growth of density perturbations in an expanding universe in the non-linear regime is investigated. The underlying equations of motion are cast in a suggestive form, and motivate a conjecture that the scaled pair velocity, $h(a,x)\equiv -[v/(\dot{a}x)]$ depends on the expansion factor $a$ and comoving coordinate $x$ only through the density contrast $\sigma(a,x)$. This leads to the result that the true, non-linear, density contrast $<(\delta\rho/\rho)^{2}_{x}>^{1/2}=\sigma(a,x)$ is a universal function of the density contrast $\sigma_L(a,l)$, computed in the linear theory and evaluated at a scale $l$ where $l=x(1+\sigma^2)^{1/3}$. This universality is supported by existing numerical simulations with scale-invariant initial conditions having different power laws. We discuss a physically motivated ansatz $h(a,x)=h[\sigma^2(a,x)]$ and use it to compute the non-linear density contrast at any given scale analytically. This provides a promising method for analysing the non-linear evolution of density perturbations in the universe and for interpreting numerical simulations. 
  We investigate the origin of the arrow of time in quantum mechanics in the context of quantum cosmology. The ``Copenhagen'' quantum mechanics of measured subsystems incorporates a fundamental arrow of time. Extending discussions of Aharonov, Bergmann and Lebovitz, Griffiths, and others we investigate a generalized quantum mechanics for cosmology that utilizes both an initial and a final density matrix to give a time-neutral formulation without a fundamental arrow of time. Time asymmetries can arise for particular universes from differences between their initial and final conditions. Theories for both would be a goal of quantum cosmology. A special initial condition and a final condition of indifference would be sufficient to explain the observed time asymmetries of the universe. In this essay we ask under what circumstances a completely time symmetric universe, with T-symmetric initial and final condition, could be consistent with the time asymmetries of the limited domain of our experience. We discuss the approach to equilibrium, the electromagnetic system, and T-violation in the weak interactions in such universes.   The decoherence of alternatives that is necessary for the prediction of probabilities in the quantum mechanics of closed systems is the origin of a fundamental limitation on quantum cosmologies with initial and final conditions. The initial and final density matrices cannot both be pure; if they are there is no decoherence and no prediction. 
  Arguments are given that time must be defined in an operative manner,i.e., by constructing devices which can serve as clocks.The investigation of such devices leads to the conclusion that there is a principal uncertainity of time if one considers periods which are not large compared with the Planck time. Thus,according to the old (classical) concept,time cannot be well-defined at this scale.The uncertainity of time leads to a breakdown of Special and General relativity in the Planck regime;the same happens with causality. We present arguments that the classical concept of time,which treats t simply as a real parameter,must be replaced by a new one. 
  We study an information-theoretic measure of uncertainty for quantum systems. It is the Shannon information $I$ of the phase space probability distribution $\la z | \rho | z \ra $, where $|z \ra $ are coherent states, and $\rho$ is the density matrix. The uncertainty principle is expressed in this measure as $I \ge 1$. For a harmonic oscillator in a thermal state, $I$ coincides with von Neumann entropy, $- \Tr(\rho \ln \rho)$, in the high-temperature regime, but unlike entropy, it is non-zero at zero temperature. It therefore supplies a non-trivial measure of uncertainty due to both quantum and thermal fluctuations. We study $I$ as a function of time for a class of non-equilibrium quantum systems consisting of a distinguished system coupled to a heat bath. We derive an evolution equation for $I$. For the harmonic oscillator, in the Fokker-Planck regime, we show that $I$ increases monotonically. For more general Hamiltonians, $I$ settles down to monotonic increase in the long run, but may suffer an initial decrease for certain initial states that undergo ``reassembly'' (the opposite of quantum spreading). Our main result is to prove, for linear systems, that $I$ at each moment of time has a lower bound $I_t^{min}$, over all possible initial states. This bound is a generalization of the uncertainty principle to include thermal fluctuations in non-equilibrium systems, and represents the least amount of uncertainty the system must suffer after evolution in the presence of an environment for time $t$. 
  Action functionals describing relativistic perfect fluids are presented. Two of these actions apply to fluids whose equations of state are specified by giving the fluid energy density as a function of particle number density and entropy per particle. Other actions apply to fluids whose equations of state are specified in terms of other choices of dependent and independent fluid variables. Particular cases include actions for isentropic fluids and pressureless dust. The canonical Hamiltonian forms of these actions are derived, symmetries and conserved charges are identified, and the boundary value and initial value problems are discussed. As in previous works on perfect fluid actions, the action functionals considered here depend on certain Lagrange multipliers and Lagrangian coordinate fields. Particular attention is paid to the interpretations of these variables and to their relationships to the physical properties of the fluid. 
  With CPT-invariant initial conditions that commute with CPT-invariant final conditions, the respective probabilities (when defined) of a set of histories and its CPT reverse are equal, giving a CPT-symmetric universe. This leads me to question whether the asymmetry of the Gell-Mann--Hartle decoherence functional for ordinary quantum mechanics should be interpreted as an asymmetry of {\it time} . 
  The Vlasov-Einstein system describes the evolution of an ensemble of particles (such as stars in a galaxy, galaxies in a galaxy cluster etc.) interacting only by the gravitational field which they create collectively and which obeys Einstein's field equations. The matter distribution is described by the Vlasov or Liouville equation for a collisionless gas. Recent investigations seem to indicate that such a matter model is particularly suited in a general relativistic setting and may avoid the formation of naked singularities, as opposed to other matter models. In the present note we consider the Vlasov-Einstein system in a spherically symmetric setting and prove the existence of static solutions which are asymptotically flat and have finite total mass and finite extension of the matter. Among these there are smooth, singularity-free solutions, which have a regular center and may have isotropic or anisotropic pressure, and solutions, which have a Schwarzschild-singularity at the center. The paper extends previous work, where smooth, globally defined solutions with regular center and isotropic pressure were considered. 
  A Theorem is proved which reduces the problem of completeness of orbits of Killing vector fields in maximal globally hyperbolic, say vacuum, space--times to some properties of the orbits near the Cauchy surface. In particular it is shown that all Killing orbits are complete in maximal developements of asymptotically flat Cauchy data, or of Cauchy data prescribed on a compact manifold. This result gives a significant strengthening of the uniqueness theorems for black holes. 
  We investigate the limit of Brans-Dicke spacetimes as the scalar field coupling constant omega tends to infinity applying a coordinate-free technique. We obtain the limits of some known exact solutions. It is shown that these limits may not correspond to similar solutions in the general relativity theory. 
  We prove that for an open system, in the Markovian regime, it is always possible to construct an infinite number of non trivial sets of histories that exactly satisfy the probability sum rules. In spite of being perfectly consistent, these sets manifest a very non--classical behavior: they are quite unstable under the addition of an extra instant to the list of times defining the history. To eliminate this feature --whose implications for the interpretation of the formalism we discuss-- and to achieve the stability that characterizes the quasiclassical domain, it is necessary to separate the instants which define the history by time intervals significantly larger than the typical decoherence time. In this case environment induced superselection is very effective and the quasiclassical domain is characterized by histories constructed with ``pointer projectors''. 
  A manifestly covariant equation is derived to describe the perturbations in a domain wall on a given background spacetime. This generalizes recent work on domain walls in Minkowski space and introduces a framework for examining the stability of relativistic bubbles in curved spacetimes. 
  This is a revised version of gr-qc/9304033 
  An earlier construction by the authors of sequences of globally regular, asymptotically flat initial data for the Einstein vacuum equations containing trapped surfaces for large values of the parameter is extended, from the time symmetric case considered previously, to the case of maximal slices. The resulting theorem shows rigorously that there exists a large class of initial configurations for non-time symmetric pure gravitational waves satisfying the assumptions of the Penrose singularity theorem and so must have a singularity to the future. 
  Contribution to the Misner Festshrift, no abstract. 
  Starting from the expression for the superdeterminant of (xI-M), where M is an arbitrary supermatrix, we propose a definition for the corresponding characteristic polynomial and we prove that each supermatrix satisfies its characteristic equation. Depending upon the factorization properties of the basic polynomials whose ratio defines the above mentioned determinant we are able to construct polynomials of lower degree which are also shown to be annihilated by the supermatrix. Some particular cases and examples are considered. 
  Karolyhazy's hazy space-time model, invented for breaking down macroscopic interferences, employs wave-like gravity disturbances. If so, then electric charges would radiate permanently. Here we discuss the observational consequences of the radiation. We find that such radiation is excluded by common experimental situations. 
  Recently, Griffiths presented a generalization of the consistent history approach to quantum mechanics. I can easily construct all possible complete families satisfying Griffiths' "noninterference conditions". Since only trivial families exist one may conclude that Griffiths' proposal has not got farther than the ordinary theory of quantum measurement. 
  This paper is generally concerned with understanding how the uncertainty principle arises in formulations of quantum mechanics, such as the decoherent histories approach, whose central goal is the assignment of probabilities to histories. We first consider histories characterized by position or momentum projections at two moments of time. Both exact and approximate (Gaussian) projections are studied. Shannon information is used as a measure of the uncertainty expressed in the probabilities for these histories. We derive a number of inequalities in which the uncertainty principle is expressed as a lower bound on the information of phase space distributions derived from the probabilities for two-time histories. We go on to consider histories characterized by position samplings at $n$ moments of time. We derive a lower bound on the information of the joint probability for $n$ position samplings. Similar bounds are derived for histories characterized by samplings of other variables. All lower bounds on the information of histories have the general form $\ln \left( V_H / V_S \right) $, where $V_H$ is a volume element of history space, which we define, and $V_S$ is the volume of that space probed by the projections. We thus obtain a concise and general form of the uncertainty principle referring directly to the histories description of the system, and making no reference to notions of phase space. 
  It is shown that Einstein-Yang-Mills-dilaton theory has a countable family of static globally regular solutions which are purely magnetic but uncharged. The discrete spectrum of masses of these solutions is bounded from above by the mass of extremal Gibbons-Maeda solution. As follows from linear stability analysis all solutions are unstable. 
  We carry out the quantization of the full type I and II Bianchi models following the non-perturbative canonical quantization program. These homogeneous minisuperspaces are completely soluble, i.e., it is possible to obtain the general solution to their classical equations of motion in an explicit form. We determine the sectors of solutions that correspond to different spacetime geometries, and prove that the parameters employed to describe the different physical solutions define a good set of coordinates in the phase space of these models. Performing a transformation from the Ashtekar variables to this set of phase space coordinates, we endow the reduced phase space of each of these systems with a symplectic structure. The symplectic forms obtained for the type I and II Bianchi models are then identified as those of the cotangent bundles over ${\cal L}^+_{(+,+)}\times S^2\times S^1$ and ${\cal L}^+_{(+,+)}\times S^1$, respectively. We construct a closed *-algebra of Dirac observables in each of these reduced phase spaces, and complete the quantization program by finding unitary irreducible representations of these algebras. The real Dirac observables are represented in this way by self-adjoint operators, and the spaces of quantum physical states are provided with a Hilbert structure. 
  Previously suggested definitions of averagely trapped surfaces are not well-defined properties of 2-surfaces, and can include surfaces in flat space-time. A natural definition of averagely trapped surfaces is that the product of the null expansions be positive on average. A surface is averagely trapped in the latter sense if and only if its area $A$ and Hawking mass $M$ satisfy the isoperimetric inequality $16\pi M^2 > A$, with similar inequalities existing for other definitions of quasi-local energy. 
  From the point of view of canonical quantum gravity, it has become imperative to find a framework for quantization which provides a {\em general} prescription to find the physical inner product, and is flexible enough to accommodate non-canonical variables. In this dissertation I consider an algebraic formulation of the Dirac approach to the quantization of constrained systems, due to A. Ashtekar. The Dirac quantization program is augmented by a general principle to find the inner product on physical states. Essentially, the Hermiticity conditions on physical operators determine this inner product. I also clarify the role in quantum theory of possible algebraic identities between the elementary variables. I use this approach to quantize various finite dimensional systems. Some of these models test the new aspects of the algebraic framework. Others bear qualitative similarities to \gr, and may give some insight into the pitfalls lurking in \qg. In (spatially compact) general relativity, the Hamiltonian is constrained to vanish. I present various approaches one can take to obtain an interpretation of the quantum theory of such ``dynamically constrained'' systems. I apply some of these ideas to the Bianchi I cosmology, and analyze the issue of the initial singularity in quantum theory. 
  The structure of the moduli spaces $\M := \A/\G$ of (all, not just flat) $SL(2,C)$ and $SU(1,1)$ connections on a n-manifold is analysed. For any topology on the corresponding spaces $\A$ of all connections which satisfies the weak requirement of compatibility with the affine structure of $\A$, the moduli space $\M$ is shown to be non-Hausdorff. It is then shown that the Wilson loop functionals --i.e., the traces of holonomies of connections around closed loops-- are complete in the sense that they suffice to separate all separable points of $\M$. The methods are general enough to allow the underlying n-manifold to be topologically non-trivial and for connections to be defined on non-trivial bundles. The results have implications for canonical quantum general relativity in 4 and 3 dimensions. 
  I present a new way to solve the Wheeler--de Witt equation using the invariance of the classical lagrangian under reparametrization. This property allows one to introduce an arbitrary function for each degree of freedom of the wave function $\Psi$: this arbitrariness can be used to fix the asymptotic behaviour of $\Psi$ so as to obtain a wave function representing a closed universe or a wormhole. These considerations are applied in detail to the Kantowsky--Sachs spacetime. 
  In this short note we show that for a Markovian open quantum system it is always possible to construct a unique set of perfectly consistent Schmidt paths, supporting quasi-classicality. Our Schmidt process, elaborated several years ago, is the $\Delta t\rightarrow 0$ limit of the Schmidt chain constructed very recently by Paz and Zurek. 
  Invariant connections with torsion on simple group manifolds $S$ are studied and an explicit formula describing them is presented. This result is used for the dimensional reduction in a theory of multidimensional gravity with curvature squared terms on $M^{4} \times S$. We calculate the potential of scalar fields, emerging from extra components of the metric and torsion, and analyze the role of the torsion for the stability of spontaneous compactification. 
  The Ashtekar-Renteln Ansatz gives the self-dual solutions to the Einstein equation. A direct generalization of the Ashtekar-Renteln An\-satz to N=1 supergravity is given both in the canonical and in the covariant formulation and a geometrical property of the solutions is pointed out. (Changes: the covariant formulation and some comments about a wave function has been added) 
  The gravitational properties of the neutrino is studied in the weak field approximation. By imposing the hermiticity condition, CPT and CP invariance on the \em tensor matrix element, we shown that the allowed gravitational form factors for Dirac and Majorana neutrinos are very different. In a CPT and CP invariant theory, the \em tensor for a Dirac neutrino of the same specie is characterized by four gravitational form factors. As a result of CPT invariance, the \em tensor for a Majorana neutrino of the same specie has five form factors. In a CP invariant theory, if the initial and final Majorana neutrinos have the same (opposite) CP parity, then only tensor (pseudo-tensor) type transition are allowed. 
  A finite dimensional system with a quadratic Hamiltonian constraint is Dirac quantized in holomorphic, antiholomorphic and mixed representations. A unique inner product is found by imposing Hermitian conjugacy relations on an operator algebra. The different representations yield drastically different Hilbert spaces. In particular, all the spaces obtained in the antiholomorphic representation violate classical expectations for the spectra of certain operators, whereas no such violation occurs in the holomorphic representation. A subset of these Hilbert spaces is also recovered in a configuration space representation. A propagation amplitude obtained from an (anti)holomorphic path integral is shown to give the matrix elements of the identity operators in the relevant Hilbert spaces with respect to an overcomplete basis of representation-dependent generalized coherent states. Relation to quantization of spatially homogeneous cosmologies is discussed in view of the no-boundary proposal of Hartle and Hawking and the new variables of Ashtekar. 
  Riemannian geometry in four dimensions naturally leads to an SL(3) connection that annihilates a basis for self-dual two-forms. Einstein's equations may be written in terms of an SO(3) connection, with SO(3) chosen as an appropriate subgroup of SL(3). We show how a set of "neighbours" of Einstein's equations arises because the subgroup may be chosen in different ways. An explicit example of a non-Einstein metric obtained in this way is given. Some remarks on three dimensional space-times are made. 
  Within the framework of the minimum quadratic Poincare gauge theory of gravity in the Riemann-Cartan spacetime we study the influence of gravitational vacuum energy density (a cosmological constant) on the dynamics of various gravitating systems. It is shown that the inclusion of the cosmological term can lead to gravitational repulsion. For some simple cases of spatially homogeneous cosmological models with radiation we obtain non-singular solutions in form of elementary functions and elliptic integrals. 
  A gravitational instanton is found that can tunnel into a new more stable vacuum phase where diffeomorphism invariance is broken and pitchfork bifurcations develop. This tunnelling process involves a double sphaleron-like transition which is associated with an extra level of quantization which is above that is contained in quantum field theory. 
  If a quantum system of Hilbert space dimension $mn$ is in a random pure state, the average entropy of a subsystem of dimension $m\leq n$ is conjectured to be $S_{m,n}=\sum_{k=n+1}^{mn}\frac{1}{k}-\frac{m-1}{2n}$ and is shown to be $\simeq \ln m - \frac{m}{2n}$ for $1\ll m\leq n$. Thus there is less than one-half unit of information, on average, in the smaller subsystem of a total system in a random pure state. 
  A formalism previously introduced by the author using tesselated Cauchy surfaces is applied to define a quantized version of gravitating point particles in 2+1 dimensions. We observe that this is the first model whose quantum version automatically discretizes time. But also spacelike distances are discretized in a very special way. 
  Using distributional techniques we calculate the energy--momentum tensor of the Schwarzschild geometry. It turns out to be a well--defined tensor--distribution concentrated on the $r=0$ region which is usually excluded from space--time. This provides a physical interpretation for the curvature of this geometry. 
  We present a class of singularity free exact cosmological solutions of Einstein's equations describing a perfect fluid with heat flow. It is obtained as generalization of the Senovilla class [1] corresponding to incoherent radiation field. The spacetime is cylindrically symmetric and globally regular. 
  Related to the classical Ashtekar Hamiltonian, there have been discoveries regarding new classical actions for gravity in (2+1)- and (3+1)-dimensions, and also generalizations of Einstein's theory of gravity. In this review, I will try to clarify the relations between the new and old actions for gravity, and also give a short introduction to the new generalizations. The new results/treatments in this review are: 1. A more detailed constraint analysis of the Hamiltonian formulation of the Hilbert- Palatini Lagrangian in (3+1)-dimensions. 2. The canonical transformation relating the Ashtekar- and the ADM-Hamiltonian in (2+1)-dimensions is given. 3. There is a discussion regarding the possibility of finding a higher dimensional Ashtekar formulation. There are also two clarifying figures (in the beginning of chapter 2 and 3, respectively) showing the relations between different action-formulations for Einstein gravity in (2+1)- and (3+1)-dimensions. 
  The purpose of this paper is to analyse, in the light of information theory and with the arsenal of (elementary) quantum mechanics (EPR correlations, copying machines, teleportation, mixing produced in sub-systems owing to a trace operation, etc.) the scenarios available on the market to resolve the so-called black-hole information paradox. We shall conclude that the only plausible ones are those where either the unitary evolution of quantum mechanics is given up, in which information leaks continuously in the course of black-hole evaporation through non-local processes, or those in which the world is polluted by an infinite number of meta-stable remnants. 
  The issue of de Sitter invariance for a massless minimally coupled scalar field is revisited. Formally, it is possible to construct a de Sitter invariant state for this case provided that the zero mode of the field is quantized properly. Here we take the point of view that this state is physically acceptable, in the sense that physical observables can be computed and have a reasonable interpretation. In particular, we use this vacuum to derive a new result: that the squared difference between the field at two points along a geodesic observer's space-time path grows linearly with the observer's proper time for a quantum state that does not break de Sitter invariance. Also, we use the Hadamard formalism to compute the renormalized expectation value of the energy momentum tensor, both in the O(4) invariant states introduced by Allen and Follaci, and in the de Sitter invariant vacuum. We find that the vacuum energy density in the O(4) invariant case is larger than in the de Sitter invariant case. 
  To investigate the cosmic no hair conjecture, we analyze numerically 1-dimensional plane symmetrical inhomogeneities due to gravitational waves in vacuum spacetimes with a positive cosmological constant. Assuming periodic gravitational pulse waves initially, we study the time evolution of those waves and the nature of their collisions. As measures of inhomogeneity on each hypersurface, we use the 3-dimensional Riemann invariant ${\cal I}\equiv {}~^{(3)\!}R_{ijkl}~^{(3)\!}R^{ijkl}$ and the electric and magnetic parts of the Weyl tensor. We find a temporal growth of the curvature in the waves' collision region, but the overall expansion of the universe later overcomes this effect. No singularity appears and the result is a ``no hair" de Sitter spacetime. The waves we study have amplitudes between $0.020\Lambda \leq {\cal I}^{1/2} \leq 125.0\Lambda$ and widths between $0.080l_H \leq l \leq 2.5l_H$, where $l_H=(\Lambda/3)^{-1/2}$, the horizon scale of de Sitter spacetime. This supports the cosmic no hair conjecture. 
  A nonperturbative approach to quantum gravity that has generated much discussion is the attempt to construct a ``loop representation." Despite it's success in linear quantum theories and a part of 2+1 quantum gravity, it has recently been noticed that difficulties arise with loop representations in a different ``sector" of 2+1 gravity. The problems are related to the use of the ``loop transform" in the construction of the loop representation. We illustrate these difficulties by exploring an analogy based on the Mellin transform which allows us to work in a context that is both mathematically and physically simple and that does not require an understanding either of loop representations or of 2+1 gravity. 
  We consider quantization of the positive curvature Friedmann cosmology in the unimodular modification of Einstein's theory, in which the spacetime four-volume appears as an explicit time variable. The Hamiltonian admits self-adjoint extensions that give unitary evolution in the Hilbert space associated with the Schr\"odinger equation. The semiclassical estimate to the no-boundary wave function of Hartle and Hawking is found. If this estimate is accurate, there is a continuous flux of probability into the configuration space from vanishing three-volume, and the no-boundary wave function evolves nonunitarily. Generalizations of these results hold in a class of anisotropic cosmologies. (Talk given at the 5th Canadian Conference on General Relativity and Relativistic Astrophysics, Waterloo, Ontario, Canada, May 1993.) 
  All three-manifolds are known to occur as Cauchy surfaces of asymptotically flat vacuum spacetimes and of spacetimes with positive-energy sources. We prove here the conjecture that general relativity does not allow an observer to probe the topology of spacetime: any topological structure collapses too quickly to allow light to traverse it. More precisely, in a globally hyperbolic, asymptotically flat spacetime satisfying the null energy condition, every causal curve from $\scri^-$ to ${\scri}^+$ is homotopic to a topologically trivial curve from $\scri^-$ to ${\scri}^+$. (If the Poincar\'e conjecture is false, the theorem does not prevent one from probing fake 3-spheres). 
  The Dirac equation is solved in the Einstein-Yang-Mills background found by Bartnik and McKinnon. We find a normalizable zero-energy fermion mode in the $s$-wave sector. As shown recently, their solution corresponds to a gravitational sphaleron which mediates transitions between topologically distinct vacua. Since the Bartnik-McKinnon solution is unstable, it will either collapse to form a black hole or radiate away its energy. In either case, as the Chern-Simons number of the configuration changes, there will be an accompanying anomalous change in fermion number. 
  If the Universe has the topology of a 3-torus ($T^3$), then it follows from recent data on large-scale temperature fluctuations $\Delta T/T$ of the cosmic microwave background that either the minimal size of the torus is at least of the order of the present cosmological horizon, or the large-scale $\Delta T/T$ pattern should have a symmetry plane (in case of the effective $T^1$ topology) or a symmetry axis (in case of the effective $T^2$ topology), the latter possibility being probably excluded by the data. 
  We do not yet know how to quantize gravity in 3+1 dimensions, but in lower dimensions we face the opposite problem: many of the approaches originally developed for (3+1)-dimensional gravity can be successfully implemented in 2+1 dimensions, but the resulting quantum theories are not all equivalent. In this talk, I discuss six such approaches --- reduced phase space ADM quantization, the Chern-Simons/connection representation, covariant canonical quantization, the loop representation, the Wheeler-DeWitt equation, and lattice methods --- for the simple case of a spacetime whose spatial topology is that of a torus. A comparison of the resulting quantum theories can provide some useful insights into the conceptual issues that underlie quantum gravity in any number of dimensions. (Talk given at the Fifth Canadian Conference on General Relativity and Relativistic Astrophysics, Waterloo, Ontario, May 1993) 
  The structure of polyhomogeneous space-times (i.e., space-times with metrics which admit an expansion in terms of $r^{-j}\log^i r$) constructed by a Bondi--Sachs type method is analysed. The occurrence of some log terms in an asymptotic expansion of the metric is related to the non--vanishing of the Weyl tensor at Scri. Various quantities of interest, including the Bondi mass loss formula, the peeling--off of the Riemann tensor and the Newman--Penrose constants of motion are re-examined in this context. 
  A simple proof of a strengthened form of the first law of black hole mechanics is presented. The proof is based directly upon the Hamiltonian formulation of general relativity, and it shows that the the first law variational formula holds for arbitrary nonsingular, asymptotically flat perturbations of a stationary, axisymmetric black hole, not merely for perturbations to other stationary, axisymmetric black holes. As an application of this strengthened form of the first law, we prove that there cannot exist Einstein-Maxwell black holes whose ergoregion is disjoint from the horizon. This closes a gap in the black hole uniqueness theorems. 
  We derive two new integral mass formulas for stationary black holes in Einstein-Yang-Mills theory. From these we derive a formula for $ \J \Omega -Q V $, from which it follows immediately that any stationary, nonrotating, uncharged black hole is static and has vanishing electric field on the static slices. In the Einstein-Maxwell case, we have, in addition, the ``generalized Smarr mass formula", for which we provide a new, simple derivation. When combined with the other two formulas, we obtain a simple proof that nonrotating Einstein-Maxwell black holes must be static and have vanishing magnetic field on the static slices. Our mass formulas also can be generalized to cases with other types of matter fields, and we describe the nature of these generalizations. 
  The "problem of time" in canonical quantum gravity refers to the difficulties involved in defining a Hilbert space structure on states -- and local observables on this Hilbert space -- for a theory in which the spacetime metric is treated as a quantum field, so no classical metrical or causal structure is present on spacetime. We describe an approach -- much in the spirit of ideas proposed by Misner, Kuchar and others -- to defining states and local observables in quantum gravity which exploits the analogy between the Hamiltonian formulation of general relativity and that of a relativistic particle. In the case ofminisuperspace models, a concrete theory is obtained which appears to be mathematically and physically viable, although it contains some radical features with regard to the presence of an "arrow of time". The viability of this approach in the case ofinfinitely many degrees of freedom rests on a number of fairlywell defined issues, which, however, remain unresolved. As a byproduct of our analysis, the theory of a relativistic particle in curved spacetime is developed. 
  It is shown that the Wheeler DeWitt constraint of canonical gravity coupled to Klein Gordon scalar fields and expressed in terms of Ashtekar's variables admits formal solutions which are parametrized by loops in the three dimensional hypersurface and which are extensions of the well known Wilson loop solutions found by Jacobson, Rovelli and Smolin. 
  We study ten-dimensional Einstein-Yang-Mills model with the space of extra dimensions being a non-symmetric homogeneous space with the invariant metric parametrized by two scales. Dimensional reduction of the model is carried out and the scalar potential of the reduced theory is calculated. In a cosmological setting minima of the potential in the radiation-dominated period that followed the inflation are found and their stability is analyzed. The compactifying solution is shown to be stable for the appropriate values of parameters. 
  Wormhole boundary conditions for the Wheeler--DeWitt equation can be derived from the path integral formulation. It is proposed that the wormhole wave function must be square integrable in the maximal analytic extension of minisuperspace. Quantum wormholes can be invested with a Hilbert space structure, the inner product being naturally induced by the minisuperspace metric, in which the Wheeler--DeWitt operator is essentially self--adjoint. This provides us with a kind of probabilistic interpretation. In particular, giant wormholes will give extremely small contributions to any wormhole state. We also study the whole spectrum of the Wheeler--DeWitt operator and its role in the calculation of Green's functions and effective low energy interactions. 
  The NGT field equations with sources are expanded first about a flat Minkowski background and then about a GR background to first-order in the antisymmetric part of the fundamental tensor, $g_{\mu\nu}$. From the general, static spherically symmetric solution of the field equation in empty space, we establish that there are two conserved charges $m$ and $\ell^2$ corresponding to the two basic gauge invariances of NGT. There is no direct contribution to the flux of gravitational waves from the antisymmetric, $g_{[\mu\nu]}$, sector in the linearized, lowest order of approximation, nor in the non-linear theory. It is demonstrated that the flux of gravitational waves is finite in magnitude and positive definite for solutions of the field equations which satisfy the boundary condition of asymptotic flatness. 
  We examine the decoherence properties of a quantum open system as modeled by a quantum optical system in the Markov regime. We look for decoherence in both the Environment Induced Superselection (EIS) and Consistent Histories (CH) frameworks. We propose a general measure of the coherence of the reduced density matrix and find that EIS decoherence occurs in a number of bases for this model. The degree of ``diagonality'' achieved increases with bath temperature. We evaluate the Decoherence Functional of Consistent Histories for coarse grained phase space two-time projected histories. Using the measures proposed by Dowker and Halliwell we find that the consistency of the histories improves with increasing bath temperature, time and final grain size and decreases with initial grain size. The peaking increases with increasing grain size and decreases with increasing bath temperature. Adopting the above proposed measure of ``coherence'' to the Decoherence Functional gives similar results. The results agree in general with expectations while the anomalous dependence of the consistency on the initial grain size is discussed. 
  Local and global gravitational effects induced by eternal vacuum domain walls are studied. We concentrate on thin walls between non-equal and non-positive cosmological constants on each side of the wall. These vacuum domain walls fall in three classes depending on the value of their energy density $\sigma$: (1)\ extreme walls with $\sigma = \sigma_{{\text{ext}}}$ are planar, static walls corresponding to supersymmetric configurations, (2)\ non-extreme walls with $\sigma = \sigma_{{\text{non}}} > \sigma_{{\text{ext}}}$ correspond to expanding bubbles with observers on either side of the wall being {\em inside\/} the bubble, and (3)\ ultra-extreme walls with $\sigma = \sigma_{{\text{ultra}}} < \sigma_{{\text{ext}}}$ represent the bubbles of false vacuum decay. On the sides with less negative cosmological constant, the extreme, non-extreme, and ultra-extreme walls exhibit no, repulsive, and attractive effective ``gravitational forces,'' respectively. These ``gravitational forces'' are global effects not caused by local curvature. Since the non-extreme wall encloses observers on both sides, the supersymmetric system has the lowest gravitational mass accessable to outside observers. It is conjectured that similar positive mass protection occurs in all physical systems and that no finite negative mass object can exist inside the universe. We also discuss the global space-time structure of these singularity free space-times and point out intriguing analogies with the causal structure of black holes. 
  We propose a graded classification of the entire field of multivector physics, including all alternative points of view. The (often tacit) postulates of different types of formulations are contrasted, summarizing their consequences. Specifically, spin-gauge formulations of gravitation and GUT which assume standard column spinors will require unnecessarily large matrix algebras. An extreme generalization is introduced, where wavefunctions are multivectors, in which multiple generations of particles naturally appear without resorting to increasing the size of the algebra. Further, this allows for two-sided (bilateral) operators, which can accomodate in excess of 10 times more gauge fields without increasing the algebraic representation. As this generalization encompasses all the essential features of the other categories, it is proposed to be the best path to new physics. [Summary of talk at 3rd International Conf on Clifford Algebras and Their Appl. in Physics, Deinze, Belgium May 1993]. 
  An algorithm is described for the construction of actions for scalar, spinor, and vector gauge fields that remains well-defined when the metric is degenerate and that involve no contravariant tensor fields. These actions produce the standard matter dynamics and coupling to gravity when tetrad is nondegenerate, but have the property that all fields that appear in them can be pulled back through an arbitrary map of degree one and that this pull back leave the action invariant when the map has degree one. 
  We show that it is possible to realize an inflationary scenario even without conversion of the false vacuum energy to radiation. Such cosmological models have a deflationary stage in which $Ha^2$ is decreasing and radiation produced by particle creation in an expanding Universe becomes dominant. The preceding inflationary stage ends since the inflaton potential becomes steep. False vacuum energy is finally (partly) converted to the inflaton kinetic energy , the potential energy rapidly decreases and the Universe comes to the deflationary stage with a scale factor $a(t) \propto t^{1/3}$. Basic features and observational consequences of this scenario are indicated. 
  We obtain analytic solutions for the density contrast and the anisotropic pressure in a multi-dimensional FRW cosmology with collisionless, massless matter. These are compared with perturbations of a perfect fluid universe. To describe the metric perturbations we use manifest gauge invariant metric potentials. The matter perturbations are calculated by means of (automatically gauge invariant) finite temperature field theory, instead of kinetic theory. (Talk given at the Journ\'ees Relativistes '93, 5 -- 7 April, Brussels, Belgium) 
  The new solution of the Einstein equations in empty space is presented. The solution is constructed using Schwarzschild solution but essentially differs from it. The basic properties of the solution are: the existence of a horizon which is a hyperboloid of one sheet moving along its axis with superluminal velocity, right signature of the metric outside the horizon and Minkovsky-flatness of it at infinity outside the horizon. There is also a discussion in the last chapter, including comparing with recent astronomical observations. 
  The global properties of spatially homogeneous cosmological models with collisionless matter are studied. It is shown that as long as the mean curvature of the hypersurfaces of homogeneity remains finite no singularity can occur in finite proper time as measured by observers whose worldlines are orthogonal to these hypersurfaces. Strong cosmic censorship is then proved for the Bianchi I, Bianchi IX and Kantowski-Sachs symmetry classes. 
  We investigate rotation and rotating structures in (2+1)-dimensional Einstein gravity. We show that rotation generally leads to pathological physical situations. 
  Following the formalism of Gell-Mann and Hartle, phenomenological equations of motion are derived from the decoherence functional formalism of quantum mechanics, using a path-integral description. This is done explicitly for the case of a system interacting with a ``bath'' of harmonic oscillators whose individual motions are neglected. The results are compared to the equations derived from the purely classical theory. The case of linear interactions is treated exactly, and nonlinear interactions are compared using classical and quantum perturbation theory. 
  We show that black holes fulfill the scaling laws arising in critical transitions. In particular, we find that in the transition from negative to positive values the heat capacities $C_{JQ}$, $C_{\Omega Q}$ and $C_{J\Phi}$ give rise to critical exponents satisfying the scaling laws. The three transitions have the same critical exponents as predicted by the universality Hypothesis. We also briefly discuss the implications of this result with regards to the connections among gravitation, quantum mechanics and statistical physics. 
  We consider the stimulated emission of gravitons from an initial state of thermal equilibrium, under the action of the cosmic gravitational background field. We find that the low-energy graviton spectrum is enhanced if compared with spontaneous creation from the vacuum; as a consequence, the scale of inflation must be lowered, in order not to exceed the observed CMB quadrupole anisotropy. This effect is particularly important for models based on a symmetry-breaking transition which require, as initial condition, a state of thermal equilibrium at temperatures of the order of the inflation scale. 
  We address the issue of recovering the time-dependent Schr\"{o}dinger equation from quantum gravity in a natural way. To reach this aim it is necessary to understand the nonoccurrence of certain superpositions in quantum gravity.   We explore various possible explanations and their relation. These are the delocalisation of interference terms through interaction with irrelevant degrees of freedom (decoherence), gravitational anomalies, and the possibility of $\theta$ states. The discussion is carried out in both the geometrodynamical and connection representation of canonical quantum gravity. 
  Recent work on $N=2$ supersymmetric Bianchi type IX cosmologies coupled to a scalar field is extended to a general treatment of homogeneous quantum cosmologies with explicitely solvable momentum constraints, i.e. Bianchi types I, II, VII, VIII besides the Bianchi type IX, and special cases, namely the Friedmann universes, the Kantowski-Sachs space, and Taub-NUT space. Besides the earlier explicit solution of the Wheeler DeWitt equation for Bianchi type IX, describing a virtual wormhole fluctuation, an additional explicit solution is given and identified with the `no-boundary state'. 
  We discuss the canonical treatment and quantization of matter coupled supergravity in three dimensions, with special emphasis on $N=2$ supergravity. We then analyze the quantum constraint algebra; certain operator ordering ambiguities are found to be absent due to local supersymmetry. We show that the supersymmetry constraints can be partially solved by a functional analog of the method of characteristics. We also consider extensions of Wilson loop integrals of the type previously found in ordinary gravity, but now with connections involving the bosonic and fermionic matter fields in addition to the gravitational connection. In a separate section of this paper, the canonical treatment and quantization of non-linear coset space sigma models are discussed in a self contained way. 
  In the present article, using a further generalization of the algebraic method of separation of variables, the Dirac equation is separated in a family of space-times where it is not possible to find a complete set of first order commuting differential operators. After separating variables, the Dirac equation is reduced to a set of coupled ordinary differential equations and some exact solutions corresponding to cosmological backgrounds and gravitational waves are computed in terms of hypergeometric functions. By passing, the Klein Gordon equation in this background field is discussed. 
  In a previous paper two of the authors (G. R. and A. D. R.) showed that there exist global, classical solutions of the spherically symmetric Vlasov-Einstein system for small initial data. The present paper continues this investigation and allows also large initial data. It is shown that if a solution of the spherically symmetric Vlasov-Einstein system develops a singularity at all then the first singularity has to appear at the center of symmetry. The result adds weight to the conjecture that cosmic censorship holds if one replaces dust as matter model for which naked singularities do form by a collisionless gas described by the Vlasov equation. The main tool is an estimate which shows that a solution is global if all the matter remains away from the center of symmetry. 
  The initial value problem for the Vlasov-Poisson system is by now well understood in the case of an isolated system where, by definition, the distribution function of the particles as well as the gravitational potential vanish at spatial infinity. Here we start with homogeneous solutions, which have a spatially constant, non-zero mass density and which describe the mass distribution in a Newtonian model of the universe. These homogeneous states can be constructed explicitly, and we consider deviations from such homogeneous states, which then satisfy a modified version of the Vlasov-Poisson system. We prove global existence and uniqueness of classical solutions to the corresponding initial value problem for initial data which represent spatially periodic deviations from homogeneous states. 
  We study two dimensional dilaton gravity and supergravity following hamiltonian methods. Firstly, we consider the structure of constraints of 2D dilaton gravity and then the 2D dilaton supergravity is obtained taking the squere root of the bosonic constraints. We integrate exactly the equations of motion in both cases and we show that the solutions of the equation of motion of 2D dilaton supergravity differs from the solutions of 2D dilaton gravity only by boundary conditions on the fermionic variables, i.e. the black holes of 2D dilaton supergravity theory are exactly the same black holes of 2D bosonic dilaton gravity modulo supersymmetry transformations. This result is the bidimensional analogue of the no-hair theorem for supergravity. 
  Pair production of Reissner-Nordstrom black holes in a magnetic field can be described by a euclidean instanton. It is shown that the instanton amplitude contains an explicit factor of $e^{A/4}$, where $A$ is the area of the event horizon. This is consistent with the hypothesis that $e^{A/4}$ measures the number of black hole states. 
  In recent work on Einstein gravity in four dimensions using the Ashtekar variables, non-local loop variables have played an important role in attempts to formulate a quantum theory. The introduction of such variables is guided by gauge invariance, and here an infinite set of loop variables is introduced for the Hamiltonian form of the Einstein-Maxwell theory. The loops that enter the description naturally are the (source free) electric field lines. These variables are invariant under spatial diffeomorphisms and they also form a closed Poisson algebra. As such they may be useful for quantization attempts and for studying classical solutions. 
  We discuss some outstanding open questions regarding the validity and uniqueness of the standard second order Newton-Einstein classical gravitational theory. On the observational side we discuss the degree to which the realm of validity of Newton's Law of Gravity can actually be extended to distances much larger than the solar system distance scales on which the law was originally established. On the theoretical side we identify some commonly accepted but actually still open to question assumptions which go into the formulating of the standard second order Einstein theory in the first place. In particular, we show that while the familiar second order Poisson gravitational equation (and accordingly its second order covariant Einstein generalization) may be sufficient to yield Newton's Law of Gravity they are not in fact necessary. The standard theory thus still awaits the identification of some principle which would then make it necessary too. We show that current observational information does not exclusively mandate the standard theory, and that the conformal invariant fourth order theory of gravity considered recently by Mannheim and Kazanas is also able to meet the constraints of data, and in fact to do so without the need for any so far unobserved non-luminous or dark matter. 
  Gauge fields are described on an Riemann-Cartan space-time by means of tensor-valued differential forms and exterior calculus. It is shown that minimal coupling procedure leads to a gauge invariant theory where gauge fields interact with torsion, and that consistency conditions for the gauge fields impose restrictions in the non-Riemannian structure of space-time. The new results differ from the well established ones obtained by using minimal coupling procedure at the action formulation. The sources of these differences are pointed out and discussed. 
  The behaviour of test fields near a compact Cauchy horizon is investigated. It is shown that solutions of nonlinear wave equations on Taub spacetime with generic initial data cannot be continued smoothly to both extensions of the spacetime through the Cauchy horizon. This is proved using an energy method. Similar results are obtained for the spacetimes of Moncrief containing a compact Cauchy horizon and for more general matter models. 
  Three different methods viz. i) a perturbative analysis of the Schr\"odinger equation ii) abstract differential geometric method and iii) a semiclassical reduction of the Wheeler-Dewitt equation, relating Pancharatnam phase to vacuum instability are discussed. An improved semiclassical reduction is also shown to yield the correct zeroth order semicalssical Einstein equations with backreaction. This constitutes an extension of our earlier discussions on the topic 
  Any quantum theory of gravity which treats the gravitational constant as a dynamical variable has to address the issue of superpositions of states corresponding to different eigenvalues. We show how the unobservability of such superpositions can be explained through the interaction with other gravitational degrees of freedom (decoherence). The formal framework is canonically quantized Jordan-Brans-Dicke theory. We discuss the concepts of intrinsic time and semiclassical time as well as the possibility of tunneling into regions corresponding to a negative gravitational constant. We calculate the reduced density matrix of the Jordan-Brans-Dicke field and show that the off-diagonal elements can be sufficiently suppressed to be consistent with experiments. The possible relevance of this mechanism for structure formation in extended inflation is briefly discussed. 
  Using a recent classification of local symmetries of the vacuum Einstein equations, it is shown that there can be no observables for the vacuum gravitational field (in a closed universe) built as spatial integrals of local functions of Cauchy data and their derivatives. 
  Two new exact analytical solutions of the euclidean Einstein equations for a minimal massless scalar field and negative cosmological constant have been obtained. These solutions are given in terms of Jacobian elliptic or circular functions, rather than hyperbolic functions, connect large asymptotic regions of maximally-symmetric anti-DeSitter metrics through a microscopic throat, and correspond to negative definite components of the Ricci tensor. Therefore, they describe wormhole-like changes of topology driven by nucleation of baby universes. The quantum state of such elliptic and circular wormholes or handles is discussed in the most interesting inner and asymptotic regions. 
  The specific nonlinear vector $\sigma$-model coupled to Einstein gravity is investigated. The model arises in the studies of the gravitating matter in non-commutative geometry. The static spherically symmetric spacetimes are identified by direct solving of the field equations. The asymptotically flat black hole with the ``non-commutative'' vector hair appears for the special choice of the integration constants, giving thus another counterexample to the famous ``no-hair'' theorem. 
  We examine gravitational waves in an isolated axi--symmetric reflexion symmetric NGT system. The structure of the vacuum field equations is analyzed and the exact solutions for the field variables in the metric tensor are found in the form of expansions in powers of a radial coordinate. We find that in the NGT axially symmetric case the mass of the system remains constant only if the system is static (as it necessarily is in the case of spherical symmetry). If the system radiates, then the mass decreases monotonically and the energy flux associated with waves is positive. 
  As shown by Ashtekar in the mid 80's, general relativity can be extended to incorporate degenerate metrics. This extension is not unique, however, as one can change the form of the hamiltonian constraints and obtain an {\it alternative} degenerate extension of general relativity that disagrees with Ashtekar's original theory when the triads vectors are degenerate. In this paper, the constraint algebra of a particular alternative theory is explicitly evaluated and compared with that of Ashtekar's original degenerate extension. A generic classification of the difference between the two theories is given in terms of the degeneracy and surface-forming properties of the triad vectors. (This classification is valid when the degeneracy and surface-forming properties of the triad vectors is the same everywhere in an open set about a point in space.) If the triad vectors are degenerate and surface-forming, then all the secondary constraints of the alternative degenerate extension are satisfied as a consequence of the primary constraints, and the constraints of this theory are weaker than those of Ashtekar's. If the degenerate triad vectors are not surface-forming, then the first secondary constraint of the alternative theory already implies equivalence with Ashtekar's degenerate extension. What happens when the degeneracy and surface-forming properties of the triad vectors change from point to point is an open question. 
  We consider chaotic inflation in the theories with the effective potentials phi^n and e^{\alpha\phi}. In such theories inflationary domains containing sufficiently large and homogeneous scalar field \phi permanently produce new inflationary domains of a similar type. We show that under certain conditions this process of the self-reproduction of the Universe can be described by a stationary distribution of probability, which means that the fraction of the physical volume of the Universe in a state with given properties (with given values of fields, with a given density of matter, etc.) does not depend on time, both at the stage of inflation and after it. This represents a strong deviation of inflationary cosmology from the standard Big Bang paradigm. We compare our approach with other approaches to quantum cosmology, and illustrate some of the general conclusions mentioned above with the results of a computer simulation of stochastic processes in the inflationary Universe. 
  We deal with the black hole information loss paradox by showing that the stimulated emission component of the black hole radiation contains information about the initial state of the system. The nonlocal behaviour that allows the recovery of information about the matter that falls behind the horizon appears in a natural way. We calculate the expectation value and probability distribution of particles at ${\cal J}^+$ for a non-vacuum initial state. The entropy of the final state is compared to that of a thermal state with the same energy per mode. We find that the information recovered about the initial state increases with the number $r$ of the initially incoming particles, reaching for example over 30\% for $r = 1000$. We point out that recovering information about the initial state, nevertheless, does not automatically imply the purity of the final state. 
  It is shown that the surface gravity and temperature of a stationary black hole are invariant under conformal transformations of the metric that are the identity at infinity. More precisely, we find a conformal invariant definition of the surface gravity of a conformal Killing horizon that agrees with the usual definition(s) for a true Killing horizon and is proportional to the temperature as defined by Hawking radiation. This result is reconciled with the intimate relation between the trace anomaly and the Hawking effect, despite the {\it non}invariance of the trace anomaly under conformal transformations. 
  We investigate two models of measuring devices designed to detect a non-relativistic free particle in a given region of spacetime. These models predict different probabilities for a free quantum particle to enter a spacetime region $R$ so that this notion is device dependent. The first model is of a von Neumann coupling which we present as a contrast to the second model. The second model is shown to be related to probabilities defined through partitions of configuration space paths in a path integral. This study thus provides insight into the physical situations to which such definitions of probabilities are appropriate. 
  We give a new definition, based on considerations of well-posedness for a certain asymptotic initial value problem, of the phase space for the radiative degrees of freedom of the gravitational field in exact General Relativity. This space fibres over the space of final states, with the fibres being the purely radiative degrees of freedom. The symplectic form is rigorously identified.   The infrared sectors are shown to be the level surfaces of a moment map of an action of the quotient group Supertranslations/Translations. A similar result holds for Electromagnetism in Minkowski space. 
  We compute the motions of null infinity to which the components of the angular momentum of the gravitational field, as defined by Penrose, are conjugate. We find that the boosts are supplemented by anomalous translations. If $c_{ab}$ is the skew bivector determining the component of the angular momentum in question, the anomaly is proportional to $$c_{ab}t^b\phi ^{ad}\, ,$$ where $t^a$ is a unit vector in the direction of the Bondi--Sachs momentum and $\phi _{ab}$ is the quadrupole moment of the shear of the cut at which the angular momentum is evaluated. This effect persists in the weak--field limit. This surprising result is a general consequence of the requirement that angular momentum be supertranslation--invariant in a quiescent regime, and not some essentially twistorial peculiarity. 
  The Ashtekar formulation of 2+1 gravity differs from the geometrodynamical and Witten descriptions when the 2-metric is degenerate. We study the phase space of 2+1 gravity in the Ashtekar formulation to understand these degenerate solutions to the field equations. In the process we find two new systems of first class constraints which describe part of the degenerate sectors of the Ashtekar formulation. One of them also generalizes the Witten constraints. Finally we argue that the Ashtekar formulation has an arbitrarily large number of degrees of freedom in contrast to the usual descriptions. TO GET THE FIGURES CONTACT Barbero@suhep.phy.syr.edu or Madhavan@suhep.phy.syr.edu 
  A semiclassical two-dimensional dilaton-gravity model is obtained by dimensional reduction of the spherically symmetric five-dimensional Einstein equations and used to investigate black hole evaporation. It is shown that this model prevents the formation of naked singularity and allows spacetime wormholes to contribute the process of formation and evaporation of black holes. 
  We derive the effective action for a domain wall with small thickness in curved spacetime and show that, apart from the Nambu term, it includes a contribution proportional to the induced curvature. We then use this action to study the dynamics of a spherical thick bubble of false vacuum (de Sitter) surrounded by an infinite region of true vacuum (Schwarzschild). 
  Problem with the figures should be corrected. Apparently a broken uuencoder was the cause. 
  Problem with the figures should be corrected. Apparently a broken uuencoder was the cause. 
  We study the geodesics of the singularity free metric considered in the preceding Paper I and show that they are complete. This once again demonstrates the absence of singularity. The geodesic completeness is established in general without reference to any particular matter distribution. The metric is globally hyperbolic and causally stable. The question of inapplicability of the powerful singularity theorems in this case is discussed. 
  We show that the metric for the singularity free family of fluid models [3] can be obtained by a simple and natural inhomogenisation and anisotropisation procedure from Friedman--Robertson--Walker metric with negative curvature. The metric is unique for cylindrically symmetric spacetime with metric potentials being separable functions of radial and time coordinates. It turns out that fluid models separate out into two classes, with $\rho \not= \mu p$ in general but $\rho = 3p$ in particular and $\rho = p$. It is shown that in both the cases radial heat flow can be incorporated without disturbing the singularity free character of the spacetime. Further by introducing massless scalar field it is possible to open out a narrow window for $\mu, 4 \geq \mu \geq 3$. 
  This paper is concerned with two questions in the decoherent histories approach to quantum mechanics: the emergence of approximate classical predictability, and the fluctuations about it necessitated by the uncertainty principle. We consider histories characterized by position samplings at $n$ moments of time. We use this to construct a probability distribution on the value of (discrete approximations to) the field equations, $F = m \ddot x + V'(x) $, at $n-2$ times. We find that it is peaked around $F=0$; thus classical correlations are exhibited. We show that the width of the peak $ \Delta F$ is largely independent of the initial state and the uncertainty principle takes the form $2 \sigma^2 \ (\Delta F)^2 \ge { \hbar^2 / t^2 } $, where $\sigma$ is the width of the position samplings, and $t$ is the timescale between projections. We determine the modifications to this result when the system is coupled to a thermal environment. We show that the thermal fluctuations become comparable with the quantum fluctuations under the same conditions that decoherence effects come into play. We also study an alternative measure of classical correlations, namely the conditional probability of finding a sequence of position samplings, given that particular initial phase space data have occurred. We use these results to address the issue of the formal interpretation of the probabilities for sequences of position samplings in the decoherent histories approach to quantum mechanics. The decoherence of the histories is also briefly discussed. 
  There exists an upper limit on the mass of black holes when the cosmological constant $\Lambda$ is positive. We study the collision of two black holes whose total mass exceeds this limit. Our investigation is based on a recently discovered exact solution describing the collision of $Q=M$ black holes with $\Lambda > 0$. The global structure of this solution is analyzed. We find that if the total mass is less than the extremal limit, then the black holes coalesce. If it is greater, then a naked singularity forms to the future of a Cauchy horizon. However, the horizon is not smooth. Generically, there is a mild curvature singularity, which still allows geodesics to be extended. The implications of these results for cosmic censorship are discussed. 
  This paper proposes to generalize the histories included in Euclidean functional integrals from manifolds to a more general set of compact topological spaces. This new set of spaces, called conifolds, includes nonmanifold stationary points that arise naturally in a semiclasssical evaluation of such integrals; additionally, it can be proven that sequences of approximately Einstein manifolds and sequences of approximately Einstein conifolds both converge to Einstein conifolds. Consequently, generalized Euclidean functional integrals based on these conifold histories yield semiclassical amplitudes for sequences of both manifold and conifold histories that approach a stationary point of the Einstein action. Therefore sums over conifold histories provide a useful and self-consistent starting point for further study of topological effects in quantum gravity. Postscript figures available via anonymous ftp at black-hole.physics.ubc.ca (137.82.43.40) in file gen1.ps. 
  The research effort reported in this paper is directed, in a broad sense, towards understanding the small-scale structure of spacetime. The fundamental question that guides our discussion is ``what is the physical content of spacetime topology?" In classical physics, if spacetime, $(X, \tau )$, has sufficiently regular topology, and if sufficiently many fields exist to allow us to observe all continuous functions on $X$, then this collection of continuous functions uniquely determines both the set of points $X$ and the topology $\tau$ on it. To explore the small-scale structure of spacetime, we are led to consider the physical fields (the observables) not as classical (continuous functions) but as quantum operators, and the fundamental observable as not the collection of all continuous functions but the local algebra of quantum field operators. In pursuing our approach further, we develop a number of generalizations of quantum field theory through which it becomes possible to talk about quantum fields defined on arbitrary topological spaces. Our ultimate generalization dispenses with the fixed background topological space altogether and proposes that the fundamental observable should be taken as a lattice (or more specifically a ``frame," in the sense of set theory) of closed subalgebras of an abstract $C^{\ast}$ algebra. Our discussion concludes with the definition and some elementary 
  The mathematical formalism for linear quantum field theory on curved spacetime depends in an essential way on the assumption of global hyperbolicity. Physically, what lie at the foundation of any formalism for quantization in curved spacetime are the canonical commutation relations, imposed on the field operators evaluated at a global Cauchy surface. In the algebraic formulation of linear quantum field theory, the canonical commutation relations are restated in terms of a well-defined symplectic structure on the space of smooth solutions, and the local field algebra is constructed as the Weyl algebra associated to this symplectic vector space. When spacetime is not globally hyperbolic, e.g. when it contains naked singularities or closed timelike curves, a global Cauchy surface does not exist, and there is no obvious way to formulate the canonical commutation relations, hence no obvious way to construct the field algebra. In a paper submitted elsewhere, we report on a generalization of the algebraic framework for quantum field theory to arbitrary topological spaces which do not necessarily have a spacetime metric defined on them at the outset. Taking this generalization as a starting point, in this paper we give a prescription for constructing the field algebra of a (massless or massive) Klein-Gordon field on an arbitrary background spacetime. When spacetime is globally hyperbolic, the theory defined by our construction coincides with the ordinary Klein-Gordon field theory on a 
  Superspace parametrized by gauge potentials instead of metric three-geometries is discussed in the context of the Ashtekar variables. Among other things, an "internal clock" for the full theory can be identified. Gauge-fixing conditions which lead to the natural geometrical separation of physical from gauge modes are derived with the use of the metric in connection-superspace. A perturbation scheme about an unconventional background which is inaccessible to conventional variables is presented. The resultant expansion retains much of the simplicity of Ashtekar's formulation of General Relativity. 
  This paper examines the issues involved with concretely implementing a sum over conifolds in the formulation of Euclidean sums over histories for gravity. The first step in precisely formulating any sum over topological spaces is that one must have an algorithmically implementable method of generating a list of all spaces in the set to be summed over. This requirement causes well known problems in the formulation of sums over manifolds in four or more dimensions; there is no algorithmic method of determining whether or not a topological space is an n-manifold in five or more dimensions and the issue of whether or not such an algorithm exists is open in four. However, as this paper shows, conifolds are algorithmically decidable in four dimensions. Thus the set of 4-conifolds provides a starting point for a concrete implementation of Euclidean sums over histories in four dimensions. Explicit algorithms for summing over various sets of 4-conifolds are presented in the context of Regge calculus. Postscript figures available via anonymous ftp at black-hole.physics.ubc.ca (137.82.43.40) in file gen2.ps. 
  We study the collapse of a circular loop of cosmic string. The gravitational field of the string is treated using the weak field approximation. The gravitational radiation from the loop is evaluated numerically. The memtric of the loop near the point of collapse is found analytically. 
  In 1990 Senovilla$^1$ obtained an interestisng cosmological solution of Einstein's equations that was free of the big-bang singularity. It represented an inhomogeneous and anisotropic cylindrical model filled with disordered radiation, ${\bf \rho = 3p}$. The model was valid for ${\bf t \rightarrow - \infty }$ to ${\bf t \rightarrow \infty}$ having all physical and geometrical invariants finite and regular for the whole of spacetime. This was the first instance of a singularity free cosmological model satisfying all the energy and causality conditions and remaining true to general relativity (GR). Subsequently a family of singularity free models has been identified${\bf ^2}$. In this letter we wish to point out that a simple and natural inhomogenisation and anisotropisation, appropriate for cylindrical symmetry, of the Friedman-Robertson-Walker (FRW) model with negative curvature leads to the same singularity free family. It consists of the complete set of singularity free general solutions of Einstein's equations for perfect fluid when cylindrically symmetric metric potentials are assumed to be separable functions of radial and time coordinates. 
  A deeper understanding of the thermal properties of black holes than we presently have depends to a large degree on obtaining a firmer grasp of the properties of the entropy. For such an understanding we must at least know the basic relations among entropy, energy, and temperature of a black hole in thermal equilibrium with quantized matter fields. Limiting attention to spherical, uncharged (``Schwarzschild") holes, we will find that the basic Bekenstein-Hawking relations have to be generalized when the hole is dressed by quantum fields. Though this fact is not surprising, the corrections contain surprises and are very instructive. My purpose here is to discuss several aspects of this problem and to display some concrete results within the framework of the semi-classical theory of quantum fields in curved spacetime. 
  The energy--momentum radiated in gravitational waves by an isolated source is given by a formula of Bondi. This formula is highly non--local: the energy--momentum is not given as the integral of a well--defined local density. It has therefore been unclear whether the Bondi formula can be used to get information from gravity--wave measurements. In this note, we obtain, from local knowledge of the radiation field, a lower bound on the Bondi flux. 
  The entropy growth in a cosmological process of pair production is completely determined by the associated squeezing parameter, and is insensitive to the number of particles in the initial state. The total produced entropy may represent a significant fraction of the entropy stored today in the cosmic black-body radiation, provided pair production originates from a change in the background metric at a curvature scale of the Planck order. 
  It is well known that one can parameterize 2-D Riemannian structures by conformal transformations and diffeomorphisms of fiducial constant curvature geometries; and that this construction has a natural setting in general relativity theory in 2-D. I will show that a similar parameterization exists for 3-D Riemannian structures, with the conformal transformations and diffeomorphisms of the 2-D case replaced by a finite dimensional group of gauge transformations. This parameterization emerges from the theory of 3-D gravity coupled to topological matter. 
  Recently Brown and York have devised a new method for defining quasilocal energy in general relativity. Their method yields expressions for the quasilocal energy and momentum surface densities associated with the two-boundary of a spacelike slice of a spatially bounded spacetime. These expressions are essentially Arnowitt-Deser-Misner variables, but with canonical conjugacy defined with respect to the time history of the two-boundary. This paper introduces Ashtekar-type variables on the time history of the two-boundary and shows that these variables lead to elegant alternative expressions for the quasilocal surface densities. In addition, it is demonstrated here that both the boundary ADM variables and the boundary Ashtekar-type variables can be incorporated into a larger framework by appealing to the tetrad-dependent Sparling differential forms. 
  The in-in effective action formalism is used to derive the semiclassical correction to Einstein's equations due to a massless scalar quantum field conformally coupled to small gravitational perturbations in spatially flat cosmological models. The vacuum expectation value of the stress tensor of the quantum field is directly derived from the renormalized in-in effective action. The usual in-out effective action is also discussed and it is used to compute the probability of particle creation. As one application, the stress tensor of a scalar field around a static cosmic string is derived and the backreaction effect on the gravitational field of the string is discussed. 
  We look at the program of modelling a subatomic particle---one having mass, charge, and angular momentum---as an interior solution joined to a classical general-relativistic Kerr-Newman exterior spacetime. We find that the assumption of stationarity upon which the validity of the Kerr-Newman exterior solution depends is in fact violated quantum mechanically for all known subatomic particles. We conclude that the appropriate stationary spacetime matched to any known subatomic particle is flat space. 
  The reduction of the dreibein formalism of 2+1 General Relativity to the holonomies is explicitly performed. We also show explicitly how to relate these holonomies to a geometry classically, and how to generate these holonomies from any initial data for 2+1 gravity obeying the constraints. 
  The commutative algebra of functions on a manifold is extended to a noncommutative algebra by considering its tensor product with the algebra of nxn complex matrices. Noncommutative geometry is used to formulate an extension of the Einstein-Hilbert action. The result is shown to be equivalent to the usual Kaluza-Klein theory with the manifold SUn as an internal space, in a truncated approximation. 
  We examine the relationship between covariant and canonical (Ashtekar/Rovelli/Smolin) loop variables in the context of BF type topological field theories in 2+1 and 3+1 dimensions, with respective gauge groups SO(2,1) and SO(3,1). The latter model can be considered as the simplest topological gravity theory in 3+1 dimensions. We carry out the canonical quantization of this model in both the connection and loop representations, for the two spatial topologies $T^3$ and $S^2\times S^1$. 
  Although cosmological solutions to Einstein's equations are known to be generically singular, little is known about the nature of singularities in typical spacetimes. It is shown here how the operator splitting used in a particular symplectic numerical integration scheme fits naturally into the Einstein equations for a large class of cosmological models and thus allows study of their approach to the singularity. The numerical method also naturally singles out the asymptotically velocity term dominated (AVTD) behavior known to be characteristic of some of these models, conjectured to describe others, and probably characteristic of a subclass of the rest. The method is first applied to the unpolarized Gowdy T$^3$ cosmology. Exact pseudo-unpolarized solutions are used as a code test and demonstrate that a 4th order accurate implementation of the numerical method yields acceptable agreement. For generic initial data, support for the conjecture that the singularity is AVTD with geodesic velocity (in the harmonic map target space) < 1 is found. A new phenomenon of the development of small scale spatial structure is also observed. Finally, it is shown that the numerical method straightforwardly generalizes to an arbitrary cosmological spacetime on $T^3 \times R$ with one spacelike U(1) symmetry. 
  Static, spherically symmetric solutions of the field equations for a particular dimensional continuation of general relativity with negative cosmological constant are studied. The action is, in odd dimensions, the Chern-Simons form for the anti-de Sitter group and, in even dimensions, the Euler density constructed with the Lorentz part of the anti-de Sitter curvature tensor. Both actions are special cases of the Lovelock action, and they reduce to the Hilbert action (with negative cosmological constant) in the lower dimensional cases $\mbox{$\cal D$}=3$ and $\mbox{$\cal D$}=4$. Exact black hole solutions characterized by mass ($M$) and electric charge ($Q$) are found. In odd dimensions a negative cosmological constant is necessary to obtain a black hole, while in even dimensions, both asymptotically flat and asymptotically anti-de Sitter black holes exist. The causal structure is analyzed and the Penrose diagrams are exhibited. 
  I discuss the relation between arbitrarily high-order theories of gravity and scalar-tensor gravity at the level of the field equations and the action. I show that $(2n+4)$-order gravity is dynamically equivalent to Brans-Dicke gravity with an interaction potential for the Brans-Dicke field and $n$ further scalar fields. This scalar-tensor action is then conformally equivalent to the Einstein-Hilbert action with $n+1$ scalar fields. This clarifies the nature and extent of the conformal equivalence between extended gravity theories and general relativity with many scalar fields. 
  We rederive the universal bound on entropy with the help of black holes while allowing for Unruh--Wald buoyancy. We consider a box full of entropy lowered towards and then dropped into a Reissner--Nordstr\"om black hole in equilibrium with thermal radiation. We avoid the approximation that the buoyant pressure varies slowly across the box, and compute the buoyant force exactly. We find, in agreement with independent investigations, that the neutral point generically lies very near the horizon. A consequence is that in the generic case, the Unruh--Wald entropy restriction is neither necessary nor sufficient for enforcement of the generalized second law. Another consequence is that generically the buoyancy makes only a negligible contribution to the energy bookeeping, so that the original entropy bound is recovered if the generalized second law is assumed to hold. The number of particle species does not figure in the entropy bound, a point that has caused some perplexity. We demonstrate by explicit calculation that, for arbitrarily large number of particle species, the bound is indeed satisfied by cavity thermal radiation in the thermodynamic regime, provided vacuum energies are included. We also show directly that thermal radiation in a cavity in $D$ dimensional space also respects the bound regardless of the value of $D$. As an application of the bound we show that it strongly restricts the information capacity of the posited black hole remnants, so that they cannot serve to resolve the information paradox. 
  The Cosmic Background Explorer (COBE) detection of microwave background anisotropies may contain a component due to gravitational waves generated by inflation. It is shown that the gravitational waves from inflation might be seen using `beam-in-space' detectors, but not the Laser Interferometer Gravity Wave Observatory (LIGO). The central conclusion, dependent only on weak assumptions regarding the physics of inflation, is a surprising one. The larger the component of the COBE signal due to gravitational waves, the {\em smaller} the expected local gravitational wave signal. 
  Expectation values of one-loop renormalized thermal equilibrium stress-energy tensors of free conformal scalars, spin-${1 \over 2}$ fermions and U(1) gauge fields on a Schwarzschild black hole background are used as sources in the semi-classical Einstein equation. The back-reaction and new equilibrium metric are solved for at $O({\hbar})$ for each spin field. The nature of the modified black hole spacetime is revealed through calculations of the effective potential for null and timelike orbits. Significant novel features affecting the motions of both massive and massless test particles show up at lowest order in $\epsilon= (M_{Pl}/M)^2 < 1$, where $M$ is the renormalized black hole mass, and $M_{Pl}$ is the Planck mass. Specifically, we find the tendency for \underline{stable} circular photon orbits, an increase in the black hole capture cross sections, and the existence of a gravitationally repulsive region associated with the black hole which is generated from the U(1) back-reaction. We also consider the back-reaction arising from multiple fields, which will be useful for treating a black hole in thermal equilibrium with field ensembles belonging to gauge theories. 
  We consider a general, classical theory of gravity in $n$ dimensions, arising from a diffeomorphism invariant Lagrangian. In any such theory, to each vector field, $\xi^a$, on spacetime one can associate a local symmetry and, hence, a Noether current $(n-1)$-form, ${\bf j}$, and (for solutions to the field equations) a Noether charge $(n-2)$-form, ${\bf Q}$. Assuming only that the theory admits stationary black hole solutions with a bifurcate Killing horizon, and that the canonical mass and angular momentum of solutions are well defined at infinity, we show that the first law of black hole mechanics always holds for perturbations to nearby stationary black hole solutions. The quantity playing the role of black hole entropy in this formula is simply $2 \pi$ times the integral over $\Sigma$ of the Noether charge $(n-2)$-form associated with the horizon Killing field, normalized so as to have unit surface gravity. Furthermore, we show that this black hole entropy always is given by a local geometrical expression on the horizon of the black hole. We thereby obtain a natural candidate for the entropy of a dynamical black hole in a general theory of gravity. Our results show that the validity of the ``second law" of black hole mechanics in dynamical evolution from an initially stationary black hole to a final stationary state is equivalent to the positivity of a total Noether flux, and thus may be intimately related to the positive energy properties of the theory. The relationship between the derivation of our formula for black hole entropy and the derivation via ``Euclidean methods" also is explained. 
  We find a supersymmetrization of the Bianchi IX cosmology in terms of Ashtekar's new variables. This provides a framework for connecting the recent results of Graham and those of Ryan and Moncrief for quantum states of this model. These states are also related with the states obtained particularizing supergravity for a minisuperspace. Implications for the general theory are also briefly discussed. 
  A quantum hamiltonian which evolves the gravitational field according to time as measured by constant surfaces of a scalar field is defined through a regularization procedure based on the loop representation, and is shown to be finite and diffeomorphism invariant. The problem of constructing this hamiltonian is reduced to a combinatorial and algebraic problem which involves the rearrangements of lines through the vertices of arbitrary graphs. This procedure also provides a construction of the hamiltonian constraint as a finite operator on the space of diffeomorphism invariant states as well as a construction of the operator corresponding to the spatial volume of the universe. 
  The closed-universe recollapse conjecture is studied for the spherically symmetric spacetimes. It is proven that there exists an upper bound to the lengths of timelike curves in any Tolman spacetime that possesses $S^3$ Cauchy surfaces and whose energy density is positive. Furthermore, an explicit bound is constructed from the initial data for such a spacetime. 
  The stability of the minisuperspace model of the early universe is studied by solving the Wheeler-DeWitt equation numerically. We consider a system of Einstein gravity with a scalar field. When we solve the Wheeler-DeWitt equation, we pick up some inhomogeneous wave modes from the infinite number of modes adequately: degrees of freedom of the superspace are restricted to a finite one. We show that the minisuperspace is stable when a scale factor ($a$) of the universe is larger than a few times of the Planck length, while it becomes unstable when $a$ is comparable to the Planck length. 
  I give an informal overview of the decoherent histories approach to quantum mechanics, due to Griffiths, to Omn\`es, and to Gell-Mann and Hartle is given. Results on the connections between decoherence, records, correlation and entropy are described. The emphasis of the presentation is on understanding the broader meaning of the conditions of consistency and decoherence, and in particular, the extent to which they permit one to assign definite properties to the system. The quantum Brownian motion model is briefly discussed. (To appear in proceedings of the workshop, "Stochastic Evolution of Quantum States in Open Systems and Measurement Processes", Budapest, March, 1993, edited by L.Diosi). 
  An extended analysis is made of the Gell-Mann and Hartle axioms for a generalised `histories' approach to quantum theory. Emphasis is placed on finding equivalents of the lattice structure that is employed in standard quantum logic. Particular attention is given to `quasi-temporal' theories in which the notion of time-evolution is less rigid than in conventional Hamiltonian physics; theories of this type are expected to arise naturally in the context of quantum gravity and quantum field theory in a curved space-time. The quasi-temporal structure is coded in a partial semi-group of `temporal supports' that underpins the lattice of history propositions. Non-trivial examples include quantum field theory on a non globally-hyperbolic spacetime, and a simple cobordism approach to a theory of quantum topology.   It is shown how the set of history propositions in standard quantum theory can be realised in such a way that each history proposition is represented by a genuine projection operator. This provides valuable insight into the possible lattice structure in general history theories, and also provides a number of potential models for theories of this type. 
  This essay argues that when measurement processes involve energies of the order of the Planck scale, the fundamental assumption of locality may no longer be a good approximation. Idealized position measurements of two distinguishable spin-$0$ particles are considered. The measurements alter the space-time metric in a fundamental manner governed by the commutation relations $[x_i\,\,p_j]= i\hbar\,\delta_{ij}$ and the classical field equations of gravitation. This {\it in-principle} unavoidable change in the space-time metric destroys the commutativity (and hence locality) of position measurement operators. 
  In canonical quantum gravity certain topological properties of 3-manifolds are of interest. This article gives an account of those properties which have so far received sufficient attention, especially those concerning the diffeomorphism groups of 3-manifolds. We give a summary of these properties and list some old and new results concerning them. The appendix contains a discussion of the group of large diffeomorphisms of the $l$-handle 3-manifold. 
  We study the propagator of a non-relativistic, non-interacting particle in any non-relativistic ``time-machine'' spacetime of the type shown in Fig.~1: an external, flat spacetime in which two spatial regions, $V_-$ at time $t_-$ and $V_+$ at time $t_+$, are connected by two temporal wormholes, one leading from the past side of $V_-$ to t the future side of $V_+$ and the other from the past side of $V_+$ to the future side of $V_-$. We express the propagator explicitly in terms of those for ordinary, flat spacetime and for the two wormholes; and from that expression we show that the propagator satisfies completeness and unitarity in the initial and final ``chronal regions'' (regions without closed timelike curves) and its propagation from the initial region to the final region is unitary. However, within the time machine it satisfies neither completeness nor unitarity. We also give an alternative proof of initial-region-to-final-region unitarity based on a conserved current and Gauss's theorem. This proof can be carried over without change to most any non-relativistic time-machine spacetime; it is the non-relativistic version of a theorem by Friedman, Papastamatiou and Simon, which says that for a free scalar field, quantum mechanical unitarity follows from the fact that the classical evolution preserves the Klein-Gordon inner product. 
  We present a number of new, exact scalar field cosmologies where the potential consists of two or more exponential terms. Such potentials are motivated by supergravity or superstring models formulated in higher dimensional spacetimes that have been compactified to (3+1)-dimensions. We have found solutions in both curved and flat Robertson Walker spacetimes. These models have a diverse range of properties and often possess several distinct phases, with a smooth transition between ordinary and inflationary expansion. While exponential potentials typically produce powerlaw inflation, we find models where the inflationary period contains eras of both powerlaw and exponential growth. 
  We use the quantum Brownian model to derive the uncertainty relation for a quantum open system in an arbitrarily-squeezed initial state interacting with an environment at finite temperature. We examine the relative importance of the quantum and thermal fluctuations in the evolution of the system towards equilibrium with the aim of clarifying the meaning of quantum, classical and thermal. We show that upon contact with the bath the system evolves from a quantum-dominated state to a thermal-dominated state in a time which is the same as the decoherence time calculated before in the context of quantum to classical transitions. We also use these results to deduce the conditions when the two basic postulates of quantum statistical mechanics become valid. 
  We expand on the idea that spacetime signature should be treated as a dynamical degree of freedom in quantum field theory. It has been argued that the probability distribution for signature, induced by massless free fields, is peaked at the Lorentzian value uniquely in D=4 dimensions. This argument is reviewed, and certain consistency constraints on the generalized signature (i.e. the tangent space metric $\eta_{ab}(x)=\mbox{diag}[e^{i\theta(x)},1,1,1]$) are derived. It is shown that only one dynamical "Wick angle" $\theta(x)$ can be introduced in the generalized signature, and the magnitude of fluctuations away from Lorentzian signature $\delta \theta = \pi - \theta$ is estimated to be of order $(l_P/R)^3$, where $l_P$ is the Planck length, and $R$ is the length scale of the Universe. For massless fields, the case of D=2 dimensions and the case of supersymmetry are degenerate, in the sense that no signature is preferred. Mass effects lift this degeneracy, and we show that a dynamical origin of Lorentzian signature is also possible for (broken) supersymmetry theories in D=6 dimensions, in addition to the more general non-supersymmetric case in D=4 dimensions. 
  A calculation of the no-boundary wave-function of the universe is put forward for a spacetime with negative curvature. A semi-classical Robertson-Walker approximation is attempted and two solutions to the field equations, one Lorentzian and the other a tunneling one are found. The regularity of those solutions are analysed explicitly, both in 2+1 and 3+1 dimensions and a conical singularity is found at the origin of the time axis, contradicting the no-boundary assumption. 
  In this note we examine some recently proposed solutions of the linearized vacuum Einstein equations. We show that such solutions are {\it not} symmetries of the Einstein equations, because of a crucial integrability condition. 
  The use of $\sum \exp(iS[x])$ as the generic form for a sum over histories in configuration space is discussed critically and placed in its proper context. The standard derivation of the sum over paths by discretizing the paths is reviewed, and it is shown that the form $\sum \exp(iS[x])$ is justified only for Schrodinger-type systems which are at most second order in the momenta. Extending this derivation to the relativistic free particle, the causal Green's function is expressed as a sum over timelike paths, and the Feynman Green's function is expressed both as a sum over paths which only go one way in time and as a sum over paths which move forward and backward in time. The weighting of the paths is shown not to be $\exp(iS[x])$ in any of these cases. The role of the inner product and the operator ordering of the wave equation in defining the sum over histories is discussed. 
  To assess the validity of the Belinskii, Khalatnikov, and Lifshitz (BKL) approximation to Mixmaster dynamics, it would be useful to evaluate the BKL discrete parameters as a byproduct of the numerical solution of Einstein's equations. An algorithm to do this and results for a typical trajectory are presented. 
  By a combination of analytical and numerical methods, the density profile of a momentarily at rest spherical star is varied, and the corresponding response in the area of the spherical shells is monitored. It is shown that the inner apparent horizon (if it exists) must lie within or at most on the star's surface, while no such restriction is found for the outer apparent horizon. However, an apparent horizon lying in the vacuum region will always have non vanishing area, as long as the ADM mass of the system is non zero. Furthermore for density profiles not decreasing inwards, it appears that all spherical trapped surfaces lie on a thick spherical shell. Finally for a uniform density star a simple criterion is found, relating density and proper radius that guarantees the presence or absence of trapped regions. 
  We investigate the space ${\cal M}$ of classical solutions to Witten's formulation of 2+1 gravity on the manifold ${\bf R} \times T^2$. ${\cal M}$ is connected, unlike the spaces of classical solutions in the cases where $T^2$ is replaced by a higher genus surface. Although ${\cal M}$ is neither Hausdorff nor a manifold, removing from ${\cal M}$ a set of measure zero yields a manifold which is naturally viewed as the cotangent bundle over a non-Hausdorff base space~${\cal B}$. We discuss the relation of the various parts of ${\cal M}$ to spacetime metrics, and various possibilities of quantizing~${\cal M}$. There exist quantizations in which the exponentials of certain momentum operators, when operating on states whose support is entirely on the part of ${\cal B}$ corresponding to conventional spacetime metrics, give states whose support is entirely outside this part of~${\cal B}$. Similar results hold when the gauge group ${\rm SO}_0(2,1)$ is replaced by ${\rm SU}(1,1)$. 
  In this article the particle creation process of scalar and spin 1/2 particles in a spatially open cosmological model associated with a universe filled by radiation and dustlike matter. The Klein-Gordon and Dirac equations are solved via separation of variables. After comparing the{\it \ in }and% {\it \ out} vacua, we obtain that the number of created particles corresponds to   Planckian and Fermi-Dirac distributions for the scalar and Dirac cases 
  This paper gives a brief description of the derivation of a composition law for the propagator of a relativistic particle, in a sum over histories quantization. The extension of this derivation to the problem of finding a composition law for quantum cosmology is also discussed. (For the proceedings of Journees Relativistes 93) 
  We apply the inverse scattering method to the midi-superspace models that are characterized by a two-parameter Abelian group of motions with two spacelike Killing vectors. We present a formulation that simplifies the construction of the soliton solutions of Belinski\v i and Zakharov. Furthermore, it enables us to obtain the zero curvature formulation for these models. Using this, and imposing periodic boundary conditions corresponding to the Gowdy models when the spatial topology is a three torus $T ^3$, we show that the equation of motion for the monodromy matrix is an evolution equation of the Heisenberg type. Consequently, the eigenvalues of the monodromy matrix are the generating functionals for the integrals of motion. Furthermore, we utilise a suitable formulation of the transition matrix to obtain explicit expressions for the integrals of motion. This involves recursion relations which arise in solving an equation of Riccati type. In the case when the two Killing vectors are hypersurface orthogonal the integrals of motion have a particularly simple form. 
  Physical situations involving multiplicative noise arise generically in cosmology and field theory. In this paper, the focus is first on exact nonlinear Langevin equations, appropriate in a cosmologica setting, for a system with one degree of freedom. The Langevin equations are derived using an appropriate time-dependent generalization of a model due to Zwanzig. These models are then extended to field theories and the generation of multiplicative noise in such a context is discussed. Important issues in both the cosmological and field theoretic cases are the fluctuation-dissipation relations and the relaxation time scale. Of some importance in cosmology is the fact that multiplicative noise can substantially reduce the relaxation time. In the field theoretic context such a noise can lead to a significant enhancement in the nucleation rate of topological defects. 
  Original abstract: Consider the worldline of a charged particle in a static spacetime. Contraction of the time-translation Killing field with the retarded electromagnetic energy-momentum tensor gives a conserved electromagnetic energy vector which can be used to define the radiated electromagnetic energy. This note points out that for a conformally flat spacetime, the radiated energy is the same as for a flat spacetime (i.e. Minkowski space). This appears to be inconsistent with an equation of motion for such particles derived by DeWitt and Brehme and later corrected by Hobbs [End of original abstract]   New abstract: Same as old abstract with last sentence deleted. The body of the paper is the same as previously. A new Appendix 2 has been added discussing implications to the previous arguments of recent work of Sonego (J. Math. Phys. 40 (1999), 3381-3394) and of Quinn and Wald (Phys. Rev. D 60 (1999), gr-qc/9610053). 
  We discuss the relevance of quantum gravitational corrections to the functional Schr\"odinger equation for the information loss paradox in black hole evaporation. These corrections are found from the Wheeler-DeWitt equation through a semiclassical expansion scheme. The dominant contribution in the final evaporation stage, when the black hole approaches the Planck regime, is a term which explicitly violates unitarity in the non-gravitational sector. While pure states remain pure, there is an increase in the degree of purity for non-pure states in this sector. This result holds irrespective of whether full quantum gravity respects unitarity or not. 
  This paper gives an elementary introduction to some of the conceptual problems of quantum cosmology. Contents: 1. Why quantum cosmology? 2. Time in quantum gravity 3.Decoherence and the recovery of the Schrodinger equation 4. The direction of time 
  We construct Nicolai maps for $N=2$ supersymmetric extensions of minisuperspace models. It is shown that Nicolai maps exist for only a very restricted set of states. In the models considered these are the two states corresponding to the empty and the filled fermion sectors. The form of the Nicolai maps in these sectors is given explicitly, and it is shown that they have a natural stochastic interpretation. This result also suggests a probabilistic interpretation of the wave function. 
  For normal thermodynamic systems superadditivity $\S$, homogeneity $\H$ and concavity $\C$ of the entropy hold, whereas for $(3+1)$-dimensional black holes the latter two properties are violated. We show that $(1+1)$-dimensional black holes exhibit qualitatively new types of thermodynamic behaviour, discussed here for the first time, in which $\C$ always holds, $\H$ is always violated and $\S$ may or may not be violated, depending of the magnitude of the black hole mass. Hence it is now seen that neither superadditivity nor concavity encapsulate the meaning of the second law in all situations. 
  We present a classification of the possible regular, spherically symmetric solutions of the Einstein-Yang-Mills system which is based on a bundle theoretical analysis for arbitrary gauge groups. It is shown that such solitons must be of magnetic type, at least if the magnetic Yang-Mills charge vanishes. Explicit expressions for the Chern-Simons numbers of these selfgravitating Yang-Mills solitons are derived, which involve only properties of irreducible root systems and some information about the asymptotics of the solutions. It turns out, as an example, that the Chern-Simons numbers are always half-integers or integers for the gauge groups $SU(n)$. Possible physical implications of these results, which are based on analogies with the unstable sphaleron solution of the electroweak theory, are briefly indicated. 
  By using the minisuperspace model for the interior metric ofstatic black holes, we solve the Wheeler-DeWitt equation to study quantum mechanics of the horizon geometry. Our basic idea is to introduce the gravitational mass and the expansions of null rays as quantum operators. Then, the exact wave function is found as a mass eigenstate, and the radius of the apparent horizon is quantum-mechanically defined. In the evolution of the metric variables, the wave function changes from a WKB solution giving the classical trajectories to a tunneling solution. By virtue of the quantum fluctuations of the metric evolution beyond the WKB approximation, we can observe a static black hole state with the apparent horizon separating from the event horizon. 
  We investigate the global structure of the space time with a spherically symmetric inhomogeneity using a metric junction, and classify all possible types. We found that a motion with a negative gravitational mass is possible although the energy condition of the matter is not violated. Using the result, formation of black hole and worm hole during the inflationary era is discussed. 
  We investigate the evolution of the apparent horizons in a numerically gererated worm hole spacetime. The behavior of the apparent horizons is affected by the dynamics of the matter field. By using the local mass of the system, we interpret the evolution of the worm hole structure. Figures are available by mail to author. 
  The quantum stress tensor $<T_{\mu\nu}>$ is calculated in the 2+1 dimensional black hole found by Banados, Teitelboim, and Zanelli. The Greens function, from which $<T_{\mu\nu}> $ is derived, is obtained by the method of images.  For the non-rotating black hole, it is shown that $<T_{\mu\nu}>$ is finite on the event horizon, but diverges at the singularity. For the rotating solution, the stress tensor is finite at the outer horizon, but diverges near the inner horizon. This suggests that the inner horizon is quantum mechanically unstable against the formation of a singularity. 
  Modes of physical fields which are located inside a horizon and which cannot be observed by a distant observer are identified with dynamical degrees of freedom of a black hole. A new invariant statistical mechanical definition of a black-hole's entropy is proposed. It is shown that the main contribution to the entropy is given by thermally excited `invisible' modes propagating in the close vicinity of the horizon. A calculation based on the proposed definition yields a value of the entropy which is in good agreement with the usually adopted value. 
  In contrast to other approaches to (2+1)-dimensional quantum gravity, the Wheeler-DeWitt equation appears to be too complicated to solve explicitly, even for simple spacetime topologies. Nevertheless, it is possible to obtain a good deal of information about solutions and their interpretation. In particular, strong evidence is presented that Wheeler-DeWitt quantization is not equivalent to reduced phase space quantization. 
  Table of contents:   Editorial 1   Correspondents 3   Some recent work in general relativistic Astrophysics 4   Two dimensional black holes 6   Resonant-mass gravitational wave detectors: an update 8   Universality and scaling in gravitational collapse 9   Gravitational Wave memories upgraded 11   Conference report: quantum aspects of black holes 13   Conference report: knots and quantum gravity 14   Conference report: deterministic chaos in GR 17 
  In a space-time with cosmological constant $\Lambda>0$ and matter satisfying the dominant energy condition, the area of a black or white hole cannot exceed $4\pi/\Lambda$. This applies to event horizons where defined, i.e. in an asymptotically deSitter space-time, and to outer trapping horizons (cf. apparent horizons) in any space-time. The bound is attained if and only if the horizon is identical to that of the degenerate `Schwarzschild-deSitter' solution. This yields a topological restriction on the event horizon, namely that components whose total area exceeds $4\pi/\Lambda$ cannot merge. We discuss the conjectured isoperimetric inequality and implications for the cosmic censorship conjecture. 
  A new definition of the Wigner function for quantum fields coupled to curved space--time and an external Yang--Mills field is studied on the example of a scalar and a Dirac fields. The definition uses the formalism of the tangent bundles and is explicitly covariant and gauge invariant. Derivation of collisionless quantum kinetic equations is carried out for both quantum fields by using the first order formalism of Duffin and Kemmer. The evolution of the Wigner function is governed by the quantum corrected Liouville--Vlasov equation supplemented by the generalized mass--shell constraint. The structure of the quantum corrections is perturbatively found in all adiabatic orders. The lowest order quantum--curvature corrections coincide with the ones found by Winter. 
  To clarify some issues raised by D'Eath's recent proposal for the physical states of $N=1$ supergravity in four dimensions, we study pure (topological) $N=2$ supergravity in three dimensions, which is formally very similar, but much easier to solve. The wave functionals solving the quantum constraints can be understood in terms of arbitrary functions on the space of moduli and supermoduli, which is not Hausdorff. We discuss the implications for the wave functionals and show that these are not amenable to expansions in fermionic coordinates, but can serve as lowest-order solutions to the quantum constraints in an expansion in $\hbar$ in more realistic theories. 
  We show that for $U(1)$-symmetric spacetimes on $S^3 \times R$ a constant of motion associated with the well known Geroch transformation, a functional $K[h_{ij},\pi^{ij}]$, quadratic in gravitational momenta, is strictly positive in an open subset of the set of all $U(1)$-symmetric initial data, and therefore not weakly zero. The Mixmaster initial data appear to be on the boundary of that set. We calculate the constant of motion perturbatively for the Mixmaster spacetime and find it to be proportional to the minisuperspace Hamiltonian to the first order in the Misner anisotropy variables, i.e. weakly zero. Assuming that $K$ is exactly zero for the Mixmaster spacetime, we show that Geroch's transformation, when applied to the Mixmaster spacetime, gives a new \mbox{$U(1)$-symmetric} solution of the vacuum Einstein equations, globally defined on \mbox{$S^2 \times S^1 \times R$},which is non-homogeneous and presumably exhibits Mixmaster-like complicated dynamical behavior. 
  A spontaneous symmetry breaking mechanism is used in quantum gravity to obtain a convergent positive definite density-matrix as the most general quantum state of Euclidean wormholes 
  By assuming that only (i) bilocal vertex operators which are diagonal with respect to the basis for local field operators, and (ii) the convergent elements with nonzero positive energy of the density matrix representing the quantum state of multiply-connected wormholes, contribute the path integral that describes the effects of wormholes on ordinary matter fields at low energy, it is obtained that the probability measure for multiply connected wormholes with nondegenerate energy spectrum is given in terms of a Planckian probability distribution for the momenta of a quantum field $\frac{1}{2}\alpha^ {2}$, where the $\alpha$'s are the Coleman parameters, rather than a classical gaussian distribution law, and that an observable classical universe can exist if, and only if, such multiply connected wormholes are allowed to occur. 
  A transfer matrix formalism applicable to certain reparametrization invariant theories, including quantum gravity, is proposed. In this formulation it is found that every stationary state in quantum gravity satisfies a Wheeler-DeWitt equation, but each with a different value of the Planck mass; the value $m_{Planck}^4$ turns out to be proportional to the eigenvalue of the evolution operator. As a consequence, the fact that the Universe is non-stationary implies that it is not in an eigenstate of Newton's constant. 
  Static, spherically symmetric solutions of the field equations for a particular dimensional continuation of general relativity with negative cosmological constant are found. In even dimensions the solution has many similarities with the \Sch\ metric. In odd dimensions, the equations of motion are explicitly anti de-Sitter invariant, and the solution is alike in many ways to the 2+1 black hole. 
  Spacetime must be foliable by spacelike surfaces for the quantum mechanics of matter fields to be formulated in terms of a unitarily evolving state vector defined on spacelike surfaces. When a spacetime cannot be foliated by spacelike surfaces, as in the case of spacetimes with closed timelike curves, a more general formulation of quantum mechanics is required. In such generalizations the transition matrix between alternatives in regions of spacetime where states {\it can} be defined may be non-unitary. This paper describes a generalized quantum mechanics whose probabilities consistently obey the rules of probability theory even in the presence of such non-unitarity. The usual notion of state on a spacelike surface is lost in this generalization and familiar notions of causality are modified. There is no signaling outside the light cone, no non-conservation of energy, no ``Everett phones'', and probabilities of present events do not depend on particular alternatives of the future. However, the generalization is acausal in the sense that the existence of non-chronal regions of spacetime in the future can affect the probabilities of alternatives today. The detectability of non-unitary evolution and violations of causality in measurement situations are briefly considered. The evolution of information in non-chronal spacetimes is described. 
  We obtain a general exact solution of the Einstein field equations for the anisotropic Bianchi type I universes filled with an exponential-potential scalar field and study their dynamics. It is shown, in agreement with previous studies, that for a wide range of initial conditions the late-time behaviour of the models is that of a power-law inflating FRW universe. This property, does not hold, in contrast, when some degree of inhomogeneity is introduced, as discussed in our following paper II. 
  We obtain exact solutions for the Einstein equations with an exponential-potential scalar field (\(V=\Lambda e^{k\phi}\)) which represent simple inhomogeneous generalizations of Bianchi I cosmologies. Studying these equations numerically we find that in most of the cases there is a certain period of inflationary behaviour for \(k^2<2\). We as well find that for \(k^2>2\) the solutions homogenize generically at late times. Yet, {\em none of the solutions} isotropize. For some particular values of the integration constants we find a multiple inflationary behaviour for which the deceleration and the inflationary phases interchange each other several times during the history of the model. 
  We studied the formation of compact bosonic objects through a dissipationless cooling mechanism. Implications of the existence of this mechanism are discussed, including the abundance of bosonic stars in the universe, and the possibility of ruling out the axion as a dark matter candidate. 
  We study the head-on collision of two equal mass, nonrotating black holes. We consider a range of cases from holes surrounded by a common horizon to holes initially separated by about $20M$, where $M$ is the mass of each hole. We determine the waveforms and energies radiated for both the $\ell = 2$ and $\ell=4$ waves resulting from the collision. In all cases studied the normal modes of the final black hole dominate the spectrum. We also estimate analytically the total gravitational radiation emitted, taking into account the tidal heating of horizons using the membrane paradigm, and other effects. For the first time we are able to compare analytic calculations, black hole perturbation theory, and strong field, nonlinear numerical calculations for this problem, and we find excellent agreement. 
  A simple spacetime wormhole, which evolves classically from zero throat radius to a maximum value and recontracts, can be regarded as one possible mode of fluctuation in the microscopic ``spacetime foam'' first suggested by Wheeler. The dynamics of a particularly simple version of such a wormhole can be reduced to that of a single quantity, its throat radius; this wormhole thus provides a ``minisuperspace model'' for a structure in Lorentzian-signature foam. The classical equation of motion for the wormhole throat is obtained from the Einstein field equations and a suitable equation of state for the matter at the throat. Analysis of the quantum behavior of the hole then proceeds from an action corresponding to that equation of motion. The action obtained simply by calculating the scalar curvature of the hole spacetime yields a model with features like those of the relativistic free particle. In particular the Hamiltonian is nonlocal, and for the wormhole cannot even be given as a differential operator in closed form. Nonetheless the general solution of the Schr\"odinger equation for wormhole wave functions, i.e., the wave-function propagator, can be expressed as a path integral. Too complicated to perform exactly, this can yet be evaluated via a WKB approximation. The result indicates that the wormhole, classically stable, is quantum-mechanically unstable: A Feynman-Kac decomposition of the WKB propagator yields no spectrum of bound states. Though an initially localized wormhole wave function may oscillate for many classical expansion/recontraction periods, it must eventually leak to large radius values. The possibility of such a mode unstable against growth, combined with 
  The most general dilaton gravity theory in 2 spacetime dimensions is considered. A Hamiltonian analysis is performed and the reduced phase space, which is two dimensional, is explicitly constructed in a suitable parametrization of the fields. The theory is then quantized via the Dirac method in a functional Schrodinger representation. The quantum constraints are solved exactly to yield the (spatial) diffeomorphism invariant quantum wave functional for all theories considered. This wave function depends explicitly on the (single) configuration space coordinate as well as on the imbedding of space into spacetime (i.e. on the choice of time). 
  In the present article we solve, via separation of variables, the massless Dirac equation in a nonstationary rotating, causal G\"odel-type cosmological universe, having a constant rotational speed in all the points of the space. We compute the frequency spectrum. We show that the spectrum of massless Dirac particles is discrete and unbounded. 
  This paper is a review of the relationship between the metric formulation of (2+1)-dimensional gravity and the loop observables of Rovelli and Smolin. I emphasize the possibility of reconstructing the geometry, via the theory of geometric structures, from the values of the loop variables. I close with a brief discussion of implications for quantization, particularly for covariant canonical approaches to quantum gravity. 
  Recently, the possibility has emerged of an early detection of astrophysical gravity waves. In certain astronomical configurations, and through a new light-deflection effect, gravity waves can cause apparent shifts in stellar angular positions as large as $10^{-7}arcsec$. In these same configurations, the magnitude of the gravity-wave-induced time-delay effect can exceed $10^{-14}$. Both these figures lie just at present-day theoretical limits of detectability. For instance, cases are described where the very faint neutron-star gravity waves could soon become detectable. The detection meant here involves direct observations of the very wave-forms. 
  : From the epoch of recombination $(z\approx 10^3)$ till today, the typical density contrasts have grown by a factor of about $10^6$ in a Friedmann universe with $\Omega=1$. However, during the same epoch the typical gravitational potential has grown only by a factor of order unity. We present theoretical arguments explaining the origin of this approximate constancy of gravitational potential. This fact can be exploited to provide a new, powerful, approximation scheme to study the formation of nonlinear structures in the universe. The essential idea of this method is to evolve the initial distribution of particles using a gravitational potential frozen in time (Frozen Potential Approximation). This approximation provides valuable insight into understanding various features of nonlinear evolution; for example, it provides a simple explanation as to why pancakes remain thin during the evolution even in the absence of any artificial, adhesion-like, damping terms. We compare the trajectories of particles in various approximations. We also discuss a few applications of the frozen potential approximation. 
  We show that the two-dimensional theory of Teitelboim and Jackiw has a black hole solution, with two surprising properties: first, it has constant curvature, and second, is free of spacetime singularities. The maximally extended spacetime consists of an infinite chain of universes connected by timelike wormholes. 
  We carry out the nonperturbative canonical quantization of several types of cosmological models that have already been studied in the geometrodynamic formulation using the complex path-integral approach. We establish a relation between the choices of complex contours in the path integral and the sets of reality conditions for which the metric representation is well defined, proving that the ambiguity in the selection of complex contours disappears when one imposes suitable reality conditions. In most of the cases, the wave functions defined by means of the path integral turn out to be non-normalizable and cannot be accepted as proper quantum states. Moreover, the wave functions of the Universe picked out in quantum cosmology by the no-boundary condition and the tunneling proposals do not belong, in general, to the Hilbert space of quantum states. Finally, we show that different sets of reality conditions can lead to equivalent quantum theories. This fact enables us to extract physical predictions corresponding to Lorentzian gravity from quantum theories constructed with other than Lorentzian reality conditions. 
  A method for calculating the retarded Green's function for the gravitational wave equation in Friedmann-Roberson-Walker spacetimes, within the formalism of linearized Einstein gravity is developed. Hadamard's general solution to Cauchy's problem for second-order, linear partial differential equations is applied to the FRW gravitational wave equation. The retarded Green's function may be calculated for any FRW spacetime, with curved or flat spatial sections, for which the functional form of the Ricci scalar curvature $R$ is known. The retarded Green's function for gravitational waves propagating through a cosmological fluid composed of both radiation and dust is calculated analytically for the first time. It is also shown that for all FRW spacetimes in which the Ricci scalar curvatures does not vanish, $R \neq 0$, the Green's function violates Huygens' principle; the Green's function has support inside the light-cone due to the scatter of gravitational waves off the background curvature. 
  The Euclidean black hole has topology $\Re^2 \times {\cal S}^{d-2}$. It is shown that -in Einstein's theory- the deficit angle of a cusp at any point in $\Re^2$ and the area of the ${\cal S}^{d-2}$ are canonical conjugates. The black hole entropy emerges as the Euler class of a small disk centered at the horizon multiplied by the area of the ${\cal S}^{d-2}$ there.These results are obtained through dimensional continuation of the Gauss-Bonnet theorem. The extension to the most general action yielding second order field equations for the metric in any spacetime dimension is given. 
  The role of space-time torsion in general relativity is reviewed in accordance with some recent results on the subject. It is shown that, according to the connection compatibility condition, the usual Riemannian volume element is not appropriate in the presence of torsion. A new volume element is proposed and used in the Lagrangian formulation for Einstein-Cartan theory of gravity.   The dynamical equations for the space-time geometry and for matter fields are obtained, and some of their new predictions and features are discussed. In particular, one has that torsion propagates and that gauge fields can interact with torsion without the breaking of gauge invariance. It is shown also that the new Einstein-Hilbert action for Einstein-Cartan theory may provide a physical interpretation for dilaton gravity in terms of the non-riemannian structure of space-time. 
  We investigate the properties of a quantum Robertson - Walker universe described by the Wheeler -- DeWitt equation. The universe is filled with a quantum Yang -- Mills uniform field. This is then a quantum mini copy of the standard model of our universe. We discuss the interpretation of the Wheeler -- DeWitt wave function using the correspondence principle to connect $\vert\psi\vert^2$ for large quantum numbers to the classical probability for a radiation dominated universe. This can be done in any temporal gauge. The correspondence principle determines the Schr\"odinger representation of the momentum associated to the gravitational degree of freedom. We also discuss the measure in the mini--superspace needed to ensure invariance of the quantum description under change of the temporal gauge. Finally, we examine the behaviour of $\vert\psi\vert^2$ in inflationary conditions. 
  Recently Penrose, Sorkin and Woolgar have developed a new technique for proving the positive mass theorem in general relativity. We extend their result to produce a new inequality relating the mass, electric and scalar charges in theories coupling to a dilaton in the usual way. Using a five dimensional formalism, our result provides new information not available from the existing techniques. The main result may be simply expressed as   M + g\Sigma \ge { \sqrt{1+g^2} |Q| \over \sqrt {4\pi G} } . where `M' is the A.D.M. mass, `Q' is the electric charge, `\Sigma' the scalar charge and `g' is the dilaton coupling parameter. 
  Starting from the Ashtekar Hamiltonian variables for general relativity, the self-dual Einstein equations (SDE) may be rewritten as evolution equations for three divergence free vector fields given on a three dimensional surface with a fixed volume element.   From this general form of the SDE, it is shown how they may be interpreted as the field equations for a two dimensional field theory. It is further shown that these equations imply an infinite number of non-local conserved currents.   A specific way of writing the vector fields allows an identification of the full SDE with those of the two dimensional chiral model, with the gauge group being the group of area preserving diffeomorphisms of a two dimensional surface. This gives a natural Hamiltonian formulation of the SDE in terms of that of the chiral model. The conservation laws using the explicit chiral model form of the equations are also given. 
  (3+1) (continuous time) Regge calculus is reduced to Hamiltonian form. The constraints are classified, classical and quantum consequences are discussed. As basic variables connection matrices and antisymmetric area tensors are used supplemented with appropriate bilinear constraints. In these variables the action can be made quasipolinomial with $\arcsin$ as the only deviation from polinomiality. In comparison with analogous formalism in the continuum theory classification of constraints changes: some of them disappear, the part of I class constraints including Hamiltonian one become II class (and vice versa, some new constraints arise and some II class constraints become I class). As a result, the number of the degrees of freedom coincides with the number of links in 3-dimensional leaf of foliation. Moreover, in empty space classical dynamics is trivial: the scale of timelike links become zero and spacelike links are constant. 
  As basic variables in general relativity (GR) are chosen antisymmetric connection and bivectors - bilinear in tetrad area tensors subject to appropriate (bilinear) constraints. In canonical formalism we get theory with polinomial constraints some of which are II class. On partial resolving the latter we get another polinomial formulation. Separating self- and antiselfdual parts of antisymmetric tensors we come to Ashtekar constraints including those known as "reality conditions" which connect self- and antiselfdual sectors of the theory. These conditions form second class system and cannot be simply imposed on quantum states (or taken as initial conditions in classical theory). Rather these should be taken into account in operator sence by forming corresponding Dirac brackets. As a result, commutators between canonical variables are no longer polinomial, and even separate treatment of self- and antiselfdual sectors is impossible. 
  An approach to black hole quantization is proposed wherein it is assumed that quantum coherence is preserved. A consequence of this is that the Penrose diagram describing gravitational collapse will show the same topological structure as flat Minkowski space. After giving our motivations for such a quantization procedure we formulate the background field approximation, in which particles are divided into "hard" particles and "soft" particles. The background space-time metric depends both on the in-states and on the out-states. We present some model calculations and extensive discussions. In particular, we show, in the context of a toy model, that the $S$-matrix describing soft particles in the hard particle background of a collapsing star is unitary, nevertheless, the spectrum of particles is shown to be approximately thermal. We also conclude that there is an important topological constraint on functional integrals. 
  The gravitational effect of vacuum polarization in space exterior to a particle in (2+1)-dimensional Einstein theory is investigated. In the weak field limit this gravitational field corresponds to an inverse square law of gravitational attraction, even though the gravitational mass of the quantum vacuum is negative. The paradox is resolved by considering a particle of finite extension and taking into account the vacuum polarization in its interior. 
  The quantization of a massless conformally coupled scalar field on the 2+1 dimensional Anti de Sitter black hole background is presented. The Green's function is calculated, using the fact that the black hole is Anti de Sitter space with points identified, and taking into account the fact that the black hole spacetime is not globally hyperbolic. It is shown that the Green's function calculated in this way is the Hartle-Hawking Green's function. The Green's function is used to compute $\langle T^\mu_\nu \rangle$, which is regular on the black hole horizon, and diverges at the singularity. A particle detector response function outside the horizon is also calculated and shown to be a fermi type distribution. The back-reaction from $\langle T_{\mu\nu} \rangle$ is calculated exactly and is shown to give rise to a curvature singularity at $r=0$ and to shift the horizon outwards. For $M=0$ a horizon develops, shielding the singularity. Some speculations about the endpoint of evaporation are discussed. 
  We show in this paper that it is possible to formulate General Relativity in a phase space coordinatized by two $SO(3)$ connections. We analyze first the Husain-Kucha\v{r} model and find a two connection description for it. Introducing a suitable scalar constraint in this phase space we get a Hamiltonian formulation of gravity that is close to the Ashtekar one, from which it is derived, but has some interesting features of its own. Among them a possible mechanism for dealing with the degenerate metrics and a neat way of writing the constraints of General Relativity. 
  We discuss tests of the Einstein Equivalence Principle due to energies which are purely quantum mechanical in origin. In particular, we consider using Lamb Shift energies to test for possible quantum violations of Local Position Invariance. (to appear in the Proceedings of the 5th Canadian Conference on General Relativity and Relativistic Astrophysics, University of Waterloo, May 13-- 15, 1993) (acknowledgements changed to correct an omission in the originally submitted version) 
  It is shown that rotational cosmological perturbations can be generated in the early Universe, similarly to gravitational waves. The generating mechanism is quantum-mechanical in its nature, and the created perturbations should now be placed in squeezed vacuum quantum states. The physical conditions under which the phenomenon can occur are formulated. The generated perturbations can contribute to the large-angular-scale anisotropy of the cosmic microwave background radiation. An exact formula is derived for the angular correlation function of the temperature variations caused by the quantum-mechanically generated rotational perturbations. The multipole expansion begins from the dipole component. The comparison with the case of gravitational waves is made. 
  We show that the quantization of spherically symmetric pure gravity can be carried out completely in the framework of Ashtekar's self-dual representation. Consistent operator orderings can be given for the constraint functionals yielding two kinds of solutions for the constraint equations, corresponding classically to globally nondegenerate or degenerate metrics. The physical state functionals can be determined by quadratures and the reduced Hamiltonian system possesses 2 degrees of freedom, one of them corresponding to the classical Schwarzschild mass squared and the canonically conjugate one representing a measure for the deviation of the nonstatic field configurations from the static Schwarzschild one. There is a natural choice for the scalar product making the 2 fundamental observables self-adjoint. Finally, a unitary transformation is performed in order to calculate the triad-representation of the physical state functionals and to provide for a solution of the appropriately regularized Wheeler-DeWitt equation. 
  The method of solution of the initial value constraints for pure canonical gravity in terms of Ashtekar's new canonical variables due to CDJ is further developed in the present paper. There are 2 new main results : 1) We extend the method of CDJ to arbitrary matter-coupling again for non-degenerate metrics : the new feature is that the 'CDJ-matrix' adopts a nontrivial antisymmetric part when solving the vector constraint and that the Klein-Gordon-field is used, instead of the symmetric part of the CDJ-matrix, in order to satisfy the scalar constraint. 2) The 2nd result is that one can solve the general initial value constraints for arbitrary matter coupling by a method which is completely independent of that of CDJ. It is shown how the Yang-Mills and gravitational Gauss constraints can be solved explicitely for the corresponding electric fields. The rest of the constraints can then be satisfied by using either scalar or spinor field momenta. This new trick might be of interest also for Yang-Mills theories on curved backgrounds. 
  To appear in proceedings of II Workshop on ``Constraints Theory and Quantisation Methods''Montepulciano (Siena) 1993}   General discussion of the constraints of 2+1 gravity, with emphasis on two approaches, namely the second order and first order formalisms, and comparison with the four dimensional theory wherever possible. Introduction to an operator algebra approach that has been developed in the last few years in collaboration with T.Regge, and discussion of the quantisation of the g=1 case. 
  Two types of self-gravitating particle solutions found in several theories with non-Abelian fields are smoothly connected by a family of non-trivial black holes. There exists a maximum point of the black hole entropy, where the stability of solutions changes. This criterion is universal, and the changes in stability follow from a catastrophe-theoretic analysis of the potential function defined by black hole entropy. 
  We consider the simplicial state-sum model of Ponzano and Regge as a path integral for quantum gravity in three dimensions.   We examine the Lorentzian geometry of a single 3-simplex and of a simplicial manifold, and interpret an asymptotic formula for $6j$-symbols in terms of this geometry. This extends Ponzano and Regge's similar interpretation for Euclidean geometry.   We give a geometric interpretation of the stationary points of this state-sum, by showing that, at these points, the simplicial manifold may be mapped locally into flat Lorentzian or Euclidean space. This lends weight to the interpretation of the state-sum as a path integral, which has solutions corresponding to both Lorentzian and Euclidean gravity in three dimensions. 
  Einstein's equations for stationary axisymmetric fields are reformulated as the equations for affine geodesics in a two--dimensional space. The affine collineations of this space are investigated and used to relate explicit solutions of Einstein's equations with different physical properties. Particularly, the solutions describing the exterior fields of a dyon and a slowly rotating body are discussed. 
  A relation expressing the covariant transformation properties of a relativistic position operator is derived. This relation differs from the one existing in the literature expressing manifest covariance by some factor ordering. The relation is derived in order for the localization of a particle to represent a space-time event. It is shown that there exists a conflict between this relation and the hermiticity of a positive energy position operator. 
  The phenomenon of gyroscopic precession is studied within the framework of Frenet-Serret formalism adapted to quasi-Killing trajectories. Its relation to the congruence vorticity is highlighted with particular reference to the irrotational congruence admitted by the stationary, axisymmetric spacetime. General precession formulae are obtained for circular orbits with arbitrary constant angular speeds. By successive reduction, different types of precessions are derived for the Kerr - Schwarzschild - Minkowski spacetime family. The phenomenon is studied in the case of other interesting spacetimes, such as the De Sitter and G\"{o}del universes as well as the general stationary, cylindrical, vacuum spacetimes. 
  We demonstrate a systematic method for solving the Hamilton-Jacobi equation for general relativity with the inclusion of matter fields. The generating functional is expanded in a series of spatial gradients. Each term is manifestly invariant under reparameterizations of the spatial coordinates (``gauge-invariant''). At each order we solve the Hamiltonian constraint using a conformal transformation of the 3-metric as well as a line integral in superspace. This gives a recursion relation for the generating functional which then may be solved to arbitrary order simply by functionally differentiating previous orders. At fourth order in spatial gradients, we demonstrate solutions for irrotational dust as well as for a scalar field. We explicitly evolve the 3-metric to the same order. This method can be used to derive the Zel'dovich approximation for general relativity. 
  We point out that in general the Reissner-Nordstr\"om (RN) charged black holes of general relativity are not solutions of the four dimensional quadratic gravitational theories. They are, e.g., exact solutions of the $R+R^2$ quadratic theory but not of a theory where a $R_{ab}R^{ab}$ term is present in the gravitational Lagrangian. In the case where such a non linear curvature term is present with sufficiently small coupling, we obtain an approximate solution for a charged black hole of charge $Q$ and mass $M$. For $Q\ll M$ the validity of this solution extends down to the horizon. This allows us to explore the thermodynamic properties of the quadratic charged black hole and we find that, to our approximation, its thermodynamics is identical to that of a RN black hole. However our black hole's entropy is not equal to the one fourth of the horizon area. Finally we extend our analysis to the rotating charged black hole and qualitatively similar results are obtained. 
  It is widely accepted that temporal asymmetry is largely a cosmological problem; the task of explaining temporal asymmetry reduces in the main to that of explaining an aspect of the condition of the early universe. However, cosmologists who discuss these issues often make mistakes similar to those that plagued nineteenth century discussions of the statistical foundations of thermodynamics. In particular, they are often guilty of applying temporal "double standards" of various kinds---e.g., in failing to recognise that certain statistical arguments apply with equal force in either temporal direction. This paper aims to clarify the issue as to what would count as adequate explanation of cosmological time asymmetry. A particular concern is the question whether it is possible to explain why entropy is low near the Big Bang without showing that it must also be low near a Big Crunch, in the event that the universe recollapses. I criticise some of the objections raised to this possibility, showing that these too often depend on a temporal double standard. I also discuss briefly some issues that arise if we take the view seriously. (Could we observe a time- reversing future, for example?) 
  Using as dynamical variable the square of the radius of the Universe, we solve analytically the Einstein equations in the framework of Robertson-Walker models where a cosmological constant describing phenomenologically the vacuum energy decays into radiation. Emphasis is put on the computation of the entropy creation. 
  I argue that the leading quantum corrections, in powers of the energy or inverse powers of the distance, may be computed in quantum gravity through knowledge of only the low energy structure of the theory. As an example, I calculate the leading quantum corrections to the Newtonian gravitational potential. 
  We show that the exponential of the Gauss (self) linking number of a knot is a solution of the Wheeler-DeWitt equation in loop space with a cosmological constant. Using this fact, it is straightforward to prove that the second coefficient of the Jones Polynomial is a solution of the Wheeler-DeWitt equation in loop space with no cosmological constant. We perform calculations from scratch, starting from the connection representation and give details of the proof. Implications for the possibility of generation of other solutions are also discussed. 
  The requirement that physical phenomena associated with gravitational collapse should be duly reconciled with the postulates of quantum mechanics implies that at a Planckian scale our world is not 3+1 dimensional. Rather, the observable degrees of freedom can best be described as if they were Boolean variables defined on a two-dimensional lattice, evolving with time. This observation, deduced from not much more than unitarity, entropy and counting arguments, implies severe restrictions on possible models of quantum gravity. Using cellular automata as an example it is argued that this dimensional reduction implies more constraints than the freedom we have in constructing models. This is the main reason why so-far no completely consistent mathematical models of quantum black holes have been found.   Essay dedicated to Abdus Salam. 
  Two formulations of quantum mechanics, inequivalent in the presence of closed timelike curves, are studied in the context of a soluable system. It illustrates how quantum field nonlinearities lead to a breakdown of unitarity, causality, and superposition using a path integral. Deutsch's density matrix approach is causal but typically destroys coherence. For each of these formulations I demonstrate that there are yet further alternatives in prescribing the handling of information flow (inequivalent to previous analyses) that have implications for any system in which unitarity or coherence are not preserved. 
  A chaotic model of the early universe within the framework of the singularity-free solutions of Einstein's equation is suggested. The evolution of our universe at its early stage, starting out as a small domain of the parent universe, is governed by the dynamics of a classical scalar field $\phi$ . If in any such domain, larger than Planck length,$\dot \phi$ happens to be very large,$\phi$ may develop a dominant inhomogeneous mode,leading to an anisotropic inflation of the universe. The particle $\phi$ is coupled to other particles, which are produced copiously after inflation and these thermalize leading to a rather low temperature universe $(T \geq 10^{4} $ Gev). The electroweak B+L Baryogenesis is assumed to account for the observed baryon asymmetry. The universe now passes through a radiation-dominated phase, leading eventually to a matter-dominated universe, which is isotropic and homogeneous. The model does not depend on the details of Planck scale physics. 
  The influence od space-time curvature on quantum matter which can be theoretically described by covariant wave equations has not been experimentally established yet. In this paper we analyse in detail the suitability of the Ramsey atom beam interferometer for the measurement of the phase shift caused by the Riemannian curvature of the earth. It appears that the detection should be possible with minor modifications of existing devices within the near future. The paper is divided into two parts. The first one is concerned with the derivation of general relativistic correction terms to the Pauli equation starting from the fully covariant Dirac equation and their physical interpretation. The inertial effects of acceleration and rotation are included. In the second part we calculate the phase shift as seen in a laboratory resting on the rotating earth and examine various possibilities to enlarge the sensitivity of the apparatus to space-time curvature. Some remarks on the Lense-Thirring effect and on gravitational waves are made. Since the two parts may be more or less interesting for physicists with different research fields they are written in such a way that each one may be read without much reference to the other one. 
  Renormalized vacuum expectation values of electromagnetic stress-energy tensor are calculated in the background spherically-symmetrical metric of the wormhole's topology. Covariant geodesic point separation method of regularization is used. Violation of the weak energy condition at the throat of wormhole takes place for geometry sufficiently close to that of infinitely long wormhole of constant radius irrespectively of the detailed form of metric. This is an argument in favour of possibility of existence of selfconsistent wormhole in empty space maintained by vacuum field fluctuations in the wormhole's background. 
  The long history of the study of quantum gravity has thrown up a complex web of ideas and approaches. The aim of this article is to unravel this web a little by analysing some of the {\em prima facie\/} questions that can be asked of almost any approach to quantum gravity and whose answers assist in classifying the different schemes. Particular emphasis is placed on (i) the role of background conceptual and technical structure; (ii) the role of spacetime diffeomorphisms; and (iii) the problem of time. 
  Some experimental implications of the recent progress on wave function collapse are calculated. Exact results are derived for the center-of-mass wave function collapse caused by random scatterings and applied to a range of specific examples. The results show that recently proposed experiments to measure the GRW effect are likely to fail, since the effect of naturally occurring scatterings is of the same form as the GRW effect but generally much stronger. The same goes for attempts to measure the collapse caused by quantum gravity as suggested by Hawking and others. The results also indicate that macroscopic systems tend to be found in states with (Delta-x)(Delta-p) = hbar/sqrt(2), but microscopic systems in highly tiltedly squeezed states with (Delta-x)(Delta-p) >> hbar. 
  This is a reply, given at the conference ``Mach's Principle" in T\"ubingen in July 1993, to the paper by Pfister (1993). Unfortunately the Pfister paper itself was not sent to gr-qc. 
  This is the reply given at the conference ``Mach's Principle" at T\"ubingen in July 1993 to the paper by Isenberg (1993a). Unfortunately the Isenberg paper itself was not submitted to gr-qc. 
  An exposition of Vassiliev invariants is given in terms of the simplest approach to the functional integral construction of link invariants from Chern-Simons theory. This approach gives the top row evaluations of Vassiliev invariants for the classical Lie algebras, and a neat point of view on the results of Bar-Natan. It also clarifies the relation between Vassiliev invariants and the extension of the bracket invariant to links with transverse double points that appears in the work of Bruegmann, Gambini and Pullin on the loop representation of quantum gravity. We see that the Vassiliev vertex is not just a transversal intersection of Wilson loops, but rather has the structure of Casimir insertion (up to first order of approximation) coming from the difference formula in the functional integral. (Figures are available from the author.) 
  In this paper, a candidate for pregeometry, Ponzano-Regge spin networks, will be examined in the context of the pregeometric philosophy of Wheeler. Ponzano and Regge were able to construct a theory for 3-dimensional quantum gravity based on 3nj-symbols, obtaining the path integral over the metric in the semiclassical limit. However, extension of this model to 4-dimensions has proven to be difficult. It will be shown that the building blocks for 4-dimensional spacetime are already present in the Ponzano-Regge formalism using a reinterpretation of the theory based on the pregeometric hypotheses of Wheeler. 
  The interaction of matter with gravity in two dimensional spacetimes can be supplemented with a geometrical force analogous to a Lorentz force produced on a surface by a constant perpendicular magnetic field. In the special case of constant curvature, the relevant symmetry does not lead to the de Sitter or the Poincar\'e algebra but to an extension of them by a central element. This richer structure suggests to construct a gauge theory of 2-D gravity that reproduces the Jackiw-Teitelboim model and the string inspired model. Moreover matter can be coupled in a gauge invariant fashion. Classical and quantized results are discussed. Based on a talk given at the XXIIIth International Conference on Differential Geometric Methods in Theoretical Physics. Ixtapa, Mexico. September 1993. 
  We consider in this paper the quantum limits for measurements on macroscopic bodies which are obtained in a novel way employing the concept of decoherence coming from an analysis of the quantum mechanics of dissipative systems. Two cases are analysed, the free mass and the harmonic oscillator, and for both systems we compare our approach with previous treatments of such limits. 
  There is a gap that has been left open since the formulation of general relativity in terms of Ashtekar's new variables namely the treatment of asymptotically flat field configurations that are general enough to be able to define the generators of the Lorentz subgroup of the asymptotical Poincar\'e group. While such a formulation already exists for the old geometrodynamical variables, up to now only the generators of the translation subgroup could be defined because the function spaces of the fields considered earlier are taken too special. The transcription of the framework from the ADM variables to Ashtekar's variables turns out not to be straightforward due to the freedom to choose the internal SO(3) frame at spatial infinity and due to the fact that the non-trivial reality conditions of the Ashtekar framework reenter the game when imposing suitable boundary conditions on the fields and the Lagrange multipliers. 
  We propose the following way of constructing quantum measure in Regge calculus: the full discrete Regge manifold is made continuous in some direction by tending corresponding dimensions of simplices to zero, then functional integral measure corresponding to the canonical quantization (with continuous coordinate playing the role of time) can be constructed. The full discrete measure is chosen so that it would result in canonical quantization one whatever coordinate is made continuous. This strategy is followed in 3D case where full discrete measure is determined in such the way practically uniquely (in fact, family of similar measures is obtained). Averaging with the help of the constructed measure gives finite expectation values for links. 
  General Relativity in three or more dimensions can be obtained by taking the limit $\omega\rightarrow\infty$ in the Brans-Dicke theory. In two dimensions General Relativity is an unacceptable theory. We show that the two-dimensional closest analogue of General Relativity is a theory that also arises in the limit $\omega\rightarrow\infty$ of the two-dimensional Brans-Dicke theory. 
  In quantum Regge calculus areas of timelike triangles possess discrete spectrum. This is because bivectors of these triangles are variables canonically conjugate to orthogonal connection matrices varying in the compact group. (The scale of quantum of this spectrum is nothing but Plankian one). This is checked in simple exactly solvable model - dimensionally reduced in some way Regge calculus. 
  We consider the production of gravitons in an inflationary cosmology by approximating each epoch of change in the equation of state as sudden, from which a simple analytic graviton mode function has been derived. We use this mode function to compute the graviton spectral energy density and the tensor-induced cosmic microwave background anisotropy. The results are then compared to the numerical calculations which incorporate a smooth radiation-matter phase transition. We find that the sudden approximation is a fairly good method. Besides, in determining the frequency range and amplitude of the mode function, we introduce a pre-inflationary radiation-dominated epoch and use a physically sensible regularization method. 
  The new method of invariant definition of the measurable angle of light deflection in the static central symmetric gravitational field is suggested. The predicted pure gravitational contribution to the deflection angle slightly differs from its classical estimate and one may hope that this discrepancy could be experimentally detected in the near future. (uuencoded,compressed PostScript, 23 p. inc. 2 fig.) 
  We solve the quantum constraints for homogeneous N=1 supergravity on 3-geometries with a Bianchi IX metric. Because these geometries admit Killing vectors with the same commutation relations as the angular momentum generators, there are two distinct definitions of homogeneity. The first of these is well-known and has been shown by D'Eath to give the wormhole state. We show that the alternative definition of homogeneity leads to the Hartle-Hawking ``no boundary" state. 
  In the (3+1)D Hamiltonian Regge calculus (one of the coordinates, $ t$, is continuous) conjugate variables are (defined on triangles of discrete 3D section $ t=const$) finite connections and antisymmetric area bivectors. The latter, however, are not independent, since triangles may have common edges. This circumstance can be taken into account with the help of the set of kinematical (that is, required to hold by definition of Regge manifold) bilinear constraints on bivectors. Some of these contain derivatives over $ t$, and taking them into account with the help of Lagrange multipliers would result in the new dynamical variables not having analogs in the continuum theory. It is shown that kinematical constraints with derivatives are consequences of eqs. of motion for Regge action supplemented with the rest of these constraints without derivatives and can be omitted; so the new dynamical variables do not appear. 
  We consider the 3-dimensional formulation of Einstein's theory for spacetimes possessing a non-null Killing field $\xi^a$. It is known that for the vacuum case some of the basic field equations are deducible from the others. It will be shown here how this result can be generalized for the case of essentially arbitrary matter fields. The systematic study of the structure of the fundamental field equations is carried out. In particular, the existence of geometrically preferred reference systems is shown. Using local coordinates of this type two approaches are presented resulting resolvent systems of partial differential equations for the basic field variables. Finally, the above results are applied for perfect fluid spacetimes describing possible equilibrium configurations of relativistic dissipative fluids. 
  For spacetimes with the topology $\IR\!\times\!T^2$, the action of (2+1)-dimensional gravity with negative cosmological constant $\La$ is written uniquely in terms of the time-independent traces of holonomies around two intersecting noncontractible paths on $T^2$. The holonomy parameters are related to the moduli on slices of constant mean curvature by a time-dependent canonical transformation which introduces an effective Hamiltonian. The quantisation of the two classically equivalent formulations differs by terms of order $O(\hbar^3)$, negligible for small $|\La|$. 
  A general dilaton gravity theory in 1+1 spacetime dimensions with a cosmological constant $\lambda$ and a new dimensionless parameter $\omega$, contains as special cases the constant curvature theory of Teitelboim and Jackiw, the theory equivalent to vacuum planar General Relativity, the first order string theory, and a two-dimensional purely geometrical theory. The equations of this general two-dimensional theory admit several different black holes with various types of singularities. The singularities can be spacelike, timelike or null, and there are even cases without singularities. Evaluation of the ADM mass, as a charge density integral, is possible in some situations, by carefully subtrating the black hole solution from the corresponding linear dilaton at infinity. 
  In this paper we report some results obtained by applying the radial gauge to 2+1 dimensional gravity. The general features of this gauge are reviewed and it is shown how they allow the general solution of the problem in terms of simple quadratures. Then we concentrate on the general stationary problem providing the explicit solving formulas for the metric and the explicit support conditions for the energy momentum tensor. The chosen gauge allows, due to its physical nature, to exploit the weak energy condition and in this connection it is proved that for an open universe conical at space infinity the weak energy condition and the absence of closed time like curves (CTC) at space infinity imply the total absence of CTC. It is pointed out how the approach can be used to examine cosmological solution in 2+1 dimensions. 
  Integral calculus on the space  of gauge equivalent connections is developed. Loops, knots, links and graphs feature prominently in this description. The framework is well--suited for quantization of diffeomorphism invariant theories of connections.   The general setting is provided by the abelian C* algebra of functions on the quotient space of connections  generated by Wilson loops (i.e., by the traces of holonomies of connections around closed loops). The representation theory of this algebra leads to an interesting and powerful ``duality'' between gauge--equivalence classes of connections and certain equivalence classes of closed loops. In particular, regular measures on (a suitable completion of) connections/gauges  are in 1--1 correspondence with certain functions of loops and diffeomorphism invariant measures correspond to (generalized) knot and link invariants. By carrying out a non--linear extension of the theory of cylindrical measures on topological vector spaces, a faithful, diffeomorphism invariant measure is introduced. This measure can be used to define the Hilbert space of quantum states in theories of connections. The Wilson--loop functionals then serve as the configuration operators in the quantum theory. 
  Near a spinning point particle in (2+1)-dimensional gravity (or near an infinitely thin, straight, spinning string in 3+1 dimensions) there is a region of space-time with closed timelike curves. Exact solutions for extended sources with apparently physically acceptable energy-momentum tensors, have produced the same exterior space-time structure. Here it is pointed out that in the case with torsion, closed timelike curves appear only for spin densities so high that the spin energy density is higher than the net effective energy density. In models without torsion, the presence of closed time-like curves is related to a heat flow of unphysical magnitude. This corroborates earlier arguments against the possibility of closed timelike curves in space-time geometries generated by physical sources. 
  The assumption that a solution to the Einstein equations is static (or stationary) very strongly constrains the asymptotic behaviour of the metric. It is shown that one need only impose very weak differentiability and decay conditions {\it a priori} on the metric for the field equations to force the metric to be analytic near infinity and to have the standard Schwarzschildian falloff. 
  Consider spherically symmetric initial data for a cosmology which, in the large, approximates an open $k = -1 ,\Lambda = 0$ Friedmann-Lema{\^\i}tre universe. Further assume that the data is chosen so that the trace of the extrinsic curvature is a constant and that the matter field is at rest at this instant of time. One expects that no trapped surfaces appear in the data if no significant clump of excess matter is to be found. This letter confirms this belief by displaying a necessary condition for the existence of trapped surfaces.This necessary condition, simply stated, says that a relatively large amount of excess matter must be concentrated in a small volume for trapped surfaces to appear. 
  Considered are I class constraints in the tetrad-connection formulation of Regge calculus. One of these is well-known Gauss law which generates rotations in the local frames associated with tetrahedrons in the continuous time 3D section. Another two types of these are new, satisfied by definition of Regge manifold and having no I class analogs in the continuum general relativity. Constraints of the first type express vanishing of the dual squares of antisymmetric tensors of the triangles in the 3D section thus ensuring each such tensor being a bivector. Constraints of the second type are trigonometric relations between areas of triangles of 3D section caused by that the set of areas is redundant as compared to the set of linklengts. 
  Multivector quantum mechanics utilizes wavefunctions which are Clifford aggregates (e.g. sum of scalar, vector, bivector). This is equivalent to multi- spinors constructed of Dirac matrices, with the representation independent form of the generators geometrically interpreted as the basis vectors of spacetime. Multiple generations of particles appear as left ideals of the algebra, coupled only by now-allowed right-side applied (dextral) operations. A generalized bilateral (two-sided operation) coupling is proposed which includes the above mentioned dextrad field, and the spin-gauge interaction as particular cases. This leads to a new principle of {\it poly-dimensional covariance}, in which physical laws are invariant under the reshuffling of coordinate geometry. Such a multigeometric superfield equation is proposed, which is sourced by a bilateral current. In order to express the superfield in representation and coordinate free form, we introduce Eddington E-F {\it double-frame} numbers. Symmetric tensors can now be represented as 4D ``dyads", which actually are elements of a global 8D Clifford algebra. As a restricted example, the dyadic field created by the Greider-Ross multivector current (of a Dirac electron) describes both electromagnetic and Morris- Greider gravitational interactions. [to appear in Proceedings edited by J. Keller, UNAM, Mexico (Kluwer Academic Publ)] 
  The Dirac quantization of a 2+1 dimensional bubble is performed. The bubble consists of a string forming a boundary between two regions of space-time with distinct geometries. The ADM constraints are solved and the coupling to the string is introduced through the boundary conditions. The wave functional is obtained and the quantum uncertainty in the radius of the ring is calculated; this uncertainty becomes large at the Planck scale. 
  We investigate certain properties of the Wheeler-DeWitt metric (for constant lapse) in canonical General Relativity associated with its non-definite nature. Contribution to the conference on Mach's principle: "From Newtons Bucket to Quantum Gravity", July 26-30 1993, Tuebingen, Germany 
  A brief review of the Wigner functions method in curved space-time. Contribution to the 3rd International Wigner Symposium, 5th-11th September 1993, Oxford, UK. 
  I show in this letter that it is possible to construct a Hamiltonian description for Lorentzian General Relativity in terms of two real $SO(3)$ connections. The constraints are simple polynomials in the basic variables. The present framework gives us a new formulation of General Relativity that keeps some of the interesting features of the Ashtekar formulation without the complications associated with the complex character of the latter. 
  Using Ashtekar variables, we analyze Lorentzian and Euclidean gravity in vacuum up to a constant conformal transformation. We prove that the reality conditions are invariant under a Wick rotation of the time, and show that the compatibility of the algebra of commutators and constraints with the involution defined by the reality conditions restricts the possible values of the conformal factor to be either real or purely imaginary. In the first case, one recovers real Lorentzian general relativity. For purely imaginary conformal factors, the classical theory can be interpreted as real Euclidean gravity. The reality conditions associated with this Euclidean theory demand the hermiticity of the Ashtekar connection, but the densitized triad is represented by an anti-Hermitian operator. We also demonstrate that the Euclidean and Lorentzian sets of reality conditions lead to inequivalent quantizations of full general relativity. As a consequence, it seems impossible to obtain Lorentzian physical predictions from the quantum theory constructed with the Euclidean reality conditions. 
  We introduce a reduced model for a real sector of complexified Ashtekar gravity that does not correspond to a subset of Einstein's gravity but for which the programme of canonical quantization can be carried out completely, both, via the reduced phase space approach or along the lines of the algebraic quantization programme.\\ This model stands in a certain correspondence to the frequently treated cylindrically symmetric waves.\\ In contrast to other models that have been looked at up to now in terms of the new variables the reduced phase space is infinite dimensional while the scalar constraint is genuinely bilinear in the momenta.\\ The infinite number of Dirac observables can be expressed in compact and explicit form in terms of the original phase space variables.\\ They turn out, as expected, to be non-local and form naturally a set of countable cardinality. 
  We present a theory of tunnelling geometries originating from the no-boundary quantum state of Hartle and Hawking. We reformulate the no-boundary wavefunction in the representation of true physical variables and calculate it in the one-loop approximation. For this purpose a special technique is developed, which reduces the formalism of complex tunnelling geometries to the real ones, and also the method of collective variables is applied, separating the macroscopic collective degrees of freedom from the microscopic modes. The quantum distribution of Lorentzian universes, defined on the space of such collective variables, incorporates the probabilty conservation and represents the partition function of quasi-DeSitter gravitational instantons weighted by their Euclidean effective action. They represent closed compact manifolds obtained by the procedure of doubling the Euclidean spacetime which nucleates the Lorentzian universes. The over-Planckian behaviour of their distribution is determined by the anomalous scaling of the theory on such instantons, which serves as a criterion for the high-energy normalizability of the no-boudary wavefunction and the validity of the semiclassical expansion. 
  The reduction algorithms for functional determinants of differential operators on spacetime manifolds of different topological types are presented, which were recently used for the calculation of the no-boundary wavefunction and the partition function of tunnelling geometries in quantum gravity and cosmology. 
  We complete a classification of junctions of two Friedmann-Robertson-Walker space-times bounded by a spherical thin wall. Our analysis covers super-horizon bubbles and thus complements the previous work of Berezin, Kuzumin and Tkachev. Contrary to sub-horizon bubbles, various topology types for super-horizon bubbles are possible, regardless of the sign of the extrinsic curvature. We also derive a formula for the peculiar velocity of a domain wall for all types of junction. 
  We study quantum mechanical and classical stability properties of Reissner-Nordstrom deSitter spacetimes, which describe black holes with mass $M$ and charge $Q$ in a background with cosmological constant $\Lambda \ge 0$. There are two sources of particle production in these spacetimes; the black hole horizon and the cosmological horizon. A scattering calculation is done to compute the Hawking radiation in these spacetimes. We find that the flux from the black hole horizon equals the flux from the cosmological horizon, if and only if $|Q|=M$, indicating that this is a state of thermodynamic equilibrium. The spectrum, however, is not thermal. We also show that spacetimes containing a number of charge equal to mass black holes with $\Lambda \ge 0$, have supercovariantly constant spinors, suggesting that they may be minimum energy states in a positive energy construction. As a first step in this direction, we present a positive energy construction for asymptotically deSitter spacetimes with vanishing charge. Because the construction depends only on a spatial slice, our result also holds for spacetimes which are asymptotically Robertson-Walker. 
  Calculating the van Vleck determinant in traversable wormhole spacetimes is an important ingredient in understanding the physical basis behind Hawking's chronology protection conjecture. This paper presents extensive computations of this object --- at least in the short--throat flat--space approximation. An important technical trick is to use an extension of the usual junction condition formalism to probe the full Riemann tensor associated with a thin shell of matter. Implications with regard to Hawking's chronology protection conjecture are discussed. Indeed, any attempt to transform a single isolated wormhole into a time machine results in large vacuum polarization effects sufficient to disrupt the internal structure of the wormhole before the onset of Planck scale physics, and before the onset of time travel. On the other hand, it is possible to set up a putative time machine built out of two or more wormholes, each of which taken in isolation is not itself a time machine. Such ``Roman configurations'' are much more subtle to analyse. For some particularly bizarre configurations (not traversable by humans) the vacuum polarization effects can be arranged to be arbitrarily small at the onset of Planck scale physics. This indicates that the disruption scale has been pushed down into the Planck slop. Ultimately, for these configurations, questions regarding the truth or falsity of Hawking's chronology protection can only be addressed by entering the uncharted wastelands of full fledged quantum gravity. 
  The gravitational field of a moving point particle is obtained in a Lorentz covariant form for both uncharged and charged cases. It is shown that the general relativistic proper time interval at the location of the particle is the same as the special relativistic one and the gravitational and electromagnetic self forces are zero. 
  It is a deceptively simple question to ask how acoustic disturbances propagate in a non--homogeneous flowing fluid. If the fluid is barotropic and inviscid, and the flow is irrotational (though it may have an arbitrary time dependence), then the equation of motion for the velocity potential describing a sound wave can be put in the (3+1)--dimensional form: d'Alembertian psi = 0. That is partial_mu(sqrt{-g} g^{mu nu} partial_nu psi)/sqrt{-g} = 0. The acoustic metric --- g_{mu nu}(t,x) --- governing the propagation of sound depends algebraically on the density, flow velocity, and local speed of sound. Even though the underlying fluid dynamics is Newtonian, non--relativistic, and takes place in flat space + time, the fluctuations (sound waves) are governed by a Lorentzian spacetime geometry. 
  In [1,2] we established and discussed the algebra of observables for $2+1$ gravity at both the classical and quantum level, and gave a systematic discussion of the reduction of the expected number of independent observables to $6g - 6 (g > 1)$. In this paper the algebra of observables for the case $g=2$ is reduced to a very simple form. A Hilbert space of state vectors is defined and its representations are discussed using a deformation of the Euler-Gamma function. The deformation parameter $\th$ depends on the cosmological and Planck's constants. 
  The Kucha\v{r} canonical transformation for vacuum geometrodynamics in the presence of cylindrical symmetry is applied to a general non-vacuum case. The resulting constraints are highly non-linear and non-local in the momenta conjugate to the Kucha\v{r} embedding variables. However, it is demonstrated that the constraints can be solved for these momenta and thus the dynamics of cylindrically symmetric models can be cast in a form suitable for the construction of a hypertime functional Schr\"odinger equation. 
  A systematic approach to the coordinate transformations in the Schwarzschild singularity problem is considered. It is shown that both singularities at r=0 and r=2m can be eliminated by suitable coordinate transformations. 
  A theoretical framework based on a simple quasi-number algebra is investigated in a treatment of space-time and gravity. 
  We propose a reduced constrained Hamiltonian formalism for the exactly soluble $B \wedge F$ theory of flat connections and closed two-forms over manifolds with topology $\Sigma^3 \times (0,1)$. The reduced phase space variables are the holonomies of a flat connection for loops which form a basis of the first homotopy group $\pi_1(\Sigma^3)$, and elements of the second cohomology group of $\Sigma^3$ with value in the Lie algebra $L(G)$. When $G=SO(3,1)$, and if the two-form can be expressed as $B= e\wedge e$, for some vierbein field $e$, then the variables represent a flat spacetime. This is not always possible: We show that the solutions of the theory generally represent spacetimes with ``global torsion''. We describe the dynamical evolution of spacetimes with and without global torsion, and classify the flat spacetimes which admit a locally homogeneous foliation, following Thurston's classification of geometric structures. 
  As gravity is a long-range force, one might a priori expect the Universe's global matter distribution to select a preferred rest frame for local gravitational physics. At the post-Newtonian approximation, two parameters suffice to describe the phenomenology of preferred-frame effects. One of them has already been very tightly constrained (|alpha_2| < 4 x 10^-7, 90% C.L.), but the present bound on the other one is much weaker (|alpha_1| < 5 x 10^-4, 90% C.L.). It is pointed out that the observation of particular orbits of artificial Earth satellites has the potential of improving the alpha_1 limits by a couple of orders of magnitude, thanks to the appearance of small divisors which enhance the corresponding preferred-frame effects. There is a discrete set of inclinations which lead to arbitrarily small divisors, while, among zero-inclination (equatorial) orbits, geostationary ones are near optimal. The main alpha_1-induced effects are: (i) a complex secular evolution of the eccentricity vector of the orbit, describable as the vectorial sum of several independent rotations; and (ii) a yearly oscillation in the longitude of the satellite. 
  Topological gravity is the reduction of Einstein's theory to spacetimes with vanishing curvature, but with global degrees of freedom related to the topology of the universe. We present an exact Hamiltonian lattice theory for topological gravity, which admits translations of the lattice sites as a gauge symmetry. There are additional symmetries, not present in Einstein's theory, which kill the local degrees of freedom. We show that these symmetries can be fixed by choosing a gauge where the torsion is equal to zero. In this gauge, the theory describes flat space-times. We propose two methods to advance towards the holy grail of lattice gravity: A Hamiltonian lattice theory for curved space-times, with first-class translation constraints. 
  The gravitational and electromagnetic fields of a moving charged spinning point particle are obtained in the Lorentz covariant form by transforming the Kerr--Newman solution in Boyer--Lindquist coordinates to the one in the coordinate system which resembles the isotropic coordinates and then covariantizing it. It is shown that the general relativistic proper time at the location of the particle is the same as the special relativistic one and the gravitational and electromagnetic self forces vanish. 
  This paper constructs the geometrically natural objects which are associated with any projection tensor field on a manifold with any affine connection. The approaches to projection tensor fields which have been used in general relativity and related theories assume normal projection tensors of co-dimension one and connections which are metric compatible and torsion-free. These assumptions fail for projections onto lightlike curves or surfaces and other situations where degenerate metrics occur as well as projections onto two-surfaces and projections onto spacetime in the higher dimensional manifolds of unified field theories. This paper removes these restrictive assumptions. One key idea is to define two different ''extrinsic curvature tensors'' which become equal for normal projections. In addition, a new family of geometrical tensors is introduced: the cross-projected curvature tensors. In terms of these objects, projection decompositions of covariant derivatives, the full Riemann curvature tensor and the Bianchi identities are obtained and applied to perfect fluids, timelike curve congruences, string congruences, and the familiar 3+1 analysis of the spacelike initial value problem of general relativity. 
  The loop representation plays an important role in canonical quantum gravity because loop variables allow a natural treatment of the constraints. In these lectures we give an elementary introduction to (i) the relevant history of loops in knot theory and gauge theory, (ii) the loop representation of Maxwell theory, and (iii) the loop representation of canonical quantum gravity. (Based on lectures given at the 117. Heraeus Seminar, Bad Honnef, Sept. 1993) 
  The standard (Euclidean) action principle for the gravitational field implies that for spacetimes with black hole topology, the opening angle at the horizon and the horizon area are canonical conjugates. It is shown that the opening angle bears the same relation to the horizon area that the time separation bears to the mass at infinity. The dependence of the wave function on this new degree of freedom is governed by an extended Wheeler-DeWitt equation. Summing over all horizon areas yields the black hole entropy. 
  We investigate axially symmetric asymptotically flat vacuum self-gravitating system. A class of initial data with apparent horizon was numerically constructed. The examined solutions satisfy the Penrose inequality. The prior analysis of a massive system and the present results suggest that either massive or sourcefree configurations fulfill the Penrose inequality. 
  The solutions of generalized Killing equation have been obtained for line element with initial $t^2 \oplus so(3)$ symmetry. The coefficients of the metric $g$ corresponding to these vector fields are written down. 
  We determine the transfer functions of two kinds of filters that can be used in the detection of continuous gravitational radiation. The first one optimizes the signal-to-noise ratio, and the second reproduces the wave with minimum error. We analyse the behaviour of these filters in connection with actual detection schemes. 
  The quantum dynamics of a two-dimensional charged spin $1/2$ particle is studied for general, symmetry--free curved surfaces and general, nonuniform magnetic fields that are, when different from zero, orthogonal to the defining two surface. Although higher Landau levels generally lose their degeneracy under such general conditions, the lowest Landau level, the ground state, remains degenerate. Previous discussions of this problem have had less generality and/or used supersymmetry, or else have appealed to very general mathematical theorems from differential geometry. In contrast our discussion relies on simple and standard quantum mechanical concepts.   The mathematical similarity of the physical problem at hand and that of a phase-space path integral quantization scheme of a general classical system is emphasized. Adopting this analogy in the general case leads to a general quantization procedure that is invariant under general coordinate transformations-- completely unlike any of the conventional quantization prescriptions -- and therefore generalizes the concept of quantization to new and hitherto inaccesible situations.   In a complementary fashion , the so-obtained picture of general quantization helps to derive useful semiclassical formulas for the Hall current in the case of a filling factor equal to one for a general surface and magnetic field. 
  A description of electromagnetism as four-dimensional spacetime structure leads to the dynamics of a charged particle being determined only by the four-vector potential and the existence of an electromagnetic field depending on the topological structure of the background spacetime. When the spacetime structure of electromagnetism is complex it is possible to connect spacetime structure and quantum physics via the method of path integration. (This paper is a consequence of an attempt to incorporate gravitation into electromagnetism by describing gravity as a coupling of two electromagnetic fields.) 
  We argue that in a nonlinear gravity theory, which according to well-known results is dynamically equivalent to a self-gravitating scalar field in General Relativity, the true physical variables are exactly those which describe the equivalent general-relativistic model (these variables are known as Einstein frame). Whenever such variables cannot be defined, there are strong indications that the original theory is unphysical. We explicitly show how to map, in the presence of matter, the Jordan frame to the Einstein one and backwards. We study energetics for asymptotically flat solutions. This is based on the second-order dynamics obtained, without changing the metric, by the use of a Helmholtz Lagrangian. We prove for a large class of these Lagrangians that the ADM energy is positive for solutions close to flat space. The proof of this Positive Energy Theorem relies on the existence of the Einstein frame, since in the (Helmholtz--)Jordan frame the Dominant Energy Condition does not hold and the field variables are unrelated to the total energy of the system. 
  We show that absence of space-like boundaries in 1+1 dimensional dilaton gravity implies a catastrophic event at the end point of black hole evaporation. The proof is completely independent of the physics at Planck scales, which suggests that the same will occur in any theory of quantum gravity which only admits trivial space-time topologies. 
  The problem of the referring of space and time relationships between physical objects in a curved space-time is discussed. The basic notions of column and weak column that could constitute the basis for consistent general relativistic description of measurements of space and time relations are introduced and the corresponding equations are derived. A column of observers is proposed to realize still vague notion of `non-inertial frame of reference' in case of curved space-time, the criterion distinguishing `inertial columns' is suggested as well. A number of solutions of column equations is described. They are applied for calculation of gravitational, inertial (or gravitational-inertial in more general situation) and ``external" forces and the strength of gravitational field in several cases of physical interest. In particular, the `general relativistic' version of Newton's law of inverse squares is derived in case of the interaction of test particle with a static central symmetric body. 
  We present a compactified version of the 3-dimensional black hole recently found by considering extra identifications and determine the analytical continuation of the solution beyond its coordinate singularity by extending the identifications to the extended region of the spacetime. In the extended region of the spacetime, we find a topology change and non-trivial closed timelike curves both in the ordinary 3-dimensional black hole and in the compactified one. Especially, in the case of the compactified 3-dimensional black hole, we show an example of topology change from one double torus to eight spheres with three punctures. 
  Mass of singularity is defined, and its relation to whether the singularity is spacelike, timelike or null is discussed for spherically symmetric spacetimes.   It is shown that if the mass of singularity is positive   (negative) the singularity is non-timelike (non-spacelike).   The connection between the sign of the mass and the force on a particle is also discussed. 
  We consider the phase-space of Yang-Mills on a cylindrical space-time ($S^1 \times {\bf R}$) and the associated algebra of gauge-invariant functions, the $T$-variables. We solve the Mandelstam identities both classically and quantum-mechanically by considering the $T$-variables as functions of the eigenvalues of the holonomy and their associated momenta. It is shown that there are two inequivalent representations of the quantum $T$-algebra. Then we compare this reduced phase space approach to Dirac quantization and find it to give essentially equivalent results. We proceed to define a loop representation in each of these two cases. One of these loop representations (for $N=2$) is more or less equivalent to the usual loop representation. 
  We use the language of squeezed states to give a systematic description of two issues in cosmological particle creation: a) Dependence of particle creation on the initial state specified. We consider in particular the number state, the coherent and the squeezed state. b) The relation of spontaneous and stimulated particle creation and their dependence on the initial state. We also present results for the fluctuations in particle number in anticipation of its relevance to defining noise in quantum fields and the vacuum susceptibility of spacetime. 
  A detailed review is given of the semiclassical approximation to quantum gravity in the canonical framework. This includes in particular the derivation of the functional Schr\"odinger equation and a discussion of semiclassical time as well as the derivation of quantum gravitational correction terms to the Schr\"odinger equation. These terms are used to calculate energy shifts for fields in De~Sitter space and non-unitary contributions in black hole evaporation. Emphasis is also put on the relevance of decoherence and correlations in semiclassical gravity. The back reaction of non-gravitational quantum fields onto the semiclassical background and the emergence of a Berry connection on superspace is also discussed in this framework. 
  This revision includes clarified exposition and simplified analysis. Solutions of the Einstein equations which are periodic and have standing gravitational waves are valuable approximations to more physically realistic solutions with outgoing waves. A variational principle is found which has the power to provide an accurate estimate of the relationship between the mass and angular momentum of the system, the masses and angular momenta of the components, the rotational frequency of the frame of reference in which the system is periodic, the frequency of the periodicity of the system, and the amplitude and phase of each multipole component of gravitational radiation. Examination of the boundary terms of the variational principle leads to definitions of the effective mass and effective angular momentum of a periodic geometry which capture the concepts of mass and angular momentum of the source alone with no contribution from the gravitational radiation. These effective quantities are surface integrals in the weak-field zone which are independent of the surface over which they are evaluated, through second order in the deviations of the metric from flat space. 
  A possible resolution of the information loss paradox for black holes is proposed in which a phase transition occurs when the temperature of an evaporating black hole equals a critical value, $T_c$, and Lorentz invariance and diffeomorphism invariance are spontaneously broken. This allows a generalization of Schr\"odinger's equation for the quantum mechanical density matrix, such that a pure state can evolve into a mixed state, because in the symmetry broken phase the conservation of energy-momentum is spontaneously violated. TCP invariance is also spontaneously broken together with time reversal invariance, allowing the existence of white holes, which are black holes moving backwards in time. Domain walls would form which separate the black holes and white holes (anti-black holes) in the broken symmetry regime, and the system could evolve into equilibrium producing a balance of information loss and gain. 
  We find general, time-dependent solutions produced by open string sources carrying no momentum flow in 2+1 dimensional gravity. The local Poincar\'e group elements associated with these solutions and the coordinate transformations that transform these solutions into Minkowski metric are obtained. We also find the relation between these solutions and the planar wall solutions in 3+1 dimensions. 
  We analyse quantum--kinetic effects in the early Universe. We show that quantum corrections to the Vlasov equation give rise to a dynamical variation of the gravitational constant. The value of the gravitational constant at the Grand Unification epoch is shown to differ from its present value to about $10^{-4} \div 10^{-3} \% $. 
  We resurrect a standard construction of analytical mechanics dating from the last century. The technique allows one to pass from any dynamical system whose first order evolution equations are known, and whose bracket algebra is not degenerate, to a system of canonical variables and a non-zero Hamiltonian that generates their evolution. We advocate using this method to infer a canonical formalism, as a prelude to quantization, for systems in which the naive Hamiltonian is constrained to vanish. The construction agrees with the usual results for gauge theories and can be applied as well to gravity, {\it even when the spatial manifold is closed.} As an example, we construct such a reduced Hamiltonian in perturbation theory around a flat background on the manifold $T^3 \times R$. The resulting Hamiltonian is positive semidefinite and agrees with the A.D.M. energy in the limit that deviations from flat space remain localized as the toroidal radii become infinite. 
  Attention is given to the interface of mathematics and physics, specifically noting that fundamental principles limit the usefulness of otherwise perfectly good mathematical general integral solutions. A new set of multivector solutions to the meta-monogenic (massive) Dirac equation is constructed which form a Hilbert space. A new integral solution is proposed which involves application of a kernel to the right side of the function, instead of to the left as usual. This allows for the introduction of a multivector generalization of the Feynman Path Integral formulation, which shows that particular ``geometric groupings'' of solutions evolve in the manner to which we ascribe the term ``quantum particle''. Further, it is shown that the role of usual $i$ is subplanted by the unit time basis vector, applied on the right side of the functions. Summary of talk, to appear in: Proceedings of the 17th Annual Lecture Series in the Mathematical Sciences, April 8-10, 1993, University of Arkansas, `Clifford Algebas in Analysis', J. Ryan, editor (CRC Press, expected 1994) 
  It is shown that a physically reasonable spacetime that is eternally inflating to the future must possess an initial singularity. 
  Two techniques for computing black hole entropy in generally covariant gravity theories including arbitrary higher derivative interactions are studied. The techniques are Wald's Noether charge approach introduced recently, and a field redefinition method developed in this paper. Wald's results are extended by establishing that his local geometric expression for the black hole entropy gives the same result when evaluated on an arbitrary cross-section of a Killing horizon (rather than just the bifurcation surface). Further, we show that his expression for the entropy is not affected by ambiguities which arise in the Noether construction. Using the Noether charge expression, the entropy is evaluated explicitly for black holes in a wide class of generally covariant theories. Further, it is shown that the Killing horizon and surface gravity of a stationary black hole metric are invariant under field redefinitions of the metric of the form $\bar{g}_{ab}\equiv g_{ab} + \Delta_{ab}$, where $\Delta_{ab}$ is a tensor field constructed out of stationary fields. Using this result, a technique is developed for evaluating the black hole entropy in a given theory in terms of that of another theory related by field redefinitions. Remarkably, it is established that certain perturbative, first order, results obtained with this method are in fact {\it exact}. The possible significance of these results for the problem of finding the statistical origin of black hole entropy is discussed.} 
  The coincidence of quantum cosmology solutions generated by solving a Euclidean version of the Hamilton-Jacobi equation for gravity and by using the complex canonical transformation of the Ashtekar variables is discussed. An examination of similar solutions for the free electromagnetic field shows that this coincidence is an artifact of the homogeneity of the cosmological space. (This paper will appear in a festchrift volume for Jerzy Plebanski) 
  The evolution of a universe with Brans-Dicke gravity and nonzero curvature is investigated here. We present the equations of motion and their solutions during the radiation dominated era. In a Friedman-Robertson-Walker cosmology we show explicitly that the three possible values of curvature $\kappa=+1,0,-1$ divide the evolution of the Brans-Dicke universe into dynamically distinct classes just as for the standard model. Subsequently we discuss the flatness problem which exists in Brans-Dicke gravity as it does in the standard model. In addition, we demonstrate a flatness problem in MAD Brans-Dicke gravity. In general, in any model that addresses the horizon problem, including inflation, there are two components to the flatness issue: i) at the Planck epoch curvature gains importance, and ii) during accelerated expansion curvature becomes less important and the universe flattens. In many cases the universe must be very flat at the Planck scale in order for the accelerated epoch to be reached, thus there can be a residual flatness problem. 
  The Wheeler-DeWitt equation is derived from the bosonic sector of the heterotic string effective action assuming a toroidal compactification. The spatially closed, higher dimensional Friedmann-Robertson-Walker (FRW) cosmology is investigated and a suitable change of variables rewrites the equation in a canonical form. Real- and imaginary-phase exact solutions are found and a method of successive approximations is employed to find more general power series solutions. The quantum cosmology of the Bianchi IX universe is also investigated and a class of exact solutions is found. 
  Time plays different roles in quantum mechanics and gravity. These roles are examined and the problems that the conflict in the roles presents for quantum gravity are briefly summarised. 
  Using the Kerr-Schild decomposition of the metric tensor that employs the algebraically special nature of the Kerr-Newman space-time family, we calculate the energy-momentum tensor. The latter turns out to be a well-defined tensor-distribution with disk-like support. 
  We propose a new representation for gauge theories and quantum gravity. It can be viewed as a generalization of the loop representation. We make use of a recently introduced extension of the group of loops into a Lie Group. This extension allows the use of functional methods to solve the constraint equations. It puts in a precise framework the regularization problems of the loop representation. It has practical advantages in the search for quantum states. We present new solutions to the Wheeler-DeWitt equation that reinforce the conjecture that the Jones Polynomial is a state of nonperturbative quantum gravity. 
  We analyze the radiative aspects of nonsymmetric gravity theory to show that, in contrast to General Relativity, its nonstationary solutions cannot simultaneously exhibit acceptable asymptotic behavior at both future and past null infinity: good behavior at future null infinity is only possible through the use of advanced potentials with concomitant unphysical behavior at past null infinity. 
  We derive an action for gravity in the framework of non-commutative geometry by using the Wodzicki residue. We prove that for a Dirac operator $D$ on an $n$ dimensional compact Riemannian manifold with $n\geq 4$, $n$ even, the Wodzicki residue Res$(D^{-n+2})$ is the integral of the second coefficient of the heat kernel expansion of $D^{2}$. We use this result to derive a gravity action for commutative geometry which is the usual Einstein Hilbert action and we also apply our results to a non-commutative extension which, is given by the tensor product of the algebra of smooth functions on a manifold and a finite dimensional matrix algebra. In this case we obtain gravity with a cosmological constant. 
  This paper contains an introduction into Ashtekar's reformulation of General Relativity in terms of connection variables. To appear in "Canonical Gravity - From Classical to Quantum", ed. by J. Ehlers and H. Friedrich, Springer Verlag (1994). 
  A simple self gravitating system --- a thin spherical shell of charged pressureless matter --- is naively quantized as a test case of quantum gravitational collapse. The model is interpreted in terms of an inner product on the positive energy states. An S-matrix is constructed describing scattering between negatively and positively infinite radius. 
  We derive the uncertainty relation for a quantum open system comprised of a Brownian particle interacting with a bath of quantum oscillators at finite temperature. We examine how the quantum and thermal fluctuations of the environment contribute to the uncertainty in the canonical variables of the system. We show that upon contact with the bath (assumed ohmic in this paper) the system evolves from a quantum-dominated state to a thermal-dominated state in a time which is the same as the decoherence time in similar models in the discussion of quantum to classical transition. This offers some insight into the physical mechanisms involved in the environment-induced decoherence process. We obtain closed analytic expressions for this generalized uncertainty relation under the conditions of high temperature and weak damping separately. We also consider under these conditions an arbitrarily-squeezed initial state and show how the squeeze parameter enters in the generalized uncertainty relation. Using these results we examine the transition of the system from a quantum pure state to a nonequilibrium quantum statistical state and to an equilibrium quantum statistical state. The three stages are marked by the decoherence time and the relaxation time respectively. With these observations we explicate the physical conditions when the two basic postulates of quantum statistical mechanics become valid. We also comment on the inappropriateness in the usage of the word classicality in many decoherence studies of quantum to classical transition. 
  The quantum Brownian motion paradigm provides a unified framework where one can see the interconnection of some basic quantum statistical processes like decoherence, dissipation, particle creation, noise and fluctuation. We treat the case where the Brownian particle is coupled linearly to a bath of time dependent quadratic oscillators. While the bath mimics a scalar field, the motion of the Brownian particle modeled by a single oscillator could be used to depict the behavior of a particle detector, a quantum field mode or the scale factor of the universe. An important result of this paper is the derivation of the influence functional encompassing the noise and dissipation kernels in terms of the Bogolubov coefficients. This method enables one to trace the source of statistical processes like decoherence and dissipation to vacuum fluctuations and particle creation, and in turn impart a statistical mechanical interpretation of quantum field processes. With this result we discuss the statistical mechanical origin of quantum noise and thermal radiance from black holes and from uniformly- accelerated observers in Minkowski space as well as from the de Sitter universe discovered by Hawking, Unruh and Gibbons-Hawking. We also derive the exact evolution operator and master equation for the reduced density matrix of the system interacting with a parametric oscillator bath in an initial squeezed thermal state. These results are useful for decoherence and backreaction studies for systems and processes of interest in semiclassical cosmology and gravity. Our model and results are also expected to be useful for related problems in quantum optics. %\pacs {05.40.+j,03.65.Sq,98.80.Cq,97.60.Lf} 
  We continue our earlier investigation of the backreaction problem in semiclassical gravity with the Schwinger-Keldysh or closed-time-path (CTP) functional formalism using the language of the decoherent history formulation of quantum mechanics. Making use of its intimate relation with the Feynman-Vernon influence functional (IF) method, we examine the statistical mechanical meaning and show the interrelation of the many quantum processes involved in the backreaction problem, such as particle creation, decoherence and dissipation. We show how noise and fluctuation arise naturally from the CTP formalism. We derive an expression for the CTP effective action in terms of the Bogolubov coefficients and show how noise is related to the fluctuations in the number of particles created. In so doing we have extended the old framework of semiclassical gravity, based on the mean field theory of Einstein equation with a source given by the expectation value of the energy-momentum tensor, to that based on a Langevin-type equation, where the dynamics of fluctuations of spacetime is driven by the quantum fluctuations of the matter field. This generalized framework is useful for the investigation of quantum processes in the early universe involving fluctuations, vacuum stability and phase transtion phenomena and the non-equilibrium thermodynamics of black holes. It is also essential to an understanding of the transition from any quantum theory of gravity to classical general relativity. \pacs{pacs numbers: 04.60.+n,98.80.Cq,05.40.+j,03.65.Sq} 
  A Poincar\'{e} gauge theory of (2+1)-dimensional gravity is developed. Fundamental gravitational field variables are dreibein fields and Lorentz gauge potentials, and the theory is underlain with the Riemann-Cartan space-time. The most general gravitational Lagrangian density, which is at most quadratic in curvature and torsion tensors and invariant under local Lorentz transformations and under general coordinate transformations, is given. Gravitational field equations are studied in detail, and solutions of the equations for weak gravitational fields are examined for the case with a static, \lq \lq spin"less point like source. We find, among other things, the following: (1)Solutions of the vacuum Einstein equation satisfy gravitational field equations in the vacuum in this theory. (2)For a class of the parameters in the gravitational Lagrangian density, the torsion is \lq \lq frozen" at the place where \lq \lq spin" density of the source field is not vanishing. In this case, the field equation actually agrees with the Einstein equation, when the source field is \lq \lq spin"less. (3)A teleparallel theory developed in a previous paper is \lq \lq included as a solution" in a limiting case. (4)A Newtonian limit is obtainable, if the parameters in the Lagrangian density satisfy certain conditions. 
  The equivalence of a conformal metric on 4-dimensional space-time and a local field of 3-dimensional subspaces of the space of 2-forms over space-time is discussed and the basic notion of transection is introduced. Corresponding relation is spread to the metric case in terms of notion of normalized ordered oriented transection field. As a result, one obtains a possibility to handle the metric geometry without any references to the metric tensor itself on a distinct base which nevertheless contains all the information on metricity. Moreover, the notion of space-time curvature is provided with its natural counterpart in the transection `language' in a form of curvature endomorphism as well. To globalize the local constructions introduced, a certain fiber bundle is defined whose sections are equivalent to normalized ordered oriented transection fields and locally to the metric tensor on space-time. The criterion distinguishing the Lorentz geometry is discussed. The resulting alternative method of the description of space-time metricity, dealing with exterior forms foliation alone, seems to be of a power compatible with one of the standard concept based on the metric tensor. 
  This supplementary part of the paper gr-qc 9312038 contains the necessary proofs of the claims stated in the main part. 
  The recent method of the description of classical fields in terms of Lagrange-Souriau form is applied to the case of source-free electromagnetic field in order to check its computational capabilities. The relevant calculations are represented in all details and yield a useful data for comparisons of the method with more usual approaches in this simple and transparent case. 
  A gauge theory of quantum gravity is formulated, in which an internal, field dependent metric is introduced which non-linearly realizes the gauge fields on the non-compact group $SL(2,C)$, while linearly realizing them on $SU(2)$. Einstein's $SL(2,C)$ invariant theory of gravity emerges at low energies, since the extra degrees of freedom associated with the quadratic curvature and the internal metric only dominate at high energies. In a fixed internal metric gauge, only the the $SU(2)$ gauge symmetry is satisfied, the particle spectrum is identified and the Hamiltonian is shown to be bounded from below. Although Lorentz invariance is broken in this gauge, it is satisfied in general. The theory is quantized in this fixed, broken symmetry gauge as an $SU(2)$ gauge theory on a lattice with a lattice spacing equal to the Planck length. This produces a unitary and finite theory of quantum gravity. 
  Two locally positive expressions for the gravitational Hamiltonian, one using 4-spinors the other special orthonormal frames, are reviewed. A new quadratic 3-spinor-curvature identity is used to obtain another positive expression for the Hamiltonian and thereby a localization of gravitational energy and positive energy proof. These new results provide a link between the other two methods. Localization and prospects for quasi-localization are discussed. 
  The additivity of classical probabilities is only the first in a hierarchy of possible sum-rules, each of which implies its successor. The first and most restrictive sum-rule of the hierarchy yields measure-theory in the Kolmogorov sense, which physically is appropriate for the description of stochastic processes such as Brownian motion. The next weaker sum-rule defines a {\it generalized measure theory} which includes quantum mechanics as a special case. The fact that quantum probabilities can be expressed ``as the squares of quantum amplitudes'' is thus derived in a natural manner, and a series of natural generalizations of the quantum formalism is delineated. Conversely, the mathematical sense in which classical physics is a special case of quantum physics is clarified. The present paper presents these relationships in the context of a ``realistic'' interpretation of quantum mechanics. 
  We discuss earlier unsuccessful attempts to formulate a positive gravitational energy proof in terms of the New Variables of Ashtekar. We also point out the difficulties of a Witten spinor type proof. We then use the special orthonormal frame gauge conditions to obtain a locally positive expression for the New Variables Hamiltonian and thereby a ``localization'' of gravitational energy as well as a positive energy proof. 
  The cosmological constant problem is examined under the assumption that the extrinsic curvature of the space-time contributes to the vacuum. A compensation mechanism based on a variable cosmological term is proposed. Under a suitable hypothesis on the behavior of the extrinsic curvature, we find that an initially large $\Lambda(t)$ rolls down rapidly to zero during the early stages of the universe. Using perturbation analysis, it is shown that such vacuum behaves essentially as a spin-2 field which is independent of the metric. 
  We study properties of steady states (states with time-independent density operators) of systems of coupled harmonic oscillators. Formulas are derived showing how adiabatic change of the Hamiltonian transforms one steady state into another. It is shown that for infinite systems, sudden change of the Hamiltonian also tends to produce steady states, after a transition period of oscillations. These naturally arising steady states are compared to the maximum-entropy state (the thermal state) and are seen not to coincide in general. The approach to equilibrium of subsystems consisting of n coupled harmonic oscillators has been widely studied, but only in the simple case where n=1. The power of our results is that they can be applied to more complex subsustems, where n>1. It is shown that the use of coupled harmonic oscillators as heat baths models is fraught with some problems that do not appear in the simple n=1 case. Specifically, the thermal states that are though to be achievable through hard-sphere collisions with heat-bath particles can generally NOT be achieved with harmonic coupling to the heat-bath particles, except approximately when the coupling is weak. 
  The behavior of gravitational wave perturbations on a locally Minkowskian spacetime background containing a planar domain wall is investigated in the gauge-invariant general relativistic framework. It is shown that for this particular background the domain wall does not emit gravitational waves spontaneously by its free oscillation in the first order, although it scatters incidental gravitational waves. 
  We apply the reduced radial gauge to give the general solution of the metric in 2+1 dimensions in term of quadratures. It allows a complete controll on the support of the source. We use the result to prove that for a general stationary universe, conical at infinity, the weak energy condition and the absence of CTC at space infinity prevent the occurrrence of any CTC. 
  We present the complete scheme of the application of the one-and two dimensional subspace and subgroups method to five-dimensional gravity with a $G_{3}$ group of motion. We do so in the space time and in the potential space formalisms. From this method one obtains the Kramer, Belinsky-Ruffini, Dobiasch-Maison, Cl\'{e}ment, Gross-Perry-Sorkin solutions etc. as special cases. 
  The tunneling approach, for entropy generation in quantum gravity, is shown to be valid when applied to 3-D general relativity. The entropy of de Sitter and Reissner-Nordstr\"om external event horizons and of the 3-D black hole obtained by Ba\~nados et. al. is rederived from tunneling of the metric to these spacetimes. The analysis for spacetimes with an external horizon is carried out in a complete analogy with the 4-D case. However, we find significant differences for the black hole. In particular the initial configuration that tunnels to a 3-D black hole may not to yield an infinitely degenerate object, as in 4-D Schwarzschild black hole. We discuss the possible relation to the evaporation of the 3-D black hole. 
  We study the quantum fermions+gravity system, that is, the gravitational counterpart of QED. We start from the standard Einstein-Weyl theory, reformulated in terms of Ashtekar variables; and we construct its non- perturbative quantum theory by extending the loop representation of general relativity. We construct the fermion equivalent to the loop variables. Not surprisingly, fermions can be incorporated in the loop representation simply by including open curves into ''Loop space''. We explicitely construct the diffeomorphism and hamiltonain operators. The first can be fully solved as in pure gravity. The second is constructed by using a background-independent regularization technique. The theory retains the clean geometrical features of the pure quantum gravity. In particular, the hamiltonian constraint admits the same simple geometrical interpretation as its pure gravity counterpart. Quite surprisingly, a simple action codes the full dynamics of the interacting theory. To unravel the dynamics of the theory we study the evolution of the fermion-gravity system in the physical-time defined by an additional coupled (''clock''-) scalar field. We construct the Hamiltonian operator that evolves the system in this physical time. We show that this Hamiltonian is finite, diffeomorphism invariant, and has a simple geometrical action. The theory fermions+gravity evolving in the clock time is finally given by the combinatorial and geometrical action of this Hamiltonian on a set of graphs with a finite number of end points. This action defines the "topological Feynman rules" of the theory. 
  We propose a 4-dimensional Kaluza-Klein approach to general relativity in the (2,2)-splitting of space-time using the double null gauge. The associated Lagrangian is equivalent to the Einstein-Hilbert Lagrangian, since it yields the same field equations as the E-H Lagrangian does. It is describable as a (1+1)-dimensional Yang-Mills type gauge theory coupled to (1+1)-dimensional matter fields, where the minimal coupling associated with the diffeomorphism group of the 2-dimensional spacelike fibre space automatically appears. Written in the first-order formalism, our Lagrangian density directly yields a non-zero local Hamiltonian density, where the associated time function is the retarded time. From this Hamiltonian density, we obtain a positive-definite local gravitational energy density. In the asymptotically flat space-times, the volume integrals of the proposed local gravitational energy density over suitable 3-dimensional hypersurfaces correctly reproduce the Bondi and the ADM surface integral, at null and spatial infinity, respectively, supporting our proposal. We also obtain the Bondi mass-loss formula as a negative-definite flux integral of a bilinear in the gravitational currents at null infinity. 
  We develop a method for solving the field equations of a quadratic gravitational theory coupled to matter. The quadratic terms are written as a function of the matter stress tensor and its derivatives in such a way to have, order by order, a set of Einstein field equations with an effective $T_{\mu\nu}$. We study the cosmological scenario recovering the de Sitter exact solution, and the first order (in the coupling constants $\alpha$ and $\beta$ appearing in the gravitational Lagrangian) solution to the gauge cosmic string metric and the charged black hole. For this last solution we discuss the consequences on the thermodynamics of black holes, and in particular, the entropy - area relation which gets additional terms to the usual ${1\over4} A$ value. 
  We demonstrate that the Penrose inequality is valid for spherically symmetric geometries even when the horizon is immersed in matter. The matter field need not be at rest. The only restriction is that the source satisfies the weak energy condition outside the horizon. No restrictions are placed on the matter inside the horizon. The proof of the Penrose inequality gives a new necessary condition for the formation of trapped surfaces. This formulation can also be adapted to give a sufficient condition. We show that a modification of the Penrose inequality proposed by Gibbons for charged black holes can be broken in early stages of gravitational collapse. This investigation is based exclusively on the initial data formulation of General Relativity. 
  We discuss various features of the dynamical system determined by the flow of null geodesic generators of Cauchy horizons. Several examples with non--trivial (``chaotic'', ``strange attractors'', etc.) global behaviour are constructed. Those examples are relevant to the ``chronology protection conjecture'', and they show that the occurrence of ``fountains'' is {\em not} a generic feature of Cauchy horizons. 
  A central object in the interpretation of quantum mechanics of closed systems with decoherent histories is the decoherence matrix. But only for a very small number of models one is able to give explicit expressions for its elements. So numerical methods are required. Unfortunately the dimensions of this matrices are usually very high, which makes also a direct numerical calculation impossible. A solution of this problem would be given by a method which only calculates the dominant matrix elements. This includes to make a decision about the dominance of an element before it will be calculated. In this paper I will develop an algorithm that combines the numerical calculation of the elements of the decoherence matrix with a permanent estimation, so that finally the dominant elements will be calculated only. As an example I apply this procedure to the Caldeira- Leggett-modell. 
  In 2+1 dimensional Chern-Simons gravity, Wilson loops in the three dimensional Anti de Sitter group, $SO(2,2)$, reproduce the spinning black hole of Ba\~nados, Teitelboim and Zanelli (BTZ) by naturally duplicating the necessary identification of points of a four dimensional globally $SO(2,2)$ invariant space in which the hole appears as an embedding. 
  ``Reports of my death are greatly exaggerated''   - Mark Twain. We consider the claim by Damour, Deser and McCarthy that nonsymmetric gravity theory has unacceptable global asymptotics. We explain why this claim is incorrect. 
  We prove that static, spherically symmetric, asymptotically flat, regular solutions of the Einstein-Yang-Mills equations are unstable for arbitrary gauge groups. The proof involves the following main steps. First, we show that the frequency spectrum of a class of radial perturbations is determined by a coupled system of radial "Schroedinger equations". Eigenstates with negative eigenvalues correspond to exponentially growing modes. Using the variational principle for the ground state it is then proven that there always exist unstable modes (at least for "generic" solitons). This conclusion is reached without explicit knowledge of the possible equilibrium solutions. 
  The energy density correlation function $C(x,y)=<\rho(x)\rho(y)>-<\rho>^2$ and its Fourier transform generated by the gravitational tidal forces of the inflationary de Sitter expansion are derived for a massless or light ($m<<H$) neutral scalar field $\Phi$ without self--interactions, minimally coupled to gravity. The field has no classical background component, $<\Phi>=0$. Every observationally relevant mode (which has today $\lambda_{phys}<H_0^{-1}$) had at early times $R/k_{phys}^2\to0$, and is taken to be initially in the Minkowski vacuum state. Our computation of $C(x,y)$, which involves four field operators, is finite and unambiguous at each step, since we use the following two tools: (1) We use a normal ordered energy density operator N$[\rho]$ and show that any normal ordering N gives the same finite result. (2) Since $C(x,y)$ has the universal $|x-y|^{-8}$ short--distance behaviour, the Fourier transform can only be performed after smearing the energy density operator in space and time with a smearing scale $\tau$. The resulting energy density fluctuations are non--Gaussian, but obey a $\chi^2$--distribution. The power spectrum $\dk$ involves a smearing scale $\tau$, and we choose $\tau=k^{-1}$. For massless scalars we obtain $k^3\dk\sim H^4k^4$ on super--horizon scales $k<H$. For light scalars with masses $m\ll H$ a plateau appears on the largest scales $k<\sqrt{m^3/H}$, the result is $k^3\dk\sim m^6H^2(k/H)^{4m^2/3H^2}$. On sub--horizon scales $k>H$ we have the universal law $k^3\dk\sim k^8$. 
  We establish the relation between the ISO(2,1) homotopy invariants and the polygon representation of (2+1)-dimensional gravity. The polygon closure conditions, together with the SO(2,1) cycle conditions, are equivalent to the ISO(2,1) cycle conditions for the representa- tions of the fundamental group in ISO(2,1). Also, the symplectic structure on the space of invariants is closely related to that of the polygon representation. We choose one of the polygon variables as internal time and compute the Hamiltonian, then perform the Hamilton-Jacobi transformation explicitly. We make contact with other authors' results for g = 1 and g = 2 (N = 0). 
  We consider the quantum dynamics of both open and closed two- dimensional universes with ``wormholes'' and particles. The wave function is given as a sum of freely propagating amplitudes, emitted from a network of mapping class images of the initial state. Interference between these amplitudes gives non-trivial scattering effects, formally analogous to the optical diffraction by a multidimensional grating; the ``bright lines'' correspond to the most probable geometries. 
  We study a general field theory of a scalar field coupled to gravitation through a quadratic Gauss-Bonnet term $\xi({\phi})$$R_{GB}^2$. We show that, under mild assumptions about the function $\xi(\phi)$, the classical solutions in a spatially flat FRW background include singularity - free solutions. 
  New exact solutions of Einstein's gravity coupled to a self-interacting conformal scalar field are derived in this work. Our approach extends a solution-generating technique originally introduced by Bekenstein for massless conformal scalar fields. Solutions are obtained for a Friedmann-Robertson-Walker geometry both for the cases of zero and non-zero curvatures, and a variety of interesting features are found. It is shown that one class of solutions tends asymptotically to a power-law inflationary behaviour $S(t)\sim t^p$ with $p>1$, while another class exhibits a late time approach to the $S(t)\sim t$ behaviour of the coasting models. Bouncing models which avoid an initial singularity are also obtained. A general discussion of the asymptotic behaviour and of the possibility of occurrence of inflation is provided. 
  Null and timelike geodesics around a 2+1 black hole are determined. Complete geodesics of both types exist in the rotating black-hole background, but not in the spinless case. Upper and lower bounds for the radial size of the orbits are given in all cases and the possibility of passing from one black hole exterior spacetime to another is discussed using the Penrose diagrams. An analysis of particle motions by means of effective potentials and orbit graphs are also included. 
  We discuss bounds on the dilaton mass, following from the cosmological amplification of the quantum fluctuations of the dilaton background, under the assumption that such fluctuations are dominant with respect to the classical background oscillations. We show that if the fluctuation spectrum grows with the frequency the bounds are relaxed with respect to the more conventional case of a flat or decreasing spectrum. As a consequence, the allowed range of masses may become compatible with models of supersymmetry breaking, and with a universe presently dominated by a relic background of dilaton dark matter. 
  We propose a quantum description of black holes. The degrees of freedom to be quantized are identified with the microscopic degrees of freedom of the horizon, and their dynamics is governed by the action of the relatistic bosonic membrane in $D=4$. We find that a consistent and plausible description emerges, both at the classical and at the quantum level. We present results for the level structure of black holes. We find a ``principal series'' of levels, corresponding to quantization of the area of the horizon. From each level of this principal series starts a quasi-continuum of levels due to excitations of the membrane. We discuss the statistical origin of the black hole entropy and the relation with Hawking radiation and with the information loss problem. The limits of validity of the membrane approach turn out to coincide with the known limits of validity of the thermodynamical description of black holes. 
  We introduce a gauge and diffeomorphism invariant theory on Yang-Mills phase space. The theory is well defined for an arbitrary gauge group with an invariant bilinear form, it contains only first class constraints, and the spacetime metric has a simple form in terms of the phase space variables. With gauge group $SO(3,C)$, the theory equals the Ashtekar formulation of gravity with a cosmological constant. For Lorentzian signature, the theory is complex, and we have not found any good reality conditions. In the Euclidean signature case, everything is real. In a weak field expansion around de Sitter spacetime, the theory is shown to give the conventional Yang-Mills theory to the lowest order in the fields. 
  We study a model for quantum gravity on a circle in which the notion of a classical metric tensor is replaced by a quantum metric with an inhomogeneous transformation law under diffeomorphisms. This transformation law corresponds to the co--adjoint action of the Virasoro algebra, and resembles that of the connection in Yang--Mills theory. The transformation property is motivated by the diffeomorphism invariance of the one dimensional Schr\"odinger equation. The quantum distance measured by the metric corresponds to the phase of a quantum mechanical wavefunction. The dynamics of the quantum gravity theory are specified by postulating a Riemann metric on the space $Q$ of quantum metrics and taking the kinetic energy operator to be the resulting laplacian on the configuration space $Q/\rm Diff_0(S^1)$. The resulting metric on the configuration space is analyzed and found to have singularities. The second--quantized Schr\"odinger equation is derived, some exact solutions are found, and a generic wavefunction behavior near one of the metric singularities is described. Finally some further directions are indicated, including an analogue of the Yamabe problem of differential geometry. 
  It is shown how coarse-graining of quantum field ^M theory in de Sitter space leads to the emergence of a classical ^M stochastic description as an effective theory in the infra-red regime. ^M The quantum state of the coarse-grained scalar field is found to be a highly^M squeezed coherent state, whose center performs a random walk on a^M bundle of classical trajectories.^M 
  We calculate the response of an ideal Michelson interferometer incorporating both dual recycling and squeezed light to gravitational waves. The photon counting noise has contributions from the light which is sent in through the input ports as well as the vacuum modes at sideband frequencies generated by the gravitational waves. The minimum detectable gravity wave amplitude depends on the frequency of the wave as well as the squeezing and recycling parameters. Both squeezing and the broadband operation of dual recycling reduce the photon counting noise and hence the two techniques can be used together to make more accurate phase measurements. The variance of photon number is found to be time-dependent, oscillating at the gravity wave frequency but of much lower order than the constant part. 
  It is shown - in Ashtekar's canonical framework of General Relativity - that spherically symmetric (Schwarzschild) gravity in 4 dimensional space-time constitutes a finite dimensional completely integrable system. Canonically conjugate observables for asymptotically flat space-times are masses as action variables and - surprisingly - time variables as angle variables, each of which is associated with an asymptotic "end" of the Cauchy surfaces. The emergence of the time observable is a consequence of the Hamiltonian formulation and its subtleties concerning the slicing of space and time and is not in contradiction to Birkhoff's theorem. The results are of interest as to the concept of time in General Relativity. They can be formulated within the ADM formalism, too. Quantization of the system and the associated Schr\"odinger equation depend on the allowed spectrum of the masses. 
  We consider multiple scalar fields coupled to gravity, with special attention given to two-field theories. First, the conditions necessary for these theories to meet solar system tests are given. Next, we investigate the cosmological evolution of the fields to see if these conditions can be met. Solutions are found in the dust era, as well as radiation and cosmological constant dominated epochs. The possibility of inflation in these theories is discussed. While power law growth of the scalar fields can yield the appropriate conditions to meet solar system constraints, these solutions are unstable. 
  The two dimensional version of the Sen connection for spinors and tensors on spacelike 2-surfaces is constructed. A complex metric $\gamma_{AB}$ on the spin spaces is found which characterizes both the algebraic and extrinsic geometrical properties of the 2-surface $\$ $. The curvature of the two dimensional Sen operator $\Delta_e$ is the pull back to $\$ $ of the anti-self-dual part of the spacetime curvature while its `torsion' is a boost gauge invariant expression of the extrinsic curvatures of $\$ $. The difference of the 2 dimensional Sen and the induced spin connections is the anti-self-dual part of the `torsion'. The irreducible parts of $\Delta_e$ are shown to be the familiar 2-surface twistor and the Weyl--Sen--Witten operators. Two Sen--Witten type identities are derived, the first is an identity between the 2 dimensional twistor and the Weyl--Sen--Witten operators and the integrand of Penrose's charge integral, while the second contains the `torsion' as well. For spinor fields satisfying the 2-surface twistor equation the first reduces to Tod's formula for the kinematical twistor. 
  Table of contents   Editorial   Correspondents   Gravity News:   Open Letter to gravitational physicists, Beverly Berger   A Missouri relativist in King Gustav's Court, Clifford Will   Gary Horowitz wins the Xanthopoulos award, Abhay Ashtekar   Research briefs:   Gamma-ray bursts and their possible cosmological implications, Peter Meszaros   Current activity and results in laboratory gravity, Riley Newman   Update on representations of quantum gravity, Donald Marolf   Ligo project report: December 1993, Rochus E. Vogt   Dark matter or new gravity?, Richard Hammond   Conference reports:   Gravitational waves from coalescing compact binaries, Curt Cutler   Mach's principle: from Newton's bucket to quantum gravity, Dieter Brill   Cornelius Lanczos international centenary conference, David Brown   Third Midwest relativity conference, David Garfinkle 
  We consider a quantum scalar field on an arbitrary gravitational background. We obtain the effective {\it in-in} equations for the gravitational fields using a covariant and non-local approximation for the effective action proposed by Vilkovisky and collaborators. {}From these equations, we compute the quantum corrections to the Newtonian potential. We find logarithmic corrections which we identify as the running of the gravitational constants. This running coincides with the renormalization group prediction only for minimal and conformal coupling. 
  Spacetime discretized in simplexes, as proposed in the pioneer work of Regge, is described in terms of selfdual variables. In particular, we elucidate the "kinematic" structure of the initial value problem, in which 3--space is divided into flat tetrahedra, paying particular attention to the role played by the reality condition for the Ashtekar variables. An attempt is made to write down the vector and scalar constraints of the theory in a simple and potentially useful way. 
  The recently constructed two dimensional Sen connection is applied in the problem of quasi-local energy-momentum in general relativity. First it is shown that, because of one of the two 2 dimensional Sen--Witten identities, Penrose's quasi-local charge integral can be expressed as a Nester--Witten integral.Then, to find the appropriate spinor propagation laws to the Nester--Witten integral, all the possible first order linear differential operators that can be constructed only from the irreducible chiral parts of the Sen operator alone are determined and examined. It is only the holomorphy or anti-holomorphy operator that can define acceptable propagation laws. The 2 dimensional Sen connection thus naturally defines a quasi-local energy-momentum, which is precisely that of Dougan and Mason. Then provided the dominant energy condition holds and the 2-sphere S is convex we show that the next statements are equivalent: i. the quasi-local mass (energy-momentum) associated with S is zero; ii.the Cauchy development $D(\Sigma)$ is a pp-wave geometry with pure radiation ($D(\Sigma)$ is flat), where $\Sigma$ is a spacelike hypersurface whose boundary is S; iii. there exist a Sen--constant spinor field (two spinor fields) on S. Thus the pp-wave Cauchy developments can be characterized by the geometry of a two rather than a three dimensional submanifold. 
  We investigate implications of decoherence for quantum systems which are classically chaotic. We show that, in open systems, the rate of von Neumann entropy production quickly reaches an asymptotic value which is: (i) independent of the system-environment coupling, (ii) dictated by the dynamics of the system, and (iii) dominated by the largest Lyapunov exponent. These results shed a new light on the correspondence between quantum and classical dynamics as well as on the origins of the ``arrow of time.'' 
  We examine the dependence of decoherence on the spectral density of the environment as well as on the initial state of the system. We use two simple examples to illustrate some important effects. The simplest derivation of the general form of the master equation for a general quantum Brownian motion is outlined in an Appendix. 
  We find a new class of exact solutions of the five-dimensional Einstein equations whose corresponding four-dimensional spacetime possesses a Schwarzschild-like behavior. The electromagnetic potential depends on a harmonic function and can be choosen to be of a monopole, dipole, etc. field. The solutions are asymptotically flat and for vanishing magnetic field the four metrics are of the Schwarzschild solution. The spacetime is singular in $r=2m$ for higher multipole moments, but regular for monopoles or vanishing magnetic fields in this point. The scalar field posseses a singular behavior. #(Preprint CINVESTAV 15/93)# 
  The grand-canonical thermodynamic potential for Bose and Fermi fields in anti-de Sitter space-time is introduced and a Mellin complex representation is given. This is used to investigate the high temperature properties of the thermal state at finite charge. The time-like nature of spatial infinity is shown to act as a boundary, and changes the free energy of the field in thermal equilibrium. 
  (Some Latex problems should be removed in this version) Fermi coordinates (FC) are supposed to be the natural extension of Cartesian coordinates for an arbitrary moving observer in curved space-time. Since their construction cannot be done on the whole space and even not in the whole past of the observer we examine which construction principles are responsible for this effect and how they may be modified. One proposal for a modification is made and applied to the observer with constant acceleration in the two and four dimensional Minkowski space. The two dimensional case has some surprising similarities to Kruskal space which generalize those found by Rindler for the outer region of Kruskal space and the Rindler wedge. In perturbational approaches the modification leads also to different predictions for certain physical systems. As an example we consider atomic interferometry and derive the deviation of the acceleration-induced phase shift from the standard result in Fermi coordinates. 
  Selection of the preferred classical set of states in the process of decoherence -- so important for cosmological considerations -- is discussed with an emphasis on the role of information loss and entropy. {\it Persistence of correlations} between the observables of two systems (for instance, a record and a state of a system evolved from the initial conditions described by that record) in the presence of the environment is used to define classical behavior. From the viewpoint of an observer (or any system capable of maintaining records) {\it predictability} is a measure of such persistence. {\it Predictability sieve} -- a procedure which employs both the statistical and algorithmic entropies to systematically explore all of the Hilbert space of an open system in order to eliminate the majority of the unpredictable and non-classical states and to locate the islands of predictability including the preferred {\it pointer basis} is proposed. Predictably evolving states of decohering systems along with the time-ordered sequences of records of their evolution define the effectively classical branches of the universal wavefunction in the context of the ``Many Worlds Interpretation". The relation between the consistent histories approach and the preferred basis is considered. It is demonstrated that histories of sequences of events corresponding to projections onto the states of the pointer basis are consistent. 
  In Einstein's gravitational theory, the spacetime is Riemannian, that is, it has vanishing torsion and vanishing nonmetricity (covariant derivative of the metric). In the gauging of the general affine group ${A}(4,R)$ and of its subgroup ${GL}(4,R)$ in four dimensions, energy--momentum and hypermomentum currents of matter are canonically coupled to the one--form basis and to the connection of a metric--affine spacetime with nonvanishing torsion and nonmetricity, respectively. Fermionic matter can be described in this framework by half--integer representations of the $\overline{SL}(4,R)$ covering subgroup. --- We set up a (first--order) Lagrangian formalism and build up the corresponding Noether machinery. For an arbitrary gauge Lagrangian, the three gauge field equations come out in a suggestive Yang-Mills like form. The conservation--type differential identities for energy--momentum and hypermomentum and the corresponding complexes and superpotentials are derived. Limiting cases such as the Einstein--Cartan theory are discussed. In particular we show, how the ${A}(4,R)$ may ``break down'' to the Poincar\'e (inhomogeneous Lorentz) group. In this context, we present explicit models for a symmetry breakdown in the cases of the Weyl (or homothetic) group, the ${SL}(4,R)$, or the ${GL}(4,R)$. 
  We construct time-dependent multi-centre solutions to three-dimensional general relativity with zero or negative cosmological constant. These solutions correspond to dynamical systems of freely falling black holes and conical singularities, with a multiply connected spacetime topology. Stationary multi-black-hole solutions are possible only in the extreme black hole case. 
  The most promising source of gravitational waves for the planned detectors LIGO and VIRGO are merging compact binaries, i.e., neutron star/neutron star (NS/NS), neutron star/black hole (NS/BH), and black hole/black-hole (BH/BH) binaries. We investigate how accurately the distance to the source and the masses and spins of the two bodies will be measured from the gravitational wave signals by the three detector LIGO/VIRGO network using ``advanced detectors'' (those present a few years after initial operation). The combination ${\cal M} \equiv (M_1 M_2)^{3/5}(M_1 +M_2)^{-1/5}$ of the masses of the two bodies is measurable with an accuracy $\approx 0.1\%-1\%$. The reduced mass is measurable to $\sim 10\%-15\%$ for NS/NS and NS/BH binaries, and $\sim 50\%$ for BH/BH binaries (assuming $10M_\odot$ BH's). Measurements of the masses and spins are strongly correlated; there is a combination of $\mu$ and the spin angular momenta that is measured to within $\sim 1\%$. We also estimate that distance measurement accuracies will be $\le 15\%$ for $\sim 8\%$ of the detected signals, and $\le 30\%$ for $\sim 60\%$ of the signals, for the LIGO/VIRGO 3-detector network. 
  We study conformal properties of the quantum kinetic equations in curved spacetime. A transformation law for the covariant Wigner function under conformal transformations of a spacetime is derived by using the formalism of tangent bundles. The conformal invariance of the quantum corrected Vlasov equation is proven. This provides a basis for generating new solutions of the quantum kinetic equations in the presence of gravitational and other external fields. We use our method to find explicit quantum corrections to the class of locally isotropic distributions, to which equilibrium distributions belong. We show that the quantum corrected stress--energy tensor for such distributions has, in general, a non--equilibrium structure. Local thermal equilibrium is possible in quantum systems only if an underlying spacetime is conformally static (not stationary). Possible applications of our results are discussed. 
  A brief review of recent research on soliton and black hole solutions of Einstein's equations with nonlinear field sources is presented and some open questions are pointed out. 
  We present a self-similar model of spherically symmetric collapse of a massless scalar field with a parameter $p$. The black hole formation is explicitly shown to occur only in the strong-field implosion of $p >1$. The field evolution in the critical limit $ p \rightarrow 1 $ is compared with numerical results found by Choptuik. 
  Histories and measures for quantum cosmology are investigated through a quantization of the Bianchi IX cosmology using path integral techniques. The result, derived in the context of Ashtekar variables, is compared with earlier work. A non-trivial correction to the measure is found, which may dominate the classical potential for universes on the Planck scale. 
  The Einstein equations for spacetimes with two commuting spacelike Killing field symmetries are studied from a Hamiltonian point of view. The complexified Ashtekar canonical variables are used, and the symmetry reduction is performed directly in the Hamiltonian theory. The reduced system corresponds to the field equations of the SL(2,R) chiral model with additional constraints.   On the classical phase space, a method of obtaining an infinite number of constants of the motion, or observables, is given. The procedure involves writing the Hamiltonian evolution equations as a single `zero curvature' equation, and then employing techniques used in the study of two dimensional integrable models. Two infinite sets of observables are obtained explicitly as functionals of the phase space variables. One set carries sl(2,R) Lie algebra indices and forms an infinite dimensional Poisson algebra, while the other is formed from traces of SL(2,R) holonomies that commute with one another. The restriction of the (complex) observables to the Euclidean and Lorentzian sectors is discussed.   It is also shown that the sl(2,R) observables can be associated with a solution generating technique which is linked to that given by Geroch. 
  The self-dual Einstein equation (SDE) is shown to be equivalent to the two dimensional chiral model, with gauge group chosen as the group of area preserving diffeomorphisms of a two dimensional surface. The approach given here leads to an analog of the Plebanski equations for general self-dual metrics, and to a natural Hamiltonian formulation of the SDE, namely that of the chiral model. 
  We give an exact spherically symmetric solution for the Einstein-scalar field system. The solution may be interpreted as an inhomogeneous dynamical scalar field cosmology. The spacetime has a timelike conformal Killing vector field and is asymptotically conformally flat. It also has black or white hole-like regions containing trapped surfaces. We describe the properties of the apparent horizon and comment on the relevance of the solution to the recently discovered critical behaviour in scalar field collapse. 
  We study a generality of an inflationary scenario by integrating the Einstein equations numerically in a plane-symmetric spacetime. We consider the inhomogeneous spacetimes due to (i) localized gravitational waves with a positive cosmological constant $\Lambda$, and (ii) an inhomogeneous inflaton field $\Phi$ with a potential $\frac12 m^2 \Phi^2$. For the case (i), we find that any initial inhomogeneities are smoothed out even if waves collide, so that we conclude that inhomogeneity due to gravitational waves do not prevent the onset of inflation. As for the case (ii), if the mean value of the inflaton field is initially as large as the condition in an isotropic and homogeneous inflationary model (i.e., the mean value is larger than several times Planck mass), the field is soon homogenized and the universe always evolves into de Sitter spacetime. These support the cosmic no hair conjecture in a planar universe. We also discuss the effects of an additional massless scalar field, which is introduced to set initial data in usual analysis. 
  An exact one parameter ($\alpha$) family of solutions representing scalar field collapse is presented. These solutions exhibit a type of critical behaviour which has been discussed by Choptuik. The three possible evolutions are outlined. For supercritical evolution (when black holes form) I show that a quantity related to the mass of the black hole exhibits a power law dependence on $\alpha$, for near critical evolution $M \simeq |\alpha-\alpha_{\rm crit}|^{1/2}$. Based on the properties of the solution some comments are also made on conjectures of Choptuik. 
  Spherical configurations that are very massive must be surrounded by apparent horizons. These in turn, when placed outside a collapsing body, have a fixed area and must propagate outward with a velocity equal to the velocity of radially outgoing photons. That proves, within the framework of the (1+3) formalism and without resorting to the Birkhoff theorem, that apparent horizons coincide with event horizons in electrovacuum. 
  We consider the specialization to spatially homogenous solutions of the   Jacobson formulation of N=1 canonical supergravity in terms of Ashtekar's new variables.   We find that the classical Poisson algebra of the supersymmetry constraints is preserved by this specialization only for Bianchi type A models. The quantization of supersymmetric Bianchi type A models is carried out in the triad representation. We find the physical states of this quantum theory. Since we are missing a suitable inner product on these physical states, our results are only formal. 
  We study the behavior of infinite systems of coupled harmonic oscillators as t->infinity, and generalize the Central Limit Theorem (CLT) to show that their reduced Wigner distributions become Gaussian under quite general conditions. This shows that generalized coherent states tend to be produced naturally. A sufficient condition for this to happen is shown to be that the spectral function is analytic and nonlinear. For a rectangular lattice of coupled oscillators, the nonlinearity requirement means that waves must be dispersive, so that localized wave-packets become suppressed. Virtually all harmonic heat-bath models in the literature satisfy this constraint, and we have good reason to believe that coherent states and their generalizations are not merely a useful analytical tool, but that nature is indeed full of them. Standard proofs of the CLT rely heavily on the fact that probability densities are non-negative. Although the CLT generally fails if the probability densities are allowed to take negative values, we show that a CLT does indeed hold for a special class of such functions. We find that, intriguingly, nature has arranged things so that all Wigner functions belong to this class. 
  We investigate the phase-space for trajectories in multi-black hole spacetimes. We find that complete, chaotic geodesics are well described by Lyapunov exponents, and that the attractor basin boundary scales as a fractal in a diffeomorphism invariant manner. 
  We show in this article how the usual hamiltonian formalism of General Relativity should be modified in order to allow the inclusion of the Euclidean classical solutions of Einstein's equations. We study the effect that the dynamical change of signature has on the superspace and we prove that it induces a passage of the signature of the supermetric from ($-+++++$) to ($+-----$). Next, all these features are more particularly studied on the example of minisuperspaces. Finally, we consider the problem of quantization of the Euclidean solutions. The consequences of different choices of boundary conditions are examined. 
  By adding the Pontrjagin topological invariant to the gauge theory of the de Sitter group proposed by MacDowell and Mansouri we obtain an action quadratic in the field-strengths, of the Chern-Simons type, from which the Ashtekar formulation is derived. 
  We show that the appropriate vacuum state for the interior of a box with reflecting walls being lowered adiabatically into a Schwarzschild black hole is the Boulware state. This is concordant with the results of Unruh and Wald, who used a different approach to obtain the stress-energy inside the box. Some comments about an entropy bound for ordinary matter, as first conjectured by Bekenstein, are presented. 
  We discuss how to fix the gauge in the canonical treatment of Lagrangians, with finite number of degrees of freedom, endowed with time reparametrization invariance. The motion can then be described by an effective Hamiltonian acting on the gauge shell canonical space. The system is then suited for quantization. We apply this treatment to the case of a Robertson--Walker metric interacting with zero modes of bosonic fields and write a \S equation for the on--shell wave function. 
  Various assumptions underlying the uniqueness theorems for black holes are discussed. Some new results are described, and various unsatisfactory features of the present theory are stressed. 
  Conformally invariant GUT-like model including gravity based on Riemann - Cartan space-time $U_4$ is considered. Cosmological scenario that follows from the model is discussed and standard quantum gravitational formalism in the Arnowitt-Deser-Misner form is developed. General formalism is then illustrated on Bianchi-IX minisuperspace cosmological model. Wave functions of the universe in the de Sitter minisuperspace model with Vilenkin and Hartle-Hawking boundary conditions are considered and corresponding probability distributions for the scalar field values are calculated. 
  We determine the wavefunction that corresponds to the exponential of the Chern-Simons action in a family of gravitational models provided with cosmological constant whose non-perturbative canonical quantization is completely known. We show that this wavefunction does not represent a proper quantum state, because it is not normalizable with respect to the unique inner product of Lorentzian gravity. 
  In this paper we develop some combinatorial models for continuous spaces. In this spirit we study the approximations of continuous spaces by graphs, molecular spaces and coordinate matrices. We define the dimension on a discrete space by means of axioms, and the axioms are based on an obvious geometrical background. This work presents some discrete models of n-dimensional Euclidean spaces, n-dimensional spheres, a torus and a projective plane. It explains how to construct new discrete spaces and describes in this connection several three-dimensional closed surfaces with some topological singularities   It also analyzes the topology of (3+1)-spacetime. We are also discussing the question by R. Sorkin [19] about how to derive the system of simplicial complexes from a system of open covering of a topological space S. 
  We show that the Wheeler-DeWitt equation with a consistent boundary condition is only compatible with an arrow of time that formally reverses in a recollapsing universe. Consistency of these opposite arrows is facilitated by quantum effects in the region of the classical turning point. Since gravitational time dilation diverges at horizons, collapsing matter must then start re-expanding ``anticausally" (controlled by the reversed arrow) before horizons or singularities can form. We also discuss the meaning of the time-asymmetric expression used in the definition of ``consistent histories". We finally emphasize that there is no mass inflation nor any information loss paradox in this scenario. 
  The $S$-matrix Ansatz for the construction of a quantum theory of black holes is further exploited. We first note that treating the metric tensor $g_{\m\n}$ as an operator rather than a background allows us to use a setting where information is not lost. But then we also observe that the 'trans-Planckian' particles (particles with kinetic energies beyond the Planck energy) need to be addressed. It is now postulated that they can be transformed into 'cis-Planckian particles' (having energies less that the Planck energy). This requires the existence of a delicate algebra of operators defined at a black hole horizon. Operators describing ingoing particles are mapped onto operators describing outgoing ones, preserving their commutator algebra. At short distance, the transverse gravitational back reaction dictates a discrete lattice of data points, and at large distance the algebra must reproduce known interactions of the Standard Model of elementary particles. It is suggested that further elaboration of these ideas requiring complete agreement with general relativity and unitarity should lead to severe restrictions concerning the inter-particle interactions. 
  Given any space-time $M$ without singularities and any event $O$, there is a natural continuous mapping $f$ of a two dimensional sphere into any space-like slice $T$ not containing $O$. The set of future null geodesics (or the set of past null geodesics) forms a 2-sphere $S^2$ and the map $f$ sends a point in $S^2$ to the point in $T$ which is the intersection of the corresponding geodesic with $T$. To require that $f$, which maps a two dimensional space into a three dimensional space, satisfy the condition that any point in the image of $f$ has an odd number of preimages, is to place a very strong condition on $f$. This is exactly what happens in any case where the odd image theorem holds for a transparent gravitational lens. It is argued here that this condition on $f$ is probably too restrictive to occur in general; and if it appears to hold in a specific example, then some $f$ should be calculated either analytically or numerically to provide either an illustrative example or counterexample. 
  The problem of the mutual attraction and joining of two black holes is of importance as both a source of gravitational waves and as a testbed of numerical relativity. If the holes start out close enough that they are initially surrounded by a common horizon, the problem can be viewed as a perturbation of a single black hole. We take initial data due to Misner for close black holes, apply perturbation theory and evolve the data with the Zerilli equation. The computed gravitational radiation agrees with and extends the results of full numerical computations. 
  Topological defects can be formed during inflation by phase transitions as well as by quantum nucleation. We study the effect of the expansion of the Universe on the internal structure of the defects. We look for stationary solutions to the field equations, i.e. solutions that depend only on the proper distance from the defect core. In the case of very thin defects, whose core dimensions are much smaller than the de Sitter horizon, we find that the solutions are well approximated by the flat space solutions. However, as the flat space thickness parameter $\delta_0$ increases we notice a deviation from this, an effect that becomes dramatic as $\delta_0$ approaches $(H)^{-1}/{\sqrt 2}$. Beyond this critical value we find no stationary solutions to the field equations. We conclude that only defects that have flat space thicknesses less than the critical value survive, while thicker defects are smeared out by the expansion. 
  We observe critical phenomena in spherical collapse of radiation fluid. A sequence of spacetimes $\cal{S}[\eta]$ is numerically computed, containing models ($\eta\ll 1$) that adiabatically disperse and models ($\eta\gg 1$) that form a black hole. Near the critical point ($\eta_c$), evolutions develop a self-similar region within which collapse is balanced by a strong, inward-moving rarefaction wave that holds $m(r)/r$ constant as a function of a self-similar coordinate $\xi$. The self-similar solution is known and we show near-critical evolutions asymptotically approaching it. A critical exponent $\beta \simeq 0.36$ is found for supercritical ($\eta>\eta_c$) models. 
  We present a method of post-Newtonian expansion to solve the homogeneous Regge-Wheeler equation which describes gravitational waves on the Schwarzschild spacetime. The advantage of our method is that it allows a systematic iterative analysis of the solution. Then we obtain the Regge-Wheeler function which is purely ingoing at the horizon in closed analytic form, with accuracy required to determine the gravitational wave luminosity to (post)$^{4}$-Newtonian order (i.e., order $v^8$ beyond Newtonian) from a particle orbiting around a Schwarzschild black hole. Our result, valid in the small-mass limit of one body, gives an important guideline for the study of coalescing compact binaries. In particular, it provides basic formulas to analytically calculate detailed waveforms and luminosity, including the tail terms to (post)$^3$-Newtonian order, which should be reproduced in any other post-Newtonian calculations. 
  We show that the quantum super-minisuperspace of N=1 supergravity with $\Lambda \ne 0 $ has no non-trivial physical states for class A Bianchi models. Hence, in super quantum cosmology, the vanishing of $\Lambda$ is a condition for the existence of the universe. We argue that this result implies that in full supergravity with $\Lambda$ there are no non-trivial physical states with a finite number of fermionic fields. We use the Jacobson canonical formulation. 
  We present a method for generating exact solutions of Einstein equations in vacuum using harmonic maps, when the spacetime possesses two commutating Killing vectors. This method consists in writing the axisymmetric stationry Einstein equations in vacuum as a harmonic map which belongs to the group SL(2,R), and decomposing it in its harmonic "submaps". This method provides a natural classification of the solutions in classes (Weil's class, Lewis' class etc). 
  We prove the instability of the gravitating regular sphaleron solutions of the $SU(2)$ Einstein-Yang-Mills-Higgs system with a Higgs doublet, by studying the frequency spectrum of a class of radial perturbations. With the help of a variational principle we show that there exist always unstable modes. Our method has the advantage that no detailed knowledge of the equilibrium solution is required. It does, however, not directly apply to black holes. 
  This is a contribution to the forthcoming book "Canonical Gravity: {}From Classical to Quantum" edited by J. Ehlers and H. Friedrich. Ashtekar's criterion for choosing an inner product in the quantisation of constrained systems is discussed. An erroneous claim in a previous paper is corrected and a cautionary example is presented. 
  We introduce a gauge and diffeomorphism invariant theory on the Yang-Mills phase space. The theory is well defined for an arbitrary gauge group with an invariant bilinear form, it contains only first class constraints, and the spacetime metric has a simple form in terms of the phase space variables. With gauge group $SO(3,C)$, the theory equals the Ashtekar formulation of gravity with a cosmological constant. For Lorentzian signature, the theory is complex, and we have not found any good reality conditions. In the Euclidean signature case, everything is real. In a weak field expansion around de Sitter spacetime, the theory is shown to give the conventional Yang-Mills theory to the lowest order in the fields. We show that the coupling to a Higgs scalar is straightforward, while the naive spinor coupling does not work. We have not found any way of including spinors that gives a closed constraint algebra. For gauge group $U(2)$, we find a static and spherically symmetric solution. 
  The curvature coordinates $T,R$ of a Schwarz\-schild spacetime are turned into canonical coordinates $T(r), {\sf R}(r)$ on the phase space of spherically symmetric black holes. The entire dynamical content of the Hamiltonian theory is reduced to the constraints requiring that the momenta $P_{T}(r), P_{\sf R}(r)$ vanish. What remains is a conjugate pair of canonical variables $m$ and $p$ whose values are the same on every embedding. The coordinate $m$ is the Schwarzschild mass, and the momentum $p$ the difference of parametrization times at right and left infinities. The Dirac constraint quantization in the new representation leads to the state functional $\Psi (m; T, {\sf R}] = \Psi (m)$ which describes an unchanging superposition of black holes with different masses. The new canonical variables may be employed in the study of collapsing matter systems. 
  Inflation is known to be generically eternal to the future: the false vacuum is thermalized in some regions of space, while inflation continues in other regions. Here, we address the question of whether inflation can also be eternal to the past. We argue that such a steady-state picture is impossible and, therefore, that inflation must have had a beginning. First, it is shown that the old inflationary model is not past-eternal. Next, some necessary conditions are formulated for inflationary spacetimes to be past-eternal and future-eternal. It is then shown that these conditions cannot simultaneously hold in physically reasonable open universes. 
  The question of the interpretation of Wheeler-DeWitt solutions in the context of cosmological models is addressed by implementing the Hamiltonian constraint as a spinor wave equation in minisuperspace. We offer a relative probability interpretation based on a non-closed vector current in this space and a prescription for a parametrisation of classical solutions in terms of classical time. Such a prescription can accommodate classically degenerate metrics describing manifolds with signature change. The relative probability density, defined in terms of a Killing vector of the Dewitt metric on minisuperspace, should permit one to identify classical loci corresponding to geometries for a classical manifold. This interpretation is illustrated in the context of a quantum cosmology model for two-dimensional dilaton gravity. 
  We analyse the classical and quantum theory of a scalar field interacting with gravitation in two dimensions. We describe a class of analytic solutions to the Wheeler-DeWitt equation from which we are able to synthesise states that give prominence to a set of classical cosmologies. These states relate in a remarkable way to the general solution of the classical field equations. We express these relations, without approximation, in terms of a metric and a closed form on the domain of quantum states. 
  The model of a stationary universe and the notion of local times presented in [10] are reviewed with some alternative formulation of the consistent unification of the Riemannian and Euclidean geometries of general relativity and quantum mechanics. The method of unification adopted in the present paper is by constructing a vector bundle $X\times R^6$ or $X\times R^4$ with $X$ being the observer's reference frame and $R^6$ or $R^4$ being the unobservable inner space(-time) within each observer's local system. Some applications of our theory to two concrete examples of human size and of cosmological size are discussed, as well as the uncertainty of time in our context is calculated. 
  The existence of a fundamental scale, a lower bound to any output of a position measurement, seems to be a model-independent feature of quantum gravity. In fact, different approaches to this theory lead to this result. The key ingredients for the appearance of this minimum length are quantum mechanics, special relativity and general relativity. As a consequence, classical notions such as causality or distance between events cannot be expected to be applicable at this scale. They must be replaced by some other, yet unknown, structure. 
  The role of topology in the perturbative solution of the Euclidean Einstein equations about flat instantons is examined. 
  Different proposals for the wave function of the universe are analyzed, with an emphasis on various forms of the tunneling proposal. The issues discussed include the equivalence of the Lorentzian path integral and outgoing-wave proposals, the definitions of the outgoing waves and of superspace boundaries, topology change and the corresponding modification of the Wheeler-DeWitt equation. Also discussed are the "generic" boundary conditions and the third quantization approach. 
  The dynamics of apparent and event horizons of various black hole spacetimes, including those containing distorted, rotating and colliding black holes, are studied. We have developed a powerful and efficient new method for locating the event horizon, making possible the study of both types of horizons in numerical relativity. We show that both the event and apparent horizons, in all dynamical black hole spacetimes studied, oscillate with the quasinormal frequency. 
  We prove that the Gauss map of a surface of constant mean curvature embedded in Minkowski space is harmonic. This fact will then be used to study 2+1 gravity for surfaces of genus higher than one. By considering the energy of the Gauss map, a canonical transform between the ADM reduced variables and holonomy variables can be constructed. This allows one to solve (in principle) for the evolution in the ADM variables without having to explicitly solve the constraints first. 
  We present a geometrical gravitational theory which reduces to Einstein's theory for weak gravitational potentials and which has a singularity-free analog of the Schwarzschild metric. 
  The theory of perturbation of Friedman-Robertson-Walker (FRW) cosmology is analysed exclusively in terms of observable quantities. Although this can be a very complete and general procedure we limit our presentation here to the case of irrotational perturbations for simplicity. We show that the electric part of Weyl conformal tensor {\bf $E$} and the shear {\bf $\Sigma$} constitute the two basic perturbed variables in terms of which all remaining observable quantities can be described. Einstein\rq s equations of General Relativity reduce to a closed set of dynamical system for {\bf $E$} and {\bf $\Sigma$}. The basis for a gauge-invariant Hamiltonian treatment of the Perturbation Theory in the FRW background is then set up. 
  We use wave packet mode quantization to compute the creation of massless scalar quantum particles in a colliding plane wave spacetime. The background spacetime represents the collision of two gravitational shock waves followed by trailing gravitational radiation which focus into a Killing-Cauchy horizon. The use of wave packet modes simplifies the problem of mode propagation through the different spacetime regions which was previously studied with the use of monocromatic modes. It is found that the number of particles created in a given wave packet mode has a thermal spectrum with a temperature which is inversely proportional to the focusing time of the plane waves and which depends on the mode trajectory. 
  The issues of scaling symmetry and critical point behavior are studied for fluctuations about extremal charged black holes. We consider the scattering and capture of the spherically symmetric mode of a charged, massive test field on the background spacetime of a black hole with charge $Q$ and mass $M$. The spacetime geometry near the horizon of a $|Q|=M$ black hole has a scaling symmetry, which is absent if $|Q|<M$, a scale being introduced by the surface gravity. We show that this symmetry leads to the existence of a self-similiar solution for the charged field near the horizon, and further, that there is a one parameter family of discretely self-similiar solutions . The scaling symmetry, or lack thereof, also shows up in correlation length scales, defined in terms of the rate at which the influence of an external source coupled to the field dies off. It is shown by constructing the Greens functions, that an external source has a long range influence on the extremal background, compared to a correlation length scale which falls off exponentially fast in the $|Q|<M$ case. Finally it is shown that in the limit of $\Delta \equiv (1-{Q^2 \overM^2} )^{1\over 2} \rightarrow 0$ in the background spacetime, that infinitesimal changes in the black hole area vary like $\Delta ^{1\over 2}$. 
  Kaluza-Klein theory admits ``bubble" configurations, in which the circumference of the fifth dimension shrinks to zero on some compact surface. A three parameter family of such bubble initial data at a moment of time-symmetry (some including a magnetic field) has been found by Brill and Horowitz, generalizing the (zero-energy) ``Witten bubble" solution. Some of these data have negative total energy. We show here that all the negative energy bubble solutions start out expanding away from the moment of time symmetry, while the positive energy bubbles can start out either expanding or contracting. Thus it is unlikely that the negative energy bubbles would collapse and produce a naked singularity. 
  The gravitational action is not always additive in the usual sense. We provide a general prescription for the change in action that results when different portions of the boundary of a spacetime are topologically identified. We discuss possible implications for the superposition law of quantum gravity. We present a definition of `generalized additivity' which does hold for arbitrary spacetime composition. 
  We review the evidence for and against the possibility that the inner singularity of a black hole contains a lightlike segment which is locally mild and characterized by mass inflation. 
  It is argued that Hawking's `greatest mistake' may not have been a mistake at all. According to the canonical quantum theory of gravity for Friedmann type universes, any time arrows of general nature can only be correlated with that of the expansion. For recollapsing universes this seems to be facilitated in part by quantum effects close to their maximum size. Because of the resulting thermodynamical symmetry between expansion and (formal) collapse, black holes must formally become `white' during the collapse phase (while physically only expansion of the universe and black holes can be observed). It is conjectured that the quantum universe remains completely singularity-free in this way (except for the homogeneous singularity) if an appropriate boundary condition for the wave function is able to exclude {\it past} singularities (as is often assumed). 
  Gravitational radiation from a binary neutron star or black hole system leads to orbital decay and the eventual coalescence of the binary's components. During the last several minutes before the binary components coalesce, the radiation will enter the bandwidth of the United States Laser Inteferometer Gravitational-wave Observatory (LIGO) and the French/Italian VIRGO gravitational radiation detector. The combination of detector sensitivity, signal strength, and source density and distribution all point to binary inspiral as the most likely candidate for observation among all the anticipated sources of gravitational radiation for LIGO/VIRGO. Here I review briefly some of the questions that are posed to theorists by the impending observation of binary inspiral. 
  Geodesics for a 5D magnetized Schwarzschild-like solution are analyzed by reducing the problem to the motion of a test particle in an effective potential. In absence of magnetic field comparison is established with Schwarzschild's geometry. Embedding diagrams are constructed in order to visualize the geometry of the metric. The study performed here is also valid, when the electromagnetic interactions are neglected, for the low energy superstring theory and the Brans-Dicke theory. 
  Possible breakdown of the universality of the gravitational couplings to different neutrino flavors can be tested in long-baseline neutrino-oscillation experiments, such as a proposed experiment at SOUDAN 2 or one at the DUMAND project. Such a breakdown could be detected with sensitivity to the order of $10^{-14}$ or $10^{-15}$. For generic case of neutrino oscillations, it is emphasized that one can identify the spin of the particle which mediates the force causing the oscillation, by investigating the energy spectrum of the outgoing charged leptons. (based on talk presented at the Workshop on General Relativity and Gravitation held at Univ. of Tokyo, January 17-20, 1994) 
  We consider the generation of entropy when particle pairs are created at a cosmological level. Making a reduction via the particle number basis, we compute the classical limit for the entropy generation due to the evolution of the matter field fluctuations (squeeze transformation), obtaining that it is linear in the squeeze parameter for a general class of initial states. We also discuss the dependence of the generated entropy on the coarse graining criteria. 
  The lensing properties of superconducting cosmic strings endowed with a time dependent pulse of lightlike current are investigated. The metric outside the core of the string belongs to the $pp$--wave class, with a deficit angle. We study the field theoretic bosonic Witten model coupled to gravity, and we show that the full metric (both outside and inside the core) is a Taub-Kerr-Shild generalization of that for the static string with no current. It is shown that the double image due to the deficit angle evolves in an unambiguous way as a pulse of lightlike current passes between the source and the observer. Observational consequences of this signature of the existence of cosmic strings are briefly discussed. 
  We describe a class of spinor-curvature identities which exist for Riemannian or Riemann-Cartan geometries. Each identity relates an expression quadratic in the covariant derivative of a spinor field with an expression linear in the curvature plus an exact differential. Certain special cases in 3 and 4 dimensions which have been or could be used in applications to General Relativity are noted. 
  The properties of canonical and microcanonical ensembles of a black hole with thermal radiation and the problem of black hole evaporation in 3-D are studied. In 3-D Einstein-anti-de Sitter gravity we have two relevant mass scales, $m_c=1/G$, and $m_p=(\hbar^2\Lambda/G)^{1/3}$, which are particularly relevant for the evaporation problem. It is argued that in the `weak coupling' regime $\Lambda<(\hbar G)^{-2}$, the end point of an evaporating black hole formed with an initial mass $m_0>m_p$, is likely to be a stable remnant in equilibrium with thermal radiation. The relevance of these results for the information problem and for the issue of back reaction is discussed. In the `strong coupling' regime, $\Lambda>(\hbar G)^{-2}$ a full fledged quantum gravity treatment is required. Since the total energy of thermal states in anti-de Sitter space with reflective boundary conditions at spatial infinity is bounded and conserved, the canonical and microcanonical ensembles are well defined. For a given temperature or energy black hole states are locally stable. In the weak coupling regime black hole states are more probable then pure radiation states. 
  We consider a general, classical theory of gravity with arbitrary matter fields in $n$ dimensions, arising from a diffeomorphism invariant Lagrangian, $\bL$. We first show that $\bL$ always can be written in a ``manifestly covariant" form. We then show that the symplectic potential current $(n-1)$-form, $\th$, and the symplectic current $(n-1)$-form, $\om$, for the theory always can be globally defined in a covariant manner. Associated with any infinitesimal diffeomorphism is a Noether current $(n-1)$-form, $\bJ$, and corresponding Noether charge $(n-2)$-form, $\bQ$. We derive a general ``decomposition formula" for $\bQ$. Using this formula for the Noether charge, we prove that the first law of black hole mechanics holds for arbitrary perturbations of a stationary black hole. (For higher derivative theories, previous arguments had established this law only for stationary perturbations.) Finally, we propose a local, geometrical prescription for the entropy, $S_{dyn}$, of a dynamical black hole. This prescription agrees with the Noether charge formula for stationary black holes and their perturbations, and is independent of all ambiguities associated with the choices of $\bL$, $\th$, and $\bQ$. However, the issue of whether this dynamical entropy in general obeys a ``second law" of black hole mechanics remains open. In an appendix, we apply some of our results to theories with a nondynamical metric and also briefly develop the theory of stress-energy pseudotensors. 
  We construct a new exact solution of Einstein's equations in vacuo in terms of Weyl canonical coordinates. This solution may be interpreted as a black hole in a space-time which is periodic in one direction and which behaves asymptotically like the Kasner solution with Kasner index equal to $4M L^{-1}$, where $L$ is the period and $M$ is the mass of the black hole. Outside the horizon, the solution is free of singularities and approaches the Schwarzschild solution as $L \rightarrow \infty$. 
  Spatially homogeneous cosmological models reduce to Hamiltonian systems in a low dimensional Minkowskian space moving on the total energy shell $H=0$. Close to the initial singularity some models (those of Bianchi type VIII and IX) can be reduced further, in a certain approximation, to a non-compact triangular billiard on a 2-dimensional space of constant negative curvature with a separately conserved positive kinetic energy. This type of billiard has long been known as a prototype chaotic dynamical system. These facts are reviewed here together with some recent results on the energy level statistics of the quantized billiard and with direct explicit semi-classical solutions of the Hamiltonian cosmological model to which the billiard is an approximation. In the case of Bianchi type IX models the latter solutions correspond to the special boundary conditions of a `no-boundary state' as proposed by Hartle and Hawking and of a `wormhole' state. 
  We consider the coupled evolution of density, (scalar) metric and dilaton perturbations in the transition from a ``stringy" phase of growing curvature and gravitational coupling to the standard radiation-dominated cosmology. We show that dilaton production, with a spectrum tilted towards large frequencies, emerges as a general property of this scenario. We discuss the frame-independence of the dilaton spectrum and of the inflationary properties of the metric background by using, as model of source, a pressureless gas of weakly interacting strings, which is shown to provide an approximate but consistent solution to the full system of background equations and string equations of motion. We combine various cosmological bounds on a growing dilaton spectrum with the bound on the dilaton mass obtained from tests of the equivalence principle, and we find allowed windows compatible with a universe presently dominated by a relic background of dilatonic dark matter. 
  A closed set of equations for the evolution of linear perturbations of homogeneous, isotropic cosmological models can be obtained in various ways. The simplest approach is to assume a macroscopic equation of state, e.g.\ that of a perfect fluid. For a more refined description of the early universe, a microscopic treatment is required. The purpose of this paper is to compare the approach based on classical kinetic theory to the more recent thermal-field-theory approach. It is shown that in the high-temperature limit the latter describes cosmological perturbations supported by collisionless, massless matter, wherein it is equivalent to the kinetic theory approach. The dependence of the perturbations in a system of a collisionless gas and a perfect fluid on the initial data is discussed in some detail. All singular and regular solutions are found analytically. 
  Inspiralling binary systems of neutron stars or black holes are promising sources of gravitational radiation detectable by large-scale laser interferometric gravitational observatories, such as the US LIGO and Italian-French VIRGO projects. Accurate theoretical gravitational-waveform templates will be needed to carry out matched filtering data analysis of the detectors' output once they are on the air by the end of this decade. For all but the final, strongly general relativistic coalescence of the two bodies, high-order post-Newtonian methods are playing a major role in the theorists' efforts to develop the needed templates. This paper discusses the foundations of this method, and provides a compendium of useful formulae and results. Figures available upon request. (Invited talk given at the 8th Nishinomiya-Yukawa Memorial Symposium, October 28, 1993.) 
  The Wilson loop functionals in terms of Ashtekar's variables were the first (formal) solutions to the quantized hamiltonian constraint of canonical gravity. Here it is shown that the same functionals also solve the supergravity constraints and some evidence is presented that they are artificially generated by multiplying the constraints by the metric determinant, which has become a widely accepted procedure. Using the same method in 2+1 dimensional gravity and supergravity leads to wrong results, e.g.~2+1 gravity is no longer a purely topological theory. As another feature of the densitized constraints it turns out that the classical theory desribed by them is not invariant under space time diffeomorphisms. 
  We use our recently proposed algebraic approach for calculating the heat kernel associated with the Laplace operator to calculate the one-loop effective action in the non-Abelian gauge theory. We consider the most general case of arbitrary space-time dimension, arbitrary compact simple gauge group and arbitrary matter and assume a covariantly constant gauge field strength of the most general form, having many independent color and space-time invariants (Savvidy type chromomagnetic vacuum) and covariantly constant scalar fields as a background. The explicit formulas for all the needed heat kernels and zeta-functions are obtained. We propose a new method to study the vacuum stability and show that the background field configurations with covariantly constant chromomagnetic fields can be stable only in the case when more than one independent field invariants are present and the values of these invariants differ not greatly from each other. The role of space-time dimension is analyzed in this connection and it is shown that this is possible only in space-times with dimensions not less than five $d\geq 5$. 
  We continue the development of the effective covariant methods for calculating the heat kernel and the one-loop effective action in quantum field theory and quantum gravity. The status of the low-energy approximation in quantum gauge theories and quantum gravity is discussed in detail on the basis of analyzing the local Schwinger - De Witt expansion. It is argued that the low-energy limit, when defined in a covariant way, should be related to background fields with covariantly constant curvature, gauge field strength and potential. Some new approaches for calculating the low-energy heat kernel assuming a covariantly constant background are proposed. The one-loop low-energy effective action in Yang-Mills theory in flat space with arbitrary compact simple gauge group and arbitrary matter on a covariantly constant background is calculated. The stability problem of the chromomagnetic (Savvidy-type) vacuum is analyzed. It is shown, that this type of vacuum structure can be stable only in the case when more than one background chromomagnetic fields are present and the values of these fields differ not greatly from each other.This is possible only in space-times of dimension not less than five $d\geq 5$. 
  We study the dynamics of multiwormhole configurations within the framework of the Euclidean Polyakov approach to string theory, incorporating a modification to the Hamiltonian which makes it impossible to interpret the Coleman Alpha parameters of the effective interactions as a quantum field on superspace, reducible to an infinite tower of fields on space-time. We obtain a Planckian probability measure for the Alphas that allows $\frac{1}{2}\alpha^{2}$ to be interpreted as the energy of the quanta of a radiation field on superspace whose values may still fix the coupling constants. 
  This report is based on the talk given by the author in the concluding session of the workshop on Canonical Methods in Classical and Quantum General Relativity, held a Bad-Honef, Germany, in September 93. It contains an assessment of the canonical approach in general, an overview of recent developments within the canonical approach based on connections and loops, and some suggestions for future work. 
  The classical value of the Hamiltonian for a system with timelike boundary has been interpreted as a quasilocal energy. This quasilocal energy is not positive definite. However, we derive a `quasilocal dominant energy condition' which is the natural consequence of the local dominant energy condition. We discuss some implications of this quasilocal energy condition. In particular, we find that it implies a `quasilocal weak energy condition'. 
  We discuss a general formalism for numerically evolving initial data in general relativity in which the (complex) Ashtekar connection and the Newman-Penrose scalars are taken as the dynamical variables. In the generic case three gauge constraints and twelve reality conditions must be solved. The analysis is applied to a Petrov type \{1111\} planar spacetime where we find a spatially constant volume element to be an appropriate coordinate gauge choice. 
  We investigate the space ${\cal M}$ of classical solutions to Witten's formulation of 2+1 gravity on the manifold ${\bf R} \times T^2$. ${\cal M}$ is connected, but neither Hausdorff nor a manifold. However, removing from ${\cal M}$ a set of measure zero yields a connected manifold which is naturally viewed as the cotangent bundle over a non-Hausdorff base space. Avenues towards quantizing the theory are discussed in view of the relation between spacetime metrics and the various parts of~${\cal M}$. (Contribution to the proceedings of the Lanczos Centenary Conference, Raleigh, NC, December 12--17, 1993.) 
  A summary of the known results on integration theory on the space of connections modulo gauge transformations is presented and its significance to quantum theories of gauge fields and gravity is discussed. The emphasis is on the underlying ideas rather than the technical subtleties. 
  Using the influence functional formalism we show how to derive a generalized Einstein equation in the form of a Langevin equation for the description of the backreaction of quantum fields and their fluctuations on the dynamics of curved spacetimes. We show how a functional expansion on the influence functional gives the cumulants of the stochastic source, and how these cumulants enter in the equations of motion as noise sources. We derive an expression for the influence functional in terms of the Bogolubov coefficients governing the creation and annihilation operators of the Fock spaces at different times, thus relating it to the difference in particle creation in different histories. We then apply this to the case of a free quantum scalar field in a spatially flat Friedmann- Robertson-Walker universe and derive the Einstein-Langevin equations for the scale factor for these semiclassical cosmologies. This approach based on statistical field theory extends the conventional theory of semiclassical gravity based on a semiclassical Einstein equation with a source given by the average value of the energy momentum tensor, thus making it possible to probe into the statistical properties of quantum fields like noise, fluctuations, entropy, decoherence and dissipation. Recognition of the stochastic nature of semiclassical gravity is an essential step towards the investigation of the behavior of fluctuations, instability and phase transition processes associated with the crossover to quantum gravity. 
  A Reference is corrected. (We derive the Fermi coordinate system of an observer in arbitrary motion in an arbitrary weak gravitational field valid to all orders in the geodesic distance from the worldline of the observer. In flat space-time this leads to a generalization of Rindler space for arbitrary acceleration and rotation. The general approach is applied to the special case of an observer resting with respect to the weak gravitational field of a static mass distribution. This allows to make the correspondence between general relativity and Newtonian gravity more precise.) 
  We investigate massless fermion production by a two-dimensional dilatonic black hole. Our analysis is based on the Bogoliubov transformation relating the outgoing fermion field observed outside the black hole horizon to the incoming field present before the black hole creation. It takes full account of the fact that the transformation is neither invertible nor unitarily implementable. The particle content of the outgoing radiation is specified by means of inclusive probabilities for the detection of sets of outgoing fermions and antifermions in given states. For states localized near the horizon these probabilities characterize a thermal equilibrium state. The way the probabilities become thermal as one approaches the horizon is discussed in detail. 
  The evolution of the reduced density operator $\rho$ of Brownian particle is discussed in single collision approach valid typically in low density gas environments. This is the first succesful derivation of quantum friction caused by {\it local} environmental interactions. We derive a Lindblad master equation for $\rho$, whose generators are calculated from differential cross section of a single collision between Brownian and gas particles, respectively. The existence of thermal equilibrium for $\rho$ is proved. Master equations proposed earlier are shown to be particular cases of our one. 
  We demonstrate a close connection between the decoherent histories (DH) approach to quantum mechanics and the quantum state diffusion (QSD) picture, for open quantum systems described by a master equation of Lindblad form. The (physically unique) set of variables that localize in the QSD picture also define an approximately decoherent set of histories in the DH approach. The degree of localization is related to the degree of decoherence, and the probabilities for histories prescribed by each approach are essentially the same. 
  We study the behaviour of scalar fields on background geometries which undergo quantum tunneling. The two examples considered are a moving mirror in flat space which tunnels through a potential barrier, and a false vacuum bubble which tunnels to form a black hole. WKB approximations to the Schrodinger and Wheeler-DeWitt equations are made, leading one to solve field equations on the Euclidean metric solution interpolating between the classically allowed geometries. The state of the field after tunneling can then be determined using the method of non-unitary Bogolubov transformations developed by Rubakov. It is shown that the effect of the tunneling is to damp any excitations initially present, and, in the case of the black hole, that the behaviour of fields on the Euclidean Kruskal manifold ensures that the late time radiation will be thermal at the Hawking temperature. 
  The existence of initial singularities in expanding universes is proved without assuming the timelike convergence condition. The assumptions made in the proof are ones likely to hold both in open universes and in many closed ones. (It is further argued that at least some of the expanding closed universes that do not obey a key assumption of the theorem will have initial singularities on other grounds.) The result is significant for two reasons: (a)~previous closed-universe singularity theorems have assumed the timelike convergence condition, and (b)~the timelike convergence condition is known to be violated in inflationary spacetimes. An immediate consequence of this theorem is that a recent result on initial singularities in open, future-eternal, inflating spacetimes may now be extended to include many closed universes. Also, as a fringe benefit, the time-reverse of the theorem may be applied to gravitational collapse. 
  The validity of the cosmic no-hair theorem is investigated in the context of Newtonian cosmology with a perfect fluid matter model and a positive cosmological constant. It is shown that if the initial data for an expanding cosmological model of this type is subjected to a small perturbation then the corresponding solution exists globally in the future and the perturbation decays in a way which can be described precisely. It is emphasized that no linearization of the equations or special symmetry assumptions are needed. The result can also be interpreted as a proof of the nonlinear stability of the homogeneous models. In order to prove the theorem we write the general solution as the sum of a homogeneous background and a perturbation. As a by-product of the analysis it is found that there is an invariant sense in which an inhomogeneous model can be regarded as a perturbation of a unique homogeneous model. A method is given for associating uniquely to each Newtonian cosmological model with compact spatial sections a spatially homogeneous model which incorporates its large-scale dynamics. This procedure appears very natural in the Newton-Cartan theory which we take as the starting point for Newtonian cosmology. 
  We reexamine non-Einsteinian effects observable in the orbital motion of low-orbit artificial Earth satellites. The motivations for doing so are twofold: (i) recent theoretical studies suggest that the correct theory of gravity might contain a scalar contribution which has been reduced to a small value by the effect of the cosmological expansion; (ii) presently developed space technologies should soon give access to a new generation of satellites endowed with drag-free systems and tracked in three dimensions at the centimeter level. Our analysis suggests that such data could measure two independent combinations of the Eddington parameters (beta - 1) and (gamma - 1) at the 10^-4 level and probe the time variability of Newton's "constant" at the d(ln G)/dt ~ 10^-13 yr^-1 level. These tests would provide well-needed complements to the results of the Lunar Laser Ranging experiment, and of the presently planned experiments aiming at measuring (gamma -1). In view of the strong demands they make on the level of non- gravitational perturbations, these tests might require a dedicated mission consisting of an optimized passive drag-free satellite. 
  We discuss the derivation of the so-called semi-classical equations for both mini-superspace and dilaton gravity. We find that there is no systematic derivation of a semi-classical theory in which quantum mechanics is formulated in a space-time that is a solution of Einstein's equation, with the expectation value of the matter stress tensor on the right-hand side. The issues involved are related to the well-known problems associated with the interpretation of the Wheeler-deWitt equation in quantum gravity, including the problem of time. We explore the de Broglie-Bohm interpretation of quantum mechanics (and field theory) as a way of spontaneously breaking general covariance, and thereby giving meaning to the equations that many authors have been using to analyze black hole evaporation. We comment on the implications for the ``information loss" problem. 
  We consider the gravitational sector of the superstring effective action with axion-matter couplings. The field equations are developed in the post-Newtonian scheme and approximate solutions for a spinning point mass and a cosmic string are presented. Furthermore, assuming vanishing axion mass and vanishing potential for the dilaton, we consider the gravitational radiation in the leading $O(\alpha'^0)$ order. We find that the total luminosity of a radiative source has monopole and dipole components besides the standard quadrupole one. These components may possibly be checked in binary systems. 
  Using the concept of open systems where the classical geometry is treated as the system and the quantum matter field as the environment, we derive a fluctuation-dissipation theorem for semiclassical cosmology. This theorem which exists under very general conditions for dissipations in the dynamics of the system, and the noise and fluctuations in the environment, can be traced to the formal mathematical relation between the dissipation and noise kernels of the influence functional depicting the open system, and is ultimately a consequence of the unitarity of the closed system. In particular, for semiclassical gravity, it embodies the backreaction effect of matter fields on the dynamics of spacetime. The backreaction equation derivable from the influence action is in the form of a Einstein-Langevin equation. It contains a dissipative term in the equation of motion for the dynamics of spacetime and a noise term related to the fluctuations of particle creation in the matter field. Using the well-studied model of a quantum scalar field in a Bianchi Type-I universe we illustrate how this Langevin equation and the noise term are derived and show how the creation of particles and the dissipation of anisotropy during the expansion of the universe can be understood as a manifestation of this fluctuation-dissipation relation. 
  Nonnegative probabilities that obey the sum rules may be assigned to a much wider family of sets of histories than decohering histories. The resulting {\it linearly positive histories} avoid the highly restrictive decoherence conditions and yet give the same probabilities when those conditions apply. Thus linearly positive histories are a broad extension of decohering histories. Moreover, the resulting theory is manifestly time-reversal invariant. 
  We have estimated higher order quantum gravity corrections to de~Sitter spacetime. Our results suggest that, while the classical spacetime metric may be distorted by the graviton self-interactions, the corrections are relatively weaker than previously thought, possibly growing like a power rather than exponentially in time. 
  For an asymptotically flat initial-data set in general relativity, the total mass-momentum may be interpreted as a Hermitian quadratic form on the complex, two-dimensional vector space of ``asymptotic spinors''. We obtain a generalization to an arbitrary initial-data set. The mass-momentum is retained as a Hermitian quadratic form, but the space of ``asymptotic spinors'' on which it is a function is modified. Indeed, the dimension of this space may range from zero to infinity, depending on the initial data. There is given a variety of examples and general properties of this generalized mass-momentum. 
  We discuss the possibility of constraining theories of gravity in which the connection is a fundamental variable by searching for observational consequences of the torsion degrees of freedom. In a wide class of models, the only modes of the torsion tensor which interact with matter are either a massive scalar or a massive spin-1 boson. Focusing on the scalar version, we study constraints on the two-dimensional parameter space characterizing the theory. For reasonable choices of these parameters the torsion decays quickly into matter fields, and no long-range fields are generated which could be discovered by ground-based or astrophysical experiments. 
  The recently suggested SEE (Satellite Energy Exchange) method of measuring the gravitational constant $G$, possible equivalence principle violation (measured by the E\"{o}tv\"{o}s parameter $\eta$) and the hypothetic 5th force parameters $\alpha$ and $\lambda$ on board a drag-free Earth's satellite is discussed and further developed. Various particle trajectories near a heavy ball are numerically simulated. Some basic sources of error are analysed. The $G$ measurement procedure is modelled by noise insertion to a ``true'' trajectory. It is concluded that the present knowledge of $G, \alpha$ (for $\lambda \geq 1$ m) and $\eta$ can be improved by at least two orders of magnitude. 
  We discuss how various properties of dilaton black holes depend on the dilaton coupling constant $a$. In particular we investigate the $a$-dependence of certain mass parameters both outside and in the extremal limit and discuss their relation to thermodynamical quantities. To further illuminate the role of the coupling constant $a$ we look at a massless point particle in a dilaton black hole geometry as well as the scattering of (neutral) fermions. In this latter case we find that the scattering potential vanishes for the zero angular momentum mode which seems to indicate a catastrophic deradiation when $a>1$. 
  We describe how the concepts of quantum open systems and the methods of closed-time-path (CTP) effective action and influence functional (IF) can be usefully applied to the analysis of statistical mechanical problems involving quantum fields in gravitation and cosmology. In the first lecture we discuss in general terms the relevance of open system concepts in the description of a variety of physical processes, and outline the basics of the CTP and IF formalisms. In the second lecture we illustrate the IF method with a model of two interacting quantum fields, deriving the influence action via a perturbative expansion involving the closed-time-path Green functions. We show how noise of quantum fields can be defined and derive a general fluctuation- dissipation relation for quantum fields. In the third lecture we discuss the problem of backreaction in semiclassical gravity with the example of a scalar field in a Bianchi Type-I universe. We show that the CTP effective action not only yields a real and causal equation of motion with a dissipative term depicting the effect of particle creation, as was found earlier, it also contains a noise term measuring the fluctuations in particle number and governing the metric fluctuations. The particle creation-backreaction problem can be understood as a manifestation of a fluctuation-dissipation relation for quantum fields in dynamic spacetimes, generalizing Sciama's observation for black hole Hawking radiation. A more complete description of semiclassical gravity is given by way of an Einstein-Langevin equation, the conventional theory based on the expectation value of the energy momentum tensor being its 
  The force acting on the charged particle moving along an arbitrary trajectory near the straight cosmic string is calculated. This interaction leads to the scattering of particles by the cosmic string. The scattering cross section is considered. 
  The validity conditions for the extended Birkhoff theorem in multidimensional gravity with $n$ internal spaces are formulated, with no restriction on space-time dimensionality and signature.  Examples of matter sources and geometries for which the theorem is valid are given. Further generalization of the theorem is discussed. 
  A cosmological model describing the evolution of $n$ Einstein spaces $(n>1)$ with $m$-component perfect-fluid matter is considered. When all spaces are Ricci-flat and for any $\alpha$-th component the pressures in all spaces are proportional to the density: $p_{i}^{(\alpha)} = (1- h_{i}^{(\alpha)}) \rho^{(\alpha)}$, $h_{i}^{(\alpha)}$ = const, the Einstein and Wheeler-DeWitt equations are integrated in the cases: i) $m=1$, for all $h_{i}^{(\alpha)}$; ii) $m > 1$, for some special sets of $h_{i}^{(\alpha)}$. For $m=1$ the quantum wormhole solutions are also obtained. \\ 
  The generators of the Poincar\'{e} symmetry of scalar electrodynamics are quantized in the functional Schr\"{o}dinger representation. We show that the factor ordering which corresponds to (minimal) Dirac quantization preserves the Poincar\'{e} algebra, but (minimal) reduced quantization does not. In the latter, there is a van Hove anomaly in the boost-boost commutator, which we evaluate explicitly to lowest order in a heat kernel expansion using zeta function regularization. We illuminate the crucial role played by the gauge orbit volume element in the analysis. Our results demonstrate that preservation of extra symmetries at the quantum level is sometimes a useful criterion to select between inequivalent, but nevertheless self-consistent, quantization schemes. 
  The Divergence Theorem as usually stated cannot be applied across a change of signature unless it is re-expressed to allow for a finite source term on the signature change surface. Consequently all conservation laws must also be `modified', and therefore insistence on conservation of matter across such a surface cannot be physically justified. The Darmois junction conditions normally ensure conservation of matter via Israel's identities for the jump in the energy-momentum density, but not when the signature changes. Modified identities are derived for this jump when a signature change occurs, and the resulting surface effects in the conservation laws are calculated. In general, physical vector fields experience a jump in at least one component, and a source term may therefore appear in the corresponding conservation law. Thus current is also not conserved. These surface effects are a consequence of the change in the character of physical law. The only way to recover standard conservation laws is to impose restrictions that no realistic cosmological model can satisfy. 
  The divergence theorem in its usual form applies only to suitably smooth vector fields. For vector fields which are merely piecewise smooth, as is natural at a boundary between regions with different physical properties, one must patch together the divergence theorem applied separately in each region. We give an elegant derivation of the resulting "patchwork divergence theorem" which is independent of the metric signature in either region, and which is thus valid if the signature changes.   (PACS numbers 4.20.Cv, 04.20.Me, 11.30.-j, 02.40.Hw) 
  Gompf's end-sum techniques are used to establish the existence of an infinity of non-diffeomorphic manifolds, all having the same trivial ${\bf R^4}$ topology, but for which the exotic differentiable structure is confined to a region which is spatially limited. Thus, the smoothness is standard outside of a region which is topologically (but not smoothly) ${\bf B^3}\times {\bf R^1}$, where ${\bf B^3}$ is the compact three ball. The exterior of this region is diffeomorphic to standard ${\bf R^1}\times {\bf S^2}\times{\bf R^1}$. In a space-time diagram, the confined exoticness sweeps out a world tube which, it is conjectured, might act as a source for certain non-standard solutions to the Einstein equations. It is shown that smooth Lorentz signature metrics can be globally continued from ones given on appropriately defined regions, including the exterior (standard) region. Similar constructs are provided for the topology, ${\bf S^2}\times {\bf R^2}$ of the Kruskal form of the Schwarzschild solution. This leads to conjectures on the existence of Einstein metrics which are externally identical to standard black hole ones, but none of which can be globally diffeomorphic to such standard objects. Certain aspects of the Cauchy problem are also discussed in terms of ${\bf R^4_\Theta}$\models which are ``half-standard'', say for all $t<0,$ but for which $t$ cannot be globally smooth. 
  The solution of topologically massive gravity with cosmological constant is reduced, for space-times with two commuting Killing vectors, to a special-relativistic dynamical problem. This approach is applied to the construction of a class of exact sourceless, horizonless solutions asymptotic to the BTZ extreme black holes. 
  In this paper we study space-times which evolve out of Cauchy data $(\Sigma,\metrict,K)$ invariant under the action of a two-dimensional commutative Lie group. Moreover $(\Sigma,\metrict,K)$ are assumed to satisfy certain completeness and asymptotic flatness conditions in spacelike directions. We show that asymptotic flatness and energy conditions exclude all topologies and group actions except for a cylindrically symmetric $\R^3$, or a periodic identification thereof along the $z$--axis. We prove that asymptotic flatness, energy conditions and cylindrical symmetry exclude the existence of compact trapped surfaces. Finally we show that the recent results of Christodoulou and Tahvildar--Zadeh concerning global existence of a class of wave--maps imply that strong cosmic censorship holds in the class of asymptotically flat cylindrically symmetric electro--vacuum space--times. 
  Simple calculations indicate that the partition function for a black hole is defined only if the temperature is fixed on a finite boundary. Consequences of this result are discussed. (Contribution to the Proceedings of the Lanczos Centenary Conference.) 
  The idea that spacetime points are to be identified by a fleet of clock--carrying particles can be traced to the earliest days of general relativity. Such a fleet of clocks can be described phenomenologically as a reference fluid. One approach to the problem of time consists in coupling the metric to a reference fluid and solving the super--Hamiltonian constraint for the momentum conjugate to the clock time variable. The resolved constraint leads to a functional Schr\"{o}dinger equation and formally to a conserved inner product. The reference fluid that is described phenomenologically as incoherent dust has the extraordinary property that the true Hamiltonian density for the coupled system depends only on the gravitational variables. The dust particles also endow space with a privileged system of coordinates that allows the supermomentum constraint to be solved explicitly. (Contribution to the Proceedings of the Lanczos Centenary Conference.) 
  Diagonal Bianchi type-IX models are studied in the quantum theory of $ N = 1 $ supergravity with a cosmological constant. It is shown, by imposing the supersymmetry and Lorentz quantum constraints, that there are no physical quantum states in this model. The $ k = + 1 $ Friedmann model in supergravity with cosmological constant does admit quantum states. However, the Bianchi type-IX model provides a better guide to the behaviour of a generic state, since more gravitino modes are available to be excited. These results indicate that there may be no physical quantum states in the full theory of $ N = 1 $ supergravity with a non-zero cosmological constant. are available to be excited. These results indicate that there may be no physical quantum states in the full theory of $ N = 1 $ supergravity with a non-zero cosmological constant. 
  Coalescing binary systems are one of the most promising sources of gravitational waves. The technique of matched filtering used in the detection of gravitational waves from coalescing binaries relies on the construction of accurate templates. Until recently filters modelled on the quadrupole or the Newtonian approximation were deemed sufficient. Recently it was shown that post-Newtonian effects contribute to a secular growth in the phase difference between the actual signal and its corresponding Newtonian template. In this paper we investigate the possibility of compensating for the phase difference caused by the post-Newtonian terms by allowing for a shift in the Newtonian filter parameters. We find that Newtonian filters perform adequately for the purpose of detecting the presence of the signal for both the initial and the advanced LIGO detectors. 
  The canonical theory of quantum gravity in the loop representation can be extended to incorporate topology change, in the simple case that this refers to the creation or annihilation of "minimalist wormholes" in which two points of the spatial manifold are identified. Furthermore, if the states of the wormholes threaded by loop states are taken to be antisymmetrized under the permutation of wormhole mouths, as required by the relation between spin and statistics, then the quantum theory of pure general relativity, without matter but with minimalist wormholes, is shown to be equivalent to the quantum theory of general relativity coupled to a single Weyl fermion field, at both the kinematical and diffeomorphism invariant levels. The correspondence is also shown to extend to the action of the dynamics generated by the Hamiltonian constraint, on a large subspace of the physical state space, and is thus conjectured to be completely general. 
  A cosmological scenario which explains the values of the parameters of the standard models of elementary particle physics and cosmology is discussed. In this scenario these parameters are set by a process analogous to natural selection which follows naturally from the assumption that the singularities in black holes are removed by quantum effects leading to the creation of new expanding regions of the universe. The suggestion of J. A. Wheeler that the parameters change randomly at such events leads naturally to the conjecture that the parameters have been selected for values that extremize the production of black holes. This leads directly to a prediction, which is that small changes in any of the parameters should lead to a decrease in the number of black holes produced by the universe. On plausible astrophysical assumptions it is found that changes in many of the parameters do lead to a decrease in the number of black holes produced by spiral galaxies. These include the masses of the proton,neutron, electron and neutrino and the weak, strong and electromagnetic coupling constants. Finally,this scenario predicts a natural time scale for cosmology equal to the time over which spiral galaxies maintain appreciable rates of star formation, which is compatible with current observations that $\Omega = .1-.2$. 
  Special theory of relativity has been formulated in a vacuum momentum-energy representation which is equivalent to Einstein special relativity and predicts just the same results as it. Although in this sense such a formulation would be at least classically useless, its consistent extension to noninertial frames produces a momentum-energy metric which behaves as a new dynamical quantity that is here interpreted in terms of a cosmological field. This new field would be complementary to gravity in that its strength varies inversely to as that of gravity does. Using a strong-field approximation, we suggest that the existence of this cosmological field would induce a shift of luminous energy which could justify the existence of all the assumed invisible matter in the universe, so as the high luminousities found in active galactic nuclei and quasars. 
  We consider notions of physical equivalence of sets of histories in the quantum mechanics of a closed system. We show first how the same set of histories can be relabeled in various ways, including the use of the Heisenberg equations of motion and of passive transformations of field variables. In the the usual approximate quantum mechanics of a measured subsystem, two observables re- presented by different Hermitian operators are physically distinguished by the different apparatus used to measure them. In the quantum mechanics of a closed system, however, any apparatus is part of the system and the notion of physically distinct situations has a different character. We show that a triple consisting of an initial condition, a Hamiltonian, and a set of histories is physically equivalent to another triple if the operators representing these initial conditions, Hamiltonians, and histories are related by any fixed unitary transformation. We apply this result to the question of whether the universe might exhibit physically inequivalent quasiclassical realms (which we earlier called quasiclassical domains), not just the one that includes familiar experience. We describe how the probabilities of alternative forms, behaviors, and evolutionary histories of information gathering and utilizing systems (IGUSes) using the usual quasiclassical realm could in principle be calculated in quantum cosmology, although it is, of course, impractical to perform the computations. We discuss how, in principle, the probabilities of occurence of IGUSes could be calculated in realms distinct from the usual quasiclassical one. We discuss how IGUSes adapted mainly to two different realms could draw inferences about each other using a hybrid realm consisting of alternatives drawn from each. 
  Black hole solutions can be used to shed light on general issues in General Relativity and Quantum Physics. Black--hole hair, entropy and naked singularities are considered here, along with some implications for the inflationary universe scenario. 
  We begin a program of work aimed at examining the interior of a rotating black hole with a non--zero cosmological constant. The generalisation of Teukolsky's equation for the radial mode functions is presented. It is shown that the energy fluxes of scalar, electromagnetic and gravity waves are regular at the Cauchy horizon whenever the surface gravity there is less than the surface gravity at the cosmological horizon. This condition is narrowly allowed, even when the cosmological constant is very small, thus permitting an observer to pass through the hole, viewing the naked singularity along the way. 
  In this paper we apply the techniques which have been developed over the last few decades for generating nontrivially new solutions of the Einstein-Maxwell equations from seed solutions for simple spacetimes. The simple seed spacetime which we choose is the "magnetic universe" to which we apply the Ehlers transformation. Three interesting non-singular metrics are generated. Two of these may be described as "rotating magnetic universes" and the third as an "evolving magnetic universe." Each is causally complete - in that all timelike and lightlike geodesics do not end in a finite time or affine parameter. We also give the electromagnetic field in each case. For the two rotating stationary cases we give the projection with respect to a stationary observer of the electromagnetic field into electric and magnetic components. 
  In this universe, governed fundamentally by quantum mechanical laws, characterized by indeterminism and distributed probabilities, classical deterministic laws are applicable over a wide range of time, place, and scale. We review the origin of these laws in the context of the quantum mechanics of closed systems, most generally, the universe as a whole. There probabilities are predicted for members of decoherent sets of alternative histories of the universe, ie ones for which the interference between pairs in the set is negligible as measured by a decoherence functional. An expansion of the decoherence functional in the separation between histories allows the form of the deterministic equations of motion to be derived for suitable coarse grainings of a class of non-relativ- istic systems, including ones with general non-linear interactions. More coarse graining is needed to achieve classical predictability than naive arguments based on the uncertainty principle would suggest. Coarse graining is needed for decoherence, and coarse graining beyond that for the inertia necessary to resist the noise that mechanisms of decoherence produce. Sets of histories governed largely by deterministic laws constitute the quasiclassical realm of everyday experience which is an emergent feature of the closed system's initial condition and Hamiltonian. We analyse the sensitivity of the existence of a quasiclassical realm to the particular form of the initial condition. We find that almost any initial condition will exhibit a quasiclassical realm of some sort, but only a small fraction of the total number of possible initial states could reproduce the everyday quasiclassical realm of our universe. (Talk given at the Lanczos Centenary Conference, North Carolina State University, December 15, 1993.) 
  The helicity flip of a spin-${\textstyle \frac{1}{2}}$ Dirac particle interacting gravitationally with a scalar field is analyzed in the context of linearized quantum gravity. It is shown that massive fermions may have their helicity flipped by gravity, in opposition to massless fermions which preserve their helicity. 
  We use the formulation of asymptotically anti-de Sitter boundary conditions given by Ashtekar and Magnon to obtain a coordinate expression for the general asymptotically AdeS metric in a neighbourhood of infinity. From this, we are able to compute the time delay of null curves propagating near infinity. If the gravitational mass is negative, so will be the time delay (relative to null geodesics at infinity) for certain null geodesics in the spacetime. Following closely an argument given by Penrose, Sorkin, and Woolgar, who treated the asymptotically flat case, we are then able to argue that a negative time delay is inconsistent with non-negative matter-energies in spacetimes having good causal properties. We thereby obtain a new positive mass theorem for these spacetimes. The theorem may be applied even when the matter flux near the boundary-at-infinity falls off so slowly that the mass changes, provided the theorem is applied in a time-averaged sense. The theorem also applies in certain spacetimes having local matter-energy that is sometimes negative, as can be the case in semi-classical gravity. 
  Within the general class of Asymptotically Anti-de Sitter spacetimes that are asymptotic to the A-de-S Schwarzschild metric, we give a simple positive mass theorem based on arguments from causal structure. A general result for all asymptotically A-de-S spacetimes will be given elsewhere. 
  In choosing a family of histories for a system, it is often convenient to choose a succession of locations in phase space, rather than configuration space, for comparison to classical histories. Although there are no good projections onto phase space, several approximate projections have been used in the past; three of these are examined in this paper. Expressions are derived for the probabilities of histories containing arbitrary numbers of projections onto phase space, and the conditions for the decoherence of these histories are studied. 
  Interpretation problems are eliminated from quantum theory by picturing a quantum history as having been sampled from a probability distribution over the set of histories which are permitted by all relevant boundary conditions. In laboratory physics, the final measurement plays a crucial role in defining the set of allowed histories by constraining the final state to be an eigenstate of the measurement operator. For the universe itself, a final boundary condition can play a similar role. Together with the special big bang initial state, the final constraint may ensure that the universe admits a classical description in the asymptotic future. Acknowledging the role of a future constraint dispels the mysteries of quantum theory without any amendment to the theory itself. 
  This paper presents the detailed, standard treatment of a simple, gauge invariant action for Weyl and Weyl-like Cartan geometries outlined in a previous paper. In addition to the familiar scalar curvature squared and Maxwell terms, the action chosen contains the logarithmic derivative of the scalar curvature combined with the intrinsic four vector (Weyl vector) in a gauge invariant fashion. This introduces higher order derivative terms directly into the action. No separate, ``matter'' fields are introduced. As the usual Weyl metric and four vector are varied, certain gauge invariant combinations of quantities arise naturally as the results are collected, provided the scalar curvature is nonzero. This paper demonstrates the general validity of these results for any gauge choice. Additionally, ``matter'' terms appear in the field equations. Furthermore, the resulting forms isolate the familiar mathematical structure of a coupled Einstein-Maxwell-Schr\"{o}dinger (relativistic) system of classical fields, with the exception of additional, second derivative terms in the stress tensor for the Schr\"{o}dinger field, and the algebraic independence of the conjugate wavefunction. This independence is found to be equivalent to the presence of a second, negative energy, Schr\"{o}dinger field. A detailed comparison is made between this model, and the standard Einstein-Maxwell-Schr\"{o}dinger field theory. The possible use of such continuum models as a basis for quantum phenomena, and some generalizations of the model are discussed. 
  In view of the well-known correspondence between gravitational fields and space-time distributions on a world manifold X, the criterion of gravitation singularities as singularities of these distributions is suggested. In the germ terms, singularities of a (3+1) distribution look locally like singularities of a foliation whose leaves are level surfaces of a real function f on X. If f is a single-valued function, changes of leave topology at critical points of f take place. In case of a multi-valued function f, one can lift the foliation to the total space of the cotangent bundle over X, then extend it over branch points of f and project this extension onto X. Singular points of this projection constitute a Lagrange map caustic by Arnol'd. 
  The area entropy $A/4$ and the related Hawking temperature in the presence of event horizons are rederived, for de Sitter and black hole topologies, as a consequence of a tunneling of the wave functional associated to the classical coupled matter and gravitational fields. The extension of the wave functional outside the barrier provides a reservoir of quantum states which allows for an additive constant to $A/4$. While, in a semi-classical analysis, this gives no new information in the de Sitter case, it yields an infinite constant in the black hole case. Evaporating black holes would then leave residual ``planckons" - Planckian remnants with infinite degeneracy. Generic planckons can neither decay into, nor be directly formed from, ordinary matter in a finite time. Such opening at the Planck scale of an infinite Hilbert space is expected to provide the ultraviolet cutoff required to render the theory finite in the sector of large scale physics. 
  The emergence of Hawking radiation from vacuum fluctuations is analyzed in conventional field theories and their energy content is defined through the Aharonov weak value concept. These fluctuations travel in flat space-time and carry transplanckian energies sharply localized on cisplanckian distances. We argue that these features cannot accommodate gravitational nonlinearities. We suggest that the very emission of Hawking photons from tamed vacuum fluctuations requires the existence of an exploding set of massive fields. These considerations corroborate some conjectures of Susskind and may prove relevant for the back-reaction problem and for the unitarity issue. 
  The metrics of gravitational shock waves for a Schwarzschild black hole in ordinary coordinates and for a Kerr black hole in Boyer-Lindquist coordinates are derived. The Kerr metric is discussed for two cases: the case of a Kerr black hole moving parallel to the rotational axis, and moving perpendicular to the rotational axis. Then, two properties from the derived metrics are investigated: the shift of a null coordinate and the refraction angle crossing the gravitational shock wave. Astrophysical applications for these metrics are discussed in short. 
  It is demonstrated that almost any S-matrix of quantum field theory in curved spaces posses an infinite set of complex poles (or branch cuts). These poles can be transformed into complex eigenvalues, the corresponding eigenvectors being Gamow vectors. All this formalism, which is heuristic in ordinary Hilbert space, becomes a rigorous one within the framework of a properly chosen rigged Hilbert space. Then complex eigenvalues produce damping or growing factors. It is known that the growth of entropy, decoherence, and the appearance of correlations, occur in the universe evolution, but only under a restricted set of initial conditions. It is proved that the damping factors allow to enlarge this set up to almost any initial conditions. 
  We present 2 recent results on the problems of time and observables in canonical gravity. (1) We cannot use parametrized field theory to solve the problem of time because, strictly speaking, general relativity is not a parametrized field theory. (2) We show that there are essentially no local observables for vacuum spacetimes. 
  A generalized symmetry of a system of differential equations is an infinitesimal transformation depending locally upon the fields and their derivatives which carries solutions to solutions. We classify all generalized symmetries of the vacuum Einstein equations in four spacetime dimensions. To begin, we analyze symmetries that can be built from the metric, curvature, and covariant derivatives of the curvature to any order; these are called natural symmetries and are globally defined on any spacetime manifold. We next classify first-order generalized symmetries, that is, symmetries that depend on the metric and its first derivatives. Finally, using results from the classification of natural symmetries, we reduce the classification of all higher-order generalized symmetries to the first-order case. In each case we find that the generalized symmetries are infinitesimal generalized diffeomorphisms and constant metric scalings. There are no non-trivial conservation laws associated with these symmetries. A novel feature of our analysis is the use of a fundamental set of spinorial coordinates on the infinite jet space of Ricci-flat metrics, which are derived from Penrose's ``exact set of fields'' for the vacuum equations. 
  Consider a spherically symmetric spacelike slice through a spherically symmetric spacetime. One can derive a universal bound for the optical scalars on any such slice. The only requirement is that the matter sources satisfy the dominant energy condition and that the slice be asymptotically flat and regular at the origin. This bound can be used to derive new conditions for the formation of apparent horizons. The bounds hold even when the matter has a distribution on a shell or blows up at the origin so as to give a conical singularity. 
  A finite subgroup of the conformal group SL(2,C) can be related to invariant polynomials on a hypersurface in C^3. The latter then carries a simple singularity, which resolves by a finite iteration of basic cycles of deprojections. The homological intersection graph of this cycles is the Dynkin graph of an ADE Lie group. The deformation of the simple singularity corresponds to ADE symmetry breaking. A 3+1-dimensional topological model of observation is constructed, transforming consistently under SL(2,C), as an evolving 3-dimensional system of world tubes, which connect ``possible points of observation". The existence of an initial singularity for the 4-dimensional space-time is related to its global topological structure. Associating the geometry of ADE singularities to the vertex structure of the topological model puts forward the conjecture on a likewise relation of inner symmetries of elementary particles to local space-time structure. 
  Conformal transformations of the following kinds are compared: (1) conformal coordinate transformations, (2) conformal transformations of Lagrangian models for a D-dimensional geometry, given by a Riemannian manifold M with metric g of arbitrary signature, and (3) conformal transformations of (mini-)superspace geometry. For conformal invariance under this transformations the following applications are given respectively: (1) Natural time gauges for multidimensional geometry, (2) conformally equivalent Lagrangian models for geometry coupled to a spacially homogeneous scalar field, and (3) the conformal Laplace operator on the $n$-dimensional manifold $M of minisuperspace for multidimensional geometry and the Wheeler de Witt equation. The conformal coupling constant xi_c is critically distinguished among arbitrary couplings xi, for both, the equivalence of Lagrangian models with D-dimensional geometry and the conformal geometry on n-dimensional minisuperspace. For dimension D=3,4,6 or 10, the critical number xi_c={D-2}/{4(D-1)} is especially simple as a rational fraction. 
  The conditions for the existence of Killing-Yano tensors, which are closely related to the appearance of non-generic world-line SUSY, are presented for static axisymmetric spacetimes. Imposing the vacuum Einstein equation, the set of solutions admitting Killing-Yano tensors is considered. In particular, it is shown that static, axisymmetric and asymptotically flat vacuum solutions admitting Killing-Yano tensors are only the Schwarzschild solution. 
  We show that in a large class of two dimensional models with conformal matter fields, the semiclassical cosmological solutions have a weak coupling singularity if the classical matter content is below a certain threshold. This threshold and the approach to the singularity are model-independent. When the matter fields are not conformally invariant, the singularity persists if the quantum state is the vacuum near the singularity, and could dissappear for other quantum states. 
  Recently it was proposed to explain the dynamical origin of the entropy of a black hole by identifying its dynamical degrees of freedom with states of quantum fields propagating in the black-hole's interior. The present paper contains the further development of this approach. The no-boundary proposal (analogous to the Hartle-Hawking no-boundary proposal in quantum cosmology) is put forward for defining the wave function of a black hole. This wave function is a functional on the configuration space of physical fields (including the gravitational one) on the three-dimensional space with the Einstein-Rosen bridge topology.It is shown that in the limit of small perturbations on the Kruskal background geometry the no-boundary wave function coincides with the Hartle-Hawking vacuum state. The invariant definition of inside and outside modes is proposed. The density matrix describing the internal state of a black hole is obtained by averaging over the outside modes. This density matrix is used to define the entropy of a black hole, which is to be divergent. It is argued that the quantum fluctuations of the horizon which are internally present in the proposed formalism may give the necessary cut-off and provide a black hole with the finite entropy. 
  We prove the theorem valid for (Pseudo)-Riemannian manifolds $V_n$: "Let $x \in V_n$ be a fixed point of a homothetic motion which is not an isometry then all curvature invariants vanish at $x$." and get the Corollary: "All curvature invariants of the plane wave metric $$ds \sp 2 \quad = \quad 2 \, du \, dv \, + \, a\sp 2 (u) \, dw \sp 2 \, + \, b\sp 2 (u) \, dz \sp 2 $$ identically vanish." Analysing the proof we see: The fact that for definite signature flatness can be characterized by the vanishing of a curvature invariant, essentially rests on the compactness of the rotation group $SO(n)$. For Lorentz signature, however, one has the non-compact Lorentz group $SO(3,1)$ instead of it. A further and independent proof of the corollary uses the fact, that the Geroch limit does not lead to a Hausdorff topology, so a sequence of gravitational waves can converge to the flat space-time, even if each element of the sequence is the same pp-wave. 
  We analyze the presumptions which lead to instabilities in theories of order higher than second. That type of fourth order gravity which leads to an inflationary (quasi de Sitter) period of cosmic evolution by inclusion of one curvature squared term (i.e. the Starobinsky model) is used as an example. The corresponding Hamiltonian formulation (which is necessary for deducing the Wheeler de Witt equation) is found both in the Ostrogradski approach and in another form. As an example, a closed form solution of the Wheeler de Witt equation for a spatially flat Friedmann model and L=R\sp 2 is found. The method proposed by Simon to bring fourth order gravity to second order can be (if suitably generalized) applied to bring sixth order gravity to second order. In the Erratum we show that a spatially flat Friedmann model need not be geodesically complete even if the scale factor a(t) is positive and smooth for all real values of the synchronized time t. 
  In this short essay we review the arguments showing that black hole entropy is, at least in part, ``entanglement entropy", i.e., missing information contained in correlations between quantum field fluctuations inside and outside the event horizon. Although the entanglement entropy depends upon the matter field content of the theory, it turns out that so does the Bekenstein-Hawking entropy $A/4\hbar G_{ren}$, in precisely the same way, because the effective gravitational constant $G_{ren}$ is renormalized by the very same quantum fluctuations. It appears most satisfactory if the entire gravitational action is ``induced", in the manner suggested by Sakharov, since then the black hole entropy is purebred entanglement entropy, rather than being hybrid with bare gravitational entropy (whatever that might be.) 
  A one parameter family of static charged black hole solutions in $(2+1)$-dimensional general relativity minimally coupled to a dilaton $\phi\propto ln({r\over\beta})$ with a potential term $e^{b\phi}\Lambda$ is obtained. Their causal strutures are investigated, and thermodynamical temperature and entropy are computed. One particular black hole in the family has the same thermodynamical properties as the Schwarzschild black hole in $3+1$ dimensions. Solutions with cosmological horizons are also discussed. Finally, a class of black holes arising from the dilaton with a negative kinetic term (tachyon) is briefly discussed. 
  A black hole solution of Einstein's field equations with cylindrical symmetry is found. Using the Hamiltonian formulation one is able to define mass and angular momentum for the cylindrical black hole through the corresponding and equivalent three dimensional theory. The causal structure is analyzed. Comments: revised version. 
  We show that if the space of physical states spanned by the wormhole wave functions can be equipped with a Hilbert structure, such a Hilbert space must coincide with that of the Lorentzian gravitational system under consideration. The physical inner product can then be determined by imposing a set of Lorentzian reality conditions. The Hilbert space of the gravitational model admits in this case a basis of wormhole solutions, and every proper quantum state can be interpreted as a superposition of wormholes. We also argue that the wave functions that form the basis of wormholes must be eigenfunctions of a complete set of compatible observables. The associated eigenvalues provide a set of well-defined wormhole parameters, in the sense that they can be employed to designate the different elements of the basis of wormholes. We analyse in detail the case of a Friedmann-Robertson-Walker spacetime minimally coupled to a massless scalar field. For this minisuperspace, we prove the validity of all the above statements and discuss various admissible choices of bases of wormhole wave functions. 
  The construction of initial-data sets representing binary black-hole configurations in quasi-circular orbits is studied in the context of the conformal-imaging formalism. An effective-potential approach for locating quasi-circular orbits is outlined for the general case of two holes of arbitrary size and with arbitrary spins. Such orbits are explicitly determined for the case of two equal-sized nonrotating holes, and the innermost stable quasi-circular orbit is located. The characteristics of this innermost orbit are compared to previous estimates for it, and the entire sequence of quasi-circular orbits is compared to results from the post-Newtonian approximation. Some aspects of the numerical evolution of such data sets are explored. 
  In this paper the quantum cosmological consequences of introducing a term cubic in the Ricci curvature scalar $R$ into the Einstein--Hilbert action are investigated. It is argued that this term represents a more generic perturbation to the action than the quadratic correction usually considered. A qualitative argument suggests that there exists a region of parameter space in which neither the tunneling nor the no-boundary boundary conditions predict an epoch of inflation that can solve the horizon and flatness problems of the big bang model. This is in contrast to the $R^2$--theory. 
  Black hole spacetimes can arise when a Liouville field is coupled to two- dimensional gravity. Exact solutions are obtained both classically and when quantum corrections due to back reaction effects are included. The black hole temperature depends upon the mass and the thermodynamic limit breaks down before evaporation of the black hole is complete, indicating that higher-loop effects must be included for a full description of the process. 
  Macroscopic traversable wormhole solutions to Einstein's field equations in $(2+1)$ and $(3+1)$ dimensions with a cosmological constant are investigated. Ensuring traversability severely constrains the material used to generate the wormhole's spacetime curvature. Although the presence of a cosmological constant modifies to some extent the type of matter permitted (for example it is possible to have a positive energy density for the material threading the throat of the wormhole in $(2+1)$ dimensions), the material must still be ``exotic'', that is matter with a larger radial tension than total mass-energy density multiplied by $c^2$. Two specific solutions are applied to the general cases and a partial stability analysis of a $(2+1)$ dimensional solution is explored. 
  Two non-static solutions for three dimensional gravity coupled to matter fields are given. One describes the collapse of radiation that results in a black hole. This is the three dimensional analog of the Vaidya metric, and is used to construct a model for mass inflation. The other describes plane gravitational waves for coupling to a massless scalar field. 
  In the framework of finite temperature conformal scalar field theory on de Sitter space-time the linearized Einstein equations for the renormalized stress tensor are exactly solved. In this theory quantum field fluctuations are concentrated near two spheres of the de Sitter radius, propagating as light wave fronts. Related cosmological aspects are shortly discussed. The analysis, performed for flat expanding universe, shows exponential damping of the back-reaction effects far from these spherical objects. The obtained solutions for the semiclassical Einstein equations in de Sitter background can be straightforwardly extended also to the anti-de Sitter geometry. 
  The closed-universe recollapse conjecture is studied for a class of closed spherically symmetric spacetimes which includes those having as a matter source: (1) a massless scalar field; (2) a perfect fluid obeying the equation of state $\rho = P$; and (3) null dust. It is proven that all timelike curves in any such spacetime must have length less than $6 \max_\Sigma(2m)$, where $m$ is the mass associated with the spheres of symmetry and $\Sigma$ is any Cauchy surface for the spacetime. The simplicity of this result leads us to conjecture that a similar bound can be established for the more general spherically symmetric spacetimes. 
  The quantum cosmology of the string effective action is considered within the context of the Bianchi class A minisuperspace. An exact unified solution is found for all Bianchi types and interpreted physically as a quantum wormhole. The solution is generalized for types ${\rm VI_0}$ and ${\rm VII_0}$. The Bianchi type IX wavefunction becomes increasingly localized around the isotropic Universe at large three-geometries. 
  We draw attention to an elementary flaw in a recently proposed experiment to measure the wave function of a single quantum system. 
  We study vector fields on a disk satisfying two types of mixed boundary conditions. These boundary conditions are selected by BRST-invariance in electrodynamics. They also appear in the de Rham complex. The manifest construction of the harmonic expansion is presented. The eigenfunctions of the vector Laplace operator are expressed in terms of fields satisfying pure Dirichlet or Robin boundary conditions. For the case of four-dimensional disk several first coefficients of the heat kernel expansion are computed. An error in the analitical expression by Branson and Gilkey is corrected. 
  Within a simple quantization scheme, observables for a large class of finite dimensional time reparametrization invariant systems may be constructed by integration over the manifold of time labels. This procedure is shown to produce a complete set of densely defined operators on a physical Hilbert space for which an inner product is identified and to provide reasonable results for simple test cases. Furthermore, many of these observables have a clear interpretation in the classical limit and we use this to demonstrate that, for a class of minisuperspace models including LRS Bianchi IX and the Kantowski-Sachs model this quantization agrees with classical physics in predicting that such spacetimes recollapse. 
  While the linearity of the Schr\"odinger equation and the superposition principle are fundamental to quantum mechanics, so are the backaction of measurements and the resulting nonlinearity. It is remarkable, therefore, that the wave-equation of systems in continuous interaction with some reservoir, which may be a measuring device, can be cast into a linear form, even after the degrees of freedom of the reservoir have been eliminated. The superposition principle still holds for the stochastic wave-function of the observed system, and exact analytical solutions are possible in sufficiently simple cases. We discuss here the coupling to Markovian reservoirs appropriate for homodyne, heterodyne, and photon counting measurements. For these we present a derivation of the linear stochastic wave-equation from first principles and analyze its physical content. 
  We attempt to a physical interpretation of some known static vacuum solutions of Einstein's equations, namely, the A and B metrics of Ehlers and Kundt. All of them have axial symmetry, so they can be transformed to the Weyl form. In Weyl coordinates $\ln\sqrt{-g_{44}}$ obeys a Laplace equation, and from this a source, called The Newtonian image source can be identified. We use the image sources to interpret the metrics. The procedure is sucessful in some cases. In others it fails because the Weyl transform does not have reasonable properties at infinity. 
  The asymptotic behavior of geometry near the boundary of maximal Cauchy development is studied using a perturbative method, which at the zeroth order reduces Einstein's equations to an exactly solvable set of equations---Einstein's equations with all ``space" derivatives dropped. The perturbative equations are solved to an {\em arbitrarily-high order} for the cosmological spacetimes admitting constant-mean-curvature foliation that ends in a crushing singularity, i..e., whose mean curvature blows up somewhere. Using a ``new" set of dynamical variables (generalized Kasner variables) restrictions on the initial data are found that make the zeroth-order term in the expansion asymptotically dominant when approaching the crushing singularity. The results obtained are in agreement with the first order results of Belinskii, Lifshitz and Khalatnikov on general velocity-dominated cosmological singularities, and, in addition, provide clearer geometrical formulation. 
  The vacuum fluctuations that induce the transitions and the thermalisation of a uniformly accelerated two level atom are studied in detail. Their energy content is revealed through the weak measurement formalism of Aharonov et al. It is shown that each time the detector makes a transition it radiates a Minkowski photon. The same analysis is then applied to the conversion of vacuum fluctuations into real quanta in the context of black hole radiation. Initially these fluctuations are located around the light like geodesic that shall generate the horizon and carry zero total energy. However upon exiting from the star they break up into two pieces one of which gradually acquires positive energy and becomes a Hawking quantum, the other, its ''partner", ends up in the singularity. As time goes by the vacuum fluctuations generating Hawking quanta have exponentially large energy densities. This implies that back reaction effects are large. 
  Aspects of the thermo-dynamics of a black hole which is either pierced by a cosmic gauge string or contains a global monopole are investigated. We also make some comments on the physical significance of the fact that the gravitational mass carried by a global monopole is negative. We note in particular that the negative monopole mass implies a gravitational super-radiance effect. 
  We study the algebra of constraints of quantum gravity in the loop representation based on Ashtekar's new variables. We show by direct computation that the quantum commutator algebra reproduces the classical Poisson bracket one, in the limit in which regulators are removed. The calculation illustrates the use of several computational techniques for the loop representation. 
  We study the suitability of complex Wilson loop variables as (generalized) coordinates on the physical phase space of $SU(2)$-Yang-Mills theory. To this end, we construct a natural one-to-one map from the physical phase space of the Yang-Mills theory with compact gauge group $G$ to a subspace of the physical configuration space of the complex $G^\C$-Yang-Mills theory. Together with a recent result by Ashtekar and Lewandowski this implies that the complex Wilson loop variables form a complete set of generalized coordinates on the physical phase space of $SU(2)$-Yang-Mills theory. They also form a generalized canonical loop algebra. Implications for both general relativity and gauge theory are discussed. 
  A complete basis of nonlocal invariants in quantum gravity theory is built to third order in spacetime curvature and matter-field strengths. The nonlocal identities are obtained which reduce this basis for manifolds with dimensionality $2\omega<6$. The present results are used in heat-kernel theory, theory of gauge fields and serve as a basis for the model-independent approach to quantum gravity and, in particular, for the study of nonlocal vacuum effects in the gravitational collapse problem. 
  The quantum gravitational scale of inflation is calculated by finding a sharp probability peak in the distribution function of chaotic inflationary cosmologies driven by a scalar field with large negative constant $\xi$ of nonminimal interaction. In the case of the no-boundary state of the universe this peak corresponds to the eternal inflation, while for the tunnelling quantum state it generates a standard inflationary scenario. The sub-Planckian parameters of this peak (the mean value of the corresponding Hubble constant ${\mbox{\boldmath $H$}}\simeq 10^{-5}m_P$, its quantum width $\Delta{\mbox{\boldmath $H$}}/{\mbox{\boldmath $H$}}\simeq 10^{-5}$ and the number of inflationary e-foldings ${\mbox{\boldmath $N$}}\simeq 60$) are found to be in good correspondence with the observational status of inflation theory, provided the coupling constants of the theory are constrained by a condition which is likely to be enforced by the (quasi) supersymmetric nature of the sub-Planckian particle physics model. 
  The trace of the heat kernel is expanded in a basis of nonlocal curvature invariants of $N$th order. The coefficients of this expansion (the nonlocal form factors) are calculated to third order in the curvature inclusive. The early-time and late-time asymptotic behaviours of the trace of the heat kernel are presented with this accuracy. The late-time behaviour gives the criterion of analyticity of the effective action in quantum field theory. The latter point is exemplified by deriving the effective action in two dimensions. 
  Standard methods of the theory of permanent state reduction are shown to offer an alternative realization of Omn\`es' project. Our proposal, as simple as Omn\`es' one, possesses closed master equation for the ensemble density operator, assuring causality. 
  The discovery by Gott of a remarkably simple spacetime with closed timelike curves (CTC's) provides a tool for investigating how the creation of time machines is prevented in classical general relativity. The Gott spacetime contains two infinitely long, parallel cosmic strings, which can equivalently be viewed as point masses in (2+1)-dimensional gravity. We examine the possibility of building such a time machine in an open universe. Specifically, we consider initial data specified on an edgeless, noncompact, spacelike hypersurface, for which the total momentum is timelike (i.e., not the momentum of a Gott spacetime). In contrast to the case of a closed universe (in which Gott pairs, although not CTC's, can be produced from the decay of stationary particles), we find that there is never enough energy for a Gott-like time machine to evolve from the specified data; it is impossible to accelerate two particles to sufficiently high velocity. Thus, the no-CTC theorems of Tipler and Hawking are enforced in an open (2+1)-dimensional universe by a mechanism different from that which operates in a closed universe. In proving our result, we develop a simple method to understand the inequalities that restrict the result of combining momenta in (2+1)-dimensional gravity. 
  We show that it is not possible to smooth out the metric on the Deutsch-Politzer time machine to obtain an everywhere non-singular asymptotically flat Lorentzian metric. 
  Two gauge and diffeomorphism invariant theories on the Yang-Mills phase space are studied. They are based on the Lie-algebras $so(1,3)$ and $\widetilde{so(3)}$ -- the loop-algebra of $so(3)$. Although the theories are manifestly real, they can both be reformulated to show that they describe complex gravity and an infinite number of copies of complex gravity, respectively. The connection to real gravity is given. For these theories, the reality conditions in the conventional Ashtekar formulation are represented by normal constraint-like terms. 
  The ADM masses of particle-like solutions to the Einstein-Yang/Mills Equations tend to 2 as the number of nodes of the solutions increases. The same result is true for black hole solutions with event horizon less than 1. For event horizon $\rho > 1$ the ADM masses converge to $\rho + \rho^{-1} .$ These statements extend and correct ``An Investigation at the Limiting Behavior of Particle-Like Solutions to the Einstein-Yang/Mills Equations and a New black Hole Solutions'', by J. A. Smoller and A. G. Wasserman, in Comm. Math. Phys., 161, 365-389, (1994). 
  We present a class of theories of two dimensional gravity which admits homogeneous and isotropic solutions that are nonsingular and asymptotically approach a FRW matter dominated universe at late times. These models are generalizations of two dimensional dilaton gravity and both vacuum solutions and those including conformally coupled matter are investigated. In each case our construction leads to an inflationary stage driven by the gravitational sector. Our work comprises a simple example of the `Nonsingular Universe' constructions of ref. [1]. 
  We present a new formula for the rate at which cosmic strings lose energy into gravitational radiation, valid for all piecewise-linear cosmic string loops. At any time, such a loop is composed of $N$ straight segments, each of which has constant velocity. Any cosmic string loop can be arbitrarily-well approximated by a piecewise-linear loop with $N$ sufficiently large. The formula is a sum of $O(N^4)$ polynomial and log terms, and is exact when the effects of gravitational back-reaction are neglected. For a given loop, the large number of terms makes evaluation ``by hand" impractical, but a computer or symbolic manipulator yields accurate results. The formula is more accurate and convenient than previous methods for finding the gravitational radiation rate, which require numerical evaluation of a four-dimensional integral for each term in an infinite sum. It also avoids the need to estimate the contribution from the tail of the infinite sum. The formula has been tested against all previously published radiation rates for different loop configurations. In the cases where discrepancies were found, they were due to errors in the published work. We have isolated and corrected both the analytic and numerical errors in these cases. To assist future work in this area, a small catalog of results for some simple loop shapes is provided. 
  In this paper we examine the change in the estimated spatial power spectra at decoupling due to the effects of our clumpy universe which modify observational distances. We find that scales at decoupling can be significantly underestmated in our approximation of neglecting the shear of the ray bundle. We compare our results with other work on lensing and speculate on the implications for structure formation. In particular we examine a proposal to use the position of the first Doppler peak to determine $\Omega$, and find that shrinking will modify the esimated curvature, so that it must be included to obtain an accurate estimate of $\Omega$. Finally we consider future applications and improvements of our results. 
  We investigate the thermodynamical properties of black holes in (3+1) and (2+1) dimensional Einstein gravity with a negative cosmological constant. In each case, the thermodynamic internal energy is computed for a finite spatial region that contains the black hole. The temperature at the boundary of this region is defined by differentiating the energy with respect to entropy, and is equal to the product of the surface gravity (divided by~$2\pi$) and the Tolman redshift factor for temperature in a stationary gravitational field. We also compute the thermodynamic surface pressure and, in the case of the (2+1) black hole, show that the chemical potential conjugate to angular momentum is equal to the proper angular velocity of the black hole with respect to observers who are at rest in the stationary time slices. In (3+1) dimensions, a calculation of the heat capacity reveals the existence of a thermodynamically stable black hole solution and a negative heat capacity instanton. This result holds in the limit that the spatial boundary tends to infinity only if the comological constant is negative; if the cosmological constant vanishes, the stable black hole solution is lost. In (2+1) dimensions, a calculation of the heat capacity reveals the existence of a thermodynamically stable black hole solution, but no negative heat capacity instanton. 
  Given that observations seem to favour a \index{density parameter} $\Omega_0<1$, corresponding to an open universe, we consider gauge\hs invariant perturbations of non\hs flat Robertson\hs Walker universes filled with a general imperfect fluid which can also be taken to represent a scalar field. Our aim is to set up the equations that govern the evolution of the density perturbations $\Delta$ so that it can be determined through a {\it first order differential equation} with a quantity $\kk$ which is conserved at any length scale, even in non\hs flat universe models, acting as a source term. The quantity $\kk$ generalizes other variables that are conserved in specific cases (for example at large scales in a flat universe) and is useful to connect different epochs in the evolution of density perturbations via a transfer function. We show that the problem of finding a conserved $\kk$ can be reduced to determining two auxiliary variables $X$ and $Y$, and illustrate the method with two simple examples. 
  Inflationary models and their claim to solve many of the outstanding problems in cosmology have been the subject of a great deal of debate over the last few years. A major sticking point has been the lack of both good observational and theoretical arguments to single out one particular model out of the many that solve these problems. Here we examine the degree of restrictiveness on the dynamical relationship between the cosmological scale factor and the inflation driving self-interaction potential of a minimally coupled scalar field, imposed by the condition that the scalar field is required to be real during a classical regime (the reality condition). We systema\-tically look at the effects of this constraint on many of the inflationary models found in the literature within the FLRW framework, and also look at what happens when physically motivated perturbations such as shear and bulk viscosity are introduced. We find that in many cases, either the models are totally excluded or the reality condition gives rise to constraints on the scale factor and on the various parameters of the model. 
  The essential role played by differentiable structures in physics is reviewed in light of recent mathematical discoveries that topologically trivial space-time models, especially the simplest one, ${\bf R^4}$, possess a rich multiplicity of such structures, no two of which are diffeomorphic to each other and thus to the standard one. This means that physics has available to it a new panoply of structures available for space-time models. These can be thought of as source of new global, but not properly topological, features. This paper reviews some background differential topology together with a discussion of the role which a differentiable structure necessarily plays in the statement of any physical theory, recalling that diffeomorphisms are at the heart of the principle of general relativity. Some of the history of the discovery of exotic, i.e., non-standard, differentiable structures is reviewed. Some new results suggesting the spatial localization of such exotic structures are described and speculations are made on the possible opportunities that such structures present for the further development of physical theories. 
  Consider multidim. universes M= R x M_1 x ... x M_n with D = 1+ d_1 .. + d_n, where M_i of dimension d_i are of have constant curvature and compact for i>1. For Lagrangian models L(R,phi) on M which depend only on Ricci curvature R and a scalar field phi, there exists an explicit description of conformal equivalence, with the minimal coupling model and the conformal coupling model as distinguished representatives of a conformal class. For the conformally coupled model we study classical solutions and their relation to solutions in the equivalent minimally coupled model. The domains of equivalence are separated by certain critical values of the scalar field phi. Furthermore the coupling constant xi of the coupling between phi and R is critical at both, the minimal value xi=0 and the conformal value xi_c={D-2}/{4(D-1)}. In different noncritical regions of $xi$ the solutions behave qualitatively different. For vanishing potential of the minimally coupled scalar field we find a multidimensional generalization of Kasner's solution. Its scale factor singularity vanishes in the conformal coupling model. Static internal spaces in the minimal model become dynamical in the conformal one. The nonsingular conformal solution has a particular interesting region, where internal spaces shrink while the external space expands. While the Lorentzian solution relates to a creation of the universe at finite scale, it Euclidean counterpart is an (instanton) wormhole. Solving the Wheeler de Witt equation we obtain the quantum counterparts to the classical solutions. A real Euclidean quantum wormhole is obtained in a special case. 
  Recently introduced classical theory of gravity in non-commutative geometry is studied. The most general (four parametric) family of $D$ dibensional static spherically symmetric spacetimes is identified and its properties are studied in detail. For wide class of the choices of parameters, the corresponding spacetimes have the structure of asymptotically flat black holes with a smooth event horizon hiding the curvature singularity. A specific attention is devoted to the behavior of components of the metric in non-commutative direction, which are interpreted as the black hole hair. 
  {\it If gravity is a metric field by Einstein, it is a Higgs field.} Gravitation theory meets spontaneous symmetry breaking in accordance with the Equivalence Principle reformulated in the spirit of Klein-Chern geometries of invariants. In gravitation theory, the structure group of the principal linear frame bundle $LX$ over a world manifold $X^4$ is reducible to the connected Lorentz group $SO(3,1)$. The physical underlying reason of this reduction is Dirac fermion matter possessing only exact Lorentz symmetries. The associated Higgs field is a tetrad gravitational field $h$ represented by a global section of the quotient $\Si$ of $LX$ by $SO(3,1)$. The feature of gravity as a Higgs field issues from the fact that, in the presence of different tetrad fields, Dirac fermion fields are described by spinor bundles associated with different reduced Lorentz subbundles of $LX$, and we have nonequivalent representations of cotangent vectors to $X^4$ by Dirac's matrices. It follows that a fermion field must be regarded only in a pair with a certain tetrad field. These pairs fail to be represented by sections of any product bundle $S\times\Si$, but sections of the composite spinor bundle $S\to\Si\to X^4$. They constitute the so-called fermion-gravitation complex where values of tetrad gravitational fields play the role of coordinate parameters, besides the familiar world coordinates. In Part 1 of the article, geometry of the fermion-gravitation complex is investigated. The goal is the total Dirac operator into which components of a connection on $S\to\Si$ along tetrad coordinate directions make contribution. The Part II will be devoted to dynamics of fermion-gravitation complex. It is a constraint system to describe which we use the covariant multisymplectic generalization of the Hamiltonian formalism when canonical momenta correspond to derivatives of fields with respect to all world coordinates, not only the time. 
  We analyze the limiting solution of the Bartnik-McKinnon family and show that its exterior is an extremal Reissner-Nordstr{\o}m black hole and not a new type of non-abelian black hole as claimed in a recent article by Smoller and Wasserman. 
  We propose that the Chern-Simons invariant of the Ashtekar-Sen connection is the natural internal time coordinate for classical and quantum cosmology. The reasons for this are a number of interesting properties of this functional, which we describe here. 1)It is a function on the gauge and diffeomorphism invariant configuration space, whose gradient is orthogonal to the two physical degrees of freedom, in the metric defined by the Ashtekar formulation of general relativity. 2)The imaginary part of the Chern-Simons form reduces in the limit of small cosmological constant, $\Lambda$, and solutions close to DeSitter spacetime, to the York extrinsic time coordinate. 3)Small matter-field excitations of the Chern-Simons state satisfy, by virtue of the quantum constraints, a functional Schroedinger equation in which the matter fields evolve on a DeSitter background in the Chern-Simons time. We then n propose this is the natural vacuum state of the theory for $\Lambda \neq 0$. 4)This time coordinate is periodic on the configuration space of Euclideanized spacetimes, due to the large gauge transformations, which means that physical expectation values for all states in non-perturbative quantum gravity will satisfy the $KMS$ condition, and may then be interpreted as thermal states. 5)Forms for the physical hamiltonians and inner product which support the proposal are suggested, and a new action principle for general relativity, as a geodesic principle on the connection superspace, is found. 
  We analyze solutions to Friedmann-Robertson-Walker cosmologies in Brans-Dicke theory, where a scalar field is coupled to gravity. Matter is modelled by a $\gamma$-law perfect fluid, including false-vacuum energy as a special case. Through a change of variables, we reduce the field equations from fourth order to second order, and they become equivalent to a two-dimensional dynamical system. We then analyze the entire solution space of this dynamical system, and find that many qualitative features of these cosmologies can be gleaned, including standard non-inflationary or extended inflationary expansion, but also including bifurcations of stable or unstable expansion or contraction, noninflationary vacuum-energy dominated models, and several varieties of ``coasting," ``bouncing," ``hesitating," and ``vacillating" universes. It is shown that inflationary dogma, which states that a universe with curvature and dominated by inflationary matter will always approach a corresponding flat-space solution at late times, does not hold in general for the scalar-tensor theory, but rather that the occurence of inflation depends upon the initial energy of the scalar field relative to the expansion rate. In the case of flat space ($k=0$), the dynamical system formalism generates some previously known exact power-law solutions. 
  The gravitaional force produced by a point particle, like the sun, in the background of the static Einstein universe is studied. Both the approximate solution in the weak field limit and exact solution are obtained. The main properties of the solution are {\it i}) near the point particle, the metric approaches the Schwarzschild one and the radius of its singularity becomes larger than that of the Schwarzschild singularity, {\it ii}) far from the point particle, the metric approaches the static Einstein closed universe. The maximum length of the equator of the universe becomes smaller than that of the static Einstein universe due to the existence of the point particle. These properties show the strong correlation betweem the particle and the universe. 
  The multidimensional charged dilatonic black hole solution with $n$ internal Ricci-flat spaces is considered. The bound on the mass of the black hole is obtained. In the strong dilatonic coupling limit the critical mass becomes zero. The case $n = \infty$ is also considered. 
  Multidimensional cosmological models with $n~(n > 1)$ Einstein spaces are discussed classically and with respect to canonical quantization. These models are integrable in the case of Ricci flat internal spaces. For negative curvature of the external space we find exact classical solutions modelling dynamical as well as spontaneous compactification of the internal spaces. Spontaneous compactification turns out to be an attractor solution. Solutions of the quantum Wheeler-DeWitt equation are also obtained. Some of them describe the tunneling process to be interpreted as the birth of the universe from ''nothing''. 
  A multidimensional cosmological model with space-time consisting of $n (n \ge 2)$ Einstein spaces $M_{i}$ is investigated in the presence of a cosmological constant $\Lambda$ and a homogeneous minimally coupled scalar field $\varphi(t)$ as a matter source. Classical and quantum wormhole solutions are obtained for $\Lambda < 0$ and all $M_{i}$ being Ricci-flat. Classical wormhole solutions are also found for $\Lambda < 0$ and only one of the $M_{i}$ being Ricci-flat for the case of spontaneous compactification of the internal dimensions with fine tuning of parameters. 
  Exact static, spherically symmetric solutions to the Einstein-Maxwell-scalar equations, with a dilatonic-type scalar-vector coupling, in $D$-dimensional gravity with a chain of $n$ Ricci-flat internal spaces are considered. Their properties and special cases are discussed. A family of multidimensional dilatonic black-hole solutions is singled out, depending on two integration constants (related to black hole mass and charge) and three free parameters of the theory (the coordinate sphere, internal space dimensions, and the coupling constant). The behaviour of the solutions under small perturbations preserving spherical symmetry, is studied. It is shown that the black-hole solutions without a dilaton field are stable, while other solutions, possessing naked singularities, are catastrophically unstable. 
  Mutidimensional cosmological models with $n\left( n\geq 2\right) $ Einstein spaces $M_i\left( i=1,\ldots ,n\right) $ are investigated. The cosmological constant and homogeneous minimally coupled scalar field as a matter sources are considered. The scalar field has a potential of general form depending on the scalar field as well as on the scale factors of $M_i$ . The general condition of existence of the solutions with spontaneous compactification is obtained and applied to physically important cases. Two particular kinds of solutions are found. One of them has a wormhole-type continuation into the Euclidean region. Another one represents the generalization of the de Sitter universe to the multidimensional model under consideration. 
  We solve perturbative constraints and eliminate gauge freedom for Ashtekar's gravity on de Sitter background. We show that the reduced phase space consists of transverse, traceless, symmetric fluctuations of the triad and of transverse, traceless, symmetric fluctuations of the connection. A part of gauge freedom corresponding to the conformal Killing vectors of the three-manifold can be fixed only by imposing conditions on Lagrange multiplier. The reduced phase space is equivalent to that of ADM gravity on the same background. 
  The first objective of this article is to show that the black hole partition function can be placed on a firm logical foundation by enclosing the black hole in a spatially finite "box" or boundary. The presence of the box has the effect of stabilizing the black hole and yields a system with a positive heat capacity. The second objective of this article is to explore the origin of black hole entropy. This is accomplished through the construction of a path integral expression for the density matrix for the gravitational field, and through an analysis of the connection between the density matrix and the black hole density of states. Our results suggest that black hole entropy can be associated with an absence of certain "inner boundary information" for the system. (Based on the talk presented by J.D. Brown at the conference "The Black Hole 25 Years After", Santiago, Chile, January 1994.) 
  We derive exact Friedmann--Robertson--Walker cosmological solutions in general scalar--tensor gravity theories, including Brans--Dicke gravity, for stiff matter or radiation. These correspond to the long or short wavelength modes respectively of massless scalar fields. If present, the long wavelength modes of such fields would be expected to dominate the energy density of the universe at early times and thus these models provide an insight into the classical behaviour of these scalar--tensor cosmologies near an initial singularity, or bounce. The particularly simple exact solutions also provide a useful example of the possible evolution of the Brans--Dicke (or dilaton) field, $\phi$, and the Brans--Dicke parameter, $\omega(\phi)$, at late times in spatially curved as well as flat universes. We also discuss the corresponding solutions in the conformally related Einstein metric. 
  To resolve infrared problems with the de~Sitter invariant vacuum, we argue that the history of the de~Sitter phase is crucial. We illustrate how either (1)~the diagonalization of the Hamiltonian for long-wavelength modes or (2)~an explicit modification of the metric in the distant past leads to natural infrared cutoffs. The former case resembles a bosonic superconductor in which graviton-pairing occurs between non-adiabatic modes. While the dynamical equations respect de~Sitter symmetry, the vacuum is not de~Sitter invariant because of the introduction of an initial condition at a finite time. The implications for the one-loop stress tensor and the production of particles are also discussed. 
  We carry out to completion the quantization of a Friedmann-Robertson-Walker model provided with a conformal scalar field, and of a Kantowski-Sachs spacetime minimally coupled to a massless scalar field. We prove that the Hilbert space determined by the reality conditions that correspond to Lorentzian gravity admits a basis of wormhole wave functions. This result implies that the vector space spanned by the quantum wormholes can be equipped with an unique inner product by demanding an adequate set of Lorentzian reality conditions, and that the Hilbert space of wormholes obtained in this way can be identified with the whole Hilbert space of physical states for Lorentzian gravity. In particular, all the normalizable quantum states can then be interpreted as superpositions of wormholes. For each of the models considered here, we finally show that the physical Hilbert space is separable by constructing a discrete orthonormal basis of wormhole solutions. 
  Multidimensional cosmological models with $n (n > 1)$ spaces of constant curvature are discussed classically and with respect to canonical quantization. These models are integrable in the case of Ricci flat internal spaces. For positive curvature in the external space we find exact solutions modelling dynamical as well as spontaneous compactification of internal spaces. 
  We analyse and develop the recent suggestion that a temporal form of quantum logic provides the natural mathematical framework within which to discuss the proposal by Gell-Mann and Hartle for a generalised form of quantum theory based on the ideas of histories and decoherence functionals. Particular stress is placed on properties of the space of decoherence functionals, including one way in which certain global and topological properties of a classical system are reflected in a quantum history theory. 
  Newman and Rovelli have used singular Hamilton-Jacobi transformations to reduce the phase space of general relativity in terms of the Ashtekar variables. Their solution of the gauge constraint cannot be inverted and indeed has no Minkowski space limit. Nonetheless, we exhibit an explicit Hamilton-Jacobi solution of all the linearized constraints. The result does not encourage an iterative solution, but it does indicate the origin of the singularity of the Newman-Rovelli result. 
  Since the gauge group underlying 2+1-dimensional general relativity is non-compact, certain difficulties arise in the passage from the connection to the loop representations. It is shown that these problems can be handled by appropriately choosing the measure that features in the definition of the loop transform. Thus, ``old-fashioned'' loop representations - based on ordinary loops - do exist. In the case when the spatial topology is that of a two-torus, these can be constructed explicitly; {\it all} quantum states can be represented as functions of (homotopy classes of) loops and the scalar product and the action of the basic observables can be given directly in terms of loops. 
  The Ernst equation is formulated on an arbitrary Riemann surface. Analytically, the problem reduces to finding solutions of the ordinary Ernst equation which are periodic along the symmetry axis. The family of (punctured) Riemann surfaces admitting a non-trivial Ernst field constitutes a ``partially discretized'' subspace of the usual moduli space. The method allows us to construct new exact solutions of Einstein's equations in vacuo with non-trivial topology, such that different ``universes'', each of which may have several black holes on its symmetry axis, are connected through necks bounded by cosmic strings. We show how the extra topological degrees of freedom may lead to an extension of the Geroch group and discuss possible applications to string theory. 
  The quasilocal energy associated with a constant stationary time slice of the   Kerr spacetime is presented. The calculations are based on a recent proposal \cite{by} in which quasilocal energy is derived from the Hamiltonian of spatially bounded gravitational systems. Three different classes of boundary surfaces for the Kerr slice are considered (constant radius surfaces, round spheres, and the ergosurface). Their embeddings in both the Kerr slice and flat three-dimensional space (required as a normalization of the energy) are analyzed. The energy contained within each surface is explicitly calculated in the slow rotation regime and its properties discussed in detail. The energy is a positive, monotonically decreasing function of the boundary surface radius. It approaches the Arnowitt-Deser-Misner (ADM) mass at spatial infinity and reduces to (twice) the irreducible mass at the horizon of the Kerr black hole. The expressions possess the correct static limit and include negative contributions due to gravitational binding. The energy at the ergosurface is compared with the energies at other surfaces. Finally, the difficulties involved in an estimation of the energy in the fast rotation regime are discussed. 
  A comprehensive analysis of general relativistic spacetimes which admit a shear-free, irrotational and geodesic timelike congruence is presented. The equations governing the models for a general energy-momentum tensor are written down. Coordinates in which the metric of such spacetimes takes on a simplified form are established. The general subcases of `zero anisotropic stress', `zero heat flux vector' and `two component fluids' are investigated. In particular, perfect fluid Friedmann-Robertson-Walker models and spatially homogeneous models are discussed. Models with a variety of physically relevant energy-momentum tensors are considered. Anisotropic fluid models and viscous fluid models with heat conduction are examined. Also, models with a perfect fluid plus a magnetic field or with pure radiation, and models with two non-collinear perfect fluids (satisfying a variety of physical conditions) are investigated. In particular, models with a (single) perfect fluid which is tilting with respect to the shear-free, vorticity-free and acceleration-free timelike congruence are discussed. 
  General relativistic anisotropic fluid models whose fluid flow lines form a shear-free, irrotational, geodesic timelike congruence are examined. These models are of Petrov type D, and are assumed to have zero heat flux and an anisotropic stress tensor that possesses two distinct non-zero eigenvalues. Some general results concerning the form of the metric and the stress-tensor for these models are established. Furthermore, if the energy density and the isotropic pressure, as measured by a comoving observer, satisfy an equation of state of the form $p = p(\mu)$, with $\frac{dp}{d\mu} \neq -\frac{1}{3}$, then these spacetimes admit a foliation by spacelike hypersurfaces of constant Ricci scalar. In addition, models for which both the energy density and the anisotropic pressures only depend on time are investigated; both spatially homogeneous and spatially inhomogeneous models are found. A classification of these models is undertaken. Also, a particular class of anisotropic fluid models which are simple generalizations of the homogeneous isotropic cosmological models is studied. 
  We consider here the gravitational collapse of a spherically symmetric inhomogeneous dust cloud described by the Tolman-Bondi models. By studying a general class of these models, we find that the end state of the collapse is either a black hole or a naked singularity, depending on the parameters of the initial density distribution, which are $\rho_{c}$, the initial central density of the massive body, and $R_0$, the initial boundary. The collapse ends in a black hole if the dimensionless quantity $\beta$ constructed out of this initial data is greater than 0.0113, and it ends in a naked singularity if $\beta$ is less than this number. A simple interpretation of this result can be given in terms of the strength of the gravitational potential at the starting epoch of the collapse. 
  Cosmic string loops are defined by a pair of periodic functions ${\bf a}$ and ${\bf b}$, which trace out unit-length closed curves in three-dimensional space. We consider a particular class of loops, for which ${\bf a}$ lies along a line and ${\bf b}$ lies in the plane orthogonal to that line. For this class of cosmic string loops one may give a simple analytic expression for the power $\gamma$ radiated in gravitational waves. We evaluate $\gamma$ exactly in closed form for several special cases: (1) ${\bf b}$ a circle traversed $M$ times; (2) ${\bf b}$ a regular polygon with $N$ sides and interior vertex angle $\pi-2\pi M/N$; (3) ${\bf b}$ an isosceles triangle with semi-angle $\theta$. We prove that case (1) with $M=1$ is the absolute minimum of $\gamma$ within our special class of loops, and identify all the stationary points of $\gamma$ in this class. 
  We establish the main features of homogeneous and isotropic dilaton, metric and Yang-Mills field configurations in a cosmological framework. Special attention is paid to the energy exchange between the dilaton and the Yang-Mills field and, in particular, a new energy exchange term is identified. Implications for the Polonyi problem in 4-dimensional string models and in dynamical supersymmetry breaking scenarios are discussed. 
  I give a brief review of the recovery of semiclassical time from quantum gravity and discuss possible extrapolations of this concept to the full theory. 
  We investigate the class of ultralocal metrics on the configuration space of canonical gravity. It is described by a parameter $\alpha$, where $\alpha=0.5$ corresponds to general relativity.   For $\alpha$ less than a critical value the signature is positive definite, while for all other values it is indefinite. We show that in the positive definite case gravity becomes repulsive. From the primordial helium abundance we find that $\alpha$ must lie between $0.4$ and $0.55$. 
  We present the gravitational action and Hamiltonian for a spatially bounded region of an eternal black hole. The Hamiltonian is of the general form $H=H_{+} - H_{-}$, where $H_{+}$ and $H_{-}$ are respectively the Hamiltonians for the regions $M_+$ and $M_-$ located in the left and right wedges of the spacetime. We construct explicitly the quasilocal energy for the system and discuss its dependence on the time direction induced at the boundaries of the manifold. This paper extends the analysis of Ref.~[1] to spacetimes possesing a bifurcation surface and two timelike boundaries. The construction suggests that an interpretation of black hole thermodynamics based on thermofield dynamics ideas can be generalized beyond perturbations to the gravitational field itself of a bounded spacetime region (based on the talk presented by E.A. Martinez at the Lake Louise Winter School on Particle Physics and Cosmology, February 20-26, 1994.) 
  Using canonical (Schrodinger) quantization of spherically symetric gravitational dust systems, we find the quasi-classical (coherent) state, |\alpha^{(s)}>, that corresponds to the classical Schwarzschild solution. We calculate the ``quasi-classical Schwarzschild mertic", which is the expectation value of the quantized metric in thhis quasi-classical state. Depending on the quantization scheme that we use, we study three different quasi- classical geometries, all of which turn out to be singularity free. Their maximal extensions are complete manifolds with no singularities, describing a tower of asymptotically flat universes connected through Planck size wormholes. 
  General relativity in three spacetime dimensions is used to explore three approaches to the ``problem of time'' in quantum gravity: the internal Schr\"odinger approach with mean extrinsic curvature as a time variable, the Wheeler-DeWitt equation, and covariant canonical quantization with ``evolving constants of motion.'' (To appear in {\em Proc.\ of the Lanczos Centenary Conference}, Raleigh, NC, December 1993.) 
  The anomalous rescaling for antisymmetric tensor fields, including gauge bosons, and Dirac fermions on Einstein spaces with boundary has been prone to errors and these are corrected here. The explicit calculations lead to some interesting identities that indicate a deeper underlying structure. 
  Some indication of conditions that are necessary for the formation of black holes from the collision of bubbles during a supercooled phase transition in the the early universe are explored. Two colliding bubbles can never form a black hole. Three colliding bubbles can refocus the energy in their walls to the extent that it becomes infinite. 
  Newton's standard theory of gravitation is reformulated as a {\it gauge} theory of the {\it extended} Galilei Group. The Action principle is obtained by matching the {\it gauge} technique and a suitable limiting procedure from the ADM-De Witt action of general relativity coupled to a relativistic mass-point. 
  In a preceding paper we developed a reformulation of Newtonian gravitation as a {\it gauge} theory of the extended Galilei group. In the present one we derive two true generalizations of Newton's theory (a {\it ten-fields} and an {\it eleven-fields} theory), in terms of an explicit Lagrangian realization of the {\it absolute time} dynamics of a Riemannian three-space. They turn out to be {\it gauge invariant} theories of the extended Galilei group in the same sense in which general relativity is said to be a {\it gauge} theory of the Poincar\'e group. The {\it ten-fields} theory provides a dynamical realization of some of the so-called ``Newtonian space-time structures'' which have been geometrically classified by K\"{u}nzle and Kucha\v{r}. The {\it eleven-fields} theory involves a {\it dilaton-like} scalar potential in addition to Newton's potential and, like general relativity, has a three-metric with {\it two} dynamical degrees of freedom. It is interesting to find that, within the linear approximation, such degrees of freedom show {\it graviton-like} features: they satisfy a wave equation and propagate with a velocity related to the scalar Newtonian potential. 
  We study the occurrence of critical phenomena in four - dimensional, rotating and charged black holes, derive the critical exponents and show that they fulfill the scaling laws. Correlation functions critical exponents and Renormalization Group considerations assign an effective (spatial) dimension, $d=2$, to the system. The two - dimensional Gaussian approximation to critical systems is shown to reproduce all the black hole's critical exponents. Higher order corrections (which are always relevant) are discussed. Identifying the two - dimensional surface with the event horizon and noting that generalization of scaling leads to conformal invariance and then to string theory, we arrive to 't Hooft's string interpretation of black holes. From this, a model for dealing with a coarse grained black hole quantization is proposed. We also give simple arguments that lead to a rough quantization of the black hole mass in units of the Planck mass, i. e. $M\simeq{1\over\sqrt{2}}M_{pl} \sqrt{l}$ with a $l$ positive integer and then, from this result, to the proportionality between quantum entropy and area. 
  Generalizing earlier results of Joshi and Dwivedi (Commun. Math. Phys. 146, 333 (1992); Lett. Math. Phys. 27, 235 (1993)), we analyze here the spherically symmetric gravitational collapse of a matter cloud with a general form of matter for the formation of a naked singularity. It is shown that this is related basically to the choice of initial data to the Einstein field equations, and would therefore occur in generic situations from regular initial data within the general context considered here, subject to the matter satisfying the weak energy condition. The condition on initial data which leads to the formation of black hole is also characterized. 
  We revisit the long standing problem of analyzing an inertial electric charge from the point of view of uniformly accelerated observers in the context of semi-classical gravity. We choose a suitable set of accelerated observers with respect to which there is no photon emission coming from the inertial charge. We discuss this result against previous claims [F. Rohrlich, Ann. Phys. (N.Y.) vol: 22, 169 (1963)]. (This Essay was awarded a Honorable Mention for 1994 by the Gravity Research Foundation.) 
  We consider general relativistic Cauchy data representing two nonspinning, equal-mass black holes boosted toward each other. When the black holes are close enough to each other and their momentum is sufficiently high, an encompassing apparent horizon is present so the system can be viewed as a single, perturbed black hole. We employ gauge-invariant perturbation theory, and integrate the Zerilli equation to analyze these time-asymmetric data sets and compute gravitational wave forms and emitted energies. When coupled with a simple Newtonian analysis of the infall trajectory, we find striking agreement between the perturbation calculation of emitted energies and the results of fully general relativistic numerical simulations of time-symmetric initial data. 
  A thorough classification of the topologies of compact homogeneous universes is given except for the hyperbolic spaces, and their global degrees of freedom are completely worked out. To obtain compact universes, spatial points are identified by discrete subgroups of the isometry group of the generalized Thurston geometries, which are related to the Bianchi and the Kantowski-Sachs-Nariai universes. Corresponding to this procedure their total degrees of freedom are shown to be categorised into those of the universal covering space and the Teichm\"uller parameters. The former are given by constructing homogeneous metrics on simply connected manifold. The Teichm\"uller spaces are also given by explicitly constructing expressions for the discrete subgroups of the isometry group. 
  The ultrarelativistic limit of the Schwarzschild and the Kerr-geometry together with their respective energy-momentum tensors is derived. The approach is based on tensor-distributions making use of the underlying Kerr-Schild structure, which remains stable under the ultrarelativistic boost. 
  When space-time is assumed to be non-Riemannian the minimal coupling procedure (MCP) is not compatible, in general, with minimal action principle (MAP). This means that the equations gotten by applying MCP to the Euler-Lagrange equations of a Lagrangian $\cal L$ do not coincide with the Euler-Lagrange equations of the Lagrangian obtained by applying MCP to $\cal L$. Such compatibility can be restored if the space-time admits a connection-compatible volume element. We show how these concepts can alter qualitatively the predictions of the Einstein-Cartan theory of gravity. 
  The question of whether an observer can escape from a black hole is addressed, using a recent general definition of a black hole in the form of a future outer trapping horizon. An observer on a future outer trapping horizon must enter the neighbouring trapped region. It is possible for the observer to subsequently escape from the trapped region. However, if the horizon separates the space-time into two disjoint components, inside and outside the horizon, then an observer inside a future outer trapping horizon cannot get outside, assuming the null energy condition. A similar confinement property holds for trapped, locally area-preserving cylinders, as suggested by Israel. 
  We have considered the quantum behavior of a conformally induced gravity in the minimal Riemann-Cartan space. The regularized one-loop effective potential considering the quantum fluctuations of the dilaton and the torsion fields in the Coleman-Weinberg sector gives a sensible phase transition for an inflationary phase in De Sitter space. For this effective potential, we have analyzed the semi-classical equation of motion of the dilaton field in the slow-rolling regime. 
  I describe the treatment of gravity as a quantum effective field theory. This allows a natural separation of the (known) low energy quantum effects from the (unknown) high energy contributions. Within this framework, gravity is a well behaved quantum field theory at ordinary energies. In studying the class of quantum corrections at low energy, the dominant effects at large distance can be isolated, as these are due to the propagation of the massless particles (including gravitons) of the theory and are manifested in the nonlocal/nonanalytic contributions to vertex functions and propagators. These leading quantum corrections are parameter-free and represent necessary consequences of quantum gravity. The methodology is illustrated by a calculation of the leading quantum corrections to the gravitational interaction of two heavy masses. 
  A proposal is made for a mathematically unambiguous treatment of evolution in the presence of closed timelike curves. In constrast to other proposals for handling the naively nonunitary evolution that is often present in such situations, this proposal is causal, linear in the initial density matrix and preserves probability. It provides a physically reasonable interpretation of invertible nonunitary evolution by redefining the final Hilbert space so that the evolution is unitary or equivalently by removing the nonunitary part of the evolution operator using a polar decomposition. 
  If the large-angular-scale anisotropy in the cosmic microwave background radiation is caused by the long-wavelength cosmological perturbations of quantum mechanical origin, they are, most likely, gravitational waves, rather than density perturbations or rotational perturbations. 
  The radial collapse of a homogeneous disk of collisionless particles can be solved analytically in Newtonian gravitation. To solve the problem in general relativity, however, requires the full machinery of numerical relativity. The collapse of a disk is the simplest problem that exhibits the two most significant and challenging features of strong-field gravitation: black hole formation and gravitational wave generation. We carry out dynamical calculations of several different relativistic disk systems. We explore the growth of ring instabilities in equilibrium disks, and how they are suppressed by sufficient velocity dispersion. We calculate wave forms from oscillating disks, and from disks that undergo gravitational collapse to black holes. Studies of disk collapse to black holes should also be useful for developing new techniques for numerical relativity, such as apparent horizon boundary conditions for black hole spacetimes. 
  It is shown here that a dynamical Planck mass can drive the scale factor of the universe to accelerate. The negative pressure which drives the cosmic acceleration is identified with the unusual kinetic energy density of the Planck field. No potential nor cosmological constant is required. This suggests a purely gravity driven, kinetic inflation. Although the possibility is not ruled out, the burst of acceleration is often too weak to address the initial condition problems of cosmology. To illustrate the kinetic acceleration, three different cosmologies are presented. One such example, that of a bouncing universe, demonstrates the additional feature of being nonsingular. The acceleration is also considered in the conformally related Einstein frame in which the Planck mass is constant. 
  Based upon the formalism recently developed by one of us (MS), we analytically perform the post-Newtonian expansion of gravitational waves from a test particle in circular orbit of radius $r_0$ around a Schwarzschild black hole of mass $M$. We calculate gravitational wave forms and luminosity up to $v^8$ order beyond Newtonian, where $v=(M/r_0)^{1/2}$. In particular, we give the exact analytical values of the coefficients of $\ln v$ terms at $v^6$ and $v^8$ orders in the luminosity and confirm the numerical values obtained previously by the other of us (HT) and Nakamura. Our result is valid in the small mass limit of one body and gives an important guideline for the gravitational wave physics of coalescing compact binaries. 
  A new scheme is proposed for dealing with the problem of singularities in General Relativity. The proposal is, however, much more general than this. It can be used to deal with manifolds of any dimension which are endowed with nothing more than an affine connection, and requires a family \calc\ of curves satisfying a {\em bounded parameter property} to be specified at the outset. All affinely parametrised geodesics are usually included in this family, but different choices of family \calc\ will in general lead to different singularity structures. Our key notion is the {\em abstract boundary\/} or {\em $a$-boundary\/} of a manifold, which is defined for any manifold \calm\ and is independent of both the affine connection and the chosen family \calc\ of curves. The $a$-boundary is made up of equivalence classes of boundary points of \calm\ in all possible open embeddings. It is shown that for a pseudo-Riemannian manifold $(\calm,g)$ with a specified family \calc\ of curves, the abstract boundary points can then be split up into four main categories---regular, points at infinity, unapproachable points and singularities. Precise definitions are also provided for the notions of a {\em removable singularity} and a {\em directional singularity}. The pseudo-Riemannian manifold will be said to be singularity-free if its abstract boundary contains no singularities. The scheme passes a number of tests required of any theory of singularities. For instance, it is shown that all compact manifolds are singularity-free, irrespective of the metric and chosen family \calc. 
  In the framework of QED we investigate the bremsstrahlung process for an electron passing by a straight static cosmic string. This process is precluded in empty Minkowski space-time by energy and momentum conservation laws. It happens in the presence of the cosmic string as a consequence of the conical structure of space, in spite of the flatness of the metric. The cross section and emitted electromagnetic energy are computed and analytic expressions are found for different energies of the incoming electron. The energy interval is divided in three parts depending on whether the energy is just above electron rest mass $M$, much larger than $M$, or exceeds $M/\delta$, with $\delta$ the string mass per unit length in Planck units. We compare our results with those of scalar QED and classical electrodynamics and also with conic pair production process computed earlier. 
  The general equations describing hydrostatic equilibrium are developed for Non-singular Gravity. A new type of astrophysical structure, a Super Dense Object (SDO) or "Dark Star", is shown to exist beyond Neutron star field strengths. These structures are intrinsically stable against gravitational collapse and represent the non-singular alternative to General Relativity's Black Holes. 
  We consider the functional Schrodinger equation for a self interacting scalar field in an expanding geometry. By performing a time dependent scale transformation on the argument of the field we derive a functional Schrodinger equation whose hamiltonian is time independent but involves a time-odd term associated to a constraint on the expansion current. We study the mean field approximation to this equation and generalize in this case, for interacting fields, the solutions worked out by Bunch and Davies for free fields. 
  The n-time generalization of the Tangherlini solution [1] is considered. The equations of geodesics for the metric are integrated. For $n = 2$ it is shown that the naked singularity is absent only for two sets of parameters, corresponding to the trivial extensions of the Tangherlini solution. The motion of a relativistic particle in the multitemporal background is considered. This motion is governed by the gravitational mass tensor. Some generalizations of the solution, including the multitemporal analogue of the Myers-Perry charged black hole solution, are obtained. 
  This paper begins with a short presentation of the Bianchi IX or ``Mixmaster'' cosmological model, and some ways of writing the Einstein equations for it. There is then an interlude describing how I came to a study of this model, and then a report of some mostly unpublished work from a Ph.\ D. thesis of D. M. (Prakash) Chitre relating approximate solutions to geodesic flows on finite volume negative curvature Riemannian manifolds, for which he could quote results on ergodicity. A final section restates studies of a zero measure set of solutions which in first approximation appear to have only a finite number of Kasner epochs before reaching the singularity. One finds no plausible case for such behavior in better approximations. 
  We propose to study the behavior of complicated numerical solutions to Einstein's equations for generic cosmologies by following the geodesic motion of a swarm of test particles. As an example, we consider a cylinder of test particles initially at rest in the plane symmetric Gowdy universe on $T^3 \times R$. For a circle of test particles in the symmetry plane, the geodesic equations predict evolution of the circle into distortions and rotations of an ellipse as well as motion perpendicular to the plane. The evolutionary sequence of ellipses depends on the initial position of the circle of particles. We display snapshots of the evolution of the cylinder. 
  We discuss the quantum mechanics and thermodynamics of the (2+1)-dimensional black hole, using both minisuperspace methods and exact results from Chern-Simons theory. In particular, we evaluate the first quantum correction to the black hole entropy. We show that the dynamical variables of the black hole arise from the possibility of a deficit angle at the (Euclidean) horizon, and briefly speculate as to how they may provide a basis for a statistical picture of black hole thermodynamics. 
  A formula is given for the variation of the Hawking energy along any one-parameter foliation of compact spatial 2-surfaces. A surface for which one null expansion is positive and the other negative has a preferred orientation, with a spatial or null normal direction being called outgoing or ingoing as the area increases or decreases respectively. A natural way to propagate such a surface through a hypersurface is to choose the foliation such that the null expansions are constant over each surface. For such uniformly expanding foliations, the Hawking energy is non-decreasing in any outgoing direction, and non-increasing in any ingoing direction, assuming the dominant energy condition. It follows that the Hawking energy is non-negative if the foliation is bounded at the inward end by either a point or a marginal surface, and in the latter case satisfies the Penrose-Gibbons isoperimetric inequality. The Bondi-Sachs energy may be expressed as a limit of the Hawking energy at conformal infinity, and the energy-variation formula reduces at conformal infinity to the Bondi-Sachs energy-loss formula. 
  In the scattering picture, thermal radiation effects may be associated with the `above barrier' reflection coefficient, which comes into play because of complex turning points. This note contains several general remarks on the application of the above statement to Schwarzschild black hole radiance 
  An extension of the Dirac procedure for the quantization of constrained systems is necessary to address certain issues that are left open in Dirac's original proposal. These issues play an important role especially in the context of non-linear, diffeomorphism invariant theories such as general relativity. Recently, an extension of the required type was proposed by one of us using algebraic quantization methods. In this paper, the key conceptual and technical aspects of the algebraic program are illustrated through a number of finite dimensional examples. The choice of examples and some of the analysis is motivated by certain peculiar problems endemic to quantum gravity. However, prior knowledge of general relativity is not assumed in the main discussion. Indeed, the methods introduced and conclusions arrived at are applicable to any system with first class constraints. In particular, they resolve certain technical issues which are present also in the reduced phase space approach to quantization of these systems. 
  We determine the different possible space-time metrics inside an infinite rotating hollow cylinder with given energy density and longitudinal and azimuthal stresses, the metric outside the cylinder being chosen of the spinning cosmic string type. The solutions we obtain for various domains of values of the cylinder parameters include a space-time with topologically Euclidean spatial sections, a black-hole solution, a quasi-regular solution, and various wormhole solutions. A solution which is regular only if the longitudinal dimension is compactified might approximately describe spontaneous compactification of the cylinder to a torus. 
  In this paper, the behavior of a spherical hole in an otherwise infinite and uniform universe is investigated. First, the Newtonian theory is developed. The concept of negative gravity, an outward gravitational force acting away from the center of the spherical hole, is presented, and the resulting expansion of the hole is investigated. Then, the same result is derived using the techniques of Einstein's theory of general relativity. The field equations are solved for an infinite uniform universe and then for an infinite universe in which matter is uniformly distributed except for a spherical hole. Negative pressure caused by negative gravity is utilized. The physical significance of the cosmological constant is explained, and a new physical concept, that of the gravitational potential of a hole, is discussed. The relationship between the Newtonian potential for a hole and the Schwarzschild solution of the field equations is explored. Finally, the geodesic equations are considered. It is shown that photons and particles are deflected away from the hole. An application of this idea is pursued, in which a new cosmology based upon expanding holes in a uniform universe is developed. The microwave background radiation and Hubble's Law, among others, are explained. Finally, current astronomical data are used to compute a remarkably accurate value of Hubble's constant, as well as estimates of the average mass density of the universe and the cosmological constant. 
  We discuss the Shannon-Wehrl entropy within the squeezing vocabulary for the cosmological and black hole particle production. 
  A b s t r a c t It will be argued that 1) the Bell inequalities are not equivalent with those inequalities derived by Pitowsky and others that indicate the Kolmogorovity of a probability model, 2) the original Bell inequalities are irrelevant to both the question of whether or not quantum mechanics is a Kolmogorovian theory as well as the problem of determinism, whereas 3) the Pitowsky type inequalities are not violated by quantum mechanics, hence 4) quantum mechanics is a Kolmogorovian probability theory, therefore, 5) it is compatible with an entirely deterministic universe. 
  We present a new method for finding principal null directions (PNDs). Because our method assumes as input the intrinsic metric and extrinsic curvature of a spacelike hypersurface, it should be particularly useful to numerical relativists. We illustrate our method by finding the PNDs of the Kastor-Traschen spacetimes, which contain arbitrarily many $Q=M$ black holes in a de Sitter back-ground. 
  The causality properties of space-time models with traversable wormholes are considered. It is shown that relativity principle cannot be applied to the motion of the wormhole's mouths in the outer space and the dynamical wormhole transformation into the time machine is impossible. The examples of both causal and noncausal space-time models with traversable wormholes are also considered. Some properties of space-time models with causality violation are briefly discussed. 
  We solve for the retarded Greens function for linearized gravity in a background with a negative cosmological constant, anti de Sitter space. In this background, it is possible for a signal to reach spatial infinity in a finite time. Therefore the form of the Greens function depends on a choice of boundary condition at spatial infinity. We take as our condition that a signal which reaches infinity should be lost, not reflected back. We calculate the Greens function associated with this condition, and show that it reproduces the correct classical solution for a point mass at the origin, the anti de Sitter-Schwarzchild solution. 
  In (2+1)-dimensional general relativity, the path integral for a manifold $M$ can be expressed in terms of a topological invariant, the Ray-Singer torsion of a flat bundle over $M$. For some manifolds, this makes an explicit computation of transition amplitudes possible. In this paper, we evaluate the amplitude for a simple topology-changing process. We show that certain amplitudes for spatial topology change are nonvanishing---in fact, they can be infrared divergent---but that they are infinitely suppressed relative to similar topology-preserving amplitudes. 
  A non-singular, static spherically symmetric solution to the nonsymmetric gravitational and electromagnetic theory field equations is derived, which depends on the four parameters m, l^2, Q and s, where m is the mass, Q is the electric charge, l^2 is the NGT charge of a body and s is a dimensionless constant. The electromagnetic field invariants are also singularity-free, so that it is possible to construct regular particle-like solutions in the theory. All the curvature invariants are finite, there are no null surfaces in the spacetime and there are no black holes. A new stable, superdense object (SDO) replaces black holes. 
  Is it possible to define what we could mean by chaos in a space-time metric (even in the simplest toy-model studies)? Is it of importance for phenomena we may search for in Nature? 
  Static solutions of the Einstein-Yang-Mills-Higgs system containing extreme black holes are studied. The field equations imply strong restrictions on boundary values of all fields at the horizon. If the Yang-Mills radial electric field $E$ is non-zero there, then all fields at the horizon take values in the centralizer of $E$. For the particular case of SU(3), there are two different kinds of centralizers: two-dimensional abelian (Cartan subalgebra) and four-dimensional (su(2)$\times$u(1)) ones. The two-dimensional centralizer admits only constant fields: even the geometry of the horizon is that of constant curvature. If the cosmological constant $\Lambda$ is negative, a two-surface of any genus is possible; for positive curvature, only spherically symmetrical horizons are allowed. For the four-dimensional centralizer, all spherically symmetrical horizons are explicitly given. 
  The world view suggested by quantum cosmology is that inflating universes with all possible values of the fundamental constants are spontaneously created out of nothing. I explore the consequences of the assumption that we are a `typical' civilization living in this metauniverse. The conclusions include inflation with an extremely flat potential and low thermalization temperature, structure formation by topological defects, and an appreciable cosmological constant. 
  It is found that the deviation of an effective potential from the quartic form is related to the metric and vector torsion dependencies of the effective potential in the vector torsion coupled conformally induced gravity. 
  The Hawking effect and the Unruh effect are two of the most important predictions in the theoretical physics of the last quarter of the 20th century. In parallel to the theoretical investigations there is great interest in the possibility of revealing effects of this type in some sort of experiments. I present a general discussion of the proposals to measure the Hawking and Unruh effects and/or their `analogues' in the laboratory, and I make brief comments on each of them. The reader may also find the various physical pictures corresponding to the two effects which were applied to more common phenomena, and vice versa 
  We re-analyze the globally neutral non-Abelian black holes and present a unified picture, classifying them into two types; Type I (black holes with massless non-Abelian field) and Type II (black holes with ``massive" non-Abelian field). For the Type II, there are two branches: The black hole in the high-entropy branch is ``stable" and almost neutral, while that in the low entropy branch, which is similar to the Type I, is unstable and locally charged. To analyze their stabilities, we adopt the catastrophe theoretic method, which reveals us a universal picture of stability of the black holes. It is shown that the isolated Type II black hole has a fold catastrophe structure.   In a heat bath system, the Type I black hole shows a cusp catastrophe, while the Type II has both fold and cusp catastrophe. 
  We consider the eigenvalues of the three-dimensional Weyl operator defined in terms of the (Euclidean) Ashtekar variables, and we study their dependence on the gravitational field. We notice that these eigenvalues can be used as gravitational variables, and derive explicit formulas for their Poisson brackets and their time evolution. 
  Gell-Mann and Hartle have proposed a significant generalisation of quantum theory with a scheme whose basic ingredients are `histories' and decoherence functionals. Within this scheme it is natural to identify the space $\UP$ of propositions about histories with an orthoalgebra or lattice. This raises the important problem of classifying the decoherence functionals in the case where $\UP$ is the lattice of projectors $\PV$ in some Hilbert space $\V$; in effect we seek the history analogue of Gleason's famous theorem in standard quantum theory. In the present paper we present the solution to this problem for the case where $\V$ is finite-dimensional. In particular, we show that every decoherence functional $d(\a,\b)$, $\a,\b\in\PV$ can be written in the form $d(\a,\b)=\tr_{\V\otimes\V}(\a\otimes\b X)$ for some operator $X$ on the tensor product space $\V\otimes\V$. 
  We prove a positive energy theorem in 2+1 dimensional gravity for open universes and any matter energy-momentum tensor satisfying the dominant energy condition. We consider on the space-like initial value surface a family of widening Wilson loops and show that the energy-momentum of the enclosed subsystem is a future directed time-like vector whose mass is an increasing function of the loop, until it reaches the value $1/4G$ corresponding to a deficit angle of $2\pi$. At this point the energy-momentum of the system evolves, depending on the nature of a zero norm vector appearing in the evolution equations, either into a time-like vector of a universe which closes kinematically or into a Gott-like universe whose energy momentum vector, as first recognized by Deser, Jackiw and 't Hooft is space-like.   This treatment generalizes results obtained by Carroll, Fahri, Guth and Olum for a system of point-like spinless particle, to the most general form of matter whose energy-momentum tensor satisfies the dominant energy condition. The treatment is also given for the anti de Sitter 2+1 dimensional gravity. 
  In the hybrid skyrmion in which an Anti-de Sitter bag is imbedded into the skyrmion configuration a S^{1}\times S^{2} membrane is lying on the compactified spatial infinity of the bag [H. Rosu, Nuovo Cimento B 108, 313 (1993)]. The connection between the quark degrees of freedom and the mesonic ones is made through the membrane, in a way that should still be clarified from the standpoint of general relativity and topology. The S^1 \times S^2 membrane as a 3-dimensional manifold is at the same time a Weyl-Einstein space. We make here an excursion through the mathematical body of knowledge in the differential geometry and topology of these spaces which is expected to be useful for hadronic membranes 
  The behaviour of magnetic monopole solutions of the Einstein-Yang-Mills-Higgs equations subject to linear spherically symmetric perturbations is studied. Using Jacobi's criterion some of the monopoles are shown to be unstable. Furthermore the numerical results and analytical considerations indicate the existence of a set of stable solutions. 
  We consider the cluster of problems raised by the relation between the notion of time, gravitational theory, quantum theory and thermodynamics; in particular, we address the problem of relating the "timelessness" of the hypothetical fundamental general covariant quantum field theory with the "evidence" of the flow of time. By using the algebraic formulation of quantum theory, we propose a unifying perspective on these problems, based on the hypothesis that in a generally covariant quantum theory the physical time-flow is not a universal property of the mechanical theory, but rather it is determined by the thermodynamical state of the system ("thermal time hypothesis"). We implement this hypothesis by using a key structural property of von Neumann algebras: the Tomita-Takesaki theorem, which allows to derive a time-flow, namely a one-parameter group of automorphisms of the observable algebra, from a generic thermal physical state. We study this time-flow, its classical limit, and we relate it to various characteristic theoretical facts, as the Unruh temperature and the Hawking radiation. We also point out the existence of a state-independent notion of "time", given by the canonical one-parameter subgroup of outer automorphisms provided by the Cocycle Radon-Nikodym theorem. 
  We extend our previous analysis of the quantum state during and after $O(4)$-symmetric bubble nucleation to the case including gravitational effects. We find that there exists a simple relationship between the case with and without gravitational effects. In a special case of a conformally coupled scalar field which is massless except on the bubble wall, the state is found to be conformally equivalent to the case without gravity. 
  We investigate the phenomenon of mass inflation in two-dimensional dilaton theories of gravity. We consider two distinct black hole spacetimes and construct the mass-inflation solution for each. Our analysis is extended to include multi-horizon spacetimes. We find that the mass function diverges in a manner quantitatively similar to its four-dimensional counterpart. 
  Observations of gravitational waves from inspiralling compact binaries using laser-interferometric detectors can provide accurate measures of parameters of the source. They can also constrain alternative gravitation theories. We analyse inspiralling compact %binaries in the context of the scalar-tensor theory of Jordan, Fierz, Brans and Dicke, focussing on the effect on the inspiral of energy lost to dipole gravitational radiation, whose source is the gravitational self-binding energy of the inspiralling bodies. Using a matched-filter analysis we obtain a bound on the coupling constant $\omega_{\rm BD}$ of Brans-Dicke theory. For a neutron-star/black-hole binary, we find that the bound could exceed the current bound of $\omega_{\rm BD}>500$ from solar-system experiments, for sufficiently low-mass systems. For a $0.7 M_\odot$ neutron star and a $3 M_\odot$ black hole we find that a bound $\omega_{\rm BD} \approx 2000$ is achievable. The bound decreases with increasing black-hole mass. For binaries consisting of two neutron stars, the bound is less than 500 unless the stars' masses differ by more than about $0.5 M_\odot$. For two black holes, the behavior of the inspiralling binary is observationally indistinguishable from its behavior in general relativity. These bounds assume reasonable neutron-star equations of state and a detector signal-to-noise ratio of 10. 
  The {\em abstract boundary\/} (or {\em {\em a\/}-boundary\/}) of Scott and Szekeres \cite{Scott94} constitutes a ``boundary'' to any $n$-dimensional, paracompact, connected, Hausdorff, $C^\infty$-manifold (without a boundary in the usual sense). In general relativity one deals with a {\em space-% time\/} $(\cM,g)$ (a 4-dimensional manifold $\cM$ with a Lorentzian metric $g$), together with a chosen preferred class of curves in $\cM$. In this case the {\em a\/}-boundary points may represent ``singularities'' or ``points at infinity''. Since the {\em a\/}-boundary itself, however, does not depend on the existence of further structure on the manifold such as a Lorentzian metric or connection, it is possible for it to be used in many contexts.   In this paper we develop some purely topological properties of abstract boundary sets and abstract boundary points ({\em a\/}-boundary points). We prove, amongst other things, that compactness is invariant under boundary set equivalence, and introduce another invariant concept ({\em isolation\/}), which encapsulates the notion that a boundary set is ``separated'' from other boundary points of the same embedding. .......   [The abstract continues in paper proper - truncated to fit here.] 
  The stability of the black hole horizon is demanded by both cosmic censorship and the generalized second law of thermodynamics. We test the consistency of these principles by attempting to exceed the black hole extremality condition in various process in which a U(1) charge is added to a nearly extreme Reissner--Nordstr\"om black hole charged with a {\it different\/} type of U(1) charge. For an infalling spherical charged shell the attempt is foiled by the self--Coulomb repulsion of the shell. For an infalling classical charge it fails because the required classical charge radius exceeds the size of the black hole. For a quantum charge the horizon is saved because in order to avoid the Landau ghost, the effective coupling constant cannot be large enough to accomplish the removal. 
  (This is a report for the Proceedings of ``Journees Relativistes 1993'' written in September 1993. Containes a short description of the results published elsewhere in the joint paper with A. Ashtekar) Integral calculus on the space of gauge equivalent connections is developed. By carring out a non-linear generalization of the theory of cylindrical measures on topological vector spaces, a faithfull, diffeomorphism invariant measure is introduced on a suitable completion of the quotient space. The strip (i.e. momentum) operators are densely-defined in the resulting Hilbert space and interact with the measure correctly, to become essentially self adjoint operators. 
  (This short article is a continuation of a longer, review work, in the same volume of Proceedings, by Ashtekar, Marolf and Mour\~ao [gr-qc/9403042]. All the details and other results are to be found in joint papers of the author with Abhay Ashtekar.) The projective limit technics derived for spaces of connections are extended to a new framework which for the associated projective limit plays a role of the differential geometry. It provides us with powerfull technics for construction and studding various operators. In particular, we introduce the commutator algebra of `vector fields', define a divergence of a vector field and find for them a quantum representation. Among the vector fields, there are operators which we identify as regularised Rovelli-Smolin loop operators linear in momenta. Another class of operators which comes out naturally are Laplace operators. 
  Pure gravity and gauge theories in two dimensions are shown to be special cases of a much more general class of field theories each of which is characterized by a Poisson structure on a finite dimensional target space. A general scheme for the quantization of these theories is formulated. Explicit examples are studied in some detail. In particular gravity and gauge theories with equivalent actions are compared. Big gauge transformations as well as the condition of metric nondegeneracy in gravity turn out to cause significant differences in the structure of the corresponding reduced phase spaces and the quantum spectra of Dirac observables. For $R^2$ gravity coupled to SU(2) Yang Mills the question of quantum dynamics (`problem of time') is addressed. [This article is a contribution to the proceedings (to appear in LNP) of the 3rd Baltic RIM Student Seminar (1993). Importance is attached to concrete examples. A more abstract presentation of the ideas underlying this article (including new developments) is found in hep-th/9405110.] 
  Bell's Theorem assumes that hidden variables are not influenced by future measurement settings. The assumption has sometimes been questioned, but the suggestion has been thought outlandish, even by the taxed standards of the discipline. (Bell thought that it led to fatalism.) The case for this reaction turns out to be surprisingly weak, however. We show that QM easily evades the standard objections to advanced action. And the approach has striking advantages, especially in avoiding the apparent conflict between Bell's Theorem and special relativity.   The second part of the paper considers the broader question as to why advanced action seems so counterintuitive. We investigate the origins of our ordinary intuitions about causal asymmetry. It is argued that the view that the past does not depend on the future is largely anthropocentric, a kind of projection of our own temporal asymmetry. Many physicists have also reached this conclusion, but have thought that if causation has no objective direction, there is no objective content to an advanced action interpretation of QM. This turns out to be a mistake. From the ordinary subjective perspective, we can distinguish two sorts of objective world: one ``looks as if'' it contains only forward causation, the other ``looks as if'' it involves a mix of backward and forward causation. This clarifies the objective core of an advanced action interpretation of QM, and shows that there is an independent symmetry argument in favour of the approach. 
     One of the reasons we expect a standard quantum mechanics, which predicts probabilities for alternatives defined on spacelike slices, to be inadequate for quantum gravity is that the notion of ``spacelike'' is ill-defined in a theory where the metric itself is behaving quantum-mechanically.  Spacetime coarse grainings--sets of alternatives defined with respect to a region extended in time as well as space--have previously been considered in the quantum mechanics of a single nonrelativistic particle.  This work examines spacetime coarse grainings in the quantum mechanics of a free *relativistic* particle, not as a theory for real particles (which are described by quantum field theory) but as a simple reparametrization-invariant theory which may model some features of quantum gravity.      We use Hartle's sum-over-histories generalized quantum mechanics formalism, in which probabilities are assigned to a set of alternatives if they decohere, i.e., if the quantum-mechanical interference between their branches of the initial state is nearly zero.  For one extremely simple set of alternatives, we calculate these branches.  Despite the nonlocality of the positive-definite version of the Klein-Gordon inner product, some initial conditions are found to give decoherence and allow the consistent assignment of probabilities. 
  The evolution of 4-dimensional and (4+d)-dimensional (d=1,2) cosmological models based on the integrable Weyl geometry are considered numerically both for empty space-time and for scalar field with non minimal coupling with gravity. 
  The classical theory of gravity predicts its own demise -- singularities. We therefore attempt to quantize gravitation, and present here a new approach to the quantization of gravity wherein the concept of time is derived by imposing the constraints as expectation-value equations over the true dynamical degrees of freedom of the gravitational field -- a representation of the underlying anisotropy of space. This self-consistent approach leads to qualitatively different predictions than the Dirac and the ADM quantizations, and in addition, our theory avoids the interpretational conundrums associated with the problem of time in quantum gravity. We briefly describe the structure of our functional equations, and apply our quantization technique to two examples so as to illustrate the basic ideas of our approach. 
  LAGEOS is an accurately-tracked, dense spherical satellite covered with 426 retroreflectors. The tracking accuracy is such as to yield a medium term (years to decades) inertial reference frame determined via relatively inexpensive observations. This frame is used as an adjunct to the more difficult and data intensive VLBI absolute frame measurements. There is a substantial secular precession of the satellite's line of nodes consistent with the classical, Newtonian precession due to the non-sphericity of the earth. Ciufolini has suggested the launch of an identical satellite (LAGEOS-3) into an orbit supplementary to that of LAGEOS-1: LAGEOS-3 would then experience an equal and opposite classical precession to that of LAGEOS-1. Besides providing a more accurate real-time measurement of the earth's length of day and polar wobble, this paired-satellite experiment would provide the first direct measurement of the general relativistic frame-dragging effect. Of the five dominant error sources in this experiment, the largest one involves surface forces on the satellite, and their consequent impact on the orbital nodal precession. The surface forces are a function of the spin dynamics of the satellite. Consequently, we undertake here a theoretical effort to model the spin ndynamics of LAGEOS. In this paper we present our preliminary results. 
  General laws of black-hole dynamics, some of which are analogous to the laws of thermodynamics, have recently been found for a general definition of black hole in terms of a future outer trapping horizon, a hypersurface foliated by marginal surfaces of a certain type. This theory is translated here into spin-coefficient language. Second law: the area form of a future outer trapping horizon is generically increasing, otherwise constant. First law: the rate of change of the area form is given by an energy flux and the trapping gravity. Zeroth law: the total trapping gravity of a compact outer marginal surface has an upper bound, attained if and only if the trapping gravity is constant. Topology law: a compact future outer marginal surface has spherical topology. Signature law: an outer trapping horizon is generically spatial, otherwise null. Trapping law: spatial surfaces sufficiently close to a compact future outer marginal surface are trapped if they lie inside the trapping horizon. Confinement law: if the interior and exterior of a future outer trapping horizon are disjoint, an observer inside the horizon cannot get outside. 
  We prove that for an orthogonal spacetime metric separable in space and time in comoving coordinates, the requirements of perfect fluid and non-singularity single out the unique family of singularity free cosmological models. Further homogeneous models could only be Bianchi I or FLRW while inhomogeneous ones can be with or without a singularity. 
  We reinterpret the well known Taub-singularity in terms of a cylinder symmetric geometry. It is shown that a cylindrical analog to the Einstein-Rosen bridge as well as a cosmic string will be present in the geometry. 
  A generalisation of Price's theorem is given for application to Inflationary Cosmologies. Namely, we show that on a Schwarzschild--de Sitter background there are no static solutions to the wave or gravitational perturbation equations for modes with angular momentum greater than their intrinsic spin. 
  A black hole considered as a part of a thermodynamical system possesses the Bekenstein-Hawking entropy $S_H =A_H /(4l_{\mbox{\scriptsize{P}}}^2)$, where $A_H$ is the area of a black hole surface and $l_{\,\mbox{\scriptsize{P}}}$ is the Planck length. Recent attempts to connect this entropy with dynamical degrees of freedom of a black hole generically did not provide the universal mechanism which allows one to obtain this exact value. We discuss the relation between the 'dynamical' contribution to the entropy and $S_H$, and show that the universality of $S_H$ is restored if one takes into account that the parameters of the internal dynamical degrees of freedom as well as their number depends on the black hole temperature. 
  An analytical expression for the third coefficient of the Jones Polynomial $P_J[\gamma,\, {\em e}^q]$ in the variable $q$ is reported. Applications of the result in Quantum Gravity are considered. 
  A new representation of Quantum Gravity is developed. This formulation is based on an extension of the group of loops. The enlarged group, that we call the Extended Loop Group, behaves locally as an infinite dimensional Lie group. Quantum Gravity can be realized on the state space of extended loop dependent wavefunctions. The extended representation generalizes the loop representation and contains this representation as a particular case. The resulting diffeomorphism and hamiltonian constraints take a very simple form and allow to apply functional methods and simplify the loop calculus. In particular we show that the constraints are linear in the momenta. The nondegenerate solutions known in the loop representation are also solutions of the constraints in the new representation. The practical calculation advantages allows to find a new solution to the Wheeler-DeWitt equation. Moreover, the extended representation puts in a precise framework some of the regularization problems of the loop representation. We show that the solutions are generalized knot invariants, smooth in the extended variables, and any framing is unnecessary. 
  {\sl A Hamiltonian framework for 2+1 dimensional gravity coupled with matter (satisfying positive energy conditions) is considered in the asymptotically flat context. It is shown that the total energy of the system is non-negative, vanishing if and only if space-time is (globally) Minkowskian. Furthermore, contrary to one's experience with usual field theories, the Hamiltonian is} {\rm bounded from above}. This is a genuinely non-perturbative result. {\sl In the presence of a space-like Killing field, 3+1 dimensional vacuum general relativity is equivalent to 2+1 dimensional general relativity coupled to certain matter fields. Therefore, our expression provides, in particular, a formula for energy per-unit length (along the symmetry direction) of gravitational waves with a space-like symmetry in 3+1 dimensions. A special case is that of cylindrical waves which have two hypersurface orthogonal, space-like Killing fields. In this case, our expression is related to the ``c-energy'' in a non-polynomial fashion. While in the weak field limit, the two agree, in the strong field regime they differ significantly. By construction, our expression yields the generator of the time-translation in the full theory, and therefore represents the physical energy in the gravitational field.} \footnote{$^1$}{This is a detailed account of the results presented in the Brill-Misner symposium at the University of Maryland in May1993} 
  We argue that in standard quantum electrodynamics radiative corrections do not lead to decoherence of unexcited atomic systems. The proposal of Santos relies upon deliberate switching on and off the vacuum interactions. 
  Guided by a Hamiltonian treatment of spherically symmetric geometry, we find a remarkably simple -- stationary, but not static -- form for the line element of Schwarzschild (and Reissner-Nordstrom) geometry. The line element continues smoothly through the horizon; by exploiting this feature we are able to give a very simple and physically transparent derivation of the Hawking radiance. We construct the complete Penrose diagram by enforcing time-reversal symmetry. Finally we outline how an improved treatment of the radiance, including effects of self-gravitation, can be obtained. 
  The proposal of the possibility of change of signature in quantum cosmology has led to the study of this phenomenon in classical general relativity theory, where there has been some controversy about what is and is not possible. We here present a new analysis of such a change of signature, based on previous studies of the initial value problem in general relativity. We emphasize that there are various continuity suppositions one can make at a classical change of signature, and consider more general assumptions than made up to now. We confirm that in general such a change can take place even when the second fundamental form of the surface of change does not vanish. 
  We point out that for a large class of parametrized theories, there is a constant in the constrained Hamiltonian which drops out of the classical equations of motion in configuration space. Examples include the mass of a relativistic particle in free fall, the tension of the Nambu string, and Newton's constant for the case of pure gravity uncoupled to matter or other fields. In the general case, the classically irrelevant constant is proportional to the ratio of the kinetic and potential terms in the Hamiltonian. It is shown that this ratio can be reinterpreted as an {\it unconstrained} Hamiltonian, which generates the usual classical equations of motion. At the quantum level, this immediately suggests a resolution of the "problem of time" in quantum gravity. We then make contact with a recently proposed transfer matrix formulation of quantum gravity and discuss the semiclassical limit. In this formulation, it is argued that a physical state can obey a (generalized) Poincar\'e algebra of constraints, and still be an approximate eigenstate of 3-geometry. Solutions of the quantum evolution equations for certain minisuperspace examples are presented. An implication of our proposal is the existence of a small, inherent uncertainty in the phenomenological value of Planck's constant. 
  Recent work on an approach to the geometrodynamics of cylindrical gravity waves in the presence of interacting scalar matter fields, based on the Kucha\v{r} hypertime formalism, is extended to the analogous spherically symmetric system. This produces a geometrodynamical formalism for spherical black holes and wormholes in which the metric variables are divided into two classes, dynamical and redundant. The redundant variables measure the embedding of a spacelike hypersurface into the spacetime, and proper time in the asymptotically flat regions. All the constraints can be explicitly solved for the momenta conjugate to the embedding variables. The dynamical variables, including an extra ADM mass for wormhole topologies, can then be considered as functionals of the redundant ones, including the proper time variable. The solution of the resulting constraint system determines the momentum conjugate to the proper time as a function of the other variables, producing Unruh's Hamiltonian formalism for the spherical black hole, whilst extending it to an arbitrary foliation choice. The resulting formalism is appropriate as a starting point for the construction of a hypertime functional Schr\"odinger equation for quantized spherically symmetric black holes and wormholes. 
  M\o ller's Tetrad Theory of Gravitation is examined with regard to the energy-momentum complex. The energy-momentum complex as well as the superpotential associated with M\o ller's theory are derived. M\o ller's field equations are solved in the case of spherical symmetry. Two different solutions, giving rise to the same metric, are obtained. The energy associated with one solution is found to be twice the energy associated with the other. Some suggestions to get out of this inconsistency are discussed at the end of the paper. 
  We study the quantization of some cosmological models within the theory of N=1 supergravity with a positive cosmological constant. We find, by imposing the supersymmetry and Lorentz constraints, that there are no physical states in the models we have considered. For the k=1 Friedmann-Robertson-Walker model, where the fermionic degrees of freedom of the gravitino field are very restricted, we have found two bosonic quantum physical states, namely the wormhole and the Hartle-Hawking state.   From the point of view of perturbation theory, it seems that the gravitational and gravitino modes that are allowed to be excited in a supersymmetric Bianchi-IX model contribute in such a way to forbid any physical solutions of the quantum constraints. This suggests that in a complete perturbation expansion we would have to conclude that the full theory of N=1 supergravity with a non-zero cosmological constant should have no physical states. 
  The general theory of N = 1 supergravity with supermatter is studied using a canonical approach. The supersymmetry and gauge constraint generators are found. The framework is applied to the study of a Friedmann minisuperspace model. We consider a Friedmann k = + 1 geometry and a family of spin-0 as well as spin-1 gauge fields together with their odd (anti-commuting) spin-1/2 partners. The quantum supersymmetry constraints give rise to a set of first-order coupled partial differential equations for the components of the wave function. As an intermediate stage in this project, we put both the spin-1 field and its fermionic partner equal to zero. The physical states of our simplified model correspond effectively to those of a mini-superspace quantum cosmological model possessing N=4 local supersymmetry coupled to complex scalars with spin-1/2 partners. The different supermatter models are given by specifying a K\"ahler metric for the scalars; the allowed quantum states then depend on the K\"ahler geometries. For the cases of spherically symmetric and flat K\"ahler geometries we find the general solution for the quantum state with a very simple form. However, although they allow a Hartle-Hawking state, they do not allow a wormhole state. 
  The phenomenon of mass inflation is shown to occur for a rotating black hole. We demonstrate this feature in $(2+1)$ dimensions by extending the charged spinning BTZ black hole to Vaidya form. We find that the mass function diverges in a manner quantitatively similar to its static counterparts in $(3+1)$, $(2+1)$ and $(1+1)$ dimensions. 
  The $n$-time generalization of Schwarzschild solution is considered. The equations of geodesics for the metric are integrated. The multitemporal analogues of Newton laws for the extended objects described by the solution are suggested. The scalar-vacuum generalization of the solution is also presented. 
  A new approach is suggested which allows to describe phenomenologically arbitrary topologies of the Universe. It consists in a generalizaton of third quantization. This quantization is carried out for the case of asymptotic closeness to a cosmological singularity. It is also pointed out that suggested approach leads to a modification of the ordinary quantum field theory. In order to show this modification we consider the example of a free massless scalar field. 
  We discuss the reduction of gravitating Chern-Simons electrodynamics with two commuting Killing vectors to a dynamical problem with four degrees of freedom, and present regular particle-like solutions. 
  This paper clarifies some aspects of Lorentzian topology change, and it extends to a wider class of spacetimes previous results of Geroch and Tipler that show that topology change is only to be had at a price. The scenarios studied here are ones in which an initial spacelike surface is joined by a connected ``interpolating spacetime'' to a final spacelike surface, possibly of different topology. The interpolating spacetime is required to obey a condition called causal compactness, a condition satisfied in a very wide range of situations. No assumption is made about the dimension of spacetime. First, it is stressed that topology change is kinematically possible; i.e., if a field equation is not imposed, it is possible to construct topology-changing spacetimes with non-singular Lorentz metrics. Simple 2-dimensional examples of this are shown. Next, it is shown that there are problems in such spacetimes: Geroch's closed-universe argument is applied to causally compact spacetimes to show that even in this wider class of spacetimes there are causality violations associated with topology change. It follows from this result that there will be causality violations if the initial (or the final) surface is not connected, even when there is no topology change. Further, it is shown that in dimensions $\geq 3$ causally compact topology-changing spacetimes cannot satisfy Einstein's equation (with a reasonable source); i.e., there are severe dynamical obstructions to topology change. This result extends a previous one due to Tipler. Like Tipler's result, it makes no assumptions about geodesic completeness; i.e., it does not permit topology change even at the price of singularities (of the standard incomplete-geodesic variety). Brief discussions are also given of ways in which the results of this paper might be circumvented. 
  The effect of decoherence is analysed for a free particle, interacting with an environment via a dissipative coupling. The interaction between the particle and the environment occurs by a coupling of the position operator of the particle with the environmental degrees of freedom. By examining the exact solution of the density matrix equation one finds that the density matrix becomes completely diagonal in momentum with time while the position space density matrix remains nonlocal. This establishes the momentum basis as the emergent 'preferred basis' selected by the environment which is contrary to the general expectation that position should emerge as the preferred basis since the coupling with the environment is via the position coordinate. 
  We discuss the fate of initial states of the cat type for the damped harmonic oscillator, mostly employing a linear version of the stochastic Schr\"odinger equation. We also comment on how such cat states might be prepared and on the relation of single realizations of the noise to single runs of experiments. 
  The observation that we are among the first $10^{11}$ or so humans reduces the prior probability that we find ourselves in a species whose total lifetime number of individuals is much higher, according to arguments of Carter, Leslie, Nielsen, and Gott. However, if we instead start with a prior probability that a history has a total lifetime number which is very large, without assuming that we are in such a history, this more basic probability is not reduced by the observation of how early in history we exist. 
  The field equations of the scalar-tensor theories of gravitation are presented in different representations, related to each other by conformal transformations of the metric. One of the representations resembles the Jordan-Brans-Dicke theory, and is the starting point for the generation of exact electrostatic and magnetostatic exterior solutions. The corresponding solutions for each specific theory can be obtained by transforming back to the original canonical representation, and the conversions are given for the theories of Jordan-Brans-Dicke, Barker, Schwinger, and conformally invariant coupling. The electrostatic solutions represent the exterior metrics and fields of configurations where the gravitational and electric equipotential surfaces have the same symmetry. A particular family of electrostatic solutions is developed, which includes as special case the spherically symmetric solutions of the scalar-tensor theories. As expected, they reduce to the well-known Reissner-Nordstr\"{o}m metric when the scalar field is set equal to a constant. The analysis of the Jordan-Brans-Dicke metric yields an upper bound for the mass-radius ratio of static stars, for a class of interior structures. 
  We present a new finite action for Einstein gravity in which the Lagrangian is quadratic in the covariant derivative of a spinor field. Via a new spinor-curvature identity, it is related to the standard Einstein-Hilbert Lagrangian by a total differential term. The corresponding Hamiltonian, like the one associated with the Witten positive energy proof is fully four-covariant. It defines quasi-local energy-momentum and can be reduced to the one in our recent positive energy proof. (Fourth Prize, 1994 Gravity Research Foundation Essay.) 
  A one-fold infinity of explicit quasi-stationary regular line elements for the Schwarzschild geometry is obtained directly from the vacuum Einstein equations. The class includes the familiar Eddington-Finkelstein coordinates, and the coordinates discussed recently by Kraus and Wilczek. 
  We reconsider space-time singularities in classical Einsteinian general relativity: with the help of several new co-ordinate systems we show that the Schwarzschild solution can be extended beyond the curvature singularity at r=0. The extension appears as an infinite covering of standard Kruskal space-time. While the two-dimensional reduction of this infinite sequence of Kruskal-Szekeres domains obtained by suppressing the angular degrees of freedom is still a topological manifold - albeit one for which the metric structure is singular on one-dimensional submanifolds - we obtain for the full four-dimensional geometry the more general structure of a stratified variety. 
  All regular and singular cosmological perturbations in a radiation dominated Einstein-de Sitter Universe with collisionless particles can be found by a generalized power series ansatz. Talk given at "Birth of the Universe and Fundamental Physics", May 1994 (Rome). 
  The connection is established between two different action principles for perfect fluids in the context of general relativity. For one of these actions, $S$, the fluid four--velocity is expressed as a sum of products of scalar fields and their gradients (the velocity--potential representation). For the other action, ${\bar S}$, the fluid four--velocity is proportional to the totally antisymmetric product of gradients of the fluid Lagrangian coordinates. The relationship between $S$ and ${\bar S}$ is established by expressing $S$ in Hamiltonian form and identifying certain canonical coordinates as ignorable. Elimination of these coordinates and their conjugates yields the action ${\bar S}$. The key step in the analysis is a point canonical transformation in which all tensor fields on space are expressed in terms of the Lagrangian coordinate system supplied by the fluid. The canonical transformation is of interest in its own right. It can be applied to any physical system that includes a material medium described by Lagrangian coordinates. The result is a Hamiltonian description of the system in which the momentum constraint is trivial. 
  We prove that static, spherically symmetric, asymptotically flat, regular solutions of the Einstein-Yang-Mills equations are unstable for arbitrary gauge groups, at least for the ``generic" case. This conclusion is derived without explicit knowledge of the possible equilibrium solutions. 
  We review the current status and prospects for the conformal invariant fourth order theory of gravity which has recently been advanced by Mannheim and Kazanas as a candidate alternative to the standard second order Einstein theory. We examine how it is possible in principle to replace the Einstein theory at all while retaining its tested features, and we appeal to the wisdom gleaned from particle physics to suggest a fully covariant, conformal invariant alternative. We explore the theory as a microscopic fundamental theory of elementary particles, to thus give curvature a more prominent role in elementary particle dynamics, and in particular in the generation both of elementary particle masses and the extended bag-like models which are thought to describe them. We discuss the degree to which conformal gravity can compete with string theory as a consistent candidate microscopic theory of gravity. Additionally, we explore the theory as a macroscopic theory of gravity where the exact, non-perturbative, classical potential is found to be of the confining $V(r)=-\beta /r+\gamma r/2$ form, a form which enables us to provide an explanation for the general systematics of galactic rotational velocity curves without the need to assume the existence of copious amounts of dark matter; with this explanation of the curves apparently not being in conflict with the current round of microlensing observations. 
  A parametric manifold can be viewed as the manifold of orbits of a (regular) foliation of a manifold by means of a family of curves. If the foliation is hypersurface orthogonal, the parametric manifold is equivalent to the 1-parameter family of hypersurfaces orthogonal to the curves, each of which inherits a metric and connection from the original manifold via orthogonal projections; this is the well-known Gauss-Codazzi formalism. We generalize this formalism to the case where the foliation is not hypersurface orthogonal. Crucial to this generalization is the notion of deficiency, which measures the failure of the orthogonal tangent spaces to be surface-forming, and which behaves very much like torsion. Some applications to initial value problems in general relativity will be briefly discussed. 
  A parametric manifold is a manifold on which all tensor fields depend on an additional parameter, such as time, together with a parametric structure, namely a given (parametric) 1-form field. Such a manifold admits natural generalizations of Lie differentiation, exterior differentiation, and covariant differentiation, all based on a nonstandard action of vector fields on functions. There is a new geometric object, called the deficiency, which behaves much like torsion, and which measures whether a parametric manifold can be viewed as a 1-parameter family of orthogonal hypersurfaces. 
  The existence of non trivial, non topological solutions in a class of induced, effective gravity models arising out of a non minimally coupled scalar field is established. We shall call such solutions ``Gravity Balls'' as the effective gravitational constant inside the soliton differs from its effective value outside. 
  In theories of gravitation in which dimensional parameters are dynamically induced, one can have non - topological - soliton solutions. This article reviews related topics connected with such solutions. The existence of such solutions in curved spacetime can give rise to halos of gravity (g-) balls with gravitational ``constant" having different values inside and outside the ball. Such g - balls can have quite interesting bearing on the dark matter problem over galactic and cluster scales. We describe the origin of such solutions. We speculate on related problems in Cosmology. Such objects would naturally occur in a large class of induced gravity models in which we have scalar fields non minimally coupled to the scalar curvature. 
  The paths on the {\bf R$^3$} real Euclidean manifold are defined as 2-dimensional simplicial strips; points are replaced by 2-simplexes and the orbits of the action of a one discrete-parameter group on the base manifold becomes a convex polyhedron attached to a 2-dimensional simplicial complex. The Lagrangian of a moving mass is proportional to the width of the path. The special relativistic form of the Lagrangian is recovered in the continuum limit, without relativistic Lorenz invariance considerations. 
  We test on the 2+1 dimensional black hole the quantization method that we have previously proposed in four dimensions. While in the latter case the horizon dynamics is described, at the effective level, by the action of a relativistic membrane, in the 2+1 case it is described by a string. We show that this approach reproduces the correct value of the temperature and entropy of the 2+1 dimensional black hole, and we write down the Schroedinger equation satisfied by the horizon wave function. 
  A general bimetric theory of gravitation is described as a linear in the second approximation. This is allowed due to the small experimental significance of the higher order terms. Solar System tests are satisfied. The theory allows black holes, which are physical singularities. The predicted black hole radius is equal to the one resulting from Einstein's theory, if the compatibility between the gauge conditions and the existence of gravitational waves is required (the Rosen-Fock metric). The quadrupolar gravitational radiation formula is regained. 
  It is shown that the metric of a massless particle obtained from boosting the Schwarzschild metric to the velocity of light, has four Killing vectors corresponding to an $E(2)\times \RR$ symmetry-group. This is in agreement with the expectations based on flat-space kinematics but is in contrast to previous statements in the literature \cite{Schueck}. Moreover, it also goes beyond the general Jordan-Ehlers-Kundt-(JEK)-classification of gravitational pp-waves as given in \cite{JEK}. 
  The integration procedure for multidimensional cosmological models with multicomponent perfect fluid in spaces of constant curvature is developed. Reduction of pseudo-Euclidean Toda-like systems to the Euclidean ones is done. Some known solutions are singled out from those obtained. The existence of the wormholes is proved. 
  We present a unified treatment of the slicing (3+1) and threading (1+3) decompositions of spacetime in terms of foliations. It is well-known how to decompose the metric and connection in the slicing picture; this is at the heart of any initial-value problem in general relativity. We describe here the analogous problem in the threading picture, recovering the recent results of Perjes on parametric manifolds. 
  We derive the global properties of static spherically symmetric solutions to the Einstein-Maxwell-dilaton system in the presence of an arbitrary exponential dilaton potential. We show that -- with the exception of a pure cosmological constant `potential' -- no asymptotically flat, asymptotically de Sitter or asymptotically anti-de Sitter solutions exist in these models. 
  The purpose of this note is to establish the basic properties--- regularity at the horizon, time independence, and thermality--- of the generalized Hartle-Hawking vacua defined in static spacetimes with bifurcate Killing horizon admitting a regular Euclidean section. These states, for free or interacting fields, are defined by a path integral on half the Euclidean section. The emphasis is on generality and the arguments are simple but formal. 
  We modify the recent analytic formula given by Allen and Casper for the rate at which piecewise linear cosmic string loops lose energy to gravitational radiation to yield the analogous analytic formula for the rate at which loops radiate momentum. The resulting formula (which is exact when the effects of gravitational back-reaction are neglected) is a sum of O(N^4) polynomial and log terms where, N is the total number of segments on the piecewise linear string loop. As illustration, we write the formula explicitly for a simple one-parameter family of loops with N=5. For most loops the large number of terms makes evaluation ``by hand" impractical, but, a computer or symbolic manipulator may by used to yield accurate results. The formula has been used to correct numerical results given in the existing literature. To assist future work in this area, a small catalog of results for a number of simple string loops is provided. 
  The Einstein-Hilbert action with a cosmological term is used to derive a new action in 1+1 spacetime dimensions. It is shown that the two-dimensional theory is equivalent to planar symmetry in General Relativity. The two-dimensional theory admits black holes and free dilatons, and has a structure similar to two-dimensional string theories. Since by construction these solutions also solve Einstein's equations, such a theory can bring two-dimensional results into the four-dimensional real world. In particular the two-dimensional black hole is also a black hole in General Relativity. 
  Decoherence of Friedmann-Robertson-Walker (FRW) geometries due to massive vector fields with $SO(3)$ global symmetry is discussed in the context of Quantum Cosmology. (Talk presented at the 3rd National Meeting of Astronomy and Astrophysics, July 1993, IST, Lisboa, Portugal. -- A more complete version will be presented at MG7 conference and in the report CERN-TH.7241/94 DAMTP R-94/22 by the same authors) 
  We express the complex potential E and the metrical fields omega and gamma of all stationary axisymmetric vacuum spacetimes that result from the application of two successive quadruple-Neugebauer (or two double-Harrison) transformations to Minkowski space in terms of data specified on the symmetry axis, which are in turn easily expressed in terms of multipole moments. Moreover, we suggest how, in future papers, we shall apply our approach to do the same thing for those vacuum solutions that arise from the application of more than two successive transformations, and for those electrovac solutions that have axis data similar to that of the vacuum solutions of the Neugebauer family. (References revised following response from referee.) 
  Generalizing a method presented in an earlier paper, we express the complex potentials E and Phi of all stationary axisymmetric electrovac spacetimes that correspond to axis data of the form E(z,0) = (U-W)/(U+W) , Phi(z,0) = V/(U+W) , where U = z^{2} + U_{1} z + U_{2} , V = V_{1} z + V_{2} , W = W_{1} z + W_{2} , in terms of the complex parameters U_{1}, V_{1}, W_{1}, U_{2}, V_{2} and W_{2}, that are directly associated with the various multipole moments. (Revised to clarify certain subtle points.) 
  The multidimensional cosmological model describing the evolution of $n$ Einstein spaces in the presence of multicomponent perfect fluid is considered. When certain restrictions on the parameters of the model are imposed, the dynamics of the model near the singularity is reduced to a billiard on the $(n-1)$-dimensional Lobachevsky space $H^{n-1}$. The geometrical criterion for the finiteness of the billiard volume and its compactness is suggested. This criterion reduces the problem to the problem of illumination of $(n-2)$-dimensional sphere $S^{n-2}$ by point-like sources. Some generalization of the considered scheme (including scalar field and quantum generalizations) are considered. 
  I present quantum wormhole solutions in the Kantowski--Sachs spacetime generated from coupling the gravitational field to the axion and EM fields. These solutions correspond to the classical ones found by Keay and Laflamme and by Cavagli\`a et al. 
  I present and discuss a class of solutions of the Wheeler-de Witt equation describing wormholes generated by coupling of gravity to the electromagnetic field for Kantowski-Sachs and Bianchi I spacetimes. Since the electric charge can be viewed as electric lines of force trapped in a finite region of spacetime, these solutions can be interpreted as the quantum corresponding of the Ein\-stein\--Ro\-sen\--Mis\-ner\--Whee\-ler electromagnetic geon. 
  Recently, the {\it spacelike} part of the $SU(2)$ Yang--Mills equations has been identified with geometrical objects of a three--dimensional space of constant Riemann--Cartan curvature. We give a concise derivation of this Ashtekar type (``inverse Kaluza--Klein") {\it mapping} by employing a $(3+1)$--decomposition of {\it Clifford algebra}--valued torsion and curvature two--forms. In the subcase of a mapping to purely axial 3D torsion, the corresponding Lagrangian consists of the translational and Lorentz {\it Chern--Simons term} plus cosmological term and is therefore of purely topological origin. 
  Gravitation theory meets spontaneous symmetry breaking when the structure group of the principal linear frame bundle $LX$ over a world manifold $X^4$ is reducible to the Lorentz group $SO(3,1)$. The physical underlying reason of this reduction is Dirac fermion matter possessing only exact Lorentz symmetries. The associated Higgs field is a tetrad gravitational field $h$ represented by a section of the quotient $\Si$ of $LX$ by $SO(3,1)$. The feature of gravity as a Higgs field issues from the fact that, in the presence of different tetrad fields, there are nonequivalent representations of cotangent vectors to $X^4$ by Dirac's matrices. It follows that fermion fields must be regarded only in a pair with a certain tetrad field. These pairs constitute the so-called fermion-gravitation complex and are represented by sections of the composite spinor bundle $S\to\Si\to X^4$ where values of tetrad gravitational fields play the role of coordinate parameters, besides familiar world coordinates. In Part I of the work [gr-qc:9405013], geometry of this composite spinor bundle has been investigated. This Part is devoted to dynamics of the fermion-gravitation complex. It is a constraint system to describe which we use the covariant multimomentum Hamiltonian formalism when canonical momenta correspond to derivatives of fields with respect to all world coordinates, not only time. On the constraint space, the canonical momenta of tetrad gravitational fields are equal to zero, otherwise in the presence of fermion fields. 
  Exact static, spherically symmetric solutions to the Einstein-Maxwell-scalar equations, with a dilatonic-type scalar-vector coupling, in $D$-dimensional gravity with a chain of $n$ Ricci-flat internal spaces are considered, with the Maxwell field potential having two nonzero components: the temporal, Coulomb-like one and the one pointing to one of the extra dimensions. The properties and special cases of the solutions are discussed, in particular, those when there are horizons in the space-time. Two types of horizons are distinguished: the conventional black-hole (BH) ones and those at which the physical section of the space-time changes its signature (the latter are called {\it T-horizons}). Two theorems are proved, one fixing the BH- and T-horizon existence conditions, the other claiming that the system under study cannot have a regular center. The stability of a selected family of solutions under spherically symmetric perturbations is studied. It is shown that only black-hole solutions are stable, while all others, in particular, those with T-horizons are unstable. 
  We propose functional approach to the stochastic inflationary universe dynamics. It is based on path integral representation of the solution to the differential equation for the scalar field probability distribution. In the saddle-point approximation scalar field probability distributions of various type are derived and the statistics of the inflationary-history-dependent functionals is developed. 
  We consider two programs for quantizing gravity in $1+1$ dimensions, which have appeared in the literature: one using a gauge--theoretic approach and the other following a more conventional ``geometric'' approach. We compare the wave functionals produced by the two different programs by finding matrix elements between the variables of the two theories. We find that the wave functionals are equivalent. 
  We show that the cosmological sphaleron of Einstein-Yang-Mills system can be produced from real tunneling geometries. The sphaleron will tend to roll down to the vacuum or pure gauge field configuration, when the universe evolves in the Lorentzian signature region with the sphaleron and the corresponding hypersurface being the initial data for the Yang-Mills field and the universe, respectively. However, we can also show that the sphaleron, although unstable, can be regarded as a pseudo-stable solution because its lifetime is even much greater than those of the universe. 
  In this paper wormhole effects on $SO(3)$ YM theory are examined. The wormhole wave functions for the scalar, the vector and the tensor expansion modes are computed assuming a small gauge coupling which leads to an effective decoupling of gravity and YM theory. These results are used to determine the wormhole vertices and the corresponding effective operators for the lowest expansion mode of each type. For the lowest scalar mode we find a renormalization of the gauge coupling from the two point function and the operators $\tr (F^3)$, $\tr (F^2\tilde{F})$ from the three point function. The two point function for the lowest vector mode contributes to the gauge coupling renormalization only whereas the lowest tensor mode can also generate higher derivative terms. 
  We apply a recent proposal for defining states and observables in quantum gravity to simple models. First, we consider a Klein-Gordon particle in an ex- ternal potential in Minkowski space and compare our proposal to the theory ob- tained by deparametrizing with respect to a time slicing prior to quantiza- tion. We show explicitly that the dynamics of the deparametrization approach depends on the time slicing. Our proposal yields a dynamics independent of the choice of time slicing at intermediate times but after the potential is turned off, the dynamics does not return to the free particle dynamics. Next we apply our proposal to the closed Robertson-Walker quantum cosmology with a massless scalar field with the size of the universe as our time variable, so the only dynamical variable is the scalar field. We show that the resulting theory has the semi-classical behavior up to the classical turning point from expansion to contraction, i.e., given a classical solution which expands for much longer than the Planck time, there is a quantum state whose dynamical evolution closely approximates this classical solution during the expansion. However, when the "time" gets larger than the classical maximum, the scalar field be- comes "frozen" at its value at the maximum expansion. We also obtain similar results in the Taub model. In an Appendix we derive the form of the Wheeler- DeWitt equation for the Bianchi models by performing a proper quantum reduc- tion of the momentum constraints; this equation differs from the usual one ob- tained by solving the momentum constraints classically, prior to quantization. 
  We study the time evolution of the reduced Wigner function for a class of quantum Brownian motion models. We derive two generalized uncertainty relations. The first consists of a sharp lower bound on the uncertainty function, $U = (\Delta p)^2 (\Delta q)^2 $, after evolution for time $t$ in the presence of an environment. The second, a stronger and simpler result, consists of a lower bound at time $t$ on a modified uncertainty function, essentially the area enclosed by the $1-\sigma$ contour of the Wigner function. In both cases the minimizing initial state is a non-minimal Gaussian pure state. These generalized uncertainty relations supply a measure of the comparative size of quantum and thermal fluctuations. We prove two simple inequalites, relating uncertainty to von Neumann entropy, and the von Neumann entropy to linear entropy. We also prove some results on the long-time limit of the Wigner function for arbitrary initial states. For the harmonic oscillator the Wigner function for all initial states becomes a Gaussian at large times (often, but not always, a thermal state). We derive the explicit forms of the long-time limit for the free particle (which does not in general go to a Gaussian), and also for more general potentials in the approximation of high temperature. 
  I review the decoherent (or consistent) histories approach to quantum mechanics, due to Griffiths, to Gell-Mann and Hartle, and to Omnes. This is an approach to standard quantum theory specifically designed to apply to genuinely closed systems, up to and including the entire universe. It does not depend on an assumed separation of classical and quantum domains, on notions of measurement, or on collapse of the wave function. Its primary aim is to find sets of histories for closed systems exhibiting negligble interference, and therefore, to which probabilities may be assigned. Such sets of histories are called consistent or decoherent, and may be manipulated according to the rules of ordinary (Boolean) logic. The approach provides a framework from which one may predict the emergence of an approximately classical domain for macroscopic systems, together with the conventional Copenhagen quantum mechanics for microscropic subsystems. In the special case in which the total closed system naturally separates into a distinguished subsystem coupled to an environment, the decoherent histories approach is closed related to the quantum state diffusion approach of Gisin and Percival. 
  The mechanism of the initial inflation of the universe is based on gravitationally coupled scalar fields $\phi$. Various scenarios are distinguished by the choice of an {\it effective self--interaction potential} $U(\phi)$ which simulates a {\it temporarily} non--vanishing {\em cosmological term}. Using the Hubble expansion parameter $H$ as a new ``time" coordinate, we can formally derive the {\it general} Robertson--Walker metric for a {\em spatially flat} cosmos. Our new method provides a classification of allowed inflationary potentials and is broad enough to embody all known {\it exact} solutions involving one scalar field as special cases. Moreover, we present new inflationary and deflationary exact solutions and can easily predict the influence of the form of $U(\phi)$ on density perturbations. 
  We consider the quantization of a general spatially homogeneous space-time belonging to an arbitrary but fixed Class A Bianchi type. Exploiting the information furnished by the quantum version of the momentum constraints, we use as variables the two simplest contractions of $C^{\alpha}_{\beta \gamma}$ and $\gamma_{\alpha \beta}$ as well as the determinant of $\gamma_{\alpha \beta}$ ; We thus arrive at an equation for the wave function in terms of these quantities. This fact enables us to treat in a uniform manner all Class A cosmologies. For these spacetimes the Langrangian used correctly reproduces Einstein's equations. We also discuss the imposition of simplifying ans\"{a}tzen at the quantum level in a way that respects the scalings of the quadratic Hamiltonian constraint. 
  The nature of the initial singularity in spatially compact plane symmetric scalar field cosmologies is investigated. It is shown that this singularity is crushing and velocity dominated and that the Kretschmann scalar diverges uniformly as it is approached. The last fact means in particular that a maximal globally hyperbolic spacetime in this class cannot be extended towards the past through a Cauchy horizon. A subclass of these spacetimes is identified for which the singularity is isotropic. 
  Various properties of the Misner-Sharp spherically symmetric gravitational energy E are established or reviewed. In the Newtonian limit of a perfect fluid, E yields the Newtonian mass to leading order and the Newtonian kinetic and potential energy to the next order. For test particles, the corresponding Hajicek energy is conserved and has the behaviour appropriate to energy in the Newtonian and special-relativistic limits. In the small-sphere limit, the leading term in E is the product of volume and the energy density of the matter. In vacuo, E reduces to the Schwarzschild energy. At null and spatial infinity, E reduces to the Bondi-Sachs and Arnowitt-Deser-Misner energies respectively. The conserved Kodama current has charge E. A sphere is trapped if E>r/2, marginal if E=r/2 and untrapped if E<r/2, where r is the areal radius. A central singularity is spatial and trapped if E>0, and temporal and untrapped if E<0. On an untrapped sphere, E is non-decreasing in any outgoing spatial or null direction, assuming the dominant energy condition. It follows that E>=0 on an untrapped spatial hypersurface with regular centre, and E>=r_0/2 on an untrapped spatial hypersurface bounded at the inward end by a marginal sphere of radius r_0. All these inequalities extend to the asymptotic energies, recovering the Bondi-Sachs energy loss and the positivity of the asymptotic energies, as well as proving the conjectured Penrose inequality for black or white holes. Implications for the cosmic censorship hypothesis and for general definitions of gravitational energy are discussed. 
  We consider the modification of the formulas for black hole radiation, due to the self-gravitation of the radiation. This is done by truncating the coupled particle-hole system to a small set of modes, that are plausibly the most significant ones, and quantizing the reduced system. In this way we find that the particles no longer move along geodesics, nor is the action along the rays zero for a massless particle. The radiation is no longer thermal, but is corrected in a definite way that we calculate. Our methods can be extended in a straightforward manner to discuss correlations in the radiation, or between incoming particles and the radiation. 
  The problem of construction of a general ihomogeneous solution of $D$-dimensional Einstein equations in the vicinity of a cosmological singularity is considered. It is shown that near the singularity a local behavior of metric functions is described by a billiard on a space of a constant negative curvature. The billiard is shown to have a finite volume and consequently to be a mixing one. Dynamics of inhomogeneities of metric is studied and it is shown that its statistical properties admit a complete description. An invariant measure describing statistics of inhomogeneities is obtained and a role of a minimally-coupled scalar field in dynamics of the inhomogeneities is also considered. 
  Quantum physics at scales large compared to the Planck scale is described in the framework of classical space-time geometries. A criterion for selecting these backgrounds out of quantized gravity is proposed. It leads to an instability of the black-hole geometry, as experienced by motion across the horizon, against emission of Hawking quanta. A phenomenological treatment of the evaporation process perceived by external observers who do not cross the event horizon is presented. Evaporation occurs within a topologically trivial ``achronon" geometrical background devoid of horizons and singularities describing a collapse frozen up to decay time scales. It is ignited as in the conventional theory from pair creation out of the vacuum of the collapsing star of mass $M$, but after a time of order $M\ln M$ the source of thermal radiation shifts gradually to the star itself. This allows for a unitary evolution except possibly for exponentially small background transition amplitudes. The emerging picture is compared with approaches of t'Hooft and Susskind and the problem of its overall quantum consistency is evoked. (Figures available upon request) 
  A finite-length magnetic vortex line solution is derived within the context of (4-dim) dilaton gravity. We approach the Bonnor metric at the Einstein-Maxwell limit, and encounter the "flux tube as (Euclidean) Kerr horizon" at the Kaluza-Klein level. Exclusively for string theory, the magnetic flux tube world-sheet exhibits a 2-dim black & white dihole structure. (The figure has been cut off, and is now available upon request from davidson@bguvms.bgu.ac.il) 
  We identify an explicit set of complete and independent Wilson loop invariants for 2+1 gravity on a three-manifold $M=\R\times\Sigma^g$, with $\Sigma^g$ a compact oriented Riemann surface of arbitrary genus $g$. In the derivation we make use of a global cross section of the $PSU(1,1)$-principal bundle over Teichm\"uller space given in terms of Fenchel-Nielsen coordinates. 
  It was noted recently that the ADM-diffeomorphism-constraint does not generate all observed symmetries for several Bianchi-models. We will suggest not to use the ADM-constraint restricted to homogeneous variables, but some equivalent which is derived from a restricted action principle. This will generate all homogeneity preserving diffeomorphisms, which will be shown to be automorphism generating vector fields, in class A and class B models. Following Dirac's constraint formalism one will naturally be restricted to the unimodular part of the automorphism group. 
  The analog of the Schwarzschild metric is explored in the context of Non-Singular Gravity. Analytic results are developed describing redshifts, curvatures and topological features of the spacetime. All curvatures and redshifts are finite so there are no Black Holes, no singularities and no Hawking radiation. 
  Inflationary cosmologies, regarded as dynamical systems, have rather simple asymptotic behavior, insofar as the cosmic baldness principle holds. Nevertheless, in the early stages of an inflationary process, the dynamical behavior may be very complex. In this paper, we show how even a simple inflationary scenario, based on Linde's ``chaotic inflation'' proposal, manifests nontrivial dynamical effects such as the breakup of invariant tori, formation of cantori and Arnol'd's diffusion. The relevance of such effects is highlighted by the fact that even the occurrence or not of inflation in a given Universe is dependent upon them. 
  We explore a simple toy model of interacting universes to establish that a small baby universe could become large ($\gg$ Planck length) if a third quantization mechanism is taken into account. 
  For a class of scalar fields including the massless Klein-Gordon field the general relativistic hyperboloidal initial value problems are equivalent in a certain sense. By using this equivalence and conformal techniques it is proven that the hyperboloidal initial value problem for those scalar fields has an unique solution which is weakly asymptotically flat. For data sufficiently close to data for flat spacetime there exist a smooth future null infinity and a regular future timelike infinity. 
  We investigate the Ponzano-Regge and Turaev-Viro topological field theories using spin networks and their $q$-deformed analogues. I propose a new description of the state space for the Turaev-Viro theory in terms of skein space, to which $q$-spin networks belong, and give a similar description of the Ponzano-Regge state space using spin networks.   I give a definition of the inner product on the skein space and show that this corresponds to the topological inner product, defined as the manifold invariant for the union of two 3-manifolds.   Finally, we look at the relation with the loop representation of quantum general relativity, due to Rovelli and Smolin, and suggest that the above inner product may define an inner product on the loop state space. 
  The generalized projection-tensor geometry introduced in an earlier paper is extended. A compact notation for families of projected objects is introduced and used to summarize the results of the previous paper and obtain fully projected decompositions of Lie derivatives of the projection tensor field, the metric and the projected parts of the metric. These results are applied to the analysis of spacetimes with isometries. The familiar cases of spacetimes with isotropic group orbits --- cosmological models and spherical symmetry --- are discussed as illustrations of the results. 
  We consider the most general dilaton gravity theory in 1+1 dimensions. By suitably parametrizing the metric and scalar field we find a simple expression that relates the energy of a generic solution to the magnitude of the corresponding Killing vector. In theories that admit black hole solutions, this relationship leads directly to an expression for the entropy $S=2\pi \tau_0/G$, where $\tau_0$ is the value of the scalar field (in this parametrization) at the event horizon. This result agrees with the one obtained using the more general method of Wald. Finally, we point out an intriguing connection between the black hole entropy and the imaginary part of the ``phase" of the exact Dirac quantum wave functionals for the theory. 
  Using the influence functional formalism, the problem of an accelerating detector in the presence of a scalar field in its ground state is considered in Minkowski space. As is known since the work of Unruh, to a quantum mechanical detector following a definite, classical acceleration, the field appears to be thermally excited. We relax the requirement of perfect classicality for the trajectory and substitute it with one of {\it derived} classicality through the criteria of decoherence. The ensuing fluctuations in temperature are then related with the time and the amplitude of excitation in the detector's internal degree of freedom. 
  Decoherence in quantum cosmology is shown to occur naturally in the presence of induced geometric gauge interactions associated with particle production.A new 'gauge '-variant form of the semiclassical Einstein equations is also presented which makes the non-gravitating character of the vacuum polarisation energy explicit. 
  In this paper we show the existence of a large class of spherically symmetric data $d$ (on a spacelike hypersurface $S$), from which a perfect fluid spacetime (surrounded by vacuum) develops. This spacetime contains an event horizon (with trapped surfaces behind it). The data $d$ are regular and {\it innociuous}, i.e. the data--surface $S$ does not contain any point of the horizon or of the trapped surface area.   We give auxiliary data on an auxiliary hypersurface $H$ and also on the star boundary; then we solve Einstein's equations for perfect fluid in the future and past of $H$. Our solution induces the above mentioned data $d$ on some chosen spacelike hypersurface $S$ in the past of $H$. By construction $H$ turns out to be the matter part of the horizon, once we attach a vacuum to our matter spacetime. Obviously, from these data $d$ on $S$ it develops (into the future) the event horizon $H$.   We solve the constraint equations for the auxiliary data posed on the null--surface $H$. This reduces the choice of these data to the choice of the density $\rho$ and $R:=[\text{curvature}]^{-1/2}$. Our data fulfil positivity of $\rho$, $2m/R=1$ (at the star boundary) and other properties. This is archieved by an algorithm, which for given $\rho $ yields $R$ (from an input parameter function $h \in C^1( \left[0,1\right],\left]0,-\infty \right[)$). 
  We consider an atom in interaction with a massless scalar quantum field. We discuss the structure of the rate of variation of the atomic energy for an arbitrary stationary motion of the atom through the quantum vacuum. Our main intention is to identify and to analyze quantitatively the distinct contributions of vacuum fluctuations and radiation reaction to the spontaneous excitation of a uniformly accelerated atom in its ground state. This gives an understanding of the role of the different physical processes underlying the Unruh effect. The atom's evolution into equilibrium and the Einstein coefficients for spontaneous excitation and spontaneous emission are calculated. 
  In this work we study a {\it gedanken} experiment constructed in order to test the cosmic censorship hypothesis and the second law of black hole thermo-dynamics. Matter with a negative gravitating energy is imagined added to a near extremal $U(1)$-charged static black hole in Einstein-Maxwell theory. The dynamics of a similar process is studied and the thermo-dynamical properties of the resulting black hole structure is discussed. A new mechanism which stabilizes black hole event horizons is shown to operate in such processes. 
  This paper studies the time-symmetry problem in quantum gravity. The issue depends critically on the choice of the quantum state and has been considered in this paper by restricting to the case of quantum wormholes. It is seen that pure states represented by a wave functional are time symmetric. However, a maximal analytic extension of the wormhole manifold is found that corresponds to a mixed state describable by a nondegenerate density-matrix functional that involves an extra quantum uncertainty for the three-metric, and is free from the divergences encountered so far in statistical states formulated in quantum gravity. It is then argued that, relative to one asymptotic region, the statistical quantum state of single Euclidean wormholes in semiclassical approximation is time-asymmetric and gives rise to a {\it topological} arrow of time which will reflect in the set of all quantum fields at low energies of the asymptotic flat region (To appear in Int. J. Mod. Phys. D) 
  A gravitational scenario is proposed where the euclidean action is invariant under the isotropic and homogeneous version of the euclidean {\it U(1)} group of local transformations of the scale factor and scalar matter field, interpreting the the trace of the second fundamental form as the gauge field. The model allows spontaneous breakdown of some involved symmetries, including possibly diffeomorphism invariance, and leads to the formation of topologiocal deffects. In particular, we consider here the case of the wormhole-induced formation of cosmic strings. (To appear in the Proceedings of the International Workshop on "Quantum Systems: New Trends and Methods", World Scientific) 
  To what extent can our limited set of observations be used to pin down the specifics of a ``Theory of Everything''? In the limit where the links are arbitrarily tenuous, a ``Theory of Everything'' might become a ``Theory of Anything''. A clear understanding of what we can and can not expect to learn about the universe is particularly important in the field of particle cosmology. The aim of this article is to draw attention to some key issues which arise in this context, in the hopes of fostering further discussion. In particular, I explore the idea that a variety of different inflaton potentials may contribute to worlds ``like ours''. A careful examination of the conditional probability questions we can ask might give a physical measure of what is ``natural'' for an inflation potential which is quite different from those previously used. 
  We analyse a simple model with just a $\Lambda$ term present. Differing results are obtained depending on the boundary conditions applied. HH boundary conditions give the factor exp(1/Lambda) but in agreement with Rubakov et al. is badly behaved for negative lambda. Tunneling boundary conditions suggests a large initial lambda. If only a Lorentzian region is considered all boundary conditions suggest a large value of Lambda. This differs from the result of Strominger for such models. 
  We consider an alternative higher order gravity theory which is non-analytic in the Ricci curvature. It has power-law inflationary behaviour. This is further evidence that inflation is a general property of higher derivative gravity theories. 
  We consider the claim of Hawking and Page that the canonical measure applied to a FRW universe with a massive scalar field can solve the flatness problem regardless of inflation. By considering a general potential we are able to understand how the ambiguity for $\Omega$ found by Page in the $R^2$ theory is present when the potential is bounded from above. We suggests reasons why such potentials are realistic. The ambiguity for the possibility of inflation present in the Gibbons- Hawking- Stewart measure can be resolved by an input from Quantum cosmology. We contrast the resulting measure obtained in this way with those obtained in the more usual approaches to quantum cosmology. The measure agrees with that found from a "classical" signature change. 
  The notions of time in the theories of Newton and Einstein are reviewed so that certain of their assumptions are clarified. These assumptions will be seen as the causes of the incompatibility between the two different ways of understanding time, and seen to be philosophical hypotheses, rather than purely scientific ones. The conflict between quantum mechanics and (general) relativity is shown to be a consequence of retaining the Newtonian conception of time in the context of quantum mechanics. As a remedy for this conflict, an alternative definition of time -- earlier presented in Kitada 1994a and 1994b -- is reviewed with less mathematics and more emphasis on its philosophical aspects. Based on this revised understanding of time it is shown that quantum mechanics and general relativity are reconciled while preserving the current mathematical formulations of both theories. 
  An overview about recent progress in the calculation of the heat kernel and the one-loop effective action in quantum gravity and gauge theories is given. We analyse the general structure of the standard Schwinger-De Witt asymptotic expansion and discuss the applicability of that to the case of strongly curved manifolds and strong background fields. We argue that the low-energy limit in gauge theories and quantum gravity, when formulated in a covariant way, should be related to background fields with covariantly constant curvature, gauge field strength and potential term. It is shown that the condition of the covariant constancy of the background curvatures brings into existence some Lie algebra. The heat kernel operator for the Laplace operator is presented then as an average over the corresponding Lie group with some nontrivial Gaussian measure. Using this representation the heat kernel diagonal is obtained. The result is expressed purely in terms of curvature invariants and is explicitly covariant. Related topics concerning the structure of symmetric spaces and the calculation of the effective action are discussed. 
  This discussion examines recent developments in the theory of a Weyl-like, Cartan geometry with natural Schr\"odinger field behavior proposed previously. In that model, very nearly exactly a coupled Einstein-Maxwell- Schr\"odinger, classical field theory emerges from a gauge invariant, purely geometric action based solely on variations of the electromagnetic potentials and the metric. In spite of this, only slight differences appear between the resulting Schr\"odinger part, and the conventional theory of the Schr\"odinger field. Close examination of the differences reveals that most are general relativistic effects which are unobservable in flat spacetime, and which are estimated to interact significantly only via their gravitational fields, or on scales comparable with neutrino interaction cross sections. The only remaining difference is that the wavefunction obeying the conjugate wave equation is not always restricted to be exactly the complex conjugate of the primary wavefunction. Generalizations of the model lead naturally to spinlike phenomena, a possible new mechanism for a theory of rest mass, and spinor connections containing the form of an SU(2) potential. 
  It is tempting to raise the issue of (metric) chaos in general relativity since the Einstein equations are a set of highly nonlinear equations which may exhibit dynamically very complicated solutions for the space-time metric. However, in general relativity it is not easy to construct indicators of chaos which are gauge-invariant. Therefore it is reasonable to start by investigating - at first - the possibility of a gauge-invariant description of local instability. In this paper we examine an approach which aims at describing the dynamics in purely geometrical terms. The dynamics is formulated as a geodesic flow through the Maupertuis principle and a criterion for local instability of the trajectories may be set up in terms of curvature invariants (e.g. the Ricci scalar) of the manifold on which geodesic flow is generated. We discuss the relation of such a criterion for local instability (negativity of the Ricci scalar) to a more standard criterion for local instability and we emphasize that no inferences can be made about global chaotic behavior from such local criteria. 
  In many interesting models, including superstring theories, a negative vacuum energy is predicted. Although this effect is usually regarded as undesirable from a cosmological point of view, we show that this can be the basis for a new approach to the cosmology of the early Universe. In the framework of quantum cosmology (in higher dimensions) when we consider a negative cosmological constant and matter that could be dust or, alternatively, coherent excitations of a scalar field, the role of cosmic time can be understood. Then we can predict the existence of a ``quantum inflationary phase'' for some dimensions and a simultaneous ``quantum deflationary phase'' for the remaining dimensions. We discuss how it may be possible to exit from this inflation-compactification era to a phase with zero cosmological constant which allows a classical description at late times. 
  The actions of the ``$R=T$'' and string-inspired theories of gravity in (1+1) dimensions are generalized into one single action which is characterized by two functions. We discuss differing interpretations of the matter stress-energy tensor, and show how two such different interpretations can yield two different sets of field equations from this action. The weak-field approximation, post-Newtonian expansion, hydrostatic equilibrium state of star and two-dimensional cosmology are studied separately by using the two sets of field equations. Some properties in the ``$R=T$'' and string-inspired theories are shown to be generic in the theory induced by the generalized action. 
  The Chern-Simons functional $S_{\rm CS}$ is an exact solution to the Ashtekar-Hamilton-Jacobi equation of general relativity with a nonzero cosmological constant. In this paper we consider $S_{\rm CS}$ in Bianchi type IX cosmology with $S^3$ spatial surfaces. We show that among the classical solutions generated by~$S_{\rm CS}$, there is a two-parameter family of Euclidean spacetimes that have a regular NUT-type closing. When two of the three scale factors are equal, these spacetimes reduce to a one-parameter family within the Euclidean Taub-NUT-de~Sitter metrics. For a nonzero cosmological constant, $\exp(iS_{\rm CS})$ therefore provides a semiclassical estimate to the Bianchi~IX no-boundary wave function in Ashtekar's variables. 
  This comment contains a suggestion for a slight modification of Israel's covariant formulation of junction conditions between two spacetimes, placing both sides on equal footing with normals having uniquely defined orientations. The signs of mass energy densities in thin shells at the junction depend not only on the orientations of the normals and it is useful therefore to discuss the sign separately. Calculations gain in clarity by not choosing the orientations in advance. Simple examples illustrate our point and complete previous classifications of spherical thin shells in spherically symmetric spacetimes relevant to cosmology. 
  A nonstatic and circularly symmetric exact solution of the Einstein equations (with a cosmological constant $\Lambda$ and null fluid) in $2+1$ dimensions is given. This is a nonstatic generalization of the uncharged spinless BTZ metric. For $\Lambda = 0 $, the spacetime is though not flat, the Kretschmann invariant vanishes. The energy, momentum, and power output for this metric are obtained. Further a static and circularly symmetric exact solution of the Einstein-massless scalar equations is given, which has a curvature singularity at $r =0$ and the scalar field diverges at $r=0$ as well as at infinity . 
  Many simulations of gravitational collapse to black holes become inaccurate before the total emitted gravitational radiation can be determined. The main difficulty is that a significant component of the radiation is still in the near-zone, strong field region at the time the simulation breaks down. We show how to calculate the emitted waveform by matching the numerical simulation to a perturbation solution when the final state of the system approaches a Schwarzschild black hole. We apply the technique to two scenarios: the head-on collision of two black holes, and the collapse of a disk to a black hole. This is the first reasonably accurate calculation of the radiation generated from colliding black holes that form from matter collapse. 
  The response of the Unruh-DeWitt type monopole detectors which were coupled to the quantum field only for a finite proper time interval is studied for inertial and accelerated trajectories, in the Minkowski vacuum in (3+1) dimensions. Such a detector will respond even while on an inertial trajctory due to the transient effects. Further the response will also depend on the manner in which the detector is switched on and off. We consider the response in the case of smooth as well as abrupt switching of the detector. The former case is achieved with the aid of smooth window functions whose width, $T$, determines the effective time scale for which the detector is coupled to the field. We obtain a general formula for the response of the detector when a window function is specified, and work out the response in detail for the case of gaussian and exponential window functions. A detailed discussion of both $T \rightarrow 0$ and $T \rightarrow \infty$ limits are given and several subtlities in the limiting procedure are clarified. The analysis is extended for detector responses in Schwarzschild and de-Sitter spacetimes in (1+1) dimensions. 
  We consider the general orthogonal metric separable in space and time variables in comoving coordinates. We then characterise perfect fluid models admitted by such a metric. It turns out that the homogeneous models can only be either FLRW or Bianchi I while the inhomogeneous ones can only admit $G_2 $ (two mutually as well as hypersurface orthogonal spacelike Killing vectors) isometry. The latter can possess singularities of various kinds or none. The non-singular family is however unique and cylindrically symmetric. 
  This is a short note on the black hole remote-sensing problem, i.e., finding out `surface' temperature distributions of various types of small (micron-sized) black holes from the spectral measurements of their Hawking grey pulses. Chen's modified Moebius inverse transform is illustrated in this context 
  The entropy of a black hole can be different from a quarter of the area even at the semiclassical level. 
  We study the head-on collision of two equal mass, nonrotating black holes. Various initial configurations are investigated, including holes which are initially surrounded by a common apparent horizon to holes that are separated by about $20M$, where $M$ is the mass of a single black hole. We have extracted both $\ell = 2$ and $\ell=4$ gravitational waveforms resulting from the collision. The normal modes of the final black hole dominate the spectrum in all cases studied. The total energy radiated is computed using several independent methods, and is typically less than $0.002 M$. We also discuss an analytic approach to estimate the total gravitational radiation emitted in the collision by generalizing point particle dynamics to account for the finite size and internal dynamics of the two black holes. The effects of the tidal deformations of the horizons are analysed using the membrane paradigm of black holes. We find excellent agreement between the numerical results and the analytic estimates. 
  The head-on collision of two nonrotating axisymmetric equal mass black holes is treated numerically. We take as initial data the single parameter family of time-symmetric solutions discovered by Misner which consists of two Einstein-Rosen bridges that can be placed arbitrarily distant from one another. A number of problems associated with previous attempts to evolve these data sets have been overcome. In this article, we discuss our choices for coordinate systems, gauges, and the numerical algorithms that we have developed to evolve this system. 
  The quasinormal modes (QNM's) of gravitational systems modeled by the Klein-Gordon equation with effective potentials are studied in analogy to the QNM's of optical cavities. Conditions are given for the QNM's to form a complete set, i.e., for the Green's function to be expressible as a sum over QNM's, answering a conjecture by Price and Husain [Phys. Rev. Lett. {\bf 68}, 1973 (1992)]. In the cases where the QNM sum is divergent, procedures for regularization are given. The crucial condition for completeness is the existence of spatial discontinuities in the system, e.g., the discontinuity at the stellar surface in the model of Price and Husain. 
  We discuss how to fix the gauge in the canonical treatment of Lagrangians, with finite number of degrees of freedom, endowed with time reparametrization invariance. The motion can then be described by an effective Hamiltonian acting on the gauge shell canonical space. The system is then suited for quantization. We apply this treatment to the case of a Robertson--Walker metric interacting with zero modes of bosonic fields and write a \S equation for the on--shell wave function (Presented at the International Workshop ``Birth of the Universe and Fundamental Physics'', Rome, May 18-21, 1994). 
  The coupling of the metric to an incoherent dust introduces into spacetime a privileged dynamical reference frame and time foliation. The comoving coordinates of the dust particles and the proper time along the dust worldlines become canonical coordinates in the phase space of the system. The Hamiltonian constraint can be resolved with respect to the momentum that is canonically conjugate to the dust time. Imposition of the resolved constraint as an operator restriction on the quantum states yields a functional Schr\"{o}dinger equation. The ensuing Hamiltonian density has an extraordinary feature: it depends only on the geometric variables, not on the dust coordinates or time. This has three important consequences. First, the functional Schr\"{o}dinger equation can be solved by separating the dust time from the geometric variables. Second, the Hamiltonian densities strongly commute and therefore can be simultaneously defined by spectral analysis. Third, the standard constraint system of vacuum gravity is cast into a form in which it generates a true Lie algebra. The particles of dust introduce into space a privileged system of coordinates that allows the supermomentum constraint to be solved explicitly. The Schr\"{o}dinger equation yields a conserved inner product that can be written in terms of either the instantaneous state functionals or the solutions of constraints. Examples of gravitational observables are given, though neither the intrinsic metric nor the extrinsic curvature are observables. Disregarding factor--ordering difficulties, the introduction of dust provides a satisfactory phenomenological approach to the problem of time in canonical quantum gravity. 
  Solutions to flat space Friedmann-Robertson-Walker cosmologies in Brans-Dicke theory with a cosmological constant are investigated. The matter is modelled as a $\gamma$-law perfect fluid. The field equations are reduced from fourth order to second order through a change of variables, and the resulting two-dimensional system is analyzed using dynamical system theory. When the Brans-Dicke coupling constant is positive $(\omega > 0)$, all initially expanding models approach exponential expansion at late times, regardless of the type of matter present. If $\omega < 0$, then a wide variety of qualitatively distinct models are present, including nonsingular ``bounce'' universes, ``vacillating'' universes and, in the special case of $\omega = -1$, models which approach stable Minkowski spacetime with an exponentially increasing scalar field at late times. Since power-law solutions do not exist, none of the models appear to offer any advantage over the standard deSitter solution of general relativity in achieving a graceful exit from inflation. 
  We examine consequences of the density matrix approach to quantum theory in the context of a model spacetime containing closed timelike curves and find that in general, an initially pure state will evolve in a nonlinear way to a mixed quantum state. CPT invariance and the implications of this nonlinearity for the statistical interpretation of quantum theory are discussed. 
  Table of contents   Editorial.   Gravity News:   Report on the APS topical group in gravitation, Beverly Berger.   Research briefs:   Gravitational microlensing and the search for dark matter, Bohdan Paczynski.   Laboratory gravity: the G mystery, Riley Newman.   LIGO project update, Stan Whitcomb.   Conference Reports   PASCOS '94, Peter Saulson.   The Vienna Meeting, P. Aichelburg, R. Beig.   The Pitt binary black hole grand challenge meeting, Jeff Winicour.   International symposium on experimental gravitation at Pakistan,Munawar Karim   10th Pacific coast gravity meeting, Jim Isenberg. 
  In usual quantum theory, the information available about a quantum system is defined in terms of the density matrix describing it on a spacelike surface. This definition must be generalized for extensions of quantum theory which do not have a notion of state on a spacelike surface. It must be generalized for the generalized quantum theories appropriate when spacetime geometry fluctuates quantum mechanically or when geometry is fixed but not foliable by spacelike surfaces. This paper introduces a four-dimensional notion of the information available about a quantum system's boundary conditions in the various sets of decohering histories it may display. The idea of spacetime information is applied in several contexts: When spacetime geometry is fixed the information available through alternatives restricted to a spacetime region is defined. The information available through histories of alternatives of general operators is compared to that obtained from the more limited coarse- grainings of sum-over-histories quantum mechanics. The definition of information is considered in generalized quantum theories. We consider as specific examples time-neutral quantum mechanics with initial and final conditions, quantum theories with non-unitary evolution, and the generalized quantum frameworks appropriate for quantum spacetime. In such theories complete information about a quantum system is not necessarily available on any spacelike surface but must be searched for throughout spacetime. The information loss commonly associated with the ``evolution of pure states into mixed states'' in black hole evaporation is thus not in conflict with the principles of generalized quantum mechanics. 
  We investigate topology-changing processes in 4-dimensional quantum gravity with a negative cosmological constant. By playing the ``gluing-polytope game" in hyperbolic geometry, we explicitly construct an instanton-like solution without singularity. Because of cusps, this solution is non-compact but has a finite volume. Then we evaluate a topology change amplitude in the WKB approximation in terms of the volume of this solution. 
  In the pure essay style (no mathematical formulas), I present a number of speculative reflections and suggestions on possible applications of mesoscopic methods and of quantum mechanical concepts to as such a complex system as the human brain. As an initial guide for this essay I used ``The Emperor's New Mind" of Roger Penrose 
  The naive calculation of black hole evaporation makes the thermal emission depend on the arbitrary high frequency behaviour of the theory where the theory is certainly wrong. Using the sonic analog to black holes-- dumb holes-- I show numerically that a change in the dispersion relation at high frequencies does not seem to alter the evaporation process, lending weight to the reality of the black hole evaporation process. I also suggest a reason for the insensitivity of the process to high frequency regime. 
  The existence and nature of singularities in locally spatially homogeneous solutions of the Einstein equations coupled to various phenomenological matter models is investigated. It is shown that, under certain reasonable assumptions on the matter, there are no singularities in an expanding phase of the evolution and that unless the spacetime is empty a contracting phase always ends in a singularity where at least one scalar invariant of the curvature diverges uniformly. The class of matter models treated includes perfect fluids, mixtures of non-interacting perfect fluids and collisionless matter. 
  A theory which claims to describe all the universe is advanced. It unifies general relativity, quantum field theory, and indeterministic conception. Basic entities are: classical metric tensor $g$, cosmic reference frame (including cosmic time $t$), operator $T$ of energy-momentum tensor, Hamiltonian $H_t$, and state vector $\Psi$. Dynamical equations are: the Einstein equation $G[g]=(\Psi,T\Psi)$ ($G$ is the Einstein tensor), the Heisenberg equation $dT/dt=i[H_t,T]$, and the condition $H_t\Psi_t=\epsilon_t \Psi_t$ arising from the cosmic energy determinacy principle advanced in the theory. The last equation describes quantum jump dynamics. Quantum jumps lead to the instantaneous transferring of action and information, which, however, neither violates the causality principle, nor contradicts quantum field theory and general relativity. The cosmic energy determinacy principle implies the eternal universe, i.e., the cyclic one without beginning and ending, the minimal energy in every cycle being finite. 
  It is proven that any spherically symmetric spacetime that possesses a compact Cauchy surface $\Sigma$ and that satisfies the dominant-energy and non-negative-pressures conditions must have a finite lifetime in the sense that all timelike curves in such a spacetime must have a length no greater than $10 \max_\Sigma(2m)$, where $m$ is the mass associated with the spheres of symmetry. This result gives a complete resolution, in the spherically symmetric case, of one version of the closed-universe recollapse conjecture (though it is likely that a slightly better bound can be established). This bound has the desirable properties of being computable from the (spherically symmetric) initial data for the spacetime and having a very simple form. In fact, its form is the same as was established, using a different method, for the spherically symmetric massless scalar field spacetimes, thereby proving a conjecture offered in that work. Prospects for generalizing these results beyond the spherically symmetric case are discussed. 
  For the simplest minisuperspace model based on a homogeneous, isotropic metric and a minimally coupled scalar field we derive analytic expressions for the caustic which separates Euklidean and Minkowskian region and its breakdown value $\p_*$. This value represents the prediction of the no-boundary wave function for the scalar field at the beginning of inflation. We use our results to search for inflationary models which can render the no-boundary wave function consistent with the requirement of a sufficiently long inflationary period. 
  We clarify the relation between gravitational entropy and the area of horizons. We first show that the entropy of an extreme Reissner-Nordstr\"om black hole is $zero$, despite the fact that its horizon has nonzero area. Next, we consider the pair creation of extremal and nonextremal black holes. It is shown that the action which governs the rate of this pair creation is directly related to the area of the acceleration horizon and (in the nonextremal case) the area of the black hole event horizon. This provides a simple explanation of the result that the rate of pair creation of non-extreme black holes is enhanced by precisely the black hole entropy. Finally, we discuss black hole $annihilation$, and argue that Planck scale remnants are not sufficient to preserve unitarity in quantum gravity. 
  We discuss some subtleties which arise in the semiclassical approximation to quantum gravity. We show that integrability conditions prevent the existence of Tomonaga-Schwinger time functions on the space of three-metrics but admit them on superspace. The concept of semiclassical time is carefully examined. We point out that central charges in the matter sector spoil the consistency of the semiclassical approximation unless the full quantum theory of gravity and matter is anomaly-free. We finally discuss consequences of these considerations for quantum field theory in flat spacetime, but with arbitrary foliations. 
  I review various proposals for the nature of black hole entropy and for the mechanism behind the operation of the generalized second law. I stress the merits of entanglement entropy {\tenit qua\/} black hole entropy, and point out that, from an operational viewpoint, entanglement entropy is perfectly finite. Problems with this identification such as the multispecies problem and the trivialization of the information puzzle are mentioned. This last leads me to associate black hole entropy rather with the multiplicity of density operators which describe a black hole according to exterior observers. I relate this identification to Sorkin's proof of the generalized second law. I discuss in some depth Frolov and Page's proof of the same law, finding it relevant only for scattering of microsystems by a black hole. Assuming that the law is generally valid I make evident the existence of the universal bound on entropy regardless of issues of acceleration buoyancy, and discuss the question of why macroscopic objects cannot emerge in the Hawking radiance. 
  It is argued heuristically -- using an ${\bf S}^3 \times {\bf S}^6$ minisuperspace model -- that there might be a fundamental quantum gravity effect stabilizing internal spaces with non-vanishing Ricci curvature. 
  The usual composition rule of independent systems, as applied to decoherent histories or to linearly positive histories, requires (at least) medium decoherence and, consequently, a second constraint for the linearly positive histories of Goldstein and Page. Other plausible classical features, valid for medium decoherence, seem anyhow to be missed by linear positive histories. 
  The objectivity of black hole entropy is discussed in the particular case of a Schwarzchild black hole. Using Jaynes' maximum entropy formalism and Euclidean path integral evaluation of partition function, it is argued that in the semiclassical limit when the fluctutation of metric is neglected, the black hole entropy of a Schwarzchild black hole is equal to the maximal information entropy of an observer whose sole knowledge of the black hole is its mass. Black hole entropy becomes a measure of number of its internal mass eigenstates in accordance with the Boltzmann principle only in the limit of negligible relative mass fluctutation. {}From the information theoretic perspective, the example of a Schwarzchild black hole seems to suggest that black hole entropy is no different from ordinary thermodynamic entropy. It is a property of the experimental data of a black hole, rather than being an intrinsic physical property of a black hole itself independent of any observer. However, it is still weakly objective in the sense that different observers given the same set of data of a black hole will measure the same maximal information entropy. 
  In 1+1 dimensions, the Wheeler-DeWitt equation cannot be imposed owing to an anomaly in its commutator with the diffeomorphism constraint. A similar obstruction prevents also a semiclassical definition of a WKB local time. (Talk given at the 7th Marcel Grossman Meeting on General Relativity, Stanford, July 28, 1994.) 
  We consider self-interacting scalar fields coupled to gravity. Two classes of exact solutions to Einstein's equations are obtained: the first class corresponds to the minimal coupling, the second one to the conformal coupling. One of the solutions is shown to describe a formation of a black hole in a cosmological setting. Some properties of this solution are described. There are two kinds of event horizons: a black hole horizon and cosmological horizons. The cosmological horizons are not smooth. There is a mild curvature singularity, which affects extended bodies but allows geodesics to be extended. It is also shown that there is a critical value for a parameter on which the solution depends. Above the critical point, the black hole singularity is hidden within a global black hole event horizon. Below the critical point, the singularity appears to be naked. The relevance to cosmic censorship is discussed. 
  Most spherical thin shells, enclosing black body radiation satisfy the dominant energy condition if they have at least $\simeq 30\%$ of the total mass-energy. Containers with less mass energy, able to sustain high pressures, contain mostly unstable radiation. If they have negligible mass energy they are unable to sustain the pressures and the radiation is unstable to gravitational collapse. Containers with black holes and radiation in thermal equilibrium, considered in the literature, are often unrealistic. 
  This paper investigates the relationship between the quasilocal energy of Brown and York and certain spinorial expressions for gravitational energy constructed from the Witten-Nester integral. A key feature of the Brown-York method for defining quasilocal energy is that it allows for the freedom to assign the reference point of the energy. When possible, it is perhaps most natural to reference the energy against flat space, i.e. assign flat-space the zero value of energy. It is demonstrated that the Witten-Nester integral when evaluated on solution spinors to the Sen-Witten equation (obeying appropriate boundary conditions) is essentially the Brown-York quasilocal energy with a reference point determined by the Sen-Witten spinors. For the case of round spheres in the Schwarzschild geometry, these spinors determine the flat-space reference point. A similar viewpoint is proposed for the Schwarzschild-case quasilocal energy of Dougan and Mason. 
  It is argued that a characteristic length may be associated with the entropic state of a spherically symmetric black hole in the cosmological context. This length is much smaller than the Schwarzschild-radius of a black hole and may act as a regulator of arbitrarily high frequencies apparently entering the usual derivation of Hawking's radiation. 
  The `runaway solutions' of the Lorentz-Dirac equation of a charged particle interacting with its own field in classical electrodynamics are well-known. This type of self accelerated phenomena also exists in the solutions of the Einstein-Maxwell equations in general relativity. In particular, runaway solutions occur in a class of simple models known as the `Asymptotically Flat Robinson-Trautman Einstein-Maxwell' (AFRTEM) spacetimes. Consequently these spacetimes cannot evolve to their unique regular steady state, viz. a charged non-rotating black hole. This seems to contradict the established results that charged non-rotating black holes are stable under first order perturbations. We show that if an AFRTEM spacetime also possesses an apparent horizon, then it has a Lyapunov functional. This suggests that the evolution equations with additional constraints arising from the apparent horizon would evolve stably to a charged non-rotating black hole. We also demonstrate that the linearised equations of these restricted spacetimes are stable and the exponentially growing dipole modes, which give rise to self accelerated motions in classical electrodynamics are also eliminated. 
  The perturbation of an exact solution exhibits a movable transcendental essential singularity, thus proving the nonintegrability. Then, all possible exact particular solutions which may be written in closed form are isolated with the perturbative Painlev\'e test; this proves the inexistence of any vacuum solution other than the three known ones. 
  Contributed talk at the Seventh Marcel Grossman Meeting on Gravity, June 24-30. A theory of evolution of the universe requires both a mutation mechanism and a selection mechanism. We believe that both can be encountered in the stochastic approach to quantum cosmology. In Brans-Dicke chaotic inflation, the quantum fluctuations of Planck mass behave as mutations, such that new inflationary domains may contain values of Planck mass that differ slightly from their parent's. The selection mechanism establishes that the value of Planck mass should be such as to increase the proper volume of the inflationary domain, which will then generate more offsprings. This mechanism predicts that the effective Planck scale at the end of inflation should be much larger than any given scale in the model. 
  The gravitational interaction between a massive particle and a spinning particle in the weak-field limit is studied. We show that a system of a spinning point-like particle and a massive rod exhibit a topological effect analogous to the electromagnetic Aharonov-Casher effect. We discuss the effect also for systems with a cosmic string instead of a massive rod and in the context of 2+1-dimensional gravity. 
  It is shown that $N^2$ is the upper limit for the number of histories in a decohering family of $N$-state quantum system. Simple criterion is found for a family of $N^2$ fine grained decohering histories of Gell-Mann and Hartle to be identical with a family of Griffiths' consistent quantum trajectories. 
  A formalism and its numerical implementation is presented which allows to calculate quantities determining the spacetime structure in the large directly. This is achieved by conformal techniques by which future null infinity ($\Scri{}^+$) and future timelike infinity ($i^+$) are mapped to grid points on the numerical grid. The determination of the causal structure of singularities, the localization of event horizons, the extraction of radiation, and the avoidance of unphysical reflections at the outer boundary of the grid, are demonstrated with calculations of spherically symmetric models with a scalar field as matter and radiation model. 
  It is shown that the only static asymptotically flat non-extrema black hole solution of the Einstein-conformally invariant scalar field equations having the scalar field bounded on the horizon, is the Schwarzschild one. Thus black holes cannot be endowed with conformal scalar hair of finite length. 
  This bibliography attempts to give a comprehensive overview of all the literature related to the Ashtekar variables. The original version was compiled by Peter H\"ubner in 1989, and it has been subsequently updated by Gabriela Gonzalez, Bernd Br\"ugmann, and Troy Schilling. Information about additional literature, new preprints, and especially corrections are always welcome. 
  An extrinsic time is identified in most isotropic and homogeneous cosmological models by matching them with the ideal clock - a parametrized system whose only "degree of freedom" is time -. Once this matching is established, the cosmological models are quantized in the same way as the ideal clock. The space of solutions of the Wheeler-DeWitt equation is turned out into a Hilbert space by inserting a time dependent operator in the inner product, yielding a unitary theory equivalent to the phase space reduced theory. 
  We argue that it is inconsistent to ignore the gravitational backreaction for on-shell superstring states at the Planck mass and beyond, and that these quantum states become Kerr black holes in the classical limit. Consequences are discussed. 
  We show that mass inflation occurs inside spinning black cosmic string, which is a solution of a low-energy effective string theory in $(3+1)$-dimensions. This confirms Poisson and Israel's conjecture that the inner mass parameter diverges even if spacetime is not spherically symmetric. 
  Homothetic scalar field collapse is considered in this article. By making a suitable choice of variables the equations are reduced to an autonomous system. Then using a combination of numerical and analytic techniques it is shown that there are two classes of solutions. The first consists of solutions with a non-singular origin in which the scalar field collapses and disperses again. There is a singularity at one point of these solutions, however it is not visible to observers at finite radius. The second class of solutions includes both black holes and naked singularities with a critical evolution (which is neither) interpolating between these two extremes. The properties of these solutions are discussed in detail. The paper also contains some speculation about the significance of self-similarity in recent numerical studies. 
  This submission to the Proceedings of the Seventh Marcel-Grossman Conference is an advertisement for the use of the ``spectral analysis inner product" for minisuperspace models in quantum gravity. 
  We describe some properties of consistent sets of histories in the Gell-Mann--Hartle formalism, and give an example to illustrate that one cannot recover the standard predictions, retrodictions and inferences of quasiclassical physics using the criterion of consistency alone. 
  The exterior solution for an arbitrary charged, massive source, is studied as a static deviation from the Reissner-Nordstr\o m metric. This is reduced to two coupled ordinary differential equations for the gravitational and electrostatic potential functions. The homogeneous equations are explicitly solved in the particular case $q^2=m^2$, obtaining a multipole expansion with radial hypergeometric dependence for both potentials. In the limiting case of a neutral source, the equations are shown to coincide with recent results by Bondi and Rindler. 
  The general solution of M\o ller's field equations in case of spherical symmetry is derived. The previously obtained solutions are verified as special cases of the general solution. 
  We describe an elementary proof that a manifold with the topology of the Politzer time machine does not admit a nonsingular, asymptotically flat Lorentz metric. 
  The Vlasov-Einstein system describes a self-gravitating, collisionless gas within the framework of general relativity. We investigate the initial value problem in a cosmological setting with spherical, plane, or hyperbolic symmetry and prove that for small initial data solutions exist up to a spacetime singularity which is a curvature and a crushing singularity. An important tool in the analysis is a local existence result with a continuation criterion saying that solutions can be extended as long as the momenta in the support of the phase-space distribution of the matter remain bounded. 
  Retrieval of classical behaviour in quantum cosmology is usually discussed in the framework of minisuperspace models in the presence of scalar fields together with the inhomogeneous modes either of the gravitational or of the scalar fields. In this work we propose alternatively a model where the scalar field is replaced by a massive vector field with global $U(1)$ or SO(3) symmetries. 
  Considerable interest has recently been expressed regarding the issue of whether or not quantum field theory on a fixed but curved background spacetime satisfies the averaged null energy condition (ANEC). A comment by Wald and Yurtsever [Phys. Rev. D43, 403 (1991)] indicates that in general the answer is no. In this note I explore this issue in more detail, and succeed in characterizing a broad class of spacetimes in which the ANEC is guaranteed to be violated. Finally, I add some comments regarding ANEC violation in Schwarzschild spacetime. 
  We apply linear perturbation theory to the study of the universality and criticality first observed by Choptuik in gravitational collapse. Since these are essentially nonlinear phenomena our attempt is only a rough approximation. In spite of this, universal behavior of the final black hole mass is observed with an exponent of 1/2, slightly higher than the observed value of 0.367. The universal behavior is rooted in the universal form that in-falling perturbations on black holes have at the horizon. 
  A simple analytic model is presented which exhibits a critical behavior in black hole formation, namely, collapse of a thin shell coupled with outgoing null fluid. It is seen that the critical behavior is caused by the gravitational nonlinearity near the event horizon. We calculate the value of the critical exponent analytically and find that it is very dependent on the coupling constants of the system. 
  We obtain the most general explicit (anti)self-dual solution of the Einstein equations. We find that any (anti)self-dual solution can be characterised by three free functions of which one is harmonic. Any stationary (anti)self-dual solution can be characterised by a harmonic function. It turns out that the form of the Gibbons and Hawking multi-center metrics is the most general stationary (anti)self-dual solution. We further note that the stationary (anti)self-dual Einstein equations can be reinterpreted as the (anti)self-dual Maxwell equations on the Euclidean background metric. 
  We discuss isometric embedding diagrams for the visualization of initial data for the problem of the head-on collision of two black holes. The problem of constructing the embedding diagrams is explicitly presented for the best studied initial data, the Misner geometry. We present a partial solution of the embedding diagrams and discuss issues related to completing the solution. 
  We demonstrate that charged null particles can be infinitely blue\-shifted in a Kerr-Newman spacetime. The surface of infinite blueshift can be outside of the ergosphere in a Kerr-Newman spacetime, and outside of the outer event horizon for a Reissner-Nordstrom spacetime. Implications for extensions of the standard model which incorporate charged neutrinos are discussed. 
  We consider a quantization of the Bianchi IX cosmological model based on taking the constraint to be a self-adjoint operator in an auxiliary Hilbert space. Using a WKB-style self-consistent approximation, the constraint chosen is shown to have only continuous spectrum at zero. Nevertheless, the auxiliary space induces an inner product on the zero-eigenvalue generalized eigenstates such that the resulting physical Hilbert space has countably infinite dimension. In addition, a complete set of gauge-invariant operators on the physical space is constructed by integrating differential forms over the spacetime. The behavior of these operators indicates that this quantization preserves Wald's classical result that the Bianchi IX spacetimes expand to a maximum volume and then recollapse. 
  In the standard treatment of the Einstein gravitational theory the energy-momentum tensor has always been taken to be composed of perfect fluid aggregates of kinematic Newtonian point test particles with fundamental mechanical masses. Moreover, this standard prescription was not revised after the discovery of the mass-generating Higgs mechanism which is known to be present in the elementary particle physics of these self-same sources and which is also required in the conformal invariant gravitational alternative being considered by Mannheim and Kazanas. In this short contribution we show that despite the presence of the Higgs mechanism, the standard geodesic motion and Euler hydrodynamics still obtain in the one-particle sector of the theory even while the overall energy-momentum tensor differs substantially from the conventional kinematic one. 
  We have developed a 3-D Eulerian hydrodynamics code to model sources of gravitational radiation. The code is written in cylindrical coordinates $(\varpi,z,\varphi)$ and has moving grids in the $\varpi$ and $z$-directions. We use Newtonian gravity and calculate the gravitational radiation in the quadrupole approximation. This code has been tested on a variety of problems to verify its accuracy and stability, and the results of these tests are reported here. 
  The presence of a horizon breaks the gauge invariance of (2+1)-dimensional general relativity, leading to the appearance of new physical states at the horizon. I show that the entropy of the (2+1)-dimensional black hole can be obtained as the logarithm of the number of these microscopic states. 
  Concepts of quantum open systems and ideas of correlation dynamics in nonequilibrium statistical mechanics, as well as methods of closed-time-path effective action and influence functional in quantum field theory can be usefully applied for the analysis of quantum statistical processes in gravitation and cosmology. We raise a few conceptual questions and suggest some new directions of research on selected currrent topics on the physics of the early universe, such as entropy generation in cosmological particle creation, quantum theory of galaxy formation, and phase transition in inflationary cosmology. 
  Using the Teukolsky and Sasaki-Nakamura equations for the gravitational perturbation of the Kerr spacetime, we calculate the post-Newtonian expansion of the energy and angular momentum luminosities of gravitational waves from a test particle orbiting around a rotating black hole up through ${\rm P^{5/2}N}$ order beyond the quadrupole formula. We apply a method recently developed by Sasaki to the case of a rotating black hole. We take into account a small inclination of the orbital plane to the lowest order of the Carter constant. The result to $ P^{3/2}N}$ order is in agreement with a similar calculation by Poisson as well as with the standard post-Newtonian calculation by Kidder et al. Using our result, we calculate the integrated phase of gravitational waves from a neutron star-neutron star binary and a black hole-neutron star binary during their inspiral stage. We find that, in both cases, spin-dependent terms in the P$^2$N and P$^{5/2}$N corrections are important to construct effective template waveforms which will be used for future laser-interferometric gravitational wave detectors. 
  The exact solutions for linear cosmological perturbations which have been obtained for collisionless relativistic matter within thermal field theory are extended to a self-interacting case. The two-loop contributions of scalar $\lambda\phi^4$ theory to the thermal graviton self-energy are evaluated, which give the $O(\lambda)$ corrections in the perturbation equations. The changes are found to be perturbative on scales comparable to or larger than the Hubble horizon, but the determination of the large-time damping behavior of subhorizon perturbations requires a resummation of thermally induced masses. 
  Using a long wavelength iteration scheme to solve Einstein's equations near the Big-Bang singularity of a universe driven by a massive scalar field, we find how big initial quasi-isotropic inhomogeneities can be before they can prevent inflation to set in. 
  We present the first calculations of the gravitational radiation produced by nonaxisymmetric dynamical instability in a rapidly rotating compact star. The star deforms into a bar shape, shedding $\sim 4\%$ of its mass and $\sim 17\%$ of its angular momentum. The gravitational radiation is calculated in the quadrupole approximation. For a mass $M \sim 1.4$ M$_{\odot}$ and radius $R \sim 10$ km, the gravitational waves have frequency $\sim 4$ kHz and amplitude $h \sim 2 \times 10^{-22}$ at the distance of the Virgo Cluster. They carry off energy $\Delta E/M \sim 0.1\%$ and radiate angular momentum $\Delta J/J \sim 0.7\%$. 
  We present here an application of a new quantization scheme. We quantize the Taub cosmology by quantizing only the anisotropy parameter $\beta$ and imposing the super-Hamiltonian constraint as an expectation-value equation to recover the relationship between the scale factor $\Omega$ and the time $t$. This approach appears to avoid the problem of time. (Paper to appear in the Seventh Marcel Grossmann Conference Proceedings.) 
  We establish the main features of homogeneous and isotropic dilaton, metric and Yang-Mills configurations in a cosmological framework. We identify a new energy exchange term between the dilaton and the Yang-Mills field which may lead to a possible solution to the Polonyi problem in 4-dimensional string models. 
  In four dimensions a Gauss-Bonnet term in the action corre- sponds to a total derivative, and it does not contribute to the classical equations of motion. For higher-dimensional geometries this term has the interesting property (shared with other dimensionally continued Euler densities) that when the action is varied with respect to the metric, it gives rise to a symmetric, covariantly conserved tensor of rank two which is a function of the metric and its first and second order derivatives. Here we review the unification of General Relativity and electromagnetism in the classical five-dimen- sional, restricted (with g_55 = 1) Kaluza-Klein model. Then we discuss the modifications of the Einstein-Maxwell theory that results from adding the Gauss-Bonnet term in the action. The resulting four-dimensional theory describes a non-linear U(1) gauge theory non-minimally coupled to gravity. For a point charge at rest, we find a perturbative solution for large distances which gives a mass-dependent correction to the Coulomb potential. Near the source we find a power-law solution which seems to cure the short-distance divergency of the Coulomb potential. Possible ways to obtain an experimen- tal upper limit to the coupling of the hypothetical Gauss- Bonnet term are also considered. 
  The general construction of extended refrence frames for noninertial observers in flat space is studied. It is shown that, if the observer moves inertially before and after an arbitrary acceleration and rotation, the region where reference frames can coincide with an inertial system is bounded for final velocities exceeding 0.6 c. 
  We examine the role of the initial density and velocity distribution in the gravitational collapse of a spherical inhomogeneous dust cloud. Such a collapse is described by the Tolman-Bondi metric which has two free functions: the `mass-function' and the `energy function', which are determined by the initial density and velocity profile of the cloud. The collapse can end in a black-hole or a naked singularity, depending on the initial parameters characterizing these profiles. In the marginally bound case, we find that the collapse ends in a naked singularity if the leading non-vanishing derivative of the density at the center is either the first one or the second one. If the first two derivatives are zero, and the third derivative non-zero, the singularity could either be naked or covered, depending on a quantity determined by the third derivative and the central density. If the first three derivatives are zero, the collapse ends in a black hole. In particular, the classic result of Oppenheimer and Snyder, that homogeneous dust collapse leads to a black hole, is recovered as a special case. Analogous results are found when the cloud is not marginally bound, and also for the case of a cloud starting from rest. We also show how the strength of the naked singularity depends on the density and velocity distribution. Our analysis generalizes and simplifies the earlier work of Christodoulou and Newman [4,5] by dropping the assumption of evenness of density functions. It turns out that relaxing this assumption allows for a smooth transition from the naked singularity phase to the black-hole phase, and also allows for the occurrence of strong curvature naked singularities. 
  It is shown that in 2+1 dimensions the Fermi-Walker gauge allows the general solution of the problem of determining the metric from the sources in terms of simple quadratures. This technique is used to solve the problem of the occurrence of closed time like curves (CTC's) in stationary solutions. In fact the Fermi-Walker gauge, due to its physical nature, allows to exploit the weak energy condition and in this connection it is proved that, both for open and closed universes with axial symmetry, the energy condition imply the total absence of closed time like curves. The extension of this theorem to the general stationary problem, in absence of axial symmetry is considered and at present the proof of such generalization is subject to some assumptions on the behavior of the determinant of the dreibeins in this gauge. 
  We consider the possibility of an alternative gravity theory explaining the dynamics of galactic systems without dark matter. From very general assumptions about the structure of a relativistic gravity theory we derive a general expression for the metric to order $(v/c)^2$. This allows us to compare the predictions of the theory with various experimental data: the Newtonian limit, light deflection and retardation, rotation of galaxies and gravitational lensing. Our general conclusion is that the possibility for any gravity theory to explain the behaviour of galaxies without dark matter is rather improbable. 
  A new class of time-symmetric solutions to the initial value constraints of vacuum General Relativity is introduced. These data are globally regular, asymptotically flat (with possibly several asymptotic ends) and in general have no isometries, but a $U(1)\times U(1)$ group of conformal isometries. After decomposing the Lichnerowicz conformal factor in a double Fourier series on the group orbits, the solutions are given in terms of a countable family of uncoupled ODEs on the orbit space. 
  We prove that the domain of outer communication of a stationary, globally hyperbolic spacetime satisfying the null energy condition must be simply connected. Under suitable additional hypotheses, this implies, in particular, that each connected component of a cross-section of the event horizon of a stationary black hole must have spherical topology. 
  It is argued that states in $N=1$ supergravity that solve all of the constraint equations cannot be bosonic in the sense of being independent of the fermionic degrees of freedom. (Based on a talk given by Miguel Ortiz at the 7th Marcel Grossmann Meeting.) 
  We investigate conservation laws in the quantum mechanics of closed systems. We review an argument showing that exact decoherence implies the exact conservation of quantities that commute with the Hamiltonian including the total energy and total electric charge. However, we also show that decoherence severely limits the alternatives which can be included in sets of histories which assess the conservation of these quantities when they are not coupled to a long-range field arising from a fundamental symmetry principle. We then examine the realistic cases of electric charge coupled to the electromagnetic field and mass coupled to spacetime curvature and show that when alternative values of charge and mass decohere, they always decohere exactly and are exactly conserved as a consequence of their couplings to long-range fields. Further, while decohering histories that describe fluctuations in total charge and mass are also subject to the limitations mentioned above, we show that these do not, in fact, restrict {\it physical} alternatives and are therefore not really limitations at all. 
  We consider fundamental problems on the understanding of the tunneling phenomena in the context of the multi-dimensional wave function. In this paper, we reconsider the quantum state after tunneling and extend our previous formalism to the case when the quantum state before tunneling is in a squeezed state. Through considering this problem, we reveal that the quantum decoherence plays a crucial role to allow us of the concise description of the quantum state after tunneling. 
  It is considered the quantum complex scalar field which obeys the authomorphic condition in the two-dimensional spacetime with closed timelike curves and the chronology horizon. The renormalized stress-energy tensor is obtained. It is shown that the value of the stress-energy tensor is regular at the chronology horizon for specific authomorphic parameters. Thus the particular example of field configuration is given for which the Hawking's chronology protection conjecture is violated. 
  We stress the strongly observer-dependent nature of the fine-grained black hole entropy as computed by usual methods. From the point of view of a fiducial observer, the 1-loop renormalized energy-momentum tensor diverges at the horizon. Considering, more consistently, the computation of the black hole entropy in the back-reaction-corrected curved background an ultraviolet cut-off, needed to regularize the entropy, naturally araises. Examples of the classical dynamics of test particles, strings and relativistic membranes in the corrected metric are also given to complete the picture. 
  We reformulate the Ponzano-Regge quantum gravity model in terms of surfaces on a 3-dimensional simplex lattice. This formulation (1) has a clear relation to the loop representation of the canonical quantum general relativity in 3-dimensions, (2) may have a 4-dimensional analogue, in contrast to the 6-j symbolic formalism of the Ponzano-Regge model, and (3) is purely a theory of surfaces, in the sense that it does not include any field variables; hence it is coordinate-free on the surface and background-free in spacetime. We discuss implications and applications of this formulation. 
  It is shown that the Lax pair equation dL/dt = [L,A] can be given a neat tensorial interpretation for finite-dimensional quadratic Hamiltonians. The Lax matrices L and A are shown to arise from third rank tensors on the configuration space. The second Lax matrix A is related to a connection which characterizes the Hamiltonian system. The Toda lattice system is used to motivate the definition of the Lax pair tensors. The possible existence of solutions to the Einstein equations having the Lax pair property is discussed. 
  It is shown -- using a FRW model with ${\bf S}^3 \times {\bf S}^6$ as spatial sections and a positive cosmological constant -- that classical signature change implies a new compactification mechanism. The internal scale factor is of the order $\Lambda^{-1/2}$, and the solutions are stable against small perturbatons. In the case of compactified ${\bf S}^6$, it is shown that the effective four-dimensional space-time metric has Lorentzian signature, undergoes exponential inflation in ${\bf S}^3$ and is unique. Speculations concerning relations to quantum cosmology and conceivable modifications are added. 
  We give in this paper a modified self-dual action that leads to the $SO(3)$-ADM formalism without having to face the difficult second class constraints present in other approaches (for example if one starts from the Hilbert-Palatini action). We use the new action principle to gain some new insights into the problem of the reality conditions that must be imposed in order to get real formulations from complex general relativity. We derive also a real formulation for Lorentzian general relativity in the Ashtekar phase space by using the modified action presented in the paper. 
  I suggest in this letter a new strategy to attack the problem of the reality conditions in the Ashtekar approach to classical and quantum general relativity. By writing a modified Hamiltonian constraint in the usual $SO(3)$ Yang-Mills phase space I show that it is possible to describe space-times with Lorentzian signature without the introduction of complex variables. All the features of the Ashtekar formalism related to the geometrical nature of the new variables are retained; in particular, it is still possible, in principle, to use the loop variables approach in the passage to the quantum theory. The key issue in the new formulation is how to deal with the more complicated Hamiltonian constraint that must be used in order to avoid the introduction of complex fields. 
  We present a unified thermodynamical description of the configurations consisting on self-gravitating radiation with or without a black hole. We compute the thermal fluctuations and evaluate where will they induce a transition from metastable configurations towards stable ones. We show that the probability of finding such a transition is exponentially small. This indicates that, in a sequence of quasi equilibrium configurations, the system will remain in the metastable states till it approaches very closely the critical point beyond which no metastable configuration exists. Near that point, we relate the divergence of the local temperature fluctuations to the approach of the instability of the whole system, thereby generalizing the usual fluctuations analysis in the cases where long range forces are present. When angular momentum is added to the cavity, the above picture is slightly modified. Nevertheless, at high angular momentum, the black hole loses most of its mass before it reaches the critical point at which it evaporates completely. 
  We reanalyze the gravitating monopole and its black hole solutions in the Einstein-Yang-Mills-Higgs system and we discuss their stabilities from the point of view of catastrophe theory. Although these non-trivial solutions exhibit fine and complicated structures, we find that stability is systematically understood via a swallow tail catastrophe. The Reissner-Nordstr\"{o}m trivial solution becomes unstable from the point where the non-trivial monopole black hole appears. We also find that, within a very small parameter range, the specific heat of a monopole black hole changes its sign . 
  The inequivalence of thermodynamical ensembles related by a Legendre transformation is manifest in self-gravitating systems and in black hole thermodynamics. Using the Poincare's method of the linear series, we describe the mathematical reasons which lead to this inequivalence which in turn induces a hierarchy of ensembles: the most stable ensemble describes the most isolated system. Moreover, we prove that one can obtain the degree of stability of all equilibrium configurations in any ensensemble related by Legendre transformations to the most stable if one knows the degree of stability in the most stable ensemble. 
  Recent work on the loop representation of quantum gravity has revealed previously unsuspected connections between knot theory and quantum gravity, or more generally, 3-dimensional topology and 4-dimensional generally covariant physics. We review how some of these relationships arise from a `ladder of field theories' including quantum gravity and BF theory in 4 dimensions, Chern-Simons theory in 3 dimensions, and the G/G gauged WZW model in 2 dimensions. We also describe the relation between link (or multiloop) invariants and generalized measures on the space of connections. In addition, we pose some research problems and describe some new results, including a proof (due to Sawin) that the Chern-Simons path integral is not given by a generalized measure. 
  The dipole coupling term between a system of N particles with total charge zero and the electromagnetic field is derived in the presence of a weak gravitational field. It is shown that the form of the coupling remains the same as in flat space-time if it is written with respect to the proper time of the observer and to the measurable field components. Some remarks concerning the connection between the minimal and the dipole coupling are given. 
  We calculate the Bogolubov coefficients for a metric which describes the snapping of a cosmic string. If we insist on a matching condition for all times {\it and} a particle interpretation, we find no particle creation. 
  We investigate the black hole solution to (2+1)-dimensional gravity coupled to topological matter, with a vanishing cosmological constant. We calculate the total energy, angular momentum and entropy of the black hole in this model and compare with results obtained in Einstein gravity. We find that the theory with topological matter reverses the identification of energy and angular momentum with the parameters in the metric, compared with general relativity, and that the entropy is determined by the circumference of the inner rather than the outer horizon. We speculate that this results from the contribution of the topological matter fields to the conserved currents. We also briefly discuss two new possible (2+1)-dimensional black holes. 
  The role of the identification of the vacuum and non-vacuum space-times in the computation of vacuum fluctuations in the presence of a cosmic string is discussed and an alternative interpretation of the renormalization is proposed. This procedure does not give rise to vacuum fluctuations. 
  We prove that, under certain conditions, the topology of the event horizon of a four dimensional asymptotically flat black hole spacetime must be a 2-sphere. No stationarity assumption is made. However, in order for the theorem to apply, the horizon topology must be unchanging for long enough to admit a certain kind of cross section. We expect this condition is generically satisfied if the topology is unchanging for much longer than the light-crossing time of the black hole. More precisely, let $M$ be a four dimensional asymptotically flat spacetime satisfying the averaged null energy condition, and suppose that the domain of outer communication $\C_K$ to the future of a cut $K$ of $\Sm$ is globally hyperbolic. Suppose further that a Cauchy surface $\Sigma$ for $\C_K$ is a topological 3-manifold with compact boundary $\partial\S$ in $M$, and $\S'$ is a compact submanifold of $\bS$ with spherical boundary in $\S$ (and possibly other boundary components in $M/\S$). Then we prove that the homology group $H_1(\Sigma',Z)$ must be finite. This implies that either $\partial\S'$ consists of a disjoint union of 2-spheres, or $\S'$ is nonorientable and $\partial\S'$ contains a projective plane. Further, $\partial\S=\partial\Ip[K]\cap\partial\Im[\Sp]$, and $\partial \Sigma$ will be a cross section of the horizon as long as no generator of $\partial\Ip[K]$ becomes a generator of $\partial\Im[\Sp]$. In this case, if $\S$ is orientable, the horizon cross section must consist of a disjoint union of 2-spheres.} 
  It has been shown by one of the authors$^1$ that in isotropic spherical coordinates there is a relation between the mass of a static spherical gravitating body and the pressure distribution inside it. In this paper the result is generalized for the case of stationary axisymmetric configurations. 
  A theory of quantum-mechanical generation of cosmological perturbations is considered. The conclusion of this study is that if the large-angular-scale anisotropy in the cosmic microwave background radiation is caused by the long-wavelength cosmological perturbations of quantum mechanical origin, they are, most likely, gravitational waves, rather than density perturbations or rotational perturbations. Some disagreements with previous publications are clarified. This contribution to the Proceedings is based on Reference~[34]. 
  In the present article we obtain, by separation of variables, an exact solution to the Dirac equation with anomalous momentum for an electrically neutral massless particle in a Bertotti-Robinson universe. We discuss the phenomenom of particle creation in this model. 
  Retrieval of classical behaviour in quantum cosmology is usually discussed in the framework of {\em midi}superspace models in the presence of scalar fields and the inhomogeneous modes corresponding either to gravitational or scalar fields. In this work, we propose an alternative model to study the decoherence of homogeneous and isotropic geometries where the scalar field is replaced by a massive vector field with a global internal symmetry. We study here the cases with $U(1)$ and $SO(3)$ global internal symmetries. The presence of a mass term breaks the conformal invariance and allows for the longitudinal modes of the spin-1 field to be present in the Wheeler-DeWitt equation. In the case of the U(1) global internal symmetry, we have only one single ``classical'' degree of freedom while in the case of the SO(3) global symmetry, we are led to consider a simple two-dimensional minisuperspace model. These minisuperspaces are shown to be equivalent to a set of coupled harmonic oscillators where the kinetic term of the longitudinal modes has a coefficient proportional to the inverse of the scale factor. The conditions for a suitable decoherence process and correlations between coordinate and momenta are established. The validity of the semi-classical Einstein equations when massive vector fields (Abelian and non-Abelian) are present is also discussed. 
  Multibubble solutions for a cosmological model which lead to thermal inflationary states due to a semi-classical tunneling of gravity are calculated. 
  We discuss the applicability of the programme of decoherence -- emergence of approximate classical behaviour through interaction with the environment -- to cases where it was suggested that the presence of symmetries would lead to exact superselection rules. For this discussion it is useful to make a distinction between pure symmetries and redundancies, which results from an investigation into the constraint equations of the corresponding theories. We discuss, in particular, superpositions of states with different charges, as well as with different masses, and suggest how the corresponding interference terms, although they exist in principle, become inaccessible through decoherence. 
  We discuss the instabilities appearing in the cosmological model with a quasi de Sitter phase following from a fourth-order gravity theory. Both the classical equation as well as the quantization in form of a Wheeler - De Witt equation are conformally related to the analogous model with Einstein's theory of gravity with a minimally coupled scalar field. Results are: 1. In the non-tachyonic case, classical fourth-order gravity is not more unstable than Einstein's theory itself. 2. The well-known classically valid conformal relation is also (at least for some typical cases) valid on the level of the corresponding Wheeler - De Witt equations, which turns out to be a non-trivial statement. (to appear in: Proc. Sem. Relativistic Astrophysics Potsdam 1994, Ed .: J. M\"ucket) 
  We present a scenario in $1 + 1$ and $3 + 1$ dimensional space time which is paradoxical in the presence of a time machine. We show that the paradox cannot be resolved and the scenario has {\em no} consistent classical solution. Since the system is macroscopic, quantisation is unlikely to resolve the paradox. Moreover, in the absence of a consistent classical solution to a macroscopic system, it is not obvious how to carry out the path integral quantisation. Ruling out, by fiat, the troublesome initial conditions will resolve the paradox, by not giving rise to it in the first place. However this implies that time machines have an influence on events, extending indefinitely into the past, and also tachyonic communication between physical events in an era when no time machine existed. If no resolution to the paradox can be found, the logical conclusion is that time machines of a certain, probably large, class cannot exist in $3 + 1$ and $1 + 1$ dimensional space time, maintaining the consistency of known physical laws. 
  A black hole solution in a teleparallel theory of (2+1)-dimensional gravity, given in a previous paper, is examined. This solution is also a solution of the three-dimensional vacuum Einstein equation with a vanishing cosmological constant. Remarkable is the fact that this solution gives a black hole in a \lq \lq flat-land" in the Einstein theory and a Newtonian limit. Coordinate transformations to \lq \lq Minkowskian" coordinates, however, are singular not only at the origin, but also on the event horizon. {\em In the three-dimensional Einstein theory, vacuum regions of space-times can be locally non-trivial}. 
  In this paper we calculate the Bogoliubov coefficients and the energy density of the stochastic gravitational wave background for a universe that undergoes inflation followed by radiation domination and matter domination, using a formalism that gives the Bogoliubov coefficients as continous functions of time. By making a reasonable assumption for the equation of state during reheating, we obtain in a natural way the expected high frequency cutoff in the spectral energy density. 
  We consider the thermodynamics of a black hole coupled to thermal radiation in a spatially finite (spherical) region. Thermodynamic state functions are derived in the canonical ensemble, defined by elements of radius $r_o$ and boundary temperature $T(r_o)$. Using recent solutions of the semi-classical back reaction problem, we compute the $O(\hbar)$ corrections to the mass of the black hole, thermal energy, the entropy and free energy due to the presence of hot conformal scalars, massless spinors and U(1) gauge quantum fields in the vicinity of the hole. The free energy is particularly important for assessing under what conditions the nucleation of black holes from hot flat space is likely to occur. 
  Pair-production of magnetic Reissner-Nordstr\"{o}m black holes (of charges $\pm q$) is studied in the next-to-leading WKB approximation. We consider generic quantum fluctuations in the corresponding instanton geometry, a detailed study of which suggests that, for sufficiently weak field $B$, the problem can be reduced to that of quantum fluctuations around a single truncated near-extremal Euclidean black hole in thermal equilibrium. Typical one-loop contributions are such that the leading WKB exponent is corrected by a small fraction $\sim \hbar /q^2$. We show that this effect is merely due to a semiclassical shift of the black hole mass-to-charge ratio that persists even in the extremal limit. We close with a few final comments. 
  A possible mathematical model has been proposed for motion of illuminated quantum particles seen by eyes or similar devices mapping the scattered light. 
  We showed several years ago that the density operator of Markovian open systems can be diagonalized continuously in time. The resulting pure state jump processes correspond to quantum trajectories proposed in recent quantum optics calculations or, at fundamental level, to exact consistent histories. 
  Vacuum cosmological models are considered in the context of a multidimensional theory of gravity with integrable Weyl geometry.  A family of exact solutions with a chain of internal spaces is obtained. Models with one internal space are considered in more detail; nonsingular models are selected. 
  The conformal equivalence of some cosmological models in Brans-Dicke theory to general relativistic cosmologies with a scalar field is discussed. In the case of radiation-dominated universes, it is shown that the presence of the scalar field has a negligible impact upon the evolution of the models in the Einstein frame. It is also shown that power-law inflation in general relativity, which is conformal to ``extended'' power-law inflation in Brans-Dicke theory, is not a unique attractor for expanding closed universes, but rather that the occurence of inflation depends upon the initial kinetic energy of the scalar field. 
  We consider the question of strong cosmic censorship in spatially compact, spatially locally homogeneous vacuum models. We show in particular that strong cosmic censorship holds in Bianchi IX vacuum space--times with spherical spatial topology. 
  The ultrarelativistic limit of the Kerr - Newman geometry is studied in detail. We find the gravitational shock wave background associated with this limit. We study the scattering of scalar fields in the gravitational shock wave geometries and compare this with the scattering by ultrarelativistic extended sources and with the scattering of fundamental strings.   We also study planckian energy string collisions in flat spacetime as the scattering of a string in the effective curved background produced by the others as the impact parameter $b$ decreases. We find the effective energy density distribution generated by these collisions. The effective metric generated by these collisions is a gravitational shock wave with profile $f(\rho)\sim p\rho^{4-D}$, for large impact parameter $b$. For intermediate $b$, $f(\rho)\sim q\rho^2$, corresponding to an extended source of momentum $q$.   We finally study the emergence of string instabilities in $D$ - dimensional black hole spacetimes and De Sitter space. We solve the first order string fluctuations around the center of mass motion at spatial infinity, near the horizon and at the spacetime singularity. We find that the time components are always well behaved in the three regions and in the three backgrounds. The radial components are unstable: imaginary frequencies develop in the oscillatory modes near the horizon, and the evolution is like $(\tau-\tau_0)^ {-P}$, $(P>0)$, near the spacetime singularity, $r\to0$, where the world - sheet time $(\tau-\tau_0)\to0$, and the proper string length grows infinitely. 
  We make some general remarks on long-ranged configurations in gauge or diffeomorphism invariant theories where the fields are allowed to assume some non vanishing values at spatial infinity. In this case the Gauss constraint only eliminates those gauge degrees of freedom which lie in the connected component of asymptotically trivial gauge transformations. This implies that proper physical symmetries arise either from gauge transformations that reach to infinity or those that are asymptotically trivial but do not lie in the connected component of transformations within that class. The latter transformations form a discrete subgroup of all symmetries whose position in the ambient group has proven to have interesting implications. We explain this for the dyon configuration in the $SO(3)$ Yang-Mills-Higgs theory, where we prove that the asymptotic symmetry group is $Z_{|m|}\times \Re$ where $m$ is the monopole number. We also discuss the application of the general setting to general relativity and show that here the only implication of discrete symmetries for the continuous part is a possible extension of the rotation group $SO(3)$ to $SU(2)$. 
  Connections are uncovered between the averaged weak (AWEC) and averaged null (ANEC) energy conditions, and quantum inequality restrictions on negative energy for free massless scalar fields. In a two-dimensional compactified Minkowski universe, we derive a covariant quantum inequality-type bound on the difference of the expectation values of the energy density in an arbitrary quantum state and in the Casimir vacuum state. From this bound, it is shown that the difference of expectation values also obeys AWEC and ANEC-type integral conditions. In contrast, it is well-known that the stress tensor in the Casimir vacuum state alone satisfies neither quantum inequalities nor averaged energy conditions. Such difference inequalities represent limits on the degree of energy condition violation that is allowed over and above any violation due to negative energy densities in a background vacuum state. In our simple two-dimensional model, they provide physically interesting examples of new constraints on negative energy which hold even when the usual AWEC, ANEC, and quantum inequality restrictions fail. In the limit when the size of the space is allowed to go to infinity, we derive quantum inequalities for timelike and null geodesics which, in appropriate limits, reduce to AWEC and ANEC in ordinary two-dimensional Minkowski spacetime. We also derive a quantum inequality bound on the energy density seen by an inertial observer in four-dimensional Minkowski spacetime. The bound implies that any inertial observer in flat spacetime cannot see an arbitrarily large negative energy density which lasts for an arbitrarily long period of time. 
  The late time behavior of waves propagating on a general curved spacetime is studied. The late time tail is not necessarily an inverse power of time. Our work extends, places in context, and provides understanding for the known results for the Schwarzschild spacetime. Analytic and numerical results are in excellent agreement. 
  When joined the unified gauge picture of fundamental interactions, the gravitation theory leads to geometry of a space-time which is far from simplicity of pseudo-Riemannian geometry of Einstein's General Relativity. This is geometry of the affine-metric composite dislocated manifolds. The goal is modification of the familiar equations of a gravitational field and entirely the new equations of its deviations. In the present brief, we do not detail the mathematics, but discuss the reasons why it is just this geometry. The major physical underlying reason lies in spontaneous symmetry breaking when the fermion matter admits only the Lorentz subgroup of world symmetries of the geometric arena. 
  A higher-order analysis of the evolution of cosmological perturbations in a Friedman universe is given by using the PMF method. The essence of the PMF approach is to choose a gauge where all fluctuations of the density, the pressure, and the four-velocity vanish. In that gauge, even in higher orders, the perturbation field equations simplify considerably; they can be decoupled and - for simple equations of state - also be solved analytically. We give the solution for the dust universe up to third order. Comparison of these solutions strongly supports the conjecture that in general instable perturbations grow much faster than they do according to the first-order analysis. However, perturbations with very large spatial extension behave differently; they grow only moderatly. Thus, an upper boundary of the region of instability seems to exist. 
  Gravitons in a squeezed vacuum state, the natural result of quantum creation in the early universe or by black holes, will introduce metric fluctuations. These metric fluctuations will introduce fluctuations of the lightcone. It is shown that when the various two-point functions of a quantized field are averaged over the metric fluctuations, the lightcone singularity disappears for distinct points. The metric averaged functions remain singular in the limit of coincident points. The metric averaged retarded Green's function for a massless field becomes a Gaussian which is nonzero both inside and outside of the classical lightcone. This implies some photons propagate faster than the classical light speed, whereas others propagate slower. The possible effects of metric fluctuations upon one-loop quantum processes are discussed and illustrated by the calculation of the one-loop electron self-energy. 
  We present the Hamiltonian, quasilocal energy, and angular momentum for a spacetime region spatially bounded by two timelike surfaces. The results are applied to the particular case of a spacetime representing an eternal black hole. It is shown that in the case when the boundaries are located in two different wedges of the Kruskal diagram, the Hamiltonian is of the form $H = H_+ - H_-$, where $H_+$ and $H_-$ are the Hamiltonian functions for the right and left wedges respectively. The application of the obtained results to the thermofield dynamics description of quantum effects in black holes is briefly discussed. 
  For a certain region of the parameter space $\{M,e,\Lambda\}$, the Cauchy horizon of a (charged) black hole residing in de Sitter space is classically stable to gravitational perturbations. This implies that, when left to its own devices, classical theory is unable to retain full predictive power: the evolution of physical fields beyond the Cauchy horizon is not uniquely determined by the initial conditions. In this paper we argue that the Cauchy horizon of a Reissner-Nordstr\"om-de Sitter black hole must always be unstable quantum mechanically. 
  Based on the analysis of two dimensional dilaton gravity we argue that the semiclassical equations of black hole formation and evaporation should not be interpreted in terms of expectation values of operators in the exact quantum theory, but rather as WKB trajectories. Thus at the semiclassical level it does not seem possible to formulate a notion of {\it quantum mechanical} information loss. 
  It is suggested that probabilities need not apply at all to matter in the physical world, which may be entirely described by the amplitudes given by the quantum mechanical state. Instead, probabilities may apply only to conscious perceptions in the mental world. Such perceptions may not form unique sequences that one could call individual minds. 
  We study the operator that corresponds to the measurement of volume, in non-perturbative quantum gravity, and we compute its spectrum. The operator is constructed in the loop representation, via a regularization procedure; it is finite, background independent, and diffeomorphism-invariant, and therefore well defined on the space of diffeomorphism invariant states (knot states). We find that the spectrum of the volume of any physical region is discrete. A family of eigenstates are in one to one correspondence with the spin networks, which were introduced by Penrose in a different context. We compute the corresponding component of the spectrum, and exhibit the eigenvalues explicitly. The other eigenstates are related to a generalization of the spin networks, and their eigenvalues can be computed by diagonalizing finite dimensional matrices. Furthermore, we show that the eigenstates of the volume diagonalize also the area operator. We argue that the spectra of volume and area determined here can be considered as predictions of the loop-representation formulation of quantum gravity on the outcomes of (hypothetical) Planck-scale sensitive measurements of the geometry of space. 
  A new version of nonsymmetric gravitational theory is presented. The field equations are expanded about the Minkowski metric, giving in lowest order the linear Einstein field equations and massive Proca field equations for the antisymmetric field $g_{[\mu\nu]}$. An expansion about an arbitrary Einstein background metric yields massive Proca field equations with couplings to only physical modes. It follows that the new version of NGT is free of ghost poles, tachyons and higher-order poles and there are no problems with asymptotic boundary conditions. A static spherically symmetric solution of the field equations in the short-range approximation is everywhere regular and does not contain a black hole event horizon. 
  Given a real-analytic manifold M, a compact connected Lie group G and a principal G-bundle P -> M, there is a canonical `generalized measure' on the space A/G of smooth connections on P modulo gauge transformations. This allows one to define a Hilbert space L^2(A/G). Here we construct a set of vectors spanning L^2(A/G). These vectors are described in terms of `spin networks': graphs phi embedded in M, with oriented edges labelled by irreducible unitary representations of G, and with vertices labelled by intertwining operators from the tensor product of representations labelling the incoming edges to the tensor product of representations labelling the outgoing edges. We also describe an orthonormal basis of spin networks associated to any fixed graph phi. We conclude with a discussion of spin networks in the loop representation of quantum gravity, and give a category-theoretic interpretation of the spin network states. 
  The role of Regge calculus as a tool for numerical relativity is discussed, and a parallelizable implicit evolution scheme described. Because of the structure of the Regge equations, it is possible to advance the vertices of a triangulated spacelike hypersurface in isolation, solving at each vertex a purely local system of implicit equations for the new edge-lengths involved. (In particular, equations of global ``elliptic-type'' do not arise.) Consequently, there exists a parallel evolution scheme which divides the vertices into families of non-adjacent elements and advances all the vertices of a family simultaneously. The relation between the structure of the equations of motion and the Bianchi identities is also considered. The method is illustrated by a preliminary application to a 600--cell Friedmann cosmology. The parallelizable evolution algorithm described in this paper should enable Regge calculus to be a viable discretization technique in numerical relativity. 
  We examine the constraints of spherically symmetric general relativity with one asymptotically flat region, exploiting both the traditional metric variables and variables constructed from the optical scalars. With respect to the latter variables, there exist two linear combinations of the Hamiltonian and momentum constraints which are related by time reversal. We introduce a one-parameter family of linear extrinsic time foliations of spacetime. The values of the parameter yielding globally valid gauges correspond to the vanishing of a timelike vector in the superspace of spherically symmetric geometries. We define a quasi-local mass on spheres of fixed proper radius which we prove is positive when the constraints are satisfied. Underpinning the proof are various local bounds on the configuration variables. We prove that a reasonable definition of the gravitational binding energy is always negative. Finally, we provide a tentative characterization of the configuration space of the theory in terms of closed bounded trajectories on the parameter space of the optical scalars. 
  We continue our investigation of the configuration space of general relativity begun in I (gr-qc/9411009). Here we examine the Hamiltonian constraint when the spatial geometry is momentarily static (MS). We show that MS configurations satisfy both the positive quasi-local mass (QLM) theorem and its converse. We derive an analytical expression for the spatial metric in the neighborhood of a generic singularity. The corresponding curvature singularity shows up in the traceless component of the Ricci tensor. We show that if the energy density of matter is monotonically decreasing, the geometry cannot be singular. A supermetric on the configuration space which distinguishes between singular geometries and non-singular ones is constructed explicitly. Global necessary and sufficient criteria for the formation of trapped surfaces and singularities are framed in terms of inequalities which relate appropriate measures of the material energy content on a given support to a measure of its volume. The strength of these inequalities is gauged by exploiting the exactly solvable piece-wise constant density star as a template. 
  It is shown that the initial singularities in spatially compact spacetimes with spherical, plane or hyperbolic symmetry admitting a compact constant mean curvature hypersurface are crushing singularities when the matter content of spacetime is described by the Vlasov equation (collisionless matter) or the wave equation (massless scalar field). In the spherically symmetric case it is further shown that if the spacetime admits a maximal slice then there are crushing singularities both in the past and in the future. The essential properties of the matter models chosen are that their energy-momentum tensors satisfy certain inequalities and that they do not develop singularities in a given regular background spacetime. 
  The degenerate Lagrangian system describing a lot of cosmological models is considered. When certain restrictions on the parameters of the model are imposed, the dynamics of the model near the "singularity" is reduced to a billiard on the Lobachevsky space. The Wheeler-DeWitt equation in the asymptotical regime is solved and a third-quantized model is suggested. 
  In Parts I and II of the work (gr-qc/9405013, 9407032), we have shown that gravity is {\it sui generis} a Higgs field corresponding to spontaneous symmetry breaking when the fermion matter admits only the Lorentz subgroup of world symmetries of the geometric arena. From the mathematical viewpoint, the Higgs nature of gravity issues from the fact that different gravitational fields are responsible for nonequivalent representations of cotangent vectors to a world manifold by $\gamma$-matrices on spinor bundles. It follows that gravitational fields fail to form an affine space modelled on a linear space of deviations of some background field. In other words, even weak gravitational fields do not satisfy the superposition principle and, in particular, can not be quantized by usual methods. At the same time, one can examine superposable deviations $\sigma$ of a gravitational field $h$ so that $h+\sigma$ fails to be a gravitational field. These deviations are provided with the adequate mathematical description in the framework of the affine group gauge theory in dislocated manifolds, and their Lagrangian densities differ from the familiar gravitational ones. They make contribution to the standard gravitational effects, e.g., modify Newton's gravitational potential. 
  Existing mathematical results are applied to the problem of classifying closed $p$-forms which are locally constructed from Lorentzian metrics on an $n$-dimensional orientable manifold $M$ ($0<p<n$). We show that the only closed, non-exact forms are generated by representatives of cohomology classes of $M$ and $(n-1)$-forms representing $n$-dimensional (with $n$ even) generalizations of the conservation of ``kink number'', which was exhibited by Finkelstein and Misner for $n=4$. The cohomology class that defines the kink number depends only on the diffeomorphism equivalence class of the metric, but a result of Gilkey implies that there is no representative of this cohomology class which is built from the metric, curvature and covariant derivatives of curvature to any finite order. 
  A solution for the problem of understanding observed rotation curves in galaxies without the introduction of dark matter halos is presented. This solution has been obtained upon considering the distribution of masses in the expanding universe, then, having a cosmological character. A formal limiting radius for galaxies depending on cosmological parameters is given. The empirical conclusions derived from the theory of M. Milgrom and J. Bekenstein arise as direct consequences of the present approach without any need of drastically modifying newtonian dynamics. 
  I show in this letter that it is possible to solve some of the constraints of the $SO(3)$-ADM formalism for general relativity by using an approach similar to the one introduced by Capovilla, Dell and Jacobson to solve the vector and scalar constraints in the Ashtekar variables framework. I discuss the advantages of using the ADM formalism and compare the result with similar proposals for different Hamiltonian formulations of general relativity. 
  Kucha\v{r} has recently given a detailed analysis of the classical and quantum geometrodynamics of the Kruskal extension of the Schwarzschild black hole. In this paper we adapt Kucha\v{r}'s analysis to the exterior region of a Schwarzschild black hole with a timelike boundary. The reduced Lorentzian Hamiltonian is shown to contain two independent terms, one from the timelike boundary and the other from the bifurcation two-sphere. After quantizing the theory, a thermodynamical partition function is obtained by analytically continuing the Lorentzian time evolution operator to imaginary time and taking the trace. This partition function is in agreement with the partition function obtained from the Euclidean path integral method; in particular, the bifurcation two-sphere term in the Lorentzian Hamiltonian gives rise to the black hole entropy in a way that is related to the Euclidean variational problem. We also outline how Kucha\v{r}'s analysis of the Kruskal spacetime can be adapted to the $\RPthree$ geon, which is a maximal extension of the Schwarzschild black hole with $\RPthree \setminus \{p\}$ spatial topology and just one asymptotically flat region. 
  We study the Geroch group in the framework of the Ashtekar formulation. In the case of the one-Killing-vector reduction, it turns out that the third column of the Ashtekar connection is essentially the gradient of the Ernst potential, which implies that the both quantities are based on the ``same'' complexification. In the two-Killing-vector reduction, we demonstrate Ehlers' and Matzner-Misner's SL(2,R) symmetries, respectively, by constructing two sets of canonical variables that realize either of the symmetries canonically, in terms of the Ashtekar variables. The conserved charges associated with these symmetries are explicitly obtained. We show that the gl(2,R) loop algebra constructed previously in the loop representation is not the Lie algebra of the Geroch group itself. We also point out that the recent argument on the equivalence to a chiral model is based on a gauge-choice which cannot be achieved generically. 
  Gravitational and massless particle radiation of straight cosmic strings with finite thickness is studied analytically. It is found that the non-linear interaction of the radiation fields emitted by a cosmic string with the ones of the string always makes the spacetime singular at the symmetry axis. The singularity is not removable and is a scalar one. 
  The local and global properties of the Goetz thick plane domain wall space-time are studied. It is found that when the surface energy of the wall is greater than a critical value $\sigma_{c}$, the space-time will be closed by intermediate singularities at a finite proper distance. A model is presented in which these singularities will give rise to scalar ones when interacting with null fluids. The maximum extension of the space-time of the wall whose surface energy is less than $\sigma_{c}$ is presented. It is shown that for certain choice of the free parameter the space-time has a black hole structure but with plane symmetry. 
  We examine the consistency of the thermodynamics of the most general class of conformally flat solution with an irrotational perfect fluid source (the Stephani Universes). For the case when the isometry group has dimension $r\ge2$, the Gibbs-Duhem relation is always integrable, but if $r<2$ it is only integrable for the particular subclass (containing FRW cosmologies) characterized by $r=1$ and by admitting a conformal motion parallel to the 4-velocity. We provide explicit forms of the state variables and equations of state linking them. These formal thermodynamic relations are determined up to an arbitrary function of time which reduces to the FRW scale factor in the FRW limit of the solutions. We show that a formal identification of this free parameter with a FRW scale factor determined by FRW dynamics leads to an unphysical temperature evolution law. If this parameter is not identified with a FRW scale factor, it is possible to find examples of solutions and formal equations of state complying with suitable energy conditions and reasonable asymptotic behavior and temperature laws. 
  We examine the consistency of the thermodynamics of irrotational and non-isentropic perfect fluids complying with matter conservation by looking at the integrability conditions of the Gibbs-Duhem relation. We show that the latter is always integrable for fluids of the following types: (a) static, (b) isentropic (admits a barotropic equation of state), (c) the source of a spacetime for which $r\ge 2$, where $r$ is the dimension of the orbit of the isometry group. This consistency scheme is tested also in two large classes of known exact solutions for which $r< 2$, in general: perfect fluid Szekeres solutions (classes I and II). In none of these cases, the Gibbs-Duhem relation is integrable, in general, though specific particular cases of Szekeres class II (all complying with $r<2$) are identified for which the integrability of this relation can be achieved. We show that Szekeres class I solutions satisfy the integrability conditions only in two trivial cases, namely the spherically symmetric limiting case and the Friedman-Roberson-Walker (FRW) cosmology. Explicit forms of the state variables and equations of state linking them are given explicitly and discussed in relation to the FRW limits of the solutions. We show that fixing free parameters in these solutions by a formal identification with FRW parameters leads, in all cases examined, to unphysical temperature evolution laws, quite unrelated to those of their FRW limiting cosmologies. 
  The transverse traceless spin-two tensor harmonics on $S^3$ and $H^3$ may be denoted by $T^{(kl)}{}_{ab}$. The index $k$ labels the (degenerate) eigenvalues of the Laplacian $\square$ and $l$ the other indices. We compute the bitensor $\sum_l T^{(kl)}{}_{ab}(x) T^{(kl)}{}_{a'b'}(x')^*$ where $x,x'$ are distinct points on a sphere or hyperboloid of unit radius. These quantities may be used to find the correlation function of a stochastic background of gravitational waves in spatially open or closed Friedman-Robertson-Walker cosmologies. 
  This paper proves a theorem about the existence of an apparent horizon in general relativity, which applies equally well to vacuum configurations and matter configurations. The theorem uses the reciprocal of the surface-to-volume ratio of a region on a space slice to measure the radius of the region, and uses the minimum value $K_{\rm min}$ of certain components of the extrinsic curvature to measure the strengh of the gravitational field in the region. The theorem proves that, if the product of the radius times $K_{\rm min}$ is larger than unity, then an apparent horizon must form, signalling the formation of a black hole. 
  We present a new numerical code that evolves a spherically symmetric configuration of collisionless matter in the Brans-Dicke theory of gravitation. In this theory the spacetime is dynamical even in spherical symmetry, where it can contain gravitational radiation. Our code is capable of accurately tracking collapse to a black hole in a dynamical spacetime arbitrarily far into the future, without encountering either coordinate pathologies or spacetime singularities. This is accomplished by truncating the spacetime at a spherical surface inside the apparent horizon, and subsequently solving the evolution and constraint equations only in the exterior region. We use our code to address a number of long-standing theoretical questions about collapse to black holes in Brans-Dicke theory. 
  We discuss a number of long-standing theoretical questions about collapse to black holes in the Brans-Dicke theory of gravitation. Using a new numerical code, we show that Oppenheimer-Snyder collapse in this theory produces black holes that are identical to those of general relativity in final equilibrium, but are quite different from those of general relativity during dynamical evolution. We find that there are epochs during which the apparent horizon of such a black hole passes {\it outside\/} the event horizon, and that the surface area of the event horizon {\it decreases\/} with time. This behavior is possible because theorems which prove otherwise assume $R_{ab}l^al^b \ge 0$ for all null vectors $l^a$. We show that dynamical spacetimes in Brans-Dicke theory can violate this inequality, even in vacuum, for any value of $\omega$. 
  A nonsymmetric gravitational theory (NGT) is presented which is free of ghost poles, tachyons and higher-order poles and there are no problems with asymptotic boundary conditions. An extended Birkhoff theorem is shown to hold for the spherically symmetric solution of the field equations. A static spherically symmetric solution in the short-range approximation, $\mu^{-1} > 2m$, is everywhere regular and does not contain a black hole event horizon. 
  Recently, it has been shown that an infinite succession of classical signature changes (''signature oscillations'') can compactify and stabilize internal dimensions, and simultaneously leads, after a coarse graining type of average procedure, to an effective (''physical'') space-time geometry displaying the usual Lorentzian metric signature. Here, we consider a minimally coupled scalar field on such an oscillating background and study its effective dynamics. It turns out that the resulting field equation in four dimensions contains a coupling to some non-metric structure, the imprint of the ''microscopic'' signature oscillations on the effective properties of matter. In a multidimensional FRW model, this structure is identical to a massive scalar field evolving in its homogeneous mode. 
  We calculate the gravitational radiation produced by the merger and coalescence of inspiraling binary neutron stars using 3-dimensional numerical simulations. The stars are modeled as polytropes and start out in the point-mass limit at wide separation. The hydrodynamic integration is performed using smooth particle hydrodynamics (SPH) with Newtonian gravity, and the gravitational radiation is calculated using the quadrupole approximation. We have run several simulations, varying both the neutron star radius and the equation of state. The resulting gravitational wave energy spectra $dE/df$ are rich in information about the hydrodynamics of merger and coalescence. In particular, our results demonstrate that detailed information on both $GM/Rc^2$ and the equation of state can in principle be extracted from the spectrum. 
  The gauge bundle of the 4-dim conformal group over an 8-dim base space, called biconformal space, is shown have a consistent interpretation as a scale-invariant phase space. Specifically, we show that a classical Hamiltonian system generates a differential geometry which is necessarily biconformal, and that the classical Hamiltonian dynamics of a point particle is equivalent to the specification of a 7-dim hypersurface in flat biconformal space together with the consequent necessary existence of a set of preferred curves. The result is centrally important for establishing the physical interpretation of conformal gauging. 
  We compare three approaches to the quantization of (2+1)-dimensional gravity with a negative cosmological constant: reduced phase space quantization with the York time slicing, quantization of the algebra of holonomies, and quantization of the space of classical solutions. The relationships among these quantum theories allow us to define and interpret time-dependent operators in the ``frozen time'' holonomy formulation. 
  The hydrostatic equilibrium of a $2+1$ dimensional perfect fluid star in asymptotically anti-de Sitter space is discussed. The interior geometry matches the exterior $2+1$ black-hole solution. An upper mass limit is found, analogous to Buchdahl's theorem in 3+1, and the possibility of collapse is discussed. The case of a uniform matter density is solved exactly and a new interior solution is presented. 
  We study the massless scalar field on asymptotically flat spacetimes with closed timelike curves (CTC's), in which all future-directed CTC's traverse one end of a handle (wormhole) and emerge from the other end at an earlier time. For a class of static geometries of this type, and for smooth initial data with all derivatives in $L_2$ on ${\cI}^{-}$, we prove existence of smooth solutions which are regular at null and spatial infinity (have finite energy and finite $L_2$-norm) and have the given initial data on $\cI^-$. A restricted uniqueness theorem is obtained, applying to solutions that fall off in time at any fixed spatial position. For a complementary class of spacetimes in which CTC's are confined to a compact region, we show that when solutions exist they are unique in regions exterior to the CTC's. (We believe that more stringent uniqueness theorems hold, and that the present limitations are our own.) An extension of these results to Maxwell fields and massless spinor fields is sketched. Finally, we discuss a conjecture that the Cauchy problem for free fields is well defined in the presence of CTC's whenever the problem is well-posed in the geometric-optics limit. We provide some evidence in support of this conjecture, and we present counterexamples that show that neither existence nor uniqueness is guaranteed under weaker conditions. In particular, both existence and uniqueness can fail in smooth, asymptotically flat spacetimes with a compact nonchronal region. 
  Recently, it has been claimed that the back reaction of vacuum polarization on a black hole spacetime naturally regularizes infinities in the black hole entropy. We examine the back reaction calculation and find no such short-distance cut-off,in contradiction with these recent claims. Moreover, the intuitive expectation that the perturbative calculation breaks down near the event horizon is confirmed. The new surface gravity diverges and the metric is degenerate at the stretched horizon. 
  I present an elementary essay on some issues related to nonrelativistic quantum mechanics, which is written in the spirit of extreme simplicity, making it an easy-to-read paper. Moreover, one can find a useful collection of ideas and opinions expressed by many well-known authors in this vast research field 
  Some recent results show that the covariant path integral and the integral over physical degrees of freedom give contradicting results on curved background and on manifolds with boundaries. This looks like a conflict between unitarity and covariance. We argue that this effect is due to the use of non-covariant measure on the space of physical degrees of freedom. Starting with the reduced phase space path integral and using covariant measure throughout computations we recover standard path integral in the Lorentz gauge and the Moss and Poletti BRST-invariant boundary conditions. We also demonstrate by direct calculations that in the approach based on Gaussian path integral on the space of physical degrees of freedom some basic symmetries are broken. 
  We study first order fluctuations of a relativistic membrane in the curved background of a black hole. The zeroth-order solution corresponds to a spherical membrane tightly covering the event horizon. We obtain a massive Klein-Gordon equation for the fluctuations of the membrane's radial coordinate on the 2+1 dimensional world-volume. We finally suggest that quantization of the fluctuations can be related to black hole's mass quantization and the corresponding entropy is computed. This entropy is proportional to the membrane area and is related to the one-loop correction to the thermodynamical entropy $A_H/4$. With regards to the membrane model for describing effectively a quantum black hole, we connect these results with previous work on critical phenomena in black hole thermodynamics. 
  In the paper we discuss the process of regularization of the Hamiltonian constraint in the Ashtekar approach to quantizing gravity. We show in detail the calculation of the action of the regulated Hamiltonian constraint on Wilson loops. An important issue considered in the paper is the closure of the constraint algebra. The main result we obtain is that the Poisson bracket between the regulated Hamiltonian constraint and the Diffeomorphism constraint is equal to a sum of regulated Hamiltonian constraints with appropriately redefined regulating functions. 
  The semi-classical back reaction to black hole evaporation (wherein the renormalized energy momentum tensor is taken as source of Einstein's equations) is analyzed in detail. It is proven that the mass of a Schwarzshild black hole decreases according to Hawking's law $dM/dt = - C/ M^2$ where $C$ is a constant of order one and that the particles are emitted with a thermal spectrum at temperature $1/8\pi M(t)$. 
  We clarify the links between a recently developped long wavelength iteration scheme of Einstein's equations, the Belinski Khalatnikov Lifchitz (BKL) general solution near a singularity and the antinewtonian scheme of Tomita's. We determine the regimes when the long wavelength or antinewtonian scheme is directly applicable and show how it can otherwise be implemented to yield the BKL oscillatory approach to a spacetime singularity. When directly applicable we obtain the generic solution of the scheme at first iteration (third order in the gradients) for matter a perfect fluid. Specializing to spherical symmetry for simplicity and to clarify gauge issues, we then show how the metric behaves near a singularity when gradient effects are taken into account. 
  A gravity-driven inflation is shown to arise from a simple higher dimensional universe. In vacuum, the shear of $n>1$ contracting dimensions is able to inflate the remaining three spatial dimensions. Said another way, the expansion of the 3-volume is accelerated by the contraction of the $n$-volume. Upon dimensional reduction, the theory is equivalent to a four dimensional cosmology with a dynamical Planck mass. A connection can therefore be made to recent examples of inflation powered by a dilaton kinetic energy. Unfortunately, the graceful exit problem encountered in dilaton cosmologies will haunt this cosmology as well. 
  An observer, situated several thousand light-years away from a radio pulsar, finds himself embedded in the diffraction pattern resulting from the propagation of the radio waves through the irregular interstellar medium. The observer's movement relative to the pattern causes an apparent scintillation of the pulsar. A binary star, situated close to the pulsar's line-of-sight, is generating relatively strong gravity waves. The rays originating from the pulsar experience a tiny periodic deflection due to the gravity waves produced by the binary star. This deflection displaces the diffraction pattern laterally in a manner that is familiar from refractive interstellar scintillation, except that this gravity wave effect is not dispersive. The displacement has the same period as the gravity waves. Its amplitude equals the product of the tiny deflection angle and the large distance from the binary star to the observer. This periodic displacement can reach a few hundred kilometers, which can be comparable to the size of the features in the diffractive pattern. Thus, there seems to be a possibility that the exceedingly faint gravity waves can manifest themselves macroscopically. Observationally, the end effect could be a substantial, deterministic alteration of the scintillation time structure. 
  Gravitational waves from inspiralling binaries are expected to be detected using a data analysis technique known as {\it matched filtering.} This technique is applicable whenever the form of the signal is known accurately. Though we know the form of the signal precisely, we will not know {\it a priori} its parameters. Hence it is essential to filter the raw output through a host of search templates each corresponding to different values of the parameters. The number of search templates needed in detecting the Newtonian waveform characterized by three independent parameters is itself several thousands. With the inclusion of post-Newtonian corrections the inspiral waveform will have four independent parameters and this, it was thought, would lead to an increase in the number of filters by several orders of magnitude---an unfavorable feature since it would drastically slow down data analysis. In this paper I show that by a judicious choice of signal parameters we can work, even when the first post-Newtonian corrections are included, with as many number of parameters as in the Newtonian case. In other words I demonstrate that the effective dimensionality of the signal parameter space does not change when first post-Newtonian corrections are taken into account. 
  We explain the collapse of the wavefunction with the notion that, in a measurement, the system observed nucleates a first order phase transition in the measuring device. The possible final states differ by the values of macroscopic observables, and their relative phase is therefore unobservable. The process is irreversible, but needs no separate postulate. 
  In this paper we present a Friedmann-Robertson-Walker cosmological model conformally coupled to a massive scalar field where the WKB approximation fails to reproduce the exact solution to the Wheeler-DeWitt equation for large Universes. The breakdown of the WKB approximation follows the same pattern than in semiclassical physics of chaotic systems, and it is associated to the development of small scale structure in the wave function. This result puts in doubt the ``WKB interpretation'' of Quantum Cosmology. 
  A general framework for integration over certain infinite dimensional spaces is first developed using projective limits of a projective family of compact Hausdorff spaces. The procedure is then applied to gauge theories to carry out integration over the non-linear, infinite dimensional spaces of connections modulo gauge transformations. This method of evaluating functional integrals can be used either in the Euclidean path integral approach or the Lorentzian canonical approach. A number of measures discussed are diffeomorphism invariant and therefore of interest to (the connection dynamics version of) quantum general relativity. The account is pedagogical; in particular prior knowledge of projective techniques is not assumed. (For the special JMP issue on Functional Integration, edited by C. DeWitt-Morette.) 
  An ADM-like Hamiltonian approach is proposed for static spherically symmetric relativistic star configurations. For a given equation of state the entire information about the model can be encoded in a certain 2-dimensional minisuperspace geometry. We derive exact solutions which arise from symmetries corresponding to linear and quadratic geodesic invariants in minisuperspace by exploiting the relation to minisuperspace Killing tensors. A classification of exact solutions having the full number of integrations constants is given according to their minisuperspace symmetry properties. In particular it is shown that Schwarzschild's exterior solution and Buchdahl's n=1 polytrope solution correspond to minisuperspaces with a Killing vector symmetry while Schwarzschild's interior solution, Whittaker's solution and Buchdahl's n=5 polytrope solution correspond to minisuperspaces with a second rank Killing tensor. New solutions filling in empty slots in this classification scheme are also given. One of these new solutions has a physically reasonable equation of state and is a generalization of Buchdahl's n=1 polytrope model. 
  From a covariant Hamiltonian formulation, using symplectic ideas, we obtain covariant quasilocal energy-momentum boundary expressions for general gravity theories. The expressions depend upon which variables are fixed on the boundary, a reference configuration and a displacement vector field. We consider applications to Einstein's theory, black hole thermodynamics and alternate spinor expressions. 
  We obtain an exact solution of the coupled Einstein-scalar-gauge field equations for a local infinitely long supermassive cosmic string. The solution correponds to that of Hiscock-Gott . The string appears to be due to the freezing of the scalar field at the null value giving rise to a constant linear energy density. 
  The internal structure of a charged spherical black hole is still a topic of debate. In a nonrotating but aspherical gravitational collapse to form a spherical charged black hole, the backscattered gravitational wave tails enter the black hole and are blueshifted at the Cauchy horizon. This has a catastrophic effect if combined with an outflux crossing the Cauchy horizon: a singularity develops at the Cauchy horizon and the effective mass inflates. Recently a numerical study of a massless scalar field in the \RN background suggested that a spacelike singularity may form before the Cauchy horizon forms. We will show that there exists an approximate analytic solution of the scalar field equations which allows the mass inflation singularity at the Cauchy horizon to exist. In particular, we see no evidence that the Cauchy horizon is preceded by a spacelike singularity. 
  I discuss a phenomenological model of the nucleon in which a small anti-de Sitter bag is placed into the Skyrmion configuration. Such a bag has a timelike boundary and allows naturally the Cheshire Cat Principle. Very important in this model is the membrane of the bag, the 3-dimensional time-space manifold S^{1}\times S^{2}, in which topological techniques will come into play 
  The influence of radiative corrections on the photon propagation in a gravitational background is investigated without the low-frequency approximation $\omega \ll m$. The conclusion is made in this way that the velocity of light can exceed unity. 
  Short remarks on the problem of assigning frequency spectra to Casimir, sonoluminescence, Hawking, Unruh, and quantum optical squeezing effects are presented 
  Requiring that the matter fields are subject to the dominant energy condition, we establish the lower bound $(4\pi)^{-1} \kappa {\cal A}$ for the total mass $M$ of a static, spherically symmetric black hole spacetime. (${\cal A}$ and $\kappa$ denote the area and the surface gravity of the horizon, respectively.) Together with the fact that the Komar integral provides a simple relation between $M - (4\pi)^{-1} \kappa A$ and the strong energy condition, this enables us to prove that the Schwarzschild metric represents the only static, spherically symmetric black hole solution of a selfgravitating matter model satisfying the dominant, but violating the strong energy condition for the timelike Killing field $K$ at every point, that is, $R(K,K) \leq 0$. Applying this result to scalar fields, we recover the fact that the only black hole configuration of the spherically symmetric Einstein-Higgs model with arbitrary non-negative potential is the Schwarzschild spacetime with constant Higgs field. In the presence of electromagnetic fields, we also derive a stronger bound for the total mass, involving the electromagnetic potentials and charges. Again, this estimate provides a simple tool to prove a ``no-hair'' theorem for matter fields violating the strong energy condition. 
  After a brief chronological sketch of developments in non-perturbative canonical quantum gravity, some of the recent mathematical results are reviewed. These include: i) an explicit construction of the quantum counterpart of Wheeler's superspace; ii) a rigorous procedure leading to the general solution of the diffeomorphism constraint in quantum geometrodynamics as well as connection dynamics; and, iii) a scheme to incorporate the reality conditions in quantum connection dynamics. Furthermore, there is a new language to formulate the central questions and techniques to answer them. These developments put the program on a sounder footing and, in particular, address certain concerns and reservations about consistency of the overall scheme. 
  Recently, Larry Ford and Tom Roman have discovered that in a flat cylindrical space, although the stress-energy tensor itself fails to satisfy the averaged null energy condition (ANEC) along the (non-achronal) null geodesics, when the ``Casimir-vacuum" contribution is subtracted from the stress-energy the resulting tensor does satisfy the ANEC inequality. Ford and Roman name this class of constraints on the quantum stress-energy tensor ``difference inequalities." Here I give a proof of the difference inequality for a minimally coupled massless scalar field in an arbitrary two-dimensional spacetime, using the same techniques as those we relied on to prove ANEC in an earlier paper with Robert Wald. I begin with an overview of averaged energy conditions in quantum field theory. 
  We give a Hamiltonian formulation of massive spin 2 theory in arbitrary Einstein space-times. We pay particular attention to Higuchi's forbidden mass range in deSitter space. 
  We prove that static, spherically symmetric, asymptotically flat soliton and black hole solutions of the Einstein-Yang-Mills equations are unstable for arbitrary gauge groups, at least for the ``generic" case. This conclusion is derived without explicit knowledge of the possible equilibrium solutions. 
  We give a formulation of the vacuum Einstein equations in terms of a set of volume-preserving vector fields on a four-manifold ${\cal M}$. These vectors satisfy a set of equations which are a generalisation of the Yang-Mills equations for a constant connection on flat spacetime. 
  A kinematical description of infinitesimal deformations of the worldsheet spanned in spacetime by a relativistic membrane is presented. This provides a framework for obtaining both the classical equations of motion and the equations describing infinitesimal deformations about solutions of these equations when the action describing the dynamics of this membrane is constructed using {\it any} local geometrical worldsheet scalars. As examples, we consider a Nambu membrane, and an action quadratic in the extrinsic curvature of the worldsheet. 
  A coupled system of non-linear partial differential equations is presented which describes non-perturbatively the evolution of deformations of a relativistic membrane of arbitrary dimension, $D$, in an arbitrary background spacetime. These equations can be considered from a formal point of view as higher dimensional analogs of the Raychaudhuri equations for point particles to which they are shown to reduce when $D=1$. For $D=1$ or $D=2$ (a string), there are no constraints on the initial data. If $D>2$, however, there will be constraints with a corresponding complication of the evolution problem. The consistent evolution of the constraints is guaranteed by an integrability condition which is satisfied when the equations of motion are satisfied. Explicit calculations are performed for membranes described by the Nambu action. 
  We suggest a method of reduction of mixed absolute and relative boundary conditions to pure ones. The case of rank two tensor is studied in detail. For four-dimensional disk the corresponding heat kernel is expressed in terms of scalar heat kernels. The result for scaling behavior $\zeta (0)$ agrees with previous calculations. 
  There exists a widespread belief that signature type change could be used to avoid spacetime singularities. We show that signature change cannot be utilised to this end unless the Einstein equation is abandoned at the suface of signature type change. We also discuss how to solve the initial value problem and show to which extent smooth and discontinuous signature changing solutions are equivalent. 
  We show that the interior of a charged, spinning black hole formed from a general axially symmetric gravitational collapse is unstable to inflation of both its mass and angular momentum parameters. Although our results are formulated in the context of $(2+1)$-dimensional black holes, we argue that they are applicable to $(3+1)$ dimensions. 
  We study the process of the evolution of the space of extra dimensions in the framework of Einstein-Yang-Mills cosmological models. It is shown that, for certain classes of models, the static compact space of extra dimensions is the attractor for a wide range of initial conditions. Also the effect of isotropization of extra dimensions in the course of evolution is demonstrated. 
  A twenty--dimensional space of charged solutions of spin--2 equations is proposed. The relation with extended (via dilatation) Poincar\'e group is analyzed. Locally, each solution of the theory may be described in terms of a potential, which can be interpreted as a metric tensor satisfying linearized Einstein equations. Globally, the non--singular metric tensor exists if and only if 10 among the above 20 charges do vanish. The situation is analogous to that in classical electrodynamics, where vanishing of magnetic monopole implies the global existence of the electro--magnetic potentials. The notion of {\em asymptotic conformal Yano--Killing tensor} is defined and used as a basic concept to introduce an inertial frame in General Relativity via asymptotic conditions at spatial infinity. The introduced class of asymptotically flat solutions is free of supertranslation ambiguities. 
  Local action principles on a manifold $\M$ are invariant (if at all) only under diffeomorphisms that preserve the boundary of $\M$. Suppose, however, that we wish to study only part of a system described by such a principle; namely, the part that lies in a bounded region $R$ of spacetime where $R$ is specified in some diffeomorphism invariant manner. In this case, a description of the physics within $R$ should be invariant under {\it all} diffeomorphisms regardless of whether they preserve the boundary of this region. The following letter shows that physics in such a region can be described by an action principle that $i$) is invariant under both diffeomorphisms which preserve the boundary of $R$ and those that do not, $ii$) leaves the dynamics of the part of the system {\it outside} the region $R$ completely undetermined, and $iii$) can be constructed without first solving the original equations of motion. 
  A quantum correction to the Brans-Dicke theory due to interactions among matter fields is calculated, resulting in violation of WEP, hence giving a constraint on the parameter $\omega$ far more stringent than accepted so far. The tentative estimate gives the lower bounds $\gsim 10^{6}$ and $\gsim 10^{8}$ for the assumed force-range $\gsim 1$m and $\gsim 1$AU, respectively. 
  It is pointed out that string-loop effects may generate matter couplings for the dilaton allowing this scalar partner of the tensorial graviton to stay massless while contributing to macroscopic gravity in a way naturally compatible with existing experimental data. Under a certain assumption of universality of the dilaton coupling functions (possibly realized through a discrete symmetry such as S-duality), the cosmological evolution drives the dilaton towards values where it decouples from matter. At the present cosmological epoch, the coupling to matter of the dilaton should be very small, but non zero. This provides a new motivation for improving the experimental tests of Einstein's Equivalence Principle. 
  Starting from the Lagrangian formulation of the Einstein equations for the vacuum static spherically symmetric metric, we develop a canonical formalism in the radial variable $r$ that is time--like inside the Schwarzschild horizon. The Schwarzschild mass turns out to be represented by a canonical function that commutes with the $r$--Hamiltonian. We investigate the Wheeler--DeWitt quantization and give the general representation for the solution as superposition of eigenfunctions of the mass operator. 
  The effect of bulk viscisity on the evolution of the homogeneous and isotropic cosmological models is considered. Solutions are found, with a barotropic equation of state, and a viscosity coefficient that is proportional to a power of the energy density of the universe. For flat space, power law expansions, related to extended inflation are found as well as exponential solutions, related to old inflation; also a solution with expansion that is an exponential of an exponential of the time is found. 
  For pure fourth order (${\cal{L}} \propto R^2$) quantum cosmology the Wheeler-DeWitt equation is solved exactly for the closed homogeneous and isotropic model. It is shown that by imposing as boundary condition that $\Psi = 0$ at the origin of the universe the wave functions behave as suggested by Vilenkin. 
  We analyze a system consisting of an oscillator coupled to a field. With the field traced out as an environment, the oscillator loses coherence on a very short {\it decoherence timescale}; but, on a much longer {\it relaxation timescale}, predictably evolves into a unique, pure (ground) state. This example of {\it re-coherence} has interesting implications both for the interpretation of quantum theory and for the loss of information during black hole evaporation. We examine these implications by investigating the intermediate and final states of the quantum field, treated as an open system coupled to an unobserved oscillator. 
  The Weyl equation (massless Dirac equation) is studied in a family of metrics of the G\"odel type. The field equation is solved exactly for one member of the family. 
  We consider perturbative solutions to the classical field equations coming from a quadratic gravitational lagrangian in four dimensions. We study the charged, spherically symmetric black hole and explicitly give corrections up to third order (in the coupling constant $\beta$ multiplying the $R_{\mu\nu}R^{\mu\nu}$ term) to the Reissner--Nordstr\"om hole metric. We consider the thermodynamics of such black holes, in particular, we compute explicitly its temperature and entropy--area relation which deviates from the usual $S=A/4$ expression. 
  We consider the minisuperspace model arising from the lowest order string effective action containing the graviton and the dilaton and study solutions of the resulting Wheeler-Dewitt equation. The scale factor duality symmetry is discussed in the context of our quantum cosmological model. 
  The process of quantum creation of a qusihomogeneous inflationary universe near a cosmological singularity is considered. It is shown that during the evolution quantum fluctuations of spatial topologies increase and the universe acquires homogeneous and isotropic character on arbitrary large distances. 
  The action principle is frequently used to derive the classical equations of motion. The action may also be used to associate group elements with curves in the space-time manifold, similar to the gauge transformations. The action principle is shown here to be an equivalence relation between the infinitesimal elements so defined for a collection of closed curves and the identity element. The action principle is then extended by requiring the equivalence of global elements with the identity and by considering all curves. The resulting equation is generalized further to include the non-Abelian gauge fields. The extended equation has an infinite number, but not all, trajectories as solutions. The properties of these paths are shown to impart wave-like properties to the particles in motion. These results provide an insight into the wave-particle duality and lead to a modified path-integral formalism. The motion of a particle is formulated within the resulting framework which yields a generalized Schrodinger equation. This equation is shown to reduce to a set of equations, one of them being the Klein-Gordon equation. 
  Two problems will be considered: the question of hidden parameters and the problem of Kolmogorovity of quantum probabilities. Both of them will be analyzed from the point of view of two distinct understandings of quantum mechanical probabilities. Our analysis will be focused, as a particular example, on the Aspect-type EPR experiment. It will be shown that the quantum mechanical probabilities appearing in this experiment can be consistently understood as conditional probabilities without any paradoxical consequences. Therefore, nothing implies in the Aspect experiment that quantum theory is incompatible with a deterministic universe. 
  The behavior of a quantum test particle satisfying the Klein-Gordon equation in a certain class of 4 dimensional stationary space-times is examined. In a space-time of a spinning cosmic string, the wave function of a particle in a box is shown to acquire a geometric phase when the box is transported around a closed path surrounding the string. When interpreted as an Aharonov-Anandan geometric phase, the effect is shown to be related to the Aharonov-Bohm effect. 
  It has been conjectured by Rovelli that there is a correspondence between the space of link classes of a Riemannian 3-manifold and the space of 3-geometries (on the same manifold). An exact statement of his conjecture will be established and then verified for the case when the 3-manifold is compact, orientable and closed. 
  A numerical simulation is performed of the gravitational collapse of a spherically symmetric scalar field. The algorithm uses the null initial value formulation of the Einstein-scalar equations, but does {\it not} use adaptive mesh refinement. A study is made of the critical phenomena found by Choptuik in this system. In particular it is verified that the critical solution exhibits periodic self-similarity. This work thus provides a simple algorithm that gives verification of the Choptuik results. 
  The field equations in the nonsymmetric gravitational theory are derived from a Lagrangian density using a first-order formalism. Using the general covariance of the Lagrangian density, conservation laws and tensor identities are derived. Among these are the generalized Bianchi identities and the law of energy-momentum conservation. The Lagrangian density is expanded to second-order, and treated as an ``Einstein plus fields'' theory. From this, it is deduced that the energy is positive in the radiation zone. 
  A dynamical model of the decaying vacuum energy is presented, which is based on Jordan-Brans-Dicke theory with a scalar field $\phi$. The solution of an evolutionary differential equation for the scalar field $\phi$ drives the vacuum energy towards a cosmological constant at the present epoch that can give for the age of the universe, $t_0\sim 13.5$ Gyr for $\Omega_0=1$, which is consistent with the age of globular clusters. 
  Using the multi-dimensional wave function formalism, we investigate the quantum state of a scalar field inside a true vacuum bubble nucleated through false vacuum decay in flat spacetime. We developed a formalism which allows us a mode-by-mode analysis. To demonstrate its advantage, we describe in detail the evolution of the quantum state during the tunneling process in terms of individual mode functions and interpret the result in the language of particle creation. The spectrum of the created particles is examined based on quantum field theory in the Milne universe. 
  A simple, general discussion of the problem of inertia is provided both in classical physics and in the quantum world. After briefly reviewing the classical principles of equivalence (weak (WEP), Einstein (EEP), strong (SEP)), I pass to a presentation of several equivalence statements in nonrelativistic quantum mechanics and for quantum field vacuum states. It is suggested that a reasonable type of preferred quantum field vacua may be considered: those possessing stationary spectra of their vacuum fluctuations with respect to accelerated classical trajectories 
  We describe the time evolution of quantum systems in a classical background space-time by means of a covariant derivative in an infinite dimensional vector bundle. The corresponding parallel transport operator along a timelike curve $\cC$ is interpreted as the time evolution operator of an observer moving along $\cC$. The holonomy group of the connection, which can be interpreted as a group of local symmetry transformations, and the set of observables have to satisfy certain consistency conditions. Two examples related to local $\mbox{SO}(3)$ and $\mbox{U}(1)$-symmetries, respectively, are discussed in detail. The theory developed in this paper may also be useful to analyze situations where the underlying space-time manifold has closed timelike curves. 
  The Segal-Bargmann transform plays an important role in quantum theories of linear fields. Recently, Hall obtained a non-linear analog of this transform for quantum mechanics on Lie groups. Given a compact, connected Lie group $G$ with its normalized Haar measure $\mu_H$, the Hall transform is an isometric isomorphism from $L^2(G, \mu_H)$ to ${\cal H}(G^{\Co})\cap L^2(G^{\Co}, \nu)$, where $G^{\Co}$ the complexification of $G$, ${\cal H}(G^{\Co})$ the space of holomorphic functions on $G^{\Co}$, and $\nu$ an appropriate heat-kernel measure on $G^{\Co}$. We extend the Hall transform to the infinite dimensional context of non-Abelian gauge theories by replacing the Lie group $G$ by (a certain extension of) the space ${\cal A}/{\cal G}$ of connections modulo gauge transformations. The resulting ``coherent state transform'' provides a holomorphic representation of the holonomy $C^\star$ algebra of real gauge fields. This representation is expected to play a key role in a non-perturbative, canonical approach to quantum gravity in 4-dimensions. 
  We solve the complex Einstein equations for Bianchi I and II models formulated in the Ashtekar variables. We then solve the reality conditions to obtain a parametrization of the space of Lorentzian solutions in terms of real canonically conjugate variables. In the Ashtekar variables, the dynamics of the universe point particle is governed by only a curved supermetric -- there is no potential term. In the usual metric formulation the particle bounces off a potential wall in flat superspace. We consider possible characterizations of this ``bounce'' in the potential-free Ashtekar variables. 
  A formalism for quantizing time reparametrization invariant dynamics is considered and applied to systems which contain an `almost ideal clock.' Previously, this formalism was successfully applied to the Bianchi models and, while it contains no fundamental notion of `time' or `evolution,' the approach does contain a notion of correlations. Using correlations with the almost ideal clock to introduce a notion of time, the work below derives the complete formalism of external time quantum mechanics. The limit of an ideal clock is found to be closely associated with the Klein-Gordon inner product and the Newton-Wigner formalism and, in addition, this limit is shown to fail for a clock that measures metric-defined proper time near a singularity in Bianchi models. 
  A review is given of some classical and quantum aspects of 2+1 dimensional gravity. 
  It is shown that in the instanton approximation the rate of creation of black holes is always enhanced by a factor of the exponential of the black hole entropy relative to the rate of creation of compact matter distributions (stars). This result holds for any generally covariant theory of gravitational and matter fields that can be expressed in Hamiltonian form. It generalizes the result obtained previously for the pair creation of magnetically charged black holes by a magnetic field in Einstein--Maxwell theory. The particular example of pair creation of electrically charged black holes by an electric field in Einstein--Maxwell theory is discussed in detail. 
  We show in the context of Einstein gravity that the removal of a spatial region leads to the appearance of an infinite set of observables and their associated edge states localized at its boundary. Such a boundary occurs in certain approaches to the physics of black holes like the one based on the membrane paradigm. The edge states can contribute to black hole entropy in these models. A ``complementarity principle" is also shown to emerge whereby certain ``edge" observables are accessible only to certain observers. The physical significance of edge observables and their states is discussed using their similarities to the corresponding quantities in the quantum Hall effect. The coupling of the edge states to the bulk gravitational field is demonstrated in the context of (2+1) dimensional gravity. 
  A new representation for canonical gravity and supergravity is presented, which combines advantages of Ashtekar's and the Wheeler~DeWitt representation: it has a nice geometric structure and the singular metric problem is absent. A formal state functional can be given, which has some typical features of a vacuum state in quantum field theory. It can be canonically transformed into the metric representation. Transforming the constraints too, one recovers the Wheeler~DeWitt equation up to an anomalous term. A modified Dirac quantization is proposed to handle possible anomalies in the constraint algebra. 
  We analyze the energy density fluctuations contributed by scalar fields $\Phi$ with vanishing expectation values, $\langle\Phi\rangle=0$, which are present in addition to the inflaton field. For simplicity we take $\Phi$ to be non--interacting and minimally coupled to gravity. We use normal ordering to define the renormalized energy density operator $\rho$, and we show that any normal ordering gives the same result for correlation functions of $\rho$. We first consider massless fields and derive the energy fluctuations in a single mode $\vk$, the two--point correlation function of the energy density, the power spectrum, and the variance of the smeared energy density, $\ddR$. Mass effects are investigated for energy fluctuations in single modes. All quantities considered are scale invariant at the second horizon crossing (Harrison--Zel'dovich type) for massless and for unstable massive fields. The magnitude of the relative fluctuations $\de\rho/\rt$ is of order $(\Hi/\Mp)^2$ in the massless case, where $\Hi$ is the Hubble constant during inflation. For an unstable field of mass $m_\Phi\ll\Hi$ with a decay rate $\Gamma_\Phi$ the magnitude is enhanced by a factor $\sqrt{m_\Phi/\Gamma_\Phi}$. Finally, the prediction for the cosmic variance of the average energy density in a sample is given in the massless case. 
  We analyze the non--Gaussian primordial fluctuations which are inescapably contributed by scalar fields $\Phi$ with vanishing expectation values, $\langle\Phi\rangle=0$, present during inflation in addition to the inflaton field. For simplicity we take $\Phi$ to be non--interacting and minimally coupled to gravity. $\Phi$ is a Gaussian variable, but the energy density fluctuations contributed by such a field are $\chi^2$--distributed. We compute the three--point function $\xxxT$ for the configuration of an equilateral triangle (with side length $\ell$) and the skewness $\dddRR$, {\it i.e.} the third moment of the one--point probability distribution of the spatially smeared energy density contrast $\de_R$, where $R$ is the smearing scale. The relative magnitudes of the non--Gaussian effects, $[\xi^{(N)}]^{1/N}/[\xi^{(2)}]^{1/2}$, do not grow in time. They are given by numerical constants of order unity, independent of the scale $\ell$. The "bi--skewness" $\dddRS$ is positive. For smearing lengths $R\ll S$ this shows that in our model (in contrast to Gaussian models) voids are more quiet than high--density regions. 
  The collapse of thin dust shells in 2+1 dimensional gravity with and without a cosmological constant in analyzed. A critical value of the shell's mass as a function of its radius and position is derived. For $\Lambda < 0$, a naked singularity or black hole forms depending on whether the shell's mass is below or just above this value. The solution space is divided into four different regions by three critical surfaces. For $\Lambda < 0$, two surfaces separate regions of black hole solutions and solutions with naked singularities, while the other surface separates regions of open and closed spaces. Near the transition between black hole and naked singularity, we find ${\cal M} \sim c_{p}(p-p^*)^{\beta}$, where $\beta=1/2$ and ${\cal M}$ is a naturally defined order parameter. We find no phase transition in crossing from an open to closed space. The critical solutions are analogous to higher dimensional extremal black holes. All four phases coexist at one point in solution space correspondiong to the static extremal solution. 
  The confrontation between Einstein's theory of gravitation and experiment is summarized. Although all current experimental data are compatible with general relativity, the importance of pursuing the quest for possible deviations from Einstein's theory is emphasized. 
  Motivated by recent studies of the one-bubble inflationary universe scenario that predicts a low density, negative curvature universe, we investigate the Euclidean vacuum mode functions of a scalar field in a spatially open chart of de Sitter space which is foliated by hyperbolic time slices. When we consider the possibility of an open inflationary universe, we are faced with the problem of the initial condition for the quantum fluctuations of the inflaton field, because the inflationary era should not last too long to lose every information of the initial condition. In the one-bubble scenario in which an open universe is created in an exponentially expanding false vacuum universe triggered by quantum decay of false vacuum, it seems natural that the initial state is the de Sitter-invariant Euclidean vacuum. Here we present explicit expressions for the Euclidean vacuum mode functions in the open chart for a scalar field with arbitrary mass and curvature coupling. 
  It is proven that the usual quadratic general-covariant Lagrangian for the Dirac field leads to a symmetric, divergence-free energy-momentum tensor in the standard Riemannian framework of space-time without torsion, provided the tetrad field components are the only quantities related to gravitation that are varied independently. 
  A particle of mass $\mu$ moves on a circular orbit of a nonrotating black hole of mass $M$. Under the restrictions $\mu/M \ll 1$ and $v \ll 1$, where $v$ is the orbital velocity, we consider the gravitational waves emitted by such a binary system. We calculate $\dot{E}$, the rate at which the gravitational waves remove energy from the system. The total energy loss is given by $\dot{E} = \dot{E}^\infty + \dot{E}^H$, where $\dot{E}^\infty$ denotes that part of the gravitational-wave energy which is carried off to infinity, while $\dot{E}^H$ denotes the part which is absorbed by the black hole. We show that the black-hole absorption is a small effect: $\dot{E}^H/\dot{E} \simeq v^8$. We also compare the wave generation formalism which derives from perturbation theory to the post-Newtonian formalism of Blanchet and Damour. Among other things we consider the corrections to the asymptotic gravitational-wave field which are due to wave-propagation (tail) effects. 
  Nicolai's theorem suggests a simple stochastic interpetation for supersymmetric Euclidean quantum theories, without requiring any inner product to be defined on the space of states. In order to apply this idea to supergravity, we first reduce to a one-dimensional theory with local supersymmetry by the imposition of homogeneity conditions. We then make the supersymmetry rigid by imposing gauge conditions, and quantise to obtain the evolution equation for a time-dependent wave function. Owing to the inclusion of a certain boundary term in the classical action, and a careful treatment of the initial conditions, the evolution equation has the form of a Fokker-Planck equation. Of particular interest is the static solution, as this satisfies all the standard quantum constraints. This is naturally interpreted as a cosmological probability density function, and is found to coincide with the square of the magnitude of the conventional wave function for the wormhole state. 
  Transverse-tracefree (TT-) tensors on $({\bf R}^3,g_{ab})$, with $g_{ab}$ an asymptotically flat metric of fast decay at infinity, are studied. When the source tensor from which these TT tensors are constructed has fast fall-off at infinity, TT tensors allow a multipole-type expansion. When $g_{ab}$ has no conformal Killing vectors (CKV's) it is proven that any finite but otherwise arbitrary set of moments can be realized by a suitable TT tensor. When CKV's exist there are obstructions -- certain (combinations of) moments have to vanish -- which we study. 
  We consider inertial and accelerated Unruh-DeWitt detectors moving in a background thermal bath and calculate their excitation rates. It is shown that for fast moving detectors such a thermal bath does not affect substantially the excitation probability. Our results are discussed in connection with a possible proposal of testing the Unruh effect in high energy particle accelerators. 
  The now-famous Majumdar-Papapetrou exact solution of the Einstein-Maxwell equations describes, in general, $N$ static, maximally charged black holes balanced under mutual gravitational and electrostatic interaction. When $N=2$, this solution defines the two-black-hole spacetime, and the relativistic two-center problem is the problem of geodesic motion on this static background. Contopoulos and a number of other workers have recently discovered through numerical experiments that in contrast with the Newtonian two-center problem, where the dynamics is completely integrable, relativistic null-geodesic motion on the two black-hole spacetime exhibits chaotic behavior. Here I identify the geometric sources of this chaotic dynamics by first reducing the problem to that of geodesic motion on a negatively curved (Riemannian) surface. 
  There are several regimes in chaotic inflationary cosmology where some part of the system is classical and some other quantum. I describe how to deal with such systems and how to disentangle their dynamics into classical behaviour and quantum corrections. I also discuss the conditions for quantum corrections to be small. 
  I comment on a number of theoretical issues related to magnetobremsstrahlung, and especially on synchrotron radiation and Unruh (temperature) radiation, that I consider of importance for the current progress towards a better understanding of the stationary features of such fundamental radiation patterns both in an accelerator context and, more generally, in the physical world 
  Enforcing the spacetime of supermassive cosmic strings to satisfy the symmetry of a gravitational topological defect, that is a spacetime kink, it is shown that the energy of these strings becomes quantized so that only defects whose internal geometry is that of a hemisphere (critical string) and that of a sphere (extreme string) are allowed. In the latter case, the exterior conical singularity becomes an event-horizon singularity which can be removed by using the Kruskal technique in the kink metric. It is also shown that the geodesically complete metric of the extreme string can accommodate closed timelike curves which are compatible with quantum theory, but cannot be noticed by any observers. The quantum string can also drive an essentially unique process of inflationary expansion in its core, without any fine tuning of the initial conditions. 
  The evolution operator for states of gauge theories in the graph representation (closely related to the loop representation of Gambini and Trias, and Rovelli and Smolin) is formulated as a weighted sum over worldsheets interpolating between initial and final graphs. As examples, lattice $SU(2)$ BF and Yang-Mills theories are expressed as worldsheet theories, and (formal) worldsheet forms of several continuum $U(1)$ theories are given. It is argued that the world sheet framework should be ideal for representing GR, at least euclidean GR, in 4 dimensions, because it is adapted to both the 4-diffeomorphism invariance of GR, and the discreteness of 3-geometry found in the loop representation quantization of the theory. However, the weighting of worldsheets in GR has not yet been found. 
  The Einstein/Maxwell equations reduce in the stationary and axially symmetric case to a harmonic map with prescribed singularities phi: R^3\Sigma -> H^2_C, where Sigma is a subset of the axis of symmetry, and H^2_C is the complex hyperbolic plane. Motivated by this problem, we prove the existence and uniqueness of harmonic maps with prescribed singularities phi: R^n\Sigma -> H, where Sigma is a submanifold of R^n of co-dimension at least 2, and H is a classical Riemannian globally symmetric space of noncompact type and rank one. This result, when applied to the black hole problem, yields solutions which can be interpreted as equilibrium configurations of multiple co-axially rotating charged black holes held apart by singular struts. 
  We advance here a new gravity quantization procedure that explicitly utilizes York's analysis of the geometrodynamic degrees of freedom. This geometrodynamic procedure of quantization is based on a separation of the true dynamic variables from the embedding parameters and a distinctly different treatment of these two kinds of variables. While the dynamic variables are quantized following the standard quantum mechanical and quantum field theoretic procedures, the embedding parameters are determined by the "classical" constraint equations in which the expectation values of the dynamic variables are substituted in place of their classical values. This self-consistent procedure of quantization leads to a linear Schrodinger equation augmented by nonlinear "classical" constraints and supplies a natural description of time evolution in quantum geometrodynamics. In particular, the procedure sheds new light on the "problems of time" in quantum gravity. 
  We point out that, contrary to signs of heat capacities, thermodynamic fluctuations are simply and unequivocally related to onset of instabilities that show up near critical points. Fluctuation theory is then applied to \Schw\ \bh s surrounded by radiation. This shows that slowly evolving \bh s along quasi-equilibrium states in cavities greater than $10^6$ Planck length will not evaporate below the critical Hawking limit temperature despite the fact that pure radiation has a much higher entropy. (CQG in press, figure request to okamoto@gprx.miz.nao.ac.jp) 
  We present analytical and numerical results for static, spherically symmetric solutions of the Einstein Yang-Mills Higgs equations corresponding to magnetic monopoles and non-abelian magnetically charged black holes. In the limit of infinite Higgs mass we give an existence proof for these solutions. The stability of the abelian extremal Reissner-Nordstrom black holes is reanalyzed. 
  We re-examine the semiclassical approximation to quantum gravity in the canonical formulation, focusing on the definition of a quasiclassical state for the gravitational field. It is shown that a state with classical correlations must be a superposition of states of the form $e^{iS}$. In terms of a reduced phase space formalism, this type of state can be expressed as a coherent superposition of eigenstates of operators that commute with the constraints and so correspond to constants of the motion. Contact is made with the usual semiclassical approximation by showing that a superposition of this kind can be approximated by a WKB state with an appropriately localised prefactor. A qualitative analysis is given of the effects of geometry fluctuations, and the possibility of a breakdown of the semiclassical approximation due to interference between neighbouring classical trajectories is discussed. It is shown that a breakdown in the semiclassical approximation can be a coordinate dependent phenomenon, as has been argued to be the case close to a black hole horizon. 
  In the Lagrangian field theory, one gets different identities for different stress energy-momentum tensors, e.g., canonical energy-momentum tensors. Moreover, these identities are not conservation laws of the above-mentioned energy-momentum tensors in general. In the framework of the multimomentum Hamiltonian formalism, we have the fundamental identity whose restriction to a constraint space can be treated the energy-momentum conservation law. In standard field models, this appears the metric energy-momentum conservation law. 
  We consider the quantum analogues of wormholes obtained by Carlini and Miji\'c (CM), who analytically continued closed universe models. To obtain wormholes when the strong energy condition ($\gamma>2/3$) is satisfied, we are able to simplify the Wheeler-DeWitt (WDW) equation by using an equivalent scalar potential which is a function of the scale factor. Such wormholes are found to be consistent with the Hawking-Page (HP) conjecture for quantum wormholes as solutions of the WDW equation. In addition to the CM type wormholes, for a scalar field realization of the potential in the WDW equation we also obtain quantum wormholes when the strong energy condition is violated. This violation can be up to an arbitrary large distance from the wormhole throat, before the violation eventually has to be relaxed in order to have a flat Euclidean space time. These results give support to the claim of HP that wormhole solutions are a fairly general property of the WDW equation. However, by allowing such solutions one might be precluding other more important properties such as a Lorentzian behaviour and a possible inflationary earlier stage of our universe. 
  We present a general scheme for the nonlinear gauge realizations of spacetime groups on coset spaces of the groups considered. In order to show the relevance of the method for the rigorous treatment of the translations in gravitational gauge theories, we apply it in particular to the affine group. This is an illustration of the family of spacetime symmetries having the form of a semidirect product $H\semidirect T$, where $H$ is the stability subgroup and $T$ are the translations . The translational component of the connection behaves like a true tensor under $H$ when coset realizations are involved. 
  We show that all Majumdar--Papapetrou electrovacuum space--times with a non--empty black hole region and with a non--singular domain of outer communications are the standard Majumdar--Papapetrou space--times. 
  We present a general framework to include ordinary fermionic matter in the metric--affine gauge theories of gravity. It is based on a nonlinear gauge realization of the affine group, with the Lorentz group as the classification subgroup of the matter and gravitational fields. 
  Various scenarios of the initial inflation of the universe are distinguished by the choice of a scalar field {\em potential} $U(\phi)$ which simulates a {\it temporarily} non--vanishing {\em cosmological term}. Our new method, which involves a reparametrization in terms of the Hubble expansion parameter $H$, provides a classification of allowed inflationary potentials and of the stability of the critical points. It is broad enough to embody all known {\it exact} solutions involving one scalar field as special cases. Inflation corresponds to the evolution of critical points of some catastrophe manifold. The coalescence of its nondegenerate critical points with the creation of a degenerate critical point corresponds the reheating phase of the universe. This is illustrated by several examples. 
  A method of quantizing parametrized systems is developed that is based on a kind of ``gauge invariant'' quantities---the so-called perennials (a perennial must also be an ``integral of motion''). The problem of time in its particular form (frozen time formalism, global problem of time, multiple choice problem) is met, as well as the related difficulty characteristic for this type of theory: the paucity of perennials. The present paper is an attempt to find some remedy in the ideas on ``forms of relativistic dynamics'' by Dirac. Some aspects of Dirac's theory are generalized to all finite-dimensional first-class parametrized systems. The generalization is based on replacing the Poicar\'{e} group and the algebra of its generators as used by Dirac by a canonical group of symmetries and by an algebra of elementary perennials. A number of insights is gained; the following are the main results. First, conditions are revealed under which the time evolution of the ordinary quantum mechanics, or a generalization of it, can be constructed. The construction uses a kind of gauge and time choice and it is described in detail. Second, the theory is structured so that the quantum mechanics resulting from different choices of gauge and time are compatible. Third, a practical way is presented of how a broad class of problems can be solved without the knowledge of explicit form of perennials. 
  The method of group quantization described in the preceeding paper I is extended so that it becomes applicable to some parametrized systems that do not admit a global transversal surface. A simple completely solvable toy system is studied that admits a pair of maximal transversal surfaces intersecting all orbits. The corresponding two quantum mechanics are constructed. The similarity of the canonical group actions in the classical phase spaces on the one hand and in the quantum Hilbert spaces on the other hand suggests how the two Hilbert spaces are to be pasted together. The resulting quantum theory is checked to be equivalent to that constructed directly by means of Dirac's operator constraint method. The complete system of partial Hamiltonians for any of the two transversal surfaces is chosen and the quantum Schr\"{o}dinger or Heisenberg pictures of time evolution are constructed. 
  The Wheeler-DeWitt equation for a class of Kantowski-Sachs like models is completely solved. The generalized models include the Kantowski-Sachs model with cosmological constant and pressureless dust. Likewise contained is a joined model which consists of a Kantowski-Sachs cylinder inserted between two FRW half--spheres. The (second order) WKB approximation is exact for the wave functions of the complete set and this facilitates the product structure of the wave function for the joined model. In spite of the product structure the wave function can not be interpreted as admitting no correlations between the different regions. This problem is due to the joining procedure and may therefore be present for all joined models. Finally, the {s}ymmetric {i}nitial {c}ondition (SIC) for the wave function is analyzed and compared with the ``no bouindary'' condition. The consequences of the different boundary conditions for the arrow of time are briefly mentioned. 
  We determine generally the spinor Green's function and the twisted spinor Green's function in an Euclidean space with a conical-type line singularity. In particular, in the neighbourhood of the point source, we expree them as a sum of the usual Euclidean spinor Green's functin and a regular term. In four dimensions, we use these determinations to calculate the vacuum energy density and the twisted one for a massless spinor field in the spacetime of a straight cosmic string. In the Minkowski spacetime, we determine explicitly the vacuum energy density for a massive twisted spinor field. 
  The microcanonical functional integral for an eternal black hole system is considered. This requires computing the microcanonical action for a spatially bounded spacetime region when its two disconnected timelike boundary surfaces are located in different wedges of the Kruskal diagram. The path integral is a sum over Lorentzian geometries and is evaluated semiclassically when its boundary data are chosen such that the system is approximated by any Lorentzian, stationary eternal black hole. This approach opens the possibility of including explicitly the internal degrees of freedom of a physical black hole in path integral descriptions of its thermodynamical properties. If the functional integral is interpreted as the density of states of the system, the corresponding entropy equals ${\cal S} = A_H/4 - A_H/4 = 0$ in the semiclassical approximation, where $A_H$ is the area of the black hole horizon. The functional integral reflects the properties of a pure state. The description of the black hole density of states in terms of the eternal black hole functional integral is also discussed. 
  We propose a Lorentz-covariant Yang-Mills ``spin-gauge'' theory, where the function valued Pauli matrices play the role of a non-scalar Higgs-field. As symmetry group we choose $SU(2) \times U(1)$ of the 2-spinors describing particle/antiparticle states. After symmetry breaking a non-scalar Lorentz-covariant Higgs-field gravity appears, which can be interpreted within a classical limit as Einstein's metrical theory of gravity, where we restrict ourselves in a first step to its linearized version. 
  An exact solution of Einstein's field equations for a point mass surrounded by a static, spherically symmetric fluid of strings is presented. The solution is singular at the origin. Near the string cloud limit there is a $1/r$ correction to Newton's force law. It is noted that at large distances and small accelerations, this law coincides with the phenomenological force law invented by Milgrom in order to explain the flat rotation curves of galaxies without introducing dark matter. When interpreted in the context of a cosmological model with a string fluid, the new solution naturally explains why the critical acceleration of Milgrom is of the same order of magnitude as the Hubble parameter. 
  We calculate the one-loop corrections to the free energy and to the entropy for fields with arbitrary spins in the space $S^1\otimes H^N$. For conformally invariant fields by means of a conformal transformation of the metric the results are valid in Rindler space with $D=N+1$ dimensions. We use the zeta regularization technique which yields an ultraviolet finite result for the entropy per unit area. The problem of the infinite area factor in the entropy which arises equally in Rindler space and in the black hole background is addressed in the light of a factor space $H^N/\Gamma$. 
  Spacetime singularities in numerical relativity can be avoided by excising a region of the computational domain from inside the apparent horizon. We report on results of such a scheme that is based on using ({\it i}) a horizon locking coordinate which locks the coordinate system to the geometry, and ({\it ii}) a finite differencing scheme which respects the causal structure of the spacetime. With this technique a black hole can be evolved accurately well beyond $t=1000M$, where $M$ is the black hole mass. 
  We have developed a powerful and efficient method for locating the event horizon of a black hole spacetime, making possible the study of the dynamics of event horizons in numerical relativity. We describe the method and apply it to a colliding black hole spacetime. 
  We discuss the use of Adaptative Mesh Refinement (AMR) techniques in dynamical black hole spacetimes. We compare results between traditional fixed grid methods and a new AMR application for the 1-D Schwarzschild case. 
  We report on an efficient method for locating the apparent horizon in numerically constructed dynamical 3D black hole spacetimes. Instead of solving the zero expansion partial differential equation, our method uses a minimization procedure. Converting the PDE problem to minimization eliminates the difficulty of imposing suitable boundary conditions for the PDE. We demonstrate the effectiveness of this method in both 2D and 3D cases. The method is also highly parallelizable for implementation in massively parallel computers. 
  Numerical relativity is finally coming of age with the development of massively parallel computers. 3D problems, which were completely intractable several years ago due to limited computer power, can now be performed with grid sizes of about $200^3$. We report on several new codes developed for solving the full 3D Einstein equations, and present results of using them to evolve black holes and gravitational waves. 
  I report on recent progress in the exciting field of Numerical Relativity, with special attention to black hole horizons. 
  Using the momentum constraint, the standard evolution system is written in a fully first order form. The class of first order invariant algebraic slicing conditions is considered. The full set of characteristic fields is explicitly given. Characteristic speeds associated to the gauge dependent eigenfields (gauge speeds) are related to light speed. 
  We showed that compact bosonic objects can be formed through a process we called gravitational cooling. A central issue in the subject of boson star is whether a classical field configuration, {\it e.g.,} one described by the Klein-Gordon equation, can collapse to form a compact star-like object, as there is apparently no dissipation in the Klein-Gordon equation. We demonstrated that there IS an efficient cooling mechanism to get rid of the kinetic energy for the formation of a compact object purely through the gravitational coupling, a mechanism universal to all self-graviting fields. Implications of this mechanism are discussed, including the abundance of bosonic stars in the universe, and the possibility of ruling out the axion as a dark matter condidate. 
  The absence of gravitational radiation in Kinnersley's ``photon rocket'' solution of Einstein's equations is clarified by studying the mathematically well-defined problem of point-like photon rockets in Minkowski space (i.e. massive particles emitting null fluid anisotro\-pically and accelerating because of the recoil). We explicitly compute the (uniquely defined) {\it linearized} retarded gravitational waves emitted by such objects, which are the coherent superposition of the gravitational waves generated by the motion of the massive point-like rocket and of those generated by the energy-momentum distribution of the photon fluid. In the special case (corresponding to Kinnersley's solution) where the anisotropy of the photon emission is purely dipolar we find that the gravitational wave amplitude generated by the energy-momentum of the photons exactly cancels the usual $1/r$ gravitational wave amplitude generated by the accelerated motion of the rocket. More general photon anisotropies would, however, generate genuine gravitational radiation at infinity. Our explicit calculations show the compatibility between the non-radiative character of Kinnersley's solution and the currently used gravitational wave generation formalisms based on post-Minkowskian perturbation theory. 
  The confrontation between General Relativity and experimental results, notably binary pulsar data, is summarized and its significance discussed. The agreement between experiment and theory is numerically very impressive. However, some recent theoretical findings (existence of non-perturbative strong-field effects, natural cosmological attraction toward zero scalar couplings) suggest that the present agreement between Einstein's theory and experiment might be a red herring and provide new motivations for improving the experimental tests of gravity. 
  The RST Model is given boundary term and Z-field so that it is well-posed and local. The Euclidean method is described for general theory and used to calculate the RST intrinsic entropy. The evolution of this entropy for the shockwave solutions is found and obeys a second law. 
  This paper studies gravitational collapse of a complex scalar field at the threshold for black hole formation, assuming that the collapse is spherically symmetric and continuously self-similar. A new solution of the coupled Einstein-scalar field equations is derived, after a small amount of numerical work with ordinary differential equations. The universal scaling and echoing behavior discovered by Choptuik in spherically symmetrical gravitational collapse appear in a somewhat different form. Properties of the endstate of the collapse are derived: The collapse leaves behind an irregular outgoing pulse of scalar radiation, with exactly flat spacetime within it. 
  We review the consistent histories formulations of quantum mechanics developed by Griffiths, Omn\`es and Gell-Mann and Hartle, and describe the classification of consistent sets. We illustrate some general features of consistent sets by a few simple lemmas and examples. We consider various interpretations of the formalism, and examine the new problems which arise in reconstructing the past and predicting the future. It is shown that Omn\`es' characterisation of true statements --- statements which can be deduced unconditionally in his interpretation --- is incorrect. We examine critically Gell-Mann and Hartle's interpretation of the formalism, and in particular their discussions of communication, prediction and retrodiction, and conclude that their explanation of the apparent persistence of quasiclassicality relies on assumptions about an as yet unknown theory of experience. Our overall conclusion is that the consistent histories approach illustrates the need to supplement quantum mechanics by some selection principle in order to produce a fundamental theory capable of unconditional predictions. 
  This is the first paper in a series on event horizons in numerical relativity. In this paper we present methods for obtaining the location of an event horizon in a numerically generated spacetime. The location of an event horizon is determined based on two key ideas: (1) integrating backward in time, and (2) integrating the whole horizon surface. The accuracy and efficiency of the methods are examined with various sample spacetimes, including both analytic (Schwarzschild and Kerr) and numerically generated black holes. The numerically evolved spacetimes contain highly distorted black holes, rotating black holes, and colliding black holes. In all cases studied, our methods can find event horizons to within a very small fraction of a grid zone. 
  It was recently shown that spacetime singularities in numerical relativity could be avoided by excising a region inside the apparent horizon in numerical evolutions. In this paper we report on the details of the implementation of this scheme. The scheme is based on using (1)~a horizon locking coordinate which locks the coordinate system to the geometry, and (2)~a finite differencing scheme which respects the causal structure of the spacetime. We show that the horizon locking coordinate can be affected by a number of shift conditions, such as a ``distance freezing'' shift, an ``area freezing'' shift, an ``expansion freezing'' shift, or the minimal distortion shift. The causal differencing scheme is illustrated with the evolution of scalar fields, and its use in evolving the Einstein equations is studied. We compare the results of numerical evolutions with and without the use of this horizon boundary condition scheme for spherical black hole spacetimes. With the boundary condition a black hole can be evolved accurately well beyond $t=1000 M$, where $M$ is the black hole mass. 
  Spherically symmetric (1D) black-hole spacetimes are considered as a test for numerical relativity. A finite difference code, based in the hyperbolic structure of Einstein's equations with the harmonic slicing condition is presented. Significant errors in the mass function are shown to arise from the steep gradient zone behind the black hole horizon, which challenge the Computational Fluid Dynamics numerical methods used in the code. The formalism is extended to moving numerical grids, which are adapted to follow horizon motion. The black hole exterior region can then be modeled with higher accuracy. 
  We present a new formulation of the Einstein equations that casts them in an explicitly first order, flux-conservative, hyperbolic form. We show that this now can be done for a wide class of time slicing conditions, including maximal slicing, making it potentially very useful for numerical relativity. This development permits the application to the Einstein equations of advanced numerical methods developed to solve the fluid dynamic equations, {\em without} overly restricting the time slicing, for the first time. The full set of characteristic fields and speeds is explicitly given. 
  We have developed a new numerical code to study the evolution of distorted, rotating black holes. We discuss the numerical methods and gauge conditions we developed to evolve such spacetimes. The code has been put through a series of tests, and we report on (a) results of comparisons with codes designed to evolve non-rotating holes, (b) evolution of Kerr spacetimes for which analytic properties are known, and (c) the evolution of distorted rotating holes. The code accurately reproduces results of the previous NCSA non-rotating code and passes convergence tests. New features of the evolution of rotating black holes not seen in non-rotating holes are identified. With this code we can evolve rotating black holes up to about $t=100M$, depending on the resolution and angular momentum. We also describe a new family of black hole initial data sets which represent rotating holes with a wide range of distortion parameters, and distorted non-rotating black holes with odd-parity radiation. Finally, we study the limiting slices for a maximally sliced rotating black hole and find good agreement with theoretical predictions. 
  We have developed a numerical code to study the evolution of distorted, rotating black holes. This code is used to evolve a new family of black hole initial data sets corresponding to distorted ``Kerr'' holes with a wide range of rotation parameters, and distorted Schwarzschild black holes with odd-parity radiation. Rotating black holes with rotation parameters as high as $a/m=0.87$ are evolved and analyzed in this paper. The evolutions are generally carried out to about $t=100M$, where $M$ is the ADM mass. We have extracted both the even- and odd-parity gravitational waveforms, and find the quasinormal modes of the holes to be excited in all cases. We also track the apparent horizons of the black holes, and find them to be a useful tool for interpreting the numerical results. We are able to compute the masses of the black holes from the measurements of their apparent horizons, as well as the total energy radiated and find their sum to be in excellent agreement with the ADM mass. 
  We present a derivation of the equation of motion for a test-particle in the framework of the nonsymmetric gravitational theory. Three possible couplings of the test-particle to the non-symmetric gravitational field are explored. The equation of motion is found to be similar in form to the standard geodesic equation of general relativity, but with an extra antisymmetric force term present. The equation of motion is studied for the case of a static, spherically symmetric source, where the extra force term is found to take the form of a Yukawa force. 
  The quantum theory of linearized perturbations of the gravitational field of a Schwarzschild black hole is presented. The fundamental operators are seen to be the perturbed Weyl scalars $\dot\Psi_0$ and $\dot\Psi_4$ associated with the Newman-Penrose description of the classical theory. Formulae are obtained for the expectation values of the modulus squared of these operators in the Boulware, Unruh and Hartle-Hawking quantum states. Differences between the renormalized expectation values of both $\bigl| \dot\Psi_0 \bigr|^2$ and $\bigl| \dot\Psi_4 \bigr|^2$ in the three quantum states are evaluated numerically. 
  Supersymmetric Wess-Zumino type models are considered as classical material media that can be interpreted as fluids of ordered strings with heat flow along the strings or a mixture of fluids of ordered strings with either a cloud of particles or a flux of directed radiation. 
We calculate the first quantum gravitational correction term to the trace anomaly in De Sitter space from the Wheeler-DeWitt equation. This is obtained through an expansion of the full wave functional for gravity and a conformally coupled scalar field in powers of the Planck mass. We also discuss a quantum gravity induced violation of unitarity and comment on its possible relevance for inflation. 
  We show that the pseudotensors of Einstein, Tolman, Landau and Lifshitz, Papapetrou, and Weinberg (ETLLPW) give the same distributions of energy, linear momentum and angular momentum, for any Kerr-Schild metric. This result generalizes a previous work by G\"urses and G\"ursey that dealt only with the pseudotensors of Einstein and Landau and Lifshitz. We compute these distributions for the Kerr-Newman and Bonnor-Vaidya metrics and find reasonable results. All calculations are performed without any approximation in Kerr-Schild Cartesian coordinates. For the Reissner-Nordstr\"{o}m metric these definitions give the same result as the Penrose quasi-local mass. For the Kerr black hole the entire energy is confined to its interior whereas for the Kerr-Newman black hole, as expected, the energy is shared by its interior as well as exterior. The total energy and angular momentum of the Kerr-Newman black hole are $M$ and $ M a$, respectively ($M$ is the mass parameter and $a$ is the rotation parameter). The energy distribution for the Bonnor-Vaidya metric is the same as the Penrose quasi-local mass obtained by Tod. 
The electromagnetic radiation that falls into a Reissner-Nordstrom black hole develops a ``blue sheet'' of infinite energy density at the Cauchy horizon. We consider classical electromagnetic fields (that were produced during the collapse and then backscattered into the black hole), and investigate the blue-sheet effects of these fields on infalling objects within a simplified model. These effects are found to be finite and even negligible for typical parameters. 
This is a contribution on the controversy about junction conditions for classical signature change. The central issue in this debate is whether the extrinsic curvature on slices near the hypersurface of signature change has to be continuous ({\it weak} signature change) or to vanish ({\it strong} signature change). Led by a Lagrangian point of view, we write down eight candidate action functionals $S_1$,\dots $S_8$ as possible generalizations of general relativity and investigate to what extent each of these defines a sensible variational problem, and which junction condition is implied. Four of the actions involve an integration over the total manifold. A particular subtlety arises from the precise definition of the Einstein-Hilbert Lagrangian density $|g|^{1/2} R[g]$. The other four actions are constructed as sums of integrals over singe-signature domains. The result is that {\it both} types of junction conditions occur in different models, i.e. are based on different first principles, none of which can be claimed to represent the ''correct'' one, unless physical predictions are taken into account. From a point of view of naturality dictated by the variational formalism, {\it weak} signature change is slightly favoured over {\it strong} one, because it requires less {\it \`a priori} restrictions for the class of off-shell metrics. In addition, a proposal for the use of the Lagrangian framework in cosmology is made. 
The coupling of gravity to dust helps to discover simple quadratic combinations of the gravitational super-Hamiltonian and supermomentum whose Poisson brackets strongly vanish. This leads to a new form of vacuum constraints which generate a true Lie algebra. We show that the coupling of gravity to a massless scalar field leads to yet another set of constraints with the same property, albeit not as simple as that based on the coupling to dust. 
The supersymmetric extension of charged point particle's motion is applied to investigate symmetries of gravitational fields and electromagnetic fields. We mainly focus on the role of the Killing- Yano tensors of both usual and generalized types. Results obtained by systematic analysis strengthen the connection of the Killing- Yano tensor and superinvariants (functions commuting with the supercharge). 
A Kantowski-Sachs cosmological model with an O(3) global defect as a source is studied in the context of the topological inflation scenario. 
  It is pointed out that the entropy of a membrane which is quantized perturbatively around a background position of fixed radius in a black hole spacetime is equal to the Bekenstein-Hawking entropy, if 1) the membrane surface is the horizon surface plus one Planck unit, and 2) its temperature is the Hawking temperature. (This is a comment on gr-qc 9411037.) 
The Lagrangian formulation of field theory does not provide any universal energy-momentum conservation law in order to analize that in gravitation theory. In Lagrangian field theory, we get different identities involving different stress energy-momentum tensors which however are not conserved, otherwise in the covariant multimomentum Hamiltonian formalism. In the framework of this formalism, we have the fundamental identity whose restriction to a constraint space can be treated the energy-momentum transformation law. This identity remains true also for gravity. Thus, the tools are at hand to investigate the energy-momentum conservation laws in gravitation theory. The key point consists in the feature of a metric gravitational field whose canonical momenta on the constraint space are equal to zero. 
In 1991 Gott presented a solution of Einstein's field equations in 2+1 dimensions with $\Lambda = 0$ that contained closed timelike curves (CTC's). This solution was remarkable because at first it did not seem to be unphysical in any other respect. Later, however, it was shown that Gott's solution is tachyonic in a certain sense. Here the case $\Lambda < 0$ is discussed. We show that it is possible to construct CTC's also in this case, in a way analogous to that used by Gott. We also show that this construction still is tachyonic. $\Lambda < 0$ means that we are dealing with Anti-de Sitter space, and since the CTC-construction necessitates some understanding of its structure, a few pages are devoted to this subject. 
We study the coupled Einstein-Klein-Gordon equations for a complex scalar field with and without a quartic self-interaction in a curvatureless Friedman-Lema\^{\i}\-tre Universe. The equations can be written as a set of four coupled first order non-linear differential equations, for which we establish the phase portrait for the time evolution of the scalar field. To that purpose we find the singular points of the differential equations lying in the finite region and at infinity of the phase space and study the corresponding asymptotic behavior of the solutions. This knowledge is of relevance, since it provides the initial conditions which are needed to solve numerically the differential equations. For some singular points lying at infinity we recover the expected emergence of an inflationary stage. 
  The Kerr solution is defined by a null congruence which is geodesic and shear free and has a singular line contained in a bounded region of space. A generalization of the Kerr congruence for nonstationary case is obtained. We find a nonstationary shear free geodesic null congruence which is generated by a given analytical complex world line. Solutions of the Einstein equations are analyzed. It is shown that there exists complex radiative solution which is generalization of the Kerr solution and the Kinnersley accelerating solution for "photon rocket". 
It is shown that the Fermi-Walker gauge allows the general solution of determining the metric given the sources, in terms of simple quadratures. We treat the general stationary problem providing explicit solving formulas for the metric and explicit support conditions for the energy momentum tensor. The same type of solution is obtained for the time dependent problem with circular symmetry. In both cases the solutions are classified in terms of the invariants of the Wilson loops outside the sources. The Fermi-Walker gauge, due to its physical nature, allows to exploit the weak energy condition and in this connection it is proved that, both for open and closed universes with rotational invariance, the energy condition imply the total absence of closed time like curves. The extension of this theorem to the general stationary problem, in absence of rotational symmetry is considered. At present such extension is subject to some assumptions on the behavior of the determinant of the dreibein in this gauge. PACS number: 0420 
We give a general derivation of the gravitational hamiltonian starting from the Einstein-Hilbert action, keeping track of all surface terms. The surface term that arises in the hamiltonian can be taken as the definition of the `total energy', even for spacetimes that are not asymptotically flat. (In the asymptotically flat case, it agrees with the usual ADM energy.) We also discuss the relation between the euclidean action and the hamiltonian when there are horizons of infinite area (e.g. acceleration horizons) as well as the usual finite area black hole horizons. Acceleration horizons seem to be more analogous to extreme than nonextreme black holes, since we find evidence that their horizon area is not related to the total entropy. 
A canonical formalism for spherical symmetry, originally developed by Kucha\v{r} to describe vacuum Schwarzschild black holes, is extended to include a spherically symmetric, massless, scalar field source. By introducing the ADM mass as a canonical coordinate on phase space, one finds that the super-Hamiltonian and supermomentum constraints for the coupled system simplify considerably. Yet, despite this simplification, it is difficult to find a functional time formalism for the theory. First, the configuration variable that played the role of time for the vacuum theory is no longer a spacetime scalar once spherically symmetric matter is coupled to gravity. Second, although it is possible to perform a canonical transformation to a new set of variables in terms of which the super-Hamiltonian and supermomentum constraints can be solved, the new time variable also fails to be a spacetime scalar. As such, our solutions suffer from the so-called {\it spacetime problem of time}. A candidate for a time variable that {\it is} a spacetime scalar is presented. Problems with turning this variable into a canonical coordinate on phase space are discussed. 
  In canonical quantum gravity, when space is a compact manifold with boundary there is a Hamiltonian given by an integral over the boundary. Here we compute the action of this `boundary Hamiltonian' on observables corresponding to open Wilson lines in the new variables formulation of quantum gravity. In cases where the boundary conditions fix the metric on the boundary (e.g., in the asymptotically Minkowskian case) one can obtain a finite result, given by a `shift operator' generating translations of the Wilson line in the direction of its tangent vector. A similar shift operator serves as the Hamiltonian constraint in Morales-T\'ecotl and Rovelli's work on quantum gravity coupled to Weyl spinors. This suggests the appearance of an induced field theory of Weyl spinors on the boundary, analogous to that considered in Carlip's work on the statistical mechanics of the 2+1-dimensional black hole. 
  A unified description is presented of the physical observables and thermodynamic variables associated with black hole solutions in generic 2-D dilaton gravity. The Dirac quantization of these theories is reviewed and an intriguing relationship between the entropy of the black hole and the WKB phase of the corresponding physical wave functionals is revealed. 
We find a class of (2+1)-dimensional spacetimes admitting Killing spinors appropriate to (2,0) adS-supergravity. The vacuum spacetimes include anti-de Sitter (adS) space and charged extreme black holes, but there are many others, including spacetimes of arbitrarily large negative energy that have only conical singularities, and the spacetimes of fractionally charged point particles. The non-vacuum spacetimes are those of self-gravitating solitons obtained by coupling (2,0) adS supergravity to sigma-model matter. We show, subject to a condition on the matter currents (satisfied by the sigma model), and a conjecture concerning global obstructions to the existence of certain types of spinor fields, that the mass of each supersymmetric spacetime saturates a classical bound, in terms of the angular momentum and charge, on the total energy of arbitrary field configurations with the same boundary conditions, although these bounds may be violated quantum mechanically. 
  An alternate Hamiltonian H different from Ostrogradski's one is found for the Lagrangian L = L(q, \dot q, \ddot q). We add a suitable divergence to L and insert a=q and b=\ddot q. Contrary to other approaches no constraint is needed because \ddot a = b is one of the canonical equations. Another canonical equation becomes equivalent to the fourth-order Euler-Lagrange equation of L. Usually, H becomes quadratic in the momenta, whereas the Ostrogradski approach has Hamiltonians always linear in the momenta. For non-linear L=F(R), G=dF/dR \ne 0 the Lagrangians L and \hat L=\hat F(\hat R) with \hat F=2R/G\sp 3-3L/G\sp 4, \hat g_{ij}=G\sp 2g_{ij} and \hat R=3R/G\sp 2 - 4L/G \sp 3 give conformally equivalent fourth-order field equations being dual to each other. This generalizes Buchdahl's result for L=R^2. The exact fourth-order gravity cosmological solutions found by Accioly and Chimento are interpreted from the viewpoint of the instability of fourth-order theories and how they transform under this duality. Finally, the alternate Hamiltonian is applied to deduce the Wheeler-De Witt equation for fourth-order gravity models more systematically than before. 
  The issue of how to deal with the modular transformations -- large diffeomorphisms -- in (2+1)-quantum gravity on the torus is discussed. I study the Chern-Simons/connection representation and show that the behavior of the modular transformations on the reduced configuration space is so bad that it is possible to rule out all finite dimensional unitary representations of the modular group on the Hilbert space of $L^2$-functions on the reduced configuration space. Furthermore, by assuming piecewise continuity for a dense subset of the vectors in any Hilbert space based on the space of complex valued functions on the reduced configuration space, it is shown that finite dimensional representations are excluded no matter what inner-product we define in this vector space. A brief discussion of the loop- and ADM-representations is also included. 
It has been shown for low-spin fields that the use of only the self-dual part of the connection as basic variable does not lead to spurious equations or inconsistencies. We slightly generalize the form of the chiral Lagrangian of half-integer spin fields and express its imaginary part in a simple form. If the imaginary part is non-vanishing, it will lead to spurious equations. As an example, for (Majorana) Rarita-Schwinger fields the equations of motion of the torsion is solved and it is shown that it vanishes owing to the Fierz identity. 
  Scalar fields coupled to gravity via $\xi R {\Phi}^2$ in arbitrary Friedmann-Robertson-Walker backgrounds can be represented by an effective flat space field theory. We derive an expression for the scalar energy density where the effective scalar mass becomes an explicit function of $\xi$ and the scale factor. The scalar quartic self-coupling gets shifted and can vanish for a particular choice of $\xi$. Gravitationally induced symmetry breaking and de-stabilization are possible in this theory. 
  Exact solutions of the Einstein-Maxwell equations that describe moving black holes in a cosmological setting are discussed with the aim of discovering the global structure and testing cosmic censorship. Continuation beyond the horizons present in these solutions is necessary in order to identify the global structure. Therefore the possibilities and methods of analytic extension of geometries are briefly reviewed. The global structure of the Reissner-Nordstr\"om-de Sitter geometry is found by these methods. When several black holes are present, the exact solution is no longer everywhere analytic, but less smooth extensions satisfying the Einstein equations everywhere are possible. Some of these provide counterexamples to cosmic censorship. 
  We consider the lagrangian $L=F(R)$ in classical (=non-quantized) two-dimensional fourth-order gravity and give new relations to Einstein's theory with a non-minimally coupled scalar field. We distinguish between scale-invariant lagrangians and scale-invariant field equations. $L$ is scale-invariant for $F = c_1 R\sp {k+1}$ and a divergence for $F=c_2 R$. The field equation is scale-invariant not only for the sum of them, but also for $F=R\ln R$. We prove this to be the only exception and show in which sense it is the limit of $\frac{1}{k} R\sp{k+1}$ as $k\to 0$. More generally: Let $H$ be a divergence and $F$ a scale-invariant lagrangian, then $L= H\ln F $ has a scale-invariant field equation. Further, we comment on the known generalized Birkhoff theorem and exact solutions including black holes. 
Using the general solution to the Einstein equations on intersecting null surfaces developed by Hayward, we investigate the non-linear instability of the Cauchy horizon inside a realistic black hole. Making a minimal assumption about the free gravitational data allows us to solve the field equations along a null surface crossing the Cauchy Horizon. As in the spherical case, the results indicate that a diverging influx of gravitational energy, in concert with an outflux across the CH, is responsible for the singularity. The spacetime is asymptotically Petrov type N, the same algebraic type as a gravitational shock wave. Implications for the continuation of spacetime through the singularity are briefly discussed. 
  We make a rigorous study of classical field equations on a 2-dimensional signature changing spacetime using the techniques of operator theory. Boundary conditions at the surface of signature change are determined by forming self-adjoint extensions of the Schr\"odinger Hamiltonian. We show that the initial value problem for the Klein--Gordon equation on this spacetime is ill-posed in the sense that its solutions are unstable. Furthermore, if the initial data is smooth and compactly supported away from the surface of signature change, the solution has divergent $L^2$-norm after finite time. 
  The rate of gravitational-wave energy loss from inspiralling binary systems of compact objects of arbitrary mass is derived through second post-Newtonian (2PN) order $O[(Gm/rc^2)^2]$ beyond the quadrupole approximation. The result has been derived by two independent calculations of the (source) multipole moments. The 2PN terms, and in particular the finite mass contribution therein (which cannot be obtained in perturbation calculations of black hole spacetimes), are shown to make a significant contribution to the accumulated phase of theoretical templates to be used in matched filtering of the data from future gravitational-wave detectors. 
  Exact black hole and cosmological solutions are obtained for a special two-dimensional dilaton-spectator ($\phi-\psi$) theory of gravity. We show how in this context any desired spacetime behaviour can be determined by an appropriate choice of a dilaton potential function $V(\phi)$ and a ``coupling function'' $l(\phi)$ in the action. We illustrate several black hole solutions as examples. In particular, asymptotically flat double- and multiple- horizon black hole solutions are obtained. One solution bears an interesting resemblance to the $2D$ string-theoretic black hole and contains the same thermodynamic properties; another resembles the $4D$ Reissner-Nordstrom solution. We find two characteristic features of all the black hole solutions. First the coupling constants in $l(\phi)$ must be set equal to constants of integration (typically the mass). Second, the spectator field $\psi$ and its derivative $\psi^{'}$ both diverge at any event horizon. A test particle with ``spectator charge" ({\it i.e.} one coupled either to $\psi$ or $\psi^{'}$), will therefore encounter an infinite tidal force at the horizon or an ``infinite potential barrier'' located outside the horizon respectively. We also compute the Hawking temperature and entropy for our solutions. In $2D$ $FRW$ cosmology, two non-singular solutions which resemble two exact solutions in $4D$ string-motivated cosmology are obtained. In addition, we construct a singular model which describes the $4D$ standard non-inflationary big bang cosmology ($big-bang\rightarrow radiation\rightarrow dust$). Motivated by the similaritiesbetween $2D$ and $4D$ gravitational field equations in $FRW$ cosmology, we briefly discuss a special $4D$ dilaton-spectator action constructed from the bosonic part of the low energy heterotic string action and 
Gravitational waves generated by inspiralling compact binaries are investigated to the second--post-Newtonian (2PN) approximation of general relativity. Using a recently developed 2PN-accurate wave generation formalism, we compute the gravitational waveform and associated energy loss rate from a binary system of point-masses moving on a quasi-circular orbit. The crucial new input is our computation of the 2PN-accurate ``source'' quadrupole moment of the binary. Tails in both the waveform and energy loss rate at infinity are explicitly computed. Gravitational radiation reaction effects on the orbital frequency and phase of the binary are deduced from the energy loss. In the limiting case of a very small mass ratio between the two bodies we recover the results obtained by black hole perturbation methods. We find that finite mass ratio effects are very significant as they increase the 2PN contribution to the phase by up to 52\%. The results of this paper should be of use when deciphering the signals observed by the future LIGO/VIRGO network of gravitational-wave detectors. 
This paper derives the expressions of the multipole moments of an isolated gravitating source with an accuracy corresponding to the second post-Newtonian (2-PN) approximation of general relativity. The moments are obtained by a procedure of matching of the external gravitational field of the source to the inner field, and are found to be given by integrals extending over the stress-energy distribution of the matter fields and the gravitational field. Although this is not manifest on their expressions, the moments have a compact support limited to the material source only (they are thus perfectly well-defined mathematically). From the multipole moments we deduce the expressions of the asymptotic gravitational waveform and associated energy generated by the source at the 2-PN approximation. This work, together with a forthcoming work devoted to the application to coalescing compact binaries, will be used in the future observations of gravitational radiation by laser interferometric detectors. 
  The compatibility between the general relativity and the mathematical property that the space-times are embedded manifolds are further examined. In particular we study the uniqueness of the signature of the embedding space for a given space-time. The interpretation of the twisting vector as a gauge potential is also implemented. 
  For arbitrary high order of the field equation one can always find examples where the de Sitter space-time is an attractor solution in the set of the spatially flat Friedmann-Robertson-Walker models. 
  Some aspects of two-dimensional gravity coupled to matter fields, especially to the Sine-Gordon-model are examined. General properties and boundary conditions of possible soliton-solutions are considered. Analytic soliton-solutions are discovered and the structure of the induced space-time geometry is discussed. These solutions have interesting features and may serve as a starting point for further investigations. 
We show that, contrary to recent criticism, our previous work yields a reasonable class of solutions for the massless scalar field in the presence of signature change. 
  We discuss two expressions for the conserved quantities (energy momentum and angular momentum) of the Poincar\'e Gauge Theory. We show, that the variations of the Hamiltonians, of which the expressions are the respective boundary terms, are well defined, if we choose an appropriate phase space for asymptotic flat gravitating systems. Furthermore, we compare the expressions with others, known from the literature. 
  It is shown that a relativistic (i.e. a Poincar{\' e} invariant) theory of extended objects (called p-branes) is not necessarily invariant under reparametrizations of corresponding $p$-dimensional worldsheets (including worldlines for $p = 0$). Consequnetly, no constraints among the dynamical variables are necessary and quantization is straightforward. Additional degrees of freedom so obtained are given a physical interpretation as being related to membrane's elastic deformations ("wiggleness"). In particular, such a more general, unconstrained theory implies as solutions also those p-brane states that are solutions of the conventional theory of the Dirac-Nambu-Goto type. 
The role of the parameter `$a$' associated with the Kerr metric which represents the angular momentum per unit mass of a rotating massive object has been converted into determining the rotationally induced quadrapole electric field outside a rotating massive object with external dipole magnetic field. A comparison of the result with that of the Newtonian case implies that the parameter `$a$' represents the angular momentum per unit mass of a rotating hollow object. 
I survey the physics of black holes in two and three spacetime dimensions, with special attention given to an understanding of their exterior and interior properties. 
We examine homogeneous but anisotropic cosmologies in scalar-tensor gravity theories, including Brans-Dicke gravity. We present a method for deriving solutions for any isotropic perfect fluid with a barotropic equation of state ($p\propto\rho$) in a spatially flat (Bianchi type~I) cosmology. These models approach an isotropic, general relativistic solution as the expansion becomes dominated by the barotropic fluid. All models that approach general relativity isotropize except for the case of a maximally stiff fluid. For stiff fluid or radiation or in vacuum we are able to give solutions for arbitrary scalar-tensor theories in a number of anisotropic Bianchi and Kantowski-Sachs metrics. We show how this approach can also be used to derive solutions from the low-energy string effective action. We discuss the nature, and possibly avoidance of, the initial singularity where both shear and non-Einstein behavior is important. 
  The averaged null energy condition has been recently shown to hold for linear quantum fields in a large class of spacetimes. Nevertheless, it is easy to show by using a simple scaling argument that ANEC as stated cannot hold generically in curved four-dimensional spacetime, and this scaling argument has been widely interpreted as a death-blow for averaged energy conditions in quantum field theory. In this note I propose a simple generalization of ANEC, in which the right-hand-side of the ANEC inequality is replaced by a finite (but in general negative) state-independent lower bound. As long as attention is focused on asymptotically well-behaved spacetimes, this generalized version of ANEC is safe from the threat of the scaling argument, and thus stands a chance of being generally valid in four-dimensional curved spacetime. I argue that when generalized ANEC holds, it has implications for the non-negativity of total energy and for singularity theorems similar to the implications of ANEC. In particular, I show that if generalized ANEC is satisfied in static traversable wormhole spacetimes (which is likely but remains to be shown), then macroscopic wormholes (but not necessarily microscopic, Planck-size wormholes) are ruled out by quantum field theory. 
Fundamental errors exist in the above-mentioned article, which attempts to justify previous erroneous claims concerning signature change. In the simplest example, the authors' proposed ``solutions'' do not satisfy the relevant equation, as may be checked by substitution. These ``solutions'' are also different to the authors' originally proposed ``solutions'', which also do not satisfy the equation. The variational equations obtained from the authors' ``actions'' are singular at the change of signature. The authors' ``distributional field equations'' are manifestly ill defined. 
Spectrum of density perturbations in the Universe generated from quantum-gravitational fluctuations in slow-roll-over inflationary scenarios with the Brans-Dicke gravity is calculated. It is shown that after inflation the isocurvature mode of perturbations may be neglected as compared to the adiabatic mode, and that an amplitude of the latter mode is not significantly different from that in the Einstein gravity. However, the account of the isocurvature mode is necessary to obtain the quantitatively correct spectrum of adiabatic perturbations. 
We consider a general situation where a charged massive scalar field $\phi(x)$ at finite temperature interacts with a magnetic flux cosmic string. We determine a general expression for the Euclidean thermal Green's function of the massive scalar field and a handy expression for a massless scalar field. With this result, we evaluate the thermal average $<\phi^{2}(x)>$ and the thermal average of the energy-momentum tensor of a nonconformal massless scalar field. 
The Lagrangean of $N=1$ supergravity is dimensionally reduced to one (time-like) dimension assuming spatial homogeneity of any Bianchi type within class A of the classification of Ellis and McCallum. The algebra of the supersymmetry generators, the Lorentz generators, the diffeomorphism generators and the Hamiltonian generator is determined and found to close. In contrast to earlier work, infinitely many physical states with non-vanishing even fermion number are found to exist in these models, indicating that minisuperspace models in supergravity may be just as useful as in pure gravity. 
  We discuss the possibility of incorporating non-Riemannian parallel transport into Regge calculus. It is shown that every Regge lattice is locally equivalent to a space of constant curvature. Therefore well known-concepts of differential geometry imply the definition of an arbitrary linear affine connection on a Regge lattice. 
We propose a new, discretized model for the study of 3+1-dimensional canonical quantum gravity, based on the classical $SL(2,\C)$-connection formulation. The discretization takes place on a topological $N^3$- lattice with periodic boundary conditions. All operators and wave functions are constructed from one-dimensional link variables, which are regarded as the fundamental building blocks of the theory. The kinematical Hilbert space is spanned by polynomials of certain Wilson loops on the lattice and is manifestly gauge- and diffeomorphism- invariant. The discretized quantum Hamiltonian $\hat H$ maps this space into itself. We find a large sector of solutions to the discretized Wheeler-DeWitt equation $\hat H\psi=0$, which are labelled by single and multiple Polyakov loops. These states have a finite norm with respect to a natural scalar product on the space of holomorphic $SL(2,\C)$-Wilson loops. We also investigate the existence of further solutions for the case of the $1^3$-lattice. - Our results provide for the first time a rigorous, regularized framework for studying non-perturbative quantum gravity. 
Table of contents:    -Editorial and Correspondents   Gravity news:    -LISA Recommended to ESA as Possible New Cornerstone Mission, Peter Bender.    -LIGO Project News, Stan Whitcomb.   Research briefs:    -Some Recent Work in General Relativistic Astrophysics, John Friedman.   -Pair Creation of Black Holes, Gary Horowitz.    -Conformal Field Equations and Global Properties of Spacetimes, Bernd Schmidt  Conference reports:    -Aspen Workshop on Numerical Investigations of Singularities in GR, Susan    Scott    -Second Annual Penn State Conference: Quantum Geometry, Abhay Ashtekar    -First Samos Meeting, Spiros Cotsakis and Dieter Brill   -Aspen Conference on Gravitational Waves and Their Detection, Syd Meshkov 
We obtain exact analytic solutions for a typica autonomous dynamical system, related to the problem of a vector field nonminimally coupled to gravity. 
We discuss some recent results on black hole thermodynamics within the context of effective gravitational actions including higher-curvature interactions. Wald's derivation of the First Law demonstrates that black hole entropy can always be expressed as a local geometric density integrated over a space-like cross-section of the horizon. In certain cases, it can also be shown that these entropy expressions satisfy a Second Law. One such simple example is considered from the class of higher curvature theories where the Lagrangian consists of a polynomial in the Ricci scalar. 
  Riemannian geometry in four dimensions, including Einstein's equations, can be described by means of a connection that annihilates a triad of two-forms (rather than a tetrad of vector fields). Our treatment of the conformal factor of the metric differs from the original presentation of this result, due to 'tHooft. In the action the conformal factor now appears as a field to be varied. 
Einstein gravity minimally coupled to a self-interacting scalar field is investigated in the static and isotropic situation. We explicitly construct in partially closed form a new black-hole solution with exponentially decaying scalar hair. The symmetric interaction potential has both signs and a triple-well shape with a smooth but non-analytic minimum at vanishing field. We present numerical data as well as double series expansions around spatial infinity. 
  On the basis of the Lie derivative method in a metric-affine space-time it is shown that in the metric-affine gravitational theory the energy-momentum conservation law and therefore the equations of the matter motion are the consequence (as in the GR) of the gravitational field equations. The possi- bility of the detection of the space-time non-metric properties is discussed. 
Recent work on the expected event rate of neutron star inspiral signals in the LIGO detector is summarized. The observed signals will be from inspirals at cosmological distances, and the important cosmological effects on the event rate and spectrum are discussed.  This paper is a contribution to the proceedings of the 17th Texas Symposium held in Munich, 12-17 December 1994. 
Fractal basin boundaries provide an important means of characterizing chaotic systems. We apply these ideas to general relativity, where other properties such as Lyapunov exponents are difficult to define in an observer independent manner. Here we discuss the difficulties in describing chaotic systems in general relativity and investigate the motion of particles in two- and three-black-hole spacetimes. We show that the dynamics is chaotic by exhibiting the basins of attraction of the black holes which have fractal boundaries. Overcoming problems of principle as well as numerical difficulties, we evaluate Lyapunov exponents numerically and find that some trajectories have a positive exponent. 
The cosmic, general analitic solutions of the Brans--Dicke Theory for the flat space of homogeneous and isotropic models containing perfect, barotropic, fluids are seen to belong to a wider class of solutions --which includes cosmological models with the open and the closed spaces of the Friedmann--Robertson--Walker metric, as well as solutions for models with homogeneous but anisotropic spaces corresponding to the Bianchi--Type metric clasification-- when all these solutions are expressed in terms of reduced variables. The existence of such a class lies in the fact that the scalar field, $\phi$, times a function of the mean scale factor or ``volume element'', $a^3 = a_1 a_2 a_3$, which depends on time and on the barotropic index of the equation of state used, can be written as a function of a ``cosmic time'' reduced in terms of another function of the mean scale factor depending itself again on the barotropic index but independent of the metrics here employed. This reduction procedure permites one to analyze if explicitly given anisotropic cosmological solutions ``isotropize'' in the course of their time evolution. For if so can happen, it could be claimed that there exists a subclass of solutions that is stable under anisotropic perturbations. 
We analyse the massless wave equation on a class of two dimensional manifolds consisting of an arbitrary number of topological cylinders connected to one or more topological spheres. Such manifolds are endowed with a degenerate (non-globally hyperbolic) metric. Attention is drawn to the topological constraints on solutions describing monochromatic modes on both compact and non-compact manifolds. Energy and momentum currents are constructed and a new global sum rule discussed. The results offer a rigorous background for the formulation of a field theory of topologically induced particle production. 
Various calculations of the $S$ matrix have shown that it seems to be non unitary for interacting fields when there are closed timelike curves. It is argued that this is because there is loss of quantum coherence caused by the fact that part of the quantum state circulates on the closed timelike curves and is not measured at infinity. A prescription is given for calculating the superscattering matrix $\$ $ on space times whose parameters can be analytically continued to obtain a Euclidean metric. It is illustrated by a discussion of a spacetime in with two disks in flat space are identified. If the disks have an imaginary time separation, this corresponds to a heat bath. An external field interacting with the heat bath will lose quantum coherence. One can then analytically continue to an almost real separation of the disks. This will give closed timelike curves but one will still get loss of quantum coherence. 
  We examine the axi-dilatonic sector of low energy string theory and demonstrate how the gravitational interactions involving the axion and dilaton fields may be derived from a geometrical action principle involving the curvature scalar associated with a non-Riemannian connection. In this geometry the antisymmetric tensor 3-form field determines the torsion of the connection on the frame bundle while the gradient of the metric is determined by the dilaton field. By expressing the theory in terms of the Levi-Civita connection associated with the metric in the ``Einstein frame'' we confirm that the field equations derived from the non-Riemannian Einstein-Hilbert action coincide with the axi-dilaton sector of the low energy effective action derived from string theory. 
  We examine a 1 parameter class of actions describing the gravitational interaction between a pair of scalar fields and Einsteinian gravitation. When the parameter is positive the theory corresponds to an axi-dilatonic sector of low energy string theory. We exploit an SL(2,R) symmetry of the theory to construct a family of electromagnetically neutral solutions with non-zero axion and dilaton charge from solutions of the Brans-Dicke theory. We also comment on solutions to the theory with negative coupling parameter. 
  We reformulate the standard local equations of general relativity for asymptotically flat spacetimes in terms of two non-local quantities, the holonomy $H$ around certain closed null loops on characteristic surfaces and the light cone cut function $Z$, which describes the intersection of the future null cones from arbitrary spacetime points, with future null infinity. We obtain a set of differential equations for $H$ and $Z$ equivalent to the vacuum Einstein equations. By finding an algebraic relation between $H$ and $Z$ this set of equations is reduced to just two coupled equations: an integro-differential equation for $Z$ which yields the conformal structure of the underlying spacetime and a linear differential equation for the ``vacuum'' conformal factor. These equations, which apply to all vacuum asymptotically flat spacetimes, are however lengthy and complicated and we do not yet know of any solution generating technique. They nevertheless are amenable to an attractive perturbative scheme which has Minkowski space as a zeroth order solution. 
  We discuss the averaging hypothesis tacitly assumed in standard cosmology. Our approach is implemented in a "3+1" formalism and invokes the coarse graining arguments, provided and supported by the real-space Renormalization Group (RG) methods. Block variables are introduced and the recursion relations written down explicitly enabling us to characterize the corresponding RG flow. To leading order, the RG flow is provided by the Ricci-Hamilton equations studied in connection with the geometry of three-manifolds. The properties of the Ricci-Hamilton flow make it possible to study a critical behaviour of cosmological models. This criticality is discussed and it is argued that it may be related to the formation of sheet-like structures in the universe. We provide an explicit expression for the renormalized Hubble constant and for the scale dependence of the matter distribution. It is shown that the Hubble constant is affected by non-trivial scale dependent shear terms, while the spatial anisotropy of the metric influences significantly the scale-dependence of the matter distribution. 
In gravitation theory, the realistic fermion matter is described by spinor bundles associated with the cotangent bundle of a world manifold $X$. In this case, the Dirac operator can be introduced. There is the 1:1 correspondence between these spinor bundles and the tetrad gravitational fields represented by sections of the quotient $\Si$ of the linear frame bundle over $X$ by the Lorentz group. The key point lies in the fact that different tetrad fields imply nonequivalent representations of cotangent vectors to $X$ by the Dirac's matrices. It follows that a fermion field must be regarded only in a pair with a certain tetrad field. These pairs can be represented by sections of the composite spinor bundle $S\to\Si\to X$ where values of tetrad fields play the role of parameter coordinates, besides the familiar world coordinates. 
  We consider quantization of the Baierlein-Sharp-Wheeler form of the gravitational action, in which the lapse function is determined from the Hamiltonian constraint. This action has a square root form, analogous to the actions of the relativistic particle and Nambu string. We argue that path-integral quantization of the gravitational action should be based on a path integrand $\exp[ \sqrt{i} S ]$ rather than the familiar Feynman expression $\exp[ i S ]$, and that unitarity requires integration over manifolds of both Euclidean and Lorentzian signature. We discuss the relation of this path integral to our previous considerations regarding the problem of time, and extend our approach to include fermions. 
The energy and particle fluxes emitted by an accelerated two level atom are analysed in detail. It is shown both perturbatively and non perturbatively that the total number of emitted photons is equal to the number of transitions characterizing thermal equilibrium thereby confirming that each internal transition is accompanied by the emission of a Minkowski quantum. The mean fluxes are then decomposed according to the final state of the atom and the notion of conditional flux is introduced. This notion is generalized so as to study the energy content of the vacuum fluctuations that induce the transitions of the accelerated atom. The physical relevance of these conditional fluxes is displayed and contact is made with the formalism of Aharonov et al. The same decomposition is then applied to isolate, in the context of black hole radiation, the energy content of the particular vacuum fluctuations which are converted into on mass shell quanta. It is shown that initially these fluctuations are located around the light like geodesic that shall generate the horizon and have exponentially large energy densities. Upon exiting from the star they break up into two pieces. The external one is red shifted and becomes an on mass shell quantum, the other, its ''partner", ends up in the singularity. We avail ourselves of this analysis to study back reaction effects to the production of a single quantum. 
  The following issue is raised and discussed; when do families of foliations by hypersurfaces on a given four dimensional manifold become the null surfaces of some unknown, but to be determined, metric $g_{ab}(x)$? It follows from these results that one can use these surfaces as fundamental variables for GR. 
Recently there has been developed a reformulation of General Relativity - referred to as {\it the null surface version of GR} - where instead of the metric field as the basic variable of the theory, families of three-surfaces in a four-manifold become basic. From these surfaces themselves, a conformal metric, conformal to an Einstein metric, can be constructed. A choice of conformal factor turns them into Einstein metrics. The surfaces are then automatically characteristic surfaces of this metric. In the present paper we explore the linearization of this {\it null surface theory} and compare it with the standard linear GR. This allows a better understanding of many of the subtle mathematical issues and sheds light on some of the obscure points of the null surface theory. It furthermore permits a very simple solution generating scheme for the linear theory and the beginning of a perturbation scheme for the full theory. 
We study the dynamics of multiwormhole configurations within the framework of the Euclidean Polyakov approach to string theory, incorporating a modification to the Hamiltonian which leads to a Planckian probability measure for the Coleman parameters $\alpha$ that allows $\frac{1}{2}\alpha^{2}$ to be interpreted as the energy of the quanta of a radiation field on superspace whose values might still fix the coupling constants. 
  We reformulate the Einstein equations as equations for families of surfaces on a four-manifold. These surfaces eventually become characteristic surfaces for an Einstein metric (with or without sources). In particular they are formulated in terms of two functions on R4xS2, i.e. the sphere bundle over space-time, - one of the functions playing the role of a conformal factor for a family of associated conformal metrics, the other function describing an S2's worth of surfaces at each space-time point. It is from these families of surfaces themselves that the conformal metric - conformal to an Einstein metric - is constructed; the conformal factor turns them into Einstein metrics. The surfaces are null surfaces with respect to this metric. 
  The critical behavior of black holes in even and odd dimensional spacetimes is studied based on Ba\~nados-Teitelboim-Zanelli (BTZ) dimensionally continued black holes. In even dimensions it is found that asymptotically flat and anti de-Sitter Reissner-Nordstr\"om black holes present up to two second order phase transitions. The case of asymptotically anti-de-Sitter Schwarzschild black holes present only one critical transition and a minimum of temperature, which occurs at the transition. Finally, it is shown that phase transitions are absent in odd dimensions. 
  The recoils of an accelerated system caused by the transitions characterizing thermal equilibrium are taken into account through the quantum character of the position of the system. The specific model is that of a two level ion accelerated by a constant electric field.   The introduction of the wave function of the center of mass position clarifies the role of the classical trajectory in the thermalization process. Then the effects of the recoils on the properties of the emitted fluxes are analyzed. The decoherence of successive emissions is manifest and simplifies the properties of the fluxes. The fundamental reason for which one cannot neglect the recoils upon considering accelerated systems is stressed and put in parallel with the emergence of "transplanckian" frequencies in Hawking radiation. 
General matterless--theories in 1+1 dimensions include dilaton gravity, Yang--Mills theory as well as non--Einsteinian gravity with dynamical torsion and higher power gravity, and even models of spherically symmetric d = 4 General Relativity. Their recent identification as special cases of 'Poisson--sigma--models' with simple general solution in an arbitrary gauge, allows a comprehensive discussion of the relation between the known absolutely conserved quantities in all those cases and Noether charges, resp. notions of quasilocal 'energy--momentum'. In contrast to Noether like quantities, quasilocal energy definitions require some sort of 'asymptotics' to allow an interpretation as a (gauge--independent) observable. Dilaton gravitation, although a little different in detail, shares this property with the other cases. We also present a simple generalization of the absolute conservation law for the case of interactions with matter of any type. 
In this paper we derive exact quantum Langevin equations for stochastic dynamics of large-scale inflation in de~Sitter space. These quantum Langevin equations are the equivalent of the Wigner equation and are described by a system of stochastic differential equations. We present a formula for the calculation of the expectation value of a quantum operator whose Weyl symbol is a function of the large-scale inflation scalar field and its time derivative. The unique solution is obtained for the Cauchy problem for the Wigner equation for large-scale inflation. The stationary solution for the Wigner equation is found for an arbitrary potential. It is shown that the large-scale inflation scalar field in de Sitter space behaves as a quantum one-dimensional dissipative system, which supports the earlier results. But the analogy with a one-dimensional model of the quantum linearly damped anharmonic oscillator is not complete: the difference arises from the new time dependent commutation relation for the large-scale field and its time derivative. It is found that, for the large-scale inflation scalar field the large time asymptotics is equal to the `classical limit'. For the large time limit the quantum Langevin equations are just the classical stochastic Langevin equations (only the stationary state is defined by the quantum field theory). 
We construct regular multi-wormhole solutions to a gravitating $\sigma$ model in three space-time dimensions, and extend these solutions to cylindrical traversable wormholes in four and five dimensions. We then discuss the possibility of identifying wormhole mouths in pairs to give rise to Wheeler wormholes. Such an identification is consistent with the original field equations only in the absence of the $\sigma$-model source, but with possible naked cosmic string sources. The resulting Wheeler wormhole space-times are flat outside the sources and may be asymptotically Minkowskian. 
Tidal and tidal-resonant effects in coalescing compact binary systems are investigated by direct numerical integration of the equations of motion. For the stars polytropic models are used. The tidal effects are found to be dominated by the (non-resonant) $f$-modes. The effect of the $g$-mode-tidal resonances is obtained. The tidal interaction is shown to be of interest especially for low-mass binaries. There exists a characteristic final plunge orbit beyond which the system cannot remain stable even if radiation reaction is not taken into account; in agreement with results obtained by Lai et al. \shortcite{Lai93}. The importance of the investigated effects for the observation of gravitational waves on Earth is discussed. 
  CARTAN is an easy-to-use symbolic, tensor component package based on the popular Mathematica program. CARTAN makes use of the powerful formalism of rigid frames, and can return results both in this frame and in the coordinate basis. CARTAN has predefined functions for computation of tensors such as Riemann, Ricci, Einstein, Weyl, as well as other tensors and invariants. CARTAN works for spaces with curvature and/or torsion with any constant signature and for any number of dimensions. A free copy of CARTAN 1.01 for Unix with the installation program is contained in the file CARTAN.FILES.uu. The documentation is found in CARTAN.DOC.uu. The program has built-in Help functions. 
  The basic ingredients of the `consistent histories' approach to a generalized quantum theory are `histories'and decoherence functionals. The main aim of this program is to find and to study the behaviour of consistent sets associated with a particular decoherence functional $d$. In its recent formulation by Isham it is natural to identify the space $\UP$ of propositions about histories with an orthoalgebra or lattice. When $\UP$ is given by the lattice of projectors $\PV$ in some Hilbert space $\V$, consistent sets correspond to certain partitions of the unit operator in $\V$ into mutually orthogonal projectors $\{\a_1,\a_2,\ldots\}$, such that the function $d(\a,\a)$ is a probability distribution on the boolean algebra generated by $\{\a_1,\a_2,\ldots\}$. Using the classification theorem for decoherence functionals, proven previously, we show that in the case where $\V$ is some separable Hilbert space there exists for each partition of the unit operator into a set of mutually orthogonal projectors, and for any probability distribution $p(\a)$ on the corresponding boolean algebra, decoherence functionals $d$ with respect to which this set is consistent and which are such that for the probability functions $d(\a,\a)=p(\a)$ holds. 
General relativity is derived from an action which is quadratic in the covariant derivative of certain spinor one-form gravitational potentials. Either a pair of 2-component spinor one-forms or a single Dirac spinor one-form can be employed. The metric is a quadratic function of these spinor one-forms. In the 2-component spinor formulation the action differs from the usual chiral action for general relativity by a total differential. In the Dirac spinor formulation the action is the real part of the former one. The Hamiltonian is related to the ones in positive energy proofs and spinorial quasilocal mass constructions. 
  The properties of static spherically symmetric black holes, which carry electric and magnetic charges, and which are coupled to the dilaton in the presence of a cosmological term (Liouville-type potential, or cosmological constant) are reviewed. 
In conventional field theories, the emission of Hawking radiation in the background of a collapsing star requires transplanckian energy fluctuations. These fluctuations are encoded in the weak values of the energy-momentum operator constructed from matrix elements between both -in and -out states. It is argued that taming of these weak values by back-reaction may lead to geometrical backgrounds which are also build from weak values of the gravitational field operators. This leads to different causal histories of the black hole as reconstructed by observers crossing the horizon at different times but reduces, in accordance with the equivalence principle, to the classical description of the collapse for the proper history of the star as recorded by an observer comoving with it. For observers never crossing the horizon, the evaporation would be interpreted within a topologically trivial ``achronon geometry" void of horizon and singularity: after the initial ignition of the radiation from pair creation out of the vacuum of the collapsing star of mass M, as in the conventional theory, the source of the thermal radiation would shift gradually to the star itself in a time at least of order $4M\ln 2M$. The burning of the star could be consistent with a quantum unitary evolution along the lines suggested by 't Hooft. A provisional formal expression of general black hole complementarity is proposed and its possible relevance for testing features of a theory of quantum gravity is suggested. 
The parameters of inspiralling compact binaries can be estimated using matched filtering of gravitational-waveform templates against the output of laser-interferometric gravitational-wave detectors. Using a recently calculated formula, accurate to second post-Newtonian (2PN) order [order $(v/c)^4$, where $v$ is the orbital velocity], for the frequency sweep ($dF/dt$) induced by gravitational radiation damping, we study the statistical errors in the determination of such source parameters as the ``chirp mass'' $\cal M$, reduced mass $\mu$, and spin parameters $\beta$ and $\sigma$ (related to spin-orbit and spin-spin effects, respectively). We find that previous results using template phasing accurate to 1.5PN order actually underestimated the errors in $\cal M$, $\mu$, and $\beta$. For two inspiralling neutron stars, the measurement errors increase by less than 16 percent. 
Purely magnetic spacetimes, in which the Riemann tensor satisfies $R_{abcd}u^bu^d=0$ for some unit timelike vector $u^a$, are studied. The algebraic consequences for the Weyl and Ricci tensors are examined in detail and consideration given to the uniqueness of $u^a$. Some remarks concerning the nature of the congruence associated with $u^a$ are made. 
We present a new class of black hole solutions in Einstein-Maxwell-dilaton gravity in $n \ge 4$ dimensions. These solutions have regular horizons and a singularity only at the origin. Their asymptotic behavior is neither asymptotically flat nor (anti-) de Sitter. Similar solutions exist for certain Liouville-type potentials for the dilaton. 
It will be shown that the truncation error for the Regge Calculus, as an approximation to Einstein's equations, varies as $O(\Delta^2)$ where $\Delta$ is the typical discretization scale. This result applies to any metric, whether or not it is a solution of the vacuum Einstein equations. It is in this sense that the Regge Calculus is not a discrete representation of Einstein's equations. A new set of equations will be presented and will be shown to have a truncation error of $O(\Delta^4)$ for exact solutions and of $O(\Delta^2)$ for any other metric. 
The convergence properties of numerical Regge calculus as an approximation to continuum vacuum General Relativity is studied, both analytically and numerically. The Regge equations are evaluated on continuum spacetimes by assigning squared geodesic distances in the continuum manifold to the squared edge lengths in the simplicial manifold. It is found analytically that, individually, the Regge equations converge to zero as the second power of the lattice spacing, but that an average over local Regge equations converges to zero as (at the very least) the third power of the lattice spacing. Numerical studies using analytic solutions to the Einstein equations show that these averages actually converge to zero as the fourth power of the lattice spacing. 
Guided by a linearized approximation to Einstein theory, an interim prescription for ``weak source of gravity'' - - in ``particle'' energy-momentum distributed along standpoint light cone - - is formulated for (classical) standpoint cosmology. 
New Planck scale physics may solve the singularity problems of classical general relativity and may lead to interesting consequences for very early Universe cosmology. Two approaches to these questions are reviewed in this article. The first is an effective action approach to including the effects of Planck scale physics in the basic framework of general relativity. It is shown that effective actions with improved singularity properties can be constructed. The second approach is based on superstring theory. A scenario which eliminates the big bang singularity and possibly explains the dimensionality of space-time is reviewed. 
A proof is given that the polar decomposition procedure for unitarity restoration works for products of invertible nonunitary operators. A brief discussion follows that the unitarity restoration procedure, applied to propagators in spacetimes containing closed timelike curves, is analogous to the original introduction by Feynman of ghosts to restore unitarity in non-abelian gauge theories. (The substance of this paper will be a note added in proof to the published version of gr-qc/9405058, to appear in Phys Rev D.) 
  We present evidence which confirms a suggestion by Susskind and Uglum regarding black hole entropy. Using a Pauli-Villars regulator, we find that 't Hooft's approach to evaluating black hole entropy through a statistical-mechanical counting of states for a scalar field propagating outside the event horizon yields precisely the one-loop renormalization of the standard Bekenstein-Hawking formula, $S=\A/(4G)$. Our calculation also yields a constant contribution to the black hole entropy, a contribution associated with the one-loop renormalization of higher curvature terms in the gravitational action. 
Exact solutions of the string equations of motion in a specific Lorentzian wormhole background are obtained. These include both closed and open string configurations. Perturbations about some of these configurations are investigated using the manifestly covariant formalism of Larsen and Frolov. Finally, the generalized Raychaudhuri equations for the corresponding string worldsheet deformations are written down and analysed briefly. 
  The de~Broglie--Bohm (pilot wave) formulation of quantum theory appears to be free from the conceptual problems specific to quantum mechanics (problem of measurement) and to quantum cosmology (problem of time). We discuss the issue of quantum equilibrium which arises within its context. We then study the extension of this formulation to the case of gravity and demonstrate that the foliation of spacetime by space-like hypersurfaces obtained during the solution of the quantum problem turns out to be distinguished in general. This means that quantum pilot wave dynamics is not invariant with respect to arbitrary change of foliation, or, in other terms, that quantum non-locality takes place. We also discuss general structure of a realistic pilot wave theory which could describe the universe. 
We consider 1 spacelike Killing vector field reductions of 4-d vacuum general relativity. We restrict attention to cases in which the manifold of orbits of the Killing field is $R^{3}$. The reduced Einstein equations are equivalent to those for Lorentzian 3-d gravity coupled to an SO(2,1) nonlinear sigma model on this manifold. We examine the theory in terms of a Hamiltonian formulation obtained via a 2+1 split of the 3-d manifold. We restrict attention to geometries which are asymptotically flat in a 2-d sense defined recently. We attempt to pass to a reduced Hamiltonian description in terms of the true degrees of freedom of the theory via gauge fixing conditions of 2-d conformal flatness and maximal slicing. We explicitly solve the diffeomorphism constraints and relate the Hamiltonian constraint to the prescribed negative curvature equation in $R^2$ studied by mathematicians. We partially address issues of existence and/or uniqueness of solutions to the various elliptic partial differential equations encountered. 
A scenario is presented, based on renormalization group (linear perturbation) ideas, which can explain the self-similarity and scaling observed in a numerical study of gravitational collapse of radiation fluid. In particular, it is shown that the critical exponent $\beta$ and the largest Lyapunov exponent ${\rm Re\, } \kappa$ of the perturbation is related by $\beta= ({\rm Re\, } \kappa) ^{-1}$. We find the relevant perturbation mode numerically, and obtain a fairly accurate value of the critical exponent $\beta \simeq 0.3558019$, also in agreement with that obtained in numerical simulation. 
  Retrieval of classical behaviour in quantum cosmology is usually discussed in the framework of minisuperspace models in the presence of scalar fields together with the inhomogeneous modes of gravitational or scalar fields. In this work we propose alternatively a model where the scalar field is replaced by a massive vector field with global U(1) or SO(3) internal symmetries. 
  The theory of N = 1 supergravity with gauged supermatter is studied in the context of a k = + 1 Friedmann minisuperspace model. It is found by imposing the Lorentz and supersymmetry constraints that there are {\seveni no} physical states in the particular SU(2) model studied. 
The effect of the recently obtained 2nd post-Newtonian corrections on the accuracy of estimation of parameters of the gravitational-wave signal from a coalescing binary is investigated. It is shown that addition of this correction degrades considerably the accuracy of determination of individual masses of the members of the binary. However the chirp mass and the time parameter in the sinal is still determined to a very good accuracy. The performance of the Newtonian filter is investigated and it is compared with performance of post-Newtonian search templates introduced recently. It is shown that both search templates can extract accurately useful information about the binary. 
By allowing the lightcones to tip over according to the conservation laws of an one-kink in static, Schwarzschild metric, we show that there also exists an instanton which represents production of pairs of chargeless, nonrotating black holes with mass $M$, joined on an interior surface beyond the horizon ar $r=M$. Evaluation of the thermal properties of each of the black hole in a pair leads one to check that each black hole is exactly the antiblack hole to the other black hole in the pair. The instantonic action has been calculated and seen to be smaller than that corresponding to pair production by factors that associate with the Bekenstein-Hawking entropy and a baby universe entropy $2pi^{2}M^{2}$. This suggests these entropies to count numbers of internal states. 
Quasinormal modes of ultra compact stars with uniform energy density have been calculated. For less compact stars, there is only one very slowly damped polar mode (corresponding to the Kelvin f-mode) for each spherical harmonic index $l$. Further long-lived modes become possible for a sufficiently compact star (roughly when $M/R \ge 1/3$). We compare the characteristic frequencies of these resonant polar modes to the axial modes first found by Chandrasekhar and Ferrari [{\em Proc. Roy. Soc. London A} {\bf 434} 449 (1991)]. We find that the two spectra approach each other as the star is made more compact. The oscillation frequencies of the corresponding polar and axial modes agree to within a percent for stars more compact than $M/R = 0.42$. At the same time, the damping times are slightly different. The results illustrate that there is no real difference between the origin of these axial and polar modes: They are essentially spacetime modes. 
The effect of the recently obtained 2nd post-Newtonian corrections on the accuracy of estimation of parameters of the gravitational-wave signal from a coalescing binary is investigated. It is shown that addition of this correction degrades considerably the accuracy of determination of individual masses of the members of the binary. However the chirp mass and the time parameter in the signal is still determined to a very good accuracy. The possibility of estimation of effects of other theories of gravity is investigated. The performance of the Newtonian filter is investigated and it is compared with performance of post-Newtonian search templates introduced recently. It is shown that both search templates can extract accurately useful information about the binary. 
The oscillation modes of a simple polytropic stellar model are studied. Using a new numerical approach (based on integration for complex coordinates) to the problem for the stellar exterior we have computed the eigenfrequencies of the highly damped w-modes. The results obtained agree well with recent ones of Leins, Nollert and Soffel (1993) Specifically, we are able to explain why several modes in this regime of the complex frequency plane could not be identified within the WKB approach of Kokkotas and Schutz (1992). Furthermore, we have established that the ``kink'' that was a prominent feature of the spectra of Kokkotas and Schutz, but did not appear in the results of Leins {\em et al.}, was a numerical artefact. Using our new numerical code we are also able to compute, for the first time, several of the slowly damped (p) modes for the considered stellar models. For very compact stars we find, somewhat surprisingly, that the damping of these modes does not decrease monotonically as one proceeds to higher oscillation frequencies. The existence of low-order modes that damp away much faster than anticipated may have implications for questions regarding stellar stability and the lifetime of gravitational-wave sources. The present results illustrate the accuracy and reliability of the complex-coordinate method and indicate that the method could prove to be of great use also in problems involving rotating stars. There is no apparent reason why the complex-coordinate approach should not extend to rotating stars, whereas it is accepted that all previous methods will fail to do so. 
  We study the selfadjoint extensions of the spatial part of the D'Alembert operator in a spacetime with two changes of signature. We identify a set of boundary conditions, parametrised by U(2) matrices, which correspond to Dirichlet boundary conditions for the fields, and from which we argue against the suggestion that regions of signature change can isolate singularities. 
  The most general two-dimensional dilaton gravity theory coupled to an Abelian gauge field is considered. It is shown that, up to spacetime diffeomorphisms and $U(1)$ gauge transformations, the field equations admit a two-parameter family of distinct, static solutions.  For theories with black hole solutions, coordinate invariant expressions are found for the energy, charge, surface gravity, Hawking temperature and entropy of the black holes. The Hawking temperature is proportional to the surface gravity as expected, and both vanish in the case of extremal black holes in the generic theory. A Hamiltonian analysis of the general theory is performed, and a complete set of (global) Dirac physical observables is obtained. The theory is then quantized using the Dirac method in the WKB approximation. A connection between the black hole entropy and the imaginary part of the WKB phase of the Dirac quantum wave functional is found for arbitrary values of the mass and $U(1)$ charge. The imaginary part of the phase vanishes for extremal black holes and for eternal, non-extremal Reissner-Nordstrom black holes. 
The non-singular, oscillating Friedman cosmology within the framework of General Relativity is considered. The general oscillatory solution given in terms of elliptic functions and the conditions for its existence are discussed. It is shown that the wall-like-matter and the small, but negative cosmological constant are required for oscillations. The oscillations can , in principle, be deep enough to allow standard hot universe processes like recombination and nucleosynthesis. It is shown that the wall-like-matter and string-like-matter can be interpreted as scalar fields with some potentials. This may give another candidate for the dark matter which may be compatible with observational data. For an exact elementary oscillatory solution it is shown that the associated scalar field potential is oscillating as well. 
We consider a multidimensional universe with the topology $M= \R\times M_1\times \cdots \times M_n$, where the $M_i$ ($i>1$) are $d_i$-dimensional Ricci flat spaces. Exploiting a conformal equivalence between minimal coupling models and conformal coupling models, we get exact solutions for such an universe filled by a conformally coupled scalar field. One of the solutions can be used to describe trapped unobservable extra dimensions. 
We consider a multidimensional model of the universe given as a $D$-dimensional geometry, represented by a Riemannian manifold $(M,g)$ with arbitrary signature of $g$, $M= \R\times M_1\times \cdots \times M_n$, where the $M_i$ of dimension $d_i$ are Einstein spaces, compact for $i>1$. For Lagrangian models $L(R,\phi)$ on $M$ which depend only on the Ricci curvature $R$ and a scalar field $\phi$, there exists a conformal equivalence with minimal coupling models. For certain nonminimal models we study classical solutions and their relation to solutions in the equivalent minimal coupling model. The domains of equivalence are separated by certain critical values of the scalar field $\phi$. Furthermore, the coupling constant $\xi$ of the coupling between $\phi$ and $R$ is critical at both, the minimal value $\xi=0$ and the conformal value $\xi_c=\frac{D-2}{4(D-1)}$. In different noncritical regions of $\xi$ the solutions behave qualitatively different. Instability can occure only in certain ranges of $\xi$. {This paper is dedicated to Prof. D. D. Ivanenko.} 
We examine the Zeroth Law and the Second Law of black hole thermodynamics within the context of effective gravitational actions including higher curvature interactions. We show that entropy can never decrease for quasi-stationary processes in which a black hole accretes positive energy matter, independent of the details of the gravitational action. Within a class of higher curvature theories where the Lagrangian consists of a polynomial in the Ricci scalar, we use a conformally equivalent theory to establish that stationary black hole solutions with a Killing horizon satisfy the Zeroth Law, and that the Second Law holds in general for any dynamical process. We also introduce a new method for establishing the Second Law based on a generalization of the area theorem, which may prove useful for a wider class of Lagrangians. Finally, we show how one can infer the form of the black hole entropy, at least for the Ricci polynomial theories, by integrating the changes of mass and angular momentum in a quasistationary accretion process. 
We use gravitational Lagrangians $R \Box \sp k R \sqrt{-g}$ and linear combinations of them; we ask under which circumstances the de Sitter space-time represents an attractor solution in the set of spatially flat Friedman models. Results are: For arbitrary $k$, i.e., for arbitrarily large order of the field equation, on can always find examples where the attractor property takes place. Such examples necessarily need a non-vanishing $R\sp 2$-term. The main formulas do not depend on the dimension, so one gets similar results also for 1+1-dimensional gravity and for Kaluza-Klein cosmology. 
  In this work we review, in the framework of the so-called brick wall model, the divergence problem arising in the one loop calculations of various thermodynamical quantities, like entropy, internal energy and heat capacity. Particularly we find that, if one imposes that entanglement entropy is equal to the Bekenstein-Hawking one, the model gives problematic results. Then a proposal of solution to the divergence problem is made following the zeroth law of black hole mechanics. 
We present a small computer algebra program for use in Maxwell's theory. The Maxwell equations and the energy-momentum current of the electromagnetic field are formulated in the language of exterior differential forms. The corresponding program can be applied (in the presence of a gravitational field) in curved Riemannian spacetime as well as in flat Minkowski spacetime in inertial or {\em non-inertial} frames. Our program is written for the computer algebra system REDUCE with the help of the EXCALC package for exterior differential forms.\par The two major advantages of this modern approch to electrodynamics --- the natural formulation, free of both metric and coordinates, and the straightforward programming of problems --- are illustrated by examining a number of examples ranging from the Coulomb field of a static point charge to the Kerr--Newman solution of the Einstein--Maxwell equations. 
These lectures briefly review our current understanding of classical and quantum gravity in three spacetime dimensions, concentrating on the quantum mechanics of closed universes and the (2+1)-dimensional black hole. Three formulations of the classical theory and three approaches to quantization are discussed in some detail, and a number of other approaches are summarized. An extensive, although by no means complete, list of references is included. (Lectures given at the First Seoul Workshop on Gravity and Cosmology, February 24-25, 1995.) 
We report on a new 3D numerical code designed to solve the Einstein equations for general vacuum spacetimes. This code is based on the standard 3+1 approach using cartesian coordinates. We discuss the numerical techniques used in developing this code, and its performance on massively parallel and vector supercomputers. As a test case, we present evolutions for the first 3D black hole spacetimes. We identify a number of difficulties in evolving 3D black holes and suggest approaches to overcome them. We show how special treatment of the conformal factor can lead to more accurate evolution, and discuss techniques we developed to handle black hole spacetimes in the absence of symmetries. Many different slicing conditions are tested, including geodesic, maximal, and various algebraic conditions on the lapse. With current resolutions, limited by computer memory sizes, we show that with certain lapse conditions we can evolve the black hole to about $t=50M$, where $M$ is the black hole mass. Comparisons are made with results obtained by evolving spherical initial black hole data sets with a 1D spherically symmetric code. We also demonstrate that an ``apparent horizon locking shift'' can be used to prevent the development of large gradients in the metric functions that result from singularity avoiding time slicings. We compute the mass of the apparent horizon in these spacetimes, and find that in many cases it can be conserved to within about 5\% throughout the evolution with our techniques and current resolution. 
  We investigate topology changing processes in the WKB approximation of four dimensional quantum cosmology with a negative cosmological constant. As Riemannian manifolds which describe quantum tunnelings of spacetime we consider constant negative curvature solutions of the Einstein equation i.e. hyperbolic geometries. Using four dimensional polytopes, we can explicitly construct hyperbolic manifolds with topologically non-trivial boundaries which describe topology changes. These instanton-like solutions are constructed out of 8-cell's, 16-cell's or 24-cell's and have several points at infinity called cusps. The hyperbolic manifolds are non-compact because of the cusps but have finite volumes. Then we evaluate topology change amplitudes in the WKB approximation in terms of the volumes of these manifolds. We find that the more complicated are the topology changes, the more likely are suppressed. 
  I review several different calculations, coming from string theory, nonperturbative quantum gravity and analyses of black holes that lead to predictions of phenomena that would uniquely be signatures of quantum gravitational effects.  These include: 1) deviations from a thermal spectra for evaporating black holes, 2) upper limits on the entropy and energy content of bounded regions, 3) suppression of ultra-high energy scattering amplitudes, consistent with a modified uncertainty principle, 4) physical volumes and areas have discrete spectra, 5), violations of $CPT$ and universal violations of $CP$, 6) otherwise inexplicable conditions on the initial state of the universe or otherwise inexplicable correlations between cosmological and microscopic parameters. Consideration of all of these together suggests the possibility of connections between perturbative and nonperturbative approaches to quantum gravity. 
In this paper we calculate the magnetic and electric self-forces, induced by the conical structure of a cosmic string space-time, on a long straight wire which presents either a constant current or a linear charge density. We also show how these self-forces are related by a Lorentz tranformation and, in this way, explain what two different inertial observers detect in their respective frames. 
We investigate the formation of trapped surfaces in cosmological spacetimes, using constant mean curvature slicing. Quantitative criteria for the formation of trapped surfaces demonstrate that cosmological regions enclosed by trapped surfaces may have matter density exceeding significantly the background matter density of the flat and homogeneous cosmological model. 
We investigate the formation of trapped surfaces in asymptotically flat spherical spacetimes, using constant mean curvature slicing. 
The importance of general relativity to the induced electric field exterior to pulsars has been investigated by assuming aligned vacuum and non-vacuum magnetosphere models. For this purpose the stationary and axisymmetric vector potential in Schwarzschild geometry has been considered and the corresponding expressions for the induced electric field due to the rotation of the magnetic dipole have been derived for both vacuum and non-vacuum conditions. Due to the change in the magnetic dipole field in curved spacetime the induced electric field also changes its magnitude and direction and increases significantly near the surface of the star. As a consequence the surface charge density, the acceleration of charged particles in vacuum magnetospheres and the space charge density in non-vacuum magnetosphere greatly increase near the surface of the star. The results provide the most general feature of the important role played by gravitation and could have several potentially important implications for the production of high-energy radiation from pulsars. 
In these lectures the properties of magnetically charged black holes are described. In addition to the standard Reissner-N\"ordstrom solution, there are new types of static black holes that arise in theories containing electrically charged massive vector mesons. These latter solutions have nontrivial matter fields outside the horizon; i.e., they are black holes with hair. While the solutions carrying unit magnetic charge are spherically symmetric, those with more than two units of magnetic charge are not even axially symmetric. These thus provide the first example of time-independent black hole solutions that have no rotational symmetry. 
  In the last decades the logico-algebraic approach to quantum mechanics turned to be a successful tool to render the quantum mechanical formalism on a steady operationalistic background. The algebraic approach to general relativity first proposed by Geroch is used to build a smearing procedure for events in the quantized theory of spacetime. It stems from the notion of basic algebra which possesses both the differential structure (as functional algebras) and non-commutativity (as algebras of observables in quantum theories). The main essence of this formalism is that it deprives the notion of spacetime as event support of its fundamental status making it relative to the measurement performed. Before the observation is executed, it is pointless to speak of events as points of spacetime: there is no underlying manifold. Only when an eigen-subalgebra of the basic algebra is outlined as eigenstate of the observable, it gives rise to its functional representation on the appropriate set thought of as spacetime. 
  The simplest solution to the black hole information loss problem is to eliminate black holes. Modifications of Einstein gravity which accomplish this are discussed and the possibility that string theory is free of black holes is considered. 
  We construct static black hole solutions that have no rotational symmetry. These arise in theories, including the standard electroweak model, that include charged vector mesons with mass $m\ne 0$. In such theories, a magnetically charged Reissner-Nordstrom black hole with horizon radius less than a critical value of the order of $m^{-1}$ is classically unstable against the development of a nonzero vector meson field just outside the horizon, indicating the existence of static black hole solutions with vector meson hair. For the case of unit magnetic charge, spherically symmetric solutions of this type have previously been studied. For other values of the magnetic charge, general arguments show that any new solution with hair cannot be spherically symmetric. In this paper we develop and apply a perturbative scheme (which may have applicability in other contexts) for constructing such solutions in the case where the Reissner-Nordstrom solution is just barely unstable. For a few low values of the magnetic charge the black holes retain a rotational symmetry about a single axis, but this axial symmetry disappears for higher charges. While the vector meson fields vanish exponentially fast at distances greater than $O(m^{-1})$, the magnetic field and the metric have higher multipole components that decrease only as powers of the distance from the black hole. 
  Numerical evidence for a cosmological version of the Bartnik-McKinnon family of particle-like solutions of the Einstein-Yang-Mills system is presented. Our solutions are also static, but space has the topology of a three-sphere. By adjusting the cosmological constant we found numerically a spherically symmetric solution which can be regarded as an excitation of the unique SO(4)-invariant solution. We expect that for each node number there exists such a solution without a cosmological horizon. 
  One can simplify the triad formulations of canonical gravity by abandoning any relation to a fixed coordinate system. That means in case of the \ADM formalism that one can determine the momentum by direct derivation of the Lagrange-3-form w.r.t the time-derivative of the triad-1-forms, thus the momentum is most naturally a 2-form. We apply this concept to the Palatini formulation where we can closely follow Dirac's concept to find and eliminate the second class constraints. Following the same way for the Ashtekar theory it will turn out to be equivalent to two successive canonical transformations where the first makes explicit use of the spatial dimension being 3 and the second is usually hidden in the use of densities. At the end we can give a simple version of the reality constraints. 
  One has not any conventional energy-momentum conservation law in Lagrangian field theory, but relations involving different stress-energy-momentum tensors associated with different connections. It is not obvious how to choose the true energy-momentum tensor. This problem is solved in the framework of the multimomentum Hamiltonian formalism which provides the adequate description of constraint field systems. The goal is that, for different solutions of the same constraint field model, one should choose different stress-energy-momentum tensors in general. Gauge theory illustrates this result. The stress-energy-momentum tensors of affine-metric gravity are examined. 
  The stability of cosmological event and Cauchy horizons of spacetimes associated with plane symmetric domain walls are studied. It is found that both horizons are not stable against perturbations of null fluids and massless scalar fields; they are turned into curvature singularities. These singularities are light-like and strong in the sense that both the tidal forces and distortions acting on test particles become unbounded when theses singularities are approached. 
  In the framework of the Connes-Lott model based on noncommutative geometry, the basic features of a gauge theory in the presence of gravity are reviewed, in order to show the possible physical relevance of this scheme for inflationary cosmology. These models naturally contain at least two scalar fields, interacting with each other whenever more than one fermion generation is assumed. In this paper we propose to investigate the behaviour of these two fields (one of which represents the distance between the copies of a two-sheeted space-time) in the early stages of the universe evolution. In particular the simplest abelian model, which preserves the main characteristics of more complicate gauge theories, is considered and the corresponding inflationary dynamics is studied. We find that a chaotic inflation is naturally favoured, leading to a field configuration in which no symmetry breaking occurs and the final distance between the two sheets of space-time is smaller the greater the number of $e$-fold in each sheet. 
  The general form of the global conservation laws for $N$-body systems in the first post-Newtonian approximation of general relativity is considered. Our approach applies to the motion of an isolated system of $N$ arbitrarily composed and shaped, weakly self-gravitating, rotating, deformable bodies and uses a framework recently introduced by Damour, Soffel and Xu (DSX). We succeed in showing that seven of the first integrals of the system (total mass-energy, total dipole mass moment and total linear momentum) can be broken up into a sum of contributions which can be entirely expressed in terms of the basic quantities entering the DSX framework: namely, the relativistic individual multipole moments of the bodies, the relativistic tidal moments experienced by each body, and the positions and orientations with respect to the global coordinate system of the local reference frames attached to each body. On the other hand, the total angular momentum of the system does not seem to be expressible in such a form due to the unavoidable presence of irreducible nonlinear gravitational effects. 
  The thermodynamic and euclidean functional integral approaches to black hole entropy are discussed. The existence of some freedom in the definition of the entropy is pointed out and the possibility of a departure from the semiclassical expression discussed in the light of quantum corrections. The semiclassical area dependence of the entropy of matter in the background of a black hole is also reviewed and shown to break down in the case of extremal black holes. The cutoff dependence is shown to be different for the extreme dilatonic and Reissner - Nordstrom black holes. 
  The time dependent quantum variational principle is emerging as an important means of studying quantum dynamics, particularly in early universe scenarios. To date all investigations have worked within a Gaussian framework. Here we present an improved method which is demonstrated to be superior to the Gaussian approach and may be naturally extended to the field--theoretic case. 
  We present the first numerical solutions of the coupled Einstein-Maxwell equations describing rapidly rotating neutron stars endowed with a magnetic field. These solutions are fully relativistic and self-consistent, all the effects of the electromagnetic field on the star's equilibrium (Lorentz force, spacetime curvature generated by the electromagnetic stress-energy) being taken into account. The magnetic field is axisymmetric and poloidal. Five dense matter equations of state are employed. The partial differential equation system is integrated by means of a pseudo-spectral method. Various tests passed by the numerical code are presented. The effects of the magnetic field on neutron stars structure are then investigated, especially by comparing magnetized and non-magnetized configurations with the same baryon number. The deformation of the star induced by the magnetic field is important only for huge values of B (B>10^{10} T). The maximum mass as well as the maximum rotational velocity are found to increase with the magnetic field. The maximum allowable poloidal magnetic field is of the order of 10^{14} T (10^{18} G) and is reached when the magnetic pressure is comparable to the fluid pressure at the centre of the star. For such values, the maximum mass of neutron stars is found to increase by 13 to 29 % (depending upon the EOS) with respect to the maximum mass of non-magnetized stars. 
  The harmonicity condition of the curvature 2-form of a pseudo- Riemannian manifold is formulated on the basis of annulment of this form by the de Rham-Lichnerowicz Laplacian. The following theorem is proved: The curvature 2-form of any Einstein manifold is harmonic. 
  The role of topology in elementary quantum physics is discussed in detail. It is argued that attributes of classical spatial topology emerge from properties of state vectors with suitably smooth time evolution. Equivalently, they emerge from considerations on the domain of the quantum Hamiltonian, this domain being often specified by boundary conditions in elementary quantum physics. Several examples are presented where classical topology is changed by smoothly altering the boundary conditions. When the parameters labelling the latter are treated as quantum variables, quantum states need not give a well-defined classical topology, instead they can give a quantum superposition of such topologies. An existing argument of Sorkin based on the spin-statistics connection and indicating the necessity of topology change in quantum gravity is recalled. It is suggested therefrom and our results here that Einstein gravity and its minor variants are effective theories of a deeper description with additional novel degrees of freedom. Other reasons for suspecting such a microstructure are also summarized. 
  We consider quantum fields around uniformly accelerated black holes. At a particular value of the acceleration, the Bogolubov transformation which would be responsible for the late-time Hawking radiation, is found to be trivial. When this happens, Hawking's thermal radiation, Doppler-shifted or not, is absent to the asymptotic inertial observers despite the nonzero Hawking temperature, while the co-moving observers find the black hole radiance exactly balanced by the acceleration heat bath. After a brief comparison to the classical system of a uniformly accelerated charge, we close with two important comments. (Phys. Rev. Lett. 75 (1995) 382) 
  The new approach to quantize the gravity based on the notion of differential algebra is suggested. It is shown that the differential geometry of this object can not be described in terms of points. The spatialization procedure giving rise to points by loosing a part of the entire structure is discussed. The counterpart of the traditional objects of differential geometry are studied. 
  Quantum fluctuations of scalar fields during inflation could determine the very large-scale structure of the universe. In the case of general scalar-tensor gravity theories these fluctuations lead to the diffusion of fundamental constants like the Planck mass and the effective Brans--Dicke parameter, $\omega$. In the particular case of Brans--Dicke gravity, where $\omega$ is constant, this leads to runaway solutions with infinitely large values of the Planck mass. However, in a theory with variable $\omega$ we find stationary probability distributions with a finite value of the Planck mass peaked at exponentially large values of $\omega$ after inflation. We conclude that general relativity is an attractor during the quantum diffusion of the fields. 
  We derive the matching conditions for cosmological perturbations in a Friedmann Universe where the equation of state undergoes a sharp jump, for instance as a result of a phase transition. The physics of the transition which is needed to follow the fate of the perturbations is clarified. We dissipate misleading statements made recently in the literature concerning the predictions of the primordial fluctuations from inflation and confirm standard results. Applications to string cosmology are considered. 
  We review and systematize recent attempts to canonically quantize general relativity in 2+1 dimensions, defined on space-times $\R\times\Sigma^g$, where $\Sigma^g$ is a compact Riemann surface of genus $g$. The emphasis is on quantizations of the classical connection formulation, which use Wilson loops as their basic observables, but also results from the ADM formulation are summarized. We evaluate the progress and discuss the possible quantum (in)equivalence of the various approaches. 
  The entropy of stationary black holes has recently been calculated by a number of different approaches. Here we compare the Noether charge approach (defined for any diffeomorphism invariant Lagrangian theory) with various Euclidean methods, specifically, (i) the microcanonical ensemble approach of Brown and York, (ii) the closely related approach of Ba\~nados, Teitelboim, and Zanelli which ultimately expresses black hole entropy in terms of the Hilbert action surface term, (iii) another formula of Ba\~nados, Teitelboim and Zanelli (also used by Susskind and Uglum) which views black hole entropy as conjugate to a conical deficit angle, and (iv) the pair creation approach of Garfinkle, Giddings, and Strominger. All of these approaches have a more restrictive domain of applicability than the Noether charge approach. Specifically, approaches (i) and (ii) appear to be restricted to a class of theories satisfying certain properties listed in section 2; approach (iii) appears to require the Lagrangian density to be linear in the curvature; and approach (iv) requires the existence of suitable instanton solutions. However, we show that within their domains of applicability, all of these approaches yield results in agreement with the Noether charge approach. In the course of our analysis, we generalize the definition of Brown and York's quasilocal energy to a much more general class of diffeomorphism invariant, Lagrangian theories of gravity. In an appendix, we show that in an arbitrary diffeomorphism invariant theory of gravity, the ``volume term" in the ``off-shell" Hamiltonian associated with a time evolution vector field $t^a$ always can be expressed as the spatial integral of $t^a {\cal C}_a$, where ${\cal C}_a = 0$ are the constraints associated with the diffeomorphism invariance. 
  We consider rotating black hole configurations of self-gravitating maps from spacetime into arbitrary Riemannian manifolds. We first establish the integrability conditions for the Killing fields generating the stationary and the axisymmetric isometry (circularity theorem). Restricting ourselves to mappings with harmonic action, we subsequently prove that the only stationary and axisymmetric, asymptotically flat black hole solution with regular event horizon is the Kerr metric. Together with the uniqueness result for non-rotating configurations and the strong rigidity theorem, this establishes the uniqueness of the Kerr family amongst all stationary black hole solutions of self-gravitating harmonic mappings. 
  In this lecture supersymmetric minisuperspace models of any Bianchi type within class A of the classification of Ellis and McCallum are considered. The algebra of the supersymmetry generators, the Lorentz generators, the diffeomorphism generators and the Hamiltonian generator is determined explicitely and found to close. Different from earlier work it is established that physical states, which are annihilated by all these generators, exist in {\it all } sectors of these models with fixed even fermion number. A state in the 4-fermion sector of the Bianchi type IX model is considered as a specific example, which satisfies the `no-boundary' condition of Hartle and Hawking. The conclusion is that supersymmetric minisuperspace models have a much richer manifold of physical states than had been recognized before. 
  An important limitation is shown in the analogy between the Aharonov-Bohm effect and the parallel transport on a cone. It illustrates a basic difference between gravity and gauge fields due to the existence of the solder form for the space-time geometry. This difference is further shown by the observability of the gravitational phase for open paths. This reinforces a previous suggestion that the fundamental variables for quantizing the gravitational field are the solder form and the connection, and not the metric. 
  We apply the algebraic quantization programme proposed by Ashtekar to the analysis of the Belinski\v{\i}-Zakharov classical spacetimes, obtained from the Kasner metrics by means of a generalized soliton transformation. When the solitonic parameters associated with this transformation are frozen, the resulting Belinski\v{\i}-Zakharov metrics provide the set of classical solutions to a gravitational minisuperspace model whose Einstein equations reduce to the dynamical equations generated by a homogeneous Hamiltonian constraint and to a couple of second-class constraints. The reduced phase space of such a model has the symplectic structure of the cotangent bundle over $I\!\!\!\,R^+\times I\!\!\!\,R^+$. In this reduced phase space, we find a complete set of real observables which form a Lie algebra under Poisson brackets. The quantization of the gravitational model is then carried out by constructing an irreducible unitary representation of that algebra of observables. Finally, we show that the quantum theory obtained in this way is unitarily equivalent to that which describes the quantum dynamics of the Kasner model. 
  We study the effect of higher-curvature terms in the string low-energy effective actions on the cosmological solutions of the theory, up to corrections quartic in the curvatures, for the bosonic and heterotic strings as well as the type II superstring. We find that cosmological solutions exist for all string types but they always disappear when the dilaton field is included, a conclusion that can be avoided if string-loop effects are taken into account. 
  We consider the influence of acceleration on the radiative energy shifts of atoms in Minkowski space. We study a two-level atom coupled to a scalar quantum field. Using a Heisenberg picture approach, we are able to separate the contributions of vacuum fluctuations and radiation reaction to the Lamb shift of the two-level atom. The resulting energy shifts for the special case of a uniformly accelerated atom are then compared with those of an atom at rest. 
  This article reviews the status of several solutions to all the constraints of quantum gravity that have been proposed in terms of loops and extended loops. We discuss pitfalls of several of the results and in particular discuss the issues of covariance and regularization of the constraints in terms of extended loops. We also propose a formalism for ``thickened out loops'' which does not face the covariance problems of extended loops and may allow to regularize expressions in a consistent manner. 
  We consider a point charge fixed in the Rindler coordinates which describe a uniformly accelerated frame. We determine an integral expression of the induced charge density due to the vacuum polarization at the first order in the fine structure constant. In the case where the acceleration is weak, we give explicitly the induced electrostatic potential. 
  A framework is introduced which explains the existence and similarities of most exact solutions of the Einstein equations with a wide range of sources for the class of hypersurface-homogeneous spacetimes which admit a Hamiltonian formulation. This class includes the spatially homogeneous cosmological models and the astrophysically interesting static spherically symmetric models as well as the stationary cylindrically symmetric models. The framework involves methods for finding and exploiting hidden symmetries and invariant submanifolds of the Hamiltonian formulation of the field equations. It unifies, simplifies and extends most known work on hypersurface-homogeneous exact solutions. It is shown that the same framework is also relevant to gravitational theories with a similar structure, like Brans-Dicke or higher-dimensional theories. 
  We point out that spacetime singularities play a useful role in gravitational theories by eliminating unphysical solutions. In particular, we argue that any modification of general relativity which is completely nonsingular cannot have a stable ground state. This argument applies both to classical extensions of general relativity, and to candidate quantum theories of gravity. 
  We treat continuous histories within the histories approach to generalised quantum mechanics. The essential tool is the `history group': the analogue, within the generalised history scheme, of the canonical group of single-time quantum mechanics. 
  A key aspect of a recent proposal for a {\em generalized loop representation} of quantum Yang-Mills theory and gravity is considered. Such a representation of the quantum theory has been expected to arise via consideration of a particular algebra of observables -- given by the traces of the holonomies of {\em generalized loops}. We notice, however, a technical subtlety, which prevents us from reaching the conclusion that the generalized holonomies are covariant with respect to small gauge transformations. Further analysis is given which shows that they are {\em not} covariant with respect to small gauge transformations; their traces are {\em not} observables of the gauge theory. This result indicates what may be a serious complication to the use of generalized loops in physics. 
  Vacuum asymptotically flat Robinson-Trautman spacetimes are a well known class of spacetimes exhibiting outgoing gravitational radiation. In this paper we describe a method of locating the past apparent horizon in these spacetimes, and discuss the properties of the horizon. We show that the past apparent horizon is non-timelike, and that its surface area is a decreasing function of the retarded time. A numerical simulation of the apparent horizon is also discussed. 
  The machinery is suggested to describe the varying spacetime topology on the level of its substitutes by finite topological spaces. 
  We study a tentative generally covariant quantum field theory, denoted the T-Theory, as a tool to investigate the consistency of quantum general relativity. The theory describes the gravitational field and a minimally coupled scalar field; it is based on the loop representation, and on a certain number of quantization choices. Four-dimensional diffeomorphism-invariant quantum transition probabilities can be computed from the theory. We present the explicit calculation of the transition probability between two volume eigenstates as an example. We discuss the choices on which the T-theory relies, and the possibilities of modifying them. 
  The abstract quantum algebra of observables for 2+1 gravity is analysed in the limit of small cosmological constant. The algebra splits into two sets with an explicit phase space representation;~one set consists of $6g-6$ {\it commuting} elements which form a basis for an algebraic manifold defined by the trace and rank identities;~the other set consists of $6g-6$ tangent vectors to this manifold. The action of the quantum mapping class group leaves the algebra and algebraic manifold invariant. The previously presented representation for $g=2$ is analysed in this limit and reduced to a very simple form. The symplectic form for $g=2$ is computed. 
  The duality found by Aharonov and Casher for topological phases in the electromagnetic field is generalized to an arbitrary linear interaction. This provides a heuristic principle for obtaining a new solution of the field equations from a known solution. This is applied to the general relativistic Sagnac phase shift due to the gravitational field in the interference of mass or energy around a line source that has angular momentum and the dual phase shift in the interference of a spin around a line mass. These topological phases are treated both in the linearized limit of general relativity and the exact solutions for which the gravitational sources are cosmic strings containing torsion and curvature, which do not have a Newtonian limit. 
  The static black hole solutions to the Einstein-Maxwell equations are all spherically symmetric, as are many of the recently discovered black hole solutions in theories of gravity coupled to other forms of matter. However, counterexamples demonstrating that static black holes need not be spherically symmetric exist in theories, such as the standard electroweak model, with electrically charged massive vector fields. In such theories, a magnetically charged Reissner-Nordstrom solution with sufficiently small horizon radius is unstable against the development of a nonzero vector field outside the horizon. General arguments show that, for generic values of the magnetic charge, this field cannot be spherically symmetric. Explicit construction of the solution shows that it in fact has no rotational symmetry at all. 
  The Einstein equation is derived from the proportionality of entropy and horizon area together with the fundamental relation $\delta Q=TdS$ connecting heat, entropy, and temperature. The key idea is to demand that this relation hold for all the local Rindler causal horizons through each spacetime point, with $\delta Q$ and $T$ interpreted as the energy flux and Unruh temperature seen by an accelerated observer just inside the horizon. This requires that gravitational lensing by matter energy distorts the causal structure of spacetime in just such a way that the Einstein equation holds. Viewed in this way, the Einstein equation is an equation of state. This perspective suggests that it may be no more appropriate to canonically quantize the Einstein equation than it would be to quantize the wave equation for sound in air. 
  We study the ground state wave function for a universe which is topologically a lens space within the Regge calculus approach. By restricting the four dimensional simplicial complex to be a cone over the boundary lens space, described by a single internal edge length, and a single boundary edge length, one can analyze in detail the analytic properties of the action in the space of complex edge lengths. The classical extrema and convergent steepest descent contours of integration yielding the wave function are found. Both the Hartle-Hawking and Linde-Vilenkin type proposals are examined and, in all cases, we find wave functions which predict Lorentzian oscillatory behaviour in the late universe. A factorization property of topology changing amplitudes within a restricted edge length class of cone type complexes is established. The behaviour of the results under subdivision of the boundary universe is also presented. 
  We carry out the canonical quantization of the Levi-Civit\`a family of static and cylindrical solutions. The reduced phase space of this family of metrics is proved to coincide with that corresponding to the Kasner model, including the associated symplectic structures, except for that the respective domains of definition of one of the phase space variables are not identical. Using this result, we are able to construct a quantum model that describes spacetimes of both the Levi-Civit\`a and the Kasner type, and in which the three-dimensional spatial topology is not uniquely fixed. Finally, we quantize to completion the subfamily of Levi-Civit\`a solutions which represent the exterior gravitational field of a straight cosmic string. These solutions are conical geometries,ie, Minkowski spacetime minus a wedge. The quantum theory obtained provides us with predictions about the angular size of this wedge. 
  This paper studies the quantization of the electromagnetic field on a flat Euclidean background with boundaries. One-loop scaling factors are evaluated for the one-boundary and two-boundary backgrounds. The mode-by-mode analysis of Faddeev-Popov quantum amplitudes is performed by using zeta-function regularization, and is compared with the space-time covariant evaluation of the same amplitudes. It is shown that a particular gauge condition exists for which the corresponding operator matrix acting on gauge modes is in diagonal form from the beginning. Moreover, various relativistic gauge conditions are studied in detail, to investigate the gauge invariance of the perturbative quantum theory. 
  The aim of the present letter is to explain the `critical behaviour' observed in numerical studies of spherically symmetric gravitational collaps of a perfect fluid. A simple expression results for the critical index $\gamma$ of the black hole mass considered as an order parameter. $\gamma$ turns out to vary strongly with the parameter $k$ of the assumed equation of state $p=k\rho$. 
  Regularity theorems are presented for cosmology and gravitational collapse in non-Riemannian gravitational theories. These theorems establish conditions necessary to allow the existence of timelike and null path complete spacetimes for matter that satisfies the positive energy condition. Non-Riemannian theories of gravity can have solutions that have a non-singular beginning of the universe, and the gravitational collapse of a star does not lead to a black hole event horizon and a singularity as a final stage of collapse. A perturbatively consistent version of nonsymmetric gravitational theory is studied that, in the long-range approximation, has a nonsingular static spherically symmetric solution which is path complete, does not have black hole event horizons and has finite curvature invariants. The theory satisfies the regularity theorems for cosmology and gravitational collapse. The elimination of black holes resolves the information loss puzzle. 
  In the framework of the teleparallel equivalent of general relativity the energy density of asymptoticaly flat gravitational fields can be naturally and unambiguously defined. Upon integration of the energy density over the whole three dimensional space we obtain the ADM energy. We use this energy density to calculate the energy inside a Schwarzschild black hole. 
  In the teleparallel equivalent of general relativity the energy density of asymptoticaly flat gravitational fields can be naturally defined as a scalar density restricted to a three dimensional spacelike hypersurface. The scalar density has simple expression in terms of the trace of the torsion tensor. Here we obtain the formal expression of the localized energy for a Kerr black hole. The energy inside a surface of constant radius can be explicitly calculated in the limit of small $a$, the specific angular momentum. Such expression turns out to be exactly the same as the one obtained by means of the method recently proposed by Brown and York. 
  A globally hyperbolic asymptotically flat spacetime is presented (having non-negative energy density and pressures) that shows that not all K(pi,1) prime factors of the Cauchy surface topology are passively censored according to asymptotic observers in contradiction to an argument of Friedman, Schleich, and Witt. 
  This paper studies the two-spinor form of the Rarita-Schwinger potentials subject to local boundary conditions compatible with local supersymmetry. The massless Rarita-Schwinger field equations are studied in four-real-dimensional Riemannian backgrounds with boundary. Gauge transformations on the potentials are shown to be compatible with the field equations providing the background is Ricci-flat, in agreement with previous results in the literature. However, the preservation of boundary conditions under such gauge transformations leads to a restriction of the gauge freedom. The recent construction by Penrose of secondary potentials which supplement the Rarita-Schwinger potentials is then applied. The equations for the secondary potentials, jointly with the boundary conditions, imply that the background four-geometry is further restricted to be totally flat. 
  By choosing an unconventional polarization of the connection phase space in (2+1)-gravity on the torus, a modular invariant quantum theory is constructed. Unitary equivalence to the ADM-quantization is shown. 
  We investigate the pair creation of charged black holes in a background with a positive cosmological constant. We consider $C$ metrics with a cosmological constant, and show that the conical singularities in the metric only disappear when it reduces to the Reissner-Nordstr\"om de Sitter metric. We construct an instanton describing the pair production of extreme black holes and an instanton describing the pair production of non-extreme black holes from the Reissner-Nordstr\"om de Sitter metric, and calculate their actions. There are a number of striking similarities between these instantons and the Ernst instantons, which describe pair production in a background electromagnetic field. We also observe that the type I instanton in the ordinary $C$ metric with zero cosmological constant is actually the Reissner-Nordstr\"om solution. 
  This paper studies the linearized gravitational field in the presence of boundaries. For this purpose, $\zeta$-function regularization is used to perform the mode-by-mode evaluation of BRST-invariant Faddeev-Popov amplitudes in the case of flat Euclidean four-space bounded by a three-sphere. On choosing the de Donder gauge-averaging term, the resulting $\zeta(0)$ value is found to agree with the space-time covariant calculation of the same amplitudes, which relies on the recently corrected geometric formulas for the asymptotic heat kernel in the case of mixed boundary conditions. Two sets of mixed boundary conditions for Euclidean quantum gravity are then compared in detail. The analysis proves that one cannot restrict the path-integral measure to transverse-traceless perturbations. By contrast, gauge-invariant amplitudes are only obtained on considering from the beginning all perturbative modes of the gravitational field, jointly with ghost modes. 
  We survey the role of stable clocks in general relativity. Clock comparisons have provided important tests of the Einstein Equivalence Principle, which underlies metric gravity. These include tests of the isotropy of clock comparisons (verification of local Lorentz invariance) and tests of the homogeneity of clock comparisons (verification of local position invariance). Comparisons of atomic clocks with gravitational clocks test the Strong Equivalence Principle by bounding cosmological variations in Newton's constant. Stable clocks also play a role in the search for gravitational radiation: comparision of atomic clocks with the binary pulsar's orbital clock has verified gravitational-wave damping, and phase-sensitive detection of waves from inspiralling compact binaries using laser interferometric gravitational observatories will facilitate extraction of useful source information from the data. Stable clocks together with general relativity have found important practical applications in navigational systems such as GPS. 
  Quantization of diffeomorphism invariant theories of connections is studied. A solutions of the diffeomorphism constraints is found. The space of solutions is equipped with an inner product that is shown to satisfy the physical reality conditions. This provides, in particular, a quantization of the Husain-Kucha\v{r} model. The main results also pave way to quantization of other diffeomorphism invariant theories such as general relativity. In the Riemannian case (i.e., signature ++++), the approach appears to contain all the necessary ingredients already. In the Lorentzian case, it will have to combined in an appropriate fashion with a coherent state transform to incorporate complex connections. 
  The stress-energy tensor of a free quantized scalar field is calculated in the extreme Reissner-Nordstr\"{o}m black hole spacetime in the zero temperature vacuum state. The stress-energy appears to be regular on the event horizon, contrary to the suggestion provided by two-dimensional calculations. An analytic calculation on the event horizon for a thermal state shows that if the temperature is nonzero then the stress-energy diverges strongly there. 
  We discuss the most general effective Lagrangian obtained from the assumption that the degrees of freedom to be quantized, in a black hole, are on the horizon. The effective Lagrangian depends only on the induced metric and the extrinsic curvature of the (fluctuating) horizon, and the possible operators can be arranged in an expansion in powers of $\mpl/M$, where $\mpl$ is the Planck mass and $M$ the black hole mass. We perform a semiclassical expansion of the action with a formalism which preserves general covariance explicitly. Quantum fluctuations over the classical solutions are described by a single scalar field living in the 2+1 dimensional world-volume swept by the horizon, with a given coupling to the background geometry. We discuss the resulting field theory and we compute the black hole entropy with our formalism. 
  The vacuum expectation value of the stress-energy tensor $\left\langle 0\left| T_{\mu\nu} \right|0\right\rangle$ is calculated in several multiply connected flat spacetimes for a massive scalar field with arbitrary curvature coupling. We find that a nonzero field mass always decreases the magnitude of the energy density in chronology-respecting manifolds such as $R^3 \times S^1$, $R^2 \times T^2$, $R^1 \times T^3$, the M\"{o}bius strip, and the Klein bottle. In Grant space, which contains nonchronal regions, whether $\left\langle 0\left| T_{\mu\nu} \right|0\right\rangle$ diverges on a chronology horizon or not depends on the field mass. For a sufficiently large mass $\left\langle 0\left| T_{\mu\nu} \right|0\right\rangle$ remains finite, and the metric backreaction caused by a massive quantized field may not be large enough to significantly change the Grant space geometry. 
  In Brans-Dicke theory the Universe becomes divided after inflation into many exponentially large domains with different values of the effective gravitational constant. Such a process can be described by diffusion equations for the probability of finding a certain value of the inflaton and dilaton fields in a physical volume of the Universe. For a typical chaotic inflation potential, the solutions for the probability distribution never become stationary but grow forever towards larger values of the fields. We show here that a non-minimal conformal coupling of the inflaton to the curvature scalar, as well as radiative corrections to the effective potential, may provide a dynamical cutoff and generate stationary solutions. We also analyze the possibility of large nonperturbative jumps of the fluctuating inflaton scalar field, which was recently revealed in the context of the Einstein theory. We find that in the Brans--Dicke theory the amplitude of such jumps is strongly suppressed. 
  It is shown how the different irreducibility classes of the energy-momentum tensor allow for a Lagrangian formulation of the gravity-matter system using a selfdual 2-form as a basic variable. It is pointed out what kind of difficulties arise when attempting to construct a pure spin-connection formulation of the gravity-matter system. Ambiguities in the formulation especially concerning the need for constraints are clarified. 
  We investigate the massless $\lambda \phi^4$ theory in de~Sitter space. It is unnatural to assume a minimally coupled interacting scalar field, since $\xi=0$ is not a fixed point of the renormalization group once interactions are included. In fact, the only case where perturbation theory can be trusted is when the field is non-minimally coupled at the minimum of the effective potential. Thus, in perturbation theory, there is no infrared divergence associated with this scalar field. 
  It is shown that the tunneling effect in quantum cosmology is possible not only at the very beginning or the very end of the evolution, but also at the moment of maximum expansion of the universe. A positive curvature expanding Friedmann universe changes its state of evolution spontaneously and completely, {\it without} any changes in the matter content, avoiding recollapse, and falling into oscillations between the nonzero values of the scale factor. On the other hand, an oscillating nonsingular universe can tunnel spontaneously to a recollapsing regime. The probability of such kind of tunneling is given explicitly. It is inversely related to the amount of nonrelativistic matter (dust), and grows from a certain fixed value to unity if the negative cosmological constant approaches zero. 
  This paper presents a complete set of quasilocal densities which describe the stress-energy-momentum content of the gravitational field and which are built with Ashtekar variables. The densities are defined on a two-surface $B$ which bounds a generic spacelike hypersurface $\Sigma$ of spacetime. The method used to derive the set of quasilocal densities is a Hamilton-Jacobi analysis of a suitable covariant action principle for the Ashtekar variables. As such, the theory presented here is an Ashtekar-variable reformulation of the metric theory of quasilocal stress-energy-momentum originally due to Brown and York. This work also investigates how the quasilocal densities behave under generalized boosts, i. e. switches of the $\Sigma$ slice spanning $B$. It is shown that under such boosts the densities behave in a manner which is similar to the simple boost law for energy-momentum four-vectors in special relativity. The developed formalism is used to obtain a collection of two-surface or boost invariants. With these invariants, one may ``build" several different mass definitions in general relativity, such as the Hawking expression. Also discussed in detail in this paper is the canonical action principle as applied to bounded spacetime regions with ``sharp corners." 
  We consider the axially symmetric coupled system of gravitation, electromagnetism and a dilaton field. Reducing from four to three dimensions, the system is described by gravity coupled to a non-linear $\sigma$-model. We find the target space isometries and use them to generate new solutions. It seems that it is only possible to generate rotating solutions from non-rotating ones for the special cases when the dilaton coupling parameter $a=0, \pm \sqrt{3}$. For those particular values, the target space symmetry is enlarged. 
  It is shown that there are static spacetimes with timelike curvature singularities which appear completely nonsingular when probed with quantum test particles. Examples include extreme dilatonic black holes and the fundamental string solution. In these spacetimes, the dynamics of quantum particles is well defined and uniquely determined. 
  We investigate the dynamics of a Yang-Mills cosmology (YMC, the FRW type spacetime) bubble in the Bartnik-McKinnon (BK) spacetimes. Because a BK spacetime can be identified to a YMC spacetime with a finite scale factor in the neighborhood of the origin, we can give a natural initial condition for the YMC bubble. The YMC bubble can smoothly emerge from the origin without an initial singularity. Under a certain condition, the bubble develops continuously and finally replaces the entire BK spacetime, the metric of which is the same as the one of the radiation dominated universe. We also discuss why an initial singularity can be avoided in the present case in spite of the singularity theorems by Hawking and Penrose. 
  Transition to the semiclassical behaviour and the decoherence process for inhomogeneous perturbations generated from the vacuum state during an inflationary stage in the early Universe are considered both in the Heisenberg and the Schr\"odinger representations to show explicitly that both approaches lead to the same prediction: the equivalence of these quantum perturbations to classical perturbations having stochastic Gaussian amplitudes and belonging to the quasi-isotropic mode. This equivalence and the decoherence are achieved once the exponentially small (in terms of the squeezing parameter $r_k$) decaying mode is neglected. In the quasi-classical limit $|r_k|\to \infty$, the perturbation mode functions can be made real by a time-independent phase rotation, this is shown to be equivalent to a fixed relation between squeezing angle and phase for all modes in the squeezed-state formalism. Though the present state of the gravitational wave background is not a squeezed quantum state in the rigid sense and the squeezing parameters loose their direct meaning due to interaction with the environment and other processes, the standard predictions for the rms values of the perturbations generated during inflation are not affected by these mechanisms (at least, for scales of interest in cosmological applications). This stochastic background still occupies a small part of phase space. 
  We study the problem of how long a journey within a black hole can last. Based on our observations, we make two conjectures. First, for observers that have entered a black hole from an asymptotic region, we conjecture that the length of their journey within is bounded by a multiple of the future asymptotic ``size'' of the black hole, provided the spacetime is globally hyperbolic and satisfies the dominant-energy and non-negative-pressures conditions. Second, for spacetimes with ${\Bbb R}^3$ Cauchy surfaces (or an appropriate generalization thereof) and satisfying the dominant energy and non-negative-pressures conditions, we conjecture that the length of a journey anywhere within a black hole is again bounded, although here the bound requires a knowledge of the initial data for the gravitational field on a Cauchy surface. We prove these conjectures in the spherically symmetric case. We also prove that there is an upper bound on the lifetimes of observers lying ``deep within'' a black hole, provided the spacetime satisfies the timelike-convergence condition and possesses a maximal Cauchy surface. Further, we investigate whether one can increase the lifetime of an observer that has entered a black hole, e.g., by throwing additional matter into the hole. Lastly, in an appendix, we prove that the surface area $A$ of the event horizon of a black hole in a spherically symmetric spacetime with ADM mass $M_{\text{ADM}}$ is always bounded by $A \le 16\pi M_{\text{ADM}}^2$, provided that future null infinity is complete and the spacetime is globally hyperbolic and satisfies the dominant-energy condition. 
  Recent research has established that nonsymmetric gravitation theories like Moffat's NGT predict that a gravitational field singles out an orthogonal pair of polarization states of light that propagate with different phase velocities. We show that a much wider class of nonmetric theories encompassed by the $\chi g$ formalism predict such violations of the Einstein equivalence principle. This gravity-induced birefringence of space implies that propagation through a gravitational field can alter the polarization of light. We use data from polarization measurements of extragalactic sources to constrain birefringence induced by the field of the Galaxy. Our new constraint is $10^8$ times sharper than previous ones. 
  I investigate the relationship between the phase space path integral in (2+1)-dimensional gravity and the canonical quantization of the corresponding reduced phase space in the York time slicing. I demonstrate the equivalence of these two approaches, and discuss some subtleties in the definition of the path integral necessary to prove this equivalence. 
  The question of the integrability of the mixmaster model of the Universe, presented as a dynamical system with finite degrees of freedom, is investigated in present paper. As far as the model belongs to the class of pseudo-Euclidean generalized Toda chains, the method of getting Kovalevskaya exponents developed for chains of Euclidean type, is used. The generalized formula of Adler and van Moerbeke for systems of an indefinite metric is obtained. There was shown that although by the formula we got integer values of Kovalevskaya exponents there were multivalued solutions, branched at particular points on a plain of complex time t. This class of solutions differs from known ones. Apparently, the system does not possess additional algebraic and one-valued first integrals because of complex and transcendental values of the exponents. In addition, a ten-dimensional mixmaster model was studied. There were not integer exponents for this case. 
  We address the role of large diffeomorphisms in Witten's 2+1 gravity on the manifold ${\bf R} \times T^2$. In a ``spacelike sector" quantum theory that treats the large diffeomorphisms as a symmetry, rather than as gauge, the Hilbert space is shown to contain no nontrivial finite dimensional subspaces that are invariant under the large diffeomorphisms. Infinite dimensional closed invariant subspaces are explicitly constructed, and the representation of the large diffeomorphisms is thus shown to be reducible. Comparison is made to Witten's theory on ${\bf R} \times \Sigma$, where $\Sigma$ is a higher genus surface. 
  A spin network is a generalization of a knot or link: a graph embedded in space, with edges labelled by representations of a Lie group, and vertices labelled by intertwining operators. Such objects play an important role in 3-dimensional topological quantum field theory, functional integration on the space A/G of connections modulo gauge transformations, and the loop representation of quantum gravity. Here, after an introduction to the basic ideas of nonperturbative canonical quantum gravity, we review a rigorous approach to functional integration on A/G in which L^2(A/G) is spanned by states labelled by spin networks. Then we explain the `new variables' for general relativity in 4-dimensional spacetime and describe how canonical quantization of gravity in this formalism leads to interesting applications of these spin network states. 
  Positive energy singularities induced by Sine-Gordon solitons in 1+1 dimensional dilaton gravity with positive and negative cosmological constant are considered. When the cosmological constant is positive, the singularities combine a white hole, a timelike singularity and a black hole joined smoothly near the soliton center. When the cosmological constant is negative, the solutions describe two timelike singularities joined smoothly near the soliton center. We describe these spacetimes and examine their evaporation in the one loop approximation.  
  We investigate the possibility that the quantum theory of gravity could be constructed discretely using algebraic methods. The algebraic tools are similar to ones used in constructing topological quantum field theories.The algebraic tools are related to ideas about the reinterpretation of quantum mechanics in a general relativistic context. 
  We continue our examination of the constraints in spherically symmetric general relativity begun in I (gr-qc/9411009) and II (gr-qc/9411010). We extend to general configurations with $J\ne 0$ the analysis of II which treated a moment of time symmetry. We exploit the one parameter family of foliations introduced in I which are linear and homogeneous in the extrinsic curvature to characterize apparent horizons and spatial singularities in the initial data. In particular, we demonstrate that these characterizations do not depend sensitively on the foliation. 
  Using the concept of real tunneling configurations (classical signature change) and nucleation energy, we explore the consequences of an alternative minimization procedure for the Euclidean action in multiple-dimensional quantum cosmology. In both standard Hartle-Hawking type as well as Coleman type wormhole-based approaches, it is suggested that the action should be minimized among configurations of equal energy. In a simplified model, allowing for arbitrary products of spheres as Euclidean solutions, the favoured space-time dimension is 4, the global topology of spacelike slices being ${\bf S}^1 \times {\bf S}^2$ (hence predicting a universe of Kantowski-Sachs type). There is, however, some freedom for a Kaluza-Klein scenario, in which case the observed spacelike slices are ${\bf S}^3$. In this case, the internal space is a product of two-spheres, and the total space-time dimension is 6, 8, 10 or 12. 
  A number of general issues relating to superluminal photon propagation in gravitational fields are explored. The possibility of superluminal, yet causal, photon propagation arises because of Equivalence Principle violating interactions induced by vacuum polarisation in QED in curved spacetime. Two general theorems are presented: first, a polarisation sum rule which relates the polarisation averaged velocity shift to the matter energy-momentum tensor and second, a `horizon theorem' which ensures that the geometric event horizon for black hole spacetimes remains a true horizon for real photon propagation in QED. A comparision is made with the equivalent results for electromagnetic birefringence and possible connections between superluminal photon propagation, causality and the conformal anomaly are exposed. 
  A positive semi-definite Euclidean action for arbitrary four-topologies can be constructed by adding appropriate Yang-Mills and topological terms to the Samuel-Jacobson-Smolin action of gravity with (anti)self-dual variables. Moreover, on-shell, the (anti)self-dual sector of the new theory corresponds precisely to all Einstein manifolds in four dimensions. The Lorentzian signature action, and its analytic continuations are also considered. A self-contained discussion is given on the effects of discrete transformations C, P and T on the Samuel-Jacobson-Smolin action, and other proposed actions which utilize self- or anti-self-dual variables as fundamental variables in the description of four-dimensional gravity. 
  We use the canonical formalism developed together with David Robinson to st= udy the Einstein equations on a null surface. Coordinate and gauge conditions = are introduced to fix the triad and the coordinates on the null surface. Toget= her with the previously found constraints, these form a sufficient number of second class constraints so that the phase space is reduced to one pair of canonically conjugate variables: $\Ac_2\and\Sc^2$. The formalism is related to both the Bondi-Sachs and the Newman-Penrose methods of studying the gravitational field at null infinity. Asymptotic solutions in the vicinity of null infinity which exclude logarithmic behavior require the connection to fall off like $1/r^3$ after the Minkowski limit. This, of course, gives the previous results of Bondi-Sachs and Newman-Penrose. Introducing terms which fall off more slowly leads to logarithmic behavior which leaves null infinity intact, allows for meaningful gravitational radiation, but the peeling theorem does not extend to $\Psi_1$ in the terminology of Newman-Penrose. The conclusions are in agreement with those of Chrusciel, MacCallum, and Singleton. This work was begun as a preliminary study of a reduced phase space for quantization of general relativity. 
  The expectation value of the stress-energy tensor $\langleT_{\mu\nu}\rangle$ of a free conformally invariant scalar field is computed in a general static two-dimensional black hole spacetime when the field is in either a zero temperature vacuum state or a thermal state at a nonzero temperature. It is found that for every static two-dimensional black hole the stress-energy diverges strongly on the event horizon unless the field is in a state at the natural black hole temperature which is defined by the surface gravity of the event horizon. This implies that both extreme and nonextreme two-dimensional black holes can only be in equilibrium with radiation at the natural black hole temperature. 
  It is likely that the observed large-angular-scale anisotropies in the microwave background radiation are induced by the cosmological perturbations of quantum-mechanical origin. Such perturbations are now placed in squeezed vacuum quantum states and, hence, are characterized by large variances of their amplitude. The statistical properties of the anisotropies should reflect the underlying statistics of the squeezed vacuum quantum states. The theoretical variances for the temperature angular correlation function are derived and described quantitatively. It is shown that they are indeed large. Unfortunately, these large theoretical statistical uncertainties will make the extraction of cosmological information from the measured anisotropies a much more difficult problem than we wanted it to be. 
  The propagation of classical gravitational waves in Bianchi Type-I universes is studied. We find that gravitational waves in Bianchi Type-I universes are not equivalent to two minimally coupled massless scalar fields as it is for the Robertson-Walker universe. Due to its tensorial nature, the gravitational wave is much more sensitive to the anisotropy of the spacetime than the scalar field is and it gains an effective mass term. Moreover, we find a coupling between the two polarization states of the gravitational wave which is also not present in the Robertson-Walker universe. 
  The electromagnetic interaction in the Einstein-Infeld-Hoffmann (EIH) equations of motion for charged particles in Einstein's Unified Field Theory is found to be {\em automatically\/} precluded by the conventional identification of the skew part of the fundamental tensor with the Faraday tensor. It is shown that an alternative identification, suggested by observations of Einstein, Bergmann and Papapetrou, would lead to the expected electromagnetic interaction, were it not for the intervention of an infelicitous (radiation) gauge. Therefore, an EIH analysis of EUFT is {\em inconclusive\/} as a test of the physical viability of the theory, and it follows that EUFT cannot be considered necessarily unphysical on the basis of such an analysis. Thus, historically, Einstein's Unified Field Theory was rejected for the wrong reason. 
  This note contains some comments on a recent paper by Friedman on two-component spinors in spacetimes which do not admit a time-orientation, and is intended to clarify the relation of the work reported in that paper to previous literature. 
  We consider the computation of the entanglement entropy in curved backgrounds with event horizons. We use a Hamiltonian approach to the problem and perform numerical computations on a spherical lattice of spacing $a$. We study the cosmological case and make explicit computations for the Friedmann-Robertson-Walker universe. Our results for a massless, minimally coupled scalar field can be summarized by $S_{ent}=0.30 r_H^2/a^2$,which resembles the flat space formula, although here the horizon radius, $r_H$, is time-dependent. 
  A central tenet of the new theory of gravity proposed by H. Yilmaz is the inclusion of a gravitational stress-energy tensor $-{t_\mu}^\nu$ along with the matter stress-energy tensor ${T_\mu}^\nu$ on the right hand side of the Einstein field equations. This change does not effect the Newtonian limit of the field equations since these terms are quadratic in potential gradients. {}From the Bianchi identities, however, important changes appear in any equations of motion consistent with these field equations. For matter described as a perfect fluid, and with Yilmaz's choice of signs when introducing these quadratic terms, we find that the Euler hydrodynamic equation in the Newtonian limit is modified to remove all gravitational forces. This allows, e.g., a solar system in which the Sun and the planets are permanently at rest, but does not explain how fluid bodies such as the Sun or Jupiter could form or be prevented from dispersing. 
  Specific examples of the generalized Raychaudhuri Equations for the evolution of deformations along families of $D$ dimensional surfaces embedded in a background $N$ dimensional spacetime are discussed. These include string worldsheets embedded in four dimensional spacetimes and two dimensional timelike hypersurfaces in a three dimensional curved background. The issue of focussing of families of surfaces is introduced and analysed in some detail. 
  The general theory of N=1 supergravity with supermatter is applied to a Bianchi type IX diagonal model. The supermatter is constituted by a complex scalar field and its spin-$1\over 2$ fermionic partners. The Lorentz invariant Ansatz for the wave function of the universe, $\Psi$, is taken to be as simple as possible in order to obtain {\it new} solutions. The wave function has a simple form when the potential energy term is set to zero. However, neither the wormhole or the Hartle-Hawking state could be found. The Ansatz for $\Psi$ used in this paper is constrasted with the more general framework of R. Graham and A. Csord\'as. 
  A simple model for the formation of a straight cosmic string, wiggly or unperturbed is considered. The gravitational field of such string is computed in the linear approximation. The vacuum expectation value of the stress tensor of a massless scalar quantum field coupled to the string gravitational field is computed to the one loop order. Finally, the back-reaction effect on the gravitational field of the string is obtained by solving perturbatively the semiclassical Einstein's equations. 
  We develop various topological notions on four-manifolds of Kleinian signature $(- - + +)$. In particular, we extend the concept of `Kleinian metric homotopy' to non-orientable manifolds. We then derive the topological obstructions to pin-Klein cobordism, for all of the pin groups. Finally, we discuss various examples and applications which arise from this work. 
  The arguments leading to the introduction of the massive Nonsymmetric Gravitational action are reviewed \cite{Moffat:1994,Moffat:1995b}, leading to an action that gives asymptotically well-behaved perturbations on GR backgrounds. Through the analysis of spherically symmetric perturbations about GR (Schwarzschild) and NGT (Wyman-type) static backgrounds, it is shown that spherically symmetric systems are not guaranteed to be static, and hence Birkhoff's theorem is not valid in NGT. This implies that in general one must consider time dependent exteriors when looking at spherically symmetric systems in NGT. For the surviving monopole mode considered here there is no energy flux as it is short ranged by construction. Further work on the spherically symmetric case will be motivated through a discussion of the possibility that there remain additional modes that do not show up in weak field situations, but nonetheless exist in the full theory and may again result in bad global asymptotics. A presentation of the action and field equations in a general frame is given in the course of the paper, providing an alternative approach to dealing with the algebraic complications inherent in NGT, as well as offering a more general framework for discussing the physics of the antisymmetric sector. 
  We introduce a new basis on the state space of non-perturbative quantum gravity. The states of this basis are linearly independent, are well defined in both the loop representation and the connection representation, and are labeled by a generalization of Penrose's spin netoworks. The new basis fully reduces the spinor identities (SU(2) Mandelstam identities) and simplifies calculations in non-perturbative quantum gravity. In particular, it allows a simple expression for the exact solutions of the Hamiltonian constraint (Wheeler-DeWitt equation) that have been discovered in the loop representation. Since the states in this basis diagnolize operators that represent the three geometry of space, such as the area and volumes of arbitrary surfaces and regions, these states provide a discrete picture of quantum geometry at the Planck scale. 
  We consider an Einstein-Hilbert-Dilaton action for gravity coupled to various types of Abelian and non-Abelian gauge fields in a spatially finite system. These include Yang-Mills fields and Abelian gauge fields with three and four-form field strengths. We obtain various quasilocal quantities associated with these fields, including their energy and angular momentum, and develop methods for calculating conserved charges when a solution possesses sufficient symmetry. For stationary black holes, we find an expression for the entropy from the micro-canonical form of the action. We also find a form of the first law of black hole thermodynamics for black holes with the gauge fields of the type considered here. 
  The energy density of asymptotically flat gravitational fields can be calculated from a simple expression involving the trace of the torsion tensor. Integration of this energy density over the whole space yields the ADM energy. Such expression can be justified within the framework of the teleparallel equivalent of general relativity, which is an alternative geometrical formulation of Einstein's general relativity. In this paper we apply this energy density to the evaluation of the energy per unit length of a class of conical defects of topological nature, which include disclinations and dislocations (in the terminology of crystallography). Disclinations correspond to cosmic strings, and for a spacetime endowed with only such a defect we obtain precisely the well known expression of energy per unit length. However for a pure spacetime dislocation the total gravitational energy is zero. 
  The eigenfunctions of the Dirac operator on spheres and real hyperbolic spaces of arbitrary dimension are computed by separating variables in geodesic polar coordinates. These eigenfunctions are used to derive the heat kernel of the iterated Dirac operator on these spaces. They are then studied as cross sections of homogeneous vector bundles, and a group-theoretic derivation of the spinor spherical functions and heat kernel is given based on Harish-Chandra's formula for the radial part of the Casimir operator. 
  It is shown how the use of the Kerr-Schild coordinate system can greatly simplify the formulation of the geodesic equation of the Schwarzschild solution. An application of this formulation to the numerical computation of the aspect of a non-rotating black hole is presented. The generalization to the case of the Kerr solution is presented too. 
  The determination of the quantum state of a single system by protective observation is used to justify operationally a formulation of quantum theory on the quantum state space (projective Hilbert space) $\cal P$. Protective observation is extended to a more general quantum theory in which the Schrodinger evolution is generalized so that it preserves the symplectic structure but not necessarily the metric in $\cal P$. The relevance of this more general evolution to the apparant collapse of the state vector during the usual measurement, and its possible connection to gravity is suggested. Some criticisms of protective observation are answered. A comparison is made between the determination of quantum states using the geometry of $\cal P$ by protective measurements, via a reconstruction theorem, and the determination of space-time points by means of the space-time geometry, via Einstein's hole argument. It is argued that a protective measurement may not determine a time average. 
  We develop the idea that, in quantum gravity where the horizon fluctuates, a black hole should have a discrete mass spectrum with concomitant line emission. Simple arguments fix the spacing of the lines, which should be broad but unblended. Assuming uniformity of the matrix elements for quantum transitions between near levels, we work out the probabilities for the emission of a specified series of quanta and the intensities of the spectral lines. The thermal character of the radiation is entirely due to the degeneracy of the levels, the same degeneracy that becomes manifest as black hole entropy. One prediction is that there should be no lines with wavelength of order the black hole size or larger. This makes it possible to test quantum gravity with black holes well above Planck scale. 
  We discuss an analytic proof of a conjecture (Nakamura) that solutions of Toda molecule equation give those of Ernst equation giving Tomimatsu-Sato solutions of Einstein equation. Using Pfaffian identities it is shown for Weyl solutions completely and for generic cases partially. 
  For an analogon of the free Wess-Zumino model on Ricci flat spacetimes, the relation between a conserved `supercurrent' and the point-separated improved energy momentum tensor is investigated and a similar relation as on Minkowski space is established. The expectation value of the latter in any globally Hadamard product state is found to be a priori finite in the coincidence limit if the theory is massive. On arbitrary globally hyperbolic spacetimes the `supercurrent' is shown to be a well defined operator valued distribution on the GNS Hilbertspace of any globally Hadamard product state. Viewed as a new field, all n-point distributions exist, giving a new example for a Wightman field on that manifold. Moreover, it is shown that this field satisfies a new wave front set spectrum condition in a non trivial way. 
  The Geometry of planar domain walls is studied. It is argued that the planar walls indeed have plane symmetry. In the Minkowski coordinates the walls are mapped into revolution paraboloids. 
  The possible cosmological effects of primordial fluctuation corrections to the evolution equation of matter obtained from the Wheeler--De Witt equation are explored. In particular, both the metric and a scalar matter field are expanded around their homogeneous values and the corrections induced on the scalar field fluctuation spectrum are perturbatively estimated. Finally, results of a preliminary numerical simulation to investigate the effects on large--scale structure formation are presented. 
  The dynamics of solutions of the Einstein-Vlasov system with Bianchi I symmetry is discussed in the case of massive or massless particles. It is shown that in the case of massive particles the solutions are asymptotic to isotropic dust solutions at late times. The initial singularity is more difficult to analyse. It is shown that the asymptotic behaviour there must be one of a small set of possibilities but it is not clear whether all of these possibilities are realized. One solution is exhibited in the case of massless particles which behaves quite differently near the singularity from any Bianchi I solution with perfect fluid as matter model. In particular the matter is not dynamically negligeable near the singularity for this solution. 
  We examine the rates at which energy and momentum are radiated into gravitational waves by a large set of realistic cosmic string loops. The string loops are generated by numerically evolving parent loops with different initial conditions forward in time until they self-intersect, fragmenting into two child loops. The fragmentation of the child loops is followed recursively until only non-self-intersecting loops remain. The properties of the final non-self-intersecting loops are found to be independent of the initial conditions of the parent loops. We have calculated the radiated energy and momentum for a total of 11,625 stable child loops. We find that the majority of the final loops do not radiate significant amounts of spatial momentum. The velocity gained due to the rocket effect is typically small compared to the center-of-mass velocity of the fragmented loops. The distribution of gravitational radiation rates in the center of mass frame of the loops, $\gamma^0 \equiv (G\mu^2)^{-1} \Delta E/\Delta \tau$, is strongly peaked in the range $\gamma^0=45-55$, however there are no loops found with $\gamma^0 < 40$. Because the radiated spatial momentum is small, the distribution of gravitational radiation rates appears roughly the same in any reference frame. We conjecture that in the center-of-mass frame there is a lower bound $\gamma^0_{\rm min}>0$ for the radiation rate from cosmic string loops. In a second conjecture, we identify a candidate for the loop with the minimal radiation rate and suggest that $\gamma^0_{\rm min}\cong 39.003$. 
  It has been shown for low-spin fields that the use of only the self-dual part of the connection as basic variable does not lead to extra conditions or inconsistencies. We study whether this is true for more general chiral action. We generalize the chiral gravitational action, and assume that half-integer spin fields are coupled with torsion linearly. The equation for torsion is solved and substituted back into the generalized chiral action, giving four-fermion contact terms. If these contact terms are complex, the imaginary part will give rise to extra conditions for the gravitational and matter field equations. We study the four-fermion contact terms taking spin-1/2 and spin-3/2 fields as examples. 
  Exact static, spherically symmetric solutions to the Einstein-Abelian gauge-dilaton equations, in $D$-dimensional gravity with a chain of $n$ Ricci-flat internal spaces are considered, with the gauge field potential having three nonzero components: the temporal, Coulomb-like one, the one pointing to one of the extra dimensions, and the one responsible for a radial magnetic field. For dilaton coupling implied by string theory an $(n+5)$-parametric family of exact solutions is obtained, while for other dilaton couplings only $(n+3)$-parametric ones. The geometric properties and special cases of the solutions are discussed, in particular, those when there are horizons in the space-time. Two types of horizons are distinguished: the conventional black-hole (BH) ones and those at which the physical section of the space-time changes its signature ({\it T-horizons}). Two theorems are proved, one fixing the BH and T-horizon existence conditions, the other discarding the possibility of a regular center. Different conformal gauges are used to characterize the system from the $D$-dimensional and 4-dimensional viewpoints. 
  Vacuum static, axially symmetric space-times in $D$-dimensional general relativity with a Ricci-flat internal space are discussed. It is shown, in particular, that some of the monopole-type solutions are free of curvature singularities and their source can be a disk membrane bounded by a ring with a string or branching type singularity. Another possibility is a wormhole configuration where a particle can penetrate to another spatial infinity by passing through a ring with a string or branching type singularity. The results apply, in particular, to vacuum and scalar-vacuum configurations in conventional general relativity. 
  Several problems in cosmology and astrophysics are described in which critical phenomena of various types may play a role. These include the organization of the disks of spiral galaxies, various aspects of the problem of structure formation in icosmology, the problem of the selection of initial conditions and parameters in particle physics and cosmology and the problem of recovering the classical limit from non-perturbative formulations of quantum gravity. 
  Graviational radiation is described by canonical Yang-Mills wave equations on the curved space-time manifold, together with evolution equations for the metric in the tangent bundle. The initial data problem is described in Yang-Mills scalar and vector potentials, resulting in Lie-constraints in addition to the familiar Gauss-Codacci relations. 
  We give an approach to studying the critical behaviour that has been observed in numerical studies of gravitational collapse. These studies suggest, among other things, that black holes initially form with infinitesimal mass. We show generally how a black hole mass formula can be extracted from a transcendental equation.    Using our approach, we give an explicit one parameter set of metrics that are asymptotically flat and describe the collapse of apriori unspecified but physical matter fields. The black hole mass formula obtained from this metric exhibits a mass gap - that is, at the onset of black hole formation, the mass is finite and non-zero. 
  We analyze the relativistic dynamical properties of Keplerian and non-Keplerian circular orbits in a general axisymmetric and stationary gravitational field, and discuss the implications for the stability of co- and counter-rotating accretion disks and tori surrounding a spinning black hole. Close to the horizon there are orbital peculiarities which can seem counterintuitive, but are elucidated by formulating the dynamics in terms of the orbital velocity actually measured by a local, zero-angular-momentum observer. 
  Witten's formulation of 2+1 gravity is investigated on the nonorientable three-manifold R x (Klein bottle). The gauge group is taken to consist of all four components of the 2+1 Poincare group. We analyze in detail several components of the classical solution space, and we show that from four of the components one can recover nondegenerate spacetime metrics. In particular, from one component we recover metrics for which the Klein bottles are spacelike. An action principle is formulated for bundles satisfying a certain orientation compatibility property, and the corresponding components of the classical solution space are promoted into a phase space. Avenues towards quantization are briefly discussed. 
  The O(N) non-linear sigma model in a $D$-dimensional space of the form ${\bf R}^{D-M} \times {\bf T}^M$, ${\bf R}^{D-M} \times {\bf S}^M$, or ${\bf T}^M \times {\bf S}^P$ is studied, where ${\bf R}^M$, ${\bf T}^M$ and ${\bf S}^M$ correspond to flat space, a torus and a sphere, respectively. Using zeta regularization and the $1/N$ expansion, the corresponding partition functions and the gap equations are obtained. Numerical solutions of the gap equations at the critical coupling constants are given, for several values of $D$. The properties of the partition function and its asymptotic behaviour for large $D$ are discussed. In a similar way, a higher-derivative non-linear sigma model is investigated too. The physical relevance of our results is discussed. 
  Quantum gravity is studied nonperturbatively in the case in which space has a boundary with finite area. A natural set of boundary conditions is studied in the Euclidean signature theory, in which the pullback of the curvature to the boundary is self-dual (with a cosmological constant). A Hilbert space which describes all the information accessible by measuring the metric and connection induced in the boundary is constructed and is found to be the direct sum of the state spaces of all $SU(2)$ Chern-Simon theories defined by all choices of punctures and representations on the spatial boundary $\cal S$. The integer level $k$ of Chern-Simons theory is found to be given by $k= 6\pi /G^2 \Lambda + \alpha$, where $\Lambda$ is the cosmological constant and $\alpha$ is a $CP$ breaking phase. Using these results, expectation values of observables which are functions of fields on the boundary may be evaluated in closed form. The Beckenstein bound and 't Hooft-Susskind holographic hypothesis are confirmed, (in the limit of large area and small cosmological constant) in the sense that once the two metric of the boundary has been measured, the subspace of the physical state space that describes the further information that the observer on the boundary may obtain about the interior has finite dimension equal to the exponent of the area of the boundary, in Planck units, times a fixed constant. Finally,the construction of the state space for quantum gravity in a region from that of all Chern-Simon theories defined on its boundary confirms the categorical-theoretic ``ladder of dimensions picture" of Crane. 
  We analyse the expression for the mass of a stationary axisymmetric configuration in general relativity obtained in our previous work [1]. From the generality of our formula and its incompatibility with the corresponding expression in Kerr space-time we argue that a stationary equilibrium distribution of a real matter cannot be a source of the Kerr metric. 
  A particle of mass $\mu$ moves on a circular orbit around a nonrotating black hole of mass $M$. Under the assumption $\mu \ll M$ the gravitational waves emitted by such a binary system can be calculated exactly numerically using black-hole perturbation theory. If, further, the particle is slowly moving, then the waves can be calculated approximately analytically, and expressed in the form of a post-Newtonian expansion. We determine the accuracy of this expansion in a quantitative way by calculating the reduction in signal-to-noise ratio incurred when matched filtering the exact signal with a nonoptimal, post-Newtonian filter. 
  Eternally inflating universes can contain large thermalized regions with different values of the constants of Nature and with different density fluctuation spectra. To find the probability for a `typical' observer to detect a certain set of constants, or a certain fluctuation spectrum, one needs to compare the volumes occupied by different types of regions. If the volumes are taken on an equal-time hypersurface, the results of such a comparison are extremely sensitive to the choice of the time variable t. Here, I propose a method of comparing the volumes which is rather insensitive to the choice of t. The method is then applied to evaluate the relative probability of different minima of the inflaton potential and the probability distribution for the density fluctuation spectra. 
  We study a two-dimensional dilaton gravity model related by a conformal transformation of the metric to the Callan-Giddings-Harvey-Strominger model. We find that most of the features and problems of the latter can be simply understood in terms of the classical and semiclassical dynamics of accelerated observers in two-dimensional Minkowski space. 
  In Rindler space, we consider the Feynman Green's functions associated with either the Fulling-Rindler vacuum or the Minkowski vacuum. In Euclidean field theory, they becomes respectively the Euclidean Green's functions $G_{\infty}$ and $G_{\2\pi}$, whose we give different suitable forms. In the case of the massive spin-$\frac{1}{2}$ field, we determine also the Euclidean spinor Green's function $S_{\infty}$ and $S_{\2\pi}$ in different suitable forms. In both cases for massless fields in four dimensions, we compute the vacuum expectation value of the energy-momentum tensor relative to the Rindler observer. 
  This paper is a continuation of the paper [V.S.Mashkevich, gr-qc/9409010]. Indeterministic quantum gravity is a theory that unifies general relativity and quantum theory involving indeterministic conception, i.e., quantum jumps. By the same token the theory claims to describe all the universe.  Spacetime is the direct product of cosmic time and space. The state of the universe is given by metric, its derivative with respect to cosmic time, and the number of an energy level. A quantum jump occurs at the tangency of two levels. Equations of motion are the restricted Einstein equation (the cosmic space part thereof) and a probability rule for the quantum jump.  Keywords: indeterminism, quantum jumps, state vector reduction, cosmology, cosmic spacetime 
  We examine quantum field theory in spacetimes that are time nonorientable but have no other causal pathology. These are Lorentzian universes-from-nothing, spacetimes with a single spacelike boundary that nevertheless have a smooth Lorentzian metric. Classically, such spacetimes are locally indistinguishable from their globally hyperbolic covering spaces. However, the construction of a quantum field theory (QFT) is more problematic. One can define a family of local algebras on an atlas of globally hyperbolic subspacetimes. But one cannot extend a generic positive linear function from a single algebra to the collection of all local algebras without violating positivity, while satisfying the physically appropriate overlap conditions. This difficulty can be overcome by restricting the size of neighborhoods so that the union of any pair is time- orientable. The structure of local algebras and states is then locally indistinguishable from that of QFT on a globally hyperbolic spacetime. But this size restriction on neighborhoods makes the structure unsatisfactory as a global field theory. The theory allows less information than QFT in a globally hyperbolic spacetime, because correlations between field operators at a pair of points are defined only if a curve joining the points lies in a single neigh- borhood. Moreover, to extend a local state to a collection of states, we use an antipodally symmetric state on the covering space, a state that would not yield a sensible state on the spacetime if all correlations could be measured. 
  We study the motion of test particle in static axisymmetric vacuum spacetimes and discuss two criteria for strong chaos to occur: (1) a local instability measured by the Weyl curvature, and (2) a tangle of a homoclinic orbit, which is closely related to an unstable periodic orbit in general relativity. We analyze several static axisymmetric spacetimes and find that the first criterion is a sufficient condition for chaos, at least qualitatively. Although some test particles which do not satisfy the first criterion show chaotic behavior in some spacetimes, these can be accounted for the second criterion. 
  For want of a more natural proposal, it is generally assumed that the back-reaction of a quantised matter field on a classical metric is given by the expectation value of its energy-momentum tensor, evaluated in a specified state. This proposal can be expected to be quite sound only when the fluctuations in the energy-momentum tensor of the quantum field are negligible. Based on this condition, a dimensionless criterion has been suggested earlier by Kuo and Ford for drawing the limits on the validity of this semiclassical theory. In this paper, we examine this criterion for the case of a toy model, constructed with two degrees of freedom and a coupling between them that exactly mimics the behaviour of a scalar field in a Friedmann universe. To reproduce the semiclassical regime of the field theory, in the toy model, one of degrees of freedom is assumed to be classical and the other quantum mechanical. Also the backreaction is assumed to be given by the expectation values of the quantum operators involved in the equations of motion for the classical system. Motivated by the same physical reasoning as Kuo and Ford, we, here, suggest another criterion, one which will be shown to perform more reliably as we evaluate these criterions for different states of the quantum system in the toy model. Finally, from the results obtained we conclude that the semiclassical theory being considered for the toy model is reliable, during all stages of its evolution, only if the quantum system is specified to be in coherent like states. The implications of these investigations on field theory are discussed. 
  We discuss the main cosmological implications of considering string-loop effects and a potential for the dilaton in the lowest order string effective action. Our framework is based on the effective model arising from regarding homogeneous and isotropic dilaton, metric and Yang-Mills field configurations. The issues of inflation, entropy crisis and the Polonyi problem as well as the problem of the cosmological constant are discussed. 
  We find evidence for the existence of solutions of the Einstein and Abelian Higgs field equations describing a black hole pierced by a Nielsen-Olesen vortex. This situation falls outside the scope of the usual no-hair arguments due to the non-trivial topology of the vortex configuration and the special properties of its energy-momentum tensor. By a combination of numerical and perturbative techniques we conclude that the black hole horizon has no difficulty in supporting the long range fields of the Nielsen Olesen string. Moreover, the effect of the vortex can in principle be measured from infinity, thus justifying its characterization as black hole ``hair". 
  A positive, diffeomorphism-invariant generalized measure on the space of metrics of a two-dimensional smooth manifold is constructed. We use the term generalized measure analogously with the generalized measures of Ashtekar and Lewandowski and of Baez. A family of actions is presented which, when integrated against this measure, give the two-dimensional axiomatic topological quantum field theories, or TQFT's, in terms of which Durhuus and Jonsson decompose every two-dimensional unitary TQFT as a direct sum. 
  We study gravitational collapse of the axion/dilaton field in classical low energy string theory, at the threshold for black hole formation. A new critical solution is derived that is spherically symmetric and continuously self-similar. The universal scaling and echoing behavior discovered by Choptuik in gravitational collapse appear in a somewhat different form. In particular, echoing takes the form of SL(2,R) rotations (cf. S-duality). The collapse leaves behind an outgoing pulse of axion/dilaton radiation, with nearly but not exactly flat spacetime within it. 
  A benchmark problem for numerical relativity has been the head-on collision of two black holes starting from the ``Misner initial data,'' a closed form momentarily stationary solution to the constraint equations with an adjustable closeness parameter $\mu_0$. We show here how an eclectic mixture of approximation methods can provide both an efficient means of determining the time development of the initial data and a good understanding of the physics of the problem. When the Misner data is chosen to correspond to holes initially very close together, a common horizon surrounds both holes and the geometry exterior to the horizon can be treated as a non-spherical perturbation of a single Schwarzschild hole. When the holes are initially well separated the problem can be treated with a different approximation scheme, ``the particle-membrane method.'' For all initial separations, numerical relativity is in principle applicable, but is costly and of uncertain accuracy. We present here a comparison of the different approaches. We compare waveforms, for $\ell=2$ and $\ell=4$ radiation, for different values of $\mu_0$, from the three different approaches to the problem. 
  We show that the partition function of the Ponzano-Regge quantum gravity model can be written as a sum over surfaces in a $(2+1)$ dimensional space-time. We suggest a geometrical meaning, in terms of surfaces, for the (regulated) divergences that appear in the partition function. 
  Ashtekar's canonical theory of classical complex Euclidean GR (no Lorentzian reality conditions) is found to be invariant under the full algebra of infinitesimal 4-diffeomorphisms, but non-invariant under some finite proper 4-diffeos when the densitized dreibein, $\tilE^a_i$, is degenerate. The breakdown of 4-diffeo invariance appears to be due to the inability of the Ashtekar Hamiltonian to generate births and deaths of $\tilE$ flux loops (leaving open the possibility that a new `causality condition' forbidding the birth of flux loops might justify the non-invariance of the theory).  A fully 4-diffeo invariant canonical theory in Ashtekar's variables, derived from Plebanski's action, is found to have constraints that are stronger than Ashtekar's for $rank\tilE < 2$. The corresponding Hamiltonian generates births and deaths of $\tilE$ flux loops.  It is argued that this implies a finite amplitude for births and deaths of loops in the physical states of quantum GR in the loop representation, thus modifying this (partly defined) theory substantially.  Some of the new constraints are second class, leading to difficulties in quantization in the connection representation. This problem might be overcome in a very nice way by transforming to the classical loop variables, or the `Faraday line' variables of Newman and Rovelli, and then solving the offending constraints.  Note that, though motivated by quantum considerations, the present paper is classical in substance. 
  We investigate whether inertial thermometers moving in a thermal bath behave as being hotter or colder. This question is directly related to the classical controversy concerning how temperature transforms under Lorentz transformations. Rather than basing our arguments on thermodynamical hypotheses, we perform straightforward calculations in the context of relativistic quantum field theory. For this purpose we use Unruh-DeWitt detectors, since they have been shown to be reliable thermometers in semi-classical gravity. We believe that our discussion helps in definitely clarifying this issue. 
  We examine from first principles one of the basic assumptions of modern quantum theories of structure formation in the early universe, i.e., the conditions upon which fluctuations of a quantum field may transmute into classical stochastic perturbations, which grew into galaxies. Our earlier works have discussed the quantum origin of noise in stochastic inflation and quantum fluctuations as measured by particle creation in semiclassical gravity. Here we focus on decoherence and the relation of quantum and classical fluctuations. Instead of using the rather ad hoc splitting of a quantum field into long and short wavelength parts, the latter providing the noise which decoheres the former, we treat a nonlinear theory and examine the decoherence of a quantum mean field by its own quantum fluctuations, or that of other fields it interacts with. This is an example of `dynamical decoherence' where an effective open quantum system decoheres through its own dynamics. The model we use to discuss fluctuation generation has the inflation field coupled to the graviton field. We show that when the quantum to classical transition is properly treated, with due consideration of the relation of decoherence, noise, fluctuation and dissipation, the amplitude of density contrast predicted falls in the acceptable range without requiring a fine tuning of the coupling constant of the inflation field ($\lambda$). The conventional treatment which requires an unnaturally small $\lambda \approx 10^{-12}$ stems from a basic flaw in naively identifying classical perturbations with quantum fluctuations. 
  The conformal anomaly for spinors and scalars on a N-dimensional hyperbolic space is calculated explicitly, by using zeta-function regularization techniques and the Selberg trace formula. In the case of conformally invariant spinors and scalars the results are very much related with those corresponding to a N-dimensional sphere. 
  The general theory of N=1 supergravity with supermatter is applied to a Bianchi type IX diagonal model. The supermatter is constituted by a complex scalar field and its spin-$1\over 2$ fermionic partners. The K\"ahler geometry is chosen to be a two-dimensional flat one. The Lorentz invariant Ansatz for the wave function of the universe is taken to be as simple as possible in order to obtain {\it new} solutions. The set of differential equations derived from the quantum constraints are analysed in two different cases: if the supermatter terms include an analytical potential or not. In the latter the wave function is found to have a simple form. 
  A formulation of Poincare symmetry as an inner symmetry of field theories defined on a fixed Minkowski spacetime is given. Local P gauge transformations and the corresponding covariant derivative with P gauge fields are introduced. The renormalization properties of scalar, spinor and vector fields in P gauge field backgrounds are determined. A minimal gauge field dynamics consistent with the renormalization constraints is given. 
  Experimental tests of homogeneous-universe classical standpoint cosmology are proposed after presentation of conceptual considerations that encourage this radical departure from the standard model. Among predictions of the new model are standpoint age equal to Hubble time, energy-density parameter $\Omega_0 = 2 - \sqrt{2} =.586$, and relations between redshift, Hubble-scale distribution of matter and galaxy luminosity and angular diameter. These latter relations coincide with those of the standard model for zero deceleration. With eye to further tests, geodesics of the non-Riemannian standpoint metric are explicitly given. Although a detailed thermodynamic ``youthful-standpoint'' approximation remains to be developed (for particle mean free path small on standpoint scale), standpoint temperature depending only on standpoint age is a natural concept, paralleling energy density and redshift that perpetuates thermal spectrum for cosmic background radiation. Prospects for primordial nucleosynthesis are promising. 
  The Kadanoff-Wilson renormalization group approach for a scalar self-interacting field theor generally coupled with gravity is presented. An average potential that monitors the fluctuations of the blocked field in different scaling regimes is constructed in a nonflat background and explicitly computed within the loop-expansion approximation for an Einstein universe. The curvature turns out to be dominant in setting the crossover scale from a double-peak and a symmetric distribution of the block variables. The evolution of all the coupling constants generated by the blocking procedure is examined: the renormalized trajectories agree with the standard perturbative results for the relevant vertices near the ultraviolet fixed point, but new effective interactions between gravity and matter are present. The flow of the conformal coupling constant is therefore analyzed in the improved scheme and the infrared fixed point is reached for arbitrary values of the renormalized parameters. 
  The thermodynamic behaviour of a relativistic perfect simple fluid obeying the equation of state $p=(\gamma-1)\rho $, where $0 \le \gamma \le 2$ is a constant, has been investigated. Particular cases include: vacuum($p=-\rho $, $\gamma=0$), a randomly oriented distribution of cosmic strings ($p=-{1 \over 3} \rho $, $\gamma =2/3$), blackbody radiation ($p={1\over 3} \rho$, $\gamma =4/3$) and stiff matter ($p=\rho$, $\gamma=2$). Fluids with $\gamma <1$ become hotter when they expand adiabatically ($T\propto V^{1- \gamma}$). By assuming that such fluids may be regarded as a kind of generalized radiation, the general Planck's type form of the spectrum is deduced. As a limiting case, a new Lorentz invariant spectrum of the vacuum which is compatible with the equation of state and other thermodynamic constraints is proposed. Some possible consequences to the early universe physics are also discussed. 
  One of the greatest challenges facing theoretical physics lies in reconciling Einstein's classical theory of gravity - general relativity - with quantum field theory. Although both theories have been experimentally supported in their respective regimes, they seem mutually incompatible. This article summarises the current status of the superstring approach to the problem, the status of the Ashtekar program, and addresses the problem of time in quantum gravity. It contains interviews with Abhay Ashtekar, Chris Isham, and Edward Witten. 
  Spatially homogeneous models in quantum supergravity with a nonvanishing cosmological constant are studied. A class of exact nontrivial solutions of the supersymmetry and Lorentz constraints is obtained in terms of the Chern-Simons action on the spatially homogeneous 3-manifold, both in Ashketar variables where the solution is explicit up to reality conditions, and, more concretely, in the tetrad-representation, where the solutions are given as integral representations differing only by the contours of integration. In the limit of a vanishing cosmological constant earlier exact solutions for Bianchi type IX models in the tetrad-representation are recovered and additional asymmetric solutions are found. 
  A class of exact solutions of the Einstein field equations representing non-static wormholes that obey the {\em weak and dominant energy conditions } is presented. Hence, in principle, these wormholes can be built with less exotic matter than the static ones. 
  The first junction conditions of spherically symmetric bubbles are solved for some cases, and whereby analytic models to the Einstein field equations are constructed. The effects of bubbles on the spacetime structure are studied and it is found that in some cases bubbles can close the spatial sector of the spacetime and turn it into a compact one, while in other cases they can give rise to wormholes. One of the most remarkable features of these wormholes is that they do not necessarily violate the weak and dominant energy condition even at the classical level. 
  The weak cosmic censorship hypothesis can be understood as a statement that there exists a global Cauchy evolution of a selfgravitating system outside an event horizon. The resulting Cauchy problem has a free null-like inner boundary. We study a selfgravitating spherically symmetric nonlinear scalar field. We show the global existence of a spacetime with a null inner boundary that initially is located outside the Schwarzschild radius or, more generally, outside an apparent horizon. The global existence of a patch of a spacetime that is exterior to an event horizon is obtained as a limiting case. 
  It is well-known that gauge fields defined on manifolds with spatial boundaries support states localized at the boundary. In this talk, we show how similar states arise in canonical gravity abd discuss their physical relevance using their analogy to quantum Hall effect. 
  Once the action for Einstein's equations is rewritten as a functional of an SO(3,C) connection and a conformal factor of the metric, it admits a family of ``neighbours'' having the same number of degrees of freedom and a precisely defined metric tensor. This paper analyzes the relation between the Riemann tensor of that metric and the curvature tensor of the SO(3) connection. The relation is in general very complicated. The Einstein case is distinguished by the fact that two natural SO(3) metrics on the GL(3) fibers coincide. In the general case the theory is bimetric on the fibers. 
  The Lagrangian formulation of classical field theories and in particular general relativity leads to a coordinate-free, fully covariant analysis of these constrained systems. This paper applies multisymplectic techniques to obtain the analysis of Palatini and self-dual gravity theories as constrained systems, which have been studied so far in the Hamiltonian formalism. The constraint equations are derived while paying attention to boundary terms, and the Hamiltonian constraint turns out to be linear in the multimomenta. The equivalence with Ashtekar's formalism is also established. The whole constraint analysis, however, remains covariant in that the multimomentum map is evaluated on {\it any} spacelike hypersurface. This study is motivated by the non-perturbative quantization program of general relativity. 
  As an example of what happens with physically relevant theories like effective gravity, we consider the covariant relativistic theory of a scalar field of arbitrarily higher differential order. A procedure based on the Legendre transformation and suitable field redefinitions allows to recast it as a theory of second order with one explicit independent field for each degree of freedom. The physical and ghost fields are then apparent. The full (classical) equivalence of both Higher and Lower Derivative versions is shown. An artifact of the method is the appearance of irrelevant spurious fields which are devoid of any dynamical content. 
  In a recent paper, Parker and Zhang [1] consider cosmological perturbations that can be possibly produced in the early Universe. This is an interesting problem having important observational implications. The paper of Parker and Zhang is transparent and honest, in the sense that the authors clearly identify what they derive and what they use from the previous literature. The final numerical estimates of the paper~[1] rely entirely on a formula which the authors take from the inflationary literature. This formula, see Eqs.~(22), (43), relates the amplitude of perturbations today with the amplitude of perturbations during the inflationary stage. The formula suggests an enormous increase of the amplitude, if the equation of state at the inflationary phase has happened to be sufficiently close to the de Sitter one. The point of my comment is that this formula is incorrect, as we will see below, and the results based on this formula cannot be trusted. 
  Within the framework of the quantum field theory at finite temperature on a conical space, we determine the Euclidean thermal spinor Green's function for a massless spinor field. We then calculate the thermal average of the energy-momentum tensor of a thermal bath of massless fermions. In the high-temperature limit, we find that the straight cosmic string does not perturb the thermal bath 
  A recent analysis of real general relativity based on multisymplectic techniques has shown that boundary terms may occur in the constraint equations, unless some boundary conditions are imposed. This paper studies the corresponding form of such boundary terms in complex general relativity, where space-time is a four-complex-dimensional complex-Riemannian manifold. A complex Ricci-flat space-time is recovered providing some boundary conditions are imposed on two-complex-dimensional surfaces. One then finds that the holomorphic multimomenta should vanish on an arbitrary three-complex-dimensional surface, to avoid having restrictions at this surface on the spinor fields which express the invariance of the theory under holomorphic coordinate transformations. The Hamiltonian constraint of real general relativity is then replaced by a geometric structure linear in the holomorphic multimomenta, and a link with twistor theory is found. Moreover, a deep relation emerges between complex space-times which are not anti-self-dual and two-complex-dimensional surfaces which are not totally null. 
  In complex general relativity, Lorentzian space-time is replaced by a four-complex-dimensional complex-Riemannian manifold, with holomorphic connection and holomorphic curvature tensor. A multisymplectic analysis shows that the Hamiltonian constraint is replaced by a geometric structure linear in the holomorphic multimomenta, providing some boundary conditions are imposed on two-complex-dimensional surfaces. On studying such boundary conditions, a link with the Penrose twistor programme is found. Moreover, in the case of real Riemannian four-manifolds, the local theory of primary and secondary potentials for gravitino fields, recently proposed by Penrose, has been applied to Ricci-flat backgrounds with boundary. The geometric interpretation of the differential equations obeyed by such secondary potentials is related to the analysis of integrability conditions in the theory of massless fields, and might lead to a better understanding of twistor geometry. Thus, new tools are available in complex general relativity and in classical field theory in real Riemannian backgrounds. 
  We investigate the spectral properties of the volume operator in quantum gravity in the framework of a previously introduced lattice discretization. The presence of a well-defined scalar product in this approach permits us to make definite statements about the hermiticity of quantum operators. We find that the spectrum of the volume operator is discrete, but that the nature of its eigenstates differs from that found in an earlier continuum treatment. 
  Several isotropic, homogeneous cosmological models containing a self-interacting minimally coupled scalar field, a perfect fluid source and cosmological constant are solved. New exact, asymptotically stable solutions with an inflationary regime or a final Friedmann stage are found for some simple, interesting potentials. It is shown that the fluid and the curvature may determine how these models evolve for large times. 
  The minimal prolongation structure for the Robinson-Trautman equations of Petrov type III is shown to always include the infinite-dimensional, contragredient algebra, K$_2$, which is of infinite growth. Knowledge of faithful representations of this algebra would allow the determination of B\"acklund transformations to evolve new solutions. 
  The union of high-energy particle theories and gravitation often gives rise to an evolving strength of gravity. The standard picture of the earliest universe would certainly deserve revision if the Planck mass, which defines the strength of gravity, varied. A notable consequence is a gravity-driven, kinetic inflation. Unlike standard inflation, there is no potential or cosmological constant. The unique elasticity in the kinetic energy of the Planck mass provides a negative pressure able to drive inflation. As the kinetic energy grows, the spacetime expands more quickly. The phenomenon of kinetic inflation has been uncovered in both string theory and Kaluza-Klein theories. The difficulty in exiting inflation in these cases is reviewed. General forms of the Planck field coupling are shown to avoid the severity of the graceful exit problem found in string and Kaluza-Klein theories. The completion of the model is foreshadowed with a suggestion for a heating mechanism to generate the hot soup of the big bang. 
  We present spherically symmetric static solutions (a particle-like solution and a black hole solution) in the Einstein-Yang-Mills system with a cosmological constant.Although their gravitational structures are locally similar to those of the Bartnik-McKinnon particles or the colored black holes, the asymptotic behavior becomes quite different because of the existence of a cosmological horizon. We also discuss their stability by means of a catastrophe theory as well as a linear perturbation analysis and find the number of unstable modes. 
  Liouville theory is shown to describe the asymptotic dynamics of three-dimensional Einstein gravity with a negative cosmological constant. This is because (i) Chern-Simons theory with a gauge group $SL(2,R) \times SL(2,R)$ on a space-time with a cylindrical boundary is equivalent to the non-chiral $SL(2,R)$ WZW model; and (ii) the anti-de Sitter boundary conditions implement the constraints that reduce the WZW model to the Liouville theory. 
  We discuss the pair creation of black holes by the breaking of a cosmic string. We obtain an instanton describing this process from the $C$ metric, and calculate its probability. This is very low for the strings that have been suggested for galaxy formation. 
  We find solutions for quantum Class A Bianchi models of the form $\rm \Psi=W e^{\pm \Phi}$ generalizing the results obtained by Moncrief and Ryan in standard quantum cosmology. For the II and IX Bianchi models there are other solutions $\rm \tilde\Phi_2$, $\rm \tilde\Phi_9$ to the Hamilton-Jacobi equation for which $\rm \Psi$ is necessarely zero, in contrast with solutions found in supersymmetric quantum cosmology. 
  We examine the effects of spin-orbit and spin-spin coupling on the inspiral of a coalescing binary system of spinning compact objects and on the gravitational radiation emitted therefrom. Using a formalism developed by Blanchet, Damour, and Iyer, we calculate the contributions due to the spins of the bodies to the symmetric trace-free radiative multipole moments which are used to calculate the waveform, energy loss, and angular momentum loss from the inspiralling binary. Using equations of motion which include terms due to spin-orbit and spin-spin coupling, we evolve the orbit of a coalescing binary and use the orbit to calculate the emitted gravitational waveform. We find the spins of the bodies affect the waveform in several ways: 1) The spin terms contribute to the orbital decay of the binary, and thus to the accumulated phase of the gravitational waveform. 2) The spins cause the orbital plane to precess, which changes the orientation of the orbital plane with respect to an observer, thus causing the shape of the waveform to be modulated. 3) The spins contribute directly to the amplitude of the waveform. We discuss the size and importance of spin effects for the case of two coalescing neutron stars, and for the case of a neutron star orbiting a rapidly rotating $10M_\odot$ black hole. 
  The effect of gravitational radiation reaction on circular orbits around a spinning (Kerr) black hole is computed to leading order in $S$ (the magnitude of the spin angular momentum of the hole) and in the strength of gravity $M/r$ (where $M$ is the mass of the black hole, $r$ is the orbital radius, and $G=c=1$). The radiation reaction makes the orbit shrink but leaves it circular, and drives the orbital plane very slowly toward antialignment with the spin of the hole: $\tan (\iota /2) = \tan (\iota_0 /2) [1+(61/72)(S/M^2) (M/r)^{3/2}]$, where $\iota$ is the angle between the normal to the orbital plane and the spin direction, and $\iota_0$ is the initial value of $\iota$, when $r$ is very large. 
  A new method of gravitational- waves detection in the $10^{-2}\div 10^{-1} Hz$ band for a space laboratory is proposed based on the use of the Kozorez effect in the magnetic interaction of superconducting solenoids. 
  We argue that production of charged black hole pairs joined by a cosmic string in the presence of a magnetic field can be analyzed using the Ernst metric. The effect of the cosmic string is to pull the black holes towards each other, opposing to the background field. An estimation of the production rate using the Euclidean action shows that the process is suppressed as compared to the formation of black holes without strings. 
  The canonical quantization of homogeneous cosmologies is considered in the high anisotropic limit. Exact wavefunctions are found in this limit when the momentum constraints are reduced at the classical level. Lorentzian solutions that represent tunnelling from classically forbidden regimes are identified. Solutions to the modified Wheeler-DeWitt equation are also found for the vacuum Bianchi IX model when a quantum reduction of the momentum constraints is considered. 
  The pulsation equations for spherically symmetric black hole and soliton solutions are brought into a standard form. The formulae apply to a large class of field theoretical matter models and can easily be worked out for specific examples. The close relation to the energy principle in terms of the second variation of the Schwarzschild mass is also established. The use of the general expressions is illustrated for the Einstein-Yang-Mills and the Einstein-Skyrme system. 
  Tangles of loops which approximate an aspect of the Kerr-Newman black hole metrics at large scales compared to the Planck length are constructed. The physical aspect the tangles approximate is discussed. This construction may be useful in the loop representation of canonical quantum gravity. Implications and applications of the tangles are remarked. 
  Quantization of the system comprising gravitational, fermionic and electromagnetic fields is developed in the loop representation. As a result we obtain a natural unified quantum theory. Gravitational field is treated in the framework of Ashtekar formalism; fermions are described by two Grassmann-valued fields. We define a $C^{*}$-algebra of configurational variables whose generators are associated with oriented loops and curves; ``open'' states -- curves -- are necessary to embrace the fermionic degrees of freedom. Quantum representation space is constructed as a space of cylindrical functionals on the spectrum of this $C^{*}$-algebra. Choosing the basis of ``loop'' states we describe the representation space as the space of oriented loops and curves; then configurational and momentum loop variables become in this basis the operators of creation and annihilation of loops and curves. The important difference of the representation constructed from the loop representation of pure gravity is that the momentum loop operators act in our case simply by joining loops in the only compatible with their orientaiton way, while in the case of pure gravity this action is more complicated. 
  The quantum cosmology of two-dimensional dilaton-gravity models is investigated. A class of models is mapped onto the constrained oscillator-ghost-oscillator model. A number of exact and approximate solutions to the corresponding Wheeler-DeWitt equation are presented. A wider class of minisuperspace models that can be solved in this fashion is identified. Supersymmetric extensions to the induced gravity theory and the bosonic string theory are then considered and closed-form solutions to the associated quantum constraints are derived. The possibility of applying the third-quantization procedure to two-dimensional dilaton-gravity is briefly discussed. 
  An algebraic classification of second order symmetric tensors in 5-dimensional Kaluza-Klein-type Lorentzian spaces is presented by using Jordan matrices. We show that the possible Segre types are $[1,1111]$, [2111], [311], [z,\bar{z},111], and the degeneracies thereof. A set of canonical forms for each Segre type is found. The possible continuous groups of symmetry for each canonical form are also studied. 
  Quantum-mechanical model of self-gravitating dust shell is considered. To clarify the relation between classical and quantum spacetime which the shell collapse form, we consider various time slicing on which quantum mechanics is developed. By considering the static time slicing which corresponds to an observer at a constant circumference radius, we obtain the wave functions of the shell motion and the discrete mass spectra which specify the global structures of spherically symmetric spacetime formed by the shell collapse. It is found that wormhole states are forbidden when the rest mass is comparable with Plank mass scale due to the zero-point quantum fluctuations. 
  It has been recently suggested that the Non-symmetric Gravitational Theory (NGT) is free of black holes. Here, we study the linear version of NGT. We find that even with spherical symmetry the skew part of the metric is generally non-static. In addition, if the skew field is initially regular, it will remain regular everywhere and, in particular, at the horizon. Therefore, in the fully-nonlinear theory, if the initial skew-field is sufficiently small, the formation of a black hole is to be anticipated. 
  We introduce an external field to calculate the quantum corrections of the 2d gravity, via trace anomaly. We show that there are black hole type solution even in the absence of matter field and cosmological constant. We also see that these solutions are similar to the ones obtained from dilaton two-dimensional gravity. 
  We study some two-dimensional dilaton gravity models using the formal theory of partial differential equations. This allows us to prove that the reduced phase space is two-dimensional without an explicit construction. By using a convenient (static) gauge we reduce the theory to coupled \ode s and we are able to derive for some potentials of interest closed-form solutions. We use an effective (particle) Lagrangian for the reduced field equations in order to quantize the system in a finite-dimensional setting leading to an exact partial differential Wheeler-DeWitt equation instead of a functional one. A WKB approximation for some quantum states is computed and compared with the classical Hamilton-Jacobi theory. The effect of minimally coupled matter is examined. 
  Spacetimes with everywhere vanishing curvature tensor, but with torsion different from zero only on world sheets that represent closed loops in ordinary space are presented, also defects along open curves with end points at infinity are studied. The case of defects along timelike loops is also considered and the geodesics in these spaces are briefly discussed. 
  Commutator anomalies obstruct solving the Wheeler-DeWitt constraint equation in Dirac quantization of quantum gravity-matter theory. When the obstruction is removed, there result quantal modifications to the constraints. The same classical theory gives rise to different quantum theories when different procedures for overcoming anomalies are implemented. 
  Some aspects of the algebraic quantization programme proposed by Ashtekar are revisited in this article. It is proved that, for systems with first-class constraints, the involution introduced on the algebra of quantum operators via reality conditions can never be projected unambiguously to the algebra of physical observables, ie, of quantum observables modulo constraints. It is nevertheless shown that, under sufficiently general assumptions, one can still induce an involution on the algebra of physical observables from reality conditions, though the involution obtained depends on the choice of particular representatives for the equivalence classes of quantum observables and this implies an additional ambiguity in the quantization procedure suggested by Ashtekar. 
  We study the Hilbert bundle description of stochastic quantum mechanics in curved spacetime developed by Prugove\v{c}ki, which gives a powerful new framework for exploring the quantum mechanical propagation of states in curved spacetime. We concentrate on the quantum transport law in the bundle, specifically on the information which can be obtained from the flat space limit. We give a detailed proof that quantum transport coincides with parallel transport in the bundle in this limit, confirming statements of Prugove\v{c}ki. We furthermore show that the quantum-geometric propagator in curved spacetime proposed by Prugove\v{c}ki, yielding a Feynman path integral-like formula involving integrations over intermediate phase space variables, is Poincar\'e gauge covariant (i.e.$\!$ is gauge invariant except for transformations at the endpoints of the path) provided the integration measure is interpreted as a ``contact point measure'' in the soldered stochastic phase space bundle raised over curved spacetime. 
  Loop variables are used to describe the presence of topological defects in spacetime. In particular we study the dependence of the holonomy transformation on angular momentum and torsion for a multi-chiral cone. We also compute the holonomies for multiple moving crossed cosmic strings and two plane topological defects-crossed by a cosmic string. 
  It is shown that topologically stable cosmic strings can, in fact, appear to end or to break, even in theories without monopoles. This can occur whenever the spatial topology of the universe is nontrivial. For the case of Abelian-Higgs strings, we describe the gauge and scalar field configurations necessary for a string to end on a black hole. We give a lower bound for the rate at which a cosmic string will break via black hole pair production, using an instanton calculation based on the Euclidean C-metric. 
  The issue concerning the existence of wormhole states in locally supersymmetric minisuperspace models with matter is addressed. Wormhole states are apparently absent in models obtained from the more general theory of N=1 supergravity with supermatter. A Hartle-Hawking type solution can be found, even though some terms (which are scalar field dependent) cannot be determined in a satisfactory way. A possible cause is investigated here. As far as the wormhole situation is concerned, we argue here that the type of Lagrange multipliers and fermionic derivative ordering one uses may make a difference. A proposal is made for supersymmetric quantum wormholes to also be invested with a Hilbert space structure, associated with a maximal analytical extension of the corresponding minisuperspace. 
  Dirac's quantization of the (2+1)-dimensional analog of Ashtekar's approach to quantum gravity is investigated. After providing a diffeomorphism-invariant regularization of the Hamiltonian constraint, we find a set of solutions to this Hamiltonian constraint which is a generalization of the solution discovered by Jacobson and Smolin. These solutions are given by particular linear combinations of the spin network states. While the classical counterparts of these solutions have degenerate metric, due to a \lq quantum effect' the area operator has nonvanishing action on these states. We also discuss how to extend our results to (3+1)-dimensions. 
  We report on a numerical study of the spherically symmetric collapse of a self-gravitating massless scalar field. Earlier results of Choptuik(1992, 1994) are confirmed. The field either disperses to infinity or collapses to a black hole, depending on the strength of the initial data. For evolutions where the strength is close to but below the strength required to form a black hole, we argue that there will be a region close to the axis where the scalar curvature and field energy density can reach arbitrarily large levels, and which is visible to distant observers 
  We reconsider the cosmic string perturbative solution to the classical fourth-order gravity field equations, obtained in Ref.\cite{CLA94}, and we obtain that static, cylindricaly symmetric gauge cosmic strings, with constant energy density, can contain only $\beta$-terms in the first order corrections to the interior gravitational field, while the exact exterior solution is a conical spacetime with deficit angle $D=8\pi\mu$. 
  The basic problem of quantum cosmology is the definition of the quantum state of the universe, with appropriate boundary conditions on Riemannian three-geometries. This paper describes recent progress in the corresponding analysis of quantum amplitudes for Euclidean Maxwell theory and linearized gravity. Within the framework of Faddeev-Popov formalism and zeta-function regularization, various choices of mixed boundary conditions lead to a deeper understanding of quantized gauge fields and quantum gravity in the presence of boundaries. 
  We consider two quantum cosmological models with a massive scalar field: an ordinary Friedmann universe and a universe containing primordial black holes. For both models we discuss the complex solutions to the Euclidean Einstein equations. Using the probability measure obtained from the Hartle-Hawking no-boundary proposal, we find that the only unsuppressed black holes start at the Planck size but can grow with the horizon scale during the roll down of the scalar field to the minimum. 
  Asymptotic expansions were first introduced by Henri Poincare in 1886. This paper describes their application to the semi-classical evaluation of amplitudes in quantum field theory with boundaries. By using zeta-function regularization, the conformal anomaly for a massless spin-${1\over 2}$ field in flat Euclidean backgrounds with boundary is obtained on imposing locally supersymmetric boundary conditions. The quantization program for gauge fields and gravitation in the presence of boundaries is then introduced by focusing on conformal anomalies for higher-spin fields. The conditions under which the covariant Schwinger-DeWitt and the non-covariant, mode-by-mode analysis of quantum amplitudes agree are described. 
  This paper reviews the progress made over the last five years in studying boundary conditions and semiclassical properties of quantum fields about 4-real-dimensional Riemannian backgrounds. For massless spin-${1\over 2}$ fields one has a choice of spectral or supersymmetric boundary conditions, and the corresponding conformal anomalies have been evaluated by using zeta-function regularization. For Euclidean Maxwell theory in vacuum, the mode-by-mode analysis of BRST-covariant Faddeev-Popov amplitudes has been performed for relativistic and non-relativistic gauge conditions. For massless spin-${3\over 2}$ fields, the contribution of physical degrees of freedom to one-loop amplitudes, and the 2-spinor analysis of Dirac and Rarita-Schwinger potentials, have been obtained. In linearized gravity, gauge modes and ghost modes in the de Donder gauge have been studied in detail. This program may lead to a deeper understanding of different quantization techniques for gauge fields and gravitation, to a new vision of gauge invariance, and to new points of view in twistor theory. 
  We show how observations of the perturbation spectra produced during inflation may be used to constrain the parameters of general scalar-tensor theories of gravity, which include both an inflaton and dilaton field. An interesting feature of these models is the possibility that the curvature perturbations on super-horizon scales may not be constant due to non-adiabatic perturbations of the two fields. Within a given model, the tilt and relative amplitude of the scalar and tensor perturbation spectra gives constraints on the parameters of the gravity theory, which may be comparable with those from primordial nucleosynthesis and post-Newtonian experiments. 
  We present a toy model approach to the canonical non-perturbative quantization of the spatially-flat Robertson-Walker Universes with cosmological constant, based on the fact that such models are exactly solvable within the framework of a simple Lagrangian formulation. The essential quantum dynamical metric-field and the corresponding Hamiltonian, explicitly derived in terms of annihilation and creation operators, point out that the Wheeler - DeWitt equation is a natural (quantum) generalization of the $G_{44}$ - Einstein equation for the classical De Sitter spacetime and selects the physical states of the quantum De Sitter Universe. As a result of the exponential universal expansion, the usual Fock states (defined as the eigenstates of the number-operator) are no longer invariant under the derived Hamiltonian. They exhibit quantum fluctuation of the energy and of the metric field which lead to a (geometrical) volume quantization. 
  In this paper the averaged weak (AWEC) and averaged null (ANEC) energy conditions, together with uncertainty principle-type restrictions on negative energy (``quantum inequalities''), are examined in the context of evaporating black hole backgrounds in both two and four dimensions. In particular, integrals over only half-geodesics are studied. We determine the regions of the spacetime in which the averaged energy conditions are violated. In all cases where these conditions fail, there appear to be quantum inequalities which bound the magnitude and extent of the negative energy, and hence the degree of the violation. The possible relevance of these results for the validity of singularity theorems in evaporating black hole spacetimes is discussed. 
  This paper describes recent progress in the analysis of relativistic gauge conditions for Euclidean Maxwell theory in the presence of boundaries. The corresponding quantum amplitudes are studied by using Faddeev-Popov formalism and zeta-function regularization, after expanding the electromagnetic potential in harmonics on the boundary 3-geometry. This leads to a semiclassical analysis of quantum amplitudes, involving transverse modes, ghost modes, coupled normal and longitudinal modes, and the decoupled normal mode of Maxwell theory. On imposing magnetic or electric boundary conditions, flat Euclidean space bounded by two concentric 3-spheres is found to give rise to gauge-invariant one-loop amplitudes, at least in the cases considered so far. However, when flat Euclidean 4-space is bounded by only one 3-sphere, one-loop amplitudes are gauge-dependent, and the agreement with the covariant formalism is only achieved on studying the Lorentz gauge. Moreover, the effects of gauge modes and ghost modes do not cancel each other exactly for problems with boundaries. Remarkably, when combined with the contribution of physical (i.e. transverse) degrees of freedom, this lack of cancellation is exactly what one needs to achieve agreement with the results of the Schwinger-DeWitt technique. The most general form of coupled eigenvalue equations resulting from arbitrary gauge-averaging functions is now under investigation. 
  We construct two possible metrics for abelian Higgs vortices with ends on black holes. We show how the detail of the vortex fields smooths out the nodal singularities which exist in the idealized metrics. A corollary is that apparently topologically stable strings might be able to split by black hole pair production. We estimate the rate per unit length by reference to related Ernst and C-metric instantons, concluding that it is completely negligible for GUT-scale strings. The estimated rate for macroscopic superstrings is much higher, although still extremely small, unless there is an early phase of strong coupling. 
  We critically examine the claim made by Burko and Ori that black holes are expected to form in nonsymmetric gravity and find their analysis to be inconclusive. Their conclusion is a result of the approximations they make, and not a consequence of the true dynamics of the theory. The approximation they use fails to capture the crucial equivalence principle violations which enable the full nonsymmetric field equations to detect and tame would-be horizons. An examination of the dynamics of the full theory reveals no indication that black holes should form. For these reasons, one cannot conclude from their analysis that nonsymmetric gravity has black holes. A definitive answers awaits a comprehensive study of gravitational collapse, using the full field equations. 
  Local boundary conditions involving field strengths and the normal to the boundary, originally studied in anti-de Sitter space-time, have been recently considered in one-loop quantum cosmology. This paper derives the conditions under which spin-raising operators preserve these local boundary conditions on a 3-sphere for fields of spin 0,1/2,1,3/2 and 2. Moreover, the two-component spinor analysis of the four potentials of the totally symmetric and independent field strengths for spin 3/2 is applied to the case of a 3-sphere boundary. It is shown that such boundary conditions can only be imposed in a flat Euclidean background, for which the gauge freedom in the choice of the potentials remains. 
  The relativistic theory of unconstrained $p$-dimensional membranes ($p$-branes) is further developed and then applied to the embedding model of induced gravity. Space-time is considered as a 4-dimensional unconstrained membrane evolving in an $N$-dimensional embedding space. The parameter of evolution or the evolution time $\tau$ is a distinct concept from the coordinate time $t = x^0$. Quantization of the theory is also discussed. A covariant functional Schr\" odinger equations has a solution for the wave functional such that it is sharply localized in a certain subspace $P$ of space-time, and much less sharply localized (though still localized) outside $P$. With the passage of evolution the region $P$ moves forward in space-time. Such a solution we interpret as incorporating two seemingly contradictory observations: (i) experiments clearly indicate that space-time is a continuum in which events are existing; (ii) not the whole 4-dimensional space-time, but only a 3-dimensional section which moves forward in time is accessible to our immediate experience. The notorious problem of time is thus resolved in our approach to quantum gravity. Finally we include sources into our unconstrained embedding model. Possible sources are unconstrained worldlines which are free from the well known problem concerning the Maxwell fields generated by charged unconstrained point particles. 
  Local boundary conditions involving field strengths and the normal to the boundary, originally studied in anti-de Sitter space-time, have been recently considered in one-loop quantum cosmology. This paper derives the conditions under which spin-lowering and spin-raising operators preserve these local boundary conditions on a 3-sphere for fields of spin 0,1/2,1,3/2 and 2. Moreover, the two-component spinor analysis of the four potentials of the totally symmetric and independent field strengths for spin 3/2 is applied to the case of a 3-sphere boundary. It is shown that such boundary conditions can only be imposed in a flat Euclidean background, for which the gauge freedom in the choice of the potentials remains. Alternative boundary conditions for supergravity involving the spinor-valued 1-forms for gravitinos and the normal to the boundary are also studied. 
  It is pointed out that if gravitational interactions among ordinary bodies propagate in extra space-time dimensions the velocity of gravitational waves in vacuum could be different from the speed of light $c$. 
  This paper studies the one-loop expansion of the amplitudes of electromagnetism about flat Euclidean backgrounds bounded by a 3-sphere, recently considered in perturbative quantum cosmology, by using zeta-function regularization. For a specific choice of gauge-averaging functional, the contributions to the full zeta value owed to physical degrees of freedom, decoupled gauge mode, coupled gauge modes and Faddeev-Popov ghost field are derived in detail, and alternative choices for such a functional are also studied. This analysis enables one to get a better understanding of different quantization techniques for gauge fields and gravitation in the presence of boundaries. 
  Zeta-function regularization is applied to complete a recent analysis of the quantized electromagnetic field in the presence of boundaries. The quantum theory is studied by setting to zero on the boundary the magnetic field, the gauge-averaging functional and hence the Faddeev-Popov ghost field. Electric boundary conditions are also studied. On considering two gauge functionals which involve covariant derivatives of the 4-vector potential, a series of detailed calculations shows that, in the case of flat Euclidean 4-space bounded by two concentric 3-spheres, one-loop quantum amplitudes are gauge independent and their mode-by-mode evaluation agrees with the covariant formulae for such amplitudes and coincides for magnetic or electric boundary conditions. By contrast, if a single 3-sphere boundary is studied, one finds some inconsistencies, i.e. gauge dependence of the amplitudes. 
  The well-known discrepancies between covariant and non-covariant formalisms in quantum field theory and quantum cosmology are analyzed by focusing on the Coulomb gauge for vacuum Maxwell theory. On studying a flat Euclidean background with boundaries, the corresponding mode-by-mode analysis of one-loop quantum amplitudes agrees with the results of the Schwinger-DeWitt technique and of mode-by-mode calculations in relativistic gauges. 
  A new, field-theory-based framework for discussing and interpreting tests of gravity, notably at the second post-Newtonian (2PN) level, is introduced. Contrary to previous frameworks which attempted at parametrizing any conceivable deviation from general relativity, we focus on the best motivated class of models, in which gravity is mediated by a tensor field together with one or several scalar fields. The 2PN approximation of these "tensor-multi-scalar" theories is obtained thanks to a diagrammatic expansion which allows us to compute the Lagrangian describing the motion of N bodies. In contrast with previous studies which had to introduce many phenomenological parameters, we find that the 2PN deviations from general relativity can be fully described by only two new 2PN parameters, epsilon and zeta, beyond the usual (Eddington) 1PN parameters beta and gamma. It follows from the basic tenets of field theory, notably the absence of negative-energy excitations, that (beta-1), epsilon and zeta (as well as any new parameter entering higher post-Newtonian orders) must tend to zero with (gamma-1). It is also found that epsilon and zeta do not enter the 2PN equations of motion of light. Therefore, light-deflection or time-delay experiments cannot probe any theoretically motivated 2PN deviation from general relativity, but they can give a clean access to (gamma-1), which is of greatest significance as it measures the basic coupling strength of matter to the scalar fields. Because of the importance of self-gravity effects in neutron stars, binary-pulsar experiments are found to constitute a unique testing ground for the 2PN structure of gravity. A simplified analysis of four binary pulsars already leads to significant constraints: |epsilon| < 7x10^-2, |zeta| < 6x10^-3. 
  A two-dimensional dilatonic black hole induced by a topological soliton is exactly solvable in the scalar field theory coupled to dilaton gravity. The Hawking radiation of the black hole is studied in the one-loop approximation with the help of the trace anomaly of energy-momentum tensors which is a geometrical invariant. The quantum theory can be also soluble in the RST scheme in order to consider the back reaction of the metric. The energy of the black hole system is calculated and the classical thunderpop energy corresponding to the soliton energy is needed to describe the final state of the black hole. Finally we discuss the possibility of conservation of the topological charge. 
  The contribution of physical degrees of freedom to the one-loop amplitudes of Euclidean supergravity is here evaluated in the case of flat Euclidean backgrounds bounded by a three-sphere, recently considered in perturbative quantum cosmology. The physical degrees of freedom (denoted by PDF) are picked out by imposing the supersymmetry constraints and choosing a gauge condition. Remarkably, for the massless gravitino field the PDF method and local boundary conditions lead to a result for the trace anomaly which is equal to the PDF value one obtains using spectral boundary conditions on a 3-sphere. 
  The discrepancy between the results of covariant and non-covariant one-loop calculations for higher-spin fields in quantum cosmology is analyzed. A detailed mode-by-mode study of perturbative quantum gravity about a flat Euclidean background bounded by two concentric 3-spheres, including non-physical degrees of freedom and ghost modes, leads to one-loop amplitudes in agreement with the covariant Schwinger-DeWitt method. This calculation provides the generalization of a previous analysis of fermionic fields and electromagnetic fields at one-loop about flat Euclidean backgrounds admitting a well-defined 3+1 decomposition. 
  The singularity structure of charged spherical collapse is studied by considering the evolution of the gravity-scalar field system. A detailed examination of the geometry at late times strongly suggests the validity of the mass-inflation scenario~\cite{PI:90}. Although the area of the two-spheres remains finite at the Cauchy horizon, its generators are eventually focused to zero radius. Thus the null, mass-inflation singularity {\em generally}\/ precedes a crushing $r=0$ singularity deep inside the black hole core. This central singularity is spacelike. 
  We study the dynamics of topological defects in the context of ``topological inflation" proposed by Vilenkin and Linde independently. Analysing the time evolution of planar domain walls and of global monopoles, we find that the defects undergo inflationary expansion if $\eta\stackrel{>}{\sim}0.33m_{Pl}$, where $\eta$ is the vacuum expectation value of the Higgs field and $m_{Pl}$ is the Planck mass. This result confirms the estimates by Vilenkin and Linde. The critical value of $\eta$ is independent of the coupling constant $\lambda$ and the initial size of the defect. Even for defects with an initial size much greater than the horizon scale, inflation does not occur at all if $\eta$ is smaller than the critical value. We also examine the effect of gauge fields for static monopole solutions and find that the spacetime with a gauge monopole has an attractive nature, contrary to the spacetime with a global monopole. It suggests that gauge fields affect the onset of inflation. 
  General aspects of vielbein representation, ADM formulation and canonical quantization of gravity are reviewed using pure gravity in three dimensions as a toy model. The classical part focusses on the role of observers in general relativity, which will later be identified with quantum observers. A precise definition of gauge symmetries and a classification of inequivalent solutions of Einstein's equations in dreibein formalism is given as well. In the quantum part the construction of the physical Hilbert space is carried out explicitly for a torus and cylinder type space manifold, which has not been done so far. Some conceptual problems of quantum gravity are discussed from the point of view of an observer sitting inside the universe. 
  The physics of quantum gravity is discussed within the framework of topological quantum field theory. Some of the principles are illustrated with examples taken from theories in which space-time is three dimensional. 
  We show that, given an arbitrary shift, the lapse $N$ can be chosen so that the extrinsic curvature $K$ of the space slices with metric $\overline g$ in arbitrary coordinates of a solution of Einstein's equations satisfies a quasi-linear wave equation. We give a geometric first order symmetric hyperbolic system verified in vacuum by $\overline g$, $K$ and $N$. We show that one can also obtain a quasi-linear wave equation for $K$ by requiring $N$ to satisfy at each time an elliptic equation which fixes the value of the mean extrinsic curvature of the space slices. 
  The evolution of physical and gauge degrees of freedom in the Einstein and Yang-Mills theories are separated in a gauge-invariant manner. We show that the equations of motion of these theories can always be written in flux-conservative first-order symmetric hyperbolic form. This dynamical form is ideal for global analysis, analytic approximation methods such as gauge-invariant perturbation theory, and numerical solution. 
  This paper studies the self-dual Einstein-Dirac theory. A generalization is obtained of the Jacobson-Smolin proof of the equivalence between the self-dual and Palatini purely gravitational actions. Hence one proves equivalence of self-dual Einstein-Dirac theory to the Einstein-Cartan-Sciama-Kibble-Dirac theory. The Bianchi symmetry of the curvature, core of the proof, now contains a non-vanishing torsion. Thus, in the self-dual framework, the extra terms entering the equations of motion with respect to the standard Einstein-Dirac field equations, are neatly associated with torsion. 
  The Hartle-Hawking `no-boundary' state is constructed explicitly for the recently developed supersymmetric minisuperspace model with non-vanishing fermion number. 
  Previous work in the literature has studied the Hamiltonian structure of an R-squared model of gravity with torsion in a closed Friedmann-Robertson-Walker universe. Within the framework of Dirac's theory, torsion is found to lead to a second-class primary constraint linear in the momenta and a second-class secondary constraint quadratic in the momenta. This paper studies in detail the same problem at a Lagrangian level, i.e. working on the tangent bundle rather than on phase space. The corresponding analysis is motivated by a more general program, aiming to obtain a manifestly covariant, multisymplectic framework for the analysis of relativistic theories of gravitation regarded as constrained systems. After an application of the Gotay-Nester Lagrangian analysis, the paper deals with the generalized method, which has the advantage of being applicable to any system of differential equations in implicit form. Multiplication of the second-order Lagrange equations by a vector with zero eigenvalue for the Hessian matrix yields the so-called first-generation constraints. Remarkably, in the cosmological model here considered, if Lagrange equations are studied using second-order formalism, a second-generation constraint is found which is absent in first-order formalism. This happens since first- and second-order formalisms are inequivalent. There are, however, no {\it a priori} reasons for arguing that one of the two is incorrect. First- and second-generation constraints are used to derive physical predictions for the cosmological model. 
  Local supersymmetry leads to boundary conditions for fermionic fields in one-loop quantum cosmology involving the Euclidean normal to the boundary and a pair of independent spinor fields. This paper studies the corresponding classical properties, i.e. the classical boundary-value problem and boundary terms in the variational problem. Interestingly, a link is found with the classical boundary-value problem when spectral boundary conditions are imposed on a 3-sphere in the massless case. Moreover, the boundary term in the action functional is derived. 
  We find a choice of variables for the 3+1 formulation of general relativity which casts the evolution equations into (flux-conservative) symmetric-hyperbolic first order form for arbitrary lapse and shift, for the first time. We redefine the lapse function in terms of the determinant of the 3-metric and a free function U which embodies the lapse freedom. By rescaling the variables with appropriate factors of 1/c, the system is shown to have a smooth Newtonian limit when the redefined lapse U and the shift are fixed by means of elliptic equations to be satisfied on each time slice. We give a prescription for the choice of appropriate initial data with controlled extra-radiation content, based on the theory of problems with different time-scales. Our results are local, in the sense that we are not concerned with the treatment of asymptotic regions. On the other hand, this local theory is all what is needed for most problems of practical numerical computation. 
  This paper continues a study on Choptuik scaling in gravitational collapse of a complex scalar field at the threshold for black hole formation. We perform a linear perturbation analysis of the previously derived complex critical solution, and calculate the critical exponent for black hole mass, $\gamma \approx 0.387106$. We also show that this critical solution is unstable via a growing oscillatory mode. 
  I review the classical and quantum properties of the (2+1)-dimensional black hole of Ba{\~n}ados, Teitelboim, and Zanelli. This solution of the Einstein field equations in three spacetime dimensions shares many of the characteristics of the Kerr black hole: it has an event horizon, an inner horizon, and an ergosphere; it occurs as an endpoint of gravitational collapse; it exhibits mass inflation; and it has a nonvanishing Hawking temperature and interesting thermodynamic properties. At the same time, its structure is simple enough to allow a number of exact computations, particularly in the quantum realm, that are impractical in 3+1 dimensions. 
  The gravitational-wave energy flux produced during the head-on infall and collision of two compact objects is calculated using two approaches: (i) a post-Newtonian method, carried to second post-Newtonian order beyond the quadrupole formula, valid for systems of arbitrary masses; and (ii) a black-hole perturbation method, valid for a test-body falling radially toward a black hole. In the test-body case, the methods are compared. The post-Newtonian method is shown to converge to the ``exact'' perturbation result more slowly than expected {\it a priori\/}. A surprisingly good approximation to the energy radiated during the infall phase, as calculated by perturbation theory, is found to be given by a Newtonian, or quadrupole, approximation combined with the exact test-body equations of motion in the Schwarzschild spacetime. 
  In this paper we discuss the quantum potential approach of Bohm in the context of quantum cosmological model. This approach makes it possible to convert the wavefunction of the universe to a set of equations describing the time evolution of the universe. Following Ashtekar et.\ al., we make use of quantum canonical transformation to cast a class of quantum cosmological models to a simple form in which they can be solved explicitly, and then we use the solutions to recover the time evolution. 
  It is shown that an article by C. W. Misner contains serious errors. In particular, the claim that the Yilmaz theory of gravitation cancels the Newtonian gravitational interaction is based on a false premise. With the correct premise the conclusion of the article regarding the absence of gravitational interactions applies to general relativity and not to the Yilmaz theory. 
  The class of spherically-symmetric thin-shell wormholes provides a particularly elegant collection of exemplars for the study of traversable Lorentzian wormholes. In the present paper we consider linearized (spherically symmetric) perturbations around some assumed static solution of the Einstein field equations. This permits us to relate stability issues to the (linearized) equation of state of the exotic matter which is located at the wormhole throat. 
  A review is given of recent research on gravitational waves from compact bodies and its relevance to the LIGO/VIRGO international network of high-frequency (10 to 10,000 Hz) gravitational-wave detectors, and to the proposed LISA system of low-frequency (0.1 to 0.0001 Hz) detectors. The sources that are reviewed are ordinary binary star systems, binaries made from compact bodies (black holes and neutron stars), the final inspiral and coalescence of compact-body binaries, the inspiral of stars and small black holes into massive black holes, the stellar core collapse that triggers supernovae, and the spin of neutron stars. This paper is adapted from a longer review article entitled ``Gravitational Waves'' (GRP-411) that the author has written for the Proceedings of the Snowmass '94 Summer Study on Particle and Nuclear Astrophysics and Cosmology. 
  Path integral methods are used to derive a general expression for the entropy of a black hole in a diffeomorphism invariant theory. The result, which depends on the variational derivative of the Lagrangian with respect to the Riemann tensor, agrees with the result obtained from Noether charge methods by Iyer and Wald. The method used here is based on the direct expression of the density of states as a path integral (the microcanonical functional integral). The analysis makes crucial use of the Hamiltonian form of the action. An algorithm for placing the action of a diffeomorphism invariant theory in Hamiltonian form is presented. Other path integral approaches to the derivation of black hole entropy include the Hilbert action surface term method and the conical deficit angle method. The relationships between these path integral methods are presented. 
  This article reviews current efforts and plans for gravitational-wave detection, the gravitational-wave sources that might be detected, and the information that the detectors might extract from the observed waves. Special attention is paid to (i) the LIGO/VIRGO network of earth-based, kilometer-scale laser interferometers, which is now under construction and will operate in the high-frequency band ($1$ to $10^4$ Hz), and (ii) a proposed 5-million-kilometer-long Laser Interferometer Space Antenna (LISA), which would fly in heliocentric orbit and operate in the low-frequency band ($10^{-4}$ to $1$ Hz). LISA would extend the LIGO/VIRGO studies of stellar-mass ($M\sim2$ to $300 M_\odot$) black holes into the domain of the massive black holes ($M\sim1000$ to $10^8M_\odot$) that inhabit galactic nuclei and quasars. 
  We consider the action principle to derive the classical, non-relativistic motion of a self-interacting particle in a 4-D Lorentzian spacetime containing a wormhole and which allows the existence of closed time-like curves. For the case of a `hard-sphere' self-interaction potential we show that the only possible trajectories (for a particle with fixed initial and final positions and which traverses the wormhole once) minimizing the classical action are those which are globally self-consistent, and that the `Principle of self-consistency' (originally introduced by Novikov) is thus a natural consequence of the `Principle of minimal action.' 
  At first we introduce the space-time manifold and we compare some aspects of Riemannian and Lorentzian geometry such as the distance function and the relations between topology and curvature. We then define spinor structures in general relativity, and the conditions for their existence are discussed. The causality conditions are studied through an analysis of strong causality, stable causality and global hyperbolicity. In looking at the asymptotic structure of space-time, we focus on the asymptotic symmetry group of Bondi, Metzner and Sachs, and the b-boundary construction of Schmidt. The Hamiltonian structure of space-time is also analyzed, with emphasis on Ashtekar's spinorial variables. Finally, the question of a rigorous theory of singularities in space-times with torsion is addressed, describing in detail recent work by the author. We define geodesics as curves whose tangent vector moves by parallel transport. This is different from what other authors do, because their definition of geodesics only involves the Christoffel symbols, though studying theories with torsion. We then prove how to extend Hawking's singularity theorem without causality assumptions to the space-time of the ECSK theory. This is achieved studying the generalized Raychaudhuri equation in the ECSK theory, the conditions for the existence of conjugate points and properties of maximal timelike geodesics. Our result can also be interpreted as a no-singularity theorem if the torsion tensor does not obey some additional conditions. Thus, it seems that the occurrence of singularities in closed cosmological models based on the ECSK theory is less generic than in general relativity. Our work should be compared with important previous papers. There are some relevant differences, because we rely on a different definition of 
  This paper compares recent approaches appearing in the literature on the singularity problem for space-times with nonvanishing torsion. 
  This paper studies necessary conditions for the existence of alpha-surfaces in complex space-time manifolds with nonvanishing torsion. For these manifolds, Lie brackets of vector fields and spinor Ricci identities contain explicitly the effects of torsion. This leads to an integrability condition for alpha-surfaces which does not involve just the self-dual Weyl spinor, as in complex general relativity, but also the torsion spinor, in a nonlinear way, and its covariant derivative. Interestingly, a particular solution of the integrability condition is given by conformally right-flat and right-torsion-free space-times. 
  This paper studies one-loop effective potential and spontaneous-symmetry-breaking pattern for SU(5) gauge theory in de Sitter space-time. Curvature effects modify the flat-space effective potential by means of a very complicated special function previously derived in the literature. An algebraic technique already developed by the first author to study spontaneous symmetry breaking of SU(n) for renormalizable polynomial potentials is here generalized, for SU(5), to the much harder case of a de Sitter background. A detailed algebraic and numerical analysis provides a better derivation of the stability of the extrema in the maximal subgroups SU(4) x U(1), SU(3) x SU(2) x U(1), SU(3) x U(1) x U(1) x R(311), SU(2) x SU(2) x U(1) x U(1) x R(2211), where R(311) and R(2211) discrete symmetries select particular directions in the corresponding two-dimensional strata. One thus obtains a deeper understanding of the result, previously found with a different numerical analysis, predicting the slide of the inflationary universe into either the SU(3) x SU(2) x U(1) or SU(4) x U(1) extremum. Interestingly, using this approach, one can easily generalize all previous results to a more complete SU(5) tree-level potential also containing cubic terms. 
  This paper studies a new set of mixed boundary conditions in Euclidean quantum gravity. These involve, in particular, Robin boundary conditions on the perturbed 3-metric and hence lead, by gauge invariance, to Robin conditions on the whole ghost 1-form. The corresponding trace anomaly is evaluated in the case of flat Euclidean 4-space bounded by a 3-sphere. In general, this anomaly differs from the ones resulting from other local or non-local boundary conditions studied in the recent literature. 
  Dirac's theory of constrained Hamiltonian systems is applied to the minimal conformally invariant SU(5) grand-unified model studied at 1-loop level in a de Sitter universe. For this model, which represents a simple and interesting example of GUT theory and at the same time is a step towards theories with larger gauge group like SO(10), second-class constraints in the Euclidean-time regime exist. In particular, they enable one to prove that, to be consistent with the experimentally established electroweak standard model and with inflationary cosmology, the residual gauge-symmetry group of the early universe, during the whole de Sitter era, is bound to be SU(3) x SU(2) x U(1). Moreover, the numerical solution of the field equations subject to second-class constraints is obtained. This confirms the existence of a sufficiently long de Sitter phase of the early universe, in agreement with the initial assumptions. 
  Evolving Lorentzian wormholes with the required matter satisfying the Energy conditions are discussed. Several different scale factors are used and the corresponding consequences derived. The effect of extra, decaying (in time) compact dimensions present in the wormhole metric is also explored and certain interesting conclusions are derived for the cases of exponential and Kaluza--Klein inflation. 
  We present a minisuperspace analysis of a class of Lorentzian wormholes that evolves quantum mechanically in a background Friedman Robertson Walker spacetime. The quantum mechanical wavefunction for these wormholes is obtained by solving the Wheeler-DeWitt equation for Einstein gravity on this minisuperspace. The time-dependent expectation value of the wormhole throat radius is calculated to lowest order in an adiabatic expansion of the Wheeler-DeWitt hamiltonian. For a radiation dominated expansion, the radius is shown to relax asymptotically to obtain a value of order the Planck length while for a deSitter background, the radius is stationary but always larger than the Planck length. These two cases are of particular relevance when considering wormholes in the early universe. 
  An analysis of cosmic string breaking with the formation of black holes attached to the ends reveals a remarkable feature: the black holes can be correlated or uncorrelated. We find that, as a consequence, the number-of-states enhancement factor in the action governing the formation of uncorrelated black holes is twice the one for a correlated pair. We argue that when an uncorrelated pair forms at the ends of the string, the physics involved is more analogous to thermal nucleation than to particle-antiparticle creation. Also, we analyze the process of intercommuting strings induced by black hole annihilation and merging. Finally, we discuss the consequences for grand unified strings. The process whereby uncorrelated black holes are formed yields a rate which significantly improves over those previously considered, but still not enough to modify string cosmology. 
  In this pedagogical note, I present a method for constructing a fully covariant normal coordinate expansion of the gauge potential on a curved space-time manifold. Although the content of this paper is elementary, the results may prove useful in some applications and have not, to the best of my knowledge, been discussed explicitly in the literature. 
  This paper studies local boundary conditions for fermionic fields in quantum cosmology, originally introduced by Breitenlohner, Freedman and Hawking for gauged supergravity theories in anti-de Sitter space. For a spin-1/2 field the conditions involve the normal to the boundary and the undifferentiated field. A first-order differential operator for this Euclidean boundary-value problem exists which is symmetric and has self-adjoint extensions. The resulting eigenvalue equation in the case of a flat Euclidean background with a three-sphere boundary of radius a is found to be: $F(E)=[J_{n+1}(Ea)]^{2}-[J_{n+2}(Ea)]^{2}=0 , \forall n \geq 0$. Using the theory of canonical products, this function F may be expanded in terms of squared eigenvalues, in a way which has been used in other recent one-loop calculations involving eigenvalues of second-order operators. One can then study the generalized Riemann zeta-function formed from these squared eigenvalues. The value of zeta(0) determines the scaling of the one-loop prefactor in the Hartle-Hawking amplitude in quantum cosmology. Suitable contour formulae, and the uniform asymptotic expansions of the Bessel functions and their first derivatives, yield for a massless Majorana field: zeta(0)=11/360. Combining this with zeta(0) values for other spins, one can then check whether the one-loop divergences in quantum cosmology cancel in a supersymmetric theory. 
  For fermionic fields on a compact Riemannian manifold with boundary one has a choice between local and non-local (spectral) boundary conditions. The one-loop prefactor in the Hartle-Hawking amplitude in quantum cosmology can then be studied using the generalized Riemann zeta-function formed from the squared eigenvalues of the four-dimensional fermionic operators. For a massless Majorana spin-1/2 field, the spectral conditions involve setting to zero half of the fermionic field on the boundary, corresponding to harmonics of the intrinsic three-dimensional Dirac operator on the boundary with positive eigenvalues. Remarkably, a detailed calculation for the case of a flat background bounded by a three-sphere yields the same value zeta(0)=11/360 as was found previously by the authors using local boundary conditions. A similar calculation for a spin-3/2 field, working only with physical degrees of freedom (and, hence, excluding gauge and ghost modes, which contribute to the full Becchi-Rouet-Stora-Tyutin-invariant amplitude), again gives a value zeta(0)=-289/360 equal to that for the natural local boundary conditions. 
  A method to determine the quantum state of a scalar field after $O(4)$-symmetric bubble nucleation has been developed recently. The method has an advantage that it concisely gives us a clear picture of the resultant quantum state. In particular, one may interpret the excitations as a particle creation phenomenon just as in the case of particle creation in curved spacetime. As an application, we investigate in detail the spectrum of quantum excitations of the tunneling field when it undergoes false vacuum decay. We consider a tunneling potential which is piece-wise quadratic, hence is simple enough to allow us an analytical treatment. We find a strong dependence of the excitation spectrum upon the shape of the potential on the true vacuum side. We then discuss features of the excitation spectrum common to general tunneling potentials not restricted to our simple model. 
  Mixed invariants are used to classify the Riemann spinor in the case of Einstein-Maxwell fields and perfect fluids. In the Einstein-Maxwell case these mixed invariants provide information as to the relative orientation of the gravitational and electromagnetic principal null directions. Consideration of the perfect fluid case leads to some results about the behaviour of the Bel-Robinson tensor regarded as a quartic form on unit timelike vectors. 
  For N=1 supergravity in 3+1 dimensions we determine the graded algebra of the quantized Lorentz generators, supersymmetry generators, and diffeo-morphism and Hamiltonian generators and find that, at least formally, it closes in the chosen operator ordering. Following our recent conjecture and generalizing an ansatz for Bianchi-type models we proposed earlier we find an explicit exact quantum solution of all constraints in the metric representation. 
  In a recent paper, we discussed the formation of black holes in non-symmetric gravity. That paper was then criticized by Cornish and Moffat. In the present paper, we address the arguments raised by Cornish and Moffat. In summary, we do not see any reason to doubt the validity of our former conclusions. 
  The large-angular-scale anisotropy of the cosmic microwave background radiation in multidimensional cosmological models (Kaluza-Klein models) is studied. Limits on parameters of the models imposed by the experimental data are obtained. It is shown that in principle there is a room for Kaluza-Klein models as possible candidates for the description of the Early Universe. However, the obtained limits are very restrictive and none of the concrete models, analyzed in the article, satisfy them. 
  We construct coordinate systems that cover all of the Reissner-Nordstroem solution with m>|q| and m=|q|, respectively. This is possible by means of elementary analytical functions. The limit of vanishing charge q provides an alternative to Kruskal which, to our mind, is more explicit and simpler. The main tool for finding these global charts is the description of highly symmetrical metrics by two-dimensional actions. Careful gauge fixing yields global representatives of the two-dimensional theory that can be rewritten easily as the corresponding four-dimensional line elements. 
  The formalism developed by Chandrasekhar for the linear polar perturbations of the Reissner-Nordstrom solution is generalized to include the case of dipole (l=1) perturbations. Then, the perturbed metric coefficients and components of the Maxwell tensor are computed. 
  Space-times which allow a slicing into homogeneous spatial hypersurfaces generalize the usual Bianchi models. One knows already that in these models the Bianchi type may change with time. Here we show which of the changes really appear. To this end we characterize the topological space whose points are the 3-dimensional oriented homogeneous Riemannian manifolds; locally isometric manifolds are considered as same. 
  This paper studies the two-spinor form of the Rarita-Schwinger potentials subject to local boundary conditions compatible with local supersymmetry. The massless Rarita-Schwinger field equations are studied in four-real-dimensional Riemannian backgrounds with boundary. Gauge transformations on the potentials are shown to be compatible with the field equations providing the background is Ricci-flat, in agreement with previous results in the literature. However, the preservation of boundary conditions under such gauge transformations leads to a restriction of the gauge freedom. The recent construction by Penrose of secondary potentials which supplement the Rarita-Schwinger potentials is then applied. The equations for the secondary potentials, jointly with the boundary conditions, imply that the background four-geometry is further restricted to be totally flat. The analysis of other gauge transformations confirms that, in the massless case, the only admissible class of Riemannian backgrounds with boundary is totally flat. 
  This paper studies the two-component spinor form of massive spin-3/2 potentials in conformally flat Einstein four-manifolds. Following earlier work in the literature, a non-vanishing cosmological constant makes it necessary to introduce a supercovariant derivative operator. The analysis of supergauge transformations of primary and secondary potentials for spin 3/2 shows that the gauge freedom for massive spin-3/2 potentials is generated by solutions of the supertwistor equations. The supercovariant form of a partial connection on a non-linear bundle is then obtained, and the basic equation of massive secondary potentials is shown to be the integrability condition on super beta-surfaces of a differential operator on a vector bundle of rank three. Moreover, in the presence of boundaries, a simple algebraic relation among some spinor fields is found to ensure the gauge invariance of locally supersymmetric boundary conditions relevant for quantum cosmology and supergravity. 
  The perturbation of the lunar motion caused by a hypothetical violation of the equivalence principle is analytically worked out in terms of power series \`a la Hill-Brown. The interaction with the quadrupolar tide is found to amplify the leading order term in the synodic range oscillation by more than 62\%. Confirming a recent suggestion of Nordtvedt, we find that this amplification has a pole singularity for an orbit beyond the lunar orbit. This singularity is shown to correspond to the critical prograde orbit beyond which, as found by H\'enon, Hill's periodic orbit becomes exponentially unstable. It is suggested that ranging between prograde and retrograde orbits around outer planets might provide future high precision orbital tests of the equivalence principle. It is argued that, within the context of string-derived non-Einsteinian theories, the theoretical significance of orbital tests of the universality of free fall is to measure the basic coupling strength of some scalar field through composition-dependent effects. Present Lunar Laser Ranging data yield the value $\bar{\gamma} = (-1.2\pm 1.7) \times 10^{-7}$ for the effective Eddington parameter $\bar{\gamma} \equiv \gamma -1$ measuring this coupling strength. 
  In a teleparallel theory of (2+1)-dimensional gravity developed in a previous paper, we examine generators of internal Lorentz transformations and of general affine coordinate transformations for static circularly symmetric exact solutions of gravitational field equation. The \lq \lq spin" angular momentum, the energy-momentum and the \lq \lq extended orbital angular momentum" are explicitly given for each solution. Also, we give a critical comment on Deser's claim that neither momentum nor boosts are definable for finite energy solutions of three-dimensional Einstein gravity. 
  After reviewing the general ideas of quantum cosmology (Wheeler-DeWitt equation, boundary conditions, interpretation of $\psi$), I discuss how these ideas can be tested observationally. Observational predictions differ for different choices of boundary conditions. With tunneling boundary conditions, $\psi$ favors initial states that lead to inflation, while with Hartle-Hawking boundary conditions it does not. This difficulty of the Hartle-Hawking wave function becomes particularly severe if the role of `inflatons' is played by the moduli fields of superstring theories. In models where the constants of Nature can take more than one set of values, $\psi$ can also determine the probability distribution for the constants. This can be done with the aid of the `principle of mediocrity' which asserts that we are a `typical' civilization in the ensemble of universes described by $\psi$. The resulting distribution favors inflation with a very flat potential, thermalization and baryogenesis at electroweak scale, a non-negligible cosmological constant, and density fluctuations seeded either by topological defects, or by quantum fluctuations in models like hybrid inflation (as long as these features are consistent with the allowed values of the constants). 
CONTENTS:    1 Introduction    2 Analytic Manifolds and Analytic Continuation of     Metrics    3 Walker's Spacetimes and their Maximal Extension    4 Global Structure of de Sitter and Reissner-Nordstr\"om-de Sitter Cosmos         4.1 Special Cases         4.2 Collapsing Dust    5 Euclidean Metrics    6 Physical Interpretation of Euclidean Solutions, and a remark about the     Gravitational Action         6.1 Thermal Interpretation         6.2 Tunneling Interpretation    7 The Multi-Black-Hole Solutions         7.1 Merging Black Holes         7.2 Continuing Beyond the Horizons    8 Naked Singularities?   References 
  Gott recently has constructed a spacetime modeled by two infinitely long, parallel cosmic strings which pass and gravitationally interact with each other. For large enough velocity, the spacetime will contain closed timelike curves. An explicit construction of the solution for a scalar field is presented in detail and a proof for the existence of such a solution is given for initial data satisfying conditions on an asymptotically null partial Cauchy surface.  Solutions to smooth operators on the covering space are invariant under the isometry are shown to be pull back of solutions of the associated operator on the base space.  Projection maps and translation operators for the covering space are developed for the spacetime, and explicit expressions for the projection operator and the isometry group of the covering space are given. It is shown that the Gott spacetime defined is a quotient space of Minkowski space by the discrete isometry subgroup of self-equivalences of the projection map. 
  Three propositions about Jordan matrices are proved and applied to algebraically classify the Ricci tensor in n-dimensional Kaluza-Klein-type spacetimes. We show that the possible Segre types are [1,1...1], [21...1], [31\ldots 1], [z\bar{z}1...1] and degeneracies thereof. A set of canonical forms for the Segre types is obtained in terms of semi-null bases of vectors. 
  We investigate a quasi-local energy naturally introduced by Kodama's prescription for a spherically symmetric space-time with a positive cosmological constant $\Lambda$. We find that this quasi-local energy is well behaved inside a cosmological horizon. However, when there is a scalar field with a long enough Compton wavelength, the quasi-local energy diverges in the course of its time evolution outside the cosmological horizon. This means that the quasi-local energy has a meaning only inside the cosmological horizon. 
  The formalism of Ashtekar and Magnon \cite{AshtekarMagnon:1975} for the definition of particles in quantum field theory in curved spacetime is further developed. The relation between basic objects of this formalism (e.g., the complex structure) and different Green functions is found. It allows one to derive composition laws for Green functions. The relation of two definitions of particles is reformulated in the formalism and the base-independent Bogoljubov transformation is expressed using quantities which are derivable directly from the ``in-out'' Green function. 
  Quantum mechanics may be formulated as {\it Sensible Quantum Mechanics} (SQM) so that it contains nothing probabilistic except conscious perceptions. Sets of these perceptions can be deterministically realized with measures given by expectation values of positive-operator-valued {\it awareness operators}. Ratios of the measures for these sets of perceptions can be interpreted as frequency-type probabilities for many actually existing sets. These probabilities generally cannot be given by the ordinary quantum ``probabilities'' for a single set of alternatives. {\it Probabilism}, or ascribing probabilities to unconscious aspects of the world, may be seen to be an {\it aesthemamorphic myth}. 
  Quantum mechanics may be formulated as SENSIBLE QUANTUM MECHANICS (SQM) so that it contains nothing probabilistic, except, in a certain frequency sense, conscious perceptions. Sets of these perceptions can be deterministically realized with measures given by expectation values of positive-operator-valued AWARENESS OPERATORS in a quantum state of the universe which never jumps or collapses. Ratios of the measures for these sets of perceptions can be interpreted as frequency-type probabilities for many actually existing sets rather than as propensities for potentialities to be actualized, so there is nothing indeterministic in SQM. These frequency-type probabilities generally cannot be given by the ordinary quantum "probabilities" for a single set of alternatives. PROBABILISM, or ascribing probabilities to unconscious aspects of the world, may be seen to be an AESTHEMAMORPHIC MYTH. No fundamental correlation or equivalence is postulated between different perceptions (each being the entirety of a single conscious experience and thus not in direct contact with any other), so SQM, a variant of Everett's "many-worlds" framework, is a "many-perceptions" framework but not a "many-minds" framework. Different detailed SQM theories may be tested against experienced perceptions by the TYPICALITIES (defined herein) they predict for these perceptions. One may adopt the CONDITIONAL AESTHEMIC PRINCIPLE: among the set of all conscious perceptions, our perceptions are likely to be typical. 
  The generating function method is applied to the trace of the heat kernel and the one-loop effective action derived from the covariant perturbation theory. The basis of curvature invariants of second order for the heat kernel (Green function) is built and simple rules for form factor manipulations are proposed. The results are checked by deriving the Schwinger-DeWitt series of the heat kernel and divergences of one-loop currents. 
  Using Wilsonian procedure (renormalization group improvement) we discuss the finite quantum corrections to black hole entropy in renormalizable theories. In this way, the Wilsonian black hole entropy is found for GUTs (of asymptotically free form, in particularly) and for the effective theory of conformal factor aiming to describe quantum gravity in infrared region. The off-critical regime (where the coupling constants are running) for effective theory of conformal factor in quantum gravity (with or without torsion) is explicitly constructed. The correspondent renormalization group equations for the effective couplings are found using Schwinger-De Witt technique for the calculation of the divergences of fourth order operator. 
  Quantum cosmology is the quantum theory of the entire universe. Although strange at first sight, it is appropriate because (1) our world appears to be fundamentally quantum, (2) the classical description of gravity breaks down at singularities it would give at the beginning of the universe, and (3) our universe has many properties that cannot be explained by a classical description of them. A quantum state of the universe should obey certain constraints, such as the Wheeler-DeWitt equations. One approach to interpreting such a state is {\it Sensible Quantum Mechanics} (SQM), in which nothing is probabilistic, except, in a certain frequency sense, conscious perceptions. Sets of these perceptions can be deterministically realized with measures given by expectation values of positive-operator-valued {\it awareness operators}. These may be defined even when the quantum state itself is not normalizable, though there still seem to be problems of divergences when one has an infinite amount of inflation, producing infinitely large spatial volumes with presumably infinite measures of perceptions. Ratios of the measures for sets of perceptions (if finite) can be interpreted as frequency-type probabilities for many actually existing sets rather than as propensities for potentialities to be actualized, so there is nothing indeterministic in a specific SQM. One can do a Bayesian analysis to test between different specific SQMs. One can also make statistical predictions of what one might perceive within a specific SQM by invoking the {\it Conditional Aesthemic Principle}: among the set of all conscious perceptions, our perceptions are likely to be typical. 
  We study the interaction of maximally-charged dilatonic black holes at low velocity. We compute the metric on moduli space for three extreme black holes under a simple constraint. The Hamiltonian of the multi-black hole system of $O(v^2)$ is also calculated for the $a=1$ and $a=1/\sqrt{3}$ cases, where $a$ is the dilaton coupling constant. The behavior of the system is discussed qualitatively. 
  We obtain a one parameter class of stationary rotating string cosmological models of which the well-known G$\ddot o $del Universe is a particular case. By suitably choosing the free parameter function, it is always possible to satisfy the energy conditions. The rotation of the model hinges on the cosmological constant which turns out to be negative. 
  We attempt to solve the Einstein equations for string dust and null flowing radiation for the general axially symmetric metric, which we believe is being done for the first time. We obtain the string-dust and radiating generalizations of the Kerr and the NUT solutions. There also occurs an interesting case of radiating string-dust which arises from string-dust generalization of Vaidya's solution of a radiating star. 
  The temperature of an extremal Reissner - Nordstrom black hole is not restricted by the requirement of absence of a conical singularity. It is demonstrated how Kruskal-like coordinates may be constructed corresponding to any temperature whatsoever. A recently discovered stringy extremal black hole which apparently has an infinite temperature is also shown to have its temperature unrestricted by conical singularity arguments. 
  The collection of all topologies on the set of three points is studied treating the topology as quantum-like observable. It turns out to be possible under the assumption of the asymmetry between the spaces of bra- and ket-vectors. The analogies between the introduced topologimeter and the Stern-Gerlach experiments are outlined. 
  The dynamics of relativistic stars and black holes are often studied in terms of the quasinormal modes (QNM's) of the Klein-Gordon (KG) equation with different effective potentials $V(x)$. In this paper we present a systematic study of the relation between the structure of the QNM's of the KG equation and the form of $V(x)$. In particular, we determine the requirements on $V(x)$ in order for the QNM's to form complete sets, and discuss in what sense they form complete sets. Among other implications, this study opens up the possibility of using QNM expansions to analyse the behavior of waves in relativistic systems, even for systems whose QNM's do {\it not} form a complete set. For such systems, we show that a complete set of QNM's can often be obtained by introducing an infinitesimal change in the effective potential. 
  It is well-known that the dominant late time behavior of waves propagating on a Schwarzschild spacetime is a power-law tail; tails for other spacetimes have also been studied. This paper presents a systematic treatment of the tail phenomenon for a broad class of models via a Green's function formalism and establishes the following. (i) The tail is governed by a cut of the frequency Green's function $\tilde G(\omega)$ along the $-$~Im~$\omega$ axis, generalizing the Schwarzschild result. (ii) The $\omega$ dependence of the cut is determined by the asymptotic but not the local structure of space. In particular it is independent of the presence of a horizon, and has the same form for the case of a star as well. (iii) Depending on the spatial asymptotics, the late time decay is not necessarily a power law in time. The Schwarzschild case with a power-law tail is exceptional among the class of the potentials having a logarithmic spatial dependence. (iv) Both the amplitude and the time dependence of the tail for a broad class of models are obtained analytically. (v) The analytical results are in perfect agreement with numerical calculations. 
  This is the first of a series of papers in which a new formulation of quantum theory is developed for totally constrained systems, that is, canonical systems in which the hamiltonian is written as a linear combination of constraints $h_\alpha$ with arbitrary coefficients. The main purpose of the present paper is to make clear that classical dynamics of a totally constrained system is nothing but the foliation of the constraint submanifold in phase space by the involutive system of infinitesimal canonical transformations $Y_\alpha$ generated by the constraint functions. From this point of view it is shown that statistical dynamics for an ensemble of a totally constrained system can be formulated in terms of a relative distribution function without gauge fixing or reduction. There the key role is played by the fact that the canonical measure in phase space and the vector fields $Y_\alpha$ induce natural conservative measures on acausal submanifolds, which are submanifolds transversal to the dynamical foliation. Further it is shown that the structure coefficients $c^\gamma_{\alpha\beta}$ defined by $\{h_\alpha,h_\beta\}=\sum_\gamma c^\gamma_{\alpha\beta}h_\gamma$ should weakly commute with $h_\alpha$, $\sum_\gamma\{h_\gamma,c^\gamma_{\alpha\beta}\}\approx0$, in order that the description in terms of the relative distribution function is consistent. The overall picture on the classical dynamics given in this paper provides the basic motivation for the quantum formulation developed in the subsequent papers. 
  In this paper a new formulation of quantum dynamics of totally constrained systems is developed, in which physical quantities representing time are included as observables. In this formulation the hamiltonian constraints are imposed on a relative probability amplitude functional $\Psi$ which determines the relative probability for each state to be observed, instead of on the state vectors as in the conventional Dirac quantization. This leads to a foliation of the state space by linear manifolds on each of which $\Psi$ is constant, and dynamics is described as linear mappings among acausal subspaces which are transversal to these linear manifolds. This is a quantum analogue of the classical statistical dynamics of totally constrained systems developed in the previous paper. It is shown that if the von Neumann algebra $\C$ generated by the constant of motion is of type I, $\Psi$ can be consistently normalizable on the acausal subspaces on which a factor subalgebra of $\C$ is represented irreducibly, and the mappings among these acausal subspaces are conformal. How the formulation works is illustrated by simple totally constrained systems with a single constraint such as the parametrized quantum mechanics, a relativistic free particle in Minkowski and curved spacetimes, and a simple minisuperspace model. It is pointed out that the inner product of the relative probability amplitudes induced from the original Hilbert space picks up a special decomposition of the wave functions to the positive and the negative frequency modes. 
  A critical presentation of Rovelli's ``evolving constants of motion'' is given. Previous criticisms by Kucha\v{r} concerning the role of factor ordering and the non-existence of observables are dealt with and shown to be unfounded. Kucha\v{r}'s criticisms that this approach does not solve the global, multiple choice or Hilbert space problems of time are confirmed, and new insight into why this is so is obtained. 
  A general definition of a clock is proposed, and the role of clocks in establishing temporal pre-conditions in quantum mechanical questions is critically discussed. The different status of clocks as used by theorists external to a system and as used by participant-observers within a system is emphasized. It is shown that the foliation of spacetime into instants of time is necessary to correctly interpret the readings of clocks and that clocks are thus insufficient to reconstruct time in the absence of such a foliation. 
  The Einstein Equivalence Principle has as one of its implications that the non-gravitational laws of physics are those of special relativity in any local freely-falling frame. We consider possible tests of this hypothesis for systems whose energies are due to radiative corrections, i.e. which arise purely as a consequence of quantum field theoretic loop effects. Specifically, we evaluate the Lamb shift transition (as given by the energy splitting between the $2S_{1/2}$ and $2P_{1/2}$ atomic states) within the context of violations of local position invariance and local Lorentz invariance, as described by the $T H \epsilon\mu$ formalism. We compute the associated red shift and time dilation parameters, and discuss how (high-precision) measurements of these quantities could provide new information on the validity of the equivalence principle. 
  This talk is devoted to the problem how to compute relative nucleation probabilities of configurations with different topology and dimension in quantum cosmology. Assuming the semiclassical approximation, the usual formula for the nucleation probability induced by the no-boundary wave function is $P_{NB}\approx\exp(-I)$, where $I$ is the Euclidean action, evaluated at a solution of the Euclidean Einstein equations with effective cosmological constant $\Lambda$. Relative probabilities of different configurations are usually compared at equal values of $\Lambda$. When configurations with different dimensions are admitted (the $n$-dimensional gravitational constant being subject to a rather mild restriction), as e.g. ${\bf S}^n$ for any $n$, this procedure leads to the prediction that the space-time dimension tends to be as large as possible. In this contribution, I would like to propose an alternative scheme, namely to compare the probabilities $P_{NB}\approx\exp(-I)$ at equal values of the {\it energy} $E$, instead of the {\it energy density} $\Lambda$. As a result, the space-time dimension settles at $n=4$. Attempts to predict the topology of the spacelike slices lead to the candidates ${\bf S}^3$ and ${\bf S}^1\times {\bf S}^2$. Since the ''process'' of nucleation (possibly connected with decoherence) is not well known in detail, we expect that either {\it both} configurations may be realized with roughly equal probability, or the {\it latter} one is favoured. Finally, we comment on the analogous situation based on the tunneling wave function. 
  Using a new regulator, we examine 't Hooft's approach for evaluating black hole entropy through a statistical-mechanical counting of states for a scalar field propagating outside the event horizon. We find that this calculation yields precisely the one-loop renormalization of the standard Bekenstein-Hawking formula, $S={\Cal A}/(4G)$. Thus our result provides evidence confirming a suggestion by Susskind and Uglum regarding black hole entropy. 
  In this paper we introduce a current-current type interaction term in the Lagrangian density of gravity coupled to complex scalar fields, in the presence of a degenerated Fermi gas. For low transferred momenta such a term, which might account for the interaction among boson and fermion constituents of compact stellar objects, is subsequently reduced to a quadratic one in the scalar sector. This procedure enforces the use of a complex radial field counterpart in the equations of motion. The real and the imaginary components of the scalar field exhibit different behaviour as the interaction increases. The results also suggest that the Bose-Fermi system undergoes a BCS-like phase transition for a suitable choice of the coupling constant. 
  The constraint hypersurfaces defining the Witten and Ashtekar formulations for 2+1 gravity are very different. In particular the constraint hypersurface in the Ashtekar case is not a manifold but consists of several sectors that intersect each other in a complicated way. The issue of how to define a consistent dynamics in such a situation is then rather non-trivial. We discuss this point by working out the details in a simplified (finite dimensional) homogeneous reduction of 2+1 gravity in the Ashtekar formulation. 
  We discuss, in the context of $N=1$ hidden sector non-minimal supergravity chaotic inflationary models, constraints on the parameters of a polynomial superpotential resulting from existing bounds on the reheating temperature and on the amplitude of the primordial energy density fluctuations as inferred from COBE. We present a specific two-parameter chaotic inflationary model which satisfies these constraints and discuss a possible scenario for adequate baryon asymmetry generation. 
  The aim of the lectures is to introduce first-year Ph.D. students and research workers to the theory of the Dirac operator, spinor techniques, and their relevance for the theory of eigenvalues in Riemannian geometry. Topics: differential operators on manifolds, index of elliptic operators, Dirac operator, index problem for manifolds with a boundary, index of the Dirac operator and anomalies, spectral asymmetry and Riemannian geometry, spectral or local boundary conditions for massless spin-1/2 fields, potentials for massless spin-3/2 fields, conformal anomalies for massless spin-1/2 fields. 
  The structure of the Cauchy horizon of a charged rotating black hole is analyzed under the combined effect of an ingoing and outgoing flux of gravitational waves. In particular, by means of an axisymmetric realization of the Ori model, the growth of the mass parameter near the Cauchy horizon is studied in the slow rotation approximation. It is shown that the mass-parameter inflates, while the angular momentum per unit mass deflates, but initial deviations from spherical symmetry survive. 
  We propose a simple approach for the radiative evolution of generic orbits around a Kerr black hole. For a scalar-field, we recover the standard results for the evolution of the energy $E$ and the azimuthal angular momentum $L_z$ . In addition, our method provides a closed expression for the evolution of the Carter constant $Q$. 
  A quantum equivalence principle is formulated by means of a gravitational phase operator which is an element of the Poincare group. This is applied to the spinning cosmic string which suggests that it may (but not necessarily) contain gravitational torsion. A new exact solution of the Einstein- Cartan-Sciama-Kibble equations for the gravitational field with torsion is obtained everywhere for a cosmic string with uniform energy density, spin density and flux. A novel effect due to the quantized gravitational field of the cosmic string on the wave function of a particle outside the string is used to argue that spacetime points are not meaningful in quantum gravity. 
  The electromagnetic field is studied in a family of exact solutions of the Einstein equations whose material content is a perfect fluid with stiff equation of state (p = $\epsilon $ ). The field equations are solved exactly for several members of the family. 
  Exact solutions of N=2 supergravity in five dimensions are found in the metric with cylindrical symmetry, a particular case corresponds to the exterior of a cosmic string. 
  We present a quasilocal formalism, based on the one proposed by Brown and York, for dilaton gravity with Yang-Mills fields. For solutions possessing sufficient symmetry, we define conserved quantities such as mass, angular momentum, and charge. We also present a micro-canonical action and use it to arrive at a quasilocal version of the first law of thermodynamics for static systems containing a black hole. 
  Zeta-function regularization is applied to evaluate the one-loop effective potential for SO(10) grand-unified theories in de Sitter cosmologies. When the Higgs scalar field belongs to the 210-dimensional irreducible representation of SO(10), attention is focused on the mass matrix relevant for the SU(3)xSU(2)xU(1) symmetry-breaking direction, to agree with low-energy phenomenology of the particle-physics standard model. The analysis is restricted to those values of the tree-level-potential parameters for which the absolute minima of the classical potential have been evaluated. As shown in the recent literature, such minima turn out to be SO(6)xSO(4)- or SU(3)xSU(2)xSU(2)xU(1)-invariant. Electroweak phenomenology is more naturally derived, however, from the former minima. Hence the values of the parameters leading to the alternative set of minima have been discarded. Within this framework, flat-space limit and general form of the one-loop effective potential are studied in detail by using analytic and numerical methods. It turns out that, as far as the absolute-minimum direction is concerned, the flat-space limit of the one-loop calculation about a de Sitter background does not change the results previously obtained in the literature, where the tree-level potential in flat space-time was studied. Moreover, when curvature effects are no longer negligible in the one-loop potential, it is found that the early universe remains bound to reach only the SO(6)xSO(4) absolute minimum. 
  By fine-tuning generic Cauchy data, critical phenomena have recently been discovered in the black hole/no black hole "phase transition" of various gravitating systems. For the spherisymmetric real scalar field system, we find the "critical" spacetime separating the two phases by demanding discrete scale-invariance, analyticity, and an additional reflection-type symmetry. The resulting nonlinear hyperbolic boundary value problem, with the rescaling factor Delta as the eigenvalue, is solved numerically by relaxation. We find Delta = 3.4439 +/- 0.0004. 
  We consider a globally hyperbolic, stationary spacetime containing a black hole but no white hole. We assume, further, that the event horizon, $\tn$, of the black hole is a Killing horizon with compact cross-sections. We prove that if surface gravity is non-zero constant throughout the horizon one can {\it globally} extend such a spacetime so that the image of $\cal N$ is a proper subset of a regular bifurcate Killing horizon in the enlarged spacetime. The necessary and sufficient conditions are given for the extendibility of matter fields to the enlarged spacetime. These conditions are automatically satisfied if the spacetime is static (and, hence ``$t$"-reflection symmetric) or stationary-axisymmetric with ``$t-\phi$" reflection isometry and the matter fields respect the reflection isometry. In addition, we prove that a necessary and sufficient condition for the constancy of the surface gravity on a Killing horizon is that the exterior derivative of the twist of the horizon Killing field vanish on the horizon. As a corollary of this, we recover a result of Carter that constancy of surface gravity holds for any black hole which is static or stationary- axisymmetric with the ``$t-\phi$" reflection isometry. No use of Einstein's equation is made in obtaining any of the above results. Taken together, these results support the view that any spacetime representing the asymptotic final state of a black hole formed by gravitational collapse may be assumed to possess a bifurcate Killing horizon or a Killing horizon with vanishing surface gravity. 
  It is proved that the Riemann tensor squared is divergent as $\tau \ra 0$ for a wide class of cosmological metrics with non-exceptional Kasner-like behaviour of scale factors as $\tau \ra 0$, where $\tau$ is synchronous time. Using this result it is shown that any non-trivial generalization of the spherically-symmetric Tangherlini solution to the case of $n$ Ricci-flat internal spaces \cite{FIM} has a divergent Riemann tensor squared as $R \ra R_0$, where $R_0$ is parameter of length of the solution. Multitemporal naked singularities are also considered. 
  We propose a realistic, spacetime interpretation of quantum theory in which reality constitutes a *single* history obeying a "law of motion" that makes definite, but incomplete, predictions about its behavior. We associate a "quantum measure" |S| to the set S of histories, and point out that |S| fulfills a sum rule generalizing that of classical probability theory. We interpret |S| as a "propensity", making this precise by stating a criterion for |S|=0 to imply "preclusion" (meaning that the true history will not lie in S). The criterion involves triads of correlated events, and in application to electron-electron scattering, for example, it yields definite predictions about the electron trajectories themselves, independently of any measuring devices which might or might not be present. (So we can give an objective account of measurements.) Two unfinished aspects of the interpretation involve *conditonal* preclusion (which apparently requires a notion of coarse-graining for its formulation) and the need to "locate spacetime regions in advance" without the aid of a fixed background metric (which can be achieved in the context of conditional preclusion via a construction which makes sense both in continuum gravity and in the discrete setting of causal set theory). 
  Two major apparently unrelated problems, that of the origin of time in the universe associated with quantum gravity and to the entropy in de Sitter cosmological models, are found to have their origin in a single physical phenomenon: the semi-classical tunneling through a classically forbidden region of the cosmological scale function. In this region there is a mixing of the states of quantum matter and those of the semi-classical gravity which produces a thermal mixture of the matter states and hence an "entropy"; this same mixing effect brings about the conversion of a parametric time variable into a physical intrinsic time. 
  We propose a model universe, in which the dimension of the space is a continuous variable, which can take any real positive number. The dynamics leads to a model in which the universe has no singularity. The difference between our model and the standard Friedman-Robertson-Walker models become effective for times much before the presently accepted age of the universe. 
  We prove that the maximal development of any spherically symmetric spacetime with collisionless matter (obeying the Vlasov equation) or a massless scalar field (obeying the massless wave equation) and possessing a constant mean curvature $S^1 \times S^2$ Cauchy surface also contains a maximal Cauchy surface. Combining this with previous results establishes that the spacetime can be foliated by constant mean curvature Cauchy surfaces with the mean curvature taking on all real values, thereby showing that these spacetimes satisfy the closed-universe recollapse conjecture. A key element of the proof, of interest in itself, is a bound for the volume of any Cauchy surface $\Sigma$ in any spacetime satisfying the timelike convergence condition in terms of the volume and mean curvature of a fixed Cauchy surface $\Sigma_0$ and the maximal distance between $\Sigma$ and $\Sigma_0$. In particular, this shows that any globally hyperbolic spacetime having a finite lifetime and obeying the timelike-convergence condition cannot attain an arbitrarily large spatial volume. 
  We treat two aspects of the physics of stationary black holes. First we prove that the proportionality, d(energy) ~ d(area) for arbitrary perturbations (``extended first law''), follows directly from an extremality theorem drawn from earlier work. Second we consider quantum fluctuations in the shape of the horizon, concluding heuristically that they exhibit a fractal character, with order lambda fluctuations occurring on all scales lambda below M^{1/3} in natural units. 
  A no-go theorem pertaining to the graceful exit problem in Pre-Big-Bang inflation is reviewed. It is shown that dilaton self-interactions and string fluid sources fail to facilitate branch changing necessary to avoid singularities. A comment on the failure of the higher genus corrections to induce graceful exit is also included. 
  This paper examines an observable consequence for the diffuse extragalactic background radiation (EGBR) of the hypothesis that if closed, our universe possesses time symmetric boundary conditions. For simplicity, attention is focused on optical wavelengths. The universe is modeled as closed Friedmann- Roberston-Walker. It is shown that, over a wide range of frequencies, electromagnetic radiation can propagate largely unabsorbed from the present epoch into the recollapsing phase, confirming and demonstrating the generality of results of Davies and Twamley. As a consequence, time symmetric boundary conditions imply that the optical EGBR is at least twice that due to the galaxies on our past light cone, and possibly considerably more. It is therefore possible to test experimentally the notion that if our universe is closed, it may be in a certain sense time symmetric. The lower bound on the "excess" EGBR in a time symmetric universe is consistent with present observations. Nevertheless, better observations and modelling may soon rule it out entirely. In addition, many physical complications arise in attempting to reconcile a transparent future light cone with time symmetric boundary conditions, thereby providing further arguments against the possibility that our universe is time symmetric. 
  Many important features of a field theory, {\it e.g.}, conserved currents, symplectic structures, energy-momentum tensors, {\it etc.}, arise as tensors locally constructed from the fields and their derivatives. Such tensors are naturally defined as geometric objects on the jet space of solutions to the field equations. Modern results from the calculus on jet bundles can be combined with a powerful spinor parametrization of the jet space of Einstein metrics to unravel basic features of the Einstein equations. These techniques have been applied to computation of generalized symmetries and ``characteristic cohomology'' of the Einstein equations, and lead to results such as a proof of non-existence of ``local observables'' for vacuum spacetimes and a uniqueness theorem for the gravitational symplectic structure. 
  Using a classical action associated to a point-particle in (1+1)-dimensions the classical string theory is derived. In connection with this result two aspects are clarified: First, the point particle in (1+1)-dimensions is not an ordinary relativistic system, but rather a some kind of a relativistic top; and second, through the quantization of such a kind of top the ordinary string theory is not obtained, but rather a $\sigma$-model associated to a non-compact group which may be understood as an extended string theory. 
  The phenomenon of black hole thermodynamics raises several deep issues which any proper theory of quantum gravity must confront: to what extent does the inclusion of the back-reaction alter the thermal character of the radiation, how can the entropy be understood from a microscopic standpoint, what is the ultimate fate of an evaporating black hole, and is the outcome reconcilable with unitary time evolution in quantum mechanics?  In the first part of this thesis, we address the issue of determining what the actual emission spectrum from a black hole is, once the gravitational field of the emitted quanta is included in a quantum mechanical manner. To make the problem tractable, we employ two important approximations: we quantize only the s-wave sector of the full theory, and we consider only single particle emission. By proceeding in the framework of a Hamiltonian path integral description of this system, we are able to integrate out the gravitational field, thereby obtaining an effective action depending only on the matter degrees of freedom. This effective action can then be second quantized in terms of new, corrected, mode solutions thus enabling the calculation of the emission spectrum from modified Bogoliubov coefficients. The results are particularly interesting in the case of emission from Reissner-Nordstrom black holes, since in the extremal limit our results are dramatically different from what a naive, and incorrect, semi-classical calculation would yield.  The other major topic which we discuss is the dynamics of quantum fields on background geometries which undergo quantum tunneling. An example of such a system which has important implications for both cosmology and quantum gravity in general, is the tunneling of a false vacuum bubble leading to the creation 
  Up to a conjecture in Riemannian geometry, we significantly strengthen a recent theorem of Eardley by proving that a compact region in an initial data surface that is collapsing sufficiently fast in comparison to its surface-to-volume ratio must contain a future trapped region. In addition to establishing this stronger result, the geometrical argument used does not require any asymptotic or energy conditions on the initial data. It follows that if such a region can be found in an asymptotically flat Cauchy surface of a spacetime satisfying the null-convergence condition, the spacetime must contain a black hole with the future trapped region therein. Further, up to another conjecture, we prove a strengthened version of our theorem by arguing that if a certain function (defined on the collection of compact subsets of the initial data surface that are themselves three-dimensional manifolds with boundary) is not strictly positive, then the initial data surface must contain a future trapped region. As a byproduct of this work, we offer a slightly generalized notion of a future trapped region as well as a new proof that future trapped regions lie within the black hole region. 
  We study a manifestly unitary formulation of 2d dilaton quantum gravity based on the reduced phase space quantization. The spacetime metric can be expanded in a formal power series of the matter energy-momentum tensor operator. This expansion can be used for calculating the quantum corrections to the classical black hole metric by evaluating the expectation value of the metric operator in an appropriate class of the physical states. When the normal ordering in the metric operator is chosen to be with respect to Kruskal vacuum, the lowest order semiclassical metric is exactly the one-loop effective action metric discovered by Bose, Parker and Peleg. The corresponding semiclassical geometry describes an evaporating black hole which ends up as a remnant. The calculation of higher order corrections and implications for the black hole fate are discussed. 
  Electrodynamics for self-interacting scalar fields in spatially flat Friedmann-Robertson-Walker space-times is studied. The corresponding one-loop field equation for the expectation value of the complex scalar field in the conformal vacuum is derived. For exponentially expanding universes, the equations for the Bogoliubov coefficients describing the coupling of the scalar field to gravity are solved numerically. They yield a non-local correction to the Coleman-Weinberg effective potential which does not modify the pattern of minima found in static de Sitter space. Such a correction contains a dissipative term which, accounting for the decay of the classical configuration in scalar field quanta, may be relevant for the reheating stage. The physical meaning of the non-local term in the semiclassical field equation is investigated by evaluating this contribution for various background field configurations. 
  The paper deals with issues pertaining the detection of gravitational waves from coalescing binaries. We introduce the application of differential geometry to the problem of optimal detection of the `chirp signal'. We have also carried out extensive Monte Carlo simulations to understand the errors in the estimation of parameters of the binary system. We find that the errors are much more than those predicted by the covariance matrix even at a high SNR of 10-15. We also introduce the idea of using the instant of coalescence rather than the time of arrival to determine the direction to the source. 
  We develop an alternative derivation of Unruh and Wald's seminal result that the absorption of a Rindler particle by a detector as described by uniformly accelerated observers corresponds to the emission of a Minkowski particle as described by inertial observers. Actually, we present it in an inverted version, namely, that the emission of a Minkowski particle corresponds in general to either the emission or the absorption of a Rindler particle. 
  We investigate two-dimensional higher derivative gravitational theories in a Riemann-Cartan framework and obtain the most general static black hole solutions in conformal coordinates. We also consider the hamiltonian formulation of the theory and discuss its symmetries, showing they are implemented by a non-linear generalizations of the 2-dimensional Poincar\'e algebra. Finally, we discuss the quantization in the Dirac formalism. 
  This paper presents a detailed discussion of the ``Newton's method'' algorithm for finding apparent horizons in 3+1 numerical relativity.   We describe a method for computing the Jacobian matrix of the finite differenced $H(h)$ function by symbolically differentiating the finite difference equations, giving the Jacobian elements directly in terms of the finite difference molecule coefficients used in computing $H(h)$. Assuming the finite differencing scheme commutes with linearization, we show how the Jacobian elements may be computed by first linearizing the continuum $H(h)$ equations, then finite differencing the linearized (continuum) equations.   We find this symbolic differentiation method of computing the $H(h)$ Jacobian to be {\em much} more efficient than the usual numerical perturbation method, and also much easier to implement than is commonly thought.   When solving the discrete $H(h) = 0$ equations, we find that Newton's method generally converges very rapidly. However, if the initial guess for the horizon position contains significant high-spatial-frequency error components, Newton's method has a small (poor) radius of convergence. This is {\em not} an artifact of insufficient resolution in the finite difference grid; rather, it appears to be caused by a strong nonlinearity in the continuum $H(h)$ function for high-spatial-frequency error components in $h$. Robust variants of Newton's method can boost the radius of convergence by O(1) factors, but the underlying nonlinearity remains, and appears to worsen rapidly with increasing initial-guess-error spatial frequency.   Using 4th~order finite differencing, we find typical accuracies for computed horizon positions in the $10^{-5}$ range for $\Delta\theta = \frac{\pi/2}{50}$. 
  This paper explores in some detail a recent proposal (the Rieffel induction/refined algebraic quantization scheme) for the quantization of constrained gauge systems. Below, the focus is on systems with a single constraint and, in this context, on the uniqueness of the construction. While in general the results depend heavily on the choices made for certain auxiliary structures, an additional physical argument leads to a unique result for typical cases. We also discuss the `superselection laws' that result from this scheme and how their existence also depends on the choice of auxiliary structures. Again, when these structures are chosen in a physically motivated way, the resulting superselection laws are physically reasonable. 
  As it is well known, the Minkowski vacuum appears thermally populated to a quantum mechanical detector on a uniformly accelerating course. We investigate how this thermal radiation may contribute to the classical nature of the detector's trajectory through the criteria of decoherence. An uncertainty-type relation is obtained for the detector involving the fluctuation in temperature, the time of flight and the coupling to the bath. 
  This article is intended to provide a pedagogical account of issues related to, and recent work on, gravitational waves from coalescing compact binaries (composed of neutron stars and/or black holes). These waves are the most promising for kilometer-size interferometric detectors such as LIGO and VIRGO. Topics discussed include: interferometric detectors and their noise; coalescing compact binaries and their gravitational waveforms; the technique of matched filtering for signal detection and measurement; waveform calculations in post-Newtonian theory and in the black-hole perturbation approach; and the accuracy of the post-Newtonian expansion. 
  We recast the tools of ``global causal analysis'' in accord with an approach to the subject animated by two distinctive features: a thoroughgoing reliance on order-theoretic concepts, and a utilization of the Vietoris topology for the space of closed subsets of a compact set. We are led to work with a new causal relation which we call $K^+$, and in terms of it we formulate extended definitions of concepts like causal curve and global hyperbolicity. In particular we prove that, in a spacetime $\M$ which is free of causal cycles, one may define a causal curve simply as a compact connected subset of $\M$ which is linearly ordered by $K^+$. Our definitions all make sense for arbitrary $C^0$ metrics (and even for certain metrics which fail to be invertible in places). Using this feature, we prove for a general $C^0$ metric, the familiar theorem that the space of causal curves between any two compact subsets of a globally hyperbolic spacetime is compact. We feel that our approach, in addition to yielding a more general theorem, simplifies and clarifies the reasoning involved. Our results have application in a recent positive energy theorem, and may also prove useful in the study of topology change. We have tried to make our treatment self-contained by including proofs of all the facts we use which are not widely available in reference works on topology and differential geometry. 
  In this review I consider several different issues related to inflation. I will begin with the wave function of the Universe. This issue is pretty old, but recently there were some new insights based on the theory of the self-reproducing inflationary universe. Then we will discuss stationarity of inflationary universe and the possibility to make predictions in the context of quantum cosmology using stochastic approach to inflation. Returning to more pragmatic aspects of inflationary theory, we will discuss inflationary models with $\Omega < 1$. Finally, we will describe several aspects of the theory of reheating of the Universe based on the effect of parametric resonance. 
  We provide a concise approach to generalized dilaton theories with and without torsion and coupling to Yang-Mills fields. Transformations on the space of fields are used to trivialize the field equations locally. In this way their solution becomes accessible within a few lines of calculation only. In this first of a series of papers we set the stage for a thorough global investigation of classical and quantum aspects of more or less all available 2D gravity-Yang-Mills models. 
  A physical interpretation of axioms of the differential structure of space-time is presented. Consequences of such interpretation for cosmic string's space-time with a scalar field are studied. It is shown that the assumption of smoothness of the scalar field leads either to modification of cosmic string's space-time global properties or to quantization of the deficit of angle: $\Delta =2\pi (1-1/n)$, $n=1,2,\dots$. 
  The metric for gravitational plane waves has very high symmetry (two spacelike commuting Killing vectors). For this high symmetry, a simple renormalization of the lapse function is found which allows the constraint algebra for canonical quantum gravity to close; also, the vector constraint has the correct form to generate spatial diffeomorphisms. A measure is constructed which respects the reality conditions, but does not yet respect the invariances of the theory. 
  Usual quantum mechanics requires a fixed, background, spacetime geometry and its associated causal structure. A generalization of the usual theory may therefore be needed at the Planck scale for quantum theories of gravity in which spacetime geometry is a quantum variable. The elements of generalized quantum theory are briefly reviewed and illustrated by generalizations of usual quantum theory that incorporate spacetime alternatives, gauge degrees of freedom, and histories that move forward and backward in time. A generalized quantum framework for cosmological spacetime geometry is sketched. This theory is in fully four-dimensional form and free from the need for a fixed causal structure. Usual quantum mechanics is recovered as an approximation to this more general framework that is appropriate in those situations where spacetime geometry behaves classically.  (Talk given at the Workshop on Physics at the Planck Scale, Puri, India, December12-21, 1994. This talk is a precis of the author's 1992 Les Houches Lectures: Spacetime Quantum Mechanics and the Quantum Mechanics of Spacetime, gr-qc/9304006). 
  We investigate cosmological models with a free scalar field and a viscous fluid. We find exact solutions for a linear and nonlinear viscosity pressure. Both yield singular and bouncing solutions. In the first regime, a de Sitter stage is asymptotically stable, while in the second case we find power-law evolutions for vanishing cosmological constant. 
  In this paper we deal with the measurement of the parameters of the gravitational wave signal emitted by a coalescing binary signal.  We present the results of Monte Carlo simulations carried out for the case of the initial LIGO, incorporating the first post-Newtonian corrections into the waveform. Using the parameters so determined, we estimate the direction to the source. We stress the use of the time-of-coalescence rather than the time-of-arrival of the signal to determine the direction of the source. We show that this can considerably reduce the errors in the determination of the direction of the source. 
  We give a brief summary of results and ongoing research in the application of linearized theory to the study of black hole collisions in the limit in which the holes start close to each other. This approximation can be a valuable tool for comparison and code-checking of full numerical relativity computations. The approximation works quite well for the head-on case and this is motivation to pursue its use in other more interesting contexts. We summarize current efforts towards establishing the domain of validity of the approximation and its use in generation and evolution of initial data for more interesting physical cases. 
  We find a family of exact solutions to the semi-classical equations (including back-reaction) of two-dimensional dilaton gravity, describing infalling null matter that becomes outgoing and returns to infinity without forming a black hole. When a black hole almost forms, the radiation reaching infinity in advance of the original outgoing null matter has the properties of Hawking radiation. The radiation reaching infinity after the null matter consists of a brief burst of negative energy that preserves unitarity and transfers information faster than the theoretical bound for positive energy. 
  In a thorough paper Kuchar has examined the canonical reduction of the most general action functional describing the geometrodynamics of the maximally extended Schwarzschild geometry. This reduction yields the true degrees of freedom for (vacuum) spherically symmetric general relativity. The essential technical ingredient in Kuchar's analysis is a canonical transformation to a certain chart on the gravitational phase space which features the Schwarzschild mass parameter $M_{S}$, expressed in terms of what are essentially Arnowitt-Deser-Misner variables, as a canonical coordinate. In this paper we discuss the geometric interpretation of Kuchar's canonical transformation in terms of the theory of quasilocal energy-momentum in general relativity given by Brown and York. We find Kuchar's transformation to be a ``sphere-dependent boost to the rest frame," where the ``rest frame'' is defined by vanishing quasilocal momentum. Furthermore, our formalism is general enough to cover the case of (vacuum) two-dimensional dilaton gravity. Therefore, besides reviewing Kucha\v{r}'s original work for Schwarzschild black holes from the framework of hyperbolic geometry, we present new results concerning the canonical reduction of Witten-black-hole geometrodynamics. 
  Based on our previous attempt, we propose a better way to understand a small but nonzero cosmological constant, as indicated by a number of recent observational studies. We re-examine the assumptions of our model of two scalar fields, trying to explain the basic mechanism resulting in a series of mini-inflations occuring nearly periodically with respect to $\ln t$ with $t$ the cosmic time. We also discuss how likely the solution of this type would be, depending on the choice of the parameters. 
  The necessary and sufficient condition for the existence of $\alpha$-surfaces in complex space-time manifolds with nonvanishing torsion is derived. For these manifolds, Lie brackets of vector fields and spinor Ricci identities contain explicitly the effects of torsion. This leads to an integrability condition for $\alpha$-surfaces which does not involve just the self-dual Weyl spinor, as in complex general relativity, but also the torsion spinor, in a nonlinear way, and its covariant derivative. A similar result also holds for four-dimensional, smooth real manifolds with a positive-definite metric. Interestingly, a particular solution of the integrability condition is given by conformally right-flat and right-torsion-free space-times. 
  We demonstrate that under plausible assumptions the entropy and temperature associated with the small oscillations on a circular loop of radius $R$ and a black hole of mass $M=R/2G$ are identical. 
  The problem of boundary conditions in a supersymmetric theory of quantum cosmology is studied, with application to the one-loop prefactor in the quantum amplitude. Our background cosmological model is flat Euclidean space bounded by a three-sphere, and our calculations are based on the generalized Riemann zeta-function. One possible set of supersymmetric local boundary conditions involves field strengths for spins 1, 3/2 and 2, the undifferentiated spin-1/2 field, and a mixture of Dirichlet and Neumann conditions for spin 0. In this case the results we can obtain are: zeta(0)=7/45 for a complex scalar field, zeta(0)=11/360 for spin 1/2, zeta(0)= -77/180 (magnetic) and 13/180 (electric) for spin 1, and zeta(0)=112/45 for pure gravity when the linearized magnetic curvature is vanishing on $S^3$. The zeta(0) values for gauge fields have been obtained by working only with physical degrees of freedom. An alternative set of boundary conditions can be motivated by studying transformation properties under local supersymmetry; these involve Dirichlet conditions for the spin-2 and spin-1 fields, a mixture of Dirichlet and Neumann conditions for spin-0, and local boundary conditions for the spin-1/2 field and the spin-3/2 potential. For the latter one finds: zeta(0)=-289/360. The full zeta(0) does not vanish in extended supergravity theories, indicating that supersymmetry is one-loop divergent in the presence of boundaries. 
  It is shown that the classical field equations pertaining to gravity coupled to other bosonic fields are equivalent to a single geodesic equation, describing the free fall of a point particle in superspace. Some implications for quantum gravity are discussed. 
  We investigate the cosmological model with complex scalar self-interacting inflaton field non-minimally coupled to gravity. The different geometries of the Euclidean classically forbidden regions are represented. The instanton solutions of the corresponding Euclidean equations of motion are found by numerical calculations. These solutions give a rather non-trivial examples of real tunnelling geometries. Possible interpretation of obtained results and their connection with inflationary cosmology is discussed. 
  We consider multidimensional cosmological models with a generalized space-time manifold M = R x M_1 ...x M_n, composed from a finite number of factor spaces M_i, i=1,..n. While usually each factor space M_i is considered to be some Riemannian space of integer dimension d_i, here it is, more generally, a fractal space, the dimension of which is a smooth function d_i(t) of time. Hence, besides the scale factor exponents ln a_i and their derivatives, we consider also the dimensions d_i of the factor spaces as classical dynamical variables. The classical equation of motions and the corresponding Wheeler-de Witt equation are set up generally, and the qualitative behaviour of the system is discussed for some specific model with 2 factor spaces. 
  Reduction to physical degrees of freedom before quantization leads to predictions for one-loop amplitudes in quantum cosmology in the presence of boundaries which disagree with the results obtained from Faddeev-Popov theory and boundary-counterterms technique. However, the mode-by-mode analysis of eigenvalue equations for gauge modes and ghost fields remains a very difficult problem. Hence the equivalence or inequivalence of various quantization and regularization techniques cannot be easily proved. 
  The quantization of classical theories that admit more than one Hamiltonian description is considered. This is done from a geometrical viewpoint, both at the quantization level (geometric quantization) and at the level of the dynamics of the quantum theory. A spin-1/2 system is taken as an example in which all the steps can be completed. It is shown that the geometry of the quantum theory imposes restrictions on the physically allowed nonstandard quantum theories. 
  In a number of papers it has been claimed that the time machine are quantum unstable, which manifests itself in the divergence of the vacuum expectation value of the stress-energy tensor $\langle{\bf T}\rangle$ near the Cauchy horizon. The expression for $\langle{\bf T}\rangle$ was found in these papers on the basis of some specific approach \cite{Fro,KimT}.\par We show that this approach is untenable in that the above expression firstly is not derived from some more fundamental and undeniable premises, as it is claimed, but rather postulated and secondly contains undefined terms, so that one can neither use nor check it. As a counterexample we cite a few cases of (two-dimensional) spacetimes containing time machines with $\langle{\bf T}\rangle$ bounded near the Cauchy horizon. 
  We perform a canonical analysis of the system of 2d vacuum dilatonic black holes. Our basic variables are closely tied to the spacetime geometry and we do not make the field redefinitions which have been made by other authors. We present a careful discssion of asymptotics in this canonical formalism. Canonical transformations are made to variables which (on shell) have a clear spacetime significance. We are able to deduce the location of the horizon on the spatial slice (on shell) from the vanishing of a combination of canonical data. The constraints dramatically simplify in terms of the new canonical variables and quantization is easy. The physical interpretation of the variable conjugate to the ADM mass is clarified. This work closely parallels that done by Kucha{\v r} for the vacuum Schwarzschild black holes and is a starting point for a similar analysis, now in progress, for the case of a massless scalar field conformally coupled to a 2d dilatonic black hole. 
  We explain how the round four-sphere can be sliced along homogeneous 3~-~manifolds of topology $S^3/D_8^*$. This defines a Euclidean Bianchi type IX model for Einstein's equations with cosmological constant. The geometric properties of this model are investigated. 
  Renormalization group techniques are used in order to obtain the improved non-local gravitational effective action corresponding to any asymptotically free GUT, up to invariants which are quadratic on the curvature. The corresponding non-local gravitational equations are written down, both for the case of asymptotically free GUTs and also for quantum R$^2$-gravity. The implications of the results when obtaining the flux of vacuum radiation through the future null infinity are briefly discussed. 
  We show that the Riemannian Kerr solutions are the only Riemannian, Ricci-flat and asymptotically flat ${\rm C}^{2}$-metrics $g_{\mu\nu}$ on a 4-dimensional complete manifold ${\cal M}$ of topology ${\rm R}^{2} \times {\rm S}^{2}$ which have (at least) a 1-parameter group of periodic isometries with  only isolated fixed points ("nuts") and with orbits of bounded length at infinity. 
  Omitting the motivations and historical connections, and also the detailed calculations, I state succinctly the principles that determine the relativistic idealization of a GPS system. These determine the results that Ashby presents in his tutorial. 
  We generalize previous work \cite{BaNa3} on the ultrarelativistic limit of the Kerr-geometry by lifting the restriction on boosting along the axis of symmetry. 
  I propose the Langevin equation for 3-geometries in the Ashtekar's formalism to describe 4D Euclidean quantum gravity, in the sense that the corresponding Fokker-Planck hamiltonian recovers the hamiltonian in 4D quantum gravity exactly. The stochastic time corresponds to the Euclidean time in the gauge, N=1 and $N^i=0$. In this approach, the time evolution in 4D quantum gravity is understood as a stochastic process where the quantum fluctuation of ` ` triad \rq\rq is characterized by the curvature at the one unit time step before. The lattice regularization of 4D quantum gravity is presented in this context. 
  In gravitation theory, a fermion field must be regarded only in a pair with a certain tetrad gravitational field. These pairs can be represented by sections of the composite spinor bundle $S\to\Si\to X^4$ where values of gravitational fields play the role of parameter coordinates, besides the familiar world coordinates. 
  Several problems in physics, in particular the averaging problem in gravity, can be described in a formalism derived from the real-space Renormalization Group (RG) methods. It is shown that the RG flow is provided by the Ricci-Hamilton equations which are thereby provided with a {\it physical} interpretation. The connection between a manifold deformation according to these equations and Thurston's conjecture is exhibited. The significance of criticality which naturally appears in this framework is briefly discussed. This article summarizes also recent work with M. Carfora. Moreover, a report on some work in progress is given and some open issues in the averaging problem pointed out. 
  The effective action for QED in curved spacetime includes equivalence principle violating interactions between the electromagnetic field and the spacetime curvature. These interactions admit the possibility of superluminal yet causal photon propagation in gravitational fields. In this paper, we extend our analysis of photon propagation in gravitational backgrounds to the Kerr spacetime describing a rotating black hole. The results support two general theorems -- a polarisation sum rule and a `horizon theorem'. The implications for the stationary limit surface bounding the ergosphere are also discussed. 
  For irrotational dust the shear tensor is consistently diagonalizable with its covariant time derivative: $\sigma_{ab}=0=\dot{\sigma}_{ab},\; a\neq b$, if and only if the divergence of the magnetic part of the Weyl tensor vanishes: $div~H = 0$. We show here that in that case, the consistency of the Ricci constraints requires that the magnetic part of the Weyl tensor itself vanishes: $H_{ab}=0$. 
  The problem of topology change description in gravitation theory is analized in detailes. It is pointed out that in standard four-dimensional theories the topology of space may be considered as a particular case of boundary conditions (or constraints). Therefore, the possible changes of space topology in (3+1)-dimensions do not admit dynamical description nor in classical nor in quantum theories and the statements about dynamical supressing of topology change have no sence. In the framework of multidimensional theories the space (and space-time) may be considered as the embedded manifolds. It give the real posibilities for the dynamical description of the topology of space or space-time. 
  We study real linear scalar field theory on two simple non-globally hyperbolic spacetimes containing closed timelike curves within the framework proposed by Kay for algebraic quantum field theory on non-globally hyperbolic spacetimes. In this context, a spacetime (M,g) is said to be `F-quantum compatible' with a field theory if it admits a *-algebra of local observables for that theory which satisfies a locality condition known as `F-locality'. Kay's proposal is that, in formulating algebraic quantum field theory on $(M,g)$, F-locality should be imposed as a necessary condition on the *-algebra of observables. The spacetimes studied are the 2- and 4-dimensional spacelike cylinders (Minkowski space quotiented by a timelike translation). Kay has shown that the 4-dimensional spacelike cylinder is F-quantum compatible with massless fields. We prove that it is also F-quantum compatible with massive fields and prove the F-quantum compatibility of the 2-dimensional spacelike cylinder with both massive and massless fields. In each case, F-quantum compatibility is proved by constructing a suitable F-local algebra. 
  We describe in detail how to eliminate nonphysical degrees of freedom in the Lagrangian and Hamiltonian formulations of a constrained system. Two important and distinct steps in our method are the fixing of ambiguities in the dynamics and the determination of inequivalent initial data. The Lagrangian discussion is novel, and a proof is given that the final number of degrees of freedom in the two formulations agrees. We give applications to reparameterization invariant theories, where we prove that one of the constraints must be explicitly time dependent. We illustrate our procedure with the examples of trajectories in spacetime and with spatially homogeneous cosmological models. Finally, we comment briefly on Dirac's extended Hamiltonian technique. 
  We discuss photon emission from particles decelerlated by the cosmic expansion. This can be interpretated as a kind of bremsstrahlung induced by the Universe geometry. In the high momentum limit its transition probability does not depend on detailed behavior of the expansion. 
  Superluminal particles are not excluded by particle physics. The apparent Lorentz invariance of the laws of physics does not imply that space-time is indeed minkowskian. Matter made of solutions of Lorentz-invariant equations would feel a relativistic space-time even if the actual space-time had a quite different geometry (f.i. a galilean space-time). If Lorentz invariance is only a property of equations describing a sector of matter at a given scale, an absolute frame (the "vacuum rest frame") may exist without contradicting the minkowskian structure felt by ordinary particles. Then c , the speed of light, will not necessarily be the only critical speed in vacuum and superluminal sectors of matter may equally exist feeling space-times with critical speeds larger than c . We present a discussion of possible cosmological implications of such a scenario, assuming that the superluminal sectors couple weakly to ordinary matter. The universality of the equivalence between inertial and gravitational mass will be lost. The Big Bang scenario will undergo important modifications, and the evolution of the Universe may be strongly influenced by superluminal particles. 
  This paper deals with the geometry of supermassive cosmic strings. We have used an approach that enforces the spacetime of cosmic strings to also satisfy the conservation laws of a cylindric gravitational topological defect, that is a spacetime kink. In the simplest case of kink number unity, the entire energy range of supermassive strings becomes then quantized so that only cylindrical defects with linear energy density $G\mu=1/4$ (critical string) and $G\mu=1/2$ (extreme string) are allowed to occur in this range. It has been seen that the internal spherical coordinate $\theta$ of the string metric embedded in an Euclidean three-space also evolves on imaginary values, leading to the creation of a covering shell of broken phase that protects the core with trapped energy, even for $G\mu=1/2$. Then the conical singularity becomes a removable horizaon singularity. We re-express the extreme string metric in the Finkelstein- McCollum standard form and remove the geodesic incompleteness by using the Kruskal technique. The z=const. sections of the resulting metric are the same as the hemispherical section of the metric of a De Sitter kink. Some physical consequences from these results, including the possibility that the extreme string drives inflation and thermal effects in its core, are also discussed. 
  In the one-loop approximation for Euclidean quantum gravity, the boundary conditions which are completely invariant under gauge transformations of metric perturbations involve both normal and tangential derivatives of the metric perturbations $h_{00}$ and $h_{0i}$, while the $h_{ij}$ perturbations and the whole ghost one-form are set to zero at the boundary. The corresponding one-loop divergency for pure gravity has been recently evaluated by means of analytic techniques. It now remains to compute the contribution of all perturbative modes of gauge fields and gravitation to the one-loop effective action for problems with boundaries. The functional determinant has a non-local nature, independently of boundary conditions. Moreover, the analysis of one-loop divergences for supergravity with non-local boundary conditions has not yet been completed and is still under active investigation. 
  The entropy of a quantum-statistical system which is classically approximated by a general stationary eternal black hole is studied by means of a microcanonical functional integral. This approach opens the possibility of including explicitly the internal degrees of freedom of a physical black hole in path integral descriptions of its thermodynamical properties. If the functional integral is interpreted as the density of states of the system, the corresponding entropy equals ${cal S} = A_H/4 - A_H/4 =0$ in the semiclassical approximation, where $A_H$ is the area of the black hole horizon. The functional integral reflects the properties of a pure state. 
  The dynamics of the fluid fields in a large class of causal dissipative fluid theories is studied. It is shown that the physical fluid states in these theories must relax (on a time scale that is characteristic of the microscopic particle interactions) to ones that are essentially indistinguishable from the simple relativistic Navier-Stokes descriptions of these states. Thus, for example, in the relaxed form of a physical fluid state the stress energy tensor is in effect indistinguishable from a perfect fluid stress tensor plus small dissipative corrections proportional to the shear of the fluid velocity, the gradient of the temperature, etc. 
  Nonspherical perturbation theory has been necessary to understand the meaning of radiation in spacetimes generated through fully nonlinear numerical relativity. Recently, perturbation techniques have been found to be successful for the time evolution of initial data found by nonlinear methods. Anticipating that such an approach will prove useful in a variety of problems, we give here both the practical steps, and a discussion of the underlying theory, for taking numerically generated data on an initial hypersurface as initial value data and extracting data that can be considered to be nonspherical perturbations. 
  We calculate, in a class of Gauge invariant functionals, by variational methods, the difference of vacuum energy between two different backgrounds: Schwarzschild and Flat Space. We perform this evaluation in an Hamiltonian formulation of Quantum Gravity by standard ''$3+1$'' decomposition. After the decomposition the scalar curvature is expanded to second order with respect to the Schwarzschild metric. We evaluate this energy difference in momentum space, in the lowest possible state (regardless of any negative mode). We find a singular behaviour in the UV-limit, due to the presence of the horizon when $r=2m.$ When $r>2m$ this singular behaviour disappears, which is in agreement with various other models presented in the literature. 
  Recent work in the literature has shown that general relativity can be formulated in terms of a jet bundle which, in local coordinates, has five entries: local coordinates on Lorentzian space-time, tetrads, connection one-forms, multivelocities corresponding to the tetrads and multivelocities corresponding to the connection one-forms. The derivatives of the Lagrangian with respect to the latter class of multivelocities give rise to a set of multimomenta which naturally occur in the constraint equations. Interestingly, all the constraint equations of general relativity are linear in terms of this class of multimomenta. This construction has been then extended to complex general relativity, where Lorentzian space-time is replaced by a four-complex-dimensional complex-Riemannian manifold. One then finds a holomorphic theory where the familiar constraint equations are replaced by a set of equations linear in the holomorphic multimomenta, providing such multimomenta vanish on a family of two-complex-dimensional surfaces. In quantum gravity, the problem arises to quantize a real or a holomorphic theory on the extended space where the multimomenta can be defined. 
  We quantize by the Dirac - Wheeler-DeWitt method the canonical formulation of the Schwarzschild black hole developed in a previous paper. We investigate the properties of the operators that generate rigid symmetries of the Hamiltonian, establish the form of the invariant measure under the rigid transformations, and determine the gauge fixed Hilbert space of states. We also prove that the reduced quantization method leads to the same Hilbert space for a suitable gauge fixing. 
  The two-dimensional CGHS model provides an interesting toy-model for the study of black hole evaporation. For this model, a quantum effective action, which incorporates Hawking radiation and backreaction, can be explicitly constructed. In this paper, we study a generalization of this effective action. In our extended model, it is possible to remove certain curvature singularities arising for the original theory. We also find that the flux of Hawking radiation is identical to that encountered in other two-dimensional models. 
  An approach to quantum gravity and cosmology is proposed based on a synthesis of four elements: 1) the Bekenstein bound and the related holographic hypothesis of 't Hooft and Susskind, 2) topological quantum field theory, 3) a new approach to the interpretational issues of quantum cosmology and 4) the loop representation formulation of non-perturbative quantum gravity. A set of postulates are described, which define a (\it pluralistic quantum cosmological theory.) These incorporates a statistical and relational approach to the interpretation problem, following proposals of Crane and Rovelli, in which there is a Hilbert space associated to each timelike boundary, dividing the universe into two parts. A quantum state of the universe is an assignment of a statistical state into each of these Hilbert spaces, subject to certain conditions of consistency which come from an analysis of the measurement problem. A proposal for a concrete realization of these postulates is described, which is based on certain results in the loop representation and topological quantum field theory, and in particular on the fact that spin networks and punctured surfaces appear in both contexts. The Capovilla-Dell-Jacobson solution of the constraints of quantum gravity are expressed quantum mechanically in the language of Chern-Simons theory, in a way that leads also to the satisfaction of the Bekenstein bound. 
  The long term perturbation of a Newtonian binary system by an incident gravitational wave is discussed in connection with the issue of gravitational ionization. The periodic orbits of the planar tidal equation are investigated and the conditions for their existence are presented. The possibility of ionization of a Keplerian orbit via gravitational radiation is discussed. 
  The spacetime singularities inside realistic black holes are sometimes thought to be spacelike and strong, since there is a generic class of solutions (BKL) to Einsteins equations with these properties. We show that null, weak singularities are also generic, in the following sense: there is a class of vacuum solutions containing null, weak singularities, depending on 8 arbitrary (up to some inequalities) analytic initial functions of 3 spatial coordinates. Since 8 arbitrary functions are needed (in the gauge used here) to span the generic solution, this class can be regarded as generic. 
  New classes of classically integrable models in the cosmological theories with a scalar field are obtained by using freedoms of defining time and fields. In particular, some models with the sum of exponential potentials in the flat spatial metric are shown to be integrable. The model with the Sine-Gordon potential can be solved in terms of analytic continuation of the non-periodic Toda field theory. 
  The evaluation of the absorption coefficients are important for particle emission caused by Hawking radiation. In the case of cosmological particle emission from the event horizon in De Sitter space, it is known that the scalar wave functions are solved in terms of Legendre functions. For fields with higher spin, the solution has been examined with low frequency approximation. We shows that the radial equations of the fields with spin $0,1/2,1$ and $2$ can be solved analytically in terms of the hypergeometric functions. We calculate the absorption probability using asymptotic expansion for high frequency limit. It turns out that the absorption coefficients are universal to all bosonic fields; They depend only on the angular momentum and not spin. In the case of spin $1/2$ fermions, we can also find non-vanishing absorption probability in contrast to the previously known result. 
  We discuss the Anderson localization of electromagnetic fields in the fluctuating plasma induced by the gravitational density perturbation before the recombination time of the Universe. Randomly distributed localized coherent electromagnetic fields emerge in the thermal equilibrium before the recombination time. We argue that the localized coherent electric fields eventually produce cosmological magnetic fields after the decoupling time. 
  Contents:   -Report from the APS Topical Group in Gravitation, Beverly Berger  -Some remarks on the passing of S. Chandrasekhar, Robert Wald  -LIGO Project Status, Syd Meshkov and Stan Whitcomb  -New Hyperbolic forms of the Einstein Equations, Andrew Abrahams  -Massless Black Holes, Gary Horowitz  -A plea for theoretical help from the gravity wave co-op, William Hamilton  -Why Quantum Cosmologists are Interested in Quantum Mechanics, James Hartle  -Probing the Early Universe with the Cosmic Microwave Background, Rahul Dave   and Paul Steinhardt  -G measurements, Riley Newman  -Is general relativity about null surfaces?, Carlo Rovelli  -ITP program solicitation, James Hartle  -7th Gregynog workshop in general relativity, Miguel Alcubierre  -Third Annual Penn State Conference: Astrophysical Sources of Gravitational   Waves, Curt Cutler  -General news from GR14, Abhay Ashtekar  -VIth Canadian General Relativity Conference, Jack Gegenberg 
  We prove that in the Hartle-Hawking approach to quantum cosmology the existence of an inflationary phase is a general property of minisuperspace models given by a closed Friedmann-Robertson-Walker universe containing a massless scalar field with a $\lambda\phi^{n}$ self-interaction. The evolution in time of the cosmic scale factor and of the scalar field in the very early universe is derived, together with the conditions to be satisfied in order to solve the horizon and flatness problems. 
  The problem of a rigorous theory of singularities in space-times with torsion is addressed. We define geodesics as curves whose tangent vector moves by parallel transport. This is different from what other authors have done, because their definition of geodesics only involves the Christoffel connection, though studying theories with torsion. We propose a preliminary definition of singularities which is based on timelike or null geodesic incompleteness, even though for theories with torsion the paths of particles are not geodesics. The study of the geodesic equation for cosmological models with torsion shows that the definition has a physical relevance. It can also be motivated, as done in the literature, remarking that the causal structure of a space-time with torsion does not get changed with respect to general relativity. We then prove how to extend Hawking's singularity theorem without causality assumptions to the space-time of the ECSK theory. This is achieved studying the generalized Raychaudhuri equation in the ECSK theory, the conditions for the existence of conjugate points and properties of maximal timelike geodesics. Hawking's theorem can be generalized, provided the torsion tensor obeys some conditions. Thus our result can also be interpreted as a no-singularity theorem if these additional conditions are not satisfied. In other words, it turns out that the occurrence of singularities in closed cosmological models based on the ECSK theory is less generic than in general relativity. Our work is to be compared with previous papers in the literature. There are some relevant differences, because we rely on a different definition of geodesics, we keep the field equations of the ECSK theory in their original form rather than casting them in a form similar to general relativity with a modified energy momentum tensor, 
  We present here a panoramic view of our unified, bi--scale theory of gravitational and strong interactions [which is mathematically analogous to the last version of N.Rosen's bi--metric theory; and yields physical results similar to strong gravity's]. This theory, developed during the last 15 years, is purely geometrical in nature, adopting the methods of General Relativity for the description of hadron structure and strong interactions. In particular, hadrons are associated with `` strong black--holes'', from the external point of view, and with ``micro--universes'', from the internal point of view. Among the results herein presented, let us mention the derivation: (i) of confinement and (ii) asymptotic freedom for the hadron constituents; (iii) of the Yukawa behaviour for the strong potential at the static limit; (iv) of the strong coupling ``constant'', and (v) of mesonic mass spectra. 
  Making use of the primordially isocurvature fluctuations, which are generated in inflationary models with multiple scalar fields, we make a phenomenological model that predicts formation of primordial black holes which can account for the massive compact halo objects recently observed. 
  We study a $R^{2}$ model of gravity with torsion in a closed Friedmann-Robertson-Walker universe. The model is cast in Hamiltonian form subtracting from the original Lagrangian the total time derivative of $f_{K}f_{R}$, where $f_{K}$ is proportional to the trace of the extrinsic curvature tensor, and $f_{R}$ is obtained differentiating the Lagrangian with respect to the highest derivative. Torsion is found to lead to a primary constraint linear in the momenta and a secondary constraint quadratic in the momenta, and the full field equations are finally worked out in detail. Problems to be studied for further research are the solution of these equations and the quantization of the model. One could then try to study a new class of quantum cosmological models with torsion. 
  Hamilton-Jacobi equation for Brans-Dicke theory is solved by using a long-wavelength approximation. We examine the non-linear evolution of the inhomogeneities in the dust fluid case and the cosmological constant case. In the case of dust fluid, it turns out that the inhomogeneities of space-time grow. In the case of cosmological constant, the inhomogeneities decay, which is consistent with the cosmic no hair conjecture. The inhomogeneities of the density perturbation and the gravitational constant behave similarly with that of space-time. 
  Gravitational instantons of Bianchi type IX space are constructed in Ashtekar's canonical formalism. Instead of solving the self-duality condition, we fully solve the constraint on the ``initial surface'' and ``Hamiltonian equations''. This formalism is applicable to the matter coupled system with cosmological constant. 
  I consider a truncation of low-energy string theory which contains two $U(1)$ gauge fields. After making some general comments on the theory, I describe a previously-obtained instanton for the pair creation of black holes when both gauge fields are non-zero, and obtain the pair creation rate by calculating its action. This calculation agrees qualitatively with the earlier calculation of the pair creation rate for black holes in Einstein-Maxwell theory. That is, the pair creation is strongly suppressed in realizable circumstances, and it reduces to the Schwinger result in the point-particle limit. The pair creation of non-extreme black holes is enhanced over that of extreme black holes by $e^{{\cal A}_{bh}/4}$. 
  A simple modification to Einstein's theory of gravity in terms of a non-Riemannian connection is examined. A new tensor-variational approach yields field equations that possess a covariance similar to the gauge covariance of electromagnetism. These equations are shown to possess solutions analogous to those found in the Einstein-Maxwell system. In particular one finds gravi-electric and gravi-magnetic charges contributing to a spherically symmetric static Reissner-Nordstr\"om metric. Such Weyl ``charges'' provide a source for the non-Riemannian torsion and metric gradient fields instead of the electromagnetic field. The theory suggests that matter may be endowed with gravitational charges that couple to gravity in a manner analogous to electromagnetic couplings in an electromagnetic field. The nature of gravitational coupling to spinor matter in this theory is also investigated and a solution exhibiting a plane-symmetric gravitational metric wave coupled via non-Riemannian waves to a propagating spinor field is presented. 
  The fraction of cosmic string loops which collapse to form black holes is estimated using a set of realistic loops generated by loop fragmentation. The smallest radius sphere into which each cosmic string loop may fit is obtained by monitoring the loop through one period of oscillation. For a loop with invariant length $L$ which contracts to within a sphere of radius $R$, the minimum mass-per-unit length $\mu_{\rm min}$ necessary for the cosmic string loop to form a black hole according to the hoop conjecture is $\mu_{\rm min} = R /(2 G L)$. Analyzing $25,576$ loops, we obtain the empirical estimate $f_{\rm BH} = 10^{4.9\pm 0.2} (G\mu)^{4.1 \pm 0.1}$ for the fraction of cosmic string loops which collapse to form black holes as a function of the mass-per-unit length $\mu$ in the range $10^{-3} \lesssim G\mu \lesssim 3 \times 10^{-2}$. We use this power law to extrapolate to $G\mu \sim 10^{-6}$, obtaining the fraction $f_{\rm BH}$ of physically interesting cosmic string loops which collapse to form black holes within one oscillation period of formation. Comparing this fraction with the observational bounds on a population of evaporating black holes, we obtain the limit $G\mu \le 3.1 (\pm 0.7) \times 10^{-6}$ on the cosmic string mass-per-unit-length. This limit is consistent with all other observational bounds. 
  The variational theory of the perfect fluid with an intrinsic hypermomentum is developed. The Lagrangian density of such fluid is stated and the equations of motion of the fluid and the evolution equation of the hypermomentum tensor are derived. The expressions of the matter currents of the fluid (the metric stress-energy 4-form, the canonical energy-momentum 3-form and the hypermomentum 3-form) are obtained. 
  An extremely simple and unified base for physics comes out by starting all over from a single postulate on the common nature of matter and stationary forms of radiation quanta. Basic relativistic, gravitational (G) and quantum mechanical properties of a standing wave particle model have been derived. This has been done from just dual properties of radiation's and strictly homogeneous relationships for nonlocal cases in G fields. This way reduces the number of independent variables and puts into relief (and avoid) important inhomogeneity errors of some G theories. It unifies and accounts for basic principles and postulates physics. The results for gravity depend on linear radiation properties but not on arbitrary field relations. They agree with the conventional tests. However they have some fundamental differences with current G theories. The particle model, at a difference of the conventional theories, also fixes well-defined cosmological and astrophysical models that are different from the rather conventional ones. They have been described and tested with the astronomical observations. These tests have been resumed in a separated work to be sent to the astro Archive. 
  The DeWitt-Schwinger proper time point-splitting procedure is applied to a massive complex scalar field with arbitrary curvature coupling interacting with a classical electromagnetic field in a general curved spacetime. The scalar field current is found to have a linear divergence. The presence of the external background gauge field is found to modify the stress-energy tensor results of Christensen for the neutral scalar field by adding terms of the form $(eF)^2$ to the logarithmic counterterms. These results are shown to be expected from an analysis of the degree of divergence of scalar quantum electrodynamics. 
  The observable Universe is described by a collection of equal mass galaxies linked into a common unit by their mutual gravitational interaction. The partition function of this system is cast in terms of Ising model spin variables and maps exactly onto a three-dimensional stochastic scalar classical field theory. The full machinery of the renormalization group and critical phenomena is brought to bear on this field theory allowing one to calculate the galaxy-galaxy correlation function, whose critical exponent is predicted to be between 1.530 to 1.862, compared to the phenomenological value of 1.6 to 1.8 
  String theory provides the only consistent framework so far that unifies all interactions including gravity. We discuss gravity and cosmology in string theory. Conventional notions from general relativity like geometry, topology etc. are well defined only as low energy approximations in string theory. At small distances physics deviates from the field theoretic intuition. We present several examples of purely stringy phenomena which imply that the physics at strong curvatures can be quite different from what one might expect from field theory. They indicate new possibilities in the context of quantum cosmology. 
  Solutions of Einstein vacuum equations, for a static pseudospherically symmetric system, are presented. They describe a naked singularity and a singular solution with many resemblances to the Schwartzschild solution but with two major differences: its static region, lying inside the null horizon, sees the singularity, and its effective gravitational field is repulsive. We shortly discuss on the phenomenological plausibility of this last solution as a self-consistent system living on a space-time domain, and discuss some features of particle geodesics in its gravitational field. 
  Cosmological solutions of Einstein equation for a \mbox{5-dimensional} space-time, in the case of a dust-filled universe, are presented. With these solutions we are able to test a hypothetical relation between the rest mass of a particle and the $5^{\rm th}$ dimension. Comparison with experiment strongly refutes the implied dependence of the rest mass on the cosmological time. 
  The Misner initial value solution for two momentarily stationary black holes has been the focus of much numerical study. We report here analytic results for an astrophysically similar initial solution, that of Brill and Lindquist (BL). Results are given from perturbation theory for initially close holes and are compared with available numerical results. A comparison is made of the radiation generated from the BL and the Misner initial values, and the physical meaning is discussed. 
  We study the problem of gauge-invariance and gauge-dependence in one-loop quantum cosmology. We formulate some requirements which should be satisfied by boundary conditions in order to give gauge-independent path integral. The case of QED is studied in some detail. We outline difficulties in gauge-invariant quantization of gravitational field in a bounded region. 
  A class of boson-fermion stars, whose spin-0 and spin-1/2 constituents interact through a U(1) current-current term in the Lagrangian density, is analysed. It is shown that it describes the low-energy behavior of a system of weakly interacting massive particles (WIMPs) from the leptonic sector of the minimal supersymmetric standard model. In this case the effective coupling constant $\lambda$ is related to the Fermi constant $G_F$. 
  We investigate the effects of general relativity upon the non-axisymmetric ``bar'' mode secular instability of rapidly rotating stars, i.e. the relativistic and compressible analog of the transition from Maclaurin spheroids to Jacobi ellipsoids. Our method consists in perturbing a stationary axisymmetric configuration, constructed by a 2-D general relativistic numerical code, and taking into account only the dominant terms in the non-axisymmetric part of the 3-D relativistic equations. For a polytropic equation of state, we have determined, as a function of the degree of relativity, the critical adiabatic index $\gamma_{\rm crit}$ above which rapidly rotating stars can break their axial symmetry. A by-product of the present study is the confirmation of the Newtonian value $\gamma_{\rm crit} = 2.238$ obtained by James (1964). We have also considered neutron star models contructed upon twelve nuclear matter equations of state taken from the literature. We found that five equations of state from this sample allow the symmetry breaking for sufficiently high rotation velocities. For the others, the Keplerian velocity (mass-shedding from the equator) is reached before the axisymmetry is broken. Rotating neutron stars that break their axial symmetry can be an important source of gravitational waves for the LIGO/VIRGO interferometric detectors. 
  I review a new (and still tentative) approach to black hole thermodynamics that seeks to explain black hole entropy in terms of microscopic quantum gravitational boundary states induced on the black hole horizon. 
  We generalize the classification of (non-vacuum) pp-waves \cite{JEK} based on the Killing-algebra of the space-time by admitting distribution-valued profile functions. Our approach is based on the analysis of the (infinite-dimensional) group of ``normal-form-preserving'' diffeomorphisms. 
  This work closes certain gaps in the literature on material reference systems in general relativity. It is shown that perfect fluids are a special case of DeWitt's relativistic elastic media and that the velocity--potential formalism for perfect fluids can be interpreted as describing a perfect fluid coupled to a fleet of clocks. A Hamiltonian analysis of the elastic media with clocks is carried out and the constraints that arise when the system is coupled to gravity are studied. When the Hamiltonian constraint is resolved with respect to the clock momentum, the resulting true Hamiltonian is found to be a functional only of the gravitational variables. The true Hamiltonian is explicitly displayed when the medium is dust, and is shown to depend on the detailed construction of the clocks. 
  We further clarify how scalar metric perturbations are amplified in an inflationary cosmology. We first construct a simple, analytic model of an inflationary cosmology in which the expansion scale factor evolves continuously from an inflationary era to a radiation-dominated era. From this model, it is clear to see how scalar perturbations are amplified. Second, we examine the recent claims of Grishchuk, and the reply by Deruelle and Mukhanov, regarding the evolution of scalar perturbations through an abrupt transition in the equation of state of the cosmological fluid. We demonstrate that the ``standard results'' regarding the amplification of scalar, density perturbations from inflation are valid. 
  The dynamics of a class of nonsymmetric gravitational theories is presented in Hamiltonian form. The derivation begins with the first-order action, treating the generalized connection coefficients as the canonical coordinates and the densitised components of the inverse of the fundamental tensor as conjugate momenta. The phase space of the symmetric sector is enlarged compared to the conventional treatments of General Relativity (GR) by a canonical pair that represents the metric density and its conjugate, removable by imposing strongly an associated pair of second class constraints and introducing Dirac brackets. The lapse and shift functions remain undetermined Lagrange multipliers that enforce the diffeomorphism constraints in the standard form of the NGT Hamiltonian. Thus the dimension of the physical constraint surface in the symmetric sector is not enlarged over that of GR. In the antisymmetric sector, all six components of the fundamental tensor contribute conjugate pairs for the massive theory, and the absence of additional constraints gives six configuration space degrees of freedom per spacetime point in the antisymmetric sector. For the original NGT action (or, equivalently, Einstein's Unified Field Theory), the U(1) invariance of the action is shown to remove one of these antisymmetric sector conjugate pairs through an additional first class constraint, leaving five degrees of freedom. The restriction of the dynamics to GR configurations is considered, as well as the form of the surface terms that a rise from the variation of the Hamiltonian. In the resulting Hamiltonian system for the massive theory, singular behavior is found in the relations that determine some of the Lagrange multipliers near GR and certain NGT spacetimes. What this implies about the dynamics of the theory is not clearly understood at 
  We investigate non-linear generalization of Maxwell theory of electromagnetic field keeping the gauge invariance of Lagrangian. New theory, which is standard Yang-Mills theory, is based on Harmonic Oscillator HO(4,R) gauge group. It's a solvable Lie group with nilpotent normal subgroup of codimension 1. We write down the Yang-Mills equation and point out their pecularities and connection with standard Maxwell theory. 
  In the present paper the idea is proposed to solve Maxwell equations for a curved hollow wave conductor by means of effective Riemannian space, in which the lines of motion of fotons are isotropic geodesies for a 4-dimensional space-time. The algorithm of constructing such a metric and curvature tensor components are written down explicitly. The result is in accordance with experiment. 
  The main topic of this talk is the Hawking effect when the black holes in question are undergoing a uniform acceleration. The semiclassical effect of the acceleration is most striking when the Hawking temperature equals the acceleration temperature. Within the usual late-time approximation, the natural black hole vacuum is then equivalent to the asymptotically empty Minkowskian vacuum, and the accelerated black hole becomes semiclassically stable against the familiar thermal evaporation. An important application of this phenomenon is found in the problem of charged black hole pair-creation. 
  A recent paper [M. Kamata and T. Koikawa, Phys. Lett. {\bf B353} (1995) 196.] claimed to obtain the charged version of the $(2+1)$-dimensional spinning $BTZ$ black hole solution by assuming a (anti-) self dual condition imposed on the electric and magnetic fields. We point out that the angular momentum and mass diverge at spatial infinity and as a consequence the solution is unphysical 
  An axially symmetric exact solution of the Einstein-Maxwell equations is obtained and is interpreted to give the gravitational and electromagnetic fields of a charged tachyon. Switching off the charge parameter yields the solution for the uncharged tachyon which was earlier obtained by Vaidya. The null surfaces for the charged tachyon are discussed. 
  Recently Nathan Rosen and the present author obtained the energy and momentum densities of cylindrical gravitational waves in Einstein's prescription and found them to be finite and reasonable. In the present paper we calculate the same in prescriptions of Tolman as well as Landau and Lifshitz and discuss the results. 
  We study the problem of test-particle motion in the Nonsymmetric Gravitational Theory (NGT) assuming the four-velocity of the particle is parallel-transported along the trajectory. The predicted motion is studied on a static, spherically symmetric background field, with particular attention paid to radial and circular motions. Interestingly, it is found that the proper time taken to travel between any two non-zero radial positions is finite. It is also found that circular orbits can be supported at lower radii than in General Relativity for certain forms of motion.  We present three interactions which could be used as alternate methods for coupling a test-particle to the antisymmetric components of the NGT field. One of these takes the form of a Yukawa force in the weak-field limit of a static, spherically symmetric field, which could lead to interesting phenomenology. 
  Thermal Wightman functions of a massless scalar field are studied within the framework of a ``near horizon'' static background model of an extremal R-N black hole. This model is built up by using global Carter-like coordinates over an infinite set of Bertotti-Robinson submanifolds glued together. The analytical extendibility beyond the horizon is imposed as constraints on (thermal) Wightman's functions defined on a Bertotti-Robinson sub manifold. It turns out that only the Bertotti-Robinson vacuum state, i.e. $T=0$, satisfies the above requirement. Furthermore the extension of this state onto the whole manifold is proved to coincide exactly with the vacuum state in the global Carter-like coordinates. Hence a theorem similar to Bisognano-Wichmann theorem for the Minkowski space-time in terms of Wightman functions holds with vanishing ``Unruh-Rindler temperature''. Furtermore, the Carter-like vacuum restricted to a Bertotti-Robinson region, resulting a pure state there, has vanishing entropy despite of the presence of event horizons. Some comments on the real extreme R-N black hole are given. 
  The ``evolving constants'' method of defining the quantum dynamics of time-reparametrization-invariant theories is investigated for a particular implementation of parametrized non-relativistic quantum mechanics (PNRQM). The wide range of time functions that are available to define evolving constants raises issues of interpretation, consistency, and the degree to which the resulting quantum theory coincides with, or generalizes, the usual non-relativistic theory. The allowed time functions must be restricted for the predictions of PNRQM to coincide with those of usual quantum theory. They must be restricted to have a notion of quantum evolution in a time-parameter connected to spacetime geometry. They must be restricted to prevent the theory from making inconsistent predictions for the probabilities of histories. Suitable restrictions can be introduced in PNRQM but these seem unlikely to apply to a reparametrization invariant theory like general relativity. 
  The quantum theory of a spatially flat Friedmann-Robertson-Walker universe with a massless scalar field as source is further investigated. The classical model is singular, and in the framework of the Arnowitt-Deser-Misner canonical quantization formalism a discussion is made of the cosmic evolution, particularly of the quantum gravitational collapse problem. It is shown that in a matter-time gauge such that time is identified with the scalar field the classical model is singular either at $t=-\infty$ or at $t=+\infty$, but the quantum model is nonsingular. The latter behavior disproves a conjecture according to which quantum cosmological singularities are predetermined on the classical level by the choice of time. 
  We derive the topological obstructions to the existence of non-Cliffordian pin structures on four-dimensional spacetimes. We apply these obstructions to the study of non-Cliffordian pin-Lorentz cobordism. We note that our method of derivation applies equally well in any dimension and in any signature, and we present a general format for calculating obstructions in these situations. Finally, we interpret the breakdown of pin structure and discuss the relevance of this to aspects of physics. 
  In this paper we discuss the quantum potential approach of Bohm in the context of quantum cosmological model. This approach makes it possible to convert the wavefunction of the universe to a set of equations describing the time evolution of the universe. Following Ashtekar et.\ al., we make use of quantum canonical transformation to cast a class of quantum cosmological models to a simple form in which they can be solved explicitly, and then we use the solutions do recover the time evolution. 
  We use the polygon representation of 2+1--dimensional gravity to explicitly carry out the canonical quantization of a universe with the topology of a torus. The mapping-class-invariant wave function for a quantum ''big bounce'', is reminiscent of the interference patterns of linear gratings. We consider the ``problem of time'' of quantum gravity: for one choice of internal time the universe recovers a semiclassical interpretation after the bounce, with a wave packet centered at a single geometry; for another choice of internal time, the quantum solutions involve interference between macroscopically distinct universes. 
  We describe how the six planned detectors (2 LIGOs, VIRGO, GEO, AIGO, TAMA) can be used to perform coincidence experiments for the detection of broadband signals from either coalescing compact binaries or burst sources. We make comparisons of the achievable sensitivities of these detectors under different optical configurations and find that a meaningful coincidence experiment for the detection of coalescing binary signals can only be performed by a network where the LIGOs and VIRGO are operated in power recycling mode and other medium scale detectors are operated in dual recycling mode. For the model of burst waveform considered by us (i.e. uniform power upto 2000Hz), we find that the relative sensitivity of the power-recycled VIRGO is quite high as compared to others with their present design parameters and thus coincidence experiment performed by including VIRGO in the network would not be a meaningful one. We also calculate optimized values for the time-delay window sizes for different possible networks. The effect of filtering on the calculation of thresholds has also been discussed. We set the thresholds for different detectors and find out the volume of sky that can be covered by different possible networks and the corresponding rate of detection of coalescing binaries in the beginning of the next century. We note that a coincidence experiment of power-recycled LIGOs and VIRGO and dual-recycled GEO and AIGO can increase the volume of the sky covered by 3.2 times as compared with only the power-recycled LIGO detectors and by 1.7 times the sky covered by the power-recycled LIGO-VIRGO network. These values are far less than the range that can be covered by only the LIGO-VIRGO network with dual recycling operation at a later stage, but the accuracy in the determination of direction, distance and other source parameters will be much 
  It has been shown that the extension of the elasticity theory in more than three dimensions allows a description of space-time as a properly stressed medium, even recovering the Minkowski metric in the case of uniaxial stress. The fundamental equation for the metric in the theory is shown to be the equilibrium equation for the medium. Examples of spherical and cylindrical symmetries in four dimensions are considered, evidencing convergencies and divergencies with the classical general relativity theory. Finally the possible meaning of the dynamics of the four dimensional elastic medium is discussed. 
  In a class of generalized Einstein's gravity theories we derive the equations and general asymptotic solutions describing the evolution of the perturbed universe in unified forms. Our gravity theory considers general couplings between the scalar field and the scalar curvature in the Lagrangian, thus includes broad classes of generalized gravity theories resulting from recent attempts for the unification. We analyze both the scalar-type mode and the gravitational wave in analogous ways. For both modes the large scale evolutions are characterized by the same conserved quantities which are valid in the Einstein's gravity. This unified and simple treatment is possible due to our proper choice of the gauges, or equivalently gauge invariant combinations. 
  The thermal properties of black holes in the presence of quantum fields can be revealed through solutions of the semi-classical Einstein equation. We present a brief but self-contained review of the main features of the semi-classical back reaction problem for a black hole in the microcanonical ensemble. The solutions, obtained for conformal scalars, massless spinors and U(1) gauge bosons, are used to calculate the $O(\hbar)$ corrections to the temperature and thermodynamical entropy of a Schwarzschild black hole. In each spin case considered, the entropy corrections $\Delta S(r)$, are positive definite and monotone increasing with increasing distance $r$ from the hole, and are of the same order as the naive flat space radiation entropy. 
  We investigate the causal structure of spacetimes $(M, g)$ for which the metric $g$ is singular on a set of points. 
  The classical and quantum dynamics of simple time-reparametrization- invariant models containing two degrees of freedom are studied in detail. Elimination of one ``clock'' variable through the Hamiltonian constraint leads to a description of time evolution for the remaining variable which is essentially equivalent to the standard quantum mechanics of an unconstrained system. In contrast to a similar proposal of Rovelli, evolution is with respect to the geometrical proper time, and the Heisenberg equation of motion is exact. The possibility of a ``test clock'', which would reveal time evolution while contributing negligibly to the Hamiltonian constraint is examined, and found to be viable in the semiclassical limit of large quantum numbers. 
  Recently, there have been several applications of differential and algebraic topology to problems concerned with the global structure of spacetimes. In this paper, we derive obstructions to the existence of spin-Lorentz and pin-Lorentz cobordisms and we show that for compact spacetimes with non-empty boundary there is no relationship between the homotopy type of the Lorentz metric and the causal structure. We also point out that spin-Lorentz and tetrad cobordism are equivalent. Furthermore, because the original work [7] on metric homotopy and causality may not be known to a wide audience, we present an overview of the results here. 

  When it comes to performing thought experiments with black holes, Einstein-Bohr like discussions have to be re-opened. For instance one can ask what happens to the quantum state of a black hole when the wave function of a single ingoing particle is replaced by an other one that is orthogonal to the first, while keeping the total energy and momentum unaffected. Observers at $t\rightarrow\infty$ will not notice any difference, or so it seems in certain calculational schemes.   If one argues that this cannot be correct for the complete theory because a black hole should behave in accordance with conventional quantum mechanics, implying a unitary evolution, one is forced to believe that local quantum field theory near the black hole horizon is very different from what had hitherto been accepted. This would give us very valuable information concerning physics in the Planck length region, notably a mathematical structure very close to that of super string theory, but it does lead to conceptual difficulties.   An approach that is somewhat related to this is to suspect a breakdown of General Relativity for quantum mechanical systems. It is to some extent unavoidable that Hilbert space is not invariant under general coordinate transformations because such transformations add and remove some states. Finally the cosmological constant problem also suggests that flat space-time has some special significance in a quantum theory. We suggest that a new causality principle could lead to further clues on how to handle this problem. 
  We suggest a method of construction of general diffeomorphism invariant boundary conditions for metric fluctuations. The case of $d+1$ dimensional Euclidean disk is studied in detail. The eigenvalue problem for the Laplace operator on metric perturbations is reduced to that on $d$-dimensional vector, tensor and scalar fields. Explicit form of the eigenfunctions of the Laplace operator is derived. We also study restrictions on boundary conditions which are imposed by hermiticity of the Laplace operator. 
  For a contravariant 4-metric which changes signature from Lorentzian to Riemannian across a spatial hypersurface, the mixed Einstein tensor is manifestly non-singular. In Gaussian normal coordinates, the metric contains a step function and the Einstein tensor contains the Dirac delta function with support at the junction. The coefficient of the Dirac function is a linear combination of the second fundamental form (extrinsic curvature) of the junction. Thus, unless the junction has vanishing extrinsic curvature, the physical interpretation of the metric is that it describes a layer of matter (with stresses but no energy or momentum) at the junction. In particular, such metrics do not satisfy the vacuum Einstein equations, nor the Einstein-Klein-Gordon equations and so on. Similarly, the d'Alembertian of a Klein-Gordon field contains the Dirac function with coefficient given by the momentum of the field. Thus, if the momentum of the field does not vanish at the junction, the physical interpretation is that there is a source (with step potential) at the junction. In particular, such fields do not satisfy the massless Klein-Gordon equation. These facts contradict claims in the literature. 
  Cosmological perturbations on a manifold admitting signature change are studied. The background solution consists in a Friedmann-Lemaitre-Robertson- Walker (FLRW) Universe filled by a constant scalar field playing the role of a cosmological constant. It is shown that no regular solution exist satisfying the junction conditions at the surface of change. The comparison with similar studies in quantum cosmology is made. 
  We introduce a condition for the strong decoherence of a set of alternative histories of a closed quantum-mechanical system such as the universe. The condition applies, for a pure initial state, to sets of homogeneous histories that are chains of projections, generally branch-dependent. Strong decoherence implies the consistency of probability sum rules but not every set of consistent or even medium decoherent histories is strongly decoherent. Two conditions characterize a strongly decoherent set of histories: (1) At any time the operators that effectively commute with generalized records of history up to that moment provide the pool from which -- with suitable adjustment for elapsed time -- the chains of projections extending history to the future may be drawn. (2) Under the adjustment process, generalized record operators acting on the initial state of the universe are approximately unchanged. This expresses the permanence of generalized records. The strong decoherence conditions (1) and (2) guarantee what we call ``permanence of the past'' -- in particular the continued decoherence of past alternatives as the chains of projections are extended into the future. Strong decoherence is an idealization capturing in a general way this and other aspects of realistic physical mechanisms that destroy interference, as we illustrate in a simple model. We discuss the connection between the reduced density matrices that have often been used to characterize mechanisms of decoherence and the more general notion of strong decoherence. The relation between strong decoherence and a measure of classicality is briefly described. 
  The Hamiltonian structure of spacetimes with two commuting Killing vector fields is analyzed for the purpose of addressing the various problems of time that arise in canonical gravity. Two specific models are considered: (i) cylindrically symmetric spacetimes, and (ii) toroidally symmetric spacetimes, which respectively involve open and closed universe boundary conditions. For each model canonical variables which can be used to identify points of space and instants of time, {\it i.e.}, internally defined spacetime coordinates, are identified. To do this it is necessary to extend the usual ADM phase space by a finite number of degrees of freedom. Canonical transformations are exhibited that identify each of these models with harmonic maps in the parametrized field theory formalism. The identifications made between the gravitational models and harmonic map field theories are completely gauge invariant, that is, no coordinate conditions are needed. The degree to which the problems of time are resolved in these models is discussed. 
  The renormalized stress-energy tensor $\langle T_{\mu\nu}\rangle$ of the quantized complex massless scalar field which obeys the automorphic condition in Misner space is obtained. It is shown that there exists the special value of the automorphic parameter for which $\langle T_{\mu\nu}\rangle$ is regular on the chronology horizon and, so, can not act as a protector of chronology through a back reaction on a spacetime metric. However, it is shown that, at the same time, the value of field square $\langle\phi^2\rangle$, which characterizes the quantum field fluctuations, is divergent on the chronology horizon. The assumption is suggested that the infinitely growing quantum field fluctuations, which appear if a (self)interaction of the scalar field is taken into account, would prevent the chronology horizon formation. 
  We review the mathematically rigorous formulation of the quantum theory of a linear field propagating in a globally hyperbolic spacetime. This formulation is accomplished via the algebraic approach, which, in essence, simultaneously admits all states in all possible (unitarily inequivalent) Hilbert space constructions. The physically nonsingular states are restricted by the requirement that their two-point function satisfy the Hadamard condition, which insures that the ultra-violet behavior of the state be similar to that of the vacuum state in Minkowski spacetime, and that the expected stress-energy tensor in the state be finite. We briefly review the Unruh and Hawking effects from the perspective of the theoretical framework adopted here. A brief discussion also is given of several open issues and questions in quantum field theory in curved spacetime regarding the treatment of ``back-reaction", the validity of some version of the ``averaged null energy condition'', and the formulation and properties of quantum field theory in causality violating spacetimes. 
  Quadratic theory of gravity is a complicated constraint system. We investigate some consequences of treating quadratic terms perturbatively (higher derivative version of backreaction effects). This approach is shown to overcome some well known problems associated with higher derivative theories, i.e., the physical gravitational degree of freedom remains unchanged from those of Einstein gravity.  Using such an interpretation of $R + \beta R^2$ gravity, we investigate a classical and Wheeler DeWitt evolution of $R + \beta R^2$ gravity for a particular sign of $\beta$, corresponding to non- tachyon case. Matter is described by a phenomenological $\rho \propto a(t)^{-n}$. It is concluded that both the Friedmann potential $U(a)$ ($ {\dot a}^2 + 2U(a) = 0 $) and the Wheeler DeWitt potential $W(a)$ ($\left[-{\partial^2\over \partial a^2} + 2W(a)\right]\psi (a) =0 $) develop repulsive barriers near $a\approx 0$ for $n>4$ (i.e., $ p > {1\over 3}\rho $). The interpretations is clear. Repulsive barrier in $U(a)$ implies that a contracting FRW universe ($k>0, k=0, k<0$) will bounce to an expansion phase without a total gravitational collapse. Repulsive barrier in $W(a)$ means that $a \approx 0$ is a classically forbidden region. Therefore, probability of finding a universe with the big bang singularity ($a=0$ ) is exponentially suppressed. 
  The effective action for gravity at high curvatures is likely to contain higher derivative terms. These corrections may have profound consequences for the singularity structure of space-time and for early Universe cosmology. In this contribution, recent work is reviewed which demonstrates that it is possible to construct a class of effective gravitational actions for which all solutions with sufficient symmetries have limited curvature and are nonsingular. Near the limiting curvature, the coupling between matter and gravity goes to zero and in this sense the theory is asymptotically free. 
  We critically reexamine the gravitational scattering of scalar particles on a global monopole studied recently. The original investigation of Mazur and Papavassiliou is extended by considering different couplings of the scalar field to the space-time curvature and by varying the dimension of the space-time where the Klein-Gordon (KG) field lives. A universal behavior of the leading term in the scattering amplitude as a function of this dimension is revealed. 
  An attempt is made to promote an analogy between quantized vortex in condensed matter and black hole: both compact objects have fermion zero modes which induce the finite temperature of these objects. The motion of the quantized vortex lines in Fermi superfluids and superconductors leads to the spectral flow of fermion zero mode. This results in finite temperature and entropy of the moving vortex. The tunneling transition rate between the fermionic levels under the influence of the vortex motion suggests the effective temperature of the vortex core $T_{\rm eff}=(2/\pi) p_Fv_L$, where $v_L$ is the velocity of the vortex with respect to the heat bath reference frame and $p_F$ the Fermi momentum. This is an analogue of the Unruh temperature of the accelerating object in the relativistic system. For the vortex ring with the radius $R$ this leads to the Hawking type temperature $T_{\rm vortex~ring}=(\hbar v_F / 2\pi R) \ln (R/r_c)$, where $v_F$ is the Fermi velocity and $r_c$ is the radius of the vortex core. The corresponding "Hawking" entropy of the vortex ring of radius $R$ and area $A=\pi R^2$ appears to be ${\cal S}_{\rm vortex ~ring}=(1/6) A p_F^2$. Similar expression but with different numerical factor is obtained for the instanton action for the quantum nucleation of the vortex loop from the homogeneous vacuum, and also for the "Bekenstein" entropy obtained by counting the number of fermionic bound states which appear when the vortex loop is created. For the superfluid $^3$He-A, where some components of the order parameter play the part of the gravitational field, the Fermi momentum $p_F$ corresponds to the Planck scale for this effective gravity. The effective action for the gravity field is obtained after integration over the fermion fields in correspondence with the Sakharov 
  We analyze the statistical mechanical properties of n-detectors in arbitrary states of motion interacting with each other via a quantum field. We use the open system concept and the influence functional method to calculate the influence of quantum fields on detectors in motion, and the mutual influence of detectors via fields. We discuss the difference between self and mutual impedance and advanced and retarded noise. The mutual effects of detectors on each other can be studied from the Langevin equations derived from the influence functional, as it contains the backreaction of the field on the system self-consistently. We show the existence of general fluctuation- dissipation relations, and for trajectories without event horizons, correlation-propagation relations, which succinctly encapsulate these quantum statistical phenomena. These findings serve to clarify some existing confusions in the accelerated detector problem. The general methodology presented here could also serve as a platform to explore the quantum statistical properties of particles and fields, with practical applications in atomic and optical physics problems. 
  The Euclidean action is calculated in the thin-wall approximation for a first-order vacuum phase transition in which the bubble appears symmetrically around either a global monopole or a gauge cosmic string. The bubble is assumed to be much larger than the core size of the monopole or string. In both cases the value of the Euclidean action is shown to be reduced below the $O(4)$ symmetric action value, indicating that the topological defects act as effective nucleation sites for vacuum decay. 
  We present a counter example to a theorem given by Amir {\em et al.} J. Math. Phys. {\bf 35}, 3005 (1994). We also comment on a misleading statements of the same reference. 
  We consider an $N$-body system of charged particle coupled to gravitational, electromagnetic, and scalar fields. The metric on moduli space for the system can be considered if a relation among the charges and mass is satisfied, which includes the BPS relation for monopoles and the extreme condition for charged black holes. Using the metric on moduli space in the long distance approximation, we study the statistical mechanics of the charged particles at low velocities. The partition function is evaluated as the leading order of the large $d$ expansion, where $d$ is the spatial dimension of the system and will be substituted finally as $d=3$. 
  In this lecture we address some topological questions connected with the existence on a general spacetime manifold of diffeomorphisms connected to the identity which reverse the time-orientation. 
  The basic properties of Poincare gauge invariant Hilbert bundles over Lorentzian manifolds are derived. Quantum connections are introduced in such bundles, which govern a parallel transport that is shown to satisfy the strong equivalence principle in the quantum regime. Path-integral expressions are presented for boson propagators in Hilbert bundles over globally hyperbolic curved spacetimes. Their Poincare gauge covariance is proven, and their special relativistic limit is examined. A method for explicitly computing such propagators is presented for the case of cosmological models with Robertson-Walker metric. 
  In this paper, we study Raychaudhuri's equation in the background of $R + \beta R^2$ gravity with a phenomenological matter ($\rho \propto a(t)^{-n}$). We conclude that even though the Strong Energy Condition (S.E.C.) for Einstein's gravity, which guarantees singularity, is $n\geq 2$ for $\rho \propto a(t)^{-n}$, a perturbative analysis of Raychaudhuri's equation in the background of $R + \beta R^2$ gravity reveals that the big bang singularity may not be guaranteed for $n > 4$. We derive the following Strong Energy Conditions for $R + \beta R^2$ ($\beta \not= 0$):  1) For $k<0$ FRW metric, S.E.C. is ($0\leq n\leq 4$)     i.e., $-\rho_n \leq p_n \leq {1\over 3}\rho_n$.  2) For $k=0$ FRW metric, S.E.C. is ( $1\leq n\leq 4$)      i.e., $-{2\over 3}\rho_n \leq p_n \leq {1\over 3}\rho_n$.  3) For $k>0$ FRW metric, S.E.C. is ($2\leq n\leq 4$)      i.e., $-{1\over 3}\rho_n \leq p_n \leq {1\over 3}\rho_n$. 
  Recent results on solutions of the Einstein equations with matter are surveyed and a number of open questions are stated. The first group of results presented concern asymptotically flat spacetimes, both stationary and dynamical. Then there is a discussion of solutions of the equations describing matter in special relativity and Newtonian gravitational theory and their relevance for general relativity. Next spatially compact solutions of the Einstein-matter equations are presented. Finally some remarks are made on the methods which have been used, and could be used in the future, to study solutions of the Einstein equations with matter. 
  As an application of Gradient Expansion (long-wavelength) approximation, we studied the inhomogeneous universe including the gravitational wave(GW). For a plane-symmetric cosmological model, we could implement the 2nd order expansion and analyze the purely non-linear perturbation of GW. The result indicates that the non-linear effect plays an important role in anisotropic stage of the universe. We also confirm that the perturbation from the linear source term in gradient expansion is consistent with the result of linear perturbation. 
  The radiation from extreme Reissner-Nordstr\"{o}m black holes is computed by explicitly considering the collapse of a spherical charged shell. No neutral scalar radiation is found but there is emission of charged particles, provided the charge to mass ratio be different from one. The absence of thermal effects is in accord with the predictions of the euclidean theory but since the body emits charged particles the entropy issue is not the same as for eternal extreme black holes. 
  Black hole solutions in the context of a generic matter-coupled two-dimensional dilaton gravity theory are discussed both at the classical and semiclassical level. Starting from general assumptions, a criterion for the existence of black holes is given. The relationship between conformal anomaly and Hawking radiation is extended to a broad class of two-dimensional dilaton gravity models. A general and simple formula relating the magnitude of the Hawking effect to the dilaton potential evaluated on the horizon is derived. 
  The principal properties of geodesic normal coordinates are the vanishing of the connection components and first derivatives of the metric components at some point. It is well-known that these hold only at points where the connection has vanishing torsion and non-metricity. However, it is shown that normal frames, possessing the essential features of normal coordinates, can still be constructed when the connection is non-Riemannian. 
  We have carried out 3-D numerical simulations of the dynamical bar instability in a rotating star and the resulting gravitational radiation using both an Eulerian code written in cylindrical coordinates and a smooth particle hydrodynamics (SPH) code. The star is modeled initially as a polytrope with index $n = 3/2$ and $T_{\rm rot}/|W| \approx 0.30$, where $T_{\rm rot}$ is the rotational kinetic energy and $|W|$ is the gravitational potential energy. In both codes the gravitational field is purely Newtonian, and the gravitational radiation is calculated in the quadrupole approximation.  We have run 3 simulations with the Eulerian code, varying the number of angular zones and the treatment of the boundary between the star and the vacuum. Using the SPH code we did 7 runs, varying the number of particles, the artificial viscosity, and the type of initial model. We compare the growth rate and rotation speed of the bar, the mass and angular momentum distributions, and the gravitational radiation quantities. We highlight the successes and difficulties of both methods, and make suggestions for future improvements. 
  We study Killing vector fields in asymptotically flat space-times. We prove the following result, implicitly assumed in the uniqueness theory of stationary black holes. If the conditions of the rigidity part of the positive energy theorem are met, then in such space-times there are no asymptotically null Killing vector fields except if the initial data set can be embedded in Minkowski space-time. We also give a proof of the non-existence of non-singular (in an appropriate sense) asymptotically flat space-times which satisfy an energy condition and which have a null ADM four-momentum, under conditions weaker than previously considered. 
  The Hessling improvement of the Haag, Narnhofer and Stein principle is analysed in the case of a massless scalar field propagating outside of an extremal R-N black hole. It is found that this sort of ``Quantum (Einstein's) Equivalence Principle'' selects only the R-N vacuum as a physically sensible state, i.e., it selects the temperature $T=0$ only. 
  It is shown that spatially flat, isotropic cosmologies derived from the Brans--Dicke gravity action exhibit a scale factor duality invariance. This classical duality is then associated with a hidden $N=2$ supersymmetry at the quantum level and the supersymmetric quantum constraints are solved exactly. These symmetries also apply to a dimensionally reduced version of vacuum Einstein gravity. 
  We simplify the warp drive space-time so that it becomes stationary and the distorsion becomes one-dimensional and static. We use this simplified warp drive space-time as a background for a photon field. We shall especially use the Drummond\&Hathrell action in order to investigate the velocity effects on photons in this background. Finally, we discuss the limitations of this model. 
  A discretized version of canonical quantum gravity proposed by Loll is investigated. After slightly modifying Loll's discretized Hamiltonian constraint, we encode its action on the spin network states in terms of combinatorial topological manipulations of the lattice loops. Using this topological formulation we find new solutions to the discretized Wheeler-Dewitt equation. These solutions have their support on the connected set of plaquettes. We also show that these solutions are not normalizable with respect to the induced heat-kernel measure on $SL(2,{\bf C})$ gauge theories. 
  The quantization of closed cosmologies makes it necessary to study squared Dirac operators on closed intervals and the corresponding quantum amplitudes. This paper proves self-adjointness of these second-order elliptic operators. 
  We discuss the two-dimensional dilaton gravity with a scalar field as the source matter. The coupling between the gravity and the scalar, massless, field is presented in an unusual form. We work out two examples of these couplings, and solutions with black-hole behaviour are discussed and compared with those found in the literature. 
  We investigate the group of large diffeomorphisms fixing a frame at a point for general closed 3-manifolds. We derive some general structural properties of these groups which relate to the picture of the manifold as being composed of extended `objects' (geons). 
  The phase space reduction of Schwarzschild black holes by Thiemann and Kastrup and by Kucha\v{r} is reexamined from a different perspective on gauge freedom. This perspective introduces additional gauge transformations which correspond to asymptotically nontrivial diffeomorphisms. Various subtleties concerning variational principles for asymptotically flat systems are addressed which allow us to avoid the usual conclusion that treating such transformations as gauge implies the vanishing of corresponding total charges. Instead, superselection rules are found for the (nonvanishing) ADM mass at the asymptotic boundaries. The addition of phenomenological clocks at each asymptotic boundary is also studied and compared with the `parametrization clocks' of Kucha\v{r}. 
  The issue concerning the existence of wormhole states in locally supersymmetric minisuperspace models with matter is addressed. Wormhole states are apparently absent in models obtained from the more general theory of N=1 supergravity with supermatter. A Hartle-Hawking type solution can be found, even though some terms (which are scalar field dependent) cannot be determined in a satisfactory way. A possible cause is investigated here. As far as the wormhole situation is concerned, we argue here that the type of Lagrange multipliers and fermionic derivative ordering used can make a difference. A proposal is made for supersymmetric quantum wormholes to also be invested with a Hilbert space structure, associated with a maximal analytical extension of the corresponding minisuperspace.is concerned, we argue here that the type of Lagrange multipliers and fermionic derivative ordering used can make a difference. A proposal is made for supersymmetric quantum wormholes to also be invested with a Hilbert space structure, associated with a maximal analytical extension of the corresponding minisuperspace. 
  We generate from the static charged BTZ black hole a family of spinning charged solutions to the Einstein-Maxwell equations in 2+1 dimensions. These solutions go over, in a suitable limit, to self-dual spinning charged solutions, which are horizonless and regular, with logarithmically divergent mass and spin. To cure this divergence, we add a topological Chern-Simons term to the gauge field action. The resulting self-dual solution is horizonless, regular, and asymptotic to the extreme BTZ black hole. 
  We show that particle production during the expansion of bubbles of true vacuum in the sea of false vacuum is possible and calculate the resulting rate. As a result the nucleated bubbles cannot expand due to the transfer of false vacuum energy to the created particles inside the bubbles. Therefore all the inflationary models dealing with the nucleation and expansion of the bubbles (including extended inflation) may not be viable. 
  We have studied various classical solutions in $R^2$ cosmology. Especially we have obtained general classical solutions in pure $R^2$\ cosmology. Even in the quantum theory, we can solve the Wheeler-DeWitt equation in pure $R^2$\ cosmology exactly. Comparing these classical and quantum solutions in $R^2$\ cosmology, we have studied the problem of time in general relativity. 
  We study the real, massive Klein-Gordon field on a $C^\infty$ globally-hyperbolic background space-time with compact Cauchy hypersurfaces. In particular, the parametrization of this system as initiated by Dirac and Kucha\v{r} is put on a rigorous basis. The discussion is focussed on the structure of the set of spacelike embeddings of the Cauchy manifold into the space-time, and on the associated $e$-tensor density bundles and their tangent and cotangent bundles. The dynamics of the field is expressed as a set of automorphisms of the space of initial data in which each pair of embeddings defines one such automorphism. Using these results, the extended phase space of the system is shown to be a weak-symplectic manifold, and the Kucha\v{r} constraint is shown to define a smooth constraint submanifold which is foliated smoothly by the constraint orbits. The pull-back of the symplectic form to the constraint surface is a presymplectic form which is singular on the tangent spaces to the constraint orbits. Thus, the geometric structure of this infinite-dimensional system is analogous to that of a finite-dimensional, first-class parametrized system, and hence many of the results for the latter can be transferred to the infinite-dimensional case without difficulty. 
  The Earth, Mars, Sun, Jupiter system allows for a sensitive test of the strong equivalence principle (SEP) which is qualitatively different from that provided by Lunar Laser Ranging. Using analytic and numerical methods we demonstrate that Earth-Mars ranging can provide a useful estimate of the SEP parameter $\eta$. Two estimates of the predicted accuracy are derived and quoted, one based on conventional covariance analysis, and another (called ``modified worst case'' analysis) which assumes that systematic errors dominate the experiment. If future Mars missions provide ranging measurements with an accuracy of $\sigma$ meters, after ten years of ranging the expected accuracy for the SEP parameter $\eta$ will be of order $(1-12)\times 10^{-4}\sigma$. These ranging measurements will also provide the most accurate determination of the mass of Jupiter, independent of the SEP effect test. 
  The Ashtekar variables have been use to find a number of exact solutions in quantum gravity and quantum cosmology. We investigate the origin of these solutions in the context of a number of canonical transformations (both complex and real) of the basic Hamiltonian variables of general relativity. We are able to present several new solutions in the minisuperspace (quantum cosmology) sector. The meaning of these solutions is then discussed. 
  We derive the so-called first law of black hole mechanics for variations about stationary black hole solutions to the Einstein--Maxwell equations in the absence of sources. That is, we prove that $\delta M=\kappa\delta A+\omega\delta J+VdQ$ where the black hole parameters $M, \kappa, A, \omega, J, V$ and $Q$ denote mass, surface gravity, horizon area, angular velocity of the horizon, angular momentum, electric potential of the horizon and charge respectively. The unvaried fields are those of a stationary, charged, rotating black hole and the variation is to an arbitrary `nearby' black hole which is not necessarily stationary. Our approach is 4-dimensional in spirit and uses techniques involving Action variations and Noether operators. We show that the above formula holds on any asymptotically flat spatial 3-slice which extends from an arbitrary cross-section of the (future) horizon to spatial infinity.(Thus, the existence of a bifurcation surface is irrelevant to our demonstration. On the other hand, the derivation assumes without proof that the horizon possesses at least one of the following two (related)properties: ($i$) it cannot be destroyed by arbitrarily small perturbations of the metric and other fields which may be present, ($ii$) the expansion of the null geodesic generators of the perturbed horizon goes to zero in the distant future.) 
  Quantum origin of the early inflationary Universe from the no-boundary and tunnelling quantum states is considered in the one-loop approximation of quantum cosmology. A universal effective action algorithm for the distribution function of chaotic inflationary cosmologies is derived for both of these states.The energy scale of inflation is calculated by finding a sharp probability peak in this distribution function for a tunnelling model driven by the inflaton field with large negative constant $\xi$ of nonminimal interaction. The sub-Planckian parameters of this peak (the mean value of the corresponding Hubble constant ${\mbox{\boldmath $H$}}\simeq 10^{-5}m_P$, its quantum width $\Delta{\mbox{\boldmath $H$}}/{\mbox{\boldmath $H$}}\simeq 10^{-5}$ and the number of inflationary e-foldings ${\mbox{\boldmath $N$}}\simeq 60$) are found to be in good correspondence with the observational status of inflation theory, provided the coupling constants of the theory are constrained by a condition which is likely to be enforced by the (quasi) supersymmetric nature of the sub-Planckian particle physics model. 
  The ADM approach to canonical general relativity combined with Dirac's method of quantizing constrained systems leads to the Wheeler-DeWitt equation. A number of mathematical as well as physical difficulties that arise in connection with this equation may be circumvented if one employs a covariant Hamiltonian method in conjunction with a recently developed, mathematically rigorous technique to quantize constrained systems using Rieffel induction. The classical constraints are cleanly separated into four components of a covariant momentum map coming from the diffeomorphism group of spacetime, each of which is linear in the canonical momenta, plus a single finite-dimensional quadratic constraint that arises in any theory, parametrized or not. The new quantization method is carried through in a minisuperspace example, and is found to produce a ``wavefunction of the universe". This differs from the proposals of both Vilenkin and Hartle-Hawking for a closed FRW universe, but happens to coincide with the latter in the open case. 
  The perennial formalism is applied to the real, massive Klein-Gordon field on a globally-hyperbolic background space-time with compact Cauchy hypersurfaces. The parametrized form of this system is taken over from the accompanying paper. Two different algebras ${\cal S}_{\text{can}}$ and ${\cal S}_{\text{loc}}$ of elementary perennials are constructed. The elements of ${\cal S}_{\text{can}}$ correspond to the usual creation and annihilation operators for particle modes of the quantum field theory, whereas those of ${\cal S}_{\text{loc}}$ are the smeared fields. Both are shown to have the structure of a Heisenberg algebra, and the corresponding Heisenberg groups are described. Time evolution is constructed using transversal surfaces and time shifts in the phase space. Important roles are played by the transversal surfaces associated with embeddings of the Cauchy hypersurface in the space-time, and by the time shifts that are generated by space-time isometries. The automorphisms of the algebras generated by this particular type of time shift are calculated explicitly. 
  The purpose of this Comment is to show that the solutions to the zero energy Schr\"odinger equations for monomial central potentials discussed in a recently published Letter, may also be obtained from the corresponding free particle solutions in a straight forwardly way, using an algorithm previously devised by us. New solutions to the zero energy Schr\"odinger equation are also exhibited. 
  We show that in all theories in which black hole hair has been discovered, the region with non-trivial structure of the non-linear matter fields must extend beyond $3/ 2$ the horizon radius, independently of all other parameters present in the theory. We argue that this is a universal lower bound that applies in every theory where hair is present. This {\it no short hair conjecture} is then put forward as a more modest alternative to the now debunked {\it no hair conjecture}. 
  The class of effective actions exactly reproducing the conformal anomaly in 4D is considered. It is demonstrated that the freedom within this class can be fixed by the choice of the conformal gauge. The conformal invariant part of the generic one-loop effective action expanded in the covariant series up to third order in the curvature is rewritten in the new conformal basis. The possible applications of the obtained results are discussed. 
  It is shown how quantum field theory at finite temperature can be used to set up self-consistent and gauge invariant equations for cosmological perturbations sustained by an ultrarelativistic plasma. While in the collisionless case, the results are equivalent to those obtained from the Einstein-Vlasov equations, weak self-interactions in the plasma turn out to require the full machinery of perturbative thermal field theories such as resummation of hard thermal loops. Nevertheless it is still possible to use the same methods that yielded exact solutions in the collisionless case. 
  The so called ''Principle of the self-consistency'' for space-time models with causality violation, which was firstly formulated by I.D.Novikov, is discussed for the test particle motion and for test scalar field. It is shown that the constraints, which provide the self-concistensy of test particle motion have pure geometrical (topological) nature. So, the recent statement that ''The Principle of self-consistensy is a consiquence of the Principle of minimal action'' is wrong. 
  The paper develops a $(2+2)$-imbedding formalism adapted to a double foliation of spacetime by a net of two intersecting families of lightlike hypersurfaces. The formalism is two-dimensionally covariant, and leads to simple, geometrically transparent and tractable expressions for the Einstein field equations and the Einstein-Hilbert action, and it should find a variety of applications. It is applied here to elucidate the structure of the characteristic initial-value problem of general relativity. 
  The aim of this paper is to show, that the 'oscillating universe' is a viable alternative to inflation. We remind that this model provides a natural solution to the flatness or entropy and to the horizon problem of standard cosmology. We study the evolution of density perturbations and determine the power spectrum in a closed universe. The results lead to constraints of how a previous cycle might have looked like. We argue that most of the radiation entropy of the present universe may have originated from gravitational entropy produced in a previous cycle.  We show that measurements of the power spectrum on very large scales could in principle decide whether our universe is closed, flat or open. 
  Three different classes of static solutions of the Einstein--Maxwell equations non--minimally coupled to a dilaton field are presented. The solutions are given in general in terms of two arbitrary harmonic functions and involve among others an arbitrary parameter which determines their applicability as charged black holes, dilaton black holes or strings. Most of the known solutions are contained as special cases and can be non--trivially generalized in different ways. 
  First order perturbations for the fields with spin on the background metric of the gravitational shock waves are discussed. Applying the Newman -- Penrose formalism, exact solutions of the perturbation equations are obtained. For particle physics, this would be one approach to the problem of scattering paticle at Planck energy. 
  The dynamics of perfect fluid spacetime geometries which exhibit {\em Local Rotational Symmetry} (LRS) are reformulated in the language of a $1+\,3$ "threading" decomposition of the spacetime manifold, where covariant fluid and curvature variables are used. This approach presents a neat alternative to the orthonormal frame formalism. The dynamical equations reduce to a set of differential relations between purely scalar quantities. The consistency conditions are worked out in a transparent way. We discuss their various subcases in detail and focus in particular on models with higher symmetries within the class of expanding spatially inhomogeneous LRS models, via a consideration of functional dependencies between the dynamical variables. 
  We discuss the definitions of standard clocks in theories of gravitation. These definitions are motivated by the invariance of actions under different gauge symmetries. We contrast the definition of a standard Weyl clock with that of a clock in general relativity and argue that the historical criticisms of theories based on non-metric compatible connections by Einstein, Pauli and others must be considered in the context of Weyl's original gauge symmetry. We argue that standard Einsteinian clocks can be defined in non-Riemannian theories of gravitation by adopting the Weyl group as a local gauge symmetry that {\it preserves the metric} and discuss the hypothesis that atomic clocks may be adopted to measure proper time in the presence of non-Riemannian gravitational fields. These ideas are illustrated in terms of a recently developed model of gravitation based on a non-Riemannian space-time geometry. 
  Using a recent thermal-field-theory approach to cosmological perturbations, the exact solutions that were found for collisionless ultrarelativistic matter are generalized to include the effects from weak self-interactions in a $\lambda\phi^4$ model through order $\lambda^{3/2}$. This includes the effects of a resummation of thermal masses and associated nonlocal gravitational vertices, thus going far beyond classical kinetic theory. Explicit solutions for all the scalar, vector, and tensor modes are obtained for a radiation-dominated Einstein-de Sitter model containing a weakly interacting scalar plasma with or without the admixture of an independent component of perfect radiation fluid. 
  Starting with a procedure for dealing with general asymptotic behaviors, we construct a quantum theory for asymptotically anti-de Sitter wormholes. We follow both the path integral formalism and the algebraic quantization program proposed by Ashtekar. By adding suitable surface terms, the Euclidean action of the asymptoically anti-de Sitter wormholes can be seen to be finite and gauge invariant. This action determines an appropriate variational problem for wormholes. We also obtain the wormhole wave functions of the gravitational model and show that all the physical states of the quantum theory are superpositions of wormhole states. 
  We consider the Hamiltonian dynamics and thermodynamics of the two-dimensional vacuum dilatonic black hole in the presence of a timelike boundary with a fixed value of the dilaton field. A~canonical transformation, previously developed by Varadarajan and Lau, allows a reduction of the classical dynamics into an unconstrained Hamiltonian system with one canonical pair of degrees of freedom. The reduced theory is quantized, and a partition function of a canonical ensemble is obtained as the trace of the analytically continued time evolution operator. The partition function exists for any values of the dilaton field and the temperature at the boundary, and the heat capacity is always positive. For temperatures higher than $\beta_c^{-1} = \hbar\lambda/(2\pi)$, the partition function is dominated by a classical black hole solution, and the dominant contribution to the entropy is the two-dimensional Bekenstein-Hawking entropy. For temperatures lower than~$\beta_c^{-1}$, the partition function remains well-behaved and the heat capacity is positive in the asymptotically flat space limit, in contrast to the corresponding limit in four-dimensional spherically symmetric Einstein gravity; however, in this limit, the partition function is not dominated by a classical black hole solution. 
  We study the even-parity $\ell=2$ perturbations of a Schwarzschild black hole to second order. The Einstein equations can be reduced to a single linear wave equation with a potential and a source term. The source term is quadratic in terms of the first order perturbations. This provides a formalism to address the validity of many first order calculations of interest in astrophysics. 
  An analysis of the action of the Hamiltonian constraint of quantum gravity on the Kauffman bracket and Jones knot polynomials is proposed. It is explicitely shown that the Kauffman bracket is a formal solution of the Hamiltonian constraint with cosmological constant ($\Lambda$) to third order in $\Lambda$. The calculation is performed in the extended loop representation of quantum gravity. The analysis makes use of the analytical expressions of the knot invariants in terms of the two and three point propagators of the Chern-Simons theory. Some particularities of the extended loop calculus are considered and the implications of the results to the case of the conventional loop representation are discussed. 
  Some time ago it was conjectured that the coefficients of an expansion of the Jones polynomial in terms of the cosmological constant could provide an infinite string of knot invariants that are solutions of the vacuum Hamiltonian constraint of quantum gravity in the loop representation. Here we discuss the status of this conjecture at third order in the cosmological constant. The calculation is performed in the extended loop representation, a generalization of the loop representation. It is shown that the the Hamiltonian does not annihilate the third coefficient of the Jones polynomal ($J_3$) for general extended loops. For ordinary loops the result acquires an interesting geometrical meaning and new possibilities appear for $J_3$ to represent a quantum state of gravity. 
  We present the GRjunction package which allows boundary surfaces and thin-shells in general relativity to be studied with a computer algebra system. Implementing the Darmois-Israel thin shell formalism requires a careful selection of definitions and algorithms to ensure that results are generated in a straight-forward way. We have used the package to correctly reproduce a wide variety of examples from the literature. We present several of these verifications as a means of demonstrating the packages capabilities. We then use GRjunction to perform a new calculation - joining two Kerr solutions with differing masses and angular momenta along a thin shell in the slow rotation limit. 
  We extend the study of the possibility to use the Schwarzschild black hole as a gravitational mirror to the more general case of an uncharged Kerr black hole. We use the null geodesic equation in the equatorial plane to prove a theorem concerning the conditions the impact parameter has to satisfy if there shall exist boomerang photons. We derive an equation for these boomerang photons and an equation for the emission angle. Finally, the radial null geodesic equation is integrated numerically in order to illustrate boomerang photons. 
  In a previous paper \cite{MakingPredictions}, a method of comparing the volumes of thermalized regions in eternally inflating universe was introduced. In this paper, we investigate the dependence of the results obtained through that method on the choice of the time variable and factor ordering in the diffusion equation that describes the evolution of eternally inflating universes. It is shown, both analytically and numerically, that the variation of the results due to factor ordering ambiguity inherent in the model is of the same order as their variation due to the choice of the time variable. Therefore, the results are, within their accuracy, free of the spurious dependence on the time parametrization. 
  A study of kinematics of a 2-body system is used to show that the Mach principle, previously rejected by general relativity, can still serve as an alternative to the concept of absolute space, if one takes into account that the background of distant stars (galaxies) determines {\it both} the inertial and the gravitational masses of a body. 
  Quantum fields propagating on a curved spacetime are investigated in terms of microlocal analysis. We discuss a condition on the wave front set for the corresponding n-point distributions, called ``microlocal spectrum condition'' ($\mu$SC). On Minkowski space, this condition is satisfied as a consequence of the usual spectrum condition. Based on Radzikowski's determination of the wave front set of the two-point function of a free scalar field, satisfying the Hadamard condition in the Kay and Wald sense, we construct in the second part of this paper all Wick polynomials including the energy-momentum tensor for this field as operator valued distributions on the manifold and prove that they satisfy our microlocal spectrum condition. 
  We describe how exactly the intra-cavity fields in a dual recycling cavity build up their power before achieving a steady state value. We restricted our analysis here to interferometers with lossless mirrors and beam-splitter. The complete series representation of the intra-cavity lights at any stage of evolution in non-steady state have been presented. 
  We combine and further develop ideas and techniques of Allen \& Ottewill, Phys. Rev.D, {\bf 42}, 2669 (1990) and Kay \& Studer Commun. Math. Phys., {\bf 139}, 103 (1991) for calculating the long range effects of cosmic string cores on classical and quantum field quantities far from an (infinitely long, straight) cosmic string. We find analytical approximations for (a) the gravity-induced ground state renormalized expectation values of $\hat\varphi^2$ and $\hat T_\mu{}^\nu$ for a non-minimally coupled quantum scalar field far from a cosmic string (b) the classical electrostatic self force on a test charge far from a superconducting cosmic string. Surprisingly -- even at cosmologically large distances -- all these quantities would be very badly approximated by idealizing the string as having zero thickness and imposing regular boundary conditions; instead they are well approximated by suitably fitted strengths of logarithmic divergence at the string core. Our formula for ${\langle {\hat \varphi}^2 \rangle}$ reproduces (with much less effort and much more generality) the earlier numerical results of Allen \& Ottewill. Both ${\langle {\hat \varphi}^2 \rangle}$ and ${\langle {\hat T}_{\mu}{}^{\nu} \rangle}$ turn out to be ``weak field topological invariants'' depending on the details of the string core only through the minimal coupling parameter ``$\xi$'' (and the deficit angle). Our formula for the self-force (leaving aside relatively tiny gravitational corrections) turns out to be attractive: We obtain, for the self-potential of a test charge $Q$ a distance $r$ from a (GUT scale) superconducting string, the formula $- Q^2/(16\epsilon_0r\ln(qr))$ where $q$ is an (in principle, computable) constant of the order of the inverse string radius. 
  Full generalization of Kasner metric for the case of $n+1$ dimensions and $m\le n+1$ essential variables is obtained. Any solution is defined by the corresponding constant matrix of Kasner parameters. This parameters form in euclidian space Casner hyperspheres and are connected by additional conditions. General properties of obtained solutions are analyzed. 
  We show that the splitting feature of the Einstein tensor, as the first term of the Lovelock tensor, into two parts (the Ricci tensor and the term proportional to the curvature scalar) {\tenit with} the trace relation between them is a common feature of {\tenpoint\it any} other homogeneous terms in the Lovelock tensor. Motivated by the principle of general invariance, we find that this property can be generalized, with the aid of a generalized trace operator which we define, for any {\tenit inhomogeneous} Euler--Lagrange expression which can be spanned linearly in terms of homogeneous tensors. As an example, we demonstrate this analogy for the Lovelock tensor. 
  Differential conservation laws in Lagrangian field theory are usually related to symmetries of a Lagrangian density and are obtained if the Lie derivative of a Lagrangian density by a certain class of vector fields on a fiber bundle vanishes. However, only two field models meet this property in fact. In gauge theory of exact internal symmetries, the Lie derivative by vertical vector fields corresponding to gauge transformations is equal to zero. The corresponding N\"oether current is reduced to a superpotential that provides invariance of the N\"oether conservation law under gauge transformations. In the gravitation theory, we meet the phenomenon of "hidden energy". Only the superpotential part of energy-momentum of gravity and matter is observed when the general covariant transformations are exact. Other parts of energy-momentum display themselves if the invariance under general covariance transformations is broken, e.g., by a background world metric. In this case, the Lie derivatives of Lagrangian densities by vector fields which call into play the stress-energy-momentum tensors fail to be equal to zero in general. We base our analysis of differential conservation laws on the canonical decomposition of the Lie derivative of a Lagrangian density $L$ by a projectable vector field on a bundle and with respect to different Lepagian equivalents of $L$. Different Lepagian equivalents lead to conserved quantities which differ from each other in superpotential terms. We have different stress-energy-momentum tensors depending on different lifts of vector fields on a base onto a bundle. Moreover, different solutions of the same Euler-Lagrange equations may require different energy-momentum tensors. We show that different stress-energy-momentum tensors differ from each other in N\"oether currents. As 
  Mechanics is developed over a differentiable manifold as space of possible positions. Time is considered to fill a one--dimensional Riemannian manifold, so having the metric as lapse. Then the system is quantized with covariant instead of partial derivatives in the Schr\"odinger operator. Consequences for quantum cosmology are shortly discussed. 
  A discursive, non-technical, analysis is made of some of the basic issues that arise in almost any approach to quantum gravity, and of how these issues stand in relation to recent developments in the field. Specific topics include the applicability of the conceptual and mathematical structures of both classical general relativity and standard quantum theory. This discussion is preceded by a short history of the last twenty-five years of research in quantum gravity, and concludes with speculations on what a future theory might look like. 
  It is shown that growing-entropy stiff-fluid Kantowski-Sachs universes become time-symmetric (if they start with time-asymmetric phase) and isotropize. Isotropization happens without any inflationary era during the evolution since there is no cosmological term here. It seems that this approach is an alternative to inflation since the universe gets bigger and bigger approaching 'flatness'. 
  We use Boulware's Hamiltonian formalism of quadratic gravity theories in order to analyze the classical behaviour of Bianchi cosmological models for a Lagrangian density containing quadratic terms in the curvature. For this purpose we define a canonical transformation which leads to a clear distinction between two main variants of the quadratic theory, namely the R-squared or conformal Lagrangian densities. In this paper we restrict the study to the first variant. For Bianchi type I and IX models we give the explicit forms of the super-Hamiltonian constraint, of the ADM Hamiltonian density and of the corresponding canonical equations. In the case of a pure R-squared theory we solve these equations analytically for Bianchi I model. For Bianchi type IX model, we reduce the first-order equations of the Hamiltonian system to three coupled second-order equations for the true physical degrees of freedom. This discussion is extended to isotropic FLRW models. 
  We extend to multidimensional cosmology Vilenkin's prescription of tunnelling from nothing for the quantum origin of the observable Universe. Our model consists of a $D+4$-dimensional spacetime of topology ${\cal R}\times {\cal S}^3 \times{\cal S}^D$, with a scalar field (``chaotic inflaton'') for the matter component. Einstein gravity and Casimir compactification are assumed. The resulting minisuperspace is 3--dimensional. Patchwise we find an approximate analytic solution of the Wheeler--DeWitt equation through which we discuss the tunnelling picture and the probability of nucleation of the classical Universe with compactifying extra dimensions. Our conclusion is that the most likely initial conditions, although they do not lead to the compactification of the internal space, still yield (power-law) inflation for the outer space. The scenario is physically acceptable because the inner space growth is limited to $\sim 10^{11}$ in 100 e-foldings of inflation, starting from the Planck scale. 
  Vacuum multidimensional cosmological models with internal spaces being compact $n$-dimensional Lie group manifolds are considered. Products of 3-spheres and $SU(3)$ manifold (a novelty in cosmology) are studied. It turns out that the dynamical evolution of the internal space drives an accelerated expansion of the external world (power law inflation). This generic solution (attractor in a phase space) is determined by the Lie group space without any fine tuning or arbitrary inflaton potentials. Matter in the four dimensions appears in the form of a number of scalar fields representing anisotropic scale factors for the internal space. Along the attractor solution the volume of the internal space grows logarithmically in time. This simple and natural model should be completed by mechanisms terminating the inflationary evolution and transforming the geometric scalar fields into ordinary particles. 
  In this paper we investigate some properties, including causality, of a particular class of relativistic dissipative fluid theories of divergence type. This set is defined as those theories coming from a statistical description of matter, in the sense that the three tensor fields appearing in the theory can be expressed as the three first momenta of a suitable distribution function. In this set of theories the causality condition for the resulting system of hyperbolic partial differential equations is very simple and allow to identify a subclass of manifestly causal theories, which are so for all states outside equilibrium for which the theory preserves this statistical interpretation condition. This subclass includes the usual equilibrium distributions, namely Boltzmann, Bose or Fermi distributions, according to the statistics used, suitably generalized outside equilibrium. Therefore this gives a simple proof that they are causal in a neighborhood of equilibrium. We also find a bigger set of dissipative divergence type theories which are only pseudo-statistical, in the sense that the third rank tensor of the fluid theory has the symmetry and trace properties of a third momentum of an statistical distribution, but the energy-momentum tensor, while having the form of a second momentum distribution, it is so for a different distribution function. This set also contains a subclass (including the one already mentioned) of manifestly causal theories. 
  We present a new class of spinning black hole solutions in $(2+1)$-dimensional general relativity minimally coupled to a dilaton with potential $e^{b\phi}\Lambda$. When $b=4$, the corresponding spinning black hole is a solution of low energy $(2+1)$-dimensional string gravity. Apart from the limiting case of the $BTZ$ black hole, these spinning black holes have no inner horizon and a curvature singularity only at the origin. We compute the mass and angular momentum parameters of the solutions at spatial infinity, as well as their temperature and entropy. 
  We consider the possibility of using measurements of anomalous magnetic moments of elementary particles as a possible test of the Einstein Equivalence Principle (EEP). For the class non-metric theories of gravity described by the \tmu formalism we find several novel mechanisms for breaking the EEP, and discuss the possibilities of setting new empirical constraints on such effects. 
  Recently a bound on negative energy densities in four-dimensional Minkowski spacetime was derived for a minimally coupled, quantized, massless, scalar field in an arbitrary quantum state. The bound has the form of an uncertainty principle-type constraint on the magnitude and duration of the negative energy density seen by a timelike geodesic observer. When spacetime is curved and/or has boundaries, we argue that the bound should hold in regions small compared to the minimum local characteristic radius of curvature or the distance to any boundaries, since spacetime can be considered approximately Minkowski on these scales. We apply the bound to the stress-energy of static traversable wormhole spacetimes. Our analysis implies that either the wormhole must be only a little larger than Planck size or that there is a large discrepancy in the length scales which characterize the wormhole. In the latter case, the negative energy must typically be concentrated in a thin band many orders of magnitude smaller than the throat size. These results would seem to make the existence of macroscopic traversable wormholes very improbable. 
  This work presents a brief discussion and a plan towards the analytical solving of Partial Differential Equations (PDEs) using symbolic computing, as well as an implementation of part of this plan as the PDEtools software-package of commands. 
  The geometro-stochastic method of quantization provides a framework for quantum general relativity, in which the principal frame bundles of local Lorentz frames that underlie the fibre-theoretical approach to classical general relativity are replaced by Poincar\'e-covariant quantum frame bundles. In the semiclassical regime for quantum field theory in curved spacetime, where the gravitational field is not quantized, the elements of these local quantum frames are generalized coherent states, which emerge naturally from phase space representations of the Poincar\'e group. Due to their informational completeness, these quantum frames are capable of taking over the role played by complete sets of observables in conventional quantum theory. The propagation of quantum-geometric fields proceeds by path integral methods, based on parallel transport along broken paths consisting of arcs of geodesics of the Levi-Civita connection. The formulation of quantum gravity within this framework necessitates the transition to quantum superframe bundles and a quantum gravitational supergroup capable of incorporating diffeomorphism invariance into the framework. This results in a geometric version of quantum gravity which shares some conceptual features with covariant as well as with canonical gravity, but which avoids the foundational and the mathematical difficulties encountered by these two approaches. 
  We give circularly symmetric solutions for null fluid collapse in 2+1-dimensional Einstein gravity with a cosmological constant. The fluid pressure $P$ and energy density $\rho$ are related by $P=k\rho$ $(k\le 1)$. The long time limit of the solutions are black holes whose horizon structures depend on the value of $k$. The $k=1$ solution is the Banados-Teitelboim-Zanelli black hole metric in the long time static limit, while the $k<1$ solutions give other, `hairy' black hole metrics in this limit. 
  We investigate the effects of the gravitational field on the quantum dynamics of non-relativistic particles. We consider N non-relativistic particles, interacting with the linearized gravitational field. Using the Feynman - Vernon influence functional technique, we trace out the graviton field, to obtain a master equation for the system of particles to first order in $G$. The effective interaction between the particles, as well as the self-interaction is non-local in time and in general non-markovian. We show that the gravitational self-interaction cannot be held responsible for decoherence of microscopic particles due to the fast vanishing of the diffusion function. For macroscopic particles though, it leads to diagonalization to the energy eigenstate basis, a desirable feature in gravity induced collapse models. We finally comment on possible applications. 
  The interaction of maximally-charged dilatonic black holes on $R^{4}\times T^{d}$ is studied in the low-velocity limit. In particular, the scattering of two black holes on $R^{4}\times S^{1}$ is investigated. 
  Some properties of cosmological models with matter creation are investigated in the framework of the Friedman-Robertson-Walker (FRW) line element. For adiabatic matter creation, as developed by Prigogine and coworkers, we derive a simple expression relating the particle number density $n$ and energy density $\rho$ which holds regardless of the matter creation rate. The conditions to generate inflation are discussed and by considering the natural phenomenological matter creation rate $\psi =3 \beta nH$, where $\beta$ is a pure number of the order of unity and $H$ is the Hubble parameter, a minimally modified hot big-bang model is proposed. The dynamic properties of such models can be deduced from the standard ones simply by replacing the adiabatic index $\gamma$ of the equation of state by an effective parameter $\gamma_{*} = \gamma (1 - \beta)$. The thermodynamic behavior is determined and it is also shown that ages large enough to agree with observations are obtained even given the high values of $H$ suggested by recent measurements. 
  Imaginary time is often used in quantum tunnelling calculations. This article advocates a conceptually sounder alternative: complex lapse. In the ``3+1'' action for the Einstein gravitational field minimally coupled to a Klein-Gordon field, allowing the lapse function to be complex yields a complex action which generates both the usual Lorentzian theory and its Riemannian analogue, and in particular allows a change of signature between the two. The action and variational equations are manifestly well defined in the Hamiltonian representation, with the momentum fields consequently being complex. The complex action interpolates between the Lorentzian and Riemannian actions as they appear formally in the respective path integrals. Thus the complex-lapse theory provides a unified basis for a path-integral quantum theory of gravity involving both Lorentzian and Riemannian aspects. A major motivation is the quantum-tunnelling scenario for the origin of the universe. Taken as an explanation for the observed quantum tunnelling of particles, the complex-lapse theory determines that the argument of the lapse for the universe now is extremely small but negative. 
  In case of the Einstein's gravitation theory and its first order Palatini reformulation, the stress-energy-momentum of gravity has been proved to reduce to the Komar superpotential. We generalize this result to the affine-metric theory of gravity in case of general connections and arbitrary Lagrangian densities invariant under general covariant transformations. In this case, the stress-energy-momentum of gravity comes to the generalized Komar superpotential depending on a Lagrangian density in a precise way. 
  Two dimensional gravity with torsion is proved to be equivalent to special types of generalized 2d dilaton gravity. E.g. in one version, the dilaton field is shown to be expressible by the extra scalar curvature, constructed for an independent Lorentz connection corresponding to a nontrivial torsion. Elimination of that dilaton field yields an equivalent torsionless theory, nonpolynomial in curvature. These theories, although locally equivalent exhibit quite different global properties of the general solution. We discuss the example of a (torsionless) dilaton theory equivalent to the $R^2 + T^2$--model. Each global solution of this model is shown to split into a set of global solutions of generalized dilaton gravity. In contrast to the theory with torsion the equivalent dilaton one exhibits solutions which are asymptotically flat in special ranges of the parameters. In the simplest case of ordinary dilaton gravity we clarify the well known problem of removing the Schwarzschild singularity by a field redefinition. 
  We generalise the equations governing relativistic fluid dynamics given by Ehlers and Ellis for general relativity, and by Maartens and Taylor for quadratic theories, to generalised $f(R)$ theories of gravity. In view of the usefulness of this alternative framework to general relativity, its generalisation can be of potential importance for deriving analogous results to those obtained in general relativity. We generalise, as an example, the results of Maartens and Taylor to show that within the framework of general $f(R)$ theories, a perfect fluid spacetime with vanishing vorticity, shear and acceleration is Friedmann--Lema\^{\i}tre--Robertson--Walker only if the fluid has in addition a barotropic equation of state. It then follows that the Ehlers--Geren--Sachs theorem and its ``almost'' extension also hold for $f(R)$ theories of gravity. 
  Exact non-static spherically symmetric solutions of the Einstein equations for a null fluid source with pressure $P$ and density $\rho$ related by $P = k\rho^a$ are given. The $a=1$ metrics are asymptotically flat for $1/2<k\le 1$ and cosmological for $0<k<1/2$. The $k=1$ metric is the known charged Vaidya solution. In general the metrics have multiple apparent horizons. In the long time limit, the asymptotically flat metrics are hairy black hole solutions that `fall between' the Schwarzschild and Reissner-Nordstrom metrics. 
  We present a formalism for studying the thermodynamics of black holes in dilaton gravity. The thermodynamic variables are defined on a quasilocal surface surrounding the black hole system and are obtained from a general class of Lagrangians involving a dilaton. The formalism thus accommodates a large number of possible theories and black hole spacetimes. Many of the thermodynamic quantities are identified from the contribution of the action on the quasilocal boundary. The entropy is found using path integral techniques, and a first law of thermodynamics is obtained. As an illustration, we calculate the thermodynamic quantities for two black hole solutions in $(1+1)$ dimensions: one obtained from a string inspired theory and the other being a Liouville black hole in the ``$R=\kappa T$'' theory with a Liouville field. 
  Hawking ``thermal'' radiation could be a means to detect black holes of micron sizes, which may be hovering through the universe. We consider these micro-black holes to be distorted by the presence of some distribution of matter representing a convolution factor for their Hawking radiation. One may hope to determine from their Hawking signals the temperature distribution of their material shells by the inverse black body problem. In 1990, Nan-xian Chen has used a so-called modified M\"{o}bius transform to solve the inverse black body problem. We discuss and apply this technique to Hawking radiation of distorted micro-black holes. Some comments on supersymmetric applications of M\"{o}bius function and transform are also added. 
  The quantum potential approach makes it possible to construct a complementary picture of quantum mechanical evolution which reminds classical equation of motion. The only difference as compared to equations of motion for the underlying classical system is the presence of an additional potential term being a functional of the real part of the wavefunction. In the present paper this approach is applied to the quantum theory of gravity based on Wheeler -- De Witt equation. We describe the derivation of the `quantum Einstein equation' and discuss the new features of their solutions. 
  The gravitational properties of a local cosmic string in the framework of scalar-tensor gravity are examined. We find the metric in the weak-field approximation and we show that, contrary to the General Relativity case, the cosmic string in scalar-tensor gravitation exerces a force on non-relativistic, neutral test particle. This force is proportional to the derivative of the conformal factor $A^{2}(\phi)$ and it is always attractive. Moreover, this force could have played an important role at the Early Universe, although nowadays it can be neglegible. It is also shown that the angular separation $\delta\varphi$ remains unaltered for scalar-tensor cosmic strings. 
  We generalise Wesson's procedure, whereby vacuum $(4+1)-$dimensional field equations give rise to $(3+1)-$dimensional equations with sources, to arbitrary dimensions. We then employ this generalisation to relate the usual $(3+1)-$dimensional vacuum field equations to $(2+1)-$dimensional field equations with sources and derive the analogues of the classes of solutions obtained by Ponce de Leon. This way of viewing lower dimensional gravity theories can be of importance in establishing a relationship between such theories and the usual 4-dimensional general relativity, as well as giving a way of producing exact solutions in $(2+1)$ dimensions that are naturally related to the vacuum $(3+1)-$dimensional solutions. An outcome of this correspondence, regarding the nature of lower dimensional gravity, is that the intuitions obtained in $(3+1)$ dimensions may not be automatically transportable to lower dimensions.  We also extend a number of physically motivated solutions studied by Wesson and Ponce de Leon to $(D+1)$ dimensions and employ the equivalence between the $(D+1)$ Kaluza-Klein theories with empty $D-$dimensional Brans-Dicke theories (with $\omega=0$) to throw some light on the solutions derived by these authors. 
  The Strong Equivalence Principle (SEP) demands, besides the validity of the Einstein Equivalence Principle, that all self-gravitating bodies feel the same acceleration in an external gravitational field. It has been found that metric theories of gravity other that than general relativity typically predict a violation of the SEP. In case of the Earth-Moon system (weak field system) this violation is called the Nordtvedt effect. It has been shown by Damour and Sch\"afer, that small-eccentricity long-orbital-period binary pulsars with a white dwarf companion provide excellent conditions to test the SEP in strong field regimes. Based on newly discovered binary pulsars this paper investigates a possible violation of the SEP in strong field regimes. New limits with an improved confidence level are presented. The results of this paper lead to constrains on the combination $\epsilon/2-\zeta$ of the only two (post)$^2$-Newtonian parameters $\epsilon$ and $\zeta$ that arise from the (post)$^2$-Newtonian approximation of the tensor-multi-scalar theory of Damour and Esposito-Far{\`e}se. 
  A naked singularity is formed by the collapse of a Sine-Gordon soliton in 1+1 dimensional dilaton gravity with a negative cosmological constant. We examine the quantum stress tensor resulting from the formation of the singularity. Consistent boundary conditions require that the incoming soliton is accompanied by a flux of incoming radiation across past null infinity, but neglecting the back reaction of the spacetime leads to the absurd conclusion that the total energy entering the system by the time the observer is able to receive information from the singularity is infinite. We conclude that the back reaction must prevent the formation of the naked singularity. 
  We present a theory of general two-point functions and of generalized free fields in d-dimensional de Sitter space-time which closely parallels the corresponding minkowskian theory. The usual spectral condition is now replaced by a certain geodesic spectral condition, equivalent to a precise thermal characterization of the corresponding ``vacuum''states. Our method is based on the geometry of the complex de Sitter space-time and on the introduction of a class of holomorphic functions on this manifold, called perikernels, which reproduce mutatis mutandis the structural properties of the two-point correlation functions of the minkowskian quantum field theory. The theory contains as basic elementary case the linear massive field models in their ``preferred'' representation. The latter are described by the introduction of de Sitter plane waves in their tube domains which lead to a new integral representation of the two-point functions and to a Fourier-Laplace type transformation on the hyperboloid. The Hilbert space structure of these theories is then analysed by using this transformation. In particular we show the Reeh-Schlieder property. For general two-point functions, a substitute to the Wick rotation is defined both in complex space-time and in the complex mass variable, and substantial results concerning the derivation of Kallen-Lehmann type representation are obtained. 
  We consider a theory in which spacetime is an n-dimensional surface $V_n$ embedded in an $N$-dimensional space $V_N$. In order to enable also the Kaluza-Klein approach we admit $n > 4$. The dynamics is given by the minimal surface action in a curved embedding space. The latter is taken, in our specific model, as being a conformally flat space. In the quantization of the model we start from a generalization of the Howe-Tucker action which depends on the embedding variables ${\eta}^a (x)$ and the (intrinsic) induced metric $g_{\mu \nu}$ on $V_n$. If in the path integral we perform only the functional integration over ${\eta}^a (x)$, we obtain the effective action which functionally depends on $g_{\mu \nu}$ and contains the Ricci scalar $R$ and its higher orders $R^2$ etc. But due to our special choice of the conformal factor in $V_N$ enterig our original action, it turns out that the effective action contains also the source term. The latter is in general that of a $p$-dimensional membrane ($p$-brane); in particular we consider the case of a point particle. Thus, starting from the basic fields ${\eta}^a (x)$, we induce not only the kinetic term for $g_{\mu \nu}$, but also the "matter" source term. 
  Kaluza-Klein theory is a 5-dimensional Einstein general relativity; it has the interest of describing on an equal footing the laws of gravitation and electromagnetism in a geometrically unified way. We present it in Chapter 1, and we generalize it by adding to the equations of the theory the Lanczos tensor (endowed with the same physical properties as the Einstein tensor but quadratic with respect to the Riemann tensor.) One has obtained in the last decade all the spherically symmetric 2-stationary solutions (independent of time and of the extra coordinate) of the ``special" Kaluza-Klein theory. The study of the stability of these solutions against radial excitations is carried out in Chapter 2. We begin by presenting the spherically symmetric 2-static solutions; then we write and separate the perturbation-linearized equations of the 5-metric. The problem of stability against small oscillations is reduced to an eigenvalue problem which we discuss in detail in the static-solution parameter space. We show that regular solutions of non-vanishing finite energy (Kaluza-Klein solitons) --with non-euclidean spatial topology-- are stable. A broad class of singular solutions, containing among others the Schwarzschild solution, are also stable. Finally our stability results are compared to those obtained previously by Tomimatsu for a less broad class of solutions. We search in Chapter 3 for other exact stationary solutions, endowed this time with cylindrical symmetry, thus actually 4-stationary (depending on only one spacelike coordinate). First we obtain by a systematic study all the 4-stationary solutions of the special theory, some of which are interpreted as neutral or charged distributional cosmic string sources. We generalize these solutions in a following section by considering the Lanczos tensor, and we find 
  Generalizations of the Black Hole geometry of Ba\~nados, Teitelboim and Zanelli (BTZ) are presented. The theory is three-dimensional vacuum Einstein theory with a negative cosmological constant. The $n$-black-hole solution has $n$ asymptotically anti-de Sitter ``exterior" regions that join in one ``interior" region. The geometry of each exterior region is identical to that of a BTZ geometry; in particular, each contains a black hole horizon that surrounds (as judged from that exterior) all the other horizons. The interior region acts as a closed universe containing $n$ black holes. The initial state and its time development are discussed in some detail for the simple case when the angular momentum parameters of all the black holes vanish. A procedure to construct $n$ black holes with angular momentum (for $n \geq 4$) is also given. 
  We investigate topology change in (1+1) dimensions by analyzing the scalar-curvature action $1/2 \int R dV$ at the points of metric-degeneration that (with minor exceptions) any nontrivial Lorentzian cobordism necessarily possesses. In two dimensions any cobordism can be built up as a combination of only two elementary types, the ``yarmulke'' and the ``trousers.'' For each of these elementary cobordisms, we consider a family of Morse-theory inspired Lorentzian metrics that vanish smoothly at a single point, resulting in a conical-type singularity there. In the yarmulke case, the distinguished point is analogous to a cosmological initial (or final) singularity, with the spacetime as a whole being obtained from one causal region of Misner space by adjoining a single point. In the trousers case, the distinguished point is a ``crotch singularity'' that signals a change in the spacetime topology (this being also the fundamental vertex of string theory, if one makes that interpretation). We regularize the metrics by adding a small imaginary part whose sign is fixed to be positive by the condition that it lead to a convergent scalar field path integral on the regularized spacetime. As the regulator is removed, the scalar density $1/2 \sqrt{-g} R$ approaches a delta-function whose strength is complex: for the yarmulke family the strength is $\beta -2\pi i$, where $\beta$ is the rapidity parameter of the associated Misner space; for the trousers family it is simply $+2\pi i$. This implies that in the path integral over spacetime metrics for Einstein gravity in three or more spacetime dimensions, topology change via a crotch singularity is exponentially suppressed, whereas appearance or disappearance of a universe via a yarmulke singularity is exponentially enhanced. 
  We study the spherically symmetric collapse of the axion/dilaton system coupled to gravity. We show numerically that the critical solution at the threshold of black hole formation is continuously self-similar. Numerical and analytical arguments both demonstrate that the mass scaling away from criticality has a critical exponent of $\gamma = 0.264$. 
  When the Penrose-Goldberg (PG) superpotential is used to compute the angular momentum of an axial symmetry, the Killing potential $Q_{(\varphi )}^{\mu\nu}$ for that symmetry is needed. Killing potentials used in the PG superpotential must satisfy Penrose's equation. It is proved for the Schwarzschild and Kerr solutions that the Penrose equation does not admit a $%Q_{(\varphi )}^{\mu\nu}$ at finite r and therefore the PG superpotential can only be used to compute angular momentum asymptotically. 
  Barbero recently suggested a modification of Ashtekar's choice of canonical variables for general relativity. Although leading to a more complicated Hamiltonian constraint this modified version, in which the configuration variable still is a connection, has the advantage of being real. In this article we derive Barbero's Hamiltonian formulation from an action, which can be considered as a generalization of the ordinary Hilbert-Palatini action. 
  Current generalizations of the classical Einstein-Hilbert Lagrangian formulation of General Relativity are reviewed. Some alternative variational principles are known to reproduce Einstein's gravitational equations, and should therefore be regarded as equivalent descriptions of the same physical model, while other variational principles ("Scalar-tensor theories" and "Higher-derivative theories") are commonly presented as truly alternative physical theories. Such theories, however, are also known to admit a reformulation which is formally identical to General Relativity (with auxiliary fields). The physical significance of this change of variables has been questioned by several authors in recent years. Here, we investigate to which extent purely affine, metric-affine, scalar-tensor and purely metric theories can be regarded as physically equivalent to GR; we show that in general this depends on which metric tensorfield is assumed to represent the true physical space-time geometry. For purely metric theories where the Lagrangian is a nonlinear function f(R) of the curvature scalar, we present an argument based on the definition of the physical energy, which leads one to regard the rescaled metric (Einstein frame) as the true physical one. As a direct consequence, the physical content of such "alternative" models is reset to coincide with General Relativity, and the "Nonlinear Gravity Theories" become nothing but exotic reformulations of General Relativity in terms of unphysical variables. 
  We investigate the possibility of testing of the Einstein Equivalence Principle (EEP) using measurements of anomalous magnetic moments of elementary particles. We compute the one loop correction for the $g-2$ anomaly within the class of non metric theories of gravity described by the \tmu formalism. We find several novel mechanisms for breaking the EEP whose origin is due purely to radiative corrections. We discuss the possibilities of setting new empirical constraints on these effects. 
  It is a well known fact that quantum fields on Minkowski spacetime are correlated for each pair of spacetime regions. In Robertson-Walker spacetimes there are spacelike separated regions with disjoint past horizons but the absence of correlations in that case was never proved. We derive in this paper formulae for correlations of quantum fields on Robertson-Walker spacetimes. Such correlations could have reasonably influenced the formation of structure in the early universe. We use methods of algebraic and constructive quantum field theory. 
  The volume operator is an important kinematical quantity in the non-perturbative approach to four-dimensional quantum gravity in the connection formulation. We give a general algorithm for computing its spectrum when acting on four-valent spin network states, evaluate some of the eigenvalue formulae explicitly, and discuss the role played by the Mandelstam constraints. 
  The ultra--high energy cosmic rays recently detected by several air shower experiments could have an extragalactic origin. In this case, the nearest active galaxy Centaurus A might be the source of the most energetic particles ever detected on Earth. We have used recent radio observations in order to estimate the arrival energy of the protons accelerated by strong shock fronts in the outer parts of this southern radio source. We expect detections corresponding to particles with energies up to $\sim 2.2 \times 10^{21}$ eV and an arrival direction of ($l \approx 310^{\circ}$, $b \approx 20^{\circ}$) in galactic coordinates. The future Southern Hemisphere Pierre Auger Observatory might provide a decisive test for extragalactic models of the origin of the ultra--high energy cosmic rays. 
  It has long been known that the search for gravitational waves from inspiraling binaries must be aided by signal processing methods, and that the technique of matched filtering is the most likely to be used for the detection of inspiral ``chirps''. This means that the output of an interferometer must be cross- correlated with many waveform templates to dig out faint signals buried in the noise. The templates are characterized by several parameters which vary continuously over some finite range; but because the amount of computing power available to perform the cross-correlations of the search templates with the output is finite, the actual templates used must be picked with certain discrete values of these parameters, which will have some finite spacing. If the spacing is too small, the number of templates (and therefore the computing power) needed to perform an on-line search becomes prohibitive; if the spacing is too large, many chirps will not be detected because the values of their parameters lie too far from those of the nearest template. In this paper I use differential geometry to extend the earlier formalism of Sathyaprakash and Dhurandhar to estimate the template spacing, number of templates, and computing power required for a single-pass, on-line search of the output of a single interferometer in terms of the fraction of events lost to parameter discretization. My formalism obtains these results with little numerical computation, and is valid for arbitrary noise spectra and template parameterizations. I find that the computing power needed for the most computationally intensive, reasonable search is of order several hundred Gigaflops. This will be feasible by the time LIGO is operational, but it is worth pursuing methods of reducing this number. 
  First order corrections to the Unruh effect are calculated from a model of an accelerated particle detector of finite mass. We show that quantum smearing of the trajectory and large recoil essentially do not modify the Unruh effect. Nevertheless, we find corrections to the thermal distribution and to the Unruh temperature. In a certain limit, when the distribution at equilibrium remains exactly thermal, the corrected temperature is found to be $T = T_U( 1 - T_U/M)$, where $T_U$ is the Unruh temperature. We estimate the consequent corrections to the Hawking temperature and the black hole entropy, and comment on the relationship to the problem of trans-planckian frequencies. 
  A manifestly diffeomorphism invariant extension of Einstein gravity is constructed, which includes singular metrics, and whose ADM formulation is Ashtekar's gravity. The latter is shown to be locally equivalent to the covariant theory. It turns out that exactly those kinds of degenerate four dimensional metrics are allowed which do not destroy the causal structure of spacetime. It is also shown that Ashtekar's gravity possesses an extension that provides a local SO(3,C) invariance, without complexifying or changing the signature of the metric. 
  We show that the covariant derivative of Dirac fermion fields in the presence of a general linear connection on a world manifold is universal for Einstein's, gauge and affine-metric gravitation theories. 
  We analyze a class of linear wave equations for odd half spin that have a well posed initial value problem. We demonstrate consistency of the equations in curved space-times. They generalize the Weyl neutrino equation. We show that there exists an associated invariant exact set of spinor fields indicating that the characteristic initial value problem on a null cone is formally solvable, even for the system coupled to general relativity. We derive the general analytic solution in flat space by means of Fourier transforms. Finally, we present a twistor contour integral description for the general analytic solution and assemble a representation of the group $O(4,4)$ on the solution space. 
  The condition for the unitarity of a quantum field is investigated in semiclassical gravity from the Wheeler-DeWitt equation. It is found that the quantum field preserves unitarity asymptotically in the Lorentzian universe, but does not preserve unitarity completely in the Euclidean universe. In particular we obtain a very simple matter field equation in the basis of the generalized invariant of the matter field Hamiltonian whose asymptotic solution is found explicitly. 
  A new asymptotic expansion method is developed to separate the Wheeler-DeWitt equation into the time-dependent Schr\"{o}dinger equation for a matter field and the Einstein-Hamilton-Jacobi equation for the gravitational field including the quantum back-reaction of the matter field. In particular, the nonadiabatic basis of the generalized invariant for the matter field Hamiltonian separates the Wheeler-DeWitt equation completely in the asymptotic limit of $m_p^2$ approaching infinity. The higher order quantum corrections of the gravity to the matter field are found. The new asymptotic expansion method is valid throughout all regions of superspace compared with other expansion methods with a certain limited region of validity. We apply the new asymptotic expansion method to the minimal FRW universe. 
  A first-order perturbation approach to $k=0$ Friedmann cosmologies filled with dust and radiation is developed. Adopting the coordinate gauge comoving with the perturbed matter, and neglecting the vorticity of the radiation, a pair of coupled equations is obtained for the trace $h$ of the metric perturbations and for the velocity potential $v$. A power series solution with upwards cutoff exists such that the leading terms for large values of the dimensionless time $\xi$ agree with the relatively growing terms of the dust solution of Sachs and Wolfe. 
  Our investigation of differential conservation laws in Lagrangian field theory is based on the first variational formula which provides the canonical decomposition of the Lie derivative of a Lagrangian density by a projectable vector field on a bundle (Part 1: gr-qc/9510061). If a Lagrangian density is invariant under a certain class of bundle isomorphisms, its Lie derivative by the associated vector fields vanishes and the corresponding differential conservation laws take place. If these vector fields depend on derivatives of parameters of bundle transformations, the conserved current reduces to a superpotential. This Part of the work is devoted to gravitational superpotentials. The invariance of a gravitational Lagrangian density under general covariant transformations leads to the stress-energy-momentum conservation law where the energy-momentum flow of gravity reduces to the corresponding generalized Komar superpotential. The associated energy-momentum (pseudo) tensor can be defined and calculated on solutions of metric and affine-metric gravitational models. 
  We present a new field theory of gravity. It incorporates a great part of General Relativity (GR) and can be interpreted in the standard geometrical way like GR as far as the interaction of matter to gravity is concerned. However, it differs from GR when treating gravity to gravity interaction. The most crucial distinction concerns the velocity of propagation of gravitational waves. Since there is a large expectation that the detection of gravitational waves will occur in the near future the question of which theory describes Nature better will probably be settled soon. 
  We show that the second coefficient of the Conway knot polynomial is annihilated by the Hamiltonian constraint of canonically quantized general relativity in the loop representation. The calculations are carried out in a fully regularized lattice framework. Crucial to the calculation is the explicit form of the skein relations of the second coefficient, which relate it to the Gauss linking number. Contrary to the lengthy formal continuum calculation, the rigorous lattice version can be summarized in a few pictures. 
  In the $R+\alpha R^2$ gravity theory, we show that if freely propagating massless particles have an almost isotropic distribution, then the spacetime is almost Friedmann-Robertson-Walker (FRW). This extends the result proved recently in general relativity ($\alpha=0$), which is applicable to the microwave background after photon decoupling. The higher-order result is in principle applicable to a massless species that decouples in the early universe, such as a relic graviton background. Any future observations that show small anisotropies in such a background would imply that the geometry of the early universe were almost FRW. 
  We consider quantum effects of an electromagnetic field in a radiation-dominated almost FRW spacetime. The dominant non-local quantum correction to the photon distribution is a quadrupole moment, corresponding to an effective anisotropic pressure in the energy-momentum tensor. 
  Isotropic inhomogeneous dust universes are analysed via observational coordinates based on the past light cones of the observer's galactic worldline. The field equations are reduced to a single first--order {\sc ode} in observational variables on the past light cone, completing the observational integration scheme. This leads naturally to an explicit exact solution which is locally nearly homogeneous (i.e. {\sc frw}), but at larger redshift develops inhomogeneity. New observational characterisations of homogeneity ({\sc frw} universes) are also given. 
  We investigate the back-reaction effect of the quantum field on the topological degrees of freedom in (2+1)-dimensional toroidal universe, ${\cal M} \simeq T^2\times {\bf R}$. Constructing a homogeneous model of the toroidal universe, we examine explicitly the back-reaction effect of the Casimir energy of a massless, conformally coupled scalar field, with a conformal vacuum. The back-reaction causes an instability of the universe: The torus becomes thinner and thinner as it evolves, while its total 2-volume (area) becomes smaller and smaller. The back-reaction caused by the Casimir energy can be compared with the influence of the negative cosmological constant: Both of them make the system unstable and the torus becomes thinner and thinner in shape. On the other hand, the Casimir energy is a complicated function of the Teichm\"uller parameters $(\tau^1, \tau^2)$ causing highly non-trivial dynamical evolutions, while the cosmological constant is simply a constant.    Since the spatial section is a 2-torus, we shall write down the partition function of this system, fixing the path-integral measure for gravity modes, with the help of the techniques developed in string theories. We show explicitly that the partition function expressed in terms of the canonical variables corresponding to the (redundantly large) original phase space, is reduced to the partition function defined in terms of the physical-phase-space variables with a standard Liouville measure. This result is compatible with the general theory of the path integral for the 1st-class constrained systems. 
  We seek an analogy of the mathematical form of the alternative form of Einstein's field equations for Lovelock's field equations. We find that the price for this analogy is to accept the existence of the trace anomaly of the energy-momentum tensor even in classical treatments. As an example, we take this analogy to any generic second order Lagrangian and exactly derive the trace anomaly relation suggested by Duff. This indicates that an intrinsic reason for the existence of such a relation should perhaps be, classically, somehow related to the covariance of the form of Einstein's equations. 
  (from the talk:) I shall here speak on gravity in (1+1)-dimensional space-time --- lineal gravity. The purpose of studying lower dimensional theories, and specifically lower dimensional gravity, is to gain insight into difficult conceptional issues, which are present and even more opaque in the physical (3+1)-dimensional world. Perhaps lessons learned in the lower-dimensional setting can be used to explicate physical problems. Moreover, if we are lucky, the lower-dimensional theories can have a direct physical relevance to modelling phenomena that is actually dynamically confined to the lower dimensionality. This is what happened with (2+1)-dimensional gravity: gravitational physics in the presence of cosmic strings (infinitely long, perpendicular to a plane) is adequately described planar gravity. Indeed the recently discussed causality puzzles raised by ``Gott time machines'' were resolved with the help of the lower-dimensional model... 
  In this article we summarize and describe the recently found transforms for theories of connections modulo gauge transformations associated with compact gauge groups. Specifically, we put into a coherent picture the so-called loop transform, the inverse loop transform, the coherent state transform and finally the Wick rotation transform which is the appropriate transform that incorporates the correct reality conditions of quantum gravity when formulated as a dynamical theory of connections while preserving the simple algebraic form of the Hamiltonian constraint. 
  What restrictions are there on a spacetime for which the Ricci curvature is such as to produce convergence of geodesics (such as the preconditions for the Singularity Theorems) but for which there are no singularities? We answer this question for a restricted class of spacetimes: static, geodesically complete, and globally hyperbolic. The answer is that, in at least one spacelike direction, the Ricci curvature must fall off at a rate inversely quadratic in a naturally-occurring Riemannian metric on the space of static observers. Along the way, we establish some global results on the static observer space, regarding its completeness and its behavior with respect to universal covering spaces. 
  The correspondence between the linear multiplicity-free unirreps of SL(4, R) studied by Ne'eman and {\~{S}}ija{\~{c}}ki and the non-linear realizations of the affine group is worked out. The results obtained clarify the inclusion of spinorial fields in a non-linear affine gauge theory of gravitation. 
  We examine the gravitational collapse of a non-linear sigma model in spherical symmetry. There exists a family of continuously self-similar solutions parameterized by the coupling constant of the theory. These solutions are calculated together with the critical exponents for black hole formation of these collapse models. We also find that the sequence of solutions exhibits a Hopf-type bifurcation as the continuously self-similar solutions become unstable to perturbations away from self-similarity. 
  Time-symmetric initial data for two-body solutions in three dimensional anti-deSitter gravity are found. The spatial geometry has constant negative curvature and is constructed as a quotient of two-dimensional hyperbolic space. Apparent horizons correspond to closed geodesics. In an open universe, it is shown that two black holes cannot exist separately, but are necessarily enclosed by a third horizon. In a closed universe, two separate black holes can exist provided there is an additional image mass. 
  An open system consisting of a scalar field bound to a Kerr black hole whose mass ($M$) and specific angular momentum ($a$) are slowly (adiabatically) perturbed is considered. The adiabatically induced phase and the conditions for the validity of the adiabatic approximation are obtained. The effect of closed cycles in parameter space ($a$, $M$ plane) on the energy levels of both stable and unstable scalar field bound states, together with other quantities of interest, is illustrated. Lastly it is noted that the black hole wavefunction will acquire an equal and opposite phase to that of matter thus leading to a change of its effective action (entropy). 
  The deviation of primordial Helium production due to a variation on the difference between the rest masses of the nucleons is presented. It is found an upper bound $\delta (M_{_n} - M_{_p}) \alt 0.129$ MeV, between the present and nucleosynthesis epochs. This bound is used to analyze Wesson's theory of gravitation; as a result, it is ruled out by observation. 
  A new approach to the algebraic classification of second order symmetric tensors in 5-dimensional space-times is presented. The possible Segre types for a symmetric two-tensor are found. A set of canonical forms for each Segre type is obtained. A theorem which collects together some basic results on the algebraic structure of the Ricci tensor in 5-dimensional space-times is also stated. 
  For various theories, in particular gauge field theories, the algebraic form of the Hamiltonian simplifies considerably if one writes it in terms of certain complex variables. Also general relativity when written in the new canonical variables introduced by Ashtekar belongs to that category, the Hamiltonian being replaced by the so-called scalar (or Wheeler-DeWitt) constraint. In order to ensure that one is dealing with the correct physical theory one has to impose certain reality conditions on the classical phase space which generally are algebraically quite complicated and render the task of finding an appropriate inner product into a difficult one. This article shows, for a general theory, that if we prescribe first a {\em canonical} complexification and second a $^*$ representation of the canonical commutation relations in which the real connection is diagonal, then there is only one choice of a holomorphic representation which incorporates the correct reality conditions {\em and} keeps the Hamiltonian (constraint) algebraically simple ! We derive a canonical algorithm to obtain this holomorphic representation and in particular explicitly compute it for quantum gravity in terms of a {\em Wick rotation transform}. 
  We show that there exists a large class of regularization schemes for probabilistic predictions in the theory of a self-reproducing inflationary univere, all of which eliminate the apparent dependence on the time reparametrization. However, all these schemes lead to different answers for relative probabilities of finding various types of post-inflationary universes. Besides, all these schemes fail to be reparametrization invariant beyond the range of the inflaton field close to end of inflation boundary. Therefore, we argue that at the current level of understanding, the simple regularization schemes associated with cutoffs at equal time hypersurfaces are as good as the recently proposed more complicated procedures which try to fix the time-reparametrization dependence. 
  Rovelli's `` quantum mechanics without time'' motivates an intrinsically time-slicing independent picture of reduced phase space quantum gravity, which may be described as ``quantization after evolution''. Sufficient criteria for carrying out quantization after evolution are developed in terms of a general concept of the classical limit of quantum mechanics. If these criteria are satisfied then it is possible to have consistent unitary evolution of operators, with respect to an infinite parameter family of time-slicings (and probably all time-slicings), with the correct classical limit. The criteria are particularly amenable to study in (2+1)-dimensional gravity, where the reduced phase space is finite dimensional. 
  In the (2+2) formulation of general relativity spacetime is foliated by a two-parameter family of spacelike 2-surfaces (instead of the more usual one-parameter family of spacelike 3-surfaces). In a partially gauge-fixed setting (double-null gauge), I write down the symplectic structure of general relativity in terms of intrinsic and extrinsic quantities associated with these 2-surfaces. This leads to an identification of the reduced phase space degrees of freedom. In particular, I show that the two physical degrees of freedom of general relativity are naturally encoded in a quantity closely related to the twist of the pair of null normals to the 2-surfaces. By considering the characteristic initial-value problem I establish a canonical transformation between these and the more usually quoted conformal 2-metric (or shear) degrees of freedom. (This paper is based on a talk given at the Fifth Midwest Relativity Conference, Milwaukee, USA.) 
  The metric for plane gravitational waves is quantized using the Ashtekar field variables. The z axis (direction of travel of the waves) is taken to be the entire real line. Solutions to the constraints are proposed; they involve open-ended flux lines running along the entire z axis. These solutions are annihilated by the constraints except at the two boundary points, where the Gauss constraint does not annihilate the solutions. This result is in sharp contrast to the situation in the general, 3+1 dimensional case without planar symmetry, where the Gauss constraints do not contribute at boundaries because the Lagrange multipliers for the Gauss constraints vanish there. The constraints annihilate the solutions if classical matter is included (so that flux lines are terminated on the matter). The SU(2) holonomy matrices used in the solutions are (2j+1) dimensional, where j may be any spin, not necessarily j = 1/2. In this respect the solutions resemble the spin network states recently constructed by Rovelli and Smolin in loop space. 
  The effect of gravitational radiation reaction on orbits around a spinning black hole is analyzed. Such orbits possess three constants of motion: $\iota$, $e$, and $a$, which correspond, in the Newtonian limit of the orbit being an ellipse, to the inclination angle of the orbital plane to the hole's equatorial plane, the eccentricity, and the semi-major axis length, respectively. First, it is argued that circular orbits ($e=0$) remain circular under gravitational radiation reaction. Second, for elliptical orbits (removing the restriction of $e=0$), the evolution of $\iota$, $e$, and $a$ is computed to leading order in $S$ (the magnitude of the spin angular momentum of the hole) and in $M/a$, where $M$ is the mass of the black hole. As $a$ decreases, $\iota$ increases and $e$ decreases. 
  I describe the history of my attempts to arrive at a discrete substratum underlying the spacetime manifold, culminating in the hypothesis that the basic structure has the form of a partial-order (i.e. that it is a causal set). 
  Einstein's field equations for spatially self-similar locally rotationally symmetric perfect fluid models are investigated. The field equations are rewritten as a first order system of autonomous ordinary differential equations. Dimensionless variables are chosen in such a way that the number of equations in the coupled system of differential equations is reduced as far as possible. The system is subsequently analyzed qualitatively for some of the models. The nature of the singularities occurring in the models is discussed. 
  The quantization of a real massless scalar field in a spacetime produced in a collision of two electromagnetic plane waves with constant wave fronts is considered. The background geometry in the interaction region, the Bell-Szekeres solution, is locally isometric to the conformally flat Bertotti-Robinson universe filled with a uniform electric field. It is shown that before the waves interact the Bogoliubov coefficients relating different observers are trivial and no vacuum polarization takes place. In the non- singular interaction region neutral scalar particles are produced with number of created particles and spectrum typical of gravitational wave collision. 
  Some exact solutions for the Einstein field equations corresponding to inhomogeneous $G_2$ cosmologies with an exponential-potential scalar field which generalize solutions obtained previously are considered. Several particular cases are studied and the properties related to generalized inflation and asymptotic behaviour of the models are discussed. 
  In the context of Kaluza-Klein theories, we consider a model in which the universe is filled with a perfect fluid described by a barotropic equation of state. An analysis of density perturbations employing the synchronous gauge shows that there are cases where these perturbations have an exponential growth during a de Sitter phase evolution in the external space. 
  The problem is discussed of whether a traveller can reach a remote object and return back sooner than a photon would when taken into account that the traveller can partly control the geometry of his world. It is argued that under some reasonable assumptions in globally hyperbolic spacetimes the traveller cannot hasten reaching the destination. Nevertheless, it is perhaps possible for him to make an arbitrarily long round-trip within an arbitrarily short (from the point of view of a terrestrial observer) time. 
  The tool of functional averaging over some ``large'' diffeomorphisms is used to describe quantum systems with constraints, in particular quantum cosmology, in the language of quantum Effective Action. Simple toy models demonstrate a supposedly general phenomenon: the presence of a constraint results in ``quantum repel'' from the classical mass shell. 
  In this article we show that one can construct initial data for the Einstein equations which satisfy the vacuum constraints. This initial data is defined on a manifold with topology $R^3$ with a regular center and is asymptotically flat. Further, this initial data will contain an annular region which is foliated by two-surfaces of topology $S^2$. These two-surfaces are future trapped in the language of Penrose. The Penrose singularity theorem guarantees that the vacuum spacetime which evolves from this initial data is future null incomplete. 
  The integrability properties of the field equation $L_{xx} = F(x)L^2$ of a spherically symmetric shear--free fluid are investigated. A first integral, subject to an integrability condition on $F(x)$, is found, giving a new class of solutions which contains the solutions of Stephani (1983) and Srivastava (1987) as special cases. The integrability condition on $F(x)$ is reduced to a quadrature which is expressible in terms of elliptic integrals in general. There are three classes of solution and in general the solution of $L_{xx} = F(x)L^2$ can only be written in parametric form. The case for which $F=F(x)$ can be explicitly given corresponds to the solution of Stephani (1983). A Lie analysis of $L_{xx} = F(x) L^2$ is also performed. If a constant $\alpha$ vanishes, then the solutions of Kustaanheimo and Qvist (1948) and of this paper are regained. For $\alpha \neq 0$ we reduce the problem to a simpler, autonomous equation. The applicability of the Painlev\'e analysis is also briefly considered. 
  We present a new generating algorithm to construct exact non static solutions of the Einstein field equations with two-dimensional inhomogeneity. Infinite dimensional families of $G_1$ inhomogeneous solutions with a self interacting scalar field, or alternatively with perfect fluid, can be constructed using this algorithm. Some families of solutions and the applications of the algorithm are discussed. 
  Among relativistic theories of gravitation the closest ones to general relativity are the scalar-tensor ones and these with Lagrangians being any function f(R) of the curvature scalar. A complete chart of relationships between these theories and general relativity can be delineated. These theories are mathematically (locally) equivalent to general relativity plus a minimally coupled self-interacting scalar field. Physically they describe a massless spin-2 field (graviton) and a spin-0 component of gravity. It is shown that these theories are either physically equivalent to general relativity plus the scalar or flat space is classically unstable (or at least suspected of being unstable). In this sense general relativity is universal: it is an isolated point in the space of gravity theories since small deviations from it either carry the same physical content as it or give rise to physically untenable theories. 
  The genuine quantum gravity effects can already be around us. It is likely that the observed large-angular-scale anisotropies in the microwave background radiation are induced by cosmological perturbations of quantum-mechanical origin. Such perturbations are placed in squeezed vacuum quantum states and, hence, are characterized by large variances of their amplitude. The statistical properties of the anisotropies should reflect the underlying statistics of the squeezed vacuum quantum states. In this paper, the theoretical variances for the temperature angular correlation function are described in detail. It is shown that they are indeed large and must be present in the observational data, if the anisotropies are truly caused by the perturbations of quantum-mechanical origin. Unfortunately, these large theoretical statistical uncertainties will make the extraction of cosmological information from the measured anisotropies a much more difficult problem than we wanted it to be. This contribution to the Proceedings is largely based on references~[42,8]. The Appendix contains an analysis of the ``standard'' inflationary formula for density perturbations. 
  In recent years a statistical mechanics description of particles, fields and spacetime based on the concept of quantum open systems and the influence functional formalism has been introduced. It reproduces in full the established theory of quantum fields in curved spacetime and contains also a microscopic description of their statistical properties, such as noise, fluctuations, decoherence, and dissipation. This new framework allows one to explore the quantum statistical properties of spacetime at the interface between the semiclassical and quantum gravity regimes, as well as important non-equilibrium processes in the early universe and black holes, such as particle creation, entropy generation, galaxy formation, Hawking radiation, gravitational collapse, backreaction and the black hole end-state and information lost issues. Here we give a summary of the theory of correlation dynamics of quantum fields and describe how this conceptual scheme coupled with scaling behavior near the infrared limit can shed light on the black hole information paradox. 
  We note that in general there exist two basic aspects in any branch of physics, including cosmology - one dealing with the attributes of basic constituents and forces of nature, the other dealing with how structures arise from them and how they evolve. Current research in quantum and superstring cosmology is directed mainly towards the first aspect, even though a viable theory of the underlying interactions is lacking. We call the attention to the development of the second aspect, i.e., on the organization and processing of the basic constituents of matter (in classical cosmology) and spacetime (in quantum cosmology). Many newly developed concepts and techniques in condensed matter physics stemming from the investigation of disordered, dynamical and complex systems can guide us in asking the right questions and formulating new solutions to existing and developing cosmological issues, thereby broadening our view of the universe both in its formative and present state. 
  Developments in theoretical cosmology in the recent decades show a close connection with particle physics, quantum gravity and unified theories. Answers or hints to many fundamental questions in cosmology like the homogeneity and isotropy of the Universe, the sources of structure formation and entropy generation, and the initial state of the Universe can be traced back to the activities of quantum fields and the dynamics of spacetime from the Grand Unification time to the Planck time at $10^{-43} sec$. A closer depiction of this primordial state of the Universe requires at least a semiclassical theory of gravity and the consideration of non-equilibrium statistical processes involving quantum fields. This critical state is intermediate between the well-known classical epoch successfully described by Einstein's Theory of General Relativity and the completely unknown realm of quantum gravity. Many issues special to this stage such as the transition from quantum to classical spacetime via decoherence, cross-over behavior at the Planck scale, tunneling and particle creation, or growth of density contrast from vacuum fluctuations share some basic concerns of mesoscopic physics for condensed matter, atoms or nuclei, in the quantum/classical and the micro/macro interfaces, or the discrete/continuum and the stochastic/ deterministic transitions. We point out that underlying these issues are three main factors: quantum coherence, fluctuations and correlation. We discuss how a deeper understanding of these aspects of fields and spacetimes can help one to address some basic problems, such as Planck scale metric fluctuations, cosmological phase transition and structure formation, and the black hole entropy, end-state and information paradox. 
  Semiclassical Einstein-Langevin equations for arbitrary small metric perturbations conformally coupled to a massless quantum scalar field in a spatially flat cosmological background are derived. Use is made of the fact that for this problem the in-in or closed time path effective action is simply related to the Feynman and Vernon influence functional which describes the effect of the ``environment'', the quantum field which is coarse grained here, on the ``system'', the gravitational field which is the field of interest. This leads to identify the dissipation and noise kernels in the in-in effective action, and to derive a fluctuation-dissipation relation. A tensorial Gaussian stochastic source which couples to the Weyl tensor of the spacetime metric is seen to modify the usual semiclassical equations which can be viewed now as mean field equations. As a simple application we derive the correlation functions of the stochastic metric fluctuations produced in a flat spacetime with small metric perturbations due to the quantum fluctuations of the matter field coupled to these perturbations. 
  Statistical mechanical concepts and processes such as decoherence, correlation, and dissipation can prove to be of basic importance to understanding some fundamental issues of quantum cosmology and theoretical physics such as the choice of initial states, quantum to classical transition and the emergence of time. Here we summarize our effort in 1) constructing a unified theoretical framework using techniques in interacting quantum field theory such as influence functional and coarse-grained effective action to discuss the interplay of noise, fluctuation, dissipation and decoherence; and 2) illustrating how these concepts when applied to quantum cosmology can alter the conventional views on some basic issues. Two questions we address are 1) the validity of minisuperspace truncation, which is usually assumed without proof in most discussions, and 2) the relevance of specific initial conditions, which is the prevailing view of the past decade. We also mention how some current ideas in chaotic dynamics, dissipative collective dynamics and complexity can alter our view of the quantum nature of the universe. 
  We describe recent attempts at discretizing canonical quantum gravity in four dimensions in terms of a connection formulation. This includes a general introduction, a comparison between the real and complex connection approach, and a discussion of some open problems. (Contribution to the proceedings of the workshop ``Recent mathematical developments in classical and quantum gravity", Sintra, Portugal, July 1995.) 
  A set of simple rules for constructing the maximal (e.g. analytic) extensions for any metric with a Killing field in an (effectively) two-dimensional spacetime is formulated. The application of these rules is extremely straightforward, as is demonstrated at various examples and illustrated with numerous figures. Despite the resulting simplicity we also comment on some subtleties concerning the concept of Penrose diagrams. Most noteworthy among these, maybe, is that (smooth) spacetimes which have both degenerate and non-degenerate (Killing) horizons do not allow for globally smooth Penrose diagrams. Physically speaking this obstruction corresponds to an infinite relative red/blueshift between observers moving across the two horizons. -- The present work provides a further step in the classification of all global solutions of the general class of two-dimensional gravity-Yang-Mills systems introduced in Part I, comprising, e.g., all generalized (linear and nonlinear) dilaton theories. In Part I we constructed the local solutions, which were found to always have a Killing field; in this paper we provide all universal covering solutions (the simply connected maximally extended spacetimes). A subsequent Part III will treat the diffeomorphism inequivalent solutions for all other spacetime topologies. -- Part II is kept entirely self-contained; a prior reading of Part I is not necessary. 
  Radiation-filled Friedmann-Robertson-Walker universes are quantized according to the Arnowitt-Deser-Misner formalism in the conformal-time gauge. Unlike previous treatments of this problem, here both closed and open models are studied, only square-integrable wave functions are allowed, and the boundary conditions to ensure self-adjointness of the Hamiltonian operator are consistent with the space of admissible wave functions. It turns out that the tunneling boundary condition on the universal wave function is in conflict with self-adjointness of the Hamiltonian. The evolution of wave packets obeying different boundary conditions is studied and it is generally proven that all models are nonsingular. Given an initial condition on the probability density under which the classical regime prevails, it is found that a closed universe is certain to have an infinite radius, a density parameter $\Omega = 1$ becoming a prediction of the theory. Quantum stationary geometries are shown to exist for the closed universe model, but oscillating coherent states are forbidden by the boundary conditions that enforce self-adjointness of the Hamiltonian operator. 
  Using a key observation due to Thiemann, a generalized Wick transform is introduced to map the constraint functionals of Riemannian general relativity to those of the Lorentzian theory, including matter sources. This opens up a new avenue within ``connection-dynamics'' where one can work, throughout, only with real variables. The resulting quantum theory would then be free of complicated reality conditions. Ramifications of this development to the canonical quantization program are discussed. 
  The interaction of a cosmic string and a maximally charged dilatonic black hole is studied in the low-velocity limit. In particular, the string-black hole scattering at a low velocity is investigated. 
  It is shown that a space-time hypersurface of a 5-dimensional Ricci-flat space-time has its energy momentum tensor algebrically related to its extrinsic curvature and to the Riemann curvature of the embedding space. It is also seen that the Einstein-Maxwell field does not arise naturally from this geometry, so that a Kaluza-Klein model based on it would require further assumptions. 
  Numerical exploration of the properties of singularities could, in principle, yield detailed understanding of their nature in physically realistic cases. Examples of numerical investigations into the formation of naked singularities, critical behavior in collapse, passage through the Cauchy horizon, chaos of the Mixmaster singularity, and singularities in spatially inhomogeneous cosmololgies are discussed. 
  We describe a numerical approach to address the BKL conjecture that the generic cosmological singularity is locally Mixmaster-like. We consider application of a symplectic PDE solver to three models of increasing complexity--spatially homogeneous (vacuum) Mixmaster cosmologies where we compare the symplectic ODE solver to a Runge-Kutta one, the (plane symmetric, vacuum) Gowdy universe on $T^3 \times R$ whose dynamical degrees of freedom satisfy nonlinearly coupled PDE's in one spatial dimension and time, and U(1) symmetric, vacuum cosmologies on $T^3 \times R$ which are the simplest spatially inhomogeneous universes in which local Mixmaster dynamics is allowed. 
  Description of the magnetic Bianchi VI$_0$ cosmologies of LeBlanc, Kerr, and Wainwright in the formalisms both of Belinskii, Khalatnikov, and Lifshitz, and of Misner allows qualitative understanding of the Mixmaster-like singularity in those models. 
  The use of time--like geodesics to measure temporal distances is better justified than the use of space--like geodesics for a measurement of spatial distances. We give examples where a ''spatial distance'' cannot be appropriately determined by the length of a space--like geodesic. 
  The following four statements have been proven decades ago already, but they continue to induce a strange feeling:   - All curvature invariants of a gravitational wave vanish - in spite of the fact that it represents a nonflat spacetime.   - The eigennullframe components of the curvature tensor (the Cartan ''scalars'') do not represent curvature scalars.    - The Euclidean topology in the Minkowski spacetime does not possess a basis composed of Lorentz--invariant neighbourhoods.   - There are points in the de Sitter spacetime which cannot be joined to each other by any geodesic.   We explain that our feeling is influenced by the compactness of the rotation group; the strangeness disappears if we fully acknowledge the noncompactness of the Lorentz group. Output: Imaginary coordinate rotations from Euclidean to Lorentzian signature are very dangerous. 
  We show that Einstein's gravitational field has zero energy, momentum, and stress. This conclusion follows directly from the gravitational field equations, in conjunction with the differential law of energy-momentum conservation $ T^{\mu\nu}_{;\nu} = 0 $. Einstein rejected this conservation law despite the fact that it is generally covariant. We trace his rejection to a misapplication of Gauss' divergence formula. Finally, we derive the formula which pertains to energy-momentum conservation. 
  Solving dynamical problems in general relativity requires the full machinery of numerical relativity. Wilson has proposed a simpler but approximate scheme for systems near equilibrium, like binary neutron stars. We test the scheme on isolated, rapidly rotating, relativistic stars. Since these objects are in equilibrium, it is crucial that the approximation work well if we are to believe its predictions for more complicated systems like binaries. Our results are very encouraging. 
  A massless scalar field minimally coupled to the gravitational field in a simplified spherical symmetry is discussed. It is shown that, in this case, the solution found by Roberts, describing a scalar field collapse, is in fact the most general one. Taking that solution as departure point, a study of the gravitational collapse for the self-similar conformal case is presented. 
  Some features of extended loops are considered. In particular, the behaviour under diffeomorphism transformations of the wavefunctions with support on the extended loop space are studied. The basis of a method to obtain analytical expressions of diffeomorphism invariants via extended loops are settled. Applications to knot theory and quantum gravity are considered. 
  When introducing special relativity, an elegant connection to familiar rules governing Galilean constant acceleration can be made, by describing first the discovery at high speeds that the clocks (as well as odometers) of different travelers may proceed at different rates. One may then show how to parameterize any given interval of constant acceleration with {\em either}: Newtonian (low-velocity approximation) time, inertial relativistic (unaccelerated observer) time, or traveler proper (accelerated observer) time, by defining separate velocities for each of these three kinematics as well. Kinematic invariance remains intact for proper acceleration since $m a_o = dE/dx$. This approach allows students to solve relativistic constant acceleration problems {\em with the Newtonian equations}! It also points up the self-contained and special nature of the accelerated-observer kinematic, with its frame-invariant time, 4-vector velocities which in traveler terms exceed Newtonian values and the speed of light, and of course relativistic momentum conservation. 
  In the present paper the determination of the {\it pp}-wave metric form the geometry of certain spacelike two-surfaces is considered. It has been shown that the vanishing of the Dougan--Mason quasi-local mass $m_{\$}$, associated with the smooth boundary $\$:=\partial\Sigma\approx S^2$ of a spacelike hypersurface $\Sigma$, is equivalent to the statement that the Cauchy development $D(\Sigma)$ is of a {\it pp}-wave type geometry with pure radiation, provided the ingoing null normals are not diverging on $\$ $ and the dominant energy condition holds on $D(\Sigma)$. The metric on $D(\Sigma)$ itself, however, has not been determined. Here, assuming that the matter is a zero-rest-mass-field, it is shown that both the matter field and the {\it pp}-wave metric of $D(\Sigma)$ are completely determined by the value of the zero-rest-mass-field on $\$ $ and the two dimensional Sen--geometry of $\$ $ provided a convexity condition, slightly stronger than above, holds. Thus the {\it pp}-waves can be characterized not only by the usual Cauchy data on a {\it three} dimensional $\Sigma$ but by data on its {\it two} dimensional boundary $\$ $ too. In addition, it is shown that the Ludvigsen--Vickers quasi-local angular momentum of axially symmetric {\it pp}-wave geometries has the familiar properties known for pure (matter) radiation. 
  The quantum gravity has great difficulties with application of the probability notion. In given article this problem is analyzed according to algorithmic viewpoint. According to A.N. Kolmogorov, the probability notion can be connected with algorithmic complexity of given object. The paper proposes an interpretation of quantum gravity, according to which an appearance of something corresponds to its Kolmogorov's algorithmic complexity. By this viewpoint the following questions are considered: the quantum transition with supplementary coordinates splitting off, the algorithmic complexity of the Schwarzschild black hole is estimated, the redefinition of the Feynman path integral, the quantum birth of the Euclidean Universe with the following changing of the metric signature. 
  We analyze the possibility of testing local Lorentz invariance from the observation of tau decays. Future prospects of probing distances below the electroweak characteristic scale are discussed. 
  We continue here the exam \cite{LuMa} of a theory of gravity that satisfies the Einstein Equivalence Principle (EEP) for any kind of matter/energy, except for the gravitational energy. This is part of a research program that intends to re-examine the standard Feynman-Deser approach of field theoretical derivation of Einstein\rq s General Relativity. The hypothesis implicit in such precedent derivations \cite{Feynman} \cite{Deser} concerns the universality of gravity interaction. Although there is a strong observational basis supporting the universality of matter to gravity interaction, there is not an equivalent situation that supports the hypothesis that gravity interacts with gravity as any other form of non-gravitational energy. We analyse here a kind of gravity-gravity interaction distinct from GR but, as we shall see, that conforms with the actual status of observation. We exhibit the gravitational field produced by a spherically symmetric static configuration as described in this field theory of gravity. The values that we obtain for the standard PPN parameters ($\alpha = \beta = \gamma = 1$) coincide with those of General Relativity. Thus, as we pointed out in a previous paper, the main different aspect of our theory and GR concerns the velocity of the gravitational waves. Since there is a large expectation that the detection of gravitational waves will occur in the near future, the question of which theory describes Nature better will probably be settled soon. 
  Ashtekar's formulation for canonical quantum gravity is known to possess the topological solutions which have their supports only on the moduli space $\CN$ of flat $SL(2,C)$ connections. We show that each point on the moduli space $\CN$ corresponds to a geometric structure, or more precisely the Lorentz group part of a family of Lorentzian structures, on the flat (3+1)-dimensional spacetime. A detailed analysis is given in the case where the spacetime is homeomorphic to $R\times T^{3}$. Most of the points on the moduli space $\CN$ yield pathological spacetimes which suffers from singularities on each spatial hypersurface or which violates the strong causality condition. There is, however, a subspace of $\CN$ on which each point corresponds to a family of regular spacetimes. 
  It is demonstrated that, in the adiabatic approximation, non-Equatorial circular orbits of particles in the Kerr metric (i.e. orbits of constant Boyer-Lindquist radius) remain circular under the influence of gravitational radiation reaction. A brief discussion is given of conditions for breakdown of adiabaticity and of whether slightly non-circular orbits are stable against the growth of eccentricity. 
  We investigate the vacuum and charged spherically symmetric static solutions of the Einstein equations on cosmological background. The background metric is not flat, but curved, with constant - curvature spatial sections. Both vacuum and charged cases contain two branches. The first branches transform into the Schwarzschild and Reissner-Nordstr\"om solutions if the background metric goes to the Minkovski one. The second branches describe wormholes and have no Einstein limit. 
  We describe a deformation of the observable algebra of quantum gravity in which the loop algebra is extended to framed loops. This allows an alternative nonperturbative quantization which is suitable for describing a phase of quantum gravity characterized by states which are normalizable in the measure of Chern-Simons theory. The deformation parameter, q, depends on the cosmological constant. The Mandelstam identities are extended to a set of relations which are governed by the Kauffman bracket so that the spin network basis is deformed to a basis of SU(2)q spin networks. Corrections to the actions of operators in non-perturbative quantum gravity may be readily computed using recoupling theory; the example of the area observable is treated here. Finally, eigenstates of the q-deformed Wilson loops are constructed, which may make possible the construction of a q-deformed connection representation through an inverse transform. 
  The construction of self-dual vortex solutions to the Chern-Simons-Higgs model (with a suitable eighth-order potential) coupled to Einstein gravity in (2 + 1) dimensions is reconsidered. We show that the self-duality condition may be derived from the sole assumption $g_{00} = 1$. Next, we derive a family of exact, doubly self-dual vortex solutions, which interpolate between the symmetrical and asymmetrical vacua. The corresponding spacetimes have two regions at spatial infinity. The eighth-order Higgs potential is positive definite, and closed timelike curves are absent, if the gravitational constant is chosen to be negative. 
  The usual equivalence between the Palatini and metric (or affinity and vielbein) formulations of Einstein theory fails in two spacetime dimensions for its ``Kaluza--Klein" reduced (as well as for its standard) version. Among the differences is the necessary vanishing of the cosmological constant in the first order forms. The purely affine Eddington formulation of Einstein theory also fails here. 
  We consider Gell-Mann and Hartle's consistent histories formulation of quantum cosmology in the interpretation in which one history, chosen randomly according to the decoherence functional probabilities, is realised from each consistent set. We show that in this interpretation, if one assumes that an observed quasiclassical structure will continue to be quasiclassical, one cannot infer that it will obey the predictions of classical or Copenhagen quantum mechanics. 
  This is a pedagogical introduction to the treatment of general relativity as a quantum effective field theory. Gravity fits nicely into the effective field theory description and forms a good quantum theory at ordinary energies. 
  The Brill-Hartle gravitational geon construct as a spherical shell of small amplitude, high frequency gravitational waves is reviewed and critically analyzed. The Regge-Wheeler formalism is used to represent gravitational wave perturbations of the spherical background as a superposition of tensor spherical harmonics and an attempt is made to build a non-singular solution to meet the requirements of a gravitational geon. High-frequency waves are seen to be a necessary condition for the geon and the field equations are decomposed accordingly. It is shown that this leads to the impossibility of forming a spherical gravitational geon. The attempted constructs of gravitational and electromagnetic geons are contrasted. The spherical shell in the proposed Brill-Hartle geon does not meet the regularity conditions required for a non-singular source and hence cannot be regarded as an adequate geon construct. Since it is the high frequency attribute which is the essential cause of the geon non-viability, it is argued that a geon with less symmetry is an unlikely prospect. The broader implications of the result are discussed with particular reference to the problem of gravitational energy. 
  The discussion is limited to first-class parametrized systems, where the definition of time evolution and observables is not trivial, and to finite dimensional systems in order that technicalities do not obscure the conceptual framework. The existence of reasonable true, or physical, degrees of freedom is rigorously defined and called {\em local reducibility}. A proof is given that any locally reducible system admits a complete set of perennials. For locally reducible systems, the most general construction of time evolution in the Schroedinger and Heisenberg form that uses only geometry of the phase space is described. The time shifts are not required to be 1symmetries. A relation between perennials and observables of the Schroedinger or Heisenberg type results: such observables can be identified with certain classes of perennials and the structure of the classes depends on the time evolution. The time evolution between two non-global transversal surfaces is studied. The problem is posed and solved within the framework of the ordinary quantum mechanics. The resulting non-unitarity is different from that known in the field theory (Hawking effect): state norms need not be preserved so that the system can be lost during the evolution of this kind. 
  The classical concept of "mass density" is not fundamental to the quantum theory of matter. Therefore, mass density cannot be the source of gravitation. Here, we treat electromagnetic energy, momentum, and stress as its source. The resulting theory predicts that the gravitational potential near any charged elementary particle is many orders of magnitude greater than the Newtonian value. 
  The fundamental theorem of submanifolds is adapted to space-times. It is shown that the integrability conditions for the existence of submanifolds of a pseudo-Euclidean space contain the Einstein and Yang-Mills equations. 
  The diagonal metric tensor whose components are functions of one spatial coordinate is considered. Einstein's field equations for a perfect-fluid source are reduced to quadratures once a generating function, equal to the product of two of the metric components, is chosen. The solutions are either static fluid cylinders or walls depending on whether or not one of the spatial coordinates is periodic. Cylinder and wall sources are generated and matched to the vacuum (Levi--Civita) space--time. A match to a cylinder source is achieved for $-\frac{1}{2}<\si<\frac{1}{2}$, where $\si$ is the mass per unit length in the Newtonian limit $\si\to 0$, and a match to a wall source is possible for $|\si|>\frac{1}{2}$, this case being without a Newtonian limit; the positive (negative) values of $\si$ correspond to a positive (negative) fluid density. The range of $\si$ for which a source has previously been matched to the Levi--Civita metric is $0\leq\si<\frac{1}{2}$ for a cylinder source. 
  We study the static and spherically symmetric exact solution of the Einstein-massless scalar equations given by Janis, Newman and Winicour. We find that this solution satisfies the weak energy condition and has strong globally naked singularity. 
  In models where the constants of Nature can take more than one set of values, the cosmological wave function $\psi$ describes an ensemble of universes with different values of the constants. The probability distribution for the constants can be determined with the aid of the `principle of mediocrity' which asserts that we are a `typical' civilization in this ensemble. I discuss the implications of this approach for inflationary scenarios, the origin of density fluctuations, and the cosmological constant. 
  The semiclassical collapse of a homogeneous sphere of dust is studied. After identifying the independent dynamical variables, the system is canonically quantised and coupled equations describing matter (dust) and gravitation are obtained. The conditions for the validity of the adiabatic (Born--Oppenheimer) and semiclassical approximations are derived. Further on neglecting back--reaction effects, it is shown that in the vicinity of the horizon and inside the dust the Wightman function for a conformal scalar field coupled to a monopole emitter is thermal at the characteristic Hawking temperature. 
  The flat-space limit of the one-loop effective potential for SO(10) GUT theories in spatially flat Friedmann-Robertson-Walker cosmologies is applied to study the dynamics of the early universe. The numerical integration of the corresponding field equations shows that, for such grand unified theories, a sufficiently long inflationary stage is achieved for suitable choices of the initial conditions. However, a severe fine tuning of these initial conditions is necessary to obtain a large $e$-fold number. In the direction with residual symmetry $SU(4)_{PS} \otimes SU(2)_{L} \otimes SU(2)_{R}$, one eventually finds parametric resonance for suitable choices of the free parameters of the classical potential. This phenomenon leads in turn to the end of inflation. 
  Classical methods of differential geometry are used to construct equations of motion for particles in quantum, electrodynamic and gravitational fields. For a five dimensional geometrical system, the equivalence principle can be extended. Local transformations generate the effects of electromagnetic and quantum fields. A combination of five dimensional coordinate transformations and internal conformal transformations leads to a quantum Kaluza-Klein metric. The theories of Weyl and Kaluza can be interrelated when charged particle quantum mechanics is included. Measurements of trajectories are made relative to an observers' space that is defined by the motion of neutral particles. It is shown that a preferred set of null geodesics describe valid classical and quantum trajectories. These are tangent to the probability density four vector. This construction establishes a generally covariant basis for geodesic motion of quantum states. 
  A theoretical study is made of conformal factors in certain types of physical theories based on classical differential geometry. Analysis of quantum versions of Weyl's theory suggest that similar field equations should be available in four, five and more dimensions. Various conformal factors are associated with the wave functions of source and test particles. This allows for quantum field equations to be developed. The curvature tensors are calculated and separated into gravitational, electromagnetic and quantum components. Both four and five dimensional covariant theories are studied. Nullity of the invariant five scalar of curvature leads to the Klein-Gordon equation. The mass is associated with an eigenvalue of the differential operator of the fifth dimension. Different concepts of interaction are possible and may apply in a quantum gravitational theory. 
  Although an important issue in canonical quantization, the problem of representing the constraint algebra in the loop representation of quantum gravity has received little attention. The only explicit computation was performed by Gambini, Garat and Pullin for a formal point-splitting regularization of the diffeomorphism and Hamiltonian constraints. It is shown that the calculation of the algebra simplifies considerably when the constraints are expressed not in terms of generic area derivatives but rather as the specific shift operators that reflect the geometric meaning of the constraints. 
  In this paper the quantization of the 2$+$1-dimensional gravity couplet to the massless Dirac field is carried out. The problem is solved by the application of the new Dynamic Quantization Method [1,2]. It is well-known that in general covariant theories such as gravitation, a Hamiltonian is any linear combination of the first class constraints, which can be considered as gauge transformation generators. To perform quantization, the Dirac field modes with gauge invariant creation and annihilation operators are selected. The regularization of the theory is made by imposing an infinite set of the second class constraints: almost all the gauge invariant creation and annihilation operators (except for a finite number) are put equal to zero. As a result the regularized theory is gauge invariant. The gauge invariant states are built by using the remained gauge invariant fermion creation operators similar to the usual construction of the states in any Fock space. The developed dynamic quantization method can construct a mathematically correct perturbation theory in a gravitational constant. 
  In this letter we propose a semi-analitical method of evaluation of the power spectrum of a circular moving Unruh-type detector using the method of residue and compare the spectrum with the already known result in the relativistic limit. 
  The approach to asymptotic electromagnetic fields introduced by Goldberg and Kerr is used to study various aspects of Lorentz Covariant Gravity. Retarded multipole moments of the source, the central objects of this study, are defined, and a sequence of conservation equations for these are derived. These equations are used extensively throughout the paper. The solution of the linearized Einstein equation is obtained in terms of the retarded moments for a general bound source, correct to $O(r^{-4})$. This is used to obtain the peeling--off of the linearized field, and to study the geometric optics approximation for the field and for the energy-momentum pseudotensor of the field. It is shown that the energy-momentum 4-vector splits into the `total radiated 4-momentum' and the `bound 4-momentum of the source', similar to the case of the electromagnetic field. In the case of a source which has only retarded pole, dipole and quadrupole moments, a decomposition into arbitrary functions of a null coordinate is obtained which allows comparison with the solutions for linearized gravity obtained by other authors. 
  A discretized version of canonical gravity in (3+1)-d introduced in a previous paper is further developed, introducing the Liouville form and the Poisson brackets, and studying them in detail in an explicit parametrization that shows the nature of the variables when the second class constraints are imposed. It is then shown that, even leaving aside the difficult question of imposing the first class constraints on the states, it is impossible to quantize the model directly, using complex variables and leaving the second class constraints to fix the metric of the quantum Hilbert, because one cannot find a metric which makes the area variables hermitean. 
  This paper relates skein spaces based on the Kauffman bracket and spin structures. A spin structure on an oriented 3-manifold provides an isomorphism between the skein space for parameter A and the skein space for parameter -A.    There is an application to Penrose's binor calculus, which is related to the tensor calculus of representations of SU(2). The perspective developed here is that this tensor calculus is actually a calculus of spinors on the plane, and the matrices a re determined by a type of spinor transport which generalises to links in any 3-manifold.    A second application shows that there is a skein space which is the algebra of functions on the set of spin structures for the 3-manifold. 
  A set of Maplev R.3 software routines, for plotting 2D/3D projections of Poincar\'e surfaces-of-section of Hamiltonian dynamical systems, is presented. The package consists of a plotting-command plus a set of facility-commands for a quick setup of the Hamilton equations of motion, initial conditions for numerical experiments, and for the zooming of plots. 
  The discrete picture of geometry arising from the loop representation of quantum gravity can be extended by a quantum deformation. The operators for area and volume defined in the q-deformation of the theory are partly diagonalized. The eigenstates are expressed in terms of q-deformed spin networks. The q-deformation breaks some of the degeneracy of the volume operator so that trivalent spin-networks have non-zero volume. These computations are facilitated by use of a technique based on the recoupling theory of SU(2)_q, which simplifies the construction of these and other operators through diffeomorphism invariant regularization procedures. 
  We review the problem of divergences in one--loop thermodynamical quantities for matter fields in thermal equilibrium on a black hole background. We discuss a number of results obtained for various thermodynamical quantities. Then we discuss the ansatz called ``literal interpretation" of zeroth law of black hole mechanics and try to explain the diseases of the conical defect procedure in light of this ansatz. Finally, an analysis of the consequences implied by our ansatz on the calculation of the partition function is made. 
  According to previous work on magnetic monopoles, static regular solutions are nonexistent if the vacuum expectation value of the Higgs field $\eta$ is larger than a critical value $\eta_{{\rm cr}}$, which is of the order of the Planck mass. In order to understand the properties of monopoles for $\eta>\eta_{{\rm cr}}$, we investigate their dynamics numerically. If $\eta$ is large enough ($\gg\eta_{{\rm cr}}$), a monopole expands exponentially and a wormhole structure appears around it, regardless of coupling constants and initial configuration. If $\eta$ is around $\eta_{{\rm cr}}$, there are three types of solutions, depending on coupling constants and initial configuration: a monopole either expands as stated above, collapses into a black hole, or comes to take a stable configuration. 
  According to previous work on magnetic monopoles, static regular solutions are nonexistent if the vacuum expectation value of the Higgs field $\eta$ is larger than a critical value $\eta_{{\rm cr}}$, which is of the order of the Planck mass. In order to understand the properties of monopoles for $\eta>\eta_{{\rm cr}}$, we investigate their dynamics numerically and classify those dynamical solutions into three types as follows. If $\eta$ is larger than another critical value $\eta_{{\rm inf}}~(>\eta_{{\rm cr}})$, a monopole inflates and a wormhole structure appears around it. In the case of $\eta_{{\rm cr}}<\eta<\eta_{{\rm inf}}$, inflation does not occur and the dynamics depend on the ratio of the Higgs self coupling constant $\lambda$ and the gauge coupling constant $e^2$: if $\lambda/e^2\stackrel{<}{\sim}1$, a monopole just shrinks and becomes a black hole; otherwise, a monopole approaches a stable configuration. 
  A consistent variational procedure applied to the gravitational action requires according to Gibbons and Hawking a certain balance between the volume and boundary parts of the action. We consider the problem of preserving this balance in the quantum effective action for the matter non-minimally coupled to metric. It is shown that one has to add a special boundary term to the matter action analogous to the Gibbons-Hawking one. This boundary term modifies the one-loop quantum corrections to give a correct balance for the effective action as well. This means that the boundary UV divergences do not require independent renormalization and are automatically renormalized simultaneously with their volume part. This result is derived for arbitrary non-minimally coupled matter. The example of 2D Maxwell field is considered in much detail. The relevance of the results obtained to the problem of the renormalization of the black hole entropy is discussed. 
  An extensive survey of gravitational-wave modes for uniform density stars is presented. The study covers stars ranging in compactness from $R/M=100$ to the limit of stability in general relativity: $R/M = 9/4$. We establish that polar and axial gravitational-wave modes exist for all these stellar models. Moreover, there are two distinct families of both axial and polar modes. We discuss the physics of these modes and argue that one family is primarily associated with the interior of the star, while the second family is mainly associated with the stellar surface. We also show that the problem of axial perturbations has all the essential features of the polar problem as far as gravitational waves are concerned. This means that the axial problem is much more important than has previously been assumed. We also find some surprising features, such as avoided crossings between the polar gravitational-wave modes and the Kelvin f-mode as the star becomes very compact. This seems to suggest that the f-mode should be considered on equal footing with the polar w-modes for ultracompact stars. All modes may have the main character of trapped modes inside the curvature potential barrier for $R/M < 3$. 
  We address two basic issues in the theory of galaxy formation from fluctuations of quantum fields: 1) the nature and origin of noise and fluctuations and 2) the conditions for using a classical stochastic equation for their description. On the first issue, we derive the influence functional for a $\lambda \phi^4 $ field in a zero-temperature bath in de Sitter universe and obtain the correlator for the colored noises of vacuum fluctuations. This exemplifies a new mechanism we propose for colored noise generation which can act as seeds for galaxy formation with non-Gaussian distributions. For the second issue, we present a (functional) master equation for the inflaton field in de Sitter universe. By examining the form of the noise kernel we study the decoherence of the long-wavelength sector and the conditions for it to behave classically. 
  We find an {\it exact} pp--gravitational wave solution of the fourth order gravity field equations. Outside the (delta--like) source this {\it not} a vacuum solution of General Relativity. It represents the contribution of the massive, $m=(-\beta)^{-1/2}$, spin--two field associated to the Ricci squared term in the gravitational Lagrangian. The fourth order terms tend to make milder the singularity of the curvature at the point where the particle is located. We generalize this analysis to $D$--dimensions, extended sources, and higher than fourth order theories. We also briefly discuss the scattering of fields by this kind of plane gravitational waves. 
  Quantum fields are investigated in the (2+1)-open-universes with non-trivial topologies by the method of images. The universes are locally de Sitter spacetime and anti-de Sitter spacetime. In the present article we study spacetimes whose spatial topologies are a torus with a cusp and a sphere with three cusps as a step toward the more general case. A quantum energy momentum tensor is obtained by the point stripping method. Though the cusps are no singularities, the latter cusps cause the divergence of the quantum field. This suggests that only the latter cusps are quantum mechanically unstable. Of course at the singularity of the background spacetime the quantum field diverges. Also the possibility of the divergence of topological effect by a negative spatial curvature is discussed. Since the volume of the negatively curved space is larger than that of the flat space, one see so many images of a single source by the non-trivial topology. It is confirmed that this divergence does not appear in our models of topologies. The results will be applicable to the case of three dimensional multi black hole\cite{BR}. 
  Recently Paw{\l}owski and R\c{a}czka proposed a unified model for the fundamental interactions which does not contain a physical Higgs field. The gravitational field equation of their model is rederived under heavy use of the computer algebra system Mathematica and its package MathTensor. 
  A Chern-Simons action for supergravity in odd-dimensional spacetimes is proposed. For all odd dimensions, the local symmetry group is a non trivial supersymmetric extension of the Poincar\'e group. In $2+1$ dimensions the gauge group reduces to super-Poincar\'e, while for $D=5$ it is super-Poincar\'e with a central charge. In general, the extension is obtained by the addition of a 1-form field which transforms as an antisymmetric fifth-rank tensor under Lorentz rotations. Since the Lagrangian is a Chern-Simons density for the supergroup, the supersymmetry algebra closes off shell without the need of auxiliary fields. 
  We show that the energy-momentum four-vector of a planet, p = mu, is conserved during geodesic motion. Therefore, there is no exchange of energy-momentum with the gravitational field. We discuss the meaning of a gravitational field which is free of energy, momentum, and stress. 
  We investigate the distribution of energy density in a stationary self-reproducing inflationary universe. We show that the main fraction of volume of the universe in a state with a given density at any given moment of proper time t is concentrated near the centers of deep exponentially wide spherically symmetric wells in the density distribution. Since this statement is very surprising and counterintuitive, we perform our investigation by three different analytical methods to verify our conclusions, and then confirm our analytical results by computer simulations. If one assumes that we are typical observers living in the universe at a given moment of time, then our results may imply that we should live near the center of a deep and exponentially large void, which we will call infloid. Validity of this particular interpretation of our results is not quite clear since it depends on the as-yet unsolved problem of measure in quantum cosmology. Therefore at the moment we would prefer to consider our results simply as a demonstration of nontrivial properties of the hypersurface of a given time in the fractal self-reproducing universe, without making any far-reaching conclusions concerning the structure of our own part of the universe. Still we believe that our results may be of some importance since they demonstrate that nonperturbative effects in quantum cosmology, at least in principle, may have significant observational consequences, including an apparent violation of the Copernican principle. 
  We review and correct the classical critical exponents characterizing the transition from negative to positive black hole's heat capacity at high charge--angular momentum. We discuss the stability properties of black holes as a thermodynamic system in equilibrium with a radiation bath (canonical ensamble) by using the Helmholtz free energy potential. We finally analytically extend the analysis to negative mass holes and study its thermodynamical stability behavior. 
  In the loop representation the quantum constraints of gravity can be solved. This fact allowed significant progress in the understanding of the space of states of the theory. The analysis of the constraints over loop dependent wavefunctions has been traditionally based upon geometric (in contrast to analytic) properties of the loops. The reason for this preferred way is twofold: for one hand the inherent difficulties associated with the analytic loop calculus, and on the other our limited knowledge about the analytic properties of knots invariants. Extended loops provide a way to overcome the difficulties at both levels. For one hand, a systematic method to construct analytic expressions of diffeomorphism invariants (the extended knots) in terms of the Chern-Simons propagators can be developed. Extended knots are simply related to ordinary knots (at least formally). The analytic expressions of knot invariants could be produced then in a generic way. On the other hand, the evaluation of the Hamiltonian over extended loop wavefunctions can be thoroughly accomplished in the extended loop framework. These two ingredients promote extended loops as a potential resort for answering important questions about quantum gravity. 
  The possible external couplings of an extended non-relativistic classical system are characterized by gauging its maximal dynamical symmetry group at the center-of-mass. The Galilean one-time and two-times harmonic oscillators are exploited as models. The following remarkable results are then obtained: 1) a peculiar form of interaction of the system as a whole with the external gauge fields; 2) a modification of the dynamical part of the symmetry transformations, which is needed to take into account the alteration of the dynamics itself, induced by the {\it gauge} fields. In particular, the Yang-Mills fields associated to the internal rotations have the effect of modifying the time derivative of the internal variables in a scheme of minimal coupling (introduction of an internal covariant derivative); 3) given their dynamical effect, the Yang-Mills fields associated to the internal rotations apparently define a sort of Galilean spin connection, while the Yang-Mills fields associated to the quadrupole momentum and to the internal energy have the effect of introducing a sort of dynamically induced internal metric in the relative space. 
  We present analytic expressions that approximate the behavior of the spacetime of a collapsing spherically symmetric scalar field in the critical regime first discovered by Choptuik. We find that the critical region of spacetime can usefully be divided into a ``quiescent'' region and an ``oscillatory'' region, and a moving ``transition edge'' that separates the two regions. We find that in each region the critical solution can be well approximated by a flat spacetime scalar field solution. A qualitative nonlinear matching of the solutions across the edge yields the right order of magnitude for the oscillations of the discretely self-similar critical solution found by Choptuik. 
  In this paper we study a new family of black hole initial data sets corresponding to distorted ``Kerr'' black holes with moderate rotation parameters, and distorted Schwarzschild black holes with even- and odd-parity radiation. These data sets build on the earlier rotating black holes of Bowen and York and the distorted Brill wave plus black hole data sets. We describe the construction of this large family of rotating black holes. We present a systematic study of important properties of these data sets, such as the size and shape of their apparent horizons, and the maximum amount of radiation that can leave the system during evolution. These data sets should be a very useful starting point for studying the evolution of highly dynamical black holes and can easily be extended to 3D. 
  We show that 't Hooft's representation of (2+1)-dimensional gravity in terms of flat polygonal tiles is closely related to a gauge-fixed version of the covariant Hamiltonian lattice theory. 't Hooft's gauge is remarkable in that it leads to a Hamiltonian which is a linear sum of vertex Hamiltonians, each of which is defined modulo $2 \pi$. A cyclic Hamiltonian implies that ``time'' is quantized. However, it turns out that this Hamiltonian is {\it constrained}. If one chooses an internal time and solves this constraint for the ``physical Hamiltonian'', the result is not a cyclic function. Even if one quantizes {\it a la Dirac}, the ``internal time'' observable does not acquire a discrete spectrum. We also show that in Euclidean 3-d lattice gravity, ``space'' can be either discrete or continuous depending on the choice of quantization. Finally, we propose a generalization of 't Hooft's gauge for Hamiltonian lattice formulations of topological gravity dimension 4. 
  In this work we find the Killing vector fields of the riemannian submanifolds of the thermodynamic phase space of an ideal gas and show that the isometry group corresponding to them is homomorphic to the euclidean group $E(2)$. We also give the embedding of these submanifolds in the euclidean space $(R^{3},\ delta)$. 
  In the present work we show that the Einstein equations on $M$ without cosmological constant and with perfect fluid as source, can be obtained from the field equations for vacuum with cosmological constant on the principal fibre bundle $P(\frac{1}{I} M,U(1))$, $M$ being the space-time and $I$ the radius of the internal space $U(1)$. 
  By investigating the canonical commutation rules for gravitating quantized particles in a 2+1 dimensional world it is found that these particles live on a space-time lattice. The space-time lattice points can be characterized by three integers. Various representations are possible, the details depending on the topology chosen for energy-momentum space. We find that an $S_2\times S_1$ topology yields a physically most interesting lattice within which first quantization of Dirac particles is possible. An $S_3$ topology also gives a lattice, but does not allow first quantized particles. 
  The problem of relativistic stellar pulsations is studied in a somewhat ad hoc approximation that ignores all fluid motions. This {\em Inverse Cowling Approximation} (ICA) is motivated by two observations: 1) For highly damped ($w$-mode) oscillations the fluid plays very little role. 2) If the fluid motion is neglected the problem for polar oscillation modes becomes similar to that for axial modes. Using the ICA we find a polar mode spectrum that has all features of the $w$-mode spectrum of the full problem. Moreover, in the limit of superdense stars, we find the ICA spectrum to be qualitatively similar to that of the axial modes. These results clearly show the importance of general relativity for the pulsation modes of compact stars, and that there are modes whose existence do not depend on motions of the fluid at all; pure ``spacetime'' modes. 
  In this paper we study the evolution of density perturbations in the framework of Phase Coupling Gravity theory at the very early universe. We show that these perturbation display an exponential-like behaviour. 
  We describe a numerical method for calculating the (3+1) dimensional general relativistic hydrodynamics of a coalescing neutron-star binary system. The relativistic field equations are solved at each time slice with a spatial 3-metric chosen to be conformally flat. Against this solution to the general relativistic field equations the hydrodynamic variables and gravitational radiation are allowed to respond. The gravitational radiation signal is derived via a multipole expansion of the metric perturbation to the hexadecapole order including both mass and current moments and a correction for the slow motion approximation. Using this expansion, the effect of gravitational radiation on the system evolution can also be recovered by introducing an acceleration term in the matter evolution. 
  The properties of gravitational kinks are studied within some simple models of two dimensional gravity. In spacetimes of cylindrical topology we prove the existence of kinks of constant curvature with arbitrary kink numbers. In $R^1\times R^1$ spacetimes $m=1$ kink solutions of the equation $R=0$ are found, whereas $|m|>1$ flat kinks are proved not to exist. We give a detailed analysis of the behaviour of gravitational kinks under coordinate transformations. Viewed as nonsingular black holes $|m|>1$ kink solutions are found within a simple dilaton gravity theory. The general form of the potential function is determined from the demand that the theory possesses an arbitrary number of inequivalent kink configurations. 
  In a recent Physical Review Letter, Wilson and Mathews presented some interesting numerical calculations of a system of two equally massive neutron stars in strong-field gravity. In particular they estimated the innermost stable circular orbit in their system. Here we point out a possibly important consequence of their results: Their calculated configurations have total angular momentum $J$ and total mass $M$ too large to form any Kerr black hole: $J>M^2$, in constrast to previous calculations of the innermost stable circular orbit. 
  We show that in all theories in which black hole hair has been discovered, the region with non-trivial structure of the non-linear matter fields must extend beyond 3/2 the horizon radius, independently of all other parameters present in the theory. We argue that this is a universal lower bound that applies in every theory where hair is present. This {\it no short hair conjecture} is then put forward as a more modest alternative to the original {\it no hair conjecture}, the validity of which now seems doubtful. 
  A new no-hair theorem is formulated which rules out a very large class of non-minimally coupled finite scalar dressing of an asymptotically flat, static, and spherically symmetric black-hole. The proof is very simple and based in a covariant method for generating solutions for non-minimally coupled scalar fields starting from the minimally coupled case. Such method generalizes the Bekenstein method for conformal coupling and other recent ones. We also discuss the role of the finiteness assumption for the scalar field. 
  We perform the canonical quantization of a relativistic spinless particle moving in a curved and static spacetime. We show that the classical theory already describes at the same time both particle and antiparticle. The analyses involves time-depending constraints and we are able to construct the two-particle Hilbert space. The requirement of a static spacetime is necessary in order to have a well defined Schr\"odinger equation and to avoid problems with vacuum instabilities. The severe ordering ambiguities we found are in essence the same ones of the well known non-relativistic case. 
  Imposing extendibility on Kasner-Fronsdal black hole local isometric embedding is equivalent to removing conic singularities in Kruskal representation. Allowing for globally non-trivial (living in $M_{5}\times S_{1}$) embeddings, parameterized by $k$, extendibility can be achieved for apparently forbidden frequencies $\omega_{1}(k)\le\omega (k)\le \omega_{2}(k)$. The Hawking-Gibbons limit, say $\displaystyle{\omega_{1,2}(0)= {1\over{4M}}}$ for Schwarzschild geometry, is respected. The corresponding Kruskal sheets are viewed as slices in some Kaluza-Klein background. Euclidean $k$ discreteness, dictated by imaginary time periodicity, is correlated with twistor flux quantization. 
  The interaction of a gravitational wave with a system made of an RLC circuit forming one end of a mechanical harmonic oscillator is investigated. We show that, in some configurations, the coherent interaction of the wave with both the mechanical oscillator and the RLC circuit gives rise to a mechanical quality factor increase of the electromagnetic signal. When this system is used as an amplifier of gravitational periodic signals in the frequency range 50-1000 Hz, at ultracryogenic temperatures and for sufficiently long integration times (up to 4 months), a sensitivity of 10^(-24)-10^(-27) on the amplitude of the metric could be achieved when thermal noise, shot noise and amplifier back--action are considered. 
  In this paper we derive an expression for the conserved Pauli-Lubanski scalar in 't Hooft's polygon approach to 2+1-dimensional gravity coupled to point particles. We find that it is represented by an extra spatial shift $\Delta$ in addition to the usual identification rule (being a rotation over the cut). For two particles this invariant is expressed in terms of 't Hooft's phase-space variables and we check its classical limit. 
  We report on a systematic study of the dynamics of gravitational waves in full 3D numerical relativity. We find that there exists an interesting regime in the parameter space of the wave configurations: a near-linear regime in which the amplitude of the wave is low enough that one expects the geometric deviation from flat spacetime to be negligible, but nevertheless where nonlinearities can excite unstable modes of the Einstein evolution equations causing the metric functions to evolve out of control. The implications of this for numerical relativity are discussed. 
  We put forth a few ideas on coordinate conditions and their implementation in numerical relativity. Coordinate conditions are important for the long time scale simulations of relativistic systems, e.g., for the determination of gravitational waveforms from astrophysical events to be measured by LIGO/VIRGO. We demonstrate the importance of, and propose methods for, the {\it active enforcement} of coordinate properties. In particular, the constancy of the determinant of the spatial 3-metric is investigated as such a property. We propose an exceedingly simple but powerful idea for implementing elliptic coordinate conditions that not only makes possible the use of complicated elliptic conditions, but is also {\it orders of magnitude} more efficient than existing methods for large scale 3D simulations. 
  We propose an exact Hamiltonian lattice theory for (2+1)-dimensional spacetimes with homogeneous curvature. By gauging away the lattice we find a generalization of the ``polygon representation'' of (2+1)-dimensional gravity. We compute the holonomies of the Lorentz connection ${\bf A}_i = \omega_i^a {\bf L}_a + e_i^a {\bf K}_a$ and find that the cycle conditions are satisfied only in the limit $\Lambda \to 0$. This implies that, unlike in (2+1)-dimensional Einstein gravity, the connection ${\bf A}$ is not flat. If one modifies the theory by taking the cycle conditions as constraints, then one finds that the constraints algebra is first-class only if the Poisson bracket structure is deformed. This suggests that a finite theory of quantum gravity would require either a modified action including higher-order curvature terms, or a deformation of the commutator structure of the metric observables. 
  In the last few years renewed interest in the 3-tensor potential $L_{abc} $ proposed by Lanczos for the Weyl curvature tensor has not only clarified and corrected Lanczos's original work, but generalised the concept in a number of ways. In this paper we carefully summarise and extend some aspects of these results, and clarify some misunderstandings in the literature. We also clarify some comments in a recent paper by Dolan and Kim; in addition, we correct some internal inconsistencies in their paper and extend their results.   The following new results are also presented. The (computer checked) complicated second order partial differential equation for the 3-potential, in arbitrary gauge, for Weyl candidates satisfying Bianchi-type equations is given -- in those $n $-dimensional spaces (with arbitrary signature) for which the potential exists; this is easily specialised to Lanczos potentials for the Weyl curvature tensor. It is found that it is {\it only} in 4-dimensional spaces (with arbitrary signature and gauge), that the non-linear terms disappear and that the awkward second order derivative terms cancel; for 4-dimensional spacetimes (with Lorentz signature), this remarkably simple form was originally found by Illge, using spinor methods. It is also shown that, for most 4-dimensional vacuum spacetimes, any 3-potential in the Lanczos gauges which satisfies a simple homogeneous wave equation must be a Lanczos potential for the Weyl curvature tensor of the background vacuum spacetime. This result is used to prove that the form of a {\it possible} Lanczos potential proposed by Dolan and Kim for a class of vacuum spacetimes is in fact a genuine Lanczos potential for these spacetimes. 
  We analyse the mathematical underpinnings of a mixed hyperbolic-elliptic form of the Einstein equations of motion in which the lapse function is determined by specified mean curvature and the shift is arbitrary. We also examine a new recently-published first order symmetric hyperbolic form of the equations of motion. This paper is dedicated to Andre Lichnerowicz on the occasion of his 80th birthday and will appear in a volume edited by G. Ferrarese. 
  This paper focuses on the imposition of boundary conditions for numerical relativity simulations of black holes. This issue is used to motivate the discussion of a new hyperbolic formulation of 3+1 general relativity. The paper will appear in the Proceedings of the Les Houches School on Astrophysical Sources of Gravitational Radiation, 1995, edited by J.-A. Marck and J.-P. Lasota to be published by Springer-Verlag. 
  In this work some proposals for black hole entropy interpretation are exposed and investigated. In particular I will firstly consider the so called ``entanglement entropy" interpretation, in the framework of the brick wall model, and the divergence problem arising in the one loop calculations of various thermodynamical quantities, like entropy, internal energy and heat capacity. It is shown that the assumption of equality of entanglement entropy and Bekenstein-Hawking one appears to give inconsistent results. These will be a starting point for a different interpretation of black hole entropy based on peculiar topological structures of manifolds with ``intrinsic" thermodynamical features. It is possible to show an exact relation between black hole gravitational entropy (tree level contribution in path integral approach) and topology of these Euclidean space-times. The expression for the Euler characteristic, through the Gauss-Bonnet integral, and the one for entropy for gravitational instantons are proposed in a form which makes the relation between these self evident. Using this relations I shall propose a generalization of Bekenstein-Hawking entropy in which the former and Euler characteristic are related in the equation: $S=\chi A/8$. The results, quoted above, are more largely exposed in previous works. Finally I'll try to expose some conclusions and hypotheses about possible further development of this research. 
  The behavior near the singularity of an isotropic, homogeneous cosmological model with a viscous fluid source is investigated. This turns out to be a relaxation dominated regime. Full extended irreversible thermodynamics is used, and comparison with results of the truncated theory is made. New singular behaviors are found and it is shown that a relaxation dominated inflationary epoch may exist for fluids with small heat capacity. 
  We derive conditions for rotating particle detectors to respond in a variety of bounded spacetimes and compare the results with the folklore that particle detectors do not respond in the vacuum state appropriate to their motion. Applications involving possible violations of the second law of thermodynamics are briefly addressed. 
  Studying two-dimensional evaporating dilatonic black holes, we show that the semiclassical approximation, based on the background field approach, is valid everywhere in regions of weak curvature (including the horizon), as long as one takes into account the effects of back-reaction of the Hawking radiation on the background geometry. 
  We examine the relative performance of algorithms for the calculation of curvature in spacetime. The classical coordinate component method is compared to two distinct versions of the Newman-Penrose tetrad approach for a variety of spacetimes, and distinct coordinates and tetrads for a given spacetime. Within the system GRTensorII, we find that there is no single preferred approach on the basis of speed. Rather, we find that the fastest algorithm is the one that minimizes the amount of time spent on simplification. This means that arguments concerning the theoretical superiority of an algorithm need not translate into superior performance when applied to a specific spacetime calculation. In all cases it is the global simplification strategy which is of paramount importance. An appropriate simplification strategy can change an untractable problem into one which can be solved essentially instantaneously. 
  The features of the fundamental thermodynamical relation (expressing entropy as function of state variables) that arise from the self-gravitating character of a system are analyzed. The models studied include not only a spherically symmetric hot matter shell with constant particle number but also a black hole characterized by a general thermal equation of state. These examples illustrate the formal structure of thermodynamics developed by Callen as applied to a gravitational configuration as well as the phenomenological manner in which Einstein equations largely determine the thermodynamical equations of state. We consider in detail the thermodynamics and quasi-static collapse of a self-gravitating shell. This includes a discussion of intrinsic stability for a one-parameter family of thermal equations of state and the interpretation of the Bekenstein bound. The entropy growth associated with a collapsing sequence of equilibrium states of a shell is computed under different boundary conditions in the quasi-static approximation and compared with black hole entropy. Although explicit expressions involve empirical coefficients, these are constrained by physical conditions of thermodynamical origin. The absence of a Gibbs-Duhem relation and the associated scaling laws for self-gravitating matter systems are presented. 
  We find a first--order partial differential equation whose solutions are all ultralocal scalar combinations of gravitational constraints with Abelian Poisson brackets between themselves. This is a generalisation of the Kucha\v{r} idea of finding alternative constraints for canonical gravity. The new scalars may be used in place of the hamiltonian constraint of general relativity and, together with the usual momentum constraints, replace the Dirac algebra for pure gravity with a true Lie algebra: the semidirect product of the Abelian algebra of the new constraint combinations with the algebra of spatial diffeomorphisms. 
  This article reports on emergence of structures in a class of alternative theories of gravity. These theories do not have any horizon, flatness, initial cosmological singularity and (possibly) quantization problems. The model is characterised by a dynamically induced gravitational constant with a ``wrong'' sign corresponding to repulsive gravitation on the large scale. A non - minimal coupling of a scalar field in the model can give rise to non - topological solitons in the theory. This results in domains (gravity - balls) inside which an effective, canonical, attractive gravitational constant is induced. We consider simulations of the formation and evolution of such solutions. Starting with a single gravity - ball, we consider its fragmentation into smaller (lower mass) balls - evolving by mutual repulsion. After several runs, we have been able to identify two parameters: the strength of the long range gravitational constant and the size of the gravity balls, which can be used to generate appropriate two point correlations of the distribution of these balls. 
  We contrast the two approaches to ``classical" signature change used by Hayward with the one used by us (Hellaby and Dray). There is (as yet) no rigorous derivation of appropriate distributional field equations. Hayward's distributional approach is based on a postulated modified form of the field equations. We make an alternative postulate. We point out an important difference between two possible philosophies of signature change --- ours is strictly classical, while Hayward's Lagrangian approach adopts what amounts to an imaginary proper ``time" on one side of the signature change, as is explicitly done in quantum cosmology. We also explain why we chose to use the Darmois-Israel type junction conditions, rather than the Lichnerowicz type junction conditions favoured by Hayward. We show that the difference in results is entirely explained by the difference in philosophy (imaginary versus real Euclidean ``time"), and not by the difference in approach to junction conditions (Lichnerowicz with specific coordinates versus Darmois with general coordinates). 
  Motivated by the recent wave of investigations on plane domain wall spacetimes with nontrivial topologies, the present paper deals with (probably) the most simple source field configuration which can generate a spatially planary symmetric static spacetime, namely a minimally coupled massless scalar field that depends only upon a spacelike coordinate, $z$. It is shown that the corresponding exact solutions $({\cal M}, {\bf{\rm g}}_{\pm})$ are algebraically special, type $D - [S - 3T]_{(11)}$, and represent globally pathologic spacetimes with a $G_{4}$ - group of motion acting on ${\bf{\rm R}}^{2} \times {\bf{\rm R}}$ orbits. In spite of the model simplicity, these $\phi$ - generated worlds possess naked timelike singularities (reached within a finite universal time by normal non-spacelike geodesics), are completely free of Cauchy surfaces and contain into the $t$ - leveled sections points which can not be jointed by ${\rm C}^{1}$ - trajectories images of oblique non-spacelike geodesics. Finally, we comment on the possibility of deriving from $({\cal M}, {\bf{\rm g}}_{\pm})$ two other physically interesting ^^ ^^ $\phi$ - generated'' spacetimes, by appropiate jonction conditions in the $(z = 0)$ - plane. 
  We show an equivalence between Dirac quantization and the reduced phase space quantization. The equivalence of the both quantization methods determines the operator ordering of the Hamiltonian. Some examples of the operator ordering are shown in simple models. 
  We discuss the issue initiated by Kucha\v{r} {\it et al}, of replacing the usual Hamiltonian constraint by alternative combinations of the gravitational constraints (scalar densities of arbitrary weight), whose Poisson brackets strongly vanish and cast the standard constraint-system for vacuum gravity into a form that generates a true Lie algebra. It is shown that any such combination---that satisfies certain reality conditions---may be derived from an action principle involving a single scalar field and a single Lagrange multiplier with a non--derivative coupling to gravity. 
  We find the most general, spherically symmetric solution in a special class of tetrad theory of gravitation. The tetrad gives the Schwarzschild metric. The energy is calculated by the superpotential method and by the Euclidean continuation method. We find that unless the time-space components of the tetrad go to zero faster than ${1/\sqrt{r}}$ at infinity, the two methods give results different from each other, and that these results differ from the gravitational mass of the central gravitating body. This fact implies that the time-space components of the tetrad describing an isolated spherical body must vanish faster than ${1/\sqrt{r}}$ at infinity. 
  An approximate form for the vacuum averaged stress-energy tensor of a conformal spin-2 quantum field on a black hole background is employed as a source term in the semiclassical Einstein equations. Analytic corrections to the Schwarzschild metric are obtained to first order in $\epsilon = {\hbar}/M^2$, where $M$ denotes the mass of the black hole. The approximate tensor possesses the exact trace anomaly and the proper asymptotic behavior at spatial infinity, is conserved with respect to the background metric and is uniquely defined up to a free parameter $\hat c_2$, which relates to the average quantum fluctuation of the field at the horizon. We are able to determine and calculate an explicit upper bound on $\hat c_2$ by requiring that the entropy due to the back-reaction be a positive increasing function in $r$. A lower bound for $\hat c_2$ can be established by requiring that the metric perturbations be uniformly small throughout the region $2M \leq r < r_o$, where $r_o$ is the radius of perturbative validity of the modified metric. Additional insight into the nature of the perturbed spacetime outside the black hole is provided by studying the effective potential for test particles in the vicinity of the horizon. 
  Existing physical theories do not predict every feature of our experience but only certain regularities of that experience. That difference between what could be observed and what can be predicted is one kind of limit on scientific knowledge. Such limits are inevitable if the world is complex and the laws governing the regularities of that world are simple. Another kind of limit on scientific knowledge arises because even simple theories may require intractable or impossible computations to yield specific predictions. A third kind of limit concerns our ability to know theories through the process of induction and test. Quantum cosmology -- that part of science concerned with the quantum origin of the universe and its subsequent evolution -- displays all three kinds of limits. This paper briefly describes quantum cosmology and discusses these limits. The place of the other sciences in this most comprehensive of physical frameworks is described. 
  The magnitudes of the external gravitational perturbations associated with the normal modes of the Sun are evaluated to determine whether these solar oscillations could be observed with the proposed Laser Interferometer Space Antenna (LISA), a network of satellites designed to detect gravitational radiation. The modes of relevance to LISA---the $l=2$, low-order $p$, $f$ and $g$-modes---have not been conclusively observed to date. We find that the energy in these modes must be greater than about $10^{30} \rm{ergs}$ in order to be observable above the LISA detector noise. These mode energies are larger than generally expected, but are much smaller than the current observational upper limits. LISA may be confusion-limited at the relevant frequencies due to the galactic background from short-period white dwarf binaries. Present estimates of the number of these binaries would require the solar modes to have energies above about $10^{33} \rm{ergs}$ to be observable by LISA. 
  Observations of binary inspiral in a single interferometric gravitational wave detector can be cataloged according to signal-to-noise ratio $\rho$ and chirp mass $\cal M$. The distribution of events in a catalog composed of observations with $\rho$ greater than a threshold $\rho_0$ depends on the Hubble expansion, deceleration parameter, and cosmological constant, as well as the distribution of component masses in binary systems and evolutionary effects. In this paper I find general expressions, valid in any homogeneous and isotropic cosmological model, for the distribution with $\rho$ and $\cal M$ of cataloged events; I also evaluate these distributions explicitly for relevant matter-dominated Friedmann-Robertson-Walker models and simple models of the neutron star mass distribution. In matter dominated Friedmann-Robertson-Walker cosmological models advanced LIGO detectors will observe binary neutron star inspiral events with $\rho>8$ from distances not exceeding approximately $2\,\text{Gpc}$, corresponding to redshifts of $0.48$ (0.26) for $h=0.8$ ($0.5$), at an estimated rate of 1 per week. As the binary system mass increases so does the distance it can be seen, up to a limit: in a matter dominated Einstein-deSitter cosmological model with $h=0.8$ ($0.5$) that limit is approximately $z=2.7$ (1.7) for binaries consisting of two $10\,\text{M}_\odot$ black holes. Cosmological tests based on catalogs of the kind discussed here depend on the distribution of cataloged events with $\rho$ and $\cal M$. The distributions found here will play a pivotal role in testing cosmological models against our own universe and in constructing templates for the detection of cosmological inspiraling binary neutron stars and black holes. 
  We show how classical spacetime emerges from quantum gravity through the study of a quantum FRW cosmological model coupled to a free massive scalar field using a new asymptotic expansion method of the Wheeler-DeWitt equation. It is shown that the coherent states of the nonadiabatic basis of a particular generalized invariant give rise to the quantum back reaction of matter field proportional to classical energy and the Einstein-Hamilton-Jacobi with matter becomes equivalent to the classical Einstein equation. 
  In this paper we will make a survey of solutions to the Wheeler-Dewitt equation which have been found up to now in Ashtekar's formulation for canonical quantum gravity. Roughly speaking they are classified into two categories, namely, Wilson-loop solutions and topological solutions. While the program of finding solutions which are composed of Wilson loops is still in its infancy, it is expected to be developed in the near future. Topological solutions are the only solutions at present which we can give their interpretation in terms of spacetime geometry. While the analysis made here is formal in the sense that we do not deal with rigorously regularized constraint equations, these topological solutions are expected to exist even in the fully regularized theory and they are considered to yield vacuum states of quantum gravity. We also make an attempt to review the spin network states as intuitively as possible. In particular, the explicit formulae for two kinds of measures on the space of spin network states are given. 
  It is assumed that the radial propagation of light with respect to the naive coordinate system of the observer is uniform and isotropic and that the physical rate of propagation of light is the same for all observers. In accelerated frames of reference, these assumptions lead to the findings that the measured value of $c$ is a function of the gravitational energy per unit mass (GEPUM) of the observer, and that this is due to the physical characteristics of the standard measuring-devices being a function of their GEPUM. The consequences of these findings include observers who at rest with respect to each other assigning different values to the same physical separation, the mixed metric tensor ${g^\mu}_\nu$ describing how gravitation affects measuring-devices, and the De-Broglie wavelength being a function of an object's GEPUM. How the measured values of various types of physical quantities are affected is described. The Schwarzschild solution is re-examined: The physical size of radial coordinate $2m$ is 0, a traveler must perceive himself to go an infinite distance to reach the radial coordinate of $2m$, and gravitational self-potential energy reaches a minimal value at the radial coordinate $3m$. Therefore, black holes do not exist in this theory. 
 Contents:    Editorial, J. Pullin   Report from the APS TGG, Beverly Berger   We hear that..., J. Pullin   LIGO project status, S. Whitcomb   General relativity survives another test, C. Will   Macroscopic deviations from Hawking radiation? L. Smolin   Event horizons and topological censorship, E. Seidel   Critical behavior in black hole collapse, J. Horne    Third Texas Workshop on 3D Numerical Relativity, P. Laguna   ICGC-95, M.A.H. MacCallum   The Josh Goldberg Symposium, P. Saulson   Summer school in Bad Honnef, H.-P. Nollert   Fifth Annual Midwest Relativity Conference, J. Romano   Volga-7 '95, A. Aminova, D. Brill. 
  The LIGO Research Community (LRC) is an independent organization of researchers interested in the scientific opportunities created by the construction and operation of the Laser Interferometer Gravitational-wave Observatory (LIGO). Membership is open to all interested individuals, irrespective of any other affiliations (including affiliation with the LIGO project, VIRGO or other gravitational-wave detector projects). The LRC has begun a study project designed to {\em identify} the ways that an operating LIGO will affect the research environment in gravitational physics, {\em decide} what we want that environment to look like, and recommend (to LIGO and the NSF) the steps to be taken now to develop that environment in the future. Contributions from LRC members and from the broader gravitational physics research community are actively solicited. 
  We consider the Hamiltonian dynamics and thermodynamics of spherically symmetric Einstein-Maxwell spacetimes with a negative cosmological constant. We impose boundary conditions that enforce every classical solution to be an exterior region of a Reissner-Nordstr\"om-anti-de Sitter black hole with a nondegenerate Killing horizon, with the spacelike hypersurfaces extending from the horizon bifurcation two-sphere to the asymptotically anti-de Sitter infinity. The constraints are simplified by a canonical transformation, which generalizes that given by Kucha\v{r} in the spherically symmetric vacuum Einstein theory, and the theory is reduced to its true dynamical degrees of freedom. After quantization, the grand partition function of a thermodynamical grand canonical ensemble is obtained by analytically continuing the Lorentzian time evolution operator to imaginary time and taking the trace. A~similar analysis under slightly modified boundary conditions leads to the partition function of a thermodynamical canonical ensemble. The thermodynamics in each ensemble is analyzed, and the conditions that the (grand) partition function be dominated by a classical Euclidean black hole solution are found. When these conditions are satisfied, we recover in particular the Bekenstein-Hawking entropy. The limit of a vanishing cosmological constant is briefly discussed.   (This paper is dedicated to Karel Kucha\v{r} on the occasion of his sixtieth birthday.) 
  Spin-orbit and spin-spin effects in the gravitational interaction are treated in a close analogy with the fine and hyperfine interactions in atoms. The proper definition of the cener-of-mass coordinate is discussed. The technique developed is applied then to the gravitational radiation of compact binary stars. Our result for the spin-orbit correction differs from that obtained by other authors. New effects possible for the motion of a spinning particle in a gravitational field are pointed out. The corresponding corrections, nonlinear in spin, are in principle of the same order of magnitude as the ordinary spin-spin interaction. 
  The energy associated with a static and spherically symmetric charged dilaton black hole is obtained for arbitrary value of the coupling parameter (which regulates the strength of the coupling of the dilaton to the Maxwell field) $\beta$. The energy distribution depends on $\beta$, whereas the total energy is independent of this and is given by the mass parameter of the black hole. 
  We present a class of solutions in Einstein-Yang-Mills-systems with arbitrary gauge groups and space-time dimensions, which are symmetric under the action of the group of spatial rotations. Our approach is based on the dimensional reduction method for gauge and gravitational fields and relates symmetric EYM-solutions to certain solutions of two-dimensional Einstein-Yang-Mills-Higgs-dilaton theory. Application of this method to four-dimensional spherically symmetric (pseudo)riemannian space-time yields, in particular, new solutions describing both a magnetic and an electric charge in the center of a black hole. Moreover, we give an example of a solution with nonabelian gauge group in six-dimensional space-time. We also comment on the stability of the obtained solutions. 
  The string model of gravitational force is proposed where the string forms the mediation of the gravitational interaction between two gravitating bodies. It reproduces the Newtonian results in the first-order approximation and it predicts in the higher-oder approximations the existence of oscillations of the gravitational field between two massive bodies. It can be easily generalized to the two-body interaction in particle physics. 
  We describe the gravitational degrees of freedom of the Schwarzschild black hole by one free variable. We introduce an equation which we suggest to be the Schroedinger equation of the Schwarzschild black hole corresponding to this model. We solve the Schroedinger equation explicitly and obtain the mass spectrum of the black hole as such as it can be observed by an observer very far away and at rest relative to the black hole. Our equation implies that there is no singularity inside the Schwarzschild black hole, and that the black hole has a certain ground state in which its mass is non-zero. 
  We study the existence and stability of spherical membranes in curved spacetimes. For Dirac membranes in the Schwarzschild--de Sitter background we find that there exists an equilibrium solution. By fine--tuning the dimensionless parameter $\Lambda M^2,$ the static membrane can be at any position outside the black hole event horizon, even at the stretched horizon, but the solution is unstable. We show that modes having $l=0$ (and for $\Lambda M^2<16/243$ also $l=1$) are responsible for the instability. We also find that spherical higher order membranes (membranes with extrinsic curvature corrections), contrary to what happens in flat Minkowski space, {\it do} have equilibrium solutions in a general curved background and, in particular, also in the ``plain'' Schwarzschild geometry (while Dirac membranes do not have equilibrium solutions there). These solutions, however, are also unstable. We shall discuss a way of by--passing these instability problems, and we also relate our results to the recent ideas of representing the black hole event horizon as a relativistic bosonic membrane. 
  In this work a proposal for definition of twistors on generic curved spaces is exposed and investigated. We consider superpositions of nearly autoparallel and nearly geodesic maps (nearly conformal maps, nc-maps) of (pseudo-)Riemannian spaces as generalizations of conformal transforms. We introduce the nearly autoparallel twistor equations being compatible on nc-flat spaces and study nearly autoparallel twistor structures generating curved spaces and vacuum Einstein equations. 
  We display a logarithmic divergence in the density matrix of a scalar field in the presence of an Einstein-Yang-Mills black hole in four dimensions. This divergence is related to a previously-found logarithmic divergence in the entropy of the scalar field, which cannot be absorbed into a renormalization of the Hawking-Bekenstein entropy of the black hole. As the latter decays, the logarithmic divergence induces a non-commutator term $\nd{\delta H}\rho$ in the quantum Liouville equation for the density matrix $\rho$ of the scalar field, leading to quantum decoherence. The order of magnitude of $\nd{\delta H}$ is $\mu^2/M_P$, where $\mu$ is the mass of the scalar particle. 
  Beginning with a special form of the Einstein-Rosen metric, we find new cosmological solutions of the Einstein equations, having two hypersurface-orthogonal Killing vectors , with ideal fluid. The equation of state is in the most cases of the form $p = \gamma \rho$. 
  We give a short outline, in Sec.\ 2, of the historical development of the gauge idea as applied to internal ($U(1),\, SU(2),\dots$) and external ($R^4,\,SO(1,3),\dots$) symmetries and stress the fundamental importance of the corresponding conserved currents. In Sec.\ 3, experimental results with neutron interferometers in the gravitational field of the earth, as inter- preted by means of the equivalence principle, can be predicted by means of the Dirac equation in an accelerated and rotating reference frame. Using the Dirac equation in such a non-inertial frame, we describe how in a gauge- theoretical approach (see Table 1) the Einstein-Cartan theory, residing in a Riemann-Cartan spacetime encompassing torsion and curvature, arises as the simplest gravitational theory. This is set in contrast to the Einsteinian approach yielding general relativity in a Riemannian spacetime. In Secs.\ 4 and 5 we consider the conserved energy-momentum current of matter and gauge the associated translation subgroup. The Einsteinian teleparallelism theory which emerges is shown to be equivalent, for spinless matter and for electromagnetism, to general relativity. Having successfully gauged the translations, it is straightforward to gauge the four-dimensional affine group $R^4 \semidirect GL(4,R)$ or its Poincar\'e subgroup $R^4\semidirect SO(1,3)$. We briefly report on these results in Sec.\ 6 (metric-affine geometry) and in Sec.\ 7 (metric-affine field equations (\ref{zeroth}, \ref{first}, \ref{second})). Finally, in Sec.\ 8, we collect some models, currently under discussion, which bring life into the metric-affine gauge framework developed. 
  We discuss the choice of the Lagrangian in the Poincar\'e gauge theory of gravity. Drawing analogies to earlier de Sitter gauge models, we point out the possibility of deriving the Einstein-Cartan Lagrangian {\it without} cosmological term from a {\it modified} quadratic curvature invariant of {\it topological} type. 
  Efficient formulae of Ricci tensor for an arbitrary diagonal metric are presented. 
  The theory of the $\kappa$-deformed Poincare algebra is applied to the analysis of various phenomena in special relativity, quantum mechanics and field theory. The method relies on the development of series expansions in $\kappa^{-1}$ of the generalised Lorentz transformations, about the special-relativistic limit. Emphasis is placed on the underlying assumptions needed in each part of the discussion, and on in principle limits for the deformation parameter, rather than on rigorous numerical bounds. In the case of the relativistic Doppler effect, and the Michelson-Morley experiment, comparisons with recent experiemntal tests yield the relatively weak lower bounds on $\kappa c$ of 90eV and 250 keV, respectively. Corrections to the Casimir effect and the Thomas precession are also discussed. 
  By imposing natural geometrical and kinematical conditions on a conformal Killing vector in Bianchi I spacetime, we show that a class of axisymmetric metrics admits a conformal motion. This class contains new exact solutions of Einstein's equations, including anisotropic radiation universes that isotropise at late times. 
  We have calculated signal-to-noise ratios for eight spherical resonant-mass antennas interacting with gravitational radiation from inspiralling and coalescing binary neutron stars and from the dynamical and secular bar-mode instability of a rapidly rotating star. We find that by using technology that could be available in the next several years, spherical antennas can detect neutron star inspiral and coalescence at a distance of 15 Mpc and the dynamical bar-mode instability at a distance of 2 Mpc. 
  While there does not at this time exist a complete canonical theory of full 3+1 quantum gravity, there does appear to be a satisfactory canonical quantization of minisuperspace models. The method requires no `choice of time variable' and preserves the systems' explicit reparametrization invariance. In the following study, this canonical formalism is used to derive a path integral for quantum minisuperspace models. As it comes from a well-defined canonical starting point, the measure and contours of integration are specified by this construction. The properties of the resulting path integral are analyzed, both exactly and in the semiclassical limit. Particular attention is paid to the role of the (unbounded) Euclidean action and Euclidean instantons are argued to contribute as $e^{-|S_E|/\hbar}$. 
  The black hole model with a self-gravitating charged spherical symmetric dust thin shell as a source is considered. The Schroedinger-type equation for such a model is derived. This equation appeared to be a finite differences equation. A theory of such an equation is developed and general solution is found and investigated in details. The discrete spectrum of the bound state energy levels is obtained. All the eigenvalues appeared to be infinitely degenerate. The ground state wave functions are evaluated explicitly. The quantum black hole states are selected and investigated. It is shown that the obtained black hole mass spectrum is compatible with the existence of Hawking's radiation in the limit of low temperatures both for large and nearly extreme Reissner-Nordstrom black holes. The above mentioned infinite degeneracy of the mass (energy) eigenvalues may appeared helpful in resolving the well known information paradox in the black hole physics. 
  It is shown that the class of algebro-geometrical (finite-gap) solutions of the Ernst equation constructed several years ago in [D.Korotkin, Theor.Math.Phys., 77 (1989), p. 1018] contains the solutions recently constructed by R.Meinel and G.Neugebauer as a subset. 
  The chiral Lagrangian including more than two spin-3/2 fields becomes complex in general when the self-dual connection satisfies its equation of motion, and the imaginary part of the chiral Lagrangian gives the additional equation for spin-3/2 fields which gives rise to the inconsistency. This inconsistency can be removed by taking the tetrad to be complex and defining the total Lagrangian which is the sum of the chiral Lagrangian and its complex conjugate. It is possible to establish the right (left)-handed supersymmetry in its total Lagrangian as in the case of the chiral Lagrangian. We also comment on the canonical formulation of the total Lagrangian. 
  We summarize the basics of the loop representation of quantum gravity and describe the main aspects of the formalism, including its latest developments, in a reorganized and consistent form. Recoupling theory, in its graphical Temperley-Lieb-Kauffman formulation, provides a powerful calculation tool in this context. We describe its application to the loop representation in detail. Using recoupling theory, we derive general expressions for the spectrum of the quantum area and the quantum volume operators. We compute several volume eigenvalues explicitly. We introduce a scalar product with respect to which area and volume are symmetric operators, and (the trivalent expansions of) the spin network states are orthogonal. 
  The two independent ``plus" and ``cross" polarization waveforms associated with the gravitational waves emitted by inspiralling, non-spinning, compact binaries are presented, ready for use in the data analysis of signals received by future laser interferometer gravitational-wave detectors such as LIGO and VIRGO. The computation is based on a recently derived expression of the gravitational field at the second-post-Newtonian approximation of general relativity beyond the dominant (Newtonian) quadrupolar field. The use of these theoretical waveforms to make measurements of astrophysical parameters and to test the nature of relativistic gravity is discussed. 
  In the context of the open inflationary universe, we calculate the amplitude of quantum fluctuations which deform the bubble shape. These give rise to scalar field fluctuations in the open Friedman-Robertson-Walker universe which is contained inside the bubble. One can transform to a new gauge in which matter looks perfectly smooth, and then the perturbations behave as tensor modes (gravitational waves of very long wavelength). For $(1-\Omega)<<1$, where $\Omega$ is the density parameter, the microwave temperature anisotropies produced by these modes are of order $\delta T/T\sim H(R_0\mu l)^{-1/2} (1-\Omega)^{l/2}$. Here, $H$ is the expansion rate during inflation, $R_0$ is the intrinsic radius of the bubble at the time of nucleation, $\mu$ is the bubble wall tension and $l$ labels the different multipoles ($l>1$). The gravitational backreaction of the bubble has been ignored. In this approximation, $G\mu R_0<<1$, and the new effect can be much larger than the one due to ordinary gravitational waves generated during inflation (unless, of course, $\Omega$ gets too close to one, in which case the new effect disappears). 
  We show how to treat the constraints and reality conditions in the $SO(3)$-ADM (Ashtekar) formulation of general relativity, for the case of a vacuum spacetime with a cosmological constant. We clarify the difference between the reality conditions on the metric and on the triad. Assuming the triad reality condition, we find a new variable, allowing us to solve the gauge constraint equations and the reality conditions simultaneously. 
  This contribution is a review of the method of isomonodromic quantization of dimensionally reduced gravity. Our approach is based on the complete separation of variables in the isomonodromic sector of the model and the related ``two-time" Hamiltonian structure. This allows an exact quantization in the spirit of the scheme developed in the framework of integrable systems. Possible ways to identify a quantum state corresponding to the Kerr black hole are discussed. In addition, we briefly describe the relation of this model with Chern Simons theory. 
  Recently, the phenomenological description of cosmological particle production processes in terms of effective viscous pressures has attracted some attention. Using a simple creation rate model we discuss the question to what extent this approach is compatible with the kinetic theory of a relativistic gas. We find the effective viscous pressure approach to be consistent with this model for homogeneous spacetimes but not for inhomogeneous ones. 
  Using the Hubble parameter as new `inverse time' coordinate ($H$-formalism), a new method of reconstructing the inflaton potential is developed also using older results which, in principle, is applicable to any order of the slow-roll approximation. In first and second order, we need three observational data as inputs: the scalar spectral index $n_s$ and the amplitudes of the scalar and the tensor spectrum. We find constraints between the values of $n_s$ and the corresponding values for the wavelength $\lambda $. By imposing a dependence $\lambda (n_s)$, we were able to reconstruct and visualize inflationary potentials which are compatible with recent COBE and other astrophysical observations. >From the reconstructed potentials, it becomes clear that one cannot find only one special value of the scalar spectral index $n_s$. 
  Based on a thoeretical model in which scalar fields play crucial roles, we propose a mechanism to better understand a cosmological constant expected to be small (nearly comparable with the critical density) but nonzero as suggested strongly by the recent observations. We emphasize that a step further is needed beyond the simplest scenario of a decaying cosmological constant, according to which the effective $\Lambda$ is not a true constant but decays like $t^{-2}$, thus explaining why any effect of $\Lambda$ which is assumed to be of the Planckian order of magnitude in the earliest Universe is at most 120 orders smaller today ($t\sim 10^{60}$ in units of the Planck time), without any extreme fine-tuning of the parameters. In addition to the overall smooth fall-off $\sim t^{-2}$, the total energy density, the sum of the contributions from the ordinary matter and the scalar fields, must show some leveling-off behavior, hence mimicking a truly constant $\Lambda$ to fit the observations. We find that the desired behavior can be obtained by introducing two scalar fields interacting with each other in a specific manner. Many aspects of this model are discussed in detail, including the real origin of the whole mechanism, and how the parameters and the initial conditions of the theory can be determined by the cosomological parameters at the present epoch as well as the past history of the Universe. 
  A class of free quantum fields defined on the Poincare' group, is described by means of their two-point vacuum expectation values. They are not equivalent to fields defined on the Minkowski spacetime and they are "elementary" in the sense that they describe particles that transform according to irreducible unitary representations of the symmetry group, given by the product of the Poincare' group and of the group SL(2, C) considered as an internal symmetry group. Some of these fields describe particles with positive mass and arbitrary spin and particles with zero mass and arbitrary helicity or with an infinite helicity spectrum. In each case the allowed SL(2, C) internal quantum numbers are specified. The properties of local commutativity and the limit in which one recovers the usual field theories in Minkowski spacetime are discussed. By means of a superposition of elementary fields, one obtains an example of a field that present a broken symmetry with respect to the group Sp(4, R), that survives in the short-distance limit. Finally, the interaction with an accelerated external source is studied and and it is shown that, in some theories, the average number of particles emitted per unit of proper time diverges when the acceleration exceeds a finite critical value. 
  Quasinormal modes have played a prominent role in the discussion of perturbations of black holes, and the question arises whether they are as significant as normal modes are for self adjoint systems, such as harmonic oscillators. They can be significant in two ways: Individual modes may dominate the time evolution of some perturbation, and a whole set of them could be used to completely describe this time evolution. It is known that quasinormal modes of black holes have the first property, but not the second. It has recently been suggested that a discontinuity in the underlying system would make the corresponding set of quasinormal modes complete. We therefore turn the Regge-Wheeler potential, which describes perturbations of Schwarzschild black holes, into a series of step potentials, hoping to obtain a set of quasinormal modes which shows both of the above properties. This hope proves to be futile, though: The resulting set of modes appears to be complete, but it does not contain any individual mode any more which is directly obvious in the time evolution of initial data. Even worse: The quasinormal frequencies obtained in this way seem to be extremely sensitive to very small changes in the underlying potential. The question arises whether - and how - it is possible to make any definite statements about the significance of quasinormal modes of black holes at all, and whether it could be possible to obtain a set of quasinormal modes with the desired properties in another way. 
  We elaborate on a proposal made by Greensite and others to solve the problem of time in quantum gravity. The proposal states that a viable concept of time and a sensible inner product can be found from the demand for the Ehrenfest equations to hold in quantum gravity. We derive and discuss in detail exact consistency conditions from both Ehrenfest equations as well as from the semiclassical approximation. We also discuss consistency conditions arising from the full field theory. We find that only a very restricted class of solutions to the Wheeler-DeWitt equation fulfills all consistency conditions. We conclude that therefore this proposal must either be abandoned as a means to solve the problem of time or, alternatively, be used as an additional boundary condition to select physical solutions from the Wheeler-DeWitt equation. 
  We construct two classes of exact solutions to the field equations of topologically massive electrodynamics coupled to topologically massive gravity in 2 + 1 dimensions. The self-dual stationary solutions of the first class are horizonless, asymptotic to the extreme BTZ black-hole metric, and regular for a suitable parameter domain. The diagonal solutions of the second class, which exist if the two Chern-Simons coupling constants exactly balance, include anisotropic cosmologies and static solutions with a pointlike horizon. 
  The aim of this letter is to indicate the differences between the Rovelli-Smolin quantum volume operator and other quantum volume operators existing in the literature. The formulas for the operators are written in a unifying notation of the graph projective framework. It is clarified whose results apply to which operators and why. 
  A set of models is considered which, in a certain sense, interpolates between 1+1 free quantum field theories on topologically distinct backgrounds. The intermediate models may be termed free quantum field theories, though they are certainly not local. Their ground state energies are computed and shown to be finite. The possible relevance to changing spacetime topologies is discussed. 
  We study the coupled Einstein-Yang-Mills-Dilaton (EYMD) equations for a Fried\-mann-Le\-mai\-tre universe with constant curvature $k=1$. Our detailed analysis is restricted to the case where the dilaton potential and the cosmological constant vanish. Also assuming a static gauge field, we present analytical and numerical results on the behavior of solutions of the EYMD equations. For different values of the dilaton coupling constant we analyze the phase portrait for the time evolution of the dilaton field and give the behavior of the scale factor. It turns out that there are no inflationary stages in this model. 
  Using the Teukolsky and Sasaki-Nakamura formalisms for the perterbations around a Kerr black hole, we calculate the energy flux of gravitational waves induced by a {\it spinning} particle of mass $\mu$ and spin $S$ moving in circular orbits near the equatorial plain of a rotating black hole of mass $M (\gg \mu)$ and spin $Ma$. The calculations are performed by using the recently developed post-Newtonian expansion technique of the Teukolsky equation. To evaluate the source terms of perturbations caused by a {\it spinning} particle, we used the equations of motion of a spinning particle derived by Papapetrou and the energy momentum tensor of a spinning particle derived by Dixon. We present the post-Newtonian formula of the gravitational wave luminosity up to the order $(v/c)^5$ beyond the quadrupole formula including the linear order of particle spin. The results obtained in this paper will be an important guideline to the post-Newtonian calculation of the inspiral of two spinning compact objects. 
  If the universe is finite and smaller than the distance to the surface of last scatter, then the signature of the topology of the universe is writ on the microwave background sky. Previous efforts to search for this topology have focused on one particular model: a toroidal flat universe. We show how both the high degree of spatial symmetry of this topology and the integrability of its geodesics make it unreliable as a paradigmatic example, and discuss why topology on scales significantly smaller than the horizon are not ruled out by previous analyses focussing on this special case. We show that in these small universes the microwave background will be identified at the intersections of the surface of last scattering as seen by different ``copies'' of the observer. Since the surface of last scattering is a sphere, these intersections will be circles, regardless of the background geometry or topology. We therefore propose a statistic that is sensitive to all small finite homogeneous topologies. Here, small means that the distance to the surface of last scatter is smaller than the ``periodicity scale'' of the universe. 
  The black hole of the widely used ordinary 2d--dilaton model (DBH) deviates from the Schwarzschild black hole (SBH) of General Relativity in one important feature: Whereas non-null extremals or geodesics show the expected incompleteness this turns out {\it not to be the case for the null extremals}. After a simple analysis in Kruskal coordinates for singularities with power behavior of this -- apparently till now overlooked -- property we discuss the global structure of a large family of generalized dilaton theories which does not only contain the DBH and SBH but also other proposed dilaton theories as special cases. For large ranges of the parameters such theories are found to be free from this defect and exhibit global SBH behavior. 
  We show that the Hamiltonian of four-dimensional Lorentzian gravity, defined on a space of real, SU(2)-valued connections, in spite of its non-polynomiality possesses a natural quantum analogue in a lattice-discretized formulation of the theory. This opens the way for a systematic investigation of its spectrum. The unambiguous and well-defined scalar product is that of the SU(2)-gauge theory. We also comment on various aspects of the continuum theory. 
  To discuss the quantum to classical transition in quantum cosmology, we study the decoherence factor and the peak of the Wigner function, which respectively represent the degree of decoherence and the degree to which the classical motion of the Universe is defined, in a Robertson-Walker universe model coupled with a scalar field. It is known that the decoherence factor is divergent in some cases. This implies that perfect decoherence occurs, and classical correlation criterion fails. In this paper we discuss the divergence of the decoherence factor in some detail and obtain the constraints for decoherence factor to be convergent, making use of the arbitrariness defining the reduced density matrix. The result is discussed in connection with the arbitrariness of {\it system/environment} splitting. 
  The ``screen mapping" introduced by Susskind to implement 't Hooft's holographic hypothesis is studied. For a single screen time, there are an infinite number of images of a black hole event horizon, almost all of which have smaller area on the screen than the horizon area. This is consistent with the focusing equation because of the existence of focal points. However, the {\it boundary} of the past (or future) of the screen obeys the area theorem, and so always gives an expanding map to the screen, as required by the holographic hypothesis. These considerations are illustrated with several axisymmetric static black hole spacetimes. 
  The Einstein's field equations of Friedmann-Robertson-Walker universes filled with a dissipative fluid described by both the {\em truncated} and {\em non-truncated} causal transport equations are analyzed using techniques from dynamical systems theory. The equations of state, as well as the phase space, are different from those used in the recent literature. In the de Sitter expansion both the hydrodynamic approximation and the non-thermalizing condition can be fulfilled simultaneously. For $\Lambda=0$ these expansions turn out to be stable provided a certain parameter of the fluid is lower than 1/2. The more general case $\Lambda>0$ is studied in detail as well. 
  Stationary perfect-fluid configurations of Einstein's theory of gravity are studied. It is assumed that the 4-velocity of the fluid is parallel to the stationary Killing field, and also that the norm and the twist potential of the stationary Killing field are functionally independent. It has been pointed out earlier by one of us (I.R.) that for these perfect-fluid geometries some of the basic field equations are invariant under an $SL(2,R)$ transformation. Here it is shown that this transformation can be applied to generate possibly new perfect-fluid solutions from existing known ones only for the case of barotropic equation of state $\rho+3p=0$. In order to study the effect of this transformation, its application to known perfect-fluid solutions is presented. In this way, different previously known solutions could be written in a singe compact form. A new derivation of all Petrov type D stationary axisymmetric rigidly rotating perfect-fluid solutions with an equation of state $\rho+3p=constant$ is given in an appendix. 
  A new functional calculus, developed recently for a fully non-perturbative treatment of quantum gravity, is used to begin a systematic construction of a quantum theory of geometry. Regulated operators corresponding to areas of 2-surfaces are introduced and shown to be self-adjoint on the underlying (kinematical) Hilbert space of states. It is shown that their spectra are {\it purely} discrete indicating that the underlying quantum geometry is far from what the continuum picture might suggest. Indeed, the fundamental excitations of quantum geometry are 1-dimensional, rather like polymers, and the 3-dimensional continuum geometry emerges only on coarse graining. The full Hilbert space admits an orthonormal decomposition into finite dimensional sub-spaces which can be interpreted as the spaces of states of spin systems. Using this property, the complete spectrum of the area operators is evaluated. The general framework constructed here will be used in a subsequent paper to discuss 3-dimensional geometric operators, e.g., the ones corresponding to volumes of regions. 
  Analytical extensions and resulting Penrose diagrams are given for the solutions of a simple 2d dilaton gravity theory describing black holes which are static as viewed by asymptotically accelerated observers. 
  The thermodynamic second law in the evaporating black hole space-time is examined in the context of two-dimensional dilaton black hole. The dynamical evolution of entropy is investigated by using the analytical perturbation method and the numerical method. It is shown that the thermodynamic second law holds in the vicinity of infalling shell %analytically and in the region far from the infalling shell %numerically in the case of semi-classical CGHS model. The analysis of the general models is also presented and its implication is discussed. 
  The model consisting of gravitational, scalar and axionic fields is considered. It is shown that the action of the Lavrelashvili-Rubakov-Tinyakov wormhole can be made arbitrarily negative by varying the parameters of the model. This means that semiclassically calculated probability of transition through this wormhole is not exponentially small (as usual) but exponentially large. 
  The vacuum Einstein equations for spacetimes with two commuting spacelike Killing field symmetries are studied using the Ashtekar variables. The case of compact spacelike hypersurfaces which are three-tori is considered, and the determinant of the Killing two-torus metric is chosen as the time gauge. The Hamiltonian evolution equations in this gauge may be rewritten as those of a modified SL(2) principal chiral model with a time dependent `coupling constant', or equivalently, with time dependent SL(2) structure constants. The evolution equations have a generalized zero-curvature formulation. Using this form, the explicit time dependence of an infinite number of spatial-diffeomorphism invariant phase space functionals is extracted, and it is shown that these are observables in the sense that they Poisson commute with the reduced Hamiltonian. An infinite set of observables that have SL(2) indices are also found. This determination of the explicit time dependence of an infinite set of spatial-diffeomorphism invariant observables amounts to the solutions of the Hamiltonian Einstein equations for these observables. 
  The relativistic free particle system in 1+1 dimensions is formulated as a ``bi-Hamiltonian system''. One Hamiltonian generates ordinary time translations, and another generates (essentially) boosts. Any observer, accelerated or not, sees evolution as the continuous unfolding of a canonical transformation which may be described using the two Hamiltonians. When the system is quantized both Hamiltonians become Hermitian operators in the standard positive definite inner product. Hence, each observer sees the evolution of the wave function as the continuous unfolding of a unitary transformation in the standard positive definite inner product. The result appears to be a consistent single particle interpretation of relativistic quantum mechanics. This interpretation has the feature that the wave function is observer dependent, and observables have non-local character, similar to what one might expect in quantum gravity. 
  The expected stress-energy tensor <T_{ab}> of quantum fields generically violates the local positive energy conditions of general relativity. However, <T_{ab}> may satisfy some nonlocal conditions such as the averaged null energy condition (ANEC), which would rule out traversable wormholes. Although ANEC holds in Minkowski spacetime, it can be violated in curved spacetimes if one is allowed to choose the spacetime and quantum state arbitrarily, without imposition of the semiclassical Einstein equation G_{ab} = 8 \pi <T_{ab}>. In this paper we investigate whether ANEC holds for solutions to this equation, by studying a free, massless scalar field with arbitrary curvature coupling in perturbation theory to second order about the flat spacetime/vacuum solution. We "reduce the order" of the perturbation equations to eliminate spurious solutions, and consider the limit in which the lengthscales determined by the incoming state are much larger than the Planck length. We also need to assume that incoming classical gravitational radiation does not dominate the first order metric perturbation. We find that although the ANEC integral can be negative, if we average the ANEC integral transverse to the geodesic with a suitable Planck scale smearing function, then a strictly positive result is obtained in all cases except for the flat spacetime/vacuum solution. This result suggests --- in agreement with conclusions drawn by Ford and Roman from entirely independent arguments --- that if traversable wormholes do exist as solutions to the semiclassical equations, they cannot be macroscopic but must be ``Planck scale''. A large portion of our paper is devoted to the analysis of general issues concerning the nature of the semiclassical Einstein equation and of prescriptions for extracting physically relevant solutions. 
  We construct a static axisymmetric wormhole from the gravitational field of two Schwarzschild particles which are kept in equilibrium by strings (ropes) extending to infinity. The wormhole is obtained by matching two three-dimensional timelike surfaces surrounding each of the particles and thus spacetime becomes non-simply connected. Although the matching will not be exact in general it is possible to make the error arbitrarily small by assuming that the distance between the particles is much larger than the radius of the wormhole mouths. Whenever the masses of the two wormhole mouths are different, causality violating effects will occur. 
  The study of dynamics in general relativity has been hampered by a lack of coordinate independent measures of chaos. Here we present a variety of invariant measures for quantifying chaotic dynamics in relativity by exploiting the coordinate independence of fractal dimensions. We discuss how preferred choices of time naturally arise in chaotic systems and how the existence of invariant signals of chaos allow us to reinstate standard coordinate dependent measures. As an application, we study the Mixmaster universes and find it to exhibit transient soft chaos. 
  Apparently, all partial differential equations that describe physical phenomena in space-time can be cast into a universal quasilinear, first-order form. In this paper, we do two things. First, we describe some broad features of systems of differential equations so formulated. Examples of such features include hyperbolicity of the equations, constraints and their roles (e.g., in connection with the initial-value formulation), how diffeomorphism freedom is manifest, and how interactions between systems arise and operate. Second, we give a number of examples that illustrate how the equations for physical systems are cast into this form. These examples suggest that the first-order, quasilinear form for a system is often not only the simplest mathematically, but also the most transparent physically. 
  Some recently discovered nonperturbative strong-field effects in tensor-scalar theories of gravitation are interpreted as a scalar analog of ferromagnetism: "spontaneous scalarization". This phenomenon leads to very significant deviations from general relativity in conditions involving strong gravitational fields, notably binary-pulsar experiments. Contrary to solar-system experiments, these deviations do not necessarily vanish when the weak-field scalar coupling tends to zero. We compute the scalar "form factors" measuring these deviations, and notably a parameter entering the pulsar timing observable gamma through scalar-field-induced variations of the inertia moment of the pulsar. An exploratory investigation of the confrontation between tensor-scalar theories and binary-pulsar experiments shows that nonperturbative scalar field effects are already very tightly constrained by published data on three binary-pulsar systems. We contrast the probing power of pulsar experiments with that of solar-system ones by plotting the regions they exclude in a generic two-dimensional plane of tensor-scalar theories. 
  We prove three theorems in general relativity which rule out classical scalar hair of static, spherically symmetric, possibly electrically charged black holes. We first generalize Bekenstein's no--hair theorem for a multiplet of minimally coupled real scalar fields with not necessarily quadratic action to the case of a charged black hole. We then use a conformal map of the geometry to convert the problem of a charged (or neutral) black hole with hair in the form of a neutral self--interacting scalar field nonminimally coupled to gravity to the preceding problem, thus establishing a no--hair theorem for the cases with nonminimal coupling parameter $\xi<0$ or $\xi\geq {1\over 2}$. The proof also makes use of a causality requirement on the field configuration. Finally, from the required behavior of the fields at the horizon and infinity we exclude hair of a charged black hole in the form of a charged self--interacting scalar field nonminimally coupled to gravity for any $\xi$. 
  We consider an inflationary model inspired in the low energy limit of string theory. In this model, the scale factor grows exponentially with time. A perturbation study is performed, and we show that there is a mode which displays an exponential growth in the perturbation of the scalar field. 
  Among plenty of applications, low-dimensional homogeneous spaces appear in cosmological models as both, classical factor spaces of multidimensional geometry and minisuperspaces in canonical quantization.  Here a new tool to restrict their continuous deformations is presented: Classifying spaces for homogeneous manifolds and their related Lie isometry deformations.  The adjoint representation of n-dimensional real Lie algebras induces a natural topology on their classifying space, which encodes the natural algebraic relationship between different Lie algebras therein. For n>1 this topology is not Hausdorffian. Even more it satisfies only the separation axiom T_0, but not T_1, i.e. there is a constant sequence which has a limit different from the members of the sequence. Such a limit is called a transition.  Recently it was found that transitions are the natural generalization and transitive completion of the well-known In\"on\"u-Wigner contractions. For n<5 the relational classifying spaces are constructed explicitly.  Calculating their characteristic scalar invariants via triad representations of the characteristic isometry, local homogeneous Riemannian 3-spaces are classified in their natural geometrical relations to each other. Their classifying space is a composition of pieces with different isometry types. Although it is Hausdorffian, different topological transitions to the same limit may induce locally non-Euclidean regions (e.g. at Bianchi tppes VII_0). 
  We compute the stress--energy operator for a scalar linear quantum field in curved space-time, modulo c-numbers. For the associated Hamiltonian operators, even those generating evolution along timelike vector fields, we find that in general on locally Fock-like (`Hadamard') representations: (a) The Hamiltonians cannot be self-adjoint operators; (b) The automorphisms of the field algebra generated by the evolution cannot be unitarily implemented; (c) The expectation values of the Hamiltonians are well-defined on a dense family of states; but (d) These expectation values are unbounded below, even for evolution along future-directed timelike vector fields and even on Hadamard states. These are all local, ultraviolet, effects. 
  In this work we study the asymptotically flat, static, and spherically symmetric black-hole solutions of the theory described by the action $$S = \int d^nx\sqrt{-g} \left\{\left(1-\xi\phi^2 \right)R - g^{\mu\nu}\partial_\mu\phi\partial_\nu\phi\right\},$$ with $n>3$ and arbitrary $\xi$. We demonstrate the absence of scalar hairs for $\xi<0$. For $\xi>\xi_c=\frac{n-2}{4(n-1)}$, we show that there is no scalar hair obeying $|\phi(r)| < 1/\sqrt{\xi}$ or $|\phi(r)| > 1/\sqrt{\xi}$. For $0<\xi<\xi_c$, we prove the absence of scalar hairs such that $|\phi(r)| < 1/\sqrt{\xi}$ or $\frac{1}{\xi} < \phi^2(r) < \frac{\xi_c}{\xi(\xi_c-\xi)}$.  
  We investigate a minisuperspace model of Einstein gravity plus dilaton that describes a static spherically symmetric configuration or a Kantowski - Sachs like universe. We develop the canonical formalism and identify canonical quantities that generate rigid symmetries of the Hamiltonian. Quantization is performed by the Dirac and the reduced methods. Both approaches lead to the same positive definite Hilbert space. 
  The vacuum cosmological model on the manifold $R \times M_1 \times \ldots \times M_n$ describing the evolution of $n$ Einstein spaces of non-zero curvatures is considered. For $n = 2$ the Einstein equations are reduced to the Abel (ordinary differential) equation and solved, when $(N_1 = $dim $ M_1, N_2 = $ dim$ M_2) = (6,3), (5,5), (8,2)$. The Kasner-like behaviour of the solutions near the singularity $t_s \to +0$ is considered ($t_s$ is synchronous time). The exceptional ("Milne-type") solutions are obtained for arbitrary $n$. For $n=2$ these solutions are attractors for other ones, when $t_s \to + \infty$. For dim $ M = 10, 11$ and $3 \leq n \leq 5$ certain two-parametric families of solutions are obtained from $n=2$ ones using "curvature-splitting" trick. In the case $n=2$, $(N_1, N_2)= (6,3)$ a family of non-singular solutions with the topology $R^7 \times M_2$ is found. 
  By turning to a differential formulation, the post-Newtonian description of metric gravitational theories (PPN formalism) has been extended to include cosmological boundary conditions. The dimensionless expansion parameter is the ratio distance $L$ (measured from the center of a selected space region) to Hubble distance $c/H_0$. The aim was to explore the significance and applicability of a Newtonian cosmology and to clarify to some extent its relation to general-relativistic cosmology. It turns out that up to post-Newtonian order two classes of gravitational theories can be distinguished, here called Machian and non-Machian. In a non-Machian theory like General Relativity the dynamics of cosmic objects within a space region $L \ll c/H_0 $ is described by the usual PPN metric set up for the objects, without introducing time-dependent Newtonian potentials at the origin of the PPN coordinate system. Such potentials of obviously cosmological origin seem to be required for the majority of (by our definition) Machian gravitational theories (including, e.g., Brans-Dicke). Conditions for a theory to be Machian or non-Machian are given in terms of algebraic relations for the PPN parameters. 
  We investigate the representation of the geometrical information of the universe in terms of the eigenvalues of the Laplacian defined on the universe. We concentrate only on one specific problem along this line: To introduce a concept of distance between universes in terms of the difference in the spectra.   We can find out such a measure of closeness from a general discussion. The basic properties of this `spectral distance' are then investigated. It can be related to a reduced density matrix element in quantum cosmology. Thus, calculating the spectral distance gives us an insight for the quantum theoretical decoherence between two universes. The spectral distance does not in general satisfy the triangular inequality, illustrating that it is not equivalent to the distance defined by the DeWitt metric on the superspace.  We then pose a question: Whether two universes with different topologies interfere with each other quantum mechanically? We concentrate on the difference in the orientabilities. Several concrete models in 2-dimension are set up, and the spectral distances between them are investigated: Tori and Klein's bottles, spheres and real projective spaces. Quite surprisingly, we find many cases of spaces with different orientabilities in which the spectral distance turns out to be very short. It may suggest that, without any other special mechanism, two such universes interfere with each other quite strongly. 
  We determine the electrostatic self-force at rest in an arbitrary static metric with cylindrical symmetry in the linear approximation in the Newtonian constant. In linearised Einstein theory, we express it in terms of the components of the energy-momentum tensor. 
  The nature of binary black hole coalescence is the final, uncharted frontier of the relativistic Kepler problem. In the United States, binary black hole coalescence has been identified as a computational ``Grand Challenge'' whose solution is the object of a coordinated effort, just reaching its half-way point, by more than two-score researchers at nearly a dozen institutions. In this report I highlight what I see as the most serious problems standing between us and a general computational solution to the problem of binary black hole coalescence:   * the computational burden associated of the problem based on reasonable extrapolations of present-day computing algorithms and near-term hardware developments;   * some of the computational issues associated with those estimates, and how, through the use of different or more sophisticated computational algorithms we might reduce the expected burden; and    * some of the physical problems associated with the development of a numerical solution of the field equations for a binary black hole system, with particular attention to work going on in, or in association with, the Grand Challenge. 
  The regularized expectation value of the stress-energy tensor for a massless bosonic or fermionic field in 1+1 dimensions is calculated explicitly for the instantaneous vacuum relative to any Cauchy surface (here a spacelike curve) in terms of the length L of the curve (if closed), the local extrinsic curvature K of the curve, its derivative K' with respect to proper distance x along the curve, and the scalar curvature R of the spacetime: T_{00} = - epsilon pi/(6L^2) - K^2/(24 pi), T_{01} = - K'/(12 pi), T_{11} = - epsilon pi/(6L^2) - K^2/(24 pi) + R/(24 pi), in an orthonormal frame with the spatial vector parallel to the curve. Here epsilon = 1 for an untwisted (i.e., periodic in x) one-component massless bosonic field or for a twisted (i.e., antiperiodic in x) two-component massless fermionic field, epsilon = -1/2 for a twisted one-component massless bosonic field, and epsilon = - 2 for an untwisted two-component massless fermionic field. The calculation uses merely the energy-momentum conservation law and the trace anomaly (for which a very simple derivation is also given herein, as well as the expression for the Casimir energy of bosonic and fermionic fields twisted by an arbitrary amount in R^{D-1} x S^1). The two coordinate and conformal invariants of a quantum state that are (nonlocally) determined by the stress-energy tensor are given. Applications to topologically modified deSitter spacetimes, to a flat cylinder, and to Minkowski spacetime are discussed. 
  We introduce a generalized tetrad which plays the role of a potential for torsion and makes torsion dynamic. Starting from the Einstein-Cartan action with torsion, we get two field equations, the Einstein equation and the torsion field equation by using the metric tensor and the torsion potential as independent variables; in the former equation the torsion potential plays the role of a matter field. We also discuss properties of local linear transformations of the torsion potential and give a simple example in which the torsion potential is described by a scalar field. 
  The solution of the 5D Kaluza-Klein's theory is obtained. This solution is a Lorentzian wormhole between two event horizons. It is shown that this solution can sew with (4D Reissner-Nordstr\"om's solution + Maxwell's electrical field) on the event horizon. Such construction is composite wormhole connecting two asymptotically flat region. From the viewpoint of infinite observer this wormhole is an electrical charge. According to J. Wheeler terminology this is "charge without charge". 
  In this essay, we introduce a new effect of gravitationally induced quantum mechanical phases in neutrino oscillations. These phases arise from an hitherto unexplored interplay of gravitation and the principle of the linear superposition of quantum mechanics. In the neighborhood of a 1.4 solar-mass neutron star, gravitationally induced quantum mechanical phases are roughly 20% of their kinematical counterparts. When this information is coupled with the mass square differences implied by the existing neutrino-oscillation data we find that the new effect may have profound consequences for type-II supernova evolution. 
  In the model of flat expansive homogeneous and isotropic relativistic universe with total zero and local non-zero energy the gravitation energy of bodies and the elecromagnetic energy of charged bodies can be localised. 
  The physical excitations entering the effective Lagrangian for quantum black holes are related to a Goldstone boson which is present in the Rindler limit and is due to the spontaneous breaking of the translation symmetry of the underlying Minkowski space. This physical interpretation, which closely parallels similar well-known results for the effective stringlike description of flux tubes in QCD, gives a physical insight into the problem of describing the quantum degrees of freedom of black holes. It also suggests that the recently suggested concept of 'black hole complementarity' emerges at the effective Lagrangian level rather than at the fundamental level. 
  The notion of vacuum fluctuations of the gravitational field plays important role in cosmology. The strong variable gravitational field of the very early Universe amplifies these fluctuations and transforms them into macroscopical cosmological perturbations. Since the underlying process is the parametric amplification, the perturbations are placed in squeezed vacuum quantum states. It is possible that we already see the manifestations of these processes in certain cosmological observations. 
  We prove two theorems which concern difficulties in the formulation of the quantum theory of a linear scalar field on a spacetime, (M,g_{ab}), with a compactly generated Cauchy horizon. These theorems demonstrate the breakdown of the theory at certain `base points' of the Cauchy horizon, which are defined as `past terminal accumulation points' of the horizon generators. Thus, the theorems may be interpreted as giving support to Hawking's `Chronology Protection Conjecture', according to which the laws of physics prevent one from manufacturing a `time machine'. Specifically, we prove: Theorem 1: There is no extension to (M,g_{ab}) of the usual field algebra on the initial globally hyperbolic region which satisfies the condition of F-locality at any base point. In other words, any extension of the field algebra must, in any globally hyperbolic neighbourhood of any base point, differ from the algebra one would define on that neighbourhood according to the rules for globally hyperbolic spacetimes. Theorem 2: The two-point distribution for any Hadamard state defined on the initial globally hyperbolic region must (when extended to a distributional bisolution of the covariant Klein-Gordon equation on the full spacetime) be singular at every base point x in the sense that the difference between this two point distribution and a local Hadamard distribution cannot be given by a bounded function in any neighbourhood (in MXM) of (x,x). Theorem 2 implies quantities such as the renormalized expectation value of \phi^2 or of the stress-energy tensor are necessarily ill-defined or singular at any base point. The proofs rely on the `Propagation of Singularities' theorems of Duistermaat and H\"ormander. 
  By taking into account both quantum mechanical and general relativistic effects, I derive an equation that describes limitations on the measurability of space-time distances as defined by a material reference system. 
  By taking into account both quantum mechanical and general relativistic effects, I derive an equation that describes some limitations on the measurability of space-time distances. I then discuss possible features of quantum gravity which are suggested by this equation. 
  We analyse the fate of density perturbation in the Brans-Dicke Theory, giving a general classification of the solutions of the perturbed equations when the scale factor of the background evolves as a power law. We study with details the cases of vacuum, inflation, radiation and incoherent matter. We find, for the a negative Brans-Dicke parameter, a significant amplification of perturbations. 
  The level of background signals in modern cryogenic resonant mass gravitational wave antenna is discussed caused by (a) the geomagnetic field pulsations and (b) an atmosferic of very low frequency band, generated by a lightning flash. The analysis of our results show that the signals of this origin will generally exceed the signals from the gravitational wave sources. To suppress these artifacts in such gravitational antenna, it is necessary to use the magnetometer included as anti-coincidence protection and a system of magnetic screens. 
  A class of $ G $-invariant Einstein-Yang-Mills (EYM) systems with cosmological constant on homogeneous spaces $ G / H $, where $ G $ is a semisimple compact Lie group, is presented. These EYM--systems can be obtained in terms of dimensional reduction of pure gravity. If $ G / H $ is a symmetric space, the EYM--system on $ G / H $ provides a static solution of the EYM--equations on spacetime $ {\Bbb R} \times G / H $. This way, in particular, a solution for an arbitrary Lie group $ F $, considered as a symmetric space, is obtained. This solution is discussed in detail for the case $ F = SU(2) $. A known analytical EYM--system on $ {\Bbb R} \times S^3 $ is recovered and it is shown - using a relation to the BPST instanton - that this solution is of sphaleron type. Finally, a relation to the distance of Bures and to parallel transport along mixed states is shown. 
  We argue that the analysis of transients, of decoherence effects, or of any breaking of the exact boost invariance of the Ernst metric shows that uniformly accelerated black holes do emit an energy flux given by the Doppler-shifted Hawking radiation, in perfect {\it agreement} to what happens for accelerated particles. 
  We suggest how to interpret the action of the quantum Hamiltonian constraint of general relativity in the loop representation as a skein relation on the space of knots. Therefore, by considering knot polynomials that are compatible with that skein relation, one guarantees that all the quantum Einstein equations are solved. We give a particular example of such invariant and discuss the consistency of the constraint algebra in this approach. 
  Analytic solutions of the Teukolsky equation in Kerr geometries are presented in the form of series of hypergeometric functions and Coulomb wave functions. Relations between these solutions are established. The solutions provide a very powerful method not only for examining the general properties of solutions and physical quantities when they are applied to, but also for numerical computations. The solutions are given in the expansion of a small parameter $\epsilon \equiv 2M\omega$, $M$ being the mass of black hole, which corresponds to Post-Minkowski expansion by $G$ and to post-Newtonian expansion when they are applied to the gravitational radiation from a particle in circular orbit around a black hole. It is expected that these solutions will become a powerful weapon to construct the theoretical template towards LIGO and VIRGO projects. 
  A new algorithm for computing the accurate values of quasinormal frequencies of extremal Reissner-Nordstr\"{o}m black holes is presented. The numerically computed values are consistent with the values earlier obtained by Leaver and those obtained through the WKB method. Our results are more precise than other results known to date.   We also find a curious fact that the resonant frequencies of gravitational waves with multi-pole index $l$ coincide with those of electromagnetic waves with multi-pole index $l-1$ in the extremal limit. 
  This paper is a continuation of the papers [gr-qc/9409010, gr-qc/9505034]. A revision of the Einstein equation shows that its dynamic incompleteness, contrary to a popular opinion, cannot be circumvented by so-called coordinate conditions. Gravidynamics, i.e., dynamics for gravitational potentials $g_{\mu\nu}$ is advanced, which differs from geometrodynamics of general relativity in that the former is based on a projected Einstein equation. Cosmic gravidynamics, due to a global structure of spacetime, is complete. The most important result is a possibility of the closed universe with a density below the critical one.   Keywords: general relativity, cosmic time, cosmic space, Einstein equation, quantum, metric, indeterministic 
  We develop a Hamiltonian formalism suitable to be applied to gauge theories in the presence of Gravitation, and to Gravity itself when considered as a gauge theory. It is based on a nonlinear realization of the Poincar\'e group, taken as the local spacetime group of the gravitational gauge theory, with $SO(3)$ as the classification subgroup. The Wigner--like rotation induced by the nonlinear approach singularizes out the role of time and allows to deal with ordinary $SO(3)$ vectors. We apply the general results to the Einstein--Cartan action. We study the constraints and we obtain Einstein's classical equations in the extremely simple form of time evolution equations of the coframe. As a consequence of our approach, we identify the gauge--theoretical origin of the Ashtekar variables. 
  In this paper we review the recent developments in the relativistic theory of stellar pulsations with special emphasis on the spacetime or w-modes. We discuss also the excitation of these modes and the information about the neutron star that we will be able to mine from their detection. The paper will appear in the Proceedings of the Les Houches School on Astrophysical Sources of Gravitational Radiation, 1995, edited by J.-A. Marck and J.-P. Lasota to be published by Springer-Verlag. 
  We adopt the point of view that (Riemannian) classical and (loop-based) quantum descriptions of geometry are macro- and micro-descriptions in the usual statistical mechanical sense. This gives rise to the notion of geometrical entropy, which is defined as the logarithm of the number of different quantum states which correspond to one and the same classical geometry configuration (macro-state). We apply this idea to gravitational degrees of freedom induced on an arbitrarily chosen in space 2-dimensional surface. Considering an `ensemble' of particularly simple quantum states, we show that the geometrical entropy $S(A)$ corresponding to a macro-state specified by a total area $A$ of the surface is proportional to the area $S(A)=\alpha A$, with $\alpha$ being approximately equal to $1/16\pi l_p^2$. The result holds both for case of open and closed surfaces. We discuss briefly physical motivations for our choice of the ensemble of quantum states. 
  The equations of the 1+3 orthonormal frame approach are explicitly presented and discussed. Natural choices of local coordinates are mentioned. A dimensionless formulation is subsequently given. It is demonstrated how one can obtain a number of interesting problems by specializing the general equations. In particular, equation systems for ``silent'' dust cosmological models also containing magnetic Maxwell fields, locally rotationally symmetric spacetime geometries and spatially homogeneous cosmological models are presented. We show that while the 3-Cotton--York tensor is zero for Szekeres dust models, it is nonzero for a generic representative within the ``silent'' class. 
  A class of exact solutions of the Wheeler-DeWitt equation for diagonal Bianchi type IX cosmologies with cosmological constant is derived in the metric representation. This class consists of all the ``topological solutions'' which are associated with the Bianchi type IX reduction of the Chern-Simons functional in Ashtekar variables. The different solutions within the class arise from the topologically inequivalent choices of the integration contours in the transformation from the Ashtekarrepresentation to the metric representation. We show how the saddle-points of the reduced Chern-Simons functional generate a complete basis of such integration contours and the associated solutions. Among the solutions we identify two, which, semi-classically, satisfy the boundary conditions proposed by Vilenkin and by Hartle and Hawking, respectively. In the limit of vanishing cosmological constant our solutions reduce to a class found earlier in special fermion sectors ofsupersymmetric Bianchi type IX models. 
  Extending a method developed by Sasaki in the Schwarzschild case and by Shibata, Sasaki, Tagoshi, and Tanaka in the Kerr case, we calculate the post-Newtonian expansion of the gravitational wave luminosities from a test particle in circular orbit around a rotating black hole up to $O(v^8)$ beyond the quadrupole formula. The orbit of a test particle is restricted on the equatorial plane. We find that spin dependent terms appear in each post-Newtonian order, and that at $O(v^6)$ they have a significant effect on the orbital phase evolution of coalescing compact binaries. By comparing the post-Newtonian formula of the luminosity with numerical results we find that, for $30M\lesssim r \lesssim 100M$, the spin dependent terms at $O(v^6)$ and $O(v^7)$ improve the accuracy of the post-Newtonian formula significantly, but those at $O(v^8)$ do not improve. 
  We study the problem of boundary terms and boundary conditions for Chern-Simons gravity in five dimensions. We show that under reasonable boundary conditions one finds an effective field theory at the four-dimensional boundary described by dilaton gravity with a Gauss-Bonnet term. The coupling of matter is also discussed. 
  Topological gravity is the reduction of general relativity to flat space-times. A lattice model describing topological gravity is developed starting from a Hamiltonian lattice version of $B\w F$ theory. The extra symmetries not present in gravity that kill the local degrees of freedom in $B\wedge F$ theory are removed. The remaining symmetries preserve the geometrical character of the lattice. Using self-dual variables, the conditions that guarantee the geometricity of the lattice become reality conditions. The local part of the remaining symmetry generators, that respect the geometricity-reality conditions, has the form of Ashtekar's constraints for GR. Only after constraining the initial data to flat lattices and considering the non-local (plus local) part of the constraints does the algebra of the symmetry generators close. A strategy to extend the model for non-flat connections and quantization are discussed. 
  Several exact, cylindrically symmetric solutions to Einstein's vacuum equations are given. These solutions were found using the connection between Yang-Mills theory and general relativity. Taking known solutions of the Yang-Mills equations (e.g. the topological BPS monopole solutions) it is possible to construct exact solutions to the general relativistic field equations. Although the general relativistic solutions were found starting from known solutions of Yang-Mills theory they have different physical characteristics. 
  It is shown, for the self-consistent system of scalar, electro-magnetic and gravitational fields in general relativity, that the equations of motion admit a special kind of solutions with spherical or cylindrical symmetry. For these solutions, the physical fields vanish and the space-time is flat outside of the critical sphere or cylinder. Therefore, the mass and the electric charge of these configurations are zero. The principal difference between droplet-like solutions with spherical symmetry and those with cylindrical one has been established. In the first case there exists a possibility of continuous transformation of droplet-like configuration into the solitonian one, when for the second case there is no such a possibility. 
  We consider the Vlasov-Poisson system in a cosmological setting and prove nonlinear stability of homogeneous solutions against small, spatially periodic perturbations in the sup-norm of the spatial mass density. This result is connected with the question of how large scale structures such as galaxies have evolved out of the homogeneous state of the early universe. 
  A definition of surface gravity at the apparent horizon of dynamical spherically symmetric spacetimes is proposed. It is based on a unique foliation by ingoing null hypersurfaces. The function parametrizing the hypersurfaces can be interpreted as the phase of a light wave uniformly emitted by some far-away static observer. The definition gives back the accepted value of surface gravity in the static case by virtue of its nonlocal character. Although the definition is motivated by the behavior of outgoing null rays, it turns out that there is a simple connection between the generalized surface gravity, the acceleration of any radially moving observer, and the observed frequency change of the infalling light signal. In particular, this gives a practical and simple method of how any geodesic observer can determine surface gravity by measuring only the redshift of the infalling light wave. The surface gravity can be expressed as an integral of matter field quantities along an ingoing null line, which shows that it is a continuous function along the apparent horizon. A formula for the area change of the apparent horizon is presented, and the possibility of thermodynamical interpretation is discussed. Finally, concrete expressions of surface gravity are given for a number of four-dimensional and two-dimensional dynamical black hole solutions. 
  Self-consistent solutions to the system of spinor and scalar field equations in General Relativity are studied for the case of Bianchi type-I space-time. It should be emphasized the absence of initial singularity for some types of solutions and also the isotropic mode of space-time in some special case. 
  Exact droplet-like solutions to the nonlinear scalar field equations have been obtained in the Robertson-Walker space-time and their linearized stability has been proved. 
  On the basis of exact solutions to the Einstein-Abelian gauge-dilaton equations in $D$-dimensional gravity, the properties of static axial configurations are discussed. Solutions free of curvature singularities are selected; they can be attributed to traversible wormholes with cosmic string-like singularities at their necks. In the presence of an electromagnetic field some of these wormholes are globally regular, the string-like singularity being replaced by a set of twofold branching points. Consequences of wormhole regularity and symmetry conditions are discussed. In particular, it is shown that (i) regular, symmetric wormholes have necessarily positive masses as viewed from both asymptotics and (ii) their characteristic length scale in the big charge limit ($GM^2 \ll Q^2$) is of the order of the ``classical radius" $Q^2/M$. 
  We investigate some modifications of the static BTZ black hole solution due to a chosen asymptotically constant dilaton/scalar. New classes of static black hole solutions are obtained. One of the solutions contains the Martinez-Zanelli conformal black hole solution as a special case. Using quasilocal formalism, we calculate their mass for a finite spatial region that contains the black hole. Their temperatures are also computed. Finally, using some of the curvature singularities as examples, we investigate whether a quantum particle behaves singularly or not. 
  The problem of topology change transitions in quantum gravity is investigated from the Wheeler-de Witt wave function point of view. It is argued that for all theories allowing wormhole effects the wave function of the universe is exponentially large. If the wormhole action is positive, one can try to overcome this difficulty by redefinition of the inner product, while for the case of negative wormhole action the more serious problems arise. 
  A closed explicit representation of the vacuum Einstein equations in terms of components of curvature 2-forms is given. The discussion is restricted to the case of non-vanishing cubic invariant of conformal curvature spinor. The complete set of algebraic and differential identities connecting particular equations is presented and their consistency conditions are analyzed. 
  We consider the supersymmetry (SUSY) transformations in the chiral Lagrangian for $N = 1$ supergravity (SUGRA) with the complex tetrad following the method used in the usual $N = 1$ SUGRA, and present the explicit form of the SUSY trasformations in the first-order form. The SUSY transformations are generated by two independent Majorana spinor parameters, which are apparently different from the constrained parameters employed in the method of the 2-form gravity. We also calculate the commutator algebra of the SUSY transformations on-shell. 
  Paradoxes that can supposedly occur if a time machine is created are discussed. It is shown that the existence of trajectories of ``multiplicity zero'' (i.e. trajectories that describe a ball hitting its younger self so that the latter cannot fall into the time machine) is not paradoxical by itself. This {\em apparent paradox} can be resolved (at least sometimes) without any harm to local physics or to the time machine. Also a simple model is adduced for which the absence of {\em true} paradoxes caused by self-interaction is proved. 
  The Wilson approximate dynamics and the Einstein dynamics are compared for binary systems. At the second post-Newtonian approximation, genuine two-body aspects are found to differ by up to 114\%. In the regime of a formal innermost stable circular orbit (ISCO) the both dynamics differ by up to 7\%. 
  The physical concept of locality is first analyzed in the special relativistic quantum regime, and compared with that of microcausality and the local commutativity of quantum fields. Its extrapolation to quantum general relativity on quantum bundles over curved spacetime is then described. It is shown that the resulting formulation of quantum-geometric locality based on the concept of local quantum frame incorporating a fundamental length embodies the key geometric and topological aspects of this concept. Taken in conjunction with the strong equivalence principle and the path-integral formulation of quantum propagation, quantum-geometric locality leads in a natural manner to the formulation of quantum-geometric propagation in curved spacetime. Its extrapolation to geometric quantum gravity formulated over quantum spacetime is described and analyzed. 
  We study the classical and quantum theory of a class of nonlinear differential equations on chronology violating spacetime models in which space consists of finitely many discrete points. Classically, in the linear and weakly nonlinear regimes (for generic choices of parameters) we prove existence and uniqueness of solutions corresponding to initial data specified before the dischronal region; however, uniqueness (but not existence) fails in the strongly coupled regime. The evolution preserves the symplectic structure.   The quantum theory is approached via the quantum initial value problem (QIVP); that is, by seeking operator-valued solutions to the equation of motion with initial data representing the canonical (anti)commutation relations. Using normal operator ordering, we construct solutions to the QIVP for both Bose and Fermi statistics (again for generic choice of parameters) and prove that these solutions are unique. For models with two spatial points, the resulting evolution is unitary; however, for a more general model the evolution fails to preserve the (anti)commutation relations and is therefore nonunitary. We show that this nonunitary evolution cannot be described using a superscattering operator with the usual properties.   We present numerical evidence to show that the bosonic quantum theory can pick out a unique classical limit for certain ranges of the coupling strength, even when there are many classical solutions. We show that the quantum theory depends strongly on the choice of operator ordering.   In addition, we show that our results differ from those obtained using the ``self-consistent path integral''. It follows that the path integral evolution does not correspond to a solution of the equation of motion. 
  Quantum noise is an important issue for advanced LIGO. Although it is in principle possible to beat the Standard Quantum Limit (SQL), no practical recipe has been found yet. This paper dicusses quantum noise in the context of speedmeter-a devise monitoring the speed of the testmass. The scheme proposed to overcome SQL in this case might be more practical than the methods based on monitoring position of the testmass. 
  We analyze the quantum cosmology of the simplest pre--big--bang model without dilaton potential. In addition to the minisuperspace variables we include inhomogeneous dilaton fluctuations and determine their wave function on a semiclassical background. This wave function is used to calculate the reduced density matrix and to find criteria for the loss of decoherence. It is shown that coherence between different backgrounds can always be achieved by a specific choice of vacua though generically decoherence can be expected. In particular, we discuss the implications of these results on the ``exit problem'' of pre--big--bang cosmology. 
  This paper derives the total power or energy loss rate generated in the form of gravitational waves by an inspiralling compact binary system to the five halves post-Newtonian (2.5PN) approximation of general relativity. Extending a recently developed gravitational-wave generation formalism valid for arbitrary (slowly-moving) systems, we compute the mass multipole moments of the system and the relevant tails present in the wave zone to 2.5PN order. In the case of two point-masses moving on a quasi-circular orbit, we find that the 2.5PN contribution in the energy loss rate is entirely due to tails. Relying on an energy balance argument we derive the laws of variation of the instantaneous frequency and phase of the binary. The 2.5PN order in the accumulated phase is significantly large, being grossly of the same order of magnitude as the previous 2PN order, but opposite in sign. However finite mass effects at 2.5PN order are small. The results of this paper should be useful when analyzing the data from inspiralling compact binaries in future gravitational-wave detectors like VIRGO and LIGO. 
  Although we know that black holes are characterized by a temperature and an entropy, we do not yet have a satisfactory microscopic ``statistical mechanical'' explanation for black hole thermodynamics. I describe a new approach that attributes the thermodynamic properties to ``would-be gauge'' degrees of freedom that become dynamical on the horizon. For the (2+1)-dimensional black hole, this approach gives the correct entropy. (Talk given at the Pacific Conference on Gravitation and Cosmology, Seoul, February 1996.) 
  This paper generalizes earlier work on Hamiltonian boundary terms by omitting the requirement that the spacelike hypersurfaces $\Sigma_t$ intersect the timelike boundary $\cal B$ orthogonally. The expressions for the action and Hamiltonian are calculated and the required subtraction of a background contribution is discussed. The new features of a Hamiltonian formulation with non-orthogonal boundaries are then illustrated in two examples. 
  We present results from a numerical study of spherically-symmetric collapse of a self-gravitating, SU(2) gauge field. Two distinct critical solutions are observed at the threshold of black hole formation. In one case the critical solution is discretely self-similar and black holes of arbitrarily small mass can form. However, in the other instance the critical solution is the n=1 static Bartnik-Mckinnon sphaleron, and black hole formation turns on at finite mass. The transition between these two scenarios is characterized by the superposition of both types of critical behaviour. 
  The existence of point symmetries in the cosmological field equations of generalized vacuum scalar--tensor theories is considered within the context of the spatially homogeneous cosmologies. It is found that such symmetries only occur in the Brans--Dicke theory when the dilaton field self--interacts. Moreover, the interaction potential of the dilaton must take the form of a cosmological constant. For the spatially flat, isotropic model, it is shown how this point symmetry may be employed to generate a discrete scale factor duality in the Brans--Dicke action. 
  This is in fact an Erratum to the paper published in Physics Letters A221 (1996) 359. The reduced-phase-space discussion remains essentially valid in spite of the fact that many equations are changed. However, the analysis based on the Wheeler-DeWitt equation should be very different from the one given in the paper, which must be disregarded. 
  The version of supergravity formulated by Ogievetsky and Sokatchev is almost identical to the conventional $N=1$ theory, except that the cosmological constant $\Lambda$ appears as a dynamical variable which is constant only by virtue of the field equations. We consider the canonical quantisation of this theory, and show that the wave function evolves with respect to a dynamical variable which can be interpreted as a cosmological time parameter. The square of the modulus of the wave function obeys a set of simple conservation equations and can be interpreted as a probability density functional. The usual problems associated with time in quantum gravity are avoided. 
  Using the optical reference geometry approach, we have derived in the following, a general expression for the ellipticity of a slowly rotating fluid configuration using Newtonian force balance equation in the conformally projected absolute 3-space, in the realm of general relativity. Further with the help of Hartle-Thorne (H-T) metric for a slowly rotating compact object, we have evaluated the centrifugal force acting on a fluid element and also evaluated the ellipticity and found that the centrifugal reversal occurs at around $R/R_s \approx 1.45$, and the ellipticity maximum at around $R/R_s \approx 2.75$. The result has been compared with that of Chandrasekhar and Miller which was obtained in the full 4-spacetime formalism. 
  The collapse of a massless scalar field in the Brans-Dicke theory of gravitation is studied in the analysis of both analytical solution and numerical one. By conformally transforming the Roberts's solution into the Brans-Dicke frame, we find for $\omega > -3/2$ that a continuous self-similarity continues and that the critical exponent does depend on $\omega$. By conformally transforming the Choptuik's solution into the Brans-Dicke frame, we find for $\omega > -3/2$ that at the critical solution shows discrete self-similarity, however, the critical exponent depends strongly on $\omega$ while the echoing parameter weakly on it. 
  Photons and thermal photons are studied in the Rindler Wedge employing Feynman's gauge and canonical quantization. A Gupta-Bleuler-like formalism is explicitly implemented. Non thermal Wightman functions and related (Euclidean and Lorentzian) Green functions are explicitly calculated and their complex time analytic structure is analyzed using the Fulling-Ruijsenaars master function. The invariance of the advanced minus retarded fundamental solution is checked and a Ward identity discussed. It is suggested the KMS condition can be implemented to define thermal states also dealing with unphysical photons. Following this way, thermal Wightman functions and related (Euclidean and Lorentzian) Green functions are built up. Their analytic structure is examined employing a thermal master function as in the non thermal case and other corresponding properties are discussed. Some subtleties arising dealing with unphysical photons in presence of the Rindler conical singularity are pointed out. In particular, a family of thermal Wightman and Schwinger functions with the same physical content is proved to exist due to a remaining static gauge ambiguity. A photon version of Bisognano-Wichmann theorem is investigated in the case of photons propagating in the Rindler Wedge employing Wightman functions. Despite of the found ambiguity in defining Rindler Green functions, the coincidence of $(\beta = 2\pi)$-Rindler Wightman functions and Minkowski Wightman functions is proved dealing with test functions related to physical photons and Lorentz photons. 
  Using as an illustrative example the p=1 operator-ordered Wheeler-DeWitt equation for a closed, radiation-filled Friedmann-Robertson-Walker universe, we introduce and discuss the supersymmetric double Darboux method in quantum cosmology. A one-parameter family of ``quantum'' universes and the corresponding ``wavefunctions of the universe" for this case are presented 
  We make a systematic investigation of stationary cylindrically symmetric solutions to the five-dimensional Einstein and Einstein-Gauss-Bonnet equations. Apart from the five-dimensional neutral cosmic string metric, we find two new exact solutions which qualify as cosmic strings, one corresponding to an electrically charged cosmic string, the other to an extended superconducting cosmic string surrounding a charged core. In both cases, test particles are deflected away from the singular line source. We extend both kinds of solutions to exact multi-cosmic string solutions. 
  A Cauchy-characteristic initial value problem for the Einstein-Klein-Gordon system with spherical symmetry is presented. Initial data are specified on the union of a space-like and null hypersurface. The development of the data is obtained with the combination of a constrained Cauchy evolution in the interior domain and a characteristic evolution in the exterior, asymptotically flat region. The matching interface between the space-like and characteristic foliations is constructed by imposing continuity conditions on metric, extrinsic curvature and scalar field variables, ensuring smoothness across the matching surface. The accuracy of the method is established for all ranges of $M/R$, most notably, with a detailed comparison of invariant observables against reference solutions obtained with a calibrated, global, null algorithm. 
  The null-surface formulation of general relativity -- recently introduced -- provides novel tools for describing the gravitational field, as well as a fresh physical way of viewing it. The new formulation provides ``local'' observables corresponding to the coordinates of points --- which constitute the spacetime manifold --- in a {\em geometrically defined chart\/}, as well as non-local observables corresponding to lightcone cuts and lightcones. In the quantum theory, the spacetime point observables become operators and the spacetime manifold itself becomes ``quantized'', or ``fuzzy''. This novel view may shed light on some of the interpretational problems of a quantum theory of gravity. Indeed, as we discuss briefly, the null-surface formulation of general relativity provides (local) geometrical quantities --- the spacetime point observables --- which are candidates for the long-sought physical operators of the quantum theory. 
  The massive nonsymmetric gravitational theory is shown to posses a linearisation instability at purely GR field configurations, disallowing the use of the linear approximation in these situations. It is also shown that arbitrarily small antisymmetric sector Cauchy data leads to singular evolution unless an ad hoc condition is imposed on the initial data hypersurface. 
  We argue that the statistical entropy relevant for the thermal interactions of a black hole with its surroundings is (the logarithm of) the number of quantum microstates of the hole which are distinguishable from the hole's exterior, and which correspond to a given hole's macroscopic configuration. We compute this number explicitly from first principles, for a Schwarzschild black hole, using nonperturbative quantum gravity in the loop representation. We obtain a black hole entropy proportional to the area, as in the Bekenstein-Hawking formula. 
  We study quantum gravitational effects on black hole radiation, using loop quantum gravity. Bekenstein and Mukhanov have recently considered the modifications caused by quantum gravity on Hawking's thermal black-hole radiation. Using a simple ansatz for the eigenstates the area, they have obtained the intriguing result that the quantum properties of geometry affect the radiation considerably, yielding a definitely non-thermal spectrum. Here, we replace the simple ansatz employed by Bekenstein and Mukhanov with the actual eigenstates of the area, computed using the loop representation of quantum gravity. We derive the emission spectra, using a classic result in number theory by Hardy and Ramanujan. Disappointingly, we do not recover the Bekenstein-Mukhanov spectrum, but --effectively-- a Hawking's thermal spectrum. The Bekenstein-Mukhanov result is therefore likely to be an artefact of the naive ansatz, rather than a robust result. The result is an example of concrete (although somewhat disappointing) application of nonperturbative quantum gravity. 
  These lectures are designed to provide a general introduction to the Einstein-Vlasov system and to the global Cauchy problem for these equations. To start with some general facts are collected and a local existence theorem for the Cauchy problem stated. Next the case of spherically symmetric asymptotically flat solutions is examined in detail. The approach taken, using maximal-isotropic coordinates, is new. It is shown that if a singularity occurs in the time evolution of spherically symmetric initial data, the first singularity (as measured by a maximal time coordinate) occurs at the centre. Then it is shown that for small initial data the solution exists globally in time and is geodesically complete. Finally, the proof of the general local existence theorem is sketched. This is intended to be an informal introduction to some of the ideas which are important in proving such theorems rather than a formal proof. 
  We discuss a gauge invariant approach to the theory of cosmological perturbations in a higher-dimensonal background. We find the normal modes which diagonalize the perturbed action, for a scalar field minimally coupled to gravity, in a higher-dimensional manifold M of the Bianchi-type I, under the assumption that the translations along an isotropic spatial subsection of M are isometries of the full, perturbed background. We show that, in the absence of scalar field potential, the canonical variables for scalar and tensor metric perturbations satisfy exactly the same evolution equation, and we discuss the possible dependence of the spectrum on the number of internal dimensions. 
  The rank--1 sector of classical Ashtekar gravity is considered, motivated by the degeneracy of the metric along the Wilson lines in quantum loop states. It is found that the lines behave like 1+1 dimensional spacetimes with a pair of massless complex fields propagating along them. The inclusion of matter and extension to supergravity are also considered. 
  In the Wheeler-DeWitt framework, by a gauge fixing procedure, we set up a scheme to recover a Schr\"odinger type equation, living in the orbits space with the {\it lapse} function as evolution parameter. By means of the associated stationary equation, we have the possibility of calculating quantum corrections to classical quantities. The Schwarzschild wormhole case is discussed as an example of application. 
  It is known from the work of Banados et al. that a space-time with event horizons (much like the Schwarzschild black hole) can be obtained from 2+1 dimensional anti-de Sitter space through a suitable identification of points. We point out that this can be done in 3+1 dimensions as well. In this way we obtain black holes with event horizons that are tori or Riemann surfaces of genus higher than one. They can have either one or two asymptotic regions. Locally, the space-time is isometric to anti-de Sitter space. 
  The non-minimal coupling of gravity to a scalar field can be transformed into a minimal coupling through a conformal transformation. We show how to connect the results of a perturbation calculation, performed around a Friedmann-Robertson-Walker background solution, before and after the conformal transformation. We work in the synchronous gauge, but we discuss the implications of employing other formalisms. 
  It is well-known that gravitationally induced vacuum polarization often violates the point-wise energy conditions and sometimes violates the averaged energy conditions. In this paper I begin a systematic attack on the question of where and by how much the various energy conditions are violated. I work in the test-field limit, and focus on conformally coupled massless scalar fields in Schwarzschild spacetime, using the Hartle--Hawking vacuum. I invoke a mixture of analytical and numerical techniques, and critically compare the qualitative behaviour to be expected from the Page approximation with that adduced from the numerical calculations of Anderson, Hiscock, and Samuel. I show that the various point-wise energy conditions are violated in a series of onion-like layers located between the unstable photon orbit and the event horizon, the sequence of violations being DEC, WEC, and (NEC+SEC). Furthermore the ANEC is violated for *some* of the null geodesics trapped in this region. Having established the basic machinery in this paper, the Boulware vacuum will be treated in a companion paper, while other exensions should be straightforward. 
  I show that in the Boulware vacuum (1) all standard (point-wise and averaged) energy conditions are violated throughout the exterior region---all the way from spatial infinity down to the event horizon, and (2) outside the event horizon the standard point-wise energy conditions are violated in a maximal manner: they are violated at all points and for all null/timelike vectors. (The region inside the event horizon is considerably messier, and of dubious physical relevance. Nevertheless the standard point-wise energy conditions also seem to be violated even inside the event horizon.) This is rather different from the case of the Hartle--Hawking vacuum, wherein violations of the energy conditions were confined to the region inside the unstable photon orbit. These calculations are for the quantum stress-energy tensor corresponding to a conformally-coupled massless scalar field in the Boulware vacuum. I work in the test-field limit, restrict attention to the Schwarzschild geometry, and invoke a mixture of analytical and numerical techniques. This *suggests* that general self-consistent solutions of semiclassical quantum gravity might *not* satisfy the energy conditions, and may in fact for certain quantum fields and certain quantum states violate *all* the energy conditions. 
  Building on a pair of earlier papers, I investigate the various point-wise and averaged energy conditions for the quantum stress-energy tensor corresponding to a conformally-coupled massless scalar field in the in the (1+1)-dimensional Schwarzschild spacetime. Because the stress-energy tensors are analytically known, I can get exact results for the Hartle--Hawking, Boulware, and Unruh vacua. This exactly solvable model serves as a useful sanity check on my (3+1)-dimensional investigations wherein I had to resort to a mixture of analytic approximations and numerical techniques. Key results in (1+1) dimensions are: (1) NEC is satisfied outside the event horizon for the Hartle--Hawking vacuum, and violated for the Boulware and Unruh vacua. (2) DEC is violated everywhere in the spacetime (for any quantum state, not just the standard vacuum states). 
  It is well known that knots are countable in ordinary knot theory. Recently, knots {\it with intersections} have raised a certain interest, and have been found to have physical applications. We point out that such knots --equivalence classes of loops in $R^3$ under diffeomorphisms-- are not countable; rather, they exhibit a moduli-space structure. We characterize these spaces of moduli and study their dimension. We derive a lower bound (which we conjecture being actually attained) on the dimension of the (non-degenerate components) of the moduli spaces, as a function of the valence of the intersection. 
  The Born-Oppenheimer approach to the matter-gravity system is illustrated and the unitary evolution for matter, in the absence of phenomena such as tunnelling or other instabilities, verified. The Born-Oppenheimer approach to the matter-gravity system is illustrated in a simple minisuperspace model and the corrections to quantum field theory on a semiclassical background exhibited. Within such a context the unitary evolution for matter, in the absence of phenomena such as tunnelling or other instabilities, is verified and compared with the results of other approaches. Lastly the simplifications associated with the use of adiabatic invariants to obtain the solution of the explicitly time dependent evolution equation for matter are evidenced. 
  In the consistent histories formulation of quantum theory, the probabilistic predictions and retrodictions made from observed data depend on the choice of a consistent set. We show that this freedom allows the formalism to retrodict contrary propositions which correspond to orthogonal commuting projections and which each have probability one. We also show that the formalism makes contrary probability one predictions when applied to Gell-Mann and Hartle's generalised time-neutral quantum mechanics. 
  An introduction into the theory of locally anisotropic spaces (modelled as vector bundles provided with compatible nonlinear and distinguished linear connections and metric structures and containing as particular cases different types of Kaluza--Klein and/or extensions of Lagrange and Finsler spaces) is presented. The conditions for consistent propagation of closed strings in locally anisotropic background spaces are analyzed. The connection between the conformal invariance, the vanishing of the renormalization group beta--function of the generalized sigma--model and field equations of locally anisotropic gravity is studied in detail. 
  The purpose of this work is to extend the formalism of stochastic calculus to the case of spaces with local anisotropy (modeled as vector bundles with compatible nonlinear and distinguished connections and metric structures and containing as particular cases different variants of Kaluza--Klein and generalized Lagrange and Finsler spaces). We shall examine nondegenerate diffusions on the mentioned spaces and theirs horizontal lifts. 
  The theory of spinors is developed for locally anisotropic (la) spaces, in brief la-spaces, which in general are modeled as vector bundles provided with nonlinear and distinguished connections and metric structures (such la-spaces contain as particular cases the Lagrange, Finsler and, for trivial nonlinear connections, Kaluza-Klein spaces). The la-spinor differential geometry is constructed. The distinguished spinor connections are studied and compared with similar ones on la-spaces. We derive the la-spinor expressions of curvatures and torsions and analyze the conditions when the distinguished torsion and nonmetricity tensors can be generated from distinguished spinor connections. The dynamical equations for gravitational and matter field la-interactions are formulated. 
  The theory of locally anisotropic superspaces (supersymmetric generalizations of various types of Kaluza--Klein, Lagrange and Finsler spaces) is laid down. In this framework we perform the analysis of construction of the supervector bundles provided with nonlinear and distinguished connections and metric structures. Two models of locally anisotropic supergravity are proposed and studied in details. 
  We formulate the theory of nearly autoparallel maps (generalizing conformal transforms) of locally anisotropic spaces and define the nearly autoparallel integration as the inverse operation to both covariant derivation and deformation of connections by nearly autoparallel maps. By using this geometric formalism we consider a variant of solution of the problem of formulation of conservation laws for locally anisotropic gravity. We note that locally anisotropic spases contain as particular cases various extensions of Kaluza--Klein, generalized Lagrange and Finsler spaces. 
  We consider the standard gauge theory of Poincar\'{e} group, realizing as a subgroup of $GL(5. R)$. The main problem of this theory was appearing of the fields connected with non-Lorentz symmetries, whose physical sense was unclear. In this paper we treat the gravitation as a Higgs-Goldstone field, and the translation gauge field as a new tensor field. The effective metric tensor in this case is hybrid of two tensor fields. In the linear approximation the massive translation gauge field can give the Yukava type correction to the Newtons potential. Also outer potentials of a sphere and ball of the same mass are different in this case. Corrections to the standard Einshtein post Newtonian formulas of the light deflection and radar echo delay is obtained. The string like solution of the nonlinear equations of the translation gauge fields is found. This objects can results a Aharonov-Bohm type effect even for the spinless particles. They can provide density fluctuations in the early universe, necessary for galaxy formations. The spherically symmetric solution of the theory is found. The translation gauge field lead to existence of a impenetrable for the matter singular surface inside the Schwarzschild sphere, which can prevent gravitational collapse of a massive body. 
  I construct a spherically symmetric solution for a massless real scalar field minimally coupled to general relativity which is discretely self-similar (DSS) and regular. This solution coincides with the intermediate attractor found by Choptuik in critical gravitational collapse. The echoing period is Delta = 3.4453 +/- 0.0005. The solution is continued to the future self-similarity horizon, which is also the future light cone of a naked singularity. The scalar field and metric are C1 but not C2 at this Cauchy horizon. The curvature is finite nevertheless, and the horizon carries regular null data. These are very nearly flat. The solution has exactly one growing perturbation mode, thus confirming the standard explanation for universality. The growth of this mode corresponds to a critical exponent of gamma = 0.374 +/- 0.001, in agreement with the best experimental value. I predict that in critical collapse dominated by a DSS critical solution, the scaling of the black hole mass shows a periodic wiggle, which like gamma is universal. My results carry over to the free complex scalar field. Connections with previous investigations of self-similar scalar field solutions are discussed, as well as an interpretation of Delta and gamma as anomalous dimensions. 
  We study the motion of a spinning test particle in Schwarzschild spacetime, analyzing the Poincar\'e map and the Lyapunov exponent. We find chaotic behavior for a particle with spin higher than some critical value (e.g. $S_{cr} \sim 0.64 \mu M$ for the total angular momentum $J=4 \mu M$), where $\mu$ and $M$ are the masses of a particle and of a black hole, respectively. The inverse of the Lyapunov exponent in the most chaotic case is about three orbital periods, which suggests that chaos of a spinning particle may become important in some relativistic astrophysical phenomena. The ``effective potential" analysis enables us to classify the particle orbits into four types as follows. When the total angular momentum $J$ is large, some orbits are bounded and the ``effective potential"s are classified into two types: (B1) one saddle point (unstable circular orbit) and one minimal point (stable circular orbit) on the equatorial plane exist for small spin; and (B2) two saddle points bifurcate from the equatorial plane and one minimal point remains on the equatorial plane for large spin. When $J$ is small, no bound orbits exist and the potentials are classified into another two types: (U1) no extremal point is found for small spin; and (U2) one saddle point appears on the equatorial plane, which is unstable in the direction perpendicular to the equatorial plane, for large spin. The types (B1) and (U1) are the same as those for a spinless particle, but the potentials (B2) and (U2) are new types caused by spin-orbit coupling. The chaotic behavior is found only in the type (B2) potential. The ``heteroclinic orbit'', which could cause chaos, is also observed in type (B2). 
  A three dimensional black hole solution of Einstein equations with negative cosmological constant coupled to a conformal scalar field is given. The solution is static, circularly symmetric, asymptotically anti-de Sitter and nonperturbative in the conformal field. The curvature tensor is singular at the origin while the scalar field is regular everywhere. The condition that the Euclidean geometry be regular at the horizon fixes the temperature to be $T=\frac{9\, r_+}{16\pi l^2}$. Using the Hamiltonian formulation including boundary terms of the Euclidean action, the entropy is found to be $\frac{2}{3}$ of the standard value ($\frac{1}{4} A$), and in agreement with the first law of thermodynamics. 
  The stochastic inflation program is a framework for understanding the dynamics of a quantum scalar field driving an inflationary phase. Though widely used and accepted, there have over recent years been serious criticisms of this theory. In this paper I will present a new theory of stochastic inflation which avoids the problems of the conventional approach. Specifically, the theory can address the quantum-to-classical transition problem, and it will be shown to lead to a dramatic easing of the fine tuning constraints that have plagued inflation theories. 
  The field equations of the new general relativity constructed by Hayashi and Shirafuji (1979), have been applied to two different geometric structures, given by Robertson (1932), in the domain of cosmology. In the first application a family of models, involving two of the parameters characterizing the field equations of the new general relativity, is obtained. In the second application the models obtained are found to involve one parameter only. The cosmological parameters in both applications are calculated and some cosmological problems are discussed in comparison with the corresponding results of other field theories. 
  It is known that the radial equation of the massless fields with spin around Kerr black holes cannot be solved by special functions. Recently, the analytic solution was obtained by use of the expansion in terms of the special functions and various astrophysical application have been discussed. It was pointed out that the coefficients of the expansion by the confluent hypergeometric functions are identical to those of the expansion by the hypergeometric functions. We explain the reason of this fact by using the integral equations of the radial equation. It is shown that the kernel of the equation can be written by the product of confluent hypergeometric functions. The integral equaton transforms the expansion in terms of the confluent hypergeometric functions to that of the hypergeometric functions and vice versa,which explains the reason why the expansion coefficients are universal. 
  The canonical quantization of $N=1$ and $N=2$ supergravity theories is reviewed in this report. A special emphasis is given to the topic of supersymmetric Bianchi class-A and FRW minisuperspaces, namely in the presence of supermatter fields. The quantization of the general theory (including supermatter) is also contemplated. The issue of quantum physical states is subsequently analysed. A discussion on further research problems still waiting to be addressed is included. An extensive and updated bibliography concludes this review. 
  We consider the possibility of discriminating different theories of gravity using a recently proposed gravitational wave detector of spherical shape. We argue that the spin content of different theories can be extracted relating the measurements of the excited spheroidal vibrational eigenmodes to the Newman-Penrose parameters. The sphere toroidal modes cannot be excited by any metric GW and can be thus used as a veto. 
  The spacetime of the metric-affine gauge theory of gravity (MAG) encompasses {\it nonmetricity} and {\it torsion} as post-Riemannian structures. The sources of MAG are the conserved currents of energy-momentum and dilation, shear and spin. We present an exact static spherically symmetric vacuum solution of the theory describing the exterior of a lump of matter carrying mass and dilation, shear and spin charges. 
  A solution of the linearized Einstein's equations for a spherically symmetric perturbation of the ultrarelativistic fluid in the homogeneous and isotropic universe is obtained. Conditions on the boundary of the perturbation are discussed. The examples of particle-like and wave-like solutions are given. 
  Several applications of spectral methods to problems related to the relativistic astrophysics of compact objects are presented. Based on a proper definition of the analytical properties of regular tensorial functions we have developed a spectral method in a general sphericallike coordinate system. The applications include the investigation of spherically symmetric neutron star collapse as well as the solution of the coupled 2D-Einstein-Maxwell equations for magnetized, rapidly rotating neutron stars. In both cases the resulting codes are efficient and give results typically several orders of magnitude more accurate than equivalent codes based on finite difference schemes. We further report the current status of a 3D-code aiming at the simulation of non-axisymmetric neutron star collapse where we have chosen a tensor based numerical scheme. 
  The nucleation and evolution of bubbles are investigated in the model of an $O(3)$-symmetric scalar field coupled to gravity in the high temperature limit. It is shown that, in addition to the well-known bubble of which the inside region is true vacuum, there exists another decay channel at high temperature which is described by a new solution such that a false vacuum region like a global monopole remains at the center of a bubble. The value of the Euclidean action of this bubble is higher than that of the ordinary bubble; however, the production rate of it can be considerable for a certain range of scalar potentials. 
  Contributions of primordial gravitational waves to the large-angular-scale anisotropies of the cosmic microwave background radiation in multidimensional cosmological models (Kaluza-Klein models) are studied. We derive limits on free parameters of the models using results of the COBE experiment and other astrophysical data. It is shown that in principle there is a room for Kaluza-Klein models as possible candidates for the description of the Early Universe. However, the obtained limits are very restrictive. Assuming that the anisotropies are mostly due to gravitational waves, none of the concrete models, analyzed in the article, satisfy them. On the other hand, if the contribution of gravitational waves is very small then a string inspired model is not ruled out. 
  It is shown that the scattering of a charged test particle by a system of four extreme Reissner-Nordtr\"om black holes is chaotic in some cases. The fractal structure of the scattering angle and time delay functions is another manifestation of the existence of a nonattracting chaotic set: fractal basin boundaries were previously known. 
  A world-wide effort is now underway to build gravitational wave detectors based on highly-sensitive laser interferometers. When data from detectors at different sites is properly combined, it will permit highly-sensitive searches for a stochastic background of relic gravitational radiation. These lectures (from the Les Houches School in October 1995) review the current status of this program, and discuss the methods by which data from different detectors can be used to make measurements of, or place limits on, a stochastic background. They also review possible cosmological sources and their potential detectability. 
  It is demonstrated how a convenient choice of the mathematical structure of the quantum cosmology superspace, precisely the definition of a convenient regular state superspace and the restriction of the dynamics to this space, yields directly to an irreversible evolution, in the classical (and semiclassical) phase of the universe, where:   Decoherence and correlations take place and therefore give origin to a classical universe. The second law of thermodynamic is demonstrated. Connection with Reichenbach branch system idea can be implemented. Some rough coincidence with observational data are obtained. The arrows of time can be correlated. Time asymmetry can be explained as a state space asymmetry (e. g. like a spontaneous symmetry breaking All these facts solve the problem of time-asymmetry and show that it is time asymmetry itself that defines the most important features of mathematical structure of superspace. 
  We present an exact stationary {\it axially symmetric} vacuum solution of metric-affine gravity (MAG) which generalises the recently reported spherically symmetric solution. Besides the metric, it carries nonmetricity and torsion as post-Riemannian geometrical structures. The parameters of the solution are interpreted as mass and angular momentum and as dilation, shear and spin charges. 
  The meaning of `tunneling' in a timeless theory such as quantum cosmology is discussed. A recent suggestion of `tunneling' of the macroscopic universe at the classical turning point is analyzed in an anisotropic and inhomogeneous toy model. This `inhomogeneous tunneling' is a local process which cannot be interpreted as a tunneling of the universe. 
  A solution of the Einstein's equations that represents the superposition of a Schwarszchild black hole with both quadrupolar and octopolar terms describing a halo is exhibited. We show that this solution, in the Newtonian limit, is an analog to the well known H\'enon-Heiles potential. The integrability of orbits of test particles moving around a black hole representing the galactic center is studied and bounded zones of chaotic behavior are found. 
  Wormholes -- solutions to the euclidean Einstein equations with non-trivial topology -- are usually assumed to make real contributions to amplitudes in quantum gravity. However, we find a negative mode among fluctuations about the Giddings-Strominger wormhole solution. Hence, the wormhole contribution to the euclidean functional integral is argued to be purely imaginary rather than real, which suggests the interpretation of the wormhole as describing the instability of a large universe against the emission of baby universes. 
  We argue that all Einstein-Maxwell or Einstein-Proca solutions to general relativity may be used to construct a large class of solutions (involving torsion and non-metricity) to theories of non-Riemannian gravitation that have been recently discussed in the literature. 
  We study space-time Killing vectors in terms of their "lapse and shift" relative to some spacelike slice. We give a necessary and sufficient condition in order for these lapse-shift pairs, which we call Killing initial data (KID'S), to form a Lie algebra under the bracket operation induced by the Lie commutator of vector fields on space-time. This result is applied to obtain a theorem on the periodicity of orbits for a class of Killing vector fields in asymptotically flat space-times. 
  The general scheme for the construction of the general-relativistic model of the magnetically driven jet is suggested. The method is based on the usage of the 3+1 MHD formalism. It is shown that the critical points of the flow and the explicit radial behavior of the physical variables may be derived through the jet ``profile function." 
  A recently proposed "gedanken experiment" [G.Z. Machabeli and A.D. Rogava. Phys. Rev. A {\bf 50}, 98 (1994)], exhibiting surprising behavior, is reexamined. A description of this behavior in terms of the laboratory inertial frame is presented, avoiding uncertainties arising due to a definition of a centrifugal force in relativity. The surprising analogy with the radial geodesic motion in Schwarzschild geometry is discovered. The definition of the centrifugal force, suggested by J.C. Miller and M.A. Abramowicz, is discussed. 
  The conventional role of spacetime geometry in the description of gravity is pointed out. Global Poincar$\acute{\mbox{e}}$ symmetry as an inner symmetry of field theories defined on a fixed Minkowski spacetime is discussed. Its extension to local {\bf P\/} gauge symmetry and the corresponding {\bf P\/} gauge fields are introduced. Their minimal coupling to matter is obtained. The scaling behaviour of the partition function of a spinor in {\bf P\/} gauge field backgrounds is computed. The corresponding renormalization constraint is used to determine a minimal gauge field dynamics. 
  The anomalous, energy dependent shift of the center of mass of an idealized, perfectly rigid, uniformly rotating hemispherical shell which is caused by the relativistic mass increase effect is investigated in detail. It is shown that a classical object on impact which has the harmonic binding force between the adjacent constituent particles has the similar effect of the energy dependent, anomalous shift of the center of mass. From these observations, the general mode of the linear acceleration is suggested to be caused by the anomalous center of mass shift whether it's due to classical or relativistic origin. The effect of the energy dependent center of mass shift perpendicular to the plane of rotation of a rotating hemisphere appears as the non zero gravitational dipole moment in general relativity. Controlled experiment for the measurement of the gravitational dipole field and its possible links to the cylindrical type line formation of a worm hole in the extreme case are suggested. The jets from the black hole accretion disc and the observed anomalous red shift from far away galaxies are considered to be the consequences of the two different aspects of the dipole gravity. 
  The purpose of this paper is to investigate a specific FRW model derived from the theory of N=1 supergravity with gauged supermatter. The supermatter content is restricted to a vector supermultiplet. This objective is particularly worthwhile. In fact, it was pointed in ref. ({\em Class. Quantum Grav. {\bf 12} {\rm (} {\rm 1995} {\rm )} {\rm 1343}}) that $\Psi = 0$ was the only allowed quantum state for N=1 supergravity with {\em generic} gauged supermatter subject to suitable FRW ans\"atze. The ans\"atze employed here for the physical variables was presented in the above reference. The corresponding Lorentz and supersymmetry quantum constraints are then derived. Non-trivial solutions are subsquently found. A no-boundary solution is identified while another state may be interpreted as a wormhole solution. In addition, the usefulness and limitations of the ans\"atze are addressed. The implications of the ans\"atze with respect to the allowed quantum states are also discussed. 
  Further to results in [9], pointing out the role of initial density and velocity distributions towards determining the final outcome of spherical dust collapse, the causal structure of singularity is examined here in terms of evolution of the apparent horizon. We also bring out several related features which throw some useful light towards understanding the nature of this singularity, including the behaviour of geodesic families coming out and some aspects related to the stability of singularity. 
  Examples in which spacetime might become non-Riemannian appear above Planck energies in string theory or, in the very early universe, in the inflationary model. The simplest such geometry is metric-affine geometry, in which {\it nonmetricity} appears as a field strength, side by side with curvature and torsion. In matter, the shear and dilation currents couple to nonmetricity, and they are its sources. After reviewing the equations of motion and the Noether identities, we study two recent vacuum solutions of the metric-affine gauge theory of gravity. We then use the values of the nonmetricity in these solutions to study the motion of the appropriate test-matter. As a Regge-trajectory like hadronic excitation band, the test matter is endowed with shear degrees of freedom and described by a world spinor. 
  Recent discoveries in differential topology are reviewed in light of their possible implications for spacetime models and related subjects in theoretical physics. Although not often noted, a particular smoothness (differentiability) structure must be imposed on a topological manifold before geometric or other structures of physical interest can be discussed. The recent discoveries of interest here are of various surprising ``exotic'' smoothness structures on topologically trivial manifolds such as ${S^7}$ and ${\bf R^4}$. Since no two of these are diffeomorphic to each other, each such manifold represents a physically distinct model of topologically trivial spacetime. That is, these are not merely different coordinate representations of a given spacetime. The path to such structures intertwines many branches of mathematics and theoretical physics (Yang-Mills and other gauge theories). An overview of these topics is provided, followed by certain results concerning the geometry and physics of such manifolds. Although exotic ${\bf R^4}$'s cannot be effectively exhibited by finite constructions, certain existence and non-existence results can be stated. For example, it is shown that the ``exoticness'' can be confined to a time-like world tube, providing a possible model for an exotic source. Other suggestions and conjectures for future research are made. 
  We overview our recent studies of cosmological models with expansion and global rotation. Problems of the early rotating models are discussed, and the class of new viable cosmologies is described in detail. Particular attention is paid to the observational effects of the cosmic rotation. 
  We assume that the Universe has a non trivial topology whose compact spatial sections have a volume significantly smaller than the horizon volume. By a topological lens effect, such a "folded" space configuration generates multiple images of cosmic sources, e.g. clusters of galaxies. We present a simple and powerful method to unveil non ambiguous observational effects, independently of the sign of the curvature and of the topological type. 
  I review some recent work in which the quantum states of string theory which are associated with certain black holes have been identified and counted. For large black holes, the number of states turns out to be precisely the exponential of the Bekenstein-Hawking entropy. This provides a statistical origin for black hole thermodynamics in the context of a potential quantum theory of gravity. 
  A model of a fluctuating lightcone due to a bath of gravitons is further investigated. The flight times of photons between a source and a detector may be either longer or shorter than the light propagation time in the background classical spacetime, and will form a Gaussian distribution centered around the classical flight time. However, a pair of photons emitted in rapid succession will tend to have correlated flight times. We derive and discuss a correlation function which describes this effect. This enables us to understand more fully the operational significance of a fluctuating lightcone. Our results may be combined with observational data on pulsar timing to place some constraints on the quantum state of cosmological gravitons. 
  The Hilbert-Palatini (HP) Lagrangian of general relativity being written in terms of selfdual and antiselfdual variables contains Ashtekar Lagrangian (which governs the dynamics of the selfdual sector of the theory on condition that the dynamics of antiselfdual sector is not fixed). We show that nonequivalence of the Ashtekar and HP quantum theories is due to the specific form (of the "loose relation" type) of constraints which relate self- and antiselfdual variables so that the procedure of (canonical) quantisation of such the theory is noncommutative with the procedure of excluding antiselfdual variables. 
  After a brief summary of the Newton-Cartan theory in a form which emphasizes its close analogy to general relativity, we illustrate the theory with selective applications in cosmology. The geometrical formulation of this nonrelativistic theory of gravity, pioneered by Cartan and further developed by various workers, leads to a conceptually sound basis of Newtonian cosmology. In our discussion of homogeneous models and cosmological perturbation theory, we stress the close relationship with their general relativistic treatments. Spatially compact flat models also fit into this framework. 
  We analyze the relationship between quasilocal masses calculated for solutions of conformally related theories. We show that the ADM mass of a static, spherically symmetric solution is conformally invariant (up to a constant factor) only if the background action functional is conformally invariant. Thus, the requirement of conformal invariance places restrictions on the choice of reference spacetimes. We calculate the mass of the black hole solutions obtained by Garfinkle, Horowitz, and Strominger (GHS) for both the string and the Einstein metrics. In addition, the quasilocal thermodynamic quantities in the string metrics are computed and discussed. 
  A complete description of dynamics of compact locally homogeneous universes is given, which, in particular, includes explicit calculations of Teichm\"uller deformations and careful counting of dynamical degrees of freedom. We regard each of the universes as a simply connected four dimensional spacetime with identifications by the action of a discrete subgroup of the isometry group. We then reduce the identifications defined by the spacetime isometries to ones in a homogeneous section, and find a condition that such spatial identifications must satisfy. This is essential for explicit construction of compact homogenoeus universes. Some examples are demonstrated for Bianchi II, VI${}_0$, VII${}_0$, and I universal covers. 
  Cosmic strings, as topological spacetime defects, show striking resemblance to defects in solid continua: distortions, which can be classified into disclinations and dislocations, are line-like defects characterized by a delta function-valued curvature and torsion distribution giving rise to rotational and translational holonomy. We exploit this analogy and investigate how distortions can be adapted in a systematic manner from solid state systems to Einstein-Cartan gravity. As distortions are efficiently described within the framework of a $SO(3) {\rlap{$\supset$}\times}} T(3)$ gauge theory of solid continua with line defects, we are led in a straightforward way to a Poincar\'e gauge approach to gravity which is a natural framework for introducing the notion of distorted spacetimes. Constructing all ten possible distorted spacetimes, we recover, inter alia, the well-known exterior spacetime of a spin-polarized cosmic string as a special case of such a geometry. In a second step, we search for matter distributions which, in Einstein-Cartan gravity, act as sources of distorted spacetimes. The resulting solutions, appropriately matched to the distorted vacua, are cylindrically symmetric and are interpreted as spin-polarized cosmic strings and cosmic dislocations. 
  The wavefunctional in quantum gravity gives an amplitude for 3-geometries and matter fields. The four-space is usually recovered in a semiclassical approximation where the gravity variables are taken to oscillate rapidly compared to matter variables; this recovers the Schrodinger evolution for the matter. We examine turning points in the gravity variables where this approximation appears to be troublesome. We investigate the effect of such a turning point on the matter wavefunction, in simple quantum mechanical models and in a closed minisuperspace cosmology. We find that after evolving sufficiently far from the turning point the matter wavefunction recovers to a form close to that predicted by the semiclassical approximation, and we compute the leading correction (from `backreaction') in a simple model. We also show how turning points can appear in the gravitational sector in dilaton gravity. We give some remarks on the behavior of the wavefunctional in the vicinity of turning points in the context of dilaton gravity black holes. 
  The kinetic energy of a local system of objects placed in a curved spacetime is gained by the subsequent acceleration of the object following the more contracted region of spacetime. Normally this happens near massive gravitating stars. However, the gravitational dipole moment has been shown to be capable of self creating asymmetrically distorted spacetime in its vicinity, therby, capable of being accelerated indefinitely following the successive self created loophole of the spacetime. Localization of this kinetic energy may be possible by designing a system that uses the artificially created gavitational dipole moments to rotate the main axis. A mechanical constraint is derived for the extraction of unlimited gravitational energy from such system. 
  We prove the theorem: The necessary and sufficient condition for a spherically symmetric spacetime to represent an isothermal perfect fluid (barotropic equation of state with density falling off as inverse square of the curvature radius) distribution without boundary is that it is conformal to the ``minimally'' curved (gravitation only manifesting in tidal acceleration and being absent in particle trajectory) spacetime. 
  Instead of conformal to flat spacetime, we take the metric conformal to a spacetime which can be thought of as ``minimally'' curved in the sense that free particles experience no gravitational force yet it has non-zero curvature. The base spacetime can be written in the Kerr-Schild form in spherical polar coordinates. The conformal metric then admits the unique three parameter family of perfect fluid solution which is static and inhomogeneous. The density and pressure fall off in the curvature radial coordinates as $R^{-2}, $ for unbounded cosmological model with a barotropic equation of state. This is the characteristic of isothermal fluid. We thus have an ansatz for isothermal perfect fluid model. The solution can also represent bounded fluid spheres. 
  We develop the formalism for canonical reduction of $(1+1)$--dimensional gravity coupled with a set of point particles by eliminating constraints and imposing coordinate conditions. The formalism itself is quite analogous to the $(3+1)$--dimensional case; however in $(1+1)$ dimensions an auxiliary scalar field is shown to have an important role. The reduced Hamiltonian is expressed as a form of spatial integral of the second derivative of the scalar field. Since in $(1+1)$ dimensions there exists no dynamical degree of freedom of the gravitational field ({\it i.e.} the transverse-traceless part of the metric tensor is zero), the reduced Hamiltonian is completely determined in terms of the particles' canonical variables (coordinates and momenta). The explicit form of the Hamiltonian is calculated both in post-linear and post-Newtonian approximations. 
  We find a one-parameter family of variables which recast the 3+1 Einstein equations into first-order symmetric-hyperbolic form for any fixed choice of gauge. Hyperbolicity considerations lead us to a redefinition of the lapse in terms of an arbitrary factor times a power of the determinant of the 3-metric; under certain assumptions, the exponent can be chosen arbitrarily, but positive, with no implication of gauge-fixing. 
  Held has proposed a coordinate- and gauge-free integration procedure within the GHP formalism built around four functionally independent zero-weighted scalars constructed from the spin coefficients and the Riemann tensor components. Unfortunately, a spacetime with Killing vectors will be unable to supply the full quota of four scalars of this type. However, for such a spacetime additional scalars are supplied by the components of the Killing vectors; by using these alongside the spin coefficients and the Riemann tensor components we have the possibility of constructing the full quota of four functionally independent zero-weighted scalars, and of exploiting Held's procedure.   As an illustration we investigate the vacuum Type N spaces admitting a Killing vector and a homothetic Killing vector. In a direct manner, we reduce the problem to a pair of ordinary differential operator `master equations', making use of a new zero-weighted GHP operator. By first rewriting the master equations as a closed set of complex first order equations, we reduce the problem to one real third order operator differential equation for a complex function of a real variable --- but with still the freedom to choose explicitly our fourth coordinate. An alternative, more algorithmic approach, using a closed chain of real first order equations for real functions, reduces the problem to the same order, but in a more natural and much more concise form. It is also outlined how the various other third order differential equations, which have been derived previously by other workers on this problem, can be deduced from our master equations. 
  Following a previous work on the quantization of a massless scalar field in a spacetime representing the head on collision of two plane waves which fucus into a Killing-Cauchy horizon, we compute the renormalized expectation value of the stress-energy tensor of the quantum field near that horizon in the physical state which corresponds to the Minkowski vacuum before the collision of the waves. It is found that for minimally coupled and conformally coupled scalar fields the respective stress-energy tensors are unbounded in the horizon. The specific form of the divergences suggests that when the semiclassical Einstein equations describing the backreaction of the quantum fields on the spacetime geometry are taken into account, the horizon will acquire a curvature singularity. Thus the Killing-Cauchy horizon which is known to be unstable under ``generic" classical perturbations is also unstable by vacuum polarization. The calculation is done following the point splitting regularization technique. The dynamical colliding wave spacetime has four quite distinct spacetime regions, namely, one flat region, two single plane wave regions, and one interaction region. Exact mode solutions of the quantum field equation cannot be found exactly, but the blueshift suffered by the initial modes in the plane wave and interaction regions makes the use of the WKB expansion a suitable method of solution. To ensure the correct regularization of the stress-energy tensor, the initial flat modes propagated into the interaction region must be given to a rather high adiabatic order of approximation. 
  We use the technique of conformal transformations to generate self-similar collapse in Brans-Dicke theory. We analyze the solutions concerning the critical behavior found recently by Choptuik. The critical exponent associated to the formation of black hole for near critical evolution is obtained. The role of the coupling parameter is discussed. 
  We show that the emission of a Minkowski particle by a general class of scalar sources as described by inertial observers corresponds to either the emission or the absorption of a Rindler particle as described by uniformly accelerated observers. Our results are discussed in connection with the current controversy whether uniformly accelerated detectors radiate. 
  General relativity does not allow one to specify the topology of space, leaving the possibility that space is multi-- rather than simply--connected. We review the main mathematical properties of multi--connected spaces, and the different tools to classify them and to analyse their properties. Following the mathematical classification, we describe the different possible muticonnected spaces which may be used to construct universe models. We briefly discuss some implications of multi--connectedness for quantum cosmology, and its consequences concerning quantum field theory in the early universe. We consider in details the properties of the cosmological models where space is multi--connected, with emphasis towards observable effects. We then review the analyses of observational results obtained in this context, to search for a possible signature of multi--connectedness, or to constrain the models. They may concern the distribution of images of cosmic objects like galaxies, clusters, quasars,..., or more global effects, mainly those concerning the Cosmic Microwave Background, and the present limits resulting from them. 
  We consider in a pedagogical fashion alterations to Newtonian gravity due to the postulate that all energy corresponds to active gravitational mass when applied to the self-energy of the gravitational field. We show why a simple addition of ${1\over c^2}$ times the gravitational field energy to the matter density in Newton's field equation is inconsistent. A consistent prescription is shown and discussed. The connection to general relativity is pointed out. 
  We study five dimensional cosmological models with four dimensional hypersufaces of the Bianchi type I and V. In this way the five dimensional vacuum field equations $\rm G_{AB} = 0$, led us to four dimensional matter equations $\rm G_{\mu\nu}=T_{\mu\nu}$ and the matter is interpreted as a purely geometrical property of a fifth dimension. Also, we find that the energy-momentum tensor induced from the fifth dimension has the structure of an imperfect fluid that has dissipative terms. 
  Post-relativistic gravity is a hidden variable theory for general relativity. It introduces the pre-relativistic notions absolute space, absolute time, and ether as hidden variables into general relativity. Evolution is defined by the equations of general relativity and the harmonic coordinate condition interpreted as a physical equation. There are minor differences in predictions compared with general relativity (i.e. trivial topology of the universe is predicted).   The unobservable absolute time is designed to solve the problem of time in quantization of general relativity. Background space and time define a Newtonian frame for the quantization of the gravitational field. By the way, a lot of other conceptual problems of quantization will be solved (i.e. no constraints, no topological foam, no black hole and bib bang singularities, natural vacuum definition for quantum fields on classical background). 
  The evolution equations of Einstein's theory and of Maxwell's theory---the latter used as a simple model to illustrate the former--- are written in gauge covariant first order symmetric hyperbolic form with only physically natural characteristic directions and speeds for the dynamical variables. Quantities representing gauge degrees of freedom [the spatial shift vector $\beta^{i}(t,x^{j})$ and the spatial scalar potential $\phi(t,x^{j})$, respectively] are not among the dynamical variables: the gauge and the physical quantities in the evolution equations are effectively decoupled. For example, the gauge quantities could be obtained as functions of $(t,x^{j})$ from subsidiary equations that are not part of the evolution equations. Propagation of certain (``radiative'') dynamical variables along the physical light cone is gauge invariant while the remaining dynamical variables are dragged along the axes orthogonal to the spacelike time slices by the propagating variables. We obtain these results by $(1)$ taking a further time derivative of the equation of motion of the canonical momentum, and $(2)$ adding a covariant spatial derivative of the momentum constraints of general relativity (Lagrange multiplier $\beta^{i}$) or of the Gauss's law constraint of electromagnetism (Lagrange multiplier $\phi$). General relativity also requires a harmonic time slicing condition or a specific generalization of it that brings in the Hamiltonian constraint when we pass to first order symmetric form. The dynamically propagating gravity fields straightforwardly determine the ``electric'' or ``tidal'' parts of the Riemann tensor. 
  We study the classical dynamics of a bosonic string in the $D$--dimensional flat Friedmann--Robertson--Walker and Schwarzschild backgrounds. We make a perturbative development in the string coordinates around a {\it null} string configuration; the background geometry is taken into account exactly. In the cosmological case we uncouple and solve the first order fluctuations; the string time evolution with the conformal gauge world-sheet $\tau$--coordinate is given by $X^0(\sigma, \tau)=q(\sigma)\tau^{1\over1+2\beta}+c^2B^0(\sigma, \tau)+\cdots$, $B^0(\sigma,\tau)=\sum_k b_k(\sigma)\tau^k$ where $b_k(\sigma)$ are given by Eqs.\ (3.15), and $\beta$ is the exponent of the conformal factor in the Friedmann--Robertson--Walker metric, i.e. $R\sim\eta^\beta$. The string proper size, at first order in the fluctuations, grows like the conformal factor $R(\eta)$ and the string energy--momentum tensor corresponds to that of a null fluid. For a string in the black hole background, we study the planar case, but keep the dimensionality of the spacetime $D$ generic. In the null string expansion, the radial, azimuthal, and time coordinates $(r,\phi,t)$ are $r=\sum_n A^1_{n}(\sigma)(-\tau)^{2n/(D+1)}~,$ $\phi=\sum_n A^3_{n}(\sigma)(-\tau)^{(D-5+2n)/(D+1)}~,$ and $t=\sum_n A^0_{n} (\sigma)(-\tau)^{1+2n(D-3)/(D+1)}~.$ The first terms of the series represent a {\it generic} approach to the Schwarzschild singularity at $r=0$. First and higher order string perturbations contribute with higher powers of $\tau$. The integrated string energy-momentum tensor corresponds to that of a null fluid in $D-1$ dimensions. As the string approaches the $r=0$ singularity its proper size grows indefinitely like $\sim(-\tau)^{-(D-3)/(D+1)}$. We end the paper giving three particular exact string solutions inside the black hole. 
  We impose in the nonsymmetric gravitational theory, by means of Lagrange multiplier fields in the action, a set of covariant constraints on the antisymmetric tensor field. The canonical Hamiltonian constraints in the weak field approximation for the antisymmetric sector yield a Hamiltonian energy bounded from below. An analysis of the Cauchy evolution, in terms of an expansion of the antisymmetric sector about a symmetric Einstein background, shows that arbitrarily small antisymmetric Cauchy data can lead to smooth evolution. 
  The pre--big bang cosmological scenario is studied within the context of the Brans--Dicke theory of gravity. An epoch of superinflationary expansion may occur in the pre--big bang phase of the Universe's history in a certain region of parameter space. Two models are considered that contain a cosmological constant in the gravitational and matter sectors of the theory, respectively. Classical pre-- and post--big bang solutions are found for both models. The existence of a curvature singularity forbids a classical transition between the two branches. On the other hand, a quantum cosmological approach based on the tunneling boundary condition results in a non--zero transition probability. The transition may be interpreted as a spatial reflection of the wavefunction in minisuperspace. 
  At present we have only the very successful but phenomenological Einstein geometrical modelling of the spacetime phenomenon. This geometrical model provides a `container' for other theories, in particular the quantum field theories. Here we report progress in developing a {\em Heraclitean Quantum System}. This is a particular pregeometric theory for space and time in which no classical or geometric structures are assumed, but rather the emergence of such phenomena is sought. 
  We formulate a ''minimal'' interpretational scheme for fairly general (minisuperspace) quantum cosmological models. Admitting as few exact mathematical structure as is reasonably possible at the fundamental level, we apply approximate WKB-techniques locally in minisuperspace in order to make contact with the realm of predictions, and propose how to deal with the problems of mode decomposition and almost-classicality without introducing further principles. In order to emphasize the general nature of approximate local quantum mechanical structures, we modify the standard WKB-expansion method so as to rely on exact congruences of classical paths, rather than a division of variables into classical and quantum. The only exact mathematical structures our interpretation needs are the space of solutions of the Wheeler-DeWitt equation and the Klein-Gordon type indefinite scalar product. The latter boils down to plus or minus the ordinary quantum mechanical scalar product in the local quantum structures. According to our approach all further structures, in particular the concepts encountered in conventional physics, such as observables, time and unitarity, are approximate. Our interpretation coincides to some extent with the standard WKB-oriented view, but the way in which the conventional concepts emerge, and the accuracy at which they are defined at all, are more transparent. A simpler book-keeping of normalization issues is automatically achieved. Applying our scheme to the Hawking model, we find hints that the no-boundary wave function predicts a cosmic catastrophe with some non-zero probability. 
  Recently the low-energy effective string theory has been used by Gasperini and Veneziano to elaborate a very interesting scenario for the early history of the universe (``birth of the universe as quantum scattering''). Here we investigate the gauge fixing and the problem of the definition of a global time parameter for this model, and we obtain the positive norm Hilbert space of states. 
  The Hawking minisuperspace model (closed FRW geometry with a homogeneous massive scalar field) provides a fairly non-trivial testing ground for fundamental problems in quantum cosmology. We provide evidence that the Wheeler-DeWitt equation admits a basis of solutions that is distinguished by analyticity properities in a large scale factor expansion. As a consequence, the space of solutions decomposes in a preferred way into two Hilbert spaces with positive and negative definite scalar product, respectively. These results may be viewed as a hint for a deeper significance of analyticity. If a similar structure exists in full (non-minisuperspace) models as well, severe implications on the foundations of quantum cosmology are to be expected. Semiclassically, the elements of the preferred basis describe contracting and expanding universes with a prescribed value of the matter (scalar field) energy. Half of the basis elements have previously been constructed by Hawking and Page in a wormhole context, and they appear in a new light here. The technical tools to arrive at these conclusions are transformations of the wave function into several alternative representations that are based on the harmonic oscillator form of the matter energy operator, and that are called oscillator, energy and Fock representation. The framework defined by these may be of some help in analyzing the Wheeler-DeWitt equation for other purposes as well. 
  It is shown that in a class of maximal globally hyperbolic spacetimes admitting two local Killing vectors, the past (defined with respect to an appropriate time orientation) of any compact constant mean curvature hypersurface can be covered by a foliation of compact constant mean curvature hypersurfaces. Moreover, the mean curvature of the leaves of this foliation takes on arbitrarily negative values and so the initial singularity in these spacetimes is a crushing singularity. The simplest examples occur when the spatial topology is that of a torus, with the standard global Killing vectors, but more exotic topologies are also covered. In the course of the proof it is shown that in this class of spacetimes a kind of positive mass theorem holds. The symmetry singles out a compact surface passing through any given point of spacetime and the Hawking mass of any such surface is non-negative. If the Hawking mass of any one of these surfaces is zero then the entire spacetime is flat. 
  The higher dimensional spherical symmetric scalar field collapse problem is studied in the light of the critical behavior in black hole formation. To make the analysis tractable, the self similarity is also imposed. By giving a new view to the self-similar scalar field collapse problem, we give the general formula for the critical exponents in higher dimensions. In the process, the explanation of the universality of the critical phenomena is given within the self-similar context. 
  A broad class of generalized Einstein's gravity can be cast into Einstein's gravity with a minimally coupled scalar field using suitable conformal rescaling of the metric. Using this conformal equivalence between the theories, we derive the equations for the background and the perturbations, and the general asymptotic solutions for the perturbations in the generalized Einstein's gravity from the simple results known in the minimally coupled scalar field. Results for the scalar and tensor perturbations can be presented in unified forms. The large scale evolutions for both modes are characterized by corresponding conserved quantities. We also present the normalization condition for canonical quantization. 
  We discuss to what extent classical singularities persist upon quantization in two simple cosmological models. 
  For Einstein's General Relativity (GR) or the alternatives suggested up to date$ the vacuum energy gravitates. We present a model where a new measure is introduced for integration of the total action in the D-dimensional space-time. This measure is built from D scalar fields $\varphi_{a}$. As a consequence of such a choice of the measure, the matter lagrangian $L_{m}$ can be changed by adding a constant while no gravitational effects, like a cosmological term, are induced. Such Non-Gravitating Vacuum Energy (NGVE) theory has an infinite dimensional symmetry group which contains volume-preserving diffeomorphisms in the internal space of scalar fields $\varphi_{a}$. Other symmetries contained in this symmetry group, suggest a deep connection of this theory with theories of extended objects. In general {\em the theory is different from GR} although for certain choices of $L_{m}$, which are related to the existence of an additional symmetry, solutions of GR are solutions of the model. This is achieved in four dimensions if $L_{m}$ is due to fundamental bosonic and fermionic strings. Other types of matter where this feature of the theory is realized, are for example: scalars without potential or subjected to nonlinear constraints, massless fermions and point particles. The point particle plays a special role, since it is a good phenomenological description of matter at large distances. de Sitter space is realized in an unconventional way, where the de Sitter metric holds, but such de Sitter space is supported by the existence of a variable scalar field which in practice destroys the maximal symmetry. The only space - time where maximal symmetry is not broken, in a dynamical sense, is Minkowski space. The theory has non trivial dynamics in 1+1 dimensions, unlike GR. 
  In this article we summarise the proceedings of the Workshop on Gravitational Waves held during ICGC-95. In the first part we present the discussions on 3PN calculations (L. Blanchet, P. Jaranowski), black hole perturbation theory (M. Sasaki, J. Pullin), numerical relativity (E. Seidel), data analysis (B.S. Sathyaprakash), detection of gravitational waves from pulsars (S. Dhurandhar), and the limit on rotation of relativistic stars (J. Friedman). In the second part we briefly discuss the contributed papers which were mainly on detectors and detection techniques of gravitational waves. 
  We present a simpler and more powerful version of the recently-discovered action principle for the motion of a spinless point particle in spacetimes with curvature and torsion. The surprising feature of the new principle is that an action involving only the metric can produce an equation of motion with a torsion force, thus changing geodesics to autoparallels. This additional torsion force arises from a noncommutativity of variations with parameter derivatives of the paths due to the closure failure of parallelograms in the presence of torsion 
  For the past decade there has been a considerable debate about the existence of chaos in the mixmaster cosmological model. The debate has been hampered by the coordinate, or observer dependence of standard chaotic indicators such as Lyapanov exponents. Here we use coordinate independent, fractal methods to show the mixmaster universe is indeed chaotic. 
  We show that in complete analogy with the usual bremsstrahlung process, when studied from the coaccelerated observer's point of view, a charge moving along the integral curves of the static Killing field in the exterior of a static black hole gives rise to the emission of zero-energy photons, induced by the thermal bath of Hawking Radiation. 
  The gravitational ionization of a Keplerian binary system via normally incident periodic gravitational radiation of definite helicity is discussed. The periodic orbits of the planar tidal equation are investigated on the basis of degenerate continuation theory. The relevance of the Kolmogorov-Arnold-Moser theory to the question of gravitational ionization is elucidated, and it is conjectured that the process of ionization is closely related to the Arnold diffusion of the perturbed system. 
  A comparison between the proposals made to measure Hawking-like effects and the Unruh effect in the laboratory is given at the level of their estimates. No satisfactory scheme exists as yet for their detection. 
  We investigate the occurrence and nature of naked singularities in the Szekeres space-times. These space-times represent irrotational dust. They do not have any Killing vectors and they are generalisations of the Tolman-Bondi-Lemaitre space-times. It is shown that in these space-times there exist naked singularities that satisfy both the limiting focusing condition and the strong limiting focusing condition. The implications of this result for the cosmic censorship hypothesis are discussed. 
  The question if conserved currents can be sensibly defined in supersymmetric minisuperspaces is investigated in this essay. The objective is to employ exclusively the differential equations obtained {\em directly} from the Lorentz and supersymmetry quantum constraints. The ``square-root'' structure of N=1 supergravity is the motivation to contemplate this tempting idea. However, it is shown that such prospect is not feasible but for some very simple scenarios. Otherwise, conserved currents (and consistent probability densities) can be derived from subsequent Wheeler-DeWitt like equations obtained from the supersymmetric algebra of constraints. 
  Quantum states of the diagonal Bianchi type IX model with negative cosmological constant $\Lambda$ are obtained by transforming the Chern-Simons solution in Ashtekar's variables to the metric representation. We apply our method developed earlier for $\Lambda>0$ and obtain five linearly independent solutions by using the complete set of topologically inequivalent integration contours in the required generalized Fourier-transformation. A caustic in minisuperspace separates two Euclidean regimes at small and large values of the scale parameter from a single classically interpretable Lorentzian regime in between, corresponding to the fact that classically these model-Universes recollapse. Just one particular solution out of the five we find gives a normalizable probability distribution on both branches of the caustic. However, in contrast to the case of positive cosmological constant, this particular solution neither satisfies the semi-classical no-boundary condition, nor does the special initial condition it picks out for $\hbar \to 0$ evolve into a classically interpretable Universe. 
  We demonstrate that the left (and right) invariant Maurer-Cartan forms for any semi-simple Lie group enable one to construct solutions of the Yang-Mills equations on the group manifold equipped with the natural Cartan-Killing metric. For the unitary unimodular groups the Yang-Mills action integral is finite for such solutions. This is explicitly exhibited for the case of $SU(3)$. 
  We examine some of the subtleties inherent in formulating a theory of spinors on a manifold with a smooth degenerate metric. We concentrate on the case where the metric is singular on a hypersurface that partitions the manifold into Lorentzian and Euclidean domains. We introduce the notion of a complex spinor fibration to make precise the meaning of continuity of a spinor field and give an expression for the components of a local spinor connection that is valid in the absence of a frame of local orthonormal vectors. These considerations enable one to construct a Dirac equation for the discussion of the behavior of spinors in the vicinity of the metric degeneracy. We conclude that the theory contains more freedom than the spacetime Dirac theory and we discuss some of the implications of this for the continuity of conserved currents. 
  The known canonical quantum theory of a spherically symmetric pure (Schwarzschild) gravitational system describes isolated black holes by plane waves exp(-iMc^2\tau/\hbar) with respect to their continuous masses M and the proper time \tau of obsevers at spatial infinity. On the other hand Bekenstein and Mukhanov postulated discrete mass levels for such black holes in the spirit of the Bohr-Sommerfeld quantisation in atomic physics. The two approaches can be related by postulating periodic boundary conditions in time for the plane waves and by identifying the period \Delta in real time with the period \Delta_H= 8\pi GM/c^3 in Euclidean time. This yields the mass spectrum M_n=(1/2)\sqrt{n}m_P, n=1,2,... . 
  We consider possible tests of the Einstein Equivalence Principle for physical systems in which quantum-mechanical vacuum energies cannot be neglected. Specific tests include a search for the manifestation of non-metric effects in Lamb-shift transitions of Hydrogenic atoms and in anomalous magnetic moments of massive leptons. We discuss how current experiments already set bounds on the violation of the equivalence principle in this sector and how new (high-precision) measurements of these quantities could provide further information to this end. 
  We obtain a two-parameter set of solutions, which represents a spherically symmetric space-time with a superposition of a neutral fluid and an electric field. The electromagnetic four-potential of this Einstein-Maxwell space-time is taken in the form A=(q/n)(r^n)dt, when n=/0 and A=q*ln(r)dt, when n=0 (where q and n are arbitrary constants) 
  Recently many people have discussed the possibility that the universe is hyperbolic and was in an inflationary phase in the early stage. Under these assumptions, it is shown that the universe cannot have compact hyperbolic time-slices. Though the universal covering space of the universe has a past Cauchy horizon and can be extended analytically beyond it, the extended region has densely many points which correspond to singularities of the compact universe. The result is essentially attributed to the ergodicity of the geodesic flow on a compact negatively curved manifold. Validity of the result is also discussed in the case of inhomogeneous universe. Relationship with the strong cosmic censorship conjecture is also discussed. 
  Inhomogeneous and anisotropic cosmologies are modeled withing the framework of scalar-tensor gravity theories. The inhomogeneities are calculated to third-order in the so-called long-wavelength iteration scheme. We write the solutions for general scalar coupling and discuss what happens to the third-order terms when the scalar-tensor solution approaches at first-order the general relativistic one. We work out in some detail the case of Brans-Dicke coupling and determine the conditions for which the anisotropy and inhomogeneity decay as time increases. The matter is taken to be that of perfect fluid with a barotropic equation of state. 
  We argue that the reasonings that underlie a recent comment by Gotay and Demaret [1, gr-qc/9605025] are defective. We maintain that, contrary to what they assert, our previous papers [2,3,4] are correct and indeed disprove their conjecture that quantum cosmological singularities are predetermined on the classical level by the choice of time. 
  A generalized dilaton action is considered of which the standard dilaton black hole and spherically reduced gravity are particular cases. The Arnowitt-Deser-Misner (ADM) and the Bondi-Sachs (BS) mass are calculated. Special attention is paid to both the asymptotic conditions for the metric as well as for the reference space-time. For the latter one we suggest a modified expression thereby obtaining a new definition of energy. Depending on the parameters of the model the Hawking radiation behaves like a positive or negative power of the mass. 
  Einstein-Vlasov system is solved for a homogeneous isotropic spacetime with positive curvature. In the case of the Universe consisting of massless particles the equation for R(t) is solved analytically. 
  We exhibit the transformation properties of the mechanical action principle under anholonomic transformations. Using the fact that spaces with torsion can be produced by anholonomic transformations we derive the correct action principle in these spaces which is quite different from the conventional action principle. 
  Quantum theory of geometry, developed recently in the framework of non-perturbative quantum gravity, is used in an attempt to explain thermodynamics of Schwarzschild black holes on the basis of a microscopical (quantum) description of the system. We work with the formulation of thermodynamics in which the black hole is enclosed by a spherical surface B and a macroscopic state of the system is specified by two parameters: the area of the boundary surface and a quasilocal energy contained within. To derive thermodynamical properties of the system from its microscopics we use the standard statistical mechanical method of Gibbs. Under a certain number of assumptions on the quantum behavior of the system, we find that its microscopic (quantum) states are described by states of quantum Chern-Simons theory defined by sets of points on B labelled with spins. The level of the Chern-Simons theory turns out to be proportional to the horizon area of black hole measured in Planck units. The statistical mechanical analysis turns out to be especially simple in the case when the entire interior of B is occupied by a black hole. We find in this case that the entropy contained within B, that is, the black hole entropy, is proportional to the horizon surface area. 
  We report an upper bound on the strain amplitude of gravitational wave bursts in a waveband from around 800Hz to 1.25kHz. In an effective coincident observing period of 62 hours, the prototype laser interferometric gravitational wave detectors of the University of Glasgow and Max Planck Institute for Quantum Optics, have set a limit of 4.9E-16, averaging over wave polarizations and incident directions. This is roughly a factor of 2 worse than the theoretical best limit that the detectors could have set, the excess being due to unmodelled non-Gaussian noise. The experiment has demonstrated the viability of the kind of observations planned for the large-scale interferometers that should be on-line in a few years time. 
  We obtain the Einstein-Maxwell equations for (2+1)-dimensional static space-time, which are invariant under the transformation $q_0=i\,q_2,q_2=i\,q_0,\alpha \rightleftharpoons \gamma$. It is shown that the magnetic solution obtained with the help of the procedure used in Ref.~\cite{Cataldo}, can be obtained from the static BTZ solution using an appropriate transformation. Superpositions of a perfect fluid and an electric or a magnetic field are separately studied and their corresponding solutions found. 
  It is well known that the inequivalent unitary irreducible representations (UIR's) of the mapping class group $G$ of a 3-manifold give rise to ``theta sectors'' in theories of quantum gravity with fixed spatial topology. In this paper, we study several families of UIR's of $G$ and attempt to understand the physical implications of the resulting quantum sectors. The mapping class group of a three-manifold which is the connected sum of $\R^3$ with a finite number of identical irreducible primes is a semi-direct product group. Following Mackey's theory of induced representations, we provide an analysis of the structure of the general finite dimensional UIR of such a group. In the picture of quantized primes as particles (topological geons), this general group-theoretic analysis enables one to draw several interesting qualitative conclusions about the geons' behavior in different quantum sectors, without requiring an explicit knowledge of the UIR's corresponding to the individual primes. 
  We give conditions to obtain cosmological asymptotic freedom in scalar-tensor theories of gravity. We show that this feature can be achieved in FRW flat spacetimes since we obtain singularity free solutions where the effective gravitational constant $G_{eff}\rightarrow 0$ for $t\rightarrow -\infty$ and, for some of them, $G_{eff}\rightarrow G_{N}$ for $t\rightarrow\infty$, where $G_{N}$ is the Newton constant. 
  A quantum frame is defined by a material object subject to the laws of quantum mechanics. The present paper studies the relations between quantum frames, which in the classical case are described by elements of the Poincare' group. The possibility of using a suitable quantum group is examined, but some arguments are given which show that a different mathematical structure is necessary. Some simple examples in lower dimensional spacetimes are treated. They indicate the necessity of taking into account some "internal" degrees of freedom of the quantum frames, that can be disregarded in a classical treatment. 
  Subject of this talk is an overview of results on self-gravitating solitons of the classical Yang-Mills-Higgs theory. One finds essentially two classes of solitons, one of them corresponding to the magnetic monopoles the other one to the sphalerons of flat space. The coupling to the gravitational field leads to new features absent in flat space. These are the gravitational instability of these solitons at the Planck scale and the existence of black holes with `non-abelian hair'' in addition to the regular solutions. 
  In 1993, a proof was published, within ``Classical and Quantum Gravity,'' that there are no regular solutions to the {\it linearized} version of the twisting, type-N, vacuum solutions of the Einstein field equations. While this proof is certainly correct, we show that the conclusions drawn from that fact were unwarranted, namely that this irregularity caused such solutions not to be able to truly describe pure gravitational waves. In this article, we resolve the paradox---since such first-order solutions must always have singular lines in space for all sufficiently large values of $r$---by showing that if we perturbatively iterate the solution up to the third order in small quantities, there are acceptable regular solutions. That these solutions become flat before they become non-twisting tells us something interesting concerning the general behavior of solutions describing gravitational radiation from a bounded source. 
  The thermodynamic behavior of vacuum decaying cosmologies is investigated within a manifestly covariant formulation. Such a process corresponds to a continuous irreversible energy flow from the vacuum component to the created matter constituents. It is shown that if the specific entropy per particle remains constant during the process, the equilibrium relations are preserved. In particular, if the vacuum decays into photons, the energy density $\rho$ and average number density of photons $n$ scale with the temperature as $\rho \sim T^{4}$ and $n \sim T^{3}$. The temperature law is determined and a generalized Planckian type form of the spectrum, which is preserved in the course of the evolution, is also proposed. Some consequences of these results for decaying vacuum FRW type cosmologies as well as for models with ``adiabatic'' photon creation are discussed. 
  A new Planckian distribution for cosmologies with photon creation is derived using thermodynamics and semiclassical considerations. This spectrum is preserved during the evolution of the universe and compatible with the present spectral shape of the cosmic microwave background radiation(CMBR). Accordingly, the widely spread feeling that cosmologies with continuous photon creation are definitely ruled out by the COBE limits on deviation of the CMBR spectrum from blackbody shape should be reconsidered. It is argued that a crucial test for this kind of cosmologies is provided by measurements of the CMBR temperature at high redshifts. For a given redshift $z$ greater than zero, the temperature is smaller than the one predicted by the standard FRW model. 
  Analytic solutions of the Regge-Wheeler equation are presented in the form of series of hypergeometric functions and Coulomb wave functions which have different regions of convergence. Relations between these solutions are established. The series solutions are given as the Post-Minkowskian expansion with respect to a parameter $\epsilon \equiv 2M\omega$, $M$ being the mass of black hole. This expansion corresponds to the post-Newtonian expansion when they are applied to the gravitational radiation from a particle in circular orbit around a black hole. These solutions can also be useful for numerical computations. 
  We consider a Hamiltonian theory of spherically symmetric vacuum Einstein gravity under Kruskal-like boundary conditions in variables associated with the Einstein-Rosen wormhole throat. The configuration variable in the reduced classical theory is the radius of the throat, in a foliation that is frozen at the left hand side infinity but asymptotically Minkowski at the right hand side infinity, and such that the proper time at the throat agrees with the right hand side Minkowski time. The classical Hamiltonian is numerically equal to the Schwarzschild mass. Within a class of Hamiltonian quantizations, we show that the spectrum of the Hamiltonian operator is discrete and bounded below, and can be made positive definite. The large eigenvalues behave asymptotically as~$\sqrt{2k}$, where $k$ is an integer. The resulting area spectrum agrees with that proposed by Bekenstein and others. Analogous results hold in the presence of a negative cosmological constant and electric charge. The classical input that led to the quantum results is discussed. 
  Originally regarded as forbidden, black hole ``hair'' are fields associated with a stationary black hole apart from the gravitational and electromagnetic ones. Several stable stationary black hole solutions with gauge or Skyrme field hair are known today within general relativity. We formulate here a ``no scalar--hair'' conjecture, and adduce some new theorems that almost establish it by ruling out - for all but a small parameter range - scalar field hair of spherical black holes in general relativity, whether the field be self--interacting, coupled to an Abelian gauge field, or nonminimally coupled to gravity. 
  A new method is presented for assigning distributional curvature, in an invariant manner, to a space-time of low differentiability, using the techniques of Colombeau's `new generalised functions'. The method is applied to show that curvature of a cone is equivalent to a delta function. The same is true under small enough perturbations. 
  Isotropic and spatially homogeneous viscous fluid cosmological models are investigated using the truncated Israel-Stewart theory of irreversible thermodynamics to model the bulk viscous pressure. The governing system of differential equations is written in terms of dimensionless variables and a set of dimensionless equations of state is then utilized to complete the system. The resulting dynamical system is analyzed using geometric techniques from dynamical systems theory to find the qualitative behaviour of the Friedmann-Robertson-Walker models with bulk viscosity. In these models there exists a free parameter such that the qualitative behaviour of the models can be quite different (for certain ranges of values of this parameter) from that found in models satisfying the Eckart theory studied previously. In addition, the conditions under which the models inflate are investigated. 
  The truncated Israel-Stewart theory of irreversible thermodynamics is used to describe the bulk viscous pressure and the anisotropic stress in a class of spatially homogeneous viscous fluid cosmological models. The governing system of differential equations is written in terms of dimensionless variables and a set of dimensionless equations of state is utilized to complete the system. The resulting dynamical system is then analyzed using standard geometric techniques. It is found that the presence of anisotropic stress plays a dominant role in the evolution of the anisotropic models. In particular, in the case of the Bianchi type I models it is found that anisotropic stress leads to models that violate the weak energy condition and to the creation of a periodic orbit in some instances. The stability of the isotropic singular points is analyzed in the case with zero heat conduction; it is found that there are ranges of parameter values such that there exists an attracting isotropic Friedmann-Robertson-Walker model. In the case of zero anisotropic stress but with non-zero heat conduction the stability of the singular points is found to be the same as in the corresponding case with zero heat conduction; hence the presence of heat conduction does not apparently affect the global dynamics of the model. 
  The Full (non--truncated) Israel--Stewart theory of bulk viscosity is applied to dissipative FRW spacetimes. Dimensionless variables and dimensionless equations of state are used to write the Einstein--thermodynamic equations as a plane autonomous system and the qualitative behaviour of this system is determined. Entropy production in these models is also discussed. 
  Using Penrose binor calculus for $SU(2)$ ($SL(2,C)$) tensor expressions, a graphical method for the connection representation of Euclidean Quantum Gravity (real connection) is constructed. It is explicitly shown that: {\it (i)} the recently proposed scalar product in the loop-representation coincide with the Ashtekar-Lewandoski cylindrical measure in the space of connections; {\it (ii)} it is possible to establish a correspondence between the operators in the connection representation and those in the loop representation. The construction is based on embedded spin network, the Penrose graphical method of $SU(2)$ calculus, and the existence of a generalized measure on the space of connections modulo gauge transformations. 
  The role of the equivalence principle in the context of non-relativistic quantum mechanics and matter wave interferometry, especially atom beam interferometry, will be discussed. A generalised form of the weak equivalence principle which is capable of covering quantum phenomena too, will be proposed. It is shown that this generalised equivalence principle is valid for matter wave interferometry and for the dynamics of expectation values. In addition, the use of this equivalence principle makes it possible to determine the structure of the interaction of quantum systems with gravitational and inertial fields. It is also shown that the path of the mean value of the position operator in the case of gravitational interaction does fulfill this generalised equivalence principle. 
  We study in this paper a new approach to the problem of relating solutions to the Einstein field equations with Riemannian and Lorentzian signatures. The procedure can be thought of as a "real Wick rotation". We give a modified action for general relativity, depending on two real parameters, that can be used to control the signature of the solutions to the field equations. We show how this procedure works for the Schwarzschild metric and discuss some possible applications of the formalism in the context of signature change, the problem of time, black hole thermodynamics... 
  A recently-developed theory of quantum general relativity provides a propagator for free-falling particles in curved spacetimes. These propagators are constructed by parallel-transporting quantum states within a quantum bundle associated to the Poincare frame bundle. We consider such parallel transport in the case that the spacetime is a classical Robertson-Walker universe. An explicit integral formula is developed which expresses the propagators for parallel transport between any two points of such a spacetime. The integrals in this formula are evaluated in closed form for a particular spatially-flat model. 
  A version of foliated spacetime is constructed in which the spatial geometry is described as a time dependent noncommutative geometry. The ADM version of the gravitational action is expressed in terms of these variables. It is shown that the vector constraint is obtained without the need for an extraneous shift vector in the action. 
  We study a behavior of quantum generalized affine parameter (QGAP), which has been recently proposed by one of the present authors, near the singularity and the event horizon in three and four spacetime dimensions in terms of a minisuperspace model of quantum gravity. It is shown that the QGAP is infinite to the singularity while it remains finite to the event horizon. This fact indicates a possible interpretation that the singularity is wiped out in quantum gravity in this particular model of black hole. 
  We study the possibility of obtaining inflationary solutions from S-dual superstring potentials. We find, in particular, that such solutions occur at the core of domain walls separating degenerate minima whose positions differ by modular transformations. 
  We study and compare the decoherent histories approach, the environment-induced decoherence and the localization properties of thesolutions to the stochastic Schr\"{o}dinger equation in quantum jump simulationand quantum state diffusion approaches, for a quantum two-level system model. We show, in particular, that there is a close connection between the decoherent histories and the quantum jump simulation, complementing a connection with the quantum state diffusion approach noted earlier by Di\'{o}si, Gisin, Halliwell and Percival. In the case of the decoherent histories analysis, the degree of approximate decoherence is discussed in detail. 
  We compute the spectrum of relic gravitons in a model of string cosmology. In the low- and in the high-frequency limits we reproduce known results. The full spectrum, however, also displays a series of oscillations which could give a characteristic signature at the planned LIGO/VIRGO detectors. For special values of the parameters of the model the signal reaches its maximum already at frequencies accessible to LIGO and VIRGO and it is close to the sensitivity of first generation experiments. 
  A local formula for the dimension of a superselection sector in Quantum Field Theory is obtained as vacuum expectation value of the exponential of the proper Hamiltonian. In the particular case of a chiral conformal theory, this provides a local analogue of a global formula obtained by Kac and Wakimoto within the context of representations of certain affine Lie algebras. Our formula is model independent and its version in general Quantum Field Theory applies to black hole thermodynamics. The relative free energy between two thermal equilibrium states associated with a black hole turns out to be proportional to the variation of the conditional entropy in different superselection sectors, where the conditional entropy is defined as the Connes-Stoermer entropy associated with the DHR localized endomorphism representing the sector. The constant of proportionality is half of the Hawking temperature. As a consequence the relative free energy is quantized proportionally to the logarithm of a rational number, in particular it is equal to a linear function the logarithm of an integer once the initial state or the final state is taken fixed. 
  We critically examine the recent claim (gr-qc/9603008) of a ``new effect'' of gravitationally induced quantum mechanical phases in neutrino oscillations. A straightforward exercise in the Schwarzschild coordinates appropriate to a spherically symmetric non-rotating star shows that, although there is a general relativistic effect of the star's gravity on neutrino oscillations, it is not of the form claimed, and is too small to be measured. 
  We review some well posed formulations of the evolution part of the Cauchy problem of General Relativity that we have recently obtained. We include also a new first order symmetric hyperbolic system based directly on the Riemann tensor and the full Bianchi identities. It has only physical characteristics and matter sources can be included. It is completely equivalent to our other system with these properties. 
  We examine the possibility that time dependence might remove the singular nature of global string spacetimes. We first show that this time dependence takes a specific form -- a de-Sitter like expansion along the length of the string and give an argument for the existence of such a solution, estimating the rate of expansion. We compare our solution to the singular Cohen-Kaplan spacetime. 
  A numerical study of the evolution of a massless scalar field in the background of rotating black holes is presented. First, solutions to the wave equation are obtained for slowly rotating black holes. In this approximation, the background geometry is treated as a perturbed Schwarzschild spacetime with the angular momentum per unit mass playing the role of a perturbative parameter. To first order in the angular momentum of the black hole, the scalar wave equation yields two coupled one-dimensional evolution equations for a function representing the scalar field in the Schwarzschild background and a second field that accounts for the rotation. Solutions to the wave equation are also obtained for rapidly rotating black holes. In this case, the wave equation does not admit complete separation of variables and yields a two-dimensional evolution equation. The study shows that, for rotating black holes, the late time dynamics of a massless scalar field exhibit the same power-law behavior as in the case of a Schwarzschild background independently of the angular momentum of the black hole. 
  We consider a static, axially symmetric, and asymptotically flat exact solution of the Einstein vacuum equations, known as the gamma metric. This is characterized by two constant parameters $m$ and $\gamma$. We find that the total energy associated with this metric is $m \gamma$. Considering the total energy to be positive, we investigate the nature of a curvature singularity $r=2m$ ($r$ is the radial coordinate) in this metric. For $\gamma < 1$, this singularity is globally visible along $\theta = 0$ as well as $\theta = \pi /2$. However, for $\gamma > 1$, this singularity is though globally naked along $\theta =\pi/2$, it is not visible (even locally) along $\theta = 0$. Thus, this exhibits ``directional nakedness'' for $\gamma > 1$. This could have implications for astrophysics. 
  Simplicial geometries are collections of simplices making up a manifold together with an assignment of lengths to the edges that define a metric on that manifold. The simplicial analogs of the Einstein equations are the Regge equations. Solutions to these equations define the semiclassical approximation to simplicial approximations to a sum-over-geometries in quantum gravity. In this paper, we consider solutions to the Regge equations with cosmological constant that give Euclidean metrics of high symmetry on a family of triangulations of CP^2 presented by Banchoff and Kuhnel. This family is characterized by a parameter p. The number of vertices grows larger with increasing p. We exhibit a solution of the Regge equations for p=2 but find no solutions for p=3. This example shows that merely increasing the number of vertices does not ensure a steady approach to a continuum geometry in the Regge calculus. 
  We consider a quantum cosmology with a massless background scalar field $\pb$ and adopt a wave packet as the wave function. This wave packet is a superposition of the WKB form wave functions, each of which has a definite momentum of the scalar field $\pb$. In this model it is shown that to trace the formalism of the WKB time is seriously difficult without introducing a complex value for a time. We define a semiclassical real time variable $\tp$ from the phase of the wave packet and calculate it explicitly. We find that, when a quantum matter field $\pq$ is coupled to the system, an approximate Schr\"odinger equation for $\pq$ holds with respect to $\tp$ in a region where the size $a$ of the universe is large and $|\pb|$ is small. 
  It is well known that Einstein gravity is non-renormalizable; however a generalized approach is proposed that leads to Einstein gravity {\it after} renormalization. This them implies that at least one candidate for quantum gravity treats all matter on an equal footing with regard to the gravitational behaviour. 
  I discuss a recent analytic proof of bypassing the no-hair conjecture for two interesting (and quite generic) cases of four-dimensional black holes: (i) black holes in Einstein-Yang-Mills-Higgs (EYMH) systems and (ii) black holes in higher-curvature (Gauss-Bonnet (GB) type) string-inspired gravity. Both systems are known to possess black-hole solutions with non-trivial scalar hair outside the horizon. The `spirit' of the no-hair conjecture, however, seems to be maintained either because the black holes are unstable (EYMH), or because the hair is of secondary type (GB), i.e. it does not lead to new conserved quantum numbers. 
  It is shown that well-known Vlasov equation can be derived by adding "hidden" degrees of freedom and subsequent quantization. The Shrodinger equation obtained in this manner coincides (in x-representation) with the kinetic equation for the original dynamical system 
  Locating apparent horizons is not only important for a complete understanding of numerically generated spacetimes, but it may also be a crucial component of the technique for evolving black-hole spacetimes accurately. A scheme proposed by Libson et al., based on expanding the location of the apparent horizon in terms of symmetric trace-free tensors, seems very promising for use with three-dimensional numerical data sets. In this paper, we generalize this scheme and perform a number of code tests to fully calibrate its behavior in black-hole spacetimes similar to those we expect to encounter in solving the binary black-hole coalescence problem. An important aspect of the generalization is that we can compute the symmetric trace-free tensor expansion to any order. This enables us to determine how far we must carry the expansion to achieve results of a desired accuracy. To accomplish this generalization, we describe a new and very convenient set of recurrence relations which apply to symmetric trace-free tensors. 
  In this paper we describe the matter-free toroidal spacetime in 't Hooft's polygon approach to 2+1-dimensional gravity (i.e. we consider the case without any particles present). Contrary to earlier results in the literature we find that it is not possible to describe the torus by just one polygon but we need at least two polygons. We also show that the constraint algebra of the polygons closes. 
  We study how the initial inhomogeneities of the spatial curvature affect the onset of inflation in the closed universe. We consider a cosmological model which contains a radiation and a cosmological constant. In order to treat the inhomogeneities in the closed universe, we improve the long wavelength approximation such that the non-small spatial curvature is tractable in the lowest order. Using the improved scheme, we show how large inhomogeneities of the spatial curvature prevent the occurrence of inflation. 
  The evolution of spin network states in loop quantum gravity can be described by introducing a time variable, defined by the surfaces of constant value of an auxiliary scalar field. We regulate the Hamiltonian, generating such an evolution, and evaluate its action both on edges and on vertices of the spin network states. The analytical computations are carried out completely to yield a finite, diffeomorphism invariant result. We use techniques from the recoupling theory of colored graphs with trivalent vertices to evaluate the graphical part of the Hamiltonian action. We show that the action on edges is equivalent to a diffeomorphism transformation, while the action on vertices adds new edges and re-routes the loops through the vertices. 
  Four--dimensional Einstein--Maxwell--dilaton--axion system restricted to space--times with one non--null Killing symmetry is formulated as the three--dimensional gravity coupled sigma--model. Several alternative representations are discussed and the associated hidden symmetries are revealed. The action of target space isometries on the initial set of (non--dualized ) variables is found. New mulicenter solutions are obtained via generating technique based on the formulation in terms of the non--dualized variables. 
  An exact solution of the source-free Kaluza-Klein field equations is presented. It is a 5D generalization of the Robinson-Trautman quasi-spherical gravitational wave with a cosmological constant. The properties of the 5D solution are briefly described. 
  This article gives an elementary overview of the end-state of gravitational collapse according to classical general relativity. The focus of discussion is the formation of black holes and naked singularities in various physically reasonable models of gravitational collapse. Possible implications for the cosmic censorship hypothesis are outlined. 
  A plane monochromatic wave will not appear monochromatic to a noninertial observer. We show that this feature leads to a `thermal' ambience in an accelerated frame {\it even in classical field theory}. When a real, monochromatic, mode of a scalar field is Fourier analyzed with respect to the proper time of a uniformly accelerating observer, the resulting power spectrum consists of three terms: (i)~a factor $(1/2)$ that is typical of the ground state energy of a quantum oscillator, (ii)~a Planckian distribution $N(\Omega)$ and---most importantly---(iii)~a term $\sqrt{N(N+1)}$, which is the root mean square fluctuations about the Planckian distribution. It is the appearance of the root mean square fluctuations that motivates us to attribute a `thermal' nature to the power spectrum. This result shows that some of the `purely' quantum mechanical results might have a classical analogue. The `thermal' ambience that we report here also proves to be a feature of observers stationed at a constant radius in the Schwarzschild and de-Sitter spacetimes. 
  In this paper, we discuss the leading order correction to the equation of motion of the particle, which presumably describes the effect of gravitational radiation reaction. We derive the equation of motion in two different ways. The first one is an extension of the well-known formalism by DeWitt and Brehme developed for deriving the equation of motion of an electrically charged particle. In contrast to the electromagnetic case, in which there are two different charges, i.e., the electric charge and the mass, the gravitational counterpart has only one charge. This fact prevents us from using the same renormalization scheme that was used in the electromagnetic case. To make clear the subtlety in the first approach, we then consider the asymptotic matching of two different schemes, i.e., the internal scheme in which the small particle is represented by a spherically symmetric black hole with tidal perturbations and the external scheme in which the metric is given by small perturbations on the given background geometry. The equation of motion is obtained from the consistency condition of the matching. We find that in both ways the same equation of motion is obtained. The resulting equation of motion is analogous to that derived in the electromagnetic case. We discuss implications of this equation of motion. 
  We argue that our equation of gravitation ( Phys.Lett. A 156 (1991) 404 ) lead in pseudo-Euclidean space-time to the finite energy of the gravitational field of a point mass. 
  We have shown that the dynamics of the scalar field $\phi (x)= ``G^{-1}(x)"$ in Brans-Dicke theories of gravity makes the surface area of the black hole horizon {\it oscillatory} during its dynamical evolution. It explicitly explains why the area theorem does not hold in Brans-Dicke theory. However, we show that there exists a certain non-decreasing quantity defined on the event horizon which is proportional to the black hole entropy for the case of stationary solutions in Brans-Dicke theory. Some numerical simulations have been demonstrated for Oppenheimer-Snyder collapse in Brans-Dicke theory. 
  Like a fairy-tale princess, trajectories around black holes can be sensitive to small disturbances. We describe how a small disturbance can lead to erratic orbits and an increased production of gravitational waves. 
  We suggest that ``free evolution'' integration schemes for the Einstein equations (that do not enforce constraints) may contain exponentially growing modes that render them useless in numerical integrations of black hole spacetimes, independently of how the equations are differenced. As an example we consider the evolution of Schwarzschild and Reissner-Nordstr\"om spacetimes in double null coordinates. 
  Lecture given at the workshop "Mathematical aspects of theories of gravitation", Stefan Banach International Mathematical Centre, 7 March 1996. A mini-introduction to critical phenomena in gravitational collapse is combined with a more detailed discussion of the regularity of the ``critical spacetimes'' dominating these phenomena. 
  Among the expected sources of gravitational waves for the Laser Interferometer Space Antenna (LISA) is the capture of solar-mass compact stars by massive black holes residing in galactic centers. We construct a simple model for such a capture, in which the compact star moves freely on a circular orbit in the equatorial plane of the massive black hole. We consider the gravitational waves emitted during the late stages of orbital evolution, shortly before the orbiting mass reaches the innermost stable circular orbit. We construct a simple model for the gravitational-wave signal, in which the phasing of the waves plays the dominant role. The signal's behavior depends on a number of parameters, including $\mu$, the mass of the orbiting star, $M$, the mass of the central black hole, and $J$, the black hole's angular momentum. We calculate, using our simplified model, and in the limit of large signal-to-noise ratio, the accuracy with which these quantities can be estimated during a gravitational-wave measurement. Our simplified model also suggests a method for experimentally testing the strong-field predictions of general relativity. 
  We construct a supersymmetric extension of the $I\big(ISO(2,1)\big)$ Chern-Simons gravity and show that certain particle-like solutions and the adS black-hole solution of this theory are supersymmetric. 
  recent theoretical results show the existence of arbitrary speeds ($0\leq v <\infty$) solutions of the wave equations of mathematical physics. Some recent experiments confirm the results for sound waves. The question arises naturally: What is the appropriate spacetime model to describe superluminal phenomena? In this paper we present a spacetime model that incorporates the valid results of Relativity Theory and yet describes coherently superluminal phenomena without paradoxes. 
  We use the image sum method to reproduce Sushkov's result that for a massless automorphic field on the initial globally hyperbolic region $IGH$ of Misner space, one can always find a special value of the automorphic parameter $\alpha$ such that the renormalized expectation value $\langle\alpha|T_{ab}|\alpha\rangle$ in the {\it Sushkov state} ``$\langle\alpha|\cdot|\alpha\rangle$'' (i.e. the automorphic generalization of the Hiscock-Konkowski state) vanishes. However, we shall prove by elementary methods that the conclusions of a recent general theorem of Kay-Radzikowski-Wald apply in this case. I.e. for any value of $\alpha$ and any neighbourhood $N$ of any point $b$ on the chronology horizon there exists at least one pair of non-null related points $(x,x') \in (N\cap IGH)\times (N\cap IGH)$ such that the renormalized two-point function of an automorphic field $G^\alpha_{\rm ren}(x,x')$ in the Sushkov state is singular. In consequence $\langle\alpha|T_{ab}|\alpha\rangle$ (as well as other renormalized expectation values such as $\langle\alpha|\phi^2|\alpha\rangle$) is necessarily singular {\it on} the chronology horizon. We point out that a similar situation (i.e. singularity {\it on} the chronology horizon) holds for states on Gott space and Grant space. 
  Due to its relatively large eccentricity and proximity to the Sun, Mercury's orbital motion provides one of the best solar-system tests of relativistic gravity. We emphasize the number of feasible relativistic gravity tests that can be performed within the context of the parameterized weak field and slow motion approximation - a usefulframework for testing modern gravitational theories in the solar system. We discuss a new approximation method, which includes two Eddington parameters $(\gamma,\beta)$, proposed for construction of the relativistic equations of motion of extended bodies. Within the present accuracy of radio measurements, we discuss the generalized Fermi-normal-like proper reference frame which is defined in the immediate vicinity of the extended compact bodies. Based on the Hermean-centric equations of motion of the spacecraft around the planet Mercury, we suggest a new test of the Strong Equivalence Principle. The corresponding experiment could be performed with the future {\it Mercury Orbiter} mission scheduled by the European Space Agency ({\small ESA}) for launch between 2006 and 2016. We discuss other relativistic effects including the perihelion advance, redshift and geodetic precession of the orbiter's orbital plane about Mercury. 
  Stationary solutions to the full non-linear topologically massive gravity (TMG) are obtained for localized sources of mass $m$ and spin $\sigma$. Our results show that the topological term induces spin and that the total spin J (which is the spin observed by an asymptotic observer) ranges from 0 to $\sigma + \frac{m}{\mu}(\frac{4\pi+m}{4\pi +2m})$ depending on the structure of the spin source (here $\mu$ is the topological mass). We find that it is inconsistent to consider actual delta function mass and spin sources. In the point-like limit, however, we find no condition constraining $m$ and $\sigma$ contrary to a previous analysis by Clement. 
  We apply the strictly isospectral technique of standard supersymmetric quantum mechanics to the Q=0 factor ordered Wheeler-DeWitt equation for the Friedmann-Robertson-Walker (FRW) minisuperspace model. The resulting strictly isospectral one-parameter families of both FRW cosmological potentials and "wavefunctions of the universe" are exhibited with relevant plots 
  We present some thoughts on how to interpret the gravitionally induced neutrino oscillation phases presented by us in our 1996 Gravity Research Foundation Essay. 
  The problem of cosmological graviton creation for homogeneous and isotropic universes with elliptical ($\vae =+1$) and hyperbolical ($\vae =-1$) geometries is addressed. The gravitational wave equation is established for a self-gravitating fluid satisfying the barotropic equation of state $p=(\gamma -1)\rho$, which is the source of the Einstein's equations plus a cosmological $\Lambda$-term. The time dependent part of this equation is exactly solved in terms of hypergeometric functions for any value of $\gamma$ and spatial curvature $\vae$. An expression representing an adiabatic vacuum state is then obtained in terms of associated Legendre functions whenever $\gamma\neq \frac{2}{3}\; \frac{(2n+1)}{(2n-1)}$, where n is an integer. This includes most cases of physical interest such as $\gamma =0,\;4/3\;,1$. The mechanism of graviton creation is reviewed and the Bogoliubov coefficients related to transitions between arbitrary cosmic eras are also explicitly evaluated. 
  The processing of the galactic coordinates of 3543 solar flares with a magnitude of 2 or more has shown that the distribution of these fast-going processes on the surface of the "nonrotating" Sun is irregular and non-random what testifies that in the near-Sun space an anisotropy takes place which practically coincides with that predicted and obtained in laboratory experiments and caused by existence of the intergalactic vector potential ${\bf A_g}$. 
  We discuss black holes in an effective theory derived from a superstring model, which includes a dilaton field, a gauge field and the Gauss-Bonnet term. Assuming U(1) or SU(2) symmetry for the gauge field, we find four types of spherically symmetric solutions, i.e., a neutral, an electrically charged, a magnetically charged and a ``colored'' black hole, and discuss their thermodynamical properties and fate via the Hawking evaporation process. For neutral and electrically charged black holes, we find critical point and a singular end point. Below the mass corresponding to the critical point, nosolution exists, while the curvature on the horizon diverges and anaked singularity appears at the singular point. A cusp structure in the mass-entropy diagram is found at the critical point and black holes on the branch between the critical and singular points become unstable. For magnetically charged and ``colored" black holes, the solution becomes singular just at the end point with a finite mass. Because the black hole temperature is always finite even at the critical point or the singular point, we may conclude that the evaporation process will not be stopped even at the critical point or the singular point, and the black hole will move to a dynamical evaporation phase or a naked singularity will appear. 
  We argue that space-time geometry is not absolute with respect to the frame of reference being used. The space-time metric differential form $ds$ in noninertial frames of reference (NIFR) is caused by the properties of the used frames in accordance with the Berkley - Leibnitz - Mach - Poincar\'{e} ideas about relativity of space and time . It is shown that the Sagnac effect and the existence of inertial forces in NIFR can be considered from this point of view. An experimental test is proposed. 
  By allowing the light cones to tip over on hypersurfaces according to the conservation laws of an one-kink in static, Schwarzschild and five-dimensional black hole metrics, we show that in the quantum regime there also exist instantons whose finite imaginary action gives the probability of accurrence of the kink metric corresponding to single chargeless, nonrotating black holes taking place in pairs, joined on an interior surface beyond the horizon, with each hole residing in a diffrent universe. Evaluation of the thermal properties of each of the black holes in a neutral pair leads one to check that to an asymptotic abserver in either universe, each black hole is exactly the quantum-mechanically defined {\it anti-black hole} to the other hole in the pair. The independent quantum states of black holes in neutral pairs have been formulated by using the path integral method, and shown to be that of a harmonic oscillator. Our results suggest that the boundary condition of a single universe in the metauniverse is that this universe can never be self-contained and must always have at least one boundary which connects it to the rest of a self-contained metauniverse. 
  The topology of space is usually assumed simply connected, but could be multi-connected. We review in the latter case the possibility that topological defects arising at high energy phase transitions might still be present and find that either they are very unlikely to form at all, or space is effectively simply connected on scales up to the horizon size. 
  We consider the model involving the oscillation of the effective gravitational constant that has been put forward in an attempt to reconcile the observed periodicity in the galaxy number distribution with the standard cosmological models. This model involves a highly nonlinear dynamics which we analyze numerically. We carry out a detailed study of the bound that nucleosynthesis imposes on this model. The analysis shows that for any assumed value for $\Omega$ (the total energy density) one can fix the value of $\Omega_{\rm bar}$ (the baryonic energy density) in such a way as to accommodate the observational constraints coming from the $^4{\rm He}$ primordial abundance. In particular, if we impose the inflationary value $\Omega=1$ the resulting baryonic energy density turns out to be $\Omega_{\rm bar}\sim 0.021$. This result lies in the very narrow range $0.016 \leq \Omega_{\rm bar} \leq 0.026$ allowed by the observed values of the primordial abundances of the other light elements. The remaining fraction of $\Omega$ corresponds to dark matter represented by a scalar field. 
  There are many indications that ordinary matter represents only a tiny fraction of the matter content of the Universe, with the remainder assumed to consist of some different type of matter, which, for various reasons must be nonluminous (dark matter). Among these indications are the inflationary scenarios which predicts that the average energy density of the Universe coincides with the so called critical value (for which the expansion never stops but the rate of expansion approaches zero at very late times). At the same time it is known (from the predictions of Big Bang nucleosynthesis on the abundances of the light elements, other than Helium) that the baryonic energy density (ordinary matter) must represent ($1.5\pm 0.5)h^{-2}$ \% (where $h$ is the Hubble constant in units of 100 km s$^{-1}$Mpc$^{-1}$) of this critical value \cite{Copi,OstStein}. We present here evidence supporting the model in which the rest of the energy density corresponds to a scalar field, which can be observed, however indirectly, in the oscillation of the effective gravitational constant, and manifests itself in the known periodicity of the number distribution of galaxies \cite{Broad,Szalay}. We analyze this model numerically and show that, the requirement that the model satisfy the bounds of light element abundances in the Universe, as predicted by Big Bang nucleosynthesis, yields a specific value for the red-shift-galactic-count oscillation amplitude compatible with that required to explain the oscillations described above \cite{hill,CritStein}, and, furthermore, yields a value for the age of the Universe compatible with standard bounds \cite{OstStein}. 
  With the concept of "discrete space-time" the space-time continuum is resolved into discrete points at the scale of the Planck length. We postulate with the "principle of the fermionic projector" that physical equations must be formulated intrinsically in discrete space-time with the projector on fermionic states. This principle combines the Pauli principle, a local gauge principle and the principle of relativity, but does not include a locality or causality principle. It is shown that a well-established limit yields the structure of a Lorentzian manifold. The equations of discrete space-time reduce to classical field equations.   This preliminary text is intended as introduction. Detailed calculations can be obtained from the author on request. 
  Using the (3+1) formalism in general relativity, we perform the post-Newtonian(PN) approximation to clarify what sort of gauge condition is suitable for numerical analysis of coalescing compact binary neutron stars and gravitational waves from them. We adopt a kind of transverse gauge condition to determine the shift vector. On the other hand, for determination of the time slice, we adopt three slice conditions(conformal slice, maximal slice and harmonic slice) and discuss their properties. Using these conditions, the PN hydrodynamic equations are obtained up through the 2.5PN order including the quadrupole gravitational radiation reaction. In particular, we describe methods to solve the 2PN tensor potential which arises from the spatial 3-metric. It is found that the conformal slice seems appropriate for analysis of gravitational waves in the wave zone and the maximal slice will be useful for describing the equilibrium configurations. The PN approximation in the (3+1) formalism will be also useful to perform numerical simulations using various slice conditions and, as a result, to provide an initial data for the final merging phase of coalescing binary neutron stars which can be treated only by fully general relativistic simulations. 
  Asymptotically flat, time-symmetric, axially symmetric and conformally flat initial data for vacuum general relativity are studied numerically on $R^3$ with the interior of a standard torus cut out. By the choice of boundary condition the torus is marginally outer trapped, and thus a surface of minimal area. Apart from pure scaling the standard tori are parameterized by a radius $a\in [0,1]$, where $a=0$ corresponds to the limit where the boundary torus degenerates to a circle and $a=1$ to a torus that touches the axis of symmetry. Noting that these tori are the orbits of a $U(1)\times U(1)$ conformal isometry allows for a simple scheme to solve the constraint, involving numerical solution of only ordinary differential equations.The tori are unstable minimal surfaces (i.e. only saddle points of the area functional) and thus can not be apparent horizons, but are always surrounded by an apparent horizon of spherical topology, which is analyzed in the context of the hoop conjecture and isoperimetric inequality for black holes. 
  In its formulation as a Chern-Simons theory, three-dimensional general relativity induces a Wess-Zumino-Witten action on spatial boundaries. Treating the horizon of the three-dimensional Euclidean black hole as a boundary, I count the states of the resulting WZW model, and show that when analytically continued back to Lorentzian signature, they yield the correct Bekenstein-Hawking entropy. The relevant states can be understood as ``would-be gauge'' degrees of freedom that become dynamical at the horizon. 
  This is a comment on a reply (gr-qc/9601040) to a comment (gr-qc/9606045) on a paper of Hellaby & Dray (gr-qc/9404001), repeating the identification of an important mistake which is still being denied by the authors: their proposed solutions do not satisfy the Einstein- Klein-Gordon equations at a change of signature. Substitution of the proposed solutions into the Einstein-Klein-Gordon equations in unit normal coordinates yields Dirac delta terms describing source layers at the junction. Hellaby & Dray's criticisms of this straightforward calculation are absurd: it does not involve "imaginary time", it does not involve a "modified form" of the field equations, and it is "purely classical". Moreover, Hellaby & Dray's latest attempt to lose the delta terms is mathematically invalid, involving division by zero and products of distributions, hinging on an identity whose incorrectness may be checked by substitution. 
  Hellaby & Dray (gr-qc/9404001) have recently claimed that matter conservation fails under a change of signature, compounding earlier claims that the standard junction conditions for signature change are unnecessary. In fact, if the field equations are satisfied, then the junction conditions and the conservation equations are satisfied. The failure is rather that the authors did not make sense of the field equations and conservation equations, which are singular at a change of signature. 
  The algebra of constraints arising in the canonical quantization of N=1 supergravity in four dimensions is investigated. Using the holomorphic action, the structure functions of the algebra are given and it is shown that the algebra does not close formally for two chosen operator orderings. 
  In this paper we investigate whether conserved currents can be sensibly defined in supersymmetric minisuperspaces. Our analysis deals with k=1 FRW and Bianchi class--A models. Supermatter in the form of scalar supermultiplets is included in the former. Moreover, we restrict ourselves to the first-order differential equations derived from the Lorentz and supersymmetry constraints. The ``square-root'' structure of N=1 supergravity was our motivation to contemplate this interesting research. We show that conserved currents cannot be adequately established except for some very simple scenarios. Otherwise, conservation equations may only be obtained from Wheeler-DeWitt--like equations, which are derived from the supersymmetric algebra of constraints. Two appendices are included. In appendix A we describe some interesting features of quantum FRW cosmologies with complex scalar fields when supersymmetry is present. In particular, we explain how the Hartle-Hawking state can now be satisfactorily identified. In appendix B we initiate a discussion about the retrieval of classical properties from supersymmetric quantum cosmologies. 
  We investigate Brans-Dicke dilaton gravity theories in 2+1 dimensions. We show that the reduced field equations for solutions with diagonal metric and depending only on one spacetime coordinate have a continuous O(2) symmetry. Using this symmetry we derive general static and cosmological solutions of the theory. The action of the discrete group O(2,Z) on the space of the solutions is discussed. Three-dimensional string effective theory and three-dimensional general relativity are discussed in detail. In particular, we find that the previously discovered black string solution is dual to a spacetime with a conical singularity. 
  Foliations by constant mean curvature hypersurfaces provide a possibility of defining a preferred time coordinate in general relativity. In the following various conjectures are made about the existence of foliations of this kind in spacetimes satisfying the strong energy condition and possessing compact Cauchy hypersurfaces. Recent progress on proving these conjectures under supplementary assumptions is reviewed. The method of proof used is explained and the prospects for generalizing it discussed. The relations of these questions to cosmic censorship and the closed universe recollapse conjecture are pointed out. 
  We use our N\"other Symmetry Approach to study the Einstein equations minimally coupled with a scalar field, in the case of Bianchi universes of class A and B. Possible cases, when such symmetries exist, are found and two examples of exact integration of the equations of motion are given in the cases of Bianchi AI and BV. 
  In this paper we give a review of the most general approach to description of reference frames, the monad formalism. This approach is explicitly general covariant at each step, permitting to use abstract representation of tensor quantities; it is applicable also to special relativity when non-inertial effects are considered in its context; moreover, it involves no hypotheses whatsoever thus being a completely natural one. For the sake of the reader's convenience, a synopsis of tensor calculus in pseudo-Riemannian space-time precedes discussion of the subject, containing expressions rarely encountered in literature but essentially facilitating the consideration. We give also a comparison of the monad formalism with the other approaches to description of reference frames in general relativity. In three chapters we consider applications of the monad formalism to general relativistic mechanics, electromagnetic and gravitational fields theory. Alongside of the general theory, which includes the monad representation of basic equations of motion of (charged) particles and of fields, several concrete solutions are provided to clarify the physical role and practical application of reference frames (e.g., cases when a rotating electrically charged fluid does not produce any electric field in its co-moving reference frame, or kinematic magnetic charges arise in a rotating frame). The cases are discussed when it is unnecessary to introduce a reference frame, and when such an introduction is essential. Special attention is dedicated to analogy between gravitation, electromagnetism and mechanics (e.g., the dragging phenomenon and existence in the Maxwell equations in rotating frames of terms of the same nature as that of the Coriolis and centrifugal forces). 
  Black holes came into existence together with the universe through the quantum process of pair creation in the inflationary era. We present the instantons responsible for this process and calculate the pair creation rate from the no boundary proposal for the wave function of the universe. We find that this proposal leads to physically sensible results, which fit in with other descriptions of pair creation, while the tunnelling proposal makes unphysical predictions. We then describe how the pair created black holes evolve during inflation. In the classical solution, they grow with the horizon scale during the slow roll-down of the inflaton field; this is shown to correspond to the flux of field energy across the horizon according to the First Law of black hole mechanics. When quantum effects are taken into account, however, it is found that most black holes evaporate before the end of inflation. Finally, we consider the pair creation of magnetically charged black holes, which cannot evaporate. In standard Einstein-Maxwell theory we find that their number in the presently observable universe is exponentially small. We speculate how this conclusion may change if dilatonic theories are applied. 
  I show that all inertial systems are not equivalent, and the Lorentz transformation is not the space-time transformation over two inertial systems moving with relative constant velocity. To do this, I consider imaginary signals travelling over any inertial system K with arbitrarily large velocities. The travelling of an imaginary signal over K is just a time lapse over K. Then I present an example to show that all coordinate systems are not equivalent when the related theory is restricted over tensor-based coordinate transformations, i.e., the genereal relativity principle is not valid. Instead of the relativity principle, I propose the twofold metric principle which may be roughly stated to assert that the set of equations H(v) describing the motion of a material body with velocity v> 0 can be obtained from the corresponding set of equations H(0) for velocity v=0 by replacing, in each differential equation in H(0), each infinitesimal time variable dt with dt / \beta(v) , each maximal velocity-critical infinitesimal length variable dr with beta(v) dr, and each zero velocity-critical infinitesimal length variable dx with dx, where beta(v) = 1/ sqrt {1- v^2/c^2}. By depending on the twofold metric principle and the energy-velocity equation, I derive beta(v)mc^2, the twofold Schwarzshild metric, the centennial procession of planatery orbits and deflection of light. We also present a reason why the Michelson-Morley experiment is observed. Several other topics are also studied. 
  We propose that the Universe created from "nothing" with relatively small particles number and quickly relaxed to quasiequilibrium state at the Planck parameters. The classic cosmological solution for this Universe with Lambda-term has two branches divided by the gap. The quantum process of tunneling between the cosmological solution branches and kinetic of the second order relativistic phase transition in supersymmetric SU(5) model on the GUT scale are investigated by numerical methods. Einstein equations was solved together with the equations of relaxation kinetics. Other quantum geometrodynamics process (the bounce from singularity) and the Wheeler- De Witt equation are investigated also. For the formation of observable particles number the model of the slowly swelling Universe in the result of the multiple reproduction of cosmological cycles is arised naturally. 
  We study transverse-tracefree (TT)-tensors on conformally flat 3-manifolds $(M,g)$. The Cotton-York tensor linearized at $g$ maps every symmetric tracefree tensor into one which is TT. The question as to whether this is the general solution to the TT-condition is viewed as a cohomological problem within an elliptic complex first found by Gasqui and Goldschmidt and reviewed in the present paper. The question is answered affirmatively when $M$ is simply connected and has vanishing 2nd de Rham cohomology. 
  A limiting diagram for the Segre classification in 5-dimensional space-times is obtained, extending a recent work on limits of the energy-momentum tensor in general relativity. Some of Geroch's results on limits of space-times in general relativity are also extended to the context of five-dimensional Kaluza-Klein space-times. 
  We study the collapse of a free scalar field in the Brans-Dicke model of gravity. At the critical point of black hole formation, the model admits two distinctive solutions dependent on the value of the coupling parameter. We find one solution to be discretely self-similar and the other to exhibit continuous self-similarity. 
  As part of an ongoing photometric survey of young Magellanic Cloud clusters we identified Be stars in NGC 1818 and a nearby smaller cluster in the Large Magellanic Cloud. The neighbouring cluster does not contain evolved stars, and its sparsely populated main sequence does not extend to stars as massive as in NGC 1818. Both clusters are younger than the surrounding field population and contain a much higher Be star fraction.   We present a table with photometry, astrometric positions, and IAU-conform identifiers of the Be stars. We compare Be star fractions in Magellanic Cloud clusters and Galactic open clusters and discuss possible constraints on Be star theories. 
  The electromagnetic field of the ultrarelativistic Reissner-Nordstrom solution shows the physically highly unsatisfactory property of a vanishing field tensor but a nonzero, i.e., delta-like, energy density. The aim of this work is to analyze this situation from a mathematical point of view, using the framework of Colombeau's theory of nonlinear generalized functions. It is shown that the physically unsatisfactory situation is mathematically perfectly defined and that one cannot aviod such situations when dealing with distributional valued field tensors. 
  This is a brief review of critical phenomena in gravitational collapse. The conceptual issues are emphasized and some directions for future research are suggested. The paper is not addressed to the experts in the field -- for them little will be new. It is rather meant to introduce others into one of the most rapidly developing areas of research in general relativity with the hope of attracting them into the subject. 
  This is a brief and updated summary of a talk given at the International Conference on Gravitation and Cosmology that took place in Poona in December 1995. It is very brief and is mostly intended as a guide to current literature, or to keep people updated only in very broad terms on the latest developments in the subject. 
  General relativity predicts that energy and momentum conservation laws hold and that preferred frames do not exist. The parametrised post-Newtonian formalism (PPN) phenomenologically quantifies possible deviations from general relativity. The PPN parameter alpha_3 (which identically vanishes in general relativity) plays a dual role in that it is associated both with a violation of the momentum conservation law, and with the existence of a preferred frame. By considering the effects of alpha_3 neq 0 in certain binary pulsar systems, it is shown that alpha_3 < 2.2 x 10^-20 (90% CL). This limit improves on previous results by several orders of magnitude, and shows that pulsar tests of alpha_3 rank (together with Hughes-Drever-type tests of local Lorentz invariance) among the most precise null experiments of physics. 
  We present a theoretical foundation for relativistic astronomical measurements in curved space-time. In particular, we discuss a new iterative approach for describing the dynamics of an astronomical N-body system. To do this, we generalize the Fock-Chandrasekhar method of the weak-field and slow-motion approximation (WFSMA) and develop a theory of relativistic reference frames (RF) for a gravitationally bounded many-extended-body problem. In any proper RF constructed in the immediate vicinity of an arbitrary body, the N-body solutions of the gravitational field equations are presented as a sum of the Riemann-flat inertial space-time, the gravitational field generated by the body itself, the unperturbed solutions for each body in the system transformed to the coordinates of this proper RF, and the gravitational interaction term. We develop the basic concept of a general theory of the celestial RFs applicable to a wide class of metric theories of gravity with an arbitrary model of matter distribution. We apply the proposed method to general relativity. Bodies are described using a perfect fluid model; as such, they possess any number of internal mass and current multipole moments which explicitly characterize their internal structure. The obtained relativistic equations of motion contain an explicit information about the coupling of the bodies proper multiple moments to the surrounding gravitational field. We further generalize the proposed method and include two Eddington parameters $(\gamma,\beta)$. This generalized approach was used to derive the relativistic equations of satellite motion in the vicinity of the extended bodies. We discuss the possible future implementation of the proposed formalism in software codes developed for solar-system orbit determination. 
  The flat inflationary dust universe with matter creation proposed by Prigogine and coworkers is generalized and its dynamical properties are reexamined. It is shown that the starting point of these models depends critically on a dimensionless parameter $\Sigma$, closely related to the matter creation rate $\psi$. For $\Sigma$ bigger or smaller than unity flat universes can emerge, respectively, either like a Big-Bang FRW singularity or as a Minkowski space-time at $t=-\infty$. The case $\Sigma=1$ corresponds to a de Sitter-type solution, a fixed point in the phase diagram of the system, supported by the matter creation process. The curvature effects have also been investigated. The inflating de Sitter is a universal attractor for all expanding solutions regardless of the initial conditions as well as of the curvature parameter. 
  The causality structure of two-dimensional manifolds with degenerate metrics is analysed in terms of global solutions of the massless wave equation. Certain novel features emerge. Despite the absence of a traditional Lorentzian Cauchy surface on manifolds with a Euclidean domain it is possible to uniquely determine a global solution (if it exists), satisfying well defined matching conditions at the degeneracy curve, from Cauchy data on certain spacelike curves in the Lorentzian region. In general, however, no global solution satisfying such matching conditions will be consistent with this data. Attention is drawn to a number of obstructions that arise prohibiting the construction of a bounded operator connecting asymptotic single particle states. The implications of these results for the existence of a unitary quantum field theory are discussed. 
  The observed CMBR dipole is generally interpreted as the consequence of the peculiar motion of the Sun with respect to the reference frame of the CMBR. This article proposes an alternative interpretation in which the observed dipole is the result of isocurvature perturbations on scales larger than the present Hubble radius. These perturbations are produced in the simplest model of double inflation, depending on three parameters. The observed dipole and quadrupole can be explained in this model, while severely constraining its parameters. 
  A nonsingular deflationary cosmology driven by adiabatic matter creation is proposed. In this scenario there is no preinflationary stage as happens in conventional inflationary models. Deflation starts from a de Sitter spacetime characterized by an arbitrary time scale $H_{I}^{-1}$, which also pins down an initial value for the temperature of the universe. The model evolves continuously towards a slightly modified Friedman-Robertson-Walker universe. The horizon and other well known problems of the standard model are then solved but, unlike in microscopic models of inflation, there is no supercooling and subsequent reheating. Entropy generation is concomitant with deflation and if $H_{I}^{-1}$ is of the order of the Planck time, the present day value of the radiation temperature is deduced. It is also shown that the ``age problem'' does not exist here. In particular, the theoretically favored FRW flat model is old enough to agree with the observations even given the high values of $H_{o}$ suggested by recent measurements. 
  The Caushy problem in the LTB model is formulated. The rules of calculating three undetermined functions which defined a solution in the LTB model are presented. One example of exact nonhomogeneous model is studied. The limit transformation to the FRW model is shown. 
  We construct a static axisymmetric wormhole from the gravitational field of two charged shells which are kept in equilibrium by their electromagnetic repulsion. For large separations the exterior tends to the Majumdar-Papapetrou spacetime of two charged particles. The interior of the wormhole is a Reissner-Nordstr\"om black hole matching to the two shells. The wormhole is traversable and connects to the same asymptotics without violation of energy conditions. However, every point in the Majumdar-Papapetrou region lies on a closed timelike curve. 
  Superoscillations can explain the arbitrarily high frequencies' paradox in black hole radiation 
  The rules of calculating three undetermined functions which defined a solution in the LTB model are used to study the class of exact nonhomogene\-ous models with $f^2(\mu) = 1$, $\Lambda = 0$. The parameter $\nu(\mu)$ defined the difference between LTB and FRW models is found out and the limit transformation to the FRW model is shown. The initial conditions are present throught density and Habble function at the moment of time $\tau = 0$. Two criteria of homogeneous of matter distribution are studied. The asimptotic of the present solution for $\tau \rightarrow +\infty$ is studied. 
  Consider any 1-parameter family of initial data such that data with parameter value p > p* form black holes, and data with p < p* do not. As p -> p* from above ("critical collapse"), the black hole mass scales as M ~ (p-p*)^gamma, where the critical exponent gamma is the same for all such families of initial data. So far critical collapse has been investigated only for initial data with zero charge and zero angular momentum. Here we allow for U(1) charge. In scalar electrodynamics coupled to gravity, with action R + |(d + iqA) phi|^2 + F^2, we consider initial data with spherical symmetry and nonvanishing charge. From dimensional analysis and a previous calculation of Lyapunov exponents, we predict that in critical collapse the black hole mass scales as M ~ (p-p*)^gamma, and the black hole charge as Q ~ (p-p*)^delta, with gamma = 0.374 +- 0.001 (as for the real scalar field), and delta = 0.883 +- 0.007. We conjecture that, where there is no mass gap, this behavior generalizes to other charged matter models, with delta \ge 2 gamma. We suggest the existence of universality classes with respect to parameters such as q. 
  The Hawking-Unruh effect of thermal radiance from a black hole or observed by an accelerated detector is usually viewed as a geometric effect related to the existence of an event horizon. Here we propose a new viewpoint, that the detection of thermal radiance in these systems is a local, kinematic effect arising from the vacuum being subjected to a relativistic exponential scale transformation. This kinematic effect alters the relative weight of quantum versus thermal fluctuations (noise) between the two vacua. This approach can treat conditions which the geometric approach cannot, such as systems which do not even have an event horizon. An example is the case of an observer whose acceleration is nonuniform or only asymptotically uniform. Since this approach is based on concepts and techniques of non-equilibrium statistical mechanics, it is more adept to dynamical problems, such as the dissipation, fluctuation, and entropy aspects of particle creation and phase transitions in black hole collapse and in the early universe. 
  In analyzing the nature of thermal radiance experienced by an accelerated observer (Unruh effect), an eternal black hole (Hawking effect) and in certain types of cosmological expansion, one of us proposed a unifying viewpoint that these can be understood as arising from the vacuum fluctuations of the quantum field being subjected to an exponential scale transformation. This viewpoint, together with our recently developed stochastic theory of particle-field interaction understood as quantum open systems described by the influence functional formalism, can be used to address situations where the spacetime possesses an event horizon only asymptotically, or none at all. Examples studied here include detectors moving at uniform acceleration only asymptotically or for a finite time, a moving mirror, and a collapsing mass. We show that in such systems radiance indeed is observed, albeit not in a precise Planckian spectrum. The deviation therefrom is determined by a parameter which measures the departure from uniform acceleration or from exact exponential expansion. These results are expected to be useful for the investigation of non-equilibrium black hole thermodynamics and the linear response regime of backreaction problems in semiclassical gravity. 
  We show that singularities necessarily occur when a boundary of causality violating set exists in a space-time under the physically suitable assumptions except the global causality condition in the Hawking-Penrose singularity theorems. Instead of the global causality condition, we impose some restrictions on the causality violating sets to show the occurrence of singularities. 
  As gravity is a long-range force, it is {\it a priori} conceivable that the Universe's global matter distribution select a preferred rest frame for local gravitational physics. At the post-Newtonian approximation, the phenomenology of preferred-frame effects is described by two parameters, $\alpha_1$ and $\alpha_2$, the second of which is already very tightly constrained. Confirming previous suggestions, we show through a detailed Hill-Brown type calculation of a perturbed lunar orbit that lunar laser ranging data have the potential of constraining $\alpha_1$ at the $10^{-4}$ level. It is found that certain retrograde planar orbits exhibit a resonant sensitivity to external perturbations linked to a fixed direction in space. The lunar orbit being quite far from such a resonance exhibits no significant enhancement due to solar tides. Our Hill-Brown analysis is extended to the perturbation linked to a possible differential acceleration toward the galactic center. It is, however, argued that there are strong {\it a priori} theoretical constraints on the conceivable magnitude of such an effect. 
  A central problem in gravitational wave research is the {\it generation problem}, i.e., the problem of relating the outgoing gravitational wave field to the structure and motion of the material source. This problem has become, in recent years, of increased interest in view of the development of a worldwide network of gravitational wave detectors. We review recent progress in {\it analytical} methods of tackling the gravitational wave generation problem. In particular, we describe recent work in an approach which consists of matching a post-Newtonian expansion of the metric near the material source with a multipolar-post-Minkowskian expansion of the external metric. The results of such analytical methods are important notably for providing accurate theoretical predictions for the most promising targets of the LIGO/VIRGO interferometric network: the ``chirp'' gravitational waveforms emitted during the radiation-reaction-driven inspiral of binary systems of compact objects (neutron stars or black holes). 
  The Teukolsky equation has long been known to lead to divergent integrals when it is used to calculate the gravitational radiation emitted when a test mass falls into a black hole from infinity. Two methods have been used in the past to remove those divergent integrals. In the first, integrations by parts are carried out, and the infinite boundary terms are simply discarded. In the second, the Teukolsky equation is transformed into another equation which does not lead to divergent integrals. The purpose of this paper is to show that there is nothing intrinsically wrong with the Teukolsky equation when dealing with non-compact source terms, and that the divergent integrals result simply from an incorrect choice of Green's function. In this paper, regularization of the Teukolsky equation is carried out in an entirely natural way which does not involve modifying the equation. 
  The confrontation between general relativity (and its theoretically most plausible deviations) and experimental or observational results is summarized. Some discussion is devoted to the various methodologies used in confronting theory and experiment. Both weak-field (solar system) and strong-field (binary pulsar) tests are discussed in detail. A special discussion is devoted to the cosmology of moduli fields, i.e. scalar fields having only gravitational-strength couplings to matter. 
  Part of the theoretical motivation for improving the present level of testing of the equivalence principle is reviewed. The general rationale for optimizing the choice of pairs of materials to be tested is presented. One introduces a simplified rationale based on a trichotomy of competing classes of theoretical models. 
  Self-interacting scalar field configurations which are non-minimally coupled ($\zeta\neq0$) to the gravity of a strictly stationary black hole with non-rotating horizon are studied. It is concluded that for analytical configurations the corresponding domain of outer communications is static. 
  A `no-hair' theorem for the Brans-Dicke field, in the domain of outer communications of a static black hole, is established, when the trace of the energy-momentum tensor, T, is a definite-sign scalar. This implies that Brans-Dicke theory of gravitation, for matter with T of definite sign, is equivalent to Einstein gravity with the constraint T=0. 
  Trans-planckian frequencies can be mimicked outside a black-hole horizon as a tail of an exponentially large amplitude wave that is mostly hidden behind the horizon. The present proposal requires implementing a final state condition. This condition involves only frequencies below the cutoff scale. It may be interpreted as a condition on the singularity. Despite the introduction of the cutoff, the Hawking radiation is restored for static observers. Freely falling observers see empty space outside the horizon, but are "heated" as they cross the horizon. 
  First-order cosmological phase transitions are considered in the models with an $O(3)$-symmetric scalar field, in the high temperature limit. It is shown that a global monopole can be produced at the center of a bubble when the bubble is nucleated. 
  It is well-known that the Einstein-Rosen solutions to the 3+1 dimensional vacuum Einstein's equations are in one to one correspondence with solutions of 2+1 dimensional general relativity coupled to axi-symmetric, zero rest mass scalar fields. We first re-examine the quanization of this midi-superspace paying special attention to the asymptotically flat boundary conditions and to certain functional analytic subtleties associated with regularization. We then use the resulting quantum theory to analyze several conceptual and technical issues of quantum gravity. 
  An investigation of the perturbations of the Reissner-Nordstr\"{o}m black hole in the $N=2$ supergravity is presented. We prove in the extremal limit that the black hole responds to the perturbation of each field in the same manner. We conjecture that we can match the modes of the graviton, gravitino and photon because of supersymmetry transformations. 
  We conjecture (analytically) and demonstrate (numerically) the existence of a fine-structure above the power-law behavior of the mass of black-holes that form in gravitational collapse of spherical massless scalar field. The fine-structure is a periodic function of the critical-separation $(p-p^*)$. We predict that the period $\varpi$ is {\it universal} and that it depends on the previous universal parameters, the critical exponent, $\beta$, and the echoing period $\Delta$ as $\varpi = \Delta /\beta$. 
  A Wheeler-Dewitt quantum constraint operator for four-dimensional, non-perturbative Lorentzian vacuum quantum gravity is defined in the continuum. The regulated Wheeler-DeWitt constraint operator is densely defined, does not require any renormalization and the final operator is anomaly-free and at least symmmetric. The technique introduced here can also be used to produce a couple of other completely well-defined regulated operators including but not exhausting a) the Euclidean Wheeler-DeWitt operator, b)the generator of the Wick rotation transform that maps solutions to the Euclidean Hamiltonian constraint to solutions to the Lorentzian Hamiltonian constraint, c) length operators, d) Hamiltonian operators of the matter sector and e) the generators of the asymptotic Poincar\'e group including the quantum ADM energy. 
  An anomaly-free operator corresponding to the Wheeler-DeWitt constraint of Lorentzian, four-dimensional, canonical, non-perturbative vacuum gravity is constructed in the continuum. This operator is entirely free of factor ordering singularities and can be defined in symmetric and non-symmetric form. We work in the real connection representation and obtain a well-defined quantum theory. We compute the complete solution to the Quantum Einstein Equations for the non-symmetric version of the operator and a physical inner product thereon. The action of the Wheeler-DeWitt constraint on spin-network states is by annihilating, creating and rerouting the quanta of angular momentum associated with the edges of the underlying graph while the ADM-energy is essentially diagonalized by the spin-network states. We argue that the spin-network representation is the ``non-linear Fock representation" of quantum gravity, thus justifying the term ``Quantum Spin Dynamics (QSD)". 
  We continue here the analysis of the previous paper of the Wheeler-DeWitt constraint operator for four-dimensional, Lorentzian, non-perturbative, canonical vacuum quantum gravity in the continuum. In this paper we derive the complete kernel, as well as a physical inner product on it, for a non-symmetric version of the Wheeler-DeWitt operator. We then define a symmetric version of the Wheeler-DeWitt operator. For the Euclidean Wheeler-DeWitt operator as well as for the generator of the Wick transform from the Euclidean to the Lorentzian regime we prove existence of self-adjoint extensions and based on these we present a method of proof of self-adjoint extensions for the Lorentzian operator. Finally we comment on the status of the Wick rotation transform in the light of the present results. 
  We derive a closed formula for the matrix elements of the volume operator for canonical Lorentzian quantum gravity in four spacetime dimensions in the continuum in a spin-network basis. We also display a new technique of regularization which is state dependent but we are forced to it in order to maintain diffeomorphism covariance and in that sense it is natural. We arrive naturally at the expression for the volume operator as defined by Ashtekar and Lewandowski up to a state independent factor. 
  We construct an operator that measures the length of a curve in four-dimensional Lorentzian vacuum quantum gravity. We work in a representation in which a $SU(2)$ connection is diagonal and it is therefore surprising that the operator obtained after regularization is densely defined, does not suffer from factor ordering singularities and does not require any renormalization. We show that the length operator admits self-adjoint extensions and compute part of its spectrum which like its companions, the volume and area operators already constructed in the literature, is purely discrete and roughly is quantized in units of the Planck length. The length operator contains full and direct information about all the components of the metric tensor which faciliates the construction of a new type of weave states which approximate a given classical 3-geometry. 
  We summarize results from a study of spherically symmetric collapse of a {\it charged} (complex) massless scalar-field \cite{Hod}. We present an analytic argument which conjecture the generalization of the mass-scaling relation and echoing phenomena, originally discovered by Choptuik, for the {\it charged} case. Furthermore, we study the behaviour of the self-similar critical solution under {\it external} perturbations -- addition of a cosmological constant $\Lambda$ and a charge-conjugation $e$. Finally, we study the scaling-relation of the black-hole charge. Using an analytic argument we conjecture that black-holes of infinitesimal mass are neutral or obey the relation $Q_{BH} \ll M_{BH}$. We verify our predictions with numerical results. 
  We establish the equality of the ADM mass and the total electric charge for asymptotically flat, static electrovac black hole spacetimes with completely degenerate, not necessarily connected horizon. 
  In this article, a developed, manufactured and tested model of a new type generator is presented, which allows to differentiate and predict, with probability close to 1, coasting characteristics of the rotor during its clockwise and counter-clockwise rotation. An explication of the effect on a base of a new interaction, different from four known ones, is provided. 
  In a previous paper, a bound on the negative energy density seen by an arbitrary inertial observer was derived for the free massless, quantized scalar field in four-dimensional Minkowski spacetime. This constraint has the form of an uncertainty principle-type limitation on the magnitude and duration of the negative energy density. That result was obtained after a somewhat complicated analysis. The goal of the current paper is to present a much simpler method for obtaining such constraints. Similar ``quantum inequality'' bounds on negative energy density are derived for the electromagnetic field, and for the massive scalar field in both two and four-dimensional Minkowski spacetime. 
  We construct the regularized Wheeler--De Witt operator demanding that the algebra of constraints of quantum gravity is anomaly free. We find that for only a small subset of all wavefunctions being integrals of scalar densities this condition can be satisfied. It turns out that the resulting operator is much simpler than the one used in \cite{JK} to find exact solutions of Wheeler--De Witt equation. We proceed to finding exact solutions of quantum gravity and we discuss their interpretation making use of the quantum potential approach to quantum theory. 
  It is well known that Einstein gravity is non-renormalizable; however this does not preclude the existence of a quantum form. 
  We obtain a system for the spatial metric and extrinsic curvature of a spacelike slice that is hyperbolic non-strict in the sense of Leray and Ohya and is equivalent to the Einstein equations. Its characteristics are the light cone and the normal to the slice for any choice of lapse and shift functions, and it admits a well-posed causal Cauchy problem in a Gevrey class of index $\alpha=2$. The system becomes quasidiagonal hyperbolic if we posit a certain wave equation for the lapse function, and we can then relate the results to our previously obtained first order symmetric hyperbolic system for general relativity. 
  The multidimensional gravity on principal bundle with structural SU(2) gauge group is considered. The Lorentzian wormhole solution disposed between two null surfaces is founded in this case. As the nondiagonal components are similar to a gauge fields (in Kaluza - Klein's sense) then in some sense this solution is dual to the black hole in 4D Einstein - Yang - MIlls gravity: 4D black hole has the stationary area outside of event horizon but multidimensional wormhole inside null surface (on which $ds^2=0$). 
  Special multi-cosmic string metrics are analytically extended to describe configurations of Wheeler-Misner wormholes and ordinary cosmic strings. I investigate in detail the case of flat, asymptotically Minkowskian, Wheeler-Misner wormhole spacetimes generated by two cosmic strings, each with tension $-1/4G$. 
  We respond to a recent paper by Rindler on the ``Anti--Machian'' nature of the Lense--Thirring effect. We remark that his conclusion depends crucially on the particular formulation of Mach's principle used. 
  We present a general framework for understanding and analyzing critical behaviour in gravitational collapse. We adopt the method of renormalization group, which has the following advantages. (1) It provides a natural explanation for various types of universality and scaling observed in numerical studies. In particular, universality in initial data space and universality for different models are understood in a unified way. (2) It enables us to perform a detailed analysis of time evolution beyond linear perturbation, by providing rigorous controls on nonlinear terms. Under physically reasonable assumptions we prove: (1) Uniqueness of the relevant mode around a fixed point implies universality in initial data space. (2) The critical exponent $\beta_{BH}$ and the unique positive eigenvalue $\kappa$ of the relevant mode is exactly related by $\beta_{BH} = \beta /\kappa$, where $\beta$ is a scaling exponent. (3) The above (1) and (2) hold also for discretely self-similar case (replacing ``fixed point'' with ``limit cycle''). (4) Universality for diffent models holds under a certain condition.   According to the framework, we carry out a rather complete (though not mathematically rigorous) analysis for perfect fluids with pressure proportional to density, in a wide range of the adiabatic index $\gamma$. The uniqueness of the relevant mode around a fixed point is established by Lyapunov analyses. This shows that the critical phenomena occurs not only for the radiation fluid but also for perfect fluids with $1 < \gamma \lesssim 1.88$. The accurate values of critical exponents are calculated for the models. 
  As is well known, structure formation in the Universe at times after decoupling can be described by hydrodynamic equations. These are shown here to be equivalent to a generalization of the stochastic Kardar--Parisi--Zhang equation with time-- dependent viscosity in epochs of dissipation. As a consequence of the Dynamical Critical Scaling induced by noise and fluctuations, these equations describe the fractal behavior (with a scale dependent fractal dimension) observed at the smaller scales for the galaxy--to--galaxy correlation function and $also$ the Harrison--Zel'dovich spectrum at decoupling. By a Renormalization Group calculation of the two--point correlation function between galaxies in the presence of (i) the expansion of the Universe and (ii) non--equilibrium, we can account, from first principles, for the main features of the observed shape of the power spectrum. 
  By considering charged black hole solutions of a one parameter family of two dimensional dilaton gravity theories, one finds the existence of quantum mechanically stable gravitational kinks with a simple mass to charge relation. Unlike their Einsteinian counterpart (i.e. extreme Reissner-Nordstr\"om), these have nonvanishing horizon surface gravity. 
  In this paper we analyse the stability of black hole Cauchy horizons arising in a class of 2d dilaton gravity models. It is shown that due to the characteristic asymptotic Rindler form of the metric of these models, time dependent gravitational perturbations generated in the external region do not necessarily blow-up when propagated along the Cauchy horizon. There exists, in fact, a region of nonzero measure in the space of the parameters characterizing the solutions such that both instability and mass inflation are avoided. This is a new result concerning asymptotically flat space-times, not shared by the well-known solutions of General Relativity. Despite this fact, however, quantum back-reaction seems to produce a scalar curvature singularity there. 
  The Gauss constraint in the extended loop representation for quantum gravity is studied. It is shown that there exists a sector of the state space that is rigorously gauge invariant without the generic convergence issues of the extended holonomies. 
  The quantum cosmological version of the multidimensional Einstein-Yang-Mills model in a $R \times S^3 \times S^d$ topology is studied in the framework of the Hartle-Hawking proposal. In contrast to previous work in the literature, we consider Yang-Mills field configurations with non-vanishing time-dependent components in both $S^3$ and $S^d$ spaces. We obtain stable compactifying solutions that do correspond to extrema of the Hartle-Hawking wave function of the Universe. Subsequently, we also show that the regions where 4-dimensional metric behaves classically or quantum mechanically (i.e. regions where the metric is Lorentzian or Euclidean) will depend on the number, $d$, of compact space dimensions. 
  We present an exact solution to the problem of the relativistic motion of 2 point masses in $(1+1)$ dimensional dilaton gravity. The motion of the bodies is governed entirely by their mutual gravitational influence, and the spacetime metric is likewise fully determined by their stress-energy. A Newtonian limit exists, and there is a static gravitational potential. Our solution gives the exact Hamiltonian to infinite order in the gravitational coupling constant. 
  It is argued that when the dimension of space is a constant integer the full set of Einstein's field equations has more information than the spatial components of Einstein's equation plus the energy conservation law. Applying the former approach to the decrumpling FRW cosmology recently proposed, it is shown that the spacetime singularity cannot be avoided and that turning points are absent. This result is in contrast to the decrumpling nonsingular spacetime model with turning points previously obtained using the latter approach. 
  In order to understand the physical effect of the brick wall boundary condition, we compute the distribution of the zero-point energy of the massless scalar fields minimally coupled to the Schwarzschild and Reissner-Nordstr\"{o}m black hole backgrounds. We find that the black hole radiation spectrum depends on the positions of the brick wall and the observer, and reveals the interference effect due to the reflected field by the brick wall. 
  We elaborate the renormalization process of entropy of a nonextremal and an extremal Reissner-Nordstr\"{o}m black hole by using the Pauli-Villars regularization method, in which the regulator fields obey either the Bose-Einstein or Fermi-Dirac distribution depending on their spin-statistics. The black hole entropy involves only two renormalization constants. We also discuss the entropy and temperature of the extremal black hole. 
  An approach to quantum cosmology, relying on strengths of both canonical and path integral formalisms, is applied to the cosmological model, Bianchi type IX. Physical quantum states are constructed on the maximal slice of the cosmological history. A path integral is derived which evolves observables off the maximal slice. This result is compared a path integral propagator derived earlier with conventional Faddeev-Poppov gauge fixing. 
  Gamma ray Bursts (GRBs) - short bursts of few hundred keV $\gamma$-rays - have fascinated astronomers since their accidental discovery in the sixties. GRBs were ignored by most relativists who did not expect that they are associated with any relativistic phenomenon. The recent observations of the BATSE detector on the Compton GRO satellite have revolutionized our ideas on these bursts and the picture that emerges shows that GRBs are the most relativistic objects discovered so far. 
  If one assumes the validity of conventional quantum field theory in the vicinity of the horizon of a black hole, one does not find a quantum mechanical description of the entire black hole that even remotely resembles that of conventional forms of matter; in contrast with matter made out of ordinary particles one finds that, even if embedded in a finite volume, a black hole would be predicted to have a strictly continuous spectrum.   Dissatisfied with such a result, which indeed hinges on assumptions concerning the horizon that may well be wrong, various investigators have now tried to formulate alternative approaches to the problem of ``quantizing" the black hole. We here review the approach based on the assumption of quantum mechanical purity and unitarity as a starting point, as has been advocated by the present author for some time, concentrating on the physics of the states that should live on a black hole horizon. The approach is shown to be powerful in not only producing promising models for the quantum black hole, but also new insights concerning the dynamics of physical degrees of freedom in ordinary flat space-time. 
  The form of Maxwell's theory is well known in the framework of general relativity, a fact that is related to the applicability of the principle of equivalence to electromagnetic phenomena. We pose the question whether this form changes if torsion and/or nonmetricity fields are allowed for in spacetime. Starting from the conservation laws of electric charge and magnetic flux, we recognize that the Maxwell equations themselves remain the same, but the constitutive law must depend on the metric and, additionally, may depend on quantities related to torsion and/or nonmetricity. We illustrate our results by putting an electric charge on top of a spherically symmetric exact solution of the metric-affine gauge theory of gravity (comprising torsion and nonmetricity). All this is compared to the recent results of Vandyck. 
  I discuss a method for obtaining the one-loop quantum corrections to the tree-level entropy for a charged Kerr black hole. Divergences which appear can be removed by renormalization of couplings in the tree-level gravitational action in a manner similar to that for a static black hole. 
  1. Introduction   2. Basics and outline of the method     2.1 Regions around an isolated source      2.2 The quadrupole formalism      2.3 Outline of the method   3. Properties of radiative gravitational fields     3.1 The field in the exterior domain (D_e)     3.2 Computing the field nonlinearities in (D_e)      3.3 The field in the far wave zone D_w      3.4 Nonlinear effects in the far-zone radiation field   4. Generation of gravitational waves     4.1 Post-Newtonian iteration of the field in D_i      4.2 The multipole moments as integrals over the source   5. Gravitational radiation reaction effects   6. Application to inspiralling compact binaries 
  Lecture given at ``Raychaudhuri session," ICGC-95 conference, Pune (India), December 1995.  Contents:  I. Introduction  II. (2+1)-Dimensional Initial Values in Stereographic Projection  A. The single, non-rotating black hole  B. Non-rotating multi-black-holes  C. ``Black Hole" Universe Initial Values  III. Time Development: Enter the Raychaudhuri Equation  IV. Time Development in Stereographic Projection  A. The BTZ black hole spacetime  B. Multi-Black-Hole Spacetimes  V. Black Holes with Angular Momentum  VI. Analogous 3+1-Dimensional Black Holes  VII. Conclusions 
  The Ernst spacetime is a solution of the Einstein-Maxwell equations describing two charged black holes accelerating apart in a uniform electric (or magnetic) field. As the field approaches a critical value, the black hole horizon appears to touch the acceleration horizon. We show that weak cosmic censorship cannot be violated by increasing the field past this critical value: The event horizon remains intact. On the other hand, strong cosmic censorship does appear to be violated in this spacetime: For a certain range of parameters, we find evidence that the inner horizon is classically stable. 
  We develop the idea that, as a result of the arbitrariness of the factor ordering in Wheeler-DeWitt equation, gauge phases can not, in general, being completely removed from the wave functional in quantum gravity. The latter may be conveniently described by means of a remnant complex term in WDW equation depending of the factor ordering. Taking this equation for granted we can obtain WKB complex solutions and, therefore, we should be able to derive a semiclassical time parameter for the Schroedinger equation corresponding to matter fields in a given classical curved space. 
  We assume that a self-gravitating thin string can be locally described by what we shall call a smoothed cone. If we impose a specific constraint on the model of the string, then its central line obeys the Nambu-Goto equations. If no constraint is added, then the worldsheet of the central line is a totally geodesic surface. 
  Two arguments for the quantization of entropy for black holes in generic 2-D dilaton gravity are summarized. The first argument is based on reduced quantization of the only physical observables in the theory, namely the black hole mass and its conjugate momentum, the Killing time separation. The second one uses the exact physical mass eigenstates for Euclidean black holes found via Dirac quantization. Both methods give the same spectrum: the black hole entropy must be quantized $S= 2\pi n/G$. 
  By writing the complete set of $3 + 1$ (ADM) equations for linearized waves, we are able to demonstrate the properties of the initial data and of the evolution of a wave problem set by Alcubierre and Schutz. We show that the gauge modes and constraint error modes arise in a straightforward way in the analysis, and are of a form which will be controlled in any well specified convergent computational discretization of the differential equations. 
  In a recent paper, general solutions for the vacuum wave functionals in the Schrodinger picture were given for a variety of classes of curved spacetimes. Here, we describe a number of simple examples which illustrate how the presence of spacetime boundaries influences the vacuum wave functional and how physical quantities are independent of the choice of spacetime foliation used in the Schrodinger approach despite the foliation dependence of the wave functionals themselves. 
  A general expression is given for the quintic Lovelock tensor as well as for the coefficient of the quintic Lovelock Lagrangian in terms of the Riemann-Christoffel and Ricci curvature tensors and the Riemann curvature scalar for n-dimensional differentiable manifolds having a general linear connection. 
  We examine the interactions of a black hole with a massless scalar field using a coordinate system which extends ingoing Eddington-Finkelstein coordinates to dynamic spherically symmetric-spacetimes. We avoid problems with the singularity by excising the region of the black hole interior to the apparent horizon. We use a second-order finite difference scheme to solve the equations. The resulting program is stable and convergent and will run forever without problems. We are able to observe quasi-normal ringing and power-law tails as well an interesting nonlinear feature. 
  Point sources in (2+1)-dimensional gravity are conical singularities that modify the global curvature of the space giving rise to self-interaction effects on classical fields. In this work we study the electrostatic self-interaction of a point charge in the presence of point masses in (2+1)-dimensional gravity with a cosmological constant. 
  The effective action for string theory which takes into account non-minimal coupling of moduli admits multi-black hole solutions. The euclidean continuation of these solutions can be interpreted as an instanton mediating the splitting and recombination of the throat of extremal magnetically charged black holes. 
  A set of Maple V R.3/4 computer algebra routines for the analytical solving of 1st. order ODEs, using Lie group symmetry methods, is presented. The set of commands includes a 1st. order ODE-solver and routines for, among other things: the explicit determination of the coefficients of the infinitesimal symmetry generator; the construction of the most general invariant 1st. order ODE under given symmetries; the determination of the canonical coordinates of the underlying invariant group; and the testing of the returned results. 
  Motions with respect to one inertial (or ``map'') frame are often described in terms of the coordinate time/velocity pair (or ``kinematic'') of the map frame itself. Since not all observers experience time in the same way, other time/velocity pairs describe map-frame trajectories as well. Such coexisting kinematics provide alternate variables to describe acceleration. We outline a general strategy to examine these. For example, Galileo's acceleration equations describe unidirectional relativistic motion {\it exactly} if one uses $V=\frac{dx}{dT}$, where $x$ is map-frame position and $T$ is clock time in a chase plane moving such that $\gamma ^{\prime }=\gamma \sqrt{\frac 2{\gamma +1}}$. Velocity in the traveler's kinematic, on the other hand, has dynamical and transformational properties which were lost by coordinate-velocity in the transition to Minkowski space-time. Its repeated appearance with coordinate time, when expressing relationships in simplest form, suggests complementarity between traveler and coordinate kinematic views. 
  In quantum field theory there is now a well developed technique, effective field theory, which allows one to obtain low energy quantum predictions in ``non-renormalizable'' theories, using only the degrees of freedom and interactions appropriate for those energies. Whether or not general relativity is truly fundamental, at low energies it is automatically described as a quantum effective field theory and this allows a consistent framework for quantum gravity at ordinary energies. I briefly describe the nature and limits of the technique. 
  A classical model for the interior structure of a Schwarzshild black hole which consists in creating multiple de Sitter universes with lightlike boundaries is proposed.The interaction of the boundaries is studied and a scenario leading to disconnected de Sitter universes is described. 
  Numerical study of order parameter dynamics in the course of second order (Landau-Ginzburg) symmetry breaking transitions shows that the density of topological defects, kinks, is proportional to the fourth root of the rate of the quench. This confirms the more general theory of domain-size evolution in the course of symmetry breaking transformations proposed by one of us \cite{Zurek85}. Using these ideas, it is possible to compute the density of topological defects from the quench timescale and from the equilibrium scaling of the correlation length and relaxation time near the critical point. 
  The special relativistic test theory of Mansouri and Sexl is sketched. Theories based on different clock synchronisations are found to be equivalent to special relativity, as regards experimental results. The conventionality of clock synchronisation is shown not to hold, by means of an example, in a simple accelerated system and through the principle of equivalence in gravitational fields, especially when the metric is not static. Experimental implications on very precise clock synchronisation on earth are discussed. 
  In the last two decades, theories explaining the same experiments as well as special relativity does, were developed by using different synchronization procedures. All of them are ether-like theories. Most authors believe these theories to be equivalent to special relativity, but no general proof was ever brought. By means of a Gedankenexperiment on light aberration, we produce strong evidence that this is the case for experiments made in inertial systems. 
  We show that the reality conditions to be imposed on Ashtekar variables to recover real gravity can be implemented as second class constraints a la Dirac. Thus, counting gravitational degrees of freedom follows accordingly. Some constraints of the real theory turn out to be non-polynomial, regardless of the form, polynomial or non-polynomial, taken for the reality conditions. We comment upon the compatibility of our approach with the recently proposed Wick transform point of view, as well as on some alternatives for dealing with such second class constraints. 
  We generalize previous \cite{AiBa2} work on the classification of ($C^\infty$) symmetries of plane-fronted waves with an impulsive profile. Due to the specific form of the profile it is possible to extend the group of normal-form-preserving diffeomorphisms to include non-smooth transformations. This extension entails a richer structure of the symmetry algebra generated by the (non-smooth) Killing vectors. 
  We define a new parameter `cumulative drag index' for a particle in circular orbit in a stationary, axisymmetric gravitational field and study its behaviour in the two well known solutions of general relativity {\it viz.}, the Kerr spacetime and the G\"odel spacetime, wherein the inertial frame dragging has an important role. As it shows similar behaviour for both co and counter rotating particles, it may indeed be an indication of the influence of the faraway universe on local physics and thus Machian. 
  Clock synchronisation is conventional when inertial systems are involved. This statement is no longer true in accelerated systems. A demonstration is given in the case of a rotating platform. We conclude that theories based on the Einstein's clock synchronisation procedure are unable to explain, for example, the Sagnac effect on the platform. Implications on very precise clock synchronisation on earth are discussed. 
  The stress-energy tensor of a quantized scalar field is computed in the reduced two-dimensional charged dilatonic black hole spacetime of Garfinkle, Horowitz, and Strominger. In order for the stress-energy of quantized fields to be regular on the event horizon in both the extreme string metric and the conformally associated physical metric, it is necessary to assign a nonzero temperature, T = (8 pi e^{phi_0} M)^{-1}, to the extreme string metric, contrary to the expectation that this horizonless spacetime would have a natural temperature of zero. 
  Assuming a cellular structure for the space-time, we propose a model in which the expansion of the universe is understood as a decrumpling process, much like the one we know from polymeric surfaces. The dimension of space is then a dynamical real variable. The generalized Friedmann equation, derived from a Lagrangian, and the generalized equation of continuity for the matter content of the universe, give the dynamics of our model universe. This leads to an oscillatory non-singular model with two turning points for the dimension of space. 
  The basic ingredients of the `consistent histories' approach to quantum theory are a space $\UP$ of `history propositions' and a space $\D$ of `decoherence functionals'. In this article we consider such history quantum theories in the case where $\UP$ is given by the set of projectors $\P(\V)$ on some Hilbert space $\V$. Using an analogue of Wigner's Theorem in the context of history quantum theories proven earlier, we develop the notion of a `symmetry of a decoherence functional' and prove that all such symmetries form a group which we call `the symmetry group of a decoherence functional'. We calculate---for the case of history quantum mechanics---some of these symmetries explicitly and relate them to some discussions that have appeared previously. 
  The basic ingredients of the `consistent histories' approach to quantum theory are a space $\UP$ of `history propositions' and a space $\D$ of `decoherence functionals'. In this article we consider such history quantum theories in the case where $\UP$ is given by the set of projectors $\P(\V)$ on some Hilbert space $\V$. We define the notion of a `physical symmetry of a history quantum theory' (PSHQT) and specify such objects exhaustively with the aid of an analogue of Wigner's theorem. In order to prove this theorem we investigate the structure of $\D$, define the notion of an `elementary decoherence functional' and show that each decoherence functional can be expanded as a certain combination of these functionals. We call two history quantum theories that are related by a PSHQT `physically equivalent' and show explicitly, in the case of history quantum mechanics, how this notion is compatible with one that has appeared previously. 
  We investigate the instability of the Cauchy horizon caused by causality violation in the compact vacuum universe with the topology $B\times {\bf S}^{1}\times {\bf R}$, which Moncrief and Isenberg considered. We show that if the occurrence of curvature singularities are restricted to the boundary of causality violating region, the whole segments of the boundary become curvature singularities. This implies that the strong cosmic censorship holds in the spatially compact vacuum space-time in the case of the causality violation. This also suggests that causality violation cannot occur for a compact universe. 
  The metric for plane gravitational waves is quantized within the Hamiltonian framework, using a Dirac constraint quantization and the self-dual field variables proposed by Ashtekar. The z axis (direction of travel of the waves) is taken to be the entire real line rather than the torus (manifold coordinatized by (z,t) is RxR rather than $S_1$ x R). Solutions to the constraints proposed in a previous paper involve open-ended flux lines running along the entire z axis, rather than closed loops of flux; consequently, these solutions are annihilated by the Gauss constraint at interior points of the z axis, but not at the two boundary points. The solutions studied in the present paper are based on closed flux loops and satisfy the Gauss constraint for all z. 
  We present detailed calculations of the quasinormal modes of Reissner-Nordstrom black holes. While the first few, slowly damped, modes depend on the charge of the black hole in a relatively simple way, we find that the rapidly damped modes show several peculiar features. The higher modes generally spiral into the value for the extreme black hole as the charge increases. We also discuss the possible existence of a purely imaginary mode for the Schwarzschild black hole: Our data suggest that there is a quasinormal mode that limits to $\omega M = -2i$ as $Q\to 0$. 
  We calculate the statistical mechanical entropy associated with boundary terms in the two-dimensional Euclidean black holes in deSitter gravity. 
  We provide physical interpretation for the four parameters of the stationary Lewis metric restricted to the Weyl class. Matching this spacetime to a completely anisotropic, rigidly rotating, fluid cilinder, we obtain from the junction conditions that one of these parameters is proportional to the vorticity of the source. From the Newtonian approximation a second parameter is found to be proportional to the energy per unit of length. The remaining two parameters may be associated to a gravitational analog of the Aharanov-Bohm effect. We prove, using the Cartan scalars, that the Weyl class metric and static Levi-Civita metric are locally equivalent, i.e., indistinguishable in terms of its curvature. 
  The physical and geometrical meaning of the four parameters of Lewis metric for the Lewis class are investigated. Matching this spacetime to a completely anisotropic, rigidly rotating, fluid cylinder, we obtain from the junction conditions that the four parameters are related to the vorticity of the source. Furthermore it is shown that one of the parameters must vanish if one wishes to reduce the Lewis class to a locally static spacetime. Using the Cartan scalars it is shown that the Lewis class does not include globally Minkowski as special class, and that it is not locally equivalent to the Levi-Civita metric. Also it is shown that, in contrast with the Weyl class, the parameter responsible for the vorticity appears explicitly in the expression for the Cartan scalars. Finally, to enhance our understanding of the Lewis class, we analyse the van Stockum metric. 
  We consider two exact solutions of Einstein's field equations corresponding to a cylinder of dust with net zero angular momentum. In one of the cases, the dust distribution is homogeneous, whereas in the other, the angular velocity of dust particles is constant [1]. For both solutions we studied the junction conditions to the exterior static vacuum Levi-Civita spacetime. From this study we find an upper limit for the energy density per unit length $\sigma$ of the source equal ${1\over 2}$ for the first case and ${1\over 4}$ for the second one. Thus the homogeneous cluster provides another example [2] where the range of $\sigma$ is extended beyond the limit value ${1\over 4}$ previously found in the literature [3,4]. Using the Cartan Scalars technics we show that the Levi-Civita spacetime gets an extra symmetry for $\sigma={1\over 2}$ or ${1\over 4}$. We also find that the cluster of homogeneous dust has a superior limit for its radius, depending on the constant volumetric energy density $\rho_0$. 
  Recently, we presented a unified way of analysing classical cosmological perturbation in generalized gravity theories. In this paper, we derive the perturbation spectrums generated from quantum fluctuations again in unified forms. We consider a situation where an accelerated expansion phase of the early universe is realized in a particular generic phase of the generalized gravity. We take the perturbative semiclassical approximation which treats the perturbed parts of the metric and matter fields as quantum mechanical operators. Our generic results include the conventional power-law and exponential inflations in Einstein's gravity as special cases. 
  A method has been recently proposed for defining an arbitrary number of differential calculi over a given noncommutative associative algebra. As an example a version of quantized space-time is considered here. It is found that there is a natural differential calculus using which the space-time is necessarily flat Minkowski space-time. Perturbations of this calculus are shown to give rise to non-trivial gravitational fields. 
  A brief summary of results on homotheties in General Relativity is given, including general information about space-times admitting an r-parameter group of homothetic transformations for r>2, as well as some specific results on perfect fluids. Attention is then focussed on inhomogeneous models, in particular on those with a homothetic group $H_4$ (acting multiply transitively) and $H_3$. A classification of all possible Lie algebra structures along with (local) coordinate expressions for the metric and homothetic vectors is then provided (irrespectively of the matter content), and some new perfect fluid solutions are given and briefly discussed. 
  The asymptotic behaviour of a class of inhomogeneous scalar field cosmologies with a Liouville type of potential is studied. We define a set of new variables for which the phase space of the system of Einstein equations is bounded. This allows us to perform a complete analysis of the evolution of these cosmologies. We also discuss the extension of the cosmic no-hair theorem. 
  We consider the action principle to derive the classical, relativistic motion of a self-interacting particle in a 4-D Lorentzian spacetime containing a wormhole and which allows the existence of closed time-like curves. In particular, we study the case of a pointlike particle subject to a `hard-sphere' self-interaction potential and which can traverse the wormhole an arbitrary number of times, and show that the only possible trajectories for which the classical action is stationary are those which are globally self-consistent. Generically, the multiplicity of these trajectories (defined as the number of self-consistent solutions to the equations of motion beginning with given Cauchy data) is finite, and it becomes infinite if certain constraints on the same initial data are satisfied. This confirms the previous conclusions (for a non-relativistic model) by Echeverria, Klinkhammer and Thorne that the Cauchy initial value problem in the presence of a wormhole `time machine' is classically `ill-posed' (far too many solutions). Our results further extend the recent claim by Novikov et al. that the `Principle of self-consistency' is a natural consequence of the `Principle of minimal action.' 
  We consider the initial value problem for a massless scalar field in the Schwarzschild geometry. When constructed using a complex-frequency approach the necessary Green's function splits into three components. We discuss all of these in some detail: 1) The contribution from the singularities (the quasinormal modes of the black hole) is approximated and the mode-sum is demonstrated to converge after a certain well defined time in the evolution. A dynamic description of the mode-excitation is introduced and tested. 2) It is shown how a straightforward low-frequency approximation to the integral along the branch cut in the black-hole Green's function leads to the anticipated power-law fall off at very late times. We also calculate higher order corrections to this tail and show that they provide an important complement to the leading order. 3) The high-frequency problem is also considered. We demonstrate that the combination of the obtained approximations for the quasinormal modes and the power-law tail provide a complete description of the evolution at late times. Problems that arise (in the complex-frequency picture) for early times are also discussed, as is the fact that many of the presented results generalize to, for example, Kerr black holes. 
  A review is made of recent efforts to define linear connections and their corresponding curvature within the context of noncommutative geometry. As an application it is suggested that it is possible to identify the gravitational field as a phenomenological manifestation of space-time commutation relations and to thereby clarify its role as an ultraviolet regularizer. 
  We present a gauge--theoretical derivation of the notion of time, suitable to describe the Hamiltonian time evolution of gravitational systems. It is based on a nonlinear coset realization of the Poincar\'e group, implying the time component of the coframe to be invariant, and thus to represent a metric time. The unitary gauge fixing of the boosts gives rise to the foliation of spacetime along the time direction. The three supressed degrees of freedom correspond to Goldstone--like fields, whereas the remaining time component is a Higgs--like boson. 
  For a canonical formulation of quantum gravity, the superspace of all possible 3-geometries on a Cauchy hypersurface of a 3+1-dimensional Lorentzian manifold plays a key role. While in the analogous 2+1-dimensional case the superspace of all Riemannian 2-geometries is well known, the structure of the superspace of all Riemannian 3-geometries has not yet been resolved at present.   In this paper, an important subspace of the latter is disentangled: The superspace of local homogenous Riemannian 3-geometries. It is finite dimensional and can be factored by conformal scale dilations, with the flat space as the center of projection. The corresponding moduli space can be represented by homothetically normalized 3-geometries.   By construction, this moduli space of the local homogenous 3-geometries is an algebraic variety. An explicit parametrization is given by characteristic scalar invariants of the Riemannian 3-geometry.   Although the moduli space is not locally Euclidean, it is a Hausdorff space. Nevertheless, its topology is compatible with the non-Hausdorffian topology of the space of all Bianchi-Lie algebras, which characterize the moduli modulo differences in their anisotropy. 
  By correlating the signals from a pair of gravitational-wave detectors, one can undertake sensitive searches for a stochastic background of gravitational radiation. If the stochastic background is anisotropic, then this correlated signal varies harmonically with the earth's rotation. We calculate how the harmonics of this varying signal are related to the multipole moments which characterize the anisotropy, and give a formula for the signal-to-noise ratio of a given harmonic. The specific case of the two LIGO (Laser Interferometric Gravitational Observatory) detectors, which will begin operation around the year 2000, is analyzed in detail. We consider two possible examples of anisotropy. If the gravitational-wave stochastic background contains a dipole intensity anisotropy whose origin (like that of the Cosmic Background Radiation) is motion of our local system, then that anisotropy will be observable by the advanced LIGO detector (with 90% confidence in one year of observation) if \Omega_{gw} > 5.3 \times 10^{-8} h_{100}^{-2}. We also study the signal produced by stochastic sources distributed in the same way as the luminous matter in the galactic disk, and in the same way as the galactic halo. The anisotropy due to sources distributed as the galactic disk or as the galactic halo will be observable by the advanced LIGO detector (with 90% confidence in one year of observation) if \Omega_{gw} > 1.8 \times 10^{-10} h_{100}^{-2} or \Omega_{gw} > 6.7 \times 10^{-8} h_{100}^{-2}, respectively. 
  A major problem in the consistent-histories approach to quantum theory is contending with the potentially large number of consistent sets of history propositions. One possibility is to find a scheme in which a unique set is selected in some way. However, in this paper we consider the alternative approach in which all consistent sets are kept, leading to a type of `many world-views' picture of the quantum theory. It is shown that a natural way of handling this situation is to employ the theory of varying sets (presheafs) on the space $\B$ of all Boolean subalgebras of the orthoalgebra $\UP$ of history propositions. This approach automatically includes the feature whereby probabilistic predictions are meaningful only in the context of a consistent set of history propositions. More strikingly, it leads to a picture in which the `truth values', or `semantic values' of such contextual predictions are not just two-valued (\ie true and false) but instead lie in a larger logical algebra---a Heyting algebra---whose structure is determined by the space $\B$ of Boolean subalgebras of $\UP$. 
  In the spirit of Sakharov's `metric elasticity' proposal, we draw a loose analogy between general relativity and the hydrodynamic state of a quantum gas. In the `top-down' approach, we examine the various conditions which underlie the transition from some candidate theory of quantum gravity to general relativity. Our emphasis here is more on the `bottom-up' approach, where one starts with the semiclassical theory of gravity and examines how it is modified by graviton and quantum field excitations near and above the Planck scale. We mention three aspects based on our recent findings: 1) Emergence of stochastic behavior of spacetime and matter fields depicted by an Einstein-Langevin equation. The backreaction of quantum fields on the classical background spacetime manifests as a fluctuation-dissipation relation. 2) Manifestation of stochastic behavior in effective theories below the threshold arising from excitations above. The implication for general relativity is that such Planckian effects, though exponentially suppressed, is in principle detectable at sub-Planckian energies. 3) Decoherence of correlation histories and quantum to classical transition. From Gell-Mann and Hartle's observation that the hydrodynamic variables which obey conservation laws are most readily decohered, one can, in the spirit of Wheeler, view the conserved Bianchi identity obeyed by the Einstein tensor as an indication that general relativity is a hydrodynamic theory of geometry. Many outstanding issues surrounding the transition to general relativity are of a nature similar to hydrodynamics and mesoscopic physics. 
  The pair creation of black holes with event horizons of non-trivial topology is described. The spacetimes are all limiting cases of the cosmological $C$ metric. They are generalizations of the $(2+1)$ dimensional black hole and have asymptotically anti de Sitter behaviour. Domain walls instantons can mediate their pair creation for a wide range of mass and charge. 
  We provide a detailed analysis of Friedmann-Robertson-Walker universes in a wide range of scalar-tensor theories of gravity. We apply solution-generating methods to three parametrised classes of scalar-tensor theory which lead naturally to general relativity in the weak-field limit. We restrict the parameters which specify these theories by the requirements imposed by the weak-field tests of gravitation theories in the solar system and by the requirement that viable cosmological solutions be obtained. We construct a range of exact solutions for open, closed, and flat isotropic universes containing matter with equation of state $p\leq \frac{1}{3}\rho$ and in vacuum. We study the range of early and late-time behaviours displayed, examine when there is a `bounce' at early times, and expansion maxima in closed models. 
  Classical mechanics and standard Copenhagen quantum mechanics respect subspace implications. For example, if a particle is confined in a particular region $R$ of space, then in these theories we can deduce that it is confined in regions containing $R$. However, subspace implications are generally violated by versions of quantum theory that assign probabilities to histories, such as the consistent histories approach. I define here a new criterion, ordered consistency, which refines the criterion of consistency and has the property that inferences made by ordered consistent sets do not violate subspace relations. This raises the question: do the operators defining our observations form an ordered consistent history? If so, ordered consistency defines a version of quantum theory with greater predictive power than the consistent histories formalism. If not, and our observations are defined by a non-ordered consistent quantum history, then subspace implications are not generally valid. 
  Spherically symmetric solutions in Brans-Dicke theory of relativity with zero coupling constant, $\omega=0$, are derived in the Schwarzschild line-element. The solutions are obtained from a cubic transition equation with one small parameter. The exterior space-time of one family of solutions is arbitrarily close to the exterior Schwarzschild space-time. These nontopological solitons have some similarity with soliton stars, and are proposed as candidates for {\em approximate black holes} for the use in numerical relativity, in particular for treatment of horizon boundary conditions. 
  Locations and orientations of current and proposed laser-interferometric gravitational wave detectors are given in tabular form. 
  We consider particle trajectories in the gravitational field of an impulsive pp-wave. Due to the distributional character of the wave profile one inevitably encounters an ambiguous point value $\theta(0)$. We show that this ambiguity may be resolved by imposing covariant constancy of the square of the tangent. Our result is consistent with Colombeau's multiplication of distributions. 
  Scalar cosmological perturbations of a weakly self-interacting plasma mixed with a perfect radiation fluid are investigated. Effects of this plasma are considered through order $\lambda^{3/2}$ of perturbative thermal-field-theory in the radiation dominated universe. The breakdown of thermal perturbation theory at vastly subhorizon scales is circumvented by a Pad\'e approximant solution. Compared to collisionless plasmas the phase speed and subhorizon damping of the plasma density perturbations are changed. An example for a self-interacting thermal field is provided by the neutrinos with effective 4-fermion interactions. 
  The exact solution in the LTB model with $f^2 = 1$, $\Lambda \ne 0$ is studied. The initial conditions for the metrical function and its derivatives generate the solution with complicated structure including the solutions like "stripping of the shell", "collapce" and "core", or "accretion". In the limit of big time the solution allows the constant Hubble function and the density, depending on time. The transformation to the FRW model is shown. Three pictures are available by e-mail. 
  We extend the analysis of black hole pair creation to include non- orientable instantons. We classify these instantons in terms of their fundamental symmetries and orientations. Many of these instantons admit the pin structure which corresponds to the fermions actually observed in nature, and so the natural objection that these manifolds do not admit spin structure may not be relevant. Furthermore, we analyse the thermodynamical properties of non-orientable black holes and find that in the non-extreme case, there are interesting modifications of the usual formulae for temperature and entropy. 
  The double series approximation method of Bonnor is a means for examining the gravitational radiation from an axisymmetric isolated source that undergoes a finite period of oscillation. It involves an expansion of the metric as a double Taylor series. Here we examine the integration procedure that is used to form an algorithmic solution to the field equations and point out the possibility of the expansion method breaking down and predicting a singularity along the axis of symmetry. We derive a condition on the solutions obtained by the double series method that must be satisfied to avoid this singularity. We then consider a source with only a quadrupole moment and verify that to fourth order in each of the expansion parameters, this condition is satisfied. This is a reassuring test of the consistency of the expansion procedure. We do, however, find that the imposition of this condition makes a physical interpretation of any but the lowest order solutions very difficult. The most obvious decomposition of the solution into a series of independent physical effects is shown not to be valid. 
  Bartnik's definition of gravitational quasilocal energy is analyzed. For a wide class of systems Bartnik's function is given by the ADM mass of some vacuous extension. As an example we calculate mass of a non central ball in Schwarzschild geometry. The ratio mass to volume becomes singular in the limit of small volumes. 
  Relying on the general theory of Lie derivatives a new geometric definition of Lie derivative for general spinor fields is given, more general than Kosmann's one. It is shown that for particular infinitesimal lifts, i.e. for Kosmann vector fields, our definition coincides with the definition given by Kosmann more than 20 years ago. 
  The extension of the general relativity theory to higher dimensions, so that the field equations for the metric remain of second order, is done through the Lovelock action. This action can also be interpreted as the dimensionally continued Euler characteristics of lower dimensions. The theory has many constant coefficients apparently without any physical meaning. However, it is possible, in a natural way, to reduce to two (the cosmological and Newton's constant) these several arbitrary coefficients, yielding a restricted Lovelock gravity. In this process one separates theories in even dimensions from theories in odd dimensions. These theories have static black hole solutions. In general relativity, black holes appear as the final state of gravitational collapse. In this work, gravitational collapse of a regular dust fluid in even dimensional restricted Lovelock gravity is studied. It is found that black holes emerge as the final state for these regular initial conditions. 
  Quantum inequality restrictions on the stress-energy tensor for negative energy are developed for three and four-dimensional static spacetimes. We derive a general inequality in terms of a sum of mode functions which constrains the magnitude and duration of negative energy seen by an observer at rest in a static spacetime. This inequality is evaluated explicitly for a minimally coupled scalar field in three and four-dimensional static Robertson-Walker universes. In the limit of vanishing curvature, the flat spacetime inequalities are recovered. More generally, these inequalities contain the effects of spacetime curvature. In the limit of short sampling times, they take the flat space form plus subdominant curvature-dependent corrections. 
  We present a classification of all global solutions (with Lorentzian signature) for any general 2D dilaton gravity model. For generic choices of potential-like terms in the Lagrangian one obtains maximally extended solutions on arbitrary non-compact two-manifolds, including various black-hole and kink configurations. We determine all physical quantum states in a Dirac approach. In some cases the spectrum of the (black-hole) mass operator is found to be sensitive to the signature of the theory, which may be relevant in view of current attempts to implement a generalized Wick-rotation in 4D quantum gravity. 
  We propose to uncover the signature of a stringy era in the primordial Universe by searching for a prominent peak in the relic graviton spectrum. This feature, which in our specific model terminates an $\omega^3$ increase and initiates an $\omega^{-7}$ decrease, is induced during the so far overlooked bounce of the scale factor between the collapsing deflationary era (or pre-Big Bang) and the expanding inflationary era (or post-Big Bang). We evaluate both analytically and numerically the frequency and the intensity of the peak and we show that they may likely fall in the realm of the new generation of interferometric detectors. The existence of a peak is at variance with ordinarily monotonic (either increasing or decreasing) graviton spectra of canonical cosmologies; its detection would therefore offer strong support to string cosmology. 
  We summarise recent work on the quantum production of black holes in the inflationary era. We describe, in simple terms, the Euclidean approach used, and the results obtained both for the pair creation rate and for the evolution of the black holes. 
  In the inflationary era, black holes came into existence together with the universe through the quantum process of pair creation. We calculate the pair creation rate from the no boundary proposal for the wave function of the universe. Our results are physically sensible and fit in with other descriptions of pair creation. The tunnelling proposal, on the other hand, predicts a catastrophic instability of de Sitter space to the nucleation of large black holes, and cannot be maintained. 
  The (single) black hole solutions of Ba\~nados, Teitelboim and Zanelli (BTZ) in 2+1 dimensional anti-de Sitter space are generalized to an arbitrary number $n$ of such black holes. The resulting multi-black-hole (MBH) spacetime is locally isometric to anti-de Sitter space, and globally it is obtained from the latter as a quotient space by means of suitable identifications. The MBH spacetime has $n$ asymptotically anti-de Sitter exterior regions, each of which has the geometry of a single BTZ black hole. These exterior regions are separated by $n$ horizons from a common interior region. This interior region can be described as a ``closed" universe containing $n$ black holes. Similar configurations in 3+1 dimensions, with horizons of toroidal and higher genus topologies, are also presented. 
  We present an anomaly-free Dirac constraint quantization of the string-inspired dilatonic gravity (the CGHS model) in an open 2-dimensional spacetime. We show that the quantum theory has the same degrees of freedom as the classical theory; namely, all the modes of the scalar field on an auxiliary flat background, supplemented by a single additional variable corresponding to the primordial component of the black hole mass. The functional Heisenberg equations of motion for these dynamical variables and their canonical conjugates are linear, and they have exactly the same form as the corresponding classical equations. A canonical transformation brings us back to the physical geometry and induces its quantization. 
  We derive the gravitational waveform and gravitational-wave energy flux generated by a binary star system of compact objects (neutron stars or black holes), accurate through second post-Newtonian order ($O[(v/c)^4] \sim O[(Gm/rc^2)^2]$) beyond the lowest-order quadrupole approximation. We cast the Einstein equations into the form of a flat-spacetime wave equation together with a harmonic gauge condition, and solve it formally as a retarded integral over the past null cone of the chosen field point. The part of this integral that involves the matter sources and the near-zone gravitational field is evaluated in terms of multipole moments using standard techniques; the remainder of the retarded integral, extending over the radiation zone, is evaluated in a novel way. The result is a manifestly convergent and finite procedure for calculating gravitational radiation to arbitrary orders in a post-Newtonian expansion. Through second post-Newtonian order, the radiation is also shown to propagate toward the observer along true null rays of the asymptotically Schwarzschild spacetime, despite having been derived using flat spacetime wave equations. The method cures defects that plagued previous ``brute- force'' slow-motion approaches to the generation of gravitational radiation, and yields results that agree perfectly with those recently obtained by a mixed post-Minkowskian post-Newtonian method. We display explicit formulae for the gravitational waveform and the energy flux for two-body systems, both in arbitrary orbits and in circular orbits. In an appendix, we extend the formalism to bodies with finite spatial extent, and derive the spin corrections to the waveform and energy loss. 
  W.Lawvere suggested a approach to differential geometry and to others mathematical disciplines closed to physics, which allows to give definitions of derivatives, tangent vectors and tangent bundles without passages to the limits. This approach is based on a idea of consideration of all settings not in sets but in some cartesian closed category E, particular in some elementary topos.    The synthetic differential geometry (SDG) is the theory developed by A.Kock in a context of Lawvere's ideas. In a base of the theory is an assumption of that a geometric line is not a filed of real numbers, but a some nondegenerate commutative ring R of a line type in E.    In this work we shall show that SDG allows to develop a Riemannian geometry and write the Einstein's equations of a field on pseudo-Riemannian formal manifold. This give a way for constructing a intuitionistic models of general relativity in suitable toposes. 
  The effect of the existence of tails on the propagation of scalar waves in curved space-time is considered via an analysis of flux integrals of the energy-stress-momentum tensor of the waves. The geometric optics approximation is formulated in terms of such flux integrals, and three examples are investigated in detail in order to determine the possible effects of wave tails. The approximation is valid for waves in Minkowski space-time (tail-free) and waves in Schwarzschild space-time (weak tails) but it is shown how the approximation can break down in a cosmological scenario due to destructive interference by strong tails. In this last situation, the waves do not radiate. 
  We study the ultraviolet divergent structures of the matter (scalar) field in a higher D-dimensional Reissner-Nordstr\"{o}m black hole and compute the matter field contribution to the Bekenstein-Hawking entropy by using the Pauli-Villars regularization method. We find that the matter field contribution to the black hole entropy does not, in general, yield the correct renormalization of the gravitational coupling constants. In particular we show that the matter field contribution in odd dimensions does not give the term proportional to the area of the black hole event horizon. 
  The two-dimensional theory of Teitelboim and Jackiw has constant and negative curvature. In spite of this, the theory admits a black hole solution with no singularities. In this work we study the thermodynamics of this black hole using York's formalism. 
  It is shown that there is an interesting interplay between self-duality, loop representation and knots invariants in the quantum theory of Maxwell fields in Minkowski space-time. Specifically, in the loop representation based on self-dual connections, the measure that dictates the inner product can be expressed as the Gauss linking number of thickened loops. 
  In this brief note we present a set of equations describing the evolution of perturbed scalar fields in a cosmological spacetime with multiple scalar fields. We take into account of the simultaneously excited full metric perturbations in the context of the uniform-curvature gauge which is known to be the best choice. The equations presented in a compact form will be useful for handling the structure formation processes under the multiple episodes of inflation. 
  A treatment in a neighborhood and at a point of the equivalence principle on the basis of derivations of the tensor algebra over a manifold is given. Necessary and sufficient conditions are given for the existence of local bases, called normal frames, in which the components of derivations vanish in a neighborhood or at a point. These frames (bases), if any, are explicitly described and the problem of their holonomicity is considered. In particular, the obtained results concern symmetric as well as nonsymmetric linear connections. 
  Inhomogeneous multidimensional cosmological models with a higher dimensional space-time manifold M=M_0 x M_1 x ... M_n are investigated under dimensional reduction to tensor-multi-scalar theories. In the Einstein conformal frame, these theories take the shape of a flat sigma-model. For the singular case where M_0 is 2-dimensional, the dimensional reduction to dilaton gravity is preformed with different distinguished representations of the action. 
  We consider dilaton--axion gravity interacting with $p\;\, U(1)$ vectors ($p=6$ corresponding to $N=4$ supergravity) in four--dimensional spacetime admitting a non--null Killing vector field. It is argued that this theory exibits features of a ``square'' of vacuum General Relativity. In the three--dimensional formulation it is equivalent to a gravity coupled $\sigma$--model with the $(4+2p)$--dimensional target space $SO(2,2+p)/(SO(2)\times SO(2+p))$. K\"ahler coordinates are introduced on the target manifold generalising Ernst potentials of General Relativity. The corresponding K\"ahler potential is found to be equal to the logarithm of the product of the four--dimensional metric component $g_{00}$ in the Einstein frame and the dilaton factor, independently on presence of vector fields. The K\"ahler potential is invariant under exchange of the Ernst potential and the complex axidilaton field, while it undergoes holomorphic/antiholomorphic transformations under general target space isometries. The ``square'' property is also manifest in the two--dimensional reduction of the theory as a matrix generalization of the Kramer--Neugebauer map. 
  In the investigation of the evolution of cosmological perturbations in inflationary universe models the behavior of perturbations during the reheating stage is the most unclear point. In particular in the early reheating phase in which a rapidly oscillating scalar field dominates the energy density, the behavior of perturbations are not known well because their evolution equation expressed in terms of the curvature perturbation becomes singular. In this paper it is shown that in spite of this singular behavior of the evolution equation the Bardeen parameter stays constant in a good accuracy during this stage for superhorizon-scale perturbations except for a sequence of negligibly short intervals around the zero points of the time derivative of the scalar field. This justifies the conventional formula relating the amplitudes of quantum fluctuations during inflation and those of adiabatic perturbations at horizon crossing in the Friedmann stage, except for possible corrections produced by the energy transfer from the scalar field to radiation in the late stage of reheating. It is further shown that outside the above sequence of time intervals the behavior of the perturbations coincides in a good accuracy with that for a perfect fluid system obtained from the original scalar field system by the WKB approximation and a spacetime averaging over a Hubble horizon scale. 
  A $2p + 5$ parametric family of black holes is constructed in dilaton--axion gravity with $p$ vector fields using a holomorphic representation of U--duality in three dimensions. The metric of the non--extremal black holes has a Reissner--Nordstr\"om type structure and generically possesses an internal horizon. However in the extremal limit the generic solution exhibits a dilatonic type null singularity. Only in the case of the orthogonal electric and magnetic charges (if $p>1$) the extremal solution may have a non--singular event horizon. 
  We construct the most general form of axially symmetric SU(2)-Yang-Mills fields in Bianchi cosmologies. The dynamical evolution of axially symmetric YM fields in Bianchi I model is compared with the dynamical evolution of the electromagnetic field in Bianchi I and the fully isotropic YM field in Friedmann-Robertson-Walker cosmologies. The stochastic properties of axially symmetric Bianchi I-Einstein-Yang-Mills systems are compared with those of axially symmetric YM fields in flat space. After numerical computation of Liapunov exponents in synchronous (cosmological) time, it is shown that the Bianchi I-EYM system has milder stochastic properties than the corresponding flat YM system. The Liapunov exponent is non-vanishing in conformal time. 
  Symmetries of generalized gravitational actions, yielding field equations which typically involve at most second-order derivatives of the metric, are considered. The field equations for several different higher-derivative theories in the first-order formalism are derived, and variations of a generic set of higher-order curvature terms appearing in string effective actions are studied. It is shown that there often exists a particular set of solutions to the field equations of pure gravity theories, consisting of different combinations of curvature tensors, which satisfies the vacuum equations with cosmological constant. Implications of generalized symmetries of the field equations derived from the superstring effective action for the cosmological constant problem are discussed. 
  A Lorentz and general co-ordinate co-variant form of canonical gravity, using Ashtekar's variables, is investigated. A co-variant treatment due to Crnkovic and Witten is used, in which a point in phase space represents a solution of the equations of motion and a symplectic functional two form is constructed which is Lorentz and general co-ordinate invariant. The subtleties and difficulties due to the complex nature of Ashtekar's variables are addressed and resolved. 
  In this paper we show how quantum corrections, although perturbatively small, may play an important role in the analysis of the existence of some classical models. This, in fact, appears to be the case of static, uniform--density models of the interior metric of cosmic strings and neutron stars. We consider the fourth order semiclassical equations and first look for perturbative solutions in the coupling constants $\alpha$ and $\beta$ of the quadratic curvature terms in the effective gravitational Lagrangian. We find that there is not a consistent solution; neither for strings nor for spherical stars. We then look for non--perturbative solutions and find an explicit approximate metric for the case of straight cosmic strings. We finally analyse the contribution of the non--local terms to the renormalized energy--momentum tensor and the possibility of this terms to allow for a perturbative solution. We explicitly build up a particular renormalized energy--momentum tensor to fulfill that end. These state--dependent corrections are found by simple considerations of symmetry, conservation law and trace anomaly, and are chosen to compensate for the local terms. However, they are not only ad hoc, but have to depend on $\alpha$ and $\beta$, what is not expected to first perturbative order. We then conclude that non--perturbative solutions are valuable for describing certain physical situations. 
  We present three reasons for rewriting the Einstein equation. The new version is physically equivalent but geometrically more clear. 1. We write $4 \pi$ instead of $8 \pi$ at the r.h.s, and we explain how this factor enters as surface area of the unit 2--sphere. 2. We define the Riemann curvature tensor and its contractions (including the Einstein tensor at the l.h.s.) with one half of its usual value. This compensates not only for the change made at the r.h.s., but it gives the result that the curvature scalar of the unit 2--sphere equals one, i.e., in two dimensions, now the Gaussian curvature and the Ricci scalar coincide. 3. For the commutator $[u,v]$ of the vector fields $u$ and $v$ we prefer to write (because of the analogy with the antisymmetrization of tensors) $$[u,v]\ = \ \frac{1}{2} \, ( \, u\, v \ - \ v \, u \,)$$ which is one half of the usual value. Then, the curvature operator defined by $$ \nabla_{[u} \ \nabla_{v]} \quad - \quad \nabla_{[u,\,v]}$$ (where $\nabla $ denotes the covariant derivative) is consistent with point 2, i.e., it equals one half of the usual value. 
  A distributional method to solve the Einstein's field equations for thin shells is formulated. The familiar field equations and jump conditions of Darmois-Israel formalism are derived. A carefull analysis of the Bianchi identities shows that, for cases under consideration, they make sense as distributions and lead to jump conditions of Darmois-Israel formalism. 
  We find that the model of a black hole plus an exterior halo of quadrupoles and octopoles recently proposed by us is more chaotic than previously detected. In fact, the quadrupolar component gives rise also to a chaotic behavior, found after further numerical search. This fact reinforces even more the role of chaos in relativistic core--halo models. 
  We discuss semiclassical states in quantum gravity corresponding to Schwarzschild as well as Reissner Nordstr\"om black holes. We show that reduced quantisation of these models is equivalent to Wheeler-DeWitt quantisation with a particular factor ordering. We then demonstrate how the entropy of black holes can be consistently calculated from these states. While this leads to the Bekenstein-Hawking entropy in the Schwarzschild and non-extreme Reissner-Nordstr\"om cases, the entropy for the extreme Reissner-Nordstr\"om case turns out to be zero. 
  I summarize the basic ideas and formalism of loop quantum gravity. I illustrate the results on the discrete aspects of quantum geometry and two applications of these results to black hole physics. In particular, I discuss in detail a derivation of the Bekenstein-Hawking formula for the entropy of a black hole from first principles. 
  We present an implementation of the loop representation of quantum gravity on a square lattice. Instead of starting from a classical lattice theory, quantizing and introducing loops, we proceed backwards, setting up constraints in the lattice loop representation and showing that they have appropriate (singular) continuum limits and algebras. The diffeomorphism constraint reproduces the classical algebra in the continuum and has as solutions lattice analogues of usual knot invariants. We discuss some of the invariants stemming from Chern--Simons theory in the lattice context, including the issue of framing. We also present a regularization of the Hamiltonian constraint. We show that two knot invariants from Chern--Simons theory are annihilated by the Hamiltonian constraint through the use of their skein relations, including intersections. We also discuss the issue of intersections with kinks. This paper is the first step towards setting up the loop representation in a rigorous, computable setting. 
  Gravitational perturbations which are present in any realistic stellar collapse to a black hole, die off in the exterior of the hole, but experience an infinite blueshift in the interior. This is believed to lead to a slowly contracting lightlike scalar curvature singularity, characterized by a divergence of the hole's (quasi-local) mass function along the inner horizon.   The region near the inner horizon is described to great accuracy by a plane wave spacetime. While Einstein's equations for this metric are still too complicated to be solved in closed form it is relatively simple to integrate them numerically.   We find for generic regular initial data the predicted mass inflation type null singularity, rather than a spacelike singularity. It thus seems that mass inflation indeed represents a generic self-consistent picture of the black hole interior. 
  Dark matter is obtained from a scalar field coupled conformally to gravitation; the scalar being a relict of Dirac's gauge function. This conformally coupled dark matter includes a gas of very light ($m\approx 2.25\times 10^{-34} eV$) neutral bosons having spin 0, as well as a time-dependent global scalar field, both pervading all of the cosmic space. The time-development of this dark matter in the expanding F-R-W universe is investigated, and an acceptable cosmological behaviour is obtained. 
  Static traversable wormhole solutions of the Einstein equations require ``exotic'' matter which violates the weak energy condition. The vacuum stress-energy of quantized fields has been proposed as the source for this matter. Using the Dewitt-Schwinger approximation, analytic expressions for the stress-energy of a quantized massive scalar field are calculated in five static spherically symmetric Lorentzian wormhole spacetimes. We find that in all cases, for both minimally and conformally coupled scalar fields, the stress-energy does not have the properties needed to support the wormhole geometry. 
  The fact that in Minkowski space, space and time are both quantized does not have to be introduced as a new postulate in physics, but can actually be derived by combining certain features of General Relativity and Quantum Mechanics. This is demonstrated first in a model where particles behave as point defects in 2 space dimensions and 1 time, and then in the real world having 3+1 dimensions. The mechanisms in these two cases are quite different, but the outcomes are similar: space and time form a (non-cummutative) lattice.   These notes are short since most of the material discussed in these lectures is based on two earlier papers by the same author (gr-qc/9601014 and gr-qc/9607022), but the exposition given in the end is new. 
  Particle production processes in the expanding universe are described within a simple kinetic model. The equilibrium conditions for a Maxwell-Boltzmann gas with variable particle number are investigated. We find that radiation and nonrelativistic matter may be in equilibrium at the same temperature provided the matter particles are created at a rate that is half the expansion rate. Using the fact that the creation of particles is dynamically equivalent to a nonvanishing bulk pressure we calculate the backreaction of this process on the cosmological dynamics. It turns out that the `adiabatic' creation of massive particles with an equilibrium distribution for the latter necessarily implies power-law inflation. Exponential inflation in this context is shown to become inconsistent with the second law of thermodynamics after a time interval of the order of the Hubble time. 
  We provide a method to determine the motion of a classical massive particle in a background geometry of 2-dimensional gravity theories, for which the Birkhoff theorem holds. In particular, we get the particle trajectory in a continuous class of 2-dimensional dilaton gravity theories that includes the Callan-Giddings-Harvey-Strominger (CGHS) model, the Jackiw-Teitelboim (JT) model, and the $d$-dimensional $s$-wave Einstein gravity. The explicit trajectory expressions for these theories are given along with the discussions on the results. 
  We study numerically the collapse of massless scalar fields in two-dimensional dilaton gravity, both classically and semiclassically. At the classical level, we find that the black hole mass scales at threshold like $M_{\rm bh} \propto |p-p^*|^{\gamma}$, where $\gamma \simeq 0.53$. At the semiclassical level, we find that in general $M_{\rm bh}$ approaches a non-zero constant as $p \rightarrow p^*$. Thus, quantum effects produce a mass gap not present classically at the onset of black hole formation. 
  The asymptotic behavior of Einstein-Rosen waves at null infinity in 4 dimensions is investigated in {\it all} directions by exploiting the relation between the 4-dimensional space-time and the 3-dimensional symmetry reduction thereof. Somewhat surprisingly, the behavior in a generic direction is {\it better} than that in directions orthogonal to the symmetry axis. The geometric origin of this difference can be understood most clearly from the 3-dimensional perspective. 
  Gravitational waves with a space-translation Killing field are considered. In this case, the 4-dimensional Einstein vacuum equations are equivalent to the 3-dimensional Einstein equations with certain matter sources. This interplay between 4- and 3- dimensional general relativity can be exploited effectively to analyze issues pertaining to 4 dimensions in terms of the 3-dimensional structures. An example is provided by the asymptotic structure at null infinity: While these space-times fail to be asymptotically flat in 4 dimensions, they can admit a regular completion at null infinity in 3 dimensions. This completion is used to analyze the asymptotic symmetries, introduce the analog of the 4-dimensional Bondi energy-momentum and write down a flux formula.   The analysis is also of interest from a purely 3-dimensional perspective because it pertains to a diffeomorphism invariant 3-dimensional field theory with {\it local} degrees of freedom, i.e., to a midi-superspace. Furthermore, due to certain peculiarities of 3 dimensions, the description of null infinity does have a number of features that are quite surprising because they do not arise in the Bondi-Penrose description in 4 dimensions. 
  We compute the complete spectrum of the area operator in the loop representation of quantum gravity, using recoupling theory. This result extends previous derivations, which did not include the ``degenerate'' sector, and agrees with the recently computed spectrum of the connection-representation area operator. 
  We discuss special supersymmetric extreme black holes in 4 and 5 dimensions which have regular horizons, non-zero entropy and can be interpreted as compactifications of BPS bound states of p-branes in 10 or 11 dimensions. 
  The main result of this paper is a proof that there are examples of spatially compact solutions of the Einstein-dust equations which only exist for an arbitrarily small amount of CMC time. While this fact is plausible, it is not trivial to prove. It is necessary to obtain a lower bound for the lapse function of a CMC foliation in a suitable class of inhomogeneous spacetimes. This bound, which shows that in these spacetimes the lapse cannot collapse in finite CMC time, may be of independent interest. This fact is contrasted with the positive results previously obtained for other matter models, e.g. collisionless matter or wave maps. 
  We show the relation between Traschen's integral equations and the energy, and ``position of the centre of mass'', of the matter perturbations in a Robertson-Walker spacetime. When the perturbations are ``localised'' we get a set of integral constraints that includes hers. We illustrate them on a simple example. 
  Including torsion in the geometric framework of the Weyl-Dirac theory we build up an action integral, and obtain from it a gauge covariant (in the Weyl sense) general relativistic massive electrodynamics. Photons having an arbitrary mass, electric, and magnetic currents (Dirac's monopole) coexist within this theory. Assuming that the space-time is torsionless, taking the photons mass zero, and turning to the Einstein gauge we obtain Maxwell's electrodynamics. 
  The quantization of the gravitational Chern-Simons coefficient is investigated in the framework of $ISO(2,1)$ gauge gravity. Some paradoxes involved are cured. The resolution is largely based on the inequivalence of $ISO(2,1)$ gauge gravity and the metric formulation. Both the Lorentzian scheme and the Euclidean scheme lead to the coefficient quantization, which means that the induced spin is not quite exotic in this context. 
  In the teleparallel equivalent of general relativity the energy density of asymptotically flat gravitational fields can be naturaly defined as a scalar density restricted to a three-dimensional spacelike hypersurface $\Sigma$. Integration over the whole $\Sigma$ yields the standard ADM energy. After establishing the reference space with zero gravitational energy we obtain the expression of the localized energy for a Kerr black hole. The expression of the energy inside a surface of constant radius can be explicitly calculated in the limit of small $a$, the specific angular momentum. Such expression turns out to be exactly the same as the one obtained by means of the method preposed recently by Brown and York. We also calculate the energy contained within the outer horizon of the black hole for {\it any} value of $a$. The result is practically indistinguishable from $E=2M_{ir}$, where $M_{ir}$ is the irreducible mass of the black hole. 
  We present first results obtained with a 3+1 dimensional adaptive mesh code in numerical general relativity. The adaptive mesh is used in conjunction with a standard ADM code for the evolution of a dynamically sliced Schwarzschild spacetime (geodesic slicing). We argue that adaptive mesh is particularly natural in the context of general relativity, where apart from adaptive mesh refinement for numerical efficiency one may want to use the built in flexibility to do numerical relativity on coordinate patches. 
  We calculate the total gravitational energy and the gravitational energy density of the de Sitter space using the definition of localized energy that arises in the framework of the teleparallel equivalent of general relativity. We find that the gravitational energy can only be defined within the cosmological horizon and is largely concentrated in regions far from the center of spherical symmetry, i.e., in the vicinity of the maximal spacelike radial coordinate $R=\sqrt{ 3 \over \Lambda}$. The smaller the cosmological constant, the farther the concentration of energy. This result complies with the phenomenological features of the de Sitter space, namely, the existence of a radial acceleration directed away from the center of symmetry experienced by a test particle in the de Sitter space. Einstein already contemplated the de Sitter solution as a world with a surface distribution of matter, a picture that is in agreement with the present analysis. 
  The automorphism invariant theory of Crawford[J. Math. Phys. 35, 2701 (1994)] has show great promise, however its application is limited by the paradigm to the domain of spin space. Our conjecture is that there is a broader principle at work which applies even to classical physics. Specifically, the laws of physics should be invariant under polydimensional transformations which reshuffle the geometry (e.g. exchanges vectors for trivectors) but preserves the algebra. To complete the symmetry, it follows that the laws of physics must be themselves polydimensional, having scalar, vector, bivector etc. parts in one multivector equation. Clifford algebra is the natural language in which to formulate this principle, as vectors/tensors were for relativity. This allows for a new treatment of the relativistic spinning particle (the Papapetrou equations) which is problematic in standard theory. In curved space the rank of the geometry will change under parallel transport, yielding a new basis for Weyl's connection and a natural coupling between linear and spinning motion. 
  The Reissner-Nordstrom-de Sitter black holes of standard Einstein-Maxwell theory with a cosmological constant have no analogue in dilatonic theories with a Liouville potential. The only exception are the solutions of maximal mass, the Charged Nariai solutions. We show that the structure of the solution space of the Dilatonic Charged Nariai black holes is quite different from the non-dilatonic case. Its dimensionality depends on the exponential coupling constants of the dilaton. We discuss the possibility of pair creating such black holes on a suitable background. We find conditions for the existence of Charged Nariai solutions in theories with general dilaton potentials, and consider specifically a massive dilaton. 
  The long term nonlinear dynamics of a Keplerian binary system under the combined influences of gravitational radiation damping and external tidal perturbations is analyzed. Gravitational radiation reaction leads the binary system towards eventual collapse, while the external periodic perturbations could lead to the ionization of the system via Arnold diffusion. When these two opposing tendencies nearly balance each other, interesting chaotic behavior occurs that is briefly studied in this paper. It is possible to show that periodic orbits can exist in this system for sufficiently small damping. Moreover, we employ the method of averaging to investigate the phenomenon of capture into resonance. 
  Recent developments in theories of non-Riemannian gravitational interactions are outlined. The question of the motion of a fluid in the presence of torsion and metric gradient fields is approached in terms of the divergence of the Einstein tensor associated with a general connection. In the absence of matter the variational equations associated with a broad class of actions involving non-Riemannian fields give rise to an Einstein-Proca system associated with the standard Levi-Civita connection. 
  The purpose of this work is to show a possible reconciliation between Mach's principle and the Schwarzschild solution, using a description of this solution in terms of a new radial cooordinate related to the behaviour of Unruh detectors. 
  We study the spectrum of primordial fluctuations in theories where the inflaton field is coupled to massless fields and/or to itself. Conformally invariant theories generically predict a scale invariant spectrum. Scales entering the theory through infrared divergences cause logarithmic corrections to the spectrum, tiltilng it towards the blue. We discuss in some detail whether these fluctuations are quantum or classical in nature. 
  The gravitational properties of the {\em only} static plane-symmetric vacuum solution of Einstein's field equations without cosmological term (Taub's solution, for brevity) are presented: some already known properties (geodesics, weak field limit and pertainment to the Schwarzschild family of spacetimes) are reviewed in a physically much more transparent way, as well as new results about its asymptotic structure, possible matchings and nature of the source are furnished. The main results point to the fact that the solution must be interpreted as representing the exterior gravitational field due to a {\em negative} mass distribution, confirming previous statements to that effect in the literature. Some analogies to Kasner's spatially homogeneous cosmological model are also referred to. 
  It is shown that in a classical spacetime with multiply connected space slices having the topology of a torus, closed timelike curves are also formed. We call these spacetime ringholes. Two regions on the torus surface can be distinguished which are separated by angular horizons. On one of such regions (that which surrounds the maximum circumference of the torus) everything happens like in spherical wormholes, but the other region (the rest of the torus surface), while still possessing a chronology horizon and non-chronal region, behaves like a coverging, rather than diverging, lens and corresponds to an energy density which is always positive for large speeds at or near the throat. It is speculated that a ringhole could be converted into a time machine to perform time travels by an observer who would never encounter any matter that violates the classical averaged weak energy condition. Based on a calculation of vacuum fluctuations, it is also seen that the angular horizons can prevent the emergence of quantum instabilities near the throat. 
  We find an anisotropic, non-supersymmetric generalization of the extreme supersymmetric domain walls of simple non-dilatonic supergravity theory. As opposed to the isotropic non- and ultra-extreme domain walls, the anisotropic non-extreme wall has the \emph{same} spatial topology as the extreme wall. The solution has naked singularities which vanish in the extreme limit. Since the Hawking temperature on the two sides is different, the generic solution is unstable to Hawking decay. 
  We critically examine the Roberts homothetic solution for the spherically symmetric Einstein-scalar field equations in double null coordinates, and show that the Roberts solution indeed solves the field equations only for one non-trivial case. We generalize this solution and discuss its relations with other known exact solutions. 
  The construction of a massive (~ 40 ton) spherical antennas for gravitational wave astronomy has been proposed by a number of groups worldwide. Summed over all five quadrupole modes, the sphere has a direction-independent cross section, permitting full-sky coverage. Combined with its enhanced sensitivity due to its multi-nature and increased mass, sphereical detectors make ideal instruments for observational astronomy. Numerical calculations have shown that gravitational-wave signals from coalescing neutron star binaries and dynamic instabilities of rapidly rotating stars in the Virgo cluster can be resolved with a spherical antenna with a near quantum-limited sensitivity. 
  An analytical model that represents the collapse of a massless scalar wave packet with continuous self-similarity is constructed, and critical phenomena are found. In the supercritical case, the mass of black holes is finite and has the form $M \propto (p - p^{*})^{\gamma}$, with $\gamma = 1/2$. 
  We study the radiation from a collision of black holes with equal and opposite linear momenta. Results are presented from a full numerical relativity treatment and are compared with the results from a ``close-slow'' approximation. The agreement is remarkable, and suggests several insights about the generation of gravitational radiation in black hole collisions. 
  The problem of topology change transitions in quantum gravity is discussed. We argue that the contribution of the Giddings-Strominger wormhole to the Euclidean path integral is pure imaginary. This is checked by two techniques: by the functional integral approach and by the analysis of the Wheeler-De Witt equation. We present also a simple quantum mechanical model which shares many features of the system consisting of parent and baby universes. In this simple model, we show that quantum coherence is completely lost and obtain the equation for the effective density matrix of the ''parent universe''. 
  The basic ingredients of the consistent histories approach to quantum mechanics are the space of histories and the space of decoherence functionals. In this work we extend the classification theorem for decoherence functionals proven by Isham, Linden and Schreckenberg to the case where the space of histories is the lattice of projection operators on an arbitrary separable or non-separable complex Hilbert space of dimension strictly greater than two. 
  Recent theoretical work determines the correct coupling constant of a scalar field to the Ricci curvature of spacetime in general relativity. The periodicity in the redshift distribution of galaxies observed by Broadhurst {\em et al.}, if genuine, determines the coupling constant in the proposed scalar field models. As a result, these observations contain important information on the problem whether general relativity is the correct theory of gravity in the region of the universe at redshifts z<0.5. 
  We uplift the static three dimensional black hole solution found by Banados, Teitelboim and Zanelli (BTZ) into four dimensional space time. In this way we obtain a black string solution with a relativistic string source, as well as a new black hole solution which is also generated by a relativistic ``stringy'' source. It is shown that when passing continuously from the region of the one dimensional parameter spaces which characterize these solutions, containing naked singular strings or naked singular ``points'', and into the region containing black strings or black holes, the metrics ``blow up'' at a critical point in both parameter spaces. We show that a similar ``separation'' mechanism can be introduced in three dimensions. In this way we also obtain a generalization of the BTZ solution. The ``separation'' mechanism, and its immediate consequences, offers an hitherto missing technical argument needed in order to exclude the naked singularities (the ``mass gap'') from the space of ``physically permissible'' solutions in three dimensional Einstein theory coupled to a negative cosmological constant.   PACS nos.: 04.40.Nr, 04.20.Cv 
  We formulate a qualitative argument, based on Heisenberg's uncertainty principle, to support the claim that when the effects of matter fields are assumed to overshadow the effects of quantum mechanics of spacetime, the discrete spectrum of black hole radiation, as such as predicted by Bekenstein's proposal for a discrete black hole area spectrum, reduces to Hawking's black-body spectrum. 
  An action for simplicial euclidean general relativity involving only left-handed fields is presented. The simplicial theory is shown to converge to continuum general relativity in the Plebanski formulation as the simplicial complex is refined. This contrasts with the Regge model for which Miller and Brewin have shown that the full field equations are much more restrictive than Einstein's in the continuum limit. The action and field equations of the proposed model are also significantly simpler then those of the Regge model when written directly in terms of their fundamental variables.   An entirely analogous hypercubic lattice theory, which approximates Plebanski's form of general relativity is also presented. 
  We clarify some points about the systems considered by Sota, Suzuki and Maeda in Class. Quantum Grav. 13, 1241 (1996). Contrary to the authors' claim for a non-homoclinic kind of chaos, we show the chaotic cases they considered are homoclinic in origin. The power of local criteria to predict chaos is once more questioned. We find that their local, curvature--based criterion is neither necessary nor sufficient for the occurrence of chaos. In fact, we argue that a merit of their search for local criteria applied to General Relativity is just to stress the weakness of locality itself, free of any pathologies related to the motion in effective Riemannian geometries. 
  The Gauss-Bonnet type identity is derived in a Weyl-Cartan space on the basis of the variational method. 
  The existence of the Pontryagin and Euler forms in a Weyl-Cartan space on the basis of the variational method with Lagrange multipliers are established. It is proved that these forms can be expressed via the exterior derivatives of the corresponding Chern-Simons terms in a Weyl-Cartan space with torsion and nonmetricity. 
  We analyze the possibility of black holes pair creation induced by three dimensional wormholes. Although this spacetime configuration is nowadays hard to suppose, it can be very important in the early universe, when the wormhole spacetime foam representation can be meaningful. We compare our approach with the no-boundary prescription of Hartle-Hawking. 
  For any asymptotically flat spacetime with a suitable causal structure obeying (a weak form of) Penrose's cosmic censorship conjecture and satisfying conditions guaranteeing focusing of complete null geodesics, we prove that active topological censorship holds. We do not assume global hyperbolicity, and therefore make no use of Cauchy surfaces and their topology. Instead, we replace this with two underlying assumptions concerning the causal structure: that no compact set can signal to arbitrarily small neighbourhoods of spatial infinity (``$i^0$-avoidance''), and that no future incomplete null geodesic is visible from future null infinity. We show that these and the focusing condition together imply that the domain of outer communications is simply connected. Furthermore, we prove lemmas which have as a consequence that if a future incomplete null geodesic were visible from infinity, then given our $i^0$-avoidance assumption, it would also be visible from points of spacetime that can communicate with infinity, and so would signify a true naked singularity. 
  Contents:  APS Topical Group in Gravitation News:  April 1997 Joint APS/AAPT Meeting  Research briefs:  GEO600 by Karsten Danzmann  Black hole microstates in string theory by Gary Horowitz  LIGO project status by Stan Whitcomb  The Hamiltonian constraint of quantum gravity and loops by John Baez  Conference reports:  International conference on gravitational waves by Valeria Ferrari  PCGM12/KKfest by Richard Price  First International LISA Symposium by Robin Stebbins  Schroedinger Institute Workshop by Abhay Ashtekar  Relativistic Astrophysics at Bad Honnef by Hans-Peter Nollert  Intermediate binary black hole workshop by Sam Finn  Quantum Gravity in the Southern Cone by Rodolfo Gambini  Report on the Spring APS Meeting by Fred Raab and Beverly Berger 
  We elaborate the recently introduced asymptotically exact semiclassical quantum gravity derived from the Wheeler-DeWitt equation by finding a particular coherent state representation of a quantum scalar field in which the back-reaction of the scalar field Hamiltonian exactly gives rise to the classical one. In this coherent state representation classical spacetime emerges naturally from semiclassical quantum gravity. 
  We describe a post-Minkowskii approximation of general relativity as a power series expansion in G, Newton's gravitational constant. Material sources are hidden behind boundaries, and only the vacuum Einstein equations are considered. An iterative procedure is described which, in one complete step, takes any approximate solution of the Einstein equations and produces a new approximation which has the error decreased by a factor of G. Each step in the procedure consists of three parts: first the equations of motion are used to update the trajectories of the boundaries; then the field equations are solved using a retarded Green's function for Minkowskii space; finally a gauge transformation is performed which makes the geometry well behaved at future null infinity. Differences between this approach to the Einstein equations and similar ones are that we use a general (non-harmonic) gauge and formulate the procedure in a constructive manner which emphasizes its suitability for implementation on a computer. 
  A propagating torsion model is derived from the requirement of compatibility between minimal action principle and minimal coupling procedure in Riemann-Cartan spacetimes. In the proposed model, the trace of the torsion tensor is derived from a scalar potential that determines the volume element of the spacetime. The equations of the model are written down for the vacuum and for various types of matter fields. Some of their properties are discussed. In particular, we show that gauge fields can interact minimally with the torsion without the breaking of gauge symmetry. 
  We compute gravitational radiation waveforms, spectra and energies for a point particle of mass $m_0$ falling from rest at radius $r_0$ into a Schwarzschild hole of mass $M$. This radiation is found to lowest order in $(m_0/M)$ with the use of a Laplace transform. In contrast with numerical relativity results for head-on collisions of equal-mass holes, the radiated energy is found not to be a monotonically increasing function of initial separation; there is a local radiated-energy maximum at $r_0\approx4.5M$. The present results, along with results for infall from infinity, provide a complete catalog of waveforms and spectra for particle infall. We give a representative sample from that catalog and an interesting observation: Unlike the simple spectra for other head-on collisions (either of particle and hole, or of equal mass holes) the spectra for $\infty>r_0>\sim5M$ show a series of evenly spaced bumps. A simple explanation is given for this. Lastly, our energy vs. $r_0$ results are compared with approximation methods used elsewhere, for small and for large initial separation. 
  A characteristic spectrum of relic gravitational radiation is produced by a period of ``stringy inflation" in the early universe. This spectrum is unusual, because the energy-density rises rapidly with frequency. We show that correlation experiments with the two gravitational wave detectors being built for the Laser Interferometric Gravitational Observatory (LIGO) could detect this relic radiation, for certain ranges of the parameters that characterize the underlying string cosmology model. 
  We study how the neutron-star equation of state affects the onset of the dynamical instability in the equations of motion for inspiraling neutron-star binaries near coalescence. A combination of relativistic effects and Newtonian tidal effects cause the stars to begin their final, rapid, and dynamically-unstable plunge to merger when the stars are still well separated and the orbital frequency is $\approx$ 500 cycles/sec (i.e. the gravitational wave frequency is approximately 1000 Hz). The orbital frequency at which the dynamical instability occurs (i.e. the orbital frequency at the innermost stable circular orbit) shows modest sensitivity to the neutron-star equation of state (particularly the mass-radius ratio, $M/R_o$, of the stars). This suggests that information about the equation of state of nuclear matter is encoded in the gravitational waves emitted just prior to the merger. 
  I consider the appearance of shocks in hyperbolic formalisms of General Relativity. I study the particular case of the Bona-Masso formalism with zero shift vector and show how shocks associated with two families of characteristic fields can develop. These shocks do not represent discontinuities in the geometry of spacetime, but rather regions where the coordinate system becomes pathological. For this reason I call them coordinate shocks. I show how one family of shocks can be eliminated by restricting the Bona-Masso slicing condition to a special case. The other family of shocks, however, can not be eliminated even in the case of harmonic slicing. I also show the results of numerical simulations in the special cases of a flat two-dimensional spacetime, a flat four-dimensional spacetime with a spherically symmetric slicing, and a spherically symmetric black hole spacetime. In all three cases coordinate shocks readily develop, confirming the predictions of the mathematical analysis. Although here I concentrate in the Bona-Masso formalism, the phenomena of coordinate shocks should arise in any other hyperbolic formalism. In particular, since the appearance of the shocks is determined by the choice of gauge, the results presented here imply that in any formalism the use of a harmonic slicing can generate shocks. 
  A purported black hole solution in (2+1)-dimensions is shown to be nothing more than flat space viewed from an accelerated frame. 
  A class of solutions of the gravitational field equations describing vacuum spacetimes outside rotating cylindrical sources is presented. A subclass of these solutions corresponds to the exterior gravitational fields of rotating cylindrical systems that emit gravitational radiation. The properties of these rotating gravitational wave spacetimes are investigated. In particular, we discuss the energy density of these waves using the gravitational stress-energy tensor. 
  We provide a physical basis for the local gravitational superenergy tensor. Furthermore, our gravitoelectromagnetic deduction of the Bel-Debever-Robinson superenergy tensor permits the identification of the gravitational stress-energy tensor. This {\it local} gravitational analog of the Maxwell stress-energy tensor is illustrated for a plane gravitational wave. 
  The rank-2 sector of classical $3+1$ dimensional Ashtekar gravity is considered. It is found that the consistency of the evolution equations with the reality of the volume requires that the 3-surface of initial data is foliated by 2-surfaces tangent to degenerate triads. In turn, the degeneracy is preserved by the evolution. The 2-surfaces behave like $2+1$ dimensional empty spacetimes with a massless complex field propagating along each of them. The results provide some evidence for the issue of evolving a non-degenerate gravitational field into a degenerate sector of Ashtekar's phase space. 
  We discuss the coupled Einstein-Klein-Gordon equations for a complex scalar field with and without a quartic self-interaction in a zero curvature Friedman-Lema\^{\i}tre Universe. The complex scalar field, as well as the metric, is decomposed in a homogeneous, isotropic part (the background) and in first order gauge invariant scalar perturbation terms. The background equations can be written as a set of four coupled first order non-linear differential equations. These equations are analyzed using modern theory of dynamical system. It is shown that, in all singular points where inflation occurs, the phase of the complex scalar field is asymptotically constant. The analysis of the first order equations is done for the inflationary phase. For the short wavelength regime the first order perturbation term of the complex scalar field is smeared out and the Bardeen potential oscillates around a nearly constant mean value. Whereas for the long wavelength regime the first order perturbed quantities increase. 
  A two-dimensional (2D) dilaton gravity model, whose static solutions have the same features of the Reissner-Nordstrom solutions, is obtained from the dimensional reduction of a four-dimensional (4D) string effective action invariant under S-duality transformations. The black hole solutions of the 2D model and their relationship with those of the 4D theory are discussed. 
  We study the head-on collision of two equal-mass momentarily stationary black holes, using black hole perturbation theory up to second order. Compared to first-order results, this significantly improves agreement with numerically computed waveforms and energy. Much more important, second-order results correctly indicate the range of validity of perturbation theory. This use of second-order, to provide ``error bars,'' makes perturbation theory a viable tool for providing benchmarks for numerical relativity in more generic collisions and, in some range of collision parameters, for supplying waveform templates for gravitational wave detection. 
  It is shown that if a generalized definition of gauge invariance is used, gauge invariant effective stress-energy tensors for gravitational waves and other gravitational perturbations can be defined in a much larger variety of circumstances than has previously been possible. In particular it is no longer necessary to average the stress-energy tensor over a region of spacetime which is larger in scale than the wavelengths of the waves and it is no longer necessary to restrict attention to high frequency gravitational waves. 
  We present a formalism to obtain equilibrium configurations of uniformly rotating fluid in the second post-Newtonian approximation of general relativity. In our formalism, we need to solve 29 Poisson equations, but their source terms decrease rapidly enough at the external region of the matter(i.e., at worst $O(r^{-4})$). Hence these Poisson equations can be solved accurately as the boundary value problem using standard numerical methods.This formalism will be useful to obtain nonaxisymmetric uniformly rotating equilibrium configurations such as synchronized binary neutron stars just before merging and the Jacobi ellipsoid. 
  We show that static sources coupled to a massless scalar field in Schwarzschild spacetime give rise to emission and absorption of zero-energy particles due to the presence of Hawking radiation. This is in complete analogy with the description of the bremsstrahlung by a uniformly accelerated charge from the coaccelerated observers' point of view. The response rate of the source is found to coincide with that in Minkowski spacetime as a function of its proper acceleration. This result may be viewed as restoration of the equivalence principle by the Hawking effect. 
  We derive the effective energy-momentum tensor for cosmological perturbations and prove its gauge-invariance. The result is applied to study the influence of perturbations on the behaviour of the Friedmann background in inflationary Universe scenarios. We found that the back reaction of cosmological perturbations on the background can become important already at energies below the self-reproduction scale. 
  When completed, the gravitational wave detectors now proposed or under construction will provide us with a perspective on the Universe fundamentally different from any we have come to know. With this new perspective comes the hope that new insights and understandings of Nature will emerge. The proposed acoustic detectors, with spherical geometries and operating at millikelvin temperatures, are particularly well-suited to study gravitational radiation in the 1--3 KHz band. In this brief report I review some sources of particular interest for these detectors. 
  We investigate the signature of the Lund-Regge metric on spaces of simplicial three-geometries which are important in some formulations of quantum gravity. Tetrahedra can be joined together to make a three-dimensional piecewise linear manifold. A metric on this manifold is specified by assigning a flat metric to the interior of the tetrahedra and values to their squared edge-lengths. The subset of the space of squared edge-lengths obeying triangle and analogous inequalities is simplicial configuration space. We derive the Lund-Regge metric on simplicial configuration space and show how it provides the shortest distance between simplicial three-geometries among all choices of gauge inside the simplices for defining this metric (Regge gauge freedom). We show analytically that there is always at least one physical timelike direction in simplicial configuration space and provide a lower bound on the number of spacelike directions. We show that in the neighborhood of points in this space corresponding to flat metrics there are spacelike directions corresponding to gauge freedom in assigning the edge-lengths. We evaluate the signature numerically for the simplicial configuration spaces based on some simple triangulations of the three-sphere (S^3) and three-torus (T^3). For the surface of a four-simplex triangulation of S^3 we find one timelike direction and all the rest spacelike over all of the simplicial configuration space. For the triangulation of T^3 around flat space we find degeneracies in the simplicial supermetric as well as a few gauge modes corresponding to a positive eigenvalue. Moreover, we have determined that some of the negative eigenvalues are physical, i.e. the corresponding eigenvectors are not generators of diffeomorphisms. We compare our results with the known properties of continuum superspace. 
  The evolution of the cosmological perturbation during the oscillatory stage of the scalar field is investigated. For the power law potential of the inflaton field, the evolution equation of the Mukhanov's gauge invariant variable is reduced to the Mathieu equation and the density perturbation grows by the parametric resonance. 
  We make a direct connection between the construction of three dimensional topological state sums from tensor categories and three dimensional quantum gravity by noting that the discrete version of the Wheeler-DeWitt equation is exactly the pentagon for the associator of the tensor category, the Biedenharn-Elliott identity. A crucial role is played by an asymptotic formula relating 6j-symbols to rotation matrices given by Edmonds. 
  A string theory in $3$ euclidean spacetime dimensions is found to describe the semiclassical behavior of a certain exact physical state of quantum general relativity in $4$ dimensions. Both the worldsheet and the three dimensional metric emerge as collective coordinates that describe a sector of the solution space of quantum general relativity. Additional collective coordinates exist which are interpreted as worldsheet degrees of freedom. The construction may be extended to the case in which the Kalb-Ramond field is included in the non-perturbative dynamics. It is possible that this mechanism is the inverse of the strong coupling limit by which some $D$ dimensional string theories are conjectured to give rise to $D+1$ dimensional field theories. 
  It is already understood that the increasing observational evidence for an open Universe can be reconciled with inflation if our horizon is contained inside one single huge bubble nucleated during the inflationary phase transition. In this frame of ideas, we show here that the probability of living in a bubble with the right $\Omega_0$ (now the observations require $\Omega_0 \approx .2$) can be comparable with unity, rather than infinitesimally small. For this purpose we modify both quantitatively and qualitatively an intuitive toy model based upon fourth order gravity. As this scheme can be implemented in canonical General Relativity as well (although then the inflation driving potential must be designed entirely ad hoc), inferring from the observations that $\Omega_0 < 1$ not only does not conflict with the inflationary paradigm, but rather supports therein the occurrence of a primordial phase transition. 
  We study spherically symmetric solutions of a four-dimensional theory of gravity with a topological action, which was constructed as a Yang-Mills theory of the Poincar\'e group and can be considered a generalization to higher dimensions of well-known two-dimensional models. We also discuss the perturbative degrees of freedom and the properties of the theory under conformal transformations. 
  It is argued that some approaches to non-perturbative quantum general relativity lack a sensible continuum limit that reproduces general relativity. The basic problem is that generic physical states lack long ranged correlations, because the form of the state allows a division into spatial regions, such that no change in the physical state in one region can be measured by observables restricted to another. These disconnected regions have generically finite expectation value of physical volume, which means that the theory has no long ranged correlations or massless particles. One consequence of this is that the $ADM$ energy is unbounded from below, at least when that is defined with respect to a natural notion of quantum asymptotic flatness and a corresponding definition of an operator that measures $E_{ADM}$ (which is given here). These problems occur in Thiemann's new formulation of quantum gravity. Related issues arise in some other approaches such as that of Borissov, Rovelli and Smolin. A new approach to the Hamiltonian constraint, which may avoid the problem of the lack of long ranged correlations, is proposed. 
  This paper is a sequel of the series of papers [gr-qc/9409010, gr-qc/9505034, gr-qc/9603022], being an immediate continuation and development of the latter of them. In the Friedmann universe, the equation $\Omega_0=2q_0$ holds ($\Omega$ is density parameter, $q$ is deceleration parameter, and subscript 0 indicates present-day values), which gives rise to the problem of the missing matter as observational data give $\Omega_0<2q_0$. In the cosmic-length universe, $\Omega_0=2q_0- L/R_0^3H_0^2$ ($R_0$ is the radius of the universe, $H_0$ is Hubble constant), which lifts the problem. The cosmic length, $L=const \approx 1/H_0$, is the infimum of the set of maximal radii of a closed universe. 
  The behavior of scalar perturbations on superhorizon scales during the reheating stage is investigated by replacing the rapidly oscillating inflaton field by a perfect fluid obtained by spacetime averaging and the WKB approximation. The influence of the energy transfer from the inflaton to radiation on the evolution of the Bardeen parameter is examined for realistic reheating processes. It is shown that the entropy perturbation generated by the energy transfer is negligibly small, and therefore the Bardeen parameter is conserved in a good accuracy during reheating. This justifies the conventional prescription relating the amplitudes of quantum fluctuations during inflation and those of adiabatic perturbations at horizon crossing in the post-Friedmann stage. 
  We show that the exact solution of the two_superbody problem in N=2 Chern Simons Supergravity in 2+1 dimensions leads to a supermultiplet of space-times. This supersymmetric space-time is characterized by the two gauge invariant observables of the super Poincare' group, which may be viewed as the Casimir invariants of an equivalent one-superbody state. The metric of this space-time supermultiplet can be cast into the form of the metric for a spinning cone in which the coordinates do not commute or for a spinning cone with an additional finite discrete dimension. Some of the interesting features of this universe and their possible physical implications are discussed in the light of a corresponding observation by Witten. 
  Employing energy-momentum pseudotensor of Einstein, we obtian the energy distribution of a dyonic dilaton black hole. The energy distribution of this black hole depends on mass $ M $, electric charge $ Q_{e} $, magnetic charge $ Q_{m} $ and asymptotic values of the dilaton $ \phi_{0} $. We also mmake some comparisons between the results of Virbhadra et. al. and ours. 
  We consider the empirical validity of the equivalence principle for non-baryonic matter. Working in the context of the TH\epsilon\mu formalism, we evaluate the constraints experiments place on parameters associated with violation of the equivalence principle (EVPs) over as wide a sector of the standard model as possible. Specific examples include new parameter constraints which arise from torsion balance experiments, gravitational red shift, variation of the fine structure constant, time-dilation measurements, and matter/antimatter experiments. We find several new bounds on EVPs in the leptonic and kaon sectors. 
  We consider in detail the problem of gauge dependence that exists in relativistic perturbation theory, going beyond the linear approximation and treating second and higher order perturbations. We first derive some mathematical results concerning the Taylor expansion of tensor fields under the action of one-parameter families (not necessarily groups) of diffeomorphisms. Second, we define gauge invariance to an arbitrary order $n$. Finally, we give a generating formula for the gauge transformation to an arbitrary order and explicit rules to second and third order. This formalism can be used in any field of applied general relativity, such as cosmological and black hole perturbations, as well as in other spacetime theories. As a specific example, we consider here second order perturbations in cosmology, assuming a flat Robertson-Walker background, giving explicit second order transformations between the synchronous and the Poisson (generalized longitudinal) gauges. 
  We present a new family of exact solutions of the Einstein equations, constructed through the Khan-Penrose procedure, that may be interpreted as representing the propagation of a pair of solitons, in the background of a plane-wave collision spacetime. The metric in the interaction region is obtained as a diagonal solitonic perturbation of Rindler's spacetime, applying the Belinskii and Zakharov Inverse Scattering Method (ISM), with two real poles and a pair of complex conjugate poles. We use a non-standard renormalization procedure, obtaining solutions that contain two more parameters than in the standard ISM. We analyze the asymptotic behaviour of the solutions in the limit where the determinant of the Killing part of the metric vanishes, finding in this limit a curvature singularity, except when the free parameters contained in the solutions satisfy a particular relation. Assuming this condition is satisfied, we show that the metric is regular in the above mentioned limit, and that the spacetimes contain in this case a Killing Cauchy horizon instead of a curvature singularity. The solutions are analytically extended through the horizon, and we find a curvature singularity in this extension, related to the presence and propagation of the complex poles. 
  A new formalism for spinors on curved spaces is developed in the framework of variational calculus on fibre bundles. The theory has the same structure of a gauge theory and describes the interaction between the gravitational field and spinors. An appropriate gauge structure is also given to General Relativity, replacing the metric field with spin frames. Finally, conserved quantities and superpotentials are calculated under a general covariant form. 
  We are able to characterize a 2--dimensional classical fluid sharing some of the same thermodynamic state functions as the Schwarzschild black hole. This phenomenological correspondence between black holes and fluids is established by means of the model liquid's pair-correlation function and the two-body atomic interaction potential. These latter two functions are calculated exactly in terms of the black hole internal (quasilocal) energy and the isothermal compressibility. We find the existence of a ``screening" like effect for the components of the liquid. 
  Cosmological solutions of the Brans-Dicke theory are investigated by including a quantum effect coming from 1-loop correction of matter fields that couple to the scalar field. As the most serious result we face a cosmological ``constant'' in the original conformal frame which is shown to be ``physical'' after a careful analysis of the time variable employed in any of the conventional approaches. We find an ``attractor'' solution featuring no expansion during the radiation-dominated eras. To evade this unacceptable consequence, we suggest to modify one of the fundamental premises of the model, rendering the scalar field almost ``invisible.'' 
  We present a new approach to the theory of static deformations of elastic test bodies in general relativity based on a generalization of the concept of frame of reference which we identify with the concept of quo-harmonic congruence. We argue on the basis of this new approach that weak gravitational plane waves do not couple to elastic bodies and therefore the latter, whatever their shape, are not suitable antennas to detect them. 
  This paper is a continuation of the papers [gr-qc/9409010, gr-qc/9505034, gr-qc/9603022, gr-qc/9609035] and is devoted to the riddle of the origin of the arrow of time. The problem of time orientation reduces to that of the difference between the past and the future. The riddle escapes solution in deterministic dynamics and in the dynamics of standard indeterministic quantum theory as well. In the dynamics of indeterministic quantum gravity, the past is reconstructible uniquely, whereas the future may be forecasted only on a probabilistic level. Thus the problems of the past and the future and, by the same token, of time orientation are solved. 
  The canonical formalism of three dimensional gravity coupled with the Dirac field is considered. We introduce complex variables to simplify the Dirac brackets of canonical variables and examine the canonical structure of the theory. We discuss the reality conditions which guarantee the equivalence between the complex and real theory. 
  The general principles and logical structure of a thermodynamic formalism that incorporates strongly self-gravitating systems are presented. This framework generalizes and simplifies the formulation of thermodynamics developed by Callen. The definition of extensive variables, the homogeneity properties of intensive parameters, and the fundamental problem of gravitational thermodynamics are discussed in detail. In particular, extensive parameters include quasilocal quantities and are naturally incorporated into a set of basic general postulates for thermodynamics. These include additivity of entropies (Massieu functions) and the generalized second law. Fundamental equations are no longer homogeneous first-order functions of their extensive variables. It is shown that the postulates lead to a formal resolution of the fundamental problem despite non-additivity of extensive parameters and thermodynamic potentials. Therefore, all the results of (gravitational) thermodynamics are an outgrowth of these postulates. The origin and nature of the differences with ordinary thermodynamics are analyzed. Consequences of the formalism include the (spatially) inhomogeneous character of thermodynamic equilibrium states, a reformulation of the Euler equation, and the absence of a Gibbs-Duhem relation. 
  Gravitational radiation reaction forces and balance equations for energy and momenta are investigated to 3/2 post-Newtonian (1.5PN) order beyond the quadrupole approximation, corresponding to the 4PN order in the equations of motion of an isolated system. By matching a post-Newtonian solution for the gravitational field inside the system to a post-Minkowskian solution (obtained in a previous work) for the gravitational field exterior to the system, we determine the 1PN relativistic corrections to the ``Newtonian" radiation reaction potential of Burke and Thorne. The 1PN reaction potential involves both scalar and vectorial components, with the scalar component depending on the mass-type quadrupole and octupole moments of the system, and the vectorial component depending in particular on the current-type quadrupole moment. In the case of binary systems, the 1PN radiation reaction potential has been shown to yield consistent results for the 3.5PN approximation in the binary's equations of motion. Adding up the effects of tails, the radiation reaction is then written to 1.5PN order. In this paper, we establish the validity to 1.5PN order, for general systems, of the balance equations relating the losses of energy, linear momentum, and angular momentum in the system to the corresponding fluxes in the radiation field far from the system. 
  For a physical interpretation of a theory of quantum gravity, it is necessary to recover classical spacetime, at least approximately. However, quantum gravity may eventually provide classical spacetimes by giving spectral data similar to those appearing in noncommutative geometry, rather than by giving directly a spacetime manifold. It is shown that a globally hyperbolic Lorentzian manifold can be given by spectral data. A new phenomenon in the context of spectral geometry is observed: causal relationships. The employment of the causal relationships of spectral data is shown to lead to a highly efficient description of Lorentzian manifolds, indicating the possible usefulness of this approach. Connections to free quantum field theory are discussed for both motivation and physical interpretation. It is conjectured that the necessary spectral data can be generically obtained from an effective field theory having the fundamental structures of generalized quantum mechanics: a decoherence functional and a choice of histories. 
  Assuming the weak energy condition, we study the nature of the non-central shell-focussing singularity which can form in the gravitational collapse of a spherical compact object in classical general relativity. We show that if the radial pressure is positive, the singularity is covered by a horizon. For negative radial pressures, the singularity will be covered if the ratio of pressure to the density is greater than -1/3 and naked if this ratio is $\leq -1/3$. 
  Taking (2+1)-dimensional pure Einstein gravity for arbitrary genus $g$ as a model, we investigate the relation between the partition function formally defined on the entire phase space and the one written in terms of the reduced phase space. In particular the case of $g=1$ is analyzed in detail.  By a suitable gauge-fixing, the partition function $Z$ basically reduces to the partition function defined for the reduced system, whose dynamical variables are $(\tau^A, p_A)$. [The $\tau^A$'s are the Teichm\"uller parameters, and the $p_A$'s are their conjugate momenta.]  As for the case of $g=1$, we find out that $Z$ is also related with another reduced form, whose dynamical variables are $(\tau^A, p_A)$ and $(V, \sigma)$. [Here $\sigma$ is a conjugate momentum to 2-volume $V$.] A nontrivial factor appears in the measure in terms of this type of reduced form. The factor turns out to be a Faddeev-Popov determinant coming from the time-reparameterization invariance inherent in this type of formulation. Thus the relation between two reduced forms becomes transparent even in the context of quantum theory.  Furthermore for $g=1$, a factor coming from the zero-modes of a differential operator $P_1$ can appear in the path-integral measure in the reduced representation of $Z$. It depends on the path-integral domain for the shift vector in $Z$: If it is defined to include $\ker P_1$, the nontrivial factor does not appear. On the other hand, if the integral domain is defined to exclude $\ker P_1$, the factor appears in the measure. This factor can depend on the dynamical variables, typically as a function of $V$, and can influence the semiclassical dynamics of the (2+1)-dimensional spacetime.  These results shall be significant from the viewpoint of quantum gravity. 
  Stationary spherically symmetric gravity is equivalent to a nonlinear coset sigma model on SL(2,R)/SO(2) coupled to a gravitational remnant. Classically there are stationary solutions besides the static Schwarzschild metric labeled by the Schwarzschild mass $m$ and the Taub-NUT charge $l$. Imposing the SL(2,R) symmetry at the quantum level the Wheeler-DeWitt equation becomes related to the Casimir operator on the coset, which makes the system amenable to exact quantization. 
  These are the proceedings of the XVIII Conference of the Indian Association for General Relativity and Gravitation (IAGRG) held at the Institute of Mathematical Sciences, Madras, INDIA during Feb. 15-17, 1996. The Conference was dedicated the late Prof. S. Chandrasekhar.   The proceedings consists of 17 articles on:      Chandrasekhar's work (N. Panchapkesan);      Vaidya-Raychaudhuri Lecture (C.V. Vishveshwara)      Gravitational waves (B.R. Iyer, R. Balasubramanian)      Gravitational Collapse (T.P. Singh)      Accretion on black hole (S. Chakrabarti)      Cosmology (D. Munshi, S. Bharadwaj, G.S. Mohanty,           P. Bhattacharjee);      Classical GR (S. Kar, D.C. Srivatsava)      Quantum aspects (J. Maharana, Saurya Das, P. Mitra,           G. Date, N.D. Hari Dass)   The body of THIS article contains ONLY the title, contents, foreword, organizing committees, preface, list of contributed talks and list of participants. The plenery talks are available at:   http://www.imsc.ernet.in/physweb/Conf/ both as post-script files of individual articles and also as .uu source files. For further information please send e-mail to shyam@imsc.ernet.in 
  Self-consistent solutions to nonlinear spinor field equations in General Relativity have been studied for the case of Bianchi type-I space-time filled with perfect fluid. The initial and the asymptotic behavior of the field functions and the metric one has been thoroughly studied. It should be emphasized the absence of initial singularity for some types of solutions and also the isotropic mode of space-time expansion in some special cases. 
  Self-consistent solutions to interacting spinor and scalar field equations in General Relativity are studied for the case of Bianchi type-I space-time filled with perfect fluid. The initial and the asymptotic behavior of the field functions and the metric one has been thoroughly studied. 
  We define the energy of a perfectly isolated system at a given retarded time as the suitable null limit of the quasilocal energy $E$. The result coincides with the Bondi-Sachs mass. Our $E$ is the lapse-unity shift-zero boundary value of the gravitational Hamiltonian appropriate for the partial system $\Sigma$ contained within a finite topologically spherical boundary $B = \partial \Sigma$. Moreover, we show that with an arbitrary lapse and zero shift the same null limit of the Hamiltonian defines a physically meaningful element in the space dual to supertranslations. This result is specialized to yield an expression for the full Bondi-Sachs four-momentum in terms of Hamiltonian values. 
  We discuss the data acquisition and analysis procedures used on the Allegro gravity wave detector, including a full description of the filtering used for bursts of gravity waves. The uncertainties introduced into timing and signal strength estimates due to stationary noise are measured, giving the windows for both quantities in coincidence searches. 
  We have developed a general method for finding apparent horizons in 3D numerical relativity. Instead of solving for the partial differential equation describing the location of the apparent horizons, we expand the closed 2D surfaces in terms of symmetric trace--free tensors and solve for the expansion coefficients using a minimization procedure. Our method is applied to a number of different spacetimes, including numerically constructed spacetimes containing highly distorted axisymmetric black holes in spherical coordinates, and 3D rotating, and colliding black holes in Cartesian coordinates. 
  We study the problem of whether the active gravitational mass of an isolated system is equal to the total energy in the tetrad theory of gravitation. The superpotential is derived using the gravitational Lagrangian which is invariant under parity operation, and applied to an exact spherically symmetric solution. Its associated energy is found equal to the gravitational mass. The field equation in vacuum is also solved at far distances under the assumption of spherical symmetry. Using the most general expression for parallel vector fields with spherical symmetry, we find that the equality between the gravitational mass and the energy is always true if the parameters of the theory $a_1$, $a_2$ and $a_3$ satisfy the condition, $(a_1+ a_2) (a_1-4a_3/9)\neq0$. In the two special cases where either $(a_1+a_2)$ or $(a_1-4a_3/9)$ is vanishing, however, this equality is not satisfied for the solutions when some components of the parallel vector fields tend to zero as $1/\sqrt{r}$ for large $r$. 
  Assuming the equivalence of FRW-cosmological models and their Newtonian counterparts, we propose using the Gauss law in arbitrary dimension a general relation between the Newtonian gravitational constant G and the gravitational coupling constant \kappa. 
  The observed microwave background anisotropies in combination with the theory of quantum mechanically generated cosmological perturbations predict a well measurable amount of relic gravitational waves in the frequency intervals tested by LISA and ground-based laser interferometers. 
  A global O$(2,2)$ symmetry is found in the Brans-Dicke theory of gravity when the dilaton is coupled to axion and moduli fields. The symmetry is broken if a cosmological constant is introduced. Within the class of spatially homogeneous Bianchi cosmologies, only the type I and V models respect the symmetry. Isotropic cosmological solutions are found for arbitrary spatial curvature. In the region of parameter space relevant to the pre-big bang scenario, the interplay between the scalar fields results in a bouncing cosmology. 
  We review the mechanism in quantum gravity whereby topological geons, particles made from non-trivial spatial topology, are endowed with nontrivial spin and statistics. In a theory without topology change there is no obstruction to ``anomalous'' spin-statistics pairings for geons. However, in a sum-over-histories formulation including topology change, we show that non-chiral abelian geons do satisfy a spin-statistics correlation if they are described by a wave function which is given by a functional integral over metrics on a particular four-manifold. This manifold describes a topology changing process which creates a pair of geons from $R^3$. 
  The static, plane symmetric solutions and cylindrically symmetric solutions of Einstein-Maxwell equations with a negative cosmological constant are investigated. These black configurations are asymptotically anti-de Sitter not only in the transverse directions, but also in the membrane or string directions. Their causal structure is similar to that of Reissner-Nordstr\"{o}m black holes, but their Hawking temperature goes with $M^{1/3}$, where $M$ is the ADM mass density. We also discuss the static plane solutions in Einstein-Maxwell-dilaton gravity with a Liouville-type dilaton potential. The presence of the dilaton field changes drastically the structure of solutions. They are asymptotically ``anti-de Sitter'' or ``de Sitter'' depending on the parameters in the theory. 
  It is investigated if massless particles can couple to scalar fields in a special relativistic theory with classical particles. The only possible obvious theory which is invariant under Lorentz transformations and reparametrization of the affine parameter leads to trivial trajectories (straight lines) for the massless case, and also the investigation of the massless limit of the massive theory shows that there is no influence of the scalar field on the limiting trajectories.   On the other hand, in contrast to this result, it is shown that massive particles are influenced by the scalar field in this theory even in the ultra-relativistic limit. 
  The tunneling approach to the wave function of the universe has been recently criticized by Bousso and Hawking who claim that it predicts a catastrophic instability of de Sitter space with respect to pair production of black holes. We show that this claim is unfounded. First, we argue that different horizon size regions in de Sitter space cannot be treated as independently created, as they contend. And second, the WKB tunneling wave function is not simply the `inverse' of the Hartle-Hawking one, except in very special cases. Applied to the related problem of pair production of massive particles, we argue that the tunneling wave function leads to a small constant production rate, and not to a catastrophy as Bousso and Hawking's argument would suggest. 
  Within the scalar-tensor theory of gravity with Higgs mechanism without Higgs particles, we prove that the excited Higgs potential (the scalar field) vanishs inside and outside of the stellar matter for static spherically symmetric configurations. The field equation for the metric (the tensorial gravitational field) turns out to be essentially the Einsteinian one. 
  The scalar background field and its consequences are discussed for the Friedmann type cosmological solutions of the scalar-tensor theory of gravity with the Higgs field of the Standard Model as the scalar gravitational field. 
  Differential properties of Klein-Gordon and electromagnetic fields on a straight cosmic string's space-time background are studied by means of methods of the differential spaces theory. It is shown that these fields are smooth on the interior of cosmic string's space-time and they loose this property at the singular boundary except for cosmic string space-times with the following deficits of angle: $\Delta=2\pi (1-1/n) $, $n=1,2,\dots$. A connection between co-called asymptotic smoothness of fields (smoothness at singularity) and the radiative Aharonov-Bohm effects of Aliev and Gal'tsov is discussed. It is also argued that the asymptotic smoothness of physical fields assumption may be treated as a kind of deficit's of angle "quantization" condition leading to the above mentioned quantization formula for $\Delta$. 
  Using the continuity of the scalar $\Psi_2$ (the mass aspect) at null infinity through $i_o$ we show that the space of radiative solutions of general relativity can be thought of a fibered space where the value of $\Psi_2$ at $i_o$ plays the role of the base space. We also show that the restriction of the available symplectic form to each ``fiber'' is degenerate. By finding the orbit manifold of this degenerate direction we obtain the reduced phase space for the radiation data. This reduced phase space posses a global structure, i.e., it does not distinguishes between future or past null infinity. Thus, it can be used as the space of quantum gravitons. Moreover, a Hilbert space can be constructed on each ``fiber'' if an appropriate definition of scalar product is provided. Since there is no natural correspondence between the Hilbert spaces of different foliations they define superselection sectors on the space of asymptotic quantum states. 
  We present a new numerical algorithm for evolving the Mixmaster spacetimes. By using symplectic integration techniques to take advantage of the exact Taub solution for the scattering between asymptotic Kasner regimes, we evolve these spacetimes with higher accuracy using much larger time steps than previously possible. The longer Mixmaster evolution thus allowed enables detailed comparison with the Belinskii, Khalatnikov, Lifshitz (BKL) approximate Mixmaster dynamics. In particular, we show that errors between the BKL prediction and the measured parameters early in the simulation can be eliminated by relaxing the BKL assumptions to yield an improved map. The improved map has different predictions for vacuum Bianchi Type IX and magnetic Bianchi Type VI$_0$ Mixmaster models which are clearly matched in the simulation. 
  The Einstein-Langevin equations take into account the backreaction of quantum matter fields on the background geometry. We present a derivation of these equations to lowest order in a covariant expansion in powers of the curvature. For massless fields, the equations are completely determined by the running coupling constants of the theory. 
  We investigate whether the gravitational thermodynamic properties of the scalar-tensor theory of gravity are affected by the conformal transformation or not. As an explicit example, we consider an electrically charged static spherical black hole in the 4-dimensional low energy effective theory of bosonic string. 
  The model independent analysis of the $dp$ elastic scattering in the collinear geometry has been performed. It is shown that the measurements of 10 polarization observables of the first and second order realize the complete experimental program on the determination of the amplitudes of the $dp$ backward elastic scattering reaction. 
  By allowing the light cones to tip over on hypersurfaces according to the conservation laws of an one-kink in static, Schwarzschild black hole metric, we show that in the quantum regime there also exist instantons whose finite imaginary action gives the probability of occurrence of the kink metric corresponding to single chargeless, nonrotating black hole taking place in pairs, the holes of each pair being joined on an interior surface, beyond the horizon. 
  It is shown via the principle of path independence that the (time gauge) constraint algebra derived in (Class. Quantum Grav. 5 (1988) pg. 1405) for vielbein General Relativity is a generic feature of any covariant theory formulated in a vielbein frame. In the process of doing so, the relationship between the coordinate and orthonormal frame algebera is made explicit. 
  The existence of a dilaton (or moduli) with gravitational-strength coupling to matter imposes stringent constraints on the allowed energy scale of cosmic strings, $\eta$. In particular, superheavy gauge strings with $\eta \sim 10^{16} GeV$ are ruled out unless the dilaton mass $m_{\phi} \gsim 100 TeV$, while the currently popular value $m_{\phi} \sim 1 TeV$ imposes the bound $\eta \lsim 3 \times 10^{11} GeV$. Similar constraints are obtained for global topological defects. Some non-standard cosmological scenarios which can avoid these constraints are pointed out. 
  We show that regularizing divergent integrals is crucially important when applied to the loop diagrams corresponding to quantum corrections to the coupling of the ``gravitational" scalar field due to the interaction among matter fields. We use the method of continuous spacetime dimensions to demonstrate that WEP is a robust property of the Brans-Dicke theory beyond the classical level, hence correcting our previous assertion of the contrary. The same technique can be used to yield the violation of WEP when applied to the scale-invariant theory, thus providing another reason for expecting fifth- force-type phenomena. 
  We consider a local cosmic string described by the Abelian-Higgs model in the framework of scalar-tensor gravities. We find the metric of the cosmic string in the weak-field approximation. The propagation of particles and light is analysed in this background. This analysis shows that the (unperturbed) cosmic string in scalar-tensor theories presents some analogous features to the wiggly cosmic string in General Relativity. 
  3-dimensional gravity coupled to Maxwell (or Klein-Gordon) fields is exactly soluble under the assumption of axi-symmetry. The solution is used to probe several quantum gravity issues. In particular, it is shown that the quantum fluctuations in the geometry are large unless the number and frequency of photons satisfy the inequality $\N(\hbar G\omega)^2 << 1$. Thus, even when there is a single photon of Planckian frequency, the quantum uncertainties in the metric are significant. Results hold also for a sector of the 4-dimensional theory (consisting of Einstein Rosen gravitational waves). 
  In this paper the relation between the choice of a differential structure and a smooth connection in the tangential bundle is discussed. For the case of an exotic $S^7$ one obtains corrections to the curvature after the change of the differential structure, which can not be neglected by a gauge transformation. In the more interesting case of four dimensions we obtain a correction of the connection constructed by intersections of embedded surfaces. This correction produce a source term in the equation of the general relativity theory which can be interpreted as the energy-momentum tensor of a embedded surface. 
  In recent years there has been some progress in the understanding of the global structure of stationary black hole space-times. In this paper we review some new results concerning the structure of stationary black hole space-times. In particular we prove a corrected version of the ``black hole rigidity theorem'', and we prove a uniqueness theorem for static black holes with degenerate connected horizons. This paper is an expanded version of a lecture given at the Journ\'ees relativistes in Ascona, May 1996. 
  We establish global extendibility (to the domain of outer communications) of locally defined isometries of appropriately regular analytic black holes. This allows us to fill a gap in the Hawking-Ellis proof of black-hole rigidity. 
  The Wheeler-DeWitt equation for the minimally coupled FRW-massive-scalar-field minisuperspace is written as a two-component Schr\"odinger equation with an explicitly `time'-dependent Hamiltonian. This reduces the solution of the Wheeler-DeWitt equation to the eigenvalue problem for a non-relativistic one-dimensional harmonic oscillator and an infinite series of trivial algebraic equations whose iterative solution is easily found. The solution of these equations yields a mode expansion of the solution of the original Wheeler-DeWitt equation. Further analysis of the mode expansion shows that in general the solutions of the Wheeler-DeWitt equation for this model are doubly graded, i.e., every solution is a superposition of two definite-parity solutions. Moreover, it is shown that the mode expansion of both even and odd-parity solutions is always infinite. It may be terminated artificially to construct approximate solutions. This is demonstrated by working out an explicit example which turns out to satisfy DeWitt's boundary condition at initial singularity. 
  Existence of global CMC foliations of constant curvature 3-dimensional maximal globally hyperbolic Lorentzian manifolds, containing a constant mean curvature hypersurface with $\genus(\Sigma) > 1$ is proved. Constant curvature 3-dimensional Lorentzian manifolds can be viewed as solutions to the 2+1 vacuum Einstein equations with a cosmological constant. The proof is based on the reduction of the corresponding Hamiltonian system in constant mean curvature gauge to a time dependent Hamiltonian system on the cotangent bundle of Teichm\"uller space. Estimates of the Dirichlet energy of the induced metric play an essential role in the proof. 
  We give an analysis of the spin-weighted Green's functions well-defined in a conical space. We apply these results in the case of a straight cosmic string and in the Rindler space in order to determine generally the Euclidean Green's functions for the massless spin 1/2 field and for the electromagnetic field. We give also the corresponding Green's functions at zero temperature. However, except for the scalar field, it seems that these Euclidean Green's functions do not correspond to the thermal Feynman Green's functions. 
  Einstein's vacuum equations are solved up to the second approximation for imploding quadrupole gravitational waves. The implosion generates a black hole singularity irrespective of the strength of the waves. 
  We consider discontinuous signature change with the weak junction condition favoured by Ellis et. al. (1992). We impose certain regularity conditions and investigate the space of solutions (considered as one-parameter families of three-dimensional Riemannian manifolds) for dust and scalar field models. 
  We extend the Colombeau algebra of generalized functions to arbitrary (infinitely differentiable, paracompact) n-dimensional manifolds M. Embedding of continuous functions and distributions is achieved with the help of a family of n-forms defined on the tangent bundle TM, which form a partition of unity upon integration over the fibres. 
  The strictly isospectral double Darboux method is applied to the quantum Taub model in order to generate a one-parameter family of strictly isospectral potentials for this case. The family we build is based on a scattering Wheeler-DeWitt solution first discussed by Ryan and collaborators that we slightly modified according to a suggestion due to Dunster. The strictly isospectral Taub potentials possess different (attenuated) scattering states with respect to the original Taub potential 
  The critical steps leading to the uniqueness theorem for the Kerr-Newman metric are reexamined in the light of the new black hole solutions with Yang-Mills and scalar hair. Various methods -- including scaling techniques, arguments based on energy conditions, conformal transformations and divergence identities -- are reviewed, and their range of application to selfgravitating scalar and non-Abelian gauge fields is discussed. In particular, the no-hair theorem is extended to harmonic mappings with arbitrary Riemannian target manifolds. (This paper is an extended version of an invited lecture held at the Journ\'ees Relativistes 96.) 
  The classical field equations of general relativity can be expressed as a single geodesic equation, describing the free fall of a point particle in superspace. Based on this formulation, a ``worldline'' quantization of gravity, analogous to the Feynman-Schwinger treatment of particle propagation, is proposed, and a hidden mass-shell parameter is identified. We consider the effective action for the supermetric, which would be induced at one loop. In certain minisuperspace models, we find that this effective action is stationary for vanishing cosmological constant. 
  The slow roll approximation is studied for cosmological models in Hyperextended Scalar-Tensor Theories of Gravity. A procedure to obtain slow roll solutions in non-minimally coupled gravity is outlined and some examples are provided. An integral condition over the functional form of the non-minimal coupling is imposed in order to obtain intermediate inflationary behavior. 
  We study a general Scalar-Tensor Theory with an arbitrary coupling funtion $\omega (\phi )$ but also an arbitrary dependence of the ``gravitational constant'' $G(\phi )$ in the cases in which either one of them, or both, do not admit an analytical inverse, as in the hyperextended inflationary scenario. We present the full set of field equations and study their cosmological behavior. We show that different scalar-tensor theories can be grouped in classes with the same solution for the scalar field. 
  A systematic way of generating sets of local boundary conditions on the gauge fields in a path integral is presented. These boundary conditions are suitable for one--loop effective action calculations on manifolds with boundary and for quantum cosmology. For linearised gravity, the general proceedure described here leads to new sets of boundary conditions. 
  A classification of the possible symmetric principal bundles with a compact gauge group, a compact symmetry group and a base manifold which is regularly foliated by the orbits of the symmetry group is derived. A generalization of Wang's theorem (classifying the invariant connections) is proven and local expressions for the gauge potential of an invariant connection are given. 
  The first law of black hole mechanics (in the form derived by Wald), is expressed in terms of integrals over surfaces, at the horizon and spatial infinity, of a stationary, axisymmetric black hole, in a diffeomorphism invariant Lagrangian theory of gravity. The original statement of the first law given by Bardeen, Carter and Hawking for an Einstein-perfect fluid system contained, in addition, volume integrals of the fluid fields, over a spacelike slice stretching between these two surfaces. When applied to the Einstein-perfect fluid system, however, Wald's methods yield restricted results. The reason is that the fluid fields in the Lagrangian of a gravitating perfect fluid are typically nonstationary. We therefore first derive a first law-like relation for an arbitrary Lagrangian metric theory of gravity coupled to arbitrary Lagrangian matter fields, requiring only that the metric field be stationary. This relation includes a volume integral of matter fields over a spacelike slice between the black hole horizon and spatial infinity, and reduces to the first law originally derived by Bardeen, Carter and Hawking when the theory is general relativity coupled to a perfect fluid. We also consider a specific Lagrangian formulation for an isentropic perfect fluid given by Carter, and directly apply Wald's analysis. The resulting first law contains only surface integrals at the black hole horizon and spatial infinity, but this relation is much more restrictive in its allowed fluid configurations and perturbations than that given by Bardeen, Carter and Hawking. In the Appendix, we use the symplectic structure of the Einstein-perfect fluid system to derive a conserved current for perturbations of this system: this current reduces to one derived ab initio for this system by Chandrasekhar and Ferrari. 
  We use a systematic construction method for invariant connections on homogeneous spaces to find the Einstein-SU(n)-Yang-Mills equations for Friedmann-Robertson-Walker and locally rotationally symmetric homogeneous cosmologies. These connections depend on the choice of a homomorphism from the isotropy group into the gauge group. We consider here the cases of the gauge group SU(n) and SO(n) where these homomorphisms correspond to unitary or orthogonal representations of the isotropy group. For some of the simpler cases the full system of the evolution equations are derived, for others we only determine the number of dynamical variables that remain after some mild fixing of the gauge. 
  We describe how the iterative technique used by Isenberg and Moncrief to verify the existence of large sets of non constant mean curvature solutions of the Einstein constraints on closed manifolds can be adapted to verify the existence of large sets of asymptotically hyperbolic non constant mean curvature solutions of the Einstein constraints. 
  The consistent histories formulation of the quantum theory of a closed system with pure initial state defines an infinite number of incompatible consistent sets, each of which gives a possible description of the physics. We investigate the possibility of using the properties of the Schmidt decomposition to define an algorithm which selects a single, physically natural, consistent set. We explain the problems which arise, set out some possible algorithms, and explain their properties with the aid of simple models. Though the discussion is framed in the language of the consistent histories approach, it is intended to highlight the difficulty in making any interpretation of quantum theory based on decoherence into a mathematically precise theory. 
  Using a simple analysis based on the measurement procedure for a quantized area we explain the 1/4 factor in the Bekenstein-Hawking black hole formula A/4 for the entropy. 
  We discuss the entropy generation in quantum tunneling of a relativistic particle under the influence of a time varying force with the help of squeezing formalism. It is shown that if one associates classical coarse grained entropy to the phase space volume, there is an inevitable entropy increase due to the changes in position and momentum variances. The entropy change can be quantified by a simple expression $\Delta S=\ln\cosh 2r$, where $r$ is the squeeze parameter measuring the "height" and "width" of the potential barrier. We suggest that the universe could have acquired its initial entropy in a quantum squeeze from "nothing" and briefly discuss the implications of our proposal. 
  Einstein equations with $T_{\mu\nu} = k_\mu k_\nu + \ell_\mu \ell_\nu$ where $k, \ell$ are null are considered with spherical symmetry and staticity. The solution has naked singularity and is not asymptotically flat. However, it may be interpreted as an envelope for any static spherical body making it more massive. Such an interpretation and some of its implications are detailed. 
  We have calculated evolution of neutron star binaries towards the coalescence driven by gravitational radiation. The hydrodynamical effects as well as the general relativistic effects are important in the final phase. All corrections up to post$^{2.5}$-Newtonian order and the tidal effect are included in the orbital motion. The star is approximated by a simple Newtonian stellar model called affine star model. Stellar spins and angular momentum are assumed to be aligned. We have showed how the internal stellar structure affects the stellar deformation, variations of the spins, and the orbital motion of the binary just before the contact. The gravitational wave forms from the last a few revolutions significantly depend on the stellar structure. 
  We study the structure of linearized field equations in $N = 1$ chiral supergravity (SUGRA) with a complex tetrad, as a preliminary to introducing additional auxiliary fields in order that the supersymmetry (SUSY) algebra close off shell. We follow the first-order formulation we have recently constructed using the method of the usual $N = 1$ SUGRA. In particular, we see how the real and imaginary parts of the complex tetrad are coupled to matter fields in the weak field approximation. Starting from the linearized (free) theory of $N = 1$ chiral SUGRA, we then construct a Lagrangian which is invariant under local SUSY transformations to zeroth order of the gravitational constant, and compare the results with the linearized field equations. 
  We give a complete classification of all connected isometry groups, together with their actions in the asymptotic region, in asymptotically flat, asymptotically vacuum space--times with timelike ADM four--momentum. 
  We present new results for pulsating stars in general relativity. First we show that the so-called gravitational-wave modes of a neutron star can be excited when a gravitational wave impinges on the star. Numerical simulations suggest that the modes may be astrophysically relevant, and we discuss whether they will be observable with future gravitational-wave detectors. We also discuss how such observations could lead to estimates of both the radius and the mass of a neutron star, and thus put constraints on the nuclear equation of state. 
  The theory obtained as a singular limit of General Relativity, if the reciprocal velocity of light is assumed to tend to zero, is known to be not exactly the Newton-Cartan theory, but a slight extension of this theory. It involves not only a Coriolis force field, which is natural in this theory (although not original Newtonian), but also a scalar field which governs the relation between Newtons time and relativistic proper time. Both fields are or can be reduced to harmonic functions, and must therefore be constants, if suitable global conditions are imposed. We assume this reduction of Newton-Cartan to Newton`s original theory as starting point and ask for a consistent post-Newtonian extension and for possible differences to usual post-Minkowskian approximation methods, as developed, for example, by Chandrasekhar. It is shown, that both post-Newtonian frameworks are formally equivalent, as far as the field equations and the equations of motion for a hydrodynamical fluid are concerned. 
  By using conformal Killing-Yano tensors, and their generalisations, we obtain scalar potentials for both the source-free Maxwell and massless Dirac equations. For each of these equations we construct, from conformal Killing-Yano tensors, symmetry operators that map any solution to another. 
  I generalize the quasilocal formulation of thermodynamics of Brown and York to include dilaton theories of gravity as well as Abelian and Yang-Mills gauge matter fields with possible dilaton couplings. The resulting formulation is applicable to a large class of theories including two-dimensional toy models and the low energy limit of string theory as well as to many types of matter such as massless scalar fields, electromagnetism, Yang-Mills fields, and matter induced cosmological constants. I use this formalism to evaluate the thermodynamic variables for several black hole spacetimes. I find that the formulation can handle black hole spacetimes that are not asymptotically flat as well as rotating black hole spacetimes, and black hole spacetimes possessing a dilaton field, an electric charge, and a magnetic charge. 
  We calculate the gravitational radiation produced by the coalescence of inspiraling binary neutron stars in the Newtonian regime using 3-dimensional numerical simulations. The stars are modeled as polytropes and start out in the point-mass regime at wide separation. The hydrodynamic integration is performed using smooth particle hydrodynamics (SPH) with Newtonian gravity, and the gravitational radiation is calculated using the quadrupole approximation. We have run a number of simulations varying the neutron star radii, equations of state, spins, and mass ratio. The resulting gravitational waveforms and spectra are rich in information about the hydrodynamics of coalescence, and show characteristic dependence on GM/Rc^2, the equation of state, and the mass ratio. 
  Dilatonic Charged Nariai instantons mediate the nucleation of black hole pairs during extended chaotic inflation. Depending on the dilaton and inflaton fields, the black holes are described by one of two approximations in the Lorentzian regime. For each case we find Euclidean solutions that satisfy the no boundary proposal. The complex initial values of the dilaton and inflaton are determined, and the pair creation rate is calculated from the Euclidean action. Similar to standard inflation, black holes are abundantly produced near the Planck boundary, but highly suppressed later on. An unusual feature we find is that the earlier in inflation that the dilatonic black holes are created, the more highly charged they can be. 
  We consider space-times with two isometries which represent gravitational waves with distinct wavefronts which propagate into exact Friedmann-Robertson-Walker (FRW) universes. The geometry of possible wavefronts is analysed in detail in all three types of FRW models. In the spatially flat and open universes, the wavefronts can be planar or cylindrical; in the closed case they are toroidal. Exact solutions are given which describe gravitational waves propagating into the FRW universes with a fluid with a stiff equation of state. It is shown that the plane-fronted waves may include impulsive or step (shock) components, while the cylindrical waves in the spatially flat and open universes and the toroidal waves in closed universes may contain steps. In general, wavefronts may exist which have an arbitrary finite degree of smoothness. In all cases, the waves are backscattered. The head-on collision of such waves is also briefly mentioned. 
  After dimensional reduction the stationary spherically symmetric sector of Einstein's gravity is identified with an SL(2,R)/SO(2) Sigma model coupled to a one dimensional gravitational remnant. The space of classical solutions consists of a one parameter family interpolating between the Schwarzschild and the Taub-NUT solution. A Dirac Quantization of this system is performed and the observables -- the Schwarzschild mass and the Taub-NUT charge operator -- are shown to be self-adjoint operators with a continuous spectrum ranging from $-\infty$ to $\infty$. The Hilbert space is constructed explicitely using a harmonic space approach. 
  The ordinary quantum theory points out that general relativity is negligible for spatial distances up to the Planck scale. Consistency in the foundations of the quantum theory requires a``soft'' spacetime structure of the general relativity at essentially longer length. However, for some reasons this appears to be not enough. A new framework (``superrelativity'') for the desirable generalization of the foundation of quantum theory is proposed. A generalized non-linear Klein-Gordon equation has been derived in order to shape a stable wave packet. 
  The aim of this article is twofold. First we examine from a new angle the question of recovery of time in quantum cosmology. We construct Green functions for matter fields from the solutions of the Wheeler De Witt equation. For simplicity we work in a mini-superspace context. By evaluating these Green functions in a first order development of the energy ``increment'' induced by matrix elements of field operators, we show that the background geometry is the solution of Einstein equations driven by the mean matter energy and that it is this background which determines the time lapses separating the field operators. Then, by studying higher order corrections, we clarify the nature of the small dimensionless parameters which guarantee the validity of the approximations used. In this respect, we show that the formal expansion in the inverse Planck mass which is sometime presented as the ``standard procedure'' is illegitimate. Secondly, by the present analysis of Green functions, we prepare the study of quantum matter transitions in quantum cosmology. In a next article, we show that the time parametrization of transition amplitudes appears for the same reasons that it appeared in this article. This proves that the background is dynamically determined by the transition under examination. 
  We describe radiative processes in Quantum Cosmology, from the solutions of the Wheeler De Witt equation. By virtue of this constraint equation, the quantum propagation of gravity is modified by the matter interaction hamiltonian at the level of amplitudes. In this we generalize previous works where gravity was coupled only to expectation values of matter operators. By a ``reduction formula'' we show how to obtain transition amplitudes from the entangled gravity+matter system. Then we show how ``each'' transition among matter constituents of the universe determines dynamically ``its'' background from which a time parameter is defined. Finally, we leave the mini-superspace context by introducing an extended formalism in which the momenta of the exchanged quanta no longer vanish. Then, the concept of spatial displacement emerges from radiative processes like the time parametrization did, thereby unifying the way by which space and time intervals are recovered in quantum cosmology. 
  The use of internal variables for the description of relativistic particles with arbitrary mass and spin in terms of scalar functions is reviewed and applied to the stochastic phase space formulation of quantum mechanics. Following Bacry and Kihlberg a four-dimensional internal spin space $\bar S$ is chosen possessing an invariant measure and being able to represent integer as well as half integer spins. $\bar S$ is a homogeneous space of the group $SL(2, C)$ parametrized in terms of spinors $\a\in C_2$ and their complex conjugates $\bar\a$. The generalized scalar quantum mechanical wave functions may be reduced to yield irreducible components of definite physical mass and spin $[m,s]$, with $m\ge 0$ and $s=0, 1/2, 1, 3/2 ...$, with spin described in terms of the usual $(2s+1)$-component fields. Viewed from the internal space description of spin this reduction amounts to a restriction of the variable $\a$ to a compact subspace of $\bar S$, i.e. a ``spin shell'' $S^2_{r=2s}$ of radius $r=2s$ in $ C_2$. This formulation of single particles or single antiparticles of type $[m,s]$ is then used to study the geometro- stochastic (i.e. quantum) propagation of amplitudes for arbitrary spin on a curved background space-time possessing a metric and axial vector torsion treated as external fields. A Poincar\'e gauge covariant path integral-like representation for the probability amplitude (generalized wave function) of a particle with arbitrary spin is derived satisfying a second order wave equation on the Hilbert bundle constructed over curved space-time. The implications for the stochastic nature of polarization effects in the presence of gravitation are pointed out and the extension to Fock bundles of bosonic and fermionic type is briefly mentioned. 
  We define a new paradigm --- postrelativity --- based on the hypothesis of a preferred hidden Newtonian frame in relativistic theories. It leads to a modification of general relativity with ether interpretation, without topological problems, black hole and big bang singularities. Semiclassical theory predicts Hawking radiation with evaporation before horizon formation. In quantum gravity there is no problem of time and topology. Configuration space and quasiclassical predictions are different from canonical quantization of general relativity. Uncertainty of the light cone or an atomic structure of the ether may solve ultraviolet problems. The similar concept for gauge fields leads to real, physical gauge potential without Faddeev-Popov ghost fields and Gribov copy problem. 
  We compute the quasinormal frequencies of the Kerr black hole using a continued fraction method. The continued fraction method first proposed by Leaver is still the only known method stable and accurate for the numerical determination of the Kerr quasinormal frequencies. We numerically obtain not only the slowly but also the rapidly damped quasinormal frequencies and analyze the peculiar behavior of these frequencies at the Kerr limit. We also calculate the algebraically special frequency first identified by Chandrasekhar and confirm that it coincide with the $n=8$ quasinormal frequency only at the Schwarzschild limit. 
  We solve the linearized Einstein equations for a specific oscillating mass distribution and discuss the usual counterarguments against the existence of observable gravitational retardations in the "near zone", where d/r << 1 (d = oscillation amplitude of the source, r = distance from the source). We show that they do not apply in the region d/r \approx 1, and prove that gravitational forces are retarded in the immediate vicinity of the source. An experiment to measure this retardation is proposed, which may provide the first direct experimental observation of propagating gravitational fields. 
  The first order corrections to the geometry of the (2+1)-dimensional black hole due to back-reaction of a massless conformal scalar field are computed. The renormalized stress energy tensor used as the source of Einstein equations is computed with the Green function for the black-hole background with transparent boundary conditions. This tensor has the same functional form as the one found in the nonperturbative case which can be exactly solved. Thus, a static, circularly symmetric and asymptotically anti-de Sitter black hole solution of the semiclassical equations is found. The corrections to the thermodynamic quantities are also computed. 
  We continue the study of the existence and stability of static spherical membrane configurations in curved spacetimes. We first consider higher order membranes described by a Lagrangian which, besides the Dirac term, includes a term proportional to the scalar curvature of the world--volume ${}^{(3)}R$. Notably, in this case, the equations of motion can be reduced to second order ones and an effective potential analysis can be made. The conditions for stability are then explicitly derived. We find a self--consistent static spherical membrane, determining the spacetime generated by the membrane itself. In this case we find, however, that the total energy of the membrane has to be negative, and no {\it stable} equilibrium can be achieved. We then generalize the discussion to a membrane described by a Lagrangian including all possible second derivative terms. We conclude the paper with some discussion on the generality of the results obtained. 
  The Levi-Civita (LC) solution is matched to a cylindrical shell of an anisotropic fluid. The fluid satisfies the energy conditions when the mass parameter $\sigma$ is in the range $0 \le \sigma \le 1$. The mass per unit length of the shell is given explicitly in terms of $\sigma$, which has a finite maximum. The relevance of the results to the non-existence of horizons in the LC solution and to gauge cosmic strings is pointed out. 
  The problem of determining the electromagnetic and gravitational ``self-force'' on a particle in a curved spacetime is investigated using an axiomatic approach. In the electromagnetic case, our key postulate is a ``comparison axiom'', which states that whenever two particles of the same charge $e$ have the same magnitude of acceleration, the difference in their self-force is given by the ordinary Lorentz force of the difference in their (suitably compared) electromagnetic fields. We thereby derive an expression for the electromagnetic self-force which agrees with that of DeWitt and Brehme as corrected by Hobbs. Despite several important differences, our analysis of the gravitational self-force proceeds in close parallel with the electromagnetic case. In the gravitational case, our final expression for the (reduced order) equations of motion shows that the deviation from geodesic motion arises entirely from a ``tail term'', in agreement with recent results of Mino et al. Throughout the paper, we take the view that ``point particles'' do not make sense as fundamental objects, but that ``point particle equations of motion'' do make sense as means of encoding information about the motion of an extended body in the limit where not only the size but also the charge and mass of the body go to zero at a suitable rate. Plausibility arguments for the validity of our comparison axiom are given by considering the limiting behavior of the self-force on extended bodies. 
  We take the general quantum constraints of N=1 supergravity in the special case of a Bianchi metric, with gravitino fields constant in the invariant basis. We construct the most general possible wave function which solves the Lorentz constraints and study the supersymmetry constraints in the Bianchi Class A Models. For the Bianchi-IX cases, both the Hartle-Hawking state and wormhole state are found to exist in the middle fermion levels. 
  We study a black hole radiation inside the apparent horizon in quantum gravity. First we perform a canonical quantization for spherically symmetric geometry where one of the spatial coordinates is dealt as the time variable since we would like to consider the interior region of a black hole. Next this rather general formalism is applied for a specific model where the ingoing Vaidya metric is used as a simple model of an evaporating black hole. Following Tomimatsu's idea, we will solve analytically the Wheeler-DeWitt equation in the vicinity of the apparent horizon and see that mass-loss rate of a black hole by thermal radiation is equal to the result obtained by Hawking in his semiclassical treatment. The present formalism may have a wide application in quantum gravity inside the horizon of a black hole such as mass inflation and strong cosmic censorship etc. 
  We present a toy model for growing wormholes as a model of effective low-energy topology changes. We study the propagation of quantum fields on a $1+1$ spacetime analogous to the trouser-leg topology change. A low-energy effective topology change is produced by a physical model which corresponds to a barrier smoothly changing the tunneling probability between two spatial regions. 
  The dynamics of gravitational waves is investigated in full 3+1 dimensional numerical relativity, emphasizing the difficulties that one might encounter in numerical evolutions, particularly those arising from non-linearities and gauge degrees of freedom. Using gravitational waves with amplitudes low enough that one has a good understanding of the physics involved, but large enough to enable non-linear effects to emerge, we study the coupling between numerical errors, coordinate effects, and the nonlinearities of the theory. We discuss the various strategies used in identifying specific features of the evolution. We show the importance of the flexibility of being able to use different numerical schemes, different slicing conditions, different formulations of the Einstein equations (standard ADM vs. first order hyperbolic), and different sets of equations (linearized vs. full Einstein equations). A non-linear scalar field equation is presented which captures some properties of the full Einstein equations, and has been useful in our understanding of the coupling between finite differencing errors and non-linearites. We present a set of monitoring devices which have been crucial in our studying of the waves, including Riemann invariants, pseudo-energy momentum tensor, hamiltonian constraint violation, and fourier spectrum analysis. 
  Gravitational radiation can be expressed in terms of an infinite series of radiative, symmetric trace-free (STF) multipole moments which can be connected to the behavior of the source. We consider a truncated model for gravitational radiation from binary systems in which each STF mass and current moment of order l is given by the lowest-order, Newtonian-like l-pole moment of the orbiting masses; we neglect post-Newtonian corrections to each STF moment. Specializing to orbits which are circular (apart from the radiation-induced inspiral), we find an explicit infinite series for the energy flux in powers of $v/c$, where v is the orbital velocity. We show that the series converges for all values $v/c < 2/e$ when one mass is much smaller than the other, and $v/c < 4/e$ for equal masses,where e is the base of natural logarithms. These values include all physically relevant values for compact binary inspiral. This convergence cannot indicate whether or not the full series (obtained from the exact moments) will converge. But if the full series does not converge, our analysis shows that this failure to converge does not originate from summing over the Newtonian part of the multipole moments. 
  We present a set of equations describing the cosmological gravitational wave in a gravity theory with quadratic order gravitational coupling terms which naturally arise in quantum correction procedures. It is known that the gravitational wave equation in the gravity theories with a general $f(R)$ term in the action leads to a second order differential equation with the only correction factor appearing in the damping term. The case for a $R^{ab} R_{ab}$ term is completely different. The gravitational wave is described by a fourth order differential equation both in time and space. However, curiously, we find that the contributions to the background evolution are qualitatively the same for both terms. 
  The quantization of gravitational waves in the Milne universe is discussed. The relation between positive frequency functions of the gravitational waves in the Milne universe and those in the Minkowski universe is clarified. Implications to the one-bubble open inflation scenario are also discussed. 
  Self-similarity in general relativity is briefly reviewed and the differences between self-similarity of the first kind and generalized self-similarity are discussed. The covariant notion of a kinematic self-similarity in the context of relativistic fluid mechanics is defined. Various mathematical and physical properties of spacetimes admitting a kinematic self-similarity are discussed. The governing equations for perfect fluid cosmological models are introduced and a set of integrability conditions for the existence of a proper kinematic self-similarity in these models is derived. Exact solutions of the irrotational perfect fluid Einstein field equations admitting a kinematic self-similarity are then sought in a number of special cases, and it is found that; (1) in the geodesic case the 3-spaces orthogonal to the fluid velocity vector are necessarily Ricci-flat and (ii) in the further specialisation to dust the differential equation governing the expansion can be completely integrated and the asymptotic properties of these solutions can be determined, (iii) the solutions in the case of zero-expansion consist of a class of shear-free and static models and a class of stiff perfect fluid (and non-static) models, and (iv) solutions in which the kinematic self-similar vector is parallel to the fluid velocity vector are necessarily Friedmann-Robertson-Walker (FRW) models. 
  Kossowski and Kriele derived boundary conditions on the metric at a surface of signature change. We point out that their derivation is based not only on certain smoothness assumptions but also on a postulated form of the Einstein field equations. Since there is no canonical form of the field equations at a change of signature, their conclusions are not inescapable. We show here that a weaker formulation is possible, in which less restrictive smoothness assumptions are made, and (a slightly different form of) the Einstein field equations are satisfied. In particular, in this formulation it is possible to have a bounded energy-momentum tensor at a change of signature without satisfying their condition that the extrinsic curvature vanish. 
  The use of proper ``time'' to describe classical ``spacetimes'' which contain both Euclidean and Lorentzian regions permits the introduction of smooth (generalized) orthonormal frames. This remarkable fact permits one to describe both a variational treatment of Einstein's equations and distribution theory using straightforward generalizations of the standard treatments for constant signature. 
  We discuss Einstein's field equations in the presence of signature change using variational methods, obtaining a generalization of the Lanczos equation relating the distributional term in the stress tensor to the discontinuity of the extrinsic curvature. In particular, there is no distributional term in the stress tensor, and hence no surface layer, precisely when the extrinsic curvature is continuous, in agreement with the standard result for constant signature. 
  We examine the effect of local matter on the chaotic behavior of a relativistic test particle in non-vacuum static axisymmetric spacetimes. We find that the sign of the sectional curvature in the geodesic deviation equation defined by the Riemann curvature does not always become a good tool to judge the occurrence of chaos in the non-vacuum case. However, we show that the locally unstable region ( LU region ) defined by the Weyl curvature can provide information about chaos even in non-vacuum spacetime as well as in vacuum spacetime. Since the Weyl tensor affects only the shear part of the geodesic congruence, it works effectively to stretch some directions of geodesic congruence, which helps to cause the chaotic behavior of geodesics. Actually, the orbit moving around an unstable periodic orbit (UPO) becomes strongly chaotic if it passes through an LU region, which means that the LU region can be used as a good tool to know in which situation the chaos by homoclinic mixing occurs around a UPO. 
  A structured collection of thought provoking conclusions about space and time is given. Using only the Compton wavelength lambda = hbar / m c and the Schwarzschild radius r_s = 2 G m / c^2, it is argued that neither the continuity of space-time nor the concepts of space-point, instant, or point particle have experimental backing at high energies. It is then deduced that Lorentz, gauge, and discrete symmetries are not precisely fulfilled in nature. In the same way, using a simple and new Gedankenexperiment, it is found that at Planck energies, vacuum is fundamentally indistinguishable from radiation and from matter. Some consequences for supersymmetry, duality, and unification are presented. 
  Gravity is treated as a stochastic phenomenon based on fluctuations of the metric tensor of general relativity. By using a (3+1) slicing of spacetime, a Langevin equation for the dynamical conjugate momentum and a Fokker-Planck equation for its probability distribution are derived. The Raychaudhuri equation for a congruence of timelike or null geodesics leads to a stochastic differential equation for the expansion parameter $\theta$ in terms of the proper time $s$. For sufficiently strong metric fluctuations, it is shown that caustic singularities in spacetime can be avoided for converging geodesics. The formalism is applied to the gravitational collapse of a star and the Friedmann-Robertson-Walker cosmological model. It is found that owing to the stochastic behavior of the geometry, the singularity in gravitational collapse and the big-bang have a zero probability of occurring. Moreover, as a star collapses the probability of a distant observer seeing an infinite red shift at the Schwarzschild radius of the star is zero. Therefore, there is a vanishing probability of a Schwarzschild black hole event horizon forming during gravitational collapse. 
  With the exception of gravitation, the known fundamental interactions of Nature are mediated by gauge fields. A comparison of the candidate groups for a gauge theory possibly describing gravitation favours the Poincar\'e group as the obvious choice. This theory gives Einstein's equations in a particular case, and Newton's law in the static non-relativistic limit, being seemingly sound at the classical level. But it comes out that it is not quantizable. The usual procedure of adding counterterms to make it a consistent and renormalizable theory leads to two possible theories, one for each of the two de Sitter groups, SO(4,1) and SO(3,2). The consequences of changing from the Poincar\'e to the de Sitter group, as well as the positive aspects, perspectives and drawbacks of the resulting theory are discussed. 
  We confirm recent numerical results of echoing and mass scaling in the gravitational collapse of a spherical Yang-Mills field by constructing the critical solution and its perturbations as an eigenvalue problem. Because the field equations are not scale-invariant, the Yang-Mills critical solution is asymptotically, rather than exactly, self-similar, but the methods for dealing with discrete self-similarity developed for the real scalar field can be generalized. We find an echoing period Delta = 0.73784 +/- 0.00002 and critical exponent for the black hole mass gamma = 0.1964 +/- 0.0007. 
  Analytical wormhole solutions in Brans-Dicke theory in the presence of matter are presented. It is shown that the wormhole throat must not be necessarily threaded with exotic matter. 
  We consider the Hamiltonian dynamics and thermodynamics of spherically symmetric spacetimes within a one-parameter family of five-dimensional Lovelock theories. We adopt boundary conditions that make every classical solution part of a black hole exterior, with the spacelike hypersurfaces extending from the horizon bifurcation three-sphere to a timelike boundary with fixed intrinsic metric. The constraints are simplified by a Kucha\v{r}-type canonical transformation, and the theory is reduced to its true dynamical degrees of freedom. After quantization, the trace of the analytically continued Lorentzian time evolution operator is interpreted as the partition function of a thermodynamical canonical ensemble. Whenever the partition function is dominated by a Euclidean black hole solution, the entropy is given by the Lovelock analogue of the Bekenstein-Hawking entropy; in particular, in the low temperature limit the system exhibits a dominant classical solution that has no counterpart in Einstein's theory. The asymptotically flat space limit of the partition function does not exist. The results indicate qualitative robustness of the thermodynamics of five-dimensional Einstein theory upon the addition of a nontrivial Lovelock term. 
  The covariant kinetic approach for the radiative plasma, a mixture of a relativistic moving gas plus radiation quanta (photons, neutrinos, or gravitons) is generalized to D spatial dimensions. The operational and physical meaning of Eckart's temperature is reexamined and the D-dimensional expressions for the transport coefficients (heat conduction, bulk and shear viscosity) are explicitly evaluated to first order in the mean free time of the radiation quanta. Weinberg's conclusion that the mixture behaves like a relativistic imperfect simple fluid (in Eckart's formulation) depends neither on the number of spatial dimensions nor on the details of the collisional term. The case of Thomson scaterring is studied in detail, and some consequences for higher dimensional cosmologies are also discussed. 
  We study in this paper different topos-theoretical approaches to the problem of construction of General Theory of Relativity. In general case the resulting space-time theory will be non-classical, different from that of the usual Einstein theory of space-time. This is a new theory of space-time, created in a purely logical manner. Four possibitities are investigated: axiomatic approach to causal theory of space-time, the smooth toposes as a models of Theory of Relativity, Synthetic Theory of Relativity, and space-time as Grothendieck topos. 
  A careful analysis of the gravitational geon solution found by Brill and Hartle is made. The gravitational wave expansion they used is shown to be consistent and to result in a gauge invariant wave equation. It also results in a gauge invariant effective stress-energy tensor for the gravitational waves provided that a generalized definition of a gauge transformation is used. To leading order this gauge transformation is the same as the usual one for gravitational waves. It is shown that the geon solution is a self-consistent solution to Einstein's equations and that, to leading order, the equations describing the geometry of the gravitational geon are identical to those derived by Wheeler for the electromagnetic geon. An appendix provides an existence proof for geon solutions to these equations. 
  A recent paper by Hartle [Phys. Rev. D 51, 1800 (1995)] proposes a definition of "spacetime information" - the information available about a quantum system's boundary conditions in the various sets of decohering histories it may display - and investigates its properties. We note here that Hartle's analysis contains errors which invalidate several of the conclusions. In particular, the proof that the proposed definition agrees with the standard definition for ordinary quantum mechanics is invalid, the evaluations of the spacetime information for time-neutral generalized quantum theories and for generalized quantum theories with non-unitary evolution are incorrect, and the argument that spacetime information is conserved on spacelike surfaces in these last theories is erroneous. We show however that the proposed definition does, in fact, agree with the standard definition for ordinary quantum mechanics. Hartle's definition relies on choosing, case by case, a class of fine-grained consistent sets of histories. We supply a general definition of the relevant class of sets that agrees with Hartle's definition in the cases explicitly considered and that generalizes to other cases. 
  The Cauchy+characteristic matching (CCM) problem for the scalar wave equation is investigated in the background geometry of a Schwarzschild black hole. Previously reported work developed the CCM framework for the coupled Einstein-Klein-Gordon system of equations, assuming a regular center of symmetry. Here, the time evolution after the formation of a black hole is pursued, using a CCM formulation of the governing equations perturbed around the Schwarzschild background. An extension of the matching scheme allows for arbitrary matching boundary motion across the coordinate grid. As a proof of concept, the late time behavior of the dynamics of the scalar field is explored. The power-law tails in both the time-like and null infinity limits are verified. 
  I describe how the states of a discrete automata with p sites, each of which may be off or on, can be represented as Majorana spinors associated to a spacetime with signature (p,p). Some ideas about the quantization of such systems are discussed and the relationship to some unconventional formulations and generalizations of quantum mechanics, particularly Jordan's spinorial quantum mechanics are pointed out. A connection is made to the problem of time and the complex numbers in quantum gravity. 
  We revisit the issue of integrability conditions for the irrotational silent cosmological models. We formulate the problem both in 1+3 covariant and 1+3 orthonormal frame notation, and show there exists a series of constraint equations that need to be satisfied. These conditions hold identically for FLRW-linearised silent models, but not in the general exact non-linear case. Thus there is a linearisation instability, and it is highly unlikely that there is a large class of silent models. We conjecture that there are no spatially inhomogeneous solutions with Weyl curvature of Petrov type I, and indicate further issues that await clarification. 
  In a cosmological context, the electric and magnetic parts of the Weyl tensor, E_{ab} and H_{ab}, represent the locally free curvature - i.e. they are not pointwise determined by the matter fields. By performing a complete covariant decomposition of the derivatives of E_{ab} and H_{ab}, we show that the parts of the derivative of the curvature which are locally free (i.e. not pointwise determined by the matter via the Bianchi identities) are exactly the symmetrised trace-free spatial derivatives of E_{ab} and H_{ab} together with their spatial curls. These parts of the derivatives are shown to be crucial for the existence of gravitational waves. 
  Recently M. Kamata and T. Koikawa claimed to obtain the charged version of the spinning BTZ black hole solution by assuming an (anti-) self dual condition imposed on the orthonormal basis components of the electric and magnetic fields. We point out that the Kamata-Koikawa field is not a solution of the Einstein-Maxwell equations and we find the correct solution of the studied problem. It is shown that a duality mapping exists among spinning solutions obtained from electrostatic and magnetostatic fields with the help of the local transformation $t\longrightarrow t-\omega \theta \theta \longrightarrow \theta -\omega t.$ 
  After a brief overview of the so-called silent models and their present status, we consider the subclass of Bianchi Type--I models with a magnetic field source. Due to the presence of the magnetic field, the initial singularity shows ``oscillatory'' features reminiscent of the Bianchi Type--IX case. The Bianchi Type--I models with a magnetic field are therefore a counterexample to the folklore that matter fields can be neglected in the vicinity of the singularity. 
  Considering quantum cosmological minisuperspace models with positive potential, we present evidence that    (i) despite common belief there are perspectives for defining a unique, naturally preferred decomposition of the space H of wave functions into two subspaces H^\pm that generalizes the concept of positive and negative frequency, and that    (ii) an underlying unitary evolution within these two subspaces exists and may be described in analogy to the representation of a geometric object in local coordinates: it is associated with the choice of a congruence of classical trajectories endowed with a suitable weight (such a setting is called WKB-branch).    The transformation properties of various quantities under a variation of the WKB-branch provide the tool for defining the decomposition. The construction leads to formal series whose actual convergence seems to require additional conditions on the model (related to global geometric issues and possibly to analyticity). It is speculated that this approach might relate to the refined algebraic quantization program. 
  We present new exact inhomogeneous vacuum cosmological solutions of Einstein's equations. They provide new information about the nature of general cosmological solutions to Einstein's equations. 
  We present the exact solution of two-body motion in (1+1) dimensional dilaton gravity by solving the constraint equations in the canonical formalism. The determining equation of the Hamiltonian is derived in a transcendental form and the Hamiltonian is expressed for the system of two identical particles in terms of the Lambert $W$ function. The $W$ function has two real branches which join smoothly onto each other and the Hamiltonian on the principal branch reduces to the Newtonian limit for small coupling constant. On the other branch the Hamiltonian yields a new set of motions which can not be understood as relativistically correcting the Newtonian motion. The explicit trajectory in the phase space $(r, p)$ is illustrated for various values of the energy. The analysis is extended to the case of unequal masses. The full expression of metric tensor is given and the consistency between the solution of the metric and the equations of motion is rigorously proved. 
  We find a self-consistent pp-gravitational shock wave solution to the semiclassical Einstein equations resulting from the $1/N$ approach to the effective action. We model the renormalized matter stress-energy-momentum tensor by $N$ massless scalar fields in the Minkowski vacuum plus a classical particle. We show that quantum effects generate a milder singularity at the position of the particle than the classical solution, but the singularity does not disappear. At large distances from the particle, the quantum correction decreases slowly, as $1/\rho^2$ ($\rho$ being the distance to the particle in the shock wave plane). We argue that this large distance correction is a necessary consequence of quantum gravity. 
  The concept of time is discussed in the context of the canonical formulation of the gravitational field. Using a hypersurface orthogonal foliation, the arbitrariness of the lapse function is eliminated and the shift vector vanishes, allowing a consistent definition of time. 
  It is shown that a transition from a multidimensional cosmological model with one internal space of the dimension d_1 to the effective tree-level bosonic string corresponds to an infinite number of the internal dimensions: d_1 -> infinity. 
  For quantum fields on a curved spacetime with an Euclidean section, we derive a general expression for the stress energy tensor two-point function in terms of the effective action. The renormalized two-point function is given in terms of the second variation of the Mellin transform of the trace of the heat kernel for the quantum fields. For systems for which a spectral decomposition of the wave opearator is possible, we give an exact expression for this two-point function. Explicit examples of the variance to the mean ratio $\Delta' = (<\rho^2>-<\rho>^2)/(<\rho>^2)$ of the vacuum energy density $\rho$ of a massless scalar field are computed for the spatial topologies of $R^d\times S^1$ and $S^3$, with results of $\Delta'(R^d\times S^1) =(d+1)(d+2)/2$, and $\Delta'(S^3) = 111$ respectively. The large variance signifies the importance of quantum fluctuations and has important implications for the validity of semiclassical gravity theories at sub-Planckian scales. The method presented here can facilitate the calculation of stress-energy fluctuations for quantum fields useful for the analysis of fluctuation effects and critical phenomena in problems ranging from atom optics and mesoscopic physics to early universe and black hole physics. 
  We consider a monopole detector interacting with a massive scalar field. The radiative processes are discussed from the accelerated frame point of view. After this, we obtain the Minkowski vacuum stress tensor measured by the accelerated observer using a non-gravitational stress tensor detector as discussed by Ford and Roman (PRD 48, 776 (1993)). Finally, we analyse radiative processes of the monopole detector travelling in a world line that is inertial in the infinite past and has a constant proper acceleration in the infinite future. 
  The analytical solutions reported in our previous paper are given as series of hypergeometric or Coulomb wave functions. By using them, we can get the Teukolsky functions analytically in a desired accuracy. For the computation, the deep understanding of their properties is necessary. We summarize the main result: The relative normalization between the solutions with a spin weight s and -s is given analytically by using the Teukolsky-Starobinsky (T-S) identities. By examining the asymptotic behaviors of our solution and combined with the T-S identities and the Wronskian, we found nontrivial identities between the sums of coefficients of the series. These identities will serve to make various expression in simpler forms and also become a powerful tool to test the accuracy of the computation. As an application, we investigated the absorption rate and the evaporation rate of black hole and obtain interesting analytic results. 
  We extend the work done for cosmic strings and show that for a more general class of locally flat metrics the one loop calculation do not introduce any new divergences to the VEV of the energy of a scalar particle. We explicitly perform the calculation for the configuration where we have one cosmic string in the presence of a dipole made out of cosmic strings. 
  When quantum mechanical and general relativistic effects are taken into account in the analysis of distance measurements, one finds a measurability bound. I observe that some of the structures that have been encountered in the literature on the Quantum $\kappa$-Poincar\`e Group naturally lead to this bound. 
  The generalized second law of black hole thermodynamics was proved by Frolov and Page for a quasi-stationary eternal black hole. However, realistic black holes arise from a gravitational collapse, and in this case their proof does not hold. In this paper we prove the generalized second law for a quasi-stationary black hole which arises from a gravitational collapse. 
  We carry out the first step of a program conceived, in order to build a realistic model, having the particle spectrum of the standard model and renormalized masses, interaction terms and couplings, etc. which include the class of quantum gravity corrections, obtained by handling gravity as an effective theory. This provides an adequate picture at low energies, i.e. much less than the scale of strong gravity (the Planck mass). Hence our results are valid, irrespectively of any proposal for the full quantum gravity as a fundamental theory. We consider only non-analytic contributions to the one-loop scattering matrix elements, which provide the dominant quantum effect at long distance. These contributions are finite and independent from the finite value of the renormalization counter terms of the effective lagrangian. We calculate the interaction of two heavy scalar particles, i.e. close to rest, due to the effective quantum gravity to the one loop order and compare with similar results in the literature. 
  Transition from quantum to semiclassical behaviour and loss of quantum coherence for inhomogeneous perturbations generated from a non-vacuum initial state in the early Universe is considered in the Heisenberg and the Schr\"odinger representations, as well as using the Wigner function. We show explicitly that these three approaches lead to the same prediction in the limit of large squeezing (i.e. when the squeezing parameter $|r_k|\to \infty$): each two-modes quantum state (k, -k) of these perturbations is equivalent to a classical perturbation that has a stochastic amplitude, obeying a non-gaussian statistics which depends on the initial state, and that belongs to the quasi-isotropic mode (i.e. it possesses a fixed phase). The Wigner function is not everywhere positive for any finite $r_k$, hence its interpretation as a classical distribution function in phase space is impossible without some coarse graining procedure. However, this does not affect the transition to semiclassical behaviour since the Wigner function becomes concentrated near a classical trajectory in phase space when $|r_k|\to \infty$ even without coarse graining. Deviations of the statistics of the perturbations in real space from a Gaussian one lie below the cosmic variance level for the N-particles initial states with N=N(|k|) but may be observable for other initial states without statistical isotropy or with correlations between different k modes. As a way to look for this effect, it is proposed to measure the kurtosis of the angular fluctuations of the cosmic microwave background temperature. 
  It is considered, in the framework of constrained systems, the quantum dynamics of non-relativistic particles moving on a d-dimensional Riemannian manifold M isometrically embedded in $R^{d+n}$. This generalizes recent investigations where M has been assumed to be a hypersurface of $R^{d+1}$. We show, contrary to recent claims, that constrained systems theory does not contribute to the elimination of the ambiguities present in the canonical and path integral formulations of the problem. These discrepancies with recent works are discussed. 
  We discuss how to describe U(N)-monopoles on the Schwarzschild and Reissner-Nordstr\"om black holes by the parameters of the moduli space of holomorphic vector bundles over S^2. For N = 2,3 we obtain such a description in an explicit form as well as the expressions for the corresponding monopole masses. This gives a possibility to adduce some reasonings in favour of existence of both a 'fine structure' for black holes and the statistical ensemble tied with it which might generate the black hole entropy. Also there arises some analogy with the famous K-theory in topology. 
  The close-limit method has given approximations in excellent agreement with those of numerical relativity for collisions of equal mass black holes. We consider here colliding holes with unequal mass, for which numerical relativity results are not available. We try to ask two questions: (i) Can we get approximate answers to astrophysical questions (ideal mass ratio for energy production, maximum recoil velocity, etc.), and (ii) can we better understand the limitations of approximation methods. There is some success in answering the first type of question, but more with the second, especially in connection with the issue of measures of the intrinsic mass of the colliding holes, and of the range of validity of the method. 
  The possibility of a gravitational phase transition, especially with respect to neutron stars is investigated. First, a semiclassical treatment is given, predicting a gravitational London penetration depth of 12km for neutron stars. Second, the problem is considered from a Ginzburg-Landau point of view. A gravitational Meissner effect, a gravitational Aharanov-Bohm type effect and a gravitational ferromagnetic type phase are predicted. Finally, a field theoretic consideration predicts a mass term for the graviton below a certain critical temperature. 
  We study the form of the Turaev-Viro partition function Z(M) for different 3-manifolds with boundary. We show that for $S^2$ boundaries Z(M) factorizes into a term which contains the boundary dependence and another which depends only on the topology of the underlying manifold. From this follows easily the formula for the connected sum of two manifolds Z(M # N). For general $T_g$ boundaries this factorization holds only in a particular case. 
  The interaction of the unboundedly blue-shifted photons of the cosmic microwave background radiation with a physical object falling towards the inner horizon of a Reissner-Nordstrom black hole is analyzed. To evaluate this interaction we consider the QED effects up to the second order in the perturbation expansion. We then extrapolate the QED effects up to a cutoff, which we introduce at the Planckian level. (Our results are not sensitive to the cutoff energy.) We find that the energy absorbed by an infalling observer is finite, and for typical parameters would not lead to a catastrophic heating. However, this interaction would almost certainly be fatal for a human being, or other living organism of similar size. On the other hand, we find that smaller objects may survive the interaction. Our results do not provide support to the idea that the Cauchy horizon is to be regarded as the boundary of spacetime. 
  A review is made of recent efforts to find relations between the commutation relations which define a noncommutative geometry and the gravitational field which remains as a shadow in the commutative limit. 
  A small fixed-mirror Michelson interferometer has been built in Frascati to experimentally study the alignment method that has been suggested for VIRGO. The experimental results fully confirm the adequacy of the method. The minimum angular misalignment that can be detected in the present set-up is 10 nrad/sqrt{Hz} 
  We apply the causal interpretation of quantum mechanics to homogeneous quantum cosmology and show that the quantum theory is independent of any time-gauge choice and there is no issue of time. We exemplify this result by studying a particular minisuperspace model where the quantum potential driven by a prescribed quantum state prevents the formation of the classical singularity, independently on the choice of the lapse function. This means that the fast-slow-time gauge conjecture is irrelevant within the framework of the causal interpretation of quantum cosmology. 
  In this comment we bring attention to the fact that when we apply the ontological interpretation of quantum mechanics, we must be sure to use it in the coordinate representation. This is particularly important when canonical tranformations that mix momenta and coordinates are present. This implies that some of the results obtained by A. B\l aut and J. Kowalski-Glikman are incorrect. 
  We give an SU(2) covariant representation of the constraints of Euclidean general relativity in the Ashtekar variables. The guiding principle is the use of triads to transform all free spatial indices into SU(2) indices. A central role is played by a special covariant derivative. The Gauss, diffeomorphism and Hamiltonian constraints become purely algebraic restrictions on the curvature and the torsion associated with this connection. We introduce coordinates on the jet space of the dynamical fields which cleanly separate the constraint and gauge directions from the true physical directions. This leads to a classification of all local diffeomorphism and Gauss invariant charges. 
  Unlike general relativity, scalar-tensor theories of gravity predict scalar gravitational waves even from a spherically symmetric gravitational collapse. We solve numerically the generation and propagation of the scalar gravitational wave from a spherically symmetric and homogeneous dust collapse under the approximation that we can neglect the back reaction of the scalar wave on the space-time, and examine how the amplitude, characteristic frequency and wave form of the observed scalar gravitational wave depend on the initial radius and mass of the dust and parameters contained in the theory. In the Brans-Dicke theory, through the observation of the scalar gravitational wave, it is possible to determine the initial radius and mass and a parameter contained in the theory. In the scalar-tensor theories, it would be possible to get the information of the first derivative of the coupling function contained in the theory because the wave form of the scalar gravitational wave greatly depends on it. 
  It is folklore knowledge amongst general relativists that horizons are well behaved, continuously differentiable hypersurfaces except perhaps on a negligible subset one needs not to bother with. We show that this is not the case, by constructing a Cauchy horizon, as well as a black hole event horizon, which contain no open subset on which they are differentiable. 
  We carry out 3-D numerical simulations of the dynamical instability in rapidly rotating stars initially modeled as polytropes with n = 1.5, 1.0, and 0.5. The calculations are done with a SPH code using Newtonian gravity, and the gravitational radiation is calculated in the quadrupole limit. All models develop the global m=2 bar mode, with mass and angular momentum being shed from the ends of the bar in two trailing spiral arms. The models then undergo successive episodes of core recontraction and spiral arm ejection, with the number of these episodes increasing as n decreases: this results in longer-lived gravitational wave signals for stiffer models. This instability may operate in a stellar core that has expended its nuclear fuel and is prevented from further collapse due to centrifugal forces. The actual values of the gravitational radiation amplitudes and frequencies depend sensitively on the radius of the star R_{eq} at which the instability develops. 
  We consider the role of the diffeomorphism constraint in the quantization of lattice formulations of diffeomorphism invariant theories of connections. It has been argued that in working with abstract lattices, one automatically takes care of the diffeomorphism constraint in the quantum theory. We use two systems in order to show that imposing the diffeomorphism constraint is imperative to obtain a physically acceptable quantum theory. First, we consider $2+1$ gravity where an exact lattice formulation is available. Next, general theories of connections for compact gauge groups are treated, where the quantum theories are known --for both the continuum and the lattice-- and can be compared. 
  The semiclassical approach to quantum gravity would yield the Schroedinger formalism for the wave function of metric perturbations or gravitons plus quantum gravity correcting terms in pure gravity; thus, in the inflationary scenario, we should expect correcting effects to the relic graviton (Zel'dovich) spectrum of the order (H/mPl)^2. 
  The observed CMBR dipole, generally interpreted as the consequence of the peculiar motion of the Sun with respect to the reference frame of the CMBR, can also be explained by the presence of ultra large scale (of the order of $100H_0^{-1}$) isocurvature perturbations. Moreover, the simplest model of inflation with several scalar fields, namely that of double inflation, with appropriate parameters, can produce such perturbations. 
  We show that the gravitational modification of the phase of a neutron beam (the COW experiment) has a classical origin, being due to the time delay which classical particles experience in traversing a background gravitational field. Similarly, we show that classical light waves also undergo a phase shift in traversing a gravitational field. We show that the COW experiment respects the equivalence principle even in the presence of quantum mechanics. 
  Our long experience with Newtonian potentials has inured us to the view that gravity only produces local effects. In this paper we challenge this quite deeply ingrained notion and explicitly identify some intrinsically global gravitational effects. In particular we show that the global cosmological Hubble flow can actually modify the motions of stars and gas within individual galaxies, and even do so in a way which can apparently eliminate the need for galactic dark matter. Also we show that a classical light wave acquires an observable, global, path dependent phase in traversing a gravitational field. Both of these effects serve to underscore the intrinsic difference between non-relativistic and relativistic gravity. 
  We show that closed, radiation-filled Friedmann-Robertson-Walker quantum universes of arbitrary factor ordering obey the Whittaker equation. We also present the formal Witten factorization as well as the double Darboux strictly isospectral scheme for the Whittaker equation 
  Using both the Born-Oppenheimer idea and the de Broglie-Bohm interpretation of wavefunction we represent in a different way the semiclassical quantum gravity from the Wheeler-DeWitt equation in an oscillating regime which can preserve completely the unitary quantum evolution of a matter field at the expense of a nonlinear gravitational field equation, but has the same asymptotic limit as the others. We apply the de Broglie-Bohm interpretation to the nonlinear gravitational field equation to develop a perturbation method to find the quantum corrections of a matter field to the gravity. The semiclassical Einstein equation with the quantum corrections is found for a minimal quantum FRW cosmological model. 
  We find exact solutions of the Einstein-Boltzmann equations with relaxational collision term in FRW and Bianchi I spacetimes. The kinematic and thermodynamic properties of the solutions are investigated. We give an exact expression for the bulk viscous pressure of an FRW distribution that relaxes towards collision-dominated equilibrium. If the relaxation is toward collision-free equilibrium, the bulk viscosity vanishes - but there is still entropy production. The Bianchi I solutions have zero heat flux and bulk viscosity, but nonzero shear viscosity. The solutions are used to construct a realisation of the Weyl Curvature Hypothesis. 
  Stationary rotating matter configurations in general relativity are considered. A formalism for general stationary space times is developed. Axisymmetric systems are discussed by the use of a nonholonomic and nonrigid frame in the three-space of the time-like Killing trajectories. Two symmetric and trace-free tensors are constructed. They characterize a class of matter states in which both the interior Schwarzschild and the Kerr solution are contained. Consistency relations for this class of perfect fluids are derived. Incompressible fluids characterized by these tensors are investigated, and one differentially rotating solution is found. 
  We define spacetimes that are asymptotically flat, except for a deficit solid angle $\alpha$, and present a definition of their ``ADM'' mass, which is finite for this class of spacetimes, and, in particular, coincides with the value of the parameter $M$ of the global monopole spacetime studied by Vilenkin and Barriola . Moreover, we show that the definition is coordinate independent, and explain why it can, in some cases, be negative. 
  We examine Carlip's derivation of the 2+1 Minkowskian black hole entropy. A simplified derivation of the boundary action -valid for any value of the level k- is given. 
  The structure of the full Einstein equations in a coordinate gauge based on expanding null hypersurfaces foliated by metric 2-spheres is explored. The simple form of the resulting equations has many applications -- in the present paper we describe the structure of timelike boundary conditions; the matching problem across null hypersurfaces; and the propagation of gravitational shocks. 
  We develop the principle of nongravitating vacuum energy, which is implemented by changing the measure of integration from $\sqrt{-g}d^{D}x$ to an integration in an internal space of $D$ scalar fields $\phi_{a}$. As a consequence of such a choice of the measure, the matter Lagrangian $L_{m}$ can be changed by adding a constant while no cosmological term is induced. Here we develop this idea to build a new theory which is formulated through the first order formalism, i.e. using vielbein $e_{a}^{\mu}$ and spin connection $\omega_{\mu}^{ab}$ (a,b=1,2,...D) as independent variables. The equations obtained from the variation of $e_{a}^{\mu}$ and the fields $\phi_{a}$ imply the existence of a nontrivial constraint. This approach can be made consistent with invariance under arbitrary diffeomorphisms in the internal space of scalar fields $\phi_{a}$ (as well as in ordinary space-time), provided that the matter model is chosen so as to satisfy the above mentioned constraint. If the matter model is not chosen so as to satisfy automatically this constraint, the diffeomorphism invariance in the internal space is broken. In this case the constraint is dynamically implemented by the degrees of freedom that become physical due to the breaking of the internal diffeomorphism invariance. However, this constraint always dictates the vanishing of the cosmological constant term and the gravitational equations in the vacuum coincide with vacuum Einstein's equations with zero cosmological constant. The requirement that the internal diffeomorphisms be a symmetry of the theory points towards the unification of forces in nature like in the Kaluza-Klein scheme. 
  We study how the initial inhomogeneities of the universe affect the onset of inflation in the closed universe. We consider the model of a chaotic inflation which is driven by a massive scalar field. In order to construct an inhomogeneous universe model, we use the long wavelength approximation ( the gradient expansion method ). We show the condition of the inhomogeneities for the universe to enter the inflationary phase. 
  We present the results of a study of the gauge dependence of spacetime perturbations. In particular, we consider gauge invariance in general, we give a generating formula for gauge transformations to an arbitrary order n, and explicit transformation rules at second order. 
  We discuss some features of Einstein-Proca gravity in D=3 and 4 space-times. Our study includes a discussion on the tree-level unitarity and on the issue of light deflection in 3D gravity in the presence of a mass term. 
  We propose a Kaluza-Klein approach to general relativity of 4-dimensional spacetimes. This approach is based on the (2,2)-splitting of a generic 4-dimensional spacetime, which is viewed as a local product of a (1+1)-dimensional base manifold and a 2-dimensional fibre space. In this Kaluza-Klein formalism we find that general relativity of 4-dimensional spacetimes can be interpreted in a natural way as a (1+1)-dimensional gauge theory. In a gauge where the (1+1)-dimensional metric can be written as the Polyakov metric, the action is describable as a (1+1)-dimensional Yang-Mills type gauge theory action coupled to a (1+1)-dimensional non-linear sigma field and a scalar field, subject to the constraint equations associated with the diffeomorphism invariance of the (1+1)-dimensional base manifold. Diffeomorphisms along the fibre directions show up as the Yang-Mills type gauge symmetries, giving rise to the Gauss-law constraints. We also present the Einstein's equations in the Polyakov gauge and discuss them from the (1+1)-dimensional gauge theory point of view. Finally we show that this Kaluza-Klein formalism is closely related to the null hypersurface formalism of general relativity. 
  We show how to construct non-spherically-symmetric extended bodies of uniform density behaving exactly as pointlike masses. These ``gravitational monopoles'' have the following equivalent properties: (i) they generate, outside them, a spherically-symmetric gravitational potential $M/|x - x_O|$; (ii) their interaction energy with an external gravitational potential $U(x)$ is $- M U(x_O)$; and (iii) all their multipole moments (of order $l \geq 1$) with respect to their center of mass $O$ vanish identically. The method applies for any number of space dimensions. The free parameters entering the construction are: (1) an arbitrary surface $\Sigma$ bounding a connected open subset $\Omega$ of $R^3$; (2) the arbitrary choice of the center of mass $O$ within $\Omega$; and (3) the total volume of the body. An extension of the method allows one to construct homogeneous bodies which are gravitationally equivalent (in the sense of having exactly the same multipole moments) to any given body. 
  We apply a technique, due to Stephani, for generating solutions of the Einstein-perfect fluid equations. This technique is similar to the vacuum solution generating techniques of Ehlers, Harrison, Geroch and others. We start with a ``seed'' solution of the Einstein-perfect fluid equations with a Killing vector. The seed solution must either have (i) a spacelike Killing vector and equation of state P=rho or (ii) a timelike Killing vector and equation of state rho+3P=0. The new solution generated by this technique then has the same Killing vector and the same equation of state. We choose several simple seed solutions with these equations of state and where the Killing vector has no twist. The new solutions are twisting versions of the seed solutions. 
  It is found that the induced gravity with conformal couplings requires the conformal invariance in both classical and quantum levels for consistency. This is also true for the induced gravity with an extended conformal coupling interacting with torsion. 
  We present three types of non-conformal symmetries for a wide class of 2D dilaton-gravity models. For the particular CGHS, or string-inspired model, a linear combination of these symmetries is conformal and turns out to be the well-known symmetry which allows to construct the exactly solvable semiclassical RST and BPP models. We show that one of these non-conformal symmetries can be converted into a conformal one by means of a suitable field redefinition involving the metric and the derivatives of the dilaton. As a consequence of this, and by defining a Polyakov-type term in terms of an invariant metric under the symmetry, we are able to provide, for a generic 2D dilaton gravity model, associated semiclassical models which are symmetry invariant. 
  Contrary to common belief, there are perspectives for generalizing the notion of positive and negative frequency in minisuperspace quantum cosmology, even when the wave equation does not admit symmetries. We outline a strategy in doing so when the potential is positive. Also, an underlying unitarity structure shows up. Starting in the framework of the Klein-Gordon type quantization, I am led to a result that relies on global features on the model, and that is possibly related to structures encountered in the refined algebraic quantization scheme. 
  We study the evolution of massless scalar waves propagating on spherically symmetric spacetimes with a non-zero cosmological constant. Considering test fields on both Schwarzschild-de Sitter and Reissner-Nordstrom-de Sitter backgrounds, we demonstrate the existence of exponentially decaying tails at late times. Interestingly the l=0 mode asymptotes to a non-zero value, contrasting the asymptotically flat situation. We also compare these results, for l=0, with a numerical integration of the Einstein-Scalar field equations, finding good agreement between the two. Finally, the significance of these results to the study of the Cauchy horizon stability in black hole-de Sitter spacetimes is discussed. 
  A multidimensional cosmological model with space-time consisting of n (n>1) Einstein spaces M_i is investigated in the presence of a cosmological constant Lambda and m homogeneous minimally coupled scalar fields as a matter source. Classes of the models integrable at classical as well as quantum levels are found. These classes are equivalent to each other. Quantum wormhole solutions are obtained for them and the procedure of the third quantization is performed. An inflationary universe arising from classically forbidden Euclidean region is investigated for a model with a cosmological constant. 
  It has been recently pointed out that a definition of the geometric entropy using the partition function in a conical space does not in general lead to a positive definite quantity. For a scalar field model with a non-minimal coupling we clarify the origin of the anomalous behavior from the viewpoint of the canonical formulation. 
  The allowed mass windows for a cosmic background of relic dilatons are estimated in the context of the pre-big bang scenario. The dilatons are produced from the quantum fluctuations of the vacuum, and the extension of the windows is controlled by the string mass scale. The possible relaxation of phenomenological bounds due to an intermediate stage of reheating is discussed. Even without such a relaxation, the allowed range of masses includes a light sector in which the dilatons are not yet decayed, and could provide the dominant contribution to the present large scale density. 
  In a previous work the Weyl-Dirac framework was generalized in order to obtain a geometrically based general relativistic theory, possessing intrinsic electric and magnetic currents and admitting massive photons. Some physical phenomena in that framework are considered. So it is shown that massive photons may exist only in presence of an intrinsic magnetic field. The role of massive photons is essential in order to get an interaction between magnetic currents. A static spherically symmetric solution is obtained. It may lead either to the Reissner-Nordstr{\o}m metric, or to the metric created by a magnetic monopole. 
  The conserved nonlocal charges generating the Geroch group with respect to the canonical Poisson structure of the Ernst equation are found. They are shown to build a quadratic Poisson algebra, which suggests to identify the quantum Geroch algebra with Yangian structures. 
  The independent dynamical variables of a collapsing homogeneous sphere of dust are canonically quantised and coupled equations describing matter (dust) and gravitation are obtained. The conditions for the validity of the adiabatic (Born-Oppenheimer) and semiclassical approximations are derived. On neglecting back-reaction effects, it is also shown that in the vicinity of the horizon and inside the dust the Wightman function for a conformal scalar field coupled to a monopole emitter is thermal at the characteristic Hawking temperature. 
  I describe a recently derived stochastic approach to inflaton dynamics which can address some serious problems associated with conventional inflationary theory. Using this theory I derive an exact solution to the stochastic dynamics for the case of a $\lambda\phi^4$ potential and use it to study the generated primordial density fluctuations. It is found that on both sub and super-horizon scales the theory predicts gaussian fluctuations to a very high accuracy along with a near scale-invariant spectrum. Of most interest is that the amplitude constraint is found to be satisfied for $\lambda\sim 10^{-5}$ rather than $\lambda\sim 10^{-14}$ of the conventional theory. This represents a dramatic easing of the fine-tuning constraints, a feature likely to generalize to a wide range of potentials. 
  Negative energy density is unavoidable in the quantum theory of field. We give a revised proof of the existence of negative energy density unambiguously for a massless scalar field. 
  We consider the general behaviour of cosmologies in Brans-Dicke theory where the dilaton is self-interacting via a potential $V(\Phi)$. We show that the general radiation universe is a two-dimensional dynamical system whereas the dust or false vacuum universe is three-dimensional. This is in contrast to the non-interacting dilaton which has uniformly a two-dimensional phase space. We find the phase spaces in each case and the general behaviour of the cosmologies. 
  Approximate solutions representing the gravitational-electrostatic balance of two arbitrary point sources in general relativity have led to contradictory arguments in the literature with respect to the condition of balance. Up to the present time, the only known exact solutions which can be interpreted as the non-linear superposition of two spherically symmetric (Reissner-Nordstrom) bodies without an intervening strut has been for critically charged masses, $M^2_i = Q^2_i$. In the present paper, an exact electrostatic solution of the Einstein-Maxwell equations representing the exterior field of two arbitrary charged Reissner-Nordstrom bodies in equilibrium is studied. The invariant physical charge for each source is found by direct integration of Maxwell's equations. The physical mass for each source is invariantly defined in a manner similar to which the charge was found. It is shown through numerical methods that balance without tension or strut can occur for non-critically charged bodies. It is demonstrated that other authors have not identified the correct physical parameters for the mass and charge of the sources. Further properties of the solution, including the multipole structure and comparison with other parameterizations, are examined. 
  It has been recently shown that, in the first order (Palatini) formalism, there is universality of Einstein equations and Komar energy-momentum complex, in the sense that for a generic nonlinear Lagrangian depending only on the scalar curvature of a metric and a torsionless connection one always gets Einstein equations and Komar's expression for the energy-momentum complex. In this paper a similar analysis (also in the framework of the first order formalism) is performed for all nonlinear Lagrangians depending on the (symmetrized) Ricci square invariant. The main result is that the universality of Einstein equations and Komar energy-momentum complex also extends to this case (modulo a conformal transformation of the metric). 
  Building on the first variational formula of the calculus of variations, one can derive the energy-momentum conservation laws from the condition of the Lie derivative of gravitation Lagrangians along vector fields corresponding to generators of general covariant transformations to be equal to zero. The goal is to construct these vector fields. In gauge gravitation theory, the difficulty arises because of fermion fields. General covariant transformations fail to preserve the Dirac spin structure $S_h$ on a world manifold $X$ which is associated with a certain tetrad field $h$. We introduce the universal Dirac spin structure $S\to\Sigma\to X$ such that, given a tetrad field $h:X\to \Sigma$, the restriction of $S$ to $h(X)$ is isomorphic to $S_h$. The canonical lift of vector fields on $X$ onto $S$ is constructed. We discover the corresponding stress-energy-momentum conservation law. The gravitational model in the presence of a background spin structure also is examined. 
  The vanishing of the electromagnetic field, for purely electric configurations of spontaneously broken Abelian models, is established in the domain of outer communications of a static asymptotically flat black hole. The proof is gauge invariant, and is accomplished without any dependence on the model. In the particular case of the Abelian Higgs model, it is shown that the only solutions admitted for the scalar field become the vacuum expectation values of the self-interaction. 
  The influence of Inflation on initial (i.e. at Planck's epoch) large anisotropy of the Universe is studied, considering a more general metric than the isotropic one: the locally rotationally symmetric (L.R.S.) Bianchi IX metric. We find, then, a large set of initial conditions of intrinsic curvature and shear allowing an inflationary epoch that make the anisotropy negligible. These are not trivial because of the non-linearity of the Einstein's equations. 
  A underlying dynamical structure for both relativity and quantum theory-``superrelativity'' has been proposed in order to overcome the well known incompatibility between these theories. The relationship between curvature of spacetime (gravity) and curvature of the projective Hilbert space of pure quantum states is established as well. 
  Geometric properties of operators of quantum Dirac constraints and physical observables are studied in semiclassical theory of generic constrained systems. The invariance transformations of the classical theory -- contact canonical transformations and arbitrary changes of constraint basis -- are promoted to the quantum domain as unitary equivalence transformations. Geometry of the quantum reduction of the Dirac formalism to the physical sector of the theory is presented in the coordinate gauges and extended to unitary momentum-dependent gauges of a general type. The operators of physical observables are constructed satisfying one-loop quantum gauge invariance and Hermiticity with respect to a physical inner product. Abelianization procedure on Lagrangian constraint surfaces of phase space is discussed in the framework of the semiclassical expansion. 
  We give a detailed presentation of a recently proposed mechanism of generating the energy scale of inflation by loop effects in quantum cosmology. We discuss the quantum origin of the early inflationary Universe from the no-boundary and tunneling quantum states and present a universal effective action algorithm for the distribution function of chaotic inflationary cosmologies in both of these states. The energy scale of inflation is calculated by finding a sharp probability peak in this distribution function for a tunneling model driven by the inflaton field with large negative constant $\xi$ of non-minimal interaction. The sub-Planckian parameters of this peak (the mean value of the corresponding Hubble constant $H\simeq 10^{-5}m_P$, its quantum width $\Delta H/H\simeq 10^{-5}$ and the number of inflationary e-foldings $N\geq 60$) are found to be in good correspondence with the observational status of inflation theory, provided the coupling constants of the theory are constrained by a condition which is likely to be enforced by the (quasi) supersymmetric nature of the sub-Planckian particle physics model. 
  We study local variations of causal curves in a space-time with respect to b-length (or generalised affine parameter length). In a convex normal neighbourhood, causal curves of maximal metric length are geodesics. Using variational arguments, we show that causal curves of minimal b-length in sufficiently small globally hyperbolic sets are geodesics. As an application we obtain a generalisation of a theorem by B. G. Schmidt, showing that the cluster curve of a partially future imprisoned, future inextendible and future b-incomplete curve must be a null geodesic. We give examples which illustrate that the cluster curve does not have to be closed or incomplete. The theory of variations developed in this work provides a starting point for a Morse theory of b-length. 
  Exact self-consistent particle-like solutions with spherical and/or cylindrical symmetry to the equations governing the interacting system of scalar, electromagnetic and gravitational fields have been obtained. As a particular case it is shown that the equations of motion admit a special kind of solutions with sharp boundary known as droplets. For these solutions, the physical fields vanish and the space-time is flat outside of the critical sphere or cylinder. Therefore, the mass and the electric charge of these configurations are zero. 
  Dilaton-axion gravity with $p U(1)$ vector fields is studied on space-times admitting a timelike Killing vector field. Three-dimensional sigma-model is derived in terms of K\"ahler geometry, and holomorphic representation of the SO(2,2+p) global symmetry is constructed. A general static black hole solution depending on $2p+5$ parameters is obtained via SO(2,2+p) covariantization of the Schwarzschild solution. The metric in the curvature coordinates looks as the variable mass Reissner-Nordstr\"om one and generically possesses two horizons. The inner horizon is pushed to the singularity if electric and magnetic SO(p) charge vectors are parallel. For non-parallel charges the inner horizon has a finite area except for an extremal limit when this property is preserved only for orthogonal charges. Extremal dyon configurations with orthogonal charges have finite horizon radii continuously varying from zero to the ADM mass. New general solution is endowed with a NUT parameter, asymptotic values of dilaton and axion, and a gauge parameter which can be used to ascribe any given value to one of scalar charges. 
  Monopole-antimonopole pairs connected by strings can be formed as topological defects in a sequence of cosmological phase transitions. Such hybrid defects typically decay early in the history of the universe but can still generate an observable background of gravitational waves. We study the spectrum of gravitational radiation from these objects both analytically and numerically, concentrating on the simplest case of an oscillating pair connected by a straight string. 
  In this lecture we describe the data analysis problem for insparlling binaries. We discuss the detection statistic, how to make realiable estimation and how to compute bias in the estimation of parameters. A combination of geometrical ideas and numerical methods are employed to estimate computational costs involved in searching for post-Newtonian wave forms. 
  We define and discuss various quantum operators that describe the geometry of spacetime in quantum general relativity. These are obtained by combining the Null-Surface Formulation of general relativity, recently developed, with asymptotic quantization. One of the operators defined describes a ``fuzzy'' quantum light cone structure. Others, denoted ``spacetime-point operators'', characterize geometrically-defined physical points. We discuss the interpretation of these operators. This seems to suggest a picture of quantum spacetime as made of ``fuzzy'' physical points. We derive the commutation algebra of the quantum spacetime point operators in the linearization around flat space. 
  We show how, for all dimensions and signatures, a symmetry operator for the massless Dirac equation can be constructed from a conformal Killing-Yano tensor of arbitrary degree. 
  Local path integral quantization of generic 2D dilaton gravity is considered. Locality means that we assume asymptotic fall off conditions for all fields. We demonstrate that in the absence of `matter' fields to all orders of perturbation theory and for all 2D dilaton theories the quantum effective action coincides with the classical one. This resolves the apparent contradiction between the well established results of Dirac quantization and perturbative (path-integral) approaches which seemed to yield non-trivial quantum corrections. For a particular case, the Jackiw--Teitelboim model, our result is even extended to the situation when a matter field is present. 
  We calculate renormalised vacuum expectation values of electromagnetic stress-energy tensor in the static spherically-symmetrical wormhole topology. We find that for metric tensor sufficiently slow varied with distance violation of the averaged weak energy condition takes place irrespectively of the detailed form of metric. This is a necessary condition for the electromagnetic vacuum to be able to support the wormhole geometry. 
  The properties of some locally rotationally symmetric (LRS) perfect fluid space-times are examined in order to demonstrate the usage of the description of geometries in terms of the Riemann tensor and a finite number of its covariant derivatives for finding solutions to Einstein's field equations. A new method is introduced, which makes it possible to choose the coordinates at any stage of the calculations. Three classes are examined, one with fluid rotation, one with spatial twist in the preferred direction and the space-time homogeneous models. It is also shown that there are no LRS space-times with dependence on one null coordinate. Using an extension of the method, we find the full metric in terms of curvature quantities for the first two classes. 
  The relationship of Choptuik scaling to the scale invariance of Einstein's equation is explored. Ordinary dynamical systems often have limit cycles: periodic orbits that are the asymptotic limit of generic solutions. We show how to separate Einstein's equation into the dynamics of the overall scale and the dynamics of the ``scale invariant'' part of the metric. Periodicity of the scale invariant part implies periodic self-similarity of the spacetime. We also analyze a toy model that exhibits many of the features of Choptuik scaling. 
  General matterless models of gravity include dilaton gravity, arbitrary powers in curvature, but also dynamical torsion. They are a special class of "Poisson-sigma-models" whose solutions are known completely, together with their general global structure. Beside the ordinary black hole, arbitrary singularity structures can be studied. It is also possible to derive an action "backwards", starting from a given manifold. The role of conservation laws, Noether charge and the quantization have been investigated. Scalar and fermionic matter fields may be included as well. 
  General relativity exhibits a unique feature not represented in standard examples of chaotic systems; it is a spacetime diffeomorphism invariant theory. Thus many characterizations of chaos do not work. It is therefore necessary to develop a definition of chaos suitable for application to general relativity. This presentation will present results towards this goal. 
  Recently an alternate technique for numerical quantum gravity, dynamical triangulation, has been developed. In this method, the geometry is varied by adding and subtracting equilateral simplices from the simplicial complex. This method overcomes certain difficulties associated with the traditional approach in Regge calculus of varying geometry by varying edge lengths. However additional complications are introduced: three of the four moves in dynamical triangulation can violate the simplicial nature of the complex. Simulations indicate that the rate of these violations is significant. Thus additional conditions must be placed on the dynamical triangulation moves to ensure that the simplicial complex and its topology are preserved. 
  We argue that the conjectured dark mater in the Universe may be endowed with a new kind of gravitational charge that couples to a short range gravitational interaction mediated by a massive vector field. A model is constructed that assimilates this concept into ideas of current inflationary cosmology. The model is also consistent with the observed behaviour of galactic rotation curves according to Newtonian dynamics. The essential idea is that stars composed of ordinary (as opposed to dark matter) experience Newtonian forces due to the presence of an all pervading background of massive gravitationally charged cold dark matter. The novel gravitational interactions are predicted to have a significant influence on pre-inflationary cosmology. The precise details depend on the nature of a gravitational Proca interaction and the description of matter. A gravitational Proca field configuration that gives rise to attractive forces between dark matter charges of like polarity exhibits homogeneous isotropic eternal cosmologies that are free of cosmological curvature singularities thus eliminating the horizon problem associated with the standard big-bang scenario. Such solutions do however admit dense hot pre-inflationary epochs each with a characteristic scale factor that may be correlated with the dark matter density in the current era of expansion. The model is based on a theory in which a modification of Einsteinian gravity at very short distances can be expressed in terms of the gradient of the Einstein metric and the torsion of a non-Riemannian connection on the bundle of linear frames over spacetime. Indeed we demonstrate that the genesis of the model resides in a remarkable simplification that occurs when one analyses the variational equations associated with a broad class of non-Riemannian actions. 
  Recently, Choquet-Bruhat and York and Abrahams, Anderson, Choquet-Bruhat, and York (AACY) have cast the 3+1 evolution equations of general relativity in gauge-covariant and causal ``first-order symmetric hyperbolic form,'' thereby cleanly separating physical from gauge degrees of freedom in the Cauchy problem for general relativity. A key ingredient in their construction is a certain wave equation which governs the light-speed propagation of the extrinsic curvature tensor. Along a similar line, we construct a related wave equation which, as the key equation in a system, describes vacuum general relativity. Whereas the approach of AACY is based on tensor-index methods, the present formulation is written solely in the language of differential forms. Our approach starts with Sparling's tetrad-dependent differential forms, and our wave equation governs the propagation of Sparling's 2-form, which in the ``time-gauge'' is built linearly from the ``extrinsic curvature 1-form.'' The tensor-index version of our wave equation describes the propagation of (what is essentially) the Arnowitt-Deser-Misner gravitational momentum. 
  Within a first-order framework, we comprehensively examine the role played by boundary conditions in the canonical formulation of a completely general two-dimensional gravity model. Our analysis particularly elucidates the perennial themes of mass and energy. The gravity models for which our arguments are valid include theories with dynamical torsion and so-called generalized dilaton theories (GDTs). Our analysis of the canonical action principle (i) provides a rigorous correspondence between the most general first-order two-dimensional Einstein-Cartan model (ECM) and GDT and (ii) allows us to extract in a virtually simultaneous manner the ``true degrees of freedom'' for both ECMs and GDTs. For all such models, the existence of an absolutely conserved (in vacuo) quantity C is a generic feature, with (minus) C corresponding to the black-hole mass parameter in the important special cases of spherically symmetric four-dimensional general relativity and standard two-dimensional dilaton gravity. The mass C also includes (minimally coupled) matter into a ``universal mass function.'' We place particular emphasis on the (quite general) class of models within GDT possessing a Minkowski-like groundstate solution (allowing comparison between $C$ and the Arnowitt-Deser-Misner mass for such models). 
  In the weak field approximation to quantum gravity, a "local" positive cosmological term mu^2(x) corresponds to a local negative squared mass term in the Lagrangian and may thus induce instability and local pinning of the gravitational field. Such a term can be produced by the coupling to an external Bose condensate. In the functional integral, the local pinning acts as a constraint on the field configurations. We discuss this model in detail and apply it to a phenomenological analysis of recent experimental results. 
  We examine here the relevance of the initial state of a collapsing dust cloud towards determining it's final fate in the course of a continuing gravitational collapse. It is shown that given any arbitrary matter distribution $M(r)$ for the cloud at the initial epoch, there is always a freedom to choose rest of the initial data, namely the initial velocities of the collapsing spherical shells, so that the collapse could result either in a black hole or a naked singularity depending on this choice. Thus, given the initial density profile, to achieve the desired end state of the gravitational collapse one has to give a suitable initial velocity to the cloud. We also characterize here a wide new family of black hole solutions resulting from inhomogeneous dust collapse. These configurations obey the usual energy conditions demanding the positivity of energy density. 
  An analogy between the subtraction procedure in the Gibbons-Hawking Euclidean path integral approach to Horizon's Thermodynamics and the Casimir effect is shown. Then a conjecture about a possible Casimir nature of the Gibbons-Hawking subtraction is made in the framework of Sakharov's induced gravity. In this framework it appears that the degrees of freedom involved in the Bekenstein-Hawking entropy are naturally identified with zero--point modes of the matter fields. Some consequences of this view are sketched. 
  The ``standard'' expressions for total energy, linear momentum and also angular momentum of asymptotically flat Bondi metrics at null infinity are also obtained from differential conservation laws on asymptotically flat backgrounds, derived from a quadratic Lagrangian density by methods currently used in classical field theory. It is thus a matter of taste and commodity to use or not to use a reference spacetime in defining these globally conserved quantities. Backgrounds lead to N\oe ther conserved currents; the use of backgrounds is in line with classical views on conservation laws. Moreover, the conserved quantities are in principle explicitly related to the sources of gravity through Einstein's equations, while standard definitions are not. The relations depend, however, on a rule for mapping spacetimes on backgrounds. 
  Conformally invariant scalar waves in black hole spacetimes which are asymptotically anti-de Sitter are investigated. We consider both the $(2+1)$-dimensional black hole and $(3+1)$-dimensional Schwarzschild-anti-de Sitter spacetime as backgrounds. Analytical and numerical methods show that the waves decay exponentially in the $(2+1)$ dimensional black hole background. However the falloff pattern of the conformal scalar waves in the Schwarzschild-anti-de Sitter background is generally neither exponential nor an inverse power rate, although the approximate falloff of the maximal peak is weakly exponential. We discuss the implications of these results for mass inflation. 
  A set of algebraic equations for the topological properties of space-time is derived, and used to extend general relativity into the Planck domain. A unique basis set of three-dimensional prime manifolds is constructed which consists of $S^3$, $S^1\times S^2$, and $T^3$. The action of a loop algebra on these prime manifolds yields topological invariants which constrain the dynamics of the four-dimensional space-time manifold. An extended formulation of Mach's principle and Einstein's equivalence of inertial and gravitational mass is proposed which leads to the correct classical limit of the theory.   It is found that the vacuum possesses four topological degrees of freedom corresponding to a lattice of three-tori. This structure for the quantum foam naturally leads to gauge groups O(n) and SU(n) for the fields, a boundary condition for the universe, and an initial state characterized by local thermal equilibrium. The current observational estimate of the cosmological constant is reproduced without fine-tuning and found to be proportional to the number of macroscopic black holes. The black hole entropy follows immediately from the theory and the quantum corrections to its Schwarzschild horizon are computed. 
  The question of whether unobserved short-wavelength modes of the gravitational field can induce decoherence in the long-wavelength modes (``the decoherence of spacetime'') is addressed using a simplified model of perturbative general relativity, related to the Nordstrom-Einstein-Fokker theory, where the metric is assumed to be conformally flat. For some long-wavelength coarse grainings, the Feynman-Vernon influence phase is found to be effective at suppressing the off-diagonal elements of the decoherence functional. The requirement that the short-wavelength modes be in a sufficiently high-temperature state places limits on the applicability of this perturbative approach. 
  Building on the "quantum inequalities" introduced by Ford, I argue that the negative local energies encountered in quantum field theory can only be observed by detectors with positive energies at least as great in magnitude. This means that operationally the total energy density must be non-negative. Like reasoning shows that, in a similar operational sense, the dominant energy condition must hold: any timelike component of the four-momentum density is positive. 
  Both real and complex connections have been used for canonical gravity: the complex connection has SL(2,C) as gauge group, while the real connection has SU(2) as gauge group. We show that there is an arbitrary parameter $\beta$ which enters in the definition of the real connection, in the Poisson brackets, and therefore in the scale of the discrete spectra one finds for areas and volumes in the corresponding quantum theory. A value for $\beta$ could be could be singled out in the quantum theory by the Hamiltonian constraint, or by the rotation to the complex Ashtekar connection. 
  We consider possible tests of the Einstein Equivalence Principle for quantum-mechanical vacuum energies by evaluating the Lamb shift transition in a class of non-metric theories of gravity described by the \tmu formalism. We compute to lowest order the associated red shift and time dilation parameters, and discuss how (high-precision) measurements of these quantities could provide new information on the validity of the equivalence principle. 
  We briefly present the supersymmetric double Darboux method and next apply it to the continuum of the quantum Taub cosmological model as a toy model in order to generate a one-parameter family of bosonic Taub potentials and the corresponding wavefunctions 
  In several recent publications Carlip, as well as Balachandran, Chandar and Momen, have proposed a statistical mechanical interpretation for black hole entropy in terms of ``would be gauge'' degrees of freedom that become dynamical on the boundary to spacetime. After critically discussing several routes for deriving a boundary action, we examine their hypothesis in the context of generic 2-D dilaton gravity. We first calculate the corresponding statistical mechanical entropy of black holes in 1+1 deSitter gravity, which has a gauge theory formulation as a BF-theory. Then we generalize the method to dilaton gravity theories that do not have a (standard) gauge theory formulation. This is facilitated greatly by the Poisson-Sigma-model formulation of these theories. It turns out that the phase space of the boundary particles coincides precisely with a symplectic leaf of the Poisson manifold that enters as target space of the Sigma-model. Despite this qualitatively appealing picture, the quantitative results are discouraging: In most of the cases the symplectic leaves are non-compact and the number of microstates yields a meaningless infinity. In those cases where the particle phase space is compact - such as, e.g., in the Euclidean deSitter theory - the edge state degeneracy is finite, but generically it is far too small to account for the semiclassical Bekenstein-Hawking entropy. 
  The eigenvalues of the Dirac operator on a curved spacetime are diffeomorphism-invariant functions of the geometry. They form an infinite set of ``observables'' for general relativity. Recent work of Chamseddine and Connes suggests that they can be taken as variables for an invariant description of the gravitational field's dynamics. We compute the Poisson brackets of these eigenvalues and find them in terms of the energy-momentum of the eigenspinors and the propagator of the linearized Einstein equations. We show that the eigenspinors' energy-momentum is the Jacobian matrix of the change of coordinates from metric to eigenvalues. We also consider a minor modification of the spectral action, which eliminates the disturbing huge cosmological term and derive its equations of motion. These are satisfied if the energy momentum of the trans Planckian eigenspinors scale linearly with the eigenvalue; we argue that this requirement approximates the Einstein equations. 
  We derive a spacetime formulation of quantum general relativity from (hamiltonian) loop quantum gravity. In particular, we study the quantum propagator that evolves the 3-geometry in proper time. We show that the perturbation expansion of this operator is finite and computable order by order. By giving a graphical representation a' la Feynman of this expansion, we find that the theory can be expressed as a sum over topologically inequivalent (branched, colored) 2d surfaces in 4d. The contribution of one surface to the sum is given by the product of one factor per branching point of the surface. Therefore branching points play the role of elementary vertices of the theory. Their value is determined by the matrix elements of the hamiltonian constraint, which are known. The formulation we obtain can be viewed as a continuum version of Reisenberger's simplicial quantum gravity. Also, it has the same structure as the Ooguri-Crane-Yetter 4d topological field theory, with a few key differences that illuminate the relation between quantum gravity and TQFT. Finally, we suggests that certain new terms should be added to the hamiltonian constraint in order to implement a ``crossing'' symmetry related to 4d diffeomorphism invariance. 
  We review here some recent results that show that inflationary cosmological models must contain initial singularities. We also present a new singularity theorem. The question of the initial singularity re-emerges in inflationary cosmology because inflation is known to be generically future-eternal. It is natural to ask, therefore, if inflationary models can be continued into the infinite past in a non-singular way. The results that we discuss show that the answer to the question is ``no.'' This means that we cannot use inflation as a way of avoiding the question of the birth of the Universe. We also argue that our new theorem suggests - in a sense that we explain in the paper - that the Universe cannot be infinitely old. 
  We study spacetime diffeomorphisms in Hamiltonian and Lagrangian formalisms of generally covariant systems. We show that the gauge group for such a system is characterized by having generators which are projectable under the Legendre map. The gauge group is found to be much larger than the original group of spacetime diffeomorphisms, since its generators must depend on the lapse function and shift vector of the spacetime metric in a given coordinate patch. Our results are generalizations of earlier results by Salisbury and Sundermeyer. They arise in a natural way from using the requirement of equivalence between Lagrangian and Hamiltonian formulations of the system, and they are new in that the symmetries are realized on the full set of phase space variables. The generators are displayed explicitly and are applied to the relativistic string and to general relativity. 
  The Tolman-Bondi (TB) model is defined up to some transformation of a co-moving coordinate but the transformation is not fixed. The use of an arbitrary co-moving system of coordinates leads to the solution dependent on three functions $f, F, {\bf F}$ which are chosen independently in applications.   The article studies the transformation rule which is given by the definition of an invariant mass. It is shown that the addition of the TB model by the definition of the transformation rule leads to the separation of the couples of functions ($f, F$) into nonintersecting classes. It is shown that every class is characterized only by the dependence of $F$ on $f$ and connected with unique system of co-moving coordinates. It is shown that the Ruban-Chernin system of coordinates corresponds to identical transformation. The dependence of Bonnor's solution on the Ruban-Chernin coordinate $M$ by means of initial density and energy distribution is studied. It is shown that the simplest flat solution is reduced to an explicit dependence on the coordinate $M$. Several examples of initial conditions and transformation rules are studied. 
  This talk is based on my work in collaboration with Thibault Damour since 1991. Unified theories, like superstrings, predict the existence of scalar partners to the graviton. Such theories of gravity can be very close to general relativity in weak-field conditions (solar-system experiments), but can deviate significantly from it in the strong-field regime (near compact bodies, like neutron stars). Binary pulsars are thus the best tools available for testing these theories. This talk presents the four main binary-pulsar experiments, and discusses the constraints they impose on a generic class of tensor-scalar theories. It is shown notably that they rule out some models which are strictly indistinguishable from general relativity in the solar system. This illustrates the qualitative difference between binary-pulsar and solar-system tests of relativistic gravity. 
  The method of averaging is used to investigate the phenomenon of capture into resonance for a model that describes a Keplerian binary system influenced by radiation damping and external normally incident periodic gravitational radiation. The dynamical evolution of the binary orbit while trapped in resonance is elucidated using the second order partially averaged system. This method provides a theoretical framework that can be used to explain the main evolutionary dynamics of a physical system that has been trapped in resonance. 
  We investigate the classical and semiclassical features of generic 2D, matter-coupled, dilaton gravity theories. In particular, we show that the mass, the temperature and the flux of Hawking radiation associated with 2D black holes are invariant under dilaton-dependent Weyl rescalings of the metric. The relationship between quantum anomalies and Hawking radiation is discussed. 
  Following de Broglie and Vigier, a fully relativistic causal interpretation of quantum mechanics is given within the context of a geometric theory of gravitation and electromagnetism. While the geometric model shares the essential principles of the causal interpretation initiated by de Broglie and advanced by Vigier, the particle and wave components of the theory are derived from the Einstein equations rather than a nonlinear wave equation. This geometric approach leads to several new features, including a solution to the de Broglie variable mass problem. 
  A quantum-mechanical Hamiltonian with a gravitational potential is derived in the framework of local times. This Hamiltonian is the one used by E. H. Lieb (Bull. Amer. Math. Soc. 22(1990), 1-49) in his explanation of stability and instability of cold stars. Our procedure deriving the Hamiltonian is based on an analysis of the process of the observation with the background of the notion of quantum-mechanical local times compatible with general theory of relativity. 
  We investigate the quantum mechanical oscillations of neutrinos propagating in weak gravitational field. The correction to the result in the flat space-time is derived. 
  We study the dynamics of a scalar inflaton field with a symmetric double--well potential and prove rigorously the existence of a limit cycle in its phase space. By using analytical and numerical arguments we show that the limit cycle is stable and give an analytical formula for its period. 
  We discuss an effect of accelerated mirrors which remained hitherto unnoticed, the formation of a field condensate near its surface for massive fields. From the view point of an observer attached to the mirror, this is effect is rather natural because a gravitational field is felt there. The novelty here is that since the effect is not observer dependent even inertial observers will detect the formation of this condensate. We further show that this localization is in agreement with Bekenstein's entropy bound. 
  The geometrical model of an electrical charge is proposed. This model has the ''nake'' charge shunted with ``fur - coat'' consisting of virtual wormholes. The 5D wormhole solution in the Kaluza - Klein's theory is the ''nake'' charge. The splitting off the supplementary coordinates happens on the two spheres (null surfaces) bounding this 5D wormhole. This allows to sew two Reissner - Nordstr\"om's black holes to it on both sides. Virtual wormholes entrap a part of the electrical force lines outcoming from ''nake'' charge. This effect can essentially decrease the charge visible at infinity up to real relation $m^2 < e^2$. The analogical construction for colour SU(2) gauge charge is made. 
  A study of an algorithm method capable to reveal anisotropic solutions of general scalar-tensor gravitation -including non-minimally couplings- is presented. It is found that it is possible to classify the behavior of the field of different scalar-tensor theories in equivalence classes, with the same classifier function that was obtained in Friedmann-Robertson-Walker models. 
  Stimulated by a scholium in Newton's Principia we find some beautiful results in classical mechanics which can be interpreted in terms of the orbits in the field of a mass endowed with a gravomagnetic monopole. All the orbits lie on cones! When the cones are slit open and flattened the orbits are exactly the ellipses and hyperbolae that one would have obtained without the gravomagnetic monopole.   The beauty and simplicity of these results has led us to explore the similar problems in Atomic Physics when the nuclei have an added Dirac magnetic monopole. These problems have been explored by others and we sketch the derivations and give details of the predicted spectrum of monopolar hydrogen.   Finally we return to gravomagnetic monopoles in general relativity. We explain why NUT space has a non-spherical metric although NUT space itself is the spherical space-time of a mass with a gravomagnetic monopole. We demonstrate that all geodesics in NUT space lie on cones and use this result to study the gravitational lensing by bodies with gravomagnetic monopoles.   We remark that just as electromagnetism would have to be extended beyond Maxwell's equations to allow for magnetic monopoles and their currents so general relativity would have to be extended to allow torsion for general distributions of gravomagnetic monopoles and their currents. Of course if monopoles were never discovered then it would be a triumph for both Maxwellian Electromagnetism and General Relativity as they stand! 
  It is well-known that some physical effects may arise in the spacetime of a straigth cosmic string due to its global conic properties. Among these effects, the vacuum polarization effect has been extensively studied in the litterature. In papers of reference [4] a more general situation has been considered in which the cosmic string carries a magnetic flux $\Phi$ and interacts with a charged scalar field. In this case, the vacuum polarization arises both via non-trivial gravitational interaction (i.e, the conical structure) and via Aharonov-Bohm interaction. In papers [4] the non vanishing VEV of the energy-momentum tensor of the scalar field were computed. However, this energy-momentum tensor should, in principle, be taken into account to determine the spacetime associated with the magnetic flux cosmic string. Using the semiclassical approach to the Einstein eqs. we find the first-order (in $\hbar$) metric associated to the cosmic string and we show that the gravitational force resulting from the backreaction of the $< T^{\mu}_{\nu} >$ is attractive or repulsive depending on whether the magnetic flux is absent or present, respectively. 
  A short review of some recent work on the problem of time and of observables for the reparametrization invariant systems is given. A talk presented at the Second Meeting on Constraint Dynamics and Quantum Gravity at Santa Marguerita Ligure, September 17--21 1996. 
  Solutions to a scalar-tensor (dilaton) quantum gravity theory, interacting with quantized matter, are described. Dirac quantization is frustrated by quantal anomalies in the constraint algebra. Progress is made only after the Wheeler--DeWitt equation is modified by quantal terms, which eliminate the anomaly. More than one modification is possible, resulting in more than one `physical' spectrum in the quantum theory, corresponding to the given classical model. 
  The conformal equivalence between Jordan frame and Einstein frame can be used in order to search for exact solutions in general theories of gravity in which scalar fields are minimally or nonminimally coupled with geometry. In the cosmological arena a relevant role is played by the time parameter in which dynamics is described. In this paper we discuss such issues considering also if cosmological Noether symmetries in the ``point--like'' Lagrangian are conformally preserved. Through this analysis and through also a careful analysis of the cosmological parameters \Omega and \Lambda, it is possible to contribute to the discussion on which is the physical system. 
  This paper has a dual purpose. One aim is to study the evolution of coherent states in ordinary quantum mechanics. This is done by means of a Hamiltonian approach to the evolution of the parameters that define the state. The stability of the solutions is studied. The second aim is to apply these techniques to the study of the stability of minisuperspace solutions in field theory. For a $\lambda \varphi^4$ theory we show, both by means of perturbation theory and rigorously by means of theorems of the K.A.M. type, that the homogeneous minisuperspace sector is indeed stable for positive values of the parameters that define the field theory. 
  The variational theory of the perfect hypermomentum fluid is developed. The new type of the generalized Frenkel condition is considered. The Lagrangian density of such fluid is stated, and the equations of motion of the fluid and the Weyssenhoff-type evolution equation of the hypermomentum tensor are derived. The expressions of the matter currents of the fluid (the canonical energy-momentum 3-form, the metric stress-energy 4-form and the hypermomentum 3-form) are obtained. The special case of the dilaton-spin fluid with intrinsic spin and dilatonic charge is considered. 
  Analytical wormhole solutions in $U_4$ theory are presented. It is discussed whether the extremely short range repulsive forces, related to the spin angular momentum of matter, could be the ``carrier'' of the exoticity that threads the wormhole throat. 
  The conditions are clarified under which regular (i.e., singularity-free) black holes can exist. It is shown that in a large class of spacetimes that satisfy the weak energy condition the existence of a regular black hole requires topology change. 
  Integrable models of dilaton gravity coupled to electromagnetic and scalar matter fields in dimensions 1+1 and 0+1 are briefly reviewed. The 1+1 dimensional integrable models are either solved in terms of explicit quadratures or reduced to the classically integrable Liouville equation. The 0+1 dimensional integrable models emerge as sectors in generally non integrable 1+1 dimensional models and can be solved in terms of explicit quadratures. The Hamiltonian formulation and the problem of quantizing are briefly discussed. Applications to gravity in any space - time dimension are outlined and a generalization of the so called `no - hair' theorem is proven using local properties of the Lagrange equations for a rather general 1+1 dimensional dilaton gravity coupled to matter. This report is based on the paper hep-th/9605008 but some simplifications, corrections and new results are added. 
  The complete class of conformally flat, pure radiation metrics is given, generalising the metric recently given by Wils. 
  We study motion in the field of two fixed centres described by a family of Einstein-dilaton-Maxwell theories. Transitions between regular and chaotic motion are observed as the dilaton coupling is varied. 
  In the present paper a geometrization of electrodynamics is proposed which makes use of a generalization of Riemannian geometry considered already by Einstein and Cartan in the 20ies. Cartan's differential forms description of a teleparallel space-time with torsion is modified by introducing distortion 1-forms which correspond to the distortion tensor in dislocation theory. Under the condition of teleparallelism, the antisymmetrized part of the distorsion 1-form approximates the electromagnetic field, whereas the antisymmetrized part of torsion contributes to the electromagnetic current. Cartan's structure equations, the Bianchi identities, Maxwell's equations and the continuity equation are thus linked in a most simple way. After these purely geometric considerations a physical interpretation, using analogies to the theory of defects in ordered media, is given. A simple defect, which is neither a dislocation nor disclination proper, appears as source of the electromagnetic field. Since this defect is rotational rather than translational, there seems to be no contradiction to Noether's theorem as in other theories relating torsion to electromagnetism. Then,congruences of defect properties and quantum behaviour that arise are discussed, supporting the hypothesis that elementary particles are topological defects of space-time. In agreement with the differential geometry results, a dimensional analysis indicates that the physical unit (lenght)^2 rather than As is the appropriate unit of the electric charge. 
  We construct two solutions of the minimally coupled Einstein-scalar field equations, representing regular deformations of Schwarzschild black holes by a self-interacting, static, scalar field. One solution features an exponentially decaying scalar field and a triple-well interaction potential; the other one is completely analytic and sprouts Coulomb-like scalar hair. Both evade the no-hair theorem by having partially negative potential, in conflict with the dominant energy condition. The linear perturbation theory around such backgrounds is developed in general, and yields stability criteria in terms of effective potentials for an analog Schr\"odinger problem. We can test for more than half of the perturbation modes, and our solutions prove to be stable against those. 
  We attempt to find a function that characterizes gravitational clumping and that increases monotonically as inhomogeneity increases. We choose $S = ln\Omega$ as the candidate ``gravitational entropy'' function, where $\Omega$ is the phase-space volume below the Hamiltonian H of the system under consideration. We compute $\Omega$ for transverse electromagnetic waves and for gravitational wave, radiation and density perturbations in an expanding FLRW universe. These calculations are carried out in the linear regime under the assumption that the phases of the oscillators comprising the system are random. Entropy is thus attributed to the lack of knowledge of the exact field configuration. We find that $\Omega$, and hence $ln\Omega$ behaves as required. We also carry out calculations for Bianchi IX cosmological models and find that, even in this homogeneous case, the function can be interpreted sensibly. We compare our results with Penrose's C^2 hypothesis. Because S is defined to resemble the fundamental statistical mechanics definition of entropy, we are able to recover the entropy in a variety of familiar circumstances including, evidently, black-hole entropy. The results point to the utility of the relativistic ADM Hamiltonian formalism in establishing a connection between general relativity and statistical mechanics, although fully nonlinear calculations will need to be performed to remove any doubt. 
  The following mechanism of action of Time machine is considered. Let space-time $<V^4, g_{ik}>$ be a leaf of a foliation F of codimension 1 in 5-dimensional Lorentz manifold $<V^5, G_{AB}>$. If the Godbillon-Vey class $GV(F) \neq 0$ then the foliation F has resilient leaves. Let $V^4$ be a resilient leaf. Hence there exists an arbitrarily small neighborhood $U_a \subset V^5$ of the event $a \in V^4$ such that $U_a \cap V^4$ consists of at least two connected components $U_a^1$ and $U_a^2$.   Remove the four-dimensional balls $B_a\subset U_a^1, B_b\subset U_a^2$, where an event $b\in U_a^2$, and join the boundaries of formed two holes by means of 4-dimensional cylinder. As result we have a four-dimensional wormhole C, which is a Time machine if b belongs to the past of event a. The past of a is lying arbitrarily nearly. The distant Past is more accessible than the near Past. It seems that real global space-time V^4 is a resilient one, i.e. is a resilient leaf of some foliation F.   It follows from the conformal Kaluza-Klein theory that the movement to the Past through four-dimensional wormhole C along geodesic with respect to metric G_{AB} requires for time machine of large energy and electric charge. 
  The pole-like accelerated expansion stages purely driven by the coupling between the gravity and the dilaton field without referring to the potential term can be realized in a class of generalized gravity theories. We consider three such scenarios based on the scalar-tensor gravity, the induced gravity and the string theory. Quantum fluctuations during the expansion stages (including more general situations) can be derived in exact analytic forms. Assuming that the pole-like acceleration stage provides a viable inflation scenario in the early universe we derive the generated classical density spectrums. The generated classical density field shows a generic tilted spectrum with $n \simeq 4$ which differs from the observed spectrum supporting $n \simeq 1$. 
  When gravitational fields are at their strongest, the evolution of spacetime is thought to be highly erratic. Over the past decade debate has raged over whether this evolution can be classified as chaotic. The debate has centered on the homogeneous but anisotropic mixmaster universe. A definite resolution has been lacking as the techniques used to study the mixmaster dynamics yield observer dependent answers. Here we resolve the conflict by using observer independent, fractal methods. We prove the mixmaster universe is chaotic by exposing the fractal strange repellor that characterizes the dynamics. The repellor is laid bare in both the 6-dimensional minisuperspace of the full Einstein equations, and in a 2-dimensional discretisation of the dynamics. The chaos is encoded in a special set of numbers that form the irrational Farey tree. We quantify the chaos by calculating the strange repellor's Lyapunov dimension, topological entropy and multifractal dimensions. As all of these quantities are coordinate, or gauge independent, there is no longer any ambiguity--the mixmaster universe is indeed chaotic. 
  It is shown that a generic black hole solution of the SU(2) Einstein-Yang-Mills equations develops a new type of an infinitely oscillating behavior near the singularity. Only for certain discrete values of the event horizon radius exceptional solutions exist, possessing an inner structure of the Schwarzschild or Reissner-Nordstrom type. 
  We investigate some properties of geometric operators in canonical quantum gravity in the connection approach \`a la Ashtekar, which are associated with volume, area and length of spatial regions. We motivate the construction of analogous discretized lattice quantities, compute various quantum commutators of the type [area,volume], [area,length] and [volume,length], and find they are generally non-vanishing.   Although our calculations are performed mostly within a lattice-regularized approach, some are -- for special, fixed spin-network configurations -- identical with corresponding continuum computations. Comparison with the structure of the discretized theory leads us to conclude that anomalous commutators may be a general feature of operators constructed along similar lines within a continuum loop representation of quantum general relativity. -- The validity of the lattice approach remains unaffected. 
  Analyzing test particles falling into a Kerr black hole, we study gravitational waves in Brans-Dicke theory of gravity. First we consider a test particle plunging with a constant azimuthal angle into a rotating black hole and calculate the waveform and emitted energy of both scalar and tensor modes of gravitational radiation. We find that the waveform as well as the energy of the scalar gravitational waves weakly depends on the rotation parameter of black hole $a$ and on the azimuthal angle.      Secondly, using a model of a non-spherical dust shell of test particles falling into a Kerr black hole, we study when the scalar modes dominate. When a black hole is rotating, the tensor modes do not vanish even for a ``spherically symmetric" shell, instead a slightly oblate shell minimizes their energy but with non-zero finite value, which depends on Kerr parameter $a$. As a result, we find that the scalar modes dominate only for highly spherical collapse, but they never exceed the tensor modes unless the Brans-Dicke parameter $\omega_{BD} \lsim 750 $ for $a/M=0.99$ or unless $\omega_{BD} \lsim 20,000 $ for $a/M=0.5$, where $M$ is mass of black hole.      We conclude that the scalar gravitational waves with $\omega_{BD} \lsim$ several thousands do not dominate except for very limited situations (observation from the face-on direction of a test particle falling into a Schwarzschild black hole or highly spherical dust shell collapse into a Kerr black hole). Therefore observation of polarization is also required when we determine the theory of gravity by the observation of gravitational waves. 
  We study the spherical gravitational collapse of a compact object under the approximation that the radial pressure is identically zero, and the tangential pressure is related to the density by a linear equation of state. It turns out that the Einstein equations can be reduced to the solution of an integral for the evolution of the area radius. We show that for positive pressure there is a finite region near the center which necessarily expands outwards, if collapse begins from rest. This region could be surrounded by an inward moving one which could collapse to a singularity - any such singularity will necessarily be covered by a horizon. For negative pressure the entire object collapses inwards, but any singularities that could arise are not naked. Thus the nature of the evolution is very different from that of dust, even when the ratio of pressure to density is infinitesimally small. 
  The topologies of event horizons are investigated. Considering the existence of the endpoint of the event horizon, it cannot be differentiable. Then there are the new possibilities of the topology of the event horizon though they are excluded in smooth event horizons. The relation between the topology of the event horizon and the endpoint of it is revealed. A torus event horizon is caused by two-dimensional endpoints. One-dimensional endpoints provide the coalescence of spherical event horizons. Moreover, these aspects can be removed by an appropriate timeslicing. The result will be useful to discuss the stability and generality of the topology of the event horizon. 
  First-order semiclassical perturbations to the Schwarzschild black hole geometry are studied within the black hole interior. The source of the perturbations is taken to be the vacuum stress-energy of quantized scalar, spinor, and vector fields, evaluated using analytic approximations developed by Page and others (for massless fields) and the DeWitt-Schwinger approximation (for massive fields). Viewing the interior as an anisotropic collapsing cosmology, we find that minimally or conformally coupled scalar fields, and spinor fields, decrease the anisotropy as the singularity is approached, while vector fields increase the anisotropy. In addition, we find that massless fields of all spins, and massive vector fields, strengthen the singularity, while massive scalar and spinor fields tend to slow the growth of curvature. 
  Exact string solutions are presented, where moduli fields are varying with time. They provide examples where a dynamical change of the topology of space is occurring. Some other solutions give cosmological examples where some dimensions are compactified dynamically or simulate pre-big bang type scenarios. Some lessons are drawn concerning the region of validity of effective theories and how they can be glued together, using stringy information in the region where the geometry and topology are not well defined from the low energy point of view. Other time dependent solutions are presented where a hierarchy of scales is absent. Such solutions have dynamics which is qualitatively different and resemble plane gravitational waves. (Talk given at the Trieste Spring School and Workshop, 1994.) 
  A new type of gauge quantum theory (superrelativity) has been proposed. This differs from ordinary gauge theories in sense that the affine connection of our theory is constructed from first derivatives of the Fubini-Study metric tensor. That is we have not merely analogy with general relativity but this construction should presumably provide a unification of general relativity and quantum theory. Here we shall discuss the physical meaning of metric properties of the projective Hilbert space and manifestation of its nontrivial physical character. 
  I discuss some aspects of a lattice approach to canonical quantum gravity in a connection formulation, discuss how it differs from the continuum construction, and compare the spectra of geometric operators - encoding information about components of the spatial metric - for some simple lattice quantum states. 
  The Cumulative Drag Index defined recently by Prasanna has been generalised to include the centrifugal acceleration. We have studied the behaviour of the drag index for the Kerr metric and the Neugebauer-Meinel metric representing a self-gravitating rotating disk and their Newtonian approximations. The similarity of the behaviour of the index for a given set of parameters both in the full and approximated forms, suggests that the index characterises an intrinsic property of spacetime with rotation. Analysing the index for a given set of parameters shows possible constraints on them. 
  We derive the wave equation obeyed by electromagnetic fields in curved spacetime. We find that there are Riemann and Ricci curvature coupling terms to the photon polarisation which result in a polarisation dependent deviation of the photon trajectories from null geodesics. Photons are found to have an effective mass in an external gravitational field and their velocity in a local inertial frame is in general less than $c$. The effective photon mass in the Schwarzschild metric is $m_\gamma = ( 2GM/r^3 )^{1/2}$ and near the horizon it is larger than the Hawking temperature of the blackhole. Our result implies that Hawking radiation of photons would not take place. We also conclude that there is no superluminal photon velocity in higher derivative gravity theories (arising from QED radiative corrections), as has been claimed in literature. We show that these erroneous claims are due to the neglect of the Riemann and Ricci coupling terms which exist in Einstein's gravity. 
  We have evaluated the centrifugal force acting on a fluid element and the ellipticity of the fluid configuration, which is slowly rotating, using the Hartle-Thorne solution for different equations of state. The centrifugal force shows a maximum in every case, whereas the reversal in sign could be seen in only one case, and the system becomes unstable in other cases. The ellipticity as calculated from the usual definition shows maxima, whereas the definition obtained from the equilibration of the inertial forces, shows a negative behaviour, indicating that the system is prolate and not oblate. This prolate shape of the configuration is similar to the one earlier found by Pfister and Braun for a rotating shell of matter, using the correct centrifugal force expression for the interior. The location of the centrifugal maxima gets farther away from the Schwarzschild radius as the equation of state gets softer. 
  We investigate the stability of a family of spherically symmetric static solutions in vacuum Brans-Dicke theory (with $\omega=0$) recently described by van Putten. Using linear perturbation theory, we find one exponentially growing mode for every member of the family of solutions, and thus conclude that the solutions are not stable. Using a previously constructed code for spherically symmetric Brans-Dicke, additional evidence for instability is provided by directly evolving the static solutions with perturbations. The full non-linear evolutions also suggest that the solutions are black-hole-threshold critical solutions. 
  We present a classification of all global solutions for generalized 2D dilaton gravity models (with Lorentzian signature). While for some of the popular choices of potential-like terms in the Lagrangian, describing, e.g., string inspired dilaton gravity or spherically reduced gravity, the possible topologies of the resulting spacetimes are restricted severely, we find that for generic choices of these `potentials' there exist maximally extended solutions to the field equations on all non-compact two-surfaces. 
  Cartan's spacetime reformulation of the Newtonian theory of gravity is a generally-covariant Galilean-relativistic limit-form of Einstein's theory of gravity known as the Newton-Cartan theory. According to this theory, space is flat, time is absolute with instantaneous causal influences, and the degenerate `metric' structure of spacetime remains fixed with two mutually orthogonal non-dynamical metrics, one spatial and the other temporal. The spacetime according to this theory is, nevertheless, curved, duly respecting the principle of equivalence, and the non-metric gravitational connection-field is dynamical in the sense that it is determined by matter distributions. Here, this generally-covariant but Galilean-relativistic theory of gravity with a possible non-zero cosmological constant, viewed as a parameterized gauge theory of a gravitational vector-potential minimally coupled to a complex Schroedinger-field (bosonic or fermionic), is successfully cast -- for the first time -- into a manifestly covariant Lagrangian form. Then, exploiting the fact that Newton-Cartan spacetime is intrinsically globally-hyperbolic with a fixed causal structure, the theory is recast both into a constraint-free Hamiltonian form in 3+1-dimensions and into a manifestly covariant reduced phase-space form with non-degenerate symplectic structure in 4-dimensions. Next, this Newton-Cartan-Schroedinger system is non-perturbatively quantized using the standard C*-algebraic technique combined with the geometric procedure of manifestly covariant phase-space quantization. The ensuing unitary quantum field theory of Newtonian gravity coupled to Galilean-relativistic matter is not only generally-covariant, but also exactly soluble. 
  We examine the metric of an isolated self-gravitating abelian-Higgs vortex in dilatonic gravity for arbitrary coupling of the vortex fields to the dilaton. We look for solutions in both massless and massive dilaton gravity. We compare our results to existing metrics for strings in Einstein and Jordan-Brans-Dicke theory. We explore the generalization of Bogomolnyi arguments for our vortices and comment on the effects on test particles. 
  Hawking radiation is derived from the existence of a euclidean instanton which lives in the euclidean black hole geometry. Upon taking into account the gravitational field of the instanton itself, its action is exactly equal to one quarter the change in the horizon area. This result also applies to the Schwinger process, the Unruh process, and particle creation in deSitter space. The implications for horizon thermodynamics are discussed. 
  We study here the structure of singularity forming in gravitational collapse of spherically symmetric inhomogeneous dust. Such a collapse is described by the Tolman-Bondi-Lema{\^i}tre metric, which is a two-parameter family of solutions to Einstein equations, characterized by two free functions of the radial coordinate, namely the `mass function' F(r) and the `energy function' f(r). The main new result here relates, in a general way, the formation of black holes and naked shell-focusing singularities resulting as the final fate of such a collapse to the generic form of regular initial data. Such a data is characterized in terms of the density and velocity profiles of the matter, specified on an initial time slice from which the collapse commences. Several issues regarding the strength and stability of these singularities, when they are naked, are examined with the help of the analysis developed here. In particular, it is seen that strong curvature naked singularities can develop from a generic form of initial data in terms of the initial density profiles for the collapsing configuration. We also establish here that similar results hold for black hole formation as well. We also discuss here the physical constraints on the initial data for avoiding shell-crossing singularities; and also the shell-focusing naked singularities, so that the collapse will necessarily end as a black hole, preserving the cosmic censorship. These results generalize several earlier works on inhomogeneous dust collapse as special cases, and provide a clearer insight into the phenomena of black hole and naked singularity formation in gravitational collapse. 
  Two different ways of quantizing the relativistic Hamiltonian for radial motion in the field of Coulomb-like potential are compared. The results depend slightly on choice of time. In the case of Lorentzian time a Sommerfeld spectrum is recovered. Application to quantum black holes gives a sqrt{n} mass spectrum with about the same numerical factors. 
  I discuss the semiclassical approximation for the Wheeler-DeWitt equation when applied to the CGHS model and spherically symmetric gravity. Special attention is devoted to the issues of Hawking radiation, decoherence of semiclassical states, and black hole entropy. 
  A second-order numerical implementation is given for recently derived nonlinear wave equations for general relativity. The Gowdy T$^3$ cosmology is used as a test bed for studying the accuracy and convergence of simulations of one-dimensional nonlinear waves. The complete freedom in space-time slicing in the present formulation is exploited to compute in the Gowdy line-element. Second-order convergence is found by direct comparison of the results with either analytical solutions for polarized waves, or solutions obtained from Gowdy's reduced wave equations for the more general unpolarized waves. Some directions for extensions are discussed. 
  The properties of multimomentum maps on null hypersurfaces, and their relation with the constraint analysis of General Relativity, are described. Unlike the case of spacelike hypersurfaces, some constraints which are second class in the Hamiltonian formalism turn out to contribute to the multimomentum map. 
  We show that the well-known most general static and spherically symmetric exact solution to the Einstein-massless scalar equations given by Wyman is the same as one found by Janis, Newman and Winicour several years ago. We obtain the energy associated with this spacetime and find that the total energy for the case of the purely scalar field is zero. 
  Two fundamental laws are needed for prediction in the universe: (1) a basic dynamical law and (2) a law for the cosmological initial condition. Quantum cosmology is the area of basic research concerned with the search for a theory of the initial cosmological state. The issues involved in this search are presented in the form of eight problems. (To appear in Physics 2001, ed. by M. Kumar and in the Proceedings of the 10th Yukawa-Nishinomiya Symposium}, November 7-8, 1996, Nishinomiya, Japan.) 
  Gravitational collapse of a spherically symmetric cloud has been extensively studied to investigate the nature of resulting singularity. However, there has been considerable debate about the admissibility of certain initial density distributions. Using the Newtonian limit of the equations governing collapse of a fluid with an equation of state $p=p(\rho)$ it is shown that the density distribution has to be even function of r in a spherically symmetric situation provided $dp/d\rho\ne0$. Implications of this result on formation of strong naked singularities are examined. 
  We develop the canonical quantization of a midisuperspace model which contains, as a subspace, a minisuperspace constituted of a Friedman-Lema\^{\i}tre-Robertson-Walker Universe filled with homogeneous scalar and dust fields, where the sign of the intrinsic curvature of the spacelike hypersurfaces of homogeneity is not specified, allowing the study of topology change in these hypersurfaces. We solve the Wheeler-DeWitt equation of the midisuperspace model restricted to this minisuperspace subspace in the semi-classical approximation. Adopting the conditional probability interpretation, we find that some of the solutions present change of topology of the homogeneous hypersurfaces. However, this result depends crucially on the interpretation we adopt: using the usual probabilistic interpretation, we find selection rules which forbid some of these topology changes. 
  The Barrab\`es-Israel theory of light-like shells in General Relativity is used to show explicitly that in general a light-like shell is accompanied by an impulsive gravitational wave. The gravitational wave is identified by its Petrov Type N contribution to a Dirac delta-function term in the Weyl conformal curvature tensor (with the delta-function singular on the null hypersurface history of the wave and shell). An example is described in which an asymptotically flat static vacuum Weyl space-time experiences a sudden change across a null hypersurface in the multipole moments of its isolated axially symmetric source. A light-like shell and an impulsive gravitational wave are identified, both having the null hypersurface as history. The stress-energy in the shell is dominated (at large distance from the source) by the jump in the monopole moment (the mass) of the source with the jump in the quadrupole moment mainly responsible for the stress being anisotropic. The gravitational wave owes its existence principally to the jump in the quadrupole moment of the source confirming what would be expected. 
  We study singular hypersurfaces in tensor multi-scalar theories of gravity. We derive in a distributional and then in an intrinsic way, the general equations of junction valid for all types of hypersurfaces, in particular for lightlike shells and write the general equations of evolution for these objects. We apply this formalism to various examples in static spherically symmetric spacetimes, and to the study of planar domain walls and plane impulsive waves. 
  Sources of predictability in the basic laws of physics are described in the most general theoretical context -- the quantum theory of the universe as a whole. (To appear in the Proceedings of the conference on Fundamental Sources of Unpredictability held at the Santa Fe Institute, March 28 to 30, 1996 to be published by Complexity.) 
  A covariant formula for conserved currents of energy, momentum and angular-momentum is derived from a general form of Noethers theorem applied directly to the Einstein-Hilbert action of classical general relativity. Energy conservation in a closed big-bang cosmology is discussed as a special case. Special care is taken to distinguish between kinematic and dynamic expressions. 
  By using the Garfinkle, Horowitz and Strominger black hole solutions as examples, we illustrate that, with respect to the reference action functional proposed by Hawking and Horowitz, the asymptotic mass parameter is not invariant between two conformally related static spherically symmetric metrics. 
  We discuss some peculiar properties of the stochastic graviton background predicted by string cosmology. At Planckian times, for the values of the parameters of the model which are more interesting for the detection in gravitational wave experiments, the number density of gravitons is parametrically large compared to Planck density, and is peaked at small energies. The large parameter is related to the duration of the string phase and is a characteristic of string cosmology. The typical interaction time is parametrically larger than the Planck or string time. Therefore the shape of the graviton spectrum is not distorted by thermal effects in the Planck era and can carry informations on the pre-big bang phase suggested by string cosmology. 
  We advocate an alternative description of canonical gravity in 3+1 dimensions, obtained by using as the basic variable a real variant of the usual Ashtekar connection variables on the spatial three-manifold. With this ansatz, no non-trivial reality conditions have to be solved, and the Hamiltonian constraint, though non-polynomial, can be quantized rigorously in a lattice regularization. 
  I review and discuss some recent developments in non-perturbative approaches to quantum gravity, with an emphasis on discrete formulations, and those coming from a classical connection description. 
  We present a numerical scheme that solves the initial value problem in full general relativity for a binary neutron star in quasi-equilibrium. While Newtonian gravity allows for a strict equilibrium, a relativistic binary system emits gravitational radiation, causing the system to lose energy and slowly spiral inwards. However, since inspiral occurs on a time scale much longer than the orbital period, we can adopt a quasi-equilibrium approximation. In this approximation, we integrate a subset of the Einstein equations coupled to the equations of relativistic hydrodynamics to solve the initial value problem for binaries of arbitrary separation, down to the innermost stable orbit. 
  In this paper a one to one correspondence is established between space-time metrics of general relativity and the wave equations of quantum mechanics. This is done by first taking the square root of the metric associated with a space and from there, passing directly to a corresponding expression in the dual space. It is shown that in the case of a massless particle, Maxwell's equation for a photon follows while in the case of a particle with mass, Dirac's equation results as a first approximation. Moreover, this one to one correspondence suggests a natural explanation of wave-particle duality. As a consequence, the distinction between quantum mechanics and classical relativistic mechanics is more clearly understood and the key role of initial conditions is emphasized. PACS NUMBERS: 03.65, 04.60 
  Recently Abrahams and Cook devised a method of estimating the total radiated energy resulting from collisions of distant black holes by applying Newtonian evolution to the holes up to the point where a common apparent horizon forms around the two black holes and subsequently applying Schwarzschild perturbation techniques . Despite the crudeness of their method, their results for the case of head-on collisions were surprisingly accurate. Here we take advantage of the simple radiated energy formula devised in the close-slow approximation for black hole collisions to test how strongly the Abrahams-Cook result depends on the choice of moment when the method of evolution switches over from Newtonian to general relativistic evolution. We find that their result is robust, not depending strongly on this choice. 
  We have developed a numerical method for evolving perturbations of rotating black holes. Solutions are obtained by integrating the Teukolsky equation written as a first-order in time, coupled system of equations, in a form that explicitly exhibits the radial characteristic directions. We follow the propagation of generic initial data through the burst, quasi-normal ringing and power-law tail phases. Future results may help to clarify the role of black hole angular momentum on signals produced during the final stages of black hole coalescence. 
  Multidimensional cosmological model with the topology M=RxM_1xM_2x...xM_n where M_i (i=1,... ,n) undergo a chain splitting into arbitrary number of compact spaces is considered. It is shown that equations of motion can be solved exactly because they depend only on the effective curvatures and dimensions and "forget" about inner topological structure. It is proved that effective cosmological action for the model with n=1 in the case of infinite splitting of the internal space coincides with the tree-level effective action for a bosonic string. 
  We present an analysis of the cosmological evolution of matter sources with small anisotropic pressures. This includes electric and magnetic fields, collisionless relativistic particles, gravitons, antisymmetric axion fields in low-energy string cosmologies, spatial curvature anisotropies, and stresses arising from simple topological defects. We calculate their evolution during the radiation and dust eras of an almost isotropic universe. In many interesting cases the evolution displays a special critical behaviour created by the non-linear evolution of the pressure and expansion anisotropies. The isotropy of the microwave background is used to place strong limits of order $\Omega _{a0}\leq 5\times 10^{-6}\Delta (1+z_{rec})^{-\Delta }$on the possible contribution of these matter sources to the total density of the universe, where $1\leq \Delta \leq 3$ characterises the anisotropic stress. The present abundance of an anisotropic stress which becomes non-relativistic at a characteristic low-energy scale is also calculated. We explain why the limits obtained from primordial nucleosynthesis are generally weaker than those imposed by the microwave background isotropy. The effect of inflation on these stresses is also calculated. 
  We estimate the expected signal-to-noise ratios (SNRs) from the three phases (inspiral,merger,ringdown) of coalescing binary black holes (BBHs) for initial and advanced ground-based interferometers (LIGO/VIRGO) and for space-based interferometers (LISA). LIGO/VIRGO can do moderate SNR (a few tens), moderate accuracy studies of BBH coalescences in the mass range of a few to about 2000 solar masses; LISA can do high SNR (of order 10^4) high accuracy studies in the mass range of about 10^5 to 10^8 solar masses. BBHs might well be the first sources detected by LIGO/VIRGO: they are visible to much larger distances (up to 500 Mpc by initial interferometers) than coalescing neutron star binaries (heretofore regarded as the "bread and butter" workhorse source for LIGO/VIRGO, visible to about 30 Mpc by initial interferometers). Low-mass BBHs (up to 50 solar masses for initial LIGO interferometers; 100 for advanced; 10^6 for LISA) are best searched for via their well-understood inspiral waves; higher mass BBHs must be searched for via their poorly understood merger waves and/or their well-understood ringdown waves. A matched filtering search for massive BBHs based on ringdown waves should be capable of finding BBHs in the mass range of about 100 to 700 solar masses out to 200 Mpc (initial LIGO interferometers), and 200 to 3000 solar masses out to about z=1 (advanced interferometers). The required number of templates is of order 6000 or less. Searches based on merger waves could increase the number of detected massive BBHs by a factor of order 10 or more over those found from inspiral and ringdown waves, without detailed knowledge of the waveform shapes, using a "noise monitoring" search algorithm. A full set of merger templates from numerical relativity could further increase the number of detected BBHs by an additional factor of up to 4. 
  It has been known classically that a star with an ergoregion but no event horizon is unstable to the emission of scalar, electromagnetic and gravitational waves. This classical ergoregion instability is characterized by complex frequency modes. We show how to canonically quantize a neutral scalar field in the presence of such unstable modes by considering a simple model for a rapidly rotating star. Some of interesting results is that there exists a physically meaningful mode decomposition including unstable normal mode solutions whose representation turns out to be a non-Fock-like Hilbert space. A ``particle" detector model placed in the in-vacuum state also shows that stars with ergoregions give rise to a spontaneous energy radiation to spatial infinity until ergoregions disappear. 
  I discuss the role played by the spin-network basis and recoupling theory (in its graphical tangle-theoretic formulation) and their use for performing explicit calculations in loop quantum gravity. In particular, I show that recoupling theory allows the derivation of explicit expressions for the eingenvalues of the quantum volume operator. An important side result of these computations is the determination of a scalar product with respect to which area and volume operators are symmetric, and the spin network states are orthonormal. 
  This pre-print contains the abstracts of seminars (including key references) presented at the ESI workshop on mathematical problems in quantum gravity held during July and August of 1996. Contributors include A. Ashtekar, J. Baez, F. Barbero, A. Barvinsky, F. Embacher, R. Gambini, D. Giulini, J. Halliwell, T. Jacobson, R. Loll, D. Marolf, K. Meissner, R. Myers, J. Pullin, M. Reisenberger, C. Rovelli, T. Strobl and T. Thiemann. While these contributions cover most of the talks given during the workshop, there were also a few additional speakers whose contributions were not received in time. 
  We discuss the classicalization of a quantum state induced by an environment in the inflationary stage of the universe. The classicalization is necessary for the homogeneous ground sate to become classical non-homogeneous one accompanied with the statistical fluctuation, which is a plausible candidate for the seeds of structure formation. Using simple models, we show that i) the two classicalization criteria, the classical correlation and quantum decoherence, are simultaneously satisfied by the environment and that ii) the power spectrum of the resultant statistical fluctuation depends upon the detail of the classicalization process. Especially, the result ii) means that, taking account of the classicalization process, the inflationary scenario does not necessarily predict the unique spectrum which is usually believed. 
  We investigate Lie symmetries of Einstein's vacuum equations in N dimensions, with a cosmological term. For this purpose, we first write down the second prolongation of the symmetry generating vector fields, and compute its action on Einstein's equations. Instead of setting to zero the coefficients of all independent partial derivatives (which involves a very complicated substitution of Einstein's equations), we set to zero the coefficients of derivatives that do not appear in Einstein's equations. This considerably constrains the coefficients of symmetry generating vector fields. Using the Lie algebra property of generators of symmetries and the fact that general coordinate transformations are symmetries of Einstein's equations, we are then able to obtain all the Lie symmetries. The method we have used can likely be applied to other types of equations. 
  This paper studies the application of multimomentum maps to the constraint analysis of general relativity on null hypersurfaces. It is shown that, unlike the case of spacelike hypersurfaces, some constraints which are second class in the Hamiltonian formalism turn out to contribute to the multimomentum map. To recover the whole set of secondary constraints found in the Hamiltonian formalism, it is necessary to combine the multimomentum map with those particular Euler-Lagrange equations which are not of evolutionary type. The analysis is performed on the outgoing null cone only. 
  Irregularities in the metric tensor of a signature-changing space-time suggest that field equations on such space-times might be regarded as distributional. We review the formalism of tensor distributions on differentiable manifolds, and examine to what extent rigorous meaning can be given to field equations in the presence of signature-change, in particular those involving covariant derivatives. We find that, for both continuous and discontinuous signature-change, covariant differentiation can be defined on a class of tensor distributions wide enough to be physically interesting. 
  Tensor distributions and their derivatives are described without assuming the presence of a metric. This provides a natural framework for discussing tensor distributions on manifolds with degenerate metrics, including in particular metrics which change signature. 
  We present a perturbative construction of the $\varphi^4$ model on a smooth globally hyperbolic space-time. Our method relies on a adaptation of the Epstein and Glaser method of renormalization to curved space-times using techniques from microlocal analysis. 
  We present a brief review of some recent results on non-abelian solitons and black holes in different theories.   Lecture given at the Simi-96, Tbilisi, Georgia, September 22-28, 1996. 
  Using the post-Newtonian (PN) expansion technique of the gravitational wave perturbation around a Schwarzschild black hole, we calculate analytically the energy flux of gravitational waves induced by a particle in circular orbits up to the 5.5PN order, i.e. $O(v^{11})$ beyond Newtonian. By comparing the formula with numerical data, we find that the error of the 5.5PN formula is about 4% when the particle is on the last stable circular orbit. We also estimate the error $\Delta N$ in the total cycle of gravitational waves from coalescing compact binaries in a laser interferometer's band produced by using the post-Newtonian approximations. We find that, as for the neutron star-black hole binaries, the 4.5PN approximation gives $\Delta N\alt1$ for a black hole of mass $M<40M_\odot$, while it gives $\Delta N\agt1$ for a black hole of mass $M>40M_{\odot}$. 
  A statistical model M is a family of probability distributions, characterised by a set of continuous parameters known as the parameter space. This possesses natural geometrical properties induced by the embedding of the family of probability distributions into the Hilbert space H. By consideration of the square-root density function we can regard M as a submanifold of the unit sphere in H. Therefore, H embodies the `state space' of the probability distributions, and the geometry of M can be described in terms of the embedding of in H. The geometry in question is characterised by a natural Riemannian metric (the Fisher-Rao metric), thus allowing us to formulate the principles of classical statistical inference in a natural geometric setting. In particular, we focus attention on the variance lower bounds for statistical estimation, and establish generalisations of the classical Cramer-Rao and Bhattacharyya inequalities. The statistical model M is then specialised to the case of a submanifold of the state space of a quantum mechanical system. This is pursued by introducing a compatible complex structure on the underlying real Hilbert space, which allows the operations of ordinary quantum mechanics to be reinterpreted in the language of real Hilbert space geometry. The application of generalised variance bounds in the case of quantum statistical estimation leads to a set of higher order corrections to the Heisenberg uncertainty relations for canonically conjugate observables. 
  This is an informal review of the formulation of canonical general relativity and of its implications for quantum gravity; the various versions are compared, both in the continuum and in a discretized approximation suggested by Regge calculus. I also show that the weakness of the link with the geometric content of the theory gives rise to what I think is a serious flaw in the claimed derivation of a discrete structure for space at the quantum level. 
  Held has proposed an integration procedure within the GHP formalism built around four real, functionally independent, zero-weighted scalars. He suggests that such a procedure would be particularly simple for the `optimal situation', when the formalism directly supplies the full quota of four scalars of this type; a spacetime without any Killing vectors would be such a situation. Wils has recently obtained a metric which he claims is the only conformally flat, pure radiation metric which is not a plane wave; this metric has been shown by Koutras to admit no Killing vectors, in general. Therefore, as a simple illustration of the GHP integration procedure, we obtain systematically the complete class of conformally flat, pure radiation metrics. Our result shows that the conformally flat, pure radiation metrics, which are not plane waves, are a larger class than Wils has obtained. 
  The notions of stress and hyperstress are anchored in 3-dimensional continuum mechanics. Within the framework of the 4-dimensional spacetime continuum, stress and hyperstress translate into the energy-momentum and the hypermomentum current, respectively. These currents describe the inertial properties of classical matter fields in relativistic field theory. The hypermomentum current can be split into spin, dilation, and shear current. We discuss the conservation laws of momentum and hypermomentum and point out under which conditions the momentum current becomes symmetric. 
  We analyze the existence of inflationary solutions in an inhomogeneous Kaluza-Klein cosmological model in 4+n dimensions. It is shown that the 5-dimensional case is the exception rather than the rule, in the sense that the system is integrable (under the assumption of the equation of state $\rho= kp$) for any value of k. It is also shown that the cases k=0 and k=1/3 are integrable if and only if n=1. 
  We present evidence that, below a certain threshold scale, the horizon of a black hole is strongly wrinkled, with its shape manifesting a self-similar (``fractal'') spectrum of fluctuations on all scales below the threshold. This threshold scale is small compared to the radius of the black hole, but still much larger than the Planck scale. If present, such fluctuations might account for a large part of the horizon entropy. 
  A new lattice based scheme for numerical relativity will be presented. The scheme uses the same data as would be used in the Regge calculus (eg. a set of leg lengths on a simplicial lattice) but it differs significantly in the way that the field equations are computed. In the new method the standard Einstein field equations are applied directly to the lattice. This is done by using locally defined Riemann normal coordinates to interpolate a smooth metric over local groups of cells of the lattice. Results for the time symmetric initial data for the Schwarzschild spacetime will be presented. It will be shown that the scheme yields second order accurate estimates (in the lattice spacing) for the metric and the curvature. It will also be shown that the Bianchi identities play an essential role in the construction of the Schwarzschild initial data. 
  Using the canonical formalism for spherically symmetric black hole inside the apparent horizon we investigate the mass inflation in the Reissner-Nordstr$\ddot o$m black hole in the framework of quantum gravity. It is shown that like in classical gravity the combination of the effects of the influx coming from the past null infinity and the outflux backscattered by the black hole's curvature causes the mass inflation even in quantum gravity. The results indicate that the effects of quantum gravity neither alter the classical picture of the mass inflation nor prevent the formation of the mass inflation singularity. 
  Entanglement entropy is often speculated as a strong candidate for the origin of the black-hole entropy. To judge whether this speculation is true or not, it is effective to investigate the whole structure of thermodynamics obtained from the entanglement entropy, rather than just to examine the apparent structure of the entropy alone or to compare it with that of the black hole entropy. It is because entropy acquires a physical significance only when it is related to the energy and the temperature of a system. From this point of view, we construct a `thermodynamics of entanglement' by introducing an entanglement energy and compare it with the black-hole thermodynamics. We consider two possible definitions of entanglement energy. Then we construct two different kinds of thermodynamics by combining each of these different definitions of entanglement energy with the entanglement entropy. We find that both of these two kinds of thermodynamics show significant differences from the black-hole thermodynamics if no gravitational effects are taken into account. These differences are in particular highlighted in the context of the third law of thermodynamics. Finally we see how inclusion of gravity alter the thermodynamics of the entanglement. We give a suggestive argument that the thermodynamics of the entanglement behaves like the black-hole thermodynamics if the gravitational effects are included properly. Thus the entanglement entropy passes a non-trivial check to be the origin of the black-hole entropy. 
  We describe the analytical extension of certain static cylindrical multi--cosmic string metrics to wormhole spacetimes with only one region at spatial infinity, and investigate in detail the geometry of asymptotically Minkowskian wormhole spacetimes generated by one or two cosmic strings. We find that such wormholes tend to lengthen rather than shorten space travel. Possible signatures of these wormholes are briefly discussed. 
  The necessary and sufficient conditions for the exactness of the semiclassical approximation for the solution of the Schr\"odinger and Klein-Gordon equations are obtained. It is shown that the existence of an exact semiclassical solution of the Schr\"odinger equation determines both the semiclassical wave function and the interaction potential uniquely up to the choice of the boundary conditions. This result also holds for the Klein-Gordon equation. Its implications for the solution of the Wheeler-DeWitt equation for the FRW scalar field minisuperspace models are discussed. In particular, exact semiclassical solutions of the Wheeler-DeWitt equation for the case of massless scalar field and exponential matter potentials are constructed. The existence of exact semiclassical solutions for polynomial matter potentials of the form $\lambda\phi^{2p}$ is also analyzed. It is shown that for p=1, 2 and 3, right-going semiclassical solutions do not exist. A generalized semiclassical perturbation expansion is also developed which is quite different from the traditional $\hbar$ and $M_p^{-1}$-expansions. 
  It is argued that the fifth coordinate should correspond to an intensive parameter rather than to rest-mass as originally proposed by Wesson. 
  The metric perturbation tensor corresponding to a transverse oscillation of spacetime is composed of products of cosines. When averaged over many wavelengths, such a metric may look either Minkowskian or Euclidean at large scales, depending on the amplitude and wavelength of the oscillation. 
  We present the first results of a self-consistent solution of the semiclassical Einstein field equations corresponding to a Lorentzian wormhole coupled to a quantum scalar field. The specific solution presented here represents a wormhole connecting two asymptotically spatially flat regions. In general, the diameter of the wormhole throat, in units of the Planck length, can be arbitrarily large, depending on the values of the scalar coupling $\xi$ and the boundary values for the shape and redshift functions. In all cases we have considered, there is a fine-structure in the form of Planck-scale oscillations or ripples superimposed on the solutions. 
  A candidate theory of gravity quantized is reviewed. 
  3+1 decompositions of differential forms on a Lorentzian manifold (M,g;+ - - -) with respect to arbitrary observer field and the decomposition of the standard operations acting on them are studied, making use of the ideas of the theory of connections on principal bundles. Simple explicit general formulas are given as well as their application to the Maxwell equations. 
  A particular Finsler-metric proposed in [1,2] and describing a geometry with a preferred null direction is characterized here as belonging to a subclass contained in a larger class of Finsler-metrics with one or more preferred directions (null, space- or timelike). The metrics are classified according to their group of isometries. These turn out to be isomorphic to subgroups of the Poincar\'e (Lorentz-) group complemented by the generator of a dilatation. The arising Finsler geometries may be used for the construction of relativistic theories testing the isotropy of space. It is shown that the Finsler space with the only preferred null direction is the anisotropic space closest to isotropic Minkowski-space of the full class discussed. 
  This work addresses a specific technical question of relevance to canonical quantization of gravity using the so-called new variables and loop-based techniques of Ashtekar, Rovelli, and Smolin. In particular, certain `superselection laws' that arise in current applications of these techniques to solving the diffeomorphism constraint are considered. Their status is elucidated by studying an analogous system: 2+1 Euclidean gravity. For that system, these superselection laws are shown to be spurious. This, however, is only a technical difficulty. The usual quantum theory may still be obtained from a loop representation and the technique known as `Refined Algebraic Quantization.' 
  Beginning with the self-dual two-forms approach to the Einstein equations, we show how, by choosing basis spinors which are proportional to solutions of the Dirac equation, we may rewrite the vacuum Einstein equations in terms of a set of divergence-free vector fields, which obey a particular set of chiral equations. Upon imposing the Jacobi identity upon these vector fields, we reproduce a previous formulation of the Einstein equations linked with a generalisation of the Yang-Mills equations for a constant connection on flat space. This formulation suggests the investigation of some new aspects of the self-dual two-forms approach. In the case of real Riemannian metrics, these vector fields have a natural interpretation in terms of the torsion of the natural almost-complex-structure on the projective spin-bundle. 
  We present a finite difference version of the eth formalism, which allows use of tensor fields in spherical coordinates in a manner which avoids polar singularities. The method employs two overlapping stereographic coordinate patches, with interpolations between the patches in the regions of overlap. It provides a new and effective computational tool for dealing with a wide variety of systems in which spherical coordinates are natural, such as the generation of radiation from an isolated source. We test the formalism with the evolution of waves in three spatial dimensions and the calculation of the curvature scalar of arbitrarily curved geometries on topologically spherical manifolds. The formalism is applied to the solution of the Robinson-Trautman equation and reveals some new features of gravitational waveforms in the nonlinear regime. 
  The question of whether unobserved short-wavelength modes of the gravitational field can induce decoherence in the long-wavelength modes (``the decoherence of spacetime'') is addressed using a scalar field toy model with some features of perturbative general relativity.   This is an abridged version of the author's paper gr-qc/9612028. 
  The back reaction of gravitational perturbations in a homogeneous background is determined by an effective energy-momentum tensor quadratic in the perturbations. We show that this nonlinear feedback effect is important in the case of long wavelength scalar perturbations in inflationary universe models. We also show how to solve an old problem concerning the gauge dependence of the effective energy-momentum tensor of perturbations. 
  Einstein's equations admit solutions corresponding to photon rockets. In these a massive particle recoils because of the anisotropic emission of photons. In this paper we ask whether rocket motion can be powered only by the emission of gravitational waves. We use the double series approximation method and show that this is possible. A loss of mass and gain in momentum arise in the second approximation because of the emission of quadrupole and octupole waves. 
  While there has been some advance in the use of Regge calculus as a tool in numerical relativity, the main progress in Regge calculus recently has been in quantum gravity. After a brief discussion of this progress, attention is focussed on two particular, related aspects. Firstly, the possible definitions of diffeomorphisms or gauge transformations in Regge calculus are examined and examples are given. Secondly, an investigation of the signature of the simplicial supermetric is described. This is the Lund-Regge metric on simplicial configuration space and defines the distance between simplicial three-geometries. Information on its signature can be used to extend the rather limited results on the signature of the supermetric in the continuum case. This information is obtained by a combination of analytic and numerical techniques. For the three-sphere and the three-torus, the numerical results agree with the analytic ones and show the existence of degeneracy and signature change. Some ``vertical'' directions in simplicial configuration space, corresponding to simplicial metrics related by gauge transformations, are found for the three-torus. 
  We prove a global existence theorem (with respect to a geometrically- defined time) for globally hyperbolic solutions of the vacuum Einstein equations which admit a $T^2$ isometry group with two-dimensional spacelike orbits, acting on $T^3$ spacelike surfaces. 
  We consider dissipative relativistic fluid theories on a fixed flat, compact, globally hyperbolic, Lorentzian manifold. We prove that for all initial data in a small enough neighborhood of the equilibrium states (in an appropriate Sobolev norm), the solutions evolve smoothly in time forever and decay exponentially to some, in general undetermined, equilibrium state. To prove this, three conditions are imposed on these theories. The first condition requires the system of equations to be symmetric hyperbolic, a fundamental requisite to have a well posed and physically consistent initial value formulation. The second condition is a generic consequence of the entropy law, and is imposed on the non principal part of the equations. The third condition is imposed on the principal part of the equations and it implies that the dissipation affects all the fields of the theory. With these requirements we prove that all the eigenvalues of the symbol associated to the system of equations of the fluid theory have strictly negative real parts, which in fact, is an alternative characterization for the theory to be totally dissipative. Once this result has been obtained, a straight forward application of a general stability theorem due to Kreiss, Ortiz, and Reula, implies the results above mentioned. 
  I present here the results of the study of the gravitational radiation generated by the infall (from rest at radius $r_0$) of a point particle of mass $m_0$ into a Schwarzschild black hole of mass $M$. We use Laplace's transform methods and find that the spectra of radiation for $\sim5M<r_0<\infty$ presents a series of evenly spaced bumps. The total radiated energy is not monotonically decreasing with $r_0$, but presents a joroba (hunch-back) at around $r_0\approx4.5M$. I finally discuss the detectability of the gravitational radiation coming from the black hole in the center of our galaxy. 
  Contents:  News:  April 1997 Joint APS/AAPT Meeting, by Beverly Berger  TGG News, by Jim Isenberg  Report from NSF, by David Berley  We hear that..., by Jorge Pullin  Research briefs:  GR in GPS, by Neil Ashby  What happens near the innermost stable circular orbit? by Doug Eardley  Conference reports:  Journees Relativistes 96, by D. Brill, M. Heusler, G. Lavrelashvili  TAMA Workshop, by Peter Saulson  Midwest gravity meeting, by Comer Duncan  OMNI-1 Workshop by N.S. Magalhaes, W. F. Velloso Jr and O.D. Aguiar  Chandra Symposium, by Robert Wald  Penn State Meeting, by Lee Smolin  Aspen Winter Conference, by Syd Meshkov 
  The translational Chern-Simons type three-form coframe torsion on a Riemann-Cartan spacetime is related (by differentiation) to the Nieh-Yan four-form. Following Chandia and Zanelli, two spaces with non-trivial translational Chern-Simons forms are discussed. We then demonstrate, firstly within the classical Einstein-Cartan-Dirac theory and secondly in the quantum heat kernel approach to the Dirac operator, how the Nieh-Yan form surfaces in both contexts, in contrast to what has been assumed previously. 
  We present analytical perturbative, and numerical solutions of the Einstein equation which describe a black hole with a nontrivial dilaton field and a purely topological gauge potential. The gauge potential has zero field strength and hence no stress-energy, but it does couple to virtual string worldsheets which wrap around the Euclidean horizon two-sphere, and generate an effective interaction in the spacetime lagrangian. We use the lagrangian with a nonstandard potential for a scalar field that reproduces the effect of the worldsheet instantons. As has been previously pointed out the topological charge Q of the gauge field can be detected by an Anharonov-Bohm type experiment using quantum strings; the classical scalar hair of the solutions here is a classical detection of Q. ADM mass, dilaton charge and Hawking temperature are calculated and compared with the known cases when appropriate. We discuss why these solutions do not violate the no-hair theorems and show that, for sufficiently small interaction coupling, the solutions are stable under linear time dependent perturbations. 
  A problem related to some Green functions found by Guimar\~aes and Linet in the manifold of a straight cosmic string continued in the Rindler space is investigated. 
  A model of the universe as a self-organized critical system is considered. The universe evolves to a state independently of the initial conditions at the edge of chaos. The critical state is an attractor of the dynamics. Random metric fluctuations exhibit noise without any characteristic length scales, and the power spectrum for the fluctuations has a self-similar fractal behavior. In the early universe, the metric fluctuations smear out the local light cones removing the horizon problem. 
  The paper presents a statistical model which reproduces the results of Monte Carlo simulations to estimate the parameters of the gravitational wave signal from a coalesing binary system. The model however is quite general and would be useful in other parameter estimation problems. 
  The principles and detection of gravitational waves by resonant antennas are briefly discussed. But the main purpose of this short note is to compare the two geometries of resonant antennas, the well-known cylindrical to the spherical type. Some features of a two sphere observatory are also discussed. 
  Black holes are thermodynamic objects, but despite recent progress, the ultimate statistical mechanical origin of black hole temperature and entropy remains mysterious. Here I summarize an approach in which the entropy is viewed as arising from ``would-be pure gauge'' degrees of freedom that become dynamical at the horizon. For the (2+1)-dimensional black hole, these degrees of freedom can be counted, and yield the correct Bekenstein-Hawking entropy; the corresponding problem in 3+1 dimensions remains open. 
  It is likely that the observed distribution of the microwave background temperature over the sky is only one realization of the underlying random process associated with cosmological perturbations of quantum-mechanical origin. If so, one needs to derive the parameters of the random process, as accurately as possible, from the data of a single map. These parameters are of the utmost importance, since our knowledge of them would help us to reconstruct the dynamical evolution of the very early Universe. It appears that the lack of ergodicity of a random process on a 2-sphere does not allow us to do this with arbitrarily high accuracy. We are left with the problem of finding the best unbiased estimators of the participating parameters. A detailed solution to this problem is presented in this article. The theoretical error bars for the best unbiased estimates are derived and discussed. 
  We argue that many future-eternal inflating spacetimes are likely to violate the weak energy condition. It is possible that such spacetimes may not enforce any of the known averaged conditions either. If this is indeed the case, it may open the door to constructing non-singular, past-eternal inflating cosmologies. Simple non-singular models are, however, unsatisfactory, and it is not clear if satisfactory models can be built that solve the problem of the initial singularity. 
  A summary is given of recent exact results concerning the functional integration measure in Regge gravity. 
  Within the Hamiltonian formulation of diffeomorphism invariant theories we address the problem of how to determine and how to reduce diffeomorphisms outside the identity component. 
  We review the remarkable relationship between the laws of black hole mechanics and the ordinary laws of thermodynamics. It is emphasized that - in analogy with the laws of thermodynamics - the validity the laws of black hole mechanics does not appear to depend upon the details of the underlying dynamical theory (i.e., upon the particular field equations of general relativity). It also is emphasized that a number of unresolved issues arise in ``ordinary thermodynamics'' in the context of general relativity. Thus, a deeper understanding of the relationship between black holes and thermodynamics may provide us with an opportunity not only to gain a better understanding of the nature of black holes in quantum gravity, but also to better understand some aspects of the fundamental nature of thermodynamics itself. 
  Various authors have shown the occurence of naked singularities and black holes in the spherical gravitational collapse of inhomogeneous dust. In a recent preprint, Antia has criticised a statement in a paper by Jhingan, Joshi and Singh on dust collapse. We show that his criticism is invalid. Antia shows that in Eulerian coordinates a series expansion for the density of a collapsing Newtonian fluid can have only even powers. However, he has overlooked the fact that Jhingan et al. have actually used Lagrangian (comoving) coordinates, and not Eulerian coordinates. As we show, in Lagrangian coordinates there is no restriction that the density have only even powers and hence his criticism is invalid. We also point out that an earlier claim by Antia on the instability of strong naked singularities in dust collapse is not supported by any concrete analysis, and is hence incorrect. 
  We review the language of differential forms and their applications to Riemannian Geometry with an orientation to General Relativity. Working with the principal algebraic and differential operations on forms, we obtain the structure equations and their symmetries in terms of a new product (the co-multiplication). It is showen how the Cartan - Grassmann algebra can be endowed with the structure of a Hopf algebra. 
  A new approach to quantum gravity is described which joins the loop representation formulation of the canonical theory to the causal set formulation of the path integral. The theory assigns quantum amplitudes to special classes of causal sets, which consist of spin networks representing quantum states of the gravitational field joined together by labeled null edges. The theory exists in 3+1, 2+1 and 1+1 dimensional versions, and may also be interepreted as a theory of labeled timelike surfaces. The dynamics is specified by a choice of functions of the labelings of d+1 dimensional simplices,which represent elementary future light cones of events in these discrete spacetimes. The quantum dynamics thus respects the discrete causal structure of the causal sets. In the 1+1 dimensional case the theory is closely related to directed percolation models. In this case, at least, the theory may have critical behavior associated with percolation, leading to the existence of a classical limit. 
  We will apply the quantum inequality type restrictions to Alcubierre's warp drive metric on a scale in which a local region of spacetime can be considered ``flat''. These are inequalities that restrict the magnitude and extent of the negative energy which is needed to form the warp drive metric. From this we are able to place limits on the parameters of the ``Warp Bubble''. It will be shown that the bubble wall thickness is on the order of only a few hundred Planck lengths. Then we will show that the total integrated energy density needed to maintain the warp metric with such thin walls is physically unattainable. 
  In the context of the vacuum polarization effect, we consider the backreaction of the energy-momentum tensor of a charged scalar field on the background metric of a cosmic string carrying a magnetic flux $\Phi$. Working within the semiclassical approach to the Einstein eqs. we find the first-order (in $\hbar$) metric associated to the magnetic flux cosmic string. We show that the contribution to the vacuum polarization effect coming from the Aharonov-Bohm interaction is larger than the one coming from the non-trivial gravitational interaction. 
  We propose a distinguished set of positive and negative energy modes of the Klein-Gordon equation as a time independent definition of the vacuum state of a quantized scalar field. 
  Gravitational collapse of a spherically symmetric cloud has been extensively studied to investigate the nature of resulting singularity. However, there has been considerable debate about the admissibility of certain initial density distributions. Using the Newtonian limit of the equations governing collapse of a fluid with an equation of state p=p(\rho) it is shown that the density distribution has to be even function of r in a spherically symmetric situation provided dp/d\rho \ne 0, even in comoving coordinates. We show that recent claim by Singh that the discrepancy pointed out earlier is due to their use of comoving coordinates is totally incorrect. It is surprising that he expects the use of comoving coordinates to make any difference in this matter. It is also argued that strong curvature naked singularities in gravitational collapse of spherically symmetric dust do not violate the cosmic censorship hypothesis. 
  The roles that spin networks play in gauge theories, quantum gravity and topological quantum field theory are briefly described, with an emphasis on the question of the relationships among them. It is argued that spin networks and their generalizations provide a language which may lead to a unification of the different approaches to quantum gravity and quantum geometry. This leads to a set of conjectures about the form of a future theory that may be simultaneously an extension of the non-perturbative quantization of general relativity and a non-perturbative formulation of string theory. 
  A time machine (TM) is constructed whose creating in contrast to all TMs known so far requires neither singularities, nor violation of the weak energy condition (WEC). The spacetime exterior to the TM closely resembles the Friedmann universe. 
  The critical behavior and phase transition in the 2+1 dimensional Ba\~nados, Teitelboim, and Zanelli (BTZ) black holes are discussed. By calculating the equilibrium thermodynamic fluctuations in the microcanonical ensemble, canonical ensemble, and grand canonical ensemble, respectively, we find that the extremal spinning BTZ black hole is a critical point, some critical exponents satisfy the scaling laws of the ``first kind'', and the scaling laws related to the correlation length suggest that the effective spatial dimension of extremal black holes is one, which is in agreement with the argument that the extremal black holes are the Bogomol'nyi saturated string states. In addition, we find that the massless BTZ black hole is a critical point of spinless BTZ black holes. 
  A new concept analogous to global hyperbolicity is introduced, based on test fields. It is shown that the space-times termed here ``curve integrable'' are globally hyperbolic in this new sense, and a plausibility argument is given suggesting that the result applies to shell crossing singularities. If the assumptions behind this last argument are valid, this provides an alternative route to the assertion that such singularities do not violate cosmic censorship. 
  We give a self-contained introduction into the metric-affine gauge theory of gravity. Starting from the equivalence of reference frames, the prototype of a gauge theory is presented and illustrated by the example of Yang-Mills theory. Along the same lines we perform a gauging of the affine group and establish the geometry of metric-affine gravity. The results are put into the dynamical framework of a classical field theory. We derive subcases of metric-affine gravity by restricting the affine group to some of its subgroups. The important subcase of general relativity as a gauge theory of translations is explained in detail. 
  In this paper we provide fully covariant proofs of some theorems on shear-free perfect fluids. In particular, we explicitly show that any shear-free perfect fluid with the acceleration proportional to the vorticity vector (including the simpler case of vanishing acceleration) must be either non-expanding or non-rotating. We also show that these results are not necessarily true in the Newtonian case, and present an explicit comparison of shear-free dust in Newtonian and relativistic theories in order to see where and why the differences appear. 
  We review here some recent developments on the issue of final fate of gravitational collapse within the framework of Einstein theory of gravity. The structure of collapsed object is discussed in terms of either a black hole or a singularity having causal connection with outside universe. Implications for cosmic censorship are discussed. 
  The black hole solutions in the higher dimensional Brans-Dicke-Maxwell theory are investigated. We find that the presence of the nontrivial scalar field depends on the spacetime dimensions (D). When D=4, the solution corresponds to the Reissner-Nordstr\"{o}m black hole with a constant scalar field. In higher dimensions (D>4), one finds the charged black hole solutions with the nontrivial scalar field. The thermal properties of the charged black holes are discussed and the reason why the nontrivial scalar field exists are explained. Also the solutions for higher dimensional Brans-Dicke theory are given for comparison. 
  In this work, we extend the analysis of Brown and York to find the quasilocal energy in a spherical box in the Schwarzschild spacetime. Quasilocal energy is the value of the Hamiltonian that generates unit magnitude proper-time translations on the box orthogonal to the spatial hypersurfaces foliating the Schwarzschild spacetime. We call this Hamiltonian the Brown-York Hamiltonian. We find different classes of foliations that correspond to time-evolution by the Brown-York Hamiltonian. We show that although the Brown-York expression for the quasilocal energy is correct, one needs to supplement their derivation with an extra set of boundary conditions on the interior end of the spatial hypersurfaces inside the hole in order to obtain it from an action principle. Replacing this set of boundary conditions with another set yields the Louko-Whiting Hamiltonian, which corresponds to time-evolution of spatial hypersurfaces in a different foliation of the Schwarzschild spacetime. We argue that in the thermodynamical picture, the Brown-York Hamiltonian corresponds to the internal energy whereas the Louko-Whiting Hamiltonian corresponds to the Helmholtz free energy of the system. Unlike what has been the usual route to black hole thermodynamics in the past, this observation immediately allows us to obtain the partition function of such a system without resorting to any kind of Euclideanization of either the Hamiltonian or the action. In the process, we obtain some interesting insights into the geometrical nature of black hole thermodynamics. 
  Einstein's field equations for stationary Bianchi type II models with a perfect fluid source are investigated. The field equations are rewritten as a system of autonomous first order differential equations. Dimensionless variables are subsequently introduced for which the reduced phase space is compact. The system is then studied qualitatively using the theory of dynamical systems. It is shown that the locally rotationally symmetric models are not asymptotically self-similar for small values of the independent , tovariable. A new exact solution is also given. 
  New features of the generalized symmetries of generic two-dimensional dilaton models of gravity are presented and invariant gravity-matter couplings are introduced. We show that there is a continuum set of Noether symmetries, which contains half a de Witt algebra. Two of these symmetries are area-preserving transformations. We show that gravity-matter couplings which are invariant under area preserving transformations only contribute to the dynamics of the dilaton-gravity sector with a reshaping of the dilaton potential. The interaction with matter by means of invariant metrics is also considered. We show in a constructive way that there are metrics which are invariant under two of the symmetries. The most general metrics and minimal couplings that fulfil this condition are found. 
  In this note I introduce the notion of the ``reliability horizon'' for semi-classical quantum gravity. This reliability horizon is an attempt to quantify the extent to which we should trust semi-classical quantum gravity, and to get a better handle on just where the Planck regime resides. I point out that the key obstruction to pushing semi-classical quantum gravity into the Planck regime is often the existence of large metric fluctuations, rather than a large back-reaction. There are many situations where the metric fluctuations become large long before the back-reaction is significant. Issues of this type are fundamental to any attempt at proving Hawking's chronology protection conjecture from first principles, since I shall prove that the onset of chronology violation is always hidden behind the reliability horizon. 
  The transmission time of an electromagnetic signal in the vicinity of the earth is calculated to c-2 and contains an orbital Sagnac term. On earth, the synchronisation of the Barycentric Coordinate Time (TCB) can be realised by atomic clocks, but not the one of Geocentric Coordinate Time (TCG). The principle of equivalence is discussed. 
  In this brief report I introduce a yet another class of geometries for which semi-classical chronology protection theorems are of dubious physical reliability. I consider a ``Roman ring'' of traversable wormholes, wherein a number of wormholes are arranged in a ring in such a manner that no subset of wormholes is near to chronology violation, though the combined system can be arbitrarily close to chronology violation. I show that (with enough wormholes in the ring) the gravitational vacuum polarization (the expectation value of the quantum stress-energy tensor) can be made arbitrarily small. In particular the back-reaction can be kept arbitrarily small all the way to the ``reliability horizon''---so that semi-classical quantum gravity becomes unreliable before the gravitational back reaction becomes large. 
  We study the dynamics of anisotropic Bianchi type-IX models with matter and cosmological constant. The models can be thought as describing the role of anisotropy in the early stages of inflation. The concurrence of the cosmological constant and anisotropy are sufficient to produce a chaotic dynamics in the gravitational degrees of freedom, connected to the presence of a critical point of saddle-center type in the phase space of the system. The invariant character of chaos is guaranteed by the topology of the cylinders emanating from unstable periodic orbits in the neighborhood of the saddle-center. We discuss a possible mechanism for amplification of specific wavelengths of inhomogeneous fluctuations in the models. A geometrical interpretation is given for Wald's inequality in terms of invariant tori and their destruction by increasing values of the cosmological constant. 
  In this work, we are investigating the problem of integrability of Bianchi class A cosmological models. This class of systems is reduced to the form of Hamiltonian systems with exponential potential forms.   The dynamics of Bianchi class A models is investigated through the Euler-Lagrange equations and geodesic equations in the Jacobi metric. On this basis, we have come to some general conclusions concerning the evolution of the volume function of 3-space of constant time. The formal and general form of this function has been found. It can serve as a controller during numerical calculations of the dynamics of cosmological models.   The integrability of cosmological models is also discussed from the points of view of different integrability criterions. We show that dimension of phase space of Bianchi class A Hamiltonian systems can be reduced by two. We prove vector field of the reduced system is polynomial and it does not admit any analytic, or even formal first integral. 
  A careful study of the induced transformations on spatial quantities due to 4-dimensional spacetime diffeomorphisms in the canonical formulation of general relativity is undertaken. Use of a general formalism, which indicates the role of the embedding variables in a transparent manner, allows us to analyse the effect of 4-dimensional diffeomorphisms more generally than is possible in the standard ADM approach. This analysis clearly indicates the assumptions which are necessary in order to obtain the ADM-Dirac constraints, and furthermore shows that there are choices, other than the ADM hamiltonian constraint, that one can make for the deformations in the ``timelike'' direction. In particular an abelian generator closely related to true time evolution appears very naturally in this framework. This generator, its relation to other abelian scalars discovered recently, and the possibilities it provides for a group theoretic quantisation of gravity are discussed. 
  The essence of the method of physics is inseparably connected with the problem of interplay between local and global properties of the universe. In the present paper we discuss this interplay as it is present in three major departments of contemporary physics: general relativity, quantum mechanics and some attempts at quantizing gravity (especially geometrodynamics and its recent successors in the form of various pregeometry conceptions). It turns out that all big interpretative issues involved in this problem point towards the necessity of changing from the standard space-time geometry to some radically new, most probably non-local, generalization. We argue that the recent noncommutative geometry offers attractive possibilities, and gives us a conceptual insight into its algebraic foundations. Noncommutative spaces are, in general, non-local, and their applications to physics, known at present, seem very promising. One would expect that beneath the Planck threshold there reigns a ``noncommutative pregeometry'', and only when crossing this threshold the usual space-time geometry emerges. 
  We present a numerical study of the time evolution of perturbations of rotating black holes. The solutions are obtained by integrating the Teukolsky equation written as a first-order in time, coupled system of equations, in a form that explicitly captures its hyperbolic structure. We address the numerical difficulties of solving the equation in its original form. We follow the propagation of generic initial data through the burst, quasinormal ringing and power-law tail phases. In particular, we calculate the effects due to the rotation of the black hole on the scattering of incident gravitational wave pulses. These results may help explain how the angular momentum of the black hole affects the gravitational waves that are generated during the final stages of black hole coalescence. 
  The ``warp drive'' metric recently presented by Alcubierre has the problem that an observer at the center of the warp bubble is causally separated from the outer edge of the bubble wall. Hence such an observer can neither create a warp bubble on demand nor control one once it has been created. In addition, such a bubble requires negative energy densities. One might hope that elimination of the first problem might ameliorate the second as well. We analyze and generalize a metric, originally proposed by Krasnikov for two spacetime dimensions, which does not suffer from the first difficulty. As a consequence, the Krasnikov metric has the interesting property that although the time for a one-way trip to a distant star cannot be shortened, the time for a round trip, as measured by clocks on Earth, can be made arbitrarily short. In our four dimensional extension of this metric, a ``tube'' is constructed along the path of an outbound spaceship, which connects the Earth and the star. Inside the tube spacetime is flat, but the light cones are opened out so as to allow superluminal travel in one direction. We show that, although a single Krasnikov tube does not involve closed timelike curves, a time machine can be constructed with a system of two non-overlapping tubes. Furthermore, it is demonstrated that Krasnikov tubes, like warp bubbles and traversable wormholes, also involve unphysically thin layers of negative energy density, as well as large total negative energies, and therefore probably cannot be realized in practice. 
  We investigate the computational requirements for all-sky, all-frequency searches for gravitational waves from spinning neutron stars, using archived data from interferometric gravitational wave detectors such as LIGO. These sources are expected to be weak, so the optimal strategy involves coherent accumulaton of signal-to-noise using Fourier transforms of long stretches of data (months to years). Earth-motion-induced Doppler shifts, and intrinsic pulsar spindown, will reduce the narrow-band signal-to-noise by spreading power across many frequency bins; therefore, it is necessary to correct for these effects before performing the Fourier transform. The corrections can be implemented by a parametrized model, in which one does a search over a discrete set of parameter values. We define a metric on this parameter space, which can be used to determine the optimal spacing between points in a search; the metric is used to compute the number of independent parameter-space points Np that must be searched, as a function of observation time T. The number Np(T) depends on the maximum gravitational wave frequency and the minimum spindown age tau=f/(df/dt) that the search can detect. The signal-to-noise ratio required, in order to have 99% confidence of a detection, also depends on Np(T). We find that for an all-sky, all-frequency search lasting T=10^7 s, this detection threshhold is at a level of 4 to 5 times h(3/yr), where h(3/yr) is the corresponding 99% confidence threshhold if one knows in advance the pulsar position and spin period. 
  Working in the Palatini formalism, we describe a procedure for constructing degenerate solutions of general relativity on 4-manifold M from certain solutions of 2-dimensional BF theory on any framed surface Sigma embedded in M. In these solutions the cotetrad field e (and thus the metric) vanishes outside a neighborhood of Sigma, while inside this neighborhood the connection A and the field E = e ^ e satisfy the equations of 4-dimensional BF theory. Moreover, there is a correspondence between these solutions and certain solutions of 2-dimensional BF theory on Sigma. Our construction works in any signature and with any value of the cosmological constant. If M = R x S for some 3-manifold S, at fixed time our solutions typically describe `flux tubes of area': the 3-metric vanishes outside a collection of thickened links embedded in S, while inside these thickened links it is nondegenerate only in the two transverse directions. We comment on the quantization of the theory of solutions of this form and its relation to the loop representation of quantum gravity. 
  Some superstring theories have more than one effective low-energy limit, corresponding to classical spacetimes with different dimensionalities. We argue that all but the 3+1-dimensional one might correspond to ``dead worlds'', devoid of observers, in which case all such ensemble theories would actually predict that we should find ourselves inhabiting a 3+1-dimensional spacetime. With more or less than one time-dimension, the partial differential equations of nature would lack the hyperbolicity property that enables observers to make predictions. In a space with more than three dimensions, there can be no traditional atoms and perhaps no stable structures. A space with less than three dimensions allows no gravitational force and may be too simple and barren to contain observers. 
  The quasi-two-dimensional modeling of the small adiabatic perturbation on the background of the stationary configuration of the selfgravitating gas with the weak transverse nonhomogeneity approximation is presented. The space periodic character of the solution of this system is proofed. 
  We claim that the existence of a mechanism such that photons may be trapped in a compact domain is not an exclusive property of gravitational forces. We show the case in which a non-linear electrodynamics allows such effect. In this latter case we should call this region an Electromagnetic Black Hole (EBH). 
  Traditional clock synchronisation on a rotating platform is shown to be incompatible with the experimentally established transformation of time. The latter transformation leads directly to solve this problem through noninvariant one-way speed of light. The conventionality of some features of relativity theory allows full compatibility with existing experimental evidence. 
  In this lecture we address the following issues in the context of string theories: i) The role played by S and T dualities in obtaining topological inflation in N=1 supergravity models, ii) A mechanism to generate the baryon asymmetry of the Universe based on the string interactions that violate CPT symmetry and iii) The quantum cosmology of the dimensionally reduced multidimensional Einstein-Yang-Mills system. 
  Several classes of conformally-flat and spherically symmetric exact solutions to the Einstein field equations coupled with either a massless scalar field or a radiation fluid are given, and their main properties are studied. It is found that some represent the formation of black holes due to the gravitational collapse of the matter fields. When the spacetimes have continuous self-similarity (CSS), the masses of black holes take a scaling form $M_{BH} \propto (P - P^{*})^{\gamma}$, where $\gamma = 0.5$ for massless scalar field and $\gamma = 1$ for radiation fluid. The reasons for the difference between the values of $\gamma$ obtained here and those obtained previously are discussed. When the spacetimes have neither CSS nor DSS (Discrete self-similarity), the masses of black holes always turn on with finite non-zero values. 
  We showed that the principle of nongravitating vacuum energy, when formulated in the first order formalism, solves the cosmological constant problem. The most appealing formulation of the theory displays a local symmetry associated with the arbitrariness of the measure of integration. This can be motivated by thinking of this theory as a direct coupling of physical degrees of freedom with a "space - filling brane" and in this case such local symmetry is related to space-filling brane gauge invariance. The model is formulated in the first order formalism using the metric and the connection as independent dynamical variables. An additional symmetry (Einstein - Kaufman symmetry) allows to eliminate the torsion which appears due to the introduction of the new measure of integration. The most successful model that implements these ideas is realized in a six or higher dimensional space-time. The compactification of extra dimensions into a sphere gives the possibility of generating scalar masses and potentials, gauge fields and fermionic masses. It turns out that remaining four dimensional space-time must have effective zero cosmological constant. 
  It is shown that there exists a range of parameters in which gravitational collapse with a spherically symmetric massive scalar field can be treated as if it were collapsing dust. This implies a criterion for the formation of black holes depending on the size and mass of the initial field configuration and the mass of the scalar field. 
  We generalize previous work on the energy-momentum tensor-distribution of the Kerr geometry by extending the manifold structure into the negative mass region. Since the extension of the flat part of the Kerr-Schild decomposition from one sheet to the double cover develops a singularity at the branch surface we have to take its non-smoothness into account. It is however possible to find a geometry within the generalized Kerr-Schild class that is in the Colombeau-sense associated to the maximally analytic Kerr-metric. 
  Building on a series of earlier papers [gr-qc/9604007, gr-qc/9604008, gr-qc/9604009], I investigate the various point-wise and averaged energy conditions in the Unruh vacuum. I consider the quantum stress-energy tensor corresponding to a conformally coupled massless scalar field, work in the test-field limit, restrict attention to the Schwarzschild geometry, and invoke a mixture of analytical and numerical techniques. I construct a semi-analytic model for the stress-energy tensor that globally reproduces all known numerical results to within 0.8%, and satisfies all known analytic features of the stress-energy tensor. I show that in the Unruh vacuum (1) all standard point-wise energy conditions are violated throughout the exterior region--all the way from spatial infinity down to the event horizon, and (2) the averaged null energy condition is violated on all outgoing radial null geodesics. In a pair of appendices I indicate general strategy for constructing semi-analytic models for the stress-energy tensor in the Hartle-Hawking and Boulware states, and show that the Page approximation is in a certain sense the minimal ansatz compatible with general properties of the stress-energy in the Hartle-Hawking state. 
  We prove that for non-linear L = L(R), the Lagrangians L and \hat L give conformally equivalent fourth-order field equations being dual to each other. The proof represents a new application of the fact that the operator <D'Alembertian minus R/6> is conformally invariant. 
  The null surface formalism of GR in three dimensions is presented, and the gauge freedom thereof, which is not just diffeomorphism, is discussed briefly. 
  We analyze the possibility of detecting, with optical methods, particle and photon trajectories predicted by general relativity for a weak spherically-symmetric gravitational field. We discuss the required sensitivities and the possibility of performing specific experiments on the Earth. 
  A new optical topology and signal readout strategy for a laser interferometer gravitational wave detector were proposed recently by Braginsky and Khalili . Their method is based on using a nonlinear medium inside a microwave oscillator to detect the gravitational-wave-induced spatial shift of the interferometer's standing optical wave. This paper proposes a quantum nondemolition (QND) scheme that could be realistically used for such a readout device and discusses a "fundamental" sensitivity limit imposed by a higher order optical effect. 
  Laboratory experiments on gravitation are usually performed with objects of constant density, so that the analysis of the forces concerns only the geometry of their shape. In an ideal experiment, the shapes of the constituent parts will be optimised to meet certain mathematical criteria, which ensure that the experiment has maximal sensitivity.   Using this idea, the author suggested an experiment to determine the departure of the gravitational force from Newton's force law [1]. The geometrical problem which has to be solved is to find two shapes which differ significantly, but have the same Newtonian potential. Essentially, the experiment determines whether the two objects are distinguishable by their gravitational force. Here, we consider the case when one of them is a round ball. The result, Theorem 1, establishes a fact which appeared in numerical simulations, that the second object has to have a hole in it. 
  We consider the gravitational collapse of a dust cloud in an asymptotically anti de Sitter spacetime in which points connected by a discrete subgroup of an isometry subgroup of anti de Sitter spacetime are identified. We find that black holes with event horizons of any topology can form from the collapse of such a cloud. The quasilocal mass parameter of such black holes is proportional to the initial density, which can be arbitrarily small. 
  We extract transition amplitudes among matter constituents of the universe from the solutions of the Wheeler De Witt equation. The physical interpretation of these solutions is then reached by an analysis of the properties of the transition amplitudes. The interpretation so obtained is based on the current carried by these solutions and confirms ideas put forward by Vilenkin. 
  Bianchi -I, -III, and FRW type models minimally coupled to a massive spatially homogeneous scalar field (i.e. a particle) are studied in the framework of semiclassical quantum gravity. In a first step we discuss the solutions of the corresponding equation for a Schr\"odinger particle propagating on a classical background. The back reaction of the Schr\"odinger particle on the classical metric is calculated by means of the Wigner function and by means of the expectation value of the energy-momentum-tensor of the field as a source. Both methods in general lead to different results. 
  Two geometrical well-posed hyperbolic formulations of general relativity are described. One admits any time-slicing which preserves a generalized harmonic condition. The other admits arbitrary time-slicings. Both systems have only the physical characteristic speeds of zero and the speed of light. 
  An idealized experiment estimating the spacetime topology is considered in both classical and quantum frameworks. The latter is described in terms of histories approach to quantum theory. A procedure creating combinatorial models of topology is suggested. The correspondence between these models and discretized spacetime models is established. 
  We consider a burst of quadrupole gravitational radiation in the presence of a large static mass $M$ situated at its source. Some of the radiation is back-scattered off the static field of the large mass, forming a wave tail. After the burst, the tail is a pure incoming wave, carrying energy back towards the source. We calculate this energy, and, in a numerical example, compare it with the outgoing wave energy. If $M$ is sufficiently large the incoming energy can equal the outgoing energy, indicating that the primary outgoing wave is completely suppressed. 
  The orbital separation of compact binary stars will shrink with time due to the emission of gravitational radiation. This inspiralling phase of a binary system's evolution generally will be very long compared to the system's orbital period, but the final coalescence may be dynamical and driven to a large degree by hydrodynamic effects, particularly if there is a critical separation at which the system becomes dynamically unstable toward merger. Indeed, if weakly relativistic systems (such as white dwarf-white dwarf binaries) encounter a point of dynamical instability at some critically close separation, coalescence may be entirely a classical, hydrodynamic process. Therefore, a proper investigation of this stage of binary evolution must include three-dimensional hydrodynamic simulations. We have constructed equilibrium sequences of synchronously rotating, equal-mass binaries in circular orbit with a single parameter - the binary separation - varying along each sequence. Sequences have been constructed with various polytropic as well as realistic white dwarf and neutron star equations of state. Using a Newtonian, finite-difference hydrodynamics code, we have examined the dynamical stability of individual models along these equilibrium sequences. Our simulations indicate that no points of instability exist on the sequences we analyzed that had relatively soft equations of state (polytropic sequences with polytropic index $n=1.0$ and 1.5 and two white dwarf sequences). However, we did identify dynamically unstable binary models on sequences with stiffer equations of state ($n=0.5$ polytropic sequence and two neutron star sequences). We thus infer that binary systems with soft equations of state are not driven to merger by a dynamical instability. 
  This is a brief contribution in which a simplified criterion of the relevance of the test-particle approximation describing motion of material near a magnetized black hole is discussed. Application to processes of the dissipative collimation of astronomical jets (as proposed by de Felice and Curir, 1992) is mentioned. 
  In the presence of a Killing symmetry, various self-gravitating field theories with massless scalars (moduli) and vector fields reduce to sigma-models, effectively coupled to 3-dimensional gravity. We argue that this particular structure of the Einstein-matter equations gives rise to quadratic relations between the asymptotic flux integrals and the area and surface gravity (Hawking temperature) of the horizon. The method is first illustrated for the Einstein-Maxwell system. A derivation of the quadratic formula is then also presented for the Einstein-Maxwell-axion-dilaton model, which is relevant to the bosonic sector of heterotic string theory. 
  The averaging problem in general relativity is briefly discussed. A new setting of the problem as that of macroscopic description of gravitation is proposed. A covariant space-time averaging procedure is described. The structure of the geometry of macroscopic space-time, which follows from averaging Cartan's structure equations, is described and the correlation tensors present in the theory are discussed. The macroscopic field equations (averaged Einstein's equations) derived in the framework of the approach are presented and their structure is analysed. The correspondence principle for macroscopic gravity is formulated and a definition of the stress-energy tensor for the macroscopic gravitational field is proposed. It is shown that the physical meaning of using Einstein's equations with a hydrodynamic stress-energy tensor in looking for cosmological models means neglecting all gravitational field correlations. The system of macroscopic gravity equations to be solved when the correlations are taken into consideration is given and described. 
  We speculate about the spacetime description due to the presence of Lorentzian wormholes (handles in spacetime joining two distant regions or other universes) in quantum gravity. The semiclassical rate of production of these Lorentzian wormholes in Reissner-Nordstr\"om spacetimes is calculated as a result of the spontaneous decay of vacuum due to a real tunneling configuration. In the magnetic case it only depends on the field theoretical fine structure constant. We predict that the quantum probability corresponding to the nucleation of such geodesically complete spacetimes should be actually negligible in our physical Universe. 
  We study the evolution of a Kerr black hole emitting scalar radiation via the Hawking process. We show that the rate at which mass and angular momentum are lost by the black hole leads to a final evolutionary state with nonzero angular momentum, namely $a/M \approx 0.555$. 
  This is a latex'd version of Robert Geroch's 1973 notes entitled 'Suggestions For Giving Talks'. 
  Static, spherically symmetric configurations of gravity with nonminimally coupled scalar fields are considered in D-dimensional space-times in the framework of generalized scalar-tensor theories. We seek special cases when the system has no naked singularity but instead forms either a black hole, or a wormhole. General conditions when this is possible are formulated. In particlar, some such special cases are indicated for multidimensional Brans-Dicke theory and for linear, massless, nonminimally coupled scalar fields. It is shown that in the Brans-Dicke theory the only black-hole solution corresponds to D=4 and the coupling constant \omega< -2, and there is a wormhole solution for \omega=0. For linear scalar fields it is shown that the only black-hole solution is the well-known one (D=4, a black hole with scalar charge), while the known 4-dimensional wormhole solution is generalized to systems with conformal coupling in arbitrary dimension. 
  A class of decoherence schemes is described for implementing the principles of generalized quantum theory in reparametrization-invariant `hyperbolic' models such as minisuperspace quantum cosmology. The connection with sum-over-histories constructions is exhibited and the physical equivalence or inequivalence of different such schemes is analyzed. The discussion focuses on comparing constructions based on the Klein-Gordon product with those based on the induced (a.k.a. Rieffel, Refined Algebraic, Group Averaging, or Spectral Analysis) inner product. It is shown that the Klein-Gordon and induced products can be simply related for the models of interest. This fact is then used to establish isomorphisms between certain decoherence schemes based on these products. 
  We obtain and study static, spherically symmetric solutions for the Einstein - generalized Maxwell field system in 2n dimensions, with possible inclusion of a massless scalar field. The generalization preserves the conformal invariance of the Maxwell field in higher dimensions. Almost all solutions exhibit naked singularities, but there are some classes of black-hole solutions. For these cases the Hawking temperature is found and its charge/mass and dimension dependence is discussed. It is shown that, unlike the previously known multidimensional black-hole solutions, in our case the Hawking temperature may infinitely grow in the extreme case (that of minimum mass for given charges). 
  It is shown that the temperature may be quantized in some spacetime due to the periodically topological structure of the Euclidean section. The quanta of the temperature is the Hawking-Unruh temperature. 
  In the appearance of absorption material, the quantum vacuum fluctuations of all kinds of fields may be smoothed out and the spacetime with time machine may be stable against vacuum fluctuations. The chronology protection conjecture might break down, and the anti-chronology protection conjecture might hold: There is no law of physics preventing the appearance of closed timelike curves. 
  We consider the domain of applicability of general relativity (GR), as a classical theory of gravity, by considering its applications to a variety of settings of physical interest as well as its relationship with real observations. We argue that, as it stands, GR is deficient whether it is treated as a microscopic or a macroscopic theory of gravity. We briefly discuss some recent attempts at removing this shortcoming through the construction of a macroscopic theory of gravity. We point out that such macroscopic extensions of GR are likely to be non-unique and involve non-Riemannian geometrical frameworks. 
  We argue that in classical and quantum theories of gravity the configuration space and Hilbert space may not be constructible through any finite procedure. If this is the case then the "problem of time" in quantum cosmology may be a pseudo-problem, because the argument that time disappears from the theory depends on constructions that cannot be realized by any finite beings that live in the universe. We propose an alternative formulation of quantum cosmological theories in which it is only necessary to predict the amplitudes for any given state to evolve to a finite number of possible successor states. The space of accessible states of the system is then constructed as the universe evolves from any initial state. In this kind of formulation of quantum cosmology time and causality are built in at the fundamental level. An example of such a theory is the recent path integral formulation of quantum gravity of Markopoulou and Smolin, but there are a wide class of theories of this type. 
  This paper treats boundary conditions on black hole horizons for the full 3+1D Einstein equations. Following a number of authors, the apparent horizon is employed as the inner boundary on a space slice. It is emphasized that a further condition is necessary for the system to be well posed; the ``prescribed curvature conditions" are therefore proposed to complete the coordinate conditions at the black hole. These conditions lead to a system of two 2D elliptic differential equations on the inner boundary surface, which coexist nicely to the 3D equation for maximal slicing (or related slicing conditions). The overall 2D/3D system is argued to be well posed and globally well behaved. The importance of ``boundary conditions without boundary values" is emphasized. This paper is the first of a series. This revised version makes minor additions and corrections to the previous version. 
  We study the effects of asymptotically anti-de Sitter wormholes in low-energy field theory and give a general prescription for obtaining the local effective interaction terms induced by them. The choice of vacuum for the matter fields selects a basis of the Hilbert space of anti-de Sitter wormholes whose elements can be interpreted as states containing a given number of particles. This interpretation is subject to the same kind of ambiguity in the definition of particle as that arising from quantum field theory in curved spacetime. 
  The general solution for non-rotating perfect-fluid spacetimes admitting one Killing vector and two conformal (non-isometric) Killing vectors spanning an abelian three-dimensional conformal algebra (C_3) acting on spacelike hypersurfaces is presented. It is of Petrov type D; some properties of the family such as matter contents are given. This family turns out to be an extension of a solution recently given in \cite{SeS} using completely different methods. The family contains Friedman-Lema\^{\i}tre-Robertson-Walker particular cases and could be useful as a test for the different FLRW perturbation schemes. There are two very interesting limiting cases, one with a non-abelian G_2 and another with an abelian G_2 acting non-orthogonally transitively on spacelike surfaces and with the fluid velocity non-orthogonal to the group orbits. No examples are known to the authors in these classes. 
  A symmetry based quantization method of reparametrization invariant systems is described; it will work for all systems that possess complete sets of perennials whose Lie algebras close and which generate a sufficiently large symmetry groups. The construction leads to a quantum theory including a Hilbert space, a complete system of operator observables and a unitary time evolution. The method is applied to the 2+1 gravity. The paper is restricted to the metric-torus sector, zero cosmological constant $\Lambda$ and it makes strong use of the so-called homogeneous gauge; the chosen algebra of perennials is that due to Martin. Two frequent problems are tackled. First, the Lie algebra of perennials does not generate a group of symmetries. The notion of group completion of a reparametrization invariant system is introduced so that the group does act; the group completion of the physical phase space of our model is shown to add only some limit points to it so that the ranges of observables are not unduly changed. Second, a relatively large number of relations between observables exists; they are transferred to the quantum theory by the well-known methods due to Kostant and Kirillov. In this way, a uniqueness of the physical representation of some extension of Martin's algebra is shown. The Hamiltonian is defined by a systematic procedure due to Dirac; for the torus sector, the result coincides with that by Moncrief. The construction may be extensible to higher genera and non-zero $\Lambda$ of the 2+1 gravity, because some complete sets of perennials are well-known and there are no obstructions to the closure of the algebra. 
  A characterisation of when wave tails are strong is proposed. The existence of a curvature induced tail (i.e. a Green's function term whose support includes the interior of the light-cone) is commonly understood to cause backscattering of the field governed by the relevant wave equation. Strong tails are characterised as those for which the purely radiative part of the field is backscattered. With this definition, it is shown that electromagnetic waves in asymptotically flat space-times and fields governed by tail-free propagation have weak tails, but minimally coupled scalar fields in a cosmological scenario have strong tails. 
  A stochastic theory of gravity is described in which the metric tensor is a random variable such that the spacetime manifold is a fluctuating physical system at a certain length scale. A general formalism is described for calculating probability densities for gravitational phenomena in a generalization of general relativity (GR), which reduces to classical GR when the magnitude of the metric fluctuations is negligible. Singularities in gravitational collapse and in big-bang cosmology have zero probability of occurring. A model of a self-organized critical universe is described which is independent of its initial conditions. 
  We point out that the inequality det E > 0 distinguishes the kinematical phase space of canonical connection gravity from that of a gauge field theory, and characterize the eigenvectors with positive, negative and zero-eigenvalue of the corresponding quantum operator in a lattice-discretized version of the theory. The diagonalization of the operator det E is simplified by classifying its eigenvectors according to the irreducible representations of the octagonal group. 
  We derive the metric for a Schwarzschild black hole with global monopole charge by relaxing asymptotic flatness of the Schwarzschild field. We then study the effect of global monopole charge on particle orbits and the Hawking radiation. It turns out that existence, boundedness and stability of circular orbits scale up by $(1-8 \pi\eta^2)^{-1}$, and the perihelion shift and the light bending by $(1-8 \pi\eta^2)^{-3/2}$, while the Hawking temperature scales down by $(1 - 8 \pi \eta^2)^2$ the Schwarzschild values. Here $\eta$ is the global charge. 
  We determine the one-loop and the two-loop back-reaction corrections in the spectrum of the Hawking radiation for the CGHS model of 2d dilaton gravity by evaluating the Bogoliubov coefficients for a massless scalar field propagating on the corresponding backgrounds. Since the back-reaction can induce a small shift in the position of the classical horizon, we find that a positive shift leads to a non-Planckian late-time spectrum, while a null or a negative shift leads to a Planckian late-time spectrum in the leading-order stationary-point approximation. In the one-loop case there are no corrections to the classical Hawking temperature, while in the two-loop case the temperature is three times greater than the classical value. We argue that these results are consistent with the behaviour of the Hawking flux obtained from the operator quantization only for the times which are not too late, in accordance with the limits of validity of the semiclassical approximation. 
  The asymptotic behaviour of vacuum Bianchi models of class A near the initial singularity is studied, in an effort to confirm the standard picture arising from heuristic and numerical approaches by mathematical proofs. It is shown that for solutions of types other than VIII and IX the singularity is velocity dominated and that the Kretschmann scalar is unbounded there, except in the explicitly known cases where the spacetime can be smoothly extended through a Cauchy horizon. For types VIII and IX it is shown that there are at most two possibilities for the evolution. When the first possibility is realized, and if the spacetime is not one of the explicitly known solutions which can be smoothly extended through a Cauchy horizon, then there are infinitely many oscillations near the singularity and the Kretschmann scalar is unbounded there. The second possibility remains mysterious and it is left open whether it ever occurs. It is also shown that any finite sequence of distinct points generated by iterating the Belinskii-Khalatnikov-Lifschitz mapping can be realized approximately by a solution of the vacuum Einstein equations of Bianchi type IX. 
  Quantum gauge theory in the connection representation uses functions of holonomies as configuration observables. Physical observables (gauge and diffeomorphism invariant) are represented in the Hilbert space of physical states; physical states are gauge and diffeomorphism invariant distributions on the space of functions of the holonomies of the edges of a certain family of graphs. Then a family of graphs embedded in the space manifold (satisfying certain properties) induces a representation of the algebra of physical observables. We construct a quantum model from the set of piecewise linear graphs on a piecewise linear manifold, and another manifestly combinatorial model from graphs defined on a sequence of increasingly refined simplicial complexes. Even though the two models are different at the kinematical level, they provide unitarily equivalent representations of the algebra of physical observables in separable Hilbert spaces of physical states (their s-knot basis is countable). Hence, the combinatorial framework is compatible with the usual interpretation of quantum field theory. 
  The canonical quantization of diffeomorphism invariant theories of connections in terms of loop variables is revisited. Such theories include general relativity described in terms of Ashtekar-Barbero variables and extension to Yang-Mills fields (with or without fermions) coupled to gravity.   It is argued that the operators induced by classical diffeomorphism invariant or covariant functions are respectably invariant or covariant under a suitable completion of the diffeomorphism group. The canonical quantization in terms of loop variables described here, yields a representation of the algebra of observables in a separable Hilbert space. Furthermore, the resulting quantum theory is equivalent to a model for diffeomorphism invariant gauge theories which replaces space with a manifestly combinatorial object. 
  A quantum-mechanical prescription of static Einstein field equation is proposed in order to construct the matter-metric eigen-states in the interior of a static Schwarzschild black hole where the signature of space-time is chosen as (--++). The spectrum of the quantum states is identified to be the integral multiples of the surface gravity. A statistical explanation of black hole entropy is given and a quantisation rule for the masses of Schwarzschild black holes is proposed. 
  Constant curvature black holes are constructed by identifying points in anti-de Sitter space. In n dimensions, the resulting topology is R^{n-1} * S_1, as opposed to the usual R^2 * S_{n-2} Schwarzschild black hole, and the corresponding causal structure is displayed by a (n-1)-dimensional picture, as opposed to the usual 2-dimensional Kruskal diagram. The five dimensional case, which can be embedded in a Chern-Simons supergravity theory, is analyzed in detail. 
  Gell-Mann and Hartle (GMH) have recently considered time-neutral cosmological models in which the initial and final conditions are independently specified, and several authors have investigated experimental tests of such models.   We point out here that GMH time-neutral models can allow superluminal signalling, in the sense that it can be possible for observers in those cosmologies, by detecting and exploiting regularities in the final state, to construct devices which send and receive signals between space-like separated points. In suitable cosmologies, any single superluminal message can be transmitted with probability arbitrarily close to one by the use of redundant signals. However, the outcome probabilities of quantum measurements generally depend on precisely which past {\it and future} measurements take place. As the transmission of any signal relies on quantum measurements, its transmission probability is similarly context-dependent. As a result, the standard superluminal signalling paradoxes do not apply. Despite their unusual features, the models are internally consistent.   These results illustrate an interesting conceptual point. The standard view of Minkowski causality is not an absolutely indispensable part of the mathematical formalism of relativistic quantum theory. It is contingent on the empirical observation that naturally occurring ensembles can be naturally pre-selected but not post-selected. 
  In the context of canonical quantum gravity in terms of Ashtekar's new variables, it is known that there exists a state that is annihilated by all the quantum constraints and that is given by the exponential of the Chern--Simons form constructed with the Asthekar connection. We make a first exploration of the transform of this state into the spin-network representation of quantum gravity. The discussion is limited to trivalent nets with planar intersections. We adapt an invariant of tangles to construct the transform and study the action of the Hamiltonian constraint on it. We show that the first two coefficients of the expansion of the invariant in terms of the inverse cosmological constant are annihilated by the Hamiltonian constraint. We also discuss issues of framing that arise in the construction. 
  We study Einstein gravity in a finite spatial region. By requiring a well-defined variational principle, we identify all local boundary conditions, derive surface observables, and compute their algebra. The observables arise as induced surface terms, which contribute to a non-vanishing Hamiltonian. Unlike the asymptotically flat case, we find that there are an infinite number of surface observables. We give a similar analysis for SU(2) BF theory. 
  This bibliography attempts to give a comprehensive overview of all the literature related to the Ashtekar connection and the Rovelli-Smolin loop variables. The original version was compiled by Peter H\"ubner in 1989, and it has been subsequently updated by Gabriela Gonzalez, Bernd Br\"ugmann, Monica Pierri, Troy Schilling, Alejandro Corichi and Christopher Beetle. Information about additional literature, new preprints, and especially corrections are always welcome. 
  The action for a relativistic free particle of mass m receives a contribution $-m R(x,y)$ from a path of length R(x,y) connecting the events $x^i$ and $y^i$. Using this action in a path integral, one can obtain the Feynman propagator for a spinless particle of mass m in any background spacetime. If one of the effects of quantizing gravity is to introduce a minimum length scale $L_P$ in the spacetime, then one would expect the segments of paths with lengths less than $L_P$ to be suppressed in the path integral. Assuming that the path integral amplitude is invariant under the `duality' transformation ${\cal R}\to L_P^2/R$, one can calculate the modified Feynman propagator in an arbitrary background spacetime. It turns out that the key feature of this modification is the following: The proper distance $(\Delta x)^2$ between two events, which are infinitesimally separated, is replaced by $\Delta x^2 + L_P^2$; that is the spacetime behaves as though it has a `zero-point length' of $L_P$. This equivalence suggests a deep relationship between introducing a `zero-point-length' to the spacetime and postulating invariance of path integral amplitudes under duality transformations. In the Schwinger's proper time description of the propagator, the weightage for a path with proper time s becomes $m(s+L_P^2/s)$ rather than as ms. As to be expected, the ultraviolet behavior of the theory is improved significantly and divergences will disappear if this modification is taken into account. Implications of this result are discussed. 
  Two-point functions of the scalar curvature for metric fluctuations on the four-sphere are analysed. The two-point function for points separated by a fixed distance and for metrics of fixed volume is calculated using spacetime foam methods. This result can be used for comparison between the continuum approach to quantum gravity and numerical quantum gravity on the lattice.   Pacs numbers: 04.60.Gw, 04.60.Nc, 04.62.+v 
  We analyze the interior geometry of static, spherically symmetric black holes of the Einstein-Yang-Mills-Higgs theory. Generically the solutions exhibit a behaviour that may be described as ``mass inflation'', although with a remarkable difference between the cases with and without a Higgs field. Without Higgs field the YM field induces a kind of cyclic behaviour leading to repeated cycles of mass inflation - taking the form of violent explosions - interrupted by quiescent periods and subsequent approaches to an almost Cauchy horizon. With the Higgs field no such cycles occur. In addition there are non-generic families with a Schwarzschild resp. Reissner-Nordstr{\o}m type singularity at r=0. 
  We study a solution of the field equations for dilatonic gravity and obtain its post-post-newtonian limit. It turns out that terms to this and higher orders in the expansion may become important in strong gravitational fields, even though the post-newtonian limit coincides with that of General Relativity. This suggests that strong gravitational fields can only be studied by exact solutions of the field equations. 
  This paper proposes a gravitodynamic theory because there are similarities between gravitational theory and electrodynamics. Based on Einstein's principle of equivalence, two coordinate conditions are proposed into the four-dimensional line element and transformations. As a consequence,the equation of motion for gravitational force or inertial force has a form similar to the equation of Lorentz force on a charge in electrodynamics. The inertial forces in auniformly rotating system are calculated, which show that the Coriolis force is produced by a magnetic-type gravitational field. We have also calculated the Sagnac effect due to the rotation. These experimental facts strongly support our proposed coordinate conditions. In addition, the gravitodynamic field equations are briefly discussed. Since only four gravitational potentials (3 + 1 split) enter the metric tensor, the gravitodynamic field equations in ``3+1 split" form would be analogous to Maxwell's equations. 
  The scattering amplitudes for the perturbed fields of the N=2 supergravity about the extreme Reissner-Nordstr\"om black hole is examined. Owing to the fact that the extreme hole is a BPS state of the theory and preserves an unbroken global supersymmetry(N=1), the scattering amplitudes of the component fields should be related to each other. In this paper, we derive the formula of the transformation of the scattering amplitudes. 
  Fisher's arrow of `time' in a cosmological phase space defined as in quantum optics (i.e., whose points are coherent states) is introduced as follows. Assuming that the phase space evolution of the universe starts from an initial squeezed cosmological state towards a final thermal one, a Fokker-Planck equation for the time-dependent, cosmological Q phase space probability distribution can be written down. Next, using some recent results in the literature, we derive an information arrow of time for the Fisher phase space cosmological entropy based on the Q function. We also mention the application of Fisher's arrow of time to stochastic inflation models 
  We study the time evolution of small classical perturbations in a gauge invariant way for a complex scalar field in the early zero curvature Friedmann-Lema\^{\i}tre universe. We, thus, generalize the analysis which has been done so far for a real scalar field. We give also a derivation of the Jeans wavenumber in the Newtonian regime starting from the general relativistic equations, avoiding the so-called Jeans swindle.   During the inflationary phase, whose length depends on the value of the bosonic charge, the behavior of the perturbations turns out to be the same as for a real scalar field. In the oscillatory phase the time evolution of the perturbations can be determined analytically as long as the bosonic charge of the corresponding background solution is sufficiently large. This is not possible for the real scalar field, since the corresponding bosonic charge vanishes. 
  We cast the four-dimensional field equations of the Nonsymmetric Gravitational Theory (NGT) into a form appropriate for numerical study. In doing so, we have restricted ourselves to spherically symmetric spacetimes, and we have kept only the Wyman sector of the theory. We investigate the well-posedness of the initial-value problem of NGT for a particular data set consisting of a pulse in the antisymmetric field on an asymptotically flat space background. We include some analytic results on the solvability of the initial-value problem which allow us to place limits on the regions of the parameter space where the initial-value problem is solvable. These results are confirmed by numerically solving the constraints. 
  We analyze a new class of static exact solutions of Einstein-Maxwell-Dilaton gravity with arbitrary scalar coupling constant $\alpha$, representing a gravitational body endowed with electromagnetic dipole moment. This class possesses mass, dipole and scalar charge parameters. A discussion of the geodesic motion shows that the scalar field interaction is so weak that it cannot be measured in gravitational fields like the sun, but it could perhaps be detected in gravitational fields like pulsars. The scalar force can be attractive or repulsive. This gives rise to the hypothesis that the magnetic field of some astrophysical objects could be fundamental. 
  We discuss an evaporation of (2+1)-dimensional black hole by using quantum gravity holding in the vicinity of the black hole horizon. It is shown that the black hole evaporates at a definite rate by emitting matters through the quantum tunneling effect. A relation of the present formalism to the black hole entropy is briefly commented. 
  We study an internal structure of (2+1)-dimensional black hole with the neutral scalar matter in the spherically symmetric geometry by using a quantum theory of gravity which holds in the both vicinities of the singularity and the apparent horizon. A special attention is paid to the quantum-mechanical behavior of the singularity in the black hole. We solve analytically the Wheeler-DeWitt equation of a minisuperspace model where the ingoing Vaidya metric is used as a simple model representing a dynamical black hole. The wave function obtained in this way leads to interesting physical phenomena such as the quantum instability of singularity and the Hawking radiation. It is also pointed out a similarity between the singularity in (2+1)-dimensional black hole and the inner Cauchy horizon in (3+1)-dimensional Reissner-Nordstrom charged black hole. 
  After a review of multi particle solutions in classical 2+1 dimensional gravity we will construct a one particle Hilbert space. As we will use a curved momentum space, the coordinates $x^\mu$ are represented as non commuting Hermitian operators on this Hilbertspace. Finally we will indicate how to construct a Schrodinger equation. 
  We find that the momentum conjugate to the relative distance between two gravitating particles in their center of mass frame is a hyperbolic angle. This fact strongly suggests that momentum space should be taken to be a hyperboloid. We investigate the effect of quantization on this curved momentum space. The coordinates are represented by non commuting, Hermitian operators on this hyperboloid. We also find that there is a smallest distance between the two particles of one half times the Planck length. 
  In the context of a gauge theory for the translation group, we have obtained, for a spinless particle, a gravitational analog of the Lorentz force. Then, we have shown that this force equation can be rewritten in terms of magnitudes related to either the teleparallel or the riemannian structures induced in spacetime by the presence of the gravitational field. In the first case, it gives a force equation, with torsion playing the role of force. In the second, it gives the usual geodesic equation of General Relativity. The main conclusion is that scalar matter is able to feel anyone of the above spacetime geometries, the teleparallel and the metric ones. Furthermore, both descriptions are found to be completely equivalent in the sense that they give the same physical trajectory for a spinless particle in a gravitational field. 
  We consider a simple, physical approach to the problem of marginally trapped surfaces in the Nonsymmetric Gravitational Theory (NGT). We apply this approach to a particular spherically symmetric, Wyman sector gravitational field, consisting of a pulse in the antisymmetric field variable. We demonstrate that marginally trapped surfaces do exist for this choice of initial data. 
  The gauge symmetries of pure Chern-Simons theories with p-form gauge fields are analyzed. It is shown that the number of independent gauge symmetries depends crucially on the parity of p. The case where p is odd appears to be a direct generalization of the p=1 case and presents the remarkable feature that the timelike diffeomorphisms can be expressed in terms of the spatial diffeomorphisms and the internal gauge symmetries. By constrast, the timelike diffeomorphisms may be an independent gauge symmetry when p is even. This happens when the number of fields and the spacetime dimension fulfills an algebraic condition which is explicitely written. 
  In this paper we study a new type of solution of the spherically symmetric, Einstein-Yang/Mills (EYM) equations with SU(2) gauge group. These solutions are well-behaved in the far-field, and have a Reissner-Nordstrom type essential singularity at the origin. These solutions display some novel features which are not present in particle-like, or black-hole solutions. Any spherically symmetric solution to the EYM equations, defined in the far-field, is either a particle-like solution, a black-hole solution, or one of these RNL solutions. 
  Several comments are given to previous proofs of the generalised second law of thermodynamics: black hole entropy plus ordinary matter entropy never decreases for a thermally closed system. Arguments in favour of its truism are given in the spirit of conventional thermodynamics. 
  Using the coherent-state representation we show that the classical Einstein equation for the FRW cosmological model with a general minimal scalar field can be derived from the semiclassical quantum Einstein equation. 
  We apply the de Broglie-Bohm interpretation to the Wheeler-DeWitt equation for the quantum FRW cosmological model with a minimal massless scalar field. We find that the quantum FRW cosmological model has quantum potential dominated solutions that avoid the initial and the final cosmological singularities. It is suggested that the quantum potential and the back-reaction of geometry and matter fields may change the property of the cosmological singularities of the Universe. 
  We consider the initial data problem for several black holes in vacuum with arbitrary momenta and spins on a three space with punctures. We compactify the internal asymptotically flat regions to obtain a computational domain without inner boundaries. When treated numerically, this leads to a significant simplification over the conventional approach which is based on throats and isometry conditions. In this new setting it is possible to obtain existence and uniqueness of solutions to the Hamiltonian constraint. 
  We study numerically the fully nonlinear gravitational collapse of a self-gravitating, minimally-coupled, massless scalar field in spherical symmetry. Our numerical code is based on double-null coordinates and on free evolution of the metric functions: The evolution equations are integrated numerically, whereas the constraint equations are only monitored. The numerical code is stable (unlike recent claims) and second-order accurate. We use this code to study the late-time asymptotic behavior at fixed $r$ (outside the black hole), along the event horizon, and along future null infinity. In all three asymptotic regions we find that, after the decay of the quasi-normal modes, the perturbations are dominated by inverse power-law tails. The corresponding power indices agree with the integer values predicted by linearized theory. We also study the case of a charged black hole nonlinearly perturbed by a (neutral) self-gravitating scalar field, and find the same type of behavior---i.e., quasi-normal modes followed by inverse power-law tails, with the same indices as in the uncharged case. 
  We calculate the angular resolution of the planned LISA detector, a space-based laser interferometer for measuring low-frequency gravitational waves from galactic and extragalactic sources. LISA is not a pointed instrument; it is an all-sky monitor with a quadrupolar beam pattern. LISA will measure simultaneously both polarization components of incoming gravitational waves, so the data will consist of two time series. All physical properties of the source, including its position, must be extracted from these time series. LISA's angular resolution is therefore not a fixed quantity, but rather depends on the type of signal and on how much other information must be extracted. Information about the source position will be encoded in the measured signal in three ways: 1) through the relative amplitudes and phases of the two polarization components, 2) through the periodic Doppler shift imposed on the signal by the detector's motion around the Sun, and 3) through the further modulation of the signal caused by the detector's time-varying orientation. We derive the basic formulae required to calculate the LISA's angular resolution $\Delta \Omega_S$ for a given source. We then evaluate $\Delta \Omega_S$ for two sources of particular interest: monchromatic sources and mergers of supermassive black holes. For these two types of sources, we calculate (in the high signal-to-noise approximation) the full variance-covariance matrix, which gives the accuracy to which all source parameters can be measured. Since our results on LISA's angular resolution depend mainly on gross features of the detector geometry, orbit, and noise curve, we expect these results to be fairly insensitive to modest changes in detector design that may occur between now and launch. We also expect that our calculations could be easily modified to apply to a modified design. 
  The exact general solution to the Einstein equations in a homogeneous Universe with a full causal viscous fluid source for the bulk viscosity index $m=1/2$ is found. We have investigated the asymptotic stability of Friedmann and de Sitter solutions, the former is stable for $m\ge 1/2$ and the latter for $m\le 1/2$. The comparison with results of the truncated theory is made. For $m=1/2$, it was found that families of solutions with extrema no longer remain in the full case, and they are replaced by asymptotically Minkowski evolutions. These solutions are monotonic. 
  We consider two kinds of higher dimensional models which upon dimensional reduction lead to Jordan-Brans-Dicke type effective actions in four dimensions with the scale factor of the extra dimensions playing the role of the JBD field. These models are characterized by the potential for the JBD field which arises from the process of dimensional reduction, and by the coupling of the inflaton sector with the JBD field in the Jordan frame. Taking into account the fact that these models allow the possibility of enough inflation and dynamical compactification of the extra dimensions, we examine in the context of these models the other conditions which need to be satisfied for a viable scenario of extended inflation. We find that the requirements of conforming to general relativity at the present epoch, and producing suitable bubble spectrums during inflation lead to constraints on the allowed values taken by the parameters of these models. A model with a ten dimensional JBD field is able to satisfy the condition for appropriate density perturbations viewed in the conformal Einstein frame, with a stringent restriction on the initial value taken by the scale factor of the extra dimensions. 
  Using an energy-momentum tensor for spinning particles due to Dixon and Bailey-Israel, we develop the post-Newtonian approximation for N spinning particles in a self-contained manner. The equations of motion are derived directly from this energy-momentum tensor. Following the formalism of Epstein- Wagoner, we also obtain the waveform and the luminosity of the gravitational wave generated by these particles. 
  By adopting Nester's higher dimensional special orthonormal frames (HSOF) the tetrad equations for vacuum gravity are put into first order symmetric hyperbolic (FOSH) form with constant coefficients, independent of any time slicing or coordinate specialization. 
  An analysis of the decoherence of quantum fluctuations shows that production of classical adiabatic density perturbations may not take place in models of power-law inflation with power 1< p < 3. Some consequences for models of extended inflation are pointed out. In general, the condition for decoherence places new constraint on inflationary models, which does not depend on often complicated subsequent evolution. 
  A general expression is given for the quartic Lovelock tensor in terms of the Riemann-Christoffel and Ricci curvature tensors and the Riemann curvature scalar for n-dimensional differentiable manifolds having a general linear connection. In addition, expressions are given (in the appendix) for the coefficient of the quartic Lovelock Lagrangian as well as for lower-order Lovelock tensors and Lovelock Lagrangian coefficients. 
  Starting from the recently obtained 2PN accurate forms of the energy and angular momentum fluxes from inspiralling compact binaries, we deduce the gravitational radiation reaction to 2PN order beyond the quadrupole approximation - 4.5PN terms in the equation of motion - using the refined balance method proposed by Iyer and Will. We explore critically the features of their construction and illustrate them by contrast to other possible variants. The equations of motion are valid for general binary orbits and for a class of coordinate gauges. The limiting cases of circular orbits and radial infall are also discussed. 
  An effective black-hole-like horizon occurs, for electromagnetic waves in matter, at a surface of singular electric and magnetic permeabilities. In a physical dispersive medium this horizon disappears for wave numbers with $k>k_c$. Nevertheless, it is shown that Hawking radiation is still emitted if free field modes with $k>k_c$ are in their ground state. 
  The exact global solution of the Einstein equations [Neugebauer & Meinel, Phys. Rev. Lett. 75 (1995) 3046] describing a rigidly rotating, self-gravitating disk is discussed. The underlying matter model is a perfect fluid in the limit of vanishing pressure. The solution represents the general-relativistic analogue of the classical Maclaurin disk. It was derived by applying solution techniques from soliton theory to the axisymmetric, stationary vacuum Einstein equations. In contrast to the Newtonian solution, there exists an upper limit for the total mass of the disk - if the angular momentum is fixed. At this limit, a transition to a rotating black hole, i.e., to the Kerr solution occurs. Another limiting procedure leads to an interesting cosmological solution. These results prove conjectures formulated by Bardeen and Wagoner more than twenty-five years ago. 
  A generalised canonical formulation of gravity is devised for foliations of spacetime with codimension $n\ge1$. The new formalism retains n-dimensional covariance and is especially suited to 2+2 decompositions of spacetime. It is also possible to use the generalised formalism to obtain boundary contributions to the 3+1 Hamiltonian. 
  A calculation of the one-mode occupation numbers for vacuum fluctuations of massive fields in De Sitter space shows that their use for the generation of classical density perturbations in inflationary cosmology very much depends on their masses and conformal couplings. A new mechanism for the decoherence of relatively massive fields has been identified. A similar analysis of the power law inflation shows that production of adiabatic density perturbations may not take place in models with power p < 3. 
  One of the necessary covariant conditions for gravitational radiation is the vanishing of the divergence of the magnetic Weyl tensor H_{ab}, while H_{ab} itself is nonzero. We complete a recent analysis by showing that in irrotational dust spacetimes, the condition div H=0 evolves consistently in the exact nonlinear theory. 
  The detection of gravitational waves from coalescing compact binaries would be a computationally intensive process if a single bank of template wave forms (i.e., a one step search) is used. In an earlier paper we had presented a detection strategy, called a two step search}, that utilizes a hierarchy of template banks. It was shown that in the simple case of a family of Newtonian signals, an on-line two step search was about 8 times faster than an on-line one step search (for initial LIGO). In this paper we extend the two step search to the more realistic case of zero spin 1.5 post-Newtonian wave forms. We also present formulas for detection and false alarm probabilities which take statistical correlations into account. We find that for the case of a 1.5 post-Newtonian family of templates and signals, an on-line two step search requires about 1/21 the computing power that would be required for the corresponding on-line one step search. This reduction is achieved when signals having strength S = 10.34 are required to be detected with a probability of 0.95, at an average of one false event per year, and the noise power spectral density used is that of advanced LIGO. For initial LIGO, the reduction achieved in computing power is about 1/27 for S = 9.98 and the same probabilities for detection and false alarm as above. 
  An update of the ODEtools Maple package, for the analytical solving of 1st and 2nd order ODEs using Lie group symmetry methods, is presented. The set of routines includes an ODE-solver and user-level commands realizing most of the relevant steps of the symmetry scheme. The package also includes commands for testing the returned results, and for classifying 1st and 2nd order ODEs. 
  We analyze three classical field theories based on the wave equation: scalar field, electrodynamics and linearized gravity. We derive certain generating formula on a hyperboloid and on a null surface for them. The linearized Einstein equations are analyzed around null infinity. It is shown how the dynamics can be reduced to gauge invariant quanitities in a quasi-local way. The quasi-local gauge-invariant ``density'' of the hamiltonian is derived on the hyperboloid and on the future null infinity. The result gives a new interpretation of the Bondi mass loss formula. We show also how to define angular momentum. Starting from affine approach for Einstein equations we obtain variational formulae for Bondi-Sachs type metrics related with energy and angular momentum generators. The original van der Burg asymptotic hierarchy is revisited and the relations between linearized and asymptotic nonlinear situations are established. We discuss also supertranslations, Newman-Penrose charges and Janis solutions. 
  In this paper we present a class of metrics to be considered as new possible sources for the Kerr metric. These new solutions are generated by applying the Newman-Janis algorithm (NJA) to any static spherically symmetric (SSS) ``seed'' metric. The continuity conditions for joining any two of these new metrics is presented. A specific analysis of the joining of interior solutions to the Kerr exterior is made. The boundary conditions used are those first developed by Dormois and Israel. We find that the NJA can be used to generate new physically allowable interior solutions. These new solutions can be matched smoothly to the Kerr metric. We present a general method for finding such solutions with oblate spheroidal boundary surfaces. Finally a trial solution is found and presented. 
  A recent proposal for quantizing gravity is investigated for self consistency. There are well-known difficulties in dealing with Einstein gravity when resorting to the perturbative techniques of quantum field theory. This however does not preclude the existence of a quantum form. This Letter is all about such a subtle but important difference. 
  A quantity which measures total intrinsic spin along the z axis is constructed for planar gravity (fields dependent on z and t only), in both the Ashtekar complex connection formalism and in geometrodynamics. The total spin is conserved but (surprisingly) is not a surface term. This constant of the motion coincides with one of four observables previously discovered by Husain and Smolin. Two more of those observables can be interpreted physically as raising and lowering operators for total spin. 
  The 3+1 Hamiltonian Einstein equations, reduced by imposing two commuting spacelike Killing vector fields, may be written as the equations of the $SL(2,R)$ principal chiral model with certain `source' terms. Using this formulation, we give a procedure for generating an infinite number of non-local constants of motion for this sector of the Einstein equations. The constants of motion arise as explicit functionals on the phase space of Einstein gravity, and are labelled by sl(2,R) indices. 
  We reexamine the large quantum gravity effects discovered by Ashtekar in the context of 2+1 dimensional gravity coupled to matter. We study an alternative one-parameter family of coherent states of the theory in which the large quantum gravity effects on the metric can be diminished, at the expense of losing coherence in the matter sector. Which set of states is the one that occurs in nature will determine if the large quantum gravity effects are actually observable as wild fluctuations of the metric or rapid loss of coherence of matter fields. 
  A short foreword has been added for the archive version of this article, which otherwise appears as originally published in 1990, except for the updating of references. The original abstract follows.   This is a critical review of the literature on many-worlds interpretations (MWI), with arguments drawn partly from earlier critiques by Bell and Stein. The essential postulates involved in various MWI are extracted, and their consistency with the evident physical world is examined. Arguments are presented against MWI proposed by Everett, Graham and DeWitt. The relevance of frequency operators to MWI is examined; it is argued that frequency operator theorems of Hartle and Farhi-Goldstone-Gutmann do not in themselves provide a probability interpretation for quantum mechanics, and thus neither support existing MWI nor would be useful in constructing new MWI. Comments are made on papers by Geroch and Deutsch that advocate MWI. It is concluded that no plausible set of axioms exists for an MWI that describes known physics. 
  We present an explicit computation of matrix elements of the hamiltonian constraint operator in non-perturbative quantum gravity. In particular, we consider the euclidean term of Thiemann's version of the constraint and compute its action on trivalent states, for all its natural orderings. The calculation is performed using graphical techniques from the recoupling theory of colored knots and links. We exhibit the matrix elements of the hamiltonian constraint operator in the spin network basis in compact algebraic form. 
  Recently a restriction ("quantum inequality-type relation") on the (renormalized) energy density measured by a static observer in a "globally static" (ultrastatic) spacetime has been formulated by Pfenning and Ford for the minimally coupled scalar field, in the extension of quantum inequality-type relation on flat spacetime of Ford and Roman. They found negative lower bounds for the line integrals of energy density multiplied by a sampling (weighting) function, and explicitly evaluate them for some specific spacetimes. In this paper, we study the lower bound on spacetimes whose spacelike hypersurfaces are compact and without boundary. In the short "sampling time" limit, the bound has asymptotic expansion. Although the expansion can not be represented by locally invariant quantities in general due to the nonlocal nature of the integral, we explicitly evaluate the dominant terms in the limit in terms of the invariant quantities. We also make an estimate for the bound in the long sampling time limit. 
  This paper examines the historical controversy over whether gravitationally bound systems, such as binary stars, experienced orbital damping due to the emission of gravitational radiation, focusing especially on the period of the 1950s, but also discussing the work of Einstein and Rosen in the 1930s on cylindrical gravitational waves and the later quadrupole formula controversy. 
  The purpose of this work is to present a model for 3D massive gravity with topological and higher-derivative terms. Causality and unitarity are discussed at tree-level. Power-counting renormalizability is also contemplated. 
  We present a 3-dimensional model for massive gravity with masses induced by topological (Chern-Simons) and Proca-like mass terms. Causality and unitarity are discussed at tree-level. Power-counting renormalizability is also contemplated. 
  Solutions are constructed to the quantum constraints for planar gravity (fields dependent on z and t only) in the Ashtekar complex connection formalism. A number of operators are constructed and applied to the solutions. These include the familiar ADM energy and area operators, as well as new operators sensitive to directionality (z+ct vs. z-ct dependence). The directionality operators are quantum analogs of the classical constraints proposed for unidirectional plane waves by Bondi, Pirani, and Robinson (BPR). It is argued that the quantum BPR constraints will predict unidirectionality reliably only for solutions which are semiclassical in a certain sense. The ADM energy and area operators are likely to have imaginary eigenvalues, unless one either shifts to a real connection, or allows the connection to occur other than in a holonomy. In classical theory, the area can evolve to zero. A quantum mechanical mechanism is proposed which would prevent this collapse. 
  We study equilibrium configurations of boson stars in the framework of general scalar-tensor theories of gravitation. We analyse several possible couplings, with acceptable weak field limit and, when known, nucleosynthesis bounds, in order to work in the cosmologically more realistic cases of this kind of theories. We found that for general scalar-tensor gravitation, the range of masses boson stars might have is comparable with the general relativistic case. We also analyse the possible formation of boson stars along different eras of cosmic evolution, allowing for the effective gravitational constant far out form the star to deviate from its current value. In these cases, we found that the boson stars masses are sensitive to this kind of variations, within a typical few percent. We also study cases in which the coupling is implicitly defined, through the dependence on the radial coordinate, allowing it to have significant variations in the radius of the structure. 
  In Rindler space, we determine in terms of special functions the expression of the static, massive scalar or vector field generated by a point source. We find also an explicit integral expression of the induced electrostatic potential resulting from the vacuum polarization due to an electric charge at rest in the Rindler coordinates. For a weak acceleration, we give then an approximate expression in the Fermi coordinates associated with the uniformly accelerated observer. 
  The Nernst formulation of the third law of ordinary thermodynamics (often referred to as the ``Nernst theorem'') asserts that the entropy, $S$, of a system must go to zero (or a ``universal constant'') as its temperature, $T$, goes to zero. This assertion is commonly considered to be a fundamental law of thermodynamics. As such, it seems to spoil the otherwise perfect analogy between the ordinary laws of thermodynamics and the laws of black hole mechanics, since rotating black holes in general relativity do not satisfy the analog of the ``Nernst theorem''. The main purpose of this paper is to attempt to lay to rest the ``Nernst theorem'' as a law of thermodynamics. We consider a boson (or fermion) ideal gas with its total angular momentum, $J$, as an additional state parameter, and we analyze the conditions on the single particle density of states, $g(\epsilon,j)$, needed for the Nernst formulation of the third law to hold. (Here, $\epsilon$ and $j$ denote the single particle energy and angular momentum.) Although it is shown that the Nernst formulation of the third law does indeed hold under a wide range of conditions, some simple classes of examples of densities of states which violate the ``Nernst theorem'' are given. In particular, at zero temperature, a boson (or fermion) gas confined to a circular string (whose energy is proportional to its length) not only violates the ``Nernst theorem'' also but reproduces some other thermodynamic properties of an extremal rotating black hole. 
  We discuss some physical consequences of what might be called ``the ultimate ensemble theory'', where not only worlds corresponding to say different sets of initial data or different physical constants are considered equally real, but also worlds ruled by altogether different equations. The only postulate in this theory is that all structures that exist mathematically exist also physically, by which we mean that in those complex enough to contain self-aware substructures (SASs), these SASs will subjectively perceive themselves as existing in a physically ``real'' world. We find that it is far from clear that this simple theory, which has no free parameters whatsoever, is observationally ruled out. The predictions of the theory take the form of probability distributions for the outcome of experiments, which makes it testable. In addition, it may be possible to rule it out by comparing its a priori predictions for the observable attributes of nature (the particle masses, the dimensionality of spacetime, etc) with what is observed. 
  We consider the classical stochastic fluctuations of spacetime geometry induced by quantum fluctuations of massless non-conformal matter fields in the Early Universe. To this end, we supplement the stress-energy tensor of these fields with a stochastic part, which is computed along the lines of the Feynman-Vernon and Schwinger-Keldysh techniques; the Einstein equation is therefore upgraded to a so called Einstein-Langevin equation. We consider in some detail the conformal fluctuations of flat spacetime and the fluctuations of the scale factor in a simple cosmological modelintroduced by Hartle, which consists of a spatially flat isotropic cosmology driven by radiation and dust. 
  The purpose of this note is to establish, in a categorical manner, the universality of the Geroch-Kronheimer-Penrose causal boundary when considering the types of causal structures that may profitably be put on any sort of boundary for a spacetime. Actually, this can only be done for the future causal boundary (or the past causal boundary) separately; furthermore, only the chronology relation, not the causality relation, is considered, and the GKP topology is eschewed. The final result is that there is a unique map, with the proper causal properties, from the future causal boundary of a spacetime onto any ``reasonable" boundary which supports some sort of chronological structure and which purports to consist of a future completion of the spacetime. Furthermore, the future causal boundary construction is categorically unique in this regard. 
  We consider multidimensional cosmologies in even-dimensional space-times (D=2n) containg perfect fluid and a multidimensional generalization of the Maxwell field, preserving its conformal invariance (the F field, an n-form). Among models with an isotropic physical 3-space some integrable cases are found: vacuum models (which are integrable in the general case) and some perfect fluid models with barotropic equations of state. All of them contain a component of the F field appearing as an additional scalar in 4 dimensions. A two-parameter family of spatially flat models and four one-parameter families, including non-spatially flat models, have been obtained (where the parameters are constants from the fluid equation of state). All these integrable models admit the inclusion of a massless scalar field or an additional fluid with ultrastiff equation of state. Basic properties of vacuum models in the physical conformal frame are outlined. 
  We illustrate the relationship between spin networks and their dual representation by labelled triangulations of space in 2+1 and 3+1 dimensions. We apply this to the recent proposal for causal evolution of spin networks. The result is labelled spatial triangulations evolving with transition amplitudes given by labelled spacetime simplices. The formalism is very similar to simplicial gravity, however, the triangulations represent combinatorics and not an approximation to the spatial manifold. The distinction between future and past nodes which can be ordered in causal sets also exists here. Spacelike and timelike slices can be defined and the foliation is allowed to vary. We clarify the choice of the two rules in the causal spin network evolution, and the assumption of trivalent spin networks for 2+1 spacetime dimensions and four-valent for 3+1. As a direct application, the problem of the exponential growth of the causal model is remedied. The result is a clear and more rigid graphical understanding of evolution of combinatorial spin networks, on which further work can be based. 
  We investigate the phenomenon of "faster than light" photons in a family of dilaton black hole spacetimes. For radially directed photons, we find that their light-cone condition is modified even though the spacetimes are spherically symmetric. They also satisfy the "horizon theorem" and the "polarization sum rule" of Shore. For orbital photons, the dilatonic effect on the modification of the light-cone condition can become more dominant than the electromagnetic and the gravitational ones as the orbit gets closer to the event horizon in the extremal or near-extremal cases. 
  We provide a fully general-relativistic treatment of cosmological perturbations in a universe permeated by a large-scale primordial magnetic field, using the Ellis-Bruni gauge-invariant formalism. The exact non-linear equations for general relativistic magnetohydrodynamic evolution are derived. A number of applications are made: the behaviour of small perturbations to Friedmann universes are studied; a comparison is made with earlier Newtonian treatments of cosmological perturbations and some effects of inflationary expansion are examined. 
  Linear topological spaces with partial ordering (linear kinematics) are studied. They are defined by a set of 8 axioms implying that topology, linear structure and ordering are compatible with each other. Most of the results are valid for both the finite-dimensional and infinite-dimensional case. In applications to physics, partial ordering is interpreted as the causal structure. Both Newtonian and the special relativity causal structures are studied, and some other possible types of causality are discussed.    Linear topological spaces with pseudometric which satisfies the time inequality instead of the triangle inequality are studied (3 axioms). Pseudometric (which is determined by a pseudonorm) is shown to define a topology on a linear space, it being a continuous mapping in this topology. Proved that for a space with pseudometric to be a linear kinematics it is necessary and sufficient that mapping of multiplication by -1 (i.e. time reversion) be continuous. Minkovskii space of the special relativity theory proved to be a pseudometric linear kinematics. 
  At the minisuperspace level of homogeneous models, the bare probability for a classical universe has a huge peak at small universes for the Hartle-Hawking `no-boundary' wavefunction, in contrast to the suppression at small universes for the `tunneling' wavefunction. If the probability distribution is cut off at the Planck density (say), this suggests that the former quantum state is inconsistent with our observations. For inhomogeneous models in which stochastic inflation can occur, it is known that the idea of including a volume factor in the observational probability distribution can lead to arbitrarily large universes' being likely. Here this idea is shown to be sufficient to save the Hartle-Hawking proposal even at the minisuperspace level (for suitable inflaton potentials), by giving it enough space to be consistent with observations. 
  Recent numerical simulations of coalescing binary neutron stars conducted by Wilson, Mathews and Marronetti (WMM) show a rising central energy density of the stars as the orbital separation shrinks, i.e. the stars are individually crushed as they near coalescence. They claim this ``star-crushing'' effect is partially due to a non-linear, first post-Newtonian order enhancement of the self-gravity of each star caused by the presence of the other star. We present a concrete calculation which shows, within general relativity, first post-Newtonian order interactions with the other star leave the central energy density unchanged as the orbital radius shrinks. The results presented here are in sharp disagreement with the WMM claim. However, alternative gravitational theories, such as Brans-Dicke theory, can exhibit a small crushing effect in the binary constituents as they near coalescence. We show that the absence of the star-crushing effect at first post-Newtonian order is related to adherence to the strong equivalence principle. 
  Recent numerical work by Wilson, Mathews, and Marronetti [J. R. Wilson, G. J. Mathews and P. Marronetti, Phys. Rev. D 54, 1317 (1996)] on the coalescence of massive binary neutron stars shows a striking instability as the stars come close together: Each star's central density increases by an amount proportional to 1/(orbital radius). This overwhelms any stabilizing effects of tidal coupling [which are proportional to 1/(orbital radius)^6] and causes the stars to collapse before they merge. Since the claimed increase of density scales with the stars' mass, it should also show up in a perturbation limit where a point particle of mass $\mu$ orbits a neutron star. We prove analytically that this does not happen; the neutron star's central density is unaffected by the companion's presence to linear order in $\mu/R$. We show, further, that the density increase observed by Wilson et. al. could arise as a consequence of not faithfully maintaining boundary conditions. 
  We derive an expression for effective gravitational mass for any closed spacelike 2-surface. This effective gravitational energy is defined directly through the geometrical quantity of the freely falling 2-surface and thus is well adapted to intuitive expectation that the gravitational mass should be determined by the motion of test body moving freely in gravitational field. We find that this effective gravitational mass has reasonable positive value for a small sphere in the non-vacuum space-times and can be negative for vacuum case. Further, this effective gravitational energy is compared with the quasi-local energy based on the $(2+2)$ formalism of the General Relativity. Although some gauge freedoms exist, analytic expressions of the quasi-local energy for vacuum cases are same as the effective gravitational mass. Especially, we see that the contribution from the cosmological constant is the same in general cases. 
  We study cosmic censorship in the Reissner-Nordstrom charged black hole by means of quantum gravity holding near the apparent horizons. We consider a gedanken experiment whether or not a black hole with the electric charge $Q$ less than the mass $M$ ($Q < M$) could increase its charge to go beyond the extremal limit $Q = M$ by absorbing the external charged matters. If the black hole charge could exceed the extremal value, a naked singularity would be liberated from the protection of the outer horizon and visible to distant observers, which means weak cosmic censorship to be violated in this process. It is shown that the black hole never exceeds the extremal black hole this way in quantum gravity as in classical general gravity. An increment of the trapped external charged matters by the black hole certainly causes the inner Cauchy horizon to approach the outer horizon, but its relative approaching speed gradually slows down and stops precisely at the extremal limit. It is quite remarkable that cosmic censorship remains true even in quantum gravity. This study is the first attempt of examining weak cosmic censorship beyond the classical analysis. 
  We analyse the effects of thermal conduction in a relativistic fluid, just after its departure from hydrostatic equilibrium, on a time scale of the order of thermal relaxation time. It is obtained that the resulting evolution will critically depend on a parameter defined in terms of thermodynamic variables, which is constrained by causality requirements. 
  We consider the perturbations of a relativistic star as an initial-value problem. Having discussed the formulation of the problem (the perturbation equations and the appropriate boundary conditions at the centre and the surface of the star) in detail we evolve the equations numerically from several different sets of initial data. In all the considered cases we find that the resulting gravitational waves carry the signature of several of the star's pulsation modes. Typically, the fluid $f$-mode, the first two $p$-modes and the slowest damped gravitational $w$-mode are present in the signal. This indicates that the pulsation modes may be an interesting source for detectable gravitational waves from colliding neutron stars or supernovae. We also survey the literature and find several indications of mode presence in numerical simulations of rotating core collapse and coalescing neutron stars. If such mode-signals can be detected by future gravitational-wave antennae one can hope to infer detailed information about neutron stars. Since a perturbation evolution should adequately describe the late time behaviour of a dynamically excited neutron star, the present work can also be used as a bench-mark test for future fully nonlinear simulations. 
  We perform fully relativistic calculations of binary neutron stars in quasi-equilibrium circular orbits. We integrate Einstein's equations together with the relativistic equation of hydrostatic equilibrium to solve the initial value problem for equal-mass binaries of arbitrary separation. We construct sequences of constant rest mass and identify the innermost stable circular orbit and its angular velocity. We find that the quasi-equilibrium maximum allowed mass of a neutron star in a close binary is slightly larger than in isolation. 
  General relativity can be formulated either as in its original geometrical version (Einstein, 1915) or as a field theory (Feynman, 1962). In the Feynman presentation of Einstein theory an hypothesis concerning the interaction of gravity to gravity, which was hidden in the original version, becomes explicit. This is nothing but the assumed extension of the validity of the equivalence principle not only for matter-gravity interaction, but also for gravity-gravity. Recently we have presented a field theory of gravity (from here on called the NDL theory) which does not contain such a hypothesis. We have shown that, for this theory, both the cosmological structure and the PPN approximation for the solar tests are satisfied.   The proposal of this paper is to go one step further and to show that NDL theory is able to solve the problem of radiation emission by a binary pulsar in the same degree of accuracy as it was done in the GR theory. In the post-Newtonian order of approximation we show that the quadrupole formula of this theory is equal to the corresponding one in general relativity. Thus, the unique actual observable distinction of these theories concerns the velocity of gravitational waves, which becomes then the true ultimate test for gravity theory. 
  We comment further on the behaviour of a heat conducting fluid when a characteristic parameter of the system approaches a critical value. 
  We investigate numerically spherically symmetric collapse of a scalar field in the semi-classical approximation. We first verify that our code reproduces the critical phenomena (the Choptuik effect) in the classical limit and black hole evaporation in the semi classical limit. We then investigate the effect of evaporation on the critical behavior. The introduction of the Planck length by the quantum theory suggests that the classical critical phenomena, which is based on a self similar structure, will disappear. Our results show that when quantum effects are not strong enough, critical behavior is observed. In the intermediate regime, evaporation is equivalent to a decrease of the initial amplitude. It doesn't change the echoing structure of near critical solutions. In the regime where black hole masses are low and the quantum effects are large, the semi classical approximation breaks down and has no physical meaning. 
  Global topological defects produce nonzero stress-energy throughout spacetime, and as a result can have observable gravitational influence on surrounding matter. Gravitational effects of global strings are used to place bounds on their cosmic abundance. The minimum separation between global strings is estimated by considering the defects' contribution to the cosmological energy density. More rigorous constraints on the abundance of global strings are constructed by examining the tidal forces such defects will have on observable astrophysical systems. The small number of observed tidally disrupted systems indicates there can be very few of these objects in the observable universe. 
  We investigate the possible gravitational redshift values for boson stars with a self-interaction, studying a wide range of possible masses. We find a limiting value of $z_{lim} \simeq 0.687$ for stable boson star configurations. We compare theoretical expectation with the observational capabilities in several different wavebands, concluding that direct observation of boson stars by this means will be extremely challenging. X-ray spectroscopy is perhaps the most interesting possibility. 
  We consider a radiation from a uniformly accelerating harmonic oscillator whose minimal coupling to the scalar field changes suddenly. The exact time evolutions of the quantum operators are given in terms of a classical solution of a forced harmonic oscillator. After the jumping of the coupling constant there occurs a fast absorption of energy into the oscillator, and then a slow emission follows. Here the absorbed energy is independent of the acceleration and proportional to the log of a high momentum cutoff of the field. The emitted energy depends on the acceleration and also proportional to the log of the cutoff. Especially, if the coupling is comparable to the natural frequency of the detector ($e^2/(4m) \sim \omega_0$) enormous energies are radiated away from the oscillator. 
  Black holes are studied in the frames of superstring theory using a non-trivial numerical integration method. A low energy string action containing graviton, dilaton, Gauss-Bonnet and Maxwell contributions is considered. Four-dimensional black hole solutions are studied inside and outside the event horizon. The internal part of the solutions is shown to have a non-trivial topology. 
  Entropy of the Kerr-Newman black hole is calculated via the brick wall method with maintaining careful attention to the contribution of superradiant scalar modes. It turns out that the nonsuperradinat and superradiant modes simultaneously contribute to the entropy with the same order in terms of the brick wall cutoff $\epsilon$. In particular, the contribution of the superradiant modes to the entropy is negative. To avoid divergency in this method when the angular velocity tends to zero, we propose to intr oduce a lower bound of angular velocity and to treat the case of the angular momentum per unit mass $a=0$ separately. Moreover, from the lower bound of the angular velocity, we obtain the $\theta$-dependence structure of the brick wall cutoff, which natu rally requires an angular cutoff $\delta$. Finally, if the cutoff values, $\epsilon$ and $\delta$, satisfy a proper relation between them, the resulting entropy satisfies the area law. 
  This paper is a continuation of the papers [gr-qc/9409010, gr-qc/9505034, gr-qc/9603022, gr-qc/9609035, gr-qc/9609046]. The introduction of a prior, i.e., predynamical global geometry of spacetime manifold is substantiated, and the geometry is specified. The manifold is an infinite four-cylinder, or tube in the five-dimensional Euclidean space, the orthogonal section of the cylinder being the unit three-sphere. Supplementary conditions for metric are introduced geometrically, coordinate-independently, as opposed to coordinate conditions. Parity and time-reversal transformations are extended to the manifold specified. PT is equivalent to a rotation through \pi about an axis orthogonal to the cylinder axis. CPT invariance is discussed.    Keywords: cosmic time, cosmic space, cylindrical manifold 
  In these lectures I propose to push Einstein's principle of coordinate independence to the extreme in order to restrict the possible form of fundamental equations of motion in physics. I start from nearly tautological system theoretic axioms. They provide a minimal amount of a priori structure which is thought to be characteristic of human thinking in general. It is shown how formal discretizations of Maxwell and Yang Mills theory in flat space and of general relativity in Ashtekar variables fit into this frame work. 
  We study a four-dimensional gauge theory of the Poincar\'e group with topological action which generalizes some well-known two-dimensional gravity models. We classify the spherically symmetric solutions and discuss the perturbative propagation of excitations around flat spacetime. 
  A formalism is presented to construct a non-perturbative Grand Unified Theory when gravitational Planck-scale phenomena are included. The fundamental object on the Planck scale is the three-torus T^3 from which the known properties of superstrings, such as the geometric action and duality, follow directly. The low energy theory is 11-dimensional and compactification to a Lorentzian four-manifold is an automatic feature of the unified model. In particular, the simply-connected K3 Calabi-Yau manifold follows naturally from the model and provides a direct link with M-theory. The high energy theory is formulated on a T^3 lattice with handles which exhibits the necessary symmetry groups for the standard model, and yields a consistent amplitude for the cosmic microwave background fluctuations. The equation of motion for the supersymmetric, unified theory is derived and leads to the Higgs field. This formulation predicts a remnant, scalar topological defect of mass < 9 m_Planck/46 from the Planck epoch, which is a candidate for dark matter. Finally, it is shown that if the universe is quantum-mechanical, its spatial dimension is equal to three and the laws of nature are Lorentz invariant when gravity can be neglected. 
  We study the effective energy-momentum tensor (EMT) for cosmological perturbations and formulate the gravitational back-reaction problem in a gauge invariant manner. We analyze the explicit expressions for the EMT in the cases of scalar metric fluctuations and of gravitational waves and derive the resulting equations of state. The formalism is applied to investigate the back-reaction effects in chaotic inflation. We find that for long wavelength scalar and tensor perturbations, the effective energy density is negative and thus counteracts any pre-existing cosmological constant. For scalar perturbations during an epoch of inflation, the equation of state is de Sitter-like. 
  This paper is a continuation of the papers [gr-qc/9409010, gr-qc/9505034, gr-qc/9603022, gr-qc/9609035, gr-qc/9609046, gr-qc/9704033]. The aim of the paper is to incorporate singularities---both local (black hole and naked singularity) and global (big bang and big crunch)---into the dynamics of indeterministic quantum gravity and cosmology. The question is whether a singularity is dynamically passable, i.e., whether a dynamical process which ends with a singularity may be extended beyond the latter. The answer is yes. A local singularity is trivially passable, while the passableness for a global singularity may invoke CPT transformation. The passableness of the singularities implies pulsating black holes and the oscillating universe. For the local singularity, the escape effect takes place: In a vicinity of the singularity, quantum matter leaves the gravitational potential well.    Keywords: tempered singularity, strong singularity, trivial passage, CPT passage, pulsating black hole, escape effect, oscillating universe 
  The paper entitled ``Against Many-Worlds Interpretations'' by A. Kent, which has recently been submitted to the e-Print archive (gr-qc/9703089) contained some misconceptions. The claims on Everett's many-worlds interpretation are quoted and answered. 
  This paper studies the perturbations of the continuously self-similar critical solution of the gravitational collapse of a massless scalar field (Roberts solution). The perturbation equations are derived and solved exactly. The perturbation spectrum is found to be not discrete, but occupying continuous region of the complex plane. The renormalization group calculation gives the value of the mass-scaling exponent equal to 1. 
  The family of Gowdy universes with the spatial topology of a three-torus is studied both classically and quantum mechanically. Starting with the Ashtekar formulation of Lorentzian general relativity, we introduce a gauge fixing procedure to remove almost all of the non-physical degrees of freedom. In this way, we arrive at a reduced model that is subject only to one homogeneous constraint. The phase space of this model is described by means of a canonical set of elementary variables. These are two real, homogeneous variables and the Fourier coefficients for four real fields that are periodic in the angular coordinate which does not correspond to a Killing field of the Gowdy spacetimes. We also obtain the explicit expressions for the line element and reduced Hamiltonian. We then proceed to quantize the system by representing the elementary variables as linear operators acting on a vector space of analytic functionals. The inner product on that space is selected by imposing Lorentzian reality conditions. We find the quantum states annihilated by the operator that represents the homogeneous constraint of the model and construct with them the Hilbert space of physical states. Finally, we derive the general form of the quantum observables of the model. 
  A summary is given of the current status and plans for gravitational-wave searches at all plausible wavelengths, from the size of the observable universe to a few kilometers. The anticipated scientific payoff from these searches is described, including expectations for detailed studies of black holes and neutron stars, high-accuracy tests of general relativity, and hopes for the discovery of exotic new kinds of objects. 
  In these lectures general relativity is outlined as the classical field theory of gravity, emphasizing physical phenomena rather than mathematical formalism. Dynamical solutions representing traveling waves as well as stationary fields like those of black holes are discussed. Their properties are investigated by studying the geodesic structure of the corresponding space-times, as representing the motion of point-like test particles. The interaction between gravitational, electro-magnetic and scalar fields is also considered. 
  The problem of search for nearly periodic gravitational wave sources in the data from laser interferometric detectors is discussed using a simple model of the signal. Accuracies of estimation of the parameters and computational requirements to do the search are assessed. 
  We derive a generalized equation for the evolution of tensor perturbations in a cosmological background, taking into account higher-curvature contributions and a tree-level coupling to the dilaton in the string frame. The equation is obtained by perturbing the gravi-dilaton string effective action, expanded up to first order in $\alpha'$. The $\alpha'$ corrections can modify the low-energy perturbation spectrum, but the modifications are shown to be small when the background curvature keeps constant in the string frame. 
  A new method is presented here for evaluating approximately the pulsation modes of relativistic stellar models. This approximation relies on the fact that gravitational radiation influences these modes only on timescales that are much longer than the basic hydrodynamic timescale of the system. This makes it possible to impose the boundary conditions on the gravitational potentials at the surface of the star rather than in the asymptotic wave zone of the gravitational field. This approximation is tested here by predicting the frequencies of the outgoing non-radial hydrodynamic modes of non-rotating stars. The real parts of the frequencies are determined with an accuracy that is better than our knowledge of the exact frequencies (about 0.01%) except in the most relativistic models where it decreases to about 0.1%. The imaginary parts of the frequencies are determined with an accuracy of approximately M/R, where M is the mass and R is the radius of the star in question. 
  The coupling of space-time torsion to the Dirac equation leads to effects on the energy levels of atoms which can be tested by Hughes-Drever type experiments. Reanalysis of these experiments carried out for testing the anisotropy of mass and anomalous spin couplings can lead to the till now tightest constraint on the axial torsion by $K \leq 1.5 . 10^{-15} m^{-1}$. 
  Standard formulation is unable to distinguish between the (+++-) and (---+) spacetime metric signatures. However, the Clifford algebras associated with each are inequivalent, R(4) in the first case (real 4 by 4 matrices), H(2) in the latter (quaternionic 2 by 2). Multivector reformulations of Dirac theory by various authors look quite inequivalent pending the algebra assumed. It is not clear if this is mere artifact, or if there is a right/wrong choice as to which one describes reality. However, recently it has been shown that one can map from one signature to the other using a "tilt transformation" [see P. Lounesto, "Clifford Algebras and Hestenes Spinors", Found. Phys. 23, 1203-1237 (1993)]. The broader question is that if the universe is signature blind, then perhaps a complete theory should be manifestly tilt covariant. A generalized multivector wave equation is proposed which is fully signature invariant in form, because it includes all the components of the algebra in the wavefunction (instead of restricting it to half) as well as all the possibilities for interaction terms. 
  We study the influence of the reheating and equality transitions on superhorizon density perturbations and gravitational waves. Recent criticisms of the `standard result' for large-scale perturbations in inflationary cosmology are rectified. The claim that the `conservation law' for the amplitude of superhorizon modes was empty is shown to be wrong. For sharp transitions, i.e. the pressure jumps, we rederive the Deruelle-Mukhanov junction conditions. For a smooth transition we correct a result obtained by Grishchuk recently. We show that the junction conditions are not crucial, because the pressure is continuous during the reheating transition. The problem occurred, because the perturbed metric was not evolved correctly through the smooth reheating transition. Finally, we derive the `standard result' within Grishchuks's smooth (reheating) transition. 
  The quantum fluctuations of horizons in Robertson-Walker universes and in the Schwarzschild spacetime are discussed. The source of the metric fluctuations is taken to be quantum linear perturbations of the gravitational field. Lightcone fluctuations arise when the retarded Green's function for a massless field is averaged over these metric fluctuations. This averaging replaces the delta-function on the classical lightcone with a Gaussian function, the width of which is a measure of the scale of the lightcone fluctuations. Horizon fluctuations are taken to be measured in the frame of a geodesic observer falling through the horizon. In the case of an expanding universe, this is a comoving observer either entering or leaving the horizon of another observer. In the black hole case, we take this observer to be one who falls freely from rest at infinity. We find that cosmological horizon fluctuations are typically characterized by the Planck length. However, black hole horizon fluctuations in this model are much smaller than Planck dimensions for black holes whose mass exceeds the Planck mass. Furthermore, we find black hole horizon fluctuations which are sufficiently small as not to invalidate the semiclassical derivation of the Hawking process. 
  The Dirac equation in Riemann-Cartan spacetimes with torsion is reconsidered. As is well-known, only the axial covector torsion $A$, a one-form, couples to massive Dirac fields. Using diagrammatic techniques, we show that besides the familiar Riemannian term only the Pontrjagin type four-form $dA\wedge dA$ does arise additionally in the chiral anomaly, but not the Nieh-Yan term $d ^* A$, as has been claimed recently. Implications for cosmic strings in Einstein-Cartan theory as well as for Ashtekar's canonical approach to quantum gravity are discussed. 
  A nonlocal form of the effective gravitational action could cure the unboundedness of euclidean gravity with Einstein action. On sub-horizon length scales the modified gravitational field equations seem compatible with all present tests of general relativity and post-Newtonian gravity. They induce a difference in the effective Newton's constant between regions of space with vanishing or nonvanishing curvature scalar (or Ricci tensor). In cosmology they may lead to a value $\Omega<1$ for the critical density after inflation. The simplest model considered here appears to be in conflict with nucleosynthesis, but generalizations consistent with all cosmological observations seem conceivable. 
  We present the Lagrangian and Hamiltonian framework which incorporates null dust as a source into canonical gravity. Null dust is a generalized Lagrangian system which is described by six Clebsch potentials of its four-velocity Pfaff form. The Dirac--ADM decomposition splits these into three canonical coordinates (the comoving coordinates of the dust) and their conjugate momenta (appropriate projections of four-velocity). Unlike ordinary dust of massive particles, null dust therefore has three rather than four degrees of freedom per space point. These are evolved by a Hamiltonian which is a linear combination of energy and momentum densities of the dust. The energy density is the norm of the momentum density with respect to the spatial metric. The coupling to geometry is achieved by adding these densities to the gravitational super-Hamiltonian and supermomentum. This leads to appropriate Hamiltonian and momentum constraints in the phase space of the system. The constraints can be rewritten in two alternative forms in which they generate a true Lie algebra. The Dirac constraint quantization of the system is formally accomplished by imposing the new constraints as quantum operator restrictions on state functionals. We compare the canonical schemes for null and ordinary dust and emhasize their differences. 
  A new type of a nonlinear gauge quantum theory (superrelativity) has been proposed. Such theory demands a radical reconstruction of both the quantum field conception and spacetime structure, and this paves presumably way to the comprehension of the quantum nature of inertia. 
  Given the present status of the problem of the electromagnetic energy tensor in matter, there is perhaps use in recalling a forgotten argument given in 1923 by W. Gordon. Let us consider a material medium which is homogeneous and isotropic when viewed in its rest frame. For such a medium, Gordon's argument allows to reduce the above mentioned problem to an analogous one, defined in a general relativistic vacuum, in presence of a suitably determined effective metric. For the latter problem the form of the Lagrangian is known already, hence the determination of the energy tensor is a straightforward matter. One just performs the Hamiltonian derivative of the Lagrangian chosen in this way with respect to the true metric. Abraham's tensor is thus selected as the electromagnetic energy tensor for a medium which is homogeneous and isotropic in its rest frame. 
  We argue that, in order to obtain decoherence of spacetime, we should consider quantum conformal metric fluctuations of spacetime. This could be the required environment in the problem of selfmeasurement of spacetime in quantum gravity. 
  We propose a model for the quantum theory of gravity. the model has diffeomorphism invariance, a natural length scale, and (plausibly) propagating modes. in the new addendum, we alter the model in a way which makes the propagating modes much easier to approach. 
  We study a coupled system of gravitational waves and a domain wall which is the boundary of a vacuum bubble in de Sitter spacetime. To treat the system, we use the metric junction formalism of Israel. We show that the dynamical degree of the bubble wall is lost and the bubble wall can oscillate only while the gravitational waves go across it. It means that the gravitational backreaction on the motion of the bubble wall can not be ignored. 
  We develop and apply a fully covariant 1+3 electromagnetic analogy for gravity. The free gravitational field is covariantly characterized by the Weyl gravito-electric and gravito-magnetic spatial tensor fields, whose dynamical equations are the Bianchi identities. Using a covariant generalization of spatial vector algebra and calculus to spatial tensor fields, we exhibit the covariant analogy between the tensor Bianchi equations and the vector Maxwell equations. We identify gravitational source terms, couplings and potentials with and without electromagnetic analogues. The nonlinear vacuum Bianchi equations are shown to be invariant under covariant spatial duality rotation of the gravito-electric and gravito-magnetic tensor fields. We construct the super-energy density and super-Poynting vector of the gravitational field as natural U(1) group invariants, and derive their super-energy conservation equation. A covariant approach to gravito-electric/magnetic monopoles is also presented. 
  We construct a new class of asymptotically flat black hole solutions in Einstein-Yang-Mills and Einstein-Yang-Mills-dilaton theory. These black hole solutions are static, and they have a regular event horizon. However, they possess only axial symmetry. Like their regular counterparts, the black hole solutions are characterized by two integers, the winding number $n$ and the node number $k$ of the gauge field functions. 
  The Wheeler-DeWitt equation is based on the use of canonical quantization rules that may be inconsistent for constrained dynamical systems, such as minisuperspaces subject to Einstein's equations. The resulting quantum dynamics has no classical limit and it suffers from the infamous ``problem of time.'' In this article, it is shown how a dynamical time (an internal clock) can be constructed by means of a Hamilton-Jacobi formalism, and then used for a consistent canonical quantization, with the correct classical limit. 
  An algebraic scheme is suggested in which discretized spacetime turns out to be a quantum observable. As an example, a toy model producing spacetimes of four points with different topologies is presented. The possibility of incorporating this scheme into the framework of non-commutative differential geometry is discussed. 
  We present the general structure of proper Ricci Collineations (RC) for type B warped space-times. Within this framework, we give a detailed description of the most general proper RC for spherically symmetric metrics. As examples, static spherically symmetric and Friedmann-Robertson-Walker space-times are considered. 
  We stress that halo dipole components are nontrivial in core-halo systems in both Newton's gravity and General Relativity. To this end, we extend a recent exact relativistic model to include also a halo dipole component. Next, we consider orbits evolving in the inner vacuum between a monopolar core and a pure halo dipole and find that, while the Newtonian dynamics is integrable, its relativistic counterpart is chaotic. This shows that chaoticity due only to halo dipoles is an intrinsic relativistic gravitational effect. 
  Rotation of a body, according to Einstein's theory of general relativity, generates a "force" on other matter; in Newton's gravitational theory only the mass of a body produces a force. This phenomenon, due to currents of mass, is known as gravitomagnetism owing to its formal analogies with magnetism due to currents of electric charge. Therefore, according to general relativity, Earth's rotation should influence the motion of its orbiting satellites. Indeed, we analysed the laser ranging observations of the orbits of the satellites LAGEOS and LAGEOS II, using a program developed at NASA/GSFC, and obtained the first direct measurement of the gravitomagnetic orbital perturbation due to the Earth's rotation, known as the Lense-Thirring effect. The accuracy of our measurement is about 25%. 
  An essential step towards the identification of a fermion mass generation mechanism at Planck scale is to analyse massive fermions in a given quantum gravity framework. In this letter the two mass terms entering the Hamiltonian constraint for the Einstein-Majorana system are studied in the loop representation of quantum gravity and fermions. One resembles a bare mass gap because it is not zero for states with zero (fermion) kinetic energy as opposite to the other that is interpreted as `dressing' the mass. The former contribution originates from (at least) triple intersections of the loop states acted on whilst the latter is traced back to every couple of coinciding end points, where fermions sit. Thus, fermion mass terms get encoded in the combinatorics of loop states. At last the possibility is discussed of relating fermion masses to the topology of space. 
  Using a PASCAL program to follow the evolution of two gravitating particles in 2+1 dimensions we find solutions in which the particles wind around one another indefinitely. As their center of mass moves `tachyonic' they form a Gott-pair. To avoid unphysical boundary conditions we consider a large but closed universe. After the particles have evolved for some time their momenta have grown very large. In this limit we quantize the model and find that both the relevant configuration variable and its conjugate momentum become discrete. 
  General relativity is a non-linear theory with the distinguishing feature that gravitational field energy also acts as gravitational charge density. In the well-known Schwarzschild solution describing field of an isolated massive body at rest, the scalar function $\phi$ characterising the field acts as a gravitational potential as well as it curves space part of spacetime. We demonstrate explicitly that it is the latter property that accounts for the non-linear (gravity as its own source) aspect which is not explicit in usual derivations. It is worth noting that the Einstein vacuum equations ultimately reduce to the Laplace equation and its first integral which fixes zero of $\phi$ at infinity. Thus the Schwarzschild field alongwith its asymptotic flat character is completely determined without application of any boundary condition by the field equations themselves. That means non-zero constant value of $\phi$ will have non-vacuous effect. It in fact produces stresses exactly of the form required to represent a global monopole. By retaining freedom of choosing zero of $\phi$, which will break asymptotic flatness, we can obtain the Schwarzschild black hole with global monopole charge. It is the non-linear aspect responsible for ``curving'' space, which has no Newtonian analogue, survives even when $\phi$ is constant but not zero. 
  The response of an interferometer changing its orientation with respect to a fixed reference frame is given in terms of the beam-pattern factors and the polarization-averaged antenna power pattern. 
  The third law of black hole dynamics states that the surface gravity (temperature) of black hole cannot be reduced to zero in finite sequence of physical interactions. We argue that the same is true when surface gravity is replaced by gravitational charge. We demonstrate that the prescribed window for infalling energy and radiation pinches off as extremality ($M^2 = a^2 + Q^2 $) is approached. 
  Black hole entropy is derived from a sum over boundary states. The boundary states are labeled by energy and momentum surface densities, and parametrized by the boundary metric. The sum over state labels is expressed as a functional integral with measure determined by the density of states. The sum over metrics is expressed as a functional integral with measure determined by the universal expression for the inverse temperature gradient at the horizon. The analysis applies to any stationary, nonextreme black hole in any theory of gravitational and matter fields. 
  I review the recent progress in providing a statistical foundation for black hole thermodynamics. In the context of string theory, one can now identify and count quantum states associated with black holes. One can also compute the analog of Hawking radiation (in a certain low energy regime) in a manifestly unitary way. Both extremal and nonextremal black holes are considered, including the Schwarzschild solution. Some implications of conjectured non-perturbative string ``duality transformations'' for the description of black hole states are also discussed. 
  Physical arguments related with the existence of black hole solutions having a non trapping interior are discussed. Massive scalar fields interacting with gravity are considered. Interior asymptotic solutions showing a scalar field approaching a constant value at the horizon are given. It is argued that the coupled Einstein-Klein-Gordon equations can be satisfied in the sense of the generalized functions after removing a particular regularization designed for matching the interior solution with an external Scwartzschild spacetime. The scalar field appears as just avoiding the appearance of closed trapped surfaces while coming from the exterior region. It also follows that the usual space integral over the temporal- temporal components of energy-momnetum tensor in the internal region just gives the total proper mass associated to the external Schwartzschild solution, as it should be expected. 
  The stochastic method based on the influence functional formalism introduced in an earlier paper to treat particle creation in near-uniformly accelerated detectors and collapsing masses is applied here to treat thermal and near-thermal radiance in certain types of cosmological expansions. It is indicated how the appearance of thermal radiance in different cosmological spacetimes and in the two apparently distinct classes of black hole and cosmological spacetimes can be understood under a unifying conceptual and methodological framework. 
  We briefly review the current status of the algebraic approach to quantum field theory on globally hyperbolic spacetimes, both axiomatic -- for general field theories, and constructive -- for a linear Klein-Gordon model. We recall the concept of F-locality, introduced in the latter context in BS Kay: Rev. Math. Phys., Special Issue, 167-195 (1992) and explain how it can be formulated at an axiomatic level for a general field theory (as a condition on algebras-with-net-structure) on both globally hyperbolic and non globally hyperbolic spacetimes. We also discuss the current status of the question whether/when algebras satisfying F-locality can exist for the Klein-Gordon model on spacetimes which are chronology violating. 
  We propose a geometric ansatz, a restriction on Euclidean / Minkowski distance in the embedding space being propotional to distance in the embedded space, to generate spacetimes with vanishing gravitational mass ($R_{ik} u^i u^k = 0, u_i u^i = 1 $). It turns out that these spacetimes can represent global monopoles and textures. Thus the ansatz is a prescription to generate zero mass spacetimes that could describe topological defects, global monopoles and textures. 
  The $\zeta$ function of a massive scalar field near a cosmic string is computed and then employed to find the vacuum fluctuation of the field. The vacuum expectation value of the energy-momentum tensor is also computed using a point-splitting approach. The obtained results could be useful also for the case of self-interacting scalar fields and for the finite-temperature Rindler space theory. 
  We consider the possibility of describing the Higgs effect in unified theories without the Higgs potential in the presence of the Einstein gravity with the conformal gravity-scalar coupling under the assumption of homogeneous matter distribution.   The scalar field values can be found from the Friedmann equations for the homogeneous Universe. The considered cosmological mechanism solves the vacuum density problem (we got $\rho_\phi^{Cosmic}=10^{-34}\rho_{cr}$ instead of $\rho_\phi^{Higgs}=10^{54}\rho_{cr}$),and exludes the monopole creation and the domain walls. 
  We demonstrate by explicit calculation of the DeWitt-like measure in two-dimensional quantum Regge gravity that it is highly non-local and that the average values of link lengths $l, <l^n>$, do not exist for sufficient high powers of $n$. Thus the concept of length has no natural definition in this formalism and a generic manifold degenerates into spikes. This might explain the failure of quantum Regge calculus to reproduce the continuum results of two-dimensional quantum gravity. It points to severe problems for the Regge approach in higher dimensions. 
  Recently E. E. Donets, D. V. Galtsov, and the author reported the results of numerical and analytical investigation of the SU(2) Einstein-Yang-Mills black hole interior solutions (gr-qc/9612067). It was shown that a generic interior solution develops a new type of an infinitely oscillating behavior with exponentially growing amplitude. Numerical data for three sequential oscillations were presented. The numerical integration technique was not discussed. Later P. Breitenlohner, G. Lavrelashvili, and D. Maison confirmed our main results (gr-qc/9703047). But they have made some misleading statements. In particular, they claimed, discussing the oscillations, that ``as one performs the numerical integration one quickly runs into serious problems...'' so that ``it is practically impossible to follow more than one or two of them numerically'' because ``the numerical integration procedure breaks down''. It is shown here that trivial logarithmic substitutions and integration along the integral curve solve these ``serious problems'' easily. 
  We examine one of the advantages of Ashtekar's formulation of general relativity: a tractability of degenerate points from the point of view of following the dynamics of classical spacetime. Assuming that all dynamical variables are finite, we conclude that an essential trick for such a continuous evolution is in complexifying variables. In order to restrict the complex region locally, we propose some `reality recovering' conditions on spacetime. Using a degenerate solution derived by pull-back technique, and integrating the dynamical equations numerically, we show that this idea works in an actual dynamical problem. We also discuss some features of these applications. 
  Traversable wormholes have traditionally been viewed as intrinsically topological entities in some multiply connected spacetime. Here, we show that topology is too limited a tool to accurately characterize a generic traversable wormhole: in general one needs geometric information to detect the presence of a wormhole, or more precisely to locate the wormhole throat. For an arbitrary static spacetime we shall define the wormhole throat in terms of a 2-dimensional constant-time hypersurface of minimal area. (Zero trace for the extrinsic curvature plus a "flare-out" condition.) This enables us to severely constrain the geometry of spacetime at the wormhole throat and to derive generalized theorems regarding violations of the energy conditions-theorems that do not involve geodesic averaging but nevertheless apply to situations much more general than the spherically symmetric Morris-Thorne traversable wormhole. [For example: the null energy condition (NEC), when suitably weighted and integrated over the wormhole throat, must be violated.] The major technical limitation of the current approach is that we work in a static spacetime-this is already a quite rich and complicated system. 
  Jordan-Brans-Dicke theories with a linearized potential for the scalar field are investigated in the framework of the stochastic approach. The fluctuations of this field are examined and their backreaction on the classical background is described. We compute the mode functions and analyze the time evolution of the variance of the stochastic ensemble corresponding to the full quantum scalar field in the pre-big-bang regime. We compute fluctuations of the term discriminating between the two branches of solutions present in the theory. We find, both analytically and upon direct integration of the stochastic equations of motion, that the dispersion of these fluctuations grows to achieve the magnitude of the term separating the two classical solutions. This means that the ensembles representing classical solutions which belong to different branches do overlap; this may provide a quantum mechanical realization at the level of field theory to change among solutions belonging to different branches. 
  We study the Hawking radiation in two dimensional dilaton black hole by means of quantum gravity holding near the apparent horizon. First of all, we construct the canonical formalism of the dilaton gravity in two dimensions. Then the Vaidya metric corresponding to the dilaton black hole is established where it is shown that the dilaton field takes a form of the linear dilaton. Based on the canonical formalism and the Vaidya metric, we proceed to analyze quantum properties of a dynamical black hole. It is found that the mass loss rate of the Hawking radiation is independent of the black hole mass and at the same time the apparent horizon recedes to the singularity as shown in other studies of two dimensional gravity. It is interesting that one can construct quantum gravity even near the curvature singularity and draw the same conclusion with respect to the Hawking radiation as the above-mentioned picture. Unfortunately, the present formalism seems to be ignorant of the contributions from the functional measures over the gravitational field, the dilaton and the ghosts. 
  We study the evaporation process of a 2D black hole in thermal equilibrium when the ingoing radiation is switched off suddenly. We also introduce global symmetries of generic 2D dilaton gravity models which generalize the extra symmetry of the CGHS model. 
  S. Chandrasekhar wrote in the prologue to his book on black holes, "The black holes of nature are the most perfect macroscopic objects there are in the universe: the only elements in their construction are our concepts of space and time." In this contribution I briefly discuss recent developments in fundamental theory and black holes that vindicate this statement in a modern perspective. I also include some of my reminiscences of Chandra. 
  Two components of spherically symmetric, inhomogeneous dust penetrating each other are introduced as a generalization of the well-known Tolman-Bondi dust solution. The field equations of this model are formulated and general properties are discussed. inhomogeneous Special solutions with additional symmetries - an extra Killing- or homothetic vector - and their matching to the corresponding Tolman-Bondi solution are investigated. 
  I propound a non-linear generalization of the Poisson equation describing a "medium" in D dimensions with a "dielectric constant" proportional to the field strength to the power D-2. It is the only conformally invariant scalar theory that is second order, and in which the scalar $phi$ couples to the sources $\rho$ via a $\phi\rho$ contact term. The symmetry is used to generate solutions for the field for some non-trivial configurations (e.g. for two oppositely charged points). Systems comprising N point charges afford further application of the symmetry. For these I derive e.g. exact expressions for the following quantities: the general two-point-charge force; the energy function and the forces in any three-body configuration with zero total charge; the few-body force for some special configurations; the virial theorem for an arbitrary, bound, many-particle system relating the time-average kinetic energy to the particle charges. Possible connections with an underlying conformal quantum field theory are mentioned. 
  The Einstein's equations with a negative cosmological constant admit solutions which are asymptotically anti-de Sitter space. Matter fields in anti-de Sitter space can be in stable equilibrium even if the potential energy is unbounded from below, violating the weak energy condition. Hence there is no fundamental reason that black hole's horizons should have spherical topology. In anti-de Sitter space the Einstein's equations admit black hole solutions where the horizon can be a Riemann surface with genus $g$. The case $g=0$ is the asymptotically anti-de Sitter black hole first studied by Hawking-Page, which has spherical topology. The genus one black hole has a new free parameter entering the metric, the conformal class to which the torus belongs. The genus $g>1$ black hole has no other free parameters apart from the mass and the charge. All such black holes exhibits a natural temperature which is identified as the period of the Euclidean continuation and there is a mass formula connecting the mass with the surface gravity and the horizon area of the black hole. The Euclidean action and entropy are computed and used to argue that the mass spectrum of states is positive definite. 
  We consider a gravitational model on a manifold M = M_0 x M_1 x...x M_n with oriented connected Einstein internal spaces M_1,...,M_n. The matter part of the action contains several scalar fields and antisymmetric forms. With Ricci-flat internal spaces, the model has a midisuperspace representation in form of a sigma-model on M_0. The latter can be used to determine exact composite electric p-brane solutions, which depend on a set of harmonic functions on M_0. 
  Although we have convincing evidence that a black hole bears an entropy proportional to its surface (horizon) area, the ``statistical mechanical'' explanation of this entropy remains unknown. Two basic questions in this connection are: what is the microscopic origin of the entropy, and why does the law of entropy increase continue to hold when the horizon entropy is included? After a review of some of the difficulties in answering these questions, I propose an explanation of the law of entropy increase which comes near to a proof in the context of the ``semi-classical'' approximation, and which also provides a proof in full quantum gravity under the assumption that the latter fulfills certain natural expectations, like the existence of a conserved energy definable at infinity. This explanation seems to require a fundamental spacetime discreteness in order for the entropy to be consistently finite, and I recall briefly some of the ideas for what the discreteness might be. If such ideas are right, then our knowledge of the horizon entropy will allow us to ``count the atoms of spacetime''. 
  I demonstrate that, under certain circumstances, regions of negative energy density can undergo gravitational collapse into a black hole. The resultant exterior black hole spacetimes necessarily have negative mass and non-trivial topology. A full theory of quantum gravity, in which topology-changing processes take place, could give rise to such spacetimes. 
  The Wick rotation in quantum field theory is considered in terms of analytical continuation in the signature matrix parameter w = eta_00. Regularization of propagators by a complex metric parameter in most cases preserves (i) the convergence of Feynmann integrals (understood as Lebesgue integrals) if the corresponding integrals of Euclidean theory are convergent; (ii) the regularity of propagators in the coordinate representation if there is regularity in the Euclidean case. The well-known covariant regularization by a complex mass does not in general satisfy these conditions. Theories with a large family of propagators regularized by complex metric were previously considered by the author, and analogues of the Bogoliubov-Parasiuk-Hepp-Zimmermann theorems were proved. [V.D.Ivashchuk, Izv. Akad. Nauk Mold. SSR, Ser. Fiz.-Tekhn. i Math. Nauk, 3 (1987), 8; 1 (1988), 10]. This paper shows that in the case of multidimensional cosmology describing the evolution of n spaces M_i, i = 1, ..., n, the Wick rotation in the minisuperspace may be performed by analytical continuation in the dimensions N_i = dim M_i or in the dimension of the time submanifold M_0. 
  A brief analysis of the dynamics of a Friedmann-Robertson-Walker universe with a conformally coupled, real, self-interacting, massive scalar field, based on the Painleve theory of differential equations, is presented. Our results complete earlier works done within the framework of Dynamical System Theory. We conclude that, in general, the system will not be integrable and that the chaos that has been found in a previous work, arises from the presence of movable logarithmic branch points in the solution in the complex plane of time. 
  For a rotating dust with a 3-dimensional symmetry group all possible metric forms can be classified and, within each class, explicitly written out. This is made possible by the formalism of Pleba\'nski based on the Darboux theorem. In the resulting coordinates, the Killing vector fields (if any exist) assume a special form. Each Killing vector field may be either spanned on the fields of velocity and rotation or linearly independent of them. By considering all such cases one arrives at the classification. With respect to the structures of the groups, this is just the Bianchi classification, but with all possible orientations of the orbits taken into account. In this paper, which is part 1 of a 3-part series, all solutions are considered for which two Killing fields are spanned on velocity and rotation. The solutions of Lanczos and G\"{o}del are identified as special cases, and their new invariant definitions are provided. In addition, a new invariant definition is given of the Ozsvath class III solution. 
  We propose a model for the geometry of a dynamical spherical shell in which the metric is asymptotically Schwarzschild, but deviates from Ricci-flatness in a finite neighbourhood of the shell. Hence, the geometry corresponds to a `hairy' black hole, with the hair originating on the shell. The metric is regular for an infalling shell, but it bifurcates, leading to two disconnected Schwarzschild-like spacetime geometries. The shell is interpreted as either collapsing matter or as Hawking radiation, depending on whether or not the shell is infalling or outgoing. In this model, the Hawking radiation results from tunnelling between the two geometries. Using this model, the back reaction correction from Hawking radiation is calculated. 
  We examine counterparts of the Reissner-Nordstrom-anti-de Sitter black hole spacetimes in which the two-sphere has been replaced by a surface Sigma of constant negative or zero curvature. When horizons exist, the spacetimes are black holes with an asymptotically locally anti-de Sitter infinity, but the infinity topology differs from that in the asymptotically Minkowski case, and the horizon topology is not S^2. Maximal analytic extensions of the solutions are given. The local Hawking temperature is found. When Sigma is closed, we derive the first law of thermodynamics using a Brown-York type quasilocal energy at a finite boundary, and we identify the entropy as one quarter of the horizon area, independent of the horizon topology. The heat capacities with constant charge and constant electrostatic potential are shown to be positive definite. With the boundary pushed to infinity, we consider thermodynamical ensembles that fix the renormalized temperature and either the charge or the electrostatic potential at infinity. Both ensembles turn out to be thermodynamically stable, and dominated by a unique classical solution. 
  The canonical quantization of the essentially nonlinear midisuperspace model describing cylindrically symmetric gravitational waves with two polarizations is presented. A Fock space type representation is constructed. It is based on a complete set of quantum observables. Physical expectation values may be calculated in arbitrary excitations of the vacuum. Our approach provides a non-linear generalization of the quantization of the collinearly polarized Einstein-Rosen gravitational waves. 
  Radiative multipole moments of scalar, electromagnetic, and linearized gravitational fields in Schwarzschild spacetime are computed to third order in v in a weak-field, slow-motion approximation, where v is a characteristic velocity associated with the motion of the source. To zeroth order in v, a radiative moment of order l is given by the corresponding source moment differentiated l times with respect to retarded time. At second order in v, additional terms appear inside the spatial integrals. These are near-zone corrections which depend on the detailed behavior of the source. At third order in v, the correction terms occur outside the spatial integrals, so that they do not depend on the detailed behavior of the source. These are wave-propagation corrections which are heuristically understood as arising from the scattering of the radiation by the spacetime curvature surrounding the source. Our calculations show that the wave-propagation corrections take a universal form which is independent of multipole order and field type. We also show that in general relativity, temporal and spatial curvatures contribute equally to the wave-propagation corrections. 
  We have shown that, as in the case of black holes, an ergosphere itself with no event horizon inside can evaporate spontaneously, giving energy radiation to spatial infinity until the ergoregion disappears. However, the feature of his quantum ergoregion instability is very much different from black hole radiation. It is rather analogous to a laser amplification. This analysis is based on the canonical quantization of a neutral scalar field in the presence of unstable modes characterized by complex frequencies in a simple model for a rapidly rotating star. 
  We obtain a class of rotating charged stationary circularly symmetric solutions of Einstein-Maxwell theory coupled to a topological mass term for the Maxwell field. These solutions are regular, have finite mass and angular momentum, and are asymptotic to the uncharged extreme BTZ black hole. 
  This paper deals with several technical issues of non-perturbative four-dimensional Lorentzian canonical quantum gravity in the continuum that arose in connection with the recently constructed Wheeler-DeWitt quantum constraint operator. 1) The Wheeler-DeWitt constraint mixes the previously discussed diffeomorphism superselection sectors which thus become spurious, 2) Thus, the inner product for diffeomorphism invariant states can be fixed by requiring that diffeomorphism group averaging is a partial isometry, 3) The established non-anomalous constraint algebra is clarified by computing commutators of duals of constraint operators, 4) The full classical constraint algebra is faithfully implemented on the diffeomorphism invariant Hilbert space in an appropriate sense, 5) The Hilbert space of diffeomorphism invariant states can be made separable if a natural new superselection principle is satisfied, 6) We propose a natural physical scalar product for quantum general relativity by extending the group average approach to the case of non-self-adjoint constraint operators like the Wheeler-DeWitt constraint and 7) Equipped with this inner product, the construction of physical observables is straightforward. 
  The quantization of Lorentzian or Euclidean 2+1 gravity by canonical methods is a well-studied problem. However, the constraints of 2+1 gravity are those of a topological field theory and therefore resemble very little those of the corresponding Lorentzian 3+1 constraints. In this paper we canonically quantize Euclidean 2+1 gravity for arbitrary genus of the spacelike hypersurface with new, classically equivalent constraints that maximally probe the Lorentzian 3+1 situation. We choose the signature to be Euclidean because this implies that the gauge group is, as in the 3+1 case, SU(2) rather than SU(1,1). We employ, and carry out to full completion, the new quantization method introduced in preceding papers of this series which resulted in a finite 3+1 Lorentzian quantum field theory for gravity. The space of solutions to all constraints turns out to be much larger than the one as obtained by traditional approaches, however, it is fully included. Thus, by suitable restriction of the solution space, we can recover all former results which gives confidence in the new quantization methods. The meaning of the remaining "spurious solutions" is discussed. 
  It is an old speculation in physics that, once the gravitational field is successfully quantized, it should serve as the natural regulator of infrared and ultraviolet singularities that plague quantum field theories in a background metric. We demonstrate that this idea is implemented in a precise sense within the framework of four-dimensional canonical Lorentzian quantum gravity in the continuum. Specifically, we show that the Hamiltonian of the standard model supports a representation in which finite linear combinations of Wilson loop functionals around closed loops, as well as along open lines with fermionic and Higgs field insertions at the end points are densely defined operators. This Hamiltonian, surprisingly, does not suffer from any singularities, it is completely finite without renormalization. This property is shared by string theory. In contrast to string theory, however, we are dealing with a particular phase of the standard model coupled to gravity which is entirely non-perturbatively defined and second quantized. 
  We quantize the generators of the little subgroup of the asymptotic Poincar\'e group of Lorentzian four-dimensional canonical quantum gravity in the continuum. In particular, the resulting ADM energy operator is densely defined on an appropriate Hilbert space, symmetric and essentially self-adjoint. Moreover, we prove a quantum analogue of the classical positivity of energy theorem due to Schoen and Yau. The proof uses a certain technical restriction on the space of states at spatial infinity which is suggested to us given the asymptotically flat structure available. The theorem demonstrates that several of the speculations regarding the stability of the theory, recently spelled out by Smolin, are false once a quantum version of the pre-assumptions underlying the classical positivity of energy theorem is imposed in the quantum theory as well. The quantum symmetry algebra corresponding to the generators of the little group faithfully represents the classical algebra. 
  We extend the recently developed kinematical framework for diffeomorphism invariant theories of connections for compact gauge groups to the case of a diffeomorphism invariant quantum field theory which includes besides connections also fermions and Higgs fields. This framework is appropriate for coupling matter to quantum gravity. The presence of diffeomorphism invariance forces us to choose a representation which is a rather non-Fock-like one : the elementary excitations of the connection are along open or closed strings while those of the fermions or Higgs fields are at the end points of the string. Nevertheless we are able to promote the classical reality conditions to quantum adjointness relations which in turn uniquely fixes the gauge and diffeomorphism invariant probability measure that underlies the Hilbert space. Most of the fermionic part of this work is independent of the recent preprint by Baez and Krasnov and earlier work by Rovelli and Morales-Tec\'otl because we use new canonical fermionic variables, so-called Grassman-valued half-densities, which enable us to to solve the difficult fermionic adjointness relations. 
  Quantum mechanics for matter fields moving in an evaporating black hole spacetime is formulated in fully four-dimensional form according to the principles of generalized quantum theory. The resulting quantum theory cannot be expressed in a 3+1 form in terms of a state evolving unitarily or by reduction through a foliating family of spacelike surfaces. That is because evaporating black hole geometries cannot be foliated by a non-singular family of spacelike surfaces. A four-dimensional notion of information is reviewed. Although complete information may not be available on every spacelike surface, information is not lost in a spacetime sense in an evaporating black hole spacetime. Rather complete information is distributed about the four-dimensional spacetime. Black hole evaporation is thus not in conflict with the principles of quantum mechanics when suitably generally stated. (Talk presented at Black Holes and Relativistic Stars: A Symposium in Honor of S. Chandrasekhar, Chicago, Dec 14-15,1996.) 
  We analyze the stability of relativistic, quasi-equilibrium binary neutron stars in synchronous circular orbit. We explore stability against radial collapse to black holes prior to merger, and against orbital plunge. We apply theorems based on turning points along uniformly rotating sequences of constant angular momentum and rest mass to locate the onset of secular instabilities. We find that inspiraling binary neutron stars are stable against radial collapse to black holes all the way down to the innermost stable circular orbit. 
  Considering the fact that the present universe might have been formed out of a system of ficticious self-gravitating particles, fermionic in nature, each of mass $m$, we are able to obtain a compact expression for the radius $R_0$ of the universe by using a model density distribution $\rho (r)$ for the particles which is singular at the origin. This singularity in $\rho (r)$ can be considered to be consistent with the socalled Big Bang theory of the universe. By assuming that Mach's principle holds good in the evolution of the universe, we determine the number of particles, $N$, of the universe and its $R_0$, which are obtained in terms of the mass $m$ of the constituent particles and the Universal Gravitational constant $G$ only. It is seen that for a mass of the constituent particles $m\simeq 1.07\times 10^{-35} g$ the age of the present universe,$\tau_0$, becomes $\tau_0 \simeq 20\times 10^9 yr$, or equivalently $R_0 \simeq 1.9\times 10^{28} cm $. For this $m$, the total number of particles costituting the present universe is found to be $N \simeq 2.4 \times 10^{91}$ and its total mass $(M \simeq 1.27916 \times 10^{23} M_{\odot})$, $M_{\odot}$ being the solar mass. All these numbers seem to be quantitatively agreeing with those evaluated from other theories. Using the present theory, we have also made an estimation of the variation of the universal gravitational constant $G$ with time which gives $({\dot G \over G}) =-9.6\times 10^{-11} yr^{-1}$. This is again in extremely good agreement with the results of some of the most recent calculations. Lastly, a plausible explanation for the Dark Matter present in today's universe is given. 
  %auto-ignore This paper has been withdrawn by the authors. 
  We consider a spacetime corresponding to uniform relativistic potential analogus to Newtonian potential as an example of ``minimally curved spacetime''. We also consider a radially symmetric analogue of the Rindler spacetime of uniform proper acceleration relative to infinity. 
  We analytically calculate the equilibrium sequence of the corotating binary stars of incompressible fluid in the first post-Newtonian(PN) approximation of general relativity. By calculating the total energy and total angular momentum of the system as a function of the orbital separation, we investigate the innermost stable circular orbit for corotating binary(we call it ISCCO). It is found that by the first PN effect, the orbital separation of the binary at the ISCCO becomes small with increase of the compactness of each star, and as a result, the orbital angular velocity at the ISCCO increases. These behaviors agree with previous numerical works. 
  Using the ellipsoidal model for the density configuration, we calculate the equilibrium sequence of the corotating binary stars of the polytropic equation of state in the first post-Newtonian approximation of general relativity. After we calibrate this model by comparing with previous numerical results, we perform the stability analysis by calculating the energy and the angular momentum of the system as a function of the orbital separation. We find that the orbital angular velocity at the energy and/or momentum minimum increases with the increase of the compactness of each star, and this fact holds irrespective of the polytropic index. These features agree with those in previous numerical works. We also show that due to the influence of the tidal field from the companion star, the central density of each star slightly decreases. 
  It is shown that there are three vacuum and one electrovacuum solutions of diagonal plane waves with M=0 and constant Maxwell scalars. Namely, these are the single wave, Stoyanov, Babala and Bell-Szekeres solutions. A comparison is made with the planar solutions of Taub. 
  We find the general solution of equations of motion for self-gravitating spherical null dust as a perturbative series in powers of the outgoing matter energy-momentum tensor, with the lowest order term being the Vaidya solution for the ingoing matter. This is done by representing the null-dust model as a 2d dilaton gravity theory, and by using a symmetry of a pure 2d dilaton gravity to fix the gauge. Quantization of this solution would provide an effective metric which includes the back-reaction for a more realistic black hole evaporation model than the evaporation models studied previously. 
  In the Ashtekar and geometrodynamic formulations of vacuum general relativity, the Euclidean and Lorentzian sectors can be related by means of the generalized Wick transform discovered by Thiemann. For some vacuum gravitational systems in which there exists an intrinsic time variable which is not invariant under constant rescalings of the metric, we show that, after such a choice of time gauge and with a certain identification of parameters, the generalized Wick transform can be understood as an analytic continuation in the explicit time dependence. This result is rigorously proved for the Gowdy model with the topology of a three-torus and for a whole class of cosmological models that describe expanding universes. In these gravitational systems, the analytic continuation that reproduces the generalized Wick transform after gauge fixing turns out to map the Euclidean line element to the Lorentzian one multiplied by an imaginary factor; this transformation rule differs from that expected for an inverse Wick rotation in a complex rescaling of the four-metric. We then prove that this transformation rule for the line element continues to be valid in the most general case of vacuum gravity with no model reduction nor gauge fixing. In this general case, it is further shown that the action of the generalized Wick transform on any function of the gravitational phase space variables, the shift vector, and the lapse function can in fact be interpreted as the result of an inverse Wick rotation and a constant, imaginary conformal transformation. 
  Colombeau's theory of generalised functions is used to calculate the contributions, at the rotation axis, to the distributional curvature for a time-dependent radiating cosmic string, and hence the mass per unit length of the string source. This mass per unit length is compared with the mass at null infinity, giving evidence for a global energy conservation law. 
  We treat the calculation of gravitational radiation using the mixed timelike-null initial value formulation of general relativity. The determination of an exterior radiative solution is based on boundary values on a timelike worldtube $\Gamma$ and on characteristic data on an outgoing null cone emanating from an initial cross-section of $\Gamma$. We present the details of a 3-dimensional computational algorithm which evolves this initial data on a numerical grid, which is compactified to include future null infinity as finite grid points. A code implementing this algorithm is calibrated in the quasispherical regime. We consider the application of this procedure to the extraction of waveforms at infinity from an interior Cauchy evolution, which provides the boundary data on $\Gamma$. This is a first step towards Cauchy-characteristic matching in which the data flow at the boundary $\Gamma$ is two-way, with the Cauchy and characteristic computations providing exact boundary values for each other. We describe strategies for implementing matching and show that for small target error it is much more computationally efficient than alternative methods. 
  We ascertain the effectiveness of the second post-Newtonian approximation to the gravitational waves emitted during the adiabatic inspiral of a compact binary system as templates for signal searches with kilometer-scale interferometric detectors. The reference signal is obtained by solving the Teukolsky equation for a small mass moving on a circular orbit around a large nonrotating black hole. Fitting factors computed from this signal and these templates, for various types of binary systems, are all above the 90% mark. According to Apostolatos' criterion, second post-Newtonian waveforms should make acceptably effective search templates. 
  The solution of the problem of describing the Friedmann observables (the Hubble law, the red shift, etc.) in quantum cosmology is proposed on the basis of the method of gaugeless Hamiltonian reduction in which the gravitational part of the energy constraint is considered as a new momentum. We show that the conjugate variable corresponding to the new momentum plays a role of the invariant time parameter of evolution of dynamical variables in the sector of the Dirac observables of the general Hamiltonian approach. Relations between these Dirac observables and the Friedmann observables of the expanding Universe are established for the standard Friedmann cosmological model with dust and radiation. The presented reduction removes an infinite factor from the functional integral, provides the normalizability of the wave function of the Universe and distinguishes the conformal frame of reference where the Hubble law is caused by the alteration of the conformal dust mass. 
  Relations between the Friedmann observables of the expanding Universe and the Dirac observables in the generalized Hamiltonian approach are established for the Friedmann cosmological model of the Universe with the field excitations imitating radiation. A full separation of the physical sector from the gauge one is fulfilled by the method of the gaugeless reduction in which the gravitational part of the energy constraint is considered as a new momentum. We show that this reduction removes an infinite factor from the Hartle -- Hawking functional integral, provides the normalizability of the Wheeler -- DeWitt wave function, clarifies its relation to the observational cosmology, and picks out a conformal frame of Narlikar. 
  Black hole horizon is usually defined as the limit for existence of timelike worldline or when a spatially bound surface turns oneway (it is crossable only in one direction). It would be insightful and physically appealing to find its characterization involving an energy consideration. By employing the Brown-York [1] quasilocal energy we propose a new and novel characterization of the horizon of static black hole. It is the surface at which the Brown-York energy equipartitions itself between the matter and potential energy. It is also equivalent to equipartitioning of the binding energy and the gravitational charge enclosed by the horizon. 
  So far all known singularity-free cosmological models are cylindrically symmetric. Here we present a new family of spherically symmetric non-singular models filled with imperfect fluid and radial heat flow, and satisfying the weak and strong energy conditions. For large $t$ anisotropy in pressure and heat flux tend to vanish leading to a perfect fluid. There is a free function of time in the model, which can be suitably chosen for non-singular behaviour and there exist multiplicity of such choices. 
  The irreducible decomposition technique is applied to the study of classical models of metric-affine gravity (MAG). The dynamics of the gravitational field is described by a 12-parameter Lagrangian encompassing a Hilbert-Einstein term, torsion and nonmetricity square terms, and one quadratic curvature piece that is built up from Weyl's segmental curvature. Matter is represented by a hyperfluid, a continuous medium the elements of which possess classical momentum and hypermomentum. With the help of irreducible decompositions, we are able to express torsion and traceless nonmetricity explicitly in terms of the spin and the shear current of the hyperfluid. Thereby the field equations reduce to an effective Einstein theory describing a metric coupled to the Weyl 1-form (a Proca-type vector field) and to a spin fluid. We demonstrate that a triplet of torsion and nonmetricity 1-forms describes the general and unique vacuum solution of the field equations of MAG. Finally, we study homogeneous cosmologies with an hyperfluid. We find that the hypermomentum affects significantly the cosmological evolution at very early stages. However, unlike spin, shear does not prevent the formation of a cosmological singularity. 
  We propose a path integral formulation of noncommutative generalizations of spacetime manifold in even dimensions, characterized by a length scale $\lambda_P$. The commutative case is obtained in the limit $\lambda_P=0$. 
  We consider vacuum polarization effect of a conformally coupled massless scalar field in the background produced by an idealized straight cosmic string. Using previous criterion we show the calculation of back reaction of the field to the metric in the context of semiclassical gravity theory is not valid , in some regions due to large quantum fluctuations in the conical space. 
  The outward-pointing principal null direction of the Schwarzschild Riemann tensor is null hypersurface-forming. If the Schwarzschild mass spontaneously jumps across one such hypersurface then the hypersurface is the history of an outgoing light-like shell. The outward-- pointing principal null direction of the Kerr Riemann tensor is asymptotically (in the neighbourhood of future null infinity) null hypersurface-forming. If the Kerr parameters of mass and angular momentum spontaneously jump across one such asymptotic hypersurface then the asymptotic hypersurface is shown to be the history of an outgoing light-like shell and a wire singularity-free spherical impulsive gravitational wave. 
  Quantum cosmology may restrict the class of gauge models which unify electroweak and strong interactions. In particular, if one studies the normalizability criterion for the one-loop wave function of the universe in a de Sitter background one finds that the interaction of inflaton and matter fields, jointly with the request of normalizability at one-loop order, picks out non-supersymmetric versions of unified gauge models. 
  The explicit relationship is determined between the interior properties of a static cylindrical matter distribution and the metric of the exterior space-time according to Einstein gravity for space-time dimensionality larger or equal to four. This is achieved through use of a coordinate system isotropic in the transverse coordinates. As a corollary, similar results are obtained for a spherical matter distribution in Brans-Dicke gravity for dimensions larger than or equal to three. The approach used here leads to consistency conditions for those parameters characterizing the exterior metric. It is shown that these conditions are equivalent to the requirement of hydrostatic equilibrium of the matter distribution (generalized Oppenheimer-Volkoff equations). These conditions lead to a consistent Newtonian limit where pressures and the gravitational constant go to zero at the same rate. 
  The Einstein's field equations of FRW universes filled with a dissipative fluid described by full theory of causal transport equations are analyzed. New exact solutions are found using a non-local transformations on the nonlinear differential equation for the Hubble factor. The stability of the de Sitter and asymptotically friedmannian solutions are analyzed using Lyapunov function method. 
  We consider anisotropic cosmological models with an universe of dimension 4 or more, factorized into n>1 Ricci-flat spaces, containing an m-component perfect fluid of m non-interacting homogeneous minimally coupled scalar fields under special conditions. We describe the dynamics of the universe: It has a Kasner-like behaviour near the singularity and isotropizes during the expansion to infinity.   Some of the considered models are integrable, and classical as well as quantum solutions are found. Some solutions produce inflation from "nothing". There exist classical asymptotically anti-de Sitter wormholes, and quantum wormholes with discrete spectrum. 
  We study some quantum mechanical aspects of dynamical black holes where the Vaidya metric is used as a model representing evaporating black holes. It is shown that in this model the Wheeler-DeWitt equation is solvable in whole region of spacetime, provided that one considers the ingoing (or outgoing) Vaidya metric and selects a suitable coordinate frame. This wave function has curious features in that near the curvature singularity it oscillates violently owing to large quantum effects while in the other regions the wave function exhibits a rather benign and completely regular behavior. The general formula concerning the black hole radiation, which reduces to the Hawking's semiclassical result when $r = 2M$ is chosen, is derived by means of purely quantum mechanical approach. The present formulation can be applied essentially to any system with a spherically symmetric black hole in an arbitrary spacetime dimension. 
  Despite its central role in modern cosmology, doubts are often expressed as to whether cosmological inflation is really a falsifiable theory. We distinguish two facets of inflation, one as a theory of initial conditions for the hot big bang and the other as a model for the origin of structure in the Universe. We argue that the latter can readily be excluded by observations, and that there are also a number of ways in which the former can find itself in conflict with observational data. Both aspects of the theory are indeed falsifiable. 
  The spontaneous loss of angular momentum of a spinning cosmic string due to particle emission is discussed. The rate of particle production between two assymptotic spacetimes: the spinning cosmic string spacetime in the infinite past and a non-spinning cosmic string spacetime in the infinite future is calculated. 
  In this essay a generalized notion of flavor-oscillation clocks is introduced. The generalization contains the element that various superimposed mass eigenstates may have different relative orientation of the component of their spin with respect to the rotational axis of the the gravitational source. It is found that these quantum mechanical clocks do not always redshift identically when moved from the gravitational environment of a non-rotating source to the field of a rotating source. The non-geometric contributions to the redshifts may be interpreted as quantum mechanically induced fluctuations over a geometric structure of space-time. 
  Can we give the graviton a mass? Does it even make sense to speak of a massive graviton? In this essay I shall answer these questions in the affirmative. I shall outline an alternative to Einstein Gravity that satisfies the Equivalence Principle and automatically passes all classical weak-field tests (GM/r approx 10^{-6}). It also passes medium-field tests (GM/r approx 1/5), but exhibits radically different strong-field behaviour (GM/r approx 1). Black holes in the usual sense do not exist in this theory, and large-scale cosmology is divorced from the distribution of matter. To do all this we have to sacrifice something: the theory exhibits {*prior geometry*}, and depends on a non-dynamical background metric. 
  Hamiltonian structures for spatially compact locally homogeneous vacuum universes are investigated, provided that the set of dynamical variables contains the \Teich parameters, parameterizing the purely global geometry. One of the key ingredients of our arguments is a suitable mathematical expression for quotient manifolds, where the universal cover metric carries all the degrees of freedom of geometrical variations, i.e., the covering group is fixed. We discuss general problems concerned with the use of this expression in the context of general relativity, and demonstrate the reduction of the Hamiltonians for some examples. For our models, all the dynamical degrees of freedom in Hamiltonian view are unambiguously interpretable as geometrical deformations, in contrast to the conventional open models. 
  Twenty years ago, by extending the Wightman axiom framework, it has been found possible to quantize only a conformal factor of the gravitational field. Gravitons being excluded from this quantum scalar field theory, numerous attempts were done to give a valuable description of what could be quantum gravity. In this talk we present a familly of Lorentz manifolds which can be foliated by isotropic hypersurfaces and pose severe restrictions on the form of the energy-momentum tensor in Einstein's equations. They can be associated to gravitational waves "without gravitons" in a vacuum described by two cosmological functions, but not to a massless particle flow. From this cross-checking with the previous remark, a "very" primordial quantum cosmological scenario is proposed. 
  Taking into account the effect of self-interaction, the dynamics of the quantum fluctuations of the inflaton field with $\lambda\phi^4$ potential is studied in detail. We find that the self interaction efficiently drives the initial pure state into a mixed one, which can be understood as a statistical ensemble. Further, the expectation value of the squared field operator is found to be converted into the variance of this statistical ensemble without giving any significant change in its amplitude. These results verify the ansatz of the quantum-to-classical transition that has been assumed in the standard evaluation of the amplitude of the primordial fluctuations of the universe. 
  A coordinate transformation is found which diagonalizes the axisymmetric pp-waves. Its effect upon concrete solutions, including impulsive and shock waves, is discussed. 
  We study nonaxisymmetric perturbations of rotating relativistic stars. modeled as perfect-fluid equilibria. Instability to a mode with angular dependence $\exp(im\phi)$ sets in when the frequency of the mode vanishes. The locations of these zero-frequency modes along sequences of rotating stars are computed in the framework of general relativity. We consider models of uniformly rotating stars with polytropic equations of state, finding that the relativistic models are unstable to nonaxisymmetric modes at significantly smaller values of rotation than in the Newtonian limit. Most strikingly, the m=2 bar mode can become unstable even for soft polytropes of index $N \leq 1.3$, while in Newtonian theory it becomes unstable only for stiff polytropes of index $N \leq 0.808$. If rapidly rotating neutron stars are formed by the accretion-induced collapse of white dwarfs, instability associated with these nonaxisymmetric, gravitational-wave driven modes may set an upper limit on neutron-star rotation. Consideration is restricted to perturbations that correspond to polar perturbations of a spherical star. A study of axial perturbations is in progress. 
  By now, many examples of naked singularities in classical general relativity are known. It may however be that a physical principle over and above the general theory prevents the occurrence of such singularities in nature. Assuming the validity of the Weyl curvature hypothesis, we propose that naked singularities are forbidden by the second law of thermodynamics. 
  Building on the universal covering group of the general linear group, we introduce the composite spinor bundle whose subbundles are Lorentz spin structures associated with different gravitational fields. General covariant transformations of this composite spinor bundle are canonically defined. 
  Barbero has generalized the Ashtekar canonical transformation to a one-parameter scale transformation $U(\iota)$ on the phase space of general relativity. Immirzi has noticed that in loop quantum gravity this transformation alters the spectra of geometrical quantities. We show that $U(\iota)$ is a canonical transformation that cannot be implement unitarily in the quantum theory. This implies that there exists a one-parameter quantization ambiguity in quantum gravity, namely a free parameter that enters the construction of the quantum theory. The purpose of this letter is to elucidate the origin and the role of this free parameter. 
  We examine the growth of the Weyl curvature in two examples of naked singularity formation in spherical gravitational collapse - dust and the Vaidya spacetime. We find that the Weyl scalar diverges along outgoing radial null geodesics as they meet the naked singularity in the past. The implications of this result for the Weyl curvature hypothesis are discussed. We mention the possibility that although classical general relativity admits naked singularity solutions arising from gravitational collapse, the second law of thermodynamics could forbid their occurrence in nature. The method can also be used to compare the relative importance of initial data and that of the energy-momentum tensor in deciding the metric solution in any general case. 
  An exact solution is found describing the collision of axisymmetric pp-waves with M=0. They are impulsive in character and their coordinate singularities become point curvature singularities at the boundaries of the interaction region. The solution is conformally flat. Concrete examples are given, involving an ultrarelativistic black hole against a burst of pure radiation or two colliding beam- like waves. 
  On the basis of the local n=2 supersymmetry we construct the supersymmetric action for a set of complex scalar supermultiplets in the FRW model. This action corresponds to the dilaton-axion and chiral components of supergravity theory. 
  We present the Heisenberg-picture approach to the quantum evolution of the scalar fields in an expanding FRW universe which incorporates relatively simply the initial quantum conditions such as the vacuum state, the thermal equilibrium state, and the coherent state. We calculate the Wightman function, two-point function, and correlation function of a massive scalar field. We find the quantum evolution of fluctuations of a self-interacting field perturbatively and discuss the renormalization of field equations. 
  A global network of laser interferometric gravitational wave detectors is projected to be in operation by around the turn of the century. Here, the noisy output of a single instrument is examined. A gravitational wave is assumed to have been detected in the data and we deal with the subsequent problem of parameter estimation. Specifically, we investigate theoretical lower bounds on the minimum mean-square errors associated with measuring the parameters of the inspiral waveform generated by an orbiting system of neutron stars/black holes. Three theoretical lower bounds on parameter estimation accuracy are considered: the Cramer-Rao bound (CRB); the Weiss-Weinstein bound (WWB); and the Ziv-Zakai bound (ZZB). We obtain the WWB and ZZB for the Newtonian-form of the coalescing binary waveform, and compare them with published CRB and numerical Monte-Carlo results. At large SNR, we find that the theoretical bounds are all identical and are attained by the Monte-Carlo results. As SNR gradually drops below 10, the WWB and ZZB are both found to provide increasingly tighter lower bounds than the CRB. However, at these levels of moderate SNR, there is a significant departure between all the bounds and the numerical Monte-Carlo results. 
  Recently, string theory has provided some remarkable new insights into the microphysics of black holes. I argue that a simple and important lesson is also provided with regards to the information loss paradox, namely, pure quantum states do not form black holes! Thus it seems black hole formation, as well as evaporation, must be understood within the framework of quantum decoherence. 
  In this paper we investigate the canonical structure of diffeomorphism invariant phase spaces for spatially locally homogeneous spacetimes with 3-dimensional compact closed spaces. After giving a general algorithm to express the diffeomorphism-invariant phase space and the canonical structure of a locally homogeneous system in terms of those of a homogeneous system on a covering space and a moduli space, we completely determine the canonical structures and the Hamiltonians of locally homogeneous pure gravity systems on orientable compact closed 3-spaces of the Thurston-type $E^3$, $\Nil$ and $\Sol$ for all possible space topologies and invariance groups. We point out that in many cases the canonical structure becomes degenerate in the moduli sectors, which implies that the locally homogeneous systems are not canonically closed in general in the full diffeomorphism-invariant phase space of generic spacetimes with compact closed spaces. 
  The observation that the 2+1 dimensional BTZ black hole can be obtained as a quotient space of anti-de Sitter space leads one to ask what causal behaviour other such quotient spaces can display. In this paper we answer this question in 2+1 and 3+1 dimensions when the identification group has one generator. Among other things we find that there does not exist any 3+1 generalization of the rotating BTZ hole. However, the non-rotating generalization exists and exhibits some unexpected properties. For example, it turns out to be non-static and to possess a non-trivial apparent horizon. 
  I present the factorization(s) of the Wheeler-DeWitt equation for vacuum FRW minisuperspace universes of arbitrary Hartle-Hawking factor ordering, including the so-called strictly isospectral supersymmetric method. By the latter means, one can introduce an infinite class of singular FRW minisuperspace wavefunctions characterized by a Darboux parameter that mathematically speaking is a Riccati integration constant, while physically determines the position of these strictly isospectral singularities on the Misner time axis 
  Scalar fields have had a long and controversial life in gravity theories, having progressed through many deaths and resurrections. The first scientific gravity theory, Newton's, was that of a scalar potential field, so it was natural for Einstein and others to consider the possibility of incorporating gravity into special relativity as a scalar theory. This effort, though fruitless in its original intent, nevertheless was useful in leading the way to Einstein's general relativity, a purely two-tensor field theory. However, a universally coupled scalar field again appeared, both in the context of Dirac's large number hypothesis and in five dimensional unified field theories as studied by Fierz, Jordan, and others. While later experimentation seems to indicate that if such a scalar exists its influence on solar system size interactions is negligible, other reincarnations have been proposed under the guise of dilatons in string theory and inflatons in cosmology. This paper presents a brief overview of this history. 
  The energy conditions of Einstein gravity (classical general relativity) are designed to extract as much information as possible from classical general relativity without enforcing a particular equation of state for the stress-energy. This systematic avoidance of the need to specify a particular equation of state is particularly useful in a cosmological setting --- since the equation of state for the cosmological fluid in a Friedmann-Robertson-Walker type universe is extremely uncertain. I shall show that the energy conditions provide simple and robust bounds on the behaviour of both the density and look-back time as a function of red-shift. I shall show that current observations suggest that the so-called strong energy condition (SEC) is violated sometime between the epoch of galaxy formation and the present. This implies that no possible combination of ``normal'' matter is capable of fitting the observational data. 
  Numerical relativity, applied to collisions of black holes, starts with initial data for black holes already in each other's strong field. The initial hypersurface data typically used for computation is based on mathematical simplifying prescriptions, such as conformal flatness of the 3-geometry and longitudinality of the extrinsic curvature. In the case of head on collisions of equal mass holes, there is evidence that such prescriptions work reasonably well, but it is not clear why, or whether this success is more generally valid. Here we study these questions by considering the ``particle limit'' for head on collisions of nonspinning holes. Einstein's equations are linearized in the mass of the small hole, and described by a single gauge invariant spacetime function psi, for each multipole. The resulting equations have been solved by numerical evolution for collisions starting from various initial separations, and the evolution is studied on a sequence of hypersurfaces. In particular, we extract hypersurface data, that is psi and its time derivative, on surfaces of constant background Schwarzschild time. These evolved data can then be compared with ``prescribed'' data, evolved data can be replaced by prescribed data on any hypersurface, and evolved further forward in time, a gauge invariant measure of deviation from conformal flatness can be evaluated, etc. The main findings of this study are: (i) For holes of unequal mass the use of prescribed data on late hypersurfaces is not successful. (ii) The failure is likely due to the inability of the prescribed data to represent the near field of the smaller hole. (iii) The discrepancy in the extrinsic curvature is more important than in the 3-geometry. (iv) The use of the more general conformally flat longitudinal data does not notably improve this picture. 
  The notion of time in cosmology is revealed through an examination of transition matrix elements of radiative processes occurring in the cosmos. To begin with, the very concept of time is delineated in classical physics in terms of correlations between the succession of configurations which describe a process and a standard trajectory called the clock. The total is an isolated system of fixed energy. This is relevant for cosmology in that the universe is an isolated system which we take to be homogeneous and isotropic. Furthermore, in virtue of the constraint which arises from reparametrization invariance of time, it has total energy zero. Therefore the momentum of the scale factor is determined from the energy of matter. In the quantum theory this is exploited through use of the WKB approximation for the wave function of the scale factor, justified for a large universe. The formalism then gives rise to matrix elements describing matter processes. These are shown to take on the form of usual time dependent quantum amplitudes wherein the temporal dependence is given by a background which is once more fixed by the total energy of matter. 
  We discuss the leading order correction to the equation of motion of a particle with spin on an arbitrary spacetime. A particle traveling in a curved spacetime is known to trace a geodesic of the background spacetime if the mass of the particle is negligibly small. For a spinning particle, it is known that there appears a term due to the coupling of the spin and the Riemann tensor of the background spacetime. Recently we have found the equation of motion of a non-spinning particle which includes the effect of gravitational radiation reaction. This paper is devoted to discussion of a consistent derivation of the equation of motion which is corrected both by the spin-Riemann coupling and the gravitational radiation reaction. 
  The geodesic equations are integrated for the Lewis metric and the effects of the different parameters appearing in the Weyl class on the motion of test particles are brought out. Particular attention deserves the appearance of a force parallel to the axial axis and without Newtonian analogue. 
  The 1--loop effective Lagrangian for a massive scalar field on an arbitrary causality violating spacetime is calculated using the methods of Euclidean quantum field theory in curved spacetime. Fields of spin 1/2, spin 1 and twisted field configurations are also considered. In general, we find that the Lagrangian diverges to minus infinity at each of the nth polarised hypersurfaces of the spacetime with a structure governed by a DeWitt-Schwinger type expansion. 
  We argue that correct account of the quantum properties of macroscopic objects which form reference frames (RF) demand the change of the standard space-time picture accepted in Quantum Mechanics. The presence of RF free quantum motion in the form of wave packet smearing results in formal nonapplicability of Galilean or Lorentz space-time transformations in this case. For the description of the particles states transformations between different quantum RF the special quantum space-time transformations are formulated. Consequently it results in corrections to Schrodinger or Klein- Gordon equations which depends on the RF mass. RF proper time becomes the operator depending of momentums spread in RF wave packet ,from the point of view of other observer. The experiments with macroscopic coherent states are proposed in which this effects can be tested.} 
  A method to regularize and renormalize the fluctuations of a quantum field in a curved background in the $\zeta$-function approach is presented. The method produces finite quantities directly and finite scale-parametrized counterterms at most. These finite couterterms are related to the presence of a particular pole of the effective-action $\zeta$ function as well as to the heat kernel coefficients. The method is checked in several examples obtaining known or reasonable results. Finally, comments are given for as it concerns the recent proposal by Frolov et al. to get the finite Bekenstein-Hawking entropy from Sakharov's induced gravity theory. 
  This is the second part of a series of 3 papers. Using the same method and the same coordinates as in part 1, rotating dust solutions of Einstein's equations are investigated that possess 3-dimensional symmetry groups, under the assumption that only one of the Killing fields is spanned on the fields of velocity $u^{\alpha}$ and rotation $w^{\alpha}$, while the other two define vectors that are linearly independent of $u^{\alpha}$ and $w^{\alpha}$ at every point of the spacetime region under consideration. The Killing fields are found and the Killing equations solved for the components of the metric tensor in every case that arises. The Einstein equations are simplified in a few cases, three (most probably) new solutions are found, and several classes of solutions known earlier are identified in the present scheme. They include those by Ozsv\'ath, Maitra, Ellis, King and Vishveshwara and Winicour.   PACS numbers 04.20.-q, 04.20.Cv, 04.20.Jb, 04.40.+c 
  The residual gauge freedom within the null quasi-spherical coordinate condition is studied, for spacetimes admitting an expanding, shear-free null foliation. The freedom consists of a boost and rotation at each coordinate sphere, corresponding to a specification of inertial frame at each sphere. Explicit formulae involving arbitrary functions of two variables are obtained for the accelerated Minkowski, Schwarzschild, and Robinson-Trautman spacetimes. These examples will be useful as test metrics in numerical relativity. 
  We consider a space-time with spatial sections isomorphic to the group manifold of SU(2). Triad and connection fluctuations are assumed to be SU(2)-invariant. Thus, they form a finite dimensional phase space. We perform non-perturbative path integral quantization of the model. Contarary to previous claims the path integral measure appeared to be non-singular near configurations admitting additional Killing vectors. In this model we are able to calculate the generating functional of Green functions of the reduced phase space variables exactly. 
  We have calculated the non-radial oscillation in slowly rotating relativistic stars with the Cowling approximation. The frequencies are compared with those based on the complete linearized equations of general relativity. It is found that the results with the approximation differ by less than about $20 %$ for typical relativistic stellar models. The approximation is more accurate for higher-order modes as in the Newtonian case. 
  We discuss the generation and evolution of density perturbations during the large curvature phase of string cosmology. We find that perturbations in the scalar components of the metric evolve with cosmic time as $\exp (\gamma H_st)$ where $H_s$ is the Hubble constant during the string phase and $\gamma$ is a constant which is determined by an algebraic equation involving all orders in the string coupling $\alpha '$. The seed for the perturbations can be provided by massive string modes. 
  We consider a curved space-time whose algebra of functions is the commutative limit of a noncommutative algebra and which has therefore an induced Poisson structure. In a simple example we determine a relation between this structure and the Riemann tensor. 
  Quantum general relativity may be considered as generally covariant QFT on differentiable manifolds, without any a priori metric structure. The kinematically covariance group acts by general diffeomorphisms on the manifold and by automorphisms on the isotonic net of *-algebras encoding the QFT, while the algebra of observables is covariant under the dynamical subgroup of the general diffeomorphism group.   Here, I focus on an algebraic implementation of the dynamical subgroup of dilations. Introducing an small and large scale cutoffs algebraically, their usual a priori conflict with general covariance is avoided. Thereby, a commutant duality between the minimal and maximal algebra is proposed. This allows to extract the modular structure, which is again related to the dilations. 
  The Einstein's equivalence principle is formulated in terms of the accuracy of measurements and its dependence of the size of the area of measurement. It is shown that different refinements of the statement 'the spacetime is locally flat' lead to different conculsions about the spacetime geometry. 
  In this paper we analyze the perturbations of the Kerr-Newman dilatonic black hole background. For this purpose we perform a double expansion in both the background electric charge and the wave parameters of the relevant quantities in the Newman-Penrose formalism. We then display the gravitational, dilatonic and electromagnetic equations, which reproduce the static solution (at zero order in the wave parameter) and the corresponding wave equations in the Kerr background (at first order in the wave parameter and zero order in the electric charge). At higher orders in the electric charge one encounters corrections to the propagations of waves induced by the presence of a non-vanishing dilaton. An explicit computation is carried out for the electromagnetic waves up to the asymptotic form of the Maxwell field perturbations produced by the interaction with dilatonic waves. A simple physical model is proposed which could make these perturbations relevant to the detection of radiation coming from the region of space near a black hole. 
  For the O(N) field theory with lambda Phi^4 self-coupling, we construct the two-particle-irreducible (2PI), closed-time-path (CTP) effective action in a general curved spacetime. From this we derive a set of coupled equations for the mean field and the variance. They are useful for studying the nonperturbative, nonequilibrium dynamics of a quantum field when full back reactions of the quantum field on the curved spacetime, as well as the fluctuations on the mean field, are required. Applications to phase transitions in the early Universe such as the Planck scale or in the reheating phase of chaotic inflation are under investigation. 
  In seeking to arrive at a theory of ``quantum gravity'', one faces several choices among alternative approaches. I list some of these ``forks in the road'' and offer reasons for taking one alternative over the other. In particular, I advocate the following: the sum-over-histories framework for quantum dynamics over the ``observable and state-vector'' framework; relative probabilities over absolute ones; spacetime over space as the gravitational ``substance'' (4 over 3+1); a Lorentzian metric over a Riemannian (``Euclidean'') one; a dynamical topology over an absolute one; degenerate metrics over closed timelike curves to mediate topology-change; ``unimodular gravity'' over the unrestricted functional integral; and taking a discrete underlying structure (the causal set) rather than the differentiable manifold as the basis of the theory.      In connection with these choices, I also mention some results from unimodular quantum cosmology, sketch an account of the origin of black hole entropy, summarize an argument that the quantum mechanical measurement scheme breaks down for quantum field theory, and offer a reason why the cosmological constant of the present epoch might have a magnitude of around $10^{-120}$ in natural units. 
  An attempt is made to supplement Carter's partial investigation of the global structure of Kerr-Newman spacetime on the symmetry axis. Namely, the global structure of \theta = const. timelike submanifolds of Kerr-Newman metric starting from the symmetry axis all the way down to the equatorial plane are studied by introducing a new time coordinate slightly different from the usual Boyer-Lindquist time coordinate. It turns out that the maximal anaytic extension of \theta = \theta_0 (0 \leq \theta_0 < \pi/2) submanifolds is the same as that of the symmetry axis first studied by Carter whereas \theta = \pi/2 equatorial plane has the topology identical to that of the Reissner-Nordstrom spacetime. General applicability of this method to Kerr-Newman-type black hole solutions in other gravity theories is discussed as well. 
  It has been argued that a black hole horizon can support the long range fields of a Nielsen-Olesen string, and that one can think of such a vortex as black hole ``hair''. In this paper, we examine the properties of an Abelian Higgs vortex in the presence of a charged black hole as we allow the hole to approach extremality. Using both analytical and numerical techniques, we show that the magnetic field lines (as well as the scalar field) of the vortex are completely expelled from the black hole in the extreme limit. This was to be expected, since extreme black holes in Einstein-Maxwell theory are known to exhibit such a ``Meissner effect'' in general. This would seem to imply that a vortex does not want to be attached to an extreme black hole. We calculate the total energy of the vortex fields in the presence of an extreme black hole. When the hole is small relative to the size of the vortex, it is energetically favoured for the hole to remain inside the vortex region, contrary to the intuition that the hole should be expelled. However, as we allow the extreme horizon radius to become very large compared to the radius of the vortex, we do find evidence of an instability. This proves that it is energetically unfavourable for a thin vortex to interact with a large extreme black hole. This would seem to dispel the notion that a black hole can support `long' abelian Higgs hair in the extreme limit. We show that these considerations do not go through in the near extreme limit. Finally, we discuss whether this has implications for strings that end at black holes. 
  The Bekenstein-Hawking ``entropy'' of a Kerr-Newman dilaton black hole is computed in a perturbative expansion in the charge-to-mass ratio. The most probable configuration for a gas of such black holes is analyzed in the microcanonical formalism and it is argued that it does not satisfy the equipartition principle but a bootstrap condition. It is also suggested that the present results are further support for an interpretation of black holes as excitations of extended objects. 
  We generalize some results of Ford and Roman constraining the possible behaviors of renormalized expected stress-energy tensors of a free massless scalar field in two dimensional Minkowski spacetime. Ford and Roman showed that the energy density measured by an inertial observer, when averaged with respect to that observers proper time by integrating against some weighting function, is bounded below by a negative lower bound proportional to the reciprocal of the square of the averaging timescale. However, the proof required a particular choice for the weighting function. We extend the Ford-Roman result in two ways: (i) We calculate the optimum (maximum possible) lower bound and characterize the state which achieves this lower bound; the optimum lower bound differs by a factor of three from the bound derived by Ford and Roman for their choice of smearing function. (ii) We calculate the lower bound for arbitrary, smooth positive weighting functions. We also derive similar lower bounds on the spatial average of energy density at a fixed moment of time. 
  Some of the recent work on quantum gravity has involved modified uncertainty relations such that the products of the uncertainties of certain pairs of observables increase with time. It is here observed that this type of modified uncertainty relations would lead to quantum decoherence, which could explain the classical behavior of macroscopic systems, and CPT violation, which coud provide the seed for the emergence of a matter-antimatter asymmetry. 
  The aim of this paper is to compute transitions amplitudes in quantum cosmology, and in particular pair creation amplitudes and radiative transitions. To this end, we apply a double adiabatic development to the solutions of the Wheeler-DeWitt equation restricted to mini-superspace wherein gravity is described by the scale factor $a$. The first development consists in working with instantaneous eigenstates, in $a$, of the matter Hamiltonian. The second development is applied to the gravitational part of the wave function and generalizes the usual WKB approximation. We then obtain an exact equation which replaces the Wheeler-DeWitt equation and determines the evolution, i.e. the dependence in $a$, of the coefficients of this double expansion. When working in the gravitational adiabatic approximation, the simplified equation delivers the unitary evolution of transition amplitudes occurring among instantaneous eigenstates. Upon abandoning this approximation, one finds that there is an additional coupling among matter states living in expanding and contracting universes. Moreover one has to face also the Klein paradox, i.e. the generation of backward waves from an initially forward wave. The interpretation and the consequences of these unusual features are only sketched in the present paper. Finally, the examples of pair creation and radiative transitions are analyzed in detail to establish when and how the above mentioned unitary evolution coincides with the Schr\" odinger evolution. 
  We consider a classical Dirac field in flat Minkowski spacetime. We perform a Gordon decomposition of its canonical energy-momentum and spin currents, respectively. Thereby we find for each of these currents a convective and a polarization piece. The polarization pieces can be expressed as exterior covariant derivatives of the two-forms $\check M_\alpha$ and $M_{\alpha\beta}=-M_{\beta\alpha}$, respectively. In analogy to the magnetic moment in electrodynamics, we identify these two-forms as gravitational moments connected with the translation group and the Lorentz group, respectively. We point out the relation between the Gordon decomposition of the energy-momentum current and its Belinfante-Rosenfeld symmetrization. In the non-relativistic limit, the translational gravitational moment of the Dirac field is found to be proportional to the spin covector of the electron. 
  We present the results of a numerical study of the fluid $f$, $p$ and the gravitational $w$ modes for increasingly relativistic nonrotating polytropes. The results for $f$ and $w$-modes are in good agreement with previous data for uniform density stars, which supports an understanding of the nature of the gravitational wave modes based on the uniform density data. We show that the $p$-modes can become extremely long-lived for some relativistic stars. This effect is attributed to the change in the perturbed density distribution as the star becomes more compact. 
  This paper studies the decay of a large, closed domain wall in a closed universe. Such walls can form in the presence of a broken, discrete symmetry. We introduce a novel process of quantum decay for such a wall, in which the vacuum fluctuates from one discrete state to another throughout one half of the universe, so that the wall decays into pure field energy. Equivalently, the fluctuation can be thought of as the nucleation of a second domain wall of zero size, followed by its growth by quantum tunnelling and its collision with the first wall, annihilating both. The barrier factor for this quantum tunneling is calculated by guessing and verifying a Euclidean instanton for the two-wall system. We also discuss the classical origin and evolution of closed, topologically spherical domain walls in the early universe, through a "budding-off" process involving closed domain walls larger than the Hubble radius. This paper is the first of a series on this subject. 
  Non-trivial solutions in string field theory may lead to the spontaneous breaking of Lorentz invariance and to new tensor-matter interactions. It is argued that requiring the contribution of the vacuum expectation values of Lorentz tensors to account for the vacuum energy up to the level that $\Omega_{0}^{\Lambda} = 0.5$ implies the new interactions range is $\lambda \sim 10^{-4} m$. These conjectured violations of the Lorentz symmetry are consistent with the most stringent experimental limits. 
  The Bondi formula for calculation of the invariant mass in the Tolman- Bondi (TB) model is interprated as a transformation rule on the set of co-moving coordinates. The general procedure by which the three arbitrary functions of the TB model are determined explicitly is presented. The properties of the TB model, produced by the transformation rule are studied. Two applications are studied: for the falling TB flat model the equation of motion of two singularities hypersurfaces are obtained; for the expanding TB flat model the dependence of size of area with friedmann-like solution on initial conditions is studied in the limit $t \to +\infty$. 
  A stability analysis of a spherically symmetric star in scalar-tensor theories of gravity is given in terms of the frequencies of quasi-normal modes. The scalar-tensor theories have a scalar field which is related to gravitation. There is an arbitrary function, the so-called coupling function, which determines the strength of the coupling between the gravitational scalar field and matter. Instability is induced by the scalar field for some ranges of the value of the first derivative of the coupling function. This instability leads to significant discrepancies with the results of binary-pulsar-timing experiments and hence, by the stability analysis, we can exclude the ranges of the first derivative of the coupling function in which the instability sets in. In this article, the constraint on the first derivative of the coupling function from the stability of relativistic stars is found. Analysis in terms of the quasi-normal mode frequencies accounts for the parameter dependence of the wave form of the scalar gravitational waves emitted from the Oppenheimer-Snyder collapse. The spontaneous scalarization is also discussed. 
  Using the quasi-Maxwell form of the vacuum Einstein equations and demanding the presence of a cylindrically symmetric radial gravomagnetic field, we find the solution to the Einstein equations which represents the gravity field of a line gravomagnetic monopole. We show that this is the generalization of the Levi-Civita's cylindrically symmetric static spacetime, in the same way that the NUT metric is the empty space generalization of the Schwarzschild metric.   Some of the features of this metric as well as its relation to other metrics are discussed. 
  A Schroedinger picture analysis of time dependent quantum oscillators, in a manner of Guth and Pi, clearly identifies two physical mechanisms for possible decoherence of vacuum fluctuations in early universe: turning of quantum oscillators upside-down, and rapid squeezing of upside-right oscillators so that certain squeezing factor diverges. In inflationary cosmology the former mechanism explains the stochastic evolution of light inflatons and the classical nature of density perturbations in most of inflationary models, while the later one is responsible for the classical evolution of relatively heavy fields, with masses in a narrow range above the Hubble parameter: 2 < (m/H_0)^2 < 9/4. The same method may be applied to study of the decoherence of quantum fluctuations in any Robertson-Walker cosmology. 
  We study the cosmological evolution of massless single-field scalar-tensor theories of gravitation from the time before the onset of $e^+e^-$ annihilation and nucleosynthesis up to the present. The cosmological evolution together with the observational bounds on the abundances of the lightest elements (those mostly produced in the early universe) place constraints on the coefficients of the Taylor series expansion of $a(\phi)$, which specifies the coupling of the scalar field to matter and is the only free function in the theory. In the case when $a(\phi)$ has a minimum (i.e., when the theory evolves towards general relativity) these constraints translate into a stronger limit on the Post-Newtonian parameters $\gamma$ and $\beta$ than any other observational test. Moreover, our bounds imply that, even at the epoch of annihilation and nucleosynthesis, the evolution of the universe must be very close to that predicted by general relativity if we do not want to over- or underproduce $^{4}$He. Thus the amount of scalar field contribution to gravity is very small even at such an early epoch. 
  We show, in complete accord with the usual Rindler picture, that detectors with constant acceleration $a$ in de Sitter (dS) and Anti de Sitter (AdS) spaces with cosmological constants $\Lambda$ measure temperatures $2\pi T=(\Lambda/3+a^{2})^{1/2}\equiv a_{5}$, the detector "5-acceleration" in the embedding flat 5-space. For dS, this recovers a known result; in AdS, where $\Lambda$ is negative, the temperature is well defined down to the critical value $a_{5}=0$, again in accord with the underlying kinematics. The existence of a thermal spectrum is also demonstrated for a variety of candidate wave functions in AdS backgrounds. 
  We study a system of self-gravitating massive fermions in the framework of the general-relativistic Thomas-Fermi model. We postulate the free energy functional and show that its extremization is equivalent to solving the Einstein's field equations. A self-gravitating fermion gas we then describe by a set of Thomas-Fermi type self-consistency equations. 
  We examine the dynamics of a self-gravitating string in the scalar-tensor theories of gravitation by considering a thin tube of matter to describe it. For a class of solutions, we obtain in the generic case that the extrinsic curvature of the world sheet of the central line is null in the limit where the radius of the string tends to zero. However, if we impose a specific constraint on the behaviour of the solution then we find that only the mean curvature of the world sheet of the central line vanishes which is just the Nambu-Goto dynamics. This analysis can include the massless dilatonic theories of gravity. 
  We propose a scheme for quantizing a scalar field over the Schwarzschild manifold including the interior of the horizon. On the exterior, the timelike Killing vector and on the horizon the isometry corresponding to restricted Lorentz boosts can be used to enforce the spectral condition. For the interior we appeal to the need for CPT invariance to construct an explicitly positive definite operator which allows identification of positive and negative frequencies. This operator is the translation operator corresponding to the inexorable propagation to smaller radii as expected from the classical metric. We also propose an expression for the propagator in the interior and express it as a mode sum. 
  The dynamics of a spherically symmetric thin shell with arbitrary rest mass and surface tension interacting with a central black hole is studied. A careful investigation of all classical solutions reveals that the value of the radius of the shell and of the radial velocity as an initial datum does not determine the motion of the shell; another configuration space must, therefore, be found. A different problem is that the shell Hamiltonians used in literature are complicated functions of momenta (non-local) and they are gauge dependent. To solve these problems, the existence is proved of a gauge invariant super-Hamiltonian that is quadratic in momenta and that generates the shell equations of motion. The true Hamiltonians are shown to follow from the super-Hamiltonian by a reduction procedure including a choice of gauge and solution of constraint; one important step in the proof is a lemma stating that the true Hamiltonians are uniquely determined (up to a canonical transformation) by the equations of motion of the shell, the value of the total energy of the system, and the choice of time coordinate along the shell. As an example, the Kraus-Wilczek Hamiltonian is rederived from the super-Hamiltonian. The super-Hamiltonian coincides with that of a fictitious particle moving in a fixed two-dimensional Kruskal spacetime under the influence of two effective potentials. The pair consisting of a point of this spacetime and a unit timelike vector at the point, considered as an initial datum, determines a unique motion of the shell. 
  A general analysis of the gravitational dynamics of a medium with a continuous distribution of vorticity indicates that the answer to the question raised in the title is affirmative, contrary to a recent claim. 
  The observed CMBR dipole is generally interpreted as a Doppler effect arising from the motion of the Earth relative to the CMBR frame. An alternative interpretation, proposed in the last years, is that the dipole results from ultra-large scale isocurvature perturbations. We examine this idea in the context of open cosmologies and show that the isocurvature interpretation is not valid in an open universe, unless it is extremely close to a flat universe, $|\Omega_0 -1|< 10^{-4}$. 
  The occurrence of chaos for test particles moving around Schwarzschild black holes perturbed by a special class of gravitational waves is studied in the context of the Melnikov method. The explicit integration of the equations of motion for the homoclinic orbit is used to reduce the application of this method to the study of simple graphics. 
  The properties of a transformation previously considered for generating new perfect-fluid solutions from known ones are further investigated. It is assumed that the four-velocity of the fluid is parallel to the stationary Killing field, and also that the norm and the twist potential of the stationary Killing field are functionally related. This case is complementary to the case studied in our previous paper. The transformation can be applied to generate possibly new perfect-fluid solutions from known ones only for the case of barotropic equation of state rho+3p=0 or, alternatively, for the case of a static spacetime. For static spacetimes our method recovers the Buchdahl transformation. It is demonstrated, moreover, that Herlt's technique for constructing stationary perfect-fluid solutions from static ones is, actually, a special case of the method considered in the present paper. 
  The structure of boundaries between degenerate and nondegenerate solutions of Ashtekar's canonical reformulation of Einstein's equations is studied. Several examples are given of such "phase boundaries" in which the metric is degenerate on one side of a null hypersurface and non-degenerate on the other side. These include portions of flat space, Schwarzschild, and plane wave solutions joined to degenerate regions. In the last case, the wave collides with a planar phase boundary and continues on with the same curvature but degenerate triad, while the phase boundary continues in the opposite direction. We conjecture that degenerate phase boundaries are always null. 
  In this review article we consider three most important sources of the gravitational field of the Early Universe: self-interacting scalar field, chiral field and gauge field. The correspondence between all of them are pointed out. More attention is payed to nonlinear scalar field source of gravity. The progress in finding the exact solutions in inflationary universe is reviewed. The basic idea of `fine turning of the potential' method is discussed and computational background is presented in details. A set of new exact solutions for standard inflationary model and conformally-flat space-times are obtained. Special attention payed to relations between `fine turning of the potential' and Barrow's approaches. As the example of a synthesis of both methods new exact solution is obtained. 
  In a stationary axisymmetric spacetime, the angular velocity of a stationary observer that Fermi-Walker transports its acceleration vector is also the angular velocity that locally extremizes the magnitude of the acceleration of such an observer, and conversely if the spacetime is also symmetric under reversing both t and phi together. Thus a congruence of Nonrotating Acceleration Worldlines (NAW) is equivalent to a Stationary Congruence Accelerating Locally Extremely (SCALE). These congruences are defined completely locally, unlike the case of Zero Angular Momentum Observers (ZAMOs), which requires knowledge around a symmetry axis. The SCALE subcase of a Stationary Congruence Accelerating Maximally (SCAM) is made up of stationary worldlines that may be considered to be locally most nearly at rest in a stationary axisymmetric gravitational field. Formulas for the angular velocity and other properties of the SCALEs are given explicitly on a generalization of an equatorial plane, infinitesimally near a symmetry axis, and in a slowly rotating gravitational field, including the weak-field limit, where the SCAM is shown to be counter-rotating relative to infinity. These formulas are evaluated in particular detail for the Kerr-Newman metric. Various other congruences are also defined, such as a Stationary Congruence Rotating at Minimum (SCRAM), and Stationary Worldlines Accelerating Radially Maximally (SWARM), both of which coincide with a SCAM on an equatorial plane of reflection symmetry. Applications are also made to the gravitational fields of maximally rotating stars, the Sun, and the Solar System. 
  The spacetime foam structure is reviewed briefly (topogical fluctuations and virtual black hole possibility; equation of state of the foam). A model of space foam at the surface of the event horizon is introduced. The model is applied to the calculus of the number of states of a black hole, of its entropy and of other thermodynamical properties. A formula for the number of microholes on the surface of the event horizon is derived. Thermodynamical properties of the event horizon are extended to thermodynamical properties of the space. On the basis of the previous results, the possibility of micro black holes creation by the Unruh Effect is investigated. 
  The exact solutions in the standard inflationary model based on the self-interacting scalar field minimally coupled to gravity are considered. The shape's freedom of the self-interacting potential $V(\phi)$ is postulated to obtain a new set of the exact solutions in the framework of Friedmann-Robertson-Walker Universes. The general solution was found in the case of power law inflation. We obtained new solutions and compared them with obtained ones earlir for the exponential type inflation. 
  It has been argued that a black hole horizon can support the long range fields of a Nielsen-Olesen string, and that one can think of such a vortex as black hole ``hair''. We show that the fields inside the vortex are completely expelled from a charged black hole in the extreme limit (but not in the near extreme limit). This would seem to imply that a vortex cannot be attached to an extreme black hole. Furthermore, we provide evidence that it is energetically unfavourable for a thin vortex to interact with a large extreme black hole. This dispels the notion that a black hole can support `long' Abelian Higgs hair in the extreme limit. We discuss the implications for strings that end at black holes, as in processes where a string snaps by nucleating black holes. 
  This paper studies the decay of a large, closed domain wall in a closed universe. Such walls can form in the presence of a broken, discrete symmetry. We study a novel process of quantum decay for such a wall, in which the vacuum fluctuates from one discrete state to another throughout one half of the universe, so that the wall decays into pure field energy. Equivalently, the fluctuation can be thought of as the nucleation of a second closed domain wall of zero size, followed by its growth by quantum tunnelling and its collision with the first wall, annihilating both. We therefore study the 2-wall system coupled to a spherically symmetric gravitational field. We derive a simple form of the 2-wall action, use Dirac quantization, obtain the 2-wall wave function for annihilation, find from it the barrier factor for this quantum tunneling, and thereby get the decay probability. This is the second paper of a series. 
  We describe the first discrete-time 4-dimensional numerical application of Regge calculus. The spacetime is represented as a complex of 4-dimensional simplices, and the geometry interior to each 4-simplex is flat Minkowski spacetime. This simplicial spacetime is constructed so as to be foliated with a one parameter family of spacelike hypersurfaces built of tetrahedra. We implement a novel two-surface initial-data prescription for Regge calculus, and provide the first fully 4-dimensional application of an implicit decoupled evolution scheme (the ``Sorkin evolution scheme''). We benchmark this code on the Kasner cosmology --- a cosmology which embodies generic features of the collapse of many cosmological models. We (1) reproduce the continuum solution with a fractional error in the 3-volume of 10^{-5} after 10000 evolution steps, (2) demonstrate stable evolution, (3) preserve the standard deviation of spatial homogeneity to less than 10^{-10} and (4) explicitly display the existence of diffeomorphism freedom in Regge calculus. We also present the second-order convergence properties of the solution to the continuum. 
  The symmetries of generic 2D dilaton models of gravity with (and without) matter are studied in some detail. It is shown that $\delta_2$, one of the symmetries of the matterless models, can be generalized to the case where matter fields of any kind are present. The general (classical) solution for some of these models, in particular those coupled to chiral matter, which generalizes the Vaidya solution of Einstein Gravity, is also given. 
  We construct exact solutions to Einstein equations which represent relativistic disks immersed into an expanding FRW Universe. It is shown that the expansion influences dynamical characteristics of the disks such as rotational curves, surface mass density, etc. The effects of the expansion is exemplified with non-static generalizations of Kuzmin-Curzon and generalized Schwarzschild disks. 
  The decay of the string perturbative vacuum into our present cosmological state is associated to the transition from a phase of growing curvature and growing dilaton, to a phase of decreasing curvature and frozen dilaton. The possible approaches to a classical and quantum description of such a transition are introduced and briefly discussed. 
  The volume operator plays a central role in both the kinematics and dynamics of canonical approaches to quantum gravity which are based on algebras of generalized Wilson loops. We introduce a method for simplifying its spectral analysis, for quantum states that can be realized on a cubic three-dimensional lattice. This involves a decomposition of Hilbert space into sectors transforming according to the irreducible representations of a subgroup of the cubic group. As an application, we determine the complete spectrum for a class of states with six-valent intersections. 
  We prove that any asymptotically flat solution to the spherically symmetric SU(2) Einstein-Yang/Mills equations is globally defined. This result applies in particular to the interior of colored black holes. 
  Cosmic strings are topological defects thought to have formed early in the life of the universe. If such objects exist, a study of their interaction with black holes is of interest. The equations of motion of a cosmic string have the form of highly non-linear wave equations. General analytic solutions, except for motion in certain backgrounds such as flat and shockwave spacetimes, remain unknown; consequently, much of the work must be carried out numerically. To do this, an implicit finite difference scheme was developed that involves solving large block tridiagonal systems. This paper discusses the numerical method, its validation against analytic and semi-analytic solutions, and preliminary results on the interaction of a cosmic string with Schwarzschild black holes. 
  We define a certain differential system on an open set of $R^6$. The system locally defines a Lorentzian 4-manifold satisfying the Einstein equations. The converse statement is indicated and its details are postponed to the furthcoming paper. 
  Given a conformally nonflat Einstein spacetime we define a fibration $P$ over it. The fibres of this fibration are elliptic curves (2-dimensional tori) or their degenerate counterparts. Their topology depends on the algebraic type of the Weyl tensor of the Einstein metric. The fibration $P$ is a double branched cover of the bundle of null direction over the spacetime and is equipped with six linearly independent 1-forms which satisfy certain relatively simple system of equations. 
  Exact solutions corresponding to spherically symmetric inhomogeneous nonstationary Tolman metrics are obtained for the self-consistent 
  It has recently been argued that non-trivial Brans-Dicke black hole solutions different from the usual Schwarzschild solution could exist. We attemt here to ``censor'' these non-trivial Brans-Dicke black hole solutions by examining their thermodynamic properties. Quantities like Hawking temperature and entropy of the black holes are computed. Analysis of the behaviors of these thermodynamic quantities appears to show that even in Brans-Dicke gravity, the usual Schwarzschild spacetime turns out to be the only physically relevant uncharged static black hole solution. 
  We analyze the coupling between the internal degrees of freedom of neutron stars in a close binary, and the stars' orbital motion. Our analysis is based on the method of matched asymptotic expansions and is valid to all orders in the strength of internal gravity in each star, but is perturbative in the ``tidal expansion parameter'' (stellar radius)/(orbital separation). At first order in the tidal expansion parameter, we show that the internal structure of each star is unaffected by its companion, in agreement with post-1-Newtonian results of Wiseman (gr-qc/9704018). We also show that relativistic interactions that scale as higher powers of the tidal expansion parameter produce qualitatively similar effects to their Newtonian counterparts: there are corrections to the Newtonian tidal distortion of each star, both of which occur at third order in the tidal expansion parameter, and there are corrections to the Newtonian decrease in central density of each star (Newtonian ``tidal stabilization''), both of which are sixth order in the tidal expansion parameter. There are additional interactions with no Newtonian analogs, but these do not change the central density of each star up to sixth order in the tidal expansion parameter. These results, in combination with previous analyses of Newtonian tidal interactions, indicate that (i) there are no large general-relativistic crushing forces that could cause the stars to collapse to black holes prior to the dynamical orbital instability, and (ii) the conventional wisdom with respect to coalescing binary neutron stars as sources of gravitational-wave bursts is correct: namely, the finite-stellar-size corrections to the gravitational waveform will be unimportant for the purpose of detecting the coalescences. 
  A time-symmetric Cauchy slice of the extended Schwarzschild spacetime can be evolved into a foliation of the $r>3m/2$-region of the spacetime by maximal surfaces with the requirement that time runs equally fast at both spatial ends of the manifold. This paper studies the behaviour of these slices in the limit as proper time-at-infinity becomes arbitrarily large and gives an analytic expression for the collapse of the lapse. 
  A fuzzy version of the ordinary round 2-sphere has been constructed with an invariant curvature. We here consider linear connections on arbitrary fuzzy surfaces of genus zero. We shall find as before that they are more or less rigidly dependent on the differential calculus used but that a large number of the latter can be constructed which are not covariant under the action of the rotation group. For technical reasons we have been forced to limit our considerations to fuzzy surfaces which are small perturbations of the fuzzy sphere. 
  The dynamic shift of the center of mass for a rotating hemisphere prompts us the question of what might be its physical consequences. Despite the fact that accelerating object is known to create gravitational field, there is no known external dynamic gravitational force from a rotating sphere where the individual mass components are in constant acceleration. However, Thirring's `induced centrifugal force' and the component of the force along the longitudinal axis inside a rotating spherical shell indicate that they are non-radiative dynamic forces which depend on $\omega^2$. In this report, Thirring's force is derived by considering the component-wise acceleration of the rotating hemisphere in the weak field approximation. This new analytic solution provides the gravitational explanation of the jet phenomena observed from the fast rotating cosmological bodies, which demands a major revision in our understanding of the universe since it suggests there exists a strong, long ranged, non-Newtonian dynamic gravitational force in our universe. This also raises an interesting question of how the strength of the dipole moment can be maximized for a given mass by configuring the specific geometrical shape of the rotating source. 
  The concepts of negative gravitational mass and gravitational repulsion are alien to general relativity. Still, we show here that small negative fluctuations - small dimples in the primordial density field - that act as if they have an effective negative gravitational mass, play a dominant role in shaping our Universe. These initially tiny perturbations repel matter surrounding them, expand and grow to become voids in the galaxy distribution. These voids - regions with a diameter of $40 \h$ Mpc which are almost devoid of galaxies - are the largest object in the Universe. 
  Inhomogeneous multidimensional cosmological models with a higher dimensional space-time manifold are investigated under dimensional reduction. In the Einstein conformal frame, small excitations of the scale factors of the internal spaces near minima of an effective potential have a form of massive scalar fields in the external space-time. Parameters of models which ensure minima of the effective potentials are obtained for particular cases and masses of gravitational excitons are estimated. 
  We perform a Hamiltonian reduction of spherically symmetric Einstein gravity with a thin dust shell of positive rest mass. Three spatial topologies are considered: Euclidean (R^3), Kruskal (S^2 x R), and the spatial topology of a diametrically identified Kruskal (RP^3 - {a point at infinity}). For the Kruskal and RP^3 topologies the reduced phase space is four-dimensional, with one canonical pair associated with the shell and the other with the geometry; the latter pair disappears if one prescribes the value of the Schwarzschild mass at an asymptopia or at a throat. For the Euclidean topology the reduced phase space is necessarily two-dimensional, with only the canonical pair associated with the shell surviving. A time-reparametrization on a two-dimensional phase space is introduced and used to bring the shell Hamiltonians to a simpler (and known) form associated with the proper time of the shell. An alternative reparametrization yields a square-root Hamiltonian that generalizes the Hamiltonian of a test shell in Minkowski space with respect to Minkowski time. Quantization is briefly discussed. The discrete mass spectrum that characterizes natural minisuperspace quantizations of vacuum wormholes and RP^3-geons appears to persist as the geometrical part of the mass spectrum when the additional matter degree of freedom is added. 
  The Kaluza-Klein formalism of the Einstein's theory, based on the (2,2)-fibration of a generic 4-dimensional spacetime, describes general relativity as a Yang-Mills gauge theory on the 2-dimensional base manifold, where the local gauge symmetry is the group of the diffeomorphisms of the 2-dimensional fibre manifold. As a way of illustrating how to use this formalism in finding exact solutions, we apply this formalism to the spherically symmetric case, and obtain the Schwarzschild solution by solving the field equations. 
  We examine the effect of spatial curvature in the pre-big-bang inflationary model suggested by string theory. We study $O(\alpha ')$ corrections and we show that, independently of the initial curvature, they lead to a phase of exponential inflation. The amount of inflation in this phase is long enough to solve the horizon and flatness problems if the evolution starts deeply into the weak coupling regime. There is a region of the parameter space of the model where such a long inflationary phase at the string scale is consistent with COBE anisotropies, millisecond pulsar timing and nucleosynthesis constraints. We discuss implications for the spectrum of relic gravitational waves at the frequencies of LIGO and Virgo. 
  The entropy of a black hole can differ from a quarter of the area of the horizon because of quantum corrections. The correction is related to the contribution to the Euclidean functional integral from quantum fluctuations but is not simply equal to the correction to the effective action. A (2+1) dimensional rotating black hole is explicitly considered. 
  We present a quantum theory of gravity which is in agreement with observation in the relativistic domain. The theory is not relativistic, but a Galilean invariant generalization of Lorentz-Poincare ether theory to quantum gravity.   If we apply the methodological rule that the best available theory has to be preferred, we have to reject the relativistic paradigm and return to Galilean invariant ether theory. 
  A quantum mechanism of the inertia generation has been proposed. The threshold of the parametric instability of ``goemetric'' Goldstone modes is treated as a rest mass. 
  It is shown analytically that an external tidal gravitational field increases the secular stability of a fully general relativistic, rigidly rotating neutron star that is near marginal stability, protecting it against gravitational collapse. This stabilization is shown to result from the simple fact that the energy $\delta M(Q,R)$ required to raise a tide on such a star, divided by the square of the tide's quadrupole moment $Q$, is a decreasing function of the star's radius $R$, $(d/dR)[\delta M(Q,R)/Q^2]<0$ (where, as $R$ changes, the star's structure is changed in accord with the star's fundamental mode of radial oscillation). If $(d/dR)[\delta M(Q,R)/Q^2]$ were positive, the tidal coupling would destabilize the star. As an application, a rigidly rotating, marginally secularly stable neutron star in an inspiraling binary system will be protected against secular collapse, and against dynamical collapse, by tidal interaction with its companion. The ``local-asymptotic-rest-frame'' tools used in the analysis are somewhat unusual and may be powerful in other studies of neutron stars and black holes interacting with an external environment. As a byproduct of the analysis, in an appendix the influence of tidal interactions on mass-energy conservation is elucidated. 
  A model is proposed in which the Hawking particles emitted by a black hole are treated as an envelope of matter that obeys an equation of state, and acts as a source in Einstein's equations. This is a crude but interesting way to accommodate for the back reaction. For large black holes, the solution can be given analytically, if the equation of state is $p=\kappa\rho$, with $0<\kappa<1$. The solution exhibits a singularity at the origin. If we assume $N$ free particle types, we can use a Hartree-Fock procedure to compute the contribution of one such field to the entropy, and the result scales as expected as $1/N$. A slight mismatch is found that could be attributed to quantum corrections to Einstein's equations, but can also be made to disappear when $\k$ is set equal to one. The case $\kappa=1$ is further analysed. 
  The two dimensional substructure of general relativity and gravity, and the two dimensional geometry of quantum effect by black hole are disclosed. Then the canonical quantization of the two dimensional theory of gravity is performed. It is shown that the resulting uncertainty relations can explain black hole quantum effects. A quantum gravitational length is also derived which can clarify the origin of Planck length. 
  We show that spinless and neutral black holes in thermal equilibrium with radiation undergo fluctuations of charge and angular momentum. The corresponding spreads for a black hole in contact with charged scalar particles is calculated. The angular momentum spread turns out to grow with the size of the cavity. Charge spread does not depend either on the size of the cavity nor on the elementary charge of the field. Then, either the scale of the elementary chage is fixed by black hole physics $\alpha \sim 1/4\pi$ or the underlying physics of these fluctuations is not the random absorption and emission of charged particles from the bath, it remains unknown and must be clarified. 
  The method of adiabatic invariants for time dependent Hamiltonians is applied to a massive scalar field in a de Sitter space-time. The scalar field ground state, its Fock space and coherent states are constructed and related to the particle states. Diverse quantities of physical interest are illustrated, such as particle creation and the way a classical probability distribution emerges for the system at late times. 
  We report the results of a theoretical and experimental study of a spherical gravitational wave antenna. We show that it is possible to understand the data from a spherical antenna with 6 radial resonant transducers attached to the surface in the truncated icosahedral arrangement. We find that the errors associated with small deviations from the ideal case are small compared to other sources of error, such as a finite signal-to-noise ratio. An in situ measurement technique is developed along with a general algorithm that describes a procedure for determining the direction of an external force acting on the antenna, including the force from a gravitational wave, using a combination of the transducer responses. The practicality of these techniques was verified on a room-temperature prototype antenna. 
  Singularities inside static spherically symmetric black holes in the SU(2) Einstein-Yang-Mills and Einstein-Yang-Mills-dilaton theories are investigated. Analytical formulas are presented describing oscillatory and power law metric behavior near spacelike singularities in generic solutions. 
  Using the Kaluza-Klein structure of stationary spacetimes, a framework for analyzing stationary perturbations of static Einstein-Yang-Mills configurations with bosonic matter fields is presented. It is shown that the perturbations giving rise to non-vanishing ADM angular momentum are governed by a self-adjoint system of equations for a set of gauge invariant scalar amplitudes. The method is illustrated for SU(2) gauge fields, coupled to a Higgs doublet or a Higgs triplet. It is argued that slowly rotating black holes arise generically in self-gravitating non-Abelian gauge theories with bosonic matter, whereas, in general, soliton solutions do not have rotating counterparts. 
  Yang-Mills color fields evolve chaotically in an anisotropically expanding universe. The chaotic behaviour differs from that found in anisotropic Mixmaster universes. The universe isotropizes at late times, approaching the mean expansion rate of a radiation-dominated universe. However, small chaotic oscillations of the shear and color stresses continue indefinitely. An invariant, coordinate-independent characterisation of the chaos is provided by means of fractal basin boundaries. 
  Four-dimensional black hole solutions generated by the low energy string effective action are investigated outside and inside the event horizon. A restriction for a minimal black hole size is obtained in the frame of the model discussed. Intersections, turning points and other singular points of the solution are investigated. It is shown that the position and the behavior of these particular points are definded by various kinds of zeros of the main system determinant. Some new aspects of the $r_s$ singularity are discussed. 
  Analytical formulas are presented describing a generic singularity inside the static spherically symmetric black holes in the SU(2) Einstein-Yang-Mills-Higgs theories with triplet or doublet Higgs field. The singularity is spacelike and exhibits a `power-low mass inflation'. Alternatively this asymptotic may be interpreted as a pointlike singularity with a non-vanishing shear in the Kantowski-Sachs anisotropic cosmology. 
  Paper withdrawn due to conceptual mistakes. A corrected version will soon be available at the gr-qc archive. 
  States of a quantum mechanical system are represented by rays in a complex Hilbert space. The space of rays has, naturally, the structure of a K\"ahler manifold. This leads to a geometrical formulation of the postulates of quantum mechanics which, although equivalent to the standard algebraic formulation, has a very different appearance. In particular, states are now represented by points of a symplectic manifold (which happens to have, in addition, a compatible Riemannian metric), observables are represented by certain real-valued functions on this space and the Schr\"odinger evolution is captured by the symplectic flow generated by a Hamiltonian function. There is thus a remarkable similarity with the standard symplectic formulation of classical mechanics. Features---such as uncertainties and state vector reductions---which are specific to quantum mechanics can also be formulated geometrically but now refer to the Riemannian metric---a structure which is absent in classical mechanics. The geometrical formulation sheds considerable light on a number of issues such as the second quantization procedure, the role of coherent states in semi-classical considerations and the WKB approximation. More importantly, it suggests generalizations of quantum mechanics. The simplest among these are equivalent to the dynamical generalizations that have appeared in the literature. The geometrical reformulation provides a unified framework to discuss these and to correct a misconception. Finally, it also suggests directions in which more radical generalizations may be found. 
  A comparative study between the metric and the teleparallel descriptions of gravitation is made for the case of a scalar field. In contrast to the current belief that only spin matter could detect the teleparallel geometry, scalar matter being able to feel the metric geometry only, we show that a scalar field is able not only to feel anyone of these geometries, but also to produce torsion. Furthermore, both descriptions are found to be completely equivalent, which means that in fact, besides coupling to curvature, a scalar field couples also to torsion. 
  The static vacuum plane spacetimes are considered, which have two non-trivial solutions: The Taub solution and the Rindler solution. Imposed reflection symmetry, we find that the source for the Taub solution does not satisfy any energy conditions, which is consistent with previous studies, while the source for the Rindler solution satisfies the weak and strong energy conditions (but not the dominant one). It is argued that the counterpart of the Einstein theory to the gravitational field of a massive Newtonian plane should be described by the Rindler solution, which represents also a uniform gravitational field. 
  The stability of the cosmological event horizons found recently by Gregory [Phys. Rev. D54, 4955 (1996)] for a class of non-static global cosmic strings is studied. It is shown that they are not stable to both test particles and physical perturbations. In particular, the back reaction of the perturbations of null dust fluids will turn them into spacetime singularities. The resulted singularities are strong in the sense that the distortion of test particles diverges logarithmically when these singular hypersurfaces are approaching. 
  Perturbations of rotating relativistic stars can be classified by their behavior under parity. For axial perturbations (r-modes), initial data with negative canonical energy is found with angular dependence $e^{im\phi}$ for all values of $m\geq 2$ and for arbitrarily slow rotation. This implies instability (or marginal stability) of such perturbations for rotating perfect fluids. This low $m$-instability is strikingly different from the instability to polar perturbations, which sets in first for large values of $m$. The timescale for the axial instability appears, for small angular velocity $\Omega$, to be proportional to a high power of $\Omega$. As in the case of polar modes, viscosity will again presumably enforce stability except for hot, rapidly rotating neutron stars. This work complements Andersson's numerical investigation of axial modes in slowly rotating stars. 
  The semiclassical collapse of a sphere of quantized dust is studied. A Born-Oppenheimer decomposition is performed for the wave function of the system and the semiclassical limit is considered for the gravitational part. The method of adiabatic invariants for time dependent Hamiltonians is then employed to find (approximate) solutions to the quantum dust equations of motions. This allows us to obtain corrections to the adiabatic approximation of the dust states associated with the time evolution of the metric. The diverse non-adiabatic corrections are generally associated with particle (dust) creation and related fluctuations. The back-reaction due to the dominant contribution to particle creation is estimated and seen to slow-down the collapse. 
  The first numerical study of axial (toroidal) pulsation modes of a slowly rotating relativistic star is presented. The calculation includes terms of first order in $\epsilon \equiv \Omega \sqrt{R^3/M}<<1$ ($R$ is the radius, $M$ is the mass and $\Omega$ is the rotation frequency of the star), and accounts for effects due to the coriolis force. Effects due to the centrifugal flattening of the star enter at order $\epsilon^2$ and are not included in the analysis. It is shown that increased rotation tends to decrease the damping times for prograde modes, while retrograde become longer lived. Specifically, we show that rotation affects the axial gravitational-wave $w$-modes in this way. We also present the first relativistic calculation of the so-called $r$-modes (analogous to Rossby waves in the Earth's oceans). These have frequencies of the same order of magnitude as the rotation frequency of the star. The presented results indicate that the $r$-modes are unstable due to the emission of gravitational radiation for \underline{all} rotating perfect fluid stars. This is interesting since the previously considered gravitational-wave instability associated with (for example) the $f$-mode of the star sets in at a critical rotation rate. Because they are unstable also for the slowest rotating stars the $r$-modes may well be of considerable astrophysical importance. 
  We consider the Wheeler-De Witt equation for canonical quantum gravity coupled to massless scalar field. After regularizing and renormalizing this equation, we find a one-parameter class of its solutions. 
  The various recent studies on the applications of the Born-Oppenheimer approach in a closed gravity matter system is examined. It is pointed out that the Born-Oppenheimer approach in the absence of an a priori time is likely to yield potentially new results. 
  In the present paper a global conformal invariant $Y$ of a closed initial data set is constructed. A spacelike hypersurface $\Sigma$ in a Lorentzian spacetime naturally inherits from the spacetime metric a differentiation ${\cal D}_e$, the so-called real Sen connection, which turns out to be determined completely by the initial data $h_{ab}$ and $\chi_{ab}$ induced on $\Sigma$, and coincides, in the case of vanishing second fundamental form $\chi_{ab}$, with the Levi-Civita covariant derivation $D_e$ of the induced metric $h_{ab}$. $Y$ is built from the real Sen connection ${\cal D}_e$ in the similar way as the standard Chern-Simons invariant is built from $D_e$. The number $Y$ is invariant with respect to changes of $h_{ab}$ and $\chi_{ab}$ corresponding to conformal rescalings of the spacetime metric. In contrast the quantity $Y$ built from the complex Ashtekar connection is not invariant in this sense. The critical points of our $Y$ are precisely the initial data sets which are locally imbeddable into conformal Minkowski space. 
  In the coming decade, gravitational waves will convert the study of general relativistic aspects of black holes and stars from a largely theoretical enterprise to a highly interactive, observational/theoretical one. For example, gravitational-wave observations should enable us to observationally map the spacetime geometries around quiescient black holes, study quantitatively the highly nonlinear vibrations of curved spacetime in black-hole collisions, probe the structures of neutron stars and their equation of state, search for exotic types of general relativistic objects such as boson stars, soliton stars, and naked singularities, and probe aspects of general relativity that have never yet been seen such as the gravitational fields of gravitons and the influence of gravitational-wave tails on radiation reaction. 
  The canonical theory of (N=1) supergravity, with a matrix representation for the gravitino covector-spinor, is applied to the Bianchi class A spatially homogeneous cosmologies. The full Lorentz constraint and its implications for the wave function of the universe are analyzed in detail. We found that in this model no physical states other than the trivial "rest frame" type occur. 
  Following a minisuperspace approach to the dynamics of a spherically symmetric shell, a reduced Lagrangian for the radial degree of freedom is derived directly from the Einstein-Hilbert action. The key feature of this new Lagrangian is its invariance under time reparametrization. Indeed, all classical and quantum dynamics is encoded in the Hamiltonian constraint that follows from that invariance. Thus, at the classical level, we show that the Hamiltonian constraint reproduces, in a simple gauge, Israel's matching condition which governs the evolution of the shell. In the quantum case, the vanishing of the Hamiltonian (in a weak sense), is interpreted as the Wheeler-DeWitt equation for the physical states, in analogy to the corresponding case in quantum cosmology. Using this equation, quantum tunneling through the classical barrier is then investigated in the WKB approximation, and the connection to vacuum decay is elucidated. 
  The near-field Lienard-Wiechert potential solution of a longitudinally oscillating electrical field produced by an oscillating charge is presented, and the results are compared to the R. P. Feynman multipole far-field solution. The results indicate that the phase speed of a longitudinally oscillating electrical field is much faster than the speed of light in the near field. A similar analysis is presented for a longitudinally oscillating gravitational field produced by a vibrating mass. The result also indicates that the phase speed of a longitudinally oscillating gravitational field is also much faster than the speed of light in the near-field. The possibility of measuring the group speed of a longitudinally oscillating electrical field and a longitudinally oscillating gravitational field, which is commonly thought to be equal to the speed of light, is now being considered. The basic idea is to amplitude-modulate the longitudinal vibration of a charge or a mass and to measure the resultant longitudinal vibration of a nearby charge or a mass due to electrical or gravitational interaction. The modulation signal can then be extracted using a diode detector and the group speed can then be determined from the oscillation frequency, the separation distance between the masses, and the measurement of the phase shift of the modulation signal. If the group speed is equal to the speed of light, then phase shifts on the order of 1 microdegree could be generated with a typical experimental set-up. An analysis using the classical definition of group velocity for a longitudinally oscillating electrical field is presented, and the results indicate that the group speed is also much faster that the speed of light in the near field, which should not be possible due to causality violation. 
  New, simple models of ``black hole interiors'', namely spherically symmetric solutions of the Einstein field equations in matter matching the Schwarzschild vacuum at spacelike hypersurfaces ``R<2M'' are constructed. The models satisfy the weak energy condition and their matter content is specified by an equation of state of the elastic type. 
  We introduce an effective Lagrangian which describes the classical and semiclassical dynamics of spherically symmetric, self-gravitating objects that may populate the Universe at large and small (Planck) scale. These include wormholes, black holes and inflationary bubbles. We speculate that such objects represent some possible modes of fluctuation in the primordial spacetime foam out of which our universe was born. Several results obtained by different methods are encompassed and reinterpreted by our effective approach. As an example, we discuss: i) the gravitational nucleation coefficient for a pair of Minkowski bubbles, and ii) the nucleation coefficient of an inflationary vacuum bubble in a Minkowski background 
  In this short note we briefly review some recent mathematical results relevant to the classical Regge Calculus evolution problem. 
  The use of Lax pair tensors as a unifying framework for Killing tensors of arbitrary rank is discussed. Some properties of the tensorial Lax pair formulation are stated. A mechanical system with a well-known Lax representation -- the three-particle open Toda lattice -- is geometrized by a suitable canonical transformation. In this way the Toda lattice is realized as the geodesic system of a certain Riemannian geometry. By using different canonical transformations we obtain two inequivalent geometries which both represent the original system. Adding a timelike dimension gives four-dimensional spacetimes which admit two Killing vector fields and are completely integrable. 
  The derivation of the transformations between inertial frames made by Mansouri and Sexl is generalised to three dimensions for an arbitrary direction of the velocity. Assuming lenght contraction and time dilation to have their relativistic values, a set of transformations kinematically equivalent to special relativity is obtained. The ``clock hypothesis'' allows the derivation to be extended to accelerated systems. A theory of inertial transformations maintaining an absolute simultaneity is shown to be the only one logically consistent with accelerated movements. Algebraic properties of these transformations are discussed. Keywords: special relativity, synchronization, one-way velocity of light, ether, clock hypothesis. 
  The conventional nature of synchronisation is discussed in inertial frames, where it is found that theories using different synchronisations are experimentally equivalent to special relativity. In contrary, in accelerated systems only a theory maintaining an absolute simultaneity is consistent with the natural behaviour of clocks. The principle of equivalence is discussed, and it is found that any synchronisation can be used locally in a freely falling frame. Whatever the choosen synchronisation, the first derivatives of the metric tensor disapear and a geodesic is locally a straight line. But it is shown that only a synchronisation maintaining an absolute simultaneity allows to define time consistently on circular orbits of a Schwarzschild metric. Key words: special and general relativity, synchronisation, one-way velocity of light, ether, principle of equivalence. 
  We study an $N + 1$ dimensional generalization of the Schwarzschild black hole from the quantum mechanical viewpoint. It is shown that the mass loss rate of this higher dimensional black hole due to the black hole radiation is proportional to $1 \over r^{N-1}$ where $r$ is the radial coordinate. This fact implies that except in four dimensions the quantum formalism developed in this letter gives a different result with respect to the mass loss rate from the semiclassical formalism where the Stefan-Bolzmann formula for a blackbody radiation is used. As shown previously in the study of the four dimensional Schwarzschild metric, the wave function in this case has also quite curious features that it is singular owing to strong quantum fluctuation of the gravitational field at the singularity while completely regular in the other regions of the spacetime. 
  We solve a continuing controversy when dealing with density fluctuations in open Friedman-Robertson-Walker universes, on the physical relevance of a class of exponential modes. We show explicitly and rigorously that these modes enter the expansion of quantum fields. In the maximally symmetric de Sitter case, encountered in inflationary models, they are excited for fields with mass below a critical value. They are seen to be responsible for the breaking of the de Sitter symmetry for a massless field. We provide an exact calculation of the power spectrum for any mass. Our method is free of the divergences that appear in earlier treatments. We extend the construction to a generic open FRW universe. 
  General relativity postulates that the gravity field is defined on a Riemannian manifold. The field equations are $R^\mu_\nu = 0$ i.e. Ricci's curvature tensor vanishes. The field equations have to be augmented by natural physical requirements like orientability, time orientability and existence of a spinorial structure. Moreover, it is impossible to define the energy of the gravity field only by the metric tensor. We suggest to impose an additional structure and consider parallelizable manifold i.e. manifolds for which a smooth field of frames exists. The derivation of the field equations is by an action principle. A Lagrangian, which is quadratic in the differentials, is defined. The minimum of the action is achieved at a $16 \times 16$ second order quasi linear system. It is of Laplacian type. The system admits a unique exact solution for a centrally symmetric, static and assimptotically flat field. The resulting metric is the celebrated Rosen metric, which is very close to the Schwarzchild metric. The two are intrinsicly different since the scalar curvature of the Schuarzschild metric is zero, in contrast to Rosen's which is positive. The suggested field admits black holes which are briefly discussed. 
  Many authors - beginning with Bekenstein - have suggested that the energy levels E_n of a quantized isolated Schwarzschild black hole have the form E_n = sigma sqrt{n} E_P, n=1,2,..., sigma =O(1), with degeneracies g^n. In the present paper properties of a system with such a spectrum, considered as a quantum canonical ensemble, are discussed: Its canonical partition function Z(g,beta=1/kT), defined as a series for g<1, obeys the 1-dimensional heat equation. It may be extended to values g>1 by means of an integral representation which reveals a cut of Z(g,beta) in the complex g-plane from g=1 to infinity. Approaching the cut from above yields a real and an imaginary part of Z. Very surprisingly, it is the (explicitly known) imaginary part which gives the expected thermodynamical properties of Schwarzschild black holes: Identifying the internal energy U with the rest energy Mc^2 requires beta to have the value (in natural units) beta = 2M(lng/sigma^2)[1+O(1/M^2)], (4pi sigma^2=lng gives Hawking's beta_H), and yields the entropy S=[lng/(4pi sigma^2)] A/4 + O(lnA), where A is the area of the horizon. 
  A new link between tetrahedra and the group SU(2) is pointed out: by associating to each face of a tetrahedron an irreducible unitary SU(2) representation and by imposing that the faces close, the concept of quantum tetrahedron is seen to emerge. The Hilbert space of the quantum tetrahedron is introduced and it is shown that, due to an uncertainty relation, the ``geometry of the tetrahedron'' exists only in the sense of ``mean geometry''. A kinematical model of quantum gauge theory is also proposed, which shares the advantages of the Loop Representation approach in handling in a simple way gauge- and diff-invariances at a quantum level, but is completely combinatorial. The concept of quantum tetrahedron finds a natural application in this model, giving a possible intepretation of SU(2) spin networks in terms of geometrical objects. 
  Global U(1) strings with cylindrical symmetry are studied in anti-de Sitter spacetime. According as the magnitude of negative cosmological constant, they form regular global cosmic strings, extremal black cosmic strings and charged black cosmic strings, but no curvature singularity is involved. The relationship between the topological charge of a neutral global string and the black hole charge is clarified by duality transformation. Physical relevance as straight string is briefly discussed. 
  Lecture notes for a 'Part III' course 'Black Holes' given in DAMTP, Cambridge. The course covers some of the developments in Black Hole physics of the 1960s and 1970s. 
  The internal thermal noise in LIGO's test masses is analyzed by a new technique, a direct application of the Fluctuation-Dissipation Theorem to LIGO's readout observable, $x(t)=$(longitudinal position of test-mass face, weighted by laser beam's Gaussian profile). Previous analyses, which relied on a normal-mode decomposition of the test-mass motion, were valid only if the dissipation is uniformally distributed over the test-mass interior, and they converged reliably to a final answer only when the beam size was a non-negligible fraction of the test-mass cross section. This paper's direct analysis, by contrast, can handle inhomogeneous dissipation and arbitrary beam sizes. In the domain of validity of the previous analysis, the two methods give the same answer for $S_x(f)$, the spectral density of thermal noise, to within expected accuracy. The new analysis predicts that thermal noise due to dissipation concentrated in the test mass's front face (e.g. due to mirror coating) scales as $1/r_0^2$, by contrast with homogeneous dissipation, which scales as $1/r_0$ ($r_0$ is the beam radius); so surface dissipation could become significant for small beam sizes. 
  Trajectories of charged particle in combined poloidal, toroidal magnetic field and rotation-induced unipolar electric field superposed in Schwarzschild background geometry have been investigated extensively in the context of accreting black holes. The main purpose of the paper is to obtain a reasonably well insight on the effect of spacetime curvature to the electromagnetic field surrounding black holes. The coupled equations of motion have been solved numerically and the results have been compared with that for flat spacetime. It is found that the toroidal magnetic field dominates the induced electric field in determining the motion of charged particles in curved spacetime. The combined electromagnetic field repels a charged particle from the vicinity of a compact massive object and deconfines the particle from its orbit. In the absence of toroidal magnetic field the particle is trapped in a closed orbit. The major role of gravitation is to reduce the radius of gyration significantly while the electric field provides an additional force perpendicular to the circular orbit. Although the effect of inertial frame dragging and the effect of magnetospheric plasma have been neglected, the results provide a reasonably well qualitative picture of the important role played by gravitation in modifying the electromagnetic field near accreting black holes and hence the results have potentially important implications on the dynamics of the fluid and the radiation spectrum associated with accreting black holes. 
  Nonextreme black hole in a cavity can achieve the extreme state with a zero surface gravity at a finite temperature on a boundary, the proper distance between the boundary and the horizon being finite. The classical geometry in this state is found explicitly for four-dimensional spherically-symmetrical and 2+1 rotating holes. In the first case the limiting geometry depends only on one scale factor and the whole Euclidean manifold is described by the Bertotti-Robinson spacetime. The general structure of a metric in the limit under consideration is also found with quantum corrections taken into account. Its angular part represents two-sphere of a constant radius. In all cases the Lorentzian counterparts of the metrics are free from singularities. 
  The unexpected dynamic shift of the center of mass for a rotating hemisphere is shown to produce the general relativistic dipole field in the macroscopic scale. This prompts us the question of what might be its cosmological implications. The uniformly rotating sphere has the effect of the latitude dependent mass density distribution as reported by Bass and Pirani which is the cause of the `induced centrifugal force' in the Thirring's geodesic equation near the center of the rotating spherical mass shell. On the other hand, one would expect the constant acceleration of the mass components may cause a general relativistic gravitational field. The component-wise accumulation of this effect has been shown to appear as the non zero gravitational dipole moment in a rotating hemispherical mass shell. The present report discusses this non-Newtonian force experienced by a gravitational dipole moment placed at the center of the two mass pole model universe and its relevance to the observed anomalous red shift from far away galaxies. 
  We show that the radial Teukolsky equation (in the frequency domain) with sources that extend to infinity has well-behaved solutions. To prove that, we follow Poisson approach to regularize the non-rotating hole, and extend it to the rotating case. To do so we use the Chandrasekhar transformation among the Teukolsky and Regge-Wheeler-like equations, and express the integrals over the source in terms of solutions to the homogeneous Regge-Wheeler-like equation, to finally regularize the resulting integral. We then discuss the applicability of these results. 
  We investigate cosmological density perturbations in a covariant and gauge-invariant formalism, incorporating relativistic causal thermodynamics to give a self-consistent description. The gradient of density inhomogeneities splits covariantly into a scalar part, equivalent to the usual density perturbations, a rotational vector part that is determined by the vorticity, and a tensor part that describes the shape. We give the evolution equations for these parts in the general dissipative case. Causal thermodynamics gives evolution equations for viscous stress and heat flux, which are coupled to the density perturbation equation and to the entropy and temperature perturbation equations. We give the full coupled system in the general dissipative case, and simplify the system in certain cases. A companion paper uses the general formalism to analyze damping of density perturbations before last scattering. 
  Dilaton black holes with a pure electric charge are considered in a framework of a grand canonical ensemble near the extreme state. It is shown that there exists such a subset of boundary data that the Hawking temperature smoothly goes to zero to an infinite value of a horizon radius but the horizon area and entropy are finite and differ from zero. In string theory the existence of a horizon in the extreme limit is due to the finiteness of a system only. 
  The barotropic ideal fluid with step and delta-function discontinuities coupled to Einstein's gravity is studied. The discontinuities represent star surfaces and thin shells; only non-intersecting discontinuity hypersurfaces are considered. No symmetry (like eg. the spherical symmetry) is assumed. The symplectic structure as well as the Lagrangian and the Hamiltonian variational principles for the system are written down. The dynamics is described completely by the fluid variables and the metric on the fixed background manifold. The Lagrangian and the Hamiltonian are given in two forms: the volume form, which is identical to that corresponding to the smooth system, but employs distributions, and the surface form, which is a sum of volume and surface integrals and employs only smooth variables. The surface form is completely four- or three-covariant (unlike the volume form). The spacelike surfaces of time foliations can have a cusp at the surface of discontinuity. Geometrical meaning of the surface terms in the Hamiltonian is given. Some of the constraint functions that result from the shell Hamiltonian cannot be smeared so as to become differentiable functions on the (unconstrained) phase space. Generalization of the formulas to more general fluid is straifgtforward. 
  This is the third and last part of a series of 3 papers. Using the same method and the same coordinates as in parts 1 and 2, rotating dust solutions of Einstein's equations are investigated that possess 3-dimensional symmetry groups, under the assumption that each of the Killing vectors is linearly independent of velocity $u^{\alpha}$ and rotation $w^{\alpha}$ at every point of the spacetime region under consideration. The Killing fields are found and the Killing equations are solved for the components of the metric tensor in every case that arises. No progress was made with the Einstein equations in any of the cases, and no previously known solutions were identified. A brief overview of literature on solutions with rotating sources is given. 
  Starting from the exact non-linear description of matter and radiation, a fully covariant and gauge-invariant formula for the observed temperature anisotropy of the cosmic microwave background (CBR) radiation, expressed in terms of the electric ($E_{ab}$) and magnetic ($H_{ab}$) parts of the Weyl tensor, is obtained by integrating photon geodesics from last scattering to the point of observation today. This improves and extends earlier work by Russ et al where a similar formula was obtained by taking first order variations of the redshift. In the case of scalar (density) perturbations, $E_{ab}$ is related to the harmonic components of the gravitational potential $\Phi_k$ and the usual dominant Sachs-Wolfe contribution $\delta T_R/\bar{T}_R\sim\Phi_k$ to the temperature anisotropy is recovered, together with contributions due to the time variation of the potential (Rees-Sciama effect), entropy and velocity perturbations at last scattering and a pressure suppression term important in low density universes. We also explicitly demonstrate the validity of assuming that the perturbations are adiabatic at decoupling and show that if the surface of last scattering is correctly placed and the background universe model is taken to be a flat dust dominated Friedmann-Robertson-Walker model (FRW), then the large scale temperature anisotropy can be interpreted as being due to the motion of the matter relative to the surface of constant temperature which defines the surface of last scattering on those scales. 
  We show that the structure of the Liouville action on a two dimensional Regge surface of the topology of the sphere and of the torus is determined by the invariance under the transformations induced by the conformal Killing vector fields and under modular transformations. 
  The expectation value of the stress-energy tensor of a free conformally invariant scalar field is computed in a two-dimensional reduction of the Alcubierre ``warp drive'' spacetime. The stress-energy is found to diverge if the apparent velocity of the spaceship exceeds the speed of light. If such behavior occurs in four dimensions, then it appears implausible that ``warp drive'' behavior in a spacetime could be engineered, even by an arbitrarily advanced civilization. 
  A simple Lorentzian vacuum wormhole solution of Brans-Dicke gravitation is presented and analysed. It is shown that such solution holds for both, the Brans-Dicke theory endowed with torsion (for a value of the coupling parameter $\omega > 1/2$) and for the vacuum -no torsion- case (for $\omega < -2$). 
  Constraints on the geometry of a static spherically symmetric black hole are obtained by requiring the spacetime curvature to be analytic at the event horizon. For a zero temperature black hole further constraints are obtained by also requiring that the semiclassical trace equation be satisfied when conformally invariant fields are present. It is found that zero temperature black holes whose sizes lie within a certain range do not exist. The range depends on the numbers and types of conformally invariant quantized fields that are present. 
  We present spherically symmetric black hole solutions for Einstein gravity coupled to anisotropic matter. We show that these black holes have arbitrarily short hair, and argue for stability by showing that they can arise from dynamical collapse. We also show that a recent `no short hair' theorem does not apply to these solutions. 
  Motion sensing in a spherical GW detector requires a multiple set of transducers attached to its surface at suitable locations. If transducers are of the resonant type then their motion couples to that of the sphere and the joint dynamics of the system has to be properly understood before reliable conclusions can be drawn from its readout. In this paper we address the problem of the coupled motion of a sphere and a set of resonators attached to it at arbitrary points. A remarkably elegant and powerful scheme emerges from the general equations which shows how coupling takes place as a power series in the small coupling constant ratio m(resonator)/M(sphere). We reasses in the new light the response of the highly symmetric TIGA and also present a new proposal (called PHC) which has less symmetry but which is based on 5 rather than 6 transducers per quadrupole mode sensed. We finally assess how the system charac- teristics are affected by slight departures from ideality, and find it to be quite robust to them. In particular we recover with full accuracy the reportedly measured frequencies of the LSU prototype TIGA. 
  For Schwarzschild space-time, distributional expressions of energy-momentum densities and of scalar concomitants of the curvature tensors are examined for a class of coordinate systems which includes those of the Schwarzschild and of Kerr-Schild types as special cases. The energy-momentum density $\tilde T_\mu^{\nu}(x)$ of the gravitational source and the gravitational energy-momentum pseudo-tensor density $\tilde t_\mu^{\nu}$ have the expressions $\tilde T_\mu^{\nu}(x) =-Mc^2\delta_\mu^0\delta_0^{\nu} \delta^{(3)}x)$ and $\tilde t_\mu^{\nu}=0$, respectively. In expressions of the curvature squares for this class of coordinate systems, there are terms like $\delta^{(3)}(x)/r^3$ and $[\delta^{(3)}(x)}]^2$, as well as other terms, which are singular at $x=0$. It is pointed out that the well-known expression $R^{\rho\sigma\mu\nu}({}) R_{\rho\sigma\mu\nu}({})$ $=48G^{2}M^{2}/c^{4}r^{6}$ is not correct, if we define $1/r^6 = \lim_{\epsilon\to 0}1/(r^2+\epsilon^2)^3$.} 
  The set of constraints under which the eigenvalues of the Dirac operator can play the role of the dynamical variables for Euclidean supergravity is derived. These constraints arise when the gauge invariance of the eigenvalues of the Dirac operator is imposed. They impose conditions which restrict the eigenspinors of the Dirac operator. 
  There is a growing literature on dyonic black holes as they appear in string theory. Here we examine the correspondence limit of a dyonic black hole which is not supersymmetric. Assuming the existence of a dyon with non-supersymmetric Kerr-Schild structure, we calculate its gravitational and electromagnetic fields and compute its mass and angular momentum to obtain a modified B.P.S. relation. The contributionn of the angular momentum to the mass appears in the condition for the appearance of a horizon. 
  Constants of motion are calculated for 2+1 dimensional gravity with topology R x T^2 and negative cosmological constant. Certain linear combinations of them satisfy the anti - de Sitter algebra so(2,2) in either ADM or holonomy variables. Quantisation is straightforward in terms of the holonomy parameters. On inclusion of the Hamiltonian three new global constants are derived and the quantum algebra extends to that of the conformal algebra so(2,3). The modular group appears as a discrete subgroup of the conformal group. Its quantum action is generated by these conserved quantities. 
  Toy models for the Hubble rate or the scalar field potential have been used to analyze the amplification of scalar perturbations through a smooth transition from inflation to the radiation era. We use a Hubble rate that arises consistently from a decaying vacuum cosmology, which evolves smoothly from nearly de Sitter inflation to radiation domination. We find exact solutions for super-horizon perturbations (scalar and tensor), and for sub-horizon perturbations in the vacuum- and radiation-dominated eras. The standard conserved quantity for super-horizon scalar perturbations is exactly constant for growing modes, and zero for the decaying modes. 
  The general properties of the gravity wave backgrounds of cosmological origin are reviewed and briefly discussed, with emphasis on the relic background expected from an early pre-big bang phase typical of string cosmology models. 
  We consider a perfectly homogeneous and isotropic universe which undergoes a sudden phase transition. If the transition produces topological defects, which we assume, perturbations in the geometry and the cosmic fluid also suddenly appear. We apply the standard general relativistic junction conditions to match the pre- and post- transition eras and thus set the initial conditions for the perturbations. We solve their evolution equations analytically in the case when the defects act as a coherent source and their density scales like the background density. We show that isocurvature as well as adiabatic perturbations are created, in a ratio which is independent of the detailed properties of the defects. We compare our result to the initial conditions currently used in the literature and show how the cosmic fluid naturally "compensates" for the presence of the defects. 
  A large variety of spacetimes---including the BTZ black holes---can be obtained by identifying points in 2+1 dimensional anti-de Sitter space by means of a discrete group of isometries. We consider all such spacetimes that can be obtained under a restriction to time symmetric initial data and one asymptotic region only. The resulting spacetimes are non-eternal black holes with collapsing wormhole topologies. Our approach is geometrical, and we discuss in detail: The allowed topologies, the shape of the event horizons, topological censorship and trapped curves. 
  The time-dependent, spherically symmetric, Wyman sector of the Unified Field Theory is shown to be equivalent to a self-gravitating scalar field with a positive-definite, repulsive self-interaction potential. A homothetic symmetry is imposed on the fundamental tensor, and the resulting autonomous system is numerically integrated. Near the critical point (between the collapsing and non-collapsing spacetimes) the system displays an approximately periodic alternation between collapsing and dispersive epochs. 
  A discussion of relativistic quantum-geometric mechanics on phase space and its generalisation to the propagation of free, massive, quantum-geometric scalar fields on curved spacetimes is given. It is shown that in an arbitrary coordinate system and frame of reference in a flat spacetime, the resulting propagator is necessarily the same as derived in the standard Minkowski coordinates up to a Lorentz boost acting on the momentum content of the field, which is therefore seen to play the role of Bogolubov transformations in this formalism. These results are explicitly demonstrated in the context of a Milne universe. 
  Einstein's general relativity with both metric and vielbein treated as independent fields is considered, demonstrating the existence of a consistent variational principle and deriving a Hamiltonian formalism that treats the spatial metric and spatial vielbein as canonical coordinates. This results in a Hamiltonian in standard form that consists of Hamiltonian and momentum constraints as well as constraints that generate spatial frame transformations---all appearing as primary, first class constraints on phase space. The formalism encompasses the standard coordinate frame and vielbein approaches to general relativity, and the constraint algebra derived herein reproduces known results in either limit. 
  The gravitational field of a global monopole in the context of Brans-Dicke theory of gravity is investigated. The space-time and the scalar field generated by the monopole are obtained by solving the field equations in the weak field approximation. A comparison is made with the corresponding results predicted by General Relativity. 
  Complete tensor-scalar and hydrodynamic equations are presented and integrated, for a self-gravitating perfect fluid. The initial conditions describe unstable-equilibrium neutron star configuration, with a polytropic equation of state. They are necessary in order to follow the gravitational collapse (including full hydrodynamics) of this star toward a black hole and to study the resulting scalar gravitational wave. The amplitude of this wave, as well as the radiated energy dramatically increase above some critical value of the parameter of the coupling function, due to the spontaneous scalarization, an effect not present in Brans-Dicke theory. In most cases, the pressure of the collapsing fluid does not have a significant impact on the resulting signal. These kind of sources are not likely to be observed by future laser interferometric detectors (such as VIRGO or LIGO) of gravitational waves, if they are located at more than a few 100 kpc. However, spontaneous scalarization could be constrained if such a gravitational collapse is detected by its quadrupolar gravitational signal, since this latter is quite lower than the monopolar one. 
  We generalize the action found by 't Hooft, which describes the gravitational interaction between ingoing and outgoing particles in the neighbourhood of a black hole. The effect of this back-reaction is that of a shock wave, and it provides a mechanism for recovering information about the momentum of the incoming particles. The new action also describes particles with transverse momenta and takes into account the transverse curvature of the hole, and has the form of a string theory action. Apart from the Polyakov term found by 't Hooft, we also find an antisymmetric tensor, which is here related to the momentum of the particles. At the quantum level, the identification between position and momentum operators leads to four non-commuting coordinates. A certain relation to M(atrix) theory is proposed. 
  We consider the horizon problem in a homogeneous but anisotropic universe (Bianchi type I). We show that the problem cannot be solved if (1) the matter obeys the strong energy condition with the positive energy density and (2) the Einstein equations hold. The strong energy condition is violated during cosmological inflation. 
  Using M{\o}ller's energy complex, we obtain the energy distributions of GHS solution and dyonic dilaton black hole solution in the dilaton gravity theory. It is confirmed that the M{\o}ller's energy complex is indeed a 3-scalar under purely spatial transformation in these energy distributions. Some interested properties of the energy distribution of dyonic black hole are disscussed. 
  We discuss the static axially symmetric regular solutions, obtained recently in Einstein-Yang-Mills and Einstein-Yang-Mills-dilaton theory [1]. These asymptotically flat solutions are characterized by the winding number $n>1$ and the node number $k$ of the purely magnetic gauge field. The well-known spherically symmetric solutions have winding number $n=1$. The axially symmetric solutions satisfy the same relations between the metric and the dilaton field as their spherically symmetric counterparts. Exhibiting a strong peak along the $\rho$-axis, the energy density of the matter fields of the axially symmetric solutions has a torus-like shape. For fixed winding number $n$ with increasing node number $k$ the solutions form sequences. The sequences of magnetically neutral non-abelian axially symmetric regular solutions with winding number $n$ tend to magnetically charged abelian spherically symmetric limiting solutions, corresponding to ``extremal'' Einstein-Maxwell-dilaton solutions for finite values of $\gamma$ and to extremal Reissner-Nordstr\o m solutions for $\gamma=0$, with $n$ units of magnetic charge. 
  The spatial gradient expansion of the generating functional was recently developed by Parry, Salopek, and Stewart to solve the Hamiltonian constraint in Einstein-Hamilton-Jacobi theory for gravitationally interacting dust and scalar fields. This expansion is used here to derive an order-by-order solution of the Hamiltonian constraint for gravitationally interacting electromagnetic and scalar fields. A conformal transformation and functional integral are used to derive the generating functional up to the terms fourth order in spatial gradients. The perturbations of a flat Friedmann-Robertson-Walker cosmology with a scalar field, up to second order in spatial gradients, are given. The application of this formalism is demonstrated in the specific example of an exponential potential. 
  The linear cosmological perturbation theory of an almost homogeneous and isotropic perfect fluid universe is reconsidered and formally simplified by introducing new covariant and gauge-invariant variables with physical interpretations on hypersurfaces of constant expansion, constant curvature or constant energy density. The existence of conserved perturbation quantities on scales larger than the Hubble scale is discussed. The quantity which is conserved on large scales in a flat background universe may be expressed in terms of the fractional, spatial gradient of the energy density on constant expansion hypersurfaces or, alternatively, with the help of expansion or curvature perturbation variables on hypersurfaces of constant energy density. For nonvanishing background curvature the perturbation dynamics is most suitably described in terms of energy density perturbations on hypersurfaces of constant curvature. 
  We show that 4--dimensional conformal field theory is most naturally formulated on Kulkarni 4--folds, i. e. real 4--folds endowed with an integrable quaternionic structure. This leads to a formalism that parallels very closely that of 2--dimensional conformal field theory on Riemann surfaces. In this framework, the notion of Fueter analyticity, the quaternionic analogue of complex analyticity, plays an essential role. Conformal fields appear as sections of appropriate either harmonic real or Fueter holomorphic quaternionic line bundles. In the free case, the field equations are statements of either harmonicity or Fueter holomorphicity of the relevant conformal fields. We obtain compact quaternionic expressions of such basic objects as the energy-momentum tensor and the gauge currents for some basic models in terms of Kulkarni geometry. We also find a concise expression of the conformal anomaly and a quaternionic 4--dimensional analogue of the Schwarzian derivative describing the covariance of the quantum energy-momentum tensor. Finally, we analyse the operator product expansions of free fields. 
  Exact solutions of Einstein's vacuum equations are considered which describe gravitational waves with distinct wavefronts. A family of such solutions presented recently in which the wavefronts have various geometries and which propagate into a number of physically significant backgrounds is here related to an integral representation which is a generalisation of the Rosen pulse solution for cylindrical waves. A nondiagonal solution is also constructed which is a generalisation of the Rosen pulse, being a cylindrical pulse wave with two states of polarization propagating into a Minkowski background. The solution is given in a complete and explicit form. A further generalisation to include electromagnetic waves with a distinct wavefront of the same type is also discussed. 
  We review the problem of finding an apparent horizon in Cauchy data (Sigma, g_ab, K_ab) in three space dimensions without symmetries. We describe a family of algorithms which includes the pseudo-spectral apparent horizon finder of Nakamura et al. and the curvature flow method proposed by Tod as special cases. We suggest that other algorithms in the family may combine the speed of the former with the robustness of the latter. A numerical implementation for Cauchy data given on a grid in Cartesian coordinates is described, and tested on Brill-Lindquist and Kerr initial data. The new algorithm appears faster and more robust than previous ones. 
  Chaos in Robertson-Walker cosmological models where gravity is coupled to one or more scalar fields has been studied by a few authors, mostly using numerical simulations. In this paper we begin a systematic study of the analytical aspect. We consider one conformally coupled scalar field and, using the fact that the model is integrable when the field is massless, we show in detail how homoclinic chaos arises for nonzero masses using a perturbative method. 
  The renormalised vacuum expectation values of massless fermion (conventionally call it neutrino) stress-energy tensor are calculated in the static spherically-symmetrical wormhole topology. Consider the case when the derivatives of metric tensor over radial distance are sufficiently small (in the scale of radius) to justify studying a few first orders of quasiclassical (WKB) expansion over derivatives. Then we find that violation of the averaged weak energy condition takes place irrespectively of the detailed form of metric. This is a necessary condition for the neutrino vacuum to be able to support the wormhole geometry. In this respect, neutrino vacuum behaves like electromagnetic one and differs from the conformal scalar vacuum which does not seem to violate energy conditions for slowly varied metric automatically, but requires self-consistent wormhole solution for this. 
  Following Finkelstein and Misner, kinks are non-trivial field configurations of a field theory, and different kink-numbers correspond to different disconnected components of the space of allowed field configurations for a given topology of the base manifold. In a theory of gravity, non-vanishing kink-numbers are associated to a twisted causal structure. In two dimensions this means, more specifically, that the light-cone tilts around (non-trivially) when going along a non-contractible non-selfintersecting loop on spacetime. One purpose of this paper is to construct the maximal extensions of kink spacetimes using Penrose diagrams. This will yield surprising insights into their geometry but also allow us to give generalizations of some well-known examples like the bare kink and the Misner torus. However, even for an arbitrary 2D metric with a Killing field we can construct continuous one-parameter families of inequivalent kinks. This result has already interesting implications in the flat or deSitter case, but it applies e.g. also to generalized dilaton gravity solutions. Finally, several coordinate systems for these newly obtained kinks are discussed. 
  In present paper we construct the classical and minisuperspace quantum models of an extended charged particle. The modelling is based on the radiation fluid singular hypersurface filled with physical vacuum. We demonstrate that both at classical and quantum levels such a model can have equilibrium states at the radius equal to the classical radius of a charged particle. In the cosmological context the model could be considered also as the primary stationary state, having the huge internal energy being nonobservable for an external observer, from which the Universe was born by virtue of the quantum tunnelling. 
  The conservative model of a black hole is advanced. The model incorporates conservation laws such as those of baryon and lepton numbers, which lifts the information loss paradox. A scenario of black hole evaporation is considered.    Keywords: entropy, emission, radiation, universe, chemical potential 
  We study global vortices coupled to (2+1) dimensional gravity with negative cosmological constant. We found nonsingular vortex solutions in $\phi^4$-theory with a broken U(1) symmetry, of which the spacetimes do not involve physical curvature singularity. When the magnitude of negative cosmological constant is larger than a critical value at a given symmetry breaking scale, the spacetime structure is a regular hyperbola, however it becomes a charged black hole when the magnitude of cosmological constant is less than the critical value. We explain through duality transformation the reason why static global vortex which is electrically neutral forms black hole with electric charge. Under the present experimental bound of the cosmological constant, implications on cosmology as a straight black cosmic string is also discussed in comparison with global U(1) cosmic string in the spacetime of the zero cosmological constant. 
  It is shown that the non-Abelian black hole solutions have stationary generalizations which are parameterized by their angular momentum and electric Yang-Mills charge. In particular, there exists a non-static class of stationary black holes with vanishing angular momentum. It is also argued that the particle-like Bartnik-McKinnon solutions admit slowly rotating, globally regular excitations. In agreement with the non-Abelian version of the staticity theorem, these non-static soliton excitations carry electric charge, although their non-rotating limit is neutral. 
  We give a general treatment of the spontaneous excitation rates and the non-relativistic Lamb shift of constantly accelerated multi-level atoms as a model for multi-level detectors. Using a covariant formulation of the dipole coupling between the atom and the electromagnetic field we show that new Raman-like transitions can be induced by the acceleration. Under certain conditions these transitions can lead to stable ground and excited states which are not affected by the non inertial motion. The magnitude of the Unruh effect is not altered by multi-level effects. Both the spontaneous excitation rates and the Lamb shift are not within the range of measurability. 
  The most important features of the proposed spherical gravitational wave detectors are closely linked with their symmetry. Hollow spheres share this property with solid ones, considered in the literature so far, and constitute an interesting alternative for the realization of an omnidirectional gravitational wave detector. In this paper we address the problem of how a hollow elastic sphere interacts with an incoming gravitational wave and find an analytical solution for its normal mode spectrum and response, as well as for its energy absorption cross sections. It appears that this shape can be designed having relatively low resonance frequencies (about 200 Hz) yet keeping a large cross section, so its frequency range overlaps with the projected large interferometers. We also apply the obtained results to discuss the performance of a hollow sphere as a detector for a variety of gravitational wave signals. 
  We present exact solutions in Einstein-Yang-Mills-Dirac theories with gauge groups SU(2) and SU(4) in Robertson-Walker space-time $R \times S^3 $, which are symmetric under the action of the group SO(4) of spatial rotations. Our approach is based on the dimensional reduction method for gauge and gravitational fields and relates symmetric solutions in EYMD theory to certain solutions of an effective dynamical system.   We interpret our solutions as cosmological solutions with an oscillating Yang-Mills field passing between topologically distinct vacua. The explicit form of the solution for spinor field shows that its energy changes the sign during the evolution of the Yang-Mills field from one vacuum to the other, which can be considered as production or annihilation of fermions.   Among the obtained solutions there is also a static sphaleron-like solution, which is a cosmological analogue of the first Bartnik-McKinnon solution in the presence of fermions. 
  In this thesis we are interested in the study of the gravitational properties of quantum bosonic strings. We start by computing the quantum energy-momentum tensor ${\hat T}^{\mu\nu}(x)$ for strings in Minkowski space-time. We perform the calculation of its expectation value for different physical string states both for open and closed bosonic strings. The states we consider are described by normalizable wave-packets in the centre of mass coordinates. Amongst our results, we find in particular that ${\hat T}^{\mu\nu}(x)$ becomes a non-local operator at the quantum level, its position appears to be smeared out by quantum fluctuations. We find that the expectation value acquires a non-zero value for both massive and massless string states. After computing $<{\hat T}^{\mu\nu}(x)>$ we proceed to calculate the gravitational field due to a quantum massless bosonic string in the framework of a weak-field approximation to Einstein's equations. We obtain a multipole expansion for the weak-field metric $h^{\mu\nu}(x)$ and present its gravitational properties, including the gravitational radiation produced by such a string. Our results are then compared to those found for classical (cosmic) strings. 
  These lectures deal with selected aspects of quantum field theory in curved spacetime including the following topics: (1) Quantization of fields on a curved background, particle creation by gravitational fields, particle creation in an expanding universe; moving mirror radiation. (2) The Hawking effect - particle creation by black holes. (3) Ultraviolet and infrared divergences, renormalization of the expectation value of the stress tensor; global symmetry breaking in curved spacetime. (4) Negative energy in quantum field theory, its gravitational effects, and inequalities which limit negative energy densities and fluxes. (5) The semiclassical theory of gravity and its limitations, breakdown of this theory due to metric fluctuations, lightcone fluctuations. 
  We consider a self-gravitating string generated by a global vortex solution in general relativity. We investigate the Einstein and field equations of a global vortex in the region of its central line and at a distance from the centre of the order of the inverse of its Higgs boson mass. By combining the two we establish by a limiting process of large Higgs mass the dynamics of a self-gravitating global string. Under our assumptions the presence of gravitation restricts the world sheet of the global string to be totally geodesic. 
  We present a new method for treating the inner Cauchy boundary of a black hole spacetime by matching to a characteristic evolution. We discuss the advantages and disadvantages of such a scheme relative to Cauchy-only approaches. A prototype code, for the spherically symmetric collapse of a self-gravitating scalar field, shows that matching performs at least as well as other approaches to handling the inner boundary. 
  The paper deals with the estimation of parameters of gravitational waves from coalescing binaries. It explains the discrepancy between the Monte carlo simulations and the covariance matrix. We include the post-Newtonian as well as the Newtonian case. 
  Three variational vector equations are derived for the extended particle-field object located on the light cone. Point sources are excluded from the pure field equations and all physical magnitudes are free from divergences. Accepting 3D intersections of 4D cone-charges, vector electrogravity explains all observed phenomena under common flat 3D and 1D intervals. External cone-charges contribute to pseudo-Riemannian metrics of proper four-spaces of charged objects, resulting in dilation-compression of their proper time rates. Photon-type gravitational radiation is not associated with metric modulation of flat, laboratory space. The Minkowski-Lorentz equation for free charges corresponds to the equivalence principle for the canonical four-space. The predicted criterion of double unification, particles with fields - gravity with electrodynamics, is confirmed in vector electrogravity. 
  The evolution of a global monopole with an inflating core is investigated. An analytic expression for the exterior metric at large distances from the core is obtained. The overall spacetime structure is studied numerically, both in vacuum and in a radiation background. 
  The variational theory of the perfect fluid with intrinsic spin and dilatonic charge (dilaton-spin fluid) is developed. The spin tensor obeys the classical Frenkel condition. The Lagrangian density of such fluid is stated, and the equations of motion of the fluid, the Weyssenhoff-type evolution equation of the spin tensor and the conservation law of the dilatonic charge are derived. The expressions of the matter currents of the fluid (the canonical energy-momentum 3-form, the metric stress-energy 4-form and the dilaton-spin momentum 3-form) are obtained. 
  A four-dimensional differentiable manifold is given with an arbitrary linear connection $\Gamma_\alpha^\beta=\Gamma_{i\alpha}^\beta dx^i$. Megged has claimed that he can define a metric $G_{\alpha\beta}$ by means of a certain integral equation such that the connection is compatible with the metric. We point out that Megged's implicite definition of his metric $G_{\alpha\beta}$ is equivalent to the assumption of a vanishing nonmetricity. Thus his result turns out to be trivial. 
  The subject of this paper are spherically symmetric thin shells made of barotropic ideal fluid and moving under the influence of their own gravitational field as well as that of a central black hole; the cosmological constant is assumed to be zero. The general super-Hamiltonian derived in a previous paper is rewritten for this spherically symmetric special case. The dependence of the resulting action on the gravitational variables is trivialized by a transformation due to Kucha\v{r}. The resulting variational principle depends only on shell variables, is reparametrization invariant, and includes both first- and second-class constraints. Several equivalent forms of the constrained system are written down. Exclusion of the second-class constraints leads to a super-Hamiltonian which appears to overlap with that by Ansoldi et al. in a quarter of the phase space. As Kucha\v{r}' variables are singular at the horizons of both Schwarzschild spacetimes inside and outside the shell, the dynamics is first well-defined only inside of 16 disjoint sectors. The 16 sectors are, however, shown to be contained in a single, connected symplectic manifold and the constraints are extended to this manifold by continuity. Poisson bracket between no two independent spacetime coordinates of the shell vanish at any intersection of two horizons. 
  The equation of perfect dilaton-spin fluid motion in the form of generalized hydrodynamic Euler-type equation in a Weyl-Cartan space is derived. The equation of motion of a test particle with spin and dilatonic charge in the Weyl-Cartan geometry background is obtained. The peculiarities of test particle motion in a Weyl-Cartan space are discussed. 
  Cosmological solutions of the Brans-Dicke theory with an added cosmological constant are investigated with an emphasis to select a conformal frame in order to implement the scenario of a decaying cosmological constant, featuring an ever growing scalar field. We focus particularly on Jordan frame, the original frame with the nonminimal coupling, and conformally transformed Einstein frame without it. For the asymptotoic attractor solutions as well as the "hesitation behavior," we find that none of these conformal frames can be accepted as the basis of analyzing primordial nucleosynthesis. As a remedy, we propose to modify the prototype BD theory, by introducing a scale-invariant scalar-matter coupling, thus making Einstein frame acceptable. The invariacne is broken as a quantum anomaly effect due to the non-gravitational interactions, entailing naturally the fifth force, characterized by a finite force-range and WEP violation. A tentative estimate shows that the theoretical prediction is roughly consistent with the observational upper bounds. Further efforts to improve the experimental accuracy is strongly encouraged. 
  The Hilbert action is derived for a simplicial geometry. I recover the usual Regge calculus action by way of a decomposition of the simplicial geometry into 4-dimensional cells defined by the simplicial (Delaunay) lattice as well as its dual (Voronoi) lattice. Within the simplicial geometry, the Riemann scalar curvature, the proper 4-volume, and hence, the Regge action is shown to be exact, in the sense that the definition of the action does not require one to introduce an averaging procedure, or a sequence of continuum metrics which were common in all previous derivations. It appears that the unity of these two dual lattice geometries is a salient feature of Regge calculus. 
  We consider the Hamiltonian dynamics of spherically symmetric Einstein gravity with a thin null-dust shell, under boundary conditions that fix the evolution of the spatial hypersurfaces at the two asymptotically flat infinities of a Kruskal-like manifold. The constraints are eliminated via a Kuchar-type canonical transformation and Hamiltonian reduction. The reduced phase space $\tilde\Gamma$ consists of two disconnected copies of $R^4$, each associated with one direction of the shell motion. The right-moving and left-moving test shell limits can be attached to the respective components of $\tilde\Gamma$ as smooth boundaries with topology $R^3$. Choosing the right-hand-side and left-hand-side masses as configuration variables provides a global canonical chart on each component of $\tilde\Gamma$, and renders the Hamiltonian simple, but encodes the shell dynamics in the momenta in a convoluted way. Choosing the shell curvature radius and the "interior" mass as configuration variables renders the shell dynamics transparent in an arbitrarily specifiable stationary gauge "exterior" to the shell, but the resulting local canonical charts do not cover the three-dimensional subset of $\tilde\Gamma$ that corresponds to a horizon-straddling shell. When the evolution at the infinities is freed by introducing parametrization clocks, we find on the unreduced phase space a global canonical chart that completely decouples the physical degrees of freedom from the pure gauge degrees of freedom. Replacing one infinity by a flat interior leads to analogous results, but with the reduced phase space $R^2 \cup R^2$. The utility of the results for quantization is discussed. 
  We further deconstruct Heraclitean Quantum Systems giving a model for a universe using pregeometric notions in which the end-game problem is overcome by means of self-referential noise. The model displays self-organisation with the emergence of 3-space and time. The time phenomenon is richer than the present geometric modelling. 
  This paper is a sequel to the series of papers [gr-qc/9409010, gr-qc/9505034, gr-qc/9603022, gr-qc/9609035, gr-qc/9609046, gr-qc/9704033, gr-qc/9704038]. Gravitational autolocalization of a body is considered. A self-consistent problem is solved: A quantum state of the center of mass of the body gives rise to a classical gravitational field, and the state, on the other hand, is an eigenstate in the field. We call a resulting solution gravilon. Gravilons are classified, and their properties are studied. Gravitational autolocalization is predominantly a macroscopic effect. The motion of a gravilon as a whole is classical. 
  The question is raised whether the unique decomposition of the physical Hilbert space, as emerging in the refined algebraic quantization of a constrained system, may be understood in terms of the old Klein-Gordon type quantization. 
  Some basic ideas of the Refined Algebraic Quantization scheme are outlined at an intuitive level, using a class of simple models with a single wave equation as quantum constraint. In addition, hints are given how the scheme is applied to more sophisticated models, and it is tried to make transparent the general pattern characterizing this method. 
  In a one parameter family of dilaton gravity theories which allow the coupling of the dilaton to gravity and to a U(1) gauge field to differ, we have found the existence of everywhere regular spacetimes describing black holes hiding expanding universes inside their horizon. 
  In a universe whose elementary constituents are point particles there does not seem to be any obvious mechanism for avoiding the initial singularities in physical quantities in the standard model of cosmology. In contrast in string theory these singularities can be absent even at the level where spacetime is treated classically. This is a consequence of the basic degrees of freedom of strings in compact spaces, which necessitate a reinterpretation of what one means by a very small universe. We discuss the basic degrees of freedom of a string at the classical and quantum level, the minimum size of strings (string uncertainty principle), the t-duality symmetry, and string thermodynamics at high energy densities, and then describe how these considerations suggest a resolution of the initial singularity problem. An effort has been made to keep this writeup self-contained and accessible to non-string theorists. 
  A brief summary of results on kinematic self-similarities in general relativity is given. Attention is focussed on locally rotationally symmetric models admitting kinematic self-similar vectors. Coordinate expressions for the metric and the kinematic self-similar vector are provided.  Einstein's field equations for perfect fluid models are investigated and all the homothetic perfect fluid solutions admitting a maximal four-parameter group of isometries are given. 
  Space-times admitting a 3-dimensional Lie group of conformal motions $C_3$ acting on null orbits are studied. Coordinate expressions for the metric and the conformal Killing vectors (CKV) are provided (irrespectively of the matter content) and then all possible perfect fluid solutions are found, although none of these verify the weak and dominant energy conditions over the whole space-time manifold. 
  The quantization of a single particle without spin in an appropriate curved space-time is considered. The Hamilton formalism on reduced space for a particle in a curved space-time is constructed and the main aspects of quantization scheme are developed. These investigations are applied to quantization of the particle Hamiltonian in an appropriate curved space-time. As an example the energy eigenvalues in Einstein universe are calculated. In the last sections o the paper approximation for small values of momenta of the results previously obtained is considered as well as quantization of polynomial Hamiltonians of general type is discussed. 
  Numerical relativity codes now being developed will evolve initial data representing colliding black holes at a relatively late stage in the collision. The choice of initial data used for code development has been made on the basis of mathematical definitiveness and usefulness for computational implementation. By using the ``particle limit'' (the limit of an extreme ratio of masses of colliding holes) we recently showed that the standard choice is not a good representation of astrophysically generated initial data. Here we show that, for the particle limit, there is a very simple alternative choice that appears to give excellent results. That choice, ``convective'' initial data is, roughly speaking, equivalent to the start of a time sequence of parameterized solutions of the Hamiltonian constraint; for a particle in circular orbit, it is the initial data of the steady state solution on any hypersurface. The implementation of related schemes for equal mass holes is discussed. 
  Here we study novel effects associated with electromagnetic wave propagation in a Robertson-Walker universe and the Schwarzschild spacetime with a small amount of metric stochasticity. We find that localization of electromagnetic waves occurs in a Robertson-Walker universe with time-independent metric stochasticity, while time-dependent metric stochasticity induces exponential instability in the particle production rate. For the Schwarzschild metric, time-independent randomness can decrease the total luminosity of Hawking radiation due to multiple scattering of waves outside the black hole and gives rise to event horizon fluctuations and thus fluctuations in the Hawking temperature. 
  We give a comparative description of monopole and electrically charged spherically symmetric dust thin shells. Herewith we consider two of the most interesting configurations: the hollow shell and shell, surrounding a body with opposite charge. The classification of shells in accordance with the types of black holes and traversable wormholes is constructed. The theorems for the parameters of turning points are proved. Also for atomlike configurations the effects of screening (electrical case) and amplification (monopole case) of the internal mass by shell charge are studied. Finally, one considers the quantum aspects; herewith, exact solutions of wave equations and bound states spectra are found. 
  We show that the algebra of discretized spatial diffeomorphism constraints in Hamiltonian lattice quantum gravity closes without anomalies in the limit of small lattice spacing. The result holds for arbitrary factor-ordering and for a variety of different discretizations of the continuum constraints, and thus generalizes an earlier calculation by Renteln. 
  In the saddle point approximation, the Euclidean path integral for quantum gravity closely resembles a thermodynamic partition function, with the cosmological constant $\Lambda$ playing the role of temperature and the ``density of topologies'' acting as an effective density of states. For $\Lambda<0$, the density of topologies grows superexponentially, and the sum over topologies diverges. In thermodynamics, such a divergence can signal the existence of a maximum temperature. The same may be true in quantum gravity: the effective cosmological constant may be driven to zero by a rapid rise in the density of topologies. 
  Nonextreme black hole in a cavity within the framework of the canonical or grand canonical ensemble can approach the extreme limit with a finite temperature measured on a boundary located at a finite proper distance from the horizon. In spite of this finite temperature, it is shown that the one-loop contribution $S_{q\text{ }}$of quantum fields to the thermodynamic entropy due to equilibrium Hawking radiation vanishes in the limit under consideration. The same is true for the finite temperature version of the Bertotti-Robinson spacetime into which a classical Reissner-Nordstr\"{o}m black hole turns in the extreme limit. The result $S_{q}=0$ is attributed to the nature of a horizon for the Bertotti-Robinson spacetime. 
  We show that an analogue of the (four dimensional) image sum method can be used to reproduce the results, due to Krasnikov, that for the model of a real massless scalar field on the initial globally hyperbolic region IGH of two-dimensional Misner space there exist two-particle and thermal Hadamard states (built on the conformal vacuum) such that the (expectation value of the renormalised) stress-energy tensor in these states vanishes on IGH. However, we shall prove that the conclusions of a general theorem by Kay, Radzikowski and Wald still apply for these states. That is, in any of these states, for any point b on the Cauchy horizon and any neighbourhood N of b, there exists at least one pair of non-null related points (x,x'), with x and x' in the intersection of IGH with N, such that (a suitably differentiated form of) its two-point function is singular. (We prove this by showing that the two-point functions of these states share the same singularities as the conformal vacuum on which they are built.) In other words, the stress-energy tensor in any of these states is necessarily ill-defined on the Cauchy horizon. 
  We consider a Hamiltonian quantum theory of spherically symmetric, asymptotically flat electrovacuum spacetimes. The physical phase space of such spacetimes is spanned by the mass and the charge parameters $M$ and $Q$ of the Reissner-Nordstr\"{o}m black hole, together with the corresponding canonical momenta. In this four-dimensional phase space, we perform a canonical transformation such that the resulting configuration variables describe the dynamical properties of Reissner-Nordstr\"{o}m black holes in a natural manner. The classical Hamiltonian written in terms of these variables and their conjugate momenta is replaced by the corresponding self-adjoint Hamiltonian operator, and an eigenvalue equation for the ADM mass of the hole, from the point of view of a distant observer at rest, is obtained. Our eigenvalue equation implies that the ADM mass and the electric charge spectra of the hole are discrete, and the mass spectrum is bounded below. Moreover, the spectrum of the quantity $M^2-Q^2$ is strictly positive when an appropriate self-adjoint extension is chosen. The WKB analysis yields the result that the large eigenvalues of the quantity $\sqrt{M^2-Q^2}$ are of the form $\sqrt{2n}$, where $n$ is an integer. It turns out that this result is closely related to Bekenstein's proposal on the discrete horizon area spectrum of black holes. 
  Spectra of density perturbations produced during chaotic inflation are calculated, taking both adiabatic and isocurvature modes into account in a class of scalar-tensor theories of gravity in which the dilaton is metrically coupled. Comparing the predicted spectrum of the cosmic microwave background radiation anisotropies with the one observed by the COBE-DMR we calculate constraints on the parameters of these theories, which turn out to be stronger by an order-of-magnitude than those obtained from post-Newtonian experiments. 
  Multidimensional cosmological model describing the evolution of one Einstein space of non-zero curvature and n Ricci-flat internal spaces is considered. The action contains several dilatonic scalar fields and antisymmetric forms. When forms are chosen to be proportional of volume forms of p-brane submanifolds of internal space manifold, the Toda-like Lagrange representation is obtained. Wheeler-De Witt equation for the model is presented. The exact solutions in classical and quantum cases are obtained when dimensions of p-branes and dilatonic couplings obey some orthogonality conditions. 
  Classical and quantum statistical mechanics are cast here in the language of projective geometry to provide a unified geometrical framework for statistical physics. After reviewing the Hilbert space formulation of classical statistical thermodynamics, we introduce projective geometry as a basis for analysing probabilistic aspects of statistical physics. In particular, the specification of a canonical polarity on $RP^{n}$ induces a Riemannian metric on the state space of statistical mechanics. In the case of the canonical ensemble, we show that equilibrium thermal states are determined by the Hamiltonian gradient flow with respect to this metric. This flow is concisely characterised by the fact that it induces a projective automorphism on the state manifold. The measurement problem for thermal systems is studied by the introduction of the concept of a random state. The general methodology is then extended to include the quantum mechanical dynamics of equilibrium thermal states. In this case the relevant state space is complex projective space, here regarded as a real manifold endowed with the natural Fubini-Study metric. A distinguishing feature of quantum thermal dynamics is the inherent multiplicity of thermal trajectories in the state space, associated with the nonuniqueness of the infinite temperature state. We are then led to formulate a geometric characterisation of the standard KMS-relation often considered in the context of $C^{*}$-algebras. The example of a quantum spin one-half particle in heat bath is studied in detail. 
  We discuss the sensitivity of the CASSINI experiments to gravitational waves emitted by the in-spiral of compact binaries. We show that the maximum distance reachable by the instrument is $\sim 100$ Mpc. In particular, CASSINI can detect massive black hole binaries with chirp mass $\simgt 10^6 \Ms$ in the Virgo Cluster with signal-to-noise ratio between 5 and 30 and possible compact objects of mass $\simgt 30 \Ms$ orbiting the massive black hole that our Galactic Centre is likely to harbour. 
  The order of the post-Newtonian expansion needed, to extract in a reliable and accurate manner the fully general relativistic gravitational wave signal from inspiralling compact binaries, is explored. A class of approximate wave forms, called P-approximants, is constructed based on the following two inputs: (a) The introduction of two new energy-type and flux-type functions e(v) and f(v), respectively, (b) the systematic use of Pade approximation for constructing successive approximants of e(v) and f(v). The new P-approximants are not only more effectual (larger overlaps) and more faithful (smaller biases) than the standard Taylor approximants, but also converge faster and monotonically. The presently available O(v/c)^5-accurate post-Newtonian results can be used to construct P-approximate wave forms that provide overlaps with the exact wave form larger than 96.5% implying that more than 90% of potential events can be detected with the aid of P-approximants as opposed to a mere 10-15 % that would be detectable using standard post-Newtonian approximants. 
  The standard approach to the numerical evolution of black hole data using the ADM formulation with maximal slicing and vanishing shift is extended to non-symmetric black hole data containing black holes with linear momentum and spin by using a time-independent conformal rescaling based on the puncture representation of the black holes. We give an example for a concrete three dimensional numerical implementation. The main result of the simulations is that this approach allows for the first time to evolve through a brief period of the merger phase of the black hole inspiral. 
  Recent progress in understanding of the internal structure of non-Abelian black holes is discussed.   Talk given at the international Workshop on The Internal Structure of Black Holes and Spacetime Singularities, Haifa, Israel, June 29 -- July 3, 1997. 
  The gravitational radiation originating from a compact binary system in circular orbit is usually expressed as an infinite sum over radiative multipole moments. In a slow-motion approximation, each multipole moment is then expressed as a post-Newtonian expansion in powers of v/c, the ratio of the orbital velocity to the speed of light. The bare multipole truncation of the radiation consists in keeping only the leading-order term in the post-Newtonian expansion of each moment, but summing over all the multipole moments. In the case of binary systems with small mass ratios, the bare multipole series was shown in a previous paper to converge for all values v/c < 2/e, where e is the base of natural logarithms. In this paper, we extend the analysis to a dressed multipole truncation of the radiation, in which the leading-order moments are corrected with terms of relative order (v/c)^2 and (v/c)^3. We find that the dressed multipole series converges also for all values v/c < 2/e, and that it coincides (within 1%) with the numerically ``exact'' results for v/c < 0.2. 
  We have examined, repeated and extended earlier numerical calculations of Berger and Moncrief for the evolution of unpolarized Gowdy T3 cosmological models. Our results are consistent with theirs and we support their claim that the models exhibit AVTD behaviour, even though spatial derivatives cannot be neglected. The behaviour of the curvature invariants and the formation of structure through evolution both backwards and forwards in time is discussed. 
  A new hybrid scheme for numerical relativity will be presented. The scheme will employ a 3-dimensional spacelike lattice to record the 3-metric while using the standard 3+1 ADM equations to evolve the lattice. Each time step will involve three basic steps. First, the coordinate quantities such as the Riemann and extrinsic curvatures are extracted from the lattice. Second, the 3+1 ADM equations are used to evolve the coordinate data, and finally, the coordinate data is used to update the scalar data on the lattice (such as the leg lengths). The scheme will be presented only for the case of vacuum spacetime though there is no reason why it could not be extended to non-vacuum spacetimes. The scheme allows any choice for the lapse function and shift vectors. An example for the Kasner $T^3$ cosmology will be presented and it will be shown that the method has, for this simple example, zero discretisation error. 
  We extend previous work on conformally covariant differential operators to consider the case of second order operators acting on symmetric traceless tensor fields. The corresponding flat space Green function is explicitly constructed and shown to be in accord with the requirements of conformal invariance. 
  We study a formulation of euclidean general relativity in which the dynamical variables are given by a sequence of real numbers $\lambda_{n}$, representing the eigenvalues of the Dirac operator on the curved spacetime. These quantities are diffeomorphism-invariant functions of the metric and they form an infinite set of ``physical observables'' for general relativity. Recent work of Connes and Chamseddine suggests that they can be taken as natural variables for an invariant description of the dynamics of gravity. We compute the Poisson brackets of the $\lambda_{n}$'s, and find that these can be expressed in terms of the propagator of the linearized Einstein equations and the energy-momentum of the eigenspinors. We show that the eigenspinors' energy-momentum is the Jacobian matrix of the change of coordinates from the metric to the $\lambda_{n}$'s. We study a variant of the Connes-Chamseddine spectral action which eliminates a disturbing large cosmological term. We analyze the corresponding equations of motion and find that these are solved if the energy momenta of the eigenspinors scale linearly with the mass. Surprisingly, this scaling law codes Einstein's equations. Finally we study the coupling to a physical fermion field. 
  Recently Bernd Schmidt has given three explicit examples of spacetimes with toroidal null infinities. In this paper all solutions with a toroidal null infinity within Schmidt's metric ansatz (polarized Gowdy models) are constructed. The members of the family are determined by two smooth functions of one variable. For the unpolarized Gowdy models the same kind of analysis carries through. 
  This paper and the others in the series challenge the standard model of the effects of gravitational lensing on observations at large distances. We show that due to the cumulative effect of lensing, areas corresponding to an observed solid angle can be quite different than would be estimated from the corresponding Friedmann-Lema\^{\i}tre model, even when averaged over large angular scales. This paper concentrates on the specific example of spherically symmetric but spatially inhomogeneous dust universes, the Lema\^{\i}tre-Tolman-Bondi models, and shows that radial lensing significantly distorts the area distance-redshift and density-redshift relations in these exact solutions compared with the standard ones for Friedmann-Lema\^{\i}tre models. Thus inhomogeneity may introduce significant errors into distance estimates based on the standard FL relations, even after all-sky averaging. In addition a useful new gauge choice is presented for these models, solving the problem of locating the past null cone exactly. 
  Assuming that spacetime tunnels -wormholes and ringholes- naturally exist in the universe, we investigate the conditions making them embeddible in Friedmann space, and the possible observable effects of these tunnels, including: lensing and frequency-shifting of emitting sources, discontinuous change of background temperature, broadening and intensity enhancement of spectral lines, so as a dramatic increase of the luminosity of any object at the tunnel's throat. 
  A limiting diagram for the Segre classification of the energy-momentum tensor is obtained and discussed in connection with a Penrose specialization diagram for the Segre types. A generalization of the coordinate-free approach to limits of Paiva et al. to include non-vacuum space-times is made. Geroch's work on limits of space-times is also extended. The same argument also justifies part of the procedure for classification of a given spacetime using Cartan scalars. 
  Much of the foundational work on quantum cosmology employs a simple minisuperspace model describing a Friedmann-Robertson-Walker universe containing a massive scalar field. We show that the classical limit of this model exhibits deterministic chaos and explore some of the consequences for the quantum theory. In particular, the breakdown of the WKB approximation calls into question many of the standard results in quantum cosmology. 
  Velocity dependent forces varying as $k(\hat{r}/r)(1 - \mu \dot{r}^2 + \gamma r \ddot{r})$ (such as Weber force), here called Weber-like forces, are examined from the point of view of energy conservation and it is proved that they are conservative if and only if $\gamma=2\mu$. As a consequence, it is shown that gravitational theories employing Weber-like forces cannot be conservative and also yield both the precession of the perihelion of Mercury as well as the gravitational deflection of light. 
  We derive the formulae of fluctuating hydrodynamics appropiate to a relativistically consistent divergence type theory, obtaining Landau - Lifshitz fluctuating hydrodynamics as a limiting case. 
  Boundary conditions and the corresponding states of a quantum field theory depend on how the horizons are taken into account. There is ambiguity as to which method is appropriate because different ways of incorporating the horizons lead to different results. We propose that a natural way of including the horizons is to first consider the Kruskal extension and then define the quantum field theory on the Euclidean section. Boundary conditions emerge naturally as consistency conditions of the Kruskal extension. We carry out the proposal for the explicit case of the Schwarzschild-de Sitter manifold with two horizons. The required period $\beta$ is the interesting condition that it is the lowest common multiple of $2\pi$ divided by the surface gravity of both horizons. Restricting the ratio of the surface gravity of the horizons to rational numbers yields finite $\beta$. The example also highlights some of the difficulties of the off-shell approach with conical singularities in the multihorizon scenario; and serves to illustrate the much richer interplay that can occur among horizons, quantum field theory and topology when the cosmological constant is not neglected in black hole processes. 
  A standard and reasonable definition of asymptotic velocity term dominance (AVTD) shows that the numerical study by Hern and Stewart (gr-qc/9708038) confirms previous results that generic Gowdy cosmologies on $T^3 \times R$ have an AVTD singularity. 
  In the framework of the teleparallel equivalent of general relativity, we study the dynamics of a gravitationally coupled electromagnetic field. It is shown that the electromagnetic field is able not only to couple to torsion, but also, through its energy-momentum tensor, to produce torsion. Furthermore, it is shown that the coupling of the electromagnetic field with torsion preserves the local gauge invariance of Maxwell's theory. 
  Newtonian theory is used to study the gravitational effects of a texture, in particular the formation of massive structures. 
  A basic arbitrariness in the determination of the topology of a manifold at the Planck length is discussed. An explicit example is given of a `smooth' change in topology from the 2-sphere to the 2-torus through a sequence of noncommuting geometries. Applications are considered to the theory of D-branes within the context of the proposed $M$(atrix) theory. 
  We study the phase space structure and the quantization of a pointlike particle in 2+1 dimensional gravity. By adding boundary terms to the first order Einstein Hilbert action, and removing all redundant gauge degrees of freedom, we arrive at a reduced action for a gravitating particle in 2+1 dimensions, which is invariant under Lorentz transformations and a group of generalized translations. The momentum space of the particle turns out to be the group manifold SL(2). Its position coordinates have non-vanishing Poisson brackets, resulting in a non-commutative quantum spacetime. We use the representation theory of SL(2) to investigate its structure. We find a discretization of time, and some semi-discrete structure of space. An uncertainty relation forbids a fully localized particle. The quantum dynamics is described by a discretized Klein Gordon equation. 
  The problem of time, considered as a problem in the usual physical context, is reflected in relation with the paper by Kauffman and Smolin (gr-qc/9703026). It is shown that the problem is a misposed problem in the sense that it was raised with a lack of the recognition of mathematically known facts. 
  We derive a general integral formula on an embedded hypersurface for general relativistic space-times. Suppose the hypersurface is foliated by two-dimensional compact ``sections'' $S_s$. Then the formula relates the rate of change of the divergence of outgoing light rays integrated over $S_s$ under change of section to geometric (convexity and curvature) properties of $S_s$ and the energy-momentum content of the space-time. We derive this formula using the Sparling-Nester-Witten identity for spinor fields on the hypersurface by appropriate choice of the spinor fields. We discuss several special cases which have been discussed in the literature before, most notably the Bondi mass loss formula. 
  We present a method to study the time variation of the orbital parameters of a Post-Keplerian binary system undergoing a generic external perturbation. The method is the relativistic extension of the planetary Lagrangian equations. The theory only assumes the smallness of the external perturbation while relativistic effects are already included in the unperturbed problem. This is the major advantage of this novel approach over classical Lagrangian methods. 
  We consider a D-dimensional self-gravitating spherically symmetric configuration of a generalized electro-magnetic n-form F and a dilatonic scalar field, admitting an interpretation in terms of intersecting p-branes. For theories with multiple times, selection rules are obtained, which obstruct the existence of p-branes in certain subspaces. General static solutions are obtained under a specific restriction on the model parameters, which corresponds to the known "intersection rules". More special families of solutions (with equal charges for some of the F-field components) are found with weakened restrictions on the input parameters. Black-hole solutions are determined, and it is shown that in the extreme limit the Hawking temperature may tend to zero, a finite value, or infinity, depending on the p-brane intersection dimension. A kind of no-hair theorem is obtained, claiming that black holes cannot coexist with a quasiscalar component of the F-field. 
  We consider a net of *-algebras, locally around any point of observation, equipped with a natural partial order related to the isotony property. Assuming the underlying manifold of the net to be a differentiable, this net shall be kinematically covariant under general diffeomorphisms. However, the dynamical relations, induced by the physical state defining the related net of (von Neumann) observables, are in general not covariant under all diffeomorphisms, but only under the subgroup of dynamical symmetries.   We introduce algebraically both, IR and UV cutoffs, and assume that these are related by a commutant duality. The latter, having strong implications on the net, allows us to identify a 1-parameter group of the dynamical symmetries with the group of outer modular automorphisms.   For thermal equilibrium states, the modular dilation parameter may be used locally to define the notions of both, time and a causal structure. 
  The geometrodynamics of the spherical gravity with a selfgravitating thin dust shell as a source is constructed. The shell Hamiltonian constraint is derived and the corresponding Schroedinger equation is obtained. This equation appeared to be a finite differences equation. Its solutions are required to be analytic functions on the relevant Riemannian surface. The method of finding discrete spectra is suggested based on the analytic properties of the solutions. The large black hole approximation is considered and the discrete spectra for bound states of quantum black holes and wormholes are found. They depend on two quantum numbers and are, in fact, quasicontinuous. 
  The cosmological sector to the full non-linear topologically massive gravity (TMG) is obtained for localized sources of mass $m$ and spin $\sigma$ besides the asymptotically spinning conical flat sector previously obtained. In a small region near but outside the sources, the metric resembles the spinning conical flat metric, but we find that the mass $m$ creates a negative deficit angle of $3m$ as opposed to $m$. Furthermore, it is not possible to recover the results of pure Einstein gravity in the limit $\mu \to \infty$ unlike the flat sector. 
  In the Euclidean path integral approach, we calculate the actions and the entropies for the Reissner-Nordstr\"om-de Sitter solutions. When the temperatures of black hole and cosmological horizons are equal, the entropy is the sum of one-quarter areas of black hole and cosmological horizons; when the inner and outer black hole horizons coincide, the entropy is only one-quarter area of cosmological horizon; and the entropy vanishes when the two black hole horizons and cosmological horizon coincide. We also calculate the Euler numbers of the corresponding Euclidean manifolds, and discuss the relationship between the entropy of instanton and the Euler number. 
  In four-dimensional spacetime, when the two-sphere of black hole event horizons is replaced by a two-dimensional hypersurface with zero or negative constant curvature, the black hole is referred to as a topological black hole. In this paper we present some exact topological black hole solutions in the Einstein-Maxwell-dilaton theory with a Liouville-type dilaton potential. 
  We investigate the possibility of having hairs on the cosmological horizon. The cosmological horizon shares similar properties of black hole horizons in the aspect of having hairs on the horizons. For those theories admitting haired black hole solutions, the nontrivial matter fields may reach and extend beyond the cosmological horizon. For Q-stars and boson stars, the matter fields cannot reach the cosmological horizon. The no short hair conjecture keeps valid, despite the asymptotic behavior (de Sitter or anti-de Sitter) of black hole solutions. We prove the no scalar hair theorem for anti-de Sitter black holes. Using the Bekenstein's identity method, we also prove the no scalar hair theorem for the de Sitter space and de Sitter black holes if the scalar potential is convex. 
  We describe the computation of the Bondi news for gravitational radiation. We have implemented a computer code for this problem. We discuss the theory behind it as well as the results of validation tests. Our approach uses the compactified null cone formalism, with the computational domain extending to future null infinity and with a worldtube as inner boundary. We calculate the appropriate full Einstein equations in computational eth form in (a) the interior of the computational domain and (b) on the inner boundary. At future null infinity, we transform the computed data into standard Bondi coordinates and so are able to express the news in terms of its standard $N_{+}$ and $N_{\times}$ polarization components. The resulting code is stable and second-order convergent. It runs successfully even in the highly nonlinear case, and has been tested with the news as high as 400, which represents a gravitational radiation power of about $10^{13}M_{\odot}/sec$. 
  We discuss black hole quantization in the Wheeler-DeWitt approach. Our consideration is based on a detailed investigation of the canonical formulation of gravity with special considerations of surface terms. Since the phase space of gravity for non-compact spacetimes or spacetimes with boundaries is ill-defined unless one takes boundary degrees of freedom into account, we give a Hamiltonian formulation of the Einstein-Hilbert-action as well as a Hamiltonian formulation of the surface terms. It then is shown how application to black hole spacetimes connects the boundary degrees of freedom with thermodynamical properties of black hole physics. Our treatment of the surface terms thereby naturally leads to the Nernst theorem. Moreover, it will produce insights into correlations between the Lorentzian and the Euclidean theory. Next we discuss quantization, which we perform in a standard manner. It is shown how the thermodynamical properties can be rediscovered from the quantum equations by a WKB like approximation scheme. Back reaction is treated by going beyond the first order approximation. We end our discussion by a rigorous investigation of the so-called BTZ solution in 2+1 dimensional gravity. 
  We describe a numerical code that solves Einstein's equations for a Schwarzschild black hole in spherical symmetry, using a hyperbolic formulation introduced by Choquet-Bruhat and York. This is the first time this formulation has been used to evolve a numerical spacetime containing a black hole. We excise the hole from the computational grid in order to avoid the central singularity. We describe in detail a causal differencing method that should allow one to stably evolve a hyperbolic system of equations in three spatial dimensions with an arbitrary shift vector, to second-order accuracy in both space and time. We demonstrate the success of this method in the spherically symmetric case. 
  Diffeomorphism freedom induces a gauge dependence in the theory of spacetime perturbations. We derive a compact formula for gauge transformations of perturbations of arbitrary order. To this end, we develop the theory of Taylor expansions for one-parameter families (not necessarily groups) of diffeomorphisms. First, we introduce the notion of knight diffeomorphism, that generalises the usual concept of flow, and prove a Taylor's formula for the action of a knight on a general tensor field. Then, we show that any one-parameter family of diffeomorphisms can be approximated by a family of suitable knights. Since in perturbation theory the gauge freedom is given by a one-parameter family of diffeomorphisms, the expansion of knights is used to derive our transformation formula. The problem of gauge dependence is a purely kinematical one, therefore our treatment is valid not only in general relativity, but in any spacetime theory. 
  Using as an example the Einstein gravity with the cosmological constant, we discuss the calculation of renormalization group functions off shell. We found, that gauge dependent terms should be absorbed by the nonlinear renormalization of metric. Nevertheless, some terms can be included in the renormalization of Newton's constant. This ambiguity in the renormalization prescription is discussed. 
  The quantum gravity processes that have taken place in the very early Universe are probably responsible for the observed large-scale cosmological perturbations. The comparison of the theory with the detected microwave background anisotropies favors the conclusion that the very early Universe was not driven by a scalar field with whichever scalar field potential. At the same time, the observations allow us to conclude that there is a good probability of a direct detection of the higher frequency relic gravitational waves with the help of the advanced laser interferometers. 
  Boson stars in zero-, one-, and two-node equilibrium states are modeled numerically within the framework of Scalar-Tensor Gravity. The complex scalar field is taken to be both massive and self-interacting. Configurations are formed in the case of a linear gravitational scalar coupling (the Brans-Dicke case) and a quadratic coupling which has been used previously in a cosmological context. The coupling parameters and asymptotic value for the gravitational scalar field are chosen so that the known observational constraints on Scalar-Tensor Gravity are satisfied. It is found that the constraints are so restrictive that the field equations of General Relativity and Scalar-Tensor gravity yield virtually identical solutions. We then use catastrophe theory to determine the dynamically stable configurations. It is found that the maximum mass allowed for a stable state in Scalar-Tensor gravity in the present cosmological era is essentially unchanged from that of General Relativity. We also construct boson star configurations appropriate to earlier cosmological eras and find that the maximum mass for stable states is smaller than that predicted by General Relativity, and the more so for earlier eras. However, our results also show that if the cosmological era is early enough then only states with positive binding energy can be constructed. 
  We extend the work done for cosmic strings on the perturbative calculation of vacuum polarization of a massless field in the space-time of multiple cosmic strings and show that for a more general class of locally flat metrics the one loop calculation do not introduce any new divergences to the VEV of the energy of a scalar particle or a spinor particle. We explicitly perform the calculation for the configuration where we have one cosmic string in the presence of a dipole made out of cosmic strings both for the scalar and the spinor cases. 
  A review is made of recent efforts to add a gravitational field to noncommutative models of space-time. Special emphasis is placed on the case which could be considered as the noncommutative analog of a parallelizable space-time. It is argued that, at least in this case, there is a rigid relation between the noncommutative structure of the space-time on the one hand and the nature of the gravitational field which remains as a `shadow' in the commutative limit on the other. 
  Quasi-toroidal oscillations in slowly rotating stars are examined in the framework of general relativity. The oscillation frequency to first order of the rotation rate is not a single value even for uniform rotation unlike the Newtonian case. All the oscillation frequencies of the r-modes are purely neutral and form a continuous spectrum limited to a certain range. The allowed frequencies are determined by the resonance condition between the perturbation and background mean flow. The resonant frequency varies with the radius according to general relativistic dragging effect. 
  Perturbative $\alpha '$ corrections to the low energy string effective action have been recently found to have a potentially regularizing effect on the singularity of the lowest order pre-big-bang solutions. Whether they actually regularize it, however, cannot be determined working at any finite order in a perturbative expansion in powers of the string constant $\alpha '$, because of scheme dependence ambiguities. Physically, these corrections are dominated by the integration over the first few massive string states. Very massive string modes, instead, can have a regularizing effect which is non-perturbative in $\alpha '$ and which basically comes from the fact that in a gravitational field with Hubble constant $H$ they are produced with an effective Hawking temperature $T=H/(2\pi)$, and an infinite production rate would occur if this temperature exceeded the Hagedorn temperature. We discuss technical and conceptual difficulties of this non-perturbative regularization mechanism. 
  One of the basic peoblems of quantum cosmology is the problem of time. Various solutions have been proposed for this problem. One approach is to use the Bohmian time. Another Approach is to use the probabilistic time which was recently introduced by Castagnino. We consider both of these definitions as generalizations of a semi-classical time and compare them for a mini-super space. 
  A multidimensional field model describing the behaviour of (at most) one Einstein space of non-zero curvature and n Ricci-flat internal spaces is considered. The action contains several dilatonic scalar fields and antisymmetric forms. The problem setting covers various problems with field dependence on a single space-time coordinate, in particular, isotropic and anisotropic homogeneous cosmologies. When the forms are chosen to be proportional to volume forms of p-brane factor spaces, a Toda-like Lagrange representation arises. Exact solutions are obtained when the p-brane dimensions and the dilatonic couplings obey some orthogonality conditions. General features and some special cases of cosmological solutions are discussed. It is shown, in particular, that all hyperbolic models with a 3-dimensional external space possess an asymptotic with the external scale factor a(t) is proportional to |t| (the cosmic time), while all internal scale factors and all scalar fields tend to finite limits. For D=11 a family of models with one 5-brane and three 2-branes is described. 
  The end state of a coalescing binary of compact objects depends strongly on the final total mass M and angular momentum J. Since gravitational radiation emission causes a slow evolution of the binary system through quasi-circular orbits down to the innermost stable one, in this paper we examine the corresponding behavior of the ratio J/M^2 which must be less than 1(G/c) or about 0.7(G/c) for the formation of a black hole or a neutron star respectively. The results show cases for which, at the end of the inspiral phase, the conditions for black hole or neutron star formation are not satisfied. The inclusion of spin effects leads us to a study of precession equations valid also for the calculation of gravitational waveforms. 
  Boundary problem for Tolman-Bondi model is formulated. One-to-one correspondence between singularities hypersurfaces and initial conditions of the Tolman-Bondi model is constructed. 
  By extending Ashtekar and Romano's definition of spacelike infinity to the timelike direction, a new definition of asymptotic flatness at timelike infinity for an isolated system with a source is proposed. The treatment provides unit spacelike 3-hyperboloid timelike infinity and avoids the introduction of the troublesome differentiability conditions which were necessary in the previous works on asymptotically flat spacetimes at timelike infinity. Asymptotic flatness is characterized by the fall-off rate of the energy-momentum tensor at timelike infinity, which makes it easier to understand physically what spacetimes are investigated. The notion of the order of the asymptotic flatness is naturally introduced from the rate. The definition gives a systematized picture of hierarchy in the asymptotic structure, which was not clear in the previous works. It is found that if the energy-momentum tensor falls off at a rate faster than $\sim t^{-2}$, the spacetime is asymptotically flat and asymptotically stationary in the sense that the Lie derivative of the metric with respect to $\ppp_t$ falls off at the rate $\sim t^{-2}$. It also admits an asymptotic symmetry group similar to the Poincar\'e group. If the energy-momentum tensor falls off at a rate faster than $\sim t^{-3}$, the four-momentum of a spacetime may be defined. On the other hand, angular momentum is defined only for spacetimes in which the energy-momentum tensor falls off at a rate faster than $\sim t^{-4}$. 
  All 1+1 dimensional dipheomorphism-invariant models can be viewed in a unified manner. This includes also general dilaton theories and especially spherically symmetric gravity (SSG) and Witten's dilatonic black hole (DBH). A common feature --- also in the presence of matter fields of any type --- is the appearance of an absolutely conserved quantity C which is determined by the influx of matter. Only for a subclass of generalized dilaton theories the singularity structure vanishes together with C. Such `physical' theories include, of course, SSG and DBH. It seems to have been overlooked until recently that the (classical) 'black hole' singularity of the DBH deviates from SSG in a physically nontrivial manner. At the quantum level for all generalized dilaton theories --- in the absence of matter --- the local quantum effects are shown to disappear. This enables us to compute e.g. the second loop order correction to the Polyakov term. For non-minimal scalar coupling we also believe to have settled the controversial issue of Hawking radiation to infinity with a somewhat puzzling result for the case of SSG. 
  If gravitation is propagated by a massive field, then the velocity of gravitational waves (gravitons) will depend upon their frequency and the effective Newtonian potential will have a Yukawa form. In the case of inspiralling compact binaries, gravitational waves emitted at low frequency early in the inspiral will travel slightly slower than those emitted at high frequency later, modifying the phase evolution of the observed inspiral gravitational waveform, similar to that caused by post-Newtonian corrections to quadrupole phasing. Matched filtering of the waveforms can bound such frequency-dependent variations in propagation speed, and thereby bound the graviton mass. The bound depends on the mass of the source and on noise characteristics of the detector, but is independent of the distance to the source, except for weak cosmological redshift effects. For observations of stellar-mass compact inspiral using ground-based interferometers of the LIGO/VIRGO type, the bound on the graviton Compton wavelength is of the order of $6 \times 10^{12}$ km, about double that from solar-system tests of Yukawa modifications of Newtonian gravity. For observations of super-massive black hole binary inspiral at cosmological distances using the proposed laser interferometer space antenna (LISA), the bound can be as large as $6 \times 10^{16}$ km. This is three orders of magnitude weaker than model-dependent bounds from galactic cluster dynamics. 
  The application of the divergence theorem in a non-coordinated basis is shown to lead to a corrected variational principle for Class B Bianchi cosmological models. This variational principle is used to construct a Hamiltonian formulation for diagonal and symmetric vacuum Class B models. [Note: This is an unpublished paper from 1984 that might be useful to anyone interested in Class B Bianchi models.] 
  Gravitational collapse of non-spherical symmetric matter leads inevitably to non-static external spacetimes. It is shown here that gravitational collapse of matter with toroidal topology in a toroidal anti-de Sitter background proceeds to form a toroidal black hole. According to the analytical model presented, the collapsing matter absorbs energy in the form of radiation (be it scalar, neutrinos, electromagnetic, or gravitational) from the exterior spacetime. Upon decompactification of one or two coordinates of the torus one gets collapsing solutions of cylindrical or planar matter onto black strings or black membranes, respectively. The results have implications on the hoop conjecture. 
  We study critical behavior in the collapse of massive spherically symmetric scalar fields. We observe two distinct types of phase transition at the threshold of black hole formation. Type II phase transitions occur when the radial extent $(\lambda)$ of the initial pulse is less than the Compton wavelength ($\mu^{-1}$) of the scalar field. The critical solution is that found by Choptuik in the collapse of massless scalar fields. Type I phase transitions, where the black hole formation turns on at finite mass, occur when $\lambda \mu \gg 1$. The critical solutions are unstable soliton stars with masses $\alt 0.6 \mu^{-1}$. Our results in combination with those obtained for the collapse of a Yang-Mills field~{[M.~W. Choptuik, T. Chmaj, and P. Bizon, Phys. Rev. Lett. 77, 424 (1996)]} suggest that unstable, confined solutions to the Einstein-matter equations may be relevant to the critical point of other matter models. 
  In our previous work, it was shown that the topology of an event horizon (EH) is determined by the past endpoints of the EH. A torus EH (the collision of two EH) is caused by the two-dimensional (one-dimensional) set of the endpoints. In the present article, we examine the stability of the topology of the EH. We see that a simple case of a single spherical EH is unstable. Furthermore, in general, an EH with handles (a torus, a double torus, ...) is structurally stable in the sense of catastrophe theory. 
  The causal structure of Einstein's evolution equations is considered. We show that in general they can be written as a first order system of balance laws for any choice of slicing or shift. We also show how certain terms in the evolution equations, that can lead to numerical inaccuracies, can be eliminated by using the Hamiltonian constraint. Furthermore, we show that the entire system is hyperbolic when the time coordinate is chosen in an invariant algebraic way, and for any fixed choice of the shift. This is achieved by using the momentum constraints in such as way that no additional space or time derivatives of the equations need to be computed. The slicings that allow hyperbolicity in this formulation belong to a large class, including harmonic, maximal, and many others that have been commonly used in numerical relativity. We provide details of some of the advanced numerical methods that this formulation of the equations allows, and we also discuss certain advantages that a hyperbolic formulation provides when treating boundary conditions. 
  Unruh's detector calculation is used to study the effect of the defect angle $\beta$ in a space-time with a cosmic string for both the excitation and deexcitation cases. It is found that a rotating detector results in a non-zero effect for both finite (small) and infinite (large) time. 
  Application of the so-called refined algebraic quantization scheme for constrained systems to the relativistic particle provides an inner product that defines a unique Fock representation for a scalar field in curved space-time. The construction can be made rigorous for a general globally hyperbolic space-time, but the quasifree state so obtained turns out to be unphysical in general. We exhibit a closely related pair of Fock representations that is also defined generically and conforms to the notion of in- and outgoing states in those situations where particle creation by the external field is expected. 
  In this work we define a new state on the Weyl algebra of the free massive scalar Klein-Gordon field on a Robertson-Walker spacetime and prove that it is a Hadamard state. The state is supposed to approximate a thermal equilibrium state on a Robertson-Walker spacetime and we call it an adiabatic KMS state. This opens the possibility to do quantum statistical mechanics on Robertson-Walker spacetimes in the algebraic framework and the analysis of the free Bose gas on Robertson-Walker spacetimes. The state reduces to an adiabatic vacuum state if the temperature is zero and it reduces to the usual KMS state if the scaling factor in the metric of the Robertson-Walker spacetime is constant.   In the second part of our work we discuss the time evolution of adiabatic KMS states. The time evolution is described in terms of semigroups. We prove the existence of a propagator on the classical phase space. This defines a time evolution on the one-particle Hilbert space. We use this time evolution to analyze the evolution of the two-point function of the KMS state. The inverse temperature change is proportional to the scale factor in the metric of the Robertson-Walker spacetime, as one expects for a relativistic Bose gas. 
  It is shown that in general relativity some static metrics are able to simulate oscillatory motions. Their form depends on two arbitrary real parameters which determine the specific oscillation modes. The conclusion is that these metrics can be used for new geometric models of closed or even open bags.   Pacs: 04.02.-q and 04.02.Jb 
  A new formulation of the Hamiltonian dynamics of the gravitational field interacting with(non-dissipative) thermo-elastic matter is discussed. It is based on a gauge condition which allows us to encode the six degrees of freedom of the ``gravity + matter''-system (two gravitational and four thermo-mechanical ones), together with their conjugate momenta, in the Riemannian metric q_{ij} and its conjugate ADM momentum P^{ij}. These variables are not subject to constraints. We prove that the Hamiltonian of this system is equal to the total matter entropy. It generates uniquely the dynamics once expressed as a function of the canonical variables. Any function U obtained in this way must fulfil a system of three, first order, partial differential equations of the Hamilton-Jacobi type in the variables (q_{ij},P^{ij}). These equations are universal and do not depend upon the properties of the material: its equation of state enters only as a boundary condition. The well posedness of this problem is proved. Finally, we prove that for vanishing matter density, the value of U goes to infinity almost everywhere and remains bounded only on the vacuum constraints. Therefore the constrained, vacuum Hamiltonian (zero on constraints and infinity elsewhere) can be obtained as the limit of a ``deep potential well'' corresponding to non-vanishing matter. This unconstrained description of Hamiltonian General Relativity can be useful in numerical calculations as well as in the canonical approach to Quantum Gravity. 
  Strong cosmic censorship holds that given suitable initial data on a spacelike hypersurface, the laws of general relativity should determine, completely and uniquely, the future evolution of the spacetime. Here it is argued that while strong cosmic censorship is enforced for all black holes residing in asymptotically flat spacetime, it is violated (within the classical formulation of general relativity) for some black holes residing in non asymptotically flat spacetime. It is suggested that the semi-classical formulation of general relativity might enforce strong cosmic censorship. 
  Contents:  *News:    April 1997 Joint APS/AAPT Meeting, by Beverly Berger    The physics survey and committee on gravitational physics, by Jim Hartle  *Research Briefs:    Instability of rotating stars to axial perturbations, by Sharon Morsink    LIGO project status, by Stan Whitcomb    The Search for Frame-Dragging, by Clifford Will  *Conference Reports:    Conference of the Southern African Relativity Society, by Nigel Bishop    II Warszaw workshop on canonical and quantum gravity, by Carlo Rovelli    Alpbach summer school on fundamental physics in space, by Peter Bender    MG8, an experimentalists' summary, by Riley Newman and Peter Saulson    Amaldi Conference on Gravitational Waves, by M. Alessandra Papa    Santa Fe workshop on simplicial quantum gravity, by Lee Smolin    VII Canadian Conference on General Relativity, by David Hobill 
  We present a mathematical characterization of hyperbolic gauge pathologies in general relativity and electrodynamics. We show how non-linear gauge terms can produce a blow-up along characteristics and how this can be identified numerically by performing convergence analysis. Finally, we show some numerical examples and discuss the profound implications this may have for the field of numerical relativity. 
  The last seven years has produced a growing body of evidence which concludes that the Cauchy horizon in black hole-de Sitter spacetimes is classically stable when the surface gravity at the cosmological event horizon is greater than that at the Cauchy horizon. That stability persists for a finite, but non-zero, region of the black hole's parameter space, $(M,Q,J,\Lambda)$, suggests that black holes immersed in de Sitter space are counter-examples to the strong cosmic censorship hypothesis.   In this review we chronicle that body of evidence and describe the first steps of a program of numerical work aimed at better understanding the interior of black hole-de Sitter spacetimes. The review ends with a speculative account of the role that future work will take. 
  We perform fully relativistic calculations of binary neutron stars in corotating, circular orbit. While Newtonian gravity allows for a strict equilibrium, a relativistic binary system emits gravitational radiation, causing the system to lose energy and slowly spiral inwards. However, since inspiral occurs on a time scale much longer than the orbital period, we can treat the binary to be in quasiequilibrium. In this approximation, we integrate a subset of the Einstein equations coupled to the relativistic equation of hydrostatic equilibrium to solve the initial value problem for binaries of arbitrary separation. We adopt a polytropic equation of state to determine the structure and maximum mass of neutron stars in close binaries for polytropic indices n=1, 1.5 and 2. We construct sequences of constant rest-mass and locate turning points along energy equilibrium curves to identify the onset of orbital instability. In particular, we locate the innermost stable circular orbit (ISCO) and its angular velocity. We construct the first contact binary systems in full general relativity. These arise whenever the equation of state is sufficiently soft >= 1.5. A radial stability analysis reveals no tendency for neutron stars in close binaries to collapse to black holes prior to merger. 
  The spectrum of gravitational waves that have been produced in inflation is modified during cosmological transitions. Large drops in the number of relativistic particles, like during the QCD transition or at $e^+e^-$ annihilation, lead to steps in the spectrum of gravitational waves. We calculate the transfer function for the differential energy density of gravitational waves for a first-order and for a crossover QCD transition. 
  Relativistic spin networks are defined by considering the spin covering of the group SO(4), SU(2) times SU(2). Relativistic quantum spins are related to the geometry of the 2-dimensional faces of a 4-simplex. This extends the idea of Ponzano and Regge that SU(2) spins are related to the geometry of the edges of a 3-simplex.   This leads us to suggest that there may be a 4-dimensional state sum model for quantum gravity based on relativistic spin networks which parallels the construction of 3-dimensional quantum gravity from ordinary spin networks. 
  We analyse the gravitational field of a global monopole within the context of low energy string gravity, allowing for an arbitrary coupling of the monopole fields to the dilaton. Both massive and massless dilatons are considered. We find that, for a massless dilaton, the spacetime is generically singular, whereas when the dilaton is massive, the monopole generically induces a long range dilaton cloud. We compare and contrast these results with the literature. 
  We justify the way of the direct quantization which means immediate quantization of a conservation law. It is shown that this approach is equivalent to introducing the super Hamiltonian on a minisuperspace in spirit of the Wheeler-DeWitt's approach. Then we will have: (i) all values are observable and have an obvious physical meaning and well-defined application domain; (ii) wave function is well-defined without time slicing and often can be exactly obtained; (iii) we can take off major mathematical troubles, and therefore, more complicate models can be considered exactly without the perturbation theory. 
  This paper contains discussion of the problem of motion of extended i.e. non point test bodies in multidimensional space. Extended bodies are described in terms of so called multipole moments. Using approximated form of equations of motion for extended bodies deviation from geodesic motion is derived. Results are applied to special form of space-time. 
  A rotating star's oblateness creates a deformation in the gravitational field outside the star, which is measured by the quadrupole-moment tensor. We consider the effect of the quadrupole moment on the orbital motion and rate of inspiral of a compact binary system, composed of neutron stars and/or black holes. We find that in the case of circular orbits, the quadrupole-monopole interaction affects the relation between orbital radius and angular velocity, and also the rate of inspiral, by a quantity of order (v/c)^4, where v is the orbital velocity and c the speed of light. 
  Numerical models of rotating neutron stars are constructed for four equations of state using the computer code RNS written by Stergioulas. For five selected values of the star's gravitational mass (in the interval between 1.0 and 1.8 solar masses) and for each equation of state, the star's angular momentum is varied from J=0 to the Keplerian limit J=J_{max}. For each neutron-star configuration we compute Q, the quadrupole moment of the mass distribution. We show that for given values of M and J, |Q| increases with the stiffness of the equation of state. For fixed mass and equation of state, the dependence on J is well reproduced with a simple quadratic fit, Q \simeq - aJ^2/M c^2, where c is the speed of light, and a is a parameter of order unity depending on the mass and the equation of state. 
  The Thompson cross section for scattering of electromagnetic waves by a free electron in an expanding universe is derived here. The equations of motion of the electron are obtained using the Einstein-Infeld-Hoffmann (EIH) surface integral method. These integrals are evaluated approximately by perturbing off an Einstein-deSitter cosmological field. It is found that the Thompson cross section varies with time as the inverse square of the cosmic scale factor R(t). 
  We investigate the cosmological perturbation of two-scalar field model during the reheating phase after inflation. Using the exact solution of the perturbation in long-wavelength limit, which is expressed in terms of the background quantities, we analyze the behavior of the metric perturbation. The oscillating inflaton field gives rise to the parametric resonance of the massless scalar field and this leads to the amplification of the iso-curvature mode of the metric perturbations. 
  The study of dynamics in general relativity has been hampered by a lack of coordinate independent measures of chaos. Here I review a variety of invariant measures for quantifying chaotic dynamics in relativity that exploit the coordinate independence of fractal dimensions and symbolic entropies. 
  The Mixmaster or Bianchi IX cosmological model has become one of the archetypal settings for studying gravitational dynamics. The past decade has seen a vigourous debate about whether or not the Mixmaster's dynamics is chaotic. In this talk we review our recent work in uncovering a chaotic invariant set of orbits in the Mixmaster phase space, and how we used this discovery to prove the dynamics is chaotic. 
  According to the theory of general relativity, the relative acceleration of masses generates gravitational radiation. Although gravitational radiation has not yet been detected, it is believed that extremely violent cosmic events, such as the collision of black holes, should generate gravity waves of sufficient amplitude to detect on earth. The massive Laser Interferometer Gravitational-wave Observatory, or LIGO, is now being constructed to detect gravity waves. Consequently there is great interest in the computer simulation of black hole collisions and similar events, based on the numerical solution of the Einstein field equations. In this note we introduce the scientific, mathematical, and computational problems and discuss the development of a computer code to solve the initial data problem for colliding black holes, a nonlinear elliptic boundary value problem posed in an unbounded three dimensional domain which is a key step in solving the full field equations. The code is based on finite elements, adaptive meshes, and a multigrid solution process. Here we will particularly emphasize the mathematical and algorithmic issues arising in the generation of adaptive tetrahedral meshes. 
  I describe the general mathematical construction and physical picture of topological black holes, which are black holes whose event horizons are surfaces of non-trivial topology. The construction is carried out in an arbitrary number of dimensions, and includes all known special cases which have appeared before in the literature. I describe the basic features of massive charged topological black holes in $(3+1)$ dimensions, from both an exterior and interior point of view. To investigate their interiors, it is necessary to understand the radiative falloff behaviour of a given massless field at late times in the background of a topological black hole. I describe the results of a numerical investigation of such behaviour for a conformally coupled scalar field. Significant differences emerge between spherical and higher genus topologies. 
  The Wheeler-DeWitt equation is investigated and used to examine a state after a quantum tunneling with gravity. To make arguments definite we treat a discretized version of the Wheeler-DeWitt equation and adopt the WKB method. We expand an Euclidean wave function around an instanton, by using a deviation equation of a vector field tangent to a congruence of instantons. The instanton around which we expand the wave function corresponds to a so-called most probable escape path (MPEP). It is shown that, when the wave function is analytically continued, the corresponding state of physical perturbations is equivalent to the vacuum state determined by positive-frequency mode functions which satisfy appropriate boundary conditions. Thus a quantum field theory is effective to investigate a state after a quantum tunneling with gravity. The effective Lagrangian describing the field theory is obtained by simply reducing the original Lagrangian to a subspace spanned by the physical perturbations. The result of this paper does not depend on the operator ordering and can be applied to all physical perturbations, including gravitational perturbations, around a general MPEP. 
  The aim of this work is to show how it is possible to build an on line whitening filter in an adaptive way. We have modeled the VIRGO noise spectrum as an autoregressive stochastic process, after a pre-filtering of the theoretical curve which flattens the low frequency part of the spectrum. We have tested some very popular adaptive algorithms, based on the gradient methods and on the least squares methods with a lattice structure filter. 
  The renormalization group method has been adapted to the analysis of the long-time behavior of non-linear partial differential equation and has demonstrated its power in the study of critical phenomena of gravitational collapse. In the present work we apply the renormalization group to the Einstein equation in cosmology and carry out detailed analysis of renormalization group flow in the vicinity of the scale invariant fixed point in the spherically symmetric and inhomogeneous dust filled universe model. 
  We consider the Hamiltonian mechanics and thermodynamics of an eternal black hole in a box of fixed radius and temperature in generic 2-D dilaton gravity. Imposing boundary conditions analoguous to those used by Louko and Whiting for spherically symmetric gravity, we find that the reduced Hamiltonian generically takes the form: $$ H(M,\phi_+) = \sigma_0 E(M,\phi_+) -{N_0\over 2\pi} S(M) $$ where $E(M,\phi_+)$ is the quasilocal energy of a black hole of mass $M$ inside a static box (surface of fixed dilaton field $\phi_+$) and $S(M)$ is the associated classical thermodynamical entropy. $\sigma_0$ and $N_0$ determine time evolution along the world line of the box and boosts at the bifurcation point, respectively. An ansatz for the quantum partition function is obtained by fixing $\sigma_0$ and $N_0$ and then tracing the operator $e^{-\beta H}$ over mass eigenstates. We analyze this partition function in some detail both generically and for the class of dilaton gravity theories that is obtained by dimensional reduction of Einstein gravity in n+2 dimensions with $S^n$ spherical symmetry. 
  We consider true vacuum bubbles generated in a first order phase transition occurring during the slow rolling era of a two field inflation: it is known that gravitational waves are produced by the collision of such bubbles. We find that the epoch of the phase transition strongly affects the characteristic peak frequency of the gravitational waves, causing an observationally interesting redshift in addition to the post-inflationary expansion. In particular it is found that a phase transition occurring typically 10$\div$20 $e-$foldings before the reheating at $kT\simeq 10^{15}$ GeV may be detected by the next Ligo gravity waves interferometers. Moreover, for recently proposed models capable of generating the observed large scale voids as remnants of the primordial bubbles (for which the characteristic wave lengths are several tens of Mpc), it is found that the level of anisotropy of the cosmic microwave background provides a deep insight upon the physical parameters of the effective Lagrangian. 
  Gravitationally coupled scalar fields, originally introduced by Jordan, Brans and Dicke to account for a non constant gravitational coupling, are a prediction of many non-Einsteinian theories of gravity not excluding perturbative formulations of String Theory. In this paper, we compute the cross sections for scattering and absorption of scalar and tensor gravitational waves by a resonant-mass detector in the framework of the Jordan-Brans-Dicke theory. The results are then specialized to the case of a detector of spherical shape and shown to reproduce those obtained in General Relativity in a certain limit. Eventually we discuss the potential detectability of scalar waves emitted in a spherically symmetric gravitational collapse. 
  Cauchy horizons are shown to be differentiable at endpoints where only a single null generator leaves the horizon. A Cauchy horizon fails to have any null generator endpoints on a given open subset iff it is differentiable on the open subset and also iff the horizon is (at least) of class C^1 on the open subset. Given the null convergence condition, a compact horizon which is of class C^2 almost everywhere has no endpoints and is (at least) of class C^1 at all points. 
  I show that a quantized Klein-Gordon field in Minkowski space obeys an `operational' weak energy condition: the energy of an isolated device constructed to measure or trap the energy in a region, plus the energy it measures or traps, cannot be negative. There are good reasons for thinking that similar results hold locally for linear quantum fields in curved space-times. A thought experiment to measure energy density is analyzed in some detail, and the operational positivity is clearly manifested.      If operational energy conditions do hold for quantum fields, then the negative energy densities predicted by theory have a will-o'-the-wisp character: any local attempt to verify a total negative energy density will be self-defeating on account of quantum measurement difficulties. Similarly, attempts to drive exotic effects (wormholes, violations of the second law, etc.) by such densities may be defeated by quantum measurement problems. As an example, I show that certain attempts to violate the Cosmic Censorship principle by negative energy densities are defeated.      These quantum measurement limitations are investigated in some detail, and are shown to indicate that space-time cannot be adequately modeled classically in negative energy density regimes. 
  It is shown how the results of Deser and Levin on the response of accelerated detectors in anti-de Sitter space can be understood from the same general perspective as other thermality results in spacetimes with bifurcate Killing horizons. 
  We study the back reaction of cosmological perturbations on the evolution of the universe. The object usually employed to describe the back reaction of perturbations is called the effective energy-momentum tensor (EEMT) of cosmological perturbations. In this formulation, the problem of the gauge dependence of the EEMT must be tackled. We advance beyond traditional results that involve only high frequency perturbations in vacuo, and formulate the back reaction problem in a gauge invariant manner for completely generic perturbations. We give a quick proof that the EEMT for high-frequency perturbations is gauge invariant which greatly simplifies the pioneering approach by Isaacson. As applications we analyze the back reaction of gravitational waves and scalar metric fluctuations in Friedmann-Robertson-Walker background spacetimes. We investigate in particular back reaction effects during inflation in the Chaotic scenario. Fluctuations with a wavelength much bigger than the Hubble radius during inflation contribute a negative energy density, and in that case back reaction counteracts any pre-existing cosmological constant. Finally, we set up the equations of motion for the back reaction on the geometry and on the matter, and show how they are perfectly consistent with the Bianchi identities and the continuity equations. 
  In the homogeneous and isotropic Friedmann-Robertson-Walker minisuperspace model, it is known that there are no Euclidean wormhole solutions in the pure gravity system. Here it is demonstrated explicitly that in Taub cosmology, which is one of the simplest anisotropic cosmology models, wormhole solutions do exist in pure general relativity in both classical and quantum contexts. 
  Construction of skeletonized path integrals for a particle moving on a curved spatial manifold is considered. As shown by DeWitt, Kuchar and others, while the skeletonized configuration space action can be written unambiguously as a sum of Hamilton principal functions, different choices of the measure will lead to different Schrodinger equations. On the other hand, the Liouville measure provides a unique measure for a skeletonized phase space path integral, but there is a corresponding ambiguity in the skeletonization of a path through phase space. A family of skeletonization rules described by Kuchar and referred to here as geodesic interpolation is discussed, and shown to behave poorly under the involution process, wherein intermediate points are removed by extremization of the skeletonized action. A new skeletonization rule, tangent interpolation, is defined and shown to possess the desired involution properties. 
  While the use of spin networks has greatly improved our understanding of the kinematical aspects of quantum gravity, the dynamical aspects remain obscure. To address this problem, we define the concept of a `spin foam' going from one spin network to another. Just as a spin network is a graph with edges labeled by representations and vertices labeled by intertwining operators, a spin foam is a 2-dimensional complex with faces labeled by representations and edges labeled by intertwining operators. Spin foams arise naturally as higher-dimensional analogs of Feynman diagrams in quantum gravity and other gauge theories in the continuum, as well as in lattice gauge theory. When formulated as a `spin foam model', such a theory consists of a rule for computing amplitudes from spin foam vertices, faces, and edges. The product of these amplitudes gives the amplitude for the spin foam, and the transition amplitude between spin networks is given as a sum over spin foams. After reviewing how spin networks describe `quantum 3-geometries', we describe how spin foams describe `quantum 4-geometries'. We conclude by presenting a spin foam model of 4-dimensional Euclidean quantum gravity, closely related to the state sum model of Barrett and Crane, but not assuming the presence of an underlying spacetime manifold. 
  We investigate the validity of the equivalence principle along paths in gravitational theories based on derivations of the tensor algebra over a differentiable manifold. We prove the existence of local bases, called normal, in which the components of the derivations vanish along arbitrary paths. All such bases are explicitly described. The holonomicity of the normal bases is considered. The results obtained are applied to the important case of linear connections and their relationship with the equivalence principle is described. In particular, any gravitational theory based on tensor derivations which obeys the equivalence principle along all paths, must be based on a linear connection. 
  Dirac fermion fields are responsible for spontaneous symmetry breaking in gauge gravitation theory because the spin structure associated with a tetrad field is not preserved under general covariant transformations. Two solutions of this problem can be suggested. (i) There exists the universal spin structure $S\to X$ such that any spin structure $S^h\to X$ associated with a tetrad field $h$ is a subbundle of the bundle $S\to X$. In this model, gravitational fields correspond to different tetrad (or metric) fields. (ii) A background tetrad field $h$ and the associated spin structure $S^h$ are fixed, while gravitational fields are identified with additional tensor fields $q^\la{}_\m$ describing deviations $\wt h^\la_a=q^\la{}_\m h^\m_a$ of $h$. One can think of $\wt h$ as being effective tetrad fields. We show that there exist gauge transformations which keep the background tetrad field $h$ and act on the effective fields by the general covariant transformation law. We come to Logunov's Relativistic Theory of Gravity generalized to dynamic connections and fermion fields. 
  We find a black hole solution with non-Abelian field in Brans-Dicke theory. It is an extension of non-Abelian black hole in general relativity. We discuss two non-Abelian fields: "SU(2)" Yang-Mills field with a mass (Proca field) and the SU(2)$\times$SU(2) Skyrme field. In both cases, as in general relativity, there are two branches of solutions, i.e., two black hole solutions with the same horizon radius. Masses of both black holes are always smaller than those in general relativity. A cusp structure in the mass-horizon radius ($M_{g}$-$r_{h}$) diagram, which is a typical symptom of stability change in catastrophe theory, does not appear in the Brans-Dicke frame but is found in the Einstein conformal frame. This suggests that catastrophe theory may be simply applied for a stability analysis as it is if we use the variables in the Einstein frame. We also discuss the effects of the Brans-Dicke scalar field on black hole structure. 
  We discuss spherically symmetric static solutions of the Einstein-Klein-Gordon equations for a real scalar field with a mass and a quartic self-interaction term. As for the massless case the solutions have a naked singularity at the origin. However, linear stability analysis shows that these solutions as well as the massless ones are dynamically unstable. 
  In this letter we show that in a Kaluza-Klein framework we can have arbitrary topology change between the macroscopic (i.e. noncompactified) spacelike 3-hypersurfaces. This is achieved by using the compactified dimensions as a catalyser for topology change. In the case of odd-dimensional spacetimes (such as the 11-dimensional M-theory) this is always possible. In the even-dimensional case, a sufficient condition is the existence of a closed, odd-dimensional manifold as a factor (such as S^1, S^3) in the Kaluza-Klein sector. Since one of the most common manifolds used for compactification is the torus T^k = S^1 \times ... \times S^1, in this case we can again induce an arbitrary topology change on the 3-hypersurfaces. 
  The formula for the area eigenvalues that was obtained by many authors within the approach known as loop quantum gravity states that each edge of a spin network contributes an area proportional to sqrt{j(j+1)} times Planck length squared to any surface it transversely intersects. However, some confusion exists in the literature as to a value of the proportionality coefficient. The purpose of this rather technical note is to fix this coefficient. We present a calculation which shows that in a sector of quantum theory based on the connection A=Gamma-gamma*K, where Gamma is the spin connection compatible with the triad field, K is the extrinsic curvature and gamma is Immirzi parameter, the value of the multiplicative factor is 8*pi*gamma. In other words, each edge of a spin network contributes an area 8*pi*gamma*l_p^2*sqrt{j(j+1)} to any surface it transversely intersects. 
  We show that it is possible to obtain inflation and also solve the cosmological constant problem. The theory is invariant under changes of the Lagrangian density $L$ to $L+const$. Then the constant part of a scalar field potential $V$ cannot be responsible for inflation. However, we show that inflation can be driven by a condensate of a four index field strength. A constraint appears which correlates this condensate to $V$. After a conformal transformation, the equations are the standard GR equations with an effective scalar field potential $V_{eff}$ which has generally an absolute minimum $V_{eff}=0$ independently of $V$ and without fine tuning. We also show that, after inflation, the usual reheating phase scenario (from oscillations around the absolute minimum) is possible. 
  The geodesic deviation equation (`GDE') provides an elegant tool to investigate the timelike, null and spacelike structure of spacetime geometries. Here we employ the GDE to review these structures within the Friedmann--Lema\^{\i}tre--Robertson--Walker (`FLRW') models, where we assume the sources to be given by a non-interacting mixture of incoherent matter and radiation, and we also take a non-zero cosmological constant into account. For each causal case we present examples of solutions to the GDE and we discuss the interpretation of the related first integrals. The de Sitter spacetime geometry is treated separately. 
  We look at the transition to the semiclassical behaviour and the decoherence process for the inhomogeneous perturbations in the inflationary universe. Two different decoherence mechanisms appear: one dynamical, accompanied with a negligible, if at all, entropy gain, and the other, effectively irreversible dephasing, due to a rapid variation in time of the off-diagonal density matrix elements in the post-inflationary epoch. We thus settle the discrepancies in the entropy content of perturbations evaluated by different authors. 
  It is shown that that the area law for the entropy of a quantum field in the Schwarzschild black hole is due to the quantum statistics. The entropies for one particle, a Boltzmann gas, a quantum mechanical gas obeying Bose-Einstein or Fermi-Dirac statistics, and a quantum field in the Schwarzschild black hole are calculated using the microcanonical ensemble approach and the brick wall method. The area law holds only when the effect of quantum statistics is dominated. 
  Solutions of Einstein's equations are discussed in which the ``gravitational force" is balanced by an electrical force, and which can serve as models for the Cavendish experiment. 
  We exploit an arbitrary extrinsic time foliation of spacetime to solve the constraints in spherically symmetric general relativity. Among such foliations there is a one parameter family, linear and homogeneous in the extrinsic curvature, which permit the momentum constraint to be solved exactly. This family includes, as special cases, the extrinsic time gauges that have been exploited in the past. These foliations have the property that the extrinsic curvature is spacelike with respect to the the spherically symmetric superspace metric. What is remarkable is that the linearity can be relaxed at no essential extra cost which permits us to isolate a large non - pathological dense subset of all extrinsic time foliations. We identify properties of solutions which are independent of the particular foliation within this subset. When the geometry is regular, we can place spatially invariant numerical bounds on the values of both the spatial and the temporal gradients of the scalar areal radius, $R$. These bounds are entirely independent of the particular gauge and of the magnitude of the sources. When singularities occur, we demonstrate that the geometry behaves in a universal way in the neighborhood of the singularity. 
  We establish necessary conditions for the appearance of both apparent horizons and singularities in the initial data of spherically symmetric general relativity when spacetime is foliated extrinsically. When the dominant energy condition is satisfied these conditions assume a particularly simple form. Let $\rho_{Max}$ be the maximum value of the energy density and $\ell$ the radial measure of its support. If $\rho_{Max}\ell^2$ is bounded from above by some numerical constant, the initial data cannot possess an apparent horizon. This constant does not depend sensitively on the gauge. An analogous inequality is obtained for singularities with some larger constant. The derivation exploits Poincar\'e type inequalities to bound integrals over certain spatial scalars. A novel approach to the construction of analogous necessary conditions for general initial data is suggested. 
  We establish sufficient conditions for the appearance of apparent horizons in spherically symmetric initial data when spacetime is foliated extrinsically. Let $M$ and $P$ be respectively the total material energy and the total material current contained in some ball of radius $\ell$. Suppose that the dominant energy condition is satisfied. We show that if $M- P \ge \ell$ then the region must possess a future apparent horizon for some non -trivial closed subset of such gauges. The same inequality holds on a larger subset of gauges but with a larger constant of proportionality which depends weakly on the gauge. This work extends substantially both our joint work on moment of time symmetry initial data as well as the work of Bizon, Malec and \'O Murchadha on a maximal slice. 
  As a contribution to the ongoing discussion of trajectories of spinless particles in spaces with torsion we show that the geometry of such spaces can be induced by embedding their curves in a euclidean space without torsion. Technically speaking, we define the tangent (velocity) space of the embedded space imposing non-holonomic constraints upon the tangent space of the embedding space. Parallel transport in the embedded space is determined as an induced parallel transport on the surface of constraints. Gauss' principle of least constraint is used to show that autoparallels realize a constrained motion that has a minimal deviation from the free, unconstrained motion, this being a mathematical expression of the principle of inertia. 
  The Hamiltonian form of the Hilbert action in the first order tetrad formalism is examined. We perform a non-linear field redefinition of the canonical variables isolating the part of the spin connection which is canonically conjugate to the tetrad. The geometrical meaning of the constraints written in these new variables is examined. 
  Gravitational field of a stationary circular cosmic string loop has been studied in the context of full nonlinear Einstein's theory of gravity. It has been assumed that the radial and tangential stresses of the loop are equal to the energy density of the string loop. An exact solution for the system has been presented which has a singularity at a finite distance from the axis,but is regular for any other distances from the axis of the loop. 
  We define under which circumstances two multi-warped product spacetimes can be considered equivalent and then we classify the spaces of constant curvature in the Euclidean and Lorentzian signature. For dimension D=2, we get essentially twelve representations, for D=3 exactly eighteen. More general, for every even D, 5D+2 cases exist, whereas for every odd D, 5D+3 cases exist. For every D, exactly one half of them has the Euclidean signature. Our definition is well suited for the discussion of multidimensional cosmological models, and our results give a simple algorithm to decide whether a given metric represents the inflationary de Sitter spacetime (in unusual coordinates) or not. 
  Assuming SO(3)-spherical symmetry, the 4-dimensional Einstein equation reduces to an equation conformally related to the field equation for 2-dimensional gravity following from the Lagrangian L = R^(1/3).     Solutions for 2-dimensional gravity always possess a local isometry because the traceless part of its Ricci tensor identically vanishes. Combining both facts, we get a new proof of Birkhoff's theorem; contrary to other proofs, no coordinates must be introduced.     The SO(m)-spherically symmetric solutions of the (m+1)-dimensional Einstein equation can be found by considering L = R^(1/m) in two dimensions. This yields several generalizations of Birkhoff's theorem in an arbitrary number of dimensions, and to an arbitrary signature of the metric. 
  We answer to question Nr. 55 [Are there pictorial examples that distinguish covariant and contravariant vectors ?] posed by D. Neuenschwander, Am. J. Phys. 65 (1), 11 (1997) 
  The spatially homogeneous, isotropic Standard Cosmological Model appears to describe our Universe reasonably well. However, Einstein's equations allow a much larger class of cosmological solutions. Theorems originally due to Penrose and Hawking predict that all such models (assuming reasonable matter properties) will have an initial singularity. The nature of this singularity in generic cosmologies remains a major open question in general relativity. Spatially homogeneous but possibly anisotropic cosmologies have two types of singularities: (1) velocity dominated---(reversing the time direction) the universe evolves to the singularity with fixed anisotropic collapse rates ; (2) Mixmaster---the anisotropic collapse rates change in a deterministically chaotic way. Much less is known about spatially inhomogeneous universes. Belinskii, Khalatnikov, and Lifshitz (BKL) claimed long ago that a generic universe would evolve toward the singularity as a different Mixmaster universe at each spatial point. We shall report on the results of a program to test the BKL conjecture numerically. Results include a new algorithm to evolve homogeneous Mixmaster models, demonstration of velocity dominance and understanding of evolution toward velocity dominance in the plane symmetric Gowdy universes (spatial dependence in one direction), demonstration of velocity dominance in polarized U(1) symmetric cosmologies (spatial dependence in two directions), and exploration of departures from velocity dominance in generic U(1) universes. 
  We consider particle dynamics in singular gravitational field. In 2d spacetime the system splits into two independent gravitational systems without singularity. Dynamical integrals of each system define $sl(2,R)$ algebra, but the corresponding symmetry transformations are not defined globally. Quantization leads to ambiguity. By including singularity one can get the global $SO(2.1)$ symmetry. Quantization in this case leads to unique quantum theory. 
  We consider the numerical evolution of dynamic black hole initial data sets with a full 3D, nonlinear evolution code. These data sets consist of single black holes distorted by strong gravitational waves, and mimic the late stages of coalescing black holes. Through comparison with results from well established axisymmetric codes, we show that these dynamic black holes can be accurately evolved. In particular, we show that with present computational resources and techniques, the process of excitation and ringdown of the black hole can be evolved, and one can now extract accurately the gravitational waves emitted from the 3D Cartesian metric functions, even though they may be buried in the metric at levels on the order of $10^{-3}$ and below. Waveforms for both the $\ell=2$ and the much more difficult $\ell=4$ modes are computed and compared with axisymmetric calculations. In addition to exploring the physics of distorted black hole data sets, and showing the extent to which the waves can be accurately extracted, these results also provide important testbeds for all fully nonlinear numerical codes designed to evolve black hole spacetimes in 3D, whether they use singularity avoiding slicings, apparent horizon boundary conditions, or other evolution methods. 
  The general solutin to the constraints that define relativistic spin networks vertices is given and their relations with 3-dimensional quantum tetrahedra is dicussed. An alternative way to handle the constraints is also presented. 
  Data analysis is the application of probability and statistics to draw inference from observation. Is a signal present or absent? Is the source an inspiraling binary system or a supernova? At what point in the sky is the radiation incident from? In these notes I want to address how two different statistical methodologies --- Bayesian and Frequentist --- approach the problem of statistical inference 
  We apply recent results in the theory of PDE, specifically in problems with two different time scales, on Einstein's equations near their Newtonian limit. The results imply a justification to Postnewtonian approximations when initialization procedures to different orders are made on the initial data. We determine up to what order initialization is needed in order to detect the contribution to the quadrupole moment due to the slow motion of a massive body as distinct from initial data contributions to fast solutions and prove that such initialization is compatible with the constraint equations. Using the results mentioned the first Postnewtonian equations and their solutions in terms of Green functions are presented in order to indicate how to proceed in calculations with this approach. 
  The decay rate for a black hole to decay nonperturbatively via tunneling is shown to be related to the Bekenstein-Hawking black hole entropy $S_{bh}$. This new physical interpretation of the black hole entropy was presented first in 1988 in this paper. I quote here the relevant statement: ``It should be noticed that the decay rate of a black hole due to nonperturbative instability is proportional to $\exp(-S_{bh}$, where $S_{bh}$ is the Bekenstein-Hawking entropy. This seems to indicate that the Bekenstein-Hawking entropy is the measure of the ability of a black hole to disappear from our Universe in the one quantum jump, with the simultaneous production of particles carrying away its total energy.'' 
  We study the quantum behaviour of Reissner-Nordstr\"om (RN) black-holes interacting with a complex scalar field. A Maxwell field is also present. Our analysis is based on M. Pollock's method and is characterized by solving a Wheeler-DeWitt equation in the proximity of an apparent horizon of the RN space-time. Subsequently, we obtain a wave-function $\Psi_{RN}[M, Q]$ representing the RN black-hole when its charge, $\mid Q \mid$, is small in comparison with its mass, $M$. We then compare quantum-mechanically the cases of $(i)$ $Q = 0$ and $(ii)$ $M \geq \mid Q \mid \neq 0 $. A special emphasis is given to the evolution of the mass-charge rate affected by Hawking radiation. 
  We study the interior structure of the Einstein-Yang-Mills-Dilaton black holes as a function of the dilaton coupling constant $\gamma\in [0,1]$. For $\gamma\neq 0$ the solutions have no internal Cauchy horizons and the field amplitudes follow a power law behavior near the singularity. As $\gamma$ decreases, the solutions develop more and more oscillation cycles in the interior region, whose number becomes infinite in the limit $\gamma\to 0$. 
  We present a method for extracting gravitational radiation from a three-dimensional numerical relativity simulation and, using the extracted data, to provide outer boundary conditions. The method treats dynamical gravitational variables as nonspherical perturbations of Schwarzschild geometry. We discuss a code which implements this method and present results of tests which have been performed with a three dimensional numerical relativity code. 
  The unification of the Einstein theory of gravity with a conformal invariant version of the standard model for electroweak interaction without the Higgs potential is considered. In this theory, a module of the Higgs field is absorbed by the scale factor component of metric so that the evolution of the Universe and the elementary particle masses have one and the same cosmological origin and the flat space limit corresponds to the $\sigma$-model version of the standard model. The red shift formula and Hubble law are obtained under the assumption of homogeneous matter distribution. We show that the considered theory leads to a very small vacuum density of the Higgs field $\rho_\phi^{Cosmic}=10^{-34}\rho_{cr}$ in contrast with the theory with the Higgs potential $\rho_\phi^{Higgs}=10^{54}\rho_{cr}$. 
  Let $(M,g)$ be a time oriented Lorentzian manifold and $d$ the Lorentzian distance on $M$. The function $\tau(q):=\sup_{p< q} d(p,q)$ is the cosmological time function of $M$, where as usual $p< q$ means that $p$ is in the causal past of $q$. This function is called regular iff $\tau(q) < \infty$ for all $q$ and also $\tau \to 0$ along every past inextendible causal curve. If the cosmological time function $\tau$ of a space time $(M,g)$ is regular it has several pleasant consequences: (1) It forces $(M,g)$ to be globally hyperbolic, (2) every point of $(M,g)$ can be connected to the initial singularity by a rest curve (i.e., a timelike geodesic ray that maximizes the distance to the singularity), (3) the function $\tau$ is a time function in the usual sense, in particular (4) $\tau$ is continuous, in fact locally Lipschitz and the second derivatives of $\tau$ exist almost everywhere. 
  Wormholes and black holes have traditionally been treated a quite separate objects with relatively little overlap. The possibility of a connection arises in that wormholes, if they exist, might have profound influence on black holes, their event horizons, and their internal structure. After discussing some connections, we embark on an overview of what can generally be said about traversable wormhole throats. We discuss the violations of the energy conditions that typically occur at and near the throat of any traversable wormhole and emphasize the generic nature of this result. We discuss the original Morris-Thorne wormhole and its generalization to a spherically symmetric time-dependent wormhole, and also discuss spherically symmetric Brans-Dicke wormholes. We also discuss the relationship with the topological censorship theorem. Finally we turn to a rather general class of wormholes that permit explicit analysis: generic static traversable wormholes (without any symmetry). We define the wormhole throat in terms of a 2--dimensional constant-time hypersurface of minimal area. (Zero trace for the extrinsic curvature plus a ``flare--out'' condition.) This enables us to derive generalized theorems regarding violations of the energy conditions---theorems that do not involve geodesic averaging but nevertheless apply to situations much more general than the spherically symmetric Morris-Thorne traversable wormhole. [For example: the null energy condition (NEC), when suitably weighted and integrated over the wormhole throat, must be violated.] 
  The well known monopole solution of Barriola and Vilenkin (BV) resulting from the breaking of a global SO(3) symmetry is extended in general relativity along with a zero mass scalar field and also in Brans-Dicke(BD) theory of gravity.In the case of BD theory, the behaviour of spacetime and other variables such as BD scalar field and the monopole energy density have been studied numerically.For monopole along with a zero mass scalar field, exact solutions are obtained and depending upon the choice of arbitary parameters, the solutions either reduce to the BV case or to a pure scalar field solution as special cases.It is interesting to note that unlike the BV case the global monopole in the BD theory does exert gravitational pull on a test particle moving in its spacetime. 
  By suitably re-scaling the conformal Einstein's equations we are able to apply recent results in the theory of PDE, and prove that they possess slow solutions in a future neighborhood of an initial surface reaching ${\cal I}^+$. The structure of the equations obtained allows to split (up to any given order) the initial data into those generating slow solutions, i.e., those driven by the sources, and those generating fast solutions, i.e., those which represent gravitational radiation with no relation to the sources. Thus effectively resulting in a proposal to prescribe initial data for solutions with no extra radiation up to the order needed for each given application. 
  We show that the 3+1 vacuum Einstein field equations in Ashtekar's variables constitutes a first order symmetric hyperbolic system for arbitrary but fixed lapse and shift fields, by suitable adding to the system terms proportional to the constraint equations. 
  We examine the possibility that Friedman-Robertson-Walker evolution is governed by an effective (rather then by the actual) energy density. A concrete example is provided by $\Lambda=0$ Regge-Teitelboim cosmology, where critical cosmology only requires subcritical matter density $(\Omega_{m}<1)$, and the age of a flat matter dominated Universe gets enhanced by a factor of 9/8. Dual to the mature dilute Universe is the embryo Universe, the evolution of both is governed by $P_{eff}=-{1/9}\rho_{eff}$. 
  A quantum mechanical description of black hole states proposed recently within non-perturbative quantum gravity is used to study the emission and absorption spectra of quantum black holes. We assume that the probability distribution of states of the quantum black hole is given by the ``area'' canonical ensemble, in which the horizon area is used instead of energy, and use Fermi's golden rule to find the line intensities. For a non-rotating black hole, we study the absorption and emission of s-waves considering a special set of emission lines. To find the line intensities we use an analogy between a microscopic state of the black hole and a state of the gas of atoms. 
  A `black hole sector' of non-perturbative canonical quantum gravity is introduced. The quantum black hole degrees of freedom are shown to be described by a Chern-Simons field theory on the horizon. It is shown that the entropy of a large non-rotating black hole is proportional to its horizon area. The constant of proportionality depends upon the Immirzi parameter, which fixes the spectrum of the area operator in loop quantum gravity; an appropriate choice of this parameter gives the Bekenstein-Hawking formula S = A/4*l_p^2. With the same choice of the Immirzi parameter, this result also holds for black holes carrying electric or dilatonic charge, which are not necessarily near extremal. 
  The problem of finding the quantum theory of the gravitational field, and thus understanding what is quantum spacetime, is still open. One of the most active of the current approaches is loop quantum gravity. Loop quantum gravity is a mathematically well-defined, non-perturbative and background independent quantization of general relativity, with its conventional matter couplings. The research in loop quantum gravity forms today a vast area, ranging from mathematical foundations to physical applications. Among the most significative results obtained are: (i) The computation of the physical spectra of geometrical quantities such as area and volume; which yields quantitative predictions on Planck-scale physics. (ii) A derivation of the Bekenstein-Hawking black hole entropy formula. (iii) An intriguing physical picture of the microstructure of quantum physical space, characterized by a polymer-like Planck scale discreteness. This discreteness emerges naturally from the quantum theory and provides a mathematically well-defined realization of Wheeler's intuition of a spacetime ``foam''. Long standing open problems within the approach (lack of a scalar product, overcompleteness of the loop basis, implementation of reality conditions) have been fully solved. The weak part of the approach is the treatment of the dynamics: at present there exist several proposals, which are intensely debated. Here, I provide a general overview of ideas, techniques, results and open problems of this candidate theory of quantum gravity, and a guide to the relevant literature. 
  A model of the gravitational dipole is proposed in a close analogy to that of the global monopole. The physical properties and the range of validity of the model are examined as is the motion of test particles in the dipole background. It is found that the metric of the gravitational dipole describes a curved space-time, so one would expect it to have a more pronounced effect on the motion of the test particles than the spinning cosmic string. It is indeed so and in the generic case the impact of repulsive centrifugal force results in a motion whose orbits when projected on the equatorial plane represent unfolding spirals or hyperbolas. Only in one special case these projections are straight lines, pretty much in a manner observed in the field of the spinning cosmic string. Even if open, the orbits are nevertheless bounded in the angular coordinate $\theta$. 
  This note summarizes a model-independent analysis of the age of the universe problem that trades off precision in favour of robustness: The energy conditions of Einstein gravity are designed to extract as much information as possible from classical general relativity without specifying a particular equation of state. This is particularly useful in a cosmological setting, where the equation of state for the cosmological fluid is extremely uncertain. The strong energy condition (SEC) provides a simple and robust bound on the behaviour of the look-back time as a function of red-shift. Observation suggests that the SEC may be violated sometime between the epoch of galaxy formation and the present. 
  Perturbation techniques can be used as an alternative to supercomputer calculations in calculating gravitational radiation emitted by colliding black holes, provided the process starts with the black holes close to each other. We give a summary of the method and of the results obtained for various initial configurations, both axisymmetric and without symmetry: Initially static, boosted towards each other, counter-rotating, or boosted at an angle (pseudo-inspiral). Where applicable, we compare the perturbation results with supercomputer calculations. 
  Theorems on the emission of massless scalar particles by the CGHS black hole are presented. The convergence of the mean number of particles created spontaneously in an arbitrary state is studied and shown to be strongly dependent on the infrared behavior of this state. A bound for this quantity is given and its asymptotic forms close to the horizon and far from the black hole are investigated. The physics of a wave packet is analysed in some detail in the black-hole background. It is also shown that for some states the mean number of created particles is not thermal close to the horizon. These states have a long queue extending far from the black hole, or are unlocalised in configuration space. 
  We report and comment upon the principal results of an investigation into the evolution of rotating black holes emitting massless scalar radiation via the Hawking process. It is demonstrated that a Kerr black hole evaporating by the emission of scalar radiation will evolve towards a state with $a \approx 0.555M$. If the initial specific angular momentum is larger than this value the hole will spin down to this value; if it is less it will spin up to this value. The addition of higher spin fields to the picture strongly suggests the final asymptotic state of a realistic evaporation process will be characterized by an $a/M = 0$. 
  We present the findings of an investigation of critical behavior in the collapse of spherically symmetric distributions of massive scalar field. Two distinct types of phase transition are observed at the verge of black hole formation and a criterion for determining when each type of transition will occur is given. 
  We summarize the results of an investigation into the late time behavior of massless scalar fields propagating on spherically symmetric black hole spacetimes with a non-zero cosmological constant. The compatibility of these results with the `minimal requirement' of Brady and Poisson is commented upon. 
  This work introduces a new space $\T'_*$ of `vertex-smooth' states for use in the loop approach to quantum gravity. Such states provide a natural domain for Euclidean Hamiltonian constraint operators of the type introduced by Thiemann (and using certain ideas of Rovelli and Smolin). In particular, such operators map $\T'_*$ into itself, and so are actual operators in this space. Their commutator can be computed on $\T'_*$ and compared with the classical hypersurface deformation algebra. Although the classical Poisson bracket of Hamiltonian constraints yields an inverse metric times an infinitesimal diffeomorphism generator, and despite the fact that the diffeomorphism generator has a well-defined non-trivial action on $\T'_*$, the commutator of quantum constraints vanishes identically for a large class of proposals. 
  We study different models of radiating slowly rotating bodies up to the first order in the angular velocity. It is shown that up to this order the evolution of the eccentricity is highly model-dependent even for very compact objects. 
  We point out several features of the quantum Hamiltonian constraints recently introduced by Thiemann for Euclidean gravity. In particular we discuss the issue of the constraint algebra and of the quantum realization of the object $q^{ab}V_b$, which is classically the Poisson Bracket of two Hamiltonians. 
  A theory is developed which attempts to reconcile the measurements of nonlocal quantum observables with special relativity and quantum mechanics. The collapse of a wave function, which coincides with a nonlocal measurement by some macroscopic measuring device, is associated with the triggering of an absorber mechanism due to the interaction of the apparatus with the charges in the rest of the universe. The standard retarded electromagnetic field plus radiation damping is converted, for a short time during the collapse of the wave function, to an advanced field plus radiation. The reversal of the arrow of time during the wave function reduction permits communication in nonlocal quantum experiments at the speed of light, resolving paradoxes associated with measurements of correlated quantum states and special relativity. The absorber mechanism and the advanced field solution are consistent with conventional Friedmann-Robertson-Walker expanding universes. 
  The ``reliability horizon'' for semi-classical quantum gravity quantifies the extent to which we should trust semi-classical quantum gravity, and gives a handle on just where the ``Planck regime'' resides. The key obstruction to pushing semi-classical quantum gravity into the Planck regime is often the existence of large metric fluctuations, rather than a large back-reaction. 
  We study the deformation quantisation (Moyal quantisation) of general constrained Hamiltonian systems. It is shown how second class constraints can be turned into first class quantum constraints. This is illustrated by the O(N) non-linear $\sigma$-model. Some new light is also shed on the Dirac bracket. Furthermore, it is shown how classical constraints not in involution with the classical Hamiltonian, can be turned into quantum constraints {\em in} involution with respect to the Hamiltonian. Conditions on the existence of anomalies are also derived, and it is shown how some kinds of anomalies can be removed. The equations defining the set of physical states are also given. It turns out that the deformation quantisation of pure Yang-Mills theory is straightforward whereas gravity is anomalous. A formal solution to the Yang-Mills quantum constraints is found. In the \small{ADM} formalism of gravity the anomaly is very complicated and the equations picking out physical states become infinite order functional differential equations, whereas the Ashtekar variables remedy both of these problems -- the anomaly becoming simply a central extension (Schwinger term) and the equations for physical states become finite order. We finally elaborate on the underlying geometrical structure and show the method to be compatible with BRST methods. 
  According to the cosmic censorship hypothesis of Penrose, naked singularities should never occur in realistic collapse situations. One of the major open problems in this context is the existence of a naked singularity in the Kerr solution with |a|>m; this singularity can be interpreted as the final product of collapse of a rapidly rotating object. Assuming that certain very general and physically reasonable conditions hold, we show here, using the global techniques, that a realistic gravitational collapse of any rotating object, which develops from a regular initial state, cannot lead to the formation of a final state resembling the Kerr solution with a naked singularity. This result supports the validity of the cosmic censorship hypothesis. 
  We present numerical results on the hydrodynamic stability of coalescing binary stars in the first post Newtonian(1PN) approximation of general relativity. We pay particular attention to the hydrodynamical instability of corotating binary stars in equilibrium states assuming the stiff polytropic equation of state with the adiabatic constant $\Gamma=3$. In previous 1PN numerical studies on corotating binary stars in equilibrium states, it was found that along the sequence of binary stars as a function of the orbital separation, they have the energy and/or angular momentum minima where the secular instability sets in, and that with increase of the 1PN correction, the orbital separation at these minima decreases while the angular velocity there increases. In this paper, to know the location of the innermost stable circular orbit(ISCO), we perform numerical simulations and find where the hydrodynamical instability along the corotating sequences of binary sets in. From the numerical results, we found that the dynamical stability limit seems to exist near the energy and/or angular momentum minima not only in the Newtonian, but also in the 1PN cases. This means that the 1PN effect of general relativity increases the angular frequency of gravitational waves at the ISCO. 
  A criterion given by Castejon-Amenedo and MacCallum (1990) for the existence of (locally) hypersurface-orthogonal generators of an orthogonally-transitive two-parameter Abelian group of motions (a $G_2I$) in spacetime is re-expressed as a test for linear dependence with constant coefficients between the three components of the metric in the orbits in canonical coordinates. In general, it is shown that such a relation implies that the metric is locally diagonalizable in canonical coordinates, or has a null Killing vector, or can locally be written in a generalized form of the `windmill' solutions characterized by McIntosh. If the orbits of the $G_2I$ have cylindrical or toroidal topology and a periodic coordinate is used, these metric forms cannot in general be realized globally as they would conflict with the topological identification. The geometry then has additional essential parameters, which specify the topological identification. The physical significance of these parameters is shown by their appearance in global holonomy and by examples of exterior solutions where they have been related to characteristics of physical sources. These results lead to some remarks about the definition of cylindrical symmetry. 
  In a recent paper Kent has pointed out that in consistent histories quantum theory it is possible, given initial and final states, to construct two different consistent families of histories, in each of which there is a proposition that can be inferred with probability one, and such that the projectors representing these two propositions are mutually orthogonal. In this note we stress that, according to the rules of consistent history reasoning two such propositions are not contrary in the usual logical sense namely, that one can infer that if one is true then the other is false, and both could be false. No single consistent family contains both propositions, together with the initial and final states, and hence the propositions cannot be logically compared. Consistent histories quantum theory is logically consistent, consistent with experiment as far as is known, consistent with the usual quantum predictions for measurements, and applicable to the most general physical systems. It may not be the only theory with these properties, but in our opinion, it is the most promising among present possibilities. 
  We present here analytical solutions of General Relativity that describe evolving wormholes with a non-constant redshift function. We show that the matter that threads these wormholes is not necessarily exotic. Finally, we investigate some issues concerning WEC violation and human traversability in these time-dependent geometries. 
  A generalized Clifford manifold is proposed in which there are coordinates not only for the basis vector generators, but for each element of the Clifford group, including the identity scalar. These new quantities are physically interpreted to represent internal structure of matter (e.g. classical or quantum spin). The generalized Dirac operator must now include differentiation with respect to these higher order geometric coordinates. In a Riemann space, where the magnitude and rank of geometric objects are preserved under displacement, these new terms modify the geodesics. One possible physical interpretation is natural coupling of the classical spin to linear motion, providing a new derivation of the Papapetrou equations. A generalized curvature is proposed for the Clifford manifold in which the connection does not preserve the rank of a multivector under parallel transport, e.g. a vector may be ``rotated'' into a scalar. 
  A systematic rotation of the plane of polarization of electromagnetic radiation propagating over cosmological distances has been claimed to be detected by Nodland and Ralston.Taking the finite rotating universe of Oszvath and Sch\"ucking as a toy model for the actual universe,we calculate the rotation of the plane of polarization of electromagnetic waves in this universe.It turns out that the observed data are consistent with the Oszvath-Sch\"ucking parameter $k=0.39$. 
  A recently proposed nonlinear transport equation is used to model bulk viscous cosmologies that may be far from equilibrium, as happens during viscous fluid inflation or during reheating. The asymptotic stability of the de Sitter and Friedmann solutions is investigated. The former is stable for bulk viscosity index $q<1$ and the latter for $q>1$. New solutions are obtained in the weakly nonlinear regime for $q=1$. These solutions are singular and some of them represent a late-time inflationary era. 
  We introduce the set of constraints the wave function of the Universe has to satisfy in order to describe an Universe undergoing through the process of spontaneous breaking of supersymmetry and discuss the way this may lead to the emergence of classical spacetime. 
  A general relativistic version of the Euler equation for perfect fluid hydrodynamics is applied to a system of two neutron stars orbiting each other. In the quasi-equilibrium phase of the evolution of this system, a first integral of motion can be derived for certain velocity fields of the neutron star fluid including the (academic) case of co-rotation with respect to the orbital motion (synchronized binaries) and the realistic case of counter-rotation with respect to the orbital motion. The velocity field leading to this latter configuration can be computed by solving three-dimensional vector and scalar Poisson equations. 
  Recently is was shown that the imaginary part of the canonical partition function of Schwarzschild black holes with an energy spectrum E_n = \sigma \sqrt{n} E_P, n= 1,2, ..., has properties which - naively interpreted - leads to the expected unusual thermodynamical properties of such black holes (Hawking temperature, Bekenstein-Hawking entropy etc). The present paper interprets the same imaginary part in the framework of droplet nucleation theory in which the rate of transition from a metastable state to a stable one is proportional to the imaginary part of the canonical partition function. The conclusions concerning the emerging thermodynamics of black holes are essentially the same as before. The partition function for black holes with the above spectrum was calculated exactly recently. It is the same as that of the primitive Ising droplet model for nucleation in 1st-order phase transitions in 2 dimensions. Thus one might learn about the quantum statistics of black holes by studying that Ising model, the exact complex free energy of which is presented here for negative magnetic fields, too. 
  We demonstrate that in the absence of `matter' fields to all orders of perturbation theory and for all 2D dilaton theories the quantum effective action coincides with the classical one. This resolves the apparent contradiction between the well established results of Dirac quantization and perturbative (path-integral) approaches which seemed to yield non-trivial quantum corrections. For the Jackiw--Teitelboim (JT) model, our result is even extended to the situation when a matter field is present. 
  The energy conditions of classical Einstein gravity fail once quantum effects are introduced. These quantum violations of the energy conditions are not subtle high-energy Planck scale effects. Rather the quantum violations of the energy conditions already occur in semiclassical quantum gravity and are first-order O(\hbar) effects. Quantum violations of the energy conditions are widespread, albeit small. 
  We analyse the evolution of the rotational type cosmological perturbation in a gravity with general quadratic order gravitational coupling terms. The result is expressed independently of the generalized nature of the gravity theory, and is simply interpreted as a conservation of the angular momentum. 
  We study solutions of the Wheeler-DeWitt equation obtained when considering homogeneous and isotropic (up to a gauge transformation) field configurations of the Einstein-Yang-Mills system in $D=4+d$ dimensions with an $R \times S^3 \times S^d$ topology and assuming the Hartle-Hawking boundary conditions. 
  This paper investigates the non-linear self-interaction of quadrupole gravitational waves generated by an isolated system. The vacuum Einstein field equations are integrated in the region exterior to the system by means of a post-Minkowskian algorithm. Specializing in the quadrupole-quadrupole interaction (at the quadratic non-linear order), we recover the known results concerning the non-local modification of the ADM mass-energy of the system accounting for the emission of quadrupole waves, and the non-local memory effect due to the re-radiation by the stress-energy distribution of linear waves. Then we compute all the local (instantaneous) terms which are associated in the quadrupole-quadrupole metric with the latter non-local effects. Expanding the metric at large distances from the system, we obtain the corresponding radiation-field observables, including all non-local and transient contributions. This permits notably the completion of the observable quadrupole moment at the 5/2 post-Newtonian order. 
  The tails of gravitational waves are caused by scattering of linear waves onto the space-time curvature generated by the total mass-energy of the source. Quite naturally, the tails of tails are caused by curvature scattering of the tails of waves themselves. The tails of tails are associated with the cubic non-linear interaction between two mass monopole moments and, dominantly, the mass quadrupole of the source. In this paper we determine the radiation field at large distances from the source for this particular monopole-monopole-quadrupole interaction. We find that the tails of tails appear at the third post-Newtonian (3PN) order beyond the usual quadrupole radiation. Motivated by the need of accurate templates to be used in the data analysis of future detectors of gravitational waves, we compute the contribution of tails, and of tails of tails, up to the 3.5PN order in the energy flux generated by inspiraling compact binaries. 
  We construct the regularised Wheeler-De Witt operator demanding that the algebra of constraints of quantum gravity is anomaly free. We find that for a subset of all wavefunctions being integrals of scalar densities this condition can be satisfied. We proceed to finding exact solutions of quantum gravity being of the form of functionals of volume and average curvature of compact three-manifold. 
  The prime candidate of LIGO/VIRGO sources of gravitational waves is the spiral in of black holes and neutron stars in compact binaries. While the early stages of the evolution of compact binaries is computable from post-Newtonian calculations, prediction of their late stages requires large scale numerical simulation. A fully covariant and strictly hyperbolic formulation for numerical relativity is described, and illustrated in a one-dimensional computation of a Gowdy-wave on the three-torus. This formulation allows foliations in full generality, in particular it poses no restriction on the lapse function. 
  By employing the Bianchi identities for the Riemann tensor in conjunction with the Einstein equations, we construct a first order symmetric hyperbolic system for the evolution part of the Cauchy problem of general relativity. In this system, the metric evolves at zero speed with respect to observers at rest in a foliation of spacetime by spacelike hypersurfaces while the curvature and connection propagate at the speed of light. The system has no unphysical characteristics, and matter sources can be included. 
  We consider the thermodynamic properties of the constant curvature black hole solution recently found by Banados. We show that it is possible to compute the entropy and the quasilocal thermodynamics of the spacetime using the Einstein-Hilbert action of General Relativity. The constant curvature black hole has some unusual properties which have not been seen in other black hole spacetimes. The entropy of the black hole is not associated with the event horizon; rather it is associated with the region between the event horizon and the observer. Further, surfaces of constant internal energy are not isotherms so the first law of thermodynamics exists only in an integral form. These properties arise from the unusual topology of the Euclidean black hole instanton. 
  We study numerically the fully nonlinear spherically-symmetric collapse of a self-gravitating, minimally-coupled, massless scalar field. Our numerical code is based on double-null coordinates and on free evolution of the metric functions and the scalar field. The numerical code is stable and second-order accurate. We use this code to study the late-time asymptotic behavior at fixed $r$ (outside the black hole), along the event horizon, and along future null infinity. In all three asymptotic regions we find that, after the decay of the quasi-normal modes, the perturbations are dominated by inverse power-law tails. The corresponding power indices agree with the integer values predicted by linearized theory. We also study the case of a charged black hole nonlinearly perturbed by a (neutral) self-gravitating scalar field, and find the same type of behavior---i.e., quasi-normal modes followed by inverse power-law tails, with the same indices as in the uncharged case. 
  We study the Cauchy horizon (CH) singularity of a spherical charged black hole perturbed nonlinearly by a self-gravitating massless scalar field. We show numerically that the singularity is weak both at the early and at the late sections of the CH, where the focusing of the area coordinate $r$ is strong. In the early section the metric perturbations vanish, and the fields behave according to perturbation analysis. We find exact analytical expressions for the gradients of $r$ and of the scalar field, which are valid at both sections. We then verify these analytical results numerically. 
  Recently a simple solution of the vacuum Einstein-Maxwell field equations was given describing a plane electromagnetic shock wave sharing its wave front with a plane gravitational impulse wave. We present here an exact solution of the vacuum Einstein-Maxwell field equations describing the head-on collision of such a wave with a plane gravitational impulse wave. The solution has the Penrose-Khan solution and a solution obtained by Griffiths as separate limiting cases. 
  Static spherically symmetric solutions in SU(N)-EYM and EYMD theories are classified by the node numbers of their non-trivial gauge field functions. With increasing node numbers, the solutions form sequences, tending to limiting solutions. The limiting solutions are based on subalgebras of $su(N)$, consisting of a neutral non-abelian part and a charged abelian part, belonging to the Cartan subalgebra 
  We discuss the recently discovered new class of globally regular and black hole solutions in Einstein-Yang-Mills and Einstein-Yang-Mills-dilaton theory. These asymptotically flat solutions are static and possess only axial symmetry. The black hole solutions possess a regular event horizon. 
  `Gravitational memory' refers to the possibility that, in cosmologies with a time-varying gravitational `constant', objects such as black holes may retain a memory of conditions at the time of their birth. We consider this phenomenon in a different physical scenario, where the objects under consideration are boson stars. We construct boson star solutions in scalar-tensor gravity theories, and consider their dependence on the asymptotic value of the gravitational strength. We then discuss several possible physical interpretations, including the concept of pure gravitational stellar evolution. 
  We describe a class of impulsive gravitational waves which propagate either in a de Sitter or an anti-de Sitter background. They are conformal to impulsive waves of Kundt's class. In a background with positive cosmological constant they are spherical (but non-expanding) waves generated by pairs of particles with arbitrary multipole structure propagating in opposite directions. When the cosmological constant is negative, they are hyperboloidal waves generated by a null particle of the same type. In this case, they are included in the impulsive limit of a class of solutions described by Siklos that are conformal to pp-waves. 
  The thermal nature of the propagator in a collapsed black-hole spacetime is shown to follow from the non-trivial topology of the configuration space in tortoise coordinates by using the path integral formalism. 
  The somewhat fragmented body of current literature analyzing the properties of test particle motion in static and stationary spacetimes and in general spacetimes is pulled together and clarified using the framework of gravitoelectromagnetism. 
  Black holes of constant curvature are constructed by identifying points in anti-de Sitter space. In n dimensions the resulting topology is R^{n-1} * S_1, as opposed to the usual R^2 * S_{n-2} Schwarzschild black hole. 
  We show that there exist maximal globally hyperbolic solutions of the Einstein-dust equations which admit a constant mean curvature Cauchy surface, but are not covered by a constant mean curvature foliation. 
  A multidimensional model with (at most) one curved factor space and n Ricci-flat internal spaces is considered, with arbitrary numbers of dilatonic scalar fields and antisymmetric forms of both electric and magnetic types, associated with p-branes in theories like M-theory. The problem setting covers, in particular, homogeneous cosmologies, static, spherically symmetric and Euclidean models. Exact solutions are obtained when the p-brane dimensions and the dilatonic couplings obey orthogonality conditions in minisuperspace. Conditions for black hole and wormhole existence among static models are formulated. For black holes, a kind of no-hair theorem, leading to F-form selection, is obtained; it is shown that even in spaces with multiple times a black hole may only exist with its unique, one-dimensional time; infinite Hawking temperature is explicitly shown to imply a curvature singularity at an assumed horizon, and such cases among extreme black-hole solutions are indicated. 
  We discuss quantum inequalities for minimally coupled scalar fields in static spacetimes. These are inequalities which place limits on the magnitude and duration of negative energy densities. We derive a general expression for the quantum inequality for a static observer in terms of a Euclidean two-point function. In a short sampling time limit, the quantum inequality can be written as the flat space form plus subdominant correction terms dependent upon the geometric properties of the spacetime. This supports the use of flat space quantum inequalities to constrain negative energy effects in curved spacetime. Using the exact Euclidean two-point function method, we develop the quantum inequalities for perfectly reflecting planar mirrors in flat spacetime. We then look at the quantum inequalities in static de~Sitter spacetime, Rindler spacetime and two- and four-dimensional black holes. In the case of a four-dimensional Schwarzschild black hole, explicit forms of the inequality are found for static observers near the horizon and at large distances. It is show that there is a quantum averaged weak energy condition (QAWEC), which states that the energy density averaged over the entire worldline of a static observer is bounded below by the vacuum energy of the spacetime. In particular, for an observer at a fixed radial distance away from a black hole, the QAWEC says that the averaged energy density can never be less than the Boulware vacuum energy density. 
  We consider the possibility of setting up a new version of Regge calculus in four dimensions with areas of triangles as the basic variables rather than the edge-lengths. The difficulties and restrictions of this approach are discussed. 
  We describe how the reduced phase space quantization of the CGHS model of 2d black hole formation allows one to calculate the backreaction up to any finite order in matter loops. We then analyze the backreaction in the Hawking radiation up to two loops. At one-loop order the Hawking temperature does not change, while at two-loop order the temperature increases three times with respect to the classical value. We argue that such a behavior is consistent with the behaviour of the operator quantization Hawking flux for times which are not too late. For very late times the two-loop flux goes to zero which indicates that the backreaction can stop the black hole evaporation. 
  Suggested theory involves a drastic revision of a role of local internal symmetries in physical concept of curved geometry. Under the reflection of fields and their dynamics from Minkowski to Riemannian space a standard gauge principle of local internal symmetries is generalized. The gravitation gauge group is proposed, which is generated by hidden local internal symmetries. The developed mechanism enables one to infer Einstein's equation of gravitation, but only with strong difference from Einstein's theory at the vital point of well-defined energy-momentum tensor of gravitational field and conservation laws. The gravitational interaction as well as general distortion of manifold G(2.2.3) with hidden group U(1) was considered. 
  In the absence of a tractable theory of quantum gravity, quantum matter field effects have been so far computed by treating gravity at the Background Field Approximation. The principle aim of this paper is to investigate the validity of this approximation which is not specific to gravity. To this end, for reasons of simplicity and clarity, we shall compare the descriptions of thermal processes induced by constant acceleration (i.e. the Unruh effect) in four dynamical frameworks. In this problem, the position of the ``heavy'' accelerated system plays the role of gravity. In the first framework, the trajectory is treated at the BFA: it is given from the outset and unaffected by radiative processes. In the second one, recoil effects induced by these emission processes are taken into account by describing the system's position by WKB wave functions. In the third one, the accelerated system is described by second quantized fields and in the fourth one, gravity is turned on. It is most interesting to see when and why transitions amplitudes evaluated in different frameworks but describing the same process do agree. It is indeed this comparison that determines the validity of the BFA. It is also interesting to notice that the abandonment of the BFA delivers new physical insights concerning the processes. For instance, in the fourth framework, the ``recoils'' of gravity show that the acceleration horizon area acts as an entropy in delivering heat to accelerated systems. 
  An example is described in which an asymptotically flat static vacuum Weyl space-time experiences a sudden change across a null hypersurface in the multipole moments of its isolated axially symmetric source. A light-like shell and an impulsive gravitational wave are identified, both having the null hypersurface as history. The stress-energy in the shell is dominated (at large distance from the source) by the jump in the monopole moment (the mass) of the source with the jump in the dipole moment mainly responsible for the stress being anisotropic. The gravitational wave owes its existence prrincipally to the jump in th quadrupole moment of the source confirming what would be expected. This serves as a model of a cataclysmic astrophysical event such as a supernova. 
  We present a general and unified formulation which can handle the classical evolution and quantum generation processes of the cosmological gravitational wave in a broad class of generalized gravity theories. Applications are made in several inflation models based on the scalar-tensor theory, the induced gravity, and the low energy effective action of string theory. The gravitational wave power spectrums based on the vacuum expectation value of the quantized fluctuating metric during the pole-like inflation stages are derived in analytic forms. Assuming that the gravity theory transits to Einstein one while the relevant scales remain in the superhorizon scale, we derive the consequent power spectrums and the directional fluctuations of the relic radiation produced by the gravitational wave. The spectrums seeded by the vacuum fluctuations in the pole-like inflation models based on the generalized gravity show a distinguished common feature which differs from the scale invariant spectrum generated in an exponential inflation in Einstein gravity which is supported by observations. 
  It is claimed that it will be exceedingly unlikely to obtain CTC's in the spacetime of a spinning EYMH-string. It is conjectured that the pathological problems concerning the induced angular momentum and helical structure of time that afflict the U(1)-gauge string, will be solved in the non-abelian YM string model. In the pure YM case we find regular solutions, which resembles the abelian counterpart solutions but without causality violating regions. Just as in the spherical symmetric case, there will be probably critical behavior of the field equations at the threshold of blackhole masses. 
  The gravitational radiation backreaction effects are considered in the Lense-Thirring approximation. New methods for parameterizing the orbit and for averaging the instantaneous radiative losses are developed. To first order in the spin S of the black hole, both in the absence and in the presence of gravitational radiation, a complete description of the test-particle orbit is given. This is achieved by two improvements over the existing descriptions. First, by introducing new angle variables with a straightforward geometrical meaning. Second, by finding a new parametrization of a generic orbit, which assures that the integration over a radial period can be done in an especially simple way, by applying the residue theorem. The instantaneous radiation losses of the system are computed using the formulation of Blanchet, Damour and Iyer(1989). All losses are given both in terms of the dynamical constants of motion and the properly defined orbital elements $a,e,\iota$ and $\Psi_0$. The radiative losses of the constants characterizing the Lense-Thirring motion, when suitably converted, are in agreement with earlier results of Kidder, Will and Wiseman(1993), Ryan(1996) and Shibata(1994). In addition, the radiative losses of two slowly changing orbital elements $\Psi_0,\Phi_0$ are given in order to complete the characterization of the orbit. 
  We present a deductive theory of space-time which is realistic, objective, and relational. It is realistic because it assumes the existence of physical things endowed with concrete properties. It is objective because it can be formulated without any reference to cognoscent subjects or sensorial fields. Finally, it is relational because it assumes that space-time is not a thing but a complex of relations among things. In this way, the original program of Leibniz is consummated, in the sense that space is ultimately an order of coexistents, and time is an order of succesives. In this context, we show that the metric and topological properties of Minkowskian space-time are reduced to relational properties of concrete things. We also sketch how our theory can be extended to encompass a Riemmanian space-time. 
  General relativistic spherically symmetric matter field with a vanishing stress energy scalar is analyzed. Procedure for generating exact solutions of the field equations for such matter distributions is given. It is further pointed out that all such type I spherically symmetric fields with distinct eignvalues in the radial two space can be treated as a mixture of isotropic and directed radiations. Various classes of exact solutions are given. Junction conditions for such a matter field to the possible exterior solutions are also discussed. 
  Choptuik has demonstrated that naked singularities can arise in gravitational collapse from smooth, asymptotically flat initial data, and that such data have codimension one in spherical symmetry. Here we show, for perfect fluid matter with equation of state $p=\rho/3$, by perturbing around spherical symmetry, that such data have in fact codimension one in the full phase space, at least in a neighborhood of spherically symmetric data. 
  The quantum black hole model with a self-gravitating spherically symmetric thin dust shell as a source is considered. The shell Hamiltonian constraint is written and the corresponding Schroedinger equation is obtained. This equation appeared to be a finite differences equation. Its solutions are required to be analytic functions on the relevant Riemannian surface. The method of finding discrete spectra is suggested based on the analytic properties of the solutions. The large black hole approximation is considered and the discrete spectra for bound states of quantum black holes and wormholes are found. They depend on two quantum numbers and are, in fact, quasi-continuous. The quantum black hole bound state depends not only on mass but also on additional quantum number, and black holes with the same mass have different quantum hairs. These hairs exhibit themselves at the Planckian distances near the black hole horizon. For the observer who can not measure the distances smaller than the Planckian length the black hole has the only parameter, its mass. The other, non-measurable parameter leads to existence of the black hole entropy. 
  We review the status of the weak cosmic censorship conjecture, which asserts, in essence, that all singularities of gravitational collapse are hidden within black holes. Although little progress has been made toward a general proof (or disproof) of this conjecture, there has been some notable recent progress in the study of some examples and special cases related to the conjecture. These results support the view that naked singularities cannot arise generically. 
  New exact solutions emerge by replacing the dust source of the Lem\^aitre-Tolman-Bondi metrics with a viscous fluid satisfying the monatomic gas equation of state. The solutions have a consistent thermodynamical interpretation. The most general transport equation of Extended Irreversible Thermodynamics is satisfied, with phenomenological coefficients bearing a close resemblance to those characterizing a non relativistic Maxwell-Bolzmann gas. 
  The analytic structure of the Regge action on a cone in $d$ dimensions over a boundary of arbitrary topology is determined in simplicial minisuperspace. The minisuperspace is defined by the assignment of a single internal edge length to all 1-simplices emanating from the cone vertex, and a single boundary edge length to all 1-simplices lying on the boundary. The Regge action is analyzed in the space of complex edge lengths, and it is shown that there are three finite branch points in this complex plane. A closed contour of integration encircling the branch points is shown to yield a convergent real wave function. This closed contour can be deformed to a steepest descent contour for all sizes of the bounding universe. In general, the contour yields an oscillating wave function for universes of size greater than a critical value which depends on the topology of the bounding universe. For values less than the critical value the wave function exhibits exponential behaviour. It is shown that the critical value is positive for spherical topology in arbitrary dimensions. In three dimensions we compute the critical value for a boundary universe of arbitrary genus, while in four and five dimensions we study examples of product manifolds and connected sums. 
  We assume that a self-gravitating string is locally described by a thin tube of matter represented by a ``smoothed conical metric''. If we impose a specific constraint on the model of string then its central line obeys the Nambu-Goto dynamics in the limit where the radius of the tube tends to zero. If no constraint is added then the world sheet of the central line is totally geodesic. 
  An approach to computing, withing the framework of distribution theory, the distributional valued energy-momentum tensor for the Schwarzschild spacetime is disscused. This approach avoids the problems associated with the regularization of singularities in the curvature tensors and shares common features with the by now standard treatment of discontinuities in General Relativity. Finally, the Reissner-Nordstrom spacetime is also considered using the same approach. 
  We present two of our efforts directed toward the numerical analysis of neutron star mergers, which are the most plausible sources for gravitational wave detectors that should begin operating in the near future. First we present Newtonian 3D simulations including radiation reaction (2.5PN) effects. We discuss the gravitational wave signals and luminosity from the merger with/without radiation reaction effects. Second we present the matching problem between post-Newtonian formulations and general relativity in numerical treatments. We prepare a spherical, static neutron star in a post-Newtonian matched spacetime, and find that discontinuities at the matching surface become smoothed out during fully relativistic evolution if we use a proper slicing condition. 
  We examine the advantages of the SO(3)-ADM (Ashtekar) formulation of general relativity, from the point of following the dynamics of the Lorentzian spacetime in direction of applying this into numerical relativity. We describe our strategy how to treat new constraints and reality conditions, together with a proposal of new variables. We show an example of passing a degenerate point in flat spacetime numerically by posing `reality recovering' conditions on spacetime. We also discuss some available advantages in numerical relativity. 
  The post-post-Newtonian (2PN) accurate mass quadrupole moment, for compact binaries of arbitrary mass ratio, moving in general orbits is obtained by the multi-polar post Minkowskian approach of Blanchet, Damour, and Iyer (BDI). Using this, for binaries in general orbits, the 2PN contributions to the gravitational waveform, and the associated far-zone energy and angular momentum fluxes are computed. For quasi-elliptic orbits, the energy and angular momentum fluxes are averaged over an orbital period, and employed to determine the 2PN corrections to the rate of decay of the orbital elements. 
  In some respects the black hole plays the same role in gravitation that the atom played in the nascent quantum mechanics. This analogy suggests that black hole mass $M$ might have a discrete spectrum. I review the physical arguments for the expectation that black hole horizon area eigenvalues are uniformly spaced, or equivalently, that the spacing between stationary black hole mass levels behaves like 1/M. This sort of spectrum has also emerged in a variety of formal approaches to black hole quantization by a number of workers (with some notable exceptions). If true, this result indicates a distortion of the semiclassical Hawking spectrum which could be observable even for macroscopic black holes. Black hole entropy suggests that the mentioned mass levels should be degenerate to the tune of an exponential in $M^2$, as first noted by Mukhanov. This has implications for the statistics of the radiation. I also discuss open questions: whether radiative decay will spread the levels beyond recognition, whether extremal black holes can be described by this scheme, etc. I then describe an elementary algebra for the relevant black hole observables, an outcome of work by Mukhanov and myself, which reproduces the uniformly spaced area spectrum. 
  The equation of motion for domain wall coupled to gravitational field is derived. The domain wall is treated as a source of gravitational field around the wall. The perturbed equation is also obtained with taking account of the gravitational back reaction on the motion of the domain wall. 
  A self-gravitating cylindrical domain wall is considered as an example of non-spherical wall to clarify the interaction between a domain wall and gravitational waves. We consider the time evolution from a momentarily static initial configuration within an infinitesimal time interval using the metric junction formalism. We found that the wall with a large initial radius radiates large amplitude of the gravitational waves and undergoes its large back reaction. 
  Gravitational wave detectors in space, particularly the LISA project, can study a rich variety of astronomical systems whose gravitational radiation is not detectable from the ground, because it is emitted in the low-frequency gravitational wave band (0.1 mHz to 1 Hz) that is inaccessible to ground-based detectors. Sources include binary systems in our Galaxy and massive black holes in distant galaxies. The radiation from many of these sources will be so strong that it will be possible to make remarkably detailed studies of the physics of the systems. These studies will have importance both for astrophysics (most notably in binary evolution theory and models for active galaxies) and for fundamental physics. In particular, it should be possible to make decisive measurements to confirm the existence of black holes and to test, with accuracies better than 1%, general relativity's description of them. Other observations can have fundamental implications for cosmology and for physical theories of the unification of forces. In order to understand these conclusions, one must know how to estimate the gravitational radiation produced by different sources. In the first part of this lecture I review the dynamics of gravitational wave sources, and I derive simple formulas for estimating wave amplitudes and the reaction effects on sources of producing this radiation. With these formulas one can estimate, usually to much better than an order of magnitude, the physics of most of the interesting low-frequency sources. In the second part of the lecture I use these estimates to discuss, in the context of the expected sensitivity of LISA, what we can learn by from observations of binary systems, massive black holes, and the early Universe itself. 
  The space-based gravitational wave detector LISA will observe in the low-frequency gravitational-wave band (0.1 mHz up to 1 Hz). LISA will search for a variety of expected signals, and when it detects a signal it will have to determine a number of parameters, such as the location of the source on the sky and the signal's polarisation. This requires pattern-matching, called matched filtering, which uses the best available theoretical predictions about the characteristics of waveforms. All the estimates of the sensitivity of LISA to various sources assume that the data analysis is done in the optimum way. Because these techniques are unfamiliar to many young physicists, I use the first part of this lecture to give a very basic introduction to time-series data analysis, including matched filtering. The second part of the lecture applies these techniques to LISA, showing how estimates of LISA's sensitivity can be made, and briefly commenting on aspects of the signal-analysis problem that are special to LISA. 
  The quantum analogue of general relativistic geometry should be implementable on smooth manifolds without an a priori metric structure, the kinematical covariance group acting by diffeomorphisms.   Here I approach quantum gravity (QG) in the view of constructive, algebraic quantum field theory (QFT). Comparing QG with usual QFT, the algebraic approach clarifies analogies and peculiarities. As usual, an isotonic net of *-algebras is taken to encode the quantum field operators. For QG, the kinematical covariance group acts via diffeomorphisms on the open sets of the manifold, and via algebraic isomorphisms on the algebras. In general, the algebra of observables is covariant only under a (dynamical) subgroup of the general diffeomorphism group.   After an algebraic implementation of the dynamical subgroup of dilations, small and large scale cutoffs may be introduced algebraically. So the usual a priori conflict of cutoffs with general covariance is avoided. Even more, these cutoffs provide a natural local cobordism for topological quantum field theory.   A new commutant duality between the minimal and maximal algebra allows to extract the modular structure from the net of algebras. The outer modular isomorphisms are then again related to dilations, which (under certain conditions) may provide a notion of time. 
  We study gravitational plane impulsive waves and electromagnetic shock waves in a scalar-tensor theory of gravity of the Brans-Dicke type. In vacuum, we present an exact solution of Brans-Dicke's field equations and give an example in which a plane impulsive gravitational wave and a null shell of matter coexist on the same hypersurface. In the homogenous case, we characterize them by their surface energy density and wave amplitude and discuss the inhomogenous case. We also give an exact solution of the Brans-Dicke's field equations in the electrovacuum case which admits a true curvature singularity and use it to built an example where a plane impulsive gravitational wave and an electromagnetic shock wave have the same null hypersurface as history of their wave fronts and propagate independently and decoupled from a null shell of matter. This last solution is shown to correspond to the space-time describing the interaction region resulting from the collision of two electromagnetic shock waves leading to the formation of two gravitational impulsive waves. The properties of this solution are discussed and compared to those of the Bell-Szekeres solution of general relativity. 
  We will describe here the structure of singularity forming in gravitational collapse of spherically symmetric inhomogeneous dust. Such a collapse is described by the Tolman-Bondi-Lema{\^i}tre metric. The main new result here relates, in a general way, the formation of black holes and naked shell-focusing singularities resulting as the final fate of such a collapse to the generic form of regular initial data. Such a data is characterized in terms of the density and velocity profiles of the matter, specified on an initial time slice from which the collapse commences. We show that given any generic density profile at the initial time slice, there exists a corresponding velocity profile which gives rise to a strong curvature naked singularity. This establishes that strong naked singularities arise for a generic density profile. We also establish here that similar results hold for black hole formation as well. Keeping the model to be spherically symmetric we also consider more general form of matter fields, i.e. equation of state $p = k\rho$. We will analyse here the nature of non-central singularity forming due to collapse of spherically symmetric perfect fluid subject to weak energy condition. 
  We apply the causal interpretation of quantum mechanics to homogeneous and isotropic quantum cosmology where the sources of the gravitational field are either dust or radiation perfect fluids. We find non-singular quantum trajectories which tends to the classical one when the scale factor becomes much larger then the Planck length. In this situation, the quantum potential becomes negligible. There are no horizons. As radiation is a good approximation for the matter content of the early universe, this result suggests that the universe can be eternal due to quantum effects. 
  We study inhomogeneous multidimensional cosmological models with a higher dimensional space-time manifold under dimensional reduction and show that small inhomogeneous excitations of the scale factors of the internal spaces near minima of effective potentials should be observable as massive scalar particles (gravitational excitons) in the external space-time. 
  We study inhomogeneous multidimensional cosmological models with a higher dimensional space-time manifold under dimensional reduction. Stability due to different types of effective potentials is analyzed for specific configurations of internal spaces. Necessary restrictions on the parameters of the models are found and masses of gravitational excitons (small inhomogeneous excitations of the scale factors of the internal spaces near minima of effective potentials) are calculated. 
  A relativistic model for the emission of gravitational waves from an initially unperturbed Schwarzschild black hole, or spherical collapsing configuration, is completely integrated. The model consists basically of gravitational perturbations of the Robinson-Trautman type on the Schwarzschild spacetime. In our scheme of perturbation, gravitational waves may extract mass from the collapsing configuration. Robinson-Trautmann perturbations also include another mode of emission of mass, which we denote shell emission mode: in the equatorial plane of the configuration, a timelike $(1+2)$ shell of matter may be present, whose stress-energy tensor is modelled by neutrinos and strings emitted radially on the shell; no gravitational waves are present in this mode. The invariant characterization of gravitational wave perturbations and of the gravitational wave zone is made through the analysis of the structure of the curvature tensor and the use of the Peeling Theorem. 
  For a quantum field in a curved background the choice of the vacuum state is crucial. We exhibit a vacuum state in which the expectation values of the energy and pressure allow an intuitive interpretation. We apply this general result to the de Sitter universe. 
  A unified first law of black-hole dynamics and relativistic thermodynamics is derived in spherically symmetric general relativity. This equation expresses the gradient of the active gravitational energy E according to the Einstein equation, divided into energy-supply and work terms. Projecting the equation along the flow of thermodynamic matter and along the trapping horizon of a blackhole yield, respectively, first laws of relativistic thermodynamics and black-hole dynamics. In the black-hole case, this first law has the same form as the first law of black-hole statics, with static perturbations replaced by the derivative along the horizon. There is the expected term involving the area and surface gravity, where the dynamic surface gravity is defined as in the static case but using the Kodama vector and trapping horizon. This surface gravity vanishes for degenerate trapping horizons and satisfies certain expected inequalities involving the area and energy. In the thermodynamic case, the quasi-local first law has the same form, apart from a relativistic factor, as the classical first law of thermodynamics, involving heat supply and hydrodynamic work, but with E replacing the internal energy. Expanding E in the Newtonian limit shows that it incorporates the Newtonian mass, kinetic energy, gravitational potential energy and thermal energy. There is also a weak type of unified zeroth law: a Gibbs-like definition of thermal equilibrium requires constancy of an effective temperature, generalising the Tolman condition and the particular case of Hawking radiation, while gravithermal equilibrium further requires constancy of surface gravity. Finally, it is suggested that the energy operator of spherically symmetric quantum gravity is determined by the Kodama vector, which encodes a dynamic time related to E. 
  In this paper we report whether conserved currents can be sensibly defined in supersymmetric minisuperspaces. Our analysis deals with a k=1 FRW model. Supermatter in the form of scalar supermultiplets is included. We show that conserved currents cannot be adequately established except for some very simple scenarios. 
  The N=2 supergravity action in D=5 is generalized by the inclusion of dimensionally continued Euler-Poincare form. The spacetime torsion implied by the Einsteinean supergravity is imposed by a Lagrangian constraint and resulting variational equations are solved for the Lagrange multipliers. The corresponding terms in the Einstein and Rarita-Schwinger field equations are determined. These indicate new types of interactions that could be included in the action to achieve local supersymmetry. 
  We study the possible existence of black holes in scalar-tensor theories of gravity in four dimensions. Their existence is verified for anomalous versions of these theories, with a negative kinetic term in the Lagrangian. The Hawking temperature T_H of these holes is zero, while the horizon area is (in most cases) infinite. It is shown that an infinite value of T_H can occur only at a curvature singularity rather than a horizon. As a special case, the Brans-Dicke theory is studied in more detail, and two kinds of infinite-area black holes are revealed, with finite and infinite proper time needed for an infalling particle to reach the horizon. 
  We examine the solutions of the equations of motion for an expanding Universe, taking into account the radiation of the inflaton field energy. We then analyze the question of the generality of inflationary solutions in this more general setting of a dissipative system. We find a surprisingly rich behavior for the solutions of the dynamical system of equations in the presence of dissipational effects. We also determine that a value of dissipation as small as $\sim 10^{-7} H$ can lead to a smooth exit from inflation to radiation. 
  We present a simple model illustrating how a highly relativistic, compact object which is stable in isolation can be driven dynamically unstable by the tidal field of a binary companion. Our compact object consists of a test-particle in a relativistic orbit about a black hole; the binary companion is a distant point mass. Our example is presented in light of mounting theoretical opposition to the possibility that sufficiently massive, binary neutron stars inspiraling from large distance can collapse to form black holes prior to merger. Our strong-field model suggests that first order post-Newtonian treatments of binaries, and stability analyses of binary equilibria based on orbit-averaged, mean gravitational fields, may not be adequate to rule out this possibility. 
  In this paper, we explicitly prove the presymplectic forms of the Palatini and Ashtekar gravity to be zero along gauge orbits of the Lorentz and diffeomorphism groups, which ensures the diffeomorphism invariance of these theories. 
  The Bowen-York initial value data typically used in numerical relativity to represent spinning black hole are not those of a constant-time slice of the Kerr spacetime. If Bowen-York initial data are used for each black hole in a collision, the emitted radiation will be partially due to the ``relaxation'' of the individual holes to Kerr form. We compute this radiation by treating the geometry for a single hole as a perturbation of a Schwarzschild black hole, and by using second order perturbation theory. We discuss the extent to which Bowen-York data can be expected accurately to represent Kerr holes. 
  Using second order black hole perturbation theory, we show that the difference between the ADM mass and the final black hole mass, computed to the lowest significant order, is equal, to the same order, to the total gravitational radiation energy, obtained applying the Landau and Lifschitz (pseudotensor) equation to the first order perturbation. This result may be considered as a consistency check for the theory. 
  We consider the motion of a spinning relativistic particle in external electromagnetic and gravitational fields, to first order in the external field, but to an arbitrary order in spin. The correct account for the spin influence on the particle trajectory is obtained with the noncovariant description of spin. Concrete calculations are performed up to second order in spin included. A simple derivation is presented for the gravitational spin-orbit and spin-spin interactions of a relativistic particle. We discuss the gravimagnetic moment (GM), a specific spin effect in general relativity. It is demonstrated that for the Kerr black hole the gravimagnetic ratio, i.e., the coefficient at the GM, equals to unity (as well as for the charged Kerr hole the gyromagnetic ratio equals to two). The equations of motion obtained for relativistic spinning particle in external gravitational field differ essentially from the Papapetrou equations. 
  In the context of the M\"{u}ller-Israel-Stewart second order phenomenological theory for dissipative fluids, we analyze the effects of thermal conduction and viscosity in a relativistic fluid, just after its departure from hydrostatic equilibrium, on a time scale of the order of relaxation times. Stability and causality conditions are contrasted with conditions for which the ''effective inertial mass'' vanishes. 
  We obtain an hybrid expression for the heat-kernel, and from that the density of the free energy, for a minimally coupled scalar field in a Schwarzschild geometry at finite temperature. This gives us the zero-point energy density as a function of the distance from the massive object generating the gravitational field. The contribution to the zero-point energy due to the curvature is extracted too, in this way arriving at a renormalised expression for the energy density (the Casimir energy density). We use this to find an expression for other physical quantities: internal energy, pressure and entropy. It turns out that the disturbance of the surrounding vacuum generates entropy. For $\beta$ small the entropy is positive for $r>2M$. We also find that the internal energy can be negative outside the horizon pointing to the existence of bound states. The total energy inside the horizon turns out to be finite but complex, the imaginary part being interpreted as responsible for particle creation. 
  We discuss (3+1) dimensional general relativistic hydrodynamic simulations of close neutron star binary systems. The relativistic field equations are solved at each time slice with a spatial 3-metric chosen to be conformally flat. Against this solution the hydrodynamic variables and gravitational radiation are allowed to respond. We have studied four physical processes which occur as the stars approach merger. These include: 1) the relaxation to a hydrodynamic state of almost no spin; 2) relativistically driven compression, heating, and neutrino emission; 3) collapse to two black holes; and 4) orbit inspiral occurring at a lower frequency than previously expected. We give a brief account of the physical origin of these effects and an explanation of why they do not appear in models based upon, 1PN hydrodynamics, a weak field multipole expansion, a tidal analysis, or a rigidly corotating velocity field. The implication of these results for gravity wave detectors is also discussed. 
  Numerical studies of the plane symmetric, vacuum Gowdy universe on $T^3 \times R$ yield strong support for the conjectured asymptotically velocity term dominated (AVTD) behavior of its evolution toward the singularity except, perhaps, at isolated spatial points. A generic solution is characterized by spiky features and apparent ``discontinuities'' in the wave amplitudes. It is shown that the nonlinear terms in the wave equations drive the system generically to the ``small velocity'' AVTD regime and that the spiky features are caused by the absence of these terms at isolated spatial points. 
  In the search for exact solutions to Einstein's field equations the main simplification tool is the introduction of spacetime symmetries. Motivated by this fact we develop a method to write the field equations for general matter in a form that fully incorporates the character of the symmetry. The method is being expressed in a covariant formalism using the framework of a double congruence. The basic notion on which it is based is that of the geometrisation of a general symmetry. As a special application of our general method we consider the case of a spacelike conformal Killing vector field on the spacetime manifold regarding special types of matter fields. New perspectives in General Relativity are discussed. 
  In the path integral expression for a Feynman propagator of a spinless particle of mass $m$, the path integral amplitude for a path of proper length ${\cal R}(x,x'| g_{\mu\nu})$ connecting events $x$ and $x'$ in a spacetime described by the metric tensor $g_{\mu\nu}$ is $\exp-[m {\cal R}(x,x'| g_{\mu\nu})]$. In a recent paper, assuming the path integral amplitude to be invariant under the duality transformation ${\cal R} \to (L_P^2/{\cal R})$, Padmanabhan has evaluated the modified Feynman propagator in an arbitrary curved spacetime. He finds that the essential feature of this `principle of path integral duality' is that the Euclidean proper distance $(\Delta x)^2$ between two infinitesimally separated spacetime events is replaced by $[(\Delta x)^2 + 4L_P^2 ]$. In other words, under the duality principle the spacetime behaves as though it has a `zero-point length' $L_P$, a feature that is expected to arise in a quantum theory of gravity. In the Schwinger's proper time description of the Feynman propagator, the weightage factor for a path with a proper time $s$ is $\exp-(m^2s)$. Invoking Padmanabhan's `principle of path integral duality' corresponds to modifying the weightage factor $\exp-(m^2s)$ to $\exp-[m^2s + (L_P^2/s)]$. In this paper, we use this modified weightage factor in Schwinger's proper time formalism to evaluate the quantum gravitational corrections to some of the standard quantum field theoretic results in flat and curved spacetimes. We find that the extra factor $\exp-(L_P^2/s)$ acts as a regulator at the Planck scale thereby `removing' the divergences that otherwise appear in the theory. Finally, we discuss the wider implications of our analysis. 
  We first generalise the standard Wigner function to Dirac fermions in curved spacetimes. Secondly, we turn to the Moyal quantisation of systems with constraints. Gravity is used as an example. 
  The thermodynamic properties of the (2+1)-dimensional non-rotating black hole of Ba\~nados, Teitelboim and Zanelli are discussed. The first quantum correction to the Bekenstein-Hawking entropy is evaluated within the on-shell Euclidean formalism, making use of the related Chern-Simons representation of the 3-dimensional gravity. Horizon and ultraviolet divergences in the quantum correction are dealt with a renormalization of the Newton constant. It is argued that the quantum correction due to the gravitational field shrinks the effective radius of a hole and becomes more and more important as soon as the evaporation process goes on, while the area law is not violated. 
  The equations governing dissipative relativistic hydrodynamics are formulated within the 3+1 approach for arbitrary spacetimes. Dissipation is accounted for by applying the theory of extended causal thermodynamics (Israel-Stewart theory). This description eliminates the causality violating infinite signal speeds present in the conventional Navier-Stokes equation. As an example we treat the astrophysically relevant case of stationary and axisymmetric spacetimes, including the Kerr metric. The equations take a simpler form whenever the inertia due to the dissipative contributions can be neglected. 
  We discuss the interpretation of the state sum of Barrett and Crane. 
  The Kerr solution is generated from the Schwarzschild solution by a simple combination of real global coordinate transformations and of invariance transformations acting on the space of stationary solutions of the Einstein-Maxwell equations. The same transformation can be used to generate a spinning field configuration from any static axisymmetric configuration. We illustrate this by generating from the continuous family of Voorhees--Zipoy vacuum solutions a family of solutions endowed with mass, angular momentum, dipole magnetic moment and quadrupole electric moment. 
  A model for a possible variable cosmic object is presented. The model consists of a massive shell surrounding a compact object. The gravitational and self-gravitational forces tend to collapse the shell, but the internal tangential stresses oppose the collapse. The combined action of the two types of forces is studied and several cases are presented. In particular, we investigate the spherically symmetric case in which the shell oscillates radially around a central compact object. 
  Initial data corresponding to spacetimes containing black holes are considered in the time symmetric case. The solutions are obtained by matching across the apparent horizon different, conformally flat, spatial metrics. The exterior metric is the vacuum solution obtained by the well known conformal imaging method. The interior metric for every black hole is regular everywhere and corresponds to a positive energy density. The resulting matched solutions cover then the whole initial (Cauchy) hypersurface, without any singularity, and can be useful for numerical applications. The simpler cases of one black hole (Schwarzschild data) or two identical black holes (Misner data) are explicitly solved. A procedure for extending this construction to the multiple black hole case is also given, and it is shown to work for all time symmetric vacuum solutions obtained by the conformal imaging method. The numerical evolution of one such 'stuffed' black hole is compared with that of a pure vacuum or 'plain' black hole in the spherically symmetric case. 
  We study the Cauchy horizon (CH) singularity of a spherical charged black hole perturbed nonlinearly by a self-gravitating massless scalar field. We show numerically that the singularity is weak both at the early and at the late sections of the CH, where the focusing of the area coordinate $r$ is strong. In the early section the metric is almost Reissner-Nordstr\"{o}m, and the fields behave according to perturbation analysis. We find exact analytical expressions for the gradients of $r$ and of the scalar field, which are valid at both sections. We then verify these analytical results numerically. 
  In this report we advance the current repertoire of quantum cosmological models to incorporate inhomogenous field modes in a supersymmetric manner. In particular, we introduce perturbations about a supersymmetric FRW model. A quantum state of our model has properties typical of the no-boundary (Hartle--Hawking) proposal. This solution may then lead to a scale--free spectrum of density perturbations. 
  The dominant topologies in the Euclidean path integral for quantum gravity differ sharply according on the sign of the cosmological constant. For $\Lambda>0$, saddle points can occur only for topologies with vanishing first Betti number and finite fundamental group. For $\Lambda<0$, on the other hand, the path integral is dominated by topologies with extremely complicated fundamental groups; while the contribution of each individual manifold is strongly suppressed, the ``density of topologies'' grows fast enough to overwhelm this suppression. The value $\Lambda=0$ is thus a sort of boundary between phases in the sum over topologies. I discuss some implications for the cosmological constant problem and the Hartle-Hawking wave function. 
  This paper has been withdrawn by the author. A revised and expanded version is gr-qc/9907028 (Phys.Rev. D60 (1999) 104043). 
  Following Grischuk and Sidorov [Phys. Rev. D 42 (1990) 3413] in putting the Bogolubov-Hawking coefficient of Schwarzschild black-holes in the squeezing perspective, we provide a short discussion of Schwarzschild black holes as radiometric standards 
  We analyze the signal processing required for the optimal detection of a stochastic background of gravitational radiation using laser interferometric detectors. Starting with basic assumptions about the statistical properties of a stochastic gravity-wave background, we derive expressions for the optimal filter function and signal-to-noise ratio for the cross-correlation of the outputs of two gravity-wave detectors. Sensitivity levels required for detection are then calculated. Issues related to: (i) calculating the signal-to-noise ratio for arbitrarily large stochastic backgrounds, (ii) performing the data analysis in the presence of nonstationary detector noise, (iii) combining data from multiple detector pairs to increase the sensitivity of a stochastic background search, (iv) correlating the outputs of 4 or more detectors, and (v) allowing for the possibility of correlated noise in the outputs of two detectors are discussed. We briefly describe a computer simulation which mimics the generation and detection of a simulated stochastic gravity-wave signal in the presence of simulated detector noise. Numerous graphs and tables of numerical data for the five major interferometers (LIGO-WA, LIGO-LA, VIRGO, GEO-600, and TAMA-300) are also given. The treatment given in this paper should be accessible to both theorists involved in data analysis and experimentalists involved in detector design and data acquisition. 
  The one-loop contributions to the entropy for a massive scalar field in a Kerr black hole are investigated using an approximation of the metric, which, after a conformal transformation, permits to work in a Rindler-like spacetime. Of course, as for the Schwarzschild case, the entropy is divergent in the proximity of the event horizon. 
  The geometry of impulsive pp-waves is explored via the analysis of the geodesic and geodesic deviation equation using the distributional form of the metric. The geodesic equation involves formally ill-defined products of distributions due to the nonlinearity of the equations and the presence of the Dirac $\de$-distribution in the space time metric. Thus, strictly speaking, it cannot be treated within Schwartz's linear theory of distributions. To cope with this difficulty we proceed by first regularizing the $\de$-singularity,then solving the regularized equation within classical smooth functions and, finally, obtaining a distributional limit as solution to the original problem. Furthermore it is shown that this limit is independent of the regularization without requiring any additional condition, thereby confirming earlier results in a mathematical rigorous fashion. We also treat the Jacobi equation which, despite being linear in the deviation vector field, involves even more delicate singular expressions, like the ``square'' of the Dirac $\de$-distribution. Again the same regularization procedure provides us with a perfectly well behaved smooth regularization and a regularization-independent distributional limit. Hence it is concluded that the geometry of impulsive pp-waves can be described consistently using distributions as long as careful regularization procedures are used to handle the ill-defined products. 
  The problems causality and causality violation in topologically nontrivial space-time models are considered. To this end the mixed boundary problem for traversable wormhole models is formulated and the influence of the boundary conditions on the causal properties of space-time is analyzed 
  An analytical scheme and a numerical method in order to study the effects of general relativity on the viscosity driven secular bar mode instability of rapidly rotating stars are presented. The approach consists in perturbing an axisymmetric and stationary configuration and studying its evolution by constructing a series of triaxial quasi-equilibrium configurations. These are obtained by solution of an approximate set of field equations where only the dominant non-axisymmetric terms are taken into account. The progress with respect to our former investigation consists in a higher relativistic order of the non-axisymmetric terms included into the computation, namely the fully three-dimensional treatment of the vector part of the space-time metric tensor as opposed to the scalar part, solely, in the former case. The scheme is applied to rotating stars built on a polytropic equation of state and compared to our previous results. The 3D-vector part turns out to inhibit the symmetry breaking efficiently. Nevertheless, the bar mode instability is still possible for an astrophysically relevant mass of M_ns=1.4 M_sun when a stiff polytropic equation of state with an adiabatic index of gamma=2.5 is employed. Triaxial neutron stars may be efficient emitters of gravitational waves and are thus potentially interesting sources for the forthcoming laser interferometric gravitational wave detectors such as LIGO, VIRGO and GEO600. From a numerical point of view, the solution of the three-dimensional minimal-distortion shift vector equation in spherical coordinates is an important achievement of our code. 
  Solutions to the Einstein equation that represent the superposition of static isolated bodies with axially symmetry are presented. The equations nonlinearity yields singular structures (strut and membranes) to equilibrate the bodies. The force on the strut like singularities is computed for a variety of situations. The superposition of a ring and a particle is studied in some detail 
  A class of metrics solving Einstein's equations with negative cosmological constant and representing rotating, topological black holes is presented. All such solutions are in the Petrov type-$D$ class, and can be obtained from the most general metric known in this class by acting with suitably chosen discrete groups of isometries. First, by analytical continuation of the Kerr-de Sitter metric, a solution describing uncharged, rotating black holes whose event horizon is a Riemann surface of arbitrary genus $g > 1$, is obtained. Then a solution representing a rotating, uncharged toroidal black hole is also presented. The higher genus black holes appear to be quite exotic objects, they lack global axial symmetry and have an intricate causal structure. The toroidal blackholes appear to be simpler, they have rotational symmetry and the amount of rotation they can have is bounded by some power of the mass. 
  It has been shown recently that within the framework of the teleparallel equivalent of general relativity (TEGR) it is possible to define the energy density of the gravitational field. The TEGR amounts to an alternative formulation of Einstein's general relativity, not to an alternative gravity theory. The localizability of the gravitational energy has been investigated in a number of space-times with distinct topologies, and the outcome of these analises agree with previously known results regarding the exact expression of the gravitational energy, and/or with the specific properties of the space-time manifold. In this article we establish a relationship between the expression for the gravitational energy density of the TEGR and the Sparling two-forms, which are known to be closely connected with the gravitational energy. We also show that our expression of energy yields the correct value of gravitational mass contained in the conformal factor of the metric field. 
  Two relativistic models for collapsing spheres at different stages of evolution, which include pre-relaxation processes, are presented. The influence of relaxation time on the outcome of evolution in both cases is exhibited and established. It is shown that relaxation processes can drastically change the final state of the collapsing system. In particular, there are cases in which the value of the relaxation time determines the bounce or the collapse of the sphere. 
  We discuss spherically symmetric, static solutions to the SU(2) sigma model on a de Sitter background. Despite of its simplicity this model reflects many of the features exhibited by systems of non-linear matter coupled to gravity e.g. there exists a countable set of regular solutions with finite energy; all of the solutions show linear instability with the number of unstable modes increasing with energy. 
  We discuss a mechanism for generating the baryon asymetry of the Universe that involves a putative violation of CPT symmetry arising from string interactions. 
  The goal of this paper is to provide a new analysis of the classical dynamics of Bianchi type I, II and IX models by applying conventional Hamiltonian methods in the language of Ashtekhar variables. We show that Bianchi type II models can be seen as a perturbation of Bianchi I ones, and integrated. Bianchi IX models can be seen, in turn, as a perturbation of Bianchi IIs, but here the integration algorithm breaks down. This is an ''interesting failure'', bringing light onto the chaotic nature of Bianchi type IX dynamics.As a by product of our analysis we filled some gaps in the literature, such us recovering the BKL map in this context. 
  We discuss the extraction of information from detected binary black hole (BBH) coalescence gravitational waves, focusing on the merger phase that occurs after the gradual inspiral and before the ringdown. Our results are: (1) If numerical relativity simulations have not produced template merger waveforms before BBH detections by LIGO/VIRGO, one can band-pass filter the merger waves. For BBHs smaller than about 40 solar masses detected via their inspiral waves, the band pass filtering signal to noise ratio indicates that the merger waves should typically be just barely visible in the noise for initial and advanced LIGO interferometers. (2) We derive an optimized (maximum likelihood) method for extracting a best-fit merger waveform from the noisy detector output; one "perpendicularly projects" this output onto a function space (specified using wavelets) that incorporates our prior knowledge of the waveforms. An extension of the method allows one to extract the BBH's two independent waveforms from outputs of several interferometers. (3) If numerical relativists produce codes for generating merger templates but running the codes is too expensive to allow an extensive survey of the merger parameter space, then a coarse survey of this parameter space, to determine the ranges of the several key parameters and to explore several qualitative issues which we describe, would be useful for data analysis purposes. (4) A complete set of templates could be used to test the nonlinear dynamics of general relativity and to measure some of the binary parameters. We estimate the number of bits of information obtainable from the merger waves (about 10 to 60 for LIGO/VIRGO, up to 200 for LISA), estimate the information loss due to template numerical errors or sparseness in the template grid, and infer approximate requirements on template accuracy and spacing. 
  We re-examine the notions of time and evolution in the light of the mathematical properties of the solutions of the Wheeler-DeWitt equation which are revealed by an extended adiabatic treatment. The main advantage of this treatment is to organize the solutions in series that make explicit both the connections with the corresponding Schr\"odinger equation as well as the modifications introduced by the quantum character of gravity. When the universe is macroscopic, the ordered character of the expansion leads to connections with the Schr\"odinger equation so precise that the interpretation of the solutions of the Wheeler-DeWitt equation is unequivocally determined. On the contrary, when the expansion behaves quantum mechanically, i.e. in the presence of backscattering, major difficulties concerning the interpretation persist. 
  Constants of motion are calculated for 2+1 dimensional gravity with topology R \times T^2 and negative cosmological constant. Certain linear combinations of them satisfy the anti - de Sitter algebra so(2,2) in either ADM or holonomy variables. Quantisation is straightforward in terms of the holonomy parameters, and the modular group is generated by these conserved quantities. On inclusion of the Hamiltonian three new global constants are derived and the quantum algebra extends to the conformal algebra so(2,3). 
  It has been recently shown that the eigenvalues of the Dirac operator can be considered as dynamical variables of Euclidean gravity. The purpose of this paper is to explore the possiblity that the eigenvalues of the Dirac operator might play the same role in the case of supergravity. It is shown that for this purpose some primary constraints on covariant phase space as well as secondary constraints on the eigenspinors must be imposed. The validity of primary constraints under covariant transport is further analyzed. It is show that in the this case restrictions on the tanget bundle and on the spinor bundle of spacetime arise. The form of these restrictions is determined under some simplifying assumptions. It is also shown that manifolds with flat curvature of tangent bundle and spinor bundle and spinor bundle satisfy these restrictons and thus they support the Dirac eigenvalues as global observables. 
  The relation between the SL(2,R)/SO(2)- and the SL(2,C)-chiral model that naturally arise within the metric respectively the Ashtekar formulation of two Killing field reduced Einstein gravity is revealed. Both chiral models turn out to be completely equivalent even though the transition from the coset- to the SL(2,C)-model is accompanied by a disappearance of the non-ultralocal terms in the Poisson brackets. 
  We derive all second post-Newtonian (2PN), non-precessional effects of spin- orbit coupling on the gravitational wave forms emitted by an inspiraling binary composed of spinning, compact bodies in a quasicircular orbit. Previous post- Newtonian calculations of spin-orbit effects (at 1.5PN order) relied on a fluid description of the spinning bodies. We simplify the calculations by introducing into post-Newtonian theory a delta-function description of the influence of the spins on the bodies' energy-momentum tensor. This description was recently used by Mino, Shibata, and Tanaka (MST) in Teukolsky-formalism analyses of particles orbiting massive black holes, and is based on prior work by Dixon. We compute the 2PN contributions to the wave forms by combining the MST energy-momentum tensor with the formalism of Blanchet, Damour, and Iyer for evaluating the binary's radiative multipoles, and with the well-known 1.5PN order equations of motion for the binary. Our results contribute at 2PN order only to the amplitudes of the wave forms. The secular evolution of the wave forms' phase, the quantity most accurately measurable by LIGO, is not affected by our results until 2.5PN order, at which point other spin-orbit effects also come into play. We plan to evaluate the entire 2.5PN spin-orbit contribution to the secular phase evolution in a future paper, using the techniques of this paper. 
  We find a simple inflationary solution in an inhomogeneous spacetime with heat flux. The heat flux obeys a causal transport equation, and counteracts the inflationary decrease of energy density. At late times, the heat flux tends to zero and the fluid approaches the equation of state $p=-\rho$. 
  We use the quantum potential approach to analyse the quantum cosmological model of the universe. The quantum potential arises from exact solutions of the full Wheeler-De Witt equation. 
  The Einstein-Langevin equation is a perturbative correction to the semiclassical Einstein equation which takes into account the lowest order quantum fluctuations of the matter stress-energy tensor. It predicts classical stochastic fluctuations in the metric field which may describe some of the remnant gravitational fluctuations after the process of environment induced decoherence driving the quantum to classical transition of gravity. 
  We model a radiating, moving black hole in terms of a worldtube-nullcone boundary value problem. We evolve this data in the region interior to the worldtube but exterior to a trapped surface by means of a characteristic evolution based upon a family of ingoing null hypersurfaces. Data on the worldtube is induced from a Schwarzschild spacetime but the worldtube is allowed to move relative to the static Schwarzschild trajectories. When the worldtube is stationary (static or rotating in place), a distorted black hole inside it evolves to equilibrium with the Schwarzschild boundary. A boost of the worldtube with respect to the Schwarzschild black hole does not affect these results. The code also stably tracks an unlimited number of orbits when the worldtube wobbles periodically. The work establishes that characteristic evolution can evolve a spacetime with a distorted black hole moving on a 3-dimensional grid with the controlled accuracy and long term stability necessary to investigate new facets of black hole physics. 
  We consider a Palatini variation on a general $N$-Dimensional second order, torsion-free dilaton gravity action and determine the resulting equations of motion. Consistency is checked by considering the restraint imposed due to invariance of the matter action under simple coordinate transformations, and the special case of N=2 is examined. We also examine a sub-class of theories whereby a Palatini variation dynamically coincides with that of the "ordinary" Hilbert variational principle; in particular we examine a generalized Brans-Dicke theory and the associated role of conformal transformations. 
  We discuss the underlying relativistic physics which causes neutron stars to compress and collapse in close binary systems as has recently been observed in numerical (3+1) dimensional general relativistic hydrodynamic simulations. We show that compression is driven by velocity-dependent relativistic hydrodynamic terms which increase the self gravity of the stars. They also produce fluid motion with respect to the corotating frame of the binary. We present numerical and analytic results which confirm that such terms are insignificant for uniform translation or when the hydrodynamics is constrained to rigid corotation.  However, when the hydrodynamics is unconstrained, the neutron star fluid relaxes to a compressed nonsynchronized state of almost no net intrinsic spin with respect to a distant observer.   We also show that tidal decompression effects are much less than the velocity-dependent compression terms. We discuss why several recent attempts to analyze this effect with constrained hydrodynamics or an analysis of tidal forces do not observe compression. We argue that an independent test of this must include unconstrained relativistic hydrodynamics to sufficiently high order that all relevant velocity-dependent terms and their possible cancellations are included. 
  The well-known treatment of asymptotically flat vacuum fields is adapted to pure radiation fields. In this approach we find a natural normalization of the radiation null vector. The energy balance at null infinity shows that the mass loss results from a linear superposition of the pure and the gravitational radiation parts. By transformation to Bondi-Sachs coordinates the Kinnersley photon rocket is found to be the only axisymmetric Robinson-Trautman pure radiation solution without gravitational radiation. 
  We obtain an expression for the active gravitational mass of a collapsing fluid distribution, which brings out the role of density inhomogeneity and local anisotropy in the fate of spherical collapse. 
  We consider a Palatini variation on a generalized Einstein-Hilbert action. We find that the Hilbert constraint, that the connection equals the Christoffel symbol, arises only as a special case of this general action, while for particular values of the coefficients of this generalized action, the connection is completely unconstrained. We discuss the relationship between this situation and that usually encountered in the Palatini formulation. 
  Solutions to the field equations of the Nonsymmetric Gravitational Theory with $g_[i0] = 0$ are obtained for the homogeneous, plane-symmetric, time-dependent case, both in vacuum and in the presence of a perfect fluid. Cosmological consequences include a dependence of the speed of light on its polarisation, as in a birefringent crystal. 
  We present a semi-analytical approach to the interaction of two (originally) Kerr black holes through a head-on collision process. An expression for the rate of emission of gravitational radiation is derived from an exact solution to the Einstein's field equations. The total amount of gravitational radiation emitted in the process is calculated and compared to current numerical investigations. We find that the spin-spin interaction increases the emission of gravitational wave energy up to 0.2% of the total rest mass. We discuss also the possibility of spin-exchange between the holes. 
  We consider T-duality of the quasilocal black hole thermodynamics for the three-dimensional low energy effective string theory. Quasilocal thermodynamic variables in the first law are explicitly calculated on a general axisymmetric three-dimensional black hole solution and corresponding dual one. Physical meaning of the dual invariance of the black hole entropy is considered in terms of the Euclidean path integral formulation. 
  We define realism using a slightly modified version of the EPR criterion of reality. This version is strong enough to show that relativity is incomplete.   We show that this definition of realism is nonetheless compatible with the general principles of causality and canonical quantum theory as well as with experimental evidence in the (special and general) relativistic domain.   We show that the realistic theories we present here, compared with the standard relativistic theories, have higher empirical content in the strong sense defined by Popper's methodology. 
  Gravitational perturbations about a Kerr black hole in the Newman-Penrose formalism are concisely described by the Teukolsky equation. New numerical methods for studying the evolution of such perturbations require not only the construction of appropriate initial data to describe the collision of two orbiting black holes, but also to know how such new data must be imposed into the Teukolsky equation. In this paper we show how Cauchy data can be incorporated explicitly into the Teukolsky equation for non-rotating black holes. The Teukolsky function $% \Psi $ and its first time derivative $\partial_t \Psi $ can be written in terms of only the 3-geometry and the extrinsic curvature in a gauge invariant way. Taking a Laplace transform of the Teukolsky equation incorporates initial data as a source term. We show that for astrophysical data the straightforward Green function method leads to divergent integrals that can be regularized like for the case of a source generated by a particle coming from infinity. 
  We consider the propagation of gravitational waves generated by slow motion sources in Coulomb type potential due to the mass of the source. Then, the formula for gravitational waveform including tail is obtained in a straightforward manner by using the spherical Coulomb function. We discuss its relation with the formula in the previous work. 
  We study the energy and momentum of an isolated system in the tetrad theory of gravitation, starting from the most general Lagrangian quadratic in torsion, which involves four unknown parameters. When applied to the static spherically symmetric case, the parallel vector fields take a diagonal form, and the field equation has an exact solution. We analyze the linearized field equation in vacuum at distances far from the isolated system without assuming any symmetry property of the system. The linearized equation is a set of coupled equations for a symmetric and skew-symmetric tensor fields, but it is possible to solve it up to $O(1/r)$ for the stationary case. It is found that the general solution contains two constants, one being the gravitational mass of the source and the other a constant vector ${\grave B_\alpha}$. The total energy is calculated from this solution and is found to be equal to the gravitational mass of the source. We also calculate the spatial momentum and find that its value coincides with the constant vector ${\grave B_\alpha}$. The linearized field equation in vacuum, which is valid at distances far from the source, does not give any information about whether the constant vector ${\grave B_\alpha}$ is vanishing or not. For a weakly gravitating source for which the field is weak everywhere, we find that the constant vector ${\grave B_\alpha}$ vanishes. 
  We match the vacuum, stationary, cylindrically symmetric solution of Einstein's field equations with $\Lambda$, in a form recently given by Santos, as an exterior to an infinite cylinder of dust cut out of a G\"{o}del universe. There are three cases, depending on the radius of the cylinder. Closed timelike curves are present in the exteriors of some of the solutions. There is a considerable similarity between the spacetimes investigated here and those of van Stockum referring to an infinite cylinder of rotating dust matched to vacuum, with $\Lambda=0$. 
  Recent numerical simulations have found that the Cauchy horizon inside spherical charged black holes, when perturbed nonlinearly by a self-gravitating, minimally-coupled, massless, spherically-symmetric scalar field, turns into a null weak singularity which focuses monotonically to $r=0$ at late times, where the singularity becomes spacelike.   Our main objective is to study this spacelike singularity. We study analytically the spherically-symmetric Einstein-Maxwell-scalar equations asymptotically near the singularity. We obtain a series-expansion solution for the metric functions and for the scalar field near $r=0$ under the simplifying assumption of homogeneity. Namely, we neglect spatial derivatives and keep only temporal derivatives. We find that there indeed exists a generic spacelike singularity solution for these equations (in the sense that the solution depends on enough free parameters), with similar properties to those found in the numerical simulations. This singularity is strong in the Tipler sense, namely, every extended object would inevitably be crushed to zero volume. In this sense this is a similar singularity to the spacelike singularity inside uncharged spherical black holes. On the other hand, there are some important differences between the two cases. Our model can also be extended to the more general inhomogeneous case.   The question of whether the same kind of singularity evolves in more realistic models (of a spinning black hole coupled to gravitational perturbations) is still an open question. 
  According to this principle (EEP), in order that the local physical laws cannot change, after changes of velocity and potentials of a measuring system, the relativistic changes of any particle and any stationary radiation (like those used to measure it) must occur in identical proportion. Thus particles and stationary radiations must have the same general physical properties. In principle more exact and better defined physical laws for particles and their gravitational (G) fields can be derived from properties of particle models made up of radiation in stationary states after using fixed reference frames that don't change in the same way as the objects. Effectively, the new laws derived in this way do correspond with relativistic quantum mechanics and with all of the G tests. The main difference with current gravity is the linearity fixed by the EEP, i.e., the G field itself has not a real field energy to exchange with the bodies and it is not a secondary source of field. G work liberates energy confined in the models stationary states. The EEP also fixes a new astrophysical context that has fundamental differences with the current ones. This one has been presented in a separated work as a test for the EEP. The whole theory,including the new universe context fixed by the EEP, was published in a book. 
  We show that anisotropic Bianchi type-IX models, with matter and cosmological constant have chaotic dynamics, connected to the presence of a saddle-center in phase space. The topology of cylinders emanating from unstable periodic orbits about the saddle-center provides an invariant characterization of chaos in the models. The model can be thought to describe the early stages of inflation, the way out to inflation being chaotic. 
  We construct initial data for several black holes with arbitrary momenta and spins by a new method that is based on a compactification of Brill-Lindquist wormholes. When treated numerically, the method leads to a significant simplification over the conventional approach which is based on throats and an isometry condition. 
  We present a new class of 3D black hole initial data sets for numerical relativity. These data sets go beyond the axisymmetric, ``gravity wave plus rotating black hole'' single black hole data sets by creating a dynamic, distorted hole with adjustable distortion parameters in 3D. These data sets extend our existing test beds for 3D numerical relativity, representing the late stages of binary black hole collisions resulting from on-axis collision or 3D spiralling coalescence, and should provide insight into the physics of such systems. We describe the construction of these sets, the properties for a number of example cases, and report on progress evolving them. 
  We present preliminary results in our long-term project of studying the evolution of matter in a dynamical spacetime. To achieve this, we have developed a new code to evolve axisymmetric initial data sets corresponding to a black hole surrounded by matter fields. The code is based on the coupling of two previously existing codes. The matter fields are evolved with a 2D shock-capturing method which uses the characteristic information of the GR hydro equations to build up a linearized Riemann solver. The spacetime is evolved with a 2D ADM code designed to evolve a wormhole in full general relativity. An example of the kind of problems we are currently investigating is the on axis collision of a star with a black hole. 
  We describe a numerical grid generating procedure to construct new classes of orthogonal coordinate systems that are specially adapted to binary black hole spacetimes. The new coordinates offer an alternative approach to the conventional \v{C}ade\v{z} coordinates, in addition to providing a potentially more stable and flexible platform to extend previous calculations of binary black hole collisions. 
  Rigidly rotating stationary matter in general relativity has been investigated by Kramer by the Ernst coordinate method. A weakness of this approach is that the Ernst potential does not exist for differential rotation. We now generalize the techniques by the use of a nonholonomic and nonrigid frame. We apply these techniques for differentially rotating perfect fluids. We construct a complex analytic tensor, characterizing the class of matter states in which both the interior Schwarzschild and the Kerr solution are contained. We derive consistency relations for this class of perfect fluids. We investigate incompressible fluids characterized by these tensors. 
  It is shown that a quantized Schwarzschild black hole, if described by a square root energy spectrum with exponential multiplicity, can be treated as a microcanonical ensemble without problem leading to the expected thermodynamical properties. 
  The recent developments of the ``connection'' and ``loop'' representations have given the possibility to show that the two representation are equivalent and that it is possible to transform any result from one representation into the other. The glue between the two representations is the loop transform. Its use, combined with the techincs Penrose's binor calculus, gives the possibility to establish the exact correspondence between operators and states in the connection representation and those in the loop representation.   The main ingredients in the prove of the equivalence are: the concept of embedded spin network, the Penrose graphical method of SU(2) calculus, and the existence of a generalized measure on the space of connections. 
  We prove that a Lanczos potential L_abc for the Weyl candidate tensor W_abcd does not generally exist for dimensions higher than four. The technique is simply to assume the existence of such a potential in dimension n, and then check the integrability conditions for the assumed system of differential equations; if the integrability conditions yield another non-trivial differential system for L_abc and W_abcd, then this system's integrability conditions should be checked; and so on. When we find a non-trivial condition involving only W_abcd and its derivatives, then clearly Weyl candidate tensors failing to satisfy that condition cannot be written in terms of a Lanczos potential L_abc. 
  We show that homogeneous and isotropic cosmological models with radiation and scalar field have, in general, two possible chaotic exits to inflation. These models may describe the early stages of inflation. The central point of our analysis is based on the possible existence of one or two saddle-centers points in the phase space of the models. 
  The interior geometry of static, spherically symmetric black holes of the Einstein-Yang-Mills-Higgs theory is analyzed. It is found that in contrast to the Abelian case generically no inner (Cauchy) horizon is formed inside non-Abelian black holes. Instead the solutions come close to a Cauchy horizon but then undergo an enormous growth of the mass function, a phenomenon which can be termed `mass inflation' in analogy to what is observed for perturbations of the Reissner-Nordstr{\o}m solution. A significant difference between the theories with and without a Higgs field is observed. Without a Higgs field the YM field induces repeated cycles of massinflation -- taking the form of violent `explosions' -- interrupted by quiescent periods and subsequent approaches to an almost Cauchy horizon. With the Higgs field no such cycles occur. Besides the generic solutions there are non-generic families with a Schwarzschild, Reissner-Nordstr{\o}m and a pseudo Reissner-Nordstr{\o}m type singularity at $r=0$. 
  In a recent work on the quantization of a massless scalar field in a particular colliding plane wave space-time, we computed the vacuum expectation value of the stress-energy tensor on the physical state which corresponds to the Minkowski vacuum before the collision of the waves. We did such a calculation in a region close to both the Killing-Cauchy horizon and the folding singularities that such a space-time contains. In the present paper, we give a suitable approximation procedure to compute this expectation value, in the conformal coupling case, throughout the causal past of the center of the collision. This will allow us to approximately study the evolution of such an expectation value from the beginning of the collision until the formation of the Killing-Cauchy horizon. We start with a null expectation value before the arrival of the waves, which then acquires nonzero values at the beginning of the collision and grows unbounded towards the Killing-Cauchy horizon. The value near the horizon is compatible with our previous result, which means that such an approximation may be applied to other colliding plane wave space-times. Even with this approximation, the initial modes propagated into the interaction region contain a function which cannot be calculated exactly and to ensure the correct regularization of the stress-energy tensor with the point-splitting technique, this function must be given up to adiabatic order four of approximation. 
  We introduce the notion of topological fragility and briefly discuss some examples from the literature. An important example of this type of fragility is the way globally anisotropic Bianchi V generalisations of the FLRW $k=-1$ model result in a radical restriction on the allowed topology of spatial sections, thereby excluding compact cosmological models with negatively curved three-sections with anisotropy. An outcome of this is to exclude chaotic mixing in such models, which may be relevant, given the many recent attempts at employing compact FLRW $k=-1$ models to produce chaotic mixing in the cosmic microwave background radiation, if the Universe turns out to be globally anisotropic. 
  A recent paper of Singal [Gen. Rel. Grav. 27 (1995), 953-967] argues that a uniformly accelerated particle does not radiate, in contradiction to the consensus of the research literature over the past 30 years. This note points out some questionable aspects of Singal's argument and shows how similar calculations can lead to the opposite conclusion. 
  The space of the solutions of Dirac's quantum constraints cannot be constructed factoring the quantum state space by the ``simple'' gauge transformations generated by the constraints. However, we show here that it can be constructed by factoring the state space by suitably defined ``complete'' gauge transformations. These are generated by the action of the quantum constraints on individual components of the quantum state. 
  The thermodynamical description of black holes is reviewed and critiqued. We present an alternative, microcanonical description of black holes and discuss the major differences. In particular the decay rates of black holes are compared in the two different pictures. 
  There has been much recent work on quantum inequalities to constrain negative energy. These are uncertainty principle-type restrictions on the magnitude and duration of negative energy densities or fluxes. We consider several examples of apparent failures of the quantum inequalities, which involve passage of an observer through regions where the negative energy density becomes singular. We argue that this type of situation requires one to formulate quantum inequalities using sampling functions with compact support. We discuss such inequalities, and argue that they remain valid even in the presence of singular energy densities. 
  A functional calculus on the space of (generalized) connections was recently introduced without any reference to a background metric. It is used to continue the exploration of the quantum Riemannian geometry. Operators corresponding to volume of three-dimensional regions are regularized rigorously. It is shown that there are two natural regularization schemes, each of which leads to a well-defined operator. Both operators can be completely specified by giving their action on states labelled by graphs. The two final results are closely related but differ from one another in that one of the operators is sensitive to the differential structure of graphs at their vertices while the second is sensitive only to the topological characteristics. (The second operator was first introduced by Rovelli and Smolin and De Pietri and Rovelli using a somewhat different framework.) The difference between the two operators can be attributed directly to the standard quantization ambiguity. Underlying assumptions and subtleties of regularization procedures are discussed in detail in both cases because volume operators play an important role in the current discussions of quantum dynamics. 
  We study analytically the features of the Cauchy horizon (CH) singularity inside a spherically-symmetric charged black hole, nonlinearly perturbed by a self-gravitating massless scalar field. We derive exact expressions for the divergence rate of the blue-shift factors, namely the derivatives in the outgoing direction of the scalar field $\Phi$ and the area coordinate $r$. Both derivatives are found to grow along the contracting CH exactly like $1/r$. Our results are valid everywhere along the CH singularity, up to the point of full focusing. These exact analytic expressions are verified numerically. 
  The multidimensional gravity on the total space of principal bundle is considered. In this theory the gauge fields arise as nondiagonal components of multidimensional metric. The spherically symmetric and cosmology solutions for gravity on SU(2) principal bundle are obtained. The static spherically symmetric solution is wormhole-like solution located between two null surfaces, in contrast to 4D Einstein-Yang-Mills theory where corresponding solution (black hole) located outside of event horizon. Cosmology solution (at least locally) has the bouncing off effect for spatial dimensions. In spirit of Einstein these solutions are vacuum solutions without matter. 
  Using the late-time expansion we calculate the leading-order coefficients describing the evolution of a massless scalar field inside a Reissner-Nordstrom black hole. These coefficients may be interpreted as the reflection and transmission coefficients for scalar-field modes propagating from the event horizon to the Cauchy horizon. Our results agree with those obtained previously by Gursel et al by a different method. 
  We demonstrate the existence of chaos in realistic models of two-field inflation. The chaotic motion takes place after the end of inflation, when the fields are free to oscillate and their motion is only lightly damped by the expansion of the universe. We then investigate whether the presence of chaos affects the predictions of two-field models, and show that chaos enhances the production of topological defects and renders the growth rate of the universe sensitively dependent upon the ``initial'' conditions at the beginning of the oscillatory era. 
  A recent investigation shows that a local gauge string with a phenomenological energy momentum tensor, as prescribed by Vilenkin, is inconsistent in Brans-Dicke theory. In this work it has been shown that such a string is consistent in a more general scalar tensor theory where $\omega$ is function of the scalar field.A set of solutions of full nonlinear Einstein's equations for interior region of such a string are presented. 
  We present a systematic expansion of all constraint equations in canonical quantum gravity up to the order of the inverse Planck mass squared. It is demonstrated that this method generates the conventional Feynman diagrammatic technique involving graviton loops and vertices. It also reveals explicitly the back reaction effects of quantized matter and graviton vacuum polarization. This provides an explicit correspondence between the frameworks of canonical and covariant quantum gravity in the semiclassical limit. 
  In this paper, we investigate the pair creation of charged and rotating black hole pairs in a background with a positive cosmological constant. Instantons to describe this situation are constructed from the Kerr-Newmann deSitter solutions to the Einstein-Maxwell equations and the actions of these instantons are calculated in order to estimate pair creation rates and the entropy of the created spacetimes. 
  We study the quantum theory of the spherically symmetric black holes. The theory yields the wave function inside the apparent horizon, where the role of time and space coordinates is interchanged. The de Broglie-Bohm interpretation is applied to the wave function and then the trajectory picture on the minisuperspace is introduced in the quantum as well as the semi-classical region. Around the horizon large quantum fluctuations on the trajectories of metrics $U$ and $V$ appear in our model, where the metrics are functions of time variable $T$ and are expressed as $ds^2=-{\alpha^2}/U dT^2 + U dR^2 + V d\Omega^2$. On the trajectories, the classical relation $U=-V^{1/2}+2Gm$ holds, and the event horizon U=0 corresponds to the classical apparent horizon on $V=2Gm$. In order to investigate the quantum fluctuation near the horizon, we study a null ray on the dBB trajectory and compare it with the one in the classical black hole geometry. 
  The use of "handy singularities" (i.e. singularities similar to those arising in the Deutch-Politzer space) enables one to avoid (almost) all known difficulties inherent usually to creation of time machines. A simple method is discussed for constructing a variety of such singularities. A few 3-dimensional examples are cited. 
  We generalize the formulation of the colliding gravitational waves to metric-affine theories and present an example of such kind of exact solutions. The plane waves are equipped with five symmetries and the resulting geometry after the collision possesses two spacelike Killing vectors. 
  We consider certain interesting processes in quantum gravity which involve a change of spatial topology. We use Morse theory and the machinery of handlebodies to characterise topology changes as suggested by Sorkin. Our results support the view that that the pair production of Kaluza-Klein monopoles and the nucleation of various higher dimensional objects are allowed transitions with non-zero amplitude. 
  Every Dirac spin structure on a world manifold is associated with a certain gravitational field, and is not preserved under general covariant transformations. We construct a composite spinor bundle such that any Dirac spin structure is its subbundle, and this bundle admits general covariant transformations. 
  We study a solution of Einstein's equations that describes a straight cosmic string with a variable angular deficit, starting with a $2 \pi$ deficit at the core. We show that the coordinate singularity associated to this defect can be interpreted as a traversible wormhole lodging at the the core of the string. A negative energy density gradually decreases the angular deficit as the distance from the core increases, ending, at radial infinity, in a Minkowski spacetime. The negative energy density can be confined to a small transversal section of the string by gluing to it an exterior Gott's like solution, that freezes the angular deficit existing at the matching border. The equation of state of the string is such that any massive particle may stay at rest anywhere in this spacetime. In this sense this is 2+1 spacetime solution. 
  The gravitational interaction of an infinitely long cosmic string with a Schwarzschild black hole is studied. We consider a straight string that is initially at a great distance and moving at some initial velocity v (0 < v < c) towards the black hole. The equations of motion of the string are solved numerically to obtain the dependence of the capture impact parameter on the initial velocity. 
  The idea that the seed primordial magnetic fields can be explained by the nonminimal coupling between gravitational and electromagnetic fields is discussed. The predicted values of the magnetic field of the spiral galaxies are in agreement with the observations. 
  Minisuperspace models derived from Kaluza-Klein theories and low energy string theory are studied. They are equivalent to one and two minimally coupled scalar fields. The general classical and quantum solutions are obtained. Gaussian superposition of WKB solutions are constructed. Contrarily to what is usually expected, these states are sharply peaked around the classical trajectories only for small values of the scale factor. This behaviour is confirmed in the framework of the causal interpretation: the Bohmian trajectories of many quantum states are classical for small values of the scale factor but present quantum behaviour when the scale factor becomes large. A consequence of this fact is that these states present an initial singularity. However, there are some particular superpositions of these wave functions which have Bohmian trajectories without singularities. There are also singular Bohmian trajectories with a short period of inflation which grow forever. We could not find any non-singular trajectory which grows to the size of our universe. 
  A topological theory for the interactions in Nature is presented. The theory derives from the cyclic properties of the topological manifold Q=2T^3 + 3S^1 x S^2 which has 23 intrinsic degrees of freedom, discrete Z_3 and Z_2 x Z_3 internal groups, an SU(5) gauge group, and an anomalous U(1) symmetry. These properties reproduce the standard model with a stable proton, a natural place for CP violation and doublet-triplet splitting. The equation of motion for the unified theory is derived and leads to a Higgs field. The thermodynamic properties of Q are discussed and yield a consistent amplitude for the cosmic microwave background fluctuations. The manifold Q possesses internal energy scales which are independent of the field theory defined on it, but which constrain the predicted mass hierarchy of such theories. In particular the electron and its neutrino are identified as ground states and their masses are predicted. The correct masses of quarks and the CKM mixing angles can be derived as well from these energy scales if one uses the anomalous U(1) symmetry. Furthermore, it is shown that if the Planck scale topology of the universe involves loops as fundamental objects, its spatial dimension is equal to three. The existence of the prime manifold T^3=S^1 x S^1 S^1 is then required for a dynamical universe, i.e. a universe which supports forces. Some links with M-theory are pointed out. 
  We derive an expression for the expansion of outgoing null geodesics in spherical dust collapse and compute the limiting value of the expansion in the approach to singularity formation. An analogous expression is derived for the spherical collapse of a general form of matter. We argue on the basis of these results that the covered as well as the naked singularity solutions arising in spherical dust collapse are stable under small changes in the equation of state. 
  We construct analytic models of incompressible, rigidly rotating stars in PN gravity and study their stability against nonaxisymmetric Jacobi-like bar modes. PN configurations are modeled by homogeneous triaxial ellipsoids and the metric is obtained as a solution of Einstein's equations in 3+1 ADM form. We use an approximate subset of the equations well-suited to numerical integration for strong field, 3D configurations in quasi--equilibrium. These equations are exact at PN order, and admit an analytic solution for homogeneous ellipsoids. In this paper we present this solution, as well as analytic functionals for the conserved global quantities, M, M_0 and J. By using a variational principle we construct sequences of axisymmetric equilibria of constant density and rest mass, i.e. the PN generalization of Maclaurin spheroids, which are compared to other PN and full relativistic sequences presented by previous authors. We then consider nonaxisymmetric ellipsoidal deformations of the configurations, holding J constant and the rotation uniform, and we locate the point at which the bar modes will be driven secularly unstable by a dissipative agent like viscosity. We find that the value of the eccentricity, as well as the ratios \Omega^2/(\pi\rho_0) and T/|W|, defined invariantly, all increase at the onset of instability as the stars become more relativistic. Since higher degrees of rotation are required to trigger a viscosity-driven bar mode as the star's compactness increases, the effect of GR is to weaken the instability, at least to PN order. This behavior is opposite to that found for secular instability via Dedekind-like modes driven unstable by gravitational radiation, supporting the suggestion that in GR, nonaxisymmetric modes driven unstable by viscosity and gravitational radiation may no longer coincide. 
  In the groupoid approach to noncommutative quantization of gravity, gravitational field is quantized in terms of a C*-algebra A of complex valued funcions on a groupoid G (with convolution as multiplication). In the noncommutative quantum gravitational regime the concepts of space and time are meaningless. We study the "emergence of time" in the transition process from the noncommutative regime to the standard space-time geometry. Precise conditions are specified under which modular groups of the von Neumann algebra generated by A can be defined. These groups are interpreted as a state depending time flow. If the above conditions are further refined one obtains a state independent time flow. We show that quantum gravitational dynamics can be expressed in terms of modular groups. 
  A lattice model for four dimensional Euclidean quantum general relativity is proposed for a simplicial spacetime. It is shown how this model can be expressed in terms of a sum over worldsheets of spin networks, and an interpretation of these worldsheets as spacetime geometries is given, based on the geometry defined by spin networks in canonical loop quantized GR. The spacetime geometry has a Planck scale discreteness which arises "naturally" from the discrete spectrum of spins of SU(2) representations (and not from the use of a spacetime lattice).   The lattice model of the dynamics is a formal quantization of the classical lattice model of \cite{Rei97a}, which reproduces, in a continuum limit, Euclidean general relativity. 
  A discussion is presented of the principle of black hole com- plementarity. It is argued that this principle could be viewed as a breakdown of general relativity, or alternatively, as the introduction of a time variable with multiple `sheets' or `branches'     A consequence of the theory is that the stress-energy tensor as viewed by an outside observer is not simply the Lorentz-transform of the tensor viewed by an ingoing observer. This can serve as a justification of a new model for the black hole atmosphere, recently re-introduced. It is discussed how such a model may lead to a dynamical description of the black hole quantum states. 
  The theory of general relativity is reformed to a genuine Yang-Mills gauge theory of the Poincar\'e group for gravity. Several pathologies of the conventional theory are thus removed, but not every GR vacuum satisfies the Y-M equations. The sector of GR solutions which survive is fully classified and it is found to include the Schwarzschild black hole. Two other solutions presented here have no GR counterpart and they describe expanding Friedmann universes with torsion which vanishes only asymptotically. They are discussed along with novel theoretical possibilities, such as a well-defined energy-momentum tensor for the gravitational field, and novel perspectives for unification and quantization. 
  We consider whether local and causal non-conservation of energy may occur in generally covariant theories with long-ranged fields (analogs of Newton's gravity) whose source is energy--momentum. We find that such a possibility exists in (1+1) dimensions. 
  The quantization of time-reparametrization invariant systems such as general relativity is plagued by an ambiguity relating to the role of time in the theory. If one parametrizes observables by the (unobservable) time, and then relies on the existence of an approximate "clock" degree of freedom to give physical meaning to the observables, one finds multiple quantum states that yield the same predictions yet interfere with each other. 
  In this paper we discuss the effect of local inhomogeneities on the global expansion of nearly FLRW universes, in a perturbative setting. We derive a generic linearized averaging operation for metric perturbations from basic assumptions, and we explicify the issue of gauge invariance. We derive a gauge invariant expression for the back-reaction of density inhomogeneities on the global expansion of perturbed FLRW spacetimes, in terms of observable quantities, and we calculate the effect quantitatively. Since we do not adopt a comoving gauge, our result incorporates the back-reaction on the metric due to scalar velocity and vorticity perturbations. The results are compared with the results by other authors in this field. 
  In this paper we investigate a parametric resonance phenomenon of a Kaluza-Klein mode in a $D$-dimensional generalized Kaluza-Klein theory. As the origin of the parametric resonance we consider a small oscillation of a scale of the compactification around a today's value of it. To make our arguments definite and for simplicity we consider two classes of models of the compactification: those by $S_{d}$ ($d=D-4$) and those by $S_{d_{1}}\times S_{d_{2}}$ ($d_1\ge d_2$, $d_{1}+d_{2}=D-4$). For these models we show that parametric resonance can occur for the Kaluza-Klein mode. After that, we give formulas of a creation rate and a number of created quanta of the Kaluza-Klein mode due to the parametric resonance, taking into account the first and the second resonance band. By using the formulas we calculate those quantities for each model of the compactification. Finally we give conditions for the parametric resonance to be efficient and discuss cosmological implications. 
  If universal quantum interaction is really connected with the coset structure of deformations of quantum states then the curvature of projective Hilbert state space should be observable. I discuss some approach to the measurement of curvature-dependent values. 
  String theory suggests the existence of gravitational-strength scalar fields (``dilaton'' and ``moduli'') whose couplings to matter violate the equivalence principle. This provides a new motivation for high-precision clock experiments, as well as a generic theoretical framework for analyzing their significance. 
  The confrontation between Einstein's gravitation theory and experimental results, notably binary pulsar data, is summarized and its significance discussed. Experiment and theory agree at the 10^{-3} level. All the basic structures of Einstein's theory (coupling of gravity to matter; propagation and self-interaction of the gravitational field, including in strong-field conditions) have been verified. However, some recent theoretical findings (cosmological relaxation toward zero scalar couplings) suggest that the present agreement between Einstein's theory and experiment might be naturally compatible with the existence of a long-range scalar contribution to gravity (such as the dilaton, or a moduli field of string theory). This provides a new theoretical paradigm, and new motivations for improving the experimental tests of gravity. Ultra-high precision tests of the Equivalence Principle appear as the most sensitive way to look for possible long-range deviations from General Relativity: they might open a low-energy window on string-scale physics. 
  We investigate thermodynamical properties of four and five dimensional black hole solutions of toroidally compactified string theory. We derive an analog of Smarr's formula and verify it directly using the metric. 
  Constraints on the geometries of static spherically symmetric black holes are obtained by requiring that the spacetime curvature be analytic at the event horizon. Further constraints are obtained by requiring that the semiclassical trace equation be satisfied in the case that only conformally invariant fields are present. It is found that there exists a range of sizes for which zero temperature black holes do not exist. The range depends on the number and types of quantized fields present. 
  A class of Riemann-Cartan G\"odel-type space-times are examined in the light of the equivalence problem techniques. The conditions for local space-time homogeneity are derived, generalizing previous works on Riemannian G\"odel-type space-times. The equivalence of Riemann-Cartan G\"odel-type space-times of this class is studied. It is shown that they admit a five-dimensional group of affine-isometries and are characterized by three essential parameters $\ell, m^2, \omega$: identical triads ($\ell, m^2, \omega$) correspond to locally equivalent manifolds. The algebraic types of the irreducible parts of the curvature and torsion tensors are also presented. 
  Through dimensional analysis, eliminating the physical time, we identify the speed of light as a dilaton field. This leads to a restmass zero, spin zero gauge field which we call the speedon field. The complete Lagrangian for gravitational, electromagnetic and speedon field interactions with a charged scalar field, representing matter, is given. We then find solutions for the gravitational-electromagnetic-speedon field equations. This then gives an expression for the speed of light. 
  There are several examples known of two dimensional spacetimes which are linearly stable when perturbed by test scalar classical fields, but which are unstable when perturbed by test scalar quantum fields. We elucidate the mechanism behind such instabilities by considering minimally coupled, massless, scalar, test quantum fields on general two dimensional spacetimes with Cauchy horizons which are classically stable. We identify a geometric feature of such spacetimes which is a necessary condition for obtaining a quantum mechanical divergence of the renormalized expected stress tensor on the Cauchy horizon for regular initial states. This feature is the divergence of the affine parameter length of a one parameter family of null geodesics which lie parallel to the Cauchy horizon, where the affine parameter normalization is determined by parallel transport along a fixed, transverse null geodesic which intersects the Cauchy horizon. (By contrast, the geometric feature of such spacetimes which underlies classical blueshift instabilities is the divergence of a holonomy operator). We show that the instability can be understood as a ``delayed blueshift'' instability, which arises from the infinite blueshifting of an energy flux which is created locally and quantum mechanically. The instability mechanism applies both to chronology horizons in spacetimes with closed timelike curves, and to the inner horizon in black hole spacetimes like two dimensional Reissner-Nordstrom-de Sitter. 
  In a class of generalized gravity theories with general couplings between the scalar field and the scalar curvature in the Lagrangian, we can describe the quantum generation and the classical evolution of both the scalar and tensor structures in a simple and unified manner. An accelerated expansion phase based on the generalized gravity in the early universe drives microscopic quantum fluctuations inside a causal domain to expand into macroscopic ripples in the spacetime metric on scales larger than the local horizon. Following their generation from quantum fluctuations, the ripples in the metric spend a long period outside the causal domain. During this phase their evolution is characterized by their conserved amplitudes. The evolution of these fluctuations may lead to the observed large scale structures of the universe and anisotropies in the cosmic microwave background radiation. 
  We present a phase-plane analysis of cosmologies containing a barotropic fluid with equation of state $p_\gamma = (\gamma-1) \rho_\gamma$, plus a scalar field $\phi$ with an exponential potential $V \propto \exp(-\lambda \kappa \phi)$ where $\kappa^2 = 8\pi G$. In addition to the well-known inflationary solutions for $\lambda^2 < 2$, there exist scaling solutions when $\lambda^2 > 3\gamma$ in which the scalar field energy density tracks that of the barotropic fluid (which for example might be radiation or dust). We show that the scaling solutions are the unique late-time attractors whenever they exist. The fluid-dominated solutions, where $V(\phi)/\rho_\gamma \to 0$ at late times, are always unstable (except for the cosmological constant case $\gamma = 0$). The relative energy density of the fluid and scalar field depends on the steepness of the exponential potential, which is constrained by nucleosynthesis to $\lambda^2 > 20$. We show that standard inflation models are unable to solve this `relic density' problem. 
  We investigate topology change in 3D. Using Morse theory and handle decomposition we find the set of elementary cobordisms for 3-manifolds. These are: (i) \O <-> S^2; (ii) \Sigma_g <-> \Sigma_{g+1}; (iii) \Sigma_{g_1} \sqcup \Sigma_{g_2} <-> \Sigma_{g_1+g_2} and they have appealing physical interpretations, e.g. Big Bang/Big Crunch, wormhole creation/annihilation and Einstein-Rosen bridge creation/annihilation, respectively. This decomposition into building blocks can be used in the path integral approach to quantum gravity in the sum over topologies. 
  The result that, for a scalar quantum field propagating on a ``trousers'' topology in 1+1 dimensions, the crotch singularity is a source for an infinite burst of energy has been used to argue against the occurrence of topology change in quantum gravity. We draw attention to a conjecture due to Sorkin that it may be the particular type of topology change involved in the trousers transition that is problematic and that other topology changes may not cause the same difficulties. The conjecture links the singular behaviour to the existence of ``causal discontinuities'' in the spacetime and relies on a classification of topology changes using Morse theory. We investigate various topology changing transitions, including the pair production of black holes and of topological geons, in the light of these ideas. 
  It is shown that the classical entropy of the extremal black hole depends on two different limits procedures. If we first take the extremal limit and then the boundary limit, the entropy is zero; if we do it the other way round, we get the Bekenstein-Hawking entropy. By means of the brick wall model, the quantum entropy of scalar field in the extremal black hole background has been calculated for the above two different limits procedures. A possible explanation which considers the quantum effect for the clash of black hole entropy in the extremal limit is given. 
  When a particle moves around a Kerr black hole, it radiates gravitational waves.Some of these waves are absorbed by the black hole. We calculate such absorption of gravitational waves induced by a particle of mass mu in a circular orbit on an equatorial plane around a Kerr black hole of mass M. We assume that the velocity of the particle v is much smaller than the speed of light c and calculate the energy absorption rate analytically. We adopt an analytic technique for the Teukolsky equation developed by Mano, Suzuki and Takasugi. We obtain the energy absorption rate to O((v/c)^8) compared to the lowest order. We find that the black hole absorption occurs at O((v/c)^5) beyond the Newtonian-quadrapole luminosity at infinity in the case when the black hole is rotating, which is O((v/c)^3) lower than the non-rotating case. Using the energy absorption rate, we investigate its effects on the orbital evolution of coalescing compact binaries. 
  A classification of the maximally extended solutions for 1+1 gravity models (comprising e.g. generalized dilaton gravity as well as models with non-trivial torsion) is presented. No restrictions are placed on the topology of the arising solutions, and indeed it is found that for generic models solutions on non-compact surfaces of arbitrary genus with an arbitrary non-zero number of holes can be obtained. The moduli space of classical solutions (solutions of the field equations with fixed topology modulo gauge transformations) is parametrized explicitly. 
  In this paper we find a self-consistent vacuum for Misner space. For this "adapted" Rindler vacuum the renormalized stress-energy tensor is zero throughout the Misner space. A point-like particle detector traveling on a timelike geodesic in a Misner space with this vacuum detects nothing. Misner space with this vacuum thus creates no problems for time travel in and of itself but a time traveler may pose a danger to himself and to the spacetime. 
  It is argued that (a) In the quantum realm test-particle masses have non-trivial observability which induces a non-geometric element in gravity, (b) Any theory of quantum gravity, on fundamental grounds, must contain an element of non-locality that makes position measurements non-commutative, and (c) The classical notion of free fall does not readily generalize to the quantum regime. 
  We present a new approach for setting initial Cauchy data for multiple black hole spacetimes. The method is based upon adopting an initially Kerr-Schild form of the metric. In the case of non-spinning holes, the constraint equations take a simple hierarchical form which is amenable to direct numerical integration. The feasibility of this approach is demonstrated by solving analytically the problem of initial data in a perturbed Schwarzschild geometry. 
  We construct the minimal off-shell formulation of N = 1 chiral supergravity (SUGRA) introducing a complex antisymmetric tensor field $B_{\mu \nu}$ and a complex axial-vector field $A_{\mu}$ as auxiliary fields. The resulting algebra of the right- and left-handed supersymmetry (SUSY) transformations closes off shell and generates chiral gauge transforamtions and vector gauge transformations in addition to the transformations which appear in the case without auxiliary fields. 
  Binary black hole interactions provide potentially the strongest source of gravitational radiation for detectors currently under development. We present some results from the Binary Black Hole Grand Challenge Alliance three- dimensional Cauchy evolution module. These constitute essential steps towards modeling such interactions and predicting gravitational radiation waveforms. We report on single black hole evolutions and the first successful demonstration of a black hole moving freely through a three-dimensional computational grid via a Cauchy evolution: a hole moving ~6M at 0.1c during a total evolution of duration ~60M. 
  Near the black hole threshold in phase space, the black hole mass as a function of the initial data shows the "critical scaling" M \simeq C (p-p_*)^\gamma, where p labels a family of initial data, p_* is the value of p at the threshold, and the critical exponent \gamma is universal for a given matter model. The black hole charge Q obeys a similar law. To complete the picture, we include angular momentum as a perturbation. For the black hole angular momentum \vec L we find the oscillating behavior \vec L \simeq Re[ (\vec A + i \vec B) (p-p_*)^{\mu+i\omega} ]. The assumptions of the calculation hold for p = \rho / 3 perfect fluid matter, and we calculate \mu \simeq 0.799 and \omega \simeq 0.231. 
  We study boson stars in Brans Dicke gravity and use them to illustrate some of the properties of three different mass definitions: the Schwarzschild mass, the Keplerian mass and the Tensor mass. We analyse the weak field limit of the solutions and show that only the Tensor mass leads to a physically reasonable definition of the binding energy. We examine numerically strong field $\omega=-1$ solutions and show how, in this extreme case, the three mass values and the conserved particle number behave as a function of the central boson field amplitude. The numerical studies imply that for $\omega=-1$, solutions with extremal Tensor mass also have extremal particle number. This is a property that a physically reasonable definition of the mass of a boson star must have, and we prove analytically that this is true for all values of $\omega$. The analysis supports the conjecture that the Tensor mass uniquely describes the total energy of an asymptotically flat solution in BD gravity. 
  With the help of a conformal, timelike Killing-vector we define generalized equilibrium states for cosmological fluids with particle production. For massless particles the generalized equilibrium conditions require the production rate to vanish and the well known ``global'' equilibrium of standard relativistic thermodynamics is recovered as a limiting case. The equivalence between the creation rate for particles with nonzero mass and an effective viscous fluid pressure follows as a consequence of the generalized equilibrium properties. The implications of this equivalence for the cosmological dynamics are discussed, including the possibility of a power-law inflationary behaviour. For a simple gas a microscopic derivation for such kind of equilibrium is given on the basis of relativistic kinetic theory. 
  The general exact solution describing the dynamics of anisotropic elastic spheres supported only by tangential stresses is reduced to a quadrature using Ori's mass-area coordinates. This leads to the explicit construction of the root equation governing the nature of the central singularity. Using this equation, we formulate and motivate on physical grounds a conjecture on the nature of this singularity. The conjecture covers a large sector of the space of initial data; roughly speaking, it asserts that addition of a tangential stress cannot undress a covered dust singularity. The root equation also allows us to analyze the case of self-similar spacetimes and to get some insight on the role of stresses in deciding the nature of the singularities in this case. 
  We construct a distinguished set of positive and negative energy modes of the Klein-Gordon equation for any Born frame of reference in Minkowski's space-time. Unlike the case of a galilean frame of reference it is unclear whether this set of modes may be an appropriate basis to define the vacuum of a quantized scalar field in an accelerated cavity. 
  We review recent progress in the study of varying constants and attempts to explain the observed values of the fundamental physical constants. We describe the variation of $G$ in Newtonian and relativistic scalar-tensor gravity theories. We highlight the behaviour of the isotropic Friedmann solutions and consider some striking features of primordial black hole formation and evaporation if $G$ varies. We discuss attempts to explain the values of the constants and show how we can incorporate the simultaneou s variations of several 'constants' exactly by using higher-dimensional unified theories. Finally, we describe some new observational limits on possible space or time variations of the fine structure constant. 
  We investigate the relativistic cosmological hydrodynamic perturbations. We present the general large scale solutions of the perturbation variables valid for the general sign of three space curvature, the cosmological constant, and generally evolving background equation of state. The large scale evolution is characterized by a conserved gauge invariant quantity which is the same as a perturbed potential (or three-space curvature) in the comoving gauge. 
  In a class of generalized gravity theories with general couplings between the scalar field and the scalar curvature in the Lagrangian, we describe the quantum generation and the classical evolution processes of both the scalar and tensor structures in a simple and unified manner. 
  The total flux of outgoing radiation in a strong gravitational field decreases due to backscattering if the sources are close to an apparent horizon. It can cause detectable changes in the shape of signals. Backscattering could well be of relevance to astrophysics and would constitute a new test of the validity of general relativity. An explicit bound for this effect is derived for scalar fields. 
  We present new results for pulsating neutron stars. We have calculated the eigenfrequencies of the modes that one would expect to be the most important gravitational-wave sources: the fundamental fluid f-mode, the first pressure p-mode and the first gravitational-wave w-mode, for twelve realistic equations of state. From this numerical data we have inferred a set of ``empirical relations'' between the mode-frequencies and the parameters of the star (the radius R and the mass M). Some of these relation prove to be surprisingly robust, and we show how they can be used to extract the details of the star (radius, mass, eos) from observed modes with errors no larger than a few percent. 
  It is shown that in the case of the spherically symmetric static backgrounds there is a gauge in which the Dirac equation is manifestly covariant under rotations. This allows us to separate the spherical variables like in the flat space-time, obtaining a pair of radial equations and a specific form of the radial scalar product. 
  A new family of analytically solvable quantum geometric models is proposed. The structure of the energy spectra as well as the form of the corresponding eigenfunctions are presented pointing out their main specific properties. 
  From data in the present we can predict the future and retrodict the past. These predictions and retrodictions are for histories -- most simply time sequences of events. Quantum mechanics gives probabilities for individual histories in a decoherent set of alternative histories. This paper discusses several issues connected with the distinction between prediction and retrodiction in quantum cosmology: the difference between classical and quantum retrodiction, the permanence of the past, why we predict the future but remember the past, the nature and utility of reconstructing the past(s), and information theoretic measures of the utility of history. (Talk presented at the Nobel Symposium: Modern Studies of Basic Quantum Concepts and Phenomena, Gimo, Sweden, June 13-17, 1997) 
  To comply with the equivalence principle in Einstein-Cartan-like theories of gravity we propose a modification of the action principle in affine flat spaces with torsion. 
  We show that the Strong Cosmic Censorship is supported by the behavior of generic solutions on the class of static spherically symmetric black holes in gravitating gauge models and their stringy generalizations. 
  We derive and discus the equations of motion for spinless matter: relativistic spinless scalar fields, particles and fluids in the recently proposed by A. Saa model of gravity with covariantly constant volume with respect to the transposed connection in Einstein-Cartan spaces.   A new interpretation of this theory as a theory with variable Plank "constant" is suggested.   We show that the consistency of the semiclassical limit of the wave equation and classical motion dictates a new definite universal interaction of torsion with massive fields. 
  We study the linearized equations describing the propagation of gravitational waves through dust. In the leading order of the WKB approximation, dust behaves as a non-dispersive, non-dissipative medium. Taking advantage of these features, we explore the possibility that a gravitational wave from a distant source gets trapped by the gravitational field of a long filament of galaxies of the kind seen in the large scale structure of the Universe. Such a waveguiding effect may lead to a huge magnification of the radiation from distant sources, thus lowering the sensitivity required for a successful detection of gravitational waves by detectors like VIRGO, LIGO and LISA. 
  We study the effect of the antisymmetric tensor field $B_{\mu\nu}$ on the large curvature phase of string cosmology. It is well-known that a non-vanishing value of $H=dB$ leads to an anisotropic expansion of the spatial dimensions. Correspondingly, in the string phase of the model, including $\alpha '$ corrections, we find anisotropic fixed points of the evolution, which act as regularizing attractors of the lowest order solutions. The attraction basin can also include isotropic initial conditions for the scale factors. We present explicit examples at order $\alpha '$ for different values of the number of spatial dimensions and for different ans\"{a}tze for $H$. 
  It is shown that in the extreme limit with a zero surface gravity but nonzero local temperature the limiting metric of a generic static black hole is determined by a metric induced on a horizon and one function of two coordinates, stress-energy tensor of a source picking up its values from a horizon. The limiting procedure is extended to rotating black holes. If the extreme limit is due to merging a black hole horizon and cosmological one both horizons are always in thermal equilibrium in this limit. This is proved for a generic case of static or axially-symmetrical rotating spacetimes. 
  The occurrence of chaos for test particles moving around a slowly rotating black hole with a dipolar halo is studied using Poincar\'e sections. We find a novel effect, particles with angular momentum opposite to the black hole rotation have larger chaotic regions in phase space than particles initially moving in the same direction. 
  We propose a mathematical structure, based on a noncommutative geometry, which combines essential aspects of general relativity and quantum mechanics, and leads to correct "limiting cases" of both these theories. We quantize a groupoid constructed on space-time rather than space-time itself. Both space and time emerge in the transition process to the commutative case. Our approach clearly suggests that quantum gravitational observables should be looked for among correlations of distant phenomena rather than among local effects. A toy model is computed (based on a finite group) which predicts the value of "cosmological constants" (in the quantum sector) which vanish when going to the standard space-time physics. 
  It is a deceptively simple question to ask how acoustic disturbances propagate in a non-homogeneous flowing fluid. This question can be answered by invoking the language of Lorentzian differential geometry: If the fluid is barotropic and inviscid, and the flow is irrotational (though possibly time dependent), then the equation of motion for the velocity potential describing a sound wave is identical to that for a minimally coupled massless scalar field propagating in a (3+1)-dimensional Lorentzian geometry. The acoustic metric governing the propagation of sound depends algebraically on the density, flow velocity, and local speed of sound. This rather simple physical system is the basis underlying a deep and fruitful analogy between the black holes of Einstein gravity and supersonic fluid flows. Many results and definitions can be carried over directly from one system to another. For example, I will show how to define the ergosphere, trapped regions, acoustic apparent horizon, and acoustic event horizon for a supersonic fluid flow, and will exhibit the close relationship between the acoustic metric for the fluid flow surrounding a point sink and the Painleve-Gullstrand form of the Schwarzschild metric for a black hole. This analysis can be used either to provide a concrete non-relativistic model for black hole physics, up to and including Hawking radiation, or to provide a framework for attacking acoustics problems with the full power of Lorentzian differential geometry. 
  We examine the reduced phase space of the Barbero-Varadarajan solutions of the Ashtekar formulation of (2+1)-dimensional general relativity on a torus. We show that it is a finite-dimensional space due to existence of an infinite dimensional residual gauge invariance which reduces the infinite-dimensional space of solutions to a finite-dimensional space of gauge-inequivalent solutions. This is in agreement with general arguments which imply that the number of physical degrees of freedom for (2+1)-dimensional Ashtekar gravity on a torus is finite. 
  We study the deformation (Moyal) quantisation of gravity in both the ADM and the Ashtekar approach. It is shown, that both can be treated, but lead to anomalies. The anomaly in the case of Ashtekar variables, however, is merely a central extension of the constraint algebra, which can be ``lifted''.   Finally we write down the equations defining physical states and comment on their physical content. This is done by defining a loop representation. We find a solution in terms of a Chern-Simons state, whose Wigner function then becomes related to BF-theory. This state exist even in the absence of a cosmological constant but only if certain extra conditions are imposed. Another solution is found where the Wigner function is a Gaussian in the momenta.   Some comments on ``quantum gravity'' in lower dimensions are also made. 
  Recently it has been argued that autoparallels should be the correct description of free particle motion in spaces with torsion, and that such trajectories can be derived from variational principles if these are suitably adapted. The purpose of this letter is to call attention to the problems that such attempts raise, namely the requirement of a more elaborate structure in order to formulate the variational principle and the lack of a Hamiltonian description for the autoparallel motion. Here is also raised the problem of how to generalize this proposed new principle to quantum mechanics and to field theory. Since all applications known of such a principle are equally well described in terms of geodesics in non-holonomic frames we conclude that there is no reason to modify the conventional variational principle that leads to geodesics. 
  In this paper, the suggested similarity between micro and macro-cosmos is extended to quantum behavior, postulating that quantum mechanics, like general relativity and classical electrodynamics, is invariant under discrete scale transformations. This hypothesis leads to a large scale quantization of angular momenta. Using the scale factor $\Lambda \sim 10^{38}$, the corresponding quantum of action, obtained by scaling the Planck constant, is close to the Kerr limit for the spin of the universe -- when this is considered as a huge rotating black-hole -- and to the spin of Godel's universe, solution of Einstein equations of gravitation. Besides, we suggest the existence of another, intermediate, scale invariance, with scale factor $\lambda \sim 10^{19}$. With this factor we obtain, from Fermi's scale, the values for the gravitational radius and for the collapse proper-time of a typical black-hole, besides the Kerr limit value for its spin. It is shown that the mass-spin relations implied by the two referred scale transformations are in accordance with Muradian's Regge-like relations for galaxy clusters and stars. Impressive results are derived when we use a $\lambda$-scaled quantum approach to calculate the mean radii of planetary orbits in solar system. Finally, a possible explanation for the observed quantization of galactic redshifts is suggested, based on the large scale quantization conjecture. 
  The task of quantizing gravity is compared with Einstein's relativization of gravity. The philosophical and physical foundations of general relativity are briefly reviewed. The Ehlers-Pirani-Schild scheme of operationally determining the geometry of space-time, using freely falling classical particle trajectories, is done using operations in an infinitesimal neighborhood around each point. The study of the free fall of a quantum wave suggests a quantum principle of equivalence. The principle of general covariance is clarified. The sign change of a Fermion field when rotated by $2\pi$ radians is used to argue for a quantum mechanical modification of space-time, which leads naturally to supersymmetry. A novel effect in quantum gravity due to the author is used to extend Einstein's hole argument to quantum gravity. This suggests a quantum principle of general covariance, according to which the fundamental laws of physics should be covariant under `quantum diffeomorphisms'. This heuristic principle implies that space-time points have no invariant meaning in quantum gravity. 
  In this Letter I point out that Hawking radiation is a purely kinematic effect that is generic to Lorentzian geometries. Hawking radiation arises for any test field on any Lorentzian geometry containing an event horizon regardless of whether or not the Lorentzian geometry satisfies the dynamical Einstein equations of general relativity. On the other hand, the classical laws of black hole mechanics are intrinsically linked to the Einstein equations of general relativity (or their perturbative extension into either semiclassical quantum gravity or string-inspired scenarios). In particular, the laws of black hole thermodynamics, and the identification of the entropy of a black hole with its area, are inextricably linked with the dynamical equations satisfied by the Lorentzian geometry: entropy is proportional to area (plus corrections) if and only if the dynamical equations are the Einstein equations (plus corrections). It is quite possible to have Hawking radiation occur in physical situations in which the laws of black hole mechanics do not apply, and in situations in which the notion of black hole entropy does not even make any sense. This observation has important implications for any derivation of black hole entropy that seeks to deduce black hole entropy from the Hawking radiation. 
  The microcanonical treatment of black holes as opposed to the canonical formulation is reviewed and some major differences are displayed. In particular the decay rates are compared in the two different pictures. 
  Extending the analysis in our previous paper, we construct the entanglement thermodynamics for a massless scalar field on the Schwarzschild spacetime. Contrary to the flat case, the entanglement energy $E_{ent}$ turns out to be proportional to area radius of the boundary if it is near the horizon. This peculiar behavior of $E_{ent}$ can be understood by the red-shift effect caused by the curved background. Combined with the behavior of the entanglement entropy, this result yields, quite surprisingly, the entanglement thermodynamics of the same structure as the black hole thermodynamics. On the basis of these results, we discuss the relevance of the concept of entanglement as the microscopic origin of the black hole thermodynamics. 
  These notes represent approximately one semester's worth of lectures on introductory general relativity for beginning graduate students in physics. Topics include manifolds, Riemannian geometry, Einstein's equations, and three applications: gravitational radiation, black holes, and cosmology. 
  We discuss the cosmological evolution of matter sources with small anisotropic pressures. This includes electric and magnetic fields, collisionless relativistic particles, gravitons, antisymmetric axion fields in low-energy string cosmologies, spatial curvature anisotropies, and stresses arising from simple topological defects. The COBE microwave sky maps are used to place strong limits on the possible contribution of these sources to the total density of the universe. We explain why the limits obtained from primordial nucleosynthesis are generally weaker than those imposed by the microwave background isotropy. The effect of inflation on all these stresses is also calculated. 
  We resolve the entire gravitational field;i.e. the Riemann curvature into its electric and magnetic parts. In general, the vacuum Einstein equation is symmetric in active and passive electric parts. However it turns out that the Schwarzschild solution, which is the unique spherically symmetric vacuum solutions can be characterised by a slightly more general equation which is not symmetric. Then the duality transformation, implying interchange of active and passive parts will relate the Schwarzschlid particle with the one with global monopole charge. That is the two are dual of each-other. It further turns out that flat spacetime is dual to massless global monopole and global texture spacetimes. 
  This article investigates the interaction of a spherically symmetric massless scalar field with a strong gravitational field. It focuses on the propagation of waves in regions outside any horizons. The two factors acting on the waves can be identified as a redshift and a backscattering. The influence of backscattering on the intensity of the outgoing radiation is studied and rigorous quantitative upper bounds obtained. These show that the total flux may be decreased if the sources are placed in a region adjoining an apparent horizon. Backscattering can be neglected in the case $2m_0 /R<< 1$, that is when the emitter is located at a distance from a black hole much larger than the Schwarzschild radius. This backscattering may have noticeable astrophysical consequences. 
  A picture of the Planck scale universe as a network of mesoscopic (with respect to that scale) resonators connected through quasi-one-dimensional waveguides is presented 
  Interior structure of non-Abelian black holes is shown to exhibit in a general case either an oscillating mass-inflationary behavior, or power-law behavior with a divergent mass function. In both cases no Cauchy horizon forms. 
  We examine the variational and conformal structures of higher order theories of gravity which are derived from a metric-connection Lagrangian that is an arbitrary function of the curvature invariants. We show that the constrained first order formalism when applied to these theories may lead consistently to a new method of reduction of order of the associated field equations. We show that the similarity of the field equations which are derived from appropriate actions via this formalism to those produced by Hilbert varying purely metric Lagrangians is not merely formal but is implied by the diffeomorphism covariant property of the associated Lagrangians. We prove that the conformal equivalence theorem of these theories with general relativity plus a scalar field, holds in the extended framework of Weyl geometry with the same forms of field and self-interacting potential but, in addition, there is a new `source term' which plays the role of a stress. We point out how these results may be further exploited and address a number of new issues that arise from this analysis. 
  We prove the cosmic no-hair conjecture for all orthogonal Bianchi cosmologies with matter in the $R+\beta R^2$ theory using the conformally equivalent Einstein field equations, with the scalar field having the full self-interacting potential, in the presence of the conformally related matter fields. We show, in particular, that the Bianchi IX universe asymptotically approaches de Sitter space provided that initially the scalar three-curvature does not exceed the potential of the scalar field associated with the conformal transformation. Our proof relies on rigorous estimates of the possible bounds of the so-called Moss-Sahni function which obeys certain differential inequalities and a non-trivial argument which connects the behaviour of that function to evolution of the spatial part of the scalar curvature. 
  We present a rigorous analysis of the role and uses of the adiabatic invariant in the Mixmaster dynamical system. We propose a new invariant for the global dynamics which in some respects has an improved behaviour over the commonly used one. We illustrate its behaviour in a number of numerical results. We also present a new formulation of the dynamics via Catastrophe Theory. We find that the change from one era to the next corresponds to a fold catastrophe, during the Kasner shifts the potential is an Implicit Function Form whereas, as the anisotropy dissipates, the Mixmaster potential must become a Morse 0--saddle. We compare and contrast our results to many known works on the Mixmaster problem and indicate how extensions could be achieved. Further exploitation of this formulation may lead to a clearer understanding of the global Mixmaster dynamics. 
  The derivation of the general solutions for stationary and static cylindrically symmetric Einstein spaces of Lewis form is revisited and the physical and geometrical meaning of the parameters appearing in the resulting solutions are investigated. It is shown that three of the parameters (and the value of the cosmological constant) are essential, of which one characterizes the local gravitational field and appears in the Cartan scalars, while the remaining two give information about the topological identification made to produce cylindrical symmetry. Other than the cosmological constant, they can be related to the parameters of the vacuum Weyl and Lewis classes, whose interpretation was previously investigated by da Silva et al. (1995a, 1995b). 
  We investigate nature of asymptotically de Sitter space-times containing a black hole. We show that if the matter fields satisfy the dominant energy condition and the cosmic censorship holds in the considering space-time, the area of the cosmological event horizon for an observer approaching a future timelike infinity does not decrease, i.e. the second law is satisfied. We also show under the same conditions that the total area of the black hole and the cosmological event horizon, a quarter of which is the total Bekenstein-Hawking entropy, is less than $12\pi/\Lambda$, where $\Lambda$ is a cosmological constant. Physical implications are also discussed. 
  The occurrence of chaos for test particles moving in a Taub-NUT spacetime with a dipolar halo perturbation is studied using Poincar\'e sections. We find that the NUT parameter (magnetic mass) attenuates the presence of chaos. 
  General Relativity is recovered from Brans-Dicke gravity in the limit of large $\omega$. In this article we investigate theories of Brans-Dicke gravity with chaotic inflation, allowing for either a constant or variable value of $\omega$, known as extended and hyperextended inflation respectively. The main focus of the paper is placed on the latter. The variation $\omega$ with respect to the Brans-Dicke field is based on higher-order corrections analogous to those of the dilaton field in string theory, following the simple principle that the Brans-Dicke and metric fields decouple asymptotically. The question addressed is whether a large value of $\omega$ is predominant in most regions of the universe, which would lead to the conclusion that a typical region is then governed by General Relativity. In these theories we find that it is possible to construct inflaton potentials that drive the evolution of the Brans-Dicke field to an appropriate range of values at the end of inflation, such that a large $\omega$ is indeed typical in an average region of the universe. However, in general this conclusion does not hold and it is shown that for a wide class of inflaton potentials General Relativity is not a priori a typical theory. 
  The probability that interferometric detectors such as LIGO and VIRGO will successfully detect inspiraling compact binaries depends in part on our knowledge of the expected gravitational wave forms.   The best approximations to the true wave forms available today are the post-Newtonian (PN) templates. In this paper we argue that these 2PN templates are accurate enough for a successful search for compact binaries with the advanced LIGO interferometer. Results are presented for the 40-meter Caltech prototype as well as for the inital and advance LIGO detectors. 
  The continuation of Misner space into the Euclidean region is seen to imply the topological restriction that the period of the closed spatial direction becomes time-dependent. This restriction results in a modified Lorentzian Misner space in which the renormalized stress-energy tensor for quantized complex massless scalar fields becomes regular everywhere, even on the chronology horizon. A quantum-mechanically stable time machine with just the sub-microscopic size may then be constructed out of the modified Misner space, for which the semiclassical Hawking's chronology protection conjecture is no longer an obstruction. 
  The Einstein equation in D dimensions, if restricted to the class of space-times possessing n = D - 2 commuting hypersurface-orthogonal Killing vectors, can be equivalently written as metric-dilaton gravity in 2 dimensions with n scalar fields. For n = 2, this results reduces to the known reduction of certain 4-dimensional metrics which include gravitational waves. Here, we give such a representation which leads to a new proof of the Birkhoff theorem for plane-symmetric space--times, and which leads to an explanation, in which sense two (spin zero-) scalar fields in 2 dimensions may incorporate the (spin two-) gravitational waves in 4 dimensions. (This result should not be mixed up with well--known analogous statements where, however, the 4-dimensional space-time is supposed to be spherically symmetric, and then, of course, the equivalent 2-dimensional picture cannot mimic any gravitational waves.) Finally, remarks on hidden symmetries in 2 dimensions are made. 
  We study on the third quantization of a Kaluza-Klein toy model. In this model time ($x$) is defined by the scale factor of universe, and the space coordinate ($y$) is defined by the ratio of the scales of the ordinary space and the internal space. We calculate the number density of the universes created from nothing and examine whether the compactification can be explained statistically by the idea of the third quantization. 
  The problem of existence of spacelike hypersurfaces with constant mean curvature in asymptotically flat spacetimes is considered for a class of asymptotically Schwarzschild spacetimes satisfying an interior condition. Using a barrier construction, a proof is given of the existence of complete hypersurfaces with constant mean cuvature which intersect null infinity in a regular cut. 
  We compute the propagation and scattering of linear gravitational waves off a Schwarzschild black hole using a numerical code which solves a generalization of the Zerilli equation to a three dimensional cartesian coordinate system. Since the solution to this problem is well understood it represents a very good testbed for evaluating our ability to perform three dimensional computations of gravitational waves in spacetimes in which a black hole event horizon is present. 
  When the term ``black hole'' was originally coined in 1968, it was immediately conjectured that the only pure vacuum equilibrium states were those of the Kerr family. Efforts to confirm this made rapid progress during the ``classical phase'' from 1968 to 1975, and some gaps in the argument have been closed during more recent years. However the presently available demonstration is still subject to undesirably restrictive assumptions such as non-degeneracy of the horizon, as well as analyticity and causality in the exterior. 
  A particulare case of the three-body problem, in the PPN formalism, is presented. The Hamiltonian function is obtained and the problem is reduced to a perturbed two-body one. 
  We present a simple intuitive derivation of the corrections to the intensity of gravitational radiation due to the so-called tail effect. 
  We study analytically the asymptotic evolution of charged fields around a Reissner-Nordstr\"om black-hole. Following the no-hair theorem we focus attention on the dynamical mechanism by which the charged hair is radiated away. We find an inverse power-law relaxation of the charged fields at future timelike infinity, along future null infinity and an oscillatory inverse power-law relaxation along the future outer horizon. We show that a charged hair is shedd slower than a neutral one. Our results are also of importance to the study of mass inflation and the stability of Cauchy horizons during a dynamical gravitational collapse of charged matter to form a charged black-hole 
  We study in a Brill-Hartle type of approximation the back-reaction of a superposition of linear gravitational waves in an Einstein-de Sitter background up to the second order in the small wave amplitudes $h_{ik}$. The wave amplitudes are assumed to form a homogeneous and isotropic stochastic process. No restriction for the wavelengths is assumed. The effective stress-energy tensor $T^{e}_{\mu\nu}$ is calculated in terms of the correlation functions of the process. We discuss in particular a situation where $T^{e}_{\mu\nu}$ is the dominant excitation of the background metric. Apart from the Tolman radiation universe, a solution with the scale factor of the de Sitter universe exists with $p = -\rho$ as effective equation of state. 
  The Euclidean quantum amplitude to go between data specified on an initial and a final hypersurface may be approximated by the tree amplitude exp(-I_{classical}/\hbar), where I_{classical} is the Euclidean action of the classical solution joining the initial and final data. In certain cases the tree amplitude is exact. We study I_{classical} hence the quantum amplitude, in the case of a spherically symmetric Riemannian gravitational field coupled to a spherically symmetric scalar field. The classical scalar field obeys an elliptic equation, which we solve using relaxation techniques, in conjunction with the field equations giving the gravitational field. An example of the transition from linearity to non-linearity is presented and power law behavior of the action is demonstrated. 
  The gravitational radiation produced by binary black holes during their inspiral, merger, and ringdown phases is a promising candidate for detection by the first or second generation of kilometer-scale interferometric gravitational wave antennas. Waveforms for the last phase, the quasinormal ringing, are well understood. I discuss the feasibility of detection of the quasinormal ringing of a black hole based on an analysis of the Caltech 40-meter interferometer data. 
  The relation between the long wavelength limit of solutions to the cosmological perturbation equations and the perturbations of solutions to the exactly homogeneous background equations is investigated for scalar perturbations on spatially flat cosmological models. It is shown that a homogeneous perturbation coincides with the long wavelength limit of some inhomogeneous perturbation only when the former satisfies an additional condition corresponding to the momentum constraint if the matter consists only of scalar fields. In contrast, no such constraint appears if the fundamental variables describing the matter contain a vector field as in the case of a fluid. Further, as a byproduct of this general analysis, it is shown that there exist two universal exact solutions to the perturbation equations in the long wavelength limit, which are expressed only in terms of the background quantities. They represent adiabatic growing and decaying modes, and correspond to the well-known exact solutions for perfect fluid systems and scalar field systems. 
  We discuss the issue of motivating the analysis of higher order gravity theories and their cosmologies and introduce a rule which states that these theories may be considered as a vehicle for testing whether certain properties may be of relevance to quantum theory. We discuss the physicality issue arising as a consequence of the conformal transformation theorem, the question of formulating a consistent first order formalism of such theories and also the isotropization problem for a class of generalized cosmologies. We point out that this field may have an important role to play in clarifying issues arising also in general relativity. 
  We describe a grid generation procedure designed to construct new classes of orthogonal coordinate systems for binary black hole spacetimes. The computed coordinates offer an alternative approach to current methods, in addition to providing a framework for potentially more stable and accurate evolutions of colliding black holes. As a particular example, we apply our procedure to generate appropriate numerical grids to evolve Misner's axisymmetric initial data set representing two equal mass black holes colliding head-on. These new results are compared with previously published calculations, and we find generally good agreement in both the waveform profiles and total radiated energies over the allowable range of separation parameters. Furthermore, because no specialized treatment of the coordinate singularities is required, these new grids are more easily extendible to unequal mass and spinning black hole collisions. 
  We study a correct statement of the scattering problem arising for quantum charged scalar particles on the Reissner-Nordstr\"{o}m black holes when taking into account the own electric field of black hole. The elements of the corresponding S-matrix are explored in the form convenient to physical applications and for applying numerical methods. Some further possible issues are outlined. 
  We establish the existence of hairy black holes in su(N) Einstein-Yang-Mills theories, described by N-1 parameters, corresponding to the nodes of the gauge field functions. 
  This is the first in a series of articles on the numerical solution of Friedrich's conformal field equations for Einstein's theory of gravity. We will discuss in this paper why one should be interested in applying the conformal method to physical problems and why there is good hope that this might even be a good idea from the numerical point of view. We describe in detail the derivation of the conformal field equations in the spinor formalism which we use for the implementation of the equations, and present all the equations as a reference for future work. Finally, we discuss the implications of the assumptions of a continuous symmetry. 
  We show that an exact solution of two-dimensional dilaton gravity with matter discovered previously exhibits an irreversible temporal flow towards flat space with a vanishing cosmological constant. This time flow is induced by the back reaction of matter on the space-time geometry. We demonstrate that the system is not in equilibrium if the cosmological constant is non-zero, whereas the solution with zero cosmological constant is stable. The flow of the system towards this stable end-point is derived from the renormalization-group flow of the Zamolodchikov function. This behaviour is interpreted in terms of non-critical Liouville string, with the Liouville field identified as the target time. 
  This is the second in a series of articles on the numerical solution of Friedrich's conformal field equations for Einstein's theory of gravity. We will discuss in this paper the numerical methods used to solve the system of evolution equations obtained from the conformal field equations. In particular we discuss in detail the choice of gauge source functions and the treatment of the boundaries. Of particular importance is the process of ``radiation extraction'' which can be performed in a straightforward way in the present formalism. 
  Certain features associated with the symmetry reduction of the vacuum Einstein equations by two commuting, space-like Killing vector fields are studied. In particular, the discussion encompasses the equations for the Gowdy $T^3$ cosmology and cylindrical gravitational waves. We first point out a relation between the $SL(2,R)$ (or SO(3)) $\sigma$ and principal chiral models, and then show that the reduced Einstein equations can be obtained from a dimensional reduction of the standard SL(2,R) sigma-model in three dimensions. The reduced equations can also be derived from the action of a `generalized' two dimensional SL(2,R) sigma-model with a time dependent constraint. We give a Hamiltonian formulation of this action, and show that the Hamiltonian evolution equations for certain phase space variables are those of a certain generalization of the principal chiral model. Using these Hamiltonian equations, we give a prescription for obtaining an infinite set of constants of motion explicitly as functionals of the space-time metric variables. 
  We outline a mission with the aim of directly detecting the gravitomagnetic field of the Earth. This mission is called Gravity Probe C. Gravity Probe C(lock) is based on a recently discovered and surprisingly large gravitomagnetic clock effect. The main idea is to compare the proper time of two standard clocks in direct and retrograde orbits around the Earth. After one orbit the proper time difference of two such clocks is predicted to be of the order of $2\times 10^{-7}$ s. The conceptual difficulty to perform Gravity Probe C is expected to be comparable to that of the Gravity Probe B--mission. 
  Numerical investigation of a class of inhomogeneous cosmological spacetimes shows evidence that at a generic point in space the evolution toward the initial singularity is asymptotically that of a spatially homogeneous spacetime with Mixmaster behavior. This supports a long-standing conjecture due to Belinskii et al. on the nature of the generic singularity in Einstein's equations. 
  We consider the radiation reaction to the motion of a point-like particle of mass $m$ and specific spin $S$ traveling on a curved background. Assuming $S=O(Gm)$ and $Gm\ll L$ where $L$ is the length scale of the background curvature, we divide the spacetime into two regions; the external region where the metric is approximated by the background metric plus perturbations due to a point-like particle and the internal region where the metric is approximated by that of a black hole plus perturbations due to the tidal effect of the background curvature, and use the technique of the matched asymptotic expansion to construct an approximate metric which is valid over the entire region. In this way, we avoid the divergent self-gravity at the position of the particle and derive the equations of motion from the consistency condition of the matching. The matching is done to the order necessary to include the effect of radiation reaction of $O(Gm)$ with respect to the background metric as well as the effect of spin-induced force. The reaction term of $O(Gm)$ is found to be completely due to tails of radiation, that is, due to curvature scattering of gravitational waves. In other words, the reaction force is found to depend on the entire history of the particle trajectory. Defining a regularized metric which consists of the back- ground metric plus the tail part of the perturbed metric, we find the equations of motion reduce to the geodesic equation on this regularized metric, except for the spin-induced force which is locally expressed in terms of the curvature and spin tensors. Some implications of the result and future issues are briefly discussed. 
  We present analytic calculations of gravitational waves from a particle orbiting a black hole. We first review the Teukolsky formalism for dealing with the gravitational perturbation of a black hole. Then we develop a systematic method to calculate higher order post-Newtonian corrections to the gravitational waves emitted by an orbiting particle. As applications of this method, we consider orbits that are nearly circular, including exactly circular ones, slightly eccentric ones and slightly inclined orbits off the equatorial plane of a Kerr black hole and give the energy flux and angular momentum flux formulas at infinity with higher order post-Newtonian corrections. Using a different method that makes use of an analytic series representation of the solution of the Teukolsky equation, we also give a post-Newtonian expanded formula for the energy flux absorbed by a Kerr black hole for a circular orbit. 
  We consider space-times which are asymptotically flat at spacelike infinity, i^0. It is well known that, in general, one cannot have a smooth differentiable structure at i^0, but have to use direction dependent structures. Instead of the oftenly used C^{>1}-differentiabel structure, we suggest a weaker differential structure, a C^{1^+} structure. The reason for this is that we have not seen any completions of the Schwarzschild space-time which is C^{>1} in both spacelike and null directions at {i^0}. In a C^{1^+} structure all directions can be treated equal, at the expense of logarithmic singularities at {i^0}. We show that, in general, the relevant part of the curvature tensor, the Weyl part, is free from these singularities, and that the (rescaled) Weyl tensor has a certain symmetry. 
  We develop the argument that initial real tunneling in quantum gravity be contemplated as a thermodynamical analogous to a black hole condensate in equilibrium with Hawking's radiation in a box. The total entropy is always maximized in the Lorentzian sector of the manifold, and, in this sense, tunneling is predicted. 
  In the spirit of general relativity, spacetime should become curved due to the presence of a particle of a given mass and charge, We try to understand this fact in the quantum theory of a thin shell of matter. It leads to a generalization of the potential energy of the shell in the semiclassical highest dominant order of Planck mass. Rather surprisingly, the quantization of charge is obtained as a consequence of the existence of bound states and the quantum of the squared bare electrical charge is $\hbar c/2$ 
  We draw attention to the possibility that inflation (i.e. accelerated expansion) might continue after the end of slow roll, during a period of fast oscillations of the inflaton field \phi . This phenomenon takes place when a mild non-convexity inequality is satisfied by the potential V(\phi). The presence of such a period of \phi-oscillation-driven inflation can substantially modify reheating scenarios.   In some models the effect of these fast oscillations might be imprinted on the primordial perturbation spectrum at cosmological scales. 
  The pre-big bang's kinetic driven inflationary mechanism is not an adequate form of inflation: the Planck length grows more rapidly than the scale factor. In order to explain our large universe, the resulting post-big bang universe requires the same unnatural constants (Planck problem) as those of any other non-inflationary big bang model. 
  Starting with a `bare' 4-dimensional differential manifold as a model of spacetime, we discuss the options one has for defining a volume element which can be used for physical theories. We show that one has to prescribe a scalar density \sigma. Whereas conventionally \sqrt{|\det g_{ij}|} is used for that purpose, with g_{ij} as the components of the metric, we point out other possibilities, namely \sigma as a `dilaton' field or as a derived quantity from either a linear connection or a quartet of scalar fields, as suggested by Guendelman and Kaganovich. 
  The dynamical evolution of self-gravitating scalar field configurations in numerical relativity is studied. The previous analysis on ground state boson stars of non-interacting fields is extended to excited states and to fields with self couplings.   Self couplings can significantly change the physical dimensions of boson stars, making them much more astrophysically interesting (e.g., having mass of order 0.1 solar mass). The stable ($S$) and unstable ($U$) branches of equilibrium configurations of boson stars of self-interacting fields are studied; their behavior under perturbations and their quasi-normal oscillation frequencies are determined and compared to the non-interacting case.   Excited states of boson stars with and without self-couplings are studied and compared. Excited states also have equilibrium configurations with $S$ and $U$ branch structures; both branches are intrinsically unstable under a generic perturbation but have very different instability time scales. We carried out a detailed study of the instability time scales of these configurations. It is found that highly excited states spontaneously decay through a cascade of intermediate states similar to atomic transitions. 
  We study the dynamics of a self-gravitating scalar field solitonic object (boson star) in the Jordan-Brans-Dicke (BD) theory of gravity. We show dynamical processes of this system such as (i) black hole formation of perturbed equilibrium configuration on an unstable branch; (ii) migration of perturbed equilibrium configuration from the unstable branch to stable branch; (iii) transition from excited state to a ground state. We find that the dynamical behavior of boson stars in BD theory is quite similar to that in general relativity (GR), with comparable scalar wave emission. We also demonstrate the formation of a stable boson star from a Gaussian scalar field packet with flat gravitational scalar field initial data. This suggests that boson stars can be formed in the BD theory in much the same way as in GR. 
  In the No-Boundary Universe a primordial black hole is created from a constrained gravitational instanton. The black hole created is immersed in the de Sitter background with a positive cosmological constant. The constrained instanton is characterized not only by the external parameters, the mass parameter, charge and angular momentum, but also by one more internal parameter, the identification period in the imaginary time coordinate. Although the period has no effect on the black hole background, its inverse is the temperature of the no-boundary state of the perturbation modes perceived by an observer. By using the Bogoliubov transformation, we show that the perturbation modes of both scalar and spinor fields are in thermal q equilibrium with the black hole background at the arbitrary temperature. However, for the two extreme cases, the de Sitter and the Nariai models, the no-boundary state remains pure. 
  The space of states and operators for a large class of background independent theories of quantum spacetime dynamics is defined. The SU(2) spin networks of quantum general relativity are replaced by labelled compact two-dimensional surfaces. The space of states of the theory is the direct sum of the spaces of invariant tensors of a quantum group G_q over all compact (finite genus) oriented 2-surfaces. The dynamics is background independent and locally causal. The dynamics constructs histories with discrete features of spacetime geometry such as causal structure and multifingered time. For SU(2) the theory satisfies the Bekenstein bound and the holographic hypothesis is recast in this formalism. 
  It is supposed that in our Universe with compactified extra dimensions (ED) the domains exist with noncompactified ED. Such domain can be a wormhole-like solution in multidimensional gravity (MD), located between two null surfaces. With the availability of compactification mechanism this MD domain can be joined on null surfaces with two black holes filled by gauge field. Solution of this kind in MD gravity on the principal bundle with structural group SU(3) is obtained. This solution is wormhole-like object located between two null surfaces $ds^2=0$. In some sense these solutions are dual to black holes: they are statical spherically symmetric solutions under null surfaces whereas black holes are statical spherically symmetric solutions outside of event horizon. 
  We briefly show how we can obtain Hamiltonians for spatially compact locally homogeneous vacuum spacetimes. The dynamical variables are categorized into the curvature parameters and the Teichm\"{u}ller parameters. While the Teichm\"{u}ller parameters usually parameterise the covering group of the spatial sections, we utilise another suitable parameterization where the universal cover metric carries all the dynamical variables and with this we reduce the Hamiltonians. For our models, all dynamical variables possess their clear geometrical meaning, in contrast to the conventional open models.} 
  The quantum theory of General Relativity at low energy exists and is of the form called "effective field theory". In this talk I describe the ideas of effective field theory and its application to General Relativity. 
  The five-dimensional (5D) Riemannian G\"odel-type manifolds are examined in light of the equivalence problem techniques, as formulated by Cartan. The necessary and sufficient conditions for local homogeneity of these 5D manifolds are derived. The local equivalence of these homogeneous Riemannian manifolds is studied. It is found that they are characterized by two essential parameters $m^2$ and $\omega $: identical pairs $(m^2, \omega)$ correspond to locally equivalent 5D manifolds. An irreducible set of isometrically nonequivalent 5D locally homogeneous Riemannian G\"odel-type metrics are exhibited. A classification of these manifolds based on the essential parameters is presented, and the Killing vector fields as well as the corresponding Lie algebra of each class are determined. It is shown that apart from the $(m^2= 4 \omega^2, \omega\not=0)$ and $(m^2\not=0, \omega = 0)$ classes the homogeneous Riemannian G\"odel-type manifolds admit a seven-parameter maximal group of isometry ($G_7$). The special class $(m^2= 4 \omega^2, \omega\not=0)$ and the degenerated G\"odel-type class $(m^2\not=0, \omega=0)$ are shown to have a $G_9$ as maximal group of motion. The breakdown of causality in these classes of homogeneous G\"odel-type manifolds are also examined. 
  If at least some Wheeler-DeWitt solutions can be interpreted as zero-energy resonances then the total s-wave cross section of the corresponding quantum universes is infinite 
  Taub numbers are studied on asymptotically flat backgrounds with Killing symmetries. When the field equations are solved for a background spacetime and higher order functional derivatives (higher order variational derivatives of the Hilbert Lagrangean) are solved for perturbations from the background, such perturbed space-times admit zeroth, first, and second order Taub numbers. Zeroth order Taub numbers are Komar constants (upto numerical factors) or Penrose-Goldberg constants of the background. For a Killing symmetry of the background, first order Taub numbers give the contribution of the linearized perturbation to the associated backgound quantity, such as the perturbing mass. Second order Taub numbers give the contribution of second order perturbations to the background quantity. The Bondi mass is a sum of first and second order Taubs numbers on a Minkowski background. 
  Within a spin-gauge theory of gravity unified with the electroweak interaction we start with totally symmetric left- and right-handed fermions and explain the parity violation by symmetry breaking in such a way that the $W^{\pm}$-bosons couple only to the left-handed leptons. Right-handed neutrinos exist and couple as the right-handed electrons only to the Z-bosons (and to gravity). The mass of the neutrinos comes out to be necessarily zero. Therefore this procedure cannot be transferred to the quarks, because then the u-quark would become massless too; for this reason parity violation with respect to the quarks is avoided. On the other hand concerning gravity the u-quark couples only with u-quarks and the d-quark only with d-quarks, however with the same strength, so that isotopic effects appear regarding the equivalence principle in such a way, that the macroscopic gravitational constant depends on the isotopic composition of the material. 
  The paper presents the conservative dynamics of two-body point-mass systems up to the third post-Newtonian order ($1/c^6$). The two-body dynamics is given in terms of a higher order ADM Hamilton function which results from a third post-Newtonian Routh functional for the total field-plus-matter system. The applied regularization procedures, together with making use of distributional differentiation of homogeneous functions, give unique results for the terms in the Hamilton function apart from the coefficient of the term $(\nu p_{i}{\pa_{i}})^2r^{-1}$. The result suggests an invalidation of the binary point-mass model at the third post-Newtonian order. 
  We showed in a previous paper that a wide class of nonmetric theories of gravity encompassed by the $\chi g$ formalism predict that the speed of light rays depends on the their polarization direction relative to directions singled out by the gravitational field. This gravity-induced birefringence of space is a violation of the Equivalence Principle and is due to the nonmetric coupling between gravity and electromagnetism. In this paper we analyze the propagation of light in the gravitational field of a rotating black hole when nonmetric couplings to curvature are included and compute the time delay between rays with orthogonal polarizations. We obtain an upper bound on the strength of QED like and CP violating curvature couplings using time delay data for pulsar PSR 1937+21. By comparison the corresponding coupling strength for QED coupling is 37 orders of magnitude less. 
  The membrane paradigm is the remarkable view that, to an external observer, a black hole appears to behave exactly like a dynamical fluid membrane, obeying such pre-relativistic equations as Ohm's law and the Navier-Stokes equation. It has traditionally been derived by manipulating the equations of motion. Here we provide an action formulation of this picture, clarifying what underlies the paradigm, and simplifying the derivations. Within this framework, we derive previous membrane results, and extend them to dyonic black hole solutions. We discuss how it is that an action can produce dissipative equations. Using a Euclidean path integral, we show that familiar semi-classical thermodynamic properties of black holes also emerge from the membrane action. Finally, in a Hamiltonian description, we establish the validity of a minimum entropy production principle for black holes. 
  It is shown how the technique of restricted path integrals (RPI) or quantum corridors (QC) may be applied for the analysis of relativistic measurements. Then this technique is used to clarify the physical nature of thermal effects as seen by an accelerated observer in Minkowski space-time (Unruh effect) and by a far observer in the field of a black hole (Hawking effect). The physical nature of the "thermal atmosphere" around the observer is analysed in three cases: a) the Unruh effect, b) an eternal (Kruskal) black hole and c) a black hole forming in the process of collapse. It is shown that thermal particles are real only in the case (c). In the case (b) they cannot be distinguished from real particles but they do not carry away mass of the black hole until some of these particles are absorbed by the far observer. In the case (a) thermal particles are virtual. 
  A spherical gravitational wave antenna is distinct from other types of gravitational wave antennas in that only a single detector is necessary to determine the direction and polarization of a gravitational wave. Zhou and Michelson showed that the inverse problem can be solved using the maximum likelihood method if the detector outputs are independent and have normally distributed noise with the same variance. This paper presents an analytic solution using only linear algebra that is found to produce identical results as the maximum likelihood method but with less computational burden. Applications of this solution to gravitational waves in alternative symmetric metric theories of gravity and impulsive excitations also are discussed. 
  It is shown that in the weak field approximation solutions of Brans-Dicke equations are simply related to the solutions of General Relativity equations for the same matter distribution. A simple method is developed which permits to obtain Brans-Dicke solutions from Einstein solutions when both theories are considered in their linearized forms. To illustrate the method some examples found in the literature are discussed. 
  Static solutions of white dwarfs with spherical symmetry and local anisotropy are studied in the post-Newtonian approximation. It is argued that the condition for equilibrium must be that the total energy is a minimum for given baryon number and the question whether there is local isotropy or anisotropy inside the star should follow from that condition, rather than be postulated "a priori". It is shown show that, in post-Newtonian gravity, there are stable configurations with local anisotropy for masses above the Chandrasekhar limit. 
  The standard General Relativity results for precession of particle orbits and for bending of null rays are derived as special cases of perturbation of a quantity that is conserved in Newtonian physics, the Runge-Lenz vector. First this method is applied to give a derivation of these General Relativity effects for the case of the spherically symmetric Schwarzschild geometry. Then the lowest order correction due to an angular momentum of the central body is considered. The results obtained are well known, but the method used is rather more efficient than that found in the standard texts, and it provides a good occasion to use the Runge-Lenz vector beyond its standard applications in Newtonian physics. 
  The static spherically symmetric ``black hole solution" of the Einstein - conformally invariant massless scalar field equations known as the BBMB ( Bocharova, , Bronikov, Melinkov, Bekenstein) black hole is critically examined. It is shown that the stress energy tensor is ill-defined at the horizon, and that its evaluation through suitable regularization yields ambiguous   results. Consequently, the configuration fails to represent a genuine black hole solution. With the removal of this solution as a counterexample to the no hair conjecture, we argue that the following appears to be true: Spherical black holes cannot carry any kind of classical scalar hair. 
  As first discovered by Choptuik, the black hole threshold in the space of initial data for general relativity shows both surprising structure and surprising simplicity. Universality, power-law scaling of the black hole mass, and scale echoing have given rise to the term "critical phenomena". They are explained by the existence of exact solutions which are attractors within the black hole threshold, that is, attractors of codimension one in phase space, and which are typically self-similar. This review gives an introduction to the phenomena, tries to summarize the essential features of what is happening, and then presents extensions and applications of this basic scenario. Critical phenomena are of interest particularly for creating surprising structure from simple equations, and for the light they throw on cosmic censorship. They may have applications in quantum black holes and astrophysics. 
  In this paper we calculate the entropy of a thin spherical shell that contracts reversibly from infinity down to its event horizon. We find that, for a broad class of equations of state, the entropy of a non-extremal shell is one-quarter of its area in the black hole limit. The considerations in this paper suggest the following operational definition for the entropy of a black hole: $S_{BH}$ is the equilibrium thermodynamic entropy that would be stored in the material which gathers to form the black hole, if all of this material were compressed into a thin layer near its gravitational radius. Since the entropy for a given mass and area is maximized for thermal equilibrium we expect that this is the maximum entropy that could be stored in the material before it crosses the horizon. In the case of an extremal black hole the shell model does not assign an unambiguous value to the entropy. 
  We discuss the new class of static axially symmetric black hole solutions obtained recently in Einstein-Yang-Mills and Einstein-Yang-Mills-dilaton theory. These black hole solutions are asymptotically flat and they possess a regular event horizon. The event horizon is almost spherically symmetric with a slight elongation along the symmetry axis. The energy density of the matter fields is angle-dependent at the horizon. The static axially symmetric black hole solutions satisfy a simple relation between mass, dilaton charge, entropy and temperature. The black hole solutions are characterized by two integers, the winding number $n$ and the node number $k$ of the purely magnetic gauge field. With increasing node number the magnetically neutral black hole solutions form sequences tending to limiting solutions with magnetic charge $n$, corresponding to Einstein-Maxwell-dilaton black hole solutions for finite dilaton coupling constant and to Reissner-Nordstr\o m black hole solutions for vanishing dilaton coupling constant. 
  The purpose of this note is to make several advances in the interpretation of the balanced state sum model by Barrett and Crane in gr-qc/9709028 as a quantum theory of gravity. First, we outline a shortcoming of the definition of the model in pointed out to us by Barrett and Baez in private communication, and explain how to correct it. Second, we show that the classical limit of our state sum reproduces the Einstein-Hilbert lagrangian whenever the term in the state sum to which it is applied has a geometrical interpretation. Next we outline a program to demonstrate that the classical limit of the state sum is in fact dominated by terms with geometrical meaning. This uses in an essential way the alteration we have made to the model in order to fix the shortcoming discussed in the first section. Finally, we make a brief discussion of the Minkowski signature version of the model. 
  The conditions of the traversable wormhole joining with the exterior space-time are considered in details and the mixed boundary problem for the Einstein equations is formulated. It is shown that, in opposite to some declarations, the conditions of the wormhole joining with the exterior space-time have non-dynamical nature and can not be defined by the physical processes. The role of these conditions in the formation of the causal structure of space-time is analyzed. It is shown that the causal structure of the wormhole-type space-time models is independent from both the interior and exterior energy-momentum tensors. This statement is illustrated in the particular case of the spherical wormhole joining with flat exterior space-time. The same conditions, which define the wormhole joining with the exterior space-time, provide the absence of paradoxes in the models with causality violation. It is pointed out, that the nature and physical interpretation of the conditions of wormhole joining with the exterior space-time and induced boundary conditions for the field variables is one of the fundamental problems, which arise in the models with causality violation. 
  The Minkowski space of special relativity can be understood as a flat 4-dimensional affine space enriched by a constant Minkowski metric. If we gauge the general affine group and `superimpose' the metric, then we arrive at the metric-affine theory of gravity (MAG). The gravitational potentials are the spacetime coframe, the metric, and the linear connection. The material energy-momentum is coupled to the coframe (and the metric), a hypothetical hypermomentum current to the connection. The hypermomentum splits in a spin, a dilation, and a shear piece. We collect some evidence in favor of the existence of a material shear current in the context of Regge type trajectories of `hadronic' matter, thus supporting the link between particle physics and MAG. 
  The variational methods implemented on a quadratic Yang-Mills type Lagrangian yield two sets of equations interpreted as the field equations and the energy-momentum tensor for the gravitational field. A covariant condition is imposed on the energy-momentum tensor to represent the radiation field. A generalized pp-wave metric is found to simultaneously satisfy both the field equations and the radiation condition. The result is compared with that of Lichn\'{e}rowicz. 
  We evaluate the influence functional for two dimensional models of dilaton gravity. This functional is exactly computed when the conformal invariance is preserved, and it can be written as the difference between the Liouville actions on each closed-time-path branch plus a boundary term. From the influence action we derive the covariant form of the semiclassical field equations. We also study the quantum to classical transition in cosmological backgrounds. In the conformal case we show that the semiclassical approximation is not valid because there is no imaginary part in the influence action. Finally we show that the inclusion of the dilaton loop in the influence functional breaks conformal invariance and ensures the validity of the semiclassical approximation. 
  We derive an integral representation which encodes all coefficients of the Riemann normal coordinate expansion, and also a closed formula for those coefficients. 
  In this paper we consider a model of Poincar\'e gauge theory (PGT) in which a translational gauge field and a Lorentz gauge field are actually identified with the Einstein's gravitational field and a pair of ``Yang-Mills'' field and its partner, respectively.In this model we re-derive some special solutions and take up one of them. The solution represents a ``Yang-Mills'' field without its partner field and the Reissner-Nordstr\"om type spacetime, which are generated by a PGT-gauge charge and its mass.It is main purpose of this paper to investigate the interaction of massless Dirac fields with those fields. As a result, we find an interesting fact that the left-handed massless Dirac fields behave in the different manner from the right-handed ones. This can be explained as to be caused by the direct interaction of Dirac fields with the ``Yang-Mills'' field. Accordingly, the phenomenon can not happen in the behavior of the neutrino waves in ordinary Reissner-Nordstr\"om geometry. The difference between left- and right-handed effects is calculated quantitatively, considering the scattering problems of the massless Dirac fields by our Reissner-Nordstr\"om type black-hole. 
  An internal singularity of a string four-dimensional black hole with second order curvature corrections is discussed. A restriction to a minimal size of a neutral black hole is obtained in the frame of the model considered. Vacuum polarization of the surrounding space-time caused by this minimal-size black hole is also discussed. 
  We study stability of a circular orbit of a spinning test particle in a Kerr spacetime. We find that some of the circular orbits become unstable in the direction perpendicular to the equatorial plane, although the orbits are still stable in the radial direction. Then for the large spin case ($S < \sim O(1)), the innermost stable circular orbit (ISCO) appears before the minimum of the effective potential in the equatorial plane disappears. This changes the radius of ISCO and then the frequency of the last circular orbit. 
  We argue that from the point of view of gauge theory and of an appropriate interpretation of the interferometer experiments with matter waves in a gravitational field, the Einstein-Cartan theory is the best theory of gravity available. Alternative viable theories are general relativity and a certain teleparallelism model. Objections of Ohanian and Ruffini against the Einstein-Cartan theory are discussed. Subsequently we list the papers which were read at the `Alternative 4D Session' and try to order them, at least partially, in the light of the structures discussed. 
  The conformal equivalence of fourth-order gravity following from a non-linear Lagrangian L(R) to theories of other types is widely known, here we report on a new conformal equivalence of these theories to theories of the same type but with different Lagrangian.      For a quantization of fourth-order theories one needs a Hamiltonian formulation of them. One of the possibilities to do so goes back to Ostrogradski in 1850. Here we present another possibility: A Hamiltonian H different from Ostrogradski's one is discussed for the Lagrangian L depending on first and second order drivatives of the position variable q. We add a suitable divergence to L. Contrary to other approaches no constraint is needed. One of the canonical equations becomes equivalent to the fourth-order Euler-Lagrange equation of L.      Finally, we discuss the stability properties of cosmological models within fourth-order gravity. 
  We study chaotic inflation driven by a real, massive, homogeneous minimally coupled scalar field in a flat Robertson-Walker spacetime. The semiclassical limit for gravity is considered, whereas the scalar field is treated quantum mechanically by the technique of invariants in order to also investigate the dynamics of the system for non-classical states of the latter. An inflationary stage is found to be possible for a large set of initial quantum states, obviously including the coherent ones. States associated with a vanishing mean value of the field (such as the vacuum and the thermal) can also lead to inflation, however for such states we cannot make a definitive prediction due to the importance of higher order corrections during inflation. The results for the above coupled system are described and their corrections evaluated perturbatively. 
  We review an attempt to set a suitable foundational principle for consistent quantization of gravity based on the canonical formulation. It requires extending the spacetime description of the relativistic postulates to also encompass an alternative formulation in momentum-energy continuum where the inertial physical laws can be equivalently described. The extension to noninertial frames breaks such an equivalence, leaving a new dynamical field which, together with gravity, allows to construct a canonical scenario where the Dirac's quantization method leads to consistent definitions of hermitian ordering for the operators of the canonical quantum theory. 
  Virbhadra and Parikh studied the energy distribution associated with stringy charged black hole in Einstein's prescription. We study the same using Tolman's energy-momentum complex and get the same result as obtained by Virbhadra and Parikh. The entire energy is confined inside the black hole. 
  We study analytically the initial value problem for a charged massless scalar-field on a Reissner-Nordstr\"om spacetime. Using the technique of spectral decomposition we extend recent results on this problem. Following the no-hair theorem we reveal the dynamical physical mechanism by which the charged hair is radiated away. We show that the charged perturbations decay according to an inverse power-law behaviour at future timelike infinity and along future null infinity. Along the future outer horizon we find an oscillatory inverse power-law relaxation of the charged fields. We find that a charged black hole becomes ``bald'' slower than a neutral one, due to the existence of charged perturbations. Our results are also important to the study of mass-inflation and the stability of Cauchy horizons during a dynamical gravitational collapse of charged matter in which a charged black-hole is formed. 
  The evolution under radiation backreaction of a binary system consisting of a black hole and a companion is studied in the limiting case when the spin of the companion is negligible compared with the spin S of the black hole. To first order in the spin, the motion of the reduced-mass particle excluding radiation effects, is characterized by three constants: the energy E, the magnitude L of the angular momentum and the projection L_S of the angular momentum along the spin S. This motion is quasiperiodic with a period determined by r_{min} and r_{max}. We introduce a new parametrization, making the integration over a period of a generic orbit especially simple. We give the averaged losses in terms of the 'constants of motion' during one period for generic orbits, to linear order in spin. 
  The lecture explains the geometric basis for the recently-discovered nonholonomic mapping principle which specifies certain laws of nature in spacetimes with curvature and torsion from those in flat spacetime, thus replacing and extending Einstein's equivalence principle. An important consequence is a new action principle for determining the equation of motion of a free spinless point particle in such spacetimes. Surprisingly, this equation contains a torsion force, although the action involves only the metric. This force changes geodesic into autoparallel trajectories, which are a direct manifestation of inertia. The geometric origin of the torsion force is a closure failure of parallelograms. The torsion force changes the covariant conservation law of the energy-momentum tensor whose new form is derived. 
  In this paper, we present results obtained from our recent studies on the location of the innermost stable circular orbit (ISCO) for binary neutron stars (BNSs) in several levels of post Newtonian (PN) approximations. We reach the following conclusion at present: (1) even in the Newtonian case, there exists the ISCO for binary of sufficiently stiff equation of state (EOS). If the mass and the radius of each star are fixed, the angular velocity at the ISCO $\Omega_{ISCO}$ is larger for softer EOS: (2) when we include the first PN correction, there appear roughly two kinds of effects. One is the effect to the self-gravity of each star of binary and the other is to the gravity acting between two stars. Due to the former one, each star of binary becomes compact and the tidal effect is less effective. As a result, $\Omega_{ISCO}$ tends to be increased. On the other hand, the latter one has the property to destabilize the binary orbit, and $\Omega_{ISCO}$ tends to be decreased. If we take into account both effects, however, the former effect is stronger than the latter one, and $\Omega_{ISCO}$ becomes large with increase of the 1PN correction: (3) the feature mentioned above is more remarkable for softer EOS if the mass and radius are fixed. This is because for softer EOS, each star has the larger central density and is susceptible to the GR correction: (4) there has been no self consistent calculation including all the 2PN effects and only exist studies in which one merely includes the effect of the 2PN gravity acting between two stars. In this case, the effect has the property to destabilize the binary orbit, so that $\Omega_{ISCO}$ is always smaller than that for the Newtonian case. If we include the PN effect of the self-gravity to each star, $\Omega_{ISCO}$ will increase. 
  Using Einstein's and Weinberg's energy complex, we evaluate the energy distribution of the vaccum nonsingularity black hole solution. The energy distribution is positive everywhere and be equal to zero at origin. 
  The equations of a resonant sphere in interaction with $N$ secondary radial oscillators (transducers) on its surface have been found in the context of Lagrangian formalism. It has been shown the possibility to exert a veto against spurious events measuring the longitudinal component of a signal. Numerical simulations has been performed, which take into account thermal noise between resonators and the sphere surface, for a particular configuration of the transducers. 
  We consider the Klein-Gordon equation in FRW-like spacetimes, with compact space sections (not necessarily isotropic neither homogeneous). The bi-scalar kernel allowing to select the positive-frequency part of any solution is developed on mode solutions, using the eigenfunctions of the three-dimensional Laplacian. Of course this kernel is not unique but, except (perhaps) when the scale factor undergoes a special law of evolution, the metric has no more symmetries (connected with the identity) than those inherited from the space sections. As a result, all admissible definitions of the positive-frequency kernel are related one to another by a unitary transformation which commutes with the connected isometries of spacetime; any such kernel is invariant under these isometries isometries. A physical interpretation is tentatively suggested. 
  A new first integral for the equations corresponding to twisting type-N vacuum gravitational fields with two non-commuting Killing vectors is introduced. A new reduction of the problem to a complex second-order ordinary differential equation is given. Alternatively, the mentioned first integral can be used in order to provide a first integral of the second-order complex equation introduced in a previous treatment of the problem. 
  We calculate the electromagnetic self-force on a stationary linear distribution of four-current in the spacetime of multiple cosmic strings. It is shown that if the current is infinitely thin and stretched along a line which is parallel to the strings the problem admits an explicit solution. 
  Spatially homogeneous but possibly anisotropic cosmologies have two main types of singularities: (1) asymptotically velocity term dominated (AVTD) - (reversing the time direction) the universe evolves to the singularity with fixed anisotropic collapse rates ; (2) Mixmaster-the anisotropic collapse rates change in a deterministicaly chaotic way. Much less is known about spatially inhomogeneous universes. It has been claimed that a generic universe would evolve toward the singularity as a different Mixmaster universe at each spatial point. I shall discuss how to predict whether a cosmology has an AVTD or Mixmaster singularity and whether or not our numerical simulations agree with these predictions. 
  A recent paper by Martin and Schwarz [1] argues that the ``standard inflationary result" has been finally proven. The result itself is formulated as: ``the closer the inflationary epoch is to the de Sitter space-time, the less important are large-scale gravitational waves in the CMBR today". Beginning from the basic equations of Grishchuk [2, 3] the authors say [1] that Grishchuk's conclusion about approximate equality of metric amplitudes for gravitational waves and density perturbations ``is wrong because the time evolution of the scalar metric perturbation through the (smooth) reheating transition was not calculated correctly". They reiterate a claim about ``big amplification" of scalar perturbations (in contrast to gravitational waves) during reheating. The authors say [1] that after appropriate correction they have recovered the "standard result" within Grishchuk's approach. It is shown in this Comment that the "big amplification" is a misinterpretation. There is no difference in the evolution of long-wavelength metric perturbations for gravitational waves and density perturbations: they both stay approximately constant. The influence of cosmological transitions on the evolution is none at all, as long as the wavelength of the perturbation is much larger than the Hubble radius. It is shown that from the approach of [2, 3] follow the conclusions of [2, 3] without change. Finally, it is argued that the ``standard inflationary result" does not follow from the correct evolution and quantum normalization of density perturbations. 
  We study first order phase transitions in the gravitational collapse of spherically symmetric skyrmions. Static sphaleron solutions are shown to play the role of critical solutions separating black-hole spacetimes from no-black-hole spacetimes. In particular, we find a new type of first order phase transition where subcritical data do not disperse but evolve towards a static regular stable solution. We also demonstrate explicitly that the near-critical solutions depart from the intermediate asymptotic regime along the unstable manifold of the critical solution. 
  Multidimensional cosmological models with a higher dimensional space-time manifold are investigated under dimensional reduction. In the Einstein conformal frame, the effective potential for the internal scale factors is obtained. The stable compactification of the internal spaces is achieved due to the Casimir effect. In the case of more than one internal space a Casimir-like ansatz for the energy density of the massless scalar field fluctuations is proposed. Stable configurations with respect to the internal scale factor excitations are found in the cases of one and two internal spaces. 
  Gravitational waves from inspiralling compact binaries can be reliably extracted from a noisy detector output only if the template used in the detection is a faithful representation of the true signal. In this article we suggest a new approach to constructing faithful signal models. 
  A brief survey of the major themes and developments of black hole thermodynamics in the 1990's is given, followed by summaries of the talks on this subject at MG8 together with a bit of commentary, and closing with a look towards the future. 
  We analyze the core dynamics of critically coupled, superheavy gauge vortices in the (2+1) dimensional Einstein-Abelian-Higgs system. By numerically solving the Eistein and field equations for various values of the symmetry breaking scale, we identify the regime in which static solutions cease to exist and topological inflation begins. We explicitly include the topological winding of the vortices into the calculation and extract the dependence on the winding of the critical scale separating the static and inflating regimes. Extrapolation of our results suggests that topological inflation might occur within high winding strings formed at the Grand Unified scale. 
  We consider the dynamics of a multi-component scalar field on super-horizon scales in the context of inflationary cosmology. We present a method to solve the perturbation equations on super- horizon scales, i.e., in the long wavelength limit, by using only the knowledge of spatially homogeneous background solutions. In doing so, we clarify the relation between the perturbation equations in the long wavelength limit and the background equations. Then as a natural extension of our formalism, we provide a strategy to study super-horizon scale perturbations beyond the standard linear perturbation theory. Namely we reformulate our method so as to take into account the nonlinear dynamics of the scalar field. 
  The electromagnetic radiation that falls into a Reissner-Nordstr\"{o}m black hole is known to develop a ``blue sheet'', namely, an infinite concentration of energy density at the Cauchy horizon. The interaction of these divergent electromagnetic fields with infalling matter was recently analyzed (L. M. burko and A. Ori, Phys. Rev. Lett. 74, 1064 (1995)). Here, we give a more detailed description of that analysis: We consider classical electromagnetic fields (that were produced during the collapse and then backscattered into the black hole), and investigate the blue-sheet effects of these fields on infalling objects within two simplified models of a classical and a quantum absorber. These effects are found to be finite and even negligible for typical parameters of a supermassive black hole. 
  Assuming that the Hamiltonian of a canonical field theory can be written in the form N H + N^i H_i, and using as the only input the actual choice of the canonical variables, we derive: (i) The algebra satisfied by H and H_i, (ii) any constraints, and (iii) the most general canonical representation for H and H_i. This completes previous work by Hojman, Kuchar and Teitelboim who had to impose a set of additional postulates, among which were the form of the canonical algebra and the requirement of path-independence of the dynamical evolution. A prominent feature of the present approach is the replacement of the equal-time Poisson bracket with one evaluated at general times. The resulting formalism is therefore an example of a classical history theory -- an interesting fact, especially in view of recent work by Isham et al. 
  Using the Hartle-Hawking no-boundary proposal for the wave function of the universe, we can study the wave function and probability of a single black hole created at the birth of the universe. The black hole originates from a constrained gravitational instanton with conical singularities. The wave function and probability of a universe with a black hole are calculated at the $WKB$ level. The probability of a black hole creation is the exponential of one quarter of the sum of areas of the black hole and cosmological horizons. One quarter of this sum is the total entropy of the universe. We show that these arguments apply to all kinds of black holes in the de Sitter space background. 
  We investigate gauge invariant cosmological perturbations in a spatially flat Friedman-Robertson-Walker universe with scalar fields. It is well known that the evolution equation for the gauge invariant quantities has exact solutions in the long-wavelength limit. We find that these gauge invariant solutions can be obtained by differentiating the background solution with respect to parameters contained in the background system. This method is very useful when we analyze the long-wavelength behavior of cosmological perturbation with multiple scalar fields. 
  We consider the possibility to use the areas of two-simplexes, instead of lengths of edges, as the dynamical variables of Regge calculus. We show that if the action of Regge calculus is varied with respect to the areas of two-simplexes, and appropriate constraints are imposed between the variations, the Einstein-Regge equations are recovered. 
  Yang's pure space equations (C.N. Yang, Phys. Rev. Lett. v.33, p.445 (1974)) generalize Einstein's gravitational equations, while coming from gauge theory. We study these equations from a number of vantage points: summarizing the work done previously, comparing them with the Einstein equations and investigating their properties. In particular, the initial value problem is discussed and a number of results are presented for these equations with common energy-momentum tensors. 
  An effective model for the spacetime foam is constructed in terms of nonlocal interactions in a classical background. In the weak-coupling approximation, the evolution of the low-energy density matrix is determined by a master equation that predicts loss of quantum coherence. Moreover, spacetime foam can be described by a quantum thermal field that, apart from inducing loss of coherence, gives rise to effects such as gravitational Lamb and Stark shifts as well as quantum damping in the evolution of the low-energy observables. 
  We investigate the phenomenon of particle creation of the massless scalar field in the model of spacetime in which, depending on the model's parameter $\aout$, the chronology horizon could be formed. The model represents a two-dimensional curved spacetime with the topology $R^1 \times S^1$ which is asymptotically flat in the past and in the future. The spacetime is globally hyperbolic and has no causal pathologies if $\aout<1$, and closed timelike curves appear in the spacetime if $\aout\ge 1$. We obtain the spectrum of created particles in the case $\aout<1$. In the limit $\aout \to 1$ this spectrum gives the number of particles created into mode $n$ near the chronology horizon. The main result we have obtained is that the number of scalar particles created into each mode as well as the full number of particles remain finite at the moment of forming of the chronology horizon. 
  We study the exact renormalization group (RG) in $R^2$-gravity in the effective average action formalism using the background field method. The truncated evolution equation (where truncation is made to low-derivatives functionals space) for such a theory in a de Sitter background leads to a set of nonperturbative RG equations for cosmological and gravitational coupling constants. The gauge dependence problem is solved by working in the physical Landau-De Witt gauge corresponding to gauge-fixing independent effective action. Approximate solution of nonperturbative RG equations reveals the appearence of antiscreening or screening behaviour of Newtonian coupling, depending on the higher-derivatives coupling constants. The existence of unstable  UV fixed points is also mentioned. 
  We have studied the effects of imperfections in spherical gravitational wave antenna on our ability to properly interpret the data it will produce. The results of a numerical simulation are reported that quantitatively describe the systematic errors resulting from imperfections in various components of the antenna. In addition, the results of measurements on a room-temperature prototype are presented that verify it is possible to accurately deconvolve the data in practice. 
  We derive the Feynman rules for the graviton in the presence of a flat Robertson-Walker background and give explicit expressions for the propagator in the physically interesting cases of inflation, radiation domination, and matter domination. The aforementioned background is generated by a scalar field source which should be taken to be dynamical. As an elementary application, we compute the corrections to the Newtonian gravitational force in the present matter dominated era and conclude -- as expected -- that they are negligible except for the largest scales. 
  All gauge theories need ``something fixed'' even as ``something changes.'' Underlying the implementation of these ideas all major physical theories make indispensable use of an elaborately designed spacetime model as the ``something fixed,'' i.e., absolute. This model must provide at least the following sequence of structures: point set, topological space, smooth manifold, geometric manifold, base for various bundles. The ``fine structure'' of spacetime inherent in this sequence is of course empirically unobservable directly, certainly when quantum mechanics is taken into account. This issue is at the basis of the difficulties in quantizing general relativity and has been approached in many different ways. Here we review an approach taking into account the non-Boolean properties of quantum logic when forming a spacetime model. Finally, we recall how the fundamental gauge of diffeomorphisms (the issue of general covariance vs coordinate conditions) raised deep conceptual problems for Einstein in his early development of general relativity. This is clearly illustrated in the notorious ``hole'' argument. This scenario, which does not seem to be widely known to practicing relativists, is nevertheless still interesting in terms of its impact for fundamental gauge issues. 
  We draw attention to a novel type of geometric gauge invariance relating the autoparallel equations of motion in different Riemann-Cartan spacetimes with each other. The novelty lies in the fact that the equations of motion are invariant even though the actions are not. As an application we use this gauge transformation to map the action of a spinless point particle in a Riemann-Cartan spacetime with a gradient torsion to a purely Riemann spacetime, in which the initial torsion appears as a nongeometric external field. By extremizing the transformed action in the usual way, we obtain the same autoparallel equations of motion as those derived in the initial spacetime with torsion via a recently-discovered variational principle. 
  A convenient formalism for averaging the losses produced by gravitational radiation backreaction over one orbital period was developed in an earlier paper. In the present paper we generalize this formalism to include the case of a closed system composed from two bodies of comparable masses, one of them having the spin S.  We employ the equations of motion given by Barker and O'Connell, where terms up to linear order in the spin (the spin-orbit interaction terms) are kept. To obtain the radiative losses up to terms linear in the spin, the equations of motion are taken to the same order. Then the magnitude L of the angular momentum L, the angle kappa subtended by S and L and the energy E are conserved. The analysis of the radial motion leads to a new parametrization of the orbit.   From the instantaneous gravitational radiation losses computed by Kidder the leading terms and the spin-orbit terms are taken. Following Apostolatos, Cutler, Sussman and Thorne, the evolution of the vectors S and L in the momentary plane spanned by these vectors is separated from the evolution of the plane in space. The radiation-induced change in the spin is smaller than the leading-order spin terms in the momentary angular momentum loss. This enables us to compute the averaged losses in the constants of motion E, L and L_S=L cos kappa. In the latter, the radiative spin loss terms average to zero. An alternative description using the orbital elements a,e and kappa is given.   The finite mass effects contribute terms, comparable in magnitude, to the basic, test-particle spin terms in the averaged losses. 
  Spacetimes which have been considered counter-examples to strong cosmic censorship are revisited. We demonstrate the classical instability of the Cauchy horizon inside charged black holes embedded in de Sitter spacetime for all values of the physical parameters. The relevant modes which maintain the instability, in the regime which was previously considered stable, originate as outgoing modes near to the black hole event horizon. This same mechanism is also relevant for the instability of Cauchy horizons in other proposed counter-examples of strong cosmic censorship. 
  The limitations of three-dimensional semi-classical gravity are explored in the context of a conformally invariant theory for a self-interacting scalar field. The analysis of the theory's scaling behaviour reveals that scalar-loop effects contribute to the conformal anomaly only at advanced orders allowing for a range of relevant energy scales which extend to those comparable to the non-perturbative scale of Planck mass. 
  We study the conditions for 2-dimensional dilaton gravity models to have dynamical formation of black holes and construct all such models. Furthermore we present a parametric representation of the general solutions of the black holes. 
  We investigate dynamical instability of a two-dimensional quantum black hole model considered by Lowe in his study of Hawking evaporation. The model is supposed to express a black hole in equilibrium with a bath of Hawking radiation. It turns out that the model has at least one instability modes for a wide range of parameters, and thus it is unstable. 
  Recently Kaniel & Itin proposed a gravitational model with the wave type equation [\square+\lambda(x)]\vartheta^\alpha=0 as vacuum field equation, where \vartheta^\alpha denotes the coframe of spacetime. They found that the viable Yilmaz-Rosen metric is an exact solution of the tracefree part of their field equation. This model belongs to the teleparallelism class of gravitational gauge theories. Of decisive importance for the evaluation of the Kaniel-Itin model is the question whether the variation of the coframe commutes with the Hodge star. We find a master formula for this commutator and rectify some corresponding mistakes in the literature. Then we turn to a detailed discussion of the Kaniel-Itin model. 
  A short review of recent results on exact solutions in multidimensional cosmology and overview of reports at workshop in MG8 (held by the author) are presented. 
  The accuracy of the post-Newtonian waveforms, both in standard and Pade form, is determined by computing their matched-filtering overlap integral with a reference waveform obtained from black-hole perturbation theory. 
  We investigate the cosmological model with the complex scalar self-interacting inflaton field non-minimally coupled to gravity. The different geometries of the Euclidean classically forbidden regions are represented. The instanton solutions of the corresponding Euclidean equations of motion are found by numerical calculations supplemented by the qualitative analysis of Lorentzian and Euclidean trajectories. The applications of these solutions to the no-boundary and tunneling proposals for the wave function of the Universe are studied. Possible interpretation of obtained results and their connection with inflationary cosmology is discussed. The restrictions on the possible values of the new quasi-fundamental constant of the theory-non-zero classical charge-- are obtained. The equations of motion for the generalized cosmological model with complex scalar field are written down and investigated. The conditions of the existence of instanton solutions corresponding to permanent values of an absolute value of scalar field are obtained. 
  We develop the General Theory of Relativity in a formalism with extended causality that describes physical interaction through discrete, transversal and localized pointlike fields. The homogeneous field equations are then solved for a finite, singularity-free, point-like field that we associate to a ``classical graviton". The standard Einstein's continuous formalism is retrieved by means of an averaging process, and its continuous solutions are determined by the chosen imposed symetry. The Schwarzschild metric is obtained by the imposition of spherical symmetry on the averaged field. 
  A short review of recent results of authors on sigma-model approach in multidimensional gravity with p-branes is presented. 
  The D-dimensional cosmological model on the manifold $M = R \times M_{1} \times M_{2}$ describing the evolution of 2 Einsteinian factor spaces, $M_1$ and $M_2$, in the presence of multicomponent perfect fluid source is considered. The barotropic equation of state for mass-energy densities and the pressures of the components is assumed in each space. When the number of the non Ricci-flat factor spaces and the number of the perfect fluid components are both equal to 2, the Einstein equations for the model are reduced to the generalized Emden-Fowler (second-order ordinary differential) equation, which has been recently investigated by Zaitsev and Polyanin within discrete-group analysis. Using the integrable classes of this equation one generates the integrable cosmological models. The corresponding metrics are presented. The method is demonstrated for the special model with Ricci-flat spaces $M_1,M_2$ and the 2-component perfect fluid source. 
  We show that a surface term should be added to the Einstein-Hilbert action in order to properly describe quantum transitions occurring around a black hole. The introduction of this boundary term has been advocated by Teitelboim and collaborators and it has been used in the computation of the black hole entropy. Here, we use it to compute the gravitational corrections to the transition amplitudes giving rise to Hawking radiation. This surface term implies that the probability to emit a particle is given by $e^{- \Delta A/4}$ where $\Delta A$ is the change in the area of the black hole horizon induced by the emission. Its inclusion at the level of the amplitudes therefore relates quantum black hole radiation to the first law of black hole dynamics. In both cases indeed, the term expressing the change in area directly results from the same boundary term introduced for the same reason: to obtain a well defined action principle. 
  We study the evolution of an evaporating rotating black hole, described by the Kerr metric, which is emitting either solely massless scalar particles or a mixture of massless scalar and nonzero spin particles. Allowing the hole to radiate scalar particles increases the mass loss rate and decreases the angular momentum loss rate relative to a black hole which is radiating nonzero spin particles. The presence of scalar radiation can cause the evaporating hole to asymptotically approach a state which is described by a nonzero value of $a_* \equiv a / M$. This is contrary to the conventional view of black hole evaporation, wherein all black holes spin down more rapidly than they lose mass. A hole emitting solely scalar radiation will approach a final asymptotic state described by $a_* \simeq 0.555$. A black hole that is emitting scalar particles and a canonical set of nonzero spin particles (3 species of neutrinos, a single photon species, and a single graviton species) will asymptotically approach a nonzero value of $a_*$ only if there are at least 32 massless scalar fields. We also calculate the lifetime of a primordial black hole that formed with a value of the rotation parameter $a_{*}$, the minimum initial mass of a primordial black hole that is seen today with a rotation parameter $a_{*}$, and the entropy of a black hole that is emitting scalar or higher spin particles. 
  We conjecture that the neutral black hole pair production is related to the vacuum fluctuation of pure gravity via the Casimir-like energy. Implications on the foam-like structure of spacetime are discussed. 
  We attempt to treat the very early Universe according to quantum mechanics. Identifying the scale factor of the Universe with the width of the wave packet associated with it, we show that there cannot be an initial singularity and that the Universe expands. Invoking the correspondence principle, we obtain the scale factor of the Universe and demonstrate that the causality problem of the standard model is solved. 
  The possibility of the cosmic string creation by the vacuum fluctuations of quantum fields in the self-consistent semiclassical theory of gravity is discussed. We use the approximate method for obtaining vacuum expectation value of the renormalized stress-energy tensor of conformally invariant quantum fields in static cylindrically symmetric spacetimes. We have obtained the particular solutions of Einstein equations for the different boundary conditions at the cylinder symmetry axis. 
  Regarding Pauli's matrices as proper Higgs fields one can deduce an effective(!) approximation for gravity in flat space. In this work we extend this approximation up to the second order. Reaching complete agreement in the special case of gravitational waves. Unification in view, we introduce isospinorial degrees of freedom. In this way the mass spectrum and chiral asymmetry can be generated with the help of an additional scalar Higgs field. The Higgs modes corresponding to gravity are discussed. 
  We investigate neutron stars in scalar-tensor theories. We examine their secular stability against spherically symmetric perturbations by use of a turning point method. For some choices of the coupling function contained in the theories, the number of the stable equilibrium solutions changes and the realized equilibrium may change discontinuously as the asymptotic value of the scalar field or total baryon number is changed continuously. The behaviour of the stable equilibrium solutions is explained by fold and cusp catastrophes. Whether the cusp catastrophe appears or not depends on the choices of the coupling function. These types of the catastrophes are structurally stable. Recently discovered spontaneous scalarization, which is non-perturbative strong-field phenomenon due to the presence of the gravitational scalar field, is well described in terms of the cusp catastrophe. 
  We study the structure and stability of the recently discussed spherically symmetric Brans-Dicke black-hole type solutions with an infinite horizon area and zero Hawking temperature, existing for negative values of the coupling constant $\omega$. These solutions split into two classes: B1, whose horizon is reached by an infalling particle in a finite proper time, and B2, for which this proper time is infinite. Class B1 metrics are shown to be extendable beyond the horizon only for discrete values of mass and scalar charge, depending on two integers $m$ and $n$. In the case of even $m-n$ the space-time is globally regular; for odd $m$ the metric changes its signature at the horizon. All spherically symmetric solutions of the Brans-Dicke theory with $\omega<-3/2$ are shown to be linearly stable against spherically symmetric \pns. This result extends to the generic case of the Bergmann-Wagoner class of scalar-tensor theories of gravity with the coupling function $\omega(\phi) < -3/2$. 
  The relation between microscopic and macroscopic entities in the generally covariant theories is considered, and it is argued that a sensible definition of the macroscopic averages requires a restriction of the allowed transformations of coordinates. Spacetime averages of the geometric objects of Einstein's unified field theory are then defined, and the reconstruction of some features of macroscopic reality from hypothetic microscopic structures is attempted. It is shown how a fluctuating microscopic behaviour of the metric field can rule the constitutive relation for electromagnetism both in vacuo and in nondispersive material media. Moreover, if both the metric and the skew tensor density that represents the electric displacement and the magnetic field are assumed to possess a wavy microscopic structure, nonvanishing generalized force densities can appear in the continuum. They originate from a resonance process, in which at least three waves need to be involved. This process only occurs if the wavevectors fulfil the three-wave resonance condition, so ubiquitous in quantum physics. The wavy behaviour of the metric is essential for the occurrence of this resonance phenomenon. 
  The Siklos class of solutions of Einstein's field equations is investigated by analytical methods. By studying the behaviour of free particles we reach the conclusion that the space-times represent exact gravitational waves propagating in the anti-de Sitter universe. The presence of a negative cosmological constant implies that the 'background' space is not asymptotically flat and requires a 'rotating' reference frames in order to fully simplify and view the behaviour of nearby test particles. The Kaigorodov space-time, which is the simplest representative of the Siklos class, is analyzed in more detail. It is argued that it may serve as a 'cosmological' analogue of the well-known homogeneous pp-waves in the flat universe. 
  If an appropriate region of Kerr-Newman space-time is removed and suitable identifications are made, the resulting space-time can be interpreted as an infinitely thin disk producing the original electromagnetic and gravitational fields. We choose the shape of the regions removed in such a way that radial pressures in the disks vanish. Even the very inner parts such as ergoregions may exist around the disks. The surface energy-momentum tensor of the disks is checked to satisfy the weak and strong energy conditions. To emphasize the reality of these sources two models of the disks are presented: (i) rotating conductive charged rings which are supported against collapse by their internal pressure, (ii) two counter-rotating streams of charged particles moving along circular electro-geodesics. All these disk sources form a three-dimensional parameter space with specific electric charge Q/M, angular momentum a/M and the size of the excluded region being the parameters. 
  Non-standard sandwich gravitational waves are constructed from the homogeneous pp vacuum solution and the motions of free test particles in the space-times are calculated explicitly. They demonstrate the caustic property of sandwich waves. By performing limits to impulsive gravitational wave it is demonstrated that the resulting particle motions are identical regardless of the ''initial'' sandwich. 
  We draw parallels between the recently introduced ``Immirzi ambiguity'' of the Ashtekar-like formulation of canonical quantum gravity and other ambiguities that appear in Yang-Mills theories, like the $\theta$ ambiguity. We also discuss ambiguities in the Maxwell case, and implication for the loop quantization of these theories. 
  We present a method for constructing equilibrium disks with net angular momentum in general relativity. The method solves the relativistic Vlasov equation coupled to Einstein's equations for the gravitational field. We apply the method to construct disks that are relativistic versions of Newtonian Kalnajs disks. In Newtonian gravity these disks are analytic, and are stable against ring formation for certain ranges of their velocity dispersion. We investigate the existence of fully general relativistic equilibrium sequences for differing values of the velocity dispersion. These models are the first rotating, relativistic disk solutions of the collisionless Boltzman equation. 
  A novel method for calculation of the motion and radiation reaction for the two-body problem (body plus particle, the small parameter m/M being the ratio of the masses) is presented. In the background curvature given by the Schwarzschild geometry rippled by gravitational waves, the geodesic equations insure the presence of radiation reaction also for high velocities and strong field. The method is generally applicable to any orbit, but radial fall is of interest due to the non-adiabatic regime (equality of radiation reaction and fall time scales), in which the particle locally and immediately reacts to the emitted radiation. The energy balance hypothesis is only used (emitted radiation equal to the variation in the kinetic energy) for determination of the 4-velocity via the Lagrangian and normalization of divergencies. The solution in time domain of the Regge-Wheeler-Zerilli-Moncrief radial wave equation determines the metric tensor expressing the polar perturbations, in terms of which the geodesic equations are written and shown herein. 
  We show that only a sector of the classical solution space of the CGHS model describes formation of black holes through collapse of matter. This sector has either right or left moving matter. We describe the sector which has left moving matter in canonical language. In the nonperturbative quantum theory all operators are expressed in terms of the matter field operator which is represented on a Fock space. We discuss existence of large quantum fluctuations of the metric operator when the matter field is approximately classical. We end with some comments which may pertain to Hawking radiation in the context of the model. 
  We study analytically the initial value problem for a self-interacting (massive) scalar-field on a Reissner-Nordstr\"om spacetime. Following the no-hair theorem we examine the dynamical physical mechanism by which the self-interacting (SI) hair decays. We show that the intermediate asymptotic behaviour of SI perturbations is dominated by an oscillatory inverse power-law decaying tail. We show that at late-times the decay of a SI hair is slower than any power-law. We confirm our analytical results by numerical simulations. 
  We study the nonlinear gravitational collapse of a charged massless scalar-field. We confirm the existence of oscillatory inverse power-law tails along future timelike infinity, future null infinity and along the future outer-horizon. The nonlinear dumping exponents are in excellent agreement with the analytically predicted ones. Our results prove the analytic conjecture according to which a charged hair decays slower than a neutral one and also suggest the occurrence of mass-inflation along the Cauchy horizon of a dynamically formed charged black-hole. 
  A thin wall approximation is exploited to describe a global monopole coupled to gravity. The core is modelled by de Sitter space; its boundary by a thin wall with a constant energy density; its exterior by the asymptotic Schwarzschild solution with negative gravitational mass $M$ and solid angle deficit, $\Delta\Omega/4\pi = 8\pi G\eta^2$, where $\eta$ is the symmetry breaking scale. The deficit angle equals $4\pi$ when $\eta=1/\sqrt{8\pi G} \equiv M_p$. We find that: (1) if $\eta <M_p$, there exists a unique globally static non-singular solution with a well defined mass, $M_0<0$. $M_0$ provides a lower bound on $M$. If $M_0<M<0$, the solution oscillates. There are no inflating solutions in this symmetry breaking regime. (2) if $\eta \ge M_p$, non-singular solutions with an inflating core and an asymptotically cosmological exterior will exist for all $M<0$. (3) if $\eta$ is not too large, there exists a finite range of values of $M$ where a non-inflating monopole will also exist. These solutions appear to be metastable towards inflation. If $M$ is positive all solutions are singular. We provide a detailed description of the configuration space of the model for each point in the space of parameters, $(\eta, M)$ and trace the wall trajectories on both the interior and the exterior spacetimes. Our results support the proposal that topological defects can undergo inflation. 
  H. Akbar-Zadeh has recently proposed (J Geom Phys 17 (1995) 342) a new geometric formulation of Einstein-Maxwell system with source in terms of what are called "Generalized Einstein manifolds". We show that, contrary to the claim, Maxwell equations have not been derived in this formulation and that, the assumed equations can be identified only as source-free Maxwell equations in the proposed geometric set up. A genuine derivation of source-free Maxwell equations is presented within the same framework. We draw a conclusion that the proposed unification scheme can pertain only to source-free situations. 
  Boson stars are descendants of the so-called geons of Wheeler, except that they are built from scalar particles instead of electromagnetic fields. If scalar fields exist in nature, such localized configurations kept together by their self-generated gravitational field can form within Einstein's general relativity. In the case of complex scalar fields, an absolutely stable branch of such non-topological solitons with conserved particle number exists. Our present surge stems from the speculative possibility that these compact objects could provide a considerable fraction of the non-baryonic part of dark matter. In any case, they may serve as a convenient "laboratory" for studying numerically rapidly rotating bodies in general relativity and the generation of gravitational waves. 
  We investigate the simplest cosmological model with the massive real scalar non-interacting inflaton field minimally coupled to gravity. The classification of trajectories in closed minisuperspace Friedmann-Robertson-Walker model is presented.The fractal nature of a set of infinitely bounced trajectories is discussed. The results of numerical calculations are compared with those obtained byperturbative analytical calculations around the exactly solvable minisuperspace cosmological model with massless scalar field. 
  We analyse the spectrum of energy density fluctuations of a dual supergravity model where the dilaton and the moduli are stabilized and sucessful inflation is achieved inside domain walls that separate different vacua of the theory. Constraints on the parameters of the superpotential are derived from the amplitude of the primordial energy density fluctuations as inferred from COBE and it is shown that the scale dependence of the tensor perturbations nearly vanishes. 
  We propose a modified form of the spontaneous birth of the universe by quantum tunneling. It proceeds through topology change and inflation, to eventually become a universe with closed spatial sections of negative curvature and nontrivial global topology. 
  We revisit the question of the imposition of initial data representing astrophysical gravitational perturbations of black holes. We study their dynamics for the case of nonrotating black holes by numerically evolving the Teukolsky equation in the time domain. In order to express the Teukolsky function Psi explicitly in terms of hypersurface quantities, we relate it to the Moncrief waveform phi_M through a Chandrasekhar transformation in the case of a nonrotating black hole. This relation between Psi and phi_M holds for any constant time hypersurface and allows us to compare the computation of the evolution of Schwarzschild perturbations by the Teukolsky and by the Zerilli and Regge-Wheeler equations. We explicitly perform this comparison for the Misner initial data in the close limit approach. We evolve numerically both, the Teukolsky (with the recent code of Ref. [1]) and the Zerilli equations, finding complete agreement in resulting waveforms within numerical error. The consistency of these results further supports the correctness of the numerical code for evolving the Teukolsky equation as well as the analytic expressions for Psi in terms only of the three-metric and the extrinsic curvature. 
  It is shown that the axial and polar perturbations of the spherically symmetric black hole can be described in a gauge-invariant way. The reduced phase space describing gravitational waves outside of the horizon is described by the gauge-invariant quantities. Both degrees of freedom fulfill generalized scalar wave equation. For the axial degree of freedom the radial part of the equation corresponds to the Regge-Wheeler result (Phys. Rev. 108, 1063-1069 (1957)) and for the polar one we get Zerilli result (Phys. Rev. D2, 2141-2160 (1970)), see also Chandrasekhar (The Mathematical Theory of Black Holes,(Clarendon Press Oxford, 1983)), Moncrief (Annals of Physics 88, 323-342 (1974)) for both. An important ingredient of the analysis is the concept of quasilocality which does duty for the separation of the angular variables in the usual approach. Moreover, there is no need to represent perturbations by normal modes (with time dependence $\exp(-ikt)$), we have fields in spacetime and the Cauchy problem for them is well defined outside of the horizon. The reduced symplectic structure explains the origin of the axial and polar invariants. It allows to introduce an energy and angular momentum for the gravitational waves which is invariant with respect to the gauge transformations. Both generators represent quadratic approximation of the ADM nonlinear formulae in terms of the perturbations of the Schwarzschild metric. We also discuss the boundary-initial value problem for the linearized Einstein equations on a Schwarzschild background outside of the horizon. 
  We report new results which establish that the accurate 3-dimensional numerical simulation of generic single-black-hole spacetimes has been achieved by characteristic evolution with unlimited long term stability. Our results cover a selection of distorted, moving and spinning single black holes, with evolution times up to 60,000M. 
  This paper gives a detailed pedagogic presentation of the central concepts underlying a new algorithm for the numerical solution of Einstein's equations for gravitation. This approach incorporates the best features of the two leading approaches to computational gravitation, carving up spacetime via Cauchy hypersurfaces within a central worldtube, and using characteristic hypersurfaces in its exterior to connect this region with null infinity and study gravitational radiation. It has worked well in simplified test problems, and is currently being used to build computer codes to simulate black hole collisions in 3-D. 
  A new approach to gravitational gauge-invariant perturbation theory begins from the fourth-order Einstein-Ricci system, a hyperbolic formulation of gravity for arbitrary lapse and shift whose centerpiece is a wave equation for curvature. In the Minkowski and Schwarzschild backgrounds, an intertwining operator procedure is used to separate physical gauge-invariant curvature perturbations from unphysical ones. In the Schwarzschild case, physical variables are found which satisfy the Regge-Wheeler equation in both odd and even parity. In both cases, the unphysical "gauge'' degrees of freedom are identified with violations of the linearized Hamiltonian and momentum constraints, and they are found to evolve among themselves as a closed subsystem. If the constraints are violated, say by numerical finite-differencing, this system describes the hyperbolic evolution of the constraint violation. It is argued that an underlying raison d'\^etre of causal hyperbolic formulations is to make the evolution of constraint violations well-posed. 
  We use the duality between the local Cartezian coordinates and the solutions of the Klein-Gordon equation to parametrize locally the spacetime in terms of wave functions and prepotentials. The components of metric, metric connection, curvature as well as the Einstein equation are given in this parametrization. We also discuss the local duality between coordinates and quantum fields and the metric in this later reparametrization. 
  The usual approaches to the definition of energy give an ambiguous result for the energy of fields in the radiating regime. We show that for a massless scalar field in Minkowski space-time the definition may be rendered unambiguous by adding the requirement that the energy cannot increase in retarded time. We present a similar theorem for the gravitational field, proved elsewhere, which establishes that the Trautman-Bondi energy is the unique (up to a multiplicative factor) functional, within a natural class, which is monotonic in time for all solutions of the vacuum Einstein equations admitting a smooth ``piece'' of conformal null infinity Scri. 
  We perform the Dirac hamiltonian analysis of a four-dimensional gauge theory of gravity with an action of topological type, which generalizes some well-known two-dimensional models. We show that, in contrast with the two-dimensional case, the theory has a non-vanishing number of dynamical degrees of freedom and that its structure is very similar to higher-dimensional Chern-Simons gravity. 
  Geometrical properties of the extreme Kerr black holes in the topological sectors of nonextreme and extreme configurations are studied. We find that the Euler characteristic plays an essential role to distinguish these two kinds of extreme black holes. The relationship between the geometrical properties and the intrinsic thermodynamics are investigated. 
  We consider the electromagnetic resolution of gravitational field. We show that under the duality transformation, in which active and passive electric parts of the Riemann curvature are interchanged, a fluid spacetime in comoving coordinates remains invariant in its character with density and pressure transforming, while energy flux and anisotropic pressure remaining unaltered. Further if fluid admits a barotropic equation of state, $p = (\gamma - 1) \rho$ where $1 \leq \gamma \leq 2$, which will transform to $p = (\frac{2 \gamma}{3 \gamma - 2} - 1) \rho$. Clearly the stiff fluid and dust are dual to each-other while $\rho + 3 p =0$, will go to flat spacetime. However the n $(\rho - 3 p = 0)$ and the deSitter $(\rho + p = 0$) universes ar e self-dual. 
  A class of exact solutions of the field equations with higher derivative terms is presented when the matter field is a pressureless null fluid plus a Maxwellian static electric component. It is found that the stable solutions are black holes in anti de Sitter background. The issue of the stability of the Cauchy horizon is discussed. 
  Numerical evidence supports the conjecture that polarized U(1) symmetric cosmologies have asymptotically velocity term dominated singularities. 
  The coupled Einstein-Dirac equations for a static, spherically symmetric system of two fermions in a singlet spinor state are derived. Using numerical methods, we construct an infinite number of soliton-like solutions of these equations. The stability of the solutions is analyzed. For weak coupling (i.e., small rest mass of the fermions), all the solutions are linearly stable (with respect to spherically symmetric perturbations), whereas for stronger coupling, both stable and unstable solutions exist. For the physical interpretation, we discuss how the energy of the fermions and the (ADM) mass behave as functions of the rest mass of the fermions. Although gravitation is not renormalizable, our solutions of the Einstein-Dirac equations are regular and well-behaved even for strong coupling. 
  We derive an exact formula for the dimensionality of the Hilbert space of the boundary states of SU(2) Chern-Simons theory, which, according to the recent work of Ashtekar et al, leads to the Bekenstein-Hawking entropy of a four dimensional Schwarzschild black hole. Our result stems from the relation between the (boundary) Hilbert space of the Chern-Simons theory with the space of conformal blocks of the Wess-Zumino model on the boundary 2-sphere. 
  We consider scattering of elastic waves on parallel wedge dislocations in the geometric theory of defects or, equivalently, scattering of point particles and light rays on cosmic strings. Dislocations are described as torsion singularities located on parallel lines, and trajectories of phonons are assumed to be the corresponding extremals. Extremals are found for arbitrary distribution of the dislocations in the monopole, dipole, and quadrupole approximation and the scattering angle is obtained. Examples of continuous distribution of wedge and edge dislocations are considered. We have found that for deficit angles close to -2\pi a star located behind a cosmic string may have any even number of images, 2,4,6,... The close relationship between dislocations and conformal maps is elucidated in detail. 
  Continuing the investigation of the simplest cosmological model with the massive real scalar non-interacting inflaton field minimally coupled to gravity we study an influence of the cosmological constant on the behaviour of trajectories in closed minisuperspace Friedmann-Robertson-Walker model. The transition from chaotic to regular behaviour for large values of cosmological constant is discussed. Combining numerical calculations with qualitative analysis both in configuration and phase space we present a convenient classification of trajectories. 
  We apply the renormalization group (RG) method to examine the observable scaling properties in Newtonian cosmology. The original scaling properties of the equations of motion in our model are modified for averaged observables on constant time slices. In the RG flow diagram, we find three robust fixed points: Einstein-de Sitter, Milne and Quiescent fixed points. Their stability (or instability) property does not change under the effect of fluctuations. Inspired by the inflationary scenario in the early Universe, we set the Einstein-de Sitter fixed point with small fluctuations, as the boundary condition at the horizon scale. Solving the RG equations under this boundary condition toward the smaller scales, we find generic behavior of observables such that the density parameter $\Omega$ decreases, while the Hubble parameter $H$ increases for smaller averaging volume. The quantitative scaling properties are analyzed by calculating the characteristic exponents around each fixed point. Finally we argue the possible fractal structure of the Universe beyond the horizon scale. 
  Following the method of Buchbinder and Lyahovich, we carry out a canonical formalism for a higher-curvature gravity in which the Lagrangian density ${\cal L}$ is given in terms of a function of the salar curvature $R$ as ${\cal L}=\sqrt{-\det g_{\mu\nu}}f(R)$. The local Hamiltonian is obtained by a canonical transformation which interchanges a pair of the generalized coordinate and its canonical momentum coming from the higher derivative of the metric. 
  We consider the dynamical stability of a class of static, spherically-symmetric solutions of the nonsymmetric gravitational theory. We numerically reproduce the Wyman solution and generate new solutions for the case where the theory has a nontrivial fundamental length scale \mu^{-1}. By considering spherically symmetric perturbations of these solutions we show that the Wyman solutions are generically unstable. 
  We construct a new class of plane-symmetric solutions possessing a curvature singularity which is null and weak, like the spacetime singularity at the Cauchy horizon of spinning (or charged) black holes. We then analyse the stability of this singularity using a rigorous non-perturbative method. We find that within the framework of (linearly-polarized) plane-symmetric spacetimes this type of null weak singularity is locally stable. Generically, the singularity is also scalar-curvature. These observations support the new picture of the null weak singularity inside spinning (or charged) black holes, which is so far established primarily on the perturbative approach. 
  When using the black hole exclusion (horizon boundary condition) technique, $K$ is usually nonzero and spatially variable, so none of the special cases of York's conformal-decomposition algorithm apply, and the full 4-vector nonlinear York equations must be solved numerically.   We discuss the construction of dynamic black hole initial data slices using this technique: We perturb a known black hole slice via some Ansatz, apply the York decomposition (using another Ansatz for the inner boundary conditions) to project the perturbed field variables back into the constraint hypersurface, and finally optionally apply a numerical 3-coordinate transformation to (eg) restore an areal radial coordinate.   In comparison to other initial data algorithms, the key advantage of this algorithm is its flexibility: $K$ is unrestricted, allowing the use of whatever slicing is most suitable for (say) a time evolution.   We have implemented this algorithm for the spherically symmetric scalar field system. We present numerical results for a number of Eddington- Finkelstein--like initial data slices containing black holes surrounded by scalar field shells. Using 4th order finite differencing with resolutions of $\Delta r/r \approx 0.02$ (0.01) near the (Gaussian) perturbations, the numerically computed energy and momentum constraints for the final slices are $\ltsim 10^{-8}$ ($10^{-9}$) and $\ltsim 10^{-9}$ ($10^{-10}$) in magnitude.   Finally, we briefly discuss the errors incurred when interpolating data from one grid to another, as in numerical coordinate transformation or horizon finding. We show that for the usual moving-local-interpolation schemes, even for smooth functions the interpolation error is not smooth. 
  Gowdy spacetimes are generalized to admit two commuting spatial "local" Killing vectors, and some new varieties of them are presented, which are all closely related to Thurston's geometries. Explicit spatial compactifications, as well as the boundary conditions for the metrics are given in a systematic way. A short comment on an implication to their dynamics toward the initial singularity is made. 
  Differential properties of a spin 2 boson field $\psi_{\mu\nu}$ describing propagation of gravitational perturbations on a straight cosmic string's space-time background are studied by means of methods of the differential spaces theory. It is shown that this field is a smooth one in the interior of cosmic string's space-time and looses this property at the singular boundary except for cosmic string space-times with the following deficits of angle: $\Delta=2\pi (1-1/n) $, $n=1,2,...$. A relationship between smoothness of $\psi_{\mu\nu}$ at the singularity and the gravitational conical bremsstrahlung effect is discussed. A physical interpretation of the smoothness notion is given. It is also argued that the assumption of smoothness of $\psi_{\mu\nu}$ at the singularity plays an equivalent role to the Aliev and Gal'tsov "quantization" condition. 
  A supersymmetric approach to string quantum cosmology based on the non-compact, global duality symmetries of the effective action is developed. An N=2 supersymmetric action is derived whose bosonic component is the Neveu-Schwarz/Neveu-Schwarz sector of the $(d+1)$--dimensional effective action compactified on a $d$--torus. A representation for the supercharges is found and the form of the zero-and one-fermion quantum states is determined. The purely bosonic component of the wavefunction is unique and manifestly invariant under the symmetry of the action. The formalism applies to a wide class of non-linear sigma-models. 
  The three dimensional black hole solution of Einstein equations with negative cosmological constant coupled to a conformal scalar field is proved to be unstable against linear circularly symmetric perturbations. 
  We consider the changes which occur in cosmological distances due to the combined effects of some null geodesics passing through low-density regions while others pass through lensing-induced caustics. This combination of effects increases observed areas corresponding to a given solid angle even when averaged over large angular scales, through the additive effect of increases on all scales, but particularly on micro-angular scales; however angular sizes will not be significantly effected on large angular scales (when caustics occur, area distances and angular-diameter distances no longer coincide). We compare our results with other works on lensing, which claim there is no such effect, and explain why the effect will indeed occur in the (realistic) situation where caustics due to lensing are significant. Whether or not the effect is significant for number counts depends on the associated angular scales and on the distribution of inhomogeneities in the universe. It could also possibly affect the spectrum of CBR anisotropies on small angular scales, indeed caustics can induce a non-Gaussian signature into the CMB at small scales and lead to stronger mixing of anisotropies than occurs in weak lensing. 
  As is well-known, Newton's gravitational theory can be formulated as a four-dimensional space-time theory and follows as singular limit from Einstein's theory, if the velocity of light tends to the infinity. Here 'singular' stands for the fact, that the limiting geometrical structure differs from a regular Riemannian space-time. Geometrically, the transition Einstein to Newton can be viewed as an 'opening' of the light cones. This picture suggests that there might be other singular limits of Einstein's theory: Let all light cones shrink and ultimately become part of a congruence of singular world lines. The limiting structure may be considered as a nullhypersurface embedded in a five-dimensional spacetime. While the velocity of light tends to zero here, all other velocities tend to the velocity of light. Thus one may speak of an ultrarelativistic limit of General Relativity. The resulting theory is as simple as Newton's gravitational theory, with the basic difference, that Newton's elliptic differential equation is replaced by essentially ordinary differential equations, with derivatives tangent to the generators of the singular congruence. The Galilei group is replaced by the Carroll group introduced by L\'evy-Leblond. We suggest to study near ultrarelativistic situations with a perturbational approach starting from the singular structure, similar to post-Newtonian expansions in the $c \to \infty$ case. 
  The cosmological particle production in a $k=0$ expanding de Sitter universe with a Hubble parameter $H_0$ is considered for various values of mass or conformal coupling of a free, scalar field. One finds that, for a minimally coupled field with mass $0 \leq m^2 < 9 H_0^2/4$ (except for $m^2= 2H_0^2$), the one-mode occupation number grows to unity soon after the physical wavelength of the mode becomes larger than the Hubble radius, and afterwards diverges as $n(t) \sim O(1)(\lambda_{phys}(t)/H_0^{-1})^{2\nu}$, where $\nu \equiv [9/4 - m^2/H_0^2]^{1/2}$. However, for a field with $m^2 > 9H_0^2/4$, the occupation number of a mode outside the Hubble radius is rapidly oscillating and bounded and does not exceed unity. These results, readily generalized for cases of a nonminimal coupling, provide a clear argument that the long-wavelength vacuum fluctuations of low-mass fields in an inflationary universe do show classical behavior, while those of heavy fields do not. The interaction or self-interaction does not appear necessary for the emergence of classical features, which are entirely due to the rapid expansion of the de Sitter background and the upside-down nature of quantum oscillators for modes outside the Hubble radius. 
  The gravitational analog of the electromagnetic Poynting vector is constructed using the field equations of general relativity in the Hilbert gauge. It is found that when the gravitational Poynting vector is applied to the solution of the linear mass quadrupole oscillator, the correct gravitational quadrupole radiation flux is obtained. Further to this, the Maxwell-like gravitational Poynting vector gives rise to Einstein's quadrupole radiation formula. The gravitational energy-momentum (pseudo) tensor obtained is symmetric and traceless. The former property allows the definition of angular momentum for the free gravitational field. 
  Quantum gravity in 2+1 dimensions with a positive cosmological constant can be represented as an SL(2,C) Chern-Simons gauge theory. The symmetric vacuum of this theory is a degenerate configuration for which the gauge fields and spacetime metric vanish, while de Sitter space corresponds to a highly excited thermal state. Carlip's approach to black hole entropy can be adapted in this context to determine the statistical entropy of de Sitter space. We find that it equals one-quarter the area of the de Sitter horizon, in agreement with the semiclassical formula. 
  We investigate the ultrarelativistic limits of dilaton black holes, black $p$-branes (strings), multi-centered dilaton black hole solutions and black $p$-brane (string) solutions when the boost velocity approaches the speed of light. For dilaton black holes and black $p$-branes (boost is along the transverse directions), the resulting geometries are gravitational shock wave solutions generated by a single particle and membrane. For the multi-centered dilaton black hole solutions and black $p$-brane solutions (boost is along the transverse directions), the limiting geometries are shock wave solutions generated by multiple particles and membranes. When the boost is along the membrane directions, for the black $p$-brane and multi-centered black $p$-brane solution, the resulting geometries describe general plane-fronted waves propagating along the membranes. The effect of the dilaton on the limit is considered. 
  The one-loop effective action for QED in curved spacetime contains equivalence principle violating interactions between the electromagnetic field and the spacetime curvature. These interactions lead to the dependence of photon velocity on the motion and polarization directions. In this paper we investigate the gravitational analogue to the electromagnetic birefringence phenomenon in the static and radiating topological black hole backgrounds, respectively. For the static topological black hole spacetimes, the velocity shift of photons is the same as the one in the Reissner-Nordstr\"om black holes. This reflects that the propagation of vacuum polarized photons is not sensitive to the asymptotic behavior and topological structure of spacetimes. For the massless topological black hole and BTZ black hole, the light cone condition keeps unchanged. In the radiating topological black hole backgrounds, the light cone condition is changed even for the radially directed photons. The velocity shifts depend on the topological structures. Due to the null fluid, the velocity shift of photons does no longer vanish at the apparent horizons as well as the event horizons. But the ``polarization sum rule'' is still valid. 
  We propose a general framework for quantum field theory on the de Sitter space-time (i.e. for local field theories whose truncated Wightman functions are not required to vanish). By requiring that the fields satisfy a weak spectral condition, formulated in terms of the analytic continuation properties of their Wightman functions, we show that a geodesical observer will detect in the corresponding ``vacuum'' a blackbody radiation at temperature T=1/(2 \pi R). We also prove the analogues of the PCT, Reeh-Schlieder and Bisognano-Wichmann theorems. 
  Recent data on supernovae favor high values of the cosmological constant. Spacetimes with a cosmological constant have non-relativistic kinematics quite different from Galilean kinematics. De Sitter spacetimes, vacuum solutions of Einstein's equations with a cosmological constant, reduce in the non-relativistic limit to Newton-Hooke spacetimes, which are non-metric homogeneous spacetimes with non-vanishing curvature. The whole non-relativistic kinematics would then be modified, with possible consequences to cosmology, and in particular to the missing-mass problem. 
  This paper constructs the multipole expansion (in general relativity) of the gravitational field generated by a slowly-moving isolated source. We introduce some definitions for the source multipole moments, valid to all orders in a post-Newtonian expansion, and depending in a well-defined way on the total stress-energy pseudo-tensor of the material and gravitational fields. Previously obtained expressions of the source multipole moments are recovered in the appropriate limits. The source moments parametrize the linearized approximation of the gravitational field exterior to the source, as computed by means of a specific post-Minkowskian algorithm defined in a previous work. Since the radiative multipole moments parametrizing the radiation field far from the source can be obtained as non-linear functionals of the source moments, the present paper permits relating the radiation field far from a slowly-moving source to the stress-energy pseudo-tensor of the source. This should be useful when comparing to theory the future observations of gravitational radiation by the LIGO and VIRGO experiments. 
  In this letter we suggest a scenario for simultaneous emission of gravitational-wave and $\gamma$-ray bursts (GRBs) from soft gamma-ray repeaters (SGRs). we argue that both of the radiations can be generated by a super-Eddington accreting neutron stars in X-ray binaries. In this model a supercritical accretion transient takes back onto the remnant star the disk leftover by the hydrodynamic instability phase of a low magnetized, rapidly rotating neutron star in a X-ray binary system. We estimate the rise timescale $\Delta t_c = 0.21 ms$, minimum mass accretion rate needed to trigger the $\gamma$-ray emission, $\dot{M}_\lambda = 4.5 \times 10^{28} g$, and its effective associated temperature $T_{eff} = 740 keV$, and the timescale for repeating a burst of $\gamma$-rays $\Delta \tau_R = 11.3 yr$. Altogether, we find the associated GW amplitude and frequency to be $h_c = 2.7 \times 10^{-23}/{(Hz)}^{1/2}$ and $f_{gw} = 966 Hz$, for a source distance $\sim 55 kpc$. Detectability of the pulses by t he forthcoming GW anntenas is discussed and found likely. 
  In this paper we have applied the generalized Kerr-Schild transformation finding a new family of stationary perfect-fluid solutions of the Einstein field equations. The procedure used combines some well-known techniques of null and timelike vector fields, from which some properties of the solutions are studied in a coordinate-free way. These spacetimes are algebraically special being their Petrov types II and D. This family includes all the classical vacuum Kerr-Schild spacetimes, excepting the plane-fronted gravitational waves, and some other interesting solutions as, for instance, the Kerr metric in the background of the Einstein Universe. However, the family is much more general and depends on an arbitrary function of one variable. 
  We present a first-quantized treatment of the back reaction on an accelerated particle detector. The accelerated detector is described as a first quantized, charged, pointlike particle in a constant field with internal energy levels. A quantum version of canonically conjugate future and past Rindler horizon operators is introduced to facilitate the calculation. The evaluated transition amplitude for detection agrees with previously obtained results. 
  We study the radiation reaction on cosmic strings due to the emission of dilatonic, gravitational and axionic waves. After verifying the (on average) conservative nature of the time-symmetric self-interactions, we concentrate on the finite radiation damping force associated with the half-retarded minus half-advanced ``reactive'' fields. We revisit a recent proposal of using a ``local back reaction approximation'' for the reactive fields. Using dimensional continuation as convenient technical tool, we find, contrary to previous claims, that this proposal leads to antidamping in the case of the axionic field, and to zero (integrated) damping in the case of the gravitational field. One gets normal positive damping only in the case of the dilatonic field. We propose to use a suitably modified version of the local dilatonic radiation reaction as a substitute for the exact (non-local) gravitational radiation reaction. The incorporation of such a local approximation to gravitational radiation reaction should allow one to complete, in a computationally non-intensive way, string network simulations and to give better estimates of the amount and spectrum of gravitational radiation emitted by a cosmologically evolving network of massive strings. 
  We investigate the ratio of gravitational binding energy to rest mass in general relativity. For N pointlike masses, an upper bound on the magnitude of this ratio can be derived using the second law of black hole dynamics. Only as N approaches infinity can it approach one. A configuration that saturates the $N = \infty$ bound is a thin spherical shell as its radius is taken to zero. This system provides, in principle, a perfectly efficient mechanism for converting rest mass into energy. We mention possible implications for the black hole information problem. 
  We consider the quantization of the midi-superspace associated with a class of spacetimes with toroidal isometries, but without the compact spatial hypersurfaces of the well-known Gowdy models. By a symmetry reduction, the phase space for the system at the classical level can be identified with that of a free massless scalar field on a fixed background spacetime, thereby providing a simple route to quantization. We are then able to study certain non-perturbative features of the quantum gravitational system. In particular, we examine the quantum geometry of the asymptotic regions of the spacetimes involved and find some surprisingly large dispersive effects of quantum gravity. 
  Starting from a microscopic approach, we develop a covariant formalism to describe a set of interacting gases. For that purpose, we model the collision term entering the Boltzmann equation for a class of interactions and then integrate this equation to obtain an effective macroscopic description. This formalism will be useful to study the cosmic microwave background non-perturbatively in inhomogeneous cosmologies. It should also be useful for the study of the dynamics of the early universe and can be applied, if one considers fluids of galaxies, to the study of structure formation. 
  We consider the problem of calculating the Gaussian curvature of a conical 2-dimensional space by using concepts and techniques of distribution theory. We apply the results obtained to calculate the Riemannian curvature of the 4-dimensional conical space-time. We show that the method can be extended for calculating the curvature of a special class of more general space-times with conical singularity. 
  The 3-level leapfrog time integration algorithm is an attractive choice for numerical relativity simulations since it is time-symmetric and avoids non-physical damping. In Newtonian problems without velocity dependent forces, this method enjoys the advantage of long term stability. However, for more general differential equations, whether ordinary or partial, delayed onset numerical instabilities can arise and destroy the solution. A known cure for such instabilities appears to have been overlooked in many application areas. We give an improved cure ("deloused leapfrog") that both reduces memory demands (important for 3+1 dimensional wave equations) and allows for the use of adaptive timesteps without a loss in accuracy. We show both that the instability arises and that the cure we propose works in highly relativistic problems such as tightly bound geodesics, spatially homogeneous spacetimes, and strong gravitational waves. In the gravitational wave test case (polarized waves in a Gowdy spacetime) the deloused leapfrog method was five to eight times less CPU costly at various accuracies than the implicit Crank-Nicholson method, which is not subject to this instability. 
  We show that there exists a choice of scalar field modes, such that the evolution of the quantum field in the zero-mass and large-mass limits is consistent with the Einstein equations for the background geometry. This choice of modes is also consistent with zero production of these particles and thus corresponds to a preferred vacuum state preserved by the evolution. In the zero-mass limit, we find that the quantum field equation implies the Einstein equation for the scale factor of a radiation-dominated universe; in the large-mass case, it implies the corresponding Einstein equation for a matter-dominated universe. Conversely, if the classical radiation-dominated or matter-dominated Einstein equations hold, there is no production of scalar particles in the zero and large mass limits, respectively. The suppression of particle production in the large mass limit is over and above the expected suppression at large mass. Our results hold for a certain class of conformally ultrastatic background geometries and therefore generalize previous results by one of us for spatially flat Robertson-Walker background geometries. In these geometries, we find that the temporal part of the graviton equations reduces to the temporal equation for a massless minimally coupled scalar field, and therefore the results for massless particle production hold also for gravitons. Within the class of modes we study, we also find that the requirement of zero production of massless scalar particles is not consistent with a non-zero cosmological constant. Possible implications are discussed. 
  This paper has been withdrawn due to crucial erros in equations (24)--(31). 
  We analyse numerically the onset of pre-big bang inflation in an inhomogeneous, spherically symmetric Universe. Adding a small dilatonic perturbation to a trivial (Milne) background, we find that suitable regions of space undergo dilaton-driven inflation and quickly become spatially flat ($\Omega \to 1$). Numerical calculations are pushed close enough to the big bang singularity to allow cross checks against previously proposed analytic asymptotic solutions. 
  In recent numerical simulations of spherically symmetric gravitational collapse a new type of critical behaviour, dominated by a sphaleron solution, has been found. In contrast to the previously studied models, in this case there is a finite gap in the spectrum of black-hole masses which is reminiscent of a first order phase transition. We briefly summarize the essential features of this phase transition and describe the basic heuristic picture underlying the numerical phenomenology. 
  According to the inflationary scenario for the very early Universe, all inhomogeneities in the Universe are of genuine quantum origin. On the other hand, looking at these inhomogeneities and measuring them, clearly no specific quantum mechanical properties are observed. We show how the transition from their inherent quantum gravitational nature to classical behaviour comes about -- a transition whereby none of the successful quantitative predictions of the inflationary scenario for the present-day universe is changed. This is made possible by two properties. First, the quantum state for the spacetime metric perturbations produced by quantum gravitational effects in the early Universe becomes very special (highly squeezed) as a result of the expansion of the Universe (as long as the wavelength of the perturbations exceeds the Hubble radius). Second, decoherence through the environment distinguishes the field amplitude basis as being the pointer basis. This renders the perturbations presently indistinguishable from stochastic classical inhomogeneities. 
  The small or zero cosmological constant, $\Lambda$, probably results from a macroscopic cancellation mechanism of the zero-point energies. However, nearby horizon surfaces any macroscopic mechanism is expected to result in imperfect cancellations. A phenomenological description is given for the residual variable cosmological constant. In the static, spherically symmetric case it produces approximate black holes. The model describes the case of exponential decay by $\Box\ln\Lambda=-3a$, were $a$ is a positive constant. 
  We consider compensated spherical lens models and the caustic surfaces they create in the past light cone. Examination of cusp and crossover angles associated with particular source and lens redshifts gives explicit lensing models that confirm previous claims that area distances can differ by substantial factors from angular diameter distances even when averaged over large angular scales. `Shrinking' in apparent sizes occurs, typically by a factor of 3 for a single spherical lens, on the scale of the cusp caused by the lens; summing over many lenses will still leave a residual effect. 
  This work consists of two distinct parts. In the first part we present a new method for solving the initial value problem of general relativity. Given any spatial metric with a surface orthogonal Killing field and two freely specified components of the extrinsic curvature we solve for extrinsic curvature's remaining components. For the second part, after noting that initial data for the Kerr spacetime can be derived within our formalism we construct data for axisymmetric configurations of spinning black holes. Though our method is limited to axisymmetry, it offers an advantage over the Bowen-York proceedure that our data approach those for Kerr holes in the limit of large separations and in the close limit. 
  We determine in closed form the general static solution with cylindrical symmetry to the Brans-Dicke equations for an energy-momentum tensor corresponding to the one of the straight U(1) global string outside the core radius assuming that the Goldstone boson field takes its asymptotic value 
  We analyze the "F-locality condition" (proposed by Kay to be a mathematical implementation of a philosophical bias related to the equivalence principle, we call it the "GH-equivalence principle"), which is often used to build a generalization of quantum field theory to non-globally hyperbolic spacetimes. In particular we argue that the theorem proved by Kay, Radzikowski, and Wald to the effect that time machines with compactly generated Cauchy horizons are incompatible with the F-locality condition actually does not support the "chronology protection conjecture", but rather testifies that the F-locality condition must be modified or abandoned. We also show that this condition imposes a severe restriction on the geometry of the world (it is just this restriction that comes into conflict with the existence of a time machine), which does not follow from the above mentioned philosophical bias. So, one need not sacrifice the GH-equivalence principle to "emend" the F-locality condition. As an example we consider a particular modification, the "MF-locality condition". The theory obtained by replacing the F-locality condition with the MF-locality condition possesses a few attractive features. One of them is that it is consistent with both locality and the existence of time machines. 
  The Bianchi IX mixmaster model is quantized in its non-diagonal form, imposing spatial diffeomorphism, time reparametrization and Lorentz invariance as constraints on physical state vectors before gauge-fixing. The result turns out to be different from quantizing the diagonal model obtained by gauge-fixing already on the classical level. For the non-diagonal model a generalized 9-dimensional Fourier transformation over a suitably chosen manifold connects the representations in metric variables and in Ashtekar variables. A space of five states in the metric representation is generated from the single physical Chern-Simons state in Ashtekar variables by choosing five different integration manifolds, which cannot be deformed into each other. For the case of a positive cosmological constant $\Lambda$ we extend our previous study of these five states for the diagonal Bianchi IX model to the non-diagonal case. It is shown that additional discrete (permutation) symmetries of physical states arise in the quantization of the non-diagonal model, which are satisfied by two of the five states connected to the Chern-Simons state. These have the characteristics of a wormhole groundstate and a Hartle-Hawking `no-boundary' state, respectively. We also exhibit a special gauge-fixing of the time reparametrization invariance of the quantized system and define an associated manifestly positive scalar product. Then the wormhole ground state is left as the only normalizable physical state connected to the Chern-Simons state. 
  The general validity of the area law for black holes is still an open problem. We first show in detail how to complete the usually incompletely stated text-book proofs under the assumption of piecewise $C^2$-smoothness for the surface of the black hole. Then we prove that a black hole surface necessarily contains points where it is not $C^1$ (called ``cusps'') at any time before caustics of the horizon generators show up, like e.g. in merging processes. This implies that caustics never disappear in the past and that black holes without initial cusps will never develop such. Hence black holes which will undergo any non-trivial processes anywhere in the future will always show cusps. Although this does not yet imply a strict incompatibility with piecewise $C^2$ structures, it indicates that the latter are likely to be physically unnatural. We conclude by calling for a purely measure-theoretic proof of the area theorem. 
  We present a new method of extracting gravitational radiation from three-dimensional numerical relativity codes and providing outer boundary conditions. Our approach matches the solution of a Cauchy evolution of Einstein's equations to a set of one-dimensional linear wave equations on a curved background. We illustrate the mathematical properties of our approach and discuss a numerical module we have constructed for this purpose. This module implements the perturbative matching approach in connection with a generic three-dimensional numerical relativity simulation. Tests of its accuracy and second-order convergence are presented with analytic linear wave data. 
  We consider the coupled Einstein-Dirac-Maxwell equations for a static, spherically symmetric system of two fermions in a singlet spinor state. Soliton-like solutions are constructed numerically. The stability and the properties of the ground state solutions are discussed for different values of the electromagnetic coupling constant. We find solutions even when the electromagnetic coupling is so strong that the total interaction is repulsive in the Newtonian limit. Our solutions are regular and well-behaved; this shows that the combined electromagnetic and gravitational self-interaction of the Dirac particles is finite. 
  For a Friedman-Robertson-Walker space-time in which the only contribution to the stress-energy tensor comes from the renormalised zero-point energy (i.e. the Casimir energy) of the fundamental fields the evolution of the universe (the scale factor) depends upon whether the universe is open, flat or closed and upon which fundamental fields inhabit the space-time. We calculate this "Casimir effect" using the heat kernel method, and the calculation is thus non-perturbative. We treat fields of spin $0,1/2,1$ coupled to the gravitational background only. The heat kernels and/or zeta-functions for the various spins are related to that of a non-minimally coupled one. A WKB approximation is used in obtaining the radial part of that heat kernel. The simulations of the resulting equations of motion seem to exclude the possibility of a closed universe, $K=+1$, as these turn out to have an overwhelming tendency towards a fast collapse - the details such as the rate of this collapse depends on the structure of the underlying quantum degrees of freedom: a non-minimal coupling to curvature accelerates the process. Only $K=-1$ and K=0 will in general lead to macroscopic universes, and of these $K=-1$ seems to be more favourable. The possibility of the scale factor being a concave rather than a convex function potentially indicates that the problem of the large Hubble constant is non-existent as the age of the universe need not be less than or equal to the Hubble time. Note should be given to the fact, however, that we are not able to pursue the numerical study to really large times neither do simulations for a full standard model. 
  In an irrotational dust universe, the locally free gravitational field is covariantly described by the gravito-electric and gravito-magnetic tensors $E_{ab}$ and $H_{ab}$. In Newtonian theory, $H_{ab}=0$ and $E_{ab}$ is the tidal tensor. Newtonian-like dust universes in general relativity (i.e. with $H_{ab}=0$, often called `silent') have been shown to be inconsistent in general and unlikely to extend beyond the known spatially homogeneous or Szekeres examples. Furthermore, they are subject to a linearization instability. Here we show that `anti-Newtonian' universes, i.e. with purely gravito-magnetic field, so that $E_{ab} = 0\neq H_{ab}$, are also subject to severe integrability conditions. Thus these models are inconsistent in general. We show also that there are no anti-Newtonian spacetimes that are linearized perturbations of Robertson-Walker universes. The only $E_{ab}=0\neq H_{ab}$ solution known to us is not a dust solution, and we show that it is kinematically G\"{o}del-like but dynamically unphysical. 
  We examine the field equations of a self-gravitating global string in low energy superstring gravity, allowing for an arbitrary coupling of the global string to the dilaton. Massive and massless dilatons are considered. For the massive dilaton the spacetime is similar to the recently discovered non-singular time-dependent Einstein self-gravitating global string, but the massless dilaton generically gives a singular spacetime, even allowing for time-dependence. We also demonstrate a time-dependent non-singular string/anti-string configuration, in which the string pair causes a compactification of two of the spatial dimensions, albeit on a very large scale. 
  This paper is a sequel to the series of papers [gr-qc/9409010, gr-qc/9505034, gr-qc/9603022, gr-qc/9609035, gr-qc/9609046, gr-qc/9704033, gr-qc/9704038, gr-qc/9708014], being an immediate continuation and supplement to the last of them, where gravitational autolocalization of a body has been considered. A resulting solution, which describes a one-place location, has been called gravilon. Here it is shown that a gravilon is the only solution, i.e., that many-place gravitational autolocalization is unreal. This is closely related to nonreality of tunneling in the conditions under consideration. 
  Contents:  *News:    Topical group news, by Jim Isenberg    April 1997 Joint APS/AAPT Meeting, GTG program, by Abhay Ashtekar    Formation of the Gravitational-Wave International Committee, by Sam Finn    The 1997 Xanthopoulos award, by Abhay Asthekar    We hear that..., by Jorge Pullin  *Research Briefs:    LIGO project update, by David Shoemaker    The search for frame-dragging by NS and BH's, by Sharon Morsink    Gamma-ray bursts, recent developments, by Peter Meszaros    Status of the Binary Black hole Grand Challenge, by Richard Matzner  *Conference Reports:    Quantum gravity at GR15, by Don Marolf    GR Classical, by John Friedman    An Experimentalist's Idiosyncratic Report on GR15, by Peter Saulson    Bangalore gravitational wave meeting, by Sharon Morsink    Bangalore quantum gravity meeting, by Domenico Giulini    Cleveland cosmology-topology workshop, by Neil Cornish    Quantum Gravity in the Southern Cone II, by Carmen Nunez    Baltimore AMS meeting, by Kirill Krasnov 
  Internal friction effects are responsible for line widening of the resonance frequencies in spherical gravitational wave detectors, and result in exponentially damped oscillations of its eigenmodes with a decay time which is proportional to the quality factor of the mode and to its inverse frequency. We study the solutions to the equations of motion for a viscoelastic spherical GW detector based on various different assumptions about the material's constituent equations. Quality factor dependence on mode frequency is determined in each case, and a discussion of its applicability to actual detectors is made. 
  We study the deflection of light (and the redshift, or integrated time delay) caused by the time-dependent gravitational field generated by a localized material source lying close to the line of sight. Our calculation explicitly takes into account the full, near-zone, plus intermediate-zone, plus wave-zone, retarded gravitational field. Contrary to several recent claims in the literature, we find that the deflections due to both the wave-zone 1/r gravitational wave and the intermediate-zone 1/r^2 retarded fields vanish exactly. The leading total time-dependent deflection caused by a localized material source, such as a binary system, is proven to be given by the quasi-static, near-zone quadrupolar piece of the gravitational field, and therefore to fall off as the inverse cube of the impact parameter. 
  I give a brief introduction to the problem of detecting gravitational radiation from neutron stars. After a review of the mechanisms by which such stars may produce radiation, I consider the different search strategies appropriate to the different kinds of sources: isolated known pulsars, neutron stars in binaries, and unseen neutron stars. The problem of an all-sky survey for unseen stars is the most taxing one that we face in analysing data from interferometers. I describe the kinds of hierarchical methods that are now being investigated to reach the maximal sensitivity, and I suggest a replacement for standard Fourier-transform search methods that requires fewer floating-point operations for Fourier-based searches over large parameter spaces, and in addition is highly parallelizable, working just as well on a loosely coupled network of workstations as on a tightly coupled parallel computer. 
  The spherically symmetric thin shells of the macroscopically stable quark-gluon matter are considered within the frameworks of the bag model and theory of discontinuities in general relativity. The equation of state for the two-dimensional matter is suggested, and its features are discussed. The exact equations of motion of such shells are obtained. Distinguishing the two cases, circumstellar and microscopical shells, we calculate the parameters of equilibrium configurations, including the conditions of decay (deconfinement). 
  This paper is a sequel to the series of papers [gr-qc/9409010, gr-qc/9505034, gr-qc/9603022, gr-qc/9609035, gr-qc/9609046, gr-qc/9704033, gr-qc/9704038, gr-qc/9708014, gr-qc/9802016]. The problem of the meaning of objective a priori probability for individual random trials without repetition is considered. A sequence of such trials, namely quantum jumps, is realized in indeterministic dynamics of the universe. A hidden selector for the quantum jumps is constructed. 
  We discuss the notion that quantum fields may induce an effective time-dependent cosmological constant which decays from a large initial value. It is shown that such cosmological models are viable in a non-de Sitter spacetime. 
  We consider some aspects of quantum field theory of a conformally coupled scalar field on the singular background obtained in the massless limit of a class of toroidal black holes. The stress-tensor and its back-reaction on the metric are computed using the point-splitting regularization, in the cases of transparent, Neumann and Dirichlet boundary conditions. We find that the quantum fluctuations generate an event horizon which hides the singularity. The resulting object can be interpreted as a long lived remnant. We discuss the relevance of this result in the context of the cosmic censorship conjecture, and in connection to the end-point of the quantum evaporation process. 
  A Green's function method is developed for solving strongly-coupled gravity and matter in the semiclassical limit. In the strong-coupling limit, one assumes that Newton's constant approaches infinity. As a result, one may neglect second order spatial gradients, and each spatial point evolves like an homogeneous universe. After constructing the Green's function solution to the Hamiltonian constraint, the momentum constraint is solved using functional methods in conjunction with the superposition principle for Hamilton-Jacobi theory. Exact and approximate solutions are given for a dust field or a scalar field interacting with gravity. 
  Rotational motion of extended celestial bodies is discussed in the framework of the Parametrized Post-Newtonian (PPN) formalism with the two parameters $\gamma$ and $\beta$. A local PPN reference system of a massive extended body being a member of a system of N massive extended bodies is constructed. In the local PPN reference system the external gravitational field manifests itself only in the form of tidal potentials. Rotational equations of motion, which are then derived in the local reference system, reveal a special term in the torque analogous to the Nordtvedt effect in the translational equations of motion: it is proportional to $4\beta-\gamma-3$, to the acceleration of the body relative to the global PPN reference system and to some quantity characterizing the distribution of inertial gravitational energy within the body. This term is a direct consequence of the violation of the Strong Equivalence Principle in alternative theories of gravitation. 
  We review curvature-based hyperbolic forms of the evolution part of the Cauchy problem of General Relativity that we have obtained recently. We emphasize first order symmetrizable hyperbolic systems possessing only physical characteristics. 
  Entanglement entropy is a statistical entropy measuring information loss due to coarse-graining corresponding to a spatial division of a system. In this paper we construct a thermodynamics (entanglement thermodynamics) which includes the entanglement entropy as the entropy variable, for a massless scalar field in Minkowski, Schwarzschild and Reissner-Nordstr{\"o}m spacetimes to understand the statistical origin of black-hole thermodynamics. It is shown that the entanglement thermodynamics in Minkowski spacetime differs significantly from black-hole thermodynamics. On the contrary, the entanglement thermodynamics in Schwarzschild and Reissner-Nordstr{\"o}m spacetimes has close relevance to black-hole thermodynamics. 
  By regarding the vacuum as a perfect fluid with equation of state p=-rho, de Sitter's cosmological model is quantized. Our treatment differs from previous ones in that it endows the vacuum with dynamical degrees of freedom. Instead of being postulated from the start, the cosmological constant arises from the degrees of freedom of the vacuum regarded as a dynamical entity, and a time variable can be naturally introduced. Taking the scale factor as the sole degree of freedom of the gravitational field, stationary and wave-packet solutions to the Wheeler-DeWitt equation are found. It turns out that states of the Universe with a definite value of the cosmological constant do not exist. For the wave packets investigated, quantum effects are noticeable only for small values of the scale factor, a classical regime being attained at asymptotically large times. 
  It is shown that the recently found non-uniqueness of the third post-Newtonian binary point-mass ADM-Hamiltonian is related to the non-uniqueness at the third post-Newtonian approximation of the applied ADM-coordinate conditions. 
  We study the role played by the compactness and the degree of connectedness in the time evolution of the energy of a radiating system in the Friedmann-Robertson-Walker (FRW) space-times whose $t=const $ spacelike sections are the Euclidean 3-manifold ${\cal R}^3$ and six topologically non-equivalent flat orientable compact multiply connected Riemannian 3-manifolds. An exponential damping of the energy $E(t)$ is present in the ${\cal R}^3$ case, whereas for the six compact flat 3-spaces it is found basically the same pattern for the evolution of the energy, namely relative minima and maxima occurring at different times (depending on the degree of connectedness) followed by a growth of $E(t)$. Likely reasons for this divergent behavior of $E(t)$ in these compact flat 3-manifolds are discussed and further developments are indicated. A misinterpretation of Wolf's results regarding one of the six orientable compact flat 3-manifolds is also indicated and rectified. 
  By using the gedanken experiments suggested by Bekenstein and Rosenzweig, we have shown that nonextremal Reissner-Nordstrom black hole cannot turn into extremal one by assimilating infalling charged particle and charged spherical shell. 
  A stability analysis is made in the context of the previously discovered non-singular cosmological solution from 1-loop corrected superstring effective action. We found that this solution has an instability in graviton mode, which is shown to have a close relation to the avoidance of initial singularity via energy condition. We also estimate the condition for the breakdown of the background solution due to the overdominance of the graviton. 
  We present a rewiew and also new possible applications of $p$-adic numbers to pre-spacetime physics. It is shown that instead of the extension $R^n\to Q_p^n$, which is usually implied in $p$-adic quantum field theory, it is possible to build a model based on the $R^n\to Q_p$, where p=n+2 extension and get rid of loop divergences. It is also shown that the concept of mass naturally arises in $p$-adic models as inverse transition probability with a dimensional constant of proportionality. 
  We consider the evaporation of the (shell focusing) naked singularity formed during the self-similar collapse of a marginally bound inhomogeneous dust cloud, in the geometric optics approximation. We show that, neglecting the back reaction of the spacetime, the radiation on $\scrip$ tends to infinity as the Cauchy Horizon is approached. Two consequences can be expected from this result: (a) that the back reaction of spacetime will be large and eventually halt the formation of a naked singularity thus preserving the Cosmic Censorship Hypothesis and (b) matter attempting to collapse into a naked singularity will radiate away energy at an intense rate, thereby possibly providing experimental signatures of quantum effects in curved spacetimes. 
  The equation of motion for a domain wall coupled to gravitational field is derived from the Nambu-Goto action. The domain wall is treated as a source of gravitational field. The perturbed equation is also obtained with gravitational back reaction on the wall motion taken into account. For general spherically symmetric background case, the equation is expressed in terms of the gauge-invariant variables. 
  The model of the homogenous and isotropic universe with two spaces is considered. The background space is a coordinate system of reference and defines the behaviour of the universe. The other space characterizes the gravity of the matter of the universe. In the presented model, the first derivative of the scale factor of the universe with respect to time is equal to the velocity of light. The density of the matter of the universe changes from the Planckian value at the Planck time to the modern value at the modern time. The model under consideration describes the universe from the Planck time to the modern time and avoids the problems of the Friedman universe such as the flatness problem and the horizon problem. 
  We discuss a dramatic difference between the description of the quantum creation of an open universe using the Hartle-Hawking wave function and the tunneling wave function. Recently Hawking and Turok have found that the Hartle-Hawking wave function leads to a universe with Omega = 0.01, which is much smaller that the observed value of Omega > 0.3. Galaxies in such a universe would be about $10^{10^8}$ light years away from each other, so the universe would be practically structureless. We will argue that the Hartle-Hawking wave function does not describe the probability of the universe creation. If one uses the tunneling wave function for the description of creation of the universe, then in most inflationary models the universe should have Omega = 1, which agrees with the standard expectation that inflation makes the universe flat. The same result can be obtained in the theory of a self-reproducing inflationary universe, independently of the issue of initial conditions. However, there exist two classes of models where Omega may take any value, from Omega > 1 to Omega << 1. 
  Quantum fluctuations for a massless scalar field in the background metric of spherical impulsive gravitational waves through Minkowski and de Sitter spaces are investigated. It is shown that there exist finite fluctuations for de Sitter space. 
  Some problems of the space-time causal structure are discussed using models with traversable wormholes. For this purpose the conditions of traversable wormhole matching with the exterior space-time are considered in detail and a mixed boundary problem for the Einstein equations is formulated and analyzed. The influence of these matching conditions on the space-time properties and causal structure is analyzed. These conditions have a non-dynamical nature and cannot be determined by any physical process. So, the causality violation cannot be a result of dynamical evolution of some initial hypersurface. It is also shown that the same conditions which determine the wormhole joining with the outer space provide the self-consistency of solutions and the absence of paradoxes in the case of causality violation. 
  The spherically symmetric thin shells of the barotropic fluids with the linear equation of state are considered within the frameworks of general relativity. We study several aspects of the shells as completely relativistic models of stars, first of all the neutron stars and white dwarfs, and circumstellar shells. The exact equations of motion of the shells are obtained. Also we calculate the parameters of the equilibrium configurations, including the radii of static shells. Finally, we study the stability of the equilibrium shells against radial perturbations. 
  The spherically symmetric singular perfect fluid shells are considered for the case of their radii being equal to the event horizon (the black shells). We study their observable masses, depending at least on the three parameters, viz., the square speed of sound in the shell, instantaneous radial velocity of the shell at a moment when it reaches the horizon, and integration constant related to surface mass density. We discuss the features of black shells depending on an equation of state. 
  We study the Einstein-Klein-Gordon equations for a convex positive potential in a Bianchi I, a Bianchi III and a Kantowski-Sachs universe. After analysing the inherent properties of the system of differential equations, the study of the asymptotic behaviors of the solutions and their stability is done for an exponential potential. The results are compared with those of Burd and Barrow. In contrast with their results, we show that for the BI case isotropy can be reached without inflation and we find new critical points which lead to new exact solutions. On the other hand we recover the result of Burd and Barrow that if inflation occurs then isotropy is always reached. The numerical integration is also done and all the asymptotical behaviors are confirmed. 
  A massless Weyl-invariant dynamics of a scalar, a Dirac spinor, and electromagnetic fields is formulated in a Weyl space, $W_4$, allowing for conformal rescalings of the metric and of all fields with nontrivial Weyl weight together with the associated transformations of the Weyl vector fields $\ka_\mu$ representing the D(1) gauge fields with D(1) denoting the dilatation group. To study the appearance of nonzero masses in the theory the Weyl-symmetry is broken explicitly and the corresponding reduction of the Weyl space $W_4$ to a pseudo-Riemannian space $V_4$ is investigated assuming the breaking to be determined by an expression involving the curvature scalar $R$ of the $W_4$ and the mass of the scalar, selfinteracting field. Thereby also the spinor field acquires a mass proportional to the modulus $\Phi$ of the scalar field in a Higgs-type mechanism formulated here in a Weyl-geometric setting with $\Phi$ providing a potential for the Weyl vector fields $\ka_\mu$. After the Weyl-symmetry breaking one obtains generally covariant and U(1) gauge covariant field equations coupled to the metric of the underlying $V_4$. This metric is determined by Einstein's equations, with a gravitational coupling constant depending on $\Phi$, coupled to the energy-momentum tensors of the now massive fields involved together with the (massless) radiation fields. 
  In this paper, a procedure which gives euclidean solutions of 3-dimensional Einstein-Yang-Mills equations when one has solutions of the Einstein equations is proposed. The method is based on reformulating Yang-Mills theory in such a way that it becomes a gravity. It is applied to find black hole solutions of the coupled Einstein-Yang-Mills equations. 
  Adapting and extending a suggestion due to Page, we define a wormhole throat to be a marginally anti-trapped surface, that is, a closed two-dimensional spatial hypersurface such that one of the two future-directed null geodesic congruences orthogonal to it is just beginning to diverge. Typically a dynamic wormhole will possess two such throats, corresponding to the two orthogonal null geodesic congruences, and these two throats will not coincide, (though they do coalesce into a single throat in the static limit). The divergence property of the null geodesics at the marginally anti-trapped surface generalizes the ``flare-out'' condition for an arbitrary wormhole. We derive theorems regarding violations of the null energy condition (NEC) at and near these throats and find that, even for wormholes with arbitrary time-dependence, the violation of the NEC is a generic property of wormhole throats. We also discuss wormhole throats in the presence of fully antisymmetric torsion and find that the energy condition violations cannot be dumped into the torsion degrees of freedom. Finally by means of a concrete example we demonstrate that even temporary suspension of energy-condition violations is incompatible with the flare-out property of dynamic throats. 
  Using an energy-momentum complex we give a physical interpretation to the constants in the well-known static spherically symmetric asymptotically flat vacuum solution to the Brans-Dicke equations. The positivity of the tensor mass puts a bound on parameters in the solution. 
  We extend previous proofs that violations of the null energy condition (NEC) are a generic and universal feature of traversable wormholes to completely non-symmetric time-dependent wormholes. We show that the analysis can be phrased purely in terms of local geometry at and near the wormhole throat, and do not have to make any technical assumptions about asymptotic flatness or other global properties. A key aspect of the analysis is the demonstration that time-dependent wormholes have two throats, one for each direction through the wormhole, and that the two throats coalesce only for the case of a static wormhole. 
  We derive a modified Buchdahl inequality for scalar-tensor theories of gravity. In general relativity, Buchdahl has shown that the maximum value of the mass-to-size ratio, $2M/R$, is 8/9 for static and spherically symmetric stars under some physically reasonable assumptions. We formally apply Buchdahl's method to scalar-tensor theories and obtain theory-independent inequalities. After discussing the mass definition in scalar-tensor theories, these inequalities are related to a theory-dependent maximum mass-to-size ratio. We show that its value can exceed not only Buchdahl's limit, 8/9, but also unity, which we call {\it the black hole limit}, in contrast to general relativity. Next, we numerically examine the validity of the assumptions made in deriving the inequalities and the applicability of our analytic results. We find that the assumptions are mostly satisfied and that the mass-to-size ratio exceeds both Buchdahl's limit and the black hole limit. However, we also find that this ratio never exceeds Buchdahl's limit when we impose the further condition, $\rho-3p\ge0$, on the density, $\rho$, and pressure, $p$, of the matter. 
  In a recent work an approximation procedure was introduced to calculate the vacuum expectation value of the stress-energy tensor for a conformal massless scalar field in the classical background determined by a particular colliding plane wave space-time. This approximation procedure consists in appropriately modifying the space-time geometry throughout the causal past of the collision center. This modification in the geometry allows to simplify the boundary conditions involved in the calculation of the Hadamard function for the quantum state which represents the vacuum in the flat region before the arrival of the waves. In the present work this approximation procedure is applied to the non-singular Bel-Szekeres solution, which describes the head on collision of two electromagnetic plane waves. It is shown that the stress-energy tensor is unbounded as the killing-Cauchy horizon of the interaction is approached and its behavior coincides with a previous calculation in another example of non-singular colliding plane wave space-time. 
  A scenario of galaxy formation is put forward which is a process of sudden condensation just after recombination. It is essentially based on the fact that the cosmic matter gas after recombination is a general relativistic Boltzmann gas which runs within a few $10^6$ years into a state very close to collision--dominated equilibrium. The mass spectrum of axially symmetric condensation "drops" extends from the lower limit M about $10^5$ solar masses to the upper limit M about $10^{12}$ solar masses. The lower limit masses are spheres whereas the upper limit masses are extremely thin pancakes. These pancakes contract within a time of about $2.5 \cdot 10^9 y$ to fastly rotating spiral galaxies with ordinary proportions. In this final state they have a redshift z about 3. At an earlier time during their contraction they are higly active and are observed with a redshift z about 5. 
  A class of exact, anisotropic cosmological solutions to the vacuum Brans-Dicke theory of gravity is considered within the context of the pre-big bang scenario. Included in this class are the Bianchi type III, V and VI_h models and the spatially isotropic, negatively curved Friedmann-Robertson-Walker universe. The effects of large anisotropy and spatial curvature are determined. In contrast to negatively curved Friedmann-Robertson-Walker model, there exist regions of the parameter space in which the combined effects of curvature and anisotropy prevent the occurrence of inflation. When inflation is possible, the necessary and sufficient conditions for successful pre-big bang inflation are more stringent than in the isotropic models. The initial state for these models is established and corresponds in general to a gravitational plane wave. 
  The concept of scale factor duality is considered within the context of the spatially homogeneous, vacuum Brans-Dicke cosmologies. In the Bianchi class A, it is found that duality symmetries exist for the types I, II, VI_0, VII_0, but not for types VIII and IX. The Kantowski-Sachs and locally rotationally symmetric Bianchi type III models also exhibit a scale factor duality, but no such symmetries are found for the Bianchi type V. In this way anisotropy and spatial curvature may have important effects on the nature of such dualities. 
  We investigate the gravitational evolution of dark matter halos made up of a massless bosonic field. The coupled Einstein-Klein-Gordon equations are solved numerically, showing that such a boson halo is stable and can be formed under a large class of initial conditions. We also present an analytical proof that such objects are stable in the Newtonian limit. In the context of boson stars made of massive scalar fields, we introduce new solutions with an oscillatory scalar field, similar to boson halos. We find that these solutions are unstable. 
  We study the quantization of a scalar field on a classical background given by the Szekeres Class of solutions, which represent the collision of two gravitational plane waves with constant polarization. These solutions consist of two approaching gravitational plane waves moving in a flat background and an interaction region which always contains a curvature singularity. Following a suitable approximate procedure, introduced in a previous paper, we propose a way to compute the vacuum expectation value of the stress-energy tensor throughout the causal past region of the collision center in the quantum state which corresponds to the vacuum before the arrival of the waves. 
  We give a short historical review of early Kaluza-Klein theories. We study various causal structures on manifolds, especially those which cannot be described by a metric tensor with signature (+---). The smooth structure (atlas) on a manifold is found to be related with its causal structure. 
  We investigate connections between pairs of (pseudo-)Riemannian metrics whose sum is a (tensor) product of a covector field with itself. A bijective mapping between the classes of Euclidean and Lorentzian metrics is constructed as a special result. The existence of such maps on a differentiable manifold is discussed. Similar relations for metrics of arbitrary signature on a manifold are considered. We point the possibility that any physical theory based on real Lorentzian metric(s) can be (re)formulated equivalently in terms of real Euclidean metric(s). 
  We discuss and investigate the problem of existence of metric-compatible linear connections for a given space-time metric which is, generally, assumed to be semi-pseudo-Riemannian. We prove that under sufficiently general conditions such connections exist iff the rank and signature of the metric are constant. On this base we analyze possible changes of the space-time signature. 
  We analyze the amplification due to so-called superradiance from the scattering of pulses off rotating black holes as a numerical time evolution problem. We consider the "worst possible case" of scalar field pulses for which superradiance effects yield amplifications $< 1%$. We show that this small effect can be isolated by numerically evolving quasi-monochromatic, modulated pulses with a recently developed Teukolsky code. The results show that it is possible to study superradiance in the time domain, but only if the initial data is carefully tuned. This illustrates the intrinsic difficulties of detecting superradiance in more general evolution scenarios. 
  In the spirit of the well-known analogy between inviscid fluids and pseudo-Riemannian manifolds we study spherical singular hypersurfaces in the static superfluid. Such hypersurfaces turn out to be the interfaces dividing the superfluid into the pairs of spherical domains, examples of which are ``superfluid A - superfluid B'' or ``impurity - superfluid'' phases. It is shown that these shells form the acoustic lenses which are the sonic counterparts of the usual optical ones. The exact equations of motion of the lens interfaces are obtained. Also some quantum aspects of the theory are considered. We calculate energy spectra for bound states of acoustic lenses in dynamical equilibrium, taking into account the analogy to a material shell model of a black hole (we consider the cases of spatial topology of a black hole and a wormhole type). 
  We perform numerical simulations of the gravitational collapse of a spherically symmetric scalar field. For those data that just barely do not form black holes we find the maximum curvature at the position of the central observer. We find a scaling relation between this maximum curvature and distance from the critical solution. The scaling relation is analogous to that found by Choptuik for black hole mass for those data that do collapse to form black holes. We also find a periodic wiggle in the scaling exponent. 
  We comment on Linde's claim that one should change the sign in the action for a Euclidean instanton in quantum cosmology, resulting in the formula $P \sim e^{+S}$ for the probability of various classical universes. There are serious problems with doing so. If one reverses the sign of the action of both the instanton and the fluctuations, the latter are unsupressed and the calculation becomes meaningless. So for a sensible result one would have to reverse the sign of the action for the background, while leaving the sign for the perturbations fixed by the usual Wick rotation. The problem with this approach is that there is no invariant way to split a given four geometry into background plus perturbations. So the prescription would have to violate general coordinate invariance. There are other indications that a sign change is problematic. With the choice $P \sim e^{+S}$ the nucleation of primordial black holes during inflation is unsuppressed, with a disastrous resulting cosmology. We regard these as compelling arguments for adhering to the usual sign given by the Wick rotation. 
  We study the head-on collision of black holes starting from unsymmetrized, Brill--Lindquist type data for black holes with non-vanishing initial linear momentum. Evolution of the initial data is carried out with the ``close limit approximation,'' in which small initial separation and momentum are assumed, and second-order perturbation theory is used. We find agreement that is remarkably good, and that in some ways improves with increasing momentum. This work extends a previous study in which second order perturbation calculations were used for momentarily stationary initial data, and another study in which linearized perturbation theory was used for initially moving holes. In addition to supplying answers about the collisions, the present work has revealed several subtle points about the use of higher order perturbation theory, points that did not arise in the previous studies. These points include issues of normalization, and of comparison with numerical simulations, and will be important to subsequent applications of approximation methods for collisions. 
  Through a constructive method it is shown that the claim advanced in recent times about a clash that should occur between the Freud and the Bianchi identities in Einstein's general theory of relativity is based on a faulty argument. 
  We consider a general non-linear sigma model coupled to Einstein gravity and show that in spherical symmetry and for a simple realization of self-similarity, the spacetime can be completely determined. We also examine some more specific matter models and discuss their relation to critical collapse. 
  We present a method for finding the eigenmodes of the Laplace operator acting on any compact manifold. The procedure can be used to simulate cosmic microwave background fluctuations in multi-connected cosmological models. Other applications include studies of chaotic mixing and quantum chaos. 
  Cylindrically reduced Einstein gravity can be regarded as an $SL(2,R)/SO(2)$ sigma model coupled to 2D dilaton gravity. By using the corresponding 2D diffeomorphism algebra of constraints and the asymptotic behaviour of the Ernst equation we show that the theory can be mapped by a canonical transformation into a set of free fields with a Minkowskian target space. We briefly discuss the quantization in terms of these free-field variables, which is considerably simpler than in the other approaches. 
  Several approaches to Hawking radiation on Schwarzschild spacetime rely in some way or another on the fact that the Kruskal manifold has two causally disconnected exterior regions. We investigate the Hawking(-Unruh) effect for a real scalar field on the $\RPthree$ geon: an inextendible, globally hyperbolic, space and time orientable eternal black hole spacetime that is locally isometric to Kruskal but contains only one exterior region. The Hartle-Hawking-like vacuum~$\hhvacgeon$, which can be characterized alternatively by the positive frequency properties along the horizons or by the complex analytic properties of the Feynman propagator, turns out to contain exterior region Boulware modes in correlated pairs, and any operator in the exterior that only couples to one member of each correlated Boulware pair has thermal expectation values in the usual Hawking temperature. Generic operators in the exterior do not have this special form; however, we use a Bogoliubov transformation, a particle detector analysis, and a particle emission-absorption analysis that invokes the analytic properties of the Feynman propagator, to argue that $\hhvacgeon$ appears as a thermal bath with the standard Hawking temperature to any exterior observer at asymptotically early and late Schwarzschild times. A~(naive) saddle-point estimate for the path-integral-approach partition function yields for the geon only half of the Bekenstein-Hawking entropy of a Schwarzschild black hole with the same ADM mass: possible implications of this result for the validity of path-integral methods or for the statistical interpretation of black-hole entropy are discussed. Analogous results hold for a Rindler observer in a flat spacetime whose global properties mimic those of the geon. 
  We construct a class of linear partial differential equations describing general perturbations of non-rotating black holes in 3D Cartesian coordinates. In contrast to the usual approach, a single equation treats all radiative $\ell -m$ modes simultaneously, allowing the study of wave perturbations of black holes with arbitrary 3D structure, as would be present when studying the full set of nonlinear Einstein equations describing a perturbed black hole. This class of equations forms an excellent testbed to explore the computational issues of simulating black spacetimes using a three dimensional adaptive mesh refinement code. Using this code, we present results from the first fully resolved 3D solution of the equations describing perturbed black holes. We discuss both fixed and adaptive mesh refinement, refinement criteria, and the computational savings provided by adaptive techniques in 3D for such model problems of distorted black holes. 
  Quantization of the time symmetric system of interacting strings requires that gravity, just as electromagnetism in Wheeler-Feynman's time symmetric electro- dynamics, also be an "adjunct field" instead of an independent entity. The "adjunct field" emerges, at a scale large compared to that of the strings, as a "statistic" that summarizes how the string positions in the underlying space- time are "compactified" into those in Minkowski space. We are able to show, WITHOUT adding a scalar curvature term to the string action, that the "adjunct gravitational field" satisfies Einstein's equation with no cosmological term. 
  Experimental tests of the suggestion that the generalization of Wheeler and Feynman's time symmetric system is the dynamical basis underlying quantum mechanics are considered. In a time-symmetric system, the instantaneous correlations exhibited by two spatially separated particles in an entangled state can be established through other particles, and can reveal advanced interaction effects. In particular, the existence of advanced gravity waves may be detectable through suitable arrangements at the Laser Interferometer Gravitation- Wave Observatory. 
  The properties of the mass-radius curves of relativistic stellar models constructed from an equation of state with a first-order phase transition are examined. It is shown that the slope of the mass-radius curve is continuous unless the discontinuity in the density at the phase transition point has a certain special value. The curve has a cusp if the discontinuity is larger than this value. The curvature of the mass-radius curve becomes singular at the point where the high density phase material first appears. This singularity makes the mass-radius curve appear on large scales to have a discontinuity in its slope at this point, even though the slope is in fact continuous on microscopic scales. Analytical formulae describing the behavior of these curves are found for the simple case of models with two-zone uniform-density equations of state. 
  A detailed study of an inhomogeneous dust cosmology contained in a $\gamma$-law family of perfect-fluid metrics recently presented by Mars and Senovilla is performed. The metric is shown to be the most general orthogonally transitive, Abelian, $G_2$ on $S_2$ solution admitting an additional homothety such that the self-similar group $H_3$ is of Bianchi type VI and the fluid flow is tangent to its orbits. The analogous cases with Bianchi types I, II, III, V, VIII and IX are shown to be impossible thus making this metric privileged from a mathematical viewpoint. The differential equations determining the metric are partially integrated and the line-element is given up to a first order differential equation of Abel type of first kind and two quadratures. The solutions are qualitatively analyzed by investigating the corresponding autonomous dynamical system. The spacetime is regular everywhere except for the big bang and the metric is complete both into the future and in all spatial directions. The energy-density is positive, bounded from above at any instant of time and with an spatial profile (in the direction of inhomogeneity) which is oscillating with a rapidly decreasing amplitude. The generic asymptotic behaviour at spatial infinity is a homogeneous plane wave. Well-known dynamical system results indicate that this metric is very likely to describe the asymptotic behaviour in time of a much more general class of inhomogeneous $G_2$ dust cosmologies. 
  It has been recently pointed out that black holes of constant curvature with a "chronological singularity" can be constructed in any spacetime dimension. These black holes share many common properties with the 2+1 black hole. In this contribution we give a brief summary of these new black holes and consider them as solutions of a Chern-Simons gravity theory. We also provide a brief introduction to some aspects of higher dimensional Chern-Simons theories. 
  We study axially symmetric static solitons of O(3) nonlinear $\sigma$ model coupled to (2+1)-dimensional anti-de Sitter gravity. The obtained solutions are not self-dual under static metric. The usual regular topological lump solution cannot form a black hole even though the scale of symmetry breaking is increased. There exist nontopological solitons of half integral winding in a given model, and the corresponding spacetimes involve charged Ba$\tilde n$ados-Teitelboim-Zanelli black holes without non-Abelian scalar hair. 
  We study the inner-structure of a charged black-hole which is formed from the gravitational collapse of a self-gravitating charged scalar-field. Starting with a regular spacetime, we follow the evolution through the formation of an apparent horizon, a Cauchy horizon and a final central singularity. We find a null, weak, mass-inflation singularity along the Cauchy horizon, which is a precursor of a strong, spacelike singularity along the $r=0$ hypersurface. 
  We summarize the state of the art of the ``close approximation'' to black hole collisions. We discuss results to first and second order in perturbation theory for head-on collisions of momentarily-stationary and non-stationary black holes and discuss the near-future prospect of non-axisymmetric collisions. 
  The effect of particle creation by nonstationary external fields is considered as a radiation effect in the expectation-value spacetime. The energy of created massless particles is calculated as the vacuum contribution in the energy-momentum tensor of the expectation value of the metric. The calculation is carried out for an arbitrary quantum field coupled to all external fields entering the general second-order equation. The result is obtained as a functional of the external fields. The paper gives a systematic derivation of this result on the basis of the nonlocal effective action. Although the derivation is quite involved and touches on many aspects of the theory, the result itself is remarkably simple. It brings the quantum problem of particle creation to the level of complexity of the classical radiation problem. For external fields like the electromagnetic or gravitational field there appears a quantity, the radiation moment, that governs both the classical radiation of waves and the quantum particle production. The vacuum radiation of an electrically charged source is considered as an example. The research is aimed at the problem of backreaction of the vacuum radiation. 
  A generally relativistic theory of thermodynamics is developed, based on four main physical principles: heat is a local form of energy, therefore described by a thermal energy tensor; conservation of mass, equivalent to conservation of heat, or the local first law; entropy is a local current; and non-destruction of entropy, or the local second law. A fluid is defined by the thermostatic energy tensor being isotropic. The entropy current is related to the other fields by certain equations, including a generalised Gibbs equation for the thermostatic entropy, followed by linear and quadratic terms in the dissipative (thermal minus thermostatic) energy tensor. Then the second law suggests certain equations for the dissipative energy tensor, generalising the Israel- Stewart dissipative relations, which describe heat conduction and viscosity including relativistic effects and relaxation effects. In the thermostatic case, the perfect-fluid model is recovered. In the linear approximation for entropy, the Eckart theory is recovered. In the quadratic approximation for entropy, the theory is similar to that of Israel & Stewart, but involving neither state-space differentials, nor a non-equilibrium Gibbs equation, nor non-material frames. Also, unlike conventional thermodynamics, the thermal energy density is not assumed to be purely thermostatic, though this is derived in the linear approximation. Otherwise, the theory reduces in the non- relativistic limit to the extended thermodynamics of irreversible processes due to Mueller. The dissipative energy density seems to be a new thermodynamical field, but also exists in relativistic kinetic theory of gases. 
  This is a report of the A3 workshop ("Mathematical Studies of Field Equations"), which was held in Poone during the GR15 Conference in December 1997. 
  We analyse in a systematic way the (non-)compact four dimensional Einstein-Weyl spaces equipped with a Bianchi metric. We show that Einstein-Weyl structures with a Class A Bianchi metric have a conformal scalar curvature of constant sign on the manifold. Moreover, we prove that most of them are conformally Einstein or conformally K\"ahler ; in the non-exact Einstein-Weyl case with a Bianchi metric of the type $VII_0, VIII$ or $IX$, we show that the distance may be taken in a diagonal form and we obtain its explicit 4-parameters expression. This extends our previous analysis, limited to the diagonal, K\"ahler Bianchi $IX$ case. 
  It is shown that the only functionals, within a natural class, which are monotonic in time for all solutions of the vacuum Einstein equations admitting a smooth ``piece'' of conformal null infinity Scri, are those depending on the metric only through a specific combination of the Bondi `mass aspect' and other next--to--leading order terms in the metric. Under the extra condition of passive BMS invariance, the unique such functional (up to a multiplicative factor) is the Trautman--Bondi energy. It is also shown that this energy remains well-defined for a wide class of `polyhomogeneous' metrics. 
  This summary was prepared for the proceedings of the GR15 conference in Pune, India in Dec. 1997. 
  We consider a (non--Riemannian) metric--affine gravity theory, in particular its nonmetricity--torsion sector ``isomorphic'' to the Einstein--Maxwell theory. We map certain Einstein--Maxwell electrovacuum solutions to it, namely the Pleba\'nski--Demia\'nski class of Petrov type D metrics. 
  The interplay of gravitation and the quantum-mechanical principle of linear superposition induces a new set of neutrino oscillation phases. These ensure that the flavor-oscillation clocks, inherent in the phenomenon of neutrino oscillations, redshift precisely as required by Einstein's theory of gravitation. The physical observability of these phases in the context of the solar neutrino anomaly, type-II supernovae, and certain atomic systems is briefly discussed. 
  A model of the universe as a very large white hole provides a useful alternative inhomogeneous theory to pit against the homogeneous standard FLRW big bang models. The white hole would have to be sufficiently large that we can fit comfortably inside the event horizon at the present time, so that the inhomogeneities of space-time are not in contradiction with current observational limits. A specific Lemaitre-Tolman model of a spherically symmetric non-rotating white hole with a few adjustable parameters is investigated. Comparison of calculated anisotropy in the Hubble flow and the CMB against observational limits constrain the parameter space. A Copernican principle would require that we are not too near the centre of the white hole. As an additional constraint this predicts a value of Omega between 0.9999 and 1. 
  We study the apparent horizon for two boosted black holes in the asymptotically de Sitter space-time by solving the initial data on a space with punctures. We show that the apparent horizon enclosing both black holes is not formed if the conserved mass of the system (Abbott-Deser mass) is larger than a critical mass. The black hole with too large AD mass therefore cannot be formed in the asymptotically de Sitter space-time even though each black hole has any inward momentum. We also discuss the dynamical meaning of AD mass by examining the electric part of the Weyl tensor (the tidal force) for various initial data. 
  We present a coordinate system for a general impulsive gravitational pp - wave in vacuum in which the metric is explicitly continuous, synchronous and "transverse". Also, it is more appropriate for investigation of particle motions. 
  The inertial and gravitational properties of intrinsic spin are discussed and some of the recent work in this area is briefly reviewed. The extension of relativistic wave equations to accelerated systems and gravitational fields is critically examined. A nonlocal theory of accelerated observers is presented and its predictions are compared with observation. 
  We show that Vassiliev invariants of knots, appropriately generalized to the spin network context, are loop differentiable in spite of being diffeomorphism invariant. This opens the possibility of defining rigorously the constraints of quantum gravity as geometrical operators acting on the space of Vassiliev invariants of spin nets. We show how to explicitly realize the diffeomorphism constraint on this space and present proposals for the construction of Hamiltonian constraints. 
  This paper consolidates noscalar hair theorem for a charged spherically symmetric black hole in four dimension in general relativity as well as in all scalar tensor theories, both minimally and nonminimally coupled, when the effective Newtonian constant of gravity is positive. However, there is an exception when the matter field itself is coupled to the scalar field, such as in dilaton gravity. 
  We test the chronology protection conjecture in classical general relativity by investigating finitely vicious space-times. First we present singularity theorems in finitely vicious space-times by imposing some restrictions on the chronology violating sets. In the theorems we can refer to the location of an occurring singularity and do not assume any asymptotic conditions such as the existence of null infinities. Further introducing the concept of a non-naked singularity, we show that a restricted class of chronology violations cannot arise if all occurring singularities are the non-naked singularities. Our results suggest that the causal feature of the occurring singularities is the key to prevent the appearance of causality violation. 
  We present an autonomous phase-plane describing the evolution of Friedmann-Robertson-Walker models containing a perfect fluid (with barotropic index gamma) in Brans-Dicke gravity (with Brans-Dicke parameter omega). We find self-similar fixed points corresponding to Nariai's power-law solutions for spatially flat models and curvature-scaling solutions for curved models. At infinite values of the phase-plane variables we recover O'Hanlon and Tupper's vacuum solutions for spatially flat models and the Milne universe for negative spatial curvature. We find conditions for the existence and stability of these critical points and describe the qualitative evolution in all regions of the (omega,gamma) parameter space for 0<gamma<2 and omega>-3/2. We show that the condition for inflation in Brans-Dicke gravity is always stronger than the general relativistic condition, gamma<2/3. 
  The quantum black hole model with a self-gravitating spherically symmetric thin dust shell as a source is considered. The shell Hamiltonian constraint is written and the corresponding Schroedinger equation is obtained. This equation appeared to be a finite differences equation. Its solutions are required to be analytic functions on the relevant Riemannian surface. The method of finding discrete spectra is suggested based on the analytic properties of the solutions. The large black hole approximation is considered and the discrete spectra for bound states of quantum black holes and wormholes are found. They depend on two quantum numbers and are, in fact, quasi-continuous. The quantum black hole bound state depends not only on mass but also on an additional quantum number, and black holes with the same mass have different quantum hairs. These hairs exhibit themselves at the Planckian distances near the black hole horizon. For the observer who can not measure the distances smaller than the Planckian length the black hole has the only parameter, its mass. The other, non-measurable, parameter leads to the quantum corrections to the black hole entropy. The quantum states with given mass are the mixed ones. It is shown that there exists the ground quantum black hole state with minimal mass equal approximately the Planckian mass. Its quantum state has zero entropy and it is a pure state. The existence of the quantum hairs may solve (at least partially) the well known information paradox in the black hole physics. 
  The problem of simplifying tensor expressions is addressed in two parts. The first part presents an algorithm designed to put tensor expressions into a canonical form, taking into account the symmetries with respect to index permutations and the renaming of dummy indices. The tensor indices are split into classes and a natural place for them is defined. The canonical form is the closest configuration to the natural configuration. In the second part, the Groebner basis method is used to simplify tensor expressions which obey the linear identities that come from cyclic symmetries (or more general tensor identities, including non-linear identities). The algorithm is suitable for implementation in general purpose computer algebra systems. Some timings of an experimental implementation over the Riemann package are shown. 
  I review the present theoretical attempts to understand the quantum properties of spacetime. In particular, I illustrate the main achievements and the main difficulties in: string theory, loop quantum gravity, discrete quantum gravity (Regge calculus, dynamical triangulations and simplicial models), Euclidean quantum gravity, perturbative quantum gravity, quantum field theory on curved spacetime, noncommutative geometry, null surfaces, topological quantum field theories and spin foam models. I also briefly review several recent advances in understanding black hole entropy and attempt a critical discussion of our present understanding of quantum spacetime. 
  The Wheeler-DeWitt equation is solved for some scalar-tensor theories of gravitation in the case of homogeneous and isotropic cosmological models.We present general solutions corresponding to cosmological term: (i)\lambda(\phi)=0$ and $(ii) \lambda(\phi)=q\phi$. 
  Clues as to the geometry of the universe are encoded in the cosmic background radiation. Hot and cold spots in the primordial radiation may be randomly distributed in an infinite universe while in a universe with compact topology distinctive patterns can be generated. With improved vision, we could actually see if the universe is wrapped into a hexagonal prism or a hyperbolic horn. We discuss the search for such geometric patterns in predictive maps of the microwave sky. 
  We investigate whether self-maintained vacuum traversible wormhole can exist described by stationary but nonstatic metric. We consider metric being the sum of static spherically symmetric one and a small nondiagonal component which describes rotation sufficiently slow to be taken into account in the linear approximation. We study semiclassical Einstein equations for this metric with vacuum expectation value of stress-energy of physical fields as the source. In suggestion that the static traversible wormhole solution exists we reveal possible azimuthal angle dependence of angular velocity of the rotation (angular velocity of the local inertial frame) that solves semiclassical Einstein equations. We find that in the macroscopic (in the Plank scale) wormhole case a rotational solution exists but only such that, first, angular velocity depends on radial coordinate only and, second, the wormhole connects the two asymptotically flat spacetimes rotating with angular velocities different in asymptotic regions. 
  We discuss the possible relevance of gravitational-wave (GW) experiments for physics at very high energy. We examine whether, from the experience gained with the computations of various specific relic GW backgrounds, we can extract statements and order of magnitude estimates that are as much as possible model-independent, and we try to distinguish between general conclusions and results related to specific cosmological mechanisms. We examine the statement that the Virgo/LIGO experiments probe the Universe at temperatures $T\sim 10^{7}-10^{10}$ GeV (or timescales $t\sim 10^{-20}-10^{-26}$ sec) and we consider the possibility that they could actually probe the Universe at much higher energy scales, including the typical scales of grand unification, string theory and quantum gravity. We consider possible scenarios, depending on how the inflationary paradigm is implemented. We discuss the prospects for detection with present and planned experiments. In particular, a second Virgo interferometer correlated with the planned one, and located within a few tens of kilometers from the first, could reach an interesting sensitivity for stochastic GWs of cosmological origin. 
  A generalisation of the asymptotic wormhole boundary condition for the case of spacetimes with a cosmological horizon is proposed. In particular, we consider de Sitter spacetime with small cosmological constant. The wave functions selected by this proposal are exponentially damped in WKB approximation when the scale factor is large but still much smaller than the horizon size. In addition, they only include outgoing gravitational modes in the region beyond the horizon. We argue that these wave functions represent quantum wormholes and compute the local effective interactions induced by them in low-energy field theory. These effective interactions differ from those for flat spacetime in terms that explicitly depend on the cosmological constant. 
  A recent investigation shows that a local gauge string with a phenomenological energy momentum tensor, as prescribed by Vilenkin, is inconsistent in Brans-Dicke theory. In this work, it has been shown that such a string is indeed consistent if one introduces time dependences in the metric. A set of solutions of full nonlinear Einstein's equations for the interior region of such a string are presented. 
  Binary systems comprising at least one neutron star contain strong gravitational field regions and thereby provide a testing ground for strong-field gravity. Two types of data can be used to test the law of gravity in compact binaries: binary pulsar observations, or forthcoming gravitational-wave observations of inspiralling binaries. We compare the probing power of these two types of observations within a generic two-parameter family of tensor-scalar gravitational theories. Our analysis generalizes previous work (by us) on binary-pulsar tests by using a sample of realistic equations of state for nuclear matter (instead of a polytrope), and goes beyond a previous study (by C.M. Will) of gravitational-wave tests by considering more general tensor-scalar theories than the one-parameter Jordan-Fierz-Brans-Dicke one. Finite-size effects in tensor-scalar gravity are also discussed. 
  We develop further our extension of the Ellis-Bruni covariant and gauge-invariant formalism to the general relativistic treatment of density perturbations in the presence of cosmological magnetic fields. We present detailed analysis of the kinematical and dynamical behaviour of perturbed magnetized FRW cosmologies containing fluid with non-zero pressure. We study the magnetohydrodynamical effects on the growth of density irregularities during the radiation era. Solutions are found for the evolution of density inhomogeneities on small and large scales in the presence of pressure, and some new physical effects are identified. 
  Inertial motion superradiance, the emission of radiation by an initially unexcited system moving inertially but superluminally through a medium, has long been known. Rotational superradiance, the amplification of radiation by a rotating rigid object, was recognized much later, principally in connection with black hole radiances. Here we review the principles of inertial motion superradiance and prove thermodynamically that the Ginzburg--Frank condition for superradiance coincides with the condition for superradiant amplification of already existing radiation. Examples we cite include a new type of black hole superradiance. We correct Zel'dovich's thermodynamic derivation of the Zel'dovich--Misner condition for rotational superradiance by including the radiant entropy in the bookkeeping . We work out in full detail the electrodynamics of a Zel'dovich rotating cylinder, including a general electrodynamic proof of the Zel'dovich--Misner condition, and explicit calculations of the superradiant gain for both types of polarization. Contrary to Zel'dovich's pessimistic conclusion we conclude that, if the cylinder is surrounded by a dielectric jacket and the whole assembly is placed inside a rotating cavity, the superradiance is measurable in the laboratory. 
  The completeness of the quasinormal modes of the wave equation with Poeschl-Teller potential is investigated. A main result is that after a large enough time $t_0$, the solutions of this equation corresponding to $C^{\infty}$-data with compact support can be expanded uniformly in time with respect to the quasinormal modes, thereby leading to absolutely convergent series. Explicit estimates for $t_0$ depending on both the support of the data and the point of observation are given. For the particular case of an ``early'' time and zero distance between the support of the data and observational point, it is shown that the corresponding series is not absolutely convergent, and hence that there is no associated sum which is independent of the order of summation. 
  The effects of asymptotically anti-de Sitter wormholes in low-energy field theory are calculated in full detail for three different matter contents: a conformal scalar field, an electromagnetic field and gravitons. There exists a close relation between the choice of vacuum for the matter fields and the selection of a basis of the Hilbert space of anti-de Sitter wormholes. In the presence of conformal matter (i.e., conformal scalar or electromagnetic fields), this relation allows us to interpret the elements of these bases as wormhole states containing a given number of particles. This interpretation is subject to the same kind of ambiguity in the definition of particle as that arising from quantum field theory in curved spacetime. In the case of gravitons, owing to the non-conformal coupling, it is not possible to describe wormhole states in terms of their particle content. 
  Starting from the assumption that vacuum states in de Sitter space look for any geodesic observer like equilibrium states with some a priori arbitrary temperature, an analysis of their global properties is carried out in the algebraic framework of local quantum physics. It is shown that these states have the Reeh-Schlieder property and that any primary vacuum state is also pure and weakly mixing. Moreover, the geodesic temperature of vacuum states has to be equal to the Gibbons-Hawking temperature and this fact is closely related to the existence of a discrete PCT-like symmetry. It is also shown that the global algebras of observables in vacuum sectors have the same structure as their counterparts in Minkowski space theories. 
  We find a class of electrically charged exact solutions for a toy model of metric-affine gravity. Their metric is of the Pleba\'nski-Demia\'nski type and their nonmetricity and torsion are represented by a triplet of covectors with dilation, shear, and spin charges. 
  A large family of solutions, representing, in general, spherically symmetric Type II fluid, is presented, which includes most of the known solutions to the Einstein field equations, such as, the monopole-de Sitter-charged Vaidya ones. 
  The stability of the cosmological event horizons (CEHs) of a class of non-static global cosmic strings is studied against perturbations of gravitational waves and massless scalar field. It is found that the perturbations of gravitational waves always turn the CEHs into non-scalar weak spacetime curvature singularities, while the ones of massless scalar field turn the CEHs either into non-scalar weak singularities or into scalar ones depending on the particular cases considered. The perturbations of test massless scalar field is also studied, and it is found that they do not always give the correct prediction. 
  We extend the Vaidya radiating metric to include both a radiation field and a string fluid. Assuming diffusive transport for the string fluid, we find new analytic solutions of Einstein's field equations. Our new solutions represent an extention of Xanthopoulos superposition. 
  This paper is a sequel to the series of papers [gr-qc/9409010, gr-qc/9505034, gr-qc/9603022, gr-qc/9609035, gr-qc/9609046, gr-qc/9704033, gr-qc/9704038, gr-qc/9708014, gr-qc/9802016, gr-qc/9802022]. We define a quantum measurement as a sequence of binary quantum jumps caused by a macroscopic apparatus. A dynamical theory of measurement is developed, the role of gravity and cosmology being emphasized. 
  It is shown that, for spherically symmetric static backgrounds, a simple reduced Dirac equation can be obtained by using the Cartesian tetrad gauge in Cartesian holonomic coordinates. This equation is manifestly covariant under rotations so that the spherical coordinates can be separated in terms of angular spinors like in special relativity, obtaining a pair of radial equations and a specific form of the radial scalar product. As an example, we analytically solve the anti-de Sitter oscillator giving the formula of the energy levels and the form of the corresponding eigenspinors. 
  We study the Husain-Kuchar model by introducing a new action principle similar to the self-dual action used in the Ashtekar variables approach to Quantum Gravity. This new action has several interesting features; among them, the presence of a scalar time variable that allows the definition of geometric observables without adding new degrees of freedom, the appearance of a natural non-degenerate four-metric and the possibility of coupling ordinary matter. 
  We study the interaction of a massless quantized spinor field with the gravitational filed of N parallel static cosmic strings by using a perturbative approach. We show that the presence of more than one cosmic string gives rise to an additional contribution to the energy density of vacuum fluctuations, thereby leading to a vacuum force attraction between two parallel cosmic strings. 
  We analyze the Hamilton-Jacobi action of gravity and matter in the limit where gravity is treated at the background field approximation. The motivation is to clarify when and how the solutions of the Wheeler-DeWitt equation lead to the Schr\"odinger equation in a given background. To this end, we determine when and how the total action, solution of the constraint equations of General Relativity, leads to the HJ action for matter in a given background. This is achieved by comparing two neighboring solutions differing slightly in their matter energy content. To first order in the change of the 3-geometries, the change of the gravitational action equals the integral of the matter energy evaluated in the background geometry. Higher order terms are governed by the ``susceptibility'' of the geometry. These classical properties also apply to quantum cosmology since the conditions which legitimize the use of WKB gravitational waves are concomitant with those governing the validity of the background field approximation. 
  The dynamics of a preinflacionary phase of the universe, and its exit to inflation, is discussed. This phase is modeled by a closed Friedmann-Robertson-Walker geometry, the matter content of which is radiation plus a scalar field minimally coupled to the gravitational field. The simple configuration, with two effective degres of freedom only, presents a very complicated dynamics connected to the existence of critical points of saddle-center type and saddle type in phase space of the system. Each of these critical points is associated to an extremum of the scalar field potential. The Topology of the phase space about the saddle-center is characterized by homoclinic cylinders emanating from unstable periodic orbits, and the transversal crossing of the cylinders, due to the non-integrability of the system, results in a chaotic dynamics. The topology of the homoclinic cylinders provides an invariant characterization of chaos. The model exhibits one or more exits to inflation, associated to one or more strong asymptotic de Sitter attractors present in phase space, but the way out from the initial singularity into any of the inflationary exits is chaotic. We discuss possible mechanisms, connected to the spectrum of inhomogeneous fluctuations in the models, which would allow us to distinguish physically the several exits to inflation. 
  In quantum cosmology, one often considers tunneling phenomena which may have occurred in the early universe. Processes requiring quantum penetration of a potential barrier include black hole pair creation and the decay of vacuum domain walls. Ideally, one calculates the rates for such processes by finding an instanton, or Euclidean solution of the field equations, which interpolates between the initial and final states. In practice, however, it has become customary to calculate such amplitudes using the No-Boundary Proposal of Hartle and Hawking. A criticism of this method is that it does not use a single path which interpolates between the initial and final states, but two disjoint instantons: One divides the probability to create the final state from nothing by the probability to create the initial state from nothing and decrees the answer to be the rate of tunneling from the initial to the final state. Here, we demonstrate the validity of this approach by constructing continuous paths connecting the ingoing and outgoing data, which may be viewed as perturbations of the set of disconnected instantons. They are off-shell, but will still dominate the path integral as they have action arbitrarily close to the no-boundary action. In this picture, a virtual domain wall, or wormhole, is created and annihilated in such a way as to interface between the disjoint instantons. Decay rates calculated using our construction differ from decay rates calculated using the No-Boundary Proposal only in the prefactor; the exponent, which usually dominates the result, remains unchanged. 
  We show that homogeneous G\"odel spacetimes need not contain closed timelike curves in low-energy-effective string theories. We find exact solutions for the G\"odel metric in string theory for the full $O(\alpha ^{\prime})$ action including both dilaton and axion fields. The results are valid for bosonic, heterotic and super-strings. To first order in the inverse string tension $\alpha ^{\prime}$, these solutions display a simple relation between the angular velocity of the G\"odel universe, $\Omega ,$ and the inverse string tension of the form $\alpha ^{\prime}=1/\Omega ^2$ in the absence of the axion field. The generalization of this relationship is also found when the axion field is present. 
  This review gives an introduction to various attempts to understand the quantum nature of black holes. The first part focuses on thermodynamics of black holes, Hawking radiation, and the interpretation of entropy. The second part is devoted to the detailed treatment of black holes within canonical quantum gravity. The last part adds a brief discussion of black holes in string theory and quantum cosmology. 
  Hawking and Turok have recently published a solution to the WKB "wave-function for the universe" which they claim leads in a natural way to an open universe as the end point of the evolution for a universe dominated by a scalar field. They furthermore argue that their solution a preferred solution under the rules of the game. This paper will, I hope, clarify their solution and the limits of validity of their argument. 
  This is a summary of the workshop A.6 on Alternative Theories of Gravity, prepared for the proceedings for the GR15 conference. 
  Singular spacetimes are a natural prediction of Einstein's theory. Most memorable are the singular centers of black holes and the big bang. However, dilatonic extensions of Einstein's theory can support nonsingular spacetimes. The cosmological singularities can be avoided by dilaton driven inflation. Furthermore, a nonsingular black hole can be constructed in two dimensions. 
  We show that gravitational radiation drives an instability in hot young rapidly rotating neutron stars. This instability occurs primarily in the l=2 r-mode and will carry away most of the angular momentum of a rapidly rotating star by gravitational radiation. On the timescale needed to cool a young neutron star to about T=10^9 K (about one year) this instability can reduce the rotation rate of a rapidly rotating star to about 0.076\Omega_K, where \Omega_K is the Keplerian angular velocity where mass shedding occurs. In older colder neutron stars this instability is suppressed by viscous effects, allowing older stars to be spun up by accretion to larger angular velocities. 
  A search for a problem free cosmology within the framework of an effective non - minimally coupled scalar tensor theory is suggested. With appropriate choice of couplings in variants of a Lee - Wick model [as also in a model supporting Q - ball solutions], non topological solutions [NTS's], varying in size upto 10 kpc to 1 Mpc can exist. We explore the properties of a ``toy'' Milne model containing a distribution of NTS domains. The interior of these domains would be regions where effective gravitational effects would be indistinguishable from those expected in standard Einstein theory. For a large class of non - minimal coupling terms and the scalar effective potential, the effective cosmological constant identically vanishes. The model passes classical cosmological tests and we describe reasons to expect it to fare well as regards nucleosynthesis and structure formation. 
  We present two statistical tests for periodicities in the time series. We apply the two tests to the data taken from Glasgow prototype interferometer in March 1996. We find that the data contain several very narrow spectral features. We investigate whether these features can be confused with gravitational wave signals from pulsars. 
  The quantum theory of gravity is considered based on the assumption that gravitational interaction occurs by means of the vector field of the Planck mass. Gravitational emission is considered as a process of the decay of proton into some matter fields at the Planck scale. Within the framework of grand unification vector field of the Planck mass may be thought of as those which realize the interaction between leptons and quarks. 
  Astronomers have discovered many potential black holes in X-ray binaries and galactic nuclei. These black holes are usually identified by the fact that they are too massive to be neutron stars. Until recently, however, there was no convincing evidence that the objects identified as black hole candidates actually have event horizons. This has changed with extensive applications of a class of accretion models for describing the flow of gas onto compact objects; for these solutions, called advection-dominated accretion flows (ADAFs), the black hole nature of the accreting star, specifically its event horizon, plays an important role. We review the evidence that, at low luminosities, accreting black holes in both X-ray binaries and galactic nuclei contain ADAFs rather than the standard thin accretion disk. 
  We solve the problem of expressing the Weyl scalars $\psi $ that describe gravitational perturbations of a Kerr black hole in terms of Cauchy data. To do so we use geometrical identities (like the Gauss-Codazzi relations) as well as Einstein equations. We are able to explicitly express $\psi $ and $\partial _t\psi $ as functions only of the extrinsic curvature and the three-metric (and geometrical objects built out of it) of a generic spacelike slice of the spacetime. These results provide the link between initial data and $\psi $ to be evolved by the Teukolsky equation, and can be used to compute the gravitational radiation generated by two orbiting black holes in the close limit approximation. They can also be used to extract waveforms from spacetimes completely generated by numerical methods. 
  We study the singularity created in the supercritical collapse of a spherical massless scalar field. We first model the geometry and the scalar field to be homogeneous, and find a generic solution (in terms of a formal series expansion) describing a spacelike singularity which is monotonic, scalar polynomial and strong. We confront the predictions of this analytical model with the pointwise behavior of fully-nonlinear and inhomogeneous numerical simulations, and find full compliance. We also study the phenomenology of the spatial structure of the singularity numerically. At asymptotically late advanced time the singularity approaches the Schwarzschild singularity, in addition to discrete points at finite advanced times, where the singularity is Schwarzschild-like. At other points the singularity is different from Schwarzschild due to the nonlinear scalar field. 
  Mishra has recently established, using a generic static metric, the relative local proper-time 3-acceleration of a test-particle in one-dimensional free fall relative to a static reference frame in any static spacetime. In this paper, on the grounds of gravitoelectromagnetism we establish, in a covariant spacetime form, the relative 4-acceleration for the general free fall, indicating its canonical representation with its 3-space cinematical content. Then we obtain the relation between this representation and the very known expression for the relative free fall acceleration in Fermi coordinates. Taking this into account, it is shown that an experiment with relativistic beams in a circular accelerator, modelled by Fermi coordinates, recently proposed by Moliner et al, can test the here established covariant result and, therefore, can also verify Mishra's formula. This possibility of experimental verification, besides its intrinsic importance, can answer a recent inquire by Vigier, related to his recent proposal of derivation of inertial forces. 
  We examine quantum properties of topological black holes which are asymptotically anti--de Sitter. First, massless scalar fields and Weyl spinors which propagate in the background of an anti--de Sitter black hole are considered in an exactly soluble two--dimensional toy model. The Boulware--, Unruh--, and Hartle--Hawking vacua are defined. The latter results to coincide with the Unruh vacuum due to the boundary conditions necessary in asymptotically adS spacetimes. We show that the Hartle--Hawking vacuum represents a thermal equilibrium state with the temperature found in the Euclidean formulation. The renormalized stress tensor for this quantum state is well--defined everywhere, for any genus and for all solutions which do not have an inner Cauchy horizon, whereas in this last case it diverges on the inner horizon. The four--dimensional case is finally considered, the equilibrium states are discussed and a luminosity formula for the black hole of any genus is obtained. Since spacelike infinity in anti--de Sitter space acts like a mirror, it is pointed out how this would imply information loss in gravitational collapse. The black hole's mass spectrum according to Bekenstein's view is discussed and compared to that provided by string theory. 
  It is shown that in 2+1 dimensional gravity an open spacetime with timelike sources and total energy momentum cannot have a stable compactly generated Cauchy horizon. This constitutes a proof of a version of Kabat's conjecture and shows, in particular, that not only a Gott pair cannot be formed from processes such as the decay of a single cosmic string as has been shown by Carroll et al., but that, in a precise sense, a time machine cannot be constructed at all. 
  We propose a method for calculating vacuum fluctuations on the background of a spherical impulsive gravitational wave which results in a finite expression for the vacuum expectation value of the stress-energy tensor. The method is based on first including a cosmological constant as an auxiliary constant. We show that the result for the vacuum expectation value of the stress-energy tensor in second-order perturbation theory is finite if both the cosmological constant and the infrared parameter tend to zero at the same rate. 
  Whenever real particle production occurs in quantum field theory, the imaginary part of the Hadamard Elementary function $G^{(1)}$ is non-vanishing. A method is presented whereby the imaginary part of $G^{(1)}$ may be calculated for a charged scalar field in a static spherically symmetric spacetime with arbitrary curvature coupling and a classical electromagnetic field $A^{\mu}$. The calculations are performed in Euclidean space where the Hadamard Elementary function and the Euclidean Green function are related by $(1/2)G^{(1)}=G_{E}$. This method uses a $4^{th}$ order WKB approximation for the Euclideanized mode functions for the quantum field. The mode sums and integrals that appear in the vacuum expectation values may be evaluated analytically by taking the large mass limit of the quantum field. This results in an asymptotic expansion for $G^{(1)}$ in inverse powers of the mass $m$ of the quantum field. Renormalization is achieved by subtracting off the terms in the expansion proportional to nonnegative powers of $m$, leaving a finite remainder known as the ``DeWitt-Schwinger approximation.'' The DeWitt-Schwinger approximation for $G^{(1)}$ presented here has terms proportional to both $m^{-1}$ and $m^{-2}$. The term proportional to $m^{-2}$ will be shown to be identical to the expression obtained from the $m^{-2}$ term in the generalized DeWitt-Schwinger point-splitting expansion for $G^{(1)}$. The new information obtained with the present method is the DeWitt-Schwinger approximation for the imaginary part of $G^{(1)}$, which is proportional to $m^{-1}$ in the DeWitt-Schwinger approximation for $G^{(1)}$ derived in this paper. 
  I review some recent progress in String/M-theory 
  Smolin has pointed out that the spin network formulation of quantum gravity will not necessarily possess the long range correlations needed for a proper classical limit; typically, the action of the scalar constraint is too local. Thiemann's length operator is used to argue for a further restriction on the action of the scalar constraint: it should not introduce new edges of color unity into a spin network, but should rather change preexisting edges by $\pm$ one unit of color. Smolin has proposed a specific ansatz for a correlated scalar constraint. This ansatz does not introduce color unity edges, but the [scalar, scalar] commutator is shown to be anomalous. In general, it will be hard to avoid anomalies, once correlation is introduced into the constraint; but it is argued that the scalar constraint may not need to be anomaly-free when acting on the kinematic basis. 
  We consider the low-energy effective string action in four dimensions including the leading order-$\alpha'$ terms. An exact homogeneous solution is obtained. It represents a non-singular expanding cosmological model in which the tensor fields tend to vanish as $t\to \infty$. The scale factor $a(t)$ of the very early universe in this model has the time dependence $a(t)^2=a_0^2+t^2$. The violation of the strong energy condition of classical General Relativity to avoid the initial singularity requires that the central charge deficit of the theory be larger than a certain value. The significance of this solution is discussed. 
  We propose two new classes of instantons which describe the tunneling and/or quantum creation of closed and open universes. The instantons leading to an open universe can be considered as generalizations of the Coleman-De-Luccia solution. They are non-singular, unlike the instantons recently studied by Hawking and Turok, whose prescription has the problem that the singularity is located on the hypersurface connecting to the Lorentzian region, which makes it difficult to remove. We argue that such singularities are harmless if they are located purely in the Euclidean region. We thus obtain new singular instantons leading to a closed universe; unlike the usual regular instantons used for this purpose, they do not require complex initial conditions. The singularity gives a boundary contribution to the action which is small for the instantons leading to sufficient inflation, but changes the sign of the action for small $\phi$ corresponding to short periods of inflation. 
  We study new separable orthogonally transitive abelian G_2 on S_2 models with two mutually orthogonal integrable Killing vector fields. For this purpose we consider separability of the metric functions in a coordinate system in which the velocity vector field of the perfect fluid does not take its canonical form, providing thereby solutions which are non-separable in comoving coordinates in general. Some interesting general features concerning this class of solutions are given. We provide a full classification for these models and present several families of explicit solutions with their properties. 
  Pre--big bang models of inflation based on string cosmology produce a stochastic gravitational wave background whose spectrum grows with decreasing wavelength, and which may be detectable using interferometers such as LIGO. We point out that the gravitational wave spectrum is closely tied to the density perturbation spectrum, and that the condition for producing observable gravitational waves is very similar to that for producing an observable density of primordial black holes. Detection of both would provide strong support to the string cosmology scenario. 
  New solutions for static non-rotating thin disks of finite radius with nonzero radial stress are studied. A method to introduce either radial pressure or radial tension is presented. The method is based on the use of conformal transformations. 
  Algebraic computing in relativity and gravitation dates back more than thirty years, but only relatively recently has hardware of sufficient power to tackle large scale calculations become commonplace. Whereas it is generally understood throughout the relativity community that there are a number of packages available, the diversity of problems that the available packages can help with is not so widely appreciated. This session was devoted to computer algebra for relativity and gravitation from the point of view of developers. In this summary I expand this to include some background and outline what is available for users in the field. 
  In the hyperbolic slicing of de Sitter space appropriate for open universe models, a curvature scale is present and supercurvature fluctuations are possible. In some cases, the expansion of a scalar field in the Bunch-Davies vacuum includes supercurvature modes, as shown by Sasaki, Tanaka and Yamamoto. We express the normalizable vacuum supercurvature modes for a massless scalar field in terms of the basis modes for the spatially-flat slicing of de Sitter space. 
  We show that, apart from the usual area operator of non-perturbative quantum gravity, there exists another, closely related, operator that measures areas of surfaces. Both corresponding classical expressions yield the area. Quantum mechanically, however, the spectra of the two operators are different, coinciding only in the limit when the spins labelling the state are large. We argue that both operators are legitimate quantum operators, and which one to use depends on the context of a physical problem of interest. Thus, for example, we argue that it is the operator proposed here that is relevant in the black hole context to measure the area of black hole horizon. We show that the difference between the two operators is due to non-commutativity that is known to arise in the quantum theory. We give a heuristic picture explaining the difference between the two area spectra in terms of quantum fluctuations of the surface whose area is being measured. 
  In recent years it has become apparent that intriguing phenomenology exists at the threshold of black hole formation in a large class of general relativistic collapse models. This phenomenology, which includes scaling, self-similarity and universality, is largely analogous to statistical mechanical critical behaviour, a fact which was first noted empirically, and subsequently clarified by perturbative calculations which borrow on ideas and techniques from dynamical systems theory and renormalization group theory. This contribution, which closely parallels my talk at the conference, consists of an overview of the considerable ``zoo''' of critical solutions which have been discovered thus far, along with a brief discussion of how we currently understand the nature of these solutions from the point of view of perturbation theory. 
  In this paper we show how the nonlocal effective action for gravity, obtained after integrating out the matter fields, can be used to compute particle production and spectra for different space-time metrics. Applying this technique to several examples, we find that the perturbative calculation of the effective action up to second order in curvatures yields exactly the same results for the total number of particles as the Bogolyubov transformations method, in the case of masless scalar fields propagating in a Robertson-Walker space-time. Using an adiabatic approximation we also obtain the corresponding spectra and compare the results with the traditional WKB approximation. 
  We present a first-order symmetric hyperbolic system in the Ashtekar formulation of general relativity for vacuum spacetime. We add terms from constraint equations to the evolution equations with appropriate combinations, which is the same technique used by Iriondo, Leguizam\'on and Reula [Phys. Rev. Lett. 79, 4732 (1997)]. However our system is different from theirs in the points that we primarily use Hermiticity of a characteristic matrix of the system to characterize our system "symmetric", discuss the consistency of this system with reality condition, and show the characteristic speeds of the system. 
  The asymptotic behaviour of a family of inhomogeneous scalar field cosmologies with exponential potential is studied. By introducing new variables we can perform an almost complete analysis of the evolution of these cosmologies. Unlike the homogeneous case (Bianchi type solutions), when k^2<2 the models do not isotropize due to the presence of the inhomogeneities 
  The evolution of a class of inhomogeneous spherically symmetric universe models possessing a varying cosmological term and a material fluid, with an adiabatic index either constant or not, is studied. 
  Quasi-spherical light cones are lightlike hypersurfaces of the Kerr geometry that are asymptotic to Minkowski light cones at infinity. We develop the equations of these surfaces and examine their properties. In particular, we show that they are free of caustics for all positive values of the Kerr radial coordinate r. Useful applications include the propagation of high-frequency waves, the definition of Kruskal-like coordinates for a spinning black hole and the characteristic initial-value problem. 
  We analyze the effects of thermal conduction in a relativistic fluid just after its departure from spherical symmetry, on a time scale of the order of relaxation time. Using first order perturbation theory, it is shown that, as in spherical systems, at a critical point the effective inertial mass density of a fluid element vanishes and becomes negative beyond that point. The impact of this effect on the reliability of causality conditions is discussed. 
  Neutron stars in binary orbit emit gravitational waves and spiral slowly together. During this inspiral, they are expected to have very little vorticity. It is in fact a good approximation to treat the system as having zero vorticity, i.e., as irrotational. Because the orbital period is much shorter than the radiation reaction time scale, it is also an excellent approximation to treat the system as evolving through a sequence of equilibrium states, in each of which the gravitational radiation is neglected. In Newtonian gravity, one can simplify the hydrodynamic equations considerably for an equilibrium irrotational binary by introducing a velocity potential. The equations reduce to a Poisson-like equation for the potential, and a Bernoulli-type integral for the density. We show that a similar simplification can be carried out in general relativity. The resulting equations are much easier to solve than other formulations of the problem. 
  The distinct models that describe spin 1 and 2 massive excitations in 2+1 dimensions are analized, showing their equivalence (between models of same spin) and analogies (between models of different spin). Topics as spontaneous symmetry breaking and anyonic behaviour, in these models, are analized. A massive gravity model is introduced. This model is diffeomophism invariant, but it is not Lorentz invariant. 
  We develop the general theory of stars in Saa's model of gravity with propagating torsion and study the basic stationary state of neutron star. Our numerical results show that the torsion force decreases the role of the gravity in the star configuration leading to significant changes in the neutron star masses depending on the equation of state of star matter. The inconsistency of the Saa's model with Roll-Krotkov-Dicke and Braginsky-Panov experiments is discussed. 
  We present relativistic hydrostatic equations for obtaining irrotational binary neutron stars in quasi equilibrium states in 3+1 formalism. Equations derived here are different from those previously given by Bonazzola, Gourgoulhon, and Marck, and have a simpler and more tractable form for computation in numerical relativity. We also present hydrostatic equations for computation of equilibrium irrotational binary stars in first post-Newtonian order. 
  We analyze black hole thermodynamics in a generalized theory of gravity whose Lagrangian is an arbitrary function of the metric, the Ricci tensor and a scalar field. We can convert the theory into the Einstein frame via a "Legendre" transformation or a conformal transformation. We calculate thermodynamical variables both in the original frame and in the Einstein frame, following the Iyer--Wald definition which satisfies the first law of thermodynamics. We show that all thermodynamical variables defined in the original frame are the same as those in the Einstein frame, if the spacetimes in both frames are asymptotically flat, regular and possess event horizons with non-zero temperatures. This result may be useful to study whether the second law is still valid in the generalized theory of gravity. 
  Despite the fact that the Schwarzschild and Kerr solutions for the Einstein equations, when written in standard Schwarzschild and Boyer-Lindquist coordinates, present coordinate singularities, all numerical studies of accretion flows onto collapsed objects have been widely using them over the years. This approach introduces conceptual and practical complications in places where a smooth solution should be guaranteed, i.e., at the gravitational radius. In the present paper, we propose an alternative way of solving the general relativistic hydrodynamic equations in background (fixed) black hole spacetimes. We identify classes of coordinates in which the (possibly rotating) black hole metric is free of coordinate singularities at the horizon, independent of time, and admits a spacelike decomposition. In the spherically symmetric, non-rotating case, we re-derive exact solutions for dust and perfect fluid accretion in Eddington-Finkelstein coordinates, and compare with numerical hydrodynamic integrations. We perform representative axisymmetric computations. These demonstrations suggest that the use of those coordinate systems carries significant improvements over the standard approach, especially for higher dimensional studies. 
  I briefly review some of the recent progress in quantum field theory in curved spacetime and other aspects of semiclassical gravity, as reported at the D3 Workshop at GR15. 
  This is a summary report of the workshop D2 in GR15. In this workshop rather diverging problems related to quantum gravity and quantum cosmology were discussed. Since it is almost impossible to summarize all these discussions in coherent and integrated expressions, in this report, I give a brief description of the content of each paper in order, with some additional comments on their backgrounds. 
  The Connes formula giving the dual description for the distance between points of a Riemannian manifold is extended to the Lorentzian case. It resulted that its validity essentially depends on the global structure of spacetime. The duality principle classifying spacetimes is introduced. The algebraic account of the theory is suggested as a framework for quantization along the lines proposed by Connes. The physical interpretation of the obtained results is discussed. 
  The basic principles of generalization of the group theoretical approach to the relativistic wave equations on curved spaces are examined. The general method of the determination of wave equations from the known symmetry group of a symmetrical curved space is described. The method of obtaining the symmetrical spaces in which invariant wave equations admit the limiting passage to the relativistic wave equations on the flat (not necessarily real) spaces is explained. Starting from the equations for massless particles and from the Dirac equation in the Minkowski space, admissible real symmetrical spaces are founded. The mentioned procedure is carried out also for the complex spaces. As the basic point, the wave equations on the flat complex space are considered. It is shown that the usual Dirac equation written down in the complex variables does not lead to any curved space. By "decomposition" of the mentioned form of the Dirac equation the equation leading to the space ${\Bbb C}{\Bbb P}^{n}$ and being alternative to the usual Dirac equation on ${\Bbb C}^{n}$ is constructed. 
  The group theoretical approach to the relativistic wave equations in the de Sitter and Anti-de Sitter spaces for spin~0 and 1/2 massive particles is considered. The invariant wave equations which determines the appropriate irreducible representations are constructed. The explicit solutions of these equations possessing a simplicity and physical transparency are obtained without use of the separation of variables method. The connection with the general-covariant approach to wave equations in a curved space is established. It is shown that the Anti-de Sitter space is the solution of the Einstein-Dirac equations. The geometrical meaning of the mass quantization in the Anti-de Sitter space and of the particle creation in de Sitter space is shown. 
  We report on the first calculations of fully relativistic binary circular orbits to span a range of separation distances from the innermost stable circular orbit (ISCO), deeply inside the strong field regime, to a distance ($\sim$ 200 km) where the system is accurately described by Newtonian dynamics. We consider a binary system composed of two identical corotating neutron stars, with 1.43 $M_\odot$ gravitational mass each in isolation. Using a conformally flat spatial metric we find solutions to the initial value equations that correspond to semi-stable circular orbits. At large distance, our numerical results agree exceedingly well with the Newtonian limit. We also present a self consistent determination of the ISCO for different stellar masses. 
  We make a detailed study of boson star configurations in Jordan--Brans--Dicke theory, studying both equilibrium properties and stability, and considering boson stars existing at different cosmic epochs. We show that boson stars can be stable at any time of cosmic history and that equilibrium stars are denser in the past. We analyze three different proposed mass functions for boson star systems, and obtain results independently of the definition adopted. We study how the configurations depend on the value of the Jordan--Brans--Dicke coupling constant, and the properties of the stars under extreme values of the gravitational asymptotic constant. This last point allows us to extract conclusions about the stability behaviour concerning the scalar field. Finally, other dynamical variables of interest, like the radius, are also calculated. In this regard, it is shown that the radius corresponding to the maximal boson star mass remains roughly the same during cosmological evolution. 
  Quantum aspects of black holes represent an important testing ground for a theory of quantum gravity. The recent success of string theory in reproducing the Bekenstein-Hawking black hole entropy formula provides a link between general relativity and quantum mechanics via thermodynamics and statistical mechanics. Here we speculate on the existence of new and unexpected links between black holes and polymers and other soft-matter systems. 
  I consider the problem of computing the mass of a charged, gravitating particle in quantum field theory. It is shown how solving for the first quantized propagator of a particle in the presence of its own potentials reproduces the gauge and general coordinate invariant sum over an infinite class of diagrams. The distinguishing feature of this class of diagrams is that all closed loops contain part of the continuous matter line running from early to late times. The next order term would have one closed loop external to the continuous matter line, and so on. I argue that the gravitational potentials in the 0-th order term may permit the formation of bound states, which would then dominate the propagator. It is conceivable that this provides an tractable technique for computing the masses of fundamental particles from first principles. It is also conceivable that the expansion in external loops permits gravity to regulate certain ultraviolet divergences. 
  We suggest that in a recently proposed framework for quantum gravity, where Vassiliev invariants span the the space of states, the latter is dramatically reduced if one has a non-vanishing cosmological constant. This naturally suggests that the initial state of the universe should have been one with $\Lambda=0$. 
  The general form of a stationary, axially symmetric traversable wormhole is discussed. This provides an explicit class of rotating wormholes that generalize the static, spherically symmetric ones first considered by Morris and Thorne. In agreement with general analyses, it is verified that such a wormhole generically violates the null energy condition at the throat. However, for suitable model wormholes, there can be classes of geodesics falling through it which do not encounter any energy-condition-violating matter. The possible presence of an ergoregion surrounding the throat is also noted. 
  The topology of the event horizon (TOEH) is usually believed to be a sphere. Nevertheless, some numerical simulations of gravitational collapse with a toroidal event horizon or the collision of event horizons are reported. Considering the indifferentiability of the event horizon (EH), we see that such non-trivial TOEHs are caused by the set of endpoints (the crease set) of the EH. The two-dimensional (one-dimensional) crease set is related to the toroidal EH (the coalescence of the EH). Furthermore, examining the stability of the structure of the endpoints, it becomes clear that the spherical TOEH is unstable under linear perturbation. On the other hand, a discussion based on catastrophe theory reveals that the TOEH with handles is stable and generic. Also, the relation between the TOEH and the hoop conjecture is discussed. It is shown that the Kastor-Traschen solution is regarded as a good example of the hoop conjecture by the discussion of its TOEH. We further conjecture that a non-trivial TOEH can be smoothed out by rough observation in its mass scale. 
  In the final stages of collapse, quantum radiation due to particle creation from a naked singularity is expected to be significantly different from black hole radiation. In certain models of collapse it has been shown that, neglecting the back reaction of spacetime, the particle flux on future null infinity grows as the inverse square of the distance from the Cauchy horizon. This is to be contrasted with the flux of radiation from a black hole, which approaches a constant (inversely proportional to the square of its mass) in the neighborhood of its event horizon. The spectrum of black hole radiation is identical to that of a black body at temperature $T = (8\pi M)^{-1}$. We derive the radiation spectrum for a naked singularity formed in the collapse of a marginally bound inhomogeneous dust cloud and show that the spectrum is not black body and admits no simple interpretation. 
  I consider certain renormalization effects in curved spacetime quantum field theory. In the very early universe these effects resemble those of a cosmological constant, while in the present universe they give rise to a significant finite renormalization of the gravitational constant. The relevant renormalization term and its relation to elementary particle masses was first found by Parker and Toms in 1985, as a consequence of the ``new partially summed form'' of the propagator in curved spacetime. The significance of the term is based on the contribution of massive particles to the vacuum. In the present universe, this renormalization term appears to account for a large part or even all of the Newtonian gravitational constant. This conjecture is testable because it relates the value of Newton's constant to the elementary particle masses. 
  We present a formulation for the internal motion of equilibrium configurations with a rotational Killing vector in general relativity. As an approximation, this formulation is applicable to investigation of the internal motion of quasi-equilibrium configurations such as binary neutron stars. Based on this simple formulation, a condition for the general relativistic counter rotation has been obtained, though in the recent work by Bonazzola, Gourgoulhon and Marck, their condition for the counter rotation is not enough to specify the internal velocity field. Under the condition given in this paper, the internal velocity field can be determined completely. Indeed, in the counter-rotating case, we have also derived Poisson equations for the internal velocity, which take tractable forms in numerical implementation. 
  A simple argument is presented in favour of the equidistant spectrum in semiclassical limit for the horizon area of a black hole. The following quantization rules for the mass $M_N$ and horizon area $A_{Nj}$ are proposed: M_N = m_p [N(N+1)]^{1/4}; A_{Nj} = 8\pi l_p^2 [\sqrt{N(N+1)} + \sqrt{N(N+1) - j(j+1)} ]. Here both $N$ and $j$ are nonnegative integers or half-integers. 
  Scalar curvature invariants are studied in type N solutions of vacuum Einstein's equations with in general non-vanishing cosmological constant Lambda. Zero-order invariants which include only the metric and Weyl (Riemann) tensor either vanish, or are constants depending on Lambda. Even all higher-order invariants containing covariant derivatives of the Weyl (Riemann) tensor are shown to be trivial if a type N spacetime admits a non-expanding and non-twisting null geodesic congruence.   However, in the case of expanding type N spacetimes we discover a non-vanishing scalar invariant which is quartic in the second derivatives of the Riemann tensor.   We use this invariant to demonstrate that both linearized and the third order type N twisting solutions recently discussed in literature contain singularities at large distances and thus cannot describe radiation fields outside bounded sources. 
  In the first part I set out some unexplored historical material about the early development of cosmic topology. In the second part I briefly comment new developments in the field since the Lachieze-Rey & Luminet report (1995), both from a theoretical and an observational point of view. 
  We argue that it is possible to assign Bondi as well as ADM four-momentum to the ultrarelativistic limit of the Schwarzschild black hole in agreement to what is expected on physical grounds: The Bondi-momentum is lightlike and equal to the ADM-momentum up to the retarded time when particle and radiation escape to infinity and drops to zero thereafter, leaving flat space behind. 
  The difference in the proper azimuthal periods of revolution of two standard clocks in direct and retrograde orbits about a central rotating mass is proportional to J/Mc^2, where J and M are, respectively, the proper angular momentum and mass of the source. In connection with this gravitomagnetic clock effect, we explore the possibility of using spaceborne standard clocks for detecting the gravitomagnetic field of the Earth. It is shown that this approach to the measurement of the gravitomagnetic field is, in a certain sense, theoretically equivalent to the Gravity Probe-B concept. 
  The intimate relations between Einstein's equation, conformal geometry, geometric asymptotics, and the idea of an isolated system in general relativity have been pointed out by Penrose many years ago. A detailed analysis of the interplay of conformal geometry with Einstein's equation allowed us to deduce from the conformal properties of the field equations a method to derive under various assumptions definite statements about the feasibility of the idea of geometric asymptotics.   More recent investigations have demonstrated the possibility to analyse the most delicate problem of the subject -- the behaviour of asymptotically flat solutions to Einstein's equation in the region where ``null infinity meets space-like infinity'' -- to an arbitrary precision. Moreover, we see now that the, initially quite abstract, analysis yields methods for dealing with practical issues. Numerical calculations of complete space-times in finite grids without cut-offs become feasible now. Finally, already at this stage it is seen that the completion of these investigations will lead to a clarification and deeper understanding of the idea of an isolated system in Einstein's theory of gravitation. In the following I wish to give a survey of the circle of ideas outlined above, emphasizing the interdependence of the structures and the naturalness of the concepts involved. 
  The Hamiltonian dynamics of spherically symmetric massive thin shells in the general relativity is studied. Two different constraint dynamical systems representing this dynamics have been described recently; the relation of these two systems is investigated. The symmetry groups of both systems are found. New variables are used, which among other things simplify the complicated system a great deal. The systems are reduced to presymplectic manifolds Gamma_1 and Gamma_2, lest non-physical aspects like gauge fixings or embeddings in extended phase spaces complicate the line of reasoning. The following facts are shown. Gamma_1 is three- and Gamma_2 is five-dimensional; the description of the shell dynamics by Gamma_1 is incomplete so that some measurable properties of the shell cannot be predicted. Gamma_1 is locally equivalent to a subsystem of Gamma_2 and the corresponding local morphisms are not unique, due to the large symmetry group of Gamma_2. Some consequences for the recent extensions of the quantum shell dynamics through the singularity are discussed. 
  We prove a novel unique continuation result for weak bisolutions to the massive Klein-Gordon equation on a 2-dimensional cylinder M. Namely, if such a bisolution vanishes in a neighbourhood of a `sufficiently large' portion of a 2-dimensional surface lying parallel to the diagonal in the product manifold of M with itself, then it is (globally) translationally invariant. The proof makes use of methods drawn from Beurling's theory of interpolation. An application of our result to quantum field theory on 2-dimensional cylinder spacetimes will appear elsewhere. 
  We consider 2-dimensional cylinder spacetimes whose metrics differ from the flat Minkowskian metric within a compact region. By choice of time orientation, these spacetimes may be regarded as either globally hyperbolic timelike cylinders or nonglobally hyperbolic spacelike cylinders. For generic metrics in our class, we classify all possible candidate quantum field algebras for massive Klein-Gordon theory which obey the F-locality condition introduced by Kay. This condition requires each point of spacetime to have an intrinsically globally hyperbolic neighbourhood, N, such that the commutator (in the candidate algebra) of fields smeared with test functions supported in N agrees with the value obtained in the usual construction of Klein-Gordon theory on N.   By considering bisolutions to the Klein-Gordon equation, we prove that generic timelike cylinders admit a unique F-local algebra -- namely the algebra obtained by the usual construction -- and that generic spacelike cylinders do not admit any F-local algebras, and are therefore non F-quantum compatible. Refined versions of our results are obtained for subclasses of metrics invariant under a symmetry group. Thus F-local field theory on 2-dimensional cylinder spacetimes essentially coincides with the usual globally hyperbolic theory. In particular the result of the author and Higuchi that the Minkowskian spacelike cylinder admits infinitely many F-local algebras is now seen to represent an anomalous case. 
  The metric describing a given finite sector of a four-dimensional asymptotically anti-de Sitter wormhole can be transformed into the metric of the time constant sections of a Tangherlini black hole in a five-dimensional anti-de Sitter spacetime when one allows light cones to tip over on the hypersurfaces according to the conservation laws of an one-kink. The resulting kinked metric can be maximally extended, giving then rise to an instantonic structure on the euclidean continuation of both the Tangherlini time and the radial coordinate. In the semiclassical regime, this kink is related to the existence of closed timelike curves. 
  We present a theoretical background for the data analysis of the gravitational-wave signals from spinning neutron stars for Earth-based laser interferometric detectors. We introduce a detailed model of the signal including both the frequency and the amplitude modulations. We include the effects of the intrinsic frequency changes and the modulation of the frequency at the detector due to the Earth motion. We estimate the effects of the star's proper motion and of relativistic corrections. Moreover we consider a signal consisting of two components corresponding to a frequency $f$ and twice that frequency. From the maximum likelihood principle we derive the detection statistics for the signal and we calculate the probability density function of the statistics. We obtain the data analysis procedure to detect the signal and to estimate its parameters. We show that for optimal detection of the amplitude modulated signal we need four linear filters instead of one linear filter needed for a constant amplitude signal. Searching for the doubled frequency signal increases further the number of linear filters by a factor of two. We indicate how the fast Fourier transform algorithm and resampling methods commonly proposed in the analysis of periodic signals can be used to calculate the detection statistics for our signal. We find that the probability density function of the detection statistics is determined by one parameter: the optimal signal-to-noise ratio. We study the signal-to-noise ratio by means of the Monte Carlo method for all long-arm interferometers that are currently under construction. We show how our analysis can be extended to perform a joint search for periodic signals by a network of detectors and we perform Monte Carlo study of the signal-to-noise ratio for a network of detectors. 
  The motion of a spherical dust cloud is described by the Lemaitre-Tolman-Bondi solution and is completely specified by initial values of distributions of the rest mass density and specific energy of the dust fluid.   From generic initial conditions of this spherically symmetric collapse, there appears a naked singularity at the symmetric center in the course of the gravitational collapse of the dust cloud. So this might be a counter example to the cosmic censorship hypothesis. To investigate the genericity of this example, we examine the stability of the `nakedness' of this singularity against odd-parity modes of non-spherical linear perturbations for the metric, i.e., linear gravitational waves. We find that the perturbations do not diverge but are well-behaved even in the neighborhood of the central naked singularity. This means that the naked singularity formation process is marginally stable against the odd-parity modes of linear gravitational waves. 
  Cosmological model based on metric of Fridmann-Robertson-Walker with permanent size and acceleration of time is considered. The problem of the dark matter is analyzed within this model . 
  In the wave equation obeyed by electromagnetic fields in curved spacetime there are Riemann and Ricci curvature coupling terms to the photon polarisation, which result in a polarisation dependent deviation of the photon trajectories from null geodesics. Photons are found to have an effective mass in an external gravitational field and their velocity in an inertial frame is in general less than $c$. A consequence of this is that the curvature corrections to the propagation of electromagnetic radiation keep the velocities subluminal provided the strong energy condition is satisfied. We further show that the claims of superluminal velocities in higher derivative gravity theories are erroneous and arise due to the neglect of Riemann and Ricci coupling terms in the wave equation, which exists in Einstein's gravity itself. 
  Multidimensional cosmological models in the presence of a bare cosmological constant and a perfect fluid are investigated under dimensional reduction to 4-dimensional effective models. Stable compactification of the internal spaces is achieved for a special class of perfect fluids. The external space behaves in accordance with the standard Friedmann model. Necessary restrictions on the parameters of the models are found to ensure dynamical behavior of the external (our) universe in agreement with observations. 
  We study the dynamics of Einstein's equations in Ashtekar's variables from the point of view of the theory of hyperbolic systems of evolution equations. We extend previous results and show that by a suitable modification of the Hamiltonian vector flow outside the sub-manifold of real and constrained solutions, a symmetric hyperbolic system is obtained for any fixed choice of lapse-shift pair, without assuming the solution to be a priori real. We notice that the evolution system is block diagonal in the pair $(\sigma^a,A_b)$, and provide explicit and very simple formulae for the eigenvector-eigenvalue pairs in terms of an orthonormal tetrad with one of its components pointing along the propagation direction. We also analyze the constraint equations and find that when viewed as functions of the extended phase space they form a symmetric hyperbolic system on their own. We also provide simple formulae for its eigenvectors-eigenvalues pairs. 
  We investigate the possibility of statistical explanation of the black hole entropy by counting quasi-bounded modes of thermal fluctuation in two dimensional black hole spacetime. The black hole concerned is quantum in the sense that it is in thermal equilibrium with its Hawking radiation. It is shown that the fluctuation around such a black hole obeys a wave equation with a potential whose peaks are located near the black hole and which is caused by quantum effect. We can construct models in which the potential in the above sense has several positive peaks and there are quai-bounded modes confined between these peaks. This suggests that these modes contribute to the black hole entropy. However it is shown that the entropy associated with these modes dose not obey the ordinary area law. Therefore we can call these modes as an additional thermal hair of the quantum black hole. 
  A family of effective actions in Hamiltonian form is derived for a self-gravitating sphere of isotropic homogeneous dust. Starting from the Einstein-Hilbert action for barotropic perfect fluids and making use of the symmetry and equation of state of the matter distribution we obtain reduced actions for two canonical variables, namely the radius of the sphere and its ADM energy, the latter being conserved along trajectories of the former. These actions differ by the value of the (conserved) geodesic energy of the radius of the sphere which defines (disconnected) classes of solutions in correspondence to the inner geometry and proper volume of the sphere. Each class is thus treated as one constrained dynamical system and the union of all classes covers the full phase space of the model. Generalization to the (inhomogeneous) Tolman model is shown to be straightforward. Quantization is also discussed. 
  Using the existence of a covariant conserved quantity on large perturbation scales in a spatially flat perfect fluid or scalar field universe, we present a general formula for gauge-invariantly defined comoving energy density perturbations which encodes the entire linear perturbation dynamics in a closed time integral. On this basis we discuss perturbation modes in different cosmological epochs. 
  Physical (and weak) regularity conditions are used to determine and classify all the possible types of spherically symmetric dust spacetimes in general relativity. This work unifies and completes various earlier results. The junction conditions are described for general non-comoving (and non-null) surfaces, and the limits of kinematical quantities are given on all comoving surfaces where there is Darmois matching. We show that an inhomogeneous generalisation of the Kantowski-Sachs metric may be joined to the Lemaitre-Tolman-Bondi metric. All the possible spacetimes are explicitly divided into four groups according to topology, including a group in which the spatial sections have the topology of a 3-torus. The recollapse conjecture (for these spacetimes) follows naturally in this approach. 
  In the coming decade, the LIGO/VIRGO/GEO network of ground-based kilometer-scale laser interferometer gravitational wave detectors will open up a new astronomical window on the Universe: gravitational waves in the frequency band 10 to 10^4 Hz. In addition, if the proposed, 5 million kilometer long, space based interferometer LISA flies, another window will be opened in the frequency band 10^(-4) to 1 Hz. I review the various possible sources that might be detected in these frequency bands, and the information that might be obtainable from observed sources. This article closely parallels my talk at the GR15 conference, but is updated to include discussions of some recent developments. 
  We construct an exact relativistic cosmology in which an inhomogeneous but isotropic local region has fractal number counts and matches to a homogeneous background at a scale of the order of $10^2$ Mpc. We show that Einstein's equations and the matching conditions imply either a nonlinear Hubble law or a very low large-scale density. 
  In this paper we design and develop several filtering strategies for the analysis of data generated by a resonant bar Gravitational Wave (GW) antenna, with the goal to assess the presence (or absence) in them of long duration monochromatic GW signals, as well as their eventual amplitude and frequency, within the sensitivity band of the detector. Such signals are most likely generated in the fast rotation of slightly asymmetric spinning stars. We shall develop the practical procedures, together with the study of their statistical properties, which will provide us with useful information on each technique's performance. The selection of candidate events will then be established according to threshold-crossing probabilities, based on the Neyman-Pearson criterion. In particular, it will be shown that our approach, based on phase estimation, presents better signal-to-noise ratio than the most common one of pure spectral analysis. 
  We show that the non flat factor of the Godel metric belongs to a one parameter family of 2+1 dimensional geometries that also includes the anti-de Sitter metric. The elements of this family allow a generalization a la Kaluza-Klein of the usual 3+1 dimensional Godel metric. Their lightcones can be viewed as deformations of the anti-de Sitter ones, involving tilting and squashing. This provides a simple geometric picture of the causal structure of these space-times, anti-de Sitter geometry appearing as the boundary between causally safe and causally pathological spaces. Furthermore, we construct a global algebraic isometric embedding of these metrics in 4+3 or 3+4 dimensional flat spaces, thereby illustrating in another way the occurrence of the closed timelike curves. 
  It is shown that, in the case of Kerr-Newman space-time, the complex electromagnetic strength E + i H and analogous complex intensity of the gravitational field share the common complexified spatial direction. 
  E. Noether's general analysis of conservation laws has to be completed in a Lagrangian theory with local gauge invariance. Bulk charges are replaced by fluxes of superpotentials. Gauge invariant bulk charges may subsist when distinguished one-dimensional subgroups are present. As a first illustration we propose a new {\it Affine action} that reduces to General Relativity upon gauge fixing the dilatation (Weyl 1918 like) part of the connection and elimination of auxiliary fields. It allows a comparison of most gravity superpotentials and we discuss their selection by the choice of boundary conditions. A second and independent application is a geometrical reinterpretation of the convection of vorticity in barotropic nonviscous fluids. We identify the one-dimensional subgroups responsible for the bulk charges and thus propose an impulsive forcing for creating or destroying selectively helicity. This is an example of a new and general Forcing Rule. 
  The Reissner--Nordstr\"{o}m family of solutions can be understood to arise from the spherically symmetric sector of a nonlinear SO(2,1)/SO(1,1) sigma model coupled to three dimensional Euclidean gravity. In this context a group theoretical quantization is performed. We identify the observables of the theory and calculate their spectra. 
  If one introduces causality into quantum cosmology, then the prescription for the no-boundary universe should be revised. We show that the thermodymanic arrow of time associated with the perturbation modes should be reversed at the maximum expansion for the oscillating Hawking model. To an observer equipped with the time arrow, the universe will terminate its evolution after an half cycle. 
  The renormalised value of $<\phi^2>$ is calculated for a massless, conformally coupled scalar field in the Hartle-Hawking vacuum state. This calculation is a first step towards the calculation of the gravitational back reaction of the field in a black cosmic string spacetime which is asymptotically anti-DeSitter and possesses a non constant dilaton field. It is found that the field is divergence free throughout the spacetime and attains its maximum value near the horizon. 
  This paper shows how to obtain the spinor field and dynamics from the vielbein and geometry of General Relativity. The spinor field is physically realized as an orthogonal transformation of the vielbein, and the spinor action enters as the requirement that the unit time form be the gradient of a scalar time field. 
  Primordial black holes may form in the early Universe, for example from the collapse of large amplitude density perturbations predicted in some inflationary models. Light black holes undergo Hawking evaporation, the energy injection from which is constrained both at the epoch of nucleosynthesis and at the present. The failure as yet to unambiguously detect primordial black holes places important constraints. In this article, we are particularly concerned with the dependence of these constraints on the model for the complete cosmological history, from the time of formation to the present. Black holes presently give the strongest constraint on the spectral index $n$ of density perturbations, though this constraint does require $n$ to be constant over a very wide range of scales. 
  For slowly rotating fluids, we establish the existence of a critical point similar to the one found for non-rotating systems. As the fluid approaches the critical point, the effective inertial mass of any fluid element decreases, vanishing at that point and changing of sign beyond it. This result implies that first order perturbative method is not always reliable to study dissipative processes ocurring before relaxation. Physical consequences that might follow from this effect are commented. 
  Recent progress in understanding the structure of cosmological singularities is reviewed. The well-known picture due to Belinskii, Khalatnikov and Lifschitz (BKL) is summarized briefly and it is discussed what existing analytical and numerical results have to tell us about the validity of this picture. If the BKL description is correct then most cosmological singularities are complicated. However there are some cases where it predicts simple singularities. These cases should be particularly amenable to mathematical investigation and the results in this direction which have been achieved so far are described. 
  The existence of a topological double-covering for the $GL(n,R)$ and diffeomorphism groups is reviewed. These groups do not have finite-dimensional faithful representations. An explicit construction and the classification of all $\bar{SL}(n,R)$, $n=3,4$ unitary irreducible representations is presented. Infinite-component spinorial and tensorial $\bar{SL}(4,R)$ fields, "manifields", are introduced. Particle content of the ladder manifields, as given by the $\bar{SL}(3,R)$ "little" group is determined. The manifields are lifted to the corresponding world spinorial and tensorial manifields by making use of generalized infinite-component frame fields. World manifields transform w.r.t. corresponding $\bar{Diff}(4,R)$ representations, that are constructed explicitly. 
  World spinors are objects that transform w.r.t. double covering group $\bar{Diff}(4,R)$ of the Group of General Coordinate Transformations. The basic mathematical results and the corresponding physical interpretation concerning these, infinite-dimensional, spinorial representations are reviewed. The role of groups $Diff(4,R)$, $GA(4,R)$, $GL(4,R)$, $SL(4,R)$, $SO(3,1)$ and the corresponding covering groups is pointed out. New results on the infinite dimensionality of spinorial representations, explicit construction of the $\bar{SL}(4,R)$ representations in the basis of finite-dimensional non-unitary $SL(2,C)$ representations, $SL(4,R)$ representation regrouping of tensorial and spinorial fields of an arbitrary spin lagrangian field theory, as well as its $SL(5,R)$ generalization in the case of infinite-component world spinor and tensor field theories are presented. 
  Non-perturbative quantum general relativity provides a possible framework to analyze issues related to black hole thermodynamics from a fundamental perspective. A pedagogical account of the recent developments in this area is given. The emphasis is on the conceptual and structural issues rather than technical subtleties. The article is addressed to post-graduate students and beginning researchers. 
  In recent years the threshold of black hole formation in spherically symmetric gravitational collapse has been studied for a variety of matter models. In this paper the corresponding issue is investigated for a matter model significantly different from those considered so far in this context. We study the transition from dispersion to black hole formation in the collapse of collisionless matter when the initial data is scaled. This is done by means of a numerical code similar to those commonly used in plasma physics. The result is that for the initial data for which the solutions were computed, most of the matter falls into the black hole whenever a black hole is formed. This results in a discontinuity in the mass of the black hole at the onset of black hole formation. 
  The mass function of primordial black holes created through the near-critical gravitational collapse is calculated in a manner fairly independent of the statistical distribution of underlying density fluctuation, assuming that it has a sharp peak on a specific scale. Comparing it with various cosmological constraints on their mass spectrum, some newly excluded range is found in the volume fraction of the region collapsing into black holes as a function of the horizon mass. 
  Multidimensional cosmological model with static internal spaces describing the evolution of an Einstein space of non-zero curvature and n internal spaces is considered. The action contains several dilatonic scalar fields and antisymmetric forms and lambda-term. When forms are chosen to be proportional to volume forms of p-brane submanifolds of internal space manifold, the Toda-like Lagrange representation arises. Exact solutions for the model are obtained, when scale factors of internal spaces are constant. It is shown that they are de Sitter or anti-de Sitter. Behaviour of cosmological constant and its generation by p-branes is demonstrated. 
  We construct a general approach to decomposition of the tangent bundle of pseudo-Riemannian manifolds into direct sums of subbundles, and the associated decomposition of geometric objects. An invariant structure {\cal H}^r defined as a set of r projection operators is used to induce decomposition of the geometric objects into those of the corresponding subbundles. We define the main geometric objects characterizing decomposition. Invariant non-holonomic generalizations of the Gauss-Codazzi-Ricci's relations have been obtained. All the known types of decomposition (used in the theory of frames of reference, in the Hamiltonian formulation for gravity, in the Cauchy problem, in the theory of stationary spaces, and so on) follow from the present work as special cases when fixing a basis and dimensions of subbundles, and parameterization of a basis of decomposition. Various methods of decomposition have been applied here for the Unified Multidimensional Kaluza-Klein Theory and for relativistic configurations of a perfect fluid. Discussing an invariant form of the equations of motion we have found the invariant equilibrium conditions and their 3+1 decomposed form. The formulation of the conservation law for the curl has been obtained in the invariant form. 
  Gravitational radiation drives an instability in the r-modes of young rapidly rotating neutron stars. This instability is expected to carry away most of the angular momentum of the star by gravitational radiation emission, leaving a star rotating at about 100 Hz. In this paper we model in a simple way the development of the instability and evolution of the neutron star during the year-long spindown phase. This allows us to predict the general features of the resulting gravitational waveform. We show that a neutron star formed in the Virgo cluster could be detected by the LIGO and VIRGO gravitational wave detectors when they reach their ``enhanced'' level of sensitivity, with an amplitude signal-to-noise ratio that could be as large as about 8 if near-optimal data analysis techniques are developed. We also analyze the stochastic background of gravitational waves produced by the r-mode radiation from neutron-star formation throughout the universe. Assuming a substantial fraction of neutron stars are born with spin frequencies near their maximum values, this stochastic background is shown to have an energy density of about 10^-9 of the cosmological closure density, in the range 20 Hz to 1 kHz. This radiation should be detectable by ``advanced'' LIGO as well. 
  In this paper we continue the analysis of our previous papers and study the affect of the existence of a non-trivial dilaton background on the propagation of electromagnetic waves in the Kerr-Newman dilatonic black hole space-time. For this purpose we again employ the double expansion in both the background electric charge and the wave parameters of the relevant quantities in the Newman-Penrose formalism and then identify the first order at which the dilaton background enters the Maxwell equations. We then assume that gravitational and dilatonic waves are negligible (at that order in the charge parameter) with respect to electromagnetic waves and argue that this condition is consistent with the solutions already found in the previous paper. Explicit expressions are given for the asymptotic behavior of scattered waves, and a simple physical model is proposed in order to test the effects. An expression for the relative intensity is obtained for Reissner-Nordstrom dilaton black holes using geometrical optics. A comparison with the approximation of geometrical optics for Kerr-Newman dilaton black holes shows that at the order to which the calculations are carried out gravitational lensing of optical images cannot probe the dilaton background. 
  We show how gauge-invariant cosmological perturbations may be constructed by an unambiguous choice of hypersurface-orthogonal time-like vector field (i.e., time-slicing). This may be defined either in terms of the metric quantities such as curvature or shear, or using some matter field. As an example, we show how linear perturbations in the covariant fluid-flow approach can then be presented in an explicitly gauge-invariant form in the coordinate based formalism. 
  I employ heuristically the strictly isospectral double Darboux method based on the general superpotential of unbroken nonrelativistic supersymmetry suggesting a few small steps of principle for extending its range of applications toward relativistic (gauge) physics. The application of the method to minisuperspace quantum cosmology is also briefly presented 
  We conduct a direct comparison of three different representative numerical codes for constructing models of rapidly rotating neutron stars in general relativity. Our aim is to evaluate the accuracy of the codes and to investigate how the accuracy is affected by the choice of interpolation, domain of integration and equation of state. In all three codes, the same physical parameters, equations of state and interpolation method are used. We construct 25 selected models for polytropic equations of state and 22 models with realistic neutron star matter equations of state. The three codes agree well with each other (typical agreement is better than 0.1% to 0.01%) for most models, except for the extreme assumption of uniform density stars. We conclude that the codes can be used for the construction of highly accurate initial data configurations for polytropes of index N > 0.5 (which typically correspond to realistic neutron stars), when the domain of integration includes all space and for realistic equations with no phase transitions. With the exception of the uniform density case, the obtained values of physical parameters for the models considered in this paper can be regarded as ``standard'' and we display them in detail for all models. 
  We briefly recall the problem of defining conserved quantities such as energy in general relativity, and the solution by introducing a symmetric background. We apply the general formalism to perturbed Robertson-Walker spacetimes with de Sitter geometry as background. We relate the obtained conserved quantities to Traschen's integral constraints and mention a few applications in cosmology. 
  We derive and discuss black-hole solutions to the gravitating O(3) $\sigma$ model in (2+1) dimensions. Three different kinds of static black holes are found. One of these resembles the static BTZ black hole, another is completely free of singularities, and the last type has the same Penrose diagram as the (3+1)-dimensional Schwarzschild black hole. We also construct static and dynamical multi-black hole systems. 
  Linde's proposal of a Euclidean path integral with the ``wrong'' sign of Euclidean action is often identified with the tunneling proposal for the wave function of the universe. However, the two proposals are in fact quite different. I illustrate the difference and point out that recent criticism by Hawking and Turok does not apply to the tunneling proposal. 
  We discuss a successful three-dimensional cartesian implementation of the Bona-Mass\'o hyperbolic formulation of the 3+1 Einstein evolution equations in numerical relativity. The numerical code, which we call ``Cactus,'' provides a general framework for 3D numerical relativity, and can include various formulations of the evolution equations, initial data sets, and analysis modules. We show important code tests, including dynamically sliced flat space, wave spacetimes, and black hole spacetimes. We discuss the numerical convergence of each spacetime, and also compare results with previously tested codes based on other formalisms, including the traditional ADM formalism. This is the first time that a hyperbolic reformulation of Einstein's equations has been shown appropriate for three-dimensional numerical relativity in a wide variety of spacetimes. 
  In the framework of extended gravity theories, we discuss the meaning of a time dependent "cosmological constant" and give a set of conditions to recover asymptotic de Sitter behaviour for a class of cosmological models independently of initial data. To this purpose we introduce a time-dependent (effective) quantity which asymptotically becomes the true cosmological constant. We will deal with scalar-tensor, fourth and higher than fourth-order theories. 
  Various formalisms proposed recently for irrotational binary systems in general relativity are compared and explicit relations between them are exhibited. It is notably shown that the formalisms of (i) Teukolsky, (ii) Shibata and (iii) Bonazzola et al. (as corrected by Asada) are equivalent, i.e. yield exactly the same solution, although the former two are simpler than the latter one. 
  We review in a pedagogical fashion the 3+1-split which serves to put Einstein's equations into the form of a dynamical system with constraints. We then discuss the constraint equations under the simplifying assumption of time-symmetry. Multi-Black-Hole data are presented and more explicitly described in the case of two holes. The effect of different topologies is emphasized. 
  Positive definiteness of the quadratic part of the action of the Hawking-Turok instanton is investigated. The Euclidean quadratic action for scalar perturbations is expressed in terms of a single gauge invariant quantity $q$. The mode functions satisfy a Schr\"odinger type equation with a potential $U$. It is shown that the potential $U$ tends to a positive constant at the regular end of the instanton. The detailed shape of $U$ depends on the initial data of the instanton, on parameters of the background scalar field potential $V$ and on a positive integer, $p$, labeling different spherical harmonics. For certain well behaved scalar field potentials it is proven analytically that for $p>1$ quadratic action is non-negative. For the lowest $p=1$ (homogeneous) harmonic numerical solution of the Schr\"odinger equation for different scalar field potentials $V$ and different initial data show that in some cases the potential $U$ is negative in the intermediate region. We investigated the monotonously growing potentials and a potential with a false vacuum. For the monotonous potentials no negative modes are found about the Hawking-Turok instanton. For a potential with the false vacuum the HT instanton is shown to have a negative mode for certain initial data. 
  Gravitational radiation arising from the inspiral and merger of binary black holes (BBH's) is a promising candidate for detection by kilometer-scale interferometric gravitational wave observatories. This paper discusses a serious obstacle to searches for such radiation and to the interpretation of any observed waves: the inability of current computational techniques to evolve a BBH through its last ~10 orbits of inspiral (~100 radians of gravitational-wave phase). A new set of numerical-relativity techniques is proposed for solving this ``Intermediate Binary Black Hole'' (IBBH) problem: (i) numerical evolutions performed in coordinates co-rotating with the BBH, in which the metric coefficients evolve on the long timescale of inspiral, and (ii) techniques for mathematically freezing out gravitational degrees of freedom that are not excited by the waves. 
  We investigate conformally coupled quantum matter fields on spherically symmetric, continuously self-similar backgrounds. By exploiting the symmetry associated with the self-similarity the general structure of the renormalized quantum stress-energy tensor can be derived. As an immediate application we consider a combination of classical, and quantum perturbations about exactly critical collapse. Generalizing the standard argument which explains the scaling law for black hole mass, $M \propto |\eta-\eta^*|^\beta$, we demonstrate the existence of a quantum mass gap when the classical critical exponent satisfies $\beta \geq 0.5$. When $\beta < 0.5$ our argument is inconclusive; the semi-classical approximation breaks down in the spacetime region of interest. 
  We present techniques and methods for analyzing the dynamics of event horizons in numerically constructed spacetimes. There are three classes of analytical tools we have investigated. The first class consists of proper geometrical measures of the horizon which allow us comparison with perturbation theory and powerful global theorems. The second class involves the location and study of horizon generators. The third class includes the induced horizon 2-metric in the generator comoving coordinates and a set of membrane-paradigm like quantities. Applications to several distorted, rotating, and colliding black hole spacetimes are provided as examples of these techniques. 
  A technical flaw was found in this paper, and it has been withdrawn. It is intended that a modified version will be submitted soon. 
  A classical continuum theory corresponding to Barrett and Crane's model of Euclidean quantum gravity is presented. The fields in this classical theory are those of SO(4) BF theory, a simple topological theory of an so(4) valued 2-form field, $B^{IJ}_{\m\n}$, and an so(4) connection. The left handed (self-dual) and right handed (anti-self-dual) components of $B$ define a left handed and a, generally distinct, right handed area for each spacetime 2-surface. The theory being presented is obtained by adding to the BF action a Lagrange multiplier term that enforces the constraint that the left handed and the right handed areas be equal. It is shown that Euclidean general relativity (GR) forms a sector of the resulting theory. The remaining three sectors of the theory are also characterized and it is shown that, except in special cases, GR canonical initial data is sufficient to specify the GR sector as well as a specific solution within this sector.   Finally, the path integral quantization of the theory is discussed at a formal level and a hueristic argument is given suggesting that in the semiclassical limit the path integral is dominated by solutions in one of the non-GR sectors, which would mean that the theory quantized in this way is not a quantization of GR. 
  Above Planck energies, the spacetime might become non--Riemannian, as it is known fron string theory and inflation. Then geometries arise in which nonmetricity and torsion appear as field strengths, side by side with curvature. By gauging the affine group, a metric affine gauge theory emerges as dynamical framework. Here, by using the harmonic map ansatz, a new class of multipole like solutions in the metric affine gravity theory (MAG) is obtained. 
  I investigate the relation between an operative definition of the area of a surface specified by matter fields and the area operators recently introduced in the canonical/loop approach to Quantum Gravity and in Rovelli's variant of the Husain-Kuchar Quantum-Gravity toy model. The results suggest that the discreteness of the spectra of the area operators might not be observable. 
  We study the structure and stability of spherically symmetric Brans-Dicke black-hole type solutions with an infinite horizon area and zero Hawking temperature, existing for negative values of the coupling constant $\omega$. These solutions split into two classes, depending on finite (B1) or infinite (B2) proper time needed for an infalling particle to reach the horizon. Class B1 metrics can be extended through the horizon only for discrete values of mass and scalar charge, depending on two integers m and n. For even m-n, the space-time is globally regular; for odd m, the metric changes its signature on the horizon but remains Lorentzian. Geodesics are smoothly continued across the horizon, but for odd m timelike geodesics become spacelike and vice versa. Causality problems, arising in some cases, are discussed. Tidal forces are shown to grow infinitely near type B1 horizons. All vacuum static, spherically symmetric solutions of the Brans-Dicke theory with $\omega<-3/2$ are found to be linearly stable against spherical perturbations. This result extends to the generic case of the Bergmann-Wagoner class of scalar-tensor theories with the coupling function $\omega(\phi) < -3/2$. 
  This is the first of a series of papers describing a numerical implementation of the conformally rescaled Einstein equation, an implementation designed to calculate asymptotically flat spacetimes, especially spacetimes containing black holes.   In the present paper we derive the new first order time evolution equations to be used in the scheme. These time evolution equations can either be written in symmetric hyperbolic or in flux-conservative form. Since the conformally rescaled Einstein equation, also called the conformal field equations, formally allow us to place the grid boundaries outside the physical spacetime, we can modify the equations near the grid boundaries and get a consistent and stable discretisation. Even if we calculate spacetimes containing black holes, there is no need for introducing artifical boundaries in the physical spacetime, which then would complicate, influence, or even exclude the computation of certain spacetime regions. 
  We investigate whether the equality found for the response of static scalar sources interacting (i) with {\em Hawking radiation in Schwarzschild spacetime} and (ii) with the Fulling-Davies-Unruh thermal bath in the Rindler wedge is maintained in the case of electric charges. We find a finite result in the Schwarzschild case, which is computed exactly, in contrast with the divergent result associated with the infrared catastrophe in the Rindler case, i.e. in the case of uniformly accelerated charges in Minkowski spacetime. Thus, the equality found for scalar sources does not hold for electric charges. 
  An exact solution for a nonrotating boson star in (2+1) dimensional gravity with a negative cosmological constant is found. The relations among mass, particle number, and radius of the (2+1) dimensional boson star are shown. 
  Computer algebra programs are presented for application in general relativity, in electrodynamics, and in gauge theories of gravity. The mathematical formalism used is the calculus of exterior differential forms, the computer algebra system applied Hearn's Reduce with Schruefer's exterior form package Excalc. As a non-trivial example we discuss a metric of Plebanski & Demianski (of Petrov type D) together with an electromagnetic potential and a triplet of post-Riemannian one-forms. This whole geometrical construct represents an exact solution of a metric-affine gauge theory of gravity. We describe a sample session and verify by computer that this exact solution fulfills the appropriate field equations.-- Computer programs are described for the irreducible decomposition of (non-Riemannian) curvature, torsion, and nonmetricity. 
  Recent measurements on a class of high-Tc superconductors (HTSC) have shown that Cooper-pairs binding may be associated to a d-wave, while in another class, d and s waves may coexist. When d-wave Cooper-pairs are injected in a superconductor that can sustain s-wave binding, d-wave pairs decay to s-wave pairs and energy is irradiated by means of gravitons. We show that in s-wave to d-wave type superconductor (SDS) junctions in an equilibrium condition no net gravitational wave energy is emitted, on the other hand under non equilibrium conditions a net gravitational wave energy is emitted by the junction. Experiments which show a gravitational interaction between inomogeneus high-Tc superconductors, under non equilibrium conditions, and test objects may be understood by accepting a possibility of emission of gravitational radiation from SDS junctions. 
  We present a completely integrable deformation of the CGHS dilaton gravity model in two dimensions. The solution is a singularity free black hole that at large distances asymptoticaly joins to the CGHS solution. 
  In this note we study the correspondence between the ``relativistic spin foam'' model introduced by Barrett, Crane and Baez and the so(4) Plebanski action. We argue that the $so(4)$ Plebanski model is the continuum analog of the relativistic spin foam model. We prove that the Plebanski action possess four phases, one of which is gravity and outline the discrepancy between this model and the model of Euclidean gravity. We also show that the Plebanski model possess another natural dicretisation and can be associate with another, new, spin foam model that appear to be the $so(4)$ counterpart of the spin foam model describing the self dual formulation of gravity. 
  The FHP algorithm allows to obtain the relativistic multipole moments of a vacuum stationary axisymmetric solution in terms of coefficients which appear in the expansion of its Ernst's potential on the symmetry axis. First of all, we will use this result in order to determine, at a certain approximation degree, the Ernst's potential on the symmetry axis of the metric whose only multipole moments are mass and angular momentum.   By using Sibgatullin's method we analyse a series of exacts solutions with the afore mentioned multipole characteristic. Besides, we present an approximate solution whose Ernst's potential is introduced as a power series of a dimensionless parameter. The calculation of its multipole moments allows us to understand the existing differences between both approximations to the proposed pure multipole solution. 
  A cryogenic spherical and omni-directional resonant-mass detector proposed by the GRAIL collaboration is described. 
  We present here three new solutions of Brans-Dicke theory for a stationary geometry with cylindrical symmetry in the presence of matter in rigid rotation with $T^\mu_\mu\neq 0$. All the solutions have eternal closed timelike curves in some region of the spacetime, the size of which depends on $\omega$. Moreover, two of them do not go over a solution of general relativity in the limit $\omega \to \infty$. 
  Generalizing earlier results on the initial data and the final fate of dust collapse, we study here the relevance of the initial state of a spherically symmetric matter cloud towards determining its end state in the course of a continuing gravitational collapse. It is shown that given an arbitrary regular distribution of matter at the initial epoch, there always exists an evolution from this initial data which would result either in a black hole or a naked singularity depending on the allowed choice of free functions available in the solution. It follows that given any initial density and pressure profiles for the cloud, there is a non-zero measure set of configurations leading either to black holes or naked singularities, subject to the usual energy conditions ensuring the positivity of energy density. We also characterize here wide new families of black hole solutions resulting from spherically symmetric collapse without requiring the cosmic censorship assumption. 
  Pre-big bang cosmology predicts tiny first-order dilaton and metric perturbations at very large scales. Here we discuss the possibility that other -- more copiously generated -- perturbations may act, at second order, as scalar seeds of large-scale structure and CMB anisotropies. We study, in particular, the cases of electromagnetic and axionic seeds. We compute the stochastic fluctuations of their energy-momentum tensor and determine the resulting contributions to the multipole expansion of the temperature anisotropy. In the axion case it is possible to obtain a flat or slightly tilted blue spectrum that fits present data consistently, both for massless and for massive (but very light) axions. 
  We examine in a cosmological context the conditions for unbroken supersymmetry in N=1 supergravity in D=10 dimensions. We show that the cosmological solutions of the equations of motion obtained considering only the bosonic sector correspond to vacuum states with spontaneous supersymmetry breaking. With a non vanishing gravitino-dilatino condensate we find a solution of the equations of motion that satisfies necessary conditions for unbroken supersymmetry and that smoothly interpolates between Minkowski space and DeSitter space with a linearly growing dilaton, thus providing a possible example of a supersymmetric and non-singular pre-big-bang cosmology. 
  String cosmology models predict a stochastic cosmic background of gravitational waves with a characteristic spectrum. I describe the background, present astrophysical and cosmological bounds on it, and outline how it may be possible to detect it with gravitational wave detectors. 
  We derive the gravitational field and equations of motion of compact binary systems up to the 5/2 post-Newtonian approximation of general relativity (where radiation-reaction effects first appear). The approximate post-Newtonian gravitational field might be used in the problem of initial conditions for the numerical evolution of binary black-hole space-times. On the other hand we recover the Damour-Deruelle 2.5PN equations of motion of compact binary systems. Our method is based on an expression of the post-Newtonian metric valid for general (continuous) fluids. We substitute into the fluid metric the standard stress-energy tensor appropriate for a system of two point-like particles. We remove systematically the infinite self-field of each particle by means of the Hadamard partie finie regularization. 
  We consider a D-dimensional cosmological model describing an evolution of Ricci-flat factor spaces, M_1,...M_n (n > 2), in the presence of an m-component perfect fluid source (n > m > 1). We find characteristic vectors, related to the matter constants in the barotropic equations of state for fluid components of all factor spaces.   We show that, in the case where we can interpret these vectors as the root vectors of a Lie algebra of Cartan type A_m=sl(m+1,C), the model reduces to the classical open m-body Toda chain.   Using an elegant technique by Anderson (J. Math. Phys. 37 (1996) 1349) for solving this system, we integrate the Einstein equations for the model and present the metric in a Kasner-like form. 
  In this paper we investigate extended inflation with an exponential potential $V(\sigma)= V_0 e^{-\kappa\sigma}$, which provides a simple cosmological scenario where the distribution of the constants of Nature is mostly determined by $\kappa$. In particular, we show that this theory predicts a uniform distribution for the Planck mass at the end of inflation, for the entire ensemble of universes that undergo stochastic inflation. Eternal inflation takes place in this scenario for a broad family of initial conditions, all of which lead up to the same value of the Planck mass at the end of inflation. The predicted value of the Planck mass is consistent with the observed value within a comfortable range of values of the parameters involved. 
  It is shown that space-time may be not only in a state which is described by Riemann geometry but also in states which are described by Finsler geometry. Transitions between various metric states of space-time have the meaning of phase transitions in its geometric structure. These transitions together with the evolution of each of the possible metric states make up the general picture of space-time manifold dynamics. 
  We study quantum corrections for the Schwarzshild black hole by considering it as a vacuum solution of a 2D dilaton gravity theory obtained by spherical reduction of 4D gravity coupled with matter. We find perturbatively the vacuum solution for the standard one-loop effective action in the case of null-dust matter and in the case of minimally coupled scalar field. The corresponding state is in both cases 2D Hartle-Hawking vacuum, and we evaluate the corresponding quantum corrections for the thermodynamical parameters of the black hole. We also find that the standard effective action does not allow boundary conditions corresponding to a 4D Hartle-Hawking vacuum state. 
  In order to obtain a well defined quantum gravity we define the spacetime in relation to the "phenomenology" of the physical interactions; however we shall to speculate with this "in General". Besides, we comment the reasons that give to the gravitational field a privilegied situation over the others. 
  A longstanding conjecture by Belinskii, Lifshitz, and Khalatnikov that the singularity in generic gravitational collapse is locally oscillatory is tested numerically in vacuum, U(1) symmetric cosmological spacetimes on $T^3 \times R$. If the velocity term dominated (VTD) solution to Einstein's equations is substituted into the Hamiltonian for the full Einstein evolution equations, one term is found to grow exponentially. This generates a prediction that oscillatory behavior involving this term and another (which the VTD solution causes to decay exponentially) should be observed in the approach to the singularity. Numerical simulations strongly support this prediction. 
  Inflation of cosmic gauge and global strings is investigated by numerically solving the combined Einstein and field equations. Above some critical symmetry-breaking scales, the strings undergo inflation along the radial direction as well as the axial direction at the core. The nonsingular nature of the spacetimes around supercritical gauge and global strings is discussed and contrasted to the singular static solutions that have been discussed in the literature. 
  The scattering of a straight, infinitely long string moving with velocity $v$ by a black hole is considered. We analyze the weak-field case, where the impact parameter ($b_{imp}$) is large, and obtain exact solutions to the equations of motion. As a result of scattering, the string is displaced in the direction perpendicular to the velocity by an amount $\Delta b\sim -2\pi GMv\gamma/c^3 -\pi (GM)^2/ (4c^3 v b_{imp})$, where $\gamma=(1-(v/c)^2)^{-1/2}$. The second term dominates at low velocities $v/c<(GM/b_{imp})^{1/2}$ . The late-time solution is represented by a kink and anti-kink, propagating in opposite directions at the speed of light, and leaving behind them the string in a new ``phase''. The solutions are applied to the problem of string capture, and are compared to numerical results. 
  The capture or scattering of an initially straight infinite test cosmic string by a Kerr-Newman black hole, or by any other small source of an electrovac gravitational field, is analyzed analytically when the string moves with initial velocity v and large impact parameter b >> M so that the string stays very nearly straight (except during the final capture process, if that occurs, or except far behind the gravitating object, if b is not much greater than the energy of the object in the frame of the string). The critical impact parameter for capture at low velocities is shown to be [(pi/2)(M^2-Q^2)/v]^{1/2}. For all larger b, the displacement of the string from the plane of the gravitating object after the scattering approaches the final value [b^2 - (pi/2)(M^2-Q^2)/v]^{1/2} - 2 pi M v/(1-v^2)^{1/2}, for any v, so long as b >> M. 
  We construct N = 2 chiral supergravity (SUGRA) which leads to Ashtekar's canonical formulation. The supersymmetry (SUSY) transformation parameters are not constrained at all and auxiliary fields are not required in contrast with the method of the two-form gravity. We also show that our formulation is compatible with the reality condition, and that its real section is reduced to the usual N = 2 SUGRA up to an imaginary boundary term. 
  Einstein's field equations are solved exactly for static charged dust distributions. These solutions generalize the Majumdar Papapetrou metrics. Maxwell's equations lead to the equality of charge and mass densities of the dust distribution. Einstein's equatins reduce to a nonlinear version of Poisson's equation. 
  I investigate the relationship between faster-than-light travel and weak-energy-condition violation, i.e., negative energy densities. In a general spacetime it is difficult to define faster-than-light travel, and I give an example of a metric which appears to allow superluminal travel, but in fact is just flat space. To avoid such difficulties, I propose a definition of superluminal travel which requires that the path to be traveled reach a destination surface at an earlier time than any neighboring path. With this definition (and assuming the generic condition) I prove that superluminal travel requires weak-energy-condition violation. 
  The thermodynamical properties of the Reissner-Nordstr\"om-anti-de Sitter black hole in the grand canonical ensemble are investigated using York's formalism. The black hole is enclosed in a cavity with finite radius where the temperature and electrostatic potential are fixed. The boundary conditions allow us to compute the relevant thermodynamical quantities, e.g. thermal energy, entropy and charge. The stability conditions imply that there are thermodynamically stable black hole solutions, under certain conditions. Instantons with negative heat capacity are also found. 
  The definition of the Riemann-Cartan space of the plane wave type is given. The condition under which the torsion plane waves exist is found. It is expressed in the form of the restriction imposed on the coupling constants of the 10-parametric quadratic gravitational Lagrangian. In the mathematical appendix the formula for commutator of the variation operator and Hodge operator is proved. This formula is applied for the variational procedure when the gravitational field equations are obtained in terms of the exterior differential forms. 
  We construct the explicit form of three almost complex structures that a Riemannian manifold with self-dual curvature admits and show that their Nijenhuis tensors vanish so that they are integrable. This proves that gravitational instantons with self-dual curvature admit hyper-K\"{a}hler structure. In order to arrive at the three vector valued 1-forms defining almost complex structure, we give a spinor description of real 4-dimensional Riemanian manifolds with Euclidean signature in terms of two independent sets of 2-component spinors. This is a version of the original Newman-Penrose formalism that is appropriate to the discussion of the mathematical, as well as physical properties of gravitational instantons. We shall build on the work of Goldblatt who first developed an NP formalism for gravitational instantons but we shall adopt it to differential forms in the NP basis to make the formalism much more compact. We shall show that the spin coefficients, connection 1-form, curvature 2-form, Ricci and Bianchi identities, as well as the Maxwell equations naturally split up into their self-dual and anti-self-dual parts corresponding to the two independent spin frames. We shall give the complex dyad as well as the spinor formulation of the almost complex structures and show that they reappear under the guise of a triad basis for the Petrov classification of gravitational instantons. Completing the work of Salamon on hyper-K\"ahler structure, we show that the vanishing of the Nijenhuis tensor for all three almost complex structures depends on the choice of a self-dual gauge for the connection which is guaranteed by virtue of the fact that the curvature 2-form is self-dual for gravitational instantons. 
  Geometrical structure of homogeneous isotropic models in the frame of the metric-affine gauge theory of gravity (MAGT) is analyzed. By using general form of gravitational Lagrangian including both a scalar curvature and various invariants quadratic in the curvature, torsion and nonmetricity tensors, gravitational equations of MAGT for homogeneous isotropic models are deduced. It is shown, that obtained gravitational equations lead to generalized cosmological Friedmann equation for the metrics by certain restrictions on indefinite parameters of gravitational Lagrangian. Isotropic models in the Weyl-cartan space-time are discussed. 
  It was long believed that the singularity inside a realistic, rotating black hole must be spacelike. However, studies of the internal geometry of black holes indicate a more complicated structure is typical. While it seems likely that an observer falling into a black hole with the collapsing star encounters a crushing spacelike singularity, an observer falling in at late times generally reaches a null singularity which is vastly different in character to the standard Belinsky, Khalatnikov and Lifschitz (BKL) spacelike singularity. In the spirit of the classic work of BKL we present an asymptotic analysis of the null singularity inside a realistic black hole. Motivated by current understanding of spherical models, we argue that the Einstein equations reduce to a simple form in the neighborhood of the null singularity. The main results arising from this approach are demonstrated using an almost plane symmetric model. The analysis shows that the null singularity results from the blueshift of the late-time gravitational wave tail; the amplitude of these gravitational waves is taken to decay as an inverse power of advanced time as suggested by perturbation theory. The divergence of the Weyl curvature at the null singularity is dominated by the propagating modes of the gravitational field. The null singularity is weak in the sense that tidal distortion remains bounded along timelike geodesics crossing the Cauchy horizon. These results are in agreement with previous analyses of black hole interiors. We briefly discuss some outstanding problems which must be resolved before the picture of the generic black hole interior is complete. 
  The role of the equivalence principle in the context of non-relativistic quantum mechanics and matter wave interferometry, especially atom beam interferometry, will be discussed. A generalised form of the weak equivalence principle which is capable of covering quantum phenomena too, will be proposed. It is shown that this generalised equivalence principle is valid for matter wave interferometry and for the dynamics of expectation values. In addition, the use of this equivalence principle makes it possible to determine the structure of the interaction of quantum systems with gravitational and inertial fields. It is also shown that the path of the mean value of the position operator in the case of gravitational interaction does fulfill this generalised equivalence principle. 
  The generalised Wick transform discovered by Thiemann provides a well-established relation between the Euclidean and Lorentzian theories of general relativity. We extend this Thiemann transform to the Ashtekar formulation for gravity coupled with spin-1/2 fermions, a non-Abelian Yang-Mills field, and a scalar field. It is proved that, on functions of the gravitational and matter phase space variables, the Thiemann transform is equivalent to the composition of an inverse Wick rotation and a constant complex scale transformation of all fields. This result holds as well for functions that depend on the shift vector, the lapse function, and the Lagrange multipliers of the Yang-Mills and gravitational Gauss constraints, provided that the Wick rotation is implemented by means of an analytic continuation of the lapse. In this way, the Thiemann transform is furnished with a geometric interpretation. Finally, we confirm the expectation that the generator of the Thiemann transform can be determined just from the spin of the fields and give a simple explanation for this fact. 
  Nonrotating and rotating black hole soltuions in (2+1) dimensions are studied in a model including a real scalar field with a simple potential coupled to gravity. 
  Rotating relativistic stars are receiving significant attention in recent years, because of the information they can yield about the equation of state of matter at extremely high densities and because they are one of the more possible sources of detectable gravitational waves. We review the latest theoretical and numerical methods for modeling rotating relativistic stars, including stars with a strong magnetic field and hot proto-neutron stars. We also review nonaxisymmetric oscillations and instabilities in rotating stars and summarize the latest developments regarding the gravitational wave-driven (CFS) instability in both polar and axial quasi-normal modes. 
  We show that the quantum dynamics of a real scalar field for a large class of potentials in the symmetric Gaussian state, where the nonperturbative quantum contributions are taken into account, can be described equivalently by a two-dimensional nonlinear dynamical system with a definite angular momentum (U(1) charge of a complex theory). It is found that the Gaussian state with a nearly minimal uncertainty and a large quantum fluctuation, as an initial condition, naturally explains the most of the essential features of the early stage of the inflationary Universe. 
  We clarify the way in which cosmological perturbations of quantum origin, produced during inflation, assume classical properties. Two features play an important role in this process: First, the dynamics of fluctuations which are presently on large cosmological scales leads to a very peculiar state (highly squeezed) that is indistinguishable, in a precise sense, from a classical stochastic process. This holds for almost all initial quantum states. Second, the process of decoherence by interaction with the environment distinguishes the field amplitude basis as a robust pointer basis. We discuss in detail the interplay between these features and use simple analogies such as the free quantum particle to illustrate the main conceptual issues. 
  We present a spherically symmetric and static exact solution of Quantum Einstein Equations. This solution is asymptotically (for large $r$) identical with the black hole solution on the anti--De Sitter background and, for some range of values of the mass possesses two horizons. We investigate thermodynamical properties of this solution. 
  Since the discovery of the large angular scale anisotropies in the microwave background radiation, the behaviour of cosmological perturbations (especially, density perturbations and gravitational waves) has been of great interest. In this study, after a detailed and rigorous treatment of the behaviour of gravitational waves in viscous cosmic media, we conclude that the damping of cosmological gravitational waves of long wavelengths is negligible for most cases of physical interest. A preliminary analysis suggests that similar results hold for density perturbations in the long wavelength limit. Therefore, long wavelength cosmological perturbations have not been practically affected by viscous processes,and are good probes of the very early Universe. 
  Stationary axisymmetric perfect fluid space-times are investigated using the curvature description of geometries. Attention is focused on space-times with a vanishing electric part of the Weyl tensor. It is shown that the only incompressible axistationary magnetic perfect fluid is the interior Schwarzschild solution. The existence of a rigidly rotating perfect fluid, generalizing the interior Schwarzschild metric is proven. Theorems are stated on Petrov types and electric/magnetic Weyl tensors. 
  We review higher-dimensional unified theories from the general relativity, rather than the particle physics side. Three distinct approaches to the subject are identified and contrasted: compactified, projective and noncompactified. We discuss the cosmological and astrophysical implications of extra dimensions, and conclude that none of the three approaches can be ruled out on observational grounds at the present time. 
  A new framework is proposed for general dynamic wormholes, unifying them with black holes. Both are generically defined locally by outer trapping horizons, temporal for wormholes and spatial or null for black and white holes. Thus wormhole horizons are two-way traversible, while black-hole and white-hole horizons are only one-way traversible. It follows from the Einstein equation that the null energy condition is violated everywhere on a generic wormhole horizon. It is suggested that quantum inequalities constraining negative energy break down at such horizons. Wormhole dynamics can be developed as for black-hole dynamics, including a reversed second law and a first law involving a definition of wormhole surface gravity. Since the causal nature of a horizon can change, being spatial under positive energy and temporal under sufficient negative energy, black holes and wormholes are interconvertible. In particular, if a wormhole's negative-energy source fails, it may collapse into a black hole. Conversely, irradiating a black-hole horizon with negative energy could convert it into a wormhole horizon. This also suggests a possible final state of black-hole evaporation: a stationary wormhole. The new framework allows a fully dynamical description of the operation of a wormhole for practical transport, including the back-reaction of the transported matter on the wormhole. As an example of a matter model, a Klein-Gordon field with negative gravitational coupling is a source for a static wormhole of Morris & Thorne. 
  When working in synchronous gauges, pseudo-tensor conservation laws are often used to set the initial conditions for cosmological scalar perturbations, when those are generated by topological defects which suddenly appear in an up to then perfectly homogeneous and isotropic universe. However those conservation laws are restricted to spatially flat (K=0) Friedmann-Lema\^\i tre spacetimes. In this paper, we first show that in fact they implement a matching condition between the pre- and post- transition eras and, in doing so, we are able to generalize them and set the initial conditions for all $K$. Finally, in the long wavelength limit, we encode them into a vector conservation law having a well-defined geometrical meaning. 
  We present the results of the computation of a twisting type N solution to vacuum Einstein equations following an iterative approach. Our results show that the higher order terms fail to provide a full exact solution with non-vanishing twist. Nevertheless, our fourth-order solution still represents a regular and twisting type N solution. 
  We find a one-parameter family of Lagrangian descriptions for classical general relativity in terms of tetrads which are not c-numbers. Rather, they obey exotic commutation relations. These noncommutative properties drop out in the metric sector of the theory, where the Christoffel symbols and the Riemann tensor are ordinary commuting objects and they are given by the usual expression in terms of the metric tensor. Although the metric tensor is not a c-number, we argue that all measurements one can make in this theory are associated with c-numbers, and thus that the common invariant sector of our one--parameter family of deformed gauge theories (for the case of zero torsion) is physically equivalent to Einstein's general relativity. 
  We present here an alternative approach to data setting for spacetimes with multiple moving black holes generalizing the Kerr-Schild form for rotating or non-rotating single black holes to multiple moving holes. Because this scheme preserves the Kerr-Schild form near the holes, it selects out the behaviour of null rays near the holes, may simplify horizon tracking, and may prove useful in computational applications. For computational evolution, a discussion of coordinates (lapse function and shift vector) is given which preserves some of the properties of the single-hole Kerr-Schild form. 
  We generalise results of Ford and Roman which place lower bounds -- known as quantum inequalities -- on the renormalised energy density of a quantum field averaged against a choice of sampling function. Ford and Roman derived their results for a specific non-compactly supported sampling function; here we use a different argument to obtain quantum inequalities for a class of smooth, even and non-negative sampling functions which are either compactly supported or decay rapidly at infinity. Our results hold in $d$-dimensional Minkowski space ($d\ge 2$) for the free real scalar field of mass $m\ge 0$. We discuss various features of our bounds in 2 and 4 dimensions. In particular, for massless field theory in 2-dimensional Minkowski space, we show that our quantum inequality is weaker than Flanagan's optimal bound by a factor of 3/2. 
  Both the non-homogeneous slowness of electromagnetic waves in gravitational fields and the frequency red shift contribute to the gravitational light bending. This twofold contribution explains the measured deflection of light rays by the Sun under Euclidean geometry of space. 
  We perform simulations of relativistic binary stars in post-Newtonian gravity to investigate their dynamical stability prior to merger against gravitational collapse in a tidal field. In general, our equations are only strictly accurate to first post-Newtonian order, but they recover full general relativity for spherical, static stars. We study both corotational and irrotational binary configurations of identical stars in circular orbits. We adopt a soft, adiabatic equation of state with $\Gamma = 1.4$, for which the onset of instability occurs at a sufficiently small value of the compaction $M/R$ that a post-Newtonian approximation is quite accurate. For such a soft equation of state there is no innermost stable circular orbit, so that we can study arbitrarily close binaries. This choice still allows us to study all the qualitative features exhibited by any adiabatic equation of state regarding stability against gravitational collapse. We demonstrate that, independent of the internal stellar velocity profile, the tidal field from a binary companion stabilizes a star against gravitational collapse. 
  It is shown that the main variable Z of the Null Surface Formulation of GR is the generating function of a constrained Lagrange submanifold that lives on the energy surface H=0 and that its level surfaces Z=const. are Legendre submanifolds on that energy surface.   The behaviour of the variable Z at the caustic points is analysed and a genralization of this variable is discussed. 
  We construct a 2+1 dimensional spacetime of constant curvature whose spatial topology is that of a torus with one asymptotic region attached. It is also a black hole whose event horizon spins with respect to infinity. An observer entering the hole necessarily ends up at a "singularity"; there are no inner horizons.   In the construction we take the quotient of 2+1 dimensional anti-de Sitter space by a discrete group Gamma. A key part of the analysis proceeds by studying the action of Gamma on the boundary of the spacetime. 
  In this paper we analyze the redshift as observed by an external observer receiving photons which terminate in the past at the naked singularity formed in a Tolman-Bondi dust collapse. Within the context of models considered here it is shown that photons emitted from a weak curvature naked singularity are always finitely redshifted to an external observer. Certain cases of strong curvature naked singularities, including the self-similar one, where the photons are infinitely redshifted are also pointed out. 
  In this paper we examine in detail the implementation, with its associated difficulties, of the Killing conditions and gauge fixing into the variational principle formulation of Bianchi-Type cosmologies. We address problems raised in the literature concerning the Lagrangian and the Hamiltonian formulations: We prove their equivalence, make clear the role of the Homogeneity Preserving Diffeomorphisms in the phase space approach, and show that the number of physical degrees of freedom is the same in the Hamiltonian and Lagrangian formulations. Residual gauge transformations play an important role in our approach, and we suggest that Poincar\'e transformations for special relativistic systems can be understood as residual gauge transformations. In Appendices, we give the general computation of the equations of motion and the Lagrangian for any Bianchi-Type vacuum metric and for spatially homogeneous Maxwell fields in a nondynamical background (with zero currents). We also illustrate our counting of degrees of freedom in an Appendix. 
  The suspension noise in interferometric gravitational wave detectors is caused by losses at the top and the bottom attachments of each suspension fiber. We use the Fluctuation-Dissipation theorem to argue that by careful positioning of the laser beam spot on the mirror face it is possible to reduce the contribution of the bottom attachment point to the suspension noise by several orders of magnitude. For example, for the initial and enhanced LIGO design parameters (i.e. mirror masses and sizes, and suspension fibers' lengths and diameters) we predict a reduction of $\sim 100$ in the "bottom" spectral density throughout the band $35-100\hbox{Hz}$ of serious thermal noise. We then propose a readout scheme which suppresses the suspension noise contribution of the top attachment point. The idea is to monitor an averaged horizontal displacement of the fiber of length $ l$; this allows one to record the contribution of the top attachment point to the suspension noise, and later subtract it it from the interferometer readout. For enhanced LIGO this would allow a suppression factor about 100 in spectral density of suspension thermal noise. 
  We present a new class of exact inflationary solutions for the evolution of a universe with spatial curvature, filled with a perfect fluid, a scalar field with potential $V_{\pm}(\phi)=\lambda(\phi^2\pm\delta^2)^2$ and a cosmological constant $\Lambda$. With the $V_+(\phi)$ potential and a negative cosmological constant, the scale factor experiments a graceful exit.   We give a brief discussion about the physical meaning of the solutions. 
  In Poincar\'e gauge theory of gravity and in $\overline{\mbox{Poincar\'e}}$ gauge theory of gravity, we give the necessary and sufficient condition in order that the Schwarzschild space-time expressed in terms of the Schwarzschild coordinates is obtainable as a torsionless exact solution of gravitational field equations with a spinless point-like source having the energy-momentum density $\widetilde{\mbox{\boldmath $T$}}_\mu^{~\nu}(x) = - Mc^2 \delta_\mu^{~0} \delta_0^{~\nu} \delta^{(3)}(\mbox{\boldmath $x$})$. Further, for the case when this condition is satisfied, the energy-momentum and the angular momentum of the Schwarzschild space-time are examined in their relations to the asymptotic forms of vierbein fields. We show, among other things, that asymptotic forms of vierbeins are restricted by requiring the equality of the active gravitational mass and the inertial mass. Conversely speaking, this equality is violated for a class of vierbeins giving the Schwarzschild metric. 
  It has been proposed to study the theory resulting from setting the gravitational constant to zero in the first order formalism for general relativity. In this letter we investigate this theory in the presence of matter fields, establish its equivalence with parametrized field theory on a flat background, and relate it to previous results in topological field theory (BF theory). 
  The dynamical process following the breaking of Weyl geometry to Riemannian geometry is considered by studying the motion of de Sitter bubbles in a Weyl vacuum. The bubbles are given in terms of an exact, spherically symmetric thin shell solution to the Einstein equations in a Weyl-Dirac theory with a time-dependent scalar field of the form beta = f(t)/r. The dynamical solutions obtained lead to a number of possible applications. An important feature of the thin shell model is the manner in which beta provides a connection between the interior and exterior geometries since information about the exterior geometry is contained in the boundary conditions for beta. 
  A relativistic model of a heat conducting collapsing star, which includes thermal pre-relaxation processes, is presented. Particular attention is paid to the influence of a given parameter defined in terms of thermodynamic variables, on the outcome of evolution. Evaluation of the system when passing through a critical value of the aforesaid parameter, does not yield evidence of anomalous behaviour. 
  In quantum field theory, there exist states in which the expectation value of the energy density for a quantized field is negative. These negative energy densities lead to many problems. Although quantum field theory introduces negative energies, it also provides constraints in the form of quantum inequalities (QI's). These uncertainty principle-type relations limit the magnitude and duration of any negative energy. We derive a general form of the QI on the energy density for both the quantized scalar and electromagnetic fields in static curved spacetimes. In the case of the scalar field, the QI can be written as the Euclidean wave operator acting on the Euclidean Green's function. Additionally, a small distance expansion on the Green's function is used to derive the QI in the short sampling time limit. It is found that the QI in this limit reduces to the flat space form with subdominant correction terms which depend on the spacetime geometry.   Several example spacetimes are studied in which exact forms of the QI's can be found. These include the three- and four-dimensional static Robertson-Walker spacetimes, flat space with perfectly reflecting mirrors, Rindler and static de Sitter space, and the spacetime outside a black hole. Finally, the application of the quantum inequalities to the Alcubierre warp drive spacetime leads to strict constraints on the thickness of the negative energy region needed to maintain the warp drive. Under these constraints, we discover that the total negative energy required exceeds the total mass of the visible universe by a hundred billion times. 
  It is pointed out that very recent results based on supernovae observations that the universe will accelerate and expand for ever with ever decreasing density have been predicted in a recent cosmological model which also deduces hitherto purely empirical features like the mysterious relation between the pion mass and the Hubble Constant. 
  A new interpretation of entanglement entropy is proposed: entanglement entropy of a pure state with respect to a division of a Hilbert space into two subspaces 1 and 2 is an amount of information, which can be transmitted through 1 and 2 from a system interacting with 1 to another system interacting with 2. The transmission medium is quantum entanglement between 1 and 2. In order to support the interpretation, suggestive arguments are given: variational principles in entanglement thermodynamics and quantum teleportation. It is shown that a quantum state having maximal entanglement entropy plays an important role in quantum teleportation. Hence, the entanglement entropy is, in some sense, an index of efficiency of quantum teleportation. Finally, implications for the information loss problem and Hawking radiation are discussed. 
  Recent work in the literature has studied a new set of local boundary conditions for the quantized gravitational field, where the spatial components of metric perturbations, and ghost modes, are subject to Robin boundary conditions, whereas normal components of metric perturbations obey Dirichlet boundary conditions. Such boundary conditions are here applied to evaluate the one-loop divergence on a portion of flat Euclidean four-space bounded by two concentric three-spheres. 
  Criteria which a space-time must satisfy to represent a point mass embedded in an open Robertson--Walker (RW) universe are given. It is shown that McVittie's solution in the case $k=0$ satisfies these criteria, but does not in the case $k=-1$. Existence of a solution for the case $k=-1$ is proven and its representation in terms of an elliptic integral is given. The following properties of this and McVittie's $k=0$ solution are studied; uniqueness, behaviour at future null infinity, recovery of the RW and Schwarzschild limits, compliance with energy conditions and the occurence of singularities. Existence of solutions representing more general spherical objects embedded in a RW universe is also proven. 
  We obtain a family of first-order symmetric hyperbolic systems for the Bianchi equations. They have only physical characteristics: the light cone and timelike hypersurfaces. In the proof of the hyperbolicity, new positivity properties of the Bel tensor are used. 
  The gravitational collapse of a complex scalar field in the harmonic map is modeled in spherical symmetry. Previous work has shown that a change of stability of the attracting critical solution occurs in parameter space from the discretely self-similarity critical (DSS) solution originally found by Choptuik to the continuously self-similar (CSS) solution found by Hirschmann and Eardley. In the region of parameter space in which the DSS is the attractor, a family of initial data is found which finds the CSS as its critical solution despite the fact that it has more than one unstable mode. An explanation of this is proposed in analogy to families that find the DSS in the region where the CSS is the attractor. 
  We study the asymptotic behavior at late times of Friedmann-Robertson-Walker (uniform density) cosmological models within scalar-tensor theories of gravity. Particularly, we analyze the late time behavior in the present (matter dominated) epoch of the universe. The result of Damour and Nordtvedt that for a massless scalar in a flat cosmology the Universe evolves towards a state indistinguishable from general relativity is generalized. We first study a massless scalar field in an open universe. It is found that, while the universe tends to approach a state with less scalar contribution to gravity, the attractor mechanism is not effective enough to drive the theory towards a final state indistinguishable from general relativity. For the self-interacting case it is found that the scalar field potential dominates the late time behavior. In most cases this makes the attractor mechanism effective, thus resulting in a theory of gravity with vanishingly small scalar contribution even for the open Universe. 
  I describe some examples in support of the conjecture that the horizon area of a near equilibrium black hole is an adiabatic invariant. These include a Schwarzschild black hole perturbed by quasistatic scalar fields (which may be minimally or nonminimally coupled to curvature), a Kerr black under the influence of scalar radiation at the superradiance treshold, and a Reissner--Nordstr\"om black hole absorbing a charge marginally. These clarify somewhat the conditions under which the conjecture would be true. The desired ``adiabatic theorem'' provides an important motivation for a scheme for black hole quantization. 
  We investigate $(n+1)$-dimensional string-dilaton cosmology with effective dilaton potential in presence of perfect-fluid matter.We get exact solutions parametrized by the constant $\gam$ of the state equation $p=(\gam-1)\rho$, the spatial dimension number $n$, the bulk of matter, and the spatial curvature constant $k$. Several interesting cosmological behaviours are selected. Finally we discuss the recovering of ordinary Einstein gravity starting from string dominated regime and a sort of asymptotic freedom due to string effective coupling. 
  Some examples in support of the conjecture that the horizon area of a near equilibrium black hole is an adiabatic invariant are described. These clarify somewhat the conditions under which the conjecture would be true. 
  We consider scalar perturbations of energy-density for a class of cosmological models where an early phase of accelerated expansion evolves, without any fine-tuning for graceful exit, towards the standard Friedman eras of observed universe. The quantum geometric procedure which generates such models agrees with results for string cosmology since it works if dynamics is dominated by a primordial fluid of extended massive objects. The main result is that characteristic scales of cosmological interest, connected with the extension of such early objects, are selected. 
  The construction of a consistent theory of quantum gravity is a problem in theoretical physics that has so far defied all attempts at resolution. One ansatz to try to obtain a non-trivial quantum theory proceeds via a discretization of space-time and the Einstein action. I review here three major areas of research: gauge-theoretic approaches, both in a path-integral and a Hamiltonian formulation, quantum Regge calculus, and the method of dynamical triangulations, confining attention to work that is strictly four-dimensional, strictly discrete, and strictly quantum in nature. 
  It is shown analytically that the Dirac equation has no normalizable, time-periodic solutions in a Reissner-Nordstrom black hole background; in particular, there are no static solutions of the Dirac equation in such a background field. The physical interpretation is that Dirac particles can either disappear into the black hole or escape to infinity, but they cannot stay on a periodic orbit around the black hole. 
  We investigate existence and properties of trapped surfaces in two models of collapsing null dust shells within the Gibbons-Penrose construction. In the first model, the shell is initially a prolate spheroid, and the resulting singularity forms at the ends first (relative to a natural time slicing by flat hyperplanes), in analogy with behavior found in certain prolate collapse examples considered by Shapiro and Teukolsky. We give an explicit example in which trapped surfaces are present on the shell, but none exist prior to the last flat slice, thereby explicitly showing that the absence of trapped surfaces on a particular, natural slicing does not imply an absence of trapped surfaces in the spacetime. We then examine a model considered by Barrabes, Israel and Letelier (BIL) of a cylindrical shell of mass M and length L, with hemispherical endcaps of mass m. We obtain a "phase diagram" for the presence of trapped surfaces on the shell with respect to essential parameters $\lambda \equiv M/L$ and $\mu \equiv m/M$. It is found that no trapped surfaces are present on the shell when $\lambda$ or $\mu$ are sufficiently small. (We are able only to search for trapped surfaces lying on the shell itself.) In the limit $\lambda \to 0$, the existence or nonexistence of trapped surfaces lying within the shell is seen to be in remarkably good accord with the hoop conjecture. 
  In this paper the isometries of the dual space were investigated. The dual structural equations of a Killing tensor of order two were found . The flat space case was analyzed in details. 
  In this paper the symmetries of the dual manifold were investigated. We found the conditions when the manifold and its dual admit the same Killing vectors and Killing-Yano tensors. In the case of an Einstein's metric $g_{\mu\nu}$ the corresponding equations for its dual were found. The examples of Kerr-Newman geometry and the separable coordinates in 1+1 dimensions were analyzed in details. 
  We present the conditions when a Killing-Yano tensor becomes a Nambu tensor. We have shown that in the flat space case all Killing-Yano tensors are Nambu tensors.  In the case of Taub-NUT metric and Kerr-Newmann metric we found that a Killing-Yano tensor of order two generate a Nambu tensor of order three. 
  We examine here the nature of the central singularity forming in the spherically symmetric collapse of a dust cloud and it is shown that this is always a strong curvature singularity where gravitational tidal forces diverge powerfully. An important consequence is that the nature of the naked singularity forming in the dust collapse turns out to be stable against the perturbations in the initial data from which the collapse commences. 
  In this paper we analyze the signal emitted by a compact binary system in the Jordan-Brans-Dicke theory. We compute the scalar and tensor components of the power radiated by the source and study the scalar waveform. Eventually we consider the detectability of the scalar component of the radiation by interferometers and resonant-mass detectors. 
  The standard tenet that Brans-Dicke theory reduces to general relativity in the omega-tends-to-infinity limit has been shown to be false when the trace of the matter energy-momentum tensor vanishes. The issue is clarified in a new approach and the asymptotic behaviour of the Brans-Dicke scalar is rigorously derived. 
  One-loop effects for quantum fields living on manifolds containing conical singularities are investigated in the context of cosmic string background and of finite-temperature theory in the Rindler wedge or outside the horizon of a Schwarzschild black-hole. 
  We extend the classical integrability of the CGHS model of 2d dilaton gravity [1] to a larger class of models, allowing the gravitational part of the action to depend more generally on the dilaton field and, simultaneously, adding fermion- and U(1)-gauge-fields to the scalar matter. On the other hand we provide the complete solution of the most general dilaton-dependent 2d gravity action coupled to chiral fermions. The latter analysis is generalized to a chiral fermion multiplet with a non-abelian gauge symmetry as well as to the (anti-)self-dual sector df = *df (df = -*df) of a scalar field f. 
  I present two examples in which the curvature singularity of a radiation-dominated Universe is regularized by (a) the repulsive effects of spin interactions, and (b) the repulsive effects arising from a breaking of the local gravitational gauge symmetry. In both cases the collapse of an initial, asymptotically flat state is stopped, and the Universe bounces towards a state of decelerated expansion. The emerging picture is typical of the pre-big bang scenario, with the main difference that the string cosmology dilaton is replaced by a classical radiation fluid, and the solutions are not duality-invariant. 
  It is shown that Julia-Zee dyons do not admit slowly rotating excitations. This is achieved by investigating the complete set of stationary excitations which can give rise to non-vanishing angular momentum. The relevant zero modes are parametrized in a gauge invariant way and analyzed by means of a harmonic decomposition. Since general arguments show that the solutions to the linearized Bogomol'nyi equations cannot contribute to the angular momentum, the relevant modes are governed by a set of electric and a set of non self-dual magnetic perturbation equations. The absence of axial dipole deformations is also established. 
  Gamma-ray bursts are believed to result from the coalescence of binary neutron stars. However, the standard proposals for conversion of the gravitational energy to thermal energy have difficulties. We show that if the merger of the two neutron stars results in a naked singularity, instead of a black hole, the ensuing quantum particle creation can provide the requisite thermal energy in a straightforward way. The back-reaction of the created particles can avoid the formation of the naked singularity predicted by the classical theory. Hence cosmic censorship holds in the quantum theory, even if it were to be violated in classical general relativity. 
  A longstanding conjecture by Belinskii, Khalatnikov, and Lifshitz that the singularity in generic gravitational collapse is spacelike, local, and oscillatory is explored analytically and numerically in spatially inhomogeneous cosmological spacetimes. With a convenient choice of variables, it can be seen analytically how nonlinear terms in Einstein's equations control the approach to the singularity and cause oscillatory behavior. The analytic picture requires the drastic assumption that each spatial point evolves toward the singularity as an independent spatially homogeneous universe. In every case, detailed numerical simulations of the full Einstein evolution equations support this assumption. 
  It is well known that the perturbation equations of massless fields for the Kerr-de Sitter geometry can be written in the form of separable equations. The equations have five definite singularities so that the analysis has been expected to be difficult. We show that these equations can be transformed to Heun's equations, for which we are able to use known technique for the analysis of the solutions. We reproduce results known previously for the Kerr geometry and de Sitter geometry in the confluent limits of the Heun's functions. Our analysis applies can be extended to Kerr-Newman-de Sitter geometry for massless fields with spin 0 and 1/2. 
  We consider Einstein Gravity coupled to dynamical matter consisting of a gauge field with any compact gauge group and minimally coupled scalar fields. We investigate the conditions under which a free specification of a spatial field configuration for the total system and its derivative with respect to coordinate-time determines a solution to the field equations (generalized thin-sandwich problem). Sufficient conditions for local solvability (in the space of fields) are established. 
  This article gives an elementary review of gravitational collapse and the cosmic censorship hypothesis. Known models of collapse resulting in the formation of black holes and naked singularities are summarized. These models, when taken together, suggest that the censorship hypothesis may not hold in classical general relativity. The nature of the quantum processes that take place near a naked singularity, and their possible implication for observations, is briefly discussed. 
  The local galactic cluster, the Great attractor, embeds us in a dimensionless gravitational potential of about - 3 x 10^{-5}. In the solar system this potential is constant to about 1 part in 10^{11}. Consequently, planetary orbits, which are determined by the gradient in the gravitational potential, remain unaffected. However, this is not so for the recently introduced flavor-oscillation clocks where the new redshift-inducing phases depend on the gravitational potential itself. On these grounds, and by studying the invariance properties of the gravitational phenomenon in the weak fields, we argue that there exists an element of incompleteness in the general-relativistic description of gravitation. An incompleteness-establishing inequality is derived and an experiment is outlined to test the thesis presented. 
  By resolving the gravitational field into electric and magnetic parts, we define an electrogravity duality transformation and discover an interesting property of the field. Under the duality transformation a vacuum/flat spacetime maps into the original spacetime with a topological defect of global monopole/texture. The elctrogravity-duality is thus a topological defect generating process. It turns out that all black hole solutions possess dual solutions that imbibe a global monopole. 
  We reply to L.P. Grishchuk's comment (gr-qc/9801011) on our paper ``The Influence of Cosmological Transitions on the Evolution of Density Perturbations'' (Phys. Rev. D 57, 3302, 1998). We show that his points of criticism are not correct. 
  We show that the equations of motion of two-dimensional dilaton gravity conformally coupled to a scalar field can be reduced to a single non-linear second-order partial differential equation when the coordinates are chosen to coincide with the two scalar fields, the matter field $f$ and the dilaton $\phi$, which are present in the theory. This result may help solve and understand two- and higher-dimensional classical and quantum gravity. 
  We investigate collapse of a spherical cloud of counter-rotating particles. An explicit solution is given using an elliptic integral. If the specific angular momentum $L(r)=O(r^2)$ at $r\to 0$, no central singularity occurs. With $L(r)$ like that, there is a finite region around the center that bounces. On the other hand, if the order of $L(r)$ is higher than that, a central singularity occurs. In marginally bound collapse with $L(r)=4F(r)$, a naked singularity occurs, where $F(r)$ is the Misner-Sharp mass. The solution for this case is expressed by elementary functions. For $ 4 <L/F<\infty$ at $r\to0$, there is a finite region around the center that bounces and a naked singularity occurs. For $ 0 \le L/F< 4$ at $r\to0$, there is no such region. The results suggests that rotation may play a crucial role on the final fate of collapse. 
  We study the influence of reheating on super-horizon density perturbations and gravitational waves. We correct wrong claims [L. P. Grishchuk Phys. Rev. D 50, 7154 (1994) and gr-qc/9801011] about the joining of perturbations at cosmological transitions and about the quantization of cosmological perturbations. 
  Two strands of observational gravitation, one the search for astrophysical evidence of primordial black holes and the other the search for gravitational waves, may combine to provide strong evidence in favour of cosmological models based on superstring theory, the leading candidate for unifying gravity with the other fundamental forces. 
  Motivated by the consistent histories approach to quantum mechanics, we examine a simple model of hydrodynamic coarse-graining for a scalar field. It consists in averaging the field over spatial regions of size L and constructing the evolution equation for the coarse grained quantities, thus identifying dissipation and noise. 
  In this essay, we assume that negative mass objects can exist in the extragalactic space and analyze the consequences of their microlensing on light from distant Active Galactic Nuclei. We find that such events have very similar features to some observed Gamma Ray Bursts and use recent satellite data to set an upper bound to the amount of negative mass in the universe. 
  It is proposed that space is a four-dimensional Euclidean space with universal time. Originally this space was filled with a uniform substance, pictured as a liquid, which at some time became supercooled. Our universe began as a nucleation event initiating a liquid to solid transition. The universe we inhabit and are directly aware of consists of only the three-dimensional expanding phase boundary. Random energy transfers to the boundary from thermal fluctuations in the adjacent bulk phases are interpreted by us as quantum fluctuations. Fermionic matter is modeled as screw dislocations; gauge bosons as phonons. Minkowski space emerges dynamically through redefining local time to be proportional to the spatial coordinate perpendicular to the boundary. Other features include a geometrical quantum gravitational theory, and an explanation of quantum measurement. 
  The calculations of Guang-Wen Ma's recent Letter [Phys.Lett. A239, (1998) 209] contain an easily detectable error which makes his conclusions irrelevant. 
  We demonstrate chaotic behavior of timelike, null and spacelike geodesics in non-homogeneous vacuum pp-wave solutions. This seems to be the first known example of a chaotic motion in exact radiative spacetime. 
  Trans-Planckian particles are elementary particles accelerated such that their energies surpass the Planck value. There are several reasons to believe that trans-Planckian particles do not represent independent degrees of freedom in Hilbert space, but they are controlled by the cis-Planckian particles. A way to learn more about the mechanisms at work here, is to study black hole horizons, starting from the scattering matrix Ansatz.   By compactifying one of the three physical spacial dimensions, the scattering matrix Ansatz can be exploited more efficiently than before. The algebra of operators on a black hole horizon allows for a few distinct representations. It is found that this horizon can be seen as being built up from string bits with unit lengths, each of which being described by a representation of the SO(2,1) Lorentz group. We then demonstrate how the holographic principle works for this case, by constructing the operators corresponding to a field in space-time. The parameter t turns out to be quantized in Planckian units, divided by the period R of the compactified dimension. 
  In this essay we show that an uncharged black-hole moving superluminally in a transparent dielectric medium violates Hawking's area theorem. The violation is overcome through the emission of radiation. Since modes cannot emerge from the black hole itself, this radiation must originate from a collective effect in the medium, in complete analogy with the Vavilov-Cherenkov effect. However, because the black-hole is uncharged, the emission mechanism must be different. We discuss the physical origin of the effect and obtain a Newtonian estimative. Then we obtain the appropriate equations in the relativistic case and show that the field which is radiated away is a combination of gravitational and electromagnetic degrees of freedom. Possible astrophysical relevance for the detection of primordial black-holes and binary systems is discussed. 
  It is shown that any second order dynamic equation on a configuration space $X$ of non-relativistic time-dependent mechanics can be seen as a geodesic equation with respect to some (non-linear) connection on the tangent bundle $TX\to X$ of relativistic velocities. Using this fact, the relationship between relativistic and non-relativistic equations of motion is studied. 
  In this paper we address the issue of determining the semiclassical threshold for black hole formation in the context of a one-parameter family of theories which continuously interpolates between the RST and BPP models. We find that the results depend significantly on the initial static configuration of the spacetime geometry before the influx of matter is turned on. In some cases there is a critical energy density, given by the Hawking rate of evaporation, as well as a critical mass $m_{cr}$ (eventually vanishing). In others there is neither $m_{cr}$ nor a critical flux. 
  The paper is devoted to the description a measurable time-interval (``proper time'') in the Hamiltonian version of general relativity with the Dirac-ADM metric. To separate the dynamical parameter of evolution from the space metric we use the Lichnerowicz conformally invariant variables. In terms of these variables GR is equivalent to the conformally invariant Penrose-Chernicov-Tagirov theory of a scalar field the role of which is played by the scale factor multiplied on the Planck constant. Identification of this scalar field with the modulus of the Higgs field in the standard model of electroweak and strong interactions allows us to formulate an example of conformally invariant unified theory where the vacuum averaging of the scalar field is determined by cosmological integrals of motion of the Universe evolution. 
  Instabilities in finite difference codes due to the singularity of spherical coordinates at the center are studied. In typical Numerical Relativity applications, standard regularization techniques by themselves do not ensure long term stability. A proposal to remedy that problem is presented, which takes advantage of redundant quantities introduced in recent hyperbolic formulations of Einstein's evolution equations. The results are discussed through the example case of a boson star, where a significant improvement in the implementation of boundary conditions is also presented. 
  We study the stability of cosmological scaling solutions within the class of spatially homogeneous cosmological models with a perfect fluid subject to the equation of state p_gamma=(gamma-1) rho_gamma (where gamma is a constant satisfying 0 < gamma < 2) and a scalar field with an exponential potential. The scaling solutions, which are spatially flat isotropic models in which the scalar field energy density tracks that of the perfect fluid, are of physical interest. For example, in these models a significant fraction of the current energy density of the Universe may be contained in the scalar field whose dynamical effects mimic cold dark matter. It is known that the scaling solutions are late-time attractors (i.e., stable) in the subclass of flat isotropic models. We find that the scaling solutions are stable (to shear and curvature perturbations) in generic anisotropic Bianchi models when gamma < 2/3. However, when gamma > 2/3, and particularly for realistic matter with gamma >= 1, the scaling solutions are unstable; essentially they are unstable to curvature perturbations, although they are stable to shear perturbations. We briefly discuss the physical consequences of these results. 
  The experimental results of the two-photon absorption(TPA) and M\"{o}ssbauer-rotor(MR) for testing the isotropy of the speed of light are explained in an ether drift model with a drift velocity of $\sim 10^{-3}c$. Further tests of the ether drift assumption are suggested. 
  Exact dynamical equations for a generic dust matter source field in a cosmological context are formulated with respect to a non-comoving Newtonian-like timelike reference congruence and investigated for internal consistency. On the basis of a lapse function $N$ (the relativistic acceleration scalar potential) which evolves along the reference congruence according to $\dot{N} = \alpha \Theta N$ ($\alpha = {const}$), we find that consistency of the quasi-Newtonian dynamical equations is not attained at the first derivative level. We then proceed to show that a self-consistent set can be obtained by linearising the dynamical equations about a (non-comoving) FLRW background. In this case, on properly accounting for the first-order momentum density relating to the non-relativistic peculiar motion of the matter, additional source terms arise in the evolution and constraint equations describing small-amplitude energy density fluctuations that do not appear in similar gravitational instability scenarios in the standard literature. 
  The equivalence principle is treated on a mathematically rigorous base on sufficiently general subsets of a differentiable manifold. This is carried out using the basis of derivations of the tensor algebra over that manifold. Necessary and/or sufficient conditions of existence, uniqueness, and holonomicity of these bases in which the components of the derivations of the tensor algebra over it vanish on these subsets, are studied. The linear connections are considered in this context. It is shown that the equivalence principle is identically valid at any point, and along any path, in every gravitational theory based on linear connections. On higher dimensional submanifolds it may be valid only in certain exceptional cases. 
  It is often taken for granted that on board a rotating disk it is possible to operate a \QTR{it}{global}3+1 splitting of space-time, such that both lengths and time intervals are \QTR{it}{uniquely} defined in terms of measurements performed by real rods and real clocks at rest on the platform. This paper shows that this assumption, although widespread and apparently trivial, leads to an anisotropy of the velocity of two light beams travelling in opposite directions along the rim of the disk; which in turn implies some recently pointed out paradoxical consequences undermining the self-consistency of the Special Theory of Relativity (SRT). A correct application of the SRT solves the problem and recovers complete internal consistency for the theory. As an immediate consequence, it is shown that the Sagnac effect only depends on the non homogeneity of time on the platform and has nothing to do with any anisotropy of the speed of light along the rim of the disk, contrary to an incorrect but widely supported idea. 
  Brans-Dicke gravity is remarkable not only in that General Relativity and Mach's Principle find a common enlarged scenario where they are mutually consistent, but also in that it provides a very interesting quantum cosmological model within the inflationary paradigm. The interplay between the Brans-Dicke scalar $\Phi$ and the inflaton field $\sigma$ plays an important r\^{o}le during the course of inflation, and although the dynamics as such is governed by the potential, the onset and the end of inflation are determined by the values of both fields jointly. The relative position of the beginning-- and end-of-inflation curves (BoI and EoI respectively) is the most relevant factor in determining the resulting quantum cosmological scenario. The classification of potentials that is given in this paper is based on the criterion of whether the BoI and EoI boundaries enclose a finite or infinite area in the ($\sigma$,$\Phi$) plane where inflation takes place. It is shown that this qualitative classification distinguishes two classes of potentials that yield very different cosmologies and it is argued that only those theories in which BoI and EoI enclose a finite area in the ($\sigma$,$\Phi$) plane are compatible with our observable universe. 
  Some general properties of local $\zeta$-function procedures to renormalize some quantities in $D$-dimensional (Euclidean) Quantum Field Theory in curved background are rigorously discussed for positive scalar operators $-\Delta + V(x)$ in general closed $D$-manifolds, and a few comments are given for nonclosed manifolds too. A general comparison is carried out with respect to the more known point-splitting procedure concerning the effective Lagrangian and the field fluctuations. It is proven that, for $D>1$, the local $\zeta$-function and point-splitting approaches lead essentially to the same results apart from some differences in the subtraction procedure of the Hadamard divergences. It is found that the $\zeta$ function procedure picks out a particular term $w_0(x,y)$ in the Hadamard expansion. Also the presence of an untrivial kernel of the operator $-\Delta +V(x)$ may produce some differences between the two analyzed approaches. Finally, a formal identity concerning the field fluctuations, used by physicists, is discussed and proven within the local $\zeta$-function approach. This is done also to reply to recent criticism against $\zeta$-function techniques. 
  We study the effective action for gravity obtained after the integration of scalar matter fields, using the local momentum representation based on the Riemann normal coordinates expansion. By considering this expansion around different space-time points, we also compute the non-local terms together with the more usual divergent ones. We discuss the applicability of our results to the calculation of particle production rates in cosmological backgrounds and compare this method with the traditional Bogolyubov transformations. 
  We study the pseudo-local gravitoelectromagnetic stress-energy tensor for an arbitrary gravitational field within the framework of general relativity. It is shown that there exists a current of gravitational energy around a rotating mass. This gravitational analog of the Poynting flux is evaluated for certain classes of observers in the Kerr field. 
  The existence of conserved quantities with a structure similar to the Newman-Penrose quantities in a polyhomogeneous space-time is addressed. The most general form for the initial data formally consistent with the polyhomogeneous setting is found. The subsequent study is done for those polyhomogeneous space-times where the leading term of the shear contains no logarithmic terms. It is found that for these space-times the original NP quantities cease to be constants, but it is still possible to construct a set of other 10 quantities that are constant. From these quantities it is possible to obtain as a particular case a conserved quantity found by Chrusciel et al. 
  We calculate the quantum stress tensor for a massless scalar field in the 2-d self-similar spherical dust collapse model which admits a naked singularity. We find that the outgoing radiation flux diverges on the Cauchy horizon. This may have two consequences. The resultant back reaction may prevent the naked singularity from forming, thus preserving cosmic censorship through quantum effects. The divergent flux may lead to an observable signature differentiating naked singularities from black holes in astrophysical observations. 
  A one-loop correction of the quasilocal energy in the Schwarzschild background, with flat space as a reference metric, is performed by means of a variational procedure in the Hamiltonian framework. We examine the graviton sector in momentum space, in the lowest possible state. An application to the black hole pair creation via the Casimir energy is presented. Implications on the foam-like scenario are discussed. 
  The emission of radiation by a uniformly accelerated charge is analyzed. According to the standard approach, a radiation is observed whenever there is a relative acceleraion between the charge and the observer. Analyzing difficulties that arose in the standard approach, we propose that a radaition is created whenever a relative acceleration between the charge and its own electric field exists. The electric field induced by a charge accelerated by an external (nongravitational) force, is not accelerated with the charge. Hence the electric field is curved in the instantanous rest frame of the accelerated charge. This curvature gives rise to a stress force, and the work done to overcome the stress force is the source of the energy carried by the radiation. In this way, the "energy balance paradox" finds its solution. 
  Gravitational waves which are smooth and contain two asymptotically flat regions are constructed from the homogeneous pp-waves vacuum solution. Motion of free test particles is calculated explicitly and the limit to an impulsive wave is also considered. 
  We consider the numerical evolution of black hole initial data sets, consisting of single black holes distorted by strong gravitational waves, with a full 3D, nonlinear evolution code. These data sets mimic the late stages of coalescing black holes. We compare various aspects of the evolution of axisymmetric initial data sets, obtained with this 3D code, to results obtained from a well established axisymmetric code. In both codes we examine and compare the behavior of metric functions, apparent horizon properties, and waveforms, and show that these dynamic black holes can be accurately evolved in 3D. In particular we show that with present computational resources and techniques, the process of excitation and ringdown of the black hole can be evolved, and one can now extract accurately the gravitational waves emitted from the 3D Cartesian metric functions, even when they carry away only a small fraction ($<< 1%$) of the rest mass energy of the system. Waveforms for both the $\ell=2$ and the much more difficult $\ell=4$ and $\ell=6$ modes are computed and compared with axisymmetric calculations. In addition to exploring the physics of distorted black hole data sets, and showing the extent to which the waves can be accurately extracted, these results also provide important testbeds for all fully nonlinear numerical codes designed to evolve black hole spacetimes in 3D, whether they use singularity avoiding slicings, apparent horizon boundary conditions, or other evolution methods. 
  We give ansatze for solving classically the initial value constraints of general relativity minimally coupled to a scalar field, electromagnetism or Yang-Mills theory. The results include both time-symmetric and asymmetric data. The time-asymmetric examples are used to test Penrose's cosmic censorship inequality. We find that the inequality can be violated if only the weak energy condition holds. 
  We find a number of complex solutions of the Einstein equations in the so-called unimodular version of general relativity, and we interpret them as saddle points yielding estimates of a gravitational path integral over a space of almost everywhere Lorentzian metrics on a spacetime manifold with topology of the "no-boundary" type. In this setting, the compatibility of the no-boundary initial condition with the definability of the quantum measure reduces reduces to the normalizability and unitary evolution of the no-boundary wave function \psi. We consider the spacetime topologies R^4 and RP^4 # R^4 within a Taub minisuperspace model with spatial topology S^3, and the spacetime topology R^2 x T^2 within a Bianchi type I minisuperspace model with spatial topology T^3. In each case there exists exactly one complex saddle point (or combination of saddle points) that yields a wave function compatible with normalizability and unitary evolution. The existence of such saddle points tends to bear out the suggestion that the unimodular theory is less divergent than traditional Einstein gravity. In the Bianchi type I case, the distinguished complex solution is approximately real and Lorentzian at late times, and appears to describe an explosive expansion from zero size at T=0. (In the Taub cases, in contrast, the only complex solution with nearly Lorentzian late-time behavior yields a wave function that is normalizable but evolves nonunitarily, with the total probability increasing exponentially in the unimodular "time" in a manner that suggests a continuous creation of new universes at zero volume.) The issue of the stability of these results upon the inclusion of more degrees of freedom is raised. 
  We examine the evolution, under gravitational radiation reaction, of slightly eccentric equatorial orbits of point particles around Kerr black holes. Our method involves numerical integration of the Sasaki-Nakamura equation. It is discovered that such orbits decrease in eccentricity throughout most of the inspiral, until shortly before the innermost stable circular orbit (ISCO), when a critical radius $r_{\text{crit}}$ is reached beyond which the inspiralling orbits increase in eccentricity. It is shown that the number of orbits remaining in this last (eccentricity increasing) phase of the inspiral is an order of magnitude less for prograde orbits around rapidly spinning black holes than for retrograde orbits. In the extreme limit of a Kerr black hole with spin parameter $a=1$, this critical radius descends into the ``throat'' of the black hole. 
  A few corrections and comments are made upon a previously published paper by the author (Gen. Rel. Gravit. 24, 199 (1992)), on the subject of cosmological models with compact spatial sections. 
  Spherically symmetric, vacuum solutions in 5D and 7D Kaluza-Klein theory are obtained. These solutions are flux tubes with constant cross-sectional size, located between (+) and (-) Kaluza-Klein's ``electrical'' and ``magnetic'' charges disposed respectively at $\pm\infty$ and filled by constant Kaluza-Klein ``electrical'' and ``magnetic'' fields. These objects are surprisingly similar to the flux tubes which forms between two monopoles in Type-II superconductors and also the hypothesized color field flux tube that is thought to form between two quarks in the QCD vacuum. 
  To write down a path integral for the Ashtekar gravity one must solve three fundamental problems. First, one must understand rules of complex contour functional integration with holomorphic action. Second, one should find which gauges are compatible with reality conditions. Third, one should evaluate the Faddeev-Popov determinant produced by these conditions. In the present paper we derive the BRST path integral for the Hilbert-Palatini gravity. We show, that for certain class of gauge conditions this path integral can be re-written in terms of the Ashtekar variables. Reality conditions define contours of integration. For our class of gauges all ghost terms coincide with what one could write naively just ignoring any Jacobian factors arising from the reality conditions. 
  In a previous paper with Gibbons [CMP 120 (1987) 295] we derived a list of three dimensional symmetric space $\sigma$-model obtained by dimensional reduction of a class of four dimensional gravity theories with abelian gauge fields and scalars. Here we give a detailed analysis of their group theoretical structure leading to an abstract parametrization in terms of `triangular' group elements. This allows for a uniform treatment of all these models. As an interesting application we give a simple derivation of a `Quadratic Mass Formula' for strictly stationary black holes. 
  Integrability in general tetrad formalisms is reviewed, following and clarifying work of Papapetrou and Edgar. The integrability conditions are (combinations of) the Bianchi equations and their consequences. The introduction of additional constraints is considered. Recent results on the conservation of constraints in the 1+3 covariant formulation of cosmology are shown to follow from the Bianchi equations 
  If one encodes the gravitational degrees of freedom in an orthonormal frame field there is a very natural first order action one can write down (which in four dimensions is known as the Goldberg action). In this essay we will show that this action contains a boundary action for certain microscopic degrees of freedom living at the horizon of a black hole, and argue that these degrees of freedom hold great promise for explaining the microstates responsible for black hole entropy, in any number of spacetime dimensions. This approach faces many interesting challenges, both technical and conceptual. 
  We develop a Hamiltonian formulation of Bianchi type-I cosmological model in conformal gravity, i.e. the theory described by a Lagrangian which involves the quadratic curvature invariant constructed from the Weyl tensor, in four dimensions. We derive the explicit forms of the super-Hamiltonian and the constraint expressing the conformal invariance of the theory, and we write down the system of canonical equations. To seek out exact solutions to this system we add extra constraints on the canonical variables and we go through a global involution algorithm that possibly leads to the closure of the constraint algebra. This enables us to extract all possible particular solutions that may be written in closed analytical form. On the other hand, probing the local analytical structure we show that the system does not possess the Painleve property (presence of movable logarithms) and that it is therefore not integrable. We stress that there is a very fruitful interplay of local integrability-related methods such as the Painleve test and global techniques such as the involution algorithm. Strictly speaking, we demonstrate that the global involution algorithm has proven to be exhaustive in the search for exact solutions. The conformal relationship of the solutions, or absence thereof, with Einstein spaces is highlighted. 
  This essay elucidates recent achievements of the "nongravitating vacuum energy" (NGVE) theory" which has the feature that a shift of the Lagrangian density by a constant does not affect dynamics. In the first order formalism, a constraint appears that enforces the vanishing of the cosmological constant \Lambda. Standard dynamics of gauge unified theories (including fermions) and their SSB appear if a four index field strength condensate is present. At a vacuum state there is exact balance to zero of the gauge fields condensate and the original scalar fields potential. As a result it is possible to combine the solution of the \Lambda problem with inflation and transition to a \Lambda =0 phase without fine tuning after a reheating period. The model opens new possibilities for a solution of the hierarchy problem. 
  Colombeau's generalized functions are used to adapt the distributional approach to singular hypersurfaces in general relativity with signature change. Equations governing the dynamics of singular hypersurface is obtained and it is shown that the stress-energy tensor of the surface can be non-vanishing. 
  We compute zero-frequency (neutral) quasi-normal f-modes of fully relativistic and rapidly rotating neutron stars, using several realistic equations of state (EOSs) for neutron star matter. The zero-frequency modes signal the onset of the gravitational radiation-driven instability. We find that the l=m=2 (bar) f-mode is unstable for stars with gravitational mass as low as 1.0 - 1.2 M_\odot, depending on the EOS. For 1.4 M_\odot neutron stars, the bar mode becomes unstable at 83 % - 93 % of the maximum allowed rotation rate. For a wide range of EOSs, the bar mode becomes unstable at a ratio of rotational to gravitational energies T/W \sim 0.07-0.09 for 1.4 M_\odot stars and T/W \sim 0.06 for maximum mass stars. This is to be contrasted with the Newtonian value of T/W \sim 0.14. We construct the following empirical formula for the critical value of T/W for the bar mode,   (T/W)_2 = 0.115 - 0.048 M / M_{max}^{sph}, which is insensitive to the EOS to within 4 - 6 %. This formula yields an estimate for the neutral mode sequence of the bar mode as a function only of the star's mass, M, given the maximum allowed mass, M_{max}^{sph}, of a nonrotating neutron star. The recent discovery of the fast millisecond pulsar in the supernova remnant N157B, supports the suggestion that a fraction of proto-neutron stars are born in a supernova collapse with very large initial angular momentum. Thus, in a fraction of newly born neutron stars the instability is a promising source of continuous gravitational waves. It could also play a major role in the rotational evolution (through the emission of angular momentum) of merged binary neutron stars, if their post-merger angular momentum exceeds the maximum allowed to form a Kerr black hole. 
  The geodesic as well as the geodesic deviation equation for impulsive gravitational waves involve highly singular products of distributions $(\theta\de$, $\theta^2\de$, $\de^2$). A solution concept for these equations based on embedding the distributional metric into the Colombeau algebra of generalized functions is presented. Using a universal regularization procedure we prove existence and uniqueness results and calculate the distributional limits of these solutions explicitly. The obtained limits are regularization independent and display the physically expected behavior. 
  Exact particle-like static, spherically and/or cylindrically symmetric solutions to the equations of interacting scalar and electromagnetic field system have been obtained within the scope of general relativity. In particular, we considered Freedman-Robertson-Walker (FRW) space-time as an external homogenous and isotropic gravitational field whereas the inhomogenous and isotropic Universe is given by the G$\ddot o$del model. The solutions obtained have been thoroughly studied for different types of interaction term. It has been shown that in FRW space-time equations with different interaction terms may have stable solutions while within the scope of G$\ddot o$del model only the droplet-like configurations may be stable, if they are located in the region where $g^{00}>0$. 
  It is shown that experiments of the Einstein-Podolski-Rosen type are the natural consequence of the groupoid approach to noncommutative unification of general relativity and quantum mechanics. The geometry of this model is determined by the noncommutative algebra of complex valued, compactly supported functions (with convolution as multiplication) on the groupoid G = E x D. In the model considered in the present paper E is the total space of the frame bundle over space-time, and D is the Lorentz group. Correlations of the EPR type should be regarded as remnants of the totally non-local physics below the Planck threshold which is modelled by a noncommutative geometry. 
  The brick-wall model seeks to explain the Bekenstein-Hawking entropy as a wall-contribution to the thermal energy of ambient quantum fields raised to the Hawking temperature. Reservations have been expressed concerning the self-consistency of this model. For example, it predicts large thermal energy densities near the wall, producing a substantial mass-correction and, presumably, a large gravitational back-reaction. We re-examine this model and conclude that these reservations are unfounded once the ground state---the Boulware state---is correctly identified. We argue that the brick-wall model and the Gibbons-Hawking instanton (which ascribes a topological origin to the Bekenstein-Hawking entropy) are mutually exclusive, alternative descriptions (complementary in the sense of Bohr) of the same physics. 
  We investigate stable central structures in multiply-connected, anti de Sitter spacetimes with spherical, planar and hyperbolic geometries. We obtain an exact solution for the pressure in terms of the radius when the density is constant. We find that, apart from the usual simply-connected spherically symmetric star with a well-behaved metric at $r=0$, the only solutions with non-singular pressure and density have a wormhole topology. However these wormhole solutions must be composed of matter which violates the weak energy condition. Admitting this type of matter, we obtain a structure which is maintained via a balance between its cohesive tension and its repulsive negative matter density. If the tension is insufficiently large, this structure can collapse to a black hole of negative mass. 
  We consider a series of distorted black hole initial data sets, and develop techniques to evolve them using the linearized equations of motion for the gravitational wave perturbations on a Schwarzschild background. We apply this to 2D and 3D distorted black hole spacetimes. In 2D, waveforms for different modes of the radiation are presented, comparing full nonlinear evolutions for different axisymmetric l-modes with perturbative evolutions. We show how axisymmetric black hole codes solving the full, nonlinear Einstein equations are capable of very accurate evolutions, and also how these techniques aid in studying nonlinear effects. In 3D we show how the initial data for the perturbation equations can be computed, and we compare with analytic solutions given from a perturbative expansion of the initial value problem. In addition to exploring the physics of these distorted black hole data sets, in particular allowing an exploration of linear, nonlinear, and mode mixing effects, this approach provides an important testbed for any fully nonlinear numerical code designed to evolve black hole spacetimes in 2D or 3D. 
  We consider the general process of pair-creation of charged rotating black holes. We find that instantons which describe this process are necessarily complex due to regularity requirements. However their associated probabilities are real, and fully consistent with the interpretation that the entropy of a charged rotating black hole is the logarithm of the number of its quantum states. 
  The coupling of a stringlike fluid with ordinary matter and gravity may lead to a closed Universe with the dynamic of an open one. This can provide an alternative solution for the age and horizon problems. A study of density perturbations of the stringlike fluid indicates the existence of instabilities in the small wavelength limit when it is employed a hydrodynamic approach. Here, we extend this study to gravitational waves, where the hydrodynamical approach plays a less important role, and we argue that traces of the existence of this fluid must be present in the anisotropies of the cosmic background radiation. 
  The standard approach to initial data for both analytic and numerical computations of black hole collisions has been to use conformally-flat initial geometry. Among other advantages, this choice allows the simple superposition of holes with arbitrary mass, location and spin. The conformally flat restriction, however, is inappropriate to the study of Kerr holes, for which the standard constant-time slice is not conformally flat. Here we point out that for axisymmetric arrangements of rotating holes, a nonconformally flat form of the 3-geometry can be chosen which allows fairly simple superposition of Kerr holes with arbitrary mass and spin. We present initial data solutions representing locally Kerr holes at large separation, and representing rotating holes close enough so that outside a common horizon the spacetime geometry is a perturbation of a single Kerr hole. 
  When ambient seismic waves pass near an interferometric gravitational-wave detector, they induce density perturbations in the earth which produce fluctuating gravitational forces on the interferometer's test masses. These forces mimic a stochastic background of gravitational waves and thus constitute noise. We compute this noise using the theory of multimode Rayleigh and Love waves propagating in a layered medium that approximates the geological strata at the LIGO sites. We characterize the noise by a transfer function $T(f) \equiv \tilde x(f)/\tilde W(f)$ from the spectrum of direction averaged ground motion $\tilde W(f)$ to the spectrum of test mass motion $\tilde x(f) = L\tilde h(f)$ (where $L$ is the length of the interferometer's arms, and $\tilde h(f)$ is the spectrum of gravitational-wave noise). This paper's primary foci are (i) a study of how $T(f)$ depends on the various seismic modes; (ii) an attempt to estimate which modes are excited at the LIGO sites at quiet and noisy times; and (iii) a corresponding estimate of the seismic gravity-gradient noise level. At quiet times the noise is below the benchmark noise level of ``advanced LIGO interferometers'' (although not by much near 10 Hz); it may significantly exceed this level at noisy times. The lower edge of our quiet-time noise is a limit beyond which there is little gain from further improvements in vibration isolation and thermal noise, unless one also reduces seismic gravity-gradient noise. Two methods of reduction are discussed: monitoring the earth's density perturbations, computing their gravitational forces, and correcting the data for those forces; and constructing narrow moats around the interferometers' test masses to shield out the fundamental-mode Rayleigh waves, which we suspect dominate at quiet times. 
  The difference in travel time of corotating and counter-rotating light waves in the field of a central massive and spinning body is studied. The corrections to the special relativistic formula are worked out in a Kerr field. Estimation of numeric values for the Earth and satellites in orbit around it show that a direct measurement is in the order of concrete possibilities. 
  We describe U(N)-monopoles (N > 1) on Kerr black holes by the parameters of the moduli space of holomorphic vector U(N)-bundles over two-sphere with the help of the Grothendieck splitting theorem. For N = 2,3 we obtain this description in an explicit form as well as the estimates for the corresponding monopole masses. This gives a possibility to adduce some reasonings in favour of existence of both a fine structure for Kerr black holes and the statistical ensemble tied with it which might generate the Kerr black hole entropy. 
  Doppler tracking of interplanetary spacecraft provides the only method presently available for broad-band searches of low frequency gravitational waves. The instruments have a peak sensitivity around the reciprocal of the round-trip light-time T of the radio link connecting the Earth to the space-probe and therefore are particularly suitable to search for coalescing binaries containing massive black holes in galactic nuclei. A number of Doppler experiments -- the most recent involving the probes ULYSSES, GALILEO and MARS OBSERVER -- have been carried out so far; moreover, in 2002-2004 the CASSINI spacecraft will perform three 40 days data acquisition runs with expected sensitivity about twenty times better than that achieved so far. Central aims of this paper are: (i) to explore, as a function of the relevant instrumental and astrophysical parameters, the Doppler output produced by in-spiral signals -- sinusoids of increasing frequency and amplitude (the so-called chirp); (ii) to identify the most important parameter regions where to concentrate intense and dedicated data analysis; (iii) to analyze the all-sky and all-frequency sensitivity of the CASSINI's experiments, with particular emphasis on possible astrophysical targets, such as our Galactic Centre and the Virgo Cluster. 
  Spherical neutron star models are studied within tensor-scalar theories of gravity. Particularly, it is shown that, under some conditions on the second derivative of the coupling function and on star's mass, for a given star there exist two strong-scalar-field solutions as well as the usual weak-field one. This last solution happens to be unstable and a star, becoming massive enough to allow for all three solutions, evolves to reach one of the strong field configurations. This transition is dynamically computed and it appears that the star radiates away the difference in energy between both states (a few $10^{-3} M_\odot c^2$) as gravitational radiation. Since part of the energy ($\sim 10^{-5} M_\odot c^2$) is injected into the star as kinetic energy, the velocity of star's surface can reach up to $10^{-2} c$. The waveform of this monopolar radiation is shown as well as the oscillations undergone by the star. These oscillations are also studied within the slowly-rotating approximation, in order to estimate an order of magnitude of the resulting quadrupolar radiation. 
  We investigate Bianchi type IX ''Mixmaster'' universes within the framework of the low-energy tree-level effective action for string theory, which (when the ''stringy'' 2-form axion potential vanishes) is formally the same as the Brans-Dicke action with $\omega =-1$. We show that, unlike the case of general relativity in vacuum, there is no Mixmaster chaos in these string cosmologies. In the Einstein frame an infinite sequence of chaotic oscillations of the scale factors on approach to the initial singularity is impossible, as it was in general relativistic Mixmaster universes in the presence of stiff -fluid matter (or a massless scalar field). A finite sequence of oscillations of the scale factors approximated by Kasner metrics is possible, but it always ceases when all expansion rates become positive. In the string frame the evolution through Kasner epochs changes to a new form which reflects the duality symmetry of the theory. Again, we show that chaotic oscillations must end after a finite time. The need for duality symmetry appears to be incompatible with the presence of chaotic behaviour as $t\to 0$. We obtain our results using the Hamiltonian qualitative cosmological picture for Mixmaster models. We mention possible relations of this picture to diffeomorphism-independent methods of measuring chaos in general relativity. Finally, we discuss the effect of inhomogeneities and higher dimensions on the possible emergence of chaos within the string cosmology. 
  The material below the crust of a neutron star is understood to be describable in terms of three principal independently moving constituents, identifiable as neutrons, protons, and electrons, of which the first two are believed to form mutually coupled bosonic condensates. The large scale comportment of such a system will be that of a positively charged superconducting superfluid in a negatively charged ``normal'' fluid background. As a contribution to the development of the theory of such a system, the present work shows how, subject to neglect of dissipative effects, it is possible to set up an elegant category of simplified but fully relativistic three-constituent superconducting superfluid models whose purpose is to provide realistic approximations for cases in which a strictly conservative treatment is sufficient. A "mesoscopic" model, describing the fluid between the vortices, is constructed, as well as a "macroscopic" model taking into account the average effect of quantised vortices. 
  The modelling of light-like signals in General Relativity taking the form of impulsive gravitational waves and light-like shells of matter is examined. Systematic deductions from the Bianchi identities are made. These are based upon Penrose's hierarchical classification of the geometry induced on the null hypersurface history of the surface by its imbedding in the space-times to the future and to the past of it. The signals are not confined to propagate in a vacuum and thus their interaction with matter (a burst of radiation propagating through a cosmic fluid, for example) is also studied. Results are accompanied by illustrative examples using cosmological models, vacuum space-times, the de sitter univers and Minkowskian space-time. 
  One of the greatest unsolved issues of the physics of this century is to find a quantum field theory of gravity. According to a vast amount of literature unification of quantum field theory and gravitation requires a gauge theory of gravity which includes torsion and an associated spin field. Various models including either massive or massless torsion fields have been suggested. We present arguments for a massive torsion field, where the probable rest mass of the corresponding spin three gauge boson is the Planck mass. 
  We calculate partition functions for lens spaces L_{p,q} up to p=8 and for genus 1 and 2 handlebodies H_1, H_2 in the Turaev-Viro framework. These can be interpreted as transition amplitudes in 3D quantum gravity. In the case of lens spaces L_{p,q} these are vacuum-to-vacuum amplitudes $\O -> \O$, whereas for the 1- and 2-handlebodies H_1, H_2 they represent genuinely topological transition amplitudes $\O -> T^2$ and $\O -> T^2 # T^2$, respectively. 
  Recently, 't Hooft's S-matrix for black hole evaporation, obtained from the gravitational interactions between the in-falling particles and Hawking radiation, has been generalised to include transverse effects. The action describing the collision turned out to be a string theory action with an antisymmetric tensor background. In this article we show that the model reproduces both the correct longitudinal and transverse dynamics, even when one goes beyond the eikonal approximation or particles collide at nonvanishing incidence angles. It also gives the correct momentum tranfer that takes place in the process. Including a curvature on the horizon provides the action with an extra term, which can be interpreted as a dilaton contribution. The amplitude of the scattering is seen to reproduce the Veneziano amplitude in a certain limit, as in earlier work by 't Hooft. The theory resembles a "holographic" field theory, in the sense that it only depends on the horizon degrees of freedom, and the in- and out-Hilbert spaces are the same. The operators representing the coordinates of in- and out-going particles are non-commuting, and Heisenberg's uncertainty principle must be corrected by a term proportional to the ratio of the ingoing momentum to the impact parameter, times Newton's constant. Reducing to 2+1 dimensions, we find that the coordinates satisfy an SO(2,1) algebra. 
  We show that under particular circumstances a general relativistic spherically symmetric bounded distribution of matter could satisfy a nonlocal equation of state. This equation relates, at a given point, the components of the corresponding energy momentum tensor not only as function at that point, but as a functional throughout the enclosed configuration. We have found that these types of dynamic bounded matter configurations, with constant gravitational potentials at the surface, admit a Conformal Killing Vector and fulfill the energy conditions for anisotropic imperfect fluids. We also present several analytical and numerical models satisfying these equations of state which collapse as reasonable radiating anisotropic spheres in general relativity. 
  The Wheeler-DeWitt equation in the minisuperspace approximation is studied in four different models. Under certain circumstances each model leads to a tunneling potential and under the same circumstances the classical version of each model leads to inflation. 
  We present results from the first fully nonlinear numerical calculations of the head--on collision of two unequal mass black holes. Selected waveforms of the most dominant l=2, 3 and 4 quasinormal modes are shown, as are the total radiated energies and recoil velocities for a range of mass ratios and initial separations. Our results validate the close and distant separation limit perturbation studies, and suggest that the head--on collision scenario is not likely to produce an astrophysically significant recoil effect. 
  The descriptions of Reissner-Nordstroem and Kerr-Newman dilatonic black holes in the Einstein frame are compared to those in the string frame. We describe various physical measurements in the two frames and show which experiments can distinguish between the two frames. In particular we discuss the gyromagnetic ratios of black holes, the decay law via Hawking radiation and the propagation of light on black hole backgrounds. 
  We suggest that a satellite with a stable atomic clock on board be sent through the Earth-Sun gravitational saddle point to experimentally determine whether Nature prefers static solutions of the field equations of General Relativity, such as the standard Schwarzschild solution, or whether Nature prefers equivalent non-static solutions. This is a test of the boundary conditions of General Relativity rather than of the field equations. The fractional difference in clock rates between the two possibilities is a part in a hundred million. This is a large and easily measurable effect. 
  Kaluza-Klein reduction of 3D gravity with minimal scalars leads to 2D dilaton-Maxwell gravity with dilaton coupled scalars. Evaluating the one-loop effective action for dilaton coupled scalars in large $N$ and s-wave approximation we apply it to study quantum evolution of BTZ black hole. It is shown that quantum-corrected BTZ BH may evaporate or else anti-evaporate similarly to 4D Nariai BH as is observed by Bousso and Hawking. Instable higher modes in the spectrum indicate also the possibility of proliferation of BTZ BH. 
  Recently Herzlich proved a Penrose-like inequality with a coefficient being a kind of a Sobolev constant. We show that this constant tends to zero for charged black holes approaching maximal Reissner-Nordstroem solutions. The method proposed by Herzlich is not appropriate for charged matter with nonzero global charge.  
  We present the first results for Cauchy nonlinear evolution of 3D, nonaxisymmetric distorted black holes. We focus on the extraction and verification of 3D waveforms determined by numerical relativity. We show that the black hole evolution can be accurately followed through the ringdown period, and comparing with a recently developed perturbative evolution technique, we show that many waveforms in the black hole spectrum of modes, such as l=2 and l=4, including weakly excited nonaxisymmetric modes with m not zero, can be accurately evolved and extracted from the full nonlinear numerical evolution. We also identify new physics contained in higher modes, due to nonlinear effects. The implications for simulations related to gravitational wave astronomy are discussed. 
  We look for four dimensional Einstein-Weyl spaces equipped with a regular Bianchi metric. Using the explicit 4-parameters expression of the distance obtained in a previous work for non-conformally-Einstein Einstein-Weyl structures, we show that only four 1-parameter families of regular metrics exist on orientable manifolds : they are all of Bianchi $IX$ type and conformally K\"ahler ; moreover, in agreement with general results, they have a positive definite conformal scalar curvature. In a Gauduchon's gauge, they are compact and we obtain their topological invariants. Finally, we compare our results to the general analyses of Madsen, Pedersen, Poon and Swann : our simpler parametrisation allows us to correct some of their assertions. 
  Conformo static charged dust distributions are investigated in the framework of General Relativity. Einstein's equations reduce to a nonlinear version of Poisson's equations and Maxwell's eqations imply the equaltiy of the charge and mass densities. An interior solution to the extreme Reissner-Nordstrom metric is given. Dust distributions concentrated on regular surfaces are discussed and a complete solution is given for a spherical thin shell. 
  Examples are presented of applications of the Lema\^{i}tre - Tolman model to problems of astrophysics and gravitation theory. They are: 1. Inferring the spatial distribution of matter by interpretation of observations; 2. Interaction of inhomogeneities in matter distribution with the CMB radiation; 3. Evolution of voids; 4. Singularities; 5. Influence of electromagnetic field on gravitational collapse. This review is meant to demonstrate that the theory is already well-prepared to meet the challenges posed by an inhomogeneous Universe. 
  We solve Einstein equations coupled to a complex scalar field with infinitely large self-interaction, degenerate fermions, and a negative cosmological constant in $(2+1)$ dimensions. Exact solutions for static boson-fermion stars are found when circular symmetry is assumed. We find that the minimum binding energy of boson-fermion star takes a negative value if the value of the cosmological constant is sufficiently small. 
  The basic framework for a systematic construction of a quantum theory of Riemannian geometry was introduced recently. The quantum versions of Riemannian structures --such as triad and area operators-- exhibit a non-commutativity. At first sight, this feature is surprising because it implies that the framework does not admit a triad representation. To better understand this property and to reconcile it with intuition, we analyze its origin in detail. In particular, a careful study of the underlying phase space is made and the feature is traced back to the classical theory; there is no anomaly associated with quantization. We also indicate why the uncertainties associated with this non-commutativity become negligible in the semi-classical regime. 
  We consider scalar perturbations of energy--density for a class of cosmological models where an early phase of accelerated expansion evolves, without any fine--tuning for graceful exit, towards the standard Friedman eras of observed universe. The geometric procedure which generates such models agrees with results for string cosmology since it works if dynamics is dominated by a primordial fluid of extended massive objects. The main result is that characteristic scales of cosmological interest, connected with the extension of such early objects, are selected.} 
  An effective time--dependent cosmological constant can be recovered for higher--order theories of gravity by extending to these ones the no--hair conjecture. The results are applied to some specific cosmological models. 
  In order to gain insight into the possible Ground State of Quantized Einstein's Gravity, we have derived a variational calculation of the energy of the quantum gravitational field in an open space, as measured by an asymptotic observer living in an asymptotically flat space-time. We find that for Quantum Gravity (QG) it is energetically favourable to perform its quantum fluctuations not upon flat space-time but around a ``gas'' of wormholes of mass m_p, the Planck mass (m_p ~= 10^{19}GeV) and average distance l_p, the Planck length a_p(a_p ~= 10^{-33}cm). As a result, assuming such configuration to be a good approximation to the true Ground State of Quantum Gravity, space-time, the arena of physical reality, turns out to be well described by Wheeler's quantum foam and adequately modeled by a space-time lattice with lattice constant l_p, the Planck lattice. 
  The linear cosmological perturbation theory of almost homogeneous and isotropic perfect fluid and scalar field universes is reconsidered and formally simplified. Using the existence of a covariant conserved quantity on large perturbation scales, a closed integral expression for comoving energy density perturbations is obtained for arbitrary equations of state. On this basis we establish a simple relation between fluid energy density perturbations at `reentry' into the horizon and the corresponding scalar field quantities at the first Hubble scale crossing during an early de Sitter phase of a standard inflationary scenario. 
  Torsion in a 5D spacetime is considered. In this case gravitation is defined by the 5D metric and the torsion. It is conjectured that torsion is connected with a spinor field. In this case Dirac's equation becomes the nonlinear Heisenberg equation. It is shown that this equation has a discrete spectrum of solutions with each solution being regular on the whole space and having finite energy. Every solution is concentrated on the Planck region and hence we can say that torsion should play an important role in quantum gravity in the formation of bubbles of spacetime foam. On the basis of the algebraic relation between torsion and the classical spinor field in Einstein-Cartan gravity the geometrical interpretation of the spinor field is considered as ``the square root'' of torsion. 
  Spacetime foam can be modeled in terms of nonlocal effective interactions in a classical nonfluctuating background. Then, the density matrix for the low-energy fields evolves, in the weak-coupling approximation, according to a master equation that contains a diffusion term. Furthermore, it is argued that spacetime foam behaves as a quantum thermal field that, apart from inducing loss of coherence, gives rise to effects such as gravitational Lamb and Stark shifts as well as quantum damping in the evolution of the low-energy observables. These effects can be, at least in principle, experimentally tested. 
  The different kinds of self-similarity in general relativity are discussed, with special emphasis on similarity of the ``first'' kind, corresponding to spacetimes admitting a homothetic vector. We then survey the various classes of self-similar solutions to Einstein's field equations and the different mathematical approaches used in studying them. We focus mainly on spatially homogenous and spherically symmetric self-similar solutions, emphasizing their possible roles as asymptotic states for more general models. Perfect fluid spherically symmetric similarity solutions have recently been completely classified, and we discuss various astrophysical and cosmological applications of such solutions. Finally we consider more general types of self-similar models. 
  Using Visser's semi-analytical model for the stress-energy tensor corresponding to the conformally coupled massless scalar field in the Unruh vacuum, we examine, by explicitly evaluating the relevant integrals over half-complete geodesics, the averaged weak (AWEC) and averaged null (ANEC) energy conditions along with Ford-Roman quantum inequality-type restrictions on negative energy in the context of four dimensional evaporating black hole backgrounds. We find that in all cases where the averaged energy conditions fail, there exist quantum inequality bounds on the magnitude and duration of negative energy densities. 
  According to a recent suggestion [1], the energy--momentum tensor for gravitating fields can be computed through a suitable rearrangement of the matter field equations, without relying on the variational definition. We show that the property observed by Accioly et al. in [1] is the consequence of a general identity, which follows from the covariance of the matter Lagrangian in much the same way as (generalized) Bianchi identities follow from the covariance of the purely gravitational Lagrangian. However, we also show that only in particular cases can this identity be used to obtain the actual form of the stress-energy tensor, while in general the method leads to ambiguities and possibly to wrong results. Moreover, in nontrivial cases the computations turn out to be more difficult than the standard variational technique. 
  We show that the class of hyperelliptic solutions to the Ernst equation (the stationary axisymmetric Einstein equations in vacuum) previously discovered by Korotkin and Neugebauer and Meinel can be derived via Riemann-Hilbert techniques. The present paper extends the discussion of the physical properties of these solutions that was begun in a Physical Review Letter, and supplies complete proofs. We identify a physically interesting subclass where the Ernst potential is everywhere regular except at a closed surface which might be identified with the surface of a body of revolution. The corresponding spacetimes are asymptotically flat and equatorially symmetric. This suggests that they could describe the exterior of an isolated body, for instance a relativistic star or a galaxy. Within this class, one has the freedom to specify a real function and a set of complex parameters which can possibly be used to solve certain boundary value problems for the Ernst equation. The solutions can have ergoregions, a Minkowskian limit and an ultrarelativistic limit where the metric approaches the extreme Kerr solution. We give explicit formulae for the potential on the axis and in the equatorial plane where the expressions simplify. Special attention is paid to the simplest non-static solutions (which are of genus two) to which the rigidly rotating dust disk belongs. 
  We examine ways to write the Choptuik critical solution as the evolution of scale invariant variables. It is shown that a system of scale invariant variables proposed by one of the authors does not evolve periodically in the Choptuik critical solution. We find a different system, based on maximal slicing. This system does evolve periodically, and may generalize to the case of axisymmetry or of no symmetry at all. 
  We consider an action which consists of two terms: the first S_{1}=\int L_{1}\Phi d^{4}x and the second S_{2}=\int L_{2}\sqrt{-g}d^{4}x where \Phi is a measure which has to be determined dynamically. S_{1} satisfies the requirement that the transformation L_{1}\to L_{1}+const does not effect equations of motion. In the first order formalism, a constraint appears which allows to solve \chi =\Phi/\sqrt{-g}. Then, in a true vacuum state (TVS), \chi\to\infty and in the conformal Einstein frame no singularities are present, the energy density of TVS is zero without fine tuning of any scalar potential in S_{1} or S_{2}. When considering only a linear potential for a scalar field \phi in S_{1}, the continuous symmetry \phi\to\phi+const is respected. Surprisingly, in this case SSB takes place while no massless ("Goldstone") boson appears. 
  The propagation of free electromagnetic radiation in the field of a plane gravitational wave is investigated. A solution is found one order of approximation beyond the limit of geometrical optics in both transverse--traceless (TT) gauge and Fermi Normal Coordinate (FNC) system. The results are applied to the study of polarization perturbations. Two experimental schemes are investigated in order to verify the possibility to observe these perturbations, but it is found that the effects are exceedingly small. 
  We analyze here the issue of local versus the global visibility of a singularity that forms in gravitational collapse of a dust cloud, which has important implications for the weak and strong versions of the cosmic censorship hypothesis. We find conditions as to when a singularity will be only locally naked, rather than being globally visible, thus preseving the weak censorship hypothesis. The conditions for formation of a black hole or naked singularity in the Szekeres quasi-spherical collapse models are worked out. The causal behaviour of the singularity curve is studied by examining the outgoing radial null geodesics, and the final outcome of collapse is related to the nature of the regular initial data specified on an initial hypersurface from which the collapse evolves. An interesting feature that emerges is the singularity in Szekeres spacetimes can be ``directionally naked''. 
  We investigate the general process of black hole pair creation in a cosmological background, considering the creation of charged and rotating black holes. We motivate the use of Kerr-Newmann-deSitter solutions to investigate this process, showing how they arise from more general C-metric type solutions that describe a pair of general black holes accelerating away from each other in a cosmological background. All possible KNdS-type spacetimes are classified and we examine whether they may be considered to be in full thermodynamic equilibrium. Instantons that mediate the creation of these space-times are constructed and we see that they are necessarily complex due to regularity requirements. Thus we argue that instantons need not always be real Euclidean solutions to the Einstein equations. Finally, we calculate the actions of these instantons and find that the standard action functional must be modified to correctly take into account the effects of the rotation. The resultant probabilities for the creation of the space-times are found to be real and consistent with the interpretation that the entropy of a charged and rotating black hole is the logarithm of the number of its quantum states. 
  Non-local boundary conditions for Euclidean quantum gravity are proposed, consisting of an integro-differential boundary operator acting on metric perturbations. In this case, the operator P on metric perturbations is of Laplace type, subject to non-local boundary conditions; by contrast, its adjoint is the sum of a Laplacian and of a singular Green operator, subject to local boundary conditions. Self-adjointness of the boundary-value problem is correctly formulated by looking at Dirichlet-type and Neumann-type realizations of the operator P, following recent results in the literature. The set of non-local boundary conditions for perturbative modes of the gravitational field is written in general form on the Euclidean four-ball. For a particular choice of the non-local boundary operator, explicit formulae for the boundary-value problem are obtained in terms of a finite number of unknown functions, but subject to some consistency conditions. Among the related issues, the problem arises of whether non-local symmetries exist in Euclidean quantum gravity. 
  The connection between four different approaches to quantization of the relativistic particle is studied: reduced phase space quantization, Dirac quantization, BRST quantization, and (BRST)-Fock quantization are each carried out. The connection to the BFV path integral in phase space is provided. In particular, it is concluded that that the full range of the lapse should be used in such path integrals. The relationship between all these approaches is established. 
  The hypothesis that the energy-momentum tensor of ordinary matter is not conserved separately, leads to a non-adiabatic expansion and, in many cases, to an Universe older than usual. This may provide a solution for the entropy and age problems of the Standard Cosmological Model. We consider two different theories of this type, and we perform a perturbative analysis, leading to analytical expressions for the evolution of gravitational waves, rotational modes and density perturbations. One of these theories exhibits satisfactory properties at this level, while the other one should be discarded. 
  We propose a new 4D state sum model, related to the balanced model, which is constructed using the octonions, or equivalently, triality. An effective continuum physical theory constructed from this model coupled to the balanced model would have a non-vanishing cosmological constant, chiral asymmetry, and a gauge group related to the octonions. 
  Reexamination of general relativistic experimental results shows the universe is governed by Einstein's static-spacetime general relativity instead of Friedmann-Lemaitre expanding-spacetime general relativity. The absence of expansion redshifts in a static-spacetime universe suggests a reevaluation of the present cosmology is needed. 
  The work argues the principle of equivalence to be a theorem and not a principle (in a sense of an axiom). It contains a detailed analysis of the concepts of normal and inertial frame of reference. The equivalence principle is proved to be valid (at every point and along every path) in any gravitational theory based on linear connections. Possible generalizations of the equivalence principle are pointed out. 
  It's shown that Saa's model of gravity with propagating torsion is inconsistent with basic solar-system gravitational experiments. 
  Negative energy densities in the Dirac field produced by state vectors that are the superposition of two single particle electron states are examined. I show that for such states the energy density of the field is not bounded from below and that the quantum inequalities derived for scalar fields are satisfied. I also show that it is not possible to produce negative energy densities in a scalar field using state vectors that are arbitrary superpositions of single particle states. 
  We take a critical look at a recent conjecture concerning the past attractor in the pre-big-bang scenario. We argue that the Milne universe is unlikely to be a general past attractor for such models and support this with a number of examples. 
  The behaviour of quantum metric perturbations produced during inflation is considered at the stage after the second Hubble radius crossing. It is shown that the classical correlation between amplitude and momentum of a perturbation mode, previously shown to emerge in the course of an effective quantum-to-classical transition, is maintained for a sufficiently long time, and we present the explicit form in which it takes place using the Wigner function. We further show with a simple diffraction experiment that quantum interference, non-expressible in terms of a classical stochastic description of the perturbations, is essentially suppressed. Rescattering of the perturbations leads to a comparatively slow decay of this correlation and to a complete stochastization of the system. 
  We show how to determine the typical phase space volume $\Gamma$ for primordial gravitational waves produced during an inflationary stage, which is invariant under squeezing. An expression for $\Gamma$ is found in the long wavelength regime. The quasi-classical entropy of a pure vacuum initial state defined as the logarithm of $\Gamma$ modulo a constant remains zero in spite of the generation of fluctuations (creation of real gravitons). 
  An explicit calculation is carried out to show that the distributional curvature of a 2-cone, calculated by Clarke et al. (1996), using Colombeau's new generalised functions is invariant under non-linear $C^\infty$ coordinate transformations. 
  We show that a gap exists in the allowed sizes of all zero temperature static spherically symmetric black holes in semiclassical gravity when only conformally invariant fields are present. The result holds for both charged and uncharged black holes. By size we mean the proper area of the event horizon. The range of sizes that do not occur depends on the numbers and types of quantized fields that are present. We also derive some general properties that both zero and nonzero temperature black holes have in all classical and semiclassical metric theories of gravity. 
  We study in the physical frame the phenomenon of spontaneous scalarization that occurs in scalar-tensor theories of gravity for compact objects. We discuss the fact that the phenomenon occurs exactly in the regime where the Newtonian analysis indicates it should not. Finally we discuss the way the phenomenon depends on the equation of state used to describe the nuclear matter. 
  It is well known that it takes matter that violates the averaged weak energy condition to hold the throat of a wormhole open. The production of such ``exotic'' matter is usually discussed within the context of quantum field theory. In this paper I show that it is possible to produce the exotic matter required to hold a wormhole open classically. This is accomplished by coupling a scalar field to matter that satisfies the weak energy condition. The energy-momentum tensor of the scalar field and the matter separately satisfy the weak energy condition, but there exists an interaction energy-momentum tensor that does not. It is this interaction energy-momentum tensor that allows the wormhole to be maintained. 
  Following a survey of the abstract boundary definition of Scott and Szekeres, a rigidity result is proved for the smooth case, showing that the topological structure of the regular part of this boundary in invariantly defined. 
  In the context of the pre-big bang scenario the large-scale CMB anisotropy can be seeded by a primordial background of very light (or massless) axion fluctuations. In that case the slope of the temperature anisotropy spectrum, allowed by present observations, defines an allowed range of values for the string mass scale. Conversely, from the theoretical expected value of the string scale we can predict the slope of the anisotropy spectrum. In both cases there is a remarkable agreement between observations and theoretical expectations. 
  We obtain the vacuum spherical symmetric solutions for the gravitational sector of a (4+d)-dimensional Kaluza-Klein theory. In the various regions of parameter space, the solutions can describe either naked singularities or black-holes or wormholes. We also derive, by performing a conformal rescaling, the corresponding picture in the four-dimensional space-time. 
  The formal Ermakov approach for empty FRW minisuperspace cosmological models of arbitrary Hartle-Hawking factor ordering and the corresponding squeezing features are briefly discussed as a possible means of describing cosmological evolution 
  Maxwell electrodynamics, considered as a source of the classical Einstein field equations, leads to the singular isotropic Friedmann solutions. We show that this singular behavior does not occur for a class of nonlinear generalizations of the electromagnetic theory. A mathematical toy model is proposed for which the analytical nonsingular extension of FRW solutions is obtained. 
  We present further evidence that the second post-Newtonian (pN) approximation to the gravitational waves emitted by inspiraling compact binaries is sufficient for the detection of these systems. This is established by comparing the 2-pN wave forms to signals calculated from black hole perturbation theory. Results are presented for different detector noise curves. We also discuss the validity of this type of analysis. 
  The quasi-local energy conservation law is derived from the vacuum Einstein's equations on the timelike boundary surface in the canonical (2,2)-formalism of general relativity. The quasi-local energy and energy flux integral agree with the standard results in the asymptotically flat limit and in spherically symmetric spacetimes. 
  We present a straightforward and self-contained introduction to the basics of the loop approach to quantum gravity, and a derivation of what is arguably its key result, namely the spectral analysis of the area operator. We also discuss the arguments supporting the physical prediction following this result: that physical geometrical quantities are quantized in a non-trivial, computable, fashion. These results are not new; we present them here in a simple form that avoids the many non-essential complications of the first derivations. 
  In this Report we outline some basic results on generalized Finsler--Kaluza--Klein gravity and locally anisotropic strings. There are investigated exact solutions for locally anisotropic Friedmann--Robertson--Walker universes and three dimensional and string black holes with generic anisotropy. 
  Although negative energy densities are predicted by relativistic quantum field theories, I present an argument that an "operational" positivity still holds: the energy in a region, plus the energy of an isolated device which traps or measures that energy, must be positive. If we assume Einstein's field equation, this means the local geometry of a negative energy-density region cannot be measured by the trajectories of test particles.    So far, all attempts to design thought-experiments to verify a classical local geometry in the negative energy-density region have failed. It seems we must impute a quantum character to such a space-time regime. 
  In this work we analyze the emission of gravitational waves from a gravitational system described by a Newtonian term plus a H\'enon-Heiles term. The main concern is to analyze how the inclusion of the Newtonian term changes the emission of gravitational waves, considering its emission in the chaotic and regular regime. 
  We argue that the conventional quantum field theory in curved spacetime has a grave drawback: The canonical commutation relations for quantum fields and conjugate momenta do not hold. Thus the conventional theory should be denounced and the related results revised. A Hamiltonian version of the canonical formalism for a free scalar quantum field is advanced, and the fundamentals of an appropriate theory are constructed. The principal characteristic feature of the theory is quantum-gravitational nonlocality: The Schroedinger field operator at time t depends on the metric at t in the whole 3-space. Applications to cosmology and black holes are given, the results being in complete agreement with those of general relativity for particles in curved spacetime. A model of the universe is advanced, which is an extension of the Friedmann universe; it lifts the problem of missing dark matter. A fundamental and shocking result is the following: There is no particle creation in the case of a free quantum field in curved spacetime; in particular, neither the expanding universe nor black holes create particles. 
  A recent cosmological model is recapitulated which deduces the correct mass, radius and age of the universe as also the Hubble constant and other well known apparently coincidental relations. It also predicts an ever expanding accelerating universe as is confirmed by latest supernovae observations. Finally the Big Bang model is recovered as a suitable limiting case. 
  We study a general field theory of a scalar field coupled to gravity through a quadratic Gauss-Bonnet term $\xi(\phi) R^2_{GB}$. The coupling function has the form $\xi(\phi)=\phi^n$, where $n$ is a positive integer. In the absence of the Gauss-Bonnet term, the cosmological solutions for an empty universe and a universe dominated by the energy-momentum tensor of a scalar field are always characterized by the occurrence of a true cosmological singularity. By employing analytical and numerical methods, we show that, in the presence of the quadratic Gauss-Bonnet term, for the dual case of even $n$, the set of solutions of the classical equations of motion in a curved FRW background includes singularity-free cosmological solutions. The singular solutions are shown to be confined in a part of the phase space of the theory allowing the non-singular solutions to fill the rest of the space. We conjecture that the same theory with a general coupling function that satisfies certain criteria may lead to non-singular cosmological solutions. 
  A spinor current-source is found in the Weyl non-Abelian gauge theory which does not contain the abstract gauge space. It is shown that the searched spinor representation can be constructed in the space of external differential forms and it is a 16-component quantity for which a gauge-invariant Lagrangian is determined. The connexion between the Weyl non-Abelian gauge potential and the Cartan torsion field and the problem of a possible manifestation of the considered interactions are considered. 
  I describe approaches to the study of black hole spacetimes via numerical relativity. After a brief review of the basic formalisms and techniques used in numerical black hole simulations, I discuss a series of calculations from axisymmetry to full 3D that can be seen as stepping stones to simulations of the full 3D coalescence of two black holes. In particular, I emphasize the interplay between perturbation theory and numerical simulation that build both confidence in present results and tools to aid and to interpret results of future simulations of black hole coalescence. 
  I review recent developments in numerical relativity, focussing on progress made in 3D black hole evolution. Progress in development of black hole initial data, apparent horizon boundary conditions, adaptive mesh refinement, and characteristic evolution is highlighted, as well as full 3D simulations of colliding and distorted black holes. For true 3D distorted holes, with Cauchy evolution techniques, it is now possible to extract highly accurate, nonaxisymmetric waveforms from fully nonlinear simulations, which are verified by comparison to pertubration theory, and with characteristic techniques extremely long term evolutions of 3D black holes are now possible. I also discuss a new code designed for 3D numerical relativity, called Cactus, that will be made public. 
  We revisit the proposed theoretical model for a small but nonzero cosmological constant which seems supported increasingly better by recent observations. The model features two scalar fields which interact with each other through a specifically chosen nonlinear potential. We find a very sensitive dependence of the solutions of the scalar field equations on the initial values. We discuss how the behavior is similar to and different from those in well-known chaotic systems, coming to suggest an interesting new type of the dissipative structure. 
  We give a brief discussion on the limitations involving the expression mu ={angular deficit}/(8*pi),(G=c=1), which relates the string linear energy density ''mu'' to the conical deficit angle. Then, we establish a new equation between the angular deficit and another physical attribute of the string which shows that the angular deficit is determined not only by the amount of proper matter of the string but also, in a Newtonian sense, by its internal gravitational field. 
  The content of gr-qc/9805003 and gr-qc/9806091 now appears in a single paper in the archive under gr-qc/9805003, since these appeared as a single publication in Physical Review Letters. 
  We discuss the implications of the proposed gravitational redshift experiment on antihydrogens. We show that the result should be the same as on hydrogens in spite of different free-fall accelerations (WEP violation) which may occur if there is a vector fifth-force field. We emphasize the experiment is unique in the sense that it tests the Equivalence Principle expressed in its ultimate form, proposed to be called UEP, directly without being disturbed by the effect of possible presence of a scalar fifth-force field. 
  We show that the response rate of (i) a static source interacting with Hawking radiation of massless scalar field in Schwarzschild spacetime (with the Unruh vacuum) and that of (ii) a uniformly accelerated source with the same proper acceleration in Minkowski spacetime (with the Minkowski vacuum) are equal. We show that this equality will not hold if the Unruh vacuum is replaced by the Hartle-Hawking vacuum. It is verified that the source responds to the Hawking radiation near the horizon as if it were at rest in a thermal bath in Minkowski spacetime with the same temperature. It is also verified that the response rate in the Hartle-Hawking vacuum approaches that in Minkowski spacetime with the same temperature far away from the black hole. Finally, we compare our results with others in the literature. 
  The problem of constructing global models describing isolated axially symmetric rotating bodies in equilibrium is analyzed. It is claimed that, whenever the global spacetime is constructed by giving boundary data on the limiting surface of the body and integrating Einstein's equations both inside and ouside the body, the problem becomes overdetermined. Similarly, when the spacetime describing the interior of the body is explicitly given,the problem of finding the exterior vacuum solution becomes overdetermined. We discuss in detail the procedure to be followed in order to construct the exterior vacuum field created by a given but arbitrary distribution of matter. Finally, the uniqueness of the exterior vacuum gravitational field is proven by exploiting the harmonic map formulation of the vacuum equations and the boundary conditions prescribed from the matching. 
  A rigidly rotating incompressible perfect fluid solution of Einstein's gravitational equations is discussed. The Petrov type is D, and the metric admits a four-parameter isometry group. The Gaussian curvature of the constant-pressure surfaces is positive and they have two ring-shaped cusps. 
  A wormhole is constructed by cutting and joining two spacetimes satisfying the low energy string equations with a dilaton field. In spacetimes described by the "string metric" the dilaton energy-momentum tensor need not satisfy the weak or dominant energy conditions. In the cases considered here the dilaton field violates these energy conditions and is the source of the exotic matter required to maintain the wormhole. There is also a surface stress-energy, that must be produced by additional matter, where the spacetimes are joined. It is shown that wormholes can be constructed for which this additional matter satisfies the weak and dominant energy conditions, so that it could be a form of "normal" matter. Charged dilaton wormholes with a coupling between the dilaton and the electromagnetic field that is more general than in string theory are also briefly discussed. 
  I describe a new approach (developed in collaboration with D.E. Holz) to calculating the statistical distributions for magnification, shear, and rotation of images of cosmological sources due to gravitational lensing by mass inhomogeneities on galactic and smaller scales. Our approach is somewhat similar to that used in ``Swiss cheese'' models, but the ``cheese'' has been completely eliminated, the matter distribution in the ``voids'' need not be spherically symmetric, the total mass in each void need equal the corresponding Robertson-Walker mass only on average, and we do not impose an ``opaque radius'' cutoff. In our approach, we integrate the geodesic deviation equation backwards in time until the desired redshift is reached, using a Monte Carlo procedure wherein each photon beam in effect ``creates its own universe'' as it propagates. Our approach fully takes into account effects of multiple encounters with gravitational lenses and is much easier to apply than ``ray shooting'' methods. 
  The singularity structure of cosmological models whose matter content consists of a scalar field with arbitrary non-negative potential is discussed. The special case of spatially flat FRW space-time is analysed in detail using a dynamical systems approach which may readily be generalised to more complicated space-times. It is shown that for a very large and natural class of models a simple and regular past asymptotic structure exists. More specifically, there exists a family of solutions which is in continuous 1-1 correspondence with the exactly integrable massless scalar field cosmologies, this correspondence being realised by a unique asymptotic approximation. The set of solutions which do not fall into this class has measure zero. The significance of this result to the cosmological initial value problem is briefly discussed. 
  We consider, in the framework of General Relativity, the linear approximation of the gravitational field of the Earth taking into account its mass, its quadrupole moment, its shape and its diurnal rotation.   We conclude that in the frame of reference co-moving with the Earth the local anisotropy of the space is of the order of $10^{-12}-10^{-13}$ and could be observed. 
  A two-way traversable wormhole solution in Brans-Dicke theory with torsion is obtained using the method of massive thin shells. The solution goes over general relativity for an infinite large value of the coupling parameter, however, the Brans-Dicke scalar could never be the ``carrier'' of exoticity that threads the wormhole throat. 
  We analyse the relationship between classical chaos and particle creation in Robertson-Walker cosmological models with gravity coupled to a scalar field. Within our class of models chaos and particle production are seen to arise in the same cases. Particle production is viewed as the seed of decoherence, which both enables the quantum to classical transition, and ensures that the correspondence between the quantum and classically chaotic models will be valid 
  Multidimensional gravity interacting with intersecting electric and magnetic $p$-branes is considered for fields depending on a single variable. Some general features of the system behaviour are revealed without solving the field equations. Thus, essential asymptotic properties of isotropic cosmologies are indicated for different signs of spatial curvature; a no-hair-type theorem and a single-time theorem for black holes are proved (the latter makes sense in models with multiple time coordinates). The validity of the general observations is verified for a class of exact solutions known for the cases when certain vectors, built from the input parameters of the model, are either orthogonal in minisuperspace, or form mutually orthogonal subsystems. From the non-existence of Lorentzian wormholes, a universal restriction is obtained, applicable to orthogonal or block-orthogonal subsystems of any $p$-brane system. 
  We present a numerical scheme for determining hyperboloidal initial data sets for the conformal field equations by using pseudo-spectral methods. This problem is split into two parts. The first step is the determination of a suitable conformal factor which transforms from an initial data set in physical space-time to a hyperboloidal hypersurface in the ambient conformal manifold. This is achieved by solving the Yamabe equation, a non-linear second order equation. The second step is a division by the conformal factor of certain fields which vanish on $\scri$, the zero set of the conformal factor. The challenge there is to numerically obtain a smooth quotient. Both parts are treated by pseudo-spectral methods. The non-linear equation is solved iteratively while the division problem is treated by transforming the problem to the coefficient space, solving it there by the QR-factorisation of a suitable matrix, and then transforming back. These hyperboloidal initial data can be used to generate general relativistic space-times by evolution with the conformal field equations. 
  We investigate, in the framework of (2+1) dimensional gravity, stationary, rotationally symmetric gravitational sources of the perfect fluid type, embedded in a space of arbitrary cosmological constant. We show that the matching conditions between the interior and exterior geometries imply restrictions on the physical parameters of the solutions. In particular, imposing finite sources and absence of closed timelike curves privileges negative values of the cosmological constant, yielding exterior vacuum geometries of rotating black hole type. In the special case of static sources, we prove the complete integrability of the field equations and show that the sources' masses are bounded from above and, for vanishing cosmological constant, generally equal to one. We also discuss and illustrate the stationary configurations by explicitly solving the field equations for constant mass--energy densities. If the pressure vanishes, we recover as interior geometries Godel like metrics defined on causally well behaved domains, but with unphysical values of the mass to angular momentum ratio. The introduction of pressure in the sources cures the latter problem and leads to physically more relevant models. 
  Pure gravitational plane waves are considered as a special case of spacetimes with two commuting spacelike Killing vector fields. Starting with a midisuperspace that describes this kind of spacetimes, we introduce gauge-fixing and symmetry conditions that remove all non-physical degrees of freedom and ensure that the classical solutions are plane waves. In this way, we arrive at a reduced model with no constraints and whose only degrees of freedom are given by two fields. In a suitable coordinate system, the reduced Hamiltonian that generates the time evolution of this model turns out to vanish, so that all relevant information is contained in the symplectic structure. We calculate this symplectic structure and particularize our discussion to the case of linearly polarized plane waves. The reduced phase space can then be described by an infinite set of annihilation and creation like variables. We finally quantize the linearly polarized model by introducing a Fock representation for these variables. 
  We give an explicit canonical transformation which transforms a generic chiral 2D dilaton gravity model into a free field theory. 
  The long-term dynamical evolution of a Keplerian binary orbit due to the emission and absorption of gravitational radiation is investigated. This work extends our previous results on transient chaos in the planar case to the three dimensional Kepler system. Specifically, we consider the nonlinear evolution of the relative orbit due to gravitational radiation damping as well as external gravitational radiation that is obliquely incident on the initial orbital plane. The variation of orbital inclination, especially during resonance capture, turns out to be very sensitive to the initial conditions. Moreover, we discuss the novel phenomenon of chaotic transition. 
  We discuss various aspects of the post-Newtonian approximation in general relativity. After presenting the foundation based on the Newtonian limit, we use the (3+1) formalism to formulate the post-Newtonian approximation for the perfect fluid. As an application we show the method for constructing the equilibrium configuration of nonaxisymmetric uniformly rotating fluid. We also discuss the gravitational waves including tail from post-Newtonian systems. 
  Colombeau's generalized functions are used to adapt the distributional approach to singular hypersurfaces in general relativity with signature change. Equations governing the dynamics of singular hypersurface is obtained and it is shown that matching leads to de Sitter space for the Lorentzian region. The matching is possible for different sections of the de Sitter hyperboloid. A relation between the radius of $S^4$, as the Euclidean manifold, and the cosmological constant leading to inflation after signature change is obtained. 
  The wave type field equation $\square \vt^a=\la \vt^a$, where $\vt^a$ is a coframe field on a space-time, was recently proposed to describe the gravity field. This equation has a unique static, spherical-symmetric, asymptotically-flat solution, which leads to the viable Yilmaz-Rosen metric. We show that the wave type field equation is satisfied by the pseudo-conformal frame if the conformal factor is determined by a scalar 3D-harmonic function. This function can be related to the Newtonian potential of classical gravity. So we obtain a direct relation between the non-relativistic gravity and the relativistic model: every classical exact solution leads to a solution of the field equation. With this result we obtain a wide class of exact, static metrics. We show that the theory of Yilmaz relates to the pseudo-conformal sector of our construction. We derive also a unique cosmological (time dependent) solution of the described type. 
  We study spatially homogeneous and isotropic solutions to the equations of motion derived from dilaton gravity, in the presence of a special combination of higher derivative terms in the gravitational action. All solutions are nonsingular. For initial conditions resembling those in the pre-big-bang scenario, there are solutions corresponding to a spatially flat, bouncing Universe originating in a dilaton-dominated contracting phase and emerging as an expanding Friedmann Universe. 
  This paper constructs continuously self-similar solution of a spherically symmetric gravitational collapse of a scalar field in n dimensions. The qualitative behavior of these solutions is explained, and closed-form answers are provided where possible. Equivalence of scalar field couplings is used to show a way to generalize minimally coupled scalar field solutions to the model with general coupling. 
  The global behavior of scalar field cosmological models with very hard potential walls is investigated via the simple example of an exponentially steep potential well. It is found that the solutions exhibit a non-trivial oscillatory behavior in the approach to an initial space-time singularity. This behavior can be interpreted as being due to the inability of the scalar field to negotiate the walls of the steep potential well. 
  Electromagnetic field produced by magnetic multipoles in hyperbolic motion is derived and compared with electromagnetic field produced by electric multipoles in hyperbolic motion. The resulting fields are related by duality symmetry. Radiative properties of these solutions are demonstrated. In the second part an analogous, uniformly accelerated source of gravitational radiation is studied, within exact Einstein's theory. Radiative characteristics of the corresponding solution as flux of the radiation and the total mass-energy of the system are calculated and graphically illustrated. 
  In recent work on black hole entropy in non-perturbative quantum gravity, an action for the black hole sector of the phase space is introduced and (partially) quantized. We give a number of observations on this and related works. In particular we show that (I) the entropy calculation applies without change to generally covariant theories having no black hole solutions, (II) the phase space constraint used to select the black hole sector is not the apparent horizon equation, which is the natural phase space constraint separating trapped and untrapped regions on an initial data surface, and (III) there appears to be at least one other phase space constraint which leads to the conclusion that the entropy associated with a bounding two-dimensional surface is proportional to its area. 
  We shall use the variational decomposition technique in order to calculate equations of motion and Noether energy-momentum complex for some classes of non-linear gravitational Lagrangians within the first-order (Palatini) formalism. In particular, a complex space-time appears as a solution of our variational problem. 
  When the topology of the universe is non trivial, it has been shown that there are constraints on the network of domain walls, cosmic strings and monopoles. I generalize these results to textures and study the cosmological implications of such constraints. I conclude that a large class of multi-connected universes with topological defects accounting for structure formation are ruled out by observation of the cosmic microwave background. 
  We have considered the divergence structure in the brick-wall model for the statistical mechanical entropy of a quantum field in thermal equilibrium with a black hole which {\it rotates}. Especially, the contribution to entropy from superradiant modes is carefully incorporated, leading to a result for this contribution which corrects some previous errors in the literature. It turns out that the previous errors were due to an incorrect quantization of the superradiant modes. Some of main results for the case of rotating BTZ black holes are that the entropy contribution from superradiant modes is positive rather than negative and also has a leading order divergence as that from nonsuperradiant modes. The total entropy, however, can still be identified with the Bekenstein-Hawking entropy of the rotating black hole by introducing a universal brick-wall cutoff. Our correct treatment of superradiant modes in the ``angular-momentum modified canonical ensemble'' also removes unnecessary introductions of regulating cutoff numbers as well as ill-defined expressions in the literature. 
  This is a summary of a talk delivered at the workshop ``Quantum gravity in the Southern Cone II''. We present a very brief review of current results on canonical quantization of general relativity using Ashtekar's variables and loop quantization. 
  The role of fermionic matter in the spectrum of the area operator is analyzed using the Baez--Krasnov framework for quantum fermions and gravity. The result is that the fermionic contribution to the area of a surface $S$ is equivalent to the contribution of purely gravitational spin network's edges tangent to $S$. Therefore, the spectrum of the area operator is the same as in the pure gravity case. 
  We construct the operator that projects on the physical states in loop quantum gravity. To this aim, we consider a diffeomorphism invariant functional integral over scalar functions. The construction defines a covariant, Feynman-like, spacetime formalism for quantum gravity and relates this theory to the spin foam models. We also discuss how expectation values of physical quantity can be computed. 
  A midi-superspace model is a field theory obtained by symmetry reduction of a parent gravitational theory. Such models have proven useful for exploring the classical and quantum dynamics of the gravitational field. I present 3 recent classes of results pertinent to canonical quantization of vacuum general relativity in the context of midi-superspace models. (1) I give necessary and sufficient conditions such that a given symmetry reduction can be performed at the level of the Lagrangian or Hamiltonian. (2) I discuss the Hamiltonian formulation of models based upon cylindrical and toroidal symmetry. In particular, I explain how these models can be identified with parametrized field theories of wave maps, thus a natural strategy for canonical quantization is available. (3) The quantization of a parametrized field theory, such as the midi-superspace models considered in (2), requires construction of a quantum field theory on a fixed (flat) spacetime that allows for time evolution along arbitrary foliations of spacetime. I discuss some recent results on the possibility of finding such a quantum field theory. 
  We show that the 1+1 dimensional reduction (i.e., the radial plane) of the Kruskal black hole can be embedded in 2+1 Minkowski spacetime and discuss how features of this spacetime can be seen from the embedding diagram. The purpose of this work is educational: The associated embedding diagrams may be useful for explaining aspects of black holes to students who are familiar with special relativity, but not general relativity. 
  After the original discovery of the Kerr metric, Newman and Janis showed that this solution could be ``derived'' by making an elementary complex transformation to the Schwarzschild solution. The same method was then used to obtain a new stationary axisymmetric solution to Einstein's field equations now known as the Kerr-newman metric, representing a rotating massive charged black hole. However no clear reason has ever been given as to why the Newman-Janis algorithm works, many physicist considering it to be an ad hoc procedure or ``fluke'' and not worthy of further investigation. Contrary to this belief this paper shows why the Newman-Janis algorithm is successful in obtaining the Kerr-Newman metric by removing some of the ambiguities present in the original derivation. Finally we show that the only perfect fluid generated by the Newman-Janis algorithm is the (vacuum) Kerr metric and that the only Petrov typed D solution to the Einstein-Maxwell equations is the Kerr-Newman metric. 
  The canonical formalism for expanding metrics scenarios is presented. Non-unitary time evolution implied by expanding geometry is described as a trajectory over unitarily inequivalent representations at different times of the canonical commutation relations. Thermal properties of inflating Universe are also discussed. 
  The Penrose-Gibbons inequality for charged black holes is proved in spherical symmetry, assuming that outside the black hole there are no current sources, meaning that the charge e is constant, with the remaining fields satisfying the dominant energy condition. Specifically, for any achronal hypersurface which is asymptotically flat at spatial or null infinity and has an outermost marginal surface of areal radius r, the asymptotic mass m satisfies 2m >= r + e^2/r. Replacing m by a local energy, the inequality holds locally outside the black hole. A recent definition of dynamic surface gravity k also satisfies inequalities 2k <= 1/r - e^2/r^3 and m >= r^2 k + e^2/r. All these inequalities are sharp in the sense that equality is attained for the Reissner-Nordstrom black hole. 
  We consider a D-dimensional cosmological model describing an evolution of (n+1) Einstein factor spaces in the theory with several dilatonic scalar fields and generalized electro-magnetic forms, admitting an interpretation in terms of intersecting p-branes. The equations of motion of the model are reduced to the Euler-Lagrange equations for the so called pseudo-Euclidean Toda-like system. We consider the case, when characteristic vectors of the model, related to p-branes configuration and their couplings to the dilatonic fields, may be interpreted as the root vectors of a Lie algebra of the type Am. The model is reduced to the open Toda chain and integrated. The exact solution is presented in the Kasner-like form. 
  We analytically study gravitational radiation from corotating binary neutron stars composed of incompressible, homogeneous fluid in circular orbits. The energy and the angular momentum loss rates are derived up to the first post-Newtonian (1PN) order beyond the quadrupole approximation including effects of the finite size of each star of binary. It is found that the leading term of finite size effects in the 1PN order is only $O(GM_{\ast}/c^2 a_{\ast})$ smaller than that in the Newtonian order, where $GM_{\ast}/c^2 a_{\ast}$ means the ratio of the gravitational radius to the mean radius of each star of binary, and the 1PN term acts to decrease the Newtonian finite size effect in gravitational radiation. 
  We show how the motion of free material test particles in arbitrary spatial flows is easily determined within the context of ordinary vector calculus. This may be useful for everyone, including engineers and other non-specialists, when thinking about gravitational problems. It already has valid application to simple problems such as the problems of motion in rotating and accelerating frames and to the gravitational problem of the single spherically symmetric attractor. When applied to the two body gravitational problem, it may help us determine the actual direction of the flow. 
  We have developed a method to study the effects of a perturbation to the motion of a test point--like object in a Schwarzschild spacetime. Such a method is the extension of the Lagrangian planetary equations of classical celestial mechanics into the framework of the full theory of general relativity. The method provides a natural approach to account for relativistic effects in the unperturbed problem in an exact way. 
  As a preliminary step towards simulating binary neutron star coalescing problem, we test a post-Newtonian approach by constructing a single neutron star model. We expand the Tolman-Oppenheimer-Volkov equation of hydrostatic equilibrium by the power of $c^{-2}$, where $c$ is the speed of light, and truncate at the various order. We solve the system using the polytropic equation of state with index $\Gamma=5/3, 2$ and 3, and show how this approximation converges together with mass-radius relations. Next, we solve the Hamiltonian constraint equation with these density profiles as trial functions, and examine the differences in the final metric. We conclude the second `post-Newtonian' approximation is close enough to describe general relativistic single star. The result of this report will be useful for further binary studies.   (Note to readers) This paper was accepted for publication in Physical Review D. [access code dsj637]. However, since I was strongly suggested that the contents of this paper should be included as a section in our group's future paper, I gave up the publication. 
  Non stationary cylindrically symmetric exact solutions of the Einstein-Maxwell equations are derived as single soliton perturbations of a Levi-Civita metric, by an application of Alekseev inverse scattering method. We show that the metric derived by L. Witten, interpreted as describing the electrogravitational field of a straight, stationary, conducting wire may be recovered in the limit of a `wide' soliton. This leads to the possibility of interpreting the solitonic solutions as representing a non stationary electrogravitational field exterior to, and interacting with, a thin, straight, superconducting cosmic string. We give a detailed discussion of the restrictions that arise when appropiate energy and regularity conditions are imposed on the matter and fields comprising the string, considered as `source', the most important being that this `source' must necessarily have a non- vanishing minimum radius. We show that as a consequence, it is not possible, except in the stationary case, to assign uniquely a current to the source from a knowledge of the electrogravitational fields outside the source. A discussion of the asymptotic properties of the metrics, the physical meaning of their curvature singularities, as well as that of some of the metric parameters, is also included. 
  We examine the effect of a non-trivial nut charge on the action of non-compact four-dimensional instantons with a U(1) isometry. If the instanton action is calculated by dimensionally reducing along the isometry, then the nut charge is found to make an explicit non-zero contribution. For metrics satisfying AF, ALF or ALE boundary conditions, the action can be expressed entirely in terms of quantities (including the nut charge) defined on the fixed point set of the isometry. A source (or sink) of nut charge also implies the presence of a Misner string coordinate singularity, which will have an important effect on the Hamiltonian of the instanton. 
  We generalize previous work on Dirac eigenvalues as dynamical variables of Euclidean supergravity. The most general set of constraints on the curvatures of the tangent bundle and on the spinor bundle of the spacetime manifold under which spacetime admits Dirac eigenvalues as observables, are derived. 
  In present paper we construct classical and quantum models of an extended charged particle. One shows that consecutive modelling can be based on the hollow thin-wall charged texture (in the hydrodynamical approach of a perfect fluid) which acquires gravitational mass due to Einstein-Maxwell interaction. We demonstrate that such a model has equilibrium states at the radius equal to the established classical radius of a charged particle. Also we consider quantum aspects of the theory and obtain the (internal) Dirac sea conception in a natural way. Besides, the phenomenological unification on the mass level of the two families of elementary particles, charged pions and electrons and positrons, evidently arises as the effect induced by classical and quantum gravity prior to Standard Model.   Finally, in the cosmological connection our model proposes the answer on the important question, what are the real sources of texture matter. Besides, the texture hypothesis means that in the early Universe the topological texture foam phase existed before the lepton-hadron one. 
  A relation between the non-perturbative loop representation space and the semi-classical loop representation space is studied. A sector of (approximate) states and a sector of operators in the non-perturbative loop representation space are made related respectively to the physical states and the basic variables of the semi-classical loop representation space through a transformation. This transformation makes a construction of graviton states within the non-perturbative theory possible although the notion of gravitons originally emerged from semi-classical perturbative treatments. This transformation is ``exact'' in the sense that it does not contain an error term, a fact contrast to the previous construction of a similar transformation. This fact allows the interpretation that these graviton states represent free gravitons even after introducing the self-interaction of gravitons into the theory; the presence of an error term of order of possible perturbative self-interaction would spoil this interpretation. The existence of such an ``exact'' relation of the two theories supports the potential ability of the non-perturbative loop representation quantum gravity to address the physics of gravitons, namely quanta for large scale small fluctuations of gravitational field in the flat background spacetime. 
  We investigate the possible existence of non-topological solitons in string-like theories, or in other completions of Einstein theory, by examining a simple extension of standard theory that describes a non-linear scalar field interacting with the Einstein, Maxwell and Weyl (dilaton) fields. The Einstein and Maxwell couplings are standard while the dilatonic coupling is taken to agree with string models. The non-linear scalar potential is quite general. It is found to be impossible to satisfy the dilatonic boundary conditions. Excluding the dilaton field we find a variety of solitonic structures differing in ways that depend on the non-linear potential. In general the excited states exhibit a discrete mass spectrum. At large distances the gravitational field approaches the Reissner-Nordstrom solution. 
  We show that the set of ambiguities in the renormalized expected stress-energy tensor allowed by the Wald axioms is much larger for a massive scalar field (an infinite number of free parameters) than for a massless scalar field (two free parameters). We also use the closed-time-path effective action formalism of Schwinger to calculate the expected value of the stress-energy tensor in the incoming vacuum state, for a massive scalar field, on any spacetime which is a linear perturbation off Minkowski spacetime. This result generalizes an earlier result of Horowitz and also Jordan in the massless case, and can be used as a testbed for comparing different calculational methods. 
  The effect of a time dependent cosmological constant is considered in a family of scalar tensor theories. Friedmann-Robertson-Walker cosmological models for vacumm and perfect fluid matter are found. They have a linear expansion factor, the so called coasting cosmology, the gravitational "constant" decreace inversely with time; this model satisfy the Dirac hipotesis. The cosmological "constant" decreace inversely with the square of time, therefore we can have a very small value for it at present time. 
  We have developed a numerical code to study the evolution of self-gravitating matter in dynamic black hole axisymmetric spacetimes in general relativity. The matter fields are evolved with a high-resolution shock-capturing scheme that uses the characteristic information of the general relativistic hydrodynamic equations to build up a linearized Riemann solver. The spacetime is evolved with an axisymmetric ADM code designed to evolve a wormhole in full general relativity. We discuss the numerical and algorithmic issues related to the effective coupling of the hydrodynamical and spacetime pieces of the code, as well as the numerical methods and gauge conditions we use to evolve such spacetimes. The code has been put through a series of tests that verify that it functions correctly. Particularly, we develop and describe a new set of testbed calculations and techniques designed to handle dynamically sliced, self-gravitating matter flows on black holes, and subject the code to these tests. We make some studies of the spherical and axisymmetric accretion onto a dynamic black hole, the fully dynamical evolution of imploding shells of dust with a black hole, the evolution of matter in rotating spacetimes, the gravitational radiation induced by the presence of the matter fields and the behavior of apparent horizons through the evolution. 
  One way the ultraviolet problem may be solved is explicit physical regularization. In this scenario, QFT is only the long distance limit of some unknown non-Poincare-invariant microscopic theory. One can ask how complex and contrived such microscopic theories should be.   We show that condensed matter in standard Newtonian framework is sufficient to obtain gravity. We derive a metrical theory of gravity with two additional to GR cosmological constants. The observable difference is similar to homogeneously distributed dark matter with p = -1/3 epsilon. resp. p = epsilon. The gravitational collapse stops before horizon formation and evaporates by Hawking radiation. The cutoff is not the Planck length, but expanding together with the universe. Thus, in some cosmological future microscopic effects become observable. 
  Gamma-ray bursts are believed to be the most luminous objects in the Universe. There has been some suggestion that these arise from quantum processes around naked singularities. The main problem with this suggestion is that all known examples of naked singularities are massless and hence there is effectively no source of energy. It is argued that a globally naked singularity coupled with quantum processes operating within a distance of the order of Planck length of the singularity will probably yield energy burst of the order of M_pc^2\approx2\times 10^{16} ergs, where M_p is the Planck mass. 
  In a recent series of papers Endean examines the properties of spatially homogeneous and isotropic (FLRW) cosmological models filled with dust in the ``conformally flat spacetime presentation of cosmology'' (CFS cosmology). This author claims it is possible to resolve a certain number of the difficulties the standard model exhibits when confronted to observations, if the theoretical predictions are obtained in the special framework of CFS cosmology. As a by-product of his analysis Endean claims that no initial (big-bang) nor final (big-crunch) singularities occur in the closed FLRW model. In this paper we show up the fallacious arguments leading to Endean's conclusions and we consistently reject his CFS cosmology. 
  We argue that compatibility with elementary particle physics requires gravitational theories with torsion to be unable to distinguish between orbital angular momentum and spin. An important consequence of this principle is that spinless particles must move along autoparallel trajectories, not along geodesics. 
  We explore the dynamical stability of the minisuperspace Hamiltonian of the Bianchi IX cosmological models, giving a gauge-invariant and unapproximated description of the full continuous dynamics, achieved through a geometrical description of the equations of motion in the framework of the theory of Finsler Spaces. The numerical integrations of the geodesics and geodesic deviation equations show clearly the absence of any "traditional" signature of Chaos, while suggesting a chaotic scattering dynamics scenario. 
  In the semiclassical quantum gravity derived from the Wheeler-DeWitt equation, the energy density of a matter field loses quantum coherence due to the induced gauge potential from the parametric interaction with gravity in a non-static spacetime. It is further shown that the energy density takes only positive values and makes superposition principle hold true. By studying a minimal massive scalar field in a FRW spacetime background, we illustrate the positivity of energy density and obtain the classical Hamiltonian of a complex field from the energy density in coherent states. 
  A semiclassical cosmological model is considered which consists of a closed Friedmann-Robertson-Walker in the presence of a cosmological constant, which mimics the effect of an inflaton field, and a massless, non-conformally coupled quantum scalar field. We show that the back-reaction of the quantum field, which consists basically of a non local term due to gravitational particle creation and a noise term induced by the quantum fluctuations of the field, are able to drive the cosmological scale factor over the barrier of the classical potential so that if the universe starts near zero scale factor (initial singularity) it can make the transition to an exponentially expanding de Sitter phase. We compute the probability of this transition and it turns out to be comparable with the probability that the universe tunnels from "nothing" into an inflationary stage in quantum cosmology. This suggests that in the presence of matter fields the back-reaction on the spacetime should not be neglected in quantum cosmology. 
  It is shown that the recently proposed interpretation of the transposed equi-affine theory of gravity as a theory with variable Plank "constant" is inconsistent with basic solar system gravitational experiments. 
  The role that the quantum properties of a gravitational wave could play in the detection of gravitational radiation is analyzed. It is not only corroborated that in the current laser-interferometric detectors the resolution of the experimental apparatus could lie very far from the corresponding quantum threshold (thus the backreaction effect of the measuring device upon the gravitational wave is negligible), but it is also suggested that the consideration of the quantum properties of the wave could entail the definition of dispersion of the measurement outputs. This dispersion would be a function not only of the sensitivity of the measuring device, but also of the interaction time (between measuring device and gravitational radiation) and of the arm length of the corresponding laser- interferometer. It would have a minimum limit, and the introduction of the current experimental parameters insinuates that the dispersion of the existing proposals could lie very far from this minimum, which means that they would show a very large dispersion. 
  Geometric $\sigma$-models have been defined as purely geometric theories of scalar fields coupled to gravity. By construction, these theories possess arbitrarily chosen vacuum solutions. Using this fact, one can build a Kaluza--Klein geometric $\sigma$-model by specifying the vacuum metric of the form $M^4\times B^d$. The obtained higher dimensional theory has vanishing cosmological constant but fails to give massless gauge fields after the dimensional reduction. In this paper, a modified geometric $\sigma$-model is suggested, which solves the above problem. 
  In 1975, Ashtekar and Magnon showed that an energy condition selects a unique quantization procedure for certain observers in general, curved spacetimes. We generalize this result in two important ways, by eliminating the need to assume a particular form for the (quantum) Hamiltonian, and by considering the surprisingly nontrivial extension to nonminimal coupling. 
  We have recently constructed a numerical code that evolves a spherically symmetric spacetime using a hyperbolic formulation of Einstein's equations. For the case of a Schwarzschild black hole, this code works well at early times, but quickly becomes inaccurate on a time scale of 10-100 M, where M is the mass of the hole. We present an analytic method that facilitates the detection of instabilities. Using this method, we identify a term in the evolution equations that leads to a rapidly-growing mode in the solution. After eliminating this term from the evolution equations by means of algebraic constraints, we can achieve free evolution for times exceeding 10000M. We discuss the implications for three-dimensional simulations. 
  Having started with the general formulation of the quantum theory of the real scalar field (QFT) in the general Riemannian space--time $ V_{1,3} $, the general--covariant quasinonrelativistic quantum mechanics of a point-like spinless particle in $ V_{1,3} $ is constructed. To this end, for any normal geodesic 1+3--foliation of $ V_{1,3} $, a space $\Phi^-$ of asymptotic in $c^{-1}$ solutions of the field equation is specified, which can be mapped to a space $\Psi$ of solutions of a Schr\"odinger equation with an (asymptotically) Hermitean hamiltonian and the Born probabilistic interpretation of the vectors of $\Psi$. The basic operators of the momentum and the spatial position of the particle acting in $\Psi$ generated by the corresponding observables of QFT include relativistic corrections, and therefore differ generally from those which follow for the geodesic motion in $ V_{1,3} $ from the canonical postulates of quantization. In particular, the operators of coordinates do not commute as well as the operators of the conjugate momenta, except the cases of Cartesian coordinates in the Minkowski space--time or of the exact nonrelativistic limit $(c^{-1} = 0)$. This approach provides QFT in the general $ V_{1,3} $ in the Fock representation with a particle interpretation based on the Born interpretation of wave functions. 
  By extending the charged Vaidya metric to cover all of spacetime, we obtain a Penrose diagram for the formation and evaporation of a charged black hole. In this construction, the singularity is time-like. The entire spacetime can be predicted from initial conditions if boundary conditions at the singularity are known. 
  A class of Riemann-Cartan G\"odel-type space-times is examined by using the equivalence problem techniques, as formulated by Fonseca-Neto et al. and embodied in a suite of computer algebra programs called TCLASSI. A coordinate-invariant description of the gravitational field for this class of space-times is presented. It is also shown that these space-times can admit a group $G_{r}$ of affine-isometric motions of dimensions $r=2, 4, 5$. The necessary and sufficient conditions for space-time (ST) homogeneity of this class of space-times are derived, extending previous works on G\"odel-type space-times. The equivalence of space-times in the ST homogeneous subclass is studied, recovering recent results under different premises. The results of the limiting Riemannian case are also recovered. 
  New exact vacuum solutions with various singularities in the plane-symmetric spacetime are shown, and they are applied to the analysis of inhomogeneous cosmological models and colliding gravitational waves. One of the singularities can be true null singularities, whose existence was locally clarified by Ori. These solutions may be interesting from the viewpoint of the variety of cosmological singularities and the instability problem of Cauchy horizons inside black holes. 
  The coupled Einstein-Yang-Mills equations on a time dependent axially symmetric spacetime are investigated, without a priori any conditions on the gauge field. There is numerical evidence for the existence of a regular solution with the desired asymptotic features. Just as in the supermassive abelian counterpart model, the formation of a singularity at finite distance of the core of the string depends critically on a parameter of the model, i.e., the constant value of one of the magnetic components of the YM potentials. The multiple-scale method could supply decisive answers concerning the stability of the solution. 
  In Einstein-Maxwell theory black holes are uniquely determined by their mass, their charge and their angular momentum. This is no longer true in Einstein-Yang-Mills theory. We discuss sequences of neutral and charged SU(N) Einstein-Yang-Mills black holes, which are static spherically symmetric and asymptotically flat, and which carry Yang-Mills hair. Furthermore, in Einstein-Maxwell theory static black holes are spherically symmetric. We demonstrate that, in contrast, SU(2) Einstein-Yang-Mills theory possesses a sequence of black holes, which are static and only axially symmetric. 
  Rapidly rotating white dwarfs in cataclysmic variable systems may be emitting gravitational radiation due to the recently discovered relativistic r-mode instability. Assuming that the four most rapidly rotating known systems are limited in rotation rate by the instability, the amplitude of the emitted gravitational waves is determined at Earth for both known rapid rotators and for a model background caused by a galactic population of such systems. The proposed LISA and OMEGA space-based interferometer gravitational wave detectors could observe such signals and determine whether the r-mode instability plays a significant role in white dwarf systems. 
  We exhibit a resonance mechanism of amplification of density perturbations in inflationary mo-dels, using a minimal set of ingredients (an effective cosmological constant, a scalar field minimally coupled to the gravitational field and matter), common to most models in the literature of inflation. This mechanism is based on the structure of homoclinic cylinders, emanating from an unstable periodic orbit in the neighborhood of a saddle-center critical point, present in the phase space of the model. The cylindrical structure induces oscillatory motions of the scales of the universe whenever the orbit visits the neighborhood of the saddle-center, before the universe enters a period of exponential expansion. The oscillations of the scale functions produce, by a resonance mechanism, the amplification of a selected wave number spectrum of density perturbations, and can explain the hierarchy of scales observed in the actual universe. The transversal crossings of the homoclinic cylinders induce chaos in the dynamics of the model, a fact intimately connected to the resonance mechanism occuring immediately before the exit to inflation. 
  The final fate of the spherically symmetric collapse of a perfect fluid which follows the $\gamma$-law equation of state and adiabatic condition is investigated. Full general relativistic hydrodynamics is solved numerically using a retarded time coordinate, the so-called observer time coordinate. Thanks to this coordinate, the causal structure of the resultant space-time is automatically constructed. Then, it is found that a globally naked, shell-focusing singularity can occur at the center from relativistically high-density, isentropic and time symmetric initial data if $\gamma \alt 1.01$ within the numerical accuracy. The result is free from the assumption of self-similarity. The upper limit of $\gamma$ with which a naked singularity can occur from generic initial data is consistent with the result of Ori and Piran based on the assumption of self-similarity. 
  Cosmological perturbations with wavelengths smaller than Hubble radius can be handled in the context of Newtonian theory with very high accuracy. The application of this Newtonian approximation, however, is restricted to nonrelativistic matter and cannot be used for relativistic matter. Recently, by modifying the continuity equation, Lima, et. al., extended the domain of applicability of Newtonian cosmology to radiation dominated phase. We adopted this continuity equation to re-examine linear cosmological perturbation theory for a two fluid universe with uniform pressure. We study the evolution equations for density contrasts and their validity in different epochs and on scales larger than Hubble radius and compare the results with the full relativistic approach. The comparison shows the high accuracy of this approximation. 
  An overview is given of the formulation of low-energy string cosmologies together with examples of particular solutions, successes and problems of the theory. 
  Hamiltonian time evolution in terms of an explicit parameter time is derived for general relativity, even when the constraints are not satisfied, from the Arnowitt-Deser-Misner-Teitelboim-Ashtekar action in which the slicing density $\alpha(x,t)$ is freely specified while the lapse $N=\alpha g^{1/2}$ is not. The constraint ``algebra'' becomes a well-posed evolution system for the constraints; this system is the twice-contracted Bianchi identity when $R_{ij}=0$. The Hamiltonian constraint is an initial value constraint which determines $g^{1/2}$ and hence $N$, given $\alpha$. 
  Recently, a variational principle has been derived from Einstein-Hilbert and a matter Lagrangian for the spherically symmetric system of a dust shell and a black hole. The so-called physical region of the phase space, which contains all physically meaningful states of the system defined by the variational principle, is specified; it has a complicated boundary. The principle is then transformed to new variables that remove some problems of the original formalism: the whole phase space is covered (in particular, the variables are regular at all horizons), the constraint has a polynomial form, and the constraint equation is uniquely solvable for two of the three conserved momenta. The solutions for the momenta are written down explicitly. The symmetry group of the system is studied. The equations of motion are derived from the transformed principle and are shown to be equivalent to the previous ones. Some lower-dimensional systems are constructed by exclusion of cyclic variables, and some of their properties are found. 
  In this paper we study the symmetries of the dual Taub-NUT metrics. Generic and non-generic symmetries of dual Taub-NUT metrics are investigated.   The existence of the Runge-Lenz type symmetry is analyzed for dual Taub-NUT metrics.   We find that in some cases the symmetries of the dual metrics are the same with the symmetries of Taub-NUT metric. 
  We present pictorial means of distinguishing contravariant vectors (or simply vectors) from covariant vectors (or linear forms). When one depicts vector as the directed segment, then the pictorial image of a linear form is a family of equidistant parallel planes with an arrow joining the neighbouring planes and showing the direction of increase of the form. First of these planes is the linear subspace of dimension two on which the linear form gives value zero. Several examples of physical quantities are given which are naturally vectors, and others which are naturally linear forms. 
  An elementary introduction is given to the problem of black hole entropy as formulated by Bekenstein and Hawking. The information theoretic basis of Bekenstein's formulation is briefly reviewed and compared with Hawking's approach. The issue of calculating the entropy by actual counting of microstates is taken up next within two currently popular approaches to quantum gravity, viz., string theory and canonical quantum gravity. The treatment of the former assay is confined to a few remarks, mainly of a critical nature, while some of the computational techniques of the latter approach are elaborated. We conclude by trying to find commonalities between these two rather disparate directions of work. 
  We investigate the innermost stable circular orbit (ISCO) of a test particle moving on the equatorial plane around rotating relativistic stars such as neutron stars. First, we derive approximate analytic formulas for the angular velocity and circumferential radius at the ISCO making use of an approximate relativistic solution which is characterized by arbitrary mass, spin, mass quadrupole, current octapole and mass $2^4$-pole moments. Then, we show that the analytic formulas are accurate enough by comparing them with numerical results, which are obtained by analyzing the vacuum exterior around numerically computed geometries for rotating stars of polytropic equation of state. We demonstrate that contribution of mass quadrupole moment for determining the angular velocity and, in particular, the circumferential radius at the ISCO around a rapidly rotating star is as important as that of spin. 
  We present results from a new technique which allows extraction of gravitational radiation information from a generic three-dimensional numerical relativity code and provides stable outer boundary conditions. In our approach we match the solution of a Cauchy evolution of the nonlinear Einstein field equations to a set of one-dimensional linear equations obtained through perturbation techniques over a curved background. We discuss the validity of this approach in the case of linear and mildly nonlinear gravitational waves and show how a numerical module developed for this purpose is able to provide an accurate and numerically convergent description of the gravitational wave propagation and a stable numerical evolution. 
  We propose a new alternative gauge for the Einstein equations instead of the de Donder gauge, which allows in the limit of weak fields a straightforward integration of these equations. The Newtonian potential plays a new interesting role in this framework. The calculations are demonstrated explicitely for 2 simple astrophysical models. 
  Analytical solutions are presented for a class of generalized r-modes of rigidly rotating uniform density stars---the Maclaurin spheroids---with arbitrary values of the angular velocity. Our analysis is based on the work of Bryan; however, we derive the solutions using slightly different coordinates that give purely real representations of the r-modes. The class of generalized r-modes is much larger than the previously studied `classical' r-modes. In particular, for each l and m we find l-m (or l-1 for the m=0 case) distinct r-modes. Many of these previously unstudied r-modes (about 30% of those examined) are subject to a secular instability driven by gravitational radiation. The eigenfunctions of the `classical' r-modes, the l=m+1 case here, are found to have particularly simple analytical representations. These r-modes provide an interesting mathematical example of solutions to a hyperbolic eigenvalue problem. 
  According to different topological configurations, we suggest that there are two kinds of extreme black holes in the nature. We find that the Euler characteristic plays an essential role to classify these two kinds of extreme black holes. For the first kind of extreme black holes, Euler characteristic is zero, and for the second kind, Euler characteristic is two or one provided they are four dimensional holes or two dimensional holes respectively. 
  Third rank Killing tensors in (1+1)-dimensional geometries are investigated and classified. It is found that a necessary and sufficient condition for such a geometry to admit a third rank Killing tensor can always be formulated as a quadratic PDE, of order three or lower, in a Kahler type potential for the metric. This is in contrast to the case of first and second rank Killing tensors for which the integrability condition is a linear PDE. The motivation for studying higher rank Killing tensors in (1+1)-geometries, is the fact that exact solutions of the Einstein equations are often associated with a first or second rank Killing tensor symmetry in the geodesic flow formulation of the dynamics. This is in particular true for the many models of interest for which this formulation is (1+1)-dimensional, where just one additional constant of motion suffices for complete integrability. We show that new exact solutions can be found by classifying geometries admitting higher rank Killing tensors. 
  We report on some recent work, which points out the relevance of future measurements of gravitational waves by resonant-mass detectors of spherical shape for theories of gravity of non-Einstein type. 
  By using Hawking's treatment as well as Zaslavskii's treatment respectively and the brick wall model, two different values of classical entropy and quantum entropy of scalar fields in the two-dimensional extreme charged dilaton black hole backgrounds have been obtained. A new divergent term emerges in the quantum entropy under the extreme limit for Zaslavskii's treatment and its connection with the phase transition has been addressed. 
  We treat energy-momentum conservation laws as particular gauge conservation laws when generators of gauge transformations are horizontal vector fields on fibre bundles. In particular, the generators of general covariant transformations are the canonical horizontal prolongations of vector fields on a world manifold. This is the case of the energy-momentum conservation laws in gravitation theories. We find that, in main gravitational models, the corresponding energy-momentum flows reduce to the generalized Komar superpotential. We show that the superpotential form of a conserved flow is the common property of gauge conservation laws if generators of gauge transformations depend on derivatives of gauge parameters. At the same time, dependence of conserved flows on gauge parameters make gauge conservation laws form-invariant under gauge transformations. 
  We study the qualitative properties of cosmological models in scalar-tensor theories of gravity by exploiting the formal equivalence of these theories with general relativity minimally coupled to a scalar field under a conformal transformation and field redefinition. In particular, we investigate the asymptotic behaviour of spatially homogeneous cosmological models in a class of scalar-tensor theories which are conformally equivalent to general relativistic Bianchi cosmologies with a scalar field and an exponential potential whose qualitative features have been studied previously. Particular attention is focussed on those scalar-tensor theory cosmological models, which are shown to be self-similar, that correspond to general relativistic models that play an important r\^{o}le in describing the asymptotic behaviour of more general models (e.g., those cosmological models that act as early-time and late-time attractors). 
  As a most promising candidate for quantum theory of the gravity, the superstring theory has attracted many researchers including cosmologists. It is expected that the cosmological initial singularity is avoided within the context of the superstring theory. Indeed, Antoniadis et. al. found an interesting example of the non-singular cosmological solutions by considering the string one-loop correction. Here, we will discuss the stability of this model and find a new kind of instability in the graviton modes. We also argue this instability will persist even in the non-perturbative regime. The instability we have found might provide a mechanism to produce the primodial black holes. 
  The application of N=2 supersymmetric quantum mechanics for the quantization of homogeneous systems coupled with gravity is discussed. Starting with the superfield formulation of an N=2 SUSY sigma model, Hermitian self-adjoint expressions for quantum Hamiltonians and Lagrangians for any signature of a sigma-model metric are obtained. This approach is then applied to coupled SU(2) Einstein-Yang-Mills (EYM) systems in axially symmetric $Bianchi$-type I, II, VIII, IX, $Kantowski-Sachs$, and closed $Friedmann-Robertson-Walker$ cosmological models. It is shown that all these models admit the embedding into the N=2 SUSY sigma model with the explicit expressions for superpotentials being direct sums of gravitational and Yang-Mills (YM) parts. In addition, the YM parts of superpotentials exactly coincide with the corresponding Chern-Simons terms. The spontaneous SUSY breaking caused by YM instantons in EYM systems is discussed in a number of examples. 
  Locally rotationally symmetric perfect fluid solutions of Einstein's gravitational equations are matched along the hypersurface of vanishing pressure with the NUT metric. These rigidly rotating fluids are interpreted as sources for the vacuum exterior which consists only of a stationary region of the Taub-NUT space-time. The solution of the matching conditions leaves generally three parameters in the global solution. Examples of perfect fluid sources are discussed. 
  In a recent paper Kr\'olak and Beem have shown differentiability of Cauchy horizons at all points of multiplicity one. In this note we give a simpler proof of this result. 
  Non-singular Bianchi type I solutions are found from the effective action with a superstring-motivated Gauss-Bonnet term. These anisotropic non-singular solutions evolve from the asymptotic Minkowski region, subsequently super-inflate, and then smoothly continue either to Kasner-type (expanding in two directions and shrinking in one direction) or to Friedmann-type (expanding in all directions) solutions. We also found a new kind of singularity which arises from the fact that the anisotropic expansion rates are multiple-valued function of time. The initial singularity in the isotropic limit of this model belongs to this new kind of singularity. In our analysis the anisotropic solutions are likely to be singular when the super-inflation is steep. 
  We explore the geometry and asymptotics of extended Racah coeffecients. The extension is shown to have a simple relationship to the Racah coefficients for the positive discrete unitary representation series of SU(1,1) which is explicitly defined. Moreover, it is found that this extension may be geometrically identified with two types of Lorentzian tetrahedra for which all the faces are timelike.   The asymptotic formulae derived for the extension are found to have a similar form to the standard Ponzano-Regge asymptotic formulae for the SU(2) 6j symbol and so should be viable for use in a state sum for three dimensional Lorentzian quantum gravity. 
  Studies of new hyperbolic systems for the Einstein evolution equations show that the ``slicing density'' $\alpha(t,x)$ can be freely specified while the lapse $N = \alpha g^{1/2}$ cannot. Implementation of this small change in the Arnowitt-Deser-Misner action principle leads to canonical equations that agree with the Einstein equations whether or not the constraints are satisfied. The constraint functions, independently of their values, then propagate according to a first order symmetric hyperbolic system whose characteristic cone is the light cone. This result follows from the twice-contracted Bianchi identity and constitutes the central content of the constraint ``algebra'' in the canonical formalism. 
  We discuss gravitomagnetism in connection with rotating cylindrical systems. In particular, the gravitomagnetic clock effect is investigated for the exterior vacuum field of an infinite rotating cylinder. The dependence of the clock effect on the Weyl parameters of the stationary Lewis metric is determined. We illustrate our results by means of the van Stockum spacetime. 
  This paper presents a critical review of particle production in an uniform electric field and Schwarzchild-like spacetimes. Both problems can be reduced to solving an effective one-dimensional Schrodinger equation with a potential barrier. In the electric field case, the potential is that of an inverted oscillator -x^2 while in the case of Schwarchild-like spacetimes, the potential is of the form -1/x^2 near the horizon. The transmission and reflection coefficients can easily be obtained for both potentials. To describe particle production, these coefficients have to be suitably interpreted. In the case of the electric field, the standard Bogoliubov coefficients can be identified and the standard gauge invariant result is recovered. However, for Schwarzchild-like spacetimes, such a tunnelling interpretation appears to be invalid. The Bogoliubov coefficients cannot be determined by using an identification process similar to that invoked in the case of the electric field. The reason for such a discrepancy appears to be that, in the tunnelling method, the effective potential near the horizon is singular and symmetric. We also provide a new and simple semi-classical method of obtaining Hawking's result in the (t,r) co-ordinate system of the usual standard Schwarzchild metric. We give a prescription whereby the singularity at the horizon can be regularised with Hawking's result being recovered. This regularisation prescription contains a fundamental asymmetry that renders both sides of the horizon dissimilar. Finally, we attempt to interpret particle production by the electric field as a tunnelling process between the two sectors of the Rindler metric. 
  A study of the high angular momentum particles 'atmosphere' near the Schwarzschild black hole horizon suggested that strong gravitational interactions occur at invariant distance of the order of $\sqrt[3]{M}$ (A. Casher et. al). We present a generalization of this result to the Kerr-Newman black hole case. It is shown that the larger charge and angular momentum black hole bears, the larger invariant distance at which strong gravitational interactions occur becomes. This invariant distance is of order $\sqrt[3]{r_+^2/\hh}.$ This implies, that the Planckian structure of the Hawking radiation of extreme black holes is completely broken. 
  The non-minimal coupling of a scalar field to the Ricci curvature in a curved spacetime is unavoidable according to several authors. The coupling constant is not a free parameter: the prescriptions for the value of the coupling constant in specific scalar field and gravity theories (in particular in general relativity) are studied. The results are applied to the most popular inflationary scenarios of cosmology and their theoretical consistence is analysed. Certain observational constraints on the coupling constant are also discussed. 
  It has been found in several papers that, because of quantum corrections, light front can propagate with superluminal velocity in gravitational fields and even in flat space-time across two conducting plates. We show that, if this is the case, closed time-like trajectories would be possible and, in particular, in certain reference frames photons could return to their source of origin before they were produced there, in contrast to the opposite claim made in the literature. 
  The coordinate invariant theory of generalised functions of Colombeau and Meril is reviewed and extended to enable the construction of multi-index generalised tensor functions whose transformation laws coincide with their counterparts in classical distribution theory. 
  The need for a mathematically rigorous quantization procedure of singular spaces and incomplete motions is pointed out in connection with quantum cosmology. We put our previous suggestion for such a procedure, based on the theory of induced representations of C*-algebras, in the light of L. Schwartz' theory of Hilbert subspaces. This turns out to account for the freedom in the induction procedure, at the same time providing a basis for generalized eigenfunction expansions pertinent to the needs of quantum cosmology. Reinforcing our previous proposal for the wave-function of the Universe, we are now able to add a concrete prescription for its calculation. 
  We describe the short-distance properties of the spacetime of a system of D-particles by viewing their matrix-valued coordinates as coupling constants of a deformed worldsheet $\sigma$-model. We show that the Zamolodchikov metric on the associated moduli space naturally encodes properties of the non-abelian dynamics, and from this we derive new spacetime uncertainty relations directly from the quantum string theory. The non-abelian uncertainties exhibit decoherence effects which suggest the interplay of quantum gravity in multiple D-particle dynamics. 
  We give a comprehensive analysis of how scalar and tensor perturbations evolve in cosmologies with a smooth transition from power-law-like and de Sitter-like inflation to a radiation era. Analytic forms for the super-horizon and sub-horizon perturbations in the inflationary and radiation dominated eras are found. 
  The role of the Equivalence Principle (EP) in classical and quantum mechanics is reviewed. It is shown that the weak EP has a counterpart in quantum theory, a Quantum Equivalence Principle (QEP). This implies that also in the quantum domain the geometrisation of the gravitational interaction is an operational procedure similar to the procedure in classical physics. This QEP can be used for showing that it is only the usual Schr\"odinger equation coupled to gravito--inertial fields which obeys our equivalence principle. In addition, the QEP applied to a generalised Pauli equation including spin results in a characterisation of the gravitational fields which can be identified with the Newtonian potential and with torsion. Also, in the classical limit it is possible to state beside the usual EP for the path an EP for the spin which again may be used for introducing torsion as a gravitational field. 
  A new version of tetrad gravity in globally hyperbolic, asymptotically flat at spatial infinity spacetimes with Cauchy surfaces diffeomorphic to $R^3$ is obtained by using a new parametrization of arbitrary cotetrads to define a set of configurational variables to be used in the ADM metric action. Seven of the fourteen first class constraints have the form of the vanishing of canonical momenta. A comparison is made with other models of tetrad gravity and with the ADM canonical formalism for metric gravity. The phase space expression of various 4-tensors is explicitly given. 
  After a study of the Hamiltonian group of gauge transformations of canonical tetrad gravity on globally hyperbolic, asymptotically flat at spatial infinity, spacetimes with Cauchy hypersurfaces $\Sigma_{\tau}$ diffeomorphic to $R^3, we find the dependence of the cotriads on $\Sigma_{\tau}$ and of their momenta on the six parameters associated with rotations and space diffeomorphisms. The choice of 3-coordinates on $\Sigma_{\tau}$ is equivalent to the parametrization of the cotriads with the last three degrees of freedom individuating the 3-geometries. The Shanmugadhasan canonical transformation, corresponding to the choice of 3-orthogonal coordinates on $\Sigma_{\tau}$ and adapted to 13 of the 14 first class constraints, and the interpretation of the gauge transformations are given. The gauge interpretation of tetrad gravity based on constraint theory implies that a "Hamiltonian kinematical gravitational field" is an equivalence class of pseudo-Riemannian spacetimes modulo the Hamiltonian group of gauge transformations: it includes a conformal 3-geometry and all the different 4-geometries (standard definition of a kinematical gravitational field, $Riem M^4/Diff M^4$) connected to it by the gauge transformations. A "Hamiltonian Einstein or dynamical gravitational field" is a kinematical one which satisfies the Hamilton-Dirac equations generated by the ADM energy: it coincides with the standard Einstein or dynamical gravitational field, namely a 4-geometry solution of Einstein's equations, since the Hilbert and ADM actions both generate Einstein's equations so that the kinematical Hamiltonian gauge transformations are dynamically restricted to the spacetime diffeomorphisms of the solutions of Einstein's equations. 
  We demonstrate in the context of the minisuperspace model consisting of a closed Friedmann-Robertson-Walker universe coupled to a scalar field that Vilenkin's tunneling wavefunction can only be consistently defined for particular choices of operator ordering in the Wheeler-DeWitt equation. The requirement of regularity of the wavefunction has the particular consequence that the probability amplitude, which has been used previously in the literature in discussions of issues such as the prediction of inflation, is likewise ill-defined for certain choices of operator ordering with Vilenkin's boundary condition. By contrast, the Hartle-Hawking no-boundary wavefunction can be consistently defined within these models, independently of operator ordering. The significance of this result is discussed within the context of the debate about the predictions of semiclassical quantum cosmology. In particular, it is argued that inflation cannot be confidently regarded as a "prediction" of the tunneling wavefunction, for reasons similar to those previously invoked in the case of the no-boundary wavefunction. A synthesis of the no-boundary and tunneling approaches is argued for. 
  A common feature of reparametrization invariant theories is the difficulty involved in identifying an appropriate evolution parameter and in constructing a Hilbert space on states. Two well known examples of such theories are the relativistic point particle and the canonical formulation of quantum gravity. The strong analogy between them (specially for minisuperspace models) is considered in order to stress the correspondence between the ``localization problem'' and the ``problem of time,'' respectively. A possible solution for the first problem was given by the proper time formulation of relativistic quantum mechanics. Thus, we extrapolate the main outlines of such a formalism to the quantum gravity framework. As a consequence, a proposal to solve the problem of time arises. 
  The perturbation theory of black holes has been useful recently for providing estimates of gravitational radiation from black hole collisions. Second order perturbation theory, relatively undeveloped until recently, has proved to be important both for providing refined estimates and for indicating the range of validity of perturbation theory. Here we present the second order formalism for perturbations of Schwarzschild spacetimes. The emphasis is on practical methods for carrying out second order computations of outgoing radiation. General issues are illustrated throughout with examples from ``close-limit'' results, perturbation calculations in which black holes start from small separation. 
  The particles of a classical relativistic gas are supposed to move under the influence of a quasilinear (in the particle four-momenta), self-interacting force inbetween elastic, binary collisions. This force which is completely fixed by the equilibrium conditions of the gas, gives rise to an effective viscous pressure on the fluid phenomenological level. Earlier results concerning the possibility of accelerated expansion of the universe due to cosmological particle production are reinterpreted. A phenomenon such as power law inflation may be traced back to specific self-interacting forces keeping the particles of a gas universe in states of generalized equilibrium. 
  Assuming the four-dimensional space-time to be a general warped product of two surfaces we reduce the four-dimensional Einstein equations to a two-dimensional problem which can be solved. All global vacuum solutions are explicitly constructed and analysed. The classification of the solutions includes the Schwarzschild, the (anti-)de Sitter, and other well-known solutions but also many exact ones whose detailed global properties to our knowledge have not been discussed before. They have a natural physical interpretation describing single or several wormholes, domain walls of curvature singularities, cosmic strings, cosmic strings surrounded by domain walls, solutions with closed timelike curves, etc. 
  A geometrical model of electric charge is proposed. This model has ``naked'' charge screened with a ``fur - coat'' consisting of virtual wormholes. The 5D wormhole solution in the Kaluza - Klein theory is the ``naked'' charge. The splitting off of the 5D dimension happens on the two spheres (null surfaces) bounding this 5D wormhole. This allows one to sew two Reissner - Nordstr\"om black holes onto it on both sides. The virtual wormholes entrap a part of the electrical flux lines coming into the ``naked'' charge. This effect essentially changes the charge visible at infinity so that it satisfies the real relation $m^2<e^2$. 
  We investigate a class of impulsive gravitational waves which propagate either in Minkowski or in the (anti-)de Sitter background. These waves are constructed as impulsive members of the Kundt class $P(\Lambda)$ of non-twisting, non-expanding type N solutions of vacuum Einstein equations with a cosmological constant $\Lambda$. We show that the only non-trivial waves of this type in Minkowski spacetime are impulsive pp-waves. For $\Lambda\not=0$ we demonstrate that the canonical subclasses of $P(\Lambda)$, which are invariantly different for smooth profiles, are all locally equivalent for impulsive profiles. Also, we present coordinate system for these impulsive solutions which is explicitly continuous. 
  There is a well-known short list of asymptotic conserved quantities for a physical system at spatial infinity. We search for new ones.This is carried outwithin the asymptotic framework of Ashtekar and Romano, in which spatial infinity is represented as a smooth boundary of space-time. We first introduce, for physical fields on space-time,a characterization of their asymptotic behavior as certain fields on this boundary. Conserved quantities at spatial infinity, in turn, are constructed from these fields. We find,in Minkowski space-time, that each of a Klein-Gordon field, a Maxwell field, and a linearized gravitational field yields an entire hierarchy of conserved quantities. Only certain quantities in this hierarchy survive into curved space-time. 
  We compute the entropy of systems of quantum particles satisfying the fractional exclusion statistics in the space-time of 2+1 dimensional black hole by using the brick-wall method. We show that the entropy of each effective quantum field theory with a Planck scale ultraviolet cutoff obeys the area law, irrespective of the angular momentum of the black hole and the statistics interpolating between Bose-Einstein and Fermi-Dirac statistics. 
  Equivalence principles are a major part of modern relativity theory. Gravitational shifts can already be calculated within the time domain as motion shifts, and we examine the consequences of reversing this argument and describing motion shifts outside the time domain, as effects of curvature associated with relative velocity. This unusual "Doppler mass shift" approach appears to resolve some of Einstein's own criticisms of the "SR+GR" model and seems to remove some barriers to the reconciliation of classical and quantum theory. The disadvantage of this model is that constant-velocity problems no longer obey Euclidean geometry. By bypassing special relativity and the special theory's flat-space assumptions, the model also suggests an alternative non-transverse frequency-shift relationship. This difference should be testable. 
  In this work the possible role that Decoherence Model could play in the emergence of the classical concept of time is analyzed. We take the case of a Mixmaster universe with small anisotropy and construct its Halliwell propagator. Afterwards we introduce in our system terms that comprise the effects of Decoherence Model. This is done by means of the so called Restricted Path Integral Formalism. We obtain Halliwell's modified propagator and find that a gauge invariant physical time emerges as consequence of this process. 
  In this paper spherically symmetric solutions to 5D Kaluza-Klein theory, with ``electric'' and/or ``magnetic'' fields are investigated. It is shown that the global structure of the spacetime depends on the relation between the ``electrical'' and ``magnetic'' Kaluza-Klein fields. For small ``magnetic'' field we find a wormhole-like solution. As the strength of the ``magnetic'' field is increased relative to the strength of the ``electrical'' field, the wormhole-like solution evolves into a finite or infinite flux tube depending on the strengths of the two fields. For the large ``electric'' field case we conjecture that this solution can be considered as the mouth of a wormhole, with the $G_{55}$, $G_{5t}$ and $G_{5\phi}$ components of the metric acting as the source of the exotic matter necessary for the formation of the wormhole's mouth. For the large ``magnetic'' field case a 5D flux tube forms, which is similar to the flux tube between two monopoles in Type-II superconductors, or the hypothesized color field flux tube between two quarks in the QCD vacuum. 
  The role of the modular group in the holonomy representation of (2+1)-dimensional quantum gravity is studied. This representation can be viewed as a "Heisenberg picture", and for simple topologies, the transformation to the ADM "Schr{\"o}dinger picture" may be found. For spacetimes with the spatial topology of a torus, this transformation and an explicit operator representation of the mapping class group are constructed. It is shown that the quantum modular group splits the holonomy representation Hilbert space into physically equivalent orthogonal ``fundamental regions'' that are interchanged by modular transformations. 
  The Hamiltonian formulation of general relativity on a null surface is established in the teleparallel geometry. No particular gauge conditons on the tetrads are imposed, such as the time gauge condition. By means of a 3+1 decomposition the resulting Hamiltonian arises as a completely constrained system. However, it is structurally different from the the standard Arnowitt-Deser-Misner (ADM) type formulation. In this geometrical framework the basic field quantities are tetrads that transform under the global SO(3,1) and the torsion tensor. 
  A recent progress in obtaining non-spherical and non-static solitons in the four-dimensional Einstein--Yang--Mills (EYM) theory is discussed, and a non-perturbative formulation of the stationary axisymmetric problem is attempted. First a 2D dilaton gravity model is derived for the spherically symmetric time-dependent configurations. Then a similar Euclidean representation is constructed for the stationary axisymmetric non-circular SU(2) EYM system using the (2+1)+1 reduction scheme suggested by Maeda, Sasaki, Nakamura and Miyama. The crucial role in this reduction is played by the extra terms entering the reduced Yang--Mills and Kaluza--Klein two-forms similarly to Chern--Simons terms in the theories with higher rank antisymmetric tensor fields. We also derive a simple 2D action describing static axisymmetric magnetic EYM configurations and discuss a possibility of existence of cylindrical EYM sphalerons. 
  We give a characterization of the central shell-focusing curvature singularity that can form in the spherical gravitational collapse of a bounded matter distribution obeying the dominant energy condition. This characterization is based on the limiting behaviour of the mass function in the neighbourhood of the singularity. Depending on the rate of growth of the mass as a function of the area radius R, the singularity may be either covered or naked. The singularity is naked if this growth rate is slower than R, covered if it is faster than R, and either naked or covered if the growth rate is same as R. 
  A modification of Kaluza-Klein theory is proposed in which, as a result of a symmetry breaking, five-dimensional space-time is partially parallelized implying the appearance of torsion fields. A naturally chosen action functional leads to the Einstein-Cartan-Maxwell theory where the electromagnetic field strength is represented by the fifth component of the torsion 2-form. Incorporation of a scalar field reveals that four-dimensional space-time torsion is not induced by the scalar field. 
  Building on the results of previous work, we demonstrate how matter fields are incorporated into the general linear frame approach to general relativity. When considering the Maxwell one-form field, we find that the system that leads naturally to canonical vierbein general relativity has the extrinsic curvature of the Cauchy surface represented by gravitational as well as non-gravitational degrees of freedom. Nevertheless the metric compatibility conditions are undisturbed, and this apparent derivative-coupling is seen to be an effect of working with (possibly orthonormal) linear frames. The formalism is adapted to consider a Dirac Fermion, where we find that a milder form of this apparent derivative-coupling appears. 
  We present a review of some recent models of gravitation theory with propagating torsion based on the use of a torsion-dilaton field and propose one more model of this type which promises to be more realistic. A proper universal self-consistent minimal action principle yields the properties of this model and predicts the interactions of torsion-dilaton field with the real matter. The new model may be compatible with the string models with dilaton field and gives a novel interpretation of the dilaton as a part of the space-time torsion. A relation with some recent models of dilatonic gravity is also possible. 
  We study the tensorial modes of the two-fluid model, where one of this fluids has an equation of state $p = - \rho/3$ (variable cosmological constant, cosmic string fluid, texture) or $p = - \rho$ (cosmological constant), while the other fluid is an ordinary matter (radiation, stiff matter, incoherent matter). In the first case, it is possible to have a closed Universe whose dynamics can be that of an open Universe providing alternative solutions for the age and horizon problems. This study of the gravitational waves is extended for all values of the effective curvature $k_{eff}=k-\frac{8\pi G}{3}\rho_{0s}$, that is, positive, negative or zero, $k$ being the curvature of the spacelike section. In the second case, we restrict ourselves to a flat spatial section. The behaviour of gravitational waves have, in each case, very particular features, that can be reflected in the anisotropy spectrum of Cosmic Microwave Background Radiation. We make also some considerations of these models as candidate to dark matter models. 
  The solutions to the Einstein-Klein-Gordon equations without a cosmological constant are investigated for an exponential potential in a Bianchi VI_0 metric. There exists a two-parameter family of solutions which have a power-law inflationary behaviour when the exponent of the potential, k, satisfies k^2<2. In addition, there exists a two-parameter family of singular solutions for all k^2 values. A simple anisotropic exact solution is found to be stable when 2<k^2. 
  For a general class of scalar--tensor gravity theories, we discuss how to recover asymptotic freedom regimes when cosmic time $t\to\pm\infty$. Such a feature means that the effective gravitational coupling $G_{eff}\to 0$, while cosmological solutions can asymptotically assume de Sitter or power--law behaviours. In our opinion, through this mechanism, it is possible to cure some shortcomings in inflationary and in string--dilaton cosmology. 
  We propose a cosmological model which could explain, in a very natural way, the apparently periodic structures of the universe, as revealed in a series of recent observations. Our point of view is to reduce the cosmological Friedman--Einstein dynamical system to a sort of Schr\"odinger equation whose bound eigensolutions are oscillating functions. Taking into account the cosmological expansion, the large scale periodic structure could be easily recovered considering the amplitudes and the correlation lengths of the galaxy clusters. 
  We discuss the phase diagram of the balls in boxes model, with a varying number of boxes. The model can be regarded as a mean-field model of simplicial gravity. We analyse in detail the case of weights of the form $p(q) = q^{-\beta}$, which correspond to the measure term introduced in the simplicial quantum gravity simulations. The system has two phases~: {\em elongated} ({\em fluid}) and {\em crumpled}. For $\beta\in (2,\infty)$ the transition between these two phases is first order, while for $\beta \in (1,2]$ it is continuous. The transition becomes softer when $\beta$ approaches unity and eventually disappears at $\beta=1$. We then generalise the discussion to an arbitrary set of weights. Finally, we show that if one introduces an additional kinematic bound on the average density of balls per box then a new {\em condensed} phase appears in the phase diagram. It bears some similarity to the {\em crinkled} phase of simplicial gravity discussed recently in models of gravity interacting with matter fields. 
  The relation between angular diameter distance and redshift in a spherically symmetric dust-shell universe is studied. This model has large inhomogeneities of matter distribution on small scales. We have discovered that the relation agrees with that of an appropriate Friedmann-Lemaitre(FL) model if we set a ``homogeneous'' expansion law and a ``homogeneous'' averaged density field. This will support the averaging hypothesis that a universe looks like a FL model in spite of small-scale fluctuations of density field, if its averaged density field is homogeneous on large scales. 
  We study the distance-redshift relation in a universe filled with point particles, and discuss what the universe looks like when we make the number of particles N very large, while fixing the averaged mass density. Using the Raychaudhuri equation and a simple analysis of the probability of strong lensing effects, we show that the statistical nature of the amplification is independent of N, and clarify the appearance of the point particle universe. 
  A new method is presented for obtaining the general conformally flat radiation metric by using the differential operators of Machado Ramos and Vickers (a generalisation of the GHP operators) which are invariant under null rotations and spin and boosts. The solution is found by constructing involutive tables of these derivatives applied to the quantities which arise in the Karlhede classification of metrics. 
  Exact Tolman solutions are used to analyse the implications if the galactic number has a fractal form out to a distance of about 150 Mpc in a universe which is homogeneous on the large scale. It is concluded that such a model requires either a non-linear Hubble law or a very low density if galaxies trace the total matter distribution. 
  It was pointed out recently [A. Kent, Phys. Rev. Lett. 78 (1997) 2874] that the consistent histories approach allows contrary inferences to be made from the same data. These inferences correspond to projections $P$ and $Q$, belonging to different consistent sets, with the properties that $PQ = QP = 0$ and $P \neq 1 -Q$. To many, this seems undesirable in a theory of physical inferences. It also raises a specific problem for the consistent histories formalism, since that formalism is set up so as to eliminate contradictory inferences, i.e. inferences $P$ and $Q$ where $P = 1 - Q$. Yet there seems to be no sensible physical distinction between contradictory and contrary inferences.   It seems particularly hard to defend the asymmetry, since (i) there is a well-defined quantum histories formalisms which admits both contradictory and contrary inferences, and (ii) there is also a well-defined formalism, based on ordered consistent sets of histories, which excludes both.   In a recent comment, Griffiths and Hartle, while accepting the validity of the examples given in the above paper, restate their own preference for the consistent histories formalism. As this brief reply explains, in so doing, they fail to address the arguments made against their approach to quantum theory. 
  Based on the recent finding that the difference in proper time of two clocks in prograde and retrograde equatorial orbits about the Earth is of the order 10^{-7}s per revolution, the possibility of detecting the terrestrial gravitomagnetic field by means of clocks carried by satellites is discussed. A mission taking advantage of this influence of the rotating Earth on the proper time is outlined and the conceptual difficulties are briefly examined. 
  In the path integral approach to false vacuum decay with the effect of gravity, there is an unsolved problem, called the negative mode problem. We show that the appearance of a supercritical supercurvature mode in the one-bubble open inflation scenario is equivalent to the existence of a negative mode around the Euclidean bounce solution. Supercritical supercurvature modes are those whose mode functions diverge exponentially for large spatial radius on the time constant hypersurface of the open universe. Then we propose a conjecture that there should be ``no supercritical supercurvature mode''. For a class of models that contains a wide variety of tunneling potentials, this conjecture is shown to be correct. 
  The renormalized expectation value of the stress energy tensor of the conformally invariant massless fields in the Unruh state in the Schwarzschild spacetime is constructed. It is achieved through solving the conservation equation in conformal space and utilizing the regularity conditions in the physical metric. The relations of obtained results to the existing approximations are analysed. 
  We study the problem of the existence of a local quantum scalar field theory in a general affine metric space that in the semiclassical approximation would lead to the autoparallel motion of wave packets, thus providing a deviation of the spinless particle trajectory from the geodesics in the presence of torsion. The problem is shown to be equivalent to the inverse problem of the calculus of variations for the autoparallel motion with additional conditions that the action (if it exists) has to be invariant under time reparametrizations and general coordinate transformations, while depending analytically on the torsion tensor. The problem is proved to have no solution for a generic torsion in four-dimensional spacetime. A solution exists only if the contracted torsion tensor is a gradient of a scalar field. The corresponding field theory describes coupling of matter to the dilaton field. 
  We consider a quantum mechanical black hole model introduced in {\it Phys.Rev.}, {\bf D57}, 1118 (1998) that consists of the selfgravitating dust shell. The Schroedinger equation for this model is a finite difference equation with the shift of the argument along the imaginary axis. Solving this equation in quasiclassical limit in complex domain leads to quantization conditions that define discrete quasiclassical mass spectrum. One of the quantization conditions is Bohr-Sommerfeld condition for the bound motion of the shell. The other comes from the requirement that the wave function is unambiguously defined on the Riemannian surface on which the coefficients of Schroedinger equation are regular. The second quantization condition remains valid for the unbound motion of the shell as well, and in the case of a collapsing null-dust shell leads to $m\sim\sqrt{k}$ spectrum. 
  We derive the coupling to torsion of massive electroweak vector bosons generated by the Higgs mechanism. 
  Within a framework of noncommutative geometry, we develop an analogue of (pseudo) Riemannian geometry on finite and discrete sets. On a finite set, there is a counterpart of the continuum metric tensor with a simple geometric interpretation. The latter is based on a correspondence between first order differential calculi and digraphs. Arrows originating from a vertex span its (co)tangent space. If the metric is to measure length and angles at some point, it has to be taken as an element of the left-linear tensor product of the space of 1-forms with itself, and not as an element of the (non-local) tensor product over the algebra of functions. It turns out that linear connections can always be extended to this left tensor product, so that metric compatibility can be defined in the same way as in continuum Riemannian geometry. In particular, in the case of the universal differential calculus on a finite set, the Euclidean geometry of polyhedra is recovered from conditions of metric compatibility and vanishing torsion. In our rather general framework (which also comprises structures which are far away from continuum differential geometry), there is in general nothing like a Ricci tensor or a curvature scalar. Because of the non-locality of tensor products (over the algebra of functions) of forms, corresponding components (with respect to some module basis) turn out to be rather non-local objects. But one can make use of the parallel transport associated with a connection to `localize' such objects and in certain cases there is a distinguished way to achieve this. This leads to covariant components of the curvature tensor which then allow a contraction to a Ricci tensor. In the case of a differential calculus associated with a hypercubic lattice we propose a new discrete analogue of the (vacuum) Einstein equations. 
  Gravitational-wave data analysis requires a detailed understanding of the highly relativistic, late stages of inspiral of neutron-star and black-hole binaries. A promising method to compute the late inspiral and its emitted waves is numerical relativity in co-rotating coordinates. The coordinates must be kept co-rotating via an appropriate choice of numerical relativity's lapse and shift functions. This article proposes a model problem for testing the ability of various lapse and shift prescriptions to keep the coordinates co-rotating. 
  We present our results on numerical study of evolution of nonlinear perturbations inside spherically symmetric black holes in the SU(2) Einstein-Yang-Mills (EYM) theory. Recent developments demonstrate a new type of the behavior of the metric for EYM black hole interiors; the generic metric exhibits an infinitely oscillating approach to the singularity, which is a spacelike but not of the mixmaster type. The evolution of various types of spherically symmetric perturbations, propagating from the internal vicinity of the external horizon towards the singularity is investigated in a self-consistent way using an adaptive numerical algorithm. The obtained results give a strong numerical evidence in favor of nonlinear stability of the generic EYM black hole interiors. Alternatively, the EYM black hole interiors of S(chwarzschild)-type, which form only a zero measure subset in the space of all internal solutions are found to be unstable and transform to the generic type as perturbations are developed. 
  The semi-classical collapse, including lowest order back-reaction, of a thin shell of self-gravitating quantized matter is illustrated. The conditions for which self-gravitating matter forms a thin shell are first discussed and an effective Lagrangian for such matter is obtained. The matter-gravity system is then quantized, the semi-classical limit for gravitation is taken and the method of adiabatic invariants is applied to the resulting time dependent matter Hamiltonian. The governing equations are integrated numerically, for suitable initial conditions, in order to illustrate the effect of back-reaction, due to the creation of matter, in slowing down the collapse near the horizon. 
  We consider a massive selfgravitating shell as a model for collapsing body and a null selfgravitating shell as a model for quanta of Hawking radiation. It is show that the mass-energy spectra for the body and the radiation do not match. The way out of this difficulty is to consider not only out-going radiation but also the ingoing one. It means that the structure of black hole is changing during its evaporation resulting in the Bekenstein-Mukhanov spectrum for large masses. 
  I discuss the no hair principle, the recently found hairy solutions, generic properties of nonvacuum spherical static black holes, and the new no scalar hair theorems. I go into the generic phenomenon of superradiance, first uniform linear motion superradiance, then Kerr black hole superradiance, and finally general rotational superradiance and its possible applications in the laboratory. I show that the horizon area of a nearly stationary black hole can be regarded as an adiabatic invariant. This invariance suggests that quantum horizon area is quantized in multiples of a basic unit. Consideration of the quantum version of the Christodoulou reversible processes provides support for this idea. Horizon area quantization dictates a definite discrete black hole mass spectrum, so that Hawking's semiclassical spectrum is predicted to be replaced by a spectrum of nearly uniformly spaced lines whose envelope is roughly Planckian. Line natural broadening seems not enough to wash out the lines. To check on the possibility of line splitting, I present a simple algebra involving, among other operators, the black hole observables. Under simple assumptions it also leads to the uniformly spaced area spectrum. 
  As a consequence of the extreme precision of the measurements it performs, an interferometric gravitational wave detector is a macroscopic apparatus for which quantum effects are not negligible. I observe that this property can be exploited to probe some aspects of the interplay between Quantum Mechanics and Gravity. 
  We show that the Dirac equation on de Sitter background can be analytically solved in a special static frame where the energy eigenspinors can be expressed in terms of usual angular spinors known from special relativity, and a pair of radial wave functions. 
  Primordial nucleosynthesis is considered a success story of the standard big bang (SBB) cosmology. The cosmological and elementary particle physics parameters are believed to be severely constrained by the requirement of correct abundances of light elements. We explore nucleosynthesis in a class of models very different from SBB. In these models the cosmological scale factor increases linearly with time right through the period during which nucleosynthesis occurs till the present. It turns out that weak interactions remain in thermal equilibrium upto temperatures which are two orders of magnitude lower than the corresponding (weak interaction decoupling) temperature in SBB. Inverse beta decay of the proton can ensure adequate production of several light elements while producing primordial metalicity much higher than that produced in SBB. Other attractive features of these models are the absence of the horizon, flatness and the age problems and consistency with classical cosmological tests. 
  We show that Raychaudhuri's recently proposed theorem on nonrotating universes cannot be used to rule out realistic singularity-free descriptions of the universe, as suggested by him in PRL 80, 654 (1998). 
  The quantum measurement problem and various unsuccessful attempts to resolve it are reviewed. A suggestion by Diosi and Penrose for the half life of the quantum superposition of two Newtonian gravitational fields is generalized to an arbitrary quantum superposition of relativistic, but weak, gravitational fields. The nature of the ``collapse'' process of the wave function is examined. 
  The concept of constrained gravitational instanton is introduced. It is used to study black hole creation. We discussed the global aspects of the scenario and the alternative tunnelings. 
  Black holes are among the most intriguing objects in modern physics. Their influence ranges from powering quasars and other active galactic nuclei, to providing key insights into quantum gravity. We review the observational evidence for black holes, and briefly discuss some of their properties. We also describe some recent developments involving cosmic censorship and the statistical origin of black hole entropy. 
  We study the phenomenon of gyroscopic precession and the analogues of inertial forces within the framework of general relativity. Covariant connections between the two are established for circular orbits in stationary spacetimes with axial symmetry. Specializing to static spacetimes, we prove that gyroscopic precession and centrifugal force both reverse at the photon orbits. Simultaneous non-reversal of these in the case of stationary spacetimes is discussed. Further insight is gained in the case of static spacetime by considering the phenomena in a spacetime conformal to the original one. Gravi-electric and gravi-magnetic fields are studied and their relation to inertial forces is established. 
  It is known that the action of Euclidean Einstein gravity is not bounded from below and that the metric of flat space does not correspond to a minimum of the action. Nevertheless, perturbation theory about flat space works well. The deep dynamical reasons for this reside in the non perturbative behaviour of the system and have been clarified in part by numerical simulations. Several open issues remain. We treat in particular those zero modes of the action for which R(x) is not identically zero, but the integral of sqrt{g(x)} R(x) vanishes. 
  Creation of a black hole in quantum cosmology is the third way of black hole formation. In contrast to the gravitational collapse from a massive body in astrophysics or from the quantum fluctuation of matter fields in the very early universe, in quantum cosmology scenario the black hole is essentially created from nothing. The black hole originates from a constrained gravitational instanton. The probability of creation for all kinds of single black holes in the Kerr- Newman-de Sitter family, at the semi-classical level, is the exponential of the total entropy of the universe, or one quarter of the sum of both the black hole and the cosmological horizon areas. The de Sitter spacetime is the most probable evolution at the Planckian era. 
  We point out the existence of a new type of growing transverse mode in the gravitational instability. This appears as a post-Newtonian effect to Newtonian dynamics. We demonstrate this existence by formulating the Lagrangian perturbation theory in the framework of the cosmological post-Newtonian approximation in general relativity. Such post-Newtonian order effects might produce characteristic appearances of large-scale structure formation, for example, through the observation of anisotropy of the cosmic microwave background radiation (CMB). 
  We investigated the behavior of an open isotropic universe generated by a scalar field which couples with background curvature nonminimally with the coupling constant $\xi$. In particular we focus on the situation where the initial value for the scalar field $\phi_{\rm in}$ is greater than the critical value ${\hat \phi}_c$=${m_p}/{\sqrt{8\pi\xi(1-6\xi)}}$. The behavior is similar to an open de Sitter universe with $k=-1$ with a negative cosmological constant $\Lambda <0$. It is found that the universe will collapse eventually to a singularity and thus has a finite extent in time in the future. Furthermore, there are some cases which shows a rebouncing behavior before the final collapse. 
  We show that two-field inflation can be followed by an era in which the field dynamics become chaotic, and discuss the possible consequences of this for two-field inflationary models. [PASCOS conference proceedings - a short version of gr-qc/9711035] 
  A class of exact regular spherically symmetric solutions to the Einstein equation obeying Dymnikova's definition of the vacuumlike state is considered. These solutions, which may be interpreted as black holes, are not only singularity free, but also do not set us thinking about the loss of information under gravitational collapse. According to the singularity theorems, the geometries introduced by these solutions inevitably have some causal pathology. However, if the vacuumlike state is reinterpreted to be a sort of `confinement' representing a particular phase of matter, this pathology under certain conditions does not involve actual causality violation. By means of that, the above class of solutions, and probably much broader variety of solutions including dynamic ones, may be incorporated into General Relativity. There are listed other phenomena, such as `hidden mass', that could help to identify the presence of vacuumlike phase. 
  We show that, contrary to a widespread belief, one can overcharge a near extremal Reissner-Nordstrom black hole by throwing in a charged particle, as long as the backreaction effects may be considered negligible. Furthermore, we find that we can make the particle's classical radius, mass, and charge, as well as the relative size of the backreaction terms arbitrarily small, by adjusting the parameters corresponding to the particle appropriately. This suggests that the question of cosmic censorship is still not wholly resolved even in this simple scenario. We contrast this with attempting to overcharge a black hole with a charged imploding shell, where we find that cosmic censorship is upheld. We also briefly comment on a number of possible extensions. 
  We present black hole uniqueness theorems for the C-metric and Ernst solution. The proof follows a similar strategy as that used to prove the uniqueness of the Kerr-Newman solution, however the presence of an acceleration horizon provides some critical differences. We also show how to derive the Bunting/Mazur result (on the positivity of a suitable divergence required in the proof) using new methods. We briefly explain the importance of the uniqueness of the Ernst solution in relation to the proposed black hole monopole pair creation mediated by the related instanton. 
  We make use of an internal symmetry of a truncation of the bosonic sector of the superstring and N=4 supergravity theories to write down an analogue of Robinson's identity for the black holes of this theory. This allows us to prove the uniqueness of a restricted class of black hole solutions. In particular, we can apply the methods of the preceding paper to prove the uniqueness of a class of accelerating black holes (the Stringy Ernst solution and Stringy C-metric) which incorporate the possibility of the black hole accelerating within an electromagnetic flux tube. These solutions and their associated uniqueness may be useful in future instanton calculations. 
  Quantum Mechanics is revisited as the appropriate theoretical framework for the description of the outcome of experiments that rely on the use of classical devices. In particular, it is emphasized that the limitations on the measurability of (pairs of conjugate) observables encoded in the formalism of Quantum Mechanics reproduce faithfully the ``classical-device limit'' of the corresponding limitations encountered in (real or gedanken) experimental setups. It is then argued that devices cannot behave classically in Quantum Gravity, and that this might raise serious problems for the search of a class of experiments described by theories obtained by ``applying Quantum Mechanics to Gravity.'' It is also observed that using heuristic/intuitive arguments based on the absence of classical devices one is led to consider some candidate Quantum-Gravity phenomena involving dimensionful deformations of the Poincare' symmetries. 
  Several proposals for Quantum Gravity involve length and area operators with discrete eigenvalues. I show that the analyses of some simple procedures for the measurement of areas and lengths suggest that this discreteness characterizing the formalism might not be observable. I also discuss a possible relation with the so-called $\kappa$ deformations of Poincare' symmetries. 
  We clarify some issues related to the evaluation of the mean value of the energy-momentum tensor for quantum scalar fields coupled to the dilaton field in two-dimensional gravity. Because of this coupling, the energy-momentum tensor for the matter is not conserved and therefore it is not determined by the trace anomaly. We discuss different approximations for the calculation of the energy-momentum tensor and show how to obtain the correct amount of Hawking radiation. We also compute cosmological particle creation and quantum corrections to the Newtonian potential. 
  Causal anomalies in two Kaluza-Klein gravity theories are examined, particularly as to whether these theories permit solutions in which the causality principle is violated. It is found that similarly to general relativity the field equations of the space-time-mass Kaluza-Klein (STM-KK) gravity theory do not exclude violation of causality of G\"odel type, whereas the induced matter Kaluza-Klein (IM-KK) gravity rules out noncausal G\"odel-type models. The induced matter version of general relativity is shown to be an efficient therapy for causal anomalies that occurs in a wide class of noncausal geometries. Perfect fluid and dust G\"odel-type solutions of the STM-KK field equations are studied. It is shown that every G\"odel-type perfect fluid solution is isometric to the unique dust solution of the STM-KK field equations. The question as to whether 5-D G\"odel-type non-causal geometries induce any physically acceptable 4-D energy-momentum tensor is also addressed. 
  This paper presents some possible features of general expressions for Lovelock tensors and for the coefficients of Lovelock Lagrangians up to the 15th order in curvature (and beyond) in terms of the Riemann-Christoffel and Ricci curvature tensors and the Riemann curvature scalar for n-dimensional differentiable manifolds having a general linear connection. 
  We present a class of exact solutions of Weyl conformal gravity, which have an interpretation as topological black holes. Solutions with negative, zero or positive scalar curvature at infinity are found, the former generalizing the well-known topological black holes in anti-de Sitter gravity. The rather delicate question of thermodynamic properties of such objects in Weyl conformal gravity is discussed; suggesting that the thermodynamics of the found solutions should be treated within the framework of gravity as an induced phenomenon, in the spirit of Sakharov's work. 
  The dynamical symmetries of hot and electrically neutral plasmas in a highly conducting medium suggest that, after the epoch of the electron-positron annihilation, magnetohydrodynamical configurations carrying a net magnetic helicity can be present. The simultaneous conservation of the magnetic flux and helicity implies that the (divergence free) field lines will possess inhomogeneous knot structures acting as source seeds in the evolution equations of the scalar, vector and tensor fluctuations of the background geometry. We give explicit examples of magnetic knot configurations with finite energy and we compute the induced metric fluctuations. Since magnetic knots are (conformally) coupled to gravity via the vertex dictated by the equivalence principle, they can imprint spikes in the gravitational wave spectrum for frequencies compatible with the typical scale of the knot corresponding, in our examples, to a present frequency range of $10^{-11}$--$10^{-12}$ Hertz. At lower frequencies the spectrum is power-suppressed and well below the COBE limit. For smaller length scales (i.e. for larger frequencies) the spectrum is exponentially suppressed and then irrelevant for the pulsar bounds. Depending upon the number of knots of the configuration, the typical amplitude of the gravitational wave logarithmic energy spectrum (in critical units) can be even four orders of magnitude larger than the usual flat (inflationary) energy spectrum generated thanks to the parametric amplification of the vacuum fluctuations. 
  We study a solution of the Einstein-Gauus-Bonnet theory in 5 dimensions coupled to a Maxwell field, whose euclidean continuation gives rise to an instanton describing black hole pair production. We also discuss the dual theory with a 3-form field coupled to gravity. 
  We develop and calibrate a new method for estimating the gravitational radiation emitted by complex motions of matter sources in the vicinity of black holes. We compute numerically the linearized curvature perturbations induced by matter fields evolving in fixed black hole backgrounds, whose evolution we obtain using the equations of relativistic hydrodynamics. The current implementation of the proposal concerns non-rotating holes and axisymmetric hydrodynamical motions. As first applications we study i) dust shells falling onto the black hole isotropically from finite distance, ii) initially spherical layers of material falling onto a moving black hole, and iii) anisotropic collapse of shells. We focus on the dependence of the total gravitational wave energy emission on the flow parameters, in particular shell thickness, velocity and degree of anisotropy. The gradual excitation of the black hole quasi-normal mode frequency by sufficiently compact shells is demonstrated and discussed. A new prescription for generating physically reasonable initial data is discussed, along with a range of technical issues relevant to numerical relativity. 
  We have investigated the behavior of three curvature invariants for Schwarzschild, Reissner-Nordstr{\o}m, Kerr, and Kerr-Newman black holes. We have also studied these invariants for a Schwarzschild-de Sitter space-time, the $\gamma$ metric, and for a 2+1 charged dimensional black hole. The invariants are $I_{1}=R_{\alpha\beta\mu\nu;\lambda}R^{\alpha\beta\mu\nu;\lambda}$, $I_{2}=R_{\mu\nu;\lambda} R^{\mu\nu;\lambda}$, and $I_{3}=C_{\alpha\beta\mu\nu;\lambda}C^{\alpha\beta\mu\nu;\lambda}$. For all but the Kerr-Newman case these invariants serve as either horizon or stationary surface detectors. The Kerr-Newman case is more complicated. We show that $I_{1}$ vanishs on the horizon in any space-time with a Schwarzschild like metric. 
  We present an exact treatment of the influences on a quantum scalar field in its Minkowski vacuum state induced by coupling of the field to a uniformly accelerated harmonic oscillator. We show that there are no radiation from the oscillator in the point of view of a uniformly accelerating observer. On the other hand, there are radiations in the point of view of an inertial observer. It is shown that Einstein-Podolsky-Rosen (EPR) like correlations of Rindler particles in Minkowski vacuum states are modified by a phase factor in front of the momentum-symmetric Rindler operators. The exact quantization of a time-dependent oscillator coupled to a massless scalar field was given. 
  The density perturbation during inflation seeds the large scale structure. We consider both new inflation-type and chaotic inflation-type potentials in the framework of Einstein-Brans-Dicke gravity. The density perturbation gives strong constraints on the parameters in these potentials. For both potentials, the constraints are not much different from those obtained in the original inflationary models by using of Einstein gravity. 
  We show the rigid singularity theorem, that is, a globally hyperbolic spacetime satisfying the strong energy condition and containing past trapped sets, either is timelike geodesically incomplete or splits isometrically as space $\times$ time. This result is related to Yau's Lorentzian splitting conjecture. 
  We study the nonequilibrium dynamics leading to the formation of topological defects in a symmetry-breaking phase transition of a quantum scalar field with \lambda\Phi^4 self-interaction in a spatially flat, radiation-dominated Friedmann-Robertson-Walker Universe. The quantum field is initially in a finite-temperature symmetry-restored state and the phase transition develops as the Universe expands and cools. We present a first-principles, microscopic approach in which the nonperturbative, nonequilibrium dynamics of the quantum field is derived from the two-loop, two-particle-irreducible closed-time-path effective action. We numerically solve the dynamical equations for the two-point function and we identify signatures of topological defects in the infrared portion of the momentum-space power spectrum. We find that the density of topological defects formed after the phase transition scales as a power law with the expansion rate of the Universe. We calculate the equilibrium critical exponents of the correlation length and relaxation time for this model and show that the power law exponent of the defect density, for both overdamped and underdamped evolution, is in good agreement with the "freeze-out" scenario of Zurek. We introduce an analytic dynamical model, valid near the critical point, that exhibits the same power law scaling of the defect density with the quench rate. By incorporating the realistic quench of the expanding Universe, our approach illuminates the dynamical mechanisms important for topological defect formation. The observed power law scaling of the defect density with the quench rate, observered here in a quantum field theory context, provides evidence for the "freeze-out" scenario in three spatial dimensions. 
  We report on the cited papers refs. 1 - 18 from the following points of view: What do we exactly know about solutions when no exact solution (in the sense of "solution in closed form") is available? In which sense do these solutions possess a singularity? In which cases do conformal relations and/or dimensional reductions simplify the deduction? Furthermore, we outline some open questions worth of being studied in future research. 
  A physical interpretation is presented of the general class of conformally flat pure radiation metrics that has recently been identified by Edgar and Ludwig. It is shown that, at least in the weak field limit, successive wave surfaces can be represented as null (half) hyperplanes rolled around a two-dimensional null cone. In the impulsive limit, the solution reduces to a pp-wave whose direction of propagation depends on retarded time. In the general case, there is a coordinate singularity which corresponds to an envelope of the wave surfaces. The global structure is discussed and a possible vacuum extension through the envelope is proposed. 
  The properties of LRS class II perfect fluid space-times are analyzed using the description of geometries in terms of the Riemann tensor and a finite number of its covariant derivatives. In this manner it is straightforward to obtain the plane and hyperbolic analogues to the spherical symmetric case. For spherically symmetric static models the set of equations is reduced to the Tolman-Oppenheimer-Volkoff equation only. Some new non-stationary and inhomogeneous solutions with shear, expansion, and acceleration of the fluid are presented. Among these are a class of temporally self-similar solutions with equation of state given by $p=(\gamma-1)\mu, 1<\gamma<2$, and a class of solutions characterized by $\sigma=-\Theta/6$. We give an example of geometry where the Riemann tensor and the Ricci rotation coefficients are not sufficient to give a complete description of the geometry. Using an extension of the method, we find the full metric in terms of curvature quantities. 
  The secular evolution of a spinning, massive binary system in eccentric orbit is analyzed, expanding and generalizing our previous treatments of the Lense-Thirring motion and the one-spin limit. The spin-orbit and spin-spin effects up to the 3/2 post-Newtonian order are considered, both in the equations of motion and in the radiative losses. The description of the orbit in terms of the true anomaly parametrization provides a simple averaging technique, based on the residue theorem, over eccentric orbits. The evolution equations of the angle variables characterizing the relative orientation of the spin and orbital angular momenta reveal a speed-up effect due to the eccentricity. The dissipative evolutions of the relevant dynamical and angular variables is presented in the form of a closed system of differential equations. 
  A generally covariant system can be deparametrized by means of an ``extrinsic'' time, provided that the metric has a conformal ``temporal'' Killing vector and the potential exhibits a suitable behavior with respect to it. The quantization of the system is performed by giving the well ordered constraint operators which satisfy the algebra. The searching of these operators is enlightned by the methods of the BRST formalism. 
  The recently introduced mechanism of flavor-oscillation clocks has been used to emphasize observability of constant gravitational potentials and thereby to question completeness of the theory of general relativity. An inequality has been derived to experimentally test the thesis presented. 
  Using the Einstein-Cartan-Dirac theory, we study the effect of torsion on neutrino oscillation. We see that torsion cannot induce neutrino oscillation, but affects it whenever oscillation exists for other reasons. We show that the torsion effect on neutrino oscillation is as important as the neutrino mass effect, whenever the ratio of neutrino number density to neutrino energy is $\sim 10^{69}$ cm$^{-3}$ /eV, or the number density of the matter is $\sim 10^{69}$ cm$^{-3}$. 
  We investigate the topological black holes in a special class of Lovelock gravity. In the odd dimensions, the action is the Chern-Simons form for the anti-de Sitter group. In the even dimensions, it is the Euler density constructed with the Lorentz part of the anti-de Sitter curvature tensor. The Lovelock coefficients are reduced to two independent parameters: cosmological constant and gravitational constant. The event horizons of these topological black holes may have constant positive, zero or negative curvature. Their thermodynamics is analyzed and electrically charged topological black holes are also considered. We emphasize the differences due to the different curvatures of event horizons. As a comparison, we also discuss the topological black holes in the higher dimensional Einstein-Maxwell theory with a negative cosmological constant. 
  Symmetries compatible with asymptotic flatness and admitting gravitational and electromagnetic radiation are studied by using the Bondi-Sachs-van der Burg formalism. It is shown that in axially symmetric electrovacuum spacetimes in which at least locally a smooth null infinity in the sense of Penrose exists, the only second allowable symmetry is either the translational symmetry or the boost symmetry. Translationally invariant spacetimes with in general a straight "cosmic string" along the axis of symmetry are non-radiative although they can have a non-vanishing news function. The boost-rotation symmetric spacetimes are radiative. They describe "uniformly accelerated charged particles" or black holes which in general may also be rotating - the axial and an additional Killing vector are not assumed to be hypersurface orthogonal. The general functional forms of both gravitational and electromagnetic news functions, and of the mass aspect and total mass of asymptotically flat boost-rotation symmetric spacetimes at null infinity are obtained. The expressions for the mass are new even in the case of vacuum boost-rotation symmetric spacetimes with hypersurface orthogonal Killing vectors. In Appendices some errors appearing in previous works are corrected. 
  Under certain circumstances, the collision of magnetic monopoles, topologically locked-in regions of false vacuum, leads to topological inflation and the creation of baby universes. The future evolution of initial data represented by the two incoming monopoles may contain a timelike singularity but this need not be the case. We discuss the global structure of the spacetime associated with monopole collisions and also that of topological inflation. We suggest that topological inflation within magnetic monopoles leads to an eternally reproducing universe. 
  Usually quantum theory is formulated in terms of the evolution of states through spacelike surfaces. However, a generalization of this formulation is needed for field theory in spacetimes not foliable by spacelike surfaces, or in quantum gravity where geometry is not definite but a quantum variable. In particular, a generalization of usual quantum theory is needed for field theory in the spacetimes that model the process of black hole evaporation. This paper discusses a spacetime generalization of usual quantum theory that is applicable to evaporating black hole spacetimes. In this generalization, information is not lost in the process of evaporation. Rather, complete information is distributed about four-dimensional spacetime. Black hole evaporation is thus not in conflict with the principles of quantum theory when suitably generally stated. (Talk presented at the 49th Yamada Conference: Black Holes and High Energy Astrophysics, Kyoto Japan, April 7-10, 1998.) 
  We address the issue of coupling variables which are essentially classical to variables that are quantum. Two approaches are discussed. In the first (based on collaborative work with L.Di\'osi), continuous quantum measurement theory is used to construct a phenomenological description of the interaction of a quasiclassical variable $X$ with a quantum variable $x$, where the quasiclassical nature of $X$ is assumed to have come about as a result of decoherence. The state of the quantum subsystem evolves according to the stochastic non-linear Schr\"odinger equation of a continuously measured system, and the classical system couples to a stochastic c-number $\x (t)$ representing the imprecisely measured value of $x$. The theory gives intuitively sensible results even when the quantum system starts out in a superposition of well-separated localized states. The second approach involves a derivation of an effective theory from the underlying quantum theory of the combined quasiclassical--quantum system, and uses the decoherent histories approach to quantum theory. 
  We discuss the issue of radiation extraction in asymptotically flat space-times within the framework of conformal methods for numerical relativity. Our aim is to show that there exists a well defined and accurate extraction procedure which mimics the physical measurement process. It operates entirely intrisically within $\scri^+$ so that there is no further approximation necessary apart from the basic assumption that the arena be an asymptotically flat space-time. We define the notion of a detector at infinity by idealising local observers in Minkowski space. A detailed discussion is presented for Maxwell fields and the generalisation to linearised and full gravity is performed by way of the similar structure of the asymptotic fields. 
  Inhomogeneous multidimensional cosmological models with a higher dimensional space-time manifold M= M_0 x M_1 ...x M_n are investigated under dimensional reduction to a D_0-dimensional effective non-minimally coupled sigma-model which generalizes the familiar Brans-Dicke model.   It is argued that the Einstein frame should be considered as the physical one. The general prescription for the Einstein frame reformulation of known solutions in the Brans-Dicke frame is given. As an example, the reformulation is demonstrated explicitly for the generalized Kasner solutions where it is shown that in the Einstein frame there are no solutions with inflation of the external space. 
  We study the evolution of quantum fluctuations of a scalar field which is coupled to the geometry, in an exponentially expanding universe.  We derive an expression for the spectrum of intrinsic perturbations, and it is shown that the intrinsic degrees of freedom are well behaved in the infra-red part of the spectrum. We conclude that quantum fluctuations do not change the dynamics of the spacetime in a way which makes its evolution non-perturbative and stochastic.  This result contradicts previous derivations which are based on the study of a quantum field on a fixed geometry. 
  In the present work, the gravitational field of superconducting cosmic string has been investigated in the context of Brans-Dicke theory of gravity. We have presented two kind of solutions for the spacetime in the far field zone of the string. When the B-D scalar field is switched off, one of the solutions reduces to the solution earlier obtained by Moss and Poletti in General Relativity. 
  We estimate the number of templates, computational power, and storage required for a one-step matched filtering search for gravitational waves from inspiraling compact binaries. These estimates should serve as benchmarks for the evaluation of more sophisticated strategies such as hierarchical searches. We use waveform templates based on the second post-Newtonian approximation for binaries composed of nonspinning compact bodies in circular orbits. We present estimates for six noise curves: LIGO (three configurations), VIRGO, GEO600, and TAMA. To search for binaries with components more massive than 0.2M_o while losing no more than 10% of events due to coarseness of template spacing, initial LIGO will require about 1*10^11 flops (floating point operations per second) for data analysis to keep up with data acquisition. This is several times higher than estimated in previous work by Owen, in part because of the improved family of templates and in part because we use more realistic (higher) sampling rates. Enhanced LIGO, GEO600, and TAMA will require computational power similar to initial LIGO. Advanced LIGO will require 8*10^11 flops, and VIRGO will require 5*10^12 flops. If the templates are stored rather than generated as needed, storage requirements range from 1.5*10^11 real numbers for TAMA to 6*10^14 for VIRGO. We also sketch and discuss an algorithm for placing the templates in the parameter space. 
  The phase perturbation arising from spin-rotation coupling is developed as a natural extension of the celebrated Sagnac effect. Experimental evidence in support of this phase shift, however, has yet to be realized due to the exceptional sensitivity required. We draw attention to the relevance of a series of experiments establishing that circularly polarized light, upon passing through a rotating half-wave plate, is changed in frequency by twice the rotation rate. These experiments may be interpreted as demonstrating the role of spin-rotation coupling in inducing this frequency shift, thus providing direct empirical verification of the coupling of the photon helicity to rotation. A neutron interferometry experiment is proposed which would be sensitive to an analogous frequency shift for fermions. In this arrangement, polarized neutrons enter an interferometer containing two spin flippers, one of which is rotating while the other is held stationary. An observable beating in the transmitted neutron beam intensity is predicted. 
  The topological structure of the electric topological current of the locally gauge invariant Maxwell-Chern-Simons Model and its bifurcation is studied. The electric topological charge is quantized in term of winding number. The Hopf indices and Brouwer degree labeled the local topological structure of the electric topological current. Using $\Phi $-mapping method and implicity theory, the electric topological current is found generating or annihilating at the limit points and splitting or merging at the bifurcate points. The total electric charge holds invariant during the evolution. 
  We reconsider the issue of proving large scale spatial homogeneity of the universe, given isotropic observations about us and the possibility of source evolution both in numbers and luminosities. Two theorems make precise the freedom available in constructing cosmological models that will fit the observations. They make quite clear that homogeneity cannot be proven without either a fully determinate theory of source evolution, or availability of distance measures that are independent of source evolution. We contrast this goal with the standard approach that assumes spatial homogeneity a priori, and determines source evolution functions on the basis of this assumption. 
  The interaction, in the long--wavelength approximation, of normal and superconducting electromagnetic circuits with gravitational waves is investigated. We show that such interaction takes place by modifying the physical parameters R, L, C of the electromagnetic devices. Exploiting this peculiarity of the gravitational field we find that a circuit with two plane and statically charged condensers set at right angles can be of interest as a detector of periodic gravitational waves. 
  Radio metric data from the Pioneer 10/11, Galileo, and Ulysses spacecraft indicate an apparent anomalous, constant, acceleration acting on the spacecraft with a magnitude $\sim 8.5\times 10^{-8}$ cm/s$^2$, directed towards the Sun. Two independent codes and physical strategies have been used to analyze the data. A number of potential causes have been ruled out. We discuss future kinematic tests and possible origins of the signal. 
  The classical Einstein--Maxwell field equations admit static horizonless wormhole solutions with only a circular cosmic string singularity. We show how to extend these static solutions to exact rotating asymptotically flat solutions. For a suitable range of parameter values, these solutions describe charged or neutral rotating closed cosmic strings, with a perimeter of the order of their Schwarzschild radius. 
  Carlip has shown that the entropy of the three-dimensional black hole has its origin in the statistical mechanics of microscopic states living at the horizon. Beginning with a certain orthonormal frame action, and applying similar methods, I show that an analogous result extends to the (Euclidean) black hole in any spacetime dimension. However, this approach still faces many interesting challenges, both technical and conceptual. 
  We propose a novel self consistent minimal coupling principle in presence of torsion dilaton field. This principle yields a new local dilatation symmetry and predicts the interactions of torsion dilaton with the real matter and with metric. The soft violation of this symmetry yields a physical dilaton and a simple relation between Cartan scalar curvature and cosmological constant in this new model of gravity with propagating torsion. Its relation with scalar-tensor theories of gravity and a possible use of torsion dilaton in the inflation scenario is discussed.   \noindent{PACS number(s): 04.50.+h, 04.40.Nr, 04.62.+v} 
  The renormalized expectation value of the stress-energy tensor of the conformally invariant massless field in the Israel-Hartle-Hawking state in the Schwarzschild spacetime is constructed. It is achieved through solving the conservation equation in conformal space and utilizing the regularity conditions in the physical metric. Specifically, the relation of the results of the present approach to the stress tensor constructed within the framework of the Hadamard renormalization is analysed. Finally, the semi-analytic models reconstructing the tangential component of the stress-energy tensor with the maximal deviation not exceeding 0.7% are constructed. 
  It is shown that the known solutions for nonexpanding impulsive gravitational waves generated by null particles of arbitrary multipole structure can be obtained by boosting the Weyl solutions describing static sources with arbitrary multipole moments, at least in a Minkowski background. We also discuss the possibility of boosting static sources in (anti-) de Sitter backgrounds, for which exact solutions are not known, to obtain the known solutions for null multipole particles in these backgrounds. 
  We present a series of test beds for numerical codes designed to find apparent horizons. We consider three apparent horizon finders that use different numerical methods: one of them in axisymmetry, and two fully three-dimensional. We concentrate first on a toy model that has a simple horizon structure, and then go on to study single and multiple black hole data sets. We use our finders to look for apparent horizons in Brill wave initial data where we discover that some results published previously are not correct. For pure wave and multiple black hole spacetimes, we apply our finders to survey parameter space, mapping out properties of interesting data sets for future evolutions. 
  We present and contrast two distinct ways of including extremal black holes in a Lorentzian Hamiltonian quantization of spherically symmetric Einstein-Maxwell theory. First, we formulate the classical Hamiltonian dynamics with boundary conditions appropriate for extremal black holes only. The Hamiltonian contains no surface term at the internal infinity, for reasons related to the vanishing of the extremal hole surface gravity, and quantization yields a vanishing black hole entropy. Second, we give a Hamiltonian quantization that incorporates extremal black holes as a limiting case of nonextremal ones, and examine the classical limit in terms of wave packets. The spreading of the packets, even the ones centered about extremal black holes, is consistent with continuity of the entropy in the extremal limit, and thus with the Bekenstein-Hawking entropy even for the extremal holes. The discussion takes place throughout within Lorentz-signature spacetimes. 
  We conclude the rigorous analysis of a previous paper concerning the relation between the (Euclidean) point-splitting approach and the local $\zeta$-function procedure to renormalize physical quantities at one-loop in (Euclidean) QFT in curved spacetime. The stress tensor is now considered in general $D$-dimensional closed manifolds for positive scalar operators $-\Delta + V(x)$. Results obtained in previous works (in the case D=4 and $V(x) =\xi R(x) + m^2$) are rigorously proven and generalized. It is also proven that, in static Euclidean manifolds, the method is compatible with Lorentzian-time analytic continuations. It is found that, for $D>1$, the result of the $\zeta$ function procedure is the same obtained from an improved version of the point-splitting method which uses a particular choice of the term $w_0(x,y)$ in the Hadamard expansion of the Green function. This point-splitting procedure works for any value of the field mass $m$. Furthermore, in the case D=4 and $V(x) = \xi R(x)+ m^2$, the given procedure generalizes the Euclidean version of Wald's improved point-splitting procedure. The found point-splitting method should work generally, also dropping the hypothesis of a closed manifold, and not depending on the $\zeta$-function procedure. This fact is checked in the Euclidean section of Minkowski spacetime for $A = -\Delta + m^2$ where the method gives rise to the correct stress tensor for $m^2 \geq 0$ automatically. 
  The spreading of quantum mechanical wave packets is studied in two cases. Firstly we look at the time behavior of the packet width of a free particle confined in the observable Universe. Secondly, by imposing the conservation of the time average of the packet width of a particle driven by a harmonic oscillator potential, we find a zero-point energy which frequency is the de Broglie frequency. 
  It is given an algorithm to obtain generalized power asymptotic expansions of the solutions of the Einstein equations arising for several homogeneous cosmological models. This allows to investigate their behavior near the initial singularity or for large times. An implementation of this algorithm in the CAS system Maple V Release 4 is described and detailed calculations for three equations are shown. 
  In Einstein-Cartan theory, by the use of the general Noether theorem, the general covariant angular-momentum conservation law is obtained with the respect to the local Lorentz transformations. The corresponding conservative Noether current is interpreted as the angular momentum tensor of the gravity-matter system including the spin density. It is pointed out that, assuming the tetrad transformation given by eq. (15), torsion tensor can not play a role in the conservation law of angular momentum. 
  We study series of the stationary solutions with asymptotic flatness properties in the Einstein-Maxwell-free scalar system because they are locally equivalent with the exterior solutions in some class of the scalar-tensor theories of gravity. First, we classify spherical exterior solutions into two types of the solutions, an apparently black hole type solution and an apparently worm hole type solution. The solutions contain three parameters, and we clarify their physical significance. Second, we reduce the field equations for the axisymmetric exterior solutions. We find that the reduced equations are partially the same as the Ernst equations. As simple examples, we derive new series of the static, axisymmetric exterior solutions, which correspond to Voorhees's solutions. We then show a non-trivial relation between the spherical exterior solutions and our new solutions. Finally, since null geodesics have conformally invariant properties, we study the local geometry of the exterior solutions by using the optical scalar equations and find some anomalous behaviors of the null geodesics. 
  The group theoretical approach to the relativistic wave equations on the real reducible spaces for spin~0, 1/2 and~1 massless particles is considered. The invariant wave equations which determine the appropriate irreducible representations are constructed. The coincidence of these equations with the general-covariant Klein-Gordon, Weyl and Maxwell equations on the corresponding spaces is shown. The explicit solutions of these equations possessing a simplicity and physical transparency are obtained in the form of so-called "plane waves" without using the method of separation of variables. The invariance properties of these "plane waves" for the spinless particles under the group $SO(3,1)$ were used for the construction of the invariant spin~0,1/2 and~1 two-point functions on the $H^3$. Secondly quantized spin~0,1/2 and~1 fields on the ${\Bbb R}^{1}\otimes H^{3}$ are constructed; their propagators which are their (anti)commutators in different points, are expressed in terms of the mentioned two-point functions. From here the ${\Bbb R}^{1}\otimes  SO(3,1)$-invariance follows. 
  We use the computer algebra system \textit{GRTensorII} to examine invariants polynomial in the Riemann tensor for class $B$ warped product spacetimes - those which can be decomposed into the coupled product of two 2-dimensional spaces, one Lorentzian and one Riemannian, subject to the separability of the coupling: $ds^2 = ds_{\Sigma_1}^2 (u,v) + C(x^\gamma)^2 ds_{\Sigma_2}^2 (\theta,\phi)$ with $C(x^\gamma)^2=r(u,v)^2 w(\theta,\phi)^2$ and $sig(\Sigma_1)=0, sig(\Sigma_2)=2\epsilon (\epsilon=\pm 1)$ for class $B_{1}$ spacetimes and $sig(\Sigma_1)=2\epsilon, sig(\Sigma_2)=0$ for class $B_{2}$. Although very special, these spaces include many of interest, for example, all spherical, plane, and hyperbolic spacetimes. The first two Ricci invariants along with the Ricci scalar and the real component of the second Weyl invariant $J$ alone are shown to constitute the largest independent set of invariants to degree five for this class. Explicit syzygies are given for other invariants up to this degree. It is argued that this set constitutes the largest functionally independent set to any degree for this class, and some physical consequences of the syzygies are explored. 
  We ask the following question: Of the exact solutions to Einstein's equations extant in the literature, how many could represent the field associated with an isolated static spherically symmetric perfect fluid source? The candidate solutions were subjected to the following elementary tests: i) isotropy of the pressure, ii) regularity at the origin, iii) positive definiteness of the energy density and pressure at the origin, iv) vanishing of the pressure at some finite radius, v) monotonic decrease of the energy density and pressure with increasing radius, and vi) subluminal sound speed. A total of 127 candidate solutions were found. Only 16 of these passed all the tests. Of these 16, only 9 have a sound speed which monotonically decreases with radius. The analysis was facilitated by use of the computer algebra system GRTensorII. 
  We re-examine the dynamical stability of the nakedly singular, static, spherical ly symmetric solutions of the Einstein-Klein Gordon system. We correct an earlier proof of the instability of these solutions, and demonstrate that there are solutions to the massive Klein-Gordon system that are perturbatively stable. 
  We studied the Einstein-Brans-Dicke cosmology in detail. The difference of the evolution of the universe is significant between Einstein-Brans-Dicke cosmology and standard big-bang model during the radiation-dominated era. The power-law evolution of the scale factor is fast enough to solve the cosmological puzzles and slow enough to avoid the graceful exit problem. However, the constraints from the satisfactory bubble distribution (beta^2>0.25) and the solar system observations (beta^2<0.002) are mutually exclusive. This suggests that this kind of inflationary model is ruled out. We also clarify the distinction between Einstein frame and Jordan frame in Brans-Dicke theory. 
  A method of solving the eikonal equation, in either flat or curved space-times, with arbitrary Cauchy data, is extended to the case of data given on a characteristic surface. We find a beautiful relationship between the Cauchy and characteristic data for the same solution, namely they are related by a Legendre transformation. From the resulting solutions, we study and describe their associated wave-front singularities. 
  We present a general class of noncolinear colliding wave solutions of the Einstein-Maxwell equations given in terms of fourth order polynomials, which in turn can be expressed through Jacobi functions depending on generalized advanced and retarded time coordinates. The solutions are characterized by six free parameters. The parameters can be chosen in such a way to avoid the generic focusing singularity 
  Nakamura, Sasaki, Tanaka, and Thorne have recently estimated the initial distribution of binary MACHOs in the galactic halo assuming that the MACHOs are primordial half solar mass black holes, and considered their coalescence as a possible source for ground-based interferometer gravitational wave detectors such as LIGO. Evolving their binary distribution forward in time to the present, the low-frequency (10^{-5} < f < 10^{-1} Hz) spectrum of gravitational waves associated with such a population of compact binaries is calculated. The resulting gravitational waves would form a strong stochastic background in proposed space interferometers such as LISA and OMEGA. Low frequency gravitational waves are likely to become a key tool for determining the properties of binaries within the dark MACHO population. 
  The level surfaces of solutions to the eikonal equation define null or characteristic surfaces. In this note we study, in Minkowski space, properties of these surfaces. In particular we are interested both in the singularities of these ``surfaces'' (which can in general self-intersect and be only piece-wise smooth) and in the decomposition of the null surfaces into a one parameter family of two-dimensional wavefronts which can also have self-intersections and singularities.   We first review a beautiful method for constructing the general solution to the flat-space eikonal equation; it allows for solutions either from arbitrary Cauchy data or for time independent (stationary) solutions of the form S=t-S_{0}(x,y,z). We then apply this method to obtain global, asymptotically spherical, null surfaces that are associated with shearing ("bad") two-dimensional cuts of null infinity; the surfaces are defined from the normal rays to the cut. This is followed by a study of the caustics and singularities of these surfaces and those of their associated wavefronts. We then treat the same set of issues from an alternative point of view, namely from Arnold's theory of generating families. This treatment allows one to deal (parametrically) with the regions of self-intersection and non-smoothness of the null surfaces, regions which are difficult to treat otherwise.   Finally we generalize the analysis of the singularities to families of solutions of the eikonal equation. 
  From a covariant Hamiltonian formulation, by using symplectic ideas, we obtain certain covariant boundary expressions for the quasilocal quantities of general relativity and other geometric gravity theories. The contribution from each of the independent dynamic geometric variables (the frame, metric or connection) has two possible covariant forms associated with the selected type of boundary condition. The quasilocal expressions also depend on a reference value for each dynamic variable and a displacement vector field. Integrating over a closed two surface with suitable choices for the vector field gives the quasilocal energy, momentum and angular momentum. For the special cases of Einstein's theory and the Poincar\'e Gauge theory our expressions are similar to some previously known expressions and give good values for the total ADM and Bondi quantities. We apply our formalism to black hole thermodynamics obtaining the first law and an associated entropy expression for these general gravity theories. For Einstein's theory our quasilocal expressions are evaluated on static spherically symmetric solutions and compared with the findings of some other researchers. The choices needed for the formalism to associate a quasilocal expression with the boundary of a region are discussed. 
  We give a general scheme for finite-differencing partial differential equations in flux-conservative form to second order, with a stencil that can be arbitrarily tilted with respect to the numerical grid, parameterized by a "tilt" vector field gamma^A. This can be used to center the numerical stencil on the physical light cone, by setting gamma^A = beta^A, where beta^A is the usual shift vector in the 3+1 split of spacetime, but other choices of the tilt may also be useful. We apply this "causal differencing" algorithm to the Bona-Masso equations, a hyperbolic and flux-conservative form of the Einstein equations, and demonstrate long term stable causally correct evolutions of single black hole systems in spherical symmetry. 
  We are presenting an algorithm capable of simplifying tensor polynomials with indices when the building tensors have index symmetry properties. These properties include simple symmetry, cyclicity and those due to the presence of partial and covariant derivatives. We are also including some examples using the Riemann tensor as a paradigm. The algorithm is part of a Mathematica package called Tools of Tensor Calculus (TTC) [web address: http://baldufa.upc.es/ttc] 
  We introduce a proposal to modify Einstein's equations by embedding them in a larger symmetric hyperbolic system. The additional dynamical variables of the modified system are essentially first integrals of the original constraints. The extended system of equations reproduces the usual dynamics on the constraint surface of general relativity, and therefore naturally includes the solutions to Einstein gravity. The main feature of this extended system is that, at least for a linearized version of it, the constraint surface is an attractor of the time evolution. This feature suggests that this system may be a useful alternative to Einstein's equations when obtaining numerical solutions to full, non-linear gravity. 
  The Einstein equations are integrated in the presence of two (incoming and outgoing) streams of null dust, under the assumptions of spherical symmetry and staticity. The solution is also written in double null and radiation coordinates and it is reinterpreted as an anisotropic fluid. Interior matching with a static fluid and exterior matching with the Vaidya solution along null hypersurfaces is discussed. The connection with two-dimensional dilaton gravity is established. 
  Both the canonical and microcanonical ensembles are utilized to study the thermodynamic and evaporation properties of a closed black cosmic string whose spacetime is asymptotically anti deSitter. There are similarities and differences to the Schwarzschild-anti deSitter and 2+1 BTZ black hole solutions. It is found that there exist regimes of black string/thermal radiation equilibrium as well as stable remnant regimes. The relevance to black hole evaporation is discussed. 
  There are good motivations for considering some type of quantum histories formalism. Several possible formalisms are known, defined by different definitions of event and by different selection criteria for sets of histories. These formalisms have a natural interpretation, according to which nature somehow chooses one set of histories from among those allowed, and then randomly chooses to realise one history from that set; other interpretations are possible, but their scientific implications are essentially the same.   The selection criteria proposed to date are reasonably natural, and certainly raise new questions. For example, the validity of ordering inferences which we normally take for granted --- such as that a particle in one region is necessarily in a larger region containing it --- depends on whether or not our history respects the criterion of ordered consistency, or merely consistency.   However, the known selection criteria, including consistency and medium decoherence, are very weak. It is not possible to derive the predictions of classical mechanics or Copenhagen quantum mechanics from the theories they define, even given observational data in an extended time interval. Attempts to refine the consistent histories approach so as to solve this problem by finding a definition of quasiclassicality have so far not succeeded.   On the other hand, it is shown that dynamical collapse models, of the type originally proposed by Ghirardi-Rimini-Weber, can be re-interpreted as set selection criteria within a quantum histories framework, in which context they appear as candidate solutions to the set selection problem. This suggests a new route to relativistic generalisation of these models, since covariant definitions of a quantum event are known. 
  We compare the exact and perturbative results in two metrics and show that the spurious effects due to the perturbation method do not survive for physically relevant quantities such as the vacuum expectation value of the stress-energy tensor. 
  We review our previous work on the the calculation of the stress-energy tensor for a scalar particle in the background metric of different types of spherical impulsive, spherical shock and plane impulsive gravitational waves. 
  The thermodynamical properties of toroidal black holes in the grand canonical ensemble are investigated using York's formalism. The black hole is enclosed in a cavity with finite radius where the temperature and electrostatic potential are fixed. The boundary conditions allow one to compute the relevant thermodynamical quantities, e.g. thermal energy, entropy and specific heat. This black hole is thermodynamically stable and dominates the grand partition function. This means that there is no phase transition, as the one encountered for spherical black holes. 
  The quadratic spinor Lagrangian is shown to be equivalent to the teleparallel / tetrad representation of Einstein's theory. An important consequence is that the energy-momentum density obtained from this quadratic spinor Lagrangian is essentially the same as the ``tensor'' proposed by Moller in 1961. 
  Contents:  Editorial  News:    Topical group news, by Jim Isenberg    Summer school in gravitational physics opportunity, by Jim Hartle    Bogart, Bergman and (Al)bert, by Clifford Will and Robert Riemer    New data-analysis subgroups of the LSC, by Eanna Flanagan    Marcel Bardon, a man of vision, by Richard Isaacson  Research Briefs:    Status of the GEO600 project, by Harold Lueck    A nonperturbative formulation of string theory?, by Gary Horowitz    TAMA project update, by Seiji Kawamura    Neohistorical approaches to quantum gravity, by Lee Smolin    LIGO project update, by David Shoemaker    Gravitational waves from neutron stars, by Eanna Flanagan  Conference reports    Perugia meeting, by Joe Kovalik    Nickel and Dime gravity meeting, by Eric Poisson    Second international LISA symposium, by Robin Stebbins    JILA meeting on seismic isolation et al., by Joe Giaime 
  An expression for the oscillatory part of an asymptotic formula for the relativistic spin network amplitude for a 4-simplex is given. The amplitude depends on specified areas for each two-dimensional face in the 4-simplex. The asymptotic formula has a contribution from each flat Euclidean metric on the 4-simplex which agrees with the given areas. The oscillatory part of each contribution is determined by the Regge calculus Einstein action for that geometry. 
  The possibility of trapped modes of gravitational waves appearing in stars with R>3M is considered. It is shown that the restriction to R<3M in previous studies of trapped modes, using uniform density models, is not essential. Scattering potentials are computed for another family of analytic stellar models showing the appearance of a deep potential well for one model with R>3M. However, the provided example is unstable, although it has a more realistic equation of state in the sense that the sound velocity is finite. On the other hand it is also shown that for some stable models belonging to the same family but having R<3M, the well is significantly deeper than that of the uniform density stars. Whether there are physically realistic equations of state which allow stable configurations with trapped modes therefore remains an open problem. 
  We describe the null geometry of a multiple black hole event horizon in terms of a conformal rescaling of a flat space null hypersurface. For the prolate spheroidal case, we show that the method reproduces the pair-of-pants shaped horizon found in the numerical simulation of the head-on-collision of black holes. For the oblate case, it reproduces the initially toroidal event horizon found in the numerical simulation of collapse of a rotating cluster. The analytic nature of the approach makes further conclusions possible, such as a bearing on the hoop conjecture. From a time reversed point of view, the approach yields a description of the past event horizon of a fissioning white hole, which can be used as null data for the characteristic evolution of the exterior space-time. 
  We analyze canonical asymptotic isometry of two dimensional AdS dilaton gravity in detail. 
  We discuss the no-hair conjecture in the presence of a cosmological constant. For the firststep the real scalar field is considered as the matter field and the spacetime is assumed to be static spherically symmetric. If the scalar field is massless or has a convex potential such as a mass term, it is proved that there is no regular black hole solution. For a general positive potential, we search for black hole solutions which support the scalar field with a double well potential, and find them by numerical calculations. The existence of such solutions depends on the values of the vacuum expectation value and the self-coupling constant of the scalar field. When we take the zero horizon radius limit, the solution becomes a boson star like solution which we found before. However new solutions are found to be unstable against the linear perturbation. As a result we can conclude that the no-scalar hair conjecture holds in the case of scalar fields with a convex or double well potential. 
  Light cones of Schwarzschild geometry are studied in connection to the Null Surface Formulation and gravitational lensing. The paper studies the light cone cut function's singularity structure, gives exact gravitational lensing equations, and shows that the "pseudo-Minkowski" coordinates are well defined within the model considered. 
  We study light propagation in the picture of semi-classical space-time that emerges in canonical quantum gravity in the loop representation. In such picture, where space-time exhibits a polymer-like structure at microscales, it is natural to expect departures from the perfect non-dispersiveness of ordinary vacuum. We evaluate these departures, computing the modifications to Maxwell's equations due to quantum gravity, and showing that under certain circumstances, non-vanishing corrections appear that depend on the helicity of propagating waves. These effects could lead to observable cosmological predictions of the discrete nature of quantum spacetime. In particular, recent observations of non-dispersiveness in the spectra of gamma-ray bursts at various energies could be used to constrain the type of semi-classical state that describes the universe. 
  First post-Newtonian (1PN) hydrostatic equations for an irrotational fluid which have been recently derived are solved for an incompressible star. The 1PN configurations are expressed as a deformation of the Newtonian irrotational Riemann ellipsoid using Lagrangian displacement vectors introduced by Chandrasekhar. For the 1PN solutions, we also calculate the luminosity of gravitational waves in the 1PN approximation using the Blanchet-Damour formalism. It is found that the solutions of the 1PN equations exhibit singularities at points where the axial ratios of semi-axes are 1:0.5244:0.6579 and 1:0.2374:0.2963, and the singularities seem to show that at the points, the irrotational Riemann ellipsoid is unstable to the deformation induced by the effect of general relativity. For stable cases (a_2/a_1 > 0.5244, where a_1 and a_2 are the semi-major and minor axes, respectively) we find that when increasing the 1PN correction, the angular velocity and total angular momentum increase, while the total energy and luminosity of gravitational waves decrease. These 1PN solutions will be useful when examining the accuracy of numerical code for obtaining relativistic irrotational stars.   We also investigate the validity of an ellipsoidal approximation, in which a 1PN solution is obtained assuming an ellipsoidal figure and neglecting the deformation. It is found that for $a_2/a_1 > 0.7$, the ellipsoidal approximation gives a fairly accurate result for the energy, angular momentum, and angular velocity, although in the approximation we cannot find the singularities. 
  Early energy-momentum investigations for gravitating systems gave reference frame dependent pseudotensors; later the quasilocal idea was developed. Quasilocal energy-momentum can be determined by the Hamiltonian boundary term, which also identifies the variables to be held fixed on the boundary. We show that a pseudotensor corresponds to a Hamiltonian boundary term. Hence they are quasilocal and acceptable; each is the energy-momentum density for a definite physical situation with certain boundary conditions. These conditions are identified for well-known pseudotensors. 
  The behavior of a quantum scalar field is studied in the metric ground state of the (2+1)-dimensional black hole of Ba\~nados, Teitelboim and Zanelli which contains a naked singularity. The one-loop BTZ partition function and the associate black hole effective entropy, the expectation value of the quantum fluctuation as well as the renormalized expectation value of the stress tensor are explicitly computed in the framework of the $\zeta$-function procedure. This is done for all values of the coupling with the curvature, the mass of the field and the temperature of the quantum state. In the massless conformally coupled case, the found stress tensor is used for determining the quantum back reaction on the metric due to the scalar field in the quantum vacuum state, by solving the semiclassical Einstein equations. It is finally argued that, within the framework of the 1/N expansion, the Cosmic Censorship Hypothesis is implemented since the naked singularity of the ground state metric is shielded by an event horizon created by the back reaction. 
  The gravitational collapse and the birth of a new universe are considered in terms of quantum mechanics. Transitions from annihilation of matter to deflation in the collapse and from inflation to creation of matter in the birth of a universe are considered. The creation of a new universe takes place in another space-time since beyond the event horizon the time coordinate is inextensible for an external observer. A reasonable probability of this creation is obtainable only for miniholes. The major part of the mass of such collapsing compact objects as stars, quasars and active nuclei of galaxies remains confined in the potential well near the vacuum state. 
  Based on the mathematical similarity between the Friedman open metric and Godel's metric in the case of nearby distances, we investigate a new scenario for the Universe's evolution, where the present Friedman universe originates from a primordial Godel universe by a phase transition during which the cosmological constant vanishes. Using Hubble's constant and the present matter density as input, we show that the radius and density of the primordial Godel universe are close, in order of magnitude, to the present values, and that the time of expansion coincides with the age of the Universe in the standard Friedman model. In addition, the conservation of angular momentum provides, in this context, a possible origin for the rotation of galaxies, leading to a relation between the masses and spins corroborated by observational data. 
  We describe the application of a gravity wave-generating technique to certain higher dimensional black holes. We find that the induced waves generically destroy the event horizon producing parallelly propagated curvature singularities. 
  It has been demonstrated, in a number of special situations, that the spin coefficients of a canonical spinor dyad can be used to define a Lanczos potential of the Weyl curvature spinor. In this paper we explore some of these potentials and show that they can be defined directly from the spinor dyad in a very simple way, but that the results do not generalize significantly, in any direct manner. A link to metric, asymmetric, curvature-free connections, which suggests a more natural relationship between the Lanczos potential and spin coefficients, is also considered. 
  We examine the accuracy of estimation of parameters of the gravitational-wave signals from spinning neutron stars that can be achieved from observations by Earth-based laser interferometers. We consider a model of the signal consisting of two narrowband components and including both phase and amplitude modulation. We calculate approximate values of the rms errors of the parameter estimators using the Fisher information matrix. We carry out extensive Monte Carlo simulations and obtain cumulative distribution functions of rms errors of astrophysically interesting parameters: amplitude of the signal, wobble angle, position of the source in the sky, frequency, and spindown coefficients. We consider both all-sky searches and directed searches. We also examine the possibility of determination of neutron star proper motion. We perform simulations for all laser-interferometric detectors that are currently under construction and for several possible lengths of the observation time and sizes of the parameter space. We find that observations of continuous gravitational-wave signals from neutron stars by laser-interferometric detectors will provide a very accurate information about their astrophysical properties. We derive several simplified models of the signal that can be used in the theoretical investigations of the data analysis schemes independently of the physical mechanisms generating the gravitational-wave signal. 
  By applying Newman's algorithm, the AdS_3 rotating black hole solution is ``derived'' from the nonrotating black hole solution of Banados, Teitelboim, and Zanelli (BTZ). The rotating BTZ solution derived in this fashion is given in ``Boyer-Lindquist-type'' coordinates whereas the form of the solution originally given by BTZ is given in a kind of an ``unfamiliar'' coordinates which are related to each other by a transformation of time coordinate alone. The relative physical meaning between these two time coordinates is carefully studied. Since the Kerr-type and Boyer-Lindquist-type coordinates for rotating BTZ solution are newly found via Newman's algorithm, next, the transformation to Kerr-Schild-type coordinates is looked for. Indeed, such transformation is found to exist. And in this Kerr-Schild-type coordinates, truely maximal extension of its global structure by analytically continuing to ``antigravity universe'' region is carried out. 
  The chaotical dynamics is studied in different Friedmann-Robertson- Walker cosmological models with scalar (inflaton) field and hydrodynamical matter. The topological entropy is calculated for some particular cases. Suggested scheme can be easily generalized for wide class of models. Different methods of calculation of topological entropy are compared. 
  General relativity can be presented in terms of other geometries besides Riemannian. In particular, teleparallel geometry (i.e., curvature vanishes) has some advantages, especially concerning energy-momentum localization and its ``translational gauge theory'' nature. The standard version is metric compatible, with torsion representing the gravitational ``force''. However there are many other possibilities. Here we focus on an interesting alternate extreme: curvature and torsion vanish but the nonmetricity $\nabla g$ does not---it carries the ``gravitational force''. This {\it symmetric teleparallel} representation of general relativity covariantizes (and hence legitimizes) the usual coordinate calculations. The associated energy-momentum density is essentially the Einstein pseudotensor, but in this novel geometric representation it is a true tensor. 
  The first law of black hole mechanics was derived by Wald in a general covariant theory of gravity for stationary variations around a stationary black hole. It is formulated in terms of Noether charges, and has many advantages. In this paper several issues are discussed to strengthen the validity of the Noether charge form of the first law. In particular, a gauge condition used in the derivation is justified. After that, we justify the generalization to non-stationary variations done by Iyer-Wald. 
  We consider a model of the Universe based on the equation of state $p=(1/3)\rho (c/F)^2$, where $F$ is the scale factor. This model behaves as an inflationary Universe from the beginning and during its early stages, and behaves as dust matter during the stages of maximum expansion. 
  We study field theory models in the context of a gravitational theory based on the requirement that the measure of integration in the action is not necessarily \sqrt{-g} but it is determined dynamically through additional degrees of freedom, like four scalar fields \phi_{a}. We study three possibilities for the general structure of the theory: (A) The total action has the form S=\int\Phi Ld^{4}x where the measure \Phi is built from the scalars \phi_{a} in such a way that the transformation L\to L+const does not effect equations of motion. Then an infinite dimensional shifts group of the measure fields (SGMF) \phi_{a} by arbitrary functions of the Lagrangian density L is a symmetry group of the action. (B) The total action has the form S=S_{1}+S_{2}, S_{1}=\int\Phi L_{1}d^{4}x, S_{2}=\int\sqrt{-g}L_{2}d^{4}x which is the only model different from (A) and invariant under SGMF (but now with f_{a}= f_{a}(L_{1})). Similarly, now only S_{1} satisfies the requirement that the transformation L_{1}\to L_{1}+const does not effect equations of motion. Both in the case (A) and in the case (B) it is assumed that L, L_{1}, L_{2} do not depend on \phi_{a}. (C) The action includes a term which breaks the SGMF symmetry. It is shown that in the first order formalism in cases (A) and (B) the CCP is solved: the effective potential vanishes in a true vacuum state (TVS) without fine tuning. In the case (C), the breaking of the SGMF symmetry induces a nonzero energy density for the TVS. 
  We demonstrate that to all large scale cosmological structures where gravitation is the only overall relevant interaction assembling the system (e.g. galaxies), there is associated a characteristic unit of action per particle whose order of magnitude coincides with the Planck action constant $h$. This result extends the class of physical systems for which quantum coherence can act on macroscopic scales (as e.g. in superconductivity) and agrees with the absence of screening mechanisms for the gravitational forces, as predicted by some renormalizable quantum field theories of gravity. It also seems to support those lines of thought invoking that large scale structures in the Universe should be connected to quantum primordial perturbations as requested by inflation, that the Newton constant should vary with time and distance and, finally, that gravity should be considered as an effective interaction induced by quantization. 
  We describe impulsive gravitational pp-waves entirely in the distributional picture. Applying Colombeau's nonlinear framework of generalized functions we handle the formally ill-defined products of distributions which enter the geodesic as well as the geodesic deviation equation. Using a universal regularization procedure we explicitly derive regularization independent distributional limits. In the special case of impulsive plane waves we compare our results with the particle motion derived from the continuous form of the metric. 
  A family of exact relativistic stellar models is described. The family generalizes Buchdahl's n=1 polytropic solution. The matter content is a perfect fluid and, excluding Buchdahl's original model, it behaves as a liquid at low pressures in the sense that the energy density is non-zero in the zero pressure limit. The equation of state has two free parameters, a scaling and a stiffness parameter. Depending on the value of the stiffness parameter the fluid behaviour can be divided in four different types. Physical quantities such as masses, radii and surface redshifts as well as density and pressure profiles are calculated and displayed graphically. Leaving the details to a later publication, it is noted that one of the equation of state types can quite accurately approximate the equation of state of real cold matter in the outer regions of neutron stars. Finally, it is observed that the given equation of state does not admit models with a conical singularity at the center. 
  We consider a nonlinear generalization of Cauchy-Riemann eqs. to the algebra of biquaternions. From here we come to "universal generating equations" (1) which deal with 2-spinor and gauge fields and form the basis of some unified algebraic field theory. For each solution of eqs.(1) the components of spinor field satisfy the eikonal and d'Alembert eqs., and the strengths of gauge field - both Maxwell and Yang-Mills eqs. We reduce eqs.(1) to that of shear-free geodesic null congruence and integrate them in twistor variables. Particles are treated as concurrent singularities of the effective metric and the electromagnetic field. For unisingular solutions the electric charge is quantized, and the metric is of Schwarzschild or Kerr type. Bisingular solutions are announced too. 
  We study the fate of gravitational collapse in presence of a cosmological constant. The junctions conditions between static and non-static space-times are deduced. Three apparent horizon are formed, but only two have physical significance, one of them being the black hole horizon and the other the cosmological horizon. The cosmological constant term slows down the collapse of matter, limiting also the size of the black hole. 
  We consider the algebra of spatial diffeomorphisms and gauge transformations in the canonical formalism of General Relativity in the Ashtekar and ADM variables. Modifying the Poisson bracket by including surface terms in accordance with our previous proposal allows us to consider all local functionals as differentiable. We show that closure of the algebra under consideration can be achieved by choosing surface terms in the expressions for the generators prior to imposing any boundary conditions. An essential point is that the Poisson structure in the Ashtekar formalism differs from the canonical one by boundary terms. 
  We study the nonspherical linear perturbations of the discretely self-similar and spherically symmetric solution for a self-gravitating scalar field discovered by Choptuik in the context of marginal gravitational collapse. We find that all nonspherical perturbations decay. Therefore critical phenomena at the threshold of gravitational collapse, originally found in spherical symmetry, will extend to (at least slightly) nonspherical initial data. 
  The head-on collision of identical neutron stars from rest at infinity requires a numerical simulation in full general relativity for a complete solution. Undaunted, we provide a relativistic, analytic argument to suggest that during the collision, sufficient thermal pressure is always generated to support the hot remnant in quasi-static stable equilibrium against collapse prior to slow cooling via neutrino emission. Our conclusion is independent of the total mass of the progenitors and holds even if the remnant greatly exceeds the maximum mass of a cold neutron star. 
  Solutions of the sourceless Einstein's equation with weak and strong cosmological constants are discussed by using In\"on\"u-Wigner contractions of the de Sitter groups and spaces. The more usual case corresponds to a weak cosmological-constant limit, in which the de Sitter groups are contracted to the Poincar\'e group, and the de Sitter spaces are reduced to the Minkowski space. In the strong cosmological-constant limit, however, the de Sitter groups are contracted to another group which has the same abstract Lie algebra of the Poincar\'e group, and the de Sitter spaces are reduced to a 4-dimensional cone-space of infinite scalar curvature, but vanishing Riemann and Ricci curvature tensors. In such space, the special conformal transformations act transitively, and the equivalence between inertial frames is that of special relativity. 
  I show that cosmological perturbation spectra produced from quantum fluctuations in massless or self-interacting scalar fields during an inflationary era remain invariant under a two parameter family of transformations of the homogeneous background fields. This relates slow-roll inflation models to solutions which may be far from the usual slow-roll limit. For example, a scale-invariant spectrum of perturbations in a minimally coupled, massless field can be produced by an exponential expansion with $a\propto e^{Ht}$, or by a collapsing universe with $a\propto (-t)^{2/3}$. 
  We consider the thermodynamics of minimally coupled massive scalar field in 3+1 dimensional constant curvature black hole background. The brick wall model of 't Hooft is used. When Scharzschild like coordinates are used it is found that apart from the usual radial brick wall cut-off parammeter an angular cut-off parameter is required to regularize the solution. Free energy of the scalar field is obtained through counting of states using the WKB approximation. It is found that the free energy and the entropy are logarithmically divergent in both the cut-off parameters. 
  In this work we check the validity of the no scalar hair theorem in charged axisymmetric stationary black holes for a wide class of scalar tensor theories. 
  We investigate geodesics in non-homogeneous vacuum pp-wave solutions and demonstrate their chaotic behavior by rigorous analytic and numerical methods. For the particular class of solutions considered, distinct "outcomes" (channels to infinity) are identified, and it is shown that the boundary between different outcomes has a fractal structure. This seems to be the first example of chaos in exact radiative spacetimes. 
  An exact solution of the current-free Einstein-Maxwell equations with the cosmological constant is presented. The solution is of Petrov type D, includes the negative cosmological constant, and could be a ``background addition'' to the present-day models of the universe. It has a surprising property such that its electromagnetic field and cosmological constant are interdependent (this constant is proportional to the energy density of this field), which may suggest a new way of measuring the constant in question. The solution describes a constant electromagnetic background with a preferred direction in the universe, and defines the entire lifetime of the universe as a simple function of the negative cosmological constant. According to our solution the absolute value of this constant should be considerably lower than that recently estimated, when astrophysical data are taken into account. Our solution is a special case of that published by Bertotti in 1959. His solution (in terms of which the cosmological constant and the background electromagnetic field are independent) and its two other special cases, i.e. the conformally flat Robinson solution (1959) and the one which is the counterpart of our solution with the positive cosmological constant, are briefly discussed. 
  Barrett and Crane have proposed a model of simplicial Euclidean quantum gravity in which a central role is played by a class of Spin(4) spin networks called "relativistic spin networks" which satisfy a series of physically motivated constraints. Here a proof is presented that demonstrates that the intertwiner of a vertex of such a spin network is uniquely determined, up to normalization, by the representations on the incident edges and the constraints. Moreover, the constraints, which were formulated for four valent spin networks only, are extended to networks of arbitrary valence, and the generalized relativistic spin networks proposed by Yetter are shown to form the entire solution set (mod normalization) of the extended constraints. Finally, using the extended constraints, the Barrett-Crane model is generalized to arbitrary polyhedral complexes (instead of just simplicial complexes) representing spacetime. It is explained how this model, like the Barret-Crane model can be derived from BF theory by restricting the sum over histories to ones in which the left handed and right handed areas of any 2-surface are equal. It is known that the solutions of classical Euclidean GR form a branch of the stationary points of the BF action with respect to variations preserving this condition. 
  Special relativity includes a concealed mechanism for reducing time-dilation effects in two mutually-receding objects. Forwarding their signals via one or more intermediate physical relay stages (a "probe chain") allows enhanced communication and propulsion efficiency. These possibilities are masked by the mathematical redefinitions of the special theory, which then assigns the velocity of the signal source a correspondingly lower value by using a velocity-addition formula. Probe chains reveal the existence of velocity-dependent curvature within inertial systems, and suggest a mechanism for indirect radiation from black holes that is strongly reminiscent of Hawking radiation. The "emitter-theory" force law is mentioned as a possible basis for a curved-space alternative to special relativity. 
  We consider the motion of a spinning relativistic particle in external electromagnetic and gravitational fields, to first order in the external field, but to an arbitrary order in spin. The noncovariant spin formalism is crucial for the correct description of the influence of the spin on the particle trajectory. We show that the true coordinate of a relativistic spinning particle is its naive, common coordinate $\r$. Concrete calculations are performed up to second order in spin included. A simple derivation is presented for the gravitational spin-orbit and spin-spin interactions of a relativistic particle. We discuss the gravimagnetic moment (GM), a specific spin effect in general relativity. It is shown that for the Kerr black hole the gravimagnetic ratio, i.e., the coefficient at the GM, equals unity (just as for the charged Kerr hole the gyromagnetic ratio equals two). The equations of motion obtained for relativistic spinning particle in external gravitational field differ essentially from the Papapetrou equations. 
  The reported anomalous acceleration may be explained as the recoil of radiated waste RTG heat scattered by the back of the high gain antenna. 
  The pre-big bang's inflationary mechanism, when allowance is made for the rapid change of Newton's constant, is not actually of pole-law form . We give examples where pole-law inflation, which requires violation of the weak-energy condition, is possible but unlikely due to its very unstable character. 
  The universal decay hypothesized in xxx.lanl.gov physics/9808051 appears to have been detected from Pioneer 10 and 11 data. The paper at xxx.lanl.gov gr-qc/9808081 entitled "Indication from Pioneer 10/11, Galileo, and Ulysses Data of an Apparent, Anomalous, Weak, Long-Range Acceleration", reports that a weak, consistent, constant acceleration toward the Sun has been observed over a period of more than ten years. The paper reports the extensive efforts to identify the cause, all without success, and implies that possible explanations are essentially exhausted. The present paper offers the explanation that the observation is actually of the cosmic decay of the universal constant, c, the speed of light. 
  We study analytically the spacelike singularity inside a spherically-symmetric, charged black hole coupled to a self-gravitating spherical massless scalar field. We assume spatial homogeneity, and find a generic solution in terms of a formal series expansion. This solution is tested against fully-nonlinear and inhomogeneous numerical simulations. We find full compliance between our analytical solution and the pointwise behavior of the singularity in the numerical simulations. This is a strong scalar-curvature monotonic spacelike singularity, which connects to a weak null singularity at asymptotically-late advanced time. 
  The computational use of Killing potentials which satisfy Penrose's equation is discussed. Penrose's equation is presented as a conformal Killing-Yano equation and the class of possible solutions is analyzed. It is shown that solutions exist in spacetimes of Petrov type O, D or N. In the particular case of the Kerr background, it is shown that there can be no Killing potential for the axial Killing vector. 
  Recently, Anderson et al. presented possible evidence for an anomalous acceleration acting on spacecrafts. Furthermore, the motions of several planets and comets are known to experience unexplained disturbances. A transneptunian comet or asteroid belt might be the common origin of these anomalies. 
  We show that the Riemannian Schwarzschild and the ``Taub-bolt'' instanton solutions are the only spaces (M,g) such that   1) M is a 4-dimensional, simply connected manifold with a Riemannian, Ricci-flat C^2-metric g which admits (at least) a 1-parameter group of isometries H without isolated fixed points on M.   2) The quotient (M L)/H (where L is the set of fixed points of H) is an asymptotically flat manifold, and the length of the Killing field corresponding to H tends to a constant at infinity. 
  It is shown that for a general nonstatic spherically symmetric metric of the Kerr-Schild class several energy-momentum complexes give the same energy distribution as in the Penrose prescription, obtained by Tod. This result is useful for investigating the Seifert conjecture for naked singularities. The naked singularity forming in the Vaidya null dust collapse supports the Seifert conjecture. Further, an example and a counterexample to this conjecture are presented in the Einstein massless scalar theory. 
  The criterion of differentiability of functions of quaternion variable is used as the basis of some algebraic field theory. Its necessary consequences are free Maxwell and Yang-Mills equations. The differentiability equations may be integrated in twistor variables and are reduced to algebraic ones. In the article we present bisingular solution and its topological modifications. Related EM-fields appear to be just the well-known Born solution and its modifications with singular structure of ring-like and toroidal topology. General problems of the algebrodynamical approach are discussed. 
  We take the first nontrivial coefficient of the Schwinger-DeWitt expansion as a leading correction to the action of the second-derivative metric-dilaton gravity. To fix the ambiguities related with an arbitrary choice of the gauge fixing condition and the parametrization for the quantum field, one has to use the classical equations of motion. As a result, the only corrections are the ones to the potential of the scalar field. It turns out that the parameters of the initial classical action may be chosen in such a way that the potential satisfies most of the conditions for successful inflation. 
  The issue of a self-consistent solution of Maxwell-Einstein equations achieves a very simple form when all quantum effects are neglected but a weak vacuum polarization due to an external magnetic field is taken into account.   From a semi-classical point of view this means to deal with an appropriate limit of the one-loop effective Lagrangian for electrodynamics. When the corresponding stress-energy tensor is considered as a source of the gravitational field a surprisingly bouncing behavior is obtained. The present toy model leads to important new features which should have taken place in the early universe. 
  The problem of the energy-momentum conservation for matter in the gravitational field is discussed on the example of the effective gravity, which arises in superfluids. The "gravitational" field experienced by the relativistic-like massless quasiparticles which form the "matter" (phonons in superfluid 4He and low-energy fermions in superfluid 3He-A), is induced by the flow of the superfluid "vacuum". It appears that the energy-momentum conservation law for quasiparticles, has the covariant form T^{\mu}_{\nu;\mu}=0. "Pseudotensor" of the energy-momentum for the "gravitational field" (superfluid condensate) appears to depend on "matter". In the presence of the stationary "gravitational" (superfluid) field the real thermodynamic temperature T is constant in the true dissipationless equilibrium state with no entropy production, while the "relativistic" temperature T/\sqrt{g_{00}} is space dependent in agreement with Tolman's law. In the presence of the event horizon the true dissipationless equilibrium state does not exist. The quasiequilibrium dissipative motion across the horizon is considered. The inflationary stage of the expansion of the Universe can be modelled using the expanding Bose-condendsate. 
  We review the similarity solutions proposed by Waylen for a regular time-dependent axisymmetric vacuum space-time, and show that the key equation introduced to solve the invariant surface conditions is related by a Baecklund transform to a restriction on the similarity variables. We further show that the vacuum space-times produced via this path automatically possess a (possibly homothetic) Killing vector, which may be time-like. 
  Anti-self-dual (ASD) Maxwell solutions on 4-dimensional hyperk\"ahler manifolds are constructed. The N=2 supersymmetric half-flat equations are derived in the context of the Ashtekar formulation of N=2 supergravity. These equations show that the ASD Maxwell solutions have a direct connection with the solutions of the reduced N=2 supersymmetric ASD Yang-Mills equations with a special choice of gauge group. Two examples of the Maxwell solutions are presented. 
  The theory of linear transports along paths in vector bundles, generalizing the parallel transports generated by linear connections, is developed. The normal frames for them are defined as ones in which their matrices are the identity matrix or their coefficients vanish. A number of results, including theorems of existence and uniqueness, concerning normal frames are derived. Special attention is paid to the important case when the bundle's base is a manifold. The normal frames are defined and investigated also for derivations along paths and along tangent vector fields in the last case. It is proved that normal frames always exist at a single point or along a given (smooth) path. On other subsets normal frames exist only as an exception if (and only if) certain additional conditions, derived here, are satisfied. Gravity physics and gauge theories are pointed out as possible fields for application of the results obtained. 
  In a cold matter universe, the linearized gravito-magnetic tensor field satisfies a transverse condition (vanishing divergence) when it is purely radiative. We show that in the nonlinear theory, it is no longer possible to maintain the transverse condition, since it leads to a non-terminating chain of integrability conditions. These conditions are highly restrictive, and are likely to hold only in models with special symmetries, such as the known Bianchi and $G_2$ examples. In models with realistic inhomogeneity, the gravito-magnetic field is necessarily non-transverse at second and higher order. 
  The paper is the first of two parts of the work devoted to the investigation of constructing quantum theory of a closed universe as a system without asynptotic states. In Part I the role of asymptotic states in quantum theory of gravity is discussed, that enables us to argue that mathematically correct quantum geometrodynamics of a closed universe has to be gauge-noninvariant. It is shown that a gauge-noninvariant quantum geometrodynamics is consistent with the Copenhagen interpretation. The proposed version of the theory is thought of as describing the integrated system ``the physical object + observation means''. It is also demonstrated that introducing the observer into the theory causes the appearance of time in it. 
  When two point particles, coupled to three dimensional gravity with a negative cosmological constant, approach each other with a sufficiently large center of mass energy, then a BTZ black hole is created. An explicit solution to the Einstein equations is presented, describing the collapse of two massless particles into a non-rotating black hole. Some general arguments imply that massive particles can be used as well, and the creation of a rotating black hole is also possible. 
  We prove non-existence of static, vacuum, appropriately regular, asymptotically flat black hole space-times with degenerate (not necessarily connected) components of the event horizon. This finishes the classification of static, vacuum, asymptotically flat domains of outer communication in an appropriate class of space-times, showing that the domains of outer communication of the Schwarzschild black holes exhaust the space of appropriately regular black hole exteriors. 
  The static and stationary C-metric are revisited in a generic framework and their interpretations studied in some detail. Specially those with two event horizons, one for the black hole and another for the acceleration. We found that: i) The spacetime of an accelerated static black hole is plagued by either conical singularities or lack of smoothness and compactness of the black hole horizon; ii) By using standard black hole thermodynamics we show that accelerated black holes have higher Hawking temperature than Unruh temperature of the accelerated frame; iii) The usual upper bound on the product of the mass and acceleration parameters <1/sqrt(27) is just a coordinate artifact. The main results are extended to accelerated rotating black holes with no significant changes. 
  Accurate Quantum Gravity calculations, based on the simplicial lattice formulation, are computationally very demanding and require vast amounts of computer resources. A custom-made 64-node parallel supercomputer capable of performing up to $2 \times 10^{10}$ floating point operations per second has been assembled entirely out of commodity components, and has been operational for the last ten months. It will allow the numerical computation of a variety of quantities of physical interest in quantum gravity and related field theories, including the estimate of the critical exponents in the vicinity of the ultraviolet fixed point to an accuracy of a few percent. 
  Relaxational effects in stellar heat transport can in many cases be significant. Relativistic Fourier-Eckart theory is inherently quasi-stationary, and cannot incorporate these effects. The effects are naturally accounted for in causal relativistic thermodynamics, which provides an improved approximation to kinetic theory. Recent results, based on perturbations of a static star, show that relaxation effects can produce a significant increase in the central temperature and temperature gradient for a given luminosity. We use a simple stellar model that allows for non-perturbative deviations from staticity, and confirms qualitatively the predictions of the perturbative models. 
  We point out that the singular instanton of Hawking-Turok type, in which the singularity occurs due to the divergence of a massless scalar field, can be generated by Euclideanized regular $p$-brane solutions in string (or M-) theory upon compactification to four dimensions. 
  Consider the definition E of quasilocal energy stemming from the Hamilton-Jacobi method as applied to the canonical form of the gravitational action. We examine E in the standard "small-sphere limit," first considered by Horowitz and Schmidt in their examination of Hawking's quasilocal mass. By the term "small sphere" we mean a cut S(r), level in an affine radius r, of the lightcone belonging to a generic spacetime point. As a power series in r, we compute the energy E of the gravitational and matter fields on a spacelike hypersurface spanning S(r). Much of our analysis concerns conceptual and technical issues associated with assigning the zero-point of the energy. For the small-sphere limit, we argue that the correct zero-point is obtained via a "lightcone reference," which stems from a certain isometric embedding of S(r) into a genuine lightcone of Minkowski spacetime. Choosing this zero-point, we find agreement with Hawking's quasilocal mass expression, up to and including the first non-trivial order in the affine radius. The vacuum limit relates the quasilocal energy directly to the Bel-Robinson tensor. 
  We describe a new technique for removing troublesome interference from external coherent signals present in the gravitational wave spectrum. The method works when the interference is present in many harmonics, as long as they remain coherent with one another. The method can remove interference even when the frequency changes. We apply the method to the data produced by the Glasgow laser interferometer in 1996 and the entire series of wide lines corresponding to the electricity supply frequency and its harmonics are removed, leaving the spectrum clean enough to detect possible signals previously masked by them. We also study the effects of the line removal on the statistics of the noise in the time domain. We find that this technique seems to reduce the level of non-Gaussian noise present in the interferometer and therefore, it can raise the sensitivity and duty cycle of the detectors. 
  We classify the possible shapes of cosmic string cusps and how they transform under Lorentz boosts. A generic cusp can be brought into a form in which the motion of the cusp tip lies in the plane of the cusp. The cusp whose motion is perpendicular to this plane, considered by some authors, is a special case and not the generic situation.   We redo the calculation of the energy in the region where the string overlaps itself near a cusp, which is the maximum energy that can be released in radiation. We take into account the motion of a generic cusp and the resulting Lorentz contraction of the string core. The result is that the energy scales as $\sqrt {rL}$ instead of the usual value of $r^{1/3} L^{2/3}$, where $r$ is the string radius and $L$ and is the typical length scale of the string. Since $r << L $ for cosmological strings, the radiation is strongly suppressed and could not be observed. 
  We consider two non-statistical definitions of entropy for dynamic (non-stationary) black holes in spherical symmetry. The first is analogous to the original Clausius definition of thermodynamic entropy: there is a first law containing an energy-supply term which equals surface gravity times a total differential. The second is Wald's Noether-charge method, adapted to dynamic black holes by using the Kodama flow. Both definitions give the same answer for Einstein gravity: one-quarter the area of the trapping horizon. 
  An effective chiral model of a plane-symmetric gravitational field is considered. Isometries of the target space of the model are described and exact solutions corresponding to the isometric ansatz method are obtained. New exact solutions are found using the method of functional parameters. The solutions obtained are B\"acklund transforms of solutions of the d'Alembert equation to those of the Einstein equations. 
  We study the dynamical effects in the scale factors due to the scalar $\phi$-field at the early stages of a supposedly anisotropic Universe expansion in connection with the problem of the initial singularity in the scalar-tensor cosmology of Jordan-Brans-Dicke. This is done by considering the behaviour of the general analytical solutions for the homogeneous model of Bianchi type VII in the vacuum case. We conclude that the Bianchi-VII$_0$ model shows an isotropic expansion and that its only physical solution is equivalent to a Friedman-Robertson-Walker spacetime whose evolution begins in a singularity and ends in another; moreover, we obtain that the general Bianchi-VII$_h$ (with $h\neq 0$) models show strong curvature singularities that produces a complete collapse of the homogeinity surfaces to a 2-plane, to a string-like one-dimensional manifold, or to a single point. 
  The popular Hamilton-Jacobi method first proposed by Brown and York for defining quasilocal quantities such as energy for spatially bound regions assumes that the spatial boundary is orthogonal to the foliation of the spacetime. Such a restriction is undesirable for both theoretical and computational reasons. We remove the orthogonality assumption and show that it is more natural to focus on the foliation of the spatial boundary rather than the foliation of the entire spatially bound region. Reference spacetimes which define additional terms in the action are discussed in detail. To demonstrate this new formulation, we calculate the quasilocal energies seen by observers who are moving with respect to a Schwarzschild black hole. 
  The paper is concerned with the Einstein equations for a spherically symmetric static distribution of anisotropic matter. The equations are cast into a system of Fuchsian type ODE for certain scalar invariants of the strain. And then the existence and regularity of this ODE is studied under general constitutive relation. In the case the constitutive relation is given by a quadratic form of strain, it is also shown that the solutions stay regular up to the boundary of the material ball. 
  For a spherically symmetric vacuum model with a negative cosmological constant, a complex constrained instanton is considered as the seed for the quantum pair creation of Schwarzschild-anti-de Sitter black holes. The relative creation probability is found to be the exponential of the negative of the black hole entropy. The black hole entropy is known to be one quarter of the black hole horizon area. In the absence of a general no-boundary proposal for open creation, the constrained instanton approach is used in treating both the open and closed pair creations of black holes. 
  In the absence of a general no-boundary proposal for open creation, the complex constrained instanton is used as the seed for the open pair creations of black holes in the Kerr-Newman-anti-de Sitter family. The relative probability of the chargeless and nonrotating black hole pair is the exponential of the negative of the entropy, and that of the charged and (or) rotating black hole pair is the exponential of the negative of one quarter of the sum of the outer and inner black hole horizon areas. 
  We study solutions of the Wheeler-DeWitt equation corresponding to an S-modular invariant N=1 supergravity model and a closed homogeneous and isotropic Friedmann-Robertson-Walker spacetime. The issues of inflation and of the cosmological constant problem are addressed with the help of the relevant wave functions. We find that topological type inflation is consistent from the quantum mechanical point of view and that a solution for the cosmological constant problem along the lines of the strong CP problem naturally arises. 
  In this paper we highlight the fact that the physical content of hyperbolic theories of relativistic dissipative fluids is, in general, much broader than that of the parabolic ones. This is substantiated by presenting an ample range of dissipative fluids whose behavior noticeably departs from Navier-Stokes'. 
  Anderson, et al. (gr-qc/9808081) have recently reported the discovery of an apparent anomalous, weak, long-range acceleration in the Pioneer 10/11 and Ulysses spacecraft. I believe that this result can be explained by non-isotropic radiative cooling of the spacecraft electronics through passive radiators on the spacecraft surface. These radiators are preferentially placed on the anti-solar side of the spacecraft to avoid heating by solar radiation. The power transmitted through these radiator panels can explain the observed acceleration within the observational errors. 
  Among all forms of routine human activity, the one which produces the strongest gravity-gradient noise in interferometric gravitational-wave detectors (e.g. LIGO) is the beginning and end of weight transfer from one foot to the other during walking. The beginning and end of weight transfer entail sharp changes (timescale tau ~ 20msec) in the horizontal jerk (first time derivative of acceleration) of a person's center of mass. These jerk pairs, occuring about twice per second, will produce gravity-gradient noise in LIGO in the frequency band 2.5 Hz <~ f <~ 1/(2 tau) ~= 25 Hz with the form sqrt{S_h(f)} \~0.6 X 10^{-23} Hz^{-1/2} (f/10Hz)^{-6} (sum_i (r_i/10m)^{-6})^{1/2}. Here the sum is over all the walking people, r_i is the distance of the i'th person from the nearest interferometer test mass, and we estimate this formula to be accurate to within a factor 3. To ensure that this noise is neglible in advanced LIGO interferometers, people should be prevented from coming nearer to the test masses than r ~= 10m. A r ~= 10m exclusion zone will also reduce to an acceptable level gravity gradient noise from the slamming of a door and the striking of a fist against a wall. The dominant gravity-gradient noise from automobiles and other vehicles is probably that from decelerating to rest. To keep this below the sensitivity of advanced LIGO interferometers will require keeping vehicles at least 30 meters from all test masses. 
  We first define what we mean by gravitational lensing equations in a general space-time. A set of exact relations are then derived that can be used as the gravitational lens equations in all physical situations. The caveat is that into these equations there must be inserted a function, a two-parameter family of solutions to the eikonal equation, not easily obtained, that codes all the relevant (conformal) space-time information for this lens equation construction. Knowledge of this two-parameter family of solutions replaces knowledge of the solutions to the geodesic equations.   The formalism is then applied to the Schwarzschild lensing problem 
  The inflaton field is assumed to possess electric charge. The effect of the charge upon inflation is studied using a parameter \omega_0 that acts as a measure of the quantity of inflaton present. It is shown that at the end of inflation the charged inflaton should produce a fractal electromagnetic structure, which could act as a seed for the development of macroscopic galactic structures. Another consequence would be the presence nowadays of residual electromagnetic fields that would encompass galaxies, clusters, and larger structures. 
  We examine the recent statement by A. Dolgov and I. Novikov that superluminal propagation of light, which is induced by the quantum corrections to the photon propagation in gravitational field, permits non-causal signal propagation. For this purpose we examine the possibility of the existence of non-causal signals in the model case, when characteristic cone of the signal is outside the usual light cone in Minkowski space-time, as it take place in the case of the photon propagation in gravitation field with quantum corrections taking into account. In such model the signal propagates along invariant intervals. It is shown that there are no non-causal signals in the considered model. The recent statement of A. Dolgov and I. Novikov was obtained by using additional supposition that velocity of the signal with respect to the emitter is independent from its motion. Such supposition does not valid in the case of the photon propagation in gravitational field. 
  We obtain an expression for the active gravitational mass of a relativistic heat conducting fluid, just after its departure from hydrostatic equilibrium, on a time scale of the order of relaxation time. It is shown that an increase of a characteristic parameter leads to larger (smaller) values of active gravitational mass of collapsing (expanding) spheres, enhacing thereby the instability of the system. 
  The initial data in the polygon approach to (2+1)D gravity coupled to point particles are constrained by the vertex equations and the particle equations. We establish the hyperbolic nature of the vertex equations and derive some consequences. In particular we are able to identify the hyperbolic group of motions as discrete analogues of the diffeomorphisms in the continuum theory. We show that particles can be included ``hyperbolically'' as well, but they spoil the gauge invariance. Finally we derive consistent sets of initial data. 
  We show that static electro-vacuum black hole space-times containing an asymptotically flat spacelike hypersurface with compact interior and with both degenerate and non-degenerate components of the event horizon do not exist, under the supplementary hypothesis that all degenerate components of the event horizon have charges of the same sign. This extends previous uniqueness theorems of Simon and Masood-ul-Alam (where only non-degenerate horizons were allowed) and Heusler (where only degenerate horizons were allowed). 
  We investigate the minimal conditions under which the creation of our universe might arise due to a "bounce" from a previous collapse, rather than an explosion from a big-bang singularity. Such a bounce is sometimes referred to as a Tolman wormhole. We subject the bounce to a general model-independent analysis along the lines of that applied to the Morris-Thorne traversable wormholes, and show that there is always an open temporal region surrounding the bounce over which the strong energy condition (SEC) must be violated. On the other hand, all the other energy conditions can easily be satisfied. In particular, we exhibit an inflation-inspired model in which a big bounce is "natural". 
  One of the main results in canonical quantum gravity is the introduction of spin network states as a basis on the space of kinematical states. To arrive at the physical state space of the theory though we need to understand the dynamics of the quantum gravitational states. To this aim we study a model in which we allow for the spins, labeling the edges of spin networks, to change according to simple rules. The gauge invariance of the theory, restricting the possible values for the spins, induces propagating modes of spin changes. We investigate these modes under various assumptions about the parameters of the model. 
  We simplify the Hitchin's description of SU(2)-invariant self-dual Einstein metrics, making use of the tau-function of related four-pole Schlesinger system. 
  We argue that ``effective'' superluminal travel, potentially caused by the tipping over of light cones in Einstein gravity, is always associated with violations of the null energy condition (NEC). This is most easily seen by working perturbatively around Minkowski spacetime, where we use linearized Einstein gravity to show that the NEC forces the light cones to contract (narrow). Given the NEC, the Shapiro time delay in any weak gravitational field is always a delay relative to the Minkowski background, and never an advance. Furthermore, any object travelling within the lightcones of the weak gravitational field is similarly delayed with respect to the minimum traversal time possible in the background Minkowski geometry. 
  We derive a new class of exact solutions characterized by the Szekeres-Szafron metrics (of class I), admitting in general no isometries. The source is a fluid with viscosity but zero heat flux (adiabatic but irreversible evolution) whose equilibrium state variables satisfy the equations of state of: (a) ultra-relativistic ideal gas, (b) non-relativistic ideal gas, (c) a mixture of (a) and (b). Einstein's field equations reduce to a quadrature that is integrable in terms of elementary functions (cases (a) and (c)) and elliptic integrals (case (b)). Necessary and sufficient conditions are provided for the viscous dissipative stress and equilibrium variables to be consistent with the theoretical framework of Extended Irreversible Thermodynamics and Kinetic Theory of the Maxwell-Boltzmann and radiative gases. Energy and regularity conditions are discussed. We prove that a smooth matching can be performed along a spherical boundary with a FLRW cosmology or with a Vaidya exterior solution. Possible applications are briefly outlined. 
  We present a model for the dark matter in spiral galaxies, which is a result of a static and axial symmetric exact solution of the Einstein-Dilaton theory. We suposse that dark matter is a scalar field endowed with a scalar potential. We obtain that a) the effective energy density goes like $1/(r^2+r_{c}^{2})$ and b) the resulting circular velocity profile of tests particles is in good agreement with the observed one. 
  For an arbitrary Tolman wormhole, unconstrained by symmetry, we shall define the bounce in terms of a three-dimensional edgeless achronal spacelike hypersurface of minimal volume. (Zero trace for the extrinsic curvature plus a "flare-out" condition.) This enables us to severely constrain the geometry of spacetime at and near the bounce and to derive general theorems regarding violations of the energy conditions--theorems that do not involve geodesic averaging but nevertheless apply to situations much more general than the highly symmetric FRW-based subclass of Tolman wormholes. [For example: even under the mildest of hypotheses, the strong energy condition (SEC) must be violated.] Alternatively, one can dispense with the minimal volume condition and define a generic bounce entirely in terms of the motion of test particles (future-pointing timelike geodesics), by looking at the expansion of their timelike geodesic congruences. One re-confirms that the SEC must be violated at or near the bounce. In contrast, it is easy to arrange for all the other standard energy conditions to be satisfied. 
  The purpose of this paper is twofold - to demonstrate that in the gravitational redshift it is the frequency a photon that is constant, and to describe the mechanism responsible for the change of its wavelength. 
  Geodesic deviation is the most basic manifestation of the influence of gravitational fields on matter. We investigate geodesic deviation within the framework of Regge calculus, and compare the results with the continuous formulation of general relativity on two different levels. We show that the continuum and simplicial descriptions coincide when the cumulative effect of the Regge contributions over an infinitesimal element of area is considered. This comparison provides a quantitative relation between the curvature of the continuous description and the deficit angles of Regge calculus. The results presented might also be of help in developing generic ways of including matter terms in the Regge equations. 
  In the very early Universe the matter may be described by the free radiation, that is by the set of massless fields with negligible interactions between them. Then the dominating quantum effect is the trace anomaly which comes from the renormalization of the conformal invariant part of the vacuum action. The anomaly-induced effective action can be found with accuracy to an arbitrary conformal functional which vanishes for the special case of the conformally flat metric. This gives the solid basis for the study of the conformally-flat cosmological solutions, first of which was discovered by Mamaev and Mostepanenko and by Starobinski in 1980. Treating the anomaly-induced action as quantum correction to the Einstein-Hilbert term we explore the possibility to have inflationary solutions, investigate their dependence on the initial data and discuss the restrictions in considering the density perturbations. The shape of inflationary solutions strongly depends on the underlying gauge model of the elementary particles physics. Two special cases are considered: Minimal Standard Model and the matter sector of N=8, D=4 supergravity. It turns out that inflation is almost inevitable consequence of the great difference between Planck mass and the mass of the heaviest massive particle. 
  We show that in a generic scalar-tensor theory of gravity, the ``referenced'' quasilocal mass of a spatially bounded region in a classical solution is invariant under conformal transformations of the spacetime metric. We first extend the Brown-York quasilocal formalism to such theories to obtain the ``unreferenced'' quasilocal mass and prove it to be conformally invariant. The appropriate reference term in this case is defined by generalizing the Hawking-Horowitz prescription, which was originally proposed for general relativity. For such a choice of reference term, the referenced quasilocal mass for a general spacetime solution is obtained. This expression is shown to be a conformal invariant provided the conformal factor is a monotonic function of the scalar field. We apply this expression to the case of static spherically symmetric solutions with arbitrary asymptotics to obtain the referenced quasilocal mass of such solutions. Finally, we demonstrate the conformal invariance of our quasilocal mass formula by applying it to specific cases of four-dimensional charged black hole spacetimes, of both the asymptotically flat and non-flat kinds, in conformally related theories. 
  Using the Harmonic map ansatz, we reduce the axisymmetric, static Einstein-Maxwell equations coupled with a magnetized perfect fluid to a set of Poisson-like equations. We were able to integrate the Poisson equations in terms of an arbitrary function $M=M(\rho,\zeta)$ and some integration constants. The thermodynamic equation restricts the solutions to only some state equations, but in some cases when the solution exists, the interior solution can be matched with the corresponding exterior one. 
  The mathematically correct approach to constructing quantum geometrodynamics of a closed universe formulated in Part I is realized on the cosmological Bianchi-IX model with scalar fields. The physical adequacy of the obtained gauge-noninvariant theory to existing concepts about possible cosmological scenarios is shown. It is demonstrated that the Wheeler-DeWitt quantum geometrodynamics based on general quantum theoretical principles with probability interpretation of a closed universe wave function does not exist. The problem of the creation of the Universe is considered as a computational problem of a quantum reduction of the singular state "Nothing" to one of possible initial physical quantum states. 
  After a review of the canonical reduction to the rest-frame Wigner-covariant instant form of standard theories in Minkowski spacetime, a new formulation of tetrad gravity is introduced. Its canonical reduction, also in presence of N scalar particles, is done. The modification of the ADM formulation to solve the deparametrization problem of general relativity (how to recover the rest-frame instant form for G=0), is presented. 
  The author has removed this paper, following the publication of a more complete version. 
  We point out that although the pioneering work of Oppenheimer and Snyder (OS), technically, indicated the formation of an event horizon for a collapsing homogeneous dust ball of mass $M_b$ and radius $R_b$, the Eqs. (32) and (36) of their paper actually demand that no trapped surface is formed $ 2GM_b/ R_b c^2 \le 1$, and $M_b \equiv 0$! Anybody may verify it. Further, by analyzing the general inhomogeneous dust solutions of Tolman in a proper physical perspective, we show that, all dust solutions are characterized by $M_b=0$. Next, for a collapsing fluid endowed with radiation pressure, we discover that the collapse equations obey the same Global Constraint $2GM /R c^2 \le 1$ and which specifically shows that, contrary to the traditional intuitive Newtonian idea, which equates the gravitational mass ($M_b$) with the fixed baryonic mass ($M_0$), the trapped surfaces are not allowed in general theory of relativity (GTR). For continued collapse, the final gravitational mss $M_f \to 0$ as $R\to 0$. Thus we confirm Einstein's and Rosen's idea that Event Horizons and Schwarzschild Singularities are unphysical and can not occur in Nature. This, in turn, implies that, if there would be any continued collapse, the initial gravitational mass energy of the fluid must be radiated away $Q\to M_i c^2$. This also implies that, gravitational collapse of massive stars should give rise to Ultra Compact Objects and emission of energy much more than what is obtained in Supernova Events 
  We study the canonical quantum theory of the Reissner-Nordstrom-de Sitter black hole(RNdS). We obtain an exact general solution of the Wheeler-DeWitt equation for the spherically symmetric geometry with electro-magnetic field. We investigate the wave function form a viewpoint of the de Broglie-Bohm interpretation. The de Broglie-Bohm interpretation introduces a rigid trajectory on the minisuperspace without assuming an outside observer or causing collapse of the wave function. In our analysis, we obtain the boundary condition for the wave function which corresponds to the classical RNdS black hole and describe the quantum fluctuations near the horizons quantitatively. 
  It is shown that if an asymptotically flat spacetime is asymptotically stationary, in the sense that $\Lie_{\xi} g_{ab}$ vanishes at the rate $\sim t^{-3}$ for asymptotically timelike vector field $\xi^a$, and the energy-momentum tensor vanishes at the rate $\sim t^{-4}$, then the spacetime is an asymptotically Schwarzschild spacetime. This gives a new aspect of the uniqueness theorem of a black hole. 
  We discuss the relationship between Schlesinger system and stationary axisymmetric Einstein's equation on the level of algebro-geometric solutions. In particular, we calculate all metric coefficients corresponding to solutions of Ernst equation in terms of theta-functions. 
  We study the quantum theory of the Einstein-Maxwell action with a cosmological term in the spherically symmetric space-time, and explored quantum black hole solutions in Reissner-Nordstrom-de Sitter geometry. We succeeded to obtain analytic solutions to satisfy both the energy and momentum constraints. 
  The relative motion of many particles can be described by the geodesic deviation equation. This can be derived from the second covariant variation of the point particle's action. It is shown that the second covariant variation of the string action leads to a string deviation equation. 
  We have used data from the TeV gamma-ray flare associated with the active galaxy Markarian 421 observed on 15 May 1996 to place bounds on the possible energy-dependence of the speed of light in the context of an effective quantum gravitational energy scale. The possibility of an observable time dispersion in high energy radiation has recently received attention in the literature, with some suggestions that the relevant energy scale could be less than the Planck mass and perhaps as low as 10^16 GeV. The limits derived here indicate this energy scale to be in excess of 4x10^16 GeV at the 95% confidence level. To the best of our knowledge, this constitutes the first convincing limit on such phenomena in this energy regime. 
  Recent work on gravitational geons is extended to examine the stability properties of gravitational and electromagnetic geon constructs. All types of geons must possess the property of regularity, self-consistency and quasi-stability on a time-scale much longer than the period of the comprising waves. Standard perturbation theory, modified to accommodate time-averaged fields, is used to test the requirement of quasi-stability. It is found that the modified perturbation theory results in an internal inconsistency. The time-scale of evolution is found to be of the same order in magnitude as the period of the comprising waves. This contradicts the requirement of slow evolution. Thus not all of the requirements for the existence of electromagnetic or gravitational geons are met though perturbation theory. From this result it cannot be concluded that an electromagnetic or a gravitational geon is a viable entity. The broader implications of the result are discussed with particular reference to the problem of gravitational energy. 
  Simple formulas are given for generating Chern-Simons basic invariant polynomials by repeated exterior differentiation for n-dimensional differentiable manifolds having a general linear connection. 
  The problem of unification of electro-magnetism and gravitation in four dimensions; some new ideas involving torsion. A metric consisting of a combination of symmetric and anti-symmetric parts is postulated and, in the framework of general covariance, used to derive the free-field electro-magnetic stress-energy tensor and the source tensor. 
  We consider for j=1/2, 3/2,... a spherically symmetric, static system of (2j+1) Dirac particles, each having total angular momentum j. The Dirac particles interact via a classical gravitational and electromagnetic field.   The Einstein-Dirac-Maxwell equations for this system are derived. It is shown that, under weak regularity conditions on the form of the horizon, the only black hole solutions of the EDM equations are the Reissner-Nordstrom solutions. In other words, the spinors must vanish identically. Applied to the gravitational collapse of a "cloud" of spin-1/2-particles to a black hole, our result indicates that the Dirac particles must eventually disappear inside the event horizon. 
  It has been recently shown that topological defects can arise in symmetry breaking models where the scalar field potential $V(\phi)$ has no minima and is a monotonically decreasing function of $|\phi|$. Here we study the gravitational fields produced by such vacuumless defects in the cases of both global and gauge symmetry breaking. We find that a global monopole has a strongly repulsive gravitational field, and its spacetime has an event horizon similar to that in de Sitter space. A gauge monopole spacetime is essentially that of a magnetically charged black hole. The gravitational field of a global string is repulsive and that of a gauge string is attractive at small distances and repulsive at large distances. Both gauge and global string spacetimes have singularities at a finite distance from the string core. 
  On the basis of an algebraic relation between torsion and a classical spinor field a new interpretation of Einstein-Cartan gravity interacting with classical spinor field is proposed. In this approach the spinor field becomes an auxiliary field and the dynamical equation for this field (the Heisenberg equation) is a dynamical, gravitational equation for torsion. The simplest version of this theory is examined where the metric degrees of freedom are frozen and only torsion plays a role. A spherically symmetric solution of this theory is examined. This solution can be interpreted, in the spirit of Wheeler's ideas of ``charge without charge'' and ``mass without mass'', as a geometrical model for an uncharged and massless particle with spin (``spin without spin''). 
  The initial-value problem is posed by giving a conformal three-metric on each of two nearby spacelike hypersurfaces, their proper-time separation up to a multiplier to be determined, and the mean (extrinsic) curvature of one slice. The resulting equations have the {\it same} elliptic form as does the one-hypersurface formulation. The metrical roots of this form are revealed by a conformal ``thin sandwich'' viewpoint coupled with the transformation properties of the lapse function. 
  We study the properties of a completely integrable deformation of the CGHS dilaton gravity model in two dimensions. The solution is shown to represent a singularity free black hole that at large distances asymptoticaly joins to the CGHS solution. 
  In present article effective nonlinear sigma model (NSM) is considered.  Einstein equation solution, corresponded to the chiral fields determined by functional parameter method, are presented.  Effective NSM of stationary axially-symmetric gravitational field is constructed. Motion equations are solved exactly by functional parameter method. Einstein equations solution are constructed.  For particular dependences of functional parameter graphics of solutions are presented. Metric coefficients behaviour is shown to be similar as functional parameter one. 
  We find that general relativity can be naturally free of cosmological singularities. Several nonsingular models are currently available that either assume ad hoc matter contents, or are nonsingular only over a sector of solution space of zero measure, or depart drastically from general relativity at high energies. After much uncertainty over whether cosmological inflation could help solve the initial-singularity problem, the prevailing belief today is that general relativistic cosmology, with inflation or without, is endemically singular. This belief was reinforced by recent singularity theorems that take account specifically of inflation. Here, a viable inflationary cosmology is worked out that is naturally free of singularities despite the fact that 1) it uses only classical general relativity, 2) it assumes only the most generic inflationary matter contents, and 3) it is a theory of the chaotic-inflation type. That type of inflation is the most widely accepted today, as it demands the least fine-tuning of initial conditions. It is also shown how, by dropping the usual simplification of minimal coupling between matter and geometry, the null energy condition can be violated and the relevant singularity theorems circumvented. 
  It is shown that the expected amplitudes and specific correlation properties of the relic (squeezed) gravitational wave background may allow the registration of the relic gravitational waves by the first generation of sensitive gravity-wave detectors. 
  This article outlines our derivation of the second order perturbations to a Schwarzschild black hole, highlighting our use of, and necessary reliance on, computer algebra. The particular perturbation scenario that is presented here is the case of the linear quadrapole seeding the second order quadrapole. This problem amounts to finding the second order Zerilli wave equation, and in particular the effective source term due to the linear quadrapole. With one minor exception, our calculations confirm the earlier findings of Gleiser, et.al. On route to these results we also illustrate that, with the aid of computer algebra, the linear Schwarzschild problem can be solved in a very direct manner (i.e., without resorting to the usual function transformations), and it is this ``direct method'' that drives the higher order perturbation analysis. The calculations were performed using the GRTensorII computer algebra package, running on the Maple V platform, along with several new Maple routines that we have written specifically for these types of problems. Although we have chosen to consider only the ``quadrapole-quadrapole'' calculation in this article, the GRTensor environment, with the inclusion of these new routines, would allow this analysis to be repeated for a far more general problem. These routines, along with Maple worksheets that reproduce our calculations, are publicly available at the GRTensor website: www.astro.queensu.ca/~grtensor . The interested reader is invited to download and use them to reproduce our results and experiment. 
  An example of application of the specialized computer algebra system Grg-EC to the searching for solutions to the source-free Maxwell and Einstein--Maxwell equations is demonstrated. The solution involving five arbitrary functions of two variables is presented in explicit form (up to quadratures). An emphasis is made on the characterizing of the software utilized. 
  We employ the extended 1+3 orthonormal frame formalism for fluid spacetime geometries $({\cal M}, {\bf g}, {\bf u})$, which contains the Bianchi field equations for the Weyl curvature, to derive a 44-D evolution system of first-order symmetric hyperbolic form for a set of geometrically defined dynamical field variables. Describing the matter source fields phenomenologically in terms of a barotropic perfect fluid, the propagation velocities $v$ (with respect to matter-comoving observers that Fermi-propagate their spatial reference frames) of disturbances in the matter and the gravitational field, represented as wavefronts by the characteristic 3-surfaces of the system, are obtained. In particular, the Weyl curvature is found to account for two (non-Lorentz-invariant) Coulomb-like characteristic eigenfields propagating with $v = 0$ and four transverse characteristic eigenfields propagating with $|v| = 1$, which are well known, and four (non-Lorentz-invariant) longitudinal characteristic eigenfields propagating with $|v| = \sfrac{1}{2}$. The implications of this result are discussed in some detail and a parallel is drawn to the propagation of irregularities in the matter distribution. In a worked example, we specialise the equations to cosmological models in locally rotationally symmetric class II and include the constraints into the set of causally propagating dynamical variables. 
  The space-time foliation Sigma compatible with the gravitational field g on a 4-manifold M determines a fibration pi of M, pi : M -> N is a surjective submersion over the 1-dimensional leaves space N. M is then written as a disjoint union of the leaves of Sigma, which are 3-dimensional spacelike surfaces on M.   The decomposition, TM=Sigma + T^0 M, also implies that we can define a lift of the curves on N to curves (non-spacelike) on M.   The stable causality condition M coincides with Sigma being a causal space-time distribution, generated by an exact timelike 1-form omega^0 = dt where t is some real function on M. In this case M is written as a disjoint union of a family of spacelike 3-surfaces of constant t, which cover D^+(S) of a initial 3-surface S of M. 
  Here I present a stationary cylindrically symmetric asymptotically Einstein static universe solution with the matter consisting of a cosmological and rotating dust term which admits predicted black hole event horizon. 
  It has been claimed that extreme black holes exhibit a phenomenon of flux expulsion for abelian Higgs vortices, irrespective of the relative width of the vortex to the black hole. Recent work by two of the authors showed a subtlety in the treatment of the event horizon, which cast doubt on this claim. We analyse in detail the vortex/extreme black hole system, showing that while flux expulsion can occur, it does not do so in all cases. We give analytic proofs for both expulsion and penetration of flux, in each case deriving a bound for that behaviour. We also present extensive numerical work backing up, and refining, these claims, and showing in detail how a vortex can end on a black hole in all situations. We also calculate the backreaction of the vortex on the geometry, and comment on the more general vortex-black hole system. 
  We study string propagation in an anisotropic, cosmological background. We solve the equations of motion and the constraints by performing a perturbative expansion of the string coordinates in powers of c^2, the world-sheet speed of light. To zeroth order the string is approximated by a tensionless string (since c is proportional to the string tension T). We obtain exact, analytical expressions for the zeroth and the first order solutions and we discuss some cosmological implications. 
  String cosmology models predict a cosmic background of gravitational waves produced during a period of dilaton-driven inflation. I describe the background, present astrophysical and cosmological bounds on it, and discuss in some detail how it may be possible to detect it with large operating and planned gravitational wave detectors. The possible use of smaller detectors is outlined. 
  We analyze if Bianchi I, V, and IX models in the Induced Gravity (IG) theory can evolve to a Friedmann--Roberson--Walker (FRW) expansion due to the non--minimal coupling of gravity and the scalar field. The analytical results that we found for the Brans-Dicke (BD) theory are now applied to the IG theory which has $\omega \ll 1$ ($\omega$ being the square ratio of the Higgs to Planck mass) in a cosmological era in which the IG--potential is not significant. We find that the isotropization mechanism crucially depends on the value of $\omega$. Its smallness also permits inflationary solutions. For the Bianch V model inflation due to the Higgs potential takes place afterwads, and subsequently the spontaneous symmetry breaking (SSB) ends with an effective FRW evolution. The ordinary tests of successful cosmology are well satisfied. 
  Many numerical codes now under development to solve Einstein's equations of general relativity in 3+1 dimensional spacetimes employ the standard ADM form of the field equations. This form involves evolution equations for the raw spatial metric and extrinsic curvature tensors. Following Shibata and Nakamura, we modify these equations by factoring out the conformal factor and introducing three ``connection functions''. The evolution equations can then be reduced to wave equations for the conformal metric components, which are coupled to evolution equations for the connection functions. We evolve small amplitude gravitational waves and make a direct comparison of the numerical performance of the modified equations with the standard ADM equations. We find that the modified form exhibits much improved stability. 
  We calculate the ground state energy of a massive scalar field in the background of a cosmic string of finite thickness (Gott-Hiscock metric). Using zeta functional regularization we discuss the renormalization and the relevant heat kernel coefficients in detail. The finite (non local) part of the ground state energy is calculated in 2+1 dimensions in the approximation of a small mass density of the string. By a numerical calculation it is shown to vanish as a function of the radius of the string and of the parameter $\xi$ of the nonconformal coupling. 
  Normally the issue or question of the time of arrival of light rays at an observer coming from a given source is associated with Fermat's Principle of Least Time which yields paths of extremal time. We here investigate a related but different problem. We consider an observer receiving light from an extended source that has propagated in an arbitrary gravitational field. It is assumed from the start that the propagation is along null geodesics. Each point of the extended source is sending out a light-cones worth of null rays and the question arises which null rays from the source arrive first at the observer. Stated in an a different fashion, a pulse of light comes from the source with a wave-front as the leading edge, which rays are associated with that leading edge. In vacuum flat-space we have, from Huygen's principle, that the rays normal to the source constitute the leading edge and hence arrive first at an observer. We here investigate this issue in the presence of a gravitational field. Though it is not obvious, since the rays bend and are focused by the gravitational field and could even cross, in fact it is the normal rays that arrive earliest. We give two proofs both involving the extemization of the time of arrival, one based on an idea of Schrodinger for the derivation of gravitational frequency shifts and the other based on V.I. Arnold's theory of generating families. 
  Unpolarized Gowdy models are inhomogeneous cosmological models that depend on time and one spatial variable and have complicated nonlinear equations of motion. There are two topologies associated with these models, a three-torus and a one-sphere cross a two-sphere. The three-torus models have been used for numerical studies because it seems difficult to find analytic solutions to their nonlinear Einstein equations. The one-sphere cross tow-sphere models have even more complicated equations, but at least one family of analytic solutions can be given as a reinterpretation of known solutions. Various properties of this family of solutions are studied. 
  I present a new method to generate rotating solutions of the Einstein--Maxwell equations from static solutions, and briefly discuss its general properties. 
  The geometry of P, the bundle of null directions over an Einstein space-time, is studied. The full set of invariants of the natural G-structure on P is constructed using the Cartan method of equivalence. This leads to an extension of P which is an elliptic fibration over the space-time. Examples are given which show that such an extension, although natural, is not unique. A reinterpretation of the Petrov classification in terms of the fibres of an extension of P is presented. 
  In the multihorizon black hole spacetimes, it is possible that there are degenerate Cauchy horizons with vanishing surface gravities. We investigate the stability of the degenerate Cauchy horizon in black hole spacetimes. Despite the asymptotic behavior of spacetimes (flat, anti-de Sitter, or de Sitter), we find that the Cauchy horizon is stable against the classical perturbations, but unstable quantum mechanically. 
  We report on general relativistic calculations of quasiequilibrium configurations of binary neutron stars in circular orbits with zero vorticity. These configurations are expected to represent realistic situations as opposed to corotating configurations. The Einstein equations are solved under the assumption of a conformally flat spatial 3-metric (Wilson-Mathews approximation). The velocity field inside the stars is computed by solving an elliptical equation for the velocity scalar potential. Results are presented for sequences of constant baryon number (evolutionary sequences). Although the central density decreases much less with the binary separation than in the corotating case, it still decreases. Thus, no tendency is found for the stars to individually collapse to black hole prior to merger. 
  An attempt is made to go beyond the standard semi-classical approximation for gravity in the Born-Oppenheimer decomposition of the wave-function in minisuperspace. New terms are included which correspond to quantum gravitational fluctuations on the background metric. Their existence renders the definition of the semi-classical limit rather delicate and can lead to the avoidance of the singularities the classical theory predicts in cosmology and in the gravitational collapse of compact objects. 
  Computations of the strong field generation of gravitational waves by black hole processes produce waveforms that are dominated by quasinormal (QN) ringing, a damped oscillation characteristic of the black hole. We describe here the mathematical problem of quantifying the QN content of the waveforms generated. This is done in several steps: (i) We develop the mathematics of QN systems that are complete (in a sense to be defined) and show that there is a quantity, the ``excitation coefficient,'' that appears to have the properties needed to quantify QN content. (ii) We show that incomplete systems can (at least sometimes) be converted to physically equivalent complete systems. Most notably, we give a rigorous proof of completeness for a specific modified model problem. (iii) We evaluate the excitation coefficient for the model problem, and demonstrate that the excitation coefficient is of limited utility. We finish by discussing the general question of quantification of QN excitations, and offer a few speculations about unavoidable differences between normal mode and QN systems. 
  Static horizonless solutions to the Einstein--Maxwell field equations, with only a circular cosmic string singularity, are extended to exact rotating asymptotically flat solutions. The possible interpretation of these field configurations as spinning elementary particles or as macroscopic rotating cosmic rings is discussed. 
  An approach to quantization of fields and gravity based on the De Donder-Weyl covariant Hamiltonian formalism is outlined. It leads to a hypercomplex extension of quantum mechanics in which the algebra of complex numbers is replaced by the space-time Clifford algebra and all space-time variables enter on equal footing. A covariant hypercomplex analogue of the Schr\"odinger equation is formulated. Elements of quantization of General Relativity within the present framework are sketched. 
  We study pair creation of Kerr-Newman-anti-de Sitter and Kerr-Newman-de Sitter black holes. 
  After providing an extensive overview of the conceptual elements -- such as Einstein's `hole argument' -- that underpin Penrose's proposal for gravitationally induced quantum state reduction, the proposal is constructively criticised. Penrose has suggested a mechanism for objective reduction of quantum states with postulated collapse time T = h/E, where E is an ill-definedness in the gravitational self-energy stemming from the profound conflict between the principles of superposition and general covariance. Here it is argued that, even if Penrose's overall conceptual scheme for the breakdown of quantum mechanics is unreservedly accepted, his formula for the collapse time of superpositions reduces to T --> oo (E --> 0) in the strictly Newtonian regime, which is the domain of his proposed experiment to corroborate the effect. A suggestion is made to rectify this situation. In particular, recognising the cogency of Penrose's reasoning in the domain of full `quantum gravity', it is demonstrated that an appropriate experiment which could in principle corroborate his argued `macroscopic' breakdown of superpositions is not the one involving non-rotating mass distributions as he has suggested, but a Leggett-type SQUID or BEC experiment involving superposed mass distributions in relative rotation. The demonstration thereby brings out one of the distinctive characteristics of Penrose's scheme, rendering it empirically distinguishable from other state reduction theories involving gravity. As an aside, a new geometrical measure of gravity-induced deviation from quantum mechanics in the manner of Penrose is proposed, but now for the canonical commutation relations [Q, P] = ih. 
  It is shown that the Levi-Civita metric can be obtained from a family of the Weyl metric, the Gamma metric, by taking the limit when the length of its Newtonian image source tends to infinity. In this process a relationship appears between two fundamental parameters of both metrics. 
  Realistic black hole collisions result in a rapidly rotating Kerr hole, but simulations to date have focused on nonrotating final holes. Using a new solution of the Einstein initial value equations we present here waveforms and radiation for an axisymmetric Kerr-hole-forming collision starting from small initial separation (the ``close limit'' approximation) of two identical rotating holes. Several new features are present in the results: (i) In the limit of small separation, the waveform is linear (not quadratic) in the separation. (ii) The waveforms show damped oscillations mixing quasinormal ringing of different multipoles. 
  A new mechanism for causing naked singularities is found in an effective superstring theory. We investigate the gravitational collapse in a spherically symmetric Einstein-Maxwell-dilaton system in the presence of a pure cosmological constant "potential", where the system has no static black hole solution. We show that once gravitational collapse occurs in the system, naked singularities necessarily appear in the sense that the field equations break down in the domain of outer communications. This suggests that in generalized theories of gravity, the non-minimally coupled fields generically cause naked singularities in the process of gravitational collapse if the system has no static or stationary black hole solution. 
  We present the exact solution to the linearized Maxwell equations in space-time slightly curved by a gravitational wave. We show that in general, even dealing with a first-order theory in the strength of the gravitational field, the solution can not be written as the sum of the flat space-time one and a weak perturbation due to the external field. Such an impossibility arises when either the frequency of the gravitational wave is too low or too high with respect to the one of the electromagnetic field. We also provide an application of the solution to the case of an electromagnetic field bounced between two parallel conducting planes. 
  It is shown that if our visible universe is a thin trapped shell in a five-dimensional universe, all matter in it may be connected almost instantaneously through the fifth dimension. What appears to be action at a distance is then understood as undetectable ultrafast communication. 
  Starting with a study of the cosmological solution to the Einstein equations for the internal spacetime of an extreme supermassive cosmic string kink, and by evaluating the probability measure for the formation of such a kink in semiclassical approximation using a minisuperspace with the appropriate symmetry, we have found a set of arguments in favor of the claim that the kinked extreme string can actually be regarded as a unbounded chain of pairs of Planck- sized universes. Once one such universe pairs is created along a primordial phase transition at the Planck scale, it undergoes an endless process of continuous self-regeneration driven by chaotic inflation in each of the universes forming the pair. 
  It is proposed that the recently reported anomalous acceleration acting on the Pioneers spacecrafts should be a consequence of the existence of some local curvature in light geodesics when using the coordinate speed of light in an expanding spacetime. This suggests taht this "Pioneer effect" is nothing else but the detection of cosmological expansion in the solar system. 
  We consider cosmology in the framework of a `material reference system' of D particles, including the effects of quantum recoil induced by closed-string probe particles. We find a time-dependent contribution to the cosmological vacuum energy, which relaxes to zero as $\sim 1/ t^2$ for large times $t$. If this energy density is dominant, the Universe expands with a scale factor $R(t) \sim t^2$. We show that this possibility is compatible with recent observational constraints from high-redshift supernovae, and may also respect other phenomenological bounds on time variation in the vacuum energy imposed by early cosmology. 
  We discuss our understanding of the equivalence principle in both classical mechanics and quantum mechanics. We show that not only does the equivalence principle hold for the trajectories of quantum particles in a background gravitational field, but also that it is only because of this that the equivalence principle is even to be expected to hold for classical particles at all. 
  The conformal algebra of a 1+3 decomposable spacetime can be computed from the conformal Killing vectors (CKV) of the 3-space. It is shown that the general form of such a 3-CKV is the sum of a gradient CKV and a Killing or homothetic 3-vector. It is proved that spaces of constant curvature always admit such conformal Killing vectors. As an example, the complete conformal algebra of a G\"odel-type spacetime is computed. Finally it is shown that this method can be extended to compute the conformal algebra of more general non-decomposable spacetimes. 
  A perfect fluid is quantized by the canonical method. The constraints are found and this allows the Dirac brackets to be calculated. Replacing the Dirac brackets with quantum commutators formally quantizes the system. There is a momentum operator in the denominator of some coordinate quantum commutators. It is shown that it is possible to multiply throughout by this momentum operator. Factor ordering differences can result in a viscosity term. The resulting quantum commutator algebra is \vspace{1.0truein} \bc$v_{4}(v_{3}v_{2}-v_{2}v_{3})=-i,$\ec \bc$v_{4}(v_{1}v_{3}-v_{3}v_{1})=-iv_{3},$\ec \bc$v_{4}v_{1}-v_{1}v_{4}=-i,$\ec \bc$v_{5}v_{2}-v_{2}v_{5}=-i$.\ec 
  For any given spacetime the choice of time coordinate is undetermined. A particular choice is the absolute time associated with a preferred vector field. Using the absolute time Hamilton's equations are   $- (\delta H_{c})/(\delta q)=\dot{\pi}+\Theta\pi,    $+ (\delta H_{c})/(\delta \pi)=\dot{q}$,   where $\Theta = V^{a}_{.;a}$ is the expansion of the vector field. Thus there is a hitherto unnoticed term in the expansion of the preferred vector field. Hamilton's equations can be used to describe fluid motion. In this case the absolute time is the time associated with the fluid's co-moving vector. As measured by this absolute time the expansion term is present. Similarly in cosmology, each observer has a co-moving vector and Hamilton's equations again have an expansion term. It is necessary to include the expansion term to quantize systems such as the above by the canonical method of replacing Dirac brackets by commutators. Hamilton's equations in this form do not have a corresponding sympletic form. Replacing the expansion by a particle number $N\equiv exp(-\int\Theta d \ta)$ and introducing the particle numbers conjugate momentum $\pi^{N}$ the standard sympletic form can be recovered with two extra fields N and $\pi^N$. Briefly the possibility of a non-standard sympletic form and the further possibility of there being a non-zero Finsler curvature corresponding to this are looked at. 
  This paper studies the non-spherical perturbations of the continuously self-similar critical solution of the gravitational collapse of a massless scalar field (the Roberts solution). The exact analysis of the perturbation equations reveals that there are no growing non-spherical perturbation modes. 
  We analyse vacuum gravitational "soliton" solutions with real poles in the cosmological context. It is well known that these solutions contain singularities on certain null hypersurfaces. Using a Kasner seed solution, we demonstrate that these may contain thin sheets of null matter or may be simple coordinate singularities, and we describe a number of possible extensions through them. 
  A consistent approach for an exhaustive solution of the problem of propagation of light rays in the field of gravitational waves emitted by a localized source of gravitational radiation is developed in the first post-Minkowskian and quadrupole approximation of General Relativity. We demonstrate that the equations of light propagation in the retarded gravitational field of an arbitrary localized source emitting quadrupolar gravitational waves can be integrated exactly. The influence of the gravitational field on the light propagation is examined not only in the wave zone but also in cases when light passes through the intermediate and near zones of the source. Explicit analytic expressions for light deflection and integrated time delay (Shapiro effect) are obtained accounting for all possible retardation effects and arbitrary relative locations of the source of gravitational waves, that of light rays, and the observer. It is shown that the ADM and harmonic gauge conditions can both be satisfied simultaneously outside the source of gravitational waves. Their use drastically simplifies the integration of light propagation equations and those for the motion of light source and observer in the field of the source of gravitational waves, leading to the unique interpretation of observable effects. The two limiting cases of small and large values of impact parameter are elaborated in more detail. Explicit expressions for Shapiro effect and deflection angle are obtained in terms of the transverse-traceless part of the space-space components of the metric tensor. We also discuss the relevance of the developed formalism for interpretation of radio interferometric and timing observations, as well as for data processing algorithms for future gravitational wave detectors. 
  For the formation of a black hole in the gravitational collapse of a massless scalar field, we calculate a critical exponent that governs the black hole angular momentum for slightly non-spherical initial data near the black hole threshold. We calculate the scaling law by second-order perturbation theory. We then use the numerical results of a previous first-order perturbative analysis to obtain the numerical value mu ~ 0.76 for the angular momentum critical exponent. A quasi-periodic fine structure is superimposed on the overall power law. 
  We study the utilization of conformal compactification within the conformal approach to solving the constraints of general relativity for asymptotically flat initial data. After a general discussion of the framework, particular attention is paid to simplifications that arise when restricting to a class of initial data which have a certain $U(1)\times U(1)$ conformal symmetry. 
  We argue that the conventional construction for quantum fields in curved spacetime has a grave drawback: It involves an uncountable set of physical field systems which are nonequivalent with respect to the Bogolubov transformations, and there is, in general, no canonical way for choosing a single system. Thus the construction does not result in a definite theory. The problem of ambiguity pertains equally both to quantum and classical fields. The canonical theory is advanced, which is based on a canonical, or natural choice of field modes. The principal characteristic feature of the theory is relativistic-gravitational nonlocality: The field at a spacetime point $(s,t)$ depends on the metric at $t$ in the whole 3-space. The most fundamental and shocking result is the following: In the case of a free field in curved spacetime, there is no particle creation. Applications to cosmology and black holes are given. The results for particle energies are in complete agreement with those of general relativity. A model of the universe is advanced, which is an extension of the Friedmann universe; it lifts the problem of missing dark matter. 
  Impulsive pp-waves are commonly described either by a distributional spacetime metric or, alternatively, by a continuous one. The transformation $T$ relating these forms clearly has to be discontinuous, which causes two basic problems: First, it changes the manifold structure and second, the pullback of the distributional form of the metric under $T$ is not well defined within classical distribution theory. Nevertheless, from a physical point of view both pictures are equivalent. In this work, after calculating $T$ als well as the ''Rosen''-form of the metric in the general case of a pp-wave with arbitrary wave profile we give a precise meaning to the term ``physically equivalent'' by interpreting $T$ as the distributional limit of a suitably regularized sequence of diffeomorphisms. Moreover, it is shown that $T$ provides an example of a generalized coordinate transformation in the sense of Colombeau's generalized functions. 
  It is proved that in experiments on or near the Earth, no anisotropy in the one-way velocity of light may be detected. The very accurate experiments which have been performed to detect such an effect are to be considered significant tests of both special relativity and the equivalence principle 
  Coordinate transformations are derived from global Minkowski coordinates to the Fermi coordinates of an observer moving in a circle in Minkowski space-time. The metric for the Fermi coordinates is calculated directly from the tensor transformation rule. The behavior of ideal clocks is examined from the observer's reference frame using the Fermi coordinates. A complicated relation exists between Fermi coordinate time and proper time on stationary clocks (in the Fermi frame) and between proper time on satellite clocks that orbit the observer. An orbital Sagnac-like effect exists for portable clocks that orbit the Fermi coordinate origin. The coordinate speed of light is isotropic but varies with Fermi coordinate position and time. The magnitudes of these kinematic effects are computed for parameters that are relevant to the Global Positioning System (GPS) and are found to be small; however, for future high-accuracy time transfer systems, these effects may be of significant magnitude. 
  On the non-minimal coupling of Riemann-flat Klein-Gordon Fields to Space-time torsion} The energy spectrum of Klein-Gordon particles is obtained via the non-minimal coupling of Klein-Gordon fields to Cartan torsion in the approximation of Riemann-flatness and constant torsion.When the mass squared is proportional to torsion coupling constant it is shown that the splitting of energy does not occur.I consider that only the vector part of torsion does not vanish and that it is constant.A torsion Hamiltonian operator is constructed.The spectrum of Klein-Gordon fields is continuos. 
  The method of Hamilton-Jacobi is used to obtain geodesics around non- Riemannian planar torsional defects.It is shown that by perturbation expansion in the Cartan torsion the geodesics obtained are parabolic curves along the plane x-z when the wall is located at the plane x-y.In the absence of defects the geodesics reduce to straight lines.The family of parabolas depend on the torsion parameter and describe a gravitationally repulsive domain wall.Torsion here plays the role of the Burgers vector in solid state physics. 
  Existence check of non-trivial, stationary axisymmetric black hole solutions in Brans-Dicke theory of gravity in different direction from those of Penrose, Thorne and Dykla, and Hawking is performed. Namely, working directly with the known explicit spacetime solutions in Brans-Dicke theory, it is found that non-trivial Kerr-Newman-type black hole solutions different from general relativistic solutions could occur for the generic Brans-Dicke parameter values -5/2\leq \omega <-3/2. Finally, issues like whether these new black holes carry scalar hair and can really arise in nature and if they can, what the associated physical implications would be are discussed carefully. 
  We present a set of equations describing the evolution of the scalar-type cosmological perturbation in a gravity with general quadratic order curvature coupling terms. Equations are presented in a gauge ready form, thus are ready to implement various temporal gauge conditions depending on the problems. The Ricci-curvature square term leads to a fourth-order differential equation for describing the spacetime fluctuations in a spatially homogeneous and isotropic cosmological background. 
  An event horizon for "relativistic" fermionic quasiparticles can be constructed in a thin film of superfluid 3He-A. The quasiparticles see an effective "gravitational" field which is induced by a topological soliton of the order parameter. Within the soliton the "speed of light" crosses zero and changes sign. When the soliton moves, two planar event horizons (black hole and white hole) appear, with a curvature singularity between them. Aside from the singularity, the effective spacetime is incomplete at future and past boundaries, but the quasiparticles cannot escape there because the nonrelativistic corrections become important as the blueshift grows, yielding "superluminal" trajectories. The question of Hawking radiation from the moving soliton is discussed but not resolved. 
  This is the first in a series of papers on the construction and validation of a three-dimensional code for general relativistic hydrodynamics, and its application to general relativistic astrophysics. This paper studies the consistency and convergence of our general relativistic hydrodynamic treatment and its coupling to the spacetime evolutions described by the full set of Einstein equations with a perfect fluid source. The numerical treatment of the general relativistic hydrodynamic equations is based on high resolution shock capturing schemes. These schemes rely on the characteristic information of the system. A spectral decomposition for general relativistic hydrodynamics suitable for a general spacetime metric is presented. Evolutions based on three different approximate Riemann solvers coupled to four different discretizations of the Einstein equations are studied and compared. The coupling between the hydrodynamics and the spacetime (the right and left hand side of the Einstein equations) is carried out in a treatment which is second order accurate in {\it both} space and time. Convergence tests for all twelve combinations with a variety of test beds are studied, showing consistency with the differential equations and correct convergence properties. The test-beds examined include shocktubes, Friedmann-Robertson-Walker cosmology tests, evolutions of self-gravitating compact (TOV) stars, and evolutions of relativistically boosted TOV stars. Special attention is paid to the numerical evolution of strongly gravitating objects, e.g., neutron stars, in the full theory of general relativity, including a simple, yet effective treatment for the surface region of the star (where the rest mass density is abruptly dropping to zero). 
  We consider a simple cosmological model in order to show the importance of unstable particle creation for the validity of the semiclassical approximation. Using the mathematical structure of rigged Hilbert spaces we show that particle creation is the seed of decoherence which enables the quantum to classical transition. 
  An exact solution of Einstein-Cartan-Maxwell (ECM) field equations representing a charged domain wall given by the jump on the electric charge and spin density across the wall is obtained from the Riemannian theory of distributions. The Gauss-Coddazzi equations are used to show that spin, charge and Cartan torsion increases the repulsive character of the domain wall. Taub and Vilenkin walls are discussed as well as their relations to wormhole geometry. The electric and torsion fields are constants at the wall. 
  A technique for generating spherically symmetric dislocation solutions of a direct Poincar\'{e} gauge theory of gravity based on homogeneous functions which makes Cartan torsion to vanish is presented.Static space supported dislocation and time dependent solutions are supplied.Photons move along geodesics in analogy to geodesics described by electrons around dislocations in solid state physics.Tachyonic sectors are also found. 
  We investigate higher than the first order gravitational perturbations in the Newman-Penrose formalism. Equations for the Weyl scalar $\psi_4,$ representing outgoing gravitational radiation, can be uncoupled into a single wave equation to any perturbative order. For second order perturbations about a Kerr black hole, we prove the existence of a first and second order gauge (coordinates) and tetrad invariant waveform, $\psi_I$, by explicit construction. This waveform is formed by the second order piece of $\psi_4$ plus a term, quadratic in first order perturbations, chosen to make $\psi_I$ totally invariant and to have the appropriate behavior in an asymptotically flat gauge. $\psi_I$ fulfills a single wave equation of the form ${\cal T}\psi_I=S,$ where ${\cal T}$ is the same wave operator as for first order perturbations and $S$ is a source term build up out of (known to this level) first order perturbations. We discuss the issues of imposition of initial data to this equation, computation of the energy and momentum radiated and wave extraction for direct comparison with full numerical approaches to solve Einstein equations. 
  Distributional sources of cosmic walls crossed by cosmic strings from Riemann-Cartan (RC) Geometry. The matter density of the planar wall is maximum at the point where the cosmic string crosses the cosmic wall. Cartan torsion is has a support on the cosmic string given by the Dirac $ \delta $-function. Off the sources are left with a torsionless vacuum. 
  We consider smooth electrovac spacetimes which represent either (A) an asymptotically flat, stationary black hole or (B) a cosmological spacetime with a compact Cauchy horizon ruled by closed null geodesics. The black hole event horizon or, respectively, the compact Cauchy horizon of these spacetimes is assumed to be a smooth null hypersurface which is non-degenerate in the sense that its null geodesic generators are geodesically incomplete in one direction. In both cases, it is shown that there exists a Killing vector field in a one-sided neighborhood of the horizon which is normal to the horizon. We thereby generalize theorems of Hawking (for case (A)) and Isenberg and Moncrief (for case (B)) to the non-analytic case. 
  In this paper we discuss the connection between the geometric and tetrad approaches in the quantum affine-metric gravity. The corresponding transition formulas are obtained at the one-loop level. As an example, the one-loop counterterms are calculated in the tetrad formalism in the theory with terms quadratic in the torsion field. This model possesses the extra local symmetries connected with transformation of the connection field. It is shown that the special gauge can be chosen so that the corresponding additional ghosts do not contribute to the one-loop divergent terms. 
  The theory of distributions in non-Riemannian spaces is used to obtain exact static thin domain wall solutions of Einstein-Cartan equations of gravity. Curvature $ \delta $-singularities are found while Cartan torsion is given by Heaviside functions. Weitzenb\"{o}ck planar walls are caracterized by torsion $\delta$-singularities and zero curvature. It is shown that Weitzenb\"{o}ck static thin domain walls do not exist exactly as in general relativity. The global structure of Weitzenb\"{o}ck nonstatic torsion walls is investigated. 
  According to previous work, topological defects expand exponentially without an end if the vacuum expectation value of the Higgs field is of the order of the Planck mass. We extend the study of inflating topological defects to the Brans-Dicke gravity. With the help of numerical simulation we investigate the dynamics and spacetime structure of a global monopole. Contrary to the case of the Einstein gravity, any inflating monopole eventually shrinks and takes a stable configuration. We also discuss cosmological constraints on the model parameters. 
  In cosmology minisuperspace models are described by nonlinear time-reparametrization invariant systems with a finite number of degrees of freedom. Often these models are not explicitly integrable and cannot be quantized exactly. Having this in mind, we present a scheme for the (approximate) quantization of perturbed, nonintegrable, time-reparametrization invariant systems that uses (approximate) gauge invariant quantities. We apply the scheme to a couple of simple quantum cosmological models. 
  The structure of the Cauchy Horizon singularity of a black hole formed in a generic collapse is studied by means of a renormalization group equation for quantum gravity. It is shown that during the early evolution of the Cauchy Horizon the increase of the mass function is damped when quantum fluctuations of the metric are taken into account. 
  We present a new analytic approach for the study of late time evolution of linear test-fields, propagating on the exterior of black holes. This method provides a calculation scheme applicable to Kerr black holes (for which case no analytic calculation of the late time tails has been presented so far). In this paper we develop the new technique and apply it to the case of massless scalar waves evolving on the background geometry of a static spherically-symmetric thin shell with a Schwarzschild exterior. The late time behavior of the scalar field at null infinity is calculated, and is explicitly related to the form of (quite arbitrary) initial data. This reproduces the well-known late time power-law decaying tails. In an accompanying paper we apply our approach to the complete Schwarzschild black hole geometry, where we obtain the familiar inverse-power late time tails at null infinity, as well as at time-like infinity and along the event horizon. A calculation of the late time power-law tails in the Kerr geometry, based on the same approach, will be presented in a forthcoming paper. 
  We apply a new analytic scheme, developed in a preceding paper, in order to calculate the late time behavior of scalar test fields evolving outside a Schwarzschild black hole. The pattern of the late time decay at future null infinity is found to be the same as in the shell toy-model studied in the preceding paper. A simple late time expansion of the scalar field is then used, relying on the results at null infinity, to construct a complete picture of the late time wave behavior anywhere outside the black hole. This reproduces the well known power-law tails at time-like infinity and along the event horizon. The main motivation for the introduction of the new approach arises from its applicability to rotating black holes, as shall be discussed in a forthcoming paper. 
  The gravitational back-reaction is calculated for the conformally invariant scalar field within a black cosmic string interior with cosmological constant. Using the perturbed metric, the gravitational effects of the quantum field are calculated. It is found that the perturbations initially strengthen the singularity. This effect is similar to the case of spherical symmetry (without cosmological constant). This indicates that the behaviour of quantum effects may be universal and not dependent on the geometry of the spacetime nor the presence of a non-zero cosmological constant. 
  We deduce from energy conservation a lower bound on the mass of any system capable of imparting a constant acceleration to a charged body. We also point out a connection between this bound and the so called dominant energy condition of general relativity. 
  In covariant metric theories of coupled gravity-matter systems the necessary and sufficient conditions ensuring the existence of a Killing vector field are investigated. It is shown that the symmetries of initial data sets are preserved by the evolution of hyperbolic systems. 
  We study analytically the asymptotic late-time evolution of realistic rotating collapse. This is done by considering the asymptotic late-time solutions of Teukolsky's master equation, which governs the evolution of gravitational, electromagnetic, neutrino and scalar perturbations fields on Kerr spacetimes. In accordance with the no-hair conjecture for rotating black-holes we show that the asymptotic solutions develop inverse power-law tails at the asymptotic regions of timelike infinity, null infinity and along the black-hole outer horizon (where the power-law behaviour is multiplied by an oscillatory term caused by the dragging of reference frames). The damping exponents characterizing the asymptotic solutions at timelike infinity and along the black-hole outer horizon are independent of the spin parameter of the fields. However, the damping exponents at future null infinity are spin dependent. The late-time tails at all the three asymptotic regions are spatially dependent on the spin parameter of the field. The rotational dragging of reference frames, caused by the rotation of the black-hole (or star) leads to an active coupling of different multipoles. 
  We define a metric theory of gravity with preferred Newtonian frame (X^i(x),T(x)) by   L = L_{GR}   + \Xi g^{mn}\delta_{ij}X^i_{,m}X^j_{,n}   - \Upsilon g^{mn}T_{,m}T_{,n}   It allows a condensed matter interpretation which generalizes LET to gravity.   The \Xi-term influences the age of the universe. \Upsilon>0 allows to avoid big bang singularity and black hole horizon formation. This solves the horizon problem without inflation. An atomic hypothesis solves the ultraviolet problem by explicit regularization. We give a prediction about cutoff length. 
  The connection between the Bondi-Sachs (BS) and the Newman-Penrose (NP) framework for the study of the asymptotics of the gravitational field is done. In particular the coordinate transformation relationg the BS luminosity parameter and the NP affine parameter is obtained. Using this coordinate transformation it is possible to express BS quantities in terms of NP quantities, and to show that if the Outgoing Radiation Condition is not satisfied then the spacetime will not decay in the way prescribed by the Peeling theorem. 
  With the help of families of density contrast indicators, we study the tendency of gravitational systems to become increasingly lumpy with time. Depending upon their domain of definition, these indicators could be local or global. We make a comparative study of these indicators in the context of inhomogeneous cosmological models of Lemaitre--Tolman and Szekeres. In particular, we look at the temporal asymptotic behaviour of these indicators and ask under what conditions, and for which class of models, they evolve monotonically in time. We find that for the case of ever-expanding models, there is a larger class of indicators that grow monotonically with time, whereas the corresponding class for the recollapsing models is more restricted. Nevertheless, in the absence of decaying modes, indicators exist which grow monotonically with time for both ever-expanding and recollapsing models simultaneously. On the other hand, no such indicators may found which grow monotonically if the decaying modes are allowed to exist. We also find the conditions for these indicators to be non-divergent at the initial singularity in both models. Our results can be of potential relevance for understanding structure formation in inhomogeneous settings and in debates regarding gravitational entropy and arrow of time. In particular, the spatial dependence of turning points in inhomogeneous cosmologies may result in multiple density contrast arrows in recollapsing models over certain epochs. We also find that different notions of asymptotic homogenisation may be deduced, depending upon the density contrast indicators used. 
  The status of experimental tests of general relativity and of theoretical frameworks for analysing them are reviewed. Einstein's equivalence principle (EEP) is well supported by experiments such as the E\"otv\"os experiment, tests of special relativity, and the gravitational redshift experiment. Future tests of EEP will search for new interactions arising from unification or quantum gravity. Tests of general relativity have reached high precision, including the light deflection, the Shapiro time delay, the perihelion advance of Mercury, and the Nordtvedt effect in lunar motion. Gravitational wave damping has been detected to half a percent using the binary pulsar, and new binary pulsar systems promise further improvements. When direct observation of gravitational radiation from astrophysical sources begins, new tests of general relativity will be possible. 
  In the context of inflationary models with a pre-inflationary stage, in which the Einstein equations are obeyed, the weak energy condition is satisfied, and spacetime topology is trivial, we argue that homogeneity on super-Hubble scales must be assumed as an initial condition. Models in which inflation arises from field dynamics in a Friedman-Robertson-Walker background fall into this class but models in which inflation originates at the Planck epoch, {\it eg.} chaotic inflation, may evade this conclusion. Our arguments rest on causality and general relativistic constraints on the structure of spacetime. We discuss modifications to existing scenarios that may avoid the need for initial large-scale homogeneity. 
  If the universe is multiply connected and small the sky shows multiple images of cosmic objects, correlated by the covering group of the 3-manifold used to model it. These correlations were originally thought to manifest as spikes in pair separation histograms (PSH) built from suitable catalogues. Using probability theory we derive an expression for the expected pair separation histogram (EPSH) in a rather general topological-geometrical-observational setting. As a major consequence we show that the spikes of topological origin in PSH's are due to translations, whereas other isometries manifest as tiny deformations of the PSH corresponding to the simply connected case. This result holds for all Robertson-Walker spacetimes and gives rise to two basic corollaries: (i) that PSH's of Euclidean manifolds that have the same translations in their covering groups exhibit identical spike spectra of topological origin, making clear that even if the universe is flat the topological spikes alone are not sufficient for determining its topology; and (ii) that PSH's of hyperbolic 3-manifolds exhibit no spikes of topological origin. These corollaries ensure that cosmic crystallography, as originally formulated, is not a conclusive method for unveiling the shape of the universe. We also present a method that reduces the statistical fluctuations in PSH's built from simulated catalogues. 
  Riemann--Hilbert techniques are used in the theory of completely integrable differential equations to generate solutions that contain a free function which can be used at least in principle to solve initial or boundary value problems. The solution of a boundary value problem is thus reduced to the identification of the jump data of the Riemann-Hilbert problem from the boundary data. But even if this can be achieved,it is very difficult to get explicit solutions since the matrix Riemann-Hilbert problem is equivalent to an integral equation. In the case of the Ernst equation (the stationary axisymmetric Einstein equations in vacuum), it was shown in a previous work that the matrix problem is gauge equivalent to a scalar problem on a Riemann surface. If the jump data of the original problem are rational functions, this surface will be compact which makes it possible to give explicit solutions in terms of hyperelliptic theta functions. In the present work, we discuss Riemann-Hilbert problems on Riemann surfaces in the framework of fibre bundles. This makes it possible to treat the compact and the non-compact case in the same setting and to apply general existence theorems. 
  The epoch when the Universe had a temperature higher than a GeV is long before any time at which we have reliable observations constraining the cosmological evolution. For example, the occurrence of a second burst of inflation (sometimes called thermal inflation) at a lower energy scale than standard inflation, or a short epoch of early matter domination, cannot be ruled out by present cosmological data. The cosmological stochastic gravitational wave background, on scales accessible to interferometer detection, is sensitive to non-standard cosmologies of this type. We consider the implications of such alternative models both for ground-based experiments such as LIGO and space-based proposals such as LISA. We show that a second burst of inflation leads to a scale-dependent reduction in the spectrum. Applied to conventional inflation, this further reduces an already disappointingly low signal. In the pre big bang scenario, where a much more potent signal is possible, the amplitude is reduced but the background remains observable by LISA in certain parameter space regions. In each case, a second epoch of inflation induces oscillatory features into the spectrum in a manner analogous to the acoustic peaks in the density perturbation spectrum. On LIGO scales, perturbations can only survive through thermal inflation with detectable amplitudes if their amplitudes were at one time so large that linear perturbation theory is inadequate. Although for an epoch of early matter domination the reduction in the expected signal is not as large as the one caused by a second burst of inflation, the detection in the context of the pre big bang scenario may not be possible since the spectrum peaks around the LIGO frequency window and for lower frequencies behaves as $f^3$. 
  Linearized four-derivative gravity with a general gauge fixing term is considered. By a Legendre transform and a suitable diagonalization procedure it is cast into a second-order equivalent form where the nature of the physical degrees of freedom, the gauge ghosts, the Weyl ghosts, and the intriguing "third ghosts", characteristic to higher-derivative theories, is made explicit. The symmetries of the theory and the structure of the compensating Faddeev-Popov ghost sector exhibit non-trivial peculiarities. 
  A review of recent investigations of black holes in higher curvature theories of gravity. 
  In this letter we summarize our analysis of Bose-Einstein condensation on closed Robertson-Walker spacetimes. In a previous work we defined an adiabatic KMS state on the Weyl-algebra of the free massive Klein-Gordon field. This state describes a free Bose gas on Robertson-Walker spacetimes. We use this state to analyze the possibility of Bose-Einstein condensation on closed Robertson-Walker spacetimes. We take into account the effects due to the finiteness of the spatial volume and show that they are not relevant in the early universe. Furthermore we show that a critical radius can be defined. The condensate disappears above the critical radius. 
  Exact particle-like static, spherically and/or cylindrically symmetric solutions to the equations of interacting scalar and electromagnetic field system have been obtained. We considered Freedman-Robertson-Walker (FRW) space-time as an external homogenous and isotropic gravitational field whereas the homogeneous and anisotropic Universe is given by the G$\ddot o$del model. Beside the usual solitonian solutions some special regular solutions know as droplets, anti-droplets and bags (confined in finite interval and having trivial value beyond it) have been obtained. It has been shown that in FRW space-time equations with different interaction terms may have stable solutions while within the scope of G$\ddot o$del model only the droplet-like and the hat-like configurations may be stable, if they are located in the region where $g^{00}>0$. 
  The dynamics of a gravitational torsion kink as a plane symmetric thick domain wall solution of Einstein-Cartan (EC) field equation is given. The spin-torsion energy has to be as high as the gravitational kink potential otherwise torsion will not contribute as an appreciable effect to domain wall.Cartan torsion also contributes to the orthonal pressure of the domain wall. 
  We give a complete description of the asymptotic behavior of a Friedmann-Robertson-Walker Universe with ``normal'' matter and a minimally coupled scalar field. We classify the conditions under which the Universe is or is not accelerating. In particular, we show that only two types of large time behavior exist: an exponential regime, and a subexponential expansion with the logarithmic derivative of the scale factor tending to zero. In the case of the subexponetial expansion the Universe accelerates when the scalar field energy density is dominant and the potential behaves in a specified manner, or if matter violates the strong energy conditon $\rho + 3p >0$. When the expansion is exponential the Universe accelerates, and the scalar field energy density is dominant. We also find that the existence of the Big Bang and a never ending expansion of the Universe constrain the equation of state of matter at large and small densities, respectively. 
  In recent years, the use of conformal transformation techniques has become widespread in the literature on gravitational theories alternative to general relativity, on cosmology, and on nonminimally coupled scalar fields. Typically, the transformation to the Einstein frame is generated by a fundamental scalar field already present in the theory. In this context, the problem of which conformal frame is the physical one has to be dealt with and, in the general case, it has been clarified only recently; the formulation of a theory in the ``new'' conformal frame leads to departures from canonical Einstein gravity. In this article, we review the literature on conformal transformations in classical gravitational theories and in cosmology, seen both as purely mathematical tools and as maps with physically relevant aspects. It appears particularly urgent to refer the analysis of experimental tests of Brans-Dicke and scalar-tensor theories of gravity, as well as the predictions of cosmological inflationary scenarios, to the physical conformal frame, in order to have a meaningful comparison with the observations. 
  In this paper we analyze the conditions when the Einstein equations with cosmological constant and matter describe (2+1)-dimensional generic locally anisotropic (la) spacetimes of generalized Finsler type. New classes of solutions for such la-spacetimes are constructed. There are investigated black la-holes with the induced from general relativity la-curvature and la-torsion and, as a particular case, the black la-hole solutions are found for teleparallel la-spaces. In a more general context we consider the la-renormalization of black hole constants via the receptivity of la-spacetimes. We speculate on the properties of (2+1)-dimensional black la-holes with unusual characteristics defined by la-interactions of matter and gravity. The thermodynamics of black la-holes is discussed in connection with a possible statistical mechanics background based on locally anisotropic variants of Chern-Simons theories. 
  According to observations, in our Universe for gravitational phenomena in a Newtonian approximation the Newtonian non-modified relations are valid. The Friedmann equations of universe dynamics describe infinite number of relativistic universe models in Newtonian approximation, but only in one of them the Newtonian non-modified relations are valid. From these facts it results that the Universe is described just by this only Friedmannian universe model with Newtonian non-modified relations. 
  We discuss some features of the Decoherent Histories approach. We consider four assumptions, the first three being in our opinion necessary for a sound interpretation of the theory, while the fourth one is accepted by the supporters of the DH approach, and we prove that they lead to a logical contradiction. We discuss the consequences of relaxing any one of them. 
  The dynamics of a class of cosmological models with collisionless matter and four Killing vectors is studied in detail and compared with that of corresponding perfect fluid models. In many cases it is possible to identify asymptotic states of the spacetimes near the singularity or in a phase of unlimited expansion. Bianchi type II models show oscillatory behaviour near the initial singularity which is, however, simpler than that of the mixmaster model. 
  An explanation is proposed for the fact that pp-waves superpose linearly when they propagate parallely, while they interact nonlinearly, scatter and form singularities or Cauchy horizons if they are antiparallel. Parallel pp-waves do interact, but a generalized gravitoelectric force is exactly cancelled by a gravitomagnetic force. In an analogy, the interaction of light beams in linearized general relativity is also revisited and clarified, a new result is obtained for photon to photon attraction, and a conjecture is proved. Given equal energy density in the beams, the light-to-light attraction is twice the matter-to-light attraction and four times the matter-to-matter attraction. 
  We describe an algebraic way to code the causal information of a discrete spacetime. The causal set C is transformed to a description in terms of the causal pasts of the events in C. This is done by an evolving set, a functor which to each event of C assigns its causal past. Evolving sets obey a Heyting algebra which is characterised by a non-standard notion of complement. Conclusions about the causal structure of the causal set can be drawn by calculating the complement of the evolving set. A causal quantum theory can be based on the quantum version of evolving sets, which we briefly discuss. 
  We present two recently obtained solutions of the Einstein equations with spherical symmetry and one additional Killing vector, describing colliding null dust streams. 
  We review and summarize our results concerning the influence of the spins of a compact binary system on the motion of the binary and on its gravitational reaction. We describe briefly our method which lead us to compute the secular changes in the post-Newtonian motion and the averaged radiative losses. Our description is valid to 1.5 post-Newtonian order. All spin-orbit and some spin-spin effects are considered which contribute at this accuracy. This approach enabled us to give both the evolutions of the constants of the nonradiative motion and of the relevant angular variables under radiation reaction. 
  A global solution of the Einstein equations is given, consisting of a perfect fluid interior and a vacuum exterior. The rigidly rotating and incompressible perfect fluid is matched along the hypersurface of vanishing pressure with the stationary part of the Taub-NUT metric. The fluid core generates a negative-mass NUT space-time. The matching procedure leaves one parameter in the global solution. 
  The spherical gravitational collapse of a compact packet consisting of perfect fluid is studied. The spacetime outside the fluid packet is described by the out-going Vaidya radiation fluid. It is found that when the collapse has continuous self-similarity the formation of black holes always starts with zero mass, and when the collapse has no self-similarity, the formation of black holes always starts with a finite non-zero mass. The packet is usually accompanied by a thin matter shell. The effects of the shell on the collapse are also studied. 
  It has been suggested that by increasing the speed of light during the early universe various cosmological problems of standard big bang cosmology can be overcome, without requiring an inflationary phase. However, we find that as the Planck length and Planck time are then made correspondingly smaller, and together with the need that the universe should not re-enter a Planck epoch, the higher $c$ models have very limited ability to resolve such problems. For a constantly decreasing $c$ the universe will quickly becomes quantum gravitationally dominated as time increases: the opposite to standard cosmology where quantum behaviour is only ascribed to early times. 
  We update our previous work on the description of twisted configurations for complex massless scalar field on the Kerr black holes as the sections of complex line bundles over the Kerr black hole topology. From physical point of view the appearance of twisted configurations is linked with the natural presence of Dirac monopoles that arise as connections in the above line bundles. We consider their description in the gauge inequivalent to the one studied previously and discuss a row of new features appearing in this gauge. 
  We study some spectral features of the one-particle electron Hamiltonian obtained by separating the Dirac equation in a Kerr-Newman black hole background. We find that the essential spectrum includes the whole real line. As a consequence, there is no gap in the spectrum and discrete eigenvalues are not allowed for any value of the black hole charge $Q$ and angular momentum $J$. Our spectral analysis will be also related to the dissipation of the black hole angular momentum and charge. 
  We consider the N-body problem in (1+1) dimensional lineal gravity. For 2 point masses (N=2) we obtain an exact solution for the relativistic motion. In the equal mass case we obtain an explicit expression for their proper separation as a function of their mutual proper time. Our solution gives the exact Hamiltonian to infinite order in the gravitational coupling constant. 
  We examine the Wheeler-DeWitt equaton for a static, eternal Schwarzschild black hole in Kucha\v r-Brown variables and obtain its energy eigenstates. Consistent solutions vanish in the exterior of the Kruskal manifold and are non-vanishing only in the interior. The system is reminiscent of a particle in a box. States of definite parity avoid the singular geometry by vanishing at the origin. These definite parity states admit a discrete energy spectrum, depending on one quantum number which determines the Arnowitt-Deser-Misner (ADM) mass of the black hole according to a relation conjectured long ago by Bekenstein, $M \sim \sqrt{n}M_p$. If attention is restricted only to these quantized energy states, a black hole is described not only by its mass but also by its parity. States of indefinite parity do not admit a quantized mass spectrum. 
  A cosmological model with variable G and Lambda is considered in the framework of Israel-Stewart-Hiscock (ISH) causal theory. Power law as well as inflationary solutions are obtained. The gravitational constant is found to increase with time. 
  Einstein's field equations for spatially self-similar spherically symmetric perfect-fluid models are investigated. The field equations are rewritten as a first-order system of autonomous differential equations. Dimensionless variables are chosen in such a way that the number of equations in the coupled system is reduced as far as possible and so that the reduced phase space becomes compact and regular. The system is subsequently analysed qualitatively with the theory of dynamical systems. 
  Einstein's field equations for timelike self-similar spherically symmetric perfect-fluid models are investigated. The field equations are rewritten as a first-order system of autonomous differential equations. Dimensionless variables are chosen in such a way that the number of equations in the coupled system is reduced as far as possible and so that the reduced phase space becomes compact and regular. The system is subsequently analysed qualitatively using the theory of dynamical systems. 
  In the first part of the present paper, we show that O(d,d)-invariance usually known in a homogeneous cosmological background written in terms of proper time can be extended to backgrounds depending on one or several coordinates (which may be any space-like or time-like coordinate(s)). In all cases, the presence of a perfect fluid is taken into account and the equivalent duality transformation in Einstein frame is explicitly given. In the second part, we present several concrete applications to some four-dimensional metrics, including inhomogeneous ones, which illustrate the different duality transformations discussed in the first part. Note that most of the dual solutions given here do not seem to be known in the literature. 
  The Bohm-de Broglie interpretation of quantum mechanics is applied to canonical quantum cosmology. It is shown that, irrespective of any regularization or choice of factor ordering of the Wheeler-DeWitt equation, the unique relevant quantum effect which does not break spacetime is the change of its signature from lorentzian to euclidean. The other quantum effects are either trivial or break the four-geometry of spacetime. A Bohm-de Broglie picture of a quantum geometrodynamics is constructed, which allows the investigation of these latter structures. For instance, it is shown that any real solution of the Wheeler-De Witt equation yields a generate four-geometry compatible with the strong gravity limit of General Relativity and the Carroll group. Due to the more detailed description of quantum geometrodynamics given by the Bohm-de Broglie interpretation, some new boundary conditions on solutions of the Wheeler-DeWitt equation must be imposed in order to preserve consistency of this finer view. 
  Spatially homogeneous cosmological models with a positive cosmological constant are investigated, using dynamical systems methods. We focus on the future evolution of these models. In particular, we address the question whether there are models within this class that are de Sitter-like in the future, but are tilted. 
  We consider a closed Friedmann-Robertson-Walker Universe driven by the back reaction from a massless, non-conformally coupled quantum scalar field. We show that the back-reaction of the quantum field is able to drive the cosmological scale factor over the barrier of the classical potential so that if the universe starts near zero scale factor (initial singularity) it can make the transition to an exponentially expanding de Sitter phase, with a probability comparable to that from quantum tunneling processes. The emphasis throughout is on the stochastic nature of back reaction, which comes from the quantum fluctuations of the fundamental fields. 
  We introduce a semiclassical Einstein-Langevin equation as a consistent dynamical equation for a first order perturbative correction to semiclassical gravity. This equation includes the lowest order quantum stress-energy fluctuations of matter fields as a source of classical stochastic fluctuations of the gravitational field. The Einstein-Langevin equation is explicitly solved around one of the simplest solutions of semiclassical gravity: Minkowski spacetime with a conformal scalar field in its vacuum state. We compute the two-point correlation function of the linearized Einstein tensor. This calculation illustrates the possibility of obtaining some ``non-perturbative'' behavior for the induced gravitational fluctuations that cannot be obtained in perturbative quantum gravity. 
  The entropy of the BTZ black hole is computed in the Ponzano-Regge formulation of three-dimensional lattice gravity. It is seen that the correct semi-classical behaviour of entropy is reproduced by states that correspond to all possible triangulations of the Euclidean black hole. 
  The radiation spectrum of a classical charged particle (electron) moving in the de Sitter universe, has been calculated. The de Sitter metric is taken in the quasi-Euclidean Robertson-Walker form. It is shown that in the de Sitter spacetime an electron radiates as if it moved in a constant and homogeneous electric field with the strength vector E collinear to the electron momentum P. 
  We introduce additional restriction into "general ether theory" - a generalization of Lorentz ether theory to gravity - which fixes the signs of the cosmological constants in this theory. This leads to an oscillating universe, thus, solves the cosmological horizon problem without inflation.   We prove the equivalence of the Lagrangian of this theory with Logunov's "relativistic theory of gravity" with massive graviton and a variant of GR with four non-standard scalar fields and negative cosmological constant.   We consider the remaining differences between these theories. 
  Bondi's approach to the construction of a coordinate system is used with a different choice of gauge, in accordance with which the radial coordinate r is an affine parameter, to cast the metric tensor into a form suitable for use with the Newman-Penrose null tetrad formalism. The choice of tetrad has the result that the equations and all the functions that appear in them are real-valued. A group classification of the Sachs equations in this gauge leads to a unique expression for the first of the five independent elements $\Psi_0$ of the Weyl spinor, and to the corresponding exact solutions for two of the metric functions on an initial null hypersurface. A proof is presented that the result for $\Psi_0$ constitutes the appropriate characteristic initial value function for all physically realistic axisymmetric, non-rotating vacuum spacetimes. Integration of the field equations on the axis of symmetry when an equatorial symmetry plane is also present, and on the equatorial plane itself when time rates of change can be neglected, shows that these data produce results consistent with Newton's laws and with the Schwarzschild solution in the appropriate limits. The solution on the axis of symmetry indicates that the Weyl curvature increases without limit between black holes as their separation decreases. 
  Strongly coupled gravitational systems describe Einstein gravity and matter in the limit that Newton's constant G is assumed to be very large. The nonlinear evolution of these systems may be solved analytically in the classical and semiclassical limits by employing a Green function analysis. Using functional methods in a Hamilton-Jacobi setting, one may compute the generating functional (`the phase of the wavefunctional') which satisfies both the energy constraint and the momentum constraint. Previous results are extended to encompass the imposition of an arbitrary initial hypersurface. A Lagrange multiplier in the generating functional restricts the initial fields, and also allows one to formulate the energy constraint on the initial hypersurface. Classical evolution follows as a result of minimizing the generating functional with respect to the initial fields. Examples are given describing Einstein gravity interacting with either a dust field and/or a scalar field. Green functions are explicitly determined for (1) gravity, dust, a scalar field and a cosmological constant and (2) gravity and a scalar field interacting with an exponential potential. This formalism is useful in solving problems of cosmology and of gravitational collapse. 
  We consider two proposals for defining black hole entropy in spherical symmetry, where the horizon is defined locally as a trapping horizon. The first case, boundary terms in a dual-null form of the reduced action in two dimensions, gives a result that is proportional to the area. The second case, Wald's Noether current method, is generalized to dynamic black holes, giving an entropy that is just the area of the trapping horizon. These results are compared with a generalized first law of thermodynamics. 
  Skew-symmetric massless fields, their potentials being $r$-forms, are close analogues of Maxwell's field (though the non-linear cases also should be considered). We observe that only two of them ($r=$2 and 3) automatically yield stress-energy tensors characteristic to normal perfect fluids. It is shown that they naturally describe both non-rotating ($r=2$) and rotating (then a combination of $r=2$ and $r=3$ fields is indispensable) general relativistic perfect fluids possessing every type of equations of state. Meanwile, a free $r=3$ field is completely equivalent to appearance of the cosmological term in Einstein's equations. Sound waves represent perturbations propagating on the background of the $r=2$ field. Some exotic properties of these two fields are outlined. 
  We define the action operator in the consistent histories formalism, as the quantum analogue of the classical action functional, for the simple harmonic oscillator case. The action operator is shown to be the generator of time transformations, and to be associated with the two types of time-evolution of the standard quantum theory: the wave-packet reduction and the unitary time-evolution. We construct the corresponding classical histories and demonstrate the relevance with the quantum histories. Finally, we show the relation of the action operator to the decoherence functional. 
  We address the issue of constructing continuous instantons representing the pair creation of black holes in a cosmological context. The recent attempt at constructing such solutions using virtual domain walls is reviewed first. We then explore the existence of continuous instantons in higher curvature gravity theories where the Lagrangian is polynomial in the Ricci scalar. Lastly, we study continuous instanton solutions of ordinary gravity coupled to the Narlikar C-field. For each theory, we first consider the case of finding continuous instanton solutions which represent the "near-annihilation" of a de Sitter universe and its subsequent recreation. In situations where these solutions exist, we then ask whether solutions can be found that represent the "near-annihilation" of a de Sitter spacetime and the subsequent creation of a pair of Schwarzschild-de Sitter black holes. 
  We show that the Duru-Kleinert fixed energy amplitude leads to the path integral for the propagation amplitude in the closed FRW quantum cosmology with scale factor as one degree of freedom. Then, using the Duru-Kleinert equivalence of corresponding actions, we calculate the tunneling rate, with exact prefactor, through the dilute-instanton approximation to first order in \hbar. 
  The tunneling rate, with exact prefactor, is calculated to first order in \hbar for a closed FRW universe filled with perfect fluid violating the strong energy condition. The calculations are performed by applying the dilute-instanton approximation on the corresponding Duru-Kleinert path integral. It is shown that a closed FRW universe filled with a perfect fluid with small violation of strong energy condition is more probable to tunnel than the same universe with large violation of strong energy condition. 
  The tunneling rate, with exact prefactor, is calculated to first order in $\hbar$ for an empty closed Friedmann-Robertson-Walker (FRW) universe with decaying cosmological term $\Lambda \sim R^{-m}$ ($R$ is the scale factor and $m$ is a parameter $0\leq m \leq 2$). This model is equivalent to a cosmology with the equation of state $p_{\chi}=(m/3 -1)\rho_{\chi}$. The calculations are performed by applying the dilute-instanton approximation on the corresponding Duru-Kleinert path integral.   It is shown that the highest tunneling rate occurs for $m=2$ corresponding to the cosmic string matter universe. The obtained most probable cosmological term, like one obtained by Strominger, accounts for a possible solution to the cosmological constant problem. 
  In this work we generalize a previously developed semiclassical approach to inflation, devoted to the analysis of the effective dynamics of coarse-grained fields, which are essential to the stochastic approach to inflation. We consider general non-trivial momentum distributions when defining these fields. The use of smooth cutoffs in momentum space avoids highly singular quantum noise correlations and allows us to consider the whole quantum noise sector when analyzing the conditions for the validity of an effective classical dynamical description of the coarse-grained field. We show that the weighting of modes has physical consequences, and thus cannot be considered as a mere mathematical artifact. In particular we discuss the exponential inflationary scenario and show that colored noises appear with cutoff dependent amplitudes. 
  We have shown recently that the gravity field phenomena can be described by a traceless part of the wave-type field equation. This is an essentially non-Einsteinian gravity model. It has an exact spherically-symmetric static solution, that yields to the Yilmaz-Rosen metric. This metric is very close to the Schwarzchild metric. The wave-type field equation can not be derived from a suitable variational principle by free variations, as it was shown by Hehl and his collaborates. In the present work we are seeking for another field equation having the same exact spherically-symmetric static solution. The differential-geometric structure on the manifold endowed with a smooth orthonormal coframe field is described by the scalar objects of anholonomity and its exterior derivative. We construct a list of the first and second order SO(1,3)-covariants (one- and two-indexed quantities) and a quasi-linear field equation with free parameters. We fix a part of the parameters by a condition that the field equation is satisfied by a quasi-conformal coframe with a harmonic conformal function. Thus we obtain a wide class of field equations with a solution that yields the Majumdar-Papapetrou metric and, in particularly, the Yilmaz-Rosen metric, that is viable in the framework of three classical tests. 
  We examine the possibility that the gravitational contribution to the entropy of a system can be identified with some measure of the Weyl curvature. In this paper we consider homothetically self-similar spacetimes. These are believed to play an important role in describing the asymptotic properties of more general models. By exploiting their symmetry properties we are able to impose significant restrictions on measures of the Weyl curvature which could reflect the gravitational entropy of a system. In particular, we are able to show, by way of a more general relation, that the most widely used "dimensionless" scalar is \textit{not} a candidate for this measure along homothetic trajectories. 
  We develop a Hamiltonian formulation of the Bianchi type I space-time in conformal gravity, i.e. the theory described by a Lagrangian that is defined by the contracted quadratic product of the Weyl tensor, in a four-dimensional space-time. We derive the explicit forms of the super-Hamiltonian and of the constraint expressing the conformal invariance of the theory and we write down the system of canonical equations. To seek out exact solutions of this system we add extra constraints on the canonical variables and we go through a global involution algorithm which eventually leads to the closure of the constraint algebra. The Painleve approach provides us with a proof of non-integrability, as a consequence of the presence of movable logarithms in the general solution of the problem. We extract all possible particular solutions that may be written in closed analytical form. This enables us to demonstrate that the global involution algorithm has brought forth the complete list of exact solutions that may be written in closed analytical form. We discuss the conformal relationship, or absence thereof, of our solutions with Einstein spaces. 
  Spontaneous breaking of Lorentz invariance compatible with observational limits may realistically take place in the context of string theories, possibly endowing the photon with a mass. In this process the conformal symmetry of the electromagnetic action is broken allowing for the possibility of generating large scale ($\sim Mpc$) magnetic fields within inflationary scenarios. We show that for reheating temperatures safe from the point of view of the gravitino and moduli problem, $T_{RH} \laq 10^{9} GeV$ for $m_{3/2} \approx 1 TeV$, the strength of the generated seed fields is, in our mechanism, consistent with amplification by the galactic dynamo processes and can be even as large as to explain the observed galactic magnetic fields through the collapse of protogalactic clouds. 
  We construct a quantum measure on the power set of non-cyclic oriented graphs of N points, drawing inspiration from 1-dimensional directed percolation. Quantum interference patterns lead to properties which do not appear to have any analogue in classical percolation. Most notably, instead of the single phase transition of classical percolation, the quantum model displays two distinct crossover points. Between these two points, spacetime questions such as "does the network percolate" have no definite or probabilistic answer. 
  We present spectral methods developed in our group to solve three-dimensional partial differential equations. The emphasis is put on equations arising from astrophysical problems in the framework of general relativity. 
  We study the scintillation produced by time-varying gravitational fields within scalar-tensor theories of gravity. The problem is treated in the geometrical optics approximation for a very distant light source emitting quasi plane monochromatic electromagnetic waves. We obtain a general formula giving the time dependence of the photon flux received by a freely falling observer. In the weak-field approximation, we show that the contribution to the scintillation effect due to the focusing of the light beam by a gravitational wave is of first order in the amplitude of the scalar perturbation. Thus scalar-tensor theories contrast with general relativity, which predicts that the only first-order effect is due to the spectral shift. Moreover, we find that the scintillation effects caused by the scalar field have a local character: they depend only on the value of the perturbation at the observer. This effect provides in principle a mean to detect the presence of a long range scalar field in the Universe, but its smallness constitutes a tremendous challenge for detection. 
  We map the general relativistic two-body problem onto that of a test particle moving in an effective external metric. This effective-one-body approach defines, in a non-perturbative manner, the late dynamical evolution of a coalescing binary system of compact objects. The transition from the adiabatic inspiral, driven by gravitational radiation damping, to an unstable plunge, induced by strong spacetime curvature, is predicted to occur for orbits more tightly bound than the innermost stable circular orbit in a Schwarzschild metric of mass M = m1 + m2. The binding energy, angular momentum and orbital frequency of the innermost stable circular orbit for the time-symmetric two-body problem are determined as a function of the mass ratio. 
  The propagation of gravitational waves or tensor perturbations in a perturbed Friedmann-Robertson-Walker universe filled with a perfect fluid is re-examined. It is shown that while the shear and magnetic part of the Weyl tensor satisfy linear, homogeneous {\it second order} wave equations, for perfect fluids with a $\gamma$\hs law equation of state satisfying $\case{2}/{3}<\gamma<2$, the electric part of the Weyl tensor satisfies a linear homogeneous {\it third order} equation. Solutions to these equations are obtained for a flat Friedmann-Robertson-Walker background and we discuss implications of this result. 
  In many circumstances the perfect fluid conservation equations can be directly integrated to give a Geometric-Thermodynamic equation: typically that the lapse $N$ is the reciprocal of the enthalphy $h$, ($ N=1/h$). This result is aesthetically appealing as it depends only on the fluid conservation equations and does not depend on specific field equations such as Einstein's. Here the form of the Geometric-Thermodynamic equation is derived subject to spherical symmetry and also for the shift-free ADM formalism. There at least three applications of the Geometric-Thermodynamic equation, the most important being to the notion of asympotic flatness and hence to spacetime exterior to a star. For asymptotic flatness one wants $h\to 0$ and $N\to 1$ simultaneously, but this is incompatible with the Geometric-Thermodynamic equation. Observational data and asymptotic flatness are discussed. It is argued that a version of Mach's principle does not allow asymptotic flatness. 
  We analyze classical and quantum dynamics of a particle in 2d spacetimes with constant curvature which are locally isometric but globally different. We show that global symmetries of spacetime specify the symmetries of physical phase-space and the corresponding quantum theory. To quantize the systems we parametrize the physical phase-space by canonical coordinates. Canonical quantization leads to unitary irreducible representations of $SO_\uparrow (2.1)$ group. 
  We present a dissipative algorithm for solving nonlinear wave-like equations when the initial data is specified on characteristic surfaces. The dissipative properties built in this algorithm make it particularly useful when studying the highly nonlinear regime where previous methods have failed to give a stable evolution in three dimensions. The algorithm presented in this work is directly applicable to hyperbolic systems proper of Electromagnetism, Yang-Mills and General Relativity theories. We carry out an analysis of the stability of the algorithm and test its properties with linear waves propagating on a Minkowski background and the scattering off a Scwharszchild black hole in General Relativity. 
  We investigate how the gravitational field generated by line sources can be characterized in Brans-Dicke theory of gravity. Adapting an approach previously developed by Israel who solved the same problem in general relativity we show that in Brans-Dicke theory's case it is possible to work out the field equations which relate the energy-momentum tensor of the source to the scalar field, the coupling constant $\omega $ and the extrinsic curvature of a tube of constant geodesic radius centered on the line in the limit when the radius shrinks to zero. In this new scenario two examples are considered and an account of the Gundlach and Ortiz solution is included. Finally, a brief discussion of how to treat thin shells in Brans-Dicke theory is given. 
  During the last twenty-five years evidence has been mounting that a black-hole surface area has a {\it discrete} spectrum. Moreover, it is widely believed that area eigenvalues are {\it uniformally} spaced. There is, however, no general agreement on the {\it spacing} of the levels. In this letter we use Bohr's correspondence principle to provide this missing link. We conclude that the area spacing of a black-hole is $4\hbar \ln 3$. This is the unique spacing consistent both with the area-entropy {\it thermodynamic} relation for black holes, with Boltzmann-Einstein formula in {\it statistical physics} and with {\it Bohr's correspondence principle}. 
  The early work of Lorentz, Abraham and others, evolved through the work of Fokker, Dirac and others to ultimately culminate in the Feynman- Wheeler direct action at a distance theory. However this theory has encountered certain conceptual difficulties like non-locality in time, self force of the electron, pre acceleration and the perfect absorption condition of Feynman and Wheeler, that is the instantaneous action of the remaining charges in the universe on the charge in question. More recently, Hoyle and Narlikar have resurrected this theory, but within the context of a Steady State or Quasi Steady State cosmology. They argue that the theory infact has a better standing than the generally accepted quantum theoretic description. In this article we consider a quantum theoretic description and a cosmology which parallels the Hoyle-Narlikar approach. This leads to a synthesis and justification of the Dirac and Feynman-Wheeler approaches, clarifying the conceptual problems in the process. We deduce a scenario with quantized space-time and a holistic cosmology, consistent with physical and astrophysical data. The non-locality is now seen to be meaningful within the minimum space-time intervals, as also the perfect absorption within the holistic description. Local realism, and the usual causal field theory are seen to have an underpinning of direct action. For example this is brought out by the virtual photons which mediate interactions in Quantum Electro Dynamics, and the emergence of the inverse square law in the above approach from a background Zero Point Field. 
  A discussion of how to calculate asymptotic expansions for polyhomogeneous spacetimes using the Newman-Penrose formalism is made. The existence of logarithmic Newman-Penrose constants for a general polyhomogeneous spacetime (i.e. a polyhomogeneous spacetime such that $\Psi_0=\O(r^{-3}\ln ^{N_3})$) is addressed. It is found that these constants exist for the generic case. 
  The dynamics of closed scalar field FRW cosmological models is studied for several types of exponentially and more than exponentially steep potentials. The parameters of scalar field potentials which allow a chaotic behaviour are found from numerical investigations. It is argued that analytical studies of equation of motion at the Euclidean boundary can provide an important information about the properties of chaotic dynamics. Several types of transition from chaotic to regular dynamics are described. 
  We demonstrate the existence of spinorial states in a theory of canonical quantum gravity without matter. This should be regarded as evidence towards the conjecture that bound states with particle properties appear in association with spatial regions of non-trivial topology. In asymptotically trivial general relativity the momentum constraint generates only a subgroup of the spatial diffeomorphisms. The remaining diffeomorphisms give rise to the mapping class group, which acts as a symmetry group on the phase space. This action induces a unitary representation on the loop state space of the Ashtekar formalism. Certain elements of the diffeomorphism group can be regarded as asymptotic rotations of space relative to its surroundings. We construct states that transform non-trivially under a $2\pi$-rotation: gravitational quantum states with fractional spin. 
  Physical properties of gravitational instantons which are derivable from minimal surfaces in 3-dimensional Euclidean space are examined using the Newman-Penrose formalism for Euclidean signature. The gravitational instanton that corresponds to the helicoid minimal surface is investigated in detail. This is a metric of Bianchi Type $VII_0$, or E(2) which admits a hidden symmetry due to the existence of a quadratic Killing tensor. It leads to a complete separation of variables in the Hamilton-Jacobi equation for geodesics, as well as in Laplace's equation for a massless scalar field. The scalar Green function can be obtained in closed form which enables us to calculate the vacuum fluctuations of a massless scalar field in the background of this instanton. 
  We show how the quantum-to-classical transition of the cosmological fluctuations produced during an inflationary stage can be described using the consistent histories approach. We identify the corresponding histories in the limit of infinite squeezing. To take the decaying mode into account, we propose an extension to coarse-grained histories. 
  It has been recently shown that in order to have Dirac eigenvalues as observables of Euclidean supergravity, certain constraints should be imposed on the covariant phase space as well as on Dirac eigenspinors. We investigate the relationships among the constraints in the first set and argue that these relationships are not linear. We also derive a set of equations expressing the linear dependency of the constraints in order that the second set of constraints be linearly independent. 
  We study the late time evolution of a class of exact anisotropic cosmological solutions of Einstein's equations, namely spatially homogeneous cosmologies of Bianchi type VII$_0$ with a perfect fluid source. We show that, in contrast to models of Bianchi type VII$_h$ which are asymptotically self-similar at late times, Bianchi VII$_0$ models undergo a complicated type of self-similarity breaking. This symmetry breaking affects the late time isotropization that occurs in these models in a significant way: if the equation of state parameter $\gamma$ satisfies $\gamma \leq 4/3$ the models isotropize as regards the shear but not as regards the Weyl curvature. Indeed these models exhibit a new dynamical feature that we refer to as Weyl curvature dominance: the Weyl curvature dominates the dynamics at late times. By viewing the evolution from a dynamical systems perspective we show that, despite the special nature of the class of models under consideration, this behaviour has implications for more general models. 
  Based on the principle of relativity, we find that the sufficient and necessary condition for the general covariance of a field theory actually requires more than the invariance of its local Langrangian density. If the spacetime is not a flat one, its derivative requirement from the analysis of the parallel transportation of tensor fields over spacetime restricts the generally supposed covariance group, the group of differmorphisms, to the group of linear coordinate transformations. Moreover, for any a field theory with linear equations of motion, it stipulates for a universal physical propagation speed of interaction over the spacetime manifold. 
  We give a physical interpretation to the multi-polar Erez-Rozen-Quevedo solution of the Einstein Equations in terms of bars. We find that each multi-pole correspond to the Newtonian potential of a bar with linear density proportional to a Legendre Polynomial. We use this fact to find an integral representation of the $\gamma$ function. These integral representations are used in the context of the inverse scattering method to find solutions associated to one or more rotating bodies each one with their own multi-polar structure. 
  I review some ways in which spacetime dimensionality enters explicitly in gravitation. In particular, I recall some unusual geometrical gravity models that are constructible in dimensions different from four, especially in D=3 where even ordinary Einstein theory is "different", e.g., fully Machian. 
  The detection of quasi-periodic sources of gravitational waves requires the accumulation of signal-to-noise over long observation times. If not removed, Earth-motion induced Doppler modulations, and intrinsic variations of the gravitational-wave frequency make the signals impossible to detect. These effects can be corrected (removed) using a parameterized model for the frequency evolution. We compute the number of independent corrections $N_p(\Delta T,N)$ required for incoherent search strategies which use stacked power spectra---a demodulated time series is divided into $N$ segments of length $\Delta T$, each segment is FFTed, the power is computed, and the $N$ spectra are summed up. We estimate that the sensitivity of an all-sky search that uses incoherent stacks is a factor of 2--4 better than would be achieved using coherent Fourier transforms; incoherent methods are computationally efficient at exploring large parameter spaces. A two-stage hierarchical search which yields another 20--60% improvement in sensitivity in all-sky searches for old (>= 1000 yr) slow (<= 200 Hz) pulsars, and for young (>= 40 yr) fast (<= 1000 Hz) pulsars. Assuming 10^{12} flops of effective computing power for data analysis, enhanced LIGO interferometers should be sensitive to: (i) Galactic core pulsars with gravitational ellipticities of $\epsilon\agt5\times 10^{-6}$ at 200 Hz, (ii) Gravitational waves emitted by the unstable r-modes of newborn neutron stars out to distances of ~8 Mpc, and (iii) neutron stars in LMXB's with x-ray fluxes which exceed $2 \times 10^{-8} erg/(cm^2 s)$. Moreover, gravitational waves from the neutron star in Sco X-1 should be detectable is the interferometer is operated in a signal-recycled, narrow-band configuration. 
  We study in this paper some filters for the detection of burst-like signals in the data of interferometric gravitational-wave detectors. We present first two general (non-linear) filters with no {\it a priori} assumption on the waveforms to detect. A third filter, a peak correlator, is also introduced and permits to estimate the gain, when some prior information is known about the waveforms. We use the catalogue of supernova gravitational-wave signals built by Zwerger and M\"uller in order to have a benchmark of the performance of each filter and to compare to the performance of the optimal filter. The three filters could be a part of an on-line triggering in interferometric gravitational-wave detectors, specialised in the selection of burst events. 
  This paper discusses the gravitational scattering of a straight, infinitely long test cosmic string by a black hole. We present numerical results that probe the two-dimensional parameter space of impact parameter and initial velocity and compare them to approximate perturbative solutions derived previously. We analyze string scattering and loop formation in the ultra-relativistic regime and compare these results with analytical results for string scattering by a gravitational shock wave. Special attention is paid to regimes where the string approaches the black hole at near-critical impact parameters. The dynamics of string scattering in this case are highly sensitive to initial data and transient phenomena arise while portions of the string dwell in the strong gravitational field near the event horizon of the black hole. The role of string tension is also investigated by comparing the scattering of a cosmic string to the scattering of a tensionless "dust" string. Finally, the problem of string capture is revisited in light of these new results, and a capture curve covering the entire velocity range ($0 < v \le c$) is given. 
  A large set of complex path solutions for the Hartle Hawking semi-classical wave function are found for an inflationary universe in the "slow roll" regime. The implication of these for the semi-classical evolution of the universe is also studied. 
  Symmetries are defined in histories-based generalized quantum mechanics paying special attention to the class of history theories admitting quasitemporal structure (a generalization of the concept of `temporal sequences' of `events' using partial semigroups) and logic structure for `single time histories'. Symmetries are classified into orthochronous (those preserving the `temporal order' of `events') and non-orthochronous. A straightforward criterion for physical equivalence of histories is formulated in terms of orthochronous symmetries; this criterion covers various notions of physical equivalence considered by Gell-Mann and Hartle as special cases. In familiar situations, a reciprocal relationship between traditional symmetries (Wigner symmetries in quantum mechanics and Borel-measurable transformations of phase space in classical mechanics) and symmetries defined in this work is established. In a restricted class of theories, a definition of conservation law is given in the history language which agrees with the standard ones in familiar situations; in a smaller subclass of theories, a Noether type theorem (implying a connection between continuous symmetries of dynamics and conservation laws) is proved. 
  Almost none of the r-modes ordinarily found in rotating stars exist, if the star and its perturbations obey the same one-parameter equation of state; and rotating relativistic stars with one-parameter equations of state have no pure r-modes at all, no modes whose limit, for a star with zero angular velocity, is a perturbation with axial parity. Similarly (as we show here) rotating stars of this kind have no pure g-modes, no modes whose spherical limit is a perturbation with polar parity and vanishing perturbed pressure and density. Where have these modes gone?   In spherical stars of this kind, r-modes and g-modes form a degenerate zero-frequency subspace. We find that rotation splits the degeneracy to zeroth order in the star's angular velocity $\Omega$, and the resulting modes are generically hybrids, whose limit as $\Omega\to 0$ is a stationary current with axial and polar parts. Because each mode has definite parity, its axial and polar parts have alternating values of $l$. We show that each mode belongs to one of two classes, axial-led or polar-led, depending on whether the spherical harmonic with lowest value of $l$ that contributes to its velocity field is axial or polar. We numerically compute these modes for slowly rotating polytropes and for Maclaurin spheroids, using a straightforward method that appears to be novel and robust. Timescales for the gravitational-wave driven instability and for viscous damping are computed using assumptions appropriate to neutron stars. 
  We consider Bondi's radiating metric in the context of the teleparallel equivalent of general relativity (TEGR). This metric describes the asymptotic form of a radiating solution of Einstein's equations. The total gravitational energy for this solution can be calculated by means of pseudo-tensors in the static case. In the nonstatic case, Bondi defines the {\it mass aspect} $m(u)$, which describes the mass of an isolated system. In this paper we express Bondi's solution in asymptotically spherical 3+1 coordinates, not in radiation coordinates, and obtain Bondi's energy in the static limit by means of the expression for the gravitational energy in the framework of the TEGR. We can either obtain the total energy or the energy inside a large (but finite) portion of the three-dimensional spacelike hypersurface, whose boundary is far away from the source. The relationship of the present energy expression with Moller's energy is established. 
  We present a class of exact cosmological solutions of the low energy string effective action in the presence of a homogeneous magnetic fields. We discuss the physical properties of the obtained (fully anisotropic) cosmologies paying particular attention to their vacuum limit and to the possible isotropization mechanisms. We argue that quadratic curvature corrections are able to isotropize fully anisotropic solutions whose scale factors describe accelerated expansion. Moreover, the degree of isotropization grows with the duration of the string phase. We follow the fate of the shear parameter in a decelerated phase where, dilaton, magnetic fields and radiation fluid are simultaneously present. In the absence of any magnetic field a long string phase immediately followed by radiation is able to erase large anisotropies. Conversely, if a short string phase is followed by a long dilaton dominated phase the anisotropies can be present, in principle, also at later times. The presence of magnetic seeds after the end of the string phase can induce further anisotropies which can be studied within the formalism reported in this paper. 
  In 1985 Goode and Wainwright devised the concept of an isotropic singularity. Since that time, numerous authors have explored the interesting consequences, in mathematical cosmology, of assuming the existence of this type of singularity. In this paper, we collate all examples of cosmological models which are known to admit an isotropic singularity, and make a number of observations regarding their general characteristics. 
  Circular and radial geodesics are studied in the spacetime described by the $\gamma$ metric. Their behaviour is compared with the spherically symmetric situation, bringing out the sensitivity of the trajectories to deviations from spherical symmetry. 
  The Dirac quantization `procedure' for constrained systems is well known to have many subtleties and ambiguities. Within this ill-defined framework, we explore the generality of a particular interpretation of the Dirac procedure known as refined algebraic quantization. We find technical conditions under which refined algebraic quantization can reproduce the general implementation of the Dirac scheme for systems whose constraints form a Lie algebra with structure constants. The main result is that, under appropriate conditions, the choice of an inner product on the physical states is equivalent to the choice of a ``rigging map'' in refined algebraic quantization. 
  This paper is a brief overview of a more extensive article recently published in Found. Phys. Lett. [2]. Apparent disagreement with experiment as well as internal inconsistencies found in the traditional analysis of relativistically rotating frames/disks are summarized. As one example, a point p at 0 degrees on the circumference of a rotating disk does not, according to the standard theory, exist at the same moment in time as the same point p at 360 degrees. This and other problems with the standard theory are completely resolved by a novel analysis that directly addresses, apparently for the first time, the non-time-orthogonal nature of rotating frames. Though ultimately consonant with the special and general theories of relativity, due to non-time-orthogonality, the analysis predicts several peculiar (i.e., not traditionally relativistic) results. For example, the local circumferential speed of light is not invariant (thereby agreeing with the Sagnac experiment), and no Lorentz contraction exists along the disk rim. Other experimental results, including time dilation, mass-energy dependence on speed, and what has heretofore been considered a "spurious" signal in the most accurate Michelson-Morley experiment performed to date, are accurately predicted. Further, the widely accepted postulate for the equivalence of co-moving inertial and non-inertial rods, used liberally with prior rotating frame analyses, is shown to be invalid for non-time-orthogonal frames. This understanding of the ramifications of non-time-orthogonality resolves paradoxes inherent in the traditional theory. 
  A static axisymmetric solution with an additional cylindrical symmetry is considered and that the matter consists in a cosmological and a dust term. 
  I review the tunneling, Hartle-Hawking and Linde proposals for the wave function of the universe and comment on the recent work on quantum creation of open universes. 
  This paper discusses particle production in Schwarzchild-like spacetimes and in an uniform electric field. Both problems are approached using the method of complex path analysis. Particle production in Schwarzchild-like spacetimes with a horizon is obtained here by a new and simple semi-classical method based on the method of complex paths. Hawking radiation is obtained in the (t,r) co-ordinate system of the standard Schwarzchild metric {\it without} requiring the Kruskal extension. The co-ordinate singularity present at the horizon manifests itself as a singularity in the expression for the semi-classical propagator for a scalar field. We give a prescription whereby this singularity is regularised with Hawking's result being recovered. In the case of the electric field, standard quantum field theoretic methods can be used to obtain particle production in a purely time-dependent gauge. In a purely space-dependent gauge, however, the tunnelling interpretation has to be resorted to. We attempt, in this paper, to provide a tunnelling description for both the time and space dependent gauges. The usefulness of such a common description becomes evident when `mixed' gauges, which are functions of both space and time variables, are analysed. We report, in this paper, certain mixed gauges which have the interesting property that mode functions in these gauges are found to be a combination of {\it elementary} functions unlike the standard modes. Finally, we present an attempt to interpret particle production by the electric field as a tunnelling process between the two sectors of the Rindler spacetime. 
  In this talk I will survey some of the basic facts about superstring theories in 10 dimensions and the dualities that relate them to M theory in 11 dimensions. Then I will mention some important unresolved issues. 
  The formulation of the Einstein field equations admitting two Killing vectors in terms of harmonic mappings of Riemannian manifolds is a subject in which Charlie Misner has played a pioneering role. We shall consider the hyperbolic case of the Einstein-Maxwell equations admitting two hypersurface orthogonal Killing vectors which physically describes the interaction of two electrovac plane waves. Following Penrose's discussion of the Cauchy problem we shall present the initial data appropriate to this collision problem. We shall also present three different ways in which the Einstein-Maxwell equations for colliding plane wave spacetimes can be recognized as a harmonic map. The goal is to cast the Einstein-Maxwell equations into a form adopted to the initial data for colliding impulsive gravitational and electromagnetic shock waves in such a way that a simple harmonic map will directly yield the metric and the Maxwell potential 1-form of physical interest.    For Charles W. Misner on his 60th birthday 
  The properties of the effective sigma-model for D-dimensional Einstein gravity based on multidimensional geometries is analyzed. Besides pure geometry, additional minimally coupled scalars and (p+2)-forms are considered which yield an extended target space after reduction to the effective D_0-dimensional geometry.   In any case the target space is a homogeneous space. The orthobrane condition guarantees the existence of exact solutions. Geometrically, it makes the target space a locally symmetric one.   New solutions with scalar fields are found, which may inflate not only in time-like but in also in additional spatial directions of the effective geometry.   Static spherically symmetric solutions with a particular configuration of intersecting electric and magnetic branes are investigated both, for the orthobrane case and for degenerated charges. In both cases T_H depends critically on the intersection dimension of the p-branes.   Finally, the role of the Einstein frame for 4-geometries is addressed, and the physical frame transformation for cosmological geometries is given. 
  Certain exotic phenomena in general relativity, such as backward time travel, appear to require the presence of matter with negative energy. While quantum fields are a possible source of negative energy densities, there are lower bounds - known as quantum inequalities - that constrain their duration and magnitude. In this paper, we derive new quantum inequalities for scalar fields in static space-times, as measured by static observers with a choice of sampling function. Unlike those previously derived by Pfenning and Ford, our results do not assume any specific sampling function. We then calculate these bounds in static three- and four-dimensional Robertson-Walker universes, the de Sitter universe, and the Schwarzschild black hole. In each case, the new inequality is stronger than that of Pfenning and Ford for their particular choice of sampling function. 
  When a charged insulating spherical shell is uniformly accelerated, an oppositely directed electric field is produced inside. Outside the field is the Born field of a uniformly accelerated charge, modified by a dipole. Radiation is produced. When the acceleration is annulled by the nearly uniform gravity field of an external shell with a 1 + beta cos theta surface distribution of mass, the differently viewed Born field is static and joins a static field outside the external shell; no radiation is produced. We discuss gravitational analogues of these phenomena. When a massive spherical shell is accelerated, an untouched test mass inside experiences a uniform gravity field and accelerates parallelly to the surrounding shell. In the strong gravity regime we illustrate these effects using exact conformastatic solutions of the Einstein-Maxwell equations with charged dust. We consider a massive charged shell on which the forces due to nearly uniform electrical and gravitational fields balance. Both fields are reduced inside by the ratio of the g_00 inside the shell to that away from it. The acceleration of a free test particle, relative to a static observer, is reduced correspondingly. We give physical explanations of these effects. 
  We study the back reaction of a thermal field in a weak gravitational background depicting the far-field limit of a black hole enclosed in a box by the Close Time Path (CTP) effective action and the influence functional method. We derive the noise and dissipation kernels of this system in terms of quantities in quasi-equilibrium, and formally prove the existence of a Fluctuation-Dissipation Relation (FDR) at all temperatures between the quantum fluctuations of the thermal radiance and the dissipation of the gravitational field. This dynamical self-consistent interplay between the quantum field and the classical spacetime is, we believe, the correct way to treat back-reaction problems. To emphasize this point we derive an Einstein-Langevin equation which describes the non-equilibrium dynamics of the gravitational perturbations under the influence of the thermal field. We show the connection between our method and the linear response theory (LRT), and indicate how the functional method can provide more accurate results than prior derivations of FDRs via LRT in the test-field, static conditions. This method is in principle useful for treating fully non-equilibrium cases such as back reaction in black hole collapse. 
  A global existence theorem, with respect to a geometrically defined time, is shown for Gowdy symmetric globally hyperbolic solutions of the Einstein-Vlasov system for arbitrary (in size) initial data. The spacetimes being studied contain both matter and gravitational waves. 
  A Lagrangian for flat domain walls in spaces with Cartan torsion and electromagnetic fields is proposed.The Lagrangian is very similar to a recently proposed Lagrangian for domain walls in a Chern-Simons electrodynamics in 2+1 dimensions.We show that in the first approximation of the torsion scalar potential the field equations are reduced to a Klein-Gordon type field equation for the torsion potential and the electromagnetic wave equation.A planar symmetric solution representing a parallel plates electric capacitor interacting with the electric field is given.The photon mass is proportional to the torsion potential and in the time dependent case the angular momentum is computed and is shown to be connected with torsion in analogy with the spin-torsion relation which appears in Einstein-Cartan gravity.When the curvature Ricci scalar is introduced we are able to show that the torsion potential can be associated with Higgs massive vectorial bosons. 
  We show spurious effects in perturbative calculations due to different orderings of inhomogeneous terms while computing corrections to Green functions for two different metrics. These effects are not carried over to physically measurable quantities like the renormalized value of the vacuum expectation value of the stress-energy tensor. 
  We study the quatum to classical transition process in the context of quantum field theory. Extending the influence functional formalism of Feynman and Vernon, we study the decoherence process for self-interacting quantum fields in flat space. We also use this formalism for arbitrary geometries to analyze the quantum to classical transition in quantum gravity. After summarizing the main results known for the quantum Brownian motion, we consider a self-interacting field theory in Minkowski spacetime. We compute a coarse grained effective action by integrating out the field modes with wavelength shorter than a critical value. From this effective action we obtain the evolution equation for the reduced density matrix (master equation). We compute the diffusion coefficients for this equation and analyze the decoherence induced on the long-wavelength modes. We generalize the results to the case of a conformally coupled scalar field in de Sitter spacetime. We show that the decoherence is effective as long as the critical wavelength is taken to be not shorter than the Hubble radius. On the other hand, we study the classical limit for scalar-tensorial models in two dimensions. We consider different couplings between the dilaton and the scalar field. We discuss the Hawking radiation process and, from an exact evaluation of the influence functional, we study the conditions by which decoherence ensures the validity of the semiclassical approximation in cosmological metrics. Finally we consider four dimensional models with massive scalar fields, arbitrary coupled to the geometry. We compute the Einstein-Langevin equations in order to study the effect of the fluctuations induced by the quantum fields on the classical geometry. 
  In this paper a new formalism based on exterior differential systems is derived for perfect-fluid spacetimes endowed with an abelian orthogonally transitive G2 group of motions acting on spacelike surfaces. This formulation allows simplifications of Einstein equations and it can be applied for different purposes. As an example a singularity-free metric is rederived in this framework. A sufficient condition for a diagonal metric to be geodesically complete is also provided. 
  We have performed a lattice field theory simulation of cusps in Abelian-Higgs cosmic strings. The results are in accord with the theory that the portion of the strings which overlaps near the cusp is released as radiation. The radius of the string cores which must touch to produce the evaporation is approximately $r = 1$ in natural units. In general, the modifications to the string shape due to the cusp may produce many cusps later in the evolution of a string loop, but these later cusps will be much smaller in magnitude and more closely resemble kinks. 
  The geometry of two infinitely long lines of mass moving in a fixed circular orbit is considered as a toy model for the inspiral of a binary system of compact objects due to gravitational radiation. The two Killing fields in the toy model are used, according to a formalism introduced by Geroch, to describe the geometry entirely in terms of a set of tensor fields on the two-manifold of Killing vector orbits. Geroch's derivation of the Einstein equations in this formalism is streamlined and generalized. The explicit Einstein equations for the toy model spacetime are derived in terms of the degrees of freedom which remain after a particular choice of gauge. 
  By decomposing the Riemann curvature into electric and magnetic parts, a duality transformation, which involves interchange of active and passive electric parts, has recently been proposed. It was shown that the Schwarzschild solution is dual to the one that describes the Schwarzschild particle with cloud of string dust or a global monopole. Following the same procedure we obtain the solution dual to the NUT spacetime. 
  We calculate the reduced density matrix for the inflaton field in a model of chaotic inflation by tracing out degrees of freedom corresponding to various bosonic fields. We find a qualitatively new contribution to the density matrix given by the Euclidean effective action of quantum fields. We regularise the ultraviolet divergences in the decoherence factor. Dimensional regularisation is shown to violate the consistency conditions for a density matrix as a bounded operator. A physically motivated conformal redefinition of the environmental fields leads to well-defined expressions. They show that due to bosonic fields the Universe acquires classical properties near the onset of inflation. 
  For vacuum Maxwell theory in four dimensions, a supplementary condition exists (due to Eastwood and Singer) which is invariant under conformal rescalings of the metric, in agreement with the conformal symmetry of the Maxwell equations. Thus, starting from the de Donder gauge, which is not conformally invariant but is the gravitational counterpart of the Lorenz gauge, one can consider, led by formal analogy, a new family of gauges in general relativity, which involve fifth-order covariant derivatives of metric perturbations. The admissibility of such gauges in the classical theory is first proven in the cases of linearized theory about flat Euclidean space or flat Minkowski space-time. In the former, the general solution of the equation for the fulfillment of the gauge condition after infinitesimal diffeomorphisms involves a 3-harmonic 1-form and an inverse Fourier transform. In the latter, one needs instead the kernel of powers of the wave operator, and a contour integral. The analysis is also used to put restrictions on the dimensionless parameter occurring in the DeWitt supermetric, while the proof of admissibility is generalized to a suitable class of curved Riemannian backgrounds. Eventually, a non-local construction is obtained of the tensor field which makes it possible to achieve conformal invariance of the above gauges. 
  We analyse when and why unitarity violations might occur in quantum cosmology restricted to minisuperspace. To this end we discuss in detail backscattering transitions between expanding and contracting solutions of the Wheeler-DeWitt equation. We first show that upon neglecting only backscattering, one obtains an intermediate regime in which matter evolves unitarily but which does not correspond to any Schr\"odinger equation in a given geometry since gravitational backreaction effects are taken into account at the quantum level. We then show that backscattering amplitudes are exponentially smaller than matter transition amplitudes. Both results follow from an adiabatic treatment valid for macroscopic universes. To understand how backscattering and the intermediate regime should be interpreted, we review the problem of electronic transitions induced by nuclear motion since it is mathematically very similar. In this problem, transition amplitudes are obtained from the conserved current. The same applies to quantum cosmology and shows that the unique consistent interpretation is based on the current when backscattering is neglected. We then review why, in a relativistic context, backscattering is interpreted as pair production whereas it is not in the non relativistic case. In each example the correct interpretation is obtained by coupling the system to an external quantum device. From the absence of such external systems in cosmology, we conclude that backscattering does not have a unique consistent interpretation in quantum cosmology. 
  The aim of this set of lectures is a systematic presentation of a 1+3 covariant approach to studying the geometry, dynamics, and observational properties of relativistic cosmological models. In giving (i) the basic 1+3 covariant relations for a cosmological fluid, the present lectures cover some of the same ground as a previous set of Carg\`{e}se lectures \cite{ell73}, but they then go on to give (ii) the full set of corresponding tetrad equations, (iii) a classification of cosmological models with exact symmetries, (iv) a brief discussion of some of the most useful exact models and their observational properties, and (v) an introduction to the gauge-invariant and 1+3 covariant perturbation theory of almost-Friedmann-Lema\^{\i}tre-Robertson-Walker universes, with a fluid description for the matter and a kinetic theory description of the radiation. 
  We briefly review some results concerning the problem of classical singularities in general relativity, obtained with the help of the theory of differential spaces. In this theory one studies a given space in terms of functional algebras defined on it. Then we present a generalization of this method consisting in changing from functional (commutative) algebras to noncommutative algebras. By representing such an algebra as a space of operators on a Hilbert space we study the existence and properties of various kinds of singular space-times. The obtained results suggest that in the noncommutative regime, supposedly reigning in the pre-Planck era, there is no distinction between singular and non-singular states of the universe, and that classical singularities are produced in the transition process from noncommutative geometry to the standard space-time physics. 
  Klein-Gordon equation with minimal coupling in FRW-like spacetimes with compact but not necessarily isotropic neither homogeneous space sections. Beside exceptional cases, the kernel is invariant under continuous isometries. 
  Some properties of cosmological models with a time variable bulk viscous coefficient in presence of adiabatic mater creation and G, c, Lambda variables are investigated in the framework of flat FRW line element. We trivially find a set of solutions through Dimensional Analysis. In all the studied cases it is found that the behaviour of these constants is inversely prportional to the cosmic time. 
  We give a spinorial set of Hamiltonian variables for General Relativity in any dimension greater than 2. This approach involves a study of the algebraic properties of spinors in higher dimension, and of the elimination of second-class constraints from the Hamiltonian theory. In four dimensions, when restricted to the positive spin-bundle, these variables reduce to the standard Ashtekar variables. In higher dimensions, the theory can either be reduced to a spinorial version of the ADM formalism, or can be left in a more general form which seems useful for the investigation of some spinorial problems such as Riemannian manifolds with reduced holonomy group. In dimensions $0 \pmod 4$, the theory may be recast solely in terms of structures on the positive spin-bundle $\mathbb{V}^+$, but such a reduction does not seem possible in dimensions $2 \pmod 4$, due to algebraic properties of spinors in these dimensions. 
  The seeds for quantum creations of universes are constrained gravitational instantons. For all compact constrained instantons with U(1) isometry, the period $\beta$ of the group parameter $\tau$ is identified as the reciprocal of the temperature. If $\beta$ remains a free parameter under the constraints, then the Euclidean action becomes the negative of the "entropy". As examples, we perform the calculations for the Taub-NUT and Taub-bolt-type models and study the quantum creation of the Taub-NUT universe. 
  We consider the class of metrics that can be obtained from those of nonextreme black holes by limiting transitions to the extreme state such that the near-horizon geometry expands into a whole manifold. These metrics include, in particular, the Rindler and Bertotti - Robinson spacetimes. The general formula for the entropy of massless radiation valid either for black-hole or for acceleration horizons is derived. It is argued that, as a black hole horizon in the limit under consideration turns into an acceleration one, the thermodynamic entropy $S_{q}$ of quantum radiation is due to the Unruh effect entirely and $S_{q}=0$ exactly. The contribution to the quasilocal energy from a given curved spacetime is equal to zero and the only nonvanishing term stems from a reference metric. In the variation procedure necessary for the derivation of the general first law, the metric on a horizon surface changes along with the boundary one, and the account for gravitational and matter stresses is an essential ingredient of the first law. This law confirms the property $S_{q}=0$. The quantum-corrected geometry of the Bertotti - Robinson spacetime is found and it is argued that backreaction of quantum fields mimics the effect of the cosmological constant $\Lambda_{eff\text{}}$ and can drastically change the character of spacetime depending on the sign of $\Lambda _{eff}$ --- for instance, turn $AdS_{2}\times S_{2}$ into $dS_{2}\times S_{2}$ or $Rindler_{2}\times S_{2}$. Two latter solutions can be thought of as the quantum versions of the cold and ultracold limits of the Reissner-Nordstrom-de Sitter metric. 
  We investigate the gravitational collapse of a spherically symmetric, perfect fluid with equation of state P = (Gamma -1)rho. We restrict attention to the ultrarelativistic (``kinetic-energy-dominated'', ``scale-free'') limit where black hole formation is anticipated to turn on at infinitesimal black hole mass (Type II behavior). Critical solutions (those which sit at the threshold of black hole formation in parametrized families of collapse) are found by solving the system of ODEs which result from a self-similar ansatz, and by solving the full Einstein/fluid PDEs in spherical symmetry. These latter PDE solutions (``simulations'') extend the pioneering work of Evans and Coleman (Gamma = 4/3) and verify that the continuously self-similar solutions previously found by Maison and Hara et al for $1.05 < Gamma < 1.89 are (locally) unique critical solutions. In addition, we find strong evidence that globally regular critical solutions do exist for 1.89 < Gamma <= 2$, that the sonic point for Gamma_dn = 1.8896244 is a degenerate node, and that the sonic points for Gamma >Gamma_dn are nodal points rather than focal points as previously reported. We also find a critical solution for Gamma = 2, and present evidence that it is continuously self-similar and Type II. Mass-scaling exponents for all of the critical solutions are calculated by evolving near-critical initial data, with results which confirm and extend previous calculations based on linear perturbation theory. Finally, we comment on critical solutions generated with an ideal-gas equation of state. 
  One of the most promising sources of gravitational radiation is coalescence of binary neutron stars or black holes. In order to study gravitational radiation at the merging phase of coalescing binary neutron stars which is called the last three-milliseconds, full general relativistic simulations are required. Since coalescence of binary stars is a completely non-axisymmetric and 3 dimensional (in space) event, which needs great powers of computers, we have to develop a method of 3D numerical relativity using a vector-parallel supercomputer. As a first step of this final goal, we started simulations using post-Newtonian hydrodynamics including radiation reaction of gravitational waves from 1989. They gave a lot of perspectives on coalescing events and gravitational radiation from them. Next we started to attack 3D, fully relativistic simulations. First, basic equations on the (3+1)-formalism of the Einstein equations are shown. As for gauge conditions, we use conformal time slicing and psuedo minimal distortion condtions at present. For these conditions as well as to solve initial value equations, we should solve some elliptic partial differential equations. It consumes the greatest part of CPU time. We show recent results of test simulations on coalescing binary neutron stars. 
  By discussing the Cauchy problem, we determine the covariant equation of the characteristic hypersurfaces in a relativistic superfluid theory. 
  We consider scalar field theory on the RP^3 de Sitter spacetime (RP3dS), which is locally isometric to de Sitter space (dS) but has spatial topology RP^3. We compare the Euclidean vacua on RP3dS and dS in terms of three quantities that are relevant for an inertial observer: (i) the stress-energy tensor; (ii) the response of an inertial monopole particle detector; (iii) the expansion of the Euclidean vacuum in terms of many-particle states associated with static coordinates centered at an inertial world line. In all these quantities, the differences between RP3dS and dS turn out to fall off exponentially at early and late proper times along the inertial trajectory. In particular, (ii) and (iii) yield at early and late proper times in RP3dS the usual thermal result in the de Sitter Hawking temperature. This conforms to what one might call an exponential law: in expanding locally de Sitter spacetimes, differences due to global topology should fall off exponentially in the proper time. 
  We construct initial data for a particular class of Brill wave metrics using Regge calculus, and compare the results to a corresponding continuum solution, finding excellent agreement. We then search for trapped surfaces in both sets of initial data, and provide an independent verification of the existence of an apparent horizon once a critical gravitational wave amplitude is passed. Our estimate of this critical value, using both the Regge and continuum solutions, supports other recent findings. 
  We propose the mechanism of quantum creation of the open Universe in the observable range of values of $\Omega$. This mechanism is based on the no-boundary quantum state with the Hawking-Turok instanton applied to the model with a strong nonminimal coupling of the inflaton field. We develop the slow roll perturbation expansion for the instanton solution and obtain a nontrivial contribution to the classical instanton action. The interplay of this classical contribution with the loop effects due to quantum effective action generates the probability distribution peak with necessary parameters of the inflation stage without invoking any anthropic considerations. In contrast with a similar mechanism for closed models, existing only for the tunneling quantum state of the Universe, the observationally justified open inflation originates from the no-boundary cosmological wavefunction. 
  The phenomenon of string spreading on the black hole horizon, as originally discussed by Susskind, is considered in the {\it exact} curved Schwarzschild background. We consider an oscillating string encircling the black hole and contracting towards the horizon. We then compute the angular and radial spreading of the string, as seen by a static observer at spatial infinity using fixed finite resolution time.    Within our case study we find that there is indeed a spreading of the string in the angular direction, such that the string eventually covers the whole horizon. However, regarding the radial direction, we find that Lorentz-contraction suppresses the radial string spreading. 
  The probability of a charged particle production by the electric field of a charged black hole depends essentially on the particle energy. This probability is found in the nonrelativistic and ultrarelativistic limits. The range of values for the mass and charge of a black hole is indicated where the discussed mechanism of radiation is dominating over the Hawking one. 
  Equilibrium states in galactic dynamics can be described as stationary solutions of the Vlasov-Poisson system, which is the non-relativistic case, or of the Vlasov-Einstein system, which is the relativistic case. To obtain spherically symmetric stationary solutions the distribution function of the particles (stars) on phase space is taken to be a function of the particle energy and angular momentum. We give a new condition on this function which guarantees that the resulting steady state has finite mass and compact support both for the non-relativistic and the relativistic case. The condition is local in the sense that only the asymptotic behaviour for energy values close to the maximal energy value in the particle distribution needs to be prescribed. 
  We define a Frame of reference as a two ingredients concept: A meta-rigid motion, which is a generalization of a Born motion, and a chorodesic synchronization, which is an adapted foliation. At the end of the line we uncover a low-level 3-dimensional geometry with constant curvature and a corresponding coordinated proper-time scale. We discuss all these aspects both from the geometrical point of view as from the point of view of some of the physical applications derived from them. 
  Assuming that the mechanism proposed by Gell-Mann and Hartle works as a mechanism for decoherence and classicalization of the metric field, we formally derive the form of an effective theory for the gravitational field in a semiclassical regime. This effective theory takes the form of the usual semiclassical theory of gravity, based on the semiclassical Einstein equation, plus a stochastic correction which accounts for the back reaction of the lowest order matter stress-energy fluctuations. 
  We consider black holes in EYM theory with a negative cosmological constant. The solutions obtained are somewhat different from those for which the cosmological constant is either positive or zero. Firstly, regular black hole solutions exist for continuous intervals of the parameter space, rather than discrete points. Secondly, there are non-trivial solutions in which the gauge field has no nodes. We show that these solutions are linearly stable. 
  A set of boundary conditions defining a non-rotating isolated horizon are given in Einstein-Maxwell theory. A space-time representing a black hole which itself is in equilibrium but whose exterior contains radiation admits such a horizon . Physically motivated, (quasi-)local definitions of the mass and surface gravity of an isolated horizon are introduced. Although these definitions do not refer to infinity, the quantities assume their standard values in Reissner-Nordstrom solutions. Finally, using these definitions, the zeroth and first laws of black hole mechanics are established for isolated horizons. 
  Chern-Simons electrodynamics in 2+1-spacetimes with torsion is investigated. We start from the usual Chern-Simons (CS) electrodynamics Lagrangian and Cartan torsion is introduced in the covariant derivative and by a direct coupling of torsion vector to the CS field. Variation of the Lagrangian with respect to torsion shows that Chern-Simons field is proportional to the product of the square of the scalar field and torsion. The electric field is proportional to torsion vector and the magnetic flux is computed in terms of the time-component of the two dimensional torsion. Contrary to early massive electrodynamics in the present model the photon mass does not depend on torsion. 
  Some standard results on the initial value problem of general relativity in matter are reviewed. These results are applied first to show that in a well defined sense, finite perturbations in the gravitational field travel no faster than light, and second to show that it is impossible to construct a warp drive as considered by Alcubierre (1994) in the absence of exotic matter. 
  Treating problems in full general relativity is highly complex and frequently approximate methods are employed to simplify the solution. We present comparative solutions of a infinitesimally thin relativistic, stationary, rigidly rotating disk obtained using the full equations and the approximate approach suggested by Wilson & Mathews. We find that the Wilson-Mathews method has about the same accuracy as the first post-Newtonian approximation. 
  I present here a new algorithm to generate families of inhomogeneous massless scalar field cosmologies. New spacetimes, having a single isometry, are generated by breaking the homogeneity of massless scalar field $G_2$ models along one direction. As an illustration of the technique I construct cosmological models which in their late time limit represent perturbations in the form of gravitational and scalar waves propagating on a non-static inhomogeneous background. Several features of the obtained metrics are discussed, such as their early and late time limits, structure of singularities and physical interpretation. 
  Exact solutions for a model with variable $G$, $\Lambda$ and bulk viscosity are obtained. Inflationary solutions with constant (de Sitter-type) and variable energy density are found. An expanding anisotropic universe is found to isotropize during its course of expansion but a static universe is not. The gravitational constant is found to increase with time and the cosmological constant decreases with time as $\Lambda \propto t^{-2}$. 
  This article reviews classical and quantum aspects of critical phenomena in gravitational collapse. We pay special attention to the origin of the scaling law for black hole mass, and to phase transitions in which black hole formation turns on at finite mass. We present some new results for perfect fluids with pressure proportional to density. 
  Chiral phase transitions driven by space-time curvature effects are investigated in de Sitter space in the supersymmetric Nambu-Jona-Lasinio model with soft supersymmetry breaking. The model is considered to be suitable for the analysis of possible phase transitions in inflationary universe. It is found that a restoration of the broken chiral symmetry takes place in two patterns for increasing curvature : the first order and second order phase transition respectively depending on initial settings of the four-body interaction parameter and the soft supersymmetry breaking parameter. The critical curves expressing the phase boundaries in these parameters are obtained. Cosmological implications of the result are discussed in connection with bubble formations and the creation of cosmic strings during the inflationary era. 
  On the basis of dynamic quantization method we build in this paper a new mathematically correct quantization scheme of gravity.  In the frame of this scheme we develop a canonical formalism in tetrad-connection variables in 4-D theory of pure gravity.  In this formalism the regularized quantized fields corresponding to the classical tetrad and connection fields are constructed. It is shown, that the regularized fields satisfy to general covariant equations of motion, which have the classical form. In order to solve these equations the iterative procedure is offered. 
  An anomaly-free quantum theory of a relativistic string is constructed in two-dimensional space-time. The states of the string are found to be similar to the states of a massless chiral quantum particle. This result is obtained by generalizing the concept of an ``operator'' in quantum field theory. 
  We consider a possibility that the entropy of a Schwarzschild black hole has two different interpretations: The black hole entropy can be understood either as an outcome of a huge degeneracy in the mass eigenstates of the hole, or as a consequence of the fact that the interior region of black hole spacetime is separated from the exterior region by a horizon. In the latter case, no degeneracy in the mass eigenstates needs to be assumed. Our investigation is based on calculations performed with Lorentzian partition functions obtained for a whole maximally extended Schwarzschild spacetime, and for its right-hand-side exterior region. To check the correctness of our analysis we reproduce, in the leading order approximation, the Bekenstein--Hawking entropy of the Schwarzschild black hole. 
  We study how fluctuations of the black hole geometry affect the properties of Hawking radiation. Even though we treat the fluctuations classically, we believe that the results so obtained indicate what might be the effects induced by quantum fluctuations in a self consistent treatment. To characterize the fluctuations, we use the model introduced by York in which they are described by an advanced Vaidya metric with a fluctuating mass. Under the assumption of spherical symmetry, we solve the equation of null outgoing rays. Then, by neglecting the greybody factor, we calculate the late time corrections to the s-wave contributions of the energy flux and the asymptotic spectrum. We find three kind of modifications. Firstly, the energy flux fluctuates around its average value with amplitudes and frequencies determined by those of the metric fluctuations. Secondly, this average value receives two positive contributions one of which can be reinterpreted as due to the `renormalisation' of the surface gravity induced by the metric fluctuations. Finally, the asymptotic spectrum is modified by the addition of terms containing thermal factors in which the frequency of the metric fluctuations acts as a chemical potential. 
  We show that significant anisotropy in electromagnetic propagation generates a distinctive signature in the microwave background. The anisotropy may be determined by looking at the cross correlator of the $E$-mode and $B$-mode polarisation spectrum. 
  Gravitational collapse of radiation in an anti-de Sitter background is studied. For the spherical case, the collapse proceeds in much the same way as in the Minkowski background, i.e., massless naked singularities may form for a highly inhomogeneous collapse, violating the cosmic censorship, but not the hoop conjecture. The toroidal, cylindrical and planar collapses can be treated together. In these cases no naked singularity ever forms, in accordance with the cosmic censorship. However, since the collapse proceeds to form toroidal, cylindrical or planar black holes, the hoop conjecture in an anti-de Sitter spacetime is violated. 
  In this thesis properties and the origin of black hole entropy are investigated from various points of view. First, laws of black hole thermodynamics are reviewed. In particular, the first and generalized second laws are investigated in detail. It is in these laws that the black hole entropy plays key roles. Next, three candidates for the origin of the black hole entropy are analyzed: the D-brane statistical-mechanics, the brick wall model, and the entanglement thermodynamics. Finally, discussions are given on semiclassical consistencies of the brick wall model and the entanglement thermodynamics and on the information loss problem. 
  We study the elongated phase of 4-D Dynamical Triangulations. In the case of the sphere topology by using the Walkup's theorem we show that the dominating configurations are stacked spheres. These stacked spheres can be mapped into tree-like graphs (branched polymers). By using Baby-Universes arguments and an antsatz on the universality class between the stacked spheres and a model coming from the theory of random surfaces we argument that this elongated phase is a trivial phase. The numerical evidence for a first order phase transition and the triviality of the elongated phase suggest that a new approach to simplicial quantum gravity might be useful. Along this line following the work of various authors we study a first order version of Regge calculus formulated as a local theory of the Poincare` group. This first order formalism has the effects of smoothing out some pathological configurations, like "spikes", which prevent the theory from having a smooth continuum limit. These confingurations are in fact in the region of large deficit angles where the first order formalism and the secon order formalism are not equivalent on lattice.  We derive the first order field equations in the approximation of "small deficit angles" and prove that (second order) Regge calculus is a solution. Successively we derive the general first order field equations by taking into account the constraints of the theory. An invariant measure for the path-integral of this theory is defined. The coupling with matter, in particular fermions, is also discussed in analogy to the continuum theory. 
  According to Birkhoff's theorem the only spherically symmetric solution of the vacuum Einstein field equations is the Schwarzschild solution. Inspite of imposing asymptotically flatness and staticness as initial conditions we obtain that these equations have general solutions with the Schwarzschild metric as merely a special and simplest form of them. It is possible to have perfect and smooth metrics with the same Newtonian and post-Newtonian limits of Schwarzschild by a convenient and correct selection. 
  We investigate the evolution of the homogeneous and isotropic universe within the framework of the effective string gravity with string-loop modifications of dilaton couplings. In the case of barotropic perfect fluid as a nongravitational source the set of cosmological equations is presented in the form of the third-order autonomous dynamical system. The cases are considered when this system is integrable in terms of integrals depending on dilaton coupling functions. Without specifying these functions we describe generic evolution of the universe, using dynamical systems methods. The critical points and their stability are found for all regions of the parameters. The qualitatively different phase space diagrams are presented for the spatially-flat, closed and open universes. The case of the tree-level models is considered separately. The issue of dilaton stabilization is discussed within the framework of Damour-Polyakov mechanism. 
  Generalising results from Godel and Chaitin in mathematics suggests that self-referential systems contain intrinsic randomness. We argue that this is relevant to modelling the universe and show how three-dimensional space may arise from a non-geometric order-disorder model driven by self-referential noise. 
  This paper is the first of two papers devoted to formulation of quantum mechanics of a particle in a normal geodesic frame of reference in the general Riemannian space-time. Here canonical quantization of geodesic motion in the 1+3-formalism is considered, the result of which will be compared in the subsequent paper II with that of the field-theoretical approach, see also gr-qc/9807030. The Schr\"odinger representation of quantum-mechanical kinematics and dynamics is presented in the general-covariant form and a physical interpretation of the state vectors and the position operators is discussed. 
  The discrepancy of $0.8 %$ between theory and the COW-experiment is interpreted. This is done by using a new path equation other than the geodesic one. It is shown that this discrepancy is possibly due to a type of interaction between the torsion, of space-time generated by the background field, and the spin of the moving neutron. The results obtained are discussed and compared with the experimental interpretation suggested by Arif et al. in 1994. As a byproduct, an upper limit is imposed on the free parameter of the new path equation used. 
  The field equations of the generalized field theory (GFT) are derived from an action principle. A comparison between (GFT), M\o ller's tetrad theory of gravitation (MTT), and general relativity is carried out regarding the Lagrangian of each theory. The results of solutions of the field equations, of each theory, are compared in case of spherical symmetry. The differences between the results are discussed and interpreted. 
  We construct analogues for the quantum phenomena of black hole radiation in the context of {\it classical field theory}. Hawking radiation from a (radially) collapsing star is mathematically equivalent to radiation from a mirror moving along a specific trajectory in Minkowski spacetime. We construct a classical analogue for this quantum phenomenon and use it to construct a classical analogue for black hole radiation. The radiation spectrum in quantum field theory has the power spectrum as its classical analogue. Monochromatic light is continually reflected off a moving mirror or the silvered surface of a collapsing star.The reflected light is fourier analysed by the observer and the power spectrum is constructed. For a mirror moving along the standard black hole trajectory,it is seen that the power spectrum has a ``thermal'' nature. Mirror-observer configurations like an inertial mirror observed in an accelerated observer's frame and an accelerated mirror observed in a Rindler frame are investigated and conditions for obtaining a ``thermal'' power spectrum are derived.The corresponding results in the black hole case are then elucidated.It is seen that a ``thermal'' spectrum can arise either due to the collapse of the star or due to the motion of the observer in the Schwarzchild spacetime. The ``temperature'' of the Planckian spectrum seen is therefore dependent on whether the star or the observer is in motion.In the latter case it is possible to obtain a ``temperature'' which is entirely independent of the mass of the star. 
  We show that the mathematics of Hawking process can be interpreted classically as the Fourier analysis of an exponentially redshifted wave mode which scatters off the black hole and travels to infinity at late times. We use this method to derive the Planckian power spectrum for Schwarzchild, Reissner-Nordstrom and Kerr black holes. 
  We show that for three dimensional space-times admitting a hypersurface orthogonal Killing vector field Deser, Jackiw and Templeton's vacuum field equations of topologically massive gravity allow only the trivial flat space-time solution. Thus spin is necessary to support topological mass. 
  In the framework of real 2-component spinors in three dimensional space-time we present a description of topologically massive gravity (TMG) in terms of differential forms with triad scalar coefficients. This is essentially a real version of the Newman-Penrose formalism in general relativity. A triad formulation of TMG was considered earlier by Hall, Morgan and Perjes, however, due to an unfortunate choice of signature some of the spinors underlying the Hall-Morgan-Perjes formalism are real, while others are pure imaginary. We obtain the basic geometrical identities as well as the TMG field equations including a cosmological constant for the appropriate signature. As an application of this formalism we discuss the Bianchi Type $VIII - IX$ exact solutions of TMG and point out that they are parallelizable manifolds. We also consider various re-identifications of these homogeneous spaces that result in black hole solutions of TMG. 
  In general relativity g_ab;c=0 implies that the wave equation (\Box^2-M)g_ab=0 always has M=0. If the underlying geometry is generalized to include non-metricity this incurs M \neq 0, and the above wave equation can be rewritten as M(x)+\td{\na}_a Q_.^a+(\ep+\fr{d}{2}-2)Q_a Q_.^a=0, where \ep=0, 1, 2, or 3, d is the dimension of the spacetime, and Q is the object of non-metricity. The consequences of this equation and the properties of M are investigated. 
  Observational evidence from variety of sources points at an accelerating universewhich is approximately spatially flat and with $(\Omega_M,\Omega_{\Lambda})_0\approx(0.3,0.7)$. We have shown that for low redshifts, $z\leq 0.2$, the metric of this cosmological model is equivalant to the de Sitter metric within one percent error. Among various coordinate descriptions for the flat de Sitter model two are most widely used, one yields a non-static and FRW type, while the other gives a static and Schwarzschild type metric. We have obtained the magnitude-redshift relation in the second coordinate frame. Our result indicates a slope of 2.5 for small z. This is in disagreement with observation which is close to the slope of 5. this test discards the second and confirms the first coordinate frame as a comoving reference frame for matter in de Sitter space. We anticipate this test disqualifies any other solution of the field equations which asymptotically approaches to the static de Sitter metric, e.g. the Schwarzschild-de Sitter metric for space around a point mass m. 
  We discuss the Carter's formula about the mankind evolution probability following the derivation proposed by Barrow and Tipler. We stress the relation between the existence of billions of galaxies and the evolution of at least one intelligent life, whose living time is not trivial, all over the Universe. We show that the existence probability and the lifetime of a civilization depend not only on the evolutionary critical steps, but also on the number of places where the life can arise. In the light of these results, we propose a stronger version of Anthropic Principle. 
  Using the fact that the null geodesics in NUT space lie on spatial cones, we consider the gravomagnetic lens effect on light rays passing a NUT deflector. We show that this effect changes the observed shape, size and orientation of a source. Compared to the Schwarzschild lens, there is an extra shear (a differential twist around the lens axis) due to the gravomagnetic field which shears the shape of the source. Gravomagnetic monopoles can thus be recognized by the spirality that they produce in the lensing pattern. All the results obtained in this case (magnification factor, orientation of images, multiplicity of images, etc.) depend on $Q$, the strength of the gravomagnetic monopole represented by NUT metric. One recovers the results of the usual Schwarzschild lens effect by putting this factor equal to zero. 
  We develop a correspondence between arbitrary tensors and matrices based on the use of Kronecker products and associated identities.   Utilizing the rules of matrix differentiation we derive the vacuum Einstein field equations as a differential-matrix equation. This formulation may facilitate their efficient use in numerical relativistic models. 
  With the basic cosmological relations that agree with the recent observations, simple expressions are suggested concerning the value of cosmological constant($\Lambda$). A large contribution of quantum vacuum to the energy momentum tensor does not agree with the observed cosmos. However,one requires the presence of positive $\Lambda$ to make the various observations consistent. After a review of the effect of cosmological constant on the geodetic motions in the Schwarzschild de Sitter spacetime, some approaches to its solutions are briefly discussed. Also suggested is the very weak limit on $\Lambda$ from the planetary perturbations. 
  Multidimensional model describing the "cosmological" and/or spherically symmetric configuration with n+1 Einstein spaces in the theory with several scalar fields and forms is considered. When electro-magnetic composite p-brane ansatz is adopted, n "internal" spaces are Ricci-flat, one space M_0 has a non-zero curvature, and all p-branes do not "live" in M_0, a class of exact solutions is obtained if certain block-orthogonality relations on p-brane vectors are imposed. A subclass of spherically-symmetric solutions containing non-extremal p-brane black holes is considered. Post-Newtonian parameters are calculated and some examples are considered. 
  A self-consistent general relativistic configuration describing a finite cross-section magnetic flux tube is constructed. The cosmic solenoid is modeled by an elastic superconductive surface which separates the Melvin core from the surrounding flat conic structure. We show that a given amount $\Phi$ of magnetic flux cannot be confined within a cosmic solenoid of circumferential radius smaller than $\frac{\sqrt{3G}}{2\pi c^2}\Phi$ without creating a conic singularity. Gauss-Codazzi matching conditions are derived by means of a self-consistent action. The source term, representing the surface currents, is sandwiched between internal and external gravitational surface terms. Surface superconductivity is realized by means of a Higgs scalar minimally coupled to projective electromagnetism. Trading the 'magnetic' London phase for a dual 'electric' surface vector potential, the generalized quantization condition reads: $e/{hc} \Phi + 1/e Q=n$ with $Q$ denoting some dual 'electric' charge, thereby allowing for a non-trivial Aharonov-Bohm effect. Our conclusions persist for dilaton gravity provided the dilaton coupling is sub-critical. 
  Within the mini-superspace model, brane-like cosmology means performing the variation with respect to the embedding (Minkowski) time $\tau$ before fixing the cosmic (Einstein) time $t$. The departure from Einstein limit is parameterized by the 'energy' conjugate to $\tau$, and characterized by a classically disconnected Embryonic epoch. In contrast with canonical quantum gravity, the wave-function of the brane-like Universe is (i) $\tau$-dependent, and (ii) vanishes at the Big Bang. Hartle-Hawking and Linde proposals dictate discrete 'energy' levels, whereas Vilenkin proposal resembles $\alpha$-particle disintegration. 
  Quantum gravity of a brane-like Universe is formulated, and its Einstein limit is approached. Regge-Teitelboim embedding of Arnowitt-Deser-Misner formalism is carried out. Invoking a novel Lagrange multiplier, accompanying the lapse function and the shift vector, we derive the quadratic Hamiltonian and the corresponding bifurcated Wheeler-Dewitt-like equation. The inclusion of arbitrary matter resembles minimal coupling. 
  The geometrization of electrodynamics is obtained by performing the complex extension of the covariant derivative operator to include the Cartan torsion vector and applying this derivative to the Ginzburg-Landau equation of superfluids and Superconductors.It is shown that the introduction of torsion makes a shift in the symmetry breaking vacuum.Torsion loops are computed from geometrical phases outside the superconductor.Inside the superconductor the torsion vanishes which represents the Meissner effect for torsion geometry. Torsion in general equals the London supercurrent.It is possible to place a limit on the size of superconductor needed to give an estimate to torsion. 
  We analyze the scalar radiation emitted from a source rotating around a Schwarzschild black hole using the framework of quantum field theory at the tree level. We show that for relativistic circular orbits the emitted power is about 20% to 30% smaller than what would be obtained in Minkowski spacetime. We also show that most of the emitted energy escapes to infinity. Our formalism can readily be adapted to investigate similar processes. 
  We present some generalizations, and novel properties, of the Bel-Robinson tensor, in the context of constructing local invariants in D=11 supergravity. 
  We investigate the decay of accelerated protons and neutrons. Calculations are carried out in the inertial and coaccelerated frames. Particle interpretation of these processes are quite different in each frame but the decay rates are verified to agree in both cases. For sake of simplicity our calculations are performed in a two-dimensional spacetime since our conclusions are not conceptually affected by this. 
  In this paper we continue to study a class of four-dimensional gravity models with n Abelian vector fields and Sp(2n)/U(n) coset of scalar fields. This class contains General Relativity (n=0) and Einstein-Maxwell dilaton-axion theory (n=1), which arizes in the low-energy limit of heterotic string theory. We perform reduction of the model with arbitrary $n$ to three dimensions and study the subgroup of non-gauge symmetries of the resulting theory. First, we find an explicit form these symmetries using Ernst matrix potential formulation. Second, we construct new matrix variable which linearly transforms under the action of the non-gauge transformations. Finally, we establish one general invariant of the non-gauge symmetry subgroup, which allow us to clarify this subgroup structure. 
  In these notes we prepare the ground for a systematic investigation into the issues of black hole fluctuations and backreaction by discussing the formulation of the problem, commenting on possible advantages and shortcomings of existing works, and introducing our own approach via a stochastic semiclassical theory of gravity based on the Einstein-Langevin equation and the fluctuation-dissipation relation for a self-consistent description of metric fluctuations and dissipative dynamics of the black hole with backreaction of its Hawking radiance. 
  Using the 1+3 formulation of stationary spacetimes we show, in the context of gravoelectromagnetism, that the plane of the polarization of light rays passing close to a black hole undergoes a rotation. We show that this rotation has the same integral form as the usual Faraday effect, i.e. it is proportional to the integral of the component of the gravomagnetic field along the propagation path. We apply this integral formula to calculate the Faraday rotation induced by the Kerr and NUT spaces using the quasi-Maxwell form of the vacuum Einstein equations. 
  This paper deals with a study of the effects that spherically symmetric first-order metric perturbations and vacuum quantum fluctuations have on the stability of the multiply connected de Sitter spacetime recently proposed by Gott and Li. It is the main conclusion of this study that although such a spacetime is stable to the classical metric perturbations for any size of the nonchronal region, it is only stable against the quantum fluctuations of vacuum if the size of the multiply connected region is of the order the Planck scale. Therefore, boundary conditions for the state of the universe based on the notion that the universe created itself in a regime where closed timelike curves were operative and stable, still appear to be physically and philosophically so well supported as are those boundary conditions relying on the notion that the universe was created out of nothing. 
  We develop the analytic and numerical tools for data analysis of the gravitational-wave signals from spinning neutron stars for ground-based laser interferometric detectors. We study in detail the statistical properties of the optimum functional that need to be calculated in order to detect the gravitational-wave signal from a spinning neutron star and estimate its parameters. We derive formulae for false alarm and detection probabilities both for the optimal and the suboptimal filters. We assess the computational requirements needed to do the signal search. We compare a number of criteria to build sufficiently accurate templates for our data analysis scheme. We verify the validity of our concepts and formulae by means of the Monte Carlo simulations. We present algorithms by which one can estimate the parameters of the continuous signals accurately. 
  We investigate the stability of cosmological scaling solutions describing a barotropic fluid with $p=(\gamma-1)\rho$ and a non-interacting scalar field $\phi$ with an exponential potential $V(\phi)=V_0\e^{-\kappa\phi}$. We study homogeneous and isotropic spacetimes with non-zero spatial curvature and find three possible asymptotic future attractors in an ever-expanding universe. One is the zero-curvature power-law inflation solution where $\Omega_\phi=1$ ($\gamma<2/3,\kappa^2<3\gamma$ and $\gamma>2/3,\kappa^2<2$). Another is the zero-curvature scaling solution, first identified by Wetterich, where the energy density of the scalar field is proportional to that of matter with $\Omega_\phi=3\gamma/\kappa^2$ ($\gamma<2/3,\kappa^2>3\gamma$). We find that this matter scaling solution is unstable to curvature perturbations for $\gamma>2/3$. The third possible future asymptotic attractor is a solution with negative spatial curvature where the scalar field energy density remains proportional to the curvature with $\Omega_\phi=2/\kappa^2$ ($\gamma>2/3,\kappa^2>2$). We find that solutions with $\Omega_\phi=0$ are never late-time attractors. 
  In the canonical quantization of gravity in terms of the Ashtekar variables one uses paths in the 3-space to construct the quantum states. Usually, one restricts oneself to families of paths admitting only finite number of isolated intersections. This assumption implies a limitation on the diffeomorphisms invariance of the introduced structures. In this work, using the previous results of Baez and Sawin, we extend the existing results to a theory admitting all the possible piecewise smooth finite paths and loops. In particular, we $(i)$ characterize the spectrum of the Ashtekar-Isham configuration space, $(ii)$ introduce spin-web states, a generalization of the spin-network states, $(iii)$ extend the diffeomorphism averaging to the spin-web states and derive a large class of diffeomorphism invariant states and finally $(iv)$ extend the 3-geometry operators and the Hamiltonian operator. 
  Misner space is generalized to have the nonorientable topology of a Klein bottle, and it is shown that in a classical spacetime with multiply connected space slices having such a topology, closed timelike curves are formed. Different regions on the Klein bottle surface can be distinguished which are separated by apparent horizons fixed at particular values of the two angular variables that eneter the metric. Around the throat of this tunnel (which we denote a Klein bottlehole), the position of these horizons dictates an ordinary and exotic matter distribution such that, in addition to the known diverging lensing action of wormholes, a converging lensing action is also present at the mouths. Associated with this matter distribution, the accelerating version of this Klein bottlehole shows four distinct chronology horizons, each with its own nonchronal region. A calculation of the quantum vacuum fluctuations performed by using the regularized two-point Hadamard function shows that each chronology horizon nests a set of polarized hypersurfaces where the renormalized momentum-energy tensor diverges. This quantum instability can be prevented if we take the accelerating Klein bottlehole to be a generalization of a modified Misner space in which the period of the closed spatial direction is time-dependent. In this case, the nonchronal regions and closed timelike curves cannot exceed a minimum size of the order the Planck scale. 
  Realizations of scale invariance are studied in the context of a gravitational theory where the action (in the first order formalism) is of the form $S = \int L_{1} \Phi d^{4}x$ + $\int L_{2}\sqrt{-g}d^{4}x$ where $\Phi$ is a density built out of degrees of freedom, the "measure fields" independent of $g_{\mu\nu}$ and matter fields appearing in $L_{1}$, $L_{2}$. If $L_{1}$ contains the curvature, scalar potential $V(\phi)$ and kinetic term for $\phi$, $L_{2}$ another potential for $\phi$, $U(\phi)$, then the true vacuum state has zero energy density, when theory is analyzed in the conformal Einstein frame (CEF), where the equations assume the Einstein form. Global Scale invariance is realized when $V(\phi)$ = $f_{1}e^{\alpha\phi}$ and $U(\phi)$ = $f_{2}e^{2\alpha\phi}$. In the CEF the scalar field potential energy $V_{eff}(\phi)$ has in, addition to a minimum at zero, a flat region for $\alpha\phi \to\infty$, with non zero vacuum energy, which is suitable for either a New Inflationary scenario for the Early Universe or for a slowly rolling decaying $\Lambda$-scenario for the late universe, where the smallness of the vacuum energy can be understood as a kind of see-saw mechanism. 
  Robinson-Trautman radiative space-times of Petrov type II with a non-vanishing cosmological constant Lambda and mass parameter m>0 are studied using analytical methods. They are shown to approach the corresponding spherically symmetric Schwarzschild-de Sitter or Schwarzschild-anti-de Sitter solution at large retarded times. Their global structure is analyzed, and it is demonstrated that the smoothness of the extension of the metrics across the horizon, as compared with the case Lambda=0, is increased for Lambda>0 and decreased for Lambda<0. For the extreme value 9Lambda m^2=1, the extension is smooth but not analytic. This case appears to be the first example of a smooth but not analytic horizon. The models with Lambda>0 exhibit explicitly the cosmic no-hair conjecture under the presence of gravitational waves. 
  We define super-energy tensors for arbitrary physical fields, including the gravitational, electromagnetic and massless scalar fields. We also define super-super-energy tensors, and so on. All these tensors satisfy the so-called "Dominant Superenergy Property" among other interesting and good properties. The possibility of interchange of superenergy between gravitational and other fields is considered. 
  Although wormholes can be treated as topological objects in spacetime and from a global point-of-view, a precise definition of what a wormhole throat is and where it can be located can be developed and treated entirely in terms of local geometry. This has the advantage of being free from unnecesary technical assumptions about asymptotic flatness, and other global properties of the spacetime containing the wormhole. We discuss our recent work proving that the violation of the null energy condition (NEC) is a generic feature of all wormholes, whether they be time-dependent or static, and demonstrate that time-dependent wormholes have two throats, one for each direction through the wormhole, which coalesce only in the static limit. 
  Einstein's equations for general relativity, when viewed as a dynamical system for evolving initial data, have a serious flaw: they cannot be proven to be well-posed (except in special coordinates). That is, they do not produce unique solutions that depend smoothly on the initial data. To remedy this failing, there has been widespread interest recently in reformulating Einstein's theory as a hyperbolic system of differential equations. The physical and geometrical content of the original theory remain unchanged, but dynamical evolution is made sound. Here we present a new hyperbolic formulation in terms of $g_{ij}$, $K_{ij}$, and $\bGam_{kij}$ that is strikingly close to the space-plus-time (``3+1'') form of Einstein's original equations. Indeed, the familiarity of its constituents make the existence of this formulation all the more unexpected. This is the most economical first-order symmetrizable hyperbolic formulation presently known to us that has only physical characteristic speeds, either zero or the speed of light, for all (non-matter) variables. This system clarifies the relationships between Einstein's original equations and the Einstein-Ricci and Frittelli-Reula hyperbolic formulations of general relativity and establishes links to other hyperbolic formulations. 
  Recent improvements in astronomical observations lead to the conclusion that the Hubble constant lies between 60 and 80 Mpc km$^{-1}$ sec$^{-1}$ and the age of the universe between 11 and 14 Gigayears. Taken together with recent observations of distant type Ia supernovae and the cosmic background radiation, these limits allow a check of the consequences of predictions made a decade ago using program universe and the combinatorial hierarchy that the ratio of baryons to photons is $1/256^4$ and of dark to baryonic matter is 12.7. We find that the restrictions on the matter content of the universe and the cosmological constant are within, and much tighter than, the limits established by conventional means. The situation is further improved if we invoke an estimate of the normalized cosmological constant made by E. D. Jones of $\Omega_{\Lambda} \sim 0.6$. This opens a ``window of opportunity'' to get the predictions of the ANPA program in front of the relevant professional community {\it before} precise observations lead to a consensus. We urge ANPA members to join us in the assault on this breach in the walls of establishment thinking. 
  Over the past six years, a detailed framework has been constructed to unravel the quantum nature of the Riemannian geometry of physical space. A review of these developments is presented at a level which should be accessible to graduate students in physics. As an illustrative application, I indicate how some of the detailed features of the micro-structure of geometry can be tested using black hole thermodynamics. Current and future directions of research in this area are discussed. 
  We discuss from a philosophical perspective the way in which the normal concept of time might be said to `emerge' in a quantum theory of gravity. After an introduction, we briefly discuss the notion of emergence, without regard to time (Section 2). We then introduce the search for a quantum theory of gravity (Section 3); and review some general interpretative issues about space, time and matter Section 4). We then discuss the emergence of time in simple quantum geometrodynamics, and in the Euclidean approach (Section 5). Section 6 concludes. 
  A thought experiment is proposed to unify quantum mechanics and general relativity. The central paradigm is that space-time {\it topology} is ultimately responsible for the Heisenberg uncertaintly principle. It is found that Plankian space-time exhibits a complicated, but also definite, multiply connected character. In this framework, an analysis of the interactions in Nature is presented.   I. The Universal ground state of the constructed theory derives from the properties of the topological manifold $Q=2T^3\oplus 3S^1\times S^2$, which has 23 intrinsic degrees of freedom, discrete $Z_3$ and $Z_2\times Z_3$ internal groups, an SU(5) gauge group, and leads to a U(1) symmetry on a lattice. The structure of $Q$ provides a unique equation motion for the mass-energy and particle rest mass wave functions. In its excited state the Universe is characterized by a lattice of three-tori, $L(T^3)$. The topological identifications present in this structure, a direct reflection of the Heisenberg uncertainty principle, provide the boundary conditions for solutions to the equation of motion, and suggest an interpretation for the conceptually difficult concept of quantum mechanical entanglement. II. In the second half of the paper the (observable) properties of $Q$ and $L(T^3)$ are investigated. One reproduces the standard model, and the theory naturally contains a Higgs field with possible inflation. The electron and its neutrino are identified as particle ground states and their masses, together with those of all other known particles, are predicted. A mass of $m_{\rm H}=131.6$ GeV is found for the Higgs boson. [Abridged] 
  In this talk I introduce the critical behavior occurring at the extremal limit of black holes. The extremal limit of black holes is a critical point and a phase transition takes place from the extremal black holes to their nonextremal counterparts. Some critical exponents satisfying the scaling laws are obtained. From the scaling laws we introduce the concept of the effective dimension of black holes and discuss the relationship between the critical behavior and the statistical interpretation of black hole entropy. 
  We study the geodesic motions of a test particle around 2+1 dimensional charged black holes. We obtain a class of exact geodesic motions for the massless test particle when the ratio of its energy and angular momentum is given by square root of cosmological constant. The other geodesic motions for both massless and massive test particles are analyzed by use of numerical method. 
  Continuously self-similar solution of spherically symmetric gravitational collapse of a scalar field is studied to investigate quantum mechanical black hole formation by tunneling in the subcritical case where, classically, the collapse does not produce a black hole. 
  A rigidity theorem that applies to smooth electrovac spacetimes which represent either (A) an asymptotically flat stationary black hole or (B) a cosmological spacetime with a compact Cauchy horizon ruled by closed null geodesics was given in a recent work \cite{frw}. Here we enlarge the framework of the corresponding investigations by allowing the presence of other type of matter fields. In the first part the matter fields are involved merely implicitly via the assumption that the dominant energy condition is satisfied. In the second part Einstein-Klein-Gordon (EKG), Einstein-[non-Abelian] Higgs (E[nA]H), Einstein-[Maxwell]-Yang-Mills-dilaton (E[M]YMd) and Einstein-Yang-Mills-Higgs (EYMH) systems are studied. The black hole event horizon or, respectively, the compact Cauchy horizon of the considered spacetimes is assumed to be a smooth non-degenerate null hypersurface. It is proven that there exists a Killing vector field in a one-sided neighborhood of the horizon in EKG, E[nA]H, E[M]YMd and EYMH spacetimes. This Killing vector field is normal to the horizon, moreover, the associated matter fields are also shown to be invariant with respect to it. The presented results provide generalizations of the rigidity theorems of Hawking (for case A) and of Moncrief and Isenberg (for case B) and, in turn, they strengthen the validity of both the black hole rigidity scenario and the strong cosmic censor conjecture of classical general relativity. 
  A massless electroweak theory for leptons is formulated in a Weyl space, W_4, yielding a Weyl invariant gauge dynamics allowing for conformal rescalings of the metric and all fields with nonvanishing Weyl weight together with the corresponding transformations of the Weyl vector fields representing the D(1) or dilatation gauge fields. To study the appearance of nonzero masses this theory is explicitly broken by a term in the Lagrangean involving the curvature scalar R of the W_4 and a mass term for the scalar field. Thereby also the gauge fields as well as the charged fermion field acquire a mass as in the standard electroweak theory. The symmetry breaking is governed by the relation D Phi^2=0, where Phi is the modulus of the scalar field and D denotes the Weyl-covariant derivative. This true symmetry reduction, establishing a scale of length in the theory, is compared to the so-called spontanous symmetry breaking in the standard electroweak theory which is, actually, the choice of a particular (nonlinear) gauge obtained by adopting an origin in the coset space representing the scalar field which is invariant under the electromagnetic gauge group. Particular attention is devoted to the appearance of Einstein's equations for the metric after the Weyl-symmetry breaking yielding a pseudo-Riemannian space V_4 from a W_4 and a scalar field with a constant modulus which in turn affects Einstein's gravitational constant in a manner comparable to the Brans-Dicke theory. 
  We consider the self-similar solutions associated with the critical behavior observed in the gravitational collapse of spherically symmetric perfect fluids with equation of state $p=\alpha\mu$. We identify for the first time the global nature of these solutions and show that it is sensitive to the value of $\alpha$. In particular, for $\alpha>0.28$, we show that the critical solution is associated with a new class of asymptotically Minkowski self-similar spacetimes. We discuss some of the implications of this for critical phenomena. 
  In a gedanken experiment in which a box initially containing energy $E$ and entropy $S$ is lowered toward a black hole and then dropped in, it was shown by Unruh and Wald that the generalized second law of black hole thermodynamics holds, without the need to assume any bounds on $S$ other than the bound that arises from the fact that entropy at a given energy and volume is bounded by that of unconstrained thermal matter. The original analysis by Unruh and Wald made the approximation that the box was ``thin'', but they later generalized their analysis to thick boxes (in the context of a slightly different process). Nevertheless, Bekenstein has argued that, for a certain class of thick boxes, the buoyancy force of the ``thermal atmosphere'' of the black hole is negligible, and that his previously postulated bound on $S/E$ is necessary for the validity of the generalized second law. In arguing for these conclusions, Bekenstein made some assumptions about the nature of unconstrained thermal matter and the location of the ``floating point'' of the box. We show here that under these assumptions, Bekenstein's bound on $S/E$ follows automatically from the fact that $S$ is bounded by the entropy of unconstrained thermal matter. Thus, a box of matter which violates Bekenstein's bound would violate the assumptions made in his analysis, rather than violate the generalized second law. Indeed, we prove here that no universal entropy bound need be hypothesized in order to ensure the validity of the generalized second law in this process. 
  During the past 30 years, research in general relativity has brought to light strong hints of a very deep and fundamental relationship between gravitation, thermodynamics, and quantum theory. The most striking indication of such a relationship comes from black hole thermodynamics, where it appears that certain laws of black hole mechanics are, in fact, simply the ordinary laws of thermodynamics applied to a system containing a black hole. This article will review the present status of black hole thermodynamics and will discuss some of the related unresolved issues concerning gravitation, thermodynamics, and quantum theory. 
  We have extended the Vaidya radiating metric to include both a radiation fluid and a string fluid. This paper expands our brief introduction to extensions of the Schwarzschild vacuum which appeared in 1998 Phys. Rev. D Vol 57, R5945. Assuming diffusive transport for the string fluid, we find new analytic solutions of Einstein's field equations. 
  We conjecture a universal upper bound to the entropy of a rotating system. The entropy bound follows from application of the generalized second law of thermodynamics to an idealized gedanken experiment in which an entropy-bearing rotating system falls into a black hole. This bound is stronger than the Bekenstein entropy bound for non-rotating systems. 
  An Euclidean approach for investigating quantum aspects of a scalar field living on a class of D-dimensional static black hole space-times, including the extremal ones, is reviewed. The method makes use of a near horizon approximation of the metric and $\zeta$-function formalism for evaluating the partition function and the expectation value of the field fluctuations $<\phi^2(x)>$. After a review of the non-extreme black hole case, the extreme one is considered in some details. In this case, there is no conical singularity, but the finite imaginary time compactification introduces a cusp singularity. It is found that the $\zeta$-function regularized partition function can be defined, and the quantum fluctuations are finite on the horizon, as soon as the cusp singularity is absent, and the corresponding temperature is T=0. 
  We discuss the dependence of the pulsation frequencies of the axial quasi-normal modes of a nonrotating neutron star upon the equation of state describing the star interior. The continued fraction method has been used to compute the complex frequencies for a set of equations of state based on different physical assumptions and spanning a wide range of stiffness. The numerical results show that the detection of axial gravitational waves would allow to discriminate between the models underlying the different equation of states, thus providing relevant information on both the structure of neutron star matter and the nature of the hadronic interactions. 
  We review the renormalization of one-loop effective action for gravity coupled to a scalar field and that of the Bekenstein-Hawking entropy of a black hole plus the statistical entropy of the scalar field. It is found that the total entropy of the black hole's geometric entropy and the statistical entropy yields the renormalized Bekenstein-Hawking area-law of black hole entropy only for even dimensional Reissner-N\"{o}rdstrom (Schwarzschild) black holes. We discuss the problem of the microscopic origin of black hole entropy in connection with the renormalization of black hole entropy. 
  We continue our study of the gravitational collapse of spherically symmetric skyrmions. For certain families of initial data, we find the discretely self-similar Type II critical transition characterized by the mass scaling exponent $\gamma \approx 0.20$ and the echoing period $\Delta \approx 0.74$. We argue that the coincidence of these critical exponents with those found previously in the Einstein-Yang-Mills model is not accidental but, in fact, the two models belong to the same universality class. 
  Using a metric perturbation method, we study gravitational waves from a test particle scattered by a spherically symmetric relativistic star. We calculate the energy spectrum and the waveform of gravitational waves for axial modes. Since metric perturbations in axial modes do not couple to the matter fluid of the star, emitted waves for a normal neutron star show only one peak in the spectrum, which corresponds to the orbital frequency at the turning point, where the gravitational field is strongest. However, for an ultracompact star (the radius $R \lesssim 3M$), another type of resonant periodic peak appears in the spectrum. This is just because of an excitation by a scattered particle of axial quasinormal modes, which were found by Chandrasekhar and Ferrari. This excitation comes from the existence of the potential minimum inside of a star. We also find for an ultracompact star many small periodic peaks at the frequency region beyond the maximum of the potential, which would be due to a resonance of two waves reflected by two potential barriers (Regge-Wheeler type and one at the center of the star). Such resonant peaks appear neither for a normal neutron star nor for a Schwarzschild black hole. Consequently, even if we analyze the energy spectrum of gravitational waves only for axial modes, it would be possible to distinguish between an ultracompact star and a normal neutron star (or a Schwarzschild black hole). 
  Spontaneous breaking of Lorentz invariance may take place in string theories, possibly endowing the photon with a mass. This leads to the breaking of the conformal symmetry of the electromagnetic action allowing for the generation within inflationary scenarios of magnetic fields over $Mpc$ scales. We show that the generated fields are consistent with amplification by the galactic dynamo processes and can be as large as to explain the observed galactic magnetic fields through the collapse of protogalactic clouds. 
  Stability and characterisitic geometrical and kinematical sizes of galaxies are strictly related to a minimal characteristic action whose value is of order $h$, the Planck constant. We infer that quantum mechanics, in some sense, determines the structure and the size of galaxies. 
  The goal of the present paper is to investigate four dimensional Kahler manifolds admitting H-projective mappings with special attention to Einstein-Kahler manifolds of this type which can be interpreted as field configurations of the gravitational instantons. 
  Critical collapse in tensor-multi-scalar gravity theories is studied, and found that for any given target space all the theories conformally related belong to the same universal class. When only one scalar field is present, the universality is extended to include a class of non-linear gravity theories. 
  All differences between the role of space and time in nature are explained by proposing the principles in which none of the space-time coordinates has an {\it a priori} special role. Spacetime is treated as a nondynamical manifold, with a fixed global ${\bf R}^D$ topology. Dynamical theory of gravity determines only the metric tensor on a fixed manifold. All dynamics is treated as a Cauchy problem, so it {\em follows} that one coordinate takes a special role. It is proposed that {\em any} boundary condition that is finite everywhere leads to a solution which is also finite everywhere. This explains the $(1,D-1)$ signature of the metric, the boundedness of energy from below, the absence of tachyons, and other related properties of nature. The time arrow is explained by proposing that the boundary condition should be ordered. The quantization is considered as a boundary condition for field operators. Only the physical degrees of freedom are quantized. 
  The quantization of gravity, and its unification with the other interactions, is one of the greatest challenges of theoretical physics. Current ideas suggest that the value of G might be related to the other fundamental constants of physics, and that gravity might be richer than the standard Newton-Einstein description. This gives added significance to measurements of G and to Cavendish-type experiments. 
  Acoustic propagation in a moving fluid provides a conceptually clean and powerful analogy for understanding black hole physics. As a teaching tool, the analogy is useful for introducing students to both General Relativity and fluid mechanics. As a research tool, the analogy helps clarify what aspects of the physics are kinematics and what aspects are dynamics. In particular, Hawking radiation is a purely kinematical effect, whereas black hole entropy is intrinsically dynamical. Finally, I discuss the fact that with present technology acoustic Hawking radiation is almost experimentally testable. 
  The first post-Newtonian (PN) hydrostatic equations for an irrotational fluid are solved for an incompressible binary system. The equilibrium configuration of the binary system is given by a small deformation from the irrotational Darwin-Riemann ellipsoid which is the solution at Newtonian order. It is found that the orbital separation at the innermost stable circular orbit (ISCO) decreases when one increases the compactness parameter $M_{\ast}/c^2 a_{\ast}$, in which $M_{\ast}$ and $a_{\ast}$ denote the mass and the radius of a star, respectively. If we compare the 1PN angular velocity of the binary system at the ISCO in units of $\sqrt{M_{\ast}/a_{\ast}^3}$ with that of Newtonian order, the angular velocity at the ISCO is almost the same value as that at Newtonian order when one increases the compactness parameter. Also, we do not find the instability point driven by the deformation at 1PN order, where a new sequence bifurcates throughout the equilibrium sequence of the binary system until the ISCO.   We also investigate the validity of an ellipsoidal approximation, in which a 1PN solution is obtained assuming an ellipsoidal figure and neglecting the deformation. It is found that the ellipsoidal approximation gives a fairly accurate result for the total energy, total angular momentum and angular velocity. However, if we neglect the velocity potential of 1PN order, we tend to overestimate the angular velocity at the ISCO regardless of the shape of the star (ellipsoidal figure or deformed figure). 
  From a recent study of a stationary cylindrical solution for a relativistic two-constituent superfluid at low temperature limit, we propose to specify this solution under the form of a relativistic generalisation of a Rankine vortex (Potential vortex whose the core has a solid body rotation).Then we establish the dynamics of the central line of this vortex by supposing that the deviation from the cylindrical configuration is weak in the neighbourhood of the core of the vortex. In "stiff" material the Nambu-Goto equations are obtained. 
  We classify all spherically symmetric perfect fluid solutions of Einstein's equations with equation of state p/mu=a which are self-similar in the sense that all dimensionless variables depend only upon z=r/t. For a given value of a, such solutions are described by two parameters and they can be classified in terms of their behaviour at large and small distances from the origin; this usually corresponds to large and small values of z but (due to a coordinate anomaly) it may also correspond to finite z. We base our analysis on the demonstration that all similarity solutions must be asymptotic to solutions which depend on either powers of z or powers of lnz. We show that there are only three similarity solutions which have an exact power-law dependence on z: the flat Friedmann solution, a static solution and a Kantowski-Sachs solution (although the latter is probably only physical for a<0). For a>1/5, there are also two families of solutions which are asymptotically (but not exactly) Minkowski: the first is asymptotically Minkowski as z tends to infinity and is described by one parameter; the second is asymptotically Minkowski at a finite value of z and is described by two parameters. A complete analysis of the dust solutions is given, since these can be written down explicitly and elucidate the link between the z>0 and z<0 solutions. Solutions with pressure are then discussed in detail; these share many of the characteristics of the dust solutions but they also exhibit new features. 
  We show that the existence of the cosmological constant can be connected to a nonminimal derivative coupling, in the action of gravity, between the geometry and the kinetic part of a given scalar field without introducing any effective potential of scalar fields. Exact solutions are given. 
  We study maximally symmetric cosmological solutions of type II supersymmetric strings in the presence of the exact quartic curvature corrections to the lowest order effective action, including loop and D-instanton effects. We find that, unlike the case of type IIA theories, de Sitter solutions exist for type IIB superstrings, a conclusion that remains valid when higher-curvature corrections are included on the basis of SL(2,Z) invariance. 
  Hyperbolic formulations of the equations of motion are essential technique for proving the well-posedness of the Cauchy problem of a system, and are also helpful for implementing stable long time evolution in numerical applications. We, here, present three kinds of hyperbolic systems in the Ashtekar formulation of general relativity for Lorentzian vacuum spacetime. We exhibit several (I) weakly hyperbolic, (II) diagonalizable hyperbolic, and (III) symmetric hyperbolic systems, with each their eigenvalues. We demonstrate that Ashtekar's original equations form a weakly hyperbolic system. We discuss how gauge conditions and reality conditions are constrained during each step toward constructing a symmetric hyperbolic system. 
  We describe a simple method of determining whether the singularity that forms in the spherically symmetric collapse of inhomogeneous dust is naked or covered. This derivation considerably simplifies the analysis given in the earlier literature, while giving the same results as have been obtained before. 
  We develop the formalism for the one-loop no-boundary state in a cosmological model with fermions. We use it to calculate the reduced density matrix for an inflaton field by tracing out the fermionic degrees of freedom, yielding both the fermionic effective action and the standard decoherence factor. We show that dimensional regularisation of ultraviolet divergences would lead to an inconsistent density matrix. Suppression of these divergences to zero is instead performed through a nonlocal Bogoliubov transformation of the fermionic variables, which leads to a consistent density matrix. The resulting degree of decoherence is less than in the case of bosonic fields. 
  A code that implements Einstein equations in the characteristic formulation in 3D has been developed and thoroughly tested for the vacuum case. Here, we describe how to incorporate matter, in the form of a perfect fluid, into the code. The extended code has been written and validated in a number of cases. It is stable and capable of contributing towards an understanding of a number of problems in black hole astrophysics. 
  In a nongeometrical interpretation of gravity, the metric $g_{\mu\nu}(x)=\eta_{\mu\nu}+\Phi_{\mu\nu}(x)$ is interpreted as an {\em effective} metric, whereas $\Phi_{\mu\nu}(x)$ is interpreted as a fundamental gravitational field, propagated in spacetime which is actually flat. Some advantages and disadvantages of such an interpretation are discussed. The main advantage is a natural resolution of the flatness problem. 
  Some properties of an exact solution due to Vaidya, describing the gravitational field produced by a point particle in the background of the static Einstein universe are examined. The maximal analytic extension and the nature of the singularities of the model are discussed. By using the Euclidean approach, some quantum aspects are analysed and the thermodynamics of this spacetime is also discussed. 
  We study the cosmology of the Brans-Dicke theory with perfect fluid type matter. In our previous work, we found exact solutions for any Brans-Dicke parameter $\omega$ and for general parameter $\gamma$ of equation of state. In this paper we further study the cosmology of these solutions by analyzing them according to their asymptotic behaviors. The cosmology is classified into 19 phases according to the values of $\gamma$ and $\omega$. The effect of the cosmological constant to the Brans-Dicke theory is a particular case of our model. We give plot of time evolution of the scale factor by numerical investigations. We also give a comparison of the solutions for the theories with and without matter. 
  We compute the energy spectra of the gravitational signals emitted when a mass m is scattered by the gravitational field of a star of mass M >> m. We show that, unlike black holes in similar processes, the quasi-normal modes of the star are excited, and that the amount of energy emitted in these modes depends on how close the exciting mass can get to the star. 
  In this paper time machines are constructed from anti-de Sitter space. One is constructed by identifying points related via boost transformations in the covering space of anti-de Sitter space and it is shown that this Misner-like anti-de Sitter space is just the Lorentzian section of the complex space constructed by Li, Xu, and Liu in 1993. The others are constructed by gluing an anti-de Sitter space to a de Sitter space, which could describe an anti-de Sitter phase bubble living in a de Sitter phase universe. Self-consistent vacua for a massless conformally coupled scalar field are found for these time machines, whose renormalized stress-energy tensors are finite and solve the semi-classical Einstein equations. The extensions to electromagnetic fields and massless neutrinos are discussed. It is argued that, in order to make the results consistent with Euclidean quantization, a new renormalization procedure for quantum fields in Misner-type spaces (Misner space, Misner-like de Sitter space, and Misner-like anti-de Sitter space) is required. Such a "self-consistent" renormalization procedure is proposed. With this renormalization procedure, self-consistent vacua exist for massless conformally coupling scalar fields, electromagnetic fields, and massless neutrinos in these Misner-type spaces. 
  Families of anisotropic and inhomogeneous string cosmologies containing non-trivial dilaton and axion fields are derived by applying the global symmetries of the string effective action to a generalized Einstein-Rosen metric. The models exhibit a two-dimensional group of Abelian isometries. In particular, two classes of exact solutions are found that represent inhomogeneous generalizations of the Bianchi type VI_h cosmology. The asymptotic behaviour of the solutions is investigated and further applications are briefly discussed. 
  A change of spatial topology in a causal, compact spacetime cannot occur when the metric is globally Lorentzian. One can however construct a causal metric from a Riemannian metric and a Morse function on the background cobordism manifold, which is Lorentzian almost everywhere except that it is degenerate at each critical point of the function. We investigate causal structure in the neighbourhood of such a degeneracy, when the auxiliary Riemannian metric is taken to be Cartesian flat in appropriate coordinates. For these geometries, we verify Borde and Sorkin's conjecture that causal discontinuity occurs if and only if the Morse index is 1 or n-1. 
  A free system, considered to be a comparison system, allows for the notion of objective existence and inertial frame. Transformations connecting inertial frames are shown to be either Lorentz or generalized Galilei. 
  A stability analysis is made for a non-singular pre-big-bang like cosmological model based on 1-loop corrected string effective action. Its homogeneous and isotropic solution realizes non-singular transition from de Sitter universe to Friedmann-like universe, via super inflation phase. We are interested in whether the non-singular nature of the solution would be stable or not in more general inhomogeneous case. Perturbative analysis is made for scalar, vector, and tensor linear perturbations, and instability is found for tensor-type perturbation. 
  We consider a model of an elementary particle as a 2 + 1 dimensional brane evolving in a 3 + 1 dimensional space. Introducing gauge fields that live in the brane as well as normal surface tension can lead to a stable "elementary particle" configuration. Considering the possibility of non vanishing vacuum energy inside the bubble leads, when gravitational effects are considered, to the possibility of a quantum decay of such "elementary particle" into an infinite universe. Some remarkable features of the quantum mechanics of this process are discussed, in particular the relation between possible boundary conditions and the question of instability towards Universe formation is analyzed. 
  The possibility of mass in the context of scale-invariant, generally covariant theories, is discussed. The realizations of scale invariance which are considered, are in the context of a gravitational theory where the action, in the first order formalism, is of the form $S = \int L_{1} \Phi d^4x$ + $\int L_{2}\sqrt{-g}d^4x$ where $\Phi$ is a density built out of degrees of freedom independent of gravity, which we call the "measure fields". For global scale invariance, a "dilaton" $\phi$ has to be introduced, with non-trivial potentials $V(\phi)$ = $f_{1}e^{\alpha\phi}$ in $L_1$ and $U(\phi)$ = $f_{2}e^{2\alpha\phi}$ in $L_2$. This leads to non-trivial mass generation and potential for $\phi$. Mass terms for an arbitrary matter field can appear in a scale invariant form both in $L_1$ and in $L_2$ where they are coupled to different exponentials of the field $\phi$. Implications of these results for cosmology having in mind in particular inflationary scenarios, models of the late universe and modified gravitational theories are discussed. 
  The Ludvigsen-Vickers and two recently suggested quasi-local spin-angular momentum expressions, based on holomorphic and anti-holomorphic spinor fields, are calculated for small spheres of radius $r$ about a point $o$. It is shown that, apart from the sign in the case of anti-holomorphic spinors in non-vacuum, the leading terms of all these expressions coincide. In non-vacuum spacetimes this common leading term is of order $r^4$, and it is the product of the contraction of the energy-momentum tensor and an average of the approximate boost-rotation Killing vector that vanishes at $o$ and of the 3-volume of the ball of radius $r$. In vacuum spacetimes the leading term is of order $r^6$, and the factor of proportionality is the contraction of the Bel-Robinson tensor and an other average of the same approximate boost-rotation Killing vector. 
  In this paper we propose a definition of quasifree Hadamard states for spinor fields on a curved space-time by specifying the Polarisation Set of the two-point function. We prove that the thermal equilibrium state on an ultrastatic space-time is Hadamard. We then construct an adiabatic vacuum state on a general globally hyperbolic Lorentz manifold using a factorisation of the spinorial Klein-Gordon operator. This state is pure. In what constitutes the main part of the paper, we show that it is also Hadamard. As a side result, we obtain the propagation of singularities of the spinorial Klein-Gordon operator. Some notation and results are collected in the Appendix. 
  We consider the ground state energy of scalar massive field in the spacetime of a pointlike global monopole. Using zeta function regularization method we obtain the heat kernel coefficients for this system. We show that the coefficient $B_1$ contains additional contribution due to the non-trivial topological structure of the spacetime. Taking into account the heat kernel coefficients we obtain that the ground state energy of the scalar field is zero. We also discuss our result using dimensional considerations. 
  Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at a moment of time symmetry. We argue that only a tetrahedral lattice can successfully reproduce the continuum solution, and develop a simplicial axisymmetric lattice based on the co-ordinate structure of the continuum metric. This is used to construct initial data for Brill waves in an otherwise flat spacetime, and for the distorted black hole spacetime of Bernstein. These initial data sets are shown to be second order accurate approximations to the corresponding continuum solutions. 
  We study the problem of detecting, and infering astrophysical information from, gravitational waves from a pulsating neutron star. We show that the fluid f and p-modes, as well as the gravitational-wave w-modes may be detectable from sources in our own galaxy, and investigate how accurately the frequencies and damping rates of these modes can be infered from a noisy gravitational-wave data stream. Based on the conclusions of this discussion we propose a strategy for revealing the supranuclear equation of state using the neutron star fingerprints: the observed frequencies of an f and a p-mode. We also discuss how well the source can be located in the sky using observations with several detectors. 
  We describe a simple dynamical model characterized by the presence of two noncommuting Hamiltonian constraints. This feature mimics the constraint structure of general relativity, where there is one Hamiltonian constraint associated with each space point. We solve the classical and quantum dynamics of the model, which turns out to be governed by an SL(2,R) gauge symmetry, local in time. In classical theory, we solve the equations of motion, find a SO(2,2) algebra of Dirac observables, find the gauge transformations for the Lagrangian and canonical variables and for the Lagrange multipliers. In quantum theory, we find the physical states, the quantum observables, and the physical inner product, which is determined by the reality conditions. In addition, we construct the classical and quantum evolving constants of the system. The model illustrates how to describe physical gauge-invariant relative evolution when coordinate time evolution is a gauge. 
  Although quantum field theory allows local negative energy densities and fluxes, it also places severe restrictions upon the magnitude and extent of the negative energy. The restrictions take the form of quantum inequalities. These inequalities imply that a pulse of negative energy must not only be followed by a compensating pulse of positive energy, but that the temporal separation between the pulses is inversely proportional to their amplitude. In an earlier paper we conjectured that there is a further constraint upon a negative and positive energy delta-function pulse pair. This conjecture (the quantum  interest conjecture) states that a positive energy pulse must overcompensate the negative energy pulse by an amount which is a monotonically increasing function of the pulse separation. In the present paper we prove the conjecture for massless quantized scalar fields in two and four-dimensional flat spacetime, and show that it is implied by the quantum inequalities. 
  In order to analyze data produced by the kilometer-scale gravitational wave detectors that will begin operation early next century, one needs to develop robust statistical tools capable of extracting weak signals from the detector noise. This noise will likely have non-stationary and non-Gaussian components. To facilitate the construction of robust detection techniques, I present a simple two-component noise model that consists of a background of Gaussian noise as well as stochastic noise bursts. The optimal detection statistic obtained for such a noise model incorporates a natural veto which suppresses spurious events that would be caused by the noise bursts. When two detectors are present, I show that the optimal statistic for the non-Gaussian noise model can be approximated by a simple coincidence detection strategy. For simulated detector noise containing noise bursts, I compare the operating characteristics of (i) a locally optimal detection statistic (which has nearly-optimal behavior for small signal amplitudes) for the non-Gaussian noise model, (ii) a standard coincidence-style detection strategy, and (iii) the optimal statistic for Gaussian noise. 
  We prove that the oft-used stationary-phase method gives a very accurate expression for the Fourier transform of the gravitational-wave signal produced by an inspiraling compact binary. We give three arguments. First, we analytically calculate the next-order correction to the stationary-phase approximation, and show that it is small. This calculation is essentially an application of the steepest-descent method to evaluate integrals. Second, we numerically compare the stationary-phase expression to the results obtained by Fast Fourier Transform. We show that the differences can be fully attributed to the windowing of the time series, and that they have nothing to do with an intrinsic failure of the stationary-phase method. And third, we show that these differences are negligible for the practical application of matched filtering. 
  The quasi-stationary superfluid state is constructed, which exhibits the event horizon and Hawking radiation. 
  In Riemann geometry, the relations among two transversal submanifolds and global manifold are discussed. By replacing the normal vector of a submanifold with the tangent vector of another submanifold, the metric tensors, Christoffel symbols and curvature tensors of the three manifolds are linked together. When the inner product of the two tangent vectors vanishes, some corollaries of these relations give the most important second fundamental form and Gauss-Codazzi equation in the conventional submanifold theory. As a special case, the global manifold is Euclidean is considered. It is pointed out that, in order to obtain the nonzero energy-momentum tensor of matter field in a submanifold, there must be the contributions of the above inner product and the other submanifold. In general speaking, a submanifold is closely related to the matter fields of the other submanifold through the above inner product. This conclusion is in agreement with the Kaluza-Klein theory and it can be applied to generalize the models of direct product of manifolds in string and D-brane theories to the more general cases --- nondirect product manifolds. 
  Hawking and the author have proposed a class of singular, finite action instantons for defining the initial conditions for inflation. Vilenkin has argued they are unacceptable. He exhibited an analogous class of asymptotically flat instantons which on the face of it lead to an instability of Minkowski space. However, all these instantons must be defined by introducing a constraint into the path integral, which is then integrated over. I show that with a careful definition these instantons do not possess a negative mode. Infinite flat space is therefore stable against decay via singular instantons. 
  We derive an equation for the acceleration of a fluid element in the spherical gravitational collapse of a bounded compact object made up of an imperfect fluid. We show that non-singular as well as singular solutions arise in the collapse of a fluid initially at rest and having only a tangential pressure. We obtain an exact solution of Einstein equations, in the form of an infinite series, for collapse under tangential pressure with a linear equation of state. We show that if a singularity forms in the tangential pressure model, the conditions for the singularity to be naked are exactly the same as in the model of dust collapse. 
  A particular choice of the time function in the recently presented spherical solution by Dadhich [1] leads to a singularity free cosmological model which oscillates between two regular states. The energy-stress tensor involves anisotropic pressure and a heat flux term but is consistent with the usual energy conditions (strong, weak and dominant). By choosing the parameters suitably one can make the model consistent with observational data. An interesting feature of the model is that it involves blue shifts as in the quasi steady state model [2] but without violating general relativity. 
  The so-called negative mode problem in the path integral approach to the false vacuum decay with the effect of gravity has been an unsolved problem. Several years ago, we proposed a conjecture which is to be proved in order to give a consistent solution to the negative mode problem. We called it the ``no-negative mode conjecture''. In the present paper, we give a proof of this conjecture for rather general models. Recently, we also proposed the ``no-supercritical supercurvature mode conjecture'' that claims the absence of supercritical supercurvature modes in the one-bubble open inflation model. In the same paper, we clarified the equivalence between the ``no-negative mode conjecture'' and the ``no-supercritical supercurvature mode conjecture''. Hence, the latter is also proved at the same time when the former is proved. 
  Friedmann-Robertson-Walker cosmological models with a massive scalar field are studied in the presence of hydrodynamical matter in the form of a perfect fluid. The ratio of the number of solutions without inflation to the total number of solutions is evaluated, depending on the fluid density. It is shown that in a closed model this ratio can reach 60%, by contrast to $\sim 30 %$ in models without fluid. 
  Of all the astronomical sources of gravitational radiation, the ringdown waveform arising from a small perturbation of a spinning black hole is perhaps the best understood: for the late stages of such a perturbation, the waveform is simply an exponentially-damped sinusoid. Searching interferometric gravitational wave antenna data for these should be relatively easy. In this paper, I present the results of a single-filter search for ringdown waveforms arising from a 50 solar mass black hole with 98% of its maximum spin angular momentum using data from the Caltech 40-meter prototype interferometer. This search illustrates the techniques that may be used in analyzing data from future kilometer-scale interferometers and describes some of the difficulties present in the analysis of interferometer data. Most importantly, it illustrates the use of coincident events in the output of two independent interferometers (here simulated by 40-meter data at two different times) to substantially reduce the spurious event rate. Such coincidences will be essential tools in future gravitational wave searches in kilometer-scale interferometers. 
  Through direct thermodynamic calculations we have shown that different classical entropies of two-dimensional extreme black holes appear due to two different treatments, namely Hawking's treatment and Zaslavskii's treatment. Geometrical and topological properties corresponding to these different treatments are investigated. Quantum entropies of the scalar fields on the backgrounds of these black holes concerning different treatments are also exhibited. Different results of entropy and geometry lead us to argue that there are two kinds of extreme black holes in the nature. Explanation of black hole phase transition has also been given from the quantum point of view. 
  When a self-gravitating body (e.g., a neutron star or black hole) interacts with an external tidal field (e.g., that of a binary companion), the interaction can do work on the body, changing its mass-energy. The details of this "tidal heating" are analyzed using the Landau-Lifshitz pseudotensor and the local asymptotic rest frame of the body. It is shown that the work done on the body is gauge-invariant, while the body-tidal-field interaction energy contained within the body's local asymptotic rest frame is gauge dependent. This is analogous to Newtonian theory, where the interaction energy is shown to depend on how one localizes gravitational energy, but the work done on the body is independent of that localization. 
  We consider static spherically symmetric solutions of the Einstein equations with cosmological constant coupled to the SU(2) Yang Mills equations. We prove that most solutions can be continued back to the origin of spherical symmetry and that the qualitative behavior of solutions near this origin does not depend on the value of the cosmological constant. 
  It is shown that the motion of a multielectron atom in an external gravitational field in a good approximation is described by system of the Mathisson-Papapetrou equations, if we put as a classical angular momentum of the atom the expectation value of the operator of the full angular momentum of the system, which includes spins of the nucleus and electrons, and orbital momentums of the electrons in the atom. 
  We study the linear metric perturbations of the Reissner-Nordstr\"{o}m solution for the case of axial perturbation modes. We find that the well-known perturbative analysis fails for the case of dipole $(l=1)$ perturbations, although valid for higher multipoles. We define new radial functions, with which the perturbation formalism is generalized to all multipole orders, including the dipole. We then complete the solution by constructing the perturbed metric and Maxwell tensors. 
  We study numerically the inhomogeneous pre-big-bang inflation in a spherically symmetric space-time. We find that a large initial inhomogeneity suppresses the onset of the pre-big-bang inflation. We also find that even if the pre-big-bang inflationary stage is realized, the initial inhomogeneities are not homogenized. Namely, during the pre-big-bang inflation ``hairs''(irregularities) do not fall, in sharp contrast to the usual (potential energy dominated) inflation where initial inhomogeneity and anisotropy are damped and thus the resulting universe is less sensitive to initial conditions. 
  We discuss the quantisation of a class of string cosmology models that are characterized by scale factor duality invariance. We compute the amplitudes for the full set of classically allowed and forbidden transitions by applying the reduce phase space and the path integral methods. We show that these approaches are consistent. The path integral calculation clarifies the meaning of the instanton-like behaviour of the transition amplitudes that has been first pointed out in previous investigations. 
  We solve the eigenvalue problem of general relativity for the case of charged black holes in two-dimensional heterotic string theory, derived by McGuigan et al. For the case of $m^{2}>q^{2}$, we find a physically acceptable time-dependent growing mode; thus the black hole is unstable. The extremal case $m^{2}=q^{2}$ is stable. 
  It is shown that during expanding phases of flat homogeneous cosmologies all small enough non-linear perturbations decay exponentially. This result holds for a large class of perfect fluid equations of state, but notably not for very ``stiff'' fluids as the pure radiation case. 
  The stochastic approach to inflation suffers from ambiguities due to the arbitrary choice of the time variable and due to the choice of the factor ordering in the corresponding Fokker-Planck equation. Here it is shown that both ambiguities can be removed if we require that the factor ordering should be set in such a way that physical results are invariant with respect to time reparametrizations. This requirement uniquely selects the so-called Ito factor ordering. Additional ambiguities associated with non-trivial kinetic terms of the scalar fields are also discussed, as well as ways of constraining these ambiguities. 
  We study the gravitational collapse of a self-gravitating charged scalar-field. Starting with a regular spacetime, we follow the evolution through the formation of an apparent horizon, a Cauchy horizon and a final central singularity. We find a null, weak, mass-inflation singularity along the Cauchy horizon, which is a precursor of a strong, spacelike singularity along the $r=0$ hypersurface. The inner black hole region is bounded (in the future) by singularities. This resembles the classical inner structure of a Schwarzschild black hole and it is remarkably different from the inner structure of a charged static Reissner-Nordstr\"om or a stationary rotating Kerr black holes. 
  The Schwarzschild-deSitter metric is the known solution of Einstein field equations with cosmological constant term for vacuum spherically symmetric space around a point mass M. Recently it has been reported that in a $Lamda$-dominant world the Schwarzschild type coordinate systems are disqualified by redshift-magnitude test as a proper frame of reference(gr-qc/9812092). We derive the solution in a FRW type coordinate system which is qualified according to the mentioned test. Asymptotically it approachs to the non-static form of deSitter metric. The obtained metric is transformable to Schwarzschild-deSitter metric. It is an analytic function of $r$ for all values except $r=0$which is singular. This is carried out with no making use of Eddington-Finkelstein coordinates and without entering any cross term in the metric. 
  We consider the time evolution of a scalar field propagating in Schwarzschild-de Sitter spacetime. At early times, the field behaves as if it were in pure Schwarzschild spacetime; the structure of spacetime far from the black hole has no influence on the evolution. In this early epoch, the field's initial outburst is followed by quasi-normal oscillations, and then by an inverse power-law decay. At intermediate times, the power-law behavior gives way to a faster, exponential decay. At late times, the field behaves as if it were in pure de Sitter spacetime; the structure of spacetime near the black hole no longer influences the evolution in a significant way. In this late epoch, the field's behavior depends on the value of the curvature-coupling constant xi. If xi is less than a critical value 3/16, the field decays exponentially, with a decay constant that increases with increasing xi. If xi > 3/16, the field oscillates with a frequency that increases with increasing xi; the amplitude of the field still decays exponentially, but the decay constant is independent of xi. 
  Contents:   Editorial   Correspondents   News:    Topical Group News, by Jim Isenberg    We hear that..., by Jorge Pullin    The Chandra Satellite, by Beverly Berger   Research Briefs:    Analytical event horizons of merging black holes, by Simonetta Frittelli    LIGO project update, by David Shoemaker    Bicentenary of the Cavendish Experiment, by Riley Newman   Conference reports:    Eighth Midwest meeting, by Richard Hammond    GWDAW 98, by Sam Finn    Bad Honnef seminar, by Alan Randall. 
  In eprint gr-qc/9901053 Gen Yoneda and Hisa-aki Shinkai have made several claims disputing our results in gr-qc/9710004, and gr-qc/9804019. We show here that these claims are not correct. 
  We obtain rigorous results concerning the evaluation of integrals on the two sphere using complex methods. It is shown that for regular as well as singular functions which admit poles, the integral can be reduced to the calculation of residues through a limiting procedure. 
  It is shown that the integrability conditions that arise in the Null Surface Formulation (NSF) of general relativity (GR) impose a field equation on the local null surfaces which is equivalent to the vanishing of the Bach tensor. This field equation is written explicitly to second order in a perturbation expansion.   The field equation is further simplified if asymptotic flatness is imposed on the underlying space-time. The resulting equation determines the global null surfaces of asymptotically flat, radiative space-times. It is also shown that the source term of this equation is constructed from the free Bondi data at future null infinity.   Possible generalizations of this field equation are analyzed. In particular we include other field equations for surfaces that have already appeared in the literature which coincide with ours at a linear level. We find that the other equations do not yield null surfaces for GR. 
  In the loop approach to quantum gravity the spectra of operators corresponding to such geometrical quantities as length, area and volume become quantized. However, the size of arising quanta of geometry in Planck units is not fixed by the theory itself: a free parameter, sometimes referred to as Immirzi parameter, is known to affect the spectrum of all geometrical operators. In this paper I propose an argument that fixes the value of this parameter. I consider rotating black holes, in particular the extremal ones. For such black holes the ``no naked singularity condition'' bounds the total angular momentum J by A_H/8 pi G, where A_H is the horizon area and G Newton's constant. A similar bound on J comes from the quantum theory. The requirement that these two bounds are the same fixes the value of Immirzi parameter to be unity. A byproduct of this argument is the picture of the quantum extremal rotating black hole in which all the spin entering the extremal hole is concentrated in a single puncture. 
  We study the two-dimensional (2D) dilatonic model describing a massless scalar field minimally coupled to the spherically reduced Einstein-Hilbert gravity. The general solution of this model is given in the case when a Killing vector is present. When interpreted in four dimensions, the solution describes either a static or a homogeneous collision of incoming and outgoing null dust streams with spherical symmetry. The homogeneous Universe is closed. 
  This is a nontechnical introduction to recent work on quantum gravity using ideas from higher-dimensional algebra. We argue that reconciling general relativity with the Standard Model requires a `background-free quantum theory with local degrees of freedom propagating causally'. We describe the insights provided by work on topological quantum field theories such as quantum gravity in 3-dimensional spacetime. These are background-free quantum theories lacking local degrees of freedom, so they only display some of the features we seek. However, they suggest a deep link between the concepts of `space' and `state', and similarly those of `spacetime' and `process', which we argue is to be expected in any background-free quantum theory. We sketch how higher-dimensional algebra provides the mathematical tools to make this link precise. Finally, we comment on attempts to formulate a theory of quantum gravity in 4-dimensional spacetime using `spin networks' and `spin foams'. 
  We introduce a formulation of Eulerian general relativistic hydrodynamics which is applicable for (perfect) fluid data prescribed on either spacelike or null hypersurfaces. Simple explicit expressions for the characteristic speeds and fields are derived in the general case. A complete implementation of the formalism is developed in the case of spherical symmetry. The algorithm is tested in a number of different situations, predisposing for a range of possible applications. We consider the Riemann problem for a polytropic gas, with initial data given on a retarded/advanced time slice of Minkowski spacetime. We compute perfect fluid accretion onto a Schwarzschild black hole spacetime using ingoing null Eddington-Finkelstein coordinates. Tests of fluid evolution on dynamic background include constant density and TOV stars sliced along the radial null cones. Finally, we consider the accretion of self-gravitating matter onto a central black hole and the ensuing increase in the mass of the black hole horizon. 
  Coalescing compact binaries with neutron star or black hole components provide the most promising sources of gravitational radiation for detection by the LIGO/VIRGO/GEO/TAMA laser interferometers now under construction. This fact has motivated several different theoretical studies of the inspiral and hydrodynamic merging of compact binaries. Analytic analyses of the inspiral waveforms have been performed in the Post-Newtonian approximation. Analytic and numerical treatments of the coalescence waveforms from binary neutron stars have been performed using Newtonian hydrodynamics and the quadrupole radiation approximation. Numerical simulations of coalescing black hole and neutron star binaries are also underway in full general relativity. Recent results from each of these approaches will be described and their virtues and limitations summarized. 
  Examples of space-times are given which contain scalar curvature singularities whereat the metric tensor is regular and continuous, but which are gravitationally strong. Thus the argument that such singularities are necessarily weak is incomplete; in particular the question of the gravitational strength of the null Cauchy horizon singularity which occurs in gravitational collapse remains open. 
  Covariant equations characterizing the strength of a singularity in spherical symmetry are derived and several models are investigated. The difference between central and non-central singularities is emphasised. A slight modification to the definition of singularity strength is suggested. The gravitational weakness of shell crossing singularities in collapsing spherical dust is proven for timelike geodesics, closing a gap in the proof. 
  In this paper we obtain the energy distribution associated with the Ernst space-time (geometry describing Schwarzschild black hole in Melvin's magnetic universe) in Einstein's prescription. The first term is the rest-mass energy of the Schwarzschild black hole, the second term is the classical value for the energy of the uniform magnetic field and the remaining terms in the expression are due to the general relativistic effect. The presence of the magnetic field is found to increase the energy of the system. 
  The variant of quasiclassical (half-quantum) theory of gravity in strong gravitational field is presented. The exact solution of the problem of the renormalized energy-momentum tensor calculation is performed in terms of non-local operator-signed function. The procedure of quasilocalization is proposed, which leads to the equations of non-equilibrium thermodynamics for temperature and curvature. The effects of induced particle creation and media polarization are taking into account and used to solve the problem of non-Einstein's branches damping. The problem of Universe creation from "nothing" is also discussed. 
  We include matter sources in Einstein's field equations and show that our recently proposed 3+1 evolution scheme can stably evolve strong-field solutions. We insert in our code known matter solutions, namely the Oppenheimer-Volkoff solution for a static star and the Oppenheimer-Snyder solution for homogeneous dust sphere collapse to a black hole, and evolve the gravitational field equations. We find that we can evolve stably static, strong-field stars for arbitrarily long times and can follow dust sphere collapse accurately well past black hole formation. These tests are useful diagnostics for fully self-consistent, stable hydrodynamical simulations in 3+1 general relativity. Moreover, they suggest a successive approximation scheme for determining gravitational waveforms from strong-field sources dominated by longitudinal fields, like binary neutron stars: approximate quasi-equilibrium models can serve as sources for the transverse field equations, which can be evolved without having to re-solve the hydrodynamical equations (``hydro without hydro''). 
  A quantum mechanical model for an N + 1 dimensional universe arising from a quantum fluctuation is outlined. (3 + 1) dimensions are a closed infinitely-expanding universe and the remaining N - 3 dimensions are compact. The (3 + 1) non-compact dimensions are modeled by quantizing a canonical Hamiltonian description of a homogeneous isotropic universe. It is assumed gravity and the strong-electro-weak (SEW) forces had equal strength in the initial state. Inflation occurred when the compact N -3 dimensional space collapsed after a quantum transition from the initial state of the univers, during its evolution to the present state where gravity is much weaker than the SEW force. The model suggests the universe has no singularities and the large size of our present universe is determined by the relative strength of gravity and the SEW force today. A small cosmological constant, resulting from the zero point energy of the scalar field corresponding to the compact dimensions, makes the model universe expand forever. 
  The logical consistency of a description of Quantum Theory in the context of General Relativity, which includes Minimal Coupling Principle, is analyzed from the point of view of Feynman's formulation in terms of path integrals. We will argue from this standpoint and using an argument that claims the incompleteness of the general-relativistic description of gravitation, which emerges as a consequence of the gravitationally induced phases of the so called flavor-oscillation clocks, that the postulates of Quantum Theory are logically incompatible with the usual Minimal Coupling Principle. It will be shown that this inconsistency could emerge from the fact that the required geometrical information to calculate the probability of finding a particle in any point of the respective manifold does not lie in a region with finite volume. Afterwards, we put forth a new Quantum Minimal Coupling Principle in terms of a restricted path integral, and along the ideas of this model not only the propagator of a free particle is calculated but we also deduce the conditions under which we recover Feynman's case for a free particle. The effect on diatomic interstellar molecules is also calculated. The already existing relation between Restricted Path Integral Formalism and Decoherence Model will enable us to connect the issue of a Quantum Minimal Coupling Principle with the collapse of the wave function. From this last remark we will claim that the geometrical structure of the involved manifold acts as, an always present, measuring device on a quantum particle. In other words, in this proposal we connect the issue of a Quantum Minimal Coupling Principle with a claim which states that gravity could be one of the physical entities driving the collapse of the wave function. 
  Careful analysis of parametrized variational principles in mechanics and field theory leads to a generalization of Einstein theory that includes a cosmological stress tensor. This generalization also follows by restricting variations of the metric in the Hilbert action to spacetime diffeomorphisms. The equation of motion for the generalized theory is the twice-contracted Bianchi identity while the field equations constitute the stress tensor of the theory. Gravity is interpreted as a cosmological fluid. 
  We develop a method for constructing asymptotic solutions of finite-difference equations and implement it to a relativistic Schroedinger equation which describes motion of a selfgravitating spherically symmetric dust shell. Exact mass spectrum of black hole formed due to the collapse of the shell is determined from the analysis of asymptotic solutions of the equation. 
  We investigate whether the generalized second law is valid, using two dimensional black hole spacetime, irrespective of models. A time derivative form of the generalized second law is formulated and it is shown that the law might become invalid. The way to resolve this difficulty is also presented and discussed. 
  The Lorentz covariant theory of propagation of light in the (weak) gravitational fields of N-body systems consisting of arbitrarily moving point-like bodies with constant masses is constructed. The theory is based on the Lienard-Wiechert presentation of the metric tensor. A new approach for integrating the equations of motion of light particles depending on the retarded time argument is applied. In an approximation which is linear with respect to the universal gravitational constant, G, the equations of light propagation are integrated by quadratures and, moreover, an expression for the tangent vector to the perturbed trajectory of light ray is found in terms of instanteneous functions of the retarded time. General expressions for the relativistic time delay, the angle of light deflection, and gravitational red shift are derived. They generalize previously known results for the case of static or uniformly moving bodies. The most important applications of the theory are given. They include a discussion of the velocity dependent terms in the gravitational lens equation, the Shapiro time delay in binary pulsars, and a precise theoretical formulation of the general relativistic algorithm of data processing of radio and optical astrometric measurements in the non-stationary gravitational field of the solar system. Finally, proposals for future theoretical work being important for astrophysical applications are formulated. 
  In the following we are going to describe how macroscopic space-time is supposed to emerge as an orderparameter manifold or superstructure floating in a stochastic discrete network structure. As in preceeding work (mentioned below), our analysis is based on the working philosophy that both physics and the corresponding mathematics have to be genuinely discrete on the primordial (Planck scale) level. This strategy is concretely implemented in the form of cellular networks and random graphs. One of our main themes is the development of the concept of physical (proto)points as densely entangled subcomplexes of the network and their respective web, establishing something like (proto)causality. It max perhaps be said that certain parts of our programme are realisations of some old and qualitative ideas of Menger and more recent ones sketched by Smolin a couple of years ago. We briefly indicate how this two-story-concept of space-time can be used to encode the (at least in our view) existing non-local aspects of quantum theory without violating macroscopic space-time causality! 
  The Hamiltonian constraint formalism is used to obtain the first explicit complete analysis of non-trivial viable dynamic modes for the Poincar\'e gauge theory of gravity. Two modes with propagating spin-zero torsion are analyzed. The explicit form of the Hamiltonian is presented. All constraints are obtained and classified. The Lagrange multipliers are derived. It is shown that a massive spin-$0^-$ mode has normal dynamical propagation but the associated massless $0^-$ is pure gauge. The spin-$0^+$ mode investigated here is also viable in general. Both modes exhibit a simple type of ``constraint bifurcation'' for certain special field/parameter values. 
  Qualitativ arguments are presented which show the incompatibility of the positive results obtaned in experiments on the gravitational redshift of photones and in experiments investigating the behavior of clocks in the gravitational field. 
  We consider the problem of the symmetry of the off-diagonal heat-kernel coefficients as well as the coefficients corresponding to the short-distance-divergent part of the Hadamard expansion in general smooth (analytic or not) manifolds. The requirement of such a symmetry played a central r\^{o}le in the theory of the point-splitting one-loop renormalization of the stress tensor in either Riemannian or Lorentzian manifolds. Actually, the symmetry of these coefficients has been assumed as a hypothesis in several papers concerning these issues without an explicit proof. The difficulty of a direct proof is related to the fact that the considered off-diagonal heat-kernel expansion, also in the Riemannian case, in principle, may be not a proper asymptotic expansion. On the other hand, direct computations of the off-diagonal heat-kernel coefficients are impossibly difficult in nontrivial cases and thus no case is known in the literature where the symmetry does not hold. By approximating $C^\infty$ metrics with analytic metrics in common (totally normal) geodesically convex neighborhoods, it is rigorously proven that, in general $C^\infty$ Riemannian manifolds, any point admits a geodesically convex neighborhood where the off-diagonal heat-kernel coefficients, as well as the relevant Hadamard's expansion coefficients, are symmetric functions of the two arguments. 
  Following a recent work in which it is shown that a spacetime admitting Lie-group actions may be disjointly decomposed into a a closed subset with no interior plus a dense finite union of open sets in each of which the character and dimension of the group orbits as well as the Petrov type are constant, the aim of this work is to include the Segre types of the Ricci tensor (and hence of the Einstein tensor) into the decomposition. We also show how this type of decomposition can be carried out for any type of property of the spacetime depending on the existence of a continuous endomorphism. 
  In this work cosmological models are considered for the low energy string cosmological effective action (tree level) in the absence of dilaton potential. A two parametric non-diagonal family of analytic solutions is found. The curvature is non singular, however the string coupling diverges exponentially. 
  This work completes the task of solving locally the Einstein-Ashtekar equations for degenerate data. The two remaining degenerate sectors of the classical 3+1 dimensional theory are considered. First, with all densitized triad vectors linearly dependent and second, with only two independent ones. It is shown how to solve the Einstein-Ashtekar equations completely by suitable gauge fixing and choice of coordinates. Remarkably, the Hamiltonian weakly Poisson commutes with the conditions defining the sectors. The summary of degenerate solutions is given in the Appendix. 
  Coupled ordinary differential equations are derived for the distant gravitational interaction of a compact object of mass M and charge Q with an initially straight, infinitely long, cosmic string of tension mu << 1/G = 1, when the relative velocities are very low compared to the speed of light c = 1. (An intermediate result of this derivation is that any localized force F(t) on the string that is confined to a single plane perpendicular to the initial string configuration gives the intersection of the string with this plane -- the point where the force is applied -- the velocity F(t)/(2 mu).)   The coupled equations are then used to calculate the critical impact parameter b for marginal gravitational capture as a function of the incident velocity v. For v<<(1-Q^2/M^2)^(1/3) mu^(2/3), so that the string acts relatively stiffly, b = (pi/4)[12 mu^3 (1-Q^2/M^2)^4]^(1/5) M v^(-7/5) + O(M v^(-1/5)). For (1-Q^2/M^2)^(1/3) mu^(2/3) << v << 1 - Q^2/M^2$, so that the string acts essentially as a test string that stays nearly straight, b = [(pi/2)(1-Q^2/M^2)]^(1/2) M v^(-1/2) + O(M v^(-2)). Between these two limits the critical impact parameter is found numerically to fit a simple algebraic combination of these two formulas to better than 99.5% accuracy. 
  A contribution of quantum vacuum to the energy momentum tensor is inevitably experienced in the present universe. One requires the presence of non-zero cosmological constant ($\Lambda$) to make the various observations consistent. A case of $\Lambda$ in the Schwarzschild de Sitter space-time shows that precession of perihelion orbit provides a sensative solar test for non-zero $\Lambda$. Application of the relations involving $\Lambda$ to the planetery perturbation indicates the values near to the present bound on $\Lambda$. Also suggested are some relations in vacuum dominated flat universe with a positive $\Lambda$. 
  We make analytic derivation for maximum masses of stable boson stars formed by scalar fields with any higher order self-interactions and show that those are equivalent to numerical results. It is shown that the contribution of the higher order self-interaction terms to the maximum mass decreases as $(m/M_p)^2$ power. 
  Non-static, spherically symmetric clusters of counter-rotating particles, of the type first introduced by Einstein, are analysed here. The initial data space can be parameterized in terms of three arbitrary functions, namely; initial density, velocity and angular momentum profiles. The final state of collapse, black hole or naked singularity, turns out to depend on the order of the first non-vanishing derivatives of such functions at the centre. The work extends recent results by Harada, Iguchi and Nakao. 
  Boundary conditions play a crucial role in the path-integral approach to quantum gravity and quantum cosmology, as well as in the current attempts to understand the one-loop semiclassical properties of quantum field theories. Within this framework, one is led to consider boundary conditions completely invariant under infinitesimal diffeomorphisms on metric perturbations. These are part of a general scheme, which can be developed for Maxwell theory, Yang-Mills Theory, Rarita-Schwinger fields and any other gauge theory. A general condition for strong ellipticity of the resulting field theory on manifolds with boundary is here proved, following recent work by the authors. The relevance for Euclidean quantum gravity is eventually discussed. 
  Assuming the space dimension is not constant, but varies with the expansion of the Universe, a Lagrangian formulation for a toy model Universe is given. There is a free paremeter in the theory, $C$, with which we can fix the dimension of space at the Planck time. The standard FRW model corresponds to the limiting case $C \to +\infty$. We study the Wheeler-De Witt equation and the wave function of the Universe and the probability density in our model Universe. In the limit of constant space dimension, our wave function approaches to the Hartle-Hawking wave function or to a modified Linde wave function. 
  This dissertation investigates three main topics, all of which dealing with alternative, higher-order gravity theories in four dimensions. Firstly, we study the variational and conformal structure of those theories. Next, we analyse their Hamiltonian formulation and in particular its relationship with the famous ADM canonical version of general relativity. Finally, we study higher-order spatially homogeneous cosmologies and exemplify how Hamiltonian methods can be utilised to simplify the analysis of the associated field equations. 
  This work addresses certain ambiguities in the Dirac approach to constrained systems. Specifically, we investigate the space of so-called ``rigging maps'' associated with Refined Algebraic Quantization, a particular realization of the Dirac scheme. Our main result is to provide a condition under which the rigging map is unique, in which case we also show that it is given by group averaging techniques. Our results comprise all cases where the gauge group is a finite-dimensional Lie group. 
  The origin and nature of time in complex systems is explored using quantum (or 'Feynman') clocks and the signals produced by them. Networks of these clocks provide the basis for the evolution of complex systems. The general concept of 'time' is translated into the 'lifetimes' of these unstable configurations of matter. 'Temporal phase transitions' mark the emergence of classical properties such as irreversibility, entropy, and thermodynamic arrows of time. It is proposed that the creation of the universe can be modeled as a quantum clock. Keywords: the problem of time, the arrow of time, time asymmetry, the many-body problem, cellular networks, complexity, the Wheeler-DeWitt equation, quantum cosmology, and instantons. 
  We consider the duality of the quasilocal black hole thermodynamics, explicitly the quasilocal black hole thermodynamic first law, in BTZ black hole solution as a special one of the three-dimensional low energy effective string theory. 
  We examine the question of collapse of Turok's two-parameter family of cosmic strings. We first perform a classification of the strings according to the specific time(s) at which the minimal string size is reached during one period. We then obtain an exact analytical expression for the probability of collapse to black holes for the Turok strings. Our result has the same general behavior as previously obtained in the literature but we find, in addition, a numerical prefactor that changes the result by three orders of magnitude. Finally we show that our careful computation of the prefactor helps to understand the discrepancy between previously obtained results and, in particular, that for "large" values of $G\mu$, there may not even be a discrepancy. We also give a simple physical argument that can immediately rule out some of the previously obtained results. 
  We compute the regularized temperature for a spacetime foam model, consisting on S^4 instantons, in quantum gravity. Assuming that thermal equilibrium takes place with some amount of radiation - with thermal fields in the SU(2)xU(1) gauge theory - we obtain the remarkable result that the squared value of this temperature exactly coincides with the electroweak coupling constant at the energy scale of the gauge bosons W. This is consistent with the classical ADM result that the electrical charge should be equal to its finite gravitational self energy. 
  The gravitational weak field of a global monopole in the Einstein-Cartan theory of gravity is investigated.To obtain this solution we assume that Cartan torsion takes the form of the Newtonian gravitational potential.From the geodesics it is possible to show that the torsionic monopole produces a repulsive gravitational force. 
  We evoke situations where large fluctuations in the entropy are induced, our main example being a spacetime containing a potential black hole whose formation depends on the outcome of a quantum mechanical event. We argue that the teleological character of the event horizon implies that the consequent entropy fluctuations must be taken seriously in any interpretation of the quantal formalism. We then indicate how the entropy can be well defined despite the teleological character of the horizon, and we argue that this is possible only in the context of a spacetime or ``histories'' formulation of quantum gravity, as opposed to a canonical one, concluding that only a spacetime formulation has the potential to compute --- from first principles and in the general case --- the entropy of a black hole. From the entropy fluctuations in a related example, we also derive a condition governing the form taken by the entropy, when it is expressed as a function of the quantal density-operator. 
  Techniques are developed here for evaluating the r-modes of rotating neutron stars through second order in the angular velocity of the star. Second-order corrections to the frequencies and eigenfunctions for these modes are evaluated for neutron star models. The second-order eigenfunctions for these modes are determined here by solving an unusual inhomogeneous hyperbolic boundary-value problem. The numerical techniques developed to solve this unusual problem are somewhat non-standard and may well be of interest beyond the particular application here. The bulk-viscosity coupling to the r-modes, which appears first at second order, is evaluated. The bulk-viscosity timescales are found here to be longer than previous estimates for normal neutron stars, but shorter than previous estimates for strange stars. These new timescales do not substantially affect the current picture of the gravitational radiation driven instability of the r-modes either for neutron stars or for strange stars. 
  The paper combines theoretical and applied ideas which have been previously considered separately into a single set of evolution equations for Numerical Relativity. New numerical ingredients are presented which avoid gauge pathologies and allow one to perform robust 3D calculations. The potential of the resulting numerical code is demonstrated by using the Schwarzschild black hole as a test-bed. Its evolution can be followed up to times greater than one hundred black hole masses. 
  The Lovelock gravity extends the theory of general relativity to higher dimensions in such a way that the field equations remain of second order. The theory has many constant coefficients with no a priori meaning. Nevertheless it is possible to reduce them to two, the cosmological constant and Newton's constant. In this process one separates theories in even dimensions from theories in odd dimensions. In a previous work gravitational collapse in even dimensions was analysed. In this work attention is given to odd dimensions. It is found that black holes also emerge as the final state of gravitational collapse of a regular dust fluid. 
  Recently Nolan constructed a spherically-symmetric spacetime admitting a spacelike curvature singularity with a regular C^0 metric. We show here that this singularity is in fact weak. 
  Some recent (1997-1998) theoretical results concerning the $\zeta$-function regularization procedure used to renormalize, at one-loop, the effective Lagrangian, the field fluctuations and the stress-tensor in curved spacetime are reviewed. 
  We explore the role of the Cremmer-Scherk mechanism in the context of low energy effective string theory by coupling the antisymmetric 3-form gauge potential to an Abelian gauge potential carrying weak hypercharge. The theory admits a class of exact self-gravitating solutions in the spontaneously broken phase in which the dual fields acquire massive perturbative modes. Despite the massive nature of these fields they admit non-perturbative progressive longitudinal modes that together with pp-type gravitational waves travel in a direction of a line source at the speed of light. 
  We study maximally symmetric cosmological solutions of type II supersymmetric strings in the presence of the exact, SL(2,Z)-invariant, higher-curvature corrections to the lowest order effective action. We find that, unlike the case of type IIA theories, de Sitter solutions exist, at all orders in $\alpha'$, for type IIB superstrings, when non-perturbative instanton-effects are included on the basis of SL(2,Z) invariance. 
  Einstein's equations for a 4+n-dimensional inhomogeneous space-time are presented, and a special family of solutions is exhibited for an arbitrary n. The solutions depend on two arbitrary functions of time. The time development of a particular member of this family is studied. This solution exhibits a singularity at t=0 and dynamical compactification of the n dimensions. It is shown that the behaviour of the system in the 4-dimensional i.e. post-compactification phase is constrained by the way in which the compactified dimensions are stabilized. The fluid that generates the solution is analyzed by means of the energy conditions. 
  The pre-big bang scenario describes the evolution of the Universe from an initial state approaching the flat, cold, empty, string perturbative vacuum. The choice of such an initial state is suggested by the present state of our Universe if we accept that the cosmological evolution is (at least partially) duality-symmetric. Recently, the initial conditions of the pre-big bang scenario have been criticized as they introduce large dimensionless parameters allowing the Universe to be "exponentially large from the very beginning". We agree that a set of initial parameters (such as the initial homogeneity scale, the initial entropy) larger than those determined by the initial horizon scale, H^{-1}, would be somewhat unnatural to start with. However, in the pre-big bang scenario, the initial parameters are all bounded by the size of the initial horizon. The basic question thus becomes: is a maximal homogeneity scale of order H^{-1} necessarily unnatural if the initial curvature is small and, consequently, H^{-1} is very large in Planck (or string) units? In the impossibility of experimental information one could exclude "a priori", for large horizons, the maximal homogeneity scale H^{-1} as a natural initial condition. In the pre-big bang scenario, however, pre-Planckian initial conditions are not necessarily washed out by inflation and are accessible (in principle) to observational tests, so that their naturalness could be also analyzed with a Bayesan approach, in terms of "a posteriori" probabilities. 
  Motivated by recent interest in black holes whose asymptotic geometry approaches that of anti-de Sitter spacetime, we give a proof of topological censorship applicable to spacetimes with such asymptotic behavior. Employing a useful rephrasing of topological censorship as a property of homotopies of arbitrary loops, we then explore the consequences of topological censorship for horizon topology of black holes. We find that the genera of horizons are controlled by the genus of the space at infinity. Our results make it clear that there is no conflict between topological censorship and the non-spherical horizon topologies of locally anti-de Sitter black holes. More specifically, let D be the domain of outer communications of a boundary at infinity ``scri.'' We show that the principle of topological censorship (PTC), that every causal curve in D having endpoints on scri can be deformed to scri, holds under reasonable conditions for timelike scri, as it is known to do for a simply connected null scri. We then show that the PTC implies that the fundamental group of scri maps, via inclusion, onto the fundamental group of D, i.e., every loop in D is homotopic to a loop in scri. We use this to determine the integral homology of preferred spacelike hypersurfaces (Cauchy surfaces or analogues thereof) in the domain of outer communications of any 4-dimensional spacetime obeying PTC. From this, we establish that the sum of the genera of the cross-sections in which such a hypersurface meets black hole horizons is bounded above by the genus of the cut of infinity defined by the hypersurface. Our results generalize familiar theorems valid for asymptotically flat spacetimes requiring simple connectivity of the domain of outer communications and spherical topology for stationary and slowly evolving black holes. 
  The gravitational radiation generated by a particle in a close unbounded orbit around a neutron star is computed as a means to study the importance of the $w$ modes of the neutron star. For simplicity, attention is restricted to odd parity (``axial'') modes which do not couple to the neutron star's fluid modes. We find that for realistic neutron star models, particles in unbounded orbits only weakly excite the $w$ modes; we conjecture that this is also the case for astrophysically interesting sources of neutron star perturbations. We also find that for cases in which there is significant excitation of quadrupole $w$ modes, there is comparable excitation of higher multipole modes. 
  The BTZ stationary black hole solution is considered and its mass and angular momentum are calculated by means of Noether theorem. In particular, relative conserved quantities with respect to a suitably fixed background are discussed. Entropy is then computed in a geometric and macroscopic framework, so that it satisfies the first principle of thermodynamics. In order to compare this more general framework to the prescription by Wald et al. we construct the maximal extension of the BTZ horizon by means of Kruskal-like coordinates. A discussion about the different features of the two methods for computing entropy is finally developed. 
  We give a summary of the status of current research in stochastic semiclassical gravity and suggest directions for further investigations. This theory generalizes the semiclassical Einstein equation to an Einstein-Langevin equation with a stochastic source term arising from the fluctuations of the energy-momentum tensor of quantum fields. We mention recent efforts in applying this theory to the study of black hole fluctuations and backreaction problems, linear response of hot flat space, and structure formation in inflationary cosmology. To explore the physical meaning and implications of this stochastic regime in relation to both classical and quantum gravity, we find it useful to take the view that semiclassical gravity is mesoscopic physics and that general relativity is the hydrodynamic limit of certain spacetime quantum substructures. Three basic issues - stochasticity, collectivity, correlations- and three processes - dissipation, fluctuations, decoherence- underscore the transformation from quantum micro structure and interaction to the emergence of classical macro structure and dynamics. We discuss ways to probe into the high energy activity from below and make two suggestions: via effective field theory and the correlation hierarchy. We discuss how stochastic behavior at low energy in an effective theory and how correlation noise associated with coarse-grained higher correlation functions in an interacting quantum field could carry nontrivial information about the high energy sector. Finally we describe processes deemed important at the Planck scale, including tunneling and pair creation, wave scattering in random geometry, growth of fluctuations and forms, Planck scale resonance states, and spacetime foams. 
  The BTZ black hole solution for (2+1)-spacetime is considered as a solution of a triad-affine theory (BCEA) in which topological matter is introduced to replace the cosmological constant in the model. Conserved quantities and entropy are calculated via Noether theorem, reproducing in a geometrical and global framework earlier results found in the literature using local formalisms. Ambiguities in global definitions of conserved quantities are considered in detail. A dual and covariant Legendre transformation is performed to re-formulate BCEA theory as a purely metric (natural) theory (BCG) coupled to topological matter. No ambiguities in the definition of mass and angular momentum arise in BCG theory. Moreover, gravitational and matter contributions to conserved quantities and entropy are isolated. Finally, a comparison of BCEA and BCG theories is carried out by relying on the results obtained in both theories. 
  By resolving the Riemann curvature relative to a unit timelike vector into electric and magnetic parts, we define a duality transformation which interchanges active and passive electric parts. It implies interchange of roles of Ricci and Einstein curvatures. Further by modifying the vacuum/flat equation we construct spacetimes dual to the Schwarzschild solution and flat spacetime. The dual spacetimes describe the original spacetimes with global monopole charge and global texture. The duality so defined is thus intimately related to the topological defects and also renders the Schwarzschild field asymptotically non-flat which augurs well with Mach's Principle. 
  The emission and absorption of gravitational waves and massless particles of an infinitely long straight cosmic string with finite thickness are studied. It is shown in a general term that the back reaction of the emission and absorption {\em always} makes the symmetry axis of the string singular. The singularity is a scalar singularity and cannot be removed. 
  This note describes fitting formulae for the gravitational waveforms generated by a rapidly rotating neutron star (e.g., newly-formed in the core collapse of a supernova) as it evolves from an initial axisymmetric configuration toward a triaxial ellipsoid. 
  The technique known as group averaging provides powerful machinery for the study of constrained systems. However, it is likely to be well defined only in a limited set of cases. Here, we investigate the possibility of using a `renormalized' group averaging in certain models. The results of our study may indicate a general connection between superselection sectors and the rate of divergence of the group averaging integral. 
  The purpose of this paper is to further investigate the solution space of self-similar spherically symmetric perfect-fluid models and gain deeper understanding of the physical aspects of these solutions. We achieve this by combining the state space description of the homothetic approach with the use of the physically interesting quantities arising in the comoving approach. We focus on three types of models. First, we consider models that are natural inhomogeneous generalizations of the Friedmann Universe; such models are asymptotically Friedmann in their past and evolve fluctuations in the energy density at later times. Second, we consider so-called quasi-static models. This class includes models that undergo self-similar gravitational collapse and is important for studying the formation of naked singularities. If naked singularities do form, they have profound implications for the predictability of general relativity as a theory. Third, we consider a new class of asymptotically Minkowski self-similar spacetimes, emphasizing that some of them are associated with the self-similar solutions associated with the critical behaviour observed in recent gravitational collapse calculations. 
  A non-static solution of Einstein's field equations of General Relativity representing the gravitational field of an axisymmetric radiation flow is obtained using the Eddington or the Kerr-Schild form for the metric. A solution obtained here manifestly corresponds to the Kerr metric with its mass-parameter, $m$, being an arbitrary function of the advanced (retarded) null-time coordinate. Then, when $m$ is constant, the solution reduces to the standard Kerr metric expressed in terms of the used null coordinate. Further, when the angular momentum parameter, $a$, a constant here, is set to zero, the solution reduces to the Vaidya metric expressed in terms of the used null-coordinate. 
  We present an analytic study of the mode-coupling phenomena for a scalar field propagating on a rotating Kerr background. Physically, this phenomena is caused by the dragging of reference frames, due to the black-hole (or star's) rotation. We find that different modes become mixed during the evolution and the asymptotic late-time tails are dominated by a mode which, in general, has an angular distribution different from the original one. We show that a rotating Kerr black hole becomes `bald' slower than a spherically-symmetric Schwarzschild black hole. 
  We consider the late-time evolution of {\it gravitational} and electromagnetic perturbations in realistic {\it rotating} Kerr spacetimes. We give a detailed analysis of the mode-coupling phenomena in rotating gravitational collapse. A consequence of this phenomena is that the late-time tail is dominated by modes which, in general, may have an angular distribution different from the original one. In addition, we show that different types of fields have {\it different} decaying rates. This result turns over the traditional belief (which has been widely accepted during the last three decades) that the late-time tail of gravitational collapse is universal. 
  In all 2d theories of gravity a conservation law connects the (space-time dependent) mass aspect function at all times and all radii with an integral of the matter fields. It depends on an arbitrary constant which may be interpreted as determining the initial value together with the initial values for the matter field. We discuss this for spherically reduced Einstein-gravity in a diagonal metric and in a Bondi-Sachs metric using the first order formulation of spherically reduced gravity, which allows easy and direct fixations of any type of gauge. The relation of our conserved quantity to the ADM and Bondi mass is investigated. Further possible applications (ideal fluid, black holes in higher dimensions or AdS spacetimes etc.) are straightforward generalizations. 
  The spinning C-metric was discovered by Plebanski and Demianski as a generalization of the standard C-metric which is known to represent uniformly accelerated non-rotating black holes. We first transform the spinning C-metric into Weyl coordinates and analyze some of its properties as Killing vectors and curvature invariants. A transformation is then found which brings the metric into the canonical form of the radiative spacetimes with the boost-rotation symmetry. By analytically continuing the metric across "acceleration horizons", two new regions of the spacetime arise in which both Killing vectors are spacelike.   We show that this metric can represent two uniformly accelerated, spinning black holes, either connected by a conical singularity, or with conical singularities extending from each of them to infinity. The radiative character of the metric is briefly discussed. 
  In continuing our series on metric-affine gravity (see Gronwald IJMP D6 (1997) 263 for Part I), we review the exact solutions in this theory. 
  Similarily as in the Ashtekar approach, the translational Chern-Simons term is, as a generating function, instrumental for a chiral reformulation of simple (N=1) supergravity. After applying the algebraic Cartan relation between spin and torsion, the resulting canonical transformation induces not only decomposition of the gravitational fields into selfdual and antiselfdual modes, but also a splitting of the Rarita-Schwinger fields into their chiral parts in a natural way. In some detail, we also analyze the consequences for axial and chiral anomalies. 
  Vacuumless defects in space-times with torsion may be obtained from vacuum defects in spacetimes without torsion.This idea is applied to planar domain walls and global monopoles.In the case of domain walls exponentially decaying Higgs type potentials are obtained.In the case of global monopoles torsion string type singularities are obtained like the string singularities in Dirac monopoles. 
  We analyze radiation reaction for synchrotron radiation by computing, via a multipole expansion, the near field evaluated on the world-line of the charge. We find that the temporal component of the self four-force agrees with the radiated power, which one calculates in the radiation zone. This is the case for each mode in the multipole decomposition. We also find agreement with the Abraham-Lorentz-Dirac equation. 
  We give new simple direct proofs in all spacetimes for the existence of asymmetric $(n,m+1)$-spinor potentials for completely symmetric $(n+1,m)$-spinors and for the existence of symmetric $(n,1)$-spinor potentials for symmetric $(n+1,0)$-spinors. These proofs introduce a `superpotential', i.e., a potential of the potential, which also enables us to get explicit statements of the gauge freedom of the original potentials. The main application for these results is the Lanczos potential $L_{ABCA'}$, of the Weyl spinor and the electromagnetic vector potential $A_{AA'}$. We also investigate the possibility of existence of a {\em symmetric} potential $H_{ABA'B'}$ for the Lanczos potential, and prove that in {\em all Einstein spacetimes} any symmetric (3,1)-spinor $L_{ABCA'}$ possesses a symmetric potential $H_{ABA'B'}$. Potentials of this type have been found earlier in investigations of some very special spinors in restricted classes of spacetimes. All of the new spinor results are translated into tensor notation, and where possible given also for four dimensional spaces of arbitrary signature. 
  We investigate the spacetime of a thick gravitating domain wall for a general potential $V(\Phi)$. Using general analytical arguments we show that all nontrivial solutions fall into two categories: those interpretable as an isolated domain wall with a cosmological event horizon, and those which are pure false vacuum de Sitter solutions. Although this latter solution is always unstable to the field rolling coherently to its true vacuum, we show that there is an additional instability to wall formation if the scalar field does not couple too strongly to gravity. Using the $\lambda \Phi^4$ and sine-Gordon models as illustrative examples, we investigate the phase space of the gravitating domain wall in detail numerically, following the solutions from weak to strong gravity. We find excellent agreement with the analytic work. Then, we analyse the domain wall in the presence of a cosmological constant finding again the two kinds of solutions, wall and de Sitter, even in the presence of a negative cosmological constant. 
  We present an analytic method for calculating the late-time tails of a linear scalar field outside a Kerr black hole. We give the asymptotic behavior at timelike infinity (for fixed $r$), at future null infinity, and along the event horizon (EH). In all three asymptotic regions we find a power-law decay. We show that the power indices describing the decay of the various modes at fixed $r$ differ from the corresponding Schwarzschild values. Also, the scalar field oscillates along the null generators of the EH (with advanced-time frequency proportional to the mode's magnetic number $m$). 
  Contrary to common belief, the standard tenet of Brans-Dicke theory reducing to general relativity when omega tends to infinity is false if the trace of the matter energy-momentum tensor vanishes. The issue is clarified in a new approach using conformal transformations. The otherwise unaccountable limiting behavior of Brans-Dicke gravity is easily understood in terms of the conformal invariance of the theory when the sources of gravity have radiation-like properties. The rigorous computation of the asymptotic behavior of the Brans-Dicke scalar field is straightforward in this new approach. 
  We explicitly show that, in the context of a recently proposed 2D dilaton gravity theory, energy conservation requires the ``natural'' Killing vector to have, asymptotically, an unusual normalization. The Hawking temperature $T_H$ is then calculated according to this prescription. 
  We investigated the stability of the spherically symmetric non-abelian (Bartnik-McKinnon) black hole solution of the SU(2) Einstein-Yang-Mills system using the multiple-scale analysis. It is found, in contrast with the vacuum situation, that a spherically symmetric oscillatory perturbation to second order cannot be constructed. The singular behavior of gravitational waves is induced by the coupling of the gravitational waves to the Yang-Mills waves. 
  Zaslavskii has suggested how to tighten Bekenstein's bound on entropy when the object is electrically charged. Recently Hod has provided a second tighter version of the bound applicable when the object is rotating. Here we derive Zaslavskii's optimized bound by considering the accretion of an ordinary charged object by a black hole. The force originating from the polarization of the black hole by a nearby charge is central to the derivation of the bound from the generalized second law. We also conjecture an entropy bound for charged rotating objects, a synthesis of Zaslavskii's and Hod's. On the basis of the no hair principle for black holes, we show that this last bound cannot be tightened further in a generic way by knowledge of ``global'' conserved charges, e.g., baryon number, which may be borne by the object. 
  We generalize simplicial minisuperspace models associated with restricting the topology of the universe to be that of a cone over a closed connected combinatorial $3-$manifold by considering the presence of a massive scalar field. By restricting all the interior edge lengths and all the boundary edge lengths to be equivalent and the scalar field to be homogenous on the $3-$space, we obtain a family of two dimensional models that include some of the most relevant triangulations of the spatial universe. After studying the analytic properties of the action in the space of complex edge lengths we determine its classical extrema.   We then obtain steepest descents contours of constant imaginary action passing through Lorentzian classical geometries yielding a convergent wavefunction of the universe, dominated by the contributions coming from these extrema. By considering these contours we justify semiclassical approximations based on those classical solutions, clearly predicting classical spacetime in the late universe. These wavefunctions are then evaluated numerically.   For all of the models examined we find wavefunctions predicting Lorentzian oscillatory behaviour in the late universe. 
  We study the existence and stability of cosmological scaling solutions of a non-minimally coupled scalar field evolving in either an exponential or inverse power law potential. We show that for inverse power law potentials there exist scaling solutions the stability of which does not depend on the coupling constant $\xi$. We then study the more involved case of exponential potentials and show that the scalar field will assymptotically behaves as a barotropic fluid when $\xi\ll 1$. The general case $\xi\not\ll 1$ is then discussed an we illustrate these results by some numerical examples. 
  We trace the origin of the cosmological constant problem to the assumption that Newton's constant $G$ sets the scale for cosmology. And then we show that once this assumption is relaxed, the very same cosmic acceleration which has served to make the cosmological constant problem so very severe instead then serves to provide us with its potential resolution. We present an alternate cosmology, one based on conformal gravity, and show that once given only that the sign of the vacuum energy density $\Lambda$ is explicitly the negative one associated with spontaneous breakdown of the scale invariance of the conformal theory (this actually being the choice of sign for $\Lambda$ which precisely leads to cosmic acceleration in conformal gravity), then that alone, no matter how big $\Lambda$ might actually be in magnitude, is sufficient to not only make its measurable contribution to current era cosmology naturally be of order one today, but to even do so in a way which is completely compatible with the recent high $z$ supernovae cosmology data. Cosmology can thus live with either a fundamental $G$ or with the large (and even potentially negative) $\Lambda$ associated with elementary particle physics phase transitions but not with both. Also, we distinguish between the free energy and the internal energy, with it being the former which determines cosmological phase transitions and the latter which is the source of the gravitational field. Then we show that, even if we make the standard ad hoc assumption that $\Lambda$ actually is quenched in standard gravity, vacuum energy is nonetheless still found to dominate standard cosmology at the time of the phase transition which produced it. However, within conformal gravity no such difficulty is encountered. 
  We calculate the renormalized vacuum average of the energy-momentum tensor of massless left-handed spinor field in the pointlike global monopole spacetime using point-separation approach. The general structure of the vacuum average of the energy-momentum tensor is obtained and expressed in terms of $<T^0_0>^{ren}$ component, explicit form of which is analyzed in great details for arbitrary solid angle deficit. 
  By analyzing the Einstein's equations for the static sphere, we find that there exists a non-singular static configuration whose radius can approach its corresponding horizon size arbitrarily. 
  We consider the conformal Einstein equations for polytropic perfect fluid cosmologies which admit an isotropic singularity. For the polytropic index gamma strictly greater than 1 and less than or equal to 2 it is shown that the Cauchy problem for these equations is well-posed, that is to say that solutions exist, are unique and depend smoothly on the data, with data consisting of simply the 3-metric of the singularity. The analogous result for gamma=1 (dust) is obtained when Bianchi type symmetry is assumed. 
  We consider the conformal Einstein equations for massless collisionless gas cosmologies which admit an isotropic singularity. After developing the general theory, we restrict to spatially-homogeneous cosmologies. We show that the Cauchy problem for these equations is well-posed with data consisting of the limiting particle distribution function at the singularity. 
  We derive a universal upper bound to the entropy of a charged system. The entropy bound follows from application of the generalized second law of thermodynamics to a gedanken experiment in which an entropy-bearing charged system falls into a charged black hole. This bound is stronger than the Bekenstein entropy bound for neutral systems. 
  Recently, we derived an improved universal upper bound to the entropy of a charged system $S \leq \pi (2E b-q^2)/ \hbar$. There was, however, some uncertainty in the value of the numerical factor which multiplies the $q^2$ term. In this paper we remove this uncertainty; we rederive this upper bound from an application of the generalized second law of thermodynamics to a gedanken experiment in which an entropy-bearing charged system falls into a Schwarzschild black hole. A crucial step in the analysis is the inclusion of the effect of the spacetime curvature on the electrostatic self-interaction of the charged system. 
  We present a mathematical framework for generating thick domain wall solutions to the coupled Einstein-scalar field equations which are (locally) plane symmetric. This approach leads naturally to two broad classes of wall-like solutions. The two classes include all previously known thick domain walls.   Although one of these classes is static and the other dynamic, the corresponding Einstein-scalar equations share the same mathematical structure independent of the assumption of any reflection symmetry. We also exhibit a class of thick static domain wall spacetimes with different asymptotic vacua. Our analyses of particle motion in such spacetimes raises the interesting possibility that static domain walls will possess a unique experimental signature. 
  We describe a procedure to identify and remove a class of non-stationary and non-harmonic interference lines from gravitational wave interferometer data.  These lines appear to be associated with the external electricity main supply, but their amplitudes are non-stationary and they do not appear at harmonics of the fundamental supply frequency. We find an empirical model able to represent coherently all the non-harmonic lines we have found in the power spectrum, in terms of an assumed reference signal of the primary supply input signal. If this signal is not available then it can be reconstructed from the same data by making use of the coherent line removal algorithm that we have described elsewhere. All these lines are broadened by frequency changes of the supply signal, and they corrupt significant frequency ranges of the power spectrum. The physical process that generates this interference is so far unknown, but it is highly non-linear and non-stationary. Using our model, we cancel the interference in the time domain by an adaptive procedure that should work regardless of the source of the primary interference. We have applied the method to laser interferometer data from the Glasgow prototype detector, where all the features we describe in this paper were observed. The algorithm has been tuned in such a way that the entire series of wide lines corresponding to the electrical interference are removed, leaving the spectrum clean enough to detect signals previously masked by them. Single-line signals buried in the interference can be recovered with at least 75 % of their original signal amplitude. 
  For smooth solutions to Maxwell's equations sourced by a smooth charge-current distribution $j_a$ in stationary, asymptotically flat spacetimes, one can prove an energy conservation theorem which asserts the vanishing of the sum of (i) the difference between the final and initial electromagnetic self-energy of the charge distribution, (ii) the net electromagnetic energy radiated to infinity (and/or into a black hole/white hole), and (iii) the total work done by the electromagnetic field on the charge distribution via the Lorentz force. A similar conservation theorem can be proven for linearized gravitational fields off of a stationary, asymptotically flat background, with the second order Einstein tensor playing the role of an effective stress-energy tensor of the linearized field. In this paper, we prove the above theorems for smooth sources and then investigate the extent to which they continue to hold for point particle sources. The ``self-energy'' of point particles is ill defined, but in the electromagnetic case, we can consider situations where, initially and finally, the point charges are stationary and in the same spatial position, so that the self-energy terms should cancel. Under certain assumptions concerning the decay behavior of source-free solutions to Maxwell's equations, we prove the vanishing of the sum of the net energy radiated to infinity and the net work done on the particle by the DeWitt-Brehme radiation reaction force. We also obtain a similar conservation theorem for angular momentum in an axisymmetric spacetime. In the gravitational case, we argue that similar conservation results should hold for freely falling point masses whose orbits begin and end at infinity. 
  After a brief review of classical probability theory (measure theory), we present an observation (due to Sorkin) concerning an aspect of probability in quantum mechanics. Following Sorkin, we introduce a generalized measure theory based on a hierarchy of ``sum-rules.'' The first sum-rule yields classical probability theory, and the second yields a generalized probability theory that includes quantum mechanics as a special case. We present some algebraic relations involving these sum-rules. This may be useful for the study of the higher-order sum-rules and possible generalizations of quantum mechanics. We conclude with some open questions and suggestions for further work. 
  We consider cosmological model in 4+1 dimensions with variable scale factor in extra dimension and static external space. The time scale factor is changing. Variations of light velocity, gravity constant, mass and pressure are determined with four-dimensional projection of this space-time. Data obtained by space probes Pioneer 10/11 and Ulysses are analyzed within the framework of this model. 
  We study the evolution of fluctuations in a universe dominated by a scalar field coupled to the Gauss-Bonnet term. During the graceful exit, we found non-negligible enhancements of both curvature perturbation and gravitational wave in the long wavelength limit, and we also found a short wavelength instability for steep background superinflation just after the completion of the graceful exit. This result for one possible graceful exit mechanism would provide a significant implication on the primordial spectrum from the string cos mology. 
  We consider the conformal Einstein equations for massless collisionless gas cosmologies which admit an isotropic singularity. It is shown that the Cauchy problem for these equations is well-posed with data consisting of the limiting particle distribution function at the singularity. 
  We present a mathematically rigorous proof that the r-mode spectrum of relativistic stars to the rotational lowest order has a continuous part. A rigorous definition of this spectrum is given in terms of the spectrum of a continuous linear operator. This study verifies earlier results by Kojima (1998) about the nature of the r-mode spectrum. 
  By using the Bondi-Sachs-van der Burg formalism we analyze the asymptotic properties at null infinity of axisymmetric electrovacuum spacetimes with a translational Killing vector and, in general, an infinite ``cosmic string'' (represented by a conical singularity) along the axis. Such spacetimes admit only a local null infinity. There is a non-vanishing news function due to the existence of the string even though there is no radiation.   We prove that if null infinity has a smooth compact cross section and the spacetime is not flat in a neighbourhood of null infinity, then the translational Killing vector must be timelike and the spacetime is stationary. The other case in which an additional symmetry of axisymmetric spacetimes admits compact cross sections of null infinity is the boost symmetry, which leads to radiative spacetimes representing ``uniformly accelerated objects''. These cases were analyzed in detail in our previous works. If the translational Killing vector is spacelike or null, corresponding to cylindrical or plane waves, some complete generators of null infinity are ``singular'' but null infinity itself can be smooth apart from these generators.   As two explicit examples of local null infinity, Schwarzschild spacetime with a string and a class of cylindrical waves with a string are discussed in detail in the Appendix. 
  We summarize the general formalism describing surface flows in three-dimensional space in a form which is suitable for various astrophysical applications. We then apply the formalism to the analysis of non-radial perturbations of self-gravitating spherical fluid shells.   Spherically symmetric gravitating shells (or bubbles) have been used in numerous model problems especially in general relativity and cosmology. A radially oscillating shell was recently suggested as a model for a variable cosmic object. Within Newtonian gravity we show that self-gravitating static fluid shells are unstable with respect to linear non-radial perturbations. Only shells (bubbles) with a negative mass (or with a charge the repulsion of which is compensated by a tension) are stable. 
  Global symmetries of the string effective action are employed to generate tilted, homogeneous Bianchi type VI_h string cosmologies from a previously known stiff perfect fluid solution to Einstein gravity. The dilaton field is not constant on the surfaces of homogeneity. The future asymptotic state of the models is interpreted as a plane wave and is itself an exact solution to the string equations of motion to all orders in the inverse string tension. An inhomogeneous generalization of the Bianchi type III model is also found. 
  Two-spinor formalism for Einstein Lagrangian is developed. The gravitational field is regarded as a composite object derived from soldering forms. Our formalism is geometrically and globally well-defined and may be used in virtually any 4m-dimensional manifold with arbitrary signature as well as without any stringent topological requirement on space-time, such as parallelizability. Interactions and feedbacks between gravity and spinor fields are considered. As is well known, the Hilbert-Einstein Lagrangian is second order also when expressed in terms of soldering forms. A covariant splitting is then analysed leading to a first order Lagrangian which is recognized to play a fundamental role in the theory of conserved quantities. The splitting and thence the first order Lagrangian depend on a reference spin connection which is physically interpreted as setting the zero level for conserved quantities. A complete and detailed treatment of conserved quantities is then presented. 
  Recently we reported that radio Doppler data generated by NASA's Deep Space Network (DSN) from the Pioneer 10 and 11 spacecraft indicate an apparent anomalous, constant, spacecraft acceleration with a magnitude $\sim 8.5\times 10^{-8}$ cm s$^{-2}$, directed towards the Sun (gr-qc/9808081). Analysis of similar Doppler and ranging data from the Galileo and Ulysses spacecraft yielded ambiguous results for the anomalous acceleration, but it was useful in that it ruled out the possibility of a systematic error in the DSN Doppler system that could easily have been mistaken as a spacecraft acceleration. Here we present some new results, including a critique suggestions that the anomalous acceleration could be caused by collimated thermal emission. Based partially on a further data for the Pioneer 10 orbit determination, the data now spans January 1987 to July 1998, our best estimate of the average Pioneer 10 acceleration directed towards the Sun is $\sim 7.5 \times 10^{-8}$ cm s$^{-2}$. 
  The existence of singularities in a closed FRW universe depends on the assumption that general relativity is valid for distances less than the Planck length. However, stationary state wave functions of the Schrodinger equation for a closed radiation-dominated FRW universe derived by Elbaz et al (General Relativity and Gravitation 29, 481, 1997) are zero at zero radius of curvature. Thus, even if general relativity is assumed valid at distances less than the Planck length, quantum mechanics seems to forbid singularities in a closed FRW universe. 
  For a quantum field living on a non - static spacetime no instantaneous Hamiltonian is definable, for this generically necessitates a choice of inequivalent representation of the canonical commutation relations at each instant of time. This fact suggests a description in terms of time - dependent Hilbert spaces, a concept that fits naturally in a (consistent) histories framework. Our primary tool for the construction of the quantum theory in a continuous -time histories format is the recently developed formalism based on the notion of the history group . This we employ to study a model system involving a 1+1 scalar field in a cavity with moving boundaries.  The instantaneous (smeared) Hamiltonian and a decoherence functional are then rigorously defined so that finite values for the time - averaged particle creation rate are obtainable through the study of energy histories. We also construct the Schwinger - Keldysh closed- time - path generating functional as a ``Fourier transform'' of the decoherence functional and evaluate the corresponding n - point functions. 
  We rephrase the derivation of black hole radiation so as to take into account, at the level of transition amplitudes, the change of the geometry induced by the emission process. This enlarged description reveals that the dynamical variables which govern the emission are the horizon area and its conjugate time variable. Their conjugation is established through the boundary term at the horizon which must be added to the canonical action of general relativity in order to obtain a well defined action principle when the area varies. These coordinates have already been used by Teitelboim and collaborators to compute the partition function of a black hole. We use them to show that the probability to emit a particle is given by $e^{- \Delta A/4}$ where $\Delta A$ is the decrease in horizon area induced by the emission. This expression improves Hawking result which is governed by a temperature (given by the surface gravity) in that the specific heat of the black hole is no longer neglected. The present derivation of quantum black hole radiation is based on the same principles which are used to derive the first law of classical black hole thermodynamics. Moreover it also applies to quantum processes associated with cosmological or acceleration horizons. These two results indicate that not only black holes but all event horizons possess an entropy which governs processes according to quantum statistical thermodynamics. 
  We study the possible existence of charged and neutral black holes in the Bergmann-Wagoner class of scalar-tensor theories (STT) of gravity in four dimensions. The existence of black holes is shown for anomalous versions of these theories, with a negative kinetic term in the Lagrangian. The Hawking temperature of these holes is zero, while the horizon area is (in most cases) infinite. As a special case, the Brans-Dicke theory is studied in more detail, and two kinds of infinite-area black holes are revealed, with finite and infinite proper time needed for an infalling particle to reach the horizon; among them, analyticity properties select a discrete subfamily of solutions, parametrized by two integers, which admit an extension beyond the horizon. The causal structure and stability of these solutions with respect to small radial perturbations is discussed. As a by-product, the stability properties of all spherically symmetric electrovacuum STT solutions are outlined. 
  The problem of the quantum modes of the scalar free field on anti-de Sitter backgrounds with an arbitrary number of space dimensions is considered. It is shown that this problem can be solved by using the same quantum numbers as those of the nonrelativistic oscillator and two parameters which give the energy quanta and respectively the ground state energy. This last one is known to be just the conformal dimension of the boundary field theory of the AdS/CFT conjecture. 
  We propose a re-formulation of the Einstein evolution equations that cleanly separates the conformal degrees of freedom and the non-conformal degrees of freedom with the latter satisfying a first order strongly hyperbolic system. The conformal degrees of freedom are taken to be determined by the choice of slicing and the initial data, and are regarded as given functions (along with the lapse and the shift) in the hyperbolic part of the evolution.   We find that there is a two parameter family of hyperbolic systems for the non-conformal degrees of freedom for a given set of trace free variables. The two parameters are uniquely fixed if we require the system to be ``consistently trace-free'', i.e., the time derivatives of the trace free variables remains trace-free to the principal part, even in the presence of constraint violations due to numerical truncation error. We show that by forming linear combinations of the trace free variables a conformal hyperbolic system with only physical characteristic speeds can also be constructed. 
  Quasinormal mode (QNM) gravitational radiation from black holes is expected to be observed in a few years. A perturbative formula is derived for the shifts in both the real and the imaginary part of the QNM frequencies away from those of an idealized isolated black hole. The formulation provides a tool for understanding how the astrophysical environment surrounding a black hole, e.g., a massive accretion disk, affects the QNM spectrum of gravitational waves. We show, in a simple model, that the perturbed QNM spectrum can have interesting features. 
  Using a recently developed perturbation theory for uasinormal modes (QNM's), we evaluate the shifts in the real and imaginary parts of the QNM frequencies due to a quasi-static perturbation of the black hole spacetime. We show the perturbed QNM spectrum of a black hole can have interesting features using a simple model based on the scalar wave equation. 
  We study the thermal evolution of a pulsar after a glitch in which the energy is released from a relative compact region. A set of relativistic thermal transport and energy balance equations is used to study the thermal evolution, without making the assumption of spherical symmetry. We use an exact cooling model to solve this set of differential equtions. Our results differ significantly from those obtained under the assumption of spherical symmetry. Even for young pulsars with a hot core like the Vela pulsar, we find that a detectable hot spot can be observed after a glitch. The results suggest that the intensity variation and the relative phases of hard X-ray emissions in different epoches can provide important information on the equation of state. 
  A system of self-gravitating massive fermions is studied in the framework of the general-relativistic Thomas-Fermi model. We study the properties of the free energy functional and its relation to Einstein's field equations. A self-gravitating fermion gas we then describe by a set of Thomas-Fermi type self-consistency equations. 
  We present several filtering methods which can be used as triggers for the detection of gravitational wave bursts in interferometric detectors. All the methods are compared to matched filtering with the help of a figure of merit based on the detection of supernovae signals simulated by Zwerger and Muller. 
  In this paper the second Lyapunov method is used to study the stability of the de Sitter phase of cosmic expansion when the source of the gravitational field is a viscous fluid. Different inflationary scenarios related with reheating and decay of mini-blackholes into radiation are investigated using an effective fluid described by time--varying thermodynamical quantities. 
  We study the role played by multiply-connectedness in the time evolution of the energy E(t) of a radiating system that lies in static flat space-time manifolds M_4 whose t=const spacelike sections M_3 are compact in at least one spatial direction. The radiation reaction equation of the radiating source is derived for the case where M_3 has any non-trivial flat topology, and an exact solution is obtained. We also show that when the spacelike sections are multiply-connected flat 3-manifolds the energy E(t) exhibits a reverberation pattern with discontinuities in the derivative of E(t) and a set of relative minima and maxima, followed by a growth of E(t). It emerges from this result that the compactness in at least one spatial direction of Minkowski space-time is sufficient to induce this type of topological reverberation, making clear that our radiating system is topologically fragile. An explicit solution of the radiation reaction equation for the case where M_3 = R^2 x S^1 is discussed, and graphs which reveal how the energy varies with the time are presented and analyzed. 
  We give an explicit expression for gravitational energy, written solely in terms of physical spacetime geometry, which in suitable limits agrees with the total Arnowitt-Deser-Misner and Trautman-Bondi-Sachs energies for asymptotically flat spacetimes and with the Abbot-Deser energy for asymptotically anti-de Sitter spacetimes. Our expression is a boundary value of the standard gravitational Hamiltonian. Moreover, although it stands alone as such, we derive the expression by picking the zero-point of energy via a ``lightcone reference.'' 
  The $\omega\to\infty$ limit of Brans-Dicke theory is studied with the help of the conformal transformation approach without resorting, however, to the conformal invariance property of this formalism, that is shown to be spurious. 
  A simple example is given to show that the gauge equivalence classes of physical states in Chern Simons theory are not in one-to-one correspondence with those of Einstein gravity in three spacetime dimensions. The two theories are therefore not equivalent. It is shown that including singular metrics into general relativity has more, and in fact a quite counter-intuitive, impact on the theory than one naively expects. 
  A point of view is presented, according to which, the well known picture with the Schwarzschild black hole in canonical general relativity is one in a whole class of conformal representations of the same physical situation, that are physically equivalent. Symmetry arguments are presented that favour a conformal picture without singularity instead of the Schwarzschild one. 
  Stability of the r-modes in rapidly rotating white dwarf stars is investigated. Improved estimates of the growth times of the gravitational-radiation driven instability in the r-modes of the observed DQ Her objects are found to be longer (probably considerably longer) than 6x10^9y. This rules out the possibility that the r-modes in these objects are emitting gravitational radiation at levels that could be detectable by LISA. More generally it is shown that the r-mode instability can only be excited in a very small subset of very hot (T>10^6K), rather massive (M>0.9M_sun) and very rapidly rotating (P_min<P<1.2P_min) white dwarf stars. Further, the growth times of this instability are so long that these conditions must persist for a very long time (t>10^9y) to allow the amplitude to grow to a dynamically significant level. This makes it extremely unlikely that the r-mode instability plays a significant role in any real white dwarf stars. 
  A scalar field nonminimally coupled to gravity is studied in the canonical framework, using self-dual variables. The corresponding constraints are first class and polynomial. To identify the real sector of the theory, reality conditions are implemented as second class constraints, leading to three real configurational degrees of freedom per space point. Nevertheless, this realization makes non-polynomial some of the constraints. The original complex symplectic structure reduces to the expected real one, by using the appropriate Dirac brackets. For the sake of preserving the simplicity of the constraints, an alternative method preventing the use of Dirac brackets, is discussed. It consists of converting all second class constraints into first class by adding extra variables. This strategy is implemented for the pure gravity case. 
  We discuss numerical solutions of Einstein's field equation describing static, spherically symmetric conglomerations of a photon gas. These equations imply a back reaction of the metric on the energy density of the photon gas according to Tolman's equation. The 3-fold of solutions corresponds to a class of physically different solutions which is parameterized by only two quantities, e.g. mass and surface temperature. The energy density is typically concentrated on a shell because the center contains a repelling singularity, which can, however, not be reached by timelike or null geodesics. The physical relevance of these solutions is completely open, although their existence may raise some doubts w.r. to the stability of black holes. 
  This is a contribution to a book on quantum gravity and philosophy. I discuss nature and origin of the problem of quantum gravity. I examine the knowledge that may guide us in addressing this problem, and the reliability of such knowledge. In particular, I discuss the subtle modification of the notions of space and time engendered by general relativity, and how these might merge into quantum theory. I also present some reflections on methodological questions, and on some general issues in philosophy of science which are are raised by, or a relevant for, the research on quantum gravity. 
  We analyse in details the problems which one faces trying to quantize a scalar field on the spacelike cylinder being the simple example of a spacetime with closed timelike curves. Our analysis brings to light the fact that the usual set of positive and negative frequency solutions of the field equation turns out to be incomplete. The consequence of this fact is that the usual formulation of quantum field theory breaks down on such a spacetime. We postulate the completeness principle and build on its basis the modified quantization procedure. As an example, the Hadamard function and $<\phi^2>$ for the scalar field on the spacelike cylinder are calculated. It is shown that the ``naive'' method of images gives the same results of calculation. 
  It is shown explicitly that in the framework of Bohmian quantum gravity, the equations of motion of the space-time metric are Einstein's equations plus some quantum corrections. It is observed that these corrections are not covariant. So that in the framework of Bohmian quantum gravity the general covariance principle breaks down at the individual level. This principle is restored at the statistical level. 
  It is shown explicitly that in the framework of Bohmian quantum gravity, the equations of motion of the space-time metric are Einstein's equations plus some quantum corrections. It is observed that these corrections are not covariant. So that in the framework of Bohmian quantum gravity the general covariance principle breaks down at the individual level. This principle is restored at the statistical level. 
  Recently\cite{BQG}, it was shown that quantum effects of matter could be identified with the conformal degree of freedom of the space-time metric. Accordingly, one can introduce quantum effects either by making a scale transformation (i.e. changing the metric), or by making a conformal transformation (i.e. changing all physical quantities). These two ways are investigated and compared. Also, it is argued that, the ultimate formulation of such a quantum gravity theory should be in the framework of the scalar-tensor theories. 
  Recently, it was shown that the quantum effects of the matter, could be used to determine the conformal degree of freedom of the space-time metric. So both gravity and quantum are geometrical features. Gravity determines the causal structure of the space-time, while quantum determines the scale of the space-time. In this article, it is shown that it is possible to use the scalar-tensor framework to build a unified theory in which both quantum and gravitational effects are present. 
  The canonical quantization of a Schwarzschild black hole yields a picture of the black hole that is shown to be equivalent to a collection of oscillators whose density of levels is commensurate with that of the statistical bootstrap model. Energy eigenstates of definite parity exhibit the Bekenstein mass spectrum, $M \sim \sqrt{\cal N} M_p$, where ${\cal N} \in {\bf N}$. From the microcanonical ensemble, we derive the statistical entropy of the black hole by explicitly counting the microstates corresponding to a macrostate of fixed total energy. 
  Although the Unruh and Hawking phenomena are commonly linked to field quantization in "accelerated" coordinates or in curved spacetimes, we argue that they are deeply rooted at the classical level. We maintain in particular that these effects should be best understood by considering how the special-relativistic notion of "particle" gets blurred when employed in theories including accelerated observers or in general-relativistic theories, and that this blurring is an instantiation of a more general behavior arising when the principle of equivalence is used to generalize classical or quantum special-relativistic theories to curved spacetimes or accelerated observers. A classical analogue of the Unruh effect, stemming from the non-invariance of the notion of "electromagnetic radiation" as seen by inertial and accelerated observers, is illustrated by means of four gedanken-experimente. The issue of energy balance in the various cases is also briefly discussed. 
  In this brief communication we show why superclusters would naturally arise in the universe. 
  We compare the different approaches presently available in literature to probe the vacuum structure of quantum fields in classical electromagnetic and gravitational backgrounds. We compare the results from the Bogolubov transformations and the effective Lagrangian approach with the response of monopole detectors (of the Unruh-DeWitt type) in non-inertial frames in flat spacetime and in inertial frames in different types of classical electromagnetic backgrounds. We also carry out such a comparison in inertial and rotating frames when boundaries are present in flat spacetime. We find that the results from these different approaches do not, in general, agree with each other. We attempt to identify the origin of these differences and then go on to discuss its implications for classical gravitational backgrounds. 
  The quantum interest conjecture of Ford and Roman states that any negative energy flux in a free quantum field must be preceded or followed by a positive flux of greater magnitude, and the surplus of positive energy grows the further the positive and negative fluxes are apart. In addition, the maximum possible separation between the positive and negative energy decreases the larger the amount of negative energy. We prove that the quantum interest conjecture holds for arbitrary fluxes of non-interacting scalar fields in 4D Minkowski spacetime, and discuss the consequences in attempting to violate the second law of thermodynamics using negative energy. We speculate that quantum interest may also hold for the Electromagnetic and Dirac fields, and might be applied to certain curved spacetimes. 
  Newtonian Cosmology is commonly used in astrophysical problems, because of its obvious simplicity when compared with general relativity. However it has inherent difficulties, the most obvious of which is the non-existence of a well-posed initial value problem. In this paper we investigate how far these problems are met by using the post-Newtonian approximation in cosmology. 
  It is argued that the ambiguity introduced by the renormalization in the effective action of a four-dimensional renormalizable quantum field theory is at most a local polynomial action of canonical dimension four. The allowed ambiguity in the expected stress-energy tensor of a massive scalar field is severely restricted by this fact. 
  This talk is based on my work in collaboration with Thibault Damour. We compare the probing power of different classes of gravity experiments: solar-system tests (weak-field regime), binary-pulsar tests (strong-field regime), and future gravitational-wave observations of inspiralling binaries (strong-field effects detected in our weak-gravitational-field conditions). This is done within the most natural class of alternative theories to general relativity, namely tensor-scalar theories, in which the gravitational interaction is mediated by one tensor field (g_munu) together with one or several scalar fields (phi). Our main conclusion is that strong-field tests are qualitatively different from weak-field experiments: They constrain theories which are strictly indistinguishable from general relativity in the solar system. We also show that binary-pulsar data are so precise that they already rule out the theories for which scalar effects could have been detected with LIGO or VIRGO. This proves that it is therefore sufficient to compute the `chirp' templates within general relativity. 
  We investigate a self-gravitating thick domain wall for a $\lambda \Phi^4$ potential. The system of scalar and Einstein equations admits two types of non-trivial solutions: domain wall solutions and false vacuum-de Sitter solutions. The existence and stability of these solutions depends on the strength of the gravitational interaction of the scalar field, which is characterized by the number $\epsilon$. For $\epsilon \ll 1$, we find a domain wall solution by expanding the equations of motion around the flat spacetime kink. For ``large'' $\epsilon$, we show analytically and numerically that only the de Sitter solution exists, and that there is a phase transition at some $\epsilon_{\rm max}$ which separates the two kinds of solution. Finally, we comment on the existence of this phase transition and its relation to the topology of the domain wall spacetime. 
  Recent work on state sum models of quantum gravity in 3 and 4 dimensions has led to interest in the `quantum tetrahedron'. Starting with a classical phase space whose points correspond to geometries of the tetrahedron in R^3, we use geometric quantization to obtain a Hilbert space of states. This Hilbert space has a basis of states labeled by the areas of the faces of the tetrahedron together with one more quantum number, e.g. the area of one of the parallelograms formed by midpoints of the tetrahedron's edges. Repeating the procedure for the tetrahedron in R^4, we obtain a Hilbert space with a basis labelled solely by the areas of the tetrahedron's faces. An analysis of this result yields a geometrical explanation of the otherwise puzzling fact that the quantum tetrahedron has more degrees of freedom in 3 dimensions than in 4 dimensions. 
  Classically, all topologies are allowed as solutions to the Einstein equations. However, one does not observe any topological structures on medium range distance scales, that is scales that are smaller than the size of the observed universe but larger than the microscopic scales for which quantum gravity becomes important. Recently, Friedman, Schleich and Witt have proven that there is topological censorship on these medium range distance scales: the Einstein equations, locally positive energy, and local predictability of physics imply that any medium distance scale topological structures cannot be seen. More precisely, we show that the topology of physically reasonable isolated systems is shrouded from distant observers, or in other words there is a topological censorship principle. 
  Recent arguments have indicated that the sum over histories formulation of quantum amplitudes for gravity should include sums over conifolds, a set of histories with more general topology than that of manifolds. This paper addresses the consequences of conifold histories in gravitational functional integrals that also include scalar fields. This study will be carried out explicitly for the generalized Hartle-Hawking initial state, that is the Hartle-Hawking initial state generalized to a sum over conifolds. In the perturbative limit of the semiclassical approximation to the generalized Hartle-Hawking state, one finds that quantum field theory on Einstein conifolds is recovered. In particular, the quantum field theory of a scalar field on de Sitter spacetime with $RP^3$ spatial topology is derived from the generalized Hartle-Hawking initial state in this approximation. This derivation is carried out for a scalar field of arbitrary mass and scalar curvature coupling. Additionally, the generalized Hartle-Hawking boundary condition produces a state that is not identical to but corresponds to the Bunch-Davies vacuum on $RP^3$ de Sitter spacetime. This result cannot be obtained from the original Hartle-Hawking state formulated as a sum over manifolds as there is no Einstein manifold with round $RP^3$ boundary. 
  We apply the scale factor duality transformations introduced in the context of the effective string theory to the anisotropic Bianchi-type models. We find dual models for all the Bianchi-types [except for types $VIII$ and $IX$] and construct for each of them its explicit form starting from the exact original solution of the field equations. It is emphasized that the dual Bianchi class $B$ models require the loss of the initial homogeneity symmetry of the dilatonic scalar field. 
  We derive again the upper entropy bound for a charged object by employing thermodynamics of the Kerr-Newman black hole linearised with respect to its electric charge 
  A general formal definition of a theory of space and time compatible with the inertia principle is given. The formal definition of reference frame and inertial equivalence between reference frames are used to construct the class of inertial frames. Then, suitable cocycle relations among the coefficients of space-time transformations between inertial frames are established. The kinematical meaning of coefficients and their reciprocity properties are discussed in some detail. Finally, a rest frame map family is introduced as the most general constitutive assumption to obtain the coefficients and to define a theory of space and time. Four meaningful examples are then presented. 
  We investigate the implications for the measurability of distances of a covariant dimensionful ``$\kappa$'' deformation of D=4 relativistic symmetries, with quantum time coordinate and modified Heisenberg algebra. We show that the structure of the deformed mass-shell condition has significant implications for measurement procedures relying on light probes, whereas in the case of heavy probes the most sizeable effect is due to the nontrivial commutation relation between three-momenta and quantum time coordinate. We argue that these findings might indicate that $\kappa$-Poincar\'e symmetries capture some aspects of the physics of the Quantum-Gravity vacuum. 
  It is well known that a spherically symmetric constant density static star, modeled as a perfect fluid, possesses a bound on its mass m by its radial size R given by 2m/R \le 8/9 and that this bound continues to hold when the energy density decreases monotonically. The existence of such a bound is intriguing because it occurs well before the appearance of an apparent horizon at m = R/2. However, the assumptions made are extremely restrictive. They do not hold in a humble soap bubble and they certainly do not approximate any known topologically stable field configuration. We show that the 8/9 bound is robust by relaxing these assumptions. If the density is monotonically decreasing and the tangential stress is less than or equal to the radial stress we show that the 8/9 bound continues to hold through the entire bulk if m is replaced by the quasi-local mass. If the tangential stress exceeds the radial stress and/or the density is not monotonic we cannot recover the 8/9 bound. However, we can show that 2m/R remains strictly bounded away from unity by constructing an explicit upper bound which depends only on the ratio of the stresses and the variation of the density. 
  Olum (PRL 81 3567-3570, 1998) has defined "superluminality" as the ability of a signal path to carry information faster than any neighbouring signal path, and has suggested that this requires a negative energy-density. However, this condition can be created without exotic matter if we are only sending information along the delivery path in one particular direction, and restrict ourselves to experiments that do not involve the speed of any counterpropagating or return-trip signals. Although negative energy-densities may be required for enhanced transit speeds in both directions along a single path at the same time, a traveller will normally not need (or want!) to travel in two opposing directions at once, so the condition of bidirectionality that gives rise to the negative energy condition may be unnecessarily restrictive. 
  Charged black holes, both spherically symmetric and rotating, in the low energy limit of string theory (Einstein-Maxwell-dilaton theory) are compared to analogous geometries in pure general relativity. We describe various physical differences and investigate some experiments which can distinguish between the two theories. In particular we discuss the gyro-magnetic ratios of rotating black holes and the propagation of light on black hole backgrounds. For the former we obtain an expression in the Einstein frame (EF) which is different from the one in the String frame (SF). This (and other results) can be used to test the stringy nature of matter. For a binary system consisting of a star and a rotating black hole, we give estimates of the damping of electro-magnetic radiation coming from the star due to the existence of a scalar component of gravity. 
  Localized astronomical sources like a double stellar system, rotating neutron star, or a massive black hole at the center of the Milky Way emit periodic gravitational waves. For a long time only a far-zone contribution of gravitational fields of the localized sources (plane-wave-front approximation) were a matter of theoretical analysis. We demonstrate how this analysis can be extended to take into account near-zone and intermediate-zone contributions as well. The formalism is used to calculate gravitational-wave corrections to the Shapiro time delay in binary pulsars and low-frequency (LF) pulsar timing noise produced by an ensemble of double stars in our galaxy. 
  In a previous work General Relativity has been presented as a microscopic theory of finite and discrete point-like fields that we associate to a classical description of gravitons. The standard macroscopic continuous field is retrieved as an average-valued field through an integration over these gravitons. Here we discuss extreme alternative (the Gauss's and the Coulomb's) ways of obtaining and interpreting the averaged fields, how they depend on the kind of measurements involved, and how do they fit with the experimental data. The field measurements in the classical tests of general relativity correspond to the Coulomb's mode whereas the determination of the overall spacetime curvature in a cosmological scale is clearly a Gauss's mode. As a natural consequence there is no missing mass and, therefore, no such a need of dark mass as the value predicted by General Relativity, in the context of the Gauss's mode, agrees with the observed one. 
  We survey some philosophical aspects of the search for a quantum theory of gravity, emphasising how quantum gravity throws into doubt the treatment of spacetime common to the two `ingredient theories' (quantum theory and general relativity), as a 4-dimensional manifold equipped with a Lorentzian metric. After an introduction, we briefly review the conceptual problems of the ingredient theories and introduce the enterprise of quantum gravity We then describe how three main research programmes in quantum gravity treat four topics of particular importance: the scope of standard quantum theory; the nature of spacetime; spacetime diffeomorphisms, and the so-called problem of time. By and large, these programmes accept most of the ingredient theories' treatment of spacetime, albeit with a metric with some type of quantum nature; but they also suggest that the treatment has fundamental limitations. This prompts the idea of going further: either by quantizing structures other than the metric, such as the topology; or by regarding such structures as phenomenological. We discuss this in Section \ref{Sec:TowardsQuST}. 
  An estimate for the classical action for a Bianchi VI_h homogeneous spatially closed cosmology is presented as a function of b, a parameter of the model that is proportional to the relative rotation of the average inertial frame and the bulk of matter in the universe. It is assumed (through the equation of state) that a relativistic early universe is followed by a matter-dominated late universe. The action is used in a saddlepoint approximation to a semiclassical estimate for the wave function in quantum cosmology to explain why our inertial frame seems not to rotate relative to the stars. The saddlepoint is at b=0, as would be expected. Application of the saddlepoint approximation leads to the result that only those classical geometries whose action differs from the saddlepoint value for the action by an amount less than Planck's constant contribute significantly to the integration to give the present value of the wave function. Using estimates for our universe implies that only those classical geometries for which the present relative rotation rate of inertial frames and matter are less than about 10^(-130) radians per year contribute significantly to the integration. This is well below the limit set by experiment. The result depends on the ratio of the Hubble distance to the Planck length, but does not depend on the details of the theory of quantum gravity. 
  The extreme smallness of both the Planck length, on the one side, and the ratio of the gravitational to the electrical forces between, say, two electrons, on the other side has led to a widespread belief that the realm of quantum gravity is beyond terrestrial experiments. A series of classical and quantum arguments are put forward to dispel this view. It is concluded that whereas the smallness of the Planck length and the ratio of gravitational to electrical forces, does play its own essential role in nature, it does not make quantum gravity a science where humans cannot venture to probe her secrets. In particular attention is drawn to the latest neutron and atomic interferometry experiments, and to gravity wave interferometers. The latter, as Giovanni Amelino-Camelia argues [Nature 398, 216 (1999)], can be treated as probes of space-time fuzziness down to Planck length for certain quantum-gravity models. 
  In this work spherically symmetric solutions to 5D Kaluza-Klein theory, with "electric" and/or "magnetic" fields are examined. Different relative strengths of the "electric" and "magnetic" charges of the solutions are studied by varying certain parameters in our metric ansatz. As the strengths of these two charges are varied the resultant spacetime exhibits an interesting "evolution". 
  The canonical ``loop'' formulation of quantum gravity is a mathematically well defined, background independent, non perturbative standard quantization of Einstein's theory of General Relativity. Some among the most meaningful results of the theory are: 1) the complete calculation of the spectrum of geometric quantities like the area and the volume and the consequent physical predictions about the structure of the space-time at the Plank scale; 2) a microscopical derivation of the Bekenstein-Hawking black-hole entropy formula. Unfortunately, despite recent results, the dynamical aspect of the theory (imposition of the Wheller-De Witt constraint) remains elusive.   After a short description of the basic ideas and the main results of loop quantum gravity we show in which sence the exponential of the super Hamiltonian constraint leads to the concept of spin foam and to a four dimensional formulation of the theory. Moreover, we show that some topological field theories as the BF theory in 3 and 4 dimension admits a spin foam formulation. We argue that the spin-foams/spin-networks formalism it is the natural framework to discuss loop quantum gravity and topological field theory. 
  We obtain an exact simple solution of the Einstein equation describing a spherically symmetric cosmological model without the big-bang or any other kind of singularity. The matter content of the model is shear free isotropic fluid with radial heat flux and it satisfies the weak and strong energy conditions. It is pressure gradient combined with heat flux that prevents occurrence of singularity. So far all known non-singular models have non-zero shear. This is the first shear free non-singular model, which is also spherically symmetric. 
  As in the grand canonical treatment of Reissner - Nordstr\"{o}m black holes in anti-de Sitter spacetime, the canonical ensemble formulation also shows that non-extremal black holes tend to have lower action than extremal ones. However, some small non-extremal black holes have higher action, leading to the possibility of transitions between non-extremal and extremal black holes. 
  The expected amplitudes and spectral slopes of relic gravitational waves, plus their specific correlation properties associated with the phenomenon of squeezing, may allow the registration of relic (squeezed) gravitational waves by the first generation of sensitive gravity-wave detectors. 
  The interferometry-based experimental tests of quantum properties of space-time which the author sketched out in a recent short Letter [Nature 398 (1999) 216] are here discussed in self-contained fashion. Besides providing detailed derivations of the results already announced in the previous Letter, some new results are also derived; in particular, the analysis is extended to a larger class of scenarios for space-time fuzziness and an absolute bound on the measurability of the amplitude of a gravity wave is obtained. It is argued that these studies could be helpful for the search of a theory describing a first stage of partial unification of Gravity and Quantum Mechanics. 
  We extend the investigation of the gravitational collapse of a spherically symmetric Yang-Mills field in Einstein gravity and show that, within the black hole regime, a new kind of critical behavior arises which separates black holes formed via Type I collapse from black holes formed through Type II collapse. Further, we provide evidence that these new attracting critical solutions are in fact the previously discovered colored black holes with a single unstable mode. 
  The computation of gravitational radiation generated by the coalescence of inspiralling binary black holes is nowdays one of the main goals of numerical relativity. Perturbation theory has emerged as an ubiquitous tool for all those dynamical evolutions where the two black holes start close enough to each other, to be treated as single distorted black hole (close limit approximation), providing at the same time useful benchmarks for full numerical simulations. Here we summarize the most recent developments to study evolutions of perturbations around rotating (Kerr) black holes. The final aim is to generalize the close limit approximation to the most general case of two rotating black holes in orbit around each other, and thus provide reliable templates for the gravitational waveforms in this regime. For this reason it has become very important to know if these predictions can actually be trusted to larger separation parameters (even in the region where the holes have distinct event horizons). The only way to extend the range of validity of the linear approximation is to develop the theory of second order perturbations around a Kerr hole, by generalizing the Teukolsky formalism. 
  We investigate the formation of a locally naked singularity in the collapse of radiation shells in an expanding Vaidya-deSitter background. This is achieved by considering the behaviour of non-spacelike and radial geodesics originating at the singularity. A specific condition is determined for the existence of radially outgoing, null geodesics originating at the singularity which, when this condition is satisfied, becomes locally naked. This condition turns out to be the same as that in the collapse of radiation shells in an asymptotically flat background. Therefore, we have, at least for the case considered here, established that the asymptotic flatness of the spacetime is not essential for the development of a locally naked singularity. Our result then unequivocally supports the view that no special role be given to asymptotic observers (or, for that matter, any set of observers) in the formulation of the Cosmic Censorship Hypothesis. 
  It is argued that the so-called holographic principle will obstruct attempts to produce physically realistic models for the unification of general relativity with quantum mechanics, unless determinism in the latter is restored. The notion of time in GR is so different from the usual one in elementary particle physics that we believe that certain versions of hidden variable theories can -- and must -- be revived. A completely natural procedure is proposed, in which the dissipation of information plays an essential role. Unlike earlier attempts, it allows us to use strictly continuous and differentiable classical field theories as a starting point (although discrete variables, leading to fermionic degrees of freedom, are also welcome), and we show how an effective Hilbert space of quantum states naturally emerges when one attempts to describe the solutions statistically. Our theory removes some of the mysteries of the holographic principle; apparently non-local features are to be expected when the quantum degrees of freedom of the world are projected onto a lower-dimensional black hole horizon. Various examples and models illustrate the points we wish to make, notably a model showing that massless, non interacting neutrinos are deterministic. 
  It was recently suggested that the gravitational action could contain a scale-dependent cosmological term, depending on the length or momentum scale characteristic of the processes under consideration. In this work we explore a simple possible consequence of this assumption. We compute the field generated in empty space by a static spherical source (the Schwarzschild metric), using the modified action. The resulting static potential turns out to contain a tiny non-Newtonian component which depends on the size of the test particles. The possible relevance of this small correction for the analysis of the recent Pioneers data [J.D. Anderson et al., Phys. Rev. Lett. 81 (1998) 2858] is briefly discussed. 
  It is well known that in four or more dimensions, there exist exotic manifolds; manifolds that are homeomorphic but not diffeomorphic to each other. More precisely, exotic manifolds are the same topological manifold but have inequivalent differentiable structures. This situation is in contrast to the uniqueness of the differentiable structure on topological manifolds in one, two and three dimensions. As exotic manifolds are not diffeomorphic, one can argue that quantum amplitudes for gravity formulated as functional integrals should include a sum over not only physically distinct geometries and topologies but also inequivalent differentiable structures. But can the inclusion of exotic manifolds in such sums make a significant contribution to these quantum amplitudes? This paper will demonstrate that it will. Simply connected exotic Einstein manifolds with positive curvature exist in seven dimensions. Their metrics are found numerically; they are shown to have volumes of the same order of magnitude. Their contribution to the semiclassical evaluation of the partition function for Euclidean quantum gravity in seven dimensions is evaluated and found to be nontrivial. Consequently, inequivalent differentiable structures should be included in the formulation of sums over histories for quantum gravity. 
  The classical first law of thermodynamic for Kerr-Newmann black hole (KNBH) is generalized to that in quantum form on event horizon. Then four quantum conservation laws on the KNBH equilibrium radiation process are derived, and Bekenstein-Hawking's relation S=A/4 is recovered. It can be argued that the classical entropy of black hole arise from the quantum entropy of field quanta or quasi-particles inside the hole. 
  This is the second paper in a series describing a numerical implementation of the conformal Einstein equation. This paper deals with the technical details of the numerical code used to perform numerical time evolutions from a "minimal" set of data.   We outline the numerical construction of a complete set of data for our equations from a minimal set of data. The second and the fourth order discretisations, which are used for the construction of the complete data set and for the numerical integration of the time evolution equations, are described and their efficiencies are compared. By using the fourth order scheme we reduce our computer resource requirements --- with respect to memory as well as computation time --- by at least two orders of magnitude as compared to the second order scheme. 
  This paper is an extended version of the talk given at 19th Texas Symposium of Relativistic Astrophysics and Cosmology, Paris, 1998. It reviews of some recent work; mathematical details are skipped. It is well-known that a choice of gauge in generally covariant models has a twofold pupose: not only to render the dynamics unique, but also to define the spacetime points. A geometric way of choosing gauge that is not based on coordinate conditions---the so-called covariant gauge fixing---is described. After a covariant gauge fixing, the dynamics is unique and the background manifold points are well-defined, but the description remains invariant with respect to all diffeomorphisms of the background manifold. Transformations between different covariant gauge fixings form the well-known Bergmann-Komar group. Each covariant gauge fixing determines a so-called Kucha\v{r} decomposition. The construction of the quantum theory is based on the Kucha\v{r} form of the action and the Dirac method of operator constraints. It is demonstrated that the Bergmann-Komar group is too large to be implementable by unitary maps in the quantum domain. 
  The paper summarizes the background of Expensive Nondecelerative Universe model and its main consequences for gravitation. Applying the Vaidya metrics, the model allows for the localization and determination of the density and quantity of gravitational energy created by a body with the mass m in the distance r. The consequences are manifested both in a macrosystem (Hawking's phenomenon of black holes evaporation) and microworld phenomenon (far-infrared spectral properties) 
  The geometric operators of area, volume, and length, depend on a fundamental length l of quantum geometry which is a priori arbitrary rather than equal to the Planck length l_P. The fundamental length l and the Immirzi parameter $\gamma$ determine each other. With any l the entropy formula is rendered most naturally in units of the length gap sqrt{{sqrt 3}/2} (sqrt{gamma} l).   Independently of the choice of l, the black hole entropy derived from quantum geometry in the limit of classical geometry is completely consistent with the Bekenstein-Hawking form.   The extremal limit of 1-puncture states of the quantum surface geometry corresponds rather to an extremal string than to a classical horizon. 
  In this talk we will argue that, when gravitons are taken into account, the solution to the semiclassical Einstein equations (SEE) is not physical. The reason is simple: any classical device used to measure the spacetime geometry will also feel the graviton fluctuations. As the coupling between the classical device and the metric is non linear, the device will not measure the `background geometry' (i.e. the geometry that solves the SEE). As a particular example we will show that a classical particle does not follow a geodesic of the background metric. Instead its motion is determined by a quantum corrected geodesic equation that takes into account its coupling to the gravitons. This analysis will also lead us to find a solution to the so-called gauge fixing problem: the quantum corrected geodesic equation is explicitly independent of any gauge fixing parameter. 
  Asymptotically flat spacetimes with one Killing vector field are considered. The Killing equations are solved asymptotically using polyhomogeneous expansions (i.e. series in powers of 1/r an ln r), and solved order by order. The solution to the leading terms of these expansions yield the asymptotic form of the Killing vector field. The possible classes of Killing fields are discussed by analysing their orbits on null infinity. The integrability conditions of the Killing equations are used to obtain constraints on the components of the Weyl tensor (\Psi_0, \Psi_1, \Psi_2) and on the shear (\sigma). The behaviour of the solutions to the constraint equations is studied. It is shown that for Killing fields that are non-supertranslational the characteristics of the constraint equations are the orbits of the restriction of the Killing field to null infinity. As an application, boost-rotation symmetric spacetimes are considered. The constraints on \Psi_0 are used to study the behaviour of the coefficients that give rise to the Newman-Penrose constants, if the spacetime is non-polyhomogeneous, or the logarithmic Newman-Penrose constants if the spacetime is polyhomogeneous. 
  Dynamical vacuum energy or quintessence, a slowly varying and spatially inhomogeneous component of the energy density with negative pressure, is currently consistent with the observational data. One potential difficulty with the idea of quintessence is that couplings to ordinary matter should be strongly suppressed so as not to lead to observable time variations of the constants of nature. We further explore the possibility of an explicit coupling between the quintessence field and the curvature. Since such a scalar field gives rise to another gravity force of long range ($\simg H^{-1}_0$), the solar system experiments put a constraint on the non-minimal coupling: $|\xi| \siml 10^{-2}$. 
  A qualitative analysis is presented for spatially flat, isotropic and homogeneous cosmologies derived from the string effective action when the combined effects of a dilaton, modulus, two-form potential and central charge deficit are included. The latter has significant effects on the qualitative dynamics. The analysis is also directly applicable to the anisotropic Bianchi type I cosmology. 
  The generalization of Sommers--Sen spinor connection for spinor fields, associated with a distribution $V_4^3$ and on this basis the equations for Weyl and Dirac null vector fields on complexificated $V_4^3$ are obtained.  We interpret the obtained results by examining the interaction of spinor fields with the inertial forces. 
  There exists a two parameter action, the variation of which produces both the geodesic equation and the geodesic deviation equation. In this paper it is shown that this action can be quantized by the canonical method, resulting in equations which generalize the Klein-Gordon equation. The resulting equations might have applications, and also show that entirely unexpected systems can be quantized. The possible applications of quantized geodesic deviation are to: i)the spreading wave packet in quantum theory, ii)and also to the one particle to many particle problem in second quantized quantum field theory. 
  Chrusciel and Galloway constructed a Cauchy horizon that is nondifferentiable on a dense set. We prove that in a certain class of Cauchy horizons densely nondifferentiable Cauchy horizons are generic. We show that our class of densely nondifferentiable Cauchy horizons implies the existence of densely nondifferentiable Cauchy horizons arising from partial Cauchy surfaces and also the existence of densely nondifferentiable black hole event horizons. 
  The main problem that we will face in the data analysis for continuous gravitational-wave sources is processing of a very long time series and a very large parameter space. We present a number of analytic and numerical tools that can be useful in such a data analysis. These consist of methods to calculate false alarm probabilities, use of probabilistic algorithms, application of signal splitting, and accurate estimation of parameters by means of optimization algorithms. 
  We develop a close-limit approximation to the head-on collision of two neutron stars similar to that used to treat the merger of black hole binaries . This approximation can serve as a useful benchmark test for future fully nonlinear studies. For neutron star binaries, the close-limit approximation involves assuming that the merged object can be approximated as a perturbed, stable neutron star during the ring-down phase of the coalescence. We introduce a prescription for the construction of initial data sets, discuss the physical plausibility of the various assumptions involved, and briefly investigate the character of the gravitational radiation produced during the merger. The numerical results show that several of the merged objects fluid pulsation modes are excited to a significant level. 
  If $\gamma$-ray bursts (GRBs) are accompanied by gravitational wave bursts (GWBs) the correlated output of two gravitational wave detectors evaluated in the moments just prior to a GRB will differ from that evaluated at times not associated with a GRB. We can test for this difference independently of any model of the GWB signal waveform. If we invoke a model for the GRB source population and GWB radiation spectral density we can find a confidence interval or upper limit on the root-mean-square GWB signal amplitude in the detector waveband. To illustrate we adopt a simple, physically motivated model and estimate that initial LIGO detector observations coincident with 1000 GRBs could lead us to exclude, with 95% confidence, associated GWBs with $h_{RMS} \gtrsim 1.7 \times 10^{-22}$. This result does not require the detector noise be Gaussian or that any inter-detector correlated noise be measured or measurable; it does not require advanced or a priori knowledge of the source waveform; and the limits obtained on the wave-strength improve with the number of observed GRBs. 
  LIGO --- The Laser Interferometer Gravitational-Wave Observatory --- is one of several large projects being undertaken in the United States, Europe and Japan to detect gravitational radiation. The novelty and precision of these instruments is such that large volumes of data will be generated in an attempt to find a small number of weak signals, which can be identified only as subtle changes in the instrument output over time. In this paper, I discuss the how the nature of the LIGO experiment determines the size of the data archive that will be produced, how the nature of the analyses that must be used to search the LIGO data for signals determines the anticipated access patterns on the archive, and how the LIGO data analysis system is designed to cope with the problems of LIGO data analysis. 
  The present analytical understanding on the nature of the singularities which form at the endstate of gravitational collapse of massive fluid bodies ("stars") is reviewed. Special emphasis is devoted to the issue of physical reasonability of the models. 
  This is not for the faint of heart, for we here provide the full details concerning the statement and proof of a generalized Geroch conjecture involving not the usual analytic functions but instead functions that are merely C^3. 
  We report on numerical results from an independent formalism to describe the quasi-equilibrium structure of nonsynchronous binary neutron stars in general relativity. This is an important independent test of controversial numerical hydrodynamic simulations which suggested that nonsynchronous neutron stars in a close binary can experience compression and even collapse prior to the last stable circular orbit. We show that the interior density indeed increases as irrotational binary neutron stars approach their last orbits for particular values of the compaction ratio. The observed compression is however at a significantly reduced level. 
  It is shown that the characteristic observed radius, velocity, and temperature of a typical galaxy can be inferred from Planck action constant through a phenomenological scaling law on all cosmological scales. 
  The goal of these lecture notes is to introduce the developing research area of gravitational-wave phenomenology. In more concrete terms, they are meant to provide an overview of gravitational-wave sources and an introduction to the interpretation of real gravitational wave detector data. They are, of course, limited in both regards. Either topic could be the subject of one or more books, and certainly more than the few lectures possible in a summer school. Nevertheless, it is possible to talk about the problems of data analysis and give something of their flavor, and do the same for gravitational wave sources that might be observed in the upcoming generation of sensitive detectors. These notes are an attempt to do just that. 
  Using optimal matched filtering, we search 25 hours of data from the LIGO 40-meter prototype laser interferometric gravitational-wave detector for gravitational-wave chirps emitted by coalescing binary systems within our Galaxy. This is the first test of this filtering technique on real interferometric data. An upper limit on the rate R of neutron star binary inspirals in our Galaxy is obtained: with 90% confidence, R< 0.5/hour. Similar experiments with LIGO interferometers will provide constraints on the population of tight binary neutron star systems in the Universe. 
  Characteristic methods show excellent promise in the evolution of single black hole spacetimes. The effective coupling with matter fields may help the numerical exploration of important astrophysical systems such as neutron star black hole binaries. To this end we investigate formalisms for numerical relativistic hydrodynamics which can be adaptable to null (characteristic) foliations of the spacetime. The feasibility of the procedure is demonstrated with one-dimensional results on the evolution of self-gravitating matter accreting onto a dynamical black hole. 
  Problems of absolute G measurements, its temporal and range variations from both experimental and theoretical points of view are discussed, and a new universal space project for measuring G, G(r) and G-dot promising an improvement of our knowledge of these quantities by 2-3 orders is advocated. 
  Quantization in the minisuperspace of non minimal scalar-tensor theories leads to a partial differential equation which is non separable. Through a conformal transformation we can recast the Wheeler-DeWitt equation in an integrable form, which corresponds to the minimal coupling case, whose general solution is known. Performing the inverse conformal transformation in the solution so found, we can construct the corresponding one in the original frame. This procedure can also be employed with the bohmian trajectories. In this way, we can study the classical limit of some solutions of this quantum model. While the classical limit of these solutions occurs for small scale factors in the Einstein's frame, it happens for small values of the scalar field non minimally coupled to gravity in the Jordan's frame, which includes large scale factors. 
  A surface theoretic view of non-perturbative quantum gravity as "spin-foams" was proposed by Baez. A possibility of constructing such a model was studied some time ago based on (2+1) dimensional general relativity as a reformulation of the Ponzano-Regge model in Riemannian spacetime. In the present work, a model based on (3+1) dimensional general relativity in Riemannian spacetime is presented. The construction is explicit and calculable in details. For a physical application, a computation formula for spacetime volume density correlations is presented. Remarks for further investigations are made. 
  If the energy momentum tensor contains bulk viscous stresses violating the dominant energy condition (DOC) the energy spectra of the relic gravitons (produced at the time of the DOC's violation) increase in frequency in a calculable way. In a general relativistic context we give examples where the DOC is only violated for a limited amount of time after which the ordinary (radiation dominated) evolution takes place. We connect our discussion to some recent remarks of Grishchuk concerning the detectability of the stochastic gravitational wave background by the forthcoming interferometric detectors. 
  By using the 't Hooft "brick wall" model and the Pauli-Villars regularization scheme we calculate the statistical-mechanical entropy arising from the minimally coupled scalar fields which rotate with the azimuthal angular velocity $\Omega_0=\Omega_H$ ($\Omega_H$ is the angular velocity of the black hole horizon) in the general four-dimensional non-extreme stationary axisymmetric black hole space-time. We also show, for the Kerr-Newman and the Einstein-Maxwell dilaton-axion black holes, that the statistical-mechanical entropy obtained from our derivation and the quantum thermodynamical entropy by the conical singularity method are equivalent. 
  The field theoretical description of the general relativity (GR) is further developed. The action for the gravitational field and its sources is given explicitely. The equations of motion and the energy-momentum tensor for the gravitational field are derived by applying the variational principle. We have succeeded in constructing the unique gravitational energy-momentum tensor which is 1) symmetric, 2) conserved due to the field equations, and 3) contains not higher than the first order derivatives of the field variables. It is shown that the Landau-Lifshitz pseudotensor is an object most closely related to the derived energy-momentum tensor. 
  We study the dynamical description of gravity, the appropriate definition of the scalar field energy-momentum tensor, and the interrelation between them in scalar-tensor theories of gravity. We show that the quantity which one would naively identify as the energy-momentum tensor of the scalar field is not appropriate because it is spoiled by a part of the dynamical description of gravity. A new connection can be defined in terms of which the full dynamical description of gravity is explicit, and the correct scalar field energy-momentum tensor can be immediately identified. Certain inequalities must be imposed on the two free functions (the coupling function and the potential) that define a particular scalar-tensor theory, to ensure that the scalar field energy density never becomes negative. The correct dynamical description leads naturally to the Einstein frame formulation of scalar-tensor gravity which is also studied in detail. 
  A classification of Brans-Dicke theories of gravitation, based on the behaviour of the dimensionless gravitational coupling constant, is given. It is noted that the discussion takes place in the current literature, about which of the two distinguished conformal frames in which scalar-tensor theories of gravity can be formulated: the Jordan frame and the Einstein frame, is the physical one, may, in most cases, be meaningless for both frames may belong to the same conformal class. It is also noted that the Jordan frame formulation of Brans-Dicke gravity with ordinary matter nonminimally coupled, that is shown to be just the Jordan frame formulation of general relativity, is scale-invariant, unlike the situation with the Jordan frame representation of Brans-Dicke gravity with matter minimally coupled (the original formulation of Brans-Dicke theory), where the presence of nonzero mass ordinary matter breaks the scale-invariance of the theory. 
  It is shown that the Hartle-Hawking state of a scalar field is a maximum of entanglement entropy in the space of pure quantum states satisfying the condition that backreaction is finite. In other words, the Hartle-Hawking state is a curved-space analogue of the EPR state, which is also a maximum of entanglement entropy. 
  The field equations of general relativity can be written as first order differential equations in the Weyl tensor, the Weyl tensor in turn can be written as a first order differential equation in a three index tensor called the Lanczos tensor. The Lanczos tensor plays a similar role in general relativity to that of the vector potential in electro-magnetic theory. The Aharonov-Bohm effect shows that when quantum mechanics is applied to electro-magnetic theory the vector potential is dynamically significant, even when the electro-magnetic field tensor $F_{ab}$ vanishes. Here it is assumed that in the quantum realm the Lanczos tensor is dynamically significant, and this leads to an attempt to quantize the gravitational field by pursuing the analogy between the vector field and the Lanczos tensor. 
  Solutions to gravity with quadratic Lagrangians are found for the simple case where the only nonconstant metric component is the lapse $N$ and the Riemann tensor takes the form $R^{t}_{.itj}=-k_{i}k_{j}, i,j=1,2,3$; thus these solutions depend on cross terms in the Riemann tensor and therefore complement the linearized theory where it is the derivatives of the Riemann tensor that matter. The relationship of this metric to the null gravitational radiation metric of Peres is given. Gravitaional energy Poynting vectors are construcetd for the solutions and one of these, based on the Lanczos tensor, supports the indication in the linearized theory that nonnull gravitational radiation can occur. 
  We propose a gauge theory of gravitation. The gauge potential is a connection of the Super SL(2,C) group. A MacDowell-Mansouri type of action is proposed where the action is quadratic in the Super SL(2,C) curvature and depends purely on gauge connection. By breaking the symmetry of the Super SL(2,C) topological gauge theory to SL(2,C), a spinor metric is naturally defined. With an auxiliary anti-commuting spinor field, the theory is reduced to general relativity. The Hamiltonian variables are related to the ones given by Ashtekar. The auxiliary spinor field plays the role of Witten spinor in the positive energy proof for gravitation. 
  We present first results of the non-linear evolution of rotating relativistic stars obtained with an axisymmetric relativistic hydrodynamics code in a fixed spacetime. As initial data we use stationary axisymmetric and perturbed configurations. We find that, in order to prevent (numerical) angular momentum loss at the surface layers of the star a high-resolution grid (or a numerical scheme that retains high order at local extrema) is needed. For non-rotating stars, we compute frequencies of radial and non-radial small-amplitude oscillations, which are in excellent agreement with linear normal mode frequencies computed in the Cowling approximation. As a first application of our code, quasi-radial modes of rapidly rotating relativistic stars are computed. By generalizing our numerical code to 3-D, we plan to study the evolution and non-linear dynamics of toroidal oscillations (r-modes) of rapidly rotating neutron stars, which are a promising source of gravitational waves. 
  I describe the current status of a collaboration with J.D. Romano, R.H. Price, and W. Krivan to model the geometry of and gravitational radiation emitted by a binary system of compact objects in the regime where non-perturbative gravitational effects exist, but the rate of inspiral is still small relative to the orbital frequency. The method of looking for a stationary spacetime which approximates the evolving solution is initially being tested on a simpler model with an additional translational symmetry. This report consists of a general description of the method, followed by summaries of three techniques in varying stages of development: the simplification of the Einstein equations in the presence of two commuting Killing vectors which form a non-orthogonally-transitive symmetry group, the boundary conditions appropriate to the balance of ingoing and outgoing radiation needed to reconcile a stationary radiating solution with conservation of energy, and the treatment of gravitational waves far from the sources as linearized perturbations to the Levi-Civita spacetime. The poster presentation with which this paper is associated is available on line at http://www-itp.unibe.ch/~whelan/poster.ps.gz and the current status of the project is described at http://www-itp.unibe.ch/~whelan/qsbi.html 
  We improve and extend Shapiro's model of a relativistic, compact object which is stable in isolation but is driven dynamically unstable by the tidal field of a binary companion. Our compact object consists of a dense swarm of test particles moving in randomly-oriented, initially circular, relativistic orbits about a nonrotating black hole. The binary companion is a distant, slowly inspiraling point mass. The tidal field of the companion is treated as a small perturbation on the background Schwarzschild geometry near the hole; the resulting metric is determined by solving the perturbation equations of Regge and Wheeler and Zerilli in the quasi-static limit. The perturbed spacetime supports Bekenstein's conjecture that the horizon area of a near-equilibrium black hole is an adiabatic invariant. We follow the evolution of the system and confirm that gravitational collapse can be induced in a compact collisionless cluster by the tidal field of a binary companion. 
  Using a complex representation of the Debney-Kerr-Schild (DKS) solutions and the Kerr theorem we analyze the boosted Kerr geometries and give the exact and explicit expressions for the metrics, the principal null congruences, the coordinate systems and the location of the singularities for arbitrary value and orientation of the boost with respect to the angular momentum. In the limiting, ultrarelativistic case we obtain light-like solutions possessing diverging and twisting principal null congruences and having, contrary to the known pp-wave limiting solutions, a non-zero value of the total angular momentum. The implications of the above results in various related fields are discussed. 
  We demonstrate that evolutions of three-dimensional, strongly non-linear gravitational waves can be followed in numerical relativity, hence allowing many interesting studies of both fundamental and observational consequences. We study the evolution of time-symmetric, axisymmetric {\it and} non-axisymmetric Brill waves, including waves so strong that they collapse to form black holes under their own self-gravity. The critical amplitude for black hole formation is determined. The gravitational waves emitted in the black hole formation process are compared to those emitted in the head-on collision of two Misner black holes. 
  The astrophysics of compact objects, which requires Einstein's theory of general relativity for understanding phenomena such as black holes and neutron stars, is attracting increasing attention. In general relativity, gravity is governed by an extremely complex set of coupled, nonlinear, hyperbolic-elliptic partial differential equations. The largest parallel supercomputers are finally approaching the speed and memory required to solve the complete set of Einstein's equations for the first time since they were written over 80 years ago, allowing one to attempt full 3D simulations of such exciting events as colliding black holes and neutron stars. In this paper we review the computational effort in this direction, and discuss a new 3D multi-purpose parallel code called ``Cactus'' for general relativistic astrophysics. Directions for further work are indicated where appropriate. 
  A single spherical antenna is capable of measuring the direction and polarization of a gravitational wave. It is possible to solve the inverse problem using only linear algebra even in the presence of noise. The simplicity of this solution enables one to explore the error on the solution using standard techniques. In this paper we derive the error on the direction and polarization measurements of a gravitational wave. We show that the solid angle error and the uncertainty on the wave amplitude are direction independent. We also discuss the possibility of determining the polarization amplitudes with isotropic sensitivity for any given gravitational wave source. 
  We calculate the flux from a spherical mirror which is expanding or contracting with nearly uniform acceleration. We find that the flux at an exterior point (which could in principle be a functional of the mirror's past history) is actually found to be a local function, depending on the first and second time derivatives of acceleration at the retarded time. 
  An open system is not conservative because energy can escape to the outside. An open system by itself is thus not conservative. As a result, the time-evolution operator is not hermitian in the usual sense and the eigenfunctions (factorized solutions in space and time) are no longer normal modes but quasinormal modes (QNMs) whose frequencies $\omega$ are complex. QNM analysis has been a powerful tool for investigating open systems. Previous studies have been mostly system specific, and use a few QNMs to provide approximate descriptions. Here we review recent developments which aim at a unifying treatment. The formulation leads to a mathematical structure in close analogy to that in conservative, hermitian systems. Many of the mathematical tools for the latter can hence be transcribed. Emphasis is placed on those cases in which the QNMs form a complete set for outgoing wavefunctions, so that in principle all the QNMs together give an exact description of the dynamics. Applications to optics in microspheres and to gravitational waves from black holes are reviewed, and directions for further development are outlined. 
  It is found that conformally coupled induced gravity with gradient torsion gives a dilaton gravity in Riemann geometry. In the Einstein frame of the dilaton gravity the conformal symmetry is hidden and a non-vanishing cosmological constant is not plausible due to the constraint of the conformal coupling. 
  A new path equation in absolute parallelism (AP) geometry is derived. The equation is a generalization of three path equations derived in a previous work. It can be considered as a geodesic equation modified by a torsion term, whose numerical coefficient jumps by steps of one half. The torsion term is parametrized using the fine structure constant. It is suggested that the new equation may describe the trajectories of spinning particles under the influence of a gravitational field, and the torsion term represents a type of interaction between the quantum spin of the moving particle and the background field.     Weak field limits of the new path equation show that the gravitational potential felt by a spinning particle is different from that felt by a spinless particle (or a macroscopic body).     As a byproduct, and in order to derive the new path equation, the AP-space is reconstructed using a new affine connexion preserving metricity. The new AP-structure has non-vanishing curvature. In certain limits, the new AP-structure can be reduced either to the ordinary Riemannian space, or to the conventional AP-space. 
  We introduce a simple and straight-forward averaging procedure, which is a generalization of one which is commonly used in electrodynamics, and show that it possesses all the characteristics we require for linearized averaging in general relativity and cosmology -- for weak-field and perturbed FLRW situations. In particular we demonstrate that it yields quantities which are approximately tensorial in these situations, and that its application to an exact FLRW metric yields another FLRW metric, to first-order in integrals over the local coordinates. Finally, we indicate some important limits of any linearized averaging procedure with respect to cosmological perturbations which are the result of averages over large amplitude small and intermediate scale inhomogeneities, and show our averaging procedure can be approximately implemented by that of Zotov and Stoeger in these cases. 
  In the first part of this paper, we show that the semiclassical Einstein-Langevin equation, introduced in the framework of a stochastic generalization of semiclassical gravity to describe the back reaction of matter stress-energy fluctuations, can be formally derived from a functional method based on the influence functional of Feynman and Vernon. In the second part, we derive a number of results for background solutions of semiclassical gravity consisting of stationary and conformally stationary spacetimes and scalar fields in thermal equilibrium states. For these cases, fluctuation-dissipation relations are derived. We also show that particle creation is related to the vacuum stress-energy fluctuations and that it is enhanced by the presence of stochastic metric fluctuations. 
  We compute the two-point function and the renormalized expectation value of the stress tensor of a quantum field interacting with a nucleating bubble. Two simple models are considered. One is the massless field in the Vilenkin-Ipser-Sikivie spacetime describing the gravitational field of a reflection symmetric domain wall. The other is vacuum decay in flat spacetime where the quantum field only interacts with the tunneling field on the bubble wall. In both cases the stress tensor is of the perfect fluid form. The assymptotic form of the equation of state are given for each model. In the VIS case, we find that $p=-(1/3)\rho$, where the energy density $\rho$ is dominated by the gradients of supercurvature modes. 
  We present in this paper a fully covariant quantization of the minimally-coupled massless field on de Sitter space. We thus obtain a formalism free of any infrared (e.g logarithmic) divergence. Our method is based on a rigorous group theoretical approach combined with a suitable adaptation (Krein spaces) of the Wightman-G\"{a}rding axiomatic for massless fields (Gupta-Bleuler scheme). We make explicit the correspondence between unitary irreducible representations of the de Sitter group and the field theory on de Sitter space-time. The minimally-coupled massless field is associated with a representation which is the lowest term of the discrete series of unitary representations of the de Sitter group. In spite of the presence of negative norm modes in the theory, no negative energy can be measured: expressions as $\le n_{k_1}n_{k_2}...|T_{00}|n_{k_1}n_{k_2}...\re$ are always positive. 
  The aim of this letter is to clarify the relationships between Hawking radiation and the scattering of light by matter falling into a black hole. To this end we analyze the S-matrix elements of a model composed of a massive infalling particle (described by a quantized field) and the radiation field. These fields are coupled by current-current interactions and propagate in the Schwarzschild geometry. As long as the photons energy is much smaller than the mass of the infalling particle, one recovers Hawking radiation since our S-matrix elements identically reproduce the Bogoliubov coefficients obtained by treating the trajectory of the infalling particle classically. But after a brief period, the energy of the `partners' of Hawking photons reaches this mass and the production of thermal photons through these interactions stops. The implications of this result are discussed. 
  We give a relativistic spin network model for quantum gravity based on the Lorentz group and its q-deformation, the Quantum Lorentz Algebra.   We propose a combinatorial model for the path integral given by an integral over suitable representations of this algebra. This generalises the state sum models for the case of the four-dimensional rotation group previously studied in gr-qc/9709028.   As a technical tool, formulae for the evaluation of relativistic spin networks for the Lorentz group are developed, with some simple examples which show that the evaluation is finite in interesting cases. We conjecture that the `10J' symbol needed in our model has a finite value. 
  Heisenberg showed in the early days of quantum theory that the uncertainty principle follows as a direct consequence of the quantization of electromagnetic radiation in the form of photons. As we show here the gravitational interaction of the photon and the particle being observed modifies the uncertainty principle with an additional term. From the modified or gravitational uncertainty principle it follows that there is an absolute minimum uncertainty in the position of any particle, of order of the Planck length. A modified uncertainty relation of this form is a standard result of superstring theory, but the derivation given here is based on simpler and rather general considerations with either Newtonian gravitational theory or general relativity theory. 
  We consider gravitational wave modes in the FRW metrics in a de Sitter phase and show that the state space splits into many unitarily inequivalent representations of the canonical commutation relations. Non-unitary time evolution is described as a trajectory in the space of the representations. The generator of time evolution is related to the entropy operator. The thermodynamic arrow of time is shown to point in the same direction of the cosmological arrow of time. The vacuum is a two-mode SU(1,1) squeezed state of thermo field dynamics. The link between expanding geometry, squeezing and thermal properties is exhibited. 
  The paper shows that, conceptually and operationally, the speed of light as measured locally in the inertial comoving frame of a point on the rim of a rotating disk, is different from the one measured globally for a round trip along the rim, obtained dividing the length of the rim (as measured in the ''relative space'' of the disk) by the time of flight of the light beam (as measured by a clock at rest on the disk). As a consequence, contrary to some recent claims, the anisotropy found in the global value, obtained by the above procedure, in no way conflicts with the local isotropy, and the internal consistency of the special relativity theory remains unchallenged. 
  We explore the relation between noncommutative geometry, in the spectral triple formulation, and quantum mechanics. To this aim, we consider a dynamical theory of a noncommutative geometry defined by a spectral triple, and study its quantization. In particular, we consider a simple model based on a finite dimensional spectral triple (A, H, D), which mimics certain aspects of the spectral formulation of general relativity. We find the physical phase space, which is the space of the onshell Dirac operators compatible with A and H. We define a natural symplectic structure over this phase space and construct the corresponding quantum theory using a covariant canonical quantization approach. We show that the Connes distance between certain two states over the algebra A (two ``spacetime points''), which is an arbitrary positive number in the classical noncommutative geometry, turns out to be discrete in the quantum theory, and we compute its spectrum. The quantum states of the noncommutative geometry form a Hilbert space K. D is promoted to an operator *D on the direct product *H of H and K. The triple (A, *H, *D) can be viewed as the quantization of the family of the triples (A, H, D). 
  The variation procedure on a teleparallel manifold is studied. The main problem is the non-commutativity of the variation with the Hodge dual map. We establish certain useful formulas for variations and restate the master formula due to Hehl and his collaborates. Our approach is different and sometimes easier for applications. By introducing the technique of the variational matrix we find necessary and sufficient conditions for commutativity (anti-commutativity) of the variation derivative with the Hodge dual operator. A general formula for the variation of the quadratic-type expression is obtained. The described variational technique are used in two viable field theories: the electro-magnetic Lagrangian on a curved manifold and the Rumpf Lagrangian of the translation invariant gravity. 
  A four-index tensor is constructed with terms both quadratic in the Riemann tensor and linear in its second derivatives, which has zero divergence for space-times with vanishing scalar curvature. This tensor reduces in vacuum to the Bel-Robinson tensor. Furthermore, the completely timelike component referred to any observer is positive, and zero if and only if the space-time is flat (excluding some unphysical space-times). We also show that this tensor is the unique that can be constructed with these properties. Such a tensor does not exist for general gravitational fields. Finally, we study this tensor in several examples: the Friedmann-Lema\^{\i}tre-Robertson-Walker space-times filled with radiation, the plane-fronted gravitational waves, and the Vaidya radiating metric. 
  String theory suggests the existence of gravitational-strength scalar fields ("dilaton" and "moduli") whose couplings to matter violate the equivalence principle. This provides a new motivation for high-precision clock experiments, as well as a generic theoretical framework for analyzing their significance. 
  A theory of gravitation in 4D is presented with strings used in the material action in $U_4$ spacetime. It is shown that the string naturally gives rise to torsion. It is also shown that the equation of motion a string follows from the Bianchi identity, gives the identical result as the Noether conservation laws, and follows a geodesic only in the lowest order approximation. In addition, the conservation laws show that strings naturally have spin, which arises not from their motion but from their one dimensional structure. 
  Minimally coupled 4D scalar fields in Schwarzschild space-time are considered. Dimensional reduction to 2D leads to a well known anomaly induced effective action, which we consider here in a local form with the introduction of auxiliary fields. Boundary conditions are imposed on them in order to select the appropriate quantum states (Boulware, Unruh annd Israel-Hartle-Hawking). The stress tensor is then calculated and its comparison with the expected 4D form turns out to be unsuccessful. We also critically discuss in some detail a recent controversial result appeared in the literature on the same topic. 
  Using large N, 4d anomaly induced one-loop effective action for conformally invariant matter (typical GUT multiplet) we study the possibility to induce the primordial spherically symmetric wormholes at the early Universe. The corresponding effective equations are obtained in two different coordinate frames. The numerical investigation of these equations is done for matter content corresponding to ${\cal N}=4$ SU(N) super Yang-Mills theory. For some choice of initial conditions, the induced wormhole solution with increasing throat radius and increasing red-shift function is found. 
  The purpose of this paper is to present a generalized hole argument for gauge field theories and their geometrical setting in terms of fiber bundles. The generalized hole argument is motivated and extended from the spacetime hole arguments which appear in spacetime theories based on differentiable manifolds such as general relativity. Analogously, the generalized hole argument rules out fiber bundle substantivalism and, thus, a relationalistic interpretation of the geometry of fiber bundle spaces is favoured. Along the way, the concept of gauge field theories will be analyzed via considering the gauge principle and thereby hopefully clarifying certain terminological ambiguities. 
  The separated radial part of a sourceless massive complex scalar field equation on the Kerr-Newman black hole background is shown to be a generalized spin-weighted spheroidal wave equation of imaginary number order. While the separated angular part is an ordinary spheroidal wave equation. General exact solutions in integral forms and in power series expansion as well as several special solutions with physical interest are given for the radial equation in the non-extreme case. In the extreme case, power series solution to the radial equation is briefly studied. Recurrence relations between coefficients in power series expansion of general solutions and connection between the radial equation are discussed in both cases. 
  It has previously been shown [W. Rudnicki, Phys. Lett. A 224, 45 (1996)] that a generic gravitational collapse cannot result in a naked singularity accompanied by closed timelike curves. An important role in this result plays the so-called inextendibility condition, which is required to hold for certain incomplete null geodesics. In this paper, a theorem is proved that establishes some relations between the inextendibility condition and the rate of growth of the Ricci curvature along incomplete null geodesics. This theorem shows that the inextendibility condition may hold for a much more general class of singularities than only those of the strong curvature type. It is also argued that some earlier cosmic censorship results obtained for strong curvature singularities can be extended to singularities corresponding to the inextendibility condition. 
  The two-point function characterizing the stress tensor fluctuations of a massless minimally coupled field for an invariant vacuum state in de Sitter spacetime is discussed. This two-point function is explicitly computed for spacelike separated points which are geodesically connected. We show that these fluctuations are as important as the expectation value of the stress tensor itself. These quantum field fluctuations will induce fluctuations in the geometry of de Sitter spacetime. This paper is a first step towards the computation of such metric fluctuations, which may be of interest for large-scale structure formation in cosmology. The relevance of our results in this context is briefly discussed. 
  We present results of numerical computations of quasiequilibrium sequences of binary neutron stars with zero vorticity, in the general relativistic framework. The Einstein equations are solved under the assumption of a conformally flat spatial 3-metric (Wilson-Mathews approximation). The evolution of the central density of each star is monitored as the orbit shrinks in response to gravitational wave emission. For a compactification ratio M/R=0.14, the central density remains rather constant (with a slight increase, below 0.1%) before decreasing. For a higher compactification ratio M/R=0.17 (i.e. stars closer to the maximum mass configuration), a very small density increase (at most 0.3%) is observed before the decrease. This effect remains within the error induced by the conformally flat approximation. It can be thus concluded that no substantial compression of the stars is found, which would have indicated a tendency to individually collapse to black hole prior to merger. Moreover, no turning point has been found in the binding energy or angular momentum along evolutionary sequences, which may indicate that these systems do not have any innermost stable circular orbit (ISCO). 
  We study the question of prompt vs. delayed collapse in the head-on collision of two neutron stars. We show that the prompt formation of a black hole is possible, contrary to a conjecture of Shapiro which claims that collapse is delayed until after neutrino cooling. We discuss the insight provided by Shapiro's conjecture and its limitation. An understanding of the limitation of the conjecture is provided in terms of the many time scales involved in the problem. General relativistic simulations in the Einstein theory with the full set of Einstein equations coupled to the general relativistic hydrodynamic equations are carried out in our study. 
  One classical theory, as determined by an equation of motion or set of classical trajectories, can correspond to many unitarily {\em in}equivalent quantum theories upon canonical quantization. This arises from a remarkable ambiguity, not previously investigated, in the construction of the classical (and hence the quantized) Hamiltonian or Lagrangian. This ambiguity is illustrated for systems with one degree of freedom: An arbitrary function of the constants of motion can be introduced into this construction. For example, the nonrelativistic and relativistic free particles follow identical classical trajectories, but the Hamiltonians or Lagrangians, and the canonically quantized versions of these descriptions, are inequivalent. Inequivalent descriptions of other systems, such as the harmonic oscillator, are also readily obtained. 
  A multi-TeraFlop/TeraByte machine will enable the application of the Einstein theory of gravity to realistic astrophysical processes. Without the computational power, the complexity of the Einstein theory restricts most studies based on it to the quasi static/linear near-Newtonian regime of the theory.   The application of the Einstein theory to realistic astrophysical processes is bound to bring deep and far-reaching scientific discoveries, and produce results that will inspire the general public. It is an essential component in developing the new frontier of gravitational wave astronomy - an exciting new window to observe our universe.   The computational requirements of carrying out numerical simulations based on the Einstein theory is discussed with an explicit example, the coalescence of a neutron star binary.   This document is prepared for presentation at the National Computational Science Alliance User Advisory Council Meeting at NSF, June 1998, in support of the funding of a NSF TeraFlop computer. 
  We explicitly determine the expression of the electrostatic potential generated by a point charge at rest in the Schwarzschild black hole pierced by a cosmic string. We can then calculate the electrostatic self-energy. From this, we find again the upper entropy bound for a charged object by employing thermodynamics of the black hole. 
  We describe numerical techniques used in the construction of our 4th order evolution for the full Einstein equations, and assess the accuracy of representative solutions. The code is based on a null gauge with a quasi-spherical radial coordinate, and simulates the interaction of a single black hole with gravitational radiation. Techniques used include spherical harmonic representations, convolution spline interpolation and filtering, and an RK4 "method of lines" evolution. For sample initial data of "intermediate" size (gravitational field with 19% of the black hole mass), the code is accurate to 1 part in 10^5, until null time z=55 when the coordinate condition breaks down. 
  The energy localization hypothesis of the author that energy is localized in non-vanishing regions of the energy-momentum tensor implies that gravitational waves do not carry energy in vacuum. If substantiated, this has significant implications for current research. Support for the hypothesis is provided by a re-examination of Eddington's classic calculation of energy loss by a spinning rod. It is emphasized that Eddington did not monitor the entire Tolman energy integral, concentrating solely upon the change of the 'kinetic' part of the energy . The 'quadrupole formula' is thus seen to measure the kinetic energy change. When the derivative of the missing stress-trace integral is computed, it is seen to cancel the Eddington term and hence the energy of the rod is conserved, in support of the localization hypothesis. The issue of initial and final states is addressed. 
  We derive the general $\Sigma_2\times S$ solution of topologically massive gravity in vacuum and in presence of a cosmological constant. The field equations reduce to three-dimensional Einstein equations and the solution has constant Ricci tensor. We briefly discuss the emergence of non-Ricci flat solutions when spin is introduced. 
  We show that well-posed, conformally-decomposed formulations of the 3+1 Einstein equations can be obtained by densitizing the lapse and by combining the constraints with the evolution equations. We compute the characteristics structure and verify the constraint propagation of these new well-posed formulations. In these formulations, the trace of the extrinsic curvature and the determinant of the 3-metric are singled out from the rest of the dynamical variables, but are evolved as part of the well-posed evolution system. The only free functions are the lapse density and the shift vector. We find that there is a 3-parameter freedom in formulating these equations in a well-posed manner, and that part of the parameter space found consists of formulations with causal characteristics, namely, characteristics that lie only within the lightcone. In particular there is a 1-parameter family of systems whose characteristics are either normal to the slicing or lie along the lightcone of the evolving metric. 
  The static cylindrically symmetric solutions of the gravitating Abelian Higgs model form a two parameter family. In this paper we give a complete classification of the string-like solutions of this system. We show that the parameter plane is composed of two different regions with the following characteristics: One region contains the standard asymptotically conic cosmic string solutions together with a second kind of solutions with Melvin-like asymptotic behavior. The other region contains two types of solutions with bounded radial extension. The border between the two regions is the curve of maximal angular deficit of $2\pi$. 
  It is proven that the relativistic charged ball with its charge less than its mass (in natural units) cannot have a non-singular static configuration while its radius approaches its external horizon size. This conclusion does not depend on the details of charge distribution and the equation of state. The involved assumptions are (1) the ball is made of perfect fluid, (2) the energy density is everywhere non-negative. 
  We compute the graviton-induced corrections to the trajectory of a classical test particle. We show that the motion of the test particle is governed by an effective action given by the expectation value (with respect to the graviton state) of the classical action. We analyze the quantum corrected equations of motion for the test particle in two particular backgrounds: a Robertson Walker spacetime and a 2+1 dimensional spacetime with rotational symmetry. In both cases we show that the quantum corrected trajectory is not a geodesic of the background metric. 
  This is the first of two papers examining the critical collapse of spherically symmetric perfect fluids with the equation of state P = (Gamma -1)rho. Here we present the equations of motion and describe a computer code capable of simulating the extremely relativistic flows encountered in critical solutions for Gamma <= 2. The fluid equations are solved using a high-resolution shock-capturing scheme based on a linearized Riemann solver. 
  Using the renormalization group method, we improved the first order solution of the long-wavelength expansion of the Einstein equation. By assuming that the renormalization group transformation has the property of Lie group, we can regularize the secular divergence caused by the spatial gradient terms and absorb it to the background seed metric. The solution of the renormalization group equation shows that the renormalized metric describes the behavior of gravitational collapse in the expanding universe qualitatively well. 
  We investigate the validity of Thorne's hoop conjecture in non-axisymmetric spacetimes by examining the formation of apparent horizons numerically. If spaces have a discrete symmetry about one axis, we can specify the boundary conditions to determine an apparent horizon even in non-axisymmetric spaces. We implement, for the first time, the ``hoop finder'' in non-axisymmetric spaces with a discrete symmetry. We construct asymptotically flat vacuum solutions at a moment of time symmetry. Two cases are examined: black holes distributed on a ring, and black holes on a spherical surface. It turns out that calculating ${\cal C}$ is reduced to solving an ordinary differential equation. We find that even in non-axisymmetric spaces the existence or nonexistence of an apparent horizon is consistent with the inequality: ${\cal C} \siml 4\pi M$. 
  By using geometric methods and superenergy tensors, we find new simple criteria for the causal propagation of physical fields in spacetimes of any dimension. The method can be applied easily to many different theories and to arbitrary fields (such as scalar or electromagnetic ones). In particular, it provides a conservation theorem of the free gravitational field in all N-dimensional spacetimes conformally related to Einstein spaces (including vacuum solutions). In the case of general relativity, our criteria provide simple proofs and a unified treatment of conservation theorems for neutrinos, photons, electrons and all other massless and massive free spin n/2 fields. The uniqueness of the solution to the field equations also follows from our treatment under certain circumstances. 
  As the mass-energy is universally self-gravitating, the gravitational binding energy must be subtracted self-consistently from its bare mass value so as to give the physical gravitational mass. Such a self-consistent gravitational self-energy correction can be made non-perturbatively by the use of a gravitational `charging' technique, where we calculate the incremental change $dm$ of the physical mass of the cosmological object, of size $r_o$ due to the accretion of a bare mass $dM$, corresponding to the gravitational coupling-in of the successive zero-point vacuum modes, i.e., of the Casimir energy, whose bare value $\Sigma_{\bf k} \hbar ck$ is infinite. Integrating the `charging' equation, $dm = dM - (3\alpha/5)Gm\Delta M/r_o c^2$, we get a gravitational mass for the cosmological object that remains finite even in the limit of the infinite zero-point vacuum energy, i.e., without any ultraviolet cut-off imposed. Here $\alpha$ is a geometrical factor of order unity. Also, setting $r_o = c/H$, the Hubble length, we get the corresponding cosmological density parameter $\Omega \simeq 1$, without any adjustable parameter. The cosmological significance of this finite and unique contribution of the otherwise infinite zero-point vacuum energy to the density parameter can hardly be overstated.  
  The confrontation between Einstein's gravitation theory and experimental results, notably binary pulsar data, is summarized and its significance discussed. Experiment and theory agree at the 10^{-3} level or better. All the basic structures of Einstein's theory (coupling of gravity to matter; propagation and self-interaction of the gravitational field, including in strong-field conditions) have been verified. However, the theoretical possibility that scalar couplings be naturally driven toward zero by the cosmological expansion suggests that the present agreement between Einstein's theory and experiment might be compatible with the existence of a long-range scalar contribution to gravity (such as the dilaton field, or a moduli field, of string theory). This provides a new theoretical paradigm, and new motivations for improving the experimental tests of gravity. 
  We examine the field equations of a self-gravitating texture in low-energy superstring gravity, allowing for an arbitrary coupling of the texture field to the dilaton. Both massive and massless dilatons are considered. For the massless dilaton non-singular spacetimes exist, but only for certain values of the coupling, dependent on the gravitational strength of the texture; moreover, this non-singular behaviour exists only in a certain frame. For the massive dilaton, the texture induces a long-range dilaton cloud, but we expect the gravitational behaviour of the defect to be similar to that found in Einstein theory. We compare these results with those found for other global topological defects. 
  We develop a framework for constructing initial data sets for perturbations about spherically symmetric matter distributions. This framework facilitates setting initial data representing astrophysical sources of gravitational radiation involving relativistic stars. The procedure is based on Lichnerowicz-York's conformal approach to solve the constraints in Einstein's equations. The correspondence of these initial data sets in terms of the standard gauge perturbation variables in the Regge-Wheeler perturbation variables is established, and examples of initial data sets of merging neutron stars under the close-limit approximation are presented. 
  Some special solutions of the Einstein-Maxwell action with a non-negative cosmological constant and a very heavy point mass particle have been obtained. The solutions correspond to static spacetime of locally constant curvature in its spatial part and a constant magnetic field of a magnetic monopole together with deficit of angle at the location of point mass. The quantum mechanics of a point particle in these spacetimes in the absence of angular deficit has been solved algebraically both relativistically and non-relativistically. It has been also shown that these 2-dimensional Hamiltonians have the degeneracy group of GL(2,c) type and para-supersymmetry of arbitrary order or shape invariance, which is originated from a SO(4,c) group. 
  A classical and quantum mechanical generalized second law of thermodynamics in cosmology implies constraints on the effective equation of state of the universe in the form of energy conditions, obeyed by many known cosmological solutions, and is compatible with entropy bounds which forbid certain cosmological singularities. In string cosmology the second law provides new information about the existence of non-singular solutions, and the nature of the graceful exit transition from dilaton-driven inflation. 
  Starting from certain causality conditions and a discrete form of general covariance, we derive a very general family of classically stochastic, sequential growth dynamics for causal sets. The resulting theories provide a relatively accessible ``half way house'' to full quantum gravity that possibly contains the latter's classical limit (general relativity). Because they can be expressed in terms of state models for an assembly of Ising spins living on the relations of the causal set, these theories also illustrate how non-gravitational matter can arise dynamically from the causal set without having to be built in at the fundamental level. Additionally, our results bring into focus some interpretive issues of importance for causal set dynamics, and for quantum gravity more generally. 
  The structure of the phase boundary between degenerate and non-degenerate regions in Ashtekar's gravity has been studied by Bengtsson and Jacobson who conjectured that the "phase boundary" should always be null. In this paper, we reformulate the reparametrization procedure in the mapping language and distinguish a phase boundary from its image. It is shown that the image has to be null, while the nullness of the phase boundary requries more suitable criterion. 
  It has been suggested that a naked singularity may be a good candidate for a strong gravitational wave burster. The naked singularity occurs in the generic collapse of an inhomogeneous dust ball. We study odd-parity mode of gravitational waves from a naked singularity of the Lema\^{\i}tre-Tolman-Bondi space-time. The wave equation for gravitational waves are solved by numerical integration using the single null coordinate. The result is that the naked singularity is not a strong source of the odd-parity gravitational radiation although the metric perturbation grows in the central region. Therefore, the Cauchy horizon in this space-time would be marginally stable against odd-parity perturbations. 
  We review the idea of stretched horizon for extremal black holes in supersymmetric string theories, and we compute it for non-supersymmetric black holes in four dimensions. Only for small masses of the order of the Veneziano wavelength is the stretched horizon bigger than the event horizon. 
  The reheating process for the inflationary scenario is investigated phenomenologically. The decay of the oscillating massive inflaton field into light bosons is modeled after an out of equilibrium mixture of interacting fluids within the framework of irreversible thermodynamics. Self-consistent, analytic results for the evolution of the main macroscopic magnitudes like temperature and particle number densities are obtained. The models for linear and quadratic decay rates are investigated in the quasiperfect regime. The linear model is shown to reheat very slowly while the quadratic one is shown to yield explosive particle and entropy production. The maximum reheating temperature is reached much faster and its magnitude is comparable with the inflaton mass. 
  The Maxwell equations are formulated on an arbitrary (1+3)-dimensional manifold. Then, imposing a (constrained) linear constitutive relation between electromagnetic field $(E,B)$ and excitation $({\cal D},{\cal H})$, we derive the metric of spacetime therefrom. 
  We argue that quantum-gravitational fluctuations in the space-time background give the vacuum non-trivial optical properties that include diffusion and consequent uncertainties in the arrival times of photons, causing stochastic fluctuations in the velocity of light ``in vacuo''. Our proposal is motivated within a Liouville string formulation of quantum gravity that also suggests a frequency-dependent refractive index of the particle vacuum. We construct an explicit realization by treating photon propagation through quantum excitations of $D$-brane fluctuations in the space-time foam. These are described by higher-genus string effects, that lead to stochastic fluctuations in couplings, and hence in the velocity of light. We discuss the possibilities of constraining or measuring photon diffusion ``in vacuo'' via $\gamma$-ray observations of distant astrophysical sources. 
  The quantum space-time model which accounts material Reference Frames (RF) quantum effects considered for flat space-time and ADM canonical gravity. As was shown by Aharonov for RF - free material object its c.m. nonrelativistic motion in vacuum described by Schrodinger wave packet evolution which modify space coordinate operator of test particle in this RF and changes its Heisenberg uncertainty relations. In the relativistic case we show that Lorentz transformations between two RFs include the quantum corrections for RFs momentum uncertainty and in general can be formulated as the quantum space-time transformations. As the result for moving RF its Lorentz time boost acquires quantum fluctuations which calculated solving relativistic Heisenberg equations for the quantum clocks models. It permits to calculate RF proper time for the arbitrary RF quantum motion including quantum gravity metrics fluctuations. Space-time structure of canonical Quantum Gravity and its observables evolution for RF proper time discussed in this quantum space-time transformations framework. 
  We obtain a characterization of the Kerr metric among stationary, asymptotically flat, vacuum spacetimes, which extends the characterization in terms of the Simon tensor (defined only in the manifold of trajectories) to the whole spacetime. More precisely, we define a three index tensor on any spacetime with a Killing field, which vanishes identically for Kerr and which coincides in the strictly stationary region with the Simon tensor when projected down into the manifold of trajectories. We prove that a stationary asymptotically flat vacuum spacetime with vanishing spacetime Simon tensor is locally isometric to Kerr. A geometrical interpretation of this characterization in terms of the Weyl tensor is also given. Namely, a stationary, asymptotically flat vacuum spacetime such that each principal null direction of the Killing form is a repeated principal null direction of the Weyl tensor is locally isometric to Kerr. 
  In Riemann-Cartan spacetimes with torsion only its axial covector piece $A$ couples to massive Dirac fields. Using renormalization group arguments, we show that besides the familiar Riemannian term only the Pontrjagin type four-form $dA\wedge dA$ does arise additionally in the chiral anomaly, but not the Nieh-Yan term $d^\star A$, as has been claimed in a recent paper [PRD 55, 7580 (1997)]. 
  The warm inflation scenario is an alternative mechanism which can explain the isotropic and homogeneous Universe which we are living in. In this work I extend a previously introduced formalism, without the restriction of slow - roll regime. Quantum to classical transition of the fluctuations is studied by means of the "transition function" here introduced. I found that the fluctuations of radiation energy density decrease with time and the thermal equilibrium at the end of inflation holds. 
  It was recently shown that the metric functions which describe a spherically symmetric space-time with vanishing radial pressure can be explicitly integrated. We investigate the nakedness and curvature strength of the shell-focusing singularity in that space-time. If the singularity is naked, the relation between the circumferential radius and the Misner-Sharp mass is given by $R\approx 2y_{0} m^{\beta}$ with $ 1/3<\beta\le 1$ along the first radial null geodesic from the singularity. The $\beta$ is closely related to the curvature strength of the naked singularity. For example, for the outgoing or ingoing null geodesic, if the strong curvature condition (SCC) by Tipler holds, then $\beta$ must be equal to 1. We define the ``gravity dominance condition'' (GDC) for a geodesic. If GDC is satisfied for the null geodesic, both SCC and the limiting focusing condition (LFC) by Kr\'olak hold for $\beta=1$ and $y_{0}\ne 1$, not SCC but only LFC holds for $1/2\le \beta <1$, and neither holds for $1/3<\beta <1/2$, for the null geodesic. On the other hand, if GDC is satisfied for the timelike geodesic $r=0$, both SCC and LFC are satisfied for the timelike geodesic, irrespective of the value of $\beta$. Several examples are also discussed. 
  A five-dimensional (5D) generalized G\"odel-type manifolds are examined in the light of the equivalence problem techniques, as formulated by Cartan. The necessary and sufficient conditions for local homogeneity of these 5D manifolds are derived. The local equivalence of these homogeneous Riemannian manifolds is studied. It is found that they are characterized by three essential parameters $k$, $m^2$ and $\omega$: identical triads $(k, m^2, \omega)$ correspond to locally equivalent 5D manifolds. An irreducible set of isometrically nonequivalent 5D locally homogeneous Riemannian generalized G\"odel-type metrics are exhibited. A classification of these manifolds based on the essential parameters is presented, and the Killing vector fields as well as the corresponding Lie algebra of each class are determined. It is shown that the generalized G\"odel-type 5D manifolds admit maximal group of isometry $G_r$ with $r=7$, $r=9$ or $r=15$ depending on the essential parameters $k$, $m^2$ and $\omega$. The breakdown of causality in all these classes of homogeneous G\"odel-type manifolds are also examined. It is found that in three out of the six irreducible classes the causality can be violated. The unique generalized G\"odel-type solution of the induced matter (IM) field equations is found. The question as to whether the induced matter version of general relativity is an effective therapy for these type of causal anomalies of general relativity is also discussed in connection with a recent article by Romero, Tavakol and Zalaletdinov. 
  Timelike and null hypersurfaces in the degenerate space-times in the Ashtekar theory are defined in the light of the degenerate causal structure proposed by Matschull. Using the new definition of null hypersufaces, the conjecture that the "phase boundary" separating the degenerate space-time region from the non-degenerate one in Ashtekar's gravity is always null is proved under certain circumstances. 
  The Hawking radiation is one of the most interesting phenomena predicted by the theory of quantum field in curved space. The origin of Hawking radiation is closely related to the fact that a particle which marginally escapes from collapsing into a black hole is observed at the future infinity with infinitely large redshift. In other words, such a particle had a very high frequency when it was near the event horizon. Motivated by the possibility that the property of Hawking radiation may be altered by some unknowned physics which may exist beyond some critical scale, Unruh proposed a model which has higher order spatial derivative terms. In his model, the effects of unknown physics are modeled so as to be suppressed for the waves with a wavelength much longer than the critical scale, $k_0^{-1}$. Surprisingly, it was shown that the thermal spectrum is recovered for such modified models. To introduce such higher order spatial derivative terms, the Lorentz invariance must be violated because one special spatial direction needs to be chosen. In previous works, the rest frame of freely-falling observers was employed as this special reference frame. Here we give an extension by allowing a more general choice of the reference frame. Developing the method taken by Corley, % and especially focusing on subluminal case, we show that the resulting spectrum of created particles again becomes the thermal one at the Hawking temperature even if the choice of the reference frame is generalized. Using the technique of the matched asymptotic expansion, we also show that the correction to the thermal radiation stays of order $k_0^{-2}$ or smaller when the spectrum of radiated particle around its peak is concerned. 
  The gravitational collapse of a triplet scalar field is examined assuming a hedgehog ansatz for the scalar field. Whereas the seminal work by Choptuik with a single, strictly spherically symmetric scalar field found a discretely self-similar (DSS) solution at criticality with echoing period $\Delta=3.44$, here a new DSS solution is found with period $\Delta=0.46$. This new critical solution is also observed in the presence of a symmetry breaking potential as well as within a global monopole. The triplet scalar field model contains Choptuik's original model in a certain region of parameter space, and hence his original DSS solution is also a solution. However, the choice of a hedgehog ansatz appears to exclude the original DSS. 
  Although there is no relative motion among different points on a rotating disc, each point belongs to a different noninertial frame. This fact, not recognized in previous approaches to the Ehrenfest paradox and related problems, is exploited to give a correct treatment of a rotating ring and a rotating disc. Tensile stresses are recovered, but, contrary to the prediction of the standard approach, it is found that an observer on the rim of the disc will see equal lengths of other differently moving objects as an inertial observer whose instantaneous position and velocity are equal to that of the observer on the rim. The rate of clocks at various positions, as seen by various observers, is also discussed. Some results are generalized for observers arbitrarily moving in a flat or a curved spacetime. The generally accepted formula for the space line element in a non-time-orthogonal frame is found inappropriate in some cases. Use of Fermi coordinates leads to the result that for any observer the velocity of light is isotropic and is equal to $c$, providing that it is measured by propagating a light beam in a small neighborhood of the observer. 
  An algebraic quantization procedure for discretized spacetime models is suggested based on the duality between finitary substitutes and their incidence algebras. The provided limiting procedure that yields conventional manifold characteristics of spacetime structures is interpreted in the algebraic quantum framework as a correspondence principle. 
  We study the averaging problem from a point of view of variation of spatial volume $V$. We show that in the space of spherically symmetric dust solutions which are regular on the spatial manifold $S^3$ the variation $\delta V$ vanishes at the Friedmann-Lemaitre-Robertson-Walker (FLRW) solution in an appropriate sense, which supports the validity of the FLRW solution as the averaged solution. We also present the second variation $\delta^2 V$, giving the leading effect of the deviation from the FLRW solution. 
  It is shown that the problem of a possible violation of the Lorentz transformations at Lorentz factors $\gamma >5\times 10^{10} ,$ indicated by the situation which has developed in the physics of ultra-high energy cosmic rays (the absence of the GZK cutoff), has a nontrivial solution. Its essence consists in the discovery of the so-called generalized Lorentz transformations which seem to correctly link the inertial reference frames at any values of $\gamma .$ Like the usual Lorentz transformations, the generalized ones are linear, possess group properties and lead to the Einstein law of addition of 3-velocities. However, their geometric meaning turns out to be different: they serve as relativistic symmetry transformations of a flat anisotropic Finslerian event space rather than of Minkowski space. Consideration is given to two types of Finsler spaces which generalize locally isotropic Riemannian space-time of relativity theory, e. g. Finsler spaces with a partially and entirely broken local 3D isotropy. The investigation advances arguments for the corresponding generalization of the theory of fundamental interactions and for a specific search for physical effects due to local anisotropy of space-time. 
  The quantum lightcone fluctuations in flat spacetimes with compactified spatial dimensions or with boundaries are examined. The discussion is based upon a model in which the source of the underlying metric fluctuations is taken to be quantized linear perturbations of the gravitational field. General expressions are derived, in the transverse trace-free gauge, for the summation of graviton polarization tensors, and for vacuum graviton two-point functions. Because of the fluctuating light cone, the flight time of photons between a source and a detector may be either longer or shorter than the light propagation time in the background classical spacetime. We calculate the mean deviations from the classical propagation time of photons due to the changes in the topology of the flat spacetime. These deviations are in general larger in the directions in which topology changes occur and are typically of the order of the Planck time, but they can get larger as the travel distance increases. 
  Vacuum Einstein theory in three spacetime dimensions is locally trivial, but admits many solutions that are globally different, particularly if there is a negative cosmological constant. The classical theory of such locally "anti-de Sitter" spaces is treated in an elementary way, using visualizable models. Among the objects discussed are black holes, spaces with multiple black holes, their horizon structure, closed universes, and the topologies that are possible. 
  Recent evidence indicates that the Universe is open, i.e., spatially hyperbolic, longstanding theoretical preferences to the contrary notwithstanding. This makes it possible to select a vacuum state, Fock space, and particle definition for a quantized field, by requiring concordance with ordinary flat-spacetime theory at late times. The particle-number basis states thus identified span the physical state space of the field at all times. This construction is demonstrated here explicitly for a massive, minimally coupled, linear scalar field in an open, radiation-dominated Friedmann-Robertson-Walker spacetime. 
  Some general expressions are given for the coefficient of the 14th Chern form in terms of the Riemann-Christoffel curvature tensor and some of its concomitants (e.g., Pontrjagin's characteristic tensors) for n-dimensional differentiable manifolds having a general linear connection. 
  General expressions are given for the coefficients of Chern forms up to the 13th order in curvature in terms of the Riemann-Christoffel curvature tensor and some of its concomitants (e.g., Pontrjagin's characteristic tensors) for n-dimensional differentiable manifolds having a general linear connection. 
  A general expression is given for the 14th Chern form in terms of simple polynomial concomitants of the curvature 2-form for n-dimensional differentiable manifolds having a general linear connection. 
  General expressions are given for Chern forms up to the 13th order in curvature in terms of simple polynomial concomitants of the curvature 2-form for n-dimensional differentiable manifolds having a general linear connection. 
  A method for finding the world function of Robertson-Walker spacetimes is presented. It is applied to find the world function for the $k=0, \ga=2$, solution. The close point approximation for the Robertson-Walker world function is calculated upto fourth order. 
  Static spherically symmetric uncoupled scalar space-times have no event horizon and a divergent Kretschmann singularity at the origin of the coordinates. The singularity is always present so that non-static solutions have been sought to see if the singularities can develop from an initially singular free space-time. In flat space-time the Klein-Gordon equation $\Box\ph=0$ has the non-static spherically symmetric solution $\ph=\si(v)/r$, where $\si(v)$ is a once differentiable function of the null coordinate $v$. In particular the function $\si(v)$ can be taken to be initially zero and then grow, thus producing a singularity in the scalar field. A similar situation occurs when the scalar field is coupled to gravity via Einstein's equations; the solution also develops a divergent Kretschmann invariant singularity, but it has no overall energy. To overcome this Bekenstein's theorems are applied to give two corresponding conformally coupled solutions. One of these has positive ADM mass and has the properties: i) it develops a Kretschmann invariant singularity, ii)it has no event horizon, iii)it has a well-defined source, iv)it has well-defined junction condition to Minkowski space-time, v)it is asymptotically flat with positive overall energy. This paper presents this solution and several other non-static scalar solutions. The properties of these solutions which are studied are limited to the following three: i)whether the solution can be joined to Minkowski space-time, ii)whether the solution is asymptotically flat, iii)and if so what the solutions' Bondi and ADM masses are. 
  Recently Bekenstein and Mayo conjectured an entropy bound for charged rotating objects. On the basis of the No-Hair principle for black holes, they speculate that this bound cannot be improved generically based on knowledge of other ``quantum numbers'', e.g. baryon number, which may be borne by the object. Here we take a first step in the proof of this conjecture. The proof make use of a gedanken experiment in which a massive object endowed with a scalar charge is lowered adiabatically towards a Schwarzschild's black hole and than dropped into the black hole from some proper distance above the horizon. Central to the proof is the intriguing fact that the self-energy of the particle receives no contribution from the scalar charge. Thus the energy with which the object is assimilated consists of its gravitational energy alone. This of course agrees with the No-scalar-Hair principle for black holes: after the object is assimilated into the black hole, any knowledge of the scalar field properties is lost. Using the GSL, we reach the conclusion that the original entropy bound was not improved by the knowledge of the scalar charge. At the end we speculate on whether or not massive vector fields may serve in the tightening of the entropy bound. 
  We consider an empty (4+1) dimensional Kaluza-Klein universe with a negative cosmological constant and a Robertson-Walker type metric. It is shown that the solutions to Einstein field equations have degenerate metric and exhibit transitioins from a Euclidean to a Lorentzian domain. We then suggest a mechanism, based on signature transition which leads to compactification of the internal space in the Lorentzian region as $a \sim |\Lambda|^{1/2}$. With the assumption of a very small value for the cosmological constant we find that the size of the universe $R$ and the internal scale factor $a$ would be related according to $Ra\sim 1$ in the Lorentzian region. The corresponding Wheeler-DeWitt equation has exact solution in the mini-superspace giving rise to a quantum state which peaks in the vicinity of the classical solutions undergoing signature transition. 
  A generalized version of the Einstein equations in the 4-index form, containing the Riemann tensor linearly, is derived. It is shown, that the gravitational energy-momentum density tensor outside a source is represented across the Weyl tensor vanishing at the 2-index contraction. The 4-index energy-momentum density tensor for matter also is constructed. 
  A global monopole in dilatonic Einstein-Cartan gravity is presented. A linearized solution representing a global monopole interacting with a massless dilaton is found where Cartan torsion does not interact with the monopole Higgs field.Computation of the geodesic equation shows that the monopole-dilaton system generates a repulsive gravitational field.The solution is shown to break the linear approximation for certain values of torsion. 
  We obtain a class of exact solutions representing null particles moving in three-dimensional (anti-) de Sitter spaces by boosting the corresponding static point source solutions given by Deser and Jackiw. In de Sitter space the resulting solution describes two null particles moving on the (circular) cosmological horizon, while in anti-de Sitter space it describes a single null particle propagating from one side of the universe to the other. We also boost the BTZ black hole solution to the ultrarelativistic limit and obtain the solution for a spinning null particle moving in anti-de Sitter space. We find that the ultrarelativistic geometry of the black hole is exactly the same as that resulting from boosting the Deser-Jackiw solution when the angular momentum of the hole vanishes. A general class of solutions is also obtained which represents several null particles propagating in the Deser-Jackiw background. The differences between the three-dimensional and four-dimensional cases are also discussed. 
  The fluctuations of the flux radiated by an evaporating black hole will be discussed. Two approaches to this problem will be adopted. In the first, the squared flux operator is defined by normal ordering. In this case, both the mean flux and the mean squared flux are well defined local quantites. It is shown that the flux undergoes large fluctuations on a time scale of the order of the black hole's mass. Thus the semiclassical theory of gravity, in which a classical gravitational field is coupled to the expectation value of the stress tensor, breaks down below this time scale. In the second approach, one does not attempt to give meaning to the squared flux as a local quantity, but only as a time-averaged quantity. In both approaches, the mean squared mass minus the square of the mean mass grows linearly in time, but four times as fast in the second approach as in the first. 
  Beginning from an effective theory in eight dimensions, Macias, Camacho and Matos proposed an effective model for the electroweak part of the Standard Model of particles in curved spacetime. Using this model, we investigate the cosmological consequences of the electroweak interaction in the early universe. We use the approximation that, near the Planck epoch, the Yang-Mills fields behave like a perfect fluid. Then we recover the field equations of inflationary cosmology, with the Higgs field directly related to the inflaton. We present some qualitative discussion about this and analyse the behavior of isospin space using some known exact solutions. 
  We examine the solution of the constraints in spherically symmetric general relativity when spacetime has a flat spatial hypersurface. We demonstrate explicitly that given one flat slice, a foliation by flat slices can be consistently evolved. We show that when the sources are finite these slices do not admit singularities and we provide an explicit bound on the maximum value assumed by the extrinsic curvature. If the dominant energy condition is satisfied, the projection of the extrinsic curvature orthogonal to the radial direction possesses a definite sign. We provide both necessary and sufficient conditions for the formation of apparent horizons in this gauge which are qualitatively identical to those established earlier for extrinsic time foliations of spacetime, Phys. Rev. D56 7658, 7666 (1997) which suggests that these conditions possess a gauge invariant validity. 
  We show that a relativistic gas may be at ``global'' equilibrium in the expanding universe for any equation of state $0 < p \leq \rho /3$, provided that the gas particles move under the influence of a self-interacting, effective one-particle force in between elastic binary collisions. In the force-free limit we recover the equilibrium conditions for ultrarelativistic matter which imply the existence of a conformal timelike Killing vector. 
  We present an example where a justified modification of the law of propagation of light in a Robertson-Walker model of the universe leads to an identification of H_0 and q_0 different from that corresponding to the usual law of propagation along null geodesics. We conclude from this example that observed values which we would associate with the values of H_0 and q_0 with the usual interpretation correspond in fact to the values of 2H_0 and $1/2 (q_0-1)$. It is therefore possible that observed values that we usually interpret as corresponding to a moderately aged universe with accelerating expansion may in fact correspond a much older universe with a decelerating expansion. 
  We study charged boson stars in scalar-tensor (ST) gravitational theories. We analyse the weak field limit of the solutions and analytically show that there is a maximum charge to mass ratio for the bosons above which the weak field solutions are not stable. This charge limit can be greater than the GR limit for a wide class of ST theories. We numerically investigate strong field solutions in both the Brans Dicke and power law ST theories. We find that the charge limit decreases with increasing central boson density. We discuss the gravitational evolution of charged and uncharged boson stars in a cosmological setting and show how, at any point in its evolution, the physical properties of the star may be calculated by a rescaling of a solution whose asymptotic value of the scalar field is equal to its initial asymptotic value. We focus on evolution in which the particle number of the star is conserved and we find that the energy and central density of the star decreases as the cosmological time increases. We also analyse the appearance of the scalarization phenomenon recently discovered for neutron stars configurations and, finally, we give a short discussion on how making the correct choice of mass influences the argument over which conformal frame, the Einstein frame or the Jordan frame, is physical. 
  We describe a hierarchical, highly parallel computer algorithm to perform searches for unknown sources of continuous gravitational waves -- spinning neutron stars in the Galaxy -- over wide areas of the sky and wide frequency bandwidths. We optimize the algorithm for an observing period of 4 months and an available computing power of 20 Gflops, in a search for neutron stars resembling millisecond pulsars. We show that, if we restrict the search to the galactic plane, the method will detect any star whose signal is stronger than 15 times the $1\sigma$ noise level of a detector over that search period. Since on grounds of confidence the minimum identifiable signal should be about 10 times noise, our algorithm does only 50% worse than this and runs on a computer with achievable processing speed. 
  Inspired by the spin geometry theorem, two operators are defined which measure angles in the quantum theory of geometry. One operator assigns a discrete angle to every pair of surfaces passing through a single vertex of a spin network. This operator, which is effectively the cosine of an angle, is defined via a scalar product density operator and the area operator. The second operator assigns an angle to two ``bundles'' of edges incident to a single vertex. While somewhat more complicated than the earlier geometric operators, there are a number of properties that are investigated including the full spectrum of several operators and, using results of the spin geometry theorem, conditions to ensure that semiclassical geometry states replicate classical angles. 
  Spin networks, essentially labeled graphs, are ``good quantum numbers'' for the quantum theory of geometry. These structures encompass a diverse range of techniques which may be used in the quantum mechanics of finite dimensional systems, gauge theory, and knot theory. Though accessible to undergraduates, spin network techniques are buried in more complicated formulations. In this paper a diagrammatic method, simple but rich, is introduced through an association of 2 by 2 matrices to diagrams. This spin network diagrammatic method offers new perspectives on the quantum mechanics of angular momentum, group theory, knot theory, and even quantum geometry. Examples in each of these areas are discussed. 
  Motivated by recent work involving the graviton-graviton tree scattering amplitude, and its twin descriptions as the square of the Bel-Robinson tensor, $B_{\m\n\a\b}$, and as the "current-current interaction" square of gravitational energy pseudo-tensors $t_{\a\b}$,we find an exact tensor-square root equality $B_{\mn\a\b} = \pa^2_\mn t_{\a\b}$, for a combination of Einstein and Landau-Lifschitz $t_\ab$, in Riemann normal coordinates. In the process, we relate, on-shell, the usual superpotential basis for classifying pseudo-tensors with one spanned by polynomials in the curvature. 
  We extend the recently proved relation between certain models of Non-Riemannian gravitation and Einstein- Proca-Weyl theories to a class of Scalar gravity theories. This is used to present a Black-Hole Dilaton solution with non-Riemannian connection. 
  We present a time-frequency method to detect gravitational wave signals in interferometric data. This robust method can detect signals from poorly modeled and unmodeled sources. We evaluate the method on simulated data containing noise and signal components. The noise component approximates initial LIGO interferometer noise. The signal components have the time and frequency characteristics postulated by Flanagan and Hughes for binary black hole coalescence. The signals correspond to binaries with total masses between $45 M_\odot$ to $70 M_\odot$ and with (optimal filter) signal-to-noise ratios of 7 to 12. The method is implementable in real time, and achieves a coincident false alarm rate for two detectors $\approx$ 1 per 475 years. At this false alarm rate, the single detector false dismissal rate for our signal model is as low as 5.3% at an SNR of 10. We expect to obtain similar or better detection rates with this method for any signal of similar power that satisfies certain adiabaticity criteria. Because optimal filtering requires knowledge of the signal waveform to high precision, we argue that this method is likely to detect signals that are undetectable by optimal filtering, which is at present the best developed detection method for transient sources of gravitational waves. 
  In this paper it is explicitly demonstrated that the energy conservation law is kept when a detector uniformly accelerated in the Minkowski vacuum is excited and emits a particle. This fact had been hidden in conventional approaches in which detectors were considered to be forced on trajectories. To lift the veil we suggest a detector model written in terms of the Minkowski coordinates. In this model the Hamiltonian of the detector involves a classical potential term instead of the detector's fixed trajectory. The transition rate agrees with the corresponding conventional one in the limit of an infinite mass detector though even then the recoil remains. 
  Gauss-Bonnet formula is used to derive a new and simple theorem of nonexistence of vacuum static nonsingular lorentzian wormholes. We also derive simple proofs for the nonexistence of lorentzian wormhole solutions for some classes of static matter such as, for instance, real scalar fields with a generic potential obeying $\phi V'(\phi) \ge 0$ and massless fermions fields. 
  Gravitational wave emission is expected to arise from a variety of astrophysical phenomena. A new generation of detectors with sensitivity consistent with expectation from such sources is being developed. The Laser Interferometer Gravitational-Wave Observatory (LIGO), one of these ambitious undertakings, is being developed by a Caltech-MIT collaboration. It consists of two widely separated interferometers, which will be used in coincidence to search for sources from compact binary systems, spinning neutron stars, supernovae and other astrophysical or cosmological phenomena that emit gravitational waves. The construction of LIGO is well underway and preparations are being made for the commissioning phase. In this lecture, I review the underlying physics of gravitational waves, review possible astrophysical and cosmological sources and discuss the LIGO interferometer status and plans. 
  We present a new set of 3.5 Post-Newtonian equations in which Newtonian hydrodynamics is coupled to the nonconservative effects of gravitational radiation emission. Our formalism differs in two significant ways from a similar 3.5 Post-Newtonian approach proposed by Blanchet (1993, 1997). Firstly we concentrate only on the radiation-reaction effects produced by a time-varying mass-current quadrupole $S_{ij}$. Secondly, we adopt a gauge in which the radiation-reaction force densities depend on the fourth time derivative of $S_{ij}$, rather than on the fifth, as in Blanchet's approach. This difference makes our formalism particularly well-suited to numerical implementation and could prove useful in performing fully numerical simulations of the recently discovered $r$-mode instability for rotating neutron stars subject to axial perturbations. 
  The formalism for histories-based generalized quantum mechanics developed in two earlier papers is applied to the treatment of histories (of particles or fields or more general objects) in curved spacetimes (which need not admit foliation in spacelike hypersurfaces). The construction of the space of temporal supports (a partial semigroup generalizing the space of finite time sequences employed in traditional temporal description of histories) employs spacelike subsets of spacetime having dimensionality less than or equal to three. Definition of symmetry is sharpened by the requirement of continuity of mappings (employing topological partial semigroups). It is shown that with this proviso, a symmetry in our formalism implies a conformal isometry of the spacetime metric. 
  Summary of abstract Field theory models including gauge theories with SSB are presented where the energy density of the true vacuum state (TVS) is zero without fine tuning. The above models are constructed in the gravitational theory where a measure of integration \Phi in the action is not necessarily \sqrt{-g} but it is determined dynamically through additional degrees of freedom. The ratio \Phi/\sqrt{-g} is a scalar field which can be solved in terms of the matter degrees of freedom due to the existence of a constraint. We study a few explicit field theory models where it is possible to combine the solution of the cosmological constant problem with: 1) possibility for inflationary scenario for the early universe; 2) spontaneously broken gauge unified theories (including fermions). The models are free from the well known problem of the usual scalar-tensor theories in what is concerned with the classical GR tests. The only difference of the field equations in the Einstein frame from the canonical equations of the selfconsistent system of Einstein's gravity and matter fields, is the appearance of the effective scalar field potential which vanishes in TVS without fine tuning. 
  Recently it has been shown that a 2+1 dimensional black hole can be created by a collapse of two colliding massless particles in otherwise empty anti-de Sitter space. Here we generalize this construction to the case of a non-zero impact parameter. The resulting spacetime, which may be regarded as a Gott universe in anti-de Sitter background, contains closed timelike curves. By treating these as singular we are able to interpret our solution as a rotating black hole, hence providing a link between the Gott universe and the BTZ black hole. When analyzing the spacetime we see how the full causal structure of the interior can be almost completely inferred just from considerations of the conformal boundary. 
  We show that the vacuum energy of a free quantized field of very low mass can significantly alter the recent expansion of the universe. The effective action of the theory is obtained from a non-perturbative sum of scalar curvature terms in the propagator. We numerically investigate the semiclassical Einstein equations derived from it. As a result of non-perturbative quantum effects, the scalar curvature of the matter-dominated universe stops decreasing and approaches a constant value. The universe in our model evolves from an open matter-dominated epoch to a mildly inflating de Sitter expansion. The Hubble constant during the present de Sitter epoch, as well as the time at which the transition occurs from matter-dominated to de Sitter expansion, are determined by the mass of the field and by the present matter density. The model provides a theoretical explanation of the observed recent acceleration of the universe, and gives a good fit to data from high-redshift Type Ia supernovae, with a mass of about 10^{-33} eV, and a current ratio of matter density to critical density, Omega_0 <0.4 . The age of the universe then follows with no further free parameters in the theory, and turns out to be greater than 13 Gyr. The model is spatially open and consistent with the possibility of inflation in the very early universe. Furthermore, our model arises from the standard renormalizable theory of a free quantum field in curved spacetime, and does not require a cosmological constant or the associated fine-tuning. 
  We consider a string-inspired, gravitational theory of scalar and electromagnetic fields and we investigate the existence of axially-symmetric, G\"{o}del-type cosmological solutions. The neutral case is studied first and an "extreme" G\"{o}del-type rotating solution, that respects the causality, is determined. The charged case is considered next and two new configurations for the, minimally-coupled to gravity, electromagnetic field are presented. Another configuration motivated by the expected distribution of currents and charges in a rotating universe is studied and shown to lead to a G\"{o}del-type solution for a space-dependent coupling function. Finally, we investigate the existence of G\"{o}del-type cosmological solutions in the framework of the one-loop corrected superstring effective action and we determine the sole configuration of the electromagnetic field that leads to such a solution. It turns out that, in all the charged cases considered, Closed Timelike Curves do appear and the causality is always violated. 
  It is known that traversible wormholes require negative energy density. We here argue how much negative energy is needed for wormholes, using a local analysis which does not assume any symmetry. and in particular allows dynamic (non-stationary) but non-degenerate wormholes. We find that wormholes require two constraints on the energy density, given by two independent components of the Einstein equation. 
  We consider one model of a black hole radiation, in which the equation of motion of a matter field is modified to cut off high frequency modes. The spectrum in the model has already been analytically derived in low frequency range, which has resulted in the Planckian distributin of the Hawking temperature. On the other hand, it has been numerically shown that its spectrum deviates from the thermal one in high frequency range. In this paper, we analytically derive the form of the deviation in the high frequency range. Our result can qualitatively explain the nature of the numerically calculated spectrum. The origin of the deviation is clarified by a simple discussion. 
  A discrete symmetry of the four-dimensional string effective action is employed to derive spatially homogeneous and inhomogeneous string cosmologies from vacuum solutions of general relativity that admit two commuting spacelike Killing vectors. In particular, a tilted Bianchi type V cosmology is generated from a vacuum type VI_h solution and a plane wave solution with a bounded and oscillating dilaton field is found from a type ${\rm VII}_h$ model. Further applications are briefly discussed. 
  The 4-index energy-momentum tensors for gravitation and matter are analyzed on the basis of new equations for the gravitational field with the Riemann tensor. Some properties of the such defined gravitational energy are discussed. 
  We extend previous analyses of soliton solutions in (4+1) gravity to new ranges of their defining parameters. The geometry, as studied using invariants, has the topology of wormholes found in (3+1) gravity. In the induced-matter picture, the fluid does not satisfy the strong energy conditions, but its gravitational mass is positive. We infer the possible existance of (4+1) wormholes which, compared to their (3+1) counterparts, are less exotic. 
  We remark that the standard black hole topology admits twisted configurations of spinor field due to existence of the twisted spinor bundles and analyse them using the Schwarzschild black hole as an example. This is physically linked with the natural presence of Dirac monopoles on black holes and entails marked modification of the Hawking radiation for spinor particles. 
  We study event horizons of non-axisymmetric black holes and show how features found in axisymmetric studies of colliding black holes and of toroidal black holes are non-generic and how new features emerge. Most of the details of black hole formation and black hole merger are known only in the axisymmetric case, in which numerical evolution has successfully produced dynamical space-times. The work that is presented here uses a new approach to construct the geometry of the event horizon, not by locating it in a given spacetime, but by direct construction. In the axisymmetric case, our method produces the familiar pair-of-pants structure found in previous numerical simulations of black hole mergers, as well as event horizons that go through a toroidal epoch as discovered in the collapse of rotating matter. The main purpose of this paper is to show how new - substantially different - features emerge in the non-axisymmetric case. In particular, we show how black holes generically go through a toroidal phase before they become spherical, and how this fits together with the merger of black holes. 
  The analytic solution of Teukolsky equation in Kerr-de Sitter and Kerr-Newman-de Sitter geometries is presented and the properties of the solution are examined. In particular, we show that our solution satisfies the Teukolsky-Starobinsky identities explicitly and fix the relative normalization between solutions with the spin weight $s$ and $-s$. 
  Massive scalar particle production, due to the anisotropic evolution of a five-dimensional spacetime, is considered in the context of a quadratic Lagrangian theory of gravity. Those particles, corresponding to field modes with non-vanishing momentum component along the fifth dimension, are created mostly in the neighbourhood of a singular epoch where only their high-frequency behaviour is of considerable importance. At the 1-loop approximation level, general renormalizability conditions on the physical quantities relevant to particle production are derived and discussed. Exact solutions of the resulting Klein-Gordon field equation are obtained and the mass-energy spectrum attributed to the scalar field due to the cosmological evolution is being investigated further. Finally, analytic expressions regarding the number and the energy density of the created particles at late times, are also derived and discussed. 
  Ten-dimensional models, arising from a gravitational action which includes terms up to the fourth order in curvature tensor, are discussed. The spacetime consists of one timelike dimension and two maximally symmetric subspaces, filled with matter in the form of an anisotropic fluid. Numerical integration of the cosmological field equations indicates that exponential, as well as power law, solutions are possible. We carry out a dynamical study of the results in the H_{ext} - H_{int} plane and confirm the existence of "attractors" in the evolution of the Universe. Those attracting points correspond to "extended De Sitter" spacetimes, in which the external space exhibits inflationary expansion, while the internal one contracts. 
  In the context of higher-dimensional cosmologies with isotropic visible and internal space and multi-perfect fluid matter, we study the conditions under which adiabatic expansion of the visoble external space is possible, when a time-dependent internal space is present. The analysis is based on a reinterpretation of the four-dimensional stress-energy tensor in the presence of the extra dimensions. This modifies the usual adiabatic energy conservation laws for the visible Universe, leading to a new type of cosmological evolution which includes large-scale entropy production in four dimensions. 
  The interaction of charged particles, moving in a uniform magnetic field, with a plane-polarized gravitational wave is considered using the Fokker-Planck- Kolmogorov (FPK) approach. By using a stochasticity criterion, we determine the exact locations in phase space, where resonance overlapping occurs. We investigate the diffusion of orbits around each primary resonance of order (m) by deriving general analytical expressions for an effective diffusion coeficient. A solution to the corresponding diffusion equation (Fokker-Planck equation) for the static case is found. Numerical integration of the full equations of motion and subsequent calculation of the diffusion coefficient verifies the analytical results. 
  Theories with varying gravitational constant $G$ have been studied since long time ago. Among them, the most promising candidates as alternatives of the standard General Relativity are known as scalar-tensor theories. They provide consistent descriptions of the observed universe and arise as the low energy limit of several pictures of unified interactions. Therefore, an increasing interest on the astrophysical consequences of such theories has been sparked over the last few years. In this essay we comment on two methodological approaches to study evolution of astrophysical objects within a varying-$G$ theory, and the particular results we have obtained for boson and white dwarf stars. 
  The theory of relativity was built up on linear Lorentz transformation. However, in his fundamental work "Theory of Space, Time and Gravitation" V.A.Fock shows that the general form of the transformation between the coordinates in the two inertial frames could be taken to be linear fractional. The implicit form of this transformation contains two constants of different space-time dimensions. They can be reduced to the constant "c" with the dimension of speed ("speed of light"), and to the constant "R" with the dimension of length (an invariant radius of the visible part of the Universe). The geometry of the "light cones" shows that "R" is a fundamental constant, but "c" depends on the time of transformation. 
  We prove that, in the non-extreme Kerr-Newman black hole geometry, the Dirac equation has no normalizable, time-periodic solutions. A key tool is Chandrasekhar's separation of the Dirac equation in this geometry. A similar non-existence theorem is established in a more general class of stationary, axisymmetric metrics in which the Dirac equation is known to be separable. These results indicate that, in contrast with the classical situation of massive particle orbits, a quantum mechanical Dirac particle must either disappear into the black hole or escape to infinity. 
  A satisfactory theory of quantum gravity may necessitate a drastic modification of our perception of space-time, by giving it a foamy structure at distances comparable to the Planck length. It is argued in this essay that the experimental detection of such structures may be a realistic possibility in the foreseeable future. After a brief review of different theoretical approaches to quantum gravity and the relationships between them, we discuss various possible experimental tests of the quantum nature of space-time. Observations of photons from distant astrophysical sources such as Gamma-Ray Bursters and laboratory experiments on neutral kaon decays may be sensitive to quantum-gravitational effects if they are only minimally suppressed. Experimental limits from the Whipple Observatory and the CPLEAR Collaboration are already probing close to the Planck scale, and significant increases in sensitivity are feasible. 
  The main theme of this survey is the equivalence statements for quantum scalar field vacuum states that have been recognized over the last couple of decades as a powerful line of reasoning when discussing the highly academic thermal-like Hawking effect and Unruh effect. An important ingredient in this framework is the concept of vacuum field noise spectrum by which one can obtain information about the curvature invariants of classical worldlines (relativistic classical trajectories). It is argued, in the spirit of the free fall type universality, that the preferred quantum field vacua with respect to accelerated worldlines should be chosen in the class of all those possessing stationary spectra for their quantum fluctuations. For scalar quantum field vacua there are six stationary cases as shown by Letaw some time ago, these are reviewed here. However, the non-stationary vacuum noises are not out of reach and can be processed by a few mathematical methods that are mentioned as well. Since the information about the kinematical curvature invariants of the worldlines is of radiometric origin, hints are given on a more useful application of such an academic formalism to radiation and beam radiometric standards at high energy accelerators and in astrophysics. The survey ends up with a quick look to related axiomatic quantum field topics and a few other recent works 
  A experimental test based on the inverse Compton effect (photon-electron collision) is proposed to check the Kinetic Quantum Gravity Theory. The experimental set-up point out a possible propulsionsystem, based on gravity control. 
  The solutions of two-dimensional gravity following from a non-linear Lagrangian L = f(R) are classified, and their symmetry and singularity properties are described. Then a conformal transformation is applied to rewrite these solutions as analogous solutions of two-dimensional Einstein-dilaton gravity and vice versa. 
  Time variation of Newtonian gravitational constant, $G$, is studied in the model universe with variable space dimension proposed recently.   Using the Lagrangian formulation of these models, we find the effective gravitational constant as a function of time. To compare it with observational data, a test theory for the time variation of $G$ is formulated. We have assumed a power law behavior of the time variation of $G$ where the exponent $\beta$ is itself time dependent. Within this test theory we are able to restrict the free parameter of the theories under consideration and give upper bounds for the space dimension at the Planck era. The time variation of $G$ at earlier times, such as the time of nucleosynthesis is also predicted which express the needs to look for related observational data. 
  Thermodynamic properties of locally anisotropic (2+1)-black holes are studied by applying geometric methods. We consider a new class of black holes with a constant in time elliptical event horizon which is imbedded in a generalized Finsler like spacetime geometry induced from Einstein gravity. The corresponding thermodymanic systems are three dimensional with entropy S being a hypersurface function on mass M, anisotropy angle $\theta$ and eccentricity of elliptic deformations $\epsilon$. Two-dimensional curved thermodynamic geometries for locally anistropic deformed black holes are constructed after integration on anisotropic parameter $\theta$. Two approaches, the first one based on two-dimensional hypersurface parametric geometry and the second one developed in a Ruppeiner-Mrugala-Janyszek fashion, are analyzed. The thermodynamic curvatures are computed and the critical points of curvature vanishing are defined. 
  We study the resonant interaction of charged particles with a gravitational wave propagating in the non-empty interstellar space in the presence of a uniform magnetic field. It is found that this interaction can be cast in the form of a parametric resonance problem which, besides the main resonance, allows for the existence of many secondary ones. Each of them is associated with a non-zero resonant width, depending on the amplitude of the wave and the energy density of the interstellar plasma. Numerical estimates of the particles' energisation and the ensuing damping of the wave are given. 
  Stimulated by the methods applied for the observational determination of masses in the central regions of the AGNs, we examine the conditions under which, in the interior of a gravitating perfect fluid source, the geodesic motions and the general relativistic hydrodynamic flows are dynamically equivalent to each other. Dynamical equivalence rests on the functional similarity between the corresponding (covariantly expressed) differential equations of motion and is obtained by conformal transformations. In this case, the spaces of the solutions of these two kinds of motion are isomorphic. In other words, given a solution to the problem "hydrodynamic flow in a perfect fluid", one can always construct a solution formally equivalent to the problem "geodesic motion of a fluid element" and vice versa. Accordingly, we show that, the observationally determined nuclear mass of the AGNs is being overestimated with respect to the real, physical one. We evaluate the corresponding mass-excess and show that it is not always negligible with respect to the mass ofthe central dark object, while, under circumstances, can be even larger than the rest-mass of the circumnuclear gas involved. 
  Quantum creation of Universes with compact spacelike sections that have curvature $k$ either closed, flat or open, i.e. $k=\pm1,0$ are studied. In the flat and open cases, the superpotential of the Wheeler De Witt equation is significantly modified, and as a result the qualitative behaviour of a typical wavefunction differs from the traditional closed case. Using regularity arguments, it is shown that the only consistent state for the wavefunction is the Tunneling one. By computing the quantum probabilities for the curvature of the sections, it is shown that quantum cosmology actually favours that the Universe be open, $k=-1$. In all cases sufficient inflation $\sim 60$ e-foldings is predicted: this is an improvement over classical measures that generally are ambiguous as to whether inflation is certain to occur. 
  Inspired by classical work of Bel and Robinson, a natural purely algebraic construction of super-energy tensors for arbitrary fields is presented, having good mathematical and physical properties. Remarkably, there appear quantities with mathematical characteristics of energy densities satisfying the dominant property, which provides super-energy estimates useful for global results and helpful in other matters. For physical fields, higher order (super)^n-energy tensors involving the field and its derivatives arise. In Special Relativity, they provide infinitely many conserved quantities. The interchange of super-energy between different fields is shown. The discontinuity propagation law in Einstein-Maxwell fields is related to super-energy tensors, providing quantities conserved along null hypersurfaces. Finally, conserved super-energy currents are found for any minimally coupled scalar field whenever there is a Killing vector. 
  We have carried out simulations of the coalescence between two relativistic clusters of collisionless particles using a 3D numerical relativity code. We have adopted a new spatial gauge condition obtained by slightly modifying the minimum distortion gauge condition proposed by Smarr and York and resulting in a simpler equation for the shift vector. Using this gauge condition, we have performed several simulations of the merger between two identical clusters in which we have varied the compaction, the type of internal motion in the clusters, and the magnitude of the orbital velocity. As a result of the coalescence, either a new rotating cluster or a black hole is formed. In the case in which a black hole is not formed, simulations could be carried out for a time much longer than the dynamical time scale, and the resulting gravitational waveforms were calculated fairly accurately: In these cases, the amplitude of gravitational waves emitted can be $\sim 10^{-18}(M/10^6M_{\odot})$ at a distance 4000Mpc, and $\sim 0.5%$ of the rest mass energy may be dissipated by the gravitational wave emission in the final phase of the merger. These results confirm that the new spatial gauge condition is promising in many problems at least up to the formation of black holes. In the case in which a black hole is formed, on the other hand, the gauge condition seems to be less adequate, but we suggest a strategy to improve it in this case. All of the results obtained confirm the robustness of our formulation and the ability of our code for stable evolution of strong gravitational fields of compact binaries. 
  The standard definition of cylindrical symmetry in General Relativity is reviewed. Taking the view that axial symmetry is an essential pre-requisite for cylindrical symmetry, it is argued that the requirement of orthogonal transitivity of the isometry group should be dropped, this leading to a new, more general definition of cylindrical symmetry. Stationarity and staticity in cylindrically symmetric spacetimes are then defined, and these issues are analysed in connection with orthogonal transitivity, thus proving some new results on the structure of the isometry group for this class of spacetimes. 
  The static electrogravitational equations are studied and it is shown that an aligned type D metric which has a Weyl-type relationship between the gravitational and electric potential has shearfree geodesic lines of force. All such fields are then found and turn out to be the fields of a charged sphere, charged infinite rod and charged infinite plate. A further solution is also found with shearing geodesic lines of force. This new solution can have $m>|e|$ or $m<|e|$, but cannot be in the Majumdar-Papapetrou class (in which $m = |e|$). It is algebraically general and has flat equipotential surfaces. 
  The non-classical features of quantum mechanics are reproduced using models constructed with a classical theory - general relativity. The inability to define complete initial data consistently and independently of future measurements, non-locality, and the non-Boolean logical structure are reproduced by these examples. The key feature of the models is the role of topology change. It is the breakdown of causal structure associated with topology change that leads to the apparently non-classical behaviour. For geons, topology change is required to describe the interaction of particles. It is therefore natural to regard topology change as an essential part of the measurement process. This leads to models in which the measurement imposes additional non-redundant boundary conditions. The initial state cannot be described independently of the measurement and there is a causal connection between the measurement and the initial state. 
  String theory can (in principle) describe gravity at all curvature scales, and can be applied to cosmology to look back in time beyond the Planck epoch. The duality symmetries of string theory suggest a cosmological picture in which the imprint of a primordial, pre-big bang phase could still be accessible to present observations. The predictive power of such a scenario relies, however, on our ability to connect in a smooth way the pre-big bang to the present cosmological regime. Classical radiation back reaction seems to play a key role to this purpose, by isotropizing and turning into a final expansion any state of anisotropic contraction possibly emerging from the pre-big bang at the string scale. 
  In this essay, I present an alternative explanation for the cosmic acceleration which appears as a consequence of recent high redshift Supernova data. In the usual interpretation, this cosmic acceleration is explained by the presence of a positive cosmological constant or vacuum energy, in the background of Friedmann models. Instead, I will consider a Local Rotational Symmetric (LRS) inhomogeneous spacetime, with a barotropic equation of state for the cosmic matter. Within this framework the kinematical acceleration of the cosmic fluid or, equivalently, the inhomogeneity of matter, is just the responsible of the SNe Ia measured cosmic acceleration. Although in our model the Cosmological Principle is relaxed, it maintains local isotropy about our worldline in agreement with the CBR experiments. 
  We study formation of black holes in the Friedmann universe. We present a formulation of the Einstein equations under the constant mean curvature time-slicing condition. Our formalism not only gives us the analytic solution of the perturbation equations for non-linear density and metric fluctuations on superhorizon scales, but also allows us to carry out a numerical relativity simulation for black hole formation after the scale of the density fluctuations is well within the Hubble horizon scale. We perform a numerical simulation of spherically symmetric black hole formation in the radiation-dominated, spatially flat background universe for a realistic initial condition supplied from the analytic solution. It is found that the initial metric perturbation has to be non-linear (the maximum value of 3D conformal factor $\psi_0$ at $t=0$ should be larger than $\sim 1.4$) for a black hole to be formed, but the threshold amplitude for black hole formation and the final black hole mass considerably depend on the initial density (or metric) profile of the perturbation: The threshold value of $\psi_0$ at $t=0$ for formation of a black hole is smaller for a high density peak surrounded by a low density region than for that surrounded by the average density region of the flat universe. This suggests that it is necessary to take into account the spatial correlation of density fluctuations in the study of primordial black hole formation. 
  We postulate that all the presently known kinematic effects on physical quantities related to a material particle (e.g., masss increase) are due to its velocity relative to surrounding matter, and not to the observer's reference frame. The minimal velocity (i.e., the velocity that minimizes these quantities) relative to a single large body being a function of the distance to and mass of the body. In consequence, the minimal velocity is a function of position, and the reference frame associated to this velocity is strictly of local validity. We further assume that, at any given point, light propagates isotropically solely in the minimal-velocity local frame existing at the point. We obtain the following results: (i) After showing the compatibility of the gravitational field eqs. with our assumptions, we find the functional dependance of the minimal velocity on the distance to and mass of a single large body. (ii) A permanent gravitational field is the convective rate of change of the minimal velocity field. (iii) A Lorentz transformation connects the values of quantities related to a particle, for two different velocities of the particle relative to its minimal-velocity local frame. However, a Lorentz transformation does not connect this frame with any other moving uniformly with respect to it. (iv) The experimentally detected effects of kinematic, as well as gravitational, mass increase and time dilation are derived. This is, they all are due to the presence of the nearby (single) large mass. (v) Fizeau's experiment, Michelson's experiment, aberration of fixed stars are taken account of. (vi) Michelson's experiment performed from an vehicle orbiting the earth, or the sun, should detect the orbital velocity of the vehicle. 
  We have considered a cosmological model with a phenomenological model for the cosmological constant of the form $\Lambda=\bt\fr{\ddot R}{R}$, $ \bt$ is a constant. For age parameter consistent with observational data the Universe must be accelerating in the presence of a positive cosmological constant. The minimum age of the Universe is $H_0^{-1}$, where $H_0$ is the present Hubble constant. The cosmological constant is found to decrease as $t^{-2}$. Allowing the gravitational constant to change with time leads to an ever increasing gravitational constant at the present epoch. In the presence of a viscous fluid this decay law for $\Lambda$ is equivalent to the one with $\Lambda=3\alpha H^2$ ($\alpha=\rm const.$) provided $\alpha=\fr{\bt}{3(\bt-2)}$. The inflationary solution obtained from this model is that of the de-Sitter type. 
  The conformally flat families of initial data typically used in numerical relativity to represent boosted black holes are not those of a boosted slice of the Schwarzschild spacetime. If such data are used for each black hole in a collision, the emitted radiation will be partially due to the ``relaxation'' of the individual holes to ``boosted Schwarzschild'' form. We attempt to compute this radiation by treating the geometry for a single boosted conformally flat hole as a perturbation of a Schwarzschild black hole, which requires the use of second order perturbation theory. In this we attempt to mimic a previous calculation we did for the conformally flat initial data for spinning holes. We find that the boosted black hole case presents additional subtleties, and although one can evolve perturbatively and compute radiated energies, it is much less clear than in the spinning case how useful for the study of collisions are the radiation estimates for the ``spurious energy'' in each hole. In addition to this we draw some lessons on which frame of reference appears as more favorable for computing black hole collisions in the close limit approximation. 
  This paper has been withdrawn by the author due to the triviality of the considered coordinate transformations. A consistent treatment, based on the extended physical radial coordinate, is presented in the publications of the author 2000 - 2003. 
  A new class of electrically charged wormholes is described in which the outer two sphere is not spanned by a compact coorientable hypersurface. These wormholes can therefore display net electric charge from the source free Maxwell's equation. This extends the work of Sorkin on non-space orientable manifolds, to spacetimes which do not admit a time orientation. The work is motivated by the suggestion that quantum theory can be explained by modelling elementary particles as regions of spacetime with non-trivial causal structure. The simplest example of an electrically charged spacetime carries a spherical symmetry. 
  Due to the resemblance between Maxwell and the gravitomagnetic equations obtained in the weak field and slow motion limit of General Relativity, one can ask if it is possible to amplify a seed intrinsic rotation or spin motion by a gravitomagetic dynamo, in analogy with the well-known dynamo effect. Using the Galilean limits of the gravitomagnetic equations, the answer to this question is negative, due to the fact that a "magnetic" Galilean limit for the gravitomagnetic equations is physically inconsistent. 
  The notion of geometrical duality is discussed in the context of both Brans-Dicke theory and general relativity. It is shown that, in some particular solutions, the spacetime singularities that arise in usual Riemannian general relativity may be avoided in its dual representation (Weyl-type general relativity). This dual representation provides a singularity-free picture of the World that is physicaly equivalent to the canonical general relativistic one. 
  There are no reasons why the energy spectra of the relic gravitons, amplified by the pumping action of the background geometry, should not increase at high frequencies. A typical example of this behavior are quintessential inflationary models where the slopes of the energy spectra can be either blue or mildly violet. In comparing the predictions of scenarios leading to blue and violet graviton spectra we face the problem of correctly deriving the sensitivities of the interferometric detectors. Indeed, the expression of the signal-to-noise ratio not only depends upon the noise power spectra of the detectors but also upon the spectral form of the signal and, therefore, one can reasonably expect that models with different spectral behaviors will produce different signal-to-noise ratios. By assuming monotonic (blue) spectra of relic gravitons we will give general expressions for the signal-to-noise ratio in this class of models. As an example we studied the case of quintessential gravitons. The minimum achievable sensitivity to $h^2_{0} \Omega_{GW}$ of different pairs of detectors is computed, and compared with the theoretical expectations. 
  The formulation of Brans-Dicke (BD) gravity with matter in the Einstein frame is realized as Einstein gravity with dilaton and dilaton coupled matter. We calculate the one-loop 4d anomaly-induced effective action due to N dilaton- coupled massless fermions on the time-dependent conformally flat background with non-trivial dilaton. Using that complete effective action the (fourth-order) quantum corrected equations of motion are derived. One special solution of these equations representing an inflationary Universe (with exponential scale factor) and (much slower) expanding BD dilaton is given. Similarly, the 2d quantum BD Universe with time-dependent dilaton is constructed. In the last case the dynamics is completely due to quantum effects. 
  Withdrawn; conclusion that the singularity is strong is incorrect. 
  We study the possibility that the vacuum energy density of scalar and internal-space gauge fields arising from the process of dimensional reduction of higher dimensional gravity theories plays the role of quintessence. We show that, for the multidimensional Einstein-Yang-Mills system compactified on a $R \times S^3 \times S^d$ topology, there are classically stable solutions such that the observed accelerated expansion of the Universe at present can be accounted for without upsetting structure formation scenarios or violating observational bounds on the vacuum energy density. 
  The search for classical or quantum combinatorial invariants of compact n-dimensional manifolds (n=3,4) plays a key role both in topological field theories and in lattice quantum gravity. We present here a generalization of the partition function proposed by Ponzano and Regge to the case of a compact 3-dimensional simplicial pair $(M^3, \partial M^3)$. The resulting state sum $Z[(M^3, \partial M^3)]$ contains both Racah-Wigner 6j symbols associated with tetrahedra and Wigner 3jm symbols associated with triangular faces lying in $\partial M^3$. The analysis of the algebraic identities associated with the combinatorial transformations involved in the proof of the topological invariance makes it manifest a common structure underlying the 3-dimensional models with empty and non empty boundaries respectively. The techniques developed in the 3-dimensional case can be further extended in order to deal with combinatorial models in n=2,4 and possibly to establish a hierarchy among such models. As an example we derive here a 2-dimensional closed state sum model including suitable sums of products of double 3jm symbols, each one of them being associated with a triangle in the surface. 
  This paper has been withdrawn by the author due to the triviality of the considered coordinate transformations. A consistent treatment, based on the extended physical radial coordinates, is presented in the publications of the author 2000 - 2003. 
  An example of a sequence of the sl(N;C) chiral fields, for N$\geq 2$, tending to the complex heavenly metric (nonlinear graviton) of the type [4]x[-] when N --> infinity is given. 
  The initial data for black hole collisions is constructed using a conformal-imaging approach and a new adaptive mesh refinement technique, a fully threaded tree (FTT). We developed a second-order accurate approach to the solution of the constraint equations on a non-uniformly refined high resolution Cartesian mesh including second-order accurate treatment of boundary conditions at the black hole throats. Results of test computations show convergence of the solution as the numerical resolution is increased. FTT-based mesh refinement reduces the required memory and computer time by several orders of magnitude compared to a uniform grid. This opens up the possibility of using Cartesian meshes for very high resolution simulations of black hole collisions. 
  We define a general class of superenergy tensors of even rank 2(n+1) for a real massive scalar field propagating in Minkowski spacetime. In the case where n=1, we establish that this class is a two-parameter family, which reduces to a unique tensor W(up to a constant factor) when the complete symmetry on the four indices is required. We show that the superenergy density $W^{\al\ba\ga\da}u_{\al}u_{\ba}u_{\ga}u_{\da}$ relative to any timelike unit vector $u$ is positive definite and that the supermomentum density $W^{\al\ba\ga\da}u_{\ba}u_{\ga}u_{\da}$ is a timelike or a null vector ($W^{\al\ba\ga\da}$ stands for W). Next, we find an infinite set of conserved tensors $U_{(p,q)}$ of rank 2+p+q, that we call weak superenergy tensors of order n when p=q=n. We show that $U_{(1,1)}$ and W yield the same total superenergy and the same total supermomentum. Then, using the canonical quantization scheme, we construct explicitly the superhamiltonian and the supermomentum operators corresponding to W and to each weak superenergy tensor $U_{(n,n)}$. Finally, we exhibit a two-parameter family of superenergy tensors for an electromagnetic field and for a gravitational field. 
  Using several approximations, we calculate an estimate of the gravitational radiation emitted when two equal mass black holes coalesce at the end of their binary inspiral. We find that about 1% of the mass energy of the pair will emerge as gravitational waves during the final ringdown and a negligible fraction of the angular momentum will be radiated. 
  Noise is often used in the study of open systems, such as in classical Brownian motion and in Quantum Dynamics, to model the influence of the environment. However generalising results from G\"{o}del and Chaitin in mathematics suggests that systems that are sufficiently rich that self-referencing is possible contain intrinsic randomness. We argue that this is relevant to modelling the universe, even though it is by definition a closed system. We show how a three-dimensional process-space may arise, as a Prigogine dissipative structure, from a non-geometric order-disorder model driven by, what is termed, self-referential noise. 
  Most of the observational claims in cosmology are based on the assumption that the universe is isotropic and homogeneous so they essentially test different types of Friedmann models. This also refers to recent observations of supernovae Ia, which, within the framework of Friedmann cosmologies give strong support to negative pressure matter and also weaken the age conflict. In this essay we drop the assumption of homogeneity, though temporarily leaving the assumption of isotropy with respect to one point, and show that supernovae data can be consistent with a model of the universe with inhomogeneous pressure known as the Stephani model. Being consistent with supernovae data we are able to get the age of the universe in this model to be about 3.8 Gyr more than in its Friedmann counterpart. 
  I show how a minor modification of the Alcubierre geometry can dramatically improve the total energy requirements for a `warp bubble' that can be used to transport macroscopic objects. A spacetime is presented for which the total negative mass needed is of the order of a few solar masses, accompanied by a comparable amount of positive energy. This puts the warp drive in the mass scale of large traversable wormholes. The new geometry satisfies the quantum inequality concerning WEC violations and has the same advantages as the original Alcubierre spacetime. 
  A property well known as the first law of black hole is a relation among infinitesimal variations of parameters of stationary black holes. We consider a dynamical version of the first law, which may be called the first law of black hole dynamics. The first law of black hole dynamics is derived without assuming any symmetry or any asymptotic conditions. In the derivation, a definition of dynamical surface gravity is proposed. In spherical symmetry it reduces to that defined recently by one of the authors (SAH). 
  We reconsider here the model where large quantum gravity effects were first found, but now in its Null Surface Formulation (NSF). We find that although the set of coherent states for $Z$, the basic variable of NSF, is as restricted as it is the one for the metric, while some type of small deviations from these states may cause huge fluctuations on the metric, the corresponding fluctuations on $Z$ remain small. 
  In loop quantum gravity we now have a clear picture of the quantum geometry of space, thanks in part to the theory of spin networks. The concept of `spin foam' is intended to serve as a similar picture for the quantum geometry of spacetime. In general, a spin network is a graph with edges labelled by representations and vertices labelled by intertwining operators. Similarly, a spin foam is a 2-dimensional complex with faces labelled by representations and edges labelled by intertwining operators. In a `spin foam model' we describe states as linear combinations of spin networks and compute transition amplitudes as sums over spin foams. This paper aims to provide a self-contained introduction to spin foam models of quantum gravity and a simpler field theory called BF theory. 
  Backgrounds are pervasive in almost every application of general relativity. Here we consider the Lagrangian formulation of general relativity for large perturbations with respect to a curved background spacetime. We show that Noether's theorem combined with Belinfante's "symmetrization" method applied to the group of displacements provide a conserved vector, a "superpotential" and a energy-momentum that are independent of any divergence added to the Hilbert Lagrangian of the perturbations. The energy-momentum is symmetrical and divergenceless only on backgrounds that are Einstein spaces in the sense of A.Z.Petrov. 
  A Hamiltonian framework is introduced to encompass non-rotating (but possibly charged) black holes that are ``isolated'' near future time-like infinity or for a finite time interval. The underlying space-times need not admit a stationary Killing field even in a neighborhood of the horizon; rather, the physical assumption is that neither matter fields nor gravitational radiation fall across the portion of the horizon under consideration. A precise notion of non-rotating isolated horizons is formulated to capture these ideas. With these boundary conditions, the gravitational action fails to be differentiable unless a boundary term is added at the horizon. The required term turns out to be precisely the Chern-Simons action for the self-dual connection. The resulting symplectic structure also acquires, in addition to the usual volume piece, a surface term which is the Chern-Simons symplectic structure. We show that these modifications affect in subtle but important ways the standard discussion of constraints, gauge and dynamics. In companion papers, this framework serves as the point of departure for quantization, a statistical mechanical calculation of black hole entropy and a derivation of laws of black hole mechanics, generalized to isolated horizons. It may also have applications in classical general relativity, particularly in the investigation of of analytic issues that arise in the numerical studies of black hole collisions. 
  We derive exact series solutions for the Wheeler-DeWitt equation corresponding to a spatially closed Friedmann-Robertson-Walker universe with cosmological constant for arbitrary operator ordering of the scale factor of the universe. The resulting wave functions are those relevant to the approximation which has been widely used in two-dimensional minisuperspace models with an inflationary scalar field for the purpose of predicting the period of inflation which results from competing boundary condition proposals for the wave function of the universe. The problem that Vilenkin's tunneling wave function is not normalizable for general operator orderings, is shown to persist for other values of the spatial curvature, and when additional matter degrees of freedom such as radiation are included. 
  The universe we observe is homogeneous on super-horizon scales, leading to the ``cosmic homogeneity problem''. Inflation alleviates this problem but cannot solve it within the realm of conservative extrapolations of classical physics. A probabilistic solution of the problem is possible but is subject to interpretational difficulties. A genuine deterministic solution of the homogeneity problem requires radical departures from known physics. 
  We define space-times which are asymptotic to radiation dominant Friedman-Robertson-Walker space-times at timelike infinity and study the asymptotic structure. We discuss the local asymptotic symmetry and give a definition of the total energy from the electric part of the Weyl tensor. 
  We suggest a technique that explicitly accounts for the structure of an initial state of quantum field in the semiclassical calculations of path integral in curved space-time, and consider decay of metastable state (conformal vacuum of scalar particles above false classical vacuum) in background de Sitter space-time as an example. Making use of this technique, we justify the Coleman-De Luccia approach to the calculation of the decay probability. We propose an interpretation of the Hawking-Moss instanton as a limiting case of constrained instantons. We find that an inverse process of the transition from true vacuum to false one is allowed in de Sitter space-time, and calculate the corresponding probability. 
  We show how a Metric Affine theory of Dilaton gravity can be reduced to an effective Riemannian Dilaton gravity model. A simple generalization of the Obukhov-Tucker-Wang theorem to Dilaton gravity is then presented. 
  We consider the cosmological model of a self-interacting $\phi^4 - \phi^2$ quantum scalar field and extend our previous results, [3], on resonant tunneling and consequent particle production, to the case of finite temperature. Using the mathematical equivalence between, the Euclidean path integral of a $\phi^4 - \phi^2$ quantum field theory (in the saddle point approximation), on one hand, and the partition function of a 4-dimensional ferromagnet (in the Ising model approximation), on the other, we derive the following results. Tunneling is a first order phase transition. The creation of metastable bound states of instanton-antinstanton pairs under the barrier ,(i.e. resonant tunneling), is the seed that gives rise to particle production. Through the application of the Lee-Yang theorem for phase transitions, (as well as demonstrating the underlying connection this has with the poles of the S-matrix element in the quantum scattering theory), we show that the fluctuations around the dominant escape paths of instantons (i.e. fluctuations of the bubble wall) with momenta comparable to the scale curvature of the bubble, drive the mechanism for resonant tunneling in false vacuum decay. We also identify the temperature dependence of the parameters in the potential term, (or equivalently, of the instanton bubbles), for a wide range of temperatures Finally, we show that the picture of a dilute instanton gas,remains valid even at finite temperatures, as this gas becomes more and more dilute with the increase of the temperature. This suppression continues until we reach the critical temperature, at which point there is only one instanton left, with an infinitely thick wall. 
  We discuss the possible influence of gravity in the neutronization process, $p^+ e^- \to n \nu_e$, which is particularly important as a cooling mechanism of neutron stars. Our approach is semiclassical in the sense that leptonic fields are quantized on a classical background spacetime, while neutrons and protons are treated as excited and unexcited nucleon states, respectively. We expect gravity to have some influence wherever the energy content carried by the in-state is barely above the neutron mass. In this case the emitted neutrinos would be soft enough to have a wavelength of the same order as the space curvature radius. 
  Models of inflationary cosmology can lead to variation of observable parameters ("constants of Nature") on extremely large scales. The question of making probabilistic predictions for today's observables in such models has been investigated in the literature. Because of the infinite thermalized volume resulting from eternal inflation, it has proven difficult to obtain a meaningful and unambiguous probability distribution for observables, in particular due to the gauge dependence. In the present paper, we further develop the gauge-invariant procedure proposed in a previous work for models with a continuous variation of "constants". The recipe uses an unbiased selection of a connected piece of the thermalized volume as sample for the probability distribution. To implement the procedure numerically, we develop two methods applicable to a reasonably wide class of models: one based on the Fokker-Planck equation of stochastic inflation, and the other based on direct simulation of inflationary spacetime. We present and compare results obtained using these methods. 
  We give a consistent description of how the inflationary Universe emerges in quantum cosmology. This involves two steps: Firstly, it is shown that a sensible probability peak can be obtained from the cosmological wave function. This is achieved by going beyond the tree level of the semiclassical expansion. Secondly, due to decoherence interference terms between different semiclassical branches are negligibly small. The results give constraints on the particle content of a unified theory. 
  We present new results concerning the existence of static, electrically charged, perfect fluid spheres that have a regular interior and are arbitrarily close to a maximally charged black-hole state. These configurations are described by exact solutions of Einstein's field equations. A family of these solutions had already be found (de Felice et al., 1995) but here we generalize that result to cases with different charge distribution within the spheres and show, in an appropriate parameter space, that the set of such physically reasonable solutions has a non zero measure. We also perform a perturbation analysis and identify the solutions which are stable against adiabatic radial perturbations. We then suggest that the stable configurations can be considered as classic models of charged particles. Finally our results are used to show that a conjecture of Kristiansson et al. (1998) is incorrect. 
  We report some new exact instantons in general relativity. These solutions are K\"ahler and fall into the symmetry classes of Bianchi types VI0 and VII0, with matter content of a stiff fluid. The qualitative behaviour of the solutions is presented, and we compare it to the known results of the corresponding self-dual Bianchi solutions. We also give axisymmetric Bianchi VII0 solutions with an electromagnetic field. 
  We show that it is not possible in the absence of dark matter to construct a four-dimensional metric that explains galactic observations. In particular, by working with an effective potential it is shown that a metric which is constructed to fit flat rotation curves in spiral galaxies leads to the wrong sign for the bending of light i.e. repulsion instead of attraction. Hence, without dark matter the motion of particles on galactic scales cannot be explained in terms of geodesic motion on a four- dimensional metric. This reveals a new bright side to dark matter: it is indispensable if we wish to retain the cherished equivalence principle. 
  This paper has been withdrawn by the author due to the triviality of the considered coordinate transformations. A consistent treatment, based on the extended physical radial coordinate, is presented in the publications of the author 2000 - 2003. 
  A contribution linear in r to the gravitational potential can be created by a suitable conformal duality transformation: the conformal factor is 1/(1+r)^2 and r will be replaced by r/(1+r), where r is the Schwarzschild radial coordinate. Thus, every spherically symmetric solution of conformal Weyl gravity is conformally related to an Einstein space. This result finally resolves a long controversy about this topic.   As a byproduct, we present an example of a spherically symmetric Einstein space which is a limit of a sequence of Schwarzschild-de Sitter space-times but which fails to be expressable in Schwarzschild coordinates. This example also resolves a long controversy. 
  5D Kaluza-Klein gravity has several nonasymptotically flat solutions which generally, possessed both electric and magnetic charges. In this paper we suggest that these solutions can act as quantum virtual handles (wormholes) in spacetime foam models. By applying a sufficently large, external electric and/or magnetic field it may be possible to ``inflate'' these solutions from a quantum to a classical state. This effect would lead to a possible experimental signal for higher dimensions in multidimensional gravity. 
  We present a simple, exact and self-consistent cosmology with a phenomenological model of quantum creation of radiation due to decay of the scalar field. The decay drives a non-isentropic inflationary epoch, which exits smoothly to the radiation era, without reheating. The initial vacuum for radiation is a regular Minkowski vacuum. The created radiation obeys standard thermodynamic laws, and the total entropy produced is consistent with the accepted value. We analyze the difference between the present model and a model with decaying cosmological constant previously considered. 
  General definitions for causal structures on manifolds of dimension d+1>2 are presented for the topological category and for any differentiable one.   Locally, these are given as cone structures via local (pointwise) homeomorphic or diffeomorphic abstraction from the standard null cone variety in R^{d+1}. Weak and strong local cone (LC) structures refer to the cone itself or a manifold thickening of the cone respectively.   After introducing cone (C-)causality, a causal complement with reasonable duality properties can be defined. The most common causal concepts of space-times are generalized to the present topological setting. A new notion of precausality precludes inner boundaries within future/past cones.   LC-structures, C-causality, a topological causal complement, and precausality may be useful tools in conformal and background independent formulations of (algebraic) quantum field theory and quantum gravity. 
  We show that any second order dynamic equation on a configuration space $X\to R$ of nonrelativistic mechanics can be seen as a geodesic equation with respect to some (nonlinear) connection on the tangent bundle $TX\to X$ of relativistic velocities. We compare relativistic and nonrelativistic geodesic equations, and study the Jacobi vector fields along nonrelativistic geodesics. 
  An analytic solution of the Regge-Wheeler (RW) equation has been found via the Frobenius method at the regular singularity of the horizon 2M, in the form of a time and radial coordinate dependent series. The RW partial differential equation, derived from the Einstein field equations, represents the first order perturbations of the Schwarzschild metric. The known solutions are numerical in time domain or approximate and asymptotic for low or high frequencies in Fourier domain. The former is of scarce relevance for comprehension of the geodesic equations for a body in the black hole field, while the latter is mainly useful for the description of the emitted gravitational radiation. Instead a time domain solution is essential for the determination of radiation reaction of the falling particle into the black hole, i.e. the influence of the emitted radiation on the motion of the perturbing mass in the black hole field. To this end, a semi-analytic solution of the inhomogeneous RW equation with the source term (Regge-Wheeler-Zerilli equation) shall be the next development. 
  For any multidimensional theory with compactified internal spaces, conformal excitations of the internal space metric result in gravitational excitons in the external spacetime. These excitations contribute either to dark matter or to cross sections of usual particles. 
  Within the context of finding the initial conditions of the universe we consider gravitational instantons falling into the Bianchi IX classification. That is, a Euclidean four-manifold with a metric that satisfies Einstein's equations with an induced metric on S^3 submanifolds that is homogeneous but anisotropic. As well as finding regular solutions to the field equations with a tunnelling scalar field, we also look at the case of singular instantons with a view to applying the results to generic potentials. The study is in agreement with the prejudice that instantons with higher symmetry have a lower Euclidean action, even when we consider the singular class of solutions. It is also found that the Euclidean action can diverge for simple potentials, showing that the Hawking Turok instanton had finite action owing to its symmetry. 
  We examine the definition S = ln Omega as a candidate "gravitational entropy" function. We calculate its behavior for gravitationl and density perturbations in closed, open and flat cosmologies and find that in all cases it increases monotonically. Using the formalism to calculate the gravitational entropy produced during inflation gives the canonical answer. We compare the behavior of S with the behavior of the square of the Weyl tensor. Applying the formalism to black holes has proven more problematical. 
  By analyzing a gedanken experiment designed to measure the distance $l$ between two spatially separated points, we find that this distance cannot be measured with uncertainty less than $(ll_P^2)^{1/3}$, considerably larger than the Planck scale $l_P$ (or the string scale in string theories), the conventional wisdom uncertainty in distance measurements. This limitation to space-time measurements is interpreted as resulting from quantum fluctuations of space-time itself. Thus, at very short distance scales, space-time is "foamy." This intrinsic foaminess of space-time provides another source of noise in the interferometers. The LIGO/VIRGO and LISA generations of gravity-wave interferometers, through future refinements, are expected to reach displacement noise levels low enough to test our proposed degree of foaminess in the structure of space-time. We also point out a simple connection to the holographic principle which asserts that the number of degrees of freedom of a region of space is bounded by the area of the region in Planck units. 
  The assimilation of a quantum (finite size) particle by a Reissner-Nordstr\"om black hole inevitably involves an increase in the black-hole surface area. It is shown that this increase can be minimized if one considers the capture of the lightest charged particle in nature. The unavoidable area increase is attributed to two physical reasons: the Heisenberg quantum uncertainty principle and a Schwinger-type charge emission (vacuum polarization). The fundamental lower bound on the area increase is $4 \hbar$, which is smaller than the value given by Bekenstein for neutral particles. Thus, this process is a better approximation to a reversible process in black-hole physics. The universality of the minimal area increase is a further evidence in favor of a uniformly spaced area spectrum for spherical quantum black holes. Moreover, this universal value is in excellent agreement with the area spacing predicted by Mukhanov and Bekenstein and independently by Hod. 
  We have studied the main features of the gravitational radiation generated by an astrophysical system constituted of three compact objects attracting one another (only via gravitational interaction) in such a manner that stable orbits do exist. We have limited our analysis to systems that can be treated with perturbative methods. We show the profile of the gravitational waves emitted by such systems. These results can be useful within the framework of the new gravitational astronomy which will be made feasible by means of the new generation of gravitational detectors such as LISA in a no longer far future. 
  When neutrinos travel through a normal matter medium, the electron neutrinos couple differently to gravity compared to the other neutrinos, due to the presence of electrons in the medium and the absence of the other charged leptons. The matter-induced gravitational couplings of the neutrinos under such conditions are calculated and their contribution to the neutrino index of refraction in the presence of a gravitational potential is determined. 
  We present the generalized Reissner-Nordstr\"om solution of the field equations of metric-affine gravity (MAG), endowed with electric and magnetic charges, as well as with gravito-electric and gravito-magnetic charges and a cosmological constant term. Moreover, the case $M=e_o$, i.e. mass equal to electric charge and $\lambda=0$, corresponds to an electrically and magnetically charged monopole. Also further multipole solutions are obtained. The charge assignments of the solutions is discussed. 
  The propagator of a spinless particle is calculated from the quantum mechanical path integral formalism in static curved spacetimes endowed with event-horizons. A toy model, the Gui spacetime, and the 2D and 4D Schwarzschild black holes are considered. The role of the topology of the coordinates configuration space is emphasised in this framework. To cover entirely the above spacetimes with a single set of coordinates, tortoise coordinates are extended to complex values. It is shown that the homotopic properties of the complex tortoise configuration space imply the thermal behaviour of the propagator in these spacetimes. The propagator is calculated when end points are located in identical or distinct spacetime regions separated by one or several event-horizons. Quantum evolution through the event-horizons is shown to be unitary in the fifth variable. 
  We study multidimensional cosmology to obtain the wavefunction of the universe using wormhole dominance proposal. Using a prescription for time we obtain the Schroedinger-Wheeler-DeWitt equation without any reference to WD equation and WKB ansatz for WD wavefunction. It is found that the Hartle-Hawking or wormhole-dominated boundary conditions serve as a seed for inflation as well as for Gaussian type ansatz to Schroedinger-Wheeler-DeWitt equation. 
  The Wheeler-DeWitt equation in quantum gravity is timeless in character. In order to discuss quantum to classical transition of the universe, one uses a time prescription in quantum gravity to obtain a time contained description starting from Wheeler-DeWitt equation and WKB ansatz for the WD wavefunction. The approach has some drawbacks. In this work, we obtain the time-contained Schroedinger-Wheeler-DeWitt equation without using the WD equation and the WKB ansatz for the wavefunction. We further show that a Gaussian ansatz for SWD wavefunction is consistent with the Hartle-Hawking or wormhole dominance proposal boundary condition. We thus find an answer to the small scale boundary conditions. 
  We start from classical Hamiltonian constraint of general relativity to obtain the Einstein-Hamiltonian-Jacobi equation. We obtain a time parameter prescription demanding that geometry itself determines the time, not the matter field, such that the time so defined being equivalent to the time that enters into the Schroedinger equation. Without any reference to the Wheeler-DeWitt equation and without invoking the expansion of exponent in WKB wavefunction in powers of Planck mass, we obtain an equation for quantum gravity in Schroedinger form containing time. We restrict ourselves to a minisuperspace description. Unlike matter field equation our equation is equivalent to the Wheeler-DeWitt equation in the sense that our solutions reproduce also the wavefunction of the Wheeler-DeWitt equation provided one evaluates the normalization constant according to the wormhole dominance proposal recently proposed by us. 
  The effects of space-time reversal (PT), proper time reversal, and charge reversal are examined in the context of general relativity. The results imply accelerated repulsion between matter with past-pointing 4-velocity and matter with future-pointing 4-velocity. Past-pointing matter is thus proposed as a dark energy candidate. 
  The nature of the classical canonical phase-space variables for gravity suggests that the associated quantum field operators should obey affine commutation relations rather than canonical commutation relations. Prior to the introduction of constraints, a primary kinematical representation is derived in the form of a reproducing kernel and its associated reproducing kernel Hilbert space. Constraints are introduced following the projection operator method which involves no gauge fixing, no complicated moduli space, nor any auxiliary fields. The result, which is only qualitatively sketched in the present paper, involves another reproducing kernel with which inner products are defined for the physical Hilbert space and which is obtained through a reduction of the original reproducing kernel. Several of the steps involved in this general analysis are illustrated by means of analogous steps applied to one-dimensional quantum mechanical models. These toy models help in motivating and understanding the analysis in the case of gravity. 
  Static solutions in spherical symmetry are found for gravitating global monopoles. Regular solutions lacking a horizon are found for $\eta < 1/\sqrt{8\pi}$, where $\eta$ is the scale of symmetry breaking. Apparently regular solutions with a horizon are found for $1/\sqrt{8\pi} \le \eta \alt \sqrt{3/8\pi}$. Though they have a horizon, they are not Schwarzschild. The solution for $\eta = 1/\sqrt{8\pi}$ is argued to have a horizon at infinity. The failure to find static solutions for $\eta > \sqrt{3/8\pi} \approx 0.3455$ is consistent with findings that topological inflation begins at $\eta \approx 0.33$. 
  For general relativistic spacetimes filled with irrotational `dust' a generalized form of Friedmann's equations for an `effective' expansion factor $a_D (t)$ of inhomogeneous cosmologies is derived. Contrary to the standard Friedmann equations, which hold for homogeneous-isotropic cosmologies, the new equations include the `backreaction effect' of inhomogeneities on the average expansion of the model. A universal relation between `backreaction' and average scalar curvature is also given. For cosmologies whose averaged spatial scalar curvature is proportional to $a_D^{-2}$, the expansion law governing a generic domain can be found. However, as the general equations show, `backreaction' acts as to produce average curvature in the course of structure formation, even when starting with space sections that are spatially flat on average. 
  The computation of the simplicial minisuperspace wavefunction in the case of anisotropic universes with a scalar matter field predicts the existence of a large classical Lorentzian universe like our own at late times 
  We discuss the issue of observables in general-relativistic perturbation theory, adopting the view that any observable in general relativity is represented by a scalar field on spacetime. In the context of perturbation theory, an observable is therefore a scalar field on the perturbed spacetime, and as such is gauge invariant in an exact sense (to all orders), as one would expect. However, perturbations are usually represented by fields on the background spacetime, and expanded at different orders into contributions that may or may not be gauge independent. We show that perturbations of scalar quantities are observable if they are first order gauge-invariant, even if they are gauge dependent at higher order. Gauge invariance to first order plays therefore an important conceptual role in the theory, for it selects the perturbations with direct physical meaning from those having only a mathematical status. The so-called ``gauge problem'', and the relationship between measured fluctuations and gauge dependent perturbations that are computed in the theory are also clarified. 
  We study in a Brill-Hartle type of approximation the back reaction of a superposition of linear gravitational waves on the mean gravitational field up to second order in the wave amplitudes. The background field is taken as an Einstein-deSitter geometry. In order to follow inflationary scenarios, the wavelengths are allowed to exceed the temporary Hubble distance. As in optical coherence theory, the wave amplitudes are considered as random variables, which form a homogeneous and isotropic stochastic process, sharing the symmetries of the background metric. The effective stress-energy tensor for the random waves is calculated in terms of correlation functions and covers subhorizon as well as superhorizon modes, the latter give in many cases negative contributions to energy density and pressure. We discuss solutions of the second-order equations including pure gravitational radiation universes. 
  Numerical simulations are performed of the approach to the singularity in Gowdy spacetimes on S2XS1XR. The behavior is similar to that of Gowdy spacetimes on T3XR. In particular, the singularity is asymptotically velocity term dominated, except at isolated points where spiky features develop. 
  This paper has been withdrawn by the author due to the triviality of the considered coordinate transformations. A consistent treatment, based on the extended physical radial coordinate, is presented in the publications of the author 2000 - 2003. 
  The b-boundary construction by B. Schmidt is a general way of providing a boundary to a manifold with connection. It has been shown to have undesirable topological properties however. C. J. S. Clarke gave a result showing that for space-times, non-Hausdorffness is to be expected in general, but the argument contains some errors. We show that under somewhat different conditions on the curvature, the b-boundary will be non-Hausdorff, and illustrate the degeneracy by applying the conditions to some well known exact solutions of general relativity. 
  This is the second in a series of papers describing a 3+1 computational scheme for the numerical simulation of dynamic black hole spacetimes. We discuss the numerical time-evolution of a given black-hole-containing initial data slice in spherical symmetry. We avoid singularities via the "black-hole exclusion" or "horizon boundary condition" technique, where the slices meet the black hole's singularity, but on each slice a spatial neighbourhood of the singularity is excluded from the domain of the numerical computations.   After first discussing some of the key design choices which arise with the black hole exclusion technique, we then give a detailed description of our numerical evolution scheme for spherically symmetric scalar field evolution, assuming that a black hole is already present on the initial slice. We use a free evolution, with Eddington-Finkelstein-like coordinates and the inner boundary placed at a fixed coordinate radius well inside the horizon.   Our numerical scheme is based on the method of lines (MOL), where spacetime PDEs are first finite differenced in space only, yielding a system of coupled ODEs for the time evolution of the field variables along the spatial-grid-point world lines. These ODEs are then time-integrated by standard methods. We use 4th order finite differencing in both space and time, with 5 and/or 6 point spatial molecules (off-centered near the grid boundaries), and a Runge-Kutta time integrator. The spatial grid is smoothly nonuniform, but not adaptive.   We present numerical black hole + scalar field evolutions showing that this scheme is stable, can evolve "forever" (we have gone to t > 4000m), and is very accurate. At a resolution Delta_r/r = 3% near the horizon, typical errors in g_ij(K_ij) at t=100m are <= 1e-5(3e-7), and the energy constraint is < 3e-5. 
  The positive energy theorem precludes the possibility of Minkowski flat space decaying by any mechanism. In certain circumstances, however, large quantum fluctuations of the gravitational field could arise---not only at the Planck scale, but also at larger scales. This is because there exists a set of localised weak field configurations which satisfy the condition int d4x sqrt{g}R = 0 and thus give a null contribution to the Einstein action. Such configurations can be constructed by solving Einstein field equations with unphysical dipolar sources. We discuss this mechanism and its modification in the presence of a cosmological term and/or an external field. 
  Local solutions of the static, spherically symmetric Einstein-Yang-Mills (EYM) equations with SU(2) gauge group are studied on the basis of dynamical systems methods. This approach enables us to classify EYM solutions in the origin neighborhood, to prove the existence of solutions with the oscillating metric as well as the existence of local solutions for all known formal power series expansions, to study the extendibility of solutions, and to find two new local singular solutions. 
  Conserved quantities are obtained and analyzed in the new models with global scale invariance recently proposed. Such models allow for non tivial scalar field potentials and masses for particles, so that the scale symmetry must be broken somehow. We get to this conclusion by showing that the infrared behavior of the conserved currents is singular so that there are no conserved charges associated with the global scale symmetry. The scale symmetry plays nevertheless a crucial role in determining the structure of the theory and it implies that in some high field regions the potentials become flat. 
  These notes address the planar gravitational wave solutions of general relativity in empty space-time, and analyze the motion of test particles in the gravitational wave field. Next we consider related solutions of the Einstein equations for the gravitational field accompanied by long-range wave fields of scalar, spinor and vector type, corresponding to particles of spin s=(0,1/2,1). The motion of test masses in the combined gravitational and scalar, spinor or vector wave fields is discussed. 
  A manifestly gauge invariant formulation of the coupling of the Maxwell theory with an Einstein Cartan geometry is given, where the space time torsion originates from a massless Kalb-Ramond field augmented by suitable U(1) Chern Simons terms.We focus on the situation where the torsion violates parity, and relate it to earlier proposals for gravitational parity violation. 
  Motivated by the strong astronomical evidences supporting that huge black-holes might inhabit the center of many active galaxies, we have studied the integrability of oblique orbits of test particles around the exact superposition of a black-hole and a thin disk. We have considered the relativistic and the Newtonian limits. Exhaustive numerical analyses were performed, and bounded zones of chaotic behavior were found for both limits. An intrinsic relativistic gravitational effect is detected: the chaoticity of trajectories that do not cross the disk. 
  We present a microscopic model for light-cone fluctuations ``in vacuo'', which incorporates a treatment of quantum-gravitational recoil effects induced by energetic particles. Treating defects in space-time as solitons in string theory, we derive an energy-dependent refractive index and a stochastic spread in the arrival times of mono-energetic photons due to quantum diffusion through space-time foam, as found previously using an effective Born-Infeld action. Distant astrophysical sources provide sensitive tests of these possible quantum-gravitational phenomena. 
  In this paper a theorem is derived in order to provide a wide sufficient condition for an orthogonally transitive cylindrical spacetime to be singularity-free. The applicability of the theorem is tested on examples provided by the literature that are known to have regular curvature invariants. 
  We discuss quantum properties of the single-exterior, geon-type black (and white) holes that are obtained from the Kruskal spacetime and the spinless Banados-Teitelboim-Zanelli hole via a quotient construction that identifies the two exterior regions. For the four-dimensional geon, the Hartle-Hawking type state of a massless scalar field is thermal in a limited sense, but there is a discrepancy between Lorentzian and Riemannian derivations of the geon entropy. For the three-dimensional geon, the state induced for a free conformal scalar field on the conformal boundary is similarly thermal in a limited sense, and the correlations in this state provide support for the holographic hypothesis in the context of asymptotically anti-de Sitter black holes in string theory. 
  The coupled Einstein-Dirac-Maxwell equations are considered for a static, spherically symmetric system of two fermions in a singlet spinor state. Stable soliton-like solutions are shown to exist, and we discuss the regularizing effect of gravity from a Feynman diagram point of view. 
  We describe twisted configurations of spinor field on the Schwarzschild and Reissner-Nordstr\"om black holes that arise due to existence of the twisted spinor bundles over the standard black hole topology. From a physical point of view the appearance of spinor twisted configurations is linked with the natural presence of Dirac monopoles that play the role of connections in the complex line bundles corresponding to the twisted spinor bundles. Possible application to the Hawking radiation is also outlined. 
  In this paper we analyze a binary system consisting of a star and a rotating black hole. The electromagnetic radiation emitted by the star interacts with the background of the black hole and stimulates the production of dilaton waves. We then estimate the energy transferred from the electromagnetic radiation of the star to the dilaton field as a function of the frequency. The resulting picture is a testable signature of the existence of dilaton fields as predicted by string theory in an astrophysical context. 
  We define a hierarchy of dynamic relaxed gas spheres as solutions of the Poisson equation coupled to a hierarchy of approximations of the Liouville equation leading, when this equation is satisfied, to the well-known isothermal gas spheres in the static case, but also to a new form of dynamic maximum relaxation. The two previous steps of the hierarchy correspond to an increasing degree of local relaxation at the center of the configuration. 
  We compute the influence action for a system perturbatively coupled to a linear scalar field acting as the environment. Subtleties related to divergences that appear when summing over all the modes are made explicit and clarified. Being closely connected with models used in the literature, we show how to completely reconcile the results obtained in the context of stochastic semiclassical gravity when using mode decomposition with those obtained by other standard functional techniques. 
  We use various results concerning isometry groups of Riemannian and pseudo-Riemannian manifolds to prove that there are spaces on which differential structure can act as a source of gravitational force (Brans conjecture). The result is important for the analysis of the possible physical meaning of differential calculus. Possible astrophysical consequences are discussed. 
  Recently in boson star models in framework of Brans-Dicke theory, three possible definitions of mass have been identified, all identical in general relativity, but different in scalar-tensor theories of gravity.It has been conjectured that it's the tensor mass which peaks, as a function of the central density, at the same location where the particle number takes its maximum.This is a very important property which is crucial for stability analysis via catastrophe theory. This conjecture has received some numerical support. Here we give an analytical proof of the conjecture in framework of the generalized scalar-tensor theory of gravity, confirming in this way the numerical calculations. 
  A new treatment of the gravitational energy on the basis of 4-index gravitational equations is reviewed. The gravitational energy for the Schwarzschild field is considered. 
  Morrow-Jones and Witt have shown that generic spatial topologies admit initial data that evolve to locally de Sitter spacetimes under Einstein's equations. We simplify their arguments, make them a little more general, and solve for the global time evolution of the wormhole initial data considered by them. Finally we give explicit examples of locally de Sitter domains of development whose universal covers cannot be embedded in de Sitter space. 
  We start a systematic study of the Lema\^{i}tre-Tolman-Bondi (LTB) model as applied to the large scale structure and its evolution. Here we study three possible initial conditions of the LTB models which are asymptotically FRW at large scales: bang time, fractal density (with fractal dimension D=2), and velocity law. Any two of these determine the third one. Fractal density and simultaneous bang time provide a quantitative estimate for the scale beyond which the deflection from the linear Hubble law is small. This border may be identified with the zero-velocity surface. For fractal density and linear Hubble law it is shown that the bang time is necessarily non-simultaneous. 
  The influence of recent detections of a finite vacuum energy ("cosmological constant") on our formulation of anthropic conjectures, particularly the so-called Final Anthropic Principle is investigated. It is shown that non-zero vacuum energy implies the onset of a quasi-exponential expansion of our causally connected domain ("the universe") at some point in the future, a stage similar to the inflationary expansion at the very beginning of time. The transition to this future inflationary phase of cosmological expansion will preclude indefinite survival of intelligent species in our domain, because of the rapid shrinking of particle horizons and subsequent depletion of energy necessary for information processes within the horizon of any observer. Therefore, to satisfy the Final Anthropic Hypothesis (reformulated to apply to the entire ensemble of universes), it is necessary to show that (i) chaotic inflation of Linde (or some similar model) provides a satisfactory description of reality, (ii) migration between causally connected domains within the multiverse is physically permitted, and (iii) the time interval left to the onset of the future inflationary phase is sufficient for development of the technology necessary for such inter-domain travel. These stringent requirements diminish the probability of the Final Anthropic Hypothesis being true. 
  The Euler-Lagrange equations for some class of gravitational actions are calculated by means of Palatini principle. Polynomial structures with Einstein metrics appear among extremals of this variational problem. 
  We review some work done with C. Rovelli on the use of the eigenvalues of the Dirac operator on a curved spacetime as dynamical variables, the main motivation coming from their invariance under the action of diffeomorphisms. The eigenvalues constitute an infinite set of ``observables'' for general relativity and can be taken as variables for an invariant description of the gravitational field dynamics. 
  We have considered a cosmological model with a cosmological constant of the form $\Lambda=3\alpha\frac{\dot R^2}{R^2}+\bt\frac{\ddot R}{R} \alpha, \bt=\rm const.$ The cosmological constant is found to decrease as $t^{-2}$ and the rate of particle creation is smaller than the Steady State value. We have found that this behavior gives $\frac{\Lambda_{Pl}}{\Lambda_p}=10^{120}$ where $\Lambda_{Pl}$ is the value of $\Lambda$ at Planck time. Solutions with $\bt=3\alpha$ in the radiation dominated era and $\bt=6\alpha$ in the matter dominated era are equivalent to the FRW results. We have found an inflationary solution of the de-Sitter type with $\bt=3-3\alpha$. Some problems of the Standard Model may be resolved with the presence of the above cosmological constant in the Einstein's equation. Since observations suggest a contribution of the vacuum energy density in the range $0.40<\Omega^\Lambda<0.76$, one gets $4<\beta<12$. If $\alpha=0$ the minimum age of the universe is found to be $H_p^{-1}$ ($H_p$ is the present Hubble constant) with $\beta=\infty$. 
  We propose novel structure formation scenarios based on a non-singular higher curvature cosmological model. The model is motivated by the $R^2$ coupling of a scalar field appearing in the string theory, and in our scenarios the universe has no beginning and no end. We give two examples with explicit parameter values which are consistent with present observations of the cosmological structure. In the first example, the origin of structures are generated as the adiabatic perturbation during the chaotic inflation, while in the second the isocurvature non-Gaussian perturbation from the superinflating era is responsible for the structure. In the second case it is possible to generate primordial supermassive blackholes whose scale is comparable to what is expected in the galactic nuclei. 
  We calculate the net change in generalized entropy occurring when one carries out the gedanken experiment in which a box initially containing energy $E$, entropy $S$ and charge $Q$ is lowered adiabatically toward a Reissner-Nordstr\"{o}m black hole and then dropped in. This is an extension of the work of Unruh-Wald to a charged system (the contents of the box possesses a charge $Q$). Their previous analysis showed that the effects of acceleration radiation prevent violation of the generalized second law of thermodynamics. In our more generic case, we show that the properties of the thermal atmosphere are equally important when charge is present. Indeed, we prove here that an equilibrium condition for the the thermal atmosphere and the physical properties of ordinary matter are sufficient to enforce the generalized second law. Thus, no additional assumptions concerning entropy bounds on the contents of the box need to be made in this process. The relation between our work and the recent works of Bekenstein and Mayo, and Hod (entropy bound for a charged system) are also discussed. 
  Classes of exact static solutions in four-dimensional Einstein-Maxwell-Dilaton gravity are found. Besides of the well-known solutions previously found in the literature, new solutions are presented.It's shown that spherically symmetric solutions, except the case of charged dilaton black hole, represent globally naked strong curvature singularities. 
  1- It is shown that the upper bound for $\alpha$ in the general solutions of spherically symmetric vacuum field equations(gr-qc/9812081,$\Lambda$=0) is nearly 10^3.This has been obtained by comparing the theoretical prediction for bending of light and precession of perihelia with observation. For a significant range of possible values of$\alpha$ ($\alpha$ >2) the metric is free of coordinate singularity. 2- It is checked that the singularity in the non-static spherically symmetric solution of Einstein field equations with $\Lambda$ (JHEP04(1999)011,$\alpha$ = 0)at the origin is intrinsic. 3- Using the techniques of these two works, ageneral class of non-static solutions is presented. They are smooth and finite everywhere and have an extension larger than Schwarzschild metric. 4- The geodesic equations of a freely material particle for the general case are solved which reveals a Schwarzschild -deSitter type potential field. 
  Various objections against Alcubierre's warp drive geometry are reviewed. Superluminal warp bubbles seem an unlikey possibility within the framework of general relativity and quantum field theory, although subluminal bubbles may still be possible. 
  We study extended theories of gravity where nonminimal derivative couplings of the form $R^{kl}\phi_{, k}\phi_{, l}$ are present in the Lagrangian. We show how and why the other couplings of similar structure may be ruled out and then deduce the field equations and the related cosmological models. Finally, we get inflationary solutions which do follow neither from any effective scalar field potential nor from a cosmological constant introduced ``by hand'', and we show the de Sitter space--time to be an attractor solution. 
  The Hamiltonian of a gravitational system defined in a region with boundary is quantized. The classical Hamiltonian, and starting point for the regularization, is required by functional differentiablity of the Hamiltonian constraint. The boundary term is the quasilocal energy of the system and becomes the ADM mass in asymptopia. The quantization is carried out within the framework of canonical quantization using spin networks. The result is a gauge invariant, well-defined operator on the Hilbert space induced from the state space on the whole spatial manifold. The spectrum is computed. An alternate form of the operator, with the correct naive classical limit, but requiring a restriction on the Hilbert space, is also defined. Comparison with earlier work and several consequences are briefly explored. 
  This paper considers metrics whose curvature tensor makes sense as a distribution. A class of such metrics, the regular metrics, was defined and studied by Geroch and Traschen. Here, we generalize their definition to form a wider class: semi-regular metrics. We then examine in detail two metrics that are semi-regular but not regular: (i) Minkowski spacetime minus a wedge and (ii) a certain traveling wave metric. 
  Grishchuk has shown that the stochastic background of gravitational waves produced by an inflationary phase in the early Universe has an unusual property: it is not a stationary Gaussian random process. Due to squeezing, the phases of the different waves are correlated in a deterministic way, arising from the process of parametric amplification that created them. The resulting random process is Gaussian but non-stationary. This provides a unique signature that could in principle distinguish a background created by inflation from stationary stochastic backgrounds created by other types of processes. We address the question: could this signature be observed with a gravitational wave detector? Sadly, the answer appears to be "no": an experiment which could distinguish the non-stationary behavior would have to last approximately the age of the Universe at the time of measurement. This rules out direct detection by ground and space based gravitational wave detectors, but not indirect detections via the electromagnetic Cosmic Microwave Background Radiation (CMBR). 
  An improved method is given for the computation of the stress-energy tensor of a quantized scalar field using adiabatic regularization. The method works for fields with arbitrary mass and curvature coupling in Robertson-Walker spacetimes and is particularly useful for spacetimes with compact spatial sections. For massless fields it yields an analytic approximation for the stress-energy tensor that is similar in nature to those obtained previously for massless fields in static spacetimes. 
  We discuss the implications of a wave function for quantum gravity, which involves nothing but 3-dimensional geometries as arguments and is invariant under general coordinate transformations. We derive an analytic wave function from the Wheeler-DeWitt equation for spherically symmetric space-time with the coordinate system arbitrary. The de Broglie-Bohm interpretation of quantum mechanics is applied to the wave function. In this interpretation, deterministic dynamics can be yielded from a wave function in fully quantum regions as well as in semiclassical ones. By introducing a coordinate system additionally, we obtain a cosmological black hole picture in compensation for the loss of general covariance. Our analysis shows that the de Broglie-Bohm interpretation gives quantum gravity an appropriate prescription to introduce coordinate systems naturally and extract information from a wave function as a result of breaking general covariance. 
  We generalize the Lagrangian of N = 1 supergravity (SUGRA) by using an arbitrary parameter $\xi$, which corresponds to the inverse of Barbero's parameter $\beta$. This generalized Lagrangian involves the chiral one as a special case of the value $\xi = \pm i$. We show that the generalized Lagrangian gives the canonical formulation of N = 1 SUGRA with the real Ashtekar variable after the 3+1 decomposition of spacetime. This canonical formulation is also derived from those of the usual N = 1 SUGRA by performing Barbero's type canonical transformation with an arbitrary parameter $\beta  (=\xi^{-1})$. We give some comments on the canonical formulation of the theory. 
  Quantum buoyancy has been proposed as the mechanism protecting the generalized second law when an entropy--bearing object is slowly lowered towards a black hole and then dropped in. We point out that the original derivation of the buoyant force from a fluid picture of the acceleration radiation is invalid unless the object is almost at the horizon, because otherwise typical wavelengths in the radiation are larger than the object. The buoyant force is here calculated from the diffractive scattering of waves off the object, and found to be weaker than in the original theory. As a consequence, the argument justifying the generalized second law from buoyancy cannot be completed unless the optimal drop point is next to the horizon. The universal bound on entropy is always a sufficient condition for operation of the generalized second law, and can be derived from that law when the optimal drop point is close to the horizon. We also compute the quantum buoyancy of an elementary charged particle; it turns out to be negligible for energetic considerations. Finally, we speculate on the significance of the absence from the bound of any mention of the number of particle species in nature. 
  A short historical review is made of some recent literature in the field of noncommutative geometry, especially the efforts to add a gravitational field to noncommutative models of space-time and to use it as an ultraviolet regulator. An extensive bibliography has been added containing reference to recent review articles as well as to part of the original literature. 
  The recent theory of 't Hooft [ Nucl. Phys. Suppl. {\bf 68}, 174 (1998)] models the black hole as a system endowed with an envelope of matter that obeys an equation of state in the form $ p=(\gamma -1)\rho$, and acts as a source in Einstein's equations. The present paper generalizes the 't Hooft theory so as to take into account a bulk viscosity $\zeta$ in the fluid. It is shown that even a slight positive value of $\zeta$ will suffice to yield complete agreement with the Hawking formula for the entropy of the black hole, if the value of the constant $\gamma$ takes a value that is slightly less than 4/3. The value $\gamma=4/3$ corresponds to a radiation fluid. 
  First a Friedmann-Robertson-Walker (FRW) universe filled with dust and a conformally invariant scalar field is quantized. For the closed model we find a discrete set of wormhole quantum states. In the case of flat spacelike sections we find states with classical behaviour at small values of the scale factor and quantum behaviour for large values of the scale factor. Next we study a FRW model with a conformally invariant scalar field and a nonvanishing cosmological constant dynamically introduced by regarding the vacuum as a perfect fluid. The ensuing Wheeler-DeWitt equation turns out to be a bona fide Schrodinger equation, and we find that there are realizable states with a definite value of the cosmological constant. Once again we find finite-norm solutions to the Wheeler-DeWitt equation with definite values of the cosmological constant that represent wormholes, suggesting that in quantum cosmological models with a simple matter content wormhole states are a common occurrence. 
  We present a set of dynamical equations based on Ashtekar's extension of the Einstein equation. The system forces the space-time to evolve to the manifold that satisfies the constraint equations or the reality conditions or both as the attractor against perturbative errors. This is an application of the idea by Brodbeck, Frittelli, Huebner and Reula who constructed an asymptotically stable (i.e., constrained) system for the Einstein equation, adding dissipative forces in the extended space. The obtained systems may be useful for future numerical studies using Ashtekar's variables. 
  We study a recently proposed horizon defining identity for certain black hole spacetimes. It relates the difference of the Brown-York quasilocal energy and the Komar charge at the horizon to the total energy of the spacetime. The Brown-York quasilocal energy is evaluated for some specific choices of spacetime foliations. With a certain condition imposed on the matter distribution, we prove this identity for spherically symmetric static black hole solutions of general relativity. For these cases, we show that the identity can be derived from a Gauss-Codacci condition that any three-dimensional timelike boundary embedded around the hole must obey. We also demonstrate the validity of the identity in other cases by explicitly applying it to several static, non-static, asymptotically flat, and asymptotically non-flat black hole solutions. These include the asymptotically FRW solutions and the case of a black hole with a global monopole charge. 
  We formulate the data analysis problem for the detection of the Newtonian coalescing-binary signal by a network of laser interferometric gravitational wave detectors that have arbitrary orientations, but are located at the same site. We use the maximum likelihood method for optimizing the detection problem. We show that for networks comprising of up to three detectors, the optimal statistic is essentially the magnitude of the network correlation vector constructed from the matched network-filter. Alternatively, it is simply a linear combination of the signal-to-noise ratios of the individual detectors. This statistic, therefore, can be interpreted as the signal-to-noise ratio of the network. The overall sensitivity of the network is shown to increase roughly as the square-root of the number of detectors in the network. We further show that these results continue to hold even for the restricted post-Newtonian filters. Finally, our formalism is general enough to be extended to address the problem of detection of such waves from other sources by some other types of detectors, e.g., bars or spheres, or even by networks of spatially well-separated detectors. 
  This is intended as an introduction to and review of the work of V, Arnold and his collaborators on the theory of Lagrangian and Legendrian submanifolds and their associated maps. The theory is illustrated by applications to Hamilton-Jacobi theory and the eikonal equation, with an emphasis on null surfaces and wavefronts and their associated caustics and singularities. 
  We develop the canonical formalism for a system of $N$ bodies in lineal gravity and obtain exact solutions to the equations of motion for N=2. The determining equation of the Hamiltonian is derived in the form of a transcendental equation, which leads to the exact Hamiltonian to infinite order of the gravitational coupling constant. In the equal mass case explicit expressions of the trajectories of the particles are given as the functions of the proper time, which show characteristic features of the motion depending on the strength of gravity (mass) and the magnitude and sign of the cosmological constant. As expected, we find that a positive cosmological constant has a repulsive effect on the motion, while a negative one has an attractive effect. However, some surprising features emerge that are absent for vanishing cosmological constant. For a certain range of the negative cosmological constant the motion shows a double maximum behavior as a combined result of an induced momentum-dependent cosmological potential and the gravitational attraction between the particles. For a positive cosmological constant, not only bounded motions but also unbounded ones are realized. The change of the metric along the movement of the particles is also exactly derived.  
  For over twenty years the possibility that the electromagnetic zero point field (ZPF) may actively accelerate electromagnetically interacting particles in regions of extremely low particle density (as those extant in intergalactic space (IGS) with n < 1 particle/m^3 has been studied and analyzed. This energizing phenomenon has been one of the few contenders for acceleration of cosmic rays (CR), particularly at ultrahigh energies. The recent finding by the AGASA collaboration (Phys. Rev. Lett., 81, 1163, 1998) that the CR energy spectrum does not display any signs of the Greisen-Zatsepin-Kuzmin cut-off (that should be present if these CR particles were indeed generated in localized ultrahigh energies CR sources, as e.g., quasars and other highly active galactic nuclei), may indicate the need for an acceleration mechanism that is distributed throughout IGS as is the case with the ZPF. Other unexplained phenomena that receive an explanation from this mechanism are the generation of X-ray and gamma-ray backgrounds and the existence of Cosmic Voids. However recently, a statistical mechanics kind of challenge to the classical (not the quantum) version of the zero-point acceleration mechanism has been posed (de la Pena and Cetto, The Quantum Dice, 1996). Here we briefly examine the consequences of this challenge and a prospective resolution. 
  Small non-spherical perturbations of a spherically symmetric but time-dependent background spacetime can be used to model situations of astrophysical interest, for example the production of gravitational waves in a supernova explosion. We allow for perfect fluid matter with an arbitrary equation of state p=p(rho,s), coupled to general relativity. Applying a general framework proposed by Gerlach and Sengupta, we obtain covariant field equations, in a 2+2 reduction of the spacetime, for the background and a complete set of gauge-invariant perturbations, and then scalarize them using the natural frame provided by the fluid. Building on previous work by Seidel, we identify a set of true perturbation degrees of freedom admitting free initial data for the axial and for the l>1 polar perturbations. The true degrees of freedom are evolved among themselves by a set of coupled wave and transport equations, while the remaining degrees of freedom can be obtained by quadratures. The polar l=0,1 perturbations are discussed in the same framework. They require gauge fixing and do not admit an unconstrained evolution scheme. 
  We report on the progress of a NASA-funded study being carried out at the Lockheed Martin Advanced Technology Center in Palo Alto and the California State University in Long Beach to investigate the proposed link between the zero-point field of the quantum vacuum and inertia. It is well known that an accelerating observer will experience a bath of radiation resulting from the quantum vacuum which mimics that of a heat bath, the so-called Davies-Unruh effect. We have further analyzed this problem of an accelerated object moving through the vacuum and have shown that the zero-point field will yield a non-zero Poynting vector to an accelerating observer. Scattering of this radiation by the quarks and electrons constituting matter would result in an acceleration-dependent reaction force that would appear to be the origin of inertia of matter (Rueda and Haisch 1998a, 1998b). In the subrelativistic case this inertia reaction force is exactly newtonian and in the relativistic case it exactly reproduces the well known relativistic extension of Newton's Law. This analysis demonstrates then that both the ordinary, F=ma, and the relativistic forms of Newton's equation of motion may be derived from Maxwell's equations as applied to the electromagnetic zero-point field. We expect to be able to extend this analysis in the future to more general versions of the quantum vacuum than just the electromagnetic one discussed herein. 
  In a recent paper (gr-qc/9903081) Choptuik, Hirschmann, and Marsa have discovered the scaling law for the lifetime of an intermediate attractor in the formation of n=1 colored black holes via fine tuning. We show that their result is in agreement with the prediction of linear perturbation analysis. We also briefly comment on the dependence of the mass gap across the threshold on the radius of the event horizon. 
  Static charged perfect fluid distributions have been studied. It is shown that if the norm of the timelike Killing vector and the electrostatic potential have the Weyl-Majumdar relation, then the background spatial metric is the space of constant curvature, and the field equations reduces to a single non-linear partial differential equation. Furthermore, if the linear equation of state for the fluid is assumed, then this equation becomes a Helmholtz equation on the space of constant curvature. Some explicit solutions are given. 
  A model unifying general relativity with quantum mechanics is further developed. It is based on a noncommutative geometry which supposedly modelled the universe in its pre-Planckian epoch. The geometry is totally nonlocal with no time and no space in their usual meaning. They emerge only in the transition process from the noncommutative epoch to the standard space-time physics. Observational aspects of this model are discussed. It is shown that various nonlocal phenomena can be explained as remnants of the primordial noncommutative epoch. In particular, we explain the Einstein-Podolsky-Rosen experiment, the horizon problem in cosmology, and the appearance of singularities in general relativity. 
  The asymptotic behaviour at late times of inhomogeneous axion-dilaton cosmologies is investigated. The space-times considered here admit two abelian space-like Killing vectors. These space-times evolve towards an anisotropic universe containing gravitational radiation. Furthermore, a peeling-off behaviour of the Weyl tensor and the antisymmetric tensor field strength is found. The relation to the pre-big-bang scenario is briefly discussed. 
  In the Starobinsky inflationary model inflation is driven by quantum corrections to the vacuum Einstein equation. We reduce the Wheeler-DeWitt equation corresponding to the Starobinsky model to a Schroedinger form containing time. The Schroedinger equation is solved with a Gaussian ansatz. Using the prescription for the normalization constant of the wavefunction given in our previous work, we show that the Gaussian ansatz demands Hawking type initial conditions for the wavefunction of the universe. The wormholes induce randomness in initial states suggesting a basis for time-contained description of the Wheeler-DeWitt equation. 
  We investigate the possibility for a direct detection by future space interferometers of the stochastic gravitational wave (GW) background generated during the inflationary stage in a class of viable $\Lambda$CDM BSI models. At frequencies around $10^{-3}$Hz, maximal values $\Omega_{gw}(\nu)\sim 3\times 10^{-15}$ are found, an improvement of about one order of magnitude compared to single-field, slow-roll inflationary models. This is presumably not sufficient in order to be probed in the near future. 
  We characterise the homogeneous and isotropic gauge invariant and quasifree states for free Dirac quantum fields on Robertson-Walker spacetimes in any even dimension. Using this characterisation, we construct adiabatic vacuum states of order $n$ corresponding to some Cauchy surface. We then show that any two such states (of sufficiently high order) are locally quasi-equivalent. We propose a microlocal version of the Hadamard condition for spinor fields on arbitrary spacetimes, which is shown to entail the usual short distance behaviour of the twopoint function. The polarisation set of these twopoint functions is determined from the Dencker connection of the spinorial Klein-Gordon operator which we show to equal the (pull-back) of the spin connection. Finally it is demonstrated that adiabatic states of infinite order are Hadamard, and that those of order $n$ correspond, in some sense, to a truncated Hadamard series and will therefore allow for a point splitting renormalisation of the expected stress-energy tensor. 
  The new, complex-dynamical mechanism of the universal gravitation naturally incorporating dynamical quantization, wave-particle duality, and relativity of physically emerging space and time (quant-ph/9902015,16) provides the realistic meaning and fundamentally substantiated modification of the Planckian units of mass, length, and time approaching them closely to the extreme values observed for already discovered elementary particles. This result suggests the important change of research strategy in high-energy/particle physics, displacing it towards the already attained energy scales and permitting one to exclude the existence of elementary objects in the inexplicably large interval of parameters separating the known, practically more than sufficient set of elementary species and the conventional, mechanistically exaggerated values of the Planckian units. This conclusion is supported by the causally complete (physically and mathematically consistent) picture of the fundamental levels of reality derived, without artificial introduction of any structure or 'principle', from the unreduced analysis of the (generic) interaction process between two primal, physically real, but a priori structureless entities, the electromagnetic and gravitational protofields. The naturally emerging phenomenon of universal dynamic redundance (multivaluedness) of interaction process gives rise to the intrinsically unified hierarchy of unreduced dynamic complexity of the world, starting from the lowest levels of elementary objects, and explains the irreducible limitations of the basically single-valued approach of the canonical science leading to the well-known 'mysteries', separations, and loss of certainty. 
  We consider 2+1 dimensional gravity with a cosmological constant, and explore a duality that exists between space-times that have the De Sitter group SO(3,1) as its local isometry group. In particular, the Lorentzian theory with a positive cosmological constant is dual to the Euclidean theory with a negative cosmological constant. We use this duality to construct a mapping between apparently unrelated space-times. More precisely, we exhibit a relation between the Euclidean BTZ family and some $T^2$-cosmological solutions, and between De-Sitter point particle space-times and the analytic continuations of Anti-De Sitter point particles. We discuss some possible applications for BH and AdS thermodynamics. 
  Inhomogeneous Nelson's diffusion in flat spacetime with a tensor of diffusion can be described as a homogeneous one in a Riemannian manifold with this tensor of diffusion as a metric tensor. The influence of matter to the energy density of the stochastic background (vacuum) is considered. It is shown that gravitation can be represented as inhomogeneity of the quantum diffusion, the Einstein equations for the metrics can be derived as the equations for the corresponding tensor of diffusion. 
  We consider homothetic maps in a family of spherical relativistic star models. A generalization of Vaidya's radiating metric provides a fluid atmosphere of radiation and strings. The similarity structure of the string fluid is investigated. 
  This paper presents the derivation of Schwinger's gauge invariant result of $Im \cal{L}_{eff}$ upto one loop approximation, for particle production in an uniform electric field through the method of complex trajectory WKB approximation (CWKB). The CWKB proposed by one of the author's \cite{bis:ijtp} looks upon particle production as due to the motion of a particle in complex space-time plane, thereby requiring tunneling paths both in space and time. Recently \cite{srini:iucaa,srini1:iucaa} there have been some efforts to calculate the reflection and transmission co-efficients for particle production in uniform electric field that differ from our expressions for the same. In this paper we clarify the confusion in this regard and establish the correctness of CWKB. 
  We study the behaviour of Dirac current in expanding spacetime with Schr{\"o}dinger and de Sitter form for the evolution of the scale-factor. The study is made to understand the particle-antiparticle rotation and the evolution of quantum vacuum leading to particle production in such spacetime. 
  We present a simple and self-consistent cosmology with a phenomenological model of quantum creation of radiation and matter due to decay of the cosmological constant $\Lambda$. The decay drives a non-isentropic inflationary epoch, which exits smoothly to the radiation-dominated era, without reheating, and then evolves to the dust era. The initial vacuum for radiation and matter is a regular Minkowski vacuum. The created radiation and matter obeys standard thermodynamic laws, and the total entropy produced is consistent with the accepted value. This paper is an extension of the model with the decaying cosmological constant considered previously. We compare our model with the quantum field theory approach to creation of particles in curved space. 
  It has been proposed that the scattering of electromagnetic zero-point radiation by accelerating objects results in a reaction force that may account, at least in part, for inertia [1,2,3]. This arises because of asymmetries in the electromagnetic zero-point field (ZPF) or electromagnetic quantum vacuum as perceived from an accelerating reference frame. In such a frame, the Poynting vector and momentum flux of the ZPF become non-zero. If one assumes that scattering of the ZPF radiation takes place at the level of quarks and electrons constituting matter, then it is possible for both Newton's equation of motion, ${\bf f}=m{\bf a}$, and its relativistic covariant generalization, ${\cal F}=d{\cal P}/d\tau$, to be obtained as a consequence of the non-zero ZPF momentum flux. We now conjecture that this scattering must take place at the Compton frequency of a particle, and that this interpretation of mass leads directly to the de Broglie relation characterizing the wave nature of that particle in motion, $\lambda_B=h/p$. This suggests a perspective on a connection between electrodynamics and the quantum wave nature of matter. Attempts to extend this perspective to other aspects of the vacuum are left for future consideration. 
  Four classes of exact solutions of Einstein-Cartan dilatonic inflationary de Sitter cosmology are given.The first is obtained from the equation of state of massless dilaton instead of an unpolarized fermion fluid used previously by Gasperini.Repulsive gravity is found in the case where dilatons are constraint by the presence of spin-torsion effects.The second and third solutions represent respectively massive dilatons in the radiation era with the massive potential and torsion kinks and finally the dust of spinning particles.Primordial spin-density fluctuations are also computed based on Primordial fluctuations of temperature obtained from COBE data.The temperature fluctuation can also be computed from the nearly flat spectrum of the gravitational waves produced during inflation and by the result that the dilaton mass would be proportional to the Hubble constant.This result agrees with the COBE data.This idea is also used to compute the spin-torsion density in the inflation era. 
  Due to the resemblance between Maxwell and the gravitomagnetic equations obtained in the weak field and slow motion limit of General Relativity, one can ask if it is possible to amplify a seed intrinsic rotation or spin motion by a gravitomagetic dynamo, in analogy with the well-known dynamo effect. Using the Galilean limits of the gravitomagnetic equations, the answer to this question is negative, due to the fact that a "magnetic" Galilean limit for the gravitomagnetic equations is physically inconsistent. Also, we prove that, in spite of some claims, a gravitational Meissner effect does not exists. 
  A purely algebraic construction of super-energy tensors for arbitrary fields is presented in any dimensions. These tensors have good mathematical and physical properties, and they can be used in any theory having as basic arena an n-dimensional manifold with a metric of Lorentzian signature. In general, the completely timelike component of these s-e tensors has the mathematical features of an energy density: they are positive definite and satisfy the dominant property. Similarly, the super-momentum vectors have mathematical properties of s-e flux vectors. The classical Bel-Robinson tensor is included in our general definition. The energy-momentum and super-energy tensors of physical fields are also obtained, and the procedure is illustrated by writing down these tensors explicitly for the cases of scalar, electromagnetic, and Proca fields. Moreover, `(super)$^k$-energy' tensors are defined and shown to be meaningful and in agreement for the different physical fields. In flat spacetimes, they provide infinitely many conserved quantities. In non-flat spacetimes conserved s-e currents are found for any minimally coupled scalar field whenever there is a Killing vector. Furthermore, the exchange of gravitational and electromagnetic super-energy is also shown by studying the propagation of discontinuities. 
  We report on numerical results from an independent formalism to describe the quasi-equilibrium structure of nonsynchronous binary neutron stars in general relativity. This is an important independent test of controversial numerical hydrodynamic simulations which suggested that nonsynchronous neutron stars in a close binary can experience compression prior to the last stable circular orbit. We show that, for compact enough stars the interior density increases slightly as irrotational binary neutron stars approach their last orbits. The magnitude of the effect, however, is much smaller than that reported in previous hydrodynamic simulations. 
  Static solutions of the electro-gravitational field equations exhibiting a functional relationship between the electric and gravitational potentials are studied. General results for these metrics are presented which extend previous work of Majumdar. In particular, it is shown that for any solution of the field equations exhibiting such a Weyl-type relationship, there exists a relationship between the matter density, the electric field density and the charge density. It is also found that the Majumdar condition can hold for a bounded perfect fluid only if the matter pressure vanishes (that is, charged dust). By restricting to spherically symmetric distributions of charged matter, a number of exact solutions are presented in closed form which generalise the Schwarzschild interior solution. Some of these solutions exhibit functional relations between the electric and gravitational potentials different to the quadratic one of Weyl. All the non-dust solutions are well-behaved and, by matching them to the Reissner-Nordstr\"{o}m solution, all of the constants of integration are identified in terms of the total mass, total charge and radius of the source. This is done in detail for a number of specific examples. These are also shown to satisfy the weak and strong energy conditions and many other regularity and energy conditions that may be required of any physically reasonable matter distribution. 
  It is shown that the dynamical evolution of perturbations on a static spacetime is governed by a standard pulsation equation for the extrinsic curvature tensor. The centerpiece of the pulsation equation is a wave operator whose spatial part is manifestly self-adjoint. In contrast to metric formulations, the curvature-based approach to gravitational perturbation theory generalizes in a natural way to self-gravitating matter fields. For a certain relevant subspace of perturbations the pulsation operator is symmetric with respect to a positive inner product and therefore allows spectral theory to be applied. In particular, this is the case for odd-parity perturbations of spherically symmetric background configurations. As an example, the pulsation equations for self-gravitating, non-Abelian gauge fields are explicitly shown to be symmetric in the gravitational, the Yang Mills, and the off-diagonal sector. 
  A continuum of new monopole and dyon solutions in the Einstein-Yang-Mills theory in asymptotically anti-de Sitter space are found. They are regular everywhere and specified with their mass, and non-Abelian electric and magnetic charges. A class of monopole solutions which have no node in non-Abelian magnetic fields are shown to be stable against spherically symmetric linear perturbations. 
  Post-Newtonian expansions of the Brill-Lindquist and Misner-Lindquist solutions of the time-symmetric two-black-hole initial value problem are derived. The static Hamiltonians related to the expanded solutions, after identifying the bare masses in both solutions, are found to differ from each other at the third post-Newtonian approximation. By shifting the position variables of the black holes the post-Newtonian expansions of the three metrics can be made to coincide up to the fifth post-Newtonian order resulting in identical static Hamiltonians up the third post-Newtonian approximation. The calculations shed light on previously performed binary point-mass calculations at the third post-Newtonian approximation. 
  We consider a quantum test particle in the background of a Newtonian gravitational field in the framework of Cartan's formulation of nonrelativistic spacetimes. We have proposed a novel quantization of a point particle which amounts to introducing its position operators as multiplication with the corresponding Riemann normal coordinates and momentum operators as infinitesimal right translation operators determined by geodesic multiplication of points of the spacetime. We present detailed calculations for the simplest model of a two-dimensional Newtonian spacetime. 
  The question is examined, whether the formally straightforward extension of Hooke's time-honoured stress-strain relation to the four dimensions of special and of general relativity can make physical sense. The four-dimensional Hooke's law is found able to account for the inertia of matter; in the flat space, slow motion approximation the field equations for the ``displacement'' four-vector field can encompass both linear elasticity and inertia. In this limit one just recovers the equations of motion of the classical theory of elasticity. 
  We consider the Einstein constraints on asymptotically euclidean manifolds $M$ of dimension $n \geq 3$ with sources of both scaled and unscaled types. We extend to asymptotically euclidean manifolds the constructive method of proof of existence. We also treat discontinuous scaled sources. In the last section we obtain new results in the case of non-constant mean curvature. 
  By a simple modification of Hawking's well-known topology theorems for black hole horizons, we find lower bounds for the areas of smooth apparent horizons and smooth cross-sections of stationary black hole event horizons of genus $g>1$ in four dimensions. For a negatively curved Einstein space, the bound is ${{4\pi (g-1)}\over {-\ell}}$ where $\ell$ is the cosmological constant of the spacetime. This is complementary to the known upper bound on the area of $g=0$ black holes in de Sitter spacetime. It also emerges that $g>1$ quite generally requires a mean negative energy density on the horizon. The bound is sharp; we show that it is saturated by certain extreme, asymptotically locally anti-de Sitter spacetimes. Our results generalize a recent result of Gibbons. 
  We demonstrate that the high isotropy of the Cosmic Microwave Background (CMB), combined with the Copernican principle, is not sufficient to prove homogeneity of the universe -- in contrast to previous results on this subject. The crucial additional factor not included in earlier work is the acceleration of the fundamental observers. We find the complete class of irrotational perfect fluid spacetimes admitting an exactly isotropic radiation field for every fundamental observer and show that are FLRW if and only if the acceleration is zero. While inhomogeneous in general, these spacetimes all possess three-dimensional symmetry groups, from which it follows that they also admit a thermodynamic interpretation. In addition to perfect fluids models we also consider multi-component fluids containing non-interacting radiation, dust and a quintessential scalar field or cosmological constant in which the radiation is isotropic for the geodesic (dust) observers. It is shown that the non-acceleration of the fundamental observers forces these spacetimes to be FLRW. While it is plausible that fundamental observers (galaxies) in the real universe follow geodesics, it is strictly necessary to determine this from local observations for the cosmological principle to be more than an assumption. We discuss how observations may be used to test this. 
  The appearance of peaks in various primordial fluctuation Fourier power spectra is a generic prediction of the inflationary scenario. We investigate whether future experiments, in particular the satellite experiment PLANCK, will be able to detect the possible appearance of these peaks in the B-mode polarization multipole power spectrum. This would yield a conclusive proof of the presence of a primordial background of gravitational waves. 
  We find a self-gravitating monopole and its black hole solution in Brans-Dicke theory. We mainly discuss the properties of these solutions in the Einstein frame and compare the solutions with those in general relativity. 
  I give a review of the conceptual issues that arise in theories of quantum cosmology. I start by emphasising some features of ordinary quantum theory that also play a crucial role in understanding quantum cosmology. I then give motivations why spacetime cannot be treated classically at the most fundamental level. Two important issues in quantum cosmology -- the problem of time and the role of boundary conditions -- are discussed at some length. Finally, I discuss how classical spacetime can be recovered as an approximate notion. This involves the application of a semiclassical approximation and the process of decoherence. The latter is applied to both global degrees of freedom and primordial fluctuations in an inflationary Universe. 
  Starting with the Hamiltonian formulation for spacetimes with two commuting spacelike Killing vectors, we construct a midisuperspace model for linearly polarized plane waves in vacuum gravity. This model has no constraints and its degrees of freedom can be interpreted as an infinite and continuous set of annihilation and creation like variables. We also consider a simplified version of the model, in which the number of modes is restricted to a discrete set. In both cases, the quantization is achieved by introducing a Fock representation. We find regularized operators to represent the metric and discuss whether the coherent states of the quantum theory are peaked around classical spacetimes. It is shown that, although the expectation value of the metric on Killing orbits coincides with a classical solution, its relative fluctuations become significant when one approaches a region where null geodesics are focused. In that region, the spacetimes described by coherent states fail to admit an approximate classical description. This result applies as well to the vacuum of the theory. 
  We have developed a new numerical scheme to obtain quasiequilibrium structures of nonaxisymmetric compact stars such as binary neutron star systems as well as the spacetime around those systems in general relativity. Concerning quasiequilibrium states of binary systems in general relativity, several investigations have been already carried out by assuming conformal flatness of the spatial part of the metric. However, the validity of the conformally flat treatment has not been fully analyzed except for axisymmetric configurations. Therefore it is desirable to solve quasiequilibrium states by developing totally different methods from the conformally flat scheme. In this paper we present a new numerical scheme to solve directly the Einstein equations for 3D configurations without assuming conformal flatness, although we make use of the simplified metric for the spacetime. This new formulation is the extension of the scheme which has been successfully applied for structures of axisymmetric rotating compact stars in general relativity. It is based on the integral representation of the Einstein equations by taking the boundary conditions at infinity into account. We have checked our numerical scheme by computing equilibrium sequences of binary polytropic star systems in Newtonian gravity and those of axisymmetric polytropic stars in general relativity. We have applied this numerical code to binary star systems in general relativity and have succeeded in obtaining several equilibrium sequences of synchronously rotating binary polytropes with the polytropic indices N = 0.0, 0.5 and 1.0. 
  A scheme for an algebraic quantization of the causal sets of Sorkin et al. is presented. The suggested scenario is along the lines of a similar algebraization and quantum interpretation of finitary topological spaces due to Zapatrin and this author. To be able to apply the latter procedure to causal sets Sorkin's `semantic switch' from `partially ordered sets as finitary topological spaces' to `partially ordered sets as locally finite causal sets' is employed. The result is the definition of `quantum causal sets'. Such a procedure and its resulting definition is physically justified by a property of quantum causal sets that meets Finkelstein's requirement from `quantum causality' to be an immediate, as well as an algebraically represented, relation between events for discrete locality's sake. The quantum causal sets introduced here are shown to have this property by direct use of a result from the algebraization of finitary topological spaces due to Breslav, Parfionov and Zapatrin. 
  Generalizing previous quantum gravity results for Schwarzschild black holes from 4 to D > 3 space-time dimensions yields an energy spectrum E_n = alpha n^{(D-3)/(D-2)} E_P, n=1,2,..., alpha = O(1), where E_P is the Planck energy in that space-time. This spectrum means that the quantized area A_{D-2}(n) of the D-2 dimensional horizon has universally the form A_{D-2} = n a_{D-2}, where a_{D-2} is essentially the (D-2)th power of the D-dimensional Planck length. Assuming that the basic area quantum has a Z(2)-degeneracy according to its two possible orientation degrees of freedom implies a degeneracy d_n = 2^n for the n-th level. The energy spectrum with such a degeneracy leads to a quantum canonical partition function which is the same as the classical grand canonical partition function of a primitive Ising droplet nucleation model for 1st-order phase transitions in D-2 spatial dimensions. The analogy to this model suggests that E_n represents the surface energy of a "droplet" of n horizon quanta. Exploiting the well-known properties of the so-called critical droplets of that model immediately leads to the Hawking temperature and the Bekenstein-Hawking entropy of Schwarzschild black holes. The values of temperature and entropy appear closely related to the imaginary part of the partition function which describes metastable states 
  The symplectic reduction of pure spherically symmetric (Schwarzschild) classical gravity in D space-time dimensions yields a 2-dimensional phase space of observables consisting of the Mass M (>0) and a canonically conjugate (Killing) time variable T. Imposing (mass-dependent) periodic boundary conditions in time on the associated quantum mechanical plane waves which represent the Schwarzschild system in the period just before or during the formation of a black hole, yields an energy spectrum of the hole which realizes the old Bekenstein postulate that the quanta of the horizon A_{D-2} are multiples of a basic area quantum. In the present paper it is shown that the phase space of such a Schwarzschild black hole in D space-time dimensions is symplectomorphic to a symplectic manifold  S={(phi in R mod 2 pi, p = A_{D-2} >0)} with the symplectic form d phi wedge d p. As the action of the group SO_+(1,2) on that manifold is transitive, effective and Hamiltonian, it can be used for a group theoretical quantization of the system. The area operator p for the horizon corresponds to the generator of the compact subgroup SO(2) and becomes quantized accordingly: The positive discrete series of the irreducible unitary representations of SO_+(1,2) yields an (horizon) area spectrum proportional k+n, where k =1,2,... characterizes the representation and n = 0,1,2,... the number of area quanta. If one employs the unitary representations of the universal covering group of SO_+(1,2) the number k can take any fixed positive real value (theta-parameter). The unitary representations of the positive discrete series provide concrete Hilbert spaces for quantum Schwarzschild black holes. 
  We use the model of L. Randall et al to investigate the stability of allowed quantum field configurations. Firstly, we find that due to the topology of this 5 dimensional model, there are 2 possible configurations of the scalar field, untwisted and twisted. They give rise to two types of instability. Secondly, when allowed to interact in the 3-brane, the untwisted field becomes unstable even if it is at the true vacuum groundstate.This instability in the 4D submanifold results results from one-loop corrections that arise from coupling with the twisted field. On the other hand, the twisted field can make the two 3-branes distinguishable by causing an energy difference between them. That is due to the antiperiodicity of the twisted fields, when rotating with $\pi$ to go from one 3-brane to the other. The energy difference between the branes renders the fifth dimension unstable. 
  We study analytically black holes pierced by a thin vortex in dilatonic gravity for an arbitrary coupling of the vortex to the dilaton in an arbitrary frame. We show that the horizon of the charged black hole supports the long-range fields of the Nielsen-Olesen vortex that can be considered as black hole hair for both massive and massless dilatons. We also prove that extremal black holes exhibit a flux expulsion phenomenon for a sufficiently thick vortex. We consider the gravitational back-reaction of the thin vortex on the spacetime geometry and dilaton, and discuss under what circumstances the vortex can be used to smooth out the singularities in the dilatonic C-metrics. The effect of the vortex on the massless dilaton is to generate an additional dilaton flux across the horizon. 
  A description and analysis are given of a ``speed meter'' for monitoring a classical force that acts on a test mass. This speed meter is based on two microwave resonators (``dual resonators''), one of which couples evanescently to the position of the test mass. The sloshing of the resulting signal between the resonators, and a wise choice of where to place the resonators' output waveguide, produce a signal in the waveguide that (for sufficiently low frequencies) is proportional to the test-mass velocity (speed) rather than its position. This permits the speed meter to achieve force-measurement sensitivities better than the standard quantum limit (SQL), both when operating in a narrow-band mode and a wide-band mode. A scrutiny of experimental issues shows that it is feasible, with current technology, to construct a demonstration speed meter that beats the wide-band SQL by a factor 2. A concept is sketched for an adaptation of this speed meter to optical frequencies; this adaptation forms the basis for a possible LIGO-III interferometer that could beat the gravitational-wave standard quantum limit h_SQL, but perhaps only by a factor 1/xi = h_SQL/h ~ 3 (constrained by losses in the optics) and at the price of a very high circulating optical power --- larger by 1/xi^2 than that required to reach the SQL. 
  The Einstein theory of general relativity provides a peculiar example of classical field theory ruled by non-linear partial differential equations. A number of supplementary conditions (more frequently called gauge conditions) have also been considered in the literature. In the present paper, starting from the de Donder gauge, which is not conformally invariant but is the gravitational counterpart of the Lorenz gauge, we consider, led by geometric structures on vector bundles, a new family of gauges in general relativity, which involve fifth-order covariant derivatives of metric perturbations. A review of recent results by the authors is presented: restrictions on the general form of the metric on the vector bundle of symmetric rank-two tensor fields over space-time; admissibility of such gauges in the case of linearized theory about flat Euclidean space; generalization to a suitable class of curved Riemannian backgrounds, by solving an integral equation. Eventually, the applications to Euclidean quantum gravity are discussed. 
  The study of the properties and dynamics of self-gravitating bosonic objects in Einstein gravity was conducted. We studied self-coupled boson stars and determined the quasinormal mode (QNM) frequencies of stable boson stars in spherical symmetry. The study was carried out in the standard Einstein theory of General Relativity and in Brans-Dicke theory. We also studied the formation of these objects in Brans-Dicke theory showing that they can form from the self-gravitation of bosonic matter. We also studied the studied the possibility of a bosonic halo surrounding galaxies.   After an extensive study in spherical symmetry we carried out numerical studies of boson star dynamics in full 3+1 dimension. One focus of the 3D study was on the validation of the numerical code constructed to solve Einstein equations with matter sources. Boson Stars do not suffer from the surface problems of neutron stars or the singularities of black holes. The code was first tested with spherical perturbations and compared with the spherical results. We determined the coordinate conditions needed to provide stable evolutions. We then went on to study their behavior under non-spherical perturbations. We reproduced the QNM frequencies of the stars, as determined by perturbation studies carried out by other groups. The energy generated by the perturbation was studied with different radiation indicators. We also observed the collapse to black holes of unstable boson-star configurations. We simulated the collision of two boson stars. This is of interest as the two body problem is as yet unresolved in general relativity. 
  This paper has been withdrawn by the author. It has come to my attention that most of the results reported here have already been obtained by Ludvigsen. 
  We conclude that Katz's proposal (anisotropic heat reflection off of the back of the spacecraft high-gain antennae, the heat coming from the RTGs) does not provide enough power and so can not explain the Pioneer anomaly. 
  We conclude that Murphy's proposal (radiation of the power of the main-bus electrical systems from the rear of the craft) can not explain the anomalous Pioneer acceleration. 
  A geometrical framework for the definition of entropy in General Relativity via Noether theorem is briefly recalled and the entropy of Taub-Bolt Euclidean solutions of Einstein equations is then obtained as an application. The computed entropy agrees with previously known results, obtained by statistical methods. It was generally believed that the entropy of Taub-Bolt solution could not be computed via Noether theorem, due to the particular structure of singularities of this solution. We show here that this is not true. The Misner string singularity is, in fact, considered and its contribution to the entropy is analyzed. As a result, in our framework entropy does not obey the "one-quarter area law" and it is not directly related to horizons, as sometimes erroneously suggested in current literature on the subject. 
  We study a model of quantum cosmology originating from a classical model of gravitation where a self interacting scalar field is coupled to gravity with the metric undergoing a signature transition. We show that there are dual classical signature changing solutions, one at large scales and the other at small scales. It is possible to fine-tune the physics in both scales with an infinitesimal effective cosmological constant. 
  Beginning with Bekenstein, many authors have considered a uniformly spaced discrete quantum spectrum for black hole horizon area. It is also believed that the huge degeneracy of these area levels corresponds to the notion of black hole entropy. Starting from these two assumptions we here infer the algebra of a Schwarzschild black hole's observables. This algebra then serves as motivation for introducing in the system's Hamiltonian an interaction term. The interaction contains the horizon area operator, which is a number operator, and its canonical conjugate, the phase operator. The Hawking radiation from a Schwarzschild black hole is seen to be a consequence of an area-phase interaction. Using this interaction we have reproduced the semi-classical result for the Hawking radiation power. Furthermore, we show that the initial state of the black hole determines the nature of its development. Thus, a state which is an area eigenstate describes a static eternal black hole, but a coherent state describes a radiating black hole. Hence, it is the observer's initial knowledge or uncertainty about the horizon area which determines the evolution. 
  We examine the motion of a relativistic charged particle in a constant magnetic field perturbed by gravitational waves incident along the direction of the magnetic field. We apply a generalized energy conservation law to compute the variations of the kinetic energy of the particle during passage of the waves. We also explicitly compute the change in the orbit due to a wave with constant curvature. 
  The detection and the research of the neutrinos background of  Universe are the attractive problems. This problems do not seem the unpromising one in the case of the high neutrinos density that is necessary for the explanation of the nucleons-antinucleons asymmetry of Universe. It was offered before to use the low energy neutrinos background of Universe for the explanation of the gravitational phenomena with the quantum position attracting the Casimir's effect for this. As a result it was connected the gravitational constant with the parameters characterizing the electroweak interactions. If now we shall be based on the results of the experements fixing the equality of the gravitation mass and the inert one then it can consider that the spectrum of the particle masses is defined by their interaction with the neutrinos background of Universe. 
  A traversable wormhole solution of general scalar-tensor field equations is presented. We have shown, after a numerical analysis for the behavior of the scalar field of Brans-Dicke theory, that the solution is completely singularity--free. Furthermore, the analysis of more general scalar field dependent coupling constants indicates that the gravitational memory phenomenon may play an important role in the fate of natural wormholes. 
  We present a general formulation of the time-dependent initial value problem for a quantum scalar field of arbitrary mass and curvature coupling in a FRW cosmological model. We introduce an adiabatic number basis which has the virtue that the divergent parts of the quantum expectation value of the energy-momentum tensor <T_ab> are isolated in the vacuum piece of <T_ab>, and may be removed using adiabatic subtraction. The resulting renormalized <T_ab> is conserved, independent of the cutoff, and has a physically transparent, quasiclassical form in terms of the average number of created adiabatic `particles'. By analyzing the evolution of the adiabatic particle number in de Sitter spacetime we exhibit the time structure of the particle creation process, which can be understood in terms of the time at which different momentum scales enter the horizon. A numerical scheme to compute <T_ab> as a function of time with arbitrary adiabatic initial states (not necessarily de Sitter invariant) is described. For minimally coupled, massless fields, at late times the renormalized <T_ab> goes asymptotically to the de Sitter invariant state previously found by Allen and Folacci, and not to the zero mass limit of the Bunch-Davies vacuum. If the mass m and the curvature coupling xi differ from zero, but satisfy m^2+xi R=0, the energy density and pressure of the scalar field grow linearly in cosmic time demonstrating that, at least in this case, backreaction effects become significant and cannot be neglected in de Sitter spacetime. 
  The results of paper [1] are generalized for vacuum type-III solutions with, in general, a non-vanishing cosmological constant Lambda. It is shown that all curvature invariants containing derivatives of the Weyl tensor vanish if a type-III spacetime admits a non-expanding and non-twisting null geodesic congruence.   A non-vanishing curvature invariant containing first derivatives of the Weyl tensor is found in the case of type-III spacetime with expansion or twist. 
  We study cosmological scenarios resulting from effective actions in four dimensions which are, under some assumptions, connected with multidimensional, supergravity and string theories. These effective actions are labeled by the parameters $\omega$, the dilaton coupling constant, and $n$ which establishes the coupling between the dilaton and a scalar field originated from the gauge field existing in the original theories. There is a large class of bouncing as well as Friedmann-like solutions. We investigate under which conditions bouncing regular solutions can be obtained. In the case of the string effective action, regularity is obtained through the inclusion of contributions from the Ramond-Ramond sector of superstring. 
  Starting from the global parametrized post-Newtonian (PPN) reference system with two PPN parameters $\gamma$ and $\beta$ we consider a space-bounded subsystem of matter and construct a local reference system for that subsystem in which the influence of external masses reduces to tidal effects. Both the metric tensor of the local PPN reference system in the first post-Newtonian approximation as well as the coordinate transformations between the global PPN reference system and the local one are constructed in explicit form. The terms proportional to $\eta=4\beta-\gamma-3$ reflecting a violation of the equivalence principle are discussed in detail. We suggest an empirical definition of multipole moments which are intended to play the same role in PPN celestial mechanics as the Blanchet-Damour moments in General Relativity. Starting with the metric tensor in the local PPN reference system we derive translational equations of motion of a test particle in that system. The translational and rotational equations of motion for center of mass and spin of each of $N$ extended massive bodies possessing arbitrary multipole structure are derived. As an application of the general equations of motion a monopole-spin dipole model is considered and the known PPN equations of motion of mass monopoles with spins are rederived. 
  Continuously self-similar (CSS) solutions for the gravitational collapse of a spherically symmetric perfect fluid, with the equation of state p=kappa rho, with 0<kappa<1 a constant, are constructed numerically and their linear perturbations, both spherical and nonspherical, are investigated. The l=1 axial perturbations admit an analytical treatment. All others are studied numerically. For intermediate equations of state, with 1/9<kappa<0.49, the CSS solution has one spherical growing mode, but no nonspherical growing modes. That suggests that it is a critical solution even in (slightly) nonspherical collapse. For this range of kappa we predict the critical exponent for the black hole angular momentum to be 5(1+3kappa)/3(1+kappa) times the critical exponent for the black hole mass. For kappa=1/3 this gives an angular momentum critical exponent of mu=0.898, correcting a previous result. For stiff equations of state, 0.49<kappa<1, the CSS solution has one spherical and several nonspherical growing modes. For soft equations of state, 0<kappa<1/9, the CSS solution has 1+3 growing modes: a spherical one, and an l=1 axial mode (with m=-1,0,1). 
  Starting from the inhomogeneous shear--free Nariai metric we show, by solving the Einstein--Klein--Gordon field equations, how a self--interacting scalar field plus a material fluid, a variable cosmological term and a heat flux can drive the universe to its currently observed state of homogeneous accelerated expansion. A quintessence scenario where power-law inflation takes place for a string-motivated potential in the late--time dominated field regime is proposed. 
  On a manifold with boundary, the constraint algebra of general relativity may acquire a central extension, which can be computed using covariant phase space techniques. When the boundary is a (local) Killing horizon, a natural set of boundary conditions leads to a Virasoro subalgebra with a calculable central charge. Conformal field theory methods may then be used to determine the density of states at the boundary. I consider a number of cases---black holes, Rindler space, de Sitter space, Taub-NUT and Taub-Bolt spaces, and dilaton gravity---and show that the resulting density of states yields the expected Bekenstein-Hawking entropy. The statistical mechanics of black hole entropy may thus be fixed by symmetry arguments, independent of details of quantum gravity. 
  The destruction of the black-hole event horizon is ruled out by both cosmic censorship and the generalized second law of thermodynamics. We test the consistency of this prediction in a (more) `dangerous' version of the gedanken experiment suggested by Bekenstein and Rosenzweig. A U(1)-charged particle is lowered {\it slowly} into a near extremal black hole which is not endowed with a U(1) gauge field. The energy delivered to the black hole can be {\it red-shifted} by letting the assimilation point approach the black-hole horizon. At first sight, therefore, the particle is not hindered from entering the black hole and removing its horizon. However, we show that this dangerous situation is excluded by a combination of {\it two} factors not considered in former gedanken experiments: the effect of the spacetime curvature on the electrostatic {\it self-interaction} of the charged system (the black-hole polarization), and the {\it finite} size of the charged body. 
  The Robertson-Walker spacetimes are conformally flat and so are conformally invariant under the action of the Lie group SO(4,2), the conformal group of Minkowski spacetime. We find a local coordinate transformation allowing the Robertson-Walker metric to be written in a manifestly conformally flat form for all values of the curvature parameter k continuously and use this to obtain the conformal Killing vectors of the Robertson-Walker spacetimes directly from those of the Minkowski spacetime. The map between the Minkowski and Robertson-Walker spacetimes preserves the structure of the Lie algebra so(4,2). Thus the conformal Killing vector basis obtained does not depend upon k, but has the disadvantage that it does not contain explicitly a basis for the Killing vector subalgebra. We present an alternative set of bases that depend (continuously) on k and contain the Killing vector basis as a sub-basis (these are compared with a previously published basis). In particular, bases are presented which include the Killing vectors for all Robertson-Walker spacetimes with additional symmetry, including the Einstein static spacetimes and the de Sitter family of spacetimes, where the basis depends on the Ricci scalar R. 
  The behavior of the constants, G,c,h,a,e,m and Lambda, considering them as variable, in the framework of a flat cosmological model with FRW symmetries described by a bulk viscous fluid and considering mechanisms of adiabatic matter creation are investigated. Within two models; one with radiation predominance and another of matter predominance, this behavior are studied. 
  We investigate Refined Algebraic Quantization (RAQ) with group averaging in a constrained Hamiltonian system with unreduced phase space T^*R^4 and gauge group SL(2,R). The reduced phase space M is connected and contains four mutually disconnected `regular' sectors with topology R x S^1, but these sectors are connected to each other through an exceptional set where M is not a manifold and where M has non-Hausdorff topology. The RAQ physical Hilbert space H_{phys} decomposes as H_{phys} = (direct sum of) H_i, where the four subspaces H_i naturally correspond to the four regular sectors of M. The RAQ observable algebra A_{obs}, represented on H_{phys}, contains natural subalgebras represented on each H_i. The group averaging takes place in the oscillator representation of SL(2,R) on L^2(R^{2,2}), and ensuring convergence requires a subtle choice for the test state space: the classical analogue of this choice is to excise from M the exceptional set while nevertheless retaining information about the connections between the regular sectors. A quantum theory with the Hilbert space H_{phys} and a finitely-generated observable subalgebra of A_{obs} is recovered through both Ashtekar's Algebraic Quantization and Isham's group theoretic quantization. 
  A formulation of Einstein equations is presented that could yield advantages in the study of collisions of binary compact objects during regimes between linear-nonlinear transitions. The key idea behind this formulation is a separation of the dynamical variables into i) a fixed conformal 3-geometry, ii) a conformal factor possessing nonlinear dynamics and iii) transverse-traceless perturbations of the conformal 3-geometry. 
  Inspiraling compact binaries have been identified as one of the most promising sources of gravitational waves for interferometric detectors. Most of these binaries are expected to have circularized by the time their gravitational waves enter the instrument's frequency band. However, the possibility that some of the binaries might still possess a significant eccentricity is not excluded. We imagine a situation in which eccentric signals are received by the detector but not explicitly searched for in the data analysis, which uses exclusively circular waveforms as matched filters. We ascertain the likelihood that these filters, though not optimal, will nevertheless be successful at capturing the eccentric signals. We do this by computing the loss in signal-to-noise ratio incurred when searching for eccentric signals with those nonoptimal filters. We show that for a binary system of a given total mass, this loss increases with increasing eccentricity. We show also that for a given eccentricity, the loss decreases as the total mass is increased. 
  Ori and Soen have proposed a spacetime which has closed causal curves on the boundary of a region of normal causality, all within a region where the weak energy condition (positive energy density) is satisfied. I analyze the causal structure of this spacetime in some simplified models, show that the Cauchy horizon is compactly generated, and argue that any attempt to build such a spacetime with normal matter might lead to singular behavior where the causality violation would otherwise take place. 
  We review the anomaly induced effective action for dilaton coupled spinors and scalars in large N and s-wave approximation. It may be applied to study the following fundamental problems: construction of quantum corrected black holes (BHs), inducing of primordial wormholes in the early Universe (this effect is confirmed) and the solution of initial singularity problem. The recently discovered anti-evaporation of multiple horizon BHs is discussed. The existance of such primordial BHs may be interpreted as SUSY manifestation. Quantum corrections to BHs thermodynamics maybe also discussed within such scheme. 
  To probe naked spacetime singularities with waves rather than with particles we study the well-posedness of initial value problems for test scalar fields with finite energy so that the natural function space of initial data is the Sobolev space. In the case of static and conformally static spacetimes we examine the essential self-adjointness of the time translation operator in the wave equation defined in the Hilbert space. For some spacetimes the classical singularity becomes regular if probed with waves while stronger classical singularities remain singular. If the spacetime is regular when probed with waves we may say that the spacetime is `globally hyperbolic.' 
  We derive a new class of exact solutions of Einstein's equations providing a physically plausible hydrodynamical description of cosmological matter in the radiative era ($10^6 K > T > 10^3 K$), between nucleosynthesis and decoupling. The solutions are characterized by the Lema\^{\i}tre-Tolman -Bondi metric with a viscous fluid source, subjected to the following conditions: (a) the equilibrium state variables satisfy the equation of state of a mixture of an ultra-relativistic and a non-relativistic ideal gases, where the internal energy of the latter has been neglected, (b) the particle numbers of the mixture components are independently conserved, (c) the viscous stress is consistent with the transport equation and entropy balance law of Extended Irreversible Thermodynamics, with the coefficient of shear viscosity provided by Kinetic Theory for the `radiative gas' model. The fulfilment of (a), (b) and (c) restricts initial conditions in terms of an initial value function, $\Delta_i^{(s)}$, related to the average of spatial gradients of the fluctuations of photon entropy per baryon in the initial hypersurface. Constraints on the observed anisotropy of the microwave cosmic radiation and the condition that decoupling occurs at $T=T_{_D}\approx 4\times 10^3$ K yield an estimated value: $|\Delta_i^{(s)}|\approx 10^{-8}$ which can be associated with a bound on promordial entropy fluctuations. The Jeans mass at decoupling is of the same order of magnitude as that of baryon dominated perturbation models ($\approx 10^{16} M_\odot$) 
  Starting from metric of the general nonextreme stationary axisymmetric black hole in four-dimensional spacetime, both statistical-mechanical and thermodynamical entropies are studied. First, by means of the "brick wall" model in which the Dirichlet condition is replaced by a scattering ansatz for the field functions at the horizon and with Pauli-Villars regularization scheme, an expression for the statistical-mechanical entropy arising from the nonminimally coupled scalar fields is obtained. Then, by using the conical singularity method Mann and Solodukhin's result for the Kerr-Newman black hole (Phys. Rev. D54, 3932(1996)) is extended to the general stationary black hole and the nonminimally coupled scalar field. We last shown by comparing the two results that the statistical-mechanical entropy and one-loop correction to the thermodynamical entropy are equivalent for coupling $\xi\leq 0$. After renormalization, a relation between the two entropies is given. 
  We study the Lorentzian static traversable wormholes coupled to quadratic scalar fields. We also obtain the solutions of the scalar fields and matters in the wormhole background and find that the minimal size of the wormhole should be quantized under the appropriate boundary conditions for the positive non-minimal massive scalar field. 
  We present a specific two-dimensional dilaton gravity model in which a black hole evaporates leaving a wormhole at the end state. As the black hole formed by infalling matter in a initially static spacetime evaporates by emitting Hawking radiation, the black hole singularity that is initially hidden behind a timelike apparent horizon meets the shrinking horizon. At this intersection point, we imposed boundary conditions which require disappearance of the black hole singularity and generation of the exotic matter which is the source of the wormhole as the end state of the black hole. These, of course, preserve energy conservation and continuity of the metric. 
  The phase structure associated with the chiral symmetry is thoroughly investigated in de Sitter spacetime in the supersymmetric Nambu-Jona-Lasinio model with supersymmetry breaking terms. The argument is given in the three and four space-time dimensions in the leading order of the 1/N expansion and it is shown that the phase characteristics of the chiral symmetry is determined by the curvature of de Sitter spacetime. It is found that the symmetry breaking takes place as the first order as well as second order phase transition depending on the choice of the coupling constant and the parameter associated with the supersymmetry breaking term. The critical curves expressing the phase boundary are obtained. We also discuss the model in the context of the chaotic inflation scenario where topological defects (cosmic strings) develop during the inflation. 
  We consider a (4+d)-dimensional spacetime broken up into a (4-n)-dimensional Minkowski spacetime (where n goes from 1 to 3) and a compact (n+d)-dimensional manifold. At the present time the n compactification radii are of the order of the Universe size, while the other d compactification radii are of the order of the Planck length. 
  In the context of a family os scalar-tensor theories with a dynamical $\Lambda$, that is a binomial on the scalar field, the cosmological equations are considered. A general barotropic state equation $p=(\gamma-1)\rho$, for a perfect fluid is used for the matter content of the Universe. Some Friedmann- Robertson-Walker exact solutions are found, they have scale factor wich shows exponential or power law dependence on time. For some models the singularity can be avoided. Cosmological parameters as $\Omega_m$, $\Omega_{\Lambda}$, $q_0$ and $t_0$ are obtained and compared with observational data. 
  Combining the second-order entropy flow vector of the causal Israel-Stewart theory with the conformal Killing-vector property of $u_{i}/T$, where $u_{i}$ is the four-velocity of the medium and T its equilibrium temperature, we investigate generalized equilibrium states for cosmological fluids with nonconserved particle number. We calculate the corresponding equilibrium particle production rate and show that this quantity is reduced compared with the results of the previously studied first-order theory. Generalized equilibrium for massive particles turns out to be compatible with a dependence $\rho \propto a ^{-2}$ of the fluid energy density $\rho$ on the scale factor a of the Robertson-Walker metric and may be regarded as a realization of so-called K-matter. 
  McVittie's solution of Einstein's field equations, representing a point mass embedded into an isotropic universe, possesses a scalar curvature singularity at proper radius $R=2m$. The singularity is space-like and precedes, in the expanding case, all other events in the space-time. It is shown here that this singularity is gravitationally weak, and the possible structure of the region $R\leq 2m$ is investigated. A characterization of this solution which does not involve asymptotics is given. 
  The null geodesic equations in the Alcubierre warp drive spacetime are numerically integrated to determine the angular deflection and redshift of photons which propagate through the distortion of the ``warp drive'' bubble to reach an observer at the origin of the warp effect. We find that for a starship with an effective warp speed exceeding the speed of light, stars in the forward hemisphere will appear closer to the direction of motion than they would to an observer at rest. This aberration is qualitatively similar to that caused by special relativity. Behind the starship, a conical region forms from within which no signal can reach the starship, an effective ``horizon''. Conversely, there is also an horizon-like structure in a conical region in front of the starship, into which the starship cannot send a signal. These causal structures are somewhat analogous to the Mach cones associated with supersonic fluid flow. The existence of these structures suggests that the divergence of quantum vacuum energy when the starship effectively exceeds the speed of light, first discovered in two dimensions, will likely be present in four dimensions also, and prevent any warp-drive starship from ever exceeding the effective speed of light. 
  A class of theories of gravitation that naturally incorporates preferred frames of reference is presented. The underlying space-time geometry consists of a partial parallelization of space-time and has properties of Riemann-Cartan as well as teleparallel geometry. Within this geometry, the kinematic quantities of preferred frames are associated with torsion fields. Using a variational method, it is shown in which way action functionals for this geometry can be constructed. For a special action the field equations are derived and the coupling to spinor fields is discussed. 
  In this paper a solution for a static spherically symmetric body is thoroughly considered in the framework of the Relativistic Theory of Gravitation. By the comparison of this solution with the Schwarzschild solution in General Relativity their substantial difference is established in the region close to the Schwarzschild sphere. Just this difference excludes the possibility of collapse to form ``black holes''. 
  We explicitly demonstrate that the known solutions for expanding impulsive spherical gravitational waves that have been obtained by a "cut and paste" method may be considered to be impulsive limits of the Robinson-Trautman vacuum type N solutions. We extend these results to all the generically distinct subclasses of these solutions in Minkowski, de Sitter and anti-de Sitter backgrounds. For these we express the solutions in terms of a continuous metric. Finally, we also extend the class of spherical shock gravitational waves to include a non-zero cosmological constant. 
  Let a differential 4D-manifold with a smooth coframe field be given. Consider the operators on it that are linear in the second order derivatives or quadratic in the first order derivatives of the coframe, both with coefficients that depend on the coframe variables. The paper exhibits the class of operators that are invariant under a general change of coordinates, and, also, invariant under the global SO(1,3)-transformation of the coframe. A general class of field equations is constructed. We display two subclasses in it. The subclass of field equations that are derivable from action principles by free variations and the subclass of field equations for which spherical-symmetric solutions, Minkowskian at infinity exist. Then, for the spherical-symmetric solutions, the resulting metric is computed. Invoking the Geodesic Postulate, we find all the equations that are experimentally (by the 3 classical tests) indistinguishable from Einstein field equations. This family includes, of course, also Einstein equations. Moreover, it is shown, explicitly, how to exhibit it. The basic tool employed in the paper is an invariant formulation reminiscent of Cartan's structural equations. The article sheds light on the possibilities and limitations of the coframe gravity. It may also serve as a general procedure to derive covariant field equations. 
  We will discuss some analogies between internal gauge theories and gravity in order to better understand the charge concept in gravity. A dimensional analysis of gauge theories in general and a strict definition of elementary, monopole, and topological charges are applied to electromagnetism and to teleparallelism, a gauge theoretical formulation of Einstein gravity.   As a result we inevitably find that the gravitational coupling constant has dimension $\hbar/l^2$, the mass parameter of a particle dimension $\hbar/l$, and the Schwarzschild mass parameter dimension l (where l means length). These dimensions confirm the meaning of mass as elementary and as monopole charge of the translation group, respectively. In detail, we find that the Schwarzschild mass parameter is a quasi-electric monopole charge of the time translation whereas the NUT parameter is a quasi-magnetic monopole charge of the time translation as well as a topological charge. The Kerr parameter and the electric and magnetic charges are interpreted similarly. We conclude that each elementary charge of a Casimir operator of the gauge group is the source of a (quasi-electric) monopole charge of the respective Killing vector. 
  The Brill-Lindquist time-symmetric initial-value solution for two uncharged black holes is rederived using the Hamiltonian constraint equation with Dirac delta distributions as a source for the binary black-hole field. The bare masses of the Brill-Lindquist black holes are introduced in a way which is applied, after straightforward modification, to the Misner-Linquist binary black-hole solution. 
  In this paper the problem of the quantum stability of the two-dimensional warp drive spacetime moving with an apparent faster than light velocity is considered. We regard as a maximum extension beyond the event horizon of that spacetime its embedding in a three-dimensional Minkowskian space with the topology of the corresponding Misner space. It is obtained that the interior of the spaceship bubble becomes then a multiply connected nonchronal region with closed timelike curves and that the most natural vacuum allows quantum fluctuations which do not induce any divergent behaviour of the re-normalized stress-energy tensor, even on the event (Cauchy) chronology horizon. In such a case, the horizon encloses closed timelike curves only at scales close to the Planck length, so that the warp drive satisfies the Ford's negative energy-time inequality. Also found is a connection between the superluminal two-dimensional warp drive space and two-dimensional gravitational kinks. This connection allows us to generalize the considered Alcubierre metric to a standard, nonstatic metric which is only describable on two different coordinate patches 
  The search for the gravitational energy-momentum tensor is often qualified as an attempt of looking for ``the right answer to the wrong question''. This position does not seem convincing to us. We think that we have found the right answer to the properly formulated question. We have further developed the field theoretical formulation of the general relativity which treats gravity as a non-linear tensor field in flat space-time. The Minkowski metric is a reflection of experimental facts, not a possible choice of the artificial ``prior geometry''. In this approach, we have arrived at the gravitational energy-momentum tensor which is: 1) derivable from the Lagrangian in a regular prescribed way, 2) tensor under arbitrary coordinate transformations, 3) symmetric in its components, 4) conserved due to the equations of motion derived from the same Lagrangian, 5) free of the second (highest) derivatives of the field variables, and 6) is unique up to trivial modifications not containing the field variables. There is nothing else, in addition to these 6 conditions, that one could demand from an energy-momentum object, acceptable both on physical and mathematical grounds. The derived gravitational energy-momentum tensor should be useful in practical applications. 
  We describe general-relativistically a spherically symmetric stationary fluid accretion onto a black hole. Relativistic effects enhance mass accretion, in comparison to the Bondi model predictions, in the case when backreaction is neglected. That enhancement depends on the adiabatic index and the asymptotic gas temperature and it can magnify accretion by one order in the ultrarelativistic regime. 
  In this paper we study the electromagnetic fields generated by a Killing vector field in vacuum space-times (Papapetrou fields). The motivation of this work is to provide new tools for the resolution of Maxwell's equations as well as for the search, characterization, and study of exact solutions of Einstein's equations. The first part of this paper is devoted to an algebraic study in which we give an explicit and covariant procedure to construct the principal null directions of a Papapetrou field. In the second part, we focus on the main differential properties of the principal directions, studying when they are geodesic, and in that case we compute their associated optical scalars. With this information we get the conditions that a principal direction of the Papapetrou field must satisfy in order to be aligned with a multiple principal direction of the Weyl tensor in the case of algebraically special vacuum space-times. Finally, we illustrate this study using the Kerr, Kasner and pp waves space-times. 
  We reconsider the Kerr metric with cosmological term $\Lambda$ imposing the condition that the angular velocity $\omega$ of the dragging of the inertial frames vanishes at spatial boundaries. Some properties of the extreme black holes in the revisited solutions are discussed. 
  We propose a method to recover the time variable and the classical evolution of the Universe from the minisuperspace wave function of the Wheeler-DeWitt equation. Defining a Hamilton-Jacobi characteristic function $W$ as the imaginary part of the $\ln \Psi$ we can recover the classical solution, and quantum corrections. The key idea is to let the energy of the Wheeler-DeWitt equation vanish only after the semiclassical limit is taken. 
  We study quantum mechanically the self-similar black hole formation by collapsing scalar field and find the wave functions that give the correct semiclassical limit. In contrast to classical theory, the wave functions for the black hole formation even in the supercritical case have not only incoming flux but also outgoing flux. From this result we compute the rate for the black hole formation. In the subcritical case our result agrees with the semiclassical tunneling rate. Furthermore, we show how to recover the classical evolution of black hole formation from the wave function by defining the Hamilton-Jacobi characteristic function as $W = \hbar {\rm Im} \ln \psi$. We find that the quantum corrected apparent horizon deviates from the classical value only slightly without any qualitative change even in the critical case. 
  Uncompactified KK universes are so intrinsically connected to the otherwise only empirically required "missing" Dark Matter (DM), that:   1) They yield a simple prediction which explains both the enigma of the extra-dimensions' (XD) unobservability and the enigma of the present DM. The two enigmas are "annihilated" into the hypothesis of "missing light", or better of "photonland". This eliminates the very need to hypothesize/search/find DM-candidates of exceptional/exotic properties to explain their darkness.   2) An early, spontaneous gravitational XD-collapse of their natural 5D-DM replaces KK's compactification mechanisms and cylindricity condition, and (partly?) eliminates the quandary of the radically insufficient density fluctuations. 
  The physical deflection angle of a light ray propagating in a space-time supplied with an asymptotically flat metric has to be expressed in terms of the impact parameter. 
  We investigate a two-component model for gravitational lenses, i.e., the fermion-fermion star as a dark matter self-gravitating system made from two kinds of fermions with different masses. We calculate the deflection angles varying from arcseconds to even degrees. There is one Einstein ring. In particular, we find three radial critical curves for radial magnifications and four or five images of a point source. These are different from the case of the one-component model such as the fermion stars and boson stars. This is due to the fermion-fermion star being a two-component concentric sphere. Our results suggest that any possible observations of the number of images more than 3 could imply a polytropic distribution of the mass inside the lens in the universe. 
  We calculate the quantum corrections of geometric and thermodynamic quantities for the Reisner-Nordstrom charged black hole, within the context of 2D spherically symmetric dilaton gravity model. Special attention is payed to the quantum corrections of the extreme Reisner-Nordstrom solution. We find a state of the extreme black hole with regular behaviour on the horizon. 
  We discuss how compactified extra dimensions may have potentially observable effects which grow as the compactification scale decreases. This arises because of lightcone fluctuations in the uncompactified dimensions which can result in the broadening of the spectral lines from distant sources. We analyze this effect in a five dimensional model, and argue that data from gamma ray burst sources require the compactification length to be greater than about $10^5$ cm in this model. 
  Motivated by results of recent analytic studies, we present a numerical investigation of the late-time dynamics of scalar test fields on Kerr backgrounds. We pay particular attention to the issue of mixing of different multipoles and their fall-off behavior at late times. Confining ourselves to the special case of axisymmetric modes with equatorial symmetry, we show that, in agreement with the results of previous work, the late-time behavior is dominated by the lowest allowed l-multipole. However the numerical results imply that, in general, the late-time fall-off of the dominating multipole is different from that in the Schwarzschild case, and seems to be incompatible with a result of a recently published analytic study. 
  Superfluid condensates are known to occur in contexts ranging from laboratory liquid helium to neutron stars, and are also likely to occur in cosmological phenomena such as axion fields. In the zero temperature limit, such condensates are describable at a mesoscopic level by irrotational configurations of simple relativistic perfect fluid models. The general mechanical properties of such models are presented here in an introductory review giving special attention to the dynamics of vorticity flux 2-surfaces and the action principles governing both individual flow trajetories and the evolution of the system as a whole. Macroscopic rotation of such a condensate requires the presence of a lattice of quantised vortex defects, whose averaged tension violates perfect fluid isotropy. It is shown that for any equation of state (relating the mass density $\rho$ to the pressure $P$) the mesoscopic perfect fluid model can be extended in a uniquely simple and natural manner to a corresponding macroscopic model (in a conformally covariant category) that represents the effects of the vortex fibration anisotropy. The limiting case of an individual vortex defect is shown to be describable by a (``global'') string type model with a variable tension ${\cal T}$ (obtained as a function of the background fluid density) whose ``vorton'' (i.e. closed loop equilibrium) states are derived as an exercise. 
  A little known theorem due to Campbell is employed to establish the local embedding of a wide class of 4-dimensional spacetimes in 5-dimensional Ricci-flat spaces. An embedding for the class of n-dimensional Einstein spaces is also found. The local nature of Campbell's theorem is highlighted by studying the embedding of some lower-dimensional spaces. 
  Hawking radiation in d=4 is regarded as a well understood quantum theoretical feature of Black Holes or of other geometric backgrounds with an event horizon. On the other hand, the dilaton theory emerging after spherical reduction and generalized dilaton theories only during the last years became the subject of numerous studies which unveiled a surprisingly difficult situation. Recently we have found some solution to the problem of Hawking flux in spherically reduced gravity which has the merit of using a minimal input. It leads to exact cancellation of negative contributions to this radiative flux, encountered in other approaches at infinity, so that our result asymptotically coincides with the one of minimally coupled scalars. The use of an integrated action is avoided - although we have been able to present also that quantity in a closed expression. This short review also summarizes and critically discusses recent activities in this field, including the problem of ``conformal frames'' for the background and questions which seem to be open in our own approach as well as in others. 
  Investigations of classical signature change have generally envisaged applications to cosmological models, usually a Friedmann-Lemaitre-Robertson-Walker model. The purpose has been to avoid the inevitable singularity of models with purely Lorentzian signature, replacing the neighbourhood of the big bang with an initial, singularity free region of Euclidean signture, and a signature change. We here show that signature change can also avoid the singularity of gravitational collapse. We investigate the process of re-birth of Schwarzschild type black holes, modelling it as a double signature change, joining two universes of Lorentzian signature through a Euclidean region which provides a `bounce'. We show that this process is viable both with and without matter present, but realistic models -- which have the signature change surfaces hidden inside the horizons -- require non-zero density. In fact the most realistic models are those that start as a finite cloud of collapsing matter, surrounded by vacuum. We consider how geodesics may be matched across a signature change surface, and conclude that the particle `masses' must jump in value. This scenario may be relevant to Smolin's recent proposal that a form of natural selection operates on the level of universes, which favours the type of universe we live in. 
  A qualitative analysis is presented for a class of homogeneous cosmologies derived from the string effective action when a cosmological constant is present in the matter sector of the theory. Such a term has significant effects on the qualitative dynamics. For example, models exist which undergo a series of oscillations between expanding and contracting phases due to the existence of a heteroclinic cycle in the phase space. Particular analytical solutions corresponding to the equilibrium points are also found. 
  It is well known that the late-time behaviour of gravitational collapse is {\it dominated} by an inverse power-law decaying tail. We calculate {\it higher-order corrections} to this power-law behaviour in a spherically symmetric gravitational collapse. The dominant ``contamination'' is shown to die off at late times as $M^2t^{-4}\ln(t/M)$. This decay rate is much {\it slower} than has been considered so far. It implies, for instance, that an `exact' (numerical) determination of the power index to within $\sim 1 %$ requires extremely long integration times of order $10^4 M$. We show that the leading order fingerprint of the black-hole electric {\it charge} is of order $Q^2t^{-4}$. 
  Primordial gravitational waves are amplified during eras when their wavelengths are pushed outside the cosmological horizon. This occurs in both inflationary and ``pre-big-bang'' or ``bounce'' cosmologies. The spectrum is expressed as a normalized energy density per unit logarithmic frequency, denoted Omega. The spectral index (logarithmic slope) of Omega is simply related to three properties of the early universe: (i) the gravitons' mean initial quantum occupation number N(n) (=1/2 for a vacuum state), where n is the (invariant) conformal frequency of the mode, and (ii) & (iii) the parameter gamma=p/rho of the cosmological equation of state during the epoch when the waves left the horizon (gamma=gamma_i) and when they reentered (gamma=gamma_f). In the case of an inflationary cosmology, the spectral index is equal to      d(ln N)/d(ln n) + 2(gamma_i + 1)/(gamma_i + 1/3)                      + 2(gamma_f - 1/3)/(gamma_f + 1/3)   and for bounce cosmologies it is equal to      d(ln N)/d(ln n) + 4(gamma_i)/(gamma_i + 1/3)                      + 2(gamma_f - 1/3)/(gamma_f + 1/3)   These expressions are compared against various more model-specific results given in the literature. 
  We develop a method for computing the free-energy of a canonical ensemble of quantum fields near the horizon of a rotating black hole. We show that the density of energy levels of a quantum field on a stationary background can be related to the density of levels of the same field on a fiducial static space-time. The effect of the rotation appears in the additional interaction of the "static" field with a fiducial abelian gauge-potential. The fiducial static space-time and the gauge potential are universal, i.e., they are determined by the geometry of the given physical space-time and do not depend on the spin of the field. The reduction of the stationary axially symmetric problem to the static one leads to a considerable simplification in the study of statistical mechanics and we use it to draw a number of conclusions. First, we prove that divergences of the entropy of scalar and spinor fields at the horizon in the presence of rotation have the same form as in the static case and can be removed by renormalization of the bare black hole entropy. Second, we demonstrate that statistical-mechanical representation of the Bekenstein-Hawking entropy of a black hole in induced gravity is universal and does not depend on the rotation. 
  The dynamics of the Bianchi IX cosmological model with minimally coupled massive real scalar field is studied. The possibility of non-singular transition from contraction to expansion is shown. A set of initial conditions that lead to non-singular solutions is studied numerically. 
  All non-twisting Petrov-type N solutions of vacuum Einstein field equations with cosmological constant Lambda are summarized. They are shown to belong either to the non-expanding Kundt class or to the expanding Robinson-Trautman class. Invariant subclasses of each class are defined and the corresponding metrics are given explicitly in suitable canonical coordinates. Relations between the subclasses and their geometrical properties are analyzed. In the subsequent paper these solutions are interpreted as exact gravitational waves propagating in de Sitter or anti-de Sitter spacetimes. 
  In a suitably chosen essentially unique frame tied to a given observer in a general spacetime, the equation of geodesic deviation can be decomposed into a sum of terms describing specific effects: isotropic (background) motions associated with the cosmological constant, transverse motions corresponding to the effects of gravitational waves, longitudinal motions, and Coulomb-type effects. Conditions under which the frame is parallelly transported along a geodesic are discussed. Suitable coordinates are introduced and an explicit coordinate form of the frame is determined for spacetimes admitting a non-twisting null congruence. Specific properties of all non-twisting type N vacuum solutions with cosmological constant Lambda (non-expanding Kundt class and expanding Robinson-Trautman class) are then analyzed. It is demonstrated that these spacetimes can be understood as exact transverse gravitational waves of two polarization modes "+" and "x", shifted by pi/4, which propagate "on" Minkowski, de Sitter, or anti-de Sitter backgrounds. It is also shown that the solutions with Lambda>0 may serve as exact demonstrations of the cosmic "no-hair" conjecture in radiative spacetimes with no symmetry. 
  The well-known ``displace, cut, and reflect'' method used to generate cold disks from given solutions of Einstein equations is extended to solutions of Einstein-Maxwell equations. Four exact solutions of the these last equations are used to construct models of hot disks with surface density, azimuthal pressure, and azimuthal current. The solutions are closely related to Kerr, Taub-NUT, Lynden-Bell-Pinault and to a one-soliton solution. We find that the presence of the magnetic field can change in a nontrivial way the different properties of the disks. In particular, the pure general relativistic instability studied by Bicak, Lynden-Bell and Katz [Phys. Rev. D47, 4334, 1993] can be enhanced or cured by different distributions of currents inside the disk. These currents, outside the disk, generate a variety of axial symmetric magnetic fields. As far as we know these are the first models of hot disks studied in the context of general relativity. 
  A cosmological model of homogeneous and isotropic spatially flat Universe with gravitating self-interacting scalar field is considered. The exact solution, admitting an analytical exit from inflationary stage into a radiation era and a matter dominated epoch, is obtained by virtue of ``fine turning of the potential'' method. We found that an inflationary stage is supported by decay of higgs bosons in the framework of the solution obtained. Freidmann's regim is associated with adiabatical expantion of the Universe, filled by the matter with special equation of state. Thus we presented the exact solution which solve the problem of transition from an inflationary to a radiation eras or long standing `exit' problem. 
  This paper deals with the problem of a point-like charged source under the influence of the external electromagnetic field in terms of perturbation theory for GR equations. It is obtained that GR, in contrast with the classical electrodynamics, in linear perturbation theory predicts an unlimited growth of the dipole perturbation. It is shown that the reason for this unlimited perturbation growth might be related to the presence of the unstable rotational perturbation mode. The analysis of the conditions under which this instability may disappear is performed. The momentum value at which the stability is reached is estimated. These estimations give the electron spin by the order of magnitude (when charge value is equal to elementary one). 
  We investigate in detail the qualitative behaviour of the class of Bianchi type B spatially homogeneous cosmological models in which the matter content is composed of two non-interacting components; the first component is described by a barotropic fluid having a gamma-law equation of state, whilst the second is a non-interacting scalar field (phi) with an exponential potential V=Lambda exp(k phi). In particular, we study the asymptotic properties of the models both at early and late times, paying particular attention on whether the models isotropize (and inflate) to the future, and we discuss the genericity of the cosmological scaling solutions. 
  Statistical entropies of a general relativistic ideal gas obeying Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac statistics are calculated in a general axisymmetry space-time of arbitrary dimension. This general formation can be used to discuss the entropy of a quantum field not only in the flat space-time but also in a curved space-time. It can also be used to compare the entropies in different dimensional space-times. Analytical expressions for the thermodynamic potentials are presented, and their behaviors in the high or low temperature approximation are discussed. The entropy of a quantum field is shown to be proportional to the volume of optical space or that of the dragged optical space only in the high temperature approximation or in the zero mass case. In the case of a black hole, the entropy of a quantum field at the Hartle-Hawking temperature is proportional to the horizon "area" if and only if the horizon is located at the light velocity surface. 
  We compute the response and the angular pattern function of an interferometer for a scalar component of gravitational radiation in Brans-Dicke theory. We examine the problem of detecting a stochastic background of scalar GWs and compute the scalar overlap reduction function in the correlation between an interferometer and the monopole mode of a resonant sphere. While the correlation between two interferometers is maximized taking them as close as possible, the interferometer-sphere correlation is maximized at a finite value of f*d, where `f' is the resonance frequency of the sphere and `d' the distance between the detectors. This defines an optimal resonance frequency of the sphere as a function of the distance. For the correlation between the Virgo interferometer located near Pisa and a sphere located in Frascati, near Rome, we find an optimal resonance frequency f=590 Hz. We also briefly discuss the difficulties in applying this analysis to the dilaton and moduli fields predicted by string theory. 
  We present new exact cosmological inhomogeneous solutions for gravity coupled to a scalar field in a general framework specified by the parameter $\lambda$. The equations of motion (and consequently the solutions) in this framework correspond either to low-energy string theory or Weyl integrable spacetime according to the sign of $\lambda$. We show that different inflationary behaviours are possible, as suggested by the study of the violation of the strong energy condition. Finally, by the analysis of certain curvature scalars we found that some of the solutions may be nonsingular. 
  For each optical topology of an interferometric gravitational wave detector, quantum mechanics dictates a minimum optical power (the ``energetic quantum limit'') to achieve a given sensitivity. For standard topologies, when one seeks to beat the standard quantum limit by a substantial factor, the energetic quantum limit becomes impossibly large. Intracavity readout schemes may do so with manageable optical powers. 
  We characterize a general solution to the vacuum Einstein equations which admits isolated horizons. We show it is a non-linear superposition -- in precise sense -- of the Schwarzschild metric with a certain free data set propagating tangentially to the horizon. This proves Ashtekar's conjecture about the structure of spacetime near the isolated horizon. The same superposition method applied to the Kerr metric gives another class of vacuum solutions admitting isolated horizons. More generally, a vacuum spacetime admitting any null, non expanding, shear free surface is characterized. The results are applied to show that, generically, the non-rotating isolated horizon does not admit a Killing vector field and a spacetime is not spherically symmetric near a symmetric horizon. 
  We show that there exists a duality between the local coordinates and the solutions of the Klein-Gordon equation in curved spacetime in the same sense as in the Minkowski spacetime. However, the duality in curved spacetime does not have the same generality as in flat spacetime and it holds only if the system satisfies certain constraints. We derive these constraints and the basic equations of duality and discuss the implications in the quantum theory. 
  It is well known that there can be negative energy densities in quantum field theory. Most of the work done in this area has involved free non-interacting systems. In this paper we show how a quantum state with negative energy density can be formulated for a Dirac field interacting with an Electromagnetic field. It will be shown that, for this case, there exist quantum states whose average energy density over an arbitrary volume is a negative number with an arbitrarily large magnitude. 
  We study analytically the Cauchy horizon singularity inside spherically-symmetric charged black holes, coupled to a spherical self-gravitating, minimally-coupled, massless scalar field. We show that all causal geodesics terminate at the Cauchy horizon at a null singularity, which is weak according to the Tipler classification. The singularity is also deformationally-weak in the sense of Ori. Our results are valid at arbitrary points along the null singularity, in particular at late retarded times, when non-linear effects are crucial. 
  A method is presented for imputing a topology for any chronological set, i.e., a set with a chronology relation, such as a spacetime or a spacetime with some sort of boundary. This topology is shown to have several good properties, such as replicating the manifold topology for a spacetime and replicating the expected topology for some simple examples of spacetime-with-boundary; it also allows for a complete categorical characterization, in topological categories, of the Future Causal Boundary construction of Geroch, Kronheimer, and Penrose, showing that construction to have a universal property for future-completing chronological sets with spacelike boundaries. Rigidity results are given for any reasonable future completion of a spacetime, in terms of the GKP boundary: In the imputed topology, any such boundary must be homeomorphic to the GKP boundary (if all points have indecomposable pasts) or to a topological quotient of a closely related boundary (if boundaries are spacelike). A large class of warped-product-type spacetimes with spacelike boundaries is examined, calculating the GKP and other possible boundaries, and showing that the imputed topology gives expected results; included among these are the Schwarzschild singularity and those Robertson-Walker singularities which are spacelike. 
  We interpret the Holographic Conjecture in terms of quantum bits (qubits). N-qubit states are associated with surfaces that are punctured in N points by spin networks' edges labeled by the spin-1/2 representation of SU(2), which are in a superposed quantum state of spin "up" and spin "down". The formalism is applied in particular to de Sitter horizons, and leads to a picture of the early inflationary universe in terms of quantum computation. A discrete micro-causality emerges, where the time parameter is being defined by the discrete increase of entropy. Then, the model is analysed in the framework of the theory of presheaves (varying sets on a causal set) and we get a quantum history. A (bosonic) Fock space of the whole history is considered. The Fock space wavefunction, which resembles a Bose-Einstein condensate, undergoes decoherence at the end of inflation. This fact seems to be responsible for the rather low entropy of our universe. 
  The constrained instanton method is used to study quantum creation of a vacuum or charged topological black hole. At the $WKB$ level, the relative creation probability is the exponential of a quarter sum of the horizon areas associated with the seed instanton. 
  The constrained instanton method is used to study quantum creation of a $BTZ$ black hole. It is found that the relative creation probability is the exponential of the negative sum of the entropy associated with the outer and inner black hole horizons. The quantum creations of the 4- or higher dimensional versions of the $BTZ$ black hole are also studied. 
  For long black holes have been considered as endowed with a definite temperature. Yet when the Schwarzschild black hole is treated as a canonical ensemble three problems arise: incompatibility with the Hawking radiation, divergence of the partition function, and a formally negative mean-square fluctuation of the energy. We solve all three problems by considering the Schwarzschild black hole as a grand canonical ensemble, with the Hamiltonian (the ADM mass) and the horizon surface area, separately, as observable parameters. The horizon area simulates the number of particles in statistical mechanics since its spectrum is assumed to be discrete and equally spaced. We obtain a logarithmic correction to the Bekenstein-Hawking entropy and a Gaussian type distribution for the energy levels. 
  This note establishes the connection between Friedrich's conformal field equations and the conformally invariant formalism of local twistors. 
  A set of boundary conditions defining an undistorted, non-rotating isolated horizon are specified in general relativity. A space-time representing a black hole which is itself in equilibrium but whose exterior contains radiation admits such a horizon. However, the definition is applicable in a more general context, such as cosmological horizons. Physically motivated, (quasi-)local definitions of the mass and surface gravity of an isolated horizon are introduced and their properties analyzed. Although their definitions do not refer to infinity, these quantities assume their standard values in the static black hole solutions. Finally, using these definitions, the zeroth and first laws of black hole mechanics are established for isolated horizons. 
  A class of stationary rigidly rotating perfect fluid coupled with non-linear electromagnetic fields was investigated. An exact solution of the Einstein equations with sources for the Carter B(+) branch was found, for the equation of state $3p + \epsilon = constant$. We use a structural function for the Born-Infeld non-linear electrodynamics which is invariant under duality rotations and a metric possessing a four- parameter group of motions. The solution is of Petrov type D and the eigenvectors of the electromagnetic field are aligned to the Debever-Penrose vectors. 
  This paper has been withdrawn due to a crucial error in combining the two probability sectors represented in equations 13 and 14. Corrected, one can recover, in the limit of no background, the results of Allen et al. The general result that an analysis allowing for a background permits a substantial improvement of the limit still holds, though the presentation in the gr-qc/9907070 is flawed. 
  A model for 2D Quantum Gravity is constructed out of the Virasoro group. To this end the quantization of the abstract Virasoro group is revisited. For the critical values of the conformal anomaly c, some quantum operators (SL(2,R) generators) lose their dynamical content (they are no longer conjugated operators). The notion of space-time itself in 2D gravity then arises as associated with this kinematical SL(2,R) symmetry. An ensemble of different copies of AdS do co-exist in this model with different weights, depending on their curvature (which is proportional to \hbar^{2}) and they are connected by gravity operators. This model suggests that, in general, quantum diffemorphisms should not be imposed as constraints to the theory, except for the classical limit. 
  We extend the quasilocal formalism of Brown and York to include electromagnetic and dilaton fields and also allow for spatial boundaries that are not orthogonal to the foliation of the spacetime. The extension allows us to study the quasilocal energy measured by observers who are moving around in a spacetime. We show that the quasilocal energy transforms with respect to boosts by Lorentz-type transformation laws. The resulting formalism can be used to study spacetimes containing electric or magnetic charge but not both, a restriction inherent in the formalism. The gauge dependence of the quasilocal energy is discussed. We use the thin shell formalism of Israel to reinterpret the quasilocal energy from an operational point of view and examine the implications for the recently proposed AdS/CFT inspired intrinsic reference terms. The distribution of energy around Reissner-Nordstr\"{o}m and naked black holes is investigated as measured by both static and infalling observers. We see that this proposed distribution matches a Newtonian intuition in the appropriate limit. Finally the study of naked black holes reveals an alternate characterization of this class of spacetimes in terms of the quasilocal energies. 
  We study quantum effects in the presence of a spherical semi-transparent mirror or a system of two concentric mirrors which expand with a constant acceleration in a flat D-dimensional spacetime. Using the Euclidean approach, we obtain expressions for fluctuations and the renormalized value of stress-energy tensor for a scalar non-minimally coupled massless field. Explicit expressions are obtained for the energy fluxes at the null infinity generated by such mirrors in the physical spacetime and their properties are discussed. 
  We show that the only Tolman models which permit a Vaidya limit are those having a dust distribution that is hollow - such as the self-similar case. Thus the naked shell-focussing singularities found in Tolman models that are dense through the origin have no Vaidya equivalent. This also casts light on the nature of the Vaidya metric. We point out a hidden assumption in Lemos' demonstration that the Vaidya metric is a null limit of the Tolman metric, and in generalising his result, we find that a different transformation of coordinates is required. 
  The expression of the vector field generator of a Ricci Collineation for diagonal, spherically symmetric and non-degenerate Ricci tensors is obtained. The resulting expressions show that the time and radial first derivatives of the components of the Ricci tensor can be used to classify the collineation, leading to 64 families.   Some examples illustrate how to obtain the collineation vector. 
  In Einstein's equation we suggest a geometrical object substituting the tensor of energy of impulse and tension. The obtained equation, together with the equation for external field, makes up the complete problem of mathematical equations of gravitation, as well as those of inertia. Based on the example of centrally symmetrical field those problems and their consequences are discussed. 
  We extend the Brown and York notion of quasilocal energy to include coupled electromagnetic and dilaton fields and also allow for spatial boundaries that are not orthogonal to the foliation of the spacetime. We investigate how the quasilocal quantities measured by sets of observers transform with respect to boosts. As a natural application of this work we investigate the naked black holes of Horowitz and Ross calculating the quasilocal energies measured by static versus infalling observers. 
  A new approach to quantize the gravitational field is presented. It is based on the observation that the quantum character of matter becomes more significant as one gets closer to the big bang. As the metric loses its meaning, it makes sense to consider Schrodinger's three generic types of manifolds - unconnected differentiable, affinely connected, and metrically connected - as a temporal sequence following the big bang. Hence one should quantize the gravitational field on general differentiable manifolds or on affinely connected manifolds. The SL(2,C) gauge theory of gravitation is employed to explore this possibility. Within this framework, the quantization itself may well be canonical. 
  We study several approaches for constructing a minimal model of Universe evolution by matching different stages of scale factor laws. We discuss the continuity in the transitions among the stages and the time variables involved. We develop a way to modelize these transitions without loss of observational predictability. The key is the scale factor and its connection with minimal but well stablished observational information (transition times and expansion ratii). This construction is clearly useful in metric perturbations type computations, but it can be applied in whatever subject dealing with a cosmology involving different evolution stages. 
  We review cosmological relativity, a new special theory of relativity that was recently developed for cosmology, and discuss in detail some of its aspects. 
  We consider some aspects of conformal symmetry in a metric-scalar-torsion system. It is shown that, for some special choice of the action, torsion acts as a compensating field and the full theory is conformally equivalent to General Relativity on classical level. Due to the introduction of torsion, this equivalence can be provided for the positively-defined gravitational and scalar actions. One-loop divergences arising from the scalar loop are calculated and both the consequent anomaly and the anomaly-induced effective action are derived. 
  The invariant projections of the energy-momentum tensors of Lagrangian densities for tensor fields over differentiable manifolds with contravariant and covariant affine connections and metrics [$(\bar{L}_n,g)$-spaces] are found by the use of an non-null (non-isotropic) contravariant vector field and its corresponding projective metrics. The notions of rest mass density, momentum density, energy current density and stress tensor are introduced as generalizations of these notions from the relativistic continuum media mechanics. The energy-momentum tensors are represented by means of the introduced notions and the corresponding identities are found. The notion of covariant differential operator along a contravariant tensor field is introduced. On its basis, as a special case, the notion of contravariant metric differential operator is proposed. The properties of the operators are considered. By the use of these operators the notion of covariant divergency of a mixed tensor field is determined. The covariant divergency of tensor fields of second rank of the types 1 and 2 is found. Invariant representations of the covariant divergency of the energy-momentum tensor are obtained by means of the projective metrics of a contravariant non-isotropic (non-null) vector field and the corresponding rest mass density, momentum density, and energy flux density. An invariant representation of the first Noether identity is found as well as relations between the covariant divergencies of the different energy-momentum tensors and their structures determining covariant local conserved quantities. 
  The method of consistent potentials is used to explain how a minimally coupled (classical) scalar field can suppress Mixmaster oscillations in the approach to the singularity of generic cosmological spacetimes. 
  It is usually believed that a function whose Fourier spectrum is bounded can vary at most as fast as its highest frequency component. This is in fact not the case, as Aharonov, Berry and others drastically demonstrated with explicit counter examples, so-called superoscillations. It has been claimed that even the recording of an entire Beethoven symphony can occur as part of a signal with 1Hz bandwidth. Bandlimited functions also occur as ultraviolet regularized fields. Their superoscillations have been suggested, for example, to resolve the transplanckian frequencies problem of black hole radiation.   Here, we give an exact proof for generic superoscillations. Namely, we show that for every fixed bandwidth there exist functions which pass through any finite number of arbitrarily prespecified points. Further, we show that, in spite of the presence of superoscillations, the behavior of bandlimited functions can be characterized reliably, namely through an uncertainty relation. This also generalizes to time-varying bandwidths. In QFT, we identify the bandwidth as the in general spatially variable finite local density of degrees of freedom of ultraviolet regularized fields. 
  We study analytically, via the Newman-Penrose formalism, the late-time decay of linear electromagnetic and gravitational perturbations along the event horizon (EH) of black holes. We first analyze in detail the case of a Schwarzschild black hole. Using a straightforward local analysis near the EH, we show that, generically, the ``ingoing'' ($s>0$) component of the perturbing field dies off along the EH more rapidly than its ``outgoing'' ($s<0$) counterpart. Thus, while along $r=const>2M$ lines both components of the perturbation admit the well-known $t^{-2l-3}$ decay rate, one finds that along the EH the $s<0$ component dies off in advanced-time $v$ as $v^{-2l-3}$, whereas the $s>0$ component dies off as $v^{-2l-4}$. We then describe the extension of this analysis to a Kerr black hole. We conclude that for axially symmetric modes the situation is analogous to the Schwarzschild case. However, for non-axially symmetric modes both $s>0$ and $s<0$ fields decay at the same rate (unlike in the Schwarzschild case). 
  It is shown that the event horizon of 4D black hole or $ds^2 = 0$ surfaces of multidimensional wormhole-like solutions reduce the amount of information necessary for determining the whole spacetime and hence satisfy the Holography principle. This leads to the fact that by matching two metrics on a $ds^2 = 0$ surface (an event horizon for 4D black holes) we can match only the metric components but not their derivatives. For example, this allows us to obtain a composite wormhole inserting a 5D wormhole-like flux tube between two Reissner-Nordstr\"om black holes and matching them on the event horizon. Using the Holography principle, the entropy of a black hole from the algorithm theory is obtained. 
  The K\"ahler metric which has been constructed by present author is used in this paper to find an exact solution of Einstein equations with energy-momentum tensor of special type. The type of $T_{ij}$ admits in particular to use it as energy-momentum tensor of charged scalar field and in case of such system the field function is determined. 
  The constructive method of conformal blocks is developed for the construction of global solutions for two-dimensional metrics having one Killing vector. The method is proved to yeild a smooth universal covering space with a smooth pseudo-Riemannian metric. The Schwarzschild,  Reisner--Nordstrom solutions, extremal black hole, dilaton black hole, and constant curvature surfaces are considered as examples. 
  We present some remarkable properties of the symmetry group for gravitational plane waves. Our main observation is that metrics with plane wave symmetry satisfy every system of generally covariant vacuum field equations except the Einstein equations. The proof uses the homothety admitted by metrics with plane wave symmetry and the scaling behavior of generally covariant field equations. We also discuss a mini-superspace description of spacetimes with plane wave symmetry. 
  There have been many attempts to understand the statistical origin of black-hole entropy. Among them, entanglement entropy and the brick wall model are strong candidates. In this paper we show a relation between entanglement entropy and the brick wall model: the brick wall model seeks the maximal value of the entanglement entropy. In other words, the entanglement approach reduces to the brick wall model when we seek the maximal entanglement entropy . 
  Regular monopole and dyon solutions to the SU(2) Einstein Yang-Mills equations in asymptotically anti-de Sitter space are discussed. A class of monopole solutions are shown to be stable against spherically symmetric linear perturbations. 
  It's shown that the rotating dilaton-axion black hole solution can be obtained from GGHS static charged dilaton black hole solution via Newman-Janis method. 
  In this paper we have solved the Bohmian equations of quantum gravity, perturbatively. Solutions up to second order are derived explicitly, but in principle the method can be used in any order. Some consequences of the solution are disscused. 
  In a previous paper we derived a post-Newtonian approximation to cosmology which, in contrast to former Newtonian and post-Newtonian cosmological theories, has a well-posed initial value problem. In this paper, this new post-Newtonian theory is compared with the fully general relativistic theory, in the context of the k = 0 Friedmann Robertson Walker cosmologies. It is found that the post-Newtonian theory reproduces the results of its general relativistic counterpart, whilst the Newtonian theory does not. 
  Transports preserving the angle between two contravariant vector fields but changing their lengths proportional to their own lengths are introduced as `conformal' transports and investigated over spaces with one affine connection and metric. They are more general than the Fermi-Walker transports. In an analogous way as in the case of Fermi-Walker transports a conformal covariant differential operator and its conformal derivative are defined and considered over spaces with one affine connection and metric. Different special types of conformal transports are determined inducing also Fermi-Walker transports for orthogonal vector fields as special cases. Conditions under which the length of a non-null contravariant vector field could swing as a homogeneous harmonic oscillator are established. The results obtained regardless of any concrete field (gravitational) theory could have direct applications in such types of theories. PACS numbers: 04.90.+e; 04.50.+h; 12.10.Gq; 02.40.Vh 
  An astrophysically realistic model of wave dynamics in black-hole spacetimes must involve a non-spherical background geometry with angular momentum. We consider the evolution of gravitational (and electromagnetic) perturbations in rotating Kerr spacetimes. We show that a rotating Kerr black hole becomes `bald' slower than the corresponding spherically-symmetric Schwarzschild black hole. Moreover, our results turn over the traditional belief (which has been widely accepted during the last three decades) that the late-time tail of gravitational collapse is universal. In particular, we show that different fields have different decaying rates. Our results are also of importance both to the study of the no-hair conjecture and the mass-inflation scenario (stability of Cauchy horizons). 
  A discussion of polyhomogeneity (asymptotic expansions in terms of $1/r$ and $\ln r$) for zero-rest-mass fields and gravity and its relation with the Newman-Penrose (NP) constants is given. It is shown that for spin-$s$ zero-rest-mass fields propagating on Minkowski spacetime, the logarithmic terms in the asymptotic expansion appear naturally if the field does not obey the ``Peeling theorem''. The terms that give rise to the slower fall-off admit a natural interpretation in terms of advanced field. The connection between such fields and the NP constants is also discussed. The case when the background spacetime is curved and polyhomogeneous (in general) is considered. The free fields have to be polyhomogeneous, but the logarithmic terms due to the connection appear at higher powers of $1/r$. In the case of gravity, it is shown that it is possible to define a new auxiliary field, regular at null infinity, and containing some relevant information on the asymptotic behaviour of the spacetime. This auxiliary zero-rest-mass field ``evaluated at future infinity ($i^+$)'' yields the logarithmic NP constants. 
  We report on our numerical implementation of fully relativistic hydrodynamics coupled to Einstein's field equations in three spatial dimensions. We briefly review several steps in our code development, including our recasting of Einstein's equations and several tests which demonstrate its advantages for numerical integrations. We outline our implementation of relativistic hydrodynamics, and present numerical results for the evolution of both stable and unstable Oppenheimer-Volkov equilibrium stars, which represent a very promising first test of our code. 
  We discuss several explicitly causal hyperbolic formulations of Einstein's dynamical 3+1 equations in a coherent way, emphasizing throughout the fundamental role of the ``slicing function,'' $\alpha$---the quantity that relates the lapse $N$ to the determinant of the spatial metric $\bar{g}$ through $N = \bar{g}^{1/2} \alpha$. The slicing function allows us to demonstrate explicitly that every foliation of spacetime by spatial time-slices can be used in conjunction with the causal hyperbolic forms of the dynamical Einstein equations. Specifically, the slicing function plays an essential role (1) in a clearer form of the canonical action principle and Hamiltonian dynamics for gravity and leads to a recasting (2) of the Bianchi identities $\nabla_\beta G^\beta\mathstrut_\alpha \equiv 0$ as a well-posed system for the evolution of the gravitational constraints in vacuum, and also (3) of $\nabla_\beta T^\beta\mathstrut_\alpha \equiv 0$ as a well-posed system for evolution of the energy and momentum components of the stress tensor in the presence of matter, (4) in an explicit rendering of four hyperbolic formulations of Einstein's equations with only physical characteristics, and (5) in providing guidance to a new ``conformal thin sandwich'' form of the initial value constraints. 
  Separation of the Dirac equation in the spacetime around a Kerr black hole into radial and angular coordinates was done by Chandrasekhar in 1976. In the present paper, we solve the radial equations in a Schwarzschild geometry semi-analytically using Wentzel-Kramers-Brillouin approximation (in short WKB) method. Among other things, we present analytical expression of the instantaneous reflection and transmission coefficients and the radial wave functions of the Dirac particles. Complete physical parameter space was divided into two parts depending on the height of the potential well and energy of the incoming waves. We show the general solution for these two regions. We also solve the equations by a Quantum Mechanical approach, in which the potential is approximated by a series of steps and found that these two solutions agree. We compare solutions of different initial parameters and show how the properties of the scattered wave depend on these parameters. 
  The problem of the physical nature and the cosmological genesis of Lambda-term is discussed. This problem can't be solved in terms of the current quantum field theory which operates with Higgs and non-perturbative vacuum condensates and takes into account the changes of these condensates during relativistic phase transitions. The problem can't be completely solved also in terms of the conventional global quantum theory: Wheeler-DeWitt quantum geometrodynamics does not describe the evolution of the Universe in time (RPT in particular). We have investigated this problem in the context of energies density of different vacuum subsystems characteristic scales of which pervaid all energetic scale of the Universe. At first the phemenological solution of Lambda-term problem and then the hypothesis about the possible structure of a new global quantum theory are proposed. The main feature of this theory is the inreversible evolution of geometry and vacuum condensates in time in the regime of their selforganization. The transformation of the cosmological constant in dynamical variable is inevitably. 
  We address some of the issues that appear in the study of back reaction in Schwarzschild backgrounds. Our main object is the effective energy-momentum tensor (EEMT) of gravitational perturbations. It is commonly held that only asymptotically flat or radiation gauges can be employed for these purposes. We show that the traditional Regge-Wheeler gauge for the perturbations of the Schwarszchild metric can also be used for computing physical quantities both at the horizon and at infinity. In particular, we find that the physically relevant components of the EEMT of gravitational perturbations have the same asymptotic behaviour as the stress-energy tensor of a scalar field in the Schwarzschild background, even though some of the metric components themselves diverge. 
  We propose comparing cosmological solutions in terms of their total spatial volumes $V(\tau)$ as functions of proper time $\tau$, assuming synchronous gauge, and with this intention evaluate the variations of $V(\tau)$ about the Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW) solutions for dust. This can be done successfully in a simple manner without solving perturbation equations. In particular, we find that first variations vanish with respect to all directions which do not possess homogeneity and isotropy preserving components; in other words, every FLRW solution is a {\it critical point} for $V(\tau)$ in the properly restricted subspace of the space of solutions. This property may support a validity of the interpretation of the FLRW solutions as constituting an averaged model. We also briefly investigate the second variations of $V(\tau)$. 
  A diffuse matter filled Type V Universe is studied. The anisotropic behaviour, the distortion caused to the CMBR and the parameter region allowed by present cosmological bounds are examined. It is shown how the overall sky pattern of temperature anisotropies changes under a non-infinitesimal spatial coordinate transformation that preserves the Type V manifest homogeneity. 
  Solutions of the wave equation in a space-time containing a thin cosmic string are examined in the context of non-linear generalised functions. Existence and uniqueness of solutions to the wave equation in the Colombeau algebra G is established for a conical space-time and this solution is shown to be associated to a distributional solution. A concept of generalised hyperbolicity, based on test fields, can be defined for such singular space-times and it is shown that a conical space-time is G-hyperbolic. 
  We study the hamiltonian and constraints of spherically symmetric dilaton gravity model. We find the ADM mass of the solution representing the Schwarzchild black hole in thermal equilibrium with the Hawking radiation. 
  It is extraordinarely difficult to detect the extremely weak gravitomagnetic (GM) field of even as large a body as the earth. To detect the GM field, the gravitational analog of an ordinary magnetic field, in a modest terrestrial laboratory should be that much more difficult. Here we show, however, that for certain superconductor configuration and topologies, it should be possible to detect a measurable GM field in the terrestrial laboratory, by using the properties of superconductors imposed by quantum mechanical requirements. In particular, we show that the GM Flux should be quantized in a superconductor with non-vanishing genus, just like the ordinary magnetic flux. And this magnetically induced, quantized GM Flux, for sufficiently high quantum number and favorable geometries, should be distinguishable from the effects produced by an ordinary magnetic field. 
  Sound wave propagation in a relativistic perfect fluid with a non-homogeneous isentropic flow is studied in terms of acoustic geometry. The sound wave equation turns out to be equivalent to the equation of motion for a massless scalar field propagating in a curved space-time geometry. The geometry is described by the acoustic metric tensor that depends locally on the equation of state and the four-velocity of the fluid. For a relativistic supersonic flow in curved space-time the ergosphere and acoustic horizon may be defined in a way analogous the non-relativistic case. A general-relativistic expression for the acoustic analog of surface gravity has been found. 
  We analyse the properties of a fluid generating a spinning cosmic string spacetime with flat limiting cases corresponding to a constant angular momentum in the infinite past and static configuration in the infinite future. The spontaneous loss of angular momentum of a spinning cosmic string due to particle emission is discussed. The rate of particle production between the spinning and non-spinning cosmic string spacetimes is calculated. 
  The cosmic censorship hypothesis introduced by Penrose thirty years ago is still one of the most important open questions in {\it classical} general relativity. The main goal of this paper is to put forward the idea that cosmic censorship is intrinsically a {\it quantum} phenomena. We construct a gedanken experiment which seems to violate the cosmic censorship principle within the purely {\it classical} framework of general relativity. We prove, however, that {\it quantum} physics restores the validity of the conjecture. It is therefore suggested that cosmic censorship might be enforced by a quantum theory of gravity. 
  We study analytically, via the Newman-Penrose formalism, the late time decay of scalar, electromagnetic, and gravitational perturbations outside a realistic rotating (Kerr) black hole. We find a power-law decay at timelike infinity, as well as at null infinity and along the event horizon (EH). For generic initial data we derive the power-law indices for all radiating modes of the various fields. We also give an exact analytic expression (accurate to leading order in 1/t) for the r-dependence of the late time tail at any r. Some of our main conclusions are: (i) For generic initial data, the late time behavior of the fields is dominated by the mode l=|s| (with s being the spin parameter), which dies off at fixed $r$ as $t^{-2|s|-3}$ --- as in the Schwarzschild background. (ii) However, other modes admit decay rates slower than in the Schwarzschild case. (iii) For s>0 fields, non-axially symmetric modes dominate the late time behavior along the EH. These modes oscillate along the null generators of the EH. 
  Boundary conditions defining a non-rotating isolated horizon are given in Einstein-Maxwell theory. A spacetime representing a black hole which itself is in equilibrium but whose exterior contains radiation admits such a horizon. Inspired by Hamiltonian mechanics, a (quasi-)local definition of isolated horizon mass is formulated. Although its definition does not refer to infinity, this mass takes the standard value in a Reissner-Nordstrom solution. Furthermore, under certain technical assumptions, the mass of an isolated horizon is shown to equal the future limit of the Bondi energy. 
  We calculate how much a first-quantized string is excited after crossing the inner horizon of charged Vaidya solutions, as a simple model of generic black holes. To quantize a string suitably, we first show that the metric is approximated by a {\it plane-wave} metric near the inner horizon when the surface gravity of the horizon $\kappa_I$ is small enough. Next, it is analytically shown that the string crossing the inner horizon is excited infinitely in an asymptotically flat spacetime, while it is finite in an asymptotically de Sitter spacetime and the string can pass across the inner horizon when $\kappa_I<2\kappa:= 2 {min}\{\kappa_B,\kappa_C \}$, where $\kappa_B$~($\kappa_C$) is the surface gravity of the black hole~(cosmological) event horizon. This implies that the strong cosmic censorship holds in an asymptotically flat spacetime, while it is violated in an asymptotically de Sitter spacetime from the point of view of string theory. 
  Exact solutions for nonexpanding impulsive waves in a background with nonzero cosmological constant are constructed using a `cut and paste' method. These solutions are presented using a unified approach which covers the cases of de Sitter, anti-de Sitter and Minkowski backgrounds. The metrics are presented in continuous and distributional forms, both of which are conformal to the corresponding metrics for impulsive pp-waves, and for which the limit as $\Lambda\to 0$ can be made explicitly. 
  The Einstein gravitational field of a material point at rest is derived anew - by a suitable limit process - from the field of a sphere of a homogeneous and incompressible fluid. This result supports clearly the thesis according to which the physically interesting singularities must correspond to the presence of matter in loco. 
  We present numerical hydrodynamical evolutions of rapidly rotating relativistic stars, using an axisymmetric, nonlinear relativistic hydrodynamics code. We use four different high-resolution shock-capturing (HRSC) finite-difference schemes (based on approximate Riemann solvers) and compare their accuracy in preserving uniformly rotating stationary initial configurations in long-term evolutions. Among these four schemes, we find that the third-order PPM scheme is superior in maintaining the initial rotation law in long-term evolutions, especially near the surface of the star. It is further shown that HRSC schemes are suitable for the evolution of perturbed neutron stars and for the accurate identification (via Fourier transforms) of normal modes of oscillation. This is demonstrated for radial and quadrupolar pulsations in the nonrotating limit, where we find good agreement with frequencies obtained with a linear perturbation code. The code can be used for studying small-amplitude or nonlinear pulsations of differentially rotating neutron stars, while our present results serve as testbed computations for three-dimensional general-relativistic evolution codes. 
  We begin this work calculating Halliwell's propagator in the case of a Mixmaster universe with small anisotropy. Afterwards in the context of the Decoherence Model we introduce in our system terms that comprise the self-measurement of the universe of this model by higher multipoles of matter. Analyzing self-measurement with the Restricted Path Integral Formalism we obtain Halliwell's modified propagator and find that a gauge invariant physical time emerges as consequence of this process. The conditions leading to Wheeler-DeWitt dynamics are also obtained. The comparison of our results with those of the isotropic case will enable us to conclude that the number of conditions to be satisfied in order to have Halliwell's regime is in the anisotropic situation bigger than in an isotropic universe. We obtain also in terms of the parameters of the measurement process an expression for the threshold in time beyond which the scale factors of this model are meaningless. 
  We present a new technique for the numerical simulation of axisymmetric systems. This technique avoids the coordinate singularities which often arise when cylindrical or polar-spherical coordinate finite difference grids are used, particularly in simulating tensor partial differential equations like those of 3+1 numerical relativity. For a system axisymmetric about the z axis, the basic idea is to use a 3-dimensional Cartesian (x,y,z) coordinate grid which covers (say) the y=0 plane, but is only one finite-difference-molecule--width thick in the y direction. The field variables in the central y=0 grid plane can be updated using normal (x,y,z)--coordinate finite differencing, while those in the y \neq 0 grid planes can be computed from those in the central plane by using the axisymmetry assumption and interpolation. We demonstrate the effectiveness of the approach on a set of fully nonlinear test computations in 3+1 numerical general relativity, involving both black holes and collapsing gravitational waves. 
  In recent work, we showed that non-perturbative vacuum effects of a very low mass particle could induce, at a redshift of order 1, a transition from a matter-dominated to an accelerating universe. In that work, we used the simplification of a sudden transition out of the matter-dominated stage and were able to fit the Type Ia supernovae (SNe-Ia) data points with a spatially-open universe. In the present work, we find a more accurate, smooth {\it spatially-flat} analytic solution to the quantum-corrected Einstein equations. This solution gives a good fit to the SNe-Ia data with a particle mass parameter $m_h$ in the range $6.40 \times 10^{-33}$ eV to $7.25 \times 10^{-33}$ eV. It follows that the ratio of total matter density (including dark matter) to critical density, $\O_0$, is in the range 0.58 to 0.15, and the age $t_0$ of the universe is in the range $8.10 h^{-1}$ Gyr to $12.2 h^{-1}$ Gyr, where $h$ is the present value of the Hubble constant, measured as a fraction of the value 100 km/(s Mpc). This spatially-flat model agrees with estimates of the position of the first acoustic peak in the small angular scale fluctuations of the cosmic background radiation, and with light-element abundances of standard big-bang nucleosynthesis. Our model has only a single free parameter, $m_h$, and does not require that we live at a special time in the evolution of the universe. 
  Majumdar--Papapetrou multi--black-hole solutions of the Einstein--Maxwell equations are considered in four and higher dimensions. The Euclidean action with boundary conditions appropriate to the canonical ensemble is shown to lead to zero entropy. 
  The true and eccentric anomaly parametrizations of the Kepler motion are generalized to quasiperiodic orbits, by considering perturbations of the radial part of the kinetic energy in a form of a series of negative powers of the orbital radius. A toolchest of methods for averaging observables as functions of the energy $E$ and angular momentum $L$ is developed. A broad range of systems governed by the generic Brumberg force and recent applications in the theory of gravitational radiation involve integrals of these functions over a period of motion. These integrals are evaluated by using the residue theorem. In the course of this work two important questions emerge: (1) When does the true and eccentric anomaly parameter exist? (2) Under what circumstances and why are the poles in the origin? The purpose of this paper is to find the answer to these queries. 
  In numerically constructing a spacetime that has an approximate timelike Killing vector, it is useful to choose spacetime coordinates adapted to the symmetry, so that the metric and matter variables vary only slowly with time in these coordinates. In particular, this is a crucial issue in numerically calculating a binary black hole inspiral. An approximate homothetic vector plays a role in critical gravitational collapse. We summarize old and new suggestions for finding such coordinates from a general point of view. We then test some of these in various toy models with spherical symmetry, including critical fluid collapse and critical scalar field collapse. 
  This contribution to the proceedings of the 1999 Canadian Conference on General Relativity and Relativistic Astrophysics is a brief exposition of earlier work, with Sumati Surya (hep-th/9805121) Amanda Peet (hep-th/9903213), addressing certain results in higher dimensional supergravity that are related to black hole no-hair theorems. Its purpose is to describe, in language appropriate for an audience of relativists, how these results can be related to the Maldacena conjecture (aka, the AdS/CFT correspondence). The end product may be taken as a new kind of quantitative evidence in support of the Maldacena conjecture. 
  We compute Teitelboim's causal propagator in the context of canonical loop quantum gravity. For the Lorentzian signature, we find that the resultant power series can be expressed as a sum over branched, colored two-surfaces with an intrinsic causal structure. This leads us to define a general structure which we call a ``causal spin foam''. We also demonstrate that the causal evolution models for spin networks fall in the general class of causal spin foams. 
  A physicaly reasonable interpretation is provided for the perfect fluid, sphericaly symmetric, conformally flat ``Stephani Universes''. The free parameters of this class of exact solutions are determined so that the ideal gas relation $p=n k T$ is identicaly fulfiled, while the full equation of state of a classical monatomic ideal gas and a matter-radiation mixture holds up to a good approximation in a near dust, matter dominated regime. Only the models having spacelike slices with positive curvature admit a regular evolution domain that avoids an unphysical singularity. In the matter dominated regime these models are dynamicaly and observationaly indistinguishable from ``standard'' FLRW cosmology with a dust source. 
  We show from one-loop quantum gravity and statistical thermodynamics that the thermodynamics of quantum foam in flat space-time and Schwarzschild space-time is exactly the same as that of Hawking-Unruh radiation in thermal equilibrium. This means we show unambiguously that Hawking-Unruh thermal radiation should contain thermal gravitons or the contribution of quantum space-time foam. As a by-product, we give also the quantum gravity correction in one-loop approximation to the classical black hole thermodynamics. 
  We revisit a model of the two-scalar system proposed previously for understanding a small but nonzero cosmological constant. The model provides solutions of the scalar-fields energy $\rho_s$ which behaves truly constant for a limited time interval rather than in the way of tracker- or scaling-type variations. This causes a mini-inflation, as indicated by recent observations. As another novel feature, $\rho_s$ and the ordinary matter density $\rho_m$ fall off always side by side, but interlacing, also like (time)$^{-2}$ as an overall behavior in conformity with the scenario of a decaying cosmological constant. A mini-inflation occurs whenever $\rho_s$ overtakes $\rho_m$, which may happen more than once, shedding a new light on the coincidence problem. We present a new example of the solution, and offer an intuitive interpretation of the mechanism of the nonlinear dynamics. We also discuss a chaos-like nature of the solution. 
  Several examples and models based on noncommutative differential calculi on commutative algebras indicate that a metric should be regarded as an element of the left-linear tensor product of the space of 1-forms with itself. We show how the metric compatibility condition with a linear connection generalizes to this framework. 
  We argue that ``effective'' superluminal travel, potentially caused by the tipping over of light cones in Einstein gravity, is always associated with violations of the null energy condition (NEC). This is most easily seen by working perturbatively around Minkowski spacetime, where we use linearized Einstein gravity to show that the NEC forces the light cones to contract (narrow). Given the NEC, the Shapiro time delay in any weak gravitational field is always a delay relative to the Minkowski background, and never an advance. Furthermore, any object travelling within the lightcones of the weak gravitational field is similarly delayed with respect to the minimum traversal time possible in the background Minkowski geometry. 
  We study a new spacetime which is shown to be the general geometrical background for Thermal Field Theories at equilibrium. The different formalisms of Thermal Field Theory are unified in a simple way in this spacetime. The set of time-paths used in the Path Ordered Method is interpreted in geometrical terms. 
  We formulate and analyze the Hamiltonian dynamics of a pair of massive spinless point particles in (2+1)-dimensional Einstein gravity by anchoring the system to a conical infinity, isometric to the infinity generated by a single massive but possibly spinning particle. The reduced phase space \Gamma_{red} has dimension four and topology R^3 x S^1. \Gamma_{red} is analogous to the phase space of a Newtonian two-body system in the centre-of-mass frame, and we find on \Gamma_{red} a canonical chart that makes this analogue explicit and reduces to the Newtonian chart in the appropriate limit. Prospects for quantization are commented on. 
  We present self-similar cosmological solutions for a barotropic fluid plus scalar field with Brans-Dicke-type coupling to the spacetime curvature and an arbitrary power-law potential energy. We identify all the fixed points in the autonomous phase-plane, including a scaling solution where the fluid density scales with the scalar field's kinetic and potential energy. This is related by a conformal transformation to a scaling solution for a scalar field with exponential potential minimally coupled to the spacetime curvature, but non-minimally coupled to the barotropic fluid. Radiation is automatically decoupled from the scalar field, but energy transfer between the field and non-relativistic dark matter can lead to a change to an accelerated expansion at late times in the Einstein frame. The scalar field density can mimic a cosmological constant even for steep potentials in the strong coupling limit. 
  We present our first successful numerical results of 3D general relativistic simulations in which the Einstein equation as well as the hydrodynamic equations are fully solved. This paper is especially devoted to simulations of test problems such as spherical dust collapse, stability test of perturbed spherical stars, and preservation of (approximate) equilibrium states of rapidly rotating neutron star and/or corotating binary neutron stars. These test simulations confirm that simulations of coalescing binary neutron stars are feasible in a numerical relativity code. It is illustrated that using our numerical code, simulations of these problems, in particular those of corotating binary neutron stars, can be performed stably and fairly accurately for a couple of dynamical timescales. These numerical results indicate that our formulation for solving the Einstein field equation and hydrodynamic equations are robust and make it possible to perform a realistic simulation of coalescing binary neutron stars for a long time from the innermost circular orbit up to formation of a black hole or neutron star. 
  The WDW equation of arbitrary Hartle-Hawking factor ordering for several minisuperspace universe models, such as the pure gravity FRW and Taub ones, is mapped onto the dynamics of corresponding classical oscillators. The latter ones are studied by the classical Ermakov invariant method, which is a natural aproach in this context. For the more realistic case of a minimally coupled massive scalar field, one can study, within the same type of approach, the corresponding squeezing features as a possible means of describing cosmological evolution. Finally, we comment on the analogy with the accelerator physics 
  The massless conformally coupled scalar field is characterized by the so-called "new improved stress-energy tensor", which is capable of classically violating the null energy condition. When coupled to Einstein gravity we find a three-parameter class of exact solutions. These exact solutions include the Schwarzschild geometry, assorted naked singularities, and a large class of traversable wormholes. 
  A specific choice of gauge is shown to imply a decoupling between the tensor and scalar components of Gravitational Radiation in the context of Brans-Dicke type theories of gravitation. The comparison of the predictions of these theories with those of General Relativity is thereby made straightforward. 
  I argue that black hole entropy counts only those states of a black hole that can influence the outside, and attempt (with only partial success) to defend this claim against various objections, all but one coming from string theory. Implications for the nature of the Bekenstein bound are discussed, and in particular the case for a holographic principle is challenged. Finally, a generalization of black hole thermodynamics to "partial event horizons" in general spacetimes without black holes is proposed. 
  A model is proposed, according to which the metric tensor field in the standard gravitational Lagrangian is decomposed into a projection (generally - with a non-zero covariant derivative) tensor field, orthogonal to an arbitrary 4-vector field and a tensor part along the same vector field. A theorem has been used, according to which the variation and the partial derivative, when applied to a tensor field, commute with each other if and only if the tensor field and its variation have zero covariant derivatives, provided also the connection variation is zero. Since the projection field obviously does not fulfill the above requirements of the ''commutation'' theorem, the exact expression for the (non-zero) commutator of the variation and the partial derivative, applied to the projection tensor field, can be found from a set of the three defining equations. The above method will be used to construct a modified variational approach in relativistic hydrodynamics, based on variation of the vector field and the projection field, the last one thus accounting for the influence of the reference system of matter (characterized by the 4-vector) on the gravitational field. 
  A simple formula is given for generating Chern characters by repeated exterior differentiation for n-dimensional differentiable manifolds having a general linear connection. 
  A sequence of generalizations of Cartan's conservation of torsion theorem is given for n-dimensional differentiable manifolds having a general linear connection. 
  A simple derivation is given of the Einstein-Maxwell field equations from the 2nd ordinary exterior differential of a precursor to the soldering form for n-dimensional differentiable manifolds having a general linear connection and in 5-dimensional general relativity in particular. 
  We give a method in which a quantum of mass equal to twice the Planck mass arises naturally. Then using Bose-Einstein statistics we derive an expression for the black hole entropy which physically tends to the Bekenstein-Hawking formula. 
  The renormalization of a scalar field theory with a quartic self-coupling (a $\lambda \phi^4$ theory) via adiabatic regularization in a general Robertson-Walker spacetime is discussed. The adiabatic counterterms are presented in a way that is most conducive to numerical computations. A variation of the adiabatic regularization method is presented which leads to analytic approximations for the energy-momentum tensor of the field and the quantum contribution to the effective mass of the mean field. Conservation of the energy-momentum tensor for the field is discussed and it is shown that the part of the energy-momentum tensor which depends only on the mean field is not conserved but the full renormalized energy-momentum tensor is conserved as expected and required by the semiclassical Einstein's equation. It is also shown that if the analytic approximations are used then the resulting approximate energy-momentum tensor is conserved. This allows a self-consistent backreaction calculation to be performed using the analytic approximations. The usefulness of the approximations is discussed. 
  Using mainly analytical arguments, we derive the exact relation $\eta_{max}=\sqrt{3/8\pi}$ for the maximal vacuum value of the Higgs field for static gravitational global monopoles. For this value, the global monopole bifurcates with the de Sitter solution obtained for vanishing Higgs field. In addition, we analyze the stability properties of the solutions. 
  We consider scattering and capture of circular cosmic strings by a Schwarzschild black hole. Although being a priori a very simple axially symmetric two-body problem, it shows all the features of chaotic scattering. In particular, it contains a fractal set of unstable periodic solutions; a so-called strange repellor. We study the different types of trajectories and obtain the fractal dimension of the basin-boundary separating the space of initial conditions according to the different asymptotic outcomes. We also consider the fractal dimension as a function of energy, and discuss the transition from order to chaos. 
  We develop a general formalism to treat, in general relativity, the linear oscillations of a two-fluid star about static (non-rotating) configurations. Such a formalism is intended for neutron stars, whose matter content can be described, as a first approximation, by a two-fluid model: one fluid is the neutron superfluid, which is believed to exist in the core and inner crust of mature neutron stars; the other fluid is a conglomerate of all other constituents (crust nuclei, protons, electrons, etc...). We obtain a system of equations which govern the perturbations both of the metric and of the matter variables, whatever the equation of state for the two fluids. As a first application, we consider the simplified case of two non-interacting fluids, each with a polytropic equation of state. We compute numerically the quasi-normal modes (i.e. oscillations with purely outgoing gravitational radiation) of the corresponding system. When the adiabatic indices of the two fluids are different, we observe a splitting for each frequency of the analogous single fluid spectrum. The analysis also substantiates the claim that w-modes are largely due to spacetime oscillations. 
  This two-part contribution to the Proceedings of the Eighth Canadian Conference on General Relativity and Relativistic Astrophysics is devoted to the evolution of a massless scalar field in two black-hole spacetimes which are not asymptotically flat.   In Part I (authored by Eric Poisson) we consider the evolution of a scalar field propagating in Schwarzschild-de Sitter spacetime. The spacetime possesses a cosmological horizon in addition to the usual event horizon. The presence of this new horizon affects the late-time evolution of the scalar field.   In part II (authored by William G. Laarakkers) we consider the evolution of a scalar field propagating in Schwarzschild-Einstein-de Sitter spacetime. The spacetime has two distinct regions: an inner black-hole region and an outer cosmological region. Early on in the evolution, the field behaves as if it were in pure Schwarzschild spacetime. Later, the field learns of the existence of the cosmological region and alters its behaviour. 
  Disks of collisionless particles are important models for certain galaxies and accretion disks in astrophysics. We present here a solution to the stationary axisymmetric Einstein equations which represents an infinitesimally thin dust disk consisting of two streams of particles circulating with constant angular velocity in opposite directions. These streams have the same density distribution but their relative density may vary continuously. In the limit of only one component of dust, we get the solution for the rigidly rotating dust disk previously given by Neugebauer and Meinel, in the limit of identical densities, the static disk of Morgan and Morgan is obtained. We discuss the Newtonian and the ultrarelativistic limit, the occurrence of ergospheres, and the regularity of the solution. 
  Coalescing compact binaries have been pointed out as the most promising source of gravitational waves for kilometer-size interferometers such as LIGO. Gravitational wave signals are extracted from the noise in the detectors by matched filtering. This technique performs really well if an a priori theoretical knowledge of the signal is available. The information known about the possible sources is used to construct a model of the expected waveforms (templates). A common assumption made when constructing templates for coalescing compact binaries is that the companions move in a quasi-circular orbit. Some scenarios, however, predict the existence of eccentric binaries. We investigate the loss in signal-to-noise ratio induced by non-optimal filtering of eccentric signals. 
  We perform numerical simulations of the critical gravitational collapse of a spherically symmetric scalar field in 6 dimensions. The critical solution has discrete self-similarity. We find the critical exponent \gamma and the self-similarity period \Delta. 
  These notes present an introduction to branes in ten and eleven dimensional supergravity and string/M-theory which is geared to an audience of traditional relativists, especially graduate students and others with little background in supergravity. They are designed as a tutorial and not as a thorough review of the subject; as a result, many topics of current interest are not addressed. However, a guide to further reading is included. The presentation begins with eleven dimensional supergravity, stressing its relation to 3+1 Einstein-Maxwell theory. The notion of Kaluza-Klein compactification is then introduced, and is used to relate the eleven dimensional discussion to supergravity in 9+1 dimensions and to string theory. The focus is on type IIA supergravity, but the type IIB theory is also addressed, as is the T-duality symmetry that relates them. Branes in both 10+1 and 9+1 dimensions are included. Finally, although the details are not discussed, a few comments are provided on the relation between supergravity and string perturbation theory and on black hole entropy. The goal is to provide traditional relativists with a kernel of knowledge from which to grow their understanding of branes and strings. 
  This paper studies near-critical evolution of the spherically symmetric scalar field configurations close to the continuously self-similar solution. Using analytic perturbative methods, it is shown that a generic growing perturbation departs from the critical Roberts solution in a universal way. We argue that in the course of its evolution, initial continuous self-similarity of the background is broken into discrete self-similarity with echoing period $\Delta = \sqrt{2}\pi = 4.44$, reproducing the symmetries of the critical Choptuik solution. 
  A simple derivation of the bound on entropy is given and the holographic principle is discussed. We estimate the number of quantum states inside space region on the base of uncertainty relation. The result is compared with the Bekenstein formula for entropy bound, which was initially derived from the generalized second law of thermodynamics for black holes. The holographic principle states that the entropy inside a region is bounded by the area of the boundary of that region. This principle can be called the kinematical holographic principle. We argue that it can be derived from the dynamical holographic principle which states that the dynamics of a system in a region should be described by a system which lives on the boundary of the region. This last principle can be valid in general relativity because the ADM hamiltonian reduces to the surface term. 
  The purpose of this paper is to evaluate the `Lorentzian pedagogy' defended by J.S. Bell in his essay ``How to teach special relativity'', and to explore its consistency with Einstein's thinking from 1905 to 1952. Some remarks are also made in this context on Weyl's philosophy of relativity and his 1918 gauge theory. Finally, it is argued that the Lorentzian pedagogy - which stresses the important connection between kinematics and dynamics - clarifies the role of rods and clocks in general relativity. 
  The Bach equation, i.e., the vacuum field equation following from the Lagrangian L=C_{ijkl}C^{ijkl}, will be completely solved for the case that the metric is conformally related to the cartesian product of two 2-spaces; this covers the spherically and the plane symmetric space-times as special subcases. Contrary to other approaches, we make a covariant 2+2-decomposition of the field equation, and so we are able to apply results from 2-dimensional gravity. Finally, some cosmological solutions will be presented and discussed. 
  Two classes of metrics obtained from non-Riemannian gravitational collapse are presented.The first is the Taub planar symmetric exact solutions of Einstein-Cartan field equations of gravity describing torsion walls which are obtained from gravitational collapse of time dependent perturbation of Riemannian Taub symmetric solutions of General Relativity.The second is a modification of the Vilenkin Riemannian planar wall which is obtained from a non-Riemannian planar distribution of spinning matter. 
  The symplectic geometry of a broad class of generally covariant models is studied. The class is restricted so that the gauge group of the models coincides with the Bergmann-Komar group and the analysis can focus on the general covariance. A geometrical definition of gauge fixing at the constraint manifold is given; it is equivalent to a definition of a background (spacetime) manifold for each topological sector of a model. Every gauge fixing defines a decomposition of the constraint manifold into the physical phase space and the space of embeddings of the Cauchy manifold into the background manifold (Kuchar decomposition). Extensions of every gauge fixing and the associated Kuchar decomposition to a neighbourhood of the constraint manifold are shown to exist. 
  Recently, Byland and Scialom studied the evolution of the Bianchi I, the Bianchi III and the Kantowski-Sachs universe on the basis of dynamical systems methods (Phys. Rev. D57, 6065 (1998), gr-qc/9802043). In particular, they have pointed out a problem to determine the stability properties of one of the degenerate critical points of the corresponding dynamical system. Here we give a solution, showing that this point is unstable both to the past and to the future. We also discuss the asymptotic behavior of the trajectories in the vicinity of another critical point. 
  The dynamical effects on the scale factors due to the scalar $\phi$-field at the early stages of a supposedly anisotropic Universe expansion in the scalar-tensor cosmology of Jordan-Brans and Dicke is studied. This universe shows an {\sl isotropic} evolution and, depending on the value of the theorie's coupling parameter $\omega$, it can begin from a singularity if $\omega>0$ and after expanding shrink to another one; or, if $\omega <0$ and $-3/2< \omega\leq -4/3$, it can evolve from a flat spatially-infinite state to a non extended singularity; or, if $ -4/3 < \omega < 0$, evolve from an extended singularity to a non singular state and, at last, proceed towards a singularity. 
  Spherically symmetric Black Holes of the Vaidya type are examined in an asymptotically de Sitter, higher dimensional spacetime. The various horizons are located. The structure and dynamics of such horizons are studied. 
  We discuss the production of multi-photons squeezed states induced by the time variation of the (Abelian) gauge coupling constant in a string cosmological context. Within a fully quantum mechanical approach we solve the time evolution of the mean number of produced photons in terms of the squeezing parameters and in terms of the gauge coupling. We compute the first (amplitude interference) and second order (intensity interference) correlation functions of the magnetic part of the photon background. The photons produced thanks to the variation of the dilaton coupling are strongly bunched for the realistic case where the growth of the dilaton coupling is required to explain the presence of large scale magnetic fields and, possibly of a Faraday rotation of the Cosmic Microwave Background. 
  A fundamental problem with attempting to quantize general relativity is its perturbative non-renormalizability. However, this fact does not rule out the possibility that non-perturbative effects can be computed, at least in some approximation. We outline a quantum field theory calculation, based on general relativity as the classical theory, which implies a phase transition in quantum gravity. The order parameters are composite fields derived from spacetime metric functions. These are massless below a critical energy scale and become massive above it. There is a corresponding breaking of classical symmetry. 
  A brief overview is presented of the basis of the electromagnetic zero-point field in quantum physics and its representation in stochastic electrodynamics. Two approaches have led to the proposal that the inertia of matter may be explained as an electromagnetic reaction force. The first is based on the modeling of quarks and electrons as Planck oscillators and the method of Einstein and Hopf to treat the interaction of the zero-point field with such oscillators. The second approach is based on analysis of the Poynting vector of the zero-point field in accelerated reference frames. It is possible to derive both Newton's equation of motion, F=ma, and its relativistic co-variant form from Maxwell's equations as applied to the zero-point field of the quantum vacuum. This appears to account, at least in part, for the inertia of matter. 
  We show how to reformulate Variable Speed of Light Theories (VSLT) in a covariant fashion as Variable Light-Cone Theories (VLCT) by introducing two vierbein bundles each associated with a distinct metric. The basic gravitational action relates to one bundle while matter propagates relative to the other in a conventional way. The variability of the speed of light is represented by the variability of the matter light-cone relative to the gravitational light-cone. The two bundles are related locally by an element M, of SL(4,R). The dynamics of the field M is that of a SL(4,R)-sigma model gauged with respect to local (orthochronous) Lorentz transformations on each of the bundles. Only the ``massless'' version of the model with a single new coupling, F, that has the same dimensions as Newton's constant $G_N$, is considered in this paper. When F vanishes the theory reduces to standard General Relativity.   We verify that the modified Bianchi identities of the model are consistent with the standard conservation law for the matter energy-momentum tensor in its own background metric.   The implications of the model for some simple applications are examined, the Newtonian limit, the flat FRW universe and the spherically symmetric static solution. 
  We propose a new numerical method to compute quasi-equilibrium sequences of general relativistic irrotational binary neutron star systems. It is a good approximation to assume that (1) the binary star system is irrotational, i.e. the vorticity of the flow field inside component stars vanishes everywhere (irrotational flow), and (2) the binary star system is in quasi-equilibrium, for an inspiraling binary neutron star system just before the coalescence as a result of gravitational wave emission. We can introduce the velocity potential for such an irrotational flow field, which satisfies an elliptic partial differential equation (PDE) with a Neumann type boundary condition at the stellar surface. For a treatment of general relativistic gravity, we use the Wilson--Mathews formulation, which assumes conformal flatness for spatial components of metric. In this formulation, the basic equations are expressed by a system of elliptic PDEs. We have developed a method to solve these PDEs with appropriate boundary conditions. The method is based on the established prescription for computing equilibrium states of rapidly rotating axisymmetric neutron stars or Newtonian binary systems. We have checked the reliability of our new code by comparing our results with those of other computations available. We have also performed several convergence tests. By using this code, we have obtained quasi-equilibrium sequences of irrotational binary star systems with strong gravity as models for final states of real evolution of binary neutron star systems just before coalescence. Analysis of our quasi-equilibrium sequences of binary star systems shows that the systems may not suffer from dynamical instability of the orbital motion and that the maximum density does not increase as the binary separation decreases. 
  An example of a teleparallel texture is given by an appropriate choice of torsion components in the tetrad frame.In the light cone limit the metric is not globally Euclidean and the spherical angles depend on torsion similarly to what happens in cosmic string space-times.In this limit torsion produces a force which decays with $r^{-3}$. 
  A generalized definition of a frame of reference in spaces with affine connections and metrics is proposed based on the set of the following differential-geometric objects:  (a) a non-null (non-isotropic) vector field,  (b) the orthogonal to the vector field sub space,  (c) an affine connection and the related to it covariant differential operator determining a transport along the given non-null vector filed.  On the grounds of this definition other definitions related to the notions of accelerated, inertial, proper accelerated and proper inertial frames of reference are introduced and applied to some mathematical models for the space-time. The auto-parallel equation is obtained as an Euler-Lagrange's equation. Einstein's theory of gravitation appears as a theory for determination of a special frame of reference (with the gravitational force as inertial force) by means of the metrics and the characteristics of a material distribution.   PACS numbers: 0490, 0450, 1210G, 0240V 
  To understand the observational properties of cosmological models, in particular, the temperature of the cosmic microwave background radiation, it is necessary to study their null geodesics. Dynamical systems theory, in conjunction with the orthonormal frame approach, has proved to be an invaluable tool for analyzing spatially homogeneous cosmologies. It is thus natural to use such techniques to study the geodesics of these models. We therefore augment the Einstein field equations with the geodesic equations, all written in dimensionless form, obtaining an extended system of first-order ordinary differential equations that simultaneously describes the evolution of the gravitational field and the behavior of the associated geodesics. It is shown that the extended system is a powerful tool for investigating the effect of spacetime anisotropies on the temperature of the cosmic microwave background radiation, and that it can also be used for studying geodesic chaos. 
  I study a semiclassical approach to warm inflation scenario introduced in previous works. In this work, I define the fluctuations for the matter field by means of a new coarse - grained field with a suppression factor G. This field describes the matter field fluctuations on the now observable scale of the universe. The power spectrum for the fluctuations of the matter field is analyzed in both, de Sitter and power - law expansions for the universe. The constraint for the spectral index gives a constraint for the mass of the matter field in the de Sitter expansion and a constraint for the friction parameter in the power - law expansion for the universe. 
  The aim of this paper is to discuss a kinematical algebraic structure of a theory of gravity, that would be unitary, renormalizable and coupled in the same manner to both spinorial and tensorial matter fields. An analysis of the common features as well as differences of the Yang-Mills theories and gauge theories of gravity is carried out. In particular, we consider the following issues: (i) Representations of the relevant global symmetry on states and on fields, (ii) Relations between the relevant global and local symmetries, (iii) Representations of the local symmetries on states and fields, (iv) Dimensional analysis of the gauge algebra generators and the number of counter terms, and (v) Coupling to the spinorial matter fields. We conclude, that various difficulties on the gravity side can be overcome by considering the below outlined Hypergravity gauge theory, that to some extend parallels the string/membrane theories. This theory is based on an infinite Lie algebraic structure of generators that transforms as "states" of the infinite-dimensional irreducible representation of the $\bar{SL}(4,R)$ subgroup of the Group of General Coordinate Transformations of R^4. The metric-affine and Poincar\'e gauge theories of gravity are obtained through a spontaneous symmetry breaking mechanism, with the metric field as nonlinear symmetry realizer. 
  A generalization of the Dirac field equation in three-dimensional Minkowski space-time to the case of the $\bar{SL}(3,R)$ $\subset$ $\bar{SA}(3,R)$ symmetry is considered. Constraints that ensure a correct physical interpretation of the corresponding particle states are presented. Dirac-like equations based on both multiplicity-free and generic infinite-component $\bar{SL}(3,R)$ representations are outlined. 
  We give a preliminary report on one of the tests we have performed of a full non-axisymmetric general relativistic code. The test considered here concerns the numerical evolution of vacuum non-axisymmetric gravitational waves and their comparison at low amplitudes with theoretical waveforms obtained from linearised theory. 
  The asymptotic behaviour of two classes of scalar field cosmological models are studied using the theory of dynamical systems: general relativistic Bianchi models containing matter and a scalar field with an exponential potential and a class of spatially homogeneous string cosmological models. The purpose of this thesis is to examine some of the outstanding problems which currently exist in cosmology, particularly regarding isotropization and inflation. It is shown that the matter scaling solutions are unstable to curvature perturbations. It is then shown that the Bianchi class B exponential potential models can alleviate the isotropy problem; an open set of models within this class do isotropize to the future. It is also shown that the presence of an interaction term in the subclass of isotropic models can lead to inflationary models with late-time oscillatory behaviour in which the matter is not driven to zero. Next, within the class of the string cosmologies studied, it is shown that there is a subclass which do not inflate at late times in the post-big bang regime. Furthermore, all string models studied typically do not have a late--time flatness problem. Indeed, it is shown that curvature typically plays an important r\^{o}le only at intermediate times in most models. It is also shown that the presence of a positive cosmological constant in the models studied can lead to interesting physical behaviour, such as multi-bouncing universes. A mathematical equivalence between general relativistic scalar field theories and scalar-tensor theories and string theories has been extensively exploited and thus the results obtained from the string analysis compliment the results obtained from the Bianchi class B exponential potential analysis. 
  Completing the results obtained in a previous paper, we prove the symmetry of Hadamard/Seeley-deWitt off-diagonal coefficients in smooth $D$-dimensional Lorentzian manifolds. To this end, it is shown that, in any Lorentzian manifold, a sort of ``local Wick rotation'' of the metric can be performed provided the metric is a locally analytic function of the coordinates and the coordinates are ``physical''. No time-like Killing field is necessary. Such a local Wick rotation analytically continues the Lorentzian metric in a neighborhood of any point, or, more generally, in a neighborhood of a space-like (Cauchy) hypersurface, into a Riemannian metric. The continuation locally preserves geodesically convex neighborhoods. In order to make rigorous the procedure, the concept of a complex pseudo-Riemannian (not Hermitian or K\"ahlerian) manifold is introduced and some features are analyzed. Using these tools, the symmetry of Hadamard/Seeley-deWitt off-diagonal coefficients is proven in Lorentzian analytical manifolds by analytical continuation of the (symmetric) Riemannian heat-kernel coefficients. This continuation is performed in geodesically convex neighborhoods in common with both the metrics. Then, the symmetry is generalized to $C^\infty$ non analytic Lorentzian manifolds by approximating Lorentzian $C^{\infty}$ metrics by analytic metrics in common geodesically convex neighborhoods. The symmetry requirement plays a central r\^{o}le in the point-splitting renormalization procedure of the one-loop stress-energy tensor in curved spacetimes for Hadamard quantum states. 
  We show that nonperturbative vacuum effects can produce a vacuum-driven transition from a matter-dominated universe to one in which the effective equation of state is that of radiation plus cosmological constant. The actual material content of the universe after the transition remains that of non-relativistic matter. This metamorphosis of the equation of state can be traced to nonperturbative vacuum effects that cause the scalar curvature to remain nearly constant at a well-defined value after the transition, and is responsible for the observed acceleration of the recent expansion of the universe. 
  An example is given of a plane topological torsion defect representing a cosmic wall double wall in teleparallel gravity.The parallel planar walls undergone a repulsive gravitational force due to Cartan torsion.This is the first example of a non-Riemannian double cosmic wall.It is shown that the walls oscillate with a speed that depends on torsion and on the surface density of the wall.Cartan torsion acts also as a damping force reducing the speed of oscillation when it is stronger. 
  Quintessence is often invoked to explain the universe acceleration suggested by the type Ia supernovae observations. The aim of this letter is to study the validity of using a constant equation of state for quintessence models. We shall show that this hypothesis strongly constraint the form of the scalar potential. 
  We prove the existence of a class of plane symmetric perfect-fluid cosmologies with a (-1/3, 2/3, 2/3) Kasner-like singularity. These solutions of the Einstein equations depend on two smooth functions of one space coordinate. They are constructed by solving a symmetric hyperbolic system of Fuchsian equations. 
  Quantum inequalities (QI's) provide lower bounds on the averaged energy density of a quantum field. We show how the QI's for massless scalar fields in even dimensional Minkowski space may be reformulated in terms of the positivity of a certain self-adjoint operator - a generalised Schroedinger operator with the energy density as the potential - and hence as an eigenvalue problem. We use this idea to verify that the energy density produced by a moving mirror in two dimensions is compatible with the QI's for a large class of mirror trajectories. In addition, we apply this viewpoint to the `quantum interest conjecture' of Ford and Roman, which asserts that the positive part of an energy density always overcompensates for any negative components. For various simple models in two and four dimensions we obtain the best possible bounds on the `quantum interest rate' and on the maximum delay between a negative pulse and a compensating positive pulse. Perhaps surprisingly, we find that - in four dimensions - it is impossible for a positive delta-function pulse of any magnitude to compensate for a negative delta-function pulse, no matter how close together they occur. 
  The spherically symmetric solutions in Weyl gravity interacting with U(1) or SU(2) gauge fields are examined. It is shown that these solutions are conformally equivalent to an infinite flux tube with constant (color) electric and magnetic fields. This allows us to say that Weyl gravity has in some sense a classical confinement mechanism. We discuss a possible connection with flux tubes in quantum chromodynamics. 
  We tudy flat Friedmann-Robertson-Walker cosmology in Brans-Dicke-type theories of gravitation with minimal coupling between the scalar field and the matter fields in the Einstein frame (general relativity with an extra scalar field) for arbitrary values of the Brans-Dicke parameter $\omega>-{3/2}$. It is shown that the cosmological singularity occuring in the Einstein frame formulation of this theory is removed in the Jordan frame in the range $-{3/2}<\omega<\leq-{4/3}$. This result is interpreted in the ligth of a viewpoint (first presented in reference gr-qc/9905071) asserting that both Jordan frame and Einstein frame formulations of general relativity are physically equivalent. The implications of the obtained result for string theory are outlined. 
  A regular vacuum solution in 5D gravity on the principal bundle with the U(1) structural group is proposed as a 4D wormhole. This solution has two null hypersurfaces where an interchange of the sign of some 5D metric components happens. For a 4D observer living on the base of this principal bundle this is a wormhole with two asymptotically flat Lorentzian (Euclidean) spacetimes connected by a Euclidean (Lorentzian) throat. The 4D Lorentzian observer sees these two null hypersurfaces as electric charges. 
  We examine the reduced phase space of the Bianchi VII_0 cosmological model, including the moduli sector. We show that the dynamics of the relevant sector of local degrees of freedom is given by a Painleve III equation. We then obtain a zero-curvature representation of this Painleve III equation by applying the Belinskii-Zakharov method to the Bianchi VII_0 model. 
  In spacetime physics, we frequently need to consider a set of all spaces (`universes') as a whole. In particular, the concept of `closeness' between spaces is essential. However, there has been no established mathematical theory so far which deals with a space of spaces in a suitable manner for spacetime physics.   Based on the scheme of the spectral representation of geometry, we construct a space of all compact Riemannian manifolds equipped with the spectral measure of closeness. We show that this space of all spaces can be regarded as a metric space. We also show other desirable properties of this space, such as the partition of unity, locally-compactness and the second countability. These facts show that this space of all spaces can be a basic arena for spacetime physics. 
  We study the stability properties of the standard ADM formulation of the 3+1 evolution equations of general relativity through linear perturbations of flat spacetime. We focus attention on modes with zero speed of propagation and conjecture that they are responsible for instabilities encountered in numerical evolutions of the ADM formulation. These zero speed modes are of two kinds: pure gauge modes and constraint violating modes. We show how the decoupling of the gauge by a conformal rescaling can eliminate the problem with the gauge modes. The zero speed constraint violating modes can be dealt with by using the momentum constraints to give them a finite speed of propagation. This analysis sheds some light on the question of why some recent reformulations of the 3+1 evolution equations have better stability properties than the standard ADM formulation. 
  A new strategy in deriving the Lense-Thirring effect, in the weak field and slow motion approximation of general relativity, on the orbital elements of a test body in the field of different central rotating sources exhibiting axial symmetry is presented. The approach adopted in the present work, in the case of a perfectly spherical source, leads for all the Keplerian orbital elements of the freely falling particle to the well known Lense-Thirring equations. The case of a central nonspherical body is also worked out. 
  The general relativistic Lense-Thirring effect can be measured by inspecting a suitable combination of the orbital residuals of the nodes of LAGEOS and LAGEOS II and the perigee of LAGEOS II. The solid and ocean Earth tides affect the recovery of the parameter by means of which the gravitomagnetic force is accounted for in the combined residuals. Thus an extensive analysis of the perturbations induced on these orbital elements by the solid and ocean Earth tides is carried out. It involves the l=2 terms for the solid tides and the l=2,3,4 terms for the ocean tides. The perigee of LAGEOS II turns out to be very sensitive to the l=3 part of the ocean tidal spectrum, contrary to the nodes of LAGEOS and LAGEOS II. The uncertainty in the solid tidal perturbations, mainly due to the Love number k2, ranges from 0.4% to 1.5%, while the ocean tides are uncertain at 5%-15% level. The obtained results are used in order to check in a preliminary way which tidal constituents the Lense-Thirring shift is sensitive to. In particular it is tested if the semisecular 18.6-year zonal tide really does not affect the combined residuals. It turns out that, if modeled at the level of accuracy worked out in the paper, the l=2,4 m=0 and also, to a lesser extent, the l=3 m=0 tidal perturbations cancel out. 
  We develop an iterative approach to gravitational lensing theory based on approximate solutions of the null geodesic equations. The approach can be employed in any space-time which is ``close'' to a space-time in which the null geodesic equations can be completely integrated, such as Minkowski space-time, Robertson-Walker cosmologies, or Schwarzschild-Kerr geometries. To illustrate the method, we construct the iterative gravitational lens equations and time of arrival equation for a single Schwarzschild lens. This example motivates a discussion of the relationship between the iterative approach, the standard thin lens formulation, and an exact formulation of gravitational lensing. 
  A brief review of the stability of rotating relativistic stars is followed by a more detailed discussion of recent work on an instability of r-modes, modes of rotating stars that have axial parity in the slow-rotation limit. These modes may dominate the spin-down of neutron stars that are rapidly rotating at birth, and the gravitational waves they emit may be detectable. 
  Gravitational-wave experiments with interferometers and with resonant masses can search for stochastic backgrounds of gravitational waves of cosmological origin. We review both experimental and theoretical aspects of the search for these backgrounds. We give a pedagogical derivation of the various relations that characterize the response of a detector to a stochastic background. We discuss the sensitivities of the large interferometers under constructions (LIGO, VIRGO, GEO600, TAMA300, AIGO) or planned (Avdanced LIGO, LISA) and of the presently operating resonant bars, and we give the sensitivities for various two-detectors correlations. We examine the existing limits on the energy density in gravitational waves from nucleosynthesis, COBE and pulsars, and their effects on theoretical predictions. We discuss general theoretical principles for order-of-magnitude estimates of cosmological production mechanisms, and then we turn to specific theoretical predictions from inflation, string cosmology, phase transitions, cosmic strings and other mechanisms. We finally compare with the stochastic backgrounds of astrophysical origin. 
  We study canonical quantization of a closed Euclidean universe with a cosmological constant and a massless scalar field. The closed Euclidean universe with an ordinary matter state can be matched at a finite radius only with the closed Lorentzian universe with the Wick-rotated exotic state. The exotic state provides the Lorentzian universe with a potential barrier extending from the cosmological singularity to the classical turning point and corresponding to the Euclidean geometry through the Wick-rotation, and avoid the singularity problem at the matching boundary. We find analytically the approximate wave functions for quantum creation of the Universe from the {\it nothingness}. We prescribe the Hartle-Hawking's no-boundary wave function, Linde's wave function and Vilenkin's tunneling wave function. In particular, we find the wave function for the Euclidean geometry, whose semiclassical solution is regular at the matching boundary with the Lorentzian geometry but singular at the cosmological singularity. 
  An exact solution of the current-free Einstein-Maxwell equations with the cosmological constant is presented. It is of Petrov type II, and its double principal null vector is geodesic, shear-free, expanding, and twisting. The solution contains five constants. Its electromagnetic field is non-null and aligned. The solution admits only one Killing vector and includes, as special cases, several known solutions. 
  Relying upon the equivalence between a gauge theory for the translation group and general relativity, a teleparallel version of the original Kaluza-Klein theory is developed. In this model, only the internal space (fiber) turns out to be five dimensional, spacetime being kept always four dimensional. A five-dimensional translational gauge theory is obtained which unifies, in the sense of Kaluza-Klein theories, gravitational and electromagnetic interactions. 
  The ``close limit,'' a method based on perturbations of Schwarzschild spacetime, has proved to be a very useful tool for finding approximate solutions to models of black hole collisions. Calculations carried out with second order perturbation theory have been shown to give the limits of applicability of the method without the need for comparison with numerical relativity results. Those second order calculations have been carried out in a fixed coordinate gauge, a method that entails conceptual and computational difficulties. Here we demonstrate a gauge invariant approach to such calculations. For a specific set of models (requiring head on collisions and quadrupole dominance of both the first and second order perturbations), we give a self contained gauge invariant formalism. Specifically, we give (i) wave equations and sources for first and second order gauge invariant wave functions; (ii) the prescription for finding Cauchy data for those equations from initial values of the first and second fundamental forms on an initial hypersurface; (iii) the formula for computing the gravitational wave power from the evolved first and second order wave functions. 
  The essence of the gravitomagnetic clock effect is properly defined showing that its origin is in the topology of world lines with closed space projections. It is shown that, in weak field approximation and for a spherically symmetric central body, the loss of synchrony between two clocks counter-rotating along a circular geodesic is proportional to the angular momentum of the source of the gravitational field. Numerical estimates are presented for objects within the solar system. The less unfavorable situation is found around Jupiter. 
  We extend our analysis for scalar fields in a Robertson-Walker metric to the electromagnetic field and Dirac fields by the method of invariants. The issue of the relation between conformal properties and particle production is re-examined and it is verified that the electromagnetic and massless spinor actions are conformal invariant, while the massless conformally coupled scalar field is not. For the scalar field case it is pointed out that the violation of conformal simmetry due to surface terms, although ininfluential for the equation of motion, does lead to effects in the quantized theory. 
  We present a fully quantum version of the holographic principle in terms of quantum systems, subsystems, and their interactions. We use the concept of environment induced decoherence to prove this principle. We discuss the conditions under which the standard (semi-classical) holographic principle is obtained from this quantum mechanical version. 
  In 1919 A. Einstein suspected first that gravitational fields could play an essential role in the structure of elementary particles. In 1937, P.A.M. Dirac found a miraculous link between the properties of the visible Universe and elementary particles. Both conjectures stayed alive through the following decades but still no final theory could be derived to this issues. The herein suggested fractal model of the Universe gives a consistent explanation to Dirac's Large Numbers Hypothesis and combines the conjectures of Einstein and Dirac. 

  We study the cosmology of the Brans-Dicke(BD) theory coupled to perfect fluid type matter. In our previous works, the case where matter is coming from the Ramond-Ramond sector of the string theory was studied. Here, we study the case where matter is coming from the NS-NS sector. Exact solutions are found and the cosmology is classified according to the values of $\gamma$, the parameter of the equation of state and $\omega$, Brans-Dicke parameter. We find taht, in string frame, there are solutions without singularity for some ranges of $\gamma$ and $\omega$. In Einstein frame, however, all solutions are singular. 
  Recent results demonstrating the chaotic behavior of geodesics in non-homogeneous vacuum pp-wave solutions are generalized. Here we concentrate on motion in non-homogeneous sandwich pp-waves and show that chaos smears as the duration of these gravitational waves is reduced. As the number of radial bounces of any geodesic decreases, the outcome channels to infinity become fuzzy, and thus the fractal structure of the initial conditions characterizing chaos is cut at lower and lower levels. In the limit of impulsive waves, the motion is fully non-chaotic. This is proved by presenting the geodesics in a simple explicit form which permits a physical interpretation, and demonstrates the focusing effect. It is shown that a circle of test particles is deformed by the impulse into a family of closed hypotrochoidal curves in the transversal plane. These are deformed in the longitudinal direction in such a way that a specific closed caustic surface is formed. 
  We apply the holographic principle to the Brans-Dicke cosmology. We analyze the holographic bound in both the Jordan and Einstein frames. The holographic bound is satisfied for both the k=0 and k=-1 universe, but it is violated for the k=1 matter dominated universe. 
  According to the general theory of relativity, kinetic energy contributes to gravitational mass. Surprisingly, the observational evidence for this prediction does not seem to be discussed in the literature. I reanalyze existing experimental data to test the equivalence principle for the kinetic energy of atomic electrons, and show that fairly strong limits on possible violations can be obtained. I discuss the relationship of this result to the occasional claim that ``light falls with twice the acceleration of ordinary matter.'' 
  A new formula for the conserved charges in 3+1 gravity for spacetimes with local AdS asymptotic geometry is proposed. It is shown that requiring the action to have an extremum for this class of asymptotia sets the boundary term that must be added to the Lagrangian as the Euler density with a fixed weight factor. The resulting action gives rise to the mass and angular momentum as Noether charges associated to the asymptotic Killing vectors without requiring specification of a reference background in order to have a convergent expression. A consequence of this definition is that any negative constant curvature spacetime has vanishing Noether charges. These results remain valid in the limit of vanishing cosmological constant. 
  A class of static Lorentzian wormholes with arbitrarily wide throats is presented in which the source of the WEC violations required by the Einstein equations is the vacuum stress-energy of the neutrino, electromagnetic, or massless scalar field. 
  We discuss a recent provocative suggestion by Amelino-Camelia and others that classical spacetime may break down into ``quantum foam'' on distance scales many orders of magnitude larger than the Planck length, leading to effects which could be detected using large gravitational wave interferometers. This suggestion is based on a quantum uncertainty limit obtained by Wigner using a quantum clock in a gedanken timing experiment. Wigner's limit, however, is based on two unrealistic and unneccessary assumptions: that the clock is free to move, and that it does not interact with the environment. Removing either of these assumptions makes the uncertainty limit invalid, and removes the basis for Amelino-Camelia's suggestion. 
  An effective action is obtained for the area and mass aspect of a thin shell of radiating self-gravitating matter. On following a mini-superspace approach, the geometry of the embedding space-time is not dynamical but fixed to be either Minkowski or Schwarzschild inside the shell and Vaidya in the external space filled with radiation. The Euler-Lagrange equations of motion are discussed and shown to entail the expected invariance of the effective Lagrangian under time-reparametrization. They are equivalent to the usual junction equations and suggest a macroscopic quasi-static thermodynamic description. 
  An exact 2-dimensional conical Riemannian defect solution of 3-dimensional Euclidean Einstein equations of stresses and defects representing a shear-free Heisenberg ferromagnet is given.The system is equivalent to the Einstein equations in vacuum.Geodesics of magnetic monopoles around the ferromagnet are also investigated. 
  A review is given of recent results about the computation of irrotational Darwin-Riemann configurations in general relativity. Such configurations are expected to represent fairly well the late stages of inspiralling binary neutron stars. 
  This is the writeup of the talk I gave at the Yukawa International Symposium at Kyoto, Japan on June 29, 1999. The talk summarizes the present status of the close limit approximation for colliding black holes. 
  Research Briefs:   Does the GSL imply an entropy bound?, by Warren G. Anderson   A lightweight review of middleweight black holes, by Ben Bromley   The physics of isolated horizons, by Daniel Sudarsky   LIGO project update, by Stan Whitcomb  Meeting reports:   Worskhop on initial value for binary black holes, by Carlos Lousto   ITP Conference on strong gravitational fields, by Don Marolf   Yukawa International Seminar, by John Friedman   Minnowbrook symposium on the structure of space-time, by Kamesh Wali   Black holes II and CCGRRA 8 by Jack Gegenberg and Gabor Kunstatter   Hartlefest & 15th Pacific Coast Gravity Meeting by Simon Ross   Second Capra workshop by Patrick Brady and Alan Wiseman   Third Edoardo Amaldi Conference, by Gabriela Gonzalez   Strings 99, by Thomas Thiemann 
  One of the fundamental problems of the theoretical physics is the search of the axioms, which ought to be the basis for the one-valued construction of Lagrangians of the relativistic fields. The creation of the gauge fields theory was the great success in the solution of this problem. The gauge formalism allowed to derive the total Lagrangians of the interacting fields from the postulated Lagrangians of the noninteracting (free) fields. We offer to do quite the reverse in consequence of what it is necessary to seek from the out set the construction principles of the total Lagrangians. By the theory construction we shall differ the wave-functions being the solutions of the differential equations (``theoretical'' functions) from the wave-functions which is constructed on the base of the experimental data possibly received by a scattering of particles (``empiric'' functions). The ``empiric'' functions are necessary only for the definition (it is possibly only approximately) of the transition operators which will affect at the ``theoretical'' functions.  This operators will be approximated the differential operators so, that the generalized variance of the differentiable  ``theoretical'' fields will be the minimal one. 
  The detection and the research of the neutrinos background of  Universe are the attractive problems. This problems do not seem the unpromising one in the case of the high neutrinos density of Universe. It is offered to use the low energy neutrinos background of Universe for the explanation of the gravitational phenomena with the quantum position attracting the Casimir's effect for this. As a result it can consider the normal matter (not neutrinos) in the capasity of the  Brownian particles by the help of which it can make the attempt to estimate the statistic characteristics of the Universe neutrinos background. 
  A discussion of asymptotic weak and strong Poincare' charges in metric gravity is given to identify the proper Hamiltonian boundary conditions. The asymptotic part of the lapse and shift functions is put equal to their analogues on Minkowski hyperplanes. By adding Dirac's ten extra variables at spatial infinity, metric gravity is extended to incorporate Dirac's ten extra first class constraints (the new ten momenta equal to the weak Poincare' charges) and this allows its deparametrization to parametrized Minkowski theories restricted to spacelike hyperplanes. The absence of supertranslations implies: i) boundary conditions identifying the family of Christodoulou-Klainermann spacetimes; ii) the restriction of foliations to those (Wigner-Sen-Witten hypersurfaces) corresponding to Wigner's hyperplanes of Minkowski rest-frame instant form. These results are generalized to tetrad gravity in the new formulation given in gr-qc/9807072, gr-qc/9807073. The evolution in the parameter labelling the leaves of the foliation is generated by the weak ADM energy. Some comments on the quantization in a completely fixed special 3-orthogonal gauge are made. 
  The iterated Crank-Nicholson method has become a popular algorithm in numerical relativity. We show that one should carry out exactly two iterations and no more. While the limit of an infinite number of iterations is the standard Crank-Nicholson method, it can in fact be worse to do more than two iterations, and it never helps. We explain how this paradoxical result arises. 
  We clarify the causal structure of an inflating magnetic monopole. The spacetime diagram shows explicitly that this model is free from ``graceful exit'' problem, while the monopole itself undergoes ``eternal inflation''. We also discuss general nature of inflationary spacetimes. 
  Gravity theories are constructed on finite groups G. A self-consistent review of the differential calculi on finite G is given, with some new developments. The example of a bicovariant differential calculus on the nonabelian finite group S_3 is treated in detail, and used to build a gravity-like field theory on S_3. 
  Despite the recent excitement over the r-mode instability of rotating stars, these modes are not yet well-understood for stellar models appropriate to neutron stars - perfect fluid models in which both the equilibrium and perturbed configurations obey the same one-parameter equation of state.   In spherical stars of this kind, r-modes and g-modes form a degenerate zero-frequency subspace. Rotation splits the degeneracy to zeroth order in the star's angular velocity $\Omega$, and the resulting modes are generically hybrids, whose limit as $\Omega\to 0$ is a stationary current with axial and polar parts. Lindblom and Ipser have recently found these hybrid rotational modes in an analytic study of the Maclaurin spheroids. This dissertation studies them in Newtonian stars (both uniform density models and $n=1$ polytropes) and reports a first computation of the hybrid rotational modes of relativistic stars. 
  A new solution for the spacetime outside the core of a U(1) static global string has been presented which is nonsingular. This is the first example of a nonsingular spacetime around a static global string.}} 
  A quantum equivalence principle is formulated by means of a gravitational phase operator which is an element of the Poincare group. This is applied to the spinning cosmic string which suggests that it may, but not necessarily, contain gravitational torsion. A new exact solution of the Einstein- Cartan-Sciama-Kibble equations for the gravitational field with torsion is obtained everywhere for a cosmic string with uniform energy density, spin density and flux. The quantization condition for fluxoid due to London and DeWitt is generalized to include the spin flux. A novel effect due to the quantized gravitational field of the cosmic string on the wave function of a particle outside the string is used to show that spacetime points are not meaningful in quantum gravity. 
  Quantization of gravity is discussed in the context of field quantization based on an analogue of canonical formalism (the De Donder-Weyl canonical theory) which does not require the space+time decomposition. Using Horava's (1991) De Donder-Weyl formulation of General Relativity we put forward a covariant generalization of the Schr\"odinger equation for the wave function of space-time and metric variables and a supplementary ``bootstrap condition'' which self-consistently incorporates the classical background space-time geometry as a quantum average and closes the system of equations. Some open questions for further research are outlined. 
  In this work we analyze the generalization of Duffin-Kemmer-Petiau equation to the case of Riemannian space-times and show that the usual results for Klein-Gordon and Proca equations in Riemannian space-times can be fully recovered when one selects, respectively, the spin 0 and 1 sectors of Duffin-Kemmer-Petiau theory. 
  The question whether gravitational waves are quantised or not can in principle be answered by the help of correlation measurements. If the gravitational waves are quantised and they are generated by the change of the background metrics then they can be in squeezed state.  In a sqeezed state there is a correlation between the phase of the wave and the quantum fluctuations. It is recommended to analyse the data to be obtained by the gravitational detectors from the point of view of such correlations. An explicit formula is derived for the squeezing parameter of the quantised gravitational waves. The head on collision of two identical black holes is analysed as a possible source of squeezed gravitational waves. 
  A local observer can measure only the values of fields at the point of his own position. By exploring the coordinate transformation between two Fermi frames, it is shown that two observers, having the same instantaneous position and velocity, will observe the same values of covariant fields at their common instantaneous position, even if they have different instantaneous accelerations. In particular, this implies that in classical physics the notion of radiation is observer independent, contrary to the conclusion of some existing papers. A "freely" falling charge in curved spacetime does not move along a geodesic and therefore radiates. The essential feature of the Unruh effect is the fact that it is based on a noninstantaneous measurement, which may also be viewed as a source of effective noncovariance of measured quantities. The particle concept in Minkowski spacetime is clarified. It is argued that the particle concept in general spacetime does not depend on the observer and that there exists a preferred coordinate frame with respect to which the particle number should be defined. 
  We show that the recent claim that the 2+1 dimensional Ashtekar formulation for General Relativity has a finite number of physical degrees of freedom is not correct. 
  Standard pedagogy treats topics in general relativity (GR) in terms of tensor formulations in curved space-time. Although mathematically straightforward, the curved space-time approach can seem abstruse to beginning students due to the degree of mathematical sophistication required. As a heuristic tool to provide insight into what is meant by a curved metric, we present a polarizable-vacuum (PV) representation of GR derived from a model by Dicke and related to the "TH-epsilon-mu" formalism used in comparative studies of gravitational theories. 
  The purpose of the paper is to develop further a projection variational approach in relativistic hydrodynamics. The approach, previously proposed in [gr-qc/9908032], is based on the variation of the vector field and the projection tensor (instead of the given metric tensor) and their first partial derivatives. The previously proved property of non-commutativity of the variation and the partial derivative in respect to the projection tensor has been used to find all the variations. Subsequently, motivated by some analogy with the well-known (3+1) ADM projection formalism, an assumption has been made about a zero-covariant derivative of the projection tensor in respect to the projection connection. The combination of the equations for the variations of the projective tensor with covariant and contravariant indices has lead to the derivation of an important and concisely written relation: the derivative of the vector field length is equal to the ''twice'' projected along the vector field initial Christoffell connection. The result is of interest due to the following reasons: 1. It is a more general one and contains in itself a well-known formulae in affine differential geometry for the so called equiaffine connections (admitting covariantly conserved tensor fields), for which the trace of the connection is equal to the gradient of the logarithm of the vector field length. 2. The additional term is the projected (with the projection tensor) initially given connection and accounts for the influence of the reference system on the change of the vector field's length, measured in this system. 3. The formulae has been obtained within the proposed formalism of non-commuting variation and partial derivative. 
  The representation theory of non-centrally extended Lie algebras of Noether symmetries, including spacetime diffeomorphisms and reparametrizations of the observer's trajectory, has recently been developped. It naturally solves some long-standing problems in quantum gravity, e.g. the role of diffeomorphisms and the causal structure, but some new questions also arise. 
  We report a new class of rotating charged solutions in 2+1 dimensions. These solutions are obtained for Einstein-Maxwell gravity coupled to a dilaton field with selfdual electromagnetic fields. The mass and the angular momentum of these solutions computed at spatial infinity are finite. The class of solutions considered here have naked singularities and are asymptotically flat. 
  A finite universe naturally supports chaotic classical motion. An ordered fractal emerges from the chaotic dynamics which we characterize in full for a compact 2-dimensional octagon. In the classical to quantum transition, the underlying fractal can persist in the form of scars, ridges of enhanced amplitude in the semiclassical wave function. Although the scarring is weak on the octagon, we suggest possible subtle implications of fractals and scars in a finite universe. 
  We describe a new statistical pattern in the chaotic dynamics of closed inflationary cosmologies, associated with the partition of the Hamiltonian rotational motion energy and hyperbolic motion energy pieces, in a linear neighborhood of the saddle-center present in the phase space of the models. The hyperbolic energy of orbits visiting a neighborhood of the saddle-center has a random distribution with respect to the ensemble of initial conditions, but the associated histograms define a statistical distribution law of the form $p(x) = C x^{-\gamma}$, for almost the whole range of hyperbolic energies considered. We present numerical evidence that $\gamma$ determines the dimension of the fractal basin boundaries in the ensemble of initial conditions. This distribution is universal in the sense that it does not depend on the parameters of the models and is scale invariant. We discuss possible physical consequences of this universality for the physics of inflation.tribution law of the form $p(x) = C x^{-\gamma}$, for almost the whole range of hyperbolic energies considered. We present numerical evidence that $\gamma$ determines the dimension of the fractal basin boundaries in the ensemble of initial conditions. This distribution is universal in the sense that it does not depend on the parameters of the models and is scale invariant. We discuss possible physical consequences of this universality for the physics of inflation. 
  Methods of dynamical systems analysis are used to show rigorously that the presence of a magnetic field orthogonal to the two commuting Killing vector fields in any spatially homogeneous Bianchi type VI_0 vacuum solution to Einstein's equation changes the evolution toward the singularity from convergent to oscillatory. In particular, it is shown that the alpha-limit set (for time direction that puts the singularity in the past) of any of these magnetic solutions contains at least two sequential Kasner points of the BKL sequence and the orbit of the transition solution between them. One of the Kasner points in the alpha-limit set is non-flat, which leads to the result that each of these magnetic solutions has a curvature singularity. 
  We have studied a cosmological model with a cosmological term of the form $\Lambda=3\alpha\fr{\dot R^2}{R^2}+\bt\fr{\ddot R}{R}+\fr{3\gamma}{R^2} \alpha, \   \bt \gamma$ are constants. The scale factor (R) is found to vary linearly with time for both radiation and matter dominated epochs.   The cosmological constant is found to decrease as $t^{-2}$ and the rate of particle creation is smaller than the Steady State value.   The model gives $\Omega^\Lambda=\fr{1}{3}$ and $\Omega^m=\fr{2}{3}$ in the present era,   $\Omega^\Lambda=\Omega^m=\fr{1}{2}$ in the radiation era.   The present age of the universe $(\rm t_p$) is found to be $\rm t_p=H_p^{-1}$ , where $\rm H_p$ is the Hubble constant.   The model is free from the main problems of the Standard Model. Since the scale factor $\rm R\propto t$ during the entire evolution of the universe the ratio of the cosmological constant at the Planck and present time is $\rm\fr{\Lambda_{Pl}}{\Lambda_p}=10^{120}$.   This decay law justifies why, today, the cosmological constant is exceedingly small. 
  The low-energy (bosonic "heterotic") string theory is interpreted as a universal limit of the Kaluza-Klein reduction when the dimension of an internal space goes to infinity. We show that such an approach is helpful in obtaining classical solutions of the string model. As a particular application, we obtain new exact static solutions for the two-dimensional effective string model. They turn out to be in agreement with the generalized no-hair conjecture, in complete analogy with the four and higher dimensional Einstein theory of gravity. 
  We present the gravitational coupling function $\omega(\phi)$ in the vacuum scalar-tensor theory as allowed by the Noether symmetry. We also obtain some exact cosmological solutions in the spatially homogeneous and isotropic background thereby showing that the attractor mechanism is not effective enough to reduce the theory to Einstein theory. It is observed that, asymptotically, the scalar tensor theory goes over to Einstein theory with finite value of $\omega$. This work thus supports earlier works in this direction. 
  A way of constructing mathematically correct quantum geometrodynamics of a closed universe is presented. The resulting theory appears to be gauge-noninvariant and thus consistent with the observation conditions of a closed universe, by that being considerably distinguished from the conventional Wheeler - DeWitt one. For the Bianchi-IX cosmological model it is shown that a normalizable wave function of the Universe depends on time, allows the standard probability interpretation and satisfies a gauge-noninvariant dynamical Schrodinger equation. The Wheeler - DeWitt quantum geometrodynamics is represented by a singular, BRST-invariant solution to the Schrodinger equation having no property of normalizability. 
  Geometric features (including convexity properties) of an exact interior gravitational field due to a self-gravitating axisymmetric body of perfect fluid in stationary, rigid rotation are studied. In spite of the seemingly non-Newtonian features of the bounding surface for some rotation rates, we show, by means of a detailed analysis of the three-dimensional spatial geodesics, that the standard Newtonian convexity properties do hold. A central role is played by a family of geodesics that are introduced here, and provide a generalization of the Newtonian straight lines parallel to the axis of rotation. 
  This is a first paper of a series in which we give some generalizations of the Obukhov theorem in the Tucker-Wang approach to Metric- Affine gravity in which we consider more general actions containing scalar and in general fields which do not depend on the metric or connection. 
  We discuss the late-time behaviour of a dynamically perturbed Kerr black hole. We present analytic results for near extreme Kerr black holes that show that the large number of virtually undamped quasinormal modes that exist for nonzero values of the azimuthal eigenvalue m combine in such a way that the field oscillates with an amplitude that decays as 1/t at late times. This prediction is verified using numerical time-evolutions of the Teukolsky equation. We argue that the observed behaviour may be relevant for astrophysical black holes, and that it can be understood in terms of the presence of a ``superradiance resonance cavity'' immediately outside the black hole. 
  The stability of binary orbits can significantly shape the gravity wave signal which future Earth-based interferometers hope to detect. The inner most stable circular orbit has been of interest as it marks the transition from the late inspiral to final plunge. We consider purely relativistic orbits beyond the circular assumption. Homoclinic orbits are of particular importance to the question of stability as they lie on the boundary between dynamical stability and instability. We identify these, estimate their rate of energy loss to gravity waves, and compute their gravitational waveforms. 
  The Chern-Simons functionals built from various connections determined by the initial data $h_{\mu\nu}$, $\chi_{\mu\nu}$ on a 3-manifold $\Sigma$ are investigated. First it is shown that for asymptotically flat data sets the logarithmic fall-off for $h_{\mu\nu}$ and $r\chi_{\mu\nu}$ is the necessary and sufficient condition of the existence of these functionals. The functional $Y_{k,l}$, built in the vector bundle corresponding to the irreducible representation of SL(2,C) labelled by (k,l), is shown to be determined by the Ashtekar-Chern-Simons functional and its complex conjugate. $Y_{k,l}$ is conformally invariant precisely in the l=k (i.e. tensor) representations. An unexpected connection with twistor theory is found: $Y_{k,k}$ can be written as the Chern-Simons functional built from the 3-surface twistor connection, and the not identically vanishing spinor parts of the 3-surface twistor curvature are given by the variational derivatives of $Y_{k,k}$ with respect to $h_{\mu\nu}$ and $\chi_{\mu\nu}$. The time derivative $\dot Y_{k,k}$ of $Y_{k,k}$ is another conformal invariant of the initial data set, and for vanishing $\dot Y_{k,k}$, in particular for all Petrov III and N spacetimes, the Chern-Simons functional is a conformal invariant of the whole spacetime. 
  We present a general procedure for constructing new Hilbert spaces for loop quantum gravity on non-compact spatial manifolds. Given any fixed background state representing a non-compact spatial geometry, we use the Gel'fand-Naimark-Segal construction to obtain a representation of the algebra of observables. The resulting Hilbert space can be interpreted as describing fluctuation of compact support around this background state. We also give an example of a state which approximates classical flat space and can be used as a background state for our construction. 
  We review the derivation of the metric for a spinning body of any shape and composition using linearized general relativity theory, and also obtain the same metric using a transformation argument. The latter derivation makes it clear that the linearized metric contains only the Eddington $\alpha$ and $\gamma$ parameters, so no new parameter is involved in frame-dragging or Lense-Thirring effects. We then calculate the precession of an orbiting gyroscope in a general weak gravitational field, described by a Newtonian potential (the gravito-electric field) and a vector potential (the gravito-magnetic field). Next we make a multipole analysis of the potentials and the precession equations, giving all of these in terms of the spherical harmonics moments of the density distribution. The analysis is not limited to an axially symmetric source, although the Earth, which is the main application, is very nearly axisymmetric. Finally we analyze the precession in regard to the Gravity Probe B (GP-B) experiment, and find that the effect of the Earth's quadrupole moment (J_2) on the geodetic precession is large enough to be measured by GP-B (a previously known result), but the effect on the Lense-Thirring precession is somewhat beyond the expected GP-B accuracy. 
  In this paper, we have reviewed the present status of the theory of equilibrium configurations of compact binary star systems in Newtonian gravity. Evolutionary processes of compact binary star systems due to gravitational wave emission can be divided into three stages according to the time scales and configurations. The evolution is quasi-stationary until a merging process starts, since the time scale of the orbital change due to gravitational wave emission is longer than the orbital period. In this stage, equilibrium sequences can be applied to evolution of compact binary star systems. Along the equilibrium sequences, there appear several critical states where some instability sets in or configuration changes drastically. We have discussed relations among these critical points and have stressed the importance of the mass overflow as well as the dynamical instability of orbital motions. Concerning the equilibrium sequences of binary star systems, we have summarized classical results of incompressible ellipsoidal configurations. Recent results of compressible binary star systems obtained by the ellipsoidal approximation and by numerical computations have been shown and discussed. It is important to note that numerical computational solutions to {\it exact equations} show that compressibility may lead realistic neutron star binary systems to mass overflows instead of dynamical disruptions for a wide range of parameters. 
  A quantum picture of the causal structure of Minkowski space M is presented. The mathematical model employed to this end is a non-classical version of the classical topos {H} of real quaternion algebras used elsewhere to organize the perceptions of spacetime events of a Boolean observer into M. Certain key properties of this new quantum topos are highlighted by contrast against the corresponding ones of its classical counterpart {H} modelling M and are seen to accord with some key features of the algebraically quantized causal set structure. 
  We study the evolution of strings in the equatorial plane of a Kerr-Newmann black hole. Writting the equations of motion and the constraints resulting from Hamilton's principle, three classes of exact solutions are presented, for a closed string, encircling the black hole. They all depend on two arbitrary integration functions and two constants. A process of extracting energy is examined for the case of one of the three families of solutions. This is the analog of the Penrose process for the case of a particle. 
  Perturbations of stars and black holes have been one of the main topics of relativistic astrophysics for the last few decades. They are of particular importance today, because of their relevance to gravitational wave astronomy. In this review we present the theory of quasi-normal modes of compact objects from both the mathematical and astrophysical points of view. The discussion includes perturbations of black holes (Schwarzschild, Reissner-Nordstr\"om, Kerr and Kerr-Newman) and relativistic stars (non-rotating and slowly-rotating). The properties of the various families of quasi-normal modes are described, and numerical techniques for calculating quasi-normal modes reviewed. The successes, as well as the limits, of perturbation theory are presented, and its role in the emerging era of numerical relativity and supercomputers is discussed. 
  Recent numerical simulations address a conjecture by Shapiro that when two neutron stars collide head-on from rest at infinity, sufficient thermal pressure may be generated to support the hot remnant in quasi-static equilibrium against collapse prior to neutrino cooling. The conjecture is meant to apply even when the total remnant mass exceeds the maximum mass of a cold neutron star. One set of simulations seems to corroborate the conjecture, while another, involving higher mass progenitors each very close to the maximum mass, does not. In both cases the total mass of the remnant exceeds the maximum mass. We point out numerical subtleties in performing such simulations when the progenitors are near the maximum mass; they can explain why the simulations might have difficulty assessing the conjecture in such high-mass cases. 
  We discuss the problem of a degenerate vierbein in the framework of gauge theories of gravitation (thus including torsion). We discuss two examples: Hanson-Regge gravitational instanton and Einstein-Rose bridge.We argue that a region of space-time with vanishing vierbein but smooth principal connection can be, in principle, detected by scattering experiments. 
  For the BTZ black hole in the Einstein gravity, a statistical entropy has been calculated to be equal to the Bekenstein-Hawking entropy. In this paper, the statistical entropy of the BTZ black hole in the higher curvature gravity is calculated and shown to be equal to the one derived by using the Noether charge method. This suggests that the equivalence of the geometrical and statistical entropies of the black hole is retained in the general diffeomorphism invariant theories of gravity. A relation between the cosmic censorship conjecture and the unitarity of the conformal field theory on the boundary of AdS is also discussed. 
  The total spacetime manifold for a Schwarzschild black hole (BH) is described by the Kruskal coordinates u=u(r,t) and v=v(r,t), where r and t are the conventional Schwarzschild radial and time coordinates respectively. The relationship between r and t for a test particle moving on a radial or non-radial geodesic is well known. Similarly, the expression for the vacuum Schwarzschild derivative for a geodesic, in terms of the constants of motion, is well known. However, the same is not true for the Kruskal coordinates. Here, we, first, explicitly obtain the value of the Kruskal coordinates on the Event Horizon as a function of the constant of motion (E=energy per unit rest mass) for a test particle on a radial geodesic by directly using the r-t relationship as obtained by Chandrasekhar and also by Misner, Thorne and Wheeler. It is found that u_H and v_H are finite for E <1. And then, we derive here the expression for the Kruskal derivative for a radial geodesic in terms of the constants of motion. In particular, it is seen that the value of |du/dv| (= 1) is regular on the Event Horizon of the Black Hole. The regular nature of the Kruskal derivative is in sharp contrast with the Schwarzschild derivative, |dt/dr| =infty, at the Event Horizon. 
  We present a quantization of the Hamiltonian and diffeomorphism constraint of canonical quantum gravity in the spin network representation. The novelty consists in considering a space of wavefunctions based on the Vassiliev knot invariants. The constraints are finite, well defined, and reproduce at the level of quantum commutators the Poisson algebra of constraints of the classical theory. A similar construction can be carried out in 2+1 dimensions leading to the correct quantum theory. 
  By application of the duality transformation, which implies interchange of active and passive electric parts of the Riemann curvature (equivalent to interchange of Ricci and Einstein tensors) it is shown that the global monopole solution in the Kaluza-Klein spacetime is dual to the corresponding vacuum solution. Further we also obtain solution dual to flat space which would in general describe a massive global monopole in 4-dimensional Euclidean space and would have massless limit analogus to the 4-dimensional dual-flat solution. 
  In the expansive nondecelerative universe model, creation of matter occurs due to which the Vaidya metrics is applied. This fact allows for localizing gravitational energy and calculating the energy of gravitational waves using an approach alternative to the well established procedure based on quadrupole formula. Rationalization of the gradual increase in entropy of the Universe using relation describing the total curvature of space-time is given too. 
  By decomposing the Riemann curvature into electric and magnetic parts, we define the gravoelectric duality transformation by interchange of active and passive electric parts which amounts to interchange of the Ricci and Einstein tensors. It turns out that the vacuum equation is duality-invariant. We obtain solutions dual to the Kerr solution by writing an effective vacuum equation in such a way that it still admits the Kerr solution but is not duality invariant. The dual equation is then solved to obtain the dual-Kerr solution which can be interpreted as the Kerr black hole sitting in a string dust universe. 
  By resolving the Riemann curvature relative to a unit timelike vector into electric and magnetic parts, we consider duality relations analogous to the electromagnetic theory. It turns out that the duality symmetry of the Einstein action implies the Einstein vacuum equation without the cosmological term. The vacuum equation is invariant under interchange of active and passive electric parts giving rise to the same vacuum solutions but the gravitational constant changes sign. Further by modifying the equation it is possible to construct interesting dual solutions to vacuum as well as to flat spacetimes. 
  A large family of inhomogeneous non-static spherically symmetric solutions of the Einstein equation for null fluid in higher dimensions has been obtained. It encompasses higher dimensional versions of many previously known solutions such as Vaidya, charged Vaidya and Husain solutions and also some new solutions representing global monopole or string dust. It turns out that physical properties of the solutions carry over to higher dimensions. 
  A generalization of the McVittie solution, representing spacetime of a mass particle placed in (n+2) dimensional Robertson-Walker universe is reported. 
  Gravitational waves in cylindrically symmetric Einstein gravity are described by an effective energy tensor with the same form as that of a massless Klein- Gordon field, in terms of a gravitational potential generalizing the Newtonian potential. Energy-momentum vectors for the gravitational waves and matter are defined with respect to a canonical flow of time. The combined energy-momentum is covariantly conserved, the corresponding charge being the modified Thorne energy. Energy conservation is formulated as the first law expressing the gradient of the energy as work and energy-supply terms, including the energy flux of the gravitational waves. Projecting this equation along a trapping horizon yields a first law of black-hole dynamics containing the expected term involving area and surface gravity, where the dynamic surface gravity is defined with respect to the canonical flow of time. A first law for dynamic cosmic strings also follows. The Einstein equation is written as three wave equations plus the first law, each with sources determined by the combined energy tensor of the matter and gravitational waves. 
  Systems of equations are invariant under "polydimensional transformations" which reshuffle the geometry such that what is a line or a plane is dependent upon the frame of reference. This leads us to propose an extension of Clifford calculus in which each geometric element (vector, bivector) has its own coordinate. A new classical action principle is proposed in which particles take paths which minimize the distance traveled plus area swept out by the spin. This leads to a solution of the 50 year old conundrum of `what is the correct Lagrangian' in which to derive the Papapetrou equations of motion for spinning particles in curved space (including torsion).      Based on talk given at: 5th International Conference on Clifford Algebras and their Applications in Mathematical Physics, Ixtapa-Zihuatanejo, Mexico, June 27-July 4, 1999. 
  Three families of exact solutions of Einstein field equations are found. Each family contains three parameters. Two of these families represent thick domain walls in a five dimensional Kaluza-Klein spacetime. The dynamical behaviour of our models is briefly discussed. The spacetime in all the cases is found to be reflection symmetric with respect to the wall. 
  The spin rate \Omega of neutron stars at a given temperature T is constrained by the interplay between gravitational-radiation instabilities and viscous damping. Navier-Stokes theory has been used to calculate the viscous damping timescales and produce a stability curve for r-modes in the (\Omega,T) plane. In Navier-Stokes theory, viscosity is independent of vorticity, but kinetic theory predicts a coupling of vorticity to the shear viscosity. We calculate this coupling and show that it can in principle significantly modify the stability diagram at lower temperatures. As a result, colder stars can remain stable at higher spin rates. 
  A new approach to the inverse-scattering technique of Alekseev is presented which permits real-pole soliton solutions of the Ernst equations to be considered. This is achieved by adopting distinct real poles in the scattering matrix and its inverse. For the case in which the electromagnetic field vanishes, some explicit solutions are given using a Minkowski seed metric. The relation with the corresponding soliton solutions that can be constructed using the Belinskii-Zakharov inverse-scattering technique is determined. 
  This paper provides a thorough introduction to the causal set hypothesis aimed at students, and other interested persons, with some knowledge of general relativity and nonrelativistic quantum mechanics. I elucidate the arguments for why the causal set structure might be the appropriate structure for a theory of quantum gravity. The logical and formal development of a causal set theory as well as a few illuminating examples are also provided. 
  The quasi-stationary method for black hole binary inspiral is an approximation for studying strong field effects while suppressing radiation reaction. In this paper we use a nonlinear scalar field toy model (i) to explain the underlying method of approximating binary motion by periodic orbits with radiation; (ii) to show how the fields in such a model are found by the solution of a boundary value problem; (iii) to demonstrate how a good approximation to the outgoing radiation can be found by finding fields with a balance of ingoing and outgoing radiation (a generalization of standing waves). 
  This paper describes that the superconducting cosmic strings can be connected to an electrically charged black hole, and can be considerd as the hair of black hole. What the no-hair theorems show is that a large amount of information is lost when a body collapses to form a black hole. In addition, the no-hair theorem has not been proved for the Yang-Mills field. This paper proves and claims that the superconducting cosmic strings can be connected to an electrically charged hole when the current inside these strings and black holes approaches the critical value. Because, this state is the final state of the gravitational collapse, and the event horizon would be destroyed in this state. Therefore, these strings should be considered as hair of the charged black holes, and may be titled as BHCS (Black Hole Connected Strings). This means that at least the charged black holes have the hair. Thus, the no-hair theorem is not applicable for the charged black holes in the state of the critical current. 
  We study the topological signature of euclidean isometries in pair separations histograms (PSH) and elucidate some unsettled issues regarding distance correlations between cosmic sources in cosmic crystallography. Reducing the noise of individual PSH's using mean pair separations histograms we show how to distinguish between topological and statistical spikes. We report results of simulations that evince that topological spikes are not enough to distinguish between manifolds with the same set of Clifford translations in their covering groups, and that they are not the only signature of topology in PSH's corresponding to euclidean small universes. We also show how to evince the topological signature due to non-translational isometries. 
  Massive spin-1/2 fields are studied in the framework of loop quantum gravity by considering a state approximating, at a length scale $\cal L$ much greater than Planck length $\ell_P=1.2\times 10^{-33}$cm, a spin-1/2 field in flat spacetime. The discrete structure of spacetime at $\ell_P$ yields corrections to the field propagation at scale $\cal L$. Next, Neutrino Bursts (${\bar p}\approx 10^5$GeV) accompaning Gamma Ray Bursts that have travelled cosmological distances, $L\approx 10^{10}$l.y., are considered. The dominant correction is helicity independent and leads to a time delay w.r.t. the speed of light, $c$, of order $({\bar p} \ell_P) L/c\approx 10^4$s. To next order in ${\bar p} \ell_P$ the correction has the form of the Gambini and Pullin effect for photons. Its contribution to time delay is comparable to that caused by the mass term. Finally, a dependence $L_{\rm os}^{-1} \propto {\bar p}^2 \ell_P$ is found for a two-flavour neutrino oscillation length. 
  To determine whether particular sources of gravitational radiation will be detectable by a specific gravitational wave detector, it is necessary to know the sensitivity limits of the instrument. These instrumental sensitivities are often depicted (after averaging over source position and polarization) by graphing the minimal values of the gravitational wave amplitude detectable by the instrument versus the frequency of the gravitational wave. This paper describes in detail how to compute such a sensitivity curve given a set of specifications for a spaceborne laser interferometer gravitational wave observatory. Minor errors in the prior literature are corrected, and the first (mostly) analytic calculation of the gravitational wave transfer function is presented. Example sensitivity curve calculations are presented for the proposed LISA interferometer. We find that previous treatments of LISA have underestimated its sensitivity by a factor of $\sqrt{3}$. 
  It is shown that the complete description of the propagation of light in a gravitational field and in non-inertial reference frames in general requires an average coordinate and an average proper velocity of light. The need for an average coordinate velocity of light in non-inertial frames is demonstrated by considering the propagation of two vertical light rays in the Einstein elevator (in addition to the horizontal ray originally discussed by Einstein). As an average proper velocity of light is implicitly used in the Shapiro time delay (as shown in the Appendix) it is explicitly derived and it is shown that for a round trip of a light signal between two points in a gravitational field the Shapiro time delay not only depends on which point it is measured at, but in the case of a parallel gravitational field it is not always a delay effect. The propagation of light in rotating frames (the Sagnac effect) is also discussed and an expression for the coordinate velocity of light is derived. The use of this coordinate velocity naturally explains why an observer on a rotating disk finds that two light signals emitted from a point on the rim of the disk and propagating in opposite directions along the rim do not arrive simultaneously at the same point. 
  We investigate the structure of the gravitational field generated by a massless particle moving on the horizon of an arbitrary (stationary) black hole. This is done by employing the generalized Kerr-Schild class where we take the null generators of the horizon as the geodetic null vector-field and a scalar function which is concentrated on the horizon. 
  A technique is described for removing interference from a signal of interest ("channel 1") which is one of a set of N time-domain instrumental signals ("channels 1 to N"). We assume that channel 1 is a linear combination of "true" signal plus noise, and that the "true" signal is not correlated with the noise. We also assume that part of this noise is produced, in a poorly-understood way, by the environment, and that the environment is monitored by channels 2 to N. Finally, we assume that the contribution of channel n to channel 1 is described by an (unknown!) linear transfer function R_n(t-t'). Our technique estimates the R_i and provides a way to subtract the environmental contamination from channel 1, giving an estimate of the "true" signal which minimizes its variance. It also provides some insights into how the environment is contaminating the signal of interest. The method is illustrated with data from a prototype interferometric gravitational-wave detector, in which the channel of interest (differential displacement) is heavily contaminated by environmental noise (magnetic and seismic noise) and laser frequency noise but where the coupling between these signals is not known in advance. 
  The analogs of r-modes in superfluid neutron stars are studied here. These modes, which are governed primarily by the Coriolis force, are identical to their ordinary-fluid counterparts at the lowest order in the small angular-velocity expansion used here. The equations that determine the next order terms are derived and solved numerically for fairly realistic superfluid neutron-star models. The damping of these modes by superfluid ``mutual friction'' (which vanishes at the lowest order in this expansion) is found to have a characteristic time-scale of about 10^4 s for the m=2 r-mode in a ``typical'' superfluid neutron-star model. This time-scale is far too long to allow mutual friction to suppress the recently discovered gravitational radiation driven instability in the r-modes. However, the strength of the mutual friction damping depends very sensitively on the details of the neutron-star core superfluid. A small fraction of the presently acceptable range of superfluid models have characteristic mutual friction damping times that are short enough (i.e. shorter than about 5 s) to suppress the gravitational radiation driven instability completely. 
  We review the possibility that quantum fluctuations in the structure of space-time at the Planck scale might be subject to experimental probes. We study the effects of space-time foam in an approach inspired by string theory, in which solitonic D-brane excitations are taken into account when considering the ground state. We model the properties of this medium by analyzing the recoil of a D particle which is induced by the scattering of a closed-string state. We find that this recoil causes an energy-dependent perturbation of the background metric, which in turn induces an energy-dependent refractive index in vacuo, and stochastic fluctuations of the light cone. We show how distant astrophysical sources such as Gamma-Ray Bursters (GRBs) may be used to test this possibility, making an illustrative analysis of GRBs whose redshifts have been measured. Within this framework, we also discuss the propagation of massive particles and the possible appearance of cosmological vacuum energy that relaxes towards zero. We also discuss D-brane recoil in models with `large' extra dimensions. 
  An attempt is made here to extend to the microscopic domain the scale invariant character of gravitation - which amounts to consider expansion as applying to any physical scale. Surprisingly, this hypothesis does not prevent the redshift from being obtained. It leads to strong restrictions concerning the choice between the presently available cosmological models and to new considerations about the notion of time. Moreover, there is no horizon problem and resorting to inflation is not necessary. 
  The observed absence of gravitational aberration requires that ``Newtonian'' gravity propagate at a speed $c_g>2\times10^{10}c$. By evaluating the gravitational effect of an accelerating mass, I show that aberration in general relativity is almost exactly canceled by velocity-dependent interactions, permitting $c_g=c$. This cancellation is dictated by conservation laws and the quadrupole nature of gravitational radiation. 
  The classical electromagnetic modes outside a long, straight, superconducting cosmic string are calculated, assuming the string to be surrounded by a superconducting cylindric surface of radius R. Thereafter, by use of a Bogoliubov-type argument, the electromagnetic energy W produced per unit length in the lowest two modes is calculated when the string is formed "suddenly". The essential new element in the present analysis as compared with prior work of Parker [Phys. Rev. Lett. {\bf 59}, 1369 (1987)] and Brevik and Toverud [Phys. Rev. D {\bf 51}, 691 (1995)], is that the radius {\it a} of the string is assumed finite, thus necessitating Neumann functions to be included in the fundamental modes. We find that the theory is changed significantly: W is now strongly concentrated in the lowest mode $(m,s)=(0,1)$, whereas the proportionality $W \propto (G\mu /t)^2$ that is characteristic for zero-width strings is found in the next mode (1,1). Here G is the gravitational constant, $\mu$ the string mass per unit length, and t the GUT time. 
  Silent universes are studied using a ``3+1'' decomposition of the field equations in order to make progress in proving a recent conjecture that the only silent universes of Petrov type I are spatially homogeneous Bianchi I models. The infinite set of constraints are written in a geometrically clear form as an infinite set of Codacci tensors on the initial hypersurface. In particular, we show that the initial data set for silent universes is ``non-contorted'' and therefore (Beig and Szabados, 1997) isometrically embeddable in a conformally flat spacetime. We prove, by making use of algebraic computing programs, that the conjecture holds in the simpler case when the spacetime is vacuum. This result points to confirming the validity of the conjecture in the general case. Moreover, it provides an invariant characterization of the Kasner metric directly in terms of the Weyl tensor. A physical interpretation of this uniqueness result is briefly discussed. 
  We consider the motion of a spinning relativistic particle with an arbitrary value of spin in external electromagnetic and gravitational fields, to first order in the external field. We use the noncovariant description of spin. An explicit expression is obtained for the interaction of second order in spin. The value of the quadrupole moment is found for which this interaction decreases when the energy grows. 
  Undulatory field functions represent a real wave only if there exists a class of infinite reference systems for which an identical wave is described by the same functional forms. 
  It is demonstrated that the sensitivity of a superconductive LC - circuit placed in a weak gravitational wave is limited by two factors. One is the quantization of the magnetic flux through the circuit, the second one is the fraction of the elementary charge (effect Laughlin - Stormer - Tsui). Application to a possibility of using a superconductive LC - circuit as a weak gravitational waves detector is discussed. 
  This short communication advances the hypothesis that the observed fractal structure of large-scale distribution of galaxies is due to a geometrical effect, which arises when observational quantities relevant for the characterization of a cosmological fractal structure are calculated along the past light cone. If this hypothesis proves, even partially, correct, most, if not all, objections raised against fractals in cosmology may be solved. For instance, under this view the standard cosmology has zero average density, as predicted by an infinite fractal structure, with, at the same time, the cosmological principle remaining valid. The theoretical results which suggest this conjecture are reviewed, as well as possible ways of checking its validity. 
  This paper deals with some two-parameter solutions to the spherically symmetric, vacuum Einstein equations which, we argue, are more general than de Sitter solution. The global structure of one such spacetimes and its extension to the multiply connected case have also been investigated. By using a six-dimensional Minkowskian embedding as its maximal extension, we check that the thermal properties of the considered solution in such an embedding space are the same as those derived by the usual Euclidean method. The stability of the generalized de Sitter space containing a black hole has been investigated as well by introducing perturbations of the Ginsparg-Perry type in first order approximation. It has been obtained that such a space perdures against the effects of these perturbations. 
  We note that Eddington's radiation damping calculation of a spinning rod fails to account for the complete mass integral as given by Tolman. The missing stress contributions precisely cancel the standard rate given by the 'quadrupole formula'. This indicates that while the usual 'kinetic' term can properly account for dynamical changes in the source, the actual mass is conserved. Hence gravity waves are not carriers of energy in vacuum. This supports the hypothesis that energy including the gravitational contribution is confined to regions of non-vanishing energy-momentum tensor $T_{ik}$.   PACS numbers: 04.20.Cv, 04.30.-w 
  It is proven that the Wahlquist perfect fluid space-time cannot be smoothly joined to an exterior asymptotically flat vacuum region. The proof uses a power series expansion in the angular velocity, to a precision of the second order. In this approximation, the Wahlquist metric is a special case of the rotating Whittaker space-time. The exterior vacuum domain is treated in a like manner. We compute the conditions of matching at the possible boundary surface in both the interior and the vacuum domain. The conditions for matching the induced metrics and the extrinsic curvatures are mutually contradictory. 
  A second-order expansion for the quantum fluctuations of the matter field was considered in the framemork of the warm inflation scenario. The friction and Hubble parameters were expanded by means of a semiclassical approach. The fluctuations of the Hubble parameter generates fluctuations of the metric. These metric fluctuations produce an effective term of curvature. The power spectrum for the metric fluctuations can be calculated on the infrared sector. 
  Geometric models of quantum relativistic rotating oscillators in arbitrary dimensions are defined on backgrounds with deformed anti-de Sitter metrics. It is shown that these models are analytically solvable, deriving the formulas of the energy levels and corresponding normalized energy eigenfunctions. An important property is that all these models have the same nonrelativistic limit, namely the usual harmonic oscillator. 
  It is shown how can be derived the normalized energy eigenspinors of the free Dirac field on anti-de Sitter spacetime, by using a Cartesian tetrad gauge where the separation of spherical variables can be done like in special relativity. 
  We discuss some ways in which topos theory (a branch of category theory) can be applied to interpretative problems in quantum theory and quantum gravity. In Section 1, we introduce these problems. In Section 2, we introduce topos theory, especially the idea of a topos of presheaves. In Section 3, we discuss several possible applications of topos theory to the problems in Section 1. In Section 4, we draw some conclusions. 
  Quantum mechanical interference of wave functions leads to some difficulties if a probability density is considered as a source of gravity. We show that an introduction of a quantum energy-momentum tensor as a source term in Einstein equations can be consistent with general relativity if the gravitational waves are quantized. 
  We extend here the canonical treatment of spherically symmetric (quantum) gravity to the most simple matter coupling, namely spherically symmetric Maxwell theory with or without a cosmological constant. The quantization is based on the reduced phase space which is coordinatized by the mass and the electric charge as well as their canonically conjugate momenta, whose geometrical interpretation is explored. The dimension of the reduced phase space depends on the topology chosen, quite similar to the case of pure (2+1) gravity. We investigate several conceptual and technical details that might be of interest for full (3+1) gravity. We use the new canonical variables introduced by Ashtekar, which simplifies the analysis tremendously. 
  There is a gap that has been left open since the formulation of general relativity in terms of Ashtekar's new variables namely the treatment of asymptotically flat field configurations that are general enough to be able to define the generators of the Lorentz subgroup of the asymptotical Poincar\'e group. While such a formulation already exists for the old geometrodynamical variables, up to now only the generators of the translation subgroup could be defined because the function spaces of the fields considered earlier are taken too special. The transcription of the framework from the ADM variables to Ashtekar's variables turns out not to be straightforward due to the a priori freedom to choose the internal SO(3) frame at spatial infinity and due to the fact that the non-trivial reality conditions of the Ashtekar framework reenter the stage when imposing suitable boundary conditions on the fields and the Lagrange multipliers. 
  In order to test the canonical quantization programme for general relativity we introduce a reduced model for a real sector of complexified Ashtekar gravity which captures important properties of the full theory. While it does not correspond to a subset of Einstein's gravity it has the advantage that the programme of canonical quantization can be carried out completely and explicitly, both, via the reduced phase space approach or along the lines of the algebraic quantization programme. This model stands in close correspondence to the frequently treated cylindrically symmetric waves. In contrast to other models that have been looked at up to now in terms of the new variables the reduced phase space is infinite dimensional while the scalar constraint is genuinely bilinear in the momenta. The infinite number of Dirac observables can be expressed in compact and explicit form in terms of the original phase space variables. They turn out, as expected, to be non-local and form naturally a set of countable cardinality. 
  The preceding talks given at this conference have dealt mainly with general ideas for, main problems of and techniques for the task of quantizing gravity canonically. Since one of the major motivations to arrange for this meeting was that it should serve as a beginner's introduction to canonical quantum gravity, we regard it as important to demonstrate the usefulness of the formalism by means of applying it to simplified models of quantum gravity, here formulated in terms of Ashtekar's new variables. From the various, completely solvable, models that have been discussed in the literature we choose those that we consider as most suitable for our pedagogical reasons, namely 2+1 gravity and the spherically symmetric model. The former model arises from a dimensional, the latter from a Killing reduction of full 3+1 gravity. While 2+1 gravity is usually treated in terms of closed topologies without boundary of the initial data hypersurface, the toplogy for the spherically symmetric system is chosen to be asymptotically flat. Finally, 2+1 gravity is more suitably quantized using the loop representation while spherically symmetric gravity is easier to quantize via the self-dual representation. Accordingly, both types of reductions, both types of topologies and both types of representations that are mainly employed in the literature in the context of the new variables come into practice. What makes the discussion especially clear is the fact that for both models the reduced phase space turns out to be finitely dimensional. 
  We present here the canonical treatment of spherically symmetric (quantum) gravity coupled to spherically symmetric Maxwell theory with or without a cosmological constant. The quantization is based on the reduced phase space which is coordinatized by the mass and the electric charge as well as their canonically conjugate momenta, whose geometrical interpretation is explored.  The dimension of the reduced phase space depends on the topology chosen, quite similar to the case of pure (2+1) gravity. We also compare the reduced phase space quantization to the algebraic quantization. Altogether, we observe that the present model serves as an interesting testing ground for full (3+1) gravity. We use the new canonical variables introduced by Ashtekar which simplifies the analysis tremendously. 
  The present article summarizes the work of the papers \cite{1} dealing with the quantization of pure gravity and gravity coupled to a Maxwell field and a cosmological constant in presence of spherical symmetry. The class of models presented is intended as an interesting testing ground for the quantization of full 3+1 gravity. We are working in Ashtekar's self-dual representation. 
  The stability of the brick wall model is analyzed in a rotating background. It is shown that in the Kerr background without horizon but with an inner boundary a scalar field has complex-frequency modes and that, however, the imaginary part of the complex frequency can be small enough compared with the Hawking temperature if the inner boundary is sufficiently close to the horizon, say at a proper altitude of Planck scale. Hence, the time scale of the instability due to the complex frequencies is much longer than the relaxation time scale of the thermal state with the Hawking temperature. Since ambient fields should settle in the thermal state in the latter time scale, the instability is not so catastrophic. Thus, the brick wall model is well defined even in a rotating background if the inner boundary is sufficiently close to the horizon. 
  This paper advances a general proposal for testing non-standard cosmological models by means of observational relations of cosmological point sources in some specific waveband, and their use in the context of the data provided by the galaxy redshift surveys, but for any cosmological metric. By starting from the general theory for observations in relativistic cosmology the equations for colour, K-correction, and number counts of cosmological point sources are discussed in the context of curved spacetimes. The number counts equation is also written in terms of the selection and luminosity functions, which provides a relativistic generalization of its Euclidean version. Since these observables were not derived in the framework of any specific cosmology, they are valid for any cosmological model. The hypotheses used in such derivation are reviewed, together with some difficulties for the practical use of those observables. 
  Using the symmetric energy-momentum complexes of Landau and Lifshitz, Papapetrou, and Weinberg we obtain the energy of the universe in anisotropic Bianchi type I cosmological models . The energy (due to matter plus field) is found to be zero and this agrees with a previous result of Banerjee and Sen who investigated this problem using the Einstein energy-momentum complex. Our result supports the importance of the energy-momentum complexes and contradicts the prevailing ``folklore'' that different energy-momentum complexes could give different and hence unacceptable energy distribution in a given space-time. The result that the total energy of the universe in these models is zero supports the viewpoint of Tryon. Rosen computed the total energy of the closed homogeneous isotropic universe and found that to be zero, which agrees with the studies of Tryon. 
  The inertial effects on neutrino oscillations induced by the acceleration and angular velocity of a reference frame are calculated. Such effects have been analyzed in the framework of the solar and atmospheric neutrino problem. 
  The generalized uncertainty principle of string theory is derived in the framework of  Quantum Geometry by taking into account the existence of an upper limit on the acceleration of massive particles. 
  We are familiar with Dirac equation in flat space by which we can investigate the behaviour of half-integral spin particle. With the introduction of general relativistic effects the form of the Dirac equation will be modified. For the cases of different background geometry like Kerr, Schwarzschild etc. the corresponding form of the Dirac equation as well as the solution will be different. In 1972, Teukolsky wrote the Dirac equation in Kerr geometry. Chandrasekhar separated it into radial and angular parts in 1976. Later Chakrabarti solved the angular equation in 1984. In 1999 Mukhopadhyay and Chakrabarti have solved the radial Dirac equation in Kerr geometry in a spatially complete manner. In this review we will discuss these developments systematically and present some solutions. 
  It is pointed out that the coupling of macroscopic test masses to the gravi-dilaton background of string theory is non geodesic, in general, and cannot be parametrized by a Brans-Dicke model of scalar-tensor gravity. The response of gravitational antennas to dilatonic waves should be analyzed through a generalized equation of geodesic deviation, taking into account the possible direct coupling of the background to the (composition-dependent) dilatonic charge of the antenna. 
  We analyse a classical model of gravitation coupled to a self interacting scalar field. We show that, within the context of this model for Robertson-Walker cosmologies, there exist solutions in the spatially non-flat cases exhibiting transitions from a Euclidean to a Lorentzian spacetime. We then discuss the conditions under which these signature changing solutions to Einstein's field equations exist. In particular, we find that an upper bound for the cosmological constant exists and that close to the signature changing hypersurface, both the scale factor and the scalar field have to be constant. Moreover we find that the signature changing solutions do not exist when the scalar field is massless. 
  We study the implications of a scalar-tensorial gravity for the metric of an isolated self-gravitating superconducting cosmic string. These modifications are induced by an arbitrary coupling of a massless scalar field to the usual tensorial field in the gravitational Lagrangian. We derive the metric in the weak-field approximation and we analyse the behaviour of light in this spacetime. We end with some discussions. 
  Shapiro put forth a conjecture stating that neutron stars in head-on collisions (infalling from infinity) will not collapse to black holes before neutrino cooling, independent of the mass of the neutron stars. In a previous paper we carried out a numerical simulation showing a counter example based on 1.4 $M_{\odot}$ neutron stars, and provided an analysis explaining why Shapiro's argument was not applicable for this case. A recent paper by Shapiro put forth an argument suggesting that numerical simulations of the 1.4 $M_{\odot}$ collisions could not disprove the conjecture with the accuracy that is presently attainable. We show in this paper that this argument is not applicable for the same reason that the Shapiro conjecture is not applicable to the 1.4 $M_{\odot}$ neutron star collision, namely, the collision is too dynamical to be treated by quasi-equilibrium arguments. 
  The recent paper gr-qc/9909017 criticizes the limit on the measurability of distances that was derived by Salecker and Wigner in the 1950s. If justified, this criticism would have important implications for all the recent studies that have used in various ways the celebrated Salecker-Wigner result, but I show here that the analysis reported in gr-qc/9909017 is incorrect. Whereas Salecker and Wigner sought an operative definition of distances suitable for the Planck regime, the analysis in gr-qc/9909017 relies on several assumptions that appear to be natural in the context of most present-day experiments but are not even meaningful in the Planck regime. Moreover, contrary to the claim made in gr-qc/9909017, a relevant quantum uncertainty which is used in the Salecker-Wigner derivation cannot be truly eliminated; unsurprisingly, it can only be traded for another comparable contribution to the total uncertainty in the measurement. I also comment on the role played by the Salecker-Wigner limit in my recent proposal of interferometry-based tests of quantum properties of space-time, which was incorrectly described in gr-qc/9909017. In particular, I emphasize that, as discussed in detail in gr-qc/9903080, only some of the quantum-gravity ideas that can be probed with modern interferometers are motivated by the Salecker-Wigner limit. The bulk of the insight we can expect from such interferometric studies concerns the properties of "foamy" models of space-time, which are intrinsically interesting independently of the Salecker-Wigner limit. 
  We study about an evaporating process of black holes in SO(3) Einstein-Yang-Mills-Higgs system. We consider a massless scalar field which couple neither with the Yang-Mills field nor with the Higgs field surrounding the black hole. We discuss differences in evaporating rate between a monopole black hole and a Reissner-Nortstr\"{o}m (RN) black hole. 
  We comment on two issues in quantum cosmology, in the context of the Wheeler-De Witt equation and wave function of the Universe: (i) arrow of time and interpretation of the wave function in the classically allowed regions; (ii) stability of an approximation of the Born-Oppenheimer type in classically forbidden regions of the scale factor. 
  Flavor oscillations of neutrinos are analyzed in the framework of Brans-Dicke theory of gravity. We find a shift of quantum mechanical phase of neutrino proportional to $G_N\Delta m^2$ and depending on the parameter $\omega$. Consequences on atmospheric, solar and astrophysical neutrinos are discussed. 
  It is difficult to choose detection thresholds for tests of non-stationarity that assume {\em a priori} a noise model if the data is statistically uncharacterized to begin with. This is a potentially serious problem when an automated analysis is required, as would be the case for the huge data sets that large interferometric gravitational wave detectors will produce. A solution is proposed in the form of a {\em robust} time-frequency test for detecting non-stationarity whose threshold for a specified false alarm rate is almost independent of the statistical nature of the ambient stationary noise. The efficiency of this test in detecting bursts is compared with that of an ideal test that requires prior information about both the statistical distribution of the noise and also the frequency band of the burst. When supplemented with an approximate knowledge of the burst duration, this test can detect, at the same false alarm rate and detection probability, bursts that are about 3 times larger in amplitude than those that the ideal test can detect. Apart from being robust, this test has properties which make it suitable as an online monitor of stationarity. 
  We present two counterexamples to the paper by Carot et al. in Gen. Rel. Grav. 1997, 29, 1223 and show that the results obtained are correct but not general. 
  The extreme Schwarzschild-de Sitter space-time is a spherically symmetric solution of Einstein's equations with a cosmological constant Lambda and mass parameter m>0 which is characterized by the condition that 9 Lambda m^2=1. The global structure of this space-time is here analyzed in detail. Conformal and embedding diagrams are constructed, and synchronous coordinates which are suitable for a discussion of the cosmic no-hair conjecture are presented. The permitted geodesic motions are also analyzed. By a careful investigation of the geodesics and the equations of geodesic deviation, it is shown that specific families of observers escape from falling into the singularity and approach nonsingular asymptotic regions which are represented by special "points" in the complete conformal diagram. The redshift of signals emitted by particles which fall into the singularity, as detected by those observers which escape, is also calculated. 
  In this summary article, we review and discuss the non-existence of stationary black hole solutions for the Einstein-Dirac-Maxwell equations. 
  The construction of conformally invariant gauge conditions for Maxwell and Einstein theories on a manifold M is found to involve two basic ingredients. First, covariant derivatives of a linear gauge (e.g. Lorenz or de Donder), completely contracted with the tensor field representing the metric on the vector bundle of the theory. Second, the addition of a compensating term, obtained by covariant differentiation of a suitable tensor field built from the geometric data of the problem. If the manifold M is endowed with an m-dimensional positive-definite metric g, the existence theorem for such a gauge in gravitational theory can be proved. If the metric g is Lorentzian, which corresponds to general relativity, some technical steps are harder, but one has again to solve integral equations on curved space-time to be able to impose such gauges. 
  We present here a relativistic theory of gravity in which the spacetime metric is derived from a single scalar field $\Phi$. The field equation, derived from a simple variational principle, is a non-linear flat-space four-dimensional wave equation which is particularly suited for numerical evolution. We demonstrate that while this theory does not generate results which are exactly identical quantitatively to those of general relativity (GR), many of the qualitative features of the full GR theory are reproduced to a reasonable approximation. The advantage of this formulation lies in the fact that 3D numerical grids can be numerically evolved in minutes or hours instead of the days and weeks required by GR, thus drastically reducing the development time of new relativistic hydrodynamical codes. Scalar gravity therefore serves as a meaningful testbed for the development of larger routines destined for use under the full theory of general relativity. 
  Data of good quality is expected from a number of gravitational wave detectors within the next two years. One of these, GEO600, has special capabilities, such as narrow-band operation. I describe here the preparations that are currently being made for the analysis of GEO600 data. 
  Studies in 1+1 dimensions suggest that causally discontinuous topology changing spacetimes are suppressed in quantum gravity. Borde and Sorkin have conjectured that causal discontinuities are associated precisely with index 1 or n-1 Morse points in topology changing spacetimes built from Morse functions. We establish a weaker form of this conjecture. Namely, if a Morse function f on a compact cobordism has critical points of index 1 or n-1, then all the Morse geometries associated with f are causally discontinuous, while if f has no critical points of index 1 or n-1, then there exist associated Morse geometries which are causally continuous. 
  The existence of Killing-Yano tensors on space-times can be probed by spinning particles. Specifically, Dirac particles possess new fermionic constants of motion corresponding to non-standard supersymmetries on the particle worldline. A geometrical duality connects space-times with Killing-Yano structure, but without torsion, to other space-times with Killing-Yano structure and torsion. A relation between the indices of the Dirac operators on the dual space-times allows to express the index on the space-time with torsion in terms of that of the space-time without torsion. 
  It is demonstrated that gravitational and inertial masses are correlated by an electromagnetic factor. From the practical point of view this is very important because it means the possibility of electromagnetic control of the gravity. Some theoretical consequences of the correlation are: incorporation of Mach's principle into Gravitation Theory; new relativistic expression for the mass ; the generalization of Newton's second law for the motion; the deduction of the differential equation for entropy directly from the Gravitation Theory. Another fundamental consequence of the mentioned correlation is that, in specific ultra-high energy conditions, the gravitational and electromagnetic fields can be described by the same Hamiltonian, i.e., in these circumstances, they are unified. Such conditions can have occurred inclusive in the Initial Universe, before the first spontaneous breaking of symmetry. 
  A simple model of spacetime foam, made by N wormholes in a semiclassical approximation, is taken under examination. The Casimir-like energy of the quantum fluctuation of such a model and its probability of being realized are computed. Implications on the Bekenstein-Hawking entropy and the cosmological constant are considered. 
  We formulate deformation of relativistic stars due to the magnetic stress, considering the magnetic fields to be perturbations from spherical stars. The ellipticity for the dipole magnetic field is calculated for some stellar models. We have found that the ellipticity becomes large with increase of a relativistic factor for the models with the same energy ratio of the magnetic energy to the gravitational energy. 
  We consider topological contributions to the action integral in a gauge theory formulation of gravity. Two topological invariants are found and are shown to arise from the scalar and pseudoscalar parts of a single integral. Neither of these action integrals contribute to the classical field equations. An identity is found for the invariants that is valid for non-symmetric Riemann tensors, generalizing the usual GR expression for the topological invariants. The link with Yang-Mills instantons in Euclidean gravity is also explored. Ten independent quadratic terms are constructed from the Riemann tensor, and the topological invariants reduce these to eight possible independent terms for a quadratic Lagrangian. The resulting field equations for the parity non-violating terms are presented. Our derivations of these results are considerably simpler that those found in the literature. 
  Spinning compact binaries are shown to be chaotic in the Post-Newtonian expansion of the two body system. Chaos by definition is the extreme sensitivity to initial conditions and a consequent inability to predict the outcome of the evolution. As a result, the spinning pair will have unpredictable gravitational waveforms during coalescence. This poses a challenge to future gravity wave observatories which rely on a match between the data and a theoretical template. 
  We study the general spherical symmetric solutions of dilaton-modulus gravity non-minimally coupled to a Maxwell field, using methods from the theory of dynamical systems. We show that the solutions can be classified by the mass, the electric charge, and a third parameter which we argue can be related to a scalar charge. The global properties of the solutions are discussed. 
  A special case of metric-affine gauge theory of gravity (MAG) is equivalent to general relativity with Proca matter as source. We study in detail a corresponding numeric solution of the Reissner-Nordstr"om type. It is static, spherically symmetric, and of electric type. In particular, this solution has no horizon, so it has a naked singularity as its origin. 
  Every (1 polarization) cylindrical wave solution of vacuum general relativity is completely determined by a corresponding axisymmetric solution to the free scalar wave equation on an auxilliary 2+1 dimensional flat spacetime. The physical metric at radius R is determined by the energy, $\gamma (R)$, of the scalar field in a box (in the flat spacetime) of radius R. In a recent work, among other important results, Ashtekar and Pierri have introduced a strategy to study the quantum geometry in this system, through a regularized quantum counterpart of $\gamma (R)$. We show that this regularized object is a densely defined symmetric operator, thereby correcting an error in their proof of this result. We argue that it admits a self adjoint extension and show that the operator, unlike its classical counterpart, is not positive. 
  A method is described by which a function defined on a cubic grid (as from a finite difference solution of a partial differential equation) can be resolved into spherical harmonic components at some fixed radius. This has applications to the treatment of boundary conditions imposed at radii larger than the size of the grid, following Abrahams, Rezzola, Rupright et al.(gr-qc/9709082}. In the method described here, the interpolation of the grid data to the integration 2-sphere is combined in the same step as the integrations to extract the spherical harmonic amplitudes, which become sums over grid points. Coordinates adapted to the integration sphere are not needed. 
  We present a three-parameter family of solutions to the stationary axisymmetric Einstein equations that describe differentially rotating disks of dust. They have been constructed by generalizing the Neugebauer-Meinel solution of the problem of a rigidly rotating disk of dust. The solutions correspond to disks with angular velocities depending monotonically on the radial coordinate; both decreasing and increasing behaviour is exhibited. In general, the solutions are related mathematically to Jacobi's inversion problem and can be expressed in terms of Riemann theta functions. A particularly interesting two-parameter subfamily represents Baecklund transformations to appropriate seed solutions of the Weyl class. 
  It is well-known that the 5D equations without sources may be reduced to the 4D ones with sources, provided an appropriate definition for the energy-momentum tensor of matter in terms of the extra part of the geometry.The advantage consists on the naturally appearance of gravitational and electromagnetic fields from this decomposition. With this ansatz an algorithm is presented, which permits to express the physical parameters in terms of gauge potentials and scalar field. An explicit form for the exterior magnetic field of neutron star in terms of the scalar field and the gauge potentials is deduced for a static, spherically-symmetric metric. 
  We consider a static, spherically symmetric system of a Dirac particle in a classical gravitational and SU(2) Yang-Mills field. We prove that the only black-hole solutions of the corresponding Einstein-Dirac-Yang/Mills equations are the Bartnik-McKinnon black-hole solutions of the SU(2) Einstein-Yang/Mills equations; thus the spinors must vanish identically. This indicates that the Dirac particles must either disappear into the black-hole or escape to infinity. 
  The exterior of a relativistic star can be modelated with the Vaidya radiating metric. It is started from the generalized Vaidya metric that allows a type II fluid and studied the conditions of generating new analytical solutions of the Einstein's field equations. It is shown that the mass parameter solution gives the classical de Sitter universe in the static case and the extended de Sitter metric coupled with a dilation scalar field in the time-dependent case. It is concluded that in the time-dependent case the atmosphere of a relativistic star consists on an anisotropic string fluid coupled with a dark matter null fluid and interpreted the scalar field as the particle that produces the dark matter. 
  In this paper is presented a general-relativistic approach of cuasi-neutral bodies endowed with magnetic field. Starting from the similarity of neutron particles with neutron stars for which the general-relativistic framework is imposed, this treatment is extended to neutrons. A class of interior solutions is derived from the Einstein field equations for a spherically-symmetric distribution of a perfect magneto-fluid in the magnetohydrodinamic approximation and a magnetic mass model is proposed. The dependence of the metric and of physical parameters on the magnetic field leads to the conclusion that the mass is entirely magnetic in origin, which implies that the field equations do not admit interior solutions if the source consist only on perfect fluid. 
  We study perturbation theory for spin foam models on triangulated manifolds. Starting with any model of this sort, we consider an arbitrary perturbation of the vertex amplitudes, and write the evolution operators of the perturbed model as convergent power series in the coupling constant governing the perturbation. The terms in the power series can be efficiently computed when the unperturbed model is a topological quantum field theory. Moreover, in this case we can explicitly sum the whole power series in the limit where the number of top-dimensional simplices goes to infinity while the coupling constant is suitably renormalized. This `dilute gas limit' gives spin foam models that are triangulation-independent but not topological quantum field theories. However, we show that models of this sort are rather trivial except in dimension 2. 
  The disclination in Lorentz space-time is studied in detail by means of topological properties of $\phi $-mapping. It is found the space-time disclination can be described in term of a Dirac spinor. The size of the disclination, which is proved to be the difference of two sets of su(2)% -like monopoles expressed by two mixed spinors, is quantized topologically in terms of topological invariants$-$winding number. The projection of space-time disclination density along an antisymmetric tensor field is characterized by Brouwer degree and Hopf index. 
  We derive the equation of motion for the relativistic compact binaries in the post-Newtonian approximation taking explicitly their strong internal gravity into account. For this purpose we adopt the method of the point particle limit where the equation of motion is expressed in terms of the surface integrals. We examine carefully the behavior of the surface integrals in the derivation. As a result, we obtain the Einstein-Infeld-Hoffman equation of motion at the first post-Newtonian (1PN) order, and a part of the 2PN order which depends on the quadrupole moments and the spins of component stars. Hence, it is found that the equation of motion in the post-Newtonian approximation is valid for the compact binaries by a suitable definition of the mass, spin and quadrupole moment. 
  The role of spin-torsion coupling to gravity is analyzed in the context of a model of chaotic inflation. The system of equations constructed from the Einstein-Cartan and inflaton field equations are studied and it is shown that spin-torsion interactions are effective only at the very first e-folds of inflation, becoming quickly negligible and, therefore, not affecting the standard inflationary scenario nor the density perturbations spectrum predictions. 
  According to the socalled "quasi-metric" framework developed elsewhere, the cosmic expansion applies directly to gravitationally bound systems. This prediction has a number of observable consequences, none of which are in conflict with observation. In this paper we compare test particle motion in the nonstatic gravitational field outside a spherically symmetric source (as predicted by a quasi-metric theory of gravity) to test particle motion in the Schwarzschild geometry. It is found that if one incorrectly uses the Schwarzschild geometry (to the relevant accuracy) to represent the nonstatic quasi-metric model, the largest errors result from the mismodelling of null paths. One consequence of this is that using electromagnetic signals to track the motion of a non-relativistic particle results in the illusion that the particle is influenced by an anomalous force of size cH (where H is the Hubble parameter) directed towards the observer. This result naturally explains the apparently anomalous force acting on the Pioneer 10/11, Galileo and Ulysses spacecraft as inferred from radiometric data. 
  General relativity is unable to determine the topology of the Universe. We propose to apply quantum approach. Quantization of dynamics of a test particle is sensitive to the spacetime topology. Presented results for a particle in de Sitter spacetimes favor a finite universe. 
  Recent studies of Type Ia Supernovae with redshifts up to about $z~\laq~1$ reveal evidence for a cosmic acceleration in the expansion of the Universe. The most straightforward explanation to account for this acceleration is a cosmological constant dominating the recent history of our Universe; however, a more interesting suggestion is to consider an evolving vacuum energy. Several proposals have been put forward along these lines, most of them in the context of General Relativity. In this work we analyse the conditions under which the dynamics of a self-interacting Brans-Dicke field can account for this accelerated expansion of the Universe. We show that accelerated expanding solutions can be achieved with a quadratic self-coupling of the Brans-Dicke field and a negative coupling constant $\omega$. 
  The completion of a network of advanced laser-interferometric gravitational-wave observatories around 2001 will make possible the study of the inspiral and coalescence of binary systems of compact objects (neutron stars and black holes), using gravitational radiation. To extract useful information from the waves, such as the masses and spins of the bodies, theoretical general relativistic gravitational waveform templates of extremely high accuracy will be needed for filtering the data, probably as accurate as $O[(v/c)^6]$ beyond the predictions of the quadrupole formula. We summarize a method, called DIRE, for Direct Integration of the Relaxed Einstein Equations, which extends and improves an earlier framework due to Epstein and Wagoner, in which Einstein's equations are recast as a flat spacetime wave equation with source composed of matter confined to compact regions and gravitational non-linearities extending to infinity. The new method is free of divergences or undefined integrals, correctly predicts all gravitational wave ``tail'' effects caused by backscatter of the outgoing radiation off the background curved spacetime, and yields radiation that propagates asymptotically along true null cones of the curved spacetime. The method also yields equations of motion through $O[(v/c)^4]$, radiation-reaction terms at $O[(v/c)^5]$ and $O[(v/c)^7]$, and gravitational waveforms and energy flux through $O[(v/c)^4]$, in agreement with other approaches. We report on progress in evaluating the $O[(v/c)^6]$ contributions. 
  5D theory is an alternative model for understanding gravitational and electromagnetic interactions together. In this work we used the correspondence between 5D Einstein field equations with cosmological constant and the 4D Einstein equations with sources. We started with the principal fiber bundle P(M/F,U(1)) metric and studied the case when the gauge potential Aa corresponds to the magnetic field: Aa =(0, 0, A3, 0). We identified the base-space with the space-time of two models of universe: the Robertson-Walker - like model and the Schwarzschild - like model. With this ansatz, we followed an algorithm that permits to express the magnetic field in terms of the gauge potential Aa and the scalar field F. This algorithm seems to work very well if the scalar field is time dependent. When we analyzed its dependence on other coordinates, new terms it comes out on the 5D field equations. We obtained the translation between the effective 4D electromagnetic potential and the 5D gauge and scalar fields. 
  A gauge and coordinate invariant perturbation theory for self-gravitating non-Abelian gauge fields is developed and used to analyze local uniqueness and linear stability properties of non-Abelian equilibrium configurations. It is shown that all admissible stationary odd-parity excitations of the static and spherically symmetric Einstein-Yang-Mills soliton and black hole solutions have total angular momentum number $\ell = 1$, and are characterized by non-vanishing asymptotic flux integrals. Local uniqueness results with respect to non-Abelian perturbations are also established for the Schwarzschild and the Reissner-Nordstr\"om solutions, which, in addition, are shown to be linearly stable under dynamical Einstein-Yang-Mills perturbations. Finally, unstable modes with $\ell = 1$ are also excluded for the static and spherically symmetric non-Abelian solitons and black holes. 
  Worldline quantum inequalities provide lower bounds on weighted averages of the renormalised energy density of a quantum field along the worldline of an observer. In the context of real, linear scalar field theory on an arbitrary globally hyperbolic spacetime, we establish a worldline quantum inequality on the normal ordered energy density, valid for arbitrary smooth timelike trajectories of the observer, arbitrary smooth compactly supported weight functions and arbitrary Hadamard quantum states. Normal ordering is performed relative to an arbitrary choice of Hadamard reference state. The inequality obtained generalises a previous result derived for static trajectories in a static spacetime. The underlying argument is straightforward and is made rigorous using the techniques of microlocal analysis. In particular, an important role is played by the characterisation of Hadamard states in terms of the microlocal spectral condition. We also give a compact form of our result for stationary trajectories in a stationary spacetime. 
  In one-loop string effective action, we study a generality of non-singular cosmological solutions found in the isotropic and homogeneous case. We discuss Bianchi I and IX type spacetimes. We find that nonsingular solutions still exist in Bianchi I model around nonsingular flat Friedmann solutions. On the other hand, we cannot find any nonsingular solutions in Bianchi IX model. Non-existence of nonsingular Bianchi IX universe may be consistent with the analysis by Kawai, Sakagami and Soda, i.e. the tensor mode perturbations against nonsingular flat Friedmann universe are unstable, because Bianchi IX model can be regarded as a closed Friedmann universe with a single gravitational wave. So based on these facts, we may conclude the nonsingular universe is found in isotropic case is generally unstable, a singularity avoidance may not work in the present model. 
  We first investigate the form the general relativity theory would have taken had the gravitational mass and the inertial mass of material objects been different. We then extend this analysis to electromagnetism and postulate an equivalence principle for the electromagnetic field. We argue that to each particle with a different electric charge-to-mass ratio in a gravitational and electromagnetic field there corresponds a spacetime manifold whose metric tensor g_{\mu\nu} describes the dynamical actions of gravitation and electromagnetism. 
  If our visible universe is considered a trapped shell in a five-dimensional hyper-universe, all matter in it may be connected by superluminal signals traveling through the fifth dimension. Events in the shell are still causal, however, the propagation of signals proceeds at different velocities depending on the fifth coordinate. 
  A spinning test particle around a Schwarzschild black hole shows a chaotic behavior, if its spin is larger than a critical value. We discuss whether or not some peculiar signature of chaos appears in the gravitational waves emitted from such a system. Calculating the emitted gravitational waves by use of the quadrupole formula, we find that the energy emission rate of gravitational waves for a chaotic orbit is about 10 times larger than that for a circular orbit, but the same enhancement is also obtained by a regular "elliptic" orbit. A chaotic motion is not always enhance the energy emission rate maximally. As for the energy spectra of the gravitational waves, we find some characteristic feature for a chaotic orbit. It may tell us how to find out a chaotic behavior of the system. Such a peculiar behavior, if it will be found, may also provide us some additional informations to determine parameters of a system such as a spin. 
  Gravitons produced from quantum vacuum fluctuations during an inflationary stage in the early Universe have zero entropy as far as they reflect the time evolution (squeezing) of a pure state, their large occupation number notwithstanding. A non-zero entropy of the gravitons (classical gravitational waves (GW) after decoherence) can be obtained through coarse graining. The latter has to be physically justified {\it and} should not contradict observational constraints. We propose two ways of coarse graining for which the fixed temporal phase of each Fourier mode of the GW background still remains observable: one based on quantum entanglement, and another one following from the presence of a secondary GW background. The proposals are shown to be mutually consistent. They lead to the result that the entropy of the primordial GW background is significantly smaller than it was thought earlier. The difference can be ascribed to the information about the regular (inflationary) initial state of the Universe which is stored in this background and which reveals itself, in particular, in the appearance of primordial peaks (acoustic peaks in the case of scalar perturbations) in the multipole spectra of the CMB temperature anisotropy and polarization. 
  We study the response and cross sections for the absorption of GW energy in a Jordan-Brans-Dicke theory by a resonant mass detector shaped as a hollow sphere. 
  We investigate the late-time evolution of the Yang-Mills field in the self-gravitating backgrounds: Schwarzschild and Reissner-Nordstr\"om spacetimes. The late-time power-law tails develop in the three asymptotic regions: the future timelike infinity, the future null infinity and the black hole horizon. In these two backgrounds, however, the late-time evolution has quantitative and qualitative differences. In the Schwarzschild black hole background, the late-time tails of the Yang-Mills field are the same as those of the neutral massless scalar field with multipole moment l=1. The late-time evolution is dominated by the spacetime curvature. When the background is the Reissner-Nordstr\"om black hole, the late-time tails have not only a smaller power-law exponent, but also an oscillatory factor. The late-time evolution is dominated by the self-interacting term of the Yang-Mills field. The cause responsible for the differences is revealed. 
  Mechanics of non-rotating black holes was recently generalized by replacing the static event horizons used in standard treatments with `isolated horizons.' This framework is extended to incorporate dilaton couplings. Since there can be gravitational and matter radiation outside isolated horizons, now the fundamental parameters of the horizon, used in mechanics, must be defined using only the local structure of the horizon, without reference to infinity. This task is accomplished and the zeroth and first laws are established. To complement the previous work, the entire discussion is formulated tensorially, without any reference to spinors. 
  We consider hairy black hole solutions of Einstein-Yang-Mills-Dilaton theory, coupled to a Gauss-Bonnet curvature term, and we study their stability under small, spacetime-dependent perturbations. We demonstrate that the stringy corrections do not remove the sphaleronic instabilities of the coloured black holes with the number of unstable modes being equal to the number of nodes of the background gauge function. In the gravitational sector, and in the limit of an infinitely large horizon, the coloured black holes are also found to be unstable. Similar behaviour is exhibited by the magnetically charged black holes while the bulk of the neutral black holes are proven to be stable under small, gauge-dependent perturbations. Finally, the electrically charged black holes are found to be characterized only by the existence of a gravitational sector of perturbations. As in the case of neutral black holes, we demonstrate that for the bulk of electrically charged black holes no unstable modes arise in this sector. 
  The Wheeler-DeWitt equation for empty FRW minisuperspace universes of Hartle-Hawking factor ordering parameter Q=0 is mapped onto the dynamics of a unit mass classical oscillator. The latter is studied by the classical Ermakov invariant method. Angle quantities are presented in the same context 
  In order to study the "problem of time", Rovelli proposed a model of a two harmonic oscillator system where one of the oscillators can be thought of as a 'clock' for the other oscillator. In this paper we examine a model where the Hamiltonian is a difference between two harmonic oscillators, and we consider one of them which has the minus sign as a 'clock'. Klauder's projection operator approach to generalized coherent states is used to define physical states and operators. The resolution of unity is derived in terms of a gauge invariant coordinate. We investigate the 'quantum clock' and show that the evolution described by it is identical to the classical motion when the energy becomes large. 
  An exact twisting type N vacuum solution is found. It has regular gauge and curvature invariants and decays to flat spacetime for big retarded times. 
  I study a stochastic approach for warm inflation considering back - reaction of the metric with the fluctuations of matter field. This formalism takes into account the local inhomogeneities fo the spacetime in a globally flat Friedmann - Robertson - Walker metric. The stochastic equations for the fluctuations of the matter field and the metric are obtained. Finally, the dynamics for the amplitude of these fluctuations in a power - law expansion for the universe are examined. 
  Dynamical systems theory is especially well-suited for determining the possible asymptotic states (at both early and late times) of cosmological models, particularly when the governing equations are a finite system of autonomous ordinary differential equations. We begin with a brief review of dynamical systems theory. We then discuss cosmological models as dynamical systems and point out the important role of self-similar models. We review the asymptotic properties of spatially homogeneous perfect fluid models in general relativity. We then discuss some results concerning scalar field models with an exponential potential (both with and without barotropic matter). Finally, we discuss some isotropic cosmological models derived from the string effective action. 
  The tired-light cosmology is considered in the framework of Kaluza-Klein theory in 5D. The solution of the five-dimensional semi-classical Einstein equations with nonzero five-dimensional energy-momentum tensor gives density of matter in the Universe well conformed to the observations. Variation of the light velocity and change of the rest energy and mass are interrelated. Variation of the Planck constant and electron charge is determined from formula for hydrogen spectral frequencies and observations of the fine-structure. Physical constants variation presents as explanation of anomalous change of the length of a received wave detected during radiometric analysis of Pioneer 10/11 spacecraft data. Contemporary measurements accuracy of the Microwave Cosmic Background doesn't allow to determine tendency of its parameters change permitting choice between stationary and expanding Universe's model. 
  We carefully investigate the gravitational equations of the brane world, in which all the matter forces except gravity are confined on the 3-brane in a 5-dimensional spacetime with $Z_2$ symmetry.   We derive the effective gravitational equations on the brane, which reduce to the conventional Einstein equations in the low energy limit. {}From our general argument we conclude that the first Randall & Sundrum-type theory (RS1) [hep-ph/9905221] predicts that the brane with the negative tension is an anti-gravity world and hence should be excluded from the physical point of view. Their second-type theory (RS2) [hep-th/9906064] where the brane has the positive tension provides the correct signature of gravity. In this latter case, if the bulk spacetime is exactly anti-de Sitter, generically the matter on the brane is required to be spatially homogeneous because of the Bianchi identities. By allowing deviations from anti-de Sitter in the bulk, the situation will be relaxed and the Bianchi identities give just the relation between the Weyl tensor and the energy momentum tensor. In the present brane world scenario, the effective Einstein equations cease to be valid during an era when the cosmological constant on the brane is not well-defined, such as in the case of the matter dominated by the potential energy of the scalar field. 
  We relate Bondi systems near space-like infinity to another type of gauge conditions. While the former are based on null infinity, the latter are defined in terms of Einstein propagation, the conformal structure, and data on some Cauchy hypersurface. For a certain class of time symmetric space-times we study an expansion which allows us to determine the behavior of various fields arising in Bondi systems in the region of space-time where null infinity touches space-like infinity. The coefficients of these expansions can be read off from the initial data. We obtain in particular expressions for the constants discovered by Newman and Penrose (NP-constants) in terms of the initial data. For this purpose we calculate a certain expansion up to 3rd order. 
  Magnetohydrodynamic (MHD) waves are analysed in the early Universe, in the inflationary era, assuming the Universe to be filled with a nonviscous fluid of the Zel'dovich type ($p=\rho$) in a metric of the de Sitter form. A spatially uniform, time dependent, magnetic field ${\bf B_0}$ is assumed to be present. The Einstein equations are first solved to give the time dependence of the scale factor, assuming that the matter density, but not the magnetic field, contribute as source terms. The various modes are thereafter analysed; they turn out to be essentially of the same kind as those encountered in conventional nongravitational MHD, although the longitudinal magnetosonic wave is not interpretable as a physical energy-transporting wave as the group velocity becomes superluminal. We determine the phase speed of the various modes; they turn out to be scale factor independent. The Alfv\'{e}n velocity of the transverse magnetohydrodynamic wave becomes extremely small in the inflationary era, showing that the wave is in practice 'frozen in'. 
  This series of lectures gives a simple and self-contained introduction to the non-perturbative and background independent loop approach of canonical quantum gravity. The Hilbert space of kinematical quantum states is constructed and a complete basis of spin network states is introduced. An application of the formalism is provided by the spectral analysis of the area operator, which is the quantum analogue of the classical area function. This leads to one of the key results of loop quantum gravity: the derivation of the discreteness of the geometry and the computation of the quanta of area. Finally, an outlock on a possible covariant formulation of the theory is given leading to a "sum over histories" approach, denoted as spin foam model. Throughout the whole lecture great significance is attached to conceptual and interpretational issues. In particular, special emphasis is given to the role played by the diffeomorphism group and the notion of observability in general relativity. 
  We develop some classical descriptions for processes in the Schwarzschild string atmosphere. These processes suggest relationships between macroscopic and microscopic scales. The classical descriptions developed in this essay highlight the fundamental quantum nature of the Schwarzschild atmospheric processes. 
  A possible solution for the problem of non-existence of universal time is given by utilizing Goedel's incompleteness theorem. 
  Recently, there has been a revival of interest in the Lanczos potential of the Weyl conformal tensor. Previous work by Novello and Neto has been done with the linearized Lanczos potential as a model of a spin-2 field, which depends on a massless limit of the field. In this paper, we look at an action based on a massless potential, and show that it is classically equivalent to the linearized regime of general relativity, without reference to a massless limit. 
  A scalar model of gravity is considered. We propose Lorentz invariant field equation $\square f = k\eta_{ab}f_{,a}f_{,b}$. The aim of this model is to get, approximately, Newton's law of gravity. It is shown that $f=-\frac 1k\ln(1-k\frac mr)$ is the unique spherical symmetric static solution of the field equation. $f$ is taken to be the field of a particle at the origin, having the mass $m$. The field of a particle moving with a constant velocity is taken to be the appropriate Lorentz transformation of $f$. The field $F$ of $N$ particles moving on trajectories ${\psi_j(t)}$ is taken to be, to first order, the superposition of the fields of the particles, where the instantaneous Lorentz transformation of the fields pertaining to the $j$-th particle is ${\dot\psi_j(t)}$. When this field is inserted to the field equation the outcome is singular at $({\psi_j(t)},t)$. The singular terms of the l.h.s. and of the r.h.s. are both $O(R^{-2})$. The only way to reduce the singularity in the field equation is by postulating Newton's law of force. It is hoped that this model will be generalized to system of equations that are covariant under general diffeomorphism. 
  We discuss the action of the configuration operators of loop quantum gravity. In particular, we derive the generalised eigenbasis for the Wilson loop operator and show that the transformation between this basis and the spin-network basis is given by an expansion in terms of Chebyshev polynomials. These results are used to construct states which approximate connections on the background 3-manifold in an analogous way that the weave states reproduce area and volumes of a given 3-metric. This should be necessary for the construction of genuine semi-classical states that are peaked both in the configuration and momentum variables. 
  The energy at null infinity is presented with the help of a simple example of a massless scalar field in Minkowski spacetime. It is also discussed for Einstein gravity. In particular, various aspects of the loss of the energy in the radiating regime are shown. 
  The Maxwell field equations relative to a uniformly accelerated frame, and the variational principle from which they are obtained, are formulated in terms of the technique of geometrical gauge invariant potentials. They refer to the transverse magnetic (TM) and the transeverse electric (TE) modes. This gauge invariant "2+2" decomposition is used to see how the Coulomb field of a charge, static in an accelerated frame, has properties that suggest features of electromagnetism which are different from those in an inertial frame. In particular, (1) an illustrative calculation shows that the Larmor radiation reaction equals the electrostatic attraction between the accelerated charge and the charge induced on the surface whose history is the event horizon, and (2) a spectral decomposition of the Coulomb potential in the accelerated frame suggests the possibility that the distortive effects of this charge on the Rindler vacuum are akin to those of a charge on a crystal lattice. 
  Conformastationary metrics have been derived by Perjes and by Israel and Wilson as source-free solutions of the Einstein-Maxwell equations. By analogy with the conformastatic metrics which have charged dust sources it was assumed that conformastationary metrics would be the external metrics of charged dust in steady motion. However for axially symmetric conformastationary metrics we show that, as well as moving dust, hoop tensions are always necessary to balance the centrifugal forces induced by the motion. Exact examples of conformastationary metrics with disk sources are worked out in full. Generalisations to non-axially symmetric conformastationary metrics are indicated. 
  Without pretending to any rigour, we find a general expression of the electrostatic self-energy in static black holes with spherical symmetry. We determine the entropy bound of a charged object by assuming the existence of thermodynamics for these black holes. By combining these two results, we show that the entropy bound does not depend on the considered black hole. 
  A handful of recent papers has been devoted to proposals of experiments capable of testing some candidate quantum-gravity phenomena. These lecture notes emphasize those aspects that are most relevant to the questions that come to mind when one is exposed for the first time to these research developments: How come theory and experiments are finally meeting in spite of all the gloomy forecasts that pervade traditional reviews? Is this a case of theorists having put forward more and more speculative ideas until a point was reached at which conventional experiments could rule out the proposed phenomena? Or has there been such a remarkable improvement in experimental techniques and ideas that we are now capable of testing plausible candidate quantum-gravity phenomena? These questions are analysed rather carefully for the recent proposals of interferometry-based tests and tests using observations of gamma rays of astrophysical origin. I also briefly discuss other proposed experiments (including tests of quantum-gravity-induced decoherence using the neutral-kaon system and accelerator tests of models with large extra dimensions). The emerging picture suggests that we are finally starting the exploration of a large class of plausible quantum-gravity effects. However, our chances to obtain positive (discovery) experimental results depend crucially on the magnitude of these effects. In most cases the level of sensitivity that the relevant experiments should achieve within a few years corresponds to effects suppressed only linearly by the Planck length. 
  We study the motion of light in the gravitational field of two Schwarzschild black holes, making the approximation that they are far apart, so that the motion of light rays in the neighborhood of one black hole can be considered to be the result of the action of each black hole separately. Using this approximation, the dynamics is reduced to a 2-dimensional map, which we study both numerically and analytically. The map is found to be chaotic, with a fractal basin boundary separating the possible outcomes of the orbits (escape or falling into one of the black holes). In the limit of large separation distances, the basin boundary becomes a self-similar Cantor set, and we find that the box-counting dimension decays slowly with the separation distance, following a logarithmic decay law. 
  A major focus of much current research in gravitation theory is on understanding how radiation reaction drives the evolution of a binary system, particularly in the extreme mass ratio limit. Such research is of direct relevance to gravitational-wave sources for space-based detectors (such as LISA). We present here a study of the radiative evolution of circular (i.e., constant Boyer-Lindquist coordinate radius), non-equatorial Kerr black hole orbits. Recent theorems have shown that, at least in an adiabatic evolution, such orbits evolve from one circular configuration into another, changing only their radius and inclination angle. This constrains the system's evolution in such a way that the change in its Carter constant can be deduced from knowledge of gravitational wave fluxes propagating to infinity and down the black hole's horizon. Thus, in this particular case, a local radiation reaction force is not needed. In accordance with post-Newtonian weak-field predictions, we find that inclined orbits radiatively evolve to larger inclination angles (although the post-Newtonian prediction overestimates the rate of this evolution in the strong field by a factor $\lesssim 3$). We also find that the gravitational waveforms emitted by these orbits are rather complicated, particularly when the hole is rapidly spinning, as the radiation is influenced by many harmonics of the orbital frequencies. 
  By studying multidimensional Kaluza-Klein theories, or gravity plus U(1) or SU(2) gauge fields it is shown that these theories possess similar flux tube solutions. The gauge field which fills the tube geometry of these solutions leads to a comparision with the flux tube structures in QCD. These solutions also carry a ``magnetic'' charge, Q, which for the SU(2) Einstein-Yang-Mills (EYM) system exhibits a dual relationship with the Yang-Mills gauge coupling, g, ($Q=1/g$). As $Q \to 0$ or $Q \to \infty$, $g \to \infty$ or $g \to 0$ respectively. Thus within this classical EYM field theory we find solutions which have features - flux tubes, magnetic charges, large value of the gauge coupling - that are similar to the key ingredients of confinement in QCD. 
  The Hamilton-Jacobi equation for the string cosmology is solved using the gradient expansion method. The zeroth order solution is taken to be the standard pre-big bang model and the second order solution is found for the dilaton and the three-metric. It indicates that corrections generated by inhomogeneities of the seed metric are suppressed near the singularity and are growing towards the asymptotic past, but corrections generated by the dilaton inhomogeneities are growing near the singularity and are suppressed in the past. Possible influences of initial metric inhomogeneities on the pre-big bang superinflation are discussed. 
  A profound relationship between the ENU (Expansive Nondecelerative Universe) and dimensionless constants of the fundamental physical interactions is presented. The contribution corrects the Dirac presumption on a time decrease of the gravitational constant G and using simple relations it precises the values of the vector bosons x and y. 
  If gravitation and electromagnetism are both described in terms of a symmetric metric tensor, then the deflection of an electron beam by a charged sphere should be different from its deflection according to the Reissner-Nordstr\"om solution of General Relativity. If such a unified description is true, the equivalence principle for the electric field implies that the photon has a nonzero effective electric charge-to-mass ratio and should be redshifted in an electric field and be deflected in a magnetic field. Experiments to test these predictions are proposed. 
  I present a new method to generate rotating solutions of the Einstein-Maxwell equations from static solutions, give several examples of its application, and discuss its general properties. 
  The global Minkowski Bessel (M-B) modes, whose explicit form allows the identification and description of the condensed vacuum state resulting from the operation of a pair of accelerated refrigerators, are introduced. They span the representation space of the unitary representation of the Poincare group on 2-D Lorentz space-time. Their three essential properties are: (1) they are unitarily related to the familiar Minkowski plane waves; (2) they form a unitary representation of the translation group on two dimensional Minkowski spacetime. (3) they are eigenfunctions of Lorentz boosts around a given reference event. In addition the global Minkowski Mellin modes are introduced. They are the singular limit of the M-B modes. This limit corresponds to the zero transverse momentum solutions to the zero rest mass wave equation.    Also introduced are the four Rindler coordinate representatives of each global mode. Their normalization and density of states are exhibited in a (semi-infinite) accelerated frame with a finite bottom. In addition we exhibit the asymptotic limit as this bottom approaches the event horizon and thereby show how a mode sum approaches a mode integral as the frame becomes bottomless. 
  We consider the quantum radiation from a partially reflecting moving mirror for the massless scalar field in 1+1 Minkowski space. Partial reflectivity is achieved by localizing a delta-type potential at the mirror's position. The radiated flux is exactly obtained for arbitrary motions as an integral functional of the mirror's past trajectory. Partial reflectivity corrections to the perfect mirror result are discussed. 
  A new form of the Kerr solution is presented. The solution involves a time coordinate which represents the local proper time for free-falling observers on a set of simple trajectories. Many physical phenomena are particularly clear when related to this time coordinate. The chosen coordinates also ensure that the solution is well behaved at the horizon. The solution is well suited to the tetrad formalism and a convenient null tetrad is presented. The Dirac Hamiltonian in a Kerr background is also given and, for one choice of tetrad, it takes on a simple, Hermitian form. 
  The canonical evolution and symmetry generators are exhibited for a Klein-Gordon (K-G) system which has been partitioned by an accelerated coordinate frame into a pair of subsystems. This partitioning of the K-G system is conveyed to the canonical generators by the eigenfunction property of the Minkowski Bessel (M-B) modes. In terms of the M-B degrees of freedom, which are unitarily related to those of the Minkowski plane waves, a near complete diagonalization of these generators can be realized. 
  The arena normally used in black holes thermodynamics was recently generalized to incorporate a broad class of physically interesting situations. The key idea is to replace the notion of stationary event horizons by that of `isolated horizons.' Unlike event horizons, isolated horizons can be located in a space-time quasi-locally. Furthermore, they need not be Killing horizons. In particular, a space-time representing a black hole which is itself in equilibrium, but whose exterior contains radiation, admits an isolated horizon. In spite of this generality, the zeroth and first laws of black hole mechanics extend to isolated horizons. Furthermore, by carrying out a systematic, non-perturbative quantization, one can explore the quantum geometry of isolated horizons and account for their entropy from statistical mechanical considerations. After a general introduction to black hole thermodynamics as a whole, these recent developments are briefly summarized. 
  The notion of inflation (past or present) in standard cosmological models is shown to be a consequence of a sufficiently high second law entropy production from the internal heating of the universal expansion. The longitudinal viscous internal heating of matter requires neither ``inflaton'' fields nor ``quintessence'' fields which in theory may induce a cosmological term into the Einstein equations. The purely thermodynamic principles required to understand inflation within the context of the standard general relativity equations will be discussed in detail. 
  The framework of quantum symmetry reduction is applied to loop quantum gravity with respect to transitively acting symmetry groups. This allows to test loop quantum gravity in a large class of minisuperspaces and to investigate its features - e.g. the discrete volume spectrum - in certain cosmological regimes. Contrary to previous studies of quantum cosmology (minisuperspace quantizations) the symmetry reduction is carried out not at the classical level but on an auxiliary Hilbert space of the quantum theory before solving the constraints. Therefore, kinematical properties like volume quantization survive the symmetry reduction. In this first part the kinematical framework, i.e. implementation of the quantum symmetry reduction and quantization of Gauss and diffeomorphism constraints, is presented for Bianchi class A models as well as locally rotationally symmetric and spatially isotropic closed and flat models. 
  Volume operators measuring the total volume of space in a loop quantum theory of cosmological models are constructed. In the case of models with rotational symmetry an investigation of the Higgs constraint imposed on the reduced connection variables is necessary, a complete solution of which is given for isotropic models; in this case the volume spectrum can be calculated explicitly. It is observed that the stronger the symmetry conditions are the smaller is the volume spectrum, which can be interpreted as level splitting due to broken symmetries. Some implications for quantum cosmology are presented. 
  We suggest an explanation of the "Pioneer effect" based on the interaction of the spacecraft with a long-range scalar field, $\phi$. The scalar field under consideration is external to gravity, coupled to the ordinary matter and undergoes obedience to the weak equivalence principle. In the weak fields limit it result a long-range acceleration $a_{P}$, asymptotically constant within the region of the solar system hitherto crossed by the spacecraft. 
  The geometrical and quantum mechanical basis for Davies' and Unruh's acceleration temperature is traced to a type of quantum mechanical (``achronal'') spin. Its existence and definition are based on pairs of causally disjoint accelerated frames. For bosons the expected spin vector of monochromatic particles is given by the ``Planckian power'' and the ``r.m.s. thermal fluctuation'' spectra. Under spacetime translation the spin direction precesses around that ``Planckian'' vector. By exhibiting the conserved achronal spin four-current, we extend the identification of achronal spin from single quanta to multiparticle systems. Total achronal spin conservation is also shown to hold, even in the presence of quadratic interactions. 
  We exhibit a purely quantum mechanical carrier of the imprints of gravitation by identifying for a relativistic charge a property which (i) is independent of its mass and (ii) expresses the Poincare invariance of spacetime in the absence of gravitation.  This carrier is a Klein-Gordon-equation-determined vector field given by the ``Planckian power'' and the ``r.m.s. thermal fluctuation'' spectra. The imprints themselves are deviations away from this vector field. 
  We discuss several aspects of cosmic censorship hypothesis. There is evidence both in favor and against the hypothesis. On one hand one can prove that cosmic censorship holds in several special cases and on the other hand there is a number of special solutions of Einstein equations in which it is violated. One way to resolve cosmic censorship problem is to test it observationally. We point out to several possibilities of such tests using present and future instruments. 
  The Riemann, Ricci and Einstein tensors for N-dimensional spherically symmetric spacetimes in various systems of coordinates are studied, and the general metric for conformally flat spacetimes is given. As an application, all the Friedmann-Robertson-Walker-like solutions for a perfect fluid with an equation of state $p = k \rho$ are found. Then, these solutions are used to model the gravitational collapse of a compact ball, by first cutting them along a timelike hypersurface and then joining them with asymptotically flat Vaidya solutions. It is found that when the collapse has continuous self-similarity, the formation of black holes always starts with zero mass, and when the collapse has no such symmetry, the formation of black holes always starts with a finite non-zero mass. 
  We start from the pure Einstein-Hilbert action in Metric Affine Gravity, with the orthonormal metric. We get an effective Levi-Civita Dilaton gravity theory in which the Dilaton field is related to the scaling of the gravitational coupling. When the Weyl symmetry is broken the resulting Einstein-Hilbert term is equivalent to the Levi-Civita one, using the projective invariance of the model, the non-metricity and torsion may be removed, so that we get a theory perfectly equivalent to general relativity. This may explain why low energy gravity is described by a Riemannian connection. 
  Within the context of quantum field theory in curved spacetimes, Hacyan and Sarmiento defined the vacuum stress-energy tensor with respect to the accelerated observer. They calculated it for uniform acceleration and circular motion, and derived that the rotating observer perceives a flux. Mane related the flux to synchrotron radiation. In order to investigate the relation between the vacuum stress and bremsstrahlung, we estimate the stress-energy tensor of the electromagnetic field generated by a point charge, at the position of the charge. We use the retarded field as a self-field of the point charge. Therefore the tensor diverges if we evaluate it as it is. Hence we remove the divergent contributions by using the expansion of the tensor in powers of the distance from the point charge. Finally, we take an average for the angular dependence of the expansion. We calculate it for the case of uniform acceleration and circular motion, and it is found that the order of the vacuum stress multiplied by $\pi\alpha$ ($\alpha=e^2/\hbar c$ is the fine structure constant) is equal to that of the self-stress. In the Appendix, we give another trial approach with a similar result. 
  A gravity theory is developed with the metric ${\hat g}_{\mu\nu}= {g}_{\mu\nu}+B\partial_\mu\phi\partial_\nu\phi$. In the present universe the additional contribution from the scalar field in the metric ${\hat g}_{\mu\nu}$ can generate an acceleration in the expansion of the universe, without negative pressure and with a zero cosmological constant. In this theory, gravitational waves will propagate at a different speed from non-gravitational waves. It is suggested that gravitational wave experiments could test this observational signature. 
  Solutions for rotating boson stars in (2+1) dimensional gravity with a negative cosmological constant are obtained numerically. The mass, particle number, and radius of the (2+1) dimensional rotating boson star are shown. Consequently we find the region where the stable boson star can exist. 
  The four Rindler quadrants of a pair of oppositely accelerated frames are identified as a (Lorentzian) Mach-Zehnder interferometer. The Rindler frequency dependence of the interference process is expressed by means of a (Lorentzian) differential cross section. The Rindler frequencies of the waves in the two acccelerated frames can be measured directly by means of a simple inertially moving detector. 
  In this paper we develop a formalism to describe a superfluid in a gravitational background. This formalism is based on a covariant generalization of the field description for a superconductor in terms of a U(1) spontaneous symmetry breaking. We study the stability of the solutions for a vortexless fluid and the force acting on vortices in the fluid, which is a generalization of the well-known flat space-time Magnus force. To clarify the development we include the explicit discussion of two particular cases, one of them of astrophysical interest. 
  Proposed space-based gravitational wave antennas involve satellites arrayed either in an equilateral triangle around the earth in the ecliptic plane (the ecliptic-plane option) or in an equilateral triangle orbiting the sun in such a way that the plane of the triangle is tilted at 60 degrees relative to the ecliptic (the precessing-plane option). In this paper, we explore the angular resolution of these two classes of detectors for two kinds of sources (essentially monochromatic compact binaries and coalescing massive-black-hole binaries) using time-domain expressions for the gravitational waveform that are accurate to 4/2 PN order. Our results display an interesting effect not previously reported in the literature, and underline the importance of including the higher-order PN terms in the waveform when predicting the angular resolution of ecliptic-plane detector arrays. 
  An overview of some tools and techniques being developed for data conditioning (regression of instrumental and environmental artifacts from the data channel), detector design evaluation (modeling the science ``reach'' of alternative detector designs and configurations), noise simulations for mock data challenges and analysis system validation, and analyses for the detection of gravitational radiation from gamma-ray burst sources. 
  In this work, I review some aspects concerning the evolution of quantum low-energy fields in a foamlike spacetime, with involved topology at the Planck scale but with a smooth metric structure at large length scales, as follows. Quantum gravitational fluctuations may induce a minimum length thus introducing an additional source of uncertainty in physics. The existence of this resolution limit casts doubts on the metric structure of spacetime at the Planck scale and opens a doorway to nontrivial topologies, which may dominate Planck scale physics. This foamlike structure of spacetime may show up in low-energy physics through loss of quantum coherence and mode-dependent energy shifts, for instance, which might be observable. Spacetime foam introduces nonlocal interactions that can be modeled by a quantum bath, and low-energy fields evolve according to a master equation that displays such effects. Similar laws are also obtained for quantum mechanical systems evolving according to good real clocks, although the underlying Hamiltonian structure in this case establishes serious differences among both scenarios. Contents.--- Quantum fluctuations of the gravitational field; Spacetime foam; Loss of quantum coherence; Quantum bath; Low-energy effective evolution; Real clocks; Conclusions. 
  We review recent developments in the method of algebro-geometric integration of integrable systems related to deformations of algebraic curves. In particular, we discuss the theta-functional solutions of Schlesinger system, Ernst equation and self-dual SU(2)-invariant Einstein equations. 
  We find two-dimensional free-field variables for D-dimensional general relativity on spacetimes with D-2 commuting spacelike Killing vector fields and non-compact spatial sections for D>4. We show that there is a canonical transformation which maps the corresponding two-dimensional dilaton gravity theory into a two-dimensional diffeomorphism invariant theory of the free-field variables. We also show that the spacetime metric components can be expressed as asymptotic series in negative powers of the dilaton, with coefficients which can be determined in terms of the free fields. 
  We describe an approach to the quantisation of (2+1)-dimensional gravity with topology R x T^2 and negative cosmological constant, which uses two quantum holonomy matrices satisfying a q-commutation relation. Solutions of diagonal and upper-triangular form are constructed, which in the latter case exhibit additional, non-trivial internal relations for each holonomy matrix. Representations are constructed and a group of transformations - a quasi-modular group - which preserves this structure, is presented. 
  This paper reviews the basic features of the theory of curvature perturbations in Kerr spacetime, which is customarily written in terms of gauge invariant components of the Weyl tensor which satisfy a perturbation equation known as the Teukolsky equation. I will describe how to evolve generic perturbations about the Kerr metric and the separable form of the wave solutions that one obtains, and the relation of the Teukolsky function to the energy of gravitational waves emitted by the black hole. A discussion of a numerical scheme to evolve perturbations as a function of time and some preliminary results of our research project implementing it for matter sources falling into the black hole is included. 
  The spherically symmetric layer of matter is considered within the frameworks of general relativity. We perform generalization of the already known theory for the case of nonconstant surface entropy and finite temperature. We also propose the minisuperspace model to determine the behaviour of temperature field and perform the Wheeler-DeWitt quantization. 
  (Foreword by translator.) The aim of present translation is to clarify the historically important question who was the pioneer in obtaining of exact static solutions of Einstein equations minimally coupled with scalar field. Usually, people cite the works by Janis, Newman, Winicour (Phys. Rev. Lett. 20 (1968) 878) and others authors whereas it is clear that JNW rediscovered (in other coordinates) the Fisher's solution which was obtained 20 years before, in 1947. Regrettably, up to now I continue to meet many papers (even very fresh ones) whose authors evidently do not know about the Fisher's work, so I try to remove this gap by virtue of present translation and putting it into the LANL e-print archive. (Original Abstract.) It is considered the scalar mesostatic field of a point source with the regard for spacetime curvature caused by this field. For the field with $\mass = 0$ the exact solution of Einstein equations was obtained. It was demonstrated that at small distance from a source the gravitational effects are so large that they cause the significant changes in behavior of meson field. In particular, the total energy of static field diverges logarithmically. 
  We generalize the idea of Vassiliev invariants to the spin network context, with the aim of using these invariants as a kinematical arena for a canonical quantization of gravity. This paper presents a detailed construction of these invariants (both ambient and regular isotopic) requiring a significant elaboration based on the use of Chern-Simons perturbation theory which extends the work of Kauffman, Martin and Witten to four-valent networks. We show that this space of knot invariants has the crucial property -from the point of view of the quantization of gravity- of being loop differentiable in the sense of distributions. This allows the definition of diffeomorphism and Hamiltonian constraints. We show that the invariants are annihilated by the diffeomorphism constraint. In a companion paper we elaborate on the definition of a Hamiltonian constraint, discuss the constraint algebra, and show that the construction leads to a consistent theory of canonical quantum gravity. 
  In a companion paper we introduced a kinematical arena for the discussion of the constraints of canonical quantum gravity in the spin network representation based on Vassiliev invariants. In this paper we introduce the Hamiltonian constraint, extend the space of states to non-diffeomorphism invariant ``habitats'' and check that the off-shell quantum constraint commutator algebra reproduces the classical Poisson algebra of constraints of general relativity without anomalies. One can therefore consider the resulting set of constraints and space of states as a consistent theory of canonical quantum gravity. 
  We point out in this work that if our recently proposed unified description of gravitation and electromagnetism through a symmetric metric tensor is true, then building in the laboratory black holes for electrons with radii r_E\ge 0.5m in air and with much smaller radii in a vacuum should be possible. 
  GRLite and GRTensorJ are first and second generation graphical user interfaces to the computer algebra system GRTensorII. Current development centers on GRTensorJ, which provides fully customizable symbolic procedures that reduce many complex calculations to "elementary functions". Although still in development, GRTensorJ, which is now available (free of charge) over the internet, is sufficiently advanced to be of interest to researchers in general relativity and related fields. 
  The main properties of the Levi-Civita solutions with the cosmological constant are studied. In particular, it is found that some of the solutions need to be extended beyond certain hypersurfaces in order to have geodesically complete spacetimes. Some extensions are considered and found to give rise to black hole structure but with plane symmetry. All the spacetimes that are not geodesically complete are Petrov type D, while in general the spacetimes are Petrov type I. 
  Numerical relativity has traditionally been pursued via finite differencing. Here we explore pseudospectral collocation (PSC) as an alternative to finite differencing, focusing particularly on the solution of the Hamiltonian constraint (an elliptic partial differential equation) for a black hole spacetime with angular momentum and for a black hole spacetime superposed with gravitational radiation. In PSC, an approximate solution, generally expressed as a sum over a set of orthogonal basis functions (e.g., Chebyshev polynomials), is substituted into the exact system of equations and the residual minimized. For systems with analytic solutions the approximate solutions converge upon the exact solution exponentially as the number of basis functions is increased. Consequently, PSC has a high computational efficiency: for solutions of even modest accuracy we find that PSC is substantially more efficient, as measured by either execution time or memory required, than finite differencing; furthermore, these savings increase rapidly with increasing accuracy. The solution provided by PSC is an analytic function given everywhere; consequently, no interpolation operators need to be defined to determine the function values at intermediate points and no special arrangements need to be made to evaluate the solution or its derivatives on the boundaries. Since the practice of numerical relativity by finite differencing has been, and continues to be, hampered by both high computational resource demands and the difficulty of formulating acceptable finite difference alternatives to the analytic boundary conditions, PSC should be further pursued as an alternative way of formulating the computational problem of finding numerical solutions to the field equations of general relativity. 
  We present a critical review of the relativistic rotation transformation of Trocheris and Takeno. A new transformation is proposed which is free from the drawbacks of the former. Some applications are presented. 
  Rindler's acceleration-induced partitioning of spacetime leads to a nature-given interferometer. It accomodates quantum mechanical and wave mechanical processes in spacetime which in (Euclidean) optics correspond to wave processes in a ``Mach-Zehnder'' interferometer: amplitude splitting, reflection, and interference. These processes are described in terms of amplitudes which behave smoothly across the event horizons of all four Rindler sectors. In this context there arises quite naturally a complete set of orthonormal wave packet histories, one of whose key properties is their "explosivity index". In the limit of low index values the wave packets trace out fuzzy world lines. By contrast, in the asymptotic limit of high index values, there are no world lines, not even fuzzy ones. Instead, the wave packet histories are those of entities with non-trivial internal collapse and explosion dynamics. Their details are described by the wave processes in the above-mentioned Mach-Zehnder interferometer. Each one of them is a double slit interference process. These wave processes are applied to elucidate the amplification of waves in an accelerated inhomogeneous dielectric. Also discussed are the properties and relationships among the transition amplitudes of an accelerated finite-time detector. 
  We compare the fully nonlinear and perturbative evolution of nonrotating black holes with odd-parity distortions utilizing the perturbative results to interpret the nonlinear results. This introduction of the second polarization (odd-parity) mode of the system, and the systematic use of combined techniques brings us closer to the goal of studying more complicated systems like distorted, rotating black holes, such as those formed in the final inspiral stage of two black holes. The nonlinear evolutions are performed with the 3D parallel code for Numerical Relativity, {Cactus}, and an independent axisymmetric code, {Magor}. The linearized calculation is performed in two ways: (a) We treat the system as a metric perturbation on Schwarzschild, using the Regge-Wheeler equation to obtain the waveforms produced. (b) We treat the system as a curvature perturbation of a Kerr black hole (but here restricted to the case of vanishing rotation parameter a) and evolve it with the Teukolsky equation The comparisons of the waveforms obtained show an excellent agreement in all cases. 
  Different geometrical and topological properties have been shown for two kinds of extreme Reissner-Nordstr$\ddot{o}$m-anti-de Sitter black holes. The relationship between the geometrical properties and the intrinsic thermodynamical properties has been made explicit. 
  We exhibit a purely quantum mechanical carrier of the imprints of gravitation by identifying for a relativistic system a property which (i) is independent of its mass and (ii) expresses the Poincare invariance of spacetime in the absence of gravitation. This carrier consists of the phase and amplitude correlations of waves in oppositely accelerating frames. These correlations are expressed as a Klein-Gordon-equation-determined vector field whose components are the ``Planckian power'' and the ``r.m.s. thermal fluctuation'' spectra. The imprints themselves are deviations away from this vector field. 
  Brady, Creighton and Thorne have argued that, in numerical relativity simulations of the inspiral of binary black holes, if one uses lapse and shift functions satisfying the ``minimal strain equations'' (MSE), then the coordinates might be kept co-rotating, the metric components would then evolve on the very slow inspiral timescale, and the computational demands would thus be far smaller than for more conventional slicing choices. In this paper, we derive simple, testable criteria for the MSE to be strongly elliptic, thereby guaranteeing the existence and uniqueness of the solution to the Dirichlet boundary value problem. We show that these criteria are satisfied in a test-bed metric for inspiraling binaries, and we argue that they should be satisfied quite generally for inspiraling binaries. If the local existence and uniqueness that we have proved holds globally, then, for appropriate boundary values, the solution of the MSE exhibited by Brady et. al. (which tracks the inspiral and keeps the metric evolving slowly) will be the unique solution and thus should be reproduced by (sufficiently accurate and stable) numerical integrations. 
  The quantum cosmology of a higher-derivative derivative gravity theory arising from the heterotic string effective action is reviewed. A new type of Wheeler-DeWitt equation is obtained when the dilaton is coupled to the quadratic curvature terms. Techniques for solving the Wheeler-DeWitt equation with appropriate boundary conditions shall be described, and implications for semiclassical theories of inflationary cosmology will be outlined. 
  We outline a different method of describing scalar field particle production in a uniform electric field. In the standard approach, the (analytically continued) harmonic oscillator paradigm is important in describing particle production. However, there is another gauge in which the particle production process has striking similarities with the one used to describe Hawking radiation in black holes. The gauge we use to describe the electric field in is the lightcone gauge, so named because the mode functions for a scalar field are found to be singular on the lightcone. We use these modes in evaluating the effective Lagrangian using the proper time technique. The key feature of this analysis is that these modes can be explicitly "normalized" by using the criterion that they reduce to the usual flat space modes in the limit of the electric field tending to zero. We find that the proper time kernel is not the same as the analytically continued oscillator kernel though the effective Lagrangian is the standard result as it should be. We also consider an example of a confined electric field system using the lightcone gauge modes. Our analysis indicates that the Bogolubov coefficients, in taking the limit to the uniform electric field case, are multiplied by energy dependent boundary factors that have not been taken into account before. 
  Intrinsic unfication of quantum theory and general relativity based on the underlying quantum dynamics of fundamental field has been proposed. 
  First experimental results of a feasibility study of a gravitational wave detector based on two coupled superconducting cavities are presented. Basic physical principles underlying the detector behaviour and sensitivity limits are discussed. The detector layout is described in detail and its rf properties are showed. The limit sensitivity to small harmonic displacements at the detection frequency (around 1 MHz) is showed. The system performance as a potential g.w. detector is discussed and future developments are foreseen. 
  We first consider the Lagrangian formulation of general relativity for perturbations with respect to a background spacetime. We show that by combining Noether's method with Belinfante's "symmetrization'' procedure we obtain conserved vectors that are independent of any divergence added to the perturbed Hilbert Lagrangian. We also show that the corresponding perturbed energy- momentum tensor is symmetrical and divergenceless but only on backgrounds that are "Einstein spaces" in the sense of A.Z. Petrov. de Sitter or anti-de Sitter and Einstein "spacetimes" are Einstein spaces but in general Friedmann-Robertson -Walker spacetimes are not. Each conserved vector is a divergence of an anti- symmetric tensor, a "superpotential". We find superpotentials which are a generalization of Papapetrou's superpotential and are rigorously linear, even for large perturbations, in terms of the inverse metric density components and their first order derivatives. The superpotentials give correct globally conserved quantities at spatial infinity. They resemble Abbott and Deser's superpotential, but give correctly the Bondi-Sachs total four-momentum at null infinity. Next we calculate conserved vectors and superpotentials for perturbations of a Friedmann-Robertson-Walker background associated with its 15 conformal Killing vectors given in a convenient form. The integral of each conserved vector in a finite volume V at a given conformal time is equal to a surface integral on the boundary of V of the superpotential. For given boundary conditions each such integral is part of a flux whose total through a closed hypersurface is equal to zero. For given boundary conditions on V, the integral can be considered as an "integral constraint" on data in the volume... 
  The effective field, which plays the part of the vierbein in general relativity, can have topologically stable surfaces, vierbein domain walls, where the effective contravariant metric is degenerate. We consider vierbein walls separating domains with the flat space-time which are not causally connected at the classical level. Possibility of the quantum mechanical connection between the domains is discussed. 
  Seismic noise will be the dominant source of noise at low frequencies for ground based gravitational wave detectors, such as LIGO now under construction. Future interferometers installed at LIGO plan to use at least a double pendulum suspension for the test masses to help filter the seismic noise. We are constructing an apparatus to use as a test bed for double pendulum design. Some of the tests we plan to conduct include: dynamic ranges of actuators, and how to split control between the intermediate mass and lower test mass; measurements of seismic transfer functions; measurements of actuator and mechanical cross couplings; and measurements of the noise from sensors and actuators. All these properties will be studied as a function of mechanical design of the double pendulum. 
  I derive the stochastic equation for the perturbations of the metric for a gauge - invariant energy - momemtum - tensor (EMT) in stochastic inflation. A quantization for the field that describes the gauge - invariant perturbations for the metric is developed. In a power - law expansion for the universe the amplitude for these perturbations on a background metric could be very important in the infrared sector. 
  Stochastic description of inflationary spacetimes emulates the growth of vacuum fluctuations by an effective stochastic ``noise field'' which drives the dynamics of the volume-smoothed inflaton. We investigate statistical properties of this field and find its correlator to be a function of distance measured in units of the smoothing length. Our results apply for a wide class of smoothing window functions and are different from previous calculations by Starobinsky and others who used a sharp momentum cutoff. We also discuss the applicability of some approximate noise descriptions to simulations of stochastic inflation. 
  The Einstein-Klein-Gordon field equations are solved in a inhomogeneous shear-free universe containing a material fluid, a self-interacting scalar field, a variable cosmological term, and a heat flux. A quintessence-dominated scenario arises with a power-law accelerated expansion compatible with the currently observed homogeneous universe. 
  The full metric describing a charged, magnetized generalization of the Tomimatsu-Sato (TS) $\delta=2$ solution is presented in a concise explicit form. We use it to investigate some physical properties of the solution; in particular, we point out the existence of naked ring singularities in the hyperextreme TS metrics, the fact previously overlooked by the researchers, and we also demonstrate that the ring singularities can be eliminated by sufficiently strong magnetic fields in the subextreme case, while in the hyperextreme case the magnetic field can move singularities to the equatorial plane. 
  We survey some known facts and open questions concerning the global properties of 3+1 dimensional spacetimes containing a compact Cauchy surface. We consider spacetimes with an $\ell$-dimensional Lie algebra of space-like Killing fields. For each $\ell \leq 3$, we give some basic results and conjectures on global existence and cosmic censorship. 
  Combining incoming and outgoing characteristic formulations can provide numerical relativists with a natural implementation of Einstein's equations that better exploits the causal properties of the spacetime and gives access to both null infinity and the interior region simultaneously (assuming the foliation is free of caustics and crossovers). We discuss how this combination can be performed and illustrate its behavior in the Einstein-Klein-Gordon field in 1D. 
  The first decade of the new millenium should see the first direct detections of gravitational waves. This will be a milestone for fundamental physics and it will open the new observational science of gravitational wave astronomy. But gravitational waves already play an important role in the modeling of astrophysical systems. I review here the present state of gravitational radiation theory in relativity and astrophysics, and I then look at the development of detector sensitivity over the next decade, both on the ground (such as LIGO) and in space (LISA). I review the sources of gravitational waves that are likely to play an important role in observations by first- and second-generation interferometers, including the astrophysical information that will come from these observations. The review covers some 10 decades of gravitational wave frequency, from the high-frequency normal modes of neutron stars down to the lowest frequencies observable from space. The discussion of sources includes recent developments regarding binary black holes, spinning neutron stars, and the stochastic background. 
  The problem of unification of Gravitation and Electromagnetism in four dimensions; some new ideas involving mixtures of commuting and anti-commuting co-ordinates. Maxwell's equations are extracted in terms of the curvature of the anti-commuting component of space-time. The profound difference in the size of the coupling constants of the two forces is interpreted in terms of the degree of expansion of the two kinds of space-time with evolution of the universe. Uncertainty in the quantum realm is interpreted in terms of an unmeasurable component of anti-commuting space-time. 
  A lower limit for a neutral black hole size is obtained in the frames of the string gravity model with the second order curvature correction. It is shown that this effect remains when the third order curvature correction is also taken into account and argued that such restriction does exist in all perturbative orders of curvature expansions. 
  We study a new minimal scalar-tensor model of gravity with Brans-Dicke factor $\omega(\Phi)\equiv 0$ and cosmological factor $\Pi(\Phi)$. The constraints on $\Pi(\Phi)$ from known gravitational experiments are derived. We show that almost any time evolution of the scale factor in a homogeneous isotropic Universe can be obtained via properly chosen $\Pi(\Phi)$ and discuss the general properties of models of this type. 
  We study boson star configurations with generic, but not non-topological, self-interaction terms, i.e. we do not restrict ourselves just to consider the standard $\lambda |\psi|^4$ interaction but more general U(1)-symmetry-preserving profiles. We find that when compared with the usual potential, similar results for masses and number of particles appear. However, changes are of order of few percent of the star masses. We explore the stability properties of the configurations, that we analyze using catastrophe theory. We also study possible observational outputs: gravitational redshifts, rotation curves of accreted particles, and lensing phenomena, and compare with the usual case. 
  The chaotic behavior in FRW cosmology with a scalar field is studied for scalar field potentials less steep than quadratic. We describe a transition to much stronger chaos for appropriate parameters of such potentials. The range of parameters which allows this transition is specified. The influence of ordinary matter on the chaotic properties of this model is also discussed. 
  We outline a method for calculating the self force (the "radiation reaction force") acting on a scalar particle in a strong field orbit in a Schwarzschild spacetime. In this method, the contribution to the self force associated with each multipole mode of the particle's field is calculated separately, and the sum over modes is then evaluated, subject to a certain regularization procedure. We present some explicit results concerning the implementation of the calculation scheme for a static particle, and also for a uniform circular motion, on the Schwarzschild background. 
  Senovilla has recently defined an algebraic construction of a superenergy tensor T{A} from any arbitrary tensor A, by structuring it as an r-fold form. This superenergy tensor satisfies automatically the dominant superenergy property. We present a more compact definition using the r-direct product Clifford algebra r-Cl(p,q). This form for the superenergy tensors allows to obtain an easy proof of the dominant superenergy property valid for any dimension. 
  We study the self forces acting on static scalar and electric test charges in the spacetime of a Schwarzschild black hole. The analysis is based on a direct, local calculation of the self forces via mode decomposition, and on two independent regularization procedures: A spatially-extended particle model method, and on a mode-sum regularization prescription. In all cases we find excellent agreement with the known exact results. 
  Several new results regarding the quantum cosmology of the quadratic gravity theory derived from the heterotic string effective action are presented. After describing techniques for solving the Wheeler-De Witt equation with appropriate boundary conditions, it is shown that this quantum cosmological model may be compared with semiclassical theories of inflationary cosmology. In particular, it should be possible to compute corrections to the standard inflationary model perturbatively about a stable exponentially expanding classical background. 
  The VIRGO superattenuator (SA) is effective in depressing the seismic noise below the thermal noise level above 4 Hz. On the other hand, the residual mirror motion associated to the SA normal modes can saturate the dynamics of the interferometer locking system. This motion is reduced implementing a wideband (DC-5 Hz) multidimensional control (the so called inertial damping) which makes use of both accelerometers and position sensors and of a DSP system. Feedback forces are exerted by coil-magnet actuators on the top of the inverted pendulum. The inertial damping is successful in reducing the mirror motion within the requirements. The results are presented. 
  A monodromy transform approach, presented in this communication, provides a general base for solution of space-time symmetry reductions of Einstein equations in all known integrable cases, which include vacuum, electrovacuum, massless Weyl spinor field and stiff matter fluid, as well as some string theory induced gravity models. It was found a special finite set of functional parameters, defined as the monodromy data for the fundamental solution of associated spectral problem. Similarly to the scattering data in the inverse scattering transform, these monodromy data can be used for characterization of any local solution of the field equations. A "direct" and "inverse" problems of such monodromy transform admit unambiguous solutions. For the linear singular integral equation with a scalar (i.e. non-matrix) kernel, which solves the inverse problem of this monodromy transform, an equivalent regularization -- a Fredholm linear integral equation of the second kind is constrcuted in several convenient forms. For arbitrary choice of the monodromy data a simple iterative method leads to an effective construction of the solution in terms of homogeneously convergent functional series. 
  The first regular exact black hole solution in General Relativity is presented. The source is a nonlinear electrodynamic field satisfying the weak energy condition, which in the limit of weak field becomes the Maxwell field. The solution corresponds to a charged black hole with |q| \leq 2 s_c m \approx 0.6 m, having the metric, the curvature invariants, and the electric field regular everywhere. 
  We report on numerical results from a revised hydrodynamic simulation of binary neutron-star orbits near merger. We find that the correction recently identified by Flanagan significantly reduces but does not eliminate the neutron-star compression effect. Although results of the revised simulations show that the compression is reduced for a given total orbital angular momentum, the inner most stable circular orbit moves to closer separation distances. At these closer orbits significant compression and even collapse is still possible prior to merger for a sufficiently soft EOS. The reduced compression in the corrected simulation is consistent with other recent studies of rigid irrotational binaries in quasiequilibrium in which the compression effect is observed to be small. Another significant effect of this correction is that the derived binary orbital frequencies are now in closer agreement with post-Newtonian expectations. 
  Obukhov spin-driven inflation in General Relativity is extended to include inflaton fields.A de Sitter phase solution is obtained and new slow-rolling conditions for the spin potential are obtained.The spin potential reduces to Obukhov result at the present epoch of the Universe where the spin density is low with comparison to the Early Universe spin densities.A relation betwenn the spin density energy and the temperature fluctuation can be obtained which allow us to determine the spin density energy in terms of the COBE data for temperature fluctuations. 
  Two procedures for obtaining (extracting and constructing) the topological signature of any multiply connected Robertson-Walker (RW) universe are presented. It is shown through computer-aided simulations that both approaches give rise to the same topological signature for a multiply connected flat RW universe. The strength of these approaches is illustrated by extracting the topological signatures of a flat ($k=0$), an elliptic ($k=1$), and a hyperbolic ($k=-1$) multiply connected RW universes. We also show how separated contributions of the covering isometries add up to form the topological signature of a RW flat universe. There emerges from our theoretical results and simulations that the topological signature arises (in the mean) even when there are just a few images for each object. It is also shown that the mean pair separation histogram technique works, and that it is a suitable approach for studying the topological signatures of RW universes as well as the role of non-translational isometries. 
  We propose a sufficient condition for a general spherical symmetric static metric to be compatible with classical tests of gravity. A 1-parametric class of such metrics are constructed.   The Schwarzschild metric as well as the Yilmaz-Rosen metric are in this class.   By computing the scalar curvature we show that the non-Schwarzschild metrics can be interpreted as close 2-branes.   All the manifolds endowed the described metrics contain in a class of pseudo-Riemannian manifolds with a scalar curvature of a fixed sign. 
  An attempt is made of giving a self-contained introduction to holomorphic ideas in general relativity, following work over the last thirty years by several authors. The main topics are complex manifolds, spinor and twistor methods, heaven spaces. 
  We study spontaneous scalarization in Scalar-Tensor boson stars. We find that scalarization does not occur in stars whose bosons have no self-interaction. We introduce a quartic self-interaction term into the boson Lagrangian and show that when this term is large, scalarization does occur. Strong self-interaction leads to a large value of the compactness (or sensitivity) of the boson star, a necessary condition for scalarization to occur, and we derive an analytical expression for computing the sensitivity of a boson star in Brans-Dicke theory from its mass and particle number. Next we comment on how one can use the sensitivity of a star in any Scalar-Tensor theory to determine how its mass changes when it undergoes gravitational evolution. Finally, in the Appendix, we derive the most general form of the boson wavefunction that minimises the energy of the star when the bosons carry a U(1) charge. 
  The relationship between the intrinsic motions of gravity, light, and time is explored in terms of the principles of energy, entropy, causality, and symmetry conservation. A conceptual mechanism for gravity and the gravitational connection between quantum mechanics and relativity is explored. A "concept equation" for the gravitational conversion of space to time is presented. 
  A recent paper (gr-qc/9909017) criticizes our work on the structure of spacetime foam. Its authors argue that the quantum uncertainty limit for the position of the quantum clock in a gedanken timing experiment, obtained by Wigner and used by us, is based on unrealistic assumptions. Here we point out some flaws in their argument. We also discuss their other comments and some other issues related to our work, including a simple connection to the holographic principle. We see no reason to change our cautious optimism on the detectability of spacetime foam with future refinements of modern gravitational-wave interferometers like LIGO/VIRGO and LISA. 
  Motivated by the possible experimental opportunities to test quantum gravity via its effects on high-energy neutrinos propagating through space-time foam, we discuss how to incorporate spin structures in our D-brane description of gravitational recoil effects in vacuo. We also point to an interesting analogous condensed-matter system. We use a suitable supersymmetrization of the Born-Infeld action for excited D-brane gravitational backgrounds to argue that energetic fermions may travel slower than the low-energy velocity of light: \delta c / c \sim -E/M. It has been suggested that Gamma-Ray Bursters may emit pulses of neutrinos at energies approaching 10^{19} eV: these would be observable only if M \gsim 10^{27} GeV. 
  We discuss the capabilities of spherical antenn\ae as single multifrequency detectors of gravitational waves. A first order theory allows us to evaluate the coupled spectrum and the resonators readouts when the first and the second quadrupole bare sphere frequencies are simultaneously selected for layout tuning. We stress the existence of non-tuning influences in the second mode coupling causing draggs in the frequency splittings. These URF effects are relevant to a correct physical description of resonant spheres, still more if operating as multifrequency appliances like our PHCA proposal. 
  Cosmology is investigated within a new, scalar theory of gravitation, which is a preferred-frame bimetric theory with flat background metric. Before coming to cosmology, the motivation for an " ether theory " is exposed at length; the investigated concept of ether is presented: it is a compressible fluid, and gravity is seen as Archimedes' thrust due to the pressure gradient in that fluid. The construction of the theory is explained and the current status of the experimental confrontation is analysed, both in some detail. An analytical cosmological solution is obtained for a general form of the energy-momentum tensor. According to that theory, expansion is necessarily accelerated, both by vacuum and even by matter. In one case, the theory predicts expansion, the density increasing without limit as time goes back to infinity. High density is thus obtained in the past, without a big-bang singularity. In the other case, the Universe follows a sequence of (non-identical) contraction-expansion cycles, each with finite maximum energy density; the current expansion phase will end by infinite dilution in some six billions of years. The density ratio of the present cycle (ratio of the maximum to current densities) is not determined by the current density and the current Hubble constant H0, unless a special assumption is made. Since cosmological redshifts approaching z = 4 are observed, the density ratio should be at least 100. From this and the estimate of H0, the time spent since the maximum density is constrained to be larger than several hundreds of billions of years. Yet if a high density ratio, compatible with the standard explanation for the light elements and the 2.7 K radiation, is assumed, then the age of the Universe is much larger still. 
  We have performed 3D numerical simulations for merger of equal mass binary neutron stars in full general relativity. We adopt a $\Gamma$-law equation of state in the form $P=(\Gamma-1)\rho\epsilon$ where P, $\rho$, $\varep$ and $\Gamma$ are the pressure, rest mass density, specific internal energy, and the adiabatic constant with $\Gamma=2$. As initial conditions, we adopt models of corotational and irrotational binary neutron stars in a quasi-equilibrium state which are obtained using the conformal flatness approximation for the three geometry as well as an assumption that a helicoidal Killing vector exists. In this paper, we pay particular attention to the final product of the coalescence. We find that the final product depends sensitively on the initial compactness parameter of the neutron stars : In a merger between sufficiently compact neutron stars, a black hole is formed in a dynamical timescale. As the compactness is decreased, the formation timescale becomes longer and longer. It is also found that a differentially rotating massive neutron star is formed instead of a black hole for less compact binary cases, in which the rest mass of each star is less than 70-80% of the maximum allowed mass of a spherical star. In the case of black hole formation, we roughly evaluate the mass of the disk around the black hole. For the merger of corotational binaries, a disk of mass $\sim 0.05-0.1M_*$ may be formed, where M_* is the total rest mass of the system. On the other hand, for the merger of irrotational binaries, the disk mass appears to be very small : < 0.01M_*. 
  We search for a real bosonic and fermionic action in four dimensions which both remain invariant under local Weyl transformations in the presence of non-metricity and contortion tensor. In the presence of the non-metricity tensor the investigation is extended to Weyl $(W_n, g)$ space-time while when the torsion is encountered we are restricted to the Riemann-Cartan $(U_4, g)$ space-time. Our results hold for a subgroup of the Weyl-Cartan $(Y_4, g)$ space-time and we also calculate extra contributions to the conformal gravity. 
  The a priori time in conventional quantum mechanics is shown to contradict the uncertainty principle. A possible solution is given. 
  In the context of the Einstein-Cartan-Dirac model, where the torsion of the space-time couples to the axial currents of the fermions, we study the effects of this quantum-gravitational interaction on a massless neutrino beam crossing through a medium with high number density of fermions at rest. We calculate the reflection amplitude and show that a specific fraction of the incident neutrinos reflects from this potential if the polarization of the medium is different from zero. We also discuss the order of magnitude of the fermionic number density in which this phenomenon is observable, in other theoretical contexts, for example the strong-gravity regime and the effective field theory approach. 
  We discuss new Majumdar-Papapetrou solutions for the 3+1 Einstein-Maxwell equations, with charged dust acting as the external source of the fields. The solutions satisfy non-linear potential equations which are related to well-known wave equations of 1+1 soliton physics. Although the matter distributions are not localised, they present central structures which may be identified with voids. 
  A naked singularity occurs in the generic collapse of an inhomogeneous dust ball. We study the even-parity mode of gravitational waves from a naked singularity of the Lema\^{\i}tre-Tolman-Bondi spacetime. The wave equations for gravitational waves are solved by numerical integration using the single null coordinate. The result implies that the metric perturbation grows when it approaches the Cauchy horizon and diverges there, although the naked singularity is not a strong source of even-parity gravitational radiation. Therefore, the Cauchy horizon in this spacetime should be unstable with respect to linear even-parity perturbations. 
  We discuss some of the issues which we encounter when we try to invoke the scalar-tensor theories of gravitation as a theoretical basis of quintessence. One of the advantages of appealing to these theories is that they allow us to implement the scenario of a ``decaying cosmological constant,'' which offers a reasonable understanding of why the observed upper bound of the cosmological constant is smaller than the theoretically natural value by as much as 120 orders of magnitude. In this context, the scalar field can be a candidate of quintessence in a broader sense. We find, however, a serious drawback in the prototype Brans-Dicke model with $\Lambda$ added; a static universe in the physical conformal frame which is chosen to have constant particle masses. We propose a remedy by modifying the matter coupling of the scalar field taking advantage of scale invariance and its breakdown through quantum anomaly. By combining this with a conjecture on another cosmological constant problem coming from the vacuum energy of matter fields, we expect a possible link between quintessence and non-Newtonian gravity featuring violation of Weak Equivalence Principle and intermediate force range, likely within the experimental constraints. A new prediction is also offered on the time-variability of the gravitational constant. 
  The basic assumption of the induced gravity approach is that Einstein theory is an effective, low energy-form of a quantum theory of constituents. In this approach the Bekenstein-Hawking entropy S^{BH} of a black hole can be interpreted as a measure of the loss of information about constituents inside the black hole horizon. To be more exact, S^{BH} is determined by quantum correlations between "observable" and "non-observable" states with positive and negative energy $\cal E$, respectively. It is important that for non-minimally coupled constituents $\cal E$ differs from the canonical Hamiltonian $\cal H$. This explains why previous definitions of the entanglement entropy in terms of $\cal H$ failed to reproduce S^{BH}. 
  It is shown that any anisotropic and inhomogeneous cosmological solution to the lowest-order, four-dimensional, dilaton-graviton string equations of motion may be employed as a seed to derive a curved, three-brane cosmological solution to five-dimensional heterotic M-theory compactified on a Calabi-Yau three-fold. This correspondence formally relates a weakly coupled string cosmology directly with a strongly coupled one. The asymptotic behaviour of a wide class of spatially homogeneous braneworlds is deduced. Similar solutions may be derived in toroidally compactified massive type IIA supergravity. 
  The model of Expansive Nondecelerative Universe exploiting the Vaidya metrics is used as a tool for unification of gravitation and strong interactions. The proposed approach stems from the capability to localize the energy of gravitational field and enables to reach a certain level in unifying the general theory of relativity and quantum chromodynamics. A relationship between the energy binding quarks and gravitational energy of virtual black holes is rationalized. 
  Circularly rotating axisymmetric perfect fluid space-times are investigated to second order in the small angular velocity. The conditions of various special Petrov types are solved in a comoving tetrad formalism. A number of theorems are stated on the possible Petrov types of various fluid models. It is shown that Petrov type II solutions must reduce to the de Sitter spacetime in the static limit. Two space-times with a physically satisfactory energy-momentum tensor are investigated in detail. For the rotating incompressible fluid, it is proven that the Petrov type cannot be D. The equation of the rotation function $\omega $ can be solved for the Tolman type IV fluid in terms of quadratures. It is also shown that the rotating version of the Tolman IV space-time cannot be Petrov type D. 
  Using the equivalence theorem for the triplet ansatz sector of metric-affine gravity (MAG) theories and the Einstein-Proca system, it is shown that the only static black hole of the triplet sector of MAG is the Schwarzschild solution, under the constraint (-4\beta_4 + k_1\beta_5/2k_0 + k_2\gamma_4/k_0)/\kappa z_4 \neq 0 on the coupling constants. For the special case (-4\beta_4 + k_1\beta_5/2k_0 + k_2\gamma_4/k_0)/\kappa z_4 = 0, it follows that the only static non-extremal black hole is the Reissner-Nordstr\"om one. The results can be extended to exclude also the existence of soliton solutions of the triplet sector of MAG. 
  The cosmic no hair conjecture is tested in the spherically symmetric Einstein-Maxwell-dilaton~(EMD) system with a positive cosmological constant $\Lambda$. Firstly, we analytically show that once gravitational collapse occurs in the massless dilaton case, the system of field equations breaks down inevitably in outer communicating regions or at the boundary provided that a future null infinity ${\cal I}^+$ exists. Next we find numerically the static black hole solutions in the massive dilaton case and investigate their properties for comparison with the massless case. It is shown that their Abbott-Deser~(AD) mass are infinite, which implies that a spacetime with finite AD mass does not approach a black hole solution after the gravitational collapse. These results suggest that ${\cal I}^+$ cannot appear in the EMD system once gravitational collapse occurs and hence the cosmic no hair conjecture is violated in both the massless and the massive cases, in contrast to general relativity. 
  The four-dimensional gauge group of general relativity corresponds to arbitrary coordinate transformations on a four-manifold. Theories of gravity with a dynamical structure remarkably like Einstein's theory can be obtained on the basis of a four-dimensional gauge group of arbitrary coordinate and conformal transformations of riemannian metrics defined on a three-manifold. This new symmetry is more restrictive and hence more predictive. Many of the difficulties that have plagued the canonical quantization of general relativity seem to vanish. 
  The hot Big-Bang standard model for the evolution of the universe, despite strong successes, lets unresolved a number of problems. One of its main drawbacks, known as the horizon problem, was until now thought to be only solvable by an inflationary scenario. Here is proposed a class of inhomogeneous models of universe, getting rid of some of the worst drawbacks of standard cosmology. The horizon problem is solved by means of an initial singularity of ``delayed'' type and without need for any inflationary phase. The flatness and cosmological constant problems disappear. 
  We first present a review, intended for classical relativists, of the ultraviolet difficulties faced by local quantum gravity theories in both the usual Einstein versions and in their supergravity extensions, at least perturbatively. These problems, present in arbitrary dimensions, are traceable to the dimensionality of the Einstein constant. We then summarize very recent results about supergravity at the highest allowed dimension, D=11, showing that also this unique model suffers from infinities already at 2 loops, despite its high degree of supersymmetry. The conclusion is that there is no viable nonghost quantum field model that includes general relativity. 
  We study the gravi-dilaton field of a Schwarzschild black hole pierced by a thin cosmic string in both massless and massive dilatonic gravity. We conclude that in the thin vortex approximation the string's spacetime is asymptotically flat with a conical deficit angle and that the inertial mass of the black hole is different from the gravitational one. We generalize our results to charged black holes. 
  We investigate the dynamical properties of a class of spatially homogeneous and isotropic cosmological models containing a barotropic perfect fluid and multiple scalar fields with independent exponential potentials. We show that the assisted inflationary scaling solution is the global late-time attractor for the parameter values for which the model is inflationary, even when curvature and barotropic matter are included. For all other parameter values the multi-field curvature scaling solution is the global late-time attractor (in these solutions asymptotically the curvature is not dynamically negligible). Consequently, we find that in general all of the scalar fields in multi-field models with exponential potentials are non-negligible in late-time behaviour, contrary to what is commonly believed. The early-time and intermediate behaviour of the models is also studied. In particular, n-scalar field models are investigated and the structure of the saddle equilibrium points corresponding to inflationary m-field scaling solutions and non-inflationary m-field matter scaling solutions are also studied (where m<n), leading to interesting transient dynamical behaviour with new physical scenarios of potential importance. 
  Motivated by the invariance of current representations of quantum gravity under diffeomorphisms much more general than isometries, the Haag-Kastler setting is extended to manifolds without metric background structure. First, the causal structure on a differentiable manifold M of arbitrary dimension (d+1>2) can be defined in purely topological terms, via cones (C-causality). Then, the general structure of a net of C*-algebras on a manifold M and its causal properties required for an algebraic quantum field theory can be described as an extension of the Haag-Kastler axiomatic framework.   An important application is given with quantum geometry on a spatial slice within the causally exterior region of a topological horizon H, resulting in a net of Weyl algebras for states with an infinite number of intersection points of edges and transversal (d-1)-faces within any neighbourhood of the spatial boundary S^2. 
  Simple changes of the radial co-ordinate deprive Kerr's spinning corpuscle of its marvellous properties. 
  In gravitational thermodynamics, the entropy of a black hole with distinct surface gravities can be evaluated in a microcanonical ensemble. At the $WKB$ level, the entropy becomes the negative of the Euclidean action of the constrained instanton, which is the seed for the black hole creation in the no-boundary universe. Using the Gauss-Bonnet theorem, we prove the quite universal formula in Euclidean quantum gravity that the entropy of a nonrotating black hole is one quarter the sum of the products of the Euler characteristics and the areas of the horizons. For Lovelock gravity, the entropy and quantum creation of a black hole are also studied. 
  By using an analytic solution of the Teukolsky equation in the Kerr-de Sitter and Kerr-Newman-de Sitter geometries, an analytic expression of the absorption rate formulae for these black holes is calculated. 
  Vacuum solutions for multidimensional gravity on the principal bundle with the SU(2) structural group as the extra dimensions are found and discussed. This generalizes the results of Ref. \cite{vds3} from U(1) to the SU(2) gauge group. The spherically symmetric solution with the off-diagonal components of the multidimensional metric is obtained. It is shown that two types of solutions exist: the first has a wormhole-like 4D base, the second is a gravitational flux tube with two color and electric charges. The solution depends only on two parameters: the values of the electric and magnetic fields at the origin. In the plane of these parameters there exists a curve separating the regions with different types of solutions. An analogy with the 5D solutions is discussed. 
  It is shown that detecting or setting an upper limit on the scalar gravitational radiation is a good experimental test of relativistic gravity theories. The relativistic tensor-field theory of gravitation is revised and it is demonstrated that the scalar monopole gravitational radiation must be added to the usual quadrupole radiation. In the case of the binary pulsar PSR 1913+16 it is predicted the existence 0.735 % excess of the gravitational radiation due to the scalar gravitational waves. 
  The dynamics of a binary system with two spinning components on an eccentric orbit is studied, with the inclusion of the spin-spin interaction terms appearing at the second post-Newtonian order. A generalized true anomaly parametrization properly describes the radial component of the motion. The average over one radial period of the magnitude of the orbital angular momentum $\bar{L}$ is found to have no nonradiative secular change. All spin-spin terms in the secular radiative loss of the energy and magnitude of orbital angular momentum are given in terms of $\bar{L}$ and other constants of the motion. Among them, self-interaction spin effects are found, representing the second post-Newtonian correction to the 3/2 post-Newtonian order Lense-Thirring approximation. 
  The Lorentz covariant theory of precise Doppler measurements (PDM) based on the retarded Li\'{e}nard-Wiechert solution of the Einstein equations is described. An exact solution of equations of light propagation in the field of arbitrary moving bodies, which drastically extends the range of applicability of the new theory of PDM, is obtained. An explicit formula for the gravitational shift of frequency is given in analytic form. The limiting cases of the Doppler observations in gravitational lensing and of the spacecraft's Doppler tracking are described in more detail. We also present the post-Newtonian theory of the PDM developed for searching relativistic effects in close optical binaries and massive planetary systems. 
  A non-singular exact black hole solution in General Relativity is presented. The source is a non-linear electromagnetic field, which reduces to the Maxwell theory for weak field. The solution corresponds to a charged black hole with |q| \leq 2s_c m \approx 0.6 m, having metric, curvature invariants, and electric field bounded everywhere. 
  We analyze the propagation of light in the context of nonlinear electrodynamics, as it occurs in modified QED vacua. We show that the corresponding characteristic equation can be described in terms of a modification of the effective geometry of the underlying spacetime structure. We present the general form for this effective geometry and exhibit some new consequences that result from such approach. 
  The holographic bound in Brans-Dicke $k=1$ matter dominated Cosmology is discussed. In this talk, both the apparent horizon and the particle horizon are taken for the holographic bound. The covariant entropy conjecture proposed by Bousso is also discussed. 
  Eternally inflating universes can contain thermalized regions with different values of the cosmological parameters. In particular, the spectra of density fluctuations should be different, because of the different realizations of quantum fluctuations of the inflaton field. I discuss a general method for calculating probability distributions for such variable parameters and analyse the density fluctuation spectrum as a specific application. 
  Thin shells in general relativity can be used both as models of collapsing objects and as probes in the space-time outside compact sources. Therefore they provide a useful tool for the analysis of the final fate of collapsing matter and of the effects induced in the matter by strong gravitational fields. We describe the radiating shell as a (second quantized) many-body system with one collective degree of freedom, the (average) radius, by means of an effective action which also entails a thermodynamic description. Then we study some of the quantum effects that occur in the matter when the shell evolves from an (essentially classical) large initial radius towards the singularity and compute the corresponding backreaction on its trajectory. 
  We overview the recently proposed mode-sum regularization prescription (MSRP) for the calculation of the local radiation-reaction forces, which are crucial for the orbital evolution of binaries. We then describe some new results which were obtained using MSRP, and discuss their importance for gravitational-wave astronomy. 
  The relation between the angular diameter distance and redshift in a spherically symmetric dust-shell universe is studied. We have discovered that the relation agrees with that of an appropriate Friedmann-Lemaitre (FL) model if we set a ``homogeneous'' expansion law and a ``homogeneous'' averaged density field. This will support the averaging hypothesis that a universe looks like a FL model in spite of small-scale fluctuations of density field, if its averaged density field is homogeneous on large scales. We also study the connection of the proper mass of a shell with the mass of gravitationally bound objects. Combining this with the results of the distance-redshift relation, we discuss an impact of the local inhomogeneities on determination of the cosmological parameters through the observation of the locally inhomogeneous universe. 
  We examine a possibility that, when a black hole is formed, the information on the collapsed star is stored as the entanglement entropy between the outside and the thin region (of the order of the Planck length) of the inside the horizon. For this reason, we call this as the entanglement entropy of the black hole ``horizon''. We construct two models, one is in the Minkowski spacetime and the other is in the Rindler wedge. To calculate the entropy explicitly, we assume that the thin regions of the order of the Planck length of the outside and inside the horizon are completely entangled by quantum effects. We also use a property of the entanglement entropy that it is symmetric under an interchange of the observed and unobserved subsystems. Our setting and this symmetric property substantially reduce the needed numerical calculation. As a result of our analysis, we can explain the Bekenstein-Hawking entropy itself (rather than its correction by matter fields) in the context of the entanglement entropy. 
  The emission of radiation from an accelerated charge is analyzed. It is found that at zero velocity, the radiation emitted from the charge imparts no counter momentum to the emitting charge, and no radiation reaction force is created by the radiation. A reaction force is created by the stress force that exists in the curved electric field of the charge, and the work done in overcoming this force is the source of the energy carried by the radiation. 
  The slow-roll approximation is the usual starting point to study the constraints imposed on the inflaton potential parameters by the observational data. We show that, for a potential exhibiting at least two extrema and giving rise to a limited inflationary period, slow-roll does not have to be taken as an additional hypothesis and is in fact forced by the constraints on the number of e-foldings. 
  After a brief survey of the appearance of quantum algebras in diverse contexts of quantum gravity, we demonstrate that the particular deformed algebras, which arise within the approach of J.Nelson and T.Regge to 2+1 anti-de Sitter quantum gravity (for space surface of genus g) and which are basic for generating the algebras of independent quantum observables, are in fact isomorphic to the nonstandard q-deformed analogues U'_q(so_n) (introduced in 1991) of Lie algebras of the orthogonal groups SO(n), with n linked to g as n=2g+2. 
  In general relativity, the notion of mass and other conserved quantities at spatial infinity can be defined in a natural way via the Hamiltonian framework: Each conserved quantity is associated with an asymptotic symmetry and the value of the conserved quantity is defined to be the value of the Hamiltonian which generates the canonical transformation on phase space corresponding to this symmetry. However, such an approach cannot be employed to define `conserved quantities' in a situation where symplectic current can be radiated away (such as occurs at null infinity in general relativity) because there does not, in general, exist a Hamiltonian which generates the given asymptotic symmetry. (This fact is closely related to the fact that the desired `conserved quantities' are not, in general, conserved!) In this paper we give a prescription for defining `conserved quantities' by proposing a modification of the equation that must be satisfied by a Hamiltonian. Our prescription is a very general one, and is applicable to a very general class of asymptotic conditions in arbitrary diffeomorphism covariant theories of gravity derivable from a Lagrangian, although we have not investigated existence and uniqueness issues in the most general contexts. In the case of general relativity with the standard asymptotic conditions at null infinity, our prescription agrees with the one proposed by Dray and Streubel from entirely different considerations. 
  Following Kottler, \'E.Cartan, and van Dantzig, we formulate the Maxwell equations in a metric independent form in terms of the field strength $F=(E,B)$ and the excitation $H=({\cal D}, {\cal H})$. We assume a linear constitutive law between $H$ and $F$. First we split off a pseudo-scalar (axion) field from the constitutive tensor; its remaining 20 components can be used to define a duality operator $^#$ for 2-forms. If we enforce the constraint $^{##}=-1$, then we can derive of that the conformally invariant part of the {\em metric} of spacetime. 
  The stability of the de Sitter era of cosmic expansion in spatially curved homogeneous isotropic universes is studied. The source of the gravitational field is an imperfect fluid such that the parameters that characterize it may change with time. In this way we extend our previous analysis for spatially-flat spaces as well as the work of Barrow. 
  We describe the design of a Kalman filter that identifies suspension violin modes in an interferometric gravitational wave detectors data channel. We demonstrate the filter's effectiveness by applying it to data taken on the LIGO~40M prototype. 
  There are two effects of extra matter fields on the Lorentzian traversable wormhole. The ``primary effect'' says that the extra matter can afford to be a part of source or whole source of the wormhole when the wormhole is being formed. Thus the matter does not affect the stability of wormhole and the wormhole is still safe. If the extra matter is extotic, it can be the whole part of the source of the wormhole.   The ``auxiliary effect'' is that the extra matter plays the role of the additional matter to the stably-existed wormhole by the other exotic matter. This additional matter will change the geometry of wormhole enough to prevent from forming the wormhole by backreaction. In the minimally coupled massless scalar field case, the self-consistent solution was found. The backreaction of the scalar field can dominate the exotic matter part so that it will hinder the formation of the wormhole. 
  Within the framework of stochastic inflationary cosmology we derive steady-state distributions P_c(V) of domains in comoving coordinates, under the assumption of slow-rolling and for two specific choices of the coarse-grained inflaton potential $V(\Phi)$. We model the process as a Starobinsky-like equation in V-space plus a time-independent source term P_w(V) which carries (phenomenologically) quantum-mechanical information drawn from either of two known solutions of the Wheeler-De Witt equation: Hartle-Hawking's and Vilenkin's wave functions. The presence of the source term leads to the existence of nontrivial steady-state distributions P^w_c(V). The relative efficiencies of both mechanisms at different scales are compared for the proposed potentials. 
  An efficient approach to tensor perturbation calculations by proper use of computer algebra methods is described, reaching the sufficient generality required for a comprehensive analysis of the Schwarzschild and Reissner-Nordstroem metric stability. 
  We analyze the computational costs of searches for continuous monochromatic gravitational waves emitted by rotating neutron stars orbiting a companion object. As a function of the relevant orbital parameters, we address the computational load involved in targeted searches, where the position of the source is known; the results are applied to known binary radio pulsars and Sco-X1. 
  We discuss how asymptotic quantities, originally introduced on null infinity in terms of Bondi-type gauge conditions, can be calculated near space-like infinity to any desired precision. 
  One of the most interesting predictions of string-inspired cosmological models is the presence of a stochastic background of relic gravitational waves in the frequency band accessible to Earth-based detectors. Here we consider a ``minimal'' class of string cosmology models and explore whether they are falsifiable by gravitational wave observations. In particular, we show that, the detectability of the signal depends crucially on the actual values of the model parameters. This feature will enable laser interferometers -- starting from the second generation of detectors -- to place stringent constraints on the theory for a fairly large range of the free parameters of the model. 
  We show that there are implications on thermodynamics that come from the existence of the initial cosmic singularity. At present time this is more a conceptual change than an observable one. However at very early cosmic times there is a big difference between the actual behavior of thermodynamic quantities and the behavior assumed in the standard cosmological model. We present the discussion of two systems: an ideal monatomic gas at present, and a photon gas at the early Universe. We show the striking result that the entropy density goes to zero as the cosmic time goes to zero. This in turn, provides an explanation for the so called horizon problem. 
  The characteristic initial value problem has been implemented as a robust computational algorithm (the PITT NULL CODE), with direct application to binary black holes. The event horizon can be analyzed by characteristic techniques as a stand-alone object using an analytic conformal model which gives new insight into the intrinsic geometry of binary black holes. When applied to a non-axisymmetric horizon, the model reveals substantially new features. Colliding black holes generically go through a toroidal phase before they become spherical. The conformal structure of the horizon supplies part of the data for a simulation of the exterior space-time and calculation of the post-merger waveforms from a binary black hole inspiral. 
  We investigate the family of electrostatic spherically symmetric solutions of the five-dimensional Kaluza-Klein theory. Both charged and neutral cases are considered. The analysis of the solutions, through their geometrical properties, reveals the existence of black holes, wormholes and naked singularities. A new class of regular solutions is identified. A monopole perturbation study of all these solutions is carried out, enabling us to prove analytically the stability of large classes of solutions. In particular, the black hole solutions are stable, while for the regular solutions the stability analysis leads to an eigenvalue problem. 
  By a weak deformation of the cylindrical symmetry of the potential vortex in a relativistic perfect isentropic fluid, we study the possible dynamics of the central line of this vortex. In "stiff" material the Nanbu-Goto equations are obtained 
  We consider the problem of evolving nonlinear initial data in the close limit regime. Metric and curvature perturbations of nonrotating black holes are equivalent to first perturbative order, but Moncrief waveform in the former case and Weyl scalar $\psi_4$ in the later differ when nonlinearities are present. For exact Misner initial data (two equal mass black holes initially at rest), metric perturbations evolved via the Zerilli equation suffer of a premature break down (at proper separation of the holes $L/M\approx2.2$) while the exact Weyl scalar $\psi_4$ evolved via the Teukolsky equation keeps a very good agreement with full numerical results up to $L/M\approx3.5$. We argue that this inequivalent behavior holds for a wider class of conformally flat initial data than those studied here. We then discuss the relevance of these results for second order perturbative computations and for perturbations to take over full numerical evolutions of Einstein equations. 
  We clarify and develop the results of a previous paper on the birth of a closed universe of negative spatial curvature and multiply connected topology. In particular we discuss the initial instanton and the second topology change in more detail. This is followed by a short discussion of the results. 
  A geometric spacetime map of the Universe is presented, addressing problems inherent in deep space observations and cosmology. Implications for the observer's perspective, the cosmological horizon problem, and the recently observed "accelerating" Universe are discussed. 
  The relation between Einstein equivalence principle and a continuous quantum measurement is analyzed in the context of the recently proposed flavor-oscillation clocks, an idea pioneered by Ahluwalia and Burgard (Gen. Rel Grav. Errata 29, 681 (1997)). We will calculate the measurement outputs if a flavor-oscillation clock, which is immersed in a gravitational field, is subject to a continuous quantum measurement. Afterwards, resorting to the weak equivalence principle, we obtain the corresponding quantities in a freely falling reference frame. Finally, comparing this last result with the measurement outputs that would appear in a Minkowskian spacetime it will be found that they do not coincide, in other words, we have a violation of Einstein equivalence principle. This violation appears in two different forms, namely: (i) the oscillation frequency in a freely falling reference frame does not match with the case predicted by general relativity, a feature previously obtained by Ahluwalia; (ii) the probability distribution of the measurement outputs, obtained by an observer in a freely falling reference frame, does not coincide with the results that would appear in the case of a Minkowskian spacetime. 
  The various schemes for studying rigidly rotating perfect fluids in general relativity are reviewed. General conclusions one may draw from these are: (i) There is a need to restrict the scope of the possible ansatze, and (ii) the angular behaviour is a valuable commodity. This latter observation follows from a large number of analytic models exhibiting a NUT-like behaviour. A method of getting around problem (ii) is presented on a simple example. To alleviate problem (i) for rigidly rotating perfect fluids, approximation schemes based on a series expansion in the angular velocity are suggested. A pioneering work, due to Hartle, explores the global properties of matched space-times to quadratic order in the angular velocity.   As a first example of the applications, it is shown that the rigidly rotating incompressible fluid cannot be Petrov type D. 
  The concept of wave field is introduced to represent oriented media. The wave field is a tensor field of second rank, and directors are its eigenvectors. This exhibition of directors defines a natural gauge group inherit in continua and allows one to derive from variational principle general relativistic and gauge invariant equations for the wave field in question. Thus, the gauge-theoretical approach to continuum with internal degrees of freedom gives unambiguous and minimally coupled theory. 
  The maximal globally hyperbolic development of non-Taub-NUT Bianchi IX vacuum initial data and of non-NUT Bianchi VIII vacuum initial data is C2 inextendible. Furthermore, a curvature invariant is unbounded in the incomplete directions of inextendible causal geodesics. 
  Stability analysis on the De Sitter universe in pure gravity theory is known to be useful in many aspects. We first show how to complete the proof of an earlier argument based on a redundant field equation. It is shown further that the stability condition applies to $k \ne 0$ Friedmann-Robertson-Walker spaces based on the non-redundant Friedmann equation derived from a simple effective Lagrangian. We show how to derive this expression for the Friedmann equation of pure gravity theory. This expression is also generalized to include scalar field interactions. 
  We investigated the SU(2) Einstein-Yang-Mills system on a time-dependent non-diagonal cylindrical symmetric space-time. From the numerical investigation, wave-like solutions are found, consistent with the familiar string-like features. They possess an angle-deficit which depends on the initial form of the magnetic component of the YM field, i.e., the number of times it crosses the r-axis. The soliton-like behavior of the gravitational and YM waves show significant differences from the ones found in the Einstein-Maxwell system. The stability of the system is analyzed using the multiple-scale method. To first order a consistent set of equations is obtained. 
  Dimensional reductions of various higher dimensional (super)gravity theories lead to effectively two-dimensional field theories described by gravity coupled G/H nonlinear sigma-models. We show that a new set of complexified variables can be introduced when G/H is a Hermitian symmetric space. This generalizes an earlier construction that grew out of the Ashtekar formulation of two Killing vector reduced pure 4d general relativity. Apart from giving some new insights into dimensional reductions of higher dimensional (super)gravity theories, these Ashtekar-type variables offer several technical advantages in the context of the exact quantization of these models. As an application, an infinite set of conserved charges is constructed. Our results might serve as a starting point for probing the quantum equivalence of the Ashtekar and the metric formalism within a non-trivial midi-superspace model of quantum gravity. 
  Semiclassical perturbations to the Reissner-Nordstrom metric caused by the presence of a quantized massive scalar field with arbitrary curvature coupling are found to first order in \epsilon = \hbar/M^2. The DeWitt-Schwinger approximation is used to determine the vacuum stress-energy tensor of the massive scalar field. When the semiclassical perturbation are taken into account, we find extreme black holes will have a charge-to-mass ratio that exceeds unity, as measured at infinity. The effects of the perturbations on the black hole temperature (surface gravity) are studied in detail, with particular emphasis on near extreme ``bare'' states that might become precisely zero temperature ``dressed'' semiclassical black hole states. We find that for minimally or conformally coupled scalar fields there are no zero temperature solutions among the perturbed black holes. 
  Several electric/magnetic charged solutions (dyons) to 5D Kaluza-Klein gravity on the principal bundle are reviewed. Here we examine the possibility that these solutions can act as quantum virtual wormholes in spacetime foam models. By applying a sufficently large, external electric and/or magnetic field it may be possible to ``inflate'' these solutions from a quantum to a classical state. This effect could lead to a possible experimental signal for higher dimensions in multidimensional gravity. 
  We introduce a new spinorial, BF-like action for the Einstein gravity. This is a first, up to our knowledge, 2-form action which describes the real, Lorentzian gravity and uses only the self-dual connection. In the generic case, the corresponding classical canonical theory is equivalent to the Einstein-Ashtekar theory plus the reality conditions. 
  We consider cosmological models in which a homogeneous isotropic universe is embedded as a 3+1 dimensional surface into a 4+1 dimensional manifold. The size of the extra dimension depends on time. It is small compared to the size of the universe only if the energy of gravitational self-interaction of the universe through the compact extra dimension dominates over all other kinds of energy. The self-interaction energy gives the main contribution into the Friedmann equation, which governs the dynamics of the scale factor of the universe. 
  The general phenomena associated with sustained resonance are studied in this paper in connection with relativistic binary pulsars. We represent such a system by two point masses in a Keplerian binary system that evolves via gravitational radiation damping as well as an external tidal perturbation. For further simplification, we assume that the external tidal perturbation is caused by a normally incident circularly polarized monochromatic gravitational wave. In this case, the second-order partially averaged equations are studied and a theorem of C. Robinson is employed to prove that for certain values of the physical parameters resonance capture followed by sustained resonance is possible in the averaged system. We conjecture that sustained resonance can occur in the physical system when the perturbing influences nearly balance each other. 
  We show the existence of spatially homogeneous but anisotropic cosmological models whose cosmic microwave background temperature is exactly isotropic at one instant of time but whose rate of expansion is highly anisotropic. The existence of these models shows that the observation of a highly isotropic cosmic microwave background temperature cannot alone be used to infer that the universe is close to a Friedmann-Lemaitre model. 
  A simple model of the brane-world cosmology has been proposed, which is characterized by four parameters, the bulk cosmological constant, the spatial curvature of the universe, the radiation strength arising from bulk space-time and the breaking parameter of $Z_2$-symmetry. The bulk space-time is assumed to be locally static five-dimensional analogue of the Schwarzschild-anti-de Sitter space-time, and then the location of three-brane is determined by metric junction. The resulting Friedmann equation recovers standard cosmology, and a new term arises if the assumption $Z_2$-symmetry is dropped, which behaves as cosmological term in the early universe, next turns to negative curvature term, and finally damps rapidly. 
  A retrospective analysis of the field theory of gravitation, describing gravitational field in the same way as other fields of matter in the flat space-time, is done. The field approach could be called "quantum gravidynamics" to distinguish it from the "geometrodynamics" or general relativity. The basic propositions and main conclusions of the field approach are discussed with reference to classical works of Birkhoff, Moshinsky, Thirring, Kalman, Feynman, Weinberg, Deser. In the case of weak fields both "gravidynamics" and "geometrodynamics" give the same predictions for classical relativistic effects. However, in the case of strong field, and taking into account quantum nature of the gravitational interaction, they are profoundly different. Contents of the paper: 1) Introduction; 2) Two ways in gravity theory: 2.1.Hypotheses of Poincar\'e and Einstein, 2.2. Gravity as a geometry of space, 2.3. Gravitation as a material field in flat space-time; 3) Classical theory of tensor field: 3.1.Works of Birkhoff and Moshinsky, 3.2.Works of Thirring and Kalman, 3.3.Thirring and Deser about identity of GR and FTG; 4) Quantum theory of tensor field; 5) Modern problems in field theory of gravitation: 5.1.Multicomponent nature of tensor field, 5.2.Choice of energy-momentum tensor of gravitational field, 5.3.Absence of black holes in FTG, 5.4.Astrophysical tests of FTG; 6) Conclusions. 
  The merger of two neutron stars has been proposed as a source of gamma-ray bursts, r-process elements, and detectable gravitational waves. Extracting information from observations of these phenomena requires fully relativistic simulations. Unfortunately, the only demonstrated method for stably evolving neutron stars requires solving elliptic equations at each time step, adding substantially to the computational resources required. In this paper we present a simpler, more efficient method. The key insight is in how we apply numerical diffusion. We perform a number of tests to validate the method and our implementation. We also carry out a very rough simulation of coalescence and extraction of the gravitational waves to show that the method is viable if realistic initial data are provided. 
  We examine the string cosmology equations with a dilaton potential in the context of the Pre-Big Bang Scenario with the desired scale factor duality, and give a generic algorithm for obtaining solutions with appropriate evolutionary properties. This enables us to find pre-big bang type solutions with suitable dilaton behaviour that are regular at $t=0$, thereby solving the graceful exit problem. However to avoid fine tuning of initial data, an `exotic' equation of state is needed that relates the fluid properties to the dilaton field. We discuss why such an equation of state should be required for reliable dilaton behaviour at late times. 
  We analyze particle dynamics on $N$ dimensional one-sheet hyperboloid embedded in $N+1$ dimensional Minkowski space. The dynamical integrals constructed by $SO_\uparrow (1,N)$ symmetry of spacetime are used for the gauge-invariant Hamiltonian reduction. The physical phase-space parametrizes the set of all classical trajectories on the hyperboloid. In quantum case the operator ordering problem for the symmetry generators is solved by transformation to asymptotic variables. Canonical quantization leads to unitary irreducible representation of $SO_\uparrow (1,N)$ group on Hilbert space $L^2(S^{N-1})$. 
  Paths in an appropriate geometry are usually used as trajectories of test particles in geometric theories of gravity. It is shown that non-symmetric geometries possess some interesting quantum features. Without carrying out any quantization schemes, paths in such geometries are naturally quantized. Two different non-symmetric geometries are examined for these features. It is proved that, whatever the non-symmetric geometry is, we always get the same quantum features. It is shown that these features appear only in the pure torsion term (the anti-symmetric part of the affine connection) of the path equations. The vanishing of the torsion leads to the disappearance of these features, regardless of the symmetric part of the connection. It is suggested that, in order to be consistent with the results of experiments and observations, torsion term in path equations should be parametrized using an appropriate parameter. 
  In this paper we study the dynamical behaviour of a simple cosmological model defined by a spatially flat Robertson-Walker geometry, conformally coupled with a massive scalar field. We determine a Lyapunov-like function for the non-linear evolution equations. From this function we prove that all the stationary solutions are unstable. We also show that all initial conditions, different from the stationary points, originate an expanding universe in the asymptotic regime, with a scale parameter $a(t)$ that goes to infinity and the scalar field $\phi (t)$ that goes to zero in an oscillatory way . We also find two asymptotic solutions, valid for sufficiently large values of time. These solutions correspond to a radiation dominated phase and to a matter dominated phase, respectively. 
  Numerical relativity is finally approaching a state where the evolution of rather general (3+1)-dimensional data sets can be computed in order to solve the Einstein equations. After a general introduction, three topics of current interest are briefly reviewed: binary black hole mergers, the evolution of strong gravitational waves, and shift conditions for neutron star binaries. 
  We present a method for calculating the self-force (the ``radiation reaction force'') acting on a charged particle moving in a strong field orbit in black hole spacetime. In this approach, one first calculates the contribution to the self-force due to each multipole mode of the particle's field. Then, the sum over modes is evaluated, subject to a certain regularization procedure. Here we develop this regularization procedure for a scalar charge on a Schwarzschild background, and present the results of its implementation for radial trajectories (not necessarily geodesic). 
  We investigate up to which extend the kinematic setting of loop quantum gravity can be fit into a diffeomorphism invariant setting of algebraic QFT generalizing the Haag-Kastler setting of Wightman type QFT. The net of local (Weyl-)algebras resulting from a spin network state of quantum geometry immediately accommodates isotony and diffeomorphism covariance, and formulation of causality becomes possible via of diffeomorphism invariant foliations of the underlying manifold by cones. On a spatial horizon, quantum geometry becomes asymptotically a genuine QFT with infinitely many degrees of freedom, if the cylinder functions' supporting graphs intersect the inner boundary spheres in an infinite number of punctures. 
  The bosonic content of string theory in curved background of multidimensional structure with p-branes has a systematic geometrical description as an effective sigma-model of gravity in lower dimension (say 3+1) with additional interacting dilatonic and p-brane fields. If the target-space is locally symmetric, solutions with intersecting p-branes can be found. Some static solutions are p-brane generalizations of black holes (including the standard Reissner-Nordstr"om class), which allow the prediction of detectable features of the higher-dimensional p-brane geometry via scaling properties of black hole thermal properties. E.g. the Hawking temperature T_H depends critically on the p-brane intersection topology. 
  The superstring and superbrane theories which include gravity as a necessary and fundamental part renew an interest to alternative representations of general relativity as well as the alternative models of gravity. We study the coframe teleparallel theory of gravity with a most general quadratic Lagrangian. The coframe field on a differentiable manifold is a basic dynamical variable. A metric tensor as well as a metric compatible connection is generated by a coframe in a unique manner. The Lagrangian is a general linear combination of Weitzenb\"{o}ck's quadratic invariants with free dimensionless parameters $\r_1,\r_2,\r_3$.   Every independent term of the Lagrangian is a global SO(1,3)-invariant 4-form. For a special choice of parameters which confirms with the local SO(1,3) invariance this theory gives an alternative description of Einsteinian gravity - teleparallel equivalent of GR.   We prove that the sign of the scalar curvature of a metric generated by a static spherical-symmetric solution depends only on a relation between the free parameters. The scalar curvature vanishes only for a subclass of models with $\r_1=0$. This subclass includes the teleparallel equivalent of GR. We obtain the explicit form of all spherically symmetric static solutions of the ``diagonal'' type to the field equations for an arbitrary choice of free parameters. We prove that the unique asymptotic-flat solution with Newtonian limit is the Schwarzschild solution that holds for a subclass of teleparallel models with $\r_1=0$. Thus the Yang-Mills-type term of the general quadratic coframe Lagrangian should be rejected. 
  A corollary of general relativity that the average velocity of light between two points in a gravitational field is anisotropic has been overlooked. It is shown that this anisotropy can be probed by an experiment which constitutes another test of general relativity. 
  We define the general Hill system and briefly analyze its dynamical behavior. A particular Hill system representing the interaction of a Keplerian binary system with a normally incident circularly polarized gravitational wave is discussed in detail. In this case, we compute the Poincar\'e-Melnikov function explicitly and determine its zeros. Moreover, we provide numerical evidence in favor of chaos in this system. The partially averaged equations for the Hill system are used to predict the regular behavior of the Keplerian orbit at resonance with the external radiation. 
  We study, using Rindler coordinates, the quantization of a charged scalar field interacting with a constant, external, electric field. First we establish the expression of the Schwinger vacuum decay rate, using the operator formalism. Then we rederive it in the framework of the Feynman path integral method. Our analysis reinforces the conjecture which identifies the zero winding sector of the Minkowski propagator with the Rindler propagator. Moreover we compute the expression of the Unruh's modes that allow to make connection between Minkowskian and Rindlerian quantization scheme by purely algebraic relations. We use these modes to study the physics of a charged two level detector moving in an electric field whose transitions are due to the exchange of charged quanta. In the limit where the Schwinger pair production mechanism of the exchanged quanta becomes negligible we recover the Boltzman equilibrium ratio for the population of the levels of the detector. Finally we explicitly show how the detector can be taken as the large mass and charge limit of an interacting fields system. 
  We study the relativistic orbit of binary black holes in systems with small mass ratio. The trajectory of the smaller object (another black hole or a neutron star), represented as a particle, is determined by the geodesic equation on the perturbed massive black hole spacetime. The particle itself generates the gravitational perturbations leading to a problem that needs regularization. Here we study perturbations around a Schwarzschild black hole using Moncrief's gauge invariant formalism. We decompose the perturbations into $\ell-$multipoles to show that all $\ell-$metric coefficients are $C^0$ at the location of the particle. Summing over $\ell$, to reconstruct the full metric, gives a formally divergent result. We succeed in bringing this sum to a generalized Riemann's $\zeta-$function regularization scheme and show that this is tantamount to subtract the $\ell\to\infty$ piece to each multipole. We explicitly carry out this regularization and numerically compute the first order geodesics. Application of this method to general orbits around rotating black holes would generate accurate templates for gravitational wave laser interferometric detectors. 
  Some nondifferentiable quantities (for example, the metric signature) can be the independent physical degrees of freedom. It is supposed that in quantum gravity these degrees of freedom can fluctuate. Two examples of such quantum fluctuation are considered: a quantum interchange of the sign of two components of the 5D metric and a quantum fluctuation between Euclidean and Lorentzian metrics. The first case leads to a spin-like structure on the throat of composite wormhole and to a possible inner structure of the string. The second case leads to a quantum birth of the non-singular Euclidean Universe with frozen $5^{th}$ dimension. The probability for such quantum fluctuations is connected with an algorithmical complexity of the Einstein equations. 
  We consider the late time behaviour of non-tilted perfect fluid Bianchi VII_0 models when the source is a radiation fluid, thereby completing the analysis of the Bianchi VII_0 models initiated by Wainwright et al in a recent paper. The models exhibit the phenomena of asymptotic self-similarity breaking and Weyl-curvature dominance at late times. The late time dynamics of the VII_0 perfect fluid models, and in particular that of the radiation fluid, is a prime example of the complexity inherent in the field equations of general relativity. 
  The approximation of the renormalized stress-energy tensor of the quantized massive scalar, spinor, and vector field in the Reissner- Nordstrom spacetime is constructed. It is achieved by functional differentiation of the lowest order of the Schwinger-DeWitt effective action involving coincidence limit of the Hadamard-Minakshisundaram-DeWitt-Seely coefficient a_{3}, and restricting thus obtained general formulas to spacetimes with vanishing curvature scalar. For the massive scalar field with arbitrary curvature coupling our results reproduce those obtained previously by Anderson, Hiscock, and Samuel by means of 6-th order WKB approximation. 
  The problem of finding an appropriate geometrical/physical index for measuring a degree of inhomogeneity for a given space-time manifold is posed. Interrelations with the problem of understanding the gravitational/informational entropy are pointed out. An approach based on the notion of approximate symmetry is proposed. A number of related results on definitions of approximate symmetries known from literature are briefly reviewed with emphasis on their geometrical/physical content. A definition of a Killing-like symmetry is given and a classification theorem for all possible averaged space-times acquiring Killing-like symmetries upon averaging out a space-time with a homothetic Killing symmetry is proved. 
  A more rigorous treatment of the Schwarzschild metric by making use of the energy-momentum tensor of a single point particle as source term shows that   $g_{00}=-\{1-\frac{2GM}{c^2r}-\frac{8G^2 M^2}{c^4 r^2}(\theta (r)-1)\}\exp [2(\theta (r)-1)]$   $g_{rr}=\{1-\frac{2GM}{c^2r}-\frac{8G^2 M^2}{c^4 r^2}(\theta (r)-1)\}^{-1}$   The existence of a discontinuity at r=0 leads to an infinite repulsive force that will change the ultimate fate of a free fall test particle to a bouncing state. 
  We derive and analyze exact static solutions to the gravitating O(3) $\sigma$ model with cosmological constant in (2+1) dimensions. Both signs of the gravitational and cosmological constants are considered. Our solutions include geodesically complete spacetimes, and two classes of black holes. 
  We introduce an original approach to geometric calculus in which we define derivatives and integrals on functions which depend on extended bodies in space--that is, paths, surfaces, and volumes etc. Though this theory remains to be fully completed, we present it at its current stage of development, and discuss it's connection to physical research, in particular its application to spinning particles in curved space. 
  A solution to the 50 year old problem of a spinning particle in curved space has been recently derived using an extension of Clifford calculus in which each geometric element has its own coordinate. This leads us to propose that all the laws of physics should obey new polydimensional metaprinciples, for which Clifford algebra is the natural language of expression, just as tensors were for general relativity. Specifically, phenomena and physical laws should be invariant under local automorphism transformations which reshuffle the physical geometry. This leads to a new generalized unified basis for classical mechanics, which includes string theory, membrane theory and the hypergravity formulation of Crawford[J. Math. Phys., {\bf 35}, 2701-2718 (1994)]. Most important is that the broad themes presented can be exploited by nearly everyone in the field as a framework to generalize both the Clifford calculus and multivector physics. 
  For an inspiraling neutron-star/black-hole binary (NS/BH), we estimate the gravity-wave frequency f_td at the onset of NS tidal disruption. We model the NS as a tidally distorted, homogeneous, Newtonian ellipsoid on a circular, equatorial geodesic around a Kerr BH. We find that f_td depends strongly on the NS radius R, and estimate that LIGO-II (ca. 2006-2008) might measure R to 15% precision at 140 Mpc (about 1 event/yr under current estimates). This suggests that LIGO-II might extract valuable information about the NS equation of state from tidal-disruption waves. 
  The main theoretical aspects of gravitomagnetism are reviewed. It is shown that the gravitomagnetic precession of a gyroscope is intimately connected with the special temporal structure around a rotating mass that is revealed by the gravitomagnetic clock effect. This remarkable effect, which involves the difference in the proper periods of a standard clock in prograde and retrograde circular geodesic orbits around a rotating mass, is discussed in detail. The implications of this effect for the notion of ``inertial dragging'' in the general theory of relativity are presented. The theory of the clock effect is developed within the PPN framework and the possibility of measuring it via spaceborne clocks is examined. 
  By requiring the linear differential operator in Newton's law of motion to be self adjoint, we obtain the field equation for the linear theory, which is the classical electrodynamics. In the process, we are also led to a fundamental universal chiral relation between electric and magnetic monopoles which implies that the two are related. Thus there could just exist only one kind of charge which is conventionally called electric. 
  We describe an efficient method of matched filtering over long (greater than 1 day) time baselines starting from Fourier transforms of short durations (roughly 30 minutes) of the data stream. This method plays a crucial role in the search algorithm developed by Schutz and Papa for the detection of continuous gravitational waves from pulsars. Also, we discuss the computational cost--saving approximations used in this method, and the resultant performance of the search algorithm. 
  We investigate the numerical stability of Cauchy evolution of linearized gravitational theory in a 3-dimensional bounded domain. Criteria of robust stability are proposed, developed into a testbed and used to study various evolution-boundary algorithms. We construct a standard explicit finite difference code which solves the unconstrained linearized Einstein equations in the 3+1 formulation and measure its stability properties under Dirichlet, Neumann and Sommerfeld boundary conditions. We demonstrate the robust stability of a specific evolution-boundary algorithm under random constraint violating initial data and random boundary data. 
  Space-based gravitational-wave interferometers such as LISA will be sensitive to the inspiral of stellar mass compact objects into black holes with masses in the range of roughly 10^5 solar masses to (a few) 10^7 solar masses. During the last year of inspiral, the compact body spends several hundred thousand orbits spiraling from several Schwarzschild radii to the last stable orbit. The gravitational waves emitted from these orbits probe the strong-field region of the black hole spacetime and can make possible high precision tests and measurements of the black hole's properties. Measuring such waves will require a good theoretical understanding of the waves' properties, which in turn requires a good understanding of strong-field radiation reaction and of properties of the black hole's astrophysical environment which could complicate waveform generation. In these proceedings, I review estimates of the rate at which such inspirals occur in the universe, and discuss what is being done and what must be done further in order to calculate the inspiral waveform. 
  New results pertaining to colored static black hole solutions to the Einstein-Yang-Mills equations are obtained. The isolated horizons framework is used to define the concept of Hamiltonian Horizon Mass of the black hole. An unexpected relation between the ADM and Horizon masses of the black hole solution with the ADM mass of the corresponding Bartnik-McKinnon soliton is found. These results can be generalized to other non-linear theories and they suggest a general testing bed for the instability of the corresponding hairy black holes. 
  A previous work on the Ermakov approach for empty FRW minisuperspace universes of Hartle-Hawking factor ordering parameter Q=0 is extended to the Q not zero cases 
  Several energy-momentum "tensors" of gravitational field are considered and compared in the lowest approximation. Each of them together with energy-momentum tensor of point-like particles satisfies the conservation laws when equations of motion of particles are the same as in general relativity. It is shown that in Newtonian approximation the considered tensors differ one from the other in the way their energy density is distributed between energy density of interaction (nonzero only at locations of particles) and energy density of gravitational field. Starting from Lorentz invariance the Lagrangians for spin-2, mass-0 field are considered. They differ only by divergences. From these Lagrangians by Belinfante-Rosenfeld procedure the energy-momentum tensors are build. Using each of these tensors in 3-graviton vertex we obtain the corresponding metric of a Newtonian center in $G^2$ approximation. Only one of these ''field-theretical''tensors (namely the half sum of Thirring tensor and tensor obtained from Lagrangian given by Misner, Thorne and Wheeler) leads to correct value of the perihelion shift. This tensor does not coincide with Weinberg`s one (directly obtainable from Einstein equation) and gives metric of a spherical body differing (in space part of metric in the first nonlinear approximation) from Schwarzschild field in harmonic coordinates. As a result a relativistic particle in such field must move note according general relativity prescriptions. This approach puts the gravitational energy-momentum tensor on the same footing as any other energy-momentum tensor. 
  The sensitivity achievable by a pair of VIRGO detectors to stochastic and isotropic gravitational wave backgrounds of cosmological origin is discussed in view of the development of a second VIRGO interferometer. We describe a semi-analytical technique allowing to compute the signal-to-noise ratio for (monotonic or non-monotonic) logarithmic energy spectra of relic gravitons of arbitrary slope. We apply our results to the case of two correlated and coaligned VIRGO detectors and we compute their achievable sensitivities. The maximization of the overlap reduction function is discussed. We focus our attention on a class of models whose expected sensitivity is more promising, namely the case of string cosmological gravitons. We perform our calculations both for the case of minimal string cosmological scenario and in the case of a non-minimal scenario where a long dilaton dominated phase is present prior to the onset of the ordinary radiation dominated phase. In this framework, we study possible improvements of the achievable sensitivities by selective reduction of the thermal contributions (pendulum and pendulum's internal modes) to the noise power spectra of the detectors. Since a reduction of the shot noise does not increase significantly the expected sensitivity of a VIRGO pair (in spite of the relative spatial location of the two detectors) our findings support the experimental efforts directed towards a substantial reduction of thermal noise. 
  In this paper the scalar-tensor theory of gravity is assumed to describe the evolution of the universe and the gravitational scalar $\phi$ is ascribed to play the role of inflaton. The theory is characterized by the specified coupling function $\omega(\phi)$ and the cosmological function $\lambda(\phi)$. The function $\lambda(\phi)$ is nearly constant for $0<\phi<0.1$ and $\lambda(1)=0$. The functions $\lambda(\phi)$ and $\omega(\phi)$ provide a double-well potential for the motion of $\phi(t)$. Inflation commences and ends naturally by the dynamics of the scalar field. The energy density of matter increases steadily during inflation. When the constant $\Gamma$ in the action is determined by the present matter density, the temperature at the end of inflation is of the order of $10^{14} GeV$ in no need of reheating. Furthermore, the gravitational scalar is just the cold dark matter that men seek for. 
  The quantum mechanical approach developed by us recently for the evolution of the universe is used to derive an alternative derivation connecting the temperature of the cosmic background radiation and the age of the universe which is found to be similar to the one obtained by Gamow long back. By assuming the age of the universe to be $\approx$ 20 billion years, we reproduce a value of $\approx$ 2.91 K for the cosmic back-ground radiation, agreeing well with the recently measured experimental value of 2.728 K. Besides, this theory enables us to calculate the photon density and entropy associated with the background radiation and the ratio of the number of photons to the number of nucleons, which quantitatively agree with the results obtained by others. 
  In the standard model of universe the increase in mass of our observed expansive Universe is explained by the assumption of emerging the matter objects on the horizon (of the most remote visibility). However, the physical analysis of the influence of this assumption on the velocity of matter objects shows unambiguously that this hypothetical assumption contradicts the theory of gravity. 
  The method of conformal blocks for construction of global solutions in gravity for a two-dimensional metric having one Killing vector field is described. 
  We consider the force acting on a spinning charged test particle (probe particle) with the mass m and the charge q in slow rotating the Kerr-Newman-deSitter(KNdS) black hole with the mass M and the charge Q. We consider the case which the spin vector of the probe particle is parallel to the angular momentum vector of the KNdS space-time. We take account of the gravitational spin-spin interaction under the slow rotating limit of the KNdS space-time. When Q=M and q=m, we show that the force balance holds including the spin-spin interaction and the motion is approximately same as that of a particle in the deSitter space-time. This force cancellation suggests the possibility of the existence of an exact solution of spinning multi-KNdS black hole. 
  A scalar, preferred-frame theory of gravitation is summarized. Space-time is endowed with both a flat metric and a curved, "physical" metric. Motion is governed by a natural extension of Newton's second law, which implies geodesic motion only for a static field. The theory predicts Schwarzschild's exterior metric in the spherical static situation. It also predicts gravitation waves with the velocity of light. The equations of motion are recast into the "flat space - uniform time" form, and compared with the geodesic equations of motion. The principles of the post-Newtonian approximation of this theory are given, including the way to account for preferred-frame effects. This approximation is then developed more particularly for photons. It is found that the preferred-frame effects do not occur in this case, nor does the difference between Newton's second law and geodesic assumption. Thus, the post-Newtonian predictions of this theory for photons are indistinguishable from the standard post-Newtonian predictions of general relativity. 
  We show that local diff-invariant free field theories in four spacetime dimensions do not have local degrees of freedom. 
  We comment on recently proposed dissipative inflationary models. It is shown that the strength of the inflationary expansion is related to a specific combination of thermodynamic variables which is known to measure the instability of self-gravitating dissipative systems. 
  Motivated by the isotropy of the CMB spectrum, all existing studies of magnetised cosmological perturbations employ FRW backgrounds. However, it is important, to know the limits of this approximation and the effects one loses by neglecting the anisotropy of the background magnetic field. We develop a new treatment, which fully incorporates the anisotropic magnetic effects by allowing for a Bianchi I background universe. The anisotropy of the unperturbed model facilitates the closer study of the coupling between magnetism and geometry. The latter leads to a curvature stress, which accelerates positively curved perturbed regions and balances the effect of magnetic pressure gradients on matter condensations. We argue that the tension carried along the magnetic force-lines is the reason behind these magneto-curvature effects. For a relatively weak field, we also compare to the results of the almost-FRW approach. We find that some of the effects identified by the FRW treatment are in fact direction dependent, where the key direction is that of the background magnetic field vector. Nevertheless, the FRW-based approach to magnetised cosmological perturbations remains an accurate approximation, particularly on large scales, when one looks at the lowest order magnetic impact on gravitational collapse. On small scales however, the accuracy of the perturbed Friedmann framework may be compromised by extra shear effects. 
  These notes provide two derivations of the Lorentz-Dirac equation. The first is patterned after Landau and Lifshitz and is based on the observation that the half-retarded minus half-advanced potential is entirely responsible for the radiation-reaction force. The second is patterned after Dirac, and is based upon considerations of energy-momentum conservation; it relies exclusively on the retarded potential. The notes conclude with a discussion of the difficulties associated with the interpretation of the Lorentz-Dirac equation as an equation of motion for a point charge. The presentation is essentially self-contained, but the reader is assumed to possess some elements of differential geometry (necessary for the second derivation only). 
  We study the quantum creation of black hole pairs in the (anti-)de Sitter space background. These black hole pairs in the Kerr-Newman family are created from constrained instantons. At the $WKB$ level, for the chargeless and nonrotating case, the relative creation probability is the exponential of (the negative of) the entropy of the universe. Also for the remaining cases of the family, the creation probability is the exponential of (the negative of) one quarter of the sum of the inner and outer black hole horizon areas. In the absence of a general no-boundary proposal for open universes, we treat the creations of the closed and the open universes in the same way. 
  It is shown some exact solutions in the Brans-Dicke (BD) theory for a Bianchi V metric having the property of inflationary expansion, graceful exit, and asymptotic evolution to a Friedmann-Robertson-Walker (FRW) open model. It is remarkable that an inflationary behaviour can occur, even without a cosmological potential or constant. However, the horizon and flatness problems cannot be solve within the standard BD theory because the inflationary period is severely restricted by the value of the BD parameter $\omega$. 
  In this work we present cosmological quantum solutions for all Bianchi Class A cosmological models obtained by means of supersymmetric quantum mechanics . We are able to write one general expression for all bosonic components occuring in the Grassmann expansion of the wave function of the Universe for this class of models. These solutions are obtained by means of a more general ansatz for the so-called master equations. 
  In this talk we discuss the quantisation of a class of string cosmology models characterised by scale factor duality invariance. The amplitudes for the full set of classically allowed and forbidden transitions are computed by applying the reduced phase space and path integral methods. In particular, the path integral calculation clarifies the meaning of the instanton-like behaviour of the transition amplitudes that has been first pointed out in previous investigations. 
  In this contribution I intend to give a summary of the new relevant results obtained by using the general superenergy tensors. After a quick review of the definition and properties of these tensors, several of their mathematical and physical applications are presented. In particular, their interest and usefulness is mentioned or explicitly analyzed in 1) the study of causal propagation of general fields; 2) the existence of an infinite number of conserved quantities in Ricci-flat spacetimes; 3) the different gravitational theories, such as Einstein's General Relativity or, say, $n=11$ supergravity; 4) the appearance of some scalars possibly related to entropy or quality factors; 5) the possibility of superenergy exchange between different physical fields and the appearance of mixed conserved currents. 
  The nexus between the gravitational field and the space-time metric was an essential element in Einstein's development of General Relativity and led him to his discovery of the field equations for the gravitational field/metric. I will argue here that the metric is in fact an inessential element of this theory and can be dispensed with entirely. Its sole function in the theory was to describe the space-time measurements made by ideal clocks and rods. However, the behavior of model clocks and measuring rods can be derived directly from the field equations of General Relativity using the Einstein-Infeld-Hoffmann (EIH) approiximation procedure. Therefore one does not need to introduce these ideal clocks and rods and hence has no need of a metric. 
  The near extremal Reissner-Nordstrom black holes in arbitrary dimensions ca be modeled by the Jackiw-Teitelboim (JT) theory. The asymptotic Virasoro symmetry of the corresponding JT model exactly reproduces, via Cardy's formula, the deviation of the Bekenstein-Hawking entropy of the Reissner-Nordstrom black holes from extremality. We also comment how can we extend this approach to investigate the evaporation process. 
  The gravitational fields of vacuumless global and gauge strings have been studied in Brans-Dicke theory under the weak field assumption of the field equations. It has been shown that both global and gauge string can have repulsive as well as attractive gravitational effect in Brans-Dicke theory which is not so in General Relativity. 
  Formulations of Eulerian general relativistic ideal hydrodynamics in conservation form are analyzed in some detail, with particular emphasis to geometric source terms. Simple linear transformations of the equations are introduced and the associated equivalence class is exploited for the optimization of such sources. A significant reduction of their complexity is readily possible in generic spacetimes. The local characteristic structure of the standard member of the equivalence class is analyzed for a general equation of state (EOS). This extends previous results restricted to the polytropic case. The properties of all other members of the class, in particular specialized forms employing Killing symmetries, are derivable from the standard form. Special classes of EOS are identified for both spacelike and null foliations, which lead to explicit inversion of the state vector and computational savings. The entire approach is equally applicable to spacelike or lightlike foliations and presents a complete proposal for numerical relativistic hydrodynamics on stationary or dynamic geometries. 
  The two known exact solutions of Einstein's field equations describing rotating objects of physical significance - a black hole and a rigidly rotating disk of dust - are discussed using a single mathematical framework related to Jacobi's inversion problem. Both solutions can be represented in such a form that they differ in the choice of a complex parameter and a real solution of the axisymmetric Laplace equation only. A recently found family of solutions describing differentially rotating disks of dust fits into the same scheme. 
  If stationary, the spectrum of vacuum field noise (VFN) is an important ingredient to get information about the curvature invariants of classical worldlines (relativistic classical trajectories). For scalar quantum field vacua there are six stationary cases as shown by Letaw some time ago, these are reviewed here. However, the non-stationary vacuum noises are not out of reach and can be processed by a few mathematical methods which I briefly comment on. Since the information about the kinematical curvature invariants of the worldlines is of radiometric origin, hints are given on a more useful application to radiation and beam radiometric standards at relativistic energies 
  The use of fused-silica ribbons as suspensions in gravitational wave interferometers can result in significant improvements in pendulum mode thermal noise. Surface loss sets a lower bound to the level of noise achievable, at what level depends on the dissipation depth and other physical parameters. For LIGO II, the high breaking strength of pristine fused silica filaments, the correct choice of ribbon aspect ratio (to minimize thermoelastic damping), and low dissipation depth combined with the other achievable parameters can reduce the pendulum mode thermal noise in a ribbon suspension well below the radiation pressure noise. Despite producing higher levels of pendulum mode thermal noise, cylindrical fiber suspensions provide an acceptable alternative for LIGO II, should unforeseen problems with ribbon suspensions arise. 
  Traditional approaches to energy-momentum localization led to reference frame dependent pseudotensors. The more modern idea is quasilocal energy-momentum. We take a Hamiltonian approach. The Hamiltonian boundary term gives not only the quasilocal values but also boundary conditions via the Hamiltonian variation boundary principle. Selecting a Hamiltonian boundary term involves several choices. We found that superpotentials can serve as Hamiltonian boundary terms, consequently pseudotensors are actually quasilocal and legitimate. Various Hamiltonian boundary term quasilocal expressions are considered including some famous pseudotensors, M{\o}ller's tetrad-teleparallel ``tensor'', Chen's covariant expressions, the expressions of Katz & coworkers, the expression of Brown & York, and some spinor expressions. We emphasize the need for identifying criteria for a good choice. 
  In the following we undertake to describe how macroscopic space-time (or rather, a microscopic protoform of it) is supposed to emerge as a superstructure of a web of lumps in a stochastic discrete network structure. As in preceding work (mentioned below), our analysis is based on the working philosophy that both physics and the corresponding mathematics have to be genuinely discrete on the primordial (Planck scale) level. This strategy is concretely implemented in the form of \tit{cellular networks} and \tit{random graphs}. One of our main themes is the development of the concept of \tit{physical (proto)points} or \tit{lumps} as densely entangled subcomplexes of the network and their respective web, establishing something like \tit{(proto)causality}. It may perhaps be said that certain parts of our programme are realisations of some early ideas of Menger and more recent ones sketched by Smolin a couple of years ago. We briefly indicate how this \tit{two-story-concept} of \tit{quantum} space-time can be used to encode the (at least in our view) existing non-local aspects of quantum theory without violating macroscopic space-time causality. 
  We use computational algorithms recently developed by us to study completely four index divergence free quadratic in Riemann tensor polynomials in GR. Some results are new and some other reproduce and/or correct known ones. The algorithms are part of a Mathematica package called Tools of Tensor Calculus (TTC)[web address: http://baldufa.upc.es/ttc] 
  In this work, I develop an alternative explanation for the acceleration of the cosmic expansion, which seems to be a result of recent high redshift Supernova data. In the current interpretation, this cosmic acceleration is explained by including a positive cosmological constant term (or vacuum energy), in the standard Friedmann models. Instead, I will consider a Locally Rotationally Symmetric (LRS) and spherically symmetric (SS), but inhomogeneous spacetime, with a barotropic perfect fluid equation of state for the cosmic matter. The congruence of matter has acceleration, shear and expansion. Within this framework the kinematical acceleration of the cosmic fluid or, equivalently, the inhomogeneity of matter, is just the responsible of the SNe Ia measured cosmic acceleration. Although in our model the Cosmological Principle is relaxed, it maintains almost isotropy about our worldline in agreement with CBR observations. 
  We are presenting an algorithm capable of simplifying tensor polynomials with indices when the building tensors have index symmetry properties. These properties include simple symmetry, cyclicity and those due to the presence of covariant derivatives. The algorithm is part of a Mathematica package called Tools of Tensor Calculus (TTC) [web address: http://baldufa.upc.es/ttc] 
  A sufficiently massive collapsing star will end its life as a spacetime singularity. The nature of the Hawking radiation emitted during collapse depends critically on whether the star's boundary conditions are such as would lead to the eventual formation of a black hole or, alternatively, to the formation of a naked singularity. This latter possibility is not excluded by the singularity theorems. We discuss the nature of the Hawking radiation emitted in each case. We justify the use of Bogoliubov transforms in the presence of a Cauchy horizon and show that if spacetime is assumed to terminate at the Cauchy horizon, the resulting spectrum is thermal, but with a temperature different from the Hawking temperature. 
  This paper is an extended version of a talk at the conference Constrained Dynamics and Quantum Gravity QG99. It reviews some work on the quantum collapse of the spherically symmetric gravitating thin shell of zero rest mass. Recent results on Kucha\v{r} decomposition are applied. The constructed version of quantum mechanics is unitary, although the shell falls under its Schwarzschild radius if its energy is high enough. Rather that a permanent black hole, something like a transient black and white hole pair seems to be created in such a case. 
  Since the last Amaldi meeting in 1997 we have learned that the r-modes of rapidly rotating neutron stars are unstable to gravitational radiation reaction in astrophysically realistic conditions. Newborn neutron stars rotating more rapidly than about 100Hz may spin down to that frequency during up to one year after the supernova that gives them birth, emitting gravitational waves which might be detectable by the enhanced LIGO interferometers at a distance which includes several supernovae per year. A cosmological background of these events may be detectable by advanced LIGO. The spins (about 300Hz) of neutron stars in low-mass x-ray binaries may also be due to the r-mode instability (under different conditions), and some of these systems in our galaxy may also produce detectable gravitational waves--see the review by G. Ushomirsky in this volume. Much work is in progress on developing our understanding of r-mode astrophysics to refine the early, optimistic estimates of the detectability of the gravitational waves. 
  We consider the effect of a positive cosmological constant on spherical gravitational collapse to a black hole for a few simple, analytic cases. We construct the complete Oppenheimer-Snyder-deSitter (OSdS) spacetime, the generalization of the Oppenheimer-Snyder solution for collapse from rest of a homogeneous dust ball in an exterior vacuum. In OSdS collapse, the cosmological constant may affect the onset of collapse and decelerate the implosion initially, but it plays a diminishing role as the collapse proceeds. We also construct spacetimes in which a collapsing dust ball can bounce, or hover in unstable equilibrium, due to the repulsive force of the cosmological constant. We explore the causal structure of the different spacetimes and identify any cosmological and black hole event horizons which may be present. 
  We analyse in a systematic way the (non-)compact n-dimensional Einstein Weyl spaces equipped with a cohomogeneity-one metric. With no compactness hypothesis, we prove that, as soon as the (n-1)-dimensional space is an homogeneous reductive Riemannian space with an unimodular group of left-acting isometries G   1)a non-exact Einstein-Weyl stucture may exist only if the (n-1)-dimensional homogeneous space G/H contains a non trivial subgroup H' that commutes with the isotropy subgroup H,   2) the extra isometry due to this Killing vector corresponds to the right-action of one of the generators of the algebra of the subgroup H'.   We also prove that the subclass with a one-dimensional subgroup H' corresponds to n-dimensional Riemannian locally conformally K\"ahler metrics, the (n-2)-dimensional space G/(HxH') being an arbitrary compact symmetric K\"ahler coset space. 
  We solve the Wheeler-DeWitt equation for the minisuperspace of a cosmological model of Bianchi type I with a minimally coupled massive scalar field $\phi$ as source by generalizing the calculation of Lukash and Schmidt [1]. Contrarily to other approaches we allow strong anisotropy. Combining analytical and numerical methods, we apply an adiabatic approximation for $\phi$, and as new feature we find a period-doubling bifurcation. This bifurcation takes place near the cosmological quantum boundary, i.e., the boundary of the quasiclassical region with oscillating $\psi$-function where the WKB-approximation is good. The numerical calculations suggest that such a notion of a ``cosmological quantum boundary'' is well-defined, because sharply beyond that boundary, the WKB-approximation is no more applicable at all. This result confirms the adequateness of the introduction of a cosmological quantum boundary in quantum cosmology. 
  Recent measurements require modifications in conventional cosmology by way of introducing components other than ordinary matter into the total energy density in the universe. On the basis of some dimensional considerations in line with quantum cosmology, Chen and Wu [W. Chen and Y. Wu, Phys. Rev. D 41, 695 (1990)] have argued that an additional component, which corresponds to an effective cosmological constant $\Lambda$ must vary as a^{-2} in the classical era. Their decaying-$\Lambda$ model assumes inflation and yields a value for q_{0}, which is not compatible with observations. We generalize this model by arguing that the Chen-Wu ansatz is applicable to the total energy density of the universe and not to $\Lambda$ alone. The resulting model, which has a coasting evolution (i.e., $a \propto t$), is devoid of the problems of horizon, flatness, monopole, cosmological constant, size, age and generation of density perturbations. However, to avoid serious contradictions with big bang nucleosynthesis, the model has to make the predictions $\Omega_{m} = 4/3$ and $\Omega_{\Lambda}=2/3$, which in turn are at variance with current observational values. 
  A simple example is given of the implementation of the usual method of asymptotic expansions for weak gravitational fields. A scalar, preferred-frame theory of gravitation is considered, but the method is general. Two kinds of asymptotic expansion are a priori possible: "post-Newtonian" (PN) or "post-Minkowskian", the latter allowing to account directly for propagation effects. However, it is shown that only the PN asymptotic expansion is compatible with the Newtonian limit. It is also shown that, in the scalar theory, there is no non-Newtonian effect (in particular, no propagation effect) up to the second order, i.e., the order 1/c^2. 
  We investigate FRW cosmological solutions in the theory of modulus field coupled to gravity through a Gauss-Bonnet term. The explicit analytical forms of nonsingular asymptotics are presented for power-law and exponentially steep modulus coupling functions. We study the influence of modulus field potential on these asymptotical regimes and find some forms of the potential which do not destroy the nonsingular behavior. In particular, we obtain that exponentially steep coupling functions arising from the string theory do not allow nonsingular past asymptotic unless modulus field potential tends to zero for modulus field $\psi \to \pm \infty$. Finally, the modification of the chaotic dynamics in the closed FRW universe due to presence of the Gauss-Bonnet term is discussed. 
  Fluctuations on de Sitter solution of Einstein-Cartan field equations are obtained in terms of the matter density primordial density fluctuations and spin-torsion density and matter density fluctuations obtained from COBE data. Einstein-de Sitter solution is shown to be unstable even in the absence of torsion.The spin-torsion density fluctuation to generate a deflationary phase is computed from the COBE data. 
  An N + 1 dimensional quantum mechanical model for the origin of the universe results in a 58 e-fold inflation and a cosmological constant/vacuum energy density of the same order of magnitude as the critical density. 
  Stars and black holes are sources of gravitational radiation in many phases of their life, and the signals they emit exhibit features that are characteristic of the generating process. Emitted since the beginning of star formation, these signals also contribute to create a stochastic background of gravitational waves. We shall show how the spectral properties of this background can be estimated in terms of the energy spectrum of each single event and of the star formation rate history, which is now deducible from astronomical observations. We shall further discuss the process of scattering of masses by stars and black holes, showing that, unlike black holes, stars emit signals that carry a clear signature of the nature of the source. 
  The detection of gravitational waves from inspiraling compact binaries using matched filtering depends crucially on the availability of accurate template waveforms. We determine whether the accuracy of the templates' phasing can be improved by solving the post-Newtonian energy balance equation numerically, rather than (as is normally done) analytically within the post-Newtonian perturbative expansion. By specializing to the limit of a small mass ratio, we find evidence that there is no gain in accuracy. 
  We discuss the case of histories labelled by a continuous time parameter in the {\em History Projection Operator} consistent-histories quantum theory. We describe how the appropriate representation of the history algebra may be chosen by requiring the existence of projection operators that represent propositions about time averages of the energy. We define the action operator for the consistent histories formalism, as the quantum analogue of the classical action functional, for the simple harmonic oscillator case. We show that the action operator is the generator of two types of time transformations that may be related to the two laws of time-evolution of the standard quantum theory: the `state-vector reduction' and the unitary time-evolution. We construct the corresponding classical histories and demonstrate the relevance with the quantum histories; we demonstrate how the requirement of the temporal logic structure of the theory is sufficient for the definition of classical histories. Furthermore, we show the relation of the action operator to the decoherence functional which describes the dynamics of the system. Finally, the discussion is extended to give a preliminary account of quantum field theory in this approach to the consistent histories formalism. 
  We develop a new algorithm for the quantisation of systems with first-class constraints. Our approach lies within the (History Projection Operator) continuous-time histories quantisation programme. In particular, the Hamiltonian treatment (either classical or quantum) of parameterised systems is characterised by the loss of the notion of time in the space of true degrees of freedom (i.e. the `problem of time'). The novel temporal structure of the HPO theory (two laws of time transformation that distinguish between the temporal logical structure and the dynamics) persists after the imposition of the constraints, hence the problem of time does not arise. We expound the algorithm for both the classical and quantum cases and apply it to simple models. 
  The exact static solutions in the higher dimensional Einstein-Maxwell-Klein- Gordon theory are investigated. With the help of the methods developed for the effective dilaton type gauge gravity models in two dimensions, we find new spherically and hyperbolically symmetric solutions which generalize the four dimensional configurations of Dereli-Eris. We show that, like in four dimensions, the non-trivial scalar field yields, in general, a naked singularity. The new solutions are compared with the higher dimensional Brans-Dicke black hole type solutions. 
  Exact solutions of Einstein's equations in 2+1-dimensional anti-de Sitter space containing any number of black holes are described. In addition to the black holes these spacetimes can possess ``internal'' structure. Accordingly the generic spacetime of this type depends on a large number of parameters. Half of these can be taken as mass parameters, and the rest as the conjugate (angular) momenta. The time development and horizon structure of some of these spacetimes are sketched. 
  We present in this paper a covariant quantization of the ``massive'' vector field on de Sitter (dS) space based on analyticity in the complexified pseudo-Riemanian manifold. The correspondence between unitary irreducible representations of the de Sitter group and the field theory on de Sitter space-time is essential in our approach. We introduce the Wightman-G\"arding axiomatic for vector field on dS space. The Hilbert space structure and the unsmeared field operators $K_\alpha(x)$ are also defined. This work is in the direct continuation of previous one concerning the scalar and the spinor cases. 
  In this letter we investigate whether the isotropy problem is naturally solved in inflationary cosmologies inspired by string theory, so called pre-big-bang cosmologies. We find that, in contrast to what happens in the more common 'potential inflation' models, initial anisotropies do not decay during pre-big-bang inflation. 
  We analyse the quantization procedure of the spinor field in the Rindler spacetime, showing the boundary conditions that should be imposed to the field, in order to have a well posed theory. We then investigate the relationship between this construction and the usual one in Minkowski spacetime. This leads to the concept of "Unruh effect", that is the thermal nature of the Minkowski vacuum state from the point of view of an accelerated observer. It is demostrated that the two constructions are qualitatively different and can not be compared and consequently the conventional interpretation of the Unruh effect is questionable. 
  The existence of a Noether symmetry for a given minisuperspace cosmological model is a sort of selection rule to recover classical behaviours in cosmic evolution since oscillatory regimes for the wave function of the universe come out. The so called Hartle criterion to select correlated regions in the configuration space of dynamical variables can be directly connected to the presence of a Noether symmetry and we show that such a statement works for generic extended theories of gravity in the framework of minisuperspace approximation. Examples and exact cosmological solutions are given for nonminimally coupled and higher--order theories. 
  Higher-order corrections of Einstein-Hilbert action of general relativity can be recovered by imposing the existence of a Noether symmetry to a class of theories of gravity where Ricci scalar R and its d'Alembertian $\Box R$ are present. In several cases, it is possible to get exact cosmological solutions or, at least, to simplify dynamics by recovering constants of motion. The main result is that a Noether vector seems to rule the presence of higher-order corrections of gravitaty. 
  We discuss reality conditions and the relation between spacetime diffeomorphisms and gauge transformations in Ashtekar's complex formulation of general relativity. We produce a general theoretical framework for the stabilization algorithm for the reality conditions, which is different from Dirac's method of stabilization of constraints. We solve the problem of the projectability of the diffeomorphism transformations from configuration-velocity space to phase space, linking them to the reality conditions. We construct the complete set of canonical generators of the gauge group in the phase space which includes all the gauge variables. This result proves that the canonical formalism has all the gauge structure of the Lagrangian theory, including the time diffeomorphisms. 
  We discuss the relation between spacetime diffeomorphisms and gauge transformations in theories of the Yang-Mills type coupled with Einstein's General Relativity. We show that local symmetries of the Hamiltonian and Lagrangian formalisms of these generally covariant gauge systems are equivalent when gauge transformations are required to induce transformations which are projectable under the Legendre map. Although pure Yang-Mills gauge transformations are projectable by themselves, diffeomorphisms are not. Instead the projectable symmetry group arises from infinitesimal diffeomorphism-inducing transformations which must depend on the lapse function and shift vector of the spacetime metric plus associated gauge transformations. Our results are generalizations of earlier results by ourselves and by Salisbury and Sundermeyer. 
  We construct explicitly generators of projectable four-dimensional diffeomorphisms and triad rotation gauge symmetries in a model of vacuum gravity where the fundamental dynamical variables in a Palatini formulation are taken to be a lapse, shift, densitized triad, extrinsic curvature, and the time-like components of the Ricci rotation coefficient. Time-foliation-altering diffeomorphisms are not by themselves projectable under the Legendre transformations. They must be accompanied by a metric- and triad-dependent triad rotation. The phase space on which these generators act includes all of the gauge variables of the model. 
  We study, in this paper, curvature inheritance symmetry (CI), $\pounds_{\xi}R_{bcd}^{a}=2\alpha R_{bcd}^{a}$, where $\alpha $ is a scalar function, for string cloud and string fluid in the context of general relativity. Also, we have obtained some result when a proper CI (i.e., $\alpha \neq 0$) is also a conformal Killing vector. 
  We consider a model in which accelerated particles experience line--elements with maximal acceleration corrections. When applied to the Schwarzschild metric, the effective field experienced by accelerated test particles contains corrections that vanish in the limit $\hbar\to 0$, but otherwise affect the behaviour of matter greatly. A new effect appears in the form of a spherical shell, external to the Schwarzschild sphere, impenetrable to classical particles. 
  The causal relation $K^+$ was introduced by Sorkin and Woolgar to extend the standard causal analysis of $C^2$ spacetimes to those that are only $C^0$. Most of their results also hold true in the case of spacetimes with degeneracies. In this paper we seek to examine $K^+$ explicitly in the case of Lorentzian topology changing Morse spacetimes containing isolated degeneracies. We first demonstrate some interesting features of this relation in globally Lorentzian spacetimes. In particular, we show that $K^+$ is robust and that it coincides with the Seifert relation when the spacetime is stably causal. Moreover, the Hawking and Sachs characterisation of causal continuity translates into a natural expression in terms of $K^+$ for general spacetimes. We then examine $K^+$ in topology changing Morse spacetimes both with and without the degeneracies and find further characterisations of causal continuity. 
  Talk given at the Conference ``Constrained Dynamics and Quantum Gravity 99'', Villasimius (Sardinia, Italy), September 13-17, 1999 
  We extract all the invariants (i.e. all the functions which do not depend on the choice of phase-space coordinates) of the dynamics of two point-masses, at the third post-Newtonian (3PN) approximation of general relativity. We start by showing how a contact transformation can be used to reduce the 3PN higher-order Hamiltonian derived by Jaranowski and Sch\"afer to an ordinary Hamiltonian. The dynamical invariants for general orbits (considered in the center-of-mass frame) are then extracted by computing the radial action variable $\oint{p_r}dr$ as a function of energy and angular momentum. The important case of circular orbits is given special consideration. We discuss in detail the plausible ranges of values of the two quantities $\oms$, $\omk$ which parametrize the existence of ambiguities in the regularization of some of the divergent integrals making up the Hamiltonian. The physical applications of the invariant functions derived here (e.g. to the determination of the location of the last stable circular orbit) are left to subsequent work. 
  We use the energy-momentum complexes of Landau and Lifshitz and Papapetrou to obtain the energy distribution in Melvin's magnetic universe. For this space-time we find that these definitions of energy give the same and convincing results. The energy distribution obtained here is the same as we obtained earlier for the same space-time using the energy-momentum complex of Einstein. These results uphold the usefulness of the energy-momentum complexes. 
  Recently proposed quantization in field theory based on an analogue of Hamiltonian formulation which treats space and time on equal footing (the so-called De Donder-Weyl theory) is applied to General Relativity in metric variables. We formulate a covariant analogue of the Schroedinger equation for the wave function of space-time and metric variables and a supplementary ``bootstrap condition'' which enables us to incorporate classical metric geometry as an approximate notion - a result of quantum averaging - in the self-consistent with the underlying quantum dynamics way. In this sense an independence of an arbitrarily chosen metric background is ensured. 
  The equations governing the evolution of non-minimally coupled scalar matter and the scale factor of a Robertson-Walker universe are derived from a minisuperspace action. As for the minimally coupled case, it is shown that the entire semiclassical dynamics can be retrieved from the Wheeler-DeWitt equation via the Born-Oppenheimer reduction, which properly yields the (time-time component of the) covariantly conserved energy-momentum tensor of the scalar field as the source term for gravity. However, for a generic coupling, the expectation value of the operator which evolves the matter state in time is not equal to the source term in the semiclassical Einstein equation for the scale factor of the universe and the difference between these two quantities is related to the squeezing and quantum fluctuations of the matter state. We also argue that matter quantum fluctuations become relevant in an intermediate regime between quantum gravity and semiclassical gravity and study several cases in detail. 
  We consider a wave-function approach to the false vacuum decay with gravity and present a new method to calculate the tunneling amplitude under the WKB approximation. The result agrees with the one obtained by the Euclidean path-integral method, but gives a much clearer interpretation of an instanton (Euclidean bounce solution) that dominates the path integral. In particular, our method is fully capable of dealing with the case of a thick wall with the radius of the bubble comparable to the radius of the instanton, thus surpassing the path-integral method whose use can be justified only in the thin-wall and small bubble radius limit. The calculation is done by matching two WKB wave functions, one with the final state and another with the initial state, with the wave function in the region where the scale factor of the metric is sufficiently small compared with the inverse of the typical energy scale of the field potential at the tunneling. The relation of the boundary condition on our wave function for the false vacuum decay with Hartle-Hawking's no-boundary boundary condition and Vilenkin's tunneling boundary condition on the wave function of the universe is also discussed. 
  We present the first results from our Post-Newtonian (PN) Smoothed Particle Hydrodynamics (SPH) code, which has been used to study the coalescence of binary neutron star (NS) systems. The Lagrangian particle-based code incorporates consistently all lowest-order (1PN) relativistic effects, as well as gravitational radiation reaction, the lowest-order dissipative term in general relativity. We test our code on sequences of single NS models of varying compactness, and we discuss ways to make PN simulations more relevant to realistic NS models. We also present a PN SPH relaxation procedure for constructing equilibrium models of synchronized binaries, and we use these equilibrium models as initial conditions for our dynamical calculations of binary coalescence. Though unphysical, since tidal synchronization is not expected in NS binaries, these initial conditions allow us to compare our PN work with previous Newtonian results.   We compare calculations with and without 1PN effects, for NS with stiff equations of state, modeled as polytropes with $\Gamma=3$. We find that 1PN effects can play a major role in the coalescence, accelerating the final inspiral and causing a significant misalignment in the binary just prior to final merging. In addition, the character of the gravitational wave signal is altered dramatically, showing strong modulation of the exponentially decaying waveform near the end of the merger. We also discuss briefly the implications of our results for models of gamma-ray bursts at cosmological distances. 
  The behaviour of the wave function of the Universe under the barrier for anisotropic cosmological Bianchi type IX model with account of influence of the scalar field is explored. In view of known difficulties with interpretation of multidimensional wave functions the method of reduction of such problems to one-dimensional is offered. For this purpose in frameworks of semiclassical approach the system of characteristics equations relative to one variable is written out. This system describe a bundle of the characteristics along which the multidimensional problem is reduced to one-dimensional one that allows to utillize the standard interpretation of the wave function as well as for usual Schrodinger equation. The obtained results for Bianchi type IX model are reduced to the following statement: the Universe tunnels through the barrier from an isotropic state with zero initial value of the scalar field and appear in classically allowed region with small anisotropy that is necessary for providing of long-lived inflation for deriving the Universe such as ours. 
  Massive black hole binary systems are among the most interesting sources for the Laser Interferometer Space Antenna (LISA); gravitational radiation emitted during the last year of in-spiral could be detectable with a very large signal-to-noise ratio for sources at cosmological distance. Here we discuss the impact of LISA for astronomy and cosmology; we review our present understanding of the relevant issues, and highlight open problems that deserve further investigations. 
  We give arguments for the existence of ``radial excitations'' of gravitational global monopoles with any number of zeros of the Higgs field and present numerical results for solutions with up to two zeros. All these solutions possess a de Sitter like cosmological horizon, outside of which they become singular. In addition we study corresponding static ``hairy'' black hole solutions, representing black holes sitting inside a global monopole core. In particular, we determine their existence domains as a function of their horizon radius rh. 
  We develop a method for constructing of the basic functions with witch to expand small perturbations of space-time in General Relativity. The method allows to obtain the tensor harmonics for perturbations of the background space-time admitting an arbitrary group of isometry, and to split the linearized Einstein equations into the irreducible combinations. The essential point of the work is the construction of the generalized Casimir operator for the underlying group, which is defined not only on vector but also on tensor fields. It is used to construct the basic functions for spaces of tensor representations of the background metric's group of isometry. The method, being general, is applied here to construction of the basic functions for the case of the three-parameter group of isometry G_3 acting on the two-dimensional non-isotropic surface of transitivity. As quick illustrations of the method we consider the well-known particular cases: cylindrical harmonic for the flat space-time, and Regge-Wheller spherical harmonics for the Schwarzschild metric. 
  Interacting white dwarf binary star systems, including helium cataclysmic variable (HeCV) systems, are expected to be strong sources of gravitational radiation, and should be detectable by proposed space-based laser interferometer gravitational wave observatories such as LISA. Several HeCV star systems are presently known and can be studied optically, which will allow electromagnetic and gravitational wave observations to be correlated. Comparisons of the phases of a gravitational wave signal and the orbital light curve from an interacting binary white dwarf star system can be used to bound the mass of the graviton. Observations of typical HeCV systems by LISA could potentially yield an upper bound on the inverse mass of the graviton as strong as $h/m_{g} = \lambda_{g} > 1 \times 10^{15}$ km ($m_{g} < 1 \times 10^{-24}$ eV), more than two orders of magnitude better than present solar system derived bounds. 
  There have been many attempts to understand the statistical origin of black-hole entropy. Among them, entanglement entropy and the brick wall model are strong candidates. In this paper, first, we show that the entanglement approach reduces to the brick wall model when we seek the maximal entanglement entropy. After that, the stability of the brick wall model is analyzed in a rotating background. It is shown that in the Kerr background without horizon but with an inner boundary a scalar field has complex-frequency modes and that, however, the imaginary part of the complex frequency can be small enough compared with the Hawking temperature if the inner boundary is sufficiently close to the horizon, say at a proper altitude of Planck scale. Hence, the brick wall model is well defined even in a rotating background if the inner boundary is sufficiently close to the horizon. These results strongly suggest that the entanglement approach is also well defined in a rotating background. 
  Fluctuations on de Sitter solution of Einstein-Cartan field equations are obtained in terms of the matter density primordial density fluctuations and spin-torsion density and matter density fluctuations obtained from COBE data. Einstein-de Sitter solution is shown to be unstable even in the absence of torsion.The spin-torsion density fluctuation is simply computed from the Einstein-Cartan equations and from COBE data. 
  By modifying the Chen and Wu ansatz, we have investigated some Friedmann models in which $\Lambda$ varies as $\rho$. In order to test the consistency of the models with observations, we study the angular size - redshift relation for 256 ultracompact radio sources selected by Jackson and Dodgson. The angular sizes of these sources were determined by using very long-baseline interferometry in order to avoid any evolutionary effects. The models fit the data very well and require an accelerating universe with a positive cosmological constant. Open, flat and closed models are almost equally probable, though the open model provides a comparatively better fit to the data. The models are found to have intermediate density and imply the existence of dark matter, though not as much as in the canonical Einstein-de Sitter model. 
  In order to test the consistency of the cosmological models with observations as well as to measure the different cosmological parameters, data on angular sizes and reshifts of ultracompact radio sources, compiled by Jackson and Dodgson, has been used recently by several authors in the models with constant $\Lambda$, concluding that a non-zero $\Lambda$ is inevitable. In our attempt to solve the cosmological constant problem, we examine this data for a variable $\Lambda$ by considering a model with a contracted Ricci-collineation along the fluid flow which demands $\Lambda$ to be variable. We find that the acceptable fits are also obtained for open models with small $\Lambda$ of either sign. The models are found to be decelerating and require intermediate density, higher than that in the models of Jackson and Dodgson but not as high as in the cannonical cold dark matter model of Kellermann. 
  We show that there are no new consistent cosmological perfect fluid solutions when in an open neighbourhood ${\cal U}$ of an event the fluid kinematical variables and the electric and magnetic Weyl curvature are all assumed rotationally symmetric about a common spatial axis, specialising the Weyl curvature tensor to algebraic Petrov type D. The consistent solutions of this kind are either locally rotationally symmetric, or are subcases of the Szekeres dust models. Parts of our results require the assumption of a barotropic equation of state. Additionally we demonstrate that local rotational symmetry of perfect fluid cosmologies follows from rotational symmetry of the Riemann curvature tensor and of its covariant derivatives only up to second order, thus strengthening a previous result. 
  We investigate excitation of Kaluza-Klein modes due to the parametric resonance caused by oscillation of radius of compactification. We consider a gravitational perturbation around a D-dimensional spacetime, which we compactify on a (D-4)-sphere to obtain a 4-dimensional theory. The perturbation includes the so-called Kaluza-Klein modes, which are massive in 4-dimension, as well as zero modes, which is massless in 4-dimension. These modes appear as scalar, vector and second-rank symmetric tensor fields in the 4-dimensional theory. Since Kaluza-Klein modes are troublesome in cosmology, quanta of these Kaluza-Klein modes should not be excited abundantly. However, if radius of compactification oscillates, then masses of Kaluza-Klein modes also oscillate and, thus, parametric resonance of Kaluza-Klein modes may occur to excite their quanta. In this paper we consider part of Kaluza-Klein modes which correspond to massive scalar fields in 4-dimension and investigate whether quanta of these modes are excited or not in the so called narrow resonance regime of the parametric resonance. We conclude that at least in the narrow resonance regime quanta of these modes are not excited so catastrophically. 
  The approach, referred to as "monodromy transform", provides some general base for solution of all known integrable space - time symmetry reductions of Einstein equations for the case of pure vacuum gravitational fields, in the presence of gravitationally interacting massless fields, as well as for some string theory induced gravity models. In this communication we present the key points of this approach, applied to Einstein equations for vacuum and to Einstein - Maxwell equations for electrovacuum fields in the cases, reducible to the known Ernst equations. Definition of the monodromy data, formulation and solution of the direct and inverse problems of the monodromy transform, a proof of existence and uniqueness of their solutions, the structure of the basic linear singular integral equations and their regularizations, which lead to the equations of (quasi-)Fredholm type are also discussed. A construction of general local solution of these equations is given in terms of homogeneously convergent functional series. 
  We find an exact solution for multi-black strings in the brane world with warped compactification. 
  The simplest (3+1)D Regge calculus model (with three-dimensional discrete space and continuous time) is considered which describes evolution of the simplest closed two-tetrahedron piecewise flat manifold in the continuous time. The measure in the path integral which describes canonical quantisation of the model in terms of area bivectors and connections as independent variables is found. It is shown that selfdual-antiselfdual splitting of the variables simplifies the integral although does not admit complete separation of (anti-)selfdual sector. 
  Dynamics of a Regge three-dimensional (3D) manifold in a continuous time is considered. The manifold is closed consisting of the two tetrahedrons with identified corresponding vertices. The action of the model is that obtained via limiting procedure from the general relativity (GR) action for the completely discrete 4D Regge calculus. It closely resembles the continuous general relativity action in the Hilbert-Palatini (HP) form but possesses finite number of the degrees of freedom. The canonical structure of the theory is described. Central point is appearance of the new relations with time derivatives not following from the Lagrangian but serving to ensure completely discrete 4D Regge calculus origin of the system. In particular, taking these into account turns out to be necessary to obtain the true number of the degrees of freedom being the number of linklengths of the 3D Regge manifold at a given moment of time. 
  An approximate solution to Einstein's equations representing two widely-separated non-rotating black holes in a circular orbit is constructed by matching a post-Newtonian metric to two perturbed Schwarzschild metrics. The spacetime metric is presented in a single coordinate system valid up to the apparent horizons of the black holes. This metric could be useful in numerical simulations of binary black holes. Initial data extracted from this metric have the advantages of being linked to the early inspiral phase of the binary system, and of not containing spurious gravitational waves. 
  We investigate the thermodynamic arrow of time in a time-symmetrically recollapsing universe by calculating quantum mechanically the entropy production of a massive scalar field. It is found that even though the Hamiltonian has a time-reversal symmetry with respect to the maximum expansion of the universe, the entropy production is generic and the total entropy of the scalar field increases monotonically. We conclude that the thermodynamic arrow of time is a universal phenomenon even in the expanding and subsequently recollapsing universe due to the parametric interaction of matter field with gravity. 
  We study the analytic structure of the S-matrix which is obtained from the reduced Wheeler-DeWitt wave function describing spherically symmetric gravitational collapse of massless scalar fields. The simple poles in the S-matrix occur in the Euclidean spacetime, and the Euclidean Wheeler-DeWitt equation is a variant of the Calogero models, which is discussed in connection with conformal mechanics and a quantum instanton. 
  We propose to describe the dynamics of a cosmological term in the spherically symmetric case by an r-dependent second rank symmetric tensor invariant under boosts in the radial direction. This proposal is based on the Petrov classification scheme and Einstein field equations in the spherically symmetric case. The inflationary equation of state p=-\rho is satisfied by the radial pressure. The tangential pressure is calculated from the conservation equation (the contracted Bianchi identity). 
  Working in the context of a Lorentz-violating extension of the standard model we show that estimates of Lorentz symmetry violation extracted from ultra-high energy cosmic rays beyond the Greisen-Kuzmin-Zatsepin (GZK) cutoff allow for setting bounds on parameters of that extension. Furthermore, we argue that a correlated measurement of the difference in the arrival time of gamma-ray photons and neutrinos emitted from active galactic nuclei or gamma-ray bursts may provide a signature of possible violation of Lorentz symmetry. We have found that this time delay is energy independent, however it has a dependence on the chirality of the particles involved. We also briefly discuss the known settings where the mechanism for spontaneous violation of Lorentz symmetry in the context of string/M-theory may take place. 
  String theory and ``quantum geometry'' have recently offered independent statistical mechanical explanations of black hole thermodynamics. But these successes raise a new problem: why should models with such different microscopic degrees of freedom yield identical results? I propose that the asymptotic behavior of the density of states at a black hole horizon may be determined by an underlying symmetry inherited from classical general relativity, independent of the details of quantum gravity. I offer evidence that a two-dimensional conformal symmetry at the horizon, with a classical central extension, may provide the needed behavior. 
  We review the present status of black hole thermodynamics. Our review includes discussion of classical black hole thermodynamics, Hawking radiation from black holes, the generalized second law, and the issue of entropy bounds. A brief survey also is given of approaches to the calculation of black hole entropy. We conclude with a discussion of some unresolved open issues. 
  A possible theoretical basis is given for propulsive force generation by both conventional and unconventional means. 
  We address some issues of topological defect inflation. (1) We clarify the causal structure of an inflating magnetic monopole. The spacetime diagram shows explicitly that this model is free from the ``graceful exit'' problem, while the monopole itself undergoes ``eternal inflation''. (2) We extend the study of inflating topological defects to Brans-Dicke gravity. Contrary to the case of Einstein gravity, any inflating monopole eventually shrinks and takes a stable configuration. (3) We reanalyze chaotic inflation with a non-minimally coupled massive scalar field. We find a new solution of domain wall inflation, which relaxes constraints on the coupling constant for successful inflation. 
  We discuss the asymptotic structure of the ultrarelativistic Schwarzschild black hole. An explicit construction for a conformal boundary both at spatial and null infinity is given together with the corresponding expressions for the ADM and Bondi four-momenta. 
  Although many people have thought that the difference between the Copenhagen and many-worlds versions of quantum theory was merely metaphysical, quantum cosmology may allow us to make a physical test to distinguish between them empirically. The difference between the two versions shows up when the various components of the wavefunction have different numbers of observers and observations. In the Copenhagen version, a random observation is selected from the sample within the component that is selected by wavefunction collapse, but in the many-worlds version, a random observation is selected from those in all components. Because of the difference in the samples, probable observations in one version can be very improbable in the other version. 
  We consider the free matter of global textures within the framework of the perfect fluid approximation in general relativity. We examine thermodynamical properties of texture matter in comparison with radiation fluid and bubble matter. Then we study dynamics of thin-wall selfgravitating texture objects, and show that classical motion can be elliptical (finite), parabolical or hyperbolical. It is shown that total gravitational mass of neutral textures in equilibrium equals to zero as was expected. Finally, we perform the Wheeler-DeWitt's minisuperspace quantization of the theory, obtain exact wave functions and discrete spectra of bound states with provision for spatial topology. 
  We prove that the area of sections of future event horizons in space-times satisfying the null energy condition is non-decreasing towards the future under any one of the following circumstances: 1) the horizon is future geodesically complete; 2) the horizon is a black hole event horizon in a globally hyperbolic space-time and there exists a conformal completion with a ``H-regular'' Scri plus; 3) the horizon is a black hole event horizon in a space-time which has a globally hyperbolic conformal completion. (Some related results under less restrictive hypotheses are also established.) This extends a theorem of Hawking, in which piecewise smoothness of the event horizon seems to have been assumed. No assumptions about the cosmological constant or its sign are made. We prove smoothness or analyticity of the relevant part of the event horizon when equality in the area inequality is attained - this has applications to the theory of stationary black holes, as well as to the structure of compact Cauchy horizons. In the course of the proof we establish several new results concerning the differentiability properties of horizons. 
  Ricci and contracted Ricci collineations of the Bianchi type II, VIII, and IX space-times, associated with the vector fields of the form (i) one component of $\xi^a(x^b)$ is different from zero and (ii) two components of $\xi^a(x^b)$ are different from zero, for $a,b=1,2,3,4$, are presented. In subcase (i.b), which is $\xi^a= (0,\xi^2(x^a),0,0)$, some known solutions are found, and in subcase (i.d), which is $\xi^a =(0,0,0,\xi^4(x^a))$, choosing $S(t)=const.\times R(t)$, the Bianchi type II, VIII, and IX space-times is reduced to the Robertson-Walker metric. 
  It made that the new symmetric property, the binary law, existed newly in our time and space at the thing except the symmetric property of the principle of general relativity which is already known in this paper clear. The introduction of this symmetric property will have made dealing with the position tensor of us handy as much as the surprise. 
  Symmetries are defined in histories-based theories paying special attention to the class of history theories admitting quasitemporal structure (a generalization of the concept of `temporal sequences' of `events' using partial semigroups) and logic structure for `single-time histories'. Symmetries are classified into orthochronous (those preserving the `temporal order' of `events') and nonorthochronous. A straightforward criterion for physical equivalence of histories is formulated in terms of orthochronous symmetries; this criterion covers various notions of physical equivalence of histories considered by Gell-Mann and Hartle as special cases. In familiar situations, a reciprocal relationship between traditional symmetries (Wigner symmetries in quantum mechanics and Borel-measurable transformations of phase space in classical mechanics) and symmetries defined in this work is established. In a restricted class of theories, a definition of conservation law is given in the history language which agrees with the standard ones in familiar situations; in a smaller subclass of theories, a Noether type theorem (implying a connection between continuous symmetries of dynamics and conservation laws) is proved. The formalism evolved is applied to histories (of particles, fields or more general objects) in general curved spacetimes. Sharpening the definition of symmetry so as to include a continuity requirement, it is shown that a symmetry in our formalism implies a conformal isometry of the spacetime metric. 
  The initial singularity problem in standard general relativity is treated on the light of a viewpoint asserting that this formulation of Einstein's theory and its conformal formulations are physically equivalent. We show that flat Friedmann-Robertson-Walker universes and open dust-filled and radiation-filled universes are singularity free when described in terms of the formulation of general relativity conformal to the canonical one. 
  This article is a guide to the literature on existence theorems for the Einstein equations which also draws attention to open problems in the field. The local in time Cauchy problem, which is relatively well understood, is treated first. Next global results for solutions with symmetry are discussed. A selection of results from Newtonian theory and special relativity which offer useful comparisons is presented. This is followed by a survey of global results in the case of small data and results on constructing spacetimes with given singularity structure. The article ends with some miscellaneous topics connected with the main theme. 
  In quantum field theory there exist states for which the energy density is negative. It is important that these negative energy densities satisfy constraints, such as quantum inequalities, to minimize possible violations of causality, the second law of thermodynamics, and cosmic censorship. In this paper I show that conformally invariant scalar and Dirac fields satisfy quantum inequalities in two dimensional spacetimes with a conformal factor that depends on $x$ only or on $t$ only. These inequalities are then applied to two dimensional black hole and cosmological spacetimes. It is shown that the bound on the negative energies diverges to minus infinity as the event horizon or initial singularity is approached. Thus, neglecting back reaction, negative energies become unconstrained near the horizon or initial singularity. The results of this paper also support the hypothesis that the quantum interest conjecture applies only to deviations from the vacuum polarization energy, not to the total energy. 
  The coupling of the electromagnetic field to gravity is an age-old problem. Presently, there is a resurgence of interest in it, mainly for two reasons: (i) Experimental investigations are under way with ever increasing precision, be it in the laboratory or by observing outer space. (ii) One desires to test out alternatives to Einstein's gravitational theory, in particular those of a gauge-theoretical nature, like Einstein-Cartan theory or metric-affine gravity. A clean discussion requires a reflection on the foundations of electrodynamics. If one bases electrodynamics on the conservation laws of electric charge and magnetic flux, one finds Maxwell's equations expressed in terms of the excitation H=(D,H) and the field strength F=(E,B) without any intervention of the metric or the linear connection of spacetime. In other words, there is still no coupling to gravity. Only the constitutive law H= functional(F) mediates such a coupling. We discuss the different ways of how metric, nonmetricity, torsion, and curvature can come into play here. Along the way, we touch on non-local laws (Mashhoon), non-linear ones (Born-Infeld, Heisenberg-Euler, Plebanski), linear ones, including the Abelian axion (Ni), and find a method for deriving the metric from linear electrodynamics (Toupin, Schoenberg). Finally, we discuss possible non-minimal coupling schemes. 
  We discuss the circumstances under which gravity might be repulsive rather than attractive. In particular we show why our standard solar system distance scale gravitational intuition need not be a reliable guide to the behavior of gravitational phenomena on altogether larger distance scales such as cosmological, and argue that in fact gravity actually gets to act repulsively on such distance scales. With such repulsion a variety of current cosmological problems (the flatness, horizon, dark matter, universe age, cosmic acceleration and cosmological constant problems) are then all naturally resolved. 
  In this paper the well known Belinskii and Zakharov soliton generating transformations of the solution space of vacuum Einstein equations with two-dimensional Abelian groups of isometries are considered in the context of the so called "monodromy transform approach", which provides some general base for the study of various integrable space - time symmetry reductions of Einstein equations. Similarly to the scattering data used in the known spectral transform, in this approach the monodromy data for solution of associated linear system characterize completely any solution of the reduced Einstein equations, and many physical and geometrical properties of the solutions can be expressed directly in terms of the analytical structure on the spectral plane of the corresponding monodromy data functions. The Belinskii and Zakharov vacuum soliton generating transformations can be expressed in explicit form (without specification of the background solution) as simple (linear-fractional) transformations of the corresponding monodromy data functions with coefficients, polynomial in spectral parameter. This allows to determine many physical parameters of the generating soliton solutions without (or before) calculation of all components of the solutions. The similar characterization for electrovacuum soliton generating transformations is also presented. 
  Combining recent techniques giving non-perturbative re-summed estimates of the damping and conservative parts of the two-body dynamics, we describe the transition between the adiabatic phase and the plunge, in coalescing binary black holes with comparable masses moving on quasi-circular orbits. We give initial dynamical data for numerical relativity investigations, with a fraction of an orbit left, and provide, for data analysis purposes, an estimate of the gravitational wave-form emitted throughout the inspiral, plunge and coalescence phases. 
  We investigate the averaging problem in cosmology as the problem of introducing a distance between spaces.   We first introduce the spectral distance, which is a measure of closeness between spaces defined in terms of the spectra of the Laplacian. Then we define a space S, the space of all spaces equipped with the spectral distance. We argue that this space S can be regarded as a metric space and that it also possess other desirable properties. These facts make the space S a suitable arena for spacetime physics.   We apply the spectral framework to the averaging problem:   We sketch the model-fitting procedure in terms of the spectral representation, and also discuss briefly how to analyze the dynamical aspects of the averaging procedure with this scheme.   In these analyses, we are naturally led to the concept of the apparatus- and the scale-dependent effective evolution of the universe. These observations suggest that the spectral scheme seems to be suitable for the quantitative analysis of the averaging problem in cosmology. 
  We study the perturbative dynamics of an infinite gravitating Nambu-Goto string within the general-relativistic perturbation framework. We develop the gauge invariant metric perturbation on a spacetime containing a self-gravitating straight string with a finite thickness and solve the linearized Einstein equation. In the thin string case, we show that the string does not emit gravitational waves by its free oscillation in the first order with respect to its oscillation amplitude, nevertheless the string actually bends when the incidental gravitational waves go through it. 
  For the BTZ black hole in the Einstein gravity, a statistical entropy has been calculated. It is equal to the Bekenstein-Hawking entropy. In this article, its statistical entropy in the higer curvature gravity is calculated and shown to be equal to one derived by using the Noether charge method. This suggests that the equivalence between the geometrical and statistical entropies of the black hole is retained in the general diffeomorphism invariant theories of gravity. A relation between the cosmic censorship conjecture and the unitarity of the conformal field theory on the boundary of AdS_3 is also discussed. 
  The solutions of the Einstein-Maxwell-Chern-Simons theory are studied in (1+2) dimensions with the self-duality condition imposed on the Maxwell field. We give a closed form of the general solution which is determined by a single function having the physical meaning of the quasilocal angular momentum of the solution. This function completely determines the geometry of spacetime, also providing the direct computation of the conserved total mass and angular momentum of the configurations. 
  A complete characterization is obtained of the asymptotic behavior of solutions of the static vacuum Einstein equations which have a (pseudo)-compact horizon or boundary and are complete away from the boundary. It is proved that the time-symmetric space-like hypersurface has only finitely many ends, each of which is either asymptotically flat (AF) or parabolic, as in the (static) Kasner metric. Examples are given with both types of behavior, together with an extensive discussion and new characterization of Weyl metrics. The asymptotics result allows one in most circumstances to drop the AF assumption from the static black hole uniqueness theorems and replace it with just a completeness assumption. 
  We show that the 5-dimensional model introduced by Randall and Sundrum is (half of) a wormhole, and that this is a general result in models of the RS type. We also discuss the gravitational trapping of a scalar particle in 5-d spacetimes. Finally, we present a simple model of brane-world cosmology in which the background is a static anti-de Sitter manifold, and the location of the two 3-branes is determined by the technique of ``surgical grafting''. 
  By applying the method of moving frames modelling one and two dimensional local anisotropies we construct new solutions of Einstein equations on pseudo-Riemannian spacetimes. The first class of solutions describes non-trivial deformations of static spherically symmetric black holes to locally anisotropic ones which have elliptic (in three dimensions) and ellipsoidal, toroidal and elliptic and another forms of cylinder symmetries (in four dimensions). The second class consists from black holes with oscillating elliptic horizons. 
  We consider perturbations of a Schwarzschild black hole that can be of both even and odd parity, keeping terms up to second order in perturbation theory, for the $\ell=2$ axisymmetric case. We develop explicit formulae for the evolution equations and radiated energies and waveforms using the Regge-Wheeler-Zerilli approach. This formulation is useful, for instance, for the treatment in the ``close limit approximation'' of the collision of counterrotating black holes. 
  It is believed that there may have been a large number of black holes formed in the very early universe. These would have quantised masses. A charged ``elementary black hole'' (with the minimum possible mass) can capture electrons, protons and other charged particles to form a ``black hole atom''. We find the spectrum of such an object with a view to laboratory and astronomical observation of them, and estimate the lifetime of the bound states. There is no limit to the charge of the black hole, which gives us the possibility of observing Z>137 bound states and transitions at the lower continuum. Negatively charged black holes can capture protons. For Z>1, the orbiting protons will coalesce to form a nucleus (after beta-decay of some protons to neutrons), with a stability curve different to that of free nuclei. In this system there is also the distinct possibility of single quark capture. This leads to the formation of a coloured black hole that plays the role of an extremely heavy quark interacting strongly with the other two quarks. Finally we consider atoms formed with much larger black holes. 
  Frequency-domain filters for time-windowed gravitational waves from inspiralling compact binaries are constructed which combine the excellent performance of our previously developed time-domain P-approximants with the analytic convenience of the stationary phase approximation without a serious loss in event rate. These Fourier-domain representations incorporate the ``edge oscillations'' due to the (assumed) abrupt shut-off of the time-domain signal caused by the relativistic plunge at the last stable orbit. These new analytic approximations, the SPP-approximants, are not only `effectual' for detection and `faithful' for parameter estimation, but are also computationally inexpensive to generate (and are `faster' by factors up to 10, as compared to the corresponding time-domain templates). The SPP approximants should provide data analysts the Fourier-domain templates for massive black hole binaries of total mass m less than about 40 solar mases, the most likely sources for LIGO and VIRGO. 
  The initial-value problem for cylindrical gravitational waves is studied through the development of the inverse scattering method scheme. The inverse scattering transform in this case can be viewed as a transformation of the Cauchy data to the data on the symmetry axis. Riemann-Hilbert problem, which serves as inverse transformation, is formulated in two different ways. We consider Einstein-Rosen waves to illustrate the method. 
  The finite part of the self-force on a static scalar test-charge outside a Schwarzschild black hole is zero. By direct construction of Hadamard's elementary solution, we obtain a closed-form expression for the minimally coupled scalar field produced by a test-charge held fixed in Schwarzschild spacetime. Using the closed-form expression, we compute the necessary external force required to hold the charge stationary. Although the energy associated with the scalar field contributes to the renormalized mass of the particle (and thereby its weight), we find there is no additional self-force acting on the charge. This result is unlike the analogous electrostatic result, where, after a similar mass renormalization, there remains a finite repulsive self-force acting on a static electric test-charge outside a Schwarzschild black hole. We confirm our force calculation using Carter's mass-variation theorem for black holes. The primary motivation for this calculation is to develop techniques and formalism for computing all forces - dissipative and non-dissipative - acting on charges and masses moving in a black-hole spacetime. In the Appendix we recap the derivation of the closed-form electrostatic potential. We also show how the closed-form expressions for the fields are related to the infinite series solutions. 
  The gravitational strength of the central singularity in spherically symmetric space-times is investigated. Necessary conditions for the singularity to be gravitationally weak are derived and it is shown that these are violated in a wide variety of circumstances. These conditions allow conclusions to be drawn about the nature of the singularity without having to integrate the geodesic equations. In particular, any geodesic with a non-zero amount of angular momentum which impinges on the singularity terminates in a strong curvature singularity. 
  Many efforts have been devoted to the studies of the phenomenology in particle physics with extra dimensions. We propose the degenerate fermion star in the five dimensions, and study what effects caused by the geometry of extra dimensions should appear in its structure. We note that Kaluza-Klein excited modes have effects for the larger scale of extra dimensions and examine the conditions on which different layers should be caused in the inside of the stars. We expound how the effects of the extra dimensions appears on physical quantities. 
  We treat the model which describes "extreme black holes" moving slowly. We derive an effective lagrangian in the low energy for this model and then investigate a statistical behavior of "extreme black holes" in the finite temperature. 
  We derive the low-energy effective action of four-dimensional gravity in the Randall-Sundrum scenario in which two 3-branes of opposite tension reside in a five-dimensional spacetime. The dimensional reduction with the Ansatz for the radion field by Charmousis et al., which solves five-dimensional linearized field equations, results in a class of scalar-tensor gravity theories. In the limit of vanishing radion fluctuations, the effective action reduces to the Brans-Dicke gravity in accord with the results of Garriga and Tanaka: Brans-Dicke gravity with the corresponding Brans-Dicke parameter $0< \omega < \infty$ (for positive tension brane) and $-3/2< \omega <0$ (for negative tension brane). In general the gravity induced a brane belongs to a class of scalar-tensor gravity with the Brans-Dicke parameter which is a function of the interval and the radion. In particular, gravity on a positive tension brane contains an attractor mechanism toward the Einstein gravity. 
  Using simulated signals and measured noise with the EXPLORER and NAUTILUS detectors we find the efficiency of signal detection and the signal arrival time dispersion versus the signal-to-noise ratio. 
  It is shown that, in contrast to the case of extreme 4d dilatonic black holes, 4d neutral dilatonic black holes with horizon singularities can not be interpreted as nonsingular nondilatonic black p-branes in (4+p) dimensions, regardless of the number of extra dimensions p. That is, extra dimensions do not remove naked singularities of 4d neutral dilatonic black holes. 
  We study the Regge-Wheeler and Zerilli equations (RWE and ZE) at the `algebraically special frequency' $\Omega$, where these equations admit an exact solution (elaborated here), generating the SUSY relationship between them. The physical significance of the SUSY generator and of the solutions at $\Omega$ in general is elucidated as follows. The RWE has no (quasinormal or total-transmission) modes at all; however, $\Omega$ is nonetheless `special' in that (a) for the outgoing wave into the horizon one has a `miraculous' cancellation of a divergence expected due to the exponential potential tail, and (b) the branch-cut discontinuity at $\omega=\Omega$ vanishes in the outgoing wave to infinity. Moreover, (a) and (b) are related. For the ZE, its only mode is the-inverse-SUSY generator, which is at the same time a quasinormal mode_and_ a total-transmission mode propagating to infinity. The subtlety of these findings (of general relevance for future study of the equations on or near the negative imaginary $\omega$-axis) may help explain why the situation has sometimes been controversial. For finite black-hole rotation, the algebraically special modes are shown to be totally transmitting, and the implied singular nature of the Schwarzschild limit is clarified. The analysis draws on a recent detailed investigation of SUSY in open systems [math-ph/9909030]. 
  Birth of the brane world is studied using the Hamiltonian approach. It is shown that an inflating brane world can be created from nothing. The wave function of the universe obtained from the Wheeler de-Witt equation and the time-dependent Schr$\ddot{o}$dinger equation for quantized scalar fields on the brane are the same as in the conventional 4-dimensional quantum cosmology if the bulk is exactly the Anti-de Sitter spacetime. The effect of the massive objects in the bulk is also discussed. This analysis tells us the presence of the extra dimension imprints a nontrivial effect on the quantum cosmology of the brane world. This fact is important for the analysis of the quantum fluctuations in the inflationary scenario of the brane world. 
  It appears difficult to construct a simple model for an open universe based on the one bubble inflationary scenario. The reason is that one needs a large mass to avoid the tunneling via the Hawking Moss solution and a small mass for successful slow-rolling. However, Rubakov and Sibiryakov suggest that the Hawking Moss solution is not a solution for the false vacuum decay process since it does not satisfy the boundary condition. Hence, we have reconsidered the arguments for the defect of the simple polynomial model. We find the valley bounce belonging to a valley line in the functional space represents the decay process instead of the Hawking Moss solution. The point is that the valley bounce gives the appropriate initial condition for the inflation. We show an open inflation model can be constructed within the polynomial form of the potential so that the fluctuations can be reconciled with the observations. 
  A procedure with a Bayesan approach for calculating upper limits to gravitational wave bursts from coincidence experiments with multiple detectors is described. 
  Standard treatments of general relativity accept the gravitational slowing of clocks as a primary phenomenon, requiring no further analysis as to cause. Rejecting this attitude, I argue that one or more of the fundamental "constants" governing the quantum mechanics of atoms must depend upon position in a gravitational field. A simple relationship governing the possible dependencies of e, h, c and m is deduced, and arguments in favor of the choice of the electron rest mass, m, are presented. The reduction of rest mass is thus taken to be the sole cause of clock slowing. Importantly, this dependency implies another effect, heretofore unsuspected, namely, the gravitational elongation of measuring rods. An alternate ("telemetric") system of measurement is introduced, leading to a metric that is conformally related to the usual proper metric. In terms of the new system, many otherwise puzzling phenomena may be simply understood. In particular,the geometry of the Schwarzschild space as described by the telemetric system differs profoundly from that described by proper measurments, leading to a very different understanding of the structure of black holes. The theory is extended to cosmology,leading to a remarkable alternate view of the structure and history of the universe. 
  We propose a definition of an exact lens equation without reference to a background spacetime, and construct the exact lens equation explicitly in the case of Schwarzschild spacetime. For the Schwarzschild case, we give exact expressions for the angular-diameter distance to the sources as well as for the magnification factor and time of arrival of the images. We compare the exact lens equation with the standard lens equation, derived under the thin-lens-weak-field assumption (where the light rays are geodesics of the background with sharp bending in the lens plane, and the gravitational field is weak), and verify the fact that the standard weak-field thin-lens equation is inadequate at small impact parameter. We show that the second-order correction to the weak-field thin-lens equation is inaccurate as well. Finally, we compare the exact lens equation with the recently proposed strong-field thin-lens equation, obtained under the assumption of straight paths but without the small angle approximation, i.e., with allowed large bending angles. We show that the strong-field thin-lens equation is remarkably accurate, even for lightrays that take several turns around the lens before reaching the observer. 
  As shown recently (W. Kummer, H. Liebl, D.V. Vassilevich, Nucl. Phys. B 544, 403 (1999)) 2d quantum gravity theories --- including spherically reduced Einstein-gravity --- after an exact path integral of its geometric part can be treated perturbatively in the loops of (scalar) matter. Obviously the classical mechanism of black hole formation should be contained in the tree approximation of the theory. This is shown to be the case for the scattering of two scalars through an intermediate state which by its effective black hole mass is identified as a ``virtual black hole''. The present discussion is restricted to minimally coupled scalars without and with mass. In the first case the probability amplitude diverges, except the black hole is ``plugged'' by a suitable boundary condition. For massive scalars a finite S-matrix element is obtained. 
  Using invariant transformations of the five-dimensional Kaluza-Klein (KK) field equations, we find a series of formulae to derive axial symmetric stationary exact solutions of the KK theory starting from static ones. The procedure presented in this work allows to derive new exact solutions up to very simple integrations. Among other results, we find exact rotating solutions containing magnetic monopoles, dipoles, quadripoles, etc., coupled to scalar and to gravitational multipole fields. 
  We analyze the classical limit of kinematic loop quantum gravity in which the diffeomorphism and hamiltonian constraints are ignored. We show that there are no quantum states in which the primary variables of the loop approach, namely the SU(2) holonomies along {\em all} possible loops, approximate their classical counterparts. At most a countable number of loops must be specified. To preserve spatial covariance, we choose this set of loops to be based on physical lattices specified by the quasi-classical states themselves. We construct ``macroscopic'' operators based on such lattices and propose that these operators be used to analyze the classical limit. Thus, our aim is to approximate classical data using states in which appropriate macroscopic operators have low quantum fluctuations.   Although, in principle, the holonomies of `large' loops on these lattices could be used to analyze the classical limit, we argue that it may be simpler to base the analysis on an alternate set of ``flux'' based operators. We explicitly construct candidate quasi-classical states in 2 spatial dimensions and indicate how these constructions may generalize to 3d. We discuss the less robust aspects of our proposal with a view towards possible modifications. Finally, we show that our proposal also applies to the diffeomorphism invariant Rovelli model which couples a matter reference system to the Hussain Kucha{\v r} model. 
  The unconstrained reduced action corresponding to the dynamics of scalar fluctuations about the Friedmann-Robertson-Walker (FRW) background is derived using Dirac's method of description of singular Lagrangian systems. The results are applied to so-called negative mode problem in description of tunneling transitions with gravity. With our special choice of physical variable, the kinetic term of the reduced action has a conventional signature for a wide class of models. In this representation, the existence of a negative mode justifying the false vacuum decay picture turns out to be manifest. We also explain how the present result becomes consistent with the previously proved ``no negative mode (supercritical supercurvature mode) theorem''. 
  Relative motion in space with multifractal time (fractional dimension of time close to integer $d_{t}=1+\epsilon (r,t), \epsilon \ll 1$) for "almost" inertial frames of reference (time is almost homogeneous and almost isotropic) is considered. Presence in such space of absolute frames of reference and violation of conservation laws (though, small because of the smallness of $\epsilon$) due to the openness of all physical systems and inhomogeneiy of time are shown. The total energy of a body moving with $v=c$ is obtained to be finite and modified Lorentz transformations are formulated. The relation for the total energy (and the whole theory) reduce to the known formula of the special relativity in case of transition to the usual time with dimension equal to one. 
  Physical phenomena caused by particle's moving faster than light in a space with multifractal time with dimension close to integer ($d_{t}=1+\epsilon(r(t),t), |\epsilon| \ll 1$ - time is almost homogeneous and almost isotropic) are considered. The presence of gravitational field is taken into account. According to the results of the developed by the author theory, a particle with the rest energy $E_{0}$ would achieve the velocity of light if given the energy of about $E \sim 10^{3}E_{0}$ 
  We present a filter for detecting gravitational wave signals from burst sources. This filter requires only minimal advance knowledge of the expected signal: i.e. the signal's frequency band and time duration. It consists of a threshold on the total power in the data stream in the specified signal band during the specified time. This filter is optimal (in the Neyman-Pearson sense) for signal searches where only this minimal information is available. 
  We obtain a (5+2)-dimensional global flat embedding of the (3+1)-dimensional curved RN-AdS space. Our results include the various limiting cases of global embedding Minkowski space (GEMS) geometries of the RN, Schwarzschild-AdS in (5+2)-dimensions, Schwarzschild in (5+1)-dimensions, purely charged space, and universal covering space of AdS in (4+1)-dimensions, through the successive truncation procedure of parameters in the original curved space. 
  As first discovered by Choptuik, the black hole threshold in the space of initial data for general relativity shows both surprising structure and surprising simplicity. Universality, power-law scaling of the black hole mass, and scale echoing have given rise to the term ``critical phenomena''. They are explained by the existence of exact solutions which are attractors within the black hole threshold, that is, attractors of codimension one in phase space, and which are typically self-similar. This review gives an introduction to the phenomena, tries to summarize the essential features of what is happening, and then presents extensions and applications of this basic scenario. Critical phenomena are of interest particularly for creating surprising structure from simple equations, and for the light they throw on cosmic censorship and the generic dynamics of general relativity. 
  The most detailed existing proposal for the structure of spacetime singularities originates in the work of Belinskii, Khalatnikov and Lifshitz. We show rigorously the correctness of this proposal in the case of analytic solutions of the Einstein equations coupled to a scalar field or stiff fluid. More specifically, we prove the existence of a family of spacetimes depending on the same number of free functions as the general solution which have the asymptotics suggested by the Belinskii-Khalatnikov-Lifshitz proposal near their singularities. In these spacetimes a neighbourhood of the singularity can be covered by a Gaussian coordinate system in which the singularity is simultaneous and the evolution at different spatial points decouples. 
  The holographic principle states that the number of degrees of freedom describing the physics inside a volume (including gravity) is bounded by the area of the boundary (also called the screen) which encloses this volume. A stronger statement is that these (quantum) degrees of freedom live on the boundary and describe the physics inside the volume completely. The obvious question is, what mechanism is behind the holographic principle. Recently, 't Hooft argued that the quantum degrees of freedom on the boundary are not fundamental. We argue that this interpretation opens up the possibility that the mapping between the theory in the bulk (the holographic theory) and the theory on the screen (the dual theory) is always given by a (generalized) procedure of stochastic quantization. We show that gravity causes differences to the situation in Minkowski/Euclidean spacetime and argue that the fictitious coordinate needed in the stochastic quantization procedure can be spatial. The diffusion coefficient of the stochastic process is in general a function of this coordinate. While a mapping of a bulk theory onto a (quantum) boundary theory can be possible, such a mapping does not make sense in spacetimes in which the area of the screen is growing with time. This is connected to the average process in the formalism of stochastic quantization. We show where the stochastic quantization procedure breaks down and argue, in agreement with `t Hooft, that the quantum degrees of freedom are not fundamental degrees of freedom. They appear as a limit of a more complex process. 
  The Penrose method for constructing spherical impulsive gravitational waves is investigated in detail, including alternative spatial sections and an arbitrary cosmological constant. The resulting waves include those that are generated by a snapping cosmic string. The method is used to construct an explicit exact solution of Einstein's equations describing the collision of two nonaligned cosmic strings in a Minkowski background which snap at their point of collision. 
  We revisit the quantization of U(1) holonomy algebras using the abelian C* algebra based techniques which form the mathematical underpinnings of current efforts to construct loop quantum gravity. In particular, we clarify the role of ``smeared loops'' and of Poincare invariance in the construction of Fock representations of these algebras. This enables us to critically re-examine early pioneering efforts to construct Fock space representations of linearised gravity and free Maxwell theory from holonomy algebras through an application of the (then current) techniques of loop quantum gravity. 
  We show how the scalar field, a candidate of quintessence, in a proposed model of the scalar-tensor theories of gravity provides a way to understand a small but nonzero cosmological constant as indicated by recent observations. A particular emphasis is placed on the effort to inherit the success of the scenario of a decaying cosmological constant. Discussions of a possible link to non-Newtonian gravity, the coincidence problem, the issue of time-variability of coupling constants as well as the chaos-like nature of the solution are also included in a new perspective. 
  The Wightman two-point function for the gravitational field in the linear approximation (the rank-2 ``massless'' tensor field) on de Sitter space has a pathological behaviour for large separated points (infrared divergence). This behaviour can be eliminated in the two-point function for the traceless part of this field if one chooses the Gupta-Bleuler vacuum. But it is not possible to do the same for the pure trace part (conformal sector). We briefly discuss the consequences of this pure trace behaviour for inflationary models. 
  We present an elementary evaluation of the surface areas of the event horizon and stationary limit surface for an uncharged Kerr black hole. The latter appears not to have been previously given in the literature, and permits us to suggest new geometrical / physical interpretations of these areas. 
  We use a conformal transformation to find solutions to the generalised scalar-tensor theory, with a coupling constant dependent on a scalar field, in an empty Bianchi type I model. We describe the dynamical behaviour of the metric functions for three different couplings: two exact solutions to the field equations and a qualitative one are found. They exhibit non-singular behaviours and kinetic inflation. Two of them admit both General Relativity and string theory in the low-energy limit as asymptotic cases. 
  Analysing the Brans-Dicke solutions for the dust phase, we show that, for negative values of $\omega$, they contain scenarios that display an initial subluminal expansion followed by an inflationary phase. We discuss these solutions with respect to the results of the observation of high redshif supernova as well as the age problem and structure formation. We stablish possible connections of these solutions with those emerging from string effective models. 
  The present talk summarizes the recently progressed state of a systematic re-evaluation of cosmological models that respect the presence of inhomogeneities. Emphasis is given to identifying the basic steps towards an effective (i.e. spatially averaged) description of structural evolution, also unfolding the various facets of a `smoothed--out' cosmology. We shall highlight some results obtained within Newtonian cosmology, discuss expansion laws in general relativity within a covariant fluid approach, and put forward some promising directions of future research. 
  Our approach views the thermodynamics and kinetics in general relativity and extended gravitational theories (with generic local anisotropy) from the perspective of the theory of stochastic differential equations on curved spaces.   Nonequilibrium and irreversible processes in black hole thermodynamics are considered. The paper summarizes the author's contribution to Journees Relativistes 99 (12--17 September 1999), Weimar, Germany. 
  An analitical approximation of $<\phi^2>$ for a massive scalar field in a zero temperature vacuum state in static spherically symmetric spacetimes is obtained. The calculations are based on the method for computing vacuum expectations values for scalar fields in general static spherically symmetric spacetimes derived by Anderson, Hiscock and Samuel [Phys. Rev. D {\bf 51}, 4337 (1995)]. The analitical approximation is used to compute $<\phi^2>$ in Schwarzschild and wormhole spacetimes. 
  There is a deep interrelationship of the General Theory of Relativity and weak interactions in the model of Expansive Nondecelerative Universe. This fact allows an independent determination of the mass of vector bosons Z and W, as well as the time of separation of electromagnetic and weak interactions. 
  The kinetic theory is formulated with respect to anholonomic frames of reference on curved spacetimes. By using the concept of nonlinear connection we develop an approach to modelling locally anisotropic kinetic processes and, in corresponding limits, the relativistic non-equilibrium thermodynamics with local anisotropy. This lead to a unified formulation of the kinetic equations on (pseudo) Riemannian spaces and in various higher dimensional models of Kaluza-Klein type and/or generalized Lagrange and Finsler spaces. The transition rate considered for the locally anisotropic transport equations is related to the differential cross section and spacetime parameters of anisotropy. The equations of states for pressure and energy in locally anisotropic thermodynamics are derived. The obtained general expressions for heat conductivity, shear and volume viscosity coefficients are applied to determine the transport coefficients of cosmic fluids in spacetimes with generic local anisotropy. We emphasize that such locally anisotropic structures are induced also in general relativity if we are modelling physical processes with respect to frames with mixed sets of holonomic and anholonomic basis vectors which naturally admits an associated nonlinear connection structure. 
  The present surge for the astrophysical relevance of boson stars stems from the speculative possibility that these compact objects could provide a considerable fraction of the non-baryonic part of dark matter within the halo of galaxies. For a very light `universal' axion of effective string models, their total gravitational mass will be in the most likely range of \sim 0.5 M_\odot of MACHOs. According to this framework, gravitational microlensing is indirectly ``weighing" the axion mass, resulting in \sim 10^{-10} eV/c^2. This conclusion is not changing much, if we use a dilaton type self-interaction for the bosons. Moreover, we review their formation, rotation and stability as likely candidates of astrophysical importance. 
  Several filtering methods for the detection of gravitational wave bursts in interferometric detectors are presented. These are simple and fast methods which can act as online triggers. All methods are compared to matched filtering with the help of a figure of merit based on the detection of supernovae signals simulated by Zwerger and Muller. 
  Only one model from an infinite number of the Friedmannian models of flat expansive isotropic and homogeneous universe satisfies the assumptions resulting from the Planck quantum hypothesis. 
  In this short survey paper, we discuss certain recent results in classical gravity. Our main attention is restricted to two topics: the positive mass conjecture and its extensions to the case with horizons, including the Penrose conjecture (Part I), and the interaction of gravity with other force fields and quantum-mechanical particles (Part II). 
  The behaviour of the flat anisotropic model of the Universe with a scalar field is explored within the framework of quantum cosmology. The principal moment of the account of an anisotropy is the presence either negative potential barrier or positive repelling wall. In the first case occur the above barrier reflection of the wave function of the Universe, in the second one there is bounce off a potential wall. The further evolution of the Universe represents an exponential inflating with fast losses of an anisotropy and approach to the standard cosmological scenario. 
  The present acceleration of the Universe strongly indicated by recent observational data can be modeled in the scope of a scalar-tensor theory of gravity. We show that it is possible to determine the structure of this theory (the scalar field potential and the functional form of the scalar-gravity coupling) along with the present density of dustlike matter from the following two observable cosmological functions: the luminosity distance and the linear density perturbation in the dustlike matter component as functions of redshift. Explicit results are presented in the first order in the small inverse Brans-Dicke parameter 1/omega. 
  We consider a spherically symmetric, static system of a Dirac particle interacting with classical gravity and an SU(2) Yang-Mills field. The corresponding Einstein-Dirac-Yang/Mills equations are derived. Using numerical methods, we find different types of soliton-like solutions of these equations and discuss their properties. Some of these solutions are stable even for arbitrarily weak gravitational coupling. 
  A quasi-spherical approximation scheme, intended to apply to coalescing black holes, allows the waveforms of gravitational radiation to be computed by integrating ordinary differential equations. 
  The continuation of the Schwarzschild metric across the event horizon is almost always (in textbooks) carried out using the Kruskal-Szekeres coordinates, in terms of which the areal radius r is defined only implicitly. We argue that from a pedagogical point of view, using these coordinates comes with several drawbacks, and we advocate the use of simpler, but equally effective, coordinate systems. One such system, introduced by Painleve and Gullstrand in the 1920's, is especially simple and pedagogically powerful; it is, however, still poorly known today. One of our purposes here is therefore to popularize these coordinates. Our other purpose is to provide generalizations to the Painleve-Gullstrand coordinates, first within the specific context of Schwarzschild spacetime, and then in the context of more general spherical spacetimes. 
  Harmonic slicing has in recent years become a standard way of prescribing the lapse function in numerical simulations of general relativity. However, as was first noticed by Alcubierre (1997), numerical solutions generated using this slicing condition can show pathological behaviour. In this paper, analytic and numerical methods are used to examine harmonic slicings of Kasner and Gowdy cosmological spacetimes. It is shown that in general the slicings are prevented from covering the whole of the spacetimes by the appearance of coordinate singularities. As well as limiting the maximum running times of numerical simulations, the coordinate singularities can lead to features being produced in numerically evolved solutions which must be distinguished from genuine physical effects. 
  The covariant scheme is proposed to couple gravity and electrodynamics in pseudo-Riemannian four-spaces with electromagnetic connections. Novel dynamics of the extended charge and electromagnetic dilation-compression of its proper time can be tested in non-relativistic experiments. The vector equations acknowledge unified photon waves without metric modulations of flat laboratory space. 
  The groupoid approach to noncommutative unification of general relativity with quantum mechanics is compared with the canonical gravity quantization. It is shown that by restricting the corresponding noncommutative algebra to its (commutative) subalgebra, which determines the space-time slicing, an algebraic counterpart of superspace (space of 3-metrics) can be obtained. It turns out that when this space-time slicing emerges the universe is already in its commutative regime. We explore the consequences of this result. 
  The variational principle for a thin dust shell in General Relativity is constructed. The principle is compatible with the boundary-value problem of the corresponding Euler-Lagrange equations, and leads to ``natural boundary conditions'' on the shell. These conditions and the gravitational field equations which follow from an initial variational principle, are used for elimination of the gravitational degrees of freedom. The transformation of the variational formula for spherically-symmetric systems leads to two natural variants of the effective action. One of these variants describes the shell from a stationary interior observer's point of view, another from the exterior one. The conditions of isometry of the exterior and interior faces of the shell lead to the momentum and Hamiltonian constraints. The canonical equivalence of the mentioned systems is shown in the extended phase space. Some particular cases are considered. 
  We consider minisuperspace models constituted of Bianchi I geometries with a free massless scalar field. The classical solutions are always singular (with the trivial exception of flat space-time), and always anisotropic once they begin anisotropic. When quantizing the system, we obtain the Wheeler-DeWitt equation as a four-dimensional massless Klein-Gordon equation. We show that there are plenty of quantum states whose corresponding bohmian trajectories may be non-singular and/or presenting large isotropic phases, even if they begin anisotropic, due to quantum gravitational effects. As a specific example, we exhibit field plots of bohmian trajectories for the case of gaussian superpositions of plane wave solutions of the Wheeler-DeWitt equation which have those properties. These conclusions are valid even in the absence of the scalar field. 
  We present calculations of gyroscope precession in spacetimes described by Levi-Civita and Lewis metrics, under different circumstances. By doing so we are able to establish a link between the parameters of the metrics and observable quantities, providing thereby a physical interpretation for those parameters, without specifying the source of the field. 
  The topos theory is a theory which is used for deciding a number of problems of theory of relativity, gravitation and quantum physics. In the article spherically symmetric solution of the vacuum Einstein equations in the Intuitionistic theory of Gravitation at different stages of smooth topos ${\bf Set}^{\bf L_{op}}$ is considered. Infinitesimal "weak" gravitational field can be strong at some stagies, for which we have the additional dimensions. For example, the cosmological constant is not constant with respect to additional dimensions. Signature of space-time metric can depend of density of vacuum and cosmological constant. 
  We present approximate analytical solutions to the Hamiltonian and momentum constraint equations, corresponding to systems composed of two black holes with arbitrary linear and angular momentum. The analytical nature of these initial data solutions makes them easier to implement in numerical evolutions than the traditional numerical approach of solving the elliptic equations derived from the Einstein constraints. Although in general the problem of setting up initial conditions for black hole binary simulations is complicated by the presence of singularities, we show that the methods presented in this work provide initial data with $l_1$ and $l_\infty$ norms of violation of the constraint equations falling below those of the truncation error (residual error due to discretization) present in finite difference codes for the range of grid resolutions currently used. Thus, these data sets are suitable for use in evolution codes. Detailed results are presented for the case of a head-on collision of two equal-mass M black holes with specific angular momentum 0.5M at an initial separation of 10M. A straightforward superposition method yields data adequate for resolutions of $h=M/4$, and an "attenuated" superposition yields data usable to resolutions at least as fine as $h=M/8$. In addition, the attenuated approximate data may be more tractable in a full (computational) exact solution to the initial value problem. 
  The stability of a spherically symmetric self-gravitating magnetic monopole is examined in the thin wall approximation: modeling the interior false vacuum as a region of de Sitter space; the exterior as an asymptotically flat region of the Reissner-Nordstr\"om geometry; and the boundary separating the two as a charged domain wall. There remains only to determine how the wall gets embedded in these two geometries. In this approximation, the ratio $k$ of the false vacuum to surface energy densities is a measure of the symmetry breaking scale $\eta$. Solutions are characterized by this ratio, the charge on the wall $Q$, and the value of the conserved total energy $M$. We find that for each fixed $k$ and $Q$ up to some critical value, there exists a unique globally static solution, with $M\simeq Q^{3/2}$; any stable radial excitation has $M$ bounded above by $Q$, the value assumed in an extremal Reissner-Nordstr\"om geometry and these are the only solutions with $M<Q$. As $M$ is raised above $Q$ a black hole forms in the exterior: (i) for low $Q$ or $k$, the wall is crushed; (ii) for higher values, it oscillates inside the black hole. If the mass is not too high these `collapsing' solutions co-exist with an inflating bounce; (iii) for $k$, $Q$ or $M$ outside the above regimes, there is a unique inflating solution. In case (i) the course of the bounce lies within a single asymptotically flat region (AFR) and it resembles closely the bounce exhibited by a false vacuum bubble (with Q=0). In cases (ii) and (iii) the course of the bounce spans two consecutive AFRs. 
  It is shown that the space-time with a conical singularity, which describes a thin cosmic string, is hyperbolic in the sense that a unique H^1 solution exists to the initial value problem for the wave equation with a certain class of initial data. 
  We have developed a general geometric treatment of the GCE valid for any stationary axisymmetric metric. The method is based on the remark that the world lines of objects rotating along spacely circular trajectories are in any case, for those kind of metrics, helices drawn on the flat bidimensional surface of a cylinder. Applying the obtained formulas to the special cases of the Kerr and weak field metric for a spinning body, known results for time delays and synchrony defects are recovered. 
  A four-parameter class of exact asymptotically flat solutions of the Einstein-Maxwell equations involving only rational functions is presented. It is able to describe the exterior field of a slowly or rapidly rotating neutron star with poloidal magnetic field. 
  A dynamical study of the generalised scalar-tensor theory in the empty Bianchi type I model is made. We use a method from which we derive the sign of the first and second derivatives of the metric functions and examine three different theories that can all tend towards relativistic behaviours at late time. We determine conditions so that the dynamic be in expansion and decelerated at late time. 
  By applying a standard solution generating technique, we transform an arbitrary vacuum Mixmaster solution on $S^3 \times {\bf R}$ to a new solution which is spatially inhomogeneous. We thereby obtain a family of exact, spatially inhomogeneous, vacuum spacetimes which exhibit Belinskii, Khalatnikov, and Lifshitz (BKL) oscillatory behavior. The solutions are constructed explicitly by performing the transformations on numerically generated, homogeneous Mixmaster solutions. Their behavior is found to be qualitatively like that seen in previous numerical simulations of generic U(1) symmetric cosmological spacetimes on $T^3 \times {\bf R}$. 
  We illustrate how Hawking's radiance from a Schwarzschild black hole is modified by the electrostatic self-interaction of the emitted charged particles. A W.K.B approximation shows that the probability for a self-interacting charged particle to propagate from the interior to the exterior of the horizon is increased relative to the corresponding probability for neutral particles. We also demonstrate how the electric potential of a charged test object in the black hole's vicinity gives rise to pair creation. We analyze this phenomenon semiclassically by considering the existence of the appropriate Klein region. Finally we discuss the possible energy source for the process. 
  We prove that the quantum stress tensor for a massless scalar field in two dimensional non-selfsimilar Tolman Bondi dust collapse and Vaidya radiation collapse models diverges on the Cauchy horizon, if the latter exists. The two dimensional model is obtained by suppressing angular co-ordinates in the corresponding four dimensional spherical model. 
  By noticing that, in open 2+1 gravity, polarized surfaces cannot converge in the presence of timelike total energy momentum (except for a rotation of 2 pi), we give a simple argument which shows that, quite generally, closed timelike curves cannot exist in the presence of such energy condition. 
  We derive the general exact vacuum metrics associated with a stationary (non static), non rotating, cylindrically symmetric source. An analysis of the geometry described by these vacuum metrics shows that they contain a subfamily of metrics that, although admitting a consistent time orientation, display "exotic" properties, such as "trapping" of geodesics and closed causal curves through every point. The possibility that such spacetimes could be generated by a superconducting string, endowed with a neutral current and momentum, has recently been considered by Thatcher and Morgan. Our results, however, differ from those found by Thatcher and Morgan, and the discrepancy is explained. We also analyze the general possibility of constructing physical sources for the exotic metrics, and find that, under certain restrictions, they must always violate the dominant energy condition (DEC). We illustrate our results by explicitly analyzing the case of concentric shells, where we find that in all cases the external vacuum metric is non exotic if the matter in the shells satisfies the DEC. 
  The various roles of boundary terms in the gravitational Lagrangian and Hamiltonian are explored. A symplectic Hamiltonian-boundary-term approach is ideally suited for a large class of quasilocal energy-momentum expressions for general relativity. This approach provides a physical interpretation for many of the well-known gravitational energy-momentum expressions including all of the pseudotensors, associating each with unique boundary conditions. From this perspective we find that the pseudotensors of Einstein and M{\o}ller (which is closely related to Komar's superpotential) are especially natural, but the latter has certain shortcomings. Among the infinite possibilities, we found that there are really only two Hamiltonian-boundary-term quasilocal expressions which correspond to {\em covariant} boundary conditions; they are respectively of the Dirichlet or Neumann type. Our Dirichlet expression coincides with the expression recently obtained by Katz and coworkers using Noether arguments and a fixed background. A modification of their argument yields our Neumann expression. 
  We consider the classical theory of the Dirac massive particle in the Riemann-Cartan spacetime. We demonstrate that the translational and the Lorentz gravitational moments, obtained by means of the Gordon type decompositions of the canonical energy-momentum and spin currents, are consistently coupled to torsion and curvature, as expected. 
  We discuss a method to analyze data from interferometric gravitational wave detectors focusing on the technique of hierarchical search to detect gravitational waves from inspiraling binaries. For this purpose, we propose new coordinates to parameterize the template space. Using our new coordinates, we develop several new techniques for two step search, which would reduce the computation cost by large amount. These techniques become more important when we need to implement a $\chi^2$-test as a detection criterion. 
  It is proved that the only geodesically complete stationary vacuum solution of the Einstein equations is the empty Minkowski space, or a quotient of it by a discrete group of isometries, generalizing a classical result of Lichnerowicz. In addition, we obtain an apriori bound on the curvature of stationary vacuum solutions away from the boundary or horizon. 
  We perform a complete rho-integration of the GHP equations for all spacetimes that admit a geodesic shear-free expanding null congruence, whose Ricci spinor is aligned to the congruence and whose Ricci scalar is constant. We also deduce the system of GHP equations after the integration is completed, and discuss a few applications. 
  Variation of the 4-D string cosmology action with dynamical torsion and massless dilatons lead to an expression of torsion in terms of massless dilatons in the case of de Sitter inflation.The solution is approximated according to the COBE data. 
  We estimate the signal-to-noise ratio for two gravitational detectors interacting with a stochastic background of massive scalar waves. We find that the present experimental level of sensitivity could be already enough to detect a signal from a light but non-relativistic component of dark matter, even if the coupling is weak enough to exclude observable deviations from standard gravitational interactions, provided the mass is not too far from the sensitivity and overlapping band of the two detectors. 
  We consider a classical condensed matter theory in a Newtonian framework where conservation laws   \partial_t \rho + \partial_i (\rho v^i) = 0    \partial_t (\rho v^j) + \partial_i(\rho v^i v^j + p^{ij}) = 0   are related with the Lagrange formalism in a natural way. For an ``effective Lorentz metric'' g_{\mu\nu} it is equivalent to a metric theory of gravity close to general relativity with Lagrangian   L = L_{GR}   - (8\pi G)^{-1}(\Upsilon g^{00}-\Xi (g^{11}+g^{22}+g^{33}))\sqrt{-g}   We consider the differences between this theory and general relativity (no nontrivial topologies, stable frozen stars instead of black holes, big bounce instead of big bang singularity, a dark matter term), quantum gravity, and the connection with realism and Bohmian mechanics. 
  We present a metric theory of gravity with Lagrangian   L = (8\pi G)^{-1}(\Xi g^{ii} - \Upsilon g^{00})\sqrt{-g} + L_{GR} + L_{matter}  motivated by classical equations      \partial_t \rho + \partial_i (\rho v^i) = 0     \partial_t (\rho v^j) + \partial_i (\rho v^i v^j + p^{ij}) = 0   for a medium in Newtonian space-time. We obtain stable ``frozen stars'' instead of black holes and a ``big bounce'' instead of a big bang singularity. 
  We show that estimates of the Lorentz symmetry violation extracted from ultra-high energy cosmic rays beyond the GZK cut-off set bounds on the parameters of a Lorentz-violating extension of the Standard Model. Moreover, we argue that correlated measurements of the difference in the arrival time of gamma-ray photons and neutrinos emitted from Active Galactic Nuclei or Gamma-Ray Bursts may provide a signature for a possible violation of the Lorentz symmetry. We find that this time delay is energy independent, but that it has a dependence on the chirality of the particles involved. 
  The semiclassical Einstein-Langevin equations which describe the dynamics of stochastic perturbations of the metric induced by quantum stress-energy fluctuations of matter fields in a given state are considered on the background of the ground state of semiclassical gravity, namely, Minkowski spacetime and a scalar field in its vacuum state. The relevant equations are explicitly derived for massless and massive fields arbitrarily coupled to the curvature. In doing so, some semiclassical results, such as the expectation value of the stress-energy tensor to linear order in the metric perturbations and particle creation effects, are obtained. We then solve the equations and compute the two-point correlation functions for the linearized Einstein tensor and for the metric perturbations. In the conformal field case, explicit results are obtained. These results hint that gravitational fluctuations in stochastic semiclassical gravity have a ``non-perturbative'' behavior in some characteristic correlation lengths. 
  The energy conditions of general relativity permit one to deduce very powerful and general theorems about the behaviour of strong gravitational fields and cosmological geometries. However, the energy conditions these theorems are based on are beginning to look a lot less secure than they once seemed: (1) there are subtle quantum effects that violate all of the energy conditions, and more tellingly (2), there are also relatively benign looking classical systems that violate all the energy conditions. This opens up a Pandora's box of rather disquieting possibilities --- everything from negative asymptotic mass, to traversable wormholes, to warp drives, up to and including time machines. 
  I develop a phenomenological approach to the description of the noise levels that the space-time foam of quantum gravity could induce in modern gravity-wave detectors. Various possibilities are considered, including white noise and random-walk noise. In particular, I find that the sensitivity level expected for the planned LIGO and VIRGO interferometers and for the next upgrade of the NAUTILUS resonant-bar detector corresponds to a white-noise level which can be naturally associated with the Planck length. 
  The paper is an introduction into General Ether Theory (GET). We start with few assumptions about an universal ``ether'' in a Newtonian space-time which fulfils  \partial_t \rho + \partial_i (\rho v^i) = 0  \partial_t (\rho v^j) + \partial_i(\rho v^i v^j + p^{ij}) = 0   For an ``effective metric'' $g_{\mu\nu}$ we derive a Lagrangian where the Einstein equivalence principle is fulfilled:   L=L_{GR}-(8\pi G)^{-1}(\Upsilon g^{00}-\Xi(g^{11}+g^{22}+g^{33}))\sqrt{-g}   We consider predictions (stable frozen stars instead of black holes, big bounce instead of big bang singularity, a dark matter term), quantization (regularization by an atomic ether, superposition of gravitational fields), related methodological questions (covariance, EPR criterion, Bohmian mechanics). 
  We study the propagation of null rays and massless fields in a black hole fluctuating geometry. The metric fluctuations are induced by a small oscillating incoming flux of energy. The flux also induces black hole mass oscillations around its average value. We assume that the metric fluctuations are described by a statistical ensemble. The stochastic variables are the phases and the amplitudes of Fourier modes of the fluctuations. By averaging over these variables, we obtain an effective propagation for massless fields which is characterized by a critical length defined by the amplitude of the metric fluctuations: Smooth wave packets with respect to this length are not significantly affected when they are propagated forward in time. Concomitantly, we find that the asymptotic properties of Hawking radiation are not severely modified. However, backward propagated wave packets are dissipated by the metric fluctuations once their blue shifted frequency reaches the inverse critical length. All these properties bear many resemblences with those obtained in models for black hole radiation based on a modified dispersion relation. This strongly suggests that the physical origin of these models, which were introduced to confront the trans-Planckian problem, comes from the fluctuations of the black hole geometry. 
  We study equilibrium configurations of boson stars in the framework of a class scalar-tensor theories of gravity with massive gravitational scalar (dilaton). In particular we investigate the influence of the mass of the dilaton on the boson star structure. We find that the masses of the boson stars in presence of dilaton are close to those in general relativity and they are sensitive to the ratio of the boson mass to the dilaton mass within a typical few percent. It turns out also that the boson star structure is mainly sensitive to the mass term of the dilaton potential rather to the exact form of the potential. 
  The notion that the topology of the universe need not be that of the universal covering space of its geometry has recently received renewed attention. Generic signatures of cosmological topology have been sought, both in the distribution of objects in the universe, and especially in the temperature fluctuations of the cosmic microwave background radiation (CMBR). One signature identified in the horn topology but hypothesized to be generic is featureless regions or flat spots in the CMBR sky. We show that typical observation points within the cusped 3-manifold m003 from the Snappea census have flat spots with an angular scale of about five degrees for $\Omega_0$=0.3. We expect that this holds for other small volume cusped manifolds with this $\Omega_0$ value. 
  In the pure Einstein-Yang-Mills theory in four dimensions there exist monopole and dyon solutions. The spectrum of the solutions is discrete in asymptotically flat or de Sitter space, whereas it is continuous in asymptotically anti-de Sitter space. The solutions are regular everywhere and specified with their mass, and non-Abelian electric and magnetic charges. In asymptotically anti-de Sitter space a class of monopole solutions have no node in non-Abelian magnetic fields, and are stable against spherically symmetric perturbations. 
  We perform a numerical analysis of the gravitational field of a global monopole coupled nonminimally to gravity, and find that, for some given nonminimal couplings (in constrast with the minimal coupling case), there is an attractive region where bound orbits exist. We exhibit the behavior of the frequency shifts that would be associated with `rotation curves' of stars in circular orbits in the spacetimes of such global monopoles. 
  The black hole combines in some sense both the ``hydrogen atom'' and the ``black-body radiation'' problems of quantum gravity. This analogy suggests that black-hole quantization may be the key to a quantum theory of gravity. During the last twenty-five years evidence has been mounting that black-hole surface area is indeed {\it quantized}, with {\it uniformally} spaced area eigenvalues. There is, however, no general agreement on the {\it spacing} of the levels. In this essay we use Bohr's correspondence principle to provide this missing link. We conclude that the fundamental area unit is $4\hbar\ln3$. This is the unique spacing consistent both with the area-entropy {\it thermodynamic} relation for black holes, with Boltzmann-Einstein formula in {\it statistical physics} and with {\it Bohr's correspondence principle}. 
  What forms will have an equations of modern physics if the dimensions of our time and space are fractional? The generalized equations enumerated by title are presented by help the generalized fractional derivatives of Riemann-Liouville. 
  The problem to estimate the background due to accidental coincidences in the search for coincidences in gravitational wave experiments is discussed. The use of delayed coincidences obtained by orderly shifting the event times of one of the two detectors is shown to be the most correct 
  A gravitationally-induced modification to de Broglie wave-particle duality is presented. At Planck scale, the gravitationally-modified matter wavelength saturates to a few times the Planck length in a momentum independent manner. In certain frameworks, this circumstance freezes neutrino oscillations in the Planck realm. This effect is apart, and beyond, the gravitational red-shift. A conclusion is drawn that in a complete theory of quantum gravity the notions of ``quantum'' and ``gravity'' shall carry new meanings -- meanings, that are yet to be deciphered from theory and observations in their entirety. 
  The operation of an interferometer for gravitational waves detection requires sophisticated feedback controls in many parts of the apparatus. The aim of this lecture is to introduce the types of problems to be faced in this line of research. The attention is focused on the "inertial damping" of the test mass suspension of the VIRGO interferometer (the superattenuator): it is a multidimensional local control aimed to reduce the residual motion of the suspended mirror associated to the normal modes of the suspension. Its performance is very important for the locking of the interferometer. 
  The semiclassical Einstein equations are solved to first order in $\epsilon = \hbar/M^2$ for the case of a Reissner-Nordstr\"{o}m black hole perturbed by the vacuum stress-energy of quantized free fields. Massless and massive fields of spin 0, 1/2, and 1 are considered. We show that in all physically realistic cases, macroscopic zero temperature black hole solutions do not exist. Any static zero temperature semiclassical black hole solutions must then be microscopic and isolated in the space of solutions; they do not join smoothly onto the classical extreme Reissner-Nordst\"{o}m solution as $\epsilon \to 0$. 
  Five cryogenic resonant gravitational antennas are now in operation. This is the first time that such a large number of high sensitive antennas are taking data and an agreement on data exchange has been signed by the responsible groups. The data exchanged will consist essentially in lists of ''candidate events''. In this paper the procedure used by the Rome group in order to obtain ''candidate events'' is presented.   Some methods of analyzing the data of the "network" of the five antennas are shown. 
  We propose an adaptive denoising scheme for poorly modeled non-Gaussian features in the gravitational wave interferometric data. Preliminary tests on real data show encouraging results. 
  We formulate the data analysis problem for the detection of the Newtonian waveform from an inspiraling compact-binary by a network of arbitrarily oriented and arbitrarily distributed laser interferometric gravitational wave detectors. We obtain for the first time the relation between the optimal statistic and the magnitude of the network correlation vector, which is constructed from the matched network-filter. This generalizes the calculation reported in an earlier work (gr-qc/9906064), where the detectors are taken to be coincident. 
  In [cond-mat/9906332; Phys. Rev. Lett. 84, 822 (2000)] and [physics/9906038; Phys. Rev. A 60, 4301 (1999)] Leonhardt and Piwnicki have presented an interesting analysis of how to use a flowing dielectric fluid to generate a so-called "optical black hole". Qualitatively similar phenomena using acoustical processes have also been much investigated. Unfortunately there is a subtle misinterpretation in the Leonhardt-Piwnicki analysis regarding these "optical black holes": While it is clear that "optical black holes" can certainly exist as theoretical constructs, and while the experimental prospects for actually building them in the laboratory are excellent, the particular model geometries that Leonhardt and Piwnicki write down as alleged examples of "optical black holes" are in fact not black holes at all. 
  A comprehensive physical theory explains all aspects of the physical universe, including quantum aspects, classical aspects, relativistic aspects, their relationships, and unification. The central nonlocality principle leads to a nonlocal geometry that explains entire quantum phenomenology, including two-slit experiment, Aspect-type experiments, quantum randomness, tunneling, etc. The infinitesimal aspect of this geometry is a usual (differential) geometry, various aspects of which are energy-momentum, spin-helicity, electric, color and flavor charges. Their interactions are governed by a mathematically automatic field equation - also a grand conservation principle. New predictions: a new particle property; bending-of-light estimates refined over relativity's; shape of the universe; a no-gravitational-singularity theorem, etc. Nonlocal physics is formulated using a nonlocal calculus and nonlocal differential equations, replacing inadequate local concepts of Newton's calculus and partial differential equations. Usual quantum formalisms follow from our theory - the latter doesn't rest on the former. 
  Initial data for black hole collisions are commonly generated using the Bowen-York approach based on conformally flat 3-geometries. The standard (constant Boyer-Lindquist time) spatial slices of the Kerr spacetime are not conformally flat, so that use of the Bowen-York approach is limited in dealing with rotating holes. We investigate here whether there exist foliations of the Kerr spacetime that are conformally flat. We limit our considerations to foliations that are axisymmetric and that smoothly reduce in the Schwarzschild limit to slices of constant Schwarzschild time. With these restrictions, we show that no conformally flat slices can exist. 
  A global model of a slowly rotating perfect fluid ball in general relativity is presented. To second order in the rotation parameter, the junction surface is an ellipsoidal cylinder. The interior is given by a limiting case of the Wahlquist solution, and the vacuum region is not asymptotically flat. The impossibility of joining an asymptotically flat vacuum region has been shown in a preceding work. 
  It is shown that, in dilute-gas Bose-Einstein condensates, there exist both dynamically stable and unstable configurations which, in the hydrodynamic limit, exhibit a behavior resembling that of gravitational black holes. The dynamical instabilities involve creation of quasiparticle pairs in positive and negative energy states, as in the well-known suggested mechanism for black hole evaporation. We propose a scheme to generate a stable sonic black hole in a ring trap. 
  By the method of rho-integration we obtain all Lanczos potentials L_{ABCA'} of the Weyl spinor that, in a certain sense, are aligned to a geodesic shear-free expanding null congruence. We also obtain all spinors H_{ABA'B'}=Q_{AB}o_{A'}o_{B'}, Q_{AB}=Q_{(AB)} satisfying nabla_{(A}{}^{B'}H_{BC)A'B'}=L_{ABCA'}. We go on to prove that H_{ABA'B'} can be chosen so that Gamma_{ABCA'}=nabla_{(A}{}^{B'} H_{B)CA'B'} defines a metric asymmetric curvature-free connection such that L_{ABCA'}=Gamma_{(ABC)A'} is a Lanczos potential that is aligned to the geodesic shear-free expanding congruence. These results are a generalization to a large class of algebraically special spacetimes (including all vacuum ones for which the principal null direction is expanding) of the curvature-free connection of the Kerr spacetime found by Bergqvist and Ludvigsen, which was used in a construction of quasi-local momentum. 
  The integration of acceleration over time before reaching the uniform velocity turns out to be the source of all the special relativity effects. It explains physical phenomena like clocks comparisons. The equations for space-time, mass and energy are presented. This phenomenon complements the explanation for the twins paradox. A Universal reference frame is obtained. 
  We consider the quantization of a massless scalar field, using the geometric optics approximation, in the background spacetime of a collapsing spherical self-similar Vaidya star, which forms a black hole or a naked singularity. We show that the outgoing radiation flux of the quantized scalar field diverges on the Cauchy horizon. The spectrum of the produced scalar partcles is non-thermal when the background develops a naked singularity. These results are analogous to those obtained for the scalar quantization on a self-similar dust cloud. 
  We examine particle production from spherical bodies collapsing into extremal Reissner-Nordstr\"om black holes. Kruskal coordinates become ill-defined in the extremal case, but we are able to find a simple generalization of them that is good in this limit. The extension allows us to calculate the late-time worldline of the center of the collapsing star, thus establishing a correspondence with a uniformly accelerated mirror in Minkowski spacetime. The spectrum of created particles associated with such uniform acceleration is nonthermal, indicating that a temperature is not defined. Moreover, the spectrum contains a constant that depends on the history of the collapsing object. At first sight this points to a violation of the no-hair theorems; however, the expectation value of the stress-energy-momentum tensor is zero and its variance vanishes as a power law at late times. Hence, both the no-hair theorems and the cosmic censorship conjecture are preserved. The power-law decay of the variance is in distinction to the exponential fall-off of a nonextremal black hole. Therefore, although the vanishing of the stress tensor's expectation value is consistent with a thermal state at zero temperature, the incipient black hole does not behave as a thermal object at any time and cannot be regarded as the thermodynamic limit of a nonextremal black hole, regardless of the fact that the final product of collapse is quiescent. 
  We study vacuum polarization of quantized massive scalar fields $\phi$ in equilibrium at black-hole temperature in Reissner-Nordstr\"{o}m background. By means of the Euclidean space Green's function we analytically derive the renormalized expression $<\phi^{2}>_{H}$ at the event horizon with the area $4\pi r_{+}^{2}$. It is confirmed that the polarization amplitude $<\phi^{2}>_{H}$ is free from any divergence due to the infinite red-shift effect. Our main purpose is to clarify the dependence of $<\phi^{2}>_{H}$ on field mass $m$ in relation to the excitation mechanism. It is shown for small-mass fields with $mr_{+}\ll1$ how the excitation of $<\phi^{2}>_{H}$ caused by finite black-hole temperature is suppressed as $m$ increases, and it is verified for very massive fields with $mr_{+}\gg1$ that $<\phi^{2}>_{H}$ decreases in proportion to $m^{-2}$ with the amplitude equal to the DeWitt-Schwinger approximation. In particular, we find a resonance behavior with a peak amplitude at $mr_{+}\simeq 0.38$ in the field-mass dependence of vacuum polarization around nearly extreme (low-temperature) black holes. The difference between Scwarzschild and nearly extreme black holes is discussed in terms of the mass spectrum of quantum fields dominant near the event horizon. 
  In this paper Einstein's field equations, for static spherically symmetric perfect fluid models with a linear barotropic equation of state, are recast into a 3-dimensional regular system of ordinary differential equations on a compact state space. The system is analyzed qualitatively, using the theory of dynamical systems, and numerically. It is shown that certain special solutions play important roles as building blocks for the solution structure in general. In particular, these special solutions determine many of the features exhibited by solutions with a regular center and large central pressure. It is also shown that the present approach can be applied to more general classes of barotropic equations of state. 
  In this paper, the gravitational field equations for static spherically symmetric perfect fluid models with a polytropic equation of state, $p=k\rho^{1+1/n}$, are recast into two complementary 3-dimensional {\it regular} systems of ordinary differential equations on compact state spaces. The systems are analyzed numerically and qualitatively, using the theory of dynamical systems. Certain key solutions are shown to form building blocks which, to a large extent, determine the remaining solution structure. In one formulation, there exists a monotone function that forces the general relativistic solutions towards a part of the boundary of the state space that corresponds to the low pressure limit. The solutions on this boundary describe Newtonian models and thus the relationship to the Newtonian solution space is clearly displayed. It is numerically demonstrated that general relativistic models have finite radii when the polytropic index $n$ satisfies $0\leq n \lesssim 3.339$ and infinite radii when $n\geq 5$. When $3.339\lesssim n<5$, there exists a 1-parameter set of models with finite radii and a finite number, depending on $n$, with infinite radii. 
  We study the classical and quantum dynamics of generally covariant theories with vanishing a Hamiltonian and with a finite number of degrees of freedom. In particular, the geometric meaning of the full solution of the relational evolution of the degrees of freedom is displayed, which means the determination of the total number of evolving constants of motion required. Also a method to find evolving constants is proposed. The generalized Heinsenberg picture needs M time variables, as opposed to the Heisenberg picture of standard quantum mechanics where one time variable t is enough. As an application, we study the parameterized harmonic oscillator and the SL(2,R) model with one physical degree of freedom that mimics the constraint structure of general relativity where a Schrodinger equation emerges in its quantum dynamics. 
  We study the possibility of applying statistical mechanics to generally covariant quantum theories with a vanishing Hamiltonian. We show that (under certain appropiate conditions) this makes sense, in spite of the absence of a notion of energy and external time. We consider a composite system formed by a large number of identical components, and apply Boltzmann's ideas and the fundamental postulates of ordinary statistical physics. The thermodynamical parameters are determined by the properties of the thermalizing interaction. We apply these ideas to a simple example, in which the component system has one physical degree of freedom and mimics the constraint algebra of general relativity. 
  The behaviour of a "test" electromagnetic field in the background of an exact gravitational plane wave is investigated in the framework of Einstein's general relativity. We have expressed the general solution to the de Rham equations as a Fourier-like integral. In the general case we have reduced the problem to a set of ordinary differential equations and have explicitly written the solution in the case of linear polarization of the gravitational wave. We have expressed our results by means of Fermi Normal Coordinates (FNC), which define the proper reference frame of the laboratory. Moreover we have provided some "gedanken experiments", showing that an external gravitational wave induces measurable effects of non tidal nature via electromagnetic interaction. Consequently it is not possible to eliminate gravitational effects on electromagnetic field, even in an arbitrarily small spatial region around an observer freely falling in the field of a gravitational wave. This is opposite to the case of mechanical interaction involving measurements of geodesic deviation effects. This behaviour is not in contrast with the principle of equivalence, which applies to arbitrarily small region of both space and time. 
  In a paper presented a few years ago, De Lorenci et al. showed, in the context of canonical quantum cosmology, a model which allowed space topology changes (Phys. Rev. D 56, 3329 (1997)). The purpose of this present work is to go a step further in that model, by performing some calculations only estimated there for several compact manifolds of constant negative curvature, such as the Weeks and Thurston spaces and the icosahedral hyperbolic space (Best space). 
  News:   TGG session in the April meeting, by Cliff Will NRC report, by Beverly Berger MG9 Travel Grant for US researchers, by Jim Isenberg Research Briefs:   How many coalescing binaries are there?, by Vicky Kalogera Recent developments in black critical phenomena, by Pat Brady Optical black holes?, by Matt Visser ``Branification:'' an alternative to compactification, by Steve Giddings Searches for non-Newtonian Gravity at Sub-mm Distances, by Riley Newman Quiescent cosmological singularities by Bernd Schmidt The debut of LIGO II, by David Shoemaker Is the universe still accelerating?, by Sean Carroll Conference reports:   Journ\' ees Relativistes Weimar 1999, by Volker Perlick The 9th Midwest Relativity Meeting, by Thomas Baumgarte 
  We employ a theorem due to Campbell to build some simple 4-dimensional cosmological models which originate from solutions describing waves propagating along the extra-dimension of a 5-dimensional Ricci-flat space. The dimensional reduction is performed in the Jordan frame according to the induced-matter theory of Wesson. 
  The classical and quantum models of the Friedmann universe originally filled with a scalar field and radiation have been studied. The radiation has been used to specify a reference frame that makes it possible to remove ambiguities in choosing the time coordinate. Solutions to the Einstein and Schroedinger equations have been studied under the assumption that the rate of scalar-field variation is much less than the rate of universe expansion (contraction). It has been shown that, under certain conditions, the quantum universe can be in quasistationary states. The probability that the universe goes over to states with large quantum numbers owing to the interaction of the scalar and gravitational fields is nonzero. It has been shown that, in the lowest state, the scale factor is on order of the Planck length. The matter- and radiation-energy densities in the Planck era have been computed. The possible scenarios of Universe evolution are discussed. 
  A highly excited Friedmann universe filled with a scalar field and radiation has been considered. On the basis of a direct solution to the quantum-mechanical problem with a well-defined time variable, it has been shown that such a universe can have features (energy density, scale factor, Hubble constant, density parameter, matter mass, equivalent number of baryons, age, dimensions of large-scale fluctuations, amplitude of fluctuations of cosmic microwave background radiation temperature) identical to those of the currently observed Universe. 
  We present a class of numerical solutions to the SU(2) nonlinear $\sigma$-model coupled to the Einstein equations with cosmological constant $\Lambda\geq 0$ in spherical symmetry. These solutions are characterized by the presence of a regular static region which includes a center of symmetry. They are parameterized by a dimensionless ``coupling constant'' $\beta$, the sign of the cosmological constant, and an integer ``excitation number'' $n$. The phenomenology we find is compared to the corresponding solutions found for the Einstein-Yang-Mills (EYM) equations with positive $\Lambda$ (EYM$\Lambda$). If we choose $\Lambda$ positive and fix $n$, we find a family of static spacetimes with a Killing horizon for $0 \leq \beta < \beta_{max}$. As a limiting solution for $\beta = \beta_{max}$ we find a {\em globally} static spacetime with $\Lambda=0$, the lowest excitation being the Einstein static universe. To interpret the physical significance of the Killing horizon in the cosmological context, we apply the concept of a trapping horizon as formulated by Hayward. For small values of $\beta$ an asymptotically de Sitter dynamic region contains the static region within a Killing horizon of cosmological type. For strong coupling the static region contains an ``eternal cosmological black hole''. 
  Small-eccentricity binary pulsars with white dwarf companions provide excellent test laboratories for various effects predicted by alternative theories of gravity, in particular tests for the emission of gravitational dipole radiation and the existence of gravitational Stark effects. We will present new limits to these effects. The statistical analysis presented here, for the first time, takes appropriately care of selection effects. 
  When quantum fields are studied on manifolds with boundary, the corresponding one-loop quantum theory for bosonic gauge fields with linear covariant gauges needs the assignment of suitable boundary conditions for elliptic differential operators of Laplace type. There are however deep reasons to modify such a scheme and allow for pseudo-differential boundary-value problems. When the boundary operator is allowed to be pseudo-differential while remaining a projector, the conditions on its kernel leading to strong ellipticity of the boundary-value problem are studied in detail. This makes it possible to develop a theory of one-loop quantum gravity from first principles only, i.e. the physical principle of invariance under infinitesimal diffeomorphisms and the mathematical requirement of a strongly elliptic theory. 
  We prove the existence of a countable family of globally regular solutions of spherically symmetric Einstein-Klein-Gordon equations. These solutions, known as mini-boson stars, were discovered numerically many years ago. 
  The range of expected amplitudes and spectral slopes of relic (squeezed) gravitational waves, predicted by theory and partially supported by observations, is within the reach of sensitive gravity-wave detectors. In the most favorable case, the detection of relic gravitational waves can be achieved by the cross-correlation of outputs of the initial laser interferometers in LIGO, VIRGO, GEO600. In the more realistic case, the sensitivity of advanced ground-based and space-based laser interferometers will be needed. The specific statistical signature of relic gravitational waves, associated with the phenomenon of squeezing, is a potential reserve for further improvement of the signal to noise ratio. 
  We investigate the propagation of electromagnetic waves through a static wormhole. It is shown that the problem can be reduced to a one-dimensional Schr\"odinger-like equation with a barrier-type potential. Using numerical methods, we calculate the transmission coefficient as a function of the energy. We also discuss the polarization of the outgoing radiation due to this gravitational scattering. 
  Partial results are obtained for Schwarzschild- like solutions in a gravity theory with action density (-g)^(1/2)[Rik^2+bR^2]. A seven parameter family of implicit solutions is found. A number of explicit solutions are also exhibited. 
  Weinberg's energy-momentum pseudotensor is obtained for Schwarzschild metric in harmonic coordinates. On the horizon it possesses unintegrable singularities. For this reason the total energy of a collapsar can't be obtained by integrating energy density over the system's volume. The implication for gravity theories is noted. A thought on how to choose unique energy-momentum tensor is given. 
  I consider a new semiclassical expansion for the inflaton field in the framework of warm inflation scenario. The fluctuations of the matter field are considered as thermally coupled with the particles of the thermal bath. This coupling parameter depends on the temperature of the bath. The power spectrum remains invariant under this new semiclassical expansion for the inflaton. However, I find that the thermal component of the amplitude for the primordial field fluctuations should be very small at the end of inflation. 
  The exact formula derived by us earlier for the entropy of a four dimensional non-rotating black hole within the quantum geometry formulation of the event horizon in terms of boundary states of a three dimensional Chern-Simons theory, is reexamined for large horizon areas. In addition to the {\it semiclassical} Bekenstein-Hawking contribution to the area obtained earlier, we find a contribution proportional to the logarithm of the area together with subleading corrections that constitute a series in inverse powers of the area. 
  It is shown that if the timelike eigenvector of the Ricci tensor be hypersurface orthogonal so that the space time allows a foliation into space sections then the space average of each of the scalar that appear in the Raychaudhuri equation vanishes provided the strong energy condition holds good. This result is presented in the form of a singularity theorem. 
  We consider exact solutions generated by the inverse scattering technique, also known as the soliton transformation. In particular, we study the class of simple real pole solutions. For quite some time, those solutions have been considered interesting as models of cosmological shock waves. A coordinate singularity on the wave fronts was removed by a transformation which induces a null fluid with negative energy density on the wave front. This null fluid is usually seen as another coordinate artifact, since there seems to be a general belief that that this kind of solution can be seen as the real pole limit of the smooth solution generated with a pair of complex conjugate poles in the transformation. We perform this limit explicitly, and find that the belief is unfounded: two coalescing complex conjugate poles cannot yield a solution with one real pole. Instead, the two complex conjugate poles go to a different limit, what we call a ``pole on a pole''. The limiting procedure is not unique; it is sensitive to how quickly some parameters approach zero. We also show that there exists no improved coordinate transformation which would remove the negative energy density. We conclude that negative energy is an intrinsic part of this class of solutions. 
  As shown by Teukolsky, the master equation governing the propagation of weak radiation in a black hole spacetime can be separated into four ordinary differential equations, one for each spacetime coordinate. (``Weak'' means the radiation's amplitude is small enough that its own gravitation may be neglected.) Unfortunately, it is difficult to accurately compute solutions to the separated radial equation (the Teukolsky equation), particularly in a numerical implementation. The fundamental reason for this is that the Teukolsky equation's potentials are long ranged. For non-spinning black holes, one can get around this difficulty by applying transformations which relate the Teukolsky solution to solutions of the Regge-Wheeler equation, which has a short-ranged potential. A particularly attractive generalization of this approach to spinning black holes for gravitational radiation (spin weight s = -2) was given by Sasaki and Nakamura. In this paper, I generalize Sasaki and Nakamura's results to encompass radiation fields of arbitrary integer spin weight, and give results directly applicable to scalar (s = 0) and electromagnetic (s = -1) radiation. These results may be of interest for studies of astrophysical radiation processes near black holes, and of programs to compute radiation reaction forces in curved spacetime. 
  The recent analysis of Markovic and Shapiro on the effect of a cosmological constant on the evolution of a spherically symmetric homogeneous dust ball is extended to include the inhomogeneous and degenerate cases. The histories are shown by way of effective potential and Penrose-Carter diagrams. 
  Ultra compact stellar models with a two-zone uniform density equation of state are considered. They are shown to provide neat examples of optical geometries exhibiting double necks, implying that the gravitational wave potential has a double well structure. 
  1. Following Rimman, Minkowski and Einstein, for the first time equations of the inert filed in the covariant form are found geometrically. 2.In the approximation of a weak field for the first time the Law of Inertia in a material space (as opposed to the absolute space) is received. A consequence is the formulation of Mach's principle. 3. Analogous to Einstein's expression for the gravitational field, Compton's formula is received for the inert field. 4. For the first time transcendental equations are received, one of the solutions of which corresponds to the value of the magnetic charge by Dirack, or to the constant of fine structure. 
  The peeling properties of a lightlike signal propagating through a general Bondi-Sachs vacuum spacetime and leaving behind another Bondi-Sachs vacuum space-time are studied. We demonstrate that in general the peeling behavior is the conventional one which is associated with a radiating isolated system and that it becomes unconventional if the asymptotically flat space-times on either side of the history of the light-like signal tend to flatness at future null infinity faster than the general Bondi-Sachs space-time. This latter situation occurs if, for example, the space-times in question are static Bondi-Sachs space- times. 
  When two non-interacting plane impulsive gravitational waves undergo a head-on collision, the vacuum interaction region between the waves after the collision contains backscattered gravitational radiation from both waves. The two systems of backscattered waves have each got a family of rays (null geodesics) associated with them. We demonstrate that if it is assumed that a parameter exists along each of these families of rays such that the modulus of the complex shear of each is equal then Einstein's vacuum field equations, with the appropriate boundary conditions, can be integrated systematically to reveal the well-known solutions in the interaction region. In so doing the mystery behind the origin of such solutions is removed. With the use of the field equations it is suggested that the assumption leading to their integration may be interpreted physically as implying that the energy densities of the two backscattered radiation fields are equal. With the use of different boundary conditions this approach can lead to new collision solutions. 
  We enunciate and prove here a generalization of Geroch's famous conjecture concerning analytic solutions of the elliptic Ernst equation. Our generalization is stated for solutions of the hyperbolic Ernst equation that are not necessarily analytic, although it can be formulated also for solutions of the elliptic Ernst equation that are nowhere axis-accessible. 
  A simple perturbation description unique for all signs of curvature, and based on the gauge-invariant formalisms is proposed to demonstrate that:  (1) The density perturbations propagate in the flat radiation-dominated universe in exactly the same way as electromagnetic or gravitational waves propagate in the epoch of the matter domination.  (2) In the open universe, sounds are dispersed by curvature. The space curvature defines the minimal frequency $\omega_{\rm c}$ below which the propagation of perturbations is forbidden.   Gaussian acoustic fields are considered and the curvature imprint in the perturbations spectrum is discussed. 
  Appealing to classical methods of order reduction, we reduce the Lifshitz system to a second order differential equation. We demonstrate its equivalence to well known gauge-invariant results. For a radiation dominated universe we express the metric and density corrections in their exact forms and discuss their acoustic character. 
  In the standard model of universe the increase in mass of our observed expansive and isotropic relativistic Universe is explained by the hypothetical assumption of matter objects emerging on the horizon (of the most remote visibility). However, the mathematical-physical analysis of the increase of Universe gauge factor shows that this hypothetical assumption is non-compatible with the variants of the standard model of universe by which - according to the standard model of universe - can be described the expansive evolution of the Universe. 
  I introduce a family of closeness functions between causal Lorentzian geometries of finite volume and arbitrary underlying topology. When points are randomly scattered in a Lorentzian manifold, with uniform density according to the volume element, some information on the topology and metric is encoded in the partial order that the causal structure induces among those points; one can then define closeness between Lorentzian geometries by comparing the sets of probabilities they give for obtaining the same posets. If the density of points is finite, one gets a pseudo-distance, which only compares the manifolds down to a finite volume scale, as illustrated here by a fully worked out example of two 2-dimensional manifolds of different topology; if the density is allowed to become infinite, a true distance can be defined on the space of all Lorentzian geometries. The introductory and concluding sections include some remarks on the motivation for this definition and its applications to quantum gravity. 
  We study the modifications on the metric of an isolated self-gravitating bosonic superconducting cosmic string in a scalar-tensor gravity in the weak-field approximation. These modifications are induced by an arbitrary coupling of a massless scalar field to the usual tensorial field in the gravitational Lagrangian. The metric is derived by means of a matching at the string radius with a most general static and cylindrically symmetric solution of the Einstein-Maxwell-scalar field equations. We show that this metric depends on five parameters which are related to the string's internal structure and to the solution of the scalar field. We compare our results with those obtained in the framework of General Relativity. 
  An analysis is given of thermoelastic noise (thermal noise due to thermoelastic dissipation) in finite sized test masses of laser interferometer gravitational-wave detectors. Finite-size effects increase the thermoelastic noise by a modest amount; for example, for the sapphire test masses tentatively planned for LIGO-II and plausible beam-spot radii, the increase is less than or of order 10 per cent. As a side issue, errors are pointed out in the currently used formulas for conventional, homogeneous thermal noise (noise associated with dissipation which is homogeneous and described by an imaginary part of the Young's modulus) in finite sized test masses. Correction of these errors increases the homogeneous thermal noise by less than or of order 5 per cent for LIGO-II-type configurations. 
  Linearly polarized cylindrical waves in four-dimensional vacuum gravity are mathematically equivalent to rotationally symmetric gravity coupled to a Maxwell (or Klein-Gordon) field in three dimensions. The quantization of this latter system was performed by Ashtekar and Pierri in a recent work. Employing that quantization, we obtain here a complete quantum theory which describes the four-dimensional geometry of the Einstein-Rosen waves. In particular, we construct regularized operators to represent the metric. It is shown that the results achieved by Ashtekar about the existence of important quantum gravity effects in the Einstein-Maxwell system at large distances from the symmetry axis continue to be valid from a four-dimensional point of view. The only significant difference is that, in order to admit an approximate classical description in the asymptotic region, states that are coherent in the Maxwell field need not contain a large number of photons anymore. We also analyze the metric fluctuations on the symmetry axis and argue that they are generally relevant for all of the coherent states. 
  This paper is part of a long term program to Cauchy-characteristic matching (CCM) codes as investigative tools in numerical relativity. The approach has two distinct features: (i) it dispenses with an outer boundary condition and replaces this with matching conditions at an interface between the Cauchy and characteristic regions, and (ii) by employing a compactified coordinate, it proves possible to generate global solutions. In this paper CCM is applied to an exact two-parameter family of cylindrically symmetric vacuum solutions possessing both gravitational degrees of freedom due to Piran, Safier and Katz. This requires a modification of the previously constructed CCM cylindrical code because, even after using Geroch decomposition to factor out the $z$-direction, the family is not asymptotically flat. The key equations in the characteristic regime turn out to be regular singular in nature. 
  We study the breakdown of conformal symmetry in a conformally invariant gravitational model. The symmetry breaking is introduced by defining a preferred conformal frame in terms of the large scale characteristics of the universe. In this context we show that a local change of the preferred conformal frame results in a Hamilton-Jacobi equation describing a particle with adjustable mass. In this equation the dynamical characteristics of the particle substantially depends on the applied conformal factor and local geometry. Relevant interpretations of the results are also discussed. 
  We establish the Hamiltonian formulation of the teleparallel equivalent of general relativity, without fixing the time gauge condition, by rigorously performing the Legendre transform. The time gauge condition, previously considered, restricts the teleparallel geometry to the three-dimensional spacelike hypersurface. Geometrically, the teleparallel geometry is now extended to the four-dimensional space-time. The resulting Hamiltonian formulation is different from the standard ADM formulation in many aspects, the main one being that the dynamics is now governed by the Hamiltonian constraint H_0 and a set of primary constraints. The vector constraint H_i is derived from the Hamiltonian constraint. The vanishing of the latter implies the vanishing of the vector constraint. 
  We consider the most general class of teleparallel theories of gravity quadratic in the torsion tensor, and carry out a detailed investigation of its Hamiltonian formulation in the time gauge. Such general class is given by a three-parameter family of theories. A consistent implementation of the Legendre transform reduces the original theory to a one-parameter theory determined in terms of first class constraints. The free parameter is fixed by requiring the Newtonian limit. The resulting theory is the teleparallel equivalent of general relativity. 
  We first review the cosmological constant problem, and then mention a conjecture of Feynman according to which the general relativistic theory of gravity should be reformulated in such a way that energy does not couple to gravity. We point out that our recent unification of gravitation and electromagnetism through a symmetric tensor has the property that the free electromagnetic energy and the vacuum energy do not contribute explicitly to the curvature of spacetime just like the free gravitational energy. Therefore in this formulation of general relativity, the vacuum energy density has its very large value today as in the early universe, while the cosmological constant does not exist at all. 
  By considering families of radial null geodesics, we study the subsets of initial data that lead to naked singularities and black holes in inhomogeneous spherical dust collapse. We introduce the notion of central homogeneity for spherical dust collapse and prove that for the occurrence of naked singularities, the initial data set must in general be centrally homogeneous. Even though mathematically this indicates that naked singularities are in general unstable, we show that centrally inhomogeneous perturbations in the initial data are not physically reasonable. This provides an example of the fact that instability in this context deduced with respect to general perturbations can become stabilised once the class of perturbations are restricted to be physical. 
  We examine the dynamics of a self-gravitating domain wall using the $\lambda \Phi^4$ model as a specific example. We find that the Nambu motion of the wall is quite generic and dominates the wall motion even in the presence of gravity. We calculate the corrections to this leading order motion, and estimate the effect of the inclusion of gravity on the dynamics of the wall. We then treat the case of a spherical gravitating thick wall as a particular example, solving the field equations and calculating the corrections to the Nambu motion analytically for this specific case. We find that the presence of gravity retards collapse in this case. 
  Symmetries of spacetimes with null dust field as a source compatible with asymptotic flatness are studied by using the Bondi-Sachs-van der Burg formalism. It is shown that in an axially symmetric spacetime with null dust field in which at least locally a smooth null infinity in the sense of Penrose exists, the only allowable additional Killing vector forming with the axial one a two-dimensional Lie algebra (the axial and the additional Killing vector are not assumed to be hypersurface orthogonal) is a supertranslational Killing vector and the gravitational field is then non-radiative (the Weyl tensor has a non-radiative character). 
  We propose the use of a gravitational uncertainty principle for gravitation. We define the corresponding gravitational Planck's constant and the gravitational quantum of mass. We define entropy in terms of the quantum of gravity with the property of having an extensive quality. The equivalent 2nd law of thermodynamics is derived, the entropy increasing linearly with cosmological time. These concepts are applied to the case of black holes, finding their entropy and discussing their radiation. 
  The weak gravitational field expansion method to account for the gravitationally induced neutrino oscillation effect is critically examined. It is shown that the splitting of the neutrino phase into a ``kinematic'' and a ``gravitational'' phase is not always possible because the relativistic factor modifies the particle interference phase splitting condition in a gravitational field. 
  We report on the existence and phenomenology of type II critical collapse within the one-parameter family of SU(2) $\sigma$-models coupled to gravity. Numerical investigations in spherical symmetry show discretely self-similar (DSS) behavior at the threshold of black hole formation for values of the dimensionless coupling constant $\ccbeta$ ranging from 0.2 to 100; at 0.18 we see small deviations from DSS. While the echoing period $\Delta$ of the critical solution rises sharply towards the lower limit of this range, the characteristic mass scaling has a critical exponent $\gamma$ which is almost independent of $\ccbeta$, asymptoting to $0.1185 \pm 0.0005$ at large $\ccbeta$. We also find critical scaling of the scalar curvature for near-critical initial data. Our numerical results are based on an outgoing-null-cone formulation of the Einstein-matter equations, specialized to spherical symmetry. Our numerically computed initial-data critical parameters $p^*$ show 2nd order convergence with the grid resolution, and after compensating for this variation in $p^*$, our individual evolutions are uniformly 2nd order convergent even very close to criticality. 
  We prove in the Tucker-Wang approach to non-Riemannian Gravity that a general homogeneous Lagrangian density in the general connection with order of homogeneity of at least two, gives no contribution to the generalised Einstein equations. Using this result other important cases are also considered. 
  The possibility of an extrinsic origin for inertial reaction forces has recently seen increased attention in the physical literature. Among theories of extrinsic inertia, the two considered by the current work are (1) the hypothesis that inertia is a result of gravitational interactions, and (2) the hypothesis that inertial reaction forces arise from the interaction of material particles with local fluctuations of the quantum vacuum. A recent article supporting the former and criticizing the latter is shown to contain substantial errors. 
  In part I, we use the post-Newtonian (pn) order of Liouville's equation to study the normal modes of oscillation of a spherically symmetric relativistic system. Perturbations that are neutral in Newtonian approximation develop into a new sequence of normal modes.   In part II, stability curve of r-modes of neutron stars are calculated by considering vorticity-shear viscosity coupling.   As an application, the loss of angular momentum through gravitational radiation, driven by the excitation of r-modes, is considered in neutron stars having rotation frequencies smaller than the associated critical frequency. 
  In the present paper we give conclusive arguments pointing at physical equivalence among conformally related metrics. Based on the argument that any consistent effective theory of spacetime must be invariant under the one-parameter group of transformations of the units of length, time and reciprocal mass, it is shown, also, that canonical general relativity is not such a consistent theory. Conformal general relativity provides a consistent formulation of the laws of gravity instead. We further extend the results of papers gr-qc/9908075 and gr-qc/9905071 to open universes by studying the Raychaudhuri equation. 
  We quantize the low-energy sector of a massless scalar field in the Reissner-Nordstrom spacetime. This allows the analysis of processes involving soft scalar particles occurring outside charged black holes. In particular, we compute the response of a static scalar source interacting with Hawking radiation using the Unruh (and the Hartle-Hawking) vacuum. This response is compared with the one obtained when the source is uniformly accelerated in the usual vacuum of the Minkowski spacetime with the same proper acceleration. We show that both responses are in general different in opposition to the result obtained when the Reissner-Nordstrom black hole is replaced by a Schwarzschild one. The conceptual relevance of this result is commented. 
  The notions of length of a vector field and cosine of the angle between two vector fields over a differentiable manifold with contravariant and covariant affine connections and metrics are introduced and considered. The change of the length of a vector field and of the angle between two vector fields along a contravariant vector field are found. The introduced notions are necessary for investigations of different types of transports over a manifold of the above mentioned type. 
  Various aspects of the Cauchy problem for the Einstein equations are surveyed, with the emphasis on local solutions of the evolution equations. Particular attention is payed to giving a clear explanation of conceptual issues which arise in this context. The question of producing reduced systems of equations which are hyperbolic is examined in detail and some new results on that subject are presented. Relevant background from the theory of partial differential equations is also explained at some length 
  We begin by reviewing the derivation of generalized Maxwell equations from an operational definition of the electromagnetic field and the most basic notions of what constitutes a dynamical field theory. These equations encompass the familiar Maxwell equations as a special case but, in other cases, can predict birefringence, charge non-conservation, wave damping and other effects that the familiar Maxwell equations do not. It follows that observational constraints on such effects can restrict the dynamics of the electromagnetic field to be very like the familiar Maxwellian dynamics, thus, providing an empirical foundation for the Maxwell equations. We discuss some specific observational results that contribute to that foundation. 
  We describe a finite-difference method for locating apparent horizons and illustrate its capabilities on boosted Kerr and Schwarzschild black holes. Our model spacetime is given by the Kerr-Schild metric. We apply a Lorentz boost to this spacetime metric and then carry out a 3+1 decomposition. The result is a slicing of Kerr/Schwarzschild in which the black hole is propagated and Lorentz contracted. We show that our method can locate distorted apparent horizons efficiently and accurately. 
  A perturbing shell is introduced as a device for studying the excitation of fluid motions in relativistic stellar models. We show that this approach allows a reasonably clean separation of radiation from the shell and from fluid motions in the star, and provides broad flexibility in the location and timescale of perturbations driving the fluid motions. With this model we compare the relativistic and Newtonian results for the generation of even parity gravitational waves from constant density models. Our results suggest that relativistic effects will not be important in computations of the gravitational emission except possibly in the case of excitation of the neutron star on very short time scales. 
  The concept of "Isolated Horizon" has been recently used to provide a full Hamiltonian treatment of black holes. It has been applied successfully to the cases of {\it non-rotating}, {\it non-distorted} black holes in Einstein Vacuum, Einstein-Maxwell and Einstein-Maxwell-Dilaton Theories. In this note, it is investigated the extent to which the framework can be generalized to the case of non-Abelian gauge theories where `hairy black holes' are known to exist. It is found that this extension is indeed possible, despite the fact that in general, there is no `canonical normalization' yielding a preferred Horizon Mass. In particular the zeroth and first laws are established for all normalizations. Colored static spherically symmetric black hole solutions to the Einstein-Yang-Mills equations are considered from this perspective. A canonical formula for the Horizon Mass of such black holes is found. This analysis is used to obtain nontrivial relations between the masses of the colored black holes and the regular solitonic solutions in Einstein-Yang-Mills theory. A general testing bed for the instability of hairy black holes in general non-linear theories is suggested. As an example, the embedded Abelian magnetic solutions are considered. It is shown that, within this framework, the total energy is also positive and thus, the solutions are potentially unstable. Finally, it is discussed which elements would be needed to place the Isolated Horizons framework for Einstein-Yang-Mills theory in the same footing as the previously analyzed cases. Motivated by these considerations and using the fact that the Isolated Horizons framework seems to be the appropriate language to state uniqueness and completeness conjectures for the EYM equations --in terms of the horizon charges--, two such conjectures are put forward. 
  Stellar pulsations in rotating relativistic stars are reviewed. Slow rotation approximation is applied to solving the Einstein equations. The rotational effects on the non-axisymmetric oscillations are explicitly shown in the polar and axial modes. 
  The decay of the string perturbative vacuum, if triggered by a suitable, duality-breaking dilaton potential, can efficiently proceed via the parametric amplification of the Wheeler-De Witt wave function in superspace, and can appropriately describe the birth of our Universe as a quantum process of pair production from the vacuum. 
  This paper discusses the somewhat unintuitive conjecture that many Lorentz-invariant many-particle models can be reinterpreted to satisfy the gtr field equations. It is shown that a careful remapping of coordinates yields a non-trivial Riemannian manifold. Furthermore an energy-momentum tensor is outlined and it is argued that it may converge to its classical counterpart in the macroscopic limit. These ideas could possibly be used to partially relieve us of some the resilient problems of adding spacetime curvature to modern QM theories. 
  The inhomogeneous cosmological model with matter in the form self-acting scalar field and perfect fluid is considered. On the basis of exact solutions is considered the evolution of density distribution of a matter in space on a background cosmological expansion by the Universe. Is shown, the first, in such model the equation of a matter state is variable in time and is closely connected to character cosmological expansions. Secondly, it is shown with point of view of the observer the Universe looks as space flat, but with effect of latent mass. This effect consists in that the mass of a perfect fluid by dynamic measurements surpasses the own mass perfect fluid that is explained by presence scalar field (20K). 
  It has been recently shown that a certain non-topological spin foam model can be obtained from the Feynman expansion of a field theory over a group. The field theory defines a natural ``sum over triangulations'', which removes the cut off on the number of degrees of freedom and restores full covariance. The resulting formulation is completely background independent: spacetime emerges as a Feynman diagram, as it did in the old two-dimensional matrix models. We show here that any spin foam model can be obtained from a field theory in this manner. We give the explicit form of the field theory action for an arbitrary spin foam model. In this way, any model can be naturally extended to a sum over triangulations. More precisely, it is extended to a sum over 2-complexes. 
  We extend previous simplicial minisuperspace models to account for arbitrary scalar coupling \eta R\phi^2. 
  Black hole generalized p-brane solutions for a wide class of intersection rules are presented. The solutions are defined on a manifold that contains a product of n - 1 Ricci-flat ``internal'' spaces. They are defined up to moduli functions H_s = H_s(R) obeying a non-linear differential equations (equivalent to Toda-type equations) with certain boundary conditions imposed. Using conjecture on polynomial structure of H_s for intersections related to Lie algebras, new A_2-dyon solutions are obtained. Two examples of these A_2-dyon solutions, i.e. dyon in D = 11 supergravity with M2 and M5 branes intersecting at a point and dyon in Kaluza-Klein theory, are considered. 
  In this paper the macroscopic Einstein and Maxwell equations for system, in which the electromagnetic interactions are dominating (for instance, the cosmological plasma before the moment of recombination), are derived. Ensemble averaging of the microscopic Einstein - Maxwell equations and the iouville equations for the random functions leads to a closed system of macroscopic Einstein - Maxwell equations and kinetic equations for one-particle distribution functions. The macroscopic Einstein equations for a relativistic plasma differ from the classical Einstein equations in that their left-hand sides contain additional terms due to particle interaction. The terms are traceless tensors with zero divergence. An explicit covariant expression for these terms is given in the form of momentum-space integrals of expressions depending on one-particles distribution functions of the interacting particles of the medium. The macroscopic Maxwell equations alsow differ from the classical macroscopic Maxwell equations in that their left-hand sides contain additional terms due to particle interaction as well the effects of general relativity. 
  One of the possible applications of macroscopic Einstein equations has been considered. So, the nonsingular isotropic and uniform cosmological model is built. The cosmological consequences of this model are agree with conclusions of standard hot model of the Universe. 
  We describe some new estimates concerning the recently proposed SEE (Satellite Energy Exchange) experiment for measuring the gravitational interaction parameters in space. The experiment entails precision tracking of the relative motion of two test bodies (a heavy "Shepherd", and a light "Particle") on board a drag-free capsule. The new estimates include (i) the sensitivity of Particle trajectories and G measurement to the Shepherd quadrupole moment uncertainties; (ii) the measurement errors of G and the strength of a putative Yukawa-type force whose range parameter $\lambda$ may be either of the order of a few metres or close to the Earth radius; (iii) a possible effect of the Van Allen radiation belts on the SEE experiment due to test body electric charging 
  We study the evolution of gravitational waves through the preheating era that follows inflation. The oscillating inflaton drives parametric resonant growth of scalar field fluctuations, and although super-Hubble tensor modes are not strongly amplified, they do carry an imprint of preheating. This is clearly seen in the Weyl tensor, which provides a covariant description of gravitational waves. 
  The contribution provides the starting points and background of the model of Expansive Nondecelerative Universe (ENU), manifests the advantage of exploitation of Vaidya metrics for the localization and quantization of gravitational energy, and offers four examples of application of the ENU model, namely energy of cosmic background radiation, energy of Z and W bosons acting in weak interactions, hyperfine splitting observed for hydrogen 1s orbital. Moreover, time evolution of vacuum permitivity and permeability is predicted.% 
  The nonminimal coupling (NMC) of the scalar field to the Ricci curvature is unavoidable in many cosmological scenarios. Inflation and quintessence models based on nonminimally coupled scalar fields are studied, with particular attention to the balance between the scalar potential and the NMC term in the action. NMC makes acceleration of the universe harder to achieve for the usual potentials, but it is beneficial in obtaining cosmic acceleration with unusual potentials. The slow-roll approximation with NMC, conformal transformation techniques, and other aspects of the physics of NMC are clarified. 
  Boundary actions for three-dimensional quantum gravity in the discretized formalism of Ponzano-Regge are studied with a view towards understanding the boundary degrees of freedom. These degrees of freedom postulated in the holography hypothesis are supposed to be characteristic of quantum gravity theories. In particular it is expected that some of these degrees of freedom reside on black hole horizons. This paper is a study of these ideas in the context of a theory of quantum gravity that requires no additional structure such as supersymmetry or special gravitational backgrounds. Lorentzian as well as Euclidean regimes are examined. Some surprising relationships to Liouville theory and string theory in AdS(3) are found. 
  We examine the propagation of gravitational waves in the new field theory of gravitation recently proposed by Novello-De Lorenci-Luciane (NDL). This examination is done on a solvable case corresponding to a spherically symmetric static configuration. We show that in NDL theory the velocity of gravitational waves is lower than light velocity. We point out some consequences of this result and suggest a possible scenario for its verification. 
  Pad\'e approximants to truncated post-Newtonian neutron star models are constructed. The Pad\'e models converge faster to the general relativistic (GR) solution than the truncated post-Newtonian ones. The evolution of initial data using the Pad\'e models approximates better the evolution of full GR initial data than the truncated Taylor models. In the absence of full GR initial data (e.g., for neutron star binaries or black hole binary systems), Pad\'e initial data could be a better option than the straightforward truncated post-Newtonian (Taylor) initial data. 
  Spin foam models are the path integral counterparts to loop quantized canonical theories. In the last few years several spin foam models of gravity have been proposed, most of which live on finite simplicial lattice spacetime. The lattice truncates the presumably infinite set of gravitational degrees of freedom down to a finite set. Models that can accomodate an infinite set of degrees of freedom and that are independent of any background simplicial structure, or indeed any a priori spacetime topology, can be obtained from the lattice models by summing them over all lattice spacetimes. Here we show that this sum can be realized as the sum over Feynman diagrams of a quantum field theory living on a suitable group manifold, with each Feynman diagram defining a particular lattice spacetime. We give an explicit formula for the action of the field theory corresponding to any given spin foam model in a wide class which includes several gravity models. Such a field theory was recently found for a particular gravity model [De Pietri et al, hep-th/9907154]. Our work generalizes this result as well as Boulatov's and Ooguri's models of three and four dimensional topological field theories, and ultimately the old matrix models of two dimensional systems with dynamical topology. A first version of our result has appeared in a companion paper [gr-qc\0002083]: here we present a new and more detailed derivation based on the connection formulation of the spin foam models. 
  The field equations for a time dependent cylindrical cosmic string coupled to gravity are reformulated in terms of geometrical variables defined on a 2+1-dimensional spacetime by using the method of Geroch decomposition. Unlike the 4-dimensional spacetime the reduced case is asymptotically flat. A numerical method for solving the field equations which involves conformally compactifying the space and including null infinity as part of the grid is described. It is shown that the code reproduces the results of a number of vacuum solutions with one or two degrees of freedom. In the final section the interaction between the cosmic string and a pulse of gravitational radiation is briefly described. This will be fully analysed in the sequel. 
  A consistent Hamiltonian formulation of the teleparallel equivalent of general relativity (TEGR) requires the theory to be invariant under the global SO(3) symmetry group, which acts on orthonormal triads in three-dimensional spacelike hypersurfaces. In the TEGR it is possible to make definite statements about the energy of the gravitational field. In this geometrical framework two sets of triads related by a local SO(3) transformation yield different descriptions of the gravitational energy. Here we consider the problem of assigning a unique set of triads to the metric tensor restricted to the three-dimensional hypersurface. The analysis is carried out in the context of Bondi's radiating metric. A simple and original expression for Bondi's news function is obtained, which allows us to carry out numerical calculations and verify that a triad with a specific asymptotic behaviour yields the minimum gravitational energy for a fixed space volume containing the radiating source. This result supports the conjecture that the requirement of a minimum gravitational energy for a given space volume singles out uniquely the correct set of orthonormal triads. 
  We consider the most general class of teleparallel gravitational {}{}theories quadratic in the torsion tensor, in three space-time dimensions, and carry out a detailed investigation of its Hamiltonian formulation in Schwinger's time gauge. This general class is given by a family of three-parameter theories. A consistent implementation of the Legendre transform reduces the original theory to a one-parameter family of theories. By calculating Poisson brackets we show explicitly that the constraints of the theory constitute a first-class set. Therefore the resulting theory is well defined with regard to time evolution. The structure of the Hamiltonian theory rules out the existence of the Newtonian limit. 
  Using approximate techniques we study the final moments of the collision of two (individually non-spinnning) black holes which inspiral into each other. The approximation is based on treating the whole space-time as a single distorted black hole. We obtain estimates for the radiated energy, angular momentum and waveforms for the gravitational waves produced in such a collision. The results can be of interest for analyzing the data that will be forthcoming from gravitational wave interferometric detectors, like the LIGO, GEO, LISA, VIRGO and TAMA projects. 
  The method based on the Horsky-Mitskievitch conjecture is applied to the Levi-Civita vacuum metric. It is shown, that every Killing vector is connected with a particular class of Einstein-Maxwell fields and each of those classes is found explicitly. Some of obtained classes are quite new. Radial geodesic motion in constructed space-times is discussed and graphically illustrated in the Appendix. 
  Different (not only by sign) affine connections are introduced for contravariant and covariant tensor fields over a differentiable manifold by means of a non-canonical contraction operator, defining the notion dual bases and commuting with the covariant and with the Lie-differential operator. Classification of the linear transports on the basis of the connections between the connections is given. Notion of relative velocity and relative acceleration for vector fields are determined. By means of these kinematic characteristics several other types of notions as shear velocity, shear acceleration, rotation velocity, rotation acceleration, expansion velocity and expansion acceleration are introduced and on their basis auto-parallel and non-isotropic (non-null) vector fields are classified. 
  In this paper we present our point of view on correct physical interpretation of the Bel-Robinson tensor within the framework of the standard General Relativity ({\bf GR}), i.e., within the framework of the {\bf GR} without supplementary elements like arbitrary vector field, distinguished tetrads field or second metric. We show that this tensor arises as a consequence of the Bianchi identities and, in a natural manner, it is linked to the differences of the canonical gravitational energy-momentum calculated in normal coordinates {\bf NC(P)}. 
  We classify all spherically symmetric dust solutions of Einstein's equations which are self-similar in the sense that all dimensionless variables depend only upon $z\equiv r/t$. We show that the equations can be reduced to a special case of the general perfect fluid models with equation of state $p=\alpha \mu$. The most general dust solution can be written down explicitly and is described by two parameters. The first one (E) corresponds to the asymptotic energy at large $|z|$, while the second one (D) specifies the value of z at the singularity which characterizes such models. The E=D=0 solution is just the flat Friedmann model. The 1-parameter family of solutions with z>0 and D=0 are inhomogeneous cosmological models which expand from a Big Bang singularity at t=0 and are asymptotically Friedmann at large z; models with E>0 are everywhere underdense relative to Friedmann and expand forever, while those with E<0 are everywhere overdense and recollapse to a black hole containing another singularity. The black hole always has an apparent horizon but need not have an event horizon. The D=0 solutions with z<0 are just the time reverse of the z>0 ones. The 2-parameter solutions with D>0 again represent inhomogeneous cosmological models but the Big Bang singularity is at $z=-1/D$, the Big Crunch singularity is at $z=+1/D$, and any particular solution necessarily spans both z<0 and z>0. While there is no static model in the dust case, all these solutions are asymptotically ``quasi-static'' at large $|z|$. As in the D=0 case, the ones with $E \ge 0$ expand or contract monotonically but the latter may now contain a naked singularity. The ones with E<0 expand from or recollapse to a second singularity, the latter containing a black hole. 
  The debate on conservation laws in general relativity eighty years ago is reviewed and restudied. The physical meaning of the identities in the conservation laws for matter plus gravitational field is reexamined and new interpretations for gravitational wave are given. The conclusions of these studies are distinct from the prevalent views, it can be demonstrated that gravitational wave does not transmit energy (and momentum) but only transmits informations. An experimental test is offered to decide which conservation laws are correct. 
  We discuss spherically symmetric perfect fluid solutions of Einstein's equations which have equation of state ($p=\alpha \mu$) and which are self-similar in the sense that all dimensionless variables depend only upon $z\equiv r/t$. For each value of $\alpha$, such solutions are described by two parameters and have now been completely classified. There is a 1-parameter family of solutions asymptotic to the flat Friedmann model at large values of z. These represent either black holes or density perturbations which grow as fast as the particle horizon; the underdense solutions may be relevant to the existence of large-scale cosmic voids. There is also a 1-parameter family of solutions asymptotic to a self-similar Kantowski-Sachs model at large z. These are probably only physically realistic for $-1<\alpha<-1/3$, in which case they may relate to the formation of bubbles in an inflationary universe. There is a 2-parameter family of solutions associated with a self-similar static solution at large z. This family contains solutions with naked singularities and this includes the ``critical'' solution discovered in recent collapse calculations for $\alpha < 0.28$. Finally, for $\alpha >1/5$, there is a family of solutions which are asymptotically Minkowski. These asymptote either to infinite z, in which case they are described by one parameter, or to a finite value of z, in which case they are described by two parameters and this includes the ``critical'' solution for $\alpha >0.28$. We discuss the stability of spherically symmetric similarity solutions to more general (non-self-similar) spherically symmetric perturbations. 
  We reconstruct the Ashtekar's canonical formulation of N = 2 supergravity (SUGRA) starting from the N = 2 chiral Lagrangian derived by closely following the method employed in the usual SUGRA. In order to get the full graded algebra of the Gauss, U(1) gauge and right-handed supersymmetry (SUSY) constraints, we extend the internal, global O(2) invariance to local one by introducing a cosmological constant to the chiral Lagrangian. The resultant Lagrangian does not contain any auxiliary fields in contrast with the 2-form SUGRA and the SUSY transformation parameters are not constrained at all. We derive the canonical formulation of the N = 2 theory in such a manner as the relation with the usual SUGRA be explicit at least in classical level, and show that the algebra of the Gauss, U(1) gauge and right-handed SUSY constraints form the graded algebra, G^2SU(2)(Osp(2,2)). Furthermore, we introduce the graded variables associated with the G^2SU(2)(Osp(2,2)) algebra and we rewrite the canonical constraints in a simple form in terms of these variables. We quantize the theory in the graded-connection representation and discuss the solutions of quantum constraints. 
  Using the weak field approximation, we can express the theory of general relativity in a Maxwell-type structure comparable to electromagnetism. We find that every electromagnetic field is coupled to a gravitoelectric and gravitomagnetic field. Acknowledging the fact that both fields originate from the same source, the particle, we can express the magnetic and electric field through their gravitational respective analogues using the proportionality coefficient k. This coefficient depends on the ratio of mass and charge and the ratio between the electromagnetic and gravitic-gravitomagnetic permittivity and permeability respectively. Although the coefficient is very small, the fact that electromagnetic fields in material media can be used to generate gravitational and gravitomagnetic fields and vice versa is not commonly known. We find that the coupling coefficient can be increased by massive ion currents, electron and nuclear spin-alignment. Advances in material sciences, cryogenic technology and high frequency electromagnetic fields in material media may lead to applications of the derived relationships. 
  We give an exact solution of the five-dimensional field equations which describes a shock wave moving in time and the extra (Kaluza-Klein) coordinate. The matter in four-dimensional spacetime is a cosmology with good physical properties. The solution suggests to us that the 4D big bang was a 5D shock wave. 
  We study $T^{3}$ Gowdy spacetimes in the Einstein-Maxwell-dilaton-axion system and show by the Fuchsian algorithm that they have in general asymptotically velocity-term dominated singularities. The families of the corresponding solutions depend on the maximum number of arbitrary functions. Although coupling of the dilaton field with the Maxwell and/or the axion fields corresponds to the ``potential'' which appears in the Hamiltonian of vacuum Bianchi IX spacetimes, our result means that the spacetimes do not become Mixmaster necessarily. 
  The theory of measurement is employed to elucidate the physical basis of general relativity. For measurements involving phenomena with intrinsic length or time scales, such scales must in general be negligible compared to the (translational and rotational) scales characteristic of the motion of the observer. Thus general relativity is a consistent theory of coincidences so long as these involve classical point particles and electromagnetic rays (geometric optics). Wave optics is discussed and the limitations of the standard theory in this regime are pointed out. A nonlocal theory of accelerated observers is briefly described that is consistent with observation and excludes the possibility of existence of a fundamental scalar field in nature. 
  Gravitational field of a nonstatic global string has been studied in the context of Brans-Dicke theory of gravity. Both the metric components and the BD scalar field are assumed to be nonseparable functions of time and space.The spacetime may or may not have any singularity at a finite distance from the string core but the singularity at a particular time always remains. It has been shown that the spacetime exhibits both outgoing and incoming gravitational radiation. 
  Leonhardt and Piwnicki reply to Visser's ``Comment on `Relativistic Effects of Light in Moving Media with Extremely Low Group Velocity' ''. 
  We give an account of the physical behaviour of a quasiparticle horizon due to non-Lorentz invariant modifications of the effective space-time experienced by the quasiparticles (``matter'') for high momenta. By introducing a ``relativistic'' conserved energy-momentum tensor, we derive quasi-equilibrium states of the fluid across the ``Landau'' quasiparticle horizon at temperatures well above the quantum Hawking temperature. Nonlinear dispersion of the quasiparticle energy spectrum is instrumental for quasiparticle communication and exchange across the horizon. It is responsible for the establishment of the local thermal equilibrium across the horizon with the Tolman temperature being inhomogeneous behind the horizon. The inhomogeneity causes relaxation of the quasi-equilibrium states due to scattering of thermal quasiparticles, which finally leads to a shrinking black hole horizon. This process serves as the classical thermal counterpart of the quantum effect of Hawking radiation and will allow for an observation of the properties of the horizon at temperatures well above the Hawking temperature. We discuss the thermal entropy related to the horizon. We find that only the first nonlinear correction to the energy spectrum is important for the thermal properties of the horizon. They are fully determined by an energy of order $E_{\rm Planck}(T/E_{\rm Planck})^{1/3}$, which is well below the Planck energy scale $E_{\rm Planck}$, so that Planck scale physics is not involved in determining thermal quantities related to the horizon. 
  In the spirit of the Newtonian theory, we characterize spherically symmetric empty space in general relativity in terms of energy density measured by a static observer and convergence density experienced by null and timelike congruences. It turns out that space surrounding a static particle is entirely specified by vanishing of energy and null convergence density. The electrograv-dual$^{1}$ to this condition would be vanishing of timelike and null convergence density which gives the dual-vacuum solution representing a Schwarzschild black hole with global monopole charge$^{2}$ or with cloud of string dust$^{3}$. Here the duality$^{1}$ is defined by interchange of active and passive electric parts of the Riemann curvature, which amounts to interchange of the Ricci and Einstein tensors. This effective characterization of stationary vacuum works for the Schwarzschild and NUT solutions. The most remarkable feature of the effective characterization of empty space is that it leads to new dual spaces and the method can also be applied to lower and higher dimensions. 
  The Lagrangian proposed by York et al. and the covariant first order Lagrangian for General Relativity are introduced to deal with the (vacuum) gravitational field on a reference background. The two Lagrangians are compared and we show that the first one can be obtained from the latter under suitable hypotheses. The induced variational principles are also compared and discussed. A conditioned correspondence among Noether conserved quantities, quasilocal energy and the standard Hamiltonian obtained by 3+1 decomposition is also established. As a result, it turns out that the covariant first order Lagrangian is better suited whenever a reference background field has to be taken into account, as it is commonly accepted when dealing with conserved quantities in non-asymptotically flat spacetimes. As a further advantage of the use of a covariant first order Lagrangian, we show that all the quantities computed are manifestly covariant, as it is appropriate in General Relativity. 
  A canonical particle definition via the diagonalisation of the Hamiltonian for a quantum field theory in specific curved space-times is presented. Within the provided approach radial ingoing or outgoing Minkowski particles do not exist. An application of this formalism to the Rindler metric recovers the well-known Unruh effect. For the situation of a black hole the Hamiltonian splits up into two independent parts accounting for the interior and the exterior domain, respectively. It turns out that a reasonable particle definition may be accomplished for the outside region only. The Hamiltonian of the field inside the black hole is unbounded from above and below and hence possesses no ground state. The corresponding equation of motion displays a linear global instability. Possible consequences of this instability are discussed and its relations to the sonic analogues of black holes are addressed. PACS-numbers: 04.70.Dy, 04.62.+v, 10.10.Ef, 03.65.Db. 
  Gravitational wave experiments will play a key role in the investigation of the frontiers of cosmology and the structure of fundamental fields at high energies, by detecting, or setting strong upper-limits to, the primordial gravitational wave background produced in the early-Universe. Here we discuss the impact of space-borne laser interferometric detectors operating in the low-frequency window ($\sim 1 \mu$Hz - 1 Hz); the aim of our analysis is to assess the detectability of a primordial background characterized by a fractional energy density $\Omega \sim 10^{-16} - 10^{-15}$, which is consistent with the prediction of "slow-roll" inflationary models. 
  The gravitational couplings of intrinsic spin are briefly reviewed. A consequence of the Dirac equation in the exterior gravitational field of a rotating mass is considered in detail, namely, the difference in the energy of a spin-1/2 particle polarized vertically up and down near the surface of a rotating body is $\hbar\Omega\sin\theta$. Here $\theta$ is the latitude and $\Omega = 2GJ/(c^2 R^3)$, where $J$ and $R$ are, respectively, the angular momentum and radius of the body. It seems that this relativistic quantum gravitational effect could be measurable in the foreseeable future. 
  Several eigenvalue equations that could describe quantum black holes have been proposed in the canonical quantum gravity approach. In this paper, we choose one of the simplest of these quantum equations to show how the usual Feynman's path integral method can be applied to obtain the corresponding statistical properties. We get a logarithmic correction to the Bekenstein-Hawking entropy as already obtained by other authors by other means. 
  The geometry of three-dimensional space guides the search for a better model than the blackhole with its unwelcome singularity. An elementary construction produces on the 4-manifold of 2-spheres in a Riemannian 3-space a space-time metric invariant under uniform conformal transformations of the 3-space. When the 3-space is Euclidean, the metric reduces to de Sitter's expanding universe metric. Generalization yields a space-time metric that retains the `exponential expansion property' of the de Sitter metric. A strictly geometric action principle gives field equations which, because they do not adhere to Einstein's early confounding of energy and inertial mass with gravitating mass, admit solutions that escape the Penrose-Hawking singularity theorems. A spherically symmetric solution that is asymptotic to the Schwarzschild blackhole metric has, in place of a horizon and a singularity, an Einstein-Rosen `bridge', or `tunnel', connecting two asymptotically Euclidean regions. On one side the gravitational center attracts, and is dark but not black; on the other side it repels, and is bright. Travel and signaling from either side to the other via the tunnel are possible. Analysis of the Einstein tensor of this `darkhole' (or `darkhole-brighthole') suggests that not all energy produces gravity, and that calling energy `negative', or its relationship to geometry `exotic', is unjustified. 
  We describe the different possibilities that a simple and apparently quite harmless classical scalar field theory provides to violate the energy conditions. We demonstrate that a non-minimally coupled scalar field with a positive curvature coupling xi>0 can easily violate all the standard energy conditions, up to and including the averaged null energy condition (ANEC). Indeed this violation of the ANEC suggests the possible existence of traversable wormholes supported by non-minimally coupled scalars. To investigate this possibility we derive the classical solutions for gravity plus a general (arbitrary xi) massless non-minimally coupled scalar field, restricting attention to the static and spherically symmetric configurations. Among these classical solutions we find an entire branch of traversable wormholes for every xi>0. (This includes and generalizes the case of conformal coupling xi=1/6 we considered in Phys. Lett. B466 (1999) 127--134; gr-qc/9908029) For these traversable wormholes to exist we demonstrate that the scalar field must reach trans-Planckian values somewhere in the geometry. We discuss how this can be accommodated within the current state of the art regarding scalar fields in modern theoretical physics. We emphasize that these scalar field theories, and the traversable wormhole solutions we derive, are compatible with all known experimental constraints from both particle physics and gravity. 
  If we consider the spacetime manifold as product of a constant curvature 2-sphere (hypersphere) and a 2-space, then solution of the Einstein equation requires that the latter must also be of constant curvature. There exist only two solutions for classical matter distribution which are given by the Nariai (anti) metric describing an Einstein space and the Bertotti - Robinson (anti) metric describing a uniform electric field. These two solutions are transformable into each other by letting the timelike convergence density change sign. The hyperspherical solution is anti of the spherical one and the vice -versa. For non classical matter, we however find a new solution, which is electrograv dual to the flat space, and describes a cloud of string dust of uniform energy density. We also discuss some interesting features of the particle motion in the Bertotti - Robinson metric. 
  We present the first results in a new program intended to make the best use of all available technologies to provide an effective understanding of waves from inspiralling black hole binaries in time for imminent observations. In particular, we address the problem of combining the close-limit approximation describing ringing black holes and full numerical relativity, required for essentially nonlinear interactions. We demonstrate the effectiveness of our approach using general methods for a model problem, the head-on collision of black holes. Our method allows a more direct physical understanding of these collisions indicating clearly when non-linear methods are important. The success of this method supports our expectation that this unified approach will be able to provide astrophysically relevant results for black hole binaries in time to assist gravitational wave observations. 
  The Arnowitt-Deser-Misner (ADM) evolution equations for the induced metric and the extrinsic-curvature tensor of the spacelike surfaces which foliate the space-time manifold in canonical general relativity are a first-order system of quasi-linear partial differential equations, supplemented by the constraint equations. Such equations are here mapped into another first-order system. In particular, an evolution equation for the trace of the extrinsic-curvature tensor K is obtained whose solution is related to a discrete spectral resolution of a three-dimensional elliptic operator P of Laplace type. Interestingly, all nonlinearities of the original equations give rise to the potential term in P. An example of this construction is given in the case of a closed Friedmann-Lemaitre-Robertson-Walker universe. Eventually, the ADM equations are re-expressed as a coupled first-order system for the induced metric and the trace-free part of K. Such a system is written in a form which clarifies how a set of first-order differential operators and their inverses, jointly with spectral resolutions of operators of Laplace type, contribute to solving, at least in principle, the original ADM system. 
  I describe a new algorithm for solving nonlinear wave equations. In this approach, evolution takes place on characteristic hypersurfaces. The algorithm is directly applicable to electromagnetic, Yang-Mills and gravitational fields and other systems described by second differential order hyperbolic equations. The basic ideas should also be applicable to hydrodynamics. It is an especially accurate and efficient way for simulating waves in regions where the characteristics are well behaved. A prime application of the algorithm is to Cauchy-characteristic matching, in which this new approach is matched to a standard Cauchy evolution to obtain a global solution. In a model problem of a nonlinear wave, this proves to be more accurate and efficient than any other present method of assigning Cauchy outer boundary conditions. The approach was developed to compute the gravitational wave signal produced by collisions of two black holes. An application to colliding black holes is presented. 
  Considering the spacetime around a rotating massif body it is seen that the time of flight of a light ray is different whether it travels on one side of the source or on the other. The difference is proportional to the angular momentum of the body. In the case that a compact rapidly rotating object is the source of a gravitational lensing effect, the contribution coming from the above mentioned gravitational Aharonov-Bohm effect should be added to the other causes of phase difference between light rays coming from different images of the same object. 
  We consider curvature invariants in the context of black hole collision simulations. In particular, we propose a simple and elegant combination of the Weyl invariants I and J, the {\sl speciality index} ${\cal S}$. In the context of black hole perturbations $\cal S$ provides a measure of the size of the distortions from an ideal Kerr black hole spacetime. Explicit calculations in well-known examples of axisymmetric black hole collisions demonstrate that this quantity may serve as a useful tool for predicting in which cases perturbative dynamics provide an accurate estimate of the radiation waveform and energy. This makes ${\cal S}$ particularly suited to studying the transition from nonlinear to linear dynamics and for invariant interpretation of numerical results. 
  There are three regimes of gravitational-radiation-reaction-induced inspiral for a compact body with mass mu, in a circular, equatorial orbit around a Kerr black hole with mass M>>mu: (i) The "adiabatic inspiral regime", in which the body gradually descends through a sequence of circular, geodesic orbits. (ii) A "transition regime", near the innermost stable circular orbit (isco). (iii) The "plunge regime", in which the body travels on a geodesic from slightly below the isco into the hole's horizon. This paper gives an analytic treatment of the transition regime and shows that, with some luck, gravitational waves from the transition might be measurable by the space-based LISA mission. 
  We show examples which reveal influences of spatial topologies to dynamics, using a class of spatially {\it closed} inhomogeneous cosmological models. The models, called the {\it locally U(1)$\times$U(1) symmetric models} (or the {\it generalized Gowdy models}), are characterized by the existence of two commuting spatial {\it local} Killing vectors. For systematic investigations we first present a classification of possible spatial topologies in this class. We stress the significance of the locally homogeneous limits (i.e., the Bianchi types or the `geometric structures') of the models. In particular, we show a method of reduction to the natural reduced manifold, and analyze the equivalences at the reduced level of the models as dynamical models. Based on these fundamentals, we examine the influence of spatial topologies on dynamics by obtaining translation and reflection operators which commute with the dynamical flow in the phase space. 
  The classical tests of general relativity - light deflection, time delay and perihelion shift - are applied, along with the geodetic precession test, to the five-dimensional extension of the theory known as Kaluza-Klein gravity, using an analogue of the four-dimensional Schwarzschild metric. The perihelion advance and geodetic precession calculations are generalized for the first time to situations in which the components of momentum and spin along the extra coordinate do not vanish. Existing data on light- bending around the Sun using long- baseline radio interferometry, ranging to Mars using the Viking lander, and the perihelion precession of Mercury all constrain a small parameter b associated with the extra part of the metric to be less than |b| < 0.07 in the solar system. An order-of-magnitude increase in sensitivity is possible from perihelion precession, if better limits on solar oblateness become available. Measurement of geodetic precession by the Gravity Probe B satellite will improve this significantly, probing values of b with an accuracy of one part in 10^4 or more. 
  Owing to its transformation property under local boosts, the Brown-York quasilocal energy surface density is the analogue of E in the special relativity formula: E^2-p^2=m^2. In this paper I will motivate the general relativistic version of this formula, and thereby arrive at a geometrically natural definition of an `invariant quasilocal energy', or IQE. In analogy with the invariant mass m, the IQE is invariant under local boosts of the set of observers on a given two-surface S in spacetime. A reference energy subtraction procedure is required, but in contrast to the Brown-York procedure, S is isometrically embedded into a four-dimensional reference spacetime. This virtually eliminates the embeddability problem inherent in the use of a three-dimensional reference space, but introduces a new one: such embeddings are not unique, leading to an ambiguity in the reference IQE. However, in this codimension-two setting there are two curvatures associated with S: the curvatures of its tangent and normal bundles. Taking advantage of this fact, I will suggest a possible way to resolve the embedding ambiguity, which at the same time will be seen to incorporate angular momentum into the energy at the quasilocal level. I will analyze the IQE in the following cases: both the spatial and future null infinity limits of a large sphere in asymptotically flat spacetimes; a small sphere shrinking toward a point along either spatial or null directions; and finally, in asymptotically anti-de Sitter spacetimes. The last case reveals a striking similarity between the reference IQE and a certain counterterm energy recently proposed in the context of the conjectured AdS/CFT correspondence. 
  It is known that the gravitational collapse of a dust ball results in naked singularity formation from an initial density profile which is physically reasonable. In this paper, we show that explosive radiation is emitted during the formation process of the naked singularity. 
  The radiative evolution of the relative orientations of the spin and orbital angular momentum vectors ${\bf S}_{{\bf 1}}, {\bf S}_{{\bf 2}}$ and ${\bf L}$, characterizing a binary system on eccentric orbit is studied up to the second post-Newtonian order. As an intermediate result, all Burke-Thorne type instantaneous radiative changes in the spins are shown to average out over a radial period. It is proved that spin-orbit and spin-spin terms contribute to the radiative angular evolution equations, while Newtonian, first and second post-Newtonian terms together with the leading order tail terms do not. In complement to the spin-orbit contribution, given earlier, the spin-spin contribution is computed and split into two-body and self-interaction parts. The latter provide the second post-Newtonian order corrections to the 3/2 order Lense-Thirring description. 
  We present a method for computing the flux of energy through a closed surface containing a gravitating system. This method, which is based on the quasilocal formalism of Brown and York, is illustrated by two applications: a calculation of (i) the energy flux, via gravitational waves, through a surface near infinity and (ii) the tidal heating in the local asymptotic frame of a body interacting with an external tidal field. The second application represents the first use of the quasilocal formalism to study a non-stationary spacetime and shows how such methods can be used to study tidal effects in isolated gravitating systems. 
  A Bianchi type -I metric of Kasner form is considered, when the space is filled with a viscous fluid. Whereas an ideal (nonviscous) fluid permits the Kasner metric to be anisotropic provided that the fluid satisfies the Zel'dovich equation of state, the viscous fluid does not permit the Kasner metric to be anisotropic at all. In the latter case, we calculate the Kasner (isotropic) metric expressed by the fluid's density, pressure, and bulk viscosity, at some chosen instant $t=t_0$. The equation of state is also calculated. The present paper is related to a recent Comment of Cataldo and del Campo [Phys. Rev. D, scheduled to April 15, 2000], on a previous work of the present authors [Phys. Rev. D {\bf 56}, 3322 (1997)]. 
  We review the framework of Refined Algebraic Quantization and the method of Group Averaging for quantizing systems with first-class constraints. Aspects and results concerning the generality, limitations, and uniqueness of these methods are discussed. 
  The formation and evolution of cosmic string wakes in the framework of a scalar-tensor gravity are investigated in this work. We consider a simple model in which cold dark matter flows past an ordinary string and we treat this motion in the Zel'dovich approximation. We make a comaprison between our results and previous results obtained in the context of General Relativity. We propose a mechanism in which the contribution of the scalar field to the evolution of the wakes may lead to a cosmological observation. 
  We derive the evolution equations for the spectra of the Universe.   Here "spectra" means the eigenvalues of the Laplacian defined on a space, which contain the geometrical information on the space.   These equations are expected to be useful to analyze the evolution of the geometrical structures of the Universe.   As an application, we investigate the time evolution of the spectral distance between two Universes that are very close to each other; it is the first necessary step for the detailed analysis of the model-fitting problem in cosmology with the spectral scheme.   We find out a universal formula for the spectral distance between two very close Universes, which turns out to be independent of the detailed form of the distance nor the gravity theory. Then we investigate its time evolution with the help of the evolution equations we derive.   We also formulate the criteria for a good cosmological model in terms of the spectral distance. 
  Within the context of a recently proposed family of stochastic dynamical laws for causal sets, one can ask whether the universe might have emerged from the quantum-gravity era with a large enough size and with sufficient homogeneity to explain its present-day large-scale structure. In general, such a scenario would be expected to require the introduction of very large or very small fundamental parameters into the theory. However, there are indications that such ``fine tuning'' is not necessary, and a large homogeneous and isotropic cosmos can emerge naturally, thanks to the action of a kind of renormalization group associated with cosmic cycles of expansion and re-contraction. 
  We study the occurrence of naked singularities in the spherically symmetric collapse of a charged null fluid in an expanding deSitter background - a piece of charged Vaidya-deSitter spacetime. The necessary conditions for the formation of a naked singularity are found. The results for the uncharged solutions can be recovered from our analysis. 
  We introduce a generalized gravitational conformal invariance in the context of non-compactified 5D Kaluza-Klein theory. It is done by assuming the 4D metric to be dependent on the extra non-compactified dimension. It is then shown that the conformal invariance in 5D is broken by taking an absolute cosmological scale $R_0$ over which the 4D metric is assumed to be dependent weakly on the 5th dimension. This is equivalent to Deser's model for the breakdown of the conformal invariance in 4D by taking a constant cosmological mass term $\mu^2\sim R_0^{-2}$ in the theory. We set the scalar field to its background cosmological value leading to Einstein equation with the gravitational constant $G_N$ and a small cosmological constant. A dual Einstein equation is also introduced in which the matter is coupled to the higher dimensional geometry by the coupling $G_N^{-1}$. Relevant interpretations of the results are also discussed. 
  It is shown analytically that every static, spherically symmetric solution to the Einstein Yang Mills equations with SU(2) gauge group that is defined in the far field has finite ADM mass. Moreover, there can be at most two horizons for such solutions. The three types of solutions possible, Bartnik-McKinnon particle-like solutions, Reissner-Nordstrom-like solutions, and black hole solutions having only one horizon are distinguished by the behavior of the metric coefficients at the origin. 
  We study the analytic structure of the S-matrix which is obtained from the reduced Wheeler-DeWitt wave function describing spherically symmetric gravitational collapse of massless scalar fields. The complex simple poles in the S-matrix lead to the wave functions that satisfy the same boundary condition as quasi-normal modes of a black hole, and correspond to the bounded states of the Euclidean Wheeler-DeWitt equation. These wave function are interpreted as quantum instantons. 
  We investigate homogeneous and isotropic cosmological models in scalar-tensor theories of gravity where two scalar fields are nonminimally coupled to the geometry. Exact solutions are found, by Noether symmetries, depending on the form of couplings and self-interaction potentials. An interesting feature is that we deal with the Brans-Dicke field and the inflaton on the same ground since both are nonminimally coupled and not distinguished {\it a priori} as in earlier models. This fact allows to improve dynamics to get successful extended inflationary scenarios. Double inflationary solutions are also discussed. 
  We study an analytical solution to the Einstein's equations in 2+1-dimensions. The space-time is dynamical and has a line symmetry. The matter content is a minimally coupled, massless, scalar field. Depending on the value of certain parameters, this solution represents three distinct space-times. The first one is flat space-time. Then, we have a big bang model with a negative curvature scalar and a real scalar field. The last case is a big bang model with event horizons where the curvature scalar vanishes and the scalar field changes from real to purely imaginary. 
  The asymptotic properties of the solutions to the Einstein-Maxwell equations with boost-rotation symmetry and Petrov type D are studied. We find series solutions to the pertinent set of equations which are suitable for a late time descriptions in coordinates which are well adapted for the description of the radiative properties of spacetimes (Bondi coordinates). By calculating the total charge, Bondi and NUT mass and the Newman-Penrose constants of the spacetimes we provide a physical interpretation of the free parameters of the solutions. Additional relevant aspects on the asymptotics and radiative properties of the spacetimes considered, such as the possible polarization states of the gravitational and electromagnetic field, are discussed through the way. 
  A previously found momentum-dependent regularization ambiguity in the third post-Newtonian two point-mass Arnowitt-Deser-Misner Hamiltonian is shown to be uniquely determined by requiring global Poincar\'e invariance. The phase-space generators realizing the Poincar\'e algebra are explicitly constructed. 
  We study here some consequences of the nonlinearities of the electromagnetic field acting as a source of Einstein's equations on the propagation of photons. We restrict to the particular case of a ``regular black hole'', and show that there exist singularities in the effective geometry. These singularities may be hidden behind a horizon or naked, according to the value of a parameter. Some unusual properties of this solution are also analyzed. 
  For more than 40 years E.Schmutzer has developed a new approach to the (5-dimensional) projective relativistic theory which he later called Projective Unified Field Theory (PUFT). In the present paper we introduce a new axiomatics for Schmutzer's theory. By means of this axiomatics we can give a new geometrical interpretation of the physical concept of the PUFT. 
  The orbital dynamics of the binary point-mass systems is ambiguous at the third post-Newtonian order of approximation. The static ambiguity is known to be related to the difference between the Brill-Lindquist solution and the Misner-Lindquist solution of the time-symmetric conformally flat initial value problem for binary black holes. The kinetic ambiguity is noticed to violate in general the standard relation between the center-of-mass velocity and the total linear momentum as demanded by global Lorentz invariance. 
  We consider maximal globally hyperbolic flat (2+1) spacetimes with compact space S of genus g>1. For any spacetime M of this type, the length of time that the events have been in existence is M defines a global time, called the cosmological time CT of M, which reveals deep intrinsic properties of spacetime. In particular, the past/future asymptotic states of the cosmological time recover and decouple the linear and the translational parts of the ISO(2,1)-valued holonomy of the flat spacetime. The initial singularity can be interpreted as an isometric action of the fundamental group of S on a suitable real tree. The initial singularity faithfully manifests itself as a lack of smoothness of the embedding of the CT level surfaces into the spacetime M. The cosmological time determines a real analytic curve in the Teichmuller space of Riemann surfaces of genus g, which connects an interior point (associated to the linear part of the holonomy) with a point on Thurston's natural boundary (associated to the initial singularity). 
  Fermat-holonomic congruences are proposed as a weaker substitute for the too restrictive class of Born-rigid motions. The definition is expressed as a set of differential equations. Integrability conditions and Cauchy data are studied. 
  Predictions of the standard thin lens approximation and a new iterative approach to gravitational lensing are compared with an ``exact'' approach in simple test cases involving one or two lenses. We show that the thin lens and iterative approaches are remarkably accurate in predicting time delays, source positions and image magnifications for a single monopole lens and combinations of two monopole lenses. In the cases studied, the iterative method provided greater accuracy than the thin lens method. We also study the accuracy of a ``2 lens, single lens plane model,'' where two monopole lenses colinear with the observer are modeled by a mass distribution in a single lens plane lying between them. We see that this model can lead to large inaccuracies in physically meaningful situations.   A previous version of this paper was published as Phys.Rev.D62, 024025, (2000) with errors in the computation of two lens comparisons. This paper corrects these errors and presents new conclusions which differ from the previous version. 
  We use the modified propagator for quantum field based on a ``principle of path integral duality" proposed earlier in a paper by Padmanabhan to investigate several results in QED. This procedure modifies the Feynman propagator by the introduction of a fundamental length scale. We use this modified propagator for the Dirac particles to evaluate the first order radiative corrections in QED. We find that the extra factor of the modified propagator acts like a regulator at the Planck scales thereby removing the divergences that otherwise appear in the conventional radiative correction calculations of QED. We find that:(i) all the three renormalisation factors $Z_1$, $Z_2$, and $Z_3$ pick up finite corrections and (ii) the modified propagator breaks the gauge invariance at a very small level of ${\mathcal{O}}(10^{-45})$. The implications of this result to generation of the primordial seed magnetic fields are discussed. 
  I describe the Kaluza-Klein approach to general relativity of 4-dimensional spacetimes. This approach is based on the (2,2)-fibration of a generic 4-dimensional spacetime, which is viewed as a local product of a (1+1)-dimensional base manifold and a 2-dimensional fibre space. It is shown that the metric coefficients can be decomposed into sets of fields, which transform as a tensor field, gauge fields, and scalar fields with respect to the infinite dimensional group of the diffeomorphisms of the 2-dimensional fibre space. I discuss a few applications of this formalism. 
  I discuss the (2,2)-formalism of general relativity based on the (2,2)-fibration of a generic 4-dimensional spacetime of the Lorentzian signature. In this formalism general relativity is describable as a Yang-Mills gauge theory defined on the (1+1)-dimensional base manifold, whose local gauge symmetry is the group of the diffeomorphisms of the 2-dimensional fibre manifold. After presenting the Einstein's field equations in this formalism, I solve them for spherically symmetric case to obtain the Schwarzschild solution. Then I discuss possible applications of this formalism. 
  Dirac equation is separable in curved space-time and its solution was found for both spherically and axially symmetric geometry. But most of the works were done without considering the charge of the black hole. Here we consider the spherically symmetric charged black hole background namely Reissner-Nordstrom black hole. Due to presence of the charge of black-hole charge-charge interaction will be important for the cases of incoming charged particle (e.g. electron, proton etc.). Therefore both gravitational and electromagnetic gauge fields should be introduced. Naturally behaviour of the particle will be changed from that in Schwarzschild geometry. We compare both the solutions. In the case of Reissner-Nordstrom black hole there is a possibility of super-radiance unlike Schwarzschild case. We also check this branch of the solution. 
  In order to give the standard scenario of the astrophysics, we study the Einstein theory with minimally coupled scalar field and the cosmological term by considering the scalar field as a candidate of the dark matter. We obtained the exact solution in the cosmological scale and the approximate gravitational potential in the galactic or solar scale. We find that the scalar field plays the role of the dark matter in some sense in the cosmological scale but it does not play the role of the dark matter in the galactic or solar scale within our approximation. 
  Exact expressions for probability densities of conjugate pair separation in euclidean isometries are obtained, for the cosmic crystallography.These are the theoretical counterparts of the mean histograms arising from computer simulation of the isometries. For completeness, also the isometries with fixed points are examined, as well as the orientation reversing isometries. 
  The dynamics of $N\geq 3$ interacting particles is investigated in the non-relativistic context of the Barbour-Bertotti theories. The reduction process on this constrained system yields a Lagrangian in the form of a Riemannian line element. The involved metric, degenerate in the flat configuration space, is the first fundamental form of the space of orbits of translations and rotations (the Leibniz group). The Riemann tensor and the scalar curvature are computed by a generalized Gauss formula in terms of the vorticity tensors of generators of the rotations. The curvature scalar is further given in terms of the principal moments of inertia of the system. Line configurations are singular for $N\neq 3$. A comparison with similar methods in molecular dynamics is traced. 
  We present a geometric approach to the three-body problem in the non-relativistic context of the Barbour-Bertotti theories. The Riemannian metric characterizing the dynamics is analyzed in detail in terms of the relative separations. Consequences of a conformal symmetry are exploited and the sectional curvatures of geometrically preferred surfaces are computed. The geodesic motions are integrated. Line configurations, which lead to curvature singularities for $N\neq 3$, are investigated. None of the independent scalars formed from the metric and curvature tensor diverges there. 
  The method of asymptotic expansions is used to build an approximation scheme relevant to celestial mechanics in relativistic theories of gravitation. A scalar theory is considered, both as a simple example and for its own sake. This theory is summarized, then the relevant boundary problem is seen to be the full initial-value problem. It is shown that, with any given system of gravitating bodies, one may associate a one-parameter family of similar systems, the parameter measuring the gravitational field-strength. After a specific change of units, the derivation of asymptotic expansions becomes straightforward. Two hypotheses could be made as to which time variable has to be used in the expansion. The first one leads to an "asymptotic" post-Newtonian approximation (PNA) with instantaneous propagation, differing from the standard PNA in that, in the asymptotic PNA, all fields are expanded. The second hypothese could lead to an "asymptotic" post-Minkowskian approximation (PMA) allowing to describe propagation effects, but it is not compatible with the Newtonian limit. It is shown that the standard PNA is not compatible with the application of the usual method of asymptotic expansions as envisaged here. 
  Boost-rotation symmetric spacetimes are the only locally asymptotically flat axially symmetric electrovacuum spacetimes with a further symmetry that are radiative. They are realized by uniformly accelerated particles of various kinds or black holes. Their general properties are summarized. Several examples of boost-rotation symmetric solutions of the Maxwell and Einstein equations are studied: uniformly accelerated electric and magnetic multipoles, the Bonnor-Swaminarayan solutions, the C-metric and the spinning C-metric. 
  It has been suggested by Israel that the Kerr singularity cannot be strong in the sense of Tipler, for it tends to cause repulsive effects. We show here that, contrary to that suggestion, nearly all null geodesics reaching this singularity do in fact terminate in Tipler's strong curvature singularity. Implications of this result are discussed in the context of an earlier cosmic censorship theorem which constraints the occurrence of Kerr-like naked singularities in generic collapse situations. 
  Gravity is one of the fundamental forces of Nature, and it is the dominant force in most astronomical systems. In common with all other phenomena, gravity must obey the principles of special relativity. In particular, gravitational forces must not be transmitted or communicated faster than light. This means that when the gravitational field of an object changes, the changes ripple outwards through space and take a finite time to reach other objects. These ripples are called gravitational radiation or gravitational waves. This article gives a brief introduction to the physics of gravitational radiation, including technical material suitable for non-specialist scientists. 
  A vector-tensor theory of gravity that was introduced in an earlier publication is analyzed in detail and its consequences for early universe cosmology are examined. The multiple light cone structure of the theory generates different speeds of gravitational and matter wave fronts, and the contraction of these light cones produces acausal, superluminary inflation that can resolve the initial value problems of cosmology. 
  We study the stability of three-dimensional numerical evolutions of the Einstein equations, comparing the standard ADM formulation to variations on a family of formulations that separate out the conformal and traceless parts of the system. We develop an implementation of the conformal-traceless (CT) approach that has improved stability properties in evolving weak and strong gravitational fields, and for both vacuum and spacetimes with active coupling to matter sources. Cases studied include weak and strong gravitational wave packets, black holes, boson stars and neutron stars. We show under what conditions the CT approach gives better results in 3D numerical evolutions compared to the ADM formulation. In particular, we show that our implementation of the CT approach gives more long term stable evolutions than ADM in all the cases studied, but is less accurate in the short term for the range of resolutions used in our 3D simulations. 
  We present a spectral method for solving elliptic equations which arise in general relativity, namely three-dimensional scalar Poisson equations, as well as generalized vectorial Poisson equations of the type $\Delta \vec{N} + \lambda \vec{\nabla}(\vec{\nabla}\cdot \vec{N}) = \vec{S}$ with $\lambda \not= -1$. The source can extend in all the Euclidean space ${\bf R}^3$, provided it decays at least as $r^{-3}$. A multi-domain approach is used, along with spherical coordinates $(r,\theta,\phi)$. In each domain, Chebyshev polynomials (in $r$ or $1/r$) and spherical harmonics (in $\theta$ and $\phi$) expansions are used. If the source decays as $r^{-k}$ the error of the numerical solution is shown to decrease at least as $N^{-2(k-2)}$, where $N$ is the number of Chebyshev coefficients. The error is even evanescent, i.e. decreases as $\exp(-N)$, if the source does not contain any spherical harmonics of index $l\geq k -3$ (scalar case) or $l\geq k-5$ (vectorial case). 
  We show that non-linear electrodynamics may induce a photon to follow a closed path in spacetime. We exhibit a specific case in which such closed lightlike curve (CLC) appears 
  We study the self force acting on a scalar charge in uniform circular motion around a Schwarzschild black hole. The analysis is based on a direct calculation of the self force via mode decomposition, and on a regularization procedure based on Ori's mode-sum regularization prescription. We find the four self-force at arbitrary radii and angular velocities (both geodesic and non-geodesic), in particular near the black hole, where general-relativistic effects are strongest, and for fast motion. We find the radial component of the self force to be repulsive or attractive, depending on the orbit. 
  Using the effective metric formalism for photons in a nonlinear electromagnetic theory, we show that a certain field configuration in Born-Infeld electromagnetism in flat spacetime can be interpreted as an ultrastatic spherically symmetric wormhole. We also discuss some properties of the effective metric that are valid for any field configuration. 
  Criteria for the existence of de Sitter inflation with dilaton fields in four-dimensional space-times with torsion is discussed.The relation between matter density perturbation and the spin-density perturbation is stablished based on this criteria.From COBE data it is shown that there is a linear relationship between the spin-torsion density and temperature of the Universe for the case where matter density dominates the kinetic part of dilaton fields. 
  The main general relativistic effects in the motion of the Moon are briefly reviewed. The possibility of detection of the solar gravitomagnetic contributions to the mean motions of the lunar node and perigee is discussed. 
  Vacuum polarization of massive scalar fields in a thermal state at arbitrary temperature is studied near the horizon of a Reissner-Nordstr\"{o}m black hole. We derived an analytic form of $<\phi^2>$ approximately in the large mass limit near the black hole horizon. We uses the zeroth order WKB approximation and power series expansion near the horizon for the Euclideanized mode function. Our formula for the vacuum polarization shows regular behavior on the horizon if the temperature of the scalar field is equal to the Hawking temperature of the black hole. The finite part of the vacuum polarization agrees with the result of the DeWitt-Schwinger approximation up to $O(m^{-4})$ which is the next leading order of the expansion. 
  A new ''twice loose shoe'' method in the Wheeler-DeWitt equation of the universe wave function on the cosmic scale factor a and a scalar field $\phi$ is suggested in this letter. We analysis the both affects come from the tunneling effect of a and the potential well effect of $\phi$, and obtain the initial values $a_0$ and $\phi_0$ about a primary closed universe which is born with the largest probability in the quantum manner. Our result is able to overcome the ''large field difficulty'' of the universe quantum creation probability with only tunneling effect. This new born universe has to suffer a startup of inflation, and then comes into the usual slow rolling inflation. The universe with the largest probability maybe has a ''gentle'' inflation or an eternal chaotic inflation, this depends on a new parameter q which describes the tunneling character. 
  Self-similar, spherically symmetric cosmological models with a perfect fluid and a scalar field with an exponential potential are investigated. New variables are defined which lead to a compact state space, and dynamical systems methods are utilised to analyse the models. Due to the existence of monotone functions global dynamical results can be deduced. In particular, all of the future and past attractors for these models are obtained and the global results are discussed. The essential physical results are that initially expanding models always evolve away from a massless scalar field model with an initial singularity and, depending on the parameters of the models, either recollapse to a second singularity or expand forever towards a flat power-law inflationary model. The special cases in which there is no barotropic fluid and in which the scalar field is massless are considered in more detail in order to illustrate the asymptotic results. Some phase portraits are presented and the intermediate dynamics and hence the physical properties of the models are discussed. 
  It is shown, that the radiation of the charge, moving with uniform acceleration or uniformly moving round a circle and also freely moving in a gravitational field, contradicts the principle of equivalence. It is also shown, that the interaction of the charges, moving with uniform acceleration or uniformly circling, which has been calculated within the framework of classical electrodynamics, leads to the violation of laws of conservation of energy, impulse and angular momentum. We have offered a method in which way to conform electrodynamics to the principle of equivalence. So in the electrodynamics, which has been conformed in such a way, all the mentioned violations of the laws of conservation are automatically removed and the stability of Rutherford's atom is explained. It is shown that the changes, which we have brought into the electrodynamics, do not contradict the results of experiments. 
  We show it is possible for the information paradox in black hole evaporation to be resolved classically. Using standard junction conditions, we attach the general closed spherically symmetric dust metric to a spacetime satisfying all standard energy conditions but with a single point future c-boundary. The resulting Omega Point spacetime, which has NO event horizons, nevertheless has black hole type trapped surfaces and hence black holes. But since there are no event horizons, information eventually escapes from the black holes. We show that a scalar quintessence field with an appropriate exponential potential near the final singularity would give rise to an Omega Point final singularity. 
  We extend recent investigations on the integrability of oblique orbits of test particles under the gravitational field corresponding to the superposition of an infinitesimally thin disk and a monopole to the more realistic case, for astrophysical purposes, of a thick disk. Exhaustive numerical analyses were performed and the robustness of the recent results is confirmed. We also found that, for smooth distributions of matter, the disk thickness can attenuate the chaotic behavior of the bounded oblique orbits. Perturbations leading to the breakdown of the reflection symmetry about the equatorial plane, nevertheless, may enhance significantly the chaotic behavior, in agreement with recent studies on oblate models. 
  The minisuperspace model of a Bianchi Type I universe with compact spatial sections is investigated. The classical solutions are brought onto a form were the difference between compact and infinite spatial sections are manifest. One of the features of the compact case is that it has a non-trivial moduli space. The solution space of the compact Bianchi Type I universe is 10 dimensional whereas the Kasner solutions only have a 1 dimensional solution space. We also include the classical solutions with dust and a cosmological constant. Solutions to the Wheeler-DeWitt equation are obtained in light of the tunneling boundary proposal by Vilenkin. Backreaction effects from a simple scalar field are also investigated. 
  Density fluctuations of fluids with negative pressure exhibit decreasing time behaviour in the long wavelength limit, but are strongly unstable in the small wavelength limit when a hydrodynamical approach is used. On the other hand, the corresponding gravitational waves are well behaved. We verify that the instabilities present in density fluctuations are due essentially to the hydrodynamical representation; if we turn to a field representation that lead to the same background behaviour, the instabilities are no more present. In the long wavelength limit, both approachs give the same results. We show also that this inequivalence between background and perturbative level is a feature of negative pressure fluid. When the fluid has positive pressure, the hydrodynamical representation leads to the same behaviour as the field representation both at the background and perturbative levels. 
  The Berry phase of mixed states, as neutrino oscillations, is calculated in a accelerating and rotating reference frame. It turns out to be depending on the vacuum mixing angle, the mass--squared difference and on the coupling between the momentum of the neutrino and the spinorial connection. Berry's phase for solar neutrinos and its geometrical aspects are also discussed. 
  Massive, spinless bosons have vanishing probability of reaching the sphere r=2M from the region r>2M when the original Schwarzschild metric is modified by maximal acceleration corrections. 
  We present a new derivation of the perturbation equations governing the oscillations of relativistic non-rotating neutron star models using the ADM-formalism. This formulation has the advantage that it immediately yields the evolution equations in a hyperbolic form, which is not the case for the Einstein field equations in their original form. We show that the perturbation equations can always be written in terms of spacetime variables only, regardless of any particular gauge. We demonstrate how to obtain the Regge-Wheeler gauge, by choosing appropriate lapse and shift. In addition, not only the 3-metric but also the extrinsic curvature of the initial slice have to satisfy certain conditions in order to preserve the Regge-Wheeler gauge throughout the evolution. We discuss various forms of the equations and show their relation to the formulation of Allen et al. New results are presented for polytropic equations of state. An interesting phenomenon occurs in very compact stars, where the first ring-down phase in the wave signal corresponds to the first quasinormal mode of an equal mass black hole, rather than to one of the proper quasinormal modes of the stellar model. A somewhat heuristic explanation to account for this phenomenon is given. For realistic equations of state, the numerical evolutions exhibit an instability, which does not occur for polytropic equations of state. We show that this instability is related to the behavior of the sound speed at the neutron drip point. As a remedy, we devise a transformation of the radial coordinate $r$ inside the star, which removes this instability and yields stable evolutions for any chosen numerical resolution. 
  The entropy for a black hole in a de Sitter space is approached within the framework of spacetime foam. A simple model, made by $N$ wormholes in a semiclassical approximation, is taken under examination to compute the entropy for such a case. An extension to the extreme case when the black hole and cosmological horizons are equal is discussed. 
  A simple model of spacetime foam, made by $N$ wormholes in a semiclassical approximation, is taken under examination. The Casimir-like energy of the quantum fluctuation of such a model and its probability of being realized are computed. Implications on the Bekenstein-Hawking entropy and the cosmological constant are considered. 
  General geodesic equations of the motion of spinning systems around the (3+1)-dimensional and (2+1)-dimensional rotating anti-de Sitter black holes have been obtained. Based upon these equations, we derived the entropy bound for a rotating system from Kerr-Anti de Sitter black holes and BTZ black holes, respectively. Our result coincides with that of Hod's derived from Kerr black hole, which shows that the entropy bound of the rotating system is neither dependent on the black hole parameters, nor on spacetime dimensions. It is a universal entropy bound. 
  In this talk I discuss pertinence of the wormholes to the problem of circumventing the light speed barrier and present a specific class of wormholes. The wormholes of this class are static and have arbitrarily wide throats, which makes them traversable. The matter necessary for these spacetimes to be solutions of the Einstein equations is shown to consist of two components, one of which satisfies the Weak energy condition and the other is produced by vacuum fluctuations of neutrino, electromagnetic (in dimensional regularization), and/or massless scalar (conformally coupled) fields. 
  We discuss two nonlocal models of electrodynamics in which the nonlocality is induced by the acceleration of the observer. Such an observer actually measures an electromagnetic field that exhibits persistent memory effects. We compare Mashhoon's model with a new ansatz developed here in the framework of charge & flux electrodynamics with a constitutive law involving the Levi-Civita connection as seen from the observer's local frame and conclude that they are in partial agreement only for the case of constant acceleration. 
  A model for 2D-quantum gravity from the Virasoro symmetry is studied. The notion of space-time naturally arises as a homogeneous space associated with the kinematical (non-dynamical) SL(2,R) symmetry in the kernel of the Lie-algebra central extension for the critical values of the conformal anomaly. The rest of the generators in the group, L_n (n>1, n<-1), mix space-times with different constant curvature. Only in the classical limit all space-times can be identified, defining a unique Minkowski space-time, and the operators L_n (n<1, n<-1) gauged away. This process entails a restriction to SL(2,R) subrepresentations, which creates a non-trivial two-dimensional symplectic classical phase space. The present model thus suggests that the role of general covariance in quantum gravity is different from that played in the classical limit. 
  The quantum modes of a new family of relativistic oscillators are studied by using the supersymmetry and shape invariance in a version suitable for (1+1) dimensional relativistic systems. In this way one obtains the Rodrigues formulas of the normalized energy eigenfunctions of the discrete spectra and the corresponding rising and lowering operators.   Pacs: 04.62.+v, 03.65.Ge 
  We study a rotating and expanding, Godel type metric, originally considered by Korotkii and Obukhov, showing that, in the limit of large times and nearby distances, it reduces to the open metric of Friedmann. In the epochs when radiation or dust matter dominate the energy density, our solutions are similar to the isotropic ones and, in what concerns processes occurring at small times, the rotation leads only to higher order corrections. At large times, the solution is dominated by a decaying positive cosmological term, with negative pressure, and necessarily describes a quasi-flat universe if the energy conditions have to be satisfied. The absence of closed time-like curves requires a superior limit for the global angular velocity, which appears as a natural explanation for the observed smallness of the present rotation. The conclusion is that the introduction of a global rotation, in addition to be compatible with observation, can enrich the standard model of the Universe, explaining issues like the origin of galaxies rotation and the quasi-flatness problem. 
  We obtain the higher dimensional global flat embeddings of static, rotating, and charged BTZ black holes. On the other hand, we also study the similar higher dimensional flat embeddings of the (2+1) de Sitter black holes which are the counterparts of the anti-de Sitter BTZ black holes. As a result, the charged dS black hole is shown to be embedded in (3+2) GEMS, contrast to the charged BTZ one having (3+3) GEMS structure. 
  The flatness and cosmological constant problems are solved with varying speed of light c, gravitational coupling strength G and cosmological parameter Lambda, by explicitly assuming energy conservation of observed matter. The present solution to the flatness problem is the same as the previous solution in which energy conservation was absent. 
  It is known by the experience gained from the gravitational wave detector proto-types that the interferometric output signal will be corrupted by a significant amount of non-Gaussian noise, large part of it being essentially composed of long-term sinusoids with slowly varying envelope (such as violin resonances in the suspensions, or main power harmonics) and short-term ringdown noise (which may emanate from servo control systems, electronics in a non-linear state, etc.). Since non-Gaussian noise components make the detection and estimation of the gravitational wave signature more difficult, a denoising algorithm based on adaptive filtering techniques (LMS methods) is proposed to separate and extract them from the stationary and Gaussian background noise. The strength of the method is that it does not require any precise model on the observed data: the signals are distinguished on the basis of their autocorrelation time. We believe that the robustness and simplicity of this method make it useful for data preparation and for the understanding of the first interferometric data. We present the detailed structure of the algorithm and its application to both simulated data and real data from the LIGO 40meter proto-type. 
  In the context of a gauge theory for the translation group, a conserved energy-momentum gauge current for the gravitational field is obtained. It is a true spacetime and gauge tensor, and transforms covariantly under global Lorentz transformations. By rewriting the gauge gravitational field equation in a purely spacetime form, it becomes the teleparallel equivalent of Einstein's equation, and the gauge current reduces to the M{\o}ller's canonical energy-momentum density of the gravitational field. 
  The current status of numerical solutions for the equations of ideal general relativistic hydrodynamics is reviewed. With respect to an earlier version of the article the present update provides additional information on numerical schemes and extends the discussion of astrophysical simulations in general relativistic hydrodynamics. Different formulations of the equations are presented, with special mention of conservative and hyperbolic formulations well-adapted to advanced numerical methods. A large sample of available numerical schemes is discussed, paying particular attention to solution procedures based on schemes exploiting the characteristic structure of the equations through linearized Riemann solvers. A comprehensive summary of astrophysical simulations in strong gravitational fields is presented. These include gravitational collapse, accretion onto black holes and hydrodynamical evolutions of neutron stars. The material contained in these sections highlights the numerical challenges of various representative simulations. It also follows, to some extent, the chronological development of the field, concerning advances on the formulation of the gravitational field and hydrodynamic equations and the numerical methodology designed to solve them. 
  The Einstein-Maxwell fields of rotating stationary sources are represented by the SU(2,1) spinor potential $\Psi_A$ satisfying \[ \nabla \cdot [\Theta ^{-1}(\Psi_A\nabla \Psi_B-\Psi_B\nabla \Psi_A)]=-2\Theta ^{-2}\vec{C}\cdot (\Psi_A\nabla \Psi_B-\Psi_B\nabla \Psi_A) \] where $\Theta =\Psi ^{\dagger }\cdot \Psi $ is the SU(2,1) norm of $\Psi $% . The Ernst potentials are expressed in terms of the spinor potential by $% {\cal E}=\frac{\Psi_1-\Psi _2}{\Psi_1+\Psi_2}$, $\Phi =\frac{\Psi_3}{% \Psi_1+\Psi_2}$ . The group invariant vector $\vec{C}=-2i\func{Im}\{\Psi ^{\dagger}\cdot \nabla \Psi \}$ is generated exclusively by the rotation of the source, hence it is appropriate to refer to $\vec{C}$ as the {\em swirl} of the field. Static fields have no swirl.   The fields with no swirl are a class generalizing the equilibrium ($| e| =m$) class of Einstein-Maxwell fields. We obtain the integrability conditions and a highly symmetrical set of field equations for this class, as well as exact solutions and an open research problem. 
  In a new model that we proposed, nonperturbative vacuum contributions to the effective action of a free quantized massive scalar field lead to a cosmological solution in which the scalar curvature becomes constant after a time $t_j$ (when the redshift $z \sim 1$) that depends on the mass of the scalar field and its curvature coupling. This spatially-flat solution implies an accelerating universe at the present time and gives a good one-parameter fit to high-redshift Type Ia supernovae (SNe-Ia) data, and the present age and energy density of the universe. Here we show that the imaginary part of the nonperturbative curvature term that causes the cosmological acceleration, implies a particle production rate that agrees with predictions of other methods and extends them to non-zero mass fields. The particle production rate is very small after the transition and is not expected to alter the nature of the cosmological solution. We also show that the equation of state of our model undergoes a transition at $t_j$ from an equation of state dominated by non-relativistic pressureless matter (without a cosmological constant) to an effective equation of state of mixed radiation and cosmological constant, and we derive the equation of state of the vacuum. Finally, we explain why nonperturbative vacuum effects of this ultralow-mass particle do not significantly change standard early universe cosmology. 
  We introduce a pregeometry employing uniform spaces over the denumerable set X of spacetime events. The discrete uniformity D_X over X is used to obtain a pregeometric model of macroscopic spacetime neighborhoods. We then use a uniformity base generated by a topological group structure over X to provide a pregeometric model of microscopic spacetime neighborhoods. Accordingly, quantum non-separability as it pertains to non-locality is understood pregeometrically as a contrast between microscopic spacetime neighborhoods and macroscopic spacetime neighborhoods. A nexus between this pregeometry and conventional spacetime physics is implied per the metric induced by D_X. A metric over the topological group Z2 x ... x Z2 is so generated. Implications for quantum gravity are enumerated. 
  Acoustic black holes are fluid dynamic analogs of general relativistic black holes, wherein the behaviour of sound waves in a moving fluid acts as an analog for scalar fields propagating in a gravitational background. Acoustic horizons possess many of the properties more normally associated with the event horizons of general relativity, up to and including Hawking radiation. They have received much attention because it would seem to be much easier to experimentally create an acoustic horizon than to create an event horizon. We wish to point out some potential difficulties (and opportunities) in actually setting up an experiment that possesses an acoustic horizon. We show that in zero-viscosity, stationary fluid flow with generic boundary conditions, the creation of an acoustic horizon is accompanied by a formally infinite ``surface gravity'', and a formally infinite Hawking flux. Only by applying a suitable non-constant external body force, and for very specific boundary conditions on the flow, can these quantities be kept finite. This problem is ameliorated in more realistic models of the fluid. For instance, adding viscosity always makes the Hawking flux finite, but greatly complicates the behaviour of the acoustic radiation --- viscosity is tantamount to explicitly breaking ``acoustic Lorentz invariance''. Thus, this issue represents both a difficulty and an opportunity --- acoustic horizons may be somewhat more difficult to form than naively envisaged, but if formed, they may be much easier to detect than one would at first suppose. 
  We show the following two extensions of the standard positive mass theorem (one for either sign): Let (N,g) and (N,g') be asymptotically flat Riemannian 3-manifolds with compact interior and finite mass, such that g and g' are twice Hoelder differentiable and related via the conformal rescaling g' = (phi^4).g, with a twice Hoelder differentiable function phi>0. Assume further that the corresponding Ricci scalars satisfy either R + (phi^4).R' >= 0 or R - (phi^4).R' >= 0. Then the corresponding masses satisfy m + m' >= 0 or m - m' >= 0, respectively. Moreover, in the case of the minus signs, equality holds iff g and g' are isometric, whereas for the plus signs equality holds iff both (N,g) and (N,g') are flat Euclidean spaces.   While the proof of the case with the minus signs is rather obvious, the case with the plus signs requires a subtle extension of Witten's proof of the standard positive mass theorem. The idea for this extension is due to Masood-ul-Alam who, in the course of an application, proved the rigidity part m + m' = 0 of this theorem, for a special conformal factor. We observe that Masood-ul-Alam's method extends to the general situation. 
  We study quantum radiation generated by an accelerated motion of a small body with a refractive index n which differes slightly from 1. To simplify calculations we consider a model with a scalar massless field. We use the perturbation expansion in a small parameter n-1 to obtain a correction to the vacuum Hadamard function for a uniformly accelerated motion of the body. We obtain the vacuum expectation for the energy density flux in the wave zone and discuss its properties. 
  A one-parameter family of time dependent adiabatic indices is introduced for any given type of cosmological fluid of constant adiabatic index by a mathematical method belonging to the class of Darboux transformations. The procedure works for zero cosmological constant at the price of introducing a new constant parameter related to the time dependence of the adiabatic index. These fluids can be the real cosmological fluids that are encountered at cosmological scales and they could be used as a simple and efficient explanation for the recent experimental findings regarding the present day accelerating universe. In addition, new types of cosmological scale factors, corresponding to these fluids, are presented 
  We investigate the occurrence and nature of a naked singularity in the gravitational collapse of an inhomogeneous dust cloud described by a non self-similar higher dimensional Tolman spacetime. The necessary condition for the formation of a naked singularity or a black hole is obtained. The naked singularities are found to be gravitationally strong in the sense of Tipler and provide another example that violates the cosmic censorship conjecture. 
  It is shown that the field equations of general relativity in the Einstein tensor form and the unimodular theory of gravity do not fulfill the correspondence principle commitment completely. The consistent formalisms are briefly discussed. 
  We investigate a triad representation of the Chern-Simons state of quantum gravity with a non-vanishing cosmological constant. It is shown that the Chern-Simons state, which is a well-known exact wavefunctional within the Ashtekar theory, can be transformed to the real triad representation by means of a suitably generalized Fourier transformation, yielding a complex integral representation for the corresponding state in the triad variables. It is found that topologically inequivalent choices for the complex integration contour give rise to linearly independent wavefunctionals in the triad representation, which all arise from the one Chern-Simons state in the Ashtekar variables. For a suitable choice of the normalization factor, these states turn out to be gauge-invariant under arbitrary, even topologically non-trivial gauge-transformations. Explicit analytical expressions for the wavefunctionals in the triad representation can be obtained in several interesting asymptotic parameter regimes, and the associated semiclassical 4-geometries are discussed. In restriction to Bianchi-type homogeneous 3-metrics, we compare our results with earlier discussions of homogeneous cosmological models. Moreover, we define an inner product on the Hilbert space of quantum gravity, and choose a natural gauge-condition fixing the time-gauge. With respect to this particular inner product, the Chern-Simons state of quantum gravity turns out to be a non-normalizable wavefunctional. 
  We present new solutions of the string cosmological effective action in the presence of a homogeneous Maxwell field with pure magnetic component. Exact solutions are derived in the case of space-independent dilaton and vanishing torsion background. In our examples the four dimensional metric is either of Bianchi-type III and VI$_{-1}$ or Kantowski-Sachs. 
  Brady, Creighton and Thorne have proposed a choice of the lapse and shift for numerical evolutions in general relativity that extremizes a measure of the rate of change of the three-metric (BCT gauge). We investigate existence and uniqueness of this gauge, and comment on its use in numerical time evolutions. 
  An implicit, fully characteristic, numerical scheme for solving the field equations of a cosmic string coupled to gravity is described. The inclusion of null infinity as part of the numerical grid allows us to apply suitable boundary conditions on the metric and matter fields to suppress unphysical divergent solutions. The code is tested by comparing the results with exact solutions, checking that static cosmic string initial data remain constant when evolved and undertaking a time dependent convergence analysis of the code. It is shown that the code is accurate, stable and exhibits clear second order convergence. The code is used to analyse the interaction between a Weber--Wheeler pulse of gravitational radiation with the string. The interaction causes the string to oscillate at frequencies inversely proportional to the masses of the scalar and vector fields of the string. After the pulse has largely radiated away the string continues to ring but the oscillations slowly decay and eventually the variables return to their static values. 
  A tensor description of perturbative Einsteinian gravity about an arbitrary background spacetime is developed. By analogy with the covariant laws of electromagnetism in spacetime, gravito-electromagnetic potentials and fields are defined to emulate electromagnetic gauge transformations under substitutions belonging to the gauge symmetry group of perturbative gravitation. These definitions have the advantage that on a flat background, with the aid of a covariantly constant timelike vector field, a subset of the linearised gravitational field equations can be written in a form that is fully analogous to Maxwell's equations (without awkward factors of 4 and extraneous tensor fields). It is shown how the remaining equations in the perturbed gravitational system restrict the time dependence of solutions to these equations and thereby prohibit the existence of propagating vector fields. The induced gravito-electromagnetic Lorentz force on a test particle is evaluated in terms of these fields together with the torque on a small gyroscope. It is concluded that the analogy of perturbative gravity to Maxwell's description of electromagnetism can be valuable for (quasi-)stationary gravitational phenomena but that the analogy has its limitations. 
  In order to reduce the Klein-Gordon equation (with minimal coupling), we introduce a generalization of the so-called "mode solutions" that are well-known in the special case of a Robertson-Walker universe. After separation of the variables, we end up with a partial differential equation in lower dimension. A reduced version of the Gordon current arises and is conserved. When the first factor-manifold is Lorentzian, distinct modes appear as mutually orthogonal in the sense of the sesquilinear form obtained from the customary Gordon current. Moreover, a sesquilinear form is defined on the space of solutions to the reduced equation. Extension of this picture to curvature coupling is possible when the second factor-manifold has a constant scalar curvature. 
  We find evidence for a continuum limit of a particular causal set dynamics which depends on only a single ``coupling constant'' $p$ and is easy to simulate on a computer. The model in question is a stochastic process that can also be interpreted as 1-dimensional directed percolation, or in terms of random graphs. 
  One of the remarkable features of black holes is that they possess a thermodynamic description, even though they do not appear to be statistical systems. We use self-gravitating magnetic monopole solutions as tools for understanding the emergence of this description as one goes from an ordinary spacetime to one containing a black hole. We describe how causally distinct regions emerge as a monopole solution develops a horizon. We define an entropy that is naturally associated with these regions and that has a clear connection with the Hawking-Bekenstein entropy in the critical black hole limit. 
  A rapidly rotating, axisymmetric star can be dynamically unstable to an m=2 "bar" mode that transforms the star from a disk shape to an elongated bar. The fate of such a bar-shaped star is uncertain. Some previous numerical studies indicate that the bar is short lived, lasting for only a few bar-rotation periods, while other studies suggest that the bar is relatively long lived. This paper contains the results of a numerical simulation of a rapidly rotating gamma=5/3 fluid star. The simulation shows that the bar shape is long lived: once the bar is established, the star retains this shape for more than 10 bar-rotation periods, through the end of the simulation. The results are consistent with the conjecture that a star will retain its bar shape indefinitely on a dynamical time scale, as long as its rotation rate exceeds the threshold for secular bar instability. The results are described in terms of a low density neutron star, but can be scaled to represent, for example, a burned-out stellar core that is prevented from complete collapse by centrifugal forces. Estimates for the gravitational-wave signal indicate that a dynamically unstable neutron star in our galaxy can be detected easily by the first generation of ground based gravitational-wave detectors. The signal for an unstable neutron star in the Virgo cluster might be seen by the planned advanced detectors. The Newtonian/quadrupole approximation is used throughout this work. 
  We present some arguments in support of a {\it zero} entropy for {\it extremal} black holes. These rely on a combination of both quantum, thermodynamic, and statistical physics arguments. This result may shed some light on the nature of these extreme objects. In addition, we show that within a {\it quantum} framework the capture of a particle by an initially extremal black hole always results with a final nonextremal black hole. 
  We investigate the quantum evolution of large black holes that nucleate spontaneously in de Sitter space. By numerical computation in the s-wave and one-loop approximations, we verify claims that such black holes can initially "anti-evaporate" instead of shrink. We show, however, that this is a transitory effect. It is followed by an evaporating phase, which we are able to trace until the black holes are small enough to be treated as Schwarzschild. Under generic perturbations, the nucleated geometry is shown to decay into a ring of de Sitter regions connected by evaporating black holes. This confirms that de Sitter space is globally unstable and fragments into disconnected daughter universes. 
  This is an introduction to quantum gravity, aimed at a fairly general audience and concentrating on what have historically two main approaches to quantum gravity: the covariant and canonical programs (string theory is not covered). The quantization of gravity is discussed by analogy with the quantization of the electromagnetic field. The conceptual and technical problems of both approaches are discussed, and the paper concludes with a discussion of evidence for quantum gravity from the rest of physics.   The paper assumes some familiarity with non-relativistic quantum mechanics, special relativity, and the Lagrangian and Hamiltonian formulations of classical mechanics; some experience with classical field theory, quantum electrodynamics and the gauge principle in electromagnetism might be helpful but is not required. No knowledge of general relativity or of quantum field theory in general is assumed. 
  From calculations of the variance of fluctuations and of the mean of the energy density of a massless scalar field in the Minkowski vacuum as a function of an intrinsic scale defined by the world function between two nearby points (as used in point separation regularization) we show that, contrary to prior claims, the ratio of variance to mean-squared being of the order unity does not imply a failure of semiclassical gravity. It is more a consequence of the quantum nature of the state of matter field than any inadequacy of the theory of spacetime with quantum matter as source. 
  The slow-roll approximation to inflation is ultimately justified by the presence of inflationary attractors for the orbits of the solutions of the dynamical equations in phase space. There are many indications that the inflaton field couples nonminimally to the spacetime curvature: the existence of attractor points for inflation with nonminimal coupling is demonstrated, subject to a condition on the inflaton potential and the value of the coupling constant. 
  Motivated by the problem of the dynamics of point-particles in high post-Newtonian (e.g. 3PN) approximations of general relativity, we consider a certain class of functions which are smooth except at some isolated points around which they admit a power-like singular expansion. We review the concepts of (i) Hadamard ``partie finie'' of such functions at the location of singular points, (ii) the partie finie of their divergent integral. We present and investigate different expressions, useful in applications, for the latter partie finie. To each singular function, we associate a partie-finie (Pf) pseudo-function. The multiplication of pseudo-functions is defined by the ordinary (pointwise) product. We construct a delta-pseudo-function on the class of singular functions, which reduces to the usual notion of Dirac distribution when applied on smooth functions with compact support. We introduce and analyse a new derivative operator acting on pseudo-functions, and generalizing, in this context, the Schwartz distributional derivative. This operator is uniquely defined up to an arbitrary numerical constant. Time derivatives and partial derivatives with respect to the singular points are also investigated. In the course of the paper, all the formulas needed in the application to the physical problem are derived. 
  We investigate the dynamics of two point-like particles through the third post-Newtonian (3PN) approximation of general relativity. The infinite self-field of each point-mass is regularized by means of Hadamard's concept of ``partie finie''. Distributional forms associated with the regularization are used systematically in the computation. We determine the stress-energy tensor of point-like particles compatible with the previous regularization. The Einstein field equations in harmonic coordinates are iterated to the 3PN order. The 3PN equations of motion are Lorentz-invariant and admit a conserved energy (neglecting the 2.5PN radiation reaction). They depend on an undetermined coefficient, in agreement with an earlier result of Jaranowski and Schaefer. This suggests an incompleteness of the formalism (in this stage of development) at the 3PN order. In this paper we present the equations of motion in the center-of-mass frame and in the case of circular orbits. 
  The stability of our vacuum is analyzed and several aspects concerning this question are reviewed. 1) In the standard Glashow-Weinberg-Salam (GWS) model we review the instability towards the formation of a bubble of lower energy density and how the rate of such bubble formation process compares with the age of the Universe for the known values of the GWS model. 2) We also review the recent work by one of us (E.I.G) concerning the vacuum instability question in the context of a model that solves the cosmological constant problem. It turns out that in such model the same physics that solves the cosmological constant problem makes the vacuum stable. 3) We review our recent work concerning the instability of elementary particle embedded in our vacuum, towards the formation of an infinite Universe. Such process is not catastrophic. It leads to a "bifurcation type" instability in which our Universe is not eaten by a bubble (instead a baby universe is born). This universe does not replace our Universe rather it disconnects from it (via a wormhole) after formation. 
  The possibility of mass in the context of scale-invariant, generally covariant theories, is discussed. Scale invariance is considered in the context of a gravitational theory where the action, in the first order formalism, is of the form $S = \int L_{1} \Phi d^4x$ + $\int L_{2}\sqrt{-g}d^4x$ where $\Phi$ is a density built out of degrees of freedom independent of the metric. For global scale invariance, a "dilaton" $\phi$ has to be introduced, with non-trivial potentials $V(\phi)$ = $f_{1}e^{\alpha\phi}$ in $L_1$ and $U(\phi)$ = $f_{2}e^{2\alpha\phi}$ in $L_2$. This leads to non-trivial mass generation and a potential for $\phi$ which is interesting for inflation. The model after ssb can be connected to the induced gravity model of Zee, which is a successful model of inflation. Models of the present universe and a natural transition from inflation to a slowly accelerated universe at late times are discussed. 
  1 Introduction 2 Multipole Decomposition 3 Source Multipole Moments 4 Post-Minkowskian Approximation 5 Radiative Multipole Moments 6 Post-Newtonian Approximation 7 Point-Particles 8 Conclusion 
  It might seem that a choice of a time coordinate in Hamiltonian formulations of general relativity breaks the full four-dimensional diffeomorphism covariance of the theory. This is not the case. We construct explicitly the complete set of gauge generators for Ashtekar's formulation of canonical gravity. The requirement of projectability of the Legendre map from configuration-velocity space to phase space renders the symmetry group a gauge transformation group on configuration-velocity variables. Yet there is a sense in which the full four-dimensional diffeomorphism group survives. Symmetry generators serve as Hamiltonians on members of equivalence classes of solutions of Einstein's equations and are thus intimately related to the so-called "problem of time" in an eventual quantum theory of gravity. 
  The issue of the transformations of units is treated, mainly, in a geometrical context. It is shown that Weyl-integrable geometry is a consistent framework for the formulation of the gravitational laws since the basic law on which this geometry rests is invariant under point-dependent transformations of units. Riemann geometry does not fulfill this requirement. Spacetime singularities are then shown to be a consequence of a wrong choice of the geometrical formulation of the laws of gravitation. This result is discussed, in particular, for the Schwrazschild black hole and for Friedmann-Robertson-Walker cosmology. Arguments are given that point at Weyl-integrable geometry as a geometry implicitly containing the quantum effects of matter. The notion of geometrical relativity is presented. This notion may represent a natural extension of general relativity to include invariance under the group of units transformations. 
  In a path-integral approach to quantum cosmology, the Lorenz gauge-averaging term is studied for Euclidean Maxwell theory on a portion of flat four-space bounded by two concentric three-spheres, but with arbitrary values of the gauge parameter. The resulting set of eigenvalue equations for normal and longitudinal modes of the electromagnetic potential cannot be decoupled, and is here studied with a Green-function method. This means that an equivalent equation for longitudinal modes is obtained which has integro-differential nature, after inverting a differential operator in the original coupled system. A complete calculational scheme is therefore obtained for the one-loop semiclassical evaluation of the wave function of the universe in the presence of gauge fields. This might also lead to a better understanding of how gauge independence is actually achieved on manifolds with boundary, whose consideration cannot be avoided in a quantum theory of the universe. 
  Contents: 1) Introduction and a few excursions [A word on the role of explicit solutions in other parts of physics and astrophysics. Einstein's field equations. "Just so" notes on the simplest solutions: The Minkowski, de Sitter and anti-de Sitter spacetimes. On the interpretation and characterization of metrics. The choice of solutions. The outline] 2) The Schwarzschild solution [Spherically symmetric spacetimes. The Schwarzschild metric and its role in the solar system. Schwarzschild metric outside a collapsing star. The Schwarzschild-Kruskal spacetime. The Schwarzschild metric as a case against Lorentz-covariant approaches. The Schwarzschild metric and astrophysics] 3) The Reissner- Nordstrom solution [Reissner-Nordstrom black holes and the question of cosmic censorship. On extreme black holes, d-dimensional black holes, string theory and "all that"] 4) The Kerr metric [Basic features. The physics and astrophysics around rotating black holes. Astrophysical evidence for a Kerr metric] 5) Black hole uniqueness and multi-black hole solutions 6) Stationary axisymmetric fields and relativistic disks [Static Weyl metrics. Relativistic disks as sources of the Kerr metric and other stationary spacetimes. Uniformly rotating disks] 7) Taub-NUT space [A new way to the NUT metric. Taub-NUT pathologies and applications] 8) Plane waves and their collisions [Plane-fronted waves. New developments and applications. Colliding plane waves] 9) Cylindrical waves [Cylindrical waves and the asymptotic structure of 3-dimensional general relativity. Cylindrical waves and quantum gravity.  Cylindrical waves: a miscellany] 10) On the Robinson-Trautman solutions 11) The boost-rotation symmetric radiative spacetimes 12) The cosmological models [Spatially homogeneous cosmologies. Inhomogeneous models] 13) Concluding remarks 
  We present a model for an inhomogeneous and anisotropic early universe filled with a nonlinear electromagnetic field of Born-Infeld (BI) type. The effects of the BI field are compared with the linear case (Maxwell). Since the curvature invaria nts are well behaved then we conjecture that our model does not present an initial big bang singularity. The existence of the BI field modifies the curvature invariants at t=0 as well as sets bounds on the amplitude of the conformal metric function 
  We obtain a geometrical condition on vacuum, stationary, asymptotically flat spacetimes which is necessary and sufficient for the spacetime to be locally isometric to Kerr. Namely, we prove a theorem stating that an asymptotically flat, stationary, vacuum spacetime such that the so-called Killing form is an eigenvector of the self-dual Weyl tensor must be locally isometric to Kerr. Asymptotic flatness is a fundamental hypothesis of the theorem, as we demonstrate by writing down the family of metrics obtained when this requirement is dropped. This result indicates why the Kerr metric plays such an important role in general relativity. It may also be of interest in order to extend the uniqueness theorems of black holes to the non-connected and to the non-analytic case. 
  In a recent paper Carot et al. considered carefully the definition of cylindrical symmetry as a specialisation of the case of axial symmetry. One of their propositions states that if there is a second Killing vector, which together with the one generating the axial symmetry, forms the basis of a two-dimensional Lie algebra, then the two Killing vectors must commute, thus generating an Abelian group. In this comment a similar result, valid under considerably weaker assumptions, is recalled: any two-dimensional Lie transformation group which contains a one-dimensional subgroup whose orbits are circles, must be Abelian. The method used to prove this result is extended to apply to three-dimensional Lie transformation groups. It is shown that the existence of a one-dimensional subgroup with closed orbits restricts the Bianchi type of the associated Lie algebra to be I (Abelian), II, III, VII_0, VIII or IX. The relationship between the present approach and that of the original paper is discussed. 
  In this paper we study a class of inhomogeneous cosmological models which is a modified version of what is usually called the Lema\^itre-Tolman model. We assume that we have a space with 2-dimensional locally homogeneous spacelike surfaces. In addition we assume they are compact. Classically we investigate both homogeneous and inhomogeneous spacetimes which this model describe. For instance one is a quotient of the AdS$_4$ space which resembles the BTZ black hole in AdS$_3$.   Due to the complexity of the model we indicate a simpler model which can be quantized easily. This model still has the feature that it is in general inhomogeneous. How this model could describe a spontaneous creation of a universe through a tunneling event is emphasized. 
  We first illustrate on a simple example how, in existing brane cosmological models, the connection of a 'bulk' region to its mirror image creates matter on the 'brane'. Next, we present a cosmological model with no $Z_2$ symmetry which is a spherical symmetric 'shell' separating two metrically different 5-dimensional anti-de Sitter regions. We find that our model becomes Friedmannian at late times, like present brane models, but that its early time behaviour is very different: the scale factor grows from a non-zero value at the big bang singularity. We then show how the Israel matching conditions across the membrane (that is either a brane or a shell) have to be modified if more general equations than Einstein's, including a Gauss-Bonnet correction, hold in the bulk, as is likely to be the case in a low energy limit of string theory. We find that the membrane can then no longer be treated in the thin wall approximation. However its microphysics may, in some instances, be simply hidden in a renormalization of Einstein's constant, in which cases Einstein and Gauss-Bonnet membranes are identical. 
  We derive constraints on the form of the renormalized stress tensor for states on Kerr space-time based on general physical principles: symmetry, the conservation equations, the trace anomaly and regularity on (sections of) the event horizon. This is then applied to the physical vacua of interest. We introduce the concept of past and future Boulware vacua and discuss the non-existence of a state empty at both scri- and scri+. By calculating the stress tensor for the Unruh vacuum at the event horizon and at infinity, we are able to check our earlier conditions. We also discuss the difficulties of defining a state equivalent to the Hartle-Hawking vacuum and comment on the properties of two candidates for this state. 
  We propose an approach for constructing spatial slices of (3+1) spacetimes with cosmological constant but without a matter content, which yields (2+1) vacuum with $\Lambda$ solutions. The reduction mechanism from (3+1) to (2+1) gravity is supported on a criterion in which the Weyl tensor components are required to vanish together with a dimensional reduction via an appropriate foliation. By using an adequate reduction mechanism from the Pleba\'nski-Carter[A] solution in (3+1) gravity, the (2+1) BTZ solution can be obtained. 
  The solution of the Lax tensor equations in the case $L_{\alpha\beta\gamma}=-L_{\beta\alpha\gamma}$ was analyzed. The Lax tensors on the dual metrics were investigated. We classified all two dimensional metrics having the symmetric Lax tensor $L_{\alpha\beta\gamma}$. The Lax tensors of the flat space, Rindler system and its dual were found. 
  There is a single negative mode in the spectrum of small perturbations about the tunneling solutions describing a metastable vacuum decay in flat spacetime. This mode is needed for consistent description of decay processes. When gravity is included the situation is more complicated. An approach based on elimination of scalar field perturbations shows no negative mode, whereas the recent approach based on elimination of gravitational perturbations indicates presence of a negative mode. In this contribution we analyse and compare the present approaches to the negative mode problem in false vacuum decay with gravity. 
  A mathematical definition of classical causality over discrete spacetime dynamics is formulated. The approach is background free and permits a definition of causality in a precise way whenever the spacetime dynamics permits. It gives a natural meaning to the concepts of cosmic time, spacelike hypersurfaces and timelike or lightlike flows without assuming the notion of a background metric. The concepts of causal propagators and the speed of causality are discussed. In this approach the concepts of spacetime and dynamics are linked in an essential and inseparable whole, with no meaning to either on its own. 
  In this paper we investigate the Bose-Einstein condensation of massive spin-1 particles in an Einstein universe. The system is considered under relativistic conditions taking into consideration the possibility of particle-antiparticle pair production. An exact expression for the charge density is obtained, then certain approximations are employed in order to obtain the solutions in closed form. A discussion of the approximations employed in this and other work is given. The effects of finite-size and spin-curvature coupling are emphasized. 
  For the plane symmetry we have found the electro-vacuum exact solutions of the Einstein-Maxwell equations and we have shown that one of them is equivalent to the McVittie solution of a charged infinite thin plane. The analytical extension has been accomplished and the Penrose conformal diagram has been obtained as well. 
  It is shown that models of elementary particles in classical general relativity (geons) will naturally have the transformation properties of a spinor if the spacetime manifold is not time orientable. From a purely pragmatic interpretation of quantum theory this explains why spinor fields are needed to represent particles. The models are based entirely on classical general relativity and are motivated by the suggestion that the lack of a time-orientation could be the origin of quantum phenomena. 
  We demonstrate that there exists an inflationary solution on the positive tension brane in the Randall-Sundrum scenario. Inflation is driven by a slow-rolling scalar field on the brane and is achieved within the perturbative limit of the radion field. We find that inflation on the positive tension brane results in a slight increase in the separation between the two branes. However, we show that the slow-roll inflation is not possible on the negative tension brane. 
  Five classes of radiative solutions of Einstein's field equations are discussed in the light of some new developments. These are plane waves and their collisions, cylindrical waves, Robinson-Trautman and type N spacetimes, boost-rotation symmetric spacetimes and generalized Gowdy-type cosmological models 
  We present a systematic study of static solutions of the vacuum Einstein equations with negative cosmological constant which asymptotically approach the generalized Kottler (``Schwarzschild--anti-de Sitter'') solution, within (mainly) a conformal framework. We show connectedness of conformal infinity for appropriately regular such space-times. We give an explicit expression for the Hamiltonian mass of the (not necessarily static) metrics within the class considered; in the static case we show that they have a finite and well defined Hawking mass. We prove inequalities relating the mass and the horizon area of the (static) metrics considered to those of appropriate reference generalized Kottler metrics. Those inequalities yield an inequality which is opposite to the conjectured generalized Penrose inequality. They can thus be used to prove a uniqueness theorem for the generalized Kottler black holes if the generalized Penrose inequality can be established. 
  We propose a mode-sum formalism for the quantization of the scalar field based on distributional modes, which are naturally associated with a slight modification of the standard plane-wave modes. We show that this formalism leads to the standard Rindler temperature result, and that these modes can be canonically defined on any Cauchy surface. 
  For space-times with two spacelike isometries, we present infinite hierarchies of exact solutions of the Einstein and Einstein--Maxwell equations as represented by their Ernst potentials. This hierarchy contains three arbitrary rational functions of an auxiliary complex parameter. They are constructed using the so called `monodromy transform' approach and our new method for the solution of the linear singular integral equation form of the reduced Einstein equations. The solutions presented, which describe inhomogeneous cosmological models or gravitational and electromagnetic waves and their interactions, include a number of important known solutions as particular cases. 
  Using simplicial minisuperspace techniques we obtain a wormhole configuration, capable of modelling both a throat between two equal Euclidean and Lorentzian universes. Computing the wavefunction associated to such configuration we conclude that for Planck-size Euclidean wormholes there is a prediction of a finite non-vanishing radius of their throat, even in the absence of matter. The wavefunction associated to their Lorentzian counterparts predicts the growth of the wormhole's throat as the two Lorentzian universes expand. 
  In this work numerical methods for solving Einstein's equations are developed and applied to the study of inhomogeneous cosmological models.   A two-dimensional computer code is described which implements two advanced numerical methods: LeVeque's multi-dimensional high-resolution integration scheme which allows accurate evolution of solutions containing discontinuities or steep gradients, and an adaptive mesh refinement (AMR) algorithm which enables the local resolution of a simulation to vary dynamically in response to the behaviour of the evolved solution.   A family of hyperbolic formulations of the Einstein equations is derived by generalization of an evolution system proposed by Frittelli and Reula, and numerical solutions produced using these formulations are compared to solutions produced using alternative reductions of the evolutions equations. Properties of the harmonic time slicing condition are also investigated, and analytic and numerical results concerning the formation of coordinate singularities are presented.   Numerical simulations are performed of planar cosmologies, described using Gowdy's reduction of the Einstein equations, and U(1)-symmetric cosmologies, described using Moncrief's reduction of the Einstein equations, with the spacetimes in both cases being vacuum. Numerical studies follow up on the work of Berger, Moncrief and co-workers, with attention being focused on the small-scale features that develop in the models and the behaviour of linear and nonlinear gravitational waves. 
  We derive the `exact' Newtonian limit of general relativity with a positive cosmological constant $\Lambda$. We point out that in contrast to the case with $\Lambda = 0 $, the presence of a positive $\Lambda$ in Einsteins's equations enforces, via the condition $| \Phi | \ll 1$, on the potential $\Phi$, a range ${\cal R}_{max}(\Lambda) \gg r \gg {\cal R}_{min} (\Lambda)$, within which the Newtonian limit is valid. It also leads to the existence of a maximum mass, ${\cal M}_{max}(\Lambda)$. As a consequence we cannot put the boundary condition for the solution of the Poisson equation at infinity. A boundary condition suitably chosen now at a finite range will then get reflected in the solution of $\Phi$ provided the mass distribution is not spherically symmetric. 
  A homothetic, static, spherically symmetric solution to the massless Einstein- Klein-Gordon equations is described. There is a curvature singularity which is central, null, bifurcate and marginally trapped. The space-time is therefore extreme in the sense of lying at the threshold between black holes and naked singularities, just avoiding both. A linear perturbation analysis reveals two types of dominant mode. One breaks the continuous self-similarity by periodic terms reminiscent of discrete self-similarity, with echoing period within a few percent of the value observed numerically in near-critical gravitational collapse. The other dominant mode explicitly produces a black hole, white hole, eternally naked singularity or regular dispersal, the latter indicating that the background is critical. The black hole is not static but has constant area, the corresponding mass being linear in the perturbation amplitudes, explicitly determining a unit critical exponent. It is argued that a central null singularity may be a feature of critical gravitational collapse. 
  A new closed-form inflationary solution is given for a hyperbolic interaction potential. The method used to arrive at this solution is outlined as it appears possible to generate additional sets of equations which satisfy the model. In addition a new form of decaying cosmological constant is presented. 
  Four classical laws of black hole thermodynamics are extended from exterior (event) horizon to interior (Cauchy) horizon. Especially, the first law of classical thermodynamics for Kerr-Newman black hole (KNBH) is generalized to those in quantum form. Then five quantum conservation laws on the KNBH evaporation effect are derived in virtue of thermodynamical equilibrium conditions. As a by-product, Bekenstein-Hawking's relation $ S=A/4 $ is exactly recovered. 
  We introduce the (2+1)-spacetimes with compact space of genus g and with r gravitating particles which arise by ``Minkowskian suspensions of flat or hyperbolic cone surfaces'', by ``distinguished deformations'' of hyperbolic suspensions and by ``patchworking'' of suspensions. Similarly to the matter-free case, these spacetimes have nice properties with respect to the canonical Cosmological Time Function. When the values of the masses are sufficiently large and the cone points are suitably spaced, the distinguished deformations of hyperbolic suspensions determine a relevant open subset of the full parameter space; this open subset is homeomorphic to the product of an Euclidean space of dimension 6g-6+2r with an open subset of the Teichm\"uller Space of Riemann surfaces of genus g with r punctures. By patchworking of suspensions one can produce examples of spacetimes which are not distinguished deformations of any hyperbolic suspensions, although they have the same masses; in fact, we will guess that they belong to different connected components of the parameter space. 
  A definition of gravitational energy is proposed for any theory described by a diffeomorphism-invariant Lagrangian. The mathematical structure is a Noether- current construction of Wald involving the boundary term in the action, but here it is argued that the physical interpretation of current conservation is conservation of energy. This leads to a quasi-local energy defined for compact spatial surfaces. The energy also depends on a vector generating a flow of time. Angular momentum may be similarly defined, depending on a choice of axial vector. For Einstein gravity: for the usual vector generating asymptotic time translations, the energy is the Bondi energy; for a stationary Killing vector, the energy is the Komar energy; in spherical symmetry, for the Kodama vector, the energy is the Misner-Sharp energy. In general, the lack of a preferred time indicates the lack of a preferred energy, reminiscent of the energy-time duality of quantum theory. 
  We propose a new line of attack to create a finite quantum theory which includes general relativity and (perhaps) the standard model in its low energy limit. The theory would emerge from the categorical approach. A structure is observed on the category of unitary representations of the lorentz group which we call hypergravity. This, combined with a study of the relationship between Feynman diagrams and words in tensor categories, leads to the proposal. 
  Fuchsian equations provide a way of constructing large classes of spacetimes whose singularities can be described in detail. In some of the applications of this technique only the analytic case could be handled up to now. This paper develops a method of removing the undesirable hypothesis of analyticity. This is applied to the specific case of the Gowdy spacetimes in order to show that analogues of the results known in the analytic case hold in the smooth case. As far as possible the likely strengths and weaknesses of the method as applied to more general problems are displayed. 
  The problem of constructing a quantum theory of gravity has been tackled with very different strategies, most of which relying on the interplay between ideas from physics and from advanced mathematics. On the mathematical side, a central role is played by combinatorial topology, often used to recover the space-time manifold from the other structures involved. An extremely attractive possibility is that of encoding all possible space-times as specific Feynman diagrams of a suitable field theory. In this work we analyze how exactly one can associate combinatorial 4-manifolds to the Feynman diagrams of certain tensor theories. 
  We study the effect of non-vanishing surface terms at spatial infinity on the dynamics of a scalar field in an open FLRW spacetime. Starting from the path-integral formulation of quantum field theory we argue that classical physics is described by field configurations which extremize the action functional in the space of field configurations for which the variation of the action is well defined. Since these field configurations are not required to vanish outside a bounded domain, there is generally a non-vanishing contribution of a surface term to the variation of the action. We then investigate whether this surface term has an effect on the dynamics of the action-extremizing field configurations. This question appears to be surprisingly nontrivial in the case of the open FLRW geometry, since surface terms tend to grow as fast as volume terms in the infinite volume limit. We find that surface terms can be important for the dynamics of the field at a classical and at a quantum level, when there are supercurvature perturbations. 
  On the mass neutrino phase calculations along both the particle geodesic line and the photon null line, there exists a double counting error--factor of 2 when comparing the geodesic phase with the null phase. For the mass neutrino propagation in the flat spacetime, we study the neutrino interference phase calculation in the Minkowski diagram and find that the double counting effect originates from despising the velocity difference between two mass neutrinos. Moreover, we compare the phase calculations among the same energy description, the same momentum description and same velocity description by means of the Minkowski diagram, and obtain the practical equivalence of these three descriptions. Further, in the curved spacetime, we also prove the existence of the double counting of the geodesic phase to the null phase. 
  The interference phase of the high energy mass neutrinos and the low energy thermal neutrons in a gravitational field are studied. For the mass neutrinos, we obtain that the phase calculated along the null is equivalent to the half phase along the geodesic in the high energy limit, which means that the correct relative phase of the mass neutrinos is either the null phase or the half geodesic phase.  Further we point out the importance of the energy condition in calculating the mass neutrino interference phase. Moreover, we apply the covariant phase to the calculation of the thermal neutron interference phase, and obtain the consistent result with that exploited in COW experiment. 
  We consider the Landau-Khalatnikov two-fluid hydrodynamics of superfluid liquid as an effective theory, which provides a self-consistent analog of Einstein equations for gravity and matter. 
  We introduce a new method to construct solutions to the constraint equations of general relativity describing binary black holes in quasicircular orbit. Black hole pairs with arbitrary momenta can be constructed with a simple method recently suggested by Brandt and Bruegmann, and quasicircular orbits can then be found by locating a minimum in the binding energy along sequences of constant horizon area. This approach produces binary black holes in a "three-sheeted" manifold structure, as opposed to the "two-sheeted" structure in the conformal-imaging approach adopted earlier by Cook. We focus on locating the innermost stable circular orbit and compare with earlier calculations. Our results confirm those of Cook and imply that the underlying manifold structure has a very small effect on the location of the innermost stable circular orbit. 
  We calculate the Noether currents and charges for Einstein-Maxwell theory using a version of the Wald approach. In spherical symmetry, the choice of time can be taken as the Kodama vector. For the static case, the resulting combined Einstein-Maxwell charge is just the mass of the black hole. Using either a classically defined entropy or the Iyer-Wald selection rules, the entropy is found to be just a quarter of the area of the trapping horizon. We propose identifying the combined Noether charge as an energy associated with the Kodama time. For the extremal black hole case, we discuss the problem of Wald's rescaling of the surface gravity to define the entropy. 
  A classical model of gravity theory with several dilatonic scalar fields and differential forms admitting an interpretation in terms of intersecting p-branes is studied in (pseudo)-Riemannian space-time $M =R_+\times S^{d_0}\times R_t\times M_2^{d_2}...\times M_n^{d_n}$ of dimension D. The equations of motion of the model are reduced to the Euler-Lagrange equations for the so-called pseudo-Euclidean Toda-like system. We suppose that the characteristic vectors related to the configuration of p-branes and their couplings to the dilatonic scalar fields may be interpreted as the root vectors of a Lie algebra of the types $A_r, B_r, C_r$. In this case the model is reducible to one of the open Toda chain's algebraic generalization and is completely integrable by the known methods. The corresponding general solutions are presented in explicit form. The particular exact solution describing a class of nonextremal black holes is obtained and analyzed. 
  The response of a massive body to gravitational waves is described on the microscopic level. The results shed a new light on the commonly used oscillator model. It is shown that apart from the non-resonant tidal motion the energy transfer from a gravitational wave to an electromagnetically coupled body is in general restricted to the surface, whereas gravitational coupling gives rise to bulk excitation of quadrupole modes, but several orders of magnitude smaller. These results do not contradict standard theory, rather present a different viewpoint. A microscopic detector making use of the effect is suggested. 
  Gauge field theories may quite generally be defined as describing the coupling of a matter-field to an interaction-field, and they are suitably represented in the mathematical framework of fiber bundles. Their underlying principle is the so-called gauge principle, which is based on the idea of deriving the coupling structure of the fields by satisfying a postulate of local gauge covariance. The gauge principle is generally considered to be sufficient to define the full structure of gauge-field theories. This paper contains a critique of this usual point of view: firstly, by emphazising an gauge theoretic conventionalism which crucially restricts the conceptual role of the gauge principle and, secondly, by introducing a new generalized equivalence principle - the identity of inertial and field charge (as generalizations of inertial and gravitational mass) - in order to give a conceptual justification for combining the equations of motion of the matter-fields and the field equations of the interaction-fields. 
  We show in this comment that in an anisotropic Bianchi type I model of the Kasner form, it is not possible to describe the growth of entropy, if we want to keep the thermodynamics together with the dominant energy conditions. This consequence disagrees with the results obtained by Brevik and Pettersen [Phys. Rev. D 56, 3322 (1997)]. 
  We find an expression for the generalized gravitational entropy of Hawking in terms of Noether charge. As an example, the entropy of the Taub-Bolt spacetime is calculated. 
  A signature changing spacetime is one where an initially Riemannian manifold with Euclidean signature evolves into the Lorentzian universe we see today. This concept is motivated by problems in causality implied by the isotropy and homogeneity of the universe. As initially time and space are indistinguishable in signature change, these problems are removed. There has been some dispute as to the nature of the junction conditions across the signature change, and in particular, whether or not the metric is continuous there. We determine to what extent the Colombeau algebra of new generalised functions resolves this dispute by analysing both types of signature change within its framework. A covariant formulation of the Colombeau algebra is used, in which the usual properties of the new generalised functions are extended. We find that the Colombeau algebra is insufficient to preclude either continuous or discontinuous signature change, and is also unable to settle the dispute over the nature of the junction conditions. 
  Axial oscillations relevant to the r-mode instability are studied with slow rotation formalism in general relativity. The approximate equation governing the oscillations is derived with second-order rotational corrections. The equation contains an effective 'viscosity-like' term, which originates from coupling to the polar g-mode displacements. The term plays a crucial role on the resonance point, where the disturbance on the rotating stars satisfies a certain condition at the lowest order equation. The effect is significant for newly born hot neutron stars, which are expected to be subject to the gravitational radiation driven instability of the r-mode. 
  I consider the COBE data coarse-grained field that characterize the now observable universe for a model of warm inflation which takes into account the thermal coupled fluctuations of the scalar field with the thermal bath. The power spectrum for both, matter and metric fluctuations are analyzed. I find that the amplitude for the fluctuations of the metric when the horizon entry, should be very small for the expected values of temperature. 
  Closed, spatially homogeneous cosmological models with a perfect fluid and a scalar field with exponential potential are investigated, using dynamical systems methods. First, we consider the closed Friedmann-Robertson-Walker models, discussing the global dynamics in detail. Next, we investigate Kantowski-Sachs models, for which the future and past attractors are determined. The global asymptotic behaviour of both the Friedmann-Robertson-Walker and the Kantowski-Sachs models is that they either expand from an initial singularity, reach a maximum expansion and thereafter recollapse to a final singularity (for all values of the potential parameter kappa), or else they expand forever towards a flat power-law inflationary solution (when kappa^2<2). As an illustration of the intermediate dynamical behaviour of the Kantowski-Sachs models, we examine the cases of no barotropic fluid, and of a massless scalar field in detail. We also briefly discuss Bianchi type IX models. 
  The covariant phase technique is used to compute the constraint algebra of the stationary axisymmetric charged black hole. A standard Virasoro subalgebra with corresponding central charge is constructed at a Killing horizon with Carlip's boundary conditions. For the Kerr-Newman black hole and the Kerr-Newman-AdS black hole, the density of states determined by conformal fields theory methods yields the statistical entropy which agrees with the Bekenstein-Hawking entropy. 
  We report the development of the first apparent horizon locator capable of finding multiple apparent horizons in a ``generic'' numerical black hole spacetime. We use a level-flow method which, starting from a single arbitrary initial trial surface, can undergo topology changes as it flows towards disjoint apparent horizons if they are present. The level flow method has two advantages: 1) The solution is independent of changes in the initial guess and 2) The solution can have multiple components. We illustrate our method of locating apparent horizons by tracking horizon components in a short Kerr-Schild binary black hole grazing collision. 
  The effects of small extra dimensions upon quantum fluctuations of the lightcone are examined. We argue that compactified extra dimensions modify the quantum fluctuations of gravitational field so as to induce lightcone fluctuations. This phenomenon can be viewed as being related to the Casimir effect. The observable manifestation of the lightcone fluctuations is broadening of spectral lines from distant sources. In this paper, we further develop the formalism used to describe the lightcone fluctuations, and then perform explicit calculations for several models with flat extra dimensions. In the case of one extra compactified dimension, we find a large effect which places severe constraints on such models. When there is more than one compactified dimension, the effect is much weaker and does not place a meaningful constraint. We also discuss some brane worlds scenarios, in which gravitons satisfy Dirichlet or Neumann boundary conditions on parallel four-dimensional branes, separated by one or more flat extra dimensions. 
  Attaining the limit of sub-microarcsecond optical resolution will completely revolutionize fundamental astrometry by merging it with relativistic gravitational physics. Beyond the sub-microarcsecond threshold, one will meet in the sky a new population of physical phenomena caused by primordial gravitational waves from early universe and/or different localized astronomical sources, space-time topological defects, moving gravitational lenses, time variability of gravitational fields of the solar system and binary stars, and many others. Adequate physical interpretation of these yet undetectable sub-microarcsecond phenomena can not be achieved on the ground of the "standard" post-Newtonian approach (PNA), which is valid only in the near-zone of astronomical objects having a time-dependent gravitational field. We describe a new, post-Minkowskian relativistic approach for modeling astrometric observations having sub-microarcsecond precision and briefly discuss the light-propagation effects caused by gravitational waves and other phenomena related to time-dependent gravitational fields. The domain of applicability of the PNA in relativistic space astrometry is explicitly outlined. 
  Causal differencing has shown to be one of the promising and successful approaches towards excising curvature singularities from numerical simulations of black hole spacetimes. So far it has only been actively implemented in the ADM and Einstein-Bianchi 3+1 formulations of the Einstein equations. Recently, an approach closely related to the ADM one, commonly referred to as as ``conformal ADM'' (CADM) has shown excellent results when modeling waves on flat spacetimes and black hole spacetimes where singularity avoiding slices are used to deal with the singularity. In these cases, the use of CADM has yielded longer evolutions and better outer boundary dependence than those obtained with the ADM one. If this success translates to the case where excision is implemented, then the CADM formulation will likely be a prime candidate for modeling generic black hole spacetimes. In the present work we investigate the applicability of causal differencing to CADM, presenting the equations in a convenient way for such a goal and compare its application with the ADM approach in 1D. 
  Quantization of general relativity in metric variables using ``precanonical'' quantization based on the De Donder-Weyl covariant Hamiltonian formulation is outlined. Elements of classical geometry needed to formulate the (Dirac-like) wave equation emerge from a self-consistency with the underlying quantum dynamics of the metric. In this sense the formulation can be viewed as independent from an arbitrarily fixed background. 
  Applying the Vaidya metrics in the model of Expansive Nondecelerative Universe (ENU) leads to compatibility of the ENU model both with the classic Newton gravitational theory and the general theory of relativity in weak fields 
  We consider the spacetimes corresponding to static Global Monopoles with interior boundaries corresponding to a Black Hole Horizon and analyze the behavior of the appropriate ADM mass as a function of the horizon radius r_H. We find that for small enough r_H, this mass is negative as in the case of the regular global monopoles, but that for large enough r_H the mass becomes positive encountering an intermediate value for which we have a Black Hole with zero ADM mass. 
  We consider the problem of late-time isotropization in spatially homogeneous but anisotropic cosmological models when the source of the gravitational field consists of two non-interacting perfect fluids -- one tilted and one non-tilted. In particular, we study irrotational Bianchi type V models. By introducing appropriate dimensionless variables, a full global understanding of the state space of the gravitational field equations becomes possible. The issue of isotropization can then be addressed in a simple fashion. We also discuss implications for the cosmic ``no-hair'' theorem for Bianchi models when part of the source is a tilted fluid. 
  We construct a weave state which approximates a degenerate 3-metric of rank 2 at large scales. It turns out that a non-degenerate metric region can be evolved from this degenerate metric by the classical Ashtekar equations, hence the degeneracy of 3-metrics is not preserved by the evolution of Ashtekar's equations. As the s-knot state corresponding to this weave is shown to solve all the quantum constraints in loop quantum gravity, a physical state in canonical quantum gravity is related to the familiar classical geometry. 
  We construct static and spherically symmetric particle-like and black hole solutions with magnetic or electric charge in the Einstein-Born-Infeld-dilaton system, which is a generalization of the Einstein-Maxwell-dilaton (EMD) system and of the Einstein-Born-Infeld (EBI) system. They have remarkable properties which are not seen for the corresponding solutions in the EBI and the EMD system. 
  We discuss the physical equivalence between the Einstein and Jordan frames in Brans-Dicke theory. The inequivalence of conformal transformed theories is clarified with the help of an old equivalence theorem of Chrisholm's. 
  The inverse scattering method is applied to the investigation of the equilibrium configuration of black holes. A study of the boundary problem corresponding to this configuration shows that any axially symmetric, stationary solution of the Einstein equations with disconnected event horizon must belong to the class of Belinskii-Zakharov solutions. Relationships between the angular momenta and angular velocities of black holes are derived. 
  A set of exact quasi-local conservation equations is derived from the Einstein's equations using the first-order Kaluza-Klein formalism of general relativity in the (2,2)-splitting of 4-dimensional spacetime. These equations are interpreted as quasi-local energy, momentum, and angular momentum conservation equations. In the asymptotic region of asymptotically flat spacetimes, it is shown that the quasi-local energy and energy-flux integral reduce to the Bondi energy and energy-flux, respectively. In spherically symmetric spacetimes, the quasi-local energy becomes the Misner-Sharp energy. Moreover, on the event horizon of a general dynamical black hole, the quasi-local energy conservation equation coincides with the conservation equation studied by Thorne {\it et al}. We discuss the remaining quasi-local conservation equations briefly. 
  Cusps of cosmic strings emit strong beams of high-frequency gravitational waves (GW). As a consequence of these beams, the stochastic ensemble of gravitational waves generated by a cosmological network of oscillating loops is strongly non Gaussian, and includes occasional sharp bursts that stand above the rms GW background. These bursts might be detectable by the planned GW detectors LIGO/VIRGO and LISA for string tensions as small as $G \mu \sim 10^{-13}$. The GW bursts discussed here might be accompanied by Gamma Ray Bursts. 
  The integrals of motion for a cylindrically symmetric stationary vortex are obtained in a covariant description of a mixture of interacting superconductors, superfluids and normal fluids. The relevant integrated stress-energy coefficients for the vortex with respect to a vortex-free reference state are calculated in the approximation of a ``stiff'', i.e. least compressible, relativistic equation of state for the fluid mixture. As an illustration of the foregoing general results, we discuss their application to some of the well known examples of ``real'' superfluid and superconducting systems that are contained as special cases. These include Landau's two-fluid model, uncharged binary superfluid mixtures, rotating conventional superconductors and the superfluid neutron-proton-electron plasma in the outer core of neutron stars. 
  The relationship between modern philosophy and physics is discussed. It is shown that the latter develops some need for a modernized metaphysics which shows up as an ultima philosophia of considerable heuristic value, rather than as the prima philosophia in the Aristotelian sense as it had been intended, in the first place. It is shown then, that it is the philosophy of Spinoza in fact, that can still serve as a paradigm for such an approach. In particular, Spinoza's concept of infinite substance is compared with the philosophical implications of the foundational aspects of modern physical theory. Various connotations of sub-stance are discussed within pre-geometric theories, especially with a view to the role of spin networks within quantum gravity. It is found to be useful to intro-duce a separation into physics then, so as to differ between foundational and empirical theories, respectively. This leads to a straightforward connection bet-ween foundational theories and speculative philosophy on the one hand, and between empirical theories and sceptical philosophy on the other. This might help in the end, to clarify some recent problems, such as the absence of time and causality at a fundamental level. It is implied that recent results relating to topos theory might open the way towards eventually deriving logic from physics, and also towards a possible transition from logic to hermeneutic. 
  A brane universe derived from the Randall-Sundrum models is considered in which an additional Misner-like periodicity is introduced in the extra direction. This model solves the ambiguity in the choice of the brane world by identifying the branes with opposite tensions, in such a way that if one enters the brane with positive tension, one finds oneself emerging from the brane with negative tension, without having experienced any tension. We show that the cosmological evolution resulting from this model matches that of the standard Friedmann scenario, at least in the radiation dominated era, and that there exist closed timelike curves only in the bulk, but not in the branes which are chronologically protected from causality violations by quantum-mechanically stable chronology horizons. 
  It is known that the graviton two-point function for the de Sitter invariant "Euclidean" vacuum in a physical gauge grows logarithmically with distance in spatially-flat de Sitter spacetime. We show that this logarithmic behaviour is a gauge artifact by explicitly demonstrating that the same behaviour can be reproduced by a pure-gauge two-point function. 
  By extending the exterior Schwarzschild spacetime in two opposite directions with the Kruskal method, we get an extension which has the same T-X spacetime diagram as has the conventional Kruskal extension, while allowing its regions I and IV to correspond to different directions of the original spacetime. We further extend the exterior Schwarzschild spacetime in all directions and get a 4-dimensional form of the Kruskal extension. The new form of extension includes the conventional one as a part of itself. From the point of view of the 4-dimensional form, region IV of the conventional extension does not belong to another universe but is a portion of the same exterior Schwarzschild spacetime that contains region I. The two regions are causally related: particles can move from one to the other. 
  The Proca wave equation describes a classical massive spin 1 particle. We analyze the gravitational interaction of this vector field. In particular, the spherically symmetric solutions of the Einstein-Proca coupled system are obtained numerically. Although at infinity the metric field approaches the usual Schwarzschild (Reissner-Nordstr\"om) limit, we demonstrate the absence of black hole type configurations. 
  A solution to the Einstein field equations that represents a rigidly rotating dust accompanied by a thin matter shell of the same type is found. 
  A general theory of frames of reference proposed in a preceding publication is considered here in the framework of the post-Newtonian approximation, assuming that the frame of reference is centered on a time-like geodesic. The problem of taking into account the rotation of the frame of reference, which is usually ignored or incorrectly oversimplified, is here discussed in detail and solved. 
  The excitation of neutron stars is expected to be an important source of gravitational radiation. Of fundamental importance is then to investigate mechanisms that trigger oscillations in neutron stars in order to characterize the emitted radiation. We present results from a numerical study of the even-parity gravitational radiation generated from a particle orbiting a neutron star. We focus our investigation on those conditions on the orbital parameters that favor the excitation of $w$-modes. We find that, for astrophysically realistic conditions, there is practically no $w$-mode contribution to the emitted radiation. Only for particles with ultra-relativistic orbital speeds $\ge 0.9c$, the $w$-mode does significantly contribute to the total emitted gravitational energy. We also stress the importance of setting consistent initial configurations. 
  In order to perform accurate and stable long-time numerical integration of the Einstein equation, several hyperbolic systems have been proposed. We here present numerical comparisons between weakly hyperbolic, strongly hyperbolic, and symmetric hyperbolic systems based on Ashtekar's connection variables. The primary advantage for using this connection formulation in this experiment is that we can keep using the same dynamical variables for all levels of hyperbolicity. Our numerical code demonstrates gravitational wave propagation in plane symmetric spacetimes, and we compare the accuracy of the simulation by monitoring the violation of the constraints. By comparing with results obtained from the weakly hyperbolic system, we observe the strongly and symmetric hyperbolic system show better numerical performance (yield less constraint violation), but not so much difference between the latter two. Rather, we find that the symmetric hyperbolic system is not always the best in numerical performances.   This study is the premier to present full numerical simulations using Ashtekar's variables. We also describe our procedures in detail. 
  We consider static spherically symmetric solutions of Einstein's equations coupled to an SU(2) Yang Mills field that are smooth at the center of spherical symmetry. We prove that with small cosmological constant there exist solutions that possess a coordinate singularity at some r that is not maximum. The singularity can be removed with a Kruskal-like change of coordinates. 
  The Einstein equations with small positive cosmological constant coupled to an SU(2) Yang Mills field admits solutions that possess a coordinate singularity at a noncritical radius. Here, we prove that these solutions are otherwise globally smooth and that they asymptotically approach Schwarzschild deSitter space with a vanishing Yang Mills field. 
  We categorize the global structure of spherically symmetric static solutions of Einstein SU(2) Yang Mills equations with positive cosmological constant that are smooth at the center of spherical symmetry. 
  Charged black hole solutions with pion hair are discussed. These can be used to study monopole black hole catalysis of proton decay. There also exist multi- black hole skyrmion solutions with BPS monopole behavior. 
  A wormhole with a quantum throat on the basis of an approximate model of the spacetime foam is presented. An effective spinor field is introduced for the description of the spacetime foam. The consequences of such model of the wormhole is preventing a "naked'' singularity in the Reissner-Nordstr\"om solution with $|e|/m > 1$. 
  We study a novel set of gravitational field configurations, called "dipolar zero modes", which give an exactly null contribution to the Einstein action and are thus candidates to become large fluctuations in the quantized theory. They are generated by static unphysical sources satisfying (up to terms of order G^2) the simple condition Int d^3x T_00(x) = 0. We give two explicit examples of virtual sources: (i) a "mass dipole" consisting of two separated mass distributions with different signs; (ii) two concentric "+/- shells". The field fluctuations can be large even at macroscopic scale. There are some, for instance, which last ~ 1 s or more and correspond to the field generated by a virtual source with size ~ 1 cm and mass ~ 10^6 g. This appears paradoxical, for several reasons, both theoretical and phenomenological. We also give an estimate of possible suppression effects following the addition to the pure Einstein action of cosmological or R^2 terms. 
  We analyse the quantization procedure of the spinor field in the Rindler spacetime, showing the boundary conditions that should be imposed to the field, in order to have a well posed theory. Because of these boundary conditions we argue that this construction and the usual one in Minkowski spacetime are qualitatively different and can not be compared and consequently the conventional interpretation of the Unruh effect, that is the thermal nature of the Minkowski vacuum state from the point of view of an accelerated observer, is questionable. We also analyse in detail the Unruh quantization scheme and we show that it is not valid in the whole Minkowski space but only in the double Rindler wedge, and it cannot be used as a basis for a quantum theoretical proof of the Unruh effect. 
  We explore the possibility to locate a brane in black hole bulks. We study explicitly the cases of BHTZ and Schwarzschild-anti de Sitter (AdS) black holes. Our result is that in these cases branes cannot be supported by brane tension alone and it is necessary to introduce other forms of matter on the brane. We find classes of perfect fluid solutions obeying to peculiar state equations. For the case of BHTZ bulk geometry the state equation takes exactly the form of a ``Chaplygin gas'', which is relevant in the brane context. In the Schwarzschild-AdS case we find new state equations which reduce to the Chaplygin form when the brane is located near the horizon. 
  An absolute lower bound on the number of templates needed to keep the fitting factor above a prescribed minimal value $\Gamma$ in correlator bank detection of (newtonian) gravitational wave chirps from unknown inspiraling compact binary sources is derived, resorting to the theory of quasi-bandlimited functions in the $L^\infty$ norm. An explicit nearly-minimum redundant cardinal-interpolation formula for the (reduced, noncoherent) correlator is introduced. Its computational burden and statistical properties are compared to those of the plain lattice of (reduced, noncoherent) correlators, for the same $\Gamma$. Extension to post-newtonian models is outlined. 
  A free scalar field minimally coupled to gravity model is quantized and the Wheeler-DeWitt equation in minisuperspace is solved analytically, exhibiting positive and negative frequency modes. The analysis is performed for positive, negative and zero values of the curvature of the spatial section. Gaussian superpositions of the modes are constructed, and the quantum bohmian trajectories are determined in the framework of the Bohm-de Broglie interpretation of quantum cosmology. Oscillating universes appear in all cases, but with a characteristic scale of the order of the Planck scale. Bouncing regular solutions emerge for the flat curvature case. They contract classically from infinity until a minimum size, where quantum effects become important acting as repulsive forces avoiding the singularity and creating an inflationary phase, expanding afterwards to an infinite size, approaching the classical expansion as long as the scale factor increases. These are non-singular solutions which are viable models to describe the early Universe. 
  Formalising the logical dependence of physical quantities on material referents of scale, I show that both Hubble's law and the cosmological constant are in fact exactly replicable by a spatial contraction of referents locally on earth, and that the Pioneer anomaly is irrefutable indication that this is the case. The formalism literally embodies Feynman's ``hot-plate'' model, predictably yielding a logical derivation of the relativity postulates, and importantly, suffices to demonstrate the inherent logical consistency of general relativity and quantum mechanics, whose foundations I have shown separately to be fundamentally computational. I further show that the spatial contraction of our referents also accounts for considerable planetary and geological data inexplicable in the standard model, and predicts that the Hubble flow appears differently or is altogether absent from platforms in deep space, depending on the local physics. 
  We propose a novel, potentially useful generating technique for constructing exact solutions of inflationary scalar field cosmologies with non-trivial potentials. The generating scheme uses the so-called superpotential and is inspired by recent studies of similar equations in supergravity. Some exact solutions are derived, and the physical meaning of the superpotential in these models is clarified. 
  We construct static and spherically symmetric black hole solutions in the Einstein-Euler-Heisenberg (EEH) system which is considered as an effective action of a superstring theory. We considered electrically charged, magnetically charged and dyon solutions. We can solve analytically for the magnetically charged case. We find that they have some remarkable properties about causality and black hole thermodynamics depending on the coupling constant of the EH theory $a$ and $b$, though they have central singularity as in the Schwarzschild black hole. 
  Many recent attempts to calculate black hole entropy from first principles rely on conformal field theory techniques. By examining the logarithmic corrections to the Cardy formula, I compute the first-order quantum correction to the Bekenstein-Hawking entropy in several models, including those based on asymptotic symmetries, horizon symmetries, and certain string theories. Despite very different physical assumptions, these models all give a correction proportional to the logarithm of the horizon size, and agree qualitatively with recent results from ``quantum geometry'' in 3+1 dimensions. There are some indications that even the coefficient of the correction may be universal, up to differences that depend on the treatment of angular momentum and conserved charges. 
  The Fresnel equation governing the propagation of electromagnetic waves for the most general linear constitutive law is derived. The wave normals are found to lie, in general, on a fourth order surface. When the constitutive coefficients satisfy the so-called reciprocity or closure relation, one can define a duality operator on the space of the two-forms. We prove that the closure relation is a sufficient condition for the reduction of the fourth order surface to the familiar second order light cone structure. We finally study whether this condition is also necessary. 
  We study classically and quantum mechanically the Euclidean geometries compatible with an open inflationary universe of a Lorentzian geometry. The Lorentzian geometry of the open universe with an ordinary matter state matches either an open or a closed Euclidean geometry at the cosmological singularity. With an exotic matter state it matches only the open Euclidean geometry and describes a genuine instanton regular at the boundary of a finite radius. The wave functions are found that describe the quantum creation of the open inflationary universe. 
  In a recent paper [1], it has been shown that negative norm states are indispensable for a fully covariant quantization of the minimally coupled scalar field in de Sitter space. Their presence, while leaving unchanged the physical content of the theory, offers an automatic and covariant renormalization of the vacuum energy divergence. This paper is a completion of our previous work. An explicit construction of the covariant two-point function of the ``massless'' minimally coupled scalar field in de Sitter space is given, which is free of any infrared divergence. The associated Schwinger commutator function and retarded Green's function are calculated in a fully gauge invariant way, which also means coordinate independent. 
  This paper develops a theory of thin shells within the context of the Einstein-Cartan theory by extending the known formalism of general relativity. In order to perform such an extension, we require the general non symmetric stress-energy tensor to be conserved leading, as Cartan pointed out himself, to a strong constraint relating curvature and torsion of spacetime. When we restrict ourselves to the class of space-times satisfying this constraint, we are able to properly describe thin shells and derive the general expression of surface stress-energy tensor both in its four-dimensional and in its three-dimensional intrinsic form. We finally derive a general family of static solutions of the Einstein-Cartan theory exhibiting a natural family of null hypersurfaces and use it to apply our formalism to the construction of a null shell of matter. 
  We discuss the gravitationally interacting system of a thick domain wall and a black hole. We numerically solve the scalar field equation in the Schwarzschild spacetime and show that there exist scalar field configurations representing thick domain walls intersecting the black hole. 
  Environment interaction may induce stochastic semiclassical dynamics in open quantum systems. In the gravitational context, stress-energy fluctuations of quantum matter fields give rise to a stochastic behaviour in the spacetime geometry. Einstein-Langevin equation is a suitable tool to take these effects into account when addressing the back-reaction problem in semiclassical gravity. We analyze within this framework the generation of gravitational fluctuations during inflation, which are of great interest for large-scale structure formation in cosmology. 
  The cosmic microwave background anisotropies produced by active seeds, such as topological defects, have been computed recently for a variety of models by a number of authors. In this paper we show how the generic features of the anisotropies caused by active, incoherent, seeds (that is the absence of acoustic peaks at small scales) can be obtained semi-analytically, without entering into the model dependent details of their formation, structure and evolution. 
  A new method of construction of integral varieties of Einstein equations in three dimensional (3D) and 4D gravity is presented whereby, under corresponding redefinition of physical values with respect to anholonomic frames of reference with associated nonlinear connections, the structure of gravity field equations is substantially simplified. It is shown that there are 4D solutions of Einstein equations which are constructed as nonlinear superpositions of soliton solutions of 2D (pseudo) Euclidean sine-Gordon equations (or of Lorentzian black holes in Jackiw-Teitelboim dilaton gravity). The Belinski-Zakharov-Meison solitons for vacuum gravitational field equations are generalized to various cases of two and three coordinate dependencies, local anisotropy and matter sources. The general framework of this study is based on investigation of anholonomic soliton-dilaton black hole structures in general relativity. We prove that there are possible static and dynamical black hole, black torus and disk/cylinder like solutions (of non-vacuum gravitational field equations) with horizons being parametrized by hypersurface equations of rotation ellipsoid, torus, cylinder and another type configurations. Solutions describing locally anisotropic variants of the Schwarzschild-- Kerr (black hole), Weyl (cylindrical symmetry) and Neugebauer--Meinel (disk) solutions with anisotropic variable masses, distributions of matter and interaction constants are shown to be contained in Einstein's gravity. It is demonstrated in which manner locally anisotropic multi-soliton-- dilaton-black hole type solutions can be generated. 
  It was shown by Hiscock that the energy-momentum tensor commonly used to model local cosmic strings in linearized Einstein gravity can be extended and used in the full theory, obtaining a metric in the exterior of the source with the same deficit angle. Here we show that this tensor is an exception within a family for which a static solution does not exist in full Einstein nor in Brans-Dicke gravity. 
  The influence of the torsion on the relative velocity and on the relative acceleration between particles (points) in spaces with an affine connection and a metric [$(L_n,g)$-spaces] and in (pseudo) Riemannian spaces with torsion ($U_n$-spaces) is considered. Necessary and sufficient conditions as well as only necessary and only sufficient conditions for vanishing deformation, shear, rotation and expansion are found. The notion of relative acceleration and the related to it notions of shear, rotation and expansion accelerations induced by the torsion are determined. It is shown that the kinematic characteristics induced by the torsion (shear acceleration, rotation acceleration and expansion acceleration) could play the same role as the kinematic characteristics induced by the curvature and can (under given conditions) compensate their action as well as the action of external forces.  The change of the rate of change of the length of a deviation vector field is given in explicit form for $(L_n,g)$- and $U_n$-spaces.       PACS numbers: 04.90+e, 04.50+h, 12.10.Gq, 03.40.-t 
  We study a static, spherically symmetric system of (2j+1) massive Dirac particles, each having angular momentum j, j=1,2,..., in a classical gravitational and SU(2) Yang-Mills field. We show that for any black hole solution of the associated Einstein-Dirac-Yang/Mills equations, the spinors must vanish identically outside of the event horizon. 
  In this paper, based on the theory of parametric resonance, we propose a cosmological criterion on ways of compactification: we rule out such a model of compactification that there exists a Kaluza-Klein mode satisfying $2m_{KK}=\omega_b$, where $m_{KK}$ is mass of the Kaluza-Klein mode and $\omega_b$ is the frequency of oscillation of the radius of the compact manifold on which extra dimensions are compactified. This is a restatement of the criterion proposed previously [S. Mukohyama, Phys. Rev. {\bf D57}, 6191 (1998)]. As an example, we consider a model of compactification by a sphere and investigate Kaluza-Klein modes of U(1) field. In this case the parametric resonance is so mild that the sphere model is not ruled out. 
  Several different methods have recently been proposed for calculating the motion of a point particle coupled to a linearized gravitational field on a curved background. These proposals are motivated by the hope that the point particle system will accurately model certain astrophysical systems which are promising candidates for observation by the new generation of gravitational wave detectors. Because of its mathematical simplicity, the analogous system consisting of a point particle coupled to a scalar field provides a useful context in which to investigate these proposed methods. In this paper, we generalize the axiomatic approach of Quinn and Wald in order to produce a general expression for the self force on a point particle coupled to a scalar field following an arbitrary trajectory on a curved background. Our equation includes the leading order effects of the particle's own fields, commonly referred to as ``self force'' or ``radiation reaction'' effects. We then explore the equations of motion which follow from this expression in the absence of non-scalar forces. 
  It is standard assertion in relativity that, subject to an energy condition and the cosmic censorship hypothesis, closed trapped surfaces are not visible from future null infinity. A proof given by Hawking & Ellis in ''The Large Scale Structure of Space-Time'' is flawed since it is formulated in terms of an inadequate definition of a weakly asymptotically simple and empty space-time. A new proof is given based on a more restrictive definition of a weakly asymptotically simple and empty space-time. 
  Using scaled variables we are able to integrate an equation valid for isotropic and anisotropic Bianchi type I, V, IX models in Brans-Dicke (BD) theory. We analyze known and new solutions for these models in relation with the possibility that anisotropic models asymptotically isotropize, and/or possess inflationary properties. In particular, a new solution of curve ($k\neq0$) Friedmann-Robertson-Walker (FRW) cosmologies in Brans-Dicke theory is analyzed. 
  The homogeneous cosmological model in GR is proposed, where the vacuum energy, which can cause the inflation, is described by tensor field rather than by commonly used in inflationary scenarios scalar field. It is shown that if the initial values of the field are sufficiently big (comparable with the Planck units), under the condition of the tensor field's slow change in the beginning the regime of the quasiexponential inflation can exist. Numerical solutions for the inflationary stage are obtained that confirm the validity of the approximate solutions. Inflation takes place under wide range of initial conditions provided that the tensor field satisfies the condition imposed on the initial values of the tensor field ${\phi^0}_0(0)\approx -{\phi^i}_i(0)$ (i=1,2,3). That condition also arises from the requirement to satisfy existing observational data. 
  We discuss the analytical determination of the location of the Last Stable Orbit (LSO) in circular general relativistic orbits of two point masses. We use several different ``resummation methods'' (including new ones) based on the consideration of gauge-invariant functions, and compare the results they give at the third post-Newtonian (3PN) approximation of general relativity. Our treatment is based on the 3PN Hamiltonian of Jaranowski and Sch\"afer. One of the new methods we introduce is based on the consideration of the (invariant) function linking the angular momentum and the angular frequency. We also generalize the ``effective one-body'' approach of Buonanno and Damour by introducing a non-minimal (i.e. ``non-geodesic'') effective dynamics at the 3PN level. We find that the location of the LSO sensitively depends on the (currently unknown) value of the dimensionless quantity $\oms$ which parametrizes a certain regularization ambiguity of the 3PN dynamics. We find, however, that all the analytical methods we use numerically agree between themselves if the value of this parameter is $\oms\simeq-9$. This suggests that the correct value of $\oms$ is near -9 (the precise value $\oms^*\equiv-{47/3}+{41/64}\pi^2=-9.3439...$ seems to play a special role). If this is the case, we then show how to further improve the analytical determination of various LSO quantities by using a ``Shanks'' transformation to accelerate the convergence of the successive (already resummed) PN estimates. 
  We study the previously constructed Riemann problem whose solutions correspond to equilibrium configurations of black holes. We evaluate the metric coefficients at the symmetry axis and the interaction force between the black holes. 
  We study the quantum cosmology of a five dimensional non-compactified Kaluza-Klein theory where the 4D metric depends on the fifth coordinate, $x^4\equiv l$. This model is effectively equivalent to a 4D non-minimally coupled dilaton field in addition to matter generated on hypersurfaces l=constant by the extra coordinate dependence in the four-dimensional metric. We show that the Vilenkin wave function of the universe is more convenient for this model as it predicts a new-born 4D universe on the $l\simeq0$ constant hypersurface. 
  The analytic proof of mode stability of the Kerr black hole was provided by Whiting. In his proof, the construction of a conserved quantity for unstable mode was crucial. We extend the method of the analysis for the Kerr-de Sitter geometry. The perturbation equations of massless fields in the Kerr-de Sitter geometry can be transformed into Heun's equations which have four regular singularities. In this paper we investigate differential and integral transformations of solutions of the equations. Using those we construct a conserved quantity for unstable modes in the Kerr-de Sitter geometry, and discuss its property. 
  We study plane-fronted electrovacuum waves in metric-affine gravity theories (MAG) with cosmological constant. Their field strengths are, on the gravitational side, curvature $R_{\alpha}{}^{\beta}$, nonmetricity $Q_{\alpha\beta}$, torsion $T^{\alpha}$ and, on the matter side, the electromagnetic field strength $F$. Our starting point is the work by Ozsv\'ath, Robinson, and R\'ozga on type N gravitational fields in general relativity as coupled to null electromagnetic fields. 
  It is shown how the relation $ds=cd\tau$ between the proper distance $s$ and the proper time $\tau$ is obtained in general relativity. A general relation in curved spacetime between $d\tau$ and $dt$ is given. This relation reduces to the special relativistic one for flat spacetime. 
  The theory of General Relativity explaines the advance of Mercury perihelion using space curvature and the Schwartzschild metric. We demonstrate that this phenomena can also be interpreted due to the cogravitational field produced by the apparent motion of the Sun around Mercury giving exactly the same estimate as derived from the Schwartzschild metric in general relativity theory. This is a surprising result because the estimate from both theoretical approaches match exactly the measured value. The discussion and implications of this result is out of the scope of the present work. 
  We show that in a pioneering paper by Polnarev and Zembowicz, some conclusions concerning the characteristics of the Turok-strings are generally not correct. In addition we show that the probability of string collapse given there, is off by a large prefactor (~1000). 
  Recently, we proposed a method for calculating the ``radiation reaction'' self-force exerted on a charged particle moving in a strong field orbit in a black hole spacetime. In this approach, one first calculates the contribution to the ``tail'' part of the self force due to each multipole mode of the particle's self field. A certain analytic procedure is then applied to regularize the (otherwise divergent) sum over modes. This involves the derivation of certain regularization parameters using local analysis of the (retarded) Green's function. In the present paper we present a detailed formulation of this mode-sum regularization scheme for a scalar charge on a class of static spherically-symmetric backgrounds (including, e.g., the Schwarzschild, Reissner-Nordstr\"{o}m, and Schwarzschild-de Sitter spacetimes). We fully implement the regularization scheme for an arbitrary radial trajectory (not necessarily geodesic) by explicitly calculating all necessary regularization parameters in this case. 
  A class of vacuum initial-data sets is described which are based on certain expressions for the extrinsic curvature first studied and employed by Bowen and York. These expressions play a role for the momentum constraint of general relativity which is analogous to the role played by the Coulomb solution for the Gauss-law constraint of electromagnetism. 
  The relevance of orbital eccentricity in the detection of gravitational radiation from (steady state) binary stars is emphasized. Computationnally effective fast and accurate)tools for constructing gravitational wave templates from binary stars with any orbital eccentricity are introduced, including tight estimation criteria of the pertinent truncation and approximation errors. 
  We examine charged static perfect fluid distributions with a dilaton field in the frame-work of general relativity. We consider the case that the Einstein equations reduce to a non-linear version of Poisson equation. We show that Maxwell equation and an equation for a dilaton imply the relation among the charge, mass and dilatonic charge densities. 
  In terms of Dirac matrices the self-dual and anti-self-dual decomposition of a conformal supergravity is given and a self-dual conformal supergravity theory is developed as a connection dynamic theory in which the basic dynamic variabes include the self-dual spin connection i.e. the Ashtekar connection rather than the triad. The Hamiltonian formulation and the constraints are obtained by using the Dirac-Bergmann algorithm.     PACS numbers: 04.20.Cv, 04.20.Fy,04.65.+e 
  This article reviews, from a global point of view, rigorous results on time independent spacetimes. Throughout attention is confined to isolated bodies at rest or in uniform rotation in an otherwise empty universe. The discussion starts from first principles and is, as much as possible, self-contained. 
  This paper reviews work, largely due to W. Simon and the author, on multipole theory of static spacetimes. The main purpose is to make this work, which lies at the interface of potential theory, conformal geometry and general relativity, known to mathematicians and to perhaps motivate them to look at the open problems which still remain. 
  Working on the approximation of low frequency, we present the light cone conditions for a class of theories constructed with the two gauge invariants of the Maxwell field without making use of average over polarization states. Different polarization states are thus identified describing birefringence phenomena. We make an application of the formalism to the case of Euler-Heisenberg effective Lagrangian and well know results are obtained. 
  The photon sphere concept in Schwarzschild space-time is generalized to a definition of a photon surface in an arbitrary space-time. A photon sphere is then defined as an SO(3)xR-invariant photon surface in a static spherically symmetric space-time. It is proved, subject to an energy condition, that a black hole in any such space-time must be surrounded by a photon sphere. Conversely, subject to an energy condition, any photon sphere must surround a black hole, a naked singularity or more than a certain amount of matter. A second order evolution equation is obtained for the area of an SO(3)-invariant photon surface in a general non-static spherically symmetric space-time. Many examples are provided. 
  The model of a relativistic free massive point particle is investigated in the context of a Hamiltonian constraint system. It is shown that de Broglie oscillation, rest mass and inertia may be described within this model of Hamiltonian constraint system. 
  In a circle (an S^1) with circumference 1 assume m objects distributed pseudo-randomly. In the universal covering R^1 assume the objects replicated accordingly, and take an interval L>1. In this interval, make the normalized histogram of the pair separations which are not an integer. The theoretical (expected) such histogram is obtained in this report, as well as its difference to a similar histogram for non-replicated objects. The whole study is of interest for the cosmic crystallography. 
  In this work we introduce two experimental proposals that could shed some light upon the inertial properties of intrinsic spin. In particular we will analyze the role that the gravitomagnetic field of the Earth could have on a quantum system with spin 1/2. We will deduce the expression for Rabi transitions, which depend, explicitly, on the coupling between the spin of the quantum system and the gravitomagnetic field of the Earth. Afterwards, the continuous measurement of the energy of the spin 1/2 system is considered, and an expression for the emerging quantum Zeno effect is obtained. Thus, it will be proved that gravitomagnetism, in connection with spin 1/2 systems, could induce not only Rabi transitions but also a quantum Zeno effect. 
  Based on some observations, the apparent energy, associated with gravity, of vacuums is defined, with that of normal vacuums to be zero and that of the vacuums losing some energy to be negative. An important application of the energy is its contribution to Einstein's equation. A cosmological model, accounting for recent observations of the accelerated expansion of the universe, in the absence of the cosmological constant, can be well constructed. In a certain case, the expansion of the universe would be decelerated at its early epoch and accelerated at its late epoch. The curvature of the universe would depend on the ratio of matter energy to total energy. The missing mass problem does no longer exist in this model. Most negative apparent energy vacuums might be contained in voids, then the spacetime of galaxy clusters or that of the solar system would not be significantly affected by this kind of energy. 
  We show that in the Maxwell-Chern-Simons theory of topologically massive electrodynamics the Dirac string of a monopole becomes a cone in anti-de Sitter space with the opening angle of the cone determined by the topological mass which in turn is related to the square root of the cosmological constant. This proves to be an example of a physical system, {\it a priory} completely unrelated to gravity, which nevertheless requires curved spacetime for its very existence. We extend this result to topologically massive gravity coupled to topologically massive electrodynamics in the framework of the theory of Deser, Jackiw and Templeton. These are homogeneous spaces with conical deficit. Pure Einstein gravity coupled to Maxwell-Chern-Simons field does not admit such a monopole solution. 
  Current methods of evolving a spacetime containing one or more black holes are plagued by instabilities that prohibit long-term evolution. Some of these instabilities may be due to the numerical method used, traditionally finite differencing. In this paper, we explore the use of a pseudospectral collocation (PSC) method for the evolution of a spherically symmetric black hole spacetime in one dimension using a hyperbolic formulation of Einstein's equations. We demonstrate that our PSC method is able to evolve a spherically symmetric black hole spacetime forever without enforcing constraints, even if we add dynamics via a Klein-Gordon scalar field. We find that, in contrast to finite-differencing methods, black hole excision is a trivial operation using PSC applied to a hyperbolic formulation of Einstein's equations. We discuss the extension of this method to three spatial dimensions. 
  Multidimensional cosmological model describing the evolution of a fluid with shear and bulk viscosity in $n$ Ricci-flat spaces is investigated. The barotropic equation of state for the density and the pressure in each space is assumed. The second equation of state is chosen in the form when the bulk and the shear viscosity coefficients are inversely proportional to the volume of the Universe. The integrability of Einstein equations reads as a colinearity constraint between vectors which are related to constant parameters in the first and second equations of state. We give exact solutions in a Kasner-like form. The processes of dynamical compactification and the entropy production are discussed. The non-singular $D$-dimensional isotropic viscous solution is singled out. 
  The set up of matched filters for the detection of gravitational waves from in-spiraling compact binaries is usually carried out using the restricted post-Newtonian approximation: the filter phase is modelled including post-Newtonian corrections, whereas the amplitude is retained at the Newtonian order. Here we investigate the effects of the introduction of post-Newtonian corrections also to the amplitude and we discuss some of the implications for signal detection and parameter estimation. 
  We consider LISA observations of in-spiral signals emitted by massive black hole binary systems in circular orbit and with negligible spins. We study the accuracy with which the source parameters can be extracted from the data stream. We show that the use of waveforms retaining post-Newtonian corrections not only to the phase but also the amplitude can drastically improve the estimation of some parameters. 
  I give a brief introduction on gravitational wave laser interferometers, possible detectable sources from the ground and noise in the detectors 
  We develop a procedure to remove interference from gravitational wave spectrum.  The method is applied to the data produced by the Glasgow laser interferometer in 1996 and all the lines corresponding to the interference with the main supply are removed. 
  We present a technique that we call coherent line removal, for removing external coherent interference from gravitational wave interferometer data. We illustrate the usefulness of this technique applying it to the the data produced by the Glasgow laser interferometer in 1996 and removing all those lines corresponding to the electricity supply frequency and its harmonics. We also find that this method seems to reduce the level of non-Gaussian noise present in the interferometer and therefore, it can raise the sensitivity and duty cycle of the detectors. 
  Using the HPO approach to consistent histories we re-derive Unruh's result that an observer constantly accelerating through the Minkowski vacuum appears to be immersed in a thermal bath. We show that propositions about any symmetry of the system always form a consistent set and that the probabilities associated with such propositions are decided by their value in the initial state. We use this fact to postulate a condition on the decoherence functional in the HPO set-up. Finally we show that the Unruh effect arises from the fact that the initial density matrix corresponding to the inertial vacuum can be written as a thermal density matrix in the Fock basis associated with the accelerating observer. 
  Space-times admitting a shear-free, irrotational, geodesic null congruence are studied. Attention is focused on those space-times in which the gravitational field is a combination of a perfect fluid and null radiation. 
  Space-times admitting a 3-dimensional Lie group of conformal motions $C_3$ acting on null orbits are studied. Coordinate expressions for the metric and the conformal Killing vectors (CKV) are then provided (irrespectively of the matter content) and all possible perfect fluid solutions are found, although none of them verifies the weak and dominant energy conditions over the whole space-time manifold. 
  Space-times admitting an $r$-parameter Lie group of homotheties are studied for $r > 2$ devoting a special attention to those representing perfect fluid solutions to Einstein's field equations. 
  We present an explicit exact solution of Einstein's equations for an inhomogeneous dust universe with cylindrical symmetry. The spacetime is extremely simple but nonetheless it has new surprising features. The universe is ``closed'' in the sense that the dust expands from a big-bang singularity but recollapses to a big-crunch singularity. In fact, both singularities are connected so that the whole spacetime is ``enclosed'' within a single singularity of general character. The big-bang is not simultaneous for the dust, and in fact the age of the universe as measured by the dust particles depends on the spatial position, an effect due to the inhomogeneity, and their total lifetime has no non-zero lower limit. Part of the big-crunch singularity is naked. The metric depends on a parameter and contains flat spacetime as a non-singular particular case. For appropriate values of the parameter the spacetime is a small perturbation of Minkowski spacetime. This seems to indicate that flat spacetime may be unstable against some global {\it non-vacuum} perturbations. 
  Spacetimes admitting a similarity group are considered. Amongst them, special attention is given to the 3-parameter ones. A classification of such spacetimes is given based on the Bianchi type of the similarity group $H_3$, and the general form of the metric is provided in each case assuming the orbits are non-null. 
  A recent result by Haggag and Hajj-Boutros is reviewed within the framework of self-similar space-times, extending, in some sense, their results and presenting a family of metrics consisting of all the static spherically symmetric perfect fluid solutions admitting a homothety. 
  Perfect fluid space-times admitting a three-dimensional Lie group of conformal motions containing a two-dimensional Abelian Lie subgroup of isometries are studied. Demanding that the conformal Killing vector be proper (i.e., not homothetic nor Killing), all such space-times are classified according to the structure of their corresponding three-dimensional conformal Lie group and the nature of their corresponding orbits (that are assumed to be non-null). Each metric is then explicitly displayed in coordinates adapted to the symmetry vectors. Attention is then restricted to the diagonal case, and exact perfect fluid solutions are obtained in both the cases in which the fluid four-velocity is tangential or orthogonal to the conformal orbits, as well as in the more general "tilting" case. 
  We describe a procedure to identify and remove a class of interference lines from gravitational wave interferometer data. We illustrate the usefulness of this technique applying it to prototype interferometer data and removing all those lines corresponding to the external electricity main supply and related features. 
  I consider a semiclassical expansion of the scalar field in the warm inflation scenario. I study the evolution for the fluctuations of the metric around the Friedmann-Robertson-Walker one. The formalism predicts that, in the power-law expansion universe, the fluctuations of the metric decreases with time. 
  This paper solves two major problems which have blocked a free-fall Equivalence-Principle (EP) in a satellite for 25 years: a semimajor-axis error between the two proof masses cannot be distinguished from an EP violation and the response to an EP violation only grows as t not t^2. Using the cancellation method described in this paper, the nonobservability problem can be suppressed and a t^2 response can be generated which lasts between 10^4 and 10^6 seconds depending on the cancellation accuracy. t^2 response times between 10^5 and 10^6 seconds are equivalent to a very tall (0.1 to 10 AU) drop tower with a constant gravitational field of 3/7 ge. 
  A theory of time as 'information' is outlined using new tools such as Feynman Clocks (FCs), Collective Excitation Networks (CENs), Sequential Excitation Networks (SENs), and Plateaus of Complexity (POCs). Applications of this approach range from the Big Bang to the emergence of 'consciousness'. 
  Brane cosmology takes the unconventional form $H\sim \sqrt{\rho}$. To recover the standard cosmology, we have to assume that the matter density is much less than the brane tension. We show that the assumption can be justified even near the end of inflation if we fine-tune the coupling constant of the inflaton potential. As a consequence, the standard cosmology is recovered after inflation. 
  In a gedankenexperiment about the generalized second law (GSL) of black hole thermodynamics, the buoyant force by black hole atmosphere (the acceleration radiation) plays an important role, and then it is significant to understand the nature of the buoyant force.   Recently, Bekenstein criticizes that the fluid approximation of the acceleration radiation which is often used in the estimation of the buoyant force is invalid for the case that the size of the target is much less than a typical wavelength of the acceleration radiation, due to the diffractive effect of wave scattering.   In this letter, we argue that even if it is correct that we should calculate the buoyant force as a wave scattering process, its implication in the GSL strongly depends on whether there exists any massless scalar field, that is, S-wave scattering. By reconsidering the diffractive effect by S-wave scattering, we show that if some massless scalar field exists, then the GSL can hold without invoking a new physics, such as an entropy bound for matter. 
  A covariant quantization of the free spinor fields (s=1/2) in 4-dimensional de Sitter (dS) space-time based on analyticity in the complexified pseudo-Riemanian manifold is presented. We define the Wigthman two-point function ${\cal W}(x,y)$, which satisfies the conditions of: a) positivity, b) locality, c) covariance, and d) normal analyticity. Then the Hilbert space structure and the field operators $\psi (f)$ are defined. A coordinate-independent formula for the unsmeared field operator $\psi (x)$ is also given. 
  We describe recent work due to Niall \'O Murchadha and the author (Phys. Rev. D57, 4728 (1998)) on the late time behaviour of the maximal foliation of the extended Schwarzschild geometry which results from evolving a time symmetric slice into the past and future, with time running equally fast at both spatial ends of the manifold. We study the lapse function of this foliation in the limit where proper time-at-infinity goes to ${+\infty}$ or ${-\infty}$ and the slices approach $r=3m/2$. 
  We extend the classical general relativistic theory of measurement to include the possibility of existence of higher dimensions. The intrusion of these dimensions in the spacetime interval implies that the inertial mass of a particle in general varies along its worldline if the observations are analyzed assuming the existence of only the four spacetime dimensions. The variations of mass and spin are explored in a simple 5D Kaluza-Klein model. 
  The origin of cosmic gamma-ray bursts remains one of the most intriguing puzzles in astronomy. We suggest that purely general relativistic effects in the collapse of massive stars could account for these bursts. The late formation of closed trapped surfaces can occur naturally, allowing the escape of huge energy from curvature-generated fireballs, before these are hidden within a black hole. 
  The program Ortocartan for algebraic calculations in relativity has just been implemented in the Codemist Standard Lisp and can now be used under the Windows 98 and Linux operating systems. The paper describes the new facilities and subprograms that have been implemented since the previous release in 1992. These are: the possibility to write the output as Latex input code and as Ortocartan's input code, the calculation of the Ellis evolution equations for the kinematic tensors of flow, the calculation of the curvature tensors from given (torsion-free) connection coefficients in a manifold of arbitrary dimension, the calculation of the lagrangian from a given metric by the Landau-Lifshitz method, the calculation of the Euler-Lagrange equations from a given lagrangian (only for sets of ordinary differential equations) and the calculation of first integrals of sets of ordinary differential equations of second order (the first integrals are assumed to be polynomials of second degree in the first derivatives of the functions). 
  We extend the canonical formalism for the motion of $N$-particles in lineal gravity to include charges. Under suitable coordinate conditions and boundary conditions the determining equation of the Hamiltonian (a kind of transcendental equation) is derived from the matching conditions for the dilaton field at the particles' position. The canonical equations of motion are derived from this determining equation.   For the equal mass case the canonical equations in terms of the proper time can be exactly solved in terms of hyperbolic and/or trigonometric functions. In electrodynamics with zero cosmological constant the trajectories for repulsive charges exhibit not only bounded motion but also a countably infinite series of unbounded motions for a fixed value of the total energy $H_{0}$, while for attractive charges the trajectories are simply periodic. When the cosmological constant $\Lambda$ is introduced, the motion for a given $\Lambda$ and $H_{0}$ is classified in terms of the charge-momentum diagram from which we can predict what type of the motion is realized for a given charge.   Since in this theory the charge of each particle appears in the form $e_{1}e_{2}$ in the determining equation, the static balance condition in 1+1 dimensions turns out to be identical with the condition in Newtonian theory. We generalize this condition to non-zero momenta, obtaining the first exact solution to the static balance problem that does not obey the Majumdar-Papapetrou condition. 
  A framework was recently introduced to generalize black hole mechanics by replacing stationary event horizons with isolated horizons. That framework is significantly extended. The extension is non-trivial in that not only do the boundary conditions now allow the horizon to be distorted and rotating, but also the subsequent analysis is based on several new ingredients. Specifically, although the overall strategy is closely related to that in the previous work, the dynamical variables, the action principle and the Hamiltonian framework are all quite different. More importantly, in the non-rotating case, the first law is shown to arise as a necessary and sufficient condition for the existence of a consistent Hamiltonian evolution. Somewhat surprisingly, this consistency condition in turn leads to new predictions even for static black holes. To complement the previous work, the entire discussion is presented in terms of tetrads and associated (real) Lorentz connections. 
  In general relativity the gravitational field is a manifestation of spacetime curvature and unlike the electromagnetic field is not a force field. A particle falling in a gravitational field is represented by a geodesic worldline which means that no force is acting on it. If the particle is at rest in a gravitational field, however, its worldline is no longer geodesic and it is subjected to a force. The nature of that force is an open question in general relativity. The aim of this paper is to outline an approach toward resolving it in the case of classical charged particles which was initiated by Fermi in 1921. 
  An so(4,C)-covariant hamiltonian formulation of a family of generalized Hilbert-Palatini actions depending on a parameter (the so called Immirzi parameter) is developed. It encompasses the Ashtekar-Barbero gravity which serves as a basis of quantum loop gravity. Dirac quantization of this system is constructed. Next we study dependence of the quantum system on the Immirzi parameter. The path integral quantization shows no dependence on it. A way to modify the loop approach in the accordance with the formalism developed here is briefly outlined. 
  We prove the existence of a class of perfect-fluid cosmologies with polarised  Gowdy symmetry and a Kasner-like singularity. These solutions of the Einstein equations depend on four free functions of one space coordinate and are constructed by solving a system of Fuchsian equations. 
  The convergence of polyhomogeneous expansions of zero-rest-mass fields in asymptotically flat spacetimes is discussed. An existence proof for the asymptotic characteristic initial value problem for a zero-rest-mass field with polyhomogeneous initial data is given. It is shown how this non-regular problem can be properly recast as a set of regular initial value problems for some auxiliary fields. The standard techniques of symmetric hyperbolic systems can be applied to these new auxiliary problems, thus yielding a positive answer to the question of existence in the original problem. 
  We consider the Cauchy problem for the massive Dirac equation in the non-extreme Kerr-Newman geometry outside the event horizon. We derive an integral representation for the Dirac propagator involving the solutions of the ODEs which arise in Chandrasekhar's separation of variables. It is proved that for initial data in L^\infty_loc near the event horizon with L^2 decay at infinity, the probability of the Dirac particle to be in any compact region of space tends to zero as t goes to infinity. This means that the Dirac particle must either disappear in the black hole or escape to infinity. 
  In this talk I review recent progresses in the detection of scalar gravitational waves. Furthermore, in the framework of the Jordan-Brans-Dicke theory, I compute the signal to noise ratio for a resonant mass detector of spherical shape and for binary sources and collapsing stars. Finally I compare these results with those obtained from laser interferometers and from Einsteinian gravity. 
  I show that a cumulative contraction or expansion must result from repetitive tidal action in a curved stress field, depending on the direction of the curvature. The resulting expansion of solid materials onboard deep space probes and the corresponding contraction on earth would be of the right magnitude to account for all aspects of the Pioneer anomaly, leading to the two component model previously proposed. Importantly, I show via signal path analysis that the anomaly mathematically implies planetary Hubble flow, and that it is predicted by the standard model equations when the cosmological constant is also taken into account at this range. Also shown is that the variations of the anomaly do not permit a different explanation. The prediction of the Hubble flow occurring as a result in the view of the shrinking observer is now fully explained from both quantum and Doppler perspectives, fundamentally challenging the cosmological ideas of the past century. Lastly, I discuss how the contraction reconciles the geological evidence of a past expansion of the earth. 
  Superfluid 3He-A gives example of how chirality, Weyl fermions, gauge fields and gravity appear in low energy corner together with corresponding symmetries, including Lorentz symmetry and local SU(N). This supports idea that quantum field theory (Standard Model or GUT) is effective theory describing low-energy phenomena. * Momentum space topology of fermionic vacuum provides topological stability of universality class of systems, where above properties appear. * BCS scheme for 3He-A incorporates both ``relativistic'' infrared regime and ultraviolet ``transplanckian'' range: subtle issues of cut-off in quantum field theory and anomalies can be resolved on physical grounds. This allows to separate ``renormalizable'' terms in action, treated by effective theory, from those obtained only in ``transPlanckian'' physics. * Energy density of superfluid vacuum within effective theory is ~ E_{Planck}^4. Stability analysis of ground state beyond effective theory leads to exact nullification of vacuum energy: equilibrium vacuum is not gravitating. In nonequilibrium, vacuum energy is of order energy density of matter. * 3He-A provides experimental prove for anomalous nucleation of fermionic charge according to Adler-Bell-Jackiw. * Helical instability in 3He-A is described by the same equations as formation of magnetic field by right electrons in Joyce-Shaposhnikov scenario. * Macroscopic parity violating effect and angular momentum paradox are both desribed by axial gravitational Chern-Simons action. * High energy dispersion of quasiparticle spectrum allow to treat problems of vacuum in presence of event horizon, etc. 
  In this thesis the Bohm-de Broglie interpretation of quantum mechanics is applied to canonical quantum gravity. It is shown that, irrespective of any regularization or choice of factor ordering of the Wheeler-DeWitt equation, the unique relevant quantum effect which does not break spacetime is the change of its signature from lorentzian to euclidean. The other quantum effects are either trivial or break the four-geometry of spacetime. A Bohm-de Broglie picture of quantum geometrodynamics is constructed, which allows the investigation of these latter structures. For instance, it is shown that any real solution of the Wheeler-De Witt equation yields a generate four-geometry compatible with the strong gravity limit of General Relativity and the Carroll group. We prove that quantum geometrodynamics in the Bohm-de Broglie interpretation is consistent for any quantum potential. As a previous step to introduce our metodology, we study the quantum theory of fields in Minkowski spacetime in the Bohm-de Broglie interpretation and exhibit a concrete example where Lorentz invariance of individual events is broken. 
  We study about an approximation method of the Hawking radiation. We analyze an massless scalar field in exotic black hole backgrounds models which have peculiar properties in black hole thermodynamics (monopole black hole in SO(3) Einstein-Yang-Mills-Higgs system and dilatoic black hole in Einstein-Maxwell-dilaton system). A scalar field is assumed not to be couple to matter fields consisting of a black hole background. Except for extreme black holes, we can well approximate the Hawking radiaition by `black body' one with Hawking temperature estimated at a radius of a critical impact parameter. 
  This paper discusses the implementation of diffeomorphism invariance in purely Hamiltonian formulations of General Relativity. We observe that, if a constrained Hamiltonian formulation derives from a manifestly covariant Lagrangian, the diffeomorphism invariance of the Lagrangian results in the following properties of the constrained Hamiltonian theory: the diffeomorphisms are generated by constraints on the phase space so that a) The algebra of the generators reflects the algebra of the diffeomorphism group. b) The Poisson brackets of the basic fields with the generators reflects the space-time transformation properties of these basic fields. This suggests that in a purely Hamiltonian approach the requirement of diffeomorphism invariance should be interpreted to include b) and not just a) as one might naively suppose. Giving up b) amounts to giving up objective histories, even at the classical level. This observation has implications for Loop Quantum Gravity which are spelled out in a companion paper. We also describe an analogy between canonical gravity and Relativistic particle dynamics to illustrate our main point. 
  This letter is a critique of Barbero's constrained Hamiltonian formulation of General Relativity on which current work in Loop Quantum Gravity is based. While we do not dispute the correctness of Barbero's formulation of general relativity, we offer some criticisms of an aesthetic nature. We point out that unlike Ashtekar's complex SU(2) connection, Barbero's real SO(3) connection does not admit an interpretation as a space-time gauge field. We show that if one tries to interpret Barbero's real SO(3) connection as a space-time gauge field, the theory is not diffeomorphism invariant. We conclude that Barbero's formulation is not a gauge theory of gravity in the sense that Ashtekar's Hamiltonian formulation is. The advantages of Barbero's real connection formulation have been bought at the price of giving up the description of gravity as a gauge field. 
  Fermi Transport is useful for describing the behaviour of spins or gyroscopes following non-geodesic, timelike world lines. However, Fermi Transport breaks down for null world lines. We introduce a transport law for polarisation vectors along non-geodesic null curves. We show how this law emerges naturally from the geometry of null directions by comparing polarisation vectors associated with two distinct null directions. We then give a spinorial treatment of this topic and make contact with the geometric phase of quantum mechanics. There are two significant differences between the null and timelike cases. In the null case (i) The transport law does not approach a unique smooth limit as the null curve approaches a null geodesic. (ii) The transport law for vectors is integrable, i.e the result depends only on the local properties of the curve and not on the entire path taken. However, the transport of spinors is not integrable: there is a global sign of topological origin. 
  It is a common assumption amongst astronomers that, in the determination of the distances of remote sources from their apparent brightness, the cumulative gravitational lensing due to the matter in all the galaxies is the same, on average, as if the matter were uniformly distributed throughout the cosmos. The validity of this assumption is considered here by way of general Newtonian perturbations of Friedman-Robertson-Walker (FRW) cosmologies. The analysis is carried out in synchronous gauge, with particular attention to an additional gauge condition that must be imposed. The mean correction to the apparent magnitude-redshift relation is obtained for an arbitrary mean density perturbation. In the case of a zero mean density perturbation, when the intergalactic matter has a dust equation of state, then there is indeed a zero-mean first order correction to the apparent magnitude-redshift relation for all redshifts. Point particle and Swiss cheese models are considered as particular cases. 
  Dimensional reduction of generalized gravity theories or string theories generically yields dilaton fields in the lower-dimensional effective theory. Thus at the level of D=4 theories, and cosmology many models contain more than just one scalar field (e.g. inflaton, Higgs, quintessence). Our present work is restricted to two-dimensional gravity theories with only two dilatons which nevertheless allow a large class of physical applications.   The notions of factorizability, simplicity and conformal simplicity, Einstein form and Jordan form are the basis of an adequate classification. We show that practically all physically motivated models belong either to the class of factorizable simple theories (e.g. dimensionally reduced gravity, bosonic string) or to factorizable conformally simple theories (e.g. spherically reduced Scalar-Tensor theories). For these theories a first order formulation is constructed straightforwardly. As a consequence an absolute conservation law can be established. 
  It has been shown recently that the classical law of elasticity, expressed in terms of the displacement three-vector and of the symmetric deformation three-tensor, can be extended to the four dimensions of special and of general relativity with a physically meaningful outcome. In fact, the resulting stress- momentum-energy tensor can provide a unified account of both the elastic and the inertial properties of uncharged matter. The extension of the displacement vector to the four dimensions of spacetime allows a further possibility. If the real displacement four-vector is complemented with an imaginary part, the resulting complex ``displacement'' four-vector allows for a complex, Hermitian generalisation of the four-dimensional Hooke's law. Let the complex, Hermitian ``stress-momentum-energy'' tensor density built in this way be subjected to the usual conservation condition. It turns out that, while the real part of the latter equation is able to account for the motion of electrically charged, elastic matter, the imaginary part of the same equation can describe the evolution of the electromagnetic field and of its sources. The Hermitian extension of Hooke's law is performed by availing of the postulate of ``transposition invariance'', introduced in 1945 by A. Einstein for finding the nonsymmetric generalisation of his theory of gravitation of 1915. 
  Classical superstring vacua have zero vacuum energy and are supersymmetric and Lorentz-invariant. We argue that all these properties may be destroyed when quantum aspects of the interactions between particles and non-perturbative vacuum fluctuations are considered. A toy calculation of string/D-brane interactions using a world-sheet approach indicates that quantum recoil effects - reflecting the gravitational back-reaction on space-time foam due to the propagation of energetic particles - induce non-zero vacuum energy that is linked to supersymmetry breaking and breaks Lorentz invariance. This model of space-time foam also suggests the appearance of microscopic event horizons. 
  A new prescription to calculate the total energies and angular momenta of asymptotically $(d+1)$-dimensional anti-de Sitter spacetimes is proposed. The method is based on an extension of the field theoretical approach to General Relativity to the case where there is an effective cosmological constant. A $(d-1)$-form $\Omega$ is exhibited which, when integrated on asymptotic $(d-1)$-dimensional boundary surfaces, yields the values of those conserved quantities. The calculations are gauge independent once asymptotic conditions are not violated . Total energies and angular momenta of some known solutions in four and five dimensions are calculated agreeing with standard results. 
  The renormalized expectation value of the energy-momentum tensor for a scalar field with any mass m and curvature coupling xi is studied for an arbitrary homogeneous and isotropic physical initial state in de Sitter spacetime. We prove quite generally that <T_{ab}> has a fixed point attractor behavior at late times, which depends only on m and xi, for any fourth order adiabatic state that is infrared finite. Specifically, when m^2 + xi R > 0, <T_{ab}> approaches the Bunch-Davies de Sitter invariant value at late times, independently of the initial state. When m = xi = 0, it approaches instead the de Sitter invariant Allen-Folacci value. When m = 0 and xi \ge 0 we show that this state independent asymptotic value of the energy-momentum tensor is proportional to the conserved geometrical tensor (3)H_{ab}, which is related to the behavior of the quantum effective action of the scalar field under global Weyl rescaling. This relationship serves to generalize the definition of the trace anomaly in the infrared for massless, non-conformal fields. In the case m^2 + xi R = 0, but m and xi separately different from zero, <T_{ab}> grows linearly with cosmic time at late times. For most values of m and xi in the tachyonic cases, m^2 + xi R < 0, <T_{ab}> grows exponentially at late cosmic times for all physically admissable initial states. 
  p-Adic and adelic generalization of ordinary quantum cosmology is considered. In [1], we have calculated p-adic wave functions for some minisuperspace cosmological models according to the "no-boundary" Hartle-Hawking proposal. In this article, applying p-adic and adelic quantum mechanics, we show existence of the corresponding vacuum eigenstates. Adelic wave function contains some information on discrete structure of space-time at the Planck scale. 
  We have investigated cosmological models with a self-interacting scalar field and a dissipative matter fluid as the sources of matter. Different variables are expressed in terms of a {\it generating function}. Exact solutions are obtained for one particular choice of the {\it generating function} The potential corresponding to this generating function is a standard tree-level potential arising in the perturbative regime in quantum field theory. With suitable choice of parameters, the scale factor in our model exhibits both inflationary behaviour in the early universe as well as an accelerating phase at late times with a decelerating period in between. It also satisfies the constraints for primeval nucleosynthesis and structure formation and seems to solve the cosmic coincidence problem. The solution exhibits a attractor nature towards a asymptotic de-sitter universe. 
  With Carlip's boundary conditions, a standard Virasoro subalgebra with corresponding central charge for stationary dilaton black hole obtained in the low-energy effective field theory describing string is constructed at a Killing horizon. The statistical entropy of stationary dilaton black hole yielded by standard Cardy formula agree with its Bekenstein-Hawking entropy only if we take period $ T$ of function $v$ as the periodicity of the Euclidean black hole. On the other hand, if we consider first-order quantum correction then the entropy contains a logarithmic term with a factor $-{1/2}$, which is different from Kaul and Majumdar's one, $-{3/2}$. We also show that the discrepancy is not just for the dilaton black hole, but for any one whose corresponding central change takes the form $\frac{c}{12}= \frac{A_H}{8\pi G}\frac{2\pi}{\kappa T}$. 
  The notion of optical geometry, introduced more than twenty years ago as a formal tool in quantum field theory on a static background, has recently found several applications to the study of physical processes around compact objects. In this paper we define optical geometry for spherically symmetric gravitational collapse, with the purpose of extending the current formalism to physically interesting spacetimes which are not conformally static. The treatment is fully general but, as an example, we also discuss the special case of the Oppenheimer-Snyder model. The analysis of the late time behaviour shows a close correspondence between the structure of optical spacetime for gravitational collapse and that of flat spacetime with an accelerating boundary. Thus, optical geometry provides a natural physical interpretation for derivations of the Hawking effect based on the ``moving mirror analogy.'' Finally, we briefly discuss the issue of back-reaction in black hole evaporation and the information paradox from the perspective of optical geometry. 
  According to the weak form of Einstein's general relativity equivalence principle, the gravitational and inertial masses are equivalent. However recent calculations (gr-qc/9910036) have revealed that they are correlated by an adimensional factor, which is equal to one in absence of radiation only. We have built an experimental system to check this unexpected theoretical result. It verifies the effects of the extra-low frequency (ELF) radiation on the gravitational mass of a body. We show that there is a direct correlation between the radiation absorbed by the body and its gravitational mass, independently of the inertial mass. This has fundamental consequences to Unified Field Theory and Quantum Cosmology. 
  We present a simple proof, using the conservation equations, that any quantum stress tensor on Kerr space-time which is isotropic in a frame which rotates rigidly with the angular velocity of the event horizon must be divergent at the velocity of light surface. We comment on our result in the light of the absence of a `true Hartle-Hawking' vacuum for Kerr. 
  The Einstein equations for one of the hypersurface-homogeneous rotating dust models are investigated. It is a Bianchi type V model in which one of the Killing fields is spanned on velocity and rotation (case 1.2.2.2 in the classification scheme of the earlier papers). A first integral of the field equations is found, and with a special value of this integral coordinate transformations are used to eliminate two components of the metric. The k = -1 Friedmann model is shown to be contained among the solutions in the limit of zero rotation. The field equations for the simplified metric are reduced to 3 second-order ordinary differential equations that determine 3 metric components plus a first integral that algebraically determines the fourth component. First derivatives of the metric components are subject to a constraint (a second-degree polynomial with coefficients depending on the functions). It is shown that the set does not follow from a Lagrangian of the Hilbert type. The group of Lie point-symmetries of the set is found, it is two-dimensional noncommutative. Finally, a method of searching for first integrals (for sets of differential equations) that are polynomials of degree 1 or 2 in the first derivatives is applied. No such first integrals exist. The method is used to find a constraint (of degree 1 in first derivatives) that could be imposed on the metric, but it leads to a vacuum solution, and so is of no interest for cosmology. 
  The formation of singularities in certain situations, such as the collapse of massive stars, is one of the unresolved issues in classical general relativity. Although no complete theory of quantum gravity exists it is often suggested that quantum gravity effects may prevent the formation of these singularities. In this article we will present arguments that a quantized theory of gravity might exhibit asymptotic freedom. Considering the similarites between non-Abelian gauge theories and general relativity it is conjectured that a quantized theory of gravity may have a coupling strength which decreases with increasing energy scale. Such a scale dependent coupling strength, could provide a concrete mechanism for preventing the formation of singularities. 
  Bekenstein's conjectured entropy bound for a system of linear size R and energy E, S < 2 pi E R, can be violated by an arbitrarily large factor, among other ways, by a scalar field having a symmetric potential allowing domain walls, and by the electromagnetic field modes between an arbitrarily large number of conducting plates. 
  In this work the possibility of detecting the presence of a Yukawa term, as an additional contribution to the usual Newtonian gravitational potential, is introduced. The central idea is to analyze the effects at quantum level employing interference patterns (at this respect the present proposal resembles the Colella, Overhauser and Werner experiment), and deduce from it the possible effects that this Yukawa term could have. We will prove that the corresponding interference pattern depends on the phenomenological parameters that define this kind of terms. Afterwards, using the so called restricted path integral formalism, the case of a particle whose position is being continuously monitored, is analyzed, and the effects that this Yukawa potential could have on the measurement outputs are obtained. This allows us to obtain another scheme that could lead to the detection of these terms. This last part also renders new theoretical predictions that could enable us to confront the restricted path integral formalism against some future experiments. 
  We reinvestigate the utility of time-independent constant mean curvature foliations for the numerical simulation of a single spherically-symmetric black hole. Each spacelike hypersurface of such a foliation is endowed with the same constant value of the trace of the extrinsic curvature tensor, $K$. Of the three families of $K$-constant surfaces possible (classified according to their asymptotic behaviors), we single out a sub-family of singularity-avoiding surfaces that may be particularly useful, and provide an analytic expression for the closest approach such surfaces make to the singularity. We then utilize a non-zero shift to yield families of $K$-constant surfaces which (1) avoid the black hole singularity, and thus the need to excise the singularity, (2) are asymptotically null, aiding in gravity wave extraction, (3) cover the physically relevant part of the spacetime, (4) are well behaved (regular) across the horizon, and (5) are static under evolution, and therefore have no ``grid stretching/sucking'' pathologies. Preliminary numerical runs demonstrate that we can stably evolve a single spherically-symmetric static black hole using this foliation. We wish to emphasize that this coordinatization produces $K$-constant surfaces for a single black hole spacetime that are regular, static and stable throughout their evolution. 
  Naked singularity occurs in the gravitational collapse of an inhomogeneous dust ball from an initial density profile which is physically reasonable. We show that explosive radiation is emitted during the formation process of the naked singularity. The energy flux is proportional to $(t_{\rm CH}-t)^{-3/2}$ for a minimally coupled massless scalar field, while is proportional to $(t_{\rm CH}-t)^{-1}$ for a conformally coupled massless scalar field, where $t_{\rm CH}-t$ is the `remained time' until the distant observer could observe the singularity if the naked singularity was formed. As a consequence, the radiated energy grows unboundedly for both scalar fields. The amount of the power and the energy depends on parameters which characterize the initial density profile but do not depend on the gravitational mass of the cloud. In particular, there is characteristic frequency $\nu_{s}$ of singularity above which the divergent energy is radiated. The energy flux is dominated by particles of which the wave length is about $t_{\rm CH}-t$ at each moment. The observed total spectrum is nonthermal, i.e., $\nu dN/d\nu \sim (\nu/\nu_{s})^{-1}$ for $\nu>\nu_{s}$. If the naked singularity formation could continue until a considerable fraction of the total energy of the dust cloud is radiated, the radiated energy would reach about $10^{54}(M/M_{\odot})$ erg. The calculations are based on the geometrical optics approximation which turns out to be consistent as a rough order estimate. The analysis does not depend on whether or not the naked singularity occurs in its exact meaning. This phenomenon may provide a new candidate for a source of ultra high energy cosmic rays or a central engine of gamma ray bursts. 
  We investigate Hawking radiation from black holes in (d+1)-dimensional anti-de Sitter space. We focus on s-waves, make use of the geometrical optics approximation, and follow three approaches to analyze the radiation. First, we compute a Bogoliubov transformation between Kruskal and asymptotic coordinates and compare the different vacua. Second, following a method due to Kraus, Parikh, and Wilczek, we view Hawking radiation as a tunneling process across the horizon and compute the tunneling probablility. This approach uses an anti-de Sitter version of a metric originally introduced by Painleve for Schwarzschild black holes. From the tunneling probability one also finds a leading correction to the semi-classical emission rate arising from the backreaction to the background geometry. Finally, we consider a spherically symmetric collapse geometry and the Bogoliubov transformation between the initial vacuum state and the vacuum of an asymptotic observer. 
  The dynamics of the Einstein-Vlasov equations for a class of cosmological models with four Killing vectors is discussed in the case of massive particles. It is shown that in all models analysed the solutions with massive particles are asymptotic to solutions with massless particles at early times. It is also shown that in Bianchi types I and II the solutions with massive particles are asymptotic to dust solutions at late times. That Bianchi type III models are also asymptotic to dust solutions at late times is consistent with our results but is not established by them. 
  We study the properties of $\hat{Q}[\omega]$ operator on the kinematical Hilbert space ${\cal H}$ for canonical quantum gravity. Its complete spectrum with respect to the spin network basis is obtained. It turns out that $\hat{Q}[\omega]$ is diagonalized in this basis, and it is a well defined self-adjoint operator on ${\cal H}$. The same conclusions are also tenable on the SU(2) gauge invariant Hilbert space with the gauge invariant spin network basis. 
  We study early universe cosmologies derived from a scalar-tensor action containing cosmological constant terms and massless fields. The governing equations can be written as a dynamical system which contains no past or future asymptotic equilibrium states (i.e. no sources nor sinks). This leads to dynamics with very interesting mathematical behaviour such as the existence of heteroclinic cycles. The corresponding cosmologies have novel characteristics, including cyclical and bouncing behaviour possibly indicating chaos. We discuss the connection between these early universe cosmologies and those derived from the low-energy string effective action. 
  There are many models relating an accretion disk of Black Hole to jet outflow. The herein heuristic model describes the continuation of an external accretion disk to an internal accretion disk for less than Black Hole horizon, and subsequent polar jet outflow along polar axis out of polar vortex wherein the event horizon is no longer descriptive. 
  The Null Surface Formulation of General Relativity is developed for 2+1 dimensional gravity. The geometrical meaning of the metricity condition is analyzed and two approaches to the derivation of the field equations are presented. One method makes explicit use of the conformal factor while the other only uses conformal information. The resulting set of equations contain the same geometrical meaning as the 4-D formulation without the technical complexities of the higher dimensional analog. A canonical family of null surfaces in this formulation, the light cone cuts of null infinity, are constructed on asymptotically flat space times and some of their kinematical aspects discussed. A particular example, which nevertheless contains most of the generic features is explicitly constructed and analyzed, revealing the behavior predicted in the full theory. 
  A simple, though rarely considered, thought experiment on relativistic rotation is described in which internal inconsistencies in the theory of relativity seem to arise. These apparent inconsistencies are resolved by appropriate insight into the nature, and unique properties, of the non-time-orthogonal rotating frame. The analysis also explains a heretofore inexplicable experimental result. 
  Evading formation of the domain walls in cosmological phase transitions is one of the key problems to be solved for getting agreement with the observed large-scale homogeneity of the Universe. The previous attempts to get around this obstacle led to imposing severe observational constraints on the parameters of the fields involved. Our aim is to show that yet another way to overcome the above problem is accounting for EPR effect. Namely, if the scalar (Higgs) field was presented by a single quantum state at the initial instant of time, then its reduction during a phase transition at some later instant should be correlated even at distances exceeding the local cosmological horizon. By considering a simplest 1D model with Z_2 Higgs field, we demonstrate that EPR effect really can substantially reduce the probability of spontaneous creation of the domain walls. 
  In this article we investigate the metric signature as a non-differentiable ({\it i.e.} discrete as opposed to continuous) degree of freedom. The specific model is a vacuum 7D Universe on the principal bundle with an SU(2) structural group. An analytical solution is found which to a 4D observer appears as a flat Universe with a fluctuating metric signature, and frozen extra dimensions with an SU(2) instanton gauge field. A piece of this solution with linear size of the Planck length ($\approx l_{Pl}$) can be considered as seeding the quantum birth of a regular Universe. A boundary of this piece can initiate the formation of a Lorentzian Universe filled with the gauge fields and in which the extra dimensions have been ``frozen''. 
  The dynamics of a relativistic particle in a Reissner-Nordstrom background is studied using Caianiello model with maximal acceleration. The behaviour of the particle, embedded in a new effective geometry, changes with respect to the classical scenario because of the formation of repulsive potential barriers near the horizon. Black hole formation by accretion of massive particles is not therefore a viable process in the model. At the same time, the naked singularity remains largely unaffected by maximal acceleration corrections. 
  In this paper we study the perturbations of the charged, dilaton black hole, described by the solution of the low energy limit of the superstring action found by Garfinkle, Horowitz and Strominger. We compute the complex frequencies of the quasi-normal modes of this black hole, and compare the results with those obtained for a Reissner-Nordstr\"{o}m and a Schwarzschild black hole. The most remarkable feature which emerges from this study is that the presence of the dilaton breaks the \emph{isospectrality} of axial and polar perturbations, which characterizes both Schwarzschild and Reissner-Nordstr\"{o}m black holes. 
  Using the earlier developed classical Hamiltonian framework as the point of departure, we carry out a non-perturbative quantization of the sector of general relativity, coupled to matter, admitting non-rotating isolated horizons as inner boundaries. The emphasis is on the quantum geometry of the horizon. Polymer excitations of the bulk quantum geometry pierce the horizon endowing it with area. The intrinsic geometry of the horizon is then described by the quantum Chern-Simons theory of a U(1) connection on a punctured 2-sphere, the horizon. Subtle mathematical features of the quantum Chern-Simons theory turn out to be important for the existence of a coherent quantum theory of the horizon geometry. Heuristically, the intrinsic geometry is flat everywhere except at the punctures. The distributional curvature of the U(1) connection at the punctures gives rise to quantized deficit angles which account for the overall curvature. For macroscopic black holes, the logarithm of the number of these horizon microstates is proportional to the area, irrespective of the values of (non-gravitational) charges. Thus, the black hole entropy can be accounted for entirely by the quantum states of the horizon geometry. Our analysis is applicable to all non-rotating black holes, including the astrophysically interesting ones which are very far from extremality. Furthermore, cosmological horizons (to which statistical mechanical considerations are known to apply) are naturally incorporated. An effort has been made to make the paper self-contained by including short reviews of the background material. 
  The conserved charges associated to gauge symmetries are defined at a boundary component of space-time because the corresponding Noether current can be rewritten on-shell as the divergence of a superpotential. However, the latter is afflicted by ambiguities. Regge and Teitelboim found a procedure to lift the arbitrariness in the Hamiltonian framework. An alternative covariant formula was proposed by one of us for an arbitrary variation of the superpotential, it depends only on the equations of motion and on the gauge symmetry under consideration. Here we emphasize that in order to compute the charges, it is enough to stay at a boundary of spacetime, without requiring any hypothesis about the bulk or about other boundary components, so one may speak of holographic charges. It is well known that the asymptotic symmetries that lead to conserved charges are really defined at infinity, but the choice of boundary conditions and surface terms in the action and in the charges is usually determined through integration by parts whereas each component of the boundary should be considered separately. We treat the example of gravity (for any space-time dimension, with or without cosmological constant), formulated as an Affine theory which is a natural generalization of the Palatini and Cartan-Weyl (vielbein) first order formulations. We then show that the superpotential associated to a Dirichlet boundary condition on the metric (the one needed to treat asymptotically flat or AdS spacetimes) is the one proposed by Katz, Bi\u{c}{\'a}k and Lynden-Bell and not that of Komar. We finally discuss the KBL superpotential at null infinity. 
  Multi-connected Universe models with space idenfication scales smaller than the size of the observable universe produce topological images in the catalogs of cosmic sources. In this review, we present the recent developments for the search of the topology of the universe focusing on three dimensional methods. We present the crystallographic method, we give a new lower bound on the size of locally Euclidean multi-connected universe model of $3000 h^{-1} \hbox{Mpc}$ based on this method and a quasar catalog, we discuss its successes and failures, and the attemps to generalise it. We finally introduce a new statistical method based on a collecting correlated pair (CCP) technique. 
  A common property of known black hole solutions in (2+1)-dimensional gravity is that they require a negative cosmological constant. In this letter, it is shown that a (2+1)-dimensional gravity theory which satisfies the dominant energy condition forbids the existence of a black hole to explain the above situation. 
  Electromagnetic waves propagate in the Schwarzschild spacetime like in a nonuniform medium with a varying refraction index. A fraction of the radiation scatters off the curvature of the geometry. The energy of the backscattered part of an initially outgoing pulse of electromagnetic radiation can be estimated, in the case of dipole radiation, by a compact formula depending on the initial energy, the Schwarzschild radius and the pulse location. The magnitude of the backscattered energy depends on the frequency spectrum of the initial configuration. This effect becomes negligible in the short wave limit, but it can be significant in the long wave regime. Similar results hold for the massless scalar fields and are expected to hold also for weak gravitational waves. 
  The sonic analog of a gravitational black hole in dilute-gas Bose-Einstein condensates is investigated. It is shown that there exist both dynamically stable and unstable configurations which, in the hydrodynamic limit, exhibit a behavior completely analogous to that of gravitational black holes. The dynamical instabilities involve creation of quasiparticle pairs in positive and negative energy states. We illustrate these features in two qualitatively different one-dimensional models, namely, a long, thin condensate with an outcoupler laser beam providing an ``atom sink,'' and a tight ring-shaped condensate. We have also simulated the creation of a stable sonic black hole by solving the Gross-Pitaevskii equation numerically for a condensate subject to a trapping potential which is adiabatically deformed. A sonic black hole could in this way be created experimentally with state-of-the-art or planned technology. 
  The radial motion of matter in a centrally symmetric gravitational field in a comoving reference frame is investigated for a realistic equation of state of matter. The dynamics of the formation of an event horizon are investigated. 
  We present calculations of the variance of fluctuations and of the mean of the energy momentum tensor of a massless scalar field for the Minkowski and Casimir vacua as a function of an intrinsic scale defined by a smeared field or by point separation. We point out that contrary to prior claims, the ratio of variance to mean-squared being of the order unity is not necessarily a good criterion for measuring the invalidity of semiclassical gravity. For the Casimir topology we obtain expressions for the variance to mean-squared ratio as a function of the intrinsic scale (defined by a smeared field) compared to the extrinsic scale (defined by the separation of the plates, or the periodicity of space). Our results make it possible to identify the spatial extent where negative energy density prevails which could be useful for studying quantum field effects in worm holes and baby universe, and for examining the design feasibility of real-life `time-machines'.  For the Minkowski vacuum we find that the ratio of the variance to the mean-squared, calculated from the coincidence limit, is identical to the value of the Casimir case at the same limit for spatial point separation while identical to the value of a hot flat space result with a temporal point-separation. We analyze the origin of divergences in the fluctuations of the energy density and discuss choices in formulating a procedure for their removal, thus raising new questions into the uniqueness and even the very meaning of regularization of the energy momentum tensor for quantum fields in curved or even flat spacetimes when spacetime is viewed as having an extended structure. 
  The possibility that Galactic halo MACHOs are white dwarfs has recently attracted much attention. Using the known properties of white dwarf binaries in the Galactic disk as a model, we estimate the possible contribution of halo white dwarf binaries to the low-frequency (10^{-5} Hz} < f < 10^{-1}Hz) gravitational wave background. Assuming the fraction of white dwarfs in binaries is the same in the halo as in the disk, we find the confusion background from halo white dwarf binaries could be five times stronger than the expected contribution from Galactic disk binaries, dominating the response of the proposed space based interferometer LISA. Low-frequency gravitational wave observations will be the key to discovering the nature of the dark MACHO binary population. 
  We propose a generalization of the isometry transformations to the geometric context of the field theories with spin where the local frames are explicitly involved. We define the external symmetry transformations as isometries combined with suitable tetrad gauge transformations and we show that these form a group which is locally isomorphic with the isometry one. We point out that the symmetry transformations that leave invariant the equations of the fields with spin have generators with specific spin terms which represent new physical observables. The examples we present are the generators of the central symmetry and those of the maximal symmetries of the de Sitter and anti-de Sitter spacetimes derived in different tetrad gauge fixings.   Pacs: 04.20.Cv, 04.62.+v, 11.30.-j 
  We revisit the issue of time in quantum geometrodynamics and suggest a quantization procedure on the space of true dynamic variables. This procedure separates the issue of quantization from enforcing the constraints caused by the general covariance symmetries. The resulting theory, unlike the standard approach, takes into account the states that are off shell with respect to the constraints, and thus avoids the problems of time. In this approach, quantum geometrodynamics, general covariance, and the interpretation of time emerge together as parts of the solution of the total problem of geometrodynamic evolution. 
  Two families of models of rotating relativistic disks based on Taub-NUT and Kerr metrics are constructed using the well-known "displace, cut and reflect" method. We find that for disks built from a generic stationary axially symmetric metric the "sound velocity", $(pressure/density)^{1/2}$, is equal to the geometric mean of the prograde and retrograde geodesic circular velocities of test particles moving on the disk. We also found that for generic disks we can have zones with heat flow. For the two families of models studied the boundaries that separate the zones with and without heat flow are not stable against radial perturbations (ring formation). 
  According to the entropy bound, the entropy of a complete physical system can be universally bounded in terms of its circumscribing radius and total gravitating energy. Page's three recent candidates for counterexamples to the bound are here clarified and refuted by stressing that the energies of all essential parts of the system must be included in the energy the bound speaks about. Additionally, in response to an oft heard claim revived by Page, I give a short argument showing why the entropy bound is obeyed at low temperatures by a complete system. Finally, I remark that Page's renewed appeal to the venerable ``many species'' argument against the entropy bound seems to be inconsistent with quantum field theory. 
  A toy calculation of string/D-particle interactions within a world-sheet approach indicates that quantum recoil effects - reflecting the gravitational back-reaction on space-time foam due to the propagation of energetic particles - induces the appearance of a microscopic event horizon, or `bubble', inside which stable matter can exist. The scattering event causes this horizon to expand, but we expect quantum effects to cause it to contract again, in a `bounce' solution. Within such `bubbles', massless matter propagates with an effective velocity that is less than the velocity of light in vacuo, which may lead to observable violations of Lorentz symmetry that may be tested experimentally. The conformal invariance conditions in the interior geometry of the bubbles select preferentially three for the number of the spatial dimensions, corresponding to a consistent formulation of the interaction of D3 branes with recoiling D particles, which are allowed to fluctuate independently only on the D3-brane hypersurface. 
  Studying the threshold of black hole formation via numerical evolution has led to the discovery of fascinating nonlinear phenomena. Power-law mass scaling, aspects of universality, and self-similarity have now been found for a large variety of models. However, questions remain. Here I briefly review critical phenomena, discuss some recent results, and describe a model which demonstrates similar phenomena without gravity. 
  Boundary conditions defining a generic isolated horizon are introduced. They generalize the notion available in the existing literature by allowing the horizon to have distortion and angular momentum. Space-times containing a black hole, itself in equilibrium but possibly surrounded by radiation, satisfy these conditions. In spite of this generality, the conditions have rich consequences. They lead to a framework, somewhat analogous to null infinity, for extracting physical information, but now in the \textit{strong} field regions. The framework also generalizes the zeroth and first laws of black hole mechanics to more realistic situations and sheds new light on the `origin' of the first law. Finally, it provides a point of departure for black hole entropy calculations in non-perturbative quantum gravity. 
  By using our recent generalization of the colliding waves concept to metric-affine gravity theories, and also our generalization of the advanced and retarded time coordinate representation in terms of Jacobi functions, we find a general class of colliding wave solutions with fourth degree polynomials in metric-affine gravity. We show that our general approach contains the standard second degree polynomials colliding wave solutions as a particular case. 
  The exact renormalization group equation for pure quantum gravity is derived for an arbitrary gauge parameter in the space-time dimension $d=4$. This equation is given by a non-linear functional differential equation for the effective average action. An action functional of the effective average action is approximated by the same functional space of the Einstein-Hilbert action.   From this approximation, $\beta$-functions for the dimensionless Newton constant and cosmological constant are derived non-perturbatively. These are used for an analysis of the phase structure and the ultraviolet non-Gaussian fixed point of the dimensionless Newton constant. This fixed point strongly depends on the gauge parameter and the cutoff function. However, this fixed point exists without these ambiguities, except for some gauges. Hence, it is possible that pure quantum gravity in $d=4$ is an asymptotically safe theory and non-perturbatively renormalizable. 
  The functional method, introduced to deal with systems endowed with a continuous spectrum, is used to study the problem of decoherence and correlations in a simple cosmological model. 
  In this letter the paper [R. Aquilano, M. Castagnino, Mod. Phys. Lett A,11, 755 (1996)] is improved by considering that the main source of entropy production are the stars photospheres. 
  In this paper we improve the results of sec. VI of paper [M. Castagnino, Phys. Rev. D 57, 750 (1998)] by considering that the main source of entropy production are the photospheres of the stars. 
  Decoherence and the approach to the classical final limit are studied in two similar cases: the Mott and the Cosmological problems. 
  Starting from a Lagrangian we perform the full constraint analysis of the Hamiltonian for General relativity in the tetrad-connection formulation for an arbitrary value of the Immirzi parameter and solve the second class constraints, presenting the theory with a Hamiltonian composed of first class constraints which are the generators of the gauge symmetries of the action. In the time gauge we then recover Barbero's formulation of gravity. 
  It is shown that general relativity coupled to nonlinear electrodynamics (NED) with the Lagrangian $L(F)$, $F = F_mn F^mn$ having a correct weak field limit, leads to nontrivial static, spherically symmetric solutions with a globally regular metric if and only if the electric charge is zero and $L(F)$ tends to a finite limit as $F \to \infty$. Properties and examples of such solutions, which include magnetic black holes and soliton-like objects (monopoles), are discussed. Magnetic solutions are compared with their electric counterparts. A duality between solutions of different theories specified in two alternative formulations of NED (called $FP$ duality) is used as a tool for this comparison. 
  The thermodynamics is extented to spacetimes with spin-torsion density.Impplications to Einstein-Cartan-de Sitter inflationary phases are discussed.A relation between the spin-torsion density,entropy and temperature is presented.A lower limit for the radius of the Universe may be obtained from the spin-torsion density and the Planck lenght. 
  An approximate model of the spacetime foam is offered in which a quantum handle (wormhole) is a 5D wormhole-like solution. Neglecting the linear sizes of the wormhole throat we can introduce a spinor field for an approximate and effective description of the foam. The definition of the spinor field can be made by a dynamic and non-dynamic ways. In the first case some field equations are used and the second case leads to superspace. It is shown that : the spacetime with the foam is similar to a dielectric with dipoles and supergravity theories with a non-minimal interaction between spinor and electromagnetic fields can be considered as an effective model for the spacetime foam. 
  Motivated by a recent study casting doubt on the correspondence between Regge calculus and general relativity in the continuum limit, we explore a mechanism by which the simplicial solutions can converge whilst the residual of the Regge equations evaluated on the continuum solutions does not. By directly constructing simplicial solutions for the Kasner cosmology we show that the oscillatory behaviour of the discrepancy between the Einstein and Regge solutions reconciles the apparent conflict between the results of Brewin and those of previous studies. We conclude that solutions of Regge calculus are, in general, expected to be second order accurate approximations to the corresponding continuum solutions. 
  Starting from an (unknown) quantum gravitational model, one can invoke a sequence of approximations to progressively arrive at quantum field theory (QFT) in curved spacetime, QFT in flat spacetime, nonrelativistic quantum mechanics and newtonian mechanics. The more exact theory can put restrictions on the range of possibilities allowed for the approximate theory which are not derivable from the latter - an example being the symmetry restrictions on the wave function for a pair of electrons. We argue that the choice of vacuum state at low energies could be such a `relic' arising from combining the principles of quantum theory and general relativity, and demonstrate this result in a simple toy model. Our analysis suggests that the wave function of the universe, when it describes the large volume limit of the universe, dynamically selects a vacuum state for matter fields - which in turn defines the concept of particle in the low energy limit. The result also has the potential for providing a concrete quantum mechanical version of Mach's principle. 
  In recent papers [1,2], it has been shown that the presence of negative norm states or negative frequency solutions are indispensable for a fully covariant quantization of the minimally coupled scalar field in de Sitter space. Their presence, while leaving unchanged the physical content of the theory, offers the advantage of eliminating any ultraviolet divergence in the vacuum energy [2] and infrared divergence in the two point function [3]. We attempt here to extend this method to the interacting quantum field in Minkowski space-time. As an illustration of the procedure, we consider the $\lambda\phi^4$ theory in Minkowski space-time. The mathematical consequences of this method is the disappearance of the ultraviolet divergence to the one-loop approximation. This means, the effect of these auxiliary negative norm states is to allow an automatic renormalization of the theory in this approximation. 
  Recently, Glass and Krisch have extended the Vaidya radiating metric to include both a radiation fluid and a string fluid [1999 Class. Quantum Grav. vol 16, 1175]. Mass diffusion in the extended Schwarzschild atmosphere was studied. The continuous solutions of classical diffusive transport are believed to describe the envelope of underlying fractal behavior. In this work we examine the classical picture at scales on which fractal behavior might be evident. 
  We emphasize that a specific aspect of quantum gravity is the absence of a super-selection rule that prevents a linear superposition of different gravitational charges. As an immediate consequence, we obtain a tiny, but observable, violation of the equivalence principle, provided, inertial and gravitational masses are not assumed to be operationally identical objects. In this framework, the cosmic gravitational environment affects local experiments. A range of terrestrial experiments, from neutron interferometry to neutrino oscillations, can serve as possible probes to study the emergent quantum aspects of gravity. 
  The joint realm of quantum mechanics and the general-relativistic description of gravitation is becoming increasingly accessible to terrestrial experiments and observations. In this essay we study the emerging indications of the violation of equivalence principle (VEP). While the solar neutrino anomaly may find its natural explanation in a VEP, the statistically significant discrepancy observed in the gravitationally induced phases of neutron interferometry seems to be the first indication of a VEP. However, such a view would seem immediately challenged by the atomic interferometry results. The latter experiments see no indications of VEP, in apparent contradiction to the neutron interferometry results. Here we present arguments that support the view that these, and related torsion pendulum experiments, probe different aspects of gravity; and that current experimental techniques, when coupled to the solar-neutrino data, may be able to explore quantum mechanically induced violations of the equivalence principle. We predict quantum violation of the equivalence principle (qVEP) for next generation of atomic interferometry experiments. The prediction entails comparing free fall of two different linear superpositions of Cesium atomic states. 
  An intriguing, and possibly significant, anomalous signal in the Brillet and Hall experiment is contrasted with a simple first order test of special relativity subsequently performed to discount that signal as spurious. Analysis of the non-time-orthogonal nature of the rotating earth frame leads to the conclusion that the latter test needed second order accuracy in order to detect the effect sought, and hence was not sufficient to discount the potential cause of the anomalous signal. The analysis also explains the results found in Sagnac type experiments wherein different media were placed in the path of the light beams. 
  The theory of spaces with different (not only by sign) contravariant and covariant affine connections and metrics [}$(\bar{L}_n,g)$\QTR{it}{-spaces] is worked out within the framework of the tensor analysis over differentiable manifolds and in a volume necessary for the further considerations of the kinematics of vector fields and the Lagrangian theory of tensor fields over}$(\bar{L}_n,g)$\QTR{it}{-spaces. The possibility of introducing affine connections (whose components differ not only by sign) for contravariant and covariant tensor fields over differentiable manifolds with finite dimensions is discussed. The action of the deviation operator, having an important role for deviation equations in gravitational physics, is considered for the case of contravariant and covariant vector fields over differentiable manifolds with different affine connections A deviation identity for contravariant vector fields is obtained. The notions covariant, contravariant, covariant projective, and contravariant projective metrics are introduced in (}$\bar{L}_n,g$\{)-spaces. The action of the covariant and the Lie differential operators on the different type of metrics is found. The notions of symmetric covariant and contravariant (Riemannian) connections are determined and presented by means of the covariant and contravariant metrics and the corresponding torsion tensors. The different types of relative tensor fields (tensor densities) as well as the invariant differential operators acting on them are considered. The invariant volume element and its properties under the action of different differential operators are investigated. 
  We obtain (3+3)- or (3+2)-dimensional global flat embeddings of four uncharged and charged scalar-tensor theories with the parameters B or L in the (2+1)-dimensions, which are the non-trivially modified versions of the Banados-Teitelboim-Zanelli (BTZ) black holes. The limiting cases B=0 or L=0 exactly are reduced to the Global Embedding Minkowski Space (GEMS) solution of the BTZ black holes. 
  Dual recycling is an advanced optical technique to enhance the signal-to-noise ratio of laser interferometric gravitational wave detectors in a limited bandwidth. To optimise the center of this band with respect to Fourier frequencies of expected gravitational wave signals detuned dual recycling has to be implemented. We demonstrated detuned dual recycling on a fully suspended 30m prototype interferometer. A control scheme that allows to tune the detector to different frequencies will be outlined. Good agreement between the experimental results and numerical simulations has been achieved. 
  The paper addresses the quantization of minisuperspace cosmological models, with application to the Taub Model. By desparametrizing the model with an extrinsic time, a formalism is developed in order to define a conserved Schr\"{o}dinger inner product in the space of solutions of the Wheeler-De Witt equation. A quantum version of classical canonical transformations is introduced for connecting the solutions of the Wheeler-De Witt equation with the wave functions of the desparametrized system. Once this correspondence is established, boundary conditions on the space of solutions of the Wheeler-De Witt equation are found to select the physical subspace. The question of defining boundary conditions on the space of solutions of the Wheeler-De Witt equation without having reduced the system is examined. 
  A very simplified model of the Universe is considered in order to propose an alternative approach to the irreversible evolution of the Universe at very early times. The entropy generation at the quantum stage can be thought as a consequence of an instability of the system. Then particle creation arises from this instability. 
  Two complementary and equally important approaches to relativistic physics are explained. One is the standard approach, and the other is based on a study of the flows of an underlying physical substratum. Previous results concerning the substratum flow approach are reviewed, expanded, and more closely related to the formalism of General Relativity. An absolute relativistic dynamics is derived in which energy and momentum take on absolute significance with respect to the substratum. Possible new effects on satellites are described. 
  We consider the interactions of a strong gravitational wave with electromagnetic fields using the 1+3 orthonormal tetrad formalism. A general system of equations are derived, describing the influence of a plane fronted parallel (pp) gravitational wave on a cold relativistic multi-component plasma. We focus our attention on phenomena that are induced by terms that are higher order in the gravitational wave amplitude. In particular, it is shown that parametric excitations of plasma oscillations takes place, due to higher order gravitational non-linearities. The implications of the results are discussed. 
  We investigated the back reaction of cosmological perturbations on the evolution of the universe using the second order perturbation of the Einstein's equation. To incorporate the back reaction effect due to the inhomogeneity into the framework of the cosmological perturbation, we used the renormalization group method. The second order zero mode solution which appears by the non-linearities of the Einstein's equation is regarded as a secular term of the perturbative expansion, we renormalized a constant of integration contained in the background solution and absorbed the secular term to this constant. For a dust dominated universe, using the second order gauge invariant quantity, we derived the renormalization group equation which determines the effective dynamics of the Friedman-Robertson-Walker universe with the back reaction effect in a gauge invariant manner. We obtained the solution of the renormalization group equation and found that perturbations of the scalar mode and the long wavelength tensor mode works as positive spatial curvature, and the short wavelength tensor mode as radiation fluid. 
  In this paper we redefine the well-known metric-affine Hilbert Lagrangian in terms of a spin-connection and a spin-tetrad. On applying the Poincar\'e-Cartan method and using the geometry of gauge-natural bundles, a global gravitational superpotential is derived. On specializing to the case of the Kosmann lift, we recover the result originally found by Kijowski (1978) for the metric (natural) Hilbert Lagrangian. On choosing a different, suitable lift, we can also recover the Nester-Witten 2-form, which plays an important role in the energy positivity proof and in many quasi-local definitions of mass. 
  The correct characterization of the concept of mass point in general relativity is a straightforward consequence of the original form of solution given by Schwarzschild to the problem of the Einstein field of a material point. 
  We investigate spherically symmetric continuously self-similar (CSS) solutions in the SU(2) sigma model coupled to gravity. Using mixed numerical and analytical methods, we provide evidence for the existence (for small coupling) of a countable family of regular CSS solutions. This fact is argued to have important implications for the ongoing studies of black hole formation in the model. 
  We consider the asymptotic behaviour of spatially homogeneous spacetimes of Bianchi type IX close to the singularity (we also consider some of the other Bianchi types, e. g. Bianchi VIII in the stiff fluid case). The matter content is assumed to be an orthogonal perfect fluid with linear equation of state and zero cosmological constant. In terms of the variables of Wainwright and Hsu, we have the following results. In the stiff fluid case, the solution converges to a point for all the Bianchi class A types. For the other matter models we consider, the Bianchi IX solutions generically converge to an attractor consisting of the closure of the vacuum type II orbits. Furthermore, we observe that for all the Bianchi class A spacetimes, except those of vacuum Taub type, a curvature invariant is unbounded in the incomplete directions of inextendible causal geodesics. 
  Liberati, Rothman and Sonego have recently showed that objects collapsing into extremal Reissner-Nordstrom black holes do not behave as thermal objects at any time in their history. In particular, a temperature, and hence thermodynamic entropy, are undefined for them. I demonstrate that the analysis goes through essentially unchanged for Kerr black holes. 
  We address the old question of whether or not a uniformly accelerated charged particle radiates, and consequently, if weak equivalence principle is violated by electrodynamics. We show that radiation has different meanings; some absolute, some relative. Detecting photons or electromagnetic waves is NOT absolute, it depends both on the electromagnetic field and on the state of motion of the antenna. An antenna used by a Rindler observer does not detect any radiation from a uniformly accelerated co-moving charged particle. Therefore, a Rindler observer cannot decide whether or not he is in an accelerated lab or in a gravitational field. We also discuss the general case. 
  The quasiregular singularities (horizons) that form in the collision of cross polarized electromagnetic waves are, as in the linear polarized case unstable. The validity of the Helliwell-Konkowski stability conjecture is tested for a number of exact back-reaction cases. In the test electromagnetic case the conjecture fails to predict the correct nature of the singularity while in the scalar field and in the null dust cases the aggrement is justified. 
  We quantize the Reissner-Nordstr\"om black hole using an adaptation of Kucha\v{r}'s canonical decomposition of the Kruskal extension of the Schwarzschild black hole. The Wheeler-DeWitt equation turns into a functional Schroedinger equation in Gaussian time by coupling the gravitational field to a reference fluid or dust. The physical phase space of the theory is spanned by the mass, $M$, the charge, $Q$, the physical radius, $R$, the dust proper time, $\tau$, and their canonical momenta. The exact solutions of the functional Schroedinger equation imply that the difference in the areas of the outer and inner horizons is quantized in integer units. This agrees in spirit, but not precisely, with Bekenstein's proposal on the discrete horizon area spectrum of black holes. We also compute the entropy in the microcanonical ensemble and show that the entropy of the Reissner-Nordstr\"om black hole is proportional to this quantized difference in horizon areas. 
  As an example of a black hole in a non-flat background a composite static spacetime is constructed. It comprises a vacuum Schwarzschild spacetime for the interior of the black hole across whose horizon it is matched on to the spacetime of Vaidya representing a black hole in the background of the Einstein universe. The scale length of the exterior sets a maximum to the black hole mass. To obtain a non-singular exterior, the Vaidya metric is matched to an Einstein universe. The behaviour of scalar waves is studied in this composite model. 
  We analyze the stress-energy tensor necessary to generate a general stationary and axisymmetric spacetime. The constraints on the geometry arising from considering a perfect fluid as a source are derived. For a fluid with a nonzero stress tensor, we obtain two necessary conditions on the metric. As an example, we show that the rotating wormhole presented in the literature can not be described by either a perfect fluid or by a fluid with anisotropic stresses. 
  We describe some relations between the long-time asymptotic behavior of the vacuum Einstein evolution equations and the geometrization of 3-manifolds. These relations are expressed in terms of evolution of CMC hypersurfaces in the vacuum space-time.Some results are also obtained on the singularity avoidance of CMC foliations. In addition, the paper describes a number of open problems relating these two areas. 
  We measured Newton's gravitational constant G using a new torsion balance method. Our technique greatly reduces several sources of uncertainty compared to previous measurements: (1) it is insensitive to anelastic torsion fiber properties; (2) a flat plate pendulum minimizes the sensitivity due to the pendulum density distribution; (3) continuous attractor rotation reduces background noise. We obtain G = (6.674215 +- 0.000092)x10^-11 m^3kg^-1s^-2; the Earth's mass is, therefore, M = (5.972245 +- 0.000082)x10^24 kg and the Sun's mass is M = (1.988435 +- 0.000027)x10^30kg. 
  We study continuous groups of generalized Kerr-Schild transformations and the vector fields that generate them in any n-dimensional manifold with a Lorentzian metric. We prove that all these vector fields can be intrinsically characterized and that they constitute a Lie algebra if the null deformation direction is fixed. The properties of these Lie algebras are briefly analyzed and we show that they are generically finite-dimensional but that they may have infinite dimension in some relevant situations. The most general vector fields of the above type are explicitly constructed for the following cases: any two-dimensional metric, the general spherically symmetric metric and deformation direction, and the flat metric with parallel or cylindrical deformation directions. 
  In this paper, solutions to the Ernst equation are investigated that depend on two real analytic functions defined on the interval [0,1]. These solutions are introduced by a suitable limiting process of Backlund transformations applied to seed solutions of the Weyl class. It turns out that this class of solutions contains the general relativistic gravitational field of an arbitrary differentially rotating disk of dust, for which a continuous transition to some Newtonian disk exists. It will be shown how for given boundary conditions (i. e. proper surface mass density or angular velocity of the disk) the gravitational field can be approximated in terms of the above solutions. Furthermore, particular examples will be discussed, including disks with a realistic profile for the angular velocity and more exotic disks possessing two spatially separated ergoregions. 
  Numerical evidence is presented for the existence of a new family of static, globally regular `cosmological' solutions of the spherically symmetric Einstein-Yang-Mills-Higgs equations. These solutions are characterized by two natural numbers ($m\geq 1$, $n\geq 0$), the number of nodes of the Yang-Mills and Higgs field respectively. The corresponding spacetimes are static with spatially compact sections with 3-sphere topology. 
  We apply Cartan's method of equivalence to construct invariants of a given null hypersurface in a Lorentzian space-time. This enables us to fully classify the internal geometry of such surfaces and hence solve the local equivalence problem for null hypersurface structures in 4-dimensional Lorentzian space-times. 
  By considering full-field string solutions of the Abelian--Higgs model, we modify the model of a fluid of strings (which is composed of Nambu strings) to obtain a model for a ``fluid of vortices.'' With this model, and following closely Soleng's proposal of a fluid of strings as the source of a Milgrom-type correction to the Newton dynamics, we determine quantitatively the modified dynamics generated by a static, spherical fluid of vortices. 
  One of the known mathematical descriptions of singularities in General Relativity is the b-boundary, which is a way of attaching endpoints to inextendible endless curves in a spacetime. The b-boundary of a manifold M with connection is constructed by forming the Cauchy completion of the frame bundle LM equipped with a certain Riemannian metric, the b-metric G. We study the geometry of (LM,G) as a Riemannian manifold in the case when the connection is the Levi-Civita connection of a Lorentzian metric g on M. In particular, we give expressions for the curvature and discuss the isometries and the geodesics of (LM,G) in relation to the geometry of (M,g). 
  This paper is concerned with two themes: imprisoned curves and the b-length functional. In an earlier paper by the author, it was claimed that an endless incomplete curve partially imprisoned in a compact set admits an endless null geodesic cluster curve. Unfortunately, the proof was flawed. We give an outline of the problem and remedy the situation by providing a proof by different methods. Next, we obtain some results concerning the structure of b-length neighbourhoods, which gives a clue to how the geometry of a spacetime is encoded in the pseudo-orthonormal frame bundle equipped with the b-metric. We also show that a previous result by the author, proving total degeneracy of a b-boundary fibre in some cases, does not apply to imprisoned curves. Finally, we correct some results in the literature linking the b-lengths of general curves in the frame bundle with the b-length of the corresponding horizontal curves. 
  In this comment we point out some problems with the approach used in the paper "Scalar fields as dark matter in spiral galaxies" in the attempt to explain the flatness of the rotation curves of galaxies. 
  It has been shown that the negative norm states necessarily appear in a covariant quantization of the free minimally coupled scalar field in de Sitter space [1,2]. In this process ultraviolet and infrared divergences have been automatically eliminated [3]. A natural renormalization of the one-loop interacting quantum field in Minkowski space-time ($\lambda\phi^4$ theory) has been achieved through the consideration of the negative norm states [4]. One-loop effective action for scalar field in a general curved space-time has been calculated by this method and a natural renormalization procedure in the one-loop approximation has been established. 
  We present a calculation of the maximum sensitivity achievable by the LIGO Gravitational wave detector in construction, due to limiting thermal noise of its suspensions. We present a method to calculate thermal noise that allows the prediction of the suspension thermal noise in all its 6 degrees of freedom, from the energy dissipation due to the elasticity of the suspension wires. We show how this approach encompasses and explains previous ways to approximate the thermal noise limit in gravitational waver detectors. We show how this approach can be extended to more complicated suspensions to be used in future LIGO detectors. 
  We consider a spacetime consisting of an empty void separated from an almost Friedmann-Lema\^\i tre-Robertson-Walker (FLRW) dust universe by a spherically symmetric, slowly rotating shell which is comoving with the cosmic dust. We treat in a unified manner all types of the FLRW universes. The metric is expressed in terms of a constant characterizing the angular momentum of the shell, and parametrized by the comoving radius of the shell. Treating the rotation as a first order perturbation, we compute the dragging of inertial frames as well as the apparent motion of distant stars within the void. Finally, we discuss, in terms of in principle measurable quantities, 'Machian' features of the model. 
  A spherical gravity wave (GW) detector, unlike interferometers and bars, is a natural multi-mode device, i.e., it is capable of independently gathering information on all five quadrupole and one monopole amplitudes of a general incoming GW. This is because the sphere's degenerate oscillation eigenmodes are uniquely matched to the structure of the GWs' Riemann tensor components. Suitable linear combinations of the system readout channels completely deconvolve the six GW amplitudes. The present article is concerned with the theoretical reasons for the remarkable properties of a spherical GW detector. The analysis proceeds from first principles and is based on essentially no 'ad hoc' hypotheses. The mathematical beauty of the theory is outstanding, and abundant detail is given for a thorough understanding of the fundamental facts and ideas. Experimental evidence of the accuracy of the model is also provided, where possible. 
  In this manuscript the authors present a detailed answer to the comment in order to avoid misunderstandings in the future. 
  Spherical dust collapse generally forms a shell focusing naked singularity at the symmetric center. This naked singularity is massless. Further the Newtonian gravitational potential and speed of the dust fluid elements are everywhere much smaller than unity until the central shell focusing naked singularity formation if an appropriate initial condition is set up. Although such a situation is highly relativistic, the analysis by the Newtonian approximation scheme is available even in the vicinity of the space-time singularity. This remarkable feature makes the analysis of such singularity formation very easy. We investigate non-spherical even-parity matter perturbations in this scheme by complementary using numerical and semi-analytical approaches, and estimate linear gravitational waves generated in the neighborhood of the naked singularity by the quadrupole formula. The result shows good agreement with the relativistic perturbation analysis recently performed by Iguchi et al. The energy flux of the gravitational waves is finite but the space-time curvature carried by them diverges. 
  We consider a model of an inhomogeneous universe including a massless scalar field, where the inhomogeneity is assumed to consist of many black holes. This model can be constructed by following Lindquist and Wheeler, which has already been investigated without including scalar field to show that an averaged scale factor coincides with that of the Friedmann model. In this work we construct the inhomogeneous universe with an massless scalar field, where we assume that the averaged scale factor and scalar field are given by those of the Friedmann model including a scalar field. All of our calculations are carried out in the framework of Brans-Dicke gravity. In constructing the model of an inhomogeneous universe, we define the mass of a black hole in the Brans-Dicke expanding universe which is equivalent to ADM mass if the mass evolves adiabatically, and obtain an equation relating our mass to the averaged scalar field and scale factor. As the results we find that the mass has an adiabatic time dependence in a sufficiently late stage of the expansion of the universe, and that the time dependence is qualitatively diffenrent according to the sign of the curvature of the universe: the mass increases decelerating in the closed universe case, is constant in the flat case and decreases decelerating in the open case. It is also noted that the mass in the Einstein frame depends on time. Our results that the mass has a time dependence should be retained even in the general scalar-tensor gravitiy with a scalar field potential. Furthermore, we discuss the relation of our results to the uniqueness theorem of black hole spacetime and gravitational memory effect. 
  We investigate the Cauchy problem for the Einstein - scalar field equations in asymptotically flat spherically symmetric spacetimes, in the standard 1+3 formulation. We prove the local existence and uniqueness of solutions for initial data given on a space-like hypersurface in the Sobolev $H_1\cap H_{1,4} $ space. Solutions exist globally if a central (integral) singularity does not form and/or outside an outgoing null hypersurface. An explicit example demonstrates that there exists a local evolution with a naked initial curvature singularity at the symmetry centre. 
  An important question in mathematical relativity theory is that of the nature of spacetime singularities. The equations of general relativity, the Einstein equations, are essentially hyperbolic in nature and the study of spacetime singularities is naturally related to blow-up phenomena for nonlinear hyperbolic systems. These connections are explained and recent progress in applying the theory of hyperbolic equations in this field is presented. A direction which has turned out to be fruitful is that of constructing large families of solutions of the Einstein equations with singularities of a simple type by solving singular hyperbolic systems. Heuristic considerations indicate, however, that the generic case will be much more complicated and require different techniques. 
  I sketch the main lines of development of the research in quantum gravity, from the first explorations in the early thirties to nowadays. 
  Charged extremal black holes cannot fully evaporate through the Hawking effect and are thus long lived. Over their lifetimes, these black holes take part in a variety of astrophysical processes, including many that lead to their eventual destruction. This paper explores the various events that shape the life of extremal black holes and calculates the corresponding time scales. 
  In the following we undertake to derive quantum theory as a stochastic low-energy and coarse-grained theory from a more primordial discrete and basically geometric theory living on the Planck scale and which (as we argue) possibly underlies also \tit{string theory}. We isolate the so-called \tit{ideal elements} which represent at the same time the cornerstones of the framework of ordinary quantum theory and show how and why they encode the \tit{non-local} aspects, being ubiquituous in the quantum realm, in a, on the surface, local way. We show that the quantum non-locality emerges in our approach as a natural consequence of the underlying \tit{two-storey} nature of space-time or the physical vacuum, that is, quantum theory turns out to be a residual effect of the geometric depth structure of space-time on the Planck scale. We indicate how the \tit{measurement problem} and the emergence of the \tit{macroscopic sub-regime} can be understood in this framework. 
  We will show that the Nariai metric, i.e. the static spherically symmetric vacuum spacetime with a cosmological constant, admits a conformally Kerr-Schild spacetime representation. We find the vacuum solutions of the Einstein-Maxwell equations for the Nariai metric using the Horsky-Mitskievich generating conjecture. 
  A system of charged bosons at finite temperature and chemical potential is studied in a general-relativistic framework. We assume that the boson fields interact only gravitationally. At sufficiently low temperature the system exists in two phases: the gas and the condensate. By studying the condensation process numerically we determine the critical temperature $T_c$ at which the condensate emerges. As the temperature decreases, the system eventually settles down in the ground state of a cold boson star. 
  A Hamiltonian system with a modified Henon-Heiles potential is investigated. This describes the motion of free test particles in vacuum gravitational pp-wave spacetimes with both quadratic ("homogeneous") and cubic ("non-homogeneous") terms in the structural function. It is shown that, for energies above a certain value, the motion is chaotic in the sense that the boundaries separating the basins of possible escapes become fractal. Similarities and differences with the standard Henon-Heiles and the monkey saddle systems are discussed. The box-counting dimension of the basin boundaries is also calculated. 
  Remarkable efforts in the study of the semi-classical regime of kinematical loop quantum gravity are currently underway. In this note, we construct a ``quasi-coherent'' weave state using Gaussian factors. In a similar fashion to some other proposals, this state is peaked in both the connection and the spin network basis. However, the state constructed here has the novel feature that, in the spin network basis, the main contribution for this state is given by the fundamental representation, independently of the value of the parameter that regulates the Gaussian width. 
  Introduction of Vaidya metrics into the Expansive Nondecelerative Universe model allows to localize the energy of gravitational field. On the assumption that there is an interaction of long-range gravitational and electromagnetic fields, the localization might be verified experimentally. In this contribution some details on such an experiment are given. 
  For stationary cylindrically symmetric solutions of the Einstein-Maxwell equation we have shown that the "charged" solutions of McCrea, Chitre et al.(CGN), Van den Bergh and Wils (VW) can be obtained from the seed metrics using generating conjecture. The McCrea "charged" solution has as a seed vacuum metric the Van Stockum solution with a Killing vector (0,0,1,0). The CGN "charged" solution and the VW "charged" solution have the static seed metrics connected by the complex substitution (t --> iz), (z --> it) and the Killing vector which is a simple linear combination of ${\partial}_{\phi}$ and ${\partial}_{t}$ Killing vectors (VW), respectively ${\partial}_{\phi}$ and ${\partial}_{z}$ Killing vectors (CGN). 
  We consider the quantum mechanics of a system consisting of two identical, Planck-size Schwarzschild black holes revolving around their common center of mass. We find that even in a very highly-excited state such a system has very sharp, discrete energy eigenstates, and the system performs very rapid transitions from a one stationary state to another. For instance, when the system is in the 100th excited state, the life times of the energy eigenstates are of the order of $10^{-30}$ s, and the energies of gravitons released in transitions between nearby states are of the order of $10^{22}$ eV. 
  Previous studies \cite{berger98a} have provided strong support for a local, oscillatory approach to the singularity in U(1) symmetric, spatially inhomogeneous vacuum cosmologies on $T^3 \times R$. The description of a vacuum Bianchi type IX, spatially homogeneous Mixmaster cosmology (on $S^3 \times R$) in terms of the variables used to describe the U(1) symmetric cosmologies indicates that the oscillations in the latter are in fact those of local Mixmaster dynamics. One of the variables of the U(1) symmetric models increases only at the end of a Mixmaster era. Such an increase therefore yields a qualitative signature for local Mixmaster dynamics in spatially inhomogeneous cosmologies. 
  The controversy between relativistic causality and quantum non-locality can be resolved by establishing the general relativistic background of quantum non-locality. 
  We report a numerical evolution of axisymmetric Brill waves. The numerical algorithm has new features, including (i) a method for keeping the metric regular on the axis and (ii) the use of coordinates that bring spatial infinity to the edge of the computational grid. The dependence of the evolved metric on both the amplitude and shape of the initial data is found. 
  We argue that the Lagrangian for gravity should remain bounded at large curvature, and interpolate between the weak-field tested Einstein-Hilbert Lagrangian L_EH = R /16 pi G and a pure cosmological constant for large R with the curvature-saturated ansatz L_cs=L_EH/ \sqrt{1+l^4 R^2}, where l is a length parameter expected to be a few orders of magnitude above the Planck length. The curvature-dependent effective gravitational constant defined by dL/dR = 1/16 pi G_eff is G_eff = G \sqrt{1+l^4 R^2}^{3}, and tends to infinity for large $R$, in contrast to most other approaches where G_eff-> 0. The theory possesses neither ghosts nor tachyons. In a curvature-saturated cosmology, the coordinates with ds^2 = a^2 [da^2/B(a) - dx^2 - dy^2- dz^2] are most convenient since the curvature scalar becomes a linear function of $B(a)$. Solutions with a big-bang singularity have a much milder behavior of the curvature than in Einstein's theory. In synchronized time, the metric is given by ds^2 = dt^2 - t^{6/5(dx^2 + dy^2+ dz^2). On the technical side we show that two different conformal transformations make L_cs asymptotically equivalent to the Gurovich-ansatz L= | R |^{4/3} on the one hand, and to Einstein's theory with a minimally coupled scalar field with self-interaction on the other. 
  We study two type effects of gravitational field on mechanical gyroscopes (i.e. rotating extended bodies). The first depends on special relativity and equivalence principle. The second is related to the coupling (i.e. a new force) between the spins of mechanical gyroscopes, which would violate the equivalent principle. In order to give a theoretical prediction to the second we suggest a spin-spin coupling model for two mechanical gyroscopes. An upper limit on the coupling strength is then determined by using the observed perihelion precession of the planet's orbits in solar system. We also give predictions violating the equivalence principle for free-fall gyroscopes . 
  Following the subtraction procedure for manifolds with boundaries, we calculate by variational methods, the Schwarzschild-Anti-de Sitter and the Anti-de Sitter space energy difference. By computing the one loop approximation for TT tensors we discover the existence of an unstable mode at zero temperature, which can be stabilized by the boundary reduction method. Implications on a foam-like space are discussed. 
  We describe an approach to the issue of the singularities of null hypersurfaces, due to the focusing of null geodesics, in the context of the recently introduced formulation of GR via null foliations. 
  Using our new Post-Newtonian SPH (smoothed particle hydrodynamics) code, we study the final coalescence and merging of neutron star (NS) binaries. We vary the stiffness of the equation of state (EOS) as well as the initial binary mass ratio and stellar spins. Results are compared to those of Newtonian calculations, with and without the inclusion of the gravitational radiation reaction. We find a much steeper decrease in the gravity wave peak strain and luminosity with decreasing mass ratio than would be predicted by simple point-mass formulae. For NS with softer EOS (which we model as simple $\Gamma=2$ polytropes) we find a stronger gravity wave emission, with a different morphology than for stiffer EOS (modeled as $\Gamma=3$ polytropes as in our previous work). We also calculate the coalescence of NS binaries with an irrotational initial condition, and find that the gravity wave signal is relatively suppressed compared to the synchronized case, but shows a very significant second peak of emission. Mass shedding is also greatly reduced, and occurs via a different mechanism than in the synchronized case. We discuss the implications of our results for gravity wave astronomy with laser interferometers such as LIGO, and for theoretical models of gamma-ray bursts (GRBs) based on NS mergers. 
  We rigorously analyze the frequency response functions and antenna sensitivity patterns of three types of interferometric detectors to scalar mode of gravitational waves which is predicted to exist in the scalar-tensor theory of gravity. By a straightforward treatment, we show that the antenna sensitivity pattern of the simple Michelson interferometric detector depends strongly on the wave length $\lambda_{\rm SGW}$ of the scalar mode of gravitational waves if $\lambda_{\rm SGW}$ is comparable to the arm length of the interferometric detector. For the Delay-Line and Fabry-Perot interferometric detectors with arm length much shorter than $\lambda_{\rm SGW}$, however, the antenna sensitivity patterns depend weakly on $\lambda_{\rm SGW}$ even though $\lambda_{\rm SGW}$ is comparable to the effective path length of those interferometers. This agrees with the result obtained by Maggiore and Nicolis. 
  We calculate the precession of a gyroscope at rest in a Bondi spacetime. It is shown that, far from the source, the leading term in the rate of precession of the gyroscope is simply expressed through the news function of the system, and vanishes if and only if there is no news. Rough estimates are presented, illustrating the order of magnitude of the expected effect for different scenarios. It is also shown from the next order term that non-radiative (but time dependent) spacetimes will produce a gyroscope precession of that order, providing thereby ``observational'' evidence for the violation of the Huygens's principle. 
  We present the details of an algorithm for the global evolution of asymptotically flat, axisymmetric spacetimes, based upon a characteristic initial value formulation using null cones as evolution hypersurfaces. We identify a new static solution of the vacuum field equations which provides an important test bed for characteristic evolution codes. We also show how linearized solutions of the Bondi equations can be generated by solutions of the scalar wave equation, thus providing a complete set of test beds in the weak field regime. These tools are used to establish that the algorithm is second order accurate and stable, subject to a Courant-Friedrichs-Lewy condition. In addition, the numerical versions of the Bondi mass and news function, calculated at scri on a compactified grid, are shown to satisfy the Bondi mass loss equation to second order accuracy. This verifies that numerical evolution preserves the Bianchi identities. Results of numerical evolution confirm the theorem of Christodoulou and Klainerman that in vacuum, weak initial data evolve to a flat spacetime. For the class of asymptotically flat, axisymmetric vacuum spacetimes, for which no nonsingular analytic solutions are known, the algorithm provides highly accurate solutions throughout the regime in which neither caustics nor horizons form. 
  It is shown that the formulation of the Einstein equations widely in use in numerical relativity, namely, the standard ADM form, as well as some of its variations (including the most recent conformally-decomposed version), suffers from a certain but standard type of ill-posedness. Specifically, the norm of the solution is not bounded by the norm of the initial data irrespective of the data. A long-running numerical experiment is performed as well, showing that the type of ill-posedness observed may not be serious in specific practical applications, as is known from many numerical simulations. 
  All diagonal proper Bianchi I space-times are determined which admit certain important symmetries. It is shown that for Homotheties, Conformal motions and Kinematic Self-Similarities the resulting space-times are defined explicitly in terms of a set of parameters whereas Affine Collineations, Ricci Collineations and Curvature Collineations, if they are admitted, they determine the metric modulo certain algebraic conditions. In all cases the symmetry vectors are explicitly computed. The physical and the geometrical consequences of the results are discussed and a new anisitropic fluid, physically valid solution which admits a proper conformal Killing vector, is given. 
  Using an effective potential method we examine binary black holes where the individual holes carry spin. We trace out sequences of quasi-circular orbits and locate the innermost stable circular orbit as a function of spin. At large separations, the sequences of quasi-circular orbits match well with post-Newtonian expansions, although a clear signature of the simplifying assumption of conformal flatness is seen. The position of the ISCO is found to be strongly dependent on the magnitude of the spin on each black hole. At close separations of the holes, the effective potential method breaks down. In all cases where an ISCO could be determined, we found that an apparent horizon encompassing both holes forms for separations well inside the ISCO. Nevertheless, we argue that the formation of a common horizon is still associated with the breakdown of the effective potential method. 
  The self-gravitating, spherically symmetric thin shells built of orbiting particles are sstudied. Two new features are found. One is the minimal possible value for an angular momentum of particles, above which elleptic orbits become possible. The second is the coexistence of both the wormhole solutions and the elleptic or hyperbolic orbits for the same values of the parameters (but different initial conditions). Possible applications of these results to astrophysics and quantum black holes are briefly discussed. 
  We explore the possibility that a charged particle moving in the gravitational field generated by a scalar star could radiate energy via a recently proposed gravitational \v{C}erenkov mechanism. We numerically prove that this is not possible for stable boson stars. We also show that soliton stars could have \v{C}erenkov radiation for particular values of the boson mass, although diluteness of the star grows and actual observational possibility decreases for the more usually discussed boson masses. These conclusions diminish, although do not completely rule out, the observational possibility of actually detecting scalar stars using this mechanism, and lead us to consider other forms, like gravitational lensing. 
  Recent scenarios of the TeV-scale brane cosmology suggest a possibility of existence in the early universe of two-dimensional topological defects: relativistic membranes. Like cosmic strings, oscillating membranes could emit gravitational radiation contributing to a stochastic background of gravitational waves. We calculate dilaton and gravitational radiation from a closed toroidal membrane excited along one homology cycle. The spectral-angular distributions of dilaton and gravitational radiation are obtained in a closed form in terms of Bessel's functions. The angular distributions are affected by oscillating factors due to an interference of radiation from different segments of the membrane. The dilaton radiation power is dominated by a few lower harmonics of the basic frequency, while the spectrum of the gravitational radiation contains also a substantial contribution from higher harmonics. The radiative lifetime of the membrane is determined by its tension and depends weakly on the ratio of two radii of the torus. Qualitatively it is equal to the ratio of the membrane area at the maximal extension to the gravitational radius of the membrane as a whole. 
  A lattice quantum gravity model in 4 dimensional Riemannian spacetime is constructed based on the SU(2) Ashtekar formulation of general relativity. This model can be understood as one of the family of models sometimes called ``spin foam models.'' A version of the action of general relativity in continuum is introduced and its lattice version is defined. A dimensionless ``(inverse) coupling'' constant is defined so that the value of the action of the model is finite per lattice point. The path integral of the model is expanded in the characters and shown to be written as a sum over surface-like excitations in spacetime. A 3 dimensional version of the model exists and can be reduced to lattice BF theory. The expectation values of some quantities are computed in 3 dimensions and the meanings of the results are discussed. Although the model is studied on a hyper cubic lattice for simplicity, it can be generalized to a randomly triangulated lattice with small modifications. 
  We investigate the causal structure of the Harada-Iguchi-Nakao (HIN)'s exact solution in detail, which describes the dynamical formation of naked singularity in the collapse of a regular spherical cluster of counterrotating particles. There are three kinds of radial null geodesics in the HIN spacetime. One is the regular null geodesics and the other two are the null geodesics which terminate at the singularity. The central massless singularity is timelike naked singularity and satisfies the strong curvature condition along the null geodesics except for the instant of singularity formation. The cluster dynamically asymptotes to the singular static Einstein cluster in which centrifugal force is balanced with gravity. The HIN solution provides an interesting example which demonstrates that collisionless particles invoke timelike naked singularity. 
  The {\it exact} formulation for the effect of the Brans-Dicke scalar field on the gravitational corrections to the Sagnac delay in the Jordan and Einstein frames is presented for the first time. The results completely agree with the known PPN factors in the weak field region. The calculations also reveal how the Brans-Dicke coupling parameter (appears in various correction terms for different types of source/observer orbits. A first order correction of roughly 2.83 x 10^{-1} fringe shift for visible light is introduced by the gravity-scalar field combination for Earth bound equatorial orbits. It is also demonstrated that the final predictions in the two frames do not differ. The effect of the scalar field on the geodetic and Lense-Thirring precession of a spherical gyroscope in circular polar orbit around the Earth is also computed with an eye towards the Stanford Gravity Probe-B experiment currently in progress. The feasibility of optical and matter-wave interferometric measurements is discussed briefly. 
  From Einstein's theory we know that besides the electromagnetic spectrum, objects like quasars, active galactic nuclei, pulsars and black holes also generate a physical signal of purely gravitational nature. The actual form of the signal is impossible to determine analytically, which lead to use of numerical methods.   Two major approaches emerged. The first one formulates the gravitational radiation problem as a standard Cauchy initial value problem, while the other approach uses a Characteristic Initial value formulation. In the strong field region, where caustics in the wavefronts are likely to form, the Cauchy formulation is more advantageous. On the other side, the Characteristic formulation is uniquely suited to study radiation problems because it describes space-time in terms of radiation wavefronts.   The fact that the advantages and disadvantages of these two systems are complementary suggests that one may want to use the two of them together. In a full nonlinear problem it would be advantageous to evolve the inner (strong field) region using Cauchy evolution and the outer (radiation) region with the Characteristic approach. Cauchy Characteristic Matching enables one to evolve the whole space-time matching the boundaries of Cauchy and Characteristic evolution. The methodology of Cauchy Characteristic Matching has been successful in numerical evolution of the spherically symmetric Klein-Gordon-Einstein field equations as well as for 3-D non-linear wave equations. In this thesis the same methodology is studied in the context of the Einstein equations. 
  The Bonnor-Swaminarayan solutions are boost-rotation symmetric space-times which describe the motion of pairs of accelerating particles which are possibly connected to strings (struts). In an explicit and unified form we present a generalised class of such solutions with a few new observations. We then investigate the possible limits in which the accelerations become unbounded. The resulting space-times represent spherical impulsive gravitational waves with snapping or expanding cosmic strings. We also obtain an exact solution for a snapping string of finite length. 
  The C-metric is usually understood as describing two black holes which accelerate in opposite directions under the action of some conical singularity. Here, we examine all the solutions of this type which represent accelerating sources and investigate the null limit in which the accelerations become unbounded. We show that the resulting space-times represent spherical impulsive gravitational waves generated by snapping or expanding cosmic strings. 
  We discuss the construction of wave packets resulting from the solutions of a class of Wheeler-DeWitt equations in Robertson-Walker type cosmologies. We present an ansatz for the initial conditions which leads to a unique determination of the expansion coefficients in the construction of the wave packets with probability distributions which, in an interesting contrast to some of the earlier works, agree well with all possible classical paths. The possible relationship between these initial conditions and signature transition in the context of classical cosmology is also discussed. 
  In special relativity, the definition of coordinate systems adapted to generic accelerated observers is a long-standing problem, which has found unequivocal solutions only for the simplest motions. We show that the Marzke-Wheeler construction, an extension of the Einstein synchronization convention, produces accelerated systems of coordinates with desirable properties: (a) they reduce to Lorentz coordinates in a neighborhood of the observers' world-lines; (b) they index continuously and completely the causal envelope of the world-line (that is, the intersection of its causal past and its causal future: for well-behaved world-lines, the entire space-time). In particular, Marzke-Wheeler coordinates provide a smooth and consistent foliation of the causal envelope of any accelerated observer into space-like surfaces.   We compare the Marzke-Wheeler procedure with other definitions of accelerated coordinates; we examine it in the special case of stationary motions, and we provide explicit coordinate transformations for uniformly accelerated and uniformly rotating observers. Finally, we employ the notion of Marzke-Wheeler simultaneity to clarify the relativistic paradox of the twins, by pinpointing the local origin of differential aging. 
  We explicitly calculate the Green functions describing quantum changes of topology in Friedman-Lemaitre-Robertson-Walker Universes whose spacelike sections are compact but endowed with distinct topologies. The calculations are performed using the long wavelength approximation at second order in the gradient expansion. We argue that complex metrics are necessary in order to obtain a non-vanishing Green functions and interpret this fact as demonstrating that a quantum topology change can be viewed as a quantum tunneling effect. We demonstrate that quantum topological transitions between curved hypersurfaces are allowed whereas no transition to or from a flat section is possible, establishing thus a selection rule. We also show that the quantum topology changes in the direction of negatively curved hypersurfaces are strongly enhanced as time goes on, while transitions in the opposite direction are suppressed. 
  A spacetime with torsion produced by a Kalb-Ramond field coupled gravitationally to the Maxwell field, in accordance with a recent proposal by two of us (PM and SS), is argued to lead to an optical activity in synchrotron radiation from cosmologically distant radio sources. We suggest that this could qualitatively explain observational data from a large number of radio sources displaying such polarization asymmetry (after eliminating effects of Faraday rotation due to magnetized galactic plasma). Possible implications for heterotic string theory are also outlined. 
  By exploring the relationship between the propagation of electromagnetic waves in a gravitational field and the light propagation in a refractive medium, it is shown that, in the presence of a positive cosmological constant, the velocity of light will be smaller than its special relativity value. Then, restricting again to the domain of validity of geometrical optics, the same result is obtained in the context of wave optics. It is argued that this phenomenon and the anisotropy in the velocity of light in a gravitational field are produced by the same mechanism. 
  We determine the expression of the electrostatic self-energy for a point charge in the static black holes with spherical symmetry having suitable properties 
  The two purposes of the paper are (1) to present a regularization of the self-field of point-like particles, based on Hadamard's concept of ``partie finie'', that permits in principle to maintain the Lorentz covariance of a relativistic field theory, (2) to use this regularization for defining a model of stress-energy tensor that describes point-particles in post-Newtonian expansions (e.g. 3PN) of general relativity. We consider specifically the case of a system of two point-particles. We first perform a Lorentz transformation of the system's variables which carries one of the particles to its rest frame, next implement the Hadamard regularization within that frame, and finally come back to the original variables with the help of the inverse Lorentz transformation. The Lorentzian regularization is defined in this way up to any order in the relativistic parameter 1/c^2. Following a previous work of ours, we then construct the delta-pseudo-functions associated with this regularization. Using an action principle, we derive the stress-energy tensor, made of delta-pseudo-functions, of point-like particles. The equations of motion take the same form as the geodesic equations of test particles on a fixed background, but the role of the background is now played by the regularized metric. 
  An outstanding problem in gravitation theory and relativistic astrophysics today is to understand the final outcome of an endless gravitational collapse. Such a continual collapse would take place when stars more massive than few times the mass of the sun collapse under their own gravity on exhausting their nuclear fuel. According to the general theory of relativity, this results either in a black hole, or a naked singularity- which can communicate with faraway observers in the universe. While black holes are (almost) being detected and are increasingly used to model high energy astrophysical phenomena, naked singularities have turned into a topic of active discussion, aimed at understanding their structure and implications. Recent developments here are reviewed, indicating future directions. 
  A general formalism is set up to analyse the response of an arbitrary solid elastic body to an arbitrary metric Gravitational Wave perturbation, which fully displays the details of the interaction antenna-wave. The formalism is applied to the spherical detector, whose sensitivity parameters are thereby scrutinised. A multimode transfer function is defined to study the amplitude sensitivity, and absorption cross sections are calculated for a general metric theory of GW physics. Their scaling properties are shown to be independent of the underlying theory, with interesting consequences for future detector design. The GW incidence direction deconvolution problem is also discussed, always within the context of a general metric theory of the gravitational field. 
  A first attempt at adding matter degrees of freedom to the two-dimensional ``vacuum'' gravity model presented in gr-qc/9907071 is analyzed in this paper. Just as in the previous pure gravity case, quantum diffeomorphism operators (constructed from a Virasoro algebra) possess a dynamical content; their gauge nature is recovered only after the classical limit. Emphasis is placed on the new physical modes modelled on a SU(1,1)-Kac-Moody algebra. The non-trivial coupling to ``gravity'' is a consequence of the natural semi-direct structure of the entire extended algebra. A representation associated with the discrete series of the rigid SU(1,1) is revisited in the light of previously neglected crucial global features which imply the appearance of an SU(1,1)-Kac-Moody fusion rule, determining the rather entangled quantum structure of the physical system. In the classical limit, an action which explicitly couples gravity and matter modes governs the dynamics. 
  Quantum geometrodynamics (QGD) in extended phase space essentially distinguished from the Wheeler - DeWitt QGD is proposed. The grounds for constructing a new version of quantum geometrodynamics are briefly discussed. The main part in the proposed version of QGD is given to the Schrodinger equation for a wave function of the Universe. The Schrodinger equation carries information about a chosen gauge condition which fixes a reference system. The reference system is represented by a continual medium that can be called ``the gravitational vacuum condencate". A solution to the Schrodinger equation contains information about the integrated system ``a physical object + observation means (the gravitational vacuum condensate)". It may be demonstrated that the gravitational vacuum condensate appears to be a cosmological evolution factor. 
  We show that quantum mechanics and general relativity limit the speed $\tilde{\nu}$ of a simple computer (such as a black hole) and its memory space $I$ to $\tilde{\nu}^2 I^{-1} \lsim t_P^{-2}$, where $t_P$ is the Planck time. We also show that the life-time of a simple clock and its precision are similarly limited. These bounds and the holographic bound originate from the same physics that governs the quantum fluctuations of space-time. We further show that these physical bounds are realized for black holes, yielding the correct Hawking black hole lifetime, and that space-time undergoes much larger quantum fluctuations than conventional wisdom claims -- almost within range of detection with modern gravitational-wave interferometers. 
  We propose a new technique for detecting gravitational waves using Quantum Entangled STate (QUEST) technology. Gravitational waves reduce the non-locality of correlated quanta controlled by Bell's inequalities, distorting quantum encryption key statistics away from a pure white noise. Gravitational waves therefore act as shadow eavesdroppers. The resulting colour distortions can, at least in principle, be separated from noise and can differentiate both deterministic and stochastic sources. 
  We present a spin foam model in which the fundamental ``bubble amplitudes'' (the analog of the one-loop corrections in quantum field theory) are finite as the cutoff is removed. The model is a natural variant of the field theoretical formulation of the Barrett-Crane model. As the last, the model is a quantum BF theory plus an implementation of the constraint that reduces BF theory to general relativity. We prove that the fundamental bubble amplitudes are finite by constructing an upper bound, using certain inequalities satisfied by the Wigner (3n)j-symbols, which we derive in the paper. Finally, we present arguments in support of the conjecture that the bubble diagrams of the model are finite at all orders. 
  We use exact general solutions for the spatially flat FRW and the anisotropic Bianchi I cosmologies to show that generically uncoupled scalar fields cooperate to make inflation more probable, while the presence of several interacting fields hinders the occurrence of the phenomenon, in accordance with previous results based on particular power-law solutions. Similar conclusions are reached in the case of Bianchi VI$_0$ spacetimes, for power-law solutions which are proved to be attractors. 
  The rigoruos mathematical theory of the coupling and response of a spherical gravitational wave detector endowed with a set of resonant transducers is presented and developed. A perturbative series in ascending powers of the square root of the ratio of the resonator to the sphere mass is seen to be the key to the solution of the problem. General layouts of arbitrary numbers of transducers can be assessed, and a specific proposal (PHC), alternative to the highly symmetric TIGA of Merkowitz and Johnson, is described in detail. Frequency spectra of the coupled system are seen to be theoretically recovered in full agreement with experimental determinations. 
  The Einstein equations for a plane-symmetric gravitational field coupled to an arbitrary nonlinear sigma model (NSM) are shown to be represented in the form of dynamical equations of a {\it generalized effective NSM}. The gravitational equations are studied in this case by the methods of analyzing NSM equations. In the case of a two-component diagonal NSM exact solutions are found by the functional parameter method. 
  A gravitating global monopole produces a repulsive grativational field outside the core in addition to a solid angular deficit in the Brans-Dicke theory. As a new feature, the angular deficit is dependent on the values of \phi_{\infty} and \omega, where \phi_{\infty} is asymptotic value of scalar field in space-like infinity and \omega is the Brans-Dicke parameter. 
  A recent dynamical formulation at derivative level $\ptl^{3}g$ for fluid spacetime geometries $({\cal M}, {\bf g}, {\bf u})$, that employs the concept of evolution systems in first-order symmetric hyperbolic format, implies the existence in the Weyl curvature branch of a set of timelike characteristic 3-surfaces associated with propagation speed $|v| = \sfrac{1}{2}$ relative to fluid-comoving observers. We show it is the physical role of the constraint equations to prevent realisation of jump discontinuities in the derivatives of the related initial data so that Weyl curvature modes propagating along these 3-surfaces cannot be activated. In addition we introduce a new, illustrative first-order symmetric hyperbolic evolution system at derivative level $\ptl^{2}g$ for baryotropic perfect fluid cosmological models that are invariant under the transformations of an Abelian $G_{2}$ isometry group. 
  The general metric for N-dimensional spherically symmetric and conformally flat spacetimes is given, and all the homogeneous and isotropic solutions for a perfect fluid with the equation of state $p = \alpha \rho$ are found. These solutions are then used to model the gravitational collapse of a compact ball. It is found that when the collapse has continuous self-similarity, the formation of black holes always starts with zero mass, and when the collapse has no such a symmetry, the formation of black holes always starts with a mass gap. 
  We present a path integral formalism for quantising gravity in the form of the spectral action. Our basic principle is to sum over all Dirac operators. The approach is demonstrated on two simple finite noncommutative geometries: the two-point space, and the matrix geometry M_2(C). On the first, the graviton is described by a Higgs field, and on the second, it is described by a gauge field. We start with the partition function and calculate the propagator and Greens functions for the gravitons. The expectation values of distances are evaluated, and we discover that distances shrink with increasing graviton excitations. We find that adding fermions reduces the effects of the gravitational field. A comparison is also made with Rovelli's canonical quantisation approach, and with his idea of spectral path integrals. We include a brief discussion on the quantisation of a Riemannian manifold. 
  In a previous paper (gr-qc/9907063) we described the early inflationary universe in terms of quantum information. In this paper, we analize those results in more detail, and we stress the fact that, during inflation, the universe can be described as a superposed state of quantum registers. The self-reduction of the superposed quantum state is consistent with the Penrose's Objective Reduction (OR) model. The quantum gravity threshold is reached at the end of inflation, and corresponds to a superposed state of 10^9 quantum registers. This is also the number of superposed tubulins-qubits in our brain, which undergo the Penrose-Hameroff's Orchestrated Objective Reduction, (Orch OR), leading to a conscious event. Then, an analogy naturally arises between the very early quantum computing universe,and our mind. 
  The accuracy of astrometric observations conducted via a space-borne optical interferometer orbiting the Earth is expected to approach a few microarcseconds. Data processing of such extremely high-precision measurements requires access to a rigorous relativistic model of light ray propagation developed in the framework of General Relativity. The data-processing of the space interferometric observations must rely upon the theory of general-relativistic transformations between the spacecraft, geocentric, and solar barycentric reference systems allowing unique and unambiguous interpretation of the stellar aberration and parallax effects. On the other hand, the algorithm must also include physically adequate treatment of the relativistic effect of light deflection caused by the spherically-symmetric (monopole-dependent) part of the gravitational field of the Sun and planets as well as the quadrupole- and spin-dependent counterparts of it. In some particular cases the gravitomagnetic field induced by the translational motion of the Sun and planets should be also taken into account for unambigious prediction of the light-ray deflection angle. In the present paper we describe the corresponding software program for taking into account all classical (proper motion, parallax, etc.) and relativistic (aberration, deflection of light) effects up to the microarcsecond threshold and demonstrate, using numerical simulations, how observations of stars and/or quasars conducted on board a space optical interferometer orbiting the Earth can be processed and disentangled. 
  We present results of numerical simulations of the formation of black holes from the gravitational collapse of a massless, minimally-coupled scalar field in 2+1 dimensional, axially-symmetric, anti de-Sitter (AdS) spacetime. The geometry exterior to the event horizon approaches the BTZ solution, showing no evidence of scalar `hair'. To study the interior structure we implement a variant of black-hole excision, which we call singularity excision. We find that interior to the event horizon a strong, spacelike curvature singularity develops. We study the critical behavior at the threshold of black hole formation, and find a continuously self-similar solution and corresponding mass-scaling exponent of approximately 1.2. The critical solution is universal to within a phase that is related to the angle deficit of the spacetime. 
  The existence of symmetries in asymptotically flat space-times are studied from the point of view of initial value problems. General necessary and sufficient (implicit) conditions are given for the existence of Killing vector fields in the asymptotic characteristic and in the hyperboloidal initial value problem (both of them are formulated on the conformally compactified space-time manifold). 
  Assuming a large-scale homogeneous magnetic field, we follow the covariant and gauge-invariant approach used by Tsagas and Barrow to describe the evolution of density and magnetic field inhomogeneities and curvature perturbations in a matter-radiation universe. We use a two parameter approximation scheme to linearize their exact non-linear general-relativistic equations for magneto-hydrodynamic evolution. Using a two-fluid approach we set up the governing equations as a fourth order autonomous dynamical system. Analysis of the equilibrium points for the radiation dominated era lead to solutions similar to the super-horizon modes found analytically by Tsagas and Maartens. We find that a study of the dynamical system in the dust-dominated era leads naturally to a magnetic critical length scale closely related to the Jeans Length. Depending on the size of wavelengths relative to this scale, these solutions show three distinct behaviours: large-scale stable growing modes, intermediate decaying modes, and small-scale damped oscillatory solutions. 
  The recent progress in string theory strongly suggests that formation and evaporation of black holes is a unitary process. This fact makes it imperative that we find a flaw in the semiclassical reasoning that implies a loss of information. We propose a new criterion that limits the domain of classical gravity: the hypersurfaces of a foliation cannot be stretched too much. This conjectured criterion may have important consequences for the early Universe. 
  The connection between multidimensional soliton equations and three-dimensional Riemann space is discussed. 
  Bekenstein and Mukhanov have put forward the idea that, in a quantum theory of gravity a black hole should have a discrete mass spectrum with a concomitant {\it discrete} line emission. We note that a direct consequence of this intriguing prediction is that, compared with blackbody radiation, black-hole radiance is {\it less} entropic. We calculate the ratio of entropy emission rate from a quantum black hole to the rate of black-hole entropy decrease, a quantity which, according to the generalized second law (GSL) of thermodynamics, should be larger than unity. Implications of our results for the GSL, for the value of the fundamental area unit in quantum gravity, and for the spectrum of massless particles in nature are discussed. 
  In order to detect the gravitomagnetic clock effect by means of two counter-orbiting satellites placed on identical equatorial and circular orbits around the Earth with radius 7000 km their radial and azimuthal positions must be known with an accuracy of delta r =10^{-1} mm and delta phi =10^{-2} mas per revolution. In this work we investigate if the radial and azimuthal perturbations induced by the dynamical and static parts of the Earth' s gravitational field meet this requirements. While the radial direction is affected only by harmonic perturbations with periods up to some tens of days, the azimuthal location is perturbed by a secular drift and very long period effects.It results that the present level of accuracy in the knowledge both of the Earth solid and ocean tides, and of the static part of the geopotential does not allow an easy detection of the gravitomagnetic clock effect, at least by using short arcs only. 
  The general relativistic Lense-Thirring effect can be detected by means of a suitable combination of orbital residuals of the laser-ranged LAGEOS and LAGEOS II satellites. While this observable is not affected by the orbital perturbation induced by the zonal Earth solid and ocean tides, it is sensitive to those generated by the tesseral and sectorial tides. The assessment of their influence on the measurement of the parameter mu, with which the gravitomagnetic effect is accounted for, is the goal of this paper. After simulating the combined residual curve by calculating accurately the mismodeling of the more effective tidal perturbations, it has been found that, while the solid tides affect the recovery of mu at a level always well below 1%, for the ocean tides and the other long-period signals Delta mu depends strongly on the observational period and the noise level: Delta mu(tides) amounts to almost 2% after 7 years. The aliasing effect of K1 l=3 p=1 tide and SRP(4241) solar radiation pressure harmonic, with periods longer than 4 years, on the perigee of LAGEOS II yield to a maximum systematic uncertainty on $\m_{LT}$ of less than 4% over different observational periods. The zonal 18.6-year tide does not affect the combined residuals. 
  Chandrasekhar separated the Dirac equation for spinning and massive particles in Kerr geometry in radial and angular parts. Chakrabarti solved the angular equation and found the corresponding eigenvalues for different Kerr parameters. The radial equations were solved asymptotically by Chandrasekhar. In the present paper, we use the WKB approximation to solve the spatially complete radial equation and calculate analytical expressions of radial wave functions for a set of Kerr and wave parameters. From these solutions we obtain local values of reflection and transmission coefficients. 
  A maximally reduced system of equations corresponding to the twisting type N Einstein metrics is given. When the cosmological constant $\lambda\to 0$ they reduce to the standard equations for the vacuum twisting type N's. All the metrics which are conformally equivalent to the twisting type N metrics and which admit 3-dimensional conformal group of symmetries are presented. In the Feferman class of metrics an example is given of a twisting type N metric which satisfies Bach's equations but is not Einstein. 
  Non-time-orthogonal analysis of rotating frames is applied to objects in gravitational orbits and found to be internally consistent. The object's surface speed about its axis of rotation, but not its orbital speed, is shown to be readily detectable by any "enclosed box" experimenter on the surface of such an object. Sagnac type effects manifest readily, but by somewhat subtle means. The analysis is extended to objects bound in non-gravitational orbit, where it is found to be fully in accord with the traditional analysis of Thomas precession. 
  We systematically study late-time tails of scalar waves propagating in neutron star spacetimes. We consider uniform density neutron stars, for which the background spacetime is analytic and the compaction of the star can be varied continously between the Newtonian limit 2M/R << 1 and the relativistic Buchdahl limit 2M/R = 8/9. We study the reflection of a finite wave packet off neutron stars of different compactions 2M/R and find that a Newtonian, an intermediate, and a highly relativistic regime can be clearly distinguished. In the highly relativistic regime, the reflected signal is dominated by quasi-periodic peaks, which originate from the wave packet bouncing back and forth between the center of the star and the maximum of the background curvature potential at R ~ 3 M. Between these peaks, the field decays according to a power-law. In the Buchdahl limit 2M/R -> 8/9 the light travel time between the center and the maximum or the curvature potential grows without bound, so that the first peak arrives only at infinitely late time. The modes of neutron stars can therefore no longer be excited in the ultra-relativistic limit, and it is in this sense that the late-time radiative decay from neutron stars looses all its features and gives rise to power-law tails reminiscent of Schwarzschild black holes. 
  New exact inflationary solutions are presented in the scalar field theory, minimally coupled to gravity, with a potential term. No use is made of the slow rollover approximation. The scale factors are completely nonsingular and the transition to the deccelerating phase is smooth. Moreover, in one of these models, asymptotically one has transition to the matter dominated Universe. 
  Two theorems related to gravitational time delay are proven. Both theorems apply to spacetimes satisfying the null energy condition and the null generic condition. The first theorem states that if the spacetime is null geodesically complete, then given any compact set $K$, there exists another compact set $K'$ such that for any $p,q \not\in K'$, if there exists a ``fastest null geodesic'', $\gamma$, between $p$ and $q$, then $\gamma$ cannot enter $K$. As an application of this theorem, we show that if, in addition, the spacetime is globally hyperbolic with a compact Cauchy surface, then any observer at sufficiently late times cannot have a particle horizon. The second theorem states that if a timelike conformal boundary can be attached to the spacetime such that the spacetime with boundary satisfies strong causality as well as a compactness condition, then any ``fastest null geodesic'' connecting two points on the boundary must lie entirely within the boundary. It follows from this theorem that generic perturbations of anti-de Sitter spacetime always produce a time delay relative to anti-de Sitter spacetime itself. 
  We apply the technique of complex paths to obtain Hawking radiation in different coordinate representations of the Schwarzschild space-time. The coordinate representations we consider do not possess a singularity at the horizon unlike the standard Schwarzschild coordinate. However, the event horizon manifests itself as a singularity in the expression for the semi-classical action. This singularity is regularized by using the method of complex paths and we find that Hawking radiation is recovered in these coordinates indicating the covariance of Hawking radiation. This also shows that there is no correspondence between the particles detected by the model detector and the particle spectrum obtained by the quantum field theoretic analysis -- a result known in other contexts as well. 
  We describe the search for a continuous signal of gravitational radiation from a rotating neutron star in the data taken by the ALLEGRO gravitational wave detector in early 1994. Since ALLEGRO is sensitive at frequencies near 1 kHz, only neutron stars with spin periods near 2 ms are potential sources. There are no known sources of this typ e for ALLEGRO, so we directed the search towards both the galactic center and the globular clus ter 47 Tucanae. The analysis puts a constraint of roughly $8 \times 10^{-24}$ at frequencies near 1 kHz on the gravitational strain emitted from pulsar spin-down in either 47 Tucanae or the galactic center. 
  Referring to a conception put forward by Stuart Kauffman in his "Investigations", it is shown how the onset of classicity can be visualized in terms of an emergent process originating in entangled ensemble states of knotted spin networks. The latter exhibit a suitable autocatalytic behaviour effectively producing knots by knots acting upon other knots. In particular, a quantum computational structure can be described underlying spin networks such that most conditions for a partial ordering are not more valid for the latter. A concep- tual argument is given then indicating that on a fundamental level, physics is non-local and a-temporal, and hence does not admit of the concept of causality. Hence, modelling the emergence of classicity in terms of a percolating web of coherence eventually decohering (in using a cellular automata architecture) does not imply the necessity of visualizing histories of directed percolation as causal sets. These aspects are compatible though with the recent concept of spin foams which do not actually imply distinguished directions of time flow on the micro-level. 
  In extended new general relativity, which is formulated as a reduction of $\bar{Poincar\'e} $gauge theory of gravity whose gauge group is the covering group of the Poincar\'e group, we study the problem of whether the total energy-momentum, total angular momentum and total charge are equal to the corresponding quantities of the gravitational source. We examine this for charged axi-symmetric solutions of gravitational field equations. Our main concern is the restriction on the asymptotic form of the gravitational field variables imposed by the requirement that physical quantities of the total system are equivalent to the corresponding quantities of the charged rotating source body. This requirement can be regarded as an equivalence principle in a generalized sense. 
  Foundations of algebrodynamics based on earlier proposed equations of biquaternionic holomorphy are briefly expounded. Free Maxwell and Yang-Mills Eqs. are satisfied identically on the solutions of primary system which is also related to the Eqs. of shear-free null congruences (SFC), and through them - to the Einstein-Maxwell electrovacuum system. Kerr theorem for SFC reduces the basic system to one algebraic equation, so that with each solution of the latter some (singular) solution of vacuum Eqs. may be associated. We present some exact solutions of basic algebraic and of related field Eqs. with compact structure of singularities of electromagnetic field, in particular having the form of figure "8" curve. Fundamental solution to primary system is analogous to the metric and fields of the Kerr-Newman solution. In addition, in the framework of algebraic dynamics the value of electric charge for this solution is strictly fixed in magnitude and may be set equal to the elementary charge. 
  The generalized Cauchy-Riemann equations (GCRE) in biquaternion algebra appear to be Lorentz-invariant. The Laplace equation is in this case replaced by a nonlinear (complexified) eikonal equation. GCRE contain the 2-spinor and the gauge structures, and their integrability conditions take the form of free-source Maxwell and Yang-Mills equations. For the value of electric charge from GCRE only the quantization rule follows, as well as the treatment of Coulomb law as a stereographic map. The equivalent geometrodynamics in a Weyl-Cartan affine space and the conjecture of a complex-quaternion structure of space-time are discussed. 
  We present a numerical method to compute quasiequilibrium configurations of close binary neutron stars in the pre-coalescing stage. A hydrodynamical treatment is performed under the assumption that the flow is either rigidly rotating or irrotational. The latter state is technically more complicated to treat than the former one (synchronized binary), but is expected to represent fairly well the late evolutionary stages of a binary neutron star system. As regards the gravitational field, an approximation of general relativity is used, which amounts to solving five of the ten Einstein equations (conformally flat spatial metric). The obtained system of partial differential equations is solved by means of a multi-domain spectral method. Two spherical coordinate systems are introduced, one centered on each star; this results in a precise description of the stellar interiors. Thanks to the multi-domain approach, this high precision is extended to the strong field regions. The computational domain covers the whole space so that exact boundary conditions are set to infinity. Extensive tests of the numerical code are performed, including comparisons with recent analytical solutions. Finally a constant baryon number sequence (evolutionary sequence) is presented in details for a polytropic equation of state with gamma=2. 
  We study the problem of quantization of thin shells in a Weyl-Dirac theory by deriving a Wheeler-DeWitt equation from the dynamics. Solutions are found which have interpretations in both cosmology and particle physics. 
  A family of spacetimes suitable for describing the interior of a non-rotational black hole is constructed. The stress-energy tensor is that of a spherically symmetric vacuum, as commonly assumed nowadays. The problem of matching the exterior with the interior region is solved exactly, without using any massive shell nor having to restrict oneself to only asymptotically well-behaved solutions, whatsoever. The main physical and geometrical properties of the resulting black hole solutions are described. As models for the interior the general solution found includes, in particular, two known previous atttempts at solving the problem. Finally, effective macroscopic properties of the solution are linked with quantization issues of the corresponding spacetime. 
  We study a generally covariant model in which local Lorentz invariance is broken "spontaneously" by a dynamical unit timelike vector field $u^a$---the "aether". Such a model makes it possible to study the gravitational and cosmological consequences of preferred frame effects, such as ``variable speed of light" or high frequency dispersion, while preserving a generally covariant metric theory of gravity. In this paper we restrict attention to an action for an effective theory of the aether which involves only the antisymmetrized derivative $\nabla_{[a}u_{b]}$. Without matter this theory is equivalent to a sector of the Einstein-Maxwell-charged dust system. The aether has two massless transverse excitations, and the solutions of the model include all vacuum solutions of general relativity (as well as other solutions). However, the aether generally develops gradient singularities which signal a breakdown of this effective theory. Including the symmetrized derivative in the action for the aether field may cure this problem. 
  Spin-1/2 particles can be used to study inertial and gravitational effects by means of interferometers, particle accelerators, and ultimately quantum systems. These studies require, in general, knowledge of the Hamiltonian and of the inertial and gravitational quantum phases. The procedure followed gives both in the low- and high-energy approximations. The latter affords a more consistent treatment of mass at high energies. The procedure is based on general relativity and on a solution of the Dirac equation that is exact to first-order in the metric deviation. Several previously known acceleration and rotation induced effects are re-derived in a comprehensive, unified way. Several new effects involve spin, electromagnetic and inertial/gravitational fields in different combinations. 
  We calculate the self force acting on a scalar particle which is falling radially into a Schwarzschild black hole. We treat the particle's self-field as a linear perturbation over the fixed Schwarzschild background. The force is calculated by numerically solving the appropriate wave equation for each mode of the field in the time domain, calculating its contribution to the self force, and summing over all modes using Ori's mode-sum regularization prescription. The radial component of the force is attractive at large distances, and becomes repulsive as the particle approaches the event horizon. 
  We study asymptotically constrained systems for numerical integration of the Einstein equations, which are intended to be robust against perturbative errors for the free evolution of the initial data. First, we examine the previously proposed "$\lambda$-system", which introduces artificial flows to constraint surfaces based on the symmetric hyperbolic formulation. We show that this system works as expected for the wave propagation problem in the Maxwell system and in general relativity using Ashtekar's connection formulation. Second, we propose a new mechanism to control the stability, which we call the ``adjusted system". This is simply obtained by adding constraint terms in the dynamical equations and adjusting its multipliers. We explain why a particular choice of multiplier reduces the numerical errors from non-positive or pure-imaginary eigenvalues of the adjusted constraint propagation equations. This ``adjusted system" is also tested in the Maxwell system and in the Ashtekar's system. This mechanism affects more than the system's symmetric hyperbolicity. 
  We consider the dynamics of electromagnetic fields in an almost-Friedmann-Robertson-Walker universe using the covariant and gauge-invariant approach of Ellis and Bruni. Focusing on the situation where deviations from the background model are generated by tensor perturbations only, we demonstrate that the coupling between gravitational waves and a weak magnetic test field can generate electromagnetic waves. We show that this coupling leads to an initial pulse of electromagnetic waves whose width and amplitude is determined by the wavelengths of the magnetic field and gravitational waves. A number of implications for cosmology are discussed, in particular we calculate an upper bound of the magnitude of this effect using limits on the quadrapole anisotropy of the Cosmic Microwave Background. 
  We propose definitions for covariance and local Lorentz invariance applicable when the speed of light $c$ is allowed to vary. They have the merit of retaining only those aspects of the usual definitions which are invariant under unit transformations, and which can therefore legitimately represent the outcome of an experiment. We then discuss some possibilities for invariant actions governing the dynamics of such theories. We consider first the classical action for matter fields and the effects of a changing $c$ upon quantization. We discover a peculiar form of quantum particle creation due to a varying $c$. We then study actions governing the dynamics of gravitation and the speed of light. We find the free, empty-space, no-gravity solution, to be interpreted as the counterpart of Minkowksi space-time, and highlight its similarities with Fock-Lorentz space-time. We also find flat-space string-type solutions, in which near the string core $c$ is much higher. We label them fast-tracks and compare them with gravitational wormholes. We finally discuss general features of cosmological and black hole solutions, and digress on the meaning of singularities in these theories. 
  We contrast the possibility of inflation starting a) from the universe's inception or b) from an earlier non-inflationary state. Neither case is ideal since a) assumes quantum mechanical reasoning is straightforwardly applicable to the early universe; while case b) requires that a singularity still be present. Further, in agreement with Vachaspati and Trodden [1] case b) can only solve the horizon problem if the non-inflationary phase has equation of state $\gamma<4/3$. 
  We examine Friedmann-Robertson-Walker models in three spacetime dimensions. The matter content of the models is composed of a perfect fluid, with a $\gamma$-law equation of state, and a homogeneous scalar field minimally coupled to gravity with a self-interacting potential whose energy density red-shifts as $a^{-2 \nu}$, where a denotes the scale factor. Cosmological solutions are presented for different range of values of $\gamma$ and $\nu$. The potential required to agree with the above red-shift for the scalar field energy density is also calculated. 
  We present a study of black hole threshold phenomena for a self-gravitating, massive complex scalar field in spherical symmetry. We construct Type I critical solutions dynamically by tuning a one-parameter family of initial data composed of a boson star and a massless real scalar field. The real field is used to perturb the boson star via a gravitational interaction which results in a {\em significant} transfer of energy. The resulting critical solutions, which show great similarity with unstable boson stars, persist for a finite time before dispersing or forming a black hole. We extend the stability analysis of Gleiser and Watkins [Nucl. Phys. B319, 733 (1989)], providing a method for calculating the radial dependence of boson star modes of nonzero frequency. We find good agreement between our critical solutions and boson star modes. For critical solutions less than 90% of the maximum boson star mass $M_{\rm max} \simeq 0.633 M_{Pl}^2/m$, a small halo of matter appears in the tail of the solution. This halo appears to be an artifact of the collision between the original boson star and the real field, and does not belong to the true critical solution. It seems that unstable boson stars are unstable to dispersal in addition to black hole formation. Given the similarity in macroscopic stability between boson and neutron stars, we suggest that neutron stars at or beyond the point of instability may also be unstable to explosion. 
  A statistical quantity suitable for distinguishing simply-connected Robertson-Walker (RW) universes is introduced, and its explicit expressions for the three possible classes of simply-connected RW universes with an uniform distribution of matter are determined. Graphs of the distinguishing mark for each class of RW universes are presented and analyzed.There sprout from our results an improvement on the procedure to extract the topological signature of multiply-connected RW universes, and a refined understanding of that topological signature of these universes studied in previous works. 
  The exterior and interior Schwarzschild solutions are rewritten replacing the usual radial variable with an angular one. This allows to obtain some results otherwise less apparent or even hidden in other coordinate systems. 
  We investigate close binary neutron stars in quasiequilibrium states in a general relativistic framework. We assume conformal flatness for the spatial metric and irrotational velocity field for the neutron stars. We adopt the polytropic equation of state. The computation is performed for the polytropic index n(=0.5, 0.66667, 0.8, 1, 1.25), and compactness of neutron stars M/R(=0.03 - 0.3). Results of this paper are as follows. (i) The sequences of the irrotational binary are always terminated at an innermost orbit where a cusp (inner Lagrange point)appears at the inner edges of the stellar surface. The binaries with cusps are found to be dynamically unstable for n=0.5 and stable for n > 0.8 irrespective of M/R < 0.2. For n=0.66667, the stability changes depending on M/R. (ii) The gravitational wave frequency at the innermost orbit turns out to be between 800 and 1500 Hz for realistic compactness 0.14 < M/R < 0.2. (iii) The ISCO for n=0.5 appears to be determined by a hydrodynamic instability for M/R < 0.2. We derive fitting formulae for the relation between the orbital angular velocity at the ISCO and the compactness to clarify it. (iv) The maximum density of neutron stars in binary systems slightly decreases with decreasing the orbital separation and hence they are stable against individual radial collapse during the inspiral. (v) q = J_tot/M_ADM^2 at the innermost orbits is always less than unity for M/R > 0.13 irrespective of n, which indicates that the realistic binary neutron stars satisfy a necessary condition (q<1) for formation of a black hole before the merger. (vi) The specific angular momentum of any mass element in irrotational binary neutron stars at the innermost orbit appears to be too small to form a disk around black holes formed after the merger. 
  Some aspects of multidimensional soliton geometry are considered. 
  In the context of non-critical Liouville strings, we clarify why we expect non-quantum-mechanical dissipative effects to be of order E^2/M_P, where E is a typical energy scale of the probe, and M_P is the Planck scale. In Liouville strings, energy is conserved {\it at best} only as a statistical average, as distinct from Lindblad systems, where it is {\it strictly} conserved at an operator level, and the magnitude of dissipative effects could only be much smaller. We also emphasize the importance of nonlinear terms in the evolution equation for the density matrix, which are important for any analysis of complete positivity. 
  Adaptive techniques are crucial for successful numerical modeling of gravitational waves from astrophysical sources such as coalescing compact binaries, since the radiation typically has wavelengths much larger than the scale of the sources. We have carried out an important step toward this goal, the evolution of weak gravitational waves using adaptive mesh refinement in the Einstein equations. The 2-level adaptive simulation is compared with unigrid runs at coarse and fine resolution, and is shown to track closely the features of the fine grid run. 
  The evolution of the closed Friedmann Universe with a packet of short scalar waves is considered with the help of the Wheeler--DeWitt equation. The packet ensures conservation of homogeneity and isotropy of the metric on average. It is shown that during tunneling the amplitudes of short waves of a scalar field can increase catastrophically promptly if their influence to the metric do not take into account. This effect is similar to the Rubakov-effect of catastrophic particle creation calculated already in 1984.   In our approach to the problem it is possible to consider self-consistent dynamics of the expansion of the Universe and amplification of short waves.  It results in a decrease of the barrier and interruption of amplification of waves, and we get an exit of the wave function from the quantum to the classically available region. 
  In most studies of equivalence principle violation by solar system bodies, it is assumed that the ratio of gravitational to inertial mass for a given body deviates from unity by a parameter Delta which is proportional to its gravitational self-energy. Here we inquire what experimental constraints can be set on Delta for various solar system objects when this assumption is relaxed. Extending an analysis originally due to Nordtvedt, we obtain upper limits on linearly independent combinations of Delta for two or more bodies from Kepler's third law, the position of Lagrange libration points, and the phenomenon of orbital polarization. Combining our results, we extract numerical upper bounds on Delta for the Sun, Moon, Earth and Jupiter, using observational data on their orbits as well as those of the Trojan asteroids. These are applied as a test case to the theory of higher-dimensional (Kaluza-Klein) gravity. The results are three to six orders of magnitude stronger than previous constraints on the theory, confirming earlier suggestions that extra dimensions play a negligible role in solar systemdynamics and reinforcing the value of equivalence principle tests as a probe of nonstandard gravitational theories. 
  The undulating metric tensors of general relativity do not possess a special and common velocity of propagation. Indeed, their velocity can even coincide with the speed of thought. 
  Motivated by a recent paper by the Potsdam numerical relativity group, we have constructed a new numerical code for hydrodynamic simulation of axisymmetric systems in full general relativity. In this code, we solve the Einstein field equation using Cartesian coordinates with appropriate boundary conditions. On the other hand, the hydrodynamic equations are solved in cylindrical coordinates. Using this code, we perform simulations to study axisymmetric collapse of rotating stars, which thereby become black holes or new compact stars, in full general relativity. To investigate the effects of rotation on the criterion for prompt collapse to black holes, we first adopt a polytropic equation of state, $P=K\rho^{\Gamma}$, where $P$, $\rho$, and $K$ are the pressure, rest mass density, and polytropic constant, with $\Gamma=2$. In this case, the collapse is adiabatic (i.e., no change in entropy), and we can focus on the bare effect of rotation. As the initial conditions, we prepare rigidly and differentially rotating stars in equilibrium and then decrease the pressure to induce collapse. In this paper, we consider cases in which $q \equiv J/M_g^2 < 1$, where $J$ and $M_g$ are the angular momentum and the gravitational mass. It is found that the criterion of black hole formation is strongly dependent on the angular momentum parameter $q$. For $q < 0.5$, the criterion is not strongly sensitive to $q$; more precisely, if the rest mass is slightly larger than the maximum allowed value of spherical stars, a black hole is formed. However, for $q \alt 1$, it changes significantly: For $q \simeq 0.9$, the maximum allowed rest mass becomes $\sim 70$ - 80% larger than that for spherical stars. 
  Gravitational-wave detectors are sensitive not only to astrophysical gravitational waves, but also to the fluctuating Newtonian gravitational forces of moving masses in the ground and air around the detector. This paper studies the gravitational effects of density perturbations in the atmosphere, and from massive airborne objects near the detector. These effects were previously considered by Saulson; in this paper I revisit these phenomena, considering transient atmospheric shocks, and the effects of sound waves or objects colliding with the ground or buildings around the test masses. I also consider temperature perturbations advected past the detector as a source of gravitational noise. I find that the gravitational noise background is below the expected noise floor even of advanced interferometric detectors, although only by an order of magnitude for temperature perturbations carried along turbulent streamlines. I also find that transient shockwaves in the atmosphere could potentially produce large spurious signals, with signal-to-noise ratios in the hundreds in an advanced interferometric detector. These signals could be vetoed by means of acoustic sensors outside of the buildings. Massive wind-borne objects such as tumbleweeds could also produce gravitational signals with signal-to-noise ratios in the hundreds if they collide with the interferometer buildings, so it may be necessary to build fences preventing such objects from approaching within about 30m of the test masses. 
  The general relativistic corrections in the equations of motion and associated energy of a binary system of point-like masses are derived at the third post-Newtonian (3PN) order. The derivation is based on a post-Newtonian expansion of the metric in harmonic coordinates at the 3PN approximation. The metric is parametrized by appropriate non-linear potentials, which are evaluated in the case of two point-particles using a Lorentzian version of an Hadamard regularization which has been defined in previous works. Distributional forms and distributional derivatives constructed from this regularization are employed systematically. The equations of motion of the particles are geodesic-like with respect to the regularized metric. Crucial contributions to the acceleration are associated with the non-distributivity of the Hadamard regularization and the violation of the Leibniz rule by the distributional derivative. The final equations of motion at the 3PN order are invariant under global Lorentz transformations, and admit a conserved energy (neglecting the radiation reaction force at the 2.5PN order). However, they are not fully determined, as they depend on one arbitrary constant, which reflects probably a physical incompleteness of the point-mass regularization. The results of this paper should be useful when comparing theory to the observations of gravitational waves from binary systems in future detectors VIRGO and LIGO. 
  A low matter density decaying vacuum cosmology is proposed on the assumption that the universe's radius is a complex quantity \hat{R} if it is regarded as having a zero energy-momentum tensor. But we find that when the radius is real, it contains matter. Using the Einstein-Hilbert action principle, the physical scale factor R(t) =|\hat{R}| is obtained as equal to (R_0^{2} + t^{2})^{1/2} with R_0 representing the finite radius of the universe at t=0. The resulting physical picture is roughly a theoretical justification of the old Ozer-Taha model. The new model is devoid of all cosmological problems. In particular, it confirms the bounds on H_p, the present value of the Hubble parameter: 0.85 < H_p t_p < 1.91 and faces no age problem. We argue that the total energy density consists of parts corresponding to relativistic/non-relativistic matter, a positive vacuum energy and a form of matter with equation of state p_K = -(1/3) rho_K (textures or generally K-matter), and the following predictions are made for the present nonrelativistic era: Omega_{M,n.rel.} \approx 2/3, Omega_{V,n.rel.} \approx 1/3, Omega_ <<1, Omega_K \approx 1, where a parameter corresponding to K-matter is taken to be unity. It is shown that the spacetime with complex metric has signature changing properties. Using quantum cosmological considerations, it is shown that the wave function is peaked about the classical contour of evolution and the minimum radius R_0 of the nonsingular model is predicted as comparable with the Planck length.   PACS No(s); 98.80 Hw, 04.20, 04.60 
  In this thesis, the implications of a new cosmological model are studied, which has features similar to that of decaying vacuum cosmologies. Decaying vacuum (or cosmological constant \Lambda) models are the results of attempts to resolve the problems that plague the standard hot big bang model - the problems which elude a satisfactory solution even after the two decades of of inflationary models, the first and much publicised cure to them. We arrive at the present model by a radically new route, which extends the idea of a signature change in the metric, a widely discussed speculation in the current literature. An alternative approach uses some dimensional considerations in line with quantum cosmology and gives an almost identical model. Both derivations involve some fundamental issues in general theory of relativity. The model has a coasting evolution (i.e., a \propto t). It claims the absence of all the aforementioned puzzles in the standard model and has very good predictions. In the first two chapters of the thesis, we review the general theory of relativity, the standard model in cosmology, its successes, the problems in it and also the most successful of those attempts to solve them, namely, the inflationary and decaying vacuum models. In the third chapter, we present and discuss the new cosmological model in detail. The fourth chapter is concerned with quantum cosmology. We briefly review the canonical quantisation programme of solving the Wheeler-DeWitt equation, then apply the procedure to our model and show that it satisfies many of the much sought-after ideals of this formalism. The last chapter of the thesis compares the new model with other ones. It also discusses the appearance of a Casimir type negative energy density in it and the prospects and challenges ahead for the model. 
  The question of a phase transition in exiting the Planck epoch of the early universe is addressed. An order parameter is proposed to help decide the issue, and estimates are made concerning its behavior. Our analysis is suggestive that a phase transition occurred. 
  Coincidences are searched with the cryogenic resonant gravitational wave detectors EXPLORER and NAUTILUS, during a period of about six months (2 June-14 December 1998) for a total measuring time of 94.5 days, with the purpose to study new algorithms of analysis, based on the physical characteristics of the detectors. 
  The Lagrangian formalism for tensor fields over differentiable manifolds with contravariant and covariant affine connections (whose components differ not only by sign) and metrics [$(\bar{L}_n,g)$-spaces] is considered. The functional variation and the Lie variation of a Lagrangian density, depending on components of tensor fields (with finite rank) and their first and second covariant derivatives are established. A variation operator is determined and the corollaries of its commutation relations with the covariant and the Lie differential operators are found. The canonical (common) method of Lagrangians with partial derivatives (MLPD) and the method of Lagrangians with covariant derivatives (MLCD) are outlined. They differ from each other by the commutation relations the variation operator has to obey with the covariant and the Lie differential operator. The canonical and covariant Euler-Lagrange equations are found as well as their corresponding $(\bar{L}_n,g)$-spaces. The energy-momentum tensors are considered on the basis of the Lie variation and the covariant Noether identities. 
  The general relativistic gravitomagnetic clock effect consists in the fact that two massive test bodies orbiting a central spinning mass in its equatorial plane along two identical circular trajectories, but in opposite directions, take different times in describing a full revolution with respect to an asymptotically inertial observer. In the field of the Earth such time shift amounts to 10^{-7} s. Detecting it by means of a space based mission with artificial satellites is a very demanding task because there are severe constraints on the precision with which the radial and azimuthal positions of a satellite must be known: delta r= 10^{-2} cm and delta phi= 10^{-2} milliarcseconds per revolution. In this paper we assess if the systematic errors induced by various non-gravitational perturbations allow to meet such stringent requirements. A couple of identical, passive laser-ranged satellites of LAGEOS type with their spins aligned with the Earth's one is considered. It turns out that all the non vanishing non-gravitational perturbations induce systematic errors in r and phi within the required constraints for a reasonable assumption of the mismodeling in some satellite's and Earth's parameters and/or by using dense satellites with small area-to-mass ratio. However, the error in the Earth's GM is by far the largest source of uncertainty in the azimuthal location which is affected at a level of 1.2 milliarcseconds per revolution. 
  We introduce, by means of the Brans-Dicke scalar field, space-time fluctuations at scale comparable to Planck length near the event horizon of a black hole and examine their dramatic effects. 
  In the framework of the Hartle-Hawking no-boundary proposal, we investigated quantum creation of the multidimensional universe with a cosmological constant ($\Lambda$) but without matter fields. We have found that the classical solutions of the Euclidean Einstein equations in this model have ``quasi-attractors'', i.e., most trajectories on the a-b plane, where a and b are the scale factors of external and internal spaces, go around a point. It is presumed that the wave function of the universe has a hump near this quasi-attractor point. In the case that both the curvatures of external and internal spaces are positive, and $\Lambda>0$, there exist Lorentzian solutions which start near the quasi-attractor, the internal space remains microscopic, and the external space evolves into our macroscopic universe. 
  In the framework of the Hartle-Hawking no-boundary proposal, we investigate quantum creation of the multidimensional universe with the cosmological constant $\Lambda$ but without matter fields. In this paper we solved the Wheeler-de Witt equation numerically. We find that the universe in which both of the spaces expand exponentially is the most probable in this model. 
  In this paper we explore how far the post-Newtonian theory goes in overcoming the difficulties associated with anisotropic homogeneous cosmologies in the Newtonian approximation. It will be shown that, unlike in the Newtonian case, the cosmological equations of the post-Newtonian approximation are much more in the spirit of general relativity with regard to the nine Bianchi types and issues of singularities. The situations of vanishing rotation and vanishing shear are treated separately. The homogeneous Bianchi I model is considered as an example of a rotation-free cosmology with anisotropy. It is found in the Newtonian approximation that there are arbitrary functions that need to be given for all time if the initial value problem is to be well-posed, while in the post-Newtonian case there is no such need. For the general case of a perfect fluid only the post-Newtonian theory can satisfactorily describe the effects of pressure. This is in accordance with findings in an earlier paper where the post-Newtonian approximation was applied to homogeneous cosmologies. For a shear-free anisotropic homogeneous cosmology the Newtonian theory of Heckmann and Sch\"ucking is explored. Comparisons with its relativistic and post-Newtonian counterparts are made. In the Newtonian theory solutions exist to which there are no analogues in general relativity. The post-Newtonian approximation may provide a way out. 
  The holographic bound that the entropy (log of number of quantum states) of a system is bounded from above by a quarter of the area of a circumscribing surface measured in Planck areas is widely regarded a desideratum of any fundamental theory, but some exceptions occur. By suitable black hole gedanken experiments I show that the bound follows from the generalized second law for two broad classes of isolated systems: generic weakly gravitating systems composed of many elementary particles, and quiescent, nonrotating strongly gravitating configurations well above Planck mass. These justify an early claim by Susskind. 
  Non-relativistic quantum theory of non-interacting particles in the spacetime containing a region with closed time-like curves (time-machine spacetime) is considered with the help of the path-integral technique. It is argued that, in certain conditions, a sort of superselection may exist for evolution of a particle in such a spacetime. All types of evolution are classified by the number $n$ defined as the number of times the particle returns back to its past. It corresponds also to the topological class of trajectories of the particle. The evolutions corresponding to different values of $n$ are non-coherent. The amplitudes corresponding to such evolutions may not be superposed. Instead of one evolution operator, as in the conventional (coherent) description, evolution of the particle is described by a family $U_n$ of partial evolution operators. This is done in analogy with the formalism of quantum theory of measurements, but with essential new features in the dischronal region (the region with closed time-like curves) of the time-machine spacetime. Partial evolution operators $U_n$ are equal to integrals over the topological classes of paths if the evolution begins and ends in the chronal regions. If the evolution begins or/and ends in the dischronal region, the integral over the topological class should be multiplied by a certain projector to give the partial evolution operator $U_n$. Thus defined partial evolution operators possess the property of generalized unitarity and multiplicativity. The part of evolution containing repeated returning backward in time cannot be factorized: all backward passages of the particle have to be considered as a single act, that cannot be presented as gradually, step by step, passing through `causal loops'. 
  Self-collision of a non-relativistic classical point-like body, or particle, in the spacetime containing closed time-like curves (time-machine spacetime) is considered. A point-like body (particle) is an idealization of a small ideal elastic billiard ball. The known model of a time machine is used containing a wormhole leading to the past. If the body enters one of the mouths of the wormhole, it emerges from another mouth in an earlier time so that both the particle and its "incarnation" coexist during some time and may collide. Such self-collisions are considered in the case when the size of the body is much less than the radius of the mouth, and the latter is much less than the distance between the mouths. Three-dimensional configurations of trajectories with a self-collision are presented. Their dynamics is investigated in detail. Configurations corresponding to multiple wormhole traversals are discussed. It is shown that, for each world line describing self-collision of a particle, dynamically equivalent configurations exist in which the particle collides not with itself but with an identical particle having a closed trajectory (Jinnee of Time Machine). 
  A link between the possibility of extending a geodessically incomplete kinked spacetime to a spacetime which is geodesically complete and the energy conditions is discussed for the case of a cylindrically-symmetric spacetime kink. It is concluded that neither the strong nor the weak energy condition can be satisfied in the four-dimensional example, though the latter condition may survive on the transversal sections of such a spacetime. It is also shown that the matter which propagates quantum-mechanically in a kinked spacetime can always be trapped by closed timelike curves, but signaling connections between that matter and any possible observer can only be made of totally incoherent radiation, so preventing observation of causality violation. 
  In a geometric unified theory there is an energy momentum equation, apart from the field equations and equations of motion. The general relativity Einstein equation with cosmological constant follows from this energy momentum equation for empty space. For non empty space we obtain a generalized Einstein equation, relating the Einstein tensor to a geometric stress energy tensor. The matching exterior solution is in agreement with the standard relativity tests. Furthermore, there is a Newtonian limit where we obtain Poisson's equation. 
  Gravitation as a fundamental interaction that governs all phenomena at large and very small scales, but still not well understood at a quantum level, is a missing cardinal link to unification of all physical interactions. Problems of the absolute G measurements and its possible time and range variations are reflections of the unification problem. Integrable multidimensional models of gravitation and cosmology make up one of the proper approaches to study basic issues and strong field objects, the Early Universe and black hole physics in particular. The choice, nature, classification and precision of determination of fundamental physical constants are described. The problem of their temporal variations is also discussed, temporal and range variations of G in particular. A need for further absolute measurements of G, its possible range and time variations is pointed out. The novel multipurpose space project SEE, aimed for measuring G and its stability in space and time 3-4 orders better than at present, may answer many important questions posed by gravitation, cosmology and unified theories. 
  We study an even dimensional manifold with a pseudo-Riemannian metric with arbitrary signature and arbitrary dimensions. We consider the Ricci flat equations and present a procedure to construct solutions to some higher (even) dimensional Ricci flat field equations from the four diemnsional Ricci flat metrics. When the four dimensional Ricci flat geometry correponds to a colliding gravitational vacuum spacetime our approach provides an exact solution to the vacuum Einstein field equations for colliding graviational plane waves in an (arbitrary) even dimensional spacetime. We give explicitly higher dimensional Szekeres metrics and study their singularity behaviors. 
  Gravity is related to gravitational mass of the bodies. According to the weak form of Einstein's General Relativity equivalence principle, the gravitational and inertial masses are equivalent. However recent calculations (gr-qc/9910036) have revealed that they are correlated by an adimensional factor, which depends on the incident radiation upon the particle. It was shown that there is a direct correlation between the radiation absorbed by the particle and its gravitational mass, independently of the inertial mass. This finding has fundamental consequences to Unified Field Theory and Quantum Cosmology. It was also shown that only in the absence of electromagnetic radiation this factor becomes equal to one and that, in specific electromagnetic conditions, it can be reduced, nullified or made negative. This means that there is the possibility of control of the gravitational mass by means of the incident radiation. This unexpected theoretical result was recently confirmed by an experiment (gr-qc/0005107). Consequently there is a strong evidence that the gravitational forces can be reduced, nullified and inverted by means of electromagnetic radiation. This means that, in practice we can produce gravitational binaries, and in this way to extract energy from a gravitational field. Here we describe a process by which energy can be extracted directly from any site of a gravitational field. 
  Causal structure of the brane universe with respect to null geodesics in the bulk spacetime is studied. It is pointed out that apparent causality violation is possible for the brane universe which contains matter energy. It is also shown that there is no 'horizon problem' in the Friedmann-Robertson-Walker brane universe. 
  A mechanical quality factor of $1.1 \times 10^{7}$ was obtained for the 199 Hz bending vibrational mode in a monocrystalline sapphire fiber at 6 K. Consequently, we confirm that pendulum thermal noise of cryogenic mirrors used for gravitational wave detectors can be reduced by the sapphire fiber suspension. 
  In order to facilitate the application of standard renormalization techniques, gravitation should be decribed, if possible, in pure connection formalism, as a Yang-Mills theory of a certain spacetime group, say the Poincare or the affine group. This embodies the translational as well as the linear connection. However, the coframe is not the standard Yang-Mills type gauge field of the translations, since it lacks the inhomogeneous gradient term in the gauge transformations. By explicitly restoring the "hidden" piece responsible for this behavior within the framework of nonlinear realizations, the usual geometrical interpretation of the dynamical theory becomes possible, and in addition one can avoid the metric or coframe degeneracy which would otherwise interfere with the integrations within the path integral. We claim that nonlinear realizations provide a general mathematical scheme clarifying the foundations of gauge theories of spacetime symmetries. When applied to construct the Yang-Mills theory of the affine group, tetrads become identified with nonlinear translational connections; the anholonomic metric does not constitute any more an independent gravitational potential, since its degrees of freedom reveal to correspond to eliminable Goldstone bosons. This may be an important advantage for quantization. 
  An explicit necessary condition for the occurrence of resonance scattering of axial gravitational waves, along with the internal trapping of null geodesics, is proposed for static spherically symmetric perfect fluid solutions to Einstein's equations. Some exact inhomogeneous solutions which exhibit this trapping are given with special attention to boundary conditions and the physical acceptability of the space times. In terms of the tenuity ($\alpha = R/M$ at the boundary) all the examples given lie in the narrow range $2.8 < \alpha < 2.9$. The tenuity can be raised to more interesting values by the addition of an envelope without altering the trapping. 
  Results are presented from high-precision computations of the orbital evolution and emitted gravitational waves for a stellar-mass object spiraling into a massive black hole in a slowly shrinking, circular, equatorial orbit. The focus of these computations is inspiral near the innermost stable circular orbit (isco)---more particularly, on orbits for which the angular velocity Omega is 0.03 < Omega/Omega_{isco} < 1. The computations are based on the Teukolsky-Sasaki-Nakamura formalism, and the results are tabulated in a set of functions that are of order unity and represent relativistic corrections to low-orbital-velocity formulas. These tables can form a foundation for future design studies for the LISA space-based gravitational-wave mission. A first survey of applications to LISA is presented: Signal to noise ratios S/N are computed and graphed as functions of the time-evolving gravitational-wave frequency for representative values of the hole's mass M and spin a and the inspiraling object's mass \mu, with the distance to Earth chosen to be r_o = 1 Gpc. These S/N's show a very strong dependence on the black-hole spin, as well as on M and \mu. A comparison with predicted event rates shows strong promise for detecting these waves, but not beyond about 1Gpc if the inspiraling object is a white dwarf or neutron star. This argues for a modest lowering of LISA's noise floor. A brief discussion is given of the prospects for extracting information from the observed waves 
  The proposed so far brane-world cosmological scenarios are concerned with (D-1)-dimensional embeddings into the D-dimensional spacetime, besides, it is supposed D=5 as a rule. However, it would be much more realistic to consider our four-Universe as 4-shell or 3-brane inside, e.g., 10-dimensional spacetime. In turn it immediately means that the theory of the $(D-D_E)$-dimensional singular embeddings, where the number of extra dimensions $D_E > 1$, is needed. Hence, the aim of this work is to provide such a theory: we construct the rigorous general theory of the induced gravity on singular submanifolds. At first, we perform the decomposition of the tangent bundle into the two subbundles which will be associated later with external and visible (with respect to some low-dimensional observer) parts of the high-$D$ manifold. Then we go to physics and perform the split of the manifold (in addition to the split of the tangent bundle) to describe both the induced internal geometry and external as-a-whole dynamics of singular embeddings, assuming matter being confined on the singular submanifold but gravity being propagated through the high-$D$ manifold. With the use of the de Rham axiomatic approach to delta-distributions we demonstrate that the four-Universe can be singularly embedded only in five- and six-dimensional space so if we want to consider its embedding in 10D then extra dimensions must be included as a product space only. We discuss the revealed generic features of the theory such as the multi-normal anisotropy, restrictions on an ambient space, reformulation of the conserved gravitational stress-energy tensor problem, etc. 
  Quantum geometrodynamics in extended phase space describes phenomenologically the integrated system ``a physical object + observation means (a gravitational vacuum condensate)''. The central place in this version of QGD belongs to the Schrodinger equation for a wave function of the Universe. An exact solution to the ``conditionally-classical'' set of equations in extended phase space for the Bianchi-IX model and the appropriate solution to the Schrodinger equation are considered. The physical adequacy of the obtained solutions to existing concepts about possible cosmological scenarios is demonstrated. The gravitational vacuum condensate is shown to be a cosmological evolution factor. 
  The general formulas of a non-rotating dynamic thin shell that connects two arbitrary cylindrical regions are given using Israel's method. As an application of them, the dynamics of a thin shell made of counter-rotating dust particles, which emits both gravitational waves and massless particles when it is expanding or collapsing, is studied. It is found that when the models represent a collapsing shell, in some cases the angular momentum of the dust particles is strong enough to halt the collapse, so that a spacetime singularity is prevented from forming, while in other cases it is not, and a line-like spacetime singularity is finally formed on the symmetry axis. 
  Cylindrical spacetimes with rotation are studied using the Newmann-Penrose formulas. By studying null geodesic deviations the physical meaning of each component of the Riemann tensor is given. These spacetimes are further extended to include rotating dynamic shells, and the general expression of the surface energy-momentum tensor of the shells is given in terms of the discontinuation of the first derivatives of the metric coefficients. As an application of the developed formulas, a stationary shell that generates the Lewis solutions, which represent the most general vacuum cylindrical solutions of the Einstein field equations with rotation, is studied by assuming that the spacetime inside the shell is flat. It is shown that the shell can satisfy all the energy conditions by properly choosing the parameters appearing in the model, provided that $ 0 \le \sigma \le 1$, where $\sigma$ is related to the mass per unit length of the shell. 
  Recent cosmological observations reveal that we are living in a flat accelerated expanding universe. In this work we have investigated the nature of the potential compatible with the power law expansion of the universe in a self interacting Brans Dicke cosmology with a perfect fluid background and have analyzed whether this potential supports the accelerated expansion. It is found that positive power law potential is relevant in this scenario and can drive accelerated expansion for negative Brans Dicke coupling parameter $\omega$. The evolution of the density perturbation is also analyzed in this scenerio and is seen that the model allows growing modes for negative $\omega$. 
  The Wheeler-DeWitt equation for the induced gravity theory is constructed in the minisuperspace approximation, and then solved using the WKB method under three types of boundary condition proposed respectively by Hartle & Hawking (``no boundary''), Linde and Vilenkin (``tunneling from nothing''). It is found that no matter how the gravitational and cosmological ``constants'' vary in the classical models, they will acquire constant values when the universe comes from quantum creation, and that, in particular, the resulting tunneling wave function under the Linde or Vilenkin boundary condition reaches its maximum value if the cosmological constant vanishes. 
  In the $\bar{\mbox{\rm Poincar\'{e}}}$ gauge theory of gravity, which has been formulated on the basis of a principal fiber bundle over the space-time manifold having the covering group of the proper orthochronous Poincar\'{e} group as the structure group, we examine the tensorial properties of the dynamical energy-momentum density ${}^{G}{\mathbf T}_{k}{}^{\mu}$ and the ` ` spin" angular momentum density ${}^{G}{\mathbf S}_{kl}{}^{\mu}$ of the gravitational field. They are both space-time vector densities, and transform as tensors under {\em global} $SL(2,C)$- transformations. Under {\em local} internal translation, ${}^{G}{\mathbf T}_{k}{}^{\mu}$ is invariant, while ${}^{G}{\mathbf S}_{kl}{}^{\mu}$ transforms inhomogeneously. The dynamical energy-momentum density ${}^{M}{\mathbf T}_{k}{}^{\mu}$ and the ` ` spin" angular momentum density ${}^{M}{\mathbf S}_{kl}{}^{\mu}$ of the matter field are also examined, and they are known to be space-time vector densities and to obey tensorial transformation rules under internal $\bar{\mbox{\rm Poincar\'{e}}}$ gauge transformations. The corresponding discussions in extended new general relativity which is obtained as a teleparallel limit of $\bar{\mbox{\rm Poincar\'{e}}}$ gauge theory are also given, and energy-momentum and ` ` spin" angular momentum densities are known to be well behaved. Namely, they are all space-time vector densities, etc. In both theories, integrations of these densities on a space-like surface give the total energy-momentum and {\em total} (={\em spin}+{\em orbital}) angular momentum for asymptotically flat space-time. The tensorial properties of canonical energy-momentum and ` ` extended orbital angular momentum" densities are also examined. 
  We derive the black hole solutions with horizons of non-trivial topology and investigate their properties in the framework of an approach to quantum gravity being an extension of Bohm's formulation of quantum mechanics. The solutions we found tend asymptotically (for large $r$) to topological black holes. We also analyze the thermodynamics of these space-times. 
  A second-order expansion for the quantum fluctuations of the matter field was considered in the framework of the warm inflation scenario. The friction and Hubble parameters were expended by means of a semiclassical approach. The fluctuations of the Hubble parameter generates fluctuations of the metric. These metric fluctuations produce an effective term of curvature. The power spectrum for the metric fluctuations can be calculated on the infrared sector. 
  A novel method, based on superpotentials is proposed for obtaining the quasi-normal modes of anti-de Sitter black holes. This is inspired by the case of the three-dimensional BTZ black hole, where the quasi-normal modes can be obtained exactly and are proportional to the surface gravity. Using this approach, the quasi-normal modes of the five dimensional Schwarzschild anti-deSitter black hole are computed numerically. The modes again seem to be proportional to the surface gravity for very small and very large black holes. They reflect the well-known instability of small black holes in anti-deSitter space. 
  Initial data are the starting point for any numerical simulation. In the case of numerical relativity, Einstein's equations constrain our choices of these initial data. We will examine several of the formalisms used for specifying Cauchy initial data in the 3+1 decomposition of Einstein's equations. We will then explore how these formalisms have been used in constructing initial data for spacetimes containing black holes and neutron stars. In the topics discussed, emphasis is placed on those issues that are important for obtaining astrophysically realistic initial data for compact binary coalescence. 
  The r-mode instability in rotating relativistic stars has been shown recently to have important astrophysical implications (including the emission of detectable gravitational radiation, the explanation of the initial spins of young neutron stars and the spin-distribution of millisecond pulsars and the explanation of one type of gamma-ray bursts), provided that r-modes are not saturated at low amplitudes by nonlinear effects or by dissipative mechanisms. Here, we present the first study of nonlinear r-modes in isentropic, rapidly rotating relativistic stars, via 3-D general-relativistic hydrodynamical evolutions. Our numerical simulations show that (1) on dynamical timescales, there is no strong nonlinear coupling of r-modes to other modes at amplitudes of order one -- unless nonlinear saturation occurs on longer timescales, the maximum r-mode amplitude is of order unity (i.e., the velocity perturbation is of the same order as the rotational velocity at the equator). An absolute upper limit on the amplitude (relevant, perhaps, for the most rapidly rotating stars) is set by causality. (2) r-modes and inertial modes in isentropic stars are predominantly discrete modes and possible associated continuous parts were not identified in our simulations. (3) In addition, the kinematical drift associated with r-modes, recently found by Rezzolla, Lamb and Shapiro (2000), appears to be present in our simulations, but an unambiguous confirmation requires more precise initial data. We discuss the implications of our findings for the detectability of gravitational waves from the r-mode instability. 
  We present a self-contained framework called Direct Integration of the Relaxed Einstein Equations (DIRE) for calculating equations of motion and gravitational radiation emission for isolated gravitating systems based on the post-Newtonian approximation. We cast the Einstein equations into their ``relaxed'' form of a flat-spacetime wave equation together with a harmonic gauge condition, and solve the equations formally as a retarded integral over the past null cone of the field point (chosen to be within the near zone when calculating equations of motion, and in the far zone when calculating gravitational radiation). The ``inner'' part of this integral(within a sphere of radius $\cal R \sim$ one gravitational wavelength) is approximated in a slow-motion expansion using standard techniques; the ``outer'' part, extending over the radiation zone, is evaluated using a null integration variable. We show generally and explicitly that all contributions to the inner integrals that depend on $\cal R$ cancel corresponding terms from the outer integrals, and that the outer integrals converge at infinity, subject only to reasonable assumptions about the past behavior of the source. The method cures defects that plagued previous ``brute-force'' slow-motion approaches to motion and gravitational radiation for isolated systems. We detail the procedure for iterating the solutions in a weak-field, slow-motion approximation, and derive expressions for the near-zone field through 3.5 post-Newtonian order in terms of Poisson-like potentials. 
  It is well known that waves propagating in a nontrivial medium develop ``tails''. However, the exact form of the late-time tail has so far been determined only for a narrow class of models. We present a systematic analysis of the tail phenomenon for waves propagating under the influence of a {\it general} scattering potential $V(x)$. It is shown that, generically, the late-time tail is determined by spatial {\it derivatives} of the potential. The central role played by derivatives of the scattering potential appears not to be widely recognized. The analytical results are confirmed by numerical calculations. 
  Relativistic hydrodynamics of an isentropic fluid in a gravitational field is considered as the particular example from the family of Lagrangian hydrodynamic-type systems which possess an infinite set of integrals of motion due to the symmetry of Lagrangian with respect to relabeling of fluid particle labels. Flows with fixed topology of the vorticity are investigated in quasi-static regime, when deviations of the space-time metric and the density of fluid from the corresponding equilibrium configuration are negligibly small. On the base of the variational principle for frozen-in vortex lines dynamics, the equation of motion for a thin relativistic vortex filament is derived in the local induction approximation. 
  In the present work we perform a phase-plane analysis of the complete dynamical system corresponding to a flat FRW cosmological models with a perfect fluid and a self-interacting scalar field and show that every positive and monotonous potential which is asymptotically exponential yields a scaling solution as a global attractor. 
  This paper describes a new class of experiments that use dispersion in optical fibers to convert the gravitational frequency shift of light into a measurable phase shift or time delay. Two conceptual models are explored. In the first model, long counter-propagating pulses are used in a vertical fiber optic Sagnac interferometer. The second model uses optical solitons in vertically separated fiber optic storage rings. We discuss the feasibility of using such an instrument to make a high precision measurement of the gravitational frequency shift of light. 
  Cosmological nucleosynthesis calculations imply that there should be both non-baryonic and baryonic dark matter. Recent data suggest that some of the non-baryonic dark matter must be "hot" (i.e. massive neutrinos) and there may also be evidence for "cold" dark matter (i.e. WIMPs). If the baryonic dark matter resides in galactic halos, it is likely to be in the form of compact objects (i.e. MACHOs) and these would probably be the remnants of a first generation of pregalactic or protogalactic Population III stars. Many candidates have been proposed - brown dwarfs, red dwarfs, white dwarfs or black holes - and at various times each of these has been in vogue. We review the many types of observations which can be used to constrain or exclude both baryonic and non-baryonic dark matter candidates. 
  We construct a black hole whose interior is the false vacuum and whose exterior is the true vacuum of a classical field theory. From the outside the metric is the usual Schwarzschild one, but from the inside the space is de Sitter with a cosmological constant determined by the energy of the false vacuum. The parameters of the field potential may allow for the false vacuum to exist for more than the present age of the universe. A potentially relevant effective field theory within the context of QCD results in a Schwarzschild radius of about 200 km. 
  The asymptotic properties of self-similar spherically symmetric perfect fluid solutions with equation of state p=alpha mu (-1<alpha<1) are described. We prove that for large and small values of the similarity variable, z=r/t, all such solutions must have an asymptotic power-law form. Some of them are associated with an exact power-law solution, in which case they are asymptotically Friedmann or asymptotically Kantowski-Sachs for 1>alpha >-1 or asymptotically static for 1>alpha >0. Others are associated with an approximate power-law solution, in which case they are asymptotically quasi-static for 1>alpha >0 or asymptotically Minkowski for 1>alpha >1/5. We also show that there are solutions whose asymptotic behaviour is associated with finite values of z and which depend upon powers of ln z. These correspond either to a second family of asymptotically Minkowski solutions for 1>alpha>1/5 or to solutions that are asymptotically Kasner for 1>alpha>-1/3. There are some other asymptotic power-law solutions associated with negative alpha, but the physical significance of these is unclear. The asymptotic form of the solutions is given in all cases, together with the number of associated parameters. 
  Taking the Randall-Sundrum models as background scenario, we derive generalized Israel-Lanczos-Sen thin-shell junction conditions for systems in which several bulk scalar fields are non-minimally coupled to gravity. We demonstrate that the form of the junction conditions (though arguably not the physics) depends on the choice of frame. We show that generally (in any frame except the Einstein frame) the presence of a thin shell induces discontinuities in the normal derivative of the scalar field, even in the absence of any direct interaction between the thin shell and the scalar field. For some exceptional scalar field configurations the discontinuities in the derivatives of the metric and the scalar fields can feed back into each other and so persist even in the absence of any thin shell of stress-energy. 
  The fine structure constant $\alpha $ includes the speed of light as given by $\alpha =\frac{e^{2}}{4\pi \epsilon_{0}c\hbar}$. It is shown here that, following a $TH\epsilon \mu $ formalism, interpreting the permittivity $\epsilon_{0}$ and permeabiliy $\mu_{0}$ of free space under Lorentz local and position invariance, this is not the case. The result is a new expression as $\alpha =\frac{e^{2}}{4\pi \hbar}$ in a new system of units for the charge that preserves local and position invariance. Hence, the speed of light does not explicitly enter in the constitution of the fine structure constant. The new expressions for the Maxwell's equations are derived and some cosmological implications discussed. 
  This author's recent proposal of interferometric tests of Planck-scale-related properties of space-time is here revisited from a strictly phenomenological viewpoint. The results announced previously are rederived using elementary dimensional considerations. The dimensional analysis is then extended to the other two classes of experiments (observations of neutral kaons at particle accelerators and observations of the gamma rays we detect from distant astrophysical sources) which have been recently considered as opportunities to explore "foamy" properties of space-time. The emerging picture suggests that there is an objective and intuitive way to connect the sensitivities of these three experiments with the Planck length. While in previous studies the emphasis was always on some quantum-gravity scenario and the analysis was always primarily aimed at showing that the chosen scenario would leave a trace in a certain class of doable experiments, the analysis here reported takes as starting point the experiments and, by relating in a direct quantitative way the sensitivities to the Planck length, provides a model-independent description of the status of Planck-length phenomenology. 
  In this paper we present classes of state sum models based on the recoupling theory of angular momenta of SU(2) (and of its q-counterpart $U_q(sl(2))$, q a root of unity). Such classes are arranged in hierarchies depending on the dimension d, and include all known closed models, i.e. the Ponzano-Regge state sum and the Turaev-Viro invariant in dimension d=3, the Crane-Yetter invariant in d=4. In general, the recoupling coefficient associated with a d-simplex turns out to be a $\{3(d-2)(d+1)/2\}j$ symbol, or its q-analog. Each of the state sums can be further extended to compact triangulations $(T^d,\partial T^d)$ of a PL-pair $(M^d,\partial M^d)$, where the triangulation of the boundary manifold is not keeped fixed. In both cases we find out the algebraic identities which translate complete sets of topological moves, thus showing that all state sums are actually independent of the particular triangulation chosen. Then, owing to Pachner's theorems, it turns out that classes of PL-invariant models can be defined in any dimension d. 
  For the example of an accelerated shell we show that omission of the energy-momentum tensor (EMT) of the body that causes the acceleration and the tensions due to this acceleration can lead to a paradoxical result; Namely, the entrainment of an inertial frame by the accelerated shell in the direction opposite to that of the acceleration. We consider several models and demonstrate that the correct result can be obtained only if all components of the full EMT are adequately taken into account, and the problem statement is physically correct. 
  A variational theory of a continuous medium is developed the elements of which carry momentum and hypermomentum (hyperfluid). It is shown that the structure of the sources in metric-affine gravity is predetermined by the conservation identities and, when using the Weyssenhoff ansatz, these explicitly yield the hyperfluid currents. 
  A hyperfluid is a classical continuous medium carrying hypermomentum. We modify the earlier developed variational approach to a hyperfluid in such a way that the Frenkel type constraints imposed on the hypermomentum current are eliminated. The resulting self-consistent model is different from the Weyssenhoff type one. The essential point is a conservation of the hypermomentum current such that the final metrical and canonical energy-momentum forms coincide. 
  Following recent studies of Ford, we suggest -- in the framework of general relativity -- an inflationary cosmological model with the self-interacting spinning matter. A generalization of the standard fluid model is discussed and estimates of the physical parameters of the evolution are given. 
  Hawking's prediction of black-hole evaporation depends on the application of known physics to fantastically high energies -- well beyond the Planck scale. Here, I show that before these extreme regimes are reached, another physical effect will intervene: the quantum backreaction on the collapsing matter and its effect on the geometry through which the quantum fields propagate. These effects are estimated by a simple thought experiment. When this is done, it appears that there are no matrix elements allowing the emission of Hawking quanta: black holes do not radiate. 
  We perform a von Neumann stability analysis on a common discretization of the Einstein equations. The analysis is performed on two formulations of the Einstein equations, namely, the standard ADM formulation and the conformal-traceless (CT) formulation. The eigenvalues of the amplification matrix are computed for flat space as well as for a highly nonlinear plane wave exact solution. We find that for the flat space initial data, the condition for stability is simply $\frac {\Delta t}{\Delta z} \leq 1$. However, a von Neumann analysis for highly nonlinear plane wave initial data shows that the standard ADM formulation is unconditionally unstable, while the conformal-traceless (CT) formulation is stable for $0.25 \leq \frac {\Delta t}{\Delta z} < 1$. 
  Recent results of quantum field theory on a curved spacetime suggest that extremal black holes are not thermal objects and that the notion of zero temperature is ill-defined for them. If this is correct, one may have to go to a full semiclassical theory of gravity, including backreaction, in order to make sense of the third law of black hole thermodynamics. Alternatively, it is possible that we shall have to drastically revise the status of extremality in black hole thermodynamics. 
  We study the r-modes and rotational ``hybrid'' modes of relativistic stars. As in Newtonian gravity, the spectrum of low-frequency rotational modes is highly sensitive to the stellar equation of state. If the star and its perturbations obey the same one-parameter equation of state (as with isentropic stars), there exist {\it no pure r-modes at all} - no modes whose limit, for a star with zero angular velocity, is an axial-parity perturbation. Rotating stars of this kind similarly have no pure g-modes, no modes whose spherical limit is a perturbation with polar parity and vanishing perturbed pressure and density. We compute the post-Newtonian corrections to the $l=m$ r-modes of isentropic and non-isentropic uniform density stars. 
  Non-Abelian Gauss law is interpreted in terms of area bits described in a local frame which fit together into closed surfaces and the Non-Abelian Stokes law in terms of length bits described in a local frame which fit together into closed loops. A new equation relating the area variables and the phase space variables (or equivalently, angular momentum variables of the lattice Yang-Mills theory and phase space variables of the continuum theory) is obtained. Canonical quantization applied to these variables implies area quantization. A complete orthonormal basis of states satisfying the Gauss constraint is obtained.It has the interpretation of quantized area bits with undefined orientations and edges but fitting together into closed surfaces. 
  We calculate the effect of gravitational wave (gw) back-reaction on realistic neutron stars (NS's) undergoing torque-free precession. By `realistic' we mean that the NS is treated as a mostly-fluid body with an elastic crust, as opposed to a rigid body. We find that gw's damp NS wobble on a timescale tau_{theta} approx 2 x 10^5 yr [10^{-7}/(DId/I_0)]^2 (kHz/ nu_s)^4, where nu_s is the spin frequency and DId is the piece of the NS's inertia tensor that "follows" the crust's principal axis (as opposed to its spin axis). We give two different derivations of this result: one based solely on energy and angular momentum balance, and another obtained by adding the Burke-Thorne radiation reaction force to the Newtonian equations of motion. This problem was treated long ago by Bertotti and Anile (1973), but their claimed result is wrong. When we convert from their notation to ours, we find that their tau_{theta} is too short by a factor of order 10^5 for typical cases of interest, and even has the wrong sign for DId negative. We show where their calculation went astray. 
  We discuss the emergence of time dilation as a normal feature expected of any system where a central processor may have to wait one or more clock cycles before concluding a local calculation. We show how the process of causal implication in a typical Newtonian cellular automaton leads naturally to Lorentz transformations and invariant causal structure. 
  We find an exact solution in closed form for the critical collapse of a scalar field with cosmological constant in 2+1 dimensions. This solution agrees with the numerical simulation done by Pretorius and Choptuik of this system. 
  A global phase time is identified for homogeneous and isotropic cosmological models yielding from the low energy effective action of closed bosonic string theory. When the Hamiltonian constraint allows for the existence of an intrinsic time, the quantum transition amplitude is obtained by means of the usual path integral procedure for gauge systems. 
  We investigate here the behavior of a few spherically symmetric static acclaimed black hole solutions in respect of tidal forces in the geodesic frame. It turns out that the forces diverge on the horizon of cold black holes (CBH) while for ordinary ones, they do not. It is pointed out that Kruskal-like extensions do not render the CBH metrics nonsingular. We present a CBH that is available in the Brans-Dicke theory for which the tidal forces do not diverge on the horizon and in that sense it is a better one. 
  The LIGO-II gravitational-wave interferometers (ca. 2006--2008) are designed to have sensitivities at about the standard quantum limit (SQL) near 100 Hz. This paper describes and analyzes possible designs for subsequent, LIGO-III interferometers that can beat the SQL. These designs are identical to a conventional broad-band interferometer (without signal recycling), except for new input and/or output optics. Three designs are analyzed: (i) a "squeezed-input interferometer" (conceived by Unruh based on earlier work of Caves) in which squeezed vacuum with frequency-dependent (FD) squeeze angle is injected into the interferometer's dark port; (ii) a "variational-output" interferometer (conceived in a different form by Vyatchanin, Matsko and Zubova), in which homodyne detection with FD homodyne phase is performed on the output light; and (iii) a "squeezed-variational interferometer" with squeezed input and FD-homodyne output. It is shown that the FD squeezed-input light can be produced by sending ordinary squeezed light through two successive Fabry-Perot filter cavities before injection into the interferometer, and FD-homodyne detection can be achieved by sending the output light through two filter cavities before ordinary homodyne detection. With anticipated technology and with laser powers comparable to that planned for LIGO-II, these interferometers can beat the amplitude SQL by factors in the range from 3 to 5, corresponding to event rate increases between ~30 and ~100 over the rate for a SQL-limited interferometer. 
  We review the motivations for the search of stochastic backgrounds of gravitational waves and we compare the experimental sensitivities that can be reached in the near future with the existing bounds and with the theoretical predictions. 
  It is shown that the interaction of helicity-1 waves of gravity and matter in a thin slab configuration produces new types of instabilities. Indeed, a transverse spin-2 helicity-1 mode interacts strongly with the shear motion of matter. This mode is unstable above a critical wavelength which reminds the Jeans wavelength but with the speed of sound interchanged by the speed of light. The two instabilities are of course different. For the case analyzed, a plane parallel configuration, Jeans instability appears through a density wave perturbation, the material collapsing into a set of plane-parallel slabs. On the other hand, the helicity-1 wave instability induces a transverse motion in the fluid that tends to shear in the material along the node of the perturbation. 
  We calculate the energy distribution in a static spherically symmetric nonsingular black hole space-time by using the Tolman's energy-momentum complex. All the calculations are performed in quasi-Cartesian coordinates. The energy distribution is positive everywhere and be equal to zero at origin. We get the same result as obtained by Y-Ching Yang by using the Einstein's and Weinberg's prescriptions. 
  I modify the quasilocal energy formalism of Brown and York into a purely Hamiltonian form. As part of the reformulation, I remove their restriction that the time evolution of the boundary of the spacetime be orthogonal to the leaves of the time foliation. Thus the new formulation allows an arbitrary evolution of the boundary which physically corresponds to allowing general motions of the set of observers making up that boundary. I calculate the rate of change of the quasilocal energy in such situations, show how it transforms with respect to boosts of the boundaries, and use the Lanczos-Israel thin shell formalism to reformulate it from an operational point of view. These steps are performed both for pure gravity and gravity with attendant matter fields. I then apply the formalism to characterize naked black holes and study their properties, investigate gravitational tidal heating, and combine it with the path integral formulation of quantum gravity to analyze the creation of pairs of charged and rotating black holes. I show that one must use complex instantons to study this process though the probabilities of creation remain real and consistent with the view that the entropy of a black hole is the logarithm of the number of its quantum states. 
  We consider general relativity with a cosmological constant as a perturbative expansion around a completely solvable diffeomorphism invariant field theory. This theory is the $\Lambda\to\infty$ limit of general relativity. This allows an explicit perturbative computational setup in which the quantum states of the theory and the classical observables can be explicitly computed. The zeroth order corresponds to highly degenerate space-times with vanishing volume. Perturbations give rise to space-times with non-vanishing volumes in a natural way. The spectrum of area- and volume-related observables constructed by coupling the theory to matter can be directly assessed. An unexpected relationship arises at a quantum level between the discrete spectrum of the volume operator and the allowed values of the cosmological constant. 
  We have recently introduced an approach for studying perturbatively classical and quantum canonical general relativity. The perturbative technique appears to preserve many of the attractive features of the non-perturbative quantization approach based on Ashtekar's new variables and spin networks. With this approach one can find perturbatively classical observables (quantities that have vanishing Poisson brackets with the constraints) and quantum states (states that are annihilated by the quantum constraints). The relative ease with which the technique appears to deal with these traditionally hard problems opens several questions about how relevant the results produced can possibly be. Among the questions is the issue of how useful are results for large values of the cosmological constant and how the approach can deal with several pathologies that are expected to be present in the canonical approach to quantum gravity. With the aim of clarifying these points, and to make our construction as explicit as possible, we study its application in several simple models. We consider Bianchi cosmologies, the asymmetric top, the coupled harmonic oscillators with constant energy density and a simple quantum mechanical system with two Hamiltonian constraints. We find that the technique satisfactorily deals with the pathologies of these models and offers promise for finding (at least some) results even for small values of the cosmological constant. Finally, we briefly sketch how the method would operate in the full four dimensional quantum general relativity case. 
  We present a second-quantized field theory of massive spin one-half particles or antiparticles in the presence of a weak gravitational field treated as a spin two external field in a flat Minkowski background. We solve the difficulties which arise from the derivative coupling and we are able to introduce an interaction picture. We derive expressions for the scattering amplitude and for the outgoing spinor to first-order. In several appendices, the link with the canonical approach in General Relativity is established and a generalized stationary phase method is used to calculate the outgoing spinor. We show how our expressions can be used to calculate and discuss phase shifts in the context of matter-wave interferometry (especially atom or antiatom interferometry). In this way, many effects are introduced in a unified relativistic framework, including spin-gravitation terms: gravitational red shift, Thomas precession, Sagnac effect, spin-rotation effect, orbital and spin Lense-Thirring effects, de Sitter geodetic precession and finally the effect of gravitational waves. A new analogy with the electromagnetic interaction is pointed out. 
  We calculate the energy distribution of an anisotropic model of universe, based on the Bianchi type I metric, in the Tolman's prescription. The energy due to the matter plus gravitational field is equal to zero. This result agrees with the results of Banerjee and Sen and Xulu. Also, our result supports the viewpoint of Tryon and Rosen. 
  We calculate the energy distribution of a dyonic dilaton black hole by using the Tolman's energy-momentum complex. All the calculations are performed in quasi-Cartesian coordinates. The energy distribution of the dyonic dilaton black hole depends on the mass, electric charge, magnetic charge and asymptotic value of the dilaton. We get the same result as obtained by Y-Ching Yang, Ching-Tzung Yeh, Rue-Ron Hsu and Chin-Rong Lee by using the Einstein's prescription. 
  The barotropic indices and the corresponding FRW scale factors of the so-called Darboux cosmological fluids are presented in the comoving time axis, which is the natural one for the phenomenology related to the cosmological data. Some useful comments on the features of the plots are included 
  In this paper we give, for the first time, a complete description of the dynamics of tilted spatially homogeneous cosmologies of Bianchi type II. The source is assumed to be a perfect fluid with equation of state $p = (\gamma -1) \mu$, where $\gamma$ is a constant. We show that unless the perfect fluid is stiff, the tilt destabilizes the Kasner solutions, leading to a Mixmaster-like initial singularity, with the tilt being dynamically significant. At late times the tilt becomes dynamically negligible unless the equation of state parameter satisfies $\gamma > {10/7}$. We also find that the tilt does not destabilize the flat FL model, with the result that the presence of tilt increases the likelihood of intermediate isotropization. 
  After motivating why the study of asymptotically flat spaces is important in loop quantum gravity, we review the extension of the standard framework of this theory to the asymptotically flat sector based on the GNS construction. In particular, we provide a general procedure for constructing new Hilbert spaces for loop quantum gravity on non-compact spatial manifolds. States in these Hilbert spaces can be interpreted as describing fluctuations around fiducial fixed backgrounds. When the backgrounds are chosen to approximate classical asymptotically flat 3-geometries this gives a natural framework in which to discuss physical applications of loop quantum gravity, especially its semi-classical limit. We present three general proposals for the construction of suitable backgrounds, including one approach that can lead to quantum gravity on anti-DeSitter space as described by the Chern-Simons state. 
  We study the spherically symmetric collapse of a real, minimally coupled, massive scalar field in an asymptotically Einstein-de Sitter spacetime background. By means of an eikonal approximation for the field and metric functions, we obtain a simple analytical criterion---involving the physical size and mass scales (the field's inverse Compton wavelength and the spacetime gravitational mass) of the initial matter configuration---for generic (non-time-symmetric) initial data to collapse to a black hole. This analytical condition can then be used to place constraints on the initial primordial black hole spectrum, by considering spherical density perturbations that re-entered the horizon during an early matter-dominated phase that immediately followed inflation. 
  E. D. Fackerell claims: 1) that Alley and Yilmaz treatment of parallel slabs in general relativity is wrong because the Yilmaz metric used is not a solution of the field equations of general relativity; 2) he also claims that the correct treatment of the parallel slab problem in general relativity must be based on the so-called Taub metric. We show below that both of Fackerell's claims are false. His first claim is based on his failure to distinguish the matter-free regions and the regions with matter. His second claim is based on his failure to recognize that for the Taub metric the left-hand side, hence also the right-hand side of the field equations, are identically zero everywhere. Thus no material systems can be treated via the Taub metric. 
  The sensitivity of a pair of VIRGO interferometers to gravitational waves backgrounds (GW) of cosmological origin is analyzed for the cases of maximal and minimal overlap of the two detectors. The improvements in the detectability prospects of scale-invariant and non-scale-invariant logarithmic energy spectra of relic GW are discussed. 
  We prove local existence and uniqueness of static spherically symmetric solutions of the Einstein-Yang-Mills equations for an arbitrary compact semisimple gauge group in the so-called regular case. By this we mean the equations obtained when the rotation group acts on the principal bundle on which the Yang-Mills connection takes its values in a particularly simple way (the only one ever considered in the literature). The boundary value problem that results for possible asymptotically flat soliton or black hole solutions is very singular and just establishing that local power series solutions exist at the center and asymptotic solutions at infinity amounts to a nontrivial algebraic problem. We discuss the possible field equations obtained for different group actions and solve the algebraic problem on how the local solutions depend on initial data at the center and at infinity. 
  We study the spacetime singularity in 2+1 dimensional AdS-scalar black hole with circular symmetry using a quasi-homogeneous model. We show that this is a spacelike, scalar curvature, deformationally strong singularity. 
  Assuming the space dimension is not constant but decreases during the expansion of the Universe, we study chaotic inflation with the potential $m^2\phi^2/2$. Our investigations are based on a model Universe with variable space dimensions. We write down field equations in the slow-roll approximation, and define slow-roll parameters by assuming the number of space dimensions decreases continuously as the Universe expands. The dynamical character of the space dimension shifts the initial and final value of the inflaton field to larger values. We obtain an upper limit for the space dimension at the Planck length. This result is in agreement with previous works for the effective time variation of the Newtonian gravitational constant in a model Universe with variable space dimensions. 
  We propose a quantum version of a gedanken experiment which supports the generalized second law of black hole thermodynamics. A quantum measurement of particles in the region outside of the event horizon decreases the entropy of the outside matter due to the entanglement of the inside and outside particle states. This decrease is compensated, however, by the increase in the detector entropy. If the detector is conditionally dropped into the black hole depending on the experimental outcome, the decrease of the matter entropy is more than compensated by the increase of the black hole entropy via the increase of the black hole mass which is ultimately attributed to the work done by the measurement. 
  Inspired by Mott's (1929) analysis of particle tracks in a cloud chamber, we consider a simple model for quantum cosmology which includes, in the total Hamiltonian, model detectors registering whether or not the system, at any stage in its entire history, passes through a series of regions in configuration space. We thus derive a variety of well-defined formulas for the probabilities for trajectories associated with the solutions to the Wheeler-DeWitt equation. The probability distribution is peaked about classical trajectories in configuration space. The ``measured'' wave functions still satisfy the Wheeler-DeWitt equation, except for small corrections due to the disturbance of the measuring device. With modified boundary conditions, the measurement amplitudes essentially agree with an earlier result of Hartle derived on rather different grounds. In the special case where the system is a collection of harmonic oscillators, the interpretation of the results is aided by the introduction of ``timeless'' coherent states -- eigenstates of the Hamiltonian which are concentrated about entire classical trajectories. 
  We present an alternative method for constructing the exact and approximate solutions of electromagnetic wave equations whose source terms are arbitrary order multipoles on a curved spacetime. The developed method is based on the higher-order Green's functions for wave equations which are defined as distributions that satisfy wave equations with the corresponding order covariant derivatives of the Dirac delta function as the source terms. The constructed solution is applied to the study of various geometric effects on the generation and propagation of electromagnetic wave tails to first order in the Riemann tensor. Generally the received radiation tail occurs after a time delay which represents geometrical backscattering by the central gravitational source. It is shown that the truly nonlocal wave-propagation correction (the tail term) takes a universal form which is independent of multipole order. In a particular case, if the radiation pulse is generated by the source during a finite time interval, the tail term after the primary pulse is entirely determined by the energy-momentum vector of the gravitational field source: the form of the tail term is independent of the multipole structure of the gravitational source. We apply the results to a compact binary system and conclude that under certain conditions the tail energy can be a noticeable fraction of the primary pulse energy. We argue that the wave tails should be carefully considered in energy calculations of such systems. 
  One possible description of the very early stages of the evolution of the universe is provided by Chaotic Inflationary Cosmology. For that model the role of the inflaton field is played by quantum gravitational effects. We study if such a picture may arise within the framework of Loop Quantum gravity by studying a simple model. While we were unable to reach definitive conclusions we believe the general approach proposed in this paper may prove fruitful in the future. 
  The aim of this paper is to enlight the emerging relevance of Quantum Information Theory in the field of Quantum Gravity. As it was suggested by J. A. Wheeler, information theory must play a relevant role in understanding the foundations of Quantum Mechanics (the "It from bit" proposal). Here we suggest that quantum information must play a relevant role in Quantum Gravity (the "It from qubit" proposal). The conjecture is that Quantum Gravity, the theory which will reconcile Quantum Mechanics with General Relativity, can be formulated in terms of quantum bits of information (qubits) stored in space at the Planck scale. This conjecture is based on the following arguments: a) The holographic principle, b) The loop quantum gravity approach and spin networks, c) Quantum geometry and black hole entropy. Here we present the quantum version of the holographic principle by considering each pixel of area of an event horizon as a qubit. This is possible if the horizon is pierced by spin networks' edges of spin 1\2, in the superposed state of spin "up" and spin "down". 
  We investigate the set of spacetime general coordinate transformations (G.C.T.) which leave the line element of a generic Bianchi Type Geometry, quasi-form invariant; i.e. preserve manifest spatial Homogeneity. We find that these G.C.T.'s, induce special time-dependent automorphic changes, on the spatial scale factor matrix $\gamma_{\alpha\beta}(t)$ -along with corresponding changes on the lapse function $N(t)$ and the shift vector $N^{\alpha}(t)$. These changes, which are Bianchi Type dependent, form a group and are, in general, different from those induced by the group SAut(G) -advocated in earlier investigations as the relevant symmetry group-, they are used to simplify the form of the line element -and thus simplify Einstein's equations as well-, without losing generality.   As far as this simplification procedure is concerned, the transformations found, are proved to be essentialy unique. For the case of Bianchi Types II and V, where the most general solutions are known -Taub's and Joseph's, respectively-, it is explicitly verified that our transformations and only those, suffice to reduce the generic line element, to the previously known forms. It becomes thus possible, -for these Types- to give in closed form, the most general solution, containing all the necessary ``gauge'' freedom. 
  In the framework of a model of minimal of dilatonic gravity (MDG) with cosmological potential we consider: the relations of MDG with nonlinear gravity and string theory; natural cosmological units, defined by cosmological constant; the properties of cosmological factor, derived from solar system and Earth-surface gravitational experiments; universal anty-gravitational interactions, induced by positive cosmological constant and by Nordtved effect; a new formulation of cosmological constant problem using the ratio of introduced cosmological action and Planck constant $\sim 10 ^{122}$;qualitative analysis of this huge number based on classical action of effective Bohr hydrogen atoms; inverse cosmological problem: to find cosmological potential which yields given evolution of the RW Universe; and comment other general properties of MDG. 
  In the framework of loop quantum cosmology anomaly free quantizations of the Hamiltonian constraint for Bianchi class A, locally rotationally symmetric and isotropic models are given. Basic ideas of the construction in (non-symmetric) loop quantum gravity can be used, but there are also further inputs because the special structure of symmetric models has to be respected by operators. In particular, the basic building blocks of the homogeneous models are point holonomies rather than holonomies necessitating a new regularization procedure. In this respect, our construction is applicable also for other (non-homogeneous) symmetric models, e.g. the spherically symmetric one. 
  Using general features of recent quantizations of the Hamiltonian constraint in loop quantum gravity and loop quantum cosmology, a dynamical interpretation of the constraint equation as evolution equation is presented. This involves a transformation from the connection to a dreibein representation and the selection of an internal time variable. Due to the discrete nature of geometrical quantities in loop quantum gravity also time turns out to be discrete leading to a difference rather than differential evolution equation. Furthermore, evolving observables are discussed in this framework which enables an investigation of physical spectra of geometrical quantities. In particular, the physical volume spectrum is proven to equal the discrete kinematical volume spectrum in loop quantum cosmology. 
  An angular momentum operator in loop quantum gravity is defined using spherically symmetric states as a non-rotating reference system. It can be diagonalized simultaneously with the area operator and has the familiar spectrum. The operator indicates how the quantum geometry of non-rotating isolated horizons can be generalized to rotating ones and how the recent computations of black hole entropy can be extended to rotating black holes. 
  Using a complex representation of the Debney-Kerr-Schild (DKS) solutions and the Kerr theorem we give a method to construct boosted Kerr geometries. In the ultrarelativistic case this method yelds twisting solutions having, contrary to the known pp-wave limiting solutions, a non-zero value of the total angular momentum. The solutions show that twist plays a crucial role in removing singularity and smoothing shock wave in the ultrarelativistic limit. Two different physical situations are discussed. 
  In gravitational thermodynamics, the origin of a black hole's entropy is the topology of its instanton or constrained instanton. We prove that the entropy of an arbitrary nonrotating black hole is one quarter the sum of the products of the Euler characteristics of its horizons with their respective areas. The Gauss-Bonnet-like form of the action is not only crucial for the evaluation, but also for the existence of the entropy. This result covers all previous results on the entropy of a nonrotating black hole with a regular instanton. The argument can be readily extended into the lower or higher dimensional model. The problem of quantum creation of such a black hole is completely resolved. 
  The excitation of the axial quasi-normal modes of a relativistic star by scattered particles is studied by evolving the time dependent perturbation equations. This work is the first step towards the understanding of more complicated perturbative processes, like the capture or the scattering of particles by rotating stars. In addition, it may serve as a test for the results of the full nonlinear evolution of binary systems. 
  In its final year of inspiral, a stellar mass ($1 - 10 M_\odot$) body orbits a massive ($10^5 - 10^7 M_\odot$) compact object about $10^5$ times, spiralling from several Schwarzschild radii to the last stable orbit. These orbits are deep in the massive object's strong field, so the gravitational waves that they produce probe the strong field nature of the object's spacetime. Measuring these waves can, in principle, be used to ``map'' this spacetime, allowing observers to test whether the object is a black hole or something more exotic. Such measurements will require a good theoretical understanding of wave generation during inspiral. In this article, I discuss the major theoretical challenges standing in the way of building such maps from gravitational-wave observations, as well as recent progress in producing extreme mass ratio inspirals and waveforms. 
  The final inspiral phase in the evolution of a compact binary consisting of black holes and/or neutron stars is among the most probable events that a network of ground-based interferometric gravitational wave detectors is likely to observe. Gravitational radiation emitted during this phase will have to be dug out of noise by matched-filtering (correlating) the detector output with a bank of several $10^5$ templates, making the computational resources required quite demanding, though not formidable. We propose an interpolation method for evaluating the correlation between template waveforms and the detector output and show that the method is effective in substantially reducing the number of templates required. Indeed, the number of templates needed could be a factor $\sim 4$ smaller than required by the usual approach, when the minimal overlap between the template bank and an arbitrary signal (the so-called {\it minimal match}) is 0.97. The method is amenable to easy implementation, and the various detector projects might benefit by adopting it to reduce the computational costs of inspiraling neutron star and black hole binary search. 
  We describe results of a numerical calculation of circularly symmetric scalar field collapse in three spacetime dimensions with negative cosmological constant. The procedure uses a double null formulation of the Einstein-scalar equations. We see evidence of black hole formation on first implosion of a scalar pulse if the initial pulse amplitude $A$ is greater than a critical value $A_*$. Sufficiently near criticality the apparent horizon radius $r_{AH}$ grows with pulse amplitude according to the formula $r_{AH} \sim (A-A_*)^{0.81}$. 
  It is well known that, when an external general relativistic (electric-type) tidal field E(t) interacts with the evolving quadrupole moment I(t) of an isolated body, the tidal field does work on the body (``tidal work'') -- i.e., it transfers energy to the body -- at a rate given by the same formula as in Newtonian theory: dW/dt = -1/2 E dI/dt. Thorne has posed the following question: In view of the fact that the gravitational interaction energy between the tidal field and the body is ambiguous by an amount of order E(t)I(t), is the tidal work also ambiguous by this amount, and therefore is the formula dW/dt = -1/2 E dI/dt only valid unambiguously when integrated over timescales long compared to that for I(t) to change substantially? This paper completes a demonstration that the answer is no; dW/dt is not ambiguous in this way. More specifically, this paper shows that dW/dt is unambiguously given by -1/2 E dI/dt independently of one's choice of how to localize gravitational energy in general relativity. This is proved by explicitly computing dW/dt using various gravitational stress-energy pseudotensors (Einstein, Landau-Lifshitz, Moller) as well as Bergmann's conserved quantities which generalize many of the pseudotensors to include an arbitrary function of position. A discussion is also given of the problem of formulating conservation laws in general relativity and the role played by the various pseudotensors. 
  We develop a general framework for effective equations of expectation values in quantum cosmology and pose for them the quantum Cauchy problem with no-boundary and tunneling wavefunctions. Cosmological configuration space is decomposed into two sectors that give qualitatively different contributions to the radiation currents in effective equations. The field-theoretical sector of inhomogeneous modes is treated by the method of Euclidean effective action, while the quantum mechanical sector of the spatially homogeneous inflaton is handled by the technique of manifest quantum reduction to gauge invariant cosmological perturbations. We apply this framework in the model with a big negative non-minimal coupling, which incorporates a recently proposed low energy (GUT scale) mechanism of the quantum origin of the inflationary Universe and study the effects of the quantum inflaton mode. 
  The stability of a Nambu-Goto membrane at the equatorial plane of the Reissner-Nordstr{\o}m-de Sitter spacetime is studied. The covariant perturbation formalism is applied to study the behavior of the perturbation of the membrane. The perturbation equation is solved numerically. It is shown that a membrane intersecting a charged black hole, including extremely charged one, is unstable and that the positive cosmological constant strengthens the instability. 
  Two metric perturbations in Einstein-Cartan cosmology are examined.The first case is the scalar mode perturbation of de Sitter metric in Einstein-Cartan cosmology.In this case small perturbations of de Sitter metric shows that this metric is unstable for a universe with spin-torsion density and dilaton fields.In the second case we show that the Friedmann metric in the same space is also unstable under the small perturbations. 
  We study the self force acting on static electric or scalar charges inside or outside a spherical, massive, thin shell. The regularization of the self force is done using the recently-proposed Mode Sum Regularization Prescription. In all cases the self force acting on the charge is repulsive. We find that in the scalar case the force is quadratic in the mass of the shell, and is a second post-Newtonian effect. For the electric case the force is linear in the shell's mass, and is a first post-Newtonian effect. When the charge is outside the shell our results correct the known zero self force in the scalar case or the known repulsive, inverse-cubic force law in the electric case, for the finite size of the shell. When the charge is near the center of the shell the charge undergoes harmonic oscillations. 
  We examine the properties of an excess power method to detect gravitational waves in interferometric detector data. This method is designed to detect short-duration (< 0.5 s) burst signals of unknown waveform, such as those from supernovae or black hole mergers. If only the bursts' duration and frequency band are known, the method is an optimal detection strategy in both Bayesian and frequentist senses. It consists of summing the data power over the known time interval and frequency band of the burst. If the detector noise is stationary and Gaussian, this sum is distributed as a chi-squared (non-central chi-squared) deviate in the absence (presence) of a signal. One can use these distributions to compute frequentist detection thresholds for the measured power. We derive the method from Bayesian analyses and show how to compute Bayesian thresholds. More generically, when only upper and/or lower bounds on the bursts duration and frequency band are known, one must search for excess power in all concordant durations and bands. Two search schemes are presented and their computational efficiencies are compared. We find that given reasonable constraints on the effective duration and bandwidth of signals, the excess power search can be performed on a single workstation. Furthermore, the method can be almost as efficient as matched filtering when a large template bank is required. Finally, we derive generalizations of the method to a network of several interferometers under the assumption of Gaussian noise. 
  We describe a simple implementation of black hole excision in 3+1 numerical relativity. We apply this technique to a Schwarzschild black hole with octant symmetry in Eddington-Finkelstein coordinates and show how one can obtain accurate, long-term stable numerical evolutions. 
  The functional potential formalism is used to analyze stationary axisymmetric spaces in the Einstein-Maxwell-Dilaton theory. Performing a Legendre transformation, a ``Hamiltonian''is obtained, which allows to rewrite the dynamical equations in terms of three complex functions only. Using an ansatz resembling the one used by the harmonic maps ansatz, we express these three functions in terms of the harmonic parameters, studying the cases where these parameters are real, and when they are complex. For each case, the set of equations in terms of these harmonic parameters is derived, and several classes of solutions to the Einstein-Maxwell with arbitrary coupling constant to a dilaton field are presented. Most of the known solutions of charged and dilatonic black holes are contained as special cases and can be non-trivially generalized in different ways. 
  In this talk I will describe some recent results on the sensitivity of resonant mass detectors shaped as a hollow sphere to scalar gravitational radiation. Detection of this type of gravitational radiation will signal deviations from Einstein's gravity at large distances. I will then discuss a class of experiments aiming at finding deviations from Einstein's gravity at distances below 1 cm. I will review the main experimental difficulties in performing such experiments and evaluate the effects to be taken in account. 
  Parallel transport of vectors in curved spacetimes generally results in a deficit angle between the directions of the initial and final vectors. We examine such holonomy in the Schwarzschild-Droste geometry and find a number of interesting features that are not widely known. For example, parallel transport around circular orbits results in a quantized band structure of holonomy invariance. We also examine radial holonomy and extend the analysis to spinors and to the Reissner-Nordstr\"om metric, where we find qualitatively different behavior for the extremal ($Q = M$) case. Our calculations provide a toolbox that will hopefully be useful in the investigation of quantum parallel transport in Hilbert-fibered spacetimes. 
  This is a review of current black-hole theory, concentrating on local, dynamical aspects. 
  A general definition of energy is given, via the N\"other theorem, for the N-body problem in (1+1) dimensional gravity. Within a first-order Lagrangian framework, the density of energy of a solution relative to a background is identified with the superpotential of the theory. For specific applications we reproduce the expected Hamiltonian for the motion of N particles in a curved spacetime. This Hamiltonian agrees with that found through an ADM-like prescription for the energy when the latter is applicable but it also extends to a wider class of solutions provided a suitable background is chosen. 
  The Hamiltonian formulation of the teleparallel equivalent of general relativity without gauge fixing has recently been established in terms of the Hamiltonian constraint and a set of six primary constraints. Altogether, they constitute a set of first class constraints. In view of the constraint structure we establish definitions for the energy, momentum and angular momentum of the gravitational field. In agreement with previous investigations, the gravitational energy-momentum density follows from a total divergence that arises in the constraints. This definition is applied successfully to the calculation of the irreducible mass of the Kerr black hole. The definition of the algular momentum of the gravitational field follows from the integral form of primary constraints that satisfy the angular momentum algebra. 
  The paper gives an introduction to the gravitational radiation theory of isolated sources and to the propagation properties of light rays in radiative gravitational fields. It presents a theoretical study of the generation, propagation, back-reaction, and detection of gravitational waves from astrophysical sources. After reviewing the various quadrupole-moment laws for gravitational radiation in the Newtonian approximation, we show how to incorporate post-Newtonian corrections into the source multipole moments, the radiative multipole moments at infinity, and the back-reaction potentials. We further treat the light propagation in the linearized gravitational field outside a gravitational wave emitting source. The effects of time delay, bending of light, and moving source frequency shift are presented in terms of the gravitational lens potential. Time delay results are applied in the description of the procedure of the detection of gravitational waves. 
  We numerically implement a quasi-spherical approximation scheme for computing gravitational waveforms for coalescing black holes, testing it against angular momentum by applying it to Kerr black holes. As error measures, we take the conformal strain and specific energy due to spurious gravitational radiation. The strain is found to be monotonic rather than wavelike. The specific energy is found to be at least an order of magnitude smaller than the 1% level expected from typical black-hole collisions, for angular momentum up to at least 70% of the maximum, for an initial surface as close as $r=3m$. 
  We study the possibility of non-singular black hole solutions in the theory of general relativity coupled to a non-linear scalar field with a positive potential possessing two minima: a `false vacuum' with positive energy and a `true vacuum' with zero energy. Assuming that the scalar field starts at the false vacuum at the origin and comes to the true vacuum at spatial infinity, we prove a no-go theorem by extending a no-hair theorem to the black hole interior: no smooth solutions exist which interpolate between the local de Sitter solution near the origin and the asymptotic Schwarzschild solution through a regular event horizon or several horizons. 
  The transverse group associated to some continuous quantum measuring processes is analyzed in the presence of nonvanishing gravitational fields. This is done considering, as an exmaple, the case of a particle whose coordinates are being monitored. Employing the so called restricted path integral formalism, it will be shown that the measuring process could always contain information concerning the gravitational field. In other words, it seems that with the presence of a measuring process the equivalence principle may, in some cases, break down. The relation between the breakdown of the equivalence principle, at quantum level, and the fact that the gravitational field could act always as a decoherence environment, is also considered. The phenomena of quantum beats of quantum optics will allow us to consider the possibility that the experimental corroboration of the equivalence principle at quantum level could be taken as an indirect evidence in favor of the quantization of the gravitational field, i.e., the quantum properties of this field avoid the violation of the equivalence principle. 
  Self-consistent system of nonlinear spinor field and Bianchi I (BI) gravitational one with time dependent gravitational constant ($G$) and cosmological constant ($\Lambda$) has been studied. The initial and the asymptotic behaviors of the field functions and the metric one have been thoroughly investigated. Given $\Lambda = \Lambda_0/\tau^2$, with $\tau = \sqrt{-g}$, $G$ has been estimated as a function of $\tau$. The role of perfect fluid at the initial state of expansion and asymptotical isotropization process of the initailly anisotropic universe has been elucidated. 
  We consider: minimal scalar-tensor model of gravity with Brans-Dicke factor $\omega(\Phi)\equiv 0$ and cosmological factor $\Pi(\Phi)$; restrictions on it from gravitational experiments; qualitative analysis of new approach to cosmological constant problem based on the huge amount of action in Universe; determination of $\Pi(\Phi)$ using time evolution of scale factor of Universe. 
  In this paper, we show that the number of hyperbolic gravitational instantons grows superexponentially with respect to volume. As an application, we show that the Hartle-Hawking wave function for the universe is infinitely peaked at a certain closed hyperbolic 3-manifold. 
  We investigate the back reaction of cosmological perturbations on an inflationary universe using the renormalization-group method. The second-order zero mode solution which appears by the nonlinearity of the Einstein equation is regarded as a secular term of a perturbative expansion, we renormalized a constant of integration contained in the background solution and absorbed the secular term to this constant in a gauge-invariant manner. The resultant renormalization-group equation describes the back reaction effect of inhomogeneity on the background universe. For scalar type classical perturbation, by solving the renormalization-group equation, we find that the back reaction of the long wavelength fluctuation works as a positive spatial curvature, and the short wavelength fluctuation works as a radiation fluid. For the long wavelength quantum fluctuation, the effect of back reaction is equivalent to a negative spatial curvature. 
  The gravitational field of a stationary circular cosmic string loop ,externally supported against collapse, is investigated in the context of Brans-Dicke theory in the weak field approximation of the field equations. The solution is quasi-conformally related to the corresponding solution in Einstein's General Relativity(GR) and goes over to the corresponding solution in GR when the Brans-Dicke parameter $\omega $ becomes infinitely large. 
  The anti-self-dual projection of the spin connections of certain four-dimensional Einstein manifolds can be Abelian in nature. These configurations signify bundle reductions. By a theorem of Kobayashi and Nomizu such a process is predicated on the existence of a covariantly constant field. It turns out that even without fundamental Higgs fields and other physical matter, gravitational self-interactions can generate this mechanism if the cosmological constant is non-vanishing. This article identifies the order parameter, and clarifies how these Abelian instanton solutions are associated with a Higgs triplet which causes the bundle reduction from SO(3) gauge group to U(1). 
  Friedmann-Robertson-Walker universes with a presently large fraction of the energy density stored in an $X$-component with $w_X<-1/3$, are considered. We find all the critical points of the system for constant equations of state in that range. We consider further several background quantities that can distinguish the models with different $w_X$ values. Using a simple toy model with a varying equation of state, we show that even a large variation of $w_X$ at small redshifts is very difficult to observe with $d_L(z)$ measurements up to $z\sim 1$. Therefore, it will require accurate measurements in the range $1<z<2$ and independent accurate knowledge of $\Omega_{m,0}$ (and/or $\Omega_{X,0}$) in order to resolve a variable $w_X$ from a constant $w_X$. 
  With a generally covariant equation of Dirac fields outside a black hole, we develop a scattering theory for massive Dirac fields. The existence of modified wave operators at infinity is shown by implementing a time-dependent logarithmic phase shift from the free dynamics to offset a long-range mass term. The phase shift we obtain is a matrix operator due to the existence of both positive and negative energy wave components. 
  On a static spacetime, the solutions of the Dirac equation are generated by a time-independent Hamiltonian. We study this Hamiltonian and characterize the split into positive and negative energy. We use it to find explicit expressions for advanced and retarded fundamental solutions and for the propagator. Finally, we use a fermion Fock space based on the positive/negative energy split to define a Dirac quantum field operator whose commutator is the propagator. 
  In this Letter we study the gravitational interactions between outgoing configurations giving rise to Hawking radiation and in-falling configurations. When the latter are in their ground state, the near horizon interactions lead to collective effects which express themselves as metric fluctuations and which induce dissipation, as in Brownian motion. This dissipation prevents the appearance of trans-Planckian frequencies and leads to a description of Hawking radiation which is very similar to that obtained from sound propagation in condensed matter models. 
  Interferometric gravitational wave detectors operate by sensing the differential light travel time between free test masses. Correspondingly, they are sensitive to anything that changes the physical distance between the test masses, including physical motion of the masses themselves. In ground-based detectors the test masses are suspended as pendula and, consequently, thermal or other excitations of the suspension wires' violin modes lead to a strong, albeit narrow-band, ``signal'' in the detector wave-band that can confound attempts to observe gravitational waves.   Here we describe the design of a Kalman filter that determines the time-dependent vibrational state of a detector's suspension ``violin'' modes from the detector output. From the estimated state we can predict that component of the detector output due to suspension excitations, thermal or otherwise, and subtractively remove those disturbances from the detector output. We demonstrate the filter's effectiveness both through numerical simulations and application to real data taken on the LIGO 40 M prototype detector. 
  It is shown how, within the framework of general relativity and without the introduction of wormholes, it is possible to modify a spacetime in a way that allows a spaceship to travel with an arbitrarily large speed. By a purely local expansion of spacetime behind the spaceship and an opposite contraction in front of it, motion faster than the speed of light as seen by observers outside the disturbed region is possible. The resulting distortion is reminiscent of the ``warp drive'' of science fiction. However, just as it happens with wormholes, exotic matter will be needed in order to generate a distortion of spacetime like the one discussed here. 
  We consider a quantized scalar field in a two-dimensional Minkowski spacetime with a moving mirror and propose a definition of moving-mirror entropy associated with temporarily inaccessible information about the future. 
  Phase space path integral is worked out in a riemannian geometry, by employing a prescription for the infinitesimal propagator that takes riemannian normal coordinates and momenta on an equal footing. The operator ordering induced by this prescription leads to the DeWitt curvature coupling in the Schrodinger equation. 
  We examine the motion of charged particles in gravitational and electro-magnetic background fields. We study in particular the deviation of world lines, describing the relative acceleration between particles on different space-time trajectories. Two special cases of background fields are considered in detail: (a) pp-waves, a combination of gravitational and electro-magnetic polarized plane waves travelling in the same direction; (b) the Reissner-Nordstr{\o}m solution. We perform a non-trivial check by computing the precession of the periastron for a charged particle in the Reissner-Nordstr{\o}m geometry both directly by solving the geodesic equation, and using the world-line deviation equation. The results agree to the order of approximation considered. 
  A five dimensional brane cosmology with non-minimally coupled scalar field to gravity has been considered in a Jordan-Brans-Dicke frame. We derive an effective four dimensional field equations on a 3+1 dimensional brane where the fifth dimension has been assumed to have an orbifold symmetry. We have noticed that the evolution equation for the matter component stuck to the brane is non-trivially coupled to the scalar field living on the brane and the bulk. Finally we discuss some cosmological consequences of this set-up. 
  We point out the existence of new effects of global spacetime expansion on local binary systems. In addition to a possible change of orbital size, there is a contribution to the precession of elliptic orbits, to be added to the well-known general relativistic effect in static spacetimes, and the eccentricity can change. Our model calculations are done using geodesics in a McVittie metric, representing a localized system in an asymptotically Robertson-Walker spacetime; we give a few numerical estimates for that case, and indicate ways in which the model should be improved. 
  Black holes have their own thermodynamics including notions of entropy and temperature and versions of the three laws. After a light introduction to black hole physics, I recollect how black hole thermodynamics evolved in the 1970's, while at the same time stressing conceptual points which were given little thought at that time, such as why the entropy should be linear in the black hole's surface area. I also review a variety of attempts made over the years to provide a statistical mechanics for black hole thermodynamics. Finally, I discuss the origin of the information bounds for ordinary systems that have arisen as applications of black hole thermodynamics. 
  We investigate Bianchi I cosmological model in the theory of a dilatonM field coupled to gravity through a Gauss-Bonnet term. Two type ofM cosmological singularity are distinguished. The former is analogous toM the Einstein gravity singularity, the latter (which does not appear inM classical General Relativity) occurs when the main determinant of theM system of field equations vanishes. An analogy between the latterM cosmological singularity and the singularity inside a black hole withM a dilatonic hair is discussed. Initial conditions, leading to theseM two types of cosmological singularity are found via numericalM integration of the equation of motion. 
  We present a spin foam formulation of Lorentzian quantum General Relativity. The theory is based on a simple generalization of an Euclidean model defined in terms of a field theory over a group. Its vertex amplitude turns out to be the one recently introduced by Barrett and Crane. As in the case of its Euclidean relatives, the model fully implements the desired sum over 2-complexes which encodes the local degrees of freedom of the theory. 
  The aim of this paper is to examine some obtained exact solutions of the Einstein-Maxwell equations, especially their properties from a chronological point of view. Each our spacetime is stationary cylindrically symmetric and it is filled up with an perfect fluid that is electrically charged. There are two classes of solutions and examples of each of them are investigated. We give examples of the first class both for the vanishing as well as for the non-vanishing Lorentz force. 
  The new Process Physics models reality as self-organising relational information and takes account of the limitations of logic, discovered by Godel and extended by Chaitin, by using the concept of self-referential noise. Space and quantum physics are emergent and unified, and described by a Quantum Homotopic Field Theory of fractal topological defects embedded in a three dimensional fractal process-space. 
  Black holes are superb sources of gravitational wave signals, for example when they are born in stellar collapse. We explore the subtleties that may emerge if mass accretion events increase significantly the mass of the black hole during its gravitational wave emission. We find the familiar damped-oscillatory radiative decay but now both decay rate and frequencies are modulated by the mass accretion rate. Any appreciable increase in the horizon mass during emission reflects on the instantaneous signal frequency, which shows a prominent negative branch in the dot[f](f) evolution diagram. The features of the frequency evolution pattern reveal key properties of the accretion event, such as the total accreted mass and the accretion rate. For slow accretion rates the frequency evolution follows verbatim the accretion rate, as expected from dimensional arguments. In view of the possibility of detection of black hole ``ringing'' by the upcoming gravitational wave experiments, the deciphering of the late time frequency dynamics may provide direct insight into otherwise obscured aspects of the black hole birth process. 
  We consider quantum birth of a hot Universe in the framework of quantum qeometrodynamics in the minisuperspace model. The energy spectrum of the Universe in the pre-de-Sitter domain naturally explains the cosmic microwave background (CMB) anisotropy. The false vacuum where the Universe tunnels from the pre-de-Sitter domain is assumed to be of a Grand Unification Theory (GUT) scale. The probability of the birth of a hot Universe from a quantum level proves to be about $10^{-10^{14}}$. In the presence of matter with a negative pressure (quintessence) it is possible for open and flat universes to be born as well as closed ones. 
  We study the third quantization of a Brans-Dicke toy model, we calculate the number density of the universes created from nothing and found that it has a Planckian form. Also, we calculated the uncertainty relation for this model by means of functional Schr"odinger equation and we found that fluctuations of the third-quantized universe field tends to a finite limit in the course of cosmic expansion. 
  The Wheeler-DeWitt equation is solved for the Bergmann-Wagoner scalar-tensor gravitational theory in the case of Friedmann-Robertson- Walker cosmological model. We present solutions for several cosmological functions: i) \lambda(\phi)=0, ii) \lambda(\phi)=3\Lambda_0\phi and iii) a more complex \lambda(\phi), that depends on the choice of the coupling function, considering closed, flat and hyperbolic Friedmann universes (k=1, 0, -1). In the first two cases we show particular quantum wormhole solutions. Also, classical solutions are considered for some scalar-tensor theories, and we study the third quantization of some minisuperspace models. 
  An analitical approximation of $<\phi^2>$ for a scalar field in a static spherically symmetric wormhole spacetime is obtained. The scalar field is assumed to be both massive and massless, with an arbitrary coupling $\xi$ to the scalar curvature, and in a zero temperature vacuum state. 
  We present a suggestion on the interpretation of canonical time evolution when gravitation is present, based on the nonlinear gauge approach to gravity. Essentially, our proposal consists of an internal-time concept, with the time variable taken from the dynamical fields characteristic of the nonlinear realization of the internal time-translational symmetry. Physical time evolution requires the latter symmetry to be broken. After disregarding other breaking mechanisms, we appeal to the Jordan-Brans-Dicke action, conveniently interpreted, to achieve that goal. We show that nontrivial time evolution follows, the special relativistic limit being recovered in the absence of gravity. 
  We show that the black hole solutions of the effective string theory action, where one-loop effects that couple the moduli to gravity via a Gauss-Bonnet term are taken into account, admit primary scalar hair. The requirement of absence of naked singularities imposes an upper bound on the scalar charges. 
  This paper is the third of a series dedicated to the study of the Delayed Big-Bang (DBB) class of inhomogeneous cosmological models of Lema\^itre-Tolman-Bondi type. In the first work, it was shown that the geometrical properties of the DBB model are such that the horizon problem can be solved, without need for any inflationary phase, for an observer situated sufficiently near the symmetry center of the model to justify the ``centered earth'' approximation. In the second work, we studied, in a peculiar subclass of the DBB models, the extent to which the values of the dipole and quadrupole moments measured in the cosmic microwave background radiation (CMBR) temperature anisotropies can support a cosmological origin. This implies a relation between the location of the observer in the universe and the model parameter value: the farther the observer from the symmetry center, the closer our current universe to a local homogeneous pattern. However, in this case, the centered earth approximation is no longer valid and the results of the first work do not apply. We show here that the horizon problem can be solved, in the DBB model, also for an off-center observer, which improves the consistency of this model regarding the assumption of a CMBR large scale anisotropy cosmological origin. 
  The Schwarzschild spacetime is for electromagnetic waves like a nonuniform medium with a varying refraction index. A fraction of an outgoing radiation scatters off the curvature of the geometry and can be intercepted by a gravitational center. The amount of the intercepted energy is bounded above by the backscattered energy of an initially outgoing pulse of electromagnetic radiation, which in turn depends on the initial energy, the Schwarzschild radius and the pulse location. Its magnitude depends on the frequency spectrum: it becomes negligible in the short wave limit but can be significant in the long wave regime. 
  This talk presents: (a) A quantum-mechanically induced violation of the principle of equivalence, and (b) Gravitationally-induced modification to the wave particle duality. In this context I note that the agreement between the predictions of general relativity and observations of the energy loss due to gravitational waves emitted by binary pulsars is just as impressive as the agreement between prediction of quantum electrodynamics and the measured value of Lamb shift in atoms. However, general relativity has not yet yielded to a successful quantised theory. There is a widespread belief that the two thories are incompatible at some deep level. The question is: where? Here, I show that the conceptual foundations of the theory of general relativity and quantum mechanics are so rich that they suggest concrete modifications into each other in the interface region. Specifically, I consider quantum states that have no classical counterpart and show that such states must carry an inherent violation of the principle of equivalence. On the other hand, I show that when gravitational effects are incorporated into the quantum measurement process one must induce a gravitationally induced modification to the de Broglie's wave-particle duality. The reported changes into the foundations of the two theories are far from in-principle modifications. These are endowed with serious implications for the understanding of the early universe and, in certain instances, can be explored in terrestrial laboratories. 
  We consider scalar-tensor theories of gravity in an accelerating universe. The equations for the background evolution and the perturbations are given in full generality for any parametrization of the Lagrangian, and we stress that apparent singularities are sometimes artifacts of a pathological choice of variables. Adopting a phenomenological viewpoint, i.e., from the observations back to the theory, we show that the knowledge of the luminosity distance as a function of redshift up to z ~ (1-2), which is expected in the near future, severely constrains the viable subclasses of scalar-tensor theories. This is due to the requirement of positive energy for both the graviton and the scalar partner. Assuming a particular form for the Hubble diagram, consistent with present experimental data, we reconstruct the microscopic Lagrangian for various scalar-tensor models, and find that the most natural ones are obtained if the universe is (marginally) closed. 
  We have measured the mechanical dissipation in a sample of fused silica drawn into a rod. The sample was hung from a multiple-bob suspension, which isolated it from rubbing against its support, from recoil in the support structure, and from seismic noise. The quality factor, Q, was measured for several modes with a high value of 57 million found for mode number 2 at 726 Hz. This result is about a factor 2 higher than previous room temperature measurements. The measured Q was strongly dependent on handling, with a pristine flame-polished surface yielding a Q 3-4 times higher than a surface which had been knocked several times against a copper tube. 
  Even when the Higgs particle is finally detected, it will continue to be a legitimate question to ask whether the inertia of matter as a reaction force opposing acceleration is an intrinsic or extrinsic property of matter. General relativity specifies which geodesic path a free particle will follow, but geometrodynamics has no mechanism for generating a reaction force for deviation from geodesic motion. We discuss a different approach involving the electromagnetic zero-point field (ZPF) of the quantum vacuum. It has been found that certain asymmetries arise in the ZPF as perceived from an accelerating reference frame. In such a frame the Poynting vector and momentum flux of the ZPF become non-zero. Scattering of this quantum radiation by the quarks and electrons in matter can result in an acceleration-dependent reaction force. Both the ordinary and the relativistic forms of Newton's second law, the equation of motion, can be derived from the electrodynamics of such ZPF-particle interactions. Conjectural arguments are given why this interaction should take place in a resonance at the Compton frequency, and how this could simultaneously provide a physical basis for the de Broglie wavelength of a moving particle. This affords a suggestive perspective on a deep connection between electrodynamics, the origin of inertia and the quantum wave nature of matter. 
  A general analysis for characterizing and classifying `isolated horizons' is presented in terms of null tetrads and spin coefficients. The freely specifiable spin coefficients corresponding to isolated horizons are identified and specific symmetry classes are enumerated. For isolated horizons admitting at least one spatial isometry, a standard set of spherical coordinates are introduced and associated metric is obtained. An angular momentum is also defined. 
  We apply the method of moving anholonomic frames, with associated nonlinear connections, in (pseudo) Riemannian spaces and examine the conditions when various types of locally anisotropic (la) structures (Lagrange, Finsler like and more general ones) could be modeled in general relativity. New classes of solutions of the Einstein equations with generic local anisotropy are constructed. We formulate the theory of nearly autoparallel (na) maps and introduce the tensorial na-integration as the inverse operation to both covariant derivation and deformation of connections by na-maps. The problem of redefinition of the Einstein gravity theory on na-backgrounds, provided with a set of na-map invariant conditions and local conservation laws, is analyzed. There are illustrated some examples of generation of vacuum Einstein fields by Finsler like metrics and chains of na-maps. 
  We analyze local anisotropies induced by anholonomic frames and associated nonlinear connections in general relativity and extensions to affine Poincare and de Sitter gauge gravity and different types of Kaluza-Klein theories. We construct some new classes of cosmological solutions of gravitational field equations describing Friedmann-Robertson-Walker like universes with rotation (ellongated and flattened) ellipsoidal or torus symmetry. 
  We exploit the possibility of existence of a repulsive gravity phase in the evolution of the Universe. A toy model with a free scalar field minimally coupled to gravity, but with the "wrong sign" for the energy and negative curvature for the spatial section, is studied in detail. The background solutions display a bouncing, non-singular Universe. The model is well-behaved with respect to tensor perturbations. But, it exhibits growing models with respect to scalar perturbations whose maximum occurs in the bouncing. Hence, large inhomogeneties are produced. At least for this case, a repulsive phase may destroy homogeneity, and in this sense it may be unstable. A newtonian analogous model is worked out; it displays qualitatively the same behaviour. The generality of this result is discussed. In particular, it is shown that the addition of an attractive radiative fluid does not change essentially the results. We discuss also a quantum version of the classical repulsive phase, through the Wheeler-de Witt equation in mini-superspace, and we show that it displays essentially the same scenario as the corresponding attractive phase. 
  Ideal rods and clocks are defined as an infinitesimal symmetry of the spacetime. Since no a priori geometric structure is considered, all the possible models of spacetime are obtained. 
  New spherically symmetric dyonic solutions, describing a wormhole-like class of spacetime configurations in five-dimensional Kaluza-Klein theory, are given in an explicit form. For this type of solution the electric and magnetic fields cause a significantly different global structure. For the electric dominated case, the solution is everywhere regular but, when the magnetic strength overcomes the electric contribution, the mouths of the wormhole become singular points. When the electric and magnetic charge parameters are identical, the throats ``degenerate'' and the solution reduces to the trivial embedding of the four-dimensional massless Reissner-Nordstr{\"o}m black hole solution. In addition, their counterparts in eleven-dimensional supergravity are constructed by a non-trivial uplifting. 
  Possible effects are considered which would be caused by a hypothetical superstrong interaction of photons or massive bodies with single gravitons of the graviton background. If full cosmological redshift magnitudes are caused by the interaction, then the luminosity distance in a flat non-expanding universe as a function of redshift is very similar to the specific function which fits supernova cosmology data by Riess et al. From another side, in this case every massive body, slowly moving relatively to the background, would experience a constant acceleration, proportional to the Hubble constant, of the same order as a small additional acceleration of Pioneer 10, 11. 
  We solve the elliptic equations associated with the Hamiltonian and momentum constraints, corresponding to a system composed of two black holes with arbitrary linear and angular momentum. These new solutions are based on a Kerr-Schild spacetime slicing which provides more physically realistic solutions than the initial data based on conformally flat metric/maximal slicing methods. The singularity/inner boundary problems are circumvented by a new technique that allows the use of an elliptic solver on a Cartesian grid where no points are excised, simplifying enormously the numerical problem. 
  I discuss a specific model of space-time foam, inspired by the modern non-perturbative approach to string theory (D-branes). The model views our world as a three brane, intersecting with D-particles that represent stringy quantum gravity effects, which can be real or virtual. In this picture, matter is represented generically by (closed or open) strings on the D3 brane propagating in such a background. Scattering of the (matter) strings off the D-particles causes recoil of the latter, which in turn results in a distortion of the surrounding space-time fluid and the formation of (microscopic, i.e. Planckian size) horizons around the defects. As a mean-field result, the dispersion relation of the various particle excitations is modified, leading to non-trivial optical properties of the space time, for instance a non-trivial refractive index for the case of photons or other massless probes. Such models make falsifiable predictions, that may be tested experimentally in the foreseeable future. I describe a few such tests, ranging from observations of light from distant gamma-ray-bursters and ultra high energy cosmic rays, to tests using gravity-wave interferometric devices and terrestrial particle physics experients involving, for instance, neutral kaons. 
  The Wheeler-DeWitt equation is considered in the context of generalized scalar-tensor theories of gravitation for Bianchi type I cosmology. Exact solutions are found for two selfinteracting potentials and arbitary coupling function. The WKB wavefunctions are obtained and a family of solutions satisfying the Hawking-Page regularity conditions of wormholes are found. 
  We present the first simulations of non-headon (grazing) collisions of binary black holes in which the black hole singularities have been excised from the computational domain. Initially two equal mass black holes $m$ are separated a distance $\approx10m$ and with impact parameter $\approx2m$. Initial data are based on superposed, boosted (velocity $\approx0.5c$) solutions of single black holes in Kerr-Schild coordinates. Both rotating and non-rotating black holes are considered. The excised regions containing the singularities are specified by following the dynamics of apparent horizons. Evolutions of up to $t \approx 35m$ are obtained in which two initially separate apparent horizons are present for $t\approx3.8m$. At that time a single enveloping apparent horizon forms, indicating that the holes have merged. Apparent horizon area estimates suggest gravitational radiation of about 2.6% of the total mass. The evolutions end after a moderate amount of time because of instabilities. 
  Taking the ${\Bbb R}^1 \times H^3$ space as an example, we develop the new method of quantization of fields over symmetric spaces. We construct the quantized massless fields of an arbitrary spin over the ${\Bbb R}^1 \times H^3$ space by the resolution over the systems of "plane waves" which are solutions of the corresponding wave equations. The propagators of these fields are ${\Bbb R}^1 \times SO(3,1)$-invariant and causal. For spin 0 and 1/2 fields the propagators are obtained in the explicit form. 
  The almost perfect correspondence between certain laws of classical black hole mechanics and the ordinary laws of thermodynamics is spoiled by the failure of the conventional back hole analogue of the third law. Our aim here is to contribute to the associated discussion by flashing light on some simple facts of black hole physics. However, no attempt is made to lay to rest the corresponding long lasting debate. Instead, merely some evidence is provided to make it clear that although the borderline between extremal and non-extremal black holes is very thin they are essentially different. Hopefully, a careful investigation of the related issues will end up with an appropriate form of the third law and hence with an unblemished setting of black hole thermodynamics. 
  In this thesis, we investigate quantum vacuum effects in the presence of gravitational fields. After discussing the general theory of vacuum effects in strong fields we apply it to the relevant issue of the interaction of the quantum vacuum with black hole geometries. In particular we consider the long-standing problem of the interpretation of gravitational entropy. After these investigations, we discuss the possible experimental tests of particle creation from the quantum vacuum. This leads us to study acoustic geometries and their way of ``simulating'' gravitational structures, such as horizons and black holes. We study the stability of these structures and the problems related to setting up experimental detection of ``phonon Hawking flux'' from acoustic horizons. This line of research then leads us to propose a new model for explaining the emission of light in the phenomenon of Sonoluminescence, based on the dynamical Casimir effect. This is possibly amenable to experimental investigation. Finally we consider high energy phenomena in the early universe. In particular we discuss inflation and possible alternative frameworks for solving the cosmological puzzles. 
  Using heat kernel techniques we show that the relation between Hawking temperature and radiation flux known from Einstein gravity in D dimensions can be reproduced from the spherically reduced action. A recent controversy regarding the D=2 anomaly for that case is discussed. The generalized effective Polyakov action in the presence of a dilaton field is presented. 
  A local conception is proposed to reconcile quantum theory with general relativity, which allows one to avoid some difficulties --- as e.g. vacuum catastrophe --- of the global approach. 
  A regularization procedure, that allows one to relate singularities of curvature to those of the Einstein tensor without some of the shortcomings of previous approaches, is proposed. This regularization is obtained by requiring that (i) the density $|det {\bf g}|^{{1/2}}G^a_{b}$, associated to the Einstein tensor $G^a_{b}$ of the regularized metric, rather than the Einstein tensor itself, be a distribution and (ii) the regularized metric be a continuous metric with a discontinuous extrinsic curvature across a non-null hypersurface of codimension one. In this paper, the curvature and Einstein tensors of the geometries associated to point sources in the 2+1-dimensional gravity and the Schwarzschild spacetime are considered. In both examples the regularized metrics are continuous regular metrics, as defined by Geroch and Traschen, with well defined distributional curvature tensors at all the intermediate steps of the calculation. The limit in which the support of these curvature tensors tends to the singular region of the original spacetime is studied and the results are contrasted with the ones obtained in previous works. 
  We study static black hole solutions in scalar tensor gravity. We present exact solutions in hiperextended models with a quadratic scalar potential. 
  We study gravitational waves from a spinning test particle scattered by a relativistic star using a perturbation method. The present analysis is restricted to axial modes. By calculating the energy spectrum, the waveforms and the total energy and angular momentum of gravitational waves, we analyze the dependence of the emitted gravitational waves on a particle spin. For a normal neutron star, the energy spectrum has one broad peak whose characteristic frequency corresponds to the angular velocity at the turning point (a periastron). Since the turning point is determined by the orbital parameter, there exists the dependence of the gravitational wave on a particle spin. We find that the total energy of $l = 2$ gravitational waves gets larger as the spin increases in the anti-parallel direction to the orbital angular momentum. For an ultracompact star, in addition to such an orbital contribution, we find the quasi-normal modes exited by a scattered particle, whose excitation rate to gravitational waves depends on the particle spin. We also discuss the ratio of the total angular momentum to the total energy of gravitational waves and explain its spin dependence. 
  Following the recent recognition of a positive value for the vacuum energy density and the realization that a simple Kantowski-Sachs model might fit the classical tests of cosmology, we study the qualitative behavior of three anisotropic and homogeneous models: Kantowski-Sachs, Bianchi type-I and Bianchi type-III universes, with dust and a cosmological constant, in order to find out which are physically permitted. We find that these models undergo isotropization up to the point that the observations will not be able to distinguish between them and the standard model, except for the Kantowski-Sachs model $(\Omega_{k_{0}}<0)$ and for the Bianchi type-III $(\Omega_{k_{0}}>0)$ with $\Omega_{\Lambda_{0}}$ smaller than some critical value $\Omega_{\Lambda_{M}}$. Even if one imposes that the Universe should be nearly isotropic since the last scattering epoch ($z\approx 1000$), meaning that the Universe should have approximately the same Hubble parameter in all directions (considering the COBE 4-Year data), there is still a large range for the matter density parameter compatible with Kantowsky-Sachs and Bianchi type-III if $|\Omega_0+\Omega_{\Lambda_0}-1|\leq \delta$, for a very small $\delta$ . The Bianchi type-I model becomes exactly isotropic owing to our restrictions and we have $\Omega_0+\Omega_{\Lambda_0}=1$ in this case. Of course, all these models approach locally an exponential expanding state provided the cosmological constant $\Omega_\Lambda>\Omega_{\Lambda_{M}}$. 
  Conditions for smooth cosmological models are set out and applied to inhomogeneous spherically symmetric models constructed by matching together different Lemaitre-Tolman-Bondi solutions to the  Einstein field equations. As an illustration the methods are applied to a collapsing dust sphere in a curved background. This describes a region which expands and then collapses to form a black hole in an  Einstein de Sitter background. We show that in all such models if there is no vacuum region then the singularity must go on accreting matter for an infinite LTB time. 
  In this work we consider a quantum analog of Newton's bucket experiment in a flat spacetime: we take an Unruh-DeWitt detector in interaction with a real massless scalar field. We calculate the detector's excitation rate when it is uniformly rotating around some fixed point and the field is prepared in the Minkowski vacuum and also when the detector is inertial and the field is in the Trocheries-Takeno vacuum state. These results are compared and the relations with a quantum analog of Mach's principle are discussed. 
  We study the feasibility of producing the graviton of the novel Kaluza-Klein theory in which there are d large compact dimensions in addition to the 4 dimensions of Minkowski spacetime. We calculate the cross section for producing such a graviton in nucleus-nucleus collisions via t-channel photon-photon fusion using the semiclassical Weizsacker-Williams method and show that it can exceed the cross section for graviton production in electron-positron scattering by several orders of magnitude. 
  Research Briefs:  Cosmic microwave background anisotropy experiments, by Sean Carroll  LISA Project Update by Bill Folkner  An update on the r-mode instability, by Nils Andersson  Laboratory experiments: news from MG9, by Riley Newman  Progress toward Commissioning the LIGO detectors, by Stan Whitcomb  160 Hours of Data Taken on TAMA300, by Seiji Kawamura  Conference reports:  Kipfest, by Richard Price  Third Capra meeting, by Eric Poisson  GR at the XIIIth Congress on Mathematical Physics, by Abhay Ashtekar  3rd International LISA Symposium, by Curt Cutler 
  Several recent possible counterexamples to the Chronology Protection Conjecture are critically examined. The ``adapted'' Rindler vacuum state constructed by Li and Gott for a conformal scalar field in Misner space is extended to nonconformally coupled and self-interacting scalar fields. For these fields, the vacuum stress-energy always diverges on the chronology horizons. The divergence of the vacuum stress-energy on Misner space chronology horizons cannot be generally avoided by choosing a Rindler-type vacuum state. 
  The vorticity of a congruence is often considered to be the rate of rotation for the precession of a gyroscope moving along a world-line belonging to that congruence. Our aim here was to determine the evolution equation for the angular momentum of a gyrosocope with respect to an arbitrary time-like congruence: i.e, a reference congruence not containing the curve described by the gyroscope. In particular, we show the specific conditions needed to support the introductory statement about the vorticity. We thus establish a well-founded theoretical description for the analysis of the precession of gyroscopes, providing suitable conclusions for possible experiments. 
  We study the ``renormalization group action'' induced by cycles of cosmic expansion and contraction, within the context of a family of stochastic dynamical laws for causal sets derived earlier. We find a line of fixed points corresponding to the dynamics of transitive percolation, and we prove that there exist no other fixed points and no cycles of length two or more. We also identify an extensive ``basin of attraction'' of the fixed points but find that it does not exhaust the full parameter space. Nevertheless, we conjecture that every trajectory is drawn toward the fixed point set in a suitably weakened sense. 
  We present a new method for generating the nonlinear gravitational wavetrain from the late inspiral (pre-coalescence) phase of a binary neutron star system by means of a numerical evolution calculation in full general relativity. In a prototype calculation, we produce 214 wave cycles from corotating polytropes, representing the final part of the inspiral phase prior to reaching the ISCO. Our method is based on the inequality that the orbital decay timescale due to gravitational radiation is much longer than an orbital period and the approximation that gravitational radiation has little effect on the structure of the stars. We employ quasi-equilibrium sequences of binaries in circular orbit for the matter source in our field evolution code. We compute the gravity-wave energy flux, and, from this, the inspiral rate, at a discrete set of binary separations. From these data, we construct the gravitational waveform as a continuous wavetrain. Finally, we discuss the limitations of our current calculation, planned improvements, and potential applications of our method to other inspiral scenarios. 
  We consider the modification of a single particle Schr\"{o}dinger equation by the inclusion of an additional gravitational self-potential term which follows from the prescription that the' mass-density'that enters this term is given by $m |\psi (\vec {r},t)|^2$, where $\psi (\vec {r}, t)$ is the wavefunction and $m$ is the mass of the particle. This leads to a nonlinear equation, the ' Newton Schrodinger' equation, which has been found to possess stationary self-bound solutions, whose energy can be determined exactly using an asymptotic method. We find that such a particle strongly violates superposition and becomes a black hole as its mass approaches the Planck mass. 
  Cosmic ray showers interacting with the resonant mass gravitational wave antenna NAUTILUS have been detected. The experimental results show large signals at a rate much greater than expected. The largest signal corresponds to an energy release in NAUTILUS of 87 TeV. We remark that a resonant mass gravitational wave detector used as particle detector has characteristics different from the usual particle detectors, and it could detect new features of cosmic rays. Among several possibilities, one can invoke unexpected behaviour of superconducting Aluminium as particle detector, producing enhanced signals, the excitation of non-elastic modes with large energy release or anomalies in cosmic rays (for instance, the showers might include exotic particles as nuclearites or Q-balls). Suggestions for explaining these observations are solicited. 
  Berry phase for a spin--1/2 particle moving in a flat spacetime with torsion is investigated in the context of the Einstein-Cartan-Dirac model. It is shown that if the torsion is due to a dense polarized background, then there is a Berry phase only if the fermion is massless and its momentum is perpendicular to the direction of the background polarization. The order of magnitude of this Berry phase is discussed in other theoretical frameworks. 
  Fermi co-ordinates are proper co-ordinates of a local observer determined by his trajectory in space-time. Two observers at different positions belong to different Fermi frames even if there is no relative motion between them. Use of Fermi co-ordinates leads to several physical conclusions related to relativistic effects seen by observers in arbitrary motion. In flat space-time, the relativistic length seen by an observer depends only on his instantaneous velocity, not on his acceleration or rotation. In arbitrary space-time, for any observer the velocity of light is isotropic and equal to $c$, provided that it is measured by propagating a light beam in a small neighbourhood of the observer. The value of a covariant field measured at the position of the observer depends only on his instantaneous position and velocity, not on his acceleration. The notion of radiation is observer independent. A "freely" falling charge in curved space-time does not move along a geodesic and therefore radiates. 
  Static, spherically symmetric solutions of axi-dilaton gravity in $D$ dimensions is given in the Brans-Dicke frame for arbitrary values of the Brans-Dicke constant $\omega$ and an axion-dilaton coupling parameter $k$. The mass and the dilaton and axion charges are determined and a BPS bound is derived. There exists a one parameter family of black hole solutions in the scale invariant limit. 
  The motion of a classical spinning test particle in the field of a weak plane gravitational wave is studied. It is found that the characteristic dimensions of the particle's orbit is sensitive to the ratio of the spin to the mass of the particle. The results are compared with the corresponding motion of a particle without spin. 
  The model of creation of observable particles and particles of the dark matter, considered to be superheavy particles, due to particle creation by the gravitational field of the Friedmann model of the early Universe is given. Estimates on the parameters of the model leading to observable values of the baryon number of the Universe and the dark matter density are made. 
  A general class of solutions is obtained which describe a spherically symmetric wormhole system. The presence of arbitrary functions allows one to describe infinitely many wormhole systems of this type. The source of the stress-energy supporting the structure consists of an anisotropic brown dwarf ``star'' which smoothly joins the vacuum and may possess an arbitrary cosmological constant. It is demonstrated how this set of solutions allows for a non-zero energy density and therefore allows positive stellar mass as well as how violations of energy conditions may be minimized. Unlike examples considered thus far, emphasis here is placed on construction by manipulating the matter field as opposed to the metric. This scheme is generally more physical than the purely geometric method. Finally, explicit examples are constructed including an example which demonstrates how multiple closed universes may be connected by such wormholes. The number of connected universes may be finite or infinite. 
  We discuss ultracompact stellar objects which have multiple necks in their optical geometry. There are in fact physically reasonable equations of state for which the number of necks can be arbitrarily large. The proofs of these statements rely on a recent regularized formulation of the field equations for static spherically symmetric models due to Nilsson and Uggla. We discuss in particular the equation of state p=\rho-\rho_s which plays a central role in this context. 
  The quantization of the most general Bianchi Type II geometry -with all six scale factors, as well as the lapse function and the shift vector, present- is considered. In an earlier work, a first reduction of the initial 6-dimensional configuration space, to a 4-dimensional one, has been achieved by the usage of the information furnished by the quantum form of the linear constraints. Further reduction of the space in which the wave function -obeying the Wheeler-DeWitt equation- lives, is accomplished by unrevealling the extra symmetries of the Hamiltonian. These symmetries appear in the form of -linear in momenta- first integrals of motion. Most of these symmetries, correspond to G.C.T.s through the action of the automorphism group. Thus, a G.C.T. invariant wave function is found, which depends on the only true degree of freedom, i.e. the unique curvature invariant, characterizing the hypersurfaces t=const. 
  The metric around a wiggly cosmic string is calculated in the linear approximation of Brans-Dicke theory of gravitation. The equations of motion for relativistic and non-relativistic particles in this metric are obtained.  Light propagation is also studied and it is shown that photon trajectories can be bounded. 
  It has been shown that classical non-minimally coupled scalar fields can violate all of the standard energy conditions in general relativity. Violations of the null and averaged null energy conditions obtainable with such fields have been suggested as possible exotic matter candidates required for the maintenance of traversable wormholes. In this paper, we explore the possibility that if such fields exist, they might be used to produce large negative energy fluxes and macroscopic violations of the generalized second law (GSL) of thermodynamics. We find that it appears to be very easy to produce large magnitude negative energy fluxes in flat spacetime. However we also find, somewhat surprisingly, that these same types of fluxes injected into a black hole do {\it not} produce violations of the GSL. This is true even in cases where the flux results in a decrease in the area of the horizon. We demonstrate that two effects are responsible for the rescue of the GSL: the acausal behavior of the horizon and the modification of the usual black hole entropy formula by an additional term which depends on the scalar field. 
  The Bardeen model -- the first regular black hole model in General Relativity -- is reinterpreted as the gravitational field of a nonlinear magnetic monopole, i.e., as a magnetic solution to Einstein equations coupled to a nonlinear electrodynamics. 
  A data-analysis strategy based on the maximum-likelihood method (MLM) is presented for the detection of gravitational waves from inspiraling compact binaries with a network of laser-interferometric detectors having arbitrary orientations and arbitrary locations around the globe. The MLM is based on the network likelihood ratio (LR), which is a function of eight signal-parameters that determine the Newtonian inspiral waveform. In the MLM-based strategy, the LR must be maximized over all of these parameters. Here, we show that it is possible to maximize it analytically over four of the eight parameters. Maximization over a fifth parameter, the time of arrival, is handled most efficiently by using the Fast-Fourier-Transform algorithm. This allows us to scan the parameter space continuously over these five parameters and also cuts down substantially on the computational costs. Maximization of the LR over the remaining three parameters is handled numerically. This includes the construction of a bank of templates on this reduced parameter space. After obtaining the network statistic, we first discuss `idealized' networks with all the detectors having a common noise curve for simplicity. Such an exercise nevertheless yields useful estimates about computational costs, and also tests the formalism developed here. We then consider realistic cases of networks comprising of the LIGO and VIRGO detectors: These include two-detector networks, which pair up the two LIGOs or VIRGO with one of the LIGOs, and the three-detector network that includes VIRGO and both the LIGOs. For these networks we present the computational speed requirements, network sensitivities, and source-direction resolutions. 
  We discuss the construction of perfect fluid stellar objects having optical geometries with multiple necks corresponding to spatially closed unstable lightlike geodesics. We prove that there exist physically reasonable models with arbitrarily many necks. We also show how a first order phase transition can give rise to quite pronounced secondary double necks. The analysis is carried out using a modification of a recent dynamical systems formulation of the TOV equations due to Nilsson and Uggla. Our reformulation allows for a very general family of equations of state including, for example, phase transitions. 
  We consider quantum geometrodynamics and parametrized quantum field theories in the framework of the Bohm-de Broglie interpretation. In the first case, and following the lines of our previous work [1], where a hamiltonian formalism for the bohmian trajectories was constructed, we show the consistency of the theory for any quantum potential, completing the scenarios for canonical quantum cosmology presented there. In the latter case, we prove the consistency of scalar field theory in Minkowski spacetime for any quantum potential, and we show, using this alternative hamiltonian method, a concrete example where Lorentz invariance of individual events is broken. 
  If the human race comes to an end relatively shortly, then we have been born at a fairly typical time in history of humanity. On the other hand, if humanity lasts for much longer and trillions of people eventually exist, then we have been born in the first surprisingly tiny fraction of all people. According to the Doomsday Argument of Carter, Leslie, Gott, and Nielsen, this means that the chance of a disaster which would obliterate humanity is much larger than usually thought. Here I argue that treating possible observers in the same way as those who actually exist avoids this conclusion. Under this treatment, it is more likely to exist at all in a race which is long-lived, as originally discussed by Dieks, and this cancels the Doomsday Argument, so that the chance of a disaster is only what one would ordinarily estimate. Treating possible and actual observers alike also allows sensible anthropic predictions from quantum cosmology, which would otherwise depend on one's interpretation of quantum mechanics. 
  We obtain a general class of exact solutions to topologically massive gravity with or without a negative cosmological constant. In the first case, we show that the solution is supersymmetric and asymptotically approaches the extremal BTZ black hole solution, while in the latter case it goes to flat space-time. 
  We consider the generalized second law of black hole thermodynamics in the light of quantum information theory, in particular information erasure and Landauer's principle (namely, that erasure of information produces at least the equivalent amount of entropy). A small quantum system outside a black hole in the Hartle-Hawking state is studied, and the quantum system comes into thermal equilibrium with the radiation surrounding the black hole. For this scenario, we present a simple proof of the generalized second law based on quantum relative entropy. We then analyze the corresponding information erasure process, and confirm our proof of the generalized second law by applying Landauer's principle. 
  The imposition of symmetries or special geometric properties on submanifolds is less restrictive than to impose them in the full space-time. Starting from this idea, in this paper we study irrotational dust cosmological models in which the geometry of the hypersurfaces generated by the fluid velocity is flat, which supposes a relaxation of the restrictions imposed by the Cosmological Principle. The method of study combines covariant and tetrad methods that exploits the geometrical and physical properties of these models. This procedure will allow us to determine all the space-times within this class as well as to study their properties. Some important consequences and applications of this study are also discussed. 
  From the equivalence principle and true gravitational (G) time dilation experiments it is concluded that ``matter is not invariable after a change of relative position with respect to other bodies''. As a general principle (GP), such variations cannot be locally detected because the basic parameters of all of the 'well-defined parts' of the instruments change, lineally, in the same proportion with respect to their original values''. Only observers that don't change of position can detect them. Thus, to relate quantities measured by observers in different G potentials they must be previously transformed after Lorenz and G transformations derived from experiments. They are account for all of the ``G tests''. However ``they are not consistent with the presumed energy exchange between the field and the bodies''. The lack of energy of the G field is justified from the GP, according to which particles models made up of photons in stationary state obey same inertial and G laws as particle. Such model has been previously tested with relativistic quantum-mechanics and all of the G tests.   PACS: 04.80.cc, 04.20.Cv, 04.80.-y, 98.80.Cq 
  A general formulation of spinor fields in Riemannian space-time is given without using vierbeins. The space-time dependence of the Dirac matrices required by the anticommutation relation {\gamma_{\mu},\gamma_{\nu}}=2g_{\mu\nu} determines the spin connection. The action is invariant under any local spin base transformations in the 32 parameter group Gl(4,c) and not just under local Lorentz transformations. The Dirac equation and the energy-momentum tensor are computed from the action. 
  In his search for a unified field theory that could undercut quantum mechanics, Einstein considered five dimensional classical Kaluza-Klein theory. He studied this theory most intensively during the years 1938-1943. One of his primary objectives was finding a non-singular particle solution. In the full theory this search got frustrated and in the x^5-independent theory Einstein, together with Pauli, argued it would be impossible to find these structures. 
  Physically admissible choice of the "essential" coordinates identified with components of the metric tensor and co-moving frame of reference reduced to the formulation of the stationary axisymmetric GR problem. Such nontraditional approach allows to obtain new 4-parametric vacuum solution. In special static case this solution becomes the set well-known GR exact solutions for degenerate vacuum gravitational fields, in some special stationary case, after transformations coincides with Kerr solution. In other special stationary case the founded solution turn into modified NUT solution. 
  We present a general formalism to treat slowly rotating general relativistic superfluid neutron stars. As a first approximation, their matter content can be described in terms of a two-fluid model, where one fluid is the neutron superfluid, which is believed to exist in the core and inner crust of mature neutron stars, and the other fluid represents a conglomerate of all other constituents (crust nuclei, protons, electrons, etc.). We obtain a system of equations, good to second-order in the rotational velocities, that determines the metric and the matter variables, irrespective of the equation of state for the two fluids. In particular, allowance is made for the so-called entrainment effect, whereby the momentum of one constituent (e.g. the neutrons) carries along part of the mass of the other constituent. As an illustration of the developed framework, we consider a simplified equation of state for which the two fluids are described by different polytropes. We determine numerically the effects of the two fluids on the rotational frame-dragging, the induced changes in the neutron and proton densities and the inertial mass, as well as the change in shape of the star. We further discuss issues regarding conservation of the two baryon numbers, the mass-shedding (Kepler) limit and chemical equilibrium. 
  We give a general class of exact solutions to the (1+2)-dimensional topologically massive gravity model coupled with Maxwell-Chern-Simons theory where a "self-duality" condition is imposed on the Maxwell field. 
  We obtain general formulae for the plus- and cross- polarized waveforms of gravitational radiation emitted by a cosmic string loop in transverse, traceless (synchronous, harmonic) gauge. These equations are then specialized to the case of piecewise linear loops, and it is shown that the general waveform for such a loop is a piecewise linear function. We give several simple examples of the waveforms from such loops. We also discuss the relation between the gravitational radiation by a smooth loop and by a piecewise linear approximation to it. 
  We present an algorithm for calculating the complete data on an event horizon which constitute the necessary input for characteristic evolution of the exterior spacetime. We apply this algorithm to study the intrinsic and extrinsic geometry of a binary black hole event horizon, constructing a sequence of binary black hole event horizons which approaches a single Schwarzschild black hole horizon as a limiting case. The linear perturbation of the Schwarzschild horizon provides global insight into the close limit for binary black holes, in which the individual holes have joined in the infinite past. In general there is a division of the horizon into interior and exterior regions, analogous to the division of the Schwarzschild horizon by the r=2M bifurcation sphere. In passing from the perturbative to the strongly nonlinear regime there is a transition in which the individual black holes persist in the exterior portion of the horizon. The algorithm is intended to provide the data sets for production of a catalog of nonlinear post-merger wave forms using the PITT null code. 
  We develop an evolution scheme, based on Sorkin algorithm, to evolve the most complex regular tridimensional polytope, the 600-cell. This application has been already studied before and all authors found a stop point for the evolution of the spatial section. In our opinion a clear and satisfactory meaning to this behaviour has not been given. In this paper we propose a reason why the evolution of the 600-cell stops when its volume is still far from 0. We find that the 600-cell meets a causality-breaking singularity of space-time. We study the nature of this singularity by embedding the 600-cell into a five-dimensional Lorentzian manifold. We fit 600-cell's evolution with a continuos metric and study it as a solution of Einstein equations. 
  We present the Fourier Transform of continuous gravitational wave for arbitrary location of detector and source and for any duration of observation time in which both rotational motion of earth about its spin axis and orbital motion around sun has been taken into account. We also give the method to account the spin down of continuous gravitational wave. 
  We investigate the effects of pair creation on the internal geometry of a black hole, which forms during the gravitational collapse of a charged massless scalar field. Classically, strong central Schwarzschild-like singularity forms, and a null, weak, mass-inflation singularity arises along the Cauchy horizon, in such a collapse. We consider here the discharge, due to pair creation, below the event horizon and its influence on the {\it dynamical formation} of the Cauchy horizon. Within the framework of a simple model we are able to trace numerically the collapse. We find that a part of the Cauchy horizon is replaced by the strong space-like central singularity. This fraction depends on the value of the critical electric field, $E_{\rm cr}$, for the pair creation. 
  In the Tucker-Wang approach to Metric Affine gravity we review some particular solutions of the Cartan equation for the non-riemannian part of the connection. As application we show how a quite general non Riemannian model gives a Proca type equation for the trace of the nonmetricity 1-forms Q. 
  We discuss the two-dimensional dilaton gravity with a scalar field as the source matter. The coupling between the gravity and the scalar, massless, field is presented in an unusual form. We work out two examples of these couplings and solutions with black-hole behaviour are discussed and compared with those found in the literature. 
  This paper summarizes the contribution presented at the {\sl IX Marcel Grossmann Meeting} (Rome, July 2000). It is stressed, in particular, that a non-relativistic background of ultra-light dilatons, produced in the context of string cosmology, could represent today a significant fraction of cold dark matter. If the dilaton mass lies within the resonant band of present gravitational antennas, a stochastic dilaton background with a nearly critical density could be visible, in principle, already at the level of sensitivity of the detectors in operation and presently under construction. 
  Unimodular relativity is a theory of gravity and space-time with a fixed absolute space-time volume element, the modulus, which we suppose is proportional to the number of microscopic modules in that volume element. In general relativity an arbitrary fixed measure can be imposed as a gauge condition, while in unimodular relativity it is determined by the events in the volume. Since this seems to break general covariance, some have suggested that it permits a non-zero covariant divergence of the material stress-energy tensor and a variable cosmological ``constant.'' In Lagrangian unimodular relativity, however, even with higher-derivatives of the gravitational field in the dynamics, the usual covariant continuity holds and the cosmological constant is still a constant of integration of the gravitational field equations. 
  The paper is devoted to the description of the reparametrization - invariant dynamics of general relativity obtained by resolving constraints and constructing equivalent unconstrained systems. The constraint-shell action reveals the "field nature" of the geometric time in general relativity. The time measured by the watch of an observer coincides with one of field variables, but not with the reparametrization-noninvariant coordinate evolution parameter.   We give new solution of such problem, as the derivation of the path integral representation of the causal Green functions in the Hamiltonian scheme of general relativity. 
  The interplay between different ground based detectors and stochastic backgrounds of relic GW is described. A simultaneous detection of GW in the kHz and in the MHz--GHz region can point towards a cosmological nature of the signal. The sensitivity of a pair of VIRGO detectors to string cosmological models is presented. The implications of microwave cavities for stochastic GW backgrounds are discussed. 
  We first consider local cosmic strings in dilaton-axion gravity and show that they are singular solutions. Then we take a supermassive Higgs limit and present expressions for the fields at far distances from the core by applying a Pecci-Quinn and a duality transformation to the dilatonic Melvin's magnetic universe. 
  The entropy-to-energy bound is examined for a quantum scalar field confined to a cavity and satisfying Robin condition on the boundary of the cavity. It is found that near certain points in the space of the parameter defining the boundary condition the lowest eigenfrequency (while non-zero) becomes arbitrarily small. Estimating, according to Bekenstein and Schiffer, the ratio $S/E$ by the $\zeta$-function, $(24\zeta (4))^{1/4}$, we compute $\zeta (4)$ explicitly and find that it is not bounded near those points that signals violation of the bound. We interpret our results as imposing certain constraints on the value of the boundary interaction and estimate the forbidden region in the parameter space of the boundary conditions. 
  The displacement fluctuations of mirrors in optomechanical devices, induced via thermal expansion by temperature fluctuations due either to thermodynamic fluctuations or to fluctuations in the photon absorption, can be made smaller than quantum fluctuations, at the low temperatures, high reflectivities and high light powers needed to readout displacements at the standard quantum limit. The result is relevant for the design of quantum limited gravitational-wave detectors, both "interferometers" and "bars", and for experiments to study directly mechanical motion in the quantum regime. 
  In a general-relativistic spacetime (Lorentzian manifold), gravitational lensing can be characterized by a lens map, in analogy to the lens map of the quasi-Newtonian approximation formalism. The lens map is defined on the celestial sphere of the observer (or on part of it) and it takes values in a two-dimensional manifold representing a two-parameter family of worldlines. In this article we use methods from differential topology to characterize global properties of the lens map. Among other things, we use the mapping degree (also known as Brouwer degree) of the lens map as a tool for characterizing the number of images in gravitational lensing situations. Finally, we illustrate the general results with gravitational lensing (a) by a static string, (b) by a spherically symmetric body, (c) in asymptotically simple and empty spacetimes, and (d) in weakly perturbed Robertson-Walker spacetimes. 
  In the Tetrad Representation of General Relativity, the energy-momentum expression, found by Moller in 1961, is a tensor wrt coordinate transformations but is not a tensor wrt local Lorentz frame rotations. This local Lorentz freedom is shown to be the same as the six parameter normalized spinor degrees of freedom in the Quadratic Spinor Representation of General Relativity. From the viewpoint of a gravitational field theory in flat space-time, these extra spinor degrees of freedom allow us to obtain a local energy-momentum density which is a true tensor over both coordinate and local Lorentz frame rotations. 
  We investigate a class of spatially compact inhomogeneous spacetimes. Motivated by Thurston's Geometrization Conjecture, we give a formulation for constructing spatially compact composite spacetimes as solutions for the Einstein equations. Such composite spacetimes are built from the spatially compact locally homogeneous vacuum spacetimes which have two commuting Killing vectors by gluing them through a timelike hypersurface admitting a homogeneous spatial slice spanned by the commuting Killing vectors. Topology of the spatial section of the timelike boundary is taken to be the torus. We also assume that the matter which will arise from the gluing is compressed on the boundary, i.e. we take the thin-shell approximation. By solving the junction conditions, we can see dynamical behavior of the connected (composite) spacetime. The Teichm\"uller deformation of the torus also can be obtained. We apply our formalism to a concrete model. The relation to the torus sum of 3-manifolds and the difficulty of this problem are also discussed. 
  Using the technique of Rindler and Perlick we calculate the total precession per revolution of a gyroscope circumventing the source of Weyl metrics. We establish thereby a link between the multipole moments of the source and an ``observable'' quantity. Special attention deserves the case of the gamma-metric. As an extension of this result we also present the corresponding expressions for some stationary space-times. 
  The metric to the two-dimensional dilaton gravity can be writen in an alternative form, similar to the two-dimensional Schwarzschild metric, and allow us the identification of some quantitiies with those equivalent in the Schwarzschild solution. This new form, howewer, presents a non-physical singularity at the horizon in the same way that in the realistic four dimensional case. We show a procedure to eliminate this horizon singularity and, as an application, the resulting metric is used to obtain the associated Hawking temperature. We discuss also some differents between this metric and the Schwarzcchild one. 
  It is suggested, that a curved 4-dimensional space-time manifold is a strained elastic plate in multidimensional embedding space-time. Its thicknesses along extradimensions are much less than 4-dimensional sizes. Reduced 4-dimensional free energy density of the strained plate in a weak strain case is similar to GR Lagrangian density of a gravitational field for the particular value of the Poisson coefficient of the plate. Dynamical equations of the theory are obtained by variation of the multidimensional free energy over displacement vector components. In general case they are inhomogeneous bewave equations. 
  This paper discusses the problem of gravitational perturbations of radiating spacetimes. We lay out the theoretical framework for describing the interaction of external gravitational fields with a radiating spacetime. This is done by deriving the field perturbation equations for a radiating metric. The equations are then specialized to a Vaidya spacetime. For the Hiscock ansatz of a linear mass model of a radiating blackhole the equations are found separable. Further, the resulting ordinary differential equations are found to admit analytic solutions. We obtain the solutions and discuss their characteristics. 
  Here we present the results of applying the generalized Riemann zeta-function regularization method to the gravitational radiation reaction problem. We analyze in detail the headon collision of two nonspinning black holes with extreme mass ratio. The resulting reaction force on the smaller hole is repulsive. We discuss the possible extensions of these method to generic orbits and spinning black holes. The determination of corrected trajectories allows to add second perturbative corrections with the consequent increase in the accuracy of computed waveforms. 
  We consider a canonical ensemble of dynamical triangulations of a 2-dimensional sphere with a hole where the number $N$ of triangles is fixed. The Gibbs factor is $\exp (-\mu \sum \deg v)$ where $\deg v$ is the degree of the vertex $v$ in the triangulation $T$. Rigorous proof is presented that the free energy has one singularity, and the behaviour of the length $m$ of the boundary undergoes 3 phases: subcritical $m=O(1)$, supercritical (elongated) with $m$ of order $N$ and critical with $m=O(\sqrt{N})$. In the critical point the distribution of $m$ strongly depends on whether the boundary is provided with the coordinate system or not. In the first case $m$ is of order $\sqrt{N}$, in the second case $m$ can have order $N^{\alpha}$ for any $0<\alpha <{1/2}$. 
  We compare the performances of the templates defined by three different types of approaches: traditional post-Newtonian templates (Taylor-approximants), ``resummed'' post-Newtonian templates assuming the adiabatic approximation and stopping before the plunge (P-approximants), and further ``resummed'' post-Newtonian templates going beyond the adiabatic approximation and incorporating the plunge with its transition from the inspiral (Effective-one-body approximants). The signal to noise ratio is significantly enhanced (mainly because of the inclusion of the plunge signal) by using these new effective-one-body templates relative to the usual post-Newtonian ones for binary masses greater than $ 30 M_\odot$, the most likely sources for initial laser interferometers. Independently of the question of the plunge signal, the comparison of the various templates confirms the usefulness of using resummation methods. The paper also summarizes the key elements of the construction of various templates and thus can serve as a resource for those involved in writing inspiral search software. 
  Based on the de Broglie-Bohm relativistic quantum theory of motion we show that the conformal formulation of general relativity, being linked with a Weyl-integrable geometry, may implicitly contain the quantum effects of matter. In this context the Mach's principle is discussed. 
  The uncertainty principle, applied naively to the test masses of a laser-interferometer gravitational-wave detector, produces a Standard Quantum Limit (SQL) on the interferometer's sensitivity. It has long been thought that beating this SQL would require a radical redesign of interferometers. However, we show that LIGO-II interferometers, currently planned for 2006, can beat the SQL by as much as a factor two over a bandwidth \Delta f \sim f, if their thermal noise can be pushed low enough. This is due to dynamical correlations between photon shot noise and radiation-pressure noise, produced by the LIGO-II signal-recycling mirror. 
  Recentely, it is shown that the quantum effects of matter determine the conformal degree of freedom of the space-time metric. This was done in the framework of a scalar-tensor theory with one scalar field. A point with that theory is that the form of quantum potential is preassumed. Here we present a scalar-tensor theory with two scalar fields, and no assumption on the form of quantum potential. It is shown that using the equations of motion one gets the correct form of quantum potential plus some corrections. 
  Recently, it is shown that, the quantum effects of matter are well described by the conformal degree of freedom of the space-time metric. On the other hand, it is a wellknown fact that according to Einstein's gravity theory, gravity and geometry are interconnected. In the new quantum gravity theory, matter quantum effects completely determine the conformal degree of freedom of the space-time metric, while the causal structure of the space-time is determined by the gravitational effects of the matter, as well as the quantum effects through back reaction effects. This idea, previousely, is realized in the framework of scalar-tensor theories. In this work, it is shown that quantum gravity theory can also be realized as a purely metric theory. Such a theory is developed, its consequences and its properties are investigated. The theory is applied, then, to black holes and the radiation-dominated universe. It is shown that the initial singularity can be avoided. 
  We derive spin-orbit coupling effects on the gravitational field and equations of motion of compact binaries in the 2.5 post-Newtonian approximation to general relativity, one PN order beyond where spin effects first appear. Our method is based on that of Blanchet, Faye, and Ponsot, who use a post-Newtonian metric valid for general (continuous) fluids and represent pointlike compact objects with a delta-function stress-energy tensor, regularizing divergent terms by taking the Hadamard finite part. To obtain post-Newtonian spin effects, we use a different delta-function stress-energy tensor introduced by Bailey and Israel. In a future paper we will use the 2.5PN equations of motion for spinning bodies to derive the gravitational-wave luminosity and phase evolution of binary inspirals, which will be useful in constructing matched filters for signal analysis. The gravitational field derived here may help in posing initial data for numerical evolutions of binary black hole mergers. 
  We study here the behaviour of non-spacelike geodesics in dust collapse models in order to understand the casual structure of the spacetime. The geodesic families coming out, when the singularity is naked, corresponding to different initial data are worked out and analyzed. We also bring out the similarity of the limiting behaviour for different types of geodesics in the limit of approach to the singularity. 
  We describe the dynamics of a cosmological term in the spherically symmetric case by an r-dependent second rank symmetric tensor \Lambda_{\mu\nu} invariant under boosts in the radial direction. The cosmological tensor \Lambda_{\mu\nu} represents the extension of the Einstein cosmological term \Lambda g_{\mu\nu} which allows a cosmological constant be variable. This possibility is based on the Petrov classification scheme and Einstein field equations in the spherically symmetric case. The inflationary equation of state is satisfied by the radial pressure p_r^{\Lambda}=-\rho^{\Lambda}. The tangential pressure is calculated from the conservation equation \Lambda^{\mu}_{\nu;\mu}=0. The solutions to the Einstein equations with cosmological term \Lambda_{\mu\nu} describe several types of globally regular self-gravitating vacuum configurations including vacuum nonsingular black holes. In this case the global structure of space-time contains an infinite set of black and white holes whose singularities are replaced with the value of cosmological constant of the scale of symmetry restoration, at the background of asymptotically flat or asymptotically de Sitter universes. We outline \Lambda white hole geometry and estimate probability of quantum birth of baby universes inside a \Lambda black hole. In the course of Hawking evaporation of a \Lambda black hole, a second-order phase transition occurs, and globally regular configuration evolves towards a self-gravitating particlelike structure at the background of the Minkowski or de Sitter space. 
  The physical meaning of the Levi-Civita spacetime for some "critical" values of the parameter sigma, is discussed in the light of gedanken experiments performed with gyroscopes circumventing the axis of symmetry. The fact that sigma=1/2 corresponds to flat space described from the point of view of an accelerated frame of reference, led us to incorporate the C metric into discussion. The interpretation of phi as an angle variable for any value of sigma, appears to be at the origen of difficulties. 
  S. Weinberg pointed out a way to introduce a cosmological term by modifying the theory of gravity. This modification would be justified if the Einstein equations with the cosmological term could be obtained in the classical limit of some physically satisfied quantum theory of gravity. We propose to consider quantum geometrodynamics in extended phase space as a candidate for such a theory. Quantum geometrodynamics in extended phase space aims at giving a selfconsistent description of the integrated system ``the physical object (the Universe) + observation means'', observation means being represented by a reference frame. The Lambda term appears in classical equations under certain gauge conditions and characterizes the state of gravitational vacuum related to a chosen reference frame. The eigenvalue spectrum of Lambda depends on a concrete cosmological model and can be found by solving the Schrodinger equation for a wave function of the Universe. The proposed version of quantum geometrodynamics enables one to make predictions concerning probable values of the Lambda term at various stages of cosmological evolution. 
  The noise kernel is the vacuum expectation value of the (operator-valued) stress-energy bi-tensor which describes the fluctuations of a quantum field in curved spacetimes. It plays the role in stochastic semiclassical gravity based on the Einstein-Langevin equation similar to the expectation value of the stress-energy tensor in semiclassical gravity based on the semiclassical Einstein equation. According to the stochastic gravity program, this two point function (and by extension the higher order correlations in a hierarchy) of the stress energy tensor possesses precious statistical mechanical information of quantum fields in curved spacetime and, by the self-consistency required of Einstein's equation, provides a probe into the coherence properties of the gravity sector (as measured by the higher order correlation functions of gravitons) and the quantum nature of spacetime. It reflects the low and medium energy (referring to Planck energy as high energy) behavior of any viable theory of quantum gravity, including string theory. It is also useful for calculating quantum fluctuations of fields in modern theories of structure formation and for backreaction problems in cosmological and black holes spacetimes.  We discuss the properties of this bi-tensor with the method of point-separation, and derive a regularized expression of the noise-kernel for a scalar field in general curved spacetimes. One collorary of our finding is that for a massless conformal field the trace of the noise kernel identically vanishes. We outline how the general framework and results derived here can be used for the calculation of noise kernels for Robertson-Walker and Schwarzschild spacetimes. 
  Globular clusters house a population of compact binaries that will be interesting gravitational wave sources for LISA. We provide estimates for the numbers of sources of several categories and discuss the sensitivity of LISA to detecting these sources. The estimated total number of detectable sources ranges from about 10 to about 1000 with gravitational wave frequencies above 1 mHz. These sources are typically undetectable by any other means and thus offer an opportunity for doing true gravitational-wave astronomy. The detection of these sources would provide information about both binary star evolution and the dynamics of globular clusters. 
  The evolution of an homogeneous and isotropic dissipative fluid is analyzed using dynamical systems techniques. The dissipation is driven by bulk viscous pressure and the truncated Israel-Stewart theory is used. Although almost all solutions inflate, we show that only few of them can be considered as physical solutions since the dominant energy condition is not satisfied. 
  We solve the wave equation for the electromagnetic field tensors associated with the CMBR photons in a universe with scalar metric perturbations. We show that the coupling of the electromagnetic fields with the curvature associated with the scalar perturbations gives rise to an optical rotation of the microwave background photons. The magnitude of the gravitationally generated V-Stokes parameter anisotropy $\Delta_V$, is however very small compared to the linear polarisation caused by Thomson scattering. 
  Starting from the work of the author in 1990 with different collaborators, essential progress in 2d gravity theories has been made. Now all such theories (and not only certain special models) can be treated at the classical as well as at the quantum level. New physical insights have been obtained, as e.g. the ``virtual black hole''. The formalism developed in this context recently also finds increasing interest in mathematical physics. 
  We present a detailed examination of the variational principle for metric general relativity as applied to a ``quasilocal'' spacetime region $\M$ (that is, a region that is both spatially and temporally bounded). Our analysis relies on the Hamiltonian formulation of general relativity, and thereby assumes a foliation of $\M$ into spacelike hypersurfaces $\Sigma$. We allow for near complete generality in the choice of foliation. Using a field--theoretic generalization of Hamilton--Jacobi theory, we define the quasilocal stress-energy-momentum of the gravitational field by varying the action with respect to the metric on the boundary $\partial\M$. The gravitational stress-energy-momentum is defined for a two--surface $B$ spanned by a spacelike hypersurface in spacetime. We examine the behavior of the gravitational stress-energy-momentum under boosts of the spanning hypersurface. The boost relations are derived from the geometrical and invariance properties of the gravitational action and Hamiltonian. Finally, we present several new examples of quasilocal energy--momentum, including a novel discussion of quasilocal energy--momentum in the large-sphere limit towards spatial infinity. 
  The constraint operators belonging to a generally covariant system are found out within the framework of the BRST formalism. The result embraces quadratic Hamiltonian constraints whose potential can be factorized as a never null function times a gauge invariant function. The building of the inner product between physical states is analyzed for systems featuring either intrinsic or extrinsic time. 
  We derive the equations corresponding to twisting type-N vacuum gravitational fields with one Killing vector and one homothetic Killing vector by using the same approach as that developed by one of us in order to treat the case with two non-commuting Killing vectors. We study the case when the homothetic parameter $\phi$ takes the value -1, which is shown to admit a reduction to a third-order real ordinary differential equation for this problem, similar to that previously obtained by one of us when two Killing vectors are present. 
  We consider here the effects of a non-vanishing cosmological term on the final fate of a spherical inhomogeneous collapsing dust cloud. It is shown that depending on the nature of the initial data from which the collapse evolves, and for a positive value of the cosmological constant, we can have a globally regular evolution where a bounce develops within the cloud. We characterize precisely the initial data causing such a bounce in terms of the initial density and velocity profiles for the collapsing cloud. In the cases otherwise, the result of collapse is either formation of a black hole or a naked singularity resulting as the end state of collapse. We also show here that a positive cosmological term can cover a part of the singularity spectrum which is visible in the corresponding dust collapse models for the same initial data. 
  The Casimir stress on a spherical shell in de Sitter background for massless scalar field satisfying Dirichlet boundary conditions on the shell is calculated. The metric is written in conformally flat form. Although the metric is time dependent no particles are created. The Casimir stress is calculated for inside and outside of the shell with different backgrounds corresponding to different cosmological constants. The detail dynamics of the bubble depends on different parameter of the model. Specifically, bubbles with true vacuum inside expand if the difference in the vacuum energies is small, otherwise they collapse. 
  An approximate model of the spacetime foam is offered in which each quantum handle (wormhole) is a 5D wormhole-like solution. A spinor field is introduced for an effective description of this foam. The topological handles of the spacetime foam can be attached either to one space or connect two different spaces. In the first case we have a wormhole with the quantum throat and such object can demonstrate a model of preventing the formation the naked singularity with relation $e > m$. In the second case the spacetime foam looks as a dielectric with quantum handles as dipoles. It is supposed that supergravity theories with a nonminimal interaction between spinor and electromagnetic fields can be considered as an effective model approximately describing the spacetime foam. 
  We have investigated the possibility of having a late time accelerated expansion phase for the universe. We have used a dissipative fluid in Brans-Dicke(BD) theory for this purpose. The model does not involve any potential for the BD scalar field. We have obtained the best fit values for the different parameters in our model by comparing our model predictions with SNIa data and the also with the data from the ultra-compact radio sources. 
  We derive the the Barrett-Crane spin foam model for Euclidean 4 dimensional quantum gravity from a discretized BF theory, imposing the constraints that reduce it to gravity at the quantum level. We obtain in this way a precise prescription of the form of the Barrett-Crane state sum, in the general case of an arbitrary manifold with boundary. In particular we derive the amplitude for the edges of the spin foam from a natural procedure of gluing different 4-simplices along a common tetrahedron. The generalization of our results to higher dimensions is also shown. 
  A study is undertaken of the gravitational collapse of spherically symmetric thick shells admitting a homothetic Killing vector field under the assumption that the energy momentum tensor corresponds to the absence of a pure outgoing component of field. The energy-momentum tensor is not specified beyond this, but is assumed to satisfy the strong and dominant energy conditions. The metric tensor depends on only one function of the similarity variable and the energy conditions identify a class of functions ${\cal F}$ to which the metric function may belong. The possible global structure of such space-times is determined, with particular attention being paid to singularities and their temporal nature (naked or censored). It is shown that there are open subsets of ${\cal F}$ which correspond to naked singularities; in this sense, such singularities are stable. Furthermore, it is shown that these singularities can arise from regular (continuous), asymptotically flat initial data which deviate from the trivial data by an arbitrarily small amount. 
  Gravitational wave detectors now under construction are sensitive to the phase of the incident gravitational waves. Correspondingly, the signals from the different detectors can be combined, in the analysis, to simulate a single detector of greater amplitude and directional sensitivity: in short, aperture synthesis. Here we consider the problem of aperture synthesis in the special case of a search for a source whose waveform is known in detail: \textit{e.g.,} compact binary inspiral. We derive the likelihood function for joint output of several detectors as a function of the parameters that describe the signal and find the optimal matched filter for the detection of the known signal. Our results allow for the presence of noise that is correlated between the several detectors. While their derivation is specialized to the case of Gaussian noise we show that the results obtained are, in fact, appropriate in a well-defined, information-theoretic sense even when the noise is non-Gaussian in character.   The analysis described here stands in distinction to ``coincidence analyses'', wherein the data from each of several detectors is studied in isolation to produce a list of candidate events, which are then compared to search for coincidences that might indicate common origin in a gravitational wave signal. We compare these two analyses --- optimal filtering and coincidence --- in a series of numerical examples, showing that the optimal filtering analysis always yields a greater detection efficiency for given false alarm rate, even when the detector noise is strongly non-Gaussian. 
  We derive the Teukolsky equation for perturbations of a Kerr spacetime when the spacetime metric is written in either ingoing or outgoing Kerr-Schild form. We also write explicit formulae for setting up the initial data for the Teukolsky equation in the time domain in terms of a three metric and an extrinsic curvature. The motivation of this work is to have in place a formalism to study the evolution in the ``close limit'' of two recently proposed solutions to the initial value problem in general relativity that are based on Kerr-Schild slicings. A perturbative formalism in horizon penetrating coordinates is also very desirable in connection with numerical relativity simulations using black hole ``excision''. 
  Inspired by the Randall-Sundrum brane-world scenario, we investigate the possibility of brane-world inflation driven not by an inflaton field on the brane, but by a bulk, dilaton-like gravitational field. As a toy model for the dilaton-like gravitational field, we consider a minimally coupled massive scalar field in the bulk 5-dimensional spacetime, and look for a perturbative solution in the anti-de Sitter (AdS) background. For an adequate range of the scalar field mass, we find a unique solution that has non-trivial dependence on the 5th dimensional coordinate and that induces slow-roll inflation on the brane. 
  We formulate a new method to calculate the gravitational reaction force on a particle of mass $\mu$ orbiting a massive black hole of mass $M$. In this formalism, the tail part of the retarded Green function, which is responsible for the reaction force, is calculated at the level of the Teukolsky equation. Our method naturally allows a systematic post-Minkowskian (PM) expansion of the tail part at short distances. As a first step, we consider the case of a Schwarzschild black hole and explicitly calculate the first post-Newtonian (1PN) tail part of the Green function. There are, however, a couple of issues to be resolved before explicitly evaluating the reaction force by applying the present method. We discuss possible resolutions of these issues. 
  Typical sources of gravitational wave bursts are supernovae, for which no accurate models exist. This calls for search methods with high efficiency and robustness to be used in the data analysis of foreseen interferometric detectors. A set of such filters is designed to detect gravitational wave burst signals. We first present filters based on the linear fit of whitened data to short straight lines in a given time window and combine them in a non linear filter named ALF. We study the performances and efficiencies of these filters, with the help of a catalogue of simulated supernova signals.   The ALF filter is the most performant and most efficient of all filters. Its performance reaches about 80% of the Optimal Filter performance designed for the same signals. Such a filter could be implemented as an online trigger (dedicated to detect bursts of unknown waveform) in interferometric detectors of gravitational waves. 
  Futher development is made of a consept of space-time as multidimensional elastic plate, proposed earlier in [20,21]. General equilibrium equations, including 4-dimensional tangent stress tensor - energy-momentum tensor of matter - are derived. Comparative analysis of multidimensional elasticity theory (MET) and GR is given. Variational principle, boundary conditions, energy-momentum tensor, matter and space-time signature are reviewed within the context of MET. 
  The general exact solution of the Einstein-Dirac equations with cosmological constant in the homogeneous Riemannian space of the Bianchi 1 type is obtained. 
  The third post-Newtonian approximation to the general relativistic dynamics of two point-mass systems has been recently derived by two independent groups, using different approaches, and different coordinate systems. By explicitly exhibiting the map between the variables used in the two approaches we prove their physical equivalence. Our map allows one to transfer all the known results of the Arnowitt-Deser-Misner (ADM) approach to the harmonic-coordinates one: in particular, it gives the value of the harmonic-coordinates Lagrangian, and the expression of the ten conserved quantities associated to global Poincar\'e invariance. 
  The present work investigates the numerical evolution of linearized oscillations of non-rotating, spherically symmetric neutron stars within the framework of general relativity. We derive the appropriate equations using the (3+1)-formalism. We first focus on the evolution of radial oscillations, which do not emit gravitational waves. We demonstrate how to handle a numerical instability that also occurs in the non-radial case, when the stellar model is constructed based on a realistic equation of state. We devise a coordinate transformation that not only removes this instability but also provides much more accurate results. [...] The main part deals with the evolution of non-radial oscillations (l >= 2) of neutron stars. Here, we compare different formulations of the equations and discuss how they have to be numerically dealt with in order to avoid instabilities at the origin. We present results for various polytropic stellar models and different initial data. [...] In the last part of this thesis we consider a physical mechanism for exciting oscillations of neutron stars. We use the time dependent gravitational field of a small point mass \mu that orbits the neutron star to induce stellar oscillations. With this particle we have a physical means which removes the arbitrariness in choosing the initial data. [...] By sampling various orbital parameters of the particle we show that in general the particle is not able to excite any w-modes. It is only for speeds very close to the speed of light that the w-mode is a significant part of the wave signal. 
  We apply the technique of complex paths to obtain Hawking radiation in different coordinate representations of the Schwarzschild space-time. The coordinate representations we consider do not possess a singularity at the horizon unlike the standard Schwarzschild coordinate. However, the event horizon manifests itself as a singularity in the expression for the semiclassical action. This singularity is regularized by using the method of complex paths and we find that Hawking radiation is recovered in these coordinates indicating the covariance of Hawking radiation as far as these coordinates are concerned. 
  The exact metric around a wiggly cosmic string is found by modifying the energy momentum-tensor of a straight infinitely thin cosmic string to include an electric current along the symmetry axis. 
  Templates used in a search for binary black holes and neutron stars in gravitational wave interferometer data will have to be computed on-line since the computational storage and retrieval costs for the template bank are too expensive. The conventional dimensionless variable $T=(c^3/Gm)t,$ where $m$ is the total mass of a binary, in the time-domain and a not-so-conventional velocity-like variable $v=(\pi Gm f)^{1/3}$ in the Fourier-domain, render the phasing of the waves independent of the total mass of the system enabling the construction of {\it mother templates} that depend only on the mass ratio of a black hole binary. Use of such mother templates in a template bank will bring about a reduction in computational costs up to a factor of 10 and a saving on storage by a factor of 100. 
  Due to the recent renewal in the interest for embedded surfaces we provide a list of commented references of interest. 
  A fluid of domain walls has an effective equation of state $p_w = - {2/3}\rho_w$. This equation of state is qualitativelly in agreement with the supernova type Ia observations. We exploit a cosmological model where the matter content is given by a dust fluid and a domain wall fluid. The process of formation of galaxies is essentially preserved. On the other hand, the behaviour of the density contrast in the ordinary fluid is highly altered when domain walls begin to dominate the matter content of the Universe. This domain wall phase occurs at relative recent era, and its possible consequences are discussed, specially concerning the Sachs-Wolfe effect. 
  In the light of the local Lorentz transformations and the general Noether theorem, a new formulate of the general covariant angular momentum conservation law in Einstein-Cartan gravitation theory is obtained, which overcomes the critical difficulty in the other formulates that the conservation law depended on the coordinative choice. 
  The auto-parallel equation over spaces with affine connections and metrics is considered as a result of the application of the method of Lagrangians with covariant derivatives (MLCD) on a given Lagrangian density. 
  We study the problem of the quantization of the massive charged Dirac field on a naked Reissner-Nordstr\"{o}m background. We show that the introduction of an anomalous magnetic moment for the electron field allows a well--defined quantum theory for the one-particle Dirac Hamiltonian, because no boundary condition on the singularity is required. This means that would-be higher order corrections can play an essential role in determining physics on the naked Reissner-Nordstr\"{o}m background and that a non-perturbative approach is required. Moreover, we show that bound states for the Dirac equation are allowed. Various aspects of the physical picture emerging from our study are also discussed, such as the possibility to obtain exotic atomic systems, the formation of black holes by electronic capture and some interesting consistency problems involving quantum gravity. 
  This paper reviews the construction of quantum field theory on a 4-dimensional spacetime by combinatorial methods, and discusses the recent developments in the direction of a combinatorial construction of quantum gravity. 
  With appropriately chosen parameters, the C-metric represents two uniformly accelerated black holes moving in the opposite directions on the axis of the axial symmetry (the z-axis). The acceleration is caused by nodal singularities located on the z-axis.      In the~present paper, geodesics in the~C-metric are examined. In general there exist three types of timelike or null geodesics in the C-metric: geodesics describing particles 1) falling under the black hole horizon; 2)crossing the acceleration horizon; and 3) orbiting around the z-axis and co-accelerating with the black holes.      Using an effective potential, it can be shown that there exist stable timelike geodesics of the third type if the product of the parameters of the C-metric, mA, is smaller than a certain critical value. Null geodesics of the third type are always unstable. Special timelike and null geodesics of the third type are also found in an analytical form. 
  This is the third paper in a series describing a numerical implementation of the conformal Einstein equation. This paper describes a scheme to calculate (three) dimensional data for the conformal field equations from a set of free functions. The actual implementation depends on the topology of the spacetime. We discuss the implementation and exemplary calculations for data leading to spacetimes with one spherical null infinity (asymptotically Minkowski) and for data leading to spacetimes with two toroidal null infinities (asymptotically A3). We also outline the (technical) modifications of the implementation needed to calculate data for spacetimes with two and more spherical null infinities (asymptotically Schwarzschild and asymptotically multiple black holes). 
  We consider a cosmological model in which our Universe is a spherically symmetric bubble wall in 5-dimensional anti-de Sitter spacetime. We argue that the bubble on which we live will undergo collisions with other similar bubbles and estimate the spectrum of such collisions. The collision rate is found to be independent of the age of our Universe. Collisions with small bubbles provide an experimental signature of this scenario, while collisions with larger bubbles would be catastrophic. 
  This talk describes the evolution of studies of chaos in Yang-Mills fields, gravity, and cosmology. The main subject is a BKL regime near the singularity $t=0$ and its survival in higher dimensions and in string theory. We also describe the recent progress in the search for particle-like solutions of the Einstein-Yang-Mills system (monopoles and dyons), colored black holes and the problem of their stability. 
  Classical black holes are defined by the property that things can go in, but don't come out. However, Stephen Hawking calculated that black holes actually radiate quantum mechanical particles. The two important ingredients that result in back hole evaporation are (1) the spacetime geometry, in particular the black hole horizon, and (2) the fact that the notion of a "particle" is not an invariant concept in quantum field theory. These notes contain a step-by-step presentation of Hawking's calculation. We review portions of quantum field theory in curved spacetime and basic results about static black hole geometries, so that the discussion is self-contained. Calculations are presented for quantum particle production for an accelerated observer in flat spacetime, a black hole which forms from gravitational collapse, an eternal Schwarzschild black hole, and charged black holes in asymptotically deSitter spacetimes. The presentation highlights the similarities in all these calculations. Hawking radiation from black holes also points to a profound connection between black hole dynamics and classical thermodynamics. A theory of quantum gravity must predicting and explain black hole thermodynamics. We briefly discuss these issues and point out a connection between black hole evaportaion and the positive mass theorems in general relativity. 
  On the basis of Lagrangian formalism of relativistic field theory post-Newtonian equations of motion for a rotating body are derived in the frame of Feynman's quantum field gravity theory (FGT) and compared with corresponding geodesic equations in general relativity (GR). It is shown that in FGT the trajectory of a rotating test body does not depend on a choice of a coordinate system. The equation of translational motion of a gyroscope is applied to description of laboratory experiments with free falling rotating bodies and rotating bodies on a balance scale. Post-Newtonian relativistic effect of periodical modulation of the orbital motion of a rotating body is discussed for the case of planets of the solar system and for binary pulsars PSR B1913+16 and PSR B1259-63. In the case of binary pulsars with known spin orientations this effect gives a possibility to measure radiuses of neutron stars. 
  Ricci collineations of the Bianchi types I and III, and Kantowski-Sachs space- times are classified according to their Ricci collineation vector (RCV) field of the form (i)-(iv) one component of $\xi^a (x^b)$ is nonzero, (v)-(x) two components of $\xi^a (x^b)$ are nonzero, and (xi)-(xiv) three components of $\xi^a (x^b)$ are nonzero. Their relation with isometries of the space-times is established. In case (v), when $det(R_{ab}) = 0$, some metrics are found under the time transformation, in which some of these metrics are known, and the other ones new. Finally, the family of contracted Ricci collineations (CRC) are presented. 
  A procedure is described for matching a given stationary axisymmetric perfect fluid solution to a not necessarily asymptotically flat vacuum exterior. Using data on the zero pressure surface, the procedure yields the Ernst potential of the matching vacuum metric on the symmetry axis. From this the full metric can be constructed using a variety of well established procedures. 
  This article analyzes why the energy-momentum tensors are not the source of gravity and the dynamical variable of gravity is the affine connection instead of the metric. We derive new gravitational equations with the dimensions of three derivatives from the gravitational action, which has the same solution with the Einstein equations in the vacuum. We also discuss the connection between the new gravitational equations and the Einstein equations. 
  In the model of Expansive Nondecelerative Universe, black hole cannot totally evaporate via quantum evaporation process proposed by Hawking. In a limiting case, an equilibrium of gravitation creation and black hole evaporation can be reached keeping the surface of its horizon constant. This conclusion is in accordance with the second law of thermodynamics. 
  The contribution provides the backround of the model of Expansive Nondecelerative Universe, rationalizes the introduction of Vaidya metrics allowing thus to localize and quantify gravitational energy. A unifying explanation of the fundamental physical interactions is accompanied by demonstration of some consequences and predictions relating to cosmological problems. 
  The energy distribution in the Kerr-Newman space-time is computed using the M{\o}ller energy-momentum complex. This agrees with the Komar mass for this space-time obtained by Cohen and de Felice. These results support the Cooperstock hypothesis. 
  A piecewise Tolman-Bondi-Lemaitre (TBL) cell-model for the universe incorporating local collapsing and expanding inhomogeneities is presented here. The cell-model is made up of TBL underdense and overdense spherical regions surrounded by an intermediate region of TBL shells embedded in an expanding universe. The cell-model generalizes the Friedmann as well as Einstein-Straus swiss-cheese models and presents a number of advantages over other models, and in particular the time evolution of the cosmological inhomogeneities is now incorporated within the scheme. Important problem of gravitational collapse of a massive dust cloud, such as a cluster of galaxies or even a massive star, in such a cosmological background is examined. It is shown that the collapsing local inhomogeneities in an expanding universe could result in either a black hole, or a naked singularity, depending on the nature of the set of initial data which consists of the matter distribution and the velocities of the collapsing shells in the cloud at the initial epoch from which the collapse commences. 
  In the recent publication (Journal of Geometry and Physics,33(2000)23-102) we demonstrated that dynamics of 2+1 gravity can be described in terms of train tracks. Train tracks were introduced by Thurston in connection with description of dynamics of surface automorphisms. In this work we provide an example of utilization of general formalism developed earlier. The complete exact solution of the model problem describing equilibrium dynamics of train tracks on the punctured torus is obtained. Being guided by similarities between the dynamics of 2d liquid crystals and 2+1 gravity the partition function for gravity is mapped into that for the Farey spin chain. The Farey spin chain partition function, fortunately, is known exactly and has been thoroughly investigated recently. Accordingly, the transition between the pseudo-Anosov and the periodic dynamic regime (in Thurston's terminology) in the case of gravity is being reinterpreted in terms of phase transitions in the Farey spin chain whose partition function is just a ratio of two Riemann zeta functions. The mapping into the spin chain is facilitated by recognition of a special role of the Alexander polynomial for knots/links in study of dynamics of self homeomorphisms of surfaces. At the end of paper, using some facts from the theory of arithmetic hyperbolic 3-manifolds (initiated by Bianchi in 1892), we develop systematic extension of the obtained results to noncompact Riemannian surfaces of higher genus. Some of the obtained results are also useful for 3+1 gravity. In particular, using the theorem of Margulis, we provide new reasons for the black hole existence in the Universe: black holes make our Universe arithmetic. That is the discrete Lie groups of motion are arithmetic. 
  We investigate the quantum mechanical wave equations for free particles of spin 0,1/2,1 in the background of an arbitrary static gravitational field in order to explicitly determine if the phase of the wavefunction is $S/\hbar = \int p_{\mu} dx^{\mu} / \hbar$, as is often quoted in the literature. We work in isotropic coordinates where the wave equations have a simple managable form and do not make a weak gravitational field approximation. We interpret these wave equations in terms of a quantum mechanical particle moving in medium with a spatially varying effective index of refraction. Due to the first order spatial derivative structure of the Dirac equation in curved spacetime, only the spin 1/2 particle has \textit{exactly} the quantum mechanical phase as indicated above. The second order spatial derivative structure of the spin 0 and spin 1 wave equations yield the above phase only to lowest order in $\hbar$. We develop a WKB approximation for the solution of the spin 0 and spin 1 wave equations and explore amplitude and phase corrections beyond the lowest order in $\hbar$. For the spin 1/2 particle we calculate the phase appropriate for neutrino flavor oscillations. 
  We describe the departure from equilibrium of matter distributions representing sources for a class of Weyl metric. It is shown that, for extremely high gravitational fields, slight deviations from spherical symmetry may enhance the stability of the system weakening thereby its tendency to a catastrophic collapse. For critical values of surface gravitational potential, in contrast with the exactly spherically symmetric case, the speed of entering the collapse regime decreases substantially, at least for specific cases. 
  It is shown that the dynamical evolution of linear perturbations on a static space-time is governed by a constrained wave equation for the extrinsic curvature tensor. The spatial part of the wave operator is manifestly elliptic and self-adjoint. In contrast to metric formulations, the curvature-based approach to gravitational perturbation theory generalizes in a natural way to self-gravitating matter fields. It is also demonstrated how to obtain symmetric pulsation equations for self-gravitating non-Abelian gauge fields, Higgs fields and perfect fluids. For vacuum fluctuations on a vacuum space-time, the Regge-Wheeler and Zerilli equations are rederived. 
  The energy distribution in the most general nonstatic spherically symmetric space-time is obtained using M{\o}ller's energy-momentum complex. This result is compared with the energy expression obtained by using the energy-momentum complex of Einstein. Some examples of energy distributions in different prescriptions are discussed. 
  We use the conformal approach to numerical relativity to evolve hyperboloidal gravitational wave data without any symmetry assumptions. Although our grid is finite in space and time, we cover the whole future of the initial data in our calculation, including future null and future timelike infinity. 
  Revised version to be published in the Proceedings of the Encuentros Relativistas Espa\~noles, September, 2000 [ http://hades.eis.uva.es/EREs2000 ] 
  We present an explicit three-parameter class of $p=\gamma \varrho$, ($-1/3\leq\gamma<1$), cosmological models admitting a two-dimensional group $G_{2}$ of isometries acting on spacelike surfaces. The family is self-similar in the sense that it has a further homothetic vector field and it contains subfamilies of both (previously unknown) tilted and non-tilted Bianchi models with that equation of state. This is the first algebraically general class of solutions of this kind including dust inhomogeneous solutions. The whole class presents a universal spacelike big-bang singularity in the finite past. More interestingly, the case $p=\varrho /3$ constitutes a new two-parameter inhomogeneous subfamily which can be viewed as a Bianchi V background with a gravitational wave travelling orthogonally to the surfaces of transitivity of the $G_{2}$ group. This wave generates the {\it inhomogeneity} of the spacetime and is related to the sound waves {\it tilting} the perfect fluid. It seems to be the first explicit exact example of a gravitational wave travelling along a homogeneous background that has a realistic equation of state $p=\varrho /3$. 
  The acceleration of the cosmic expansion has been discovered as a consequence of redshift Supernovae data. In the usual way, this cosmic acceleration is explained by the presence of a positive cosmological constant or quantum vacuum energy, in the background of standard Friedmann models. Recently, looking for an alternative explanation, I have considered an inhomogeneous barotropic spherically symmetric spacetime. Obviously, in this inhomogeneous model the philosophical cosmological principle is not verified. Within this framework, the kinematical acceleration of the cosmic fluid or, equivalently, the inhomogeneity of matter, is just the responsible of the SNe Ia measured cosmic acceleration. Moreover, this model gives rise to a generalized Hubble law with two anisotropic terms (dipole acceleration and quadrupole shear), besides the expansion one. The dipole term of this generalized Hubble law could explain, in a cosmological setting, the observed large scale flow of matter, without to have recourse to peculiar velocity-type newtonian models which assume a Doppler dipole. 
  In this paper we show that the spatially homogeneous Bianchi type I and Kantowski-Sachs cosmologies derived from the Brans-Dicke theory of gravity admit a supersymmetric extension at the quantum level. Global symmetries in the effective one-dimensional actions characterize both classical and quantum solutions. A wide family of exact wavefunctions satisfying the supersymmetric constraints are found. A connection with quantum wormholes is briefly discussed. 
  We study the supersymmetry of the radial problems of the models of quantum relativistic rotating oscillators in arbitrary dimensions, defined as Klein-Gordon fields in backgrounds with deformed anti-de Sitter metrics. It is pointed out that the shape invariance of the supersymmetric partner radial potentials leads to simple operators forms of the Rodrigues formulas for the normalized radial wave functions. 
  It has been asserted in the literature that the analogy between the linear and first order slow motion approximation of Einstein equations of General Relativity (gravitomagnetic equations) and the Maxwell-Lorentz equations of electrodynamics breaks down if the gravitational potentials are time dependent.  In this work, we show that this assertion is not correct and it has arisen from an incorrect limit of the usual harmonic gauge condition, which drastically changes the physical content of the gravitomagnetic equations. 
  In this work, I present a generalized linear Hubble law for a barotropic spherically symmetric inhomogeneous spacetime, which is in principle compatible with the acceleration of the cosmic expansion obtained as a result of high redshift Supernovae data. The new Hubble function, defined by this law, has two additional terms besides an expansion one, similar to the usual volume expansion one of the FLRW models, but now due to an angular expansion. The first additional term is dipolar and is a consequence of the existence of a kinematic acceleration of the observer, generated by a negative gradient of pressure or of mass-energy density. The second one is quadrupolar and due to the shear. Both additional terms are anisotropic for off-centre observers, because of to their dependence on a telescopic angle of observation. This generalized linear Hubble law could explain, in a cosmological setting, the observed large scale flow of matter, without to have recourse to peculiar velocity-type newtonian models. It is pointed out also, that the matter dipole direction should coincide with the CBR dipole one. 
  The leading mass dependence of the wave function phase is calculated in the presence of gravitational interactions. The conditions under which this phase contains terms depending on both the square of the mass and the gravitational constant are determined. The observability of such terms is briefly discussed. 
  In this paper, we investigate the general case of black hole on horizon by the method of Damour-Ruffini-Zhao. The proof of identification of the location of horizons determined both by Damour-Ruffini-Zhao's (D-R-Z) method and by equation of null super-surface is given. The formula of temperature on the horizon for general black holes is obtained and is successfully checked by a variety of models of black holes. 
  The relation between the technique of conformal flat and Damour-Ruffini-Zhao's method is investigated in this paper. It is pointed out that the two methods give the same results when the metric has the form $g_{\alpha\beta=0},$ with $\alpha=0,1$ and $\beta=2,3$. It is indicated that the two methods are not equivalent for general case. 
  We show that General Relativity (GR) with cosmological constant may be formulated as a rather simple constrained SO(D-1,2) (or SO(D,1))-Yang-Mills (YM) theory. Furthermore, the spin connections of the Cartan-Einstein formulation for GR appear as solutions of a genuine SO(D-1,1)-YM.   We also present a theory of gravity with torsion as the most natural extension of this result. The theory comes out to be strictly an YM-theory upon relaxation of a suitable constraint. This work sets out to enforce the close connection between YM theories and GR by means of a new construction. 
  A new interpretation of compensate effect is presented. The Hawking effect in general space-time can be taken as a compensate effect of the scale transformation of coordinate time on the horizon in generalized tortoise coordinates transformation. It is proved that the Hawking temperature is the pure gauge of compensate field in tortoise coordinates. This interpretation does not refer to a zero-temperature space-time. 
  The formation of energetic rings of matter in a Kerr spacetime with an outward pointing acceleration field does not appear to have previously been noted as a relativistic effect. In this paper we show that such rings are a gravimagneto effect with no Newtonian analog, and that they do not occur in the static limit. The energy efficiency of these rings can, depending of the strength of the acceleration field, be much greater than that of Keplerian disks. The rings rotate in a direction opposite to that of compact star about which they form. The size and energy efficiency of the rings depend on the fundamental parameters of the spacetime as well as the strength the acceleration field. 
  We demonstrate how the Einstein's equations for the $D$-dimensional spherical gravity can be written in the covariant vector-like form. These equations reveal easily the causal structure of curved spherically symmetric manifolds and may appear useful in investigation of different models of brane universes. 
  A class of radiative solutions of Einstein's field equations with a negative cosmological constant and a pure radiation is investigated. The space-times, which generalize the Defrise solution, represent exact gravitational waves which interact with null matter and propagate in the anti-de Sitter universe. Interestingly, these solutions have homogeneous and non-singular wave-fronts for all freely moving observers. We also study properties of sandwich and impulsive waves which can be constructed in this class of space-times. 
  This is a review devoted to some results of Algebraic Programming (Computer Algebra) used in treating several problems of general relativity, based mainly on already published articles. The article contains the talk given by the author at The Albert Einstein Institut, Max Planck Institut fur Gravitationstheorie, Golm, Germany, september 2000 
  Black hole entropy and its relation to the horizon area are considered. More precisely, the conditions and specifications that are expected to be required for the assignment of entropy, and the consequences that these expectations have when applied to a black hole are explored. In particular, the following questions are addressed: When do we expect to assign an entropy?; when are entropy and area proportional? and, what is the nature of the horizon? It is concluded that our present understanding of black hole entropy is somewhat incomplete, and some of the relevant issues that should be addressed in pursuing these questions are pointed out. 
  The important studies of Peebles, and Bond and Efstathiou have led to the formula C_l = const/[l(l +1)] aimed at describing the lower order multipoles of the CMBR temperature variations caused by density perturbations with the flat spectrum. Clearly, this formula requires amendments, as it predicts an infinitely large monopole C_0, and a dipole moment C_1 only 6/2 times larger than the quadrupole C_2, both predictions in conflict with observations. We restore the terms omitted in the course of the derivation of this formula, and arrive at a new expression. According to the corrected formula, the monopole moment is finite and small, while the dipole moment is sensitive to short-wavelength perturbations, and numerically much larger than the quadrupole, as one would expect on physical grounds. At the same time, the function l(l +1)C_l deviates from a horizontal line and grows with l, for l \geq 2. We show that the inclusion of the modulating (transfer) function terminates the growth and forms the first peak, recently observed. We fit the theoretical curves to the position and height of the first peak, as well as to the observed dipole, varying three parameters: red-shift at decoupling, red-shift at matter-radiation equality, and slope of the primordial spectrum. It appears that there is always a deficit, as compared with the COBE observations, at small multipoles, l \sim 10. We demonstrate that a reasonable and theoretically expected amount of gravitational waves bridges this gap at small multipoles, leaving the other fits as good as before. We show that the observationally acceptable models permit somewhat `blue' primordial spectra. This allows one to avoid the infra-red divergence of cosmological perturbations, which is otherwise present. 
  We present new solutions to the Einstein-Maxwell equations representing a class of charged distorted black holes. These solutions are static-axisymmetric and are generalizations of the distorted black hole solutions studied by Geroch and Hartle. Physically, they represent a charged black hole distorted by external matter fields. We discuss the zeroth and first law for these black holes. The first law is proved in two different forms, one motivated by the isolated horizon framework and the other using normalizations at infinity. 
  The dimension of the kernels of the edth and edth-prime operators on closed, orientable spacelike 2-surfaces with arbitrary genus is calculated, and some of its mathematical and physical consequences are discussed. 
  We extend the induced matter model, previously applied to a variety of isotropic cases, to a generalization of Bianchi type-I anisotropic cosmologies. The induced matter model is a 5D Kaluza-Klein approach in which assumptions of compactness are relaxed for the fifth coordinate, leading to extra geometric terms. One interpretation of these extra terms is to identify them as an ``induced matter'' contribution to the stress-energy tensor. In similar spirit, we construct a five dimensional metric in which the spatial slices possess Bianchi type-I geometry. We find a set of solutions for the five dimensional Einstein equations, and determine the pressure and density of induced matter. We comment on the long-term dynamics of the model, showing that the assumption of positive density leads to the contraction over time of the fifth scale factor. 
  We obtain the energy distribution of a Schwarzschild black hole in a magnetic universe in the Tolman prescription. 
  In this work we have investigated the possibility of having a late time accelerated phase of the universe, suggested by recent supernova observation, in the context of Brans Dicke (BD) theory with a symmetry breaking potential and a matter field. We find that a perfect fluid matter field (pressureless and with pressure) cannot support this acceleration but a fluid with dissipative pressure can drive this late time acceleration. We have also calculated some cosmological parameters in our model to match with observations. 

  We obtain the energy of a conformal scalar dyon black hole (CSD) by using the energy-momentum complexes of Tolman and M{\o}ller. The total gravitational energy is given by the CSD charge in the both prescriptions. 
  We show how to generalize the classical electric-magnetic decomposition of the Maxwell or the Weyl tensors to arbitrary fields described by tensors of any rank in general $n$-dimensional spacetimes of Lorentzian signature. The properties and applications of this decomposition are reviewed. In particular, the definition of tensors quadratic in the original fields and with important positivity properties is given. These tensors are usually called "super-energy" (s-e) tensors, they include the traditional energy-momentum, Bel and Bel-Robinson tensors, and satisfy the so-called Dominant Property, which is a straightforward generalization of the classical dominant energy condition satisfied by well-behaved energy-momentum tensors. We prove that, in fact, any tensor satisfying the dominant property can be decomposed as a finite sum of the s-e tensors. Some remarks about the conservation laws derivable from s-e tensors, with some explicit examples, are presented. Finally, we will show how our results can be used to provide adequate generalizations of the Rainich conditions in general dimension and for any physical field. 
  We obtain the integral formulae for computing the tetrads and metric components in Riemann normal coordinates and Fermi coordinate system of an observer in arbitrary motion. Our approach admits essential enlarging the range of validity of these coordinates. The results obtained are applied to the geodesic deviation in the field of a weak plane gravitational wave and the computation of plane-wave metric in Fermi normal coordinates. 
  The approximate renormalized stress-energy tensor of the quantized massive conformally coupled scalar field in the spacetime of electrically charged nonlinear black hole is constructed. It is achieved by functional differentiation of the lowest order of the DeWitt-Schwinger effective action involving coincidence limit of the Hadamard-Minakshisundaram-DeWitt-Seely coefficient $a_{3}.$ The result is compared with the analogous result derived for the Reissner-Nordstr\"om black hole. It is shown that the most important differences occur in the vicinity of the event horizon of the black hole near the extremality limit. The structure of the nonlinear black hole is briefly studied by means of the Lambert functions. 
  We study in detail the equations of the geodesic deviation in multidimensional theories of Kaluza-Klein type. We show that their 4-dimensional space-time projections are identical with the equations obtained by direct variation of the usual geodesic equation in the presence of the Lorentz force, provided that the fifth component of the deviation vector satisfies an extra constraint derived here. 
  Absolute parallelism (AP) geometry is frequently used for physical applications. Although it is wider than Riemannian geometry, it has two main defects. The first is that its path equation does not represent physical trajectories of any test particle. The second is the identical vanishing of its curvature tensor. The present work shows that parameterizing this geometry would solve the two problems. Furthermore, the resulting parameterized AP-structure is more general than both the conventional AP-structure and the Riemannian structure. Also, it is shown that it can be reduced to one or the other, of these two geometric structures, in some special cases. The structure obtained is more appropriate for physical applications, especially in constructing field theories gauging gravity. 
  A typical stellar mass black hole with a lighter companion is shown to succumb to a chaotic precession of the orbital plane. As a result, the optimal candidates for the direct detection of gravitational waves by Earth based interferometers can show irregular modulation of the waveform during the last orbits before plunge. The precession and the subsequent modulation of the gravitational radiation depends on the mass ratio, eccentricity, and spins. The smaller the mass of the companion, the more prominent the effect of the precession. The most important parameters are the spin magnitudes and misalignments. If the spins are small and nearly aligned with the orbital angular momentum, then there will be no chaotic precession while increasing both the spin magnitudes and misalignments increases the erratic precession. A large eccentricity can be induced by large, misaligned spins but does not seem to be required for chaos. An irregular precession of the orbital plane will generate irregularities in the gravitational wave frequency but may have a lesser effect on the total number of cycles observed. 
  There are known models of spherical gravitational collapse in which the collapse ends in a naked shell-focusing singularity for some initial data. If a massless scalar field is quantized on the classical background provided by such a star, it is found that the outgoing quantum flux of the scalar field diverges in the approach to the Cauchy horizon. We argue that the semiclassical approximation (i.e. quantum field theory on a classical curved background) used in these analyses ceases to be valid about one Planck time before the epoch of naked singularity formation, because by then the curvature in the central region of the star reaches Planck scale. It is shown that during the epoch in which the semiclassical approximation is valid, the total emitted energy is about one Planck unit, and is not divergent. We also argue that back reaction in this model does not become important so long as gravity can be treated classically. It follows that the further evolution of the star will be determined by quantum gravitational effects, and without invoking quantum gravity it is not possible to say whether the star radiates away on a short time scale or settles down into a black hole state. 
  In this review we summarize the current understanding of the gravitational-wave driven instability associated with the so-called r-modes in rotating neutron stars. We discuss the nature of the r-modes, the detailed mechanics of the instability and its potential astrophysical significance. In particular we discuss results regarding the spin-evolution of nascent neutron stars, the detectability of r-mode gravitational waves and mechanisms limiting the spin-rate of accreting neutron stars in binary systems. 
  In this paper, we study by a functional method the vacuum instability of a charged scalar field, when it is quantized in the background of the Reissner-Nordtrom black hole; we also show that the first stage of the evaporation process of the black hole can be driven by a Schwinger-like effect. 
  The relativistic 2-body problem, much like the non-relativistic one, is reduced to describing the motion of an effective particle in an external field. The concept of a relativistic reduced mass and effective particle energy introduced some 30 years ago to compute relativistic corrections to the Balmer formula in quantum electrodynamics, is shown to work equally well for classical electromagnetic and gravitational interaction. The results for the gravitational 2-body problem have more than academic interest since they apply to the study of binary pulsars that provide precision tests for general relativity. They are compared with recent results derived by other methods. 
  A natural mapping of paths in a curved space onto the paths in the corresponding (tangent) flat space may be used to reduce the curved-space-time path integral to the flat-space-time path integral. The dynamics of the particle in a curved space-time is expressed then in terms of an integral over paths in the flat (Minkowski) space-time. This may be called quantum equivalence principle. Contrary to the known DeWitt's definition of a curved-space path integral, the present definition leads to the covariant equation of motion without a scalar curvature term. The reduction of a curved-space path integral to the flat-space path integral may be expressed in terms of a representation of the path group. With the help of this representation all the results may be generalized to the case of an arbitrary external field. 
  We quantise General Relativity for a class of energy-momentum-stress tensors. 
  All orientation preserving isometries of the hyperbolic three-space are studied, and the probability density of conjugate pair separations for each isometry is presented. The study is relevant for the cosmic crystallography, and is the theoretical counterpart of the mean histograms arising from computer simulations of the isometries. 
  This paper is motivated by the current development of several space missions (e.g. ACES on International Space Station) that will fly on Earth orbit laser cooled atomic clocks, providing a time-keeping accuracy of the order of 5~10^{-17} in fractional frequency. We show that to such accuracy, the theory of frequency transfer between Earth and Space must be extended from the currently known relativistic order 1/c^2 (which has been needed in previous space experiments such as GP-A) to the next relativistic correction of order 1/c^3. We find that the frequency transfer includes the first and second-order Doppler contributions, the Einstein gravitational red-shift and, at the order 1/c^3, a mixture of these effects. As for the time transfer, it contains the standard Shapiro time delay, and we present an expression also including the first and second-order Sagnac corrections. Higher-order relativistic corrections, at least O(1/c^4), are numerically negligible for time and frequency transfers in these experiments, being for instance of order 10^{-20} in fractional frequency. Particular attention is paid to the problem of the frequency transfer in the two-way experimental configuration. In this case we find a simple theoretical expression which extends the previous formula (Vessot et al. 1980) to the next order 1/c^3. In the Appendix we present the detailed proofs of all the formulas which will be needed in such experiments. 
  A class of exact solutions of Einstein's equations is analysed which describes uniformly accelerating charged black holes in an asymptotically de Sitter universe. This is a generalisation of the C-metric which includes a cosmological constant. The physical interpretation of the solutions is facilitated by the introduction of a new coordinate system for de Sitter space which is adapted to accelerating observers in this background. The solutions considered reduce to this form of the de Sitter metric when the mass and charge of the black holes vanish. 
  Recently the energy emission from a naked singularity forming in spherical dust collapse has been investigated. This radiation is due to the particle creation in a curved spacetime. In this discussion, the central role is played by the mapping formula between the incoming and the outgoing null coordinates. For the self-similar model, this mapping formula has been derived analytically. But for the model with $C^{\infty}$ density profile, the mapping formula has been obtained only numerically. In the present paper, we argue that the singular nature of the mapping is determined by the local geometry around the point at which the singularity is first formed. If this is the case, it would be natural to expect that the mapping formula can be derived analytically. In the present paper, we analytically rederive the same mapping formula for the model with $C^{\infty}$ density profile that has been earlier derived using a numerical technique. 
  We analyse here the gravitational collapse of directed null radiation in a background with a constant potential such as one produced by a star system like galaxy in which the collapsing object is immersed. Both naked singularities and black holes are shown to be developing as the final outcome of the collapse. An interesting feature that emerges is that a part of the naked singularity spectrum in collapsing Vaidya region gets covered in the corresponding dual-Vaidya region, which corresponds to the Vaidya directed null radiation sitting in constant potential bath. The implications of such a result towards the issue of stability of naked singularities are discussed. 
  Single pulse of null dust and colliding null dusts both transform a regular horizon into a space-like singularity in the space of colliding waves. The local isometry between such space-times and black holes extrapolates these results to the realm of black holes. However, inclusion of particular scalar fields instead of null dusts creates null singularities rather than space-like ones on the inner horizons of black holes. 
  In this paper it is implemented how to make compatible the boundary conditions and the gauge fixing conditions for complex general relativity written in terms of Ashtekar variables using the Henneaux-Teitelboim-Vergara approach. Moreover, it is found that at first order in the gauge parameters, the Hamiltonian action is (on shell) fully gauge-invariant under the gauge symmetry generated by the first class constraints in the case when spacetime $\cal M$ has the topology ${\cal M}= R \times \Sigma$ and $\Sigma$ has no boundary. Thus, the statement that the constraints linear in the momenta do not contribute to the boundary terms is right, but only in the case when $\Sigma$ has no boundary. 
  The significant discussion about the possible chaotic behavior of the mixmaster cosmological model due to Cornish and Levin [J.N. Cornish and J.J. Levin, Phys. Rev. Lett. 78 (1997) 998; Phys. Rev. D 55 (1997) 7489] is revisited. We improve their method by correcting nontrivial oversights that make their work inconclusive to precisely confirm their result: ``The mixmaster universe is indeed chaotic''. 
  We present the concept of a sensitive AND broadband resonant mass gravitational wave detector. A massive sphere is suspended inside a second hollow one. Short, high-finesse Fabry-Perot optical cavities read out the differential displacements of the two spheres as their quadrupole modes are excited. At cryogenic temperatures one approaches the Standard Quantum Limit for broadband operation with reasonable choices for the cavity finesses and the intracavity light power. A molybdenum detector of overall size of 2 m, would reach spectral strain sensitivities of 2x10^-23/Sqrt{Hz} between 1000 Hz and 3000 Hz. 
  The cosmic censorship hypothesis introduced by Penrose thirty years ago is still one of the most important open questions in {\it classical} general relativity. In this essay we put forward the idea that cosmic censorship is intrinsically a {\it quantum gravity} phenomena. To that end we construct a gedanken experiment in which cosmic censorship is violated within the purely {\it classical} framework of general relativity. We prove, however, that {\it quantum} effects restore the validity of the conjecture. This suggests that classical general relativity is inconsistent and that cosmic censorship might be enforced only by a quantum theory of gravity. 
  We investigate a $D$ dimensional generalization of the Schroedinger-Newton equations, which purport to describe quantum state reduction as resulting from gravitational effects. For a single particle, the system is a combination of the Schroedinger and Poisson equations modified so that the probability density of the wavefunction is the source of the potential in the Schroedinger equation. For spherically symmetric wavefunctions, a discrete set of energy eigenvalue solutions emerges for dimensions $D<6$, accumulating at D=6. Invoking Heisenberg's uncertainty principle to assign timescales of collapse correspoding to each energy eigenvalue, we find that these timescales may vary by many orders of magnitude depending on dimension. For example, the time taken for the wavefunction of a free neutron in a spherically symmetric state to collapse is many orders of magnitude longer than the age of the universe, whereas for one confined to a box of picometer-sized cross-sectional dimensions the collapse time is about two weeks. 
  We explicitly show that in (2+1) dimensions the general solution of the Einstein equations with negative cosmological constant on a neigbourhood of timelike spatial infinity can be obtained from BTZ metrics by coordinate transformations corresponding geometrically to deformations of their spatial infinity surface. Thus, whatever the topology and geometry of the bulk, the metric on the timelike extremities is BTZ. 
  We describe a general method of obtaining the constraints between area variables in one approach to area Regge calculus, and illustrate it with a simple example. The simplicial complex is the simplest tessellation of the 4-sphere. The number of independent constraints on the variations of the triangle areas is shown to equal the difference between the numbers of triangles and edges, and a general method of choosing independent constraints is described. The constraints chosen by using our method are shown to imply the Regge equations of motion in our example. 
  Transports preserving the angle between two contravariant vector fields but changing their lengths proportional to their own lengths are introduced as ''conformal'' transports and investigated over spaces with contravariant and covariant affine connections (whose components differ not only by sign) and metrics. They are more general than the Fermi-Walker transports. In an analogous way as in the case of Fermi-Walker transports a conformal covariant differential operator and its conformal derivative are defined and considered over the above mentioned spaces. Different special types of conformal transports are determined inducing also Fermi-Walker transports for orthogonal vector fields as special cases. Conditions under which the length of a non-null contravariant vector field could swing as a homogeneous harmonic oscillator are established. The results obtained regardless of any concrete field (gravitational) theory could have direct applications in such types of theories.   PACS numbers: 04.90.+e; 04.50.+h; 12.10.Gq; 02.40.Vh 
  Quantum Electrodynamics (QED) has been so successful a theory that it is taken as a model for the production of further quantum theories. However, when the prescription for quantising electromagnetic interactions that so successfully resulted in QED is applied to General Relativity the theory obtained is not renormalizable. We derive a different method of quantising classical electromagnetism which also results in QED. We call the method the versatile method. We then apply the versatile method to General Relativity, in particular the Einstein equation which equates a geometrical description derivable from the metric to the energy-momentum-stress tensor, or as we shall call it the matter tensor of the matter field. The method can be applied provided that there is always a reference frame, which may differ with location and time, where the matter tensor can be reduced to a mass density with the other elements zero. We call such matter tensors simple. This restriction means that the tensor can be put into one to one correspondence with the Dirac current. When the versatile method of quantising a classical theory is applied to General Relativity the theory that results is renormalizable. It is in fact isomorphic to QED, provided that the temporal and a spatial co-ordinate are exchanged. 
  Lanczos-Lovelock theories of gravity, in its first order version, are studied on asymptotically locally anti de Sitter spaces. It is shown that thermodynamics satisfies the standard behavior and an expression for entropy is found for this formalism. Finally a short analysis of the algebra of conserved charges is displayed. 
  A generalization of two recently proposed general relativity Hamiltonians, to the case of a general (d+1)-dimensional dilaton gravity theory in a manifold with a timelike or spacelike outer boundary, is presented. 
  It is shown that only in the space-times admitting a 1+3-foliation by flat Cauchy hypesurfaces (i.e., in the Bianchi I type space-times the isotropic version of which the spatially flat Friedmann-Robertson-Walker space-times are) the canonical quantization of geodesic motion and quantum-mechanics obtained as an asymptotics of the quantum theory of scalar field lead to the same canonical commutation relations (CCR). Otherwise, the field-theoretical approach leads to a deformation of CCR (particularly, operators of coordinates do not commute), and the Principle of Correspondence is broken in a sense. Thus, the spatially flat cosmology is distinguished intrinsically in the quantum theory. 
  This preprint contains a description of a package for Mathematica called EinS. This package allows one to perform various calculations with indexed objects. 
  The basic physical structure of the relativistic theory of gravitation is discussed. The significant role that the Hypothesis of Locality plays in relativity theory is elucidated via the phenomenon of spin-rotation coupling. The limitations of this hypothesis are critically examined. A nonlocal theory of accelerated observers is presented and some of its observational consequences are described. 
  Gravitoelectromagnetism is briefly reviewed and some recent developments in this topic are discussed. The stress-energy content of the gravitoelectromagnetic field is described from different standpoints. In particular, the gravitational Poynting flux is analyzed and it is shown that there exists a steady flow of gravitational energy circulating around a rotating mass. 
  We give a critical analysis of projective relativity theory. Examining Kaluza's own intention and the following development by Klein, Jordan, Pauli, Thiry, Ludwig and others, we conclude that projective relativity was abused in its own terms. Much more in the case of newer higher dimensional Kaluza--Klein theories with non-Abelian gauge groups. Reviewing the projective formulation of the Jordan isomorphy theorem yields some hints how one can proceed in a different direction. We can interpret the condition R5_{\mu\nu}=0 not as a field equation in a 5-dimensional Riemannian space, e.g. as vacuum Einstein-Hilbert equation, but can (or should) interpret it as a geometrical object, a null-quadric. Projective aspects of quantum (field) theory are discussed under this viewpoint. 
  Penrose's identification with warp provides the general framework for constructing the continuous form of impulsive gravitational wave metrics. We present the 2-component spinor formalism for the derivation of the full family of impulsive spherical gravitational wave metrics which brings out the power in identification with warp and leads to the simplest derivation of exact solutions. These solutions of the Einstein vacuum field equations are obtained by cutting Minkowski space into two pieces along a null cone and re-identifying them with warp which is given by an arbitrary non-linear holomorphic transformation. Using 2-component spinor techniques we construct a new metric describing an impulsive spherical gravitational wave where the vertex of the null cone lies on a world-line with constant acceleration. 
  We reformulate the dynamics of homogeneous cosmologies with a scalar field matter source with an arbitrary self-interaction potential in the language of jet bundles and extensions of vector fields. In this framework, the Bianchi-scalar field equations become subsets of the second Bianchi jet bundle, $J^2$, and every Bianchi cosmology is naturally extended to live on a variety of $J^2$. We are interested in the existence and behaviour of extensions of arbitrary Bianchi-Lie and variational vector fields acting on the Bianchi variety and accordingly we classify all such vector fields corresponding to both Bianchi classes $A$ and $B$. We give examples of functions defined on Bianchi jet bundles which are constant along some Bianchi models (first integrals) and use these to find particular solutions in the Bianchi total space. We discuss how our approach could be used to shed new light to questions like isotropization and the nature of singularities of homogeneous cosmologies by examining the behaviour of the variational vector fields and also give rise to interesting questions about the `evolution' and nature of the cosmological symmetries themselves. 
  We determine the path of the light around a dielectric vortex described by the relativistic vortex flow of a perfect fluid. 
  We apply the results of singularity analysis to the isotropic cosmological models in general relativity and string theory with a variety of matter terms. For some of these models the standard Painlev\'{e} test is sufficient to demonstrate integrability or nonintegrability in the sense of Painlev\'{e}. For others of these models it is necessary to use a less algorithmic procedure. 
  In the framework of inflationary cosmology I study some aspects of nonequilibrium thermodynamics for the matter field fluctuations. The thermodynamic analysis is developed for de Sitter and power - law expansions of the universe. In both cases, I find that the heat capacity is negative leading respectively, to exponential and superexponential growth for the number of states in the infrared sector for de Sitter and power-law expansions of the universe. The spectrum for the matter field fluctuations can be understood from the background effective temperature when the horizon entry. 
  We consider some aspects of the global evolution problem of Hamiltonian homogeneous, anisotropic cosmologies derived from a purely quadratic action functional of the scalar curvature. We show that models can isotropize in the positive asymptotic direction and that quadratic diagonal Bianchi IX models do not recollapse and may be regular initially. Although the global existence and isotropization results we prove hold quite generally, they are applied to specific Bianchi models in an attempt to describe how certain dynamical properties uncommon to the general relativity case, become generic features of these quadratic universes. The question of integrability of the models is also considered. Our results point to the fact that the more general models are not integrable in the sense of Painlev\'e and for the Bianchi IX case this may be connected to the validity of a BKL oscillatory picture on approach to the singularity in sharp contrast with other higher order gravity theories that contain an Einstein term and show a monotonic evolution towards the initial singularity. 
  The detection of some tiny gravitomagnetic effects in the field of the Earth by means of artificial satellites is a very demanding task because of the various other perturbing forces of gravitational and non-gravitational origin acting upon them. Among the gravitational perturbations a relevant role is played by the Earth solid and ocean tides. In this communication I outline their effects on the detection of the Lense-Thirring drag of the orbits of LAGEOS and LAGEOS II, currently analyzed, and the proposed GP-C experiment devoted to the measurement of the clock effect. 
  In a recent paper Carot et al. considered the definition of cylindrical symmetry as a specialisation of the case of axial symmetry. One of their propositions states that if there is a second Killing vector, which together with the one generating the axial symmetry, forms the basis of a two-dimensional Lie algebra, then the two Killing vectors must commute, thus generating an Abelian group. In this paper a similar result, valid under considerably weaker assumptions, is derived: any two-dimensional Lie transformation group which contains a one-dimensional subgroup whose orbits are circles, must be Abelian. The method used to prove this result is extended to apply three-dimensional Lie transformation groups. It is shown that the existence of a one-dimensional subgroup with closed orbits restricts the Bianchi type of the associated Lie algebra to be I, II, III, VII_0, VIII or IX. Some results on n-dimensional Lie groups are also derived and applied to show there are severe restrictions on the structure of the allowed four-dimensional Lie transformation groups compatible with cyclic symmetry. 
  We show that essentially pure classical thermodynamics is sufficient to determine Bekenstein's formula for the black hole's entropy, $S=\eta A$. We base our reasoning on the minimal assumption that since black body radiation is describable by classical thermodynamics, so is the complete black hole-Hawking radiation system. Furthermore, we argue that any non-linear correction to the black hole entropy must be quantum mechanical in nature. The proportionality coefficient, $\eta=1/4\ell_P^2$, must be calculated within a semi-classical or full-fledged quantum mechanical framework. 
  This is a chapter on Black-hole Scattering that was commissioned for an Encyclopaedia on Scattering edited by Pike and Sabatier, to be published by Academic Press. The chapter surveys wave propagation in black-hole spacetimes, diffraction effects in wave scattering, resonances, quasinormal modes and related topics. 
  We analyze prospects for the use of Bose-Einstein condensates as condensed-matter systems suitable for generating a generic ``effective metric'', and for mimicking kinematic aspects of general relativity. We extend the analysis due to Garay et al, [gr-qc/0002015, gr-qc/0005131]. Taking a long term view, we ask what the ultimate limits of such a system might be. To this end, we consider a very general version of the nonlinear Schrodinger equation (with a 3-tensor position-dependent mass and arbitrary nonlinearity). Such equations can be used for example in discussing Bose-Einstein condensates in heterogeneous and highly nonlinear systems. We demonstrate that at low momenta linearized excitations of the phase of the condensate wavefunction obey a (3+1)-dimensional d'Alembertian equation coupling to a (3+1)-dimensional Lorentzian-signature ``effective metric'' that is generic, and depends algebraically on the background field. Thus at low momenta this system serves as an analog for the curved spacetime of general relativity. In contrast at high momenta we demonstrate how to use the eikonal approximation to extract a well-controlled Bogoliubov-like dispersion relation, and (perhaps unexpectedly) recover non-relativistic Newtonian physics at high momenta. Bose-Einstein condensates appear to be an extremely promising analog system for probing kinematic aspects of general relativity. 
  In a viscous Bianchi type I metric of the Kasner form, it is well known that it is not possible to describe an anisotropic physical model of the universe, which satisfies the second law of thermodynamics and the dominant energy condition (DEC) in Einstein's theory of gravity. We examine this problem in scalar-tensor theories of gravity. In this theory we show that it is possible to describe the growth of entropy, keeping the thermodynamics and the dominant energy condition. 
  A quasi-equilibrium (QE) computational scheme was recently developed in general relativity to calculate the complete gravitational wavetrain emitted during the inspiral phase of compact binaries. The QE method exploits the fact that the the gravitational radiation inspiral timescale is much longer than the orbital period everywhere outside the ISCO. Here we demonstrate the validity and advantages of the QE scheme by solving a model problem in relativistic scalar gravitation theory. By adopting scalar gravitation, we are able to numerically track without approximation the damping of a simple, quasi-periodic radiating system (an oscillating spherical matter shell) to final equilibrium, and then use the exact numerical results to calibrate the QE approximation method. In particular, we calculate the emitted gravitational wavetrain three different ways: by integrating the exact coupled dynamical field and matter equations, by using the scalar-wave monopole approximation formula (corresponding to the quadrupole formula in general relativity), and by adopting the QE scheme. We find that the monopole formula works well for weak field cases, but fails when the fields become even moderately strong. By contrast, the QE scheme remains quite reliable for moderately strong fields, and begins to breakdown only for ultra-strong fields. The QE scheme thus provides a promising technique to construct the complete wavetrain from binary inspiral outside the ISCO, where the gravitational fields are strong, but where the computational resources required to follow the system for more than a few orbits by direct numerical integration of the exact equations are prohibitive. 
  We introduce two new families of solutions to the vacuum Einstein equations with negative cosmological constant in 5 dimensions. These solutions are static black holes whose horizons are modelled on the 3-geometries nilgeometry and solvegeometry. Thus the horizons (and the exterior spacetimes) can be foliated by compact 3-manifolds that are neither spherical, toroidal, hyperbolic, nor product manifolds, and therefore are of a topological type not previously encountered in black hole solutions. As an application, we use the solvegeometry solutions to construct Bianchi VI$_{-1}$ braneworld cosmologies. 
  We recall the main features of metric vacuum fluctuations which have the global property int d^4x sqrt{g(x)} R(x) = 0, even though R(x) does not vanish locally. We stress that these fluctuations could mediate an anomalous coupling between the gravitational field and coherent matter. Some new issues are discussed: (1) these fluctuations still imply that <T_{mu nu}(x)>=0; (2) they are not extrema of the action; (3) for finite duration, their volume in phase space is not zero; (4) vacuum fluctuations of this kind are not allowed in QED; (5) their null-action property is a nonperturbative feature; (6) any -real- pure e.m. field generates zero-modes of this kind, too, up to terms of order G^2. 
  Recent modern space missions deliver invaluable information about origin of our universe, physical processes in the vicinity of black holes and other exotic astrophysical objects, stellar dynamics of our galaxy, etc. On the other hand, space astrometric missions make it possible to determine with unparalleled precision distances to stars and cosmological objects as well as their physical characteristics and positions on the celestial sphere. Permanently growing accuracy of space astronomical observations and the urgent need for adequate data processing algorithms require corresponding development of an adequate theory of reference frames along with unambiguous description of propagation of light rays from a source of light to observer. Such a theory must be based on the Einstein's general relativity and account for numerous relativistic effects both in the solar system and outside of its boundary. The main features of the relativistic theory of reference frames are presented in this work. A hierarchy of the frames is described starting from the perturbed cosmological Friedmann-Robertson-Walker metric and going to the observer's frame through the intermediate barycentric and geocentric frames in the solar system. Microarcsecond astrometry and effects of propagation of light rays in time-dependent gravitational fields are discussed as well. 
  The detection of continuous gravitational-wave signals requires to account for the motion of the detector with respect to the solar system barycenter in the data analysis. In order to search efficiently for such signals by means of the fast Fourier transform the data needs to be transformed from the topocentric time to the barycentric time by means of resampling. The resampled data form a non-stationary random process. In this communication we prove that this non-stationary random process is mathematically well defined, and show that generalizations of the fundamental results for stationary processes, like Wiener-Khintchine theorem and Cram\`{e}r representation, exist. 
  We present several recent results concerning Cauchy and event horizons. In the first part of the paper we review the differentiablity properties of the Cauchy and the event horizons. In the second part we discuss compact Cauchy horizons and summarize their main properties. 
  It is well known that matched filtering techniques cannot be applied for searching extensive parameter space volumes for continuous gravitational wave signals. This is the reason why alternative strategies are being pursued. Hierarchical strategies are best at investigating a large parameter space when there exist computational power constraints. Algorithms of this kind are being implemented by all the groups that are developing software for analyzing the data of the gravitational wave detectors that will come online in the next years. In this talk we will report about the hierarchical Hough transform method that the GEO 600 data analysis team at the Albert Einstein Institute is developing. The three step hierarchical algorithm has been described elsewhere. In this talk we will focus on some of the implementational aspects we are currently concerned with. 
  The perturbative modes propagating along an infinite string are investigated within the framework of the gauge invariant perturbation formalism on a spacetime containing a self-gravitating straight string with a finite thickness. These modes are not included in our previous analysis. We reconstruct the perturbation formalism to discuss these modes and solve the linearized Einstein equation within the first order with respect to the string oscillation amplitude. In the thin string case, we show that the oscillations of an infinite string must involve the propagation of cosmic string traveling wave. 
  There is introduced a class of barotropic equations of state (EOS) which become polytropic of index $n = 5$ at low pressure. One then studies asymptotically flat solutions of the static Einstein equations coupled to perfect fluids having such an EOS. It is shown that such solutions, in the same manner as the vacuum ones, are conformally smooth or analytic at infinity, when the EOS is smooth or analytic, respectively. 
  We present a spinfoam formulation of Lorentzian quantum General Relativity. The theory is based on a simple generalization of an Euclidean model defined in terms of a field theory over a group. The model is an extension of a recently introduced Lorentzian model, in which both timelike and spacelike components are included. The spinfoams in the model, corresponding to quantized 4-geometries, carry a natural non-perturbative local causal structure induced by the geometry of the algebra of the internal gauge (sl(2,C)). Amplitudes can be expressed as integrals over the spacelike unit-vectors hyperboloid in Minkowski space, or the imaginary Lobachevskian space. 
  Thermal fluctuations for a massive scalar field in the Rindler wedge are obtained by applying the point-splitting procedure to the zero temperature Feynman propagator in a conical spacetime. Renormalization is implemented by removing the zero temperature contribution. It is shown that for a field of non vanishing mass the thermal fluctuations, when expressed in terms of the local temperature, do not have Minkowski form. As a by product, Minkowski vacuum fluctuations seen by an uniformly accelerated observer are determined and confronted with the literature. 
  A point particle of mass m moving on a geodesic creates a perturbation h, of the spacetime metric g, that diverges at the particle. Simple expressions are given for the singular m/r part of h and its quadrupole distortion caused by the spacetime. Subtracting these from h leaves a remainder h^R that is C^1. The self-force on the particle from its own gravitational field corrects the worldline at O(m) to be a geodesic of g+h^R. For the case that the particle is a small non-rotating black hole, an approximate solution to the Einstein equations is given with error of O(m^2) as m approaches 0. 
  By a choice of new variables the pressure isotropy condition for spherically symmetric static perfect fluid spacetimes can be made a quadratic algebraic equation in one of the two functions appearing in it. Using the other variable as a generating function, the pressure and the density of the fluid can be expressed algebraically by the function and its derivatives. One of the functions in the metric can also be expressed similarly, but to obtain the other function, related to the redshift factor, one has to perform an integral. Conditions on the generating function ensuring regularity and physicality near the center are investiagted. Two everywhere physically well behaving example solutions are generated, one representing a compact fluid body with a zero pressure surface, the other an infinite sphere. 
  In this paper we address both to the problem of identifying the noise Power Spectral Density of interferometric detectors by parametric techniques and to the problem of the whitening procedure of the sequence of data. We will concentrate the study on a Power Spectral Density like the one of the Italian-French detector VIRGO and we show that with a reasonable finite number of parameters we succeed in modeling a spectrum like the theoretical one of VIRGO, reproducing all its features. We propose also the use of adaptive techniques to identify and to whiten on line the data of interferometric detectors. We analyze the behavior of the adaptive techniques in the field of stochastic gradient and in the  Least Squares ones. 
  The gravitational-radiation-induced inspiral of a binary system of compact objects is considered. A scheme is described to model the regime in which the gravitational interaction is too strong to use weak-field approximation methods, but the time scale for decay of the orbits is still long compared to the orbital period, by numerically solving for a stationary spacetime which approximates the slowly evolving one. Equilibrium is to be maintained in the radiating system by imposing a balance of incoming and outgoing radiation at large distances. Numerical results from non-linear scalar field theory have shown that such an approach can be effective modelling a slowly evolving solution to a wave equation. 
  It is argued that the diffeomorphism on the horizontal sphere can be regarded as a nontrivial asymptotic isometry of the Schwarzschild black hole. We propose a new boundary condition of asymptotic metrics near the horizon and show that the condition admits the local time-shift and diffeomorphism on the horizon as the asymptotic symmetry. 
  In association with the Blanford-Znajek mechanism for rotational energy extraction from Kerr black holes, it is of some interest to explore how much of magnetic flux can actually penetrate the horizon at least in idealized situations. For completely uncharged Kerr hole case, it has been known for some time that the magnetic flux gets entirely expelled when the hole is maximally-rotating. In the mean time, it is known that when the rotating hole is immersed in an originally uniform magnetic field surrounded by an ionized interstellar medium (plasma), which is a more realistic situation, the hole accretes certain amount of electric charge. In the present work, it is demonstrated that as a result of this accretion charge small enough not to disturb the geometry, the magnetic flux through this slightly charged Kerr hole depends not only on the hole's angular momentum but on the hole's charge as well such that it never vanishes for any value of the hole's angular momentum. 
  Conditions for the existence of shear-free and expansion-free non-null vector fields in spaces with affine connections and metrics are found. On their basis Weyl's spaces with shear-free and expansion-free conformal Killing vectors are considered. The necessary and sufficient conditions are found under which a free spinless test particle could move in spaces with affine connections and metrics on a curve described by means of an auto-parallel equation. In Weyl's spaces with Weyl's covector, constructed by the use of a dilaton field, the dilaton field appears as a scaling factor for the rest mass density of the test particle.   PACS numbers: 02.40.Ky, 04.20.Cv, 04.50.+h, 04.90.+e 
  We obtain N = 3 chiral supergravity (SUGRA) compatible with the reality condition by applying the prescription of constructing the chiral Lagrangian density from the usual SUGRA. The $N = 3$ chiral Lagrangian density in first-order form, which leads to the Ashtekar's canonical formulation, is determined so that it reproduces the second-order Lagrangian density of the usual SUGRA especially by adding appropriate four-fermion contact terms. We show that the four-fermion contact terms added in the first-order chiral Lagrangian density are the non-minimal terms required from the invariance under first-order supersymmetry transformations. We also discuss the case of higher N theories, especially for N = 4 and N = 8. 
  In terms of the Painlev{\'e}-Gullstrand-Lema{\^\i}tre coordinates a rather general scenario for the gravitational collapse of an object and the subsequent formation of a horizon is described by a manifestly $C^\infty$-metric. For a 1+1 dimensional model of the collapse the leading contributions to the Bogoliubov coefficients are calculated explicitely and the Hawking temperature is recovered. But depending on the particular dynamics of the collapse the final state represents either evaporation or anti-evaporation. The generalization of the calculation to 3+1 dimensions is outlined and possible implications are addressed. PACS-numbers: 04.70.Dy, 04.70.-s, 04.62.+v. 
  This paper summarizes the contribution presented at the IX Marcel Grossmann Meeting (Rome, July 2000). A simple model of spacetime foam, made by $N$ Schwarzschild wormholes in a semiclassical approximation, is here proposed. The Casimir-like energy of the quantum fluctuation of such a model and its probability of being realized are computed. Implications on the Bekenstein-Hawking entropy and the cosmological constant are considered. A proposal for an alternative foamy model formed by $N$ Schwarzschild-Anti-de Sitter wormholes is here considered. 
  We discuss scenarios in which the galactic dark matter in spiral galaxies is described by a long range coherent field which settles in a stationary configuration that might account for the features of the galactic rotation curves. The simplest possibility is to consider scalar fields, so we discuss in particular, two mechanisms that would account for the settlement of the scalar field in a non-trivial configuration in the absence of a direct coupling of the field with ordinary matter: topological defects, and spontaneous scalarization. 
  We consider the problem of uniqueness of certain simultaneity structures in flat spacetime. Absolute simultaneity is specified to be a non-trivial equivalence relation which is invariant under the automorphism group Aut of spacetime. Aut is taken to be the identity-component of either the inhomogeneous Galilei group or the inhomogeneous Lorentz group. Uniqueness of standard simultaneity in the first, and absence of any absolute simultaneity in the second case are demonstrated and related to certain group theoretic properties. Relative simultaneity with respect to an additional structure X on spacetime is specified to be a non-trivial equivalence relation which is invariant under the subgroup in Aut that stabilises X. Uniqueness of standard Einstein simultaneity is proven in the Lorentzian case when X is an inertial frame. We end by discussing the relation to previous work of others. 
  Refractive gravitational waves are a generalisation of impulsive waves on a null hypersurface in which the metric is discontinuous but a weaker continuity condition for areas holds. A simple example of a plane wave is examined in detail and two arguments are given that this should be considered a solution of Einstein's vacuum field equations. The study of these waves is motivated by quantum gravity, where the refractive plane waves are considered as elementary quantum fluctuations and the `area geometry' of a null hypersurface plays a primary role. 
  Using two recent techniques giving non-perturbative re-summed estimates of the damping and of the conservative part of the dynamics of two-body systems, we describe the transition between adiabatic inspiral and plunge in binary non-spinning black holes moving along quasi-circular orbits. 
  In a recent paper (gr-qc/0002007), Anderson, Hiscock and Taylor claimed that "in all physically realistic cases, macroscopic zero temperature black hole solutions do not exist." We show this conclusion was reached on the basis of an incorrect calculation. 
  We will present results of a long term stable evolution, to $t=1000m$, of a maximally sliced Schwarzschild blackhole using a smooth lattice method. 
  Convexity and convex functions play an important role in theoretical physics. To initiate a study of the possible uses of convex functions in General Relativity, we discuss the consequences of a spacetime $(M,g_{\mu \nu})$ or an initial data set $(\Sigma, h_{ij}, K_{ij})$ admitting a suitably defined convex function. We show how the existence of a convex function on a spacetime places restrictions on the properties of the spacetime geometry. 
  The Weyl anomaly problem is treated within a purely geometrical context. Arguments are given that hint at a possible classical origin of the conformal anomaly in the Riemannian nature of the background geometry where the matter fields play out their dynamics. Some considerations allowing for a possible resolution of the Weyl anomaly problem are briefly outlined. Following the spirit of the standard model of the fundamental interactions, it is argued that the Weyl anomaly should be a consequence of the breaking of the gauge symmetry at some stage during the evolution of the universe. 
  Vaidya metric as an integral part of the Expansive Nondecelerative Universe (ENU) model enables to localize the energy of gravitational field and, subsequently, to find a deep interrelationship between quantum mechanics and the general theory of relativity. In the present paper, stemming from the ENU model, ionisation energy and energy of hyperfine splitting of the hydrogen atom, energy of the elementary quantum of action, as well as the proton and electron mass are independently expressed through the mass of the planckton, Z and W bosons and fundamental constants. 
  We prove that a certain spinfoam model for euclidean quantum general relativity, recently defined, is finite: all its all Feynman diagrams converge. The model is a variant of the Barrett-Crane model, and is defined in terms of a field theory over SO(4) X SO(4) X SO(4) X SO(4). 
  We investigate the evolution of different measures of ``Gravitational Entropy'' in Bianchi type I and Lema\^itre-Tolman universe models.   A new quantity behaving in accordance with the second law of thermodynamics is introduced. We then go on and investigate whether a quantum calculation of initial conditions for the universe based upon the Wheeler-DeWitt equation supports Penrose's Weyl Curvature Conjecture, according to which the Ricci part of the curvature dominates over the Weyl part at the initial singularity of the universe. The theory is applied to the Bianchi type I universe models with dust and a cosmological constant and to the Lema\^itre-Tolman universe models. We investigate two different versions of the conjecture. First we investigate a local version which fails to support the conjecture. Thereafter we construct a non-local entity which shows more promising behaviour concerning the conjecture. 
  Algorithms are developed for generating a class of exact braneworld cosmologies, where a self-interacting scalar field is confined to a positive-tension brane embedded in a bulk containing a negative cosmological constant. It is assumed that the five-dimensional Planck scale exceeds the brane tension but is smaller than the four-dimensional Planck mass. It is shown that the field equations can be expressed as a first-order system. A number of solutions to the equations of motion are found. The potential resulting in the perfect fluid model is identified. 
  In this paper we demonstrate that subsequent application of Lorentz transformations to the cylindrical coordinates on a rotating disk leaves the Euclidean metric invariant. Therefore, the geometry on rotating disk is the Euclidean geometry, and any experiment which do not involve tidal forces or Coriolis forces cannot identify either the disk rotates or not. We also show that, from the point of view of external inertial observer, the difference in the transit times for the light running along a circle of radius R in the opposite directions (with respect to the rotation) is 2w/c^2 S, where S is the area the circle, and w is the angle velocity. 
  We examine the scalar sector of the covariant graviton two-point function in de Sitter spacetime. This sector consists of the pure-trace part and another part described by a scalar field. We show that it does not contribute to two-point functions of gauge-invariant quantities. We also demonstrate that the long-distance growth present in some gauges is absent in this sector for a wide range of gauge parameters. 
  A Lagrangian from which derive the third post-Newtonian (3PN) equations of motion of compact binaries (neglecting the radiation reaction damping) is obtained. The 3PN equations of motion were computed previously by Blanchet and Faye in harmonic coordinates. The Lagrangian depends on the harmonic-coordinate positions, velocities and accelerations of the two bodies. At the 3PN order, the appearance of one undetermined physical parameter \lambda reflects an incompleteness of the point-mass regularization used when deriving the equations of motion. In addition the Lagrangian involves two unphysical (gauge-dependent) constants r'_1 and r'_2 parametrizing some logarithmic terms. The expressions of the ten Noetherian conserved quantities, associated with the invariance of the Lagrangian under the Poincar\'e group, are computed. By performing an infinitesimal ``contact'' transformation of the motion, we prove that the 3PN harmonic-coordinate Lagrangian is physically equivalent to the 3PN Arnowitt-Deser-Misner Hamiltonian obtained recently by Damour, Jaranowski and Sch\"afer. 
  This paper is mostly a collection of ideas already published by various authors, some of them even a long time ago. Its intention is to bring the reader to know some rather unknown papers of different fields that merit interest and to show some relations between them the author claims to have observed. In the first section, some comments on old unresolved problems in theoretical physics are collected. In the following, I shall explain what relation exists between Feynman graphs and the teleparallel theory of Einstein and Cartan in the late 1920s, and the relation of both to the theories of the incompressible aether around 1840. Reviewing these developments, we will have a look at the continuum theory of dislocations developed by Kroener in the 1950s and some techniques of differential geometry and topology relevant for a modern description of defects in continous media. I will then illustrate some basic concepts of nonlinear continuum mechanics and discuss applications to the above theories. By doing so, I hope to attract attention to the possible relevance of these facts for `fundamental' physics. 
  The aim of this note is to clarify the structure of nontrivial global solutions with nonpositive ADM mass for the static, spherically symmetric Einstein-Yang-Mills equations with SU(2) gauge group. The presented numerical results demonstrate that these solutions have zero number of nodes of the gauge field function. This is a feature that is present neither for particlelike nor for black hole solutions. 
  We calculate the energy distribution of a charged regular black hole by using the energy-momentum complexes of Einstein and M{\o}ller. 
  We study fine differentiability properties of horizons. We show that the set of end points of generators of a n-dimensional horizon H (which is included in a (n+1)-dimensional space-time M) has vanishing n-dimensional Hausdorff measure. This is proved by showing that the set of end points of generators at which the horizon is differentiable has the same property. For 1\le k\le n+1 we show (using deep results of Alberti) that the set of points where the convex hull of the set of generators leaving the horizon has dimension k is ``almost a C^2 manifold of dimension n+1-k'': it can be covered, up to a set of vanishing (n+1-k)-dimensional Hausdorff measure, by a countable number of C^2 manifolds. We use our Lorentzian geometry results to derive information about the fine differentiability properties of the distance function and the structure of cut loci in Riemannian geometry. 
  We introduce a complete gauge fixing for cylindrical spacetimes in vacuo that, in principle, do not contain the axis of symmetry. By cylindrically symmetric we understand spacetimes that possess two commuting spacelike Killing vectors, one of them rotational and the other one translational. The result of our gauge fixing is a constraint-free model whose phase space has four field-like degrees of freedom and that depends on three constant parameters. Two of these constants determine the global angular momentum and the linear momentum in the axis direction, while the third parameter is related with the behavior of the metric around the axis. We derive the explicit expression of the metric in terms of the physical degrees of freedom, calculate the reduced equations of motion and obtain the Hamiltonian that generates the reduced dynamics. We also find upper and lower bounds for this reduced Hamiltonian that provides the energy per unit length contained in the system. In addition, we show that the reduced formalism constructed is well defined and consistent at least when the linear momentum in the axis direction vanishes. Furthermore, in that case we prove that there exists an infinite number of solutions in which all physical fields are constant both in the surroundings of the axis and at sufficiently large distances from it. If the global angular momentum is different from zero, the isometry group of these solutions is generally not orthogonally transitive. Such solutions generalize the metric of a spinning cosmic string in the region where no closed timelike curves are present. 
  We consider self-interacting, perturbative quantum field theory in a curved spacetime background with a Killing vector field. We show that the action of this spacetime symmetry on interacting field operators can be implemented by a Noether charge which arises as a surface integral over the time-component of the interacting Noether current-density associated with the Killing field. The proof of this involves the demonstration of a corresponding set of Ward identities. Our work is based on the perturbative construction by Brunetti and Fredenhagen (Commun.Math.Phys. 208 (2000) 623-661) of self-interacting quantum field theories in general globally hyperbolic spacetimes. 
  We discuss the Gupta-Bleuler quantization of the free electromagnetic field outside static black holes in the Boulware vacuum. We use a gauge which reduces to the Feynman gauge in Minkowski spacetime. We also discuss its relation with gauges used previously. Then we apply the low-energy sector of this field theory to investigate some low-energy phenomena. First, we discuss the response rate of a static charge outside the Schwarzschild black hole in four dimensions. Next, motivated by string physics, we compute the absorption cross sections of low-energy plane waves for the Schwarzschild and extreme Reissner-Nordstr\"om black holes in arbitrary dimensions higher than three. 
  We derive a transformation from the usual ADM metric-extrinsic curvature variables on the phase space of Schwarzschild black holes, to new canonical variables which have the interpretation of Kruskal coordinates. We explicitly show that this transformation is non-singular, even at the horizon. The constraints of the theory simplify in terms of the new canonical variables and are equivalent to the vanishing of the canonical momenta. Our work is based on earlier seminal work by Kuchar in which he reconstructed curvature coordinates and a mass function from spherically symmetric canonical data. The key feature in our construction of a nonsingular canonical transformation to Kruskal variables, is the scaling of the curvature coordinate variables by the mass function rather than by the mass at left spatial infinity. 
  We have developped a procedure for the search of periodic signals in the data of gravitational wave detectors. We report here the analysis of one year of data from the resonant detector Explorer, searching for pulsars located in the Galactic Center (GC). No signals with amplitude greater than $\bar{h}= 2.9~10^{-24}$, in the range 921.32-921.38 Hz, were observed using data collected over a time period of 95.7 days, for a source located at $\alpha=17.70 \pm 0.01$ hours and $\delta=-29.00 \pm 0.05$ degrees. Our procedure can be extended for any assumed position in the sky and for a more general all-sky search, even with a frequency correction at the source due to the spin-down and Doppler effects. 
  We examine the dynamical implications of an interaction between some of the fluid components of the universe. We consider the combination of three matter components, one of which is a perfect fluid and the other two are interacting. The interaction term generalizes the cases found in scalar field cosmologies with an exponential potential. We find that attracting scaling solutions are obtained in several regions of parameter space, that oscillating behaviour is possible, and that new curvature scaling solutions exist. We also discuss the inflationary behaviour of the solutions and present some of the constraints on the strength of the coupling, namely those arising from nucleosynthesis. 
  We calculate the fluctuations in the current and energy densities for the case of a quantized, minimally coupled, massless, complex scalar field around a straight and infinitesimally thin cosmic string carrying magnetic flux. At zero temperature, we evaluate the fluctuations in the current and energy densities for arbitrary flux and deficit angle. At a finite temperature, we evaluate the fluctuations in the energy density for the special case wherein the flux is absent and the deficit angle equals $\pi$. We find that, quite generically, the dimensionless ratio of the variance to the mean-squared values of the current and energy densities are of order unity which suggests that the fluctuations around cosmic strings can be considered to be large. 
  We show that the non-linear evolution of long wavelength perturbations may be important in a wide class of inflationary scenarios. We develop a solution for the evolution of such nonlinear perturbations which is exact to first order in a gradient expansion. As a first application, we demonstrate that in single field models of inflation there can be no parametric amplification of super-Hubble modes during reheating. We consider the implications of the solution for recent discussions of the back-reaction effect of long wavelength perturbations on the background geometry, give a new derivation of the equation of motion of stochastic inflation, and demonstrate that if the (generalized) slow-rolling condition is not satisfied, then inevitably long wavelength vector modes for gravitational fluctuations will be generated. 
  We develop a kind of pregeometry consisting of a web of overlapping fuzzy lumps which interact with each other. The individual lumps are understood as certain closely entangled subgraphs (cliques) in a dynamically evolving network which, in a certain approximation, can be visualized as a time-dependent random graph. This strand of ideas is merged with another one, deriving from ideas, developed some time ago by Menger et al, that is, the concept of probabilistic- or random metric spaces, representing a natural extension of the metrical continuum into a more microscopic regime. It is our general goal to find a better adapted geometric environment for the description of microphysics. In this sense one may it also view as a dynamical randomisation of the causal-set framework developed by e.g. Sorkin et al. In doing this we incorporate, as a perhaps new aspect, various concepts from fuzzy set theory. 
  Sum-over-histories quantization of particle-like theory in curved space is discussed. It is reviewed that the propagator satisfies the Schrodinger equation respective wave equation with a Laplace-like operator. The exact dependence of the operator on the choice of measure is shown.   Next, modifications needed for a manifold with a boundary are introduced, and the exact form of the equation for the propagator is derived. It is shown that the Laplace-like operator contains some distributional terms localized on the boundary. These terms induce proper boundary conditions for the propagator. This choice of boundary conditions is explained as a consequence of a measurement of particles on the boundary.   The interaction with sources inside of the domain and sources on the boundary is also discussed. 
  The radion on the de Sitter brane is investigated at the linear perturbation level, using the covariant curvature tensor formalism developed by Shiromizu, Maeda and Sasaki. It is found that if there is only one de Sitter brane with positive tension, there is no radion and thus the ordinary Einstein gravity is recoverd on the brane other than corrections due to the massive Kaluza-Klein modes. As a by-product of using the covariant curvature tensor formalism, it is immediately seen that the cosmological scalar, vector and tensor type perturbations all have the same Kaluza-Klein spectrum. On the other hand, if there are two branes with positive and negative tensions, the gravity on each brane takes corrections from the radion mode in addition to the Kaluza-Klein modes and the radion is found to have a negative mass-squared proportional to the curvature of the de Sitter brane, in contrast to the flat brane case in which the radion mass vanishes and degenerates with the 4-dimensional graviton modes. To relate our result with the metric perturbation approach, we derive the second order action for the brane displacement. We find that the radion identified in our approach indeed corresponds to the relative displacement of the branes in the Randall-Sundrum gauge and describes the scalar curvature perturbations of the branes in the gaussian normal coordinates around the branes. Implications to the inflationary brane universe are briefly discussed. 
  In the context of the teleparallel equivalent of general relativity, we show that the energy-momentum density for the gravitational field can be described by a true spacetime tensor. It is also invariant under local (gauge) translations of the tangent space coordinates, but transforms covariantly only under global Lorentz transformations. When the gauge gravitational field equation is written in a purely spacetime form, it becomes the teleparallel equivalent of Einstein's equation, and we recover M{\o}ller's expression for the canonical gravitational energy-momentum pseudotensor. 
  We consider a family of cylindrical spacetimes endowed with angular momentum that are solutions to the vacuum Einstein equations outside the symmetry axis. This family was recently obtained by performing a complete gauge fixing adapted to cylindrical symmetry. In the present work, we find boundary conditions that ensure that the metric arising from this gauge fixing is well defined and that the resulting reduced system has a consistent Hamiltonian dynamics. These boundary conditions must be imposed both on the symmetry axis and in the region far from the axis at spacelike infinity. Employing such conditions, we determine the asymptotic behaviour of the metric close to and far from the axis. In each of these regions, the approximate metric describes a conical geometry with a time dislocation. In particular, around the symmetry axis the effect of the singularity consists in inducing a constant deficit angle and a timelike helical structure. Based on these results and on the fact that the degrees of freedom in our family of metrics coincide with those of cylindrical vacuum gravity, we argue that the analysed set of spacetimes represent cylindrical gravitational waves surrounding a spinning cosmic string. For any of these spacetimes, a prediction of our analysis is that the wave content increases the deficit angle at spatial infinity with respect to that detected around the axis. 
  Properties of the horizon mass of hairy black holes are discussed with emphasis on certain subtle and initially unexpected features. A key property suggests that hairy black holes may be regarded as `bound states' of ordinary black holes without hair and colored solitons. This model is then used to predict the qualitative behavior of the horizon properties of hairy black holes, to provide a physical `explanation' of their instability and to put qualitative constraints on the end point configurations that result from this instability. The available numerical calculations support these predictions. Furthermore, the physical arguments are robust and should be applicable also in more complicated situations where detailed numerical work is yet to be carried out. 
  A coherent superposition of N Schwarzschild wormholes is proposed as a model for spacetime foam. Following the subtraction procedure for manifolds with boundaries, we calculate by variational methods the Casimir energy. A proposal for an alternative foamy model formed by N Schwarzschild-Anti-de Sitter wormholes is here considered. Finally, a conjecture about the foam evolution is proposed. 
  We propose a possible measurement of the time variability of the vacuum energy using strong gravitational lensing events. As an example we take an Einstein cross lens HST 14176+5226 and demonstrate that the measurement of the velocity dispersion with the accuracy of $\pm$ 5 km/sec will have a chance to determine the time dependence of the vacuum energy as well as the density parameter with the accuracy of order 0.1 if one fixes the lens model. 
  The recently introduced Isolated Horizons (IH) formalism has become a powerful tool for realistic black hole physics. In particular, it generalizes the zeroth and first laws of black hole mechanics in terms of quasi-local quantities and serves as a starting point for quantum entropy calculations. In this note we consider theories which admit hair, and analyze some new results that the IH provides, when considering solitons and stationary solutions. Furthermore, the IH formalism allows to state uniqueness conjectures (i.e. horizon `no-hair conjectures') for the existence of solutions. 
  We consider the problem of searching for continuous gravitational wave sources orbiting a companion object. This issue is of particular interest because the LMXB's, and among them Sco X-1, might be marginally detectable with 2 years coherent observation time by the Earth-based laser interferometers expected to come on line by 2002, and clearly observable by the second generation of detectors. Moreover, several radio pulsars, which could be deemed to be CW sources, are found to orbit a companion star or planet, and the LIGO/VIRGO/GEO network plans to continuously monitor such systems. We estimate the computational costs for a search launched over the additional five parameters describing generic elliptical orbits using match filtering techniques. These techniques provide the optimal signal-to-noise ratio and also a very clear and transparent theoretical framework. We provide ready-to-use analytical expressions for the number of templates required to carry out the searches in the astrophysically relevant regions of the parameter space, and how the computational cost scales with the ranges of the parameters. We also determine the critical accuracy to which a particular parameter must be known, so that no search is needed for it. In order to disentangle the computational burden involved in the orbital motion of the CW source, from the other source parameters (position in the sky and spin-down), and reduce the complexity of the analysis, we assume that the source is monochromatic and its location in the sky is exactly known. The orbital elements, on the other hand, are either assumed to be completely unknown or only partly known. We apply our theoretical analysis to Sco X-1 and the neutron stars with binary companions which are listed in the radio pulsar catalogue. 
  We show that the scalar-tensor $\sigma$-model action is conformally equivalent to general relativity with a minimally coupled wavemap with a particular target metric. Inflation on the source manifold is then shown to occur in a natural way due both to the arbitrary curvature couplings and the wavemap self-interactions. 
  The fundamentals of the teleparallel equivalent of general relativity are presented, and its main properties described. In particular, the field equations, the definition of an energy--momentum density for the gravitational field, the teleparallel version of the equivalence principle, and the dynamical role played by torsion as compared to the corresponding role played by curvature in general relativity, are discussed in some details. 
  The exact solutions of Einstein - Yang - Mills and interacting with SO (3) - Yang-Mills field nonlinear scalar field equations in a class of spatially homogeneous cosmological Friedmann models are obtained. 
  A very brief review is given of the current state of research in quantum gravity. Over the past fifteen years, two approaches have emerged as the most promising paths to a quantum theory of gravity: string theory and quantum geometry. I will discuss the main achievements and open problems of each of these approaches, and compare their strengths and weaknesses. 
  Singularity theorems of general relativity utilize the notion of causal geodesic incompleteness as a criterion of the presence of a spacetime singularity. The incompleteness of a causal curve implies the end and/or beginning of the existence of a particle, which is an event. In the commonly accepted approach, singularities are not incorporated into spacetime. Thus spacetime turns out to be event-incomplete. With creation from nothing, singularities are sources of lawlessness. A straightforward way around those conceptual problems consists in including metric singularities in spacetime and then matching metrics and causal geodesics at the singularities. To this end, a spacetime manifold is assumed to be unboundable, so that singularities may only be interior. The matching the geodesics is achieved through weakening conditions for their smoothness. This approach is applied to a black-white hole and a big crunch-bang. 
  We consider cosmological consequences of a conformal-invariant unified theory which is dynamically equivalent to general relativity and is given in a space with the geometry of similarity. We show that the conformal-invariant theory offers new explanations for to such problems as the arrow of time, initial cosmic data, dark matter and accelerating evolution of the universe in the dust stage. 
  We review the toroidal, cylindrical and planar black hole solutions in anti-de Sitter spacetimes and present their properties. 
  We present a new survey of the radial oscillation modes of neutron stars. This study complements and corrects earlier studies of radial oscillations. We present an extensive list of frequencies for the most common equations of state and some more recent ones. In order to check the accuracy, we use two different numerical schemes which yield the same results. The stimulation for this work comes from the need of the groups that evolve the full nonlinear Einstein equation to have reliable results from perturbation theory for comparison. 
  Within the framework of general relativity, in some cases at least, it is a delicate and interesting question just what it means to say that an extended body is or is not "rotating". It is so for two reasons. First, one can easily think of different criteria of rotation. Though they agree if the background spacetime structure is sufficiently simple, they do not do so in general. Second, none of the criteria fully answers to our classical intuitions. Each one exhibits some feature or other that violates those intuitions in a significant and interesting way. The principal goal of the paper is to make the second claim precise in the form of a modest no-go theorem. 
  The classical central charge for the higher curvature gravity in 3-dimensions is calculated using the Legendre transformation method. The statistical entropy of BTZ black hole is derived by the Cardy's formula and the result completely coincides with the Iyer-Wald formula for the geometrical entropy. This coincidence suggests the generalized AdS/CFT correspondence. 
  We present tests and results of a new axisymmetric, fully general relativistic code capable of solving the coupled Einstein-matter system for a perfect fluid matter field. Our implementation is based on the Bondi metric, by which the spacetime is foliated with a family of outgoing light cones. We use high-resolution shock-capturing schemes to solve the fluid equations. The code can accurately maintain long-term stability of a spherically symmetric, relativistic, polytropic equilibrium model of a neutron star. In axisymmetry, we demonstrate global energy conservation of a perturbed neutron star in a compactified spacetime, for which the total energy radiated away by gravitational waves corresponds to a significant fraction of the Bondi mass. 
  Using traversable wormholes as theoretical background, we revisit a deep question of general relativity: Does a uniformly accelerated charged particle radiate? We particularize to the recently proposed gravitational \v{C}erenkov radiation, that happens when the spatial part of the Ricci tensor is negative. If $^{^{(3+1)}}R^i_{\phantom{i}i}< 0$, the matter threading the gravitational field violates the weak energy condition. In this case, the effective refractive index for light is bigger than 1, i.e. particles propagates, in that medium, faster than photons. This leads to a violation of the equivalence principle. 
  We present the canonical and quantum cosmological investigation of a four-dimensional, spatially flat, Friedmann-Robertson-Walker (FRW) model that is derived from the bosonic Neveu-Schwarz/Neveu-Schwarz sector of the low-energy M-theory effective action. We discuss in detail the phase space of the classical theory. We find the quantum solutions of the model and obtain the positive norm Hilbert space of states. Finally, the correspondence between wave functions and classical solutions is outlined. 
  The large scale interferometric gravitational wave detectors consist of Fabry-Perot cavities operating at very high powers ranging from tens of kW to MW for next generations. The high powers may result in several nonlinear effects which would affect the performance of the detector. In this paper, we investigate the effects of radiation pressure, which tend to displace the mirrors from their resonant position resulting in the detuning of the cavity. We observe a remarkable effect, namely, that the freely hanging mirrors gain energy continuously and swing with increasing amplitude. It is found that the `time delay', that is, the time taken for the field to adjust to its instantaneous equilibrium value, when the mirrors are in motion, is responsible for this effect. This effect is likely to be important in the optimal operation of the full-scale interferometers such as VIRGO and LIGO. 
  In this work, the interaction of electromagnetic fields with a rotating (Kerr) black hole is explored in the context of Born-Infeld (BI) theory of electromagnetism instead of standard Maxwell theory and particularly BI theory versions of the four horizon boundary conditions of Znajek and Damour are derived. Naturally, an issue to be addressed is then whether they would change from the ones given in Maxwell theory context and if they would, how. Interestingly enough, as long as one employs the same local null tetrad frame as the one adopted in the works by Damour and by Znajek to read out physical values of electromagnetic fields and fictitious surface charge and currents on the horizon, it turns out that one ends up with exactly the same four horizon boundary conditions despite the shift of the electrodynamics theory from a linear Maxwell one to a highly non-linear BI one. Close inspection reveals that this curious and unexpected result can be attributed to the fact that the concrete structure of BI equations happens to be such that it is indistinguishable at the horizon to a local observer, say, in Damour's local tetrad frame from that of standard Maxwell theory. 
  Energy-momentum (and angular momentum) for the Metric-Affine Gravity theory is considered from a Hamiltonian perspective (linked with the Noether approach). The important roles of the Hamiltonian boundary term and the many choices involved in its selection-which give rise to many different definitions-are emphasized. For each choice one obtains specific boundary conditions along with a value for the quasilocal, and (with suitable asymptotic behavior) total (Bondi and ADM) energy-momentum and angular momentum. Applications include the first law of black hole thermodynamics-which identifies a general expression for the entropy. Prospects for a positive energy proof are considered and quasilocal values for some solutions are presented. 
  Quantization in the mini-superspace of a gravity system coupled to a perfect fluid, leads to a solvable model which implies singularity free solutions through the construction of a superposition of the wavefunctions. We show that such models are equivalent to a classical system where, besides the perfect fluid, a repulsive fluid with an equation of state $p_Q = \rho_Q$ is present. This leads to speculate on the true nature of this quantization procedure. A perturbative analysis of the classical system reveals the condition for the stability of the classical system in terms of the existence of an anti-gravity phase. 
  The Euler-Lagrange equations of motion for the most general Ricci type gravitational Lagrangians are derived by means of a purely metric formalism. 
  I discuss recent work (gr-qc/0001047) on non-oscillatory singularities in four dimensional space-times with scalar field or stiff fluid matter, in the context of the BKL proposal. 
  We examine the behavior of an anisotropic brane-world in the presence of inflationary scalar fields. We show that, contrary to naive expectations, a large anisotropy does not adversely affect inflation. On the contrary, a large initial anisotropy introduces more damping into the scalar field equation of motion, resulting in greater inflation. The rapid decay of anisotropy in the brane-world significantly increases the class of initial conditions from which the observed universe could have originated. This generalizes a similar result in general relativity. A unique feature of Bianchi I brane-world cosmology appears to be that for scalar fields with a large kinetic term the initial expansion of the universe is quasi-isotropic. The universe grows more anisotropic during an intermediate transient regime until anisotropy finally disappears during inflationary expansion. 
  We study a generalized version of the Hamiltonian constraint operator in nonperturbative loop quantum gravity. The generalization is based on admitting arbitrary irreducible SU(2) representations in the regularization of the operator, in contrast to the original definition where only the fundamental representation is taken. This leads to a quantization ambiguity and to a family of operators with the same classical limit. We calculate the action of the Euclidean part of the generalized Hamiltonian constraint on trivalent states, using the graphical notation of Temperley-Lieb recoupling theory. We discuss the relation between this generalization of the Hamiltonian constraint and crossing symmetry. 
  We introduce the idea of {\it shape parameters} to describe the shape of the pencil of rays connecting an observer with a source lying on his past lightcone. On the basis of these shape parameters, we discuss a setting of image distortion in a generic (exact) spacetime, in the form of three {\it distortion parameters}. The fundamental tool in our discussion is the use of geodesic deviation fields along a null geodesic to study how source shapes are propagated and distorted on the path to an observer. We illustrate this non-perturbative treatment of image distortion in the case of lensing by a Schwarzschild black hole. We conclude by showing that there is a non-perturbative generalization of the use of Fermat's principle in lensing in the thin-lens approximation. 
  In a previous article concerning image distortion in non-perturbative gravitational lensing theory we described how to introduce shape and distortion parameters for small sources. We also showed how they could be expressed in terms of the scalar products of the geodesic deviation vectors of the source's pencil of rays in the past lightcone of an observer. In the present work we give an alternative approach to the description of the shape and distortion parameters and their evolution along the null geodesic from the source to the observer, but now in terms of the optical scalars (the convergence and shear of null vector field of the observer's lightcone) and the associated optical equations, which relate the optical scalars to the curvature of the spacetime. 
  The development of both ground- and space-based gravitational wave detectors provides new opportunities to observe the radiation from binaries containing neutron stars and black holes. Numerical simulations in 3-D are essential for calculating the coalescence waveforms, and comprise some of the most challenging problems in astrophysics today. This article briefly reviews the current status of efforts to calculate black hole and neutron star coalescences, and highlights challenges for the future. 
  A special-relativistic scalar-vector theory of gravitation is presented which mimics an important class of solutions of Einstein's gravitational field equations. The theory includes solutions equivalent to Schwarzschild, Kerr, Reissner-Nordstroem, and Friedman metrics as well as to gravitational waves. In fact, all the empirical tests until now due to general relativity can be explained within this flat spacetime theory. 
  We study the transition from the full quantum mechanical description of physical systems to an approximate classical stochastic one. Our main tool is the identification of the closed-time-path (CTP) generating functional of Schwinger and Keldysh with the decoherence functional of the consistent histories approach. Given a degree of coarse-graining in which interferences are negligible, we can explicitly write a generating functional for the effective stochastic process in terms of the CTP generating functional. This construction gives particularly simple results for Gaussian processes. The formalism is applied to simple quantum systems, quantum Brownian motion, quantum fields in curved spacetime. Perturbation theory is also explained. We conclude with a discussion on the problem of backreaction of quantum fields in spacetime geometry. 
  Refined Algebraic Quantization and Group Averaging are powerful methods for quantizing constrained systems. They give constructive algorithms for generating observables and the physical inner product. This work outlines the current status of these ideas with an eye toward quantum gravity. The main goal is provide a description of outstanding problems and possible research topics in the field. 
  The results on chaos in FRW cosmology with a massive scalar field are extended to another scalar field potential. It is shown that for sufficiently steep potentials the chaos disappears. A simple and rather accurate analytical criterion for the chaos to disappear is given. On the contrary, for gently sloping potentials the transition to a strong chaotic regime can occur. Two examples, concerning asymptotically flat and Damour-Mukhanov potentials are given. 
  This talk reviews the constraints imposed by binary-pulsar data on gravity theories, and notably on "scalar-tensor" theories which are the most natural alternatives to general relativity. Because neutron stars have a strong gravitational binding energy, binary-pulsar tests are qualitatively different from solar-system experiments: They have the capability of probing models which are indistinguishable from general relativity in weak gravitational field conditions. Besides the two most precise binary-pulsar experiments, in the systems B1913+16 and B1534+12, we also present the results of the various "null" tests of general relativity provided by several neutron star-white dwarf binaries, notably those of gravitational radiation damping. [The main interest of this very short paper is its figure, which also takes into account the "strong equivalence principle" tests.] 
  This talk is based on my work in collaboration with B. Boisseau, D. Polarski, and A.A. Starobinsky. The most natural and best-motivated alternatives to general relativity are the so-called "scalar-tensor" theories, in which the gravitational interaction is mediated not only by a (spin-2) graviton, but also by a (spin-0) scalar field. We study quintessence in this general framework, and show that the microscopic Lagrangian of the theory can be unambiguously reconstructed from two observable cosmological functions of the redshift: the luminosity distance and the linear density perturbation of dustlike matter. We also analyze the constraints imposed on the theory by the knowledge of only the first of these functions, as it will probably be available sooner with a good accuracy. [The main interest of this very short paper is to give the equations in the "Brans-Dicke" parametrization.] 
  We study plane-fronted electrovacuum waves in metric-affine gravity (MAG) with cosmological constant in the triplet ansatz sector of the theory. Their field strengths are, on the gravitational side, curvature $R_{\alpha}{}^{\beta}$, nonmetricity $Q_{\alpha\beta}$, torsion $T^{\alpha}$ and, on the matter side, the electromagnetic field strength $F$. Here we basically present, after a short introduction into MAG and its triplet subcase, the results of earlier joint work with Garcia, Macias, and Socorro. Our solution is based on an exact solution of Ozsvath, Robinson, and Rozga describing type N gravitational fields in general relativity as coupled to electromagnetic null-fields. 
  For dynamical systems of dimension three or more the question of integrability or nonintegrability is extended by the possibility of chaotic behaviour in the general solution. We determine the integrability of isotropic cosmological models in general relativity and string theory with a variety of matter terms, by a performance of the Painlev\'{e} analysis in an effort to examine whether or not there exists a Laurent expansion of the solution about a movable pole which contains the number of arbitrary constants necessary for a general solution. 
  Global problems associated with the transformation from the Arnowitt, Deser and Misner (ADM) to the Kucha\v{r} variables are studied. Two models are considered: The Friedmann cosmology with scalar matter and the torus sector of the 2+1 gravity. For the Friedmann model, the transformations to the Kucha\v{r} description corresponding to three different popular time coordinates are shown to exist on the whole ADM phase space, which becomes a proper subset of the Kucha\v{r} phase spaces. The 2+1 gravity model is shown to admit a description by embedding variables everywhere, even at the points with additional symmetry. The transformation from the Kucha\v{r} to the ADM description is, however, many-to-one there, and so the two descriptions are inequivalent for this model, too. The most interesting result is that the new constraint surface is free from the conical singularity and the new dynamical equations are linearization stable. However, some residual pathology persists in the Kucha\v{r} description. 
  We investigate the family of electrostatic spherically symmetric solutions of the five-dimensional Kaluza-Klein theory. Besides black holes and wormholes, a new class of geodesically complete solutions is identified. A monopole perturbation is carried out, enabling us to prove analytically the stability of a large class of solutions, including all black holes and neutral solutions. 
  We propose a new presentation of the Demia\'{n}ski-Newman (DN) solution of the axisymmetric Einstein equations. We introduce new dimensionless parameters $p$, $q$ and $s$, but keeping the Boyer-Lindquist coordinate transformation used for the Kerr metric in the Ernst method. The family of DN metrics is studied and it is shown that the main role of $s$ is to determine the singularities, which we obtain by calculating the Riemann tensor components and the invariants of curvature. So, $s$ reveals itself as the parameter of the singular rings on the inner ergosphere. 
  We found a consistent equation of reheating after inflation, which shows that for small quantum fluctuations the frequencies of resonance are slighted different from the standard ones. Quantum interference is taken into account and we found that at large fluctuations the process mimics very well the usual parametric resonance but proceed in a different dynamical way. The analysis is made in a toy quantum mechanical model and we discuss further its extension to quantum field theory. 
  The work on black holes immersed in external stationary magnetic fields is reviewed in both test-field approximation and within exact solutions. In particular we pay attention to the effect of the expulsion of the flux of external fields across charged and rotating black holes which are approaching extremal states. Recently this effect has been shown to occur for black hole solutions in string theory and Kaluza-Klein theory. 
  The curved spacetime surrounding a rotating black hole dramatically alters the structure of nearby electromagnetic fields. The Wald field which is an asymptotically uniform magnetic field aligned with the angular momentum of the hole provides a convenient starting point to analyze the effects of radiative corrections on electrodynamics in curved spacetime. Since the curvature of the spacetime is small on the scale of the electron's Compton wavelength, the tools of quantum field theory in flat spacetime are reliable and show that a rotating black hole immersed in a magnetic field approaching the quantum critical value of $B_k=m^2 c^3/(e\hbar) \approx 4.4 \times 10^{13}$~G $\approx 1.3\times10^{-11}$ cm$^{-1}$ is unstable. Specifically, a maximally rotating three-solar-mass black hole immersed in a magnetic field of $2.3 \times 10^{12}$~G would be a copious producer of electron-positron pairs with a luminosity of $3 \times 10^{52}$ erg s$^{-1}$. 
  In vacuum space-times the exterior derivative of a Killing vector field is a two-form that satisfies Maxwell equations without electromagnetic sources. Using the algebraic structure of this two-form we have set up a new formalism for the study of vacuum space-times with an isometry. 
  We construct exact static, axisymmetric solutions of Einstein-Maxwell-dilaton gravity presenting distorted charged dilaton black holes. The thermodynamics of such distorted black holes is also discussed. 
  We analyzed 6 hours of data from the TAMA300 detector by matched filtering, searching for gravitational waves from inspiraling compact binaries. We incorporated a two-step hierarchical search strategy in matched filtering. We obtained an upper limit of 0.59/hour (C.L.=90%) on the event rate of inspirals of compact binaries with mass between 0.3M_solar and 10M_solar and with signal-to-noise ratio greater than 7.2. The distance of 1.4M_solar (0.5M_solar) binaries which produce the signal-to-noise ratio 7.2 was estimated to be 6.2kpc (2.9kpc) when the position of the source on the sky and the inclination angle of the binaries were optimal. 
  Conditions for the existence of a gyroscope in spaces with affine connections and metrics are found. They appear as special types of Fermi-Walker transports for vector fields, lying in a subspace, orthogonal to the velocity vector field of an observer.   PACS numbers: 04.20Cv, 04.90.+e, 04.50.+h, 02.40.Ky 
  When using black hole excision to numerically evolve a fully generic black hole spacetime, most 3-D 3+1 codes use an $xyz$-topology (spatial) grid. In such a grid, an $r = \constant$ excision surface must be approximated by an irregular and non-smooth "staircase-shaped" excision grid boundary, which may introduce numerical instabilities into the evolution. In this paper I describe an alternate scheme, which uses multiple grid patches, each with topology $\{r \times ({\rm angular coordinates})\}$, to cover the slice outside the $r = \constant$ excision surface. The excision grid boundary is now smooth, so the evolution should be less prone to instabilities. With 4th order finite differencing, this code evolves Kerr initial data to ${\sim} 60M$ using the ADM equations; I'm currently implementing the BSSN equations in it in the hope that this will improve the stability. 
  Space gravitational wave detectors employing laser interferometry between free-flying spacecraft differ in many ways from their laboratory counterparts. Among these differences is the fact that, in space, the end-masses will be moving relative to each other. This creates a problem by inducing a Doppler shift between the incoming and outgoing frequencies. The resulting beat frequency is so high that its phase cannot be read to sufficient accuracy when referenced to state-of-the-art space-qualified clocks. This is the problem that is addressed in this paper. We introduce a set of time-domain algorithms in which the effects of clock jitter are exactly canceled. The method employs the two-color laser approach that has been previously proposed, but avoids the singularities that arise in the previous frequency-domain algorithms. In addition, several practical aspects of the laser and clock noise cancellation schemes are addressed. 
  The new millennium will see the upcoming of several ground-based interferometric gravitational wave antennas. Within the next decade a space-based antenna may also begin to observe the distant Universe. These gravitational wave detectors will together operate as a network taking data continuously for several years, watching the transient and continuous phenomena occurring in the deep cores of astronomical objects and dense environs of the early Universe where gravity was extremely strong and highly non-linear. The network will listen to the waves from rapidly spinning non-axisymmetric neutron stars, normal modes of black holes, binary black hole inspiral and merger, phase transitions in the early Universe, quantum fluctuations resulting in a characteristic background in the early Universe. The gravitational wave antennas will open a new window to observe the dark Universe unreachable via other channels of astronomical observations. 
  Deviation equation of Synge and Schild has been investigated over spaces with affine connections and metrics. It is shown that the condition for the vanishing of the Lie derivative of a vector field along a given non-null (non-isotropic) vector field u for obtaining this equation is only a sufficient (but not necessary) condition. By means of the vector field u and the projective metric (orthogonal to it) projected deviation equations of Synge and Schild have been obtained for a vector field, orthogonal to the given vector field u, as well as for the square of its length. For a given non-isotropic, auto-parallel and normalized vector field u this equation could have some simple solutions.   PACS numbers: 02.90; 04.50+h; 04.90.+e: 04.30.+x 
  We present a new approach to the study of vacuum spacetimes with a Killing symmetry. The central quantity in this approach is the exterior derivative of the Killing vector field, which is a test electromagnetic field. Considering the algebraic structure of this quantity we get a new view of the integrability conditions, which provides a natural way of studying the connections between the algebraic structure of the spacetime and properties of the Killing symmetry. 
  Based on the periastron precession model to account for kHz QPO of the binary X-ray neutron star, proposed by Stella and Vietri, we ascribe the 15-60 Hz Quasi Periodic Oscillation (QPO) to the periastron precession frequency of the orbiting accreted matter at the boundary of magnetosphere-disk of X-ray neutron star (NS). The obtained conclusions include: all QPO frequencies increase with increasing the accretion rate. The theoretical relations between 15-60 Hz QPO (HBO) frequency and the twin kHz QPOs are similar to the measured empirical formula. Further, the better fitted NS mass by the proposed model is about 1.9 solar masses for the detected LMXBs. 
  2+1 gravity for spacetimes with topology RxT^2 has been much studied. We add a description of how to extend these spacetimes across a Cauchy horizon into a region where the torus becomes Lorentzian. The result is a one parameter family of tori given by a geodesic in the "Teichmueller space" of Lorentzian tori. We describe this in detail. We also point out that if the modular group is regarded as part of the gauge group then these spacetimes offer a nice toy model for the dynamics of Bianchi IX models; in the region where the tori are spacelike the dynamics is described exactly by a hyperbolic billiard. On the other hand the modular group acts ergodically on the Teichmueller space of Lorentzian tori. 
  We present a general method to obtain static anisotropic spherically symmetric solutions, satisfying a nonlocal equation of state, from known density profiles. This equation of state describes, at a given point, the components of the corresponding energy-momentum tensor not only as a function at that point, but as a functional throughout the enclosed configuration. In order to establish the physical aceptability of the proposed static family of solutions satisfying nonlocal equation of state,\textit{}we study the consequences imposed by the junction and energy conditions for anisotropic fluids in bounded matter distribution. It is shown that a general relativistic spherically symmetric bounded distributions of matter, at least for certain regions, could satisfy a nonlocal equation of state. 
  Relativistic prescription is used to study the slow rotation of stars composed by self-gravitating bosons and fermions (fermions may be considered as neutrons). Previous results demand that purely boson stars are unable to display slow rotation, if one uses relativistic prescription with classical scalar fields. In contrast to this, the present work shows that a combined boson-neutron star in its ground-state can rotate. Their structure and stability are analysed under slow rotation approximations. 
  The wave function for the matter field fluctuations in the infrared sector is studied within the framework of inflationary cosmology. These fluctuations are described by a coarse-grained field which takes into account only the modes with wavelength much bigger than the size of the Hubble horizon. The case of a power-law expanding universe is considered and it is found that the relevant phase-space remains coherent under certain circumstances. In this case the classical stochastic treatment for matter field fluctuations is not valid, however, for $p > 4.6$, the system loses its coherence and a classical stochastic approximation is allowed. 
  We investigate dominant late-time tail behaviors of massive scalar fields in nearly extreme Reissner-Nordstr\"{o}m background. It is shown that the oscillatory tail of the scalar fields has the decay rate of $t^{-5/6}$ at asymptotically late times. The physical mechanism by which the asymptotic $t^{-5/6}$ tail yields and the relation between the field mass and the time scale when the tail begins to dominate, are discussed in terms of resonance backscattering due to spacetime curvature. 
  We prove the existence of a family of initial data for the Einstein vacuum equation which can be interpreted as the data for two Kerr-like black holes in arbitrary location and with spin in arbitrary direction. When the mass parameter of one of them is zero, this family reduces exactly to the Kerr initial data. The existence proof is based on a general property of the Kerr metric which can be used in other constructions as well. Further generalizations are also discussed. 
  The Maxwell field of a charge e which experiences an impulsive acceleration or deceleration is constructed explicitly by subdividing Minkowskian space-time into two halves bounded by a future null-cone and then glueing the halves back together with appropriate matching conditions. The resulting retarded radiation can be viewed as instantaneous electromagnetic bremsstrahlung. If we similarly consider a spherically symmetric, moving gravitating mass, to experience an impulsive deceleration, as viewed by a distant observer, then this is accompanied by the emission of a light-like shell whose total energy measured by this observer is the same as the kinetic energy of the source before it stops. This phenomenon is a recoil effect which may be thought of as a limiting case of a Kinnersley rocket. 
  We prove the existence of a generalization of Kelvin's circulation theorem in general relativity which is applicable to perfect isentropic magnetohydrodynamic flow. The argument is based on a new version of the Lagrangian for perfect magnetohydrodynamics. We illustrate the new conserved circulation with the example of a relativistic magnetohydrodynamic flow possessing three symmetries. 
  Claims of a general Weyl invariance of an arbitrary 2D dilaton theory are critically discussed. 
  In the framework of the spacetime with torsion, we obtain the flavor evolution equation of the mass neutrino oscillation in vacuum. A comparison with the result of general relativity case, it shows that the flavor evolutionary equations in Riemann spacetime and Weitzenb\"ock spacetimes are equivalent in the spherical symmetric Schwarzschild spacetime, but turns out to be different in the case of the axial symmetry. 
  It is discussed that the usual Heisenberg commutation relation (CR) is not a proper relation for massless particles and then an alternative is obtained. The canonical quantization of the free electromagnetic(EM)fields based on the field theoretical generalization of this alternative is carried out. Without imposing the normal ordering condition,the vacuum energy is automatically zero.This can be considered as a solution to the EM fields vacuum catastrophe and a step toward managing the cosmologial constant problem at least for the EM fields contribution to the state of vacuum. 
  The detection of signals with varying frequency is important in many areas of physics and astrophysics. The current work was motivated by a desire to detect gravitational waves from the binary inspiral of neutron stars and black holes, a topic of significant interest for the new generation of interferometric gravitational wave detectors such as LIGO. However, this work has significant generality beyond gravitational wave signal detection.   We define a Fast Chirp Transform (FCT) analogous to the Fast Fourier Transform (FFT). Use of the FCT provides a simple and powerful formalism for detection of signals with variable frequency just as Fourier transform techniques provide a formalism for the detection of signals of constant frequency. In particular, use of the FCT can alleviate the requirement of generating complicated families of filter functions typically required in the conventional matched filtering process. We briefly discuss the application of the FCT to several signal detection problems of current interest. 
  Recently 2+1 dimensional gravity theory, especially ${\rm AdS_3}$ has been studied extensively. It was shown to be equivalent to the 2+1 Chern-Simon theory and has been investigated to understand the black hole thermodynamics, i.e. Hawking temperature and others. The purpose of this report is to investigate the canonical formalism of the original 2+1 Einstein gravity theory instead of the Chern-Simon theory. For the spherically symmetric space-time, local conserved quantities(local mass and angular momentum) are introduced and using them canonical quantum theory is defined. Constraints are imposed on state vectors and solved analytically. The strategy to obtain the solution is followed by our previous work. 
  In a recent work (Torres, Capozziello and Lambiase, Physical Review D62, 104012 (2000)), it was shown that a supermassive boson star could provide an alternative model for the galactic center, usually assumed as a black hole. Here we comment on some of the possibilities to actually detect this object, and how can it be distinguished from the standard and other alternative models. 
  We study the formation of central naked singularities in spherical dust collapse with a cosmological constant. We find that the central curvature singularity is locally naked, Tipler strong, and generic, in the sense that it forms from a non-zero-measure set of regular initial data. We also find that the Weyl and Ricci curvature scalars diverge at the singularity, with the former dominating over the latter, thereby signaling the non-local origin of the singularity. 
  We consider the quasi-classical model of the spin-free configuration on the basis of the self-gravitating spherical dust shell in General Relativity. For determination of the energy spectrum of the stationary states on the basis of quasi-classical quantization rules it is required to carry out some regularization of the system. It is realized by an embedding of the initial system in the extended system with rotation. Then, the stationary states of the spherical shells are S-states of the system with the intrinsic momentum. The quasi-classical treatment of a stability of the configuration is associated with the Langer modification of a square of the quantum mechanical intrinsic momentum. It gives value of critical bare mass of the shell determining threshold of stability. For the shell with the bare mass smaller or equal to the Planck's mass, the energy spectra of bound states are found. We obtain the expression for tunneling probability of the shell and construct the quasi-classical model of the pair creation and annihilation of the shells. 
  The purpose of this paper is twofold. First, we will present recent results on the data processing for LISA, including algorithms for elimination of clock jitter noise and discussion of the generation of the data averages that will eventually need to be telemetered to the ground. Second, we will argue, based partly on these results, that a laser interferometer tracking system (LITS) that employs independent lasers in each spacecraft is preferable for reasons of simplicity to that in which the lasers in two of the spacecraft are locked to the incoming beam from the third. 
  Discrete approaches to gravity, both classical and quantum, are reviewed briefly, with emphasis on the method using piecewise-linear spaces. Models of 3-dimensional quantum gravity involving 6j-symbols are then described, and progress in generalising these models to four dimensions is discussed, as is the relationship of these models in both three and four dimensions to topological theories. Finally, the repercussions of the generalisations are explored for the original formulation of discrete gravity using edge-length variables. 
  This article provides a self contained overview of the geometry and dynamics of relativistic brane models, of the category that includes point particle, string, and membrane representations for phenomena that can be considered as being confined to a worldsheet of the corresponding dimension (respectively one, two, and three) in a thin limit approximation in an ordinary 4 dimensional spacetime background. This category also includes ``brane world'' models that treat the observed universe as a 3-brane in 5 or higher dimensional background. The first sections are concerned with purely kinematic aspects: it is shown how, to second differential order, the geometry (and in particular the inner and outer curvature) of a brane worldsheet of arbitrary dimension is describable in terms of the first, second, and third fundamental tensor. The later sections show how -- to lowest order in the thin limit -- the evolution of such a brane worldsheet will always be governed by a simple tensorial equation of motion whose left hand side is the contraction of the relevant surface stress tensor $ bar T^{\mu\nu}$ with the (geometrically defined) second fundamental tensor $K_{\mu\nu}{^\rho}$, while the right hand side will simply vanish in the case of free motion and will otherwise be just the orthogonal projection of any external force density that may happen to act on the brane. 
  The definition of the Einstein 3-form G_a is motivated by means of the contracted 2nd Bianchi identity. This definition involves at first the complete curvature 2-form. The 1-form L_a is defined via G_a = L^b \wedge #(o_b \wedge o_a). Here # denotes the Hodge-star, o_a the coframe, and \wedge the exterior product. The L_a is equivalent to the Einstein 3-form and represents a certain contraction of the curvature 2-form. A variational formula of Salgado on quadratic invariants of the L_a 1-form is discussed, generalized, and put into proper perspective. 
  A geometric prequantization formula for the Poisson-Gerstenhaber bracket of forms found within the DeDonder-Weyl Hamiltonian formalism earlier is presented. The related aspects of covariant geometric quantization of field theories are sketched. In particular, the importance of the framework of Clifford and spinor bundles and superconnections in this context is underlined. 
  We apply the point-splitting method to investigate vacuum fluctuations in the nonglobally hyperbolic background of a spinning cosmic string. Implementing renormalization by removing the Minkowski contribution it is shown that although the Green function in the literature satisfies the usual Green function equation, it leads to pathological physical results. 
  We show that the Husain-Kuchar model can be described in the framework of BF theories. This is a first step towards its quantization by standard perturbative QFT techniques or the spin-foam formalism introduced in the space-time description of General Relativity and other diff-invariant theories. The actions that we will consider are similar to the ones describing the BF-Yang-Mills model and some mass generating mechanisms for gauge fields. We will also discuss the role of diffeomorphisms in the new formulations that we propose. 
  It is proposed to describe a teleparallel structure as a combination of a Riemannian and a symplectic structure. The correspondent invariance group is an intersection of the orthogonal and the symplectic groups. For a 4D manifold it turns to be the basic group of the electro-week interaction $U(1)\times SU(2).$ 
  A new method for derivation of the equation of motion from the field equation is proposed. The problem of embedding the singularities in a field satisfying the field equations is discussed. 
  We report on a new behavior found in numerical simulations of spherically symmetric gravitational collapse in self-gravitating SU(2) sigma models at intermediate gravitational coupling constants: The critical solution (between black hole formation and dispersion) closely approximates the continuously self-similar (CSS) solution for a finite time interval, then departs from this, and then returns to CSS again. This cycle repeats several times, each with a different CSS accumulation point. We have preliminary evidence that this same critical solution is also discretely self-similar (DSS) between the CSS episodes, but with an echoing period $\Delta$ which varies during the evolution. 
  Based on principles of the Expansive Nondecelerative Universe model that enables to quantify and localize the gravitational energy density, and stemming from the see-saw mechanism, the mass of electron, muon and tau neutrinos are determined in an independent way. 
  If the Schwarzschild black-hole is moving rectilinearly with uniform 3-velocity and suddenly stops, according to a distant observer, then we demonstrate that this observer will see a spherical light--like shell or "relativistic fireball" radiate outwards with energy equal to the original kinetic energy of the black-hole. 
  The approach of metric-affine field theory is to define spacetime as a real oriented 4-manifold equipped with a metric and an affine connection. The 10 independent components of the metric tensor and the 64 connection coefficients are the unknowns of the theory. We write the Yang-Mills action for the affine connection and vary it both with respect to the metric and the connection. We find a family of spacetimes which are stationary points. These spacetimes are waves of torsion in Minkowski space. We then find a special subfamily of spacetimes with zero Ricci curvature; the latter condition is the Einstein equation describing the absence of sources of gravitation. A detailed examination of this special subfamily suggests the possibility of using it to model the neutrino. Our model naturally contains only two distinct types of particles which may be identified with left-handed neutrinos and right-handed antineutrinos. 
  Recent work alludes to various `controversies' associated with signature change in general relativity. As we have argued previously, these are in fact disagreements about the (often unstated) assumptions underlying various possible approaches. The choice between approaches remains open. 
  It is argued that the Brans-Dicke theory may explain the present accelerated expansion of the universe without resorting to a cosmological constant or quintessence matter 
  Quantum mechanics is now 100 years old and still going strong. Combining general relativity with quantum mechanics is the last hurdle to be overcome in the "quantum revolution". 
  We study the vacuum polarization effect in the spacetime generated by a magnetic flux cosmic string in the framework of a scalar-tensor gravity. The vacuum expectation values of the energy-momentum tensor of a conformally coupled scalar field are calculated. The dilaton's contribution to the vacuum polarization effect is shown explicitly. 
  I show that it is possible to formulate the Relativity postulates in a way that does not lead to inconsistencies in the case of space-times whose short-distance structure is governed by an observer-independent length scale. The consistency of these postulates proves incorrect the expectation that modifications of the rules of kinematics involving the Planck length would necessarily require the introduction of a preferred class of inertial observers. In particular, it is possible for every inertial observer to agree on physical laws supporting deformed dispersion relations of the type $E^2- c^2 p^2- c^4 m^2 + f(E,p,m;L_p)=0$, at least for certain types of $f$. 
  This paper explores properties of the instantaneous ergo surface of a Kerr black hole. The surface area is evaluated in closed form. In terms of the mass ($m$) and angular velocity ($a$), to second order in $a$, the area of the ergo surface is given by $16 \pi m^2 + 4 \pi a^2$ (compared to the familiar $16 \pi m^2 - 4 \pi a^2$ for the event horizon). Whereas the total curvature of the instantaneous event horizon is $4 \pi$, on the ergo surface it ranges from $4 \pi$ (for $a=0$) to 0 (for $a=m$) due to conical singularities on the axis ($\theta=0,\pi$) of deficit angle $2 \pi (1-\sqrt{1-(a/m)^2})$. A careful application of the Gauss-Bonnet theorem shows that the ergo surface remains topologically spherical. Isometric embeddings of the ergo surface in Euclidean 3-space are defined for $0 \leq a/m \leq 1$ (compared to $0 \leq a/m \leq \sqrt{3}/2$ for the horizon). 
  LeMa\^\i tre-Tolman-Bondi models of spherical dust collapse have been used and continue to be used extensively to study various stellar collapse scenarios. It is by now well-known that these models lead to the formation of black holes and naked singularities from regular initial data. The final outcome of the collapse, particularly in the event of naked singularity formation, depends very heavily on quantum effects during the final stages. These quantum effects cannot generally be treated semi-classically as quantum fluctuations of the gravitational field are expected to dominate before the final state is reached. We present a canonical reduction of LeMa\^\i tre-Tolman-Bondi space-times describing the marginally bound collapse of inhomogeneous dust, in which the physical radius, $R$, the proper time of the collapsing dust, $\tau$, and the mass function, $F$, are the canonical coordinates, $R(r)$, $\tau(r)$ and $F(r)$ on the phase space. Dirac's constraint quantization leads to a simple functional (Wheeler-DeWitt) equation. The equation is solved and the solution can be employed to study some of the effects of quantum gravity during gravitational collapse with different initial conditions. 
  There are documents which show that Wolfgang Pauli developed in 1953 the first consistent generalization of the five-dimensional theory of Kaluza, Klein, Fock and others to a higher dimensional internal space. Because he saw no way to give masses to the gauge bosons, he refrained from publishing his results formally. 
  We consider a Hamiltonian quantum theory of stationary spacetimes containing a Kerr-Newman black hole. The physical phase space of such spacetimes is just six-dimensional, and it is spanned by the mass $M$, the electric charge $Q$ and angular momentum $J$ of the hole, together with the corresponding canonical momenta. In this six-dimensional phase space we perform a canonical transformation such that the resulting configuration variables describe the dynamical properties of Kerr-Newman black holes in a natural manner. The classical Hamiltonian written in terms of these variables and their conjugate momenta is replaced by the corresponding self-adjoint Hamiltonian operator and an eigenvalue equation for the Arnowitt-Deser-Misner (ADM) mass of the hole, from the point of view of a distant observer at rest, is obtained. In a certain very restricted sense, this eigenvalue equation may be viewed as a sort of "Schr\"odinger equation of black holes". Our "Schr\"odinger equation" implies that the ADM mass, electric charge and angular momentum spectra of black holes are discrete, and the mass spectrum is bounded from below. Moreover, the spectrum of the quantity $M^2-Q^2-a^2$, where $a$ is the angular momentum per unit mass of the hole, is strictly positive when an appropriate self-adjoint extension is chosen. The WKB analysis yields the result that the large eigenvalues of $M$, $Q$ and $a$ are of the form $\sqrt{2n}$, where $n$ is an integer. It turns out that this result is closely related to Bekenstein's proposal on the discrete horizon area spectrum of black holes. 
  In Paper I in this series we constructed evolution equations for the complete gauge-invariant linear perturbations of a time-dependent spherically symmetric perfect fluid spacetime. A key application of this formalism is the interior of a collapsing star. Here we derive boundary conditions at the surface of the star, matching the interior perturbations to the well-known perturbations of the vacuum Schwarzschild spacetime outside the star. 
  We calculate the self-force experienced by a point scalar charge, a point electric charge, and a point mass moving in a weakly curved spacetime characterized by a time-independent Newtonian potential. The self-forces are calculated by first computing the retarded Green's functions for scalar, electromagnetic, and (linearized) gravitational fields in the weakly curved spacetime, and then evaluating an integral over the particle's past world line. In all three cases the self-force contains both a conservative and a nonconservative (radiation-reaction) part. The conservative part of the self-force is directly related to the presence of matter in the spacetime. The radiation-reaction part of the self-force, on the other hand, is insensitive to the presence of matter. Our result for the gravitational self-force is disturbing: a radiation-reaction force should not appear in the equations of motion at this level of approximation, and it should certainly not give rise to radiation antidamping. The last section of this paper attempts to make sense of this result by placing it in the context of the post-Newtonian N-body problem. 
  We first show how, from the general 3rd order ODE of the form z'''=F(z,z',z'',s), one can construct a natural Lorentzian conformal metric on the four-dimensional space (z,z',z'',s). When the function F(z,z',z'',s) satisfies a special differential condition of the form, U[F]=0, the conformal metric possesses a conformal Killing field, xi = partial with respect to s, which in turn, allows the conformal metric to be mapped into a three dimensional Lorentzian metric on the space (z,z',z'') or equivalently, on the space of solutions of the original differential equation. This construction is then generalized to the pair of differential equations, z_ss = S(z,z_s,z_t,z_st,s,t) and z_tt = T(z,z_s,z_t,z_st,s,t), with z_s and z_t, the derivatives of z with respect to s and t. In this case, from S and T, one can again, in a natural manner, construct a Lorentzian conformal metric on the six dimensional space (z,z_s,z_t,z_st,s,t). When the S and T satisfy equations analogous to U[F]=0, namely equations of the form M[S,T]=0, the 6-space then possesses a pair of conformal Killing fields, xi =partial with respect to s and eta =partial with respect to t which allows, via the mapping to the four-space of z, z_s, z_t, z_st and a choice of conformal factor, the construction of a four-dimensional Lorentzian metric. In fact all four-dimensional Lorentzian metrics can be constructed in this manner. This construction, with further conditions on S and T, thus includes all (local) solutions of the Einstein equations. 
  We perform a numerical study of the critical regime for the general relativistic collapse of collisionless matter in spherical symmetry. The evolution of the matter is given by the Vlasov equation (or Boltzmann equation) and the geometry by Einstein's equations. This system of coupled differential equations is solved using a particle-mesh (PM) method. This method approximates the distribution function which describes the matter in phase space with a set of particles moving along the characteristics of the Vlasov equation. The individual particles are allowed to have angular momentum different from zero but the total angular momentum has to be zero to retain spherical symmetry.   In accord wih previous work by Rein, Rendall and Schaeffer, our results give some indications that the critical behaviour in this model is of Type I (the smallest black hole in each family has a finite mass). For the families of initial data that we have studied it seems that in the critical regime the solution is a static spacetime with non-zero radial momentum for the individual particles. We have also found evidence for scaling laws for the time that the critical solutions spend in the critical regime. 
  Any theory that states that the basic laws of physics are time-symmetric must be strictly deterministic. Only determinism enables time reversal of entropy increase. A contradiction therefore arises between two statements of Hawking. A simulation of a system under time reversal shows how an intrinsic time arrow re-emerges, destroying the time reversal, when even slight failure of determinism occurs. 
  Rebilas argues that time-reversal can occur even in an indeterministic system. This hypothesis is untestable, hence lying beyond physics. 
  Three families of exact solutions for 2-dimensional gravity minimally coupled to electrodynamics are obtained in the context of ${\cal R}=T$ theory. It is shown, by supersymmetric formalism of quantum mechanics, that the quantum dynamics of a neutral bosonic particle on static backgrounds with both varying curvature and electric field is exactly solvable. 
  The nature of superluminal photon propagation in the gravitational field describing radiation from a time-dependent, isolated source (the Bondi-Sachs metric) is considered in an effective theory which includes interactions which violate the strong equivalence principle. Such interactions are, for example, generated by vacuum polarisation in conventional QED in curved spacetime. The relation of the resulting light-cone modifications to the Peeling Theorem for the Bondi-Sachs spacetime is explained. 
  The Arnowitt-Deser-Misner (ADM) equations are deeply intertwined with discrete spectral resolutions of an elliptic operator of Laplace type associated with the spacelike hypersurfaces which foliate the space-time manifold, and the non-linearities of the four-dimensional hyperbolic theory are mapped into the potential term occurring in this operator. The ADM equations are here re-expressed as a coupled first-order system for the induced metric and the trace-free part of the extrinsic-curvature tensor, and their formulation in terms of integral equations is studied. 
  The gravitomagnetic clock effect and the Sagnac effect for circularly rotating orbits in stationary axisymmetric spacetimes are studied from a relative observer point of view, clarifying their relationships and the roles played by special observer families. In particular Semer\'ak's recent characterization of extremely accelerated observers in terms of the two-clock clock effect is shown to be complemented by a similarly special property of the single-clock clock effect. 
  Following an earlier suggestion of the authors(gr-qc/9607030), we use some basic properties of Euclidean black hole thermodynamics and the quantum mechanics of systems with periodic phase space coordinate to derive the discrete two-parameter area spectrum of generic charged spherically symmetric black holes in any dimension. For the Reissner-Nordstrom black hole we get $A/4G\hbar=\pi(2n+p+1)$, where the integer p=0,1,2,.. gives the charge spectrum, with $Q=\pm\sqrt{\hbar p}$. The quantity $\pi(2n+1)$, n=0,1,... gives a measure of the excess of the mass/energy over the critical minimum (i.e. extremal) value allowed for a given fixed charge Q. The classical critical bound cannot be saturated due to vacuum fluctuations of the horizon, so that generically extremal black holes do not appear in the physical spectrum. Consistency also requires the black hole charge to be an integer multiple of any fundamental elementary particle charge: $Q= \pm me$, m=0,1,2,.... As a by-product this yields a relation between the fine structure constant and integer parameters of the black hole -- a kind of the Coleman big fix mechanism induced by black holes. In four dimensions, this relationship is $e^2/\hbar=p/m^2$ and requires the fine structure constant to be a rational number. Finally, we prove that the horizon area is an adiabatic invariant, as has been conjectured previously. 
  We consider a modified ``Swiss cheese'' model in the Brans-Dicke theory, and discuss the evolution of black holes in the expanding universe. We define the black hole radius by the Misner-Sharp mass and find the time evolution for dust and vacuum universes. 
  The integral formulation of Maxwell's equations expressed in terms of an arbitrary observer family in a curved spacetime is developed and used to clarify the meaning of the lines of force associated with observer-dependent electric and magnetic fields. 
  By using the Newman-Penrose formalism and 't Hooft brick-wall model, the quantum entropies of the Kerr-Newman black hole due to the Dirac and electromagnetic fields are calculated and the effects of the spins of the photons and Dirac particles on the entropies are investigated. It is shown that the entropies depend only on the square of the spins of the particles and the contribution of the spins is dependent on the rotation of the black hole, except that different fields obey different statistics. 
  Detecting binary black holes in interferometer data requires an accurate knowledge of the orbital phase evolution of the system. From the point of view of data analysis one also needs fast algorithms to compute the templates that will employed in searching for black hole binaries. Recently, there has been progress on both these fronts: On the one hand, re-summation techniques have made it possible to accelerate the convergence of poorly convergent asymptotic post-Newtonian series and derive waveforms beyond the conventional adiabatic approximation. We now have a waveform model that extends beyond the inspiral regime into the plunge phase followed by the quasi-normal mode ringing. On the other hand, explicit Fourier domain waveforms have been derived that make the generation of waveforms fast enough so as not to be a burden on the computational resources required in filtering the detector data. These new developments should make it possible to efficiently and reliably search for black hole binaries in data from first interferometers. 
  We calculate the electrostatic potential generated by a point charge in the space-time of Reissner-Nordstrom with a conical defect. An expression for the self-energy is also presented. 
  A qualitative analysis of a scalar-tensor cosmological model, with an exponential potential for the scalar field, is performed. The phase diagram for the flat case is constructed. It is shown that solutions with an initial and final inflationary behaviour appear. The conditions for which the scenario favored by supernova type Ia observations becomes an attractor in the space of the solutions are established. 
  In vacuum space-times the exterior derivative of a Killing vector field is a 2-form (named here as the Papapetrou field) that satisfies Maxwell's equations without electromagnetic sources. In this paper, using the algebraic structure of the Papapetrou field, we will set up a new formalism for the study of vacuum space-times with an isometry, which is suitable to investigate the connections between the isometry and the Petrov type of the space-time. This approach has some advantages, among them, it leads to a new classification of these space-times and the integrability conditions provide expressions that determine completely the Weyl curvature. These facts make the formalism useful for application to any problem or situation with an isometry and requiring the knowledge of the curvature. 
  A nonpertubative approach to quantum gravity using precanonical field quantization originating from the covariant De Donder-Weyl Hamiltonian formulation which treats space and time variables on equal footing is presented. A generally covariant ``multi-temporal'' generalized Schroedinger equation on the finite dimensional space of metric and space-time variables is obtained. An important ingredient of the formulation is the ``bootstrap condition'' which introduces a classical space-time geometry as an approximate concept emerging as the quantum average in a self-consistent with the underlying quantum dynamics manner. An independence of the theory from an arbitrarily fixed background is ensured in this way. The prospects and unsolved problems of precanonical quantization of gravity are outlined. 
  We discuss the problem of the stability of the isotropy of the universe in the space of ever-expanding spatially homogeneous universes with a compact spatial topology. The anisotropic modes which prevent isotropy being asymptotically stable in Bianchi-type $VII_h$ universes with non-compact topologies are excluded by topological compactness. Bianchi type $V$ and type $VII_h$ universes with compact topologies must be exactly isotropic. In the flat case we calculate the dynamical degrees of freedom of Bianchi-type $I$ and $VII_0$ universes with compact 3-spaces and show that type $VII_0$ solutions are more general than type $I$ solutions for systems with perfect fluid, although the type $I$ models are more general than type $VII_0$ in the vacuum case. For particular topologies the 4-velocity of any perfect fluid is required to be non-tilted. Various consequences for the problems of the isotropy, homogeneity, and flatness of the universe are discussed. 
  Hawking's black hole information puzzle highlights the incompatibility between our present understanding of gravity and quantum physics. However, Hawking's prediction of black-hole evaporation is at a semiclassical level. One therefore suspects some modifications of the character of the radiation when quantum properties of the {\it black hole itself} are properly taken into account. In fact, during the last three decades evidence has been mounting that, in a quantum theory of gravity black holes may have a discrete mass spectrum, with concomitant {\it discrete} line emission. A direct consequence of this intriguing prediction is that, compared with blackbody radiation, black-hole radiance is {\it less} entropic, and may therefore carry a significant amount of {\it information}. Using standard ideas from quantum information theory, we calculate the rate at which information can be recovered from the black-hole spectral lines. We conclude that the information that was suspected to be lost may gradually leak back, encoded into the black-hole spectral lines. 
  Gravitational radiation is locally defined where the wavefronts are roughly spherical. A local energy tensor is defined for the gravitational radiation. Including this energy tensor as a source in the truncated Einstein equations describes gravitational radiation reaction, such as back-reaction on a roughly spherical black hole. The energy-momentum in a canonical frame is covariantly conserved. The strain to be measured by a distant detector is simply defined. 
  Following the subtraction procedure for manifolds with boundaries, we calculate by variational methods, the Schwarzschild-de Sitter and the de Sitter space energy difference. By computing the one loop approximation for TT tensors we discover the existence of an unstable mode even for the non-degenerate case. This result seems to be in agreement with the sub-maximal black hole pair creation of Bousso-Hawking. The instability can be eliminated by the boundary reduction method. Implications on a foam-like space are discussed. 
  We present results for two colliding black holes (BHs), with angular momentum, spin, and unequal mass. For the first time gravitational waveforms are computed for a grazing collision from a full 3D numerical evolution. The collision can be followed through the merger to form a single BH, and through part of the ringdown period of the final BH. The apparent horizon is tracked and studied, and physical parameters, such as the mass of the final BH, are computed. The total energy radiated in gravitational waves is shown to be consistent with the total mass of the spacetime and the final BH mass. The implication of these simulations for gravitational wave astronomy is discussed. 
  The problem of construction of a continuous (macroscopic) matter model for a given point-like (microscopic) matter distribution in general relativity is formulated. The existing approaches are briefly reviewed and a physical analogy with the similar problem in classical macroscopic electrodynamics is pointed out. The procedure by Szekeres in the linearized general relativity on Minkowski background to construct a tensor of gravitational quadruple polarization by applying Kaufman's method of molecular moments for derivation of the polarization tensor in macroscopic electrodynamics and to derive an averaged field operator by utilizing an analogy between the linearized Bianchi identities and Maxwell equations, is analyzed. It is shown that the procedure has some inconsistencies, in particular, (1) it has only provided the terms linear in perturbations for the averaged field operator which do not contribute into the dynamics of the averaged field, and (2) the analogy between electromagnetism and gravitation does break upon averaging. A macroscopic gravity approach in the perturbation theory up to the second order on a particular background space-time taken to be a smooth weak gravitational field is applied to write down a system of macroscopic field equations: Isaacson's equations with a source incorporating the quadruple gravitational polarization tensor, Isaacson's energy-momentum tensor of gravitational waves and energy-momentum tensor of gravitational molecules and corresponding equations of motion. A suitable set of material relations which relate all the tensors is proposed. 
  We construct stationary black holes in SU(2) Einstein-Yang-Mills theory, which carry angular momentum and electric charge. Possessing non-trivial non-abelian magnetic fields outside their regular event horizon, they represent non-perturbative rotating hairy black holes. 
  The diagonalization of the metrical Hamiltonian of a scalar field with an arbitrary coupling with a curvature in N-dimensional homogeneous isotropic space is performed. The energy spectrum of the corresponding quasiparticles is obtained. The energies of the quasiparticles corresponding to the diagonal form of the canonical Hamiltonian are calculated. The modified energy-momentum tensor with the following properties is constructed. It coincides with the metrical energy-momentum tensor for conformal scalar field. Under its diagonalization the energies of relevant particles of a nonconformal field coincide to the oscillator frequencies and the density of such particles created in a nonstationary metric is finite. It is shown that the Hamiltonian calculated with the modified energy-momentum tensor can be constructed as a canonical Hamiltonian under the special choice of variables. 
  Zero point quantum fluctuations as seen from non-inertial reference frames are of interest for several reasons. In particular, because phenomena such as Unruh radiation (acceleration radiation) and Hawking radiation (quantum leakage from a black hole) depend intrinsically on both quantum zero-point fluctuations and some appropriate notion of an accelerating vacuum state, any experimental test of zero-point fluctuations in non-inertial frames is implicitly a test of the foundations of quantum field theory, and the Unruh and Hawking effects 
  A core-collapse supernova might produce large amplitude gravitational waves if, through the collapse process, the inner core can aquire enough rotational energy to become dynamically unstable. In this report I present the results of 3-D numerical simulations of core collapse supernovae. These simulations indicate that for some initial conditions the post-collapse inner core is indeed unstable. However, for the cases considered, the instability does not produce a large gravitational-wave signal. 
  The dynamics of a universe dominated by a self-interacting nonminimally coupled scalar field are considered. The structure of the phase space and complete phase portraits are given. New dynamical behaviors include superinflation ($\dot{H}>0$), avoidance of big bang singularities through classical birth of the universe, and spontaneous entry into and exit from inflation. This model is promising for describing quintessence as a nonminimally coupled scalar field. 
  We show that the principle of least action is generally inconsistent with the usual Kaluza-Klein program, the higher dimensional Einstein-Hilbert action being unbounded from below. This inconsistency is also present in other theories with higher dimensions like supergravity. Hence, we conclude to the necessity of an external scalar field to compensate this flaw. 
  There are models of gravitational collapse in classical general relativity which admit the formation of naked singularities as well as black holes. These include fluid models as well as models with scalar fields as matter. Even if fluid models were to be regarded as unphysical in their matter content, the remaining class of models (based on scalar fields) generically admit the formation of visible regions of finite but arbitrarily high curvature. Hence it is of interest to ask, from the point of view of astrophysics, as to what a stellar collapse leading to a naked singularity (or to a visible region of very high curvature) will look like, to a far away observer. The emission of energy during such a process may be divided into three phases - (i) the classical phase, during which matter and gravity can both be treated according to the laws of classical physics, (ii) the semiclassical phase, when gravity is treated classically but matter behaves as a quantum field, and (iii) the quantum gravitational phase. In this review, we first give a summary of the status of naked singularities in classical relativity, and then report some recent results comparing the semiclassical phase of black holes with the semiclassical phase of spherical collapse leading to a naked singularity. In particular, we ask how the quantum particle creation during the collapse leading to a naked singularity compares with the Hawking radiation from a star collapsing to form a black hole. It turns out that there is a fundamental difference between the two cases. A spherical naked star emits only about one Planck energy during its semiclassical phase, and the further evolution can only be determined by the laws of quantum gravity. This contrasts with the semiclassical evaporation of a black hole. 
  An observer surrounded by sufficiently small spherical light sources at a fixed distance will see a pattern of elliptical images distributed over the sky, owing to the distortion effect (shearing effect) of the spacetime geometry upon light bundles. In lowest non-trivial order with respect to the distance, this pattern is completely determined by the conformal curvature tensor (Weyl tensor) at the observation event. In this paper we derive formulas that allow to calculate these distortion patterns in terms of the Newman-Penrose formalism. Then we represent the distortion patterns graphically for all Petrov types, and we discuss their dependence on the velocity of the observer. 
  We give a derivation of general relativity and the gauge principle that is novel in presupposing neither spacetime nor the relativity principle. We consider a class of actions defined on superspace with two key properties. The first is 3-coordinate invariance. This is the only postulated symmetry, and it leads to the standard momentum constraint. The second property is that the Lagrangian is constructed from a `local' square root of an expression quadratic in the velocities, `local' because it is taken before integration over 3-space. It gives rise to quadratic constraints that do not correspond to any symmetry and are not, in general, propagated by the Euler-Lagrange equations. Therefore these actions are internally inconsistent. Only one action of this form is well behaved: the Baierlein-Sharp-Wheeler reparametrisation-invariant action for GR.   From this viewpoint, spacetime symmetry is emergent. It appears as a `hidden' symmetry in the (underdetermined) solutions of the evolution equations, without being manifestly coded into the action itself. In addition, propagation of the constraints acts as a striking selection mechanism beyond pure gravity. If a scalar field is included in the configuration space, it must have the same characteristic speed as gravity. Thus Einstein causality emerges. Finally, self-consistency requires that any 3-vector field must satisfy Einstein causality, the equivalence principle and, in addition, the Gauss constraint. Therefore we recover the standard (massless) Maxwell equations. 
  A new Lagrange formalism based on the use of a single scalar (d+1)X(d+1+n) matrix potential is developed for the low-energy heterotic string theory with n U(1) gauge fields compactified from d+3 to 3 dimensions on a torus. This formalism also includes three pairs on-shell defined scalar and vector matrix potentials of the dimensions (d+1)X(d+1), (d+1)X(d+1+n) and (d+1+n)X(d+1+n). All these potentials undergo linear transformations when the group of charging symmetries acts. 
  Using the linearized theory of general relativity, the gravitomagnetic analogue of the Barnett effect is derived. Further theoretical and experimental investigation is recommended, due to the expected macroscopic values of the gravitomagnetic field involved in this effect, and to the constraints which would appear on quantum theories of gravity, currently under development, in case of non detection of the predicted phenomena. 
  New techniques to evaluate the clock effect using light are described. These are based on the flatness of the cylindrical surface containing the world lines of the rays constrained to move on circular trajectories about a spinning mass. The effect of the angular momentum of the source is manifested in the fact that inertial observers must be replaced by local non rotating observers. Starting from this an exact formula for circular trajectories is found. Numerical estimates for the Earth environment show that light would be a better probe than actual clocks to evidence the angular momentum influence. The advantages of light in connection with some principle experiments are shortly reviewed. 
  The Zipoy-Voorhees family of static, axisymmetric vacuum solutions forms an interesting family in that it contains the Schwarzschild black hole excepting which all other members have naked singularity. We analyze some properties of the region near singularity by studying a natural family of 2-surfaces. We establish that these have the topology of the 2-sphere by an application of the Gauss-Bonnet theorem. By computing the area, we establish that the singular region is `point-like'. Isometric embedding of these surfaces in the three dimensional Euclidean space is used to distinguish the two types of deviations from spherical symmetry. 
  It is proposed that the event horizon of a black hole is a quantum phase transition of the vacuum of space-time analogous to the liquid-vapor critical point of a bose fluid. The equations of classical general relativity remain valid arbitrarily close to the horizon yet fail there through the divergence of a characteristic coherence length. The integrity of global time, required for conventional quantum mechanics to be defined, is maintained. The metric inside the event horizon is different from that predicted by classical general relativity and may be de Sitter space. The deviations from classical behavior lead to distinct spectroscopic and bolometric signatures that can, in principle, be observed at large distances from the black hole. 
  We find that the continuous matter fields are ill-defined in Regge calculus in the physical 4D theory since the corresponding effective action has infinite terms unremovable by the UV renormalisation procedure. These terms are connected with the singular nature of the curvature distribution in Regge calculus, namely, with the presence in d>2 dimensions of the (d-3)-dimensional simplices where the (d-2)-dimensional ones carrying different conical singularities are meeting. Possible resolution of this difficulty is discretisation of matter fields in Regge background. 
  Second rank non-degenerate Killing tensors for some subclasses of spacetimes admitting parallel null one-planes are investigated. Lichn\'erowicz radiation conditions are imposed to provide a physical meaning to spacetimes whose metrics are described through their associated second rank Killing tensors. Conditions under which the dual spacetimes retain the same physical properties are presented. 
  The functional integral measure in the 4D Regge calculus normalised w.r.t. the DeWitt supermetric on the space of metrics is considered. The Faddeev-Popov factor in the measure is shown according to the previous author's work on the continuous fields in Regge calculus to be generally ill-defined due to the conical singularities. Possible resolution of this problem is discretisation of the gravity ghost (gauge) field by, e.g., confining ourselves to the affine transformations of the affine frames in the simplices. This results in the singularity of the functional measure in the vicinity of the flat background, where part of the physical degrees of freedom connected with linklengths become gauge ones. 
  It is shown that a minimally coupled scalara field in Brans-Dicke theory yields a non-decelerated expansion for the present universe for open, flat and closed Friedmann-Robertson-Walker models. 
  We explore the possibility of having a good description of classical signature change in the brane scenario. 
  We contribute to the subject of the physical interpretation of exact solutions by characterizing them through a systematic study in terms of unambiguous physical concepts coming from systems in linearized gravity. We use the physical meaning of the leading order behavior of the Weyl spinor components $\Psi_0^0$, $\Psi_1^0$ and $\Psi_2^0$ and of the Maxwell spinor components $\phi_0^0$ and $\phi_1^0$ and integrate from future null infinity inwards the exact field equations. In this way it is assigned an unambiguous physical meaning to exact solutions. We indicate a method to generalize the procedure to radiating spacetimes. 
  A covariant criterion for the Cherenkov radiation emission in the field of a non-linear gravitational wave is considered in the framework of exact integrable models of particle dynamics and electromagnetic wave propagation. It is shown that vacuum interacting with curvature can give rise to Cherenkov radiation. The conically shaped spatial distribution of radiation is derived and its basic properties are discussed. 
  The gravitational effects in the relativistic quantum mechanics are investigated. The exact Foldy-Wouthuysen transformation is constructed for the Dirac particle coupled to the static spacetime metric. As a direct application, we analyze the non-relativistic limit of the theory. The new term describing the specific spin (gravitational moment) interaction effect is recovered in the Hamiltonian. The comparison of the true gravitational coupling with the purely inertial case demonstrates that the spin relativistic effects do not violate the equivalence principle for the Dirac fermions. 
  We study the relativistic quantum mechanical scattering of a bosonic particle by an infinite straight cosmic string, considering the non-minimal coupling between the bosonic field and the scalar curvature. In this case, an effective two-dimensional delta-function interaction takes place besides the usual topological scattering and a renormalization procedure is necessary in order to treat the problem that appears in connection with the delta-function. 
  The asymptotic behavior of the scalar field and its physical meaning are discussed for T=0 and T\neq 0 for the large enough coupling parameter \omega. The special character of the Brans-Dicke theory is also discussed for local and cosmological problems in comparison with general relativity and the selection rules are introduced respectivley. The scalar field by locally-distributed matter should exhibit the asymptotic behavior \phi = <\phi> +O(1/\omega) because of the presence of cosmological matter. The scalar field of a proper cosmological solution should have the asymptotic form \phi =O(\rho /\omega) and should converge to zero in the continuous limit \rho /\omega \to 0. 
  We consider the dynamics of a spatially flat universe dominated by a self-interacting nonminimally coupled scalar field. The structure of the phase space and complete phase portraits for the conformal coupling case are given. It is shown that the non-minimal coupling modifies drastically the dynamics of the universe. New cosmological behaviors are identified, including superinflation ($\dot{H}>0$), avoidance of big bang singularities through classical birth of the universe from empty Minkowski space, and spontaneous entry into and exit from inflation. The relevance of this model to the description of quintessence is discussed. 
  We consider five-dimensional cylindric spacetime $V^5$ with foliation of codimension ~1. The leaves of this foliation are four-dimensional  "parallel" universes. The metric of five-dimensional spacetime and induced metrics of four-dimensional universes are flat. The "large" fluctuations of the 5-metric are studied. These fluctuations depend only on the coordinate $x^0$, and under these fluctuations the curvature of $V^5$ is not zero. The contribution of the fluctuations in the Feynman path integral over five-dimensional trajectories doesn't change the amplitude of the probability of the real physical four-dimensional universe. Moreover the large fluctuations of 5-metric $G_{AB}$ are large fluctuations for physical four-dimensional universe $V^4$ and change signature of $V^4$. The change of the signature from $<+--->$ to $<---->$ and inversely occurs in the all 3-dimensional space simultaneously (in absolute time) and can take arbitrarily large period of time. 
  We present a characteristic algorithm for computing the perturbation of a Schwarzschild spacetime by means of solving the Teukolsky equation. We implement the algorithm as a characteristic evolution code and apply it to compute the advanced solution to a black hole collision in the close approximation. The code successfully tracks the initial burst and quasinormal decay of a black hole perturbation through 10 orders of magnitude and tracks the final power law decay through an additional 6 orders of magnitude. Determination of the advanced solution, in which ingoing radiation is absorbed by the black hole but no outgoing radiation is emitted, is the first stage of a two stage approach to determining the retarded solution, which provides the close approximation waveform with the physically appropriate boundary condition of no ingoing radiation. 
  We develop a set of data analysis tools for a realistic all-sky search for continuous gravitational-wave signals. The methods that we present apply to data from both the resonant bar detectors that are currently in operation and the laser interferometric detectors that are in the final stages of construction and commissioning. We show that with our techniques we shall be able to perform an all-sky 2-day long coherent search of the narrow-band data from the resonant bar EXPLORER with no loss of signals with the dimensionless amplitude greater than $2.8\times10^{-23}$. 
  The earlier proposed conditions of (bi)quaternionic differentiability are nonlinear, give rize to the 2-spinor and the self-dual gauge structures and may be considered as the it generating system of equations (GSE) with respect to the source-free Maxwell, Yang-Mills and eikonal equations. We present the general solution of the GSE in terms of twistor variables, analize its rather specific gauge symmetry and demonstrate the relation of GSE to the equations of shear-free null congruences and, consequently, - to effective metrics of Kerr- Shild type. The concept of particles as singularities of physical fields associated with the solutions of GSE is developed 
  We present a characteristic algorithm for computing the perturbations of a Schwarzschild spacetime by means of solving the Teukolsky equations. Our methods and results are expected to have direct bearing on the study of binary black holes presently underway using a fully {\em nonlinear} characteristic code \cite{Gomez98a}. 
  Finite dimensional models that mimic the constraint structure of Einstein's General Relativity are quantized in the framework of BRST and Dirac's canonical formalisms. The first system to be studied is one featuring a constraint quadratic in the momenta (the "super-Hamiltonian") and a set of constraints linear in the momenta (the "supermomentum" constraints). The starting point is to realize that the ghost contributions to the supermomentum constraint operators can be read in terms of the natural volume induced by the constraints in the orbits. This volume plays a fundamental role in the construction of the quadratic sector of the nilpotent BRST charge. It is shown that the quantum theory is invariant under scaling of the super-Hamiltonian. As long as the system has an intrinsic time, this property translates in a contribution of the potential to the kinetic term. In this aspect, the results substantially differ from other works where the scaling invariance is forced by introducing a coupling to the curvature. The contribution of the potential, far from being unnatural, is beautifully justified in the light of the Jacobi's principle. Then, it is shown that the obtained results can be extended to systems with extrinsic time. In this case, if the metric has a conformal temporal Killing vector and the potential exhibits a suitable behavior with respect to it, the role played by the potential in the case of intrinsic time is now played by the norm of the Killing vector. Finally, the results for the previous cases are extended to a system featuring two super-Hamiltonian constraints. This step is extremely important due to the fact that General Relativity features an infinite number of such constraints satisfying a non trivial algebra among themselves. 
  We define the cosmological parameters $H_{c,0}$, $\Omega_{m,c}$ and $\Omega_{\Lambda, c}$ within the Conformal Cosmology as obtained by the homogeneous approximation to the conformal-invariant generalization of Einstein's General Relativity theory. We present the definitions of the age of the universe and of the luminosity distance in the context of this approach. A possible explanation of the recent data from distant supernovae Ia without a cosmological constant is presented. 
  This study toward quantum gravity (QG) introduces an SU(N) gauge theory with the \Theta vacuum term as a trial theory. Newton gravitation constant G_N is realized as the effective coupling constant for a massive graviton, G_N /\sqrt{2} = g_f g_g^2/8 M_G^2 \simeq 10^{-38} GeV^{-2} with the gauge boson mass M_G = M_{Pl} \simeq 10^{19} GeV, the gravitational coupling constant g_g, and the gravitational factor g_f. This scheme postulates the effective cosmological constant as the effective vacuum energy represented by massive gauge bosons, \Lambda_e = 8 \pi G_N M_G^4, and provides a plausible explanation for the small cosmological constant at the present epoch \Lambda_0 \simeq 10^{-84} GeV^2 and the large value at the Planck epoch \Lambda_{Pl} \simeq 10^{38} GeV^2; the condensation of the singlet gauge field <\phi> triggers the current anomaly and subtracts the gauge boson mass, M_G^2 = M_{Pl}^2 - g_f g_g^2 <\phi>^2 = g_f g_g^2 (A_{0}^2 - <\phi>^2), as the vacuum energy. Relations among QG, general relativity, and Newtonian mechanics are discussed. 
  This study toward quantum gravity (QG) introduces an SU(N) gauge theory with the \Theta vacuum term for gravitational interactions, which leads to a group SU(2)_L x U(1)_Y x SU(3)_C for weak and strong interactions through dynamical spontaneous symmetry breaking (DSSB). Newton gravitation constant G_N and the effective cosmological constant are realized as the effective coupling constant and the effective vacuum energy, respectively, due to massive gauge bosons. A gauge theory relevant for the non-zero gauge bosons, 10^{-12} GeV, and the massless gauge boson (photon) is predicted as a new dynamics for the universe expansion: this is supported by the repulsive force, indicated in BUMERANG-98 and MAXIMA-1 experiments, and cosmic microwave background radiation. Under the constraint of the flat universe, \Omega = 1 - 10^{-61}, the large cosmological constant in the early universe becomes the source of the exponential expansion in 10^{30} order as expected in the inflation theory, nearly massless gauge bosons are regarded as strongly interacting mediators of dark matter, and the baryon asymmetry is related to the DSSB mechanism. 
  This is an introductory set of lecture notes on quantum cosmology, given in 1995 to an audience with interests ranging from astronomy to particle physics. Topics covered: 1. Introduction: 1.1 Quantum cosmology and quantum gravity; 1.2 A brief history of quantum cosmology. 2. Hamiltonian formulation of general relativity: 2.1 The 3+1 decomposition; 2.2 The action. 3. Quantisation: 3.1 Superspace; 3.2 Canonical quantisation; 3.3 Path integral quantisation; 3.4 Minisuperspace; 3.5 The WKB approximation; 3.6 Probability measures; 3.7 Minisuperspace for the Friedmann universe with massive scalar field. 4. Boundary Conditions: 4.1 The no-boundary proposal; 4.2 The tunneling proposal. 5. The predictions of quantum cosmology: 5.1 The period of inflation; 5.2 The origin of density perturbations; 5.3 The arrow of time. 
  Everybody knows what the classical black holes are. In short, this is a spacetime region beyond the so-called event horizon. The notion of the event horizon is mathematically well defined. The situation with a definition of quantum black hole is not so clear. The problem is that the classical event horizon can be defined only globally, i.e. in order to be sure we have a black hole we would need an infinite time interval. But, in classical physics we have trajectories off all the particles and equations of motion for all the fields and can, in principle, construct some ideal models for the gravitational collapse and study the black hole formation under different conditions. 
  D-dimensional cosmological model describing the evolution of a perfect fluid with negative pressure (x-fluid) and a fluid possessing both shear and bulk viscosity in n Ricci-flat spaces is investigated. The second equations of state are chosen in some special form of metric dependence of the shear and bulk viscosity coefficients. The equations of motion are integrated and the dynamical properties of the exact solutions are studied. It is shown the possibility to resolve the cosmic coincidence problem when the x-fluid plays role of quintessence and the viscous fluid is used as cold dark matter. 
  In this note we prove a theorem on non-vacuum initial data for general relativity. The result presents a ``rigidity phenomenon'' for the extrinsic curvature, caused by the non-positive scalar curvature.   More precisely, we state that in the case of asymptotically flat non-vacuum initial data if the metric has everywhere non-positive scalar curvature then the extrinsic curvature cannot be compactly supported. 
  We present the explicit formulae which allow to transform the general solution of the 6D Kaluza--Klein theory on a 3--torus into the special solution of the 6D bosonic string theory on a 3--torus as well as into the general solution of the 5D bosonic string theory on a 2--torus. We construct a new family of the extremal solutions of the 3D chiral equation for the SL(4,R)/SO(4) coset matrix and interpret it in terms of the component fields of these three duality related theories. 
  We formulate conditions on the geometry of a non-expanding horizon $\Delta$ which are sufficient for the space-time metric to coincide on $\Delta$ with the Kerr metric. We introduce an invariant which can be used as a measure of how different the geometry of a given non-expanding horizon is from the geometry of the Kerr horizon. Directly, our results concern the space-time metric at $\IH$ at the zeroth and the first orders. Combained with the results of Ashtekar, Beetle and Lewandowski, our conditions can be used to compare the space-time geometry at the non-expanding horizon with that of Kerr to every order. The results should be useful to numerical relativity in analyzing the sense in which the final black hole horizon produced by a collapse or a merger approaches the Kerr horizon. 
  If one surrounds a black hole with a perfectly reflecting shell and adiabatically squeezes the shell inward, one can increase the black hole area A to exceed four times the total entropy S, which stays fixed during the process. A can be made to exceed 4S by a factor of order unity before the one enters the Planck regime where the semiclassical approximation breaks down. One interpretation is that the black hole entropy resides in its thermal atmosphere, and the shell restricts the atmosphere so that its entropy is less than A/4. 
  This article synthesises and extends recent work on the cosmological consequences of dropping the usual Z_2 reflection symmetry postulate in brane world scenarios. It is observed that for a cosmological model of homogeneous isotropic type, the relevant generalised Birkhoff theorem establishing staticity of the external vacuum in the maximally symmetric ``bulk'' outside a freely moving world brane will remain valid for the case of motion that is forced by minimal (generalised Wess Zumino type) coupling to an external antisymmetric gauge field provided its kinetic action contribution has the usual homogeneous quadratic form. This means that the geometry on each side of the brane worldsheet will still be of the generalised Schwarzschild anti de Sitter type. The usual first integrated Friedmann equation for the Hubble expansion rate can thereby be straightforwardly generalised by inclusion of new terms involving 2 extra parameters respectively measuring the strength of the gauge coupling and the degree of deviation from reflection symmetry. Some conceivable phenomenological implications are briefly outlined, and corresponding limitations are derived for possible values of relevant parameters. 
  The equations of motion of massive particles in GR are completely determined by the field equation. We utilize the particular form of Einstein's field equation and propose for the $N$-body problem of the equations that are Lorentz invariant a novel algorithm for the derivation of the equations of motion from the field equations. It is: 1. Compute a static, spherically symmetric solution of the field equation. It will be singular at the origin. This will be taken to be the field generated by a single particle. 2. Move the solution on a trajectory $ {\psi(t)}$ and apply the instantaneous Lorentz transformation based on instantaneous velocity $\dot{\psi}(t)$. 3. Take, as first approximation, the field generated by $N$ particles to be the superposition of the fields generated by the single particles. 4. Compute the leading part of the equation. Hopefully, only terms that involves $\ddot{\psi}(t)$ will be dominant. This is the ``inertial'' part. 5. Compute by the quadratic part of the equation. This is the agent of the ``force''. 6. Equate for each singularity, the highest order terms of the singularities that came from the linear part and the quadratic parts, respectively. This is an equation between the inertial part and the force. The algorithm was applied to Einstein equations. The approximate evolution of scalar curvature lends, in turn, to an invariant scalar equation. The algorithm for it did produce Newton's law of gravitation. This is, also, the starting point for the embedding the trajectories in a common field. 
  We use a metric of the type Friedmann-Robertson-Walker to obtain new exact solutions of Einstein equations for a scalar and massive field. The solutions have a permanent or transitory inflationary behavior. 
  Canonical quantization of the polarized Gowdy midi-superspace with a 3-torus spatial topology is carried out. As in an earlier work on the Einstein-Rosen cylindrical waves, symmetry reduction is used to cast the original problem in 4-dimensional space-times to a 3-dimensional setting. To our knowledge, this is the first complete, systematic treatment of the Gowdy model in the geometrodynamical setting. 
  Twins travelling at constant relative velocity will each see the other's time dilate leading to the apparent paradox that each twin believes the other ages more slowly. In a finite space, the twins can both be on inertial, periodic orbits so that they have the opportunity to compare their ages when their paths cross. As we show, they will agree on their respective ages and avoid the paradox. The resolution relies on the selection of a preferred frame singled out by the topology of the space. 
  In this paper we consider the collision of spinning holes using first order perturbation theory of black holes (Teukolsky formalism). With these results (along with ones, we published in the past) one can predict the properties of the gravitational waves radiated from the late stage inspiral of two spinning, equal mass black holes. Also we note that the energy radiated by the head-on collision of two spinning holes with spins (that are equal and opposite) aligned along the common axis is more than the case in which the spins are perpendicular to the axis of the collision. 
  We show that the bubbles $S^2\times S^2$can be created from vacuum fluctuation in certain De Sitter universe, so the space-time foam-like structure might really be constructed from bubbles of $S^2\times S^2$ in the very early inflating phase of our universe. But whether such foam-like structure persisted during the later evolution of the universe is a problem unsolved now. 
  A homogeneous and isotropic model of the Universe is considered in the framework of the five-dimensional Projective Unified Field Theory in which the gravitation is described by both space-time curvature and some hypothetical scalar field (sigma-field). We propose a generation method for obtaining exact solutions. New exact Friedmann-like solutions for a dust model and inflationary solutions are found. It is shown that in the framework of exponential type inflation we obtain a natural explanation of why at present we do not observe sigma-field effects or why these effects are negligible. 
  Sufficient conditions for the well-posedness of the initial value problem for the scalar wave equation are obtained in space-times with hypersurface singularities 
  We study the evolution of tensor metric fluctuations in a class of non-singular models based on the string effective action, by including in the perturbation equation the higher-derivative and loop corrections needed to regularise the background solutions. We discuss the effects of such higher-order corrections on the final graviton spectrum, and we compare the results of analytical and numerical computations. 
  The spacetime Ehlers group, which is a symmetry of the Einstein vacuum field equations for strictly stationary spacetimes, is defined and analyzed in a purely spacetime context (without invoking the projection formalism). In this setting, the Ehlers group finds its natural description within an infinite dimensional group of transformations that maps Lorentz metrics into Lorentz metrics and which may be of independent interest. The Ehlers group is shown to be well defined independently of the causal character of the Killing vector (which may become null on arbitrary regions). We analyze which global conditions are required on the spacetime for the existence of the Ehlers group. The transformation law for the Weyl tensor under Ehlers transformations is explicitly obtained. This allows us to study where, and under which circumstances, curvature singularities in the transformed spacetime will arise. The results of the paper are applied to obtain a local characterization of the Kerr-NUT metric. 
  Based on a geometrical property which holds both for the Kerr metric and for the Wahlquist metric we argue that the Kerr metric is a vacuum subcase of the Wahlquist perfect-fluid solution. The Kerr-Newman metric is a physically preferred charged generalization of the Kerr metric. We discuss which geometric property makes this metric so special and claim that a charged generalization of the Wahlquist metric satisfying a similar property should exist. This is the Wahlquist-Newman metric, which we present explicitly in this paper. This family of metrics has eight essential parameters and contains the Kerr-Newman-de Sitter and the Wahlquist metrics, as well as the whole Pleba\'nski limit of the rotating C-metric, as particular cases. We describe the basic geometric properties of the Wahlquist-Newman metric, including the electromagnetic field and its sources, the static limit of the family and the extension of the spacetime across the horizon. 
  The cosmological origin of $\gamma$-ray bursts (GRBs) is now commonly accepted and, according to several models for the central engine, GRB sources should also emit at the same time gravitational waves bursts (GWBs). We have performed two correlation searches between the data of the resonant gravitational wave detector AURIGA and GRB arrival times collected in the BATSE 4B catalog. No correlation was found and an upper limit \bbox{$h_{\text{RMS}} \leq 1.5 \times 10^{-18}$} on the averaged amplitude of gravitational waves associated with $\gamma$-ray bursts has been set for the first time. 
  An unusual set of orbits about extreme Kerr black holes resides at the Boyer-Lindquist radius $r = M$, the coordinate of the hole's event horizon. These ``horizon skimming'' orbits have the property that their angular momentum $L_z$ {\it increases} with inclination angle, opposite to the familiar behavior one encounters at larger radius. In this paper, I show that this behavior is characteristic of a larger family of orbits, the ``nearly horizon skimming'' (NHS) orbits. NHS orbits exist in the very strong field of any black hole with spin $a\agt 0.952412M$. Their unusual behavior is due to the locking of particle motion near the event horizon to the hole's spin, and is therefore a signature of the Kerr metric's extreme strong field. An observational hallmark of NHS orbits is that a small body spiraling into a Kerr black hole due to gravitational-wave emission will be driven into orbits of progressively smaller inclination angle, toward the equator. This is in contrast to the ``normal'' behavior. For circular orbits, the change in inclination is very small, and unlikely to be of observational importance. I argue that the change in inclination may be considerably larger when one considers the evolution of inclined eccentric orbits. If this proves correct, then the gravitational waves produced by evolution through the NHS regime may constitute a very interesting and important probe of the strong-field nature of rotating black holes. 
  In this comment, I argue that chaotic effects in binary black hole inspiral will not strongly impact the detection of gravitational waves from such systems. 
  It is well known that Einstein General Relativity can be expressed covariantly in bi-metric spacetime, without the uncertainties which arise from the effects of gravitational energy-momentum pseudo-tensors. However the effect that the Strong Principle of Equivalence (SPOE) has on the curved and flat bi-metric spacetime tetrad structure has not been fully explored. In the context of bi-metric General Relativity the (SPOE) requires that: a) in the absence of gravitation due to spacetime curvature a Global Inertial Cartesian Minkowski (GICM) spacetime frame exists in which Special Relativity is valid and, b) in the presence of gravitation due to spacetime curvature a bi-metric Local Free Fall frame (LFF) must always exist. These two conditions required by the (SPOE) are then shown to imply that a symmetric gravitational potential tensor must exist in the bi-metric spacetime which is connected exponentially to the tetrad inner product between the curved symmetric tetrads and the flat background symmetric tetrads. It is then shown that this (SPOE) generated gravitational potential tensor has the property of exponentially connecting the curved metric tensor with the flat background metric tensor in a manner which allows the condition of "Light Cone Causality" (LCC) to be dynamically satisfied in the bi-metric spacetime. Substitution of this (LCC) conserving exponential metric into the bi-metric Einstein field equations, subject to an appropriate choice of tensor gauge conditions, yields an equivalent set of (LCC) conserving bi-metric Einstein field equations for the gravitational potential tensor, the full implications of which will be explored in future papers. 
  Incorporation of the Vaidya metric in the model of Expansive Nondecelerative Universe allows to precisely localize gravitational energy for weak fields and obtain the components of the Einstein energy-momentum pseudotensor for strong gravitational fields. The components are identical to those calculated by Virbhadra. 
  Incorporation of the Vaidya metric in the model of Expansive Nondecelerative Universe allows to localize the energy density of gravitational field that, subsequently, enables to determine the upper limit of stars mass. The upper limit decreases with cosmological time and at present is close to 30-fold of our Sun mass. 
  We briefly review past applications of Regge calculus in classical numerical relativity, and then outline a programme for the future development of the field. We briefly describe the success of lattice gravity in constructing initial data for the head-on collision of equal mass black holes, and discuss recent results on the efficacy of Regge calculus in the continuum limit. 
  The exact charged rotating solution of 2+1 Einstein-Maxwell equations with $\Lambda$ term is obtained and its properties outlined. It generalizes the Cataldo-Cruz-del Campo-Garc{\'\i}a relativistic charged massive black hole on the 2+1 anti-de Sitter cosmological background. We show that rotating solutions correspond to inhomogeneous field equations, thus presence of sources in 2+1 Maxwell's equations cannot be identified with existence of a charge distribution. Instead, these sources are related to the 2+1 Machian 2-form field, and the overall Lagrangian structure of the rotating system is reconstructed. 
  The tiny general relativistic Lense-Thirring effect can be measured by means of a suitable combination of the orbital residuals of the nodes of LAGEOS and LAGEOS II and the perigee of LAGEOS II. This observable is affected, among other factors, by the Earth' s solid and ocean tides. They induce long-period orbital perturbations that, over observational periods of few years, may alias the detection of the gravitomagnetic secular trend of interest. In this paper we calculate explicitly the most relevant tidal perturbations acting upon LAGEOSs and assess their influence on the detection of the Lense-Thirring effect. The present day level of knowledge of the solid and ocean tides allow us to conclude that their influence on it ranges from almost 4% over 4 years to less than 2% over 7 years. 
  Using the quantum Hamiltonian for a gravitational system with boundary, we find the partition function and derive the resulting thermodynamics. The Hamiltonian is the boundary term required by functional differentiability of the action for Lorentzian general relativity. In this model, states of quantum geometry are represented by spin networks. We show that the statistical mechanics of the model reduces to that of a simple non-interacting gas of particles with spin. Using both canonical and grand canonical descriptions, we investigate two temperature regimes determined by the fundamental constant in the theory, m. In the high temperature limit (kT > m), the model is thermodynamically stable. For low temperatures (kT < m) and for macroscopic areas of the bounding surface, the entropy is proportional to area (with logarithmic correction), providing a simple derivation of the Bekenstein-Hawking result. By comparing our results to known semiclassical relations we are able to fix the fundamental scale. Also in the low temperature, macroscopic limit, the quantum geometry on the boundary forms a `condensate' in the lowest energy level (j=1/2). 
  Two operators for quantum gravity, angle and quasilocal energy, are briefly reviewed. The requirements to model semi-classical angles are discussed. To model semi-classical angles it is shown that the internal spins of the vertex must be very large, ~10^20. 
  We consider a general two-dimensional gravity model minimally or nonminimally coupled to a scalar field. The canonical form of the model is elucidated, and a general solution of the equations of motion in the massless case is reviewed. In the presence of a scalar field all geometric fields (zweibein and Lorentz connection) are excluded from the model by solving exactly their Hamiltonian equations of motion. In this way the effective equations of motion and the corresponding effective action for a scalar field are obtained. It is written in a Minkowskian space-time and does not include any geometric variables. The effective action arises as a boundary term and is nontrivial both for open and closed universes. The reason is that unphysical degrees of freedom cannot be compactly supported because they must satisfy the constraint equation. As an example we consider spherically reduced gravity minimally coupled to a massless scalar field. The effective action is used to reproduce the Fisher and Roberts solutions. 
  We present a gauge theory of the super SL(2,C) group. The gauge potential is a connection of the Super SL(2,C) group. A MacDowell-Mansouri type of action is proposed where the action is quadratic in the Super SL(2,C) curvature and depends purely on gauge connection. By breaking the symmetry of the Super SL(2,C) topological gauge theory to SL(2,C), a metric is naturally defined. 
  Using the Noether Charge formulation, we study a perturbation of the conserved gravitating system. By requiring the boundary term in the variation of the Hamiltonian to depend only on the symplectic structure, we propose a general prescription for defining quasi-local ``conserved quantities'' (i.e. in the situation when the gravitating system has a non-vanishing energy flux). Applications include energy-momentum and angular momentum at spatial and null infinity, asymptotically anti-deSitter spacetimes, and thermodynamics of the isolated horizons. 
  We study symmetry properties of the Einstein-Maxwell theory nonminimaly coupled to the dilaton field. We consider a static case with pure electric (magnetic) Maxwell field and show that the resulting system becomes a nonlinear sigma-model wich possesses a chiral representation. We construct the corresponding chiral matrix and establish a representation which is related to the pair of Ernst-like potentials. These potentials are used for separation of the symmetry group into the gauge and nongauge (charging) sectors. New variables, which linearize the action of charging symmetries, are also established; a solution generation technique based on the use of charging symmetries is formulated. This technique is used for generation of the elecricaly (magneticaly) charged dilatonic fields from the static General Relativity ones. 
  The inverse scattering problem method application to construction of exact solution for Maxwell dilaton gravity system ia considered. By use of Belinsky and Zakharov L - A pair the solution of the theory is constructed. The rotating Kerr - like configuration with NUT - parameter is obtained. 
  Torsion appears in literature in quite different forms. Generally, spin is considered to be the source of torsion, but there are several other possibilities in which torsion emerges in different contexts. In some cases a phenomenological counterpart is absent, in some other cases torsion arises from sources without spin as a gradient of a scalar field. Accordingly, we propose two classification schemes. The first one is based on the possibility to construct torsion tensors from the product of a covariant bivector and a vector and their respective space-time properties. The second one is obtained by starting from the decomposition of torsion into three irreducible pieces. Their space-time properties again lead to a complete classification. The classifications found are given in a U_4, a four dimensional space-time where the torsion tensors have some peculiar properties. The irreducible decomposition is useful since most of the phenomenological work done for torsion concerns four dimensional cosmological models. In the second part of the paper two applications of these classification schemes are given. The modifications of energy-momentum tensors are considered that arise due to different sources of torsion. Furthermore, we analyze the contributions of torsion to shear, vorticity, expansion and acceleration. Finally the generalized Raychaudhuri equation is discussed. 
  We investigate the nature of ordinary cosmic vortices in some scalar-tensor extensions of gravity. We find solutions for which the dilaton field condenses inside the vortex core. These solutions can be interpreted as raising the degeneracy between the eigenvalues of the effective stress-energy tensor, namely the energy per unit length U and the tension T, by picking a privileged spacelike or timelike coordinate direction; in the latter case, a phase frequency threshold occurs that is similar to what is found in ordinary neutral current-carrying cosmic strings. We find that the dilaton contribution for the equation of state, once averaged along the string worldsheet, vanishes, leading to an effective Nambu-Goto behavior of such a string network in cosmology, i.e. on very large scales. It is found also that on small scales, the energy per unit length and tension depend on the string internal coordinates in such a way as to permit the existence of centrifugally supported equilibrium configuration, also known as vortons, whose stability, depending on the very short distance (unknown) physics, can lead to catastrophic consequences on the evolution of the Universe. 
  Static black holes of dilaton-axion gravity become singular in the extreme limit, which prevents a direct determination of their near-horizon geometry. This is addressed by first taking the near-horizon limit of extreme rotating NUT-less black holes, and then going to the static limit. The resulting four-dimensional geometry may be lifted to a Bertotti-Robinson-like solution of six-dimensional vacuum gravity, which also gives the near-horizon geometry of extreme Kaluza-Klein black holes in five dimensions. 
  Levin (gr-qc/9910040) has shown that spinning compact binaries can be chaotic at second post-Newtonian order. However, when higher order dissipational effects are included, the dynamics will no longer be chaotic, though the evolution may still be unpredictable in a practical sense. I discuss some of the additional work that needs to be done to decide how this unpredictability might affect gravitational wave detectors such as LIGO. 
  We present a review of the spin and statistics of topological geons, particles in 3+1 quantum gravity. They can have half-odd-integral spin and fermionic statistics and since the underlying gravitational field is tensorial and bosonic, this is an example of ``emergent'' non-trivial spin and statistics as displayed by familiar non-gravitating objects such as skyrmions. We give the topological background and show that in a ``canonical'' quantization of gravity there is no spin-statistics correlation for topological geons. Allowing the topology of space to change, for example in a sum-over-histories approach, raises the possibility that a spin-statistics correlation can be recovered for geons. We review a conjectured set of rules powerful enough to give such a spin-statistics correlation for all topological geons. These would appear to rule out the possibility of parastatistics and may rule out spinorial and fermionic geons altogether. 
  In a multidimensional model with several scalar fields and an m-form we deal with classical spherically symmetric solutions with one (electric or magnetic) p-brane and Ricci-flat internal spaces and the corresponding solutions to the Wheeler--DeWitt (WDW) equation. Classical black holes are considered and their quantum analogues (e.g. for M2 and M5 extremal solutions in D =11 supergravity, electric and magnetic charges in D=4 gravity) are suggested when the curvature coupling in the WDW equation is zero. 
  Black hole p-brane solutions for a wide class of intersection rules are considered. The solutions are defined on a manifold which contains a product of n-1 Ricci-flat "internal'' spaces. The post-Newtonian parameters "beta" and "gamma" corresponding to a 4-dimensional section of the metric for general intersection rules are studied. It is shown that "beta" does not depend but "gamma" depends on brane intersections. For "block-orthogonal" intersection rules spherically symmetric solutions are considered, and explicit relations for post-Newtonian parameters are obtained. The bounds on parameters of solutions following from observational restrictions in the Solar system are presented. 
  Newtonian point mass binaries can be brought into arbitrarily close circular orbits. Neutron stars and black holes, however, are extended, relativistic objects. Both finite size and relativistic effects make very close orbits unstable, so that there exists an innermost stable circular orbit (ISCO). We illustrate the physics of the ISCO in a simple model problem, and review different techniques which have been employed to locate the ISCO in black hole and neutron star binaries. We discuss different assumptions and approximations, and speculate on how differences in the results may be explained and resolved. 
  A brief review of the modern state of quantum cosmology is presented as a theory of quantum initial conditions for inflationary scenario. The no-boundary and tunneling states of the Universe are discussed as a possible source of probability peaks in the distribution of initial data for inflation. It is emphasized that in the tree-level approximation the existence of such peaks is in irreconcilable contradiction with the slow roll regime -- the difficulty that is likely to be solved only on account of quantum gravitational effects. The low-energy (typically GUT scale) mechanism of quantum origin of the inflationary Universe with observationally justified parameters is presented for closed and open inflation models with a strong non-minimal coupling. 
  We discuss a sequence of numerically constructed geometries describing binary black hole event horizons -- providing the necessary input for characteristic evolution of the exterior spacetime. Our sequence approaches a single Schwarzschild horizon as one limiting case and also includes cases where the horizon's crossover surface is not hidden by a marginally anti-trapped surface (MATS). 
  In a special class of globally hyperbolic, topologically trivial, asymptotically flat at spatial infinity spacetimes selected by the requirement of absence of supertranslations (compatible with Christodoulou-Klainermann spacetimes) it is possible to define the {\it rest-frame instant form} of ADM canonical gravity by using Dirac's strategy of adding ten extra variables at spatial infinity and ten extra first class constraints implying the gauge nature of these variables. The final canonical Hamiltonian is the weak ADM energy and a discussion of the Hamiltonian gauge transformations generated by the eight first class ADM constraints is given. When there is matter and the Newton constant is switched off, one recovers the description of the matter on the Wigner hyperplanes of the rest-frame instant form of dynamics in Minkowski spacetime. 
  We analyze the effective action describing the linearised gravitational self-action for a classical superconducting string in a curved spacetime. It is shown that the divergent part of the effective action is equal to zero for the both Nambu-Goto and chiral superconducting string. 
  We present a cosmological model for early stages of the universe on the basis of a Weyl-Cartan spacetime. In this model, torsion $T^{\alpha}$ and nonmetricity $Q_{\alpha \beta}$ are proportional to the vacuum polarization. Extending earlier work of one of us (RT), we discuss the behavior of the cosmic scale factor and the Weyl 1-form in detail. We show how our model fits into the more general framework of metric-affine gravity (MAG). 
  We derive a detection method for a stochastic background of gravitational waves produced by events where the ratio of the average time between events to the average duration of an event is large. Such a signal would sound something like popcorn popping. Our derivation is based on the somewhat unrealistic assumption that the duration of an event is smaller than the detector time resolution. 
  We calculate the quasinormal modes and associated frequencies of the Banados, Zanelli and Teitelboim (BTZ) non-rotating black hole. This black hole lives in 2+1-dimensions in an asymptotically anti-de Sitter spacetime. We obtain exact results for the wavefunction and quasi normal frequencies of scalar, electromagnetic and Weyl (neutrino) perturbations. 
  The most general Lagrangian for dynamical torsion theory quadratic in curvature and torsion is considered. We impose two simple and physically reasonable constraints on the solutions of the equations of motion: (i) there must be solutions with zero curvature and nontrivial torsion and (ii) there must be solutions with zero torsion and non covariantly constant curvature. The constraints reduce the number of independent coupling constants from ten to five. The resulting theory contains Einstein's general relativity and Weitzenbock's absolute parallelism theory as the two sectors. 
  A combination of qualitative analysis and numerical study indicates that vacuum $T^2$ symmetric spacetimes are, generically, oscillatory. 
  Starting from the Frolov-Zel'nikov stress-energy tensor of quantum massive fields in the Schwarzschild background, we recover the contribution $S_{q}$ of these field into the entropy of a black hole. For fermions with the spin $%s=1/2$ $S_{q}>0$, for scalar fields $S_{q}>0$ provided the coupling parameter is restricted to some interval, and $S_{q}<0$ for vector fields. The appearance of negative values of $S_{q}$ is attributed to the fact that in the situation under discussion there are no real quanta to contribute to the entropy, so $S_{q}$ is due to vacuum polarization entirely and has nothing to do with the statistical-mechanical entropy. We also consider the spacetime with an acceleration horizon - the Bertotti-Robison spacetime - and show that $S_{q}=0$ for massive fields similarly to what was proved earlier for massless fields. 
  In gr-qc/9908036 [Phys. Lett. A 265 (2000) 1] a new method was given which naturally led to a quantum of mass equal to twice the Planck mass. In the present note which, for convenience, we write formally as a continuation of that paper, we show that with spin one of the mass quantum, the physical entropy of a rotating black hole is also given by the Bekenstein-Hawking formula. 
  We analize the relational quantum evolution of generally covariant systems in terms of Rovelli's evolving constants of motion and the generalized Heisenberg picture. In order to have a well defined evolution, and a consistent quantum theory, evolving constants must be self-adjoint operators. We show that this condition imposes strong restrictions to the choices of the clock variables. We analize four cases. The first one is non- relativistic quantum mechanics in parametrized form. We show that, for the free particle case, the standard choice of time is the only one leading to self-adjoint evolving constants. Secondly, we study the relativistic case. We show that the resulting quantum theory is the free particle representation of the Klein Gordon equation in which the position is a perfectly well defined quantum observable. The admissible choices of clock variables are the ones leading to space-like simultaneity surfaces. In order to mimic the structure of General Relativity we study the SL(2R) model with two Hamiltonian constraints. The evolving constants depend in this case on three independent variables. We show that it is possible to find clock variables and inner products leading to a consistent quantum theory. Finally, we discuss the quantization of a constrained model having a compact constraint surface. All the models considered may be consistently quantized, although some of them do not admit any time choice such that the equal time surfaces are transversal to the orbits. 
  The inclusion of a flat metric tensor in gravitation permits the formulation of a gravitational stress-energy tensor and the formal derivation of general relativity from a linear theory in flat spacetime. Building on the works of Kraichnan and Deser, we present such a derivation using universal coupling and gauge invariance.   Next we slightly weaken the assumptions of universal coupling and gauge invariance, obtaining a larger ``slightly bimetric'' class of theories, in which the Euler-Lagrange equations depend only on a curved metric, matter fields, and the determinant of the flat metric. The theories are equivalent to generally covariant theories with an arbitrary cosmological constant and an arbitrarily coupled scalar field, which can serve as an inflaton or dark matter.   The question of the consistency of the null cone structures of the two metrics is addressed. A difficulty for Logunov's massive gravitation on this front is noted. 
  Recent developments in string theory have led to 5-dimensional warped spacetime models in which standard-model fields are confined to a 3-brane (the observed universe), while gravity can propagate in the fifth dimension. Gravity is localized near the brane at low energies, even if the extra dimension is noncompact. A review is given of the classical geometry and dynamics of these brane-world models. The field equations on the brane modify the general relativity equations in two ways: local 5-D effects are imprinted on the brane as a result of its embedding, and are significant at high energies; nonlocal effects arise from the 5-D Weyl tensor. The Weyl tensor transmits tidal (Coulomb), gravitomagnetic and gravitational wave effects to the brane from the 5-D nonlocal gravitational field. Local high-energy effects modify the dynamics of inflation, and increase the amplitude of scalar and tensor perturbations generated by inflation. Nonlocal effects introduce new features in cosmological perturbations. They induce a non-adiabatic mode in scalar perturbations and massive modes in vector and tensor perturbations, and they can support vector perturbations even in the absence of matter vorticity. In astrophysics, local and nonlocal effects introduce fundamental changes to gravitational collapse and black hole solutions. 
  We consider ``brane universe'' with nonzero tension in the models with large extra dimensions. We find exact solutions of higher-dimensional Einstein equations with single flat Minkowsky brane of arbitrary large tension (or brane cosmological constant) and compact extra dimensions. The brane curves the bulk space-time in a small region around it. There is no fine tuning of energy scales in our model. 
  Symmetric states are defined in the kinematical sector of loop quantum gravity and applied to spherical symmetry and homogeneity. Consequences for the physics of black holes and cosmology are discussed. 
  A two-components mixture fluid which complies with the gamma law is considered in the framework of inflation with finite temperature. The model is developed for a quartic scalar potential without symmetry breaking. The radiation energy density is assumed to be zero when inflation starts and remains below the GUT temperature during the inflationary stage. Furthermore, provides the necessary number of e-folds and sufficient radiation energy density to GUT baryogenesis can take place near the minimum energetic configuration. 
  We present approximate analytical solutions to the Hamiltonian and momentum constraint equations, corresponding to systems composed of two black holes with arbitrary linear and angular momentum. The analytical nature of these initial data solutions makes them easier to implement in numerical evolutions than the traditional numerical approach of solving the elliptic equations derived from the Einstein constraints. 
  We study the spherical collapse of a perfect fluid with an equation of state $P=k\rho$ by full general relativistic numerical simulations. For $0<k\alt 0.036$, it has been known that there exists a general relativistic counterpart of the Larson-Penston self-similar Newtonian solution. The numerical simulations strongly suggest that, in the neighborhood of the center, generic collapse converges to this solution in an approach to a singularity and that self-similar solutions other than this solution, including a ``critical solution'' in the black hole critical behavior, are relevant only when the parameters which parametrize initial data are fine-tuned. This result is supported by a mode analysis on the pertinent self-similar solutions. Since a naked singularity forms in the general relativistic Larson-Penston solution for $0<k\alt0.0105$, this will be the most serious known counterexample against cosmic censorship. It also provides strong evidence for the self-similarity hypothesis in general relativistic gravitational collapse. The direct consequence is that critical phenomena will be observed in the collapse of isothermal gas in Newton gravity, and the critical exponent $\gamma$ will be given by $\gamma\approx 0.11$, though the order parameter cannot be the black hole mass. 
  The initial data of the gravitational field produced by a loop thick string is considered. We show that a thick loop is not a geodesic on the initial hypersurface, while a loop conical singularity is. This suggests that there is the ``{\it critical thickness}'' of a string, at which the linear perturbation theory with a flat space background fails to describe the gravity of a loop cosmic string. Using the above initial data, we also show that the linear perturbation around flat space is plausible if the string thickness is larger than $\sim5\times10^{-3}a$, where $a$ is the curvature radius of the loop. 
  The Wheeler-DeWitt equation for the Bianchi Class A cosmological models is expressed generally in terms of the second-order differential equation like the Klein-Gordon equation. To obtain the positive-definite probability density, a new method extending the Dirac-Square-Root formalism, which factorizes the Wheeler-DeWitt equation into the first-order differential equation using the Pauli matrices, is investigated. The solutions to the Dirac type equation thus obtained are expressed in terms of two-component spinor form. The probability density defined by the solution is positive-definite and there is a conserved current. The newly found spin-like degree of freedom causes the universe to go through an early quantum stage of evolution with agitated anisotropy-oscillation like Zitterbewegung. 
  The spin-free binary-inspiral parameter-space introduced by Tanaka and Tagoshi to construct a uniformly-spaced lattice of templates at (and possibly beyond) $2.5PN$ order is shown to work for all first generation interferometric gravitational wave antennas. This allows to extend the minimum-redundant cardinal interpolation techniques of the correlator bank developed by the Authors to the highest available order PN templates. The total number of 2PN templates to be computed for a minimal match $\Gamma=0.97$ is reduced by a factor 4, as in the 1PN case. 
  This is a contribution to MG9 session BHT4. Certain geometrically distinguished frame on a non-expanding horizon and in its space-time neighborhood, as well as the Bondi-like coordinates are constructed. The construction provides free degrees of freedom, invariants, and the existence conditions for a Killing vector field. The reported results come from the joint works with Ashtekar and Beetle. 
  This is a contribution to the MG9 session QG1-a. A new quantum representation for the Lorentzian gravity is created from the Pullin vaccum by the operators assigned to 2-complexes. The representation uses the original, spinorial Ashtekar variables, the reality conditions are well posed and Thiemann's Hamiltonian is well defined. The results on the existence of a suitable Hilbert product are partial. They were derived in collaboration with Abhay Ashtekar. 
  We investigate the scale-dependence of Eulerian volume averages of scalar functions on Riemannian three-manifolds. We propose a complementary view of a Lagrangian scaling of variables as opposed to their Eulerian averaging on spatial domains. This program explains rigorously the origin of the Ricci deformation flow for the metric, a flow which, on heuristic grounds, has been already suggested as a possible candidate for averaging the initial data set for cosmological spacetimes. 
  One interesting class of gravitational radiation sources includes rapidly rotating astrophysical objects that encounter dynamical instabilities. We have carried out a set of simulations of rotationally induced instabilities in differentially rotating polytropes. An $n$=1.5 polytrope with the Maclaurin rotation law will encounter the $m$=2 bar instability at $T/|W| \gtrsim 0.27$. Our results indicate that the remnant of this instability is a persistent bar-like structure that emits a long-lived gravitational radiation signal. Furthermore, dynamical instability is shown to occur in $n$=3.33 polytropes with the $j$-constant rotation law at $T/|W| \gtrsim 0.14$. In this case, the dominant mode of instability is $m$=1. Such instability may allow a centrifugally-hung core to begin collapsing to neutron star densities on a dynamical timescale. If it occurs in a supermassive star, it may produce gravitational radiation detectable by LISA. 
  Using an orthonormal Lorentz frame approach to axistationary perfect fluid spacetimes, we have formulated the necessary and sufficient equations as a first order system, and investigated the integrability conditions of this set of equations. The integrability conditions are helpful tools when it comes to check the consequences and/or compatibility of certain simplifying assumptions, e.g. Petrov types. Furthermore, using this method, a relation between the fluid shear and vorticity is found for barotropic fluids. We collect some results concerning Petrov types, and it is found that an incompressible axistationary perfect fluid must be of Petrov type I. 
  We present an exact collapsing solution to 2+1 gravity with a negative cosmological constant minimally coupled to a massless scalar field, which exhibits physical properties making it a candidate critical solution. We discuss its global causal structure and its symmetries in relation with those of the corresponding continously self-similar solution derived in the $\Lambda=0$ case. Linear perturbations on this background lead to approximate black hole solutions. The critical exponent is found to be $\gamma = 2/5$. 
  The final burst of gravitational radiation emitted by coalescing binary neutron stars carries direct information about the neutron star fluid, and, in particular, about the equation of state of nuclear matter at extreme densities. The final merger may also be accompanied by a detectable electromagnetic signal, such as a gamma-ray burst. In this paper, we summarize the results of theoretical work done over the past decade that has led to a detailed understanding of this hydrodynamic merger process for two neutron stars, and we discuss the prospects for the detection and physical interpretation of the gravity wave signals by ground-based interferometers such as LIGO. We also present results from our latest post-Newtonian SPH calculations of binary neutron star coalescence, using up to 10^6 SPH particles to compute with higher spatial resolution than ever before the merger of an initially irrotational system. We discuss the detectability of our calculated gravity wave signals based on power spectra. 
  Here we place the Latex typeset of the paper M. Pavsic, Phys. Lett. A116 (1986) 1-5. In the paper we presented the picture that our spacetime is a 3-brane moving in a higher dimensional space. The dynamical equations were derived from the action which is just that for the usual Dirac-Nambu-Goto $p$-brane. We also considered the case where not only one, but many branes of various dimensionalities are present, and showed that their intersections with the 3-brane manifest as matter in 4-dimensional spacetime. We considered a particular case, where the intersections behaved as point particles, and found out that they follow the geodesics on the 3-brane worldsheet (identified with our spacetime). In a series of subsequent papers the original idea has been further improved and developped. This is discussed in a note at the end, where it is also pointed out that such a model resolves the problem of massive matter confinement on the brane, recently discussed by Rubakov et al. and Mueck et al. 
  Previous work in the literature has studied gravitational radiation in black-hole collisions at the speed of light. In particular, it had been proved that the perturbative field equations may all be reduced to equations in only two independent variables, by virtue of a conformal symmetry at each order in perturbation theory. The Green function for the perturbative field equations is here analyzed by studying the corresponding second-order hyperbolic operator with variable coefficients, instead of using the reduction method from the retarded flat-space Green function in four dimensions. After reduction to canonical form of this hyperbolic operator, the integral representation of the solution in terms of the Riemann function is obtained. The Riemann function solves a characteristic initial-value problem for which analytic formulae leading to the numerical solution are derived. 
  A covariant and invariant theory of navigation in curved space-time with respect to electromagnetic beacons is written in terms of J. L. Synge's two-point invariant world function. Explicit equations are given for navigation in space-time in the vicinity of the Earth in Schwarzschild coordinates and in rotating coordinates. The restricted problem of determining an observer's coordinate time when their spatial position is known is also considered. 
  We present solution generating methods which allow to construct exact static solutions to the equations of four-dimensional Einstein-Maxwell-Dilaton gravity starting with arbitrary static solutions to the pure vacuum Einstein equations, Einstein-dilaton or Einstein-Maxwell equations. 
  Scalar fields non--minimally coupled to (2+1)-gravity, in the presence of cosmological constant term, are considered. Non-minimal couplings are described by the term $\zeta R \Psi^2$ in the Lagrangian. Within a class of static circularly symmetric space-times, it is shown that the only existing physically relevant solutions are the anti-de Sitter space-time for $\zeta=0$, and the Martinez-Zanelli black hole for $\zeta=1/8$. We obtain also two new solutions with non-trivial scalar field, for $\zeta=1/6$ and $\zeta=1/8$ respectively, nevertheless, the corresponding space-times can be reduced, via coordinate transformations, to the standard anti-de Sitter space. 
  I describe the first steps in the construction of semiclassical states for non-perturbative canonical quantum gravity using ideas from classical, Riemannian statistical geometry and results from quantum geometry of spin network states. In particular, I concentrate on how those techniques are applied to the construction of random spin networks, and the calculation of their contribution to areas and volumes. 
  Using the analytic, global solution for the rigidly rotating disc of dust as a starting point, an iteration scheme is presented for the calculation of an arbitrary coefficient in the post-Newtonian (PN) approximation of this solution. The coefficients were explicitly calculated up to the 12th PN level and are listed in this paper up to the 4th PN level. The convergence of the series is discussed and the approximation is found to be reliable even in highly relativistic cases. Finally, the ergospheres are calculated at increasing orders of the approximation and for increasingly relativistic situations. 
  A quantum causal topology is presented. This is modeled after a non-commutative scheme type of theory for the curved finitary spacetime sheaves of the non-abelian incidence Rota algebras that represent `gravitational quantum causal sets'. The finitary spacetime primitive algebra scheme structures for quantum causal sets proposed here are interpreted as the kinematics of a curved and reticular local quantum causality. Dynamics for quantum causal sets is then represented by appropriate scheme morphisms, thus it has a purely categorical description that is manifestly `gauge-independent'. Hence, a schematic version of the Principle of General Covariance of General Relativity is formulated for the dynamically variable quantum causal sets. We compare our non-commutative scheme-theoretic curved quantum causal topology with some recent $C^{*}$-quantale models for non-abelian generalizations of classical commutative topological spaces or locales, as well as with some relevant recent results obtained from applying sheaf and topos-theoretic ideas to quantum logic proper. Motivated by the latter, we organize our finitary spacetime primitive algebra schemes of curved quantum causal sets into a topos-like structure, coined `quantum topos', and argue that it is a sound model of a structure that Selesnick has anticipated to underlie Finkelstein's reticular and curved quantum causal net. At the end we conjecture that the fundamental quantum time-asymmetry that Penrose has expected to be the main characteristic of the elusive `true quantum gravity' is possibly of a kinematical or structural rather than of a dynamical character, and we also discuss the possibility of a unified description of quantum logic and quantum gravity in quantum topos-theoretic terms. 
  We construct static and spherically symmetric particle-like and black hole solutions with magnetic and/or electric charge in the Einstein-Born-Infeld-dilaton-axion system, which is a generalization of the Einstein-Maxwell-dilaton-axion (EMDA) system and of the Einstein-Born-Infeld (EBI) system. They have remarkable properties which are not seen for the corresponding solutions in the EMDA and the EBI system. 
  Main results concerning allowable additional symmetries of axially symmetric electrovacuum spacetimes are summarized. These are translational Killing vectors and the boost Killing vector. However, this is only the boost symmetry that does not exclude radiation and permit a spacetime to be asymptotically flat with global null infinity. 
  It is well known that all curvature invariants of the order zero vanish for type-III and type-N vacuum spacetimes. We briefly summarize properties of higher order curvature invariants for these spacetimes. 
  Static, cylindrically symmetric solutions to nonlinear scalar-Einstein equations are considered. Regularity conditions on the symmetry axis and flat or string asymptotic conditions are formulated in order to select soliton-like solutions. Some non-existence theorems are proved, in particular, theorems asserting (i) the absence of black-hole and wormhole-like cylindrically symmetric solutions for any static scalar fields minimally coupled to gravity and (ii) the absence of solutions with a regular axis for scalar fields with the Lagrangian $L=F(I)$, $I=\phi^\alpha \phi_\alpha$, for any function $F(I)$ possessing a correct weak field limit. Exact solutions for scalar fields with an arbitrary potential function $V(\phi)$ are obtained by quadratures and are expressed in a parametric form in a few ways, where the parameter may be either the coordinate $x$, or the $\phi$ field, or one of the metric coefficients. Soliton-like solutions are shown to exist only with $V(\phi)$ having a variable sign. Some explicit examples of solutions (including a soliton-like one) and their flat-space limit are discussed.} 
  A two-component formulation of the Klein-Gordon equation is used to investigate the cyclic and noncyclic adiabatic geometric phases due to spatially homogeneous (Bianchi) cosmological models. It is shown that no adiabatic geometric phases arise for Bianchi type I models. For general Bianchi type IX models the problem of the adiabatic geometric phase is shown to be equivalent to the one for nuclear quadrupole interactions of a spin. For these models nontrivial non-Abelian adiabatic geometrical phases may occur in general. 
  Paper withdrawn. Replaced by by gr-qc/0212077 . 
  As a consequence of gravitomagnetism, which is a fundamental weak-field prediction of general relativity and ubiquitous in gravitational phenomena, clocks show a difference in their proper periods when moving along identical orbits in opposite directions about a spinning mass. This time shift is induced by the rotation of the source and may be used to verify the existence of the terrestrial gravitomagnetic field by means of orbiting clocks. A possible mission scenario is outlined with emphasis given to some of the major difficulties which inevitably arise in connection with such a venture. 
  For a spatially flat Friedmann model with line element $ds^2=a^2 [ da^2/B(a)-dx^2-dy^2-dz^2 ] $, the 00-component of the Einstein field equation reads $8\pi G T_{00}=3/a^2$ containing no derivative. For a nonlinear Lagrangian ${\cal L}(R)$, we obtain a second--order differential equation for $B$ instead of the expected fourth-order equation. We discuss this equation for the curvature-saturated model proposed by Kleinert and Schmidt. Finally, we argue that asymptotic freedom $G_{{\rm eff}}^{-1}\to 0$ is fulfilled in curvature-saturated gravity. 
  The work on black holes immersed in external fields is reviewed in both test-field approximation and within exact solutions. In particular we pay attention to the effect of the expulsion of the flux of external fields across charged and rotating black holes which are approaching extremal states. Recently this effect has been shown to occur for black hole solutions in string theory. We also discuss black holes surrounded by rings and disks and rotating black holes accelerated by strings. 
  We develop the formalism required to study the nonlinear interaction of modes in rotating Newtonian stars in the weakly nonlinear regime. The formalism simplifies and extends previous treatments. At linear order, we elucidate and extend slightly a formalism due to Schutz, show how to decompose a general motion of a rotating star into a sum over modes, and obtain uncoupled equations of motion for the mode amplitudes under the influence of an external force. Nonlinear effects are added perturbatively via three-mode couplings. We describe a new, efficient way to compute the coupling coefficients, to zeroth order in the stellar rotation rate, using spin-weighted spherical harmonics.  We apply this formalism to derive some properties of the coupling coefficients relevant to the nonlinear interactions of unstable r-modes in neutron stars, postponing numerical integrations of the coupled equations of motion to a later paper. From an astrophysical viewpoint, the most interesting result of this paper is that many couplings of r-modes to other rotational modes (modes with zero frequencies in the non-rotating limit) are small: either they vanish altogether because of various selection rules, or they vanish to lowest order in the angular velocity. In zero-buoyancy stars, the coupling of three r-modes is forbidden entirely and the coupling of two r-modes to one hybrid rotational mode vanishes to zeroth order in rotation frequency. In incompressible stars, the coupling of any three rotational modes vanishes to zeroth order in rotation frequency. 
  From the relativistic law of motion we attempt to deduce the field theories corresponding to the force law being linear and quadratic in 4-velocity of the particle. The linear law leads to the vector gauge theory which could be the abelian Maxwell electrodynamics or the non-abelian Yang-Mills theory. On the other hand the quadratic law demands spacetime metric as its potential which is equivalent to demanding the Principle of Equivalence. It leads to the tensor theory of gravitational field -- General Relativity. It is remarkable that a purely dynamical property of the force law leads uniquely to the corresponding field theories. 
  It is shown that a change in the signature of the space-time metric together with compactification of internal dimensions could occure in a six-dimensional cosmological model. We also show that this is due to interaction with Maxwell fields having support in the internal part of the space-time. 
  The dynamical equations which are basic for the description of the dynamics of quantum felds in arbitrary space--time geometries, can be derived from the requirements of a unique deterministic evolution of the quantum fields, the superposition principle, a finite propagation speed, and probability conservation. We suggest and describe observations and experiments which are able to test the unique deterministic evolution and analyze given experimental data from which restrictions of anomalous terms violating this basic principle can be concluded. One important point is, that such anomalous terms are predicted from loop gravity as well as from string theories. Most accurate data can be obtained from future astrophysical observations. Also, laboratory tests like spectroscopy give constraints on the anomalous terms. 
  We examine some kinds of discrete symmetries which are dynamically preserved, using the (generalized) Gowdy models of the first kind. 
  General relativity can be formally derived as a flat spacetime theory, but the consistency of the resulting curved metric's light cone with the flat metric's null cone has not been adequately considered. If the two are inconsistent, then gravity is not just another field in flat spacetime after all.  Here we discuss recent progress in describing the conditions for consistency and prospects for satisfying those conditions. 
  Topological defects experimentally induced by rotational dynamics in a continuous media replicate the coherent structure features of cosmic strings as well as hadrons. 
  Paralleling the formal derivation of general relativity as a flat spacetime theory, we introduce in addition a preferred temporal foliation. The physical interpretation of the formalism is considered in the context of 5-dimensional ``parametrized'' and 4-dimensional preferred frame contexts. In the former case, we suggest that our earlier proposal of unconcatenated parametrized physics requires that the dependence on $\tau$ be rather slow. In the 4-dimensional case, we consider and tentatively reject several areas of physics that might require a preferred foliation, but find a need for one in the process (``flowing'') theory of time. We then suggest why such a foliation might reasonably be unobservable. 
  Classical singularities inside black holes in the Einstein-Yang-Mills theory exhibit unusual features. Only for discrete values of the black hole mass one encounters singularities of the Schwarzschild type (timelike) and the Reissner-Nordstrom type (spacelike). For a generic mass the approach to singularity is not smooth: the metric oscillates with an infinitely growing amplitude and decreasing period. In spite of some similarity with the BKL oscillations, here the behavior is not chaotic. However the oscillation amplitude exceeds classical limits after few cycles, so the question arises how this behavior gets modified by quantum effects. We discuss this issue both in the framework of QFT and in the string theory. 
  Born-Infeld generalization of the Yang-Mills action suggested by the superstring theory gives rise to modification of previously known as well as to some new classical soliton solutions. Earlier it was shown that within the model with the usual trace over the group generators classical glueballs exist which form an infinite sequence similar to the Bartnik-McKinnon family of the Einstein-Yang-Mills solutions. Here we give the generalization of this result to the 'realistic' model with the symmetrized trace and show the existence of excited monopoles (in presence of triplet Higgs) which can be regarded as a non-linear superposition of monopoles and sphalerons. 
  Smooth spacetimes with a compact Cauchy horizon ruled by closed null geodesics are considered. The compact Cauchy horizon is assumed to be non-degenerate. Then, supporting the validity of Penrose's strong cosmic censor hypothesis, the existence of a smooth Killing vector field in a neighbourhood of the horizon on the Cauchy development side is shown. 
  The quantization of Class A Bianchi Type VI and VII geometries -with all six scale factors, as well as the lapse function and the shift vector present- is considered. A first reduction of the initial 6-dimensional configuration space is achieved by the usage of the information furnished by the quantum form of the linear constraints. Further reduction of the space in which the wave function -obeying the Wheeler-DeWitt equation- lives, is accomplished by revealing a classical integral of motion, tantamount to an extra symmetry of the corresponding classical Hamiltonian. This symmetry generator -member of a larger group- is linear in momenta and corresponds to G.C.T.s through the action of the automorphism group -especially through the action of the outer automorphism subgroup. Thus, a G.C.T. invariant wave function is found, which depends on one combination of the two curvature invariants --which uniquely and irreducibly characterizes the hypersurfaces t=const. 
  In the present work the detection, by means of a nondemolition measurement, of a Yukawa term, coexisting simultaneously with gravity, has been considered. In other words, a nondemolition variable for the case of a particle immersed in a gravitational field containing a Yukawa term is obtained. Afterwards the continuous monitoring of this nondemolition parameter is analyzed, the corresponding propagator is evaluated, and the probabilities associated with the possible measurement outputs are found. The relevance of these kind of proposals in connection with some unified theories of elementary particles has been underlined. 
  The axial modes for non-barotropic relativistic rotating neutron stars with uniform angular velocity are studied, using the slow-rotation formalism together with the low-frequency approximation, first investigated by Kojima. The time independent form of the equations leads to a singular eigenvalue problem, which admits a continuous spectrum. We show that for $l=2$, it is nevertheless also possible to find discrete mode solutions (the $r$-modes). However, under certain conditions related to the equation of state and the compactness of the stellar model, the eigenfrequency lies inside the continuous band and the associated velocity perturbation is divergent; hence these solutions have to be discarded as being unphysical. We corroborate our results by explicitly integrating the time dependent equations. For stellar models admitting a physical $r$-mode solution, it can indeed be excited by arbitrary initial data. For models admitting only an unphysical mode solution, the evolutions do not show any tendency to oscillate with the respective frequency. For higher values of $l$, it seems that in certain cases there are no mode solutions at all. 
  We consider the statistical mechanics of a general relativistic one-dimensional self-gravitating system. The system consists of $N$-particles coupled to lineal gravity and can be considered as a model of $N$ relativistically interacting sheets of uniform mass. The partition function and one-particle distitrubion functions are computed to leading order in $1/c$ where $c$ is the speed of light; as $c\to\infty$ results for the non-relativistic one-dimensional self-gravitating system are recovered. We find that relativistic effects generally cause both position and momentum distribution functions to become more sharply peaked, and that the temperature of a relativistic gas is smaller than its non-relativistic counterpart at the same fixed energy. We consider the large-N limit of our results and compare this to the non-relativistic case. 
  The evaluation of relativistic spin networks plays a fundamental role in the Barrett-Crane state sum model of Lorentzian quantum gravity in 4 dimensions. A relativistic spin network is a graph labelled by unitary irreducible representations of the Lorentz group appearing in the direct integral decomposition of the space of L^2 functions on three-dimensional hyperbolic space. To `evaluate' such a spin network we must do an integral; if this integral converges we say the spin network is `integrable'. Here we show that a large class of relativistic spin networks are integrable, including any whose underlying graph is the 4-simplex (the complete graph on 5 vertices). This proves a conjecture of Barrett and Crane, whose validity is required for the convergence of their state sum model. 
  We study a minisuperspace quantum cosmology for a 2+1 dimensional de Sitter universe and find the wave function both exactly and in WKB approximation. Then we extend the model to a canonically quantized field theory for quantum gravity, i.e., a midisuperspace, and obtain the wave functional of the resulting field theory in the saddle point approximation. It is shown that these two approaches yield different results. 
  A formal correspondence is established between the curvature theory of generalized implicit hypersurfaces, electromagnetism as expressed in terms of exterior differential systems, and thermodynamics. Starting with a generalized implicit surface whose normal field is represented by an exterior differential 1-form, it is possible to deduce the curvature invariants of the implicit surface and to construct a globally closed vector density in terms of the Jacobian properties of the normal field. When the closed vector density is assigned the role of an intrinsic charge current density, and the components of the normal field are assigned the roles of the electromagnetic potentials, the theory is formally equivalent to an exterior differential system that generates the PDE's of both the Maxwell Faraday equations and the Maxwell Ampere equations. The interaction energy density between the potentials and the induced closed charge current density is exactly the similarity curvature invariant of highest degree (N-1) for the implicit surface. Although developed without direct contact with M-brane theory, these ideas of generalized implicit surfaces should have application to the study of p-branes that can have multiple components and envelopes. The theory suggests that gravitational collapse of mass energy density should include terms that involve the interaction between charge-current densities and electromagnetic potentials. 
  Thermal fluctuations of a massive scalar field in the Rindler wedge have been recently obtained. As a by product, the Minkowski vacuum fluctuations seen by a uniformly accelerated observer have been determined and confronted with the corresponding Minkowski thermal fluctuations of the same field, seen by an inertial observer. Since some of the calculations of this previous work have not been detailed on it, and they present some important subtleties, they are explicitly done here. These subtleties have to do with the leading order behaviour of certain parameter dependent integrals. Some of the leading order expansions are derived using the Riemann-Lebesgue lemma. 
  The condensed matter examples, in which the effective gravity appears in the low-energy corner as one of the collective modes of quantum vacuum, provide a possible answer to the question, why the vacuum energy is so small. This answer comes from the fundamental ``trans-Planckian'' physics of quantum liquids. In the effective theory of the low energy degrees of freedom the vacuum energy density is proportional to the fourth power of the corresponding ``Planck'' energy appropriate for this effective theory. However, from the exact ``Theory of Everything'' of the quantum liquid it follows that its vacuum energy density is exactly zero without fine tuning, if: there are no external forces acting on the liquid; there are no quasiparticles which serve as matter; no space-time curvature; and no boundaries which give rise to the Casimir effect. Each of these four factors perturbs the vacuum state and induces the nonzero value of the vacuum energy density of order of energy density of the perturbation. This is the reason, why one must expect that in each epoch the vacuum energy density is of order of matter density of the Universe, or/and of its curvature, or/and of the energy density of smooth component -- the quintessence. 
  Machian solutions of which the scalar field exhibits the asymptotic behavior $\phi =O(\rho /\omega)$ are generally explored for the homogeneous and isotropic universe in the Brans-Dicke theory. It is shown that the Machian solution is unique for the closed and the open space. Such a solution is restricted to one that satisfies the relation $GM/c^{2}a=const$, which is fixed to $\pi $ in the theory for the closed model. Another type of solution satisfying $\phi =O(\rho /\omega)$ with the arbitrary coupling constant $% \omega $ is obtained for the flat space. This solution has the scalar field $% \phi \propto \rho t^{2}$ and also keeps the relation $GM/c^{2}R=const$ all the time. This Machian relation and the asymptotic behavior $\phi =O(\rho /\omega)$ is equivalent to each other in the Brans-Dicke theory. 
  We present a dynamical analysis of the (classical) spatially flat and negative curved Friedmann-Lameitre-Robertson-Walker (FLRW) universes evolving, (by assumption) close to the thermodynamic equilibrium, in presence of a particles creation process, described by means of a realiable phenomenological approach, based on the application to the comoving volume (i. e. spatial volume of unit comoving coordinates) of the theory for open thermodynamic systems. In particular we show how, since the particles creation phenomenon induces a negative pressure term, then the choice of a well-grounded ansatz for the time variation of the particles number, leads to a deep modification of the very early standard FLRW dynamics. More precisely for the considered FLRW models, we find (in addition to the limiting case of their standard behaviours) solutions corresponding to an early universe characterized respectively by an "eternal" inflationary-like birth and a spatial curvature dominated singularity. In both these cases the so-called horizon problem finds a natural solution. 
  We study the equation of motion appropriate to an inspiralling binary star system whose constituent stars have strong internal gravity. We use the post-Newtonian approximation with the strong field point particle limit by which we can introduce into general relativity a notion of a point-like particle with strong internal gravity without using Dirac delta distribution. Besides this limit, to deal with strong internal gravity we express the equation of motion in surface integral forms and calculate these integrals explicitly. As a result we obtain the equation of motion for a binary of compact bodies accurate through the second and half post-Newtonian (2.5 PN) order. This equation is derived in the harmonic coordinate. Our resulting equation perfectly agrees with Damour and Deruelle 2.5 PN equation of motion. Hence it is found that the 2.5 PN equation of motion is applicable to a relativistic compact binary. 
  We investigate properties of r-modes characterized by regular eigenvalue problem in slowly rotating relativistic polytropes. Our numerical results suggest that discrete r-mode solutions for the regular eigenvalue problem exist only for restricted polytropic models. In particular the r-mode associated with l=m=2, which is considered to be the most important for gravitational radiation driven instability, do not have a discrete mode as solutions of the regular eigenvalue problem for polytropes having the polytropic index N > 1.18 even in the post-Newtonian order. Furthermore for a N=1 polytrope, which is employed as a typical neutron star model, discrete r-mode solutions for regular eigenvalue problem do not exist for stars whose relativistic factor M/R is larger than about 0.1. Here M and R are stellar mass and stellar radius, respectively. 
  We have developed a scheme for reducing LIGO suspension thermal noise close to violin-mode resonances. The idea is to monitor directly the thermally-induced motion of a small portion of (a ``point'' on) each suspension fiber, thereby recording the random forces driving the test-mass motion close to each violin-mode frequency. One can then suppress the thermal noise by optimally subtracting the recorded fiber motions from the measured motion of the test mass, i.e., from the LIGO output. The proposed method is a modification of an analogous but more technically difficult scheme by Braginsky, Levin and Vyatchanin for reducing broad-band suspension thermal noise. The efficiency of our method is limited by the sensitivity of the sensor used to monitor the fiber motion. If the sensor has no intrinsic noise (i.e. has unlimited sensitivity), then our method allows, in principle, a complete removal of violin spikes from the thermal-noise spectrum. We find that in LIGO-II interferometers, in order to suppress violin spikes below the shot-noise level, the intrinsic noise of the sensor must be less than \~2*10^{-13}cm/sqrt(Hz). This sensitivity is two orders of magnitude greater than that of currently available sensors. 
  Recent observations suggest that blobs of matter are ejected with ultra-relativistic speeds in various astrophysical phenomena such as supernova explosions, quasars, and microquasars. In this paper we analyze the gravitational radiation emitted when such an ultra-relativistic blob is ejected from a massive object. We express the gravitational wave by the metric perturbation in the transverse-traceless gauge, and calculate its amplitude and angular dependence. We find that in the ultra-relativistic limit the gravitational wave has a wide angular distribution, like $1+\cos\theta$. The typical burst's frequency is Doppler shifted, with the blue-shift factor being strongly beamed in the forward direction. As a consequence, the energy flux carried by the gravitational radiation is beamed. In the second part of the paper we estimate the anticipated detection rate of such bursts by a gravitational-wave detector, for blobs ejected in supernova explosions. Dar and De Rujula recently proposed that ultra-relativistic blobs ejected from the central core in supernova explosions constitute the source of Gamma-ray bursts. Substituting the most likely values of the parameters as suggested by their model, we obtain an estimated detection rate of about 1 per year by the advanced LIGO-II detector. 
  We compute the spectral distribution of the quantum fluctuations of the vacuum, amplified by inflation, after an arbitrary number of background transitions. Using a graphic representation of the process we find that the final spectrum can be completely determined trough a synthetic set of working rules, and a list of simple algebraic computations. 
  The behaviour of Jacobi fields along a time-like geodesic running into an isotropic singularity is studied. It is shown that the Jacobi fields are crushed to zero length at a rate which is the same in every direction orthogonal to the geodesic. We show by means of a counter-example that this crushing effect depends crucially on a technicality of the definition of isotropic singularities, and not just on the uniform degeneracy of the metric at the singularity. 
  In this paper we propose a model for the formation of the cosmological voids. We show that cosmological voids can form directly after the collapse of extremely large wavelength perturbations into low-density black holes or cosmological black holes (CBH). Consequently the voids are formed by the comoving expansion of the matter that surrounds the collapsed perturbation. It follows that the universe evolves, in first approximation, according to the Einstein-Straus cosmological model. We discuss finally the possibility to detect the presence of these black holes through their weak and strong lensing effects and their influence on the cosmic background radiation. 
  A new 5-dimensional Classical Unified Field Theory of Kaluza-Klein type is formulated using 2 separate scalar fields which are related in such a way as to make the 5-dimensional matter-geometry coupling parameter constant. It is shown that this procedure solves the problem of the variability of the gravity coupling parameter without having to assume a conformal invariance. The corresponding Field equations are discussed paying particular attention to the possible induction of scalar field gradients by Electromagnetic Fields. A new correspondence limit in which the field equations lead to the usual Einstein-Maxwell equations is obtained. This limit does not require the usual condition that the usual scalar field be constant. 
  Starting from delay equations that model field retardation effects, we study the origin of runaway modes that appear in the solutions of the classical equations of motion involving the radiation reaction force. When retardation effects are small, we argue that the physically significant solutions belong to the so-called slow manifold of the system and we identify this invariant manifold with the attractor in the state space of the delay equation. We demonstrate via an example that when retardation effects are no longer small, the motion could exhibit bifurcation phenomena that are not contained in the local equations of motion. 
  Evolution of electric and magnetic fields in dielectric media, driven by the influence of a strong gravitational wave, is considered for four exactly integrable models. It is shown that the gravitational wave field gives rise to new effects and to singular behaviour in the electromagnetic field. 
  We derive matter collineations of the Bianchi types I, II, III, VIII and IX, and Kantowski-Sachs spacetimes. It is found that matter collineations turn out similar to Ricci collineations with different constranit equations. We solve the constraint equation for a particular case and obtain three cosmological models which represent perfect fluid dust solutions. 
  A diagramatic heat kernel expansion technique is presented. The method is especially well suited to the small-derivative expansion of the heat kernel, but it can also be used to reproduce the results obtained by the approach known as covariant perturbation theory. The new technique gives an expansion for the heat kernel at coincident points. It can also be used to obtain the derivative of the heat kernel and this is useful for evaluating the expectation values of the stress-energy tensor. 
  The dynamical consequences of a bimetric scalar-tensor theory of gravity with a dynamical light speed are investigated in a cosmological setting. The model consists of a minimally-coupled self-gravitating scalar field coupled to ordinary matter fields in the standard way through the metric: $\metric_{\mu\nu}+B\partial_\mu\phi\partial_\nu\phi$. We show that in a universe with matter that has a radiation-dominated equation of state, the model allows solutions with a de Sitter phase that provides sufficient inflation to solve the horizon and flatness problems. This behaviour is achieved without the addition of a potential for the scalar field, and is shown to be largely independent of its introduction. We therefore have a model that is fundamentally different than the potential-dominated, slowly-rolling scalar field of the standard models inflationary cosmology. The speed of gravitational wave propagation is predicted to be significantly different from the speed of matter waves and photon propagation in the early universe. 
  The Einstein field equations for several cosmological models reduce to polynomial systems of ordinary differential equations. In this paper we shall concentrate our attention to the spatially homogeneous diagonal G_2 cosmologies. By using Darboux's theory in order to study ordinary differential equations in the complex projective plane CP^2 we solve the Bianchi V models totally. Moreover, we carry out a study of Bianchi VI models and first integrals are given in particular cases. 
  A canonical quantization of two-dimensional gravity minimally coupled to real scalar and spinor Majorana fields is presented. The physical state space of the theory is completely described and calculations are also made of the average value of the metric tensor relative to states close to the ground state 
  Starting from quoted papers it show that, applying Mach's principle, on obtains transient mass fluctuation by varying proper energy. It is show why the experiments performed until now measured a smaller effect than was predicted. On establish conditions for using gravitational energy by means of transient mass fluctuations. On suggest a new type of experiment for verify the effect of transient mass fluctuation. 
  The Machian cosmological solution satisfying $\phi =O(\rho /\omega)$ for the perfect-fluid with negative pressure is discussed. When the coefficient of the equation of state $\gamma \to -1/3$, the gravitational constant approaches to constant. If we assume the present mass density $\rho_{0}\sim \rho_{c}$ (critical density), the parameter $\epsilon$ ($\gamma =(\epsilon -1)/3$) has a value of order $10^{-3}$ to support the present gravitational constant. The closed model is valid for $\omega <-3/2\epsilon$ and exhibits the slow accelerating expansion. We understand why the coupling parameter $| \omega |$ is so large ($\omega \sim -10^{3}$). The time-variation of the gravitational constant $| \dot{G}/G| \sim 10^{-13} yr^{-1}$ at present is derived in this model. 
  Using a qualitative analysis based on the Hamiltonian formalism and the orthonormal frame representation we investigate whether the chaotic behaviour which occurs close to the initial singularity is still present in the far future of Bianchi VIII models. We describe some features of the vacuum Bianchi VIII models at late times which might be relevant for studying the nature of the future asymptote of the general vacuum inhomogeneous solution to the Einstein field equations. 
  We reduce Einstein's field equations for the interior of a uniformly rotating, axisymmetric perfect fluid to a system of six second order partial differential equations for the pressure p the energy density $\mu$ and four dependent variables.Four of these equations do not depend on p and $\mu$ and the other two determine p and $\mu$. 
  The time evolution of black holes involves both the canonical equations of quantum gravity and the statistical mechanics of Hawking radiation, neither of which contains a time variable. In order to introduce the time, we apply the semiclassical approximation to the Hamiltonian constraint on the apparent horizon and show that, when the backreaction is included, it suggests the existence of a long-living remnant, similarly to what is obtained in the microcanonical picture for the Hawking radiation. 
  The idea of extra-dimensions has recently gone through a renewal with the hypothesis, suggested by recent developments in string theory, that ordinary matter is confined to a sub-space, called brane, embedded in a higher dimensional spacetime. I summarize here some consequences in cosmology of this type of models. The most remarkable aspect is that the Friedmann laws, which govern the expansion of the Universe, are modified. An important direction of research is the study of cosmological perturbations and the possible signature of extra-dimensions in cosmological observations. 
  Neutron stars are believed to contain (neutron and proton) superfluids. I will give a summary of a macroscopic description of the interior of neutron stars, in a formulation which is general relativistic. I will also present recent results on the oscillations of neutron stars, with superfluidity explicitly taken into account, which leads in particular to the existence of a new class of modes. 
  In this lecture I build up the motivation for relativity and gravitation based on general principles and common sense considerations which should fall in the sphere of appreciation of a general reader. There is a novel way of looking at things and understanding them in a more direct physical terms which should be of interest to fellow relativists as well as physicists in general. 
  A generic prediction of inflation is that the thermalized region we inhabit is spatially infinite. Thus, it contains an infinite number of regions of the same size as our observable universe, which we shall denote as $\O$-regions. We argue that the number of possible histories which may take place inside of an $\O$-region, from the time of recombination up to the present time, is finite. Hence, there are an infinite number of $\O$-regions with identical histories up to the present, but which need not be identical in the future. Moreover, all histories which are not forbidden by conservation laws will occur in a finite fraction of all $\O$-regions. The ensemble of $\O$-regions is reminiscent of the ensemble of universes in the many-world picture of quantum mechanics. An important difference, however, is that other $\O$-regions are unquestionably real. 
  A different approach has been used to evaluate the momentum imparted by the gravitational waves with spherical wavefronts. It is shown that the results obtained for momentum coincide with those already available in the literature. 
  It has long been thought that the sensitivity of laser interferometric gravitational-wave detectors is limited by the free-mass standard quantum limit, unless radical redesigns of the interferometers or modifications of their input/output optics are introduced. Within a fully quantum-mechanical approach we show that in a second-generation interferometer composed of arm cavities and a signal recycling cavity, e.g., the LIGO-II configuration, (i) quantum shot noise and quantum radiation-pressure-fluctuation noise are dynamically correlated, (ii) the noise curve exhibits two resonant dips, (iii) the Standard Quantum Limit can be beaten by a factor of 2, over a frequency range \Delta f/f \sim 1, but at the price of increasing noise at lower frequencies. 
  Gravitational radiation with roughly spherical wavefronts, produced by roughly spherical black holes or other astrophysical objects, is described by an approximation scheme. The first quasi-spherical approximation, describing radiation propagation on a background, is generalized to include additional non-linear effects, due to the radiation itself. The gravitational radiation is locally defined and admits an energy tensor, satisfying all standard local energy conditions and entering the truncated Einstein equations as an effective energy tensor. This second quasi-spherical approximation thereby includes gravitational radiation reaction, such as the back-reaction on the black hole. With respect to a canonical flow of time, the combined energy-momentum of the matter and gravitational radiation is covariantly conserved. The corresponding Noether charge is a local gravitational mass-energy. Energy conservation is formulated as a local first law relating the gradient of the gravitational mass to work and energy-supply terms, including the energy flux of the gravitational radiation. Zeroth, first and second laws of black-hole dynamics are given, involving a dynamic surface gravity. Local gravitational-wave dynamics is described by a non-linear wave equation. In terms of a complex gravitational- radiation potential, the energy tensor has a scalar-field form and the wave equation is an Ernst equation, holding independently at each spherical angle. The strain to be measured by a distant detector is simply defined. 
  We study the collapse of a self-gravitating and radiating shell. Matter constituting the shell is quantized and the construction is viewed as a semiclassical model of possible black hole formation. It is shown that the shell internal degrees of freedom are excited by the quantum non-adiabaticity of the collapse and, consequently, on coupling them to a massless scalar field, the collapsing matter emits a burst of coherent (thermal) radiation. 
  Numerical Relativity is concerned with solving the Einstein equations, as well as any field or matter equations on curved space-time, by means of computer calculations. The methods developed for this purpose up to now, as well as the addressed physical problems are getting more numerous every day. This is a brief summary of the presentations which have been given during the 9th Marcel Grossman meeting, which took place in Rome, from 2nd to 8th of July 2000. Many different fields have been addressed, from pure numerics and applied mathematics to neutron star properties and gravitational wave astronomy. 
  In Randall-Sundrum type braneworld cosmologies, the dynamical equations on the three-brane differ from the general relativity equations by terms that carry the effects of embedding and of the free gravitational field in the five-dimensional bulk. In a FRW ansatze for the metric, we present two methods for deriving inflationary solutions to the covariant non-linear dynamical equations for the gravitational and matter fields on the brane. In the first approach we examine the constraints on the dynamical relationship between the cosmological scale factor and the scalar field driving self-interaction potential, imposed by the weak energy condition. We then investigate inflationary solutions obtained from a scalar field superpotential. Both these techniques for solving the braneworld field equations are illustrated by flat curvature models. 
  This article is dedicated to the memory of Dennis Sciama. It revisits a series of issues to which he devoted much time and effort, regarding the relationship between local physics and the large scale structure of the universe - in particular, Olber's paradox, Mach's principle, and the various arrows of time. Thus the focus is various ways in which local physics is influenced by the universe itself. 
  We present a Bayesian approach to the problem of determining parameters for coalescing binary systems observed with laser interferometric detectors. By applying a Markov Chain Monte Carlo (MCMC) algorithm, specifically the Gibbs sampler, we demonstrate the potential that MCMC techniques may hold for the computation of posterior distributions of parameters of the binary system that created the gravity radiation signal. We describe the use of the Gibbs sampler method, and present examples whereby signals are detected and analyzed from within noisy data. 
  We present an algorithm for the detection of periodic sources of gravitational waves with interferometric detectors that is based on a special symmetry of the problem: the contributions to the phase modulation of the signal from the earth rotation are exactly equal and opposite at any two instants of time separated by half a sidereal day; the corresponding is true for the contributions from the earth orbital motion for half a sidereal year, assuming a circular orbit. The addition of phases through multiplications of the shifted time series gives a demodulated signal; specific attention is given to the reduction of noise mixing resulting from these multiplications. We discuss the statistics of this algorithm for all-sky searches (which include a parameterization of the source spin-down), in particular its optimal sensitivity as a function of required computational power. Two specific examples of all-sky searches (broad-band and narrow-band) are explored numerically, and their performances are compared with the stack-slide technique (P. R. Brady, T. Creighton, Phys. Rev. D, 61, 082001). 
  Spin-Polarised cylinders with and without axial magnetic fields are obtained as particular families of solutions of Einstein-Cartan gravity (EC).The first solution represents a spin-polarised cylinder in teleparallel gravity.The second solution is a magnetized solution representing a spin-polarised cylinder where the magnetic fields and spins are distributed along the infinite axis of the cylinder.Altough it seems that the first solution is less realist than the second it could be obtained by shielding the magnetic fields with a superconductor.The second solution is computed by taking into account the Ritter et al. experiment with the test spin-polarized mass to test spin dependent forces.Ritter experiment deals with a test mass with $>10^{23}$ spin polarized electrons which leads to a spin density of $10^{-4}gcm^{-1}s^{-1}$. 
  Clairin's method of obtaining B\"{a}cklund transformations is applied to Einstein's field equations for the interior of a uniformly rotating stationary axisymmetric perfect fluid. It is shown that for arbitrary pressure $p$ and mass density $\mu$ the method does not give non-trivial B\"{a}cklund transformations, while if $\mu + 3p =0$ it gives the transformation of Ehlers. 
  We present a method developed to deal with electromagnetic wave propagation inside a material medium that reacts, in general, non-linearly to the field strength. We work in the context of Maxwell' s theory in the low frequency limit and obtain a geometrical representation of light paths for each case presented. The isotropic case and artificial birefringence caused by an external electric field are analyzed as an application of the formalism and the effective geometry associated to the wave propagation is exhibited. 
  It is shown how to generate three-dimensional Einstein-Maxwell fields from known ones in the presence of a hypersurface-orthogonal non-null Killing vector field. The continuous symmetry group is isomorphic to the Heisenberg group including the Harrison-type transformation. The symmetry of the Einstein-Maxwell-dilaton system is also studied and it is shown that there is the $SL(2,{\bf R})$ transformation between the Maxwell and the dilaton fields. This $SL(2,{\bf R})$ transformation is identified with the Geroch transformation of the four-dimensional vacuum Einstein equation in terms of the Ka{\l}uza-Klein mechanism. 
  We consider a modified ``Swiss cheese'' model in Brans-Dicke theory, and use it to discuss the evolution of black holes in an expanding universe. We define the black hole radius by the Misner-Sharp mass and find their exact time evolutions for dust and vacuum universes of all curvatures. 
  We present a new solution to dilaton-axion gravity which looks like a rotating Bertotti-Robinson (BR) Universe. It is supported by an homogeneous Maxwell field and a linear axion and can be obtained as a near-horizon limit of extremal rotating dilaton-axion black holes. It has the isometry $SL(2,R)\times U(1)$ where U(1) is the remnant of the SO(3) symmetry of BR broken by rotation, while $SL(2,R)$ corresponds to the $AdS_2$ sector which no longer factors out of the full spacetime. Alternatively our solution can be obtained from the D=5 vacuum counterpart to the dyonic BR with equal electric and magnetic field strengths. The derivation amounts to smearing it in D=6 and then reducing to D=4 with dualization of one Kaluza-Klein two-form in D=5 to produce an axion. Using a similar dualization procedure, the rotating BR solution is uplifted to D=11 supergravity. We show that it breaks all supersymmetries of N=4 supergravity in D=4, and that its higher dimensional embeddings are not supersymmetric either. But, hopefully it may provide a new arena for corformal mechanics and holography. Applying a complex coordinate transformation we also derive a BR solution endowed with a NUT parameter. 
  Contrary to what is asserted in a recent paper by Kostadt and Liu ("Causality and stability of the relativistic diffusion equation"), experiments can tell apart (and in fact do) hyperbolic theories from parabolic theories of dissipation. It is stressed that the existence of a non--negligible relaxation time does not imply for the system to be out of the hydrodynamic regime. 
  We perform a numerical free evolution of a selfgravitating, spherically symmetric scalar field satisfying the wave equation. The evolution equations can be written in a very simple form and are symmetric hyperbolic in Eddington-Finkelstein coordinates. The simplicity of the system allow to display and deal with the typical gauge instability present in these coordinates. The numerical evolution is performed with a standard method of lines fourth order in space and time. The time algorithm is Runge-Kutta while the space discrete derivative is symmetric (non-dissipative). The constraints are preserved under evolution (within numerical errors) and we are able to reproduce several known results. 
  Based on the observation that the moduli of a link variable on a cyclic group modify Connes' distance on this group, we construct several action functionals for this link variable within the framework of noncommutative geometry. After solving the equations of motion, we find that one type of action gives nontrivial vacuum solution for gravity on this cyclic group in a broad range of coupling constants and that such solutions can be expressed with Chebyshev's polynomials. 
  Gravitomagnetic clock effects for circularly rotating orbits in black hole spacetimes are studied from a relative observer point of view, clarifying the roles played by special observer families. 
  The gravitation field of the flat plate was investigated. It have been shown that there exist the internal solution of Einstein equations sewed together with external one, which described a ''homogeneous'' gravitational field. 
  We study the propagation of gravitational waves in a collisionless plasma with an external magnetic field parallel to the direction of propagation. Due to resonant interaction with the plasma particles the gravitational wave experiences cyclotron damping or growth, the latter case being possible if the distribution function for any of the particle species deviates from thermodynamical equilibrium. Furthermore, we examine how the damping and dispersion depends on temperature and on the ratio between the cyclotron- and gravitational wave frequency. The presence of the magnetic field leads to different dispersion relations for different polarizations, which in turn imply Faraday rotation of gravitational waves. 
  We address the question of universes inside a $\Lambda$ black hole which is described by a spherically symmetric globally regular solution to the Einstein equations with a variable cosmological term $\Lambda_{\mu\nu}$, asymptotically $\Lambda g_{\mu\nu}$ as $r\to 0$ with $\Lambda$ of the scale of symmetry restoration. Global structure of spacetime contains an infinite sequence of black and white holes, vacuum regular cores and asymptotically flat universes. Regular core of a $\Lambda$ white hole models the initial stages of the Universe evolution. In this model it starts from a nonsingular nonsimultaneous big bang, which is followed by a Kasner-type anisotropic expansion. Creation of a mass occurs mostly at the anisotropic stage of quick decay of the initial vacuum energy. We estimate also the probability of quantum birth of baby universes inside a $\Lambda$ black hole due to quantum instability of the de Sitter vacuum. 
  Using a recently developed perturbation formalism based on curvature quantities, we investigate the linear stability of black holes and solitons with Yang-Mills hair and a negative cosmological constant. We show that those solutions which have no linear instabilities under odd- and even- parity spherically symmetric perturbations remain stable under odd-parity, linear, non-spherically symmetric perturbations. 
  Stochastic semiclassical gravity is a theory for the interaction of gravity with quantum matter fields which goes beyond the semiclassical limit. The theory predicts stochastic fluctuations of the classical gravitational field induced by the quantum fluctuations of the stress energy tensor of the matter fields. Here we use an axiomatic approach to introduce the Einstein-Langevin equations as the consistent set of dynamical equations for a first order perturbative correction to semiclassical gravity and review their main features. We then describe the application of the theory in a simple chaotic inflationary model, where the fluctuations of the inflaton field induce stochastic fluctuations in the gravitational field. The correlation functions for these gravitational fluctuations lead to an almost Harrison-Zel'dovich scale invariant spectrum at large scales, in agreement with the standard theories for structure formation. A summary of recent results and other applications of the theory is also given. 
  The Einstein equations for a perfect fluid spatially homogeneous spacetime are studied in a unified manner by retaining the generality of certain parameters whose discrete values correspond to the various Bianchi types of spatial homogeneity. A parameter-dependent decomposition of the metric variables adapted to the symmetry breaking effects of the nonabelian Bianchi types on the "free dynamics" leads to a reduction of the equations of motion for those variables to a 2-dimensional time-dependent Hamiltonian system containing various time-dependent potentials which are explicitly described and diagrammed. These potentials are extremely useful in deducing the gross features of the evolution of the metric variables. 
  We calculate energy and momentum for a class of cylindrical rotating gravitational waves using Einstein and Papapetrou's prescriptions. It is shown that the results obtained are reduced to the special case of the cylindrical gravitational waves already available in the literature. 
  We study the coalescence of non-spinning binary black holes from near the innermost stable circular orbit down to the final single rotating black hole. We use a technique that combines the full numerical approach to solve Einstein equations, applied in the truly non-linear regime, and linearized perturbation theory around the final distorted single black hole at later times. We compute the plunge waveforms which present a non negligible signal lasting for $t\sim 100M$ showing early non-linear ringing, and we obtain estimates for the total gravitational energy and angular momentum radiated. 
  We summarize a recently proposed concrete programme for investigating the (semi)classical limit of canonical, Lorentzian, continuum quantum general relativity in four spacetime dimensions. The analysis is based on a novel set of coherent states labelled by graphs. These fit neatly together with an Infinite Tensor Product (ITP) extension of the currently used Hilbert space. The ITP construction enables us to give rigorous meaning to the infinite volume (thermodynamic) limit of the theory which has been out of reach so far. 
  We consider cosmological consequences of a conformal invariant formulation of Einstein's General Relativity where instead of the scale factor of the spatial metrics in the action functional a massless scalar (dilaton) field occurs which scales all masses including the Planck mass. Instead of the expansion of the universe we get the Hoyle-Narlikar type of mass evolution, where the temperature history of the universe is replaced by the mass history. We show that this conformal invariant cosmological model gives a satisfactory description of the new supernova Ia data for the effective magnitude - redshift relation without a cosmological constant and make a prediction for the high-redshift behavior which deviates from that of standard cosmology for $z>1.7$. 
  The existence of Friedmann limits is systematically investigated for all the hypersurface-homogeneous rotating dust models, presented in previous papers by this author. Limiting transitions that involve a change of the Bianchi type are included. Except for stationary models that obviously do not allow it, the Friedmann limit expected for a given Bianchi type exists in all cases. Each of the 3 Friedmann models has parents in the rotating class; the k = +1 model has just one parent class, the other two each have several parent classes. The type IX class is the one investigated in 1951 by Goedel. For each model, the consecutive limits of zero rotation, zero tilt, zero shear and spatial isotropy are explicitly calculated. 
  The program of quantizing the gravitational field with the help of affine field variables is continued. For completeness, a review of the selection criteria that singles out the affine fields, the alternative treatment of constraints, and the choice of the initial (before imposition of the constraints) ultralocal representation of the field operators is initially presented. As analogous examples demonstrate, the introduction and enforcement of the gravitational constraints will cause sufficient changes in the operator representations so that all vestiges of the initial ultralocal field operator representation disappear. To achieve this introduction and enforcement of the constraints, a well characterized phase space functional integral representation for the reproducing kernel of a suitably regularized physical Hilbert space is developed and extensively analyzed. 
  This paper explores the effects that magnetic fields have on the viscous boundary layers (VBLs) that can form in neutron stars at the crust-core interface, and it investigates the VBL damping of the gravitational-radiation driven r-mode instability. Approximate solutions to the magnetohydrodynamic equations valid in the VBL are found for ordinary-fluid neutron stars. It is shown that magnetic fields above 10^9 Gauss significantly change the structure of the VBL, and that magnetic fields decrease the VBL damping time. Furthermore, VBL damping completely suppresses the r-mode instability for B >= 10^{12} Gauss. Thus, magnetic fields will profoundly affect the VBL damping of the r-mode instability in hot young pulsars (that are cool enough to have formed a solid crust). One can speculate that magnetic fields can affect the VBL damping of this instability in LMXBs and other cold old pulsars (if they have sufficiently large internal fields). 
  Long gamma-ray bursts (GRBs) are probably powered by high-angular momentum black hole-torus systems in suspended accretion. The torus will radiate gravitational waves as non-axisymmetric instabilities develop. The luminosity in gravitational-wave emissions is expected to compare favorably with the observed isotropic equivalent luminosity in GRB-afterglow emissions. This predicts that long GRBs are potentially the most powerful LIGO/VIRGO burst-sources in the Universe. Their frequency-dynamics is characterized by a horizontal branch in the $\dot{f}(f)-$diagram. 
  News:    APS Prize on gravitation, by Cliff Will    TGG elections, by David Garfinkle    We hear that... by Jorge Pullin   Research Briefs:    Experimental Unruh radiation?, by Matt Visser    Why is the universe accelerating?, by Beverly Berger    The Lazarus Project, by Richard Price    LIGO locks its first detector!, by Stan Whitcomb    Progress on the nonlinear r-mode problem, by Keith Lockitch   Conference reports:    Analog models of general relativity, by Matt Visser    Astrophysical Sources of Gravitational radiation by Joan Centrella    Numerical relativity at the 20th Texas meeting, by Pablo Laguna 
  Angular fluctuations of suspended mirrors in gravitational wave interferometers are a source of noise both for the locking and the operation of the detectors. We describe here some of the sources of these fluctuations and methods for the estimation of their order of magnitude. 
  I show that the recent calculation of Moffatt's regarding the viscous dissipation of a spinning coin overlooked the importance of the finite width of the viscous boundary layer. My new estimates are more in accord with that observed. I also point out that the frequency ``chirp'' of the specially designed toy ``Euler's Disk'' is similar to that expected during the last few minutes of the life of a coalescing binary of two neutron stars. As such, this toy is an excellent desktop demonstration for the expected phenomena. 
  We prove the existence of a large class of asymptotically flat initial data with non-vanishing mass and angular momentum for which the metric and the extrinsic curvature have asymptotic expansions at space-like infinity in terms of powers of a radial coordinate. 
  A mathematical formalism for treating spacetime topology as a quantum observable is provided. We describe spacetime foam entirely in algebraic terms. To implement the correspondence principle we express the classical spacetime manifold of general relativity and the commutative coordinates of its events by means of appropriate limit constructions. 
  For general relativistic spacetimes filled with an irrotational perfect fluid a generalized form of Friedmann's equations governing the expansion factor of spatially averaged portions of inhomogeneous cosmologies is derived. The averaging problem for scalar quantities is condensed into the problem of finding an `effective equation of state' including kinematical as well as dynamical `backreaction' terms that measure the departure from a standard FLRW cosmology. Applications of the averaged models are outlined including radiation-dominated and scalar field cosmologies (inflationary and dilaton/string cosmologies). In particular, the averaged equations show that the averaged scalar curvature must generically change in the course of structure formation, that an averaged inhomogeneous radiation cosmos does not follow the evolution of the standard homogeneous-isotropic model, and that an averaged inhomogeneous perfect fluid features kinematical `backreaction' terms that, in some cases, act like a free scalar field source. The free scalar field (dilaton) itself, modelled by a `stiff' fluid, is singled out as a special inhomogeneous case where the averaged equations assume a simple form. 
  Algebraically special gravitational fields are described using algebraic and differential invariants of the Weyl tensor. A type III invariant is also given and calculated for Robinson-Trautman spaces. 
  A number of background independent quantizations procedures have recently been employed in 4d nonperturbative quantum gravity. We investigate and illustrate these techniques and their relation in the context of a simple 2d topological theory. We discuss canonical quantization, loop or spin network states, path integral quantization over a discretization of the manifold, spin foam formulation, as well as the fully background independent definition of the theory using an auxiliary field theory on a group manifold. While several of these techniques have already been applied to this theory by Witten, the last one is novel: it allows us to give a precise meaning to the sum over topologies, and to compute background-independent and, in fact, "manifold-independent" transition amplitudes. These transition amplitudes play the role of Wightman functions of the theory. They are physical observable quantities, and the canonical structure of the theory can be reconstructed from them via a C* algebraic GNS construction. We expect an analogous structure to be relevant in 4d quantum gravity. 
  In these notes we present a summary of existing ideas about phase transitions of black hole spacetimes in semiclassical gravity and offer some thoughts on three possible scenarios by which these transitions could take place. Our first theme is ilustrated by a quantum atomic black hole system, generalizing to finite-temperature a model originally offered by Bekenstein. In this equilibrium atomic model, the black hole phase transition is realized as the abrupt excitation of a high energy state, suggesting analogies with the study of two-level atoms. Our second theme argues that the black hole system shares similarities with the defect-mediated Kosterlitz-Thouless transition in condensed matter. These similarities suggest that the black hole phase transition may be more fully understood by focusing upon the dynamics of black holes and white holes, the spacetime analogy of vortex and anti-vortex topological defects. Finally we compare the black hole phase transition to another transition driven by an exponentially increasing density of states, the Hagedorn transition first found in hadron physics in the context of dual models or the old string theory. In modern string theory, the Hagedorn transition is linked by the Maldacena conjecture to the Hawking-Page black hole phase transition in Anti-deSitter space, as observed by Witten. Understanding the dynamics of the Hagedorn transition may thus yield insight into the dynamics of the black hole phase transition. We argue that characteristics of the Hagedorn transition are already contained within classical string systems where a nonperturbative and dynamical analysis is possible. 
  Singularities associated with an incomplete space-time (S) are not well-defined until a boundary is attached to it. Moreover, each boundary gives rise to a different singularity structure for the resulting total space-time (TST). Since S is compatible with a variety of boundaries, it therefore does not represent a unique universe, but instead corresponds to a family of universes, one for each possible boundary.   It is shown that in the case of Weyl's space-time for a point-mass with nonzero cosmological constant, the boundary which he attached is invalid, and when the correct one is attached, the resulting TST is inextendible. This implies that the Lake-Roeder black hole cannot be produced by gravitational collapse. 
  Singularities associated with an incomplete space-time (S) are not uniquely defined until a boundary is attached to it. [The resulting space-time-with-boundary will be termed a "total" space-time (TST).] Since an incomplete space-time is compatible with a variety of boundaries, it follows that S does not represent a unique universe but instead corresponds to a family of universes, one for each of the distinct TSTs. It is shown here that the boundary attached to the Reissner-Nordstrom space-time for a point charge is invalid for q^2 < m^2. When the correct boundary is used, the resulting TST is inextendible. This implies that the Graves-Brill black hole cannot be produced by gravitational collapse. The same is true of the Kruskal-Fronsdal black hole for the point mass, and for those black holes which reduce to the latter for special values of their parameters. 
  The historical postulates for the point mass are shown to be satisfied by an infinity of space-times, differing as to the limiting acceleration of a radially approaching test particle. Taking this limit to be infinite gives Schwarzschild's result, which for a point mass at x = y = z = 0 has C(0+) = a^2, where a = 2m and C(r) denotes the coefficient of the angular terms in the polar form of the metric. Hilbert's derivation used the variable r* =[C(0+)]^1/2. For Hilbert, however, C was unknown, and thus could not be used to determine r*(0). Instead he asserted, in effect, that r* = (x^2 + y^2 + z^2)^1/2, which places the point mass at r* = 0. Unfortunately, this differs from the value (a) obtained by substituting Schwarzschild's C into the expression for r*(0), and since C(0+) is a scaler invariant, it follows that Hilbert's assertion is invalid. Owing to this error, in each spatial section of Hilbert's space-time, the boundary (r* = a) corresponding to r = 0 is no longer a point, but a two-sphere. This renders his space-time analytically extendible, and as shown by Kruskal and Fronsdal, its maximal extension contains a black hole. Thus the Kruskal-Fronsdal black hole is merely an artifact of Hilbert's error. 
  The field equations of Mannheim's theory of conformal gravity with dynamic mass generation are solved numerically in the interior and exterior regions of a model spherically symmetric sun with matched boundary conditions at the surface. The model consists of a generic fermion field inside the sun, and a scalar Higgs field in both the interior and exterior regions. From the conformal geodesic equations it is shown how an asymptotic gradient in the Higgs field causes an anomalous radial acceleration in qualitative agreement with that observed on the Pioneer 10/11, Galileo, and Ulysses spacecraft. At the same time the standard solar system tests of general relativity are preserved within the limits of observation. 
  A cyclic nature of quantum mechanical clock is discussed as ``quantization of time." Quantum mechanical clock is seen to be equivalent to the relativistic classical clock. 
  The condition for the vanishing of the Weyl tensor is integrated in the spherically symmetric case. Then, the resulting expression is used to find new, conformally flat, interior solutions to Einstein equations for locally anisotropic fluids. The slow evolution of these models is contrasted with the evolution of models with similar energy density or radial pressure distribution but non-vanishing Weyl tensor, thereby bringing out the different role played by the Weyl tensor, the local anisotropy of pressure and the inhomogeneity of the energy density in the collapse of relativistic spheres. 
  Self-consistent solutions to nonlinear spinor field equations in General Relativity are studied for the case of Bianchi type-I space-time. It has been shown that introduction of $\Lambda$-term in the Lagrangian generates oscillations of the Bianchi type-I model. 
  In order to get geodesically complete Reissner - Nordstroem space - times, it is necessary to identify pairs of singular points. This can be done in such a way that "wormholes" are created which generate electric field lines without any charge. Finally, it is shown that it is possible to glue these space - times not in the singularities r=0, but at some r>0. The surface energy generated by this gluing is exotic, but tends to zero in the limit r -> 0. 
  There is a theorem known as a Virial theorem that restricts the possible existence of non-trivial static solitary waves with scalar fields in a flat space-time with 3 or more spatial dimensions. This raises the following question: Does the analogous curved space-time version hold?. We investigate the possibility of solitons in a 4-D curved space-time with a simple model using numerical analysis. We found that there exists a static solution of the proposed non linear wave equation. This proves that in curved space-time the possibilities of solitonic solutions is enhanced relative to the flat space-time case. 
  The Becci-Rouet-Stora-Tyutin (BRST) operator quantization of a finite-dimensional gauge system featuring two quadratic super Hamiltonian and m linear supermomentum constraints is studied as a model for quantizing generally covariant gauge theories. The proposed model ``completely'' mimics the constraint algebra of General Relativity. The Dirac constraint operators are identified by realizing the BRST generator of the system as a Hermitian nilpotent operator, and a physical inner product is introduced to complete a consistent quantization procedure. 
  The quantum fluctuation of the stress tensor of a quantum field are discussed, as are the resulting spacetime metric fluctuations. Passive quantum gravity is an approximation in which gravity is not directly quantized, but fluctuations of the spacetime geometry are driven by stress tensor fluctuations. We discuss a decomposition of the stress tensor correlation function into three parts, and consider the physical implications of each part. The operational significance of metric fluctuations and the possible limits of validity of semiclassical gravity are discussed. 
  Anisotropic generalization of Randall and Sundrum brane-world model is considered. A new class of exact solutions for brane and bulk geometry is found; it is related to anisotropic Kasner solution. In view of this, the old question of isotropy of initial conditions in cosmology rises once again in the brane-world context. 
  A summary is given of some results and perspectives of the hamiltonian ADM approach to 2+1 dimensional gravity. After recalling the classical results for closed universes in absence of matter we go over the the case in which matter is present in the form of point spinless particles. Here the maximally slicing gauge proves most effective by relating 2+1 dimensional gravity to the Riemann- Hilbert problem. It is possible to solve the gravitational field in terms of the particle degrees of freedom thus reaching a reduced dynamics which involves only the particle positions and momenta. Such a dynamics is proven to be hamiltonian and the hamiltonian is given by the boundary term in the gravitational action. As an illustration the two body hamiltonian is used to provide the canonical quantization of the two particle system. 
  An introductory guide to mathematical cosmology is given focusing on the issue of the genericity of various important results which have been obtained during the last thirty or so years. Some of the unsolved problems along with certain new and potentially powerful methods which may be used for future progress are also given from a unified perspective. 
  We construct a dilatonic two-dimensional model of a charged black hole. The classical solution is a static charged black hole, characterized by two parameters, $m$ and $q$, representing the black hole's mass and charge. Then we study the semiclassical effects, and calculate the evaporation rate of both $m$ and $q$, as a function of these two quantities. Analyzing this dynamical system, we find two qualitatively different regimes, depending on the electromagnetic coupling constant $g_{A}$. If the latter is greater than a certain critical value, the charge-to-mass ratio decays to zero upon evaporation. On the other hand, for $g_{A}$ smaller than the critical value, the charge-to-mass ratio approaches a non-zero constant that depends on $g_{A}$ but not on the initial values of $m$ and $q$. 
  We give the formulation of the gravitational lensing theory in the strong field limit for a Schwarzschild black hole as a counterpart to the weak field approach. It is possible to expand the full black hole lens equation to work a simple analytical theory that describes at a high accuracy degree the physics in the strong field limit. In this way, we derive compact and reliable mathematical formulae for the position of additional critical curves, relativistic images and their magnification, arising in this limit. 
  It is shown that the cosmological singularity in isotropic minisuperspaces is naturally removed by quantum geometry. Already at the kinematical level, this is indicated by the fact that the inverse scale factor is represented by a bounded operator even though the classical quantity diverges at the initial singularity. The full demonstation comes from an analysis of quantum dynamics. Because of quantum geometry, the quantum evolution occurs in discrete time steps and does not break down when the volume becomes zero. Instead, space-time can be extended to a branch preceding the classical singularity independently of the matter coupled to the model. For large volume the correct semiclassical behavior is obtained. 
  In the context of the teleparallel equivalent of general relativity, we obtain the tetrad and the torsion fields of the stationary axisymmetric Kerr spacetime. It is shown that, in the slow rotation and weak field approximations, the axial-vector torsion plays the role of the gravitomagnetic component of the gravitational field, and is thus the responsible for the Lense-Thirring effect. 
  It is well-known that the equations for a simple fluid can be cast into what is called their Lagrange formulation. We introduce a notion of a generalized Lagrange formulation, which is applicable to a wide variety of systems of partial differential equations. These include numerous systems of physical interest, in particular, those for various material media in general relativity. There is proved a key theorem, to the effect that, if the original (Euler) system admits an initial-value formulation, then so does its generalized Lagrange formulation. 
  In the paper we show that the real gravitational waves which have $R_{iklm}\not= 0$ always carry energy-momentum and angular momentum. Our proof uses canonical superenergy and supermomentum tensors for gravitational field. 
  We propose a novel BF-type formulation of real four-dimensional gravity, which generalizes previous models. In particular, it allows for an arbitrary Immirzi parameter. We also construct the analogue of the Urbantke metric for this model. 
  A new version of tetrad gravity in globally hyperbolic, asymptotically flat at spatial infinity spacetimes with Cauchy surfaces diffeomorphic to $R^3$ is obtained by using a new parametrization of arbitrary cotetrads to define a set of configurational variables to be used in the ADM metric action. Seven of the fourteen first class constraints have the form of the vanishing of canonical momenta. A comparison is made with other models of tetrad gravity and with the ADM canonical formalism for metric gravity. 
  Regarding the spin-up of Schwarzschild-Kerr and Banados-Teitelboim-Zanelli systems as a symmetry breaking phase transition, critical exponents are evaluated and compared with classical Landau predictions. We suggest a definition of isothermal compressibity which is independent of spin direction. We find identical exponents for both systems, and possible universality in the phase transitions of these systems. 
  We study by computational means the dynamical stability against bar-mode deformation of rapidly and differentially rotating stars in a post-Newtonian approximation of general relativity. We vary the compaction of the star $M/R$ (where $M$ is the gravitational mass and $R$ the equatorial circumferential radius) between 0.01 and 0.05 to isolate the influence of relativistic gravitation on the instability. For compactions in this moderate range, the critical value of $\beta = T/W$ for the onset of the dynamical instability (where $T$ is the rotational kinetic energy and W the gravitational binding energy) slightly decreases from $\sim$ 0.26 to $\sim$ 0.25 with increasing compaction for our choice of the differential rotational law. Combined with our earlier findings based on simulations in full general relativity for stars with higher compaction, we conclude that relativistic gravitation enhances the dynamical bar-mode instability, i.e. the onset of instability sets in for smaller values of $\beta$ in relativistic gravity than in Newtonian gravity. We also find that once a triaxial structure forms after the bar-mode perturbation saturates in dynamically unstable stars, the triaxial shape is maintained, at least for several rotational periods. To check the reliability of our numerical integrations, we verify that the general relativistic Kelvin-Helmholtz circulation is well-conserved, in addition to rest-mass and total mass-energy, linear and angular momentum. Conservation of circulation indicates that our code is not seriously affected by numerical viscosity. We determine the amplitude and frequency of the quasi-periodic gravitational waves emitted during the bar formation process using the quadrupole formula. 
  In this paper, the backreaction to the traversable Lorentzian wormhole spacetime by the scalar field or electric charge is considered to find the exact solutions. The charges play the role of the additional matter to the static wormhole which is already constructed by the exotic matter. The stability conditions for the wormhole with scalar field and electric charge are found from the positiveness and flareness for the wormhole shape function. 
  In this paper, we study the scattering problem of the scalar wave in the traversable Lorentzian wormhole geometry. The potentials and Schr\"odinger-like equations are found in cases of the static uncharged and the charged wormholes. The differential scattering cross sections are determined by the phase shift of the asymptotic wave function in low frequency limit. It is also found that the cross section for charged wormhole is smaller than that for uncharged case by the reduction of the throat size due to the charge effect. 
  This is the second in a series of papers on the construction of explicit solutions to the stationary axisymmetric Einstein equations which can be interpreted as counter-rotating disks of dust. We discuss the class of solutions to the Einstein equations for disks with constant angular velocity and constant relative density which was constructed in the first part. The metric for these spacetimes is given in terms of theta functions on a Riemann surface of genus 2. We discuss the metric functions at the axis of symmetry and the disk. Interesting limiting cases are the Newtonian limit, the static limit, and the ultra-relativistic limit of the solution in which the central redshift diverges. 
  Motivated by a recent work by Terashima (Phys. Rev. D60 084001), we revisit the fluctuation-dissipation (FD) relation between the dissipative coefficient of a detector and the vacuum noise of fields in curved spacetime. In an explicit manner we show that the dissipative coefficient obtained from classical equations of motion of the detector and the scalar (or Dirac) field satisfies the FD relation associated with the vacuum noise of the field, which demonstrates that the Terashima's prescription works properly in the $N$-dimensional de Sitter spacetime. This practice is useful not only to reconfirm the validity of the use of the retarded Green function to evaluate the dissipative coefficient from the classical equations of motion but also to understand why the derivation works properly, which is discussed in connection with previous investigations on the basis on the Kubo-Martin-Schwinger (KMS) condition. Possible application to black hole spacetime is also briefly discussed. 
  Hawking effect of Dirac particles in a variable-mass Kerr space-time is investigated by using method of the generalized tortoise coordinate transformation. The location and the temperature of event horizon of the non-stationary Kerr black hole are derived. It is shown that the temperature and the shape of event horizon depend not only on the time but also on the polar angle. However, our results demonstrate that the Fermi-Dirac spectrum displays a new spin-rotation effect which is absent from that of Bose-Einstein distribution. 
  The energy-momentum tensor of the Li\'enard-Wiechert field is split into bound and emitted parts in the Rindler frame, by generalizing the reasoning of Teitelboim applied in the inertial frame. Our analysis proceeds by invoking the concept of ``energy'' defined with respect to the Killing vector field attached to the frame. We obtain the radiation formula in the Rindler frame (the Rindler version of the Larmor formula), and it is found that the radiation power is proportional to the square of acceleration $\alpha^\mu$ of the charge relative to the Rindler frame. This result leads us to split the Li\'enard-Wiechert field into a part II', which is linear in $\alpha^\mu$, and a part I', which is independent of $\alpha^\mu$. By using these, we split the energy-momentum tensor into two parts. We find that these are properly interpreted as the emitted and bound parts of the tensor in the Rindler frame. In our identification of radiation, a charge radiates neither in the case that the charge is fixed in the Rindler frame, nor in the case that the charge satisfies the equation $\alpha^\mu=0$. We then investigate this equation. We consider four gedanken experiments related to the observer dependence of the concept of radiation. 
  This is the first in a series of papers on the construction of explicit solutions to the stationary axisymmetric Einstein equations which describe counter-rotating disks of dust. These disks can serve as models for certain galaxies and accretion disks in astrophysics. We review the Newtonian theory for disks using Riemann-Hilbert methods which can be extended to some extent to the relativistic case where they lead to modular functions on Riemann surfaces. In the case of compact surfaces these are Korotkin's finite gap solutions which we will discuss in this paper. On the axis we establish for general genus relations between the metric functions and hence the multipoles which are enforced by the underlying hyperelliptic Riemann surface. Generalizing these results to the whole spacetime we are able in principle to study the classes of boundary value problems which can be solved on a given Riemann surface. We investigate the cases of genus 1 and 2 of the Riemann surface in detail and construct the explicit solution for a family of disks with constant angular velocity and constant relative energy density which was announced in a previous Physical Review Letter. 
  We analyse the Schwarzschild solution in the context of the historical development of its present use, and explain the invariant definition of a singular surface at the Schwarzschild's radius, that can be applied to the Kerr-Newman solution too. 
  I review the development of numerical evolution codes for general relativity based upon the characteristic initial value problem. Progress is traced from the early stage of 1D feasibility studies to current 3D black codes that simulate binary black holes. A prime application of characteristic evolution is Cauchy-characteristic matching, which is also reviewed. 
  Cosmic strings inside Schwarzschild black holes in teleparallel gravity are considered.Torsion flux inside the black hole is compute like a torsion vortex on a superfluid.Since some components of torsion are singular on Schwarzschild horizon and others remain finite we compute a torsion invariant to decide whether the torsion is singular and where torsion singularities located.It is found out that as in the curvature case of Einstein's black hole the event horizon is not a mere coordinate singularity for torsion although a true torsion singularity is found at the center of the teleparallel black hole.Torsion flux vanishes along the cosmic string itself.It is shown from Cartan equation in differential forms that the spins inside the black hole are polarized along the torsion string.Torsion string seems to be confined inside the black hole. 
  Motion of an ultra-relativistic perfect fluid in space-time with the Kasner metrics is investigated by the Hamiltonian method. It is found that in the limit of small times a tendency takes place to formation of strong inhomogeneities in matter distribution. In the case of slow flows the effect of non-stationary anisotropy on dynamics of sound waves and behaviour of frozen-in vortices is considered. It is shown that hydrodynamics of slow vortices on the static homogeneous background is equivalent to the usual Eulerian incompressible hydrodynamics, but in the presence of an external non-stationary strain velocity field. 
  For causal graphs we propose a definition of proper time which for small scales is based on the concept of volume, while for large scales the usual definition of length is applied. The scale where the change from "volume" to "length" occurs is related to the size of a dynamical clock and defines a natural cut-off for this type of clock. By changing the cut-off volume we may probe the geometry of the causal graph on different scales and therey define a continuum limit. This provides an alternative to the standard coarse graining procedures. For regular causal lattice (like e.g. the 2-dim. light-cone lattice) this concept can be proven to lead to a Minkowski structure. An illustrative example of this approach is provided by the breather solutions of the Sine-Gordon model on a 2-dimensional light-cone lattice. 
  A teleparallel G\"{o}del universe is shown to lead to a simple and natural relation between Cartan torsion and the global rotation of the universe.This is straightforward if one uses the formalism of Cartan's calculus of exterior differential forms.It is possible to place a limit on the global rotation from the knowledge of torsion and the converse is also true.The Obukhov recent limit of $\frac{\omega}{H_{0}}=1$ leads to the well know value for cosmological torsion of $10^{-17} cm^{-1}$.COBE constraints on the temperature anisotropy allows us to obtain limits on the torsion fluctuation in G\"{o}del's universe. 
  Some of the parameters we call ``constants of Nature'' may in fact be variables related to the local values of some dynamical fields. During inflation, these variables are randomized by quantum fluctuations. In cases when the variable in question (call it $\chi$) takes values in a continuous range, all thermalized regions in the universe are statistically equivalent, and a gauge invariant procedure for calculating the probability distribution for $\chi$ is known. This is the so-called ``spherical cutoff method''. In order to find the probability distribution for $\chi$ it suffices to consider a large spherical patch in a single thermalized region. Here, we generalize this method to the case when the range of $\chi$ is discontinuous and there are several different types of thermalized region. We first formulate a set of requirements that any such generalization should satisfy, and then introduce a prescription that meets all the requirements. We finally apply this prescription to calculate the relative probability for different bubble universes in the open inflation scenario. 
  Often it is asserted that only by using of the symmetric Landau-Lifschitz energy-momentum complex one is able to formulate a conserved angular momentum complex in General Relativity ({\bf GR}). Obviously, it is an uncorrect statement. For example, years ago, Bergmann and Thomson have given other, very useful expression on angular momentum. This expression is closely joined to the non-symmetric, Einstein canonical energy-momentum complex. In the paper we review the Bergmann-Thomson angular momentum complex and compare it with that of given by Landau and Lifschitz. 
  We review the canonical superenergy tensor and the canonical angular supermomentum tensors in general relativity and calculate them for space-time homogeneous G\"odel universes to show that both of these tensors do not, in general, vanish. We consider both an original dust-filled pressureless acausal G\"odel model of 1949 and a scalar-field-filled causal G\"odel model of Rebou\c cas and Tiomno. For the acausal model, the non-vanishing components of superenergy of matter are different from those of gravitation. The angular supermomentum tensors of matter and gravitation do not vanish either which simply reflects the fact that G\"odel universe rotates. However, the axial (totally antisymmetric) and vectorial parts of supermomentum tensors vanish. It is interesting that superenergetic quantities are {\it sensitive} to causality in a way that superenergy density $_g S_{00}$ of gravitation in the acausal model is {\it positive}, while superenergy density $_g S_{00}$ in the causal model is {\it negative}. That means superenergetic quantities might serve as criterion of causality in cosmology and prove useful. 
  We obtain observational upper bounds on a class of quantum gravity related birefringence effects, by analyzing the presence of linear polarization in the optical and ultraviolet spectrum of some distant sources. In the notation of Gambini and Pullin we find $\chi < 10^{-3}$. 
  Closed Time-like curves (CTC) in Cosmic strings in teleparallel $T_{4}$ gravity are forbidden.This result shown here in $T_{4}$ was shown by Soleng (Phys.Rev.D49 (1994)1124) also to be valid in Einstein-Cartan (EC) gravity.Here we show that in $T_{4}$ to allow for CTC we are also led to a lower bound on the angular momentum of the cosmic string.This result is obtained by matching the interior $T_{4}$ solution to a General Relativity (GR) vacuum solution.One of the main differences of the present report and the one by Soleng is that here the interior symmetric solution does not have necessary polarized spins but only Cartan torsion in the spirit of teleparallelism.Torsion flux is computed and it is show that the center of cylinder singularity corresponds to a 2+1 spacetime rotating point particle in $T_{4}$.Therefore the possibility of building time machines seems to be strongly constraint than in the case of EC gravity. 
  Static spherically symmetric solution in a background spacetime with torsion is derived explicitly. The torsion considered here is identified with the field strength of a second rank antisymmetric tensor field namely the Kalb-Ramond field and the proposed solution therefore has much significance in a string inspired gravitational field theory. 
  This is the third in a series of papers on the construction of explicit solutions to the stationary axisymmetric Einstein equations which can be interpreted as counter-rotating disks of dust. We discuss the physical properties of a class of solutions to the Einstein equations for disks with constant angular velocity and constant relative density which was constructed in the first part. The metric for these spacetimes is given in terms of theta functions on a Riemann surface of genus 2. It is parameterized by two physical parameters, the central redshift and the relative density of the two counter-rotating streams in the disk. We discuss the dependence of the metric on these parameters using a combination of analytical and numerical methods. Interesting limiting cases are the Maclaurin disk in the Newtonian limit, the static limit which gives a solution of the Morgan and Morgan class and the limit of a disk without counter-rotation. We study the mass and the angular momentum of the spacetime. At the disk we discuss the energy-momentum tensor, i.e. the angular velocities of the dust streams and the energy density of the disk. The solutions have ergospheres in strongly relativistic situations. The ultrarelativistic limit of the solution in which the central redshift diverges is discussed in detail: In the case of two counter-rotating dust components in the disk, the solutions describe a disk with diverging central density but finite mass. In the case of a disk made up of one component, the exterior of the disks can be interpreted as the extreme Kerr solution. 
  A locally finite, causal and quantal substitute for a locally Minkowskian principal fiber bundle $\cal{P}$ of modules of Cartan differential forms $\omg$ over a bounded region $X$ of a curved $C^{\infty}$-smooth differential manifold spacetime $M$ with structure group ${\bf G}$ that of orthochronous Lorentz transformations $L^{+}:=SO(1,3)^{\uparrow}$, is presented. ${\cal{P}}$ is the structure on which classical Lorentzian gravity, regarded as a Yang-Mills type of gauge theory of a $sl(2,\com)$-valued connection 1-form $\cal{A}$, is usually formulated. The mathematical structure employed to model this replacement of ${\cal{P}}$ is a principal finitary spacetime sheaf $\vec{\cal{P}}_{n}$ of quantum causal sets $\amg_{n}$ with structure group ${\bf G}_{n}$, which is a finitary version of the group ${\bf G}$ of local symmetries of General Relativity, and a finitary Lie algebra ${\bf g}_{n}$-valued connection 1-form ${\cal{A}}_{n}$ on it, which is a section of its sub-sheaf $\amg^{1}_{n}$. ${\cal{A}}_{n}$ is physically interpreted as the dynamical field of a locally finite quantum causality, while its associated curvature ${\cal{F}}_{n}$, as some sort of `finitary Lorentzian quantum gravity. 
  We present the construction of the partition function of 3-dimensional gravity in the Lorentzian regime as a state sum model over a triangulation. This generalize the work of Ponzano and Regge to the case of Lorentzian signature. 
  We solve the geodesic deviation equations for the orbital motions in the Schwarzschild metric which are close to a circular orbit. It turns out that in this particular case the equations reduce to a linear system, which after diagonalization describes just a collection of harmonic oscillators, with two characteristic frequencies. The new geodesic obtained by adding this solution to the circular one, describes not only the linear approximation of Kepler's laws, but gives also the right value of the perihelion advance (in the limit of almost circular orbits). We derive also the equations for higher-order deviations and show how these equations lead to better approximations, including the non-linear effects. The approximate orbital solutions are then inserted into the quadrupole formula to estimate the gravitational radiation from non-circular orbits. 
  We study the scattering of massless scalar waves by a Kerr black hole by letting plane monochromatic waves impinge on the black hole. We calculate the relevant scattering phase-shifts using the  Pruefer phase-function method, which is computationally efficient and reliable also for high frequencies and/or large values of the angular multipole indices (l,m). We use the obtained phase-shifts and the partial-wave approach to determine differential cross sections and deflection functions. Results for off-axis scattering (waves incident along directions misaligned with the black hole's rotation axis) are obtained for the first time. Inspection of the off-axis deflection functions reveals the same scattering phenomena as in Schwarzschild scattering. In particular, the cross sections are dominated by the glory effect and the forward (Coulomb) divergence due to the long-range nature of the gravitational field. In the rotating case the overall diffraction pattern is ``frame-dragged'' and as a result the glory maximum is not observed in the exact backward direction. We discuss the physical reason for this behaviour, and explain it in terms of the distinction between prograde and retrograde motion in the Kerr gravitational field. Finally, we also discuss the possible influence of the so-called superradiance effect on the scattered waves. 
  Rapporteur's Introduction to the GT8 session of the Ninth Marcel Grossmann Meeting (Rome, 2000); to appear in the Proceedings. 
  In pre-big-bang string cosmology one uses a phase of dilaton-driven inflation to stretch an initial (microscopic) spatial patch to the (much larger) size of the big-bang fireball. We show that the dilaton-driven inflationary phase does not naturally iron out the initial classical tensor inhomogeneities unless the initial value of the string coupling is smaller than 10^(-35). 
  The effect of the expanding universe on planetary motion is considered to first order in the Hubble constant H. Orbital elements are shown to be unaffected, but there is a small change in the connection between planetary proper time and coordinate time. This can produce an apparent anomalous acceleration in velocities inferred from echo-ranging, but the effect is too small by many orders of magnitude to account for the Pioneer 10/11 anomaly. 
  Self-dual solitons of Chern-Simons Higgs theory are examined in curved spacetime. We derive duality transformation of the Einstein Chern-Simons Higgs theory within path integral formalism and study various aspects of dual formulation including derivation of Bogomolnyi type bound. We find all possible rotationally-symmetric soliton configurations carrying magnetic flux and angular momentum when underlying spatial manifolds of these objects comprise a cone, a cylinder, and a two sphere. 
  The Machian cosmological solution satisfying $\phi =O(\rho /\omega)$ is discussed for the homogeneous and isotropic universe with a perfect fluid (with negative pressure) in the generalized scalar-tensor theory of gravitation. We propose $\omega (\phi)=\eta /(\xi -2)$ for the coupling function in the Machian point of view. The parameter $\xi$ varies in time very slowly from $\xi =0$ to $\xi =2$ because of the physical evolution of matter in the universe. When $\xi \to 2$, the coupling function diverges to $-\infty$ and the scalar field $\phi$ converges to $G_{\infty}^{-1}$. The present mass density is precisely predicted if the present time of the universe is given. We obtain $\rho_{0}=1.6\times 10^{-29} g.cm^{-3}$ for $t_{0}=1.5\times 10^{10} yr$. The universe shows the slowly decelerating expansion for the time-varying coupling function. 
  In this work we establish a relationship between Cartan's geometric approach to third order ODEs and the 3-dim Null Surface Formulation (NSF). We then generalize both constructions to allow for caustics and singularities that necessarily arise in these formalisms. 
  We use Padoa's principle of independence of primitive symbols in axiomatic systems in order to discuss the mathematical role of time and space-time in some classical physical theories. We show that time is eliminable in Newtonian mechanics and that space-time is also dispensable in Hamiltonian mechanics, Maxwell's electromagnetic theory, the Dirac electron, classical gauge fields, and general relativity. 
  The notion of finitary spacetime sheaves is introduced based on locally finite approximations of the continuous topology of a bounded region of a spacetime manifold. Finitary spacetime sheaves are seen to be sound mathematical models of approximations of continuous spacetime observables. 
  We consider the effect of backreaction of quantized massive fields on the metric of extreme black holes (EBH). We find the analytical approximate expression for the stress-energy tensor for a scalar (with an arbitrary coupling), spinor and vector fields near an event horizon. We show that, independent of a concrete type of EBH, the energy measured by a freely falling observer is finite on the horizon, so that quantum backreaction is consistent with the existence of EBH. For the Reissner-Nordstrom EBH with a total mass M_{tot} and charge Q we show that for all cases of physical interest M_{tot}< Q. We also discuss different types of quantum-corrected Bertotti-Robinson spacetimes, find for them exact self-consistent solutions and consider situations in which tiny quantum corrections lead to the qualitative change of the classical geometry and topology. In all cases one should start not from a classical background with further adding quantum corrections but from the quantum-corrected self-consistent geometries from the very beginning. 
  We study the transition amplitudes in state-sum models of quantum gravity in D=2,3,4 spacetime dimensions by using the field theory over a Lie group formulation. By promoting the group theory Fourier modes into creation and annihilation operators we construct a Fock space for the quantum field theory whose Feynman diagrams give the transition amplitudes. By making products of the Fourier modes we construct operators and states representing the spin networks associated to triangulations of spatial boundaries of a triangulated spacetime manifold. The corresponding spin network amplitudes give the state-sum amplitudes for triangulated manifolds with boundaries. We also show that one can introduce a discrete time evolution operator, where the time is given by the number of D-simplices in the triangulation, or equivalently by the number of the vertices in the Feynman diagram. The corresponding transition amplitude is a finite sum of Feynman diagrams, and in this way one avoids the problem of infinite amplitudes caused by summing over all possible triangulations. 
  The semiclassical Einstein equations are solved to first order in $\epsilon = \hbar/M^2$ for the case of an extreme or nearly extreme Reissner-Nordstr\"{o}m black hole perturbed by the vacuum stress-energy of quantized free fields. It is shown that, for realistic fields of spin 0, 1/2, or 1, any zero temperature black hole solution to the equations must have an event horizon at $r_h < |Q|$, with $Q$ the charge of the black hole. It is further shown that no black hole solutions with $r_h < |Q|$ can be obtained by solving the semiclassical Einstein equations perturbatively. 
  Vacuum gravitational fields invariant for a bidimensional non Abelian Lie algebra of Killing fields, are explicitly described. They are parameterized either by solutions of a transcendental equation (the tortoise equation) or by solutions of a linear second order differential equation on the plane. Gravitational fields determined via the tortoise equation, are invariant for a 3-dimensional Lie algebra of Killing fields with bidimensional leaves. Global gravitational fields out of local ones are also constructed. 
  Contents. 1. Introduction. 2. Kinematics of a Material Medium: Material Representation. 3. Kinematics of a Material Medium: Convected Differentials. 4. Kinematics of a Perfect Elastic Medium. 5. Small Gravitational Perturbations of an Elastic Medium. 
  Several recent surprises appear dramatically to have improved the likelihood that the spin of rapidly rotating, newly formed neutron stars (and, possibly, of old stars spun up by accretion) is limited by a nonaxisymmetric instability driven by gravitational waves. Except for the earliest part of the spin-down, the axial l=m=2 mode (an r-mode) dominates the instability, and the emitted waves may be observable by detectors with the sensitivity of LIGO II. A review of these hopeful results is followed by a discussion of constraints on the instability set by dissipative mechanisms, including viscosity, nonlinear saturation, and energy loss to a magnetic field driven by differential rotation. 
  One can get the impression from the Reissner-Nordstrom solution of Einstein's equations that the charge of a body reduces its gravitational field. This looks surprising since the energy of the electrostatic field surrounding a charged body, must contribute positively, as an additional, "electromagnetic mass", to the gravitational field produced by the body. We resolve this puzzle by showing that the mass M in the Reissner-Nordstrom solution is not the "bare mass" of the body, but its "renormalized mass". I. e. M, in addition to the bare mass, includes the total electromagnetic mass of the body. But at finite distances from the body only a part of the electromagnetic mass contributes to the gravitational field. That is why the gravity of a charged body is determined by the quantity smaller than M. 
  We consider a quantum gravity register that is a particular quantum memory register which grows with time, and whose qubits are pixels of area of quantum de Sitter horizons. At each time step, the vacuum state of this quantum register grows because of the uncertainty in quantum information induced by the vacuum quantum fluctuations. The resulting virtual states, (responsible for the speed up of growth, i.e., inflation), are operated on by quantum logic gates and transformed into qubits. The model of quantum growing network (QGN) described here is exactly solvable, and (apart from its cosmological implications), can be regarded as the first attempt toward a future model for the quantum World-Wide Web. We also show that the bound on the speed of computation, the bound on clock precision, and the holographic bound, are saturated by the QGN. 
  The cosmological constant $(1/2)\lambda_{1}\phi_{, \mu}\phi ^{, \mu}/\phi ^{2}$ is introduced to the generalized scalar-tensor theory of gravitation with the coupling function $\omega (\phi)=\eta /(\xi -2)$ and the Machian cosmological solution satisfying $\phi =O(\rho /\omega)$ is discussed for the homogeneous and isotropic universe with a perfect fluid (with negative pressure). We require the closed model and the negative coupling function for the attractive gravitational force. The constraint $% \omega (\phi)<-3/2$ for $0\leqq \xi <2$ leads to $\eta >3$. If $\lambda_{1}<0$ and $0\leqq -\eta /\lambda_{1}<2$, the universe shows the slowly accelerating expansion. The coupling function diverges to $-\infty $ and the scalar field $\phi $ converges to $G_{\infty}^{-1}$ when $\xi \to 2$ ($t\to +\infty $). The cosmological constant decays in proportion to $t^{-2}$. Thus the Machian cosmological model approaches to the Friedmann universe in general relativity with $\ddot{a}=0$, $\lambda =0$, and $p=-\rho /3$ as $t\to +\infty $. General relativity is locally valid enough at present. 
  We consider a FRW cosmological model with an exotic fluid known as Chaplygin gas. We show that the resulting evolution of the universe is not in disagreement with the current observation of cosmic acceleration. The model predict an increasing value for the effective cosmological constant. 
  A Riemann problem with prescribed initial conditions will produce one of three possible wave patterns corresponding to the propagation of the different discontinuities that will be produced once the system is allowed to relax. In general, when solving the Riemann problem numerically, the determination of the specific wave pattern produced is obtained through some initial guess which can be successively discarded or improved. We here discuss a new procedure, suitable for implementation in an exact Riemann solver in one dimension, which removes the initial ambiguity in the wave pattern. In particular we focus our attention on the relativistic velocity jump between the two initial states and use this to determine, through some analytic conditions, the wave pattern produced by the decay of the initial discontinuity. The exact Riemann problem is then solved by means of calculating the root of a nonlinear equation. Interestingly, in the case of two rarefaction waves, this root can even be found analytically. Our procedure is straightforward to implement numerically and improves the efficiency of numerical codes based on exact Riemann solvers. 
  The design and test of a detector of small harmonic displacements is presented. The detector is based on the principle of the parametric conversion of power between the resonant modes of two superconducting coupled microwave cavities. The work is based on the original ideas of Bernard, Pegoraro, Picasso and Radicati, who, in 1978, suggested that superconducting coupled cavities could be used as sensitive detectors of gravitational waves, and on the work of Reece, Reiner and Melissinos, who, {in 1984}, built a detector of this kind. They showed that an harmonic modulation of the cavity length l produced an energy transfer between two modes of the cavity, provided that the frequency of the modulation was equal to the frequency difference of the two modes. They achieved a sensitivity to fractional deformations of dl/l~10^{-17} Hz^{-1/2}. We repeated the Reece, Reiner and Melissinos experiment, and with an improved experimental configuration and better cavity quality, increased the sensitivity to dl/l~10^{-20} Hz^{-1/2}. In this paper the basic principles of the device are discussed and the experimental technique is explained in detail. Possible future developments, aiming at gravitational waves detection, are also outlined. 
  Two possible interpretations of FRW cosmologies (perfect fluid or dissipative fluid)are considered as consecutive phases of the system. Necessary conditions are found, for the transition from perfect fluid to dissipative regime to occur, bringing out the conspicuous role played by a particular state of the system (the ''critical point ''). 
  We study the self force acting on a particle endowed with scalar charge, which is held static (with respect to an undragged, static observer at infinity) outside a stationary, axially-symmetric black hole. We find that the acceleration due to the self force is in the same direction as the black hole's spin, and diverges when the particle approaches the outer boundary of the black hole's ergosphere. This acceleration diverges more rapidly approaching the ergosphere's boundary than the particle's acceleration in the absence of the self force. At the leading order this self force is a (post)$^2$-Newtonian effect. For scalar charges with high charge-to-mass ratio, the acceleration due to the self force starts dominating over the regular acceleration already far from the black hole. The self force is proportional to the rate at which the black hole's rotational energy is dissipated. This self force is local (i.e., only the Abraham-Lorentz-Dirac force and the local coupling to Ricci curvature contribute to it). The non-local, tail part of the self force is zero. 
  Analyzing the static spherically symmetric and rotating ellipsoid solutions in the Newtonian limit of Jordan, Brance - Dicke theory we find the following. In empty space scalar-tensor theories have trivial solution of field equation with constant scalar potential (efficient value of gravitation constant). In this case no celestial-mechanical experiments to reveal a difference between scalar-tensor theories and Einstein theory is not presented possible. However, scalar field, inside the matter, has characteristics like gravitation permeability of material similar electromagnetic permeability of material in Maxwell theories of electromagnetism. Investigation of obtained exact solutions for given functions of a matter distributions in the Newtonian limit of Jordan, Brance - Dicke theory show the efficient value of gravitation constant depends on density of matter, sizes and form of object, as well as on the value of theories coupling constant. That for example led to weakening gravitation force in the central regions of a Galaxies. This assumption constitutes the way to explain observed rotation curves of Galaxies without using cold dark matter. 
  Strictly respecting the Einstein equations and supposing space-time is a medium, we derive the deformation of this medium by gravity. We derive the deformation in case of infinite plane, Robertson-Walker manifold, Schwarzschild manifold and gravitational waves. Some singularities are removed or changed. We call this procedure renormalization of gravity. We show that some results following from the classical gravity must be modified. 
  A partially first-order form of the characteristic formulation is introduced to control the accuracy in the computation of gravitational waveforms produced by highly distorted single black hole spacetimes. Our approach is to reduce the system of equations to first-order differential form on the angular derivatives, while retaining the proven radial and time integration schemes of the standard characteristic formulation. This results in significantly improved accuracy over the standard mixed-order approach in the extremely nonlinear post-merger regime of binary black hole collisions. 
  We calculate the asymptotic behavior of the curvature scalar $(Riemann)^2$ near the null weak singularity at the inner horizon of a generic spinning black hole, and show that this scalar oscillates infinite number of times while diverging. The dominant parallelly-propagated Riemann components oscillate in a similar manner. This oscillatory behavior, which is in a remarkable contrast to the monotonic mass-inflation singularity in spherical charged black holes, is caused by the dragging of inertial frames due to the black- hole's spin. 
  The Machian cosmological solution satisfying $\phi =O(\rho /\omega)$ in the generalized scalar-tensor theory of gravitation with the varying cosmological constant is summarized. The scalar field $\phi $ with the exponential potential is introduced as dark matter and the barotropic evolution of matter in the universe is discussed. As the universe expands, the coefficient $\gamma $ of the equation of state approaches to -1/3 and the coupling function $\omega (\phi)$ diverges to $-\infty $. 
  A few years ago, Cornish, Spergel and Starkman (CSS), suggested that a multiply connected ``small'' Universe could allow for classical chaotic mixing as a pre-inflationary homogenization process. The smaller the volume, the more important the process. Also, a smaller Universe has a greater probability of being spontaneously created. Previously DeWitt, Hart and Isham (DHI) calculated the Casimir energy for static multiply connected flat space-times. Due to the interest in small volume hyperbolic Universes (e.g. CSS), we generalize the DHI calculation by making a a numerical investigation of the Casimir energy for a conformally coupled, massive scalar field in a static Universe, whose spatial sections are the Weeks manifold, the smallest Universe of negative curvature known. In spite of being a numerical calculation, our result is in fact exact. It is shown that there is spontaneous vacuum excitation of low multipolar components. 
  A new solution for the stationary closed world with rigid rotation is obtained for the spinning fluid source. It is found that the spin and vorticity are locally balanced. This model qualitatively shows that the local rotation of the cosmological matter can be indeed related to the global cosmic vorticity, provided the total angular momentum of the closed world is vanishing. 
  In the framework of metric-free electrodynamics, we start with a {\em linear} spacetime relation between the excitation 2-form $H = ({\cal D}, {\cal H})$ and the field strength 2-form $F = ({E,B})$. This linear relation is constrained by the so-called closure relation. We solve this system algebraically and extend a previous analysis such as to include also singular solutions. Using the recently derived fourth order {\em Fresnel} equation describing the propagation of electromagnetic waves in a general {\em linear} medium, we find that for all solutions the fourth order surface reduces to a light cone. Therefrom we derive the corresponding metric up to a conformal factor. 
  The obstruction for the existence of an energy momentum tensor for the gravitational field is connected with differential-geometric features of the Riemannian manifold. It has not to be valid for alternative geometrical structures. In this article a general 3-parameter class of teleparallel models is considered. The field equation turns out to have a form completely similar to the Maxwell field equation $d*\F^a=\T^a$. By applying the Noether procedure, the source 3-form $\T^a$ is shown to be connected with the diffeomorphism invariance of the Lagrangian. Thus the source of the coframe field is interpreted as the total conserved energy-momentum current of the system. A reduction of the conserved current to the Noether current and the Noether charge for the coframe field is provided. An energy-momentum tensor for the coframe field is defined in a diffeomorphism invariant and a translational covariant way. The total energy-momentum current of a system is conserved. Thus a redistribution of the energy-momentum current between material and coframe (gravity) field is possible in principle, unlike as in GR. The energy-momentum tensor is calculated for various teleparallel models: the pure Yang-Mills type model, the anti-Yang-Mills type model and the generalized teleparallel equivalent of GR. The latter case can serve as a very close alternative to the GR description of gravity. 
  We generalize to the case of spinning black holes a recently introduced ``effective one-body'' approach to the general relativistic dynamics of binary systems. The combination of the effective one-body approach, and of a Pad\'e definition of some crucial effective radial functions, is shown to define a dynamics with much improved post-Newtonian convergence properties, even for black hole separations of the order of $6 GM / c^2$. We discuss the approximate existence of a two-parameter family of ``spherical orbits'' (with constant radius), and, of a corresponding one-parameter family of ``last stable spherical orbits'' (LSSO). These orbits are of special interest for forthcoming LIGO/VIRGO/GEO gravitational wave observations. It is argued that for most (but not all) of the parameter space of two spinning holes the effective one-body approach gives a reliable analytical tool for describing the dynamics of the last orbits before coalescence. This tool predicts, in a quantitative way, how certain spin orientations increase the binding energy of the LSSO. This leads to a detection bias, in LIGO/VIRGO/GEO observations, favouring spinning black hole systems, and makes it urgent to complete the conservative effective one-body dynamics given here by adding (resummed) radiation reaction effects, and by constructing gravitational waveform templates that include spin effects. Finally, our approach predicts that the spin of the final hole formed by the coalescence of two arbitrarily spinning holes never approaches extremality. 
  We present a brief synopsis of related work (gr-qc/0007039), describing a study of black hole threshold phenomena for a self-gravitating, massive complex scalar field in spherical symmetry. We construct Type I critical solutions dynamically by tuning a one-parameter family of initial data consisting of a boson star and a massless real scalar field, and numerically evolving this data. The resulting critical solutions appear to correspond to boson stars on the unstable branch, as we show via comparisons between our simulations and perturbation theory. For low-mass critical solutions, we find small ``halos'' of matter in the tails of the solutions, and these distort the profiles which otherwise agree with unstable boson stars. These halos seem to be artifacts of the collisions between the original boson stars and the massless fields, and do not appear to belong to the true critical solutions. From this study, it appears that unstable boson stars are unstable to dispersal (``explosion'') in addition to black hole formation. Given the similarities in macroscopic stability between boson stars and neutron stars, we suggest that similar phenomena could occur in models of neutron stars. 
  We complete a metric-free axiomatic framework for electrodynamics by introducing the appropriate energy-momentum current Sigma of the electromagnetic field. We start from the Lorentz force density and motivate the form of Sigma. Then we postulate it (fourth axiom) and discuss its properties. In particular, it is found that Sigma is traceless and invariant under an electric-magnetic reciprocity transformation. By using the Maxwell-Lorentz spacetime relation (fifth axiom), Sigma is also shown to be symmetric, that is, it has 9 independent components 
  A simple and surprisingly realistic model of the origin of the universe can be developed using the Friedmann equation from general relativity, elementary quantum mechanics, and the experimental values of h, c, G and the proton mass. The model assumes there are N space dimensions (with N > 6) and the potential constraining the radius r of the invisible N -3 compact dimensions varies as r^4. In this model, the universe has zero total energy and is created from nothing. There is no initial singularity. If space-time is eleven dimensional, as required by M theory, the scalar field corresponding to the size of the compact dimensions inflates the universe by about 26 orders of magnitude (60 e-folds). If the Hubble constant is 65 km/sec Mpc, the energy density of the scalar field after inflation results in Omega-sub-Lambda = 0.68, in agreement with recent astrophysical observations. 
  We describe the ``deflationary'' evolution from an initial de Sitter phase to a subsequent Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW) period as a specific non-equilibrium configuration of a self-interacting gas. The transition dynamics corresponds to a conformal, timelike symmetry of an ``optical'' metric, characterized by a refraction index of the cosmic medium which continously decreases from a very large initial value to unity in the FLRW phase. 
  The basics of the relativistic astrophysics including the celestial mechanics in weak field, black holes and cosmological models are illustrated and analyzed by means of Maple 6 
  We re-express gravitational wave results in terms of post-Newtonian parameters. Using these expressions, and some simplifying assumptions, we compute that in a favorable case, i.e. a ten solar-mass black hole spiraling in to a 10^6 solar-mass black hole, LISA observations will be able to constrain at the 10% level or better a single combination of post-post-Newtonian parameters one order higher than those already constrained by solar system evidence. This significant constraint will be possible even if the signal-to-noise level is so low that the signal can only be found by matched filtering, and hence only deviations between alternate signal interpretations of order one half cycle or more can be detected. 
  That preferred-frame theory accounts for special relativity and reduces to it if the gravitation field cancels. Starting from an interpretation of gravity as a pressure force, it is based on just one scalar field. This scalar gives the relation between the flat "background" metric and the curved "physical" metric, due to an equivalence principle between the absolute effects of motion and gravitation. The scalar is also a potential for the gravity acceleration vector. Motion is governed by an extension of the special-relativistic form of Newton's second law. This provides a new equation for continuum dynamics, that gives the gravitational modification of Maxwell's equations, consistent with photon dynamics. The same effects on light rays as in GR are predicted at the post-Newtonian approximation (PNA). An asymptotic PNA is being studied, in order to build a consistent celestial mechanics in the theory. The cosmic acceleration is predicted and nonsingular cosmological models are obtained. 
  Black hole mechanics was recently extended by replacing the more commonly used event horizons in stationary space-times with isolated horizons in more general space-times (which may admit radiation arbitrarily close to black holes). However, so far the detailed analysis has been restricted to non-rotating black holes (although it incorporated arbitrary distortion, as well as electromagnetic, Yang-Mills and dilatonic charges). We now fill this gap by first introducing the notion of isolated horizon angular momentum and then extending the first law to the rotating case. 
  The contribution provides a comparison of consequences stemming from D-brane theories and Expansive Nondecelerative Universe model, and calls attention to coincidence of the results arising from the mentioned approaches to a description of the Universe. It follows from the comparison that the effects of quantum gravitation should appear at the energy near to 2 TeV. 
  Stemming from relationships between a number of information describing a system and entropy content of the system it is possible to determine maximal cosmological time. The contribution manifests a compatibility of the superstring theory and the model of Expansive Nondecelerative Universe. 
  It is known that General Relativity ({\bf GR}) uses Lorentzian Manifold $(M_4;g)$ as a geometrical model of the physical space-time. $M_4$ means here a four-dimensional differentiable manifold endowed with Lorentzian metric $g$. The metric $g$ satisfies Einstein equations. Since the 1970s many authors have tried to generalize this geometrical model of the physical space-time by introducing torsion and even more general metric-affine geometry. In this paper we discuss status of torsion in the theory of gravity. At first, we emphasize that up to now we have no experimental evidence for the existence of torsion in Nature. Contrary, the all experiments performed in weak gravitational field (Solar System) or in strong regime (binary pulsars) and tests of the Einstein Equivalence Principle ({\bf EEP}) confirmed {\bf GR} and Lorentzian manifold $(M_4;g)$ as correct geometrical model of the physical space-time. Then, we give theoretical arguments against introducing of torsion into geometrical model of the physical space-time. At last, we conclude that the general-relativistic model of the physical space-time is sufficient and it seems to be the most satisfactory. 
  We prove the existence of a family of initial data for the Einstein vacuum equation which can be interpreted as the data for two Kerr-like black holes in arbitrary location and with spin in arbitrary direction. This family of initial data has the following properties: (i) When the mass parameter of one of them is zero or when the distance between them goes to infinity, it reduces exactly to the Kerr initial data. (ii) When the distance between them is zero, we obtain exactly a Kerr initial data with mass and angular momentum equal to the sum of the mass and angular momentum parameters of each of them. The initial data depends smoothly on the distance, the mass and the angular momentum parameters. 
  In order to perform accurate and stable long-term numerical integration of the Einstein equations, several hyperbolic systems have been proposed. We here report our numerical comparisons between weakly hyperbolic, strongly hyperbolic, and symmetric hyperbolic systems based on Ashtekar's connection variables. The primary advantage for using this connection formulation is that we can keep using the same dynamical variables for all levels of hyperbolicity.   We also study asymptotically constrained systems, "$\lambda$-system" and "adjusted system", for numerical integration of the Einstein equations, which are intended to be robust against perturbative errors for the free evolution of the initial data. These systems are tested in the Maxwell system and in the Ashtekar's system. This mechanism affects more than the system's symmetric hyperbolicity.   (This workshop contribution is the summary of our gr-qc/0005003 [CQG 17 (2000) 4799] and gr-qc/0007034 [CQG 18 (2001) 441].) 
  The current important issue in numerical relativity is to determine which formulation of the Einstein equations provides us with stable and accurate simulations. Based on our previous work on "asymptotically constrained" systems, we here present constraint propagation equations and their eigenvalues for the Arnowitt-Deser-Misner (ADM) evolution equations with additional constraint terms (adjusted terms) on the right hand side. We conjecture that the system is robust against violation of constraints if the amplification factors (eigenvalues of Fourier-component of the constraint propagation equations) are negative or pure-imaginary. We show such a system can be obtained by choosing multipliers of adjusted terms. Our discussion covers Detweiler's proposal (1987) and Frittelli's analysis (1997), and we also mention the so-called conformal-traceless ADM systems. 
  Gravitational waves in the linear approximation propagate in the Schwarzschild spacetime similarly as electromagnetic waves. A fraction of the radiation scatters off the curvature of the geometry. The energy of the backscattered part of an initially outgoing pulse of the quadrupole gravitational radiation is estimated by compact formulas depending on the initial energy, the Schwarzschild radius, and the location and width of the pulse. The backscatter becomes negligible in the short wavelength regime. 
  We study the behaviour of spin-half particles in curved space-time. Since Dirac equation gives the dynamics of spin-half particles, we mainly study the Dirac equation in Schwarzschild, Kerr, Reissner-Nordstr\"om geometry. Due to the consideration of existence of black hole in space-time (the curved space-time), particles are influenced and equation will be modified. As a result the solution will be changed from that due to flat space. 
  We discuss the possible detection of a stochastic background of massive, non-relativistic scalar particles, through the cross correlation of the two LIGO interferometers in the initial, enhanced and advanced configuration. If the frequency corresponding to the mass of the scalar field lies in the detector sensitivity band, and the non-relativistic branch of the spectrum gives a significant contribution to energy density required to close the Universe, we find that the scalar background can induce a non-negligible signal, in competition with a possible signal produced by a stochastic background of gravitational radiation. 
  The status of experimental tests of general relativity and of theoretical frameworks for analysing them are reviewed. Einstein's equivalence principle (EEP) is well supported by experiments such as the E\"otv\"os experiment, tests of special relativity, and the gravitational redshift experiment. Future tests of EEP and of the inverse square law will search for new interactions arising from unification or quantum gravity. Tests of general relativity at the post-Newtonian level have reached high precision, including the light deflection, the Shapiro time delay, the perihelion advance of Mercury, and the Nordtvedt effect in lunar motion. Gravitational wave damping has been detected to half a percent using the binary pulsar, and new binary pulsar systems may yield further improvements. When direct observation of gravitational radiation from astrophysical sources begins, new tests of general relativity will be possible. 
  The master equation for a linear open quantum system in a general environment is derived using a stochastic approach. This is an alternative derivation to that of Hu, Paz and Zhang, which was based on the direct computation of path integrals, or to that of Halliwell and Yu, based on the evolution of the Wigner function for a linear closed quantum system. We first show by using the influence functinal formalism that the reduced Wigner function for the open system coincides with a distribution function resulting from averaging both over the initial conditions and the stochastic source of a formal Langevin equation. The master equation for the reduced Wigner function can then be deduced as a Fokker-Planck equation obtained from the formal Langevin equation. 
  We explore what restrictions may impose the second law of thermodynamics on varying speed of light theories. We find that the attractor scenario solving the flatness problem is consistent with the generalized second law at late time. 
  Equations, describing the curvature and torsion of general metric-affine space G4 or (in accordance with the unified field theory) the distribution and motion of matter, are obtained. Solutions of the equations for spherically symmetric stationary model and uniform isotropic model are given for pure gravitational field and massless fluid with spin. 
  Traditionally it is assumed that the Casimir vacuum pressure does not depend on the ultraviolet cut-off. There are, however, some arguments that the effect actually depends on the regularization procedure and thus on the trans-Planckian physics. We provide the condensed matter example where the Casimir forces do explicitly depend on the microscopic (correspondingly trans-Planckian) physics due to the mesoscopic finite-N effects, where N is the number of bare particles in condensed matter (or correspondingly the number of the elements comprising the quantum vacuum). The finite-N effects lead to mesoscopic fluctuations of the vacuum pressure. The amplitude of the mesoscopic flustuations of the Casimir force in a system with linear dimension L is larger by the factor N^{1/3}\sim L/a than the traditional value of the Casimir force given by effective theory, where a is the interatomic distance which plays the role of the Planck length. 
  We study equilibrium sequences of close binary systems composed of identical polytropic stars in Newtonian gravity. The solving method is a multi-domain spectral method which we have recently developed. An improvement is introduced here for accurate computations of binary systems with stiff equation of state ($\gamma > 2$). The computations are performed for both cases of synchronized and irrotational binary systems with adiabatic indices $\gamma=3,~2.5,~2.25,~2$ and 1.8. It is found that the turning points of total energy along a constant-mass sequence appear only for $\gamma \ge 1.8$ for synchronized binary systems and $\gamma \ge 2.3$ for irrotational ones. In the synchronized case, the equilibrium sequences terminate by the contact between the two stars. On the other hand, for irrotational binaries, it is found that the sequences terminate at a mass shedding limit which corresponds to a detached configuration. 
  We study the possible asymptotically flat perturbations of Robinson-Trautman spacetimes. We differentiate between algebraically special perturbations and general perturbations. The equations that determine physically realistic spacetimes with angular momentum are presented. 
  We calculate the one-loop divergences for quantum gravity with cosmological constant, using new parametrization of quantum metric. The conformal factor of the metric is treated as an independent variable. As a result the theory possesses an additional degeneracy and one needs an extra conformal gauge fixing. We verify the on shell independence of the divergences from the parameter of the conformal gauge fixing, and find a special conformal gauge in which the divergences coincide with the ones obtained by t'Hooft and Veltman (1974). Using conformal invariance of the counterterms one can restore the divergences for the conformal metric-scalar gravity. 
  This is a brief introduction to general relativity, designed for both students and teachers of the subject. While there are many excellent expositions of general relativity, few adequately explain the geometrical meaning of the basic equation of the theory: Einstein's equation. Here we give a simple formulation of this equation in terms of the motion of freely falling test particles. We also sketch some of its consequences, and explain how the formulation given here is equivalent to the usual one in terms of tensors. Finally, we include an annotated bibliography of books, articles and websites suitable for the student of relativity. 
  In the framework of chaotic inflation we study the case of a massive scalar field using a semiclassical approach. We derive the energy density fluctuations in the infrared sector generated by matter field and gauge-invariant metric fluctuations by means of a different method. We find that the super Hubble density fluctuations increase during inflation for a massive scalar field, in agreement with the result obtained in a previous work in which a power-law expanding universe was considered. 
  The gravitational neutrino oscillation problem is studied by considering the Dirac Hamiltonian in a Riemann-Cartan space-time and calculating the dynamical phase. Torsion contributions which depend on the spin direction of the mass eigenstates are found. These effects are of the order of Planck scales. 
  The compatibility axiom in Ehlers, Pirani and Schild's (EPS) constructive axiomatics of the space-time geometry that uses light rays and freely falling particles with high velocity, is replaced by several constructions with low velocity particles only. For that purpose we describe in a space-time with a conformal structure and an arbitrary path structure the radial acceleration, a Coriolis acceleration and the zig-zag construction. Each of these quantities give effects whose requirement to vanish can be taken as alternative version of the compatibility axiom of EPS. The procedural advantage lies in the fact, that one can make null-experiments and that one only needs low velocity particles to test the compatibility axiom. We show in addition that Perlick's standard clock can exist in a Weyl space only. 
  The classical and quantum dynamics of the Friedmann-Robertson-Walker Universe with massless scalar and massive fermion matter field as a source is discussed in the framework of the Dirac generalized Hamiltonian formalism. The Hamiltonian reduction of this constrained system is realized for two cases of minimal and conformal coupling between gravity and matter. It is shown that in both cases for all values of curvature, of maximally symmetric space there exists a time independent reduced local Hamiltonian which describes the dynamics of the cosmic scale factor. The relevance of conformal time-like Killing vector fields in FRW space-time to the existence of time independent Hamiltonian and the corresponding notion of conserved energy is discussed. The extended quantization with the Wheeler-deWitt equation is compared with the canonical quantization of unconstrained system. It is shown that quantum observables treated as expectation values of the Dirac observables properly describe the original classical theory. 
  Starting from the experimental fact that light propagates over a closed path at speed c (L/c law), we show to what extent the isotropy of the speed of light can be considered a matter of convention. We prove the consistence of anisotropic and inhomogenous conventions, limiting the allowed possibilities. All conventions lead to the same physical theory even if its formulation can change in form. The mathematics involved is that of gauge theories and the choice of a simultaneity convention is interpreted as a choice of the gauge. Moreover we prove that a Euclidean space where the L/c law holds, gives rise to a spacetime with Minkowskian causal structure, and we exploit the consequences for the causal approach to the conventionality of simultaneity. 
  A theory of gravity with torsion is examined in which the torsion tensor is constructed from the exterior derivative of an antisymmetric rank two potential plus the dual of the gradient of a scalar field. Field equations for the theory are derived by demanding that the action be stationary under variations with respect to the metric, the antisymmetric potential, and the scalar field. A material action is introduced and the equations of motion are derived. The correct conservation law for rotational angular momentum plus spin is observed to hold in this theory. 
  Current physics is faced with the fundamental problem of unifying quantum theory and general relativity, which would have resulted in quantum gravity. The main effort to construct the latter has been bent on quantizing spacetime structure, in particular metric. Meanwhile, taking account of the indeterministic aspect of the quantum description of matter, which manifests itself in quantum jumps, essentially affects classical spacetime structure and the Einstein equation. Quantum jumps give rise to a family of sets of simultaneous events, which implies the existence of universal cosmological time. In view of the jumps, the requirement for metric and its time derivative to be continuous implies that the Einstein equation should involve pseudomatter along with matter. Pseudomatter manifests itself only in gravitational effects, being thereby an absolutely dark ``matter''. 
  Recent proposals for improved optical tests of Special Relativity have renewed interest in the interpretation of such tests. In this paper we discuss the interpretation of modern realizations of the Michelson-Morley experiment in the context of a new model of electrodynamics featuring a vector-valued photon mass. This model is gauge invariant, unlike massive-photon theories based on the Proca equation, and it predicts anisotropy of both the speed of light and the electric field of a point charge. The latter leads to an orientation dependence of the length of solid bodies which must be accounted for when interpreting the results of a Michelson-Morley experiment. Using a simple model of ionic solids we show that, in principle, the effect of orientation dependent length can conspire to cancel the effect of an anisotropic speed of light in a Michelson-Morley experiment, thus, complicating the interpretation of the results. 
  We consider the Lema\^{\i}tre-Tolman-Bondi metric with an inhomogeneous viscous fluid source satisfying the equation of state of an interactive mixture of radiation and matter. Assuming conditions prior to the decoupling era, we apply Extended Irreversible Thermodynamcs (EIT) to this mixture. Using the full transport equation of EIT we show that the relaxation time of shear viscosity can be several orders of magnitude larger than the Thomson collision time between photons and electrons. A comparison with the ``truncated'' transport equation for these models reveals that the latter cannot describe properly the decoupling of matter and radiation 
  We study the late-time behaviour of a dynamically perturbed rapidly rotating black hole. Considering an extreme Kerr black hole, we show that the large number of virtually undamped quasinormal modes (that exist for nonzero values of the azimuthal eigenvalue m) combine in such a way that the field (as observed at infinity) oscillates with an amplitude that decays as 1/t at late times. This is in clear contrast with the standard late time power-law fall-off familiar from studies of non-rotating black holes. This long-lived oscillating ``tail'' will, however, not be present for non-extreme (presumably more astrophysically relevant) black holes, for which we find that many quasinormal modes (individually excited to a very small amplitude) combine to give rise to an exponentially decaying field. This result could have implications for the detection of gravitational-wave signals from rapidly spinning black holes, since the required theoretical templates need to be constructed from linear combinations of many modes. Our main results are obtained analytically, but we support the conclusions with numerical time-evolutions of the Teukolsky equation. These time-evolutions provide an interesting insight into the notion that the quasinormal modes can be viewed as waves trapped in the spacetime region outside the horizon. They also suggest that a plausible mechanism for the behaviour we observe for extreme black holes is the presence of a ``superradiance resonance cavity'' immediately outside the black hole. 
  I discuss J. Barbour's Machian theories of dynamics, and his proposal that a Machian perspective enables one to solve the problem of time in quantum geometrodynamics (by saying that there is no time). I concentrate on his recent book 'The End of Time' (1999). 
  Several situations of physical importance may be modelled by linear quantum fields propagating in fixed spacetime-dependent classical background fields. For example, the quantum Dirac field in a strong and/or time-dependent external electromagnetic field accounts for the creation of electron-positron pairs out of the vacuum. Also, the theory of linear quantum fields propagating on a given background curved spacetime is the appropriate framework for the derivation of black-hole evaporation (Hawking effect) and for studying the question whether or not it is possible in principle to manufacture a time-machine. It is a well-established metatheorem that any question concerning such a linear quantum field may be reduced to a definite question concerning the corresponding classical field theory (i.e. linear hyperbolic PDE with non-constant coefficients describing the background in question) -- albeit not necessarily a question which would have arisen naturally in a purely classical context. The focus in this talk will be on the covariant Klein-Gordon equation in a fixed curved background, although we shall draw on analogies with other background field problems and with the time-dependent harmonic oscillator. The aim is to give a sketch-impression of the whole subject of Quantum Field Theory in Curved Spacetime, focussing on work with which the author has been personally involved, and also to mention some ideas and work-in-progress by the author and collaborators towards a new "semi-local" vacuum construction for this subject. A further aim is to introduce, and set into context, some recent advances in our understanding of the general structure of quantum fields in curved spacetimes which rely on classical results from microlocal analysis. 
  The bending of light in Kottler space (the Schwarzschild vacuum with cosmological constant) is examined. Unlike the advance of the perihelion, the cosmological constant produces no change in the bending of light. In this note we examine the conditions under which this statement holds. 
  In this article we examine a class of wormhole and flux tube like solutions to 5D vacuum Einstein equations. These solutions possess generic local anisotropy, and their local isotropic limit is shown to be conformally equivalent to the spherically symmetric 5D solutions of gr-qc/9807086. The anisotropic solutions investigated here have two physically distinct signatures: First, they can give rise to angular-dependent, anisotropic ``electromagnetic'' interactions. Second, they can result in a gravitational running of the ``electric'' and ``magnetic'' charges of the solutions. This gravitational running of the electromagnetic charges is linear rather than logarithmic, and could thus serve as an indirect signal for the presence of higher dimensions. The local anisotropy of these solutions is modeled using the technique of anholonomic frames with respect to which the metrics are diagonalized. If holonomic coordinates frames were used then such metrics would have off-diagonal components. 
  We investigate the generalized second law for two-dimensional black holes in equilibrium (Hartle-Hawking) and nonequilibrium (Unruh) with the heat bath surrounding the black holes. We obtain a simple expression for the change of total entropy in terms of covariant thermodynamic variables, which is valid not only for the Hartle-Hawking state but also for the Unruh state up to leading order, without assuming a quasi-stationary evolution of the black holes. Using this expression, it is shown that the rate of local entropy production is non-negative in the two-dimensional black hole systems. 
  We investigate the general asymptotic behaviour of Friedman-Robertson-Walker (FRW) models with an inflaton field, scalar-tensor FRW cosmological models and diagonal Bianchi-IX models by means of Liapunov's method. This method provides information not only about the asymptotic stability of a given equilibrium point but also about its basin of attraction. This cannot be obtained by the usual methods found in the literature, such as linear stability analysis or first order perturbation techniques. Moreover, Liapunov's method is also applicable to non-autonomous systems. We use this advantadge to investigate the mechanism of reheating for the inflaton field in FRW models. 
  The Davies-Fulling (DF) model describes the scattering of a massless field by a non-inertial mirror in two dimensions. In this paper, we generalize this model in two different ways. First, we consider partially reflecting mirrors. We show that the Bogoliubov coefficients relating inertial modes can be expressed in terms of the frequency dependent reflection factor which is specified in the rest frame of the mirror and the transformation from the inertial modes to the modes at rest with respect to the mirror. In this perspective, the DF model is simply the limiting case when this factor is unity for all frequencies. In the second part, we introduce an alternative model which is based on self-interactions described by an action principle. When the coupling is constant, this model can be solved exactly and gives rise to a partially reflecting mirror. The usefulness of this dynamical model lies in the possibility of switching off the coupling between the mirror and the field. This allows to obtain regularized expressions for the fluxes in situations where they are singular when using the DF model. Two examples are considered. The first concerns the flux induced by the disappearance of the reflection condition, a situation which bears some analogies with the end of the evaporation of a black hole. The second case concerns the flux emitted by a uniformly accelerated mirror. 
  The trapped $w$-modes of stars with a first order phase transition (a density discontinuity) are computed and the excitation of some of the modes of these stars by a perturbing shell is investigated. Attention is restricted to odd parity (``axial'') $w$-modes. With $R$ the radius of the star, $M$ its mass, $R_{i}$ the radius of the inner core and $M_{i}$ the mass of such core, it is shown that stars with $R/M\geq 5$ can have several trapped $w$-modes, as long as $R_{i}/M_{i}<2.6$. Excitation of the least damped $w$-mode is confirmed for a few models. All of these stars can only exist however, for values of the ratio between the densities of the two phases, greater than $\sim 46$. We also show that stars with a phase transition and a given value of $R/M$ can have far more trapped modes than a homogeneous single density star with the same value of $R/M$, provided both $R/M$ and $R_{i}/M_{i}$ are smaller than 3. If the phase transition is very fast, most of the stars with trapped modes are unstable to radial oscillations. We compute the time of instability, and find it to be comparable to the damping of the $w$-mode excited in most cases where $w$-mode excitation is likely. If on the other hand the phase transition is slow, all the stars are stable to radial oscillations. 
  A seemingly natural mechanism is proposed, that could stop the gravitational collapse of a very massive body. Without needing to change the concept of the collapsing process itself, that is, without invoking thin layers nor resorting to asymptoticity (as has been usually done in the literature), it is proven that a model can be built in which the quantum vacuum is able to produce a negative stress that may stop the collapse of the black hole, reaching a final state of the spacetime structure that is a static de Sitter model. The solution is found by looking into a generic family of spacetimes: that of maximal spherically symmetric ones expanded by a geodesic radial null one-form from flat spacetime. They are called here GRNSS spaces, and are proven to constitute a distinguished family of Kerr-Schild metrics. The models considered previously in the literature are easily recovered in this approach, which yields, moreover, an infinite set of possible candidates for the interior of the black hole. First steps towards their semiclassical quantization are undertaken. It is shown that the quantization protocol may be here more easily carried out than within conformal field theory. 
  A family of spacetimes suitable for describing the matter conditions of a static, spherically symmetric quantum vacuum is studied, as well as its reliability for describing a regular model for the interior of a semiclassical, static black hole ---without ever invoking a mass shell for the final object. In paper I, this condition was seen to limit the search to only one, distinguished family, that was investigated in detail. Here it will be proven that, aside from being mathematically generic (in its uniqueness), this family exhibits beautiful physical properties, that one would reasonably demand from a collapse process, including the remarkable result that isotropization may take place conveniently far from the (unavoidable) regularization scale. The analysis is also extended in order to include the possibility of a stringy core, always within the limits imposed by the semiclassical approach to gravitation. This constitutes a first approximation to the final goal of trying to characterize a regular, self-gravitating, stringy black hole. 
  We exhibit a simple and explicit formula for the metric of an arbitrary static spherically symmetric perfect fluid spacetime. This class of metrics depends on one freely specifiable monotone non-increasing generating function. We also investigate various regularity conditions, and the constraints they impose. Because we never make any assumptions as to the nature (or even the existence) of an equation of state, this technique is useful in situations where the equation of state is for whatever reason uncertain or unknown.   To illustrate the power of the method we exhibit a new form of the ``Goldman--I'' exact solution and calculate its total mass. This is a three-parameter closed-form exact solution given in terms of algebraic combinations of quadratics. It interpolates between (and thereby unifies) at least six other reasonably well-known exact solutions. 
  We show that the behaviour of outgoing radial null geodesic congruence on the apparent horizon is related to the property of nakedness in spherical dust collapse justifying the difference between the Penrose diagrams in the naked and covered dust collapse scenarios. We provide arguments suggesting that the relationship could be generally valid. 
  Modern formulations of equivalence principles provide the foundation for an efficient approach to understanding and organizing the structural features of gravitation field theories. Since theories' predictions reflect differences in their structures, principles of equivalence also support an efficient experimental strategy for testing gravitation theories and for exploring the range of conceivable gravitation physics. These principles focus attention squarely on empirical consequences of the fundamental structural differences that distinguish one gravitation theory from another. Interestingly, the variety of such consequences makes it possible to design and perform experiments that test equivalence principles stringently but do so in markedly different ways than the most familiar experimental tests. 
  We present an integral formulation of observer-dependent Maxwell's equations in curved spacetime and give a classical interpretation of them. 
  Scaling behavior in the moduli space of monopole and dyon solutions in the Einstein-Yang-Mills theory in the asymptotically anti-de Sitter space is derived. The mass of monopoles and dyons scales with respect to their magnetic and electric charges, independent of the values of the cosmological constant and gauge coupling constant. The stable monopole and dyon solutions are approximated by solutions in the fixed anti-de Sitter spacetime. Unstable solutions can be viewed as the Bartnik-McKinnon solutions dressed with monopole and dyon solutions in the fixed anti-de Sitter space. 
  If our Universe is a three-brane embedded in a five-dimensional anti-deSitter spacetime, in which matter is confined to the brane and gravity inhabits an infinite bulk space, then the causal propagation of luminous and gravitational signals is in general different. A gravitational signal traveling between two points on the brane can take a ``shortcut'' through the bulk, and appear quicker than a photon traveling between the same two points along a geodesic on the brane. Similarly, in a given time interval, a gravitational signal can propagate farther than a luminous signal. We quantify this effect, and analyze the impact of these shortcuts through the fifth dimension on cosmology. 
  We study quantum radiation generated by a uniformly accelerated motion of small spherical mirrors. To obtain Green's function for a scalar massless field we use Wick's rotation. In the Euclidean domain the problem is reduced to finding an electric potential in 4D flat space in the presence of a metallic toroidal boundary. The latter problem is solved by a separation of variables. After performing an inverse Wick's rotation we obtain the Hadamard function in the wave-zone regime and use it to calculate the vacuum fluctuations and the vacuum expectation for the energy density flux in the wave zone. 
  We derive a generic identity which holds for the metric (i.e. variational) energy-momentum tensor under any field transformation in any generally covariant classical Lagrangian field theory. The identity determines the conditions under which a symmetry of the Lagrangian is also a symmetry of the energy-momentum tensor. It turns out that the stress tensor acquires the symmetry if the Lagrangian has the symmetry in a generic curved spacetime. In this sense a field theory in flat spacetime is not self-contained. When the identity is applied to the gauge invariant spin-two field in Minkowski space, we obtain an alternative and direct derivation of a known no-go theorem: a linear gauge invariant spin-2 field, which is dynamically equivalent to linearized General Relativity, cannot have a gauge invariant metric energy-momentum tensor. This implies that attempts to define the notion of gravitational energy density in terms of the metric energy--momentum tensor in a field-theoretical formulation of gravity must fail. 
  Computations in the calculable small coupling regime of string theories and the general consensus that no new physics has to be invoked in continuing to the large coupling (black hole) regime, suggest the following picture. Quantum states are not black holes even if energy distributions would suggest so. Black holes appear as macrostates, i.e. with the same procedure that blures the quantum coherence of microstates. It is also discussed how a spacetime description - and thus geometry, causal properties and event horizons - may stem from decoherence in the pregeometric approach represented by string theories. 
  In order to have well defined rules for the perturbative calculation of quantities of interest in an interacting quantum field theory in curved spacetime, it is necessary to construct Wick polynomials and their time ordered products for the noninteracting theory. A construction of these quantities has recently been given by Brunetti, Fredenhagen, and Kohler, and by Brunetti and Fredenhagen, but they did not impose any ``locality'' or ``covariance'' condition in their constructions. As a consequence, their construction of time ordered products contained ambiguities involving arbitrary functions of spacetime point rather than arbitrary parameters. In this paper, we construct an ``extended Wick polynomial algebra''-large enough to contain the Wick polynomials and their time ordered products. We then define the notion of a {\it local, covariant quantum field}, and seek a definition of {\it local} Wick polynomials and their time ordered products as local, covariant quantum fields. We impose scaling requirements on our local Wick polynomials and their time ordered products as well as certain additional requirements-such as commutation relations with the free field and appropriate continuity properties under variations of the spacetime metric. For a given polynomial order in powers of the field, we prove that these conditions uniquely determine the local Wick polynomials and their time ordered products up to a finite number of parameters. (These parameters correspond to the usual renormalization ambiguities occurring in Minkowski spacetime together with additional parameters corresponding to the coupling of the field to curvature.) We also prove existence of local Wick polynomials. However, the issue of existence of local time ordered products is deferred to a future investigation 
  It is thought that a stochastic background of gravitational waves was produced during the formation of the universe. A great deal could be learned by measuring this Cosmic Gravitational-wave Background (CGB), but detecting the CGB presents a significant technological challenge. The signal strength is expected to be extremely weak, and there will be competition from unresolved astrophysical foregrounds such as white dwarf binaries. Our goal is to identify the most promising approach to detect the CGB. We study the sensitivities that can be reached using both individual, and cross-correlated pairs of space based interferometers. Our main result is a general, coordinate free formalism for calculating the detector response that applies to arbitrary detector configurations. We use this general formalism to identify some promising designs for a GrAvitational Background Interferometer (GABI) mission. Our conclusion is that detecting the CGB is not out of reach. 
  The rotating disk problem is analyzed on the premise that proper interpretation of experimental evidence leads to the conclusion that the postulates upon which relativity theory is based, particularly the invariance of the speed of light, are not applicable to rotating frames. Different postulates based on the Sagnac experiment are proposed, and from these postulates a new relativistic theory of rotating frames is developed following steps similar to those initially followed by Einstein for rectilinear motion. The resulting theory agrees with all experiments, resolves problems with the traditional approach to the rotating disk, and exhibits both traditionally relativistic and non-relativistic characteristics. Of particular note, no Lorentz contraction exists on the rotating disk circumference, and the disk surface, contrary to the assertions of Einstein and others, is found to be Riemann flat. The variable speed of light found in the Sagnac experiment is then shown to be characteristic of non-time-orthogonal reference frames, of which the rotating frame is one. In addition, the widely accepted postulate for the equivalence of inertial and non-inertial standard rods with zero relative velocity, used liberally in prior rotating disk analyses, is shown to be invalid for such frames. Further, the new theory stands alone in correctly predicting what was heretofore considered a "spurious" non-null effect on the order of 10^-13 found by Brillet and Hall in the most accurate Michelson-Morley type test to date. The presentation is simple and pedagogic in order to make it accessible to the non-specialist.   Key words: relativistic, rotating disk, Sagnac, rotating frame, non-time-orthogonal frame. 
  Dimensional scales are examined in an extended 3+1 Vaidya atmosphere surrounding a Schwarzschild source. At one scale, the Vaidya null fluid vanishes and the spacetime contains only a single spherical 2-surface. Both of these behaviors can be addressed by including higher dimensions in the spacetime metric. 
  We numerically verify the analysis of the "expanding horizon" theory of Susskind in relation to the 't Hooft holographic conjecture. By using a numerical simulation to work out the image formed by two black holes upon a screen very far away, it is seen that it is impossible for a horizon to hide behind another. We also compute the intensity distribution of such an arrangement. 
  We discuss possible observational manifestations of static, spherically symmetric solutions of a class of multidimensional theories of gravity, which includes the low energy limits of supergravities and superstring theories as special cases. We discuss the choice of a physical conformal frame to be used for the description of observations. General expressions are given for (i) the Eddington parameters $\beta$ and $\gamma$, characterizing the post-Newtonian gravitational field of a central body, (ii) p-brane black hole temperatures in different conformal frames and (iii) the Coulomb law modified by extra dimensions. It is concluded, in particular, that $\beta$ and $\gamma$ depend on the integration constants and can be therefore different for different central bodies. If, however, the Einstein frame is adopted for describing observations, we always obtain $\gamma=1$. The modified Coulomb law is shown to be independent of the choice of a 4-dimensional conformal frame. We also argue the possible existence of specific multidimensional objects, T-holes, potentially observable as bodies with mirror surfaces. 
  Some interesting consequences of spacelike matter shells are presented, in particular the possibility of travelling through Cauchy horizons and violating the strong cosmic censorship principle. These show that the weak energy condition does not guarentee cosmic censorship. 
  We study the generalised constrained BF theory described in gr-qc/0102073 in order to introduce the Immirzi parameter in spin foam models. We show that the resulting spin foam model is still based on simple representations and that the generalised BF action is simply a deformation of the Barrett-Crane model. The Immirzi parameter doesn't change the representations used in the spin foam model, so it doesn't affect the geometry of the model. However we show how it may still appear as a factor in the area spectrum. 
  In this work we derive the evolution equation for the density contrast considering open system cosmology, where the influence of adiabatic particle production process on the dynamic of a homogeneous and isotropic is investigated within a manifestly covariant formulation. As application we derive the solution for two sources, one of them is a generalization of Prigogine's model. Then we establish the condition for the reach of the non-linear regime for the density contrast which turns out to be a necessary condition for the structure formation. 
  We study the fate of density perturbations in an Universe dominate by the Chaplygin gas, which exhibit negative pressure. We show that it is possible to obtain the value for the density contrast observed in large scale structure of the Universe by fixing a free parameter in the equation of state of this gas. The negative character of pressure must be significant only very recently. 
  What can we learn about quantum gravity from a simple toy model, without actually quantizing it? The toy model consists of a finite number of point particles, coupled to three dimensional Einstein gravity. It has finitely many physical degrees of freedom. These are basically the relative positions of the particles in spacetime and the conjugate momenta. The resulting reduced phase space is derived from Einstein gravity as a topological field theory. The crucial point is thereby that we do not make any a priori assumptions about this phase space, except that the dynamics of the gravitational field is defined by the Einstein Hilbert action. This already leads to some interesting features of the reduced phase space, such as a non-commutative structure of spacetime when the model is quantized. 
  We study a system of two pointlike particles coupled to three dimensional Einstein gravity. The reduced phase space can be considered as a deformed version of the phase space of two special-relativistic point particles in the centre of mass frame. When the system is quantized, we find some possibly general effects of quantum gravity, such as a minimal distances and a foaminess of the spacetime at the order of the Planck length. We also obtain a quantization of geometry, which restricts the possible asymptotic geometries of the universe. 
  We investigate the asymptotic tail behavior of massive scalar fields in Schwarzschild background. It is shown that the oscillatory tail of the scalar field has the decay rate of $t^{-5/6}$ at asymptotically late times, and the oscillation with the period $2\pi/m$ for the field mass $m$ is modulated by the long-term phase shift. These behaviors are qualitatively similar to those found in nearly extreme Reissner-Nordstr\"{o}m background, which are discussed in terms of a resonant backscattering due to the space-time curvature. 
  Many cosmological scenarios envisage either a bounce of the universe at early times, or collapse of matter locally to form a black hole which re-expands into a new expanding universe region. Energy conditions preclude this happening for ordinary matter in general relativistic universes, but scalar or dilatonic fields can violate some of these conditions, and so could possibly provide bounce behaviour. In this paper we show that such bounces cannot occur in Kantowski-Sachs models without violating the {\it reality condition} $\dot{\phi}^2\geq 0$. This also holds true for other isotropic spatially homogenous Bianchi models, with the exception of closed Friedmann-Robertson-Walker and Bianchi IX models; bounce behaviour violates the {\em weak energy condition} $\rho\geq 0$ and $\rho+p\geq 0$. We turn to the Randall-Sundrum type braneworld scenario for a possible resolution of this problem. 
  One of the most famous classical tests of General Relativity is the gravitoelectric secular advance of the pericenter of a test body in the gravitational field of a central mass. In this paper we explore the possibility of performing a measurement of the gravitoelectric pericenter advance in the gravitational field of the Earth by analyzing the laser-ranged data to some existing, or proposed, laser-ranged geodetic satellites. At the present level of knowledge of various error sources, the relative precision obtainable with the data from LAGEOS and LAGEOS II, suitably combined, is of the order of $10^{\rm -3}$. Nevertheless, these accuracies could sensibly be improved in the near future when the new data on the terrestrial gravitational field from the CHAMP and GRACE missions will be available. The use of the perigee of LARES (LAser RElativity Satellite), in the context of a suitable combination of orbital residuals including also LAGEOS II, should further raise the precision of the measurement. As a secondary outcome of the proposed experiment, with the so obtained value of $\ppn$ and with $\et=4\beta-\gamma-3$ from Lunar Laser Ranging it could be possible to obtain an estimate of the PPN parameters $\gamma$ and $\beta$ at the $10^{-2}-10^{-3}$ level. 
  Warm inflation is an interesting possibility of describing the early universe, whose basic feature is the absence, at least in principle, of a preheating or reheating phase. Here we analyze the dynamics of warm inflation generalizing the usual slow-roll parameters that are useful for characterizing the inflationary phase. We study the evolution of entropy and adiabatic perturbations, where the main result is that for a very small amount of dissipation the entropy perturbations can be neglected and the purely adiabatic perturbations will be responsible for the primordial spectrum of inhomogeneities. Taking into account the COBE-DMR data of the cosmic microwave background anisotropy as well as the fact that the interval of inflation for which the scales of astrophysical interest cross outside the Hubble radius is about 50 e-folds before the end of inflation, we could estimate the magnitude of the dissipation term. It was also possible to show that at the end of inflation the universe is hot enough to provide a smooth transition to the radiation era. 
  We investigate the dynamical formation and evaporation of a spherically symmetric charged black hole. We study the self-consistent one loop order semiclassical back-reaction problem. To this end the mass-evaporation is modeled by an expectation value of the stress-energy tensor of a neutral massless scalar field, while the charge is not radiated away. We observe the formation of an initially non extremal black hole which tends toward the extremal black hole $M=Q$, emitting Hawking radiation. If also the discharge due to the instability of vacuum to pair creation in strong electric fields occurs, then the black hole discharges and evaporates simultaneously and decays regularly until the scale where the semiclassical approximation breaks down. We calculate the rates of the mass and the charge loss and estimate the life-time of the decaying black holes. 
  We consider the optical Sagnac effect, when the fictitious gravitational field simulates the reflections from the mirrors. It is shown that no contradiction exists between the conclusions of the laboratory and rotated observers. Because of acting of gravity-like Coriolis force the trajectories of co- and anti-rotating photons have different radii in the rotating reference frame, while in the case of the equal radius the effective gravitational potentials for the photons have to be different. 
  Colliding Einstein-Maxwell-Scalar fields need not necessarily doomed to become in a spacelike singularity. Examples are given in which null singularities emerge as intermediate stages between a spacelike singularity and a regular horizon. 
  The shear dynamics in Bianchi I cosmological model on the brane with a perfect fluid (the equation of state is $p=(\gamma-1) \mu$) is studied. It is shown that for $1 < \gamma < 2$ the shear parameter has maximum at some moment during a transition period from nonstandard to standard cosmology. An exact formula for the matter density $\mu$ in the epoch of maximum shear parameter as a function of the equation of state is obtained. 
  The system of Einstein-Maxwell equations for fields mentioned in the title is simplified. Known pure radiation solutions are systematized and new solutions are given by separating the variables. 
  Pure-radiation solutions are found, exploiting the analogy with the Euler- Darboux equation for aligned colliding plane waves and the Euler-Tricomi equation in hydrodynamics of two-dimensional flow. They do not depend on one of the spacelike coordinates and comprise the Hauser solution as a special subcase. 
  New expanding, axisymmetric pure-radiation solutions are found, exploiting the analogy with the Euler-Darboux equation for aligned colliding plane waves. 
  In the context of the two-fluid model introduced to tame the transplanckian problem of black hole physics, the inflaton field of the chaotic inflation scenario is identified with the fluctuation of the density of modes. Its mass comes about from the exchange of degrees of freedom between the two fluids. 
  We have investigated the cosmological scenarios with a four dimensional effective action which is connected with multidimensional, supergravity and string theories. The solution for the scale factor is such that initially universe undergoes a decelerated expansion but in late times it enters into the accelerated expansion phase. Infact, it asymptotically becomes a de-Sitter universe. The dilaton field in our model is a decreasing function of time and it becomes a constant in late time resulting the exit from the scalar tensor theory to the standard Einstein's gravity. Also the dilaton field results the existence of a positive cosmological constant in late times. 
  Numerical codes based on a direct implementation of the standard ADM formulation of Einstein's equations have generally failed to provide long-term stable and convergent evolutions of black hole spacetimes when excision is used to remove the singularities. We show that, for the case of a single black hole in spherical symmetry, it is possible to circumvent these problems by adding to the evolution equations terms involving the constraints, thus adjusting the standard ADM system. We investigate the effect that the choice of the lapse and shift has on the stability properties of numerical simulations and thus on the form of the added constraint term. To facilitate this task, we introduce the concept of quasi well-posedness, a version of well-posedness suitable for ADM-like systems involving second-order spatial derivatives. 
  Several recent approaches to black hole entropy obtain the density of states from the central charge of a Liouville theory. If Liouville theory is coupled to a dynamical spacetime background, however, the classical central charge vanishes. I show that the central charge can be restored by introducing appropriate constraints, which may be interpreted as fall-off conditions at a boundary such as a black hole horizon. 
  Motivated by brane cosmology we solve the Einstein equations with a time dependent cosmological constant. Assuming that at an early epoch the vacuum energy scales as $1/logt $, we show that the universe passes from a fast growing phase (inflation) to an expanding phase in a natural way. 
  In numerical studies of Gowdy spacetimes evidence has been found for the development of localized features (spikes) involving large gradients near the singularity. The rigorous mathematical results available up to now did not cover this kind of situation. In this work we show the existence of large classes of Gowdy spacetimes exhibiting features of the kind discovered numerically. These spacetimes are constructed by applying certain transformations to previously known spacetimes without spikes. It is possible to control the behaviour of the Kretschmann scalar near the singularity in detail. This curvature invariant is found to blow up in a way which is non-uniform near the spike in some cases. When this happens it demonstrates that the spike is a geometrically invariant feature and not an artefact of the choice of variables used to parametrize the metric. We also identify another class of spikes which are artefacts. The spikes produced by our method are compared with the results of numerical and heuristic analyses of the same situation. 
  Space-time is spherically symmetric if it admits the group of SO(3) as a group of isometries,with the group orbits spacelike two-surfaces. These orbits are necessarily two-surface of constant positive curavture. One commonly chooses coordinate ${t,r,\theta,\phi}$ so that the group orbits become surfaces ${t,r = const.}$ and the radial coordinate r is defined by the requirment that $4\pi r^2$ is the area of these spacelike two-surfaces with the range of zero to infinity. According to the Birkhoff's theorem upon the above assumptions, the Schwarzschild metric is the only soltion of the vacuum Einstein field equations. Our aim is to reconsider the solution of the sperically symmetric vacuum Einstein field equtions by regarding a weaker requirement. We admit the evident fact that in the completely empty sace the radial coordinate r may be defined so that $4\pi r^2$ becomes the area of spacelike two-surfaces ${t,r = const.}$ with the range of zero to infinity. This is not necessarily to be true in the presence of a material point mass M. It turns out that inspite of imposing asymptotically flatness and staticness as initial conditions the equations have general classes of solutions which the Schwarzschild metric is the only member of them which has an intrinsic singularity at the location of the point mass M. The area of ${t,r = const.}$ is $4\pi(r +\alpha M)^2$ in one class and $4\pi(r^2 + a_1 Mr +a_2 M^2)$ in the other class while the center of symmetry is at r = 0. 
  A gauge-independent, invariant theory of linear scalar perturbations of inflation and gravitational fields has been created. This invariant theory allows one to compare gauges used in the work of other researchers and to find the unambiguous criteria to separate the physical and coordinate effects. It is shown, in particular, that the so-called longitudial gauge, commonly used when considering inflation instability, leads to a fundamental overestimation of the effect because of non-physical perturbations of the proper time in the frame of reference specified by this gauge. Back reaction theories employing this sort of gauge [1] also involve coordinate effects. The invariant theory created here shows that the classical Lifshitz (1946) [2] gauge does not lead to non-physical perturbations of the proper time and can be used to analyze the inflation regime and the back reaction of perturbations on this regime properly. The first theory of back reaction on background of all types of perturbations (scalar, vector and tensor) based on this gauge was published in 1975 [3] and has been applied recently to the inflation [4]. The investigation of long-length perturbations, which characterize the stability of the inflationary process, and quantum fluctuations, which form the Harrison-Zel'dovich spectrum at the end of inflation, is performed in the invariant form. The invariant theory proposed allows one to examine the effect of quantum fluctuations on the inflationary stage when the periodic regime changes to an aperiodic one. That only the invariant theory must be used to analyze space experiments is one of the conclusions of the present work. 
  We use the manifestly Lorentz covariant canonical formalism to evaluate eigenvalues of the area operator acting on Wilson lines. To this end we modify the standard definition of the loop states to make it applicable to the present case of non-commutative connections. The area operator is diagonalized by using the usual shift ambiguity in definition of the connection. The eigenvalues are then expressed through quadratic Casimir operators. No dependence on the Immirzi parameter appears. 
  We consider black-brane spacetimes that have at least one spatial translation Killing field that is tangent to the brane. A new parameter, the tension of a spacetime, is defined. The tension parameter is associated with spatial translations in much the same way that the ADM mass is associated with the time translation Killing field. In this work, we explore the implications of the spatial translation symmetry for small perturbations around a background black brane. For static charged black branes we derive a law which relates the tension perturbation to the surface gravity times the change in the the horizon area, plus terms that involve variations in the charges and currents. We find that as a black brane evaporates the tension decreases. We also give a simple derivation of a first law for black brane spacetimes. These constructions hold when the background stress-energy is governed by a Hamiltonian, and the results include arbitrary perturbative stress-energy sources. 
  Bianchi VIII vacuum solutions to Einstein's equations are causally geodesically complete to the future, given an appropriate time orientation, and the objective of this article is to analyze the asymptotic behaviour of solutions in this time direction. For the Bianchi class A spacetimes, there is a formulation of the field equations that was presented in an article by Wainwright and Hsu, and we analyze the asymptotic behaviour of solutions in these variables. We also try to give the analytic results a geometric interpretation by analyzing how a normalized version of the Riemannian metric on the spatial hypersurfaces of homogeneity evolves. 
  It is argued that the Rindler quantization is not a correct approach to study the effects of acceleration on quantum fields. First, the "particle"-detector approach based on the Minkowski quantization is not equivalent to the approach based on the Rindler quantization. Second, the event horizon, which plays the essential role in the Rindler quantization, cannot play any physical role for a local noninertial observer. 
  We study physical properties of conformal initial value data for single and binary black hole configurations obtained using conformal-imaging and conformal-puncture methods. We investigate how the total mass M_tot of a dataset with two black holes depends on the configuration of linear or angular momentum and separation of the holes. The asymptotic behavior of M_tot with increasing separation allows us to make conclusions about an unphysical ``junk'' gravitation field introduced in the solutions by the conformal approaches. We also calculate the spatial distribution of scalar invariants of the Riemann tensor which determine the gravitational tidal forces. For single black hole configurations, these are compared to known analytical solutions. Spatial distribution of the invariants allows us to make certain conclusions about the local distribution of the additional field in the numerical datasets. 
  We show that using an adequate coordinate transformation the charged regular black hole solution given by Ayon-Beato and Garcia can be put in the Kerr-Schild form. Then we use this metric in Kerr-Schild Cartesian coordinates with a result given by Virbhadra and obtain the energy distribution associated with this. 
  Using the notion of a general conical defect, the Regge Calculus is generalized by allowing for dislocations on the simplicial lattice in addition to the usual disclinations. Since disclinations and dislocations correspond to curvature and torsion singularities, respectively, the method we propose provides a natural way of discretizing gravitational theories with torsion degrees of freedom like the Einstein-Cartan theory. A discrete version of the Einstein-Cartan action is given and field equations are derived, demanding stationarity of the action with respect to the discrete variables of the theory. 
  It is argued that, at least for the case of Navier-Stokes fluids, the so-called hyperbolic theories of dissipation are not viable. 
  We establish a formal relationship between stationary axisymmetric spacetimes and $T^3$ Gowdy cosmological models which allows us to derive several preliminary results about the generation of exact cosmological solutions and their possible behavior near the initial singularity. In particular, we argue that it is possible to generate a Gowdy model from its values at the singularity and that this could be used to construct cosmological solutions with any desired spatial behavior at the Big Bang. 
  The cosmological creation of primordial vector bosons and fermions is described in the Standard Model of strong and electro-weak interactions given in a space-time with the relative standard of measurement of geometric intervals. Using the reparametrization - invariant perturbation theory and the holomorphic representation of quantized fields we derive equations for the Bogoliubov coefficients and distribution functions of created particles. The main result is the intensive cosmological creation of longitudinal Z and W bosons (due to their mass singularity) by the universe in the rigid state. We introduce the hypothesis that the decay of the primordially created vector bosons is the origin of the Cosmic Microwave Background radiation. 
  We demonstrate that the emergence of a curved spacetime ``effective Lorentzian geometry'' is a common and generic result of linearizing a field theory around some non-trivial background. This investigation is motivated by considering the large number of ``analog models'' of general relativity that have recently been developed based on condensed matter physics, and asking whether there is something more fundamental going on. Indeed, linearization of a classical field theory (a field theoretic ``normal mode analysis'') results in fluctuations whose propagation is governed by a Lorentzian-signature curved spacetime ``effective metric''. For a single scalar field, this procedure results in a unique effective metric, which is quite sufficient for simulating kinematic aspects of general relativity (up to and including Hawking radiation). Quantizing the linearized fluctuations, the one-loop effective action contains a term proportional to the Einstein--Hilbert action, suggesting that while classical physics is responsible for generating an ``effective geometry'', quantum physics can be argued to induce an ``effective dynamics''. The situation is strongly reminiscent of Sakharov's ``induced gravity'' scenario, and suggests that Einstein gravity is an emergent low-energy long-distance phenomenon that is insensitive to the details of the high-energy short-distance physics. (We mean this in the same sense that hydrodynamics is a long-distance emergent phenomenon, many of whose predictions are insensitive to the short-distance cutoff implicit in molecular dynamics.) 
  The Tolman VII solution is an exact static spherically symmetric perfect fluid solution of Einstein's equations that exhibits a surprisingly good approximation to a neutron star. We show that this solution exhibits trapped null orbits in a causal region even for a tenuity (total radius to mass ratio) $> 3$. In this region the dynamical part of the potential for axial w - modes dominates over the centrifugal part. 
  We further develop a model unifying general relativity with quantum mechanics proposed in our earlier papers (J. Math. Phys. 38, 5840 (1998); 41, 5168 (2000)). The model is based on a noncommutative algebra $A$ defined on a groupoid $\Gamma = E \times G$ where $E$ is the total space of a fibre bundle over space-time and $G$ a Lie group acting on $E$. In this paper, the algebra $A$ is defined in such a way that the model works also if $G$ is a noncompact group. Differential algebra based on derivations of this algebra is elaborated which allows us to construct a "noncommutative general relativity". The left regular representation of the algebra $A$ in a Hilbert space leads to the quantum sector of our model. Its position and momentum representations are discussed in some detail. It is shown that the model has correct correspondence with the standard theories: with general relativity, by restricting the algebra $A$ to a subset of its center; with quantum mechanics, by changing from the groupoid $\Gamma $ to its algebroid; with classical mechanics, by changing from the groupoid $\Gamma $ to its tangent groupoid. We also construct a noncommutative Fock space based on the proposed model. 
  We use the Bergmann energy-momentum complex to calculate the energy of a charged regular black hole. The energy distribution is the same as we obtained in the Einstein prescription. Also, we get the expression of the energy in the Bergmann prescription for a general spherically symmetric space-time of the Kerr-Schild class. 
  The expectation that it should not be possible to gain experimental insight on the structure of space-time at Planckian distance scales has been recently challenged by several studies. With respect to space-time fluctuations, one of the conjectured features of quantum-gravity foam, the experiments that have the best sensitivity are the ones which were originally devised for searches of the classical-physics phenomenon of gravity waves. In experiments searching for classical gravity waves the presence of space-time fluctuations would introduce a source of noise just like the ordinary (non-gravitational) quantum properties of the photons composing the laser beam used in interferometry introduce a source of noise. Earlier studies of the noise induced by quantum properties of space-time have shown that certain simple pictures of fluctuations of space-time occuring genuinely at the Planck scale would lead to an observably large effect. Experimentalists would benefit from the guidance of detailed description of this noise, but quantum-gravity theories are not yet developed to the point of allowing such detailed analysis of physical processes. I propose a new phenomenological approach to the description of foam-induced noise. 
  In the vicinity of merging neutron strar binaries or supernova remnants, gravitational waves can interact with the prevailing strong magnetic fields. The resulting partial conversion of gravitational waves into electromagnetic (radio) waves might prove to be an indirect way of detecting gravitational waves from such sources.   Another interesting interaction considered in this article is the excitation of magnetosonic plasma waves by a gravitational wave passing through the surrounding plasma. The transfer of gravitational wave energy into the plasma might help to fuel the `fireball' of electromagnetic radiation observed in gamma ray bursts. In the last section of the article, a dispersion relation is derived for such magnetosonic plasma waves driven by a gravitational wave. 
  We consider the four dimensional discontinuity generated by two identical pieces of a five dimensional space pasted along their edge (that is a "brane" in a "$Z_2$-symmetric" "bulk"). Using a four plus one decomposition of the Riemann tensor we write the equations for gravity on the brane and recover in a simple manner a number of known "brane world" scenarios. We study under which conditions these equations reduce, exactly or approximately, to the four dimensional Einstein equations. We conclude that if the bulk is imposed to be only an Einstein space near the brane, Einstein's equations can be recovered approximately on the brane, but if it is imposed to be strictly anti-de Sitter space then the Einstein equations cannot hold, even approximately, on a quasi-Minkowskian brane, unless matter obeys a very contrived equation of state. 
  We discuss the equations of motion of test particles for a version of Kaluza-Klein theory where the cylinder condition is not imposed. The metric tensor of the five-dimensional manifold is allowed to depend on the fifth coordinate. This is the usual working scenario in brane-world, induced-matter theory and other Kaluza-Klein theories with large extra dimensions. We present a new version for the fully covariant splitting of the 5D equations. We show how to change the usual definition of various physical quantities in order to make physics in 4D invariant under transformations in 5D. These include the redefinition of the electromagnetic tensor, force and Christoffel symbols. With our definitions, each of the force terms in the equation of motion is gauge invariant and orthogonal to the four-velocity of the particle. The "hidden" parameter associated with the rate of motion along the extra dimension is identified with the electric charge, regardless of whether there is an electromagnetic field or not. In addition, for charged particles, the charge-to-mass ratio should vary. Therefore, the motion of a charged particle should differ from the motion of a neutral particle, with the same initial mass and energy, even in the absence of electromagnetic field. These predictions have important implications and could in principle be experimentally detected. 
  We study a time-dependent 5D metric which contains a static 4D sub-metric whose 3D part is spherically symmetric. An expansion in the metric coefficient allow us to obtain close-to Schwarzschild approximation to a class of spherically-symmetric solutions. Using Campbell's embedding theorem and the induced-matter formalism we obtain two 4D solutions. One describes a source with the stiff equation of state believed to be applicable to dense astrophysical objects, and the other describes a spherical source with a radial heat flow. 
  We derive an exact expression for the partition function of the Euclidean BTZ black hole. Using this, we show that for a black hole with large horizon area, the correction to the Bekenstein-Hawking entropy is $-3/2 log(Area)$, in agreement with that for the Schwarzschild black hole obtained in the canonical gravity formalism and also in a Lorentzian computation of BTZ black hole entropy. We find that the right expression for the logarithmic correction in the context of the BTZ black hole comes from the modular invariance associated with the toral boundary of the black hole. 
  The tensorial form of the Lax pair equations are given in a compact and geometrically transparent form in the presence of Cartan's torsion tensor. Three-dimensional spacetimes admitting Lax tensors are analyzed in detail. Solutions to Lax tensor equations include interesting examples as separable coordinates and the Toda lattice. 
  In an accompanying paper, we have formulated two types of regulariz_ation methods to calculate the scalar self-force on a particle of charge $q$ moving around a black hole of mass $M$, one of which is called the ``power expansion regularization''. In this paper, we analytically evaluate the self-force (which we also call the reaction force) to the third post-Newtonian (3PN) order on the scalar particle in circular orbit around a Schwarzschild black hole by using the power expansion regularization. It is found that the $r$-component of the self-force arises at the 3PN order, whereas the $t$- and $\phi$-components, which are due to the radiation reaction, appear at the 2PN and 1.5PN orders, respectively. 
  We consider a class of condensed matter theories in a Newtonian framework with a Lagrange formalism related in a natural way with the classical conservation laws \partial_t \rho + \partial_i (\rho v^i) = 0 \partial_t (\rho v^j) + \partial_i (\rho v^i v^j + p^{ij}) = 0 We show that for an algebraically defined ``effective Lorentz metric'' g_{\mu\nu} and ``effective matter fields'' \phi these theories are equivalent to material models of a metric theory of gravity with Lagrangian L = L_{GR} + L_{matter}   - (8\pi G)^{-1}(\Upsilon g^{00}-\Xi (g^{11}+g^{22}+g^{33}))\sqrt{-g} which fulfils the Einstein equivalence principle and leads to the Einstein equations in the limit \Xi,\Upsilon\to 0. 
  We consider the effect that gravity has when one tries to set up a constant background form field. We find that in analogy with the Melvin solution, where magnetic field lines self-gravitate to form a flux-tube, the self-gravity of the form field creates fluxbranes. Several exact solutions are found corresponding to different transverse spaces and world-volumes, a dilaton coupling is also considered. 
  We study the gravitational perturbations of thick domain walls. The refraction index and spin properties of the solutions interior to the wall are analyzed in detail. It is shown that the gravitational waves suffer a refraction process by domain walls. The reflection and transmission coefficients are derived in the thin wall limit. In relation to the spin content, it is shown that the ``$\times$'' helicity 2 gravitational wave mode maintains in the domain wall the same polarization state as in vacuum. On the contrary, the ``+'' mode, of pure helicity 2 in vacuum, is contaminated inside the wall with a spin 0 state, as well as with spin 2, helicity 0 and 1 states. 
  The maximal symmetry, or Perfect Cosmological Principle(PCP), that prevents AdS type spaces from degenerating into anti-inflationary collapse is argued to be unphysical. For example, the simple requirement that brane-bulk models should be the result of having evolved from even more energetic string phenomena picks out a preferred time direction.  We question whether quantum cosmological reasoning can be applied in any meaningful way to obtain, what are essentially, classical constructs . An alternative scheme is to more readily accept the PCP and allow the branes to also become eternal. A perpetually expanding and contracting brane model could be driven by the presence of charged black holes in the AdS bulk, that effectively violates the weak-energy condition as singularities are approached. This can be contrasted with the so-called Ekpyrotic universe which also closely accepts the PCP. This being broken only by occasional collisions between branes, that can then simulate a big bang cosmology. 
  The Schwarzschild black hole can be viewed as the special case of the marginally bound Lema\^\i tre-Tolman-Bondi models of dust collapse which corresponds to a constant mass function. We have presented a midi-superspace quantization of this model for an arbitrary mass-function in a separate publication. In this communication we show that our solution leads both to Bekenstein's area spectrum for black holes as well as to the black hole entropy, which, in this context, is naturally interpreted as the loss of information of the original matter distribution within the collapsing dust cloud. 
  The local and global properties of the Levi-Civita (LC) solutions coupled with an electromagnetic field are studied and some limits to the vacuum LC solutions are given. By doing such limits, the physical and geometrical interpretations of the free parameters involved in the solutions are made clear. Sources for both the LC vacuum solutions and the LC solutions coupled with an electromagnetic field are studied, and in particular it is found that all the LC vacuum solutions with $\sigma \ge 0$ can be produced by cylindrically symmetric thin shells that satisfy all the energy conditions, weak, dominant, and strong. When the electromagnetic field is present, the situation changes dramatically. In the case of a purely magnetic field, all the solutions with $\sigma \ge 1/\sqrt{8}$ or $\sigma \le - 1/\sqrt{8}$ can be produced by physically acceptable cylindrical thin shells, while in the case of a purely electric field, no such shells are found for any value of $\sigma$. 
  To compare two space-times on large domains, and in particular the global structure of their manifolds, requires using identical frames of reference and associated coordinate conditions. In this paper we use and compare two classes of time-like congruences and corresponding adapted coordinates: the harmonic and quo-harmonic classes. Besides the intrinsic definition and some of their intrinsic properties and differences we consider with some detail their differences at the level of the linearized approximation of the field equations. The hard part of this paper is an explicit and general determination of the harmonic and quo-harmonic coordinates adapted to the stationary character of three well-know metrics, Schwarzschild's, Curzon's and Kerr's, to order five of their asymptotic expansions. It also contains some relevant remarks on such problems as defining the multipoles of vacuum solutions or matching interior and exterior solutions. 
  We extend previous work on 3D black hole excision to the case of distorted black holes, with a variety of dynamic gauge conditions that are able to respond naturally to the spacetime dynamics. We show that the combination of excision and gauge conditions we use is able to drive highly distorted, rotating black holes to an almost static state at late times, with well behaved metric functions, without the need for any special initial conditions or analytically prescribed gauge functions. Further, we show for the first time that one can extract accurate waveforms from these simulations, with the full machinery of excision and dynamic gauge conditions. The evolutions can be carried out for long times, far exceeding the longevity and accuracy of even better resolved 2D codes. While traditional 2D codes show errors in quantities such as apparent horizon mass of over 100% by t = 100M, and crash by t = 150M, with our new techniques the same systems can be evolved for hundreds of M's in full 3D with errors of only a few percent. 
  Some conceptual issues concerning general invariant theories, with special emphasis on general relativity, are analyzed. The common assertion that observables must be required to be gauge invariant is examined in the light of the role played by a system of observers. Some features of the reduction of the gauge group are discussed, including the fact that in the process of a partial gauge fixing the reduction at the level of the gauge group and the reduction at the level of the variational principle do not commute. Distinctions between the mathematical and the physical concept of gauge symmetry are discussed and illustrated with examples. The limit from general relativity to special relativity is considered as an example of a gauge group reduction that is allowed in some specific physical circumstances. Whether and when the Poincar\'e group must be considered as a residual gauge group will come out as a result of our analysis, that applies, in particular, to asymptotically flat spaces. 
  The paper contains a proposed experiment for testing the gravitomagnetic effect on the propagation of light around a rotating mass. The idea is to use a rotating spherical laboratory-scale shell, around which two mutually orthogonal lightguides are wound acting as the arms of an interferometer. Numerical estimates show that time of flight differences between the equatorial and polar guides could be in the order of $\sim 10^{-20}$ s, actually detectable with sensitivity perfectly comparable with those expected in gravitational wave detection experiments. 
  We investigate the Church-Kalm\'ar-Kreisel-Turing Theses concerning theoretical (necessary) limitations of future computers and of deductive sciences, in view of recent results of classical general relativity theory.   We argue that (i) there are several distinguished Church-Turing-type Theses (not only one) and (ii) validity of some of these theses depend on the background physical theory we choose to use. In particular, if we choose classical general relativity theory as our background theory, then the above mentioned limitations (predicted by these Theses) become no more necessary, hence certain forms of the Church-Turing Thesis cease to be valid (in general relativity). (For other choices of the background theory the answer might be different.)   We also look at various ``obstacles'' to computing a non-recursive function (by relying on relativistic phenomena) published in the literature and show that they can be avoided (by improving the ``design'' of our future computer). We also ask ourselves, how all this reflects on the arithmetical hierarchy and the analytical hierarchy of uncomputable functions. 
  The non-linear generation of harmonics in gravitational perturbations of black holes is explored using numerical relativity based on an in-going light-cone framework. Localised, finite, perturbations of an isolated black hole are parametrised by amplitude and angular harmonic form. The response of the black hole spacetime is monitored and its harmonic content analysed to identify the strength of the non-linear generation of harmonics as a function of the initial data amplitude. It is found that overwhelmingly the black hole responds at the harmonic mode perturbed, even for spacetimes with 10% of the black hole mass radiated. The relative efficiencies of down and up-scattering in harmonic space are computed for a range of couplings. Down-scattering, leading to smoothing out of angular structure is found to be equally or more efficient than the up-scatterings that would lead to increased rippling. The details of this non-linear balance may form the quantitative mechanism by which black holes avoid fission even for arbitrary strong distortions. 
  Gravitational redshifts of neutron stars have a theoretical upper limit of z=0.62. Also, it is generally believed that neutron stars have magnetic fields on the order of ten to the twelfth to ten to the thirteenth G. A previously predicted electromagnetic time dilation effect has been shown to correctly predict decay lifetimes of muons bound to atomic nuclei. In this paper it is shown that the electromagnetic time dilation effect, along with the gravitational time dilation effect, can produce total neutron star redshifts that are substantially larger than 0.62. For instance, the redshift can cutoff radiation for B on the order of ten to the thirteenth G. Consequently, we can have a neutron star that is unobservable except for localized surface regions where the cutoff condition does not hold. Assuming coherent radiation, a surface region of this kind that does not include the star's rotation axis will emit a lighthouse type beam. Since the magnetic field in these regions will usually be strong enough to cause significant redshifts, and there is no reason to expect these regions to always be of constant size, shape, or field strength, this model explains the predominance of radio waves, and the existence of pulse variations, (e.g. nulling and drifting) in pulsars. 
  The strong beams of high-frequency gravitational waves (GW) emitted by cusps and kinks of cosmic strings are studied in detail. As a consequence of these beams, the stochastic ensemble of GW's generated by a cosmological network of oscillating loops is strongly non Gaussian, and includes occasional sharp bursts that stand above the ``confusion'' GW noise made of many smaller overlapping bursts. Even if only 10% of all string loops have cusps these bursts might be detectable by the planned GW detectors LIGO/VIRGO and LISA for string tensions as small as $G \mu \sim 10^{-13}$. In the implausible case where the average cusp number per loop oscillation is extremely small, the smaller bursts emitted by the ubiquitous kinks will be detectable by LISA for string tensions as small as $G \mu \sim 10^{-12}$. We show that the strongly non Gaussian nature of the stochastic GW's generated by strings modifies the usual derivation of constraints on $G \mu$ from pulsar timing experiments. In particular the usually considered ``rms GW background'' is, when $G \mu \gaq 10^{-7}$, an overestimate of the more relevant confusion GW noise because it includes rare, intense bursts. The consideration of the confusion GW noise suggests that a Grand Unified Theory (GUT) value $ G \mu \sim 10^{-6}$ is compatible with existing pulsar data, and that a modest improvement in pulsar timing accuracy could detect the confusion noise coming from a network of cuspy string loops down to $ G \mu \sim 10^{-11}$. The GW bursts discussed here might be accompanied by Gamma Ray Bursts. 
  We have obtained the most general solution of the Einstein vacuum equation for the axially symmetric stationary metric in which both the  Hamilton-Jacobi equation for particle motion and the Klein - Gordon equation are separable. It can be interpreted to describe the gravitational field of a rotating dyon, a particle endowed with both gravoelectric (mass) and gravomagnetic (NUT parameter) charge. Further, there also exists a duality relation between the two charges and the radial and the polar angle coordinates which keeps the solution invariant. The solution can however be transformed into the known Kerr - NUT solution indicating its uniqueness under the separability of equations of motion. 
  In this letter we study adiabatic anisotropic matter filled Bianchi type I models of the Kasner form together with the cosmological holographic bound. We find that the dominant energy condition and the holographic bound give precisely the same constraint on the scale factor parameters that appear in the metric. 
  The starting point of this work is the principle that all movement of particles and photons must follow geodesics of a 4-dimensional space where time intervals are always a measure on geodesic arc lengths. The last 3 coordinates (alpha = 1,2,3) are immediately associated with the usual physical space coordinates, while the first coordinate (\alpha=0) is later found to be related to proper time. Avoiding the virtually hopeless effort to prove the initial hypothesis, the work goes through several examples of increasing complexity, to show that it is plausible. Starting with special relativity it is shown that there is perfect mapping between the geodesics on Minkowski space-time and on this alternative space. The discussion than follows through light propagation in a refractive medium, and some cases of gravitation, including Schwartzschild's outer metric. The last part of the presentation is dedicated to electromagnetic interaction and Maxwell's equations, showing that there is a particular solution where one of the space dimensions is eliminated and the geodesics become equivalent to light rays in geometrical optics. A very brief discussion is made of the implications for wave-particle duality and quantization. 
  We investigate the weak decay of uniformly {\em accelerated protons} in the context of {\em standard} Quantum Field Theory. Because the mean {\em proper} lifetime of a particle is a scalar, the same value for this observable must be obtained in the inertial and coaccelerated frames. We are only able to achieve this equality by considering the Fulling-Davies-Unruh effect. This reflects the fact that the Fulling-Davies-Unruh effect is mandatory for the consistency of Quantum Field Theory. There is no question about its existence provided one accepts the validity of standard Quantum Field Theory in flat spacetime. 
  Upper limits for the mass-radius ratio are derived for arbitrary general relativistic matter distributions in the presence of a cosmological constant. General restrictions for the red shift and total energy (including the gravitational contribution) for compact objects in the Schwarzschild-de Sitter geometry are also obtained in terms of the cosmological constant and of the mean density of the star. 
  One of the widespread confusions concerning the history of the 1887 Michelson-Morley experiment has to do with the initial explanation of this celebrated null result due independently to FitzGerald and Lorentz. In neither case was a strict, longitudinal length contraction hypothesis invoked, as is commonly supposed. Lorentz postulated, particularly in 1895, any one of a certain family of possible deformation effects for rigid bodies in motion, including purely transverse alteration, and expansion as well as contraction; FitzGerald may well have had the same family in mind. A careful analysis of the Michelson-Morley experiment (which reveals a number of serious inadequacies in many text-book treatments) indeed shows that strict contraction is not required. 
  A Randall-Sundrum type brane-cosmological model in which slow-roll inflation on the brane is driven solely by a bulk scalar field was recently proposed by Himemoto and Sasaki. We analyze their model in detail and calculate the quantum fluctuations of the bulk scalar field $\phi$ with $m^2=V''(\phi)$. We decompose the bulk scalar field into the infinite mass spectrum of 4-dimensional fields; the field with the smallest mass-square, called the zero-mode, and the Kaluza-Klein modes above it with a mass gap. We find the zero-mode dominance of the classical solution holds if $|m^2|\bar\ell^2\ll1$, where $\bar{\ell}$ is the curvature radius of the effectively anti-de Sitter bulk, but it is violated if $|m^2|\bar\ell^2\gg1$, though the violation is very small. Then we evaluate the vacuum expectation value $<\delta\phi^2>$ on the brane. We find the zero-mode contribution completely dominates if $|m^2|\bar{\ell}^2\ll 1$ similar to the case of classical background. In contrast, we find the Kaluza-Klein contribution is small but non-negligible if the value of $|m^2|\bar{\ell}^2$ is large. 
  We study a family of physical observable quantities in quantum gravity. We denote them W functions, or n-net functions. They represent transition amplitudes between quantum states of the geometry, are analogous to the n-point functions in quantum field theory, but depend on spin networks with n connected components. In particular, they include the three-geometry to three-geometry transition amplitude. The W functions are scalar under four-dimensional diffeomorphism, and fully gauge invariant. They capture the physical content of the quantum gravitational theory. We show that W functions are the natural n-point functions of the field theoretical formulation of the gravitational spin foam models. They can be computed from a perturbation expansion, which can be interpreted as a sum-over-four-geometries. Therefore the W functions bridge between the canonical (loop) and the covariant (spinfoam) formulations of quantum gravity. Following Wightman, the physical Hilbert space of the theory can be reconstructed from the W functions, if a suitable positivity condition is satisfied. We compute explicitly the W functions in a ``free'' model in which the interaction giving the gravitational vertex is shut off, and we show that, in this simple case, we have positivity, the physical Hilbert space of the theory can be constructed explicitly and the theory admits a well defined interpretation in terms of diffeomorphism invariant transition amplitudes between quantized geometries. 
  When the mass of one of the two bodies tends to zero, Weyl's definition of the gravitational force in an axially symmetric, static two-body solution can be given an invariant formulation in terms of a force four-vector. The norm of this force is calculated for Bach's two-body solution, that is known to be in one-to-one correspondence with Schwarzschild's original solution when one of the two masses l, l' is made to vanish. In the limit when, say, l' goes to zero, the norm of the force divided by l' and calculated at the position of the vanishing mass is found to coincide with the norm of the acceleration of a test body kept at rest in Schwarzschild's field. Both norms happen thus to grow without limit when the test body (respectively the vanishing mass l') is kept at rest in a position closer and closer to Schwarzschild's two-surface. 
  We present an effective four-dimensional formulation of the laws of gravity that respects the main features of a higher (five)-dimensional scenario of Randall-Sundrum type. The geometrical structure of the theory is that of a Weyl-integrable configuration. Standard general relativity over Riemann geometry is recovered through breaking of conformal symmetry. The singularity problem is treated. The local problem in the case of the static, spherically-symmetric Schwarzschild metric and the cosmological issue, in the case of Friedmann-Robertson-Walker perfect fluid filled universe, are treated separately. Vanishing of the spacetime singularities for some values of the free parameter of the theory is achieved. The implications of the results obtained for brane stabilization in the higher-dimensional structure are briefly discussed. 
  It appears to follow from the Reissner-Nordstrom solution of Einstein's equations that the charge of a body reduces its gravitational field. In a recent note Hushwater offered an explanation of this apparent paradox. His explanation, however, raises more questions than solves since it implies that the active gravitational mass of a charged body is distance-dependent and therefore is not equal to its inertial mass. 
  We consider scalar and spinor particles in the spacetime of a domain wall in the context of low energy effective string theories, such as the generalized scalar-tensor gravity theories. This class of theories allows for an arbitrary coupling of the wall and the (gravitational) scalar field. First, we derive the metric of a wall in the weak-field approximation and we show that it depends on the wall's surface energy density and on two post-Newtonian parameters. Then, we solve the Klein-Gordon and the Dirac equations in this spacetime. We obtain the spectrum of energy eigenvalues and the current density in the scalar and spinor cases, respectively. We show that these quantities, except in the case of the energy spectrum for a massless spinor particle, depend on the parameters that characterize the scalar-tensor domain wall. 
  In this paper, static spacetimes with a topological structure of R^2 \times N is studied, where N is an arbitrary manifold. Well known Schwarzschild spacetime and Reissner-Nordstrom spacetime are special cases. It is shown that the existence of a constant and positive surface gravity $\kappa$ ensures the existence of the Killing horizon, with the cross section homeomorphic to N. 
  Detecting a stationary, stochastic gravitational wave signal is complicated by impossibility of observing the detector noise independently of the signal. One consequence is that we require at least two detectors to observe the signal, which will be apparent in the cross-correlation of the detector outputs. A corollary is that there remains a systematic error, associated with the possible presence of correlated instrumental noise, in any observation aimed at estimating or limiting a stochastic gravitational wave signal. Here we describe a method of identifying this systematic error by varying the orientation of one of the detectors, leading to separate and independent modulations of the signal and noise contribution to the cross-correlation. Our method can be applied to measurements of a stochastic gravitational wave background by the ALLEGRO/LIGO Livingston Observatory detector pair. We explore -- in the context of this detector pair -- how this new measurement technique is insensitive to a cross-correlated detector noise component that can confound a conventional measurement. 
  The inspiral of a ``small'' ($\mu \sim 1-100 M_\odot$) compact body into a ``large'' ($M \sim 10^{5-7} M_\odot$) black hole is a key source of gravitational radiation for the space-based gravitational-wave observatory LISA. The waves from such inspirals will probe the extreme strong-field nature of the Kerr metric. In this paper, I investigate the properties of a restricted family of such inspirals (the inspiral of circular, inclined orbits) with an eye toward understanding observable properties of the gravitational waves that they generate. Using results previously presented to calculate the effects of radiation reaction, I assemble the inspiral trajectories (assuming that radiation reacts adiabatically, so that over short timescales the trajectory is approximately geodesic) and calculate the wave generated as the compact body spirals in. I do this analysis for several black hole spins, sampling a range that should be indicative of what spins we will encounter in nature. The spin has a very strong impact on the waveform. In particular, when the hole rotates very rapidly, tidal coupling between the inspiraling body and the event horizon has a very strong influence on the inspiral time scale, which in turn has a big impact on the gravitational wave phasing. The gravitational waves themselves are very usefully described as ``multi-voice chirps'': the wave is a sum of ``voices'', each corresponding to a different harmonic of the fundamental orbital frequencies. Each voice has a rather simple phase evolution. Searching for extreme mass ratio inspirals voice-by-voice may be more effective than searching for the summed waveform all at once. 
  Technical discussions of the Laser Interferometer Gravitational Wave Observatory (LIGO) sensitivity often focus on its effective sensitivity to gravitational waves in a given band; nevertheless, the goal of the LIGO Project is to ``do science.'' Exploiting this new observational perspective to explore the Universe is a long-term goal, toward which LIGO's initial instrumentation is but a first step. Nevertheless, the first generation LIGO instrumentation is sensitive enough that even non-detection --- in the form of an upper limit --- is also informative. In this brief article I describe in quantitative terms some of the science we can hope to do with first and future generation LIGO instrumentation: it short, the ``science reach'' of the detector we are building and the ones we hope to build. 
  We study a generalized action for gravity as a constrained BF theory, and its relationship with the Plebanski action. We analyse the discretization of the constraints and the spin foam quantization of the theory, showing that it leads naturally to the Barrett-Crane spin foam model for quantum gravity. Our analysis holds true in both the Euclidean and Lorentzian formulation. 
  A point particle treatment to the statistical mechanics of BPS black holes in Einstein-Maxwell-dilaton theory is developed. Because of the absence of the static potential, the canonical partition function for $N$ BPS black holes can be expressed by the volume of the moduli space for them. We estimate the equation of state for a classical gas of BPS black holes by Pad\'e approximation and find that the result agrees with the one obtained by the mean-field approximation. 
  We present technical results which extend previous work and show that the cosmological constant of general relativity is an artefact of the reduction to 4D of 5D Kaluza-Klein theory (or 10D superstrings and 11D supergravity). We argue that the distinction between matter and vacuum is artificial in the context of ND field theory. The concept of a cosmological ``constant'' (which measures the energy density of the vacuum in 4D) should be replaced by that of a series of variable fields whose sum is determined by a solution of ND field equations in a well-defined manner. 
  Quantum liquids, in which an effective Lorentzian metric and thus some kind of gravity gradually arise in the low-energy corner, are the objects where the problems related to the quantum vacuum can be investigated in detail. In particular, they provide the possible solution of the cosmological constant problem: why the vacuum energy is by 120 orders of magnitude smaller than the estimation from the relativistic quantum field theory. The almost complete cancellation of the cosmological constant does not require any fine tuning and comes from the fundamental ``trans-Planckian'' physics of quantum liquids. The remaining vacuum energy is generated by the perturbations of quantum vacuum caused by matter (quasiparticles), curvature, and other possible sources, such as smooth component -- the quintessence. This provides the possible solution of another cosmological constant problem: why the present cosmological constant is on the order of the present matter density of the Universe. We discuss here some properties of the quantum vacuum in quantum liquids: the vacuum energy under different conditions; excitations above the vacuum state and the effective acoustic metric for them provided by the motion of the vacuum; Casimir effect, etc. 
  We investigate spherically symmetric and static gravitational fields representing black hole configurations in the framework of metric-affine gauge theories of gravity (MAG) in the presence of different matter fields. It is shown that in the triplet ansatz sector of MAG, black hole configurations in the presence of non-Abelian matter fields allow the existence of black hole hair. We analyze several cases of matter fields characterized by the presence of hair and for all of them we show the validity of the no short hair conjecture. 
  We use qualitative arguments combined with numerical simulations to argue that, in the approach to the singularity in a vacuum solution of Einstein's equations with $T^2$ isometry, the evolution at a generic point in space is an endless succession of Kasner epochs, punctuated by bounces in which either a curvature term or a twist term becomes important in the evolution equations for a brief time. Both curvature bounces and twist bounces may be understood within the context of local mixmaster dynamics although the latter have never been seen before in spatially inhomogeneous cosmological spacetimes. 
  We analyze parallel transport of a vector field around an equatorial orbit in Kerr and stationary axisymmetric spacetimes that are reflection symmetric about their equatorial planes. As in Schwarzschild spacetime, there is a band structure of holonomy invariance. The new feature introduced by rotation is a shift in the timelike component of the vector, which is the holonomic manifestation of the gravitomagnetic clock effect. 
  We argue that the geodesic hypothesis based on the autoparalllels of the Levi-Civita connection may need refinement in the scalar- tensor theories of gravity. Based on a reformulation of the Brans- Dicke theory in terms of a connection with torsion determined dynamically in terms of the gradient of the Brans-Dicke scalar field, we compute the perihelion shift in the orbit of Mercury on the alternative hypothesis that its worldline is an autoparallel of a connection with torsion. If the Brans-Dicke scalar field couples significantly to matter and test particles move on such worldlines, the current time keeping methods based on the conventional geodesic hypothesis may need refinement. 
  The quantum theory of U(1) connections admits a diffeomorphism invariant representation in which the electric flux through any surface is quantized. This representation is the analog of the representation of quantum SU(2) theory used in loop quantum gravity. We investigate the relation between this representation, in which the basic excitations are `polymer-like', and the Fock representation, in which the basic excitations are wave-like photons. We show that normalizable states in the Fock space are associated with `distributional' states in the quantized electric flux representation. This work is motivated by the question of how wave-like gravitons in linearised gravity arise from polymer-like states in non-perturbative loop quantum gravity. 
  We show that the characteristic sizes of astrophysical and cosmological structures, where gravity is the only overall relevant interaction assembling the system, have a phenomenological relation to the microscopic scales whose order of magnitude is essentially ruled by the Compton wavelength of the proton. This result agrees with the absence of screening mechanisms for the gravitational interaction and could be connected to the presence of Yukawa correcting terms in the Newtonian potential which introduce typical interaction lengths. Furthermore, we are able to justify, in a straightforward way, the Sanders--postulated mass of a vector boson considered in order to obtain the characteristic sizes of galaxies. 
  We show that the relativistic analogue of the two types of time translation in a non-relativistic history theory is the existence of two distinct Poincar\'{e} groups. The `internal' Poincar\'{e} group is analogous to the one that arises in the standard canonical quantisation scheme; the `external' Poincar\'{e} group is similar to the group that arises in a Lagrangian description of the standard theory. In particular, it performs explicit changes of the spacetime foliation that is implicitly assumed in standard canonical field theory. 
  The quantum entropy of the Kerr black hole arising from gravitational perturbation is investigated by using Null tetrad and \'t Hooft\'s brick-wall model. It is shown that effect of the graviton\'s spins on the subleading correction is dependent of the square of the spins and the angular momentum per unit mass of the black hole, and contribution of the logarithmic term to the entropy will be positive, zero, and negative for different value of $a/r_+$. 
  Hawking effect of Dirac particles in a variable-mass Kerr space-time is investigated by using a method called as the generalized tortoise coordinate transformation. The location and the temperature of the event horizon of the non-stationary Kerr black hole are derived. It is shown that the temperature and the shape of the event horizon depend not only on the time but also on the angle. However, the Fermi-Dirac spectrum displays a residual term which is absent from that of Bose-Einstein distribution. 
  The Hawking radiation of Dirac particles in a charged Vaidya - de Sitter black hole is investigated by using the method of generalized tortoise coordinate transformation. It is shown that the Hawking radiation of Dirac particles does not exist for $P_1, Q_2$ components, but for $P_2, Q_1$ components it does. Both the location and the temperature of the event horizon change with time. The thermal radiation spectrum of Dirac particles is the same as that of Klein-Gordon particles. 
  We show that the normalized Lorentzian state sum is finite on any triangulation. It thus provides a candidate for a perturbatively finite quantum theory of general relativity in four dimensions with Lorentzian signature. 
  We consider a model in which accelerated particles experience line--elements with maximal acceleration corrections that are introduced by means of successive approximations. It is shown that approximations higher than the first need not be considered. The method is then applied to the Kerr metric. The effective field experienced by accelerated test particles contains corrections that vanish in the limit $\hbar\to 0$, but otherwise affect the behaviour of matter greatly. The corrections generate potential barriers that are external to the horizon and are impervious to classical particles. 
  In this paper we study several means of compensating for thermal lensing which, otherwise, should be a source of concern for future upgrades of interferometric detectors of gravitational waves. The methods we develop are based on the principle of heating the cold parts of the mirrors. We find that thermal compensation can help a lot but can not do miracles. It seems finally that the best strategy for future upgrades (``advanced configurations'') is maybe to use thermal compensation together with another substrate materials than Silica, for example Sapphire. 
  We explain how the Universe was created with no expenditure of energy or initial mass. 
  We derive a geometrical version of the Regge-Wheeler and Zerilli equations, which allows us to study gravitational perturbations on an arbitrary spherically symmetric slicing of a Schwarzschild black hole. We explain how to obtain the gauge-invariant part of the metric perturbations from the amplitudes obeying our generalized Regge-Wheeler and Zerilli equations and vice-versa. We also give a general expression for the radiated energy at infinity, and establish the relation between our geometrical equations and the Teukolsky formalism. The results presented in this paper are expected to be useful for the close-limit approximation to black hole collisions, for the Cauchy perturbative matching problem, and for the study of isolated horizons. 
  The first British Gravity Meeting was held at the University of Southampton, UK, on 27/28 March. 47 10-minute plenary talks were given. Here are the abstracts in the order given. A brief conference report will also appear in Matters of Gravity. 
  We present a detailed description of techniques developed to combine 3D numerical simulations and, subsequently, a single black hole close-limit approximation. This method has made it possible to compute the first complete waveforms covering the post-orbital dynamics of a binary black hole system with the numerical simulation covering the essential non-linear interaction before the close limit becomes applicable for the late time dynamics. To determine when close-limit perturbation theory is applicable we apply a combination of invariant a priori estimates and a posteriori consistency checks of the robustness of our results against exchange of linear and non-linear treatments near the interface. Once the numerically modeled binary system reaches a regime that can be treated as perturbations of the Kerr spacetime, we must approximately relate the numerical coordinates to the perturbative background coordinates. We also perform a rotation of a numerically defined tetrad to asymptotically reproduce the tetrad required in the perturbative treatment. We can then produce numerical Cauchy data for the close-limit evolution in the form of the Weyl scalar $\psi_4$ and its time derivative $\partial_t\psi_4$ with both objects being first order coordinate and tetrad invariant. The Teukolsky equation in Boyer-Lindquist coordinates is adopted to further continue the evolution. To illustrate the application of these techniques we evolve a single Kerr hole and compute the spurious radiation as a measure of the error of the whole procedure. We also briefly discuss the extension of the project to make use of improved full numerical evolutions and outline the approach to a full understanding of astrophysical black hole binary systems which we can now pursue. 
  Our previous analyses of radio Doppler and ranging data from distant spacecraft in the solar system indicated that an apparent anomalous acceleration is acting on Pioneer 10 and 11, with a magnitude a_P ~ 8 x 10^{-8} cm/s^2, directed towards the Sun (anderson,moriond). Much effort has been expended looking for possible systematic origins of the residuals, but none has been found. A detailed investigation of effects both external to and internal to the spacecraft, as well as those due to modeling and computational techniques, is provided. We also discuss the methods, theoretical models, and experimental techniques used to detect and study small forces acting on interplanetary spacecraft. These include the methods of radio Doppler data collection, data editing, and data reduction.   There is now further data for the Pioneer 10 orbit determination. The extended Pioneer 10 data set spans 3 January 1987 to 22 July 1998. [For Pioneer 11 the shorter span goes from 5 January 1987 to the time of loss of coherent data on 1 October 1990.] With these data sets and more detailed studies of all the systematics, we now give a result, of a_P = (8.74 +/- 1.33) x 10^{-8} cm/s^2. (Annual/diurnal variations on top of a_P, that leave a_P unchanged, are also reported and discussed.) 
  Old and new puzzles of cosmology are reexamined from the point of view of quantum theory of the universe developed here. It is shown that in proposed approach the difficulties of the standard cosmology do not arise. The theory predicts the observed dimensions of the nonhomogeneities of matter density and the amplitude of the fluctuations of the cosmic background radiation temperature in the Universe and points to a new quantum mechanism of their origin. It allows to obtain the value of the deceleration parameter which is in good agreement with the recent SNe Ia measurements. The theory explains the large value of entropy of the Universe and describes other parameters. 
  We describe some new estimates concerning the recently proposed SEE (Satellite Energy Exchange) experiment for measuring the gravitational interaction parameters in space. The experiment entails precision tracking of the relative motion of two test bodies (a heavy "Shepherd", and a light "Particle") on board a drag-free space capsule. The new estimates include (i) the sensitivity of Particle trajectories and G measurement to the Shepherd quadrupole moment uncertainties; (ii) the measurement errors of G and the strength of a putative Yukawa-type force whose range parameter \lambda may be either of the order of a few meters or close to the Earth radius; (iii) a possible effect of the Van Allen radiation belts on the SEE experiment due to test body electric charging. The main conclusions are that (i) the SEE concept may allow one to measure G with an uncertainty smaller than 10^{-7} and a progress up to 2 orders of magnitude is possible in the assessment of the hypothetic Yukawa forces and (ii) van Allen charging of test bodies is a problem of importance but it may be solved by the existing methods. 
  A new satellite based test of Special and General Relativity is proposed. For the Michelson-Morley experiment we expect an improvement of at least three orders of magnitude, and for the Kennedy-Thorndike experiment an improvement of more than one order of magnitude. Furthermore, an improvement by two orders of the test of the universality of the gravitational red shift by comparison of an atomic clock with an optical clock is projected.    The tests are based on ultrastable optical cavities, an atomic clock and a comb generator. 
  The canonical analysis of the (anti-) self-dual action for gravity supplemented with the (anti-) self-dual Pontrjagin term is carried out. The effect of the topological term is to add a `magnetic' term to the original momentum variable associated with the self-dual action leaving the Ashtekar connection unmodified. In the new variables, the Gauss constraint retains its form, while both vector and Hamiltonian constraints are modified. This shows, the contribution of the Euler and Pontrjagin terms is not the same as that coming from the term associated with the Barbero-Immirzi parameter, and thus the analogy between the theta-angle in Yang-Mills theory and the Barbero-Immirzi parameter of gravity is not appropriate. 
  Geometric sigma models are purely geometric theories of scalar fields coupled to gravity. Geometrically, these scalars represent the very coordinates of space-time, and, as such, can be gauged away. A particular theory is built over a given metric field configuration which becomes the vacuum of the theory. Kaluza-Klein theories of the kind have been shown to be free of the classical cosmological constant problem, and to give massless gauge fields after dimensional reduction. In this paper, the consistency of dimensional reduction, as well as the stability of the internal excitations, are analyzed. Choosing the internal space in the form of a group manifold, one meets no inconsistencies in the dimensional reduction procedure. As an example, the SO(n) groups are analyzed, with the result that the mass matrix of the internal excitations necessarily possesses negative modes. In the case of coset spaces, the consistency of dimensional reduction rules out all but the stable mode, although the full vacuum stability remains an open problem. 
  Any three-dimensional Riemannian metric can be locally obtained by deforming a constant curvature metric along one direction. The general interest of this result, both in geometry and physics, and related open problems are stressed. 
  We report the analysis we made on data taken by Caltech 40-meter prototype interferometer to identify the noise power spectral density and to whiten the sequence of noise. We concentrate our study on data taken in November 1994, in particular we analyzed two frames of data: the 18nov94.2.frame and the 19nov94.2.frame.  We show that it is possible to whiten these data, to a good degree of whiteness, using a high order whitening filter. Moreover we can choose to whiten only restricted band of frequencies around the region we are interested in, obtaining a higher level of whiteness. 
  Loop quantum cosmology is shown to provide both the dynamical law and initial conditions for the wave function of a universe by one discrete evolution equation. Accompanied by the condition that semiclassical behavior is obtained at large volume, a unique wave function is predicted. 
  Studies of black hole formation from gravitational collapse have revealed interesting non-linear phenomena at the threshold of black hole formation. In particular, in 1993 Choptuik studied the collapse of a massless scalar field with spherical symmetry and found some behaviour, which is quite similar to the critical phenomena well-known in {\em Statistical Mechanics} and {\em Quantum Field Theory}. Universality and echoing of the critical solution and power-law scaling of the black hole masses have given rise to the name {\em Critical Phenomena in Gravitational Collapse}. Choptuik's results were soon confirmed both numerically and semi-analytically, and have extended to various other matter fields.   In this paper, we shall give a brief introduction to this fascinating and relatively new area, and provides an updated publication list. An analytical "toy" model of critical collapse is presented, and some current investigations are given. 
  Source-less wave equations are derived for massless scalar, neutrino and electromagnetic perturbations of a radiating Kerr space-time, and the Hawking radiation of massless particles with spin $s =0, 1/2$ and 1 in this geometry is investigated by using a method of the generalized tortoise coordinate transformation. An extra interaction between the spin of particles and the rotation of the hole displays in the thermal spectra of Hawking radiation of massless particles with spin $s = 1/2, 1$ in the evaporating Kerr space-time. The character of such effect is its obvious dependence on different helicity states of particles with higher spin. 
  A quantum-field model of the conformally flat space is formulated using a standard field-theoretical technique, a probability interpretation and a way to establish the classical limit. The starting point is the following: after conformal transformation of the Einstein -- Hilbert action, the conformal factor represents a scalar field with the negative kinetical term and the self-interaction inspired by the cosmological constant. (It has been found that quanta of such action have a negative value as a sequence of the negative energy.) The metric energy-momentum tensor of this scalar field is proportional to the Einstein tensor for the initial metric. Therefore, a vacuum state of the field is treated as a classical space. In such vacuum the zero mode is a scale factor of the flat Friedmann Universe. It is shown that conformal factor may be viewed as a an inflaton field, and its small non-homogeneities represent gauge invariant scalar metric perturbations. 
  Chromaticity effects introduced by the finite source size in microlensing events by presumed natural wormholes are studied. It is shown that these effects provide a specific signature that allow to discriminate between ordinary and negative mass lenses through the spectral analysis of the microlensing events. Both galactic and extragalactic situations are discussed. 
  In this paper we apply the concept of radar time (popularised by Bondi in his work on k-calculus) to the well-known relativistic twin `paradox'. Radar time is used to define hypersurfaces of simultaneity for a class of travelling twins, from the `Immediate Turn-around' case, through the `Gradual Turn-around' case, to the `Uniformly Accelerating' case. We show that this definition of simultaneity is independent of choice of coordinates, and assigns a unique time to any event (with which the travelling twin can send and receive signals), resolving some common misconceptions. 
  We find equations of particle motion from the point of view of observer on a rotating disk, and demonstrate that a particle moving along a rotating disk is influenced by forces arising from geometry. They can be considered as analogs of the centrifugal forces and the Coriolis forces. 
  In Brodbeck et al 1999 it has been shown that the linearised time evolution equations of general relativity can be extended to a system whose solutions asymptotically approach solutions of the constraints. In this paper we extend the non-linear equations in similar ways and investigate the effect of various possibilities by numerical means. Although we were not able to make the constraint submanifold an attractor for all solutions of the extended system, we were able to significantly reduce the growth of the numerical violation of the constraints. Contrary to our expectations this improvement did not imply a numerical solution closer to the exact solution, and therefore did not improve the quality of the numerical solution. 
  The transition between (non supersymmetric) quantum string states and Schwarzschild black holes is discussed. This transition occurs when the string coupling $g^2$ (which determines Newton's constant) increases beyond a certain critical value $g_c^2$. We review a calculation showing that self-gravity causes a typical string state of mass $M$ to shrink, as the string coupling $g^2$ increases, down to a compact string state whose mass, size, entropy and luminosity match (for the critical value $g_c^2 \sim (M \sqrt{\alpha'})^{-1}$) those of a Schwarzschild black hole. This confirms the idea (proposed by several authors) that the entropy of black holes can be accounted for by counting string states. The level spacing of the quantum states of Schwarzschild black holes is expected to be exponentially smaller than their radiative width. This makes it very difficult to conceive (even Gedanken) experiments probing the discreteness of the quantum energy levels of black holes. 
  We show that representations of the group of spacetime diffeomorphism and the Dirac algebra both arise in a phase-space histories version of canonical general relativity. This is the general-relativistic analogue of the novel time structure introduced previously in history theory: namely, the existence in non-relativistic physics of two types of time translation; and the existence in relativistic field theory of two distinct Poincare groups. 
  In this work we obtain a nondemolition variable for the case in which a charged particle moves in the electric and gravitational fields of a spherical body. Afterwards we consider the continuous monitoring of this nondemolition parameter, and calculate along the ideas of the so called restricted path integral formalism, the corresponding propagator. Using these results the probabilities associated with the possible measurement outputs are evaluated. The limit of our results, as the resolution of the measuring device goes to zero, is analyzed, and the dependence of the corresponding propagator upon the strength of the electric and gravitational fields are commented. The role that mass plays in the corresponding results, and its possible connection with the equivalence principle at quantum level, are studied. 
  The post-Newtonian expansion appears to be a relevant tool for predicting the gravitational waveforms generated by some astrophysical systems such as binaries. In particular, inspiralling compact binaries are well-modelled by a system of two point-particles moving on a quasi-circular orbit whose decay by emission of gravitational radiation is described by a post-Newtonian expansion. In this paper we summarize the basics of the computation by means of a series of multipole moments of the exterior field generated by an isolated source in the post-Newtonian approximation. This computation relies on an ansatz of matching the exterior multipolar field to the inner field of a slowly-moving source. The formalism can be applied to point-particles at the price of a further ansatz, that the infinite self-field of point-particles can be regularized in a certain way. As it turns out, the concept of point-particle requires a precise definition in high post-Newtonian approximations of general relativity. 
  Contents: I. Introduction; II. Summary of optimal signal filtering; III. Newtonian binary polarization waveforms; IV. Newtonian orbital phase evolution; V. Post-Newtonian wave-generation; VI. Inspiral binary waveform. 
  We calculate the gravitational radiation emitted by an infinite cosmic string with two oppositely moving wave-trains, in the small amplitude approximation. After comparing our result to the previously studied cases we extend the results to a new regime where the wavelengths of the opposing wave-trains are very different. We show that in this case the amount of power radiated vanishes exponentially. This means that small excitations moving in only one direction may be very long lived, and so the size of the smallest scales in a string network might be much smaller than what one would expect from gravitational back reaction. This result allows for a potential host of interesting cosmological possibilities involving ultra-high energy cosmic rays, gamma ray bursts and gravitational wave bursts. 
  I propose a phenomenological description of space-time foam and discuss the experimental limits that are within reach of forthcoming experiments. 
  Exact solutions of the gravitational field equations for a mixture of a null charged strange quark fluid and radiation are obtained in a Vaidya space-time. The conditions for the formation of a naked singularity are analyzed by considering the behavior of radial geodesics originating from the central singularity. 
  Painleve-Gullstrand metric of the black hole allows to discuss the fermion zero modes inside the hole. The statistical mechanics of the fermionic microstates can be responsible for the black hole thermodynamics. Fermion zero modes also lead to quantization of the horizon area. 
  We consider a system of nonlinear spinor and scalar fields with minimal coupling in general relativity. The nonlinearity in the spinor field Lagrangian is given by an arbitrary function of the invariants generated from the bilinear spinor forms $S= {\bar \psi} \psi$ and $P=i {\bar \psi} \gamma^5 \psi$; the scalar Lagrangian is chosen as an arbitrary function of the scalar invariant ${\Upsilon} = {\phi}_{,\alpha}{\phi}^{,\alpha}$, that becomes linear at ${\Upsilon} \to 0$. The spinor and the scalar fields in question interact with each other by means of a gravitational field which is given by a plane-symmetric metric. Exact plane-symmetric solutions to the gravitational, spinor and scalar field equations have been obtained. Role of gravitational field in the formation of the field configurations with limited total energy, spin and charge has been investigated. Influence of the change of the sign of energy density of the spinor and scalar fields on the properties of the configurations obtained has been examined. It has been established that under the change of the sign of the scalar field energy density the system in question can be realized physically iff the scalar charge does not exceed some critical value. In case of spinor field no such restriction on its parameter occurs. In general it has been shown that the choice of spinor field nonlinearity can lead to the elimination of scalar field contribution to the metric functions, but leaving its contribution to the total energy unaltered.   Key words: Nonlinear spinor field (NLSF), nonlinear scalar field, plane-symmetric metric PACS: 03.65.P, 04.20.H 
  A rank-n tensor on a Lorentzian manifold V whose contraction with n arbitrary causal future directed vectors is non-negative is said to have the dominant property. These tensors, up to sign, are called causal tensors, and we determine their general properties in dimension N. We prove that rank-2 tensors which map the null cone on itself are causal. It is known that, to any tensor A on V there is a corresponding ``superenergy'' (s-e) tensor T{A} which always has the dominant property. We prove that, conversely, any symmetric rank-2 tensor with the dominant property can be written in a canonical way as a sum of N s-e tensors of simple forms. We show that the square of any rank-2 s-e tensor is proportional to the metric if N<5, and that this holds for the s-e tensor of any simple form for arbitrary N. Conversely, we prove that any symmetric rank-2 tensor T whose square is proportional to the metric must be, up to sign, the s-e of a simple p-form, and that the trace of T determines the rank p of the form. This generalises, both with respect to N and the rank p, the classical algebraic Rainich conditions, which are necessary and sufficient conditions for a metric to originate in some physical field, and has a geometric interpretation: the set of s-e tensors of simple forms is precisely the set of tensors which preserve the null cone and its time orientation. It also means that all involutory Lorentz transformations (LT) can be represented as s-e tensors of simple forms, and that any rank-2 s-e tensor is the sum of at most N conformally involutory LT. Non-symmetric null cone preserving maps are shown to have a causal symmetric part and are classified according to the null eigenvectors of the skew-symmetric part. We thus obtain a complete classification of all conformal LT and singular null cone preserving maps on V. 
  The Bel tensor is divergence-free in some important cases leading to the existence of conserved currents associated to Killing vectors analogously to those of the energy-momentum tensor. When the divergence of the Bel tensor does not vanish one can study the interchange of some quantities between the gravitational and other fields obtaining mixed total conserved currents. Nevertheless, the Bel currents are shown to be conserved (independently of the matter content) if the Killing vectors satisfy some very general conditions. These properties are similar to some very well known statements for the energy-momentum tensor. 
  We enumerate all possible types of spacetime causal structures that can appear in static, spherically symmetric configurations of a self-gravitating, real, nonlinear, minimally coupled scalar field \phi in general relativity, with an arbitrary potential V(\phi), not necessarily positive-definite. It is shown that a variable scalar field adds nothing to the list of possible structures with a constant \phi field, namely, Minkowski (or AdS), Schwarzschild, de Sitter and Schwarzschild - de Sitter. It follows, in particular, that, whatever is V(\phi), this theory does not admit regular black holes with flat or AdS asymptotics. It is concluded that the only possible globally regular, asymptotically flat solutions are solitons with a regular center, without horizons and with at least partly negative potentials V(\phi). Extension of the results to more general field models is discussed. 
  In this paper, we study some interesting properties of a spherically symmetric oscillating soliton star made of a real time-dependent scalar field which is called an oscillaton. The known final configuration of an oscillaton consists of a stationary stage in which the scalar field and the metric coefficients oscillate in time if the scalar potential is quadratic. The differential equations that arise in the simplest approximation, that of coherent scalar oscillations, are presented for a quadratic scalar potential. This allows us to take a closer look at the interesting properties of these oscillating objects. The leading terms of the solutions considering a quartic and a cosh scalar potentials are worked in the so called stationary limit procedure. This procedure reveals the form in which oscillatons and boson stars may be related and useful information about oscillatons is obtained from the known results of boson stars. Oscillatons could compete with boson stars as interesting astrophysical objects, since they would be predicted by scalar field dark matter models. 
  We study some aspects of three-dimensional gravity by extending Jackiw's scalar theory to (2+1)-dimensions and find a black hole solution. We show that in in general this teory does not possess a Newtonian limit except for special metric configurations. 
  Some properties of the 4-dim Riemannian spaces with metrics $$ ds^2=2(za_3-ta_4)dx^2+4(za_2-ta_3)dxdy+2(za_1-ta_2)dy^2+2dxdz+2dydt $$ associated with the second order nonlinear differential equations $$ y''+a_{1}(x,y){y'}^3+3a_{2}(x,y){y'}^2+3a_{3}(x,y)y'+a_{4}(x,y)=0 $$ with arbitrary coefficients $a_{i}(x,y)$ and 3-dim Einstein-Weyl spaces connected with dual equations $$ b''=g(a,b,b') $$ where the function $g(a,b,b')$ satisfied the partial differential equation $$ g_{aacc}+2cg_{abcc}+2gg_{accc}+c^2g_{bbcc}+2cgg_{bccc}+ g^2g_{cccc}+(g_a+cg_b)g_{ccc}-4g_{abc}- $$ $$  -4cg_{bbc} -cg_{c}g_{bcc}- 3gg_{bcc}-g_cg_{acc}+ 4g_cg_{bc}-3g_bg_{cc}+6g_{bb} =0 $$ are considered. 
  We study a scalar field in curved space in three dimensions. We obtain a static perturbative solution and show that this solution satisfies the exact equations in the asymptotic region at infinity. The new solution gives rise to a singularity in the curvature scalar at the origin. Our solution, however, necessitates the excising the region near the origin, thus this naked singularity is avoided. 
  Barbour, Hawking, Misner and others have argued that time cannot play an essential role in the formulation of a quantum theory of cosmology. Here we present three challenges to their arguments, taken from works and remarks by Kauffman, Markopoulou and Newman. These can be seen to be based on two principles: that every observable in a theory of cosmology should be measurable by some observer inside the universe, and all mathematical constructions necessary to the formulation of the theory should be realizable in a finite time by a computer that fits inside the universe. We also briefly discuss how a cosmological theory could be formulated so it is in agreement with these principles. 
  Some problems have been found as to the definition of entropy of black hole being applied to the extremal Kerr-Newman case, which has conflicts with the third law of thermodynamics. We have proposed a new modification for the near extremal one, which not only obeys the third law, but also does not conflict with other results in black hole thermodynamics. Then we proved that the inner horizon has temperature and proposed that the inner horizon contributes to the entropy of the near extremal one so that the entropy of it has a modified form and vanishes at absolute zero temperature. 
  In this paper we study how all the physical "constants" vary in the framework described by a model in which we have taken into account the generalize conservation principle for its stress-energy tensor. This possibility enable us to take into account the adiabatic matter creation in order to get rid of the entropy problem. We try to generalize this situation by contemplating multi-fluid components. To validate all the obtained results we explore the possibility of considering the variation of the"constants" in the quantum cosmological scenario described by the Wheeler-DeWitt equation. For this purpose we explore the Wheeler-DeWitt equation in different contexts but from a dimensional point of view. We end by presenting the Wheeler-DeWitt equation in the case of considering all the constants varying. The quantum potential is obtained and the tunneling probability is studied. 
  We study the back-reaction effects of the finite-temperature scalar field and the photon field in the background of an Einstein universe. In each case we find a relation between the temperature of the universe and its radius. These relations exhibit a minimum radius below which no self-consistent solution for the Einstein field equation can be found. A maximum temperature marks the transition from the vacuum dominated era to the radiation dominated era. An interpretation to this behavior in terms of Bose-Einstein condensation in the case of the scalar field is given. 
  According to the Special Theory of Relativity, a rotating magnetic dielectric cylinder in an axial magnetic field should exhibit a contribution to the radial electric potential that is associated with the motion of the material's magnetic dipoles. In 1913 Wilson and Wilson reported a measurement of the potential difference across a magnetic dielectric constructed from wax and steel balls. Their measurement has long been regarded as a verification of this prediction. In 1995 Pelligrini and Swift questioned the theoretical basis of experiment. In particular, they pointed out that it is not obvious that a rotating medium may be treated as if each point in the medium is locally inertial. They calculated the effect in the rotating frame and predicted a potential different from both Wilson's theory and experiment. Subsequent analysis of the experiment suggests that Wilson's experiment does not distinguish between the two predictions due to the fact that their composite steel-wax cylinder is conductive in the regions of magnetization. We report measurements of the radial voltage difference across various rotating dielectric cylinders, including a homogeneous magnetic material (YIG), to unambiguously test the competing calculations. Our results are compatible with the traditional treatment of the effect using a co-moving locally inertial reference frame, and are incompatible with the predictions based on the model of Pelligrini and Swift. 
  A new expression for the spin connection of teleparallel gravity is proposed, given by minus the contorsion tensor plus a zero connection. The corresponding minimal coupling is covariant under local Lorentz transformation, and equivalent to the minimal coupling prescription of general relativity. With this coupling prescription, therefore, teleparallel gravity turns out to be fully equivalent to general relativity, even in the presence of spinor fields. 
  We show that that vector field-based models of the ether generically do not have a Hamiltonian that is bounded from below in a flat spacetime. We also demonstrate that these models possess multiple light cones in flat or curved spacetime, and that the non-lightlike characteristic is associated with an ether degree of freedom that will tend to form shocks. Since the field equations (and propagation speed) of this mode is singular when the timelike component of the ether vector field vanishes, we demonstrate that linearized analyses about such configurations cannot be trusted to produce robust approximations to the theory. 
  Using the framework for solving the spherically symmetric field equations in observational coordinates given in Araujo and Stoeger (1999), their formulation and solution in the perturbed FLRW sperically symmetric case with observational data representing galaxy redshifts, number counts and observer area distances, both as functions of redshift on our past light cone, are presented. The importance of the central conditions, those which must hold on our world line C, is emphasized. In detailing the solution for these perturbations, we discuss the gauge problem and its resolution in this context, as well as how errors and gaps in the data are propagated together with the genuine perturbations. This will provide guidance for solving, and interpreting the solutions of the more complicated general perturbation problem with observational data on our past light cone. 
  Recent observations suggest that the ratio of the total density to the critical density of the universe, $\Omega_0$, is likely to be very close to one, with a significant proportion of this energy being in the form of a dark component with negative pressure. Motivated by this result, we study the question of observational detection of possible non-trivial topologies in universes with $\Omega_0 \sim 1$, which include a cosmological constant. Using a number of indicators we find that as $\Omega_0 \to 1$, increasing families of possible manifolds (topologies) become either undetectable or can be excluded observationally. Furthermore, given a non-zero lower bound on $|\Omega_0 - 1|$, we can rule out families of topologies (manifolds) as possible candidates for the shape of our universe. We demonstrate these findings concretely by considering families of topologies and employing bounds on cosmological parameters from recent observations. We find that given the present bounds on cosmological parameters, there are families of both hyperbolic and spherical manifolds that remain undetectable and families that can be excluded as the shape of our universe. These results are of importance in future search strategies for the detection of the shape of our universe, given that there are an infinite number of theoretically possible topologies and that the future observations are expected to put a non-zero lower bound on $|\Omega_0 - 1|$ which is more accurate and closer to zero. 
  It has been suggested that wormholes and other non-trivial geometrical structures might have been formed during the quantum cosmological era ($t\sim 10^{-43}$s). Subsequent inflation of the universe might have enlarged these structures to macroscopic sizes. In this paper, spherical geometrical structures in an inflationary RW background are derived from the Einstein equations, using a constraint on the energy-momentum tensor which is an extension of the one expected for inflation. The possibility of dynamical wormholes and other spherical structures are explored in the framework of the solutions. 
  The electrostatic potential of any test charge distribution in Schwarzschild space with boundary values is derived. We calculate the Green's function, generalize the second Green's identity for p-forms and find the general solution. Boundary value problems are solved. With a multipole expansion the asymptotic property for the field of any charge distribution is derived. It is shown that one produces a Reissner--Nordstrom black hole if one lowers a test charge distribution slowly toward the horizon. The symmetry of the distribution is not important. All the multipole moments fade away except the monopole. A calculation of the gravitationally induced electrostatic self-force on a pointlike test charge distribution held stationary outside the black hole is presented. 
  We compare classical and quantum dynamics of a particle in the de Sitter spacetimes with different topologies to show that the result of quantization strongly depends on global properties of a classical system. We present essentially self-adjoint representations of the algebra of observables for each system. Quantization based on global properties of a classical system accounts properly its symmetries. 
  In this paper we investigate a class of (d+1) dimensional cosmological models with a cosmological constant possessing an R^d simply transitive symmetry group and show that it can be written in a form that manifests the effect of a permutation symmetry. We investigate the solution orbifold and calculate the probability of a certain number of dimensions that will expand or contract. We use this to calculate the probabilities up to dimension d=5. 
  Perfect fluid spacetimes admitting a kinematic self-similarity of infinite type are investigated. In the case of plane, spherically or hyperbolically symmetric space-times the field equations reduce to a system of autonomous ordinary differential equations. The qualitative properties of solutions of this system of equations, and in particular their asymptotic behavior, are studied. Special cases, including some of the invariant sets and the geodesic case, are examined in detail and the exact solutions are provided. The class of solutions exhibiting physical self-similarity are found to play an important role in describing the asymptotic behavior of the infinite kinematic self-similar models. 
  A description of the canonical formulation of lineal gravity minimally coupled to N point particles in a circular topology is given. The Hamiltonian is found to be equal to the time-rate of change of the extrinsic curvature multiplied by the proper circumference of the circle. Exact solutions for pure gravity and for gravity coupled to a single particle are obtained. The presence of a single particle significantly modifies the spacetime evolution by either slowing down or reversing the cosmological expansion of the circle, depending upon the choice of parameters. 
  We extend the quasilocal (metric-based) Hamiltonian formulation of general relativity so that it may be used to study regions of spacetime with null boundaries. In particular we use this generalized Brown-York formalism to study the physics of isolated horizons. We show that the first law of isolated horizon mechanics follows directly from the first variation of the Hamiltonian. This variation is not restricted to the phase space of solutions to the equations of motion but is instead through the space of all (off-shell) spacetimes that contain isolated horizons. We find two-surface integrals evaluated on the horizons that are consistent with the Hamiltonian and which define the energy and angular momentum of these objects. These are closely related to the corresponding Komar integrals and for Kerr-Newman spacetime are equal to the corresponding ADM/Bondi quantities. Thus, the energy of an isolated horizon calculated by this method is in agreement with that recently calculated by Ashtekar and collaborators but not the same as the corresponding quasilocal energy defined by Brown and York. Isolated horizon mechanics and Brown-York thermodynamics are compared. 
  Results on the observables of the Euclidean supergravity in terms of the Dirac eigenvalues are briefly revisited. 
  I study the gauge-invariant fluctuations of the metric during inflation. In the infrared sector the metric fluctuations can be represented by a coarse-grained field. We can write a Schroedinger equation for the coarse-grained metric fluctuations which is related to an effective Hamiltonian for a time dependent parameter of mass harmonic oscillator with a stochastic external force. I study the wave function for a power-law expanding universe. I find that the phase space of the quantum state for super Hubble scalar metric perturbations loses its coherence at the end of inflation. This effect is a consequence of interference between the super Hubble metric perturbations and its canonical conjugate variable, which is produced by the interaction of the coarse-grained scalar metric fluctuation with the environment. 
  We present a stationary axisymmetric solution belonging to Carter's family [A] of spaces and representing an anisotropic fluid configuration. 
  It is shown that a recently proposed model for the gravitational interaction in non relativistic quantum mechanics may turn to be relevant to the derivation of the second law of thermodynamics. In particular, the spreading of the probability density of the center of mass of an isolated macroscopic body does not imply delocalization of the wave function, but on the contrary it corresponds to an entropy growth. 
  The accuracy reached in the past few years by Satellite Laser Ranging (SLR) allows for measuring even tiny features of the Earth's gravitational field predicted by Einstein's General Relativity by means of artificial satellites. The gravitomagnetic dragging of the orbit of a test body is currently under measurement by analyzing a suitable combination of the orbital residuals of LAGEOS and LAGEOS II. The lower bound of the error in this experiment amount to 12.92%. It is due to the mismodeling in the even zonal harmonics of the geopotential which are the most important sources of systematic error. A similar approach could be used in order to measure the relativistic gravitoelectric pericenter shift in the field of the Earth with a lower bound of the systematic relative error of 0.6% due to the even zonal harmonics as well. The inclusion of the ranging data to the Japanese passive geodetic satellite Ajisai would improve such limits to 10.78% and 0.08% respectively and would allow to improve the accuracy in the determination of the PPN parameters beta and gamma. 
  We consider principal moments of inertia of axisymmetric, magnetically deformed stars in the context of general relativity. The general expression for the moment of inertia with respect to the symmetric axis is obtained. The numerical estimates are derived for several polytropic stellar models. We find that the values of the principal moments of inertia are modified by a factor of 2 at most from Newtonian estimates. 
  The Einstein-Straus spacetime describes a nonrotating black hole immersed in a matter-dominated cosmology. It is constructed by scooping out a spherical ball of the dust and replacing it with a vacuum region containing a black hole of the same mass. The metric is smooth at the boundary, which is comoving with the rest of the universe. We study the evolution of a massless scalar field in the Einstein-Straus spacetime, with a special emphasis on its late-time behavior. This is done by numerically integrating the scalar wave equation in a double-null coordinate system that covers both portions (vacuum and dust) of the spacetime. We show that the field's evolution is governed mostly by the strong concentration of curvature near the black hole, and the discontinuity in the dust's mass density at the boundary; these give rise to a rather complex behavior at late times. Contrary to what it would do in an asymptotically-flat spacetime, the field does not decay in time according to an inverse power-law. 
  We analyze how the thermal history of the universe is influenced by the statistical description, assuming a deviation from the usual Bose-Einstein, Fermi-Dirac and Boltzmann-Gibbs distribution functions. These deviations represent the possible appearance of non-extensive effects related with the existence of long range forces, memory effects, or evolution in fractal or multi-fractal space. In the early universe, it is usually assumed that the distribution functions are the standard ones. Then, considering the evolution in a larger theoretical framework will allow to test this assumption and to place limits to the range of its validity. The corrections obtained will change with temperature, and consequently, the bounds on the possible amount of non-extensivity will also change with time. We generalize results which can be used in other contexts as well, as the Boltzmann equation and the Saha law, and provide an estimate on how known cosmological bounds on the masses of neutrinos are modified by a change in the statistics. We particularly analyze here the recombination epoch, making explicit use of the chemical potentials involved in order to attain the necessary corrections. All these results constitute the basic tools needed for placing bounds on the amount of non-extensivity that could be present at different eras and will be later used to study primordial nucleosynthesis. 
  If the universe is slightly non-extensive, and the distribution functions are not exactly given by those of Boltzmann-Gibbs, the primordial production of light elements will be non-trivially modified. In particular, the principle of detailed balance (PDB), of fundamental importance in the standard analytical analysis, is no longer valid, and a non-extensive correction appears. This correction is computed and its influence is studied and compared with previous works, where, even when the universe was considered as an slightly non-extensive system, the PDB was assumed valid. We analytically track the formation of Helium and Deuterium, and study the kind of deviation one could expect from the standard regime. The correction to the capture time, the moment in which Deuterium can no longer be substantially photo-disintegrated, is also presented. This allows us to take into account the process of the free decay of neutrons, which was absent in all previous treatments of the topic. We show that even when considering a first (linear) order correction in the quantum distribution functions, the final output on the primordial nucleosynthesis yields cannot be reduced to a linear correction in the abundances. We finally obtain new bounds upon the non-extensive parameter, both comparing the range of physical viability of the theory, and using the latest observational data. 
  Analytic spherically symmetric solutions of the Einstein field equations coupled with a perfect fluid and with self-similarities of the zeroth, first and second kinds, found recently by Benoit and Coley [Class. Quantum Grav. {\bf 15}, 2397 (1998)], are studied, and found that some of them represent gravitational collapse. When the solutions have self-similarity of the first (homothetic) kind, some of the solutions may represent critical collapse but in the sense that now the "critical" solution separates the collapse that forms black holes from the collapse that forms naked singularities. The formation of such black holes always starts with a mass gap, although the "critical" solution has homothetic self-similarity. The solutions with self-similarity of the zeroth and second kinds seem irrelevant to critical collapse. Yet, it is also found that the de Sitter solution is a particular case of the solutions with self-similarity of the zeroth kind, and that the Schwarzschild solution is a particular case of the solutions with self-similarity of the second kind with the index $\alpha = 3/2$. 
  We compare two advanced designs for gravitational-wave antennas in terms of their ability to detect two possible gravitational wave sources. Spherical, resonant mass antennas and interferometers incorporating resonant sideband extraction (RSE) were modeled using experimentally measurable parameters. The signal-to-noise ratio of each detector for a binary neutron star system and a rapidly rotating stellar core were calculated. For a range of plausible parameters we found that the advanced LIGO interferometer incorporating RSE gave higher signal-to-noise ratios than a spherical detector resonant at the same frequency for both sources. Spheres were found to be sensitive to these sources at distances beyond our galaxy. Interferometers were sensitive to these sources at far enough distances that several events per year would be expected. 
  It is shown that a recently proposed model for the gravitational interaction in non relativistic quantum mechanics is the instantaneous action at a distance limit of a field theoretic model containing a negative energy field. It reduces to the Schroedinger-Newton theory in a suitable mean field approximation. While both the exact model and its approximation lead to estimates for localization lengths, only the former gives rise to an explicit non unitary dynamics accounting for the emergence of the classical behavior of macroscopic bodies. 
  Strong field (exact) solutions of the gravitational field equations of General Relativity in the presence of a Cosmological Constant are investigated. In particular, a full exact solution is derived within the inhomogeneous Szekeres-Szafron family of space-time line element with a nonzero Cosmological Constant. The resulting solution connects, in an intrinsic way, General Relativity with the theory of modular forms and elliptic curves. The homogeneous FLRW limit of the above space-time elements is recovered and we solve exactly the resulting Friedmann Robertson field equation with the appropriate matter density for generic values of the Cosmological Constant %Lambda and curvature constant K. A formal expression for the Hubble constant is derived. The cosmological implications of the resulting non-linear solutions are systematically investigated. Two particularly interesting solutions i) the case of a flat universe K=0, Lambda not= 0 and ii) a case with all three cosmological parameters non-zero, are described by elliptic curves with the property of complex multiplication and absolute modular invariant j=0 and 1728, respectively. The possibility that all non-linear solutions of General Relativity are expressed in terms of theta functions associated with Riemann-surfaces is discussed. 
  We prove uniqueness of static, asymptotically flat spacetimes with non-degenerate black holes for three special cases of Einstein-Maxwell-dilaton theory: For the coupling ``$\alpha = 1$'' (which is the low energy limit of string theory) on the one hand, and for vanishing magnetic or vanishing electric field (but arbitrary coupling) on the other hand. Our work generalizes in a natural, but non-trivial way the uniqueness result obtained by Masood-ul-Alam who requires both $\alpha = 1$ and absence of magnetic fields, as well as relations between the mass and the charges. Moreover, we simplify Masood-ul-Alam's proof as we do not require any non-trivial extensions of Witten's positive mass theorem. We also obtain partial results on the uniqueness problem for general harmonic maps. 
  In this paper we follow a new approach for particle creation by a localized strong gravitational field. The approach is based on a definition of the physical vacuum drawn from Heisenberg uncertainty principle. Using the fact that the gravitational field red-shifts the frequency modes of the vacuum, a condition on the minimum stregth of the gravitational field required to achieve real particle creation is derived. Application of this requirement on a Schwartzchid black hole resulted in deducing an upper limit on the region, outside the event horizon, where real particles can be created. Using this regional upper limit, and considering particle creation by black holes as a consequence of the Casimir effect, with the assumption that the created quanta are to be added to the initial energy, we deduce a natural power law for the development of the event horizon, and consequently a logarithmic law for the area spectrum of an inflating black hole. Application of the results on a cosmological model shows that if we start with a Planck-dimensional black hole, then through the process of particle creation we end up with a universe having the presently estimated critical density. Such a universe will be in a state of eternal inflation. 
  The $TH\epsilon\mu$ formalism was developed to study nonmetric theories of gravitation. In this letter we show that theories that violate Local Lorentz Invariance (LLI) or Local Position Invariance (LPI) also violate charge conservation. Using upper bounds on this violation we can put very stringent limits to violations of Einstein Equivalence Principle (EEP). These limits, in turn, severely restrict string-based models of low energy physics. 
  We show that if all observers see an isotropic cosmic microwave background in an expanding geodesic perfect fluid spacetime within a scalar-tensor theory of gravity, then that spacetime must be isotropic and spatially homogeneous. This result generalizes the Ehlers-Geren-Sachs Theorem of General Relativity, and serves to underpin the important result that any evolving cosmological model in a scalar-tensor theory that is compatible with observations must be almost Friedmann-Lemaitre-Robertson-Walker. 
  We have analyzed a nonsingular model with a variable cosmological term following the Carvalho {\it et al}. ansatz. The model was shown to approximate to the model of Freese {\it et al}. in one direction and to the \"{O}zer-Taha in the other. We have then included the effect of viscosity in this cosmology, as this effect has not been considered before. The analysis showed that this viscous effect could be important with a present contribution to the cosmic pressure, at most, of order of that of radiation. The model puts a stronger upper bound on the baryonic matter than that required by the standard model. A variable gravitational and cosmological constant were then introduced in a scenario which conserves the energy and momentum in the presence of bulk viscosity. The result of the analysis reveals that various models could be viscous. A noteworthy result is that some nonsingular closed models evolve asymptotically into a singular viscous one. The considered models solve for many of the standard model problems. Though the introduction of bulk viscosity results in the creation of particles, this scenario conserves energy and momentum. As in the standard model the entropy remains constant. We have not explained the generation of bulk viscosity but some workers attributes this to neutrinos. Though the role of viscosity today is minute it could, nevertheless, have had an important contribution at early times. We have shown that these models encompass many of the old and recently proposed models, in particular, Brans-Dicke, Dirac, Freese {\it et al}., Berman, Abdel Rahman and Kalligas {\it et al}. models. Hence we claim that the introduction of bulk viscosity enriches the adopted cosmology. 
  In the near future we will witness the coming to a full operational regime of laser interferometers and resonant mass detectors of spherical shape. In this work we study the sensitivity of pairs of such gravitational wave detectors to a scalar stochastic background of gravitational waves. Our computations are carried out both for minimal and non minimal coupling of the scalar fields. 
  A method is presented for solving the characteristic initial value problem for the collision and subsequent nonlinear interaction of plane gravitational or gravitational and electromagnetic waves in a Minkowski background. This method generalizes the monodromy transform approach to fields with nonanalytic behaviour on the characteristics inherent to waves with distinct wave fronts. The crux of the method is in a reformulation of the main nonlinear symmetry reduced field equations as linear integral equations whose solutions are determined by generalized (``dynamical'') monodromy data which evolve from data specified on the initial characteristics (the wavefronts). 
  We find an exact solution to the charged 2-body problem in $(1+1)$ dimensional lineal gravity which provides the first example of a relativistic system that generalizes the Majumdar-Papapetrou condition for static balance. 
  We present a new many-parameter family of hyperbolic representations of Einstein's equations, which we obtain by a straightforward generalization of previously known systems. We solve the resulting evolution equations numerically for a Schwarzschild black hole in three spatial dimensions, and find that the stability of the simulation is strongly dependent on the form of the equations (i.e. the choice of parameters of the hyperbolic system), independent of the numerics. For an appropriate range of parameters we can evolve a single 3D black hole to $t \simeq 600 M$ -- $1300 M$, and are apparently limited by constraint-violating solutions of the evolution equations. We expect that our method should result in comparable times for evolutions of a binary black hole system. 
  We present the Lagrangian whose corresponding action is the trace K action for General Relativity. Although this Lagrangian is second order in the derivatives, it has no second order time derivatives and its behaviour at space infinity in the asymptotically flat case is identical to other alternative Lagrangians for General Relativity, like the gamma-gamma Lagrangian used by Einstein. We develop some elements of the variational principle for field theories with boundaries, and apply them to second order Lagrangians, where we stablish the conditions -- proposition 1 -- for the conservation of the Noether charges. From this general approach a pre-symplectic form is naturally obtained that features two terms, one from the bulk and another from the boundary. When applied to the trace K Lagrangian, we recover a pre-symplectic form first introduced using a different approach. We prove that all diffeomorphisms satisfying certain restrictions at the boundary -- that keep room for a realization of the Poincar\'e group -- will yield Noether conserved charges. In particular, the computation of the total energy gives, in the asymptotically flat case, the ADM result. 
  The principal purpose of this paper is to study the effect of an impulsive light-like signal on neighbouring test particles. Such a signal can in general be unambiguously decomposed into a light-like shell of null matter and an impulsive gravitational wave. Our results are: (a) If there is anisotropic stress in the light-like shell then test particles initially moving in the signal front are displaced out of this 2-surface after encountering the signal; (b) For a light-like shell with no anisotropic stress accompanying a gravitational wave the effect of the signal on test particles moving in the signal front is to displace them relative to each other with the usual distortion due to the gravitational wave diminished by the presence of the light-like shell. An explicit example for a plane-fronted signal is worked out. 
  In the s-wave approximation the 4D Einstein gravity with scalar fields can be reduced to an effective 2D dilaton gravity coupled nonminimally to the matter fields. We study the leading order (tree level) vertices. The 4-particle matrix element is calculated explicitly. It is interpreted as scattering with formation of a virtual black hole state. As one novel feature we predict the gravitational decay of s-waves. 
  Parity nonconservation in the beta-decay processes is considered as fundamental property of weak interactions. Nevertheless, this property can be treated as anomaly, because in fundamental interactions of the rest types parity is conserved. Analogously, anomaly in the short-duration strong-current pulse discharges is well known. The essence of this phenomenon consists in generation of local high-temperature plasma formations (LHTF) with the typical values of its thermodynamical parameters exceeding those related to the central section of a discharge. In this paper, an attempt is undertaken to treat these anomalies as manifestation of fundamental properties of gravitational emission. Some consequences of this assumption can be tested in the beta-decay experiments as well as in the experiments with short-duration z-pinch-type pulse discharges. 
  The paper is based on the recently proposed 4-dimensional optical space theory and draws some of its consequences for gravitation. Starting with the discussion of central movement, the paper proceeds to establish the a metric compatible with Newtonian mechanics which can be accommodated by the new theory and finds a correction term which can be neglected in most practical circumstances. Being effective in the very short range, the correction term affects substantially the results when continuous mass distributions are considered. The main consequence is the possibility of explaining the orbital speeds found around galaxies, without the need to appeal for a lot of dark matter. The speed of gravity is also discussed and the theory is found compatible with a gravitational speed equal to the speed of light. On the subject of black holes, it is suggested that they are just a possibility but not a geometric inevitability. 
  Starting from the hypothesis of scaling solutions, the general exact form of the scalar field potential is found. In the case of two fluids, it turns out to be a negative power of hyperbolic sine. In the case of three fluids the analytic form is not found, but is obtained by quadratures. 
  We show how to use dimensional regularization to determine, within the Arnowitt-Deser-Misner canonical formalism, the reduced Hamiltonian describing the dynamics of two gravitationally interacting point masses. Implementing, at the third post-Newtonian (3PN) accuracy, our procedure we find that dimensional continuation yields a finite, unambiguous (no pole part) 3PN Hamiltonian which uniquely determines the heretofore ambiguous ``static'' parameter: namely, $\omega_s=0$. Our work also provides a remarkable check of the perturbative consistency (compatibility with gauge symmetry) of dimensional continuation through a direct calculation of the ``kinetic'' parameter $\omega_k$, giving the unique answer compatible with global Poincar\'e invariance ($\omega_k={41/24}$) by summing $\sim50$ different dimensionally continued contributions. 
  The conclusions obtained in gr-qc/0101067 are shown to be valid also if the full 2.5PN expansion of the chirp phase is used. 
  We propose a practical scheme for calculating the local gravitational self-force experienced by a test mass particle moving in a black hole spacetime. The method---equally effective for either weak or strong field orbits---employs the {\em mode-sum regularization scheme} previously developed for a scalar toy model. The starting point for the calculation, in this approach, is the formal expression for the regularized self-force derived by Mino et al. (and, independently, by Quinn and Wald), which involves a worldline integral over the tail part of the retarded Green's function. This force is decomposed into multipole (tensor harmonic) modes, whose sum is subjected to a carefully designed regularization procedure. This procedure involves an analytic derivation of certain ``regularization parameters'' by means of a local analysis of the Green's function. This manuscript contains the following main parts: (1) Introduction of the mode sum scheme as applied to the gravitational case. (2) Two simple cases studied: the test case of a static particle in flat spacetime, and the case of a particle at a turning point of a radial geodesic in Schwarzschild spacetime. In both cases we derive all necessary regularization parameters. (3) An Analytic foundation is set for applying the scheme in more general cases. (In this paper, the mode sum scheme is formulated within the harmonic gauge. The implementation of the scheme in other gauges shall be discussed in a separate, forthcoming paper.) 
  We analyze a class of 5D non-compact warped-product spaces characterized by metrics that depend on the extra coordinate via a conformal factor. Our model is closely related to the so-called canonical coordinate gauge of Mashhoon et al. We confirm that if the 5D manifold in our model is Ricci-flat, then there is an induced cosmological constant in the 4D sub-manifold. We derive the general form of the 5D Killing vectors and relate them to the 4D Killing vectors of the embedded spacetime. We then study the 5D null geodesic paths and show that the 4D part of the motion can be timelike -- that is, massless particles in 5D can be massive in 4D. We find that if the null trajectories are affinely parameterized in 5D, then the particle is subject to an anomalous acceleration or fifth force. However, this force may be removed by reparameterization, which brings the correct definition of the proper time into question. Physical properties of the geodesics -- such as rest mass variations induced by a variable cosmological ``constant'', constants of the motion and 5D time-dilation effects -- are discussed and are shown to be open to experimental or observational investigation. 
  Major changes in Sec. II (with much stronger estimates), minor changes in remaining sections, a new title. Deleted Appendix, added 2 new references. 
  We consider a generalization of the interior Schwarzschild solution that we match to the exterior one to build global C^1 models that can have arbitrary large mass, or density, with arbitrary size. This is possible because of a new insight into the problem of localizing the center of symmetry of the models and the use of principal transformations to understand the structure of space. 
  We show how the Weyl formalism allows metrics to be written down which correspond to arbitrary numbers of collinear accelerating neutral black holes in 3+1 dimensions. The black holes have arbitrary masses and different accelerations and share a common acceleration horizon. In the general case, the black holes are joined by cosmic strings or struts that provide the necessary forces that, together with the inter black hole gravitational attractions, produce the acceleration. In the cases of two and three black holes, the parameters may be chosen so that the outermost black hole is pulled along by a cosmic string and the inner black holes follow behind accelerated purely by gravitational forces. We conjecture that similar solutions exist for any number of black holes. 
  In previous work the authors analysed the global properties of an approximate model of radiation damping for charged particles. This work is put into context and related to the original motivation of understanding approximations used in the study of gravitational radiation damping. It is examined to what extent the results obtained previously depend on the particular model chosen. Comparisons are made with other models for gravitational and electromagnetic fields. The relation of the kinetic model for which theorems were proved to certain many-particle models with radiation damping is exhibited. 
  We calculate the thermal noise in half-infinite mirrors containing a layer of arbitrary thickness and depth made of excessively lossy material but with the same elastic material properties as the substrate. For the special case of a thin lossy layer on the surface of the mirror, the excess noise scales as the ratio of the coating loss to the substrate loss and as the ratio of the coating thickness to the laser beam spot size. Assuming a silica substrate with a loss function of 3x10-8 the coating loss must be less than 3x10-5 for a 6 cm spot size and a 7 micrometers thick coating to avoid increasing the spectral density of displacement noise by more than 10%. A similar number is obtained for sapphire test masses. 
  We show that the mean-square displacement of a charged oscillator due to the zero point oscillations of the radiation field is unique in the sense that it is very sensitive to the value of the bare mass of the charge. Thus, a controlled experiment using gravitational wave detectors could lead to a determination of the electron bare mass and shed some light on quantum electrodynamic theory. We also speculate that the irregular signals of non-gravitational origin often observed in gravitational wave bar detectors could be caused by stray charges and that such charges could also adversely affect LIGO and other such detectors 
  If the universe has a nontrivial shape (topology) the sky may show multiple correlated images of cosmic objects. These correlations can be couched in terms of distance correlations. We propose a statistical quantity which can be used to reveal the topological signature of any Robertson-Walker (RW) spacetime with nontrivial topology. We also show through computer-aided simulations how one can extract the topological signatures of flat, elliptic, and hyperbolic RW universes with nontrivial topology. 
  If the topology of the universe is compact we show how it significantly changes our assessment of the naturalness of the observed structure of the universe and the likelihood of its present state of high isotropy and near flatness arising from generic initial conditions. We also identify the most general cosmological models with compact space. 
  We consider the formulation and some elaboration of p-adic and adelic quantum cosmology. The adelic generalization of the Hartle-Hawking proposal does not work in models with matter fields. p-Adic and adelic minisuperspace quantum cosmology is well defined as an ordinary application of p-adic and adelic quantum mechanics. It is illustrated by a few minisuperspace cosmological models in one, two and three minisuperspace dimensions. As a result of p-adic quantum effects and the adelic approach, these models exhibit some discreteness of the minisuperspace and cosmological constant. In particular, discreteness of the de Sitter space and its cosmological constant is emphasized. 
  Derived from semi-classical quantum field theory in curved spacetime, Unruh effect was known as a quantum effect. We find that there does exist a classical correspondence of this effect in electrodynamics. The thermal nature of the vacuum in correlation function for the uniformly accelerated detector is coming from the non-linear relationship between the proper time and the propagating length of the electromagnetic wave. Both the Coulomb field of the detector itself and the radiation supporting the detector's uniformly accelerating motion contribute to the non-vanishing vacuum energy. From this observation we conclude that Unruh temperature experienced by a uniformly accelerated classical electron has no additional effects to Born's solution for laboratory observers far away from the classical electron. 
  We study the evolution of universe with a single scalar field of constant potential minimally coupled to gravity in the brane world cosmology.We find an exact inflationary solution which is not in slow roll.We discuss the limiting cases of the solution.We show that at late times the solution is asymptotic to the de Sitter solution independently of the brane tension. For $t\to 0$ the solution leads to singularity but the nature of the approach to singularity depends upon the brane tension. 
  We constructed a model where the central core of the universe is a modified Gidding-Strominger wormhole and surrounding the core is a Robertson-Walker Universe with k=0. They are separated by a thin wall which does not allow the content of the inner core to travel to the outer universe. But this wall allows the pressure of the inner core to be transferred to the outer physical universe. Assuming that the fluid density of the physical universe is practically independent of time, we have calculated the Hubble constant and the deacceleration parameter, qo, of the physical universe at the present time. The Hubble constant comes out to be positive, whereas qo becomes negative. The negative signature of this deacceleration parameter conforms to present experimental data. 
  We consider several kinds of quintessence models in the framework of scale factor duality. We show that this symmetry exists only for a very small number of quintessence potentials. We then apply the duality transformations found to several analytical solutions. It turns out that, in some cases, the presence of the potential allows a smooth connection between the pre- and the post-Big Bang phases. This may be a first step toward the resolution of the singularity problem. 
  The holographic principle has revealed that physical systems in 3-D space, black holes included, are basically two-dimensional as far as their information content is concerned. This conclusion is complemented by one sketched here: as far as entropy or information flow is concerned, a black hole behaves as a one-dimensional channel. We define a channel in flat spacetime in thermodynamic terms, and contrast it with common entropy emitting systems. A black hole is more like the former: its entropy output is related to the emitted power as it would be for a one-dimensional channel, and disposal of an information stream down a black hole is limited by the power invested in the same way as for a one-dimensional channel. 
  We investigate the late-time tails of self-interacting (massive) scalar fields in the spacetime of dilaton black hole. Following the no hair theorem we examine the mechanism by which self-interacting scalar hair decay. We revealed that the intermediate asymptotic behavior of the considered field perturbations is dominated by an oscillatory inverse power-law decaying tail. The numerical simulations showed that at the very late-time massive self-interacting scalar hair decayed slower than any power law. 
  On the basis of the relativistic mass-energy concept we found that a proper mass of a test particle in a gravitational field depends on a potential energy, hence, a freely falling particle has a varying proper mass. Consequently, a multitude of freely falling reference frames cannot be regarded as a multitude of equivalent inertial reference frames. There is a class of experiments, in which an inner observer can distinguish between the state of free fall in a gravitational field and the state of free space by detecting the effect of a proper mass variation. If so, a demonstration of a violation of the Equivalence Principle is possible. It is shown that a variant of the classical Pound-Rebka-Snider experiment on a photon frequency shift in a gravitational field, if conducted in a freely falling laboratory, would be such a test.   Abbreviation: SRT- the Special Relativity Theory, GRT- the General Relativity Theory, EP - the Equivalence Principle, PRS - the Pound-Rebka-Snider (experiment) 
  A static Friedmann brane in a 5-dimensional bulk (Randall-Sundrum type scenario) can have a very different relation between the density, pressure, curvature and cosmological constant than in the case of the general relativistic Einstein static universe. In particular, static Friedmann branes with zero cosmological constant and 3-curvature, but satisfying rho>0 and rho+3p>0, are possible. Furthermore, we find static Friedmann branes in a bulk that satisfies the Einstein equations but is not Schwarzschild-anti de Sitter or its specializations. In the models with negative bulk cosmological constant, a positive brane tension leads to negative density and 3-curvature. 
  Technical results are presented on motion in N(>4)D manifolds to clarify the physics of Kaluza-Klein theory, brane theory and string theory. The so-called canonical or warp metric in 5D effectively converts the manifold from a coordinate space to a momentum space, resulting in a new force (per unit mass) parallel to the 4D velocity. The form of this extra force is actually independent of the form of the metric, but for an unbound particle is tiny because it is set by the energy density of the vacuum or cosmological constant. It can be related to a small change in the rest mass of a particle, and can be evaluated in two convenient gauges relevant to gravitational and quantum systems. In the quantum gauge, the extra force leads to Heisenberg's relation between increments in the position and momenta. If the 4D action is quantized then so is the higher-dimensional part, implying that particle mass is quantized, though only at a level of 10^{-65} gm or less which is unobservably small. It is noted that massive particles which move on timeline paths in 4D can move on null paths in 5D. This agrees with the view from inflationary quantum field theory, that particles acquire mass dynamically in 4D but are intrinsically massless. A general prescription for dynamics is outlined, wherein particles move on null paths in an N(>4)D manifold which may be flat, but have masses set by an embedded 4D manifold which is curved. 
  It is shown that the finite speed of gravity affects very-long baseline interferometric observations of quasars during the time of their line-of-sight close angular encounter with Jupiter. The next such event will take place in 2002, September 8. The present Letter suggests a new experimental test of general relativity in which the effect of propagation of gravity can be directly measured by very-long baseline interferometry as an excess time delay in addition to the logarithmic Shapiro time delay (Shapiro, I. I., 1964, Phys. Rev. Lett., 13, 789). 
  Motivated by the parallelism existing between the puzzles of classical physics at the beginning of the XXth century and the current paradoxes in the search of a quantum theory of gravity, we give, in analogy with Planck's black body radiation problem, a solution for the exact Hawking flux of evaporating Reissner-Nordstrom black holes. Our results show that when back-reaction effects are fully taken into account the standard picture of black hole evaporation is significantly altered, thus implying a possible resolution of the information loss problem. 
  The synthesis of helium in the early Universe depends on many input parameters, including the value of the gravitational coupling during the period when the nucleosynthesis takes place. We compute the primordial abundance of helium as function of the gravitational coupling, using a semi-analytical method, in order to track the influence of $G$ in the primordial nucleosynthesis. To be specific, we construct a cosmological model with varying $G$, using the Brans-Dicke theory. The greater the value of $G$ at nucleosynthesis period, the greater the abundance of helium predicted. Using the observational data for the abundance of the primordial helium, constraints for the time variation of $G$ are established. 
  We review recent progress in our understanding of the physics of black holes. In particular, we discuss the ideas from string theory that explain the entropy of black holes from a counting of microstates of the hole, and the related derivation of unitary Hawking radiation from such holes. 
  New type III and type N approximate solutions which are regular in the linear approximation are shown to exist. For that, we use complex transformations on self-dual Robinson-Trautman metrics rather then the classical approach. The regularity criterion is the boundedness and vanishing at infinity of a scalar obtained by saturating the Bel-Robinson tensor of the first approximation by a time-like vector which is constant with respect to the zeroth approximation. 
  We consider a class of exact solutions which represent nonexpanding impulsive waves in backgrounds with nonzero cosmological constant. Using a convenient 5-dimensional formalism it is shown that these spacetimes admit at least three global Killing vector fields. The same geometrical approach enables us to find all geodesics in a simple explicit form and describe the effect of impulsive waves on test particles. Timelike geodesics in the axially-symmetric Hotta-Tanaka spacetime are studied in detail. It is also demonstrated that for vanishing cosmological constant, the symmetries and geodesics reduce to those for well-known impulsive pp-waves. 
  Some years ago, Lovelock showed that a number of apparently unrelated familiar tensor identities had a common structure, and could all be considered consequences in n-dimensional space of a pair of fundamental identities involving trace-free (p,p)-forms where 2p >= n$. We generalise Lovelock's results, and by using the fact that associated with any tensor in n-dimensional space there is associated a fundamental tensor identity obtained by antisymmetrising over n+1 indices, we establish a very general 'master' identity for all trace-free (k,l)-forms. We then show how various other special identities are direct and simple consequences of this master identity; in particular we give direct application to Maxwell, Lanczos, Ricci, Bel and Bel-Robinson tensors, and also demonstrate how relationships between scalar invariants of the Riemann tensor can be investigated in a systematic manner. 
  The inverse scale factor, which in classical cosmological models diverges at the singularity, is quantized in isotropic models of loop quantum cosmology by using techniques which have been developed in quantum geometry for a quantization of general relativity. This procedure results in a bounded operator which is diagonalizable simultaneously with the volume operator and whose eigenvalues are determined explicitly. For large scale factors (in fact, up to a scale factor slightly above the Planck length) the eigenvalues are close to the classical expectation, whereas below the Planck length there are large deviations leading to a non-diverging behavior of the inverse scale factor even though the scale factor has vanishing eigenvalues. This is a first indication that the classical singularity is better behaved in loop quantum cosmology. 
  A solution of the sourceless Einstein's equation with an infinite value for the cosmological constant \Lambda is discussed by using Inonu-Wigner contractions of the de Sitter groups and spaces. When \Lambda --> infinity, spacetime becomes a four-dimensional cone, dual to Minkowski space by a spacetime inversion. This inversion relates the four-cone vertex to the infinity of Minkowski space, and the four-cone infinity to the Minkowski light-cone. The non-relativistic limit c --> infinity is further considered, the kinematical group in this case being a modified Galilei group in which the space and time translations are replaced by the non-relativistic limits of the corresponding proper conformal transformations. This group presents the same abstract Lie algebra as the Galilei group and can be named the conformal Galilei group. The results may be of interest to the early Universe Cosmology. 
  A recent proposal that gravity theory is an emergent phenomenon also entails the possibility of photon decay near the Schwarzschild event horizon. We present a possible mechanism for such decay, which utilizes a dimensional reduction near the horizon. 
  We provide an in depth study of the theoretical peculiarities that arise in effective negative mass lensing, both for the case of a point mass lens and source, and for extended source situations. We describe novel observational signatures arising in the case of a source lensed by a negative mass. We show that a negative mass lens produces total or partial eclipse of the source in the umbra region and also show that the usual Shapiro time delay is replaced with an equivalent time gain. We describe these features both theoretically, as well as through numerical simulations. We provide negative mass microlensing simulations for various intensity profiles and discuss the differences between them. The light curves for microlensing events are presented and contrasted with those due to lensing produced by normal matter. Presence or absence of these features in the observed microlensing events can shed light on the existence of natural wormholes in the Universe. 
  For generalized coordinate systems, the numerical values of vector and tensor components do not generally equal the physical values, i.e., the values one would measure with standard physical instruments. Hence, calculating physical components from coordinate components is important for comparing experiment with theory. Surprisingly, however, this calculational method is not widely known among physicists, and is rarely taught in relativity courses, though it is commonly employed in at least one other field (applied mechanics.) Different derivations of this method, ranging from elementary to advanced level, are presented. The result is then applied to clarify the oftentimes confusing issue of whether or not the speed of light in non-inertial frames is equal to c. 
  Various theoretical obstacles are associated with a homogeneous and isotropic distribution of ``charge'' which is subject to a repulsive, long-range force. We show how these can be overcome, for all practical purposes, by the simple device of endowing the particle which carries the force with a small mass. The resulting situation may be relevant to a phase of cosmological acceleration which is triggered by the approach to masslessness of such a force carrier. 
  It is shown that the conclusion regarding the existence of a scalar hair for a black hole in a nonminimally coupled self interacting scalar tensor theory can be drawn from the that for a scalar field minimally coupled to gravity by means of a conformal transformation. 
  We examine generalizations of the five-dimensional canonical metric by including a dependence of the extra coordinate in the four-dimensional metric. We discuss a more appropriate way to interpret the four-dimensional energy-momentum tensor induced from the five-dimensional space-time and show it can lead to quite different physical situations depending on the interpretation chosen. Furthermore, we show that the assumption of five-dimensional null trajectories in Kaluza-Klein gravity can correspond to either four-dimensional massive or null trajectories when the path parameterization is chosen properly. Retaining the extra-coordinate dependence in the metric, we show the possibility of a cosmological variation in the rest masses of particles and a consequent departure from four-dimensional geodesic motion by a geometric force. In the examples given, we show that at late times it is possible for particles traveling along 5D null geodesics to be in a frame consistent with the induced matter scenario. 
  Analytical and numerical solutions for the integral curves of the velocity field (streamlines) of a steady-state flow of an ideal fluid with $p = \rho$ equation of state are presented. The streamlines associated with an accelerate black hole and a rigid sphere are studied in some detail, as well as, the velocity fields of a black hole and a rigid sphere in an external dipolar field (constant acceleration field). In the latter case the dipole field is produced by an axially symmetric halo or shell of matter. For each case the fluid density is studied using contour lines. We found that the presence of acceleration is detected by these contour lines. As far as we know this is the first time that the integral curves of the velocity field for accelerate objects and related spacetimes are studied in general relativity. 
  It is postulated that the action of the FRW-universe is the cosmological term of Einsteins theory (no curvature term - R0 Cosmology). The expansion equation emerging from the embedding of this most simple brane world with variable speed of light is deduced. The universal dimensionless coupling constant of gravity is addressed. Some implications on the deep problems of cosmology are discussed. 
  We study an analytical solution to the Einstein's equations in 2+1-dimensions, representing the self-similar collapse of a circularly symmetric, minimally coupled, massless, scalar field. Depending on the value of certain parameters, this solution represents the formation of black holes. Since our solution is asymptotically flat, our black holes do not have the BTZ space-time as their long time limit. They represent a new family of black holes in 2+1-dimensions. 
  In this thesis special emphasis is put on the quantization of the spherically reduced Einstein-massless-Klein-Gordon model using a first order approach for geometric quantities, because phenomenologically it is probably the most relevant of all dilaton models with matter. After a Hamiltonian BRST analysis path integral quantization is performed using temporal gauge for the Cartan variables. Retrospectively, the simpler Faddeev-Popov approach turns out to be sufficient. It is possible to eliminate all unphysical and geometric quantities establishing a non-local and non-polynomial action depending solely on the scalar field and on some integration constants, fixed by suitable boundary conditions on the asymptotic effective line element.   Then, attention is turned to the evaluation of the (two) lowest order tree vertices, explicitly assuming a perturbative expansion in the scalar field being valid. Each of them diverges, but unexpected cancellations yield a finite S-matrix element when both contributions are summed. The phenomenon of a "virtual black hole" -- already encountered in the simpler case of minimally coupled scalars in two dimensions -- occurs, as the study of the (matter dependent) metric reveals. A discussion of the scattering amplitude leads to the prediction of gravitational decay of spherical waves, a novel physical phenomenon. Several possible extensions conclude this dissertation. 
  The general relativistic gravitomagnetic clock effect consists in the fact that two point particles orbiting a central spinning object along identical, circular equatorial geodesic paths, but in opposite directions, exhibit a time difference in describing a full revolution. It turns out that the particle rotating in the same sense of the central body is slower than the particle rotating in the opposite sense. In this paper it is proposed to measure such effect in an Earth laboratory experiment involving interferometry of slow neutrons. With a sphere of 2.5 cm radius and spinning at 4.3 x 10^4 rad/s as central source, and using neutrons with wavelength of 1 Angstrom it should be possible to obtain, for a given sense of rotation of the central source, a phase shift of 0.18 rad, well within the experimental sensitivity. By reversing the sense of rotation of the central body it should be possible to obtain a 0.06 fringe shift. 
  In reference gr-qc/0104036 a four-dimensional effective theory of gravity embeddable in a five-dimensional "distorted" Randall-Sundrum brane scenario was derived. The present paper is aimed at the application of such a theory to describe physics in an open Friedmann-Robertson-Walker (Weyl-symmetric) universe. It is shown that regular bouncing universes arise for a given range of the free parameter of the theory. 
  We re-examine the Lem\^aitre-Tolman-Bondi (LTB) solutions with a dust source admitting symmetry centers. We consider as free parameters of the solutions the initial value functions: $Y_i$, $\rho_i$ and $\Ri$, obtained by restricting the curvature radius, $Y\equiv \sqrt{g_{\theta\theta}}$, the rest mass density, $\rho$, and the 3-dimensional Ricci scalar of the rest frames, $\R$, to an arbitrary regular Cauchy hypersurface, $\Ti$, marked by constant cosmic time ($t=t_i$). Using $Y_i$ to fix the radial coordinate and the topology (homeomorphic class) of $\Ti$, and scaling the time evolution in terms of an adimensional scale factor $y=Y/Y_i$, we show that the dynamics, regularity conditions and geometric features of the models are determined by $\rho_i$, $\Ri$ and by suitably constructed volume averages and contrast functions expressible in terms of invariant scalars defined in $\Ti$. These quantities lead to a straightforward characterization of initial conditions in terms of the nature of the inhomogeneity of $\Ti$, as density and/or curvature overdensities (``lumps'') and underdensities (''voids'') around a symmetry center. In general, only models with initial density and curvature lumps evolve without shell crossing singularities, though special classes of initial conditions, associated with a simmultaneous big bang, allow for a regular evolution for initial density and curvature voids. Specific restrictions are found so that a regular evolution for $t>t_i$ is possible for initial voids. A step-by-step guideline is provided for using the new variables in the construction of LTB models and for plotting all relevant quantities. 
  We discuss the possibility that "quintessential effects", recently displayed by large scale observations, may be consistently described in the context of the low-energy string effective action, and we suggest a possible approach to the problem of the cosmic coincidence based on the link between the strength of the dilaton couplings and the cosmological state of our Universe. 
  We give a careful general relativistic and (1+3)-covariant analysis of cosmological peculiar velocities induced by matter density perturbations in the presence of a cosmological constant. In our quasi-Newtonian approach, constraint equations arise to maintain zero shear of the non-comoving fundamental worldlines which define a Newtonian-like frame, and these lead to the (1+3)-covariant dynamical equations, including a generalized Poisson-type equation. We investigate the relation between peculiar velocity and peculiar acceleration, finding the conditions under which they are aligned. In this case we find (1+3)-covariant relativistic generalizations of well-known Newtonian results. 
  We define the {\it rest-frame instant form} of tetrad gravity restricted to Christodoulou-Klainermann spacetimes. After a study of the Hamiltonian group of gauge transformations generated by the 14 first class constraints of the theory, we define and solve the multitemporal equations associated with the rotation and space diffeomorphism constraints, finding how the cotriads and their momenta depend on the corresponding gauge variables. This allows to find quasi-Shanmugadhasan canonical transformation to the class of 3-orthogonal gauges and to find the Dirac observables for superspace in these gauges.   The construction of the explicit form of the transformation and of the solution of the rotation and supermomentum constraints is reduced to solve a system of elliptic linear and quasi-linear partial differential equations. We then show that the superhamiltonian constraint becomes the Lichnerowicz equation for the conformal factor of the 3-metric and that the last gauge variable is the momentum conjugated to the conformal factor. The gauge transformations generated by the superhamiltonian constraint perform the transitions among the allowed foliations of spacetime, so that the theory is independent from its 3+1 splittings. In the special 3-orthogonal gauge defined by the vanishing of the conformal factor momentum we determine the final Dirac observables for the gravitational field even if we are not able to solve the Lichnerowicz equation. The final Hamiltonian is the weak ADM energy restricted to this completely fixed gauge. 
  We investigate the end state of gravitational collapse of null fluid in higher dimensional space-times. Both naked singularities and black holes are shown to be developing as final outcome of the collapse. The naked singularity spectrum in collapsing Vaidya region (4D) gets covered with increase in dimensions and hence higher dimensions favor black hole in comparison to naked singularity. The Cosmic Censorship Conjecture will be fully respected for a space of infinite dimension. 
  In this article on the basis of a new definition of spacetime symmetry, which is in accordance with the symmetry of the curvature invariants, we investigate exact vacuum solutions of Einstein field equations corresponding to both static and stationary plane symmetric spacetimes using the concepts of the (1+3)-decomposition or threading formalism. Demanding the presence of a plane symmetric gravitomagnetic field we find a family of two parameter (m and l) solutions, every member of which being the plane symmetric analogue of NUT space. 
  Arguments are presented which show that the recent proposals of using neutrons to test gravitomagnetic effects on Earth are impracticable. 
  Non-invariant solutions of the Boyer-Finley equation determine exact solutions of the Einstein field equations with only one rotational Killing vector. For the case of Euclidean signature such a metric was constructed by Calderbank and Tod. Recently Martina, Sheftel and Winternitz applied the method of group foliation to the Boyer-Finley equation and reproduced the Calderbank-Tod solution together with new solutions. We point out that in the case of ultra-hyperbolic signature there exist three inequivalent forms of metric. Only one of these can be obtained by analytic continuation from the Calderbank-Tod solution whereas the other two are new. 
  I present a formulation of the second law of thermodynamics in the presence of black holes which makes use of the efficiency of an ideal machine extracting heat cyclically from a black hole. The Carnot coefficient is found and it is shown to be a simple function of the mass. 
  Recent observations of the Universe have led to a conclusion suppressing an up-to-now supposed deceleration of the Universe caused by attractive gravitational forces. Contrary, there is a renaissance of the cosmological member lambda and introduction of enigmatic repulsive dark energy in attempts to rationalize a would-be acceleration of the Universe expansion. It is documented that the model of Expansive Nondecelerative Universe is capable to offer acceptable answers to the questions on the Universe expansion, state equations of the Universe, the parameter omega, the cosmological member lambda without any necessity to introduce new strange kinds of matter or energy being in accord with the fundamental conservation laws and generally accepted parameters of the Universe. 
  From the partition function of canonical ensemble we derive the entropy of the de Sitter space by anti-Wick rotation. And then from the one-loop bubble $S^2\times S^2$ created from vacuum fluctuation in de Sitter background space, we obtain the one-loop quantum correction to the entropy of de Sitter space. 
  The three-dimensional static and circularly symmetric solution of the Einstein-Born-Infeld-dilaton system is derived. The solutions corresponding to low energy string theory are investigated in detail, which include black hole solutions if the cosmological constant is negative and the mass parameter exceeds a certain critical value. Some differences between the Born-Infeld nonlinear electrodynamics and the Maxwell electrodynamics are revealed. 
  The purpose of this letter is to point out an argument which may ultimately lead to a rigorous proof of the Penrose inequality in the general case. The argument is a variation of Geroch's original proposal for a proof of the positive energy theorem which was later adapted by Jang and Wald to apply to initial data sets containing apparent horizons. The new input is to dispense with the a priori restriction to an initial data set and to use the four-dimensional structure of spacetime in an essential way. 
  We survey the application of computer algebra in the context of gravitational theories. After some general remarks, we show of how to check the second Bianchi-identity by means of the Reduce package Excalc. Subsequently we list some computer algebra systems and packages relevant to applications in gravitational physics. We conclude by presenting a couple of typical examples. 
  A large class of Type II fluid solutions to Einstein field equations in N-dimensional spherical spacetimes is found, wich includes most of the known solutions. A family of the generalized collapsing Vaidya solutions with homothetic self-similarity, parametrized by a constant $\lambda$, is studied, and found that when $\lambda$ $>$ $\lambda_c(N)$, the collapse always forms black holes, and when $\lambda$ $<$ $\lambda_c(N)$, it always forms naked singularities, where $\lambda_c(N)$ is function of the spacetime dimension N only. 
  Arguments are made in favor of broadening the scope of the various approaches to splitting spacetime into a single common framework in which measured quantities, derivative operations, and adapted coordinate systems are clearly understood in terms of associated test observer families. This ``relativity of splitting formalisms" for fully nonlinear gravitational theory has been tagged with the name "gravitoelectromagnetism" because of the well known analogy between its linearization and electromagnetism, and it allows relationships between the various approaches to be better understood and makes it easier to extrapolate familiarity with one approach to the others. This is important since particular problems or particular features of those problems in gravitational theory are better suited to different approaches, and the present barriers between the proponents of each individual approach sometimes prevent the best match from occurring. 
  A quantum physical projector is proposed for generally covariant theories which are derivable from a Lagrangian. The projector is the quantum analogue of the integral over the generators of finite one-parameter subgroups of the gauge symmetry transformations which are connected to the identity. Gauge variables are retained in this formalism, thus permitting the construction of spacetime area and volume operators in a tentative spacetime loop formulation of quantum general relativity. 
  The multipolar-post-Minkowskian approach to gravitational radiation is applied to the problem of the generation of waves by the compact binary inspiral. We investigate specifically the third post-Newtonian (3PN) approximation in the total energy flux. The new results are the computation of the mass quadrupole moment of the binary to the 3PN order, and the current quadrupole and mass octupole to the 2PN order. Wave tails and tails of tails in the far zone are included up to the 3.5PN order. The recently derived 3PN equations of binary motion are used to compute the time-derivatives of the moments. We find perfect agreement to the 3.5PN order with perturbation calculations of black holes in the test-mass limit for one body. Technical inputs in our computation include a model of point particles for describing the compact objects, and the Hadamard self-field regularization. Because of a physical incompleteness of the Hadamard regularization at the 3PN order, the energy flux depends on one unknown physical parameter, which is a combination of a parameter \lambda in the equations of motion, and a new parameter \theta coming from the quadrupole moment. 
  The inspiral of compact binaries, driven by gravitational-radiation reaction, is investigated through 7/2 post-Newtonian (3.5PN) order beyond the quadrupole radiation. We outline the derivation of the 3.5PN-accurate binary's center-of-mass energy and emitted gravitational flux. The analysis consistently includes the relativistic effects in the binary's equations of motion and multipole moments, as well as the contributions of tails, and tails of tails, in the wave zone. However the result is not fully determined because of some physical incompleteness, present at the 3PN order, of the model of point-particle and the associated Hadamard-type self-field regularization. The orbital phase, whose prior knowledge is crucial for searching and analyzing the inspiral signal, is computed from the standard energy balance argument. 
  Gravitational wave detectors will need optimal signal-processing algorithms to extract weak signals from the detector noise. Most algorithms designed to date are based on the unrealistic assumption that the detector noise may be modeled as a stationary Gaussian process. However most experiments exhibit a non-Gaussian ``tail'' in the probability distribution. This ``excess'' of large signals can be a troublesome source of false alarms. This article derives an optimal (in the Neyman-Pearson sense, for weak signals) signal processing strategy when the detector noise is non-Gaussian and exhibits tail terms. This strategy is robust, meaning that it is close to optimal for Gaussian noise but far less sensitive than conventional methods to the excess large events that form the tail of the distribution. The method is analyzed for two different signal analysis problems: (i) a known waveform (e.g., a binary inspiral chirp) and (ii) a stochastic background, which requires a multi-detector signal processing algorithm. The methods should be easy to implement: they amount to truncation or clipping of sample values which lie in the outlier part of the probability distribution. 
  The entropy of a spherically symmetric distribution of matter in self-equilibrium is calculated. When gravitational effects are neglected, the entropy of the system is proportional to its volume. As effects due to gravitational self-interactions become more important, the entropy acquires a correction term and is no longer purely volume scaling. In the limit that the boundary of the system approaches its event horizon, the total entropy of the system is proportional to its area. The scaling laws of the system's thermodynamical quantities are identical to those of a black hole, even though the system does not possess an event horizon. 
  In this paper we have investigated the pure quantum solutions of Bohmian quantum gravity. By pure quantum solution we mean a solution in which the quantum potential cannot be ignored with respect to the classical potential, especially in Bohmian quantum gravity we are interested in the case where these two potentials are equal in their magnitude and in fact their sum is zero. Such a solutions are obtained both using the perturbation and using the linear field approximation. 
  We study the quasi-normal modes (QNM) of electromagnetic and gravitational perturbations of a Schwarzschild black hole in an asymptotically Anti-de Sitter (AdS) spacetime. Some of the electromagnetic modes do not oscillate, they only decay, since they have pure imaginary frequencies. The gravitational modes show peculiar features: the odd and even gravitational perturbations no longer have the same characteristic quasinormal frequencies. There is a special mode for odd perturbations whose behavior differs completely from the usual one in scalar and electromagnetic perturbation in an AdS spacetime, but has a similar behavior to the Schwarzschild black hole in an asymptotically flat spacetime: the imaginary part of the frequency goes as 1/r+, where r+ is the horizon radius. We also investigate the small black hole limit showing that the imaginary part of the frequency goes as r+^2. These results are important to the AdS/CFT conjecture since according to it the QNMs describe the approach to equilibrium in the conformal field theory. 
  There are investigated such cosmological models which instead of the usual spatial homogeneity property only fulfil the condition that in a certain synchronized system of reference all spacelike sections t = const. are homogeneous manifolds. This allows time-dependent changes of the BIANCHI type. Discussing differential geometrical theorems it is shown which of them are permitted. Besides the trivial case of changing into type I there exist some possible changes between other types. However, physical reasons like energy inequalities partially exclude them. 
  A family of cosmological models is considered which in a certain synchronized system of reference possess flat slices t = const. They are generated from the Einstein-de Sitter universe by a suitable transformation. Under physically reasonable presumptions these transformed models fulfil certain energy conditions. 
  For a thin shell, the intrinsic 3-pressure will be shown to be analogous to -A, where A is the classical surface tension: First, interior and exterior Schwarzschild solutions will be matched together such that the surface layer generated at the common boundary has no gravitational mass; then its intrinsic 3-pressure represents a surface tension fulfilling Kelvin's relation between mean curvature and pressure difference in the Newtonian limit. Second, after a suitable definition of mean curvature, the general relativistic analogue to Kelvin's relation will be proven to be contained in the equation of motion of the surface layer. 
  For field equations of 4th order, following from a Lagrangian `Ricci scalar plus Weyl scalar', it is shown (using methods of non-standard analysis) that in a neighbourhood of Minkowski space there do not exist regular static spherically symmetric solutions. With that (besides the known local expansions about r = 0 and r = infinity resp.) for the first time a global statement on the existence of such solutions is given. Finally, this result will be discussed in connection with Einstein's particle programme. 
  We show that solutions of the Bach equation exist which are not conformal Einstein spaces. 
  The Newman-Penrose constants of the spacetime corresponding to the development of the Brill-Lindquist initial data are calculated by making use of a particular representation of spatial infinity due to H. Friedrich. The Brill-Lindquist initial data set represents the head-on collision of two non-rotating black holes. In this case one non-zero constant is obtained. Its value is given in terms of the product of the individual masses of the black holes and the square of a distance parameter separating the two black holes. This constant retains its value all along null infinity, and therefore it provides information about the late time evolution of the collision process. In particular, it is argued that the magnitude of the constants provides information about the amount of residual radiation contained in the spacetime after the collision of the black holes. 
  In order to achieve full detection sensitivity at low frequencies, the mirrors of interferometric gravitational wave detectors must be isolated from seismic noise. The VIRGO vibration isolator, called 'superattenuator', is fully effective at frequencies above 4 Hz. Nevertheless, the residual motion of the mirror at the mechanical resonant frequencies of the system are too large for the interferometer locking system and must be damped. A multidimensional feedback system, using inertial sensors and digital processing, has been designed for this purpose. An experimental procedure for determining the feedback control of the system has been defined. In this paper a full description of the system is given and experimental results are presented. 
  A new development of the ``monodromy transform'' method for analysis of hyperbolic as well as elliptic integrable reductions of Einstein equations is presented. Compatibility conditions for some alternative representations of the fundamental solutions of associated linear systems with spectral parameter in terms of a pair of dressing (``scattering'') matrices give rise to a new set of linear (quasi-Fredholm) integral equations equivalent to the symmetry reduced Einstein equations. Unlike previously derived singular integral equations constructed with the use of conserved (nonevolving) monodromy data on the spectral plane for the fundamental solutions of associated linear systems, the scalar kernels of the new equations include another kind of functional parameters -- the evolving (``dynamical'') monodromy data for the scattering matrices. For hyperbolic reductions, in the context of characteristic initial value problem these data are determined completely by the characteristic initial data for the fields. In terms of solutions of the new integral equations the field components are expressed in quadratures. 
  An exact class of solutions of the 5D vacuum Einstein field equations (EFEs) is obtained. The metric coefficients are found to be non-separable functions of time and the extra coordinate $l$ and the induced metric on $l$ = constant hypersurfaces has the form of a Friedmann-Robertson-Walker cosmology. The 5D manifold and 3D and 4D submanifolds are in general curved, which distinguishes this solution from previous ones in the literature. The singularity structure of the manifold is explored: some models in the class do not exhibit a big bang, while other exhibit a big bang and a big crunch. For the models with an initial singularity, the equation of state of the induced matter evolves from radiation like at early epochs to Milne-like at late times and the big bang manifests itself as a singular hypersurface in 5D. The projection of comoving 5D null geodesics onto the 4D submanifold is shown to be compatible with standard 4D comoving trajectories, while the expansion of 5D null congruences is shown to be in line with conventional notions of the Hubble expansion. 
  The continuum and semiclassical limits of isotropic, spatially flat loop quantum cosmology are discussed, with an emphasis on the role played by the Barbero-Immirzi parameter \gamma in controlling space-time discreteness. In this way, standard quantum cosmology is shown to be the simultaneous limit \gamma \to 0, j \to \infty of loop quantum cosmology. Here, j is a label of the volume eigenvalues, and the simultaneous limit is technically the same as the classical limit \hbar \to 0, l \to \infty of angular momentum in quantum mechanics. Possible lessons for semiclassical states at the dynamical level in the full theory of quantum geometry are mentioned. 
  Brane worlds are theories with extra spatial dimensions in which ordinary matter is localized on a (3+1) dimensional submanifold. Such theories could have interesting consequences for particle physics and gravitational physics. In this essay we concentrate on the cosmological constant (CC) problem in the context of brane worlds. We show how extra-dimensional scenarios may violate Lorentz invariance in the gravity sector of the effective 4D theory, while particle physics remains unaffected. In such theories the usual no-go theorems for adjustment of the CC do not apply, and we indicate a possible explanation of the smallness of the CC. Lorentz violating effects would manifest themselves in gravitational waves travelling with a speed different from light, which can be searched for in gravitational wave experiments. 
  This is a brief summary of lectures given at the Fourth Mexican School on Gravitation and Mathematical Physics. The lectures gave an introduction to branes in eleven-dimensional supergravity and in type IIA supergravities in ten-dimensions. Charge conservation and the role of the so-called `Chern-Simons terms' were emphasized. Known exact solutions were discussed and used to provide insight into the question `Why don't fundamental strings fall off of D-branes,' which is often asked by relativists. The following is a brief overview of the lectures with an associated guide to the literature. 
  Making use of the classical Binet's equation a general procedure to obtain the gravitational force corresponding to an arbitrary 4-dimensional spacetime is presented. This method provides, for general relativistic scenarios, classics expressions that may help to visualize certain effects that Newton's theory can not explain. In particular, the force produced by a gravitational field which source is spherically symmetrical (Schwarzschild's spacetime) is obtained. Such expression uses a redefinition of the classical reduced mass, in the limit case it can be reduced to Newton's Universal Law of Gravitation and it produces two different orbital velocities for test particles that asimptotically coincide with the Newtonian one. PACS: 04.25.Nx, 95.10.Ce, 95.30.Sf. Keywords: Universal gravitational law, perihelionshift, Schwarzschild potential, reduced mass. 
  We investigated the possibility that nonlinear gravitational effects influence the preheating era after inflation, using numerical solutions of the inhomogeneous Einstein field equations. We compared our results to perturbative calculations and to solutions of the nonlinear field equations in a rigid (unperturbed) spacetime, in order to isolate gravitational phenomena. We confirm the broad picture of preheating obtained from the nonlinear field equations in a rigid background, but find gravitational effects have a measurable impact on the dynamics. The longest modes in the simulation grow much more rapidly in the relativistic calculation than with a rigid background. We used the Weyl tensor to quantify the departure from homogeneity in the universe. We saw no evidence for the sort of gravitational collapse that leads to the formation of primordial black holes. 
  In the literature about the Randall-Sundrum scenario one finds on one hand that there exist (small) corrections to Newton's law of gravity on the brane, and on another that the exact (and henceforth linearized) Einstein equations can be recovered on the brane. The explanation for these seemingly contradictory results is that the behaviour of the bulk far from the brane is different in both models. We show that explicitely in this paper. 
  We present a streamlined calculation of the Kretschmann scalar for 5D Vacua considered recently by Fukui et al. (gr-qc/0105112). 
  We consider a version of Kaluza-Klein theory where the cylinder condition is not imposed. The metric is allowed to have explicit dependence on the "extra" coordinate(s). This is the usual scenario in brane-world and space-time-matter theories. We extend the usual discussion by considering five-dimensional metrics with off-diagonal terms. We replace the condition of cylindricity by the requirement that physics in four-dimensional space-time should remain invariant under changes of coordinates in the five-dimensional bulk. This invariance does not eliminate physical effects from the extra dimension but separates them from spurious geometrical ones. We use the appropriate splitting technique to construct the most general induced energy-momentum tensor, compatible with the required invariance. It generalizes all previous results in the literature. In addition, we find two four-vectors, J_{m}^{mu} and J_{e}^{mu}, induced by off-diagonal metrics, that separately satisfy the usual equation of continuity in 4D. These vectors appear as source-terms in equations that closely resemble the ones of electromagnetism. These are Maxwell-like equations for an antisymmetric tensor {F-hat}_{mu nu} that generalizes the usual electromagnetic one. This generalization is not an assumption, but follows naturally from the dimensional reduction. Thus, if {F-hat}_{mu nu} could be identified with the electromagnetic tensor, then the theory would predict the existence of classical magnetic charge and current. The splitting formalism used allows us to construct 4D physical quantities from five-dimensional ones, in a way that is independent on how we choose our space-time coordinates from those of the bulk. 
  We discuss a scenario in which extra dimensional effects allow a scalar field with a steep potential to play the dual role of the inflaton as well as dark energy (quintessence). The post-inflationary evolution of the universe in this scenario is generically characterised by a `kinetic regime' during which the kinetic energy of the scalar field greatly exceeds its potential energy resulting in a `stiff' equation of state for scalar field matter $P_\phi \simeq \rho_\phi$. The kinetic regime precedes the radiation dominated epoch and introduces an important new feature into the spectrum of relic gravity waves created quantum mechanically during inflation. The gravity wave spectrum increases with wavenumber for wavelengths shorter than the comoving horizon scale at the commencement of the radiative regime. This `blue tilt' is a generic feature of models with steep potentials and imposes strong constraints on a class of inflationary braneworld models. Prospects for detection of the gravity wave background by terrestrial and space-borne gravity wave observatories such as LIGO II and LISA are discussed. 
  Here we study the behaviour of spin 0 sector of the DKP field in spaces with torsion. First we show that in a Riemann-Cartan manifold the DKP field presents an interaction with torsion when minimal coupling is performed, contrary to the behaviour of the KG field, a result that breaks the usual equivalence between the DKP and the KG fields.   Next we analyse the case of Teleparallel Equivalent of General Relativity Weitzenbock manifold, showing that in this case there is a perfect agreement between KG and DKP fields. The origins of both results are also discussed. 
  Black hole interiors (the $T$-domain) are studied here in great detail. Both the {\em general} and particular $T$-domain solutions are presented including non-singular ones. Infinitely many local $T$-domain solutions may be modeled with this scheme. The duality between the $T$ and $R$ domains is presented. It is demonstrated how generally well behaved $R$-domain solutions will give rise to exotic phases of matter when collapsed inside the event horizon. However, as seen by an external observer, the field is simply that of the Schwarzschild vacuum with well behaved mass term and no evidence of this behaviour may be observed. A singularity theorem is also presented which is independent of energy conditions. 
  This is a summary of a course given at the Fourth Mexican School on Gravitation and Mathematical Physics on some aspects of PBB cosmology. After introductory remarks the lectures concentrate on some amusing consequences derived from the symmetries of the string theory with respect to such classical concepts as isotropy and homogeneity. The extra dimensions and the symmetries of the M theory are further applied to show that the classical singularities might be just physically irrelevant. In the final lecture a model universe is "produced" from "almost nothing" and it is argued that initial plane waves are thermodynamically natural state for the universe to emerge from. 
  In this essay we marshal evidence suggesting that Einstein gravity may be an emergent phenomenon, one that is not ``fundamental'' but rather is an almost automatic low-energy long-distance consequence of a wide class of theories. Specifically, the emergence of a curved spacetime ``effective Lorentzian geometry'' is a common generic result of linearizing a classical scalar field theory around some non-trivial background. This explains why so many different ``analog models'' of general relativity have recently been developed based on condensed matter physics; there is something more fundamental going on. Upon quantizing the linearized fluctuations around this background geometry, the one-loop effective action is guaranteed to contain a term proportional to the Einstein--Hilbert action of general relativity, suggesting that while classical physics is responsible for generating an ``effective geometry'', quantum physics can be argued to induce an ``effective dynamics''. This physical picture suggests that Einstein gravity is an emergent low-energy long-distance phenomenon that is insensitive to the details of the high-energy short-distance physics. 
  This paper will present the exact solution for the stress-energy tensor of a spherical matter shell of finite thickness that will patch together different metrics at the boundaries of the shell. The choice of vacuum field solutions for the shell exterior and hollow interior that we make will allow us to manipulate the inertial state of an object within the shell. The choice will cause it to be in a state of acceleration with the shell relative to an external observer for an indefinite time. The stress-energy tensor's solution results in zero ship frame energy requirements, and only finite stress requirements, and we show how any WEC violation can be avoided. 
  I have recently shown that it is possible to formulate the Relativity postulates in a way that does not lead to inconsistencies in the case of space-times whose structure is governed by observer-independent scales of both velocity and length. Here I give an update on the status of this proposal, including a brief review of some very recent developments. I also emphasize the role that one of the kappa-Poincare' Hopf algebras could play in the realization of a particular example of the new type of postulates. I show that the new ideas on Relativity require us to extend the set of tools provided by kappa-Poincare' and to revise our understanding of certain already available tools, such as the energy-momentum coproduct. 
  The observations of photons from the BL Lac object Mk501 with energies above 10 TeV and of cosmic rays with energies above the GZK threshold appear to be inconsistent with conventional theories. Remarkably, among the recent new-physics proposals of solutions of these threshold paradoxes a prominent role has been played by proposals based on quantum properties of space-time. While the experimental evidence (and theory work attempting to interpret it) is much too preliminary to justify any serious hopes that we might have stumbled upon the first manifestation of a "quantum gravity", the fact that for the first time phenomenological models involving quantum-gravity ideas are competing on level ground with other new-physics proposals clearly marks the beginning of a new stage of quantum-gravity research. I emphasize one important aspect of this new phenomenology: combining the determination of the relevant thresholds with data on the time/energy structure of gamma-ray bursts it is possible to distinguish between alternative quantum-gravity scenarios. This point is illustrated focusing on 3 specific scenarios: dispersion-inducing space-time foam, string-theory-motivated non-commutative space-time, and this author's recent proposal of a relativistic theory in which the Planck length has the role of fundamental observer-independent minimum length. 
  The inflation-free solution of problems of the modern cosmology (horizon, cosmic initial data, Planck era, arrow of time, singularity,homogeneity, and so on) is considered in the conformal-invariant unified theory given in the space with geometry of similarity where we can measure only the conformal-invariant ratio of all quantities. Conformal General Relativity is defined as the $SU_c(3)\times SU(2)\times U(1)$-Standard Model where the dimensional parameter in the Higgs potential is replaced by a dilaton scalar field described by the negative Penrose-Chernikov-Tagirov action. Spontaneous SU(2) symmetry breaking is made on the level of the conformal-invariant angle of the dilaton-Higgs mixing, and it allows us to keep the structure of Einstein's theory with the equivalence principle. We show that the lowest order of the linearized equations of motion solves the problems mentioned above and describes the Cold Universe Scenario with the constant temperature T and z-history of all masses with respect to an observable conformal time. A new fact is the intensive cosmic creation of $W,Z$-vector bosons due to their mass singularity. In the rigid state, this effect is determined by the integral of motion $(m_w^2H_{\rm hubble})^{1/3}=2.7 K k_B$ that coincides with the CMB temperature and has the meaning of the primordial Hubble parameter. The created bosons are enough to consider their decay as an origin of the CMB radiation and all observational matter with the observational element abundances and the baryon asymmetry. Recent Supernova data on the relation between the luminosity distance and redshift (including the point $z=1.7$) do not contradict the dominance of the rigid state of the dark matter in the Conformal Cosmology. 
  Observations of an apparent acceleration in the expansion rate of the universe, derived from measurements of high-redshift supernovae, have been used to support the hypothesis that the universe is permeated by some form of dark energy. We show that an alternative cosmological model, based on a linearly expanding universe with Omega=1, can fully account for these observations. This model is also able to resolve the other problems associated with the standard Big Bang model. A scale-invariant form of the field equations of General Relativity is postulated, based on the replacement of the Newtonian gravitational constant by an formulation based explicitly on Mach's principle. We derive the resulting dynamical equations, and show that their solutions describe a linearly expanding universe. Potential problems with non-zero divergencies in the modified field equations are shown to be resolved by adopting a radically different definition of time, in which the unit of time is a function of the scale-factor. We observe that the effects of the modified field equations are broadly equivalent to a Newtonian gravitational constant that varies both in time and in space, and show that this is also equivalent to Varying Speed of Light (VSL) theories using a standard definition of time. Some of the implications of this observation are discussed in relation to Black Holes, Planck scale phenomena, and the ultimate fate of the Universe. 
  We consider the internal structure of the Skyrme black hole under a static and spherically symmetric ansatz. $@u8(Be concentrate on solutions with the node number one and with the "winding" number zero, where there exist two solutions for each horizon radius; one solution is stable and the other is unstable against linear perturbation. We find that a generic solution exhibits an oscillating behavior near the sigularity, as similar to a solution in the Einstein-Yang-Mills (EYM) system, independently to stability of the solution. Comparing it with that in the EYM system, this oscillation becomes mild because of the mass term of the Skyrme field. We also find Schwarzschild-like exceptional solutions where no oscillating behavior is seen. Contrary to the EYM system where there is one such solution branch if the node number is fixed, there are two branches corresponding to the stable and the unstable ones. 
  A combination of analytic and numerical methods has yielded a clear understanding of the approach to the singularity in spatially inhomogeneous cosmologies. Strong support is found for the longstanding claim by Belinskii, Khalatnikov, and Lifshitz that the collapse is dominated by local Kasner or Mixmaster behavior. The Method of Consistent Potentials is used to establish the consistency of asymptotic velocity term dominance (AVTD) (local Kasner behavior) in that no terms in Einstein's equations will grow exponentially when the VTD solution, obtained by neglecting all terms containing spatial derivatives, is substituted into the full equations. When the VTD solution is inconsistent, the exponential terms act dynamically as potentials either to drive the system into a consistent AVTD regime or to maintain an oscillatory approach to the singularity. 
  Anderson, et al., find the measured trajectories of Pioneer 10 and 11 spacecraft deviate from the trajectories computed from known forces acting on them. This unmodelled acceleration can be accounted for by non-isotropic radiation of spacecraft heat. This explanation was first proposed by Murphy, but Anderson, et al. felt it could not explain the observed effect. This paper includes new calculations on the expected magnitude of this effect, based on the relative emissivities of the different sides of the spacecraft, as estimated from the known spacecraft construction. The calculations indicate the proposed effect can account for most, if not all, of the unmodelled acceleration. 
  The derivation of the Demianski-Newman solution within the framework of the Ernst complex formalism is considered. We show that this solution naturally arises as a two-soliton specialization of the axisymmetric multi-soliton electrovacuum metric, and we work out the full set of the corresponding metrical fields and electromagnetic potentials. Some limits and physical properties of the DN space-time are briefly discussed. 
  The spectrum of the area operator for a Schwarzschild black hole in loop quantum gravity is fixed by the demand that the entropy of a black hole is maximum. This paper has been withdrawn by the author, due a crucial error in the derivation. 
  Everyday experience with centrifugal forces has always guided thinking on the close relationship between gravitational forces and accelerated systems of reference. Once spatial gravitational forces and accelerations are introduced into general relativity through a splitting of spacetime into space-plus-time associated with a family of test observers, one may further split the local rest space of those observers with respect to the direction of relative motion of a test particle world line in order to define longitudinal and transverse accelerations as well. The intrinsic covariant derivative (induced connection) along such a world line is the appropriate mathematical tool to analyze this problem, and by modifying this operator to correspond to the observer measurements, one understands more clearly the work of Abramowicz et al who define an ``optical centrifugal force'' in static axisymmetric spacetimes and attempt to generalize it and other inertial forces to arbitrary spacetimes. In a companion article the application of this framework to some familiar stationary axisymmetric spacetimes helps give a more intuitive picture of their rotational features including spin precession effects, and puts related work of de Felice and others on circular orbits in black hole spacetimes into a more general context. 
  The tools developed in a preceding article for interpreting spacetime geometry in terms of all possible space-plus-time splitting approaches are applied to circular orbits in some familiar stationary axisymmetric spacetimes. This helps give a more intuitive picture of their rotational features including spin precession effects, and puts related work of Abramowicz, de Felice, and others on circular orbits in black hole spacetimes into a more general context. 
  We present a new approach to the problem of binary black holes in the pre-coalescence stage, i.e. when the notion of orbit has still some meaning. Contrary to previous numerical treatments which are based on the initial value formulation of general relativity on a (3-dimensional) spacelike hypersurface, our approach deals with the full (4-dimensional) spacetime. This permits a rigorous definition of the orbital angular velocity. Neglecting the gravitational radiation reaction, we assume that the black holes move on closed circular orbits, which amounts to endowing the spacetime with a helical Killing vector. We discuss the choice of the spacetime manifold, the desired properties of the spacetime metric, as well as the choice of the rotation state for the black holes. As a simplifying assumption, the space 3-metric is approximated by a conformally flat one. The problem is then reduced to solving five of the ten Einstein equations, which are derived here, as well as the boundary conditions on the black hole surfaces and at spatial infinity. We exhibit the remaining five Einstein equations and propose to use them to evaluate the error induced by the conformal flatness approximation. The orbital angular velocity of the system is computed through a requirement which reduces to the classical virial theorem at the Newtonian limit. 
  We present the first results from a new method for computing spacetimes representing corotating binary black holes in circular orbits. The method is based on the assumption of exact equilibrium. It uses the standard 3+1 decomposition of Einstein equations and conformal flatness approximation for the 3-metric. Contrary to previous numerical approaches to this problem, we do not solve only the constraint equations but rather a set of five equations for the lapse function, the conformal factor and the shift vector. The orbital velocity is unambiguously determined by imposing that, at infinity, the metric behaves like the Schwarzschild one, a requirement which is equivalent to the virial theorem. The numerical scheme has been implemented using multi-domain spectral methods and passed numerous tests. A sequence of corotating black holes of equal mass is calculated. Defining the sequence by requiring that the ADM mass decrease is equal to the angular momentum decrease multiplied by the orbital angular velocity, it is found that the area of the apparent horizons is constant along the sequence. We also find a turning point in the ADM mass and angular momentum curves, which may be interpreted as an innermost stable circular orbit (ISCO). The values of the global quantities at the ISCO, especially the orbital velocity, are in much better agreement with those from third post-Newtonian calculations than with those resulting from previous numerical approaches. 
  We present an algorithm for solving the general relativistic initial value equations for a corotating polytropic star in quasicircular orbit with a nonspinning black hole. The algorithm is used to obtain initial data for cases where the black hole mass is 1, 3, and 10 times larger than the mass of the star. By analyzing sequences of constant baryon mass, constant black hole mass initial data sets and carefully monitoring the numerical error, we find innermost stable circular orbit (ISCO) configuration for these cases. While these quasiequilibrium, conformally flat sequences of initial data sets are not true solutions of the Einstein equations (each set, however, solves the full initial value problem), and thus, we do not expect the ISCO configurations found here to be completely consistent with the Einstein equations, they will be used as convenient starting points for future numerical evolutions of the full 3+1 Einstein equations. 
  We study the evaporation of black holes in space-times with extra dimensions of size L. We show that the luminosity is greatly damped when the horizon becomes smaller than L and black holes born with an initial size smaller than L are almost stable. This effect is due to the dependence of both the occupation number density of Hawking quanta and the grey-body factor of a black hole on the dimensionality of space. 
  We derive the light deflection caused by the screw dislocation in space-time. The derivation is based on the idea that space-time is a medium which can be deformed by gravity and that the deformation of space-time is equivalent to the existence of gravity. 
  We discuss the question of whether the existence of singularities is an intrinsic property of 4D spacetime. Our hypothesis is that singularities in 4D are induced by the separation of spacetime from the other dimensions. We examine this hypothesis in the context of the so-called canonical or warp metrics in 5D. These metrics are popular because they provide a clean separation between the extra dimension and spacetime. We show that the spacetime section, in these metrics, inevitably becomes singular for some finite (non-zero) value of the extra coordinate. This is true for all canonical metrics that are solutions of the field equations in space-time-matter theory. This is a geometrical singularity in 5D, but appears as a physical one in 4D. At this singular hypersurface, the determinant of the spacetime metric becomes zero and the curvature of the spacetime blows up to infinity. These results are consistent with our hypothesis. 
  Models with a dynamic cosmological term \Lambda (t) are becoming popular as they solve the cosmological constant problem in a natural way. Instead of considering any ad-hoc assumption for the variation of \Lambda, we consider a particular symmetry, the contracted Ricci-collineation along the fluid flow, in Einstein's theory. We show that apart from having interesting properties, this symmetry does require \Lambda to be a function of the scale factor of the Robertson-Walker metric. In order to test the consistency of the resulting model with observations, we study the magnitude-redshift relation for the type Ia supernovae data from Perlmutter et al. The data fit the model very well and require a positive non-zero \Lambda and a negative deceleration parameter. The best-fitting flat model is obtained as \Omega_0 \approx 0.5 with q_0 \approx -0.2. 
  Generalized slow roll conditions and parameters are obtained for a general form of scalar-tensor theory (with no external sources), having arbitrary functions describing a nonminimal gravitational coupling F(\phi), a Kahler-like kinetic function k(\phi), and a scalar potential V(\phi). These results are then used to analyze a simple toy model example of chaotic inflation with a single scalar field \phi and a standard Higgs potential and a simple gravitational coupling function. In this type of model inflation can occur with inflaton field values at an intermediate scale of roughly 10^{11} GeV when the particle physics symmetry breaking scale is approximately 1 TeV, provided that the theory is realized within the Jordan frame. If the theory is realized in the Einstein frame, however, the intermediate scale inflation does not occur. 
  This is just to point out that the Nariai metric is the first example of the singularity free expanding perfect fluid cosmological model satisfying the weak energy condition, $\rho>0, \rho+p=0$. It is a conformally non-flat Einstein space. 
  The Causal Set hypothesis asserts that spacetime, ultimately, is discrete and its underlying structure is that of a locally finite partial ordered set, and macroscopic causality reflects a deeper notion of order in terms of which all the geometrical structure of spacetime must find their ultimate expression. After reviewing the main aspects of Causal Sets Kinematics, and the recently developed Stochastic Dynamics. We concentrate on possible implications in the fields of cosmology and black holes. In the context of black hole, we propose a possible interpretation of the entropy as the number of links crossing the horizon. 
  The Bach equation and the equation of geometrodynamics are based on two quite different physical motivations, but in both approaches, the conformal properties of gravitation plays the key role. In this paper we present an analysis of the relation between these two equations and show that the solutions of the equation of geometrodynamics are of a more general nature. We show the following non-trivial result: there exists a conformally invariant Lagrangian, whose field equation generalizes the Bach equation and has as solutions those Ricci tensors which are solutions to the equation of geometrodynamics. 
  We investigate the initial-boundary value problem for linearized gravitational theory in harmonic coordinates. Rigorous techniques for hyperbolic systems are applied to establish well-posedness for various reductions of the system into a set of six wave equations. The results are used to formulate computational algorithms for Cauchy evolution in a 3-dimensional bounded domain. Numerical codes based upon these algorithms are shown to satisfy tests of robust stability for random constraint violating initial data and random boundary data; and shown to give excellent performance for the evolution of typical physical data. The results are obtained for plane boundaries as well as piecewise cubic spherical boundaries cut out of a Cartesian grid. 
  The closed de Sitter universe is used to present a way to deal with the deparametrization and quantization of cosmological models with extrinsic time. 
  We provide a Hamiltonian analysis of the Mixmaster Universe dynamics showing the covariant nature of its chaotic behavior with respect to any choice of time variable. We construct the appropriate invariant measure for the system (which relies on the existence of an ``energy-like'' constant of motion) without fixing the time gauge, i.e. the corresponding lapse function. The key point in our analysis consists of introducing generic Misner-Chitr\'e-like variables containing an arbitrary function, whose specification allows one to set up the same dynamical scheme in any time gauge. 
  A basic problem that confronts the standard cosmological models is the problem of initial singularity characterised by infinite material density, infinite temperature and infinite spacetime curvature. The inevitable existence of such a phase of the universe may be considered to be one of the major drawbacks of Einstein's field equations. To some extent inflation models ameliorate this. In the present work we postulate that whenever matter (radiation) arises in flat spacetime, it introduces curvature and causes a repulsive interaction to develop. A modified energy momentum tensor is introduced towards this end, which invokes the temperature, entropy and a cosmic scalar field. This redefinition of the energy momentum distribution when applied to the early universe dynamics has the effect of producing a non-singular initial behaviour. the repulsive interaction introduces some features of an accelarating universe. In this model all cosmological parameters are mathematically well behaved. There is no flatness problem. Temperature at the beginning of this universe is 10^32 kelvin. 
  I review some recent results on black holes with hair. I focus on the magnetically charged black holes in spontaneously broken Yang-Mills-Higgs theories and on the related self-gravitating magnetic monopoles. Some implications for black hole thermodynamics are discussed. 
  We present an analysis of a n-dimensional vacuum Einstein field equations in which 4-dimensional space-time which is described by a Friedmann Robertson-Walker (FRW) metric and that of the extra dimensions by a Kasner type Euclidean metric. The field equations are interpreted as four dimensional Einstein equations with effective matter properties. We obtain solutions for the cases when the effective matter is described by imperfect fluid. We consider the theories of imperfect fluid given by Eckart, truncated Israel-Stewart and full Israel-Stewart theories and obtain cosmological solutions for a flat model of the universe. 
  We study in detail the modes of a classical scalar field on a Kerr-Newman-anti-de Sitter (KN-AdS) black hole. We construct sets of basis modes appropriate to the two possible boundary conditions (``reflective'' and ``transparent'') at time-like infinity, and consider whether super-radiance is possible. If we employ ``reflective'' boundary conditions, all modes are non-super-radiant. On the other hand, for ``transparent'' boundary conditions, the presence of super-radiance depends on our definition of positive frequency. For those KN-AdS black holes having a globally time-like Killing vector, the natural choice of positive frequency leads to no super-radiance. For other KN-AdS black holes, there is a choice of positive frequency which gives no super-radiance, but for other choices there will, in general, be super-radiance. 
  This article gives the construction and complete classification of all three-dimensional spherical manifolds, and orders them by decreasing volume, in the context of multiconnected universe models with positive spatial curvature. It discusses which spherical topologies are likely to be detectable by crystallographic methods using three-dimensional catalogs of cosmic objects. The expected form of the pair separation histogram is predicted (including the location and height of the spikes) and is compared to computer simulations, showing that this method is stable with respect to observational uncertainties and is well suited for detecting spherical topologies. 
  We discuss the consequences of the incorrectness [see the Erratum in Phys. Rev D 49, 1145 (1994)] of that paper and add two related remarks. The scope of this comment is to encourage further research on: `Which of the conformally equivalent metrics is the physical one?' 
  We generalize the well-known analogies between m^2 \phi^2 and R + R^2 theories to include the selfinteraction \lambda \phi^4-term for the scalar field. It turns out to be the R + R^3 Lagrangian which gives an appropriate model for it. Considering a spatially flat Friedman cosmological model, common and different properties of these models are discussed, e.g., by linearizing around a ground state the masses of the resp. spin 0-parts coincide. Finally, we prove a general conformal equivalence theorem between a Lagrangian L = L(R), L'L'' \ne 0, and a minimally coupled scalar field in a general potential. 
  For the non-tachyonic curvature squared action we show that the expanding Bianchi-type I models tend to the dust-filled Einstein-de Sitter model for t tending to infinity if the metric is averaged over the typical oscillation period. Applying a conformal equivalence between curvature squared action and a minimally coupled scalar field (which holds for all dimensions > 2) the problem is solved by discussing a massive scalar field in an anisotropic cosmological model. 
  The weak-field slow-motion limit of fourth-order gravity will be discussed. 
  This article investigates asymmetric brane-world scenarios in the limit when the bulk gravity is negligible. We show that, even when true self gravity is negligible, local mass concentrations will be subject to a mutual attraction force which simulates the effect of Newtonian gravity in the non-relativistic limit. Cosmological and also post-Newtonian constraints are examined. 
  The differential cross section for scattering of a Dirac particle in a black hole background is found. The result is the gravitational analog of the Mott formula for scattering in a Coulomb background. The equivalence principle is neatly embodied in the cross section, which depends only on the incident velocity, and not the particle mass. The low angle limit agrees with classical calculations based on the geodesic equation. The calculation employs a well-defined iterative scheme which can be extended to higher orders. Repeating the calculation in different gauges shows that our result for the cross section is gauge-invariant and highlights the issues involved in setting up a sensible iterative scheme. 
  Numerical arguments are presented for the existence of spherically symmetric regular and black hole solutions of the EYMH equations with a negative cosmological constant. These solutions approach asymptotically the anti-de Sitter spacetime. The main properties of the solutions and the differences with respect to the asymptotically flat case are discussed. The instability of the gravitating sphaleron solutions is also proven. 
  We investigate higher dimensional cosmological models in the semiclassical approximation with Hartle-Hawking Boundary conditions, assuming a gravitational action which is described by the scalar curvature with a cosmological constant. In the framework the probability for quantum creation of an inflationary universe with a pair of black holes in a multidimensional universe is evaluated. The probability for creation of a universe with a spatial section with $S^{1}XS^{D -2}$ topology is then compared with that of a higher dimensional de Sitter universe with $S^{D -1}$ spatial topology. It is found that a higher dimensional universe with a product space with primordial black holes pair is less probable to nucleate when the extra dimensions scale factors do not vary in an inflating universe. 
  We present a stationary axially symmetric two parameter vacuum solution which could be considered as ``dual'' to the Kerr solution. It is obtained by removing the mass parameter from the function of the radial coordinate and introducing a dimensionless parameter in the function of the angle coordinate in the metric functions. It turns out that it is in fact the massless limit of the Kerr - NUT solution. 
  The numerous ways of introducing spatial gravitational forces are fit together in a single framework enabling their interrelationships to be clarified. This framework is then used to treat the ``acceleration equals force" equation and gyroscope precession, both of which are then discussed in the post-Newtonian approximation, followed by a brief examination of the Einstein equations themselves in that approximation. 
  Using recent observational constraints on cosmological density parameters, together with recent mathematical results concerning small volume hyperbolic manifolds, we argue that, by employing pattern repetitions, the topology of nearly flat small hyperbolic universes can be observationally undetectable. This is important in view of the facts that quantum cosmology may favour hyperbolic universes with small volumes, and from the expectation coming from inflationary scenarios, that $\Omega_0$ is likely to be very close to one. 
  Light experiences a non-uniformly moving medium as an effective gravitational field, endowed with an effective metric tensor $\tilde{g}^{\mu \nu}=\eta^{\mu \nu}+(n^2-1)u^\mu u^\nu$, $n$ being the refractive index and $u^\mu$ the four-velocity of the medium. Leonhardt and Piwnicki [Phys. Rev. A {\bf 60}, 4301 (1999)] argued that a flowing dielectric fluid of this kind can be used to generate an 'optical black hole'. In the Leonhardt-Piwnicki model, only a vortex flow was considered. It was later pointed out by Visser [Phys. Rev. Lett. {\bf 85}, 5252 (2000)] that in order to form a proper optical black hole containing an event horizon, it becomes necessary to add an inward radial velocity component to the vortex flow. In the present paper we undertake this task: we consider a full spiral flow, consisting of a vortex component plus a radially infalling component. Light propagates in such a dielectric medium in a way similar to that occurring around a rotating black hole. We calculate, and show graphically, the effective potential versus the radial distance from the vortex singularity, and show that the spiral flow can always capture light in both a positive, and a negative, inverse impact parameter interval. The existence of a genuine event horizon is found to depend on the strength of the radial flow, relative to the strength of the azimuthal flow. A limitation of our fluid model is that it is nondispersive. 
  The dominant contributions from a discrete gravitational interaction produce the standard potential as an effective continuous field. The sub-dominant contributions are, in a first approximation, linear on n, the accumulated number of (discrete) interaction events along the test-body trajectory. For a nearly radial trajectory n is proportional to the traversed distance and its effects may have been observed as the Pioneer anomalous constant radial acceleration, which cannot be observed on the nearly circular planetary orbits. 
  The dominant contributions from a discrete gravitational interaction produce the standard potential as an effective continuous field. The sub-dominant contributions are, in a first approximation, linear on $n$, the accumulated number of (discrete) interaction events along the test-body trajectory. For a nearly radial trajectory $n$ is proportional to the transversed distance and its effects may have been observed as the Pioneer anomalous constant radial acceleration, which cannot be observed on the nearly circular planetary orbits. Here we give calculation details of the alternative II, discussed in gr-qc/0106046. 
  Closed time-like curves naturally appear in a variety of chronology-violating space-times. In these space-times, the Principle of Self-Consistency demands an harmony between local and global affairs that excludes grandfather-like paradoxes. However, self-existing objects trapped in CTCs are not seemingly avoided by the standard interpretation of this principle, usually constrained to a dynamical framework. In this paper we discuss whether we are committed to accept an ontology with self-existing objects if CTCs actually occur in the universe. In addition, the epistemological status of the Principle of Self-Consistency is analyzed and a discussion on the information flux through CTCs is presented. 
  Within the framework of the Projective Unified Field Theory the distribution of a dark matter gas around a central body is calculated. As a result the well-known formulas of the Newtonian gravitational interaction are altered. This dark matter effect leads to an additional radial force (towards the center) in the equation of motion of a test body, being used for the explanation of the so-called ``Pioneer effect'', measured in the solar system, but without a convincing theoretical basis up to now. Further the relationship of the occurring new force to the so-called ``fifth force'' is discussed. 
  The dark matter accretion theory (around a central body) of the author on the basis of his 5-dimensional Projective Unified Field Theory (PUFT) is applied to the orbital motion of stars around the center of the Galaxy. The departure of the motion from Newtonian mechanics leads to approximately flat rotation curves being in rough accordance with the empirical facts. The spirality of the motion is investigated. 
  Perfect fluid Friedmann-Robertson-Walker quantum cosmological models for an arbitrary barotropic equation of state $p = \alpha\rho$ are constructed using Schutz's variational formalism. In this approach the notion of time can be recovered. By superposition of stationary states, finite-norm wave-packet solutions to the Wheeler-DeWitt equation are found. The behaviour of the scale factor is studied by applying the many-worlds and the ontological interpretations of quantum mechanics. Singularity-free models are obtained for $\alpha < 1$. Accelerated expansion at present requires $- 1/3 > \alpha > - 1$. 
  Two-forms in Minkowski space-time may be considered as generators of Lorentz transformations. Here, the covariant and general expression for the composition law (Baker-Campbell-Hausdorff formula) of two Lorentz transformations in terms of their generators is obtained. Every subalgebra of the Lorentz algebra of such generators, up to one, may be generated by a sole pair of generators. When the subalgebra is known, the above BCH formula for the two two-forms simplifies. Its simplified expressions for all such subalgebras are also given. 
  The transformation properties of the gravitational energy-momentum in the teleparallel gravity are analyzed. It is proved that the gravitational energy-momentum in the teleparallel gravity can be expressed in terms of the Lorentz gauge potential, and therefore is not covariant under local Lorentz transformations. On the other hand, it can also be expressed in terms of the translation gauge field strength, and therefore is covariant under general coordinate transformations. A simplified Hamiltonian formulation of the teleparallel gravity is given. Its constraint algebra has the same structure as that of general relativity, which indicates the equivalence between the teleparallel gravity and general relativity in the Hamiltonian formulation. 
  In this note, we consider the response of a uniformly accelerated monopole detector that is coupled non-linearly to the nth power of a quantum scalar field in (D+1)-dimensional flat spacetime. We show that, when (D+1) is even, the response of the detector in the Minkowski vacuum is characterized by a Bose-Einstein factor for all n. Whereas, when (D+1) is odd, we find that a Fermi-Dirac factor appears in the detector response when n is odd, but a Bose-Einstein factor arises when n is even. 
  The causality principle for the Relativistic Theory of Gravitation (RTG) is presented. It is a straightforward consequence of the RTG basic postulates. The necessary conditions for physical solutions of the gravitational field equations to be fulfilled are given. 
  The r-mode frequencies of the Tolman VII solution for the slowly rotating non-barotropic approximation within the low frequency regime are estimated. The relativistic correction to Newtonian r-mode calculations is shown as function of the tenuity $\frac{R}{M}$ and is shown to be significant only for very compact neutron stars. 
  The generalization to N colors of a recently proposed non unitary two color model for the gravitational interaction in non relativistic quantum mechanics is considered. The N->Infinity limit is proven to be equivalent to the Schroedinger-Newton model, which, though sharing localization properties with the N=2 model, cannot produce decoherence. 
  The random superposition of many weak sources will produce a stochastic background of gravitational waves that may dominate the response of the LISA (Laser Interferometer Space Antenna) gravitational wave observatory. Unless something can be done to distinguish between a stochastic background and detector noise, the two will combine to form an effective noise floor for the detector. Two methods have been proposed to solve this problem. The first is to cross-correlate the output of two independent interferometers. The second is an ingenious scheme for monitoring the instrument noise by operating LISA as a Sagnac interferometer. Here we derive the optimal orbital alignment for cross-correlating a pair of LISA detectors, and provide the first analytic derivation of the Sagnac sensitivity curve. 
  Let us abandon for a moment the strict epistemological standpoint of quantum field theory, that eventually comes to declare nonsensical any question about the photon posed outside the quantum theoretical framework. We can then avail of the works by Whittaker et al. and by Synge about the particle and the wave model of the photon in the vacuum of general relativity. We can also rely on important results found by Gordon and by Pham Mau Quan: thanks to Gordon's discovery of an effective metric these authors have been able to reduce to the vacuum case several problems of the electromagnetic theory of dielectrics.     The joint use of these old findings allows one to conclude that a quantum theoretical photon in an isotropic dielectric has a classical simile only if the dielectric is also homogeneous. 
  We investigate the occurrence of naked singularities in the spherically symmetric, plane symmetric and cylindrically symmetric collapse of charged null fluid in an anti-de Sitter background. The naked singularities are found to be strong in Tipler's sense and thus violate the cosmic censorship conjecture, but not hoop conjecture. 
  A large class of vacuum space-times is constructed in dimension 4+1 from hyperboloidal initial data sets which are not small perturbations of empty space data. These space-times are future geodesically complete, smooth up to their future null infinity, and extend as vacuum space-times through their Cauchy horizon. Dimensional reduction gives non-vacuum space-times with the same properties in 3+1 dimensions. 
  The gravitational waveforms of a chaotic system will exhibit sensitive dependence on initial conditions. The waveforms of nearby orbits decohere on a timescale fixed by the largest Lyapunov exponent of the orbit. The loss of coherence has important observational consequences for systems where the Lyapunov timescale is short compared to the chirp timescale. Detectors that rely on matched filtering techniques will be unable to detect gravitational waves from these systems. 
  A massless scalar field without self interaction and string coupled to gravity is quantized in the framework of quantum cosmology using the Bohm-de Broglie interpretation. Gaussian superpositions of the quantum solutions of the corresponding Wheeler-DeWitt equation in minisuperspace are constructed. The bohmian trajectories obtained exhibit a graceful exit from the inflationary Pre-Big Bang epoch to the decelerated expansion phase. 
  A new gauge-invariant approach for describing cosmological perturbations is developed. It is based on a physically motivated splitting of the stress-energy tensor of the perturbation into two parts - the bare perturbation and the complementary perturbation associated with stresses in the background gravitational field induced by the introduction of the bare perturbation. The complementary perturbation of the stress-energy tensor is explicitly singled out and taken to the left side of the perturbed Einstein equations so that the bare stress-energy tensor is the sole source for the perturbation of the metric tensor and both sides of these equations are gauge invariant with respect to infinitesimal coordinate transformations. For simplicity we analyze the perturbations of the spatially-flat Friedmann-Lemaitre-Robertson-Walker dust model. A cosmological gauge can be chosen such that the equations for the perturbations of the metric tensor are completely decoupled for the h_{00}, h_{0i}, and h_{ij} metric components and explicitly solvable in terms of retarded integrals. 
  In this work, the quantization of the most general Bianchi Type I geometry, with and without a cosmological constant, is considered. In the spirit of identifying and subsequently removing as many gauge degrees of freedom as possible, a reduction of the initial 6--dimensional configuration space is presented. This reduction is achieved by imposing as additional conditions on the wave function, the quantum version of the --linear in momenta-- classical integrals of motion (conditional symmetries). The vector fields inferred from these integrals induce, through their integral curves, motions in the configuration space which can be identified to the action of the automorphism group of Type I, i.e. $GL(3,\Re)$. Thus, a wave function depending on one degree of freedom, namely the determinant of the scale factor matrix, is found. A measure for constructing the Hilbert space is proposed. This measure respects the above mentioned symmetries, and is also invariant under the classical property of covariance under arbitrary scalings of the Hamiltonian (quadratic constraint). 
  Spin-rotation coupling, or Mashhoon effect, is a phenomenon associated with rotating observers. We show that the effect exists and plays a fundamental role in the determination of the anomalous magnetic moment of the muon. 
  We have shown the classical correspondence of Unruh effect in the classical relativistic electron theory in our previous work (gr-qc/0105051). Here we demonstrate the analogy between the classical relativistic electron theory and the classical Unruh-DeWitt type monopole detector theory. The field configuration generated by a uniformly accelerated detector is worked out. The classical correspondence of Unruh effect for scalar fields is shown by calculating the modified energy density for the scalar field around the detector. We conclude that a classical monopole detector cannot find any evidence about its acceleration unless it has a finite size. 
  We apply the method of moving anholonomic frames in order to construct new classes of solutions of the Einstein equations on (2+1)-dimensional pseudo-Riemannian spaces. There are investigated black holes with deformed horizons and renormalized locally anisotropic constants. We speculate on properties of such anisotropic black holes with characteristics defined by anholonomic frames and anisotropic interactions of matter and gravity. The thermodynamics of locally anisotropic black holes is discussed in connection with a possible statistical mechanics background based on locally anisotropic variants of Chern-Simons theories. 
  We study new classes three dimensional black hole solutions of Einstein equations written in two holonomic and one anholonomic variables with respect to anholonomic frames Thermodynamic properties of such (2+1)-black holes with generic local anisotropy (having elliptitic horizons) are studied by applying geometric methods. The corresponding thermodynamic systems are three dimensional with entropy S being a hypersurface function on mass M, anisotropy angle $\theta$ and eccentricity of elliptic deformations $\epsilon >.$ Two-dimensional curved thermodynamic geometries for locally anistropic deformed black holes are constructed after integration on anisotropic parameter $\theta$. Two approaches, the first one based on two-dimensional hypersurface parametric geometry and the second one developed in a Ruppeiner-Mrugala-Janyszek fashion, are analyzed. The thermodynamic curvatures are computed and the critical points of curvature vanishing are defined. 
  By a suitable transformation, we derive the rotating Goedel universe from a static one and we show, how rotation may be implemented geometrically. The rotation law turns out to be a differential one. By increasing distance from the rotation axis the velocity of a rotating point will exceed the velocity of light and the cosmos has a cut-off radius. Thus, closed time-like curves do not occur in the Goedel universe. 
  We investigate both the ``physical process'' version of the first law and the second law of black hole thermodynamics for charged and rotating black holes. We begin by deriving general formulas for the first order variation in ADM mass and angular momentum for linear perturbations off a stationary, electrovac background in terms of the perturbed non-electromagnetic stress-energy, $\delta T_{ab}$, and the perturbed charge current density, $\delta j^a$. Using these formulas, we prove the "physical process version" of the first law for charged, stationary black holes. We then investigate the generalized second law of thermodynamics (GSL) for charged, stationary black holes for processes in which a box containing charged matter is lowered toward the black hole and then released (at which point the box and its contents fall into the black hole and/or thermalize with the ``thermal atmosphere'' surrounding the black hole). Assuming that the thermal atmosphere admits a local, thermodynamic description with respect to observers following orbits of the horizon Killing field, and assuming that the combined black hole/thermal atmosphere system is in a state of maximum entropy at fixed mass, angular momentum, and charge, we show that the total generalized entropy cannot decrease during the lowering process or in the ``release process''. Consequently, the GSL always holds in such processes. No entropy bounds on matter are assumed to hold in any of our arguments. 
  Computer simulations are enabling researchers to investigate systems which are extremely difficult to handle analytically. In the particular case of General Relativity, numerical models have proved extremely valuable for investigations of strong field scenarios and been crucial to reveal unexpected phenomena. Considerable efforts are being spent to simulate astrophysically relevant simulations, understand different aspects of the theory and even provide insights in the search for a quantum theory of gravity. In the present article I review the present status of the field of Numerical Relativity, describe the techniques most commonly used and discuss open problems and (some) future prospects. 
  We consider both mode calculations and time evolutions of axial r-modes for relativistic uniformly rotating non-barotropic neutron stars, using the slow-rotation formalism, in which rotational corrections are considered up to linear order in the angular velocity \Omega. We study various stellar models, such as uniform density models, polytropic models with different polytropic indices n, and some models based on realistic equations of state. For weakly relativistic uniform density models, and polytropes with small values of n, we can recover the growth times predicted from Newtonian theory when standard multipole formulae for the gravitational radiation are used. However, for more compact models, we find that relativistic linear perturbation theory predicts a weakening of the instability compared to the Newtonian results. When turning to polytropic equations of state, we find that for certain ranges of the polytropic index n, the r-mode disappears, and instead of a growth, the time evolutions show a rapid decay of the amplitude. This is clearly at variance with the Newtonian predictions. It is, however, fully consistent with our previous results obtained in the low-frequency approximation. 
  In this paper we investigate the general features of "Oscillatory Inflation". In adiabatic approximation, we derive a general formula for the number of e-foldings $\tilde{N}$ which reduces to the standard expression in case of the slow role approximation and leads to the Damour-Mukhanov type expression for the slowly varying adiabatic index. We apply our result to the logarithmic potential and arrive at a simple and elegant formula for the number of e-foldings. We evolve the field equations numerically and observe a remarkable agreement with the analytical result. 
  Why does {\bf F} equal m{\bf a} in Newton's equation of motion? How does a gravitational field produce a force? Why are inertial mass and gravitational mass the same? It appears that all three of these seemingly axiomatic foundational questions have an answer involving an identical physical process: interaction between the electromagnetic quantum vacuum and the fundamental charged particles (quarks and electrons) constituting matter. All three of these effects and equalities can be traced back to the appearance of a specific asymmetry in the otherwise uniform and isotropic electromagnetic quantum vacuum. This asymmetry gives rise to a non-zero Poynting vector from the perspective of an accelerating object. We call the resulting energy-momentum flux the {\it Rindler flux}. The key insight is that the asymmetry in an accelerating reference frame in flat spacetime is identical to that in a stationary reference frame (one that is not falling) in curved spacetime. Therefore the same Rindler flux that creates inertial reaction forces also creates weight. All of this is consistent with the conceptualizaton and formalism of general relativity. What this view adds to physics is insight into a specific physical process creating identical inertial and gravitational forces from which springs the weak principle of equivalence. What this view hints at in terms of advanced propulsion technology is the possibility that by locally modifying either the electromagnetic quantum vacuum and/or its interaction with matter, inertial and gravitational forces could be modified. 
  We investigate properties of $r$-mode instability in slowly rotating relativistic polytropes. Inside the star slow rotation and low frequency formalism that was mainly developed by Kojima is employed to study axial oscillations restored by Coriolis force. At the stellar surface, in order to take account of gravitational radiation reaction effect, we use a near-zone boundary condition instead of the usually imposed boundary condition for asymptotically flat spacetime. Due to the boundary condition, complex frequencies whose imaginary part represents secular instability are obtained for discrete $r$-mode oscillations in some polytropic models. It is found that such discrete $r$-mode solutions can be obtained only for some restricted polytropic models. Basic properties of the solutions are similar to those obtained by imposing the boundary condition for asymptotically flat spacetime. Our results suggest that existence of a continuous part of spectrum cannot be avoided even when its frequency becomes complex due to the emission of gravitational radiation. 
  An evolution scheme is developed, based on Sorkin algorithm, to evolve the most complex regular tridimensional polytope, the 600-cell. The solution of 600-cell, already studied before, is generalized by allowing a larger number of free variables. The singularities of Robertson-Walker (RW) metric are studied and a reason is given why the evolution of the 600-cell stops when its volume is still far from zero. A fit of 600-cell's evolution with a continuos metric is studied by writing a metric generalizing Friedmann's and including the 600-cell evolution too. The result is that the 600-cell meets a causality-breaking point of space-time. We also shortly discuss the way matter is introduced in Regge calculus. 
  The present contribution deals with thermodynamic aspects of the model of Expansive Nondecelerative Universe. In this model, in the matter era a dependence T^CBR ~ E^CBR ~ a(exp -3/4) holds for the energy of cosmic background radiation, E^CBR and its temperature T^CBR, while the proportionality of the energy density epsilon^CBR to the gauge factor a can be expressed as epsilon^CBR ~ a(exp -3). The given relationships comply with experimental observations of the cosmic background radiation as well as with a surprising finding that the Universe expansion is not decelerated by gravitational forces. It is rationalized that the specific entropy is proportional to a(exp -1/4), i.e. it is gradually decreasing in time. 
  We consider 5D spaces which admit the most symmetric 3D subspaces. 5D vacuum Einstein equations are constructed and 5D analog of the mass function is found. The corresponding conservation law leads to 5D analog of Birkhoff's theorem. Hence the cylinder condition is dynamically implemented for the considered spaces. For some obtained metrics a period of space with respect to the fifth coordinate was found. The problem of the dynamical degrees of freedom of the fields system obtained in the process of dimensional reduction is discussed, and the problem of their interpretation is considered. One can think that the parametrization of the scalar field and 4D metric leading to the conformally invariant 4D theory for interacting gravitational and scalar fields is most natural and adequate. 
  In the current standard viewpoint small black holes are believed to emit radiation as black bodies at the Hawking temperature, at least until they reach Planck size, after which their fate is open to conjecture. A cogent argument against the existence of remnants is that, since no evident quantum number prevents it, black holes should radiate completely away to photons and other ordinary stable particles and vacuum, like any unstable quantum system. Here we argue the contrary, that the generalized uncertainty principle may prevent their total evaporation in exactly the same way that the uncertainty principle prevents the hydrogen atom from total collapse: the collapse is prevented, not by symmetry, but by dynamics, as a minimum size and mass are approached. 
  It is well known that massive black holes may form through the gravitational collapse of a massive astrophysical body. Less known is the fact that a black hole can be produced by the quantum process of pair creation in external fields. These black holes may have a mass much lower than their astrophysical counterparts. This mass can be of the order of Planck mass so that quantum effects may be important. This pair creation process can be investigated semiclassically using non-perturbative instanton methods, thus it may be used as a theoretical laboratory to obtain clues for a quantum gravity theory. In this work, we review briefly the history of pair creation of particles and black holes in external fields. In order to present some features of the euclidean instanton method which is used to calculate pair creation rates, we study a simple model of a scalar field and propose an effective one-loop action for a two-dimensional soliton pair creation problem. This action is built from the soliton field itself and the soliton charge is no longer treated as a topological charge but as a Noether charge. The results are also valid straightforwardly to the problem of pair creation rate of domain walls in dimensions greater than 2. 
  We construct the Hadamard Green's function by using the eigenfunction, which are obtained by solving the wave equation for the massless conformal scalar field on the S^n-1 of a n-dimensional closed, static universe. We also consider the half space case with both the Dirichlet and the Neumann boundary conditions. Solving of eigenfunction and eigenvalues of the corresponding field equation is interesting since the Casimir energy could be calculated analytically by various methods. 
  Gravitational collapse of radiation shells in a non self-similar higher dimensional spherically symmetric spacetime is studied. Strong curvature naked singularities form for a highly inhomogeneous collapse, violating the cosmic censorship conjecture. As a special case, self similar models can be constructed. 
  The Isolated Horizon formalism, together with a simple phenomenological model for colored black holes was recently used to predict a formula for the ADM mass of the solitons of the EYM system in terms of horizon properties of black holes {\it for all} values of the horizon area. In this note, this formula is tested numerically --up to a large value of the area-- for spherically symmetric solutions and shown to yield the known masses of the solitons 
  There is a tendency to write the equations of general relativity as a first order symmetric system of time dependent partial differential equations. However, for numerical reasons, it might be advantageous to use a second order formulation like one obtained from the ADM equations. Unfortunately, the type of the ADM equations is not well understood and therefore we shall discuss, in the next section, the concept of wellposedness. We have to distinguish between weakly and strongly hyperbolic systems. Strongly hyperbolic systems are well behaved even if we add lower order terms. In contrast; for every weakly hyperbolic system we can find lower order terms which make the problem totally illposed. Thus, for weakly hyperbolic systems, there is only a restricted class of lower order perturbations which do not destroy the wellposedness. To identify that class can be very difficult, especially for nonlinear perturbations. In Section 3 we will show that the ADM equations, linearized around flat with constant lapse function and shift vector, are only weakly hyperbolic. However, we can use the trace of the metric as a lapse function to make the equations into a strongly second order hyperbolic system. Using simple models we shall, in section 4, demonstrate that approximations of second order equations have better accuracy properties than the corresponding approximations of first order equations. Also, we avoid spurious waves which travel against the characteristic direction. In the last section we discuss some difficulties connected with the preservation of constraints. 
  I study a stochastic approach to the recently introduced fresh inflation model for super Hubble scales. I find that the state loses its coherence at the end of the fresh inflationary period as a consequence of the damping of the interference function in the reduced density matrix. This fact should be a consequence of a) the relative evolutions of both the scale factor and the horizon and b) the additional thermal and dissipative effects. This implies a relevant difference with respect to supercooled inflationary scenarios which require a very rapid expansion of the scale factor to give the decoherence of super Hubble fluctuations. 
  The asymptotic method of post-Newtonian (PN) expansion for weak gravitational fields, recently developed, is compared with the standard method of PN expansion, in the particular case of a massive test particle moving along a geodesic line of a weak Schwarzschild field. First, the expression of the active mass in Schwarzschild's solution is given for a barotropic perfect fluid, both for general relativity (GR) and for an alternative, scalar theory. The principle of the asymptotic method is then recalled and the PN expansion of the active mass is derived. The PN correction to the active mass is made of the Newtonian elastic energy, augmented, for the scalar theory, by a term due to the self-reinforcement of the gravitational field. Third, two equations, both correct to first order, are derived for the geodesic motion of a mass particle: a "standard" one and an "asymptotic" one. Finally, the difference between the solutions of these two equations is numerically investigated in the case of Mercury. The asymptotic solution deviates from the standard one like the square of the time elapsed since the initial time. This is due to a practical shortcoming of the asymptotic method, which is shown to disappear if one reinitializes the asymptotic problem often enough. Thus, both methods are equivalent in the case investigated. In a general case, the asymptotic method seems more natural. 
  We present results for r-modes of relativistic nonbarotropic stars. We show that the main differential equation, which is formally singular at lowest order in the slow-rotation expansion, can be regularized if one considers the initial value problem rather than the normal mode problem. However, a more physically motivated way to regularize the problem is to include higher order terms. This allows us to develop a practical approach for solving the problem and we provide results that support earlier conclusions obtained for uniform density stars. In particular, we show that there will exist a single r-mode for each permissible combination of $l$ and $m$. We discuss these results and provide some caveats regarding their usefulness for estimates of gravitational-radiation reaction timescales. The close connection between the seemingly singular relativistic r-mode problem and issues arising because of the presence of corotation points in differentially rotating stars is also clarified. 
  We consider strings with the Nambu action as extremal surfaces in a given space-time, thus, we ignore their back reaction. Especially, we look for strings sharing one symmetry with the underlying space-time. If this is a non-null symmetry, the problem of determining the motion of the string can be dimensionally reduced. We get exact solutions for the following cases: straight and circle-like strings in a Friedmann background, straight strings in an anisotropic Kasner background, different types of strings in the metric of a gravitational wave. The solutions will be discussed. 
  A scheme is discussed for embedding n-dimensional, Riemannian manifolds in an (n+1)-dimensional Einstein space. Criteria for embedding a given manifold in a spacetime that represents a solution to Einstein's equations sourced by a massless scalar field are also discussed. The embedding procedures are illustrated with a number of examples. 
  This is an introduction to spin foam models for non-perturbative quantum gravity, an approach that lies at the point of convergence of many different research areas, including loop quantum gravity, topological quantum field theories, path integral quantum gravity, lattice field theory, matrix models, category theory, statistical mechanics. We describe the general formalism and ideas of spin foam models, the picture of quantum geometry emerging from them, and give a review of the results obtained so far, in both the Euclidean and Lorentzian case. We focus in particular on the Barrett-Crane model for 4-dimensional quantum gravity. 
  We study the localization of fermions on a brane embedded in a space-time with $AdS_n \times M^k$ geometry. Quantum numbers of localized fermions are associated with their rotation momenta around the brane. Fermions with different quantum numbers have different higher-dimensional profiles. Fermion masses and mixings, which are proportional to the overlap of higher-dimensional profiles of the fermions, depend on the fermion quantum numbers. 
  In present work the generalization of Einstein's special theory of relativity on 5-dimentional space is considered, in which as fifth coordinates we consider the interval s of a particle. 5-dimentional vectors in this space are isotropic both for mass, and for massless of particles.   In extended space, offered by the authors, there is a possibility in addition with usual Lorentz transformations, to enter two new transformations, therefore massive and massless of a particle can reversible converted to each other. 
  In this paper we model the gravitational wave emission of a freely precessing neutron star. The aim is to estimate likely source strengths, as a guide for gravitational wave astronomers searching for such signals. We model the star as a partly elastic, partly fluid body with quadrupolar deformations of its moment of inertia tensor. The angular amplitude of the free precession is limited by the finite breaking strain of the star's crust. The effect of internal dissipation on the star is important, with the precession angle being rapidly damped in the case of a star with an oblate deformation. We then go on to study detailed scenarios where free precession is created and/or maintained by some astrophysical mechanism. We consider the effects of accretion torques, electromagnetic torques, glitches and stellar encounters. We find that the mechanisms considered are either too weak to lead to a signal detectable by an Advanced LIGO interferometer, or occur too infrequently to give a reasonable event rate. We therefore conclude that, using our stellar model at least, free precession is not a good candidate for detection by the forthcoming laser interferometers. 
  The Klein-Gordon equation is a useful test arena for quantum cosmological models described by the Wheeler-DeWitt equation. We use the decoherent histories approach to quantum theory to obtain the probability that a free relativistic particle crosses a section of spacelike surface. The decoherence functional is constructed using path integral methods with initial states attached using the (positive definite) ``induced'' inner product between solutions to the constraint equation. The notion of crossing a spacelike surface requires some attention, given that the paths in the path integral may cross such a surface many times, but we show that first and last crossings are in essence the only useful possibilities. Different possible results for the probabilities are obtained, depending on how the relativistic particle is quantized (using the Klein-Gordon equation, or its square root, with the associated Newton-Wigner states). In the Klein-Gordon quantization, the decoherence is only approximate, due to the fact that the paths in the path integral may go backwards and forwards in time. We compare with the results obtained using operators which commute with the constraint (the ``evolving constants'' method). 
  Structure formation within the Lemaitre-Tolman model is investigated in a general manner. We seek models such that the initial density perturbation within a homogeneous background has a smaller mass than the structure into which it will develop, and the perturbation then accretes more mass during evolution. This is a generalisation of the approach taken by Bonnor in 1956. It is proved that any two spherically symmetric density profiles specified on any two constant time slices can be joined by a Lemaitre-Tolman evolution, and exact implicit formulae for the arbitrary functions that determine the resulting L-T model are obtained. Examples of the process are investigated numerically. 
  The correct quantum description for a curvature squared term in the action can be obtained by casting the action in the canonical form with the introduction of a variable which is the negative of the first derivative of the field variable appearing in the action, only after removing the total derivative terms from the action. We present the Wheeler-DeWitt equation and obtain the expression for the probability density and current density from the equation of continuity. Furthermore, in the weak energy limit we obtain the classical Einstein equation. Finally we present a solution of the wave equation. 
  The possibility of detecting the gravitomagnetic clock effect using artificial Earth satellites provides the incentive to develop a more intuitive approach to its derivation. We first consider two test electric charges moving on the same circular orbit but in opposite directions in orthogonal electric and magnetic fields and show that the particles take different times in describing a full orbit. The expression for the time difference is completely analogous to that of the general relativistic gravitomagnetic clock effect in the weak-field and slow-motion approximation. The latter is obtained by considering the gravitomagnetic force as a small classical non-central perturbation of the main central Newtonian monopole force. A general expression for the clock effect is given for a spherical orbit with an arbitrary inclination angle. This formula differs from the result of the general relativistic calculations by terms of order c^{-4}. 
  It is shown that unlike Einstein's gravity quadratic gravity produces dispersive photon propagation. The energy-dependent contribution to the deflection of photons passing by the Sun is computed and subsequently the angle at which the visible spectrum would be spread over is plotted as a function of the $R_{\mu\nu}^2-$sector mass. 
  The general solution of the gravitational field equations in the flat Friedmann-Robertson-Walker geometry is obtained in the framework of the full Israel-Stewart-Hiscock theory for a bulk viscous stiff cosmological fluid, with bulk viscosity coefficient proportional to the energy density. 
  A new cosmological solution of the gravitational field equations in the generalized Randall-Sundrum model for an anisotropic brane with Bianchi I geometry and with perfect fluid as matter sources is presented. The matter is described by a scalar field. The solution admits inflationary era and at a later epoch the anisotropy of the universe washes out. We obtain two classes of cosmological scenario, in the first case universe evolves from singularity and in the second case universe expands without singularity. 
  We study the collapse of a self-gravitating thick shell of bosons coupled to a scalar radiation field. Due to the non-adiabaticity of the collapse, the shell (quantum) internal degrees of freedom absorb energy from the (classical) gravitational field and are excited. The excitation energy is then emitted in the form of bursts of (thermal) radiation and the corresponding backreaction on the trajectory is estimated. 
  The so called gamma metric corresponds to a two-parameter family of axially symmetric, static solutions of Einstein's equations found by Bach. It contains the Schwarzschild solution for a particular value of one of the parameters, that rules a deviation from spherical symmetry.    It is shown that there is invariantly definable singular behaviour beyond the one displayed by the Kretschmann scalar when a unique, hypersurface orthogonal, timelike Killing vector exists. In this case, a particle can be defined to be at rest when its world-line is a corresponding Killing orbit. The norm of the acceleration on such an orbit proves to be singular not only for metrics that deviate from Schwarzschild's metric, but also on approaching the horizon of Schwarzschild metric itself, in contrast to the discontinuous behaviour of the curvature scalar. 
  Working with electrodynamics in the geometrical optics approximation we derive the expression representing an effectively curved geometry which guides the propagation of electromagnetic waves in material media whose physical properties depend on an external electric field. The issue of birefringence is addressed, and the trajectory of the extraordinary ray is explicitly worked out. Quite general curves are obtained for the path of the light ray by suitably setting an electric field. 
  Exact models for Bianchi VI0 spacetimes with multiple scalar fields with exponential potentials have been derived and analysed. It has been shown that these solutions, when they exist, attract neighbouring solutions in the two cases corresponding to interacting and non-interacting fields. Unlike the results obtained in a previous work dealing with the late-time inflationary behaviour of Bianchi VI0 cosmologies, the knowledge of exact solutions has made possible to study in detail the occurrence of inflation before the asymptotic regime. As happened in preceding works, here as well inflation is more likely to happen with a higher number of non-interacting fields or a lower number of interacting scalar fields. 
  I provide a prescription to define space, at a given moment, for an arbitrary observer in an arbitrary (sufficiently regular) curved space-time. This prescription, based on synchronicity (simultaneity) arguments, defines a foliation of space-time, which corresponds to a family of canonically associated observers. It provides also a natural global reference frame (with space and time coordinates) for the observer, in space-time (or rather in the part of it which is causally connected to him), which remains Minkowskian along his world-line. This definition intends to provide a basis for the problem of quantization in curved space-time, and/or for non inertial observers.   Application to Mikowski space-time illustrates clearly the fact that different observers see different spaces. It allows, for instance, to define space everywhere without ambiguity, for the Langevin observer (involved in the Langevin pseudoparadox of twins). Applied to the Rindler observer (with uniform acceleration) it leads to the Rindler coordinates, whose choice is so justified with a physical basis. This leads to an interpretation of the Unruh effect, as due to the observer's dependence of the definition of space (and time). This prescription is also applied in cosmology, for inertial observers in the Friedmann - Lemaitre models: space for the observer appears to differ from the hypersurfaces of homogeneity, which do not obey the simultaneity requirement. I work out two examples: the Einstein - de Sitter model, in which space, for an inertial observer, is not flat nor homogeneous, and the de Sitter case. 
  Upper limits for the mass-radius ratio and total charge are derived for stable charged general relativistic matter distributions. For charged compact objects the mass-radius ratio exceeds the value 4/9 corresponding to neutral stars. General restrictions for the redshift and total energy (including the gravitational contribution) are also obtained. 
  One obtains a Maxwell-like structure of gravitation by applying the weak-field approximation to the well accepted theory of general relativity or by extending Newton's laws to time-dependent systems. This splits gravity in two parts, namely a gravitoelectric and gravitomagnetic (or cogravitational) one. Due to the obtained similar structure between gravitation and electromagnetism, one can express one field by the other one using a coupling constant depending on the mass to charge ratio of the field source. Calculations of induced gravitational fields using state-of-the-art fusion plasmas reach only accelerator threshold values for laboratory testing. Possible amplification mechanisms are mentioned in the literature and need to be explored. The possibility of using the principle of equivalence in the weak field approximation to induce non-Newtonian gravitational fields and the influence of electric charge on the free fall of bodies are also investigated, leading to some additional experimental recommendations. 
  Self-consistent solutions to the nonlinear spinor field equations in General Relativity has been studied for the case of Bianchi type-I (B-I) space-time. It has been shown that, for some special type of nonliearity the model provides regular solution, but this singularity-free solutions are attained at the cost of broken dominant energy condition in Hawking-Penrose theorem. It has also been shown that the introduction of $\Lambda$-term in the Lagrangian generates oscillations of the B-I model, which is not the case in absence of $\Lambda$ term. Moreover, for the linear spinor field, the $\Lambda$ term provides oscillatory solutions, those are regular everywhere, without violating dominant energy condition.   Key words: Nonlinear spinor field (NLSF), Bianch type -I model (B-I), $\Lambda$ term   PACS 98.80.C Cosmology 
  The gravitational Poynting vector provides a mechanism for the transfer of gravitational energy to a system of falling objects. In the following we will show that the gravitational poynting vector together with the gravitational Larmor theorem also provides a mechanism to explain how massive bodies acquire rotational kinetic energy when external mechanical forces are applied on them. 
  The gravitational Poynting vector provides a mechanism for the transfer of gravitational energy to a system of falling objects. In the following we will show that the gravitational poynting vector together with the electromagnetic Poynting vector provides a mechanism to explain how massive electrically charged bodies acquire kinetic energy during a free fall. We will demontrate that falling electrically charged masses violate the Galilean law of universal free fall. An experiment is suggested to confirm or not the predicted phenomena. 
  Conformal mappings of surfaces of constant mean curvature onto compact bounded background spaces are constructed for Minkowski space-time and for Schwarzschild black hole spacetimes. In the black hole example, it is found that initial data on these CMC surfaces can be regular on the compact background space only when a certain condition is satisfied. That condition implies that the shift vector points inward from all parts of the boundary of the compact background. It also implies that the second fundamental form of these surfaces can never be isotropic when black holes are present. 
  We argue that the geodesic hypothesis based on auto-parallels of the Levi-Civita connection may need refinement in theories of gravity with additional scalar fields. This argument is illustrated with a re-formulation of the Brans-Dicke theory in terms of a spacetime connection with torsion determined dynamically in terms of the gradient of the Brans-Dicke scalar field. The perihelion shift in the orbit of Mercury is calculated on the alternative hypothesis that its world-line is an auto-parallel of such a connection. If scalar fields couple significantly to matter and spinless test particles move on such world-lines, current time keeping methods based on the conventional geodesic hypothesis may need refinement. 
  We study Cauchy initial data for asymptotically flat, stationary vacuum space-times near space-like infinity. The fall-off behavior of the intrinsic metric and the extrinsic curvature is characterized. We prove that they have an analytic expansion in powers of a radial coordinate. The coefficients of the expansion are analytic functions of the angles. This result allow us to fill a gap in the proof found in the literature of the statement that all asymptotically flat, vacuum stationary space-times admit an analytic compactification at null infinity. Stationary initial data are physical important and highly non-trivial examples of a large class of data with similar regularity properties at space-like infinity, namely, initial data for which the metric and the extrinsic curvature have asymptotic expansion in terms of powers of a radial coordinate. We isolate the property of the stationary data which is responsible for this kind of expansion. 
  Gravitational wave emission is considered to be the driving force for the evolution of short-period cataclysmic binary stars, making them a potential test for the validity of General Relativity. In spite of continuous refinements of the physical description, a 10% mismatch exists between the theoretical minimum period ($P_{\rm turn} \simeq 70$ min) and the short-period cut-off ($P_{\rm min} \simeq 80$ min) observed in the period distribution for cataclysmic variable binaries. A possible explanation for this mismatch was associated with the use of the Roche model. We here present a systematic comparison between self-consistent, numerically constructed sequences of hydrostatic models of binary stars and Roche models of semi-detached binaries. On the basis of our approach, we also derive a value for the minimum period of cataclysmic variable binaries. The results obtained through the comparison indicate that the Roche model is indeed very good, with deviations from the numerical solution which are of a few percent at most. Our results therefore suggest that additional sources of angular momentum loss or alternative explanations need to be considered in order to justify the mismatch. 
  A special arrangement of spinning strings with dislocations similar to a von K\'arm\'an vortex street is studied. We numerically solve the geodesic equations for the special case of a test particle moving along twoinfinite rows of pure dislocations and also discuss the case of pure spinning defects. 
  Using the force-susceptibility formalism of linear quantum measurements, we study the dynamics of signal recycled interferometers, such as LIGO-II. We show that, although the antisymmetric mode of motion of the four arm-cavity mirrors is originally described by a free mass, when the signal-recycling mirror is added to the interferometer, the radiation-pressure force not only disturbs the motion of that ``free mass'' randomly due to quantum fluctuations, but also and more fundamentally, makes it respond to forces as though it were connected to a spring with a specific optical-mechanical rigidity. This oscillatory response gives rise to a much richer dynamics than previously known for SR interferometers, which enhances the possibilities for reshaping the noise curves and, if thermal noise can be pushed low enough, enables the standard quantum limit to be beaten. We also show the possibility of using servo systems to suppress the instability associated with the optical-mechanical interaction without compromising the sensitivity of the interferometer. 
  The data from Pioneer 10 and 11 shows an anomalous, constant, Doppler frequency drift that can be interpreted as an acceleration directed towards the Sun of a_P = (8.74 \pm 1.33) x 10^{-8} cm/s^2. Although one can consider a new physical origin for the anomaly, one first must investigate the contributions of the prime candidates, which are systematics generated on board. Here we expand upon previous analyses of thermal systematics. We demonstrate that thermal models put forth so far are not supported by the analyzed data. Possible ways to further investigate the nature of the anomaly are proposed. 
  Space-time--time couples Kaluza's five-dimensional geometry with Weyl's conformal space-time geometry to produce an extension that goes beyond what either of those theories can achieve by itself. Kaluza's ``cylinder condition'' is replaced by an ``exponential expansion constraint'' that causes translations along the secondary time dimension to induce both the electromagnetic gauge transformations found in the Kaluza and the Weyl theories and the metrical gauge transformations unique to the Weyl theory, related as Weyl had postulated. A space-time--time geodesic describes a test particle whose rest mass, space-time momentum, and electric charge q, all defined kinematically, evolve in accord with definite dynamical laws. Its motion is governed by four apparent forces: the Einstein gravitational force, the Lorentz electromagnetic force, a force proportional to the electromagnetic potential, and a force proportional to a scalar field's gradient d(ln phi). The test particles exhibit quantum behavior: (1) they appear and disappear in full-blown motion at definite events; (2) all that share an event E of appearance or disappearance do so with the same charge magnitude |q| = phi(E); (3) conservation of space-time--time momentum at such an event entails conservation of electric charge in addition to conservation of space-time momentum, among the participating particles; (4) at such events the d(ln phi) force infinitely dominates the other three --- this strongly biases the appearance and disappearance events to be concentrated deep in the discretely spaced potential wells of ln phi, and sparse elsewhere. 
  A basic extension of the exterior part of the extreme Reissner-Nordstroem solution in terms of a continuous metric and gauge potential is constructed. This extension is not smooth at the null hypersurface given by the Cauchy-Killing horizon which separates isometric copies of the exterior metric. The Maxwell-Einstein system of equations is satisfied only in a weak sense. The manifold is topologically incomplete and the spherical symmetry is globally broken down to an axial symmetry. This behaviour can be attributed to the effect of a 'topological string', in the sense of a infinitesimally thin closed stringlike object 'sitting on the rim' of the black hole and holding it open by means of an accompanying impulsive gravitational wave. The resulting differentiable manifold and the corresponding horizons are not anymore simply connected, being 'pierced' by the strings. 
  We show that it is possible to obtain credible static anisotropic spherically symmetric matter configurations starting from known density profiles and satisfying a nonlocal equation of state. These particular types of equation of state describe, at a given point, the components of the corresponding energy-momentum tensor not only as a function at that point, but as a functional throughout the enclosed configuration. To establish the physical plausibility of the proposed family of solutions satisfying nonlocal equation of state, we study the constraints imposed by the junction and energy conditions on these bounded matter distributions.   We also show that it is possible to obtain physically plausible static anisotropic spherically symmetric matter configurations, having nonlocal equations of state\textit{,}concerning the particular cases where the radial pressure vanishes and, other where the tangential pressures vanishes. The later very particular type of relativistic sphere with vanishing tangential stresses is inspired by some of the models proposed to describe extremely magnetized neutron stars (magnetars) during the transverse quantum collapse. 
  We discuss the problem of applicability of Coordinate Systems (or Frames) that determine (t,x,y,z) values - the initial notions for most physical theories. Equipment that measure these values - Clocks and Meters - are based at Reference System and are the primary measuring units. We discus when and why physical phenomena might prevent to provide measurements of (t,x,y,z). We show that Temperature may be the factor that can significantly influence on the measurements of (t,x,y,z) by Reference System and that action may violate the usage of Coordinate System. We discuss possible origin of such unmovable Temperature and assume that it should be Gravity. 
  Recent studies have raised doubts about the occurrence of r modes in Newtonian stars with a large degree of differential rotation. To assess the validity of this conjecture we have solved the eigenvalue problem for Rossby-Haurwitz waves (the analogues of r waves on a thin-shell) in the presence of differential rotation. The results obtained indicate that the eigenvalue problem is never singular and that, at least for the case of a thin-shell, the analogues of r modes can be found for arbitrarily large degrees of differential rotation. This work clarifies the puzzling results obtained in calculations of differentially rotating axi-symmetric Newtonian stars. 
  In this essay it will be shown that Decoherence Model and Einstein Equivalence Principle are conceptually incompatible. In other words, assuming only the validity of the Weak Equivalence Principle the present work concludes that we face two possibilities: (i) if Decoherence Model provides a correct description of nature at quantum level, then there are systems which violate Local Position Invariance, or, (ii) if all the postulates behind Einstein Equivalence Principle are valid, even on quantum realm, then Decoherence Model breaks down in curved spacetimes. Finally, the present results are confronted against Schiff's conjecture. 
  In rotating viscous fluid stars, tidal torque leads to an exchange of spin and orbital angular momentum. The horizon of a black hole has an effective viscosity that is large compared to that of stellar fluids, and an effective tidal torque may lead to important effects in the strong field interaction at the endpoint of the inspiral of two rapidly rotating holes. In the most interesting case both holes are maximally rotating and all angular momenta (orbital and spins) are aligned. We point out here that in such a case (i) the transfer of angular momentum may have an important effect in modifying the gravitational wave ``chirp'' at the endpoint of inspiral. (ii) The tidal transfer of spin energy to orbital energy may increase the amount of energy being radiated. (iii) Tidal transfer in such systems may provide a mechanism for shedding excess angular momentum. We argue that numerical relativity, the only tool for determining the importance of tidal torque, should be more specifically focused on binary configurations with aligned, large, angular momenta. 
  We will present results of a numerical integration of a maximally sliced Schwarzschild black hole using a smooth lattice method. The results show no signs of any instability forming during the evolutions to t=1000m. The principle features of our method are i) the use of a lattice to record the geometry, ii) the use of local Riemann normal coordinates to apply the 1+1 ADM equations to the lattice and iii) the use of the Bianchi identities to assist in the computation of the curvatures. No other special techniques are used. The evolution is unconstrained and the ADM equations are used in their standard form. 
  In the continuum the Bianchi identity implies a relationship between different components of the curvature tensor, thus ensuring the internal consistency of the gravitational field equations. In this paper an exact form for the Bianchi identity in Regge's discrete formulation of gravity is derived, by considering appropriate products of rotation matrices constructed around null-homotopic paths. It implies an algebraic relationship between deficit angles belonging to neighboring hinges. As in the continuum, the derived identity is valid for arbitrarily curved manifolds, without a restriction to the weak field, small curvature limit, but is in general not linear in the curvatures. 
  It is shown that almost all known solutions of the kind mentioned in the title are easily derived in a unified manner when a simple ansatz is imposed on the metric. The Whittaker solution is an exception, replaced by a new solution with the same equation of state. 
  Gravitational waves are generated during first-order phase transitions, either by turbolence or by bubble collisions. If the transition takes place at temperatures of the order of the electroweak scale, the frequency of these gravitational waves is today just within the band of the planned space interferometer LISA. We present a detailed analysis of the production of gravitational waves during an electroweak phase transition in different supersymmetric models where, contrary to the case of the Standard Model, the transition can be first order. We find that the stochastic background of gravitational waves generated by bubble nucleation can reach a maximum value h0^2 Omega_{gw} of order 10^{-10} - 10^{-11}, which is within the reach of the planned sensitivity of LISA, while turbolence can even produce signals at the level h0^2 Omega_{gw} \sim 10^{-9}. These values of h0^2 Omega_{gw} are obtained in the regions of the parameter space which can account for the generation of the baryon asymmetry at the electroweak scale. 
  This is a short summary of my lectures given at the Fourth Mexican School on Gravitation and Mathematical Physics. These lectures gave a brief introduction to black holes in string theory, in which I primarily focussed on describing some of the recent calculations of black hole entropy using the statistical mechanics of D-brane states. The following overview will also provide the interested students with an introduction to the relevant literature. 
  A generalized covariant method of analysis applicable to frames for which time is not orthogonal to space, such as spacetime around a star possessing angular momentum or on a rotating disk, is presented. Important aspects of such an analysis are shown to include i) use of the physically relevant contravariant or covariant component form for a given vector/tensor, ii) conversion of physical (measured) components to generalized coordinate components prior to tensor analysis, iii) use of generalized covariant constitutive equations during tensor analysis, and iv) conversion of coordinate components back to physical components after tensor analysis. The method is then applied to electrodynamics in a rotating frame, and shown to predict the results of the Wilson/Wilson and Roentgen/Eichenwald experiments. 
  We consider the covariant graviton propagator in de Sitter spacetime in a gauge with two parameters, alpha and beta, in the Euclidean approach. We give an explicit form of the propagator with a particular choice of beta but with arbitrary value of alpha. We confirm that two-point functions of local gauge-invariant quantities do not increase as the separation of the two points becomes large. 
  The separated radial part of a massive complex scalar wave equation in the Kerr-Sen geometry is shown to satisfy the generalized spheroidal wave equation which is, in fact, a confluent Heun equation up to a multiplier. The Hawking evaporation of scalar particles in the Kerr-Sen black hole background is investigated by the Damour-Ruffini-Sannan's method. It is shown that quantum thermal effect of the Kerr-Sen black hole has the same characteras that of the Kerr-Newman black hole. 
  This paper considers some physically interesting cosmological dynamical systems in the FRW-scalarfield category which are examined for integrability according to the criterion of Painlev\'e. In the literature these systems have been examined from the point of view of dynamical systems and the results from the two disparate methods of analysis are compared. This allows some more general comments to be made on the use of the Painlev\'e method in covariant systems. 
  We consider space-times which in addition to admitting an isolated horizon also admit Killing horizons with or without an event horizon. We show that an isolated horizon is a Killing horizon provided either (1) it admits a stationary neighbourhood or (2) it admits a neighbourhood with two independent, commuting Killing vectors. A Killing horizon is always an isolated horizon. For the case when an event horizon is definable, all conceivable relative locations of isolated horizon and event horizons are possible. Corresponding conditions are given. 
  Using gyroscopes we generalize results, obtained for the gravitomagnetic clock effect in the particular case when the exterior spacetime is produced by a rotating dust cylinder, to the case when the vacuum spacetime is described by the general cylindrically symmetric Lewis spacetime. Results are contrasted with those obtained for the Kerr spacetime. 
  In this paper we present a new approach for studying the dynamics of spatially inhomogeneous cosmological models with one spatial degree of freedom. By introducing suitable scale-invariant dependent variables we write the evolution equations of the Einstein field equations as a system of autonomous partial differential equations in first-order symmetric hyperbolic format, whose explicit form depends on the choice of gauge. As a first application, we show that the asymptotic behaviour near the cosmological initial singularity can be given a simple geometrical description in terms of the local past attractor on the boundary of the scale-invariant dynamical state space. The analysis suggests the name ``asymptotic silence'' to describe the evolution of the gravitational field near the cosmological initial singularity. 
  Quantum gravitational back-reaction offers a simultaneous explanation for why the cosmological constant is so small and a natural model of inflation in which scalars play no role. In this talk I review previous work and present a simple model of the mechanism in which the induced stress tensor behaves like negative vacuum energy with a density proportional to $-\Lambda/{8\pi G} \cdot (G \Lambda)^2 \cdot H t$. The model also highlights the essential role of causality in back-reaction. 
  To bridge the gap between background independent, non-perturbative quantum gravity and low energy physics described by perturbative field theory in Minkowski space-time, Minkowskian Fock states are located, analyzed and used in the background independent framework. This approach to the analysis of semi-classical issues is motivated by recent results of Varadarajan. As in that work, we use the simpler U(1) example to illustrate our constructions but, in contrast to that work, formulate the theory in such a way that it can be extended to full general relativity. 
  A detailed study of the Counter-Rotating Model (CRM) for generic finite static axially symmetric thin disks with nonzero radial pressure is presented. We find a general constraint over the counter-rotating tangential velocities needed to cast the surface energy-momentum tensor of the disk as the superposition of two counter-rotating perfect fluids. We also found expressions for the energy density and pressure of the counter-rotating fluids. Then we shown that, in general, there is not possible to take the two counter-rotating fluids as circulating along geodesics neither take the two counter-rotating tangential velocities as equal and opposite. An specific example is studied where we obtain some CRM with well defined counter-rotating tangential velocities and stable against radial perturbations. The CRM obtained are in agree with the strong energy condition, but there are regions of the disks with negative energy density, in violation of the weak energy condition. 
  The collapse of spherical neutron stars is studied in General Relativity. The initial state is a stable neutron star to which an inward radial kinetic energy has been added through some velocity profile. For two different equations of state and two different shapes of velocity profiles, it is found that neutron stars can collapse to black holes for high enough inward velocities, provided that their masses are higher than some minimal value, depending on the equation of state. For a polytropic equation of state of the form $p=K\rho^\gamma $, with $\gamma = 2$ it is found to be $1.16 (\frac{K}{0.1})^{0.5} \msol$, whereas for a more realistic one, it reads $0.36 \msol $. In some cases of collapse forming a black hole, part of the matter composing the initial neutron star can be ejected through a shock, leaving only a fraction of the initial mass to form a black hole. Therefore, black holes of very small masses can be formed and, in particular, the mass scaling relation, as a function of initial velocity, takes the form discovered by Choptuik for critical collapses. 
  We compute the energy spectra of the gravitational signals emitted when a pointlike mass moves on a closed orbit around a non rotating neutron star, inducing a perturbation of its gravitational field and its internal structure. The Einstein equations and the hydrodynamical equations are perturbed and numerically integrated in the frequency domain. The results are compared with the energy spectra computed by the quadrupole formalism which assumes that both masses are pointlike, and accounts only for the radiation emitted because the orbital motion produces a time dependent quadrupole moment. The results of our perturbative approach show that, in general, the quadrupole formalism overestimates the amount of emitted radiation, especially when the two masses are close. However, if the pointlike mass is allowed to move on an orbit so tight that the keplerian orbital frequency resonates with the frequency of the fundamental quasi-normal mode of the star (2w_K=w_f), this mode can be excited and the emitted radiation can be considerably larger than that computed by the quadrupole approach. 
  Primary features of a new cosmological model, which is based on conjectures about an existence of the graviton background and superstrong gravitational quantum interaction, are considered. An expansion of the universe is impossible in such the model because of deceleration of massive objects by the graviton background, which is similar to the one for the NASA deep space probes Pioneer 10, 11. Redshifts of remote objects are caused in the model by interaction of photons with the graviton background, and the Hubble constant depends on an intensity of interaction and an equivalent temperature of the graviton background. Virtual massive gravitons would be dark matter particles. They transfer energy, lost by luminous matter radiation, which in a final stage may be collected with black holes and other massive objects. 
  Torsion detection from totally skew symmetric torsion waves in the context of teleparallel gravity is discussed. A gedanken experiment to detect Cartan's contortion based on a circle of particles not necessarily spinning is proposed. It is shown that by making use of previous value of contortion at the surface of the Earth computed by Nitsch of $10^{-24} s^{-1}$ a relative displacement of $10^{-21}$ is obtained which is of the order of the gravitational wave of $10^{-3}Hz$. Since LISA has been designed to work in the mHz regime this GW detector could be used for an indirect detection of torsion in $T_{4}$. 
  First, the relation between black holes and limitations on information of other systems is developed. After reviewing the relation of entropy to information, we derive the entropy bound, review its applications to cosmology and its extensions to higher dimensions, and discuss why black holes behave as 1-D objects when emitting entropy. We also discuss fundamental limitations on the information of pulses in curved space, and on the rate of disposal of information into a black hole. We then move on to a discussion of quantum black holes motivated by the adiabatic invariance of horizon area of classical holes. We develop an algebraic formalism based on symmetry which gives information on the area (or mass) spectrum of quantum black holes, and on the degeneracy of the levels. This last turns out to be consistent with the horizon area-black hole entropy proportionality while leaving room for corrections. 
  An invariant description of Bianchi Homogeneous (B.H.) 3-spaces is presented, by considering the action of the Automorphism Group on the configuration space of the real, symmetric, positive definite, $3\times 3$ matrices. Thus, the gauge degrees of freedom are removed and the remaining (gauge invariant) degrees, are the --up to 3-- curvature invariants. An apparent discrepancy between this Kinematics and the Quantum Hamiltonian Dynamics of the lower Class A Bianchi Types, occurs due to the existence of the Outer Automorphism Subgroup. This discrepancy is satisfactorily removed by exploiting the quantum version of some classical integrals of motion (conditional symmetries) which are recognized as corresponding to the Outer Automorphisms. 
  The formulation of the initial value problem for the Einstein equations is at the heart of obtaining interesting new solutions using numerical relativity and still very much under theoretical and applied scrutiny. We develop a specialised background geometry approach, for systems where there is non-trivial a priori knowledge about the spacetime under study. The background three-geometry and associated connection are used to express the ADM evolution equations in terms of physical non-linear deviations from that background. Expressing the equations in first order form leads naturally to a system closely linked to the Einstein-Christoffel system, introduced by Anderson and York, and sharing its hyperbolicity properties. We illustrate the drastic alteration of the source structure of the equations, and discuss why this is likely to be numerically advantageous. 
  The form of the coupling of the scalar field with gravity and the potential have been found by applying Noether theorem to two dimensional minisuperspaces in induced gravity model. It has been observed that though the forms thus obtained are consistent with all the equations $\pounds_{X}L=0$, yet they do not satisfy the field equations for $k=\pm 1$, in Robertson-Walker model. It has been pointed out that this is not due to the degeneracy of the Lagrangian, since this problem does not appear in $k=0$ case.It has also been shown that though Noether theorem fails to extract any symmetry from the Lagrangian of such model for $k=\pm 1$, symmetry exists, which can easily be found by studying the continuity equation. 
  The forms of coupling of the scalar field with gravity, appearing in the induced theory of gravity, and the potential are found in the Kantowski-Sachs model under the assumption that the Lagrangian admits Noether symmetry. The form thus obtained makes the Lagrangian degenerate. The constrained dynamics thus evolved due to such degeneracy has been analysed and a solution has also been presented which is inflationary in behaviour. It has further been shown that there exists other technique to explore the dynamical symmetries of the Lagrangian and that is simply by inspecting the field equations. Through this method, Noether along with some other dynamical symmetries are found, which do not make the Lagrangian degenerate. 
  We analyze spherical dust collapse with non-vanishing radial pressure, $\Pi$, and vanishing tangential stresses. Considering a barotropic equation of state, $\Pi=\gamma\rho$, we obtain an analytical solution in closed form---which is exact for $\gamma=-1,0$, and approximate otherwise---near the center of symmetry (where the curvature singularity forms). We study the formation, visibility, and curvature strength of singularities in the resulting spacetime. We find that visible, Tipler strong singularities can develop from generic initial data. Radial pressure alters the spectrum of possible endstates for collapse, increasing the parameter space region that contains no visible singularities, but cannot by itself prevent the formation of visible singularities for sufficiently low values of the energy density. Known results from pressureless dust are recovered in the $\gamma=0$ limit. 
  We investigate vacuum solutions of Einstein's equation for a universe with an S^1 topology of time. Such a universe behaves like a time-machine and has geodesics which coincide with closed time-like curves (CTCs). A system evolving along a CTC experiences the Loschmidt velocity reversion and undergoes a recurrence commensurate with the universal period. We indicate why this universe is free of the causality paradoxes, usually associated with CTCs. 
  We explore how the gravitational self force (or ``radiation reaction'' force), acting on a pointlike test particle in curved spacetime, is modified in a gauge transformation. We derive the general transformation law, describing the change in the self force in terms of the infinitesimal displacement vector associated with the gauge transformation. Based on this transformation law, we extend the regularization prescription by Mino et al. and Quinn and Wald (originally formulated within the harmonic gauge) to an arbitrary gauge. Then we extend the method of mode-sum regularization (which provides a practical means for calculating the regularized self force and was recently applied to the harmonic-gauge gravitational self force) to an arbitrary gauge. We find that the regularization parameters involved in this method are gauge-independent. We also explore the gauge transformation of the self force from the harmonic gauge to the Regge-Wheeler gauge and to the radiation gauge, focusing attention on the regularity of these gauge transformations. We conclude that the transformation of the self force to the Regge-Wheeler gauge in Schwarzschild spacetime is regular for radial orbits and irregular otherwise, whereas the transformation to the radiation gauge is irregular for all orbits. 
  In this work the possibility of detecting a non--Newtonian contribution to the gravitational potential by means of its effects upon the first and second--order coherence properties of light is analyzed. It will be proved that, in principle, the effects of a fifth force upon the correlation functions of electromagnetic radiation could be used to detect the existence of new forces. Some constraints upon the experimental parameters will also be deduced. 
  We consider the classical theory of a gravitational field with spin 2 and non-vanishing (Pauli-Fierz) mass in flat spacetime, coupled to electromagnetism and point particles. We establish the law of light propagation and calculate the amount of deflection in the background of a spherically symmetric gravitational field. As the mass tends to zero, the deflection is shown to converge to 3/4 of the value predicted by the massless theory (linearized General Relativity), even though the spherically symmetric solution of the gravitational field equations has no regular limit. This confirms an old argument of van Dam and Veltman on a purely classical level, but also shows its subtle nature. 
  It is shown that the RTG predicts an opportunity of the intensive production of gravitons at the early stage of evolution of the homogeneous isotropic Universe. A hypothesis is suggested that the produced gas of gravitons could be just the ``dark matter'' which presently manifests itself as a ``missing mass'' in our Universe. 
  It is shown that different approaches towards the solution of the Einstein equations for a static spherically symmetric perfect fluid with a gamma-law equation of state lead to an Abel differential equation of the second kind. Its only integrable cases at present are flat spacetime, de Sitter solution and its Buchdahl transform, Einstein static universe and the Klein-Tolman solution. 
  The instability of r-mode oscillations in rapidly rotating neutron stars has attracted attention as a potential mechanism for producing high frequency, almost periodic gravitational waves. The analyses carried so far have shown the existence of these modes and have considered damping by shear and bulk viscosity. However, the magnetohydrodynamic coupling of the modes with a stellar magnetic field and its role in the damping of the instability has not been fully investigated yet. Following our introductory paper (Rezzolla, Lamb and Shapiro 2000), we here discuss in more detail the existence of secular higher-order kinematical effects which will produce toroidal fluid drifts. We also define the sets of equations that account for the time evolution of the magnetic fields produced by these secular velocity fields and show that the magnetic fields produced can reach equipartition in less than a year. The full numerical calculations as well as the evaluation of the impact of strong magnetic fields on the onset and evolution of the r-mode instability will be presented in a companion paper. 
  The evolution of the r-mode instability is likely to be accompanied by secular kinematic effects which will produce differential rotation with large scale drifts of fluid elements, mostly in the azimuthal direction. As first discussed by Rezzolla, Lamb and Shapiro 2000, the interaction of these secular velocity fields with a pre-existing neutron star magnetic field could result in the generation of intense and large scale toroidal fields. Following their derivation in the companion paper, we here discuss the numerical solution of the evolution equations for the magnetic field. The values of the magnetic fields obtained in this way are used to estimate the conditions under which the r-mode instability might be prevented or suppressed. We also assess the impact of the generation of large magnetic fields on the gravitational wave detectability of r-mode unstable neutron stars. Our results indicate that the signal to noise ratio in the detection of gravitational waves from the r-mode instability might be considerably decreased if the latter develops in neutron stars with initial magnetic fields larger than 10^10 G. 
  We consider the propagation of massive-particle de Broglie waves in a static, isotropic metric in general relativity. We demonstrate the existence of an index of refraction that governs the waves and that has all the properties of a classical index of refraction. We confirm our interpretation with a WKB solution of the general-relativistic Klein-Gordon equation. Finally, we make some observations on the significance of the optical action. 
  We present an expression for the Weyl-Weyl two-point function in de Sitter spacetime, based on a recently calculated covariant graviton two-point function with one gauge parameter. We find that the Weyl-Weyl two-point function falls off with distance like r^{-4}, where r is spacelike coordinate separation between the two points. 
  Black hole solutions with nonspherical event horizon topology are shown to exist in an Einstein-Yang-Mills theory with negative cosmological constant. The main characteristics of the solutions are presented and differences with respect to the spherically symmetric case are studied. The stability of these configurations is also addressed. 
  Quantum gravitational fluctuations of the space-time background, described by virtual D branes, may induce neutrino oscillations if a tiny violation of the Lorentz invariance (or a violation of the equivalence principle) is imposed. In this framework, the oscillation length of massless neutrinos turns out to be proportional to M/E^2, where E is the neutrino energy and M is the mass scale characterizing the topological fluctuations in the vacuum. Such a functional dependence on the energy is the same obtained in the framework of loop quantum gravity. 
  We consider the extension of classical history theory to the massive vector field and electromagnetism. It is argued that the action of the two Poincare groups introduced by Savvidou suggests that the history fields should have five components. The extra degrees of freedom introduced to make the fields five-dimensional result in an extra pair of second class constraints in the case of the massive vector field, and in an extended gauge group in the case of electromagnetism. The total gauge transformations depend on two arbitrary parameters, and contain `internal' and `external' U(1) gauge transformations as subgroups. 
  Boost-rotation symmetric vacuum spacetimes with spinning sources which correspond to gravitational field of uniformly accelerated spinning "particles" are studied. Regularity conditions and asymptotic properties are analyzed. News functions are derived by transforming the general spinning boost-rotation symmetric vacuum metric to Bondi-Sachs coordinates. 
  We study the asymptotic stability of de Sitter spacetime with respect to non-linear perturbations, by considering second order perturbations of a flat Robertson-Walker universe with dust and a positive cosmological constant. Using the synchronous comoving gauge we find that, as in the case of linear perturbations, the non-linear perturbations also tend to constants, asymptotically in time. Analysing curvature and other spacetime invariants we show, however, that these quantities asymptotically tend to their de Sitter values, thus demonstrating that the geometry is indeed locally asymptotically de Sitter, despite the fact that matter inhomogeneities tend to constants in time. Our results support the inflationary picture of frozen amplitude matter perturbations that are stretched outside the horizon, and demonstrate the validity of the cosmic no-hair conjecture in the nonlinear inhomogeneous settings considered here. 
  Canonical quantization of an action containing curvature squared term requires introduction of an auxiliary variable. Boulware etal[1] prescribed a technique to choose such a variable, by taking derivative of an action with respect to the highest derivative of the field variable, present in the action.It has been shown that[2] this technique can even be applied in situations where introduction of auxiliary variables are not at all required, leading to wrong Wheeler-deWitt equation. It has also been pointed out that[2] Boulware etal's[1] prescription should be taken up only after removing all removable total derivative terms from the action. Once this is done only a unique description of quantum dynamics would emerge. For curvature squared term this technique yields, for the first time, a quantum mechanical probability interpretation of quantum cosmology, and an effective potential, whose extremization leads to Einstein's equation. We conclude that Einstein-Hilbert action should essentially be modified by at least a curvature squared term to get a quantum mechanical formulation of quantuum cosmology and hence extend our previous work[2] for such an action along with a scalar field. 
  We investigate the quantum area operator in the loop approach based on the Lorentz covariant hamiltonian formulation of general relativity. We show that there exists a two-parameter family of Lorentz connections giving rise to Wilson lines which are eigenstates of the area operator. For each connection the area spectrum is evaluated. In particular, the results of the su(2) approach turn out to be included in the formalism. However, only one connection from the family is a spacetime connection ensuring that the 4d diffeomorphism invariance is preserved under quantization. It leads to the area spectrum independent of the Immirzi parameter. As a consequence, we conclude that the su(2) approach must be modified accordingly to the results obtained since it breaks one of the classical symmetries. 
  We investigate thermodynamic properties of two types of asymptotically anti-de Sitter spacetimes: black holes and singular scalar field spacetimes. We describe the possibility that thermodynamic phase transitions can transform one spacetime into another, suggesting that black holes can radiate to naked singularities. 
  We evaluate the local contribution g_[mu nu]L of coherent matter with lagrangian density L to the vacuum energy density. Focusing on the case of superconductors obeying the Ginzburg-Landau equation, we express the relativistic invariant density L in terms of low-energy quantities containing the pairs density. We discuss under which physical conditions the sign of the local contribution of the collective wave function to the vacuum energy density is positive or negative. Effects of this kind can play an important role in bringing about local changes in the amplitude of gravitational vacuum fluctuations - a phenomenon reminiscent of the Casimir effect in QED. 
  A general recipe to define, via Noether theorem, the Hamiltonian in any natural field theory is suggested. It is based on a Regge-Teitelboim-like approach applied to the variation of Noether conserved quantities. The Hamiltonian for General Relativity in presence of non-orthogonal boundaries is analysed and the energy is defined as the on-shell value of the Hamiltonian. The role played by boundary conditions in the formalism is outlined and the quasilocal internal energy is defined by imposing metric Dirichlet boundary conditions. A (conditioned) agreement with previous definitions is proved. A correspondence with Brown-York original formulation of the first principle of black hole thermodynamics is finally established. 
  A quantum inequality for the quantized electromagnetic field is developed for observers in static curved spacetimes. The quantum inequality derived is a generalized expression given by a mode function expansion of the four-vector potential, and the sampling function used to weight the energy integrals is left arbitrary up to the constraints that it be a positive, continuous function of unit area and that it decays at infinity. Examples of the quantum inequality are developed for Minkowski spacetime, Rindler spacetime and the Einstein closed universe. 
  We perform a numerical study of the critical regime at the threshold of black hole formation in the spherically symmetric, general relativistic collapse of collisionless matter. The coupled Einstein-Vlasov equations are solved using a particle-mesh method in which the evolution of the phase-space distribution function is approximated by a set of particles (or, more precisely, infinitesimally thin shells) moving along geodesics of the spacetime. Individual particles may have non-zero angular momenta, but spherical symmetry dictates that the total angular momentum of the matter distribution vanish. In accord with previous work by Rein et al, our results indicate that the critical behavior in this model is Type I; that is, the smallest black hole in each parametrized family has a finite mass. We present evidence that the critical solutions are characterized by unstable, static spacetimes, with non-trivial distributions of radial momenta for the particles. As expected for Type I solutions, we also find power-law scaling relations for the lifetimes of near-critical configurations as a function of parameter-space distance from criticality. 
  We intend to clarify the interplay between boundary terms and conformal transformations in scalar-tensor theories of gravity. We first consider the action for pure gravity in five dimensions and show that, on compactifing a la Kaluza-Klein to four dimensions, one obtains the correct boundary terms in the Jordan (or String) Frame form of the Brans-Dicke action. Further, we analyze how the boundary terms change under the conformal transformations which lead to the Pauli (or Einstein) frame and to the non-minimally coupled massless scalar field. In particular, we study the behaviour of the total energy in asymptotically flat space-times as it results from surface terms in the Hamiltonian formalism. 
  The scalar and electromagnetic fields produced by the geodesic and uniformly accelerated discrete charges in de Sitter spacetime are constructed by employing the conformal relation between de Sitter and Minkowski space.   A special attention is paid to new effects arising in spacetimes which, like de Sitter space, have spacelike conformal infinities. Under the presence of particle and event horizons, purely retarded fields (appropriately defined) become necessarily singular or even cannot be constructed at the "creation light cones" -- future light cones of the "points" at which the sources "enter" the universe. We construct smooth (outside the sources) fields involving both retarded and advanced effects, and analyze the fields in detail in case of (i) scalar monopoles, (ii) electromagnetic monopoles, and (iii) electromagnetic rigid and geodesic dipoles. 
  We present an approximate analysis of a nonlinear effect of parametric oscillatory instability in FP interferometer. The basis for this effect is the excitation of the additional (Stokes) optical mode and of the mirror's elastic mode when the optical energy stored in the FP resonator main mode exceeds the certain threshold. This effect is undesirable in laser gravitational wave antennae because it may create a specific upper limit for the value of energy stored in FP resonator. In order to avoid it the detailed analysis of the mirror's elastic modes and FP resonator optical modes structure is necessary. 
  As a black hole in a binary spirals in gradually from large separation, energy and angular momentum flow not only to infinity but also into or out of the hole. In addition, the hole's horizon area increases slowly during this process. In this paper, the changes in the black hole's mass, spin, and horizon area during inspiral are calculated for a hole in a circular binary with a companion body of possibly comparable mass. When the binary is composed of equal-mass black holes that have spins aligned with the orbital angular momentum and are rapidly rotating (with spins 99.8 percent of their maximal values), it is found that the fractional increase in the surface area of each hole's horizon is one percent by the time the binary spirals down to a separation b of 6M (where M is the binary's total mass), and seven percent down to b=2M. The flow of energy and angular momentum into the black holes' horizons changes the number of gravitational-wave cycles in the LIGO band by no more than a tenth of a cycle by the time the binary reaches b=2M. The results obtained in this paper are relevant for the detection and analysis of gravitational waves from binary systems containing a black hole. 
  Detecting gravitational wave bursts (characterised by short durations and poorly modelled waveforms) requires to have coincidences between several interferometric detectors in order to reject non-stationary noise events. As the wave amplitude seen in a detector depends on its location with respect to the source direction and as the signal to noise ratio of these bursts are expected to be low, coincidences between antennas may not be so likely. This paper investigates this question from a statistical point of view by using a simple model of a network of detectors; it also estimates the timing precision of a detection in an interferometer which is an important issue for the reconstruction of the source location, based on time delays. 
  We investigate the orbits of compact binary systems during the final inspiral period before coalescence by integrating numerically the second-order post-Newtonian equations of motion. We include spin-orbit and spin-spin coupling terms, which, according to a recent study by Levin [J. Levin, Phys. Rev. Lett. 84, 3515 (2000)], may cause the orbits to become chaotic. To examine this claim, we study the divergence of initially nearby phase-space trajectories and attempt to measure the Lyapunov exponent gamma. Even for systems with maximally spinning objects and large spin-orbit misalignment angles, we find no chaotic behavior. For all the systems we consider, we can place a strict lower limit on the divergence time t_L=1/gamma that is many times greater than the typical inspiral time, suggesting that chaos should not adversely affect the detection of inspiral events by upcoming gravitational-wave detectors. 
  The starting point of this work is the principle that all movement of particles and photons in the observable Universe must follow geodesics of a 4-dimensional space where time intervals are always a measure of geodesic arc lengths, i.e. $c^2(\mathrm{d}t)^2 = g_{\alpha \beta} \mathrm{d} x^\alpha \mathrm{d} x^\beta$, with $c$ is the speed of light in vacuum, $t$ time, $g_{\alpha \beta}$ and the metric tensor; $x^\alpha$ represents any of 4 space coordinates. The last 3 coordinates $(\alpha = 1,2,3)$ are immediately associated with the usual physical space coordinates, while the first coordinate $(\alpha=0)$ is later found to be related to \emph{proper time}. The work shows that this principle is applicable in several important situations and suggests that the underlying principle can, in fact, be used universally. Starting with special relativity it is shown that there is perfect mapping between the geodesics on Minkowski space-time and on this alternative space. The discussion than follows through light propagation in a refractive medium, and some cases of gravitation, including Schwartzschild's outer metric. The last part of the presentation is dedicated to electromagnetic interaction and Maxwell's equations, showing that there is a particular solution where one of the space dimensions is eliminated and the geodesics become equivalent to light rays in geometrical optics. A very brief discussion is made of the implications for wave-particle duality and quantization. 
  Electromagnetic rigidity which exists in large-scale optical resonators if pumping frequency is detuned from the eigenfrequency of resonator have sophisticated spectral dependence which allows to obtain sensitivity better than the Standard Quantum Limits both for the free test mass and the harmonic oscillator. 
  Big bang of the Friedmann-Robertson-Walker (FRW)-brane universe is studied. In contrast to the spacelike initial singularity of the usual FRW universe, the initial singularity of the FRW-brane universe is point-like from the viewpoint of causality including gravitational waves propagating in the bulk. Existence of null singularities (seam singuralities) is also shown in the flat and open FRW-brane universe models. 
  I show that a Planck-scale deformation of the relativistic dispersion relation, which has been independently considered in the quantum-gravity literature, can explain the surprising results of three classes of experiments: (1) observations of cosmic rays above the expected GZK limit, (2) observations of multi-TeV photons from the BL Lac object Markarian 501, (3) studies of the longitudinal development of the air showers produced by ultra-high-energy hadronic particles. Experiments now in preparation, such as the ones planned for the GLAST space telescope, will provide an independent test of this solution of the three experimental paradoxes. 
  We study the occurrence of shell crossing in spherical weakly charged dust collapse in the presence of a non-vanishing cosmological constant. We find that shell crossing always occurs from generic time-symmetric regular initial data, near the center of the matter configuration. For non-time-symmetric initial data, the occurrence---or lack thereof---of shell crossing is determined by the initial velocity profile, for a given mass and charge distribution. Physically reasonable initial data inevitably leads to shell crossing (near the center) before the minimum bounce radius is reached. 
  We study Levi-Civita metric for values of its sigma parameter in the range 0< sigma<infinity. We show that the value sigma=1/2 makes the axial and angular coordinates switch meaning. We present its geodesics and a physical source satisfying the energy conditions for all the range of sigma. This source allows us to obtain an energy per unit length which agrees with the behaviour of the geodesics and the fact that the solution has no event horizon. 
  The purpose of this paper is to propose the implementation of some methods from algebraic geometry in the theory of gravitation, and more especially in the variational formalism. It has been assumed that the metric tensor depends on two vector fields, defined on a manifold, and also that the gravitational Lagrangian depends on the metric tensor and its first and second differentials (instead on the partial or covariant derivatives, as usually assumed).   Assuming also different operators of variation and differentiation, it has been shown that the first variation of the gravitational Lagrangian can be represented as a third-rank polynomial in respect to the variables, defined in terms of the variated or differentiated vector fields. Therefore, the solution of the variational problem is found to be equivalent to finding all the variables - elements of an algebraic variety, which satisfy the algebraic equation. 
  We present an elementary account of mathematical cosmology through a series of important unsolved problems. We introduce the fundamental notion of `a cosmology' and focus on the issue of singularities as a theme unifying many current, seemingly unrelated trends of this subject. We discuss problems associated with the definition and asymptotic structure of the notion of cosmological solution and also problems related to the qualification of approximations and to the ranges of validity of given cosmologies. 
  Motivated by the recent attention on superluminal phenomena, we investigate the compatibility between faster-than-c propagation and the fundamental principles of relativity and causality. We first argue that special relativity can easily accommodate -- indeed, does not exclude -- faster-than-c signalling at the kinematical level. As far as causality is concerned, it is impossible to make statements of general validity, without specifying at least some features of the tachyonic propagation. We thus focus on the Scharnhorst effect (faster-than-c photon propagation in the Casimir vacuum), which is perhaps the most plausible candidate for a physically sound realization of these phenomena. We demonstrate that in this case the faster-than-c aspects are ``benign'' and constrained in such a manner as to not automatically lead to causality violations. 
  Anderson, et al. find the measured trajectories of Pioneer 10 and 11 spacecraft deviate from the trajectories computed from known forces acting on them. This unmodelled acceleration (and the less well known, but similar, unmodelled torque) can be accounted for by non-isotropic radiation of spacecraft heat. Various forms of non-isotropic radiation were proposed by Katz, Murphy, and Scheffer, but Anderson, et al. felt that none of these could explain the observed effect. This paper calculates the known effects in more detail and considers new sources of radiation, all based on spacecraft construction. These effects are then modelled over the duration of the experiment. The model reproduces the acceleration from its appearance at a heliocentric distance of 5 AU to the last measurement at 71 AU to within 10 percent. However, it predicts a larger decrease in acceleration between intervals I and III of the Pioneer 10 observations than is observed. This is a 2 sigma discrepancy from the average of the three analyses (SIGMA, CHASMP, and Markwardt). A more complex (but more speculative) model provides a somewhat better fit. Radiation forces can also plausibly explain the previously unmodelled torques, including the spindown of Pioneer 10 that is directly proportional to spacecraft bus heat, and the slow but constant spin-up of Pioneer 11. In any case, by accounting for the bulk of the acceleration, the proposed mechanism makes it much more likely that the entire effect can be explained without the need for new physics. 
  We present a rich class of exact solutions which contains radiation-dominated and matter-dominated models for the early and late universe. They include a variable cosmological ``constant'' which is derived from a higher dimension and manifests itself in spacetime as an energy density for the vacuum. This is in agreement with observational data and is compatible with extensions of general relativity to string and membrane theory. Our solutions are also typified by a non-singular ``big bounce'' (as opposed to a singular big bang), where matter is created as in inflationary cosmology. 
  The Cauchy problem is considered for the massive Dirac equation in the non-extreme Kerr-Newman geometry, for smooth initial data with compact support outside the event horizon and bounded angular momentum. We prove that the Dirac wave function decays in L^\infty_loc at least at the rate t^{-5/6}. For generic initial data, this rate of decay is sharp. We derive a formula for the probability p that the Dirac particle escapes to infinity. For various conditions on the initial data, we show that p=0,1 or 0<p<1. The proofs are based on a refined analysis of the Dirac propagator constructed in gr-qc/0005088. 
  In this article a new stationary cylindrically symmetric solution of the Einstein's field equations with cosmological constant and Time machine is given. The garavitational field is created by ideal liquid with three massless scalar fields or by ideal liquid with electric-magnetic field and massless scalar field. 
  We investigate asymptotic symmetries regularly defined on spherically symmetric Killing horizons in the Einstein theory with or without the cosmological constant. Those asymptotic symmetries are described by asymptotic Killing vectors, along which the Lie derivatives of perturbed metrics vanish on a Killing horizon. We derive the general form of asymptotic Killing vectors and find that the group of the asymptotic symmetries consists of rigid O(3) rotations of a horizon two-sphere and supertranslations along the null direction on the horizon, which depend arbitrarily on the null coordinate as well as the angular coordinates. By introducing the notion of asymptotic Killing horizons, we also show that local properties of Killing horizons are preserved under not only diffeomorphisms but also non-trivial transformations generated by the asymptotic symmetry group. Although the asymptotic symmetry group contains the $\mathit{Diff}(S^1)$ subgroup, which results from the supertranslations dependent only on the null coordinate, it is shown that the Poisson bracket algebra of the conserved charges conjugate to asymptotic Killing vectors does not acquire non-trivial central charges. Finally, by considering extended symmetries, we discuss that unnatural reduction of the symmetry group is necessary in order to obtain the Virasoro algebra with non-trivial central charges, which will not be justified when we respect the spherical symmetry of Killing horizons. 
  It can be shown that negative energy requirements within the Alcubierre spacetime can be greatly reduced when one introduces a lapse function into the Einstein tensor. Thereby reducing the negative energy requirements of the warp drive spacetime arbitrarily as a function of A(ct,r_s). With this function new quantum inequality restrictions are investigated in a general form. Finally a pseudo method for controlling a warp bubble at a velocity greater than that of light is presented. 
  We study the quasi-normal modes (QNM) of scalar, electromagnetic and gravitational perturbations of black holes in general relativity whose horizons have toroidal, cylindrical or planar topology in an asymptotically anti-de Sitter (AdS) spacetime. The associated quasinormal frequencies describe the decay in time of the corresponding test field in the vicinities of the black hole. In terms of the AdS/CFT conjecture, the inverse of the frequency is a measure of the dynamical timescale of approach to thermal equilibrium of the corresponding conformal field theory. 
  The behaviors of quantum stress tensor for the scalar field on the classical background of spherical dust collapse is studied. In the previous works diverging flux of quantum radiation was predicted. We use the exact expressions in a 2D model formulated by Barve et al. Our present results show that the back reaction does not become important during the semiclassical phase. The appearance of the naked singularity would not be affected by this quantum field radiation. To predict whether the naked singularity explosion occurs or not we need the theory of quantum gravity. We depict the generation of the diverging flux inside the collapsing star. The quantum energy is gathered around the center positively. This would be converted to the diverging flux along the Cauchy horizon. The ingoing negative flux crosses the Cauchy horizon. The intensity of it is divergent only at the central naked singularity. This diverging negative ingoing flux is balanced with the outgoing positive diverging flux which propagates along the Cauchy horizon. After the replacement of the naked singularity to the practical high density region the instantaneous diverging radiation would change to more milder one with finite duration. 
  We examine theories of gravity which include finitely many coupled scalar fields with arbitrary couplings to the curvature (wavemaps). We show that the most general scalar-tensor $\sigma$-model action is conformally equivalent to general relativity with a minimally coupled wavemap with a particular target metric. Inflation on the source manifold is then shown to occur in a novel way due to the combined effect of arbitrary curvature couplings and wavemap self-interactions. A new interpretation of the conformal equivalence theorem proved for such `wavemap-tensor' theories through brane-bulk dynamics is also discussed. 
  Nonthermal radiation of a Kerr black hole is considered as tunneling of created particles through an effective Dirac gap. In the leading semiclassical approximation this approach is applicable to bosons as well. Our semiclassical results for photons and gravitons do not contradict those obtained previously. For neutrinos the result of our accurate quantum mechanical calculation is about two times larger than the previous one. 
  We discuss a model describing exactly a thin spherically symmetric shell of matter with zero rest mass. We derive the reduced formulation of this system in which the variables are embeddings, their conjugate momenta, and Dirac observables. A non-perturbative quantum theory of this model is then constructed, leading to a unitary dynamics. As a consequence of unitarity, the classical singularity is fully avoided in the quantum theory. 
  We analyze the signal-to-noise ratio for a relic background of scalar gravitational radiation composed of massive, non-relativistic particles, interacting with the monopole mode of two resonant spherical detectors. We find that the possible signal is enhanced with respect to the differential mode of the interferometric detectors. This enhancement is due to: {\rm (a)} the absence of the signal suppression, for non-relativistic scalars, with respect to a background of massless particles, and {\rm (b)} for flat enough spectra, a growth of the signal with the observation time faster than for a massless stochastic background. 
  A one-parameter family of time-symmetric initial data for the radial infall of a particle into a Schwarzschild black hole is constructed within the framework of black-hole perturbation theory. The parameter measures the amount of gravitational radiation present on the initial spacelike surface. These initial data sets are then evolved by integrating the Zerilli-Moncrief wave equation in the presence of the particle. Numerical results for the gravitational waveforms and their power spectra are presented; we show that the choice of initial data strongly influences the waveforms, both in their shapes and their frequency content. We also calculate the total energy radiated by the particle-black-hole system, as a function of the initial separation between the particle and the black hole, and as a function of the choice of initial data. Our results confirm that for large initial separations, a conformally-flat initial three-geometry minimizes the initial gravitational-wave content, so that the total energy radiated is also minimized. For small initial separations, however, we show that the conformally-flat solution no longer minimizes the energy radiated. 
  We present a method for generating solutions in some scalar-tensor theories with a minimally coupled massless scalar field or irrotational stiff perfect fluid as a source. The method is based on the group of symmetries of the dilaton-matter sector in the Einstein frame. In the case of Barker's theory the dilaton-matter sector possesses SU(2) group of symmetries. In the case of Brans-Dicke and the theory with "conformal coupling", the dilaton- matter sector has $SL(2,R)$ as a group of symmetries. We describe an explicit algorithm for generating exact scalar-tensor solutions from solutions of Einstein-minimally-coupled-scalar-field equations by employing the nonlinear action of the symmetry group of the dilaton-matter sector. In the general case, when the Einstein frame dilaton-matter sector may not possess nontrivial symmetries we also present a solution generating technique which allows us to construct exact scalar-tensor solutions starting with the solutions of Einstein-minimally-coupled-scalar-field equations. As an illustration of the general techniques, examples of explicit exact solutions are constructed. In particular, we construct inhomogeneous cosmological scalar-tensor solutions whose curvature invariants are everywhere regular in space-time. A generalization of the method for scalar-tensor-Maxwell gravity is outlined. 
  In Schwarzschild spacetime the value $r=3m$ of the radius coordinate is characterized by three different properties: (a) there is a ``light sphere'', (b) there is ``centrifugal force reversal'', (c) it is the upper limiting radius for a non-transparent Schwarschild source to act as a gravitational lens that produces infinitely many images. In this paper we prove a theorem to the effect that these three properties are intimately related in {\em any} spherically symmetric static spacetime. We illustrate the general results with some examples including black-hole spacetimes and Morris-Thorne wormholes. 
  A theorem about local in time existence of spacelike foliations with prescribed mean curvature in cosmological spacetimes will be proved. The time function of the foliation is geometrically defined and fixes the diffeomorphism invariance inherent in general foliations of spacetimes. Moreover, in contrast to the situation of the more special constant mean curvature foliations, which play an important role in the global analysis of spacetimes, this theorem overcomes the existence problem arising from topological restrictions for surfaces of constant mean curvature. 
  A 4-dimensional Lorentzian static space-time is equivalent to 3-dimensional Euclidean gravity coupled to a massless Klein-field. By canonically quantizing the coupling model in the framework of loop quantum gravity, we obtain a quantum theory which actually describes quantized static space-times. The Kinematical Hilbert space is the product of the Hilbert space of gravity with that of imaginary scalar fields. It turns out that the Hamiltonian constraint of the 2+1 model corresponds to a densely defined operator in the underlying Hilbert space, and hence it is finite without renormalization. As a new point of view, this quantized model might shed some light on a few physical problems concerning quantum gravity. 
  We investigate barotropic perfect fluid cosmologies which admit an isotropic singularity. From the General Vorticity Result of Scott, it is known that these cosmologies must be irrotational. In this paper we prove, using two different methods, that if we make the additional assumption that the perfect fluid is shear-free, then the fluid flow must be geodesic. This then implies that the only shear-free, barotropic, perfect fluid cosmologies which admit an isotropic singularity are the FRW models. 
  We study the interaction of massless scalar fields with self-gravitating neutron stars by means of fully dynamic numerical simulations of the Einstein-Klein-Gordon perfect fluid system. Our investigation is restricted to spherical symmetry and the neutron stars are approximated by relativistic polytropes. Studying the nonlinear dynamics of isolated neutron stars is very effectively performed within the characteristic formulation of general relativity, in which the spacetime is foliated by a family of outgoing light cones. We are able to compactify the entire spacetime on a computational grid and simultaneously impose natural radiative boundary conditions and extract accurate radiative signals. We study the transfer of energy from the scalar field to the fluid star. We find, in particular, that depending on the compactness of the neutron star model, the scalar wave forces the neutron star either to oscillate in its radial modes of pulsation or to undergo gravitational collapse to a black hole on a dynamical timescale. The radiative signal, read off at future null infinity, shows quasi-normal oscillations before the setting of a late time power-law tail. 
  Variational principle for a solid in classical mechanics is formulated in terms of a thin elastic 4D bar strain in Minkowsky events space of special relativity. It is shown, that the sum of elastic 4-energies of weak twist and bending under some identifications takes the form of classical non-relativistic action for a solids dynamics. The necessary conditions on 4D bar parameters and elastic constants, providing validity of Newton mechanics, are found. 
  The issue of the local visibility of the shell-focussing singularity in marginally bound spherical dust collapse is considered from the point of view of the existence of future-directed null geodesics with angular momentum which emanate from the singularity. The initial data (i.e. the initial density profile) at the onset of collapse is taken to be of class $C^3$. Simple necessary and sufficient conditions for the existence of a naked singularity are derived in terms of the data. It is shown that there exist future-directed non-radial null geodesics emanating from the singularity if and only if there exist future-directed radial null geodesics emanating from the singularity. This result can be interpreted as indicating the robustness of previous results on radial geodesics, with respect to the presence of angular momentum. 
  The thin string limit of Cosmic Strings is investigated using a description in terms of Colombeau's theory of nonlinear generalised functions. It is shown that in this description the energy-momentum tensor has a well defined thin string limit. Furthermore the deficit angle of the conical spacetime that one obtains in the limit may be given in terms of the distributional energy-momentum tensor. On the other hand it is only in the special case of critical coupling that the energy-momentum tensor defined in the Colombeau algebra is associated to a conventional distribution. The asymptotics of both the matter and gravitational field are investigated in the thin string limit and it is shown how this leads to the `conical approximation' which is valid outside the inner core of the string. 
  What are the implications if the total 'information' in the universe is conserved? Black holes might be 'logic gates' recomputing the 'lost information' from incoming 'signals' from outside their event horizons into outgoing 'signals' representing evaporative or radiative decay 'products' of the reconfiguration process of the black hole quantum logic 'gate'. Apparent local imbalances in the information flow can be corrected by including the effects of the coupling of the vacuum 'reservoir' of information as part of the total information involved in any evolutionary process. In this way perhaps the 'vacuum' computes the future of the observable universe. 
  We investigate the occurrence and nature of a naked singularity in the gravitational collapse of an inhomogeneous dust cloud described by higher dimensional Tolman-Bondi space-times. The naked singularities are found to be gravitationally strong in the sense of Tipler. Higher dimensions seem to favour black holes rather than naked singularities. 
  A wave-front in a space-time $\cal M$ is a family of null geodesics orthogonal to a smooth spacelike two-surface in $\cal M$; it is of some interest to know how a wave-front can fail to be a smoothly immersed surface in $\cal M$. In this paper we see that the space of null geodesics $\cal N$ of $\cal M$, considered as a contact manifold, provides a natural setting for an efficient study of the stable singularities arising in the time evolution of wave-fronts. 
  We investigate how braneworld gravity affects gravitational collapse and black hole formation by studying Oppenheimer-Snyder-like collapse on a Randall-Sundrum type brane. Without making any assumptions about the bulk, we prove a no-go theorem: the exterior spacetime on the brane cannot be static, which is in stark contrast with general relativity. We also consider the role of Kaluza-Klein energy density in collapse, using a toy model. 
  What is the shape of space in a spacetime? One way of addressing this issue is to consider edgeless spacelike submanifolds of the spacetime. An alternative is to foliate the spacetime by timelike curves and consider the quotient obtained by identifying points on the same timelike curve. In this article we investigate each of these notions and obtain conditions such that it yields a meaningful shape of space. We also consider the relationship between these two notions and find conditions for the quotient space to be diffeomorphic to any edgeless spacelike hypersurface. In particular, we find conditions in which merely local behavior (being spacelike) combined with the correct behavior on the homotopy level guarantees that a putative shape of space really is precisely that. 
  The sensitivity achievable by a pair of VIRGO detectors to stochastic and isotropic gravitational wave backgrounds produced in pre-big-bang models is discussed in view of the development of a second VIRGO interferometer. We describe a semi-analytical technique allowing to compute the signal-to-noise ratio for (monotonic or non-monotonic) logarithmic energy spectra of relic gravitons of arbitrary slope. We apply our results to the case of two correlated and coaligned VIRGO detectors and we compute their achievable sensitivities. We perform our calculations both for the usual case of minimal string cosmological scenario and in the case of a non-minimal scenario where a long dilaton dominated phase is present prior to the onset of the ordinary radiation dominated phase. In this framework, we investigate possible improvements of the achievable sensitivities by selective reduction of the thermal contributions (pendulum and pendulum's internal modes) to the noise power spectra of the detectors. Since a reduction of the shot noise does not increase significantly the expected sensitivity of a VIRGO pair (in spite of the relative spatial location of the two detectors) our findings support the experimental efforts directed towards a substantial reduction of thermal noise. 
  We consider a late-time cosmological model based on a recent proposal that the infinite-bare-coupling limit of superstring/M-theory exists and has good phenomenological properties, including a vanishing cosmological constant, and a massless, decoupled dilaton. As it runs away to $+ \infty$, the dilaton can play the role of the quintessence field recently advocated to drive the late-time accelerated expansion of the Universe. If, as suggested by some string theory examples, appreciable deviations from General Relativity persist even today in the dark matter sector, the Universe may smoothly evolve from an initial "focusing" stage, lasting untill radiation--matter equality, to a "dragging" regime, which eventually gives rise to an accelerated expansion with frozen $\Omega(\rm{dark energy})/\Omega(\rm{dark matter})$. 
  In this article a new solution of the Einstein-Dirac's equations is presented. There are ghost spinors, i.e. the stress-energy tensor is equal to zero and the current of these fields is non-zero vector. Last the ghost neutrino was found. These ghost spinors and shadow particles of Deutsch are identified. And in result the ghost spinors have a physical interpretation and solutions of the field equations for shadow electrons as another shadow particles are found. 
  The Bach tensor and a vector which generates conformal symmetries allow a conserved four-current to be defined. The Bach four-current gives rise to a quasilocal two-surface expression for power per luminosity distance in the Vaidya exterior of collapsing fluid interiors. This is interpreted in terms of entropy generation. 
  Hawking evaporation of Dirac particles and scalar fields in a Vaidya-type black hole is investigated by the method of generalized tortoise coordinate transformation. It is shown that Hawking radiation of Dirac particles does not exist for $P_1, Q_2$ components but for $P_2, Q_1$ components in any Vaidya-type black holes. Both the location and the temperature of the event horizon change with time. The thermal radiation spectrum of Dirac particles is the same as that of Klein-Gordon particles. We demonstrates that there is no new quantum ergosphere effect in the thermal radiation of Dirac particles in any spherically symmetry black holes. 
  The multidimensional gravity on the principal bundle with the SU(2) gauge group is considered. The numerical investigation of the spherically symmetric metrics with the center of symmetry is made. The solution of the gravitational equations depends on the boundary conditions of the ``SU(2) gauge potential'' (off-diagonal metric components) at the symmetry center and on the type of symmetry (symmetrical or antisymmetrical) of these potentials. In the chosen range of the boundary conditions it is shown that there are two types of solutions: wormhole-like and flux tube. The physical application of such kind of solutions as quantum handles in a spacetime foam is discussed. 
  The incompatibility between the treatment of time in the classical and in the quantum theory results in the so-called problem of time in canonical quantum gravity. For this reason, attempts have been made to devise algorithms of quantization which accomodate the covariance of the classical theory from the outset. One of the most prominent of these attempts is based on the notion of continuous histories (Isham and Linden) in the context of the consistent histories approach to quantum theory (Griffiths, Omnes, Gell-Mann and Hartle). By the term continuous histories it is implied that the canonical fields and the symplectic structure of the theory depend on time as well as space. The aim of this thesis (in the form it was submitted to the University of London, February 2000) is to show that, even at the purely classical level, a history approach has several advantages (compared to its equal-time counterpart) when it comes to discussing spacetime issues. This is illustrated here by reframing and generalizing the derivation of geometrodynamics from first principles (Hojman, Kuchar, Teitelboim) in the language of the history phase space. 
  The structure of the history phase space $\cal G$ of a covariant field system and its history group (in the sense of Isham and Linden) is analyzed on an example of a bosonic string. The history space $\cal G$ includes the time map $\sf T$ from the spacetime manifold (the two-sheet) $\cal Y$ to a one-dimensional time manifold $\cal T$ as one of its configuration variables. A canonical history action is posited on $\cal G$ such that its restriction to the configuration history space yields the familiar Polyakov action. The standard Dirac-ADM action is shown to be identical with the canonical history action, the only difference being that the underlying action is expressed in two different coordinate charts on $\cal G$. The canonical history action encompasses all individual Dirac-ADM actions corresponding to different choices $\sf T$ of foliating $\cal Y$. The history Poisson brackets of spacetime fields on $\cal G$ induce the ordinary Poisson brackets of spatial fields in the instantaneous phase space ${\cal G}_{0}$ of the Dirac-ADM formalism. The canonical history action is manifestly invariant both under spacetime diffeomorphisms Diff$\cal Y$ and temporal diffeomorphisms Diff$\cal T$. Both of these diffeomorphisms are explicitly represented by symplectomorphisms on the history phase space $\cal G$. The resulting classical history phase space formalism is offered as a starting point for projection operator quantization and consistent histories interpretation of the bosonic string model. 
  Maxwell's equations can be obtained in generalized coordinates by considering the electromagnetic field as an external agent. The work here presented shows how to obtain the electrodynamics for a charged particle in generalized coordinates eliminating the concept of external force. Based on Kaluza's formalism, the one here presented extends the 5x5 metric into a 6x6 space-time giving enough room to include magnetic monopoles in a very natural way. 
  We show that the black hole perturbations of the Hayward static solution to the massless Einstein-Klein-Gordon equations are actually gauge artifacts resulting from the linearization of a coordinate transformation. 
  We study the curvature strength and visibility of the central singularity arising in Tolman-Bondi-de Sitter collapse. We find that the singularity is visible and Tipler strong along an infinite number of timelike geodesics, independently of the initial data, and thus stable against perturbations of the latter. 
  It has been shown [1,2] that the electromagnetic quantum vacuum makes a contribution to the inertial mass, $m_i$, in the sense that at least part of the inertial force of opposition to acceleration, or inertia reaction force, springs from the electromagnetic quantum vacuum. As experienced in a Rindler constant acceleration frame the electromagnetic quantum vacuum mainfests an energy-momentum flux which we call the Rindler flux (RF). The RF, and its relative, Unruh-Davies radiation, both stem from event-horizon effects in accelerating reference frames. The force of radiation pressure produced by the RF proves to be proportional to the acceleration of the reference frame, which leads to the hypothesis that at least part of the inertia of an object should be due to the interaction of its quarks and electrons with the RF. We demonstrate that this quantum vacuum inertia hypothesis is consistent with general relativity (GR) and that it answers a fundamental question left open within GR, viz. is there a physical mechanism that generates the reaction force known as weight when a specific non-geodesic motion is imposed on an object? The quantum vacuum inertia hypothesis provides such a mechanism, since by assuming the Einstein principle of local Lorentz-invariance (LLI), we can immediately show that the same RF arises due to curved spacetime geometry as for acceleration in flat spactime. Thus the previously derived expression for the inertial mass contribution from the electromagnetic quantum vacuum field is exactly equal to the corresponding contribution to the gravitational mass, $m_g$. Therefore, within the electromagnetic quantum vacuum viewpoint proposed in [1,2], the Newtonian weak equivalence principle, $m_i=m_g$, ensues in a straightforward manner. 
  We present an exact solution of the averaged Einstein's field equations in the presence of two real scalar fields and a component of dust with spherical symmetry. We suggest that the space-time found provides the characteristics required by a galactic model that could explain the supermassive central object and the dark matter halo at once, since one of the fields constitutes a central oscillaton surrounded by the dust and the other scalar field distributes far from the coordinate center and can be interpreted as a halo. We show the behavior of the rotation curves all along the background. Thus, the solution could be a first approximation of a ``long exposition photograph'' of a galaxy. 
  In abstract Yang-Mills theory the standard instanton construction relies on the Hodge star having real eigenvalues which makes it inapplicable in the Lorentzian case. We show that for the affine connection an instanton-type construction can be carried out in the Lorentzian setting. The Lorentzian analogue of an instanton is a spacetime whose connection is metric compatible and Riemann curvature irreducible ("pseudoinstanton"). We suggest a metric-affine action which is a natural generalization of the Yang-Mills action and for which pseudoinstantons are stationary points. We show that a spacetime with a Ricci flat Levi-Civita connection is a pseudoinstanton, so the vacuum Einstein equation is a special case of our theory. We also find another pseudoinstanton which is a wave of torsion in Minkowski space. Analysis of the latter solution indicates the possibility of using it as a model for the neutrino. 
  The presence of Killing-Yano tensors implies the existence of non-generic supercharges in spinning point particle theories on curved backgrounds. Dual metrics are defined through their associated non-degenerate Killing tensors of valence two. Siklos spacetimes, which are the only non-trivial Einstein spaces conformal to non-flat pp-waves are investigated in regards to the existence of their corresponding Killing and Killing-Yano tensors. It is found that under some restrictions, pp-wave metrics and Siklos spacetimes admit dual metrics and non-generic supercharges. Possible significance of those dual spacetimes are discussed. 
  We present a family of solutions for the axisymmetric Plebanski-Demianski metric and other corresponding reduced metrics. We also present the black hole characteristics using a new set of parameters for Kerr-Newman metric. 
  Using technique of supersymmetric quantum mechanics we present new cosmological quantum solution, in the regime for FRW cosmological model using a barotropic perfect fluid as matter field. 
  The space of continuous states of perturbative interacting quantum field theories in globally hyperbolic curved spacetimes is determined. Following Brunetti and Fredenhagen, we first define an abstract algebra of observables which contains the Wick-polynomials of the free field as well as their time-ordered products, and hence, by the well-known rules of perturbative quantum field theory, also the observables (up to finite order) of interest for the interacting quantum field theory. We then determine the space of continuous states on this algebra. Our result is that this space consists precisely of those states whose truncated n-point functions of the free field are smooth for all n not equal to two, and whose two-point function has the singularity of a Hadamard fundamental form. A crucial role in our analysis is played by the positivity property of states. On the technical side, our proof involves functional analytic methods, in particular the methods of microlocal analysis. 
  We consider a version of Palais' Principle of Symmetric Criticality (PSC) that is applicable to the Lie symmetry reduction of Lagrangian field theories. PSC asserts that, given a group action, for any group-invariant Lagrangian the equations obtained by restriction of Euler-Lagrange equations to group-invariant fields are equivalent to the Euler-Lagrange equations of a canonically defined, symmetry-reduced Lagrangian. We investigate the validity of PSC for local gravitational theories built from a metric. It is shown that there are two independent conditions which must be satisfied for PSC to be valid. One of these conditions, obtained previously in the context of transverse symmetry group actions, provides a generalization of the well-known unimodularity condition that arises in spatially homogeneous cosmological models. The other condition seems to be new. The conditions that determine the validity of PSC are equivalent to pointwise conditions on the group action alone. These results are illustrated with a variety of examples from general relativity. It is straightforward to generalize all of our results to any relativistic field theory. 
  We present an argument that, for a large class of possible dynamics, a canonical quantization of gravity will satisfy the Bekenstein-Hawking entropy-area relation. This result holds for temperatures low compared to the Planck temperature and for boundaries with areas large compared to Planck area. We also relate our description, in terms of a grand canonical ensemble, to previous geometric entropy calculations using area ensembles. 
  The possible amplification of gauge invariant metric fluctuations in the infrared sector are very important during reheating stage of inflation. In this stage the inflaton oscillates arount the minimum of the scalar potential. The evolution for super Hubble scales gauge invariant metric fluctuations can be studied by means of the Bardeen parameter. For a massive scalar field with a quadratic potential for a nonzero cosmological constant. I find that (in the reheating regime and for super Hubble scales), the Sasaki-Mukhanov parameter oscillates with amplitude constant such that there is no amplification of $Q$ during reheating. 
  Non-time-orthogonal frame analysis is applied to determine the frequency and wavelength of light as observed i) in a relativistically rotating frame when emission is from a source fixed in the non-rotating frame, ii) in a non-rotating frame when emission is from a source fixed in the rotating frame, and iii) when both source and observer are fixed in the rotating frame and the source emission direction varies with respect to the rotating frame. Appropriate Doppler effects are demonstrated, and second order differences from translating (time-orthogonal) frame analysis are noted. 
  We suggest that all horizons of spacetime, no matter whether they are black hole, Rindler or de Sitter horizons, have certain microscopic properties in common. We propose that these propertues may be used as the starting points, or postulates, of a microscopic theory of gravity. 
  In this article we use the idea of algorithmic complexity (AC) to study various cosmological scenarios, and as a means of quantizing the gravitational interaction. We look at 5D and 7D cosmological models where the Universe begins as a higher dimensional Planck size spacetime which fluctuates between Euclidean and Lorentzian signatures. These fluctuations are governed by the AC of the two different signatures. At some point a transition to a 4D Lorentzian signature Universe occurs, with the extra dimensions becoming ``frozen'' or non-dynamical. We also apply the idea of algorithmic complexity to study composite wormholes, the entropy of blackholes, and the path integral for quantum gravity. 
  A general system of equations is derived, using the 1+3 orthonormal tetrad formalism, describing the influence of a plane-fronted-parallel gravitational wave on a warm relativistic two-component plasma. We focus our attention on phenomena that are induced by terms that are higher order in the gravitational wave amplitude. In particular, it is shown that parametric excitations of ion-acoustic waves takes place, due to these higher order gravitational non-linearities. The implications of the results are discussed. 
  The problem of reconciling general relativity and quantum theory has fascinated and bedeviled physicists for more than 70 years. Despite recent progress in string theory and loop quantum gravity, a complete solution remains out of reach. I review the status of the continuing effort to quantize gravity, emphasizing the underlying conceptual issues and the various attempts to come to grips with them. 
  In 1973, E. T. Newman considered the holomorphic extension \tilde E(x+iy) of the Coulomb field E(x) in R^3. By analyzing its multipole expansion, he showed that the real and imaginary parts of \tilde E(x+iy), viewed as functions of x for fixed y, are the electric and magnetic fields generated by a spinning ring of charge R. This represents the electromagnetic part of the Kerr-Newman solution to the Einstein-Maxwell equations. As already pointed out by Newman and Janis in 1965, this interpretation is somewhat problematic since the fields are double-valued. To make them single-valued, a branch cut must be introduced so that R is replaced by a charged disk D having R as its boundary. In the context of curved spacetime, D becomes a spinning disk of charge and mass representing the singularity of the Kerr-Newman solution. Here we confirm the above interpretation of the real and imaginary parts of \tilde E(x+iy) by computing the charge- and current densities directly as distributions in R^3 supported in the source disk D. This shows in particular that D spins rigidly at the critical rate, so that its rim R moves at the speed of light.   It is a pleasure to thank Ted Newman, Andrzej Trautman and Iwo Bialinicki-Birula for many instructive discussions, particularly in Warsaw and during a visit to Pittsburgh. 
  A method is presented, which can generate solutions of the Hermitian theory of relativity from known solutions of the general theory of relativity, when the latter depend on three co-ordinates and are invariant under reversal of the fourth one. 
  Nature abhors an infinity. The limits of general relativity are often signaled by infinities: infinite curvature as in the center of a black hole, the infinite energy of the singular big bang. We might be inclined to add an infinite universe to the list of intolerable infinities. Theories that move beyond general relativity naturally treat space as finite. In this review we discuss the mathematics of finite spaces and our aspirations to observe the finite extent of the universe in the cosmic background radiation. 
  A Darboux-transformed surface gravitational acceleration of the constant gravitational acceleration for a body endowed with an atmospheric layer is shown to turn the atmospheric free fall with quadratic resistance in the opposite motion, i.e., a free rising. Although the atmosphere of such a body may look completely normal, it is the time dependence of its gravitational field that produces this type of motion. The result is a consequence of general, one-parameter-dependent Darboux transformations in mathematical physics 
  When the behaviour of the singularities, which are used to represent masses, charges or currents in exact solutions to the field equations of the Hermitian theory of relativity, is restricted by a no-jump rule, conditions are obtained, which determine the relative positions of masses, charges and currents. Due to these conditions the Hermitian theory of relativity appears to provide a unified description of gravitational, colour and electromagnetic forces. 
  Recently, branes in supergravity have become an indispensable tool even for traditional relativists. The purpose of this manuscript is to provide a pedagogical account of the brane technology so that the relativists can use branes in their study. The type IIA supergravity theory is mainly discussed and other cases are briefly mentioned. The repulson singularity is also explained as an interesting application. 
  The search for a quantum theory of gravity is one of the major challenges facing theoretical physics today. While no complete theory exists, a promising avenue of research is the loop quantum gravity approach. In this approach, quantum states are represented by spin networks, essentially graphs with weighted edges. Since general relativity predicts the structure of space, any quantum theory of gravity must do so as well; thus, "spatial observables" such as area, volume, and angle are given by the eigenvalues of Hermitian operators on the spin network states. We present results obtained in our investigations of the angle and volume operators, two operators which act on the vertices of spin networks. We find that the minimum observable angle is inversely proportional to the square root of the total spin of the vertex, a fairly slow decrease to zero. We also present numerical results indicating that the angle operator can reproduce the classical angle distribution. The volume operator is significantly harder to investigate analytically; however, we present analytical and numerical results indicating that the volume of a region scales as the 3/2 power of its bounding surface, which corresponds to the classical model of space. 
  An action principle of singular hypersurfaces in general relativity and scalar-tensor type theories of gravity in the Einstein frame is presented without assuming any symmetry. The action principle is manifestly doubly covariant in the sense that coordinate systems on and off a hypersurface are disentangled and can be independently specified. It is shown that, including variation of the metric, the position of the hypersurface and matter fields, the variational principle gives the correct set of equations of motion: the Einstein equation off the hypersurface, Israel's junction condition in a doubly covariant form and equations of motion of matter fields including the scalar fields. The position of the hypersurface measured from one side of the hypersurface and that measured from another side can be independently variated as required by the double covariance. 
  It is shown that the precession of a gyroscope can be used to elucidate the nature of the smoothness of the null infinity of an asymptotically flat spacetime (describing an isolated body). A model for which the effects of precession in the non-smooth null infinity case are of order $r^{-2}\ln r$ is proposed. By contrast, in the smooth version the effects are of order $r^{-3}$. This difference should provide an effective criterion to decide on the nature of the smoothness of null infinity. 
  In this work the photon equation (massless Duffin-Kemmer-Petiau equation) is written expilicitly for general type of stationary G\"{o}del space-times and is solved exactly for G\"{o}del-type and G\"{o}del space-times. Harmonic oscillator behaviour of the solutions is discussed and energy spectrum of photon is obtained. 
  It is discussed how systems of quantum-correlated (entangled)particles or atoms behave in external gravitational fields and what gravitational effects may exist in such systems. An experimental setup is proposed which improves the sensitivity of the Ramsey interferometer by the usage of quantum-correlated atoms. Entanglement of $n$ atoms improves the sensitivity to small phase shifts in $n^2$ times. This scheme may be used for observing gravity-induced phase shifts in laboratory. 
  Einstein's unified field theory is extended by the addition of matter terms in the form of a symmetric energy tensor and of two conserved currents. From the field equations and from the conservation identities emerges the picture of a gravoelectrodynamics in a dynamically polarizable Riemannian continuum. Through an approximate calculation exploiting this dynamical polarizability it is argued that ordinary electromagnetism may be contained in the theory. 
  The quantization of gravity coupled to a perfect fluid model leads to a Schr\"odinger-like equation, where the matter variable plays the role of time. The wave function can be determined, in the flat case, for an arbitrary barotropic equation of state $p = \alpha\rho$; solutions can also be found for the radiative non-flat case. The wave packets are constructed, from which the expectation value for the scale factor is determined. The quantum scenarios reveal a bouncing Universe, free from singularity. We show that such quantum cosmological perfect fluid models admit a universal classical analogue, represented by the addition, to the ordinary classical model, of a repulsive stiff matter fluid. The meaning of the existence of this universal classical analogue is discussed. The quantum cosmological perfect fluid model is, for a flat spatial section, formally equivalent to a free particle in ordinary quantum mechanics, for any value of $\alpha$, while the radiative non-flat case is equivalent to the harmonic oscillator. The repulsive fluid needed to reproduce the quantum results is the same in both cases. 
  Anderson, et al., find the measured trajectories of Pioneer 10 and 11 spacecraft deviate from the trajectories computed from known forces acting on them. This unmodelled acceleration can be accounted for by non-isotropic radiation of spacecraft heat. Various forms of non-isotropic radiation were proposed by Katz, Murphy, and Scheffer, but Anderson, et al. felt that none of these could explain the observed effect. This paper calculates the known effects in more detail and considers new sources of radiation, all based on spacecraft construction. These effects are then modelled over the duration of the experiment. The model provides a reasonable fit to the acceleration from its appearance at a heliocentric distance of 5 AU to the last measurement at 71 AU, but overpredicts by 9% the decrease in acceleration between intervals I and III of the Pioneer 10 observations. (For comparison, the two different measurements of the effect (SIGMA and CHASMP) themselves differ by 4% in interval III.) In any case, by accounting for the bulk of the acceleration, the proposed mechanism makes it much more likely that the entire effect can be explained without the need for new physics. 
  HH-spaces, i.e., complex spacetimes, of Petrov type NxN are determined by a trio of pde's for two functions, lambda and a, of three independent variables (and also two gauge functions, chosen to be two of the independent variables if one prefers). As in common integrable systems, these form a second order, linear system for lambda; howver, here the integrability conditions, involving a, are more complicated than is common. Therefore, with the hope of finding new solutions, these equations are now constrained to also admit both one and two homothetic or Killing vectors. The case with one Killing and one homothetic vector reduces these equations to two ode's for two unknown functions of the one remaining variable.   In addition, we also describe in detail the explicit forms of the metric, tetrad, connections, and curvature for twisting HH-spaces of Petrov type NxN, modulo the determining equations. This simplifies considerably the process of obtaining these details cleanly from earlier articles on the subject, thus simplifying access to the research area. 
  Numerical relativity describes a discrete initial value problem for general relativity. A choice of gauge involves slicing space-time into space-like hypersurfaces. This introduces past and future gauge relative to the hypersurface of present time. Here, we propose solving the discretized Einstein equations with a choice of gauge in the future and a dynamical gauge in the past. The method is illustrated on a polarized Gowdy wave. 
  Following the general formalism presented by Rezzolla, Ahmedov and Miller (MNRAS, 322, 723 2001), we here derive analytic solutions of the electromagnetic fields equations in the internal and external background spacetime of a slowly rotating highly conducting magnetized neutron star. The star is assumed to be isolated and in vacuum, with a dipolar magnetic field not aligned with the axis of rotation. Our results indicate that the electromagnetic fields of a slowly rotating neutron star are modified by general relativistic effects arising from both the monopolar and the dipolar parts of the gravitational field. The results presented here differ from the ones discussed by Rezzolla, Ahmedov and Miller (MNRAS, 322, 723 2001) mainly in that we here consider the interior magnetic field to be dipolar with the same radial dependence as the external one. While this assumption might not be a realistic one, it should be seen as the application of our formalism to a case often discussed in the literature. 
  The stability of transparent spherically symmetric thin shells (and wormholes) to linearized spherically symmetric perturbations about static equilibrium is examined. This work generalizes and systematizes previous studies and explores the consequences of including the cosmological constant. The approach shows how the existence (or not) of a domain wall dominates the landscape of possible equilibrium configurations. 
  We review the current status of the singularity problem in string theory for non-experts. After the problem is discussed from the point of view of supergravity, we discuss classic examples and recent examples of singularity resolution in string theory. 
  We calculate quasi-normal f- and g-modes of a neutron star with density discontinuity, which may appear in a phase transition at extreme high density. We find that discontinuity will reflect largely on the f-mode, and that the g-mode could also be important for a less massive star. 
  Given a (d+1)-dimensional spacetime (M,g), one can consider the set N of all its null geodesics. If (M,g) is globally hyperbolic then this set is naturally a smooth (2d-1)-manifold. The sky of an event x in M is the set X of all null geodesics through x, and is an embedded submanifold of N diffeomorphic to S^{d-1}. Low conjectured that if d=2 then x,y are causally related in M iff X,Y are linked in N. We prove Low's conjecture for a (large) class of static spacetimes. 
  We consider discretizations of the Einstein action of general relativity such that the resulting discrete equations of motion form a consistent constrained system. Upon ``spin foam'' quantization of the system, consistency allows a natural way of recovering the correct semi-classical theory. A consistent set of approximations to the Einstein equations could also have implications for numerical relativity and for the construction of approximate classical observables for the theory. 
  Spin-polarised cylindrically symmetric solution are shown not to be compatible with teleparallel gravity.This can be done in two distinct manners.The first is to show that not all components of the orthonormal tetrad (OT).It is argue however that this result maybe done by a bad choice of the spin distribution along the cylinder.A cylindrically symmetric Riemannian solution is obtained which represents a conical geometry of defects. 
  It is shown that the results of the paper by Contreras et al. [Contreras, G., Nunez, L. A., Percoco, U. "Ricci Collineations for Non-degenerate, Diagonal and Spherically Symmetric Ricci Tensors" (2000) Gen. Rel. Grav. 32, 285-294] concerning the Ricci Collineations in spherically symmetric space-times with non-degenerate and diagonal Ricci tensor do not cover all possible cases. Furthermore the complete algebra of Ricci Collineations of certain Robertson-Walker metrics of vanishing spatial curvature are given. 
  All hypersurface homogeneous locally rotationally symmetric spacetimes which admit conformal symmetries are determined and the symmetry vectors are given explicitly. It is shown that these spacetimes must be considered in two sets. One set containing Ellis Class II and the other containing Ellis Class I, III LRS spacetimes. The determination of the conformal algebra in the first set is achieved by systematizing and completing results on the determination of CKVs in 2+2 decomposable spacetimes. In the second set new methods are developed. The results are applied to obtain the classification of the conformal algebra of all static LRS spacetimes in terms of geometrical variables. Furthermore all perfect fluid nontilted LRS spacetimes which admit proper conformal symmetries are determined and the physical properties some of them are discussed. 
  A new class of spacetime defect solutions of Einstein Field equations of Edelen's direct Poincar\'{e} Gauge Field theory without biaxial symmetry is presented. The interior solution describes a core of defects where curvature vanishes and Cartan torsion is nonvanishing. Outside the core (in vacuum) the solution represents a spacetime with vanishing curvature and torsion describing a nontrivial topological defect solution of Einstein equations of gravity. Our solution corresponds to a very weak strenght of Tachyons can be found far away from the core defect. 
  We evaluate the energy distributions of the Dymnikova space-time using the Weinberg, Papapetrou, and M{\o}ller energy-momentum complexes. This result sustain the importance of the energy-momentum complexes in the evaluation of the energy distribution of a given space-time. To compare the energy distributions obtained by using several definitions, these results show that the Einstein, Tolman, and Weinberg energy complexes are the same in Schwarzschild Cartesian coordinates, but the Papapetrou and the M{\o}ller are not. 
  Gibbs measure is one of the central objects of the modern probability, mathematical statistical physics and euclidean quantum field theory. Here we define and study its natural generalization for the case when the space, where the random field is defined is itself random. Moreover, this randomness is not given apriori and independently of the configuration, but rather they depend on each other, and both are given by Gibbs procedure; We call the resulting object a Gibbs family because it parametrizes Gibbs fields on different graphs in the support of the distribution. We study also quantum (KMS) analog of Gibbs families. Various applications to discrete quantum gravity are given. 
  Following the methods developed by Corley and Jacobson, we consider qualitatively the issue of Hawking radiation in the case when the dispersion relation is dictated by quantum kappa-Poincare algebra. This relation corresponds to field equations that are non-local in time, and, depending on the sign of the parameter kappa, to sub- or superluminal signal propagation. We also derive the conserved inner product, that can be used to count modes, and therefore to obtain the spectrum of black hole radiation in this case. 
  We consider compact binary systems, modeled in general relativity as vacuum or perfect-fluid spacetimes with a helical Killing vector k^\alpha, heuristically, the generator of time-translations in a corotating frame. Systems that are stationary in this sense are not asymptotically flat, but have asymptotic behavior corresponding to equal amounts of ingoing and outgoing radiation. For black-hole binaries, a rigidity theorem implies that the Killing vector lies along the horizon's generators, and from this one can deduce the zeroth law (constant surface gravity of the horizon). Remarkably, although the mass and angular momentum of such a system are not defined, there is an exact first law, relating the change in the asymptotic Noether charge to the changes in the vorticity, baryon mass, and entropy of the fluid, and in the area of black holes.   Binary systems with M\Omega small have an approximate asymptopia in which one can write the first law in terms of the asymptotic mass and angular momentum. Asymptotic flatness is precise in two classes of solutions used to model binary systems: spacetimes satisfying the post-Newtonian equations, and solutions to a modified set of field equations that have a spatially conformally flat metric. (The spatial conformal flatness formalism with helical symmetry, however, is consistent with maximal slicing only if replaces the extrinsic curvature in the field equations by an artificially tracefree expression in terms of the shift vector.) For these spacetimes, nearby equilibria whose stars have the same vorticity obey the relation \delta M = \Omega \delta J, from which one can obtain a turning point criterion that governs the stability of orbits. 
  We study classical and quantum self-similar collapses of a massless scalar field in higher dimensions, and examine how the increase in the number of dimensions affects gravitational collapse and black hole formation. Higher dimensions seem to favor formation of black hole rather than other final states, in that the initial data space for black hole formation enlarges as dimension increases. On the other hand, the quantum gravity effect on the collapse lessens as dimension increases. We also discuss the gravitational collapse in a brane world with large but compact extra dimensions. 
  We propose an experimental model using the Laval nozzle of a sonic analogue of the Hawking radiation. We observe a power spectrum of the out-going wave emitted from the vicinity a sonic horizon in place of a created particle number. Our treatment is based on classical theory, and it will make experiments to be easier. This experimental possibility is a great advantage of our model. 
  We show that 5-dimensional Kaluza-Klein graviton stresses can slow the decay of shear anisotropy on the brane to observable levels, and we use cosmic microwave background anisotropies to place limits on the initial anisotropy induced by these stresses. An initial shear to Hubble distortion of only \sim 10^{-3}\Omega_0h_0^2 at the 5D Planck time would allow the observed large-angle CMB signal to be a relic mainly of KK tidal effects. 
  The renormalization group method is applied to the study of homogeneous and flat Friedmann-Robertson-Walker type Universes, filled with a causal bulk viscous cosmological fluid. The starting point of the study is the consideration of the scaling properties of the gravitational field equations, of the causal evolution equation of the bulk viscous pressure and of the equations of state. The requirement of scale invariance imposes strong constraints on the temporal evolution of the bulk viscosity coefficient, temperature and relaxation time, thus leading to the possibility of obtaining the bulk viscosity coefficient-energy density dependence. For a cosmological model with bulk viscosity coefficient proportional to the Hubble parameter, we perform the analysis of the renormalization group flow around the scale invariant fixed point, therefore obtaining the long time behavior of the scale factor. 
  We use null hypersurface techniques in a new approach to calculate the retarded waveform from a binary black hole merger in the close approximation. The process of removing ingoing radiation from the system leads to two notable features in the shape of the close approximation waveform for a head-on collision of black holes: (i) an initial quasinormal ringup and (ii) weak sensitivity to the parameter controlling the collision velocity. Feature (ii) is unexpected and has the potential importance of enabling the design of an efficient template for extracting the gravitational wave signal from the noise. 
  A complete formalism for constructing initial data representing black-hole binaries in quasi-equilibrium is developed. Radiation reaction prohibits, in general, true equilibrium binary configurations. However, when the timescale for orbital decay is much longer than the orbital period, a binary can be considered to be in quasi-equilibrium. If each black hole is assumed to be in quasi-equilibrium, then a complete set of boundary conditions for all initial data variables can be developed. These boundary conditions are applied on the apparent horizon of each black hole, and in fact force a specified surface to be an apparent horizon. A global assumption of quasi-equilibrium is also used to fix some of the freely specifiable pieces of the initial data and to uniquely fix the asymptotic boundary conditions. This formalism should allow for the construction of completely general quasi-equilibrium black hole binary initial data. 
  The parametrized system called ``ideal clock'' is turned into an ordinary gauge system and quantized by means of a path integral in which canonical gauges are admissible. Then the possibility of applying the results to obtain the transition amplitude for empty minisuperspaces, and the restrictions arising from the topology of the constraint surface, are studied by matching the models with the ideal clock. A generalization to minisuperspaces with true degrees of freedom is also discussed. 
  The action functional of the anisotropic Kantowski--Sachs cosmological model is turned into that of an ordinary gauge system. Then a global phase time is identified for the model by imposing canonical gauge conditions, and the quantum transition amplitude is obtained by means of the usual path integral procedure of Fadeev and Popov. 
  Gauge invariance of systems whose Hamilton-Jacobi equation is separable is improved by adding surface terms to the action fuctional. The general form of these terms is given for some complete solutions of the Hamilton-Jacobi equation. The procedure is applied to the relativistic particle and toy universes, which are quantized by imposing canonical gauge conditions in the path integral; in the case of empty models, we first quantize the parametrized system called ``ideal clock'', and then we examine the possibility of obtaining the amplitude for the minisuperspaces by matching them with the ideal clock. The relation existing between the geometrical properties of the constraint surface and the variables identifying the quantum states in the path integral is discussed. 
  Simple cosmological models are used to show that gravitation can be quantized as an ordinary gauge system if the Hamilton-Jacobi equation for the model under consideration is separable. In this situation, a canonical transformation can be performed such that in terms of the new variables the model has a linear and homogeneous constraint, and therefore canonical gauges are admissible in the path integral. This has the additional practical advantage that gauge conditions that do not generate Gribov copies are then easy to choose. 
  Homogeneous and isotropic cosmological models whose Hamilton-Jacobi equation is separable are deparametrized by turning their action functional into that of an ordinary gauge system. Canonical gauge conditions imposed on the gauge system are used to define a global phase time in terms of the canonical coordinates and momenta of the minisuperspaces. The procedure clearly shows how the geometry of the constraint surface restricts the choice of time; the consequences that this has on the path integral quantization are discussed. 
  With the question, ``Can relativistic charged spheres form extremal black holes?" in mind, we investigate the properties of such spheres from a classical point of view. The investigation is carried out numerically by integrating the Oppenheimer-Volkov equation for relativistic charged fluid spheres and finding interior Reissner-Nordstr\"om solutions for these objects. We consider both constant density and adiabatic equations of state, as well as several possible charge distributions, and examine stability by both a normal mode and an energy analysis. In all cases, the stability limit for these spheres lies between the extremal ($Q = M$) limit and the black hole limit ($R = R_+$). That is, we find that charged spheres undergo gravitational collapse before they reach $Q = M$, suggesting that extremal Reissner-Nordtr\"om black holes produced by collapse are ruled out. A general proof of this statement would support a strong form of the cosmic censorship hypothesis, excluding not only stable naked singularities, but stable extremal black holes. The numerical results also indicate that although the interior mass-energy $m(R)$ obeys the usual $m/R < 4/9$ stability limit for the Schwarzschild interior solution, the gravitational mass $M$ does not. Indeed, the stability limit approaches $R_+$ as $Q \to M$. In the Appendix we also argue that Hawking radiation will not lead to an extremal Reissner-Nordstr\"om black hole. All our results are consistent with the third law of black hole dynamics, as currently understood. 
  In order to account for the observable Universe, any comprehensive theory or model of cosmology must draw from many disciplines of physics, including gauge theories of strong and weak interactions, the hydrodynamics and microphysics of baryonic matter, electromagnetic fields, and spacetime curvature, for example. Although it is difficult to incorporate all these physical elements into a single complete model of our Universe, advances in computing methods and technologies have contributed significantly towards our understanding of cosmological models, the Universe, and astrophysical processes within them. A sample of numerical calculations (and numerical methods) applied to specific issues in cosmology are reviewed in this article: from the Big Bang singularity dynamics to the fundamental interactions of gravitational waves; from the quark-hadron phase transition to the large scale structure of the Universe. The emphasis, although not exclusively, is on those calculations designed to test different models of cosmology against the observed Universe. 
  We review the enhancon mechanism proposed by Johnson, Peet and Polchinski. If we consider the D6-brane wrapped on K3, then there appear a naked singularity called ``repulson'' in the supergravity solution. But this singularity is resolved by a shell structure called ``enhancon''. In the interior of the enhancon, the abelian gauge symmetry is enhanced to a nonabelian one, and ordinary supergravity is no more reliable. We also review the interpretation of enhancon as fuzzy sphere. This paper is the contribution to the proceedings of "Frontier of Cosmology and Gravitation", April 25-27 2001, YITP. 
  Laboratory-based optical analogs of astronomical objects such as black holes rely on the creation of light with an extremely low or even vanishing group velocity (slow light). These brief notes represent a pedagogical attempt towards elucidating this extraordinary form of light. This paper is a contribution to the book Artificial Black Holes edited by Mario Novello, Matt Visser and Grigori Volovik. The paper is intended as a primer, an introduction to the subject for non-experts, not as a detailed literature review. 
  We consider the two-body problem in post-Newtonian approximations of general relativity. We report the recent results concerning the equations of motion, and the associated Lagrangian formulation, of compact binary systems, at the third post-Newtonian order (1/c^6 beyond the Newtonian acceleration). These equations are necessary when constructing the theoretical templates for searching and analyzing the gravitational-wave signals from inspiralling compact binaries in VIRGO-type experiments. 
  For the minimally coupled scalar field in Einstein's theory of gravitation we look for the space of solutions within the class of closed Friedmann universe models. We prove that D = 1 or D > 1, where D is the (fractal) dimension of the set of solutions which can be integrated up to t to infinity. (D > 0 was conjectured by PAGE (1984)). We discuss concepts like ``the probability of the appearance of a sufficiently long inflationary phase" and argue that it is primarily a probability measure q in the space V of solutions (and not in the space of initial conditions) which has to be applied. q is naturally defined for Bianchi-type I cosmological models because V is a compact cube. The problems with the closed Friedmann model (which led to controversial claims in the literature) will be shown to originate from the fact that V has a complicated non-compact non-Hausdorff Geroch topology: no natural definition of q can be given. We conclude: the present state of our universe can be explained by models of the type discussed, but thereby the anthropic principle cannot be fully circumvented. 
  We show evidence for a relationship between chaos and parametric resonance both in a classical system and in the semiclassical process of particle creation. We apply our considerations in a toy model for preheating after inflation. 
  The space of all Riemannian metrics is infinite-dimensional. Nevertheless a great deal of usual Riemannian geometry can be carried over. The superspace of all Riemannian metrics shall be endowed with a class of Riemannian metrics; their curvature and invariance properties are discussed. Just one of this class has the property to bring the lagrangian of General Relativity into the form of a classical particle's motion. The signature of the superspace metric depends in a non-trivial manner on the signature of the original metric, we derive the corresponding formula. Our approach is a local one: the essence is a metric in the space of all symmetric rank-two tensors, and then the space becomes a warped product of the real line with an Einstein space. 
  We construct a one-parameter family of exact time-dependent solutions to 2+1 gravity with a negative cosmological constant and a massless minimally coupled scalar field as source. These solutions present a continuously self-similar (CSS) behaviour near the central singularity, as observed in critical collapse, and an asymptotically AdS behaviour at spatial infinity. We consider the linear perturbation analysis in this background, and discuss the crucial question of boundary conditions. These are tested in the special case where the scalar field decouples and the linear perturbations describe exactly the small-mass static BTZ black hole. In the case of genuine scalar perturbations, we find a growing mode with a behavior characteristic of supercritical collapse, the spacelike singularity and apparent horizon appearing simultaneously and evolving towards the AdS boundary. Our boundary conditions lead to the value of the critical exponent $\gamma = 0.4$. 
  It is shown that photon shot noise and radiation-pressure back-action noise are the sole forms of quantum noise in interferometric gravitational wave detectors that operate near or below the standard quantum limit, if one filters the interferometer output appropriately. No additional noise arises from the test masses' initial quantum state or from reduction of the test-mass state due to measurement of the interferometer output or from the uncertainty principle associated with the test-mass state. Two features of interferometers are central to these conclusions: (i) The interferometer output (the photon number flux N(t) entering the final photodetector) commutes with itself at different times in the Heisenberg Picture, [N(t), N(t')] = 0, and thus can be regarded as classical. (ii) This number flux is linear in the test-mass initial position and momentum operators x_o and p_o, and those operators influence the measured photon flux N(t) in manners that can easily be removed by filtering -- e.g., in most interferometers, by discarding data near the test masses' 1 Hz swinging freqency. The test-mass operators x_o and p_o contained in the unfiltered output N(t) make a nonzero contribution to the commutator [N(t), N(t')]. That contribution is cancelled by a nonzero commutation of the photon shot noise and radiation-pressure noise, which also are contained in N(t). This cancellation of commutators is responsible for the fact that it is possible to derive an interferometer's standard quantum limit from test-mass considerations, and independently from photon-noise considerations. These conclusions are true for a far wider class of measurements than just gravitational-wave interferometers. To elucidate them, this paper presents a series of idealized thought experiments that are free from the complexities of real measuring systems. 
  We consider the spatially flat Friedmann model. For a(t) = t^p, especially, if p is larger or equal to 1, this is called power-law inflation. For the Lagrangian L = R^m with p = - (m - 1)(2m - 1)/(m - 2), power-law inflation is an exact solution, as it is for Einstein gravity with a minimally coupled scalar field Phi in an exponential potential V(Phi) = exp(mu Phi) and also for the higher-dimensional Einstein equation with a special Kaluza-Klein ansatz. The synchronized coordinates are not adapted to allow a closed-form solution, so we use another gauge. Finally, special solutions for the closed and open Friedmann model are found. 
  We consider the Newtonian limit of the theory based on the Lagrangian L = R + \sum a_k R \Box^k R. The gravitational potential of a point mass turns out to be a combination of Newtonian and Yukawa terms. For sixth-order gravity the coefficients are calculated explicitly. For the general case one gets as a result: The the potential is always unbounded near the origin. 
  Some mathematical errors of the paper commented upon [W.-M. Suen, Phys. Rev. D 40, (1989) 315] are corrected. 
  We present an example that non-isometric space-times with non-vanishing curvature scalar cannot be distinguished by curvature invariants. 
  We answer the following question: Let l, m, n be arbitrary real numbers. Does there exist a 3-dimensional homogeneous Riemannian manifold whose eigenvalues of the Ricci tensor are just l, m and n ? 
  In the quest of the critical solution for scalar field collapse in 2+1 gravity with a negative cosmological constant, we present a one parameter family of solutions with continuous self similar (CSS) behaviour near the central singularity. We also discuss linear perturbations on this background, leading to black hole formation, and determine the critical exponent. 
  Interior perfect fluid solutions for the Reissner-Nordstrom metric are studied on the basis of a new classification scheme. It specifies which two of the fluid's characteristics are given functions and picks up accordingly one of the three main field equations, the other two being universal. General formulae are found for charged de Sitter solutions, the case of constant energy component of the energy-momentum tensor, the case of known pressure (including charged dust) and the case of linear equation of state. Explicit new global solutions, mainly in elementary functions, are given as illustrations. Known solutions are briefly reviewed and corrected. 
  The Gowdy spacetimes are vacuum solutions of Einstein's equations with two commuting Killing vectors having compact spacelike orbits with T^3, S^2xS^1 or S^3 topology. In the case of T^3 topology, Kichenassamy and Rendall have found a family of singular solutions which are asymptotically velocity dominated by construction. In the case when the velocity is between zero and one, the solutions depend on the maximal number of free functions. We consider the similar case with S^2xS^1 or S^3 topology, where the main complication is the presence of symmetry axes. We use Fuchsian techniques to show the existence of singular solutions similar to the T^3 case. We first solve the analytic case and then generalise to the smooth case by approximating smooth data with a sequence of analytic data. However, for the metric to be smooth at the axes, the velocity must be 1 or 3 there, which is outside the range where the constructed solutions depend on the full number of free functions. A plausible explanation is that in general a spiky feature may develop at the axis, a situation which is unsuitable for a direct treatment by Fuchsian methods. 
  We study the {\em propagation of electromagnetic waves} in a spacetime devoid of a metric but equipped with a {\em linear} electromagnetic spacetime relation $H\sim\chi\cdot F$. Here $H$ is the electromagnetic excitation $({\cal D},{\cal H})$ and $F$ the field strength $(E,B)$, whereas $\chi$ (36 independent components) characterizes the electromagnetic permittivity/permeability of spacetime. We derive analytically the corresponding Fresnel equation and show that it is always quartic in the wave covectors. We study the `Fresnel tensor density' ${\cal G}^{ijkl}$ as (cubic) function of $\chi$ and identify the leading part of $\chi$ (20 components) as indispensable for light propagation. Upon requiring electric/magnetic reciprocity of the spacetime relation, the leading part of $\chi$ induces the {\em light cone} structure of spacetime (9 components), i.e., the spacetime metric up to a function. The possible existence of an Abelian {\em axion} field (1 component of $\chi$) and/or of a {\em skewon} field (15 components) and their effect on light propagation is discussed in some detail. The newly introduced skewon field is expected to be T-odd and related to dissipation. 
  We investigate the covariant Hamiltonian symplectic structure of General Relativity for spatially bounded regions of spacetime with a fixed time-flow vector. For existence of a well-defined Hamiltonian variational principle taking into account a spatial boundary, it is necessary to modify the standard Arnowitt-Deser-Misner Hamiltonian by adding a boundary term whose form depends on the spatial boundary conditions for the gravitational field. The most general mathematically allowed boundary conditions and corresponding boundary terms are shown to be determined by solving a certain equation obtained from the symplectic current pulled back to the hypersurface boundary of the spacetime region. A main result is that we obtain a covariant derivation of Dirichlet, Neumann, and mixed type boundary conditions on the gravitational field at a fixed boundary hypersurface, together with the associated Hamiltonian boundary terms. As well, we establish uniqueness of these boundary conditions under certain assumptions motivated by the form of the symplectic current. Our analysis uses a Noether charge method which extends and unifies several results developed in recent literature for General Relativity. As an illustration of the method, we apply it to the Maxwell field equations to derive allowed boundary conditions and boundary terms for existence of a well-defined Hamiltonian variational principle for an electromagnetic field in a fixed spatially bounded region of Minkowski spacetime. 
  We continue a previous analysis of the covariant Hamiltonian symplectic structure of General Relativity for spatially bounded regions of spacetime. To allow for near complete generality, the Hamiltonian is formulated using any fixed hypersurface, with a boundary given by a closed spacelike 2-surface. A main result is that we obtain Hamiltonians associated to Dirichlet and Neumann boundary conditions on the gravitational field coupled to matter sources, in particular a Klein-Gordon field, an electromagnetic field, and a set of Yang-Mills-Higgs fields. The Hamiltonians are given by a covariant form of the Arnowitt-Deser-Misner Hamiltonian modified by a surface integral term that depends on the particular boundary conditions. The general form of this surface integral involves an underlying ``energy-momentum'' vector in the spacetime tangent space at the spatial boundary 2-surface. We give examples of the resulting Dirichlet and Neumann vectors for topologically spherical 2-surfaces in Minkowski spacetime, spherically symmetric spacetimes, and stationary axisymmetric spacetimes. Moreover, we show the relation between these vectors and the ADM energy-momentum vector for a 2-surface taken in a limit to be spatial infinity in asymptotically flat spacetimes. We also discuss the geometrical properties of the Dirichlet and Neumann vectors and obtain several striking results relating these vectors to the mean curvature and normal curvature connection of the 2-surface. Most significantly, the part of the Dirichlet vector normal to the 2-surface depends only the spacetime metric at this surface and thereby defines a geometrical normal vector field on the 2-surface. Properties and examples of this normal vector are discussed. 
  Numerical solutions of the Einstein-Yang-Mills equations with a negative cosmological constant are constructed. These axially symmetric solutions approach asymptotically the anti-de Sitter spacetime and are regular everywhere. They are characterized by the winding number $n>1$, the mass and the non-Abelian magnetic charge. The main properties of the solutions and the differences with respect to the asymptotically flat case are discussed. The existence of axially symmetric monopole and dyon solutions in fixed anti-de Sitter spacetime is also discussed. 
  The hoop conjecture is well confirmed in momentarily static spaces, but it has not been investigated systematically for the system with relativistic motion. To confirm the hoop conjecture for non-time-symmetric initial data, we consider the initial data of two colliding black holes with momentum and search an apparent horizon that encloses two black holes. In testing the hoop conjecture, we use two definitions of gravitational mass : one is the ADM mass and the other is the quasi-local mass defined by Hawking. Although both definitions of gravitational mass give fairly consistent picture of the hoop conjecture, the hoop conjecture with the Hawking mass can judge the existence of an apparent horizon for wider range of parameters of the initial data compared to the ADM mass. 
  The field equations of a special class of tetrad theory of gravitation have been applied to tetrad space having three unknown functions of radial coordinate. The spherically symmetric vacuum stress-energy momentum tensor with one assumption concerning its specific form generates two non-trivial different exact analytic solutions for these field equations. For large $r$, the exact analytic solutions coincide with the Schwarzschild solution, while for small $r$, they behave in a manner similar to the de Sitter solution and describe a spherically symmetric black hole singularity free everywhere. The solutions obtained give rise to two different tetrad structures, but having the same metric, i.e., a static spherically symmetric nonsingular black hole metric. We then, calculated the energy associated with these two exact analytic solutions using the superpotential method. We find that unless the time-space components of the tetrad go to zero faster than $ \displaystyle{1 \over \sqrt{r}}$ at infinity, the two solutions give different results. This fact implies that the time-space components of the tetrad must vanish faster than $\displaystyle{1 \over \sqrt{r}}$ at infinity. 
  We study the collision of two slowly rotating, initially non boosted, black holes in the close limit. A ``punctures'' modification of the Bowen - York method is used to construct conformally flat initial data appropriate to the problem. We keep only the lowest nontrivial orders capable of giving rise to radiation of both gravitational energy and angular momentum. We show that even with these simplifications an extension to higher orders of the linear Regge-Wheeler-Zerilli black hole perturbation theory, is required to deal with the evolution equations of the leading contributing multipoles. This extension is derived, together with appropriate extensions of the Regge-Wheeler and Zerilli equations. The data is numerically evolved using these equations, to obtain the asymptotic gravitational wave forms and amplitudes. Expressions for the radiated gravitational energy and angular momentum are derived and used together with the results of the numerical evolution to provide quantitative expressions for the relative contribution of different terms, and their significance is analyzed. 
  I study the dynamical effects due to the Brans-Dicke scalar $\phi$-field at the early stages of a supposedly anisotropic Universe expansion in the scalar-tensor cosmology of Jordan-Brans-Dicke. This is done considering the behaviour of the general solutions for the homogeneous model of Bianchi type VII in the vacuum case. I conclude that the Bianchi-VII$_0$ model shows an isotropic expansion and that its only physical solution is equivalent to a Friedman-Robertson-Walker spacetime whose evolution can, depending on the value of the JBD coupling constant, begin in a singularity and, after expanding (inflating, if $\omega>0$), shrink to another, or starting in a non-singular state, collapse to a singularity. I also conclude that the general Bianchi-VII$_h$ (with $h\neq 0$) models show strong curvature singularities producing a complete collapse of the homogeinity surfaces to 2D-manifolds, to 1D-manifolds or to single points. Our analysis depends crucially on the introduction of the so-called intrinsic time, $\Phi$, as the product of the JBD scalar field $\phi$ times a mean scale factor $a^3=a_1a_2a_3$, in which the finite-cosmological-time evolution of this universe unfolds into an infinite $\Phi$-range. These universes isotropize from an anisotropic initial state, thence I conclude that they are stable against anisotropic perturbations. 
  Considering the definition of inertial forces acting on a test particle, following non-circular geodesics, in static and stationary space times we show that the centrifugal force reversal occurs only in the case of particles following prograde orbits around black holes. We first rewrite the covariant expressions for the acceleration components in terms of the lapse function, shift vector and the 3-metric $\gm_{ij}$, using the ADM 3+1 splitting and use these, for different cases as given by pure radial motion, pure azimuthal motion and the general non-circular motion. It is found that the reversal occurs only when the azimuthal angular velocity of the particle supersedes the radial velocity, which indeed depends upon the physical parameters $E$, $l$ and the Kerr parameter $a$. 
  It has been claimed that the Lemaitre-Tolman-Bondi-de Sitter solution always admits future-pointing radial time-like geodesics emerging from the shell-focussing singularity, regardless of the nature of the (regular) initial data. This is despite the fact that some data rule out the emergence of future pointing radial null geodesics. We correct this claim and show that in general in spherical symmetry, the absence of radial null geodesics emerging from a central singularity is sufficient to prove that the singularity is censored. 
  In this work we study the trajectories of test particles in a geometry that is the nonlinear electromagnetic generalization of the Reissner-Nordstrom solution. The studied spacetime is a Einstein-Born-Infeld solution, nonsingular outside a regular event horizon and characterized by three parameters: mass $M$, charge $Q$ and the Born-Infeld parameter $b$ related to the magnitude of the electric field at the origin. Asymptotically it is a Reissner-Nordstrom solution 
  If the spatial curvature of the universe is positive, then the curvature term will always dominate at early enough times in a slow-rolling inflationary epoch. This enhances inflationary effects and hence puts limits on the possible number of e-foldings that can have occurred, independently of what happened before inflation began and in particular without regard for what may have happened in the Planck era. We use a simple multi-stage model to examine this limit as a function of the present density parameter $\Omega_0$ and the epoch when inflation ends. 
  We show that in the case of positively-curved Friedmann-Lema\^{\i}tre universes $(k=+1)$, an inflationary period in the early universe will for most initial conditions not solve the horizon problem, no matter how long inflation lasts. It will only do so for cases where inflation starts in an almost static state, corresponding to an extremely high value of $\Omega_{\Lambda}$, $\Omega_{\Lambda} \gg 1$, at the beginning of inflation. For smaller values, it is not possible to solve the horizon problem because the relevant integral asymptotes to a finite value (as happens also in the de Sitter universe in a $k=+1$ frame). Thus, for these cases, the causal problems associated with the near-isotropy of the Cosmic Background Radiation have to be solved already in the Planck era. Furthermore both compact space sections and event horizons will exist in these universes even if the present cosmological constant dies away in the far future, raising potential problems for M-theory as a theory of gravity. 
  We study the occurrence, visibility, and curvature strength of singularities in dust-containing Szekeres spacetimes (which possess no Killing vectors) with a positive cosmological constant. We find that such singularities can be locally naked, Tipler strong, and develop from a non-zero-measure set of regular initial data. When examined along timelike geodesics, the singularity's curvature strength is found to be independent of the initial data. 
  Gravitational waves from binary neutron stars in quasiequilibrium circular orbits are computed using an approximate method which we propose in this paper. In the first step of this method, we prepare general relativistic irrotational binary neutron stars in a quasiequilibrium circular orbit, neglecting gravitational waves. We adopt the so-called conformal flatness approximation for a three-metric to obtain the quasiequilibrium states in this paper. In the second step, we compute gravitational waves, solving linear perturbation equations in the background spacetime of the quasiequilibrium states. Comparing numerical results with post Newtonian waveforms and luminosity of gravitational waves from two point masses in circular orbits, we demonstrate that this method can produce accurate waveforms and luminosity of gravitational waves. It is shown that the effects of tidal deformation of neutron stars and strong general relativistic gravity modify the post Newtonian results for compact binary neutron stars in close orbits. We indicate that the magnitude of a systematic error in quasiequilibrium states associated with the conformal flatness approximation is fairly large for close and compact binary neutron stars. Several formulations for improving the accuracy of quasiequilibrium states are proposed. 
  We prove some theorems characterizing the global properties of static, spherically symmetric configurations of a self-gravitating real scalar field in general relativity (GR) in various dimensions, with an arbitrary potential $V$, not necessarily positive-definite. The results are extended to sigma models, scalar-tensor and curvature-nonlinear theories of gravity. We show that the list of all possible types of space-time causal structure in the models under study is the same as for a constant scalar field, namely, Minkowski (or AdS), Schwarzschild, de Sitter and Schwarzschild - de Sitter, and all horizons are simple. In particular, these theories do not admit regular black holes with any asymptotics. Some special features of (2+1)D gravity are revealed. We give examples of two types of asymptotically flat configurations with positive mass in GR, admitted by the above theorems: (i) a black hole with nontrivial ``scalar hair'' and (ii) a particlelike solution with a regular centre; in both cases, the potential $V$ must be at least partly negative. We also discuss the global effects of conformal mappings that connect different theories. Such effects are illustrated for solutions with a conformal scalar field in GR. 
  The theory of evolution equations has been applied in various ways in general relativity. Following some general considerations about this, some illustrative examples of the use of ordinary differential equations in general relativity are presented. After this recent applications of Fuchsian equations are described, with particular attention to work on the structure of singularities of solutions of the Einstein equations coupled to a massless scalar field. Next the relations between analytical and numerical studies of the Einstein equations are discussed. Finally an attempt is made to identify fruitful directions for future research within the analytic approach to the study of the Einstein equations. 
  We define the time travel paradox in physical terms and prove its existence by constructing an explicit example. We argue further that in theories -- such as general relativity -- where the spacetime geometry is subject to nothing but differential equations and initial data no paradoxes arise. 
  We introduce a pregeometry that provides a metric and dimensionality over a Borel set (Wheeler's "bucket of dust") without assuming probability amplitudes for adjacency. Rather, a non-trivial metric is produced over a Borel set X per a uniformity base generated via the discrete topological group structures over X. We show that entourage multiplication in this uniformity base mirrors the underlying group structure. One may exploit this fact to create an entourage sequence of maximal length whence a fine metric structure. Unlike the statistical approaches of graph theory, this method can suggest dimensionality over low-order sets. An example over Z2 x Z4 produces 3-dimensional polyhedra embedded in E4. 
  Global harmonic coordinates for the complete Schwarzschild metric are found for a more general case than that addressed in a previous work by Quan-Hui Liu. The supplementary constant that appears, in addition to the mass, is related to the stress quadrupole moment of a singular energy--momentum tensor. Similar calculations are also carried out in q--harmonic coordinates. 
  We evolve a scalar field in a fixed Kerr-Schild background geometry to test simple $(3+1)$-dimensional algorithms for singularity excision. We compare both centered and upwind schemes for handling the shift (advection) terms, as well as different approaches for implementing the excision boundary conditions, for both static and boosted black holes. By first determining the scalar field evolution in a static frame with a $(1+1)$-dimensional code, we obtain the solution to very high precision. This solution then provides a useful testbed for simulations in full $(3+1)$ dimensions. We show that some algorithms which are stable for non-boosted black holes become unstable when the boost velocity becomes high. 
  Starting from a spherically symmetric tetrad with three unknown functions of the radial coordinate, a general solution of M{\o}ller's field equations in case of spherical symmetry nonsingular black hole is derived. The previously obtained solutions are verified as special cases of the general solution. The general solution is characterized by an arbitrary function and two constants of integration. The general solution gives no more than the spherically symmetric nonsingular black hole solution. The energy content of the general solution depends on the asymptotic behavior of the arbitrary function, and is different from the standard one. 
  Contents:   We hear that... by Jorge Pullin  Reflections of a decade:   Matters of Gravity and the Topical Group in Gravitation, by Beverly Berger   10 Years in Gravitational Wave Detection, by Peter Saulson   Ten years of general relativity, some reflections, by Carlo Rovelli   Tabletop gravity experiments, by Jens Gundlach   String Theory: the past ten years, by Gary Horowitz  Conference reports:   Fourth Capra Meeting on Radiation Reaction, by Lior Burko   Workshop on Numerical Relativity, by Michael Koppitz   Workshop on Canonical & Quantum Gravity III by J. Lewandowski   and J. Wisniewski 
  A new solution for the endpoint of gravitational collapse is proposed. By extending the concept of Bose-Einstein condensation to gravitational systems, a cold, compact object with an interior de Sitter condensate phase and an exterior Schwarzschild geometry of arbitrary total mass M is constructed. These are separated by a phase boundary with a small but finite thickness of fluid with eq. of state p=+\rho, replacing both the Schwarzschild and de Sitter classical horizons. The new solution has no singularities, no event horizons, and a global time. Its entropy is maximized under small fluctuations and is given by the standard hydrodynamic entropy of the thin shell, instead of the Bekenstein-Hawking entropy. Unlike black holes, a collapsed star of this kind is thermodynamically stable and has no information paradox. 
  Lie transformation groups containing a one-dimensional subgroup acting cyclically on a manifold are considered. The structure of the group is found to be considerably restricted by the existence of a one-dimensional subgroup whose orbits are circles. The results proved do not depend on the dimension of the manifold nor on the existence of a metric, but merely on the fact that the Lie group acts globally on the manifold. Firstly some results for the general case of an $m+1$-dimensional Lie group are derived: those commutators of the associated Lie algebra involving the generator of the cyclic subgroup, $X_0$ say, are severely restricted and, in a suitably chosen basis, take a simple form. The Jacobi identities involving $X_0$ are then applied to show there are further restrictions on the structure of the Lie algebra. All Lie algebras of dimensions 2 and 3 compatible with cyclic symmetry are obtained. In the two-dimensional case the group must be Abelian. For the three-dimensional case, the Bianchi type of the Lie algebra must be I, II, III, VII$_0$, VIII or IX and furthermore in all cases the vector $X_0$ forms part of a basis in which the algebra takes its canonical form. Finally four-dimensional Lie algebras compatible with cyclic symmetry are considered and the results are related to the Petrov-Kruchkovich classification of all four-dimensional Lie algebras. 
  The conformal Killing equations for the most general (non-plane wave) conformally flat pure radiation field are solved to find the conformal Killing vectors. As expected fifteen independent conformal Killing vectors exist, but in general the metric admits no Killing or homothetic vectors. However for certain special cases a one-dimensional group of homotheties or motions may exist and in one very special case, overlooked by previous investigators, a two-dimensional homethety group exists. No higher dimensional groups of motions or homotheties are admitted by these metrics. 
  We consider a cosmological setting for which the currently expanding era is preceded by a contracting phase, that is, we assume the Universe experienced at least one bounce. We show that scalar hydrodynamic perturbations lead to a singular behavior of the Bardeen potential and/or its derivatives (i.e. the curvature) for whatever Universe model for which the last bounce epoch can be smoothly and causally joined to the radiation dominated era. Such a Universe would be filled with non-linear perturbations long before nucleosynthesis, and would thus be incompatible with observations. We therefore conclude that no observable bounce could possibly have taken place in the early universe if Einstein gravity together with hydrodynamical fluids is to describe its evolution, and hence, under these conditions, that the Universe has always expanded. 
  We present an anisotropic cosmological model based on a new exact solution of Einstein equations. The matter content consists of an anisotropic scalar field minimally coupled to gravity and of two isotropic perfect fluids that represent dust matter and radiation. The spacetime is described by a spatially homogeneous, Bianchi type III metric with a conformal expansion. The model respects the evolution of the scale factor predicted by standard cosmology, as well as the isotropy of the cosmic microwave background. Remarkably, the introduction of the scalar field, apart from explaining the spacetime anisotropy, gives rise to an energy density that is close to the critical density. As a consequence, the model is quasiflat during the entire history of the universe. Using these results, we are also able to construct approximate solutions for shear-free cosmological models with rotation. We finally carry out a quantitative discussion of the validity of such solutions, showing that our approximations are acceptably good if the angular velocity of the universe is within the observational bounds derived from rotation of galaxies. 
  One can increase one-quarter the area of a black hole, A/4, to exceed the total thermodynamic entropy, S, by surrounding the hole with a perfectly reflecting shell and adiabatically squeezing it inward. A/4 can be made to exceed S by a factor of order unity before the shell enters the Planck regime, though practical limitations are much more restrictive. One interpretation is that the black hole entropy resides in its thermal atmosphere, and the shell restricts the atmosphere so that its entropy is less than A/4. 
  We study the colour changes induced by blending in a wormhole-like microlensing scenario with extended sources. The results are compared with those obtained for limb darkening. We assess the possibility of an actual detection of the colour curve using the difference image analysis method. 
  A stability criterion is derived for self-similar solutions with perfect fluids which obey the equation of state $P=k\rho$ in general relativity. A wide class of self-similar solutions turn out to be unstable against the so-called kink mode. The criterion is directly related to the classification of sonic points. The criterion gives a sufficient condition for instability of the solution. For a transonic point in collapse, all primary-direction nodal-point solutions are unstable, while all secondary-direction nodal-point solutions and saddle-point ones are stable against the kink mode. The situation is reversed in expansion. Applications are the following: the expanding flat Friedmann solution for $1/3 \le k < 1$ and the collapsing one for $0< k \le 1/3$ are unstable; the static self-similar solution is unstable; nonanalytic self-similar collapse solutions are unstable; the Larson-Penston (attractor) solution is stable for this mode for $0<k\alt 0.036$, while it is unstable for $0.036\alt k $; the Evans-Coleman (critical) solution is stable for this mode for $0<k\alt 0.89$, while it is unstable for $0.89\alt k$. The last application suggests that the Evans-Coleman solution for $0.89\alt k $ is {\em not critical} because it has at least two unstable modes. 
  Charged perfect fluid with vanishing Lorentz force and massless scalar field is studied in the case of stationary cylindrically symmetric spacetime. The scalar field can depend both on radial and longitudinal coordinates. Solutions are found and classified according to scalar field gradient and magnetic field relationship. Their physical and geometrical properties are examined and discussion of particular cases, directly generalizing G\"{o}del-type spacetimes, is presented. 
  We calculate the bounds which could be placed on scalar-tensor theories of gravity of the Jordan, Fierz, Brans and Dicke type by measurements of gravitational waveforms from neutron stars (NS) spiralling into massive black holes (MBH) using LISA, the proposed space laser interferometric observatory. Such observations may yield significantly more stringent bounds on the Brans-Dicke coupling parameter \omega than are achievable from solar system or binary pulsar measurements. For NS-MBH inspirals, dipole gravitational radiation modifies the inspiral and generates an additional contribution to the phase evolution of the emitted gravitational waveform. Bounds on \omega can therefore be found by using the technique of matched filtering. We compute the Fisher information matrix for a waveform accurate to second post-Newtonian order, including the effect of dipole radiation, filtered using a currently modeled noise curve for LISA, and determine the bounds on \omega for several different NS-MBH canonical systems. For example, observations of a 1.4 solar mass NS inspiralling to a 1000 solar mass MBH with a signal-to-noise ratio of 10 could yield a bound of \omega > 240,000, substantially greater than the current experimental bound of \omega > 3000. 
  We establish a general gluing theorem for constant mean curvature solutions of the vacuum Einstein constraint equations. This allows one to take connected sums of solutions or to glue a handle (wormhole) onto any given solution. Away from this handle region, the initial data sets we produce can be made as close as desired to the original initial data sets. These constructions can be made either when the initial manifold is compact or asymptotically Euclidean or asymptotically hyperbolic, with suitable corresponding conditions on the extrinsic curvature. In the compact setting a mild nondegeneracy condition is required. In the final section of the paper, we list a number ways this construction may be used to produce new types of vacuum spacetimes. 
  The Chern-Simons exact solution of four-dimensional quantum gravity with nonvanishing cosmological constant is presented in metric variable as the partition function of a Chern-Simons theory with nontrivial source. The perturbative expansion is given, and the wave function is computed to the lowest order of approximation for the Cauchy surface which is topologically a 3-sphere. The state is well-defined even at degenerate and vanishing values of the dreibein. Reality conditions for the Ashtekar variables are also taken into account; and remarkable features of the Chern-Simons state and their relevance to cosmology are pointed out. 
  The recently suggested quasi-local spin-angular momentum expressions, based on the Bramson superpotential and on the holomorphic or anti-holomorphic spinor fields, are calculated for large spheres near the future null infinity of asymptotically flat Einsten-Maxwell spacetimes. It is shown that although the expression based on the anti-holomorphic spinors is finite and unambiguously defined only in the center-of-mass frame (i.e. it diverges in general), the corresponding Pauli-Lubanski spin is always finite, free of ambiguities, and is built only from the gravitational data. Thus it defines a gravitational spin expression at the future null infinity. The construction based on the holomorphic spinors diverges in presence of outgoing gravitational radiation. For stationary spacetimes both constructions reduce to the `standard' expression. 
  Hollands and Wald's technique based on *-algebras of Wick products of field operators is strightforwardly generalized to define the stress-energy tensor operator in curved globally hyperbolic spacetimes. In particular, the locality and covariance requirement is generalized to Wick products of differentiated quantum fields. The proposed stress-energy tensor operator is conserved and reduces to the classical form if field operators are replaced by classical fields satisfying the equation of motion. The definition is based on the existence of convenient counterterms given by certain local Wick products of differentiated fields. They are independent from the arbitrary length scale (and any quantum state) and they classically vanish on solutions of field equation. The averaged stress-energy tensor with respect to Hadamard quantum states can be obtained by an improved point-splitting renormalization procedure which makes use of the nonambiguous part of the Hadamard parametrix only that is completely determined by the local geometry and the parameters which appear in the Klein-Gordon operator. The averaged stress-energy tensor also coincides with that found by employing the local $\zeta$-function approach. 
  The close agreement between the predictions of dynamical general relativity for the radiated power of a compact binary system and the observed orbital decay of the binary pulsars PSR B1913+16 and PSR B1534+12 allows us to bound the graviton mass to be less than 7.6 x 10^{-20} eV with 90% confidence. This bound is the first to be obtained from dynamic, as opposed to static-field, relativity. The resulting limit on the graviton mass is within two orders of magnitude of that from solar system measurements, and can be expected to improve with further observations. 
  An approximate model of a spacetime foam is presented. It is supposed that in the spacetime foam each quantum handle is like to an electric dipole and therefore the spacetime foam is similar to a dielectric. If we neglect of linear sizes of the quantum handle then it can be described with an operator containing a Grassman number and either a scalar or a spinor field. For both fields the Lagrangian is presented. For the scalar field it is the dilaton gravity + electrodynamics and the dilaton field is a dielectric permeability. The spherically symmetric solution in this case give us the screening of a bare electric charge surrounded by a polarized spacetime foam and the energy of the electric field becomes finite one. In the case of the spinor field the spherically symmetric solution give us a ball of the polarized spacetime foam filled with the confined electric field. It is shown that the full energy of the electric field in the ball can be very big. 
  We investigate what are the key physical features that cause the development of a naked singularity, rather than a black hole, as the end-state of spherical gravitational collapse. We show that sufficiently strong shearing effects near the singularity delay the formation of the apparent horizon. This exposes the singularity to an external observer, in contrast to a black hole, which is hidden behind an event horizon due to the early formation of an apparent horizon. 
  In this paper we shall address this problem: Is quantum gravity constraints algebra closed and what are the quantum Einstein equations. We shall investigate this problem in the de-Broglie--Bohm quantum theory framework. It is shown that the constraint algebra is weakly closed and the quantum Einstein's equations are derived. 
  We give a mathematical framework to describe the evolution of an open quantum systems subjected to finitely many interactions with classical apparatuses. The systems in question may be composed of distinct, spatially separated subsystems which evolve independently but may also interact. This evolution, driven both by unitary operators and measurements, is coded in a precise mathematical structure in such a way that the crucial properties of causality, covariance and entanglement are faithfully represented. We show how our framework may be expressed using the language of (poly)categories and functors. Remarkably, important physical consequences - such as covariance - follow directly from the functoriality of our axioms.   We establish strong links between the physical picture we propose and linear logic. Specifically we show that the refined logical connectives of linear logic can be used to describe the entanglements of subsystems in a precise way. Furthermore, we show that there is a precise correspondence between the evolution of a given system and deductions in a certain formal logical system based on the rules of linear logic.   This framework generalizes and enriches both causal posets and the histories approach to quantum mechanics. 
  In this short review we discuss the relevance of ongoing research into stellar oscillations and associated instabilities for the detection of gravitational waves and the future field of ``gravitational-wave astronomy''. 
  We investigate the question, "how does time flow?" and show that time may change by inversions as well. We discuss its implications to a simple class of linear systems. Instead of introducing any unphysical behaviour, inversions can lead to a new multi- time scale evolutionary path for the linear system exhibiting late time stochastic fluctuations. We explain how stochastic behaviour is injected into the linear system as a combined effect of an uncertainty in the definition of inversion and the irrationality of the golden mean number. We also give an ansatz for the nonlinear stochastic behaviour of (fractal) time which facilitates us to estimate the late and short time limits of a two-time correlation function relevant for the stochastic fluctuations in linear systems. These fluctuations are shown to enjoy generic 1/f spectrum. The implicit functional definition of the fractal time is shown to satisfy the differential equation dx=dt. We also discuss the relevance of intrinsic time in the present formalism, study of which is motivated by the issue of time in quantum gravity. 
  Within the context of loop quantum gravity there are several operators which measure geometry quantities. This work examines two of these operators, volume and angle, to study quantum geometry at a single spin network vertex - ``an atom of geometry.'' Several aspects of the angle operator are examined in detail including minimum angles, level spacing, and the distribution of angles. The high spin limit of the volume operator is also studied for monochromatic vertices. The results show that demands of the correct scaling relations between area and volume and requirements of the expected behavior of angles in three dimensional flat space require high-valence vertices with total spins of approximately 10^20. 
  The full quantum mechanical collapse of a small relativistic dust shell is studied analytically, asymptotically and numerically starting from the exact finite dimensional classical reduced Hamiltonian recently derived by H\'aj{\'\i}\v{c}ek and Kucha\v{r}. The formulation of the quantum mechanics encounters two problems. The first is the multivalued nature of the Hamiltonian and the second is the construction of an appropriate self adjoint momentum operator in the space of the shell motion which is confined to a half line. The first problem is solved by identifying and neglecting orbits of small action in order to obtain a single valued Hamiltonian. The second problem is solved by introducing an appropriate lapse function. The resulting quantum mechanics is then studied by means of analytical and numerical techniques. We find that the region of total collapse has very small probability. We also find that the solution concentrates around the classical Schwarzschild radius. The present work obtains from first principles a quantum mechanics for the shell and provides numerical solutions, whose behavior is explained by a detailed WKB analysis for a wide class of collapsing shells. 
  The canonical quantization of the most general minisuperspace actions --i.e. with all six scale factor as well as the lapse function and the shift vector present-- describing the vacuum type II, VI and VII geometries, is considered. The reduction to the corresponding physical degrees of freedom is achieved through the usage of the linear constraints as well as the quantum version of the entire set of classical integrals of motion. 
  The problems encountered in trying to quantize the various cosmological models, are brought forward by means of a concrete example. The Automorphism groups are revealed as the key element through which G.C.T.'s can be used for a general treatment of these problems. At the classical level, the time dependent automorphisms lead to significant simplifications of the line element for the generic spatially homogeneous geometry, without loss of generality. At the quantum level, the ''frozen'' automorphisms entail an important reduction of the configuration space --spanned by the 6 components of the scale factor matrix-- on which the Wheeler-DeWitt equation, is to be based. In this spirit the canonical quantization of the most general minisuperspace actions --i.e. with all six scale factor as well as the lapse function and the shift vector present-- describing the vacuum type II, I geometries, is considered. The reduction to the corresponding physical degrees of freedom is achieved through the usage of the linear constraints as well as the quantum version of the entire set of all classical integrals of motion. 
  It was shown, that the presence of the so-called X-matter with an equation of state, which lies between limits of the strong and weak energy conditions, allows the variety of the cosmological scenarios in the relativistic theory of gravity. In spite of the fixed negative sign of the cosmological term in the field equations with massive graviton, it is possible to obtain the solutions with accelerated and complicated loitering expansion of the universe. The numerical estimation of the universe's age agrees with the modern observational data if the upper limit of the graviton's mass is 10^{-71} g 
  This paper initiates a program which seeks to study the allowed spatial distributions of negative energy density in quantum field theory. Here we deal with free fields in Minkowski spacetime. Known restrictions on time integrals of the energy density along geodesics, the averaged weak energy condition and quantum inequalities are reviewed. These restrictions are then used to discuss some possible constraints on the allowable spatial distributions of negative energy. We show how some geometric configurations can either be ruled out or else constrained. We also construct some explicit examples of allowed distributions. Several issues related to the allowable spatial distributions are also discussed. These include spacetime averaged quantum inequalities in two-dimensional spacetime, the failure of generalizations of the averaged weak energy condition to piecewise geodesics, and the issue of when the local energy density is negative in the frame of all observers. 
  We investigate the possibility of generating sizeable dipole radiations in relativistic theories of gravity. Optimal parameters to observe their effects through the orbital period decay of binary star systems are discussed. Constraints on gravitational couplings beyond general relativity are derived. 
  The Equivalence Principle (EP) is not one of the ``universal'' principles of physics (like the Action Principle). It is a heuristic hypothesis which was introduced by Einstein in 1907, and used by him to construct his theory of General Relativity. In modern language, the (Einsteinian) EP consists in assuming that the only long-range field with gravitational-strength couplings to matter is a massless spin-2 field. Modern unification theories, and notably String Theory, suggest the existence of new fields (in particular, scalar fields: ``dilaton'' and ``moduli'') with gravitational-strength couplings. In most cases the couplings of these new fields ``violate'' the EP. If the field is long-ranged, these EP violations lead to many observable consequences (variation of ``constants'', non-universality of free fall, relative drift of atomic clocks,...). The best experimental probe of a possible violation of the EP is to compare the free-fall acceleration of different materials. 
  The structure of phase space is determined for spatially compact and locally homogeneous universe models with fluid. Analysis covers models with all possible space topologies except for those covered by S^3, H^3 or S^2xR which have no moduli freedom. We show that space topology significantly affects the number of dynamical degrees of freedom of the system. In particular, we give a detailed proof of the result that for the systems modeled on the Thurston types H^2xR and SL_2, which have locally the Bianchi type III or VIII symmetry, the number of dynamical degrees of freedom increases without bound when the space topology becomes more and more complicated, which was first pointed out by Koike, Tanimoto and Hosoya in an incomplete form. 
  We present a new derivation of the equations governing the oscillations of slowly rotating relativistic stars. Previous investigations have been mostly carried out in the Regge-Wheeler gauge. However, in this gauge the process of linearizing the Einstein field equations leads to perturbation equations which as such cannot be used to perform numerical time evolutions. It is only through the tedious process of combining and rearranging the perturbation variables in a clever way that the system can be cast into a set of hyperbolic first order equations, which is then well suited for the numerical integration. The equations remain quite lengthy, and we therefore rederive the perturbation equations in a different gauge, which has been first proposed by Battiston et al. (1970). Using the ADM formalism, one is immediately lead to a first order hyperbolic evolution system, which is remarkably simple and can be numerically integrated without many further manipulations. Moreover, the symmetry between the polar and the axial equations becomes directly apparent. 
  Quantization of gravitational field in the neighbourhood of arbitrary nontrivial solution of Einstein equations is considered, the 2nd order of perturbation theory is calculated. The expression for quantum corrections of the field operator and explicit view of Hamiltonian are represented. 
  Quantization of gravitational field in the neighbourhood of exact solution of Einstein equation is considered. The method of Bogoliubov group variables is used. 
  I explore the possibility that the laws of physics might be laws of inference rather than laws of nature. What sort of dynamics can one derive from well-established rules of inference? Specifically, I ask: Given relevant information codified in the initial and the final states, what trajectory is the system expected to follow? The answer follows from a principle of inference, the principle of maximum entropy, and not from a principle of physics. The entropic dynamics derived this way exhibits some remarkable formal similarities with other generally covariant theories such as general relativity. 
  A two-parameter family of spherically symmetric, static Lorentzian wormholes is obtained as the general solution of the equation $\rho=\rho_t=0$, where $\rho = T_{ij} u^iu^j$, $\rho_t = (T_{ij} - {1\over2} T g_{ij}) u^iu^j$, and $u^i u_i =- 1$. This equation characterizes a class of spacetimes which are ``self dual'' (in the sense of electrogravity duality). The class includes the Schwarzschild black hole, a family of naked singularities, and a disjoint family of Lorentzian wormholes, all of which have vanishing scalar curvature (R=0). Properties of these spacetimes are discussed. Using isotropic coordinates we delineate clearly the domains of parameter space for which wormholes, nakedly singular spacetimes and the Schwarzschild black hole can be obtained. A model for the required ``exotic'' stress-energy is discussed, and the notion of traversability for the wormholes is also examined. 
  This paper sets out to explain: 1. Why the speed of light c is a constant and is the maximum speed at which any moving entity can travel. 2. Why time elapsed is different for a moving entity relative to a stationary entity. 3. Why there has been confusion between the wave and particle nature of an entity. 4. The relation between the speed of light c, Planck's constant k and time 5. An expression for Mass using this notation 6. The derivation for De Broglie's theorem 7. Indeterminacy of position 
  The Riccati equation for the Hubble parameter H of barotropic FRW cosmologies in conformal time for \kappa \neq 0 spatial geometries and in comoving time for the \kappa =0 geometry, respectively, is generalized to odd Grassmannian time parameters. We obtain a system of simple differential equations for the four supercomponents (two of even type and two of odd type) of the Hubble superfield function {\cal H} that is explicitly solved. The second even Hubble component does not have an evolution governed by general relativity although there are effects of the latter upon it 
  We show in the framework of Pfaff systems theory, the functional dependences of the general analytic solutions of a suitable system of involutive differential equations describing the differences between the analytic solutions of the conformal and "Poincar\'e" Lie equations. Then we ascribe to the infinitesimal variations of the parametrizing functionals some physical meanings as the electromagnetic and gravitation potentials. We also deduce their corresponding fields of interactions together with the differential equations they must satisfy. Then we discuss on various possible physical interpretations. 
  We report on thermal noise from the internal friction of dielectric coatings made from alternating layers of Ta2O5 and SiO2 deposited on fused silica substrates. We present calculations of the thermal noise in gravitational wave interferometers due to optical coatings, when the material properties of the coating are different from those of the substrate and the mechanical loss angle in the coating is anisotropic. The loss angle in the coatings for strains parallel to the substrate surface was determined from ringdown experiments. We measured the mechanical quality factor of three fused silica samples with coatings deposited on them. The loss angle of the coating material for strains parallel to the coated surface was found to be (4.2 +- 0.3)*10^(-4) for coatings deposited on commercially polished slides and (1.0 +- 0.3)*10^{-4} for a coating deposited on a superpolished disk. Using these numbers, we estimate the effect of coatings on thermal noise in the initial LIGO and advanced LIGO interferometers. We also find that the corresponding prediction for thermal noise in the 40 m LIGO prototype at Caltech is consistent with the noise data. These results are complemented by results for a different type of coating, presented in a companion paper. 
  Interferometric gravitational wave detectors use mirrors whose substrates are formed from materials of low intrinsic mechanical dissipation. The two most likely choices for the test masses in future advanced detectors are fused silica or sapphire. These test masses must be coated to form mirrors, highly reflecting at 1064nm. We have measured the excess mechanical losses associated with adding dielectric coatings to substrates of fused silica and calculate the effect of the excess loss on the thermal noise in an advanced interferometer. 
  We prove local existence and uniqueness of static spherically symmetric solutions of the Einstein-Yang-Mills equations for any action of the rotation group (or SU(2)) by automorphisms of a principal bundle over space-time whose structure group is a compact semisimple Lie group G. These actions are characterized by a vector in the Cartan subalgebra of g and are called regular if the vector lies in the interior of a Weyl chamber. In the irregular cases (the majority for larger gauge groups) the boundary value problem that results for possible asymptotically flat soliton or black hole solutions is more complicated than in the previously discussed regular cases. In particular, there is no longer a gauge choice possible in general so that the Yang-Mills potential can be given by just real-valued functions. We prove the local existence of regular solutions near the singularities of the system at the center, the black hole horizon, and at infinity, establish the parameters that characterize these local solutions, and discuss the set of possible actions and the numerical methods necessary to search for global solutions. That some special global solutions exist is easily derived from the fact that su(2) is a subalgebra of any compact semisimple Lie algebra. But the set of less trivial global solutions remains to be explored. 
  We extend Campbell-Magaard embedding theorem by proving that any n-dimensional semi-Riemannian manifold can be locally embedded in an (n+1)-dimensional Einstein space. We work out some examples of application of the theorem and discuss its relevance in the context of modern higher-dimensional spacetime theories. 
  We derive the gravitational and electrostatic self-energies of a particle at rest in the background of a cosmic dispiration (topological defect), finding that the particle may experience potential steps, well potentials or potential barriers depending on the nature of the interaction and also on certain properties of the defect. The results may turn out to be useful in cosmology and condensed matter physics. 
  The dynamical instability of new-born neutron stars is studied by evolving the linearized hydrodynamical equations. The neutron stars considered in this paper are those produced by the accretion induced collapse of rigidly rotating white dwarfs. A dynamical bar-mode (m=2) instability is observed when the ratio of rotational kinetic energy to gravitational potential energy $\beta$ of the neutron star is greater than the critical value $\beta_d \approx 0.25$. This bar-mode instability leads to the emission of gravitational radiation that could be detected by gravitational wave detectors. However, these sources are unlikely to be detected by LIGO II interferometers if the event rate is less than $10^{-6}$ per year per galaxy. Nevertheless, if a significant fraction of the pre-supernova cores are rapidly rotating, there would be a substantial number of neutron stars produced by the core collapse undergoing bar-mode instability. This would greatly increase the chance of detecting the gravitational radiation. 
  A possible Yang-Mills like lagrangian formulation for gravity is explored. The starting point consists on two next assumptions. First, the metric is assumed as a real map from a given gauge group. Second, a gauge invariant lagrangian density is considered with the condition that it is related to the Einstein one up to a bound term. We study a stationary solution of the abelian case for the spherical symmetry, which is connected to the M\"oller's Maxwell like formulation for gravity. Finally, it is showed the consistence of this formulation with the Newtonian limit. 
  The Hawking radiation of Dirac particles in an arbitrarily rectilinearly accelerating Kinnersley black hole is studied by using a method of the generalized tortoise coordinate transformation. Both the location and the temperature of the event horizon depend on the time and polar angle. The Hawking thermal radiation spectrum of Dirac particles is also derived.   PACS numbers: 97.60.Lf, 04.70.Dy 
  Introducing an effective refraction index of an isotropic cosmic medium, we investigate the cosmological fluid dynamics which is consistent with a conformal, timelike symmetry of a corresponding ``optical'' metric. We demonstrate that this kind of symmetry is compatible with the existence of a negative viscous pressure and, consequently, with cosmological entropy production. We establish an exactly solvable model according to which the viscous pressure is a consequence of a self-interacting one-particle force which is self-consistently exerted on the microscopic particles of a relativistic gas. Furthermore, we show that a sufficiently high decay rate of the refraction index of an ultrarelativistic cosmic medium results in an inflationary expansion of the universe. 
  This article investigates an extension of General Relativity based upon a class of lifted metrics on the cotangent bundle of space-time. The dynamics of the theory is determined by a fixed section of the cotangent bundle, representing the momentum of a fluid flow, and Einstein's equations for the fluid applied to the induced space-time metric on the submanifold of the cotangent bundle defined by the image of the section. This construction is formally analogous to the extension of Galilean Relativity by Special Relativity, and is shown to reduce to General Relativity as the gravitational constant approaches zero. By examining the consequences of the model for homogeneous cosmologies, it is demonstrated that this construction globalizes the equivalence principle, in that, the perfect fluid model of Special Relativity is sufficient to predict both the inflationary and the current era. 
  We examine the Hartle-Hawking no-boundary initial state for the Ponzano-Regge formulation of gravity in three dimensions. We consider the behavior of conditional probabilities and expectation values for geometrical quantities in this initial state for a simple minisuperspace model consisting of a two-parameter set of anisotropic geometries on a 2-sphere boundary. We find dependence on the cutoff used in the construction of Ponzano-Regge amplitudes for expectation values of edge lengths. However, these expectation values are cutoff independent when computed in certain, but not all, conditional probability distributions. Conditions that yield cutoff independent expectation values are those that constrain the boundary geometry to a finite range of edge lengths. We argue that such conditions have a correspondence to fixing a range of local time, as classically associated with the area of a surface for spatially closed cosmologies. Thus these results may hint at how classical spacetime emerges from quantum amplitudes. 
  We investigate brane-worlds with a pure magnetic field and a perfect fluid. We extend earlier work to brane-worlds, and find new properties of the Bianchi type I brane-world. We find new asymptotic behaviours on approach to the singularity and classify the critical points of the dynamical phase space. It is known that the Einstein equations for the magnetic Bianchi type I models are in general oscillatory and are believed to be chaotic, but in the brane-world model this chaotic behaviour does not seem to be possible. 
  A unified approach to regular interiors of black holes with smooth matter distributions in the core region is given. The approach is based on a class of Kerr-Schild metrics representing minimal deformations of the Kerr-Newman solution, and allows us to give a common treatment for (charged and uncharged) rotating and nonrotating black holes. It is shown that the requirement of smoothness of the source constraints the structure of the core region in many respects: in particular, for Schwarzschild holes a de Sitter core can be selected, which is surrounded by a smooth shell giving a leading contribution to the total mass of the source. In the rotating, noncharged case the source has a similar structure, taking the form of a (anisotropic and rotating) de Sitter-like core surrounded by a rotating elliptic shell. The Kerr singular ring is regularized by anisotropic matter rotating in the equatorial plane, so that the negative sheet of the Kerr geometry is absent. In the charged case the sources take the form of ``bags'', which can have de Sitter or anti de Sitter interiors and a smooth domain wall boundary, with a tangential stress providing charge confinement. The ADM and Tolman relations are used to calculate the total mass of the sources. 
  From a particularly simple solution of the Ernst equation, we build a solution of the vacuum stationary axisymmetric Einstein equations depending on three parameters. The parameters are associated to the total mass of the source and its angular momentum. The third parameter produces a topological deformation of the ergosphere making it a two-sheet surface, and for some of its values forbids the Penrose process. 
  A mechanism is introduced to reduce a large cosmological constant to a sufficiently small value consistent with observational upper limit. The basic ingradient in this mechanism is a distinction which has been made between the two unit systems used on cosmology and particle physics. We have used a conformal invariant gravitational model to define a particular conformal frame in terms of the large scale properties of the universe. It is then argued that the contributions of mass scales in particle physics to the vacuum energy density should be considered in a different conformal frame. In this manner a cancellation mechanism is presented in which the conformal factor plays a key role to relax the large effective cosmological constant. 
  Pattern matching techniques like matched filtering will be used for online extraction of gravitational wave signals buried inside detector noise. This involves cross correlating the detector output with hundreds of thousands of templates spanning a multi-dimensional parameter space, which is very expensive computationally. A faster implementation algorithm was devised by Mohanty and Dhurandhar [1996] using a hierarchy of templates over the mass parameters, which speeded up the procedure by about 25 to 30 times. We show that a further reduction in computational cost is possible if we extend the hierarchy paradigm to an extra parameter, namely, the time of arrival of the signal. In the first stage, the chirp waveform is cut-off at a relatively low frequency allowing the data to be coarsely sampled leading to cost saving in performing the FFTs. This is possible because most of the signal power is at low frequencies, and therefore the advantage due to hierarchy over masses is not compromised. Results are obtained for spin-less templates up to the second post-Newtonian (2PN) order for a single detector with LIGO I noise power spectral density. We estimate that the gain in computational cost over a flat search is about 100. 
  We discuss the evolution of the Universe from what might be called its quantum origin. We apply the uncertainty principle to the origin of the Universe with characteristic time scale equal to the Planck time to obtain its initial temperature and density. We establish that the subsequent evolution obeying the Einstein equation gives the present temperature of the microwave background close to the observed value. The same origin allows the possibility that the Universe started with exactly the critical density, Omega =1, and remained at the critical density during evolution. Many other important features of the observed Universe, including homogeneity and isotropy, Hubble's constant at origin, its minimum age, present density etc. are all predictions of our theory. We discuss also the testability of our theory. 
  The relativistic viscous fluid equations describing the outflow of high temperature matter created via Hawking radiation from microscopic black holes are solved numerically for a realistic equation of state. We focus on black holes with initial temperatures greater than 100 GeV and lifetimes less than 6 days. The spectra of direct photons and photons from $\pi^0$ decay are calculated for energies greater than 1 GeV. We calculate the diffuse gamma ray spectrum from black holes distributed in our galactic halo. However, the most promising route for their observation is to search for point sources emitting gamma rays of ever-increasing energy. 
  Massive spin 2 theories in flat or cosmological ($\Lambda \ne 0$) backgrounds are subject to discontinuities as the masses tend to zero. We show and explain physically why their Newtonian limits do not inherit this behaviour. On the other hand, conventional ``Newtonian cosmology'', where $\Lambda $ is a constant source of the potential, displays discontinuities: e.g. for any finite range, $\Lambda$ can be totally removed. 
  Simple arguments related to the entropy of black holes strongly constrain the spectrum of the area operator for a Schwarzschild black hole in loop quantum gravity. In particular, this spectrum is fixed completely by the assumption that the black hole entropy is maximum. Within the approach discussed, one arrives in loop quantum gravity at a quantization rule with integer quantum numbers $n$ for the entropy and area of a black hole. 
  The geometry of a two-dimensional surface in a curved space can be most easily visualized by using an isometric embedding in flat three-dimensional space. Here we present a new method for embedding surfaces with spherical topology in flat space when such a embedding exists. Our method is based on expanding the surface in spherical harmonics and minimizing for the differences between the metric on the original surface and the metric on the trial surface in the space of the expansion coefficients. We have applied this method to study the geometry of back hole horizons in the presence of strong, non-axisymmetric, gravitational waves (Brill waves). We have noticed that, in many cases, although the metric of the horizon seems to have large deviations from axisymmetry, the intrinsic geometry of the horizon is almost axisymmetric. The origin of the large apparent non-axisymmetry of the metric is the deformation of the coordinate system in which the metric was computed. 
  We demonstrate how the solution to an exterior Dirichlet boundary value problem of the axisymmetric, stationary Einstein equations can be found in terms of generalized solutions of the Backlund type. The proof that this generalization procedure is valid is given, which also proves conjectures about earlier representations of the gravitational field corresponding to rotating disks of dust in terms of Backlund type solutions. 
  We investigate the stability of self-similar solutions for a gravitationally collapsing isothermal sphere in Newtonian gravity by means of a normal mode analysis. It is found that the Hunter series of solutions are highly unstable, while neither the Larson-Penston solution nor the homogeneous collapse one have an analytic unstable mode. Since the homogeneous collapse solution is known to suffer the kink instability, the present result and recent numerical simulations strongly support a proposition that the Larson-Penston solution will be realized in astrophysical situations. It is also found that the Hunter (A) solution has a single unstable mode, which implies that it is a critical solution associated with some critical phenomena which are analogous to those in general relativity. The critical exponent $\gamma$ is calculated as $\gamma\simeq 0.10567$. In contrast to the general relativistic case, the order parameter will be the collapsed mass. In order to obtain a complete picture of the Newtonian critical phenomena, full numerical simulations will be needed. 
  We sketch the results of calculations of the quasinormal frequencies of the electrically charged dilaton black hole. At the earlier phase of evaporation (Q is less than 0.7-0.8M), the dilaton black hole "rings" with the complex frequencies which differ negligibly from those of the Reissner-Nordstrom black hole. The spectrum of the frequencies weakly depends upon the dilaton coupling. 
  The Lemaitre and Schwarzschild analytical solutions for a relativistic spherical body of constant density are linked together through the use of the Weyl quadratic invariant. The critical radius for gravitational collapse of an incompressible fluid is shown to vary continuously from 9/8 of the Schwarzschild radius to the Schwarzschild radius itself while the internal pressures become locally anisotropic. 
  Action at a distance in Newtonian physics is replaced by finite propagation speeds in classical physics, the physics defined by the field theories of Maxwell and Einstein. As a result, the differential equations of motion in Newtonian physics are replaced in classical physics by functional differential equations, where the delay associated with the finite propagation speed (the speed of light) is taken into account. Newtonian equations of motion, with post-Newtonian corrections, are often used to approximate the functional differential equations of motion. Some mathematical issues related to the problem of extracting the ``correct'' approximate Newtonian equations of motion are discussed. 
  We analyze the arguments allegedly supporting the so-called Self-Indication Assumption (SIA), as an attempt to reject counterintuitive consequences of the Doomsday Argument of Carter, Leslie, Gott and others. Several arguments purportedly supporting this assumption are demonstrated to be either flawed or, at best, inconclusive. Therefore, no compelling reason for accepting SIA exists so far, and it should be regarded as an ad hoc hypothesis with several rather strange and implausible physical and epistemological consequences. Accordingly, if one wishes to reject the controversial consequences of the Doomsday Argument, a route different from SIA has to be found. 
  I present a complete set of gauge invariant observables, in the context of general relativity coupled with a minimal amount of realistic matter (four particles). These observables have a straightforward and realistic physical interpretation. In fact, the technology to measure them is realized by the Global Positioning System: they are defined by the physical reference system determined by GPS readings. The components of the metric tensor in this physical reference system are gauge invariant quantities and, remarkably, their evolution equations are local. 
  The late stage of the inspiral of two black holes may have important non-Newtonian effects that are unrelated to radiation reaction. To understand these effects we approximate a slowly inspiralling binary by a stationary solution to Einstein's equations in which the holes orbit eternally. Radiation reaction is nullified by specifying a boundary condition at infinity containing equal amounts of ingoing and outgoing radiation. The computational problem is then converted from an evolution problem with initial data to a boundary value problem. In addition to providing an approximate inspiral waveform via extraction of the outgoing modes, our approximation can give alternative initial data for numerical relativity evolution. We report results on simplified models and on progress in building 3D numerical solutions. 
  The sO(3) and the Lorentz algebra symmetries breaking with gauge curvatures are studied by means of a covariant Hamiltonian. The restoration of these algebra symmetries in flat and curved spaces is performed and led to the apparition of a monopole field. Then in the context of the Lorentz algebra we consider an application to the gravitoelectromagnetism theory. In this last case a qualitative relation giving a mass spectrum for dyons is established. 
  We model the adiabatic inspiral of relativistic binary neutron stars in a quasi-equilibrium (QE) approximation, and compute the gravitational wavetrain from the late phase of the inspiral. We compare corotational and irrotational sequences and find a significant difference in the inspiral rate, which is almost entirely caused by differences in the binding energy. We also compare our results with those of a point-mass post-Newtonian calculation. We illustrate how the late inspiral wavetrain computed with our QE numerical scheme can be matched to the subsequent plunge and merger waveform calculated with a fully relativistic hydrodynamics code. 

  In order to find a way to have a better formulation for numerical evolution of the Einstein equations, we study the propagation equations of the constraints based on the Arnowitt-Deser-Misner formulation. By adjusting constraint terms in the evolution equations, we try to construct an "asymptotically constrained system" which is expected to be robust against violation of the constraints, and to enable a long-term stable and accurate numerical simulation. We first provide useful expressions for analyzing constraint propagation in a general spacetime, then apply it to Schwarzschild spacetime. We search when and where the negative real or non-zero imaginary eigenvalues of the homogenized constraint propagation matrix appear, and how they depend on the choice of coordinate system and adjustments. Our analysis includes the proposal of Detweiler (1987), which is still the best one according to our conjecture but has a growing mode of error near the horizon. Some examples are snapshots of a maximally sliced Schwarzschild black hole. The predictions here may help the community to make further improvements. 
  We address the question of radiation emission from both perfect and dispersive mirrors following prescribed relativistic trajectories. The trajectories considered are asymptotically inertial: the mirror starts from rest and eventually reverts to motion at uniform velocity. This enables us to provide a description in terms of in and out states. We calculate exactly the Bogolubov alpha and beta coefficients for a specific form of the trajectory, and stress the analytic properties of the amplitudes and the constraints imposed by unitarity. A formalism for the description of emission of radiation from a dispersive mirror is presented. 
  We address the question of radiation emission from a perfect mirror that starts from rest and follows the trajectory z=-ln(cosht) till t->Infinity. We show that a correct derivation of the black body spectrum via the calculation of the Bogolubov amplitudes requires consideration of the whole trajectory and not just of its asymptotic part. 
  As the first generation of laser interferometric gravitational wave detectors near operation, research and development has begun on increasing the instrument's sensitivity while utilizing the existing infrastructure. In the Laser Interferometer Gravitational Wave Observatory (LIGO), significant improvements are being planned for installation in ~2007, increasing strain sensitivity through improved suspensions and test mass substrates, active seismic isolation, and higher input laser power. Even with the highest quality optics available today, however, finite absorption of laser power within transmissive optics, coupled with the tremendous amount of optical power circulating in various parts of the interferometer, result in critical wavefront deformations which would cripple the performance of the instrument. Discussed is a method of active wavefront correction via direct thermal actuation on optical elements of the interferometer. A simple nichrome heating element suspended off the face of an affected optic will, through radiative heating, remove the gross axisymmetric part of the original thermal distortion. A scanning heating laser will then be used to remove any remaining non-axisymmetric wavefront distortion, generated by inhomogeneities in the substrate's absorption, thermal conductivity, etc. A proof-of-principle experiment has been constructed at MIT, selected data of which are presented. 
  Many inflating spacetimes are likely to violate the weak energy condition, a key assumption of singularity theorems. Here we offer a simple kinematical argument, requiring no energy condition, that a cosmological model which is inflating -- or just expanding sufficiently fast -- must be incomplete in null and timelike past directions. Specifically, we obtain a bound on the integral of the Hubble parameter over a past-directed timelike or null geodesic. Thus inflationary models require physics other than inflation to describe the past boundary of the inflating region of spacetime. 
  This paper presents both a numerical method for general relativity and an application of that method. The method involves the use of harmonic coordinates in a 3+1 code to evolve the Einstein equations with scalar field matter. In such coordinates, the terms in Einstein's equations with the highest number of derivatives take a form similar to that of the wave equation. The application is an exploration of the generic approach to the singularity for this type of matter. The preliminary results indicate that the dynamics as one approaches the singularity is locally the dynamics of the Kasner spacetimes. 
  We give a Hamiltonian definition of mass for spacelike hypersurfaces in space-times with metrics which are asymptotic to the anti-de Sitter one, or to a class of generalizations thereof. We show that our definition provides a geometric invariant for a spacelike hypersurface embedded in a space-time. 
  The effect of the angular momentum density of a gravitational source on the times of flight of light rays in an interferometer is analyzed. The calculation is made imagining that the interferometer is at the equator of the gravity source and, as long as possible, the metric, provided it is stationary and axisymmetric, is not approximated. Finally, in order to evaluate the size of the effect in the case of the Earth a weak field approximation is introduced. For laboratory scales and non-geodesic paths the correction turns out to be comparable with the sensitivity expected in gravitational waves interferometric detectors, whereas it drops under the threshold of detectability when using free (geodesic) light rays. 
  We have studied classical and quantum solutions of 2+1 dimensional Einstein gravity theory. Quantum theory is defined through the local conserved angular momentum and mass operators in the case of spherically symmetric space-time. The de Broglie-Bohm interpretation is applied for the wave function and we derive the differential equations for metrics. Metrics including quantum effect are obtained in solving these equations and we compare them with classical metrics. Especially the quantum effect on the closed de Sitter universe is evaluated quantitatively. 
  Einstein's equations are solved for spherically symmetric universes composed of dust with tangential pressure provided by angular momentum, L(R), which differs from shell to shell. The metric is given in terms of the shell label, R, and the proper time, tau, experienced by the dust particles. The general solution contains four arbitrary functions of R - M(R), L(R), E(R) and r(0,R). The solution is described by quadratures, which are in general elliptic integrals. It provides a generalization of the Lemaitre-Tolman-Bondi solution. We present a discussion of the types of solution, and some examples. The relationship to Einstein clusters and the significance for gravitational collapse is also discussed. 
  A complete Lagrangian and Hamiltonian description of the theory of self-gravitating light-like matter shells is given in terms of gauge-independent geometric quantities. For this purpose the notion of an extrinsic curvature for a null-like hypersurface is discussed and the corresponding Gauss-Codazzi equations are proved. These equations imply Bianchi identities for spacetimes with null-like, singular curvature. Energy-momentum tensor-density of a light-like matter shell is unambiguously defined in terms of an invariant matter Lagrangian density. Noether identity and Belinfante-Rosenfeld theorem for such a tensor-density are proved. Finally, the Hamiltonian dynamics of the interacting system: ``gravity + matter'' is derived from the total Lagrangian, the latter being an invariant scalar density. 
  One of the types of signals for which the LIGO interferometric gravitational wave detectors will search is a stochastic background of gravitational radiation. We review the technique of searching for a background using the optimally-filtered cross-correlation statistic, and describe the state of plans to perform such cross-correlations between the two LIGO interferometers as well as between LIGO and other gravitational-wave detectors, in particular the preparation of software to perform such data analysis. 
  Using one arm of the Michelson interferometer and the power recycling mirror of the interferometric gravitational wave detector GEO600, we created a Fabry-Perot cavity with a length of 1200 m. The main purpose of this experiment was to gather first experience with the main optics, its suspensions and the corresponding control systems. The residual displacement of a main mirror is about 150 nm rms. By stabilising the length of the 1200 m long cavity to the pre-stabilised laser beam we achieved an error point frequency noise of 0.1 mHz/sqrt(Hz) at 100 Hz Fourier frequency. In addition we demonstrated the reliable performance of all included subsystems by several 10-hour-periods of continuous stable operation. Thus the full frequency stabilisation scheme for GEO600 was successfully tested. 
  Einstein's equivalence principle in classical physics is a rule stating that the effect of gravitation is locally equivalent to the acceleration of an observer. The principle determines the motion of test particles uniquely (modulo very broad general assumptions). We show that the same principle applied to a quantum particle described by a wave function on a Newtonian gravitational background determines its motion with a similar degree of uniqueness. 
  We discuss the nature of the radial and azimuthal components of centrifugal force associated with fluid flows in the equatorial plane of black hole space times. The equations of motion are solved for the radial and azimuthal components of the 3-velocity V^i which are then used in evaluating the nature of the various components of inertial accelerations. It is shown that the reversal of centrifugal force is governed mainly by the dominance of the azimuthal velocity and the reversal occurs for r, mostly at 2m <~ r <~ 3m, depending upon the boundary condition. 
  The photon equation (massless Duffin-Kemmer-Petiau equation) is studied in a nonstationary rotating, causal Godel-type cosmological universe. The spinor solution is found exactly. It is shown that frequency spectrum is discrete and unbounded. 
  A simple recipe is given for constructing a maximally sparse regular lattice of spin-free post-1PN gravitational wave chirp templates subject to a given minimal match constraint, using Tanaka-Tagoshi coordinates. 
  Irrotational dust solutions of Einstein's equations are suitable models to describe the general-relativistic aspects of the gravitational instability mechanism for the formation of cosmic structures. In this paper we study their state space by considering the local initial-value problem formulated in the covariant fluid approach. We consider a wide range of models, from homogeneous and isotropic to highly inhomogeneous irrotational dust models, showing how they constitute equilibrium configurations (invariant sets) of the dynamics. Moreover, we give the characterization of such configurations, which provides an initial-data characterization of the models under consideration. 
  In this paper explore the relation between covariant and canonical approaches to quantum gravity and $BF$ theory. We will focus on the dynamical triangulation and spin-foam models, which have in common that they can be defined in terms of sums over space-time triangulations. Our aim is to show how we can recover these covariant models from a canonical framework by providing two regularisations of the projector onto the kernel of the Hamiltonian constraint. This link is important for the understanding of the dynamics of quantum gravity. In particular, we will see how in the simplest dynamical triangulations model we can recover the Hamiltonian constraint via our definition of the projector. Our discussion of spin-foam models will show how the elementary spin-network moves in loop quantum gravity, which were originally assumed to describe the Hamiltonian constraint action, are in fact related to the time-evolution generated by the constraint. We also show that the Immirzi parameter is important for the understanding of a continuum limit of the theory. 
  A sufficient condition for the validity of Cosmic Censorship in spherical gravitational collapse is formulated and proved. The condition relies on an attractive mathematical property of the apparent horizon, which holds if ''minimal'' requirements of physical reasonableness are satisfied by the matter model. 
  Ricci collineations and Ricci inheritance collineations of Friedmann-Robertson-Walker spacetimes are considered. When the Ricci tensor is non-degenerate, it is shown that the spacetime always admits a fifteen parameter group of Ricci inheritance collineations; this is the maximal possible dimension for spacetime manifolds. The general form of the vector generating the symmetry is exhibited. It is also shown, in the generic case, that the group of Ricci collineations is six-dimensional and coincides with the isometry group. In special cases the spacetime may admit either one or four proper Ricci collineations in addition to the six isometries. These special cases are classified and the general form of the vector fields generating the Ricci collineations is exhibited. When the Ricci tensor is degenerate, the groups of Ricci inheritance collineations and Ricci collineations are infinite-dimensional. General forms for the generating vectors are obtained. Similar results are obtained for matter collineations and matter inheritance collineations. 
  The probability for quantum creation of an inflationary universe with a pair of black holes in higher derivative theories has been studied. Considering a gravitational action which includes quadratic ($\alpha R^{2}$) and/or cubic term ($\beta R^{3}$) in scalar curvature in addition to a cosmological constant ($\Lambda$) in semiclassical approximation with Hartle-Hawking boundary condition, the probability has been evaluated. The action of the instanton responsible for creating such a universe, with spatial section with $S^{1}XS^{2}$ topology, is found to be less than that with a spatial $S^{3}$ topology, unless $\alpha < - \frac{1}{8 \Lambda}$ in $R^{2}$-theory. In the $R^{3}$ theory, however, there exists a set of solutions without a cosmological constant when $\beta R^{2} = 1$ and $\alpha = - 3 \sqrt{\beta}$ which admit primordial black holes (PBH) pair in an inflationary universe scenario. We note further that when $\beta R^{2} \neq 1$, one gets PBH pairs in the two cases : (i) with $\alpha$ and $\Lambda$ both positive and (ii) with $\Lambda$ positive and $\alpha$ negative satisfying a constraint $6 | \alpha | \Lambda > 1$. However, the relative probability for creation of an inflationary universe with a pair of black holes in the $R^{3}$-theory suppresses when $\alpha > - 2 \sqrt{\beta} $ or $|\alpha| < 2 \sqrt{\beta} $. However, if the above constraints are relaxed one derives interesting results leading to a universe with PBH in $R^{3}$-theory without cosmological constant. PACS No(s). : 04.20.Jb, 04.60.+n, 98.80.Hw 
  We present an exact analytical solution of the Einstein equations with cosmological constant in a spatially flat Robertson-Walker metric. This is interpreted as an isotropic Lemaitre-type version of the cosmological Friedmann model. Implications in the recent discovered cosmic acceleration of the universe and in the theory of an inflationary model of the universe are in view. Some properties of this solution are pointed out as a result of numerical investigations of the model. 
  The article presents a series of numerical simulations of exact solutions of the Einstein equations performed using the Cactus code, a complete 3-dimensional machinery for numerical relativity. We describe an application (``thorn'') for the Cactus code that can be used for evolving a variety of exact solutions, with and without matter, including solutions used in modern cosmology for modeling the early stages of the universe. Our main purpose has been to test the Cactus code on these well-known examples, focusing mainly on the stability and convergence of the code. 
  A study of the Aichelburg--Sexl boost of the Schwarzschild field is described in which the emphasis is placed on the field (curvature tensor) with the metric playing a secondary role. This is motivated by a description of the Coulomb field of a charged particle viewed by an observer whose speed relative to the charge approaches the speed of light. Our approach is exemplified by carrying out an Aichelburg-- Sexl type boost on the Weyl vacuum gravitational field due to an isolated axially symmetric source. Detailed calculations of the boosts transverse and parallel to the symmetry axis are given and the results, which differ significantly, are discussed. 
  The present paper is a continuation of our work on curved finitary spacetime sheaves of incidence algebras and treats the latter along Cech cohomological lines. In particular, we entertain the possibility of constructing a non-trivial de Rham complex on these finite dimensional algebra sheaves along the lines of the first author's axiomatic approach to differential geometry via the theory of vector and algebra sheaves. The upshot of this study is that important `classical' differential geometric constructions and results usually thought of as being intimately associated with smooth manifolds carry through, virtually unaltered, to the finitary-algebraic regime with the help of some quite universal, because abstract, ideas taken mainly from sheaf-cohomology as developed in the first author's Abstract Differential Geometry theory. At the end of the paper, and due to the fact that the incidence algebras involved have been previously interpreted as quantum causal sets, we discuss how these ideas may be used in certain aspects of current research on discrete Lorentzian quantum gravity. 
  This is an introduction to the by now fifteen years old research field of canonical quantum general relativity, sometimes called "loop quantum gravity". The term "modern" in the title refers to the fact that the quantum theory is based on formulating classical general relativity as a theory of connections rather than metrics as compared to in original version due to Arnowitt, Deser and Misner. Canonical quantum general relativity is an attempt to define a mathematically rigorous, non-perturbative, background independent theory of Lorentzian quantum gravity in four spacetime dimensions in the continuum. The approach is minimal in that one simply analyzes the logical consequences of combining the principles of general relativity with the principles of quantum mechanics. The requirement to preserve background independence has lead to new, fascinating mathematical structures which one does not see in perturbative approaches, e.g. a fundamental discreteness of spacetime seems to be a prediction of the theory providing a first substantial evidence for a theory in which the gravitational field acts as a natural UV cut-off. An effort has been made to provide a self-contained exposition of a restricted amount of material at the appropriate level of rigour which at the same time is accessible to graduate students with only basic knowledge of general relativity and quantum field theory on Minkowski space. 
  We discuss the distinction between the notion of partial observable and the notion of complete observable. Mixing up the two is frequently a source of confusion. The distinction bears on several issues related to observability, such as (i) whether time is an observable in quantum mechanics, (ii) what are the observables in general relativity, (iii) whether physical observables should or should not commute with the Wheeler-DeWitt operator in quantum gravity. We argue that the extended configuration space has a direct physical interpretation, as the space of the partial observables. This space plays a central role in the structure of classical and quantum mechanics and the clarification of its physical meaning sheds light on this structure, particularly in context of general covariant physics. 
  Acoustic analogues of black holes (dumb holes) are generated when a supersonic fluid flow entrains sound waves and forms a trapped region from which sound cannot escape. The surface of no return, the acoustic horizon, is qualitatively very similar to the event horizon of a general relativity black hole. In particular Hawking radiation (a thermal bath of phonons with temperature proportional to the ``surface gravity'') is expected to occur. In this note we consider quasi-one-dimensional supersonic flow of a Bose--Einstein condensate (BEC) in a Laval nozzle (converging-diverging nozzle), with a view to finding which experimental settings could magnify this effect and provide an observable signal. We identify an experimentally plausible configuration with a Hawking temperature of order 70 n K; to be contrasted with a condensation temperature of the order of 90 n K. 
  The universality of Martinez's conjecture, which states that the quasilocal energy of a black hole at the outer horizon reduces to twice its irreducible mass, or equivalently, to $\sqrt{A/4\pi}$ ($A$ is the area of the black hole), is investigated by calculating Brown-York quasilocal energies for stationary black holes in heterotic string theory, e. g., for the stationary Kaluza-Klein black hole, the rotating Cveti$\check{c}$-Youm black hole, the stationary axisymmetric Einstein-Maxwell-dilaton-axion black hole, and the Kerr-Sen black hole. It is shown that Martinez's conjecture can be extended from general relativity to heterotic string theory since the quasilocal energies of these stationary black holes tend to their Arnowitt-Dener-Misner masses at spatial infinity, and reduce to $\sqrt{A/4\pi}$ at the event horizons. 
  We investigate the properties of relativistic $r$-modes of slowly rotating neutron stars by using a relativistic version of the Cowling approximation. In our formalism, we take into account the influence of the Coriolis like force on the stellar oscillations, but ignore the effects of the centrifugal like force. For three neutron star models, we calculated the fundamental $r$-modes with $l'=m=2$ and 3. We found that the oscillation frequency $\bar\sigma$ of the fundamental $r$-mode is in a good approximation given by $\bar\sigma\approx \kappa_0 \Omega$, where $\bar\sigma$ is defined in the corotating frame at the spatial infinity, and $\Omega$ is the angular frequency of rotation of the star. The proportional coefficient $\kappa_0$ is only weakly dependent on $\Omega$, but it strongly depends on the relativistic parameter $GM/c^2R$, where $M$ and $R$ are the mass and the radius of the star. All the fundamental $r$-modes with $l'=m$ computed in this study are discrete modes with distinct regular eigenfunctions, and they all fall in the continuous part of the frequency spectrum associated with Kojima's equation (Kojima 1998). These relativistic $r$-modes are obtained by including the effects of rotation higher than the first order of $\Omega$ so that the buoyant force plays a role, the situation of which is quite similar to that for the Newtonian $r$-modes. 
  Hawking evaporation of photons in a variable-mass Kerr space-time is investigated by using a method of the generalized tortoise coordinate transformation. The blackbody radiant spectrum of photons displays a new spin-rotation coupling effect obviously dependent on different helicity states of photons. 
  This is the report of the "Quantum General Relativity" session, at the 16th International Conference on General Relativity & Gravitation, held on July 15th to 21st 2001, in Durban, South Africa. The report will appear on the Proceedings of the conference. Comments and criticisms are welcome: they will be taken into account for revising the text before the publication. 
  We extend a coherent network data-analysis strategy developed earlier for detecting Newtonian waveforms to the case of post-Newtonian (PN) waveforms. Since the PN waveform depends on the individual masses of the inspiraling binary, the parameter-space dimension increases by one from that of the Newtonian case. We obtain the number of templates and estimate the computational costs for PN waveforms: For a lower mass limit of a solar mass, for LIGO-I noise, and with 3% maximum mismatch, the online computational speed requirement for single detector is a few Gflops; for a two-detector network it is hundreds of Gflops and for a three-detector network it is tens of Tflops. Apart from idealistic networks, we obtain results for realistic networks comprising of LIGO and VIRGO. Finally, we compare costs incurred in a coincidence detection strategy with those incurred in the coherent strategy detailed above. 
  It is well known that every Killing vector is a Ricci and Matter collineation. Therefore the metric, the Ricci tensor and the energy-momentum tensor are all members of a large family of second order symmetric tensors which are invariant under a common group of symmetries. This family is described by a generic metric which is defined from the symmetry group of the space-time metric. The proper Ricci and Matter (inheritance) collineations are the (conformal) Killing vectors of the generic metric which are not (conformal) Killing vectors of the space-time metric. Using this observation we compute the Ricci and Matter inheritance collineations of the Robertson-Walker space-times and we determine the Ricci and Matter collineations without any further calculations. It is shown that these higher order symmetries can be used as supplementary conditions to produce an equation of state which is compatible with the Geometry and the Physics of the Robertson-Walker space-times. 
  It seems to be generally accepted that apparently anomalous cosmological observations, such as accelerating expansion, etc., necessarily are inconsistent with standard general relativity and standard matter sources. Following the suggestions of S{\l}adkowski, we point out that in addition to exotic theories and exotic matter there is another possibility. We refer to exotic differential structures on ${\mathbb R}^4$ which could be the source of the observed anomalies without changing the Einstein equations or introducing strange forms of matter. 
  The amplitude for a spin foam in the Barrett-Crane model of Riemannian quantum gravity is given as a product over its vertices, edges and faces, with one factor of the Riemannian 10j symbols appearing for each vertex, and simpler factors for the edges and faces. We prove that these amplitudes are always nonnegative for closed spin foams. As a corollary, all open spin foams going between a fixed pair of spin networks have real amplitudes of the same sign. This means one can use the Metropolis algorithm to compute expectation values of observables in the Riemannian Barrett-Crane model, as in statistical mechanics, even though this theory is based on a real-time (e^{iS}) rather than imaginary-time (e^{-S}) path integral. Our proof uses the fact that when the Riemannian 10j symbols are nonzero, their sign is positive or negative depending on whether the sum of the ten spins is an integer or half-integer. For the product of 10j symbols appearing in the amplitude for a closed spin foam, these signs cancel. We conclude with some numerical evidence suggesting that the Lorentzian 10j symbols are always nonnegative, which would imply similar results for the Lorentzian Barrett-Crane model. 
  The 10j symbol is a spin network that appears in the partition function for the Barrett-Crane model of Riemannian quantum gravity. Elementary methods of calculating the 10j symbol require order(j^9) or more operations and order(j^2) or more space, where j is the average spin. We present an algorithm that computes the 10j symbol using order(j^5) operations and order(j^2) space, and a variant that uses order(j^6) operations and a constant amount of space. An implementation has been made available on the web. 
  The issues related to bringing together the principles of general relativity and quantum theory are discussed. After briefly summarising the points of conflict between the two formalisms I focus on four specific themes in which some contact has been established in the past between GR and quantum field theory: (i) The role of planck length in the microstructure of spacetime (ii) The role of quantum effects in cosmology and origin of the universe (iii) The thermodynamics of spacetimes with horizons and especially the concept of entropy related to spacetime geometry (iv) The problem of the cosmological constant. 
  This is the second in a series of papers on the construction and validation of a three-dimensional code for the solution of the coupled system of the Einstein equations and of the general relativistic hydrodynamic equations, and on the application of this code to problems in general relativistic astrophysics. In particular, we report on the accuracy of our code in the long-term dynamical evolution of relativistic stars and on some new physics results obtained in the process of code testing. The tests involve single non-rotating stars in stable equilibrium, non-rotating stars undergoing radial and quadrupolar oscillations, non-rotating stars on the unstable branch of the equilibrium configurations migrating to the stable branch, non-rotating stars undergoing gravitational collapse to a black hole, and rapidly rotating stars in stable equilibrium and undergoing quasi-radial oscillations. The numerical evolutions have been carried out in full general relativity using different types of polytropic equations of state using either the rest-mass density only, or the rest-mass density and the internal energy as independent variables. New variants of the spacetime evolution and new high resolution shock capturing (HRSC) treatments based on Riemann solvers and slope limiters have been implemented and the results compared with those obtained from previous methods. Finally, we have obtained the first eigenfrequencies of rotating stars in full general relativity and rapid rotation. A long standing problem, such frequencies have not been obtained by other methods. Overall, and to the best of our knowledge, the results presented in this paper represent the most accurate long-term three-dimensional evolutions of relativistic stars available to date. 
  A determination is made of the radiation emitted by a linearly uniformly accelerated uncharged dipole transmitter. It is found that, first of all, the radiation rate is given by the familiar Larmor formula, but it is augmented by an amount which becomes dominant for sufficiently high acceleration. For an accelerated dipole oscillator, the criterion is that the center of mass motion become relativistic within one oscillation period. The augmented formula and the measurements which it summarizes presuppose an expanding inertial observation frame. A static inertial reference frame will not do. Secondly, it is found that the radiation measured in the expanding inertial frame is received with 100% fidelity. There is no blueshift or redshift due to the accelerative motion of the transmitter. Finally, it is found that a pair of coherently radiating oscillators accelerating (into opposite directions) in their respective causally disjoint Rindler-coordinatized sectors produces an interference pattern in the expanding inertial frame. Like the pattern of a Young double slit interferometer, this Rindler interferometer pattern has a fringe spacing which is inversely proportional to the proper separation and the proper frequency of the accelerated sources. The interferometer, as well as the augmented Larmor formula, provide a unifying perspective. It joins adjacent Rindler-coordinatized neighborhoods into a single spacetime arena for scattering and radiation from accelerated bodies. 
  We investigate the threshold of gravitational collapse with angular momentum, under the assumption that the critical solution is spherical and self-similar and has two growing modes, namely one spherical mode and one axial dipole mode (threefold degenerate). This assumption holds for perfect fluid matter with the equation of state p=kappa rho if the constant kappa is in the range 0<kappa<1/9. There is a region in the space of initial data where the mass and angular momentum of the black hole created in the collapse are given in terms of the initial data by two universal critical exponents and two universal functions of one argument. These expressions are similar to those for the correlation length and the magnetization in a ferromagnet near its critical point, as a function of the temperature and the external magnetic field. We discuss qualitative features of the scaling functions, and hence of critical collapse with high angular momentum. 
  Two essential properties of energy-momentum tensors T_{\mu\nu} are their positivity and conservation. This is mathematically formalized by, respectively, an energy condition, as the dominant energy condition, and the vanishing of their divergence \nabla^\mu T_{\mu\nu}=0. The classical Bel and Bel-Robinson superenergy tensors, generated from the Riemann and Weyl tensors, respectively, are rank-4 tensors. But they share these two properties with energy momentum tensors: the Dominant Property (DP) and the divergence-free property in the absence of sources (vacuum). Senovilla defined a universal algebraic construction which generates a basic superenergy tensor T{A} from any arbitrary tensor A. In this construction the seed tensor A is structured as an r-fold multivector, which can always be done. The most important feature of the basic superenergy tensors is that they satisfy automatically the DP, independently of the generating tensor A. In a previous paper we presented a more compact definition of T{A} using the r-fold Clifford algebra. This form for the superenergy tensors allowed to obtain an easy proof of the DP valid for any dimension. In this paper we include this proof. We explain which new elements appear when we consider the tensor T{A} generated by a non-degree-defined r-fold multivector A and how orthogonal Lorentz transformations and bilinear observables of spinor fields are included as particular cases of superenergy tensors. We find some sufficient conditions for the seed tensor A, which guarantee that the generated tensor T{A} is divergence-free. These sufficient conditions are satisfied by some physical fields, which are presented as examples. 
  Conformal collineations (a generalization of conformal motion) and Ricci inheritance collineations, defined by $\pounds_\xi R_{ab}=2\alpha R_{ab}$, for string cloud and string fluids in general relativity are studied. By investigating the kinematical and dynamical properties of such fluids and using the field equations, some recent studies on the restrictions imposed by conformal collineations are extended, and new results are found. 
  For the last fifteen years, the limiting noise source at the low frequency end of the sensitivity window for space gravitational wave detectors has been expected to be the confusion background of overlapping galactic binary stars. Here, we present results of a study that investigates the correlation between binary star signals and conclude that there is a spatial filter in the position-dependent Doppler shift of each binary that sharply reduces the contribution of the galactic binary confusion to the noise in the detector when a monochromatic source is being detected. The sensitivity is thus determined by the instrument alone, and the confusion noise may effectively be ignored. 
  Gravitational wave astronomy will require the coordinated analysis of data from the global network of gravitational wave observatories. Questions of how to optimally configure the global network arise in this context. We have elsewhere proposed a formalism which is employed here to compare different configurations of the network, using both the coincident network analysis method and the coherent network analysis method. We have constructed a network model to compute a figure-of-merit based on the detection rate for a population of standard-candle binary inspirals. We find that this measure of network quality is very sensitive to the geographic location of component detectors under a coincident network analysis, but comparatively insensitive under a coherent network analysis. 
  This is a summary of a talk given at the CP01 meeting on possible Lorentz anomalies in canonical quantum gravity. It briefly reviews some initial explorations on the subject that have taken place recently, and should be only be seen as a short pointer to the literature on the subject, mostly for outsiders. 
  General relativity predicts that two freely counter-revolving test particles in the exterior field of a central rotating mass take different periods of time to complete the same full orbit; this time difference leads to the gravitomagnetic clock effect. The effect has been derived for circular equatorial orbits; moreover, it has been extended via azimuthal closure to spherical orbits around a slowly rotating mass. In this work, a general formula is derived for the main gravitomagnetic clock effect in the case of slow motion along an arbitrary {\it elliptical} orbit in the exterior field of a slowly rotating mass. Some of the implications of this result are briefly discussed. 
  Fully covariant wave equations predict the existence of a class of inertial-gravitational effects that can be tested experimentally. In these equations inertia and gravity appear as external classical fields, but, by conforming to general relativity, provide very valuable information on how Einstein's views carry through in the world of the quantum. 
  We obtain minimal (2+1) and (2+2) dimensional global flat embeddings of uncharged and charged dilatonic black holes in (1+1) dimensions, respectively. Moreover, we obtain the Hawking temperatures and the black hole ones of these dilatonic black holes. However, even though the minimal flat embedding structures are mathematically meaningful, through this minimal embeddings the proper entropies are shown to be unattainable, contrast to the cases of other black holes in (2+1) or much higher dimensions. 
  We obtain the energy distribution of the gamma metric using the energy-momentum complex of M{\o}ller. The result is the same as obtained by Virbhadra in the Weinberg prescription. 
  The Cotton-York and Simon-Mars tensors in stationary vacuum spacetimes are studied in the language of the congruence approach pioneered by Hawking and Ellis. Their relationships with the Papapetrou field defined by the stationary Killing congruence and with a recent characterization of the Kerr spacetime in terms of the alignment between of the principal null directions of the Weyl tensor with those of the Papapetrou field are also investigated in this more transparent language. 
  We show that the sum over geometries in the Lorentzian 4-D state sum model for quantum GR in [1] includes terms which correspond to geometries on manifolds with conical singularities. Natural approximations suggest that they can be interpreted as gauge bosond for the standard model, plus fermions plus matter. 
  We consider a model of the state evolution of relativistic vector bosons, which includes both the dynamical equations for the particle four-velocity and the equations for the polarization four-vector evolution in the field of a nonlinear plane gravitational wave. In addition to the gravitational minimal coupling, tidal forces linear in curvature tensor are suggested to drive the particle state evolution. The exact solutions of the evolutionary equations are obtained. Birefringence and tidal deviations from the geodesic motion are discussed. 
  Recent observational indications of an accelerating universe enhance the interest in studying models with a cosmological constant. We investigate cosmological expansion (FRW metric) with $\Lambda>0$ for a general linear equation of state $p=w\rho$, $w>-1$, so that the interplay between cosmological vacuum and quintessence is allowed, as well.   Four closed-form solutions (flat universe with any $w$, and $w=1/3$, $-1/3,  -2/3$) are given, of which the last one appears to be new. For the open universe a simple relation between solutions with different parameters is established: it turns out that a solution with some $w$ and (properly scaled) $\Lambda$ is expressed algebraically via another solution with special different values of these parameters.   The expansion becomes exponential at large times, and the amplitude at the exponent depends on the parameters. We study this dependence in detail, deriving various representations for the amplitude in terms of integrals and series. The closed-form solutions serve as benchmarks, and the solution transformation property noted above serves as a useful tool. Among the results obtained, one is that for the open universe with relatively small cosmological constant the amplitude is independent of the equation of state. Also, estimates of the cosmic age through the observable ratio $\Omega_\Lambda/\Omega_M$ and parameter $w$ are given; when inverted, they provide an estimate of $w$, i. e., the state equation, through the known age of the universe. 
  We examine the question of whether violation of 4D physics is an inevitable consequence of existence of an extra non-compactified dimension. Recent investigations in membrane and Kaluza-Klein theory indicate that when the metric of the spacetime is allowed to depend on the extra coordinate, i.e., the cilindricity condition is dropped, the equation describing the trajectory of a particle in one lower dimension has an extra force with some abnormal properties. Among them, a force term parallel to the four-velocity of the particle and, what is perhaps more surprising,   $u_{\mu}f^{\mu} \neq u^{\mu}f_{\mu}$. These properties violate basic concepts in 4D physics. In this paper we argue that these abnormal properties are {\em not} consequence of the extra dimension, but result from the formalism used. We propose a new definition for the force, from the extra dimension, which is free of any contradictions and consistent with usual 4D physics. We show, using warp-products metrics, that this new definition is also more consistent with our physical intuition. The effects of this force could be detected observing objects moving with high speed, near black holes and/or in cosmological situations. 
  A brief synopsis of recent conceptions and results, the current status and future outlook of our research program of applying sheaf and topos-theoretic ideas to quantum gravity and quantum logic is presented. 
  The notions of time in the theories of Newton and Einstein are reviewed so that the difficulty which impedes the unification of quantum mechanics (QM) and general relativity (GR) is clarified. It is seen that GR by itself contains an intrinsic difficulty relating to the definition of local clocks, as well as that GR still requires a kind of absolute that can serve as an objective reference standard. We present a new understanding of time, which gives a consistent definition of a local time associated with each local system in a quantum mechanical way, so that it serves the requirements of both GR as well as QM. As a consequence, QM and GR are reconciled while preserving the current mathematical formulations of both theories. 
  As a continuation of Part I [8], a more precise formulation of local time and local system is given. The observation process is reflected in order to give a relation between the classical physics for centers of mass of local systems and the quantum mechanics inside a local system. The relation will give a unification of quantum mechanics and general relativity in some cases. The existence of local time and local motion is proved so that the stationary nature of the universe is shown to be consistent with the local motion. 
  From the simple Lagrangian the equations of motion for the particle with spin are derived. The spin is shown to be conserved on the particle world-line. In the absence of a spin the equation coincides with that of a geodesic. The equations of motion are valid for massless particles as well, since mass does not enter the equations explicitely. 
  It is known that spherically symmetric spacetimes admit flat spacelike foliations. We point out a simple method of seeing this result via the Hamiltonian constraints of general relativity. The method yields explicit formulas for the extrinsic curvatures of the slicings. 
  We consider the dynamics of a causal bulk viscous cosmological fluid filled constantly decelerating Bianchi type I space-time. The matter component of the Universe is assumed to satisfy a linear barotropic equation of state and the state equation of the small temperature Boltzmann gas. The resulting cosmological models satisfy the condition of smallness of the viscous stress. The time evolution of the relaxation time, temperature, bulk viscosity coefficient and comoving entropy of the dissipative fluid is also obtained. 
  It is argued that the thermal nature of Hawking radiation arises solely due to decoherence. Thereby any information-loss paradox is avoided because for closed systems pure states remain pure. The discussion is performed for a massless scalar field in the background of a Schwarzschild black hole, but the arguments should hold in general. The result is also compared to and contrasted with the situation in inflationary cosmology. 
  A decoupled system of hyperbolic partial differential equations for linear perturbations around any spatially flat FRW universe is obtained for a wide class of perturbations. The considered perturbing energy momentum-tensors can be expressed as the sum of the perturbation of a minimally coupled scalar field plus an arbitrary (weak) energy-momentum tensor which is covariantly conserved with respect to the background. The key ingredient in obtaining the decoupling of the equations is the introduction of a new covariant gauge which plays a similar role as harmonic gauge does for perturbations around Minkowski space-time. The case of universes satysfying a linear equation of state is discussed in particular, and closed analytic expressions for the retarded Green's functions solving the de Sitter, dust and radiation dominated cases are given. 
  We investigate in 4 spacetime dimensions, all the consistent deformations of the lagrangian ${\cal L}_2+{\cal L}_{{3/2}}$, which is the sum of the Pauli-Fierz lagrangian ${\cal L}_2$ for a free massless spin 2 field and the Rarita-Schwinger lagrangian ${\cal L}_{{3/2}}$ for a free massless spin 3/2 field. Using BRST cohomogical techniques, we show, under the assumptions of locality, Poincar\'e invariance, conservation of the number of gauge symmetries and the number of derivatives on each fields, that N=1 D=4 supergravity is the only consistent interaction between a massless spin 2 and a massless spin 3/2 field. We do not assume general covariance. This follows automatically, as does supersymmetry invariance. Various cohomologies related to conservations laws are also given. 
  The gravitational redshift formula is usually derived in the geometric optics approximation. In this note we consider an exact formulation of the problem in the Schwarzschild space-time, with the intention to clarify under what conditions this redshift law is valid. It is shown that in the case of shocks the radial component of the Poynting vector can scale according to the redshift formula, under a suitable condition. If that condition is not satisfied, then the effect of the backscattering can lead to significant modifications. The obtained results imply that the energy flux of the short wavelength radiation obeys the standard gravitational redshift formula while the energy flux of long waves can scale differently, with redshifts being dependent on the frequency. 
  A model of space-time foam, made by $N$ wormholes is considered. The Casimir energy leading to such a model is computed by means of the phase shift method which is in agreement with the variational approach used in Refs.[9-14]. The collection of Schwarzschild and Reissner-Nordstr\"{o}m wormholes are separately considered to represent the foam. The Casimir energy shows that the Reissner-Nordstr\"{o}m wormholes cannot be used to represent the foam. 
  A new spherically-symmetric solution is determined in a noncompactified Kaluza-Klein theory in which a time character is ascribed to the fifth coordinate. This solution contains two independent parameters which are related with mass and electric charge. The solution exhibits a Schwarzschild radius and represents a generalization of the Schwarzschild solution in four dimensions. The parameter of the solution connected with the electric charge depends on the derivative of the fifth (second time) coordinate with respect to the ordinary time coordinate. It is shown that the perihelic motion in four-dimensional relativity has a counterpart in five dimensions in the perinucleic motion of a negatively-charged particle. If the quantization conditions of the older quantum theory are applied to that motion, an analogue of the fine-structure formula of atomic spectra is obtained. 
  We assume that the fourdimensional quantum gravity is scale invariant at short distances. We show through a simple scaling argument that correlation functions of quantum fields interacting with gravity have a universal (more regular) short distance behavior. 
  We model quantum space-time on the Planck scale as dynamical networks of elementary relations or time dependent random graphs, the time dependence being an effect of the underlying dynamical network laws. We formulate a kind of geometric renormalisation group on these (random) networks leading to a hierarchy of increasingly coarse-grained networks of overlapping lumps. We provide arguments that this process may generate a fixed limit phase, representing our continuous space-time on a mesoscopic or macroscopic scale, provided that the underlying discrete geometry is critical in a specific sense (geometric long range order). Our point of view is corroborated by a series of analytic and numerical results, which allow to keep track of the geometric changes, taking place on the various scales of the resolution of space-time. Of particular conceptual importance are the notions of dimension of such random systems on the various scales and the notion of geometric criticality. 
  Based on work of Derrick, Coll, and Morales, we define a `symmetric' null coframe with {\it four real null covectors}. We show that this coframe is closely related to the GPS type coordinates recently introduced by Rovelli. 
  Since the event horizon of a black hole is a surface of infinite redshift, it might be thought that Hawking radiation would be highly sensitive to Lorentz violation at high energies. In fact, the opposite is true for subluminal dispersion. For superluminal dispersion, however, the outgoing black hole modes emanate from the singularity in a state determined by unknown quantum gravity processes. 
  The CGHS two-dimensional dilaton gravity model is generalized to include a ghost Klein-Gordon field, i.e. with negative gravitational coupling. This exotic radiation supports the existence of static traversible wormhole solutions, analogous to Morris-Thorne wormholes. Since the field equations are explicitly integrable, concrete examples can be given of various dynamic wormhole processes, as follows. (i) Static wormholes are constructed by irradiating an initially static black hole with the ghost field. (ii) The operation of a wormhole to transport matter or radiation between the two universes is described, including the back-reaction on the wormhole, which is found to exhibit a type of neutral stability. (iii) It is shown how to maintain an operating wormhole in a static state, or return it to its original state, by turning up the ghost field. (iv) If the ghost field is turned off, either instantaneously or gradually, the wormhole collapses into a black hole. 
  This work investigates some global questions about cosmological spacetimes with two dimensional spherical, plane and hyperbolic symmetry containing matter. The result is, that these spacetimes admit a global foliation by prescribed mean curvature surfaces, which extends at least towards a crushing singularity. The time function of the foliation is geometrically defined and unique up to the choice of an initial Cauchy surface.   This work generalizes a similar analysis on constant mean curvature foliations and avoids the topological obstructions arising from the existence problem. 
  This second part is devoted to the investigation of global properties of Prescribed Mean Curvature (PMC) foliations in cosmological spacetimes with local $U(1) \times U(1)$ symmetry and matter described by the Vlasov equation. It turns out, that these spacetimes admit a global foliation by PMC surfaces, as well, but the techniques to achieve this goal are more complex than in the cases considered in part I. 
  We study spherically symmetric solutions to the Einstein field equations under the assumption that the space-time may possess an arbitrary number of spatial dimensions. The general solution of Synge is extended to describe systems of any dimension. Arbitrary dimension analogues of known four dimensional solutions are also presented, derived using the above scheme. Finally, we discuss the requirements for the existence of Birkhoff's theorems in space-times of arbitrary dimension with or without matter fields present. Cases are discussed where the assumptions of the theorem are considerably weakened yet the theorem still holds. We also discuss where the weakening of certain conditions may cause the theorem to fail. 
  We consider static axially symmetric Einstein-Yang-Mills black holes in the isolated horizon formalism. The mass of these hairy black holes is related to the mass of the corresponding particle-like solutions by the horizon mass. The hairy black holes violate the ``quasi-local uniqueness conjecture'', based on the horizon charges. 
  We evolve the binary black hole initial data family proposed by Bishop {\em et al.} in the limit in which the black holes are close to each other. We present an exact solution of the linearized initial value problem based on their proposal and make use of a recently introduced generalized formalism for studying perturbations of Schwarzschild black holes in arbitrary coordinates to perform the evolution. We clarify the meaning of the free parameters of the initial data family through the results for the radiated energy and waveforms from the black hole collision. 
  It is commonly believed that Alcubierre's warp drive works by contracting space in front of the warp bubble and expanding space behind it. We show that this expansion/contraction is but a marginal consequence of the choice made by Alcubierre, and explicitly construct a similar spacetime where no contraction/expansion occurs. Global and optical properties of warp drive spacetimes are also discussed. 
  In the context of a model of space-time foam, made by $N$ wormholes we discuss the possibility of having a foam formed by different configurations. An equivalence between Schwarzschild and Schwarzschild-Anti-de Sitter wormholes in terms of Casimir energy is shown. An argument to discriminate which configuration could represent a foamy vacuum coming from Schwarzschild black hole transition frequencies is used. The case of a positive cosmological constant is also discussed. Finally, a discussion involving charged wormholes leads to the conclusion that they cannot be used to represent a ground state of the foamy type. 
  We study the dynamical evolution of perturbations in the gravitational field of a collapsing fluid star. Specifically, we consider the initial value problem for a massless scalar field in a spacetime similar to the Oppenheimer-Snyder collapse model, and numerically evolve in time the relevant wave equation. Our main objective is to examine whether the phenomenon of parametric amplification, known to be responsible for the strong amplification of primordial perturbations in the expanding Universe, can efficiently operate during gravitational collapse. Although the time-varying gravitational field inside the star can, in principle, support such a process, we nevertheless find that the perturbing field escapes from the star too early for amplification to become significant. To put an upper limit in the efficiency of the amplification mechanism (for a scalar field) we furthermore consider the case of perturbations trapped inside the star for the entire duration of the collapse. In this extreme case, the field energy is typically amplified at the level ~ 1% when the star is about to cross its Schwarszchild radius. Significant amplification is observed at later stages when the star has even smaller radius. Therefore, the conclusion emerging from our simple model is that parametric amplification is unlikely to be of significance during gravitational collapse. Further work, based on more realistic collapse models, is required in order to fully assess the astrophysical importance of parametric amplification. 
  It is shown that a spacetime with collisionless matter evolving from data on a compact Cauchy surface with hyperbolic symmetry can be globally covered by compact hypersurfaces on which the mean curvature is constant and by compact hypersurfaces on which the area radius is constant. Results for the related cases of spherical and plane symmetry are reviewed and extended. The prospects of using the global time coordinates obtained in this way to investigate the global geometry of the spacetimes concerned are discussed. 
  The static string-like solutions of the Abelian Higgs model coupled to dilaton gravity are analyzed and compared to the non-dilatonic case. Except for a special coupling between the Higgs Lagrangian and the dilaton, the solutions are flux tubes that generate a non-asymptotically flat geometry. Any point in parameter space corresponds to two branches of solutions with two different asymptotic behaviors. Unlike the non-dilatonic case, where one branch is always asymptotically conic, in the present case the asymptotic behavior changes continuously along each branch. 
  An important issue in the dynamics of neutron star binaries is whether tidal interaction can cause the individual stars to collapse into black holes during inspiral. To understand this issue better, we study the dynamics of a cluster of collisionless particles orbiting a non-rotating black hole, which is part of a widely separated circular binary. The companion body's electric- and magnetic-type tidal fields distort the black hole and perturb the cluster, eventually causing the cluster to collapse into the hole as the companion spirals in under the influence of gravitational radiation reaction. We find that magnetic-type tidal forces do not significantly influence the evolution of the cluster as a whole. However, individual orbits can be strongly affected by these forces. For example, some orbits are destabilized due to the addition of magnetic-type tidal forces. We find that the most stable orbits are close to the companion's orbital plane and retrograde with respect to the companion's orbit. 
  Whereas semiclassical gravity is based on the semiclassical Einstein equation with sources given by the expectation value of the stress-energy tensor of quantum fields, stochastic semiclassical gravity is based on the Einstein- Langevin equation, which has in addition sources due to the noise kernel. The noise kernel is the expectation value of the stress-energy bi-tensor which describes the fluctuations of quantum matter fields. Here, we first give an axiomatic derivation of the Einstein-Langevin equations and then show how they can also be derived by the original method based on the influence functional. As a first application we solve these equations and compute the two- point correlation functions for the linearized Einstein tensor and for the metric perturbations in a Minkowski background. We then turn to the important issue of the validity of semiclassical gravity by examining the criteria based on the ratio of the variance of fluctuations of the stress-energy tensor of a quantum field to its mean. We show a calculation of these quantities for a massless scalar field in the Minkowski and the Casimir vacua as a function of an intrinsic scale defined by introducing a smeared field or by point separation. Contrary to prior claims, the ratio of variance to mean-squared being of the order unity does not necessarily imply the failure of semiclassical gravity. Next we use the point-separation method to derive a general expression for a regularized noise-kernel for quantum fields in general curved spacetimes. Using these expressions one could investigate a host of important problems in early universe and black hole physics. 
  A new formalism for construction of the stationary solutions is developed for the four-dimensional gravity coupled to the dilaton, Kalb-Ramond and two Maxwell fields in a low-energy heterotic string theory form. The result of generation is automatically invariant in respect to subgroup of the stationary charging symmetry transformations; the generation can be started from the stationary Einstein-Maxwell fields. The formalism is given both in real and new compact complex form, the result of maximal symmetry extension of the stationary Einstein-Maxwell theory to discussing string gravity model is explicitly written down. 
  A bosonic sector of the four-dimensional low-energy heterotic string theory with two Abelian gauge fields is considered in the stationary case. A new 4X4 unitary null-curvature matrix representation of the theory is derived and the corresponding formulation based on the use of a new 2X2 Ernst type matrix complex potential is developed. The group of hidden symmetries is described and classified in the matrix-valued quasi General Relativity form. A subgroup of charging symmetries is constructed and representation which transforms linearly under the action of this symmetry subgroup is established. Also the solution generation procedure based on the application of the total charging symmetry subgroup to the stationary Einstein-Maxwell theory is analyzed. 
  A new formalism for generation of solutions in the consistently truncated low-energy bosonic string theory is developed. This formalism gives a correspondence of the projection type between the theories toroidally compactified from the diverse to three dimensions. Taking the stationary Einstein-dilaton gravity as the theory with a lower dimension, we generate its bosonic string theory extension and calculate the bosonic string theory solution corresponding to the Kerr-NUT one modified by the presence of the charged dilaton field. 
  We construct a new solution subspace for the bosonic string theory toroidally compactified to 3 dimensions. This subspace corresponds to the complex harmonic scalar field coupled to the effective 3--dimensional gravity. We calculate a class of the asymptotically flat and free of the Dirac string peculiarity solutions which describes a Kalb--Ramond dipole source with the generally nontrivial dilaton charge. 
  An action for 3+1-dimensional supergravity genuinely invariant under the Poincare supergroup is proposed. The construction of the action is carried out considering a bosonic lagrangian invariant under both local Lorentz rotations and local Poincare translations as well as under diffeomorphism, and therefore the Poincare algebra closes off-shell. Since the lagrangian is invariant under the Poincare supergroup, the supersymmetry algebra closes off shell without the need of auxiliary fields. 
  The central principle of affine quantum gravity is securing and maintaining the strict positivity of the matrix $\{\hg_{ab}(x)\}$ composed of the spatial components of the local metric operator. On spectral grounds, canonical commutation relations are incompatible with this principle, and they must be replaced by noncanonical, affine commutation relations. Due to the partial second-class nature of the quantum gravitational constraints, it is advantageous to use the recently developed projection operator method, which treats all quantum constraints on an equal footing. Using this method, enforcement of regularized versions of the gravitational operator constraints is formulated quite naturally by means of a novel and relatively well-defined functional integral involving only the same set of variables that appears in the usual classical formulation. It is anticipated that skills and insight to study this formulation can be developed by studying special, reduced-variable models that still retain some basic characteristics of gravity, specifically a partial second-class constraint operator structure. Although perturbatively nonrenormalizable, gravity may possibly be understood nonperturbatively from a hard-core perspective that has proved valuable for specialized models. Finally, developing a procedure to pass to the genuine physical Hilbert space involves several interconnected steps that require careful coordination. 
  Presented cosmological model is 3D brane world sheet moved in extra dimension with variable scale factor. Analysis of the geodesic motion of the test particle gives settle explanation of the Pioneer effect. It is found that for considered metric the solution of the semi-classical Einstein equations with various parameters conforms to isotropic expanded and anisotropic stationary universe. 
  The second post-Newtonian (2PN) contribution to the `plus' and `cross' gravitational wave polarizations associated with gravitational radiation from non-spinning, compact binaries moving in elliptic orbits is computed. The computation starts from our earlier results on 2PN generation, crucially employs the 2PN accurate generalized quasi-Keplerian parametrization of elliptic orbits by Damour, Sch\"afer and Wex and provides 2PN accurate expressions modulo the tail terms for gravitational wave polarizations incorporating effects of eccentricity and periastron precession. 
  Propagation of light in the gravitational field of self-gravitating spinning bodies moving with arbitrary velocities is discussed. The gravitational field is assumed to be "weak" everywhere. Equations of motion of a light ray are solved in the first post-Minkowskian approximation that is linear with respect to the universal gravitational constant $G$. We do not restrict ourselves with the approximation of gravitational lens so that the solution of light geodesics is applicable for arbitrary locations of source of light and observer. This formalism is applied for studying corrections to the Shapiro time delay in binary pulsars caused by the rotation of pulsar and its companion. We also derive the correction to the light deflection angle caused by rotation of gravitating bodies in the solar system (Sun, planets) or a gravitational lens. The gravitational shift of frequency due to the combined translational and rotational motions of light-ray-deflecting bodies is analyzed as well. We give a general derivation of the formula describing the relativistic rotation of the plane of polarization of electromagnetic waves (Skrotskii effect). This formula is valid for arbitrary translational and rotational motion of gravitating bodies and greatly extends the results of previous researchers. Finally, we discuss the Skrotskii effect for gravitational waves emitted by localized sources such as a binary system. The theoretical results of this paper can be applied for studying various relativistic effects in microarcsecond space astrometry and developing corresponding algorithms for data processing in space astrometric missions such as FAME, SIM, and GAIA. 
  We present a scenario for galaxy formation based on the hypothesis of scalar field dark matter. We interpret galaxy formation through the collapse of a scalar field fluctuation. We find that a cosh potential for the self-interaction of the scalar field provides a reasonable scenario for galactic formation, which is in agreement with cosmological observations and phenomenological studies in galaxies. 
  We present a class of exact solutions of Einstein's gravitational field equations describing spherically symmetric and static anisotropic stellar type configurations. The solutions are obtained by assuming a particular form of the anisotropy factor. The energy density and both radial and tangential pressures are finite and positive inside the anisotropic star. Numerical results show that the basic physical parameters (mass and radius) of the model can describe realistic astrophysical objects like neutron stars. 
  A general recipe proposed elsewhere to define, via Noether theorem, the variation of energy for a natural field theory is applied to Einstein-Maxwell theory. The electromagnetic field is analysed in the geometric framework of natural bundles. Einstein-Maxwell theory turns then out to be natural rather than gauge-natural. As a consequence of this assumption a correction term \a la Regge-Teitelboim is needed to define the variation of energy, also for the pure electromagnetic part of the Einstein-Maxwell Lagrangian. Integrability conditions for the variational equation which defines the variation of energy are analysed in relation with boundary conditions on physical data. As an application the first law of thermodynamics for rigidly rotating horizons is obtained. 
  The results of the paper of Verlinde [hep-th/0008140], discussing the holographic principle in a radiation dominated universe, are extended when allowing the cosmic fluid to possess a bulk viscosity. This corresponds to a non-conformally invariant theory. The generalization of the Cardy-Verlinde entropy formula to the case of a viscous universe seems from a formal point of view to be possible, although we question on physical grounds some elements in this kind of theory, especially the manner in which the Casimir energy is evaluated. Our discussion suggests that for non-conformally invariant theories the holographic definition of Casimir energy should be modified. 
  It is shown that the Chern-Simons functional, built in the spinor representation from the initial data on spacelike hypersurfaces, is invariant with respect to infinitesimal conformal rescalings if and only if the vacuum Einstein equations are satisfied. As a consequence, we show that in the phase space the Hamiltonian constraint of vacuum general relativity is the Poisson bracket of the imaginary part of this Chern-Simons functional and Misner's time (essentially the 3-volume). Hence the vacuum Hamiltonian constraint is the condition on the canonical variables that the imaginary part of the Chern- Simons functional be constant along the volume flow. The vacuum momentum constraint can also be reformulated in a similar way as a (more complicated) condition on the change of the imaginary part of the Chern-Simons functional along the flow of York's time. 
  We consider the class of space-time defects investigated by Puntigam and Soleng. These defects describe space-time dislocations and disclinations (cosmic strings), and are in close correspondence to the actual defects that arise in crystals and metals. It is known that in such materials dislocations and disclinations require a small and large amount of energy, respectively, to be created. The present analysis is carried out in the context of the teleparallel equivalent of general relativity (TEGR). We evaluate the gravitational energy of these space-time defects in the framework of the TEGR and find that there is an analogy between defects in space-time and in continuum material systems: the total gravitational energy of space-time dislocations and disclinations (considered as idealized defects) is zero and infinit, respectively. 
  A simple ordinary differential equation is derived governing the red-shifts of wave-fronts propagating through a non-stationary spherically symmetric space-time. Approach to an event horizon corresponds to approach to a fixed point; in general, the phase portrait of the equation illuminates the qualitative features of the geometry. In particular, the asymptotics of the red-shift as a horizon is approached, a critical ingredient of Hawking's prediction of radiation from black holes, are easily brought out. This asympotic behavior has elements in common with the universal behavior near phase transitions in statistical physics. The validity of the Unruh vacuum for the Hawking process can be understood in terms of this universality. The concept of surface gravity is extended to to non-stationary spherically symmetric black holes. Finally, it is shown that in the non-stationary case, Hawking's predicted flux of radiation from a black hole would be modified. 
  The characteristics of the memory of accelerated motion in Minkowski spacetime are discussed within the framework of the nonlocal theory of accelerated observers. Two types of memory are distinguished: kinetic and dynamic. We show that only kinetic memory is acceptable, since dynamic memory leads to divergences for nonuniform accelerated motion. 
  The possibility that a charged particle propagating in a gravitational field described by Brans-Dicke theory of gravity could emit Cerenkov radiation is explored. This process is kinematically allowed depending on parameters occurring in the theory. The Cerenkov effect disappears as the BD parameter omega tends to inftinity, i.e. in the limit in which the Einstein theory is recovered, giving a signature to probe the validity of the Brans-Dicke theory. 
  The Einstein evolution equations are studied in a gauge given by a combination of the constant mean curvature and spatial harmonic coordinate conditions. This leads to a coupled quasilinear elliptic--hyperbolic system of evolution equations. We prove that the Cauchy problem is locally strongly well--posed and that a continuation principle holds. For initial data satisfying the Einstein constraint and gauge conditions, the solutions to the elliptic-hyperbolic system defined by the gauge fixed Einstein evolution equations are shown to give vacuum spacetimes. 
  We discuss the possibility that future gravitational-wave detectors may be able to detect various modes of oscillation of old, cold neutron stars. We argue that such detections would provide unique insights into the superfluid nature of neutron star cores, and could also lead to a much improved understanding of pulsar glitches. Our estimates are based on a detector configuration with several narrowbanded (cryogenic) interferometers operating as a "xylophone" which could lead to high sensitivity at high frequencies. We also draw on recent advances in our understanding of the dynamics of pulsating superfluid neutron star cores. 
  An example of discontinuity of the energy-momentum tensor moving at superluminal velocity is discussed. It is shown that the gravitational Mach cone is formed. The power spectrum of the corresponding Cherenkov radiation is evaluated. 
  A state sum model based on the group SU(1,1) is defined. Investigations of its geometry and asymptotics suggest it is a good candidate for modelling (2+1) Lorentzian quantum gravity. 
  The well known general relativistic Lense-Thirring drag of the orbit of a test particle in the stationary field of a central slowly rotating body is generated, in the weak-field and slow-motion approximation of General Relativity, by a gravitomagnetic Lorentz-like acceleration in the equations of motion of the test particle. In it the gravitomagnetic field is due to the central body's angular momentum supposed to be constant. In the context of the gravitational analogue of the Larmor theorem, such acceleration looks like a Coriolis inertial term in an accelerated frame. In this paper the effect of the variation in time of the central body's angular momentum on the orbit of a test mass is considered. It can be shown that it is analogue to the inertial acceleration due to the time derivative of the angular velocity vector of an accelerated frame. The possibility of detecting such effect in the gravitational field of the Earth with LAGEOS-like satellites is investigated. It turns out that the orbital effects are far too small to be measured. 
  In this paper the possibility of measuring some general relativistic effects in the gravitational field of the Moon via selenodetic missions, with particular emphasis to the future Japanese SELENE mission, is investigated. For a typical selenodetic orbital configuration the post-Newtonian Lense-Thirring gravitomagnetic and the Einstein's gravitoelectric effects on the satellites orbits are calculated and compared to the present-day orbit accuracy of lunar missions. It turns out that for SELENE's Main Orbiter, at present, the gravitoelectric periselenium shift, which is the largest general relativistic effect, is 1 or 2 orders of magnitude smaller than the experimental sensitivity. The systematic error induced by the mismodelled classical periselenium precession due to the first even zonal harmonic J2 of the Moon's non-spherical gravitational potential is 3 orders of magnitude larger than the general relativistic gravitoelectric precession. The estimates of this work could be used for future lunar missions having as their goals relativistic measurements as well. 
  Process Physics models reality as self-organising relational or semantic information using a self-referentially limited neural network model. This generalises the traditional non-process syntactical modelling of reality by taking account of the limitations and characteristics of self-referential syntactical information systems, discovered by Goedel and Chaitin, and the analogies with the standard quantum formalism and its limitations. In process physics space and quantum physics are emergent and unified, and time is a distinct non-geometric process. Quantum phenomena are caused by fractal topological defects embedded in and forming a growing three-dimensional fractal process-space. Various features of the emergent physics are briefly discussed including: quantum gravity, quantum field theory, limited causality and the Born quantum measurement metarule, inertia, time-dilation effects, gravity and the equivalence principle, a growing universe with a cosmological constant, black holes and event horizons, and the emergence of classicality. 
  The current observational and experimental bounds on time variation of the constants of Nature are briefly reviewed. 
  We consider the evolution of a flat Friedmann-Robertson-Walker Universe, filled with a causal bulk viscous cosmological fluid, in the presence of variable gravitational and cosmological constants. The basic equation for the Hubble parameter, generalizing the evolution equation in the case of constant gravitational coupling and cosmological term, is derived, under the supplementary assumption that the total energy of the Universe is conserved. By assuming that the cosmological constant is proportional to the square of the Hubble parameter and a power law dependence of the bulk viscosity coefficient, temperature and relaxation time on the energy density of the cosmological fluid, two classes of exact solutions of the field equations are obtained. In the first class of solutions the Universe ends in an inflationary era, while in the second class of solutions the expansion of the Universe is non-inflationary for all times. In both models the cosmological "constant" is a decreasing function of time, while the gravitational "constant" increases in the early period of evolution of the Universe, tending in the large time limit to a constant value. 
  The scenario that the Universe contracts towards a big crunch and then undergoes a transition to expanding Universe in envisaged in the quantum string cosmology approach. The Wheeler-De Witt equation is solved exactly for an exponential dilaton potential. S-duality invariant cosmological effective action, for type IIB theory, is considered to derive classical solutions and solve WDW equations. 
  This paper is a continuation of earlier work where a classical history theory of pure electrodynamics was developed in which the the history fields have \emph{five} components. The extra component is associated with an extra constraint, thus enlarging the gauge group of histories electrodynamics. In this paper we quantise the classical theory developed previously by two methods. Firstly we quantise the reduced classical history space, to obtain a reduced quantum history theory. Secondly we quantise the classical BRST-extended history space, and use the BRST charge to define a `cohomological' quantum history theory. Finally we show that the reduced history theory is isomorphic, as a history theory, to the cohomological history theory. 
  In a previous paper (gr-qc/0103002), the inflationary universe was described as a quantum growing network (QGN). Here, we propose our view of the QGN as the "ultimate Internet", as it saturates the quantum limits to computation. Also, we enlight some features of the QGN which are related to: i) the problem of causality at the Planck scale, ii) the quantum computational aspects of spacetime foam and decoherence, iii) the cosmological constant problem, iv) the "information loss" puzzle. The resulting picture is a self-organizing system of ultimate Internet-universes. 
  I study a thermodynamical approach to scalar metric perturbations during the inflationary stage. In the power-law expanding universe here studied, I find a negative heat capacity as a manifestation of superexponential growing for the number of states in super Hubble scales. The power spectrum depends on the Gibbons-Hawking and Hagedorn temperatures. 
  We study density perturbations in several cosmological models motivated by string theory. The evolution and the spectra of curvature perturbations ${\cal R}$ are analyzed in the Ekpyrotic scenario and nonsingular string cosmologies. We find that these string-inspired models generally exhibit blue spectra in contrast to standard slow-roll inflationary scenarios. We also clarify the parameter range where ${\cal R}$ is enhanced on superhorizon scales. 
  We discuss the global properties of static, spherically symmetric configurations of a self-gravitating real scalar field $\phi$ in general relativity (GR), scalar-tensor theories (STT) and high-order gravity ($L=f(R)$) in various dimensions. In GR, for fields with arbitrary potentials $V(\phi)$, not necessarily positive-definite, it is shown that the list of all possible types of space-time causal structure in the models under consideration is the same as the one for $\phi = const$. In particular, there are no regular black holes with any asymptotics. These features are extended to STT and $f(R)$ theories, connected with GR by conformal mappings, unless there is a conformal continuation, i.e., a case when a singularity in a solution of GR maps to a regular surface in an alternative theory, and the solution is continued through such a surface. This effect is exemplified by exact solutions in GR with a massless conformal scalar field, considered as a special STT. Necessary conditions for the existence of a conformal continuation are found; they only hold for special choices of STT and high-order theories of gravity. 
  A new class of exact solutions is presented which describes impulsive waves propagating in the Nariai universe. It is constructed using a six-dimensional embedding formalism adapted to the background. Due to the topology of the latter, the wave front consists of two non-expanding spheres. Special sub-classes representing pure gravitational waves (generated by null particles with an arbitrary multipole structure) or shells of null dust are analyzed in detail. Smooth isometries of the metrics are briefly discussed. Furthermore, it is shown that the considered solutions are impulsive members of a more general family of radiative Kundt spacetimes of type-II. A straightforward generalization to impulsive waves in the anti-Nariai and Bertotti-Robinson backgrounds is described. For a vanishing cosmological constant and electromagnetic field, results for well known impulsive pp-waves are recovered. 
  We present a practical method for calculating the local gravitational self-force (often called ``radiation-reaction force'') for a pointlike particle orbiting a Schwarzschild black hole. This is an implementation of the method of {\it mode-sum regularization}, in which one first calculates the (finite) contribution to the force due to each individual multipole mode of the perturbation, and then applies a certain regularization procedure to the mode sum. Here we give the values of all the ``regularization parameters'' required for implementing this regularization procedure, for any geodesic orbit in Schwarzschild spacetime. 
  Applying standard mathematical methods, it is explicitly shown how the Riccati equation for the Hubble parameter H(\eta) of barotropic open FRW cosmologies is connected with a Korteweg-de Vries equation for adiabatic index solitons. It is also shown how one can embed a discrete sequence of adiabatic indices of the type n^2({3/2}\gamma -1)^2 (\gamma \neq 2/3) in the sech FRW adiabatic index soliton 
  This is the first paper in a series aimed to implement boundary conditions consistent with the constraints' propagation in 3D numerical relativity. Here we consider spherically symmetric black hole spacetimes in vacuum or with a minimally coupled scalar field, within the Einstein-Christoffel symmetric hyperbolic formulation of Einstein's equations. By exploiting the characteristic propagation of the main variables and constraints, we are able to single out the only free modes at the outer boundary for these problems. In the vacuum case a single free modes exists which corresponds to a gauge freedom, while in the matter case an extra mode exists which is associated with the scalar field. We make use of the fact that the EC formulation has no superluminal characteristic speeds to excise the singularity. We present a second-order, finite difference discretization to treat these scenarios, where we implement these constraint-preserving boundary conditions, and are able to evolve the system for essentially unlimited times. As a test of the robustness of our approach, we allow large pulses of gauge and scalar field enter the domain through the outer boundary. We reproduce expected results, such as trivial (in the physical sense) evolution in the vacuum case (even in gauge- dynamical simulations), and the tail decay for the scalar field. 
  The oft-neglected issue of the causal structure in the flat spacetime approach to Einstein's theory of gravity is considered. Consistency requires that the flat metric's null cone be respected, but this does not happen automatically. After reviewing the history of this problem, we introduce a generalized eigenvector formalism to give a kinematic description of the relation between the two null cones, based on the Segre' classification of symmetric rank 2 tensors with respect to a Lorentzian metric. Then we propose a method to enforce special relativistic causality by using the naive gauge freedom to restrict the configuration space suitably. A set of new variables just covers this smaller configuration space and respects the flat metric's null cone automatically. In this smaller space, gauge transformations do not form a group, but only a groupoid. Respecting the flat metric's null cone ensures that the spacetime is globally hyperbolic, indicating that the Hawking black hole information loss paradox does not arise. 
  We present a new 3D SPH code which solves the general relativistic field + hydrodynamics equations in the conformally flat approximation. Several test cases are considered to test different aspects of the code. We finally apply then the code to the coalescence of a neutron star binary system. The neutron stars are modeled by a polytropic equation of state (EoS) with adiabatic indices $\Gamma=2.0$, $\Gamma=2.6$ and $\Gamma=3.0$. We calculate the gravitational wave signals, luminosities and frequency spectra by employing the quadrupole approximation for emission and back reaction in the slow motion limit. In addition, we consider the amount of ejected mass. 
  How to make compatible both boundary and gauge conditions for generally covariant theories using the gauge symmetry generated by first class constraints is studied. This approach employs finite gauge transformations in contrast with previous works which use infinitesimal ones. Two kinds of variational principles are taken into account; the first one features non-gauge-invariant actions whereas the second includes fully gauge-invariant actions. Furthermore, it is shown that it is possible to rewrite fully gauge-invariant actions featuring first class constraints quadratic in the momenta into first class constraints linear in the momenta (and homogeneous in some cases) due to the full gauge invariance of their actions. This shows that the gauge symmetry present in generally covariant theories having first class constraints quadratic in the momenta is not of a different kind with respect to the one of theories with first class constraints linear in the momenta if fully gauge-invariant actions are taken into account for the former theories. These ideas are implemented for the parametrized relativistic free particle, parametrized harmonic oscillator, and the SL(2,R) model. 
  In this review we examine the dynamics and gravitational wave detectability of rotating strained neutron stars. The discussion is divided into two halves: triaxial stars, and precessing stars. We summarise recent work on how crustal strains and magnetic fields can sustain triaxiality, and suggest that Magnus forces connected with pinned superfluid vortices might contribute to deformation also. The conclusions that could be drawn following the successful gravitational wave detection of a triaxial star are discussed, and areas requiring further study identified. The latest ideas regarding free precession are then outlined, and the recent suggestion of Middleditch et al (2000a,b) that the remnant of SN1987A contains a freely precessing star, spinning-down by gravitational wave energy loss, is examined critically. We describe what we would learn about neutron stars should the gravitational wave detectors prove this hypothesis to be correct. 
  We describe a new interactive database (GRDB) of geometric objects in the general area of differential geometry. Database objects include, but are not restricted to, exact solutions of Einstein's field equations. GRDB is designed for researchers (and teachers) in applied mathematics, physics and related fields. The flexible search environment allows the database to be useful over a wide spectrum of interests, for example, from practical considerations of neutron star models in astrophysics to abstract space-time classification schemes. The database is built using a modular and object-oriented design and uses several Java technologies (e.g. Applets, Servlets, JDBC). These are platform-independent and well adapted for applications developed to run over the World Wide Web. GRDB is accompanied by a virtual calculator (GRTensorJ), a graphical user interface to the computer algebra system GRTensorII used to perform online coordinate, tetrad or basis calculations. The highly interactive nature of GRDB allows for systematic internal self-checking and a minimization of the required internal records. 
  Torsion gravitational effects in the quantum interference of charged particles are investigated. The influence of axial torsion in the Schiff-Banhill effect (SB) inside a metallic shell is given. The effect of torsion on the surface of the earth on (SB) experiment is estimated. Torsion gravity effects on the Sagnac phase-shift of neutron interferometry are also computed. 
  We consider a generalization of the Lemaitre-Tolman-Bondi (LTB) solutions by keeping the LTB metric but replacing its dust matter source by an imperfect fluid with anisotropic pressure $\Pi_{ab} $. Assuming that total matter-energy density $\rho$ is the sum of a rest mass term, $\rhom$, plus a radiation $\rhor=3p$ density where $p$ is the isotropic pressure, Einstein's equations are fully integrated without having to place any previous assumption on the form of $\Pi_{ab} $. Three particular cases of interest are contained: the usual LTB dust solutions (the dust limit), a class of FLRW cosmologies (the homogeneous limit) and of the Vaydia solution (the vacuum limit). Initial conditions are provided in terms of suitable averages and contrast functions of the initial densities of $\rhom, \rhor$ and the 3-dimensional Ricci scalar along an arbitrary initial surface $t=t_i$. We consider the source of the models as an interactive radiation-matter mixture in local thermal equilibrium that must be consistent with causal Extended Irreversible Thermodynamics (hence $\Pi_{ab} $ is shear viscosity). Assuming near equilibrium conditions associated with small initial density and curvature contrasts, the evolution of the models is qualitatively similar to that of adiabatic perturbations on a matter plus radiation FLRW background. We show that initial conditions exist that lead to thermodynamically consistent models, but only for the full transport equation of Extended Irreversible Thermodynamics. These interactive mixtures provide a reasonable approximation to a dissipative 'tight coupling' characteristic of radiation-matter mixtures in the radiative pre-decoupling era. 
  We study the relationship between space-time-matter (STM) and brane theories. These two theories look very different at first sight, and have different motivation for the introduction of a large extra dimension. However, we show that they are equivalent to each other. First we demonstrate that STM predicts local and non-local high-energy corrections to general relativity in 4D, which are identical to those predicted by brane-world models. Secondly, we notice that in brane models the usual matter in 4D is a consequence of the dependence of five-dimensional metrics on the extra coordinate. If the 5D bulk metric is independent of the extra dimension, then the brane is void of matter. Thus, in brane theory matter and geometry are unified, which is exactly the paradigm proposed in STM. Consequently, these two 5D theories share the same concepts and predict the same physics. This is important not only from a theoretical point of view, but also in practice. We propose to use a combination of both methods to alleviate the difficult task of finding solutions on the brane. We show an explicit example that illustrate the feasibility of our proposal. 
  We consider covariant metric theories of coupled gravity-matter systems satisfying the following two conditions: First, it is assumed that, by a hyperbolic reduction process, a system of first order symmetric hyperbolic partial differential equations can be deduced from the matter field equations. Second, gravity is supposed to be coupled to the matter fields by requiring that the Ricci tensor is a smooth function of the basic matter field variables and the metric. It is shown then that the ``time'' evolution of these type of gravity-matter systems preserves the symmetries of initial data specifications. 
  We consider the collision of self-gravitating n-branes in a (n+2)-dimensional spacetime. We show that there is a geometrical constraint which can be expressed as a simple sum rule for angles characterizing Lorentz boosts between branes and the intervening spacetime regions. This constraint can then be re-interpreted as either energy or momentum conservation at the collision. 
  In this first article of a series on alternative cosmological models we present an extended version of a cosmological model in Weyl-Cartan spacetime. The new model can be viewed as a generalization of a model developed earlier jointly with Tresguerres. Within this model the non-Riemannian quantities, i.e. torsion $T^{\alpha}$ and nonmetricity $Q_{\alpha \beta}$, are proportional to the Weyl 1-form. The hypermomentum $\Delta_{\alpha \beta}$ depends on our ansatz for the nonmetricity and vice versa. We derive the explicit form of the field equations for different cases and provide solutions for a broad class of parameters. We demonstrate that it is possible to construct models in which the non-Riemannian quantities die out with time. We show how our model fits into the more general framework of metric-affine gravity (MAG). 
  Analyzing two simple experimental situations we show that from Newton's law of gravitation and Special Relativity it follows that the motion of particle in an external gravitational field can be described in terms of effective spatial fields which satisfy Maxwell-like system of equations and propagate with the speed of light. The description is adequate in a linear approximation in gravitational field and in a first order in v^2/c^2. 
  In it's usual presentation, classical mechanics appears to give time a very special role. But it is well known that mechanics can be formulated so as to treat the time variable on the same footing as the other variables in the extended configuration space. Such covariant formulations are natural for relativistic gravitational systems, where general covariance conflicts with the notion of a preferred physical-time variable. The standard presentation of quantum mechanics, in turns, gives again time a very special role, raising well known difficulties for quantum gravity. Is there a covariant form of (canonical) quantum mechanics? We observe that the preferred role of time in quantum theory is the consequence of an idealization: that measurements are instantaneous. Canonical quantum theory can be given a covariant form by dropping this idealization. States prepared by non-instantaneous measurements are described by "spacetime smeared states". The theory can be formulated in terms of these states, without making any reference to a special time variable. The quantum dynamics is expressed in terms of the propagator, an object covariantly defined on the extended configuration space. 
  The integration of the Einstein equations split into the solution of constraints on an initial space like 3 - manifold, an essentially elliptic system, and a system which will describe the dynamical evolution, modulo a choice of gauge. We prove in this paper that the simplest gauge choice leads to a system which is causal, but hyperbolic non strict in the sense of Leray - Ohya. We review some strictly hyperbolic systems obtained recently. 
  From the equivalence principle (EP) and experiments on gravitational (G) time dilation (GTD) it is proved that the standards of observers located in different ``distances'' from the earth are physically different with respect to each other. Thus the current mathematical relationships between their measurements are physically inhomogeneous. This has caused fundamental errors in gravitation, cosmology and astrophysics. The true transformations between basic parameters of bodies located in different field positions, derived from just experimental facts, are used to test fundamental hypotheses in current literature. G fields do not exchange energy with bodies and radiation, but just momentum. The G energy comes the bodies. The average relative distances and cosmological redshifts in the universe cannot change after universe expansion because the average increase of distances (G potentials) would change the sizes of particles in identical proportion. Locally, atoms must be evolving, indefinitely, in closed cycles between states of gas and linear black hole. The last ones, after recovering energy, must explode thus regenerating gas. Most of the universe must be in state of black galaxy cooled down by linear black holes. They must account for the CMBR. The new scenarios are also explained by using particle models consistent with the EP. 
  We study the behavior of non-relativistic quantum particles interacting with different potentials in the space-times generated by a cosmic string and a global monopole. We find the energy spectra in the presence of these topological defects and show how they differ from their free space-time values. 
  The consistent histories approach to quantum mechanics is traditionally based on linearly ordered sequences of events. We extend the histories formalism to sets of events whose causal ordering is described by directed acyclic graphs. The need for a global time is eliminated and our construction reflects the causal structure faithfully. 
  A distinguishable physical property between a naked singularity and a black-hole, formed during a gravitational collapse has important implications for both experimental and theoretical relativity. We examine the energy radiated during the spherically symmetric Gravitational collapse in this context within the framework of general relativity. It is shown that total energy radiated(Mass Loss) during the collapse ending in a naked singularity scenario cannot be more than the case when a collapse scenario ends in a black-hole. In cases of interest(for example stars having same mass, size, and internal composition) considered here the total energy released in the collapse ending in a black hole can be considerably more than the case otherwise. 
  We provide a simple proof of the topological invariance of the Turaev-Viro model (corresponding to simplicial 3d pure Euclidean gravity with cosmological constant) by means of a novel diagrammatic formulation of the state sum models for quantum BF-theories. Moreover, we prove the invariance under more general conditions allowing the state sum to be defined on arbitrary cellular decompositions of the underlying manifold. Invariance is governed by a set of identities corresponding to local gluing and rearrangement of cells in the complex. Due to the fully algebraic nature of these identities our results extend to a vast class of quantum groups. The techniques introduced here could be relevant for investigating the scaling properties of non-topological state sums, being proposed as models of quantum gravity in 4d, under refinement of the cellular decomposition. 
  We present a general approach to the analysis of gauge stability of 3+1 formulations of General Relativity (GR). Evolution of coordinate perturbations and the corresponding perturbations of lapse and shift can be described by a system of eight quasi-linear partial differential equations. Stability with respect to gauge perturbations depends on a choice of gauge and a background metric, but it does not depend on a particular form of a 3+1 system if its constrained solutions are equivalent to those of the Einstein equations. Stability of a number of known gauges is investigated in the limit of short-wavelength perturbations. All fixed gauges except a synchronous gauge are found to be ill-posed. A maximal slicing gauge and its parabolic extension are shown to be ill-posed as well. A necessary condition is derived for well-posedness of metric-dependent algebraic gauges. Well-posed metric-dependent gauges are found, however, to be generally unstable. Both instability and ill-posedness are associated with perturbations of physical accelerations of reference frames. 
  The instability in the r-modes of rotating neutron stars can (in principle) emit substantial amounts of gravitational radiation (GR) which might be detectable by LIGO and similar detectors. Estimates are given here of the detectability of this GR based the non-linear simulations of the r-mode instability by Lindblom, Tohline and Vallisneri. The burst of GR produced by the instability in the rapidly rotating 1.4 solar mass neutron star in this simulation is fairly monochromatic with frequency near 960 Hz and duration about 100 s. A simple analytical expression is derived here for the optimal S/N for detecting the GR from this type of source. For an object located at a distance of 20 Mpc we estimate the optimal S/N to be in the range 1.2 to about 12.0 depending on the LIGO II configuration. 
  Hollow spheres have the same theoretical capabilities as the usual solid ones, since they share identical symmetries. The hollow sphere is however more flexible, as thickness is an additional parameter one can vary to approach given specifications. I will briefly discuss the more relevant properties of the hollow sphere as a GW detector (frequencies, cross sections), and suggest some scenarios where it can generate significant astrophysical information. 
  The final state--black hole or naked singularity--of the gravitational collapse of a marginally bound matter configuration in the presence of tangential stresses is classified, in full generality, in terms of the initial data and equation of state. If the tangential pressure is sufficiently strong, configurations that would otherwise evolve to a spacelike singularity, result in a locally naked singularity, both in the homogeneous and in the general, inhomogeneous density case. 
  In this talk I show how to canonically quantize a massless scalar field in the background of a Schwarzschild black hole in Lema\^itre coordinates and then present a simplified derivation of Hawking radiation based upon this procedure. The key result of quantization procedure is that the Hamiltonian of the system is explicitly time dependent and so problem is intrinsically non-static. From this it follows that, although a unitary time-development operator exists, it is not useful to talk about vacuum states; rather, one should focus attention on steady state phenomena such as the Hawking radiation. In order to clarify the approximations used to study this problem I begin by discussing the related problem of the massless scalar field theory calculated in the presence of a moving mirror. 
  Stationary axisymmetric spacetimes containing a pair of oppositely-rotating periodically-intersecting circular geodesics allow the study of various so-called `clock effects' by comparing either observer or geodesic proper time periods of orbital circuits defined by the observer or the geodesic crossing points. This can be extended from a comparison of clocks to a comparison of parallel transported vectors, leading to the study of special elements of the spacetime holonomy group. The band of holonomy invariance found for a dense subset of special geodesic orbits outside a certain radius in the static case does not exist in the nonstatic case. In the Kerr spacetime the dimensionless frequencies associated with parallel transport rotations can be expressed as ratios of the proper and average coordinate periods of the circular geodesics. 
  A local current of particle density for scalar fields in curved background is constructed. The current depends on the choice of a two-point function. There is a choice that leads to local non-conservation of the current in a time-dependent gravitational background, which describes local particle production consistent with the usual global description based on the Bogoliubov transformation. Another choice, which might be the most natural one, leads to the local conservation of the current. 
  A complete description of the linearized gravitational field on a flat background is given in terms of gauge-independent quasilocal quantities. This is an extension of the results from gr-qc/9801068. Asymptotic spherical quasilocal parameterization of the Weyl field and its relation with Einstein equations is presented. The field equations are equivalent to the wave equation. A generalization for Schwarzschild background is developed and the axial part of gravitational field is fully analyzed. In the case of axial degree of freedom for linearized gravitational field the corresponding generalization of the d'Alembert operator is a Regge-Wheeler equation. Finally, the asymptotics at null infinity is investigated and strong peeling property for axial waves is proved. 
  The low energy physics as predicted by strings can be expressed in two (conformally related) different variables, usually called {\em frames}. The problem is raised as to whether it is physically possible in some situations to tell one from the other. 
  The paper contains a proposed experiment for testing the angular momentum effect on the propagation of light around a rotating mass. The idea is to use a rotating spherical laboratory-scale shell, around which two mutually orthogonal light guides are wound acting as the arms of an interferometer. Numerical estimates show that time of flight differences between the equatorial and polar guides could be in the order of $\sim 10^{-20}$ s per loop. Using a few thousands loops the time difference is brought in the range of feasible interference measurements. 
  We discuss the dynamics of linear, scalar perturbations in an almost Friedmann-Robertson-Walker braneworld cosmology of Randall-Sundrum type II using the 1+3 covariant approach. We derive a complete set of frame-independent equations for the total matter variables, and a partial set of equations for the non-local variables which arise from the projection of the Weyl tensor in the bulk. The latter equations are incomplete since there is no propagation equation for the non-local anisotropic stress. We supplement the equations for the total matter variables with equations for the independent constituents in a cold dark matter cosmology, and provide solutions in the high and low-energy radiation-dominated phase under the assumption that the non-local anisotropic stress vanishes. These solutions reveal the existence of new modes arising from the two additional non-local degrees of freedom. Our solutions should prove useful in setting up initial conditions for numerical codes aimed at exploring the effect of braneworld corrections on the cosmic microwave background (CMB) power spectrum. As a first step in this direction, we derive the covariant form of the line of sight solution for the CMB temperature anisotropies in braneworld cosmologies, and discuss possible mechanisms by which braneworld effects may remain in the low-energy universe. 
  Dirac's large number hypothesis is motivated by certain scaling transformations that relate the parameters of macro and microphysics. We show that these relations can actually be explained in terms of the holographic $N$ bound conjectured by Bousso and a series of purely cosmological observations, namely, that our universe is spatially homogeneous, isotropic, and flat to a high degree of approximation and that the cosmological constant dominates the energy density at present. 
  We study the renormalized energy-momentum tensor (EMT) of the inflaton fluctuations in rigid space-times during the slow-rollover regime for chaotic inflation with a mass term. We use dimensional regularization with adiabatic subtraction and introduce a novel analytic approximation for the inflaton fluctuations which is valid during the slow-rollover regime. Using this approximation we find a scale invariant spectrum for the inflaton fluctuations in a rigid space-time, and we confirm this result by numerical methods. The resulting renormalized EMT is covariantly conserved and agrees with the Allen-Folacci result in the de Sitter limit, when the expansion is exactly linearly exponential in time. We analytically show that the EMT tensor of the inflaton fluctuations grows initially in time, but saturates to the value H^2 H(0)^2, where H is the Hubble parameter and H(0) is its value when inflation has started. This result also implies that the quantum production of light scalar fields (with mass smaller or equal to the inflaton mass) in this model of chaotic inflation depends on the duration of inflation and is larger than the usual result extrapolated from the de Sitter result. 
  The obstruction for the existence of an energy momentum tensor for the gravitational field is connected with differential-geometric features of the Riemannian manifold. It has not to be valid for alternative geometrical structures. A teleparallel manifold is defined as a parallelizable differentiable 4D-manifold endowed with a class of smooth coframe fields related by global Lorentz, i.e., SO(1,3) transformations. In this article a general free parametric class of teleparallel models is considered. It includes a 1-parameter subclass of viable models with the Schwarzschild coframe solution. A new form of the coframe field equation is derived from the general teleparallel Lagrangian by introducing the notion of a 3-parameter conjugate field strength $\F^a$. The field equation turns out to have a form completely similar to the Maxwell field equation $d*\F^a=\T^a$. By applying the Noether procedure, the source 3-form $\T^a$ is shown to be connected with the diffeomorphism invariance of the Lagrangian. Thus the source $\T^a$ of the coframe field is interpreted as the total conserved energy-momentum current. The energy-momentum tensor for coframe is defined. The total energy-momentum current of a system of a coframe and a material fields is conserved. Thus a redistribution of the energy-momentum current between a material and a coframe (gravity) fields is possible in principle, unlike as in the standard GR. For special values of parameters, when the GR is reinstated, the energy-momentum tensor gives up the invariant sense, i.e., becomes a pseudo-tensor. Thus even a small-parametric change of GR turns it into a well defined Lagrangian theory. 
  Is there a version of the notions of "state" and "observable" wide enough to apply naturally and in a covariant manner to relativistic systems? I discuss here a tentative answer. 
  We describe the Kerr black hole in the ingoing and outgoing Kerr-Schild horizon penetrating coordinates. Starting from the null vector naturally defined in these coordinates, we construct the null tetrad for each case, as well as the corresponding geometrical quantities allowing us to explicitly derive the field equations for the ${\Psi_0}^{(1)}$ and ${\Psi_4}^{(1)}$ perturbed scalar projections of the Weyl tensor, including arbitrary source terms. This perturbative description, including arbitrary sources, described in horizon penetrating coordinates is desirable in several lines of research on black holes, and contributes to the implementation of a formalism aimed to study the evolution of the space time in the region where two black holes are close. 
  Using a recently developed perturbation formalism based on curvature quantities, we complete our investigation of the linear stability of black holes and solitons with Yang-Mills hair and a negative cosmological constant. We show that those solutions which have no linear instabilities under odd- and even-parity spherically symmetric perturbations remain stable under even-parity, linear, non-spherically symmetric perturbations. Together with the result from a previous work, we have therefore established the existence of stable hairy black holes and solitons with anti-de Sitter asymptotic. 
  This paper presents an efficient technique for finding Killing, homothetic, or even proper conformal Killing vectors in the Newman-Penrose (NP) formalism. Leaning on, and extending, results previously derived in the GHP formalism we show that the (conformal) Killing equations can be replaced by a set of equations involving the commutators of the Lie derivative with the four NP differential operators applied to the four coordinates.   It is crucial that these operators refer to a preferred tetrad relative to the (conformal) Killing vectors, a notion to be defined. The equations can then be readily solved for the Lie derivative of the coordinates, i.e. for the components of the (conformal) Killing vectors. Some of these equations become trivial if some coordinates are chosen intrinsically (where possible), i.e. if they are somehow tied to the Riemann tensor and its covariant derivatives.   If part of the tetrad, i.e. part of null directions and gauge, can be defined intrinsically then that part is generally preferred relative to any Killing vector. This is also true relative to a homothetic vector or a proper conformal Killing vector provided we make a further restriction on that intrinsic part of the tetrad. If because of null isotropy or gauge isotropy, where part of the tetrad cannot even in principle be defined intrinsically, the tetrad is defined only up to (usually) one null rotation parameter and/or a gauge factor, then the NP-Lie equations become slightly more involved and must be solved for the Lie derivative of the null rotation parameter and/or of the gauge factor as well. However, the general method remains the same and is still much more efficient than conventional methods.   Several explicit examples are given to illustrate the method. 
  We describe the quantum and classical radiation by a uniformly accelerating point source in terms of the elementary processes of absorption and emission of Rindler scalar photons of the Fulling-Davies-Unruh bath observed by a co-accelerating observer.To this end we compute the emission rate by a DeWitt detector of a Minkowski scalar particle with defined transverse momentum per unit of proper time of the source and we show that it corresponds to the induced absorption or spontaneous and induced emission of Rindler photons from the thermal bath. We then take what could be called the inert limit of the DeWitt detector by considering the limit of zero gap energy. As suggested by DeWitt, we identify in this limit the detector with a classical point source and verify the consistency of our computation with the classical result. Finally, we study the behavior of the emission rate in D space-time dimensions in connection with the so called apparent statistics inversion. 
  In the quest to develop viable designs for third-generation optical interferometric gravitational-wave detectors (e.g. LIGO-III and EURO), one strategy is to monitor the relative momentum or speed of the test-mass mirrors, rather than monitoring their relative position. This paper describes and analyzes the most straightforward design for a {\it speed meter interferometer} that accomplishes this -- a design (due to Braginsky, Gorodetsky, Khalili and Thorne) that is analogous to a microwave-cavity speed meter conceived by Braginsky and Khalili. A mathematical mapping between the microwave speed meter and the optical interferometric speed meter is developed and is used to show (in accord with the speed being a Quantum Nondemolition [QND] observable) that {\it in principle} the interferometric speed meter can beat the gravitational-wave standard quantum limit (SQL) by an arbitrarily large amount, over an arbitrarily wide range of frequencies, and can do so without the use of squeezed vacuum or any auxiliary filter cavities at the interferometer's input or output. However, {\it in practice}, to reach or beat the SQL, this specific speed meter requires exorbitantly high input light power. The physical reason for this is explored, along with other issues such as constraints on performance due to optical dissipation. This analysis forms a foundation for ongoing attempts to develop a more practical variant of an interferometric speed meter and to combine the speed meter concept with other ideas to yield a promising LIGO-III/EURO interferometer design that entails low laser power. 
  One method of studying the asymptotic structure of spacetime is to apply Penrose's conformal rescaling technique. In this setting, the Einstein equations for the metric and the conformal factor in the unphysical spacetime degenerate where the conformal factor vanishes, namely at the boundary representing null infinity. This problem can be avoided by means of a technique of H. Friedrich, which replaces the Einstein equations in the unphysical spacetime by an equivalent system of equations which is regular at the boundary. The initial value problem for these equations produces a system of constraint equations known as the conformal constraint equations. This work describes some of the properties of the conformal constraint equations and develops a perturbative method of generating solutions near flat space under certain simplifying assumptions. 
  Hawking evaporation of Klein-Gordon and Dirac particles in a non-stationary Kerr-Newman space-time is investigated by using a method of generalized tortoise coordinate transformation. The location and the temperature of the event horizon of a non-stationary Kerr-Newman black hole are derived. It is shown that the temperature and the shape of the event horizon depend not only on the time but also on the angle. However, the Fermionic spectrum of Dirac particles displays a new spin-rotation coupling effect which is absent from that of Bosonic distribution of scalar particles. The character of this effect is its obvious dependence on different helicity states of particles spin-1/2.   PACS numbers: 04.70.Dy, 97.60.Lf 
  Hawking evaporation of photons in a Vaidya-de Sitter black hole is investigated by using the method of generalized tortoise coordinate transformation. Both the location and the temperature of the event horizon depend on the time. It is shown that Hawking radiation of photons exists only for the complex Maxwell scalar $\phi_0$ in the advanced Eddington-Finkelstein coordinate system. This asymmetry of Hawking radiation for different components of Maxwell fields probably arises from the asymmetry of spacetime in the advanced Eddington-Finkelstein coordinate system. It is shown that the black body radiant spectrum of photons resembles that of Klein-Gordon particles.   PACS numbers: 04.70.Dy, 97.60.Lf 
  A general relation for the total angular momentum of a regular solution of the Einstein-Yang-Mills-Higgs equations is derived. Two different physical configurations, rotating dyons and rotating magnetic dipoles are discussed as particular cases. The issue of rotating pure Einstein-Yang-Mills regular solutions is addressed as well. Based on the results, we conjecture the absence of rotating regular solitons with a net magnetic charge. 
  We discuss the dynamics of a quintessence model involving two coupled scalar fields. The model presents two types of solutions, namely solutions that correspond to eternal and transient acceleration of the universe. In both cases, we obtain values for the cosmological parameters that satisfy current observational bounds as well as the nucleosynthesis constraint on the quintessence energy density. 
  Models of eternal inflation predict a stochastic self-similar geometry of the universe at very large scales and allow existence of points that never thermalize. I explore the fractal geometry of the resulting spacetime, using coordinate-independent quantities. The formalism of stochastic inflation can be used to obtain the fractal dimension of the set of eternally inflating points (the ``eternal fractal''). I also derive a nonlinear branching diffusion equation describing global properties of the eternal set and the probability to realize eternal inflation. I show gauge invariance of the condition for presence of eternal inflation. Finally, I consider the question of whether all thermalized regions merge into one connected domain. Fractal dimension of the eternal set provides a (weak) sufficient condition for merging. 
  We have previously proposed that asymptotically AdS 3D wormholes and black holes can be analytically continued to the Euclidean signature. The analytic continuation procedure was described for non-rotating spacetimes, for which a plane t=0 of time symmetry exists. The resulting Euclidean manifolds turned out to be handlebodies whose boundary is the Schottky double of the geometry of the t=0 plane. In the present paper we generalize this analytic continuation map to the case of rotating wormholes. The Euclidean manifolds we obtain are quotients of the hyperbolic space by a certain quasi-Fuchsian group. The group is the Fenchel-Nielsen deformation of the group of the non-rotating spacetime. The angular velocity of an asymptotic region is shown to be related to the Fenchel-Nielsen twist. This solves the problem of classification of rotating black holes and wormholes in 2+1 dimensions: the spacetimes are parametrized by the moduli of the boundary of the corresponding Euclidean spaces. We also comment on the thermodynamics of the wormhole spacetimes. 
  We present the Fourier Transform of a continuous gravitational wave. We have analysed the data set for one day observation time and our analysis is applicable for arbitrary location of detector and source. We have taken into account the effects arising due to rotational as well as orbital motions of the earth. 
  Our analysis shows how the covariant chaotic behavior characterizing the evolution of the Mixmaster cosmology near the initial singularity can be taken as the semiclassical limit in the canonical quantization performed by the corresponding Hamiltonian representation. 
  Questioning the experimental basis of continuous descriptions of fundamental interactions we discuss classical gravity as an effective continuous first-order approximation of a discrete interaction. The sub-dominant contributions produce a residual interaction that may be repulsive and whose physical meaning is of a correction of the excess contained in the continuous approximation. These residual interactions become important (or even dominate) at asymptotical conditions of very large distances from where there are data (rotation curves of galaxies, inflation, accelerated expansion, etc) and cosmological theoretical motivations that suggest new physics (new forms of interactions) or new forms (dark) of matter and energy. We show that a discrete picture of the world (of matter and of its interactions) produce, as an approximation, the standard continuous picture and more. The flat rotation curve of galaxies, for example, may have a simple and natural explanation. 
  We analyze here the structure of non-radial nonspacelike geodesics terminating in the past at a naked singularity formed as the end state of inhomogeneous dust collapse. The spectrum of outgoing nonspacelike null geodesics is examined analytically. The local and global visibility of the singularity is also examined by integrating numerically the null geodesics equations. The possible implications of existence of such families towards the appearance of the star in late stages of gravitational collapse are considered. It is seen that the outgoing non-radial geodesics give an appearance to the naked central singularity as that of an expanding ball whose radius reaches a maximum before the star goes within its apparent horizon. The radiated energy (along the null geodesics) is shown to decay very sharply in the neighbourhood of the singularity. Thus the total energy escaping via non-radial null geodesics from the naked central singularity vanishes in the scenario considered here. 
  Irrespective of local conditions imposed on the metric, any extendible spacetime U has a maximal extension containing no closed causal curves outside the chronological past of U. We prove this fact and interpret it as impossibility (in classical general relativity) of the time machines, insofar as the latter are defined to be causality-violating regions created by human beings (as opposed to those appearing spontaneously). 
  We consider the quantum vacuum of fermionic field in the presence of a black-hole background as a possible candidate for the stabilized black hole. The stable vacuum state (as well as thermal equilibrium states with arbitrary temperature) can exist if we use the Painlev\'e-Gullstrand description of the black hole, and the superluminal dispersion of the particle spectrum at high energy, which is introduced in the free-falling frame. Such choice is inspired by the analogy between the quantum vacuum and the ground state of quantum liquid, in which the event horizon for the low-energy fermionic quasiparticles also can arise. The quantum vacuum is characterized by the Fermi surface, which appears behind the event horizon. We do not consider the back reaction, and thus there is no guarantee that the stable black hole exists. But if it does exist, the Fermi surface behind the horizon would be the necessary attribute of its vacuum state. We also consider exact discrete spectrum of fermions inside the horizon which allows us to discuss the problem of fermion zero modes. 
  We discuss astrophysical implications of $\kappa$-Minkowski space-time, in which there appears space-time noncommutativity. We first derive a velocity formula for particles based on the motion of a wave packet. The result is that a massless particle moves at a constant speed as in the usual Minkowski space-time, which implies that an arrival time analysis by $\gamma$-rays from Markarian (Mk) 421 does not exclude space-time noncommutativity. Based on this observation, we analyze reaction processes in $\kappa$-Minkowski space-time which are related to the puzzling detections of extremely high-energy cosmic rays above the Greisen-Zatsepin-Kuzmin cutoff and of high-energy ($\sim$20 TeV) $\gamma$-rays from Mk 501. 
  We have developed a new tool for numerical work in General Relativity: GRworkbench. While past tools have been ad hoc, GRworkbench closely follows the framework of Differential Geometry to provide a robust and general way of computing on analytically defined space-times. We discuss the relationship between Differential Geometry and C++ classes in GRworkbench, and demonstrate their utility. 
  We discuss and prove a theorem which asserts that any n-dimensional semi-Riemannian manifold can be locally embedded in a (n+1)-dimensional space with a non-degenerate Ricci tensor which is equal, up to a local analytic diffeomorphism, to the Ricci tensor of an arbitrary specified space. This may be regarded as a further extension of the Campbell-Magaard theorem. We highlight the significance of embedding theorems of increasing degrees of generality in the context of higher dimensional spacetimes theories and illustrate the new theorem by establishing the embedding of a general class of Ricci-flat spacetimes. 
  In a previous paper [gr-qc/0104001; Class. Quant. Grav. 18 (2001) 3595-3610] we have shown that the occurrence of curved spacetime ``effective Lorentzian geometries'' is a generic result of linearizing an arbitrary classical field theory around some non-trivial background configuration. This observation explains the ubiquitous nature of the ``analog models'' for general relativity that have recently been developed based on condensed matter physics. In the simple (single scalar field) situation analyzed in our previous paper, there is a single unique effective metric; more complicated situations can lead to bi-metric and multi-metric theories. In the present paper we will investigate the conditions required to keep the situation under control and compatible with experiment -- either by enforcing a unique effective metric (as would be required to be strictly compatible with the Einstein Equivalence Principle), or at the worst by arranging things so that there are multiple metrics that are all ``close'' to each other (in order to be compatible with the {\Eotvos} experiment). The algebraically most general situation leads to a physical model whose mathematical description requires an extension of the usual notion of Finsler geometry to a Lorentzian-signature pseudo-Finsler geometry; while this is possibly of some interest in its own right, this particular case does not seem to be immediately relevant for either particle physics or gravitation. The key result is that wide classes of theories lend themselves to an effective metric description. This observation provides further evidence that the notion of ``analog gravity'' is rather generic. 
  We study a fundamental issue in cosmology: Whether we can rely on a cosmological model to understand the real history of the Universe. This fundamental, still unresolved issue is often called the ``model-fitting problem (or averaging problem) in cosmology''. Here we analyze this issue with the help of the spectral scheme prepared in the preceding studies.   Choosing two specific spatial geometries that are very close to each other, we investigate explicitly the time evolution of the spectral distance between them; as two spatial geometries, we choose a flat 3-torus and a perturbed geometry around it, mimicking the relation of a ``model universe'' and the ``real Universe''. Then we estimate the spectral distance between them and investigate its time evolution explicitly. This analysis is done efficiently by making use of the basic results of the standard linear structure-formation theory.   We observe that, as far as the linear perturbation of geometry is valid, the spectral distance does not increase with time prominently,rather it shows the tendency to decrease. This result is compatible with the general belief in the reliability of describing the Universe by means of a model, and calls for more detailed studies along the same line including the investigation of wider class of spacetimes and the analysis beyond the linear regime. 
  The nature of time vis-a-vis relativity is critically examined. Based on the author's space-time interaction hypothesis of late 1970's, cosmological model with time-varying velocity of light is discussed. 
  We extend the direct quantum approach of the standard FRW cosmology from 4D to 5D and obtain a Hamiltonian formulation for a wave-like 5D FRW cosmology. Using a late-time approximation we isolate out a y-part from the full wave function of the 5D Universe. Then we find that the compactness of the fifth dimension y yields a quantized spectrum for the momentum $P_{5}$ along the fifth dimension, and we show that the whole space-part of the wave function of the 5D Universe satisfies a two-dimensional Schr\"odinger equation. 
  The problem of fixing measure in the path integral for the Regge-discretised gravity is considered from the viewpoint of it's "best approximation" to the already known formal continuum general relativity (GR) measure. A rigorous formulation may consist in comparing functional Fourier transforms of the measures, i.e. characteristic or generating functionals, and requiring these to coincide on some dense set in the functional space. The possibility for such set to exist is due to the Regge manifold being a particular case of general Riemannian one (Regge calculus is a minisuperspace theory). The two versions of the measure are obtained depending on what metric tensor, covariant or contravariant one, is taken as fundamental field variable. The closed expressions for the measure are obtained in the two simple cases of Regge manifold. These turn out to be quite reasonable one of them indicating that appropriately defined continuum limit of the Regge measure would reproduce the original continuum GR measure. 
  We develop a theory in which relic gravitational waves and primordial density perturbations are generated by strong variable gravitational field of the early Universe. The generating mechanism is the superadiabatic (parametric) amplification of the zero-point quantum oscillations. The generated fields have specific statistical properties of squeezed vacuum quantum states. Macroscopically, squeezing manifests itself in a non-stationary character of variances and correlation functions of the fields, the periodic structures of the metric power spectra, and, as a consequence, in oscillatory behavior of the higher order multipoles C_l of the cosmic microwave background anisotropy. We start with the gravitational wave background and then apply the theory to primordial density perturbations. We derive an analytical formula for the positions of peaks and dips in the angular power spectrum l(l+1)C_l as a function of l. This formula shows that the values of l at the peak positions are ordered in the proportion 1:3:5:..., whereas at the dips they are ordered as 1:2:3:.... We compare the derived positions with the actually observed features, and find them to be in reasonably good agreement. It appears that the observed structure is better described by our analytical formula based on the (squeezed) metric perturbations associated with the primordial density perturbations, rather than by the acoustic peaks reflecting the existence of plasma sound waves at the last scattering surface. We formulate a forecast for other features in the angular power spectrum, that may be detected by the advanced observational missions, such as MAP and PLANCK. We tentatively conclude that the observed structure is a macroscopic manifestation of squeezing in the primordial metric perturbations. 
  We consider spherically symmetric matter configurations on a four dimensional "brane" embedded in a five dimensional $Z_2$-symmetric "bulk". We write the junction conditions between the interior and exterior of these "stars", treat a couple of static examples in order to point out the differences with ordinary four dimensional Einstein gravity, consider briefly a collapse situation and conclude with the importance of a global view including asymptotic and regularity conditions in the bulk. 
  The aim of this paper is to present a governing equation for first order axial metric perturbations of general, not necessarily static, spherically symmetric spacetimes. Under the non-restrictive assumption of axisymmetric perturbations, the governing equation is shown to be a two-dimensional wave equation where the wave function serves as a twist potential for the axisymmetry generating Killing vector. This wave equation can be written in a form which is formally a very simple generalization of the Regge-Wheeler equation governing the axial perturbations of a Schwarzschild black hole, but in general the equation is accompanied by a source term related to matter perturbations. The case of a viscous fluid is studied in particular detail. 
  Geometrical structures intrinsic to non-expanding, weakly isolated and isolated horizons are analyzed and compared with structures which arise in other contexts within general relativity, e.g., at null infinity. In particular, we address in detail the issue of singling out the preferred normals to these horizons required in various applications. This work provides powerful tools to extract invariant, physical information from numerical simulations of the near horizon, strong field geometry. While it complements the previous analysis of laws governing the mechanics of weakly isolated horizons, prior knowledge of those results is not assumed. 
  A simple model of spacetime foam, made by two different types of wormholes in a semiclassical approximation, is taken under examination: one type is a collection of $N_{w}$ Schwarzschild wormholes, while the other one is made by Schwarzschild-Anti-de Sitter wormholes. The area quantization related to the entropy via the Bekenstein-Hawking formula hints a possible selection between the two configurations. Application to the charged black hole are discussed. 
  The differential acceleration between a rotating mechanical gyroscope and a non-rotating one is directly measured by using a double free-fall interferometer, and no apparent differential acceleration has been observed at the relative level of 2x10{-6}. It means that the equivalence principle is still valid for rotating extended bodies, i.e., the spin-gravity interaction between the extended bodies has not been observed at this level. Also, to the limit of our experimental sensitivity, there is no observed asymmetrical effect or anti-gravity of the rotating gyroscopes as reported by hayasaka et al. 
  As a candidate for dark matter in galaxies, we study an SU(3) triplet of complex scalar fields which are non-minimally coupled to gravity. In the spherically symmetric static spacetime where the flat rotational velocity curves of stars in galaxies can be explained, we find simple solutions of scalar fields with SU(3) global symmetry broken to U(1) X U(1), in an exponential scalar potential, which will be useful in a quintessence model of the late-time acceleration of the Universe. 
  We examine the holography bound suggested by Bousso in his covariant entropy conjecture, and argue that it is violated because his notion of light sheet is too generous. We suggest its replacement by a weaker bound. 
  It is known that the Einstein field equations in five dimensions admit more general spherically symmetric black holes on the brane than four-dimensional general relativity. We propose two families of analytic solutions (with g_tt\not=-1/g_rr), parameterized by the ADM mass and the PPN parameter beta, which reduce to Schwarzschild for beta=1. Agreement with observations requires |\beta-1| |\eta|<<1. The sign of eta plays a key role in the global causal structure, separating metrics which behave like Schwarzschild (eta<0) from those similar to Reissner-Nordstroem (eta>0). In the latter case, we find a family of black hole space-times completely regular. 
  To resolve some unphysical interpretations related to velocity measurements by static observers, we discuss the use of generalized observer sets, give a prescription for defining the speed of test particles relative to those observers and show that, for any locally inertial frame, the speed of a freely falling material particle is always less than the speed of light at the Schwarzschild black hole surface. 
  Covariant structure of the self-force of a particle in a general curved background has been made clear in the cases of scalar [Quinn], electromagnetic [DeWittBrehme], and gravitational charges [QuinnWald]. Namely, what we need is the part of the self-field that is non-vanishing off and within the past light-cone of particle's location, the so-called tail. The radiation reaction force in the absence of external fields is entirely contained in the tail. In this paper, we develop mathematical tools for the regularization and propose a practical method to calculate the self-force of a particle orbiting a Schwarzschild black hole. 
  For the Dirac particle in the rotational system, the rotation induced inertia effect is analogously treated as the modification of the "spin connection" on the Dirac equation in the flat spacetime, which is determined by the equivalent tetrad. From the point of view of parallelism description of spacetime, the obtained torsion axial-vector is just the rotational angular velocity, which is included in the "spin connection". Furthermore the axial-vector spin coupling induced spin precession is just the rotation-spin(1/2) interaction predicted by Mashhoon. Our derivation treatment is straightforward and simplified in the geometrical meaning and physical conception, however the obtained conclusions are consistent with that of the other previous work. 
  A new class of non-static higher dimensional vacuum solutions in space-time -mass (STM) theory of gravity is found. This solution represent expanding universe without big bang singularity and the higher dimension of these models shrinks as they expands. 
  The generality of inflation in closed FRW Universe is studied for the models with a scalar field on a brane and with a complex scalar field. The results obtained are compared with the previously known results for the model with a scalar field and a perfect fluid. The influence of the measure chosen in the initial condition space on the ratio of inflationary solution is described. 
  We propose a global minimal embedding of the Schwarzschild theory in a five-dimensional flat space by using two surfaces. Covariant field equations are deduced for the gravitational forces. 
  Previous analysis about the deparametrization and path integral quantization of cosmological models are extended to models which do not admit an intrinsic time. The formal expression for the transition amplitude is written down for the Taub anisotropic universe with a clear notion of time. The relation existing between the deparametrization associated to gauge fixation required in the path integral approach and the procedure of reduction of the Wheeler-De Witt equation is also studied. 
  We study the occurrence of naked singularities in the spherically symmetric collapse of radiation shells in a higher dimensional spacetime. The necessary conditions for the formation of a naked singularity or a black hole are obtained. The naked singularities are found to be strong in the Tipler's sense and thus violating cosmic censorship conjecture. 
  The only known general base to eliminate the vacuum divergencies of quantized matter fields in quantum geometrodynamics is the fermion-boson supersymmetry. The topological effect of the closed Universe -- discretization of the vacuum fluctuations spectra -- allows to formulate the conditions of cancellation of the divergencies. In the center of attention of this work is the fact that these conditions result in the considerable restrictions on the gauge and factor-ordering ambiguities peculiar to the equations of the theory and, in the limits of the isotropic model of the Universe, remove these ambiguities completely. 
  We have reinvestigated the quintessence model with minimally coupled scalar field in the context of recent Supernova observation at $z=1.7$. By assuming the form of the scale factor which gives both the early time deceleration and late time acceleration, consistent with the observations, we show that one needs a double exponential potential. We have also shown that the equation of state and the behaviour of dark energy density are reasonably consistent with earlier constraints obtained by different authors. This work shows again the importance of double exponential potential for a quintessence field. 
  Higher dimensional solutions are obtained for a homogeneous, spatially isotropic cosmological model in Wesson theory of gravitation. Some cosmological parameter are also calculated for this model. 
  We study the Robertson-Walker type model in the Lyttleton-Bondi universe in five dimensional general theory of relativity. Some exact and physical properties of solutions are discussed. 
  We investigate how the accuracy and stability of numerical relativity simulations of 1D colliding plane waves depends on choices of equation formulations, gauge conditions, boundary conditions, and numerical methods, all in the context of a first-order 3+1 approach to the Einstein equations, with basic variables some combination of first derivatives of the spatial metric and components of the extrinsic curvature tensor. Hyperbolic schemes, specifically variations on schemes proposed by Bona and Masso and Anderson and York, are compared with variations of the Arnowitt-Deser-Misner formulation. Modifications of the three basic schemes include raising one index in the metric derivative and extrinsic curvature variables and adding a multiple of the energy constraint to the extrinsic curvature evolution equations. Redundant variables in the Bona-Masso formulation may be reset frequently or allowed to evolve freely. Gauge conditions which simplify the dynamical structure of the system are imposed during each time step, but the lapse and shift are reset periodically to control the evolution of the spacetime slicing and the longitudinal part of the metric. We show that physically correct boundary conditions, satisfying the energy and momentum constraint equations, generically require the presence of some ingoing eigenmodes of the characteristic matrix. Numerical methods are developed for the hyperbolic systems based on decomposing flux differences into linear combinations of eigenvectors of the characteristic matrix. These methods are shown to be second-order accurate, and in practice second-order convergent, for smooth solutions, even when the eigenvectors and eigenvalues of the characteristic matrix are spatially varying. 
  In this work some aspects of the detection of certain general relativistic effects in the weak gravitational field of the Earth via laser-ranged data to some existing or proposed geodetic satellites are examined. The focus is on the Lense-Thirring drag of the orbit of a test body, the gravitomagnetic clock effect and the gravitoelectric perigee shift. The impact of some sources of systematic errors is investigated. An experiment whose goal is the measurement of the PPN parameters beta and gamma in the terrestrial field with LAGEOS satellites at a level of 10^(-3)is presented. A modified version of the proposed LARES mission is examined. 
  The coframe (teleparallel) description of gravity is known as a viable alternative to GR. One of advantages of this model is the existence of a conserved energy-momentum current witch is covariant under all symmetries of the three-parameter Lagrangian. In this paper we study the relation between the covector valued current and the energy-momentum tensor. Algebraic properties of the conserved current for different values of parameters are derived. It is shown that the tensor corresponding to the coframe current is traceless and, in contrast to the electromagnetic field, has in general a non vanishing antisymmetric part. The symmetric part is also non zero for all values of the parameters. Consequently, the conserved current involves the energy-momentum as well as the rotational (spin) properties of the field. 
  A generalization of the Dirac equation to the case of affine symmetry, with SL(4,R) replacing SO(1,3), is considered. A detailed analysis of a Dirac-type Poincare-covariant equation for any spin j is carried out, and the related general interlocking scheme fulfilling all physical requirements is established. Embedding of the corresponding Lorentz fields into infinite-component SL(4,R) fermionic fields, the constraints on the SL(4,R) vector-operator generalizing Dirac's gamma matrices, as well as the minimal coupling to (Metric-)Affine gravity are studied. Finally, a symmetry breaking scenario for SA(4,R) is presented which preserves the Poincare symmetry. 
  Starting from Ooguri's construction for $BF$ theory in three (and four) dimensions, we show how to construct a well defined theory with an infinite number of degrees of freedom. The spin network states that are kept invariant by the evolution operators of the theory are exact solutions of the Hamiltonian constraint of quantum gravity proposed by Thiemann. The resulting theory is the first example of a well defined, finite, consistent, spin-foam based theory in a situation with an infinite number of degrees of freedom. Since it solves the quantum constraints of general relativity it is also a candidate for a theory of quantum gravity. It is likely, however, that the solutions constructed correspond to a spurious sector of solutions of the constraints. The richness of the resulting theory makes it an interesting example to be analyzed by forthcoming techniques that construct the semi-classical limit of spin network quantum gravity. 
  It is argued that a fundamental time asymmetry could arise from the global structure of the space manifold. The proposed mechanism relies on the CPT anomaly of certain chiral gauge theories defined over a multiply connected space manifold. The resulting time asymmetry (microscopic arrow of time) is illustrated by a simple thought experiment. The effect could, in principle, play a role in determining the initial conditions of the big bang. 
  Here we describe a stationary cylindrically symmetric solution of Einstein's equation with matter consisting of a positive cosmological and rotating dust term. The solution approaches Einstein static universe solution. 
  The long standing problem of the ordering ambiguity in the definition of the Hamilton operator for a point particle in curved space is naturally resolved by using the powerful geometric calculus based on Clifford Algebra. The momentum operator is defined to be the vector derivative (the gradient) multiplied by $-i$; it can be expanded in terms of basis vectors $\gamma_\mu$ as $p = -i \gamma^\mu \p_\mu$. The product of two such operators is unambiguous, and such is the Hamiltonian which is just the D'Alambert operator in curved space; the curvature scalar term is not present in the Hamiltonian if we confine our consideration to scalar wave functions only. It is also shown that $p$ is Hermitian and self-adjoint operator: the presence of the basis vectors $\gamma^\mu$ compensates the presence of $\sqrt{|g|}$ in the matrix elements and in the scalar product. The expectation value of such operator follows the classical geodetic line. 
  We present new numerical algorithms for the coupled Einstein-perfect fluid system in axisymmetry. Our framework uses a foliation based on a family of light cones, emanating from a regular center, and terminating at future null infinity. This coordinate system is well adapted to the study of the dynamical spacetimes associated with isolated relativistic compact objects such as neutron stars. In particular, the approach allows the unambiguous extraction of gravitational waves at future null infinity and avoids spurious outer boundary reflections. The code can accurately maintain long-term stability of polytropic equilibrium models of relativistic stars. We demonstrate global energy conservation in a strongly perturbed neutron star spacetime, for which the total energy radiated away by gravitational waves corresponds to a significant fraction of the Bondi mass. As a first application we present results in the study of pulsations of axisymmetric relativistic stars, extracting the frequencies of the different fluid modes in fully relativistic evolutions of the Einstein-perfect fluid system and making a first comparison between the gravitational news function and the predicted wave using the approximations of the quadrupole formula. 
  The extension of the Campbell-Magaard embedding theorem to general relativity with minimally-coupled scalar fields is formulated and proven. The result is applied to the case of a self-interacting scalar field for which new embeddings are found, and to Brans-Dicke theory. The relationship between Campbell-Magaard theorem and the general relativity, Cauchy and initial value problems is outlined. 
  Gravitational wave detectors capable of making astronomical observations could begin to operate within the next year, and over the next 10 years they will extend their reach out to cosmological distances, culminating in the space mission LISA. A prime target of these observatories will be binary systems, especially those whose orbits shrink measurably during an observation period. These systems are standard candles, and they offer independent ways of measuring cosmological parameters. LISA in particular could identify the epoch at which star formation began and, working with telescopes making electromagnetic observations, measure the Hubble flow at redshifts out to 4 or more with unprecedented accuracy. 
  A static conformally flat spherically symmetric perfect fluid cosmological model in Lyra geometry is investigatd. 
  Equipped with the tools of (spherically reduced) dilaton gravity in first order formulation and with the results for the lowest order S-matrix for s-wave gravitational scattering (P. Fischer, D. Grumiller, W. Kummer, and D. Vassilevich, gr-qc/0105034) new properties of the ensuing cross-section are discussed. We find CPT invariance, despite of the non-local nature of our effective theory and discover pseudo-self-similarity in its kinematic sector.   After presenting the Carter-Penrose diagram for the corresponding virtual black hole geometry we encounter distributional contributions to its Ricci-scalar and a vanishing Einstein-Hilbert action for that configuration. Finally, a comparison is done between our (Minkowskian) virtual black hole and Hawking's (Euclidean) virtual black hole bubbles. 
  We derive an inflationary exponential model in a Brans-Dicke generalized theory, where the coupling constant is in fact variable. 
  We find a static solution for the scale-factor in a Brans-Dicke generalized theory where the scalar field and the coupling constant vary with time. We find also that in the early Universe there may be amplification of gravitational waves. 
  We show that Gravitational Waves are exponetially amplified in the inflationary phase in Brans-Dicke theory, so that it would be possible to detect them and in this way verify several features of physical reality. 
  We determine a cosmological model that include a positive acceleration of the Universe and a time decreasing fine structure constant. The present day deceleration parameter is estimated by us, according to our model and the available experimental data . 
  By employing Dirac LNH, and a further generalization by Berman (GLNH), we estimate how should vary the total number of nucleons, the energy density, Newton Gravitational constant, the cosmological constant, the magnetic permeability and electric permitivity, of the Universe,in order to account for the experimentally observed time variation of the fine structure constant. As a bonus,we find an acceptable value for the deceleration parameter of the present Universe, compatible with the Supernovae observations. 
  We define the critical coordinate velocity v_c. A particle moving radially in Schwarzschild background with this velocity, v_c= c/sqrt 3, is neither accelerated, nor decelerated if gravitational field is weak, r_g << r, where r_g is the gravitational radius, while r is the current one.   We find that the numerical coincidence of v_c with velocity of sound in ultrarelativistic plasma, u_s, is accidental, since two velocities are different if the number of spatial dimensions is not equal to 3 
  The effects that the structure of a neutron star would have on the gravitational emission of a binary system are studied in a perturbative regime, and in the frequency domain. Assuming that a neutron star is perturbed by a point mass moving on a close, circular orbit, we solve the equations of stellar perturbations in general relativity to evaluate the energy lost by the system in gravitational waves. We compare the energy output obtained for different stellar models with that found by assuming that the perturbed object is a black hole with the same mass, and we discuss the role played by the excitation of the stellar modes. Ouresults indicate that the stellar structure begins to affect the emitted power when the orbital velocity is v >0.2c (about 185 Hz for a binary system composed of two canonical neutron stars). We show that the differences between different stellar models and a black hole are due mainly to the excitation of the quasinormal modes of the star. Finally, we discuss to what extent and up to which distance the perturbative approach can be used to describe the interaction of a star and a pointlike massive body. 
  We study the decoherence process associated with the scattering of stochastic gravitational waves. We discuss the case of macroscopic systems, such as the planetary motion of the Moon around the Earth, for which gravitational scattering is found to dominate decoherence though it has a negligible influence on damping. This contrast is due to the very high effective temperature of the background of gravitational waves in our galactic environment. 
  The issues of quintessence and cosmic acceleration can be discussed in the framework of higher order curvature and torsion theories of gravity. We can define effective pressure and energy density directly connected to the curvature or to the torsion fields and then ask for the conditions to get an accelerated expansion. Exact accelerated expanding solutions can be achieved for several fourth order curvature or torsion theories so that we obtain an alternative scheme to the standard quintessence scalar field, minimally coupled to gravity, usually adopted. We discuss also conformal transformations in order to see the links of quintessence between the Jordan and Einstein frames. Furthermore, we take into account a torsion fluid whose effects become relevant at large scale. Specifically, we investigate a model where a totally antisymmetric torsion field is taken into account discussing the conditions to obtain quintessence. We obtain exact solutions also in this case where dust dominated Friedmann behavior is recovered as soon as torsion effects are not relevant. 
  Dark energy/matter unification is first demonstrated within the framework of a simplified model.  Geodetic evolution of a cosmological constant dominated bubble Universe, free of genuine matter, is translated into a specific FRW cosmology whose effectively induced dark component highly resembles the cold dark matter ansatz. The realistic extension constitutes a dark soliton which bridges past (radiation and/or matter dominated) and future (cosmological constant dominated)  Einstein regimes; its experimental signature is a moderate redshift dependent cold dark matter deficiency function. 
  We establish the existence of local, covariant time ordered products of local Wick polynomials for a free scalar field in curved spacetime. Our time ordered products satisfy all of the hypotheses of our previous uniqueness theorem, so our construction essentially completes the analysis of the existence, uniqueness and renormalizability of the perturbative expansion for nonlinear quantum field theories in curved spacetime. As a byproduct of our analysis, we derive a scaling expansion of the time ordered products about the total diagonal that expresses them as a sum of products of polynomials in the curvature times Lorentz invariant distributions, plus a remainder term of arbitrary low scaling degree. 
  The usual scenario of ``eternal inflation'' involves an approximately de Sitter spacetime undergoing upward fluctuations of the local expansion rate $H$. This spacetime requires frequent violations of the Null Energy Condition (NEC). We investigate the fluctuations of the energy-momentum tensor of the scalar field in de Sitter space as a possible source of such violations. We find that fluctuations of the energy-momentum tensor smeared in space and time are well-defined and may provide the NEC violations. Our results for slow-roll inflation are consistent with the standard calculations of inflationary density fluctuations. In the diffusive regime where quantum fluctuations dominate the slow-roll evolution, the magnitude of smeared energy-momentum tensor fluctuations is large enough to create frequent NEC violations. 
  In this thesis a non-standard geometric framework, the "quasi-metric" framework (QMF), is used to define relativistic space-time. The QMF consists of a 4-dimensional space-time manifold equipped with a one-parameter family of Lorentzian 4-metrics parameterized by a (unique) global time function. Equivalently the QMF may be described as a 4-dimensional submanifold of a 5-dimensional product manifold equipped with a degenerate metric, where the global time function represents one extra degenerate time dimension. A symmetric and linear connection compatible with the non-degenerate piece of the degenerate metric is defined, yielding equations of motion. These equations are identical to the geodesic equation obtained from the connection. The role of the degenerate dimension is to describe global scale changes between gravitational and non-gravitational systems. In particular this yields an alternative description of the expansion of the Universe. In this thesis a quasi-metric theory of gravity is constructed. The field equations have only one dynamical degree of freedom coupled explicitly to matter, but there is also a second, implicit dynamical degree of freedom. The existence of an implicit coupling makes the field equations unsuitable for a standard PPN-analysis. This implies that the experimental status of the theory is not completely clear at this point in time. But the non-metric part of the theory may be tested rather independently. That is, the theory predicts that vacuum gravitational fields and gravitationally bound bodies made of ideal gas expand like the expansion of the Universe. Several observations suggest this; e.g. the "Pioneer effect", the spin-down of the Earth, palaeo-tidal records etc. Thus quasi-metric relativity has experimental support where metric theories fail. 
  Condensed matter systems, such as acoustics in flowing fluids, light in moving dielectrics, or quasiparticles in a moving superfluid, can be used to mimic aspects of general relativity. More precisely these systems (and others) provide experimentally accessible models of curved-space quantum field theory. As such they mimic kinematic aspects of general relativity, though typically they do not mimic the *dynamics*. Although these analogue models are thereby limited in their ability to duplicate all the effects of Einstein gravity they nevertheless are extremely important -- they provide black hole analogues (some of which have already been seen experimentally) and lead to tests of basic principles of curved-space quantum field theory. Currently these tests are still in the realm of *gedanken-experiments*, but there are plausible candidate models that should lead to laboratory experiments in the not too distant future. 
  We illustrate and emphasize the relevance of hyperbolic theories of dissipation in different physical scenarios. Particular attention is paid to self-gravitating systems where the relaxation time may become large enough as to require a description of the transient regime. It is argued that even outside that regime, hyperbolic theories may be needed to provide an accurate description of dissipative processes. 
  The model of Expansive Nondecelerative Universe leads to a conclusion stating that at the end of radiation era the Jeans mass was equal to the upper mass limit of a black hole and, at the same time, the effective gravitational range of nucleons was identical to their Compton wavelength. At that time nucleons started to exert gravitational impact on their environment which enabled to large scale structures become formed. Moreover, it is shown that there is a deep relationships between the inertial mass of various leptons and bosons and that such relations can be extended also into the realm of other kinds of elementary particles. 
  The Schr\"odinger equations for the Coulomb and the Harmonic oscillator potentials are solved in the cosmic-string conical space-time. The spherical harmonics with angular deficit are introduced.  The algebraic construction of the harmonic oscillator eigenfunctions is performed through the introduction of non-local ladder operators. By exploiting the hidden symmetry of the two-dimensional harmonic oscillator the eigenvalues for the angular momentum operators in three dimensions are reproduced.  A generalization for N-dimensions is performed for both Coulomb and harmonic oscillator problems in angular deficit space-times.  It is thus established the connection among the states and energies of both problems in these topologically non-trivial space-times. 
  The connection between gravity and thermodynamics is explored. Examining a perfect fluid in gravitational equilibrium we find that the entropy is extremal only if Einstein's equations are satisfied. Conversely, one can derive part of Einstein's equations from ordinary thermodynamical considerations. This allows the theory of this system to be recast in such a way that a sector of general relativity is purely thermodynamical and should not be quantized. 
  The partition function of the SO(4)- or Spin(4)-symmetric Euclidean Barrett-Crane model can be understood as a sum over all quantized geometries of a given triangulation of a four-manifold. In the original formulation, the variables of the model are balanced representations of SO(4) which describe the quantized areas of the triangles. We present an exact duality transformation for the full quantum theory and reformulate the model in terms of new variables which can be understood as variables conjugate to the quantized areas. The new variables are pairs of S^3-values associated to the tetrahedra. These S^3-variables parameterize the hyperplanes spanned by the tetrahedra (locally embedded in R^4), and the fact that there is a pair of variables for each tetrahedron can be viewed as a consequence of an SO(4)-valued parallel transport along the edges dual to the tetrahedra. We reconstruct the parallel transport of which only the action of SO(4) on S^3 is physically relevant and rewrite the Barrett-Crane model as an SO(4) lattice BF-theory living on the 2-complex dual to the triangulation subject to suitable constraints whose form we derive at the quantum level. Our reformulation of the Barrett-Crane model in terms of continuous variables is suitable for the application of various analytical and numerical techniques familiar from Statistical Mechanics. 
  Axially symmetric, stationary solutions of the Einstein-Maxwell equations with disconnected event horizon are studied by developing a method of explicit integration of the corresponding boundary-value problem. This problem is reduced to non-leaner system of algebraic equations which gives relations between the masses, the angular momenta, the angular velocities, the charges, the distance parameters, the values of the electromagnetic field potential at the horizon and at the symmetry axis. A found solution of this system for the case of two charged non-rotating black holes shows that in general the total mass depends on the distance between black holes. Two-Killing reduction procedure of the Einstein-Maxwell equations is also discussed. 
  This paper explores the interaction of weak gravitational fields with slender elastic materials in space and estimates their sensitivities for the detection of gravitational waves with frequencies between $10^{-4}$ and 1 Hz. The dynamic behaviour of such slender structures is ideally suited to analysis by the simple theory of Cosserat rods. Such a description offers a clean conceptual separation of the vibrations induced by bending, shear, twist and extension and the response to gravitational tidal accelerations can be reliably estimated in terms of the constitutive properties of the structure. The sensitivity estimates are based on a truncation of the theory in the presence of thermally induced homogeneous Gaussian stochastic forces. 
  This talk reports on recent progress toward the semiglobal study of asymptotically flat spacetimes within numerical relativity. The development of a 3D solver for asymptotically Minkowski-like hyperboloidal initial data has rendered possible the application of Friedrich's conformal field equations to astrophysically interesting spacetimes. As a first application, the whole future of a hyperboloidal set of weak initial data has been studied, including future null and timelike infinity. Using this example we sketch the numerical techniques employed and highlight some of the unique capabilities of the numerical code. We conclude with implications for future work. 
  In this article we compute the density of scalar and Dirac particles created by a cosmological anisotropic Bianchi type I universe in the presence of a time varying electric field. We show that the particle distribution becomes thermal when one neglects the electric interaction. 
  Session A5 on numerical methods contained talks on colliding black holes, critical phenomena, investigation of singularities, and computer algebra. 
  We analyze a bifurcation phenomenon associated with critical gravitational collapse in a family of self-gravitating SU(2) $\sigma$-models. As the dimensionless coupling constant decreases, the critical solution changes from discretely self-similar (DSS) to continuously self-similar (CSS). Numerical results provide evidence for a bifurcation which is analogous to a heteroclinic loop bifurcation in dynamical systems, where two fixed points (CSS) collide with a limit cycle (DSS) in phase space as the coupling constant tends to a critical value. 
  We construct the general spherically symmetric and self-similar solution of the Einstein-Vlasov system (collisionless matter coupled to general relativity) with massless particles, under certain regularity conditions. Such solutions have a curvature singularity by construction, and their initial data on a Cauchy surface to the past of the singularity can be chosen to have compact support in momentum space. They can also be truncated at large radius so that they have compact support in space, while retaining self-similarity in a central region that includes the singularity. However, the Vlasov distribution function can not be bounded. As a simpler illustration of our techniques and notation we also construct the general spherically symmetric and static solution, for both massive and massless particles. 
  It is shown that the effective inertial mass density of a dissipative fluid just after leaving the equilibrium, on a time scale of the order of relaxation time, reduces by a factor which depends on dissipative variables. Prospective applications of this result to cosmological and astrophysical scenarios are discussed. 
  We make use of Dirac LNH and results for a time varying fine structure constant in order to derive possible laws of variation for speed of light, the number of nucleons in the Universe, energy density and gravitational constant. By comparing with experimental bounds on G variation, we find that the deceleration paramenter of the present Universe is negative. This is coherent with recent Supernovae observations. 
  While general relativity possesses local Lorentz invariance, both canonical quantum gravity and string theory suggest that Lorentz invariance may be broken at high energies. Broken Lorentz invariance has also been postulated as an explanation for astrophysical anomalies such as the missing GZK cutoff. Therefore, we seek an effective field theory description of gravity where Lorentz invariance is broken. We will construct a candidate theory and then briefly discuss some of the implications. 
  We consider the problem of detecting a burst signal of unknown shape. We introduce a statistic which generalizes the excess power statistic proposed by Flanagan and Hughes and extended by Anderson et al. The statistic we propose is shown to be optimal for arbitrary noise spectral characteristic, under the two hypotheses that the noise is Gaussian, and that the prior for the signal is uniform. The statistic derivation is based on the assumption that a signal affects only affects N samples in the data stream, but that no other information is a priori available, and that the value of the signal at each sample can be arbitrary. We show that the proposed statistic can be implemented combining standard time-series analysis tools which can be efficiently implemented, and the resulting computational cost is still compatible with an on-line analysis of interferometric data. We generalize this version of an excess power statistic to the multiple detector case, also including the effect of correlated noise. We give full details about the implementation of the algorithm, both for the single and the multiple detector case, and we discuss exact and approximate forms, depending on the specific characteristics of the noise and on the assumed length of the burst event. As a example, we show what would be the sensitivity of the network of interferometers to a delta-function burst. 
  The Hawking radiation of Dirac particles in an arbitrarily rectilinearly accelerating Kinnersley black hole with electro-magnetic charge and cosmological constant is investigated by using method of the generalized tortoise coordinate transformation. Both the location and the temperature of the event horizon depend on the time and the polar angle. The Hawking thermal radiation spectrum of Dirac particles is also derived.   PACS numbers: 04.70.Dy, 97.60.Lf 
  In this we briefly cover the covariant approach to describe inertial forces in general relativity and in particular look at the behaviour of the cumulative drag index for stationary Kerr geometry. 
  We discuss fermion zero modes within the 3+1 brain -- the domain wall between the two vacua in 4+1 spacetime. We do not assume relativistic invariance in 4+1 spacetime, or any special form of the 4+1 action. The only input is that the fermions in bulk are fully gapped and are described by nontrivial momentum-space topology. Then the 3+1 wall between such vacua contains chiral 3+1 fermions. The bosonic collective modes in the wall form the gauge and gravitational fields. In principle, this universality class of fermionic vacua can contain all the ingredients of the Standard Model and gravity. 
  We have computed the eigenfrequencies of f-modes for a constant-rest-mass sequences of rapidly rotating relativistic inviscid stars in differential rotation. The frequencies have been calculated neglecting the metric perturbations (the relativistic Cowling approximation) and expressed as a function of the ratio between the rotational kinetic energy and the absolute value of the gravitational energy of the stellar model beta=T/|W|. The zeros and the end-points of these sequences mark respectively the onset of the secular instability driven by gravitational radiation-reaction and the maximum value of beta at which an equilibrium model exists. In differentially rotating stars the secular stability limits appear at a beta larger than those found for uniformly rotating stars. Differential rotation, on the other hand, also allows for the existence of equilibrium models at values of beta larger than those for uniformly rotating stars, moving the end-point of the sequences to larger beta. As a result, for some degrees of differential rotation, the onset of the secular instability for f-modes is generally favoured by the presence of differential rotation. 
  By comparing the observed orbital decay of the binary pulsars PSRB1913+16 and PSRB1534+12 to that predicted by general relativity due to gravitational-wave emission, we are able to bound the mass of the graviton to be less than $7.6\times10^{-20} \text{eV}/c^2$ at 90% confidence. This is the first such bound to be derived from dynamic gravitational fields. It is approximately two orders of magnitude weaker than the static-field bound from solar system observations, and will improve with further observations. 
  We present a solution generating technique for anisotropic fluids which preserves specific Killing symmetries. Anisotropic matter distributions that can be used with the one parameter Ehlers-Geroch transform are discussed. Example spacetimes that support the appropriate anisotropic stress-energy are found and the transformation applied. The 3+1 black string solution is one of the spacetimes with the appropriate matter distribution. Use of the transform with a black string seed is discussed. 
  We study the evolution of a flat Friedmann-Robertson- Walker Universe, filled with a bulk viscous cosmological fluid, in the presence of variable gravitational and cosmological constants. The dimensional analysis of the model suggest a proportionality between the bulk viscous pressure of the dissipative fluid and the energy density. With the use of this assumption and with the choice of the standard equations of state for the bulk viscosity coefficient, temperature and relaxation time, the general solution of the field equations can be obtained, with all physical parameters having a power-law time dependence. The symmetry analysis of this model, performed by using Lie group techniques, confirms the unicity of the solution for this functional form of the bulk viscous pressure. 
  Since 1978 superconducting coupled cavities have been proposed as a sensitive detector of gravitational waves. The interaction of the gravitational wave with the cavity walls, and the esulting motion, induces the transition of some energy from an initially excited cavity mode to an empty one. The energy transfer is maximum when the frequency of the wave is equal to the frequency difference of the two cavity modes. In 1984 Reece, Reiner and Melissinos built a detector of the type proposed, and used it as a transducer of harmonic mechanical motion, achieving a sensitivity to fractional deformations of the order dx/x ~ 10^(-18). In this paper the working principles of the detector are discussed and the last experimental results summarized. New ideas for the development of a realistic gravitational waves detector are considered; the outline of a possible detector design and its expected sensitivity are also shown. 
  We describe some specific quantum black hole model. It is pointed out that the origin of a black hole entropy is the very process of quantum gravitational collapse. The quantum black hole mass spectrum is extracted from the mass spectrum of the gravitating source. The classical analog of quantum black hole is constructed. 
  We present a family of time-dependent solutions to 2+1 gravity with negative cosmological constant and a massless scalar field as source. These solutions are continuously self-similar near the central singularity. We analyze linear perturbations of these solutions, and discuss the subtle question of boundary conditions. We find two growing modes, one of which corresponds to the linearization of static singular solutions, while the other describes black hole formation. 
  Spherically symmetric, null dust clouds, like their time-like counterparts, may collapse classically into black holes or naked singularities depending on their initial conditions. We consider the Hamiltonian dynamics of the collapse of an arbitrary distribution of null dust, expressed in terms of the physical radius, $R$, the null coordinates, $V$ for a collapsing cloud or $U$ for an expanding cloud, the mass function, $m$, of the null matter, and their conjugate momenta. This description is obtained from the ADM description by a Kucha\v{r}-type canonical transformation. The constraints are linear in the canonical momenta and Dirac's constraint quantization program is implemented. Explicit solutions the constraints are obtained for both expanding and contracting null dust clouds with arbitrary mass functions. 
  This is a survey of a new type of relativistic space-time framework; the "quasi-metric" framework. See the abstract of gr-qc/0111110 to get some idea of its geometrical basis. A theory of gravity consistent with the quasi-metric framework is presented. It is not yet clear whether or not this theory is viable, but in its non-metric sector it makes successful predictions from first principles where metric theories fail. 
  In models of spacetime that are the product of a four-dimensional spacetime with an ``extra'' dimension, there is the possibility that the extra dimension will collapse to zero size, forming a singularity. We ask whether this collapse is likely to destroy the spacetime. We argue, by an appeal to the four-dimensional cosmic censorship conjecture, that--at least in the case when the extra dimension is homogeneous--such a collapse will lead to a singularity hidden within a black string. We also construct explicit initial data for a spacetime in which such a collapse is guaranteed to occur and show how the formation of a naked singularity is likely avoided. 
  Based on the recently proposed scenario of inflation driven by a bulk scalar field in the braneworld of the Randall-Sundrum (RS) type, we investigate the dynamics of a bulk scalar field on the inflating braneworld. We derive the late time behavior of the bulk scalar field by analyzing the property of the retarded Green function. We find that the late time behavior is basically dominated by a single (or a pair of) pole(s) in the Green function irrespective of the initial condition and of the signature of $m^{2}=V''(\phi)$, where $V(\phi)$ is the potential of the bulk scalar field. Including the lowest order back-reaction to the geometry, this late time behavior can be well approximated by an effective 4-dimensional scalar field with $m^2_{\mathrm{eff}}=m^2/2$. The mapping to the 4-dimensional effective theory is given by a simple scaling of the potential with a redefinition of the field. Our result supports the picture that the scenario of inflation driven by a bulk scalar field works in a quite similar way to that in the standard 4-dimensional cosmology. 
  This is the first monograph on the geometry of anisotropic spinor spaces and its applications in modern physics. The main subjects are the theory of gravity and matter fields in spaces provided with off--diagonal metrics and associated anholonomic frames and nonlinear connection structures, the algebra and geometry of distinguished anisotropic Clifford and spinor spaces, their extension to spaces of higher order anisotropy and the geometry of gravity and gauge theories with anisotropic spinor variables. The book summarizes the authors' results and can be also considered as a pedagogical survey on the mentioned subjects. 
  The behavior of a arbitrary coupled quantum scalar field is studied in the background of the G\"odel spacetime. Closed forms are derived for the effective action and the vacuum expectation value of quadratic field fluctuations by using $\zeta$-function regularization. Based on these results, we argue that causality violation presented in this spacetime can not be removed by quantum effects. 
  We examine several higher spin modes of the Poincar\'e gauge theory (PGT) of gravity using the Hamiltonian analysis. The appearance of certain undesirable effects due to non-linear constraints in the Hamiltonian analysis are used as a test. We find that the phenomena of field activation and constraint bifurcation both exist in the pure spin 1 and the pure spin 2 modes. The coupled spin-$0^-$ and spin-$2^-$ modes also fail our test due to the appearance of constraint bifurcation. The ``promising'' case in the linearized theory of PGT given by Kuhfuss and Nitsch (KRNJ86) likewise does not pass. From this analysis of these specific PGT modes we conclude that an examination of such nonlinear constraint effects shows great promise as a strong test for this and other alternate theories of gravity. 
  We obtain the most general solution of the Einstein electro - vacuum equation for the stationary axially symmetric spacetime in which the Hamilton-Jacobi and Klein - Gordon equations are separable. The most remarkable feature of the solution is its invariance under the duality transformation involving mass and NUT parameter, and the radial and angle coordinates. It is the general solution for a rotating (gravitational dyon) particle which is endowed with both gravoelectric and gravomagnetic charges, and there exists a duality transformation from one to the other. It also happens to be a transform of the Kerr - NUT solution. Like the Kerr family, it is also possible to make this solution radiating which asymptotically conforms to the Vaidya null radiation. 
  The differential cross-sections for scattering of gravitons into photons on bosons and fermions are calculated in linearized quantum gravity. They are found to be strongly peaked in the forward direction and become constant at high energies. Numerically, they are very small as expected for such gravitational interactions. 
  In this paper, we find all the Conformal Killing Vectors (CKVs) and their Lie Algebra for the recently reported [cqg1] spherically symmetric, shear-free separable metric spacetimes with non-vanishing energy or heat flux. We also solve the geodesic equatios of motion for the spacetime under consideration. 
  In this paper, we investigate the nature of the singularity in the spherically symmetrical, shear-free, gravitational collapse of a star with heat flux using a separable metric [cqg1]. For any non-singular, regular, radial density profile for a star described by this metric, eq. (2.1), the singularity of the gravitational collapse is not naked locally. Our results here unequivocally support the Strong Cosmic Censorship Hypothesis. 
  Self-similar spacetimes are of importance to cosmology and to gravitational collapse problems. We show that self-similarity or the existence of a homothetic Killing vector field for spherically symmetric spacetimes implies the separability of the spacetime metric in terms of the co-moving coordinates and that the metric is, uniquely, the one recently reported in [cqg1]. The spacetime, in general, has non-vanishing energy-flux and shear. The spacetime admits matter with any equation of state. 
  We deal with the problem of identifying a background structure and its perturbation in tetrad theories of gravity. Starting from a peculiar trivial principal bundle we define a metric which depends only on the gauge connection. We find the allowed four-dimensional structure groups; two of them turn out to be the translation group T_4 and the unitary group U(2). When the curvature vanishes the metric reduces to its background form which coincides with Minkowski flat metric for the T_4 case and with the Einstein static universe metric for the U(2) case. The perturbation has a coordinate independent definition and allows for the introduction of observables distinguished from those obtained from the metric alone. Finally, we show that any teleparallel theory of gravity, and hence general relativity, can be considered as a gauge theory over the groups introduced. 
  The talks presented in the string theory and supergravity session of the GR16 conference in Durban, South Africa are described below for the proceedings. 
  Over the last three years, a number of fundamental physical issues were addressed in loop quantum gravity. These include: A statistical mechanical derivation of the horizon entropy, encompassing astrophysically interesting black holes as well as cosmological horizons; a natural resolution of the big-bang singularity; the development of spin-foam models which provide background independent path integral formulations of quantum gravity and `finiteness proofs' of some of these models; and, the introduction of semi-classical techniques to make contact between the background independent, non-perturbative theory and the perturbative, low energy physics in Minkowski space. These developments spring from a detailed quantum theory of geometry that was systematically developed in the mid-nineties and have added a great deal of optimism and intellectual excitement to the field.   The goal of this article is to communicate these advances in general physical terms, accessible to researchers in all areas of gravitational physics represented in this conference. 
  A simplified Randall-Sundrum-like model in 6 dimensions is discussed. The extra two dimensions correspond to the cone. The effective four-dimensional scalar self-interacting theory is studied at one-loop level. The contributions due to 6-dimensional parameters in four-dimensional beta-functions appear. Using such beta-functions the one-loop effective potential is calculated. The possibility of spontaneous symmetry breaking due to extra dimensions is demonstrated. 
  Centre manifold theory is applied to some dynamical systems arising from spatially homogeneous cosmological models. Detailed information is obtained concerning the late-time behaviour of solutions of the Einstein equations of Bianchi type III with collisionless matter. In addition some statements in the literature on solutions of the Einstein equations coupled to a massive scalar field are proved rigorously. 
  We derive a low temperature effective action for the order parameter in a symmetrized phase A of helium 3, where the Fermi velocity equals the transversal velocity of low energy fermionic quasiparticles. The effective action has a form of the electromagnetic action. This analog electromagnetism is a part of the program to derive analog gravity and the standard model as a low energy effective theory in a condensed matter system. For the analog gauge field to satisfy the Maxwell equations interactions in $^3$He require special tuning that leads to the symmetric case. 
  We show that, at first order in the angular velocity, the general relativistic description of Rossby-Haurwitz waves (the analogues of r-waves on a thin shell) can be obtained from the corresponding Newtonian one after a coordinate transformation. As an application, we show that the results recently obtained by Rezzolla and Yoshida (2001) in the analysis of Newtonian Rossby-Haurwitz waves of a slowly and differentially rotating, fluid shell apply also in General Relativity, at first order in the angular velocity. 
  We introduce a quantum volume operator $K$ in three--dimensional Quantum Gravity by taking into account a symmetrical coupling scheme of three SU(2) angular momenta. The spectrum of $K$ is discrete and defines a complete set of eigenvectors which is alternative with respect to the complete sets employed when the usual binary coupling schemes of angular momenta are considered. Each of these states, that we call quantum bubbles, represents an interference of extended configurations which provides a rigorous meaning to the heuristic notion of quantum tetrahedron. We study the generalized recoupling coefficients connecting the symmetrical and the binary basis vectors, and provide an explicit recursive solution for such coefficients by analyzing also its asymptotic limit. 
  We consider simple hydrodynamical models of galactic dark matter in which the galactic halo is a self-gravitating and self-interacting gas that dominates the dynamics of the galaxy. Modeling this halo as a sphericaly symmetric and static perfect fluid satisfying the field equations of General Relativity, visible barionic matter can be treated as ``test particles'' in the geometry of this field. We show that the assumption of an empirical ``universal rotation curve'' that fits a wide variety of galaxies is compatible, under suitable approximations, with state variables characteristic of a non-relativistic Maxwell-Boltzmann gas that becomes an isothermal sphere in the Newtonian limit. Consistency criteria lead to a minimal bound for particle masses in the range $30 \hbox{eV} \leq m \leq 60 \hbox{eV}$ and to a constraint between the central temperature and the particles mass. The allowed mass range includes popular supersymmetric particle candidates, such as the neutralino, axino and gravitino, as well as lighter particles ($m\approx$ keV) proposed by numerical N-body simulations associated with self-interactive CDM and WDM structure formation theories. 
  In this work, the experiment is discussed on the verification of the principle of universality of gravitational interactions and some related problems of gravity theory and physics of elementary particles. The meaning of this proposal lies in the fact that the self-consistency of General Relativity, as it turns out, presuppose the existence of the nongravitating form of energy. Theory predicts that electrons are particles that transfer the nongravitating form of energy. 
  Within the framework of thermo-field-dynamics (TFD), the information-entropies associated with the measurements of position and momentum for one-dimensional Rindler oscillator are derived, and the connection between its information-entropy and thermal fluctuation is obtained. A conclusion is drawn that the thermal fluctuation leads to the loss of information. 
  This work is devoted to a mathematical analysis of the distributional Schwarzschild geometry. The Schwarzschild solution is extended to include the singularity; the energy momentum tensor becomes a delta-distribution supported at r=0. Using generalized distributional geometry in the sense of Colombeau's (special) construction the nonlinearities are treated in a mathematically rigorous way. Moreover, generalized function techniques are used as a tool to give a unified discussion of various approaches taken in the literature so far; in particular we comment on geometrical issues. 
  In a preceding paper [T. Hirayama, Prog. Theor. Phys. 106 (2001), 71], the power of the classical radiation emitted by a moving charge was evaluated in the Rindler frame. In this paper, we give a simpler derivation of this radiation formula, including an estimation of the directional dependence of the radiation. We find that the splitting of the energy-momentum tensor into a bound part I' and an emitted part II' is consistent with the three conditions introduced in the preceding paper, also for each direction within the future light cone. 
  We prove that spacetimes satisfying the vacuum Einstein equations on a manifold of the form $\Sigma \times U(1)\times R$ where $\Sigma $ is a compact surface of genus $G>1$ and where the Cauchy data is invariant with respect to U(1) and sufficiently small exist for an infinite proper time in the expanding direction. 
  Fundamental theories, like strings, supergravity, Kaluza-Klein, lead after dimensional reduction and a suitable choice of field configurations, to an effective action in four dimensions where gravity is coupled non-mininally to one scalar field, and minimally to another scalar field. These scalar fields couple non-trivially between themselves. A radiation field is also considered in this effective action. All the possibilities connected to those fundamental theories are labeled by two parameters $n$ (related to the non-trivial coupling of the scalar fields) and $\omega$, connected to the coupling of the Brans-Dicke like field to gravity. Exact solutions are found, exhibiting a singulariy-free behaviour, from the four dimensional point of view, for some values of those parameter. The flatness and horizon problems for these solutions are also analyzed. It is discussed to which extent the solutions found are non-singular from the point of view of the original frames. This reveals to be a much complex problem. 
  In the canonical approach to general relativity it is customary to parametrize the phase space by initial data on spacelike hypersurfaces. However, if one seeks a theory dealing with observations that can be made by a single localized observer, it is natural to use a different description of the phase space. This results in a different set of Dirac observables from that appearing in the conventional formulation. It also suggests a possible solution to the problem of time, which has been one of the obstacles to the development of a satisfactory quantum theory of gravity. 
  In the spherically symmetric case the dominant energy condition together with the requirements of regularity at the center, asymptotic flatness and fineteness of the ADM mass, defines the family of asymptotically flat globally regular solutions to the Einstein minimally coupled equations which includes the class of metrics asymptotically de Sitter at approaching the regular center. The source term corresponds to an r-dependent cosmological term given by the second rank symmetric tensor invariant under boosts in the radial direction and evolving from de Sitter vacuum in the origin to Minkowski vacuum at infinity. Space-time symmetry changes smoothly from the de Sitter group at the center to the Lorentz group at infinity through the radial boosts in between. The standard formula for the ADM mass relates it to the de Sitter vacuum replacing a central singularity at the scale of symmetry restoration. For masses exceeding a certain critical value m_{crit} de Sitter-Schwarzschild geometry describes a vacuum nonsingular black hole, while beyond m_{crit} it describes a G-lump which is a vacuum selfgravitating particlelike structure without horizons. Quantum energy spectrum of G-lump is shifted down by the binding energy, and zero-point vacuum mode is fixed at the value corresponding to the Hawking temperature from the de Sitter horizon. 
  At a fixed point in spacetime (say, x_0), gravitational phase space consists of the space of symmetric matrices F^{ab} [corresponding to the canonical momentum pi^{ab}(x_0) and of symmetric matrices {G_{ab}}[corresponding to the canonical metric g_{ab}(x_0), where 1 \leq a,b \leq n, and, crucially, the matrix {G_{ab}} is necessarily positive definite, i.e. \sum u^a G_{ab}u^b > 0 whenever \sum (u^a)^2 > 0. In an alternative quantization procedure known as Metrical Quantization, the first and most important ingredient is the specification of a suitable metric on classical phase space. Our choice of phase space metrics, guided by a recent study of Affine Quantum Gravity, leads to gravitational phase space geometries which possess constant scalar curvature and may be regarded as higher dimensional analogs of the Poincare plane, which applies when n=1. This result is important because phase spaces endowed with such symmetry lead naturally via the procedures of Metrical Quantization to acceptable Hilbert spaces of high dimension. 
  By incorporating spinning particles into the framework of classical General Relativity, the theory is changed insofar, as, though using holonome coordinates, the connexion becomes asymmetrical. This implies, that partial derivatives do not commute any longer. Hence, the class of functions under consideration has to be extended. A non-minimal extension leads to the possibility of spacetime warps for spinning particles. 
  Vacuum polarization of a massive scalar field in the background of a two-dimensional version of a spinning cosmic string is investigated. It is shown that when the `radius of the universe' is such that spacetime is globally hyperbolic the vacuum fluctuations are well behaved, diverging though on the `chronology horizon'. Naive use of the formulae when spacetime is nonglobally hyperbolic leads to unphysical results. It is also pointed out that the set of normal modes used previously in the literature to address the problem gives rise to two-point functions which do not have a Hadamard form, and therefore are not physically acceptable. Such normal modes correspond to a locally (but not globally) Minkowski time, which appears to be at first sight a natural choice of time to implement quantization. 
  The equations of motion of two point masses have recently been derived at the 3PN approximation of general relativity. From that work we determine the location of the innermost circular orbit or ICO, defined by the minimum of the binary's 3PN energy as a function of the orbital frequency for circular orbits. We find that the post-Newtonian series converges well for equal masses. Spin effects appropriate to corotational black-hole binaries are included. We compare the result with a recent numerical calculation of the ICO in the case of two black holes moving on exactly circular orbits (helical symmetry). The agreement is remarkably good, indicating that the 3PN approximation is adequate to locate the ICO of two black holes with comparable masses. This conclusion is reached with the post-Newtonian expansion expressed in the standard Taylor form, without using resummation techniques such as Pad\'e approximants and/or effective-one-body methods. 
  Two long-standing problems with the post-Newtonian approximation for isolated slowly-moving systems in general relativity are: (i) the appearance at high post-Newtonian orders of divergent Poisson integrals, casting a doubt on the soundness of the post-Newtonian series; (ii) the domain of validity of the approximation which is limited to the near-zone of the source, and prevents one, a priori, from incorporating the condition of no-incoming radiation, to be imposed at past null infinity. In this article, we resolve the problem (i) by iterating the post-Newtonian hierarchy of equations by means of a new (Poisson-type) integral operator that is free of divergencies, and the problem (ii) by matching the post-Newtonian near-zone field to the exterior field of the source, known from previous work as a multipolar-post-Minkowskian expansion satisfying the relevant boundary conditions at infinity. As a result, we obtain an algorithm for iterating the post-Newtonian series up to any order, and we determine the terms, present in the post-Newtonian field, that are associated with the gravitational-radiation reaction onto an isolated slowly-moving matter system. 
  The physical basis of the standard theory of general relativity is examined and a nonlocal theory of accelerated observers is described that involves a natural generalization of the hypothesis of locality. The nonlocal theory is confronted with experiment via an indirect approach. The implications of the results for gravitation are briefly discussed. 
  Cancellation of laser frequency noise in interferometers is crucial for attaining the requisite sensitivity of the triangular 3-spacecraft LISA configuration. Raw laser noise is several orders of magnitude above the other noises and thus it is essential to bring it down to the level of other noises such as shot, acceleration, etc. Since it is impossible to maintain equal distances between spacecrafts, laser noise cancellation must be achieved by appropriately combining the six beams with appropriate time-delays. It has been shown in several recent papers that such combinations are possible. In this paper, we present a rigorous and systematic formalism based on algebraic geometrical methods involving computational commutative algebra, which generates in principle {\it all} the data combinations cancelling the laser frequency noise. The relevant data combinations form the first module of syzygies, as it is called in the literature of algebraic geometry. The module is over a polynomial ring in three variables, the three variables corresponding to the three time-delays around the LISA triangle. Specifically, we list several sets of generators for the module whose linear combinations with polynomial coefficients generate the entire module. We find that this formalism can also be extended in a straight forward way to cancel Doppler shifts due to optical bench motions. The two modules are infact isomorphic.   We use our formalism to obtain the transfer functions for the six beams and for the generators. We specifically investigate monochromatic gravitational wave sources in the LISA band and carry out the maximisiation over linear combinations of the generators of the signal-to-noise ratios with the frequency and source direction angles as parameters. 
  The dynamical system constituted by two spherically symmetric thin shells and their own gravitational field is studied. The shells can be distinguished from each other, and they can intersect. At each intersection, they exchange energy on the Dray, 't Hooft and Redmount formula. There are bound states: if the shells intersect, one, or both, external shells can be bound in the field of internal shells. The space of all solutions to classical dynamical equations has six components; each has the trivial topology but a non trivial boundary. Points within each component are labeled by four parameters. Three of the parameters determine the geometry of the corresponding solution spacetime and shell trajectories and the fourth describes the position of the system with respect to an observer frame. An account of symmetries associated with spacetime diffeomorphisms is given. The group is generated by an infinitesimal time shift, an infinitesimal dilatation and a time reversal. 
  The study of the two shell system started in our first paper ``Pair of null gravitating shells I'' (gr-qc/0112060) is continued. An action functional for a single shell due to Louko, Whiting and Friedman is generalized to give appropriate equations of motion for two and, in fact, any number of spherically symmetric null shells, including the cases when the shells intersect. In order to find the symplectic structure for the space of solutions described in paper I, the pull back to the constraint surface of the Liouville form determined by the action is transformed into new variables. They consist of Dirac observables, embeddings and embedding momenta (the so-called Kucha\v{r} decomposition). The calculation includes the integration of a set of coupled partial differential equations. A general method of solving the equations is worked out. 
  The study of the two-shell system started in ``Pair of null gravitating shells I and II'' (gr-qc/0112060--061) is continued. The pull back of the Liouville form to the constraint surface, which contains complete information about the Poisson brackets of Dirac observables, is computed in the singular double-null Eddington-Finkelstein (DNEF) gauge. The resulting formula shows that the variables conjugate to the Schwarzschild masses of the intershell spacetimes are simple combinations of the values of the DNEF coordinates on these spacetimes at the shells. The formula is valid for any number of in- and out-going shells. After applying it to the two-shell system, the symplectic form is calculated for each component of the physical phase space; regular coordinates are found, defining it as a symplectic manifold. The symplectic transformation between the initial and final values of observables for the shell-crossing case is written down. 
  Combining general relativity and gravitational gauge theory, the cosmological constant is determined theoretically. The cosmological constant is related to the average vacuum energy of gravitational gauge field. Because the vacuum energy of gravitational gauge field is negative, the cosmological constant is positive, which generates repulsive force on stars to make the expansion rate of the Universe accelerated. A rough estimation of it gives out its magnitude of the order of $10^{-52} m^{-2}$, which is well constant with experimental results. 
  We show that all known naked singularities in spherically symmetric self-similar spacetimes arise as a result of singular initial matter distribution. This is a result of the peculiarity of the coordinate transformation that takes these spacetimes into a separable form. Therefore, these examples of naked singularities are of no apparent consequence to astrophysical observations or theories. 
  In this work the approach for a description of the physical systems about whiches it is not possible to get basicly the total information is suggested.  As a result the differential equations, the solutions of which characterize the physical system, must bieng obtained from the demand of the minimality of the generalized variance.  As the understanding of the particles trajectory absents in the quantum theory we shall consider that the Riemannian space-time is the effective one, postulating the metric tensor on the base of the reduced density matrix of the gravitational fields. 
  We investigate the Schwarzschild-Anti-deSitter (SAdS) and SdS BH thermodynamics in 5d higher derivative gravity. The interesting feature of higher derivative gravity is the possibility for negative (or zero) SdS (or SAdS) BH entropy which depends on the parameters of higher derivative terms. The appearence of negative entropy may indicate a new type instability where a transition between SdS (SAdS) BH with negative entropy to SAdS (SdS) BH with positive entropy would occur or where definition of entropy should be modified. 
  The initial data of gravity for a cylindrical matter distribution confined on the brane is studied in the framework of the single brane Randall-Sundrum scenario. In this scenario, 5-dimensional aspect of gravity appears in the short range gravitational interaction. We found that the sufficiently thin configuration of matter leads to the formation of the marginal surface even if the configuration is infinitely long. This means that the hoop conjecture proposed by Thorne does not hold in the Randall-Sundrum scenario; Even if a mass $M$ does not get compacted into a region whose circumference in every direction is ${\cal C}\le 4\pi GM$, black holes with horizons can form in the Randall-Sundrum scenario. 
  The dynamics of our universe is characterised by the density parameters for cosmological constant ($\Omega_V$), nonbaryonic darkmatter($\Omega_{\rm wimp}$), radiation ($\Omega_R$) and baryons ($\Omega_B$). To these parameters -- which describe the smooth background universe -- one needs to add at least another dimensionless number ($\sim 10^{-5}$) characterising the strength of primordial fluctuations in the gravitational potential, in order to ensure formation of structures by gravitational instability.   I discuss several issues related to the description of the universe in terms of these numbers and argue that we do not yet have a fundamental understanding of these issues. 
  The potentials of spin-weighted wave equations in various Kerr-Newman black holes are analyzed. They all form singular potential barriers at the event horizon. Applying the WKB approximation it is shown that no particle can tunnel out of the interior of a static black hole. However, photons inside a non-extremely rotating Kerr black hole may tunnel out into the outer space, whereas neutrinos, electrons, and gravitons may not. If the rotation is extremal, any particle may tunnel out, under restrictive conditions. It is unknown whether photons and gravitons may tunnel out if the black hole is charged and rotating. 
  What is the nature - continuous or discrete - of matter and of its fundamental interactions? The physical meaning, the properties and the consequences of a discrete scalar field are discussed; limits for the validity of a mathematical description of fundamental physics in terms of continuum fields are a natural outcome of discrete fields with discrete interactions. Two demarcating points (a near and a far) define a domain where no difference between the discrete and the standard continuum field formalisms can be experimentally detected. Discrepancies, however, can be observed as a continuous-interaction is always stronger below the near point and weaker above the far point than a discrete one. The connections between the discrete scalar field and gravity from general relativity are discussed. Whereas vacuum solutions of general relativity can be retrieved from discrete scalar field solutions, this cannot be extended to solutions in presence of massive sources as they require a true tensor metric field. Contact is made, on passing, with the problem of dark matter and the rotation curve of galaxies, with inflation and the accelerated expansion, the apparent anomaly in the Pioneer spacecraft acceleration, the quantum Hall effect, high-$T_{c}$ superconductivity, quark confinement, and with Tsallis generalized one-parameter statistics as some possible manifestation of discrete interaction and of an essentially discrete world. 
  It has been shown in the presence of the cosmological constant any solution of the Einstein field equations that asymptotically approaches to the static deSitter metric does not correspond to an observer of a comoving frame but it should approach to the deSitter metric in the form of Robertson-Walker metric (gr-qc/9812092). For the case of Schwarzschild-deSitter its proper form had been derived(gr-qc/9902009,JHEP04(1999)011). Here we are going to present our derivations for a proper form of the Kerr-deSitter metric. This has been done by considering the stationary axially-symmetric spacetime in which motion of particle is integrable. That is the Hamilton-Jacobi and Klein-Gordon equations are separable. As in the Schwarzschild-deSitter case it does not possess any event horizon without imposing any additional conditions. Its intrinsic singularity and surfaces of infinite redshifts remain the same as common Kerr solutions. 
  In loop quantum gravity in the connection representation, the quantum configuration space $\bar{\mathcal{A}/\mathcal{G}}$, which is a compact space, is much larger than the classical configuration space $\mathcal{A}/% \mathcal{G}$ of connections modulo gauge transformations. One finds that $% \bar{\mathcal{A}/\mathcal{G}}$ is homeomorphic to the space $Hom(% \mathcal{L}_{\ast},G))/Ad$. We give a new, natural proof of this result, suggesting the extension of the hoop group $\mathcal{L}_{\ast}$ to a larger, compact group $\mathcal{M}(\mathcal{L}_{\ast})$ that contains $% \mathcal{L}_{\ast}$ as a dense subset. This construction is based on almost periodic functions. We introduce the Hilbert algebra $L_{2}(\mathcal{M}(% \mathcal{L}_{\ast}))$ of $\mathcal{M}(\mathcal{L}_{\ast})$ with respect to the Haar measure $\xi $ on $\mathcal{M}(\mathcal{L}_{\ast})$. The measure $% \xi $ is shown to be invariant under 3-diffeomorphisms. This is the first step in a proof that $L_{2}(\mathcal{M}(\mathcal{L}_{\ast}))$ is the appropriate Hilbert space for loop quantum gravity in the loop representation. In a subsequent paper, we will reinforce this claim by defining an extended loop transform and its inverse. 
  Density perturbations in the flat (K=0) Robertson-Walker universe with radiation ($p=\epsilon/3$) and positive cosmological constant ($\Lambda>0$) are investigated. The phenomenon of anomalous dispersion of acoustic waves on $\Lambda$ is discussed. 
  The maximum entropy of a quantized surface is demonstrated to be proportional to the surface area in the classical limit. The result is valid in loop quantum gravity, and in a somewhat more general class of approaches to surface quantization. The maximum entropy is calculated explicitly for some specific cases. 
  We study the dominant late-time behaviors of massive scalar fields in static and spherically symmetric spacetimes. Considering the field evolution in the far zone where the gravitational field is weak, we show under which conditions the massive field oscillates with an amplitude that decays slowly as $t^{-5/6}$ at very late times, as previously found in (say) the Schwarzschild case. Our conclusion is that this long-lived oscillating tail is generally observed at timelike infinity in black hole spacetimes, while it may not be able to survive if the central object is a normal star. We also discuss that such a remarkable backscattering effect is absent for the field near the null cone at larger spatial distances. 
  We explore the possibility that spacetime horizons in 4D general relativity can be treated as manifestations of higher dimensions that induce fields on our 4D spacetime. In this paper we discuss the black hole event horizon, as an example (we leave the cosmological case for future discussion). Starting off from the field equations of gravity in 5D and some conditions on the metric we construct a spacetime whose imbedding is a 4D generalization of the Schwarzchild metric. The external region of the imbedded spacetime is found to contain two distinct fields. We discuss the properties of the fields and the potential implications. Taken as they are, the results suggest that the collapse of matter to form a horizon may have non-local consequences on the geometry of spacetime. In general, the use of horizon-confined mass as a coordinate suggests three potential features of our universe. The first is that the observed 4D spacetime curvature and ordinary matter fields may be hybrid features of 5D originating from the mixing of coordinates. Secondly, because the fifth coordinate induces physical fields on the 4D hyperface, the global metric of the universe may not be asymptotically flat. And finally, associating matter with an independent dimension points towards a theory of nature that is scale invariant. 
  We discuss how the vector nature of magnetic fields, and the geometrical interpretation of gravity introduced by general relativity, lead to a special coupling between magnetism and spacetime curvature. This magneto-geometrical interaction effectively transfers the tension properties of the field into the spacetime fabric, triggering a variety of effects with potentially far-reaching implications. 
  In this talk work done by our group on cosmic topology is reviewed. It ranges from early attempts to solve a famous controversy about quasars through the multiplicity of images, to quantum cosmology in this context and an application to QED renormalization. 
  When a potential for a scalar field has two local minima there arise spherical shell-type solutions of the classical field equations due to gravitational attraction. We establish such solutions numerically in a space which is asymptotically de Sitter. It generically arises when the energy scale characterizing the scalar field potential is much less than the Planck scale. It is shown that the mirror image of the shell appears in the other half of the Penrose diagram. The configuration is smooth everywhere with no physical singularity. 
  We discuss some aspects of the gravitational interaction of the relativistic quantum particles with spin 1/2. The exact Foldy-Wouthuysen transformation is constructed for the Dirac particle coupled to the static spacetime metric. The quasi-relativistic limit of the theory is then analyzed. Using the analogous method, we obtain the exact Cini-Touschek transformation and discuss the ultra-relativistic limit of the fermion theory. We show that the Foldy-Wouthuysen transformation is not uniquely defined, and the corresponding ambiguity is deeply rooted in the relativistic quantum theory. 
  We derive the equations of motion for binary systems of compact bodies in the post-Newtonian (PN) approximation to general relativity. Results are given through 2PN order (order (v/c)^4 beyond Newtonian theory), and for gravitational radiation reaction effects at 2.5PN and 3.5PN orders. The method is based on a framework for direct integration of the relaxed Einstein equations (DIRE) developed earlier, in which the equations of motion through 3.5PN order can be expressed in terms of Poisson-like potentials that are generalizations of the instantaneous Newtonian gravitational potential, and in terms of multipole moments of the system and their time derivatives. All potentials are well defined and free of divergences associated with integrating quantities over all space. Using a model of the bodies as spherical, non-rotating fluid balls whose characteristic size s is small compared to the bodies' separation r, we develop a method for carefully extracting only terms that are independent of the parameter s, thereby ignoring tidal interactions, spin effects, and internal self-gravity effects. Through 2.5PN order, the resulting equations agree completely with those obtained by other methods; the new 3.5PN back-reaction results are shown to be consistent with the loss of energy and angular momentum via radiation to infinity. 
  The wave function for the quadratic gravity theory derived from the heterotic string effective action is deduced to first order in ${{e^{-\Phi}}\over {g_4^2}}$ by solving a perturbed second-order Wheeler-DeWitt equation, assuming that the potential is slowly varying with respect to $\Phi$. Predictions for inflation based on the solution to the second-order Wheeler-DeWitt equation continue to hold for this theory. It is shown how formal expressions for the average paths in minisuperspace $\{< a(t) >, < \Phi(t)> \}$ determine the shifts from the classical solutions to $a_{cl}(t)$ and $\Phi_{cl}(t)$, which occur only at third order in the expansion of the integrals representing the expectation values. 
  The generic crease set of an event horizon possesses anisotropic structure though most of black holes are dynamically stable. This fact suggests that a generic almost spherical black hole has a very crumpled crease set in a microscopic scale though the crease set is similar to a point-wise crease set in a macroscopic scale. In the present article, we count the number of such micro-states of an almost spherical black hole by analogy with an elastic chain polymer. This estimation of black hole entropy reproduces the well-known Bekenstein-Hawking entropy of a Schwarzschild black hole. 
  A recently proposed algebraic representation of the causal set model of the small-scale structure of space-time of Sorkin et al. is briefly reviewed and expanded. The algebraic model suggested, called quantum causal set, is physically interpreted as a locally finite, causal and quantal version of the kinematical structure of general relativity: the 4-dimensional Lorentzian space-time manifold and its continuous local orthochronous Lorentz symmetries. We discuss various possible dynamical scenarios for quantum causal sets mainly by using sheaf-theoretic ideas, and we entertain the possibility of constructing an inherently finite and genuinely smooth space-time background free quantum theory of gravity. At the end, based on the quantum causal set paradigm, we anticipate and roughly sketch out a potential future development of a noncommutative topology, sheaf and topos theory suitable for quantum space-time structure and its dynamics. 
  It is shown that, contrary to what is normally expected, it is possible to have angular momentum effects on the geometry of space time at the laboratory scale, much bigger than the purely Newtonian effects. This is due to the fact that the ratio between the angular momentum of a body and its mass, expressed as a length, is easily greater than the mass itself, again expressed as a length. 
  We discuss properties of conformal geodesics on general, vacuum, and warped product space-times and derive a system of conformal deviation equations. The results are used to show how to construct on the Schwarzschild-Kruskal space-time global conformal Gauss coordinates which extends smoothly and without degeneracy to future and past null infinity. 
  Electromagnetism in an inhomogeneous dielectric medium at rest is described using the methods of differential geometry. In contrast to a general relativistic approach the electromagnetic fields are discussed in three-dimensional space only. The introduction of an appropriately chosen three-dimensional metric leads to a significant simplification of the description of light propagation in an inhomogeneous medium: light rays become geodesics of the metric and the field vectors are parallel transported along the rays. The new metric is connected to the usual flat space metric diag[1,1,1] via a conformal transformation leading to new, effective values of the medium parameters leading to an effective constant value of the index of refraction n=1. The corresponding index of refraction is thus constant and so is the effective velocity of light. Space becomes effectively empty but curved. All deviations from straight line propagation are now due to curvature. The approach is finally used for a discussion of the Riemann-Silberstein vector, an alternative, complex formulation of the electromagnetic fields. 
  In this letter we consider the Einsteinian strengths and dynamical degrees of freedom for quadratic gravity. We show that purely metric quadratic gravity theories are much stronger in Einsteinian sense than the competitive quadratic gravity theories which admit torsion. 
  Gravity is treated as manifestation of bending of 4D plate at the variational functionals level. Some estimates of elastic constants of space-time are made. Field lagrangians and Einstein equations are discussed in view point of the approach. 
  The question of existence of general, asymptotically flat radiative spacetimes and examples of explicit classes of radiative solutions of Einstein's field equations are discussed in the light of some new developments. The examples are cylindrical waves, Robinson-Trautman and type N spacetimes and especially boost-rotation symmetric spacetimes representing uniformly accelerated particles or black holes. 
  This is the account of the workshop Exact solutions and their interpretation at the 16-th International Conference on General Relativity and Gravitation held in Durban, July 15-21, 2001. Work reported in 32 oral contributions spanned a wide variety of topics, ranging from exact radiative spacetimes to cosmological solutions. Two invited review talks, on the role of exact solutions in string theory and in cosmology, are also described. 
  Several recent studies have concerned the faith of classical symmetries in quantum space-time. In particular, it appears likely that quantum (discretized, noncommutative,...) versions of Minkowski space-time would not enjoy the classical Lorentz symmetries. I compare two interesting cases: the case in which the classical symmetries are "broken", i.e. at the quantum level some classical symmetries are lost, and the case in which the classical symmetries are "deformed", i.e. the quantum space-time has as many symmetries as its classical counterpart but the nature of these symmetries is affected by the space-time quantization procedure. While some general features, such as the emergence of deformed dispersion relations, characterize both the symmetry-breaking case and the symmetry-deformation case, the two scenarios are also characterized by sharp differences, even concerning the nature of the new effects predicted. I illustrate this point within an illustrative calculation concerning the role of space-time symmetries in the evaluation of particle-decay amplitudes. The results of the analysis here reported also show that the indications obtained by certain dimensional arguments, such as the ones recently considered in hep-ph/0106309 may fail to uncover some key features of quantum space-time symmetries. 
  2+1 gravity coupled to a massless scalar field has an initial singularity when the spatial slices are compact. The quantized model is used here to investigate several issues of quantum gravity. The spectrum of the volume operator is studied at the initial singularity. The energy spectrum is obtained. Dynamics of the universe is also investigated. 
  The paper considers the problem of finding the metric of space time around a rotating, weakly gravitating body. Both external and internal metric tensors are consistently found, together with an appropriate source tensor. All tensors are calculated at the lowest meaningful approximation in a power series. The two physical parameters entering the equations (the mass and the angular momentum per unit mass) are assumed to be such that the mass effects are negligible with respect to the rotation effects. A non zero Riemann tensor is obtained. The order of magnitude of the effects at the laboratory scale is such as to allow for experimental verification of the theory. 
  Up to now, the only known exact Foldy- Wouthuysen transformation (FWT) in curved space is that concerning Dirac particles coupled to static spacetime metrics. Here we construct the exact FWT related to a real spin-0 particle for the aforementioned spacetimes. This exact transformation exists independently of the value of the coupling between the scalar field and gravity. Moreover, the gravitational Darwin term written for the conformal coupling is one third of the relevant term in the fermionic case. 
  Quasinormal mode spectra for gravitational perturbations of black holes in four dimensional de Sitter and anti-de Sitter space are investigated. The anti-de Sitter case is relevant to the ADS-CFT correspondence in superstring theory. The ADS-CFT correspondence suggests a prefered set of boundary conditions. 
  It is well-known that waves propagating under the influence of a scattering potential develop ``tails''. However, the study of late-time tails has so far been restricted to time-independent backgrounds. In this paper we explore the late-time evolution of spherical waves propagating under the influence of a {\it time-dependent} scattering potential. It is shown that the tail structure is modified due to the temporal dependence of the potential. The analytical results are confirmed by numerical calculations. 
  We show that the existence of appropriate spatial homothetic Killing vectors is directly related to the separability of the metric functions for axially symmetric spacetimes. The density profile for such spacetimes is (spatially) arbitrary and admits any equation of state for the matter in the spacetime. When used for studying axisymmetric gravitational collapse, such solutions do not result in a locally naked singularity. 
  We investigate the evolution of the scale factor in a cosmological model in which the cosmological constant is given by the scalar arisen by the contraction of the stress-energy tensor. 
  We study the phenomenon of mass loss by a scalar charge -- a point particle that acts a source for a noninteracting scalar field -- in an expanding universe. The charge is placed on comoving world lines of two cosmological spacetimes: a de Sitter universe, and a spatially-flat, matter-dominated universe. In both cases, we find that the particle's rest mass is not a constant, but that it changes in response to the emission of monopole scalar radiation by the particle. In de Sitter spacetime, the particle radiates all of its mass within a finite proper time. In the matter-dominated cosmology, this happens only if the charge of the particle is sufficiently large; for smaller charges the particle first loses some of its mass, but then regains it all eventually. 
  We study the phenomenon of mass loss by a scalar charge -- a point particle which acts as a source for a (non-interacting) scalar field -- in (1+1)-dimensional and (2+1)-dimensional flat spacetime. We find that such particles are unstable against self interaction: they lose their mass through the emission of monopole radiation. This is in sharp contrast with scalar charges in (3+1)-dimensional flat spacetime, where no such phenomenon occurs. 
  Spacetime is composed of a fluctuating arrangement of bubbles or loops called spacetime foam, or quantum foam. We use the holographic principle to deduce its structure, and show that the result is consistent with gedanken experiments involving spacetime measurements. We propose to use laser-based atom interferometry techniques to look for spacetime fluctuations. Our analysis makes it clear that the physics of quantum foam is inextricably linked to that of black holes. A negative experimental result, therefore, might have non-trivial ramifications for semiclassical gravity and black hole physics. 
  The Thomas precession is calculated using three different transformations to the rotating frame. It is shown that for sufficiently large values of $v/c$, important differences in the predicted angle of precession appear, depending on the transformation used. For smaller values of $v/c$ these differences might be measured by extending the time of observation. 
  For asymptotically flat initial data of Einstein's equations satisfying an energy condition, we show that the Penrose inequality holds between the ADM mass and the area of an outermost apparent horizon, if the data are restricted suitably. We prove this by generalizing Geroch's proof of monotonicity of the Hawking mass under a smooth inverse mean curvature flow, for data with non-negative Ricci scalar. Unlike Geroch we need not confine ourselves to minimal surfaces as horizons. Modulo smoothness issues we also show that our restrictions on the data can locally be fulfilled by a suitable choice of the initial surface in a given spacetime. 
  Physical interpretation of some stationary and non-stationary regions of the spinning C-metric is presented. They represent different spacetime regions of a uniformly accelerated Kerr black hole.    Stability of geodesics corresponding to equilibrium points in a general stationary spacetime with an additional symmetry is also studied and results are then applied to the spinning C-metric. 
  We describe a formalism and numerical approach for studying spherically symmetric scalar field collapse for arbitrary spacetime dimension d and cosmological constant Lambda. The presciption uses a double null formalism, and is based on field redefinitions first used to simplify the field equations in generic two-dimensional dilaton gravity. The formalism is used to construct code in which d and Lambda are input parameters. The code reproduces known results in d = 4 and d = 6 with Lambda = 0. We present new results for d = 5 with zero and negative Lambda. 
  It is known that Lorentzian wormholes must be threaded by matter that violates the null energy condition. We phenomenologically characterize such exotic matter by a general class of microscopic scalar field Lagrangians and formulate the necessary conditions that the existence of Lorentzian wormholes imposes on them. Under rather general assumptions, these conditions turn out to be strongly restrictive. The most simple Lagrangian that satisfies all of them describes a minimally coupled massless scalar field with a reversed sign kinetic term. Exact, non-singular, spherically symmetric solutions of Einstein's equations sourced by such a field indeed describe traversable wormhole geometries. These wormholes are characterized by two parameters: their mass and charge. Among them, the zero mass ones are particularly simple, allowing us to analytically prove their stability under arbitrary space-time dependent perturbations. We extend our arguments to non-zero mass solutions and conclude that at least a non-zero measure set of these solutions is stable. 
  One can try to define the theory of quantum gravity as the sum over geometries. In two dimensions the sum over {\it Euclidean} geometries can be performed constructively by the method of {\it dynamical triangulations}. One can define a {\it proper-time} propagator. This propagator can be used to calculate generalized Hartle-Hawking amplitudes and it can be used to understand the the fractal structure of {\it quantum geometry}. In higher dimensions the philosophy of defining the quantum theory, starting from a sum over Euclidean geometries, regularized by a reparametrization invariant cut off which is taken to zero, seems not to lead to an interesting continuum theory. The reason for this is the dominance of singular Euclidean geometries. Lorentzian geometries with a global causal structure are less singular. Using the framework of dynamical triangulations it is possible to give a constructive definition of the sum over such geometries, In two dimensions the theory can be solved analytically. It differs from two-dimensional Euclidean quantum gravity, and the relation between the two theories can be understood. In three dimensions the theory avoids the pathologies of three-dimensional Euclidean quantum gravity. General properties of the four-dimensional discretized theory have been established, but a detailed study of the continuum limit in the spirit of the renormalization group and {\it asymptotic safety} is till awaiting. 
  Exact solutions exist which describe impulsive gravitational waves propagating in Minkowski, de Sitter, or anti-de Sitter universes. These may be either nonexpanding or expanding. Both cases in each background are reviewed here from a unified point of view. All the main methods for their construction are described systematically: the Penrose "cut and paste" method, explicit construction of continuous coordinates, distributional limits of sandwich waves, embedding from higher dimensions, and boosts of sources or limits of infinite accelerations.   Attention is concentrated on the most interesting specific solutions. In particular, the nonexpanding impulsive waves that are generated by null multipole particles are described. These generalize the well-known Aichelburg-Sexl and Hotta-Tanaka monopole solutions. Also described are the expanding spherical impulses that are generated by snapping and colliding strings. Geodesics and some other properties of impulsive wave spacetimes are also summarized. 
  Ng and van Dam have argued that quantum theory and general relativity give a lower bound of L^{1/3} L_P^{2/3} on the uncertainty of any distance, where L is the distance to be measured and L_P is the Planck length. Their idea is roughly that to minimize the position uncertainty of a freely falling measuring device one must increase its mass, but if its mass becomes too large it will collapse to form a black hole. Here we show that one can go below the Ng-van Dam bound by attaching the measuring device to a massive elastic rod. Relativistic limitations on the rod's rigidity, together with the constraint that its length exceeds its Schwarzschild radius, imply that zero-point fluctuations of the rod give an uncertainty greater than or equal to L_P. 
  The quantum fluctuation of the relative location of two (n-1)-dimensional de Sitter branes (i.e., of n spacetime dimensions) embedded in the (n+1)-dimensional anti-de Sitter bulk, which we shall call the quantum radion, is investigated at the linear perturbation level. The quantization of the radion is done by deriving the effective action of the radion. Assuming the positive tension brane is our universe, the effect of the quantum radion is evaluated by using the effective Einstein equations on the brane in which the radion contributes to the effective energy momentum tensor at the linear order of the radion amplitude. Specifically, the rms effective energy density arising from the quantum radion is compared with the background energy density. It is found out that this ratio remains small for reasonable values of the parameters of the model even without introducing a stabilizing mechanism for radion, although the radion itself has a negative mass squared and is unstable. The reason behind this phenomenon is also discussed. 
  In order to shed some light in the meaning of the relativistic multipolar expansions we consider different static solutions of the axially symmetric vacuum Einstein equations that in the non relativistic limit have same Newtonian moments. The motion of test particles orbiting around different deformed attraction centers with the same Newtonian limit is studied paying special attention to the advance of the perihelion. We find discrepancies in the fourth order of the dimensionless parameter (mass of the attraction center)/(semilatus rectum). An evolution equation for the difference of the radial coordinate due to the use of different general relativistic multipole expansions is presented. 
  The issues of quintessence and cosmic acceleration can be discussed in the framework of higher order theories of gravity. We can define effective pressure and energy density directly connected to the Ricci scalar of curvature of a generic fourth order theory and then ask for the conditions to get an accelerated expansion. Exact accelerated expanding solutions can be achieved for several fourth order theories so that we get an alternative scheme to the standard quintessence scalar field, minimally coupled to gravity, usually adopted. We discuss also conformal transformations in order to see the links of quintessence between the Jordan and Einstein frames. 
  We investigate classical formation of a D-dimensional black hole in a high energy collision of two particles. The existence of an apparent horizon is related to the solution of an unusual boundary-value problem for Poisson's equation in flat space. For sufficiently small impact parameter, we construct solutions giving such apparent horizons in D=4. These supply improved estimates of the classical cross-section for black hole production, and of the mass of the resulting black holes. We also argue that a horizon can be found in a region of weak curvature, suggesting that these solutions are valid starting points for a semiclassical analysis of quantum black hole formation. 
  The late-time tail behaviors of massive scalar fields are examined analytically in the background of a black hole with a global monopole. It is found that the presence of a solid deficit angle in the background metric makes the massive scalar fields decay faster in the intermediate times. However, the asymptotically late-time tail is not affected and it has the same decay rate of $t^{-5/6}$ as in the Schwarzschild and nearly extreme Reissner-Nordstr\"{o}m backgrounds. 
  We investigate the perihelion shift of the planetary motion and the bending of starlight in the Schwarzschild field modified by the presence of a $\Lambda$-term plus a conical defect. This analysis generalizes an earlier result obtained by Islam (Phys. Lett. A 97, 239, 1983) to the case of a pure cosmological constant. By using the experimental data we obtain that the parameter $\epsilon$ characterizing the conical defect is less than $10^{-9}$ and $10^{-7}$, respectively, on the length scales associated with such phenomena. In particular, if the defect is generated by a cosmic string, these values correspond to limits on the linear mass densities of $10^{19}g/cm$ and $10^{21}g/cm$, respectively. 
  A strict derivation of the Schwarzschild metric, based solely on Newton's law of free fall and the equivalence principle, is presented. In the light of it, regarding Schwarzschild's coordinates as representing the point of view of a distant observer resting relative to the source of a centrally symmetric gravitational field, proves illegitimate. Such point of view is better represented by the Painleve-Gullstrand system of coordinates, which agrees with Schwarzschild's system with respect to its spatial coordinates and time scale, but disagrees with respect to the relation of simultaneity. A duality of the Schwarzschild solution and its time-irreversibility is suggested. The physical meaning of the coordinate singularity at the Schwarzschild radius is clarified. 
  We report the experimental observation of the photothermal effect. The measurements are performed by modulating the laser power absorbed by the mirrors of two high-finesse Fabry-Perot cavities. The results are very well described by a recently proposed theoretical model [M. Cerdonio, L. Conti, A. Heidmann and M. Pinard, Phys. Rev. D 63 (2001) 082003], confirming the correctness of such calculations. Our observations and quantitative characterization of the photothermal effect demonstrate its critical importance for high sensitivity interferometric displacement measurements, as those necessary for gravitational wave detection. 
  State-dependent gauge principle invoked to realize the relativity to a measuring device, has been proposed. Self-consistent global (cosmic) potential forms the state space of the fundamental field and its connection, agreed with Fubini-Study metric of $CP(N-1)$, serves as state-dependent gauge potential. In this framework the linearity of the ordinary quantum mechanics appears as a `tangent approximation' to the totally nonlinear underlying pre-dynamical `functional' field theory on $CP(N-1)$. 
  Using our new post-Newtonian (PN) smoothed particle hydrodynamics (SPH) code, we have studied numerically the mergers of neutron star binaries with irrotational initial configurations. Here we describe a new method for constructing numerically accurate initial conditions for irrotational binary systems with circular orbits in PN gravity. We then compute the 3D hydrodynamic evolution of these systems until the two stars have completely merged, and we determine the corresponding GW signals. We present results for systems with different binary mass ratios, and for neutron stars represented by polytropes with $\Gamma=2$ or $\Gamma=3$. Compared to mergers of corotating binaries, we find that irrotational binary mergers produce similar peak GW luminosities, but they shed almost no mass at all to large distances. The dependence of the GW signal on numerical resolution for calculations performed with N>10^5 SPH particles is extremely weak, and we find excellent agreement between runs utilizing N=10^5 and N=10^6 SPH particles (the largest SPH calculation ever performed to study such irrotational binary mergers). We also compute GW energy spectra based on all calculations reported here and in our previous works. We find that PN effects lead to clearly identifiable features in the GW energy spectrum of binary neutron star mergers, which may yield important information about the nuclear equation of state at extreme densities. 
  Spacetimes admitting appropriate spatial homothetic Killing vectors are called spatially homothetic spacetimes. Such spacetimes conform to the fact that gravity has no length-scale for matter inhomogeneities. The matter density for such spacetimes is (spatially) arbitrary and the matter generating the spacetime admits {\it any} equation of state. Spatially homothetic spacetimes necessarily possess energy-momentum fluxes. We first discuss spherically symmetric and axially symmetric examples of such spacetimes that do not form naked singularities for regular initial data. We then show that the Cosmic Censorship Hypothesis is {\em equivalent} to the statement that gravity has no length-scale for matter properties. 
  In flat space-time, sigma-model strings and textures are both unstable to collapse and subsequent decay. With sufficient cosmological expansion, however, they are stable in a generalized sense: a small perturbation will cause them to change their shape, but they do not decay. The current rate of expansion is sufficient to stabilize strings, but not textures. 
  Cylindrically symmetric stationary spacetimes are examined in the framework of string-inpired generalized theory of gravity. In four dimensions this theory contains a dilatonic scalar field in addition to gravity. A charged perfect fluid representing fermionic matter is also considered. Explicit solution is given and a discussion of the geometrical properties of the solutions found is carried out. 
  Schwarzschild's actual exterior solution (Gs) is resurrected and together with the manifold M is shown to constitute a space-time possessing all the properties historically thought to be required of a point mass. On the other hand, the metric that today is ascribed to Schwarzschild, but which was in fact first obtained by Droste and Weyl, is shown to give rise to a space-time that is neither equivalent to Schwarzschild's nor derivable from the "historical" properties of a point mass. Consequently, the point-mass interpretation of the Kruskal-Fronsdal space-time (Mw, Gkf) can no longer be justified on the basis that it is an extension of Droste and Weyl's space-time. If such an interpretation is to be maintained, it can only be done by showing that the properties of (Mw, Gkf) are more in accord with what a point-mass space-time should possess than those of (M, Gs). To do this, one must first explain away three seeming incongruities associated with (Mw, Gkf): its global nonstationarity, the two-dimensional nature of the singularity, and the fact that for a finite interval of time it has no singularity at all. Finally, some of the consequences of choosing (M,Gs) as a model of a point-mass are discussed. 
  Axisymmetric spacetimes with a conformal symmetry are studied and it is shown that, if there is no further conformal symmetry, the axial Killing vector and the conformal Killing vector must commute. As a direct consequence, in conformally stationary and axisymmetric spacetimes, no restriction is made by assuming that the axial symmetry and the conformal timelike symmetry commute. Furthermore, we prove that in axisymmetric spacetimes with another symmetry (such as stationary and axisymmetric or cylindrically symmetric spacetimes) and a conformal symmetry, the commutator of the axial Killing vector with the two others mush vanish or else the symmetry is larger than that originally considered. The results are completely general and do not depend on Einstein's equations or any particular matter content. 
  We present a detailed analytical study of spherically symmetric self-similar solutions in the SU(2) sigma model coupled to gravity. Using a shooting argument we prove that there is a countable family of solutions which are analytic inside the past self-similarity horizon. In addition, we show that for sufficiently small values of the coupling constant these solutions possess a regular future self-similarity horizon and thus are examples of naked singularities. One of the solutions constructed here has been recently found as the critical solution at the threshold of black hole formation. 
  The fundamental metrics, which describe any static three-dimensional Einstein-Maxwell spacetime (depending only on a unique spacelike coordinate), are found. In this case there are only three independent components of the electromagnetic field: two for the vector electric field and one for the scalar magnetic field. It is shown that we can not have any superposition of these components of the electric and magnetic fields in this kind of static gravitational field. One of the electrostatic Einstein-Maxwell solutions is related to the magnetostatic solution by a duality mapping, while the second electrostatic gravitational field must be solved separately. Solutions induced by the more general (2+1)-Maxwell tensor on the static cylindrically symmetric spacetimes are studied and it is shown that all of them are also connected by duality mappings. 
  The first-order general relativistic theory of a generic dissipative (heat-conducting, viscous, particle-creating) fluid is rediscussed from a unified covariant frame-independent point of view. By generalizing some previous works in the literature, we derive a formula for the temperature variation rate, which is valid both in Eckart's (particle) and in the Landau-Lifshitz (energy) frames. Particular attention is paid to the case of gravitational particle creation and its possible cross-effect with the bulk viscosity mechanism. 
  In this paper, we analyze the conditions for convergence toward General Relativity of scalar-tensor gravity theories defined by an arbitrary coupling function $\alpha$ (in the Einstein frame). We show that, in general, the evolution of the scalar field $(\phi)$ is governed by two opposite mechanisms: an attraction mechanism which tends to drive scalar-tensor models toward Einstein's theory, and a repulsion mechanism which has the contrary effect. The attraction mechanism dominates the recent epochs of the universe evolution if, and only if, the scalar field and its derivative satisfy certain boundary conditions. Since these conditions for convergence toward general relativity depend on the particular scalar-tensor theory used to describe the universe evolution, the nucleosynthesis bounds on the present value of the coupling function, $\alpha_0$, strongly differ from some theories to others. For example, in theories defined by $\alpha \propto \mid\phi\mid$ analytical estimates lead to very stringent nucleosynthesis bounds on $\alpha_0$ ($\lesssim 10^{-19}$). By contrast, in scalar-tensor theories defined by $\alpha \propto \phi$ much larger limits on $\alpha_0$ ($\lesssim 10^{-7}$) are found. 
  Part A of this article is devoted to the general investigation of the gravitational-wave emission by post-Newtonian sources. We show how the radiation field far from the source, as well as its near-zone inner gravitational field, can (in principle) be calculated in terms of the matter stress-energy tensor up to any order in the post-Newtonian expansion. Part B presents some recent applications to the problems of the dynamics and gravitational-wave flux of compact binary systems. The precision reached in these developments corresponds to the third post-Newtonian approximation. 
  We show that a reformulation of the ADM equations in general relativity, which has dramatically improved the stability properties of numerical implementations, has a direct analogue in classical electrodynamics. We numerically integrate both the original and the revised versions of Maxwell's equations, and show that their distinct numerical behavior reflects the properties found in linearized general relativity. Our results shed further light on the stability properties of general relativity, illustrate them in a very transparent context, and may provide a useful framework for further improvement of numerical schemes. 
  We use the Kerr-Schild type Teukolsky equation (horizon penetrating) to evolve binary black hole initial data as proposed by Bishop {\em et al.} in the close limit. Our results are in agreement with those recently obtained by Sarbach {\em et al.} from the Zerilli equation evolution of the same initial data. 
  This paper is concerned with several not-quantum aspects of black holes, with emphasis on theoretical and mathematical issues related to numerical modeling of black hole space-times. Part of the material has a review character, but some new results or proposals are also presented. We review the experimental evidence for existence of black holes. We propose a definition of black hole region for any theory governed by a symmetric hyperbolic system of equations. Our definition reproduces the usual one for gravity, and leads to the one associated with the Unruh metric in the case of Euler equations. We review the global conditions which have been used in the Scri-based definition of a black hole and point out the deficiencies of the Scri approach. Various results on the structure of horizons and apparent horizons are presented, and a new proof of semi-convexity of horizons based on a variational principle is given. Recent results on the classification of stationary singularity-free vacuum solutions are reviewed. Two new frameworks for discussing black holes are proposed: a "naive approach", based on coordinate systems, and a "quasi-local approach", based on timelike boundaries satisfying a null convexity condition. Some properties of the resulting black holes are established, including an area theorem, topology theorems, and an approximation theorem for the location of the horizon. 
  We study imbedded hypersurfaces in spacetime whose causal character is allowed to change from point to point. Inherited geometrical structures on these hypersurfaces are defined by two methods: first, the standard rigged connection induced by a rigging vector (a vector not tangent to the hypersurface anywhere); and a second, more physically adapted, where each observer in spacetime induces a new type of connection that we call the rigged metric connection. The generalisation of the Gauss and Codazzi equations are also given. With the above machinery, we attack the problem of matching two spacetimes across a general hypersurface. It is seen that the preliminary junction conditions allowing for the correct definition of Einstein's equations in the distributional sense reduce to the requirement that the first fundamental form of the hypersurface be continuous. The Bianchi identities are then proven to hold in the distributional sense. Next, we find the proper junction conditions which forbid the appearance of singular parts in the curvature. Finally, we derive the physical implications of the junction conditions: only six independent discontinuities of the Riemann tensor are allowed. These are six matter discontinuities at non-null points of the hypersurface. For null points, the existence of two arbitrary discontinuities of the Weyl tensor (together with four in the matter tensor) are also allowed. 
  By treating the real Maxwell Field and real linearized Einstein equations as being imbedded in complex Minkowski space, one can interpret magnetic moments and spin-angular momentum as arising from a charge and mass monopole source moving along a complex world line in the complex Minkowski space. In the circumstances where the complex center of mass world-line coincides with the complex center of charge world-line, the gyromagnetic ratio is that of the Dirac electron. 
  This Living Review updates a previous version which its itself an update of a review article. Numerical exploration of the properties of singularities could, in principle, yield detailed understanding of their nature in physically realistic cases. Examples of numerical investigations into the formation of naked singularities, critical behavior in collapse, passage through the Cauchy horizon, chaos of the Mixmaster singularity, and singularities in spatially inhomogeneous cosmologies are discussed. 
  We prove global hyperbolicity of spacetimes under generic regularity conditions on the metric. We then show that these spacetimes are timelike and null geodesically complete if the gradient of the lapse and the extrinsic curvature $K$ are integrable. This last condition is required only for the tracefree part of $K$ if the universe is expanding. 
  We outline the class of globally regular spherically symmetric solutions to the minimally coupled GR equations asymptotically de Sitter in the origin and asymptotically Schwarzschild at infinity. A source term connects smoothly de Sitter vacuum at the regular center with Minkowski vacuum at infinity and corresponds to anisotropic spherically symmetric vacuum defined macroscopically by the algebraic structure of its stress-energy tensor invariant under boosts in the radial direction. De Sitter-Schwarzschild geometry describes a vacuum nonsingular black hole which evolves, in the course of Hawking evaporation, towards a self-gravitating particle-like structure without horizons, G-lump. Space-time symmetry changes smoothly from the de Sitter group in the center to the Lorentz group at infinity, and the standard formula for the ADM mass relates it to the de Sitter vacuum replacing a singularity at the scale of symmetry restoration. This class of metrics is easily extended to the case of a nonzero background cosmological constant. A source term connects then smoothly two de Sitter vacua with different values of cosmological constant which makes possible to associate anisotropic spherically symmetric vacuum with an r-dependent cosmological term. 
  According to general relativity, the interaction of a matter field with gravitation requires the simultaneous introduction of a tetrad field, which is a field related to translations, and a spin connection, which is a field assuming values in the Lie algebra of the Lorentz group. These two fields, however, are not independent. By analyzing the constraint between them, it is concluded that the relevant local symmetry group behind general relativity is provided by the Lorentz group. Furthermore, it is shown that the minimal coupling prescription obtained from the Lorentz covariant derivative coincides exactly with the usual coupling prescription of general relativity. Instead of the tetrad, therefore, the spin connection is to be considered as the fundamental field representing gravitation. 
  In the context of the two fluid model of space-time fluctuations proposed to tame the transplanckian problem encountered in black hole physics, it is postulated that the inflaton is the fluctuation of mode density, ``the vapor component'' of the model. The mass of the inflaton is occasioned by the exchange of degrees of freedom between the ``vapor'' and the ``liquid'', the planckian ``soup'' in which usual ``cisplanckian'' fields propagate. This exchange between vacuum fluctuations is modeled after its counterpart in the real world i.e. black hole evaporation. In order of magnitude, a very rough semiquantitative estimate, would situate the mass somewhere between $10^{-10}$ and $10^{-5}$ planck masses, the largest uncertainty being the mass of the planckian black hole fluctuation i.e. the entropy that one ascribes to it. 
  We show that a flowing dielectric medium with a linear response to an external electric field can be used to generate an analog geometry that has many of the formal properties of a Schwarzschild black hole for light rays, in spite of birefringence. We also discuss the possibility of generating these analog black holes in the laboratory. 
  We obtain an explicit solution of the momentum constraint for conformally flat, maximal slicing, initial data which gives an alternative to the purely longitudinal extrinsic curvature of Bowen and York. The new solution is related, in a precise form, with the extrinsic curvature of a Kerr slice. We study these new initial data representing spinning black holes by numerically solving the Hamiltonian constraint. They have the following features: i) Contain less radiation, for all allowed values of the rotation parameter, than the corresponding single spinning Bowen-York black hole. ii) The maximum rotation parameter $J/m^2$ reached by this solution is higher than that of the purely longitudinal solution allowing thus to describe holes closer to a maximally rotating Kerr one. We discuss the physical interpretation of these properties and their relation with the weak cosmic censorship conjecture. Finally, we generalize the data for multiple black holes using the ``puncture'' and isometric formulations. 
  Using a quantum mechanical approach, we show that in a gravitational-wave interferometer composed of arm cavities and a signal recycling cavity, e.g., the LIGO-II configuration, the radiation-pressure force acting on the mirrors not only disturbs the motion of the free masses randomly due to quantum fluctuations, but also and more fundamentally, makes them respond to forces as though they were connected to an (optical) spring with a specific rigidity. This oscillatory response gives rise to a much richer dynamics than previously known, which enhances the possibilities for reshaping the LIGO-II's noise curves. However, the optical-mechanical system is dynamically unstable and an appropriate control system must be introduced to quench the instability. 
  Using the 3+1 formalism of general relativity we obtain the equations governing the dynamics of spherically symmetric spacetimes with arbitrary sources. We then specialize for the case of perfect fluids accompanied by a flow of interacting massless or massive particles (e.g. neutrinos) which are described in terms of relativistic transport theory. We focus in three types of coordinates: 1) isotropic gauge and maximal slicing, 2) radial gauge and polar slicing, and 3) isotropic gauge and polar slicing. 
  The brane world based on the 6D gravitational model is examined. It is regarded as a higher dimensional version of the 5D model by Randall and Sundrum >. The obtained analytic solution is checked by the numerical method. The mass hierarchy is examined. Especially the {\it geometrical see-saw} mass relation, between the Planck mass, the cosmological constant, and the neutrino mass, is suggested. Comparison with the 5D model is made. 
  Stationary and axisymmetric perfect-fluid metrics are studied under the assumption of the existence of a conformal Killing vector field and in the general case of differential rotation. The possible Lie algebras for the conformal group and corresponding canonical line-elements are explicitly given. It turns out that only four different cases appear, the abelian and other three called I, II and III. We explicitly find all the solutions in the abelian and I cases. For the abelian case the general solution depends on an arbitrary function of a single variable and the perfect fluid satisfies the equation of state rho = p+const. This class of metrics is the one presented recently by one of us. The general solution for case I is a new Petrov type D metric, with the velocity vector outside the 2-space spanned by the two principal null directions and a barotropic equation of state rho +3p=0. For the cases II and III, the general solution has been found only under the further assumption of a natural separation of variables Ansatz. The conformal Killing vectors in the solutions that come out here are, in fact, homothetic. No barotropic equation of state exists in these metrics unless for a new Petrov type D solution belonging to case II and with rho +3p=0 which cannot be interpreted as an axially symmetric solution and such that the velocity vector points in the direction of one of the Killing vectors. This solution has the previously unknown curious property that both commuting Killing vectors are timelike everywhere. 
  An overview is provided of the singularity theorems in cosmological contexts at a level suitable for advanced graduate students. The necessary background from tensor and causal geometry to understand the theorems is supplied, the mathematical notion of a cosmology is described in some detail and issues related to the range of validity of general relativity are also discussed. 
  This is an elementary geometrical proof of Birkhoff theorem. It is hardly important, but the pictures behind are quite nice. 
  One of the most intriguing problem of modern physics is the question of the endpoint of black hole evaporation. Based on Einstein-dilaton-Gauss-Bonnet four dimensional string gravity model we show that black holes do not disappear and that the end of the evaporation process leaves some relic. The possibility of experimental detection of the remnant black holes is investigated. If they really exist, such objects could be a considerable part of the non baryonic dark matter in our Universe. 
  Inspired by quantum cosmology, in which the wave function of the universe is annihilated by the total Hamiltonian, we consider the internal dynamics of a simple particle system in an energy eigenstate. Such a system does not possess a uniquely defined time parameter and all physical questions about it must be posed without reference to time. We consider in particular the question, what is the probability that the system's trajectory passes through a set of regions of configuration space without reference to time? We first consider the classical case, where the answer has a variety of forms in terms of a phase space probability distribution function. We then consider the quantum case, and we analyze this question using the decoherent histories approach to quantum theory, adapted to questions which do not involve time. When the histories are decoherent, the probabilities approximately coincide with the classical case, with the phase space probability distribution replaced by the Wigner function of the quantum state. For some initial states, decoherence requires an environment, and we compute the required influence functional and examine some of its properties. Special attention is given to the inner product used in the construction (the induced or Rieffel inner product), the construction of class operators describing the histories, and the extent to which reparametrization invariance is respected. Our results indicate that simple systems without an explicit time parameter may be quantized using the decoherent histories approach, and the expected classical limit extracted. The results support, for simple models, the usual heuristic proposals for the probability distribution function associated with a semiclassical wave function satisfying the Wheeler-DeWitt equation. 
  Gravitational stability of torsion and inflaton field in a four-dimensional spacetime de Sitter solution in scalar-tensor cosmology where Cartan torsion propagates is investigated in detail. Inflaton and torsion evolution equations are derived by making use of a Lagrangean method. Stable and unstable modes for torsion and inflatons are found to be dependent of the background torsion and inflaton fields. Present astrophysical observations favour a stable mode for torsion since this would explain why no relic torsion imprint has been found on the Cosmic Background Radiation in the universe. 
  A covariant scheme for material coupling with $GL(N,R)$ gauge formulation of gravity is studied. We revisit a known idea of a Yang-Mills type construction, where quadratical power of cosmological constant have to be considered in consistence with vacuum Einstein's gravity. Then, matter coupling with gravity is introduced and some constraints on fields and background appear. Finally, exploring the N=3 case we elucidate that introduction of auxiliary fields decreases the number of these constraints. 
  We study Majumdar-Papapetrou solutions for the 3+1 Einstein-Maxwell equations, with charged dust acting as the external source for the fields. The spherically symmetric solution of G\"{u}rses is considered in detail. We introduce new parameters that simplify the construction of class $C^1$, singularity-free geometries. The arising sources are bounded or unbounded, and the redshift of light signals allows an observer at spatial infinity to distinguish these cases. We find out an interesting affinity between the conformastatic metric and some homothetic, matter and Ricci collineations. The associated non-Noetherian symmetries provide us with distinctive solutions that can be used to construct non-singular sources for Majumdar-Papapetrou spacetimes.} 
  A number of recent works in E-print arXiv have addressed the foundation of gauge gravitation theory again. As is well known, differential geometry of fibre bundles provides the adequate mathematical formulation of classical field theory, including gauge theory on principal bundles. Gauge gravitation theory is formulated on the natural bundles over a world manifold whose structure group is reducible to the Lorentz group. It is the metric-affine gravitation theory where a metric (tetrad) gravitational field is a Higgs field. 
  The stability under small time perturbations of the dilatonic black hole solution near the determinant curvature singularity is proved. This fact gives the additional arguments that the investigated topological configuration can realise in nature. In the frames of this model primordial black hole remnants are examined as time stable objects, which can form an significant part of a dark matter in the Universe. 
  If we replace the general spacetime group of diffeomorphisms by transformations taking place in the tangent space, general relativity can be interpreted as a gauge theory, and in particular as a gauge theory for the Lorentz group. In this context, it is shown that the angular momentum and the energy-momentum tensors of a general matter field can be obtained from the invariance of the corresponding action integral under transformations taking place, not in spacetime, but in the tangent space, in which case they can be considered as gauge currents. 
  We extend the lattice gauge theory-type derivation of the Barrett-Crane spin foam model for quantum gravity to other choices of boundary conditions, resulting in different boundary terms, and re-analyze the gluing of 4-simplices in this context. This provides a consistency check of the previous derivation. Moreover we study and discuss some possible alternatives and variations that can be made to it and the resulting models. 
  We present a new non-diagonal G2 inhomogeneous perfect-fluid solution with barotropic equation of state p=rho and positive density everywhere. It satisfies the global hyperbolicity condition and has no curvature singularity anywhere. This solution is very simple in form and has two arbitrary constants. 
  We present a clear-cut example of the importance of the functorial approach of gauge-natural bundles and the general theory of Lie derivatives for classical field theory, where the sole correct geometrical formulation of Einstein (-Cartan) gravity coupled with Dirac fields gives rise to an unexpected indeterminacy in the concept of conserved quantities. 
  In this paper we consider the possibility of measuring the corrections induced by the square of the parameter a_g of the Kerr metric to the general relativistic deflection of electromagnetic waves and time delay in an Earth based experiment. It turns out that, while at astronomical scale the well known gravitoelectric effects are far larger than the gravitomagnetic ones, at laboratory scale the situation is reversed: the gravitomagnetic effects exceed definitely the gravitoelectric ones which are totally negligible. By using a small rapidly rotating sphere as gravitating source on the Earth the deflection of a grazing light ray amounts to 10^{-13} rad and the time delay is proportional to 10^{-23} s. These figures are determined by the upper limit in the attainable values of a_g due to the need of preventing the body from exploding under the action of the centrifugal forces. Possible criticisms to the use of the Kerr metric at a_g^2 level are discussed. 
  In this paper we investigate the possibility of constraining the hypothesis of a fifth force at the length scale of two Earth's radii by investigating the effects of a Yukawa gravitational potential on the orbits of the laser--ranged LAGEOS satellites. The existing constraints on the Yukawa coupling $\alpha$, obtained by fitting the LAGEOS orbit, are of the order of | \alpha | < 10^{-5}-10^{-8} for distances of the order of 10^9 cm. Here we show that with a suitable combination of the orbital residuals of the perigee \omega of LAGEOS II and the nodes \Omega of LAGEOS II and LAGEOS it should be possible to constrain \alpha at a level of 4 X 10^{-12} or less. Various sources of systematic errors are accounted for, as well. Their total impact amounts to 1 X 10^{-11} during an observational time span of 5 years. In the near future, when the new data on the terrestrial gravitational field will be available from the CHAMP and GRACE missions, these limits will be further improved. The use of the proposed LARES laser--ranged satellite would yield an experimental accuracy in constraining \alpha of the order of 1 X 10^{-12}. 
  In this paper we prove that in a stationary axisymmetric SU(2) Einstein-Yang-Mills theory the most reasonable circularity conditions that can be considered for the Yang-Mills fields imply in fact that the field is of embedded Abelian type, or else that the metric is not asymptotically flat. 
  Static, spherically symmetric, traversable wormholes, induced by massless, nonminimally coupled scalar fields in general relativity, are shown to be unstable under spherically symmetric perturbations. The instability is related to blowing-up of the effective gravitational constant on a certain sphere. 
  Brief comments on a plausible holographic relationship between the opposite rotational dragging effect of a (2+1)-dimensional rotating de Sitter space and the non-unitarity of a boundary conformal field theory are given. In addition to the comments, we study how the opposite rotational dragging effect affects the statistical-mechanical quantities in the rotating de Sitter space in comparison with a BTZ black hole. 
  I review briefly, primarily for relativists, a series of recent results, obtained with A. Waldron, on the novel behavior of massive higher (s>1) spin systems in constant curvature backgrounds. We find that the cosmological constant Lambda, together with the mass parameter, define a "phase plane" in which partially massless gauge invariant lines separate allowed regions from forbidden, non-unitary, ones. These lines represent short multiplet systems, with missing lower helicities, removed by novel local gauge invariances, and (despite having nonvanishing m) propagating on the light cone. In the limit of an infinite tower of these higher spin bosons and fermions, unitarity requires Lambda to vanish. 
  This thesis describes the application of numerical techniques to solve Einstein's field equations in three distinct cases. First we present the first long-term stable second order convergent Cauchy characteristic matching code in cylindrical symmetry including both gravitational degrees of freedom. Compared with previous work we achieve a substantial simplification of the evolution equations as well as the relations at the interface by factoring out the z-Killing direction via the Geroch decomposition in both the Cauchy and the characteristic region. In the second part we numerically solve the equations for static and dynamic cosmic strings of infinite length coupled to gravity and provide the first fully non-linear evolutions of cosmic strings in curved spacetimes. The inclusion of null infinity as part of the numerical grid allows us to apply suitable boundary conditions on the metric and the matter fields to suppress unphysical divergent solutions. The code is used to study the interaction between a Weber-Wheeler pulse of gravitational radiation with an initially static string. In the final part of the thesis we probe a new numerical approach for highly accurate evolutions of non-linear neutron star oscillations in the case of radial oscillations of spherically symmetric stars. For this purpose we view the evolution of the physical quantities as deviations from a static equilibrium configuration and reformulate the equations in a fully non-linear perturbative form. The high accuracy of the new scheme enables us to study the non-linear coupling of eigenmodes over a wide range of initial amplitudes. 
  We investigate the Hilbert space in the Lorentz covariant approach to loop quantum gravity. We restrict ourselves to the space where all area operators are simultaneously diagonalizable, assuming that it exists. In this sector quantum states are realized by a generalization of spin network states based on Lorentz Wilson lines projected on irreducible representations of an SO(3) subgroup. The problem of infinite dimensionality of the unitary Lorentz representations is absent due to this projection. Nevertheless, the projection preserves the Lorentz covariance of the Wilson lines so that the symmetry is not broken. Under certain conditions the states can be thought as functions on a homogeneous space. We define the inner product as an integral over this space. With respect to this inner product the spin networks form an orthonormal basis in the investigated sector. We argue that it is the only relevant part of a larger state space arising in the approach. The problem of the noncommutativity of the Lorentz connection is solved by restriction to the simple representations. The resulting structure shows similarities with the spin foam approach. 
  We find all the perfect fluid G2 diagonal cosmologies with the property that the quotient of the norms of the two orthogonal Killing vectors is constant along each fluid world-line. We find four different families depending each one on two or three arbitrary parameters which satisfy that the metric coefficients are not separable functions. Some physical properties of these solutions including energy conditions, kinematical quantities, Petrov type, the existence and nature of the singularities and whether they contain Friedman-Robertson-Walker cosmologies as particular cases are also included. 
  The loop quantum gravity technique is applied to the free bosonic string. A Hilbert space similar to loop space in loop quantum gravity as well as representations of diffeomorphism and hamiltonian constraints on it are constructed. The string in this representation can be viewed as a set of interacting relativistic particles each carrying a certain momentum. Two different regularizations of the hamiltonian constraint are proposed. The first of them is anomaly-free and give rise to interaction very similar to that of two dimensional $\phi^4$-model. The second version of hamiltonian constraint is similar to $\phi^3$-model and contains an anomaly. A possible relation of these two models to the conventional quantization of the string based on Fock space representation is discussed. 
  It is shown that for realistic anisotropic star models the surface redshift can not exceed the values 3.842 or 5.211 when the tangential pressure satisfies the strong or the dominant energy condition respectively. Both values are higher than 2, the bound in the perfect fluid case. 
  We show that there exist asymptotically flat almost-smooth initial data for Einstein-perfect fluid's equation that represent an isolated liquid-type body. By liquid-type body we mean that the fluid energy density has compact support and takes a strictly positive constant value at its boundary. By almost-smooth we mean that all initial data fields are smooth everywhere on the initial hypersurface except at the body boundary, where tangential derivatives of any order are continuous at that boundary.   PACS: 04.20.Ex, 04.40.Nr, 02.30.Jr 
  Barbour, Foster and \'{O} Murchadha have recently developed a new framework, called here {\it{the 3-space approach}}, for the formulation of classical bosonic dynamics. Neither time nor a locally Minkowskian structure of spacetime are presupposed. Both arise as emergent features of the world from geodesic-type dynamics on a space of 3-dimensional metric--matter configurations. In fact gravity, the universal light cone and Abelian gauge theory minimally coupled to gravity all arise naturally through a single common mechanism. It yields relativity -- and more -- without presupposing relativity. This paper completes the recovery of the presently known bosonic sector within the 3-space approach. We show, for a rather general ansatz, that 3-vector fields can interact among themselves only as Yang--Mills fields minimally coupled to gravity. 
  We analyze free elementary particles with rest mass $m$ and total energy $E < m c^2$ in the Rindler wedge, outside Reissner-Nordstrom black holes and in the spacetime of relativistic (and non-relativistic) stars, and use Unruh-DeWitt-like detectors to calculate the associated particle detection rate in each case. The (mean) particle position is identified with the spatial average of the excitation probability of the detectors, which are supposed to cover the whole space. Our results are shown to be in harmony with General Relativity classical predictions. Eventually we reconcile our conclusions with Earth-based experiments which are in good agreement with $E \geq m c^2$. 
  I obtain an exact, axially symmetric, stationary solution of Einstein's equations for two massless spinning particles. The term representing the spin-spin interaction agrees with recently published approximate work. The spin-spin force appears to be proportional to the inverse fourth power of the coordinate distance between the particles. 
  The axes of gyroscopes experimentally define non-rotating frames.   But what physical cause governs the time-evolution of gyroscope axes?   Starting from an unperturbed, spatially flat FRW cosmology, we consider cosmological vorticity perturbations (i.e. vector perturbations, rotational perturbations) at the linear level. We ask: Will cosmological rotational perturbations drag the axis of a gyroscope relative to the directions (geodesics) to galaxies beyond the rotational perturbation? We cast the laws of Gravitomagnetism into a form showing clearly the close correspondence with the laws of ordinary magnetism. Our results are: 1) The dragging of a gyroscope axis by rotational perturbations beyond the $\dot{H}$ radius (H = Hubble constant) is exponentially suppressed. 2) If the perturbation is a homogeneous rotation inside a radius significantly larger than the $\dot{H}$ radius, then the dragging of the gyroscope axis by the rotational perturbation is exact for any equation of state for cosmological matter. 3) The time-evolution of a gyroscope axis exactly follows a specific average of the matter inside the $\dot{H}$ radius for any equation of state. In this precise sense Mach's Principle follows from cosmology with Einstein Gravity. 
  We examine the excitation of a uniformly accelerated DeWitt-Takagi detector coupled quadratically to a Majorana-Dirac field. We obtain the transition probability from the ground state of the detector and the vacuum state of the field to an excited state with the emission of a Minkowski pair of quanta, in terms of elementary processes of absorption and scattering of Rindler quanta from the Fulling-Davies-Unruh thermal bath in the co-accelerated frame. 
  The gravitational field exterior respectively interior to a spherically symmetric, isolated body made of perfect fluid is examined within the quasi-metric framework (QMF). It is required that the gravitational field is "metrically static", meaning that it is static except for the effects of the global cosmic expansion on the spatial geometry. Dynamical equations for the gravitational field are set up and an exact solution is found for the exterior part. Besides, equations of motion applying to inertial test particles moving in the exterior gravitational field are set up. By construction the gravitational field of the system is not static with respect to the cosmic expansion. This means that the radius of the source increases and that distances between circular orbits of inertial test particles increase according to the Hubble law. Moreover it is shown that if this model of an expanding gravitational field is taken to represent the gravitational field of the Sun (or isolated planetary systems), this has no serious consequences for observational aspects of planetary motion. On the contrary some observational facts of the Earth-Moon system are naturally explained within the QMF. Finally the QMF predicts a secular increase of the gravitational "constant" G. But this secular change is neither present in the Newtonian limit of the quasi-metric equations of motion nor in the Newtonian limit of the quasi-metric field equations valid inside metrically static sources. Thus standard interpretations of space experiments testing the secular variation of G are explicitly theory dependent and do not apply to the QMF. 
  A recently reported discrepancy between experimental and theoretical values of the muon's g-2 factor is interpreted as due to small violations of the conservation of P and T in the spin-rotation coupling. The experiments place an upper limit on these violations and on the weight change of spinning gyroscopes. 
  Pushing forward the similitudes between the gravitational collapse and the expansion of the universe (in the reversed sense of time), it should be expected that, during the collapse, eventually, a spacetime domain would be reached where attained energy scales are very high. In consequence some of the compactified extra dimensions may be decompactified and some presently broken symmetries may be restored. A more fundamental theory (of which Einstein's theory is a symmetry broken phase) is then expected to take account of further description of the collapse. I propose a simple (classical) model for the description of the late stages of the gravitational collapse: A non-Riemannian, scale-invariant version of 5-dimensional Kaluza-Klein theory in which the standard Riemann structure of the higher-dimensional manifold is replaced by a Weyl-integrable one. A class of solutions, that generalize the "soliton" one by Gross and Perry and Davidson and Owen, is found. This class contains both naked singularities and wormhole solutions. On physical grounds it is argued that a wormhole is the most reasonable destiny of the gravitational collapse. 
  The Noether current and its variation relation with respect to diffeomorphism invariance of gravitational theories have been derived from the horizontal variation and vertical-horizontal bi-variation of the Lagrangian, respectively. For Einstein's GR in the stationary, axisymmetric black holes, the mass formula in vacuum can be derived from this Noether current although it definitely vanishes. This indicates that the mass formula of black holes is a vanishing Noether charge in this case. The first law of black hole thermodynamics can also be derived from the variation relation of this vanishing Noether current. 
  In the paper published in Phys. Lett. A245} (1998) 31, Barros and Romero demonstrated that, in the weak-field approximation, solutions to the Brans-Dicke equations are related to the solutions of General Relativity for the same matter distributions. In the present work, we enphasize this result and we extend it to for generalized scalar-tensor theories in which the parameter $\omega$ is no longer a constant but an arbitrary function of the (gravitational) scalar field. 
  This paper provides a review of some recent issues on the Mixmaster dynamics concerning the features of its stochasticity. After a description of the geometrical structure characterizing the homogeneous cosmological models in the Bianchi classification and the Belinsky-Khalatnikov-Lifshitz piecewise representation of the types VIII and IX oscillatory regime, we face the question regarding the time covariance of the resulting chaos as viewed in terms of continuous Misner-Chitr\'e like variables. Finally we show how in the statistical mechanics framework the Mixmaster chaos raises as semiclassical limit of the quantum dynamics in the Planckian era. 
  We provide a Hamiltonian analysis of the Mixmaster Universe dynamics on the base of a standard Arnowitt-Deser-Misner Hamiltonian approach, showing the covariant nature of its chaotic behaviour with respect to the choice of any time variable, from the point of view either of the dynamical systems theory, either of the statistical mechanics one. 
  A study of the linearised gravitational field (spin 2 zero-rest-mass field) on a Minkowski background close to spatial infinity is done. To this purpose, a certain representation of spatial infinity in which it is depicted as a cylinder is used. A first analysis shows that the solutions generically develop a particular type of logarithmic divergence at the sets where spatial infinity touches null infinity. A regularity condition on the initial data can be deduced from the analysis of some transport equations on the cylinder at spatial infinity. It is given in terms of the linearised version of the Cotton tensor and symmetrised higher order derivatives, and it ensures that the solutions of the transport equations extend analytically to the sets where spatial infinity touches null infinity. It is later shown that this regularity condition together with the requirement of some particular degree of tangential smoothness ensures logarithm-free expansions of the time development of the linearised gravitational field close to spatial and null infinities. 
  We study an analytical solution to the Einstein's equations in 2+1-dimensions, representing the self-similar collapse of a circularly symmetric, minimally coupled, massless, scalar field. Depending on the value of certain parameters, this solution represents the formation of naked singularities. Since our solution is asymptotically flat, these naked singularities may be relevant for the weak cosmic censorship conjecture in 2+1-dimensions. 
  We prove that all spherically symmetric static spacetimes which are both regular at r=0 and satisfying the single energy condition rho + p_r + p_t >= 0 cannot contain any black hole region (equivalently, they must satisfy 2m/r <= 1 everywhere). This result holds even when the spacetime is allowed to contain a finite number of matching hypersurfaces. This theorem generalizes a result by Baumgarte and Rendall when the matter contents of the space-time is a perfect fluid and also complements their results in the general non-isotropic case. 
  A metric-field approach to gravitation is presented. It is based on an idea of dependency of space-time properties on measuring instruments. Some bimetric equations that realize this idea are considered. They were tested by the binary pulsar PSR1913+16. The spherically - symmetric solution of the equations has no event horizon and no physical singularity in the center. The proper energy of a point particle is finite. There can exist supermassive compact configurations of degenerated Fermi-gas which can be identified with observed objects in galactic centers. The problem of the Universe acceleration has a natural explanation. 
  Gravity does not provide any scale for matter properties. We argue that this is also the implication of Mach's hypothesis of the relativity of inertia. The most general spacetime compatible with this property of gravity is that admitting three, independent spatial homothetic Killing vectors generating an arbitrary function of each one of the three spatial coordinates. The matter properties for such a spacetime are (spatially) arbitrary and the matter generating the spacetime admits {\it any} equation of state. This is also the most general spacetime containing the weak gravity physics in its entirety. This spacetime is machian in that it is {\em globally} degenerate for anti-machian situations such as vacuum, a single matter particle etc. and, hence, has no meaning in the absence of matter. 
  It is shown that the exact Foldy-Wouthuysen transformation for spin-0 particles on spacetimes described by the metrics $ds^2 = V^2 dt^2 - W^2 d {\bf{x}}^2$, where $V=V({\bf{x}})$ and $W=W({\bf{x}})$, only exists if the scalar field is nonminimally coupled to the Ricci scalar field with a coupling constant equal to 1/6. The nonminimal coupling term, in turn, does not violate the equivalence principle. As an application we obtain the nonrelativistic Foldy-Wouthuysen Hamiltonian concerning the general solution to the linearized field equations of higher-derivative gravity for a static pointlike source in the Teyssandier gauge. 
  I study the possibility of baryogenesis can take place in fresh inflation. I find that it is possible that violation of baryon number conservation can occur during the period out-of-equilibrium in this scenario. Indeed, baryogenesis could be possible before the thermal equilibrium is restored at the end of fresh inflation. 
  Following the spirit of a previous work of ours, we investigate the group of those General Coordinate Transformations (GCTs) which preserve manifest spatial homogeneity. In contrast to the case of Bianchi Type Models we, here, permit an isometry group of motions $G_{4}=SO(3)\otimes T_{r}$, where $T_{r}$ is the translations group, along the radial direction, while SO(3) acts multiply transitively on each hypersurface of simultaneity $\Sigma_{t}$. The basis 1-forms, can not be invariant under the action of the entire isometry group and hence produce an Open Lie Algebra. In order for these GCTs to exist and have a non trivial, well defined action, certain integrability conditions have to be satisfied; their solutions, exhibiting the maximum expected ``gauge'' freedom, can be used to simplify the generic, spatially homogeneous, line element. In this way an alternative proof of the generality of the Kantowski-Sachs (KS) vacuum is given, while its most general, manifestly homogeneous, form is explicitly presented. 
  In the isotropic quantum cosmological perfect fluid model, the initial singularity can be avoided, while the classical behaviour is recovered asymptotically. We verify if initial anisotropies can also be suppressed in a quantum version of a classical anisotropic model where gravity is coupled to a perfect fluid. Employing a Bianchi I cosmological model, we obtain a "Schr\"odinger-like" equation where the matter variables play de role of time. This equation has a hyperbolic signature. It can be explicitly solved and a wave packet is constructed. The expectation value of the scale factor, evaluated in the spirit of the many-worlds interpretation, reveals an isotropic Universe. On the other hand, the bohmian trajectories indicate the existence of anisotropies. This is an example where the Bohm-de Broglie and the many-worlds interpretations are not equivalent. It is argued that this inequivalence is due to the hyperbolic structure of the "Schr\"odinger-like" equation. 
  We give relations for the embedding of spatially-flat Friedmann-Robertson-Walker cosmological models of Einstein's theory in flat manifolds of the type used in Kaluza-Klein theory. We present embedding diagrams that depict different 4D universes as hypersurfaces in a higher dimensional flat manifold. The morphology of the hypersurfaces is found to depend on the equation of state of the matter. The hypersurfaces possess a line-like curvature singularity infinitesimally close to the $t = 0^+$ 3-surface, where $t$ is the time expired since the big bang. The family of timelike comoving geodesics on any given hypersurface is found to have a caustic on the singular line, which we conclude is the 5D position of the point-like big bang. 
  We analyse the distributional thin wall limit of self gravitating scalar field configurations representing thick domain wall geometries. We show that thick wall solutions can be generated by appropiate scaling of the thin wall ones, and obtain an exact solution for a domain wall that interpolates between AdS_4 asymptotic vacua and has a well-defined thin wall limit.Solutions representing scalar field configurations obtained via the same scaling but that do not have a thin wall limit are also presented. 
  In the bimetric scalar-tensor gravitational theory there are two frames associated with the two metrics {\hat g}_{\mu\nu} and g_{\mu\nu}, which are linked by the gradients of a scalar field \phi. The choice of a comoving frame for the metric {\hat g}_{\mu\nu} or g_{\mu\nu} has fundamental consequences for local observers in either metric spacetimes, while maintaining diffeomorphism invariance. When the metric g_{\mu\nu} is chosen to be associated with comoving coordinates, then the speed of light varies in the frame with the metric {\hat g}_{\mu\nu}. Observers in this frame see the dimming of supernovae because of the increase of the luminosity distance versus red shift, due to an increasing speed of light in the early universe. Moreover, in this frame the scalar field \phi describes a dark energy component in the Friedmann equation for the cosmic scale without acceleration. If we choose {\hat g}_{\mu\nu} to be associated with comoving coordinates, then an observer in the g_{\mu\nu} metric frame will observe the universe to be accelerating and the supernovae will appear to be farther away. The theory predicts that the gravitational constant G can vary in spacetime, while the fine-structure constant \alpha=e^2/\hbar c does not vary. The problem of cosmological horizons as viewed in the two frames is discussed. 
  We review some old and new results about strict and non strict hyperbolic formulations of the Einstein equations. 
  We establish a variant, which has the advantage of introducing only physical characteristics, of the symmetric quasi linear first order system given by H.\ Friedrich for the evolution equations of gravitating fluid bodies in General Relativity which can be important to solve realistic problems. We explicit the conditions under which the system is hyperbolic and admits a well posed Cauchy problem. 
  In this work, we discuss the quantum mechanics on the moduli space consisting of two maximally charged dilaton black holes. We study the quantum effects resulting from the different structure of the moduli space geometry in the scattering process. 
  The article reviews the current status of a theoretical approach to the problem of the emission of gravitational waves by isolated systems in the context of general relativity. Part A of the article deals with general post-Newtonian sources. The exterior field of the source is investigated by means of a combination of analytic post-Minkowskian and multipolar approximations. The physical observables in the far-zone of the source are described by a specific set of radiative multipole moments. By matching the exterior solution to the metric of the post-Newtonian source in the near-zone we obtain the explicit expressions of the source multipole moments. The relationships between the radiative and source moments involve many non-linear multipole interactions, among them those associated with the tails (and tails-of-tails) of gravitational waves. Part B of the article is devoted to the application to compact binary systems. We present the equations of binary motion, and the associated Lagrangian and Hamiltonian, at the third post-Newtonian (3PN) order beyond the Newtonian acceleration. The gravitational-wave energy flux, taking consistently into account the relativistic corrections in the binary moments as well as the various tail effects, is derived through 3.5PN order with respect to the quadrupole formalism. The binary's orbital phase, whose prior knowledge is crucial for searching and analyzing the signals from inspiralling compact binaries, is deduced from an energy balance argument. 
  Using numerical calculations, we compare three versions of the Barrett-Crane model of 4-dimensional Riemannian quantum gravity. In the version with face and edge amplitudes as described by De Pietri, Freidel, Krasnov, and Rovelli, we show the partition function diverges very rapidly for many triangulated 4-manifolds. In the version with modified face and edge amplitudes due to Perez and Rovelli, we show the partition function converges so rapidly that the sum is dominated by spin foams where all the spins labelling faces are zero except for small, widely separated islands of higher spin. We also describe a new version which appears to have a convergent partition function without drastic spin-zero dominance. Finally, after a general discussion of how to extract physics from spin foam models, we discuss the implications of convergence or divergence of the partition function for other aspects of a spin foam model. 
  The role of the SO(2,1) symmetry in General Relativity is analyzed. Cosmological solutions of Einstein field equations invariant with respect to a space-like Lie algebra G_r, with r between 3 and 6 and containing so(2,1) as a subalgebra, are also classified. 
  We compute spectra, waveforms, angular distribution and total gravitational energy of the gravitiational radiation emitted during the radial infall of a massless particle into a Schwarzschild black hole. Our fully relativistic approach shows that (i) less than 50% of the total energy radiated to infinity is carried by quadrupole waves, (ii) the spectra is flat, and (iii) the zero frequency limit agrees extremely well with a prediction by Smarr. This process may be looked at as the limiting case of collisions between massive particles traveling at nearly the speed of light, by identifying the energy $E$ of the massless particle with $m_0 \gamma$, $m_0$ being the mass of the test particle and $\gamma$ the Lorentz boost parameter. We comment on the implications for the two black hole collision at nearly the speed of light process, where we obtain a 13.3% wave generation efficiency, and compare our results with previous results by D'Eath and Payne. 
  Quantized orbital structures are typical for many aspects of classical gravity (Newton's as well as Einstein's). The astronomical phenomenon of orbital resonances is a well-known example. Recently, Rothman, Ellis and Murugan (2001) discussed quantized orbital structures in the novel context of a holonomy invariance of parallel transport in Schwarzschild geometry. We present here yet another example of quantization of orbits, reflecting both orbital resonances and holonomy invariance. This strong-gravity effect may already have been directly observed as the puzzling kilohertz quasi-periodic oscillations (QPOs) in the X-ray emission from a few accreting galactic black holes and several neutron stars. 
  It will be shown that while horizons do not exist for warp drive spacetimes traveling at subluminal velocities horizons begin to develop when a warp drive spacetime reaches luminal velocities. However it will be shown that the control region of a warp drive ship lie within the portion of the warped region that is still causally connected to the ship even at superluminal velocities, therefore allowing a ship to slow to subluminal velocities. Further it is shown that the warped regions which are causally disconnected from a warp ship have no correlation to the ship velocity. 
  Metric perturbations the stability of solution of Einstein-Cartan cosmology (ECC) are given. The first addresses the stability of solutions of Einstein-Cartan (EC) cosmological model against Einstein static universe background. In this solution we show that the metric is stable against first-order perturbations and correspond to acoustic oscillations. The second example deals with the stability of de Sitter metric also against first-order perturbations. Torsion and shear are also computed in these cases. The resultant perturbed anisotropic spacetime with torsion is only de Sitter along one direction or is unperturbed along one direction and perturbed against the other two. Cartan torsion contributes to the frequency of oscillations in the model. Therefore gravitational waves could be triggered by the spin-torsion scalar density . 
  A simple model of spacetime foam, made by $N$ Reissner-Nordstr\"{o}m wormholes with a magnetic and electric charge in a semiclassical approximation, is taken under examination. The Casimir-like energy of the quantum fluctuation of such a model is computed and compared with that one obtained with a foamy space modeled by $N$ Schwarzschild wormholes. The comparison leads to the conclusion that a foamy spacetime cannot be considered as a collection of $N$ Reissner-Nordstr\"{o}m wormholes but that such a collection can be taken as an excited state of the foam. 
  There is proved an existence theorem, in the Newtonian theory, for static, self-gravitating bodies composed of elastic material. The theorem covers the case where these bodies are small, but allows them to have arbitrary shape. 
  A kinetic theory of relativistic gases in a two-dimensional space is developed in order to obtain the equilibrium distribution function and the expressions for the fields of energy per particle, pressure, entropy per particle and heat capacities in equilibrium. Furthermore, by using the method of Chapman and Enskog for a kinetic model of the Boltzmann equation the non-equilibrium energy-momentum tensor and the entropy production rate are determined for a universe described by a two-dimensional Robertson-Walker metric. The solutions of the gravitational field equations that consider the non-equilibrium energy-momentum tensor - associated with the coefficient of bulk viscosity - show that opposed to the four-dimensional case, the cosmic scale factor attains a maximum value at a finite time decreasing to a "big crunch" and that there exists a solution of the gravitational field equations corresponding to a "false vacuum". The evolution of the fields of pressure, energy density and entropy production rate with the time is also discussed. 
  We describe how a spin-foam state sum model can be reformulated as a quantum field theory of spin networks, such that the Feynman diagrams of that field theory are the spin-foam amplitudes. In the case of open spin networks, we obtain a new type of state-sum models, which we call the matter spin foam models. In this type of state-sum models, one labels both the faces and the edges of the dual two-complex for a manifold triangulation with the simple objects from a tensor category. In the case of Lie groups, such a model corresponds to a quantization of a theory whose fields are the principal bundle connection and the sections of the associated vector bundles. We briefly discuss the relevance of the matter spin foam models for quantum gravity and for topological quantum field theories. 
  We consider the existence of Einstein-Maxwell-dilaton plus fluid system for the case of stationary cylindrically symmetric spacetimes. An exact inhomogeneous $ \epsilon$-order solution is found, where the parameter $\epsilon$ parametrizes the non-minimally coupled electromagnetic field. Some its physical attributes are investigated and a connection with already known G\"odel-type solution is given. It is shown that the found solution also survives in the string-inspired charged gravity framework. We find that a magnetic field has positive influence on the chronology violation unlike the dilaton influence. 
  We present herewith certain thoughts on the important subject of nowadays physics, pertaining to the so-called ``singularities'', that emanated from looking at the theme in terms of ADG (: abstract differential geometry). Thus, according to the latter perspective, we can involve ``singularities'' in our arguments, while still employing fundamental differential-geometric notions such as connections, curvature, metric and the like, retaining also the form of standard important relations of the classical theory (e.g. Einstein and/or Yang-Mills equations, in vacuum), even within that generalized context of ADG. To wind up, we can extend (in point of fact, {calculate) over singularities classical differential-geometric relations/equations, without altering their forms and/or changing the standard arguments; the change concerns thus only the way, we employ the usual differential geometry of smooth manifolds, so that the base ``space'' acquires now quite a secondary role, not contributing at all (!) to the differential-geometric technique/mechanism that we apply. Thus, the latter by definition refers directly to the objects being involved--the objects that ``live on that space'', which by themselves are not, of course, ipso facto ``singular''! 
  An asymptotic framework is defined for the small parameter eta which quantifies a good separation between the extended bodies that make a weakly gravitating system. This is introduced within an alternative scalar theory of gravitation, though it may be defined similarly in other theories. This framework allows one to truncate the translational equations of motion at any well-defined order. Here, the post-Newtonian (PN) equations valid in the scalar theory are truncated beyond the order eta^3. The PN approximation scheme used is the asymptotic scheme, that expands all fields. To get the explicit form of the equations of motion for the mass centers, the bodies are assumed spherical, merely for calculating the PN corrections. It is found that, due to the use of the asymptotic PN scheme, the internal structure of the bodies does play a role in the equations of motion. 
  A framework based on an extension of Kaluza's original idea of using a five dimensional space to unify gravity with electromagnetism is used to analyze Maxwell\'{}s field equations. The extension consists in the use of a six dimensional space in which all equations of electromagnetism may be obtained using only Einstein's field equation. Two major advantages of this approach to electromagnetism are discussed, a full symmetric derivation for the wave equations for the potentials and a natural inclusion of magnetic monopoles without using any argument based on singularities. 
  Contrary to established beliefs, spacetime may not be time-orientable. By considering an experimental test of time orientability it is shown that a failure of time-orientability of a spacetime region would be indistinguishable from a particle antiparticle annihilation event. 
  In this talk r-form fields in spacetimes of any dimension D are considered (r<D). The weak-field Newtonian-type limit of Einstein's equations, in general, with relativistic sources is studied in the static case yielding a revision of the equivalence principle (intrinsically relativistic sources generate twice stronger gravitational fields and hyperrelativistic sources, e.g., the stiff matter, generate four times stronger fields than non-relativistic sources). It is shown that analogues of electromagnetic field, strictly speaking, exist only in even-dimensional spacetimes. In (2+1)-dimensional spacetime, the field traditionally interpreted as "magnetic" turns out to be in fact a perfect fluid, and "electric", a perverse fluid (this latter concept arises inevitably in the r-form description of fluids for any D, and we consider here perverse fluids in (3+1)-dimensional spacetime too). New exact solutions of (2+1)-dimensional Einstein's equations with perfect and perverse fluids are obtained, and it is shown that in this case there exists a vast family of static solutions for non-coherent dust, in a sharp contrast to the (3+1)-dimensional case. New general interpretation of the cosmological term in D-dimensional Einstein's equations is given via the (D-1)-form field, and it is shown that this field is as well responsible (as this is the case in 3+1 dimensions) for rotation of perfect fluids [(D-2)-form fields], thus the "source" term in the corresponding field equations has to be interpreted as the rotation term. 
  A physical interpretation of the C-metric with a negative cosmological constant $\Lambda$ is suggested. Using a convenient coordinate system it is demonstrated that this class of exact solutions of Einstein's equations describes uniformly accelerating (possibly charged) black holes in anti-de Sitter universe. Main differences from the analogous de Sitter case are emphasised. 
  What is the nature of the energy spectrum of a black hole ? The algebraic approach to black hole quantization requires the horizon area eigenvalues to be equally spaced. As stressed long ago by by Mukhanov, such eigenvalues must be exponentially degenerate with respect to the area quantum number if one is to understand black hole entropy as reflecting degeneracy of the observable states. Here we construct the black hole states by means of a pair of "creation operators" subject to a particular simple algebra, a slight generalization of that for the harmonic oscillator. We then prove rigorously that the n-th area eigenvalue is exactly 2 raised to the n-fold degenerate. Thus black hole entropy qua logarithm of the number of states for fixed horizon area comes out proportional to that area. 
  For binary black holes the lapse function corresponding to the Brill-Lindquist initial value solution for uncharged black holes is given in analytic form under the maximal slicing condition. In the limiting case of very small ratio of mass to separation between the black holes the surface defined by the zero value of the lapse function coincides with the minimal surfaces around the singularities. 
  We study the stationary and axisymmetric non-convective differentially rotating perfect-fluid solutions of Einstein's field equations admitting one conformal symmetry. We analyse the two inequivalent Lie algebras not exhaustively considered in Mars and Senovilla, 1994, and show that the general solution for each Lie algebra depends on one arbitrary function of one of the coordinates while a set of three ordinary differential equations for four unknowns remains to be solved. The conformal Killing vector of these solutions is necessarily homothetic. We summarize in a table all the possible solutions for all the allowed Lie algebras and also add a corrigendum to an erroneous statement in the paper quated above concerning the differentially rotating character of one of the solutions presented 
  In the next few years, the first detections of gravity-wave signals using Earth-based interferometric detectors will begin to provide precious new information about the structure and dynamics of compact bodies such as neutron stars. The intrinsic weakness of gravity-wave signals requires a proactive approach to modeling the prospective sources and anticipating the shape of the signals that we seek to detect. Full-blown 3-D numerical simulations of the sources are playing and will play an important role in planning the gravity-wave data-analysis effort. I review some recent analytical and numerical work on neutron stars as sources of gravity waves. 
  We report the reduction of the thermal lensing in cryogenic sapphire mirrors, which is planed to be used in the Large scale Cryogenic Gravitational wave Telescope (LCGT) project. We measured three key parameters of sapphire substrate for thermal lensing at cryogenic temperature. They are optical absorption coefficient, thermal conductivity and temperature coefficient of refractive index at cryogenic temperature. On basis of these measurements, we estimated the shot noise sensitivity of the interferometer with thermal lensing by using a wave-front tracing simulation. We found that thermal lensing in cryogenic sapphire mirrors is negligible. 
  In this work we solve Dirac equation by using the method of seperation of variables. Then we analyzed the particle creation process. To compute the density number of particles created Bogoliubov transformation technique is used. 
  The field equations of the new general relativity (NGR) have been applied to an absolute parallelism space having three unknown functions of the radial coordinate. The field equations have been solved using two different schemes. In the first scheme, we used the conventional procedure used in orthodox general relativity. In the second scheme, we examined the effect of signature change. The latter scheme gives a solution which is different from the Schwarzschild one. In both methods we find solution of the field equations under the same constraint imposed on the parameters of the theory. We also calculated the energy associated with the solutions in the two cases using the superpotential method. We found that the energetic content of one of the solutions is different from that of the other. A comparison between the two solutions obtained in the present work and a third one obtained by Hayashi and Shirafuji (1979) shows that the change of the signature may give rise to new physics. 
  The nonconformal scalar field is considered in N-dimensional space-time with metric which includes, in particular, the cases of nonhomogeneous spaces and anisotropic spaces of Bianchi type-I. The modified Hamiltonian is constructed. Under the diagonalization of it the energy of quasiparticles is equal to the oscillator frequency of the wave equation. The density of particles created by nonstationary metric is investigated. It is shown that the densities of conformal and nonconformal particles created in Friedmann radiative-dominant Universe coincide. 
  The gravitational collapse of an infinite cylindrical thin shell of generic matter in an otherwise empty spacetime is considered. We show that geometries admitting two hypersurface orthogonal Killing vectors cannot contain trapped surfaces in the vacuum portion of spacetime causally available to geodesic timelike observers. At asymptotic future null infinity, however, congruences of outgoing radial null geodesics become marginally trapped, due to convergence induced by shear caused by the interaction of a transverse wave component with the geodesics. The matter shell itself is shown to be always free of trapped surfaces, for this class of geometries. Finally, two simplified matter models are analytically examined. For one model, the weak energy condition is shown to be a necessary condition for collapse to halt; for the second case, it is a sufficient condition for collapse to be able to halt. 
  We report a three parameter family of solutions for dilaton gravity in 2+1 dimensions with finite mass and finite angular momentum. These solutions are obtained by a compactification of vacuum solutions in 3+1 dimensions with cylindrical symmetry. One class of solutions corresponds to conical singularities and the other leads to curvature singularities. 
  Community News:    Center for Gravitational Wave Physics, by Sam Finn    Perimeter Institute for Theoretical Physics, by Lee Smolin   Research Briefs:    Detector and Data Developments within GEO 600, by Alicia Sintes    The Virtual Data Grid and LIGO, by Pat Brady and Manuela Campanelli    1000 hours of data and first lock of the recycled TAMA300, by Seiji Kawamura    LIGO Takes Some Data!, by Stan Whitcomb    Quantum gravity: progress from an unexpected direction, by Matt Visser   Conference report:    Gravitational-wave phenomenology at PennState, by Nils Andersson 
  Starting from a gauge invariant treatment of perturbations an analytical expression for the spectrum of long wavelength density perturbations in warm inflation is derived. The adiabatic and entropy modes are exhibited explicitly. As an application of the analytical results, we determined the observational constraint for the dissipation term compatible with COBE observation of the cosmic microwave radiation anisotropy for some specific models. In view of the results the feasibility of warm inflation is discussed. 
  We study the general dynamics of the spherically symmetric gravitational collapse of a massless scalar field. We apply the Galerkin projection method to transform a system of partial differential equations into a set of ordinary differential equations for modal coefficients, after a convenient truncation procedure, largely applied to problems of turbulence. In the present case, we have generated a finite dynamical system that reproduces the essential features of the dynamics of the gravitational collapse, even for a lower order of truncation. Each initial condition in the space of modal coefficients corresponds to a well definite spatial distribution of scalar field. Numerical experiments with the dynamical system show that depending on the strength of the scalar field packet, the formation of black-holes or the dispersion of the scalar field leaving behind flat spacetime are the two main outcomes. We also found numerical evidence that between both asymptotic states, there is a critical solution represented by a limit cycle in the modal space with period $\Delta u \approx 3.55$. 
  The dynamics of a general Bianchi IX model with three scale factors is examined. The matter content of the model is assumed to be comoving dust plus a positive cosmological constant. The model presents a critical point of saddle-center-center type in the finite region of phase space. This critical point engenders in the phase space dynamics the topology of stable and unstable four dimensional tubes $R \times S^3$, where $R$ is a saddle direction and $S^3$ is the manifold of unstable periodic orbits in the center-center sector. A general characteristic of the dynamical flow is an oscillatory mode about orbits of an invariant plane of the dynamics which contains the critical point and a Friedmann-Robertson-Walker (FRW) singularity. We show that a pair of tubes (one stable, one unstable) emerging from the neighborhood of the critical point towards the FRW singularity have homoclinic transversal crossings. The homoclinic intersection manifold has topology $R \times S^2$ and is constituted of homoclinic orbits which are bi-asymptotic to the $S^3$ center-center manifold. This is an invariant signature of chaos in the model, and produces chaotic sets in phase space. The model also presents an asymptotic DeSitter attractor at infinity and initial conditions sets are shown to have fractal basin boundaries connected to the escape into the DeSitter configuration (escape into inflation), characterizing the critical point as a chaotic scatterer. 
  A method for studying the causal structure of space-time evolution systems is presented. This method, based on a generalization of the well known Riemann problem, provides intrinsic results which can be interpreted from the geometrical point of view. A one-parameter family of hyperbolic evolution systems is presented and the physical relevance of their characteristic speeds and eigenfields is discussed. The two degrees of freedom corresponding to gravitational radiation are identified in an intrinsic way, independent of the space coordinate system. A covariant interpretation of these degrees of freedom is provided in terms of the geometry of the wave fronts. The requirement of a consistent geometrical interpretation of the gravitational radiation degrees of freedom is used to solve the ordering ambiguity that arises when obtaining first order evolution systems from the second order field equations. This achievement provides a benchmark which can be used to check both the existing and future first order hyperbolic formalisms for Numerical Relativity. 
  According to a simple model of inertia a Machianized theory of special and general relativity named as relational relativity is presented. 
  Here we are interested to study the spin-1 particle i.e., electro-magnetic wave in curved space-time, say around black hole. After separating the equations into radial and angular parts, writing them according to the black hole geometry, say, Kerr black hole we solve them analytically. Finally we produce complete solution of the spin-1 particles around a rotating black hole namely in Kerr geometry. Obviously there is coupling between spin of the electro-magnetic wave and that of black hole when particles propagate in that space-time. So the solution will be depending on that coupling strength. This solution may be useful to study different other problems where the analytical results are needed. Also the results may be useful in some astrophysical contexts. 
  A general, iterative, method for the description of evolving self-gravitating relativistic spheres is presented. Modeling is achieved by the introduction of an ansatz, whose rationale becomes intelligible and finds full justification within the context of a suitable definition of the post--quasistatic approximation. As examples of application of the method we discuss three models, in the adiabatic case. 
  We report a new family of solutions to Einstein-Maxwell-dilaton gravity in 2+1 dimensions and Einstein-Maxwell gravity with cylindrical symmetry in 3+1 dimensions. A set of static charged solutions in 2+1 dimensions are obtained by a compactification of charged solutions in 3+1 dimensions with cylindrical symmetry. These solutions contain naked singularities for certain values of the parameters considered. New rotating charged solutions in 2+1 dimensions and 3+1 dimensions are generated treating the static charged solutions as seed metrics and performing $SL(2;R)$ transformations. 
  I review recent work on massive higher (s>1) spins in constant curvature (deSitter) spaces. Some of the novel properties that emerge are: partial masslessness and new local gauge invariances, unitarily forbidden ranges of mass, correlation between fermions/bosons and nagative/positive cosmological constant Lambda and finally the consistency requirement that in the limit of infinite spin towers, Lambda must tend to zero. 
  We consider the problem of uniqueness of the kernel in the nonlocal theory of accelerated observers. In a recent work, we showed that the convolution kernel is ruled out as it can lead to divergences for nonuniform accelerated motion. Here we determine the general form of bounded continuous kernels and use observational data regarding spin-rotation coupling to argue that the kinetic kernel given by $K(\tau ,\tau')=k(\tau')$ is the only physically acceptable solution. 
  We are entering an era where the numerical construction of generic spacetimes is becoming a reality. The use of computer simulations, in principle, allows us to solve Einstein equations in their full generality and unravel important messages so far hidden in the theory. Despite the problems still found in present applications, significant progress is being achieved in a number of areas where simulations are on the verge of providing useful physical information. This article reviews recent progress made in the field, paying particular attention to black hole spacetimes and discuss open problems and prospects for the future. 
  The paper establishes the result that solutions of the type described in the title of the article are only those that have been already presented in the literature. The procedure adopted in the paper is somewhat novel - while the usual practice is to display an exact solution and then to examine whether it is singularity free, the present paper discovers the conditions which a singularity free solution of the desired type must satistfy. There is no attempt to obtain exact solutions. Simply, the conditions that were ad-hoc introduced in the deduction of singularity free solutions are here shown to follow from the requirement of non-singularity. 
  We consider the interaction of gravity, as expressed by Einstein's Equations of General Relativity, to other force fields. We describe some recent results, discussing both the mathematics, and the physical interpretations. These results concern both elementary particles, as well as cosmological models. (This paper describes joint work variously done with with F. Finster, N. Kamran, B. Temple, and S.-T. Yau.) 
  The generalization of the concept of homogeneous gravitational field from Classical Mechanics was considered in the framework of Einstein's General Relativity by Bogorodskii. In this paper, I look for such a generalization in the framework of the Relativistic Theory of Gravitation. There exist a substantial difference between the solutions in these two theories. Unfortunately, the solution obtained according to the Relativistic Theory of Gravitation can't be accepted because it doesn't fulfill the Causality Principle in this theory. So, it remains open in RTG the problem of finding a generalization of the classical concept of homogeneous gravitational field. 
  Traversible wormhole space-times are found as static, spherically symmetric solutions to the Einstein equations with ingoing and outgoing pure ghost radiation, i.e. pure radiation with negative energy density. Switching off the radiation causes the wormhole to collapse to a Schwarzschild black hole. 
  We study the Einstein field equations for spacetimes admitting a maximal two-dimensional abelian group of isometries acting orthogonally transitively on spacelike surfaces and, in addition, with at least one conformal Killing vector. The three-dimensional conformal group is restricted to the case when the two-dimensional abelian isometry subalgebra is an ideal and it is also assumed to act on non-null hypersurfaces (both, spacelike and timelike cases are studied). We consider both, diagonal and non-diagonal metrics and find all the perfect-fluid solutions under these assumptions (except those already known). We find four families of solutions, each one containing arbitrary parameters for which no differential equations remain to be integrated. We write the line-elements in a simplified form and perform a detailed study for each of these solutions, giving the kinematical quantities of the fluid velocity vector, the energy-density and pressure, values of the parameters for which the energy conditions are fulfilled everywhere, the Petrov type, the singularities in the spacetimes and the Friedmann-Lemaitre-Robertson-Walker metrics contained in each family. 
  Embedding of the brane metric into Euclidean (2+4)-space is found. Brane geometry can be visualized as the surface of the hyper-sphere in six dimensions which 'radius' is governed by the cosmological constant. Minkowski space in this picture is lied on the intersection of this surface with the plane formed by the extra space-like and time-like coordinates. 
  In this paper, I determine the electrogravitational field produced by a charged mass point according to the Relativistic Theory of Gravitation. The Causality Principle in the Relativistic Theory of Gravitation will play a very important part in finding this field. The analytical form and the domain of definition, i.e the gravitational radius of the obtained solution, differ from that given by Einstein's General Relativity Theory. 
  This chapter is concerned with the question: how do gravitational waves (GWs) interact with their detectors? It is intended to be a theory review of the fundamental concepts involved in interferometric and acoustic (Weber bar) GW antennas. In particular, the type of signal the GW deposits in the detector in each case will be assessed, as well as its intensity and deconvolution. Brief reference will also be made to detector sensitivity characterisation, including very summary data on current state of the art GW detectors. 
  We consider the scenario emerging from the dynamics of a generalized $d$-brane in a $(d+1, 1)$ spacetime. The equation of state describing this system is given in terms of the energy density, $\rho$, and pressure, $p$, by the relationship $p = - A/\rho^{\alpha}$, where $A$ is a positive constant and $0 < \alpha \le 1$. We discuss the conditions under which homogeneity arises and show that this equation of state describes the evolution of a universe evolving from a phase dominated by non-relativistic matter to a phase dominated by a cosmological constant via an intermediate period where the effective equation of state is given by $p = \alpha \rho$. 
  A general class of solutions of Einstein's equation for a slowly rotating fluid source, with supporting internal pressure, is matched using Lichnerowicz junction conditions, to the Kerr metric up to and including first order terms in angular speed parameter. It is shown that the match applies to any previously known non-rotating fluid source made to rotate slowly for which a zero pressure boundary surface exists. The method is applied to the dust source of Robertson-Walker and in outline to an interior solution due to McVittie describing gravitational collapse. The applicability of the method to additional examples is transparent. The differential angular velocity of the rotating systems is determined and the induced rotation of local inertial frame is exhibited. 
  We investigate the cosmology of the two-dimensional Jackiw-Teitelboim model. Since the matter coupling is not defined uniquely, we consider two possible choices. The dilaton field plays an important role in the discussion of the properties of the solutions. In particular, the possibility of universes having a finite initial size emerges. 
  We study nature of singularities in anisotropic string-inspired cosmological models in the presence of a Gauss-Bonnet term. We analyze two string gravity models-- dilaton-driven and modulus-driven cases-- in the Bianchi type-I background without an axion field. In both scenarios singularities can be classified in two ways- the determinant singularity where the main determinant of the system vanishes and the ordinary singularity where at least one of the anisotropic expansion rates of the Universe diverges. In the dilaton case, either of these singularities inevitably appears during the evolution of the system. In the modulus case, nonsingular cosmological solutions exist both in asymptotic past and future with determinant $D=+\infty$ and D=2, respectively. In both scenarios nonsingular trajectories in either future or past typically meet the determinant singularity in past/future when the solutions are singular, apart from the exceptional case where the sign of the time-derivative of dilaton is negative. This implies that the determinant singularity may play a crucial role to lead to singular solutions in an anisotropic background. 
  The electromagnetic fields generated by a ring current around a Kerr black hole have been found. The acceleration of a charged particle by a force electric field along the rotation axis is investigated in the constructed model, as applied to the astrophysics of quasars. 
  Confirming previous heuristic analyses \`a la Belinskii-Khalatnikov-Lifshitz, it is rigorously proven that certain ``subcritical'' Einstein-matter systems exhibit a monotone, generalized Kasner behaviour in the vicinity of a spacelike singularity. The D-dimensional coupled Einstein-dilaton-p-form system is subcritical if the dilaton couplings of the p-forms belong to some dimension dependent open neighbourhood of zero, while pure gravity is subcritical if D is greater than or equal to 11. Our proof relies, like the recent theorem dealing with the (always subcritical) Einstein-dilaton system, on the use of Fuchsian techniques, which enable one to construct local, analytic solutions to the full set of equations of motion. The solutions constructed are ``general'' in the sense that they depend on the maximal expected number of free functions. 
  Quantum thermal effect of Dirac particles in an arbitrarily accelerating Kinnersley black hole is investigated by using the method of generalized tortoise coordinate transformation. Both the location and the temperature of the event horizon depend on the advanced time and the angles. The Hawking thermal radiation spectrum of Dirac particles contains a new term which represents the interaction between particles with spin and black holes with acceleration. This spin-acceleration coupling effect is absent from the thermal radiation spectrum of scalar particles. 
  We study the generation of a gravitational wave (GW) background produced from a population of core-collapse supernovae, which form black holes in scenarios of structure formation of the Universe. We obtain, for example, that a pre-galactic population of black holes, formed at redshifts $z\simeq 30-10$, could generate a stochastic GW background with a maximum amplitude of $h_{\rm BG}\simeq 10^{-24}$ in the frequency band $\nu_{\rm obs}\simeq 30-470 {\rm Hz}$ (considering a maximum efficiency of generation of GWs, namely, $\epsilon_{\rm GW}=7\times 10^{-4}$). In particular, we discuss what astrophysical information could be obtained from a positive, or even a negative, detection of such a GW background produced in scenarios such as those studied here. One of them is the possibility of obtaining the initial and final redshifts of the emission period from the observed spectrum of GWs. 
  This contribution summarizes some recent work on gravitational-wave astronomy and, especially, on the generation and detection of relic gravitational waves. We begin with a brief discussion of astrophysical sources of gravitational waves that are likely to be detected first by the coming in operation laser interferometers, such as GEO, LIGO, VIRGO. Then, we proceed to relic gravitational waves emphasizing their quantum-mechanical origin and the inevitability of their existence. Combining the theory with available observations, we discuss the prospects of direct detection of relic gravitational waves. A considerable part of the paper is devoted to comparison of relic gravitational waves with the density perturbations of quantum-mechanical origin. It is shown how the phenomenon of squeezing of quantum-mechanically generated cosmological perturbations manifests itself in the periodic structures of the metric power spectra and in the oscillatory behaviour of the CMBR multipoles $C_l$ as a function of $l$. The cosmological importance of the theoretically calculated (statistical) dipole moment $C_1$ is stressed. The paper contains also some comments on the damage to gravitational-wave research inflicted by the "standard inflationary result". We conclude with the (now common) remarks on the great scientific importance of the continuing effort to observe relic gravitational waves, directly or indirectly. 
  The classical boundary-value problem of the Einstein field equations is studied with an arbitrary cosmological constant, in the case of a compact ($S^{3}$) boundary given a biaxial Bianchi-IX positive-definite three-metric, specified by two radii $(a,b).$ For the simplest, four-ball, topology of the manifold with this boundary, the regular classical solutions are found within the family of Taub-NUT-(anti)de Sitter metrics with self-dual Weyl curvature. For arbitrary choice of positive radii $(a,b),$ we find that there are three solutions for the infilling geometry of this type. We obtain exact solutions for them and for their Euclidean actions. The case of negative cosmological constant is investigated further. For reasonable squashing of the three-sphere, all three infilling solutions have real-valued actions which possess a ``cusp catastrophe'' structure with a non-self-intersecting ``catastrophe manifold'' implying that the dominant contribution comes from the unique real positive-definite solution on the ball. The positive-definite solution exists even for larger deformations of the three-sphere, as long as a certain inequality between $a$ and $b$ holds. The action of this solution is proportional to $-a^{3}$ for large $a (\sim b)$ and hence larger radii are favoured. The same boundary-value problem with more complicated interior topology containing a ``bolt'' is investigated in a forthcoming paper. 
  We present a class of simple scalar-tensor models of gravity with one scalar field (dilaton $\Phi$) and only one unknown function (cosmological potential $U(\Phi)$). These models might be considered as a stringy inspired ones with broken SUSY. They have the following basic properties: 1) Positive dilaton mass, $m_\Phi$, and positive cosmological constant $\Lambda$, define two extremely different scales. The models under consideration are consistent with the known experimental facts if $m_\Phi > 10^{-3} eV$ and $\Lambda=\Lambda^{obs}\sim 10^{-56} cm^{-2}$. 2) Einstein week equivalence principle is strictly satisfied and extended to scalar-tensor theories of gravity using a novel form of principle of "constancy of fundamental constants". 3) The dilaton plays simultaneously role of inflation field and quintessence field and yields a sequential hyper-inflation with graceful exit to asymptotic de Sitter space-time which is an attractor, and is approached as $\exp(-\sqrt{3\Lambda^{obs}} ct/2)$. The time duration of inflation is $\Delta t_{infl} \sim m_\Phi^{-1}$. 4) Ultra-high frequency ($\omega_\Phi \sim m_\Phi$) dilatonic oscillations take place in asymptotic regime. 5) No fine tuning. (The Robertson-Walker solutions of general type have the above properties.) 6) A novel adjustment mechanism for cosmological constant problem seems to be possible: the huge value of cosmological constant in the stringy frame is re-scaled to its observed value by dilaton after transition to phenomenological frame. 
  We use the M{\o}ller energy-momentum complex to calculate the energy of the Melvin magnetic universe. The energy distribution depends on the magnetic field. 
  Using a non-Riemannian geometry that is adapted to the 4+1 decomposition of space-time in Kaluza-Klein theory, the translational part of the connection form is related to the electromagnetic vector potential and a Stueckelberg scalar. The consideration of a five-dimensional gravitational action functional that shares the symmetries of the chosen geometry leads to a unification of the four-dimensional cosmological term and a mass term for the vector potential. 
  Isotropic models in loop quantum cosmology allow explicit calculations, thanks largely to a completely known volume spectrum, which is exploited in order to write down the evolution equation in a discrete internal time. Because of genuinely quantum geometrical effects the classical singularity is absent in those models in the sense that the evolution does not break down there, contrary to the classical situation where space-time is inextendible. This effect is generic and does not depend on matter violating energy conditions, but it does depend on the factor ordering of the Hamiltonian constraint. Furthermore, it is shown that loop quantum cosmology reproduces standard quantum cosmology and hence (e.g., via WKB approximation) to classical behavior in the large volume regime where the discreteness of space is insignificant. Finally, an explicit solution to the Euclidean vacuum constraint is discussed which is the unique solution with semiclassical behavior representing quantum Euclidean space. 
  The notions of temperature, entropy and `evaporation', usually associated with spacetimes with horizons, are analyzed using general approach and the following results, applicable to different spacetimes, are obtained at one go. (i) The concept of temperature associated with the horizon is derived in a unified manner and is shown to arise from purely kinematic considerations. (ii) QFT near any horizon is mapped to a conformal field theory without introducing concepts from string theory. (iii) For spherically symmetric spacetimes (in D=1+3) with a horizon at r=l, the partition function has the generic form $Z\propto \exp[S-\beta E]$, where $S= (1/4) 4\pi l^2$ and $|E|=(l/2)$. This analysis reproduces the conventional result for the blackhole spacetimes and provides a simple and consistent interpretation of entropy and energy for deSitter spacetime. (iv) For the Rindler spacetime the entropy per unit transverse area turns out to be (1/4) while the energy is zero. (v) In the case of a Schwarzschild black hole there exist quantum states (like Unruh vacuum) which are not invariant under time reversal and can describe blackhole evaporation. There also exist quantum states (like Hartle-Hawking vacuum) in which temperature is well-defined but there is no flow of radiation to infinity. In the case of deSitter universe or Rindler patch in flat spacetime, one usually uses quantum states analogous to Hartle-Hawking vacuum and obtains a temperature without the corresponding notion of evaporation. It is, however, possible to construct the analogues of Unruh vacuum state in the other cases as well. Associating an entropy or a radiating vacuum state with a general horizon raises conceptual issues which are briefly discussed. 
  I illustrate a simple hamiltonian formulation of general relativity, derived from the work of Esposito, Gionti and Stornaiolo, which is manifestly 4d generally covariant and is defined over a finite dimensional space. The spacetime coordinates drop out of the formalism, reflecting the fact that they are not related to observability. The formulation can be interpreted in terms of Toller's reference system transformations, and provides a physical interpretation for the spinnetwork to spinnetwork transition amplitudes computable in principle in loop quantum gravity and in the spin foam models. 
  Euclidean continuation of several Lorentzian spacetimes with horizons requires treating the Euclidean time coordinate to be periodic with some period $\beta$.   Such spacetimes (Schwarzschild, deSitter,Rindler .....) allow a temperature $T=\beta^{-1}$ to be associated with the horizon. I construct a canonical ensemble of a subclass of such spacetimes with a fixed value for $\beta$ and evaluate the partition function $Z(\beta)$. For spherically symmetric spacetimes with a horizon at r=a, the partition function has the generic form $Z\propto \exp[S-\beta E]$, where $S= (1/4) 4\pi a^2$ and $|E|=(a/2)$. Both S and E are determined entirely by the properties of the metric near the horizon. This analysis reproduces the conventional result for the blackhole spacetimes and provides a simple and consistent interpretation of entropy and energy for deSitter spacetime. For the Rindler spacetime the entropy per unit transverse area turns out to be (1/4) while the energy is zero. The implications are discussed. 
  In this paper it is shown that a Lanczos potential for the Weyl curvature tensor does not exist for all spaces of dimension $n\geq 7$. 
  The linear approximation of scalar-tensor theories of gravity is obtained in the physical (Jordan) frame under the 4+0 (covariant) and 3+1 formalisms. Then the weak-field limit is analyzed and the conditions leading to significant deviations of the $1/r^{2}$ Newton's law of gravitation are discussed. Finally, the scalarization effects induced by these theories in extended objects are confronted within the weak-field limit. 
  A recent paper by Castagnino, Giacomini and Lara concludes that there is no chaos in a conformally coupled closed Friedmann-Robertson-Walker universe, which is in apparent contradiction with previous works. We point out that although nonchaotic the quoted system is nonintegrable. 
  The causality principle imposes the constraints on the cosmological parameters in the relativistic theory of gravitation. As a result, X-matter causes the quite definite cosmological scenario with the alternate acceleration and deceleration and the final recollapse 
  The historical origins of Fermi-Walker transport and Fermi coordinates and the construction of Fermi-Walker transported frames in black hole spacetimes are reviewed. For geodesics this transport reduces to parallel transport and these frames can be explicitly constructed using a Killing-Yano tensor as shown by Marck. For accelerated or geodesic circular orbits in such spacetimes, both parallel and Fermi-Walker transported frames can be given, and allow one to study circular holonomy and related clock and spin transport effects. In particular the total angle of rotation that a spin vector undergoes around a closed loop can be expressed in a factored form, where each factor is due to a different relativistic effect, in contrast with the usual sum of terms decomposition. Finally the Thomas precession frequency is shown to be a special case of the simple relationship between the parallel transport and Fermi-Walker transport frequencies for stationary circular orbits. 
  The gravitational field of a particle of small mass \mu moving through curved spacetime is naturally decomposed into two parts each of which satisfies the perturbed Einstein equations through O(\mu). One part is an inhomogeneous field which, near the particle, looks like the \mu/r field distorted by the local Riemann tensor; it does not depend on the behavior of the source in either the infinite past or future. The other part is a homogeneous field and includes the ``tail term''; it completely determines the self force effects of the particle interacting with its own gravitational field, including radiation reaction. Self force effects for scalar, electromagnetic and gravitational fields are all described in this manner. 
  The existence of static and axially symmetric regions in a Friedman-Lemaitre cosmology is investigated under the only assumption that the cosmic time and the static time match properly on the boundary hypersurface. It turns out that the most general form for the static region is a two-sphere with arbitrarily changing radius which moves along the axis of symmetry in a determined way. The geometry of the interior region is completely determined in terms of background objects. When any of the most widely used energy-momentum contents for the interior region is imposed, both the interior geometry and the shape of the static region must become exactly spherically symmetric. This shows that the Einstein-Straus model, which is the generally accepted answer for the null influence of the cosmic expansion on the local physics, is not a robust model and it is rather an exceptional and isolated situation. Hence, its suitability for solving the interplay between cosmic expansion and local physics is doubtful and more adequate models should be investigated. 
  A theorem stated by Raychaudhuri which claims that the only physical non-singular cosmological models are comprised in the Ruiz-Senovilla family is shown to be incorrect. An explicit counterexample is provided and the failure of the argument leading to the theorem is explicitly pointed out. 
  Generalized Friedmann equations governing the cosmological evolution inside a thick brane embedded in a five-dimensional Anti-de Sitter spacetime are derived. These equations are written in terms of four-dimensional effective brane quantities obtained by integrating, along the fifth dimension, over the brane thickness. In the case of a Randall-Sundrum type cosmology, different limits of these effective quantities are considered yielding cosmological equations which interpolate between the thin brane limit (governed by unconventional brane cosmology), and the opposite limit of an ``infinite'' brane thickness corresponding to the familiar Kaluza-Klein approach. In the more restrictive case of a Minkowski bulk, it is shown that no effective four-dimensional reduction is possible in the regimes where the brane thickness is not small enough. 
  The dynamical parameters conventionally used to specify the orbit of a test particle in Kerr spacetime are the energy $E$, the axial component of the angular momentum, $L_{z}$, and Carter's constant $Q$. These parameters are obtained by solving the Hamilton-Jacobi equation for the dynamical problem of geodesic motion. Employing the action-angle variable formalism, on the other hand, yields a different set of constants of motion, namely, the fundamental frequencies $\omega_{r}$, $\omega_{\theta}$ and $\omega_{\phi}$ associated with the radial, polar and azimuthal components of orbital motion. These frequencies, naturally, determine the time scales of orbital motion and, furthermore, the instantaneous gravitational wave spectrum in the adiabatic approximation. In this article, it is shown that the fundamental frequencies are geometric invariants and explicit formulas in terms of quadratures are derived. The numerical evaluation of these formulas in the case of a rapidly rotating black hole illustrates the behaviour of the fundamental frequencies as orbital parameters such as the semi-latus rectum $p$, the eccentricity $e$ or the inclination parameter $\theta_{-}$ are varied. The limiting cases of circular, equatorial and Keplerian motion are investigated as well and it is shown that known results are recovered from the general formulas. 
  We present here a simple model of radiative gravitational collapse with radial heat flux which describes qualitatively the stages close to the formation of a superdense cold star. Starting with a static general solution for a cold star, the model can generate solutions for the earlier evolutionary stages. The temporal evolution of the model is specified by solving the junction conditions appropriate for radiating gravitational collapse. 
  The classical Rainich(-Misner-Wheeler) theory gives necessary and sufficient conditions on an energy-momentum tensor $T$ to be that of a Maxwell field (a 2-form) in four dimensions. Via Einstein's equations these conditions can be expressed in terms of the Ricci tensor, thus providing conditions on a spacetime geometry for it to be an Einstein-Maxwell spacetime. One of the conditions is that $T^2$ is proportional to the metric, and it has previously been shown in arbitrary dimension that any tensor satisfying this condition is a superenergy tensor of a simple $p$-form. Here we examine algebraic Rainich conditions for general $p$-forms in higher dimensions and their relations to identities by antisymmetrisation. Using antisymmetrisation techniques we find new identities for superenergy tensors of these general (non-simple) forms, and we also prove in some cases the converse; that the identities are sufficient to determine the form. As an example we obtain the complete generalisation of the classical Rainich theory to five dimensions. 
  Numerical solutions to the nonlinear sigma model (NLSM), a wave map from 3+1 Minkowski space to S^3, are computed in three spatial dimensions (3D) using adaptive mesh refinement (AMR). For initial data with compact support the model is known to have two regimes, one in which regular initial data forms a singularity and another in which the energy is dispersed to infinity. The transition between these regimes has been shown in spherical symmetry to demonstrate threshold behavior similar to that between black hole formation and dispersal in gravitating theories. Here, I generalize the result by removing the assumption of spherical symmetry. The evolutions suggest that the spherically symmetric critical solution remains an intermediate attractor separating the two end states. 
  Recent observations seem to indicate that we live in a universe whose spatial sections are nearly or exactly flat. Motivated by this we study the problem of observational detection of the topology of universes with flat spatial sections. We first give a complete description of the diffeomorphic classification of compact flat 3-manifolds, and derive the expressions for the injectivity radii, and for the volume of each class of Euclidean 3-manifolds. There emerges from our calculations the undetectability conditions for each (topological) class of flat universes. To illustrate the detectability of flat topologies we construct toy models by using an assumption by Bernshtein and Shvartsman which permits to establish a relation between topological typical lengths to the dynamics of flat models. 
  We present the formalism for the covariant treatment of gravitational radiation in a magnetized environment and discuss the implications of the field for gravity waves in the cosmological context. Our geometrical approach brings to the fore the tension properties of the magnetic force lines and reveals their intricate interconnection to the spatial geometry of a magnetised spacetime. We show how the generic anisotropy of the field can act as a source of gravitational wave perturbations and how, depending on the spatial curvature distortion, the magnetic tension can boost or suppress waves passing through a magnetized region. 
  We present a new solver for coupled nonlinear elliptic partial differential equations (PDEs). The solver is based on pseudo-spectral collocation with domain decomposition and can handle one- to three-dimensional problems. It has three distinct features. First, the combined problem of solving the PDE, satisfying the boundary conditions, and matching between different subdomains is cast into one set of equations readily accessible to standard linear and nonlinear solvers. Second, touching as well as overlapping subdomains are supported; both rectangular blocks with Chebyshev basis functions as well as spherical shells with an expansion in spherical harmonics are implemented. Third, the code is very flexible: The domain decomposition as well as the distribution of collocation points in each domain can be chosen at run time, and the solver is easily adaptable to new PDEs. The code has been used to solve the equations of the initial value problem of general relativity and should be useful in many other problems. We compare the new method to finite difference codes and find it superior in both runtime and accuracy, at least for the smooth problems considered here. 
  Identifying an appropriate set of ``observables'' is a nontrivial task for most approaches to quantum gravity. We describe how it may be accomplished in the context of a recently proposed family of stochastic (but classical) dynamical laws for causal sets. The underlying idea should work equally well in the quantum case. 
  In this talk I review a series of recent conceptual developments at the interface of the quantum and gravitational realms. Wherever possible, I comment on the possibility to probe the interface experimentally. It is concluded that the underlying spacetime for a quantum theory of gravity must be non-commutative, that wave-particle duality suffers significant modification at the Planck scale, and that the latter forbids probing spacetime below Planck length. Furthermore, study of quantum test particles in classical and quantum sources of gravity puts forward theoretical challenges and new experimental possibilities. It is suggested that existing technology may allow to probe gravitationally-modified wave particle duality in the laboratory. 
  Exact solution are obtained for a homogeneous spacially isotropic cosmological model in a matter free space with or without cosmological consant for a n-dimensional Kaluza-Klein type of metric in the rest mass varying theory of gravity proposed by Wesson[1983]. The behavior of the model is discussed. 
  A new procedure of the linear position measurement which allows to obtain sensitivity better than the Standard Quantum Limit and close to the Energetic Quantum Limit is proposed and analyzed in details. Proposed method is based on the principles of stroboscopic quantum measurement and variation quantum measurement and allows to avoid main disadvantages of both these procedures. This method can be considered as a good candidate for use as a local position meter in the ``intracavity'' topologies of the laser gravitational-wave antennae. 
  New numerical methods have been applied in relativity to obtain a numerical evolution of Einstein equations much more robust and stable. Starting from 3+1 formalism and with the evolution equations written as a FOFCH (first-order flux conservative hyperbolic) system, advanced numerical methods from CFD (Computational Fluid Dynamics) have been successfully applied. A flux limiter mechanism has been implemented in order to deal with steep gradients like the ones usually associated with black hole spacetimes. As a test bed, the method has been applied to 3D metrics describing propagation of nonlinear gauge waves. Results are compared with the ones obtained with standard methods, showing a great increase in both robustness and stability of the numerical algorithm. 
  We find the general solution for the spacetimes describing the interior of static black holes with an equation of state of the type $ T^0_0 = T^1_1 $ ($T$ being the stress-energy tensor). This form is the one expected from taking into account different quantum effects associated with strong gravitational fields. We recover all the particular examples found in the literature. We remark that all the solutions found follow the natural scheme of an interior core linked smoothly with the exterior solution by a transient region. We also discuss their local energy properties and give the main ideas involved in a possible generalization of the scheme, in order to include other realistic types of sources. 
  In this paper we look for static, purely magnetic, nonabelian solutions with unusual topology in the context of N=4 Freedman-Schwarz supergravity in four dimensions.Two new exact solutions satisfying first order Bogomol'nyi equations are discussed.The main characteristics of the general solutions are presented and differences with respect to the spherically symmetric case are studied. We argue that all solutions present naked singularities. 
  Penrose [1] has emphasized how the initial big bang singularity requires a special low entropy state. We address how recent brane cosmological schemes address this problem and whether they offer any apparent resolution. Pushing the start time back to $t=-\infty$ or utilizing maximally symmetric AdS spaces simply exacerbates or transfers the problem.  Because the entropy of de Sitter space is $S\leq 1/\Lambda$, using the present acceleration of the universe as a low energy $(\Lambda\sim 10^{-120}$) inflationary stage, as in cyclic ekpyrotic models, produces a gravitational heat death after one cycle. Only higher energy driven inflation, together with a suitable, quantum gravity holography style, restriction on {\em ab initio} degrees of freedom, gives a suitable low entropy initial state. We question the suggestion that a high energy inflationary stage could be naturally reentered by Poincare recurrence within a finite causal region of an accelerating universe.  We further give a heuristic argument that so-called eternal inflation is not consistent with the 2nd law of thermodynamics within a causal patch. 
  There is strong evidence indicating that the particular form used to recast the Einstein equation as a 3+1 set of evolution equations has a fundamental impact on the stability properties of numerical evolutions involving black holes and/or neutron stars. Presently, the longest lived evolutions have been obtained using a parametrized hyperbolic system developed by Kidder, Scheel and Teukolsky or a conformal-traceless system introduced by Baumgarte, Shapiro, Shibata and Nakamura. We present a new conformal-traceless system. While this new system has some elements in common with the Baumgarte-Shapiro-Shibata-Nakamura system, it differs in both the type of conformal transformations and how the non-linear terms involving the extrinsic curvature are handled. We show results from 3D numerical evolutions of a single, non-rotating black hole in which we demonstrate that this new system yields a significant improvement in the life-time of the simulations. 
  Near an event horizon, the action of general relativity acquires a new asymptotic conformal symmetry. Using two-dimensional dilaton gravity as a test case, I show that this symmetry results in a chiral Virasoro algebra with a calculable classical central charge, and that Cardy's formula for the density of states reproduces the Bekenstein-Hawking entropy. This result lends support to the notion that the universal nature of black hole entropy is controlled by conformal symmetry near the horizon. 
  An improved version of the ``optical bar'' intracavity readout scheme for gravitational-wave antennae is considered. We propose to call this scheme ``optical lever'' because it can provide significant gain in the signal displacement of the local mirror similar to the gain which can be obtained using ordinary mechanical lever with unequal arms. In this scheme displacement of the local mirror can be close to the signal displacement of the end mirrors of hypothetical gravitational-wave antenna with arm lengths equal to the half-wavelength of the gravitational wave. 
  We study the response of switched particle detectors to static negative energy densities and negative energy fluxes. It is demonstrated how the switching leads to excitation even in the vacuum and how negative energy can lead to a suppression of this excitation. We obtain quantum inequalities on the detection similar to those obtained for the energy density by Ford and co-workers and in an `operational' context by Helfer. We revisit the question `Is there a quantum equivalence principle?' in terms of our model. Finally, we briefly address the issue of negative energy and the second law of thermodynamics. 
  We prove the uniqueness theorem for asymptotically flat static vacuum black hole solutions in higher dimensional space-times. We also construct infinitely many non-asymptotically flat regular static black holes on the same spacetime manifold with the same spherical topology. 
  We embed the Schwarzschild interior solution in a five-dimensional flat space and show that the systems of the interior and the exterior solution are based on the same geometrical principles. It turns out that the energy tensor of the matter has its origin in the five-dimensional structure of the geometry and is built up by the generalized second fundamental forms. Thus, the matter is already geometrized. The Codazzi equations are the field equations for the matter field. 
  We present a method for the calculation of the gravitational back reaction cutoff on the smallest scales of cosmic string networks taking into account that not all modes on strings interact with all other modes. This results in a small scale structure cutoff that is sensitive to the initial spectrum of perturbations present on strings. From a simple model, we compute the cutoffs in radiation- and matter-dominated universes. 
  The recent observations of type Ia supernovae strongly support that the universe is accelerating now and decelerated in the recent past. By assuming a general relation between the quintessence potential and the quintessence kinetic energy, a general relation is found between the quintessence energy density and the scale factor. The potential includes both the hyperbolic and the double exponential potentials. A detailed analysis of the transition from the deceleration phase to the acceleration phase is then performed. We show that the current constraints on the transition time, the equation of state and the energy density of the quintessence field are satisfied in the model. 
  Field equations for n-frames h_a{}^\mu that are possible in the theory of absolute parallelism (AP) are considered. The methods of compatibility (or formal integrability) theory enable us to find the non-Lagrangian equation having unusual kind of compatibility conditions, guaranteed by two (not one) identities. This 'unique equation' was not noted explicitly in the classification by Einstein and Mayer of compatible second order equations of AP.   It is shown that some equations of AP (including 'unique equation') can be written in a trilinear form that contains only the matrix of frame density (of some weight) H_a{}^\mu and its derivatives and not inverse (coframe density) matrix. The equations are still regular and involutive for degenerate but finite matrices H_a{}^\mu if rank H_a{}^\mu > 1. 
  A local criteria for the existence of an accelerated frame of reference is found. An accelerated frame of reference could exist in all regions where a non-null (non-isotropic) vector field does not degenerate in a null (isotropic) vector field. 
  Recently found accelarated expansion of our Universe is due to the presence of a new kind of matter called "$\Lambda$ - field" or quintessence. The limitations on its equation of state are found from the fact of its unclustering at all scales much smaller than the cosmological horizon. It is discussed how these limitations affect on the possibility to approximate the accelarated expansion by such cosmological models as the model of Chaplygin gas. 
  We derive the Teixeira, Wolk and Som method, for obtaining electrostatic solutions from given vacuum solutions, in its inverse form. Then we use it to obtain the geometrical mass $M_S$ in the Schwarzschild spacetime, and we find $M_S^2=M^2-Q^2$, where $M$ and $Q$ are, respectively, the mass and charge parameters of the Reissner-Nordstr\"om spacetime. We compare $M_S$ to the corresponding active gravitational mass and mass function. 
  This article is a guide to theorems on existence and global dynamics of solutions of the Einstein equations. It draws attention to open questions in the field. The local in time Cauchy problem, which is relatively well understood, is surveyed. Global results for solutions with various types of symmetry are discussed. A selection of results from Newtonian theory and special relativity which offer useful comparisons is presented. Treatments of global results in the case of small data and results on constructing spacetimes with prescribed singularity structure are given. A conjectural picture of the asymptotic behaviour of general cosmological solutions of the Einstein equations is built up. Some miscellaneous topics connected with the main theme are collected in a separate section. 
  Employing the PPN formalism the gravitomagnetic field in different metric theories is considered in the analysis of the LAGEOS results. It will be shown that there are several models that predict exactly the same effect that general relativity comprises. In other words, these Earth satellites results can be taken as experimental evidence that the orbital angular momentum of a body does indeed generate space--time geometry, notwithstanding they do not endow general relativity with an outstanding status among metric theories. Additionally the coupling spin--gravitomagnetic field is analyzed with the introduction of the Rabi transitions that this field produces on a quantum system with spin 1/2. Afterwards, a continuous measurement of the energy of this system is introduced, and the consequences upon the corresponding probabilities of the involved gravitomagnetic field will be obtained. Finally, it will be proved that these proposals allows us, not only to confront against future experiments the usual assumption of the coupling spin--gravotimagnetism, but also to measure some PPN parameters and to obtain functional dependences among them. 
  For the past two decades, Einstein's Hole Argument (which deals with the apparent indeterminateness of general relativity due to the general covariance of the field equations) and its resolution in terms of Leibniz equivalence (the statement that Riemannian geometries related by active diffeomorphisms represent the same physical solution) have been the starting point for a lively philosophical debate on the objectivity of the point-events of space-time. It seems that Leibniz equivalence makes it impossible to consider the points of the space-time manifold as physically individuated without recourse to dynamical individuating fields. Various authors have posited that the metric field itself can be used in this way, but nobody so far has considered the problem of explicitly distilling the metrical fingerprint of point-events from the gauge-dependent components of the metric field. Working in the Hamiltonian formulation of general relativity, and building on the results of Lusanna and Pauri (2002), we show how Bergmann and Komar's intrinsic pseudo-coordinates (based on the value of curvature invariants) can be used to provide a physical individuation of point-events in terms of the true degrees of freedom (the Dirac observables) of the gravitational field, and we suggest how this conceptual individuation could in principle be implemented with a well-defined empirical procedure. We argue from these results that point-events retain a significant kind of physical objectivity. 
  Progress in the new information-theoretic process physics is reported in which the link to the phenomenology of general relativity is made. In process physics the fundamental assumption is that reality is to be modelled as self-organising semantic (or internal or relational) information using a self-referentially limited neural network model. Previous progress in process physics included the demonstration that space and quantum physics are emergent and unified, with time a distinct non-geometric process, that quantum phenomena are caused by fractal topological defects embedded in and forming a growing three-dimensional fractal process-space, which is essentially a quantum foam. Other features of the emergent physics were: quantum field theory with emergent flavour and confined colour, limited causality and the Born quantum measurement metarule, inertia, time-dilation effects, gravity and the equivalence principle, a growing universe with a cosmological constant, black holes and event horizons, and the emergence of classicality. Here general relativity and the technical language of general covariance is seen not to be fundamental but a phenomenological construct, arising as an amalgam of two distinct phenomena: the `gravitational' characteristics of the emergent quantum foam for which `matter' acts as a sink, and the classical `spacetime' measurement protocol, but with the later violated by quantum measurement processes. Quantum gravity, as manifested in the emergent Quantum Homotopic Field Theory of the process-space or quantum foam, is logically prior to the emergence of the general relativity phenomenology, and cannot be derived from it. 
  Basic notions and mathematical tools in continuum media mechanics are recalled. The notion of exponent of a covariant differential operator is introduced and on its basis the geometrical interpretation of the curvature and the torsion in $(\bar{L}_n,g)$-spaces is considered. The Hodge (star) operator is generalized for $(\bar{L}_n,g)$-spaces. The kinematic characteristics of a flow are outline in brief. PACS numbers: 11.10.-z; 11.10.Ef; 7.10.+g; 47.75.+f; 47.90.+a; 83.10.Bb 
  Basic notions of continuous media mechanics are introduced for spaces with affine connections and metrics. The physical interpretation of the notion of relative velocity is discussed. The notions of deformation velocity tensor, shear velocity, rotation (vortex) velocity, and expansion velocity are introduced. Different types of flows are considered.   PACS numbers: 11.10.-z; 11.10.Ef; 7.10.+g; 47.75.+f; 47.90.+a; 83.10.Bb 
  A set of observables is described for the topological quantum field theory which describes quantum gravity in three space-time dimensions with positive signature and positive cosmological constant. The simplest examples measure the distances between points, giving spectra and probabilities which have a geometrical interpretation. The observables are related to the evaluation of relativistic spin networks by a Fourier transform. 
  It was believed that when gravitational, electromagnetic and scalar waves interact, a spacelike curvature singularity or Cauchy horizon develops because of mutual focusing. We show with an exact solution that the collision of Einstein-Maxwell-Scalar fields, in contrast to previous studies, predicts singularities on null surfaces and that this is a transition phase between spacelike singularities and regular horizons. Divergences of tidal forces in the null singularities is shown to be weaker relative to the spacelike ones. Using the local isometry between colliding plane waves and black holes, we show that the inner horizon of Reissner-Nordstrom black hole transforms into a null singularity when a particular scalar field is coupled to it. We also present an analytic exact solution, which represents a Reissner-Nordstrom black hole with scalar hair in between the ergosphere. 
  The prior knowledge of the gravitational waveform from compact binary systems makes matched filtering an attractive detection strategy. This detection method involves the filtering of the detector output with a set of theoretical waveforms or templates. One of the most important factors in this strategy is knowing how many templates are needed in order to reduce the loss of possible signals. In this study we calculate the number of templates and computational power needed for a one-step search for gravitational waves from inspiralling binary systems. We build on previous works by firstly expanding the post-Newtonian waveforms to 2.5-PN order and secondly, for the first time, calculating the number of templates needed when using P-approximant waveforms. The analysis is carried out for the four main first-generation interferometers, LIGO, GEO600, VIRGO and TAMA. As well as template number, we also calculate the computational cost of generating banks of templates for filtering GW data. We carry out the calculations for two initial conditions. In the first case we assume a minimum individual mass of $1 M_{\odot}$ and in the second, we assume a minimum individual mass of $5 M_{\odot}$. We find that, in general, we need more P-approximant templates to carry out a search than if we use standard PN templates. This increase varies according to the order of PN-approximation, but can be as high as a factor of 3 and is explained by the smaller span of the P-approximant templates as we go to higher masses. The promising outcome is that for 2-PN templates the increase is small and is outweighed by the known robustness of the 2-PN P-approximant templates. 
  I describe the construction of a large class of asymptotically flat initial data with non-vanishing mass and angular momentum for which the metric and the extrinsic curvature have asymptotic expansions at space-like infinity in terms of powers of a radial coordinate. I emphasize the motivations and the main ideas behind the proofs. 
  Fayos and Sopuerta have recently set up a formalism for studying vacuum spacetimes with an isometry, a formalism that is centred around the bivector corresponding to the Killing vector and that adapts the tetrad to the bivector. Steele has generalized their approach to include the homothetic case. Here, we generalize this formalism to arbitrary spacetimes and to homothetic and conformal Killing vectors but do not insist on aligning the tetrad with the bivector. The most efficient way to use the formalism to find conformal Killing vectors (proper or not) of a given spacetime is to combine it with the notion of a preferred tetrad. A metric by Kimura is used as an illustrative example. 
  We investigate the back reaction of cosmological perturbations on the evolution of the Universe using the renormalization group method. Starting from the second order perturbed Einstein's equation, we renormalize a scale factor of the Universe and derive the evolution equation for the effective scale factor which includes back reaction due to inhomogeneities of the Universe. The resulting equation has the same form as the standard Friedman-Robertson-Walker equation with the effective energy density and pressure which represent the back reaction effect. 
  Since 1978 superconducting coupled cavities have been proposed as sensitive detector of gravitational waves. The interaction of the gravitational wave with the cavity walls, and the resulting motion, induces the transition of some electromagnetic energy from an initially excited cavity mode to an empty one. The energy transfer is maximum when the frequency of the wave is equal to the frequency difference of the two cavity modes. In this paper the basic principles of the detector are discussed. The interaction of a gravitational wave with the cavity walls is studied in the proper reference frame of the detector, and the coupling between two electromagnetic normal modes induced by the wall motion is analyzed in detail. Noise sources are also considered; in particular the noise coming from the brownian motion of the cavity walls is analyzed. Some ideas for the developement of a realistic detector of gravitational waves are discussed; the outline of a possible detector design and its expected sensitivity are also shown. 
  We present an exact solution of Einstein equations that describes a Bianchi type III spacetime with conformal expansion. The matter content is given by an anisotropic scalar field and two perfect fluids representing dust and isotropic radiation. Based on this solution, we construct a cosmological model that respects the evolution of the scale factor predicted in standard cosmology. 
  The effects of spacetime quantization on black hole and big bang/big crunch singularities can be studied using new tools from (2+1)-dimensional quantum gravity. I investigate effects of spacetime quantization on singularities of the (2+1)-dimensional BTZ black hole and the (2+1)-dimensional torus universe. Hosoya has considered the BTZ black hole, and using a ``quantum generalized affine parameter'' (QGAP), has shown that, for some specific paths, quantum effects ``smear'' the singularity. Using generic gaussian wave functions, I show that both BTZ black hole and the torus universe contain families of paths that still reach the singularities with a finite QGAP, suggesting that singularities persist in quantum gravity. More realistic calculations, using modular invariant wave functions of Carlip and Nelson for the torus universe, further support this conclusion. 
  The Ashtekar-Barbero formulation of general relativity admits a one-parameter family of canonical transformations that preserves the expressions of the Gauss and diffeomorphism constraints. The loop quantization of the connection formalism based on each of these canonical sets leads to different predictions. This phenomenon is called the Immirzi ambiguity. It has been recently argued that this ambiguity could be generalized to the extent of a spatially dependent function, instead of a parameter. This would ruin the predictability of loop quantum gravity. We prove that such expectations are not realized, so that the Immirzi ambiguity introduces exclusively a freedom in the choice of a real number. 
  The deformation equation and its integrability condition (Bianchi identity) of a non-associative deformation in operad algebra are found. Their relation to the theory of gravity is discussed. 
  I explain the geometric basis for the recently-discovered nonholonomic mapping principle which permits deriving laws of nature in spacetimes with curvature and torsion from those in flat spacetime, thus replacing and extending Einstein's equivalence principle. As an important consequence, it yields a new action principle for determining the equation of motion of a free spinless point particle in such spacetimes. Surprisingly, this equation contains a torsion force, although the action involves only the metric. This force makes trajectories autoparallel rather than geodesic. Its geometric origin is the closure failure of parallelograms in the presence of torsion, A simple generalization of the mapping principle transforms path integrals from flat spacetimes to those with curvature and torsion, thus playing the role of a quantum equivalence principle, applicable at present only to spaces with gradient torsion. 
  Binary black holes are the most promising candidate sources for the first generation of earth-based interferometric gravitational-wave detectors. We summarize and discuss the state-of-the-art analytic techniques developed during the last years to better describe the late dynamical evolution of binary black holes of comparable masses. 
  We revisit the problem of PBH mass evolution in the radiation-dominated era. We solve the complete differential equation in the semiclassical regime with absorption and evaporation terms and show that PBHs can gain very little mass, if at all, in this era. Relativistic proper motion of PBHs respect to the CMBR, as a possible loophole in the growth argument, is shown to be unlikely. Finally we demonstrate that PBHs can not remain in thermodynamical equilibrium with the ambient radiation, and therefore initially non-evaporating black holes must enter the evaporating regime, supporting several efforts to look for observational signatures. 
  Within the framework of a model Universe with variable space dimension, we study chaotic inflation with the potential $m^2\phi^2/2$, and calculate the dynamical solutions of the inflaton field, variable space dimension, scale factor, and their interdependence during the inflationary epoch. We show that the characteristic of the variability of the space dimension causes the inflationary epoch in the variable space dimension to last longer than the inflationary epoch in the constant space dimension. 
  The quantization of the extended canonical momentum in quantum materials including the effects of gravitational drag is applied successively to the case of a multiply connected rotating superconductor and superfluid. Experiments carried out on rotating superconductors, based on the quantization of the magnetic flux in rotating superconductors, lead to a disagreement with the theoretical predictions derived from the quantization of a canonical momentum without any gravitomagnetic term. To what extent can these discrepancies be attributed to the additional gravitomagnetic term of the extended canonical momentum? This is an open and important question. For the case of multiply connected rotating neutral superfluids, gravitational drag effects derived from rotating superconductor data appear to be hidden in the noise of present experiments according to a first rough analysis. 
  Vacuum polarisation in QED in a background gravitational field induces interactions which effectively violate the strong equivalence principle and affect the propagation of light. In the low frequency limit, Drummond and Hathrell have shown that this mechanism leads to superluminal photon velocities. To confront this phenomenon with causality, however, it is necessary to extend the calculation of the phase velocity $\vp(\w)$ to high frequencies, since it is $\vp(\infty)$ which determines the characteristics of the effective wave equation and thus the causal structure. In this paper, we use a recently constructed expression, valid to all orders in a derivative expansion, for the effective action of QED in curved spacetime to determine the frequency dependence of the phase velocity and investigate whether superluminal velocities indeed persist in the high frequency limit. 
  The gravitomagnetic corrections to the Keplerian period for a circular, geodesic orbit of a test particle in a polar plane containing the proper angular momentum J of a central rotating body are considered. 
  We formulate the problem of finding the low-energy limit of spin foam models as a coarse-graining problem in the sense of statistical physics. This suggests that renormalization group methods may be used to find that limit. However, since spin foams are models of spacetime at Planck scale, novel issues arise: these microscopic models are sums over irregular, background-independent lattices. We show that all of these issues can be addressed by the recent application of the Kreimer Hopf algebra for quantum field theory renormalization to non-perturbative statistical physics. The main difference from standard renormalization group is that the Hopf algebra executes block transformations in parts of the lattice only but in a controlled manner so that the end result is a fully block-transformed lattice. 
  We performed 3D numerical simulations of the merger of equal-mass binary neutron stars in full general relativity using a new large scale supercomputer. We take the typical grid size as (505,505,253) for (x,y,z) and the maximum grid size as (633,633,317). These grid numbers enable us to put the outer boundaries of the computational domain near the local wave zone and hence to calculate gravitational waveforms of good accuracy (within $\sim 10%$ error) for the first time. To model neutron stars, we adopt a $\Gamma$-law equation of state in the form $P=(\Gamma-1)\rho\epsilon$, where P, $\rho$, $\varep$ and $\Gamma$ are the pressure, rest mass density, specific internal energy, and adiabatic constant. It is found that gravitational waves in the merger stage have characteristic features that reflect the formed objects. In the case that a massive, transient neutron star is formed, its quasi-periodic oscillations are excited for a long duration, and this property is reflected clearly by the quasi-periodic nature of waveforms and the energy luminosity. In the case of black hole formation, the waveform and energy luminosity are likely damped after a short merger stage. However, a quasi-periodic oscillation can still be seen for a certain duration, because an oscillating transient massive object is formed during the merger. This duration depends strongly on the initial compactness of neutron stars and is reflected in the Fourier spectrum of gravitational waves. To confirm our results and to calibrate the accuracy of gravitational waveforms, we carried out a wide variety of test simulations, changing the resolution and size of the computational domain. 
  The Mathisson-Papapetrou-Dixon (MPD) equations for the motion of electrically neutral massive spinning particles are analysed, in the pole-dipole approximation, in an Einstein-Maxwell plane-wave background spacetime. By exploiting the high symmetry of such spacetimes these equations are reduced to a system of tractable ordinary differential equations. Classes of exact solutions are given, corresponding to particular initial conditions for the directions of the particle spin relative to the direction of the propagating background fields. For Einstein-Maxwell pulses a scattering cross section is defined that reduces in certain limits to those associated with the scattering of scalar and Dirac particles based on classical and quantum field theoretic techniques. The relative simplicity of the MPD approach and its use of macroscopic spin distributions suggests that it may have advantages in those astrophysical situations that involve strong classical gravitational and electromagnetic environments. 
  We develop a general formalism to treat, in general relativity, the nonradial oscillations of a superfluid neutron star about static (non-rotating) configurations. The matter content of these stars can, as a first approximation, be described by a two-fluid model: one fluid is the neutron superfluid, which is believed to exist in the core and inner crust of mature neutron stars; the other fluid is a conglomerate of all charged constituents. We use a system of equations that governs the perturbations both of the metric and of the matter variables, whatever the equation of state for the two fluids. The entrainment effect is explicitly included. We also allow for an outer envelope composed of ordinary fluid. We derive and implement the junction conditions for the metric and matter variables at the core/envelope interface. We investigate how the quasinormal modes of a superfluid star are affected by changes in the entrainment parameter, and unveil a series of avoided crossings between the various modes. We provide a proof that all modes of a two-fluid star must radiate gravitationally. We also discuss the future detectability of pulsations in a superfluid star and argue that it may be possible to use gravitational-wave data to constrain the parameters of superfluid neutron stars. 
  The paper considers the spectrum of axial perturbations of slowly uniformly rotating general relativistic stars in the framework of Y. Kojima. In a first step towards a full analysis only the evolution equations are treated but not the constraint. Then it is found that the system is unstable due to a continuum of non real eigenvalues. In addition the resolvent of the associated generator of time evolution is found to have a special structure which was discussed in a previous paper. From this structure it follows the occurrence of a continuous part in the spectrum of oscillations at least if the system is restricted to a finite space as is done in most numerical investigations. Finally, it can be seen that higher order corrections in the rotation frequency can qualitatively influence the spectrum of the oscillations. As a consequence different descriptions of the star which are equivalent to first order could lead to different results with respect to the stability of the star. 
  New nondiagonal $G_{2}$ inhomogeneous cosmological solutions are presented in a wide range of scalar-tensor theories with a stiff perfect fluid as a matter source. The solutions have no big-bang singularity or any other curvature singularities. The dilaton field and the fluid energy density are everywhere regular, too. The geodesic completeness of the solutions is investigated. 
  We use the Fuchsian algorithm to study the behavior near the singularity of certain families of U(1) Symmetric solutions of the vacuum Einstein equations (with the U(1) isometry group acting spatially). We consider an analytic family of polarized solutions with the maximum number of arbitrary functions consistent with the polarization condition (one of the ``gravitational degrees of freedom'' is turned off) and show that all members of this family are asymptotically velocity term dominated (AVTD) as one approaches the singularity. We show that the same AVTD behavior holds for a family of ``half polarized'' solutions, which is defined by adding one extra arbitrary function to those characterizing the polarized solutions. (The full set of non-polarized solutions involves two extra arbitrary functions). We begin to address the issue of whether AVTD behavior is independent of the choice of time foliation by showing that indeed AVTD behavior is seen for a wide class of choices of harmonic time in the polarized and half-polarized (U(1) Symmetric vacuum) solutions discussed here. \ 
  We determine the number of functionally independent components of tensors involving higher-order derivatives of a Riemannian metric. 
  The first step in the building of a spacetime solution of Einstein's gravitational field equations via the initial value formulation is finding a solution of the Einstein constraint equations. We recall the conformal method for constructing solutions of the constraints and we recall what it tells us about the parameterization of the space of such solutions. One would like to know how to construct solutions which model particular physical phenomena. One useful step towards this goal is learning how to glue together known solutions of the constraint equations. We discuss recent results concerning such gluing. 
  The universe, as a closed system, is for all time in a state with a determinate value of energy, i.e., in an eigenstate of the Hamiltonian. That is the principle of cosmic energy determinacy. The Hamiltonian depends on cosmic time through metric. Therefore there are confluence and branch points of energy levels. At branch points, quantum jumps must happen to prevent the violation of energy determinacy. Thus quantum jumps are a reaction against the propensity of the universe dynamics to that violation. On the basis of this idea, an internally consistent quantum jump dynamics is developed. 
  Using Einstein-Maxwell theory I investigate the gravitational field generated by an electric charge and a magnetic dipole, both held in fixed positions, but spinning with prescribed angular momenta. There is a conical singularity between them representing a strut balancing the gravitational attraction of their masses. However, there is in general another singularity, which I call a torsion singularity. I interpret this as a couple needed to maintain the spins at their prescribed values. It vanishes when the parameters obey a certain formula.   A conclusion of the work is that the charge and the magnet must spin relative to one another unless constrained by a couple. 
  I derive an exact, static, axially symmetric solution of the Einstein-Maxwell equations representing two massless magnetic dipoles, and compare it with the corresponding solution of Einstein's equations for two massless spinning particles (see gr-qc/0201094). I then obtain an exact stationary solution of the Einstein-Maxwell equations representing two massless spinning magnets in balance. The conclusion is that the spin-spin force is analogous to the force between two magnetic dipoles, but of opposite sign, and that the latter agrees with the classical value in the first approximation. 
  We report a new result on the nice section construction used in the definition of rest frame systems in general relativity. This construction is needed in the study of non trivial gravitational radiating systems. We prove existence, regularity and non-self-crossing property of solutions of the nice section equation for general asymptotically flat space times. This proves a conjecture enunciated in a previous work. 
  In this paper we study the strong gravitational lensing scenario where the lens is a Reissner-Nordstrom black hole. We obtain the basic equations and show that, as in the case of Schwarzschild black hole, besides the primary and secondary images, two infinite sets of relativistic images are formed. We find analytical expressions for the positions and amplifications of the relativistic images. The formalism is applied to the case of a low-mass black hole placed at the galactic halo. 
  In this paper we calculate explicitly the secular classical precessions of the node \Omega and the perigee \omega of an Earth artificial satellite induced by the static, even zonal harmonics of the geopotential up to degree l=20. Subsequently, their systematic errors induced by the mismodelling in the even zonal geopotential coefficients J_l are compared to the general relativistic secular gravitomagnetic and gravitoelectric precessions of the node and the perigee of the existing laser-ranged geodetic satellites and of the proposed LARES. 
  A unified framework for black holes and traversible wormholes is described, where both are locally defined by outer trapping horizons, two-way traversible for wormholes and one-way traversible for black or white holes. In a two-dimensional dilaton gravity model, examples are given of: construction of wormholes from black holes; operation of wormholes for transport, including back-reaction; maintenance of an operating wormhole; and collapse of wormholes to black holes. In spherically symmetric Einstein gravity, several exotic matter models supporting wormhole solutions are proposed: ghost scalar fields, exotic fluids and pure ghost radiation. 
  We study oscillations of slowly rotating relativistic barotropic as well as non-barotropic polytropic stars in the Cowling approximation, including first order rotational corrections. By taking into account the coupling between the polar and axial equations, we find that, in contrast to previous results, the $m=2$ $r$ modes are essentially unaffected by the continuous spectrum and exist even for very relativistic stellar models. We perform our calculations both in the time and frequency domain. In order to numerically solve the infinite system of coupled equations, we truncate it at some value $l_{\rm max}$. Although the time dependent equations can be numerically evolved without any problems, the eigenvalue equations possess a singular structure, which is related to the existence of a continuous spectrum. This prevents the numerical computation of an eigenmode if its eigenfrequency falls inside the continuous spectrum. The properties of the latter depend strongly on the cut-off value $l_{\rm max}$ and it can consist of several either disconnected or overlapping patches, which are the broader the more relativistic the stellar model is. By discussing the dependence of the continuous spectrum as a function of both the cut-off value $l_{\rm max}$ and the compactness $M/R$, we demonstrate how it affects the inertial modes. Through the time evolutions we are able to show that some of the inertial modes can actually exist inside the continuous spectrum, but some cannot. For more compact and therefore more relativistic stellar models, the width of the continuous spectrum strongly increases and as a consequence, some of the inertial modes, which exist in less relativistic stars, disappear. 
  We construct non-trivial vacuum space-times with a global Scri. The construction proceeds by proving extension results across compact boundaries for initial data sets, adapting the gluing arguments of Corvino and Schoen. Another application of the extension results is existence of initial data which are exactly Schwarzschild both near infinity and near each of the connected component of the apparent horizon. 
  In the framework of functional integration the non-leading terms to leading eikonal behavior of the Planckian-energy scattering amplitude are calculated by the straight-line path approximation. We show that the allowance for the first-order correction terms leads to the appearance of retardation effect. The singular character of the correction terms at short distances is also noted, and they may be lead ultimately to the appearance of non-eikonal contributions to the scattering amplitudes. 
  We solve the five dimensional vacuum Einstein equations for several kinds of anisotropic geometries. We consider metrics in which the spatial slices are characterized as Bianchi types-II and V, and the scale factors are dependent both on time and a non-compact fifth coordinate. We examine the behavior of the solutions we find, noting for which parameters they exhibit contraction over time of the fifth scale factor, leading naturally to dimensional reduction. We explore these within the context of the induced matter model: a Kaluza-Klein approach that associates the extra geometric terms due to the fifth coordinate with contributions to the four dimensional stress-energy tensor. 
  Does the measurement of a quantum system necessarily break Lorentz invariance? We present a simple model of a detector that measures the spacetime localization of a relativistic particle in a Lorentz invariant manner. The detector does not select a preferred Lorentz frame as a Newton-Wigner measurement would do. The result indicates that there exists a Lorentz invariant notion of quantum measurement and sheds light on the issue of the localization of a relativistic particle. The framework considered is that of single-particle mechanics as opposed to field theory. The result may be taken as support for the interpretation postulate of the spacetime-states formulation of single-particle quantum theory. 
  The Goldberg-Sachs theorem has been very useful in constructing algebraically special exact solutions of Einstein vacuum equation. Most of the physical meaningful vacuum exact solutions are algebraically special. We show that the Goldberg-Sachs theorem is not true in linearized gravity. This is a remarkable result, which gives light on the understanding of the physical meaning of the linearized solutions. 
  The goal of this work is two-fold. In the first part of this paper we regard classical Plebanski's action as a BF action supplemented by constraints. We introduce a spin foam model for Riemannian general relativity by systematically implementing these constraints as restrictions on paths in the state-sum of the BF theory. The spin foam model obtained is precisely the Barrett-Crane model. This provides a clear-cut connection of the model with a simplicial action.   In the second part of the paper we study the quantization of the effective action corresponding to the degenerate sectors of Plebanski's theory and obtain a very simple spin foam model. This model turns out to be precisely the one introduced by De Pietri et al. as an alternative to the one proposed by Barrett and Crane. 
  Using the analogy between a shrinking fluid vortex (`draining bathtub'), modelled as a (2+1) dimensional fluid flow with a sink at the origin, and a rotating (2+1) dimensional black hole with an ergosphere, it is shown that a scalar sound wave is reflected from such a vortex with an {\it amplification} for a specific range of frequencies of the incident wave, depending on the angular velocity of rotation of the vortex. We discuss the possibility of observation of this phenomenon, especially for inviscid fluids like liquid HeII, where vortices with quantized angular momentum may occur. 
  We study the effects of the trans-Planckian dispersion relation on the spectrum of the primordial density perturbations during inflation. In contrast to the earlier analyses, we do not assume any specific form of the dispersion relation and allow the initial state of the field to be arbitrary. We obtain the spectrum of vacuum fluctuations of the quantum field by considering a scalar field satisfying the linear wave equation with higher spatial derivative terms propagating in the de Sitter space-time. We show that the power spectrum does not strongly depend on the dispersion relation and that the form of the dispersion relation does not play a significant role in obtaining the corrections to the scale invariant spectrum. We also show that the signatures of the deviations from the flat scale-invariant spectrum from the CMBR observations due to quantum gravitational effects cannot be differentiated from the standard inflationary scenario with an arbitrary initial state. 
  Perturbation theory of rotating black holes is usually described in terms of Weyl scalars $\psi_4$ and $\psi_0$, which each satisfy Teukolsky's complex master wave equation and respectively represent outgoing and ingoing radiation. On the other hand metric perturbations of a Kerr hole can be described in terms of (Hertz-like) potentials $\Psi$ in outgoing or ingoing {\it radiation gauges}. In this paper we relate these potentials to what one actually computes in perturbation theory, i.e $\psi_4$ and $\psi_0$. We explicitly construct these relations in the nonrotating limit, preparatory to devising a corresponding approach for building up the perturbed spacetime of a rotating black hole. We discuss the application of our procedure to second order perturbation theory and to the study of radiation reaction effects for a particle orbiting a massive black hole. 
  Lee. et.al. (1976) analysed the bimetric theory with the help of parameterized post Newtonian (PPN) formalism. They found that the post Newtonian limit of the theory is identical with that of general theory of relativity except for their PPN parameter $\alpha_{2}$, on the basis of cosmological considerations. In the present paper it is pointed out that feasibility of such considerations are doubtful in five dimensional bimetric theory of relativity. As the universe is unique and is governed by physical laws, many different cosmologies are possible. Examples are given for some possible cosmological models, which are different, that those of Lee. et.al. This work is an extension in five dimension of a similar one obtained earlier by Rosen (1977) for four dimensional space-time. 
  A new definition of a strong curvature singularity is proposed. This definition is motivated by the definitions given by Tipler and Krolak, but is significantly different and more general. All causal geodesics terminating at these new singularities, which we call generalized strong curvature singularities, are classified into three possible types; the classification is based on certain relations between the curvature strength of the singularities and the causal structure in their neighborhood. A cosmic censorship theorem is formulated and proved which shows that only one class of generalized strong curvature singularities, corresponding to a single type of geodesics according to our classification, can be naked. Implications of this result for the cosmic censorship hypothesis are indicated. 
  We point out the relation between the photon rocket spacetimes and the Robinson Trautman geometries. This allows a discussion of the issues related to the distinction between the gravitational and matter energy radiation that appear in these metrics in a more geometrical way, taking full advantage of their asymptotic properties at null infinity to separate the Weyl and Ricci radiations, and to clearly establish their gravitational energy content. We also give the exact solution for the generalized photon rockets. 
  We highlight the fact that the lack of scale invariance in the gravitational field equations of General Relativity results from the underlying assumption that the appropriate scale for the gravitational force should be linked to the atomic scale. We show that many of the problems associated with cosmology and quantum gravity follow directly from this assumption. An alternative scale invariant paradigm is proposed, in which the appropriate scale for General Relativity takes the Universe as its baseline, and the gravitational force does not have any fixed relationship to forces that apply on the atomic scale. It is shown that this gives rise to a quasi-static universe, and that the predicted behaviour of this model can resolve most of the problems associated with the standard Big Bang model. The replacement of Newton's gravitational constant in the quasi-static model by a scale-dependent re-normalisation factor is also able to account for a number of astronomical observations that would otherwise require ad-hoc explanations. Some of the implications of scale invariant gravity for Planck scale physics, quantum cosmology, and the nature of time are discussed. 
  Spacetimes with collisionless matter evolving from data on a compact Cauchy surface with hyperbolic symmetry are shown to be timelike and null geodesically complete in the expanding direction, provided the data satisfy a certain size restriction. 
  A one-parameter family of static and spherically symmetric solutions to Einstein equations with a traceless energy-momentum tensor is found. When the nonzero parameter $\beta$ lies in the open interval $(0,1)$ one obtains traversable Lorentzian wormholes. One also obtains naked singularities when either $\beta < 0$ or $\beta > 1$ and the Schwarzschild black hole for $\beta = 1$. 
  We consider massive spin 1 fields, in Riemann-Cartan space-times, described by Duffin-Kemmer-Petiau theory. We show that this approach induces a coupling between the spin 1 field and the space-time torsion which breaks the usual equivalence with the Proca theory, but that such equivalence is preserved in the context of the Teleparallel Equivalent of General Relativity. 
  A new version of the Teukolksy Master Equation, describing any massless field of different spin $s=1/2,1,3/2,2$ in the Kerr black hole, is presented here in the form of a wave equation containing additional curvature terms. These results suggest a relation between curvature perturbation theory in general relativity and the exact wave equations satisfied by the Weyl and the Maxwell tensors, known in the literature as the de Rham-Lichnerowicz Laplacian equations. We discuss these Laplacians both in the Newman-Penrose formalism and in the Geroch-Held-Penrose variant for an arbitrary vacuum spacetime. Perturbative expansion of these wave equations results in a recursive scheme valid for higher orders. This approach, apart from the obvious implications for the gravitational and electromagnetic wave propagation on a curved spacetime, explains and extends the results in the literature for perturbative analysis by clarifying their true origins in the exact theory. 
  Interior perfect fluid solutions for the Reissner-Nordstrom metric are studied on the basis of a new classification scheme. General formulas are found in many cases. Explicit new global solutions are given as illustrations. Known solutions are briefly reviewed. 
  We report on calculations of the total gravitational energy radiated in the head-on black hole collision, where we use the geometry of the Robinson-Trautman metrics. 
  The problem of constructing a quantum theory of gravity is considered from a novel viewpoint. It is argued that any consistent theory of gravity should incorporate a relational character between the matter constituents of the theory.  In particular, the traditional approach of quantizing a space-time metric is criticized and two possible avenues for constructing a satisfactory theory are put forward. 
  The Jacobi equation in pseudo-Riemannian geometry determines the linearized geodesic flow. The linearization ignores the relative velocity of the geodesics. The generalized Jacobi equation takes the relative velocity into account; that is, when the geodesics are neighboring but their relative velocity is arbitrary the corresponding geodesic deviation equation is the generalized Jacobi equation. The Hamiltonian structure of this nonlinear equation is analyzed in this paper. The tidal accelerations for test particles in the field of a plane gravitational wave and the exterior field of a rotating mass are investigated. In the latter case, the existence of an attractor of uniform relative radial motion with speed $2^{-1/2}c\approx 0.7 c$ is pointed out. The astrophysical implications of this result for the terminal speed of a relativistic jet is briefly explored. 
  We study the apparition of event horizons in accelerated expanding cosmologies. We give a graphical and analytical representation of the horizons using proper distances to coordinate the events. Our analysis is mainly kinematical. We show that, independently of the dynamical equations, all the event horizons tend in the future infinity to a given expression depending on the scale factor that we call asymptotic horizon. We also encounter a subclass of accelerating models without horizon. When the ingoing null geodesics do not change concavity in its cosmic evolution we recover the de Sitter and quintessence-Friedmann-Robertson-Walker models. 
  The notion of center of mass for an isolated system has been previously encoded in the definition of the so called nice sections. In this article we present a generalization of the proof of existence of solutions to the linearized equation for nice sections, and formalize a local existence proof of nice sections relaxing the radiation condition. We report on the differentiable and non-self-crossing properties of this family of solutions. We also give a proof of the global existence of nice sections. 
  Upcoming gravitational wave-experiments promise a window for discovering new physics in astronomy. Detection sensitivity of the broadband laser interferometric detectors LIGO/VIRGO may be enhanced by matched filtering with accurate wave-form templates. Where analytic methods break down, we have to resort to numerical relativity, often in Hamiltonian or various hyperbolic formulations. Well-posed numerical relativity requires consistency with the elliptic constraints of energy and momentum conservation. We explore this using a choice of gauge in the future and a dynamical gauge in the past. Applied to a polarized Gowdy wave, this enables solving {\em all} ten vacuum Einstein equations. Evolution of the Schwarzschild metric in 3+1 and, more generally, sufficient conditions for well-posed numerical relativity continue to be open challenges. 
  We find analytical solutions describing the collapse of an infinitely long cylindrical shell of counter-rotating dust. We show that--for the classes of solutions discussed herein--from regular initial data a curvature singularity inevitably develops, and no apparent horizons form, thus in accord with the spirit of the hoop conjecture. 
  We derive some more results on the nature of the singularities arising in the collapse of inhomogeneous dust spheres. (i) It is shown that there are future-pointing radial and non-radial time-like geodesics emerging from the singularity if and only if there are future-pointing radial null geodesics emerging from the singularity. (ii) Limits of various space-time invariants and other useful quantities (relating to Thorne's point-cigar-barrel-pancake classification and to isotropy/entropy measures) are studied in the approach to the singularity. (iii) The topology of the singularity is studied from the point of view of ideal boundary structure. In each case, the different nature of the visible and censored region of the singularity is emphasized. 
  Physics, as known from our local, around--earth experience, meets some of its applicability limits at the time just preceding the period of primeval nucleosynthesis. Attention is focussed here on the effects of the nucleon size. Radiation--belonging nucleons are found to produce an extremely high pressure at kT of the order of some tens or hundreds of MeV. Quark deconfinement at higher energies would not change the results. 
  A proposal of the cosmological origin of Higgs particles is given. We show, that the Higgs field could be created from the vacuum quantum conformal fluctuation of Anti-de Sitter space-time, the spontaneous breaking of vacuum symmetry, and the mass of Higgs particle are related to the cosmological constant of our universe,especially the theoretical estimated mass m$_{H}$ of Higgs particles is m$_{H}=\sqrt{-2\mu ^{2}}$ =$\sqrt{|\Lambda /\pi}$. 
  We study the asymptotic behaviour of scaling solutions with a dissipative fluid and we show that, contrary to recent claims, the existence of stable accelerating attractor solution which solves the `energy' coincidence problem depends crucially on the chosen equations of state for the thermodynamical variables. We discuss two types of equations of state, one which contradicts this claim, and one which supports it. 
  After the gravity induced on the brane in the Randall-Sundrum (RS) infinite braneworld is briefly reviewed, we discuss the possibility that black holes evaporate as a result of classical evolution in this model based on the AdS/CFT correspondence. If this possibility is really the case, the existence of long-lived solar mass black holes will give the strongest constraint on the bulk curvature radius. At the same time, we can propose a new method to simulate the evaporation of a 4D black hole due to the Hawking radiation as a 5D process. 
  In this work we study an anisotropic model of general relativity based on the framework of Finsler geometry. The observed anisotropy of the microwave background radiation is incorporated in the Finslerian structure of space-time. We also examine the electromagnetic (e.m.) field equations in our space. As a result a modified wave equation of e.m. waves yields. 
  We calculate the energy-distribution for an axially symmetric scalar field in the M{\o}ller prescription. The total energy is given by the parameter m of the space-time. 
  We compare the results of constructing binary black hole initial data with three different decompositions of the constraint equations of general relativity. For each decomposition we compute the initial data using a superposition of two Kerr-Schild black holes to fix the freely specifiable data. We find that these initial-data sets differ significantly, with the ADM energy varying by as much as 5% of the total mass. We find that all initial-data sets currently used for evolutions might contain unphysical gravitational radiation of the order of several percent of the total mass. This is comparable to the amount of gravitational-wave energy observed during the evolved collision. More astrophysically realistic initial data will require more careful choices of the freely specifiable data and boundary conditions for both the metric and extrinsic curvature. However, we find that the choice of extrinsic curvature affects the resulting data sets more strongly than the choice of conformal metric. 
  We study eccentric equatorial orbits of a test-body around a Kerr black hole under the influence of gravitational radiation reaction. We have adopted a well established two-step approach: assuming that the particle is moving along a geodesic (justifiable as long as the orbital evolution is adiabatic) we calculate numerically the fluxes of energy and angular momentum radiated to infinity and to the black hole horizon, via the Teukolsky-Sasaki-Nakamura formalism. We can then infer the rate of change of orbital energy and angular momentum and thus the evolution of the orbit. The orbits are fully described by a semi-latus rectum $p$ and an eccentricity $e$. We find that while, during the inspiral, $e$ decreases until shortly before the orbit reaches the separatrix of stable bound orbits (which is defined by $p_{s}(e)$), in many astrophysically relevant cases the eccentricity will still be significant in the last stages of the inspiral. In addition, when a critical value $p_{crit}(e)$ is reached, the eccentricity begins to increase as a result of continued radiation induced inspiral. The two values $p_{s}$, $p_{crit}$ (for given $e$) move closer to each other, in coordinate terms, as the black hole spin is increased, as they do also for fixed spin and increasing eccentricity. Of particular interest are moderate and high eccentricity orbits around rapidly spinning black holes, with $p(e) \approx p_{s}(e)$. We call these ``zoom-whirl'' orbits, because of their characteristic behaviour involving several revolutions around the central body near periastron. Gravitational waveforms produced by such orbits are calculated and shown to have a very particular signature. Such signals may well prove of considerable astrophysical importance for the future LISA detector. 
  The complete solution of the vacuum Kerr-Schild equations in general relativity is presented, including the space-times with a curved background metric. The corresponding result for a flat background has been obtained by Kerr. 
  The Kerr-Schild pencil of metrics $g_{ab}+\La l_al_b$ is investigated in the generic case when it maps an arbitrary vacuum space-time with metric $g_{ab}$ to a vacuum space-time. The theorem is proved that this generic case, with the field $l$ shearing, does not contain the shear-free subclass as a smooth limit. It is shown that one of the K\'ota-Perj\'es metrics is a solution in the shearing class. 
  Purely time dependent solutions in four-dimensional Einstein-Cartan-Kalb-Ramond (ECKR) theory of gravity are shown not to be possible leading to a trivial vanishing of all Kalb-Ramond fields. This result seems to contradict previously results obtained by SenGupta et al. (Class. and Quantum Gravity 19(2002)677) where de Sitter spacetime solution is found as an example of opticaly active spacetime. It seems that only inhomogeneous KR fields are possible in four-dimensional torsioned spacetimes. 
  The complete solution of Einstein's gravitational equations with a vacuum-vacuum Kerr-Schild pencil of metrics $g_{ab}+V l_al_b$ is obtained. Our result generalizes the solution of the Kerr-Schild problem with a flat metric $g_{ab}$ (represented by the Kerr theorem) to the case when $g_{ab}$ is the metric of a curved space-time. 
  The Kerr-Schild pencil of metrics $\tilde g_{ab}=g_{ab}+V l_al_b$, with $g_{ab}$ and $\tilde g_{ab}$ satisfying the vacuum Einstein equations, is investigated in the case when the null vector $l$ has vanishing twist. This class of Kerr-Schild metrics contains two solutions: the Kasner metric and a metric wich can be obtained from the Kasner metric by a complex coordinate transformation. Both are limiting cases of the K\'ota-Perj\'es metrics. The base space-time is a pp-wave. 
  The news function providing some relevant information about angular distribution of gravitational radiation in axisymmetric black hole collisions at the speed of light had been evaluated in the literature by perturbation methods, after inverting second-order hyperbolic operators with variable coefficients in two independent variables. More recent work has related the appropriate Green function to the Riemann function for such a class of hyperbolic operators in two variables. The present paper obtains an improvement in the evaluation of the coefficients occurring in the second-order equation obeyed by the Riemann function, which might prove useful for numerical purposes. Eventually, we find under which conditions the original Green-function calculation reduces to finding solutions of an inhomogeneous second-order ordinary differential equation with a non-regular singular point. 
  Using a recent technique, proposed by Eardley and Giddings, we extend their results to the high-energy collision of two beams of massless particles, i.e. of two finite-front shock waves. Closed (marginally) trapped surfaces can be determined analytically in several cases even for collisions at non-vanishing impact parameter in D\ge 4 space-time dimensions. We are able to confirm and extend earlier conjectures by Yurtsever, and to deal with arbitrary axisymmetric profiles, including an amusing case of ``fractal'' beams. We finally discuss some implications of our results in high-energy experiments and in cosmology. 
  Extending the study of spherically symmetric metrics satisfying the dominant energy condition and exhibiting singularities of power-law type initiated in SI93, we identify two classes of peculiar interest: focusing timelike singularity solutions with the stress-energy tensor of a radiative perfect fluid (equation of state: $p={1\over 3} \rho$) and a set of null singularity classes verifying identical properties. We consider two important applications of these results: to cosmology, as regards the possibility of solving the horizon problem with no need to resort to any inflationary scenario, and to the Strong Cosmic Censorship Hypothesis to which we propose a class of physically consistent counter-examples. 
  Connes' functional formula of the Riemannian distance is generalized to the Lorentzian case using the so-called Lorentzian distance, the d'Alembert operator and the causal functions of a globally hyperbolic spacetime. As a step of the presented machinery, a proof of the almost-everywhere smoothness of the Lorentzian distance considered as a function of one of the two arguments is given. Afterwards, using a $C^*$-algebra approach, the spacetime causal structure and the Lorentzian distance are generalized into noncommutative structures giving rise to a Lorentzian version of part of Connes' noncommutative geometry. The generalized noncommutative spacetime consists of a direct set of Hilbert spaces and a related class of $C^*$-algebras of operators. In each algebra a convex cone made of self-adjoint elements is selected which generalizes the class of causal functions. The generalized events, called {\em loci}, are realized as the elements of the inductive limit of the spaces of the algebraic states on the $C^*$-algebras. A partial-ordering relation between pairs of loci generalizes the causal order relation in spacetime. A generalized Lorentz distance of loci is defined by means of a class of densely-defined operators which play the r\^ole of a Lorentzian metric. Specializing back the formalism to the usual globally hyperbolic spacetime, it is found that compactly-supported probability measures give rise to a non-pointwise extension of the concept of events. 
  In the framework of generally covariant (pre-metric) electrodynamics (``charge & flux electrodynamics''), the Maxwell equations can be formulated in terms of the electromagnetic excitation $H=({\cal D}, {\cal H})$ and the field strength $F=(E,B)$. If the spacetime relation linking $H$ and $F$ is assumed to be {\em linear}, the electromagnetic properties of (vacuum) spacetime are encoded into 36 components of the vacuum constitutive tensor density $\chi$. We study the propagation of electromagnetic waves and find that the metric of spacetime emerges eventually from the principal part $^{(1)}\chi$ of $\chi$ (20 independent components). In this article, we concentrate on the remaining skewon part $^{(2)}\chi$ (15 components) and the axion part $^{(3)}\chi$ (1 component). The skewon part, as we'll show for the first time, can be represented by a 2nd rank traceless tensor $\not S_i{}^j$. By means of the Fresnel equation, we discuss how this tensor disturbs the light cones. Accordingly, this is a mechanism for violating Lorentz invariance and time symmetry. In contrast, the (abelian) axion part $^{(3)}\chi$ does {\em not} interfere with the light cones. 
  We consider the D dimensional Einstein Maxwell theory with a null fluid in the Kerr-Schild Geometry. We obtain a complete set of differential conditions that are necessary for finding solutions. We examine the case of vanishing pressure and cosmological constant in detail. For this specific case, we give the metric, the electromagnetic vector potential and the fluid energy density. This is, in fact, the generalization of the well known Bonnor-Vaidya solution to arbitrary D dimensions. We show that due to the acceleration of charged sources, there is an energy flux in $D \ge 4$ dimensions and we give the explicit form of this energy flux formula. 
  We describe an experiment to measure the mass of the Milky Way galaxy. The experiment is based on calculated light travel times along orthogonal directions in the Schwarzschild metric of the Galactic center. We show that the difference is proportional to the Galactic mass. We apply the result to light travel times in a 10cm Michelson type interferometer located on Earth. The mass of the Galactic center is shown to contribute 10^-6 to the flat space component of the metric. An experiment is proposed to measure the effect. 
  A new proposal for the implementation of LARES mission is presented. In particular, a new observable is proposed and alternative scenarios are discussed. 
  This paper gives a self-contained, elementary, and largely pictorial statement of Einstein's equation. 
  We prove that given a solution of the Einstein equations $g_{ab}$ for the matter field $T_{ab}$, an autoparallel null vector field $l^{a}$ and a solution $(l_{a}l_{c}, \mathcal{T}_{ac})$ of the linearized Einstein equation on the given background, the Kerr-Schild metric $g_{ac}+\lambda l_{a}l_{c}$ ($\lambda $ arbitrary constant) is an exact solution of the Einstein equation for the energy-momentum tensor $T_{ac}+\lambda \mathcal{T}_{ac}+\lambda ^{2}l_{(a}\mathcal{T}_{c)b}l^{b}$. The mixed form of the Einstein equation for Kerr-Schild metrics with autoparallel null congruence is also linear. Some more technical conditions hold when the null congruence is not autoparallel. These results generalize previous theorems for vacuum due to Xanthopoulos and for flat seed space-time due to G\"{u}rses and G\"{u}rsey. 
  We present a new formulation of the multipolar expansion of an exact boundary condition for the wave equation, which is truncated at the quadrupolar order. Using an auxiliary function, that is the solution of a wave equation on the sphere defining the outer boundary of the numerical grid, the absorbing boundary condition is simply written as a perturbation of the usual Sommerfeld radiation boundary condition. It is very easily implemented using spectral methods in spherical coordinates. Numerical tests of the method show that very good accuracy can be achieved and that this boundary condition has the same efficiency for dipolar and quadrupolar waves as the usual Sommerfeld boundary condition for monopolar ones. This is of particular importance for the simulation of gravitational waves, which have dominant quadrupolar terms, in General Relativity. 
  It is shown that the general scalar tensor cosmologies may explain all the current cosmological observations without the need of invoking any ad hoc missing energy density. The explanation is based entirely on the internal dynamics of the theories. Two important predictions of the present analysis are: the universe is tending towards a matter dominated state with the dimunition of the dark energy component and the acceleration of the universe, if any, is slowing down with time. 
  According to Schroedinger's ideas, classical dynamics of point particles should correspond to the " geometrical optics " limit of a linear wave equation, in the same way as ray optics is the limit of wave optics. It is shown that, using notions of modern wave theory, the " geometrical optics " analogy leads to the correspondence between a classical Hamiltonian H and a " quantum " wave equation in a natural and general way. In particular, the correspondence is unambiguous also in the case where H contains mixed terms involving momentum and position. In the line of Schroedinger's ideas, it is also attempted to justify the occurrence, in QM, of eigenvalues problems, not merely for energy, but also for momentum. It is shown that the wave functions of pure momentum states can be defined in a physically more satisfying way than by assuming plane waves. In the case of a spatially uniform force field, such momentum states have a singularity and move undeformed according to Newton's second law. The mentioned unambiguous correspondence allows to uniquely extend the Klein-Gordon relativistic wave equation to the case where a constant gravitational field is present. It is argued that Schroedinger's wave mechanics can be extended to the case with a variable gravitational field only if one accepts that the wave equation is a preferred-frame one. From this viewpoint, generally-covariant extensions of the wave equations of QM seem rather formal. Finally, it is conjectured that there is no need for a quantum gravity. 
  Within the framework of generally covariant (pre-metric) electrodynamics, we specify a local vacuum spacetime relation between the excitation $H=({\cal D},{\cal H})$ and the field strength $F=(E,B)$. We study the propagation of electromagnetic waves in such a spacetime by Hadamard's method and arrive, with the constitutive tensor density $\kappa\sim\partial H/\partial F$, at a Fresnel equation which is algebraic of 4th order in the wave covector. We determine how the different pieces of $\kappa$, in particular the axion and the skewon pieces, affect the propagation of light. 
  We present a new hydro code based on spectral methods using spherical coordinates. The first version of this code aims at studying time evolution of inertial modes in slowly rotating neutron stars. In this article, we introduce the anelastic approximation, developed in atmospheric physics, using the mass conservation equation to discard acoustic waves. We describe our algorithms and some tests of the linear version of the code, and also some preliminary linear results. We show, in the Newtonian framework with differentially rotating background, as in the relativistic case with the strong Cowling approximation, that the main part of the velocity quickly concentrates near the equator of the star. Thus, our time evolution approach gives results analogous to those obtained by Karino {\it et al.} \cite{karino01} within a calculation of eigenvectors. Furthermore, in agreement with the work of Lockitch {\it et al.} \cite{lockandf01}, we found that the velocity seems to always get a non-vanishing polar part. 
  Global properties of vacuum static, spherically symmetric configurations are studied in a general class of scalar-tensor theories (STT) of gravity in various dimensions. The conformal mapping between the Jordan and Einstein frames is used as a tool. Necessary and sufficient conditions are found for the existence of solutions admitting a conformal continuation (CC). The latter means that a singularity in the Einstein-frame manifold maps to a regular surface S_(trans) in the Jordan frame, and the solution is then continued beyond this surface. S_(trans) can be an ordinary regular sphere or a horizon. In the second case, S_(trans) proves to connect two epochs of a Kantowski-Sachs type cosmology. It is shown that, in an arbitrary STT, with arbitrary potential functions $U(\phi)$, the list of possible types of causal structures of vacuum space-times is the same as in general relativity with a cosmological constant. This is true even for conformally continued solutions. It is found that when S_(trans) is an ordinary sphere, one of the generic structures appearing as a result of CC is a traversable wormhole. Two explicit examples are presented: a known solution illustrating the emergence of singularities and wormholes, and a nonsingular 3-dimensional model with an infinite sequence of CCs. 
  Several numerical relativity groups are using a modified ADM formulation for their simulations, which was developed by Nakamura et al (and widely cited as Baumgarte-Shapiro-Shibata-Nakamura system). This so-called BSSN formulation is shown to be more stable than the standard ADM formulation in many cases, and there have been many attempts to explain why this re-formulation has such an advantage. We try to explain the background mechanism of the BSSN equations by using eigenvalue analysis of constraint propagation equations. This analysis has been applied and has succeeded in explaining other systems in our series of works. We derive the full set of the constraint propagation equations, and study it in the flat background space-time. We carefully examine how the replacements and adjustments in the equations change the propagation structure of the constraints, i.e. whether violation of constraints (if it exists) will decay or propagate away. We conclude that the better stability of the BSSN system is obtained by their adjustments in the equations, and that the combination of the adjustments is in a good balance, i.e. a lack of their adjustments might fail to obtain the present stability. We further propose other adjustments to the equations, which may offer more stable features than the current BSSN equations. 
  Basic notions of continuous media mechanics are introduced for spaces with affine connections and metrics. The physical interpretation of the notion of relative acceleration is discussed. The notions of deformation acceleration, shear acceleration, rotation (vortex) acceleration, and expansion acceleration are introduced. Their corresponding notions, generated by the torsion and curvature, are considered. A classification is proposed for auto-parallel vector fields with different kinematic characteristics. Relations between the kinematic characteristics of the relative acceleration and these of the relative velocity are found. A summary of the introduced and considered notions is given. A classification is proposed related to the kinematic characteristics of the relative velocity and the kinematic characteristics related to the relative acceleration. PACS numbers: 11.10.-z; 11.10.Ef; 7.10.+g; 47.75.+f; 47.90.+a; 83.10.Bb 
  Basic notions of continuous media mechanics are introduced for spaces with affine connections and metrics. Stress (tension) tensors are considered, obtained by the use of the method of Lagrangians with covariant derivatives (MLCD). On the basis of the covariant Noether's identities for the energy-momentum tensors, Navier-Stokes' identities are found and generalized Navier-Cauchy as well as Navier-Stokes' equations are investigated over spaces with affine connections and metrics. PACS numbers: 11.10.-z; 11.10.Ef; 7.10.+g; 47.75.+f; 47.90.+a; 83.10.Bb 
  Quantum thermal effect of Weyl neutrinos in a rectilinearly non-uniformly accelerating Kinnersley black hole is investigated by using the generalized tortoise coordinate transformation. The equation that determines the location, the Hawking temperature of the event horizon and the thermal radiation spectrum of neutrinos are derived. Our results show that the location and the temperature of the event horizon depend not only on the time but also on the angle. 
  The energy-momentum complexes of Einstein, Landau-Lifshitz, Papapetrou, and Weinberg give the same and meaningful results for the energy and momentum of the Bonnor spacetime describing the gravitational field of a stationary beam of light. The results support the Cooperstock hypothesis. 
  A causal set C can describe a discrete spacetime, but this discrete spacetime is not quantum, because C is endowed with Boolean logic, as it does not allow cycles. In a quasi-ordered set Q, cycles are allowed. In this paper, we consider a subset QC of a quasi-ordered set Q, whose elements are all the cycles. In QC, which is endowed with quantum logic, each cycle of maximal outdegree N in a node, is associated with N entangled qubits. Then QC describes a quantum computing spacetime. This structure, which is non-local and non-casual, can be understood as a proto-spacetime. Micro-causality and locality can be restored in the subset U of Q whose elements are unentangled qubits which we interpret as the states of quantum spacetime. The mapping of quantum spacetime into proto-spacetime is given by the action of the XOR gate. Moreover, a mapping is possible from the Boolean causal set into U by the action of the Hadamard gate. In particular, the causal order defined on the elements of U induces the causal evolution of spin networks. 
  Gravitational collapse is one of the most fruitful subjects in gravitational physics. It is well known that singularity formation is inevitable in complete gravitational collapse. It was conjectured that such a singularity should be hidden by horizons if it is formed from generic initial data with physically reasonable matter fields. Many possible counterexamples to this conjecture have been proposed over the past three decades, although none of them has proved to be sufficiently generic. In these examples, there appears a singularity that is not hidden by horizons. This singularity is called a `naked singularity.' The appearance of a naked singularity represents the formation of an observable high-curvature, strong-gravity region. In this paper we review examples of naked singularity formation and recent progress in research of observable physical processes - gravitational radiation and quantum particle creation - from a forming naked singularity. 
  Within the weak-field, post-Newtonian approximation of the metric theories of gravity, we determine the one-way time transfer up to the order 1/c^4, the unperturbed term being of order 1/c, and the frequency shift up to the order 1/c^4. We adapt the method of the world-function developed by Synge to the Nordtvedt-Will PPN formalism. We get an integral expression for the world-function up to the order 1/c^3 and we apply this result to the field of an isolated, axisymmetric rotating body. We give a new procedure enabling to calculate the influence of the mass and spin multipole moments of the body on the time transfer and the frequency shift up to the order 1/c^4. We obtain explicit formulas for the contributions of the mass, of the quadrupole moment and of the intrinsic angular momentum. In the case where the only PPN parameters different from zero are beta and gamma, we deduce from these results the complete expression of the frequency shift up to the order 1/c^4. We briefly discuss the influence of the quadrupole moment and of the rotation of the Earth on the frequency shifts in the ACES mission. 
  In the context of string theory, it is possible to explain the microscopic origin of the entropy of certain black holes in terms of D-brane systems. To date, most of the cases studied in detail refer to extremal (supersymmetric) or near-extremal black holes. In this work we propose a microscopic model for certain black branes (extended versions of black holes) which would apply to cases arbitrarily far from extremality, including the Schwarzschild case. The model is based on a system of D-branes and anti-D-branes, and is able to reproduce several properties of the corresponding supergravity solution. In particular, the microscopic entropy agrees with supergravity, except for a factor of 2^{p/p+1}, where p is the dimension of the brane. 
  We compare recent numerical results, obtained within a ``helical Killing vector'' (HKV) approach, on circular orbits of corotating binary black holes to the analytical predictions made by the effective one body (EOB) method (which has been recently extended to the case of spinning bodies). On the scale of the differences between the results obtained by different numerical methods, we find good agreement between numerical data and analytical predictions for several invariant functions describing the dynamical properties of circular orbits. This agreement is robust against the post-Newtonian accuracy used for the analytical estimates, as well as under choices of resummation method for the EOB ``effective potential'', and gets better as one uses a higher post-Newtonian accuracy. These findings open the way to a significant ``merging'' of analytical and numerical methods, i.e. to matching an EOB-based analytical description of the (early and late) inspiral, up to the beginning of the plunge, to a numerical description of the plunge and merger. We illustrate also the ``flexibility'' of the EOB approach, i.e. the possibility of determining some ``best fit'' values for the analytical parameters by comparison with numerical data. 
  Superconductors will be considered as macroscopic quantum gravitational antennas and transducers, which can directly convert upon reflection a beam of quadrupolar electromagnetic radiation into gravitational radiation, and vice versa, and thus serve as practical laboratory sources and receivers of microwave and other radio-frequency gravitational waves. An estimate of the transducer conversion efficiency on the order of unity comes out of the Ginzburg-Landau theory for an extreme type II, dissipationless superconductor with minimal coupling to weak gravitational and electromagnetic radiation fields, whose frequency is smaller than the BCS gap frequency, thus satisfying the quantum adiabatic theorem. The concept of ``the impedance of free space for gravitational plane waves'' is introduced, and leads to a natural impedance-matching process, in which the two kinds of radiation fields are impedance-matched to each other around a hundred coherence lengths beneath the surface of the superconductor. A simple, Hertz-like experiment has been performed to test these ideas, and preliminary results will be reported. (PACS nos.: 03.65.Ud, 04.30.Db, 04.30.Nk, 04.80.Nn, 74.60-w, 74.72.Bk) 
  On this paper we consider the classical wormhole solution of the Born-Infeld scalar field. The corresponding classical wormhole solution can be obtained analytically for both very small and large $\dot{\phi}$. At the extreme limits of small $\dot{\phi}$ the wormhole solution has the same format as one obtained by Giddings and Strominger[10]. At the extreme limits of large $\dot{\phi}$ the wormhole solution is a new one. The wormhole wavefunctions can also be obtained for both very small and large $\dot{\phi}$. These wormhole wavefunctions are regarded as solutions of quantum-mechanical Wheeler--Dewitt equation with certain boundary conditions. 
  $\omega(\phi) \to \infty$ limit of scalar tensor theories are studied for traceless matter source. It is shown that the limit $\omega(\phi) \to \infty$ does not reduce a scalar tensor theory to general relativity.   An exact radiation solution of scalar tensor cosmology under Nordtvedt conditions is obtained for flat Friedmann universe. 
  We study the conservation laws in the teleparallel theory with a positive cosmological constant, an extension of the teleparallel theory possessing solutions with de Sitter asymptotics. Demanding that the canonical generators of the asymptotic symmetry are well defined, we obtain their improved form, which defines the conserved charges of the theory. The physical interpretation of the results is discussed. 
  Both the static and homogeneous metrics describing the spherically symmetric gravitational field of a crossflow of incoming and outgoing null dust streams are generalized for the case of the two-component ghost radiation. Static solutions represent either naked singularities or the wormholes recently found by Hayward. The critical value of the parameter separating the two possibilities is given. The wormhole is allowed to have positive mass. The homogeneous solutions are open universes. 
  In this article we carefully distinguish the notion of bi-refringence (a polarization-dependent doubling in photon propagation speeds) from that of bi-metricity (where the two photon polarizations ``see'' two distinct metrics). We emphasise that these notions are logically distinct, though there are special symmetries in ordinary (3+1)-dimensional nonlinear electrodynamics which imply the stronger condition of bi-metricity.   To illustrate this phenomenon we investigate a generalized version of (3+1)-dimensional nonlinear electrodynamics, which permits the inclusion of arbitrary inhomogeneities and background fields. [For example dielectrics (a la Gordon), conductors (a la Casimir), and gravitational fields (a la Landau--Lifshitz).] It is easy to demonstrate that the generalized theory is bi-refringent: In (3+1) dimensions the Fresnel equation, the relationship between frequency and wavenumber, is always quartic. It is somewhat harder to show that in some cases (eg, ordinary nonlinear electrodynamics) the quartic factorizes into two quadratics thus providing a bi-metric theory. Sometimes the quartic is a perfect square, implying a single unique effective metric. We investigate the generality of this factorization process. 
  According to recent astrophysical observations the large scale mean pressure of our present universe is negative suggesting a positive cosmological constant like term. This article addresses the question of whether non-perturbative effects of self-interacting quantum fields in curved space-times may yield a significant contribution. Focusing on the trace anomaly of quantum chromo-dynamics (QCD), a preliminary estimate of the expected order of magnitude yields a remarkable coincidence with the empirical data, indicating the potential relevance of this effect. PACS: 04.62.+v, 12.38.Aw, 12.38.Lg, 98.80.Es. 
  A general formalism for understanding the thermodynamics of horizons in spherically symmetric spacetimes is developed. The formalism reproduces known results in the case of black hole spacetimes. But its power lies in being able to handle more general situations like: (i) spacetimes which are not asymptotically flat (like the de Sitter spacetime) and (ii) spacetimes with multiple horizons having different temperatures (like the Schwarzschild-de Sitter spacetime) and provide a consistent interpretation for temperature, entropy and energy. I show that it is possible to write Einstein's equations for a spherically symmetric spacetime in the form $TdS-dE=PdV$ near {\it any} horizon of radius $a$ with $S=(1/4)(4\pi a^2), |E| = (a/2)$ and the temperature $T$ determined from the surface gravity at the horizon. The pressure $P$ is provided by the source of the Einstein's equations and $dV$ is the change in the volume when the horizon is displaced infinitesimally. The same results can be obtained by evaluating the quantum mechanical partition function {\it without using Einstein's equations or WKB approximation for the action}. Both the classical and quantum analysis provide a simple and consistent interpretation of entropy and energy for de Sitter spacetime as well as for $(1+2)$ dimensional gravity. For the Rindler spacetime the entropy per unit transverse area turns out to be $(1/4)$ while the energy is zero. The approach also shows that the de Sitter horizon -- like the Schwarzschild horizon -- is effectively one dimensional as far as the flow of information is concerned, while the Schwarzschild-de Sitter, Reissner-Nordstrom horizons are not. The implications for spacetimes with multiple horizons are discussed. 
  The current observations seem to suggest that the universe has a positive cosmological constant of the order of $H_0^2$ while the most natural value for the cosmological constant will be $L_P^{-2}$ where $L_P = (G\hbar/c^3)^{1/2}$ is the Planck length. This reduction of the cosmological constant from $L_P^{-2} $ to $L_P^{-2}(L_PH_0)^2$ may be interpreted as due to the ability of quantum micro structure of spacetime to readjust itself and absorb bulk vacuum energy densities. Being a quantum mechanical process, such a cancellation cannot be exact and the residual quantum fluctuations appear as the ``small'' cosmological constant. I describe the features of a toy model for the spacetime micro structure which could allow for the bulk vacuum energy densities to be canceled leaving behind a small residual value of the the correct magnitude. Some other models (like the ones based on canonical ensemble for the four volume or quantum fluctuations of the horizon size) lead to an insignificantly small value of $H_0^2(L_PH_0)^n$ with $n=0.5-1$ showing that obtaining the correct order of magnitude for the residual fluctuations in the cosmological constant is a nontrivial task, becaue of the existence of the small dimensionless number $H_0L_P$ . 
  As recently found by Youm [hep-th/0201268], the entropy of the universe will no longer be expressible in the conventional Cardy-Verlinde form if one relaxes the radiation dominance state equation and instead assumes a more general equation of the form p=(\gamma-1)\rho, with \gamma a constant. We show that Youm's generalized entropy formula remains valid when the cosmic fluid is no longer ideal, but is allowed to possess a constant bulk viscosity \zeta. We supply our analysis with some numerical estimates, thus calculating the scale factor a(t) for a k=0 universe, and also calculate via a perturbative expansion in \zeta the magnitude of the viscosity-induced correction to the scale factor if the universe is radiation dominated. 
  This is a brief survey of the current status of Stephen Hawking's ``chronology protection conjecture''. That is: ``Why does nature abhor a time machine?'' I'll discuss a few examples of spacetimes containing ``time machines'' (closed causal curves), the sorts of peculiarities that arise, and the reactions of the physics community. While pointing out other possibilities, this article concentrates on the possibility of ``chronology protection''. As Stephen puts it:  ``It seems that there is a Chronology Protection Agency which prevents the appearance of closed timelike curves and so makes the universe safe for historians.'' 
  The determination of the geodesics in Scharzschild black hole and de Sitter space. 
  We develop the dynamics of the chiral superconducting membranes(with null current) in an alternative geometrical approach. Besides of this, we show the equivalence of the resulting description to the one known Dirac-Nambu-Goto (DNG) case. Integrability for chiral string model is obtained using a proposed light-cone gauge. In a similar way, domain walls are integrated by means of a simple ansatz. 
  This article outlines the search for an exact general relativistic description of the exterior (vacuum) gravitational field of a rotating spheroidal black hole surrounded by a realistic axially symmetric disc of matter. The problem of multi-body stationary spacetimes is first exposed from the perspective of the relativity theory and astrophysics, listing the basic methods employed and results obtained. Then, basic formulas for stationary axisymmetric solutions are summarized. Remaining sections review what we have learnt with Miroslav Zacek and Tomas Zellerin about certain static and stationary situations recently. Although the survey part is quite general, the list of references cannot be complete. Our main desideratum was the informative value rather than originality -- novelties have been preferred, mainly reviews and those with detailed introductions. 
  Kaluza's mertic with the cylinder condition is considered without the weak gravitational field approximation. It is shown that these hypoteses lead to a non-gauge-invariant electromagnetic theory in a curved space-time. The problem of electro-gravitational unification is considered from this point of view. 
  We consider a new set of effects arising from the quantum gravity corrections to the propagation of fields, associated with fluctuations of the spacetime geometry. Using already existing experimental data, we can put bounds on these effects that are more stringent by several orders of magnitude than those expected to be obtained in astrophysical observations. In fact these results can be already interpreted as questioning the whole scenario of linear (in $l_P$) corrections to the dispersion relations for free fields in Lorentz violating theories. 
  We study the wave propagation in nonlinear electrodynamical models. Particular attention is paid to the derivation and the analysis of the Fresnel equation for the wave covectors. For the class of general nonlinear Lagrangian models, we demonstrate how the originally quartic Fresnel equation factorizes, yielding the generic birefringence effect. We show that the closure of the effective constitutive (or jump) tensor is necessary and sufficient for the absence of birefringence, i.e., for the existence of a unique light cone structure. As another application of the Fresnel approach, we analyze the light propagation in a moving isotropic nonlinear medium. The corresponding effective constitutive tensor contains non-trivial skewon and axion pieces. For nonmagnetic matter, we find that birefringence is induced by the nonlinearity, and derive the corresponding optical metrics. 
  The properties of the thermal radiation are discussed by using the new equation of state density motivated by the generalized uncertainty relation in the quantum gravity. There is no burst at the last stage of the emission of a Schwarzshild black hole. When the new equation of state density is utilized to investigate the entropy of a scalar field outside the horizon of a static black hole, the divergence appearing in the brick wall model is removed, without any cutoff. The entropy proportional to the horizon area is derived from the contribution of the vicinity of the horizon. 
  A new class of solutions of the Einstein field equations in spherical symmetry is found. The new solutions are mathematically described as the metrics admitting separation of variables in area-radius coordinates. Physically, they describe the gravitational collapse of a class of anisotropic elastic materials. Standard requirements of physical acceptability are satisfied, in particular, existence of an equation of state in closed form, weak energy condition, and existence of a regular Cauchy surface at which the collapse begins. The matter properties are generic in the sense that both the radial and the tangential stresses are non vanishing, and the kinematical properties are generic as well, since shear, expansion, and acceleration are also non-vanishing. As a test-bed for cosmic censorship, the nature of the future singularity forming at the center is analyzed as an existence problem for o.d.e. at a singular point using techniques based on comparison theorems, and the spectrum of endstates - blackholes or naked singularities - is found in full generality. Consequences of these results on the Cosmic Censorship conjecture are discussed. 
  Building on general formulas obtained from the approximate renormalized effective action, the approximate stress-energy tensor of the quantized massive scalar field with arbitrary curvature coupling in the spacetime of charged black hole being a solution of coupled equations of nonlinear electrodynamics and general relativity is constructed and analysed. It is shown that in a few limiting cases, the analytical expressions relating obtained tensor to the general renormalized stress-energy tensor evaluated in the geometry of the Reissner-Nordstr\"{o}m black hole could be derived. A detailed numerical analysis with special emphasis put on the minimal coupling is presented and the results are compared with those obtained earlier for the conformally coupled field. Some novel features of the renormalized stress-energy tensor are discussed. 
  We study the dynamics of test particles and pointlike gyroscopes in 5D manifolds like those used in the Randall-Sundrum brane world and non-compact Kaluza-Klein models. Our analysis is based on a covariant foliation of the manifold using 3+1 dimensional spacetime slices orthogonal to the extra dimension, and is hence similar to the ADM 3+1 split in ordinary general relativity. We derive gauge invariant equations of motion for freely-falling test particles in the 5D and 4D affine parameterizations and contrast these results with previous work concerning the so-called ``fifth force''. Motivated by the conjectured localization of matter fields on a 3-brane, we derive the form of the classical non-gravitational force required to confine particles to a 4D hypersurface and show that the resulting trajectories are geometrically identical to the spacetime geodesics of Einstein's theory. We then discuss the issue of determining the 5D dynamics of a torque-free spinning body in the point-dipole approximation, and then perform a covariant (3+1)+1 decomposition of the relevant formulae (i.e. the 5D Fermi-Walker transport equation) for the cases of freely-falling and hypersurface-confined point gyroscopes. In both cases, the 4D spin tensor is seen to be subject to an anomalous torque. We solve the spin equations for a gyroscope confined to a single spacetime section in a simple 5D cosmological model and observe a cosmological variation of the magnitude and orientation of the 4D spin. 
  Under the assumption that the cosmological constant vanishes in the true ground state with lowest possible energy density, we argue that the observed small but finite vacuum-like energy density can be explained if we consider a theory with two or more degenerate perturbative vacua, which are unstable due to quantum tunneling, and if we still live in one of such states. An example is given making use of the topological vacua in non-Abelian gauge theories. 
  This work is concerned with the finiteness problem for static, spherically symmetric perfect fluids in both Newtonian Gravity and General Relativity. We derive criteria on the barotropic equation of state guaranteeing that the corresponding perfect fluid solutions possess finite/infinite extent. In the Newtonian case, for the large class of monotonic equations of state, and in General Relativity we improve earlier results. 
  The Hamiltonian formulation of the teleparallel equivalent of general relativity is considered. Definitions of energy, momentum and angular momentum of the gravitational field arise from the integral form of the constraint equations of the theory. In particular, the gravitational energy-momentum is given by the integral of scalar densities over a three-dimensional spacelike hypersurface. The definition for the gravitational energy is investigated in the context of the Kerr black hole. In the evaluation of the energy contained within the external event horizon of the Kerr black hole we obtain a value strikingly close to the irreducible mass of the latter. The gravitational angular momentum is evaluated for the gravitational field of a thin, slowly rotating mass shell. 
  Solutions describing the gravitational collapse of asymptotically flat cylindrical and prolate shells of (null) dust are shown to admit globally naked singularities. 
  In Newton's and in Einstein's theory we give criteria on the equation of state of a barotropic perfect fluid which guarantee that the corresponding one-parameter family of static, spherically symmetric solutions has finite extent. These criteria are closely related to ones which are known to ensure finite or infinite extent of the fluid region if the assumption of spherical symmetry is replaced by certain asymptotic falloff conditions on the solutions. We improve this result by relaxing the asymptotic asumptions. Our conditions on the equation of state are also related to (but less restrictive than) ones under which it has been shown in Relativity that static, asymptotically flat fluid solutions are spherically symmetric. We present all these results in a unified way. 
  Recent solutions to the Einstein Field Equations involving negative energy densities, i.e., matter violating the weak-energy-condition, have been obtained, namely traversable wormholes, the Alcubierre warp drive and the Krasnikov tube. These solutions are related to superluminal travel, although locally the speed of light is not surpassed. It is difficult to define faster-than-light travel in generic space-times, and one can construct metrics which apparently allow superluminal travel, but are in fact flat Minkowski space-times. Therefore, to avoid these difficulties it is important to provide an appropriate definition of superluminal travel. 
  We have investigated spherically symmetric spacetimes which contain a perfect fluid obeying the polytropic equation of state and admit a kinematic self-similar vector of the second kind which is neither parallel nor orthogonal to the fluid flow. We have assumed two kinds of polytropic equations of state and shown in general relativity that such spacetimes must be vacuum. 
  Higher-dimensional braneworld models which contain both bulk and brane curvature terms in the action admit cosmological singularities of rather unusual form and nature. These `quiescent' singularities, which can occur both during the contracting as well as the expanding phase, are characterised by the fact that while the matter density and Hubble parameter remain finite, all higher derivatives of the scale factor ($\stackrel{..}{a}$, $\stackrel{...}{a}$ etc.) diverge as the cosmological singularity is approached. The singularities are the result of the embedding of the (3+1)-dimensional brane in the bulk and can exist even in an empty homogeneous and isotropic (FRW) universe. The possibility that the present universe may expand into a singular state is discussed. 
  We give a higher even dimensional extension of vacuum colliding gravitational plane waves with the combinations of collinear and non-collinear polarized four-dimensional metric. The singularity structure of space-time depends on the parameters of the solution. 
  In this paper, we give a complete classification of vacuum branes, i.e., everywhere umbilical time-like hypersurfaces whose extrinsic curvature is a constant multiple of the induced metric, K_mn=k g_mn, in D-dimensional static spacetimes with spatial symmetry G(D-2,K), where G(n,K) is the isometry group of an n-dimensional space with constant sectional curvature K. D>=4 is assumed. It is shown that all possible configurations of a brane are invariant under an isometry subgroup G(D-3,K') for some K'>= K. In particular, configurations of a brane with non-zero k are always G(D-2,K) invariant, except for those in five special one-parameter families of spacetimes. Further, such G(D-2,K)-invariant configurations are allowed only in spacetimes whose Ricci tensors are isotropic in the two planes orthogonal to each G(D-2,K)-orbit, or for special values of k, which do not exist in generic cases. On the basis of these results, we prove the non-existence of a vacuum brane with black hole geometry in static bulk spacetimes with spatial symmetry G(D-2,K). We also discuss mathematical implications of these results. 
  This talk reports on the status of an approach to the numerical study of isolated systems with the conformal field equations. We first describe the algorithms used in a code which has been developed at AEI in the last years, and discuss a milestone result obtained by Huebner. Then we present more recent results as examples to sketch the problems we face in the conformal approach to numerical relativity and outline a possible roadmap toward making this approach a practical tool. 
  By Fuchsian techniques, a large family of Gowdy vacuum spacetimes have been constructed for which one has detailed control over the asymptotic behaviour. In this paper we formulate a condition on initial data yielding the same form of asymptotics. 
  We generalize for the case of arbitrary hydrodynamical matter the quasi-isotropic solution of Einstein equations near cosmological singularity, found by Lifshitz and Khalatnikov in 1960 for the case of radiation-dominated universe. It is shown that this solution always exists, but dependence of terms in the quasi-isotropic expansion acquires a more complicated form. 
  We study N=1 Supergravity inflation in the context of the braneworld scenario. Particular attention is paid to the problem of the onset of inflation at sub-Planckian field values and the ensued inflationary observables. We find that the so-called $\eta$-problem encountered in supergravity inspired inflationary models can be solved in the context of the braneworld scenario, for some range of the parameters involved. Furthermore, we obtain an upper bound on the scale of the fifth dimension, $M_5 \lsim 10^{-3} M_P$, in case the inflationary potential is quadratic in the inflaton field, $\phi$. If the inflationary potential is cubic in $\phi$, consistency with observational data requires that $M_5 \simeq 9.2 \times 10^{-4} M_P$. 
  We investigate the effect of the radion on cosmological perturbations in the brane world. The S^1/Z_2 compactified 5D Anti-de Sitter spacetime bounded by positive and negative tension branes is considered. The radion is the relative displacement of the branes in this model. We find two different kinds of the radion at the linear perturbation order for a cosmological brane. One describes a "fluctuation" of the brane which does not couple to matter on the brane. The other describes a "bend" of the brane which couples to the matter. The bend determines the curvature perturbation on the brane. At large scales, the radion interacts with anisotopic perturbations in the bulk. By solving the bulk anisotropic perturbations, large scale metric perturbations and anisotropies of the Cosmic Microwave Background (CMB) on the positive tension brane are calculated. We find an interesting fact that the radion contributes to the CMB anisotropies. The observational consequences of these effects are discussed. 
  A null path in 5D can appear as a timelike path in 4D, and for a certain gauge in 5D the motion of a massive particle in 4D obeys the usual quantization rule with an uncertainty-type relation. Generalizations of this result are discussed in regard to induced-matter and membrane theory. 
  Preliminary version No.~2 of the lecture notes for the talk ``Quantum theory of gravitational collapse'' given at the 271. WE-Heraeus-Seminar ``Aspects of Quantum Gravity'' at Bad Honnef, 25 February--1 March 2002 
  We study the propagation of gravitational waves carrying arbitrary information through isotropic cosmologies. The waves are modelled as small perturbations of the background Robertson-Walker geometry. The perfect fluid matter distribution of the isotropic background is, in general, modified by small anisotropic stresses. For pure gravity waves, in which the perturbed Weyl tensor is radiative (i.e. type N in the Petrov classification), we construct explicit examples for which the presence of the anisotropic stress is shown to be essential and the histories of the wave-fronts in the background Robertson-Walker geometry are shear-free null hypersurfaces. The examples derived in this case are analogous to the Bateman waves of electromagnetic theory. 
  Over the last few years part of the quantum-gravity community has adopted a more optimistic attitude toward the possibility of finding experimental contexts providing insight on non-classical properties of spacetime. I review those quantum-gravity phenomenology proposals which were instrumental in bringing about this change of attitude, and I discuss the prospects for the short-term future of quantum-gravity phenomenology. 
  We present a new approach for finding conservation laws in the perturbation theory of black holes which applies for the more general cases of non-Hermitian equations governing the perturbations. The approach is based on a general result which establishes that a covariantly conserved current can be obtained from a solution of any system of homogeneous linear differential equations and a solution of the adjoint system. It is shown that the results obtained from the present approach become essentially the same (with some diferences) to those obtained by means of the traditional methods in the simplest black hole geometry corresponding to the Schwarzschild space-time. The future applications of our approach for studying the perturbations of black hole space-time in string theory is discussed. 
  A non-perturbative canonical quantization of Gowdy $T^3$ polarized models carried out recently is considered. This approach profits from the equivalence between the symmetry reduced model and 2+1 gravity coupled to a massless real scalar field. The system is partially gauge fixed and a choice of internal time is performed, for which the true degrees of freedom of the model reduce to a massless free scalar field propagating on a 2-dimensional expanding torus. It is shown that the symplectic transformation that determines the classical dynamics cannot be unitarily implemented on the corresponding Hilbert space of quantum states. The implications of this result for both quantization of fields on curved manifolds and physically relevant questions regarding the initial singularity are discussed. 
  Local continuity equations involving background fields and variantions of the fields, are obtained for a restricted class of solutions of the Einstein-Maxwell and Einstein-Weyl theories using a new approach based on the concept of the adjoint of a differential operator. Such covariant conservation laws are generated by means of decoupled equations and their adjoints in such a way that the corresponding covariantly conserved currents possess some gauge-invariant properties and are expressed in terms of Debye potentials. These continuity laws lead to both a covariant description of bilinear forms on the phase space and the existence of conserved quantities. Differences and similarities with other approaches and extensions of our results are discussed. 
  The plenary lectures, parallel talks, oral presentations and panel contributions on Cosmology and Gravitation presented during the XXII Brazilian National Meeting on Particles and Fields are briefly reviewed. Some remarks on the area are also presented. 
  We investigate the radiation emitted by a uniformly moving charged scalar particle in the space-time of a point-like global monopole. We calculate the total energy radiated by the particle and the corresponding spectrum, for small solid angle deficit. We show that the radiated energy is proportional to the cube of the velocity of the particle and to the cube of the Lorenz factor, in the non-relativistic and ultra-relativistic cases, respectively. 
  I discuss the conformal approach to the numerical simulation of radiating isolated systems in general relativity. The method is based on conformal compactification and a reformulation of the Einstein equations in terms of rescaled variables, the so-called ``conformal field equations'' developed by Friedrich. These equations allow to include ``infinity'' on a finite grid, solving regular equations, whose solutions give rise to solutions of the Einstein equations of (vacuum) general relativity. The conformal approach promises certain advantages, in particular with respect to the treatment of radiation extraction and boundary conditions. I will discuss the essential features of the analytical approach to the problem, previous work on the problem - in particular a code for simulations in 3+1 dimensions, some new results, open problems and strategies for future work. 
  We present a new formulation to deal with the consistency problem of a massive spin-2 field in a curved spacetime. Using Fierz variables to represent the spin-2 field, we show how to avoid the arbitrariness and inconsistency that exists in the standard formulation of spin-2 field coupled to gravity. The superiority of the Fierz frame appears explicitly in the combined set of equations for spin-2 field and gravity: it preserves the standard Einstein equations of motion. 
  In this work we consider Randall-Sundrum braneworld type scenarios, in which the spacetime is described by a five-dimensional manifold with matter fields confined in a domain wall or three-brane. We present the results of a systematic analysis, using dynamical systems techniques, of the qualitative behaviour of Friedmann-Lemaitre-Robertson-Walker type models, whose matter is described by a scalar field with an exponential potential. We construct the state spaces for these models and discuss how their structure changes with respect to the general-relativistic case, in particular, what new critical points appear and their nature and the occurrence of bifurcation. 
  We investigate the apparent horizon formation for high-energy head-on collisions of particles in multi-dimensional spacetime. The apparent horizons formed before the instance of particle collision are obtained analytically. Using these solutions, we discuss the feature of the apparent horizon formation in the multi-dimensional spacetime from the viewpoint of the hoop conjecture. 
  This contribution consists of two parts. In the first part, I review the tunneling approach to quantum cosmology and comment on the alternative approaches. In the second part, I discuss the relation between quantum cosmology and eternal inflation. In particular, I discuss whether or not we need quantum cosmology in the light of eternal inflation, and whether or not quantum cosmology makes any testable predictions. 
  Sakharov's 1967 notion of ``induced gravity'' is currently enjoying a significant resurgence. The basic idea, originally presented in a very brief 3-page paper with a total of 4 formulas, is that gravity is not ``fundamental'' in the sense of particle physics. Instead it was argued that gravity (general relativity) emerges from quantum field theory in roughly the same sense that hydrodynamics or continuum elasticity theory emerges from molecular physics. In this article I will translate the key ideas into modern language, and explain the various versions of Sakharov's idea currently on the market. 
  Stationary extended frames in general relativity are considered. The requirement of stationarity allows to treat the spacetime as a principal fiber bundle over the one-dimensional group of time translations. Over this bundle a connection form establishes the simultaneity between neighboring events accordingly with the Einstein synchronization convention. The mathematics involved is that of gauge theories where a gauge choice is interpreted as a global simultaneity convention. Then simultaneity in non-stationary frames is investigated: it turns to be described by a gauge theory in a fiber bundle without structure group, the curvature being given by the Fr\"olicher-Nijenhuis bracket of the connection. The Bianchi identity of this gauge theory is a differential relation between the vorticity field and the acceleration field. In order for the simultaneity connection to be principal, a necessary and sufficient condition on the 4-velocity of the observers is given. 
  Stability arguments suggest that the Kaluza-Klein (KK) internal scalar field, $\Phi$, should be coupled to some external fields. An external bulk real scalar field, $\psi$, minimally coupled to gravity is proved to be satisfactory. At low temperature, the coupling of $\psi$ to the electromagnetic (EM) field allows $\Phi$ to be much stronger coupled to the EM field than in the genuine five dimensional KK theory. It is shown that the coupling of $\Phi$ to the geomagnetic field may explain the observed dispersion in laboratory measurements of the (effective) gravitational constant. The analysis takes into account the spatial variations of the geomagnetic field. Except the high PTB value, the predictions are found in good agreement with all of the experimental data. 
  We derive a simple form for the propagator of a massless, minimally coupled scalar in a locally de Sitter geometry of arbitrary spacetime dimension. We then employ it to compute the fully renormalized stress tensor at one and two loop orders for a massless, minimally coupled phi^4 theory which is released in Bunch-Davies vacuum at t=0 in co-moving coordinates. In this system the uncertainty principle elevates the scalar above the minimum of its potential, resulting in a phase of super-acceleration. With the non-derivative self-interaction the scalar's breaking of de Sitter invariance becomes observable. It is also worth noting that the weak energy condition is violated on cosmological scales. An interesting subsidiary result is that canceling overlapping divergences in the stress tensor requires a conformal counterterm which has no effect on purely scalar diagrams. 
  Using an axial parallel vector field we obtain two exact solutions of a vacuum gravitational field equations. One of the exact solutions gives the Schwarzschild metric while the other gives the Kerr metric. The parallel vector field of the Kerr solution have an axial symmetry. The exact solution of the Kerr metric contains two constants of integration, one being the gravitational mass of the source and the other constant $h$ is related to the angular momentum of the rotating source, when the spin density ${S_{i j}}^\mu$ of the gravitational source satisfies $\partial_\mu {S_{i j}}^\mu=0$. The singularity of the Kerr solution is studied. 
  Loop quantum gravity is based on a classical formulation of 3+1 gravity in terms of a real SU(2) connection. Linearization of this classical formulation about a flat background yields a description of linearised gravity in terms of a {\em real} $U(1)\times U(1)\times U(1)$ connection. A `loop' representation, in which holonomies of this connection are unitary operators, can be constructed. These holonomies are not well defined operators in the standard graviton Fock representation. We generalise our recent work on photons and U(1) holonomies to show that Fock space gravitons are associated with distributional states in the $U(1)\times U(1)\times U(1)$ loop representation. Our results may illuminate certain aspects of the much deeper (and as yet unkown,) relation between gravitons and states in nonperturbative loop quantum gravity.   This work leans heavily on earlier seminal work by Ashtekar, Rovelli and Smolin (ARS) on the loop representation of linearised gravity using {\em complex} connections. In the last part of this work, we show that the loop representation based on the {\em real} $U(1)\times U(1)\times U(1)$ connection also provides a useful kinematic arena in which it is possible to express the ARS complex connection- based results in the mathematically precise language currently used in the field. 
  First-order hyperbolic systems are promising as a basis for numerical integration of Einstein's equations. In previous work, the lapse and shift have typically not been considered part of the hyperbolic system and have been prescribed independently. This can be expensive computationally, especially if the prescription involves solving elliptic equations. Therefore, including the lapse and shift in the hyperbolic system could be advantageous for numerical work. In this paper, two first-order symmetrizable hyperbolic systems are presented that include the lapse and shift as dynamical fields and have only physical characteristic speeds. 
  We describe a kinetic theory approach to quantum gravity -- by which we mean a theory of the microscopic structure of spacetime, not a theory obtained by quantizing general relativity. A figurative conception of this program is like building a ladder with two knotted poles: quantum matter field on the right and spacetime on the left. Each rung connecting the corresponding knots represent a distinct level of structure. The lowest rung is hydrodynamics and general relativity; the next rung is semiclassical gravity, with the expectation value of quantum fields acting as source in the semiclassical Einstein equation. We recall how ideas from the statistical mechanics of interacting quantum fields helped us identify the existence of noise in the matter field and its effect on metric fluctuations, leading to the establishment of the third rung: stochastic gravity, described by the Einstein-Langevin equation. Our pathway from stochastic to quantum gravity is via the correlation hierarchy of noise and induced metric fluctuations. Three essential tasks beckon: 1) Deduce the correlations of metric fluctuations from correlation noise in the matter field; 2) Reconstituting quantum coherence -- this is the reverse of decoherence -- from these correlation functions 3) Use the Boltzmann-Langevin equations to identify distinct collective variables depicting recognizable metastable structures in the kinetic and hydrodynamic regimes of quantum matter fields and how they demand of their corresponding spacetime counterparts. This will give us a hierarchy of generalized stochastic equations -- call them the Boltzmann-Einstein hierarchy of quantum gravity -- for each level of spacetime structure, from the macroscopic (general relativity) through the mesoscopic (stochastic gravity) to the microscopic (quantum gravity). 
  We undertake to show how the relativistic Finslerian Metric Function (FMF) should arise under uni-directional violation of spatial isotropy, keeping the condition that the indicatrix (mass-shell) is a space of constant negative curvature. By evaluating respective Finslerian tetrads, and treating them consistently as the bases of inertial reference frame (RF), the generalized Finslerian kinematic transformations follow in a convenient explicit form. The concomitant Finslerian relativistic relations generalize their Lorentzian prototypes through the presence of one characteristic parameter g, so that the constraints on the parameter may be found in future high-precision post-Lorentzian experiments. As the associated Finslerian Hamiltonian function is also obtainable in a clear explicit form, convenient prospects for the Finslerian extension of particle dynamics are also opened. Additionally, the Finslerian extension of the general-relativistic Schwarzschild metric can unambiguously be proposed. 
  We investigate the dynamics of a Bianchi I brane Universe in the presence of a nonlocal anisotropic stress ${\cal P}_{\mu\nu}$ proportional to a "dark energy" ${\cal U}$. Using this ansatz for the case ${\cal U} > 0$ we prove that if a matter on a brane satisfies the equation of state $p=(\gamma-1)\rho$ with $\gamma \le 4/3$ then all such models isotropize. For $\gamma > 4/3$ anisotropic future asymptotic states are found. We also describe the past asymptotic regimes for this model. 
  We study the quantum vacuum fluctuations around closed Friedmann-Robertson-Walker (FRW) radiation-filled universes with nonvanishing cosmological constant. These vacuum fluctuations are represented by a conformally coupled massive scalar field and are treated in the lowest order of perturbation theory. In the semiclassical approximation, the perturbations are governed by differential equations which, properly linearized, become generalized Lame equations. The wave function thus obtained must satisfy appropriate regularity conditions which ensure its finiteness for every field configuration. We apply these results to asymptotically anti de-Sitter Euclidean wormhole spacetimes and show that there is no catastrophic particle creation in the Euclidean region, which would lead to divergences of the wave function. 
  In Newtonian and relativistic hydrodynamics the Riemann problem consists of calculating the evolution of a fluid which is initially characterized by two states having different values of uniform rest-mass density, pressure and velocity. When the fluid is allowed to relax, one of three possible wave-patterns is produced, corresponding to the propagation in opposite directions of two nonlinear hydrodynamical waves. New effects emerge in a special relativistic Riemann problem when velocities tangential to the initial discontinuity surface are present. We show that a smooth transition from one wave-pattern to another can be produced by varying the initial tangential velocities while otherwise maintaining the initial states unmodified. These special relativistic effects are produced by the coupling through the relativistic Lorentz factors and do not have a Newtonian counterpart. 
  We calculate the back reaction of cosmological perturbations on a general relativistic variable which measures the local expansion rate of the Universe. Specifically, we consider a cosmological model in which matter is described by a single field. We analyze back reaction both in a matter dominated Universe and in a phase of scalar field-driven chaotic inflation. In both cases, we find that the leading infrared terms contributing to the back reaction vanish when the local expansion rate is measured at a fixed value of the matter field which is used as a clock, whereas they do not appear to vanish if the expansion rate is evaluated at a fixed value of the background time. We discuss possible implications for more realistic models with a more complicated matter sector. 
  Adopting thin film brick-wall model, we calculate the entropy of a nonuniformly rectilinearly accelerating non-stationary black hole expressed by Kinnersley metric. Because the black hole is accelerated, the event horizon is axisymmetric. The different points of horizon surface may have different temperature. We calculate the temperature and the entropy density at every point of the horizon at first, then we obtain the total entropy through integration, which is proportional to the aera of event horizon as the same as the stationary black holes. It is shown that the black hole entropy may be regarded as the entropy of quantum fields just on the surface of event horizon. 
  It is known that de Sitter spacetime can be seen as the solution of field equation for completely isotropic matter. In the present paper a new class of exact solutions in spherical symmetry is found and discussed, such that the energy--momentum tensor has two 2--dimensional distinct isotropic subspaces. 
  A growing number of studies is being devoted to the identification of plausible quantum properties of spacetime which might give rise to observably large effects. The literature on this subject is now relatively large, including studies in string theory, loop quantum gravity and noncommutative geometry. It is useful to divide the various proposals into proposals involving a systematic quantum-gravity effect (an effect that would shift the main/average prediction for a given observable quantity) and proposals involving a non-systematic quantum-gravity effect (an effect that would introduce new fundamental uncertanties in some observable quantity). The case of quantum-gravity-induced particle-production-threshold anomalies, a much studied example of potentially observable quantum-gravity effect, is here used as an example to illustrate the differences to be expected between systematic and non-systematic effects. 
  A generalization of the thermodynamic uncertainty relations is proposed. It is done by introducing of an additional term proportional to the interior energy into the standard thermodynamic uncertainty relation that leads to existence of the lower limit of inverse temperature. 
  The study of acoustic black holes has been undertaken to provide new insights about the role of high frequencies in black hole evaporation. Because of the infinite gravitational redshift from the event horizon, Hawking quanta emerge from configurations which possessed ultra high (trans-Planckian) frequencies. Therefore Hawking radiation cannot be derived within the framework of a low energy effective theory; and in all derivations there are some assumptions concerning Planck scale physics. The analogy with condensed matter physics was thus introduced to see if the asymptotic properties of the Hawking phonons emitted by an acoustic black hole, namely stationarity and thermality, are sensitive to the high frequency physics which stems from the granular character of matter and which is governed by a non-linear dispersion relation. In 1995 Unruh showed that they are not sensitive in this respect, in spite of the fact that phonon propagation near the (acoustic) horizon drastically differs from that of photons. In 2000 the same analogy was used to establish the robustness of the spectrum of primordial density fluctuations in inflationary models. This analogy is currently stimulating research for experimenting Hawking radiation. Finally it could also be a useful guide for going beyond the semi-classical description of black hole evaporation. 
  Roughly a dozen X-ray binaries are presently known in which the compact accreting primary stars are too massive to be neutron stars. These primaries are identified as black holes, though there is as yet no definite proof that any of the candidate black holes actually possesses an event horizon. We discuss how Type I X-ray bursts may be used to verify the presence of the event horizon in these objects. Type I bursts are caused by thermonuclear explosions when gas accretes onto a compact star. The bursts are commonly seen in many neutron star X-ray binaries, but they have never been seen in any black hole X-ray binary. Our model calculations indicate that black hole candidates ought to burst frequently if they have surfaces. Based on this, we argue that the lack of bursts constitutes strong evidence for the presence of event horizons in these objects. 
  Vast amounts of entropy are produced in black hole formation, and the amount of entropy stored in supermassive black holes at the centers of galaxies is now much greater than the entropy free in the rest of the universe. Either mergers involved in forming supermassive black holes are rare,or the holes must be very efficient at capturing nearly all the entropy generated in the process.  We argue that this information can be used to constrain supermassive black hole production, and may eventually provide a check on numerical results for mergers involving black holes. 
  The initial data sets for the five-dimensional Einstein equation have been examined. The system is designed such that the black hole ($\simeq S^3$) or the black ring ($\simeq S^2\times S^1$) can be found. We have found that the typical length of the horizon can become arbitrarily large but the area of characteristic closed two-dimensional submanifold of the horizon is bounded above by the typical mass scale. We conjecture that the isoperimetric inequality for black holes in $n$-dimensional space is given by $V_{n-2} \lesssim GM$, where $V_{n-2}$ denotes the volume of typical closed $(n-2)$-section of the horizon and $M$ is typical mass scale, rather than $C\lesssim (GM)^{1/(n-2)}$ in terms of the hoop length $C$, which holds only when $n=3$. 
  We propose a quantitative test for the validity of the semi-classical approximation in gravity, namely that the solutions to the semi-classical equations should be stable to linearized perturbations, in the sense that no gauge invariant perturbation should become unbounded in time. We show that a self-consistent linear response analysis of these perturbations based upon an invariant effective action principle involves metric fluctuations about the mean semi-classical geometry and brings in the two-point correlation function of the quantum energy-momentum tensor in a natural way. The properties of this correlation function are discussed and it is shown on general grounds that it contains no state-dependent divergences and requires no new renormalization counterterms beyond those required in the leading order semi-classical approximation. 
  Ghost neutrinos in radiative Kerr spacetime endowed with totally skew-symmetric Cartan contortion is presented. The computations are made by using the Newman-Penrose (NP) calculus. The model discussed here maybe useful in several astrophysical applications specially in black hole astrophysics. 
  We present a simple sandwich gravitational wave of the Robinson-Trautman family. This is interpreted as representing a shock wave with a spherical wavefront which propagates into a Minkowski background minus a wedge. (i.e. the background contains a cosmic string.) The deficit angle (the tension) of the string decreases through the gravitational wave, which then ceases. This leaves an expanding spherical region of Minkowski space behind it. The decay of the cosmic string over a finite interval of retarded time may be considered to generate the gravitational wave. 
  Disturbing of a spacetime geometry may result in the appearance of an oscillating and damped radiation - the so-called quasinormal modes. Their periods of oscillations and damping coefficients carry unique information about the mass and the angular momentum, that would allow one to identify the source of the gravitational field. In this talk we present recent bounds on the diffused energy, applicable to the Schwarzschild spacetime, that give also rough estimates of the energy of excited quasinormal modes. 
  In this report, which is an extended version of that appearing in the Proceedings of GR16, I will give a summary of the main topics covered in Session A.3. on mathematical relativity at GR16, Durban. The summary is mainly based on extended abstracts submitted by the speakers. I would like to thank all participants for their contributions and help with this summary. 
  The Hawking radiation forms the essential basis of the black hole thermodynamics. The black hole thermodynamics denotes a nice correspondence between black hole kinematics and the laws of ordinary thermodynamics, but has been so far considered only in an asymptotically flat case. Does such the correspondence rely strongly on the feature of the gravity vanishing at the infinity? In order to resolve this question, it should be considered for the first to extend the Hawking radiation to a case with a dynamical boundary condition like an expanding universe. Therefore the Hawking radiation in an expanding universe is discussed in this paper. As a concrete model of a black hole in an expanding universe, we use the swiss cheese universe which is the spacetime including a Schwarzschild black hole in the Friedmann-Robertson-Walker universe. Further for simplicity, our calculation is performed in two dimension. The resultant spectrum of the Hawking radiation measured by a comoving observer is generally different from a thermal one. We find that the qualitative behavior of the non-thermal spectrum is of dumping oscillation as a function of the frequency measured by the observer, and that the intensity of the Hawking radiation is enhanced by the presence of a cosmological expansion. It is appropriate to say that it a black hole with an asymptotically flat boundary condition stays in a lowest energy thermal equilibrium state, and that, once a black hole is put into an expanding universe, it is excited to a non-equilibrium state and emits its mass energy with stronger intensity than a thermal one. 
  In this paper, the cosmic no hair theorem for anisotropic Bianchi models which admit inflation with a scalar field is studied in the framework of Brane world. It is found that all Bianchi models except Bianchi type IX, transit to an inflationary regime with vanishing anisotropy. In the Brane world, anisotropic universe approaches the inflationary era much faster than that in the general theory of relativity. The form of the potential does not affect the evolution in the inflationary epoch. However, the late time behaviour is controlled by a constant additive factor in the potential for the inflaton field. 
  We review current best estimates of the strength and detectability of the gravitational waves from a variety of sources, for both ground-based and space-based detectors, and we describe the information carried by the waves. 
  We find the general solution of the Einstein equation for spherically symmetric collapse of Type II fluid (null strange quark fluid) in higher dimensions. It turns out that the nakedness and curvature strength of the shell focusing singularities carry over to higher dimensions. However, there is shrinkage of the initial data space for a naked singularity of the Vaidya collapse due to the presence of strange quark matter. 
  A satisfactory theory of quantum gravity will very likely require modification of our classical perception of space-time, perhaps by giving it a 'foamy' structure at scales of order the Planck length. This is expected to modify the propagation of photons and other relativistic particles such as neutrinos, such that they will experience a non-trivial refractive index even in vacuo. The implied spontaneous violation of Lorentz invariance may also result in alterations of kinematical thresholds for key astrophysical processes involving high energy cosmic radiation. We discuss experimental probes of these possible manifestations of the fundamental quantum nature of space-time using observations of distant astrophysical sources such as gamma-ray bursts and active galactic nuclei. 
  We derive the explicit values of all regularization parameters (RP) for a scalar particle in an arbitrary geodesic orbit around a Schwarzschild black hole. These RP are required within the previously introduced mode-sum method, for calculating the local self-force acting on the particle. In this method one first calculates the (finite) contribution to the self-force due to each individual multipole mode of the particle's field, and then applies a certain regularization procedure to the mode sum, involving the RP. The explicit values of the RP were presented in a recent Letter [Phys. Rev. Lett. {\bf 88}, 091101 (2002)]. Here we give full details of the RP derivation in the scalar case. The calculation of the RP in the electromagnetic and gravitational cases will be discussed in an accompanying paper. 
  In a recently proposed scenario, where the dilaton decouples while cosmologically attracted towards infinite bare string coupling, its residual interactions can be related to the amplitude of density fluctuations generated during inflation, and are large enough to be detectable through a modest improvement on present tests of free-fall universality. Provided it has significant couplings to either dark matter or dark energy, a runaway dilaton can also induce time-variations of the natural "constants" within the reach of near-future experiments. 
  Classically general covariance is found from the idea that a vector is a physical quantity which exists independently of choice of coordinate system and is unchanged by a change of coordinate system. It is often assumed that there exists some form of absolute mathematical space or space-time, and that in a flat space approximation vectors can be imagined between defined points in this space-time, much as we can imagine an arrowed line drawn on a piece of paper. However, while classical vector quantities can be represented on paper, in the quantum domain physical quantities do not in general exist with precise values except in measurement; a change of apparatus, for example by rotating it, may affect the outcome of the measurement, so the condition for general covariance does not apply. The purpose of this paper is to re-examine covariance within the context of an orthodox, Dirac-Von Neumann interpretation of quantum mechanics, to replace it with a new condition, here called quantum covariance, and to show that quantum covariance is the required condition for the unification of general relativity with quantum mechanics for non-interacting particles. 
  In the present paper, we give some theorems representing ridigity of a vacuum brane in static bulk spacetimes. As an application, we show that a static bulk spacetime with dimension D>3 and spatial symmetry IO(D-2), O(D-1) or O_+(D-2,1) does not allow a vacuum brane with a black hole on it. We also show that if a static bulk spacetime with dimension D>4 satsifying the vacuum Einstein equations can be foliated by a continuous family of vacuum branes with asymptotically constant curvature, it is a black string solution. 
  We present a consistent extended-object approach for determining the self force acting on an accelerating charged particle. In this approach one considers an extended charged object of finite size $\epsilon $, and calculates the overall contribution of the mutual electromagnetic forces. Previous implementations of this approach yielded divergent terms $\propto 1/\epsilon $ that could not be cured by mass-renormalization. Here we explain the origin of this problem and fix it. We obtain a consistent, universal, expression for the extended-object self force, which conforms with Dirac's well known formula. 
  The aim of this contribution is to provide a short introduction to recently investigated models in which our accessible universe is a four-dimensional submanifold, or brane, embedded in a higher dimensional spacetime and ordinary matter is trapped in the brane. I focus here on the gravitational and cosmological aspects of such models with a single extra-dimension. 
  The (zeroth-order) energy of a particle in the background of a black hole is given by Carter's integrals. However, exact calculations of a particle's {\it self-energy} (first-order corrections) are still beyond our present reach in many situations. In this paper we use Hawking's area theorem in order to derive bounds on the self-energy of a particle in the vicinity of a black hole. Furthermore, we show that self-energy corrections {\it must} be taken into account in order to guarantee the validity of Penrose cosmic censorship conjecture. 
  By allowing for non zero vacuum expectation values for some of the fields that appear in the Hamiltonian constraint of canonical general relativity a time variable, with usual properties, can be identified; the constraint plays the role of the ordinary Hamiltonian. The energy eigenvalues contribute to the variation of the scale parameter similarly to the way matter density does. For a universe described by a superposition of eigenstates or by a thermodynamic ensemble the dominant contribution comes from energy, or equivalently effective matter density, of the same order as the vacuum energy (cosmological constant). This may explain the observed ``coincidence'' of these two values. 
  String theory is a promising candidate for a fundamental quantum theory of all interactions including Einstein gravity. Some solutions in string theory can be interpreted as black holes. Using the semi-analytic method and WKB method,the quasinormal modes(QNMs)of 1+1 dimensional black hole in string theory are studied.The QNMs of 1+3 dimensional black hole in string theory are also calculated through numerical approach.The numerical investigation has shown that the late time gravitational oscillation of the black hole under an external perturbaton is dominated by certain QNMs. 
  Binary black hole systems in the pre-coalescence stage are numerically constructed by demanding that the associated spacetime admits a helical Killing vector. Comparison with third order post-Newtonian calculations indicates a rather good agreement until the innermost stable circular orbit. 
  This article investigates the computation of the eigenmodes of the Laplacian operator in multi-connected three-dimensional spherical spaces. General mathematical results and analytical solutions for lens and prism spaces are presented. Three complementary numerical methods are developed and compared with our analytic results and previous investigations. The cosmological applications of these results are discussed, focusing on the cosmic microwave background (CMB) anisotropies. In particular, whereas in the Euclidean case too small universes are excluded by present CMB data, in the spherical case there will always exist candidate topologies even if the total energy density parameter of the universe is very close to unity. 
  At a time when uninhibited speculation about negative tension -- and by implication negative mass density -- world branes has become commonplace, it seems worthwhile to call attention to the risk involved in sacrificing traditional energy positivity postulates such as are required for the classical vacuum stability theorem of Hawking and Ellis. As well as recapitulating the technical content of this reassuring (when applicable) theorem, the present article provides a new, rather more economical proof. 
  We study the possibility of generalising the Einstein--Straus model to anisotropic settings, by considering the matching of locally cylindrically symmetric static regions to the set of $G_4$ on $S_3$ locally rotationally symmetric (LRS) spacetimes. We show that such matchings preserving the symmetry are only possible for a restricted subset of the LRS models in which there is no evolution in one spacelike direction. These results are applied to spatially homogeneous (Bianchi) exteriors where the static part represents a finite bounded interior region without holes. We find that it is impossible to embed finite static strings or other locally cylindrically symmetric static objects (such as bottle or coin-shaped objects) in reasonable Bianchi cosmological models, irrespective of the matter content. Furthermore, we find that if the exterior spacetime is assumed to have a perfect fluid source satisfying the dominant energy condition, then only a very particular family of LRS stiff fluid solutions are compatible with this model.   Finally, given the interior/exterior duality in the matching procedure, our results have the interesting consequence that the Oppenheimer-Snyder model of collapse cannot be generalised to such anisotropic cases. 
  On the basis of a "local" principle of equivalence of general relativity, we consider a navigation in a kind of "4D-ocean" involving measurements of conformally invariant physical properties only. Then, applying the Pfaff theory for PDE to a particular conformally equivariant system of differential equations, we show the dependency of any kind of function describing "spacetime waves", with respect to 20 parametrizing functions. These latter, appearing in a linear differential Spencer sequence and determining gauge fields of deformations relatively to "ship-metrics" or to "flat spacetime ocean metrics", may be ascribed to unified electromagnetic and gravitational waves. The present model is based neither on a classical gauge theory of gravitation or a gravitation theory with torsion, nor on any Kaluza-Klein or Weyl type unifications, but rather on a post-Newtonian approach of gravitation in a four dimensional conformal Cosserat spacetime. 
  The causal structure of space-time offers a natural notion of an opposite or orthogonal in the logical sense, where the opposite of a set is formed by all points non time-like related with it. We show that for a general space-time the algebra of subsets that arises from this negation operation is a complete orthomodular lattice, and thus has several of the properties characterizing the algebra physical propositions in quantum mechanics. We think this fact could be used to investigate causal structure in an algebraic context. As a first step in this direction we show that the causal lattice is in addition atomic, find its atoms, and give necesary and sufficient conditions for ireducibility. 
  The images of many distant galaxies are displaced, distorted and often multiplied by the presence of foreground massive galaxies near the line of sight; the foreground galaxies act as gravitational lenses. Commonly, the lens equation, which relates the placement and distortion of the images to the real source position in the thin-lens scenario, is obtained by extremizing the time of arrival among all the null paths from the source to the observer (Fermat's principle). We show that the construction of envelopes of certain families of null surfaces consitutes an alternative variational principle or version of Fermat's principle that leads naturally to a lens equation in a generic spacetime with any given metric. We illustrate the construction by deriving the lens equation for static asymptotically flat thin lens spacetimes. As an application of the approach, we find the bending angle for moving thin lenses in terms of the bending angle for the same deflector at rest. Finally we apply this construction to cosmological spacetimes (FRW) by using the fact they are all conformally related to Minkowski space. 
  In a previous paper (gr-qc/0105100) we derived a set of near-optimal signal detection techniques for gravitational wave detectors whose noise probability distributions contain non-Gaussian tails. The methods modify standard methods by truncating sample values which lie in those non-Gaussian tails. The methods were derived, in the frequentist framework, by minimizing false alarm probabilities at fixed false detection probability in weak signal limit. For stochastic signals, the resulting statistic consisted of a sum of an auto-correlation term and a cross-correlation term; it was necessary to discard by hand the auto-correlation term to obtain the correct, generalized cross-correlation statistic. In the present paper, we present an alternative Bayesian derivation of the same signal detection techniques. We compute the probability that a signal is present in the data, in the limit where the signal-to-noise ratio squared per frequency bin is small, where the integrated signal-to-noise ratio is large compared to one, and where the total probability in the non-Gaussian tail part of the noise distribution is small. We show that, for each model considered, the resulting probability is to a good approximation a monotonic function of the detection statistic derived in the previous paper. Moreover, for stochastic signals, the new Bayesian derivation automatically eliminates the problematic auto-correlation term. 
  We present a framework, based on the null-surface formulation of general relativity, for discussing the dynamics of Fermat potentials for gravitational lensing in a generic situation without approximations of any kind. Additionally, we derive two lens equations: one for the case of thick compact lenses and the other one for lensing by gravitational waves. These equations in principle generalize the astrophysical scheme for lensing by removing the thin-lens approximation while retaining the weak fields. 
  The relativistic invariant zeta-function approach to computation of the vacuum energy contribution to cosmological constant is discussed. It is shown that this value is determined by the fourth power of the quantized field mass, while the dependence from the large mass scale is only logarithmic. This value is compared to the result obtained in the dimensional regularization scheme which also satisfies the relativistic invariance condition, and found to be the same up to irrelevant finite terms. The consequences of the renormalization group invariance are also briefly discussed. 
  We consider a massless scalar field propagating in a weakly curved spacetime whose metric is a solution to the linearized Einstein field equations. The spacetime is assumed to be stationary and asymptotically flat, but no other symmetries are imposed -- the spacetime can rotate and deviate strongly from spherical symmetry. We prove that the late-time behavior of the scalar field is identical to what it would be in a spherically-symmetric spacetime: it decays in time according to an inverse power-law, with a power determined by the angular profile of the initial wave packet (Price falloff theorem). The field's late-time dynamics is insensitive to the nonspherical aspects of the metric, and it is governed entirely by the spacetime's total gravitational mass; other multipole moments, and in particular the spacetime's total angular momentum, do not enter in the description of the field's late-time behavior. This extended formulation of Price's falloff theorem appears to be at odds with previous studies of radiative decay in the spacetime of a Kerr black hole. We show, however, that the contradiction is only apparent, and that it is largely an artifact of the Boyer-Lindquist coordinates adopted in these studies. 
  Starting with an exact and simple geodesic, we generate approximate geodesics by summing up higher-order geodesic deviations within a General Relativistic setting, without using Newtonian and post-Newtonian approximations. We apply this method to the problem of closed orbital motion of test particles in the Kerr metric space-time. With a simple circular orbit in the equatorial plane taken as the initial geodesic we obtain finite eccentricity orbits in the form of Taylor series with the eccentricity playing the role of small parameter. The explicit expressions of these higher-order geodesic deviations are derived using successive systems of linear equations with constant coefficients, whose solutions are of harmonic oscillator type. This scheme gives best results when applied to the orbits with low eccentricities, but with arbitrary values of $(GM/Rc^2)$, smaller than 1/6 in the Schwarzschild limit. 
  A new family of five dimensional, R=0 braneworlds with asymmetric warp factors is proposed. Beginning with the invariance of the Ricci scalar for the general class of asymmetrically warped spacetimes we, subsequently specialise to the R=0 case. Solutions are obtained by choosing a particular relation (involving a parameter $\nu$) between the warp factors. Symmetric warping arises as a special case (particular value of $\nu$). Over a range of values of $\nu$ the energy conditions for the matter stress energy are found to hold good. It turns out that the energy density and pressures required to support these spacetimes decay as the inverse square of the fifth (extra) coordinate. The projection of this bulk stress--energy (for symmetric warping) on the 3--brane yields an effective cosmological constant. We conclude with brief comments on spacetimes with constant Ricci scalar and the extension of our results to diverse dimensions. 
  The Immirzi ambiguity arises in loop quantum gravity when geometric operators are represented in terms of different connections that are related by means of an extended Wick transform. We analyze the action of this transform in gravity coupled with matter fields and discuss its analogy with the Wick rotation on which the Thiemann transform between Euclidean and Lorentzian gravity is based. In addition, we prove that the effect of this extended Wick transform is equivalent to a constant scale transformation as far as the symplectic structure and kinematical constraints are concerned. This equivalence is broken in the dynamical evolution. Our results are applied to the discussion of the black hole entropy in the limit of large horizon areas. We first argue that, since the entropy calculation is performed for horizons of fixed constant area, one might in principle choose an Immirzi parameter that depends on this quantity. This would spoil the linearity with the area in the entropy formula. We then show that the Immirzi parameter appears as a constant scaling in all the steps where dynamical information plays a relevant role in the entropy calculation. This fact, together with the kinematical equivalence of the Immirzi ambiguity with a change of scale, is used to preclude the potential non-linearity of the entropy on physical grounds. 
  We show that Einstein's gravity coupled to a non-minimally coupled scalar field is stable even for high values of the scalar field, when the sign of the Einstein-Hilbert action is reversed. We also discuss inflationary solutions and a possible new mechanism of reheating. 
  The gravitational field of a rigidly rotating perfect fluid cylinder with gamma- law equation of state is found analytically. The solution has two parameters and is physically realistic for gamma in the interval (1.41,2]. Closed timelike curves always appear at large distances. 
  We consider the generalized set of theories of gravitation whose Lagrangians contain the term $R^{2}$ : $L=\sqrt{-g}(R+\beta R^{2})$. Inserting the RW metric with an imposed non-singular and inflationary behaviour of the scale factor $a(t)$, and using a arbitrary perfect fluid, we study the properties of $\rho $ and $p$ in this context. By requiring the positivity of the energy density, as well as real and finite velocity of sound, we can obtain the range of values of $\beta $ that ensure the inflationary behaviour and absence of singularity. 
  The method of Lagrangians with covariant derivative (MLCD) is applied to a special type of Lagrangian density depending on scalar and vector fields as well as on their first covariant derivatives. The corresponding Euler-Lagrange's equations and energy-momentum tensors are found on the basis of the covariant Noether's identities. 
  In this paper we review Penrose's Weyl curvature conjecture which states that the concept of gravitational entropy and the Weyl tensor is somehow linked, at least in a cosmological setting. We give a description of a certain entity constructed from the Weyl tensor, from the very early history of our universe until the present day. Inflation is an important mechanism in our early universe for homogenisation and isotropisation, and thus it must cause large effects upon the evolution of the gravitational entropy. Therefore the effects from inflationary fluids and a cosmological constant are studied in detail. 
  We investigate the possible total radiated energy produced by a binary black hole system containing non-vanishing total angular momentum. For the scenearios considered we find that the total radiated energy does not exceed 1%. Additionally we explore the gravitational radiation field and the variation of angular momentum in the process. 
  In the context of the Hamiltonian formulation of the teleparallel equivalent of general relativity we compute the gravitational energy of Kerr and Kerr Anti-de Sitter (Kerr-AdS) space-times. The present calculation is carried out by means of an expression for the energy of the gravitational field that naturally arises from the integral form of the constraint equations of the formalism. In each case, the energy is exactly computed for finite and arbitrary spacelike two-spheres, without any restriction on the metric parameters. In particular, we evaluate the energy at the outer event horizon of the black holes. 
  Space-time--time is a natural hybrid of Kaluza's five-dimensional geometry and Weyl's conformal space-time geometry. Translations along the secondary time dimension produce the electromagnetic gauge transformations of Kaluza--Klein theory and the metric gauge transformations of Weyl theory, related as Weyl postulated. Geometrically, this phenomenon resides in an exponential-expansion producing ``conformality constraint'', which replaces Kaluza's ``cylinder condition''. The curvature tensors exhibit a wealth of ``interactions'' among geometrical entities with physical interpretations. Unique to the conformally constrained geometry is a sectionally isotropic, ultralocally determined ``residual curvature'', useful in construction of an action density for field equations. A space-time--time geodesic describes a test particle whose rest mass m and electric charge q evolve according to definite laws. The particls's motion is governed by four apparent forces: the Einstein gravitational force, the Lorentz electromagnetic force, a force proportional to the electromagnetic potential, and a force proportional to a gradient d(ln phi), where the scalar field phi is essentially the space-time--time residual radius of curvature. The particle appears suddenly at an event E1 with q = -phi(E1) and vanishes at an event E2 with q = phi(E2). At E1 and E2 the phi-force infinitely dominates the others, causing E1 and E2 to tend to occur near where phi has an extreme value; application to the modeling of orbital transitions of atomic electrons suggests itself. The equivalence of a test particle's inertial mass and its passive gravitational mass follows from the gravitational force's proportionality to m. No connection is apparent between m and active gravitational mass or between q and active electric charge, nor does the theory seem to require any. 
  In the article {\it Gen. Rel. Grav.} {\bf 32}, 1633 (2000), by J. G. Pereira and C. M. Zhang, the special relativity energy-momentum tensor was used to discuss the neutrino phase-splitting in a weak gravitational field. However, it would be more appropriate to use the general relativity energy-momentum tensor. When we do that, as we are going to see, some results change, but the basic conclusion remains the same. 
  Gravitational field of a cylindrical Nambu-Goto wall in the vacuum spacetime is considered in order to clarify the interaction between Nambu-Goto membranes and gravitational waves. If one neglects the emission of gravitational waves by the wall motion, the spacetime becomes singular. It is also shown that the emission of gravitational waves does occur by the motion of the cylindrical wall if the initial data is singularity free. The energy loss rate due to radiation of gravitational waves agrees with that estimated from the test wall motion and the quadrupole formula for the gravitational wave emission. This is quite different from the oscillatory behavior of gravitating Nambu-Goto membranes: the presence of gravity induces the wall to lose its dynamical degree of freedom. 
  We write down a quantum gravity equation which generalizes the Wheeler-DeWitt one in view of including a time dependence in the wave functional. The obtained equation provides a consistent canonical quantization of the 3-geometries resulting from a ``gauge-fixing'' (3 + 1)-slicing of the space-time.   Our leading idea relies on a criticism to the possibility that, in a quantum space-time, the notion of a (3 + 1)-slicing formalism (underlying the Wheeler-DeWitt approach) has yet a precise physical meaning. As solution to this problem we propose of adding to the gravity-matter action the so-called {\em kinematical action} (indeed in its reduced form, as implemented in the quantum regime), and then we impose the new quantum constraints.   As consequence of this revised approach, the quantization procedure of the 3-geometries takes place in a fixed reference frame and the wave functional acquires a time evolution along a one-parameter family of spatial hypersurfaces filling the space-time. We show how the states of the new quantum dynamics can be arranged into an Hilbert space, whose associated inner product induces a conserved probability notion for the 3-geometries.   Finally, since the constraints we quantize violate the classical symmetries (i. e. the vanishing nature of the super-Hamiltonian), then a key result is to find a (non-physical) restriction on the initial wave functional phase, ensuring that general relativity outcomes when taking the appropriate classical limit. However we propose a physical interpretation of the kinematical variables which, based on the analogy with the so-called {\em Gaussian reference fluid}, makes allowance even for such classical symmetry violation. 
  We present a new approximate method for constructing gravitational radiation driven inspirals of test-bodies orbiting Kerr black holes. Such orbits can be fully described by a semi-latus rectum $p$, an eccentricity $e$, and an inclination angle $\iota$; or, by an energy $E$, an angular momentum component $L_z$, and a third constant $Q$. Our scheme uses expressions that are exact (within an adiabatic approximation) for the rates of change ($\dot{p}$, $\dot{e}$, $\dot{\iota}$) as linear combinations of the fluxes ($\dot{E}$, $\dot{L_z}$, $\dot{Q}$), but uses quadrupole-order formulae for these fluxes. This scheme thus encodes the exact orbital dynamics, augmenting it with approximate radiation reaction. Comparing inspiral trajectories, we find that this approximation agrees well with numerical results for the special cases of eccentric equatorial and circular inclined orbits, far more accurate than corresponding weak-field formulae for ($\dot{p}$, $\dot{e}$, $\dot{\iota}$). We use this technique to study the inspiral of a test-body in inclined, eccentric Kerr orbits. Our results should be useful tools for constructing approximate waveforms that can be used to study data analysis problems for the future LISA gravitational-wave observatory, in lieu of waveforms from more rigorous techniques that are currently under development. 
  We extend our approach for the exact solution of the Riemann problem in relativistic hydrodynamics to the case in which the fluid velocity has components tangential to the initial discontinuity. As in one-dimensional flows, we here show that the wave-pattern produced in a multidimensional relativistic Riemann problem can be predicted entirely by examining the initial conditions. Our method is logically very simple and allows for a numerical implementation of an exact Riemann solver which is both straightforward and computationally efficient. The simplicity of the approach is also important for revealing special relativistic effects responsible for a smooth transition from one wave-pattern to another when the tangential velocities in the initial states are suitably varied. While the content of this paper is focussed on a flat spacetime, the local Lorentz invariance allows its use also in fully general relativistic calculations. 
  We define a class of condensed matter theories in a Newtonian framework with a Lagrange formalism so that a variant of Noether's theorem gives the classical conservation laws:   \partial_t \rho + \partial_i (\rho v^i) &= &0 \partial_t (\rho v^j) + \partial_i(\rho v^i v^j + p^{ij}) &= &0.   We show that for the metric $g_{\mu\nu}$ defined by \hat{g}^{00} = g^{00} \sqrt{-g} &= &\rho \hat{g}^{i0} = g^{i0} \sqrt{-g} &= &\rho v^i \hat{g}^{ij} = g^{ij} \sqrt{-g} &= &\rho v^i v^j + p^{ij} these theories are equivalent to a metric theory of gravity with Lagrangian L = L_{GR} + L_{matter}(g_{\mu\nu},\phi^m) - (8\pi G)^{-1} (\Upsilon g^{00}-\Xi \delta_{ij}g^{ij})\sqrt{-g}. with covariant $L_{matter}$, which defines a generalization of the Lorentz ether to gravity. Thus, the Einstein equivalence may be derived from simple condensed matter axioms. The Einstein equations appear in a natural limit $\Xi,\Upsilon\to 0$. 
  We review the recent developments in superstrings. We start with a brief summary of various consistent superstring theories and discuss T-duality which necessarily leads to the presence of D-branes. The properties of D-branes are summarized and we discuss how these suggest the existence of 11-dimensional quantum theory, M-theory, which is believed to give rise to various superstrings as perturbative expansions around particular backgrounds in the theory. We also discuss the interpretation of brane solutions as black holes in string theories and statistical explanation of Bekenstein-Hawking entropy. The idea behind this interpretation is that there is a fundamental duality between closed (gravity) and open (gauge theory) string degrees of freedom, one of whose manifestation is what is kown as AdS/CFT correspondence. The idea is used to discuss the greybody factors for BTZ black holes. Finally the entropy of various balck holes are discussed in connection with Cardy-Verlinde formula. 
  An example illustrating a continuum spin foam framework is presented. This covariant framework induces the kinematics of canonical loop quantization, and its dynamics is generated by a {\em renormalized} sum over colored polyhedra.   Physically the example corresponds to 3d gravity with cosmological constant. Starting from a kinematical structure that accommodates local degrees of freedom and does not involve the choice of any background structure (e. g. triangulation), the dynamics reduces the field theory to have only global degrees of freedom. The result is {\em projectively} equivalent to the Turaev-Viro model. 
  We present a fully nonlinear calculation of the waveform of the gravitational radiation emitted in the fission of a vacuum white hole. At early times, the waveforms agree with close-approximation perturbative calculations but they reveal dramatic time and angular dependence in the nonlinear regime. The results pave the way for a subsequent computation of the radiation emitted after a binary black hole merger. 
  Talk at the 25th Johns Hopkins Workshop "2001: A Relativistic Spacetime Odyssey", Firenze September 2001. 
  A numerical procedure is described for the maximization of the energy diffusion due to the backscattering of the gravitational radiation. The obtained maxima are solutions dominated by low frequency waves. They give rise to robust gravitational ringing, with amplitudes of the order of the original signal. 
  We study numerically the stability of Morris & Thorne's first traversible wormhole, shown previously by Ellis to be a solution for a massless ghost Klein-Gordon field. Our code uses a dual-null formulation for spherically symmetric space-time integration, and the numerical range covers both universes connected by the wormhole. We observe that the wormhole is unstable against Gaussian pulses in either exotic or normal massless Klein-Gordon fields. The wormhole throat suffers a bifurcation of horizons and either explodes to form an inflationary universe or collapses to a black hole, if the total input energy is respectively negative or positive. As the perturbations become small in total energy, there is evidence for critical solutions with a certain black-hole mass or Hubble constant. The collapse time is related to the initial energy with an apparently universal critical exponent. For normal matter, such as a traveller traversing the wormhole, collapse to a black hole always results. However, carefully balanced additional ghost radiation can maintain the wormhole for a limited time. The black-hole formation from a traversible wormhole confirms the recently proposed duality between them. The inflationary case provides a mechanism for inflating, to macroscopic size, a Planck-sized wormhole formed in space-time foam. 
  Quantum phenomena such as vacuum polarisation in curved spacetime induce interactions between photons and gravity with quite striking consequences, including the violation of the strong equivalence principle and the apparent prediction of `superluminal' photon propagation. These quantum interactions can be encoded in an effective action. In this paper, we extend previous results on the effective action for QED in curved spacetime due to Barvinsky, Vilkovisky and others and present a new, local effective action valid to all orders in a derivative expansion, as required for a full analysis of the quantum theory of high-frequency photon propagation in gravitational fields. 
  We compute the gravitational self-force (or ``radiation reaction'' force) acting on a particle falling radially into a Schwarzschild black hole. Our calculation is based on the ``mode-sum'' method, in which one first calculates the individual $\ell$-multipole contributions to the self-force (by numerically integrating the decoupled perturbation equations) and then regularizes the sum over modes by applying a certain analytic procedure. We demonstrate the equivalence of this method with the $\zeta-$function scheme. The convergence rate of the mode-sum series is considerably improved here (thus notably reducing computational requirements) by employing an analytic approximation at large $\ell$. 
  Maximally dissipative boundary conditions are applied to the initial-boundary value problem for Einstein's equations in harmonic coordinates to show that it is well-posed for homogeneous boundary data and for boundary data that is small in a linearized sense. The method is implemented as a nonlinear evolution code which satisfies convergence tests in the nonlinear regime and is robustly stable in the weak field regime. A linearized version has been stably matched to a characteristic code to compute the gravitational waveform radiated to infinity. 
  Gamma-Ray Bursts (GRBs) - short bursts of 100-1MeV photons arriving from random directions in the sky are probably the most relativistic objects discovered so far. Still, somehow they did not attract the attention of the relativistic community. In this short review I discuss briefly GRB observations and show that they lead us to the fireball model - GRBs involve macroscopic relativistic motion with Lorentz factors of a few hundred or more. I show that GRB sources involve, most likely, new born black holes, and their progenitors are Supernovae or neutron star mergers. I show that both GRB progenitors and the process of GRB itself produce gravitational radiation and I consider the possibility of detecting this emission. Finally I show that GRBs could serve as cosmological indicators that could teach us about the high redshift ($z \approx 5-15$) dark ages of the universe. 
  In the present paper, we briefly review recent studies of second-order gravitational perturbations in braneworld models. After we consider the possibility of pathological behavior of gravity at higher orders of perturbation, second-order perturbations are discussed in Randall-Sundrum braneworld models. Because the mass spectra in these braneworld models are different, we analyze them using different approaches. In the respective models, it is discussed that 4D Einstein gravity is approximately recovered at level of second-order perturbation, although there are some exceptional cases in the two-brane model. 
  Vacuum gravitational fields invariant for a non Abelian Lie algebra generated by two Killing fields whose commutator is light-like are analyzed. It is shown that they represent nonlinear gravitational waves obeying to two nonlinear superposition laws. The energy and the polarization of this family of waves are explicitely evaluated. 
  We numerically reanalyze static and spherically symmetric black hole solutions in an Einstein-Maxwell-dilaton system with a dilaton potential $m_{d}^{2}\phi^{2}$. We investigate thermodynamic properties for various dilaton coupling constants and find that thermodynamic properties change at a critical dilaton mass $m_{d,crit}$. For $m_{d}\geq m_{d,crit}$, the black hole becomes an extreme solution for a nonzero horizon radius $r_{h,ex}$ as the Reissner-Nordstr\"om black hole. However, if $m_{d}$ is nearly equal to $m_{d,crit}$, there appears a solution of smaller horizon radius than $r_{h,ex}$. For $m_{d}<m_{d,crit}$, a solution continues to exist until the horizon approaches zero. The Hawking temperature in the zero horizon limit resembles that of a massless dilaton black hole for arbitrary dilaton coupling constant. 
  We give an explicit construction of a positive-definite invariant inner-product for the Klein-Gordon fields, thus solving the old problem of the probability interpretation of Klein-Gordon fields without having to restrict to the subspaces of the positive-frequency solutions. Our method has a much wider domain of application and may be used to obtain the most general invariant inner-product on the solution space of a broad class of Klein-Gordon type evolution equations. We explore its consequences for the solutions of the Wheeler-DeWitt equation associated with the FRW-massive-real-scalar-field models. 
  The original Casimir effect results from the difference in the vacuum energies of the electromagnetic field, between that in a region of space with boundary conditions and that in the same region without boundary conditions. In this paper we develop the theory of a similar situation, involving a scalar field in spacetimes with compact spatial sections of negative spatial curvature. 
  Here we derive the mass formulae for a cylindrical black hole solution with positive cosmological constant and surrounded by dust. The expressions are generalising those found by Smarr for the mass and momentum of a Kerr black hole. 
  In this second part of our series of articles on alternative cosmological models we investigate the observational consequences for the new Weyl-Cartan model proposed earlier. We review the derivation of the magnitude-redshift relation within the standard FLRW model and characterize its dependence on the underlying cosmological model. With this knowledge at hand we derive the magnitude-redshift relation within our new Weyl-Cartan model. We search for the best-fit parameters by using the combined data set of 92 SNe of type Ia as compiled by Wang, which is based on recent supernova data of Perlmutter et al. and Riess et al. Additionally, we compare our best-fit parameters with the results of several other groups which performed similar analysis within the standard cosmological model as well as in non-standard models. 
  It has been shown that inclusion of higher order curvature invariant terms in the Robertson-Walker minisuperspace model of the Einstein-Hilbert action leads to Schrodinger like equation, whose corresponding effective action is hermitian. Thus, it is possible to write the continuity equation in a straight forward manner, which reveals a quantum mechanical probability interpretation of the theory. 
  The question is examined of a mirror which starts from rest and either (i) accelerates for some time and eventually reverts to motion at constant velocity, or (ii) continues accelerating forever. A sharp distinction is made between cases (i) and (ii) concerning the spectrum of the emitted radiation, and the qualitative difference between the two cases is pointed out. The Bogolubov coefficients are calculated for a trajectory of type (i). A type (ii) trajectory is entirely unphysical as far as any realistic mirror is concerned, however it is of interest in that it has been used as a simple analog of black hole collapse. The spectrum emitted for the type (ii) trajectory z=-ln(cosht) is examined and it is shown that it is indeed that of a black body. Inconsistencies in previous derivations of the above result are pointed out. 
  Spherically symmetric solutions with a flux of electric and magnetic fields in Kaluza-Klein gravity are considered. It is shown that under the condition (electric charge $q$) $\approx$ (magnetic charge $Q$) ($q>Q$) these solutions are like a flux tube stretched between two Universes. The longitudinal size of this tube depends on the value of $\delta = 1 - Q/q$. The cross section of the tube can be chosen $\approx l_{Pl}$. In this case this flux tube looks like a 1-dimensional object (thread) stretched between two Universes. The propagation of gravitational waves on the thread is considered. The corresponding equations are very close to the classical string equations. This result allows us to say that the thread between two Universes is similar to the string attached to two $D-$branes. 
  In a recent paper we have suggested that a formulation of quantum mechanics should exist, which does not require the concept of time, and that the appropriate mathematical language for such a formulation is noncommutative differential geometry. In the present paper we discuss this formulation for the free point particle, by introducing a commutation relation for a set of noncommuting coordinates. The sought for background independent quantum mechanics is derived from this commutation relation for the coordinates. We propose that the basic equations are invariant under automorphisms which map one set of coordinates to another- this is a natural generalization of diffeomorphism invariance when one makes a transition to noncommutative geometry. The background independent description becomes equivalent to standard quantum mechanics if a spacetime manifold exists, because of the proposed automorphism invariance. The suggested basic equations also give a quantum gravitational description of the free particle. 
  This letter deals with an analysis of the space-time static metric that corresponds to a quintessential state equation with constant characteristic parameter. Following a procedure parallel to as it is used in the case of de Sitter space, we have tried to generalize the metric components that correspond to the quintessential case to also embrace black-hole terms and shown that this is not possible. We argue therefore that, in the absence of a cosmological constant, black holes seem to be prevented in a cosmological space-time which is asymptotically accelerating when the acceleration is driven by a quintessence-like field. 
  Inflationary models are generally credited with explaining the large scale homogeneity, isotropy, and flatness of our universe as well as accounting for the origin of structure (i.e., the deviations from exact homogeneity) in our universe. We argue that the explanations provided by inflation for the homogeneity, isotropy, and flatness of our universe are not satisfactory, and that a proper explanation of these features will require a much deeper understanding of the initial state of our universe. On the other hand, inflationary models are spectacularly successful in providing an explanation of the deviations from homogeneity. We point out here that the fundamental mechanism responsible for providing deviations from homogeneity -- namely, the evolutionary behavior of quantum modes with wavelength larger than the Hubble radius -- will operate whether or not inflation itself occurs. However, if inflation did not occur, one must directly confront the issue of the initial state of modes whose wavelength was larger than the Hubble radius at the time at which they were "born". Under some simple hypotheses concerning the "birth time" and initial state of these modes (but without any "fine tuning"), it is shown that non-inflationary fluid models in the extremely early universe would result in the same density perturbation spectrum and amplitude as inflationary models, although there would be no "slow roll" enhancement of the scalar modes. 
  Analysis of the radio tracking data from the Pioneer 10/11 spacecraft has consistently indicated the presence of an anomalous small Doppler frequency drift. The drift can be interpreted as being due to a constant acceleration of a_P= (8.74 +/- 1.33) x 10^{-8} cm/s^2 directed towards the Sun. Although it is suspected that there is a systematic origin to the effect, none has been found. The nature of this anomaly has become of growing interest in the fields of relativistic cosmology, astro- and gravitational physics as well as in the areas of spacecraft design and high-precision navigation. We present a concept for a designated deep-space mission to test the discovered anomaly. A number of critical requirements and design considerations for such a mission are outlined and addressed. 
  Generalizing previous work we propose how to superpose spinning black holes in a Kerr-Schild initial slice. This superposition satisfies several physically meaningful limits, including the close and the far ones. Further we consider the close limit of two black holes with opposite angular momenta and explicitly solve the constraint equations in this case. Evolving the resulting initial data with a linear code, we compute the radiated energy as a function of the masses and the angular momenta of the black holes. 
  This paper analyses the relativistic stellar aberration requirements for the Space Interferometry Mission (SIM). We address the issue of general relativistic deflection of light by the massive self-gravitating bodies. Specifically, we present estimates for corresponding deflection angles due to the monopole components of the gravitational fields of a large number of celestial bodies in the solar system. We study the possibility of deriving an additional navigational constraints from the need to correct for the gravitational bending of light that is traversing the solar system. It turns out that positions of the outer planets presently may not have a sufficient accuracy for the precision astrometry. However, SIM may significantly improve those simply as a by-product of its astrometric program. We also consider influence of the higher gravitational multipoles, notably the quadrupole and the octupole ones, on the gravitational bending of light. Thus, one will have to model and account for their influence while observing the sources of interest in the close proximity of some of the outer planets, notably the Jupiter and the Saturn. Results presented here are different from the ones obtained elsewhere by the fact that we specifically account for the differential nature of the future SIM astrometric campaign (e.g. observations will be made over the instrument's field of regard with the size of 15$^\circ$). This, in particular, lets us to obtain a more realistic estimate for the accuracy of determination of the parameterized post-Newtonian (PPN) parameter $\gamma$. Thus, based on a very conservative assumptions, we conclude that accuracy of $\sigma_\gamma \sim 10^{-5}$ is achievable in the experiments conducted in the solar gravity field. 
  We address the issue of relativistic stellar aberration requirements for the Space Interferometry Mission (SIM). Motivated by the importance of this issue for SIM, we have considered a problem of relative astrometric observations of two stars separated by angle $\theta$ on the sky with a single baseline interferometer. While a definitive answer on the stellar aberration issue may be obtained only in numerical simulations based on the accurate astrometric model of the instrument, one could still derive realistic conclusions by accounting for the main expected properties of SIM. In particular, we have analysied how the expected astrometric accuracy of determination of positions, parallaxes and proper motions will constrain the accuracy of the spaceraft navigation. We estimated the astrometric errors introduced by imperfect metrology (variations of the calibration term across the tile of interest), errors in the baseline length estimations, and those due to orbital motion of the spacecraft. We also estimate requirements on the data sampling rate necessary to apply on-board in order to correct for the stellar aberration. We have shown that the worst case observation scenario is realized for the motion of the spacecraft in the direction perpendicular to the tile. This case of motion will provide the most stringent requirement on the accuracy of knowledge of the velocity's magnitude. We discuss the implication of the results obtained for the future mission analysis. 
  This paper is a continuation of our previous analysis (i.e. Turyshev 2002a, 2002b) of the relativistic stellar aberration requirements for the Space Interferometry Mission (SIM). Here we have considered a problem of how the expected astrometric accuracy of parallax determination will constrain the accuracy of the spacecraft navigation. We show that effect of the spacecraft's navigational errors on the accuracy of parallax determination with SIM will be negligible. We discuss the implication of the results obtained for the future mission analysis. 
  We discuss an equivalence between the Baumgarte-Shapiro-Shibata-Nakamura (BSSN) formulation of the Einstein evolution equations, a subfamiliy of the Kidder--Scheel--Teukolsky formulation, and other strongly or symmetric hyperbolic first order systems with fixed shift and densitized lapse. This allows us to show under which conditions the BSSN system is, in a sense to be discussed, hyperbolic. This desirable property may account in part for the empirically observed better behavior of the BSSN formulation in numerical evolutions involving black holes. 
  We study the behavior in the remote past and future of solutions of an equation of motion for charged particles proposed by F. Rohrlich, for the special case in which the motion is in one spatial dimension.   We show that if an external force is applied for a finite time, some solutions exhibit the property of ``preacceleration'', meaning that the particle accelerates before the force is applied, but that there do exist solutions without preacceleration. However, most solutions without preacceleration exhibit ``postacceleration'' into the infinite future (i.e., the particle accelerates after the force is removed). Some may consider such behavior as sufficiently "unphysical" to rule out the equation.   More encouragingly, we show that analogs of the unphysical ``runaway'' solutions of the Lorentz-Dirac equation do not occur for solutions of Rohrlich's equation. We show that when the external force eventually vanishes, the proper acceleration vanishes asymptotically in the future, and the coordinate velocity becomes asymptotically constant. 
  The tension, if not outright inconsistency, between quantum physics and general relativity is one of the great problems facing physics at the turn of the millennium. Most often, the problems arising in merging Einstein gravity and quantum physics are viewed as Planck scale issues (10^{19} GeV, 10^{-34} m, 10^{-45} s), and so safely beyond the reach of experiment. However, over the last few years it has become increasingly obvious that the difficulties are more widespread: There are already serious problems of deep and fundamental principle at the semi-classical level, and worse, certain classical systems (inspired by quantum physics, but in no sense quantum themselves) exhibit seriously pathological behaviour. One manifestation of these pathologies is in the so-called ``energy conditions'' of general relativity. Patching things up in the gravity sector opens gaping holes elsewhere; and some ``fixes'' are more radical than the problems they are supposed to cure. 
  Motivated by various theoretical arguments that the Planck energy (Ep - 10^19 GeV) - should herald departures from Lorentz invariance, and the possibility of testing these expectations in the not too distant future, two so-called "Doubly Special Relativity" theories have been suggested -- the first by Amelino-Camelia (DSR1) and the second by Smolin and Magueijo (DSR2). These theories contain two fundamental scales -- the speed of light and an energy usually taken to be Ep. The symmetry group is still the Lorentz group, but in both cases acting nonlinearly on the energy-momentum sector. Accordingly, since energy and momentum are no longer additive quantities, finding their values for composite systems (and hence finding the correct conservation laws) is a nontrivial matter. Ultimately it is these possible deviations from simple linearly realized relativistic kinematics that provide the most promising observational signal for empirically testing these models. Various investigations have narrowed the conservation laws down to two possibilities per DSR theory. We derive unique exact results for the energy-momentum of composite systems in both DSR1 and DSR2, and indicate the general strategy for arbitrary nonlinear realizations of the Lorentz group. 
  Several years ago the so-called quantum geometrodynamics in extended phase space was proposed. The main role in this version of quantum geometrodynamics is given to a wave function that carries information about geometry of the Universe as well as about a reference frame in which this geometry is studied. We consider the evolution of a physical object (the Universe) in ``physical'' subspace of extended configurational space, the latter including gauge and ghost degrees of freedom. A measure of the ``physical'' subspace depends on a chosen reference frame, in particular, a small variation of a gauge-fixing function results in changing the measure. Thus, a transition to another gauge condition (another reference frame) leads to non-unitary transformation of a physical part of the wave function. From the viewpoint of the evolution of the Universe in the ``physical'' subspace a transition to another reference frame is an irreversible process that may be important when spacetime manifold has a nontrivial topology. 
  A pair of simple wave equations is presented for the symmetric gravitational and electromagnetic perturbations of a charged black hole. One of the equations is uncoupled, and the other has a source term given by the solution of the first equation. The derivation is presented in full detail for either axially symmetric or stationary perturbations, and is quite straightforward. This result is expected to have important applications in astrophysical models. 
  A scenario where inflation emerges as a response to protect the holographic principle is described. A two fluid model in a closed universe inflation picture is assumed, and a possible explanation for secondary exponential expansion phases as those currently observed is given. 
  We study a classical reparametrization-invariant system, in which ``time'' is not a priori defined. It consists of a nonrelativistic particle moving in five dimensions, two of which are compactified to form a torus. There, assuming a suitable potential, the internal motion is ergodic or more strongly irregular. We consider quasi-local observables which measure the system's ``change'' in a coarse-grained way. Based on this, we construct a statistical timelike parameter, particularly with the help of maximum entropy method and Fisher-Rao information metric. The emergent reparametrization-invariant ``time'' does not run smoothly but is simply related to the proper time on the average. For sufficiently low energy, the external motion is then described by a unitary quantum mechanical evolution in accordance with the Schr\"odinger equation. 
  We unearth spacetime structure of massive vector bosons, gravitinos, and gravitons. While the curvatures associated with these particles carry a definite spin, the underlying potentials cannot be, and should not be, interpreted as single spin objects. For instance, we predict that a spin measurement in the rest frame of a massive gravitino will yield the result 3/2 with probability one half, and 1/2 with probability one half. The simplest scenario leaves the Riemannian curvature unaltered; thus avoiding conflicts with classical tests of the theory of general relativity. However, the quantum structure acquires additional contributions to the propagators, and it gives rise to additional phases. 
  We study the stability properties of the Kidder-Scheel-Teukolsky (KST) many-parameter formulation of Einstein's equations for weak gravitational waves on flat space-time from a continuum and numerical point of view. At the continuum, performing a linearized analysis of the equations around flat spacetime, it turns out that they have, essentially, no non-principal terms. As a consequence, in the weak field limit the stability properties of this formulation depend only on the level of hyperbolicity of the system. At the discrete level we present some simple one-dimensional simulations using the KST family. The goal is to analyze the type of instabilities that appear as one changes parameter values in the formulation. Lessons learnt in this analysis can be applied in other formulations with similar properties. 
  In this paper we show that the claims in [Class. Quantum Grav. 19 (2002) 3067, gr-qc/0203081] related to our analysis in [Phys. Rev. D 62, 063508 (2000), astro-ph/0005070] are wrong. 
  Einstein believed that Mach's principle should play a major role in finding a meaningful spacetime geometry, though it was discovered later that his field equations gave some solutions which were not Machian. It is shown, in this essay, that the kinematical $\Lambda$ models, which are invoked to solve the cosmological constant problem, are in fact consistent with Mach's ideas. One particular model in this category is described which results from the microstructure of spacetime and seems to explain the current observations successfully and also has some benefits over the conventional models. This forces one to think whether the Mach's ideas and the cosmological constant are interrelated in some way. 
  In this work we study the particle production in time dependent periodic potential using the method of complex time WKB (CWKB) approximation. In the inflationary cosmology at the end of inflationary stage, the potential becomes time dependent as well as periodic. Reheating occurs due to particle production by the oscillating inflaton field. Using CWKB we obtain almost identical results on catastrophic particle production as obtained by others. 
  In this paper we investigate the phenomenon of particle production of massles scalar field, in a model of spacetime where the chronology horizon could be formrd, using the method of complex time WKB approximation (CWKB). For the purpose, we take two examples in a model of spacetime, one already discussed by Sushkov, to show that the mode of particle production near chronology horizon possesses the similar characteristic features as are found while discussing particle production in time dependent curved background. We get identical results as that obtained by Sushkov in this direction. We find, in both the examples studied, that the total number of particles remain finite at the moment of the formation of the chronology horizon. 
  We discuss the decay of accelerated protons and illustrate how the Fulling-Davies-Unruh effect is indeed mandatory to maintain the consistency of standard Quantum Field Theory. The confidence level of the Fulling-Davies-Unruh effect must be the same as that of Quantum Field Theory itself. 
  The foundations are laid for the numerical computation of the actual worldline for a particle orbiting a black hole and emitting gravitational waves. The essential practicalities of this computation are here illustrated for a scalar particle of infinitesimal size and small but finite scalar charge. This particle deviates from a geodesic because it interacts with its own retarded field $\psi^\ret$. A recently introduced Green's function $G^\SS$ precisely determines the singular part, $\psi^\SS$, of the retarded field. This part exerts no force on the particle. The remainder of the field $\psi^\R = \psi^\ret - \psi^\SS$ is a vacuum solution of the field equation and is entirely responsible for the self-force. A particular, locally inertial coordinate system is used to determine an expansion of $\psi^\SS$ in the vicinity of the particle. For a particle in a circular orbit in the Schwarzschild geometry, the mode-sum decomposition of the difference between $\psi^\ret$ and the dominant terms in the expansion of $\psi^\SS$ provide a mode-sum decomposition of an approximation for $\psi^\R$ from which the self-force is obtained. When more terms are included in the expansion, the approximation for $\psi^\R$ is increasingly differentiable, and the mode-sum for the self-force converges more rapidly. 
  It is shown that within conformally flat stationary axisymmetric spacetimes, besides of the static family, there exists a new class of metrics, which is always stationary and axisymmetric. All these spacetimes, the static and the stationary ones, are endowed with an arbitrary function depending on the two non--Killingian coordinates. The explicit form of this function can be determined once the coupled matter, i.e., the energy--momentum tensor is given. One might hope possible extensions of this result to black holes on two--branes in four dimensions. 
  The Lense-Thirring effect is currently being measured by means of a combination of the orbital residuals of the nodes of LAGEOS and LAGEOS II and the perigee of LAGEOS II. The total error is of the order of 20%. The most insidious systematic error is due to the mismodelled even zonal harmonics of the geopotential and amounts to 12.9%, according to EGM96 model. The role and the importance of the LAGEOS--LAGEOS II Lense--Thirring experiment is investigated. Using other suitable combinations with orbital elements of the other existing laser-ranged satellites does not yield significative improvements except for one combination including the nodes of LAGEOS, LAGEOS II and Ajisai and the perigee of LAGEOS II. The related systematic error due to the mismodelled even zonal part of the geopotential reduces to almost 10.7%. 
  We propose a method of determining solutions to the constraint equations of General Relativity approximately describing binary black holes in quasi-stationary circular orbits. Black holes with arbitrary linear momenta are constructed in the manner suggested by Brandt and Brugmann. The quasi-stationary circular orbits are determined by local minima in the ADM mass in a manner similar to Baumgarte and Cook; however, rather than fixing the area of the apparent horizon, we fix the value of the bare masses of the holes. We numerically generate an evolutionary sequence of quasi-stationary circular orbits up to and including the innermost stable circular orbit. We compare our results with post-Newtonian expectations as well as the results of Cook and Baumgarte. We also generate additional numerical results describing the dynamics of the geometry due to the emission of gravitational radiation. 
  I describe the conformal method for constructing solutions of the hyperboloidal constraint equations as well as the conditions needed on the free data in order to have regularity up to boundary for the solutions to the constraint equations. A brief discussion of the Einstein evolution equations in the unphysical setting is given. 
  The identification of a cosmic scale function with the volume integral of a spacelike hypersurface defines the cosmic evolution in General Relativity as a collective motion along a geodesic in the field space of the metric components, considered as the coset of the affine group over the Lorentz one.  The Friedmann equations are derived out of the homogeneous approximation by the Gibbs averaging exact equations over the relative constant spatial volume.  A direct correspondence between the collective cosmic motion and Special  Relativity is established, to solve the problem of time and energy by analogy with the solution of this problem for a relativistic particle by Poincare and Einstein.  A geometrical time interval is introduced into quantum theory of the relativistic collective motion by the canonical Levi-Civita -- type transformation in agreement with the correspondence principle with quantum field theory. In this context the problem of quantum cosmological creation of visible matter is formulated.  We show that latest observational data can testify to the relative measurement standard, and the cosmic evolution as an inertial motion along geodesic in the field space. 
  Micron-sized black holes do not necessarily have a constant horizon temperature distribution. The black hole remote-sensing problem means to find out the `surface' temperature distribution of a small black hole from the spectral measurement of its (Hawking) grey pulse. This problem has been previously considered by Rosu, who used Chen's modified Moebius inverse transform. Here, we hint on a Ramanujan generalization of Chen's modified Moebius inverse transform that may be considered as a special wavelet processing of the remote-sensed grey signal coming from a black hole or any other distant grey source 
  We present new many-parameter families of strongly and symmetric hyperbolic formulations of Einstein's equations that include quite general algebraic and live gauge conditions for the lapse. The first system that we present has 30 variables and incorporates an algebraic relationship between the lapse and the determinant of the three metric that generalizes the densitized lapse prescription. The second system has 34 variables and uses a family of live gauges that generalizes the Bona-Masso slicing conditions. These systems have free parameters even after imposing hyperbolicity and are expected to be useful in 3D numerical evolutions. We discuss under what conditions there are no superluminal characteristic speeds. 
  Both from gravitational (G) experiments and from a new theoretical approach based on a particle model it is proved that the classical invariability of the bodies, after a change of relative rest-position with respect to other bodies, it is not true. The same holds for the traditional hypotheses based on the classical one. The new relationships are strictly linear. From them it is proved that a universe expansion must be associated with a G expansion of every particle in it, in just the same proportion. It does not change the relative distances, indefinitely. From the relative viewpoint, globally, the universe must be rather static. According to the new cosmic scenario, galaxies must be evolving, indefinitely, in rather closed cycles between luminous and black states. The new kind of linear black hole must absorb radiation until it can explode after releasing new H gas that would trigger new luminous period of star clusters and galaxies. Statistically, most of the galaxies must be in cool states. The last ones should account for all of them, the higher velocities of the galaxies in clusters, the radiation coming from intergalactic space, including the low temperature black-body background observed in the CMBR. 
  We extend Dirac's approach about the quantization of the electric charge to the case of gravitational configurations. The spacetime curvature is used to define a phase-like object which allows us to extract information about the behavior of the corresponding spacetime. We show that all spacetimes that satisfy certain simple symmetry condition and for which the Petrov type is the same whitin a specific region, quantization conditions can be derived that impose constraints on the possible values of the parameters entering the respective metrics. As a general result we obtain that for the gravitational configurations described by those metrics, the behavior under rotations can be only of bosonic or fermionic nature. 
  We explore the possibility that the reported time variation of the fine structure constant $\alpha$ is due to a coupling between electromagnetism and gravitation. This may reconcile the claimed results on $\alpha$ with the upper limit from the Oklo naturel Uranium fission reactor. 
  I provide a general proof of the conjecture that one can attribute an entropy to the area of {\it any} horizon. This is done by constructing a canonical ensemble of a subclass of spacetimes with a fixed value for the temperature $T=\beta^{-1}$ and evaluating the {\it exact} partition function $Z(\beta)$. For spherically symmetric spacetimes with a horizon at $r=a$, the partition function has the generic form $Z\propto \exp[S-\beta E]$, where $S= (1/4) 4\pi a^2$ and $|E|=(a/2)$. Both $S$ and $E$ are determined entirely by the properties of the metric near the horizon. This analysis reproduces the conventional result for the black-hole spacetimes and provides a simple and consistent interpretation of entropy and energy for De Sitter spacetime. For the Rindler spacetime the entropy per unit transverse area turns out to be $(1/4)$ while the energy is zero. Further, I show that the relationship between entropy and area allows one to construct the action for the gravitational field on the bulk and thus the full theory. In this sense, gravity is intrinsically holographic. 
  The field equations derived from the low energy string effective action with a matter tensor describing a perfect fluid with a barotropic equation of state are solved iteratively using the long-wavelength approximation, i.e. the field equations are expanded by the number of spatial gradients. In the zero order, a quasi-isotropic solution is presented and compared with the general solution of the pure dilaton gravity. Possible cosmological models are analyzed from the point of view of the pre-big bang scenario. The second order solutions are found and their growing and decaying parts are studied. 
  The Einstein-conformally coupled scalar field system is studied in the presence of a cosmological constant. We consider a massless or massive scalar field with no additional self-interaction, and spherically symmetric black hole geometries. When the cosmological constant is positive, no scalar hair can exist and the only solution is the Schwarzschild-de Sitter black hole. When the cosmological constant is negative, stable scalar field hair exists provided the mass of the scalar field is not too large. 
  This paper withdrawn -- it has been revised and merged with gr-qc/0205067 (Conservation laws in "Doubly Special Relativities"; Simon Judes and Matt Visser) 
  For the non-rotating BTZ black hole, the distributional curvature tensor field is found. It is shown to have singular parts proportional to a $\delta$-distribution with support at the origin. This singularity is related, through Einstein field equations, to a point source. Coordinate invariance and independence on the choice of differentiable structure of the results are addressed. 
  This paper describes the Fortran 77 code SIMU, version 1.1, designed for numerical simulations of observational relations along the past null geodesic in the Lemaitre-Tolman-Bondi (LTB) spacetime. SIMU aims at finding scale invariant solutions of the average density, but due to its full modularity it can be easily adapted to any application which requires LTB's null geodesic solutions. In version 1.1 the numerical output can be read by the GNUPLOT plotting package to produce a fully graphical output, although other plotting routines can be easily adapted. Details of the code's subroutines are discussed, and an example of its output is shown. 
  In a Friedmann-Robertson-Walker (FRW) cosmological model with zero spatial curvature, we consider the interaction of the gravitational waves with the plasma in the presence of a weak magnetic field. Using the relativistic hydromagnetic equations it is verified that large amplitude magnetosonic waves are excited, assuming that both, the gravitational field and the weak magnetic field do not break the homogeneity and isotropy of the considered FRW spacetime. 
  We generalize the quantum spinor wave equation for photon into the curved space-time and discuss the solutions of this equation in Robertson-Walker space-time and compare them with the solution of the Maxwell equations in the same space-time. 
  In this paper we discuss the consequences of a Killing symmetry on the local geometrical structure of four-dimensional spacetimes. We have adopted the point of view introduced in recent works where the exterior derivative of the Killing plays a fundamental role. Then, we study some issues related with this approach and clarify why in many circumstances its use has advantages with respect to other approaches. We also extend the formalism developed in the case of vacuum spacetimes to the general case of an arbitrary energy-momentum content. Finally, we illustrate our framework with the case of spacetimes with a gravitating electromagnetic field. 
  It is demonstrated that gravity waves of a flowing fluid in a shallow basin can be used to simulate phenomena around black holes in the laboratory. Since the speed of the gravity waves as well as their high-wavenumber dispersion (subluminal vs. superluminal) can be adjusted easily by varying the height of the fluid (and its surface tension) this scenario has certain advantages over the sonic and dielectric black hole analogs, for example, although its use in testing quantum effects is dubious. It can be used to investigate the various classical instabilities associated with black (and white) holes experimentally, including positive and negative norm mode mixing at horizons. PACS: 04.70.-s, 47.90.+a, 92.60.Dj, 04.80.-y. 
  This is a written version of the review talk given at the meeting on "Interface of Gravitational and Quantum Realms" at IUCAA, Pune during December 2001. The talk reviewed the recent work of Martin Bojowald on Loop Quantum Cosmology. 
  A family of black-hole solutions in the model with 1-component perfect fluid is obtained. The metric of any solution contains (n -1) Ricci-flat "internal space" metrics and for certain equations of state coincides with the metric of black brane (or black hole) solution in the model with antisymmetric form. Certain examples (e.g. imitating M2 and M5 black branes) are considered. The post-Newtonian parameters beta and gamma corresponding to the 4-dimensional section of the metric are calculated. 
  We perform some numerical study of the secular triaxial instability of rigidly rotating homogeneous fluid bodies in general relativity. In the Newtonian limit, this instability arises at the bifurcation point between the Maclaurin and Jacobi sequences. It can be driven in astrophysical systems by viscous dissipation. We locate the onset of instability along several constant baryon mass sequences of uniformly rotating axisymmetric bodies for compaction parameter $M/R = 0-0.275$. We find that general relativity weakens the Jacobi like bar mode instability, but the stabilizing effect is not very strong. According to our analysis the critical value of the ratio of the kinetic energy to the absolute value of the gravitational potential energy $(T/|W|)_{\rm crit}$ for compaction parameter as high as 0.275 is only 30% higher than the Newtonian value. The critical value of the eccentricity depends very weakly on the degree of relativity and for $M/R=0.275$ is only 2% larger than the Newtonian value at the onset for the secular bar mode instability. We compare our numerical results with recent analytical investigations based on the post-Newtonian expansion. 
  An heuristic semiclassical procedure that incorporates quantum gravity induced corrections in the description of photons and spin 1/2 fermions is reviewed. Such modifications are calculated in the framework of loop quantum gravity and they arise from the granular structure of space at short distances. The resulting effective theories are described by power counting nonrenormalizable actions which exhibit Lorentz violations at Planck length scale. The modified Maxwell and Dirac equations lead to corrections of the energy momentum relations for the corresponding particle at such scale. An action for the relativistic point particle exhibiting such modified dispersion relations is constructed and the first steps towards the study of a consistent coupling between these effective theories are presented. 
  I propose a new mechanism to account for the observed tiny but finite dark energy in terms of a non-Abelian Higgs theory, which has infinitely many perturbative vacua characterized by a winding number, in the framework of inflationary cosmology. Inflation homogenizes field configuration and practically realizes a perturbative vacuum with vanishing winding number, which is expressed by a superposition of eigenstates of the Hamiltonian with different vacuum energy density. As a result, we naturally find a nonvanishing vacuum energy density with fairly large probability, under the assumption that the cosmological constant vanishes in some vacuum state. Since the predicted magnitude of dark energy is exponentially suppressed by the instanton action, we can fit observation without introducing any tiny parameters. 
  The motivations for investigating a theory of gravitation based on a concept of "ether" are discussed-- a crucial point is the existence of an alternative interpretation of special relativity, named the Lorentz-Poincar\'e ether theory. The basic equations of one such theory of gravity, based on just one scalar field, are presented. To check this theory in celestial mechanics, an "asymptotic" scheme of post-Newtonian (PN) approximation is summarized and its difference with the standard PN scheme is emphasized. The derivation of PN equations of motion for the mass centers, based on the asymptotic scheme, is outlined. They are implemented for the major bodies of the solar system and the prediction for Mercury is compared with an ephemeris based on general relativity. 
  We argue that, when the gravity effect is included, the generalized uncertainty principle (GUP) may prevent black holes from total evaporation in a similar way that the standard uncertainty principle prevents the hydrogen atom from total collapse. Specifically we invoke the GUP to obtain a modified Hawking temperature, which indicates that there should exist non-radiating remnants (BHR) of about Planck mass. BHRs are an attractive candidate for cold dark matter. We investigate an alternative cosmology in which primordial BHRs are the primary source of dark matter. 
  We find the perturbation spectrum of a family of spherically symmetric and continuously self-similar (CSS) exact solutions that appear to be relevant for the critical collapse of scalar field matter in 2+1 spacetime dimensions. The rate of exponential growth of the unstable perturbation yields the critical exponent. Our results are compared to the numerical simulations of Pretorius and Choptuik and are inconclusive: We find a CSS solution with exactly one unstable mode, which suggests that it may be the critical solution, but another CSS solution which has three unstable modes fits the numerically found critical solution better. 
  A Planck-scale minimal observable length appears in many approaches to quantum gravity. It is sometimes argued that this minimal length might conflict with Lorentz invariance, because a boosted observer could see the minimal length further Lorentz contracted. We show that this is not the case within loop quantum gravity. In loop quantum gravity the minimal length (more precisely, minimal area) does not appear as a fixed property of geometry, but rather as the minimal (nonzero) eigenvalue of a quantum observable. The boosted observer can see the same observable spectrum, with the same minimal area. What changes continuously in the boost transformation is not the value of the minimal length: it is the probability distribution of seeing one or the other of the discrete eigenvalues of the area. We discuss several difficulties associated with boosts and area measurement in quantum gravity. We compute the transformation of the area operator under a local boost, propose an explicit expression for the generator of local boosts and give the conditions under which its action is unitary. 
  We investigate spherically symmetric self-similar solutions in Brans-Dicke theory. Assuming a perfect fluid with the equation of state $p=(\gamma-1)\mu (1 \le \gamma<2)$, we show that there are no non-trivial solutions which approach asymptotically to the flat Friedmann-Robertson-Walker spacetime if the energy density is positive. This result suggests that primordial black holes in Brans-Dicke theory cannot grow at the same rate as the size of the cosmological particle horizon. 
  This thesis details an effort to generate astrophysically interesting solutions to the two-body problem in General Relativity. The thesis consists of two main parts. The first part presents an analytical variational principle for describing binary neutron stars undergoing irrotational fluid flow. The variational principle is a powerful tool for generating accurate estimates of orbital parameters for neutron stars in quasi-equilibrium orbits. The second part of the thesis details the numerical application of the variational principle by solving the initial value problem for binary black holes in quasi- equilibrium circular orbits. The analysis draws from the novel ``puncture'' method of describing the black holes, and relies on nonlinear adaptive multigrid techniques for generating numerical results. We arrive at two important conclusions. First, the analytical variational principle describing binary neutron stars in irrotational motion provides a road map for future numerical simulations, and also lends credence to previous simulations by other authors. Second, the numerical application and description of binary black holes in quasi-equilibrium circular orbits simplifies the analyses of previous authors, and allows for the imposition of realistic boundary data in simulations with relatively high grid densities. Both the variational principle and its application are used to generate accurate estimates of the orbital parameters. 
  Traditionally, higher-dimensional cosmological models have sought to provide a description of the fundamental forces in terms of a unifying geometrical construction. In this essay we discuss how, in their present incarnation, higher-dimensional `braneworld' models might provide answers to a number of cosmological puzzles including the issue of dark energy and the nature of the big-bang singularity. 
  Runaway solutions can be avoided in fourth order gravity by a doubling of the matter operator algebra with a symmetry constraint with respect to the exchange of observable and hidden degrees of freedom together with the change in sign of the ghost and the dilaton fields. The theory is classically equivalent to Einstein gravity, while its non-unitary Newtonian limit is compatible with the wavelike properties of microscopic particles and the classical behavior of macroscopic bodies, as well as with a trans-Planckian regularization of collapse singularities. A unified reading of ordinary and black hole entropy emerges as entanglement entropy with hidden degrees of freedom. 
  Basis and limitations of singularity theorems for Gravity are examined. As singularity is a critical situation in course of time, study of time paths, in full generality of Equivalence principle, provides two mechanisms to prevent singularity. Resolution of singular Time translation generators into space of its orbits, and essential higher dimensions for Relativistic particle interactions has facets to resolve any real singularity problem. Conceptually, these varied viewpoints have a common denominator: arbitrariness in the definition of `energy' intrinsic to the space of operation in each case, so as to render absence of singularity a tautology for self-consistency of the systems. 
  We construct a model of a relativistic fireball or light--like shell of matter by considering a spherically symmetric moving gravitating mass experiencing an impulsive deceleration to rest. We take this event to be followed by the mass undergoing a deformation leading to the emission of gravitational radiation while it returns to a spherically symmetric state. We find that the fireball is accompanied by an impulsive gravitational wave. 
  We present the full implementation of a room-temperature gravitational wave bar detector equipped with an opto-mechanical readout. The mechanical vibrations are read by a Fabry--Perot interferometer whose length changes are compared with a stable reference optical cavity by means of a resonant laser. The detector performance is completely characterized in terms of spectral sensitivity and statistical properties of the fluctuations in the system output signal. The new kind of readout technique allows for wide-band detection sensitivity and we can accurately test the model of the coupled oscillators for thermal noise. Our results are very promising in view of cryogenic operation and represent an important step towards significant improvements in the performance of massive gravitational wave detectors. 
  A multidimensional cosmological model with space-time consisting of n (n>1) Einstein spaces M_i is investigated in the presence of a cosmological constant Lambda and a homogeneous minimally coupled free scalar field. Generalized de Sitter solution was found for Lambda > 0 and Ricci-flat external space for the case of static internal spaces with fine tuning of parameters. 
  It is possible that null paths in 5D appear as the timelike paths of massive particles in 4D, where there is an oscillation in the fifth dimension around the hypersurface we call spacetime. A particle in 5D may be regarded as multiply imaged in 4D, and the 4D weak equivalence principle may be regarded as a symmetry of the 5D metric. 
  GR can be interpreted as a theory of evolving 3-geometries. A recent such formulation, the 3-space approach of Barbour, Foster and \'{O} Murchadha, also permits the construction of a limited number of other theories of evolving 3-geometries, including conformal gravity and strong gravity. In this paper, we use the 3-space approach to construct a 1-parameter family of theories which generalize strong gravity. The usual strong gravity is the strong-coupled limit of GR, which is appropriate near singularities and is one of very few regimes of GR which is amenable to quantization. Our new strong gravity theories are similar limits of scalar-tensor theories such as Brans--Dicke theory, and are likewise appropriate near singularities. They represent an extension of the regime amenable to quantization, which furthermore spans two qualitatively different types of inner product.   We find that strong gravity theories permit coupling only to ultralocal matter fields and that they prevent gauge theory. Thus in the classical picture, gauge theory breaks down (rather than undergoing unification) as one approaches the GR initial singularity. 
  For Lee Smolin, our universe is only one in a much larger cosmos (the Multiverse) - a member of a growing community of universes, each one being born in a bounce following the formation of a black hole. In the course of this, the values of the free parameters of the physical laws are reprocessed and slightly changed. This leads to an evolutionary picture of the Multiverse, where universes with more black holes have more descendants. Smolin concludes, that due to this kind of Cosmological Natural Selection our own universe is the way it is. The hospitality for life of our universe is seen as an offshot of this self-organized process. - This paper outlines Smolin's hypothesis, its strength, weakness and limits, its relationship to the anthropic principle and evolutionary biology, and comments on the hypothesis from different points of view: physics, biology, philosophy of science, philosophy of nature, and metaphysics. Some of the main points are: (1) There is no necessary connection between black holes and life. In principle, life and Cosmological Natural Selection could be independent of each other. Smolin might explain the so-called fine-tuning of physical constants, but life remains an epiphenomenon. (2) The Darwinian analogy is an inadequate model transfer. The fitness of Smolin's universes is not constrained by its environment, but by only one internal factor: the numbers of black holes. Furthermore, although Smolin's universes have different reproduction rates, they are not competing against each other. (3) Smolin's central claim cannot be falsified. 
  The motion of spinless particles in Riemann-Cartan (RC) $U_{4}$ and teleparallel spacetimes is revisited on the light of gravitational waves hitting on the spinless test particles. It is shown that in the case Cartan contortion is totally skew-symmetric the spinless particles follow geodesics but if this symmetry is dropped they are able to follow nongeodesic worldlines in torsionic background. The case of totally skew contortion and spinning particles may appear either in $T_{4}$ or $U_{4}$ spacetimes. We consider $T_{4}$ nongeodesic motion of spinless test particles in the field of gravitational waves (GW). It is shown that the general contortion obeys a tensor wave constraint and the analysis of a ring of spinless test particles as in General Relativity (GR) hit by the GW leads to damping contortion effects on the nongeodesic motion is undertaken. It is also shown that the contortion and gravitational waves possess a difference of phase of $\frac{\pi}{2}$ and a contortion at the surface of the Earth of $10^{-24} s^{-1}$ computed by Nitsch in the realm of teleparallelism is used to obtain a deviation of the lenght of the separation of test spinless particles compatible with the GR result. To obtain this result data from GW LISA detector is used. 
  The talk centers around the question: Can general-relativistic description of physical reality be considered complete? On the way I argue how -- unknown to many a physicists, even today -- the ``forty orders of magnitude argument'' against quantum gravity phenomenology was defeated more than a quarter of a century ago, and how we now stand at the possible verge of detecting a signal for the spacetime foam, and studying the gravitationally-modified wave particle duality using superconducting quantum interference devices. 
  We investigate the problem of detecting gravitational waves from binaries of nonspinning black holes with masses m = 5--20 Msun, moving on quasicircular orbits, which are arguably the most promising sources for first-generation ground-based detectors. We analyze and compare all the currently available post--Newtonian approximations for the relativistic two-body dynamics; for these binaries, different approximations predict different waveforms. We then construct examples of detection template families that embed all the approximate models, and that could be used to detect the true gravitational-wave signal (but not to characterize accurately its physical parameters). We estimate that the fitting factor for our detection families is >~0.95 (corresponding to an event-rate loss <~15%) and we estimate that the discretization of the template family, for ~10^4 templates, increases the loss to <~20%. 
  We discuss the canonical quantization of systems formulated on discrete space-times. We start by analyzing the quantization of simple mechanical systems with discrete time. The quantization becomes challenging when the systems have anholonomic constraints. We propose a new canonical formulation and quantization for such systems in terms of discrete canonical transformations. This allows to construct, for the first time, a canonical formulation for general constrained mechanical systems with discrete time. We extend the analysis to gauge field theories on the lattice. We consider a complete canonical formulation, starting from a discrete action, for lattice Yang--Mills theory discretized in space and Maxwell theory discretized in space and time. After completing the treatment, the results can be shown to coincide with the results of the traditional transfer matrix method. We then apply the method to BF theory, yielding the first lattice treatment for such a theory ever. The framework presented deals directly with the Lorentzian signature without requiring an Euclidean rotation. The whole discussion is framed in such a way as to provide a formalism that would allow a consistent, well defined, canonical formulation and quantization of discrete general relativity, which we will discuss in a forthcoming paper. 
  We develop a steady-state analytical and numerical model of the optical response of power-recycled Fabry-Perot Michelson laser gravitational-wave detectors to thermal focusing in optical substrates. We assume that the thermal distortions are small enough that we can represent the unperturbed intracavity field anywhere in the detector as a linear combination of basis functions related to the eigenmodes of one of the Fabry-Perot arm cavities, and we take great care to preserve numerically the nearly ideal longitudinal phase resonance conditions that would otherwise be provided by an external servo-locking control system. We have included the effects of nonlinear thermal focusing due to power absorption in both the substrates and coatings of the mirrors and beamsplitter, the effects of a finite mismatch between the curvatures of the laser wavefront and the mirror surface, and the diffraction by the mirror aperture at each instance of reflection and transmission. We demonstrate a detailed numerical example of this model using the MATLAB program Melody for the initial LIGO detector in the Hermite-Gauss basis, and compare the resulting computations of intracavity fields in two special cases with those of a fast Fourier transform field propagation model. Additional systematic perturbations (e.g., mirror tilt, thermoelastic surface deformations, and other optical imperfections) can be included easily by incorporating the appropriate operators into the transfer matrices describing reflection and transmission for the mirrors and beamsplitter. 
  I analyze the deformation of Lorentz symmetry that holds in certain noncommutative spacetimes and the way in which Lorentz symmetry is broken in other noncommutative spacetimes. I also observe that discretization of areas does not necessarily require departures from Lorentz symmetry. This is due to the fact that Lorentz symmetry has no implications for exclusive measurement of the area of a surface, but it governs the combined measurements of the area and the velocity of a surface. In a quantum-gravity theory Lorentz symmetry can be consistent with area discretization, but only when the observables ``area of the surface" and "velocity of the surface" enjoy certain special properties. I argue that the status of Lorentz symmetry in the loop-quantum-gravity approach requires careful scrutiny, since areas are discretized within a formalism that, at least presently, does not include an observable "velocity of the surface". In general it may prove to be very difficult to reconcile Lorentz symmetry with area discretization in theories of canonical quantization of gravity, because a proper description of Lorentz symmetry appears to require that the fundamental/primary role be played by the surface's world-sheet, whose "projection" along the space directions of a given observer describes the observable area, whereas the canonical formalism only allows the introduction as primary entities of observables defined at a fixed (common) time, and the observers that can be considered must share that time variable. 
  We have studied the science rationale, goals and requirements for a mission aimed at using the gravitational lensing from the Sun as a way of achieving high angular resolution and high signal amplification. We find that such a mission concept is plagued by several practical problems. Most severe are the effects due to the plasma in the solar atmosphere which cause refraction and scattering of the propagating rays. These effects limit the frequencies that can be observed to those above $\sim$100-200 GHz and moves the optical point outwards beyond the vacuum value of $\geq$ 550 AU. Density fluctuations in the inner solar atmosphere will further cause random pathlength differences for different rays. The corrections for the radiation from the Sun itself will also be a major challenge at any wavelength used. Given reasonable constraints on the spacecraft (particularly in terms of size and propulsion) source selection as well as severe navigational constraints further add to the difficulties for a potential mission. 
  As a consequence of Birkhoff's theorem, the exterior gravitational field of a spherically symmetric star or black hole is always given by the Schwarzschild metric. In contrast, the exterior gravitational field of a rotating (axisymmetric) star differs, in general, from the Kerr metric, which describes a stationary, rotating black hole.   In this paper, I discuss the possibility of a quasi-stationary transition from rotating equilibrium configurations of normal matter to rotating black holes. 
  We show, by using Regge calculus, that the entropy of any finite part of a Rindler horizon is, in the semi-classical limit, one quarter of the area of that part. We argue that this result implies that the entropy associated with any horizon of spacetime is, in semi-classical limit, one quarter of its area. As an example, we derive the Bekenstein-Hawking entropy law for the Schwarzschild black hole. 
  We propose a method to extend into the bulk asymptotically flat static spherically symmetric brane-world metrics. We employ the multipole (1/r) expansion in order to allow exact integration of the relevant equations along the (fifth) extra coordinate and make contact with the parameterized post-Newtonian formalism. We apply our method to three families of solutions previously appeared as candidates of black holes in the brane world and show that the shape of the horizon is very likely a flat ``pancake'' for astrophysical sources. 
  We compute the one loop vacuum polarization from massless, minimally coupled scalar QED in a locally de Sitter background. Gauge invariance is maintained through the use of dimensional regularization, whereas conformal invariance is explicitly broken by the scalar kinetic term as well as through the conformal anomaly. A fully renormalized result is obtained. The one loop corrections to the linearized, effective field equations do not vanish when evaluated on-shell. In fact the on-shell one loop correction depends quadratically on the inflationary scale factor, similar to a photon mass. The contribution from the conformal anomaly is insignificant by comparison. 
  Static, spherically symmetric, traversable wormhole solutions with electric or magnetic charges are shown to exist in general relativity in the presence of scalar fields nonminimally coupled to gravity. These wormholes, however, turn out to be unstable under spherically symmetric perturbations. The instability is related to blowing-up of the effective gravitational constant on a certain sphere. 
  It is shown that the Stelle-West Grignani-Nardelli-formalism allows, both when odd dimensions and when even dimensions are considered, constructing actions for higher dimensional gravity invariant under local Lorentz rotations and under local Poincar\`{e} translations. It is also proved that such actions have the same coefficients as those obtained by Troncoso and Zanelli in ref. Class. Quantum Grav. 17 (2000) 4451. 
  Regarding metric fluctuations as generating {\it roughness} on the fabric of the otherwise smooth vacuum, it is shown that in its simplest form, the effect can be described by the scalar $\phi^4$ model. The model exhibits a second order phase transition between a smooth (low-temperature) phase and a rough (high-temperature) one, corroborating the absence of metric fluctuations at low energies. In the rough phase near the critical point, vacuum is characterized by a power-law behavior for the fluctuating field with critical exponent $\beta \approx 0.33$. 
  We show that a system of a domain wall coupled to a scalar field has static negative energy density at certain distances from the domain wall. This system provides a simple, explicit example of violation of the averaged weak energy condition and the quantum inequalities by interacting quantum fields. Unlike idealized systems with boundary conditions or external background fields, this calculation is implemented precisely in renormalized quantum field theory with the energy necessary to support the background field included self-consistently. 
  We explore the dynamics of the r-modes in accreting neutron stars in two ways. First, we explore how dissipation in the magneto-viscous boundary layer (MVBL) at the crust-core interface governs the damping of r-mode perturbations in the fluid interior. Two models are considered: one assuming an ordinary-fluid interior, the other taking the core to consist of superfluid neutrons, type II superconducting protons, and normal electrons. We show, within our approximations, that no solution to the magnetohydrodynamic equations exists in the superfluid model when both the neutron and proton vortices are pinned. However, if just one species of vortex is pinned, we can find solutions. When the neutron vortices are pinned and the proton vortices are unpinned there is much more dissipation than in the ordinary-fluid model, unless the pinning is weak. When the proton vortices are pinned and the neutron vortices are unpinned the dissipation is comparable or slightly less than that for the ordinary-fluid model, even when the pinning is strong. We also find in the superfluid model that relatively weak radial magnetic fields ~ 10^9 G (10^8 K / T)^2 greatly affect the MVBL, though the effects of mutual friction tend to counteract the magnetic effects. Second, we evolve our two models in time, accounting for accretion, and explore how the magnetic field strength, the r-mode saturation amplitude, and the accretion rate affect the cyclic evolution of these stars. If the r-modes control the spin cycles of accreting neutron stars we find that magnetic fields can affect the clustering of the spin frequencies of low mass x-ray binaries (LMXBs) and the fraction of these that are currently emitting gravitational waves. 
  We study the dynamical instability against bar-mode deformation of differentially rotating stars. We performed numerical simulation and linear perturbation analysis adopting polytropic equations of state with the polytropic index $n=1$. It is found that rotating stars of a high degree of differential rotation are dynamically unstable even for the ratio of the kinetic energy to the gravitational potential energy of $O(0.01)$. Gravitational waves from the final nonaxisymmetric quasistationary states are calculated in the quadrupole formula. For rotating stars of mass $1.4M_{\odot}$ and radius several 10 km, gravitational waves have frequency several 100 Hz and effective amplitude $\sim 5 \times 10^{-22}$ at a distance of $\sim 100$ Mpc. 
  I present a fast algorithm to find apparent horizons. This algorithm uses an explicit representation of the horizon surface, allowing for arbitrary horizon resolutions and, in principle, shapes. Novel in this approach is that the tensor quantities describing the horizon live directly on the horizon surface, yet are represented using Cartesian coordinate components. This eliminates coordinate singularities, and leads to an efficient implementation. The apparent horizon equation is then solved as a nonlinear elliptic equation with standard methods. I explain in detail the coordinate systems used to store and represent the tensor components of the intermediate quantities, and describe the grid boundary conditions and the treatment of the polar coordinate singularities. Last I give as examples apparent horizons for single and multiple black hole configurations. 
  We undertake to develop a successful framework for commutative-associative hypercomplex numbers with the view to explicate and study associated geometric and generalized-relativistic concepts, basing on an interesting possibility to introduce appropriate multilinear metric forms in the treatment. The scalar polyproduct, which extends the ordinary scalar product used in bilinear (Euclidean and pseudo-Euclidean) theories, has been proposed and applied to be a generalized metric base for the approach. A fundamental concept of multilinear isometry is proposed. This renders possible to muse upon various relativistic physical applications based on anisotropic {\it versus} ordinary spatially-rotational case. 
  Resorting to Berry's phase, a new idea to detect, at quantum level, the gravitomagnetic field of any metric theory of gravity, is put forward.  It is found in this proposal that the magnitude of the gravitomagnetic field appears only in the definition of the adiabatic regime, but not in the magnitude of the emerging geometric phase. In other words, the physical parameter to be observed does not involve, in a direct way, (as in the usual proposals) the tiny magnitude of the gravitomagnetic field. 
  Considering the existence of nonconformal stochastic fluctuations in the metric tensor a generalized uncertainty principle and a deformed dispersion relation (associated to the propagation of photons) are deduced. Matching our model with the so called quantum kappa--Poincare group will allow us to deduce that the fluctuation--dissipation theorem could be fulfilled without needing a restoring mechanism associated with the intrinsic fluctuations of spacetime. In other words, the loss of quantum information is related to the fact that the spacetime symmetries are described by the quantum kappa--Poincare group, and not by the usual Poincare symmetries. An upper bound for the free parameters of this model will also be obtained. 
  Spin-2, spin-1 and spin-0 modes in linearised teleparallelism are obtained where the totally skew-symmetric complex contortion tensor generates scalar torsion waves and the symmetric contortion in the last two indices generates gravitational waves as gravitational perturbations of flat spacetime with contortion tensor. A gedanken experiment with this gravitational-torsion wave hitting a ring of spinless particles is proposed which allows us to estimate the contortion of the Earth by making use of data from LISA GW detector. This value coincides with previous value obtained by Nitsch in teleparallelism using another type of experiment. 
  We present a coordinate-independent method for extracting mass (M) and angular momentum (J) of a black hole in numerical simulations. This method, based on the isolated horizon framework, is applicable both at late times when the black hole has reached equilibrium, and at early times when the black holes are widely separated. We show how J and M can be determined in numerical simulations in terms of only those quantities which are intrinsic to the apparent horizon. We also present a numerical method for finding the rotational symmetry vector field (required to calculate J) on the horizon. 
  This Thesis concerns a thin fluid shell embedded in its own gravitational field. The starting point is a work of Hajicek and Kijowski, where the hamiltonian formalism for shell(s) (with no symmetry) in Einstein gravity is developed. An open problem at the end of that paper is to show how the hamiltonian formalism defines a regular constrained system: the hamiltonian and the constraints must be differentiable functionals on the phase space, so that their Poisson Brackets are well defined objects. On the contrary, some constraints at the shell result to be non differentiable functionals on the phase space. This problem is tackled, in the present thesis, by following the reduction procedure suggested by Teitelboim and Henneaux: the singular constraints are solved and the solution is substituted back into the hamiltonian. The resulting hamiltonian is shown to lead to equivalent dynamics, without singular constraints. Besides, the final reduced system (hamiltonian plus canonical constraints) is shown to be fully differentiable on the reduced phase space. 
  In this paper we consider the collision of black holes with parallel spins using first order perturbation theory of rotating black holes (Teukolsky formalism). The black holes are assumed to be close to each other, initially non boosted and spinning slowly. We estimate the properties of the gravitational radiation released from such an collision. The same problem was studied recently by Gleiser {\em et al.} in the context of the Zerilli perturbation formalism and our results for waveforms, energy and angular momentum radiated agree very well with the results presented in that work. 
  We measured forces applied by an actuator with a YBCO film at near 77 K for the Large-scale Cryogenic Gravitational-wave Telescope (LCGT) project. An actuator consisting of both a YBCO film of 1.6 micrometers thickness and 0.81 square centimeters area and a solenoid coil exerted a force of up to 0.2 mN on a test mass. The presented actuator system can be used to displace the mirror of LCGT for fringe lock of the interferometer. 
  We study numerically the evolution of spactime, and in particular of a spacetime singularity, inside a black hole under a class of perturbations of non-compact support. We use a very simplified toy model of a spherical charged black hole which is perturbed nonlinearly by a self-gravitating, spherical scalar field. The latter grows logarithmically with advanced time along an outgoing characteristic hypersurface. We find that for that class of perturbations a portion of the Cauchy horizon survives as a non-central, null singularity. 
  Static cylindrical shells made of various types of matter are studied as sources of the vacuum Levi-Civita metrics. Their internal physical properties are related to the two essential parameters of the metrics outside. The total mass per unit length of the cylinders is always less than 1/4. The results are illustrated by a number of figures. 
  We analyze the horizon and geodesic structure of a class of 4D off--diagonal metrics with deformed spherical symmetries, which are exact solutions of the vacuum Einstein equations with anholonomic variables. The maximal analytic extension of the ellipsoid type metrics are constructed and the Penrose diagrams are analyzed with respect to adapted frames. We prove that for small deformations (small eccentricities) there are such metrics that the geodesic behaviour is similar to the Schwarzcshild one. We conclude that some vacuum static and stationary ellipsoid configurations may describe black ellipsoid objects. 
  The horizon and geodesic structure of static configurations generated by anisotropic conformal transforms of the Schwarzschild metric is analyzed. We construct the maximal analytic extension of such off--diagonal vacuum metrics and conclude that for small deformations there are different classes of vacuum solutions of the Einstein equations describing "black ellipsoid" objects. This is possible because, in general, for off--diagonal metrics with deformed non--spherical symmetries and associated anholonomic frames the conditions of the uniqueness black hole theorems do not hold. 
  We study the perturbations of two classes of static black ellipsoid solutions of four dimensional vacuum Einstein equations. Such solutions are described by generic off--diagonal metrics which are generated by anholonomic transforms of diagonal metrics. The analysis is performed in the approximation of small eccentricity deformations of the Schwarzschild solution. We conclude that such anisotropic black hole objects may be stable with respect to the perturbations parametrized by the Schrodinger equations in the framework of the one--dimensional inverse scattering theory. 
  The orbital motion of the Laser Interferometer Space Antenna (LISA) produces amplitude, phase and frequency modulation of a gravitational wave signal. The modulations have the effect of spreading a monochromatic gravitational wave signal across a range of frequencies. The modulations encode useful information about the source location and orientation, but they also have the deleterious affect of spreading a signal across a wide bandwidth, thereby reducing the strength of the signal relative to the instrument noise. We describe a simple method for removing the dominant, Doppler, component of the signal modulation. The demodulation reassembles the power from a monochromatic source into a narrow spike, and provides a quick way to determine the sky locations and frequencies of the brightest gravitational wave sources. 
  In some kinds of classical dilaton theory there exist black holes with (i) infinite horizon area $A$ or infinite $F$ (the coefficient at curvature in Lagrangian) and (ii) zero Hawking temperature $T_{H}$. For a generic static black hole, without an assumption about spherical symmetry, we show that infinite $A$ is compatible with a regularity of geometry in the case $T_{H}=0$ only. We also point out that infinite $T_{H}$ is incompatible with the regularity of a horizon of a generic static black hole, both for finite or infinite $A$. Direct application of the standard Euclidean approach in the case of an infinite ''effective'' area of the horizon $A_{eff}=AF$ leads to inconsistencies in the variational principle and gives for a black hole entropy $S$ an indefinite expression, formally proportional to $T_{H}A_{eff}$. We show that treating a horizon as an additional boundary (that is, adding to the action some terms calculated on the horizon) may restore self-consistency of the variational procedure, if $F$ near the horizon grows not too rapidly. We apply this approach to Brans-Dicke black holes and obtain the same answer S=0 as for ''usual'' (for example, Reissner-Nordstr\"{o}m) extreme classical black holes. We also consider the exact solution for a conformal coupling, when $A$ is finite but $F$ diverges and find that in the latter case both the standard and modified approach give rise to an infinite action. Thus, this solution represents a rare exception of a black hole without nontrivial thermal properties. 
  Rotating thin-shell-like sources for the stationary cylindrically symmetric vacuum solutions (Lewis) are constructed and studied. It is found, by imposing the non existence of timelike curves in the exterior of the shell, and that the source satisfies the weak, dominant and strong energy conditions that the parameters, commonly denoted by $a$ and $\sigma$, are restricted to $0 \le \sigma \le 1/4$ when $a > 0$, or $1/4 \le \sigma \le 1/2$ when $a < 0$. 
  A particular approach to topology change in quantum gravity is reviewed, showing that several aspects of Stephen's work are intertwined with it in an essential way. Speculations are made on possible implications for the causal set approach to quantum gravity. 
  We consider the application of the Aichelburg-Sexl boost to plane and line distributions of matter. Our analysis shows that for a domain wall the space-time after the boost is flat except on a null hypersurface which is the history of a null shell. For a cosmic string we study the influence of the boost on the conical singularity and give the new value of the conical deficit. 
  A relativistic sub-picosecond model of gravitational time delay in radio astronomical observations is worked out and a new experimental test of general relativity is discussed in which the effect of retardation of gravity associated with its finite speed can be observed. As a consequence, the speed of gravity can be measured by differential VLBI observations. Retardation in propagation of gravity is a central part of the Einstein theory of general relativity which has not been tested directly so far. The idea of the proposed gravitational experiment is based on the fact that gravity in general relativity propagates with finite speed so that the deflection of light caused by the body must be sensitive to the ratio of the body's velocity to the speed of gravity. The interferometric experiment can be performed, for example, during the very close angular passage of a quasar by Jupiter. Due to the finite speed of gravity and orbital motion of Jupiter, the variation in its gravitational field reaches observer on Earth not instantaneously but at the retarded instant of time and should appear as a velocity-dependent excess time delay in addition to the well-known Shapiro delay, caused by the static part of the Jupiter's gravitational field. Such Jupiter-QSO encounter events take place once in a decade. The next such event will occur on September 8, 2002 when Jupiter will pass by quasar J0842+1835 at the angular distance 3.7 arcminutes. If radio interferometric measurement of the quasar coordinates in the sky are done with the precision of a few picoseconds (about 5 microarcseconds) the effect of retardation of gravity and its speed of propagation may be measured with an accuracy about 10%. 
  We consider the detection and initial guess problems for the LISA gravitational wave detector. The detection problem is the problem of how to determine if there is a signal present in instrumental data and how to identify it. Because of the Doppler and plane-precession spreading of the spectral power of the LISA signal, the usual power spectrum approach to detection will have difficulty identifying sources. A better method must be found. The initial guess problem involves how to generate {\it a priori} values for the parameters of a parameter-estimation problem that are close enough to the final values for a linear least-squares estimator to converge to the correct result. A useful approach to simultaneously solving the detection and initial guess problems for LISA is to divide the sky into many pixels and to demodulate the Doppler spreading for each set of pixel coordinates. The demodulated power spectra may then be searched for spectral features. We demonstrate that the procedure works well as a first step in the search for gravitational waves from monochromatic binaries. 
  Using ideas employed in higher dimensional gravity, non-expanding, weakly isolated and isolated horizons are introduced and analyzed in 2+1 dimensions. While the basic definitions can be taken over directly from higher dimensions, their consequences are somewhat different because of the peculiarities associated with 2+1 dimensions. Nonetheless, as in higher dimensions, we are able to: i) analyze the horizon geometry in detail; ii) introduce the notions of mass, charge and angular momentum of isolated horizons using geometric methods; and, iii) generalize the zeroth and the first laws of black hole mechanics. The Hamiltonian methods also provide, for the first time, expressions of total angular momentum and mass of charged, rotating black holes and their relation to the analogous quantities defined at the horizon. We also construct the analog of the Newman-Penrose framework in 2+1 dimensions which should be useful in a wide variety of problems in 2+1 dimensional gravity. 
  The formulae for Planck length, Hawking temperature and Unruh-Davies temperature are derived by using only laws of classical physics together with the Heisenberg principle. Besides, it is shown how the Hawking relation can be deduced from the Unruh relation by means of the principle of equivalence; the deep link between Hawking effect and Unruh effect is in this way clarified. 
  Assuming a flat Friedmann-Robertson-Walker cosmology with a single perfect fluid, we propose a pressure-density ratio that evolves as a specific universal function of the scale parameter. We show that such a ratio can indeed be consistent with several observational constraints including those pertaining to late-time accelerated expansion. Generic dynamical scalar field models of Dark energy (with quadratic kinetic terms in their Lagrange density) are shown to be in accord with the proposed equation-of-state ratio, provided the current matter density parameter $\Omega_{m0} < 0.23$ - a value {\it not} in agreement with recent measurements. 
  It is shown that optical geometry of the Reissner-Nordstrom exterior metric can be embedded in a hyperbolic space all the way down to its outer horizon. The adopted embedding procedure removes a breakdown of flat-space embeddings which occurs outside the horizon, at and below the Buchdahl-Bondi limit (R/M=9/4 in the Schwarzschild case). In particular, the horizon can be captured in the optical geometry embedding diagram. Moreover, by using the compact Poincare ball representation of the hyperbolic space, the embedding diagram can cover the whole extent of radius from spatial infinity down to the horizon. Attention is drawn to advantages of such embeddings in an appropriately curved space: this approach gives compact embeddings and it distinguishes clearly the case of an extremal black hole from a non-extremal one in terms of topology of the embedded horizon. 
  We consider a minisuperspace model for a closed universe with small and positive cosmological constant, filled with a massive scalar field conformally coupled to gravity. In the quantum version of this model, the universe may undergo a tunneling transition through an effective barrier between regions of small and large scale factor. We solve numerically the minisuperspace Wheeler--De Witt equation with tunneling boundary conditions for the wave function of the universe, and find that tunneling in quantum cosmology is quite different from that in quantum mechanics. Namely, the matter degree of freedom gets excited under the barrier, provided its interaction with the scale factor is not too weak, and makes a strong back reaction onto tunneling. In the semiclassical limit of small values of cosmological constant, the matter energy behind the barrier is close to the height of the barrier: the system ``climbs up'' the barrier, and then evolves classically from its top. These features are even more pronounced for inhomogeneous modes of matter field. Extrapolating to field theory we thus argue that high momentum particles are copiously created in the tunneling process. Nevertheless, we find empirical evidence for the semiclassical-type scaling with the cosmological constant of the wave function under and behind the barrier. 
  We obtain a new exact equilibrium solution to the N-body problem in a one-dimensional relativistic self-gravitating system. It corresponds to an expanding/contracting spacetime of a circle with N bodies at equal proper separations from one another around the circle. Our methods are straightforwardly generalizable to other dilatonic theories of gravity, and provide a new class of solutions to further the study of (relativistic) one-dimensional self-gravitating systems. 
  The issue of whether some manifestations of gravitation in the quantum domain, are indicative or not of a non-geometrical aspect in gravitation is discussed. We examine some examples that have been considered in this context, providing a critical analysis of previous interpretations. The analysis of these examples is illustrative about certain details in the interpretation of quantum mechanics. We conclude that there are, at this time, no indications of such departure from the geometrical character of gravitation. 
  Dynamic equations that are the simplest conformally invariant generalization of Einstein equations with cosmological term are considered. Dimensions and Weyl weights of the additional geometrical fields (the vector and the antisymmetric tensor) appearing in the scheme are such, that they admit an unexpected interpretation. It is proved that the fields can be interpreted as observed, generated by bispinor degrees of freedom. The vacuum polarization density matrix leads to different probabilities of different helicity particle generation. 
  We derive and study optimal and nearly-optimal strategies for the detection of sinusoidal signals hidden in additive (Gaussian and non-Gaussian) noise. Such strategies are an essential part of algorithms for the detection of the gravitational Continuous Wave  (CW) signals produced by pulsars. Optimal strategies are derived for the case where the signal phase is not known and the product of the signal frequency and the observation time is non-integral. 
  The experimentally determined Sagnac fringe shift dependency on angular velocity and enclosed area is derived from the rotating reference frame using non-time-orthogonal tensor analysis. The relationship for the most general case, in which the area enclosed is not circular and does not have the axis of rotation passing through its center, is determined. It is submitted that this quantitative result, along with a related thought experiment, can not be found using the conventional approach of local co-moving Lorentz frames. 
  We prove that there are no restrictions on the spatial topology of asymptotically flat solutions of the vacuum Einstein equations in (n+1)-dimensions. We do this by gluing a solution of the vacuum constraint equations on an arbitrary compact manifold to an asymptotically Euclidean solution of the constraints on R^n. For any compact manifold which does not admit a metric of positive scalar curvature, this provides for the existence of asymptotically flat vacuum spacetimes with no maximal slices. Our main theorem is a special case of a more general gluing construction for nondegenerate solutions of the vacuum constraint equations which have some restrictions on the mean curvature, but for which the mean curvature is not necessarily constant. This generalizes the construction [IMP] (gr-qc/0109045), which is restricted to constant mean curvature data. 
  The Einstein evolution equations may be written in a variety of equivalent analytical forms, but numerical solutions of these different formulations display a wide range of growth rates for constraint violations. For symmetric hyperbolic formulations of the equations, an exact expression for the growth rate is derived using an energy norm. This expression agrees with the growth rate determined by numerical solution of the equations. An approximate method for estimating the growth rate is also derived. This estimate can be evaluated algebraically from the initial data, and is shown to exhibit qualitatively the same dependence as the numerically-determined rate on the parameters that specify the formulation of the equations. This simple rate estimate therefore provides a useful tool for finding the most well-behaved forms of the evolution equations. 
  We study a time-dependent and spherically-symmetric solution with a star-like source. We show that this solution can be interpreted as an exterior solution of a contracting star which has a decreasing temperature and is immersed in a homogenous and isotropic background radiation. Distribution of the temperature in the fields and close-to Schwarzschild approximation of the solution are studied. By identifying the radiation with the cosmic background one, we find that the close-to-Schwarzschild approximate solution is valid in a wide range in our solar system. Possible experimental tests of the solution are discussed briefly. 
  Recently, substantial amount of activity in Quantum General Relativity (QGR) has focussed on the semiclassical analysis of the theory. In this paper we want to comment on two such developments: 1) Polymer-like states for Maxwell theory and linearized gravity constructed by Varadarajan which use much of the Hilbert space machinery that has proved useful in QGR and 2) coherent states for QGR, based on the general complexifier method, with built-in semiclassical properties. We show the following: A) Varadarajan's states {\it are} complexifier coherent states. This unifies all states constructed so far under the general complexifier principle. B) Ashtekar and Lewandowski suggested a non-Abelean generalization of Varadarajan's states to QGR which, however, are no longer of the complexifier type. We construct a new class of non-Abelean complexifiers which come close to the one underlying Varadarajan's construction. C) Non-Abelean complexifiers close to Varadarajan's induce new types of Hilbert spaces which do not support the operator algebra of QGR. The analysis suggests that if one sticks to the present kinematical framework of QGR and if kinematical coherent states are at all useful, then normalizable, graph dependent states must be used which are produced by the complexifier method as well. D) Present proposals for states with mildened graph dependence, obtained by performing a graph average, do not approximate well coordinate dependent observables. However, graph dependent states, whether averaged or not, seem to be well suited for the semiclassical analysis of QGR with respect to coordinate independent operators. 
  This paper analyses in quantitative detail the effect caused by a moving mass on a spherical gravitational wave detector. This applies to situations where heavy traffic or similar disturbances happen near the GW antenna. Such disturbances result in quadrupole tidal stresses in the antenna mass, and they therefore precisely fake a real gravitational signal. The study shows that there always are characteristic frequencies, depending on the motion of the external masses, at which the fake signals are most intense. It however appears that, even at those frequencies, fake signals should be orders of magnitude below the sensitivity curve of an optimised detector, in likely realistic situations. 
  The dimensional reduction of black hole solutions in four-dimensional (4D) general relativity is performed and new 3D black hole solutions are obtained. Considering a 4D spacetime with one spacelike Killing vector, it is possible to split the Einstein-Hilbert-Maxwell action with a cosmological term in terms of 3D quantities. Definitions of quasilocal mass and charges in 3D spacetimes are reviewed. The analysis is then particularized to the toroidal charged rotating anti-de Sitter black hole. The reinterpretation of the fields and charges in terms of a three-dimensional point of view is given in each case, and the causal structure analyzed. 
  New results on finite density of particle creation for nonconformal massive scalar particles in Friedmann Universe as well as new counterterms in dimensions higher than 5 are presented. Possible role of creation of superheavy particles for explaining observable density of visible and dark matter is discussed. 
  Gravitational wave emission from the gravitational collapse of massive stars has been studied for more than three decades. Current state of the art numerical investigations of collapse include those that use progenitors with realistic angular momentum profiles, properly treat microphysics issues, account for general relativity, and examine non--axisymmetric effects in three dimensions. Such simulations predict that gravitational waves from various phenomena associated with gravitational collapse could be detectable with advanced ground--based and future space--based interferometric observatories. 
  We bring out here the role of initial data in causing the black hole and naked singularity phases as the final end state of a continual gravitational collapse. The collapse of a type I general matter field is considered, which includes most of the known physical forms of matter. It is shown that given the distribution of the density and pressure profiles at the initial surface from which the collapse evolves, there is a freedom in choosing rest of the free functions, such as the velocities of the collapsing shells, so that the end state could be either a black hole or a naked singularity depending on this choice. It is thus seen that it is the initial data that determines the end state of spherical collapse in terms of these outcomes, and we get a good picture of how these phases come about. 
  We speculate that the universe is filled with stars composed of electromagnetic and dilaton fields which are the sources of the powerful gamma-ray bursts impinging upon us from all directions of the universe. We calculate soliton-like solutions of these fields and show that their energy can be converted into a relativistic plasma in an explosive way. As in classical detonation theory the conversion proceeds by a relativistic self-similar solution for a spherical detonation wave which extracts the energy from the scalar field via a plasma in the wave front. 
  Simple linear second-order differential equations have been written down for FRW cosmologies with barotropic fluids by Faraoni. His results have been extended by Rosu, who employed techniques belonging to nonrelativistic supersymmetry to obtain time-dependent adiabatic indices. Further extensions are presented here using the known connection between the linear second-order differential equations and Dirac-like equations in the same supersymmetric context. These extensions are equivalent to adding an imaginary part to the adiabatic index which is proportional to the mass parameter of the Dirac spinor. The natural physical interpretation of the imaginary part is related to the particular dissipation and instabilities of the barotropic FRW hydrodynamics that are introduced by means of this supersymmetric scheme 
  A possible generalization of plane fronted waves with parallel rays (gpp-wave) fall into a more general class of metrics admitting parallel null 1-planes. For gpp-wave metric, the zero-curvature condition is given, the Killing-Yano tensors of order two and three are found and the corresponding Killing tensors are constructed. Henceforth, the compatibility between geometric duality and non-generic symmetries is presented. 
  Spontaneous global symmetry breaking of O(3) scalar field gives rise to point-like topological defects, global monopoles. By taking into account self-gravity,the qualitative feature of the global monopole solutions depends on the vacuum expectation value v of the scalar field. When v < sqrt{1 / 8 pi}, there are global monopole solutions which have a deficit solid angle defined at infinity. When sqrt{1 / 8 pi} <= v < sqrt{3 / 8 pi}, there are global monopole solutions with the cosmological horizon, which we call the supermassive global monopole. When v >= sqrt{3 / 8 pi}, there is no nontrivial solution. It was shown that all of these solutions are stable against the spherical perturbations. In addition to the global monopole solutions, the de Sitter solutions exist for any value of v. They are stable against the spherical perturbations when v <= sqrt{3 / 8 pi}, while unstable for v > sqrt{3 / 8 pi}. We study polar perturbations of these solutions and find that all self-gravitating global monopoles are stable even against polar perturbations, independently of the existence of the cosmological horizon, while the de Sitter solutions are always unstable. 
  We propose a new test of Einstein's theory of gravitation. It concerns the velocity distribution of low-energy particles in a spherically symmetric gravitational field. 
  We consider the response of a uniformly accelerated monopole detector that is coupled non-linearly to the nth power of a quantum scalar field in (D+1)-dimensional flat spacetime. We show that, when (D+1) is even, the response of the detector in the Minkowski vacuum is characterized by a Bose-Einstein factor for all n. Whereas, when (D+1) is odd, we find that a Fermi-Dirac factor appears in the detector response when n is odd, but a Bose-Einstein factor arises when n is even. We emphasize the point that, since, along the accelerated trajectory, the Wightman function and, as a result, the (2n)-point function satisfy the Kubo-Martin-Schwinger condition (as required for a scalar field) in all dimensions, the appearance of a Fermi-Dirac factor (instead of the expected Bose-Einstein distribution) for odd (D+1) and n reflects a peculiar feature of the detector rather than imply a fundamental change in field theory. 
  Topological solutions in the (2+1)-dimensional Einstein theory of gravity are studied within the ADM canonical formalism. It is found that a conical singularity appears in the closed de Sitter universe solution as a topological defect in the case of the Einstein theory with a cosmological constant. Quantum effects on the conical singularity are studied using the de Broglie-Bohm interpretation. Finite quantum tunneling effects are obtained for the closed de Sitter universe, while no quantum effects are obtained for an open universe. 
  Using a representation of spatial infinity based in the properties of conformal geodesics, the first terms of an expansion for the Bondi mass for the development of time symmetric, conformally flat initial data are calculated. As it is to be expected, the Bondi mass agrees with the ADM at the sets where null infinity ``touches'' spatial infinity. The second term in the expansion is proportional to the sum of the squared norms of the Newman-Penrose constants of the spacetime. In base of this result it is argued that these constants may provide a measure of the incoming radiation contained in the spacetime. This is illustrated by means of the Misner and Brill-Lindquist data sets. 
  We show that NS's with large toroidal B-fields tend naturally to evolve into potent gravitational-wave (gw) emitters. The toroidal field B_t tends to distort the NS into a prolate shape, and this magnetic distortion can easily dominate over the oblateness ``frozen into'' the NS crust. An elastic NS with frozen-in B-field of this magnitude is clearly secularly unstable: the wobble angle between the NS's angular momentum J^i and the star's magnetic axis n_B^i grow on a dissipation timescale until J^i and n_B^i are orthogonal. This final orientation is clearly the optimal one for gravitational-wave (gw) emission. The basic cause of the instability is quite general, so we conjecture that the same final state is reached for a realistic NS. Assuming this, we show that for LMXB's with B_t of order 10^{13}G, the spindown from gw's is sufficient to balance the accretion torque--supporting a suggestion by Bildsten. The spindown rates of most millisecond pulsars can also be attributed to gw emission sourced by toroidal B-fields, and both these sources could be observed by LIGO II. While the first-year spindown of a newborn NS is most likely dominated by em processes, reasonable values of B_t and the (external) dipolar field B_d can lead to detectable levels of gw emission, for a newborn NS in our own galaxy. 
  The spherically symmetric dust model of Lemaitre-Tolman can describe wormholes, but the causal communication between the two asymptotic regions through the neck is even less than in the vacuum (Schwarzschild-Kruskal-Szekeres) case. We investigate the anisotropic generalisation of the wormhole topology in the Szekeres model. The function E(r, p, q) describes the deviation from spherical symmetry if \partial_r E \neq 0, but this requires the mass to be increasing with radius, \partial_r M > 0, i.e. non-zero density. We investigate the geometrical relations between the mass dipole and the locii of apparent horizon and of shell-crossings. We present the various conditions that ensure physically reasonable quasi-spherical models, including a regular origin, regular maxima and minima in the spatial sections, and the absence of shell-crossings. We show that physically reasonable values of \partial_r E \neq 0 cannot compensate for the effects of \partial_r M > 0 in any direction, so that communication through the neck is still worse than the vacuum.  We also show that a handle topology cannot be created by identifying hypersufaces in the two asymptotic regions on either side of a wormhole, unless a surface layer is allowed at the junction. This impossibility includes the Schwarzschild-Kruskal-Szekeres case. 
  Some typical quantization ambiguities of quantum geometry are studied within isotropic models. Since this allows explicit computations of operators and their spectra, one can investigate the effects of ambiguities in a quantitative manner. It is shown that those ambiguities do not affect the fate of the classical singularity, demonstrating that the absence of a singularity in loop quantum cosmology is a robust implication of the general quantization scheme. The calculations also allow conclusions about modified operators in the full theory. In particular, using holonomies in a non-fundamental representation of SU(2) to quantize connection components turns out to lead to significant corrections to classical behavior at macroscopic volume for large values of the spin of the chosen representation. 
  Quantum geometry predicts that a universe evolves through an inflationary phase at small volume before exiting gracefully into a standard Friedmann phase. This does not require the introduction of additional matter fields with ad hoc potentials; rather, it occurs because of a quantum gravity modification of the kinetic part of ordinary matter Hamiltonians. An application of the same mechanism can explain why the present-day cosmological acceleration is so tiny. 
  It has long been recognized that lattice gauge theory formulations, when applied to general relativity, conflict with the invariance of the theory under diffeomorphisms. Additionally, the traditional lattice field theory approach consists in fixing the gauge in a Euclidean action, which does not appear appropriate for general relativity. We analyze discrete lattice general relativity and develop a canonical formalism that allows to treat constrained theories in Lorentzian signature space-times. The presence of the lattice introduces a ``dynamical gauge'' fixing that makes the quantization of the theories conceptually clear, albeit computationally involved. Among other issues the problem of a consistent algebra of constraints is automatically solved in our approach. The approach works successfully in other field theories as well, including topological theories like BF theory. We discuss a simple cosmological application that exhibits the quantum elimination of the singularity at the big bang. 
  We develop a new method for determining the gravitationally induced quantum phase shift for a particle propagating in a stationary weak gravitational field, in the framework of the linear approximation. This method is based on the properties of the Synge's world-function. The result is applied to the neutrino oscillations in the field of an isolated, axisymetric rotating body. The gravitational field is described using the linearized part of the post-Newtonian metric yielded by the standard Nordtvedt-Will formalism. An explicit calculation of the phase shift is performed for neutrinos produced inside a homogeneous nearly spherical rotating body. Our general formulae are valid even for nonradial propagations. The gravitational effects found here involve only the PPN parameter $\gamma$. 
  General formulation of geometrization matter problem by scalar field $\phi =\sqrt{-G_{55}}$ with the help of possibilities of classical 5-D Kaluza-Klein theory is given. Mathematical integrability conditions for such geometrization for the case of perfect fluid are derived. 
  A variation of Hawking's idea about Euclidean origin of a nonsingular birth of the Universe is considered. It is assumed that near to zero moment $t = 0$ fluctuations of a metric signature are possible. 
  A model of three-body motion is developed which includes the effects of gravitational radiation reaction. The radiation reaction due to the emission of gravitational waves is the only post-Newtonian effect that is included here. For simplicity, all of the motion is taken to be planar. Two of the masses are viewed as a binary system and the third mass, whose motion will be a fixed orbit around the center-of-mass of the binary system, is viewed as a perturbation. This model aims to describe the motion of a relativistic binary pulsar that is perturbed by a third mass. Numerical integration of this simplified model reveals that given the right initial conditions and parameters one can see resonances. These (m,n) resonances are defined by the resonance condition, $m\omega=2n\Omega$, where $m$ and $n$ are relatively prime integers and $\omega$ and $\Omega$ are the angular frequencies of the binary orbit and third mass orbit, respectively. The resonance condition consequently fixes a value for the semimajor axis of the binary orbit for the duration of the resonance; therefore, the binary energy remains constant on the average while its angular momentum changes during the resonance. 
  In a previous investigation, a model of three-body motion was developed which included the effects of gravitational radiation reaction. The aim was to describe the motion of a relativistic binary pulsar that is perturbed by a third mass and look for resonances between the binary and third mass orbits. Numerical integration of an equation of relative motion that approximates the binary gives evidence of such resonances. These $(m:n)$ resonances are defined for the present purposes by the resonance condition, $m\omega=2n\Omega$, where $m$ and $n$ are relatively prime integers and $\omega$ and $\Omega$ are the angular frequencies of the binary orbit and third mass orbit, respectively. The resonance condition consequently fixes a value for the semimajor axis $a$ of the binary orbit for the duration of the resonance because of the Kepler relationship $\omega=a^{-3/2}$. This paper outlines a method of averaging developed by Chicone, Mashhoon, and Retzloff which renders a nonlinear system that undergoes resonance capture into a mathematically amenable form. This method is applied to the present system and one arrives at an analytical solution that describes the average motion during resonance. Furthermore, prominent features of the full nonlinear system, such as the frequency of oscillation and antidamping, accord with their analytically derived formulae. 
  We study the evolution of the Weyl curvature invariant in all spatially homogeneous universe models containing a non-tilted gamma-law perfect fluid. We investigate all the Bianchi and Thurston type universe models and calculate the asymptotic evolution of Weyl curvature invariant for generic solutions to the Einstein field equations. The influence of compact topology on Bianchi types with hyperbolic space sections is also considered. Special emphasis is placed on the late-time behaviour where several interesting properties of the Weyl curvature invariant occur. The late-time behaviour is classified into five distinctive categories. It is found that for a large class of models, the generic late-time behaviour the Weyl curvature invariant is to dominate the Ricci invariant at late times. This behaviour occurs in universe models which have future attractors that are plane-wave spacetimes, for which all scalar curvature invariants vanish. The overall behaviour of the Weyl curvature invariant is discussed in relation to the proposal that some function of the Weyl tensor or its invariants should play the role of a gravitational 'entropy' for cosmological evolution. In particular, it is found that for all ever-expanding models the measure of gravitational entropy proposed by Gron and Hervik increases at late times. 
  The analysis of how a stochastic background of gravitational radiation interacts with a spherical detector is given in detail, which leads to explicit expressions for the system response functions, as well as for the cross-correlation matrix of different readout channels. It is shown that distinctive features of GW induced random detector excitations, relative to locally generated noise, are in practice insufficient to separate the signal from the noise by means of a single sphere, if prior knowledge on the GW spectral density is nil. The situation significantly improves when such previous knowledge is available, due to the omnidirectionality and multimode capacities of a spherical GW antenna. 
  We apply Feynman's principle, ``The same equations have the same solutions'', to Kepler's problem and show that Newton's dynamics in a properly curved 3-D space is identical with that described by Einstein's theory in the 3-D optical geometry of Schwarzschild's spacetime. For this reason, rather unexpectedly, Newton's formulae for Kepler's problem, in the case of nearly circular motion in a static, spherically spherical gravitational potential accurately describe strong field general relativistic effects, in particular vanishing of the radial epicyclic frequency at the marginally stable orbit. 
  Based on an analysis of the entropy associated to the vacuum quantum fluctuations, we show that the holographic principle, applied to the cosmic scale, constitutes a possible explanation for the observed value of the cosmological constant, theoretically justifying a relation proposed 35 years ago by Zel'dovich. Furthermore, extending to the total energy density the conjecture by Chen and Wu, concerning the dependence of the cosmological constant on the scale factor, we show that the holographic principle may also lie at the root of the coincidence between the matter density in the universe and the vacuum energy density. 
  The possibility of quantum creation of a dilatonic AdS Universe is discussed.Without dilaton it is known that quantum effects lead to the annihilation of AdS Universe. We consider the role of the form for the dilatonic potential in the quantum creation of a dilatonic AdS Universe. Using the conformal anomaly for dilaton coupled scalar, the anomaly induced action and the equations of motion are obtained. The anomaly induced action is added to classical dilaton gravity action. The solutions of the full theory which correspond to quantum-corrected AdS Universe are given for number of dilatonic potentials. 
  The quantum interest conjecture of Ford and Roman asserts that any negative-energy pulse must necessarily be followed by an over-compensating positive-energy one within a certain maximum time delay. Furthermore, the minimum amount of over-compensation increases with the separation between the pulses. In this paper, we first study the case of a negative-energy square pulse followed by a positive-energy one for a minimally coupled, massless scalar field in two-dimensional Minkowski space. We obtain explicit expressions for the maximum time delay and the amount of over-compensation needed, using a previously developed eigenvalue approach. These results are then used to give a proof of the quantum interest conjecture for massless scalar fields in two dimensions, valid for general energy distributions. 
  Encountered in the literature generalisations of general relativity to independent area variables are considered, the discrete (generalised Regge calculus) and continuum ones. The generalised Regge calculus can be either with purely area variables or, as we suggest, with area tensor-connection variables. Just for the latter, in particular, we prove that in analogy with corresponding statement in ordinary Regge calculus (by Feinberg, Friedberg, Lee and Ren), passing to the (appropriately defined) continuum limit yields the generalised continuum area tensor-connection general relativity. 
  Using the extension of the standard Hawking-Hartle prescription for defining a wave function for the universe, we show that it is possible, given a suitable form for the scalar field potential, to have the universe begin at its largest size and thereafter contract, with the growth of perturbations proceeding from small, at the largest size, to largest when the universe is small. This solution would dominate the wave function by an exponentially large amount if one chooses the Hartle Hawking prescription for the wave-function, but is exponentially sub-dominant for the Linde-Vilenkin prescription. 
  The main purpose of this article is to guide the reader to theorems on global properties of solutions to the Einstein-Vlasov system. This system couples Einstein's equations to a kinetic matter model. Kinetic theory has been an important field of research during several decades where the main focus has been on nonrelativistic- and special relativistic physics, e.g. to model the dynamics of neutral gases, plasmas and Newtonian self-gravitating systems. In 1990 Rendall and Rein initiated a mathematical study of the Einstein-Vlasov system. Since then many theorems on global properties of solutions to this system have been established. The Vlasov equation describes matter phenomenologically and it should be stressed that most of the theorems presented in this article are not presently known for other such matter models (e.g. fluid models). The first part of this paper gives an introduction to kinetic theory in non-curved spacetimes and then the Einstein-Vlasov system is introduced. We believe that a good understanding of kinetic theory in non-curved spacetimes is fundamental in order to get a good comprehension of kinetic theory in general relativity. 
  We present three-dimensional, {\it non-axisymmetric} distorted black hole initial data which generalizes the axisymmetric, distorted, non-rotating [Bernstein93a] and rotating [Brandt94a] single black hole data developed by Bernstein, Brandt, and Seidel. These initial data should be useful for studying the dynamics of fully 3D, distorted black holes, such as those created by the spiraling coalescence of two black holes. We describe the mathematical construction of several families of such data sets, and show how to construct numerical solutions. We survey quantities associated with the numerically constructed solutions, such as ADM masses, apparent horizons, measurements of the horizon distortion, and the maximum possible radiation loss ($MRL$). 
  In this review we describe a non-trivial relationship between perturbative gauge theory and gravity scattering amplitudes. At the semi-classical or tree level, the scattering amplitudes of gravity theories in flat space can be expressed as a sum of products of well defined pieces of gauge theory amplitudes. These relationships were first discovered by Kawai, Lewellen and Tye in the context of string theory, but hold more generally. In particular, they hold for standard Einstein gravity. A method based on D-dimensional unitarity can then be used to systematically construct all quantum loop corrections order-by-order in perturbation theory using as input the gravity tree amplitudes expressed in terms of gauge theory ones. More generally, the unitarity method provides a means for perturbatively quantizing massless gravity theories without the usual formal apparatus associated with the quantization of constrained systems. As one application, this method was used to demonstrate that maximally supersymmetric gravity is less divergent in the ultraviolet than previously thought. 
  Numerical relativity has faced the problem that standard 3+1 simulations of black hole spacetimes without singularity excision and with singularity avoiding lapse and vanishing shift fail after an evolution time of around 30-40M due to the so-called slice stretching. We discuss lapse and shift conditions for the non-excision case that effectively cure slice stretching and allow run times of 1000M and more. 
  In this letter we further analyze the concept of the use of a pair of identical geodetic satellites placed in identical orbits except for the inclinations which should be supplementary for measuring the Lense-Thirring effect in the terrestrial gravitational field. It turns out that not only the sum of the nodes, as already proposed for the LAGEOS-LARES mission, but also the difference of the perigees could be fruitfully adopted. 
  In this paper we present a rather extensive error budget for the difference of the perigees of a pair of supplementary SLR satellites aimed to the detection of the Lense-Thirring effect. 
  The physical situation of the collision and subsequent interaction of plane gravitational waves in a Minkowski background gives rise to a well-posed characteristic initial value problem in which initial data are specified on the two null characteristics that define the wavefronts. In this paper, we analyse how the Abel transform method can be used in practice to solve this problem for the linear case in which the polarization of the two gravitational waves is constant and aligned. We show how the method works for some known solutions, where problems arise in other cases, and how the problem can always be solved in terms of an infinite series if the spectral functions for the initial data can be evaluated explicitly. 
  We study the matching of a general spherically symmetric spacetime with a Vaidya-Reissner-Nordstrom solution. To that end, we study the properties of spherically symmetric electromagnetic fields and develop the proper gravitational and electromagnetic junction conditions. We prove that generic spacetimes can be matched to a Vaidya-Reissner-Nordstrom solution or one of its specializations, and that these matchings have clear physical interpretations. Furthermore, the non-spacelike nature of the matching hypersurface is proved under very general hypotheses. We obtain the fundamental result that any spherically symmetric body, be it in evolution or not, has un upper limit for the total net electric charge that carries. 
  We show that the introduction of triangulations with variable connectivity and fluctuating egde-lengths (Random Regge Triangulations) allows for a relatively simple and direct analyisis of the modular properties of 2 dimensional simplicial quantum gravity. In particular, we discuss in detail an explicit bijection between the space of possible random Regge triangulations (of given genus g and with N vertices) and a suitable decorated version of the (compactified) moduli space of genus g Riemann surfaces with N punctures. Such an analysis allows us to associate a Weil-Petersson metric with the set of random Regge triangulations and prove that the corresponding volume provides the dynamical triangulation partition function for pure gravity. 
  General Relativity is contaminated with non-trivial geometries which generate closed timelike curves. These apparently violate causality, producing time-travel paradoxes. We shall briefly discuss these geometries and analyze some of their physical aspects. 
  The interaction between cosmic rays and the gravitational wave bar detector NAUTILUS is experimentally studied with the aluminum bar at temperature of T=1.5 K. The results are compared with those obtained in the previous runs when the bar was at T=0.14 K. The results of the run at T = 1.5 K are in agreement with the thermo-acoustic model; no large signals at unexpected rate are noticed, unlike the data taken in the run at T = 0.14 K. The observations suggest a larger efficiency in the mechanism of conversion of the particle energy into vibrational mode energy when the aluminum bar is in the superconductive status. 
  We demonstrate the existence of shear-free cosmological models with rotation and expansion which support the inflationary scenarios. The corresponding metrics belong to the family of spatially homogeneous models with the geometry of the closed universe (Bianchi type IX). We show that the global vorticity does not prevent the inflation and even can accelerate it. 
  Unlike ground-based interferometric gravitational wave detectors, large space-based systems will not be rigid structures. When the end-stations of the laser interferometer are freely flying spacecraft, the armlengths will change due to variations in the spacecraft positions along their orbital trajectories, so the precise equality of the arms that is required in a laboratory interferometer to cancel laser phase noise is not possible. However, using a method discovered by Tinto and Armstrong, a signal can be constructed in which laser phase noise exactly cancels out, even in an unequal arm interferometer. We examine the case where the ratio of the armlengths is a variable parameter, and compute the averaged gravitational wave transfer function as a function of that parameter. Example sensitivity curve calculations are presented for the expected design parameters of the proposed LISA interferometer, comparing it to a similar instrument with one arm shortened by a factor of 100, showing how the ratio of the armlengths will affect the overall sensitivity of the instrument. 
  We investigate the limitations of length measurements by accelerated observers in Minkowski spacetime brought about via the hypothesis of locality, namely, the assumption that an accelerated observer at each instant is equivalent to an otherwise identical momentarily comoving inertial observer. We find that consistency can be achieved only in a rather limited neighborhood around the observer with linear dimensions that are negligibly small compared to the characteristic acceleration length of the observer. 
  The polarized Gowdy ${\bf T}^3$ vacuum spacetimes are characterized, modulo gauge, by a ``point particle'' degree of freedom and a function $\phi$ that satisfies a linear field equation and a non-linear constraint. The quantum Gowdy model has been defined by using a representation for $\phi$ on a Fock space $\cal F$. Using this quantum model, it has recently been shown that the dynamical evolution determined by the linear field equation for $\phi$ is not unitarily implemented on $\cal F$. In this paper: (1) We derive the classical and quantum model using the ``covariant phase space'' formalism. (2) We show that time evolution is not unitarily implemented even on the physical Hilbert space of states ${\cal H} \subset {\cal F}$ defined by the quantum constraint. (3) We show that the spatially smeared canonical coordinates and momenta as well as the time-dependent Hamiltonian for $\phi$ are well-defined, self-adjoint operators for all time, admitting the usual probability interpretation despite the lack of unitary dynamics. 
  A new approach to the phenomenon of large numbers coincidence leads to unexpected results. No matter how strange it might sound, the exact value of cosmological parameters and their analytical expression through fundamental constants have been founded. The basis for obtaining these unusual results is the equality of the fundamental Large Number to the exponent of the inverse value of the fine structure constant. 
  We present a phase-plane analysis of cosmologies containing a scalar field $\phi$ with an exponential potential $V \propto \exp(-\lambda \kappa \phi)$ where $\kappa^2 = 8\pi G$ and $V$ may be positive or negative. We show that power-law kinetic-potential scaling solutions only exist for sufficiently flat ($\lambda^2<6$) positive potentials or steep ($\lambda^2>6$) negative potentials. The latter correspond to a class of ever-expanding cosmologies with negative potential. However we show that these expanding solutions with a negative potential are to unstable in the presence of ordinary matter, spatial curvature or anisotropic shear, and generic solutions always recollapse to a singularity. Power-law kinetic-potential scaling solutions are the late-time attractor in a collapsing universe for steep negative potentials (the ekpyrotic scenario) and stable against matter, curvature or shear perturbations. Otherwise kinetic-dominated solutions are the attractor during collapse (the pre big bang scenario) and are only marginally stable with respect to anisotropic shear. 
  We examine here in what way the pressures affect the final fate of a continual gravitational collapse. It is shown that the presence of a non-vanishing pressure gradient in the collapsing cloud determines directly the epoch of formation of trapped surfaces and the apparent horizon, thus changing the causal structure in the vicinity of singularity. 
  End state of gravitational collapse and the related cosmic censorship conjecture continue to be amongst the most important open problems in gravitation physics today. My purpose here is to bring out several aspects related to gravitational collapse and censorship, which may help towards a better understanding of the issues involved. Possible physical constraints on gravitational collapse scenarios are considered. It is concluded that the best hope for censorship lies in analyzing the genericity and stability properties of the currently known classes of collapse models which lead to the formation of naked singularities, rather than black holes, as the final state of collapse and which develop from a regular initial data. 
  A pair of wave equations is presented for the gravitational and electromagnetic perturbations of a charged black hole. One of the equations is uncoupled and determines the propagation of the electromagnetic perturbation. The other is for the propagation of the shear of a principal null direction and has a source term given by the solution of the first equation. This result is expected to have important applications in astrophysical models. 
  In brane-world theory in five dimensions, the bulk metric is usually written in gaussian coordinates, where $g_{4\mu} = 0$ and $g_{44} = - 1$. However, the choice $g_{44} = - 1$ is an external condition, not a requirement of the field equations. In this paper we study the consequences of having $g_{44} = \epsilon \Phi^2$, where $\epsilon = \pm 1$ and $\Phi$ is a scalar function varying with time, $\dot{\Phi} \neq 0$. This varying field entails the possibility of variable fundamental physical "constants". These variations are different from those predicted in scalar-tensor and multidimensional theories. We solve the five-dimensional equations for a {\em fixed} brane and use the brane-world paradigm to determine the fundamental parameters in the theory, which are the vacuum energy $\sigma$, the gravitational coupling $G$ and the cosmological term $\Lambda_{(4)}$. We present specific models where these physical quantities are variable functions of time. Different scenarios are possible but we discuss with some detail a model for which $\dot{G}/G \sim H$ and $\Lambda_{(4)} \sim H^2$, which seems to be favored by observations. Our results are not in contradiction to previous ones in the literature. In particular, to those where the brane is described as a domain wall moving in a static $Sch-AdS$ bulk. Indeed these latter models in RS scenarios describe the same spacetime as other solutions (with fixed brane) in gaussian coordinates with $\dot{\Phi} = 0 $. We conclude that the introduction of a time-varying $\Phi$ in brane-world theory yields a number of models that show variation in the fundamental physical "constants" and exhibit reasonable physical properties. 
  We write a first order symmetric hyperbolic system coupling the Riemann with the dynamical acceleration of a relativistic fluid. W determine the associated, coupled, Bel - Robinson energy, and the integral equality that it satisfies. 
  We construct a Dirac equation that is consistent with one of the recently-proposed schemes for a "doubly-special relativity", a relativity with both an observer-independent velocity scale (still naturally identified with the speed-of-light constant) and an observer-independent length/momentum scale (possibly given by the Planck length/momentum). We find that the introduction of the second observer-independent scale only induces a mild deformation of the structure of Dirac spinors. We also show that our modified Dirac equation naturally arises in constructing a Dirac equation in the kappa-Minkowski noncommutative spacetime. Previous, more heuristic, studies had already argued for a possible role of doubly-special relativity in kappa-Minkowski, but remained vague on the nature of the consistency requirements that should be implemented in order to assure the observer-independence of the two scales. We find that a key role is played by the choice of a differential calculus in kappa-Minkowski. A much-studied choice of the differential calculus does lead to our doubly-special relativity Dirac equation, but a different scenario is encountered for another popular choice of differential calculus. 
  We present a Master equation for description of fermions and bosons for special relativities with two invariant scales (c and lambda_P). We introduce canonically-conjugate variables (chi^0, chi) to (epsilon, pi) of Judes-Visser. Together, they bring in a formal element of linearity and locality in an otherwise non-linear and non-local theory. Special relativities with two invariant scales provide all corrections, say, to the standard model of the high energy physics, in terms of one fundamental constant, lambda_P. It is emphasized that spacetime of special relativities with two invariant scales carries an intrinsic quantum-gravitational character. 
  In this paper we look at the gravitational spin--spin interaction between macroscopic astronomical bodies. In particular, we calculate their post--Newtonian orbital effects of order $\mathcal{O}(c^{-2})$ on the trajectory of a spinning particle with proper angular momentum ${\bf s}$ moving in the external gravitomagnetic field generated by a central spinning mass with proper angular momentum ${\bf J}$. It turns out that, at order $\mathcal{O}(e)$ in the orbiter's eccentricity, the eccentricity the pericenter and the mean anomaly rates of the moving particle are affected by long--term harmonic effects. If, on one hand, they are undetectable in the Solar System, on the other, maybe that in an astrophysical context like that of the binary millisecond pulsars there will be some hopes of measuring them in the future. 
  It is shown that screening the background of super-strong interacting gravitons creates for any pair of bodies as an attraction force as well an repulsion force due to pressure of gravitons. For single gravitons, these forces are approximately balanced, but each of them is much bigger than a force of Newtonian attraction. If single gravitons are pairing, a body attraction force due to pressure of such graviton pairs is twice exceeding a corresponding repulsion force under the condition that graviton pairs are destructed by collisions with a body. If the considered quantum mechanism of classical gravity is realized in the nature, then an existence of black holes contradicts to Einstein's equivalence principle. In such the model, Newton's constant is proportional to $H^{2}/T^{4},$ where $H$ is the Hubble constant, $T$ is an equivalent temperature of the graviton background. The estimate of the Hubble constant is obtained $H=2.14 \cdot 10^{-18} s^{-1}$ (or $66.875 km \cdot s^{-1} \cdot Mpc^{-1}$). 
  ECG Stueckelberg (1905-1984) often published important theories years before those who would receive the Nobel Prize for their discoery. Perhpas other jewels may remain the work of this prescient genius. In a short paper in 1957--coming shortly on the heels of the suggestion and experimental confirmation of parity non-conservation by the weak force--Stueckelberg noted that sine there exist completely antisymmetric four-tensors (e) with covariant derivative zero, non-parity conserving forces can readily be accomodated by general relativity as the product of such a tensor and the non-parity conserving terms. Stueckelberg also pointed out that in classical general relativity the covariant derivative, D, of e is De=(-g)^1/2*d[e(-g)^-1/2]=0, where g is the determinant of the metric tensor, and d is an ordinary derivative. Since g is non-zero, if g is constant, e is a constant and mean proper particle lifetimes are the same in all Lorenz frames. I point out that Stueckelberg's paper has three further deep and important implications. (1) We see how gravity-- matter--can influence particle lifetimes by forcing the e tensor to be non-constant to have to "adjust" so that d[e(-g)^-1/2] is zero, and we can get some quantitative feel for the magnitude of this effect. (2) If any difference in the lifetimes of particles and antiparticles are to occur, in the absence of Lorenz invariance violation, it would be in regions where dg is non-zero. (3) If indeed, differences in particle and antiparticle lifetimes are found, before they can be ascibed to effects of quantum gravity, supersymmetry, or string theory, the effects of classical gravity need to be accounted for. 
  Many efforts have been devoted to the studies of the phenomenology in particle physics with extra dimensions. We propose degenerate fermion stars with extra dimensions and study what features characterized by the size of extra dimensions should appear in its structure. We find that Kaluza-Klein excited modes arise for the larger scale of extra dimensions and examine the conditions on which different layers should be caused in the inside of the stars. We expound how the extra dimensions affect on physical quantities. 
  In this Letter we consider the radial infall along the symmetry axis of an ultra-relativistic point particle into a rotating Kerr black hole. We use the Sasaki-Nakamura formalism to compute the waveform, energy spectra and total energy radiated during this process. We discuss possible connections between these results and the black hole-black hole collision at the speed of light process. 
  The scalar and electromagnetic fields of charges uniformly accelerated in de Sitter spacetime are constructed. They represent the generalization of the Born solutions describing fields of two particles with hyperbolic motion in flat spacetime. In the limit Lambda -> 0, the Born solutions are retrieved. Since in the de Sitter universe the infinities I^+- are spacelike, the radiative properties of the fields depend on the way in which a given point of I^+- is approached. The fields must involve both retarded and advanced effects: Purely retarded fields do not satisfy the constraints at the past infinity I^-. 
  With the goal of taking a step toward the construction of astrophysically realistic initial data for numerical simulations of black holes, we for the first time derive a family of fully general relativistic initial data based on post-2-Newtonian expansions of the 3-metric and extrinsic curvature without spin. It is expected that such initial data provide a direct connection with the early inspiral phase of the binary system. We discuss a straightforward numerical implementation, which is based on a generalized puncture method. Furthermore, we suggest a method to address some of the inherent ambiguity in mapping post-Newtonian data onto a solution of the general relativistic constraints. 
  Assuming a Friedmann universe which evolves with a power-law scale factor, $a=t^{n}$, we analyse the phase space of the system of equations that describes a time-varying fine structure 'constant', $\alpha$, in the Bekenstein-Sandvik-Barrow-Magueijo generalisation of general relativity. We have classified all the possible behaviours of $\alpha (t)$ in ever-expanding universes with different $n$ and find new exact solutions for $\alpha (t)$. We find the attractors points in the phase space for all $n$. In general, $\alpha $ will be a non-decreasing function of time that increases logarithmically in time during a period when the expansion is dust dominated ($n=2/3$), but becomes constant when $n>2/3$. This includes the case of negative-curvature domination ($n=1$). $\alpha $ also tends rapidly to a constant when the expansion scale factor increases exponentially. A general set of conditions is established for $\alpha $ to become asymptotically constant at late times in an expanding universe. 
  The gravitational field of a rigidly rotating cylinder of charged dust is found analytically. The general and all regular solutions are divided into three classes. The acceleration and the vorticity of the dust are given, as well as the conditions for the appearance of closed timelike curves. 
  A simple yet systematic new algorithm to investigate the global structure of Kerr-Newman spacetime is suggested. Namely, the global structure of \theta=const. timelike submanifolds of Kerr-Newman metric are studied by introducing a new time coordinate slightly different from the usual Boyer-Lindquist time coordinate. In addition, it is demonstrated that the possible causality violation thus far regarded to occur near the ring singularity via the development of closed timelike curves there is not really an unavoidable pathology which has plagued the Kerr-Newman solution but simply a gauge (coordinate) artifact as it disappears upon transforming from Boyer-Lindquist to the new time coordinate. This last point appears to lend support to the fact that, indeed, the Kerr-Newman spacetime is a legitimate solution to represent the interior as well as the exterior regions of a rotating, charged black hole spacetime. 
  We discuss the method of calculating the reflection coefficient using complex trajectory WKB (CWKB) approximation. This enables us to give an interpretation of non-reflecting nature of the potential under certain conditions and clarify some points, reported incorrectly elsewhere [vs:ejp] for the potential $U(x)=-U_0cosh^2(x/a)$. We show that the repeated reflectios between the turning points are essential, which most authors overlooked, in obtaining the non-reflecting c ondition. We find that the considered repeated reflection paths are in conformity with Bogolubov transformation technique. We discuss the implications of the results when applied to the particle production scenario, considering $x$ as a time variable and also stress the cosmological implications of the result with reference to radiation domonated and de Sitter spacetime. 
  Binary systems of rapidly spinning compact objects, such as black holes or neutron stars, are prime targets for gravitational wave astronomers. The dynamics of these systems can be very complicated due to spin-orbit and spin-spin couplings. Contradictory results have been presented as to the nature of the dynamics. Here we confirm that the dynamics - as described by the second post-Newtonian approximation to general relativity - is chaotic, despite claims to the contrary. When dissipation due to higher order radiation reaction terms are included, the chaos is dampened. However, the inspiral-to-plunge transition that occurs toward the end of the orbital evolution does retain an imprint of the chaotic behaviour. 
  It is shown that an embedding of the general relativity $4-$space into a flat $12-$space gives a model of gravitation with the global $U(1)-$symmetry and the discrete $D_{1}-$one. The last one may be transformed into the $SU(2)-$symmetry of the unified model, and the demand of independence of $U(1)-$ and $SU(2)-$transformations leads to the estimate $\sin^{2}\theta_{min}=0,20$ where $\theta_{min}$ is an analog of the Weinberg angle of the standard model. 
  It is often the case in numerical relativity that schemes that are known to be convergent for well posed systems are used in evolutions of weakly hyperbolic (WH) formulations of Einstein's equations. Here we explicitly show that with several of the discretizations that have been used through out the years, this procedure leads to non-convergent schemes. That is, arbitrarily small initial errors are amplified without bound when resolution is increased, independently of the amount of numerical dissipation introduced. The lack of convergence introduced by this instability can be particularly subtle, in the sense that it can be missed by several convergence tests, especially in 3+1 dimensional codes. We propose tests and methods to analyze convergence that may help detect these situations. 
  We investigate the large scale cosmological perturbation in the Universe with multiple perfect fluids. Using the long-wavelength approximation with Hamilton-Jacobi method, we derive the formula for the gauge invariant comoving curvature perturbation. As an application of our approach, we examine the large scale perturbation in a brane cosmology. 
  In a recent Letter, Schnittman and Rasio argue that they have ruled out chaos in compact binary systems since they find no positive Lyapunov exponents. In stark constrast, we find that the chaos discovered in the original paper under discussion, J.Levin, PRL, 84 3515 (2000), is confirmed by the presence of positive Lyapunov exponents. 
  Phasing formulas in a recent paper of ours' (Phys. Rev. D 63, 044023-1 (2001) are updated taking into account the recent 3.5PN results. Some misprints in the published version of the recent paper are also corrected. 
  The set of all possible spherically symmetric magnetic static Einstein-Yang-Mills field equations for an arbitrary compact semi-simple gauge group $G$ was classified in two previous papers. Local analytic solutions near the center and a black hole horizon as well as those that are analytic and bounded near infinity were shown to exist. Some globally bounded solutions are also known to exist because they can be obtained by embedding solutions for the $G=SU(2)$ case which is well understood. Here we derive some asymptotic properties of an arbitrary global solution, namely one that exists locally near a radial value $r_{0}$, has positive mass $m(r)$ at $r_{0}$ and develops no horizon for all $r>r_{0}$. The set of asymptotic values of the Yang-Mills potential (in a suitable well defined gauge) is shown to be finite in the so-called regular case, but may form a more complicated real variety for models obtained from irregular rotation group actions. 
  We utilise a form for the Hubble parameter to generate a number of solutions to the Einstein field equations with variable cosmological constant and variable gravitational constant in the presence of a bulk viscous fluid. The Hubble law utilised yields a constant value for the deceleration parameter. A new class of solutions is presented in the Robertson-Walker spacetimes. The coefficient of bulk viscosity is assumed to be a power function of the mass density. For a class of solutions, the deceleration parameter is negative which is consistent with the supernovae Ia observations. 
  In this paper general solutions are found for domain walls in Lyra geometry in the plane symmetric spacetime metric given by Taub. Expressions for the energy density and pressure of domain walls are derived in both cases of uniform and time varying displacement field $\beta$. It is also shown that the results obtained by Rahaman et al [IJMPD, {\bf 10}, 735 (2001)] are particular case of our solutions. Finally, the geodesic equations and acceleration of the test particle are discussed. 
  Evolution of an anisotropic universe described by a Bianchi type I (BI) model in presence of nonlinear spinor field has been studied by us recently in a series of papers. On offer the Bianchi models, those are both inhomogeneous and anisotropic. Within the scope of Bianchi type VI (BVI) model the self-consistent system of nonlinear spinor and gravitational fields are considered. The role of inhomogeneity in the evolution of spinor and gravitational field is studied. 
  A general relativistic wave equation is written to deal with electromagnetic waves in a metric of the form Bianchi type III. We obtain the exact form of this equation in a second order form. On the other hand, by using spinor form of the Maxwell equations the propagation problem is reduced to the solution of the second order differential equation of complex combination of electric and magnetic fields. For this two different approach we obtain spinors in terms of field strength tensor. 
  Barber's second self creation theory with bulk viscous fluid source for an LRS Bianchi type-I metric is considered by using deceleration parameter to be constant where the metric potentials are taken as function of $x$ and $t$. The coefficient of bulk viscosity is assumed to be a power function of the mass density. Some physical and geometrical features of the models are discussed 
  It is well known that the charged scalar perturbations of the Reissner-Nordstrom metric will decay slower at very late times than the neutral ones, thereby dominating in the late time signal. We show that at the stage of quasinormal ringing, on the contrary, the neutral perturbations will decay slower for RN, RNAdS and dilaton black holes. The QN frequencies of the nearly extreme RN black hole have the same imaginary parts (damping times) for charged and neutral perturbations. An explanation of this fact is not clear but, possibly, is connected with the Choptuik scaling. 
  We investigate the late time behavior of particle creation from an extremal Reissner-Nordstrom (RN) black hole formed by gravitational collapse. We calculate explicitly the particle flux associated with a massless scalar field at late times after the collapse. Our result shows that the expected number of particles in any wave packet spontaneously created from the ``in'' vacuum state approaches zero faster than any inverse power of time. This result confirms the traditional belief that extremal black holes do not emit particles. We also calculate the expectation value of the stress energy tensor in a 1+1 RN black hole and show that it also drops to zero at late times. Some comments on previous work by other authors are provided. 
  In this article and a companion paper we address the question of how one might obtain the semiclassical limit of ordinary matter quantum fields (QFT) propagating on curved spacetimes (CST) from full fledged Quantum General Relativity (QGR), starting from first principles. We stress that we do not claim to have a satisfactory answer to this question, rather our intention is to ignite a discussion by displaying the problems that have to be solved when carrying out such a program. In the present paper we propose a scheme that one might follow in order to arrive at such a limit. We discuss the technical and conceptual problems that arise in doing so and how they can be solved in principle. As to be expected, completely new issues arise due to the fact that QGR is a background independent theory. For instance, fundamentally the notion of a photon involves not only the Maxwell quantum field but also the metric operator - in a sense, there is no photon vacuum state but a "photon vacuum operator"! While in this first paper we focus on conceptual and abstract aspects, for instance the definition of (fundamental) n-particle states (e.g. photons), in the second paper we perform detailed calculations including, among other things, coherent state expectation values and propagation on random lattices. These calculations serve as an illustration of how far one can get with present mathematical techniques. Although they result in detailed predictions for the size of first quantum corrections such as the gamma-ray burst effect, these predictions should not be taken too seriously because a) the calculations are carried out at the kinematical level only and b) while we can classify the amount of freedom in our constructions, the analysis of the physical significance of possible choices has just begun. 
  The present paper is the companion of [1] in which we proposed a scheme that tries to derive the Quantum Field Theory (QFT) on Curved Spacetimes (CST) limit from background independent Quantum General Relativity (QGR). The constructions of [1] make heavy use of the notion of semiclassical states for QGR. In the present paper, we employ the complexifier coherent states for QGR recently proposed by Thiemann and Winkler as semiclassical states, and thus fill the general formulas obtained in [1] with life. We demonstrate how one can, under some simplifying assumptions, explicitely compute expectation values of the operators relevant for the gravity-matter Hamiltonians of [1] in the complexifier coherent states. These expectation values give rise to effective matter Hamiltonians on the background on which the gravitational coherent state is peaked and thus induce approximate notions of n-particle states and matter propagation on fluctuating spacetimes. We display the details for the scalar and the electromagnetic field. The effective theories exhibit two types of corrections as compared to the the ordinary QFT on CST. The first is due to the quantum fluctuations of the gravitational field, the second arises from the fact that background independence forces both geometry and matter to propagate on a spacetime that is the product of the real line and a (random) graph. Finally we obtain explicit numerical predictions for non-standard dispersion relations for the scalar and the electromagnetic field. They should, however, not be taken too seriously, due to the many ambiguities in our scheme, the analysis of the physical significance of which has only begun. We show however, that one can classify these ambiguities at least in broad terms. 
  The structure of the thermal equilibrium state of a weakly interacting Bose gas is of current interest. We calculate the density matrix of that state in two ways. The most effective method, in terms of yielding a simple, explicit answer, is to construct a generating function within the traditional framework of quantum statistical mechanics. The alternative method, arguably more interesting, is to construct the thermal state as a vector state in an artificial system with twice as many degrees of freedom. It is well known that this construction has an actual physical realization in the quantum thermodynamics of black holes, where the added degrees of freedom correspond to the second sheet of the Kruskal manifold and the thermal vector state is a state of the Unruh or the Hartle-Hawking type. What is unusual about the present work is that the Bogolubov transformation used to construct the thermal state combines in a rather symmetrical way with Bogolubov's original transformation of the same form, used to implement the interaction of the nonideal gas in linear approximation. In addition to providing a density matrix, the method makes it possible to calculate efficiently certain expectation values directly in terms of the thermal vector state of the doubled system. 
  If the gravitational field is quantized, then a solution of Einstein's field equations is a valid cosmological model only if it corresponds to a classical limit of a quantum cosmology. To determine which solutions are valid requires looking at quantum cosmology in a particular way. Because we infer the geometry by measurements on matter, we can represent the amplitude for any measurement in terms of the amplitude for the matter fields, allowing us to integrate out the gravitational degrees of freedom. Combining that result with a path-integral representation for quantum cosmology leads to an integration over 4-geometries. Even when a semiclassical approximation for the propagator is valid, the amplitude for any measurement includes an integral over the gravitational degrees of freedom. The conditions for a solution of the field equations to be a classical limit of a quantum cosmology are: (1) The effect of the classical action dominates the integration, (2) the action is stationary with respect to variation of the gravitational degrees of freedom, and (3) only one saddlepoint contributes significantly to each integration. 
  Riemannian Killing conserved currents are used to find the energy spectra of classical spinning particles moving around a spinning string in Einstein-Cartan (EC) theory of gravity. It is shown that a continuos spectrum is obtained for planar open orbits the energy shift between the upper and lower bounds obtained for circular planar orbits. A spin-spin effect between the spin of the test particle and the spin of the spinning string contributes to the energy spectrum splitting. 
  A systematic asymptotic expansion is developed for the gravitational wave degrees of freedom of a class of expanding, vacuum Gowdy cosmological spacetimes. In the wave map description of these models, the evolution of the gravitational wave amplitudes defines an orbit in the target space. The circumference of this orbit decays to zero indicating that the asymptotic spacetime is spatially homogeneous. A prescription is given to identify the asymptotic cosmological model for the gravitational wave degrees of freedom using the asymptotic point in the target space. The remaining metric function of the asymptotic cosmological model, found by solving the constraints, is determined by an effective energy of the gravitational waves rather than from the asymptotic point in the target space. 
  The helicity modification of light polarization which is induced by the gravitational deflection from a classical heavy rotating body, like a star or a planet, is considered. The expression of the helicity asymmetry is derived; this asymmetry signals the gravitationally induced spin transfer from the rotating body to the scattered photons. 
  We apply standard post-Newtonian methods in general relativity to locate the innermost circular orbit (ICO) of irrotational and corotational binary black-hole systems. We find that the post-Newtonian series converges well when the two masses are comparable. We argue that the result for the ICO which is predicted by the third post-Newtonian (3PN) approximation is likely to be very close to the ``exact'' solution, within 1% of fractional accuracy or better. The 3PN result is also in remarkable agreement with a numerical calculation of the ICO in the case of two corotating black holes moving on exactly circular orbits. The behaviour of the post-Newtonian series suggests that the gravitational dynamics of two bodies of comparable masses does not resemble that of a test particle on a Schwarzschild background. This leads us to question the validity of some post-Newtonian resummation techniques that are based on the idea that the field generated by two black holes is a deformation of the Schwarzschild space-time. 
  A free massless scalar field is coupled to homogeneous and isotropic loop quantum cosmology. The coupled model is investigated in the vicinity of the classical singularity, where discreteness is essential and where the quantum model is non-singular, as well as in the regime of large volumes, where it displays the expected semiclassical features. The particular matter content (massless, free scalar) is chosen to illustrate how the discrete structure regulates pathological behavior caused by kinetic terms of matter Hamiltonians (which in standard quantum cosmology lead to wave functions with an infinite number of oscillations near the classical singularity). Due to this modification of the small volume behavior the dynamical initial conditions of loop quantum cosmology are seen to provide a meaningful generalization of DeWitt's initial condition. 
  We have studied the properties of the static, spherically symmetric solution of Jordan, Brans-Dicke theory. An exact interior solution for standard space-time line element in the Schwarzschild form is obtained. 
  It is shown that unlike the perfect fluid case, anisotropic fluids (principal stresses unequal) may be geodesic, without this implying the vanishing of (spatial) pressure gradients. Then the condition of vanishing four acceleration is integrated in non-comoving coordinates. The resulting models are necessarily dynamic, and the mass function is expressed in terms of the fluid velocity as measured by a locally Minkowskian observer. An explicit example is worked out. 
  We propose a spin foam model for pure gauge fields coupled to Riemannian quantum gravity in four dimensions. The model is formulated for the triangulation of a four-manifold which is given merely combinatorially. The Riemannian Barrett--Crane model provides the gravity sector of our model and dynamically assigns geometric data to the given combinatorial triangulation. The gauge theory sector is a lattice gauge theory living on the same triangulation and obtains from the gravity sector the geometric information which is required to calculate the Yang--Mills action. The model is designed so that one obtains a continuum approximation of the gauge theory sector at an effective level, similarly to the continuum limit of lattice gauge theory, when the typical length scale of gravity is much smaller than the Yang--Mills scale. 
  We construct rotating hairy black holes in SU(2) Einstein-Yang-Mills theory. These stationary axially symmetric black holes are asymptotically flat. They possess non-trivial non-Abelian gauge fields outside their regular event horizon, and they carry non-Abelian electric charge. In the limit of vanishing angular momentum, they emerge from the neutral static spherically symmetric Einstein-Yang-Mills black holes, labelled by the node number of the gauge field function. With increasing angular momentum and mass, the non-Abelian electric charge of the solutions increases, but remains finite. The asymptotic expansion for these black hole solutions includes non-integer powers of the radial variable. 
  Hamiltonian mechanics of field theory can be formulated in a generally covariant and background independent manner over a finite dimensional extended configuration space. The physical symplectic structure of the theory can then be defined over a space G of three-dimensional surfaces without boundary, in the extended configuration space. These surfaces provide a preferred over-coordinatization of phase space. I consider the covariant form of the Hamilton-Jacobi equation on G, and a canonical function S on G which is a preferred solution of the Hamilton-Jacobi equation. The application of this formalism to general relativity is equivalent to the ADM formalism, but fully covariant. In the quantum domain, it yields directly the Ashtekar-Wheeler-DeWitt equation. Finally, I apply this formalism to discuss the partial observables of a covariant field theory and the role of the spin networks --basic objects in quantum gravity-- in the classical theory. 
  The cosmological constant Lambda, which has seemingly dominated the primaeval Universe evolution and to which recent data attribute a significant present-time value, is shown to have an algebraic content: it is essentially an eigenvalue of a Casimir invariant of the Lorentz group which acts on every tangent space. This is found in the context of de Sitter spacetimes but, as every spacetime is a 4-manifold with Minkowski tangent spaces, the result suggests the existence of a "skeleton" algebraic structure underlying the geometry of general physical spacetimes. Different spacetimes come from the "fleshening" of that structure by different tetrad fields. Tetrad fields, which provide the interface between spacetime proper and its tangent spaces, exhibit to the most the fundamental role of the Lorentz group in Riemannian spacetimes, a role which is obscured in the more usual metric formalism. 
  We present a procedure that allows the construction of the metric perturbations and electromagnetic four-potential, for gravitational and electromagnetic perturbations produced by sources in Kerr spacetime. This may include, for example, the perturbations produced by a point particle or an extended object moving in orbit around a Kerr black hole. The construction is carried out in the frequency domain. Previously, Chrzanowski derived the vacuum metric perturbations and electromagnetic four-potential by applying a differential operator to a certain potential $\Psi $. Here we construct $\Psi $ for inhomogeneous perturbations, thereby allowing the application of Chrzanowski's method. We address this problem in two stages: First, for vacuum perturbations (i.e. pure gravitational or electromagnetic waves), we construct the potential from the modes of the Weyl scalars $\psi_{0}$ or $\phi_{0}$. Second, for perturbations produced by sources, we express $\Psi $ in terms of the mode functions of the source, i.e. the energy-momentum tensor $T_{\alpha \beta}$ or the electromagnetic current vector $J_{\alpha}$. 
  In a recent paper (hep-th/0103228) a new initial value formulation of fermionic QFT was presented that is applicable to an arbitrary observer in any electromagnetic background. This approach suggests a consistent particle interpretation at all times, with the concept of `radar time' used to generalise this interpretation to an arbitrarily moving observer. In the present paper we extend this formalism to allow for gravitational backgrounds. The observer-dependent particle interpretation generalises Gibbons' definition to non-stationary spacetimes. This allows any observer to be considered, providing a particle interpretation that depends {\it only} on the observer's motion and the background, not on any choice of coordinates or gauge, or on details of their particle detector. Consistency with known results is demonstrated for the cases of Rindler space and deSitter space. Radar time is also considered for an arbitrarily moving observer in an arbitrary 1+1 dimensional spacetime, and for a comoving observer in a 3+1 dimensional FRW universe with arbitrary scale factor $a(t)$. Finite volume measurements and their fluctuations are also discussed, allowing one to say with definable precision where and when the particles are observed. 
  Recent theoretical developments have generated a strong interest in the ``brane-world'' picture, which assumes that ordinary matter is trapped in a three-dimensional submanifold, usually called brane, embedded in a higher dimensional space. The purpose of this review is to introduce some basic results concerning gravity in these models and then to present various aspects of the cosmology in a brane-universe. 
  We summarize recent results concerning the evolution of second order perturbations in flat dust irrotational FLRW models with $\Lambda\ne 0$. We show that asymptotically these perturbations tend to constants in time, in agreement with the cosmic no-hair conjecture. We solve numerically the second order scalar perturbation equation, and very briefly discuss its all time behaviour and some possible implications for the structure formation. 
  I give a short non-technical review of the results obtained in recent work on "Doubly Special Relativity", the relativistic theories in which the rotation/boost transformations between inertial observers are characterized by two observer-independent scales (the familiar velocity scale, $c$, and a new observer-independent length/momentum scale, naturally identified with the Planck length/momentum). I emphasize the aspects relevant for the search of a solution to the cosmic-ray paradox. 
  We discuss the question of how the number of dimensions of space and time can influence the equilibrium configurations of stars. We find that dimensionality does increase the effect of mass but not the contribution of the pressure, which is the same in any dimension. In the presence of a (positive) cosmological constant the condition of hydrostatic equilibrium imposes a lower limit on mass and matter density. We show how this limit depends on the number of dimensions and suggest that $\Lambda > 0$ is more effective in 4D than in higher dimensions. We obtain a general limit for the degree of compactification (gravitational potential on the boundary) of perfect fluid stars in $D$-dimensions. We argue that the effects of gravity are stronger in 4D than in any other number of dimensions. The generality of the results is also discussed. 
  Following Barrow, and Barrow and collaborators, we find a cosmological JBD model, with varying speed of light and varying fine structure constant, where the deceleration parameter is -1,causing acceleration of the Universe.Indeed, we have an exponential inflationary phase. Plancks time, energy, length,etc.,might have had different numerical values in the past, than those available in the litterature, due to the varying values for speed of light, and gravitational constant. 
  Considering plane gravitational waves propagating through flat spacetime, it is shown that curvatures experienced both in the starting point and during their arrival at the earth can cause a considerable shift in the frequencies as measured by earth and space-based detectors. 
  Runaway solutions can be avoided in fourth order gravity by a doubling of the matter operator algebra with a symmetry constraint with respect to the exchange of observable and hidden degrees of freedom together with the change in sign of the ghost and the dilaton fields. The theory is classically equivalent to Einstein gravity, while its non-unitary Newtonian limit is shown to lead to a sharp transition, around $10^{11}$ proton masses, from the wavelike properties of microscopic particles to the classical behavior of macroscopic bodies, as well as to a trans-Planckian regularization of collapse singularities. A unified reading of ordinary and black hole entropy emerges as entanglement entropy with hidden degrees of freedom. The emergent picture gives a substantial agreement with B-H entropy and Hawking temperature. 
  We consider quantum theoretical effects of the sudden change of the boundary conditions which mimics the occurrence of naked singularities. For a simple demonstration, we study a massless scalar field in $(1 + 1)$-dimensional Minkowski spacetime with finite spatial interval. We calculate the vacuum expectation value of the energy-momentum tensor and explicitly show that singular wave or {\em thunderbolt} appears along the Cauchy horizon. The thunderbolt possibly destroys the Cauchy horizon if its backreaction on the geometry is taken into account, leading to quantum restoration of the global hyperbolicity. The result of the present work may also apply to the situation that a closed string freely oscillating is traveling to a brane and changes itself to an open string pinned-down by the ends satisfying the Dirichlet boundary conditions on the brane. 
  Causal diamond-shaped subsets of space-time are naturally associated with operator algebras in quantum field theory, and they are also related to the Bousso covariant entropy bound. In this work we argue that the net of these causal sets to which are assigned the local operator algebras of quantum theories should be taken to be non orthomodular if there is some lowest scale for the description of space-time as a manifold. This geometry can be related to a reduction in the degrees of freedom of the holographic type under certain natural conditions for the local algebras. A non orthomodular net of causal sets that implements the cutoff in a covariant manner is constructed. It gives an explanation, in a simple example, of the non positive expansion condition for light-sheet selection in the covariant entropy bound. It also suggests a different covariant formulation of entropy bound. 
  We consider a five dimensional vacuum cosmology with Bianchi type-IX spatial geometry and an extra non-compact coordinate. Finding a new class of solutions, we examine and rule out the possibility of deterministic chaos. We interpret this result within the context of induced matter theory. 
  A class of spacetimes (comprising the Alcubierre bubble, Krasnikov tube, and a certain type of wormholes) is considered that admits `superluminal travel' in a strictly defined sense. Such spacetimes (they are called `shortcuts' in this paper) were suspected to be impossible because calculations based on `quantum inequalities' suggest that their existence would involve Planck-scale energy densities and hence unphysically large values of the `total amount of negative energy' E_tot. I argue that the spacetimes of this type may not be unphysical at all. By explicit examples I prove that: 1) the relevant quantum inequality does not (always) imply large energy densities; 2) large densities may not lead to large values of E_tot; 3) large E_tot, being physically meaningless in some relevant situations, does not necessarily exclude shortcuts. 
  The complex time WKB approximation is an effective tool in studying particle production in curved spacetime. We use it in this work to understand the formation of classical condensate in expanding de Sitter spacetime. The CWKB leads to the emergence of thermal spectrum that depends crucially on horizons (as in de Sitter spacetime) or observer dependent horizons (as in Rindler spacetime). A connection is sought between the horizon and the formation of classical condensate. We concentrate on de Sitter spacetime and study the cosmological perturbation of $k=0$ mode with various values of $m/H_0$. We find that for a minimally coupled free scalar field for $m^2/H_0^2<2$, the one-mode occupation number grows more than unity soon after the physical wavelength of the mode crosses the Hubble radius and soon after diverges as $N(t)\sim O(1)[\lambda_{phys}(t)/{H_0^{-1}}]^{2\sqrt{\nu^2-1/4}}$, where $\nu\equiv (9/4 -m^2/{H_0^2})^{1/2}$. The results substantiates the previous works in this direction. We also find the correct oscillation and behaviour of $N(z)$ at small $z$ from a single expression using CWKB approximation for various values of $m/H_0$. We also discuss decoherence in relation to the formation of classical condensate. We also find that the squeezed state formalism and CWKB method give identical results. 
  Classical instability in fourth order gravity is cured at the expense of unitarity. The appearance of hidden degrees of freedom replicating those of ordinary matter allows for ordinary thermodynamic entropy and black hole entropy to be identified with von Neumann entropy. The emergent picture gives a substantial agreement with B-H entropy and Hawking temperature. 
  Conditions for the existence of flows with non-null shear-free and expansion-free velocities in spaces with affine connections and metrics are found. On their basis, generalized Weyl's spaces with shear-free and expansion-free conformal Killing vectors as velocity's vectors of spinless test particles moving in a Weyl's space are considered. The necessary and sufficient conditions are found under which a free spinless test particle could move in spaces with affine connections and metrics on a curve described by means of an auto-parallel equation. In Weyl's spaces with Weyl's covector, constructed by the use of a dilaton field, the dilaton field appears as a scaling factor for the rest mass density of the test particle. PACS numbers: 02.40.Ky, 04.20.Cv, 04.50.+h, 04.90.+e 
  Within the framework of geodetic brane gravity, the Universe is described as a 4-dimensional extended object evolving geodetically in a higher dimensional flat background. In this paper, by introducing a new pair of canonical fields {lambda, P_{lambda}}, we derive the quadratic Hamiltonian for such a brane Universe; the inclusion of matter then resembles minimal coupling. Second class constraints enter the game, invoking the Dirac bracket formalism. The algebra of the first class constraints is calculated, and the BRST generator of the brane Universe turns out to be rank-1. At the quantum level, the road is open for canonical and/or functional integral quantization. The main advantages of geodetic brane gravity are: (i) It introduces an intrinsic, geometrically originated, 'dark matter' component, (ii) It offers, owing to the Lorentzian bulk time coordinate, a novel solution to the 'problem of time', and (iii) It enables calculation of meaningful probabilities within quantum cosmology without any auxiliary scalar field. Intriguingly, the general relativity limit is associated with lambda being a vanishing (degenerate) eigenvalue. 
  We consider the problem of searching for gravitational waves emitted during the inspiral phase of binary systems when the orbital plane precesses due to relativistic spin-orbit coupling. Such effect takes place when the spins of the binary members are misaligned with respect to the orbital angular momentum. As a first step we assess the importance of precession specifically for the first-generation of LIGO detectors. We investigate the extent of the signal-to-noise ratio reduction and, hence, detection rate that occurs when precession effects are not accounted for in the template waveforms. We restrict our analysis to binary systems that undergo the so-called simple precession and have a total mass close to 10 solar mass. We find that for binary systems with rather high mass ratios (e.g., a 1.4 solar mass neutron star and a 10 solar mass black hole) the detection rate can decrease by almost an order of magnitude. Current astrophysical estimates of the rate of binary inspiral events suggest that LIGO could detect at most a few events per year, and therefore the reduction of the detection rate even by a factor of a few is critical. In the second part of our analysis, we examine whether the effect of precession could be included in the templates by capturing the main features of the phase modulation through a small number of extra parameters. Specifically we examine and tested for the first time the 3-parameter family suggested by Apostolatos. We find that, even though these ``mimic'' templates improve the detection rate, they are still inadequate in recovering the signal-to-noise ratio at the desired level. We conclude that a more complex template family is needed in the near future, still maintaining the number of additional parameters as small as possible in order to reduce the computational costs. 
  It is shown that the Newtonian limit of a stable realization of HD gravity leads to a sharp transition, around 10^{11} proton masses, from the wavelike properties of microscopic particles to the classical behaviour of macroscopic bodies. Besides, due to nonunitarity, a pure state is expected to evolve into a microcanonical ensamble leading to thermal equilibrium even for truly closed systems. 
  This note deals with the possibility of non-trivial cosmological solutions given by quantum corrections in the framework of the Jackiw-Telteiboim model to the bidimensional gravity. The resulting model shows that the quantum corrections transform, in some cases, the classical solution into a more interesting one with initial singularity. 
  This paper contains a review of the theory and practice of gravitomagnetism, with particular attention to the different and numerous proposals which have been put forward to experimentally or observationally verify its effects. The basics of the gravitoelectromagnetic form of the Einstein equations is expounded. Then the Lense-Thirring and clock effects are described, reviewing the essentials of the theory. Space based and Earth based experiments are listed. Other effects, such as the coupling of gravitomagnetism with spin, are described and orders of magnitude are considered to give an idea of the feasibility of actual experiments. 
  A method for computing the stress-energy tensor for the quantized, massless, spin 1/2 field in a general static spherically symmetric spacetime is presented. The field can be in a zero temperature state or a non-zero temperature thermal state. An expression for the full renormalized stress-energy tensor is derived. It consists of a sum of two tensors both of which are conserved. One tensor is written in terms of the modes of the quantized field and has zero trace. In most cases it must be computed numerically. The other tensor does not explicitly depend on the modes and has a trace equal to the trace anomaly. It can be used as an analytic approximation for the stress-energy tensor and is equivalent to other approximations that have been made for the stress-energy tensor of the massless spin 1/2 field in static spherically symmetric spacetimes. 
  A new Michelson-Morley experiment is proposed by measuring the beat frequency of two near degenerate modes with orthogonal propagation in a single spherical resonator. The unique properties of the experiment allow: 1. Substantial common mode rejection of some noise sources: 2. Simple calculation of the signal if Special Relativity is violated. We show that optimum filtering may be used to increase the signal to noise ratio, and to search for a preferred direction of the speed of light. Using this technique we show that a sensitivity limit of order 7.10^-19 is possible by integrating data over one month. We propose methods to veto systematic effects by correlating the output of more than one experiment. 
  It was shown that if in Quantum Theory a fundamental length exists and a well-known measurement procedure is used, then the density matrix at the Planck scale cannot be defined in the usual way, because in this case density matrix trace is strongly less than one.  Density matrix must be changed by a progenitrix or as we call it throughout this paper, density pro-matrix. This pro-matrix is a deformed density matrix, which at low energy limit turns to usual one.  Below the explicit form of the deformation is described. Implications of obtained results are summarized as well as their application to the interpretation of Information Paradox on the Black Holes. 
  The unexpected discovery of hairy black hole solutions in theories with scalar fields simply by considering asymptotically Anti de-Sitter, rather than asymptotically flat, boundary conditions is analyzed in a way that exhibits in a clear manner the differences between the two situations.   It is shown that the trivial Schwarzschild Anti de Sitter becomes unstable in some of these situations, and the possible relevance of this fact for the ADS/CFT conjecture is pointed out. 
  It is shown that, contrary to previous claims, a scalar tensor theory of Brans-Dicke type provides a relativistic generalization of Newtonian gravity in 2+1 dimensions. The theory is metric and test particles follow the space-time geodesics. The static isotropic solution is studied in vacuum and in regions filled with an incompressible perfect fluid. It is shown that the solutions can be consistently matched at the matter vacuum interface, and that the Newtonian behavior is recovered in the weak field regime. 
  We discuss some computational problems associated to matched filtering of experimental signals from gravitational wave interferometric detectors in a parallel-processing environment. We then specialize our discussion to the use of the APEmille and apeNEXT processors for this task. Finally, we accurately estimate the performance of an APEmille system on a computational load appropriate for the LIGO and VIRGO experiments, and extrapolate our results to apeNEXT. 
  The invariance under unitary representations of the conformal group SL(2,R) of a quantum particle is rigorously investigated in two-dimensional spacetimes containing Killing horizons using DFF model. The limit of the near-horizon approximation is considered. If the Killing horizon is bifurcate the conformal symmetry is hidden, i.e. it does not arise from geometrical spacetime isometries, but the whole Hilbert space turns out to be an irreducible unitary representation of SL(2,R) and the time evolution is embodied in the unitary representation. In this case the symmetry does not depend on the mass of the particle and, if the representation is faithful, the conformal observable K shows thermal properties. If the Killing horizon is nonbifurcate the conformal symmetry is manifest, i.e. it arises from geometrical spacetime isometries. The SL(2,R) representation which arises from the geometry selects a hidden conformal representation. Also in that case the Hilbert space is an irreducible representation of SL(2,R) and the group conformal symmetries embodies the time evolution with respect to the local Killing time. However no thermal properties are involved. The conformal observable K gives rise to Killing time evolution of the quantum state with respect to another global Killing time present in the manifold. Mathematical proofs about the developed machinery are supplied and features of the operator H_g = -({d^2}/{dx^2})+ ({g}/{x^2}), with g=-1/4 are discussed. It is proven that a statement, used in the recent literature, about the spectrum of self-adjoint extensions of H_g is incorrect. 
  The Klein-Gordon-Einstein equations of classical real scalar fields have time-dependent solutions (periodic in time). We show that quantum real scalar fields can form non-oscillating (static) solitonic objects, which are quite similar to the solutions describing boson stars formed with classical and quantum complex scalar fields (the latter will be studied in this paper). We numerically analyze the difference between them concerning the mass of boson stars. On the other hand, we suggest an interesting test (a viable process that the boson star may undergo in the early universe) for the formation of boson stars. That is, it is questioned that after a second-order phase transition (a simple toy model will be considered here), what is the fate of the boson star composed of quantum real scalar field. 
  In this paper Quantum Mechanics with Fundamental Length is chosen as Quantum Mechanics at Planck's scale. This is possible due to the presence in the theory of General Uncertainty Relations (GUR). Here Quantum Mechanics with Fundamental Length is obtained as a deformation of Quantum Mechanics. The distinguishing feature of the proposed approach in comparison with previous ones, lies on the fact that here density matrix subjects to deformation whereas so far commutators have been deformed. The density matrix obtained by deformation of quantum-mechanical density one is named throughout this paper density pro-matrix, which at low energy limit turns to the density matrix. This transition corresponds to non-unitary one from Quantum Mechanics with GUR to Quantum mechanics. Below the implications of obtained results are enumerated.New view on the Black Holes Information Paradox are discussed 
  To measure the standard quantum limit (SQL) a high quality transducer must be coupled to a high quality mechanical system. Due to its monolithic nature, the monolithic sapphire transducer (MST) has high quality factors for both types of resonances. Single loop suspension is shown to yield a mechanical quality factor of 6.10^8 at 4 K. From standard analysis we show the MST has the potential to measure noise fluctuations of the mechanical oscillator at the SQL. also, we point out a new way to determine if the transducer back action is quantum limited. We show that if the fluctuations are at the quantum limit, then the amplitude of the oscillation will be amplified by the ratio of the ringdown time to the measurement time, which is an inherently easier measurement. 
  I examine the structure of the deformed Lorentz transformations in one of the recently-proposed schemes with two observer-independent scales. I develop a technique for the analysis of general combinations of rotations and deformed boosts. In particular, I verify explicitly that the transformations form group. 
  This paper studies wormhole solutions to Einstein gravity with an arbitrary number of time dependent compact dimensions and a matter-vacuum boundary. A new gauge is utilized which is particularly suited for studies of the wormhole throat. The solutions possess arbitrary functions which allow for the description of infinitely many wormhole systems of this type and, at the stellar boundary, the matter field is smoothly joined to vacuum. It turns out that the classical vacuum structure differs considerably from the four dimensional theory and is therefore studied in detail. The presence of the vacuum-matter boundary and extra dimensions places interesting restrictions on the wormhole. For example, in the static case, the radial size of a weak energy condition (WEC) respecting throat is restricted by the extra dimensions. There is a critical dimension, D=5, where this restriction is eliminated. In the time dependent case, one \emph{cannot} respect the WEC at the throat as the time dependence actually tends the solution towards WEC violation. This differs considerably from the static case and the four dimensional case. 
  We investigate the intrinsic parity of black holes. It appears that discrete symmetries require the black hole Hilbert space to be larger than suggested by the usual quantum numbers M (mass), Q (charge) and J (angular momentum). Recent results on black hole production in trans-Planckian scattering lead to gravitational effects which do not decouple from low-energy physics. Dispersion relations incorporating these effects imply that the semi-classical black hole spectrum is similar in parity even and odd channels. This result can be generalized to other discrete and continuous symmetries. 
  In this paper we classify all 4+1 cosmological models where the spatial hypersurfaces are connected and simply connected homogeneous Riemannian manifolds. These models come in two categories, multiply transitive and simply transitive models. There are in all five different multiply transitive models which cannot be considered as a special case of a simply transitive model. The classification of simply transitive models, relies heavily upon the classification of the four dimensional (real) Lie algebras. For the orthogonal case, we derive all the equations of motion and give some examples of exact solutions. Also the problem of how these models can be compactified in context with the Kaluza-Klein mechanism, is addressed. 
  Dynamical horizons are considered in full, non-linear general relativity. Expressions of fluxes of energy and angular momentum carried by gravitational waves across these horizons are obtained. Fluxes are local, the energy flux is positive and change in the horizon area is related to these fluxes. The flux formulae also give rise to balance laws analogous to the ones obtained by Bondi and Sachs at null infinity and provide generalizations of the first and second laws of black hole mechanics. 
  In this paper, we consider the Casimir energy of massless scalar field which satisfy Dirichlet boundary condition on a spherical shell. Outside the shell, the spacetime is assumed to be described by the Schwarzschild metric, while inside the shell it is taken to be the flat Minkowski space. Using zeta function regularization and heat kernel coefficients we isolate the divergent contributions of the Casimir energy inside and outside the shell, then using the renormalization procedure of the bag model the divergent parts are cancelled, finally obtaining a renormalized expression for the total Casimir energy. 
  This work is devoted to the discussion of an idea that gravitational interactions might be residual interactions of strong and electromagnetic interactions. Then, absence of the carriers of the gravitational interactions finds a natural explanation in the framework this idea. Besides, since masses (charges of the gravitational interactions) of particles are generated in strong, electromagnetic (and possibly, in other) interactions and if masses of the particles will be not generated in these interactions (i.e. $m \equiv 0$), then the gravitational interactions do not appear. That is also indirect confirmation of the considered idea. Connections between charge of the gravitational and other interactions are considered. 
  We consider Einstein Born-Infeld theory with a null fluid in Kerr-Schild Geometry. We find accelerated charge solutions of this theory. Our solutions reduce to the Plebanski solution when the acceleration vanishes and to the Bonnor-Vaidya solution as the Born-Infeld parameter b goes to infinity. We also give the explicit form of the energy flux formula due to the acceleration of the charged sources. 
  In the search for a covariant formulation for Loop Quantum Gravity, spin foams have arised as the corresponding discrete space-time structure and, among the different models, the Barrett-Crane model seems the most promising. Here, we study its boundary states and introduce cylindrical functions on both the Lorentz connection and the time normal to the studied hypersurface. We call them projected cylindrical functions and we explain how they would naturally arise in a covariant formulation of Loop Quantum Gravity. 
  The hypothesis that the Lorentz transformations may be modified at Planck scale energies is further explored. We present a general formalism for theories which preserve the relativity of inertial frames with a non-linear action of the Lorentz transformations on momentum space. Several examples are discussed in which the speed of light varies with energy and elementary particles have a maximum momenta and/or energy. Energy and momentum conservation are suitably generalized and a proposal is made for how the new transformation laws apply to composite systems. We then use these results to explain the ultra high energy cosmic ray anomaly and we find a form of the theory that explains the anomaly, and leads also to a maximum momentum and a speed of light that diverges with energy. We finally propose that the spatial coordinates be identified as the generators of translation in Minkowski spacetime. In some examples this leads to a commutative geometry, but with an energy dependent Planck constant. 
  In this letter we provide a reformulation of the strong cosmic censor conjecture taking into account recent results on Malament--Hogarth space-times. We claim that the strong version of the cosmic censor conjecture can be formulated by postulating that a physically relevant space-time is either globally hyperbolic or possesses the Malament--Hogarth property. But it is known that a Malament--Hogarth space-time in principle is capable for performing non-Turing computations such as checking consistency of ZFC set theory. In this way we get an intimate conjectured link between the cosmic censorship scenario and computability theory. 
  The second of the annual BritGrav meetings on current research in Gravitational Physics in Britain took place at the School of Mathematical Sciences of Queen Mary, University of London on June 10/11, 2002. 47 plenary talks of 12min duration were given. We make available the abstracts of the talks and the references to the electronic preprints at arXiv.org where they exist. 
  It is generally known that linear (free) field theories are one of the few QFT that are exactly soluble. In the Schroedinger functional description of a scalar field on flat Minkowski spacetime and for flat embeddings, it is known that the usual Fock representation is described by a Gaussian measure. In this paper, arbitrary globally hyperbolic space-times and embeddings of the Cauchy surface are considered. The classical structures relevant for quantization are used for constructing the Schroedinger representation in the general case. It is shown that in this case, the measure is also Gaussian. Possible implications for the program of canonical quantization of midisuperspace models are pointed out. 
  Granted the post-Lorentzian relativistic kinematic transformations are described in the Finslerian framework, the uniformity between the actual light velocity anisotropy change and the anisotropic deformation of measuring rods can be the reason proper for the null results of the Michelson-Morley-type experiments at the first-order level. 
  The interaction energy between two black holes at large separation distance is calculated. The first term in the expansion corresponds to the Newtonian interaction between the masses. The second term corresponds to the spin-spin interaction. The calculation is based on the interaction energy defined on the two black holes initial data. No test particle approximation is used. The relation between this formula and cosmic censorship is discussed. 
  Several aspects of scalar field dynamics on a brane which differs from corresponding regimes in the standard cosmology are investigated. We consider asymptotic solution near a singularity, condition for inflation and bounces and some detail of chaotic behavior in the brane model. Each results are compared with those known in the standard cosmology. 
  In this paper we discuss stability properties of various discretizations for axisymmetric systems including the so called cartoon method which was proposed by Alcubierre, Brandt et.al. for the simulation of such systems on Cartesian grids. We show that within the context of the method of lines such discretizations tend to be unstable unless one takes care in the way individual singular terms are treated. Examples are given for the linear axisymmetric wave equation in flat space. 
  Some issues in the numerical treatment of the conformal field equations are discussed. In particular, the problem of obtaining smooth initial data for the hyperboloidal initial value problem is described and solution methods are presented. 
  We describe in this article a new code for evolving axisymmetric isolated systems in general relativity. Such systems are described by asymptotically flat space-times which have the property that they admit a conformal extension. We are working directly in the extended `conformal' manifold and solve numerically Friedrich's conformal field equations, which state that Einstein's equations hold in the physical space-time. Because of the compactness of the conformal space-time the entire space-time can be calculated on a finite numerical grid. We describe in detail the numerical scheme, especially the treatment of the axisymmetry and the boundary. 
  The Einstein initial-value equations in the extrinsic curvature (Hamiltonian) representation and conformal thin sandwich (Lagrangian) representation are brought into complete conformity by the use of a decomposition of symmetric tensors which involves a weight function. In stationary spacetimes, there is a natural choice of the weight function such that the transverse traceless part of the extrinsic curvature (or canonical momentum) vanishes. 
  Numerical studies of the gravitational collapse of a stiff (P=rho) fluid have found the now familiar critical phenomena, namely scaling of the black hole mass with a critical exponent and continuous self-similarity at the threshold of black hole formation. Using the equivalence of an irrotational stiff fluid to a massless scalar field, we construct the critical solution as a scalar field solution by making a self-similarity ansatz. We find evidence that this solution has exactly one growing perturbation mode; both the mode and the critical exponent, gamma ~ 0.94, derived from its eigenvalue agree with those measured in perfect fluid collapse simulations. We explain why this solution is seen as a critical solution in stiff fluid collapse but not in scalar field collapse, and conversely why the scalar field critical solution is not seen in stiff fluid collapse, even though the two systems are locally equivalent. 
  A self-consistent field method is developed, which can be used to construct models of differentially rotating stars to first post-Newtonian order. The rotation law is specified by the specific angular momentum distribution j(m), where m is the baryonic mass fraction inside the surface of constant specific angular momentum. The method is then used to compute models of the nascent neutron stars resulting from the accretion induced collapse of white dwarfs. The result shows that the ratios of kinetic energy to gravitational binding energy of the relativistic models are slightly smaller than the corresponding values of the Newtonian models. 
  We present the first computations of quasiequilibrium binary neutron stars with different mass components in general relativity, within the Isenberg-Wilson-Mathews approximation. We consider both cases of synchronized rotation and irrotational motion. A polytropic equation of state is used with the adiabatic index gamma=2. The computations have been performed for the following combinations of stars: (M/R)_{star 1} vs. (M/R)_{star 2} = 0.12 vs. (0.12, 0.13, 0.14), 0.14 vs. (0.14, 0.15, 0.16), 0.16 vs. (0.16, 0.17, 0.18), and 0.18 vs. 0.18, where (M/R) denotes the compactness parameter of infinitely separated stars of the same baryon number. It is found that for identical mass binary systems there is no turning point of the binding energy (ADM mass) before the end point of the sequence (mass shedding point) in the irrotational case, while there is one before the end point of the sequence (contact point) in the synchronized case. On the other hand, in the different mass case, the sequence ends by the tidal disruption of the less massive star (mass shedding point). It is then more difficult to find a turning point in the ADM mass. Furthermore, we find that the deformation of each star depends mainly on the orbital separation and the mass ratio and very weakly on its compactness. On the other side, the decrease of the central energy density depends on the compactness of the star and not on that of the companion. 
  The slow-rotation approximation of Hartle is developed to a setting where a charged rotating fluid is present. The linearized Einstein-Maxwell equations are solved on the background of the Reissner-Nordstrom space-time in the exterior electrovacuum region. The theory is put to action for the charged generalization of the Wahlquist solution found by Garcia. The Garcia solution is transformed to coordinates suitable for the matching and expanded in powers of the angular velocity. The two domains are then matched along the zero pressure surface using the Darmois-Israel procedure. We prove a theorem to the effect that the exterior region is asymptotically flat if and only if the parameter C_{2}, characterizing the magnitude of an external magnetic field, vanishes. We obtain the form of the constant C_{2} for the Garcia solution. We conjecture that the Garcia metric cannot be matched to an asymptotically flat exterior electrovacuum region even to first order in the angular velocity. This conjecture is supported by a high precision numerical analysis. 
  According to very recent developments of the LARES mission, which would be devoted to the measurement of the general relativistic Lense--Thirring effect in the gravitational field of the Earth with Satellite Laser Ranging, it seems that the LARES satellite might be finally launched in a polar, low--altitude orbit by means of a relatively low--cost rocket. The observable would be the node only. In this letter we critically analyze this scenario. 
  We consider a situation in which two metrics are joined at a null hypersurface. It often occurs that the union of the two metrics gives rise to a Ricci tensor that contains a term proportional to a Dirac delta-function supported on the hypersurface. This singularity is associated with a thin distribution of matter on the hypersurface, and following Barrabes and Israel, we seek to determine its stress-energy tensor in terms of the geometric properties of the null hypersurface. While our treatment here does not deviate strongly from their previous work, it offers a simplification of the computational operations involved in a typical application of the formalism, and it gives rise to a stress-energy tensor that possesses a more recognizable phenomenology. Our reformulation of the null-shell formalism makes systematic use of the null generators of the singular hypersurface, which define a preferred flow to which the flow of matter can be compared. This construction provides the stress-energy tensor with a simple characterization in terms of a mass density, a mass current, and an isotropic pressure. Our reformulation also involves a family of freely-moving observers that intersect the surface layer and perform measurements on it. This construction gives operational meaning to the stress-energy tensor by fixing the argument of the delta-function to be proper time as measured by these observers. 
  The nonlinear aspect of gravitational wave generation that produces power at harmonics of the orbital frequency, above the fundamental quadrupole frequency, is examined to see what information about the source is contained in these higher harmonics. We use an order (4/2) post-Newtonian expansion of the gravitational wave waveform of a binary system to model the signal seen in a spaceborne gravitational wave detector such as the proposed LISA detector. Covariance studies are then performed to determine the ultimate accuracy to be expected when the parameters of the source are fit to the received signal. We find three areas where the higher harmonics contribute crucial information that breaks degeneracies in the model and allows otherwise badly-correlated parameters to be separated and determined. First, we find that the position of a coalescing massive black hole binary in an ecliptic plane detector, such as OMEGA, is well-determined with the help of these harmonics. Second, we find that the individual masses of the stars in a chirping neutron star binary can be separated because of the mass dependence of the harmonic contributions to the wave. Finally, we note that supermassive black hole binaries, whose frequencies are too low to be seen in the detector sensitivity window for long, may still have their masses, distances, and positions determined since the information content of the higher harmonics compensates for the information lost when the orbit-induced modulation of the signal does not last long enough to be apparent in the data. 
  This paper describes an approach that uses flat-spacetime dimension estimators to estimate the manifold dimension of causal sets that can be faithfully embedded into curved spacetimes. The approach is invariant under coarse graining and can be implemented independently of any specific curved spacetime. Results are given based on causal sets generated by random sprinklings into conformally flat spacetimes in 2, 3, and 4 dimensions, as well as one generated by a percolation dynamics. 
  We study the space geometry of a rotating disk both from a theoretical and operational approach, in particular we give a precise definition of the space of the disk, which is not clearly defined in the literature. To this end we define an extended 3-space, which we call relative space: it is recognized as the only space having an actual physical meaning from an operational point of view, and it is identified as the 'physical space of the rotating platform'. Then, the geometry of the space of the disk turns out to be non Euclidean, according to the early Einstein's intuition; in particular the Born metric is recovered, in a clear and self consistent context. Furthermore, the relativistic kinematics reveals to be self consistent, and able to solve the Ehrenfest's paradox without any need of dynamical considerations or ad hoc assumptions. 
  An implicit fundamental assumption in relativistic perturbation theory is that there exists a parametric family of spacetimes that can be Taylor expanded around a background. The choice of the latter is crucial to obtain a manageable theory, so that it is sometime convenient to construct a perturbative formalism based on two (or more) parameters. The study of perturbations of rotating stars is a good example: in this case one can treat the stationary axisymmetric star using a slow rotation approximation (expansion in the angular velocity Omega), so that the background is spherical. Generic perturbations of the rotating star (say parametrized by lambda) are then built on top of the axisymmetric perturbations in Omega. Clearly, any interesting physics requires non-linear perturbations, as at least terms lambda Omega need to be considered. In this paper we analyse the gauge dependence of non-linear perturbations depending on two parameters, derive explicit higher order gauge transformation rules, and define gauge invariance. The formalism is completely general and can be used in different applications of general relativity or any other spacetime theory. 
  A program was recently initiated to bridge the gap between the Planck scale physics described by loop quantum gravity and the familiar low energy world. We illustrate the conceptual problems and their solutions through a toy model: quantum mechanics of a point particle. Maxwell fields will be discussed in the second paper of this series which further develops the program and provides details. 
  We derive the quantization of action, particle number, and electric charge in a Lagrangian spin bundle over M equivalent M_# union D_J, Penrose's conformal compactification of Minkowsky space, with the world tubes of massive particles removed. 
  In classical Kaluza-Klein theory, with compactified extra dimensions and without scalar field, the rest mass as well as the electric charge of test particles are constants of motion. We show that in the case of a large extra dimension this is no longer so. We propose the Hamilton-Jacobi formalism, instead of the geodesic equation, for the study of test particles moving in a five-dimensional background metric. This formalism has a number of advantages: (i) it provides a clear and invariant definition of rest mass, without the ambiguities associated with the choice of the parameters used along the motion in 5D and 4D, (ii) the electromagnetic field can be easily incorporated in the discussion, and (iii) we avoid the difficulties associated with the "splitting" of the geodesic equation. For particles moving in a general 5D metric, we show how the effective rest mass, as measured by an observer in 4D, varies as a consequence of the large extra dimension. Also, the fifth component of the momentum changes along the motion. This component can be identified with the electric charge of test particles. With this interpretation, both the rest mass and the charge vary along the trajectory. The constant of motion is now a combination of these quantities. We study the cosmological variations of charge and rest mass in a five-dimensional bulk metric which is used to embed the standard k = 0 FRW universes. The time variations in the fine structure "constant" and the Thomson cross section are also discussed. 
  The problems of hazardous radiation and collisions with matter on a warp driven ship pose considerable obstacles to this possibility for interstellar travel. A solution to these problems lies in the Broeck metric. It is demonstrated that both threats to the ship will be greatly reduced. It is also shown that the horizon problem no longer exists. 
  We define and study a new kind of relation between two diffeomorphic Lorentzian manifolds called {\em causal relation}, which is any diffeomorphism characterized by mapping every causal vector of the first manifold onto a causal vector of the second. We perform a thorough study of the mathematical properties of causal relations and prove in particular that two given Lorentzian manifolds (say $V$ and $W$) may be causally related only in one direction (say from $V$ to $W$, but not from $W$ to $V$). This leads us to the concept of causally equivalent (or {\em isocausal} in short) Lorentzian manifolds as those mutually causally related. This concept is more general and of a more basic nature than the conformal relationship, because we prove the remarkable result that a conformal relation $\f$ is characterized by the fact of being a causal relation of the {\em particular} kind in which both $\f$ and $\f^{-1}$ are causal relations. For isocausal Lorentzian manifolds there are one-to-one correspondences, which sometimes are non-trivial, between several classes of their respective future (and past) objects. A more important feature of isocausal Lorentzian manifolds is that they satisfy the same causality constraints. This indicates that the causal equivalence provides a possible characterization of the {\it basic causal structure}, in the sense of mutual causal compatibility, for Lorentzian manifolds. Thus, we introduce a partial order for the equivalence classes of isocausal Lorentzian manifolds providing a classification of spacetimes in terms of their causal properties, and a classification of all the causal structures that a given fixed manifold can have. A full abstract inside the paper. 
  Important characteristics of the loop approach to quantum gravity are a specific choice of the algebra A of observables and of a representation of A on a measure space over the space of generalized connections. This representation is singled out by its elegance and diffeomorphism covariance.   Recently, in the context of the quest for semiclassical states, states of the theory in which the quantum gravitational field is close to some classical geometry, it was realized that it might also be worthwhile to study different representations of the algebra A of observables.   The content of the present note is the observation that under some mild assumptions, the mathematical structure of representations of A can be analyzed rather effortlessly, to a certain extent: Each representation can be labeled by sets of functions and measures on the space of (generalized) connections that fulfill certain conditions.   These considerations are however mostly of mathematical nature. Their physical content remains to be clarified, and physically interesting examples are yet to be constructed. 
  In this work we investigate the question, under what conditions Hilbert spaces that are induced by measures on the space of generalized connections carry a representation of certain non-Abelian analogues of the electric flux. We give the problem a precise mathematical formulation and start its investigation. For the technically simple case of U(1) as gauge group, we establish a number of ``no-go theorems'' asserting that for certain classes of measures, the flux operators can not be represented on the corresponding Hilbert spaces.   The flux-observables we consider play an important role in loop quantum gravity since they can be defined without recurse to a background geometry, and they might also be of interest in the general context of quantization of non-Abelian gauge theories. 
  The Bazanski approach, for deriving the geodesic equations in Riemannian geometry, is generalized in the absolute parallelism geometry. As a consequence of this generalization three path equations are obtained. A striking feature in the derived equations is the appearance of a torsion term with a numerical coefficients that jumps by a step of one half from equation to another. This is tempting to speculate that the paths in absolute parallelism geometry might admit a quantum feature. 
  The gedanken-experiment of Einstein's lift is analyzed in order of determining whether the free-falling observer inside the lift can detect the eventual topological non-triviality of space-time 
  We investigate the interaction of the gravitational field with a quantum particle. First, we give the proof of the equality of the inertial and the gravitational mass for the nonrelativistic quantum particle, independently of the equivalence principle. Second, we show that the macroscopic body cannot be described by the many-particle Quantum Mechanics. As an important tool we generalize the Bargmann's theory of ray representations and explain the connection with the state vector reduction problem. The Penrose's hypothesis is discussed, i.e. the hypothesis that the gravitational field may influence the state vector reduction. 
  The scalar normal modes of higher dimensional gravitating kink solutions are derived. By perturbing to second order the gravity and matter parts of the action in the background of a five-dimensional kink, the effective Lagrangian of the scalar fluctuations is derived and diagonalized in terms of a single degree of freedom which invariant under infinitesimal diffeomorphisms. The spectrum of the normal modes is discussed and applied to the analysis of short distance corrections to Newton law. 
  The Finslerian post-Lorentzian kinematic transformations can explicitly be obtained under uni-directional breakdown of spatial isotropy, provided that the requirement that the relativistic unit hypersurface (indicatrix or mass shell) be a space of constant negative curvature is still fulfilled. The method consists in evaluating respective Finslerian tetrads and then treating them as the bases of inertial reference frames. The Transport Synchronization has rigorously been proven, which opens up the ways proper to favour the concept of one-way light velocity. Transition to the Hamiltonian treatment is straightforward, so that the Finslerian transformation laws for momenta and frequences, as well as due Finslerian corrections to Doppler effect, become clear. An important common feature of the ordinary pseudo-Euclidean theory of special relativity and of the Finslerian relativistic approach under study is that they both endeavour to establish a universal prescription for applying the theory to systems in differing states of motion. 
  We have found the graviton contribution to the one-loop quantum correction to the Newton law. This correction results in interaction decreasing with distance as 1/r^3 and is dominated numerically by the graviton contribution. The previous calculations of this contribution to the discussed effect are demonstrated to be incorrect. 
  In this paper we obtain the Fourier Transform of a continuous gravitational wave. We have analysed the data set for (i) one year observation time and (ii) arbitrary observation time, for arbitrary location of detector and source taking into account the effects arising due to rotational as well as orbital motion of the earth. As an application of the transform we considered spin down and N-component signal analysis. 
  We classify all spherically symmetric spacetimes admitting a kinematic self-similar vector of the second, zeroth or infinite kind. We assume that the perfect fluid obeys either a polytropic equation of state or an equation of state of the form $p=K\mu$, where $p$ and $\mu$ are the pressure and the energy density, respectively, and $K$ is a constant. We study the cases in which the kinematic self-similar vector is not only ``tilted'' but also parallel or orthogonal to the fluid flow. We find that, in contrast to Newtonian gravity, the polytropic perfect-fluid solutions compatible with the kinematic self-similarity are the Friedmann-Robertson-Walker solution and general static solutions. We find three new exact solutions which we call the dynamical solutions (A) and (B) and $\Lambda$-cylinder solution, respectively. 
  We consider the problem of critical gravitational collapse of a scalar field in 2+1 dimensions with spherical (circular) symmetry. After surveying all the analytic, continuously self-similar solutions and considering their global structure, we examine their perturbations with the intent of understanding which are the critical solutions with a single unstable mode. The critical solution which we find is the one which agrees most closely with that found in numerical evolutions. However, the critical exponent which we find does not seem to agree with the numerical result. 
  We will expose a preliminary study on the feasibility of an experiment leading to a direct measurement of the gravitomagnetic field generated by the rotational motion of the Earth. This measurement would be achieved by means of an appropriate coupling of a TELEscope and a Foucault PENdulum in a laboratory on ground, preferably at the SOUTH pole. An experiment of this kind was firstly proposed by Braginski, Polnarev and Thorne, 18 years ago, but it was never re-analyzed. 
  A scientific analysis of the conditions under which gravity could be controlled and the implications that an hypothetical manipulation of gravity would have for known schemes of space propulsion have been the scope of a recent study carried out for the European Space Agency. The underlying fundamental physical principles of known theories of gravity were analysed and shown that even if gravity could be modified it would bring somewhat modest gains in terms of launching of spacecraft and no breakthrough for space propulsion. 
  A model for a noncommutative scalar field coupled to gravity is proposed via an extension of the Moyal product. It is shown that there are solutions compatible with homogeneity and isotropy to first non-trivial order in the perturbation of the star-product, with the gravity sector described by a flat Robertson-Walker metric. We show that in the slow-roll regime of a typical chaotic inflationary scenario, noncommutativity has negligible impact. 
  We report on a new two-parameter class of cosmological solutions to the Einstein-Maxwell equations. The solutions have everywhere regular curvature invariants. We prove that the solutions are geodesically complete and globally hyperbolic. 
  The dynamics of collapsing and expanding cylindrically symmetric gravitational and matter fields with lightlike wave-fronts is studied in General Relativity, using the Barrabes-Israel method. As an application of the general formulae developed, the collapse of a matter field that satisfies the condition R_{AB}g^{AB} = 0, (A, B = z, phi), in an otherwise flat spacetime background is studied. In particular, it is found that the gravitational collapse of a purely gravitational wave or a null dust fluid cannot be realized in a flat spacetime background. The studies are further specified to the collapse of purely gravitational waves and the general conditions for such collapse are found. It is shown that after the waves arrive at the axis, in general, part of them is reflected to spacelike infinity along the future light cone, and part of it is focused to form spacetime singularities on the symmetry axis. The cases where the collapse does not result in the formation of spacetime singularities are also identified. 
  We investigate a particle velocity in the $\kappa$-Minkowski space-time, which is one of the realization of a noncommutative space-time. We emphasize that arrival time analyses by high-energy $\gamma$-rays or neutrinos, which have been considered as powerful tools to restrict the violation of Lorentz invariance, are not effective to detect space-time noncommutativity. In contrast with these examples, we point out a possibility that {\it low-energy massive particles} play an important role to detect it. 
  We present cylindrically symmetric, static solutions of the Einstein field equations around a line singularity such that the energy momentum tensor corresponds to infinitely thin photonic shells. Positivity of the energy density of the thin shell and the line singularity is discussed. It is also shown that thick shells containing mostly radiation are possible by a numerical solution. 
  The 24 components of the relativistic spin tensor consist of 3+3 basic spin fields and 9+9 constitutive fields. Empirically only 3 basic spin fields and 9 constitutive fields are known. This empirem can be expressed by two spin axioms, one of them identifying 3 spin fields, and the other one 9 constitutive fields to each other. This identification by the spin axioms is material-independent and does not mix basic spin fields with constitutive properties. The approaches to the Weyssenhoff fluid and the Dirac-electron fluid found in literature are discussed with regard to these spin axioms. The conjecture is formulated, that another reduction from 6 to 3 basic spin fields which does not obey the spin axioms introduces special material properties by not allowed mixing of constitutive and basic fields. 
  An E\"otv\"os experiment to test the weak equivalence principle (WEP) for zero-point vacuum energy is proposed using a satellite. Following the suggestion of Ross for a terrestrial experiment of this type, the acceleration of a spherical test mass of aluminum would be compared with that of a similar test mass made from another material. The estimated ratio of the zero-point vacuum energy density inside the aluminum sphere to the rest mass energy density is ~ 1.6 X 10^{-14}, which would allow a 1% resolution of a potential WEP violation observed in a satellite mission test that had a baseline sensitivity to WEP violations of ~ 10^{-16}. An observed violation of the WEP for vacuum energy density would constitute a significant clue as to the origin of the cosmological constant and the source of dark energy, and test a recently proposed resolution of the cosmological constant problem, based on a model of nonlocal quantum gravity and quantum field theory. 
  No, it cannot in the following sense if a self-gravitating vacuum brane is concerned. Once we write down the full set of linear perturbation equations of the system containing a self-gravitating brane, we will see that such a brane does not have its own dynamical degrees of freedom independent of those of gravitational waves which propagate in the surrounding spacetime. This statement seems to contradict with our intuition that a brane fluctuates freely on a given background spacetime in the lowest order approximation. Based on this intuition, we usually think that the dynamics of a brane can be approximately described by the equations derived from the Nambu-Goto action. In this paper we fill the gap residing between these two descriptions, showing that the dynamics of a self-gravitating brane is in fact similar to that described by a non-gravitating brane on a fixed background spacetime when the weak backreaction condition we propose in this paper is satisfied. 
  Stochastic gravitational waves (SGW) can be detected by measuring a cross-correlation of two or more gravitational wave (GW) detectors. In this paper we describe an optimal SGW search technique in the wavelet domain. It uses a sign correlation test, which allows calculation of the cross- correlation significance for non-Gaussian data. We also address the problem of correlated noise for the GW detectors. A method that allows calculation of the cross-correlation variance, when data is affected by correlated noise, is developed. As a part of the optimal search technique a robust estimator for detector noise spectral amplitude is introduced. It is not sensitive to outliers and allows application of the search technique to non-stationary data. 
  Quantum gravity has been so elusive because we have tried to approach it by two paths which can never meet: quantum mechanics and general relativity. These contradict each other not only in superdense regimes, but also in the vacuum.   We explore a straight road to quantum gravity here--the one mandated by Clifford-algebra covariance. This bridges the gap from microscales--where the massive Dirac propagator is a sum over null zig-zags--to macroscales--where we see the energy-momentum current, *T and the resulting Einstein curvature, *G. For massive particles, *T flows in the "cosmic time" direction, y^0--centrifugally in an expanding universe.   Neighboring centrifugal currents of *T present opposite spacetime vorticities *G to the boundaries of each others' worldtubes, so they advect--i.e. attract, as we show here by integrating a Spin^c-4 Lagrangian by parts in the spinfluid regime.   This boundary integral not only explains why stress-energy *T is the source for gravitational curvature *G, but also gives a value for the gravitational constant, kappa(x^0) that depends on the current scale factor of our expanding Friedmann 3-brane. On the microscopic scale, quantum gravity appears naturally as the statistical mechanics of null zig-zags of massive particles in "imaginary time," y^0. 
  The scattering of a straight, infinitely long string by a rotating black hole is considered. We assume that a string is moving with velocity v and that initially the string is parallel to the axis of rotation of the black hole. We demonstrate that as a result of scattering, the string is displaced in the direction perpendicular to the velocity by an amount kappa(v,b), where b is the impact parameter. The late-time solution is represented by a kink and anti-kink, propagating in opposite directions at the speed of light, and leaving behind them the string in a new ``phase''. We present the results of the numerical study of the string scattering and their comparison with the weak-field approximation, valid where the impact parameter is large, b/M >> 1, and also with the scattering by a non-rotating black hole which was studied in earlier works. 
  The Riemannian 10j symbols are spin networks that assign an amplitude to each 4-simplex in the Barrett-Crane model of Riemannian quantum gravity. This amplitude is a function of the areas of the 10 faces of the 4-simplex, and Barrett and Williams have shown that one contribution to its asymptotics comes from the Regge action for all non-degenerate 4-simplices with the specified face areas. However, we show numerically that the dominant contribution comes from degenerate 4-simplices. As a consequence, one can compute the asymptotics of the Riemannian 10j symbols by evaluating a `degenerate spin network', where the rotation group SO(4) is replaced by the Euclidean group of isometries of R^3. We conjecture formulas for the asymptotics of a large class of Riemannian and Lorentzian spin networks in terms of these degenerate spin networks, and check these formulas in some special cases. Among other things, this conjecture implies that the Lorentzian 10j symbols are asymptotic to 1/16 times the Riemannian ones. 
  We compute the overlap function between Post-Newtonian (PN) templates and gravitational signals emitted by binary systems composed of one neutron star and one point mass, obtained by a perturbative approach. The calculations are performed for different stellar models and for different detectors, to estimate how effectual and faithful the PN templates are, and to establish whether effects related to the internal structure of neutron stars may possibly be extracted by the matched filtering technique. 
  This paper has been withdrawn by the author, due to an error in the proof of lemma 7.4. However, numerical evidence strongly suggest that this lemma is true. 
  The radiation emitted by charged, scalar particles in a Schwarzschild field with maximal acceleration corrections is calculated classically and in the tree approximation of quantum field theory. In both instances the particles emit radiation that has characteristics similar to those of gamma-ray bursters. 
  The gravitational radiation degrees of freedom of freedom are described in the framework of the 3+1 decomposition of spacetime. The relationship with eigenfields of the Kidder-Scheel-Teukolsky (KST) equations is established. This relationship is used to fix a parameter in the KST equations which is related to the ordering ambiguity of space derivatives in the Ricci tensor, which is inherent to first order evolution systems, like the ones currently used in Numerical Relativity applications. 
  In the literature, the matchings between spacetimes have been most of the times implicitly assumed to preserve some of the symmetries of the problem involved. But no definition for this kind of matching was given until recently. Loosely speaking, the matching hypersurface is restricted to be tangent to the orbits of a desired local group of symmetries admitted at both sides of the matching and thus admitted by the whole matched spacetime. This general definition is shown to lead to conditions on the properties of the preserved groups. First, the algebraic type of the preserved group must be kept at both sides of the matching hypersurface. Secondly, the orthogonal transivity of two-dimensional conformal (in particular isometry) groups is shown to be preserved (in a way made precise below) on the matching hypersurface. This result has in particular direct implications on the studies of axially symmetric isolated bodies in equilibrium in General Relativity, by making up the first condition that determines the suitability of convective interiors to be matched to vacuum exteriors. The definition and most of the results presented in this paper do not depend on the dimension of the manifolds involved nor the signature of the metric, and their applicability to other situations and other higher dimensional theories is manifest. 
  A space-based superconducting gravitational low-frequency wave detector is considered. Sensitivity of the detector is sufficient to use the detector as a partner of other contemporary low-frequency detectors like LIGO and LISA. This device can also be very useful for experimental study of other effects predicted by theories of gravitation. 
  We solve the Klein-Gordon and Dirac equations in an open cosmological universe with a partially horn topology in the presence of a time dependent magnetic field. Since the exact solution cannot be obtained explicitly for arbitrary time-dependence of the field, we discuss the asymptotic behavior of the solutions with the help of the relativistic Hamilton-Jacobi equation. 
  The development of quantum cosmology, in which Stephen Hawking played a crucial role, has frequently encountered substantial conceptual and technical difficulties related to the problem of time in quantum gravity and to general issues concerning the foundations of quantum theory. In this contribution to Stephen's 60th Birthday Conference, I describe some recent work in which the decoherent histories approach to quantum theory is used to quantize simple cosmological models and perhaps shed some light on some of these difficulties. 
  The Davies-Fulling model describes the scattering of a massless field by a moving mirror in 1+1 dimensions. When the mirror travels under uniform acceleration, one encounters severe problems which are due to the infinite blue shift effects associated with the horizons. On one hand, the Bogoliubov coefficients are ill-defined and the total energy emitted diverges. On the other hand, the instantaneous mean flux vanishes. To obtained well-defined expressions we introduce an alternative model based on an action principle. The usefulness of this model is to allow to switch on and off the interaction at asymptotically large times. By an appropriate choice of the switching function, we obtain analytical expressions for the scattering amplitudes and the fluxes emitted by the mirror. When the coupling is constant, we recover the vanishing flux. However it is now followed by transients which inevitably become singular when the switching off is performed at late time. Our analysis reveals that the scattering amplitudes (and the Bogoliubov coefficients) should be seen as distributions and not as mere functions. Moreover, our regularized amplitudes can be put in a one to one correspondence with the transition amplitudes of an accelerated detector, thereby unifying the physics of uniformly accelerated systems. In a forthcoming article, we shall use our scattering amplitudes to analyze the quantum correlations amongst emitted particles which are also ill-defined in the Davies-Fulling model in the presence of horizons. 
  An analog of black hole can be realized in the low-temperature laboratory. The horizon can be constructed for the `relativistic' ripplons (surface waves) living on the brane. The brane is represented by the interface between two superfluid liquids, 3He-A and 3He-B, sliding along each other without friction. Similar experimental arrangement has been recently used for the observation and investigation of the Kelvin-Helmholtz type of instability in superfluids (cond-mat/0111343). The shear-flow instability in superfluids is characterized by two critical velocities. The lowest threshold measured in recent experiments (cond-mat/0111343) corresponds to appearance of the ergoregion for ripplons. In the modified geometry this will give rise to the black-hole event horizon in the effective metric experienced by ripplons. In the region behind the horizon, the brane vacuum is unstable due to interaction with the higher-dimensional world of bulk superfluids. The time of the development of instability can be made very long at low temperature. This will allow us to reach and investigate the second critical velocity -- the proper Kelvin-Helmholtz instability threshold. The latter corresponds to the singularity inside the black hole, where the determinant of the effective metric becomes infinite. 
  The relations for G-dot in multidimensional model with Ricci-flat internal space and multicomponent perfect fluid are obtained. A two-component example: dust + 5-brane, is considered. 
  The Robinson-Trautman type N solutions, which describe expanding gravitational waves, are investigated for all possible values of the cosmological constant Lambda and the curvature parameter epsilon. The wave surfaces are always (hemi-)spherical, with successive surfaces displaced in a way which depends on epsilon. Explicit sandwich waves of this class are studied in Minkowski, de Sitter or anti-de Sitter backgrounds. A particular family of such solutions which can be used to represent snapping or decaying cosmic strings is considered in detail, and its singularity and global structure is presented. 
  We calculate dimensional reduction of gravitational flux tube solutions in the scheme of Kaluza-Klein theory. The fifth dimension is compacified to a region of Planck size. Assuming the width of the tube to be also Planck size we obtain string-like object with physical fields originated from an initial 5D metric. The dynamics of these fields is inner one. 
  One of the conceptual tensions between quantum mechanics (QM) and general relativity (GR) arises from the clash between the spatial nonseparability of entangled states in QM, and the complete spatial separability of all physical systems in GR, i.e., between the nonlocality implied by the superposition principle, and the locality implied by the equivalence principle. Experimental consequences of this conceptual tension will be explored for macroscopically coherent quantum fluids, such as superconductors, superfluids, and atomic Bose-Einstein condensates (BECs), subjected to tidal and Lense-Thirring fields arising from gravitational radiation. A Meissner-like effect is predicted, in which the Lense-Thirring field is expelled from the bulk of a quantum fluid. Superconductors are predicted to be macroscopic quantum gravitational antennas and transducers, which can directly convert upon reflection a beam of quadrupolar electromagnetic radiation into gravitational radiation, and vice versa, and thus serve as both sources and receivers of gravitational waves. An estimate of the transducer conversion efficiency on the order of unity comes out of the Ginzburg-Landau theory for an extreme type II, dissipationless superconductor with minimal coupling to weak gravitational and electromagnetic radiation fields, whose frequency is smaller than the BCS gap frequency, thus satisfying the quantum adiabatic theorem. The concept of ``the impedance of free space for gravitational plane waves'' is introduced, and leads to a natural impedance-matching process, in which the two kinds of radiation fields are impedance-matched to each other around a hundred coherence lengths beneath the surface of the superconductor. A simple, Hertz-like experiment has been performed to test these ideas, and preliminary results will be reported. 
  We study the effects of an external magnetic field, which is assumed to be uniform at infinity, on the marginally stable circular motion of charged particles in the equatorial plane of a rotating black hole. We show that the magnetic field has its greatest effect in enlarging the region of stability towards the event horizon of the black hole. Using the Hamilton-Jacobi formalism we obtain the basic equations governing the marginal stability of the circular orbits and their associated energies and angular momenta. As instructive examples, we review the case of the marginal stability of the circular orbits in the Kerr metric, as well as around a Schwarzschild black hole in a magnetic field. For large enough values of the magnetic field around a maximally rotating black hole we find the limiting analytical solutions to the equations governing the radii of marginal stability. We also show that the presence of a strong magnetic field provides the possibility of relativistic motions in both direct and retrograde innermost stable circular orbits around a Kerr black hole. 
  We investigate circularly symmetric static solutions in three-dimensional gravity with a minimally coupled massive scalar field. We integrate numerically the field equations assuming asymptotic flatness, where black holes do not exist and a naked singularity is present. We also give a brief review on the massless cases with cosmological constant. 
  The interesting early history of the cosmological term is reviewed, beginning with its introduction by Einstein in 1917 and ending with two papers of Zel'dovich, shortly before the advent of spontaneously broken gauge theories. Beside classical aspects, I shall also mention some unpublished early remarks by Pauli on possible contributions of vacuum energies in quantum field theory. 
  We estimate the possible variations of the gravitational constant G in the framework of a generalized (Bergmann-Wagoner-Nordtvedt) scalar-tensor theory of gravity on the basis of the field equations, without using their special solutions. Specific estimates are essentially related to the values of other cosmological parameters (the Hubble and acceleration parameters, the dark matter density etc.), but the values of G-dot/G compatible with modern observations do not exceed 10^{-12}. 
  Homodyne detection is one of the ways to circumvent the standard quantum limit for a gravitational wave detector. In this paper it will be shown that the same quantum-non-demolition effect using homodyne detection can be realized by heterodyne detection with unbalanced RF sidebands. Furthermore, a broadband quantum-non-demolition readout scheme can also be realized by the unbalanced sideband detection. 
  A cosmological model in which primordial black holes (PBHs) are present in the cosmic fluid at some instant t=t_0 is investigated. The time t_0 is naturally identified with the end of the inflationary period. The PBHs are assumed to be nonrelativistic in the comoving fluid, to have the same mass, and may be subject to evaporation for t>t_0. Our present work is related to an earlier paper of Zimdahl and Pavon [Phys. Rev. D {\bf 58}, 103506 (1998)], but in contradistinction to these authors we assume that the (negative) production rate of the PBHs is zero. This assumption appears to us to be more simple and more physical. Consequences of the formalism are worked out. In particular, the four-divergence of the entropy four-vector in combination with the second law in thermodynamics show in a clear way how the the case of PBH evaporation corresponds to a production of entropy. Accretion of radiation onto the black holes is neglected. We consider both a model where two different sub-fluids interact, and a model involving one single fluid only. In the latter case an effective bulk viscosity naturally appears in the formalism. 
  We argue that the complex numbers are an irreducible object of quantum probability. This can be seen in the measurements of geometric phases that have no classical probabilistic analogue. Having complex phases as primitive ingredient implies that we need to accept non-additive probabilities. This has the desirable consequence of removing constraints of standard theorems about the possibility of describing quantum theory with commutative variables. Motivated by the formalism of consistent histories and keeping an analogy with the theory of stochastic processes, we develop a (statistical) theory of quantum processes. They are characterised by the introduction of a "density matrix" on phase space paths -thus including phase information- and fully reproduce quantum mechanical predictions. In this framework wecan write quantum differential equations, that could be interpreted as referring to a single system (in analogy to Langevin's equation). We describe a reconstruction theorem by which a quantum process can yield the standard Hilbert space structure if the Markov property is imposed. Finally, we discuss the relevance of our iresults for the interpretation of quantum theory (a sample space if possible if probabilities are non-additive) and quantum gravity (the Hilbert space arises after the consideration of a background causal structure). 
  We derive all the axi-symmetric, vacuum and electrovac extremal isolated horizons. It turns out that for every horizon in this class, the induced metric tensor, the rotation 1-form potential and the pullback of the electromagnetic field necessarily coincide with those induced by the monopolar, extremal Kerr-Newman solution on the event horizon. We also discuss the general case of a symmetric, extremal isolated horizon. In particular, we analyze the case of a two-dimensional symmetry group generated by two null vector fields. Its relevance to the classification of all the symmetric isolated horizons, including the non-extremal once, is explained. 
  In this work the algebra of charges of diffeomorphisms at the horizon of generic black holes is analyzed within first order gravity. This algebra reproduces the algebra of diffeomorphisms at the horizon, (Diff(S^1)), without central extension. 
  We study the orbital evolution of a radiation-damped binary in the extreme mass ratio limit, and the resulting waveforms, to one order beyond what can be obtained using the conservation laws approach. The equations of motion are solved perturbatively in the mass ratio (or the corresponding parameter in the scalar field toy model), using the self force, for quasi-circular orbits around a Schwarzschild black hole. This approach is applied for the scalar model. Higher-order corrections yield a phase shift which, if included, may make gravitational-wave astronomy potentially highly accurate. 
  Using a new approach, we propose an analog of the Fizeau effect for massive and massless particles in an effective optical medium derived from the static, spherically symmetric gravitational field. The medium is naturally perceived as a dispersive medium by matter de Broglie waves. Several Fresnel drag coefficients are worked out, with appropriate interpretations of the wavelengths used. In two cases, it turns out that the coefficients become independent of the wavelength even if the equivalent medium itself is dispersive. A few conceptual issues are also addressed in the process of derivation. It is shown that some of our results complement recent works dealing with real fluid or optical black holes. 
  In the context of the teleparallel equivalent of general relativity, the Weitzenbock manifold is considered as the limit of a suitable sequence of discrete lattices composed of an increasing number of smaller an smaller simplices, where the interior of each simplex (Delaunay lattice) is assumed to be flat. The link lengths between any pair of vertices serve as independent variables, so that torsion turns out to be localized in the two dimensional hypersurfaces (dislocation triangle, or hinge) of the lattice. Assuming that a vector undergoes a dislocation in relation to its initial position as it is parallel transported along the perimeter of the dual lattice (Voronoi polygon), we obtain the discrete analogue of the teleparallel action, as well as the corresponding simplicial vacuum field equations. 
  The Bianchi type I cosmological model is brought into a form where the evolution of observables is governed by the unconstrained Hamiltonian that coincides with the Hamiltonian describing the relative motion of particles in the integrable three-body hyperbolic Euler-Calogero-Sutherland system. 
  The doomsday argument is a probabilistic argument that claims to predict the total lifetime of the human race. By examining the case of an individual lifetime, I conclude that the argument is fundamentally related to consciousness. I derive a reformulation stating that an infinite conscious lifetime is not possible even in principle. By considering a hypothetical conscious computer, running a non-terminating program, I deduce that consciousness cannot be generated by a single set of deterministic laws. Instead, I hypothesize that consciousness is generated by a superposition of brain states that is simultaneously associated with many quasi-classical histories, each following a different set of deterministic laws. I generalize the doomsday argument and discover that it makes no prediction in this case. Thus I conclude that the very fact of our consciousness provides us with evidence for a many-worlds interpretation of reality in which our future is not predictable using anthropic reasoning. 
  We present three different theoretically foreseen, but unusual, astrophysical situations where the gravitational lens equation ends up being the same, thus producing a degeneracy problem. These situations are (a) the case of gravitational lensing by exotic stresses (matter violating the weak energy condition and thus having a negative mass, particular cases of wormholes solutions can be used as an example), (b) scalar field gravitational lensing  (i.e. when considering the appearance of a scalar charge in the lensing scenario), and (c) gravitational lensing in closed universes (with antipodes).The reasons that lead to this degeneracy in the lens equations, the possibility of actually encountering it in the real universe, and eventually the ways to break it, are discussed. 
  By using the thermodynamic theory of irreversible processes and Einstein general relativity, a cosmological model is proposed where the early universe is considered as a mixture of a scalar field with a matter field. The scalar field refers to the inflaton while the matter field to the classical particles. The irreversibility is related to a particle production process at the expense of the gravitational energy and of the inflaton energy. The particle production process is represented by a non-equilibrium pressure in the energy-momentum tensor. The non-equilibrium pressure is proportional to the Hubble parameter and its proportionality factor is identified with the coefficient of bulk viscosity. The dynamic equations of the inflaton and the Einstein field equations determine the time evolution of the cosmic scale factor, the Hubble parameter, the acceleration and of the energy densities of the inflaton and matter. Among other results it is shown that in some regimes the acceleration is positive which simulates an inflation. Moreover, the acceleration decreases and tends to zero in the instant of time where the energy density of matter attains its maximum value. 
  Owing to Earth's rotation a free-fall body would move in an elliptical orbit rather than along a straight line forward to the center of the Earth. In this paper on the basis of the theory for spin-spin coupling between macroscopic rotating bodies we study violation of the equivalence principle from long-distance free-fall experiments by means of a rotating ball and a non-rotating sell. For the free-fall time of 40 seconds, the difference between the orbits of the two free-fall bodies is of the order of 10^{-9}cm which could be detected by a SQUID magnetometer owing to such a magnetometer can be used to measure displacements as small as 10^{-13} centimeters. 
  Dirac equation is written in a non-Riemannian spacetime with torsion and non-metricity by lifting the connection from the tangent bundle to the spinor bundle over spacetime. Foldy-Wouthuysen transformation of the Dirac equation in a Schwarzschild background spacetime is considered and it is shown that both the torsion and non-metricity couples to the momentum and spin of a massive, spinning particle. However, the effects are small to be observationally significant. 
  We discuss the motion of neutral and charged particles in a plane electro-magnetic wave and its accompanying gravitational field. 
  We consider the 3-body problem in relativistic lineal gravity and obtain an exact expression for its Hamiltonian and equations of motion. While general-relativistic effects yield more tightly-bound orbits of higher frequency compared to their non-relativistic counterparts, as energy increases we find in the equal-mass case no evidence for either global chaos or a breakdown from regular to chaotic motion, despite the high degree of non-linearity in the system. We find numerical evidence for a countably infinite class of non-chaotic orbits, yielding a fractal structure in the outer regions of the Poincare plot. 
  We construct a particular class of quantum states for a massless, minimally coupled free scalar field which are of the form of a superposition of the vacuum and multi-mode two-particle states. These states can exhibit local negative energy densities. Furthermore, they can produce an arbitrarily large amount of negative energy in a given region of space at a fixed time. This class of states thus provides an explicit counterexample to the existence of a spatially averaged quantum inequality in four-dimensional spacetime. 
  I perform an independent analysis of radio Doppler tracking data from the Pioneer 10 spacecraft for the time period 1987-1994. All of the tracking data were taken from public archive sources, and the analysis tools were developed independently by myself. I confirm that an apparent anomalous acceleration is acting on the Pioneer 10 spacecraft, which is not accounted for by present physical models of spacecraft navigation. My best fit value for the acceleration, including corrections for systematic biases and uncertainties, is (8.60 +/- 1.34) x 10^{-8} cm s^{-2}, directed towards the Sun. This value compares favorably to previous results. I examine the robustness of my result to various perturbations of the analysis method, and find agreement to within +/- 5%. The anomalous acceleration is reasonably constant with time, with a characteristic variation time scale of > 70 yr. Such a variation timescale is still too short to rule out on-board thermal radiation effects, based on this particular Pioneer 10 data set. 
  We study cosmological braneworld models with a single timelike extra dimension. Such models admit the intriguing possibility that a contracting braneworld experiences a natural bounce without ever reaching a singular state. This feature persists in the case of anisotropic braneworlds under some additional and not very restrictive assumptions. Generalizing our study to braneworld models containing an induced brane curvature term, we find that a FRW-type singularity is once again absent if the bulk extra dimension is timelike. In this case, the universe either has a non-singular origin or commences its expansion from a quasi-singular state during which both the Hubble parameter and the energy density and pressure remain finite while the curvature tensor diverges. The non-singular and quasi-singular behaviour which we have discovered differs both qualitatively and quantitatively from what is usually observed in braneworld models with spacelike extra dimensions and could have interesting cosmological implications. 
  By mapping the signal-recycling (SR) optical configuration to a three-mirror cavity, and then to a single detuned cavity, we express SR optomechanical dynamics, input--output relation and noise spectral density in terms of only three characteristic parameters: the (free) optical resonant frequency and decay time of the entire interferometer, and the laser power circulating in arm cavities. These parameters, and therefore the properties of the interferometer, are invariant under an appropriate scaling of SR-mirror reflectivity, SR detuning, arm-cavity storage time and input power at beamsplitter. Moreover, so far the quantum-mechanical description of laser-interferometer gravitational-wave detectors, including radiation-pressure effects, was only obtained at linear order in the transmissivity of arm-cavity internal mirrors. We relax this assumption and discuss how the noise spectral densities change. 
  In the quest to develop viable designs for third-generation optical interferometric gravitational-wave detectors (e.g., LIGO-III and EURO), one strategy is to monitor the relative momentum or speed of the test-mass mirrors, rather than monitoring their relative position. A previous paper analyzed a straightforward but impractical design for a {\it speed-meter interferometer} that accomplishes this. This paper describes some practical variants of speed-meter interferometers. Like the original interferometric speed meter, these designs {\it in principle} can beat the gravitational-wave standard quantum limit (SQL) by an arbitrarily large amount, over an arbitrarily wide range of frequencies. These variants essentially consist of a Michelson interferometer plus an extra "sloshing" cavity that sends the signal back into the interferometer with opposite phase shift, thereby cancelling the position information and leaving a net phase shift proportional to the relative velocity. {\it In practice}, the sensitivity of these variants will be limited by the maximum light power $W_{\rm circ}$ circulating in the arm cavities that the mirrors can support and by the leakage of vacuum into the optical train at dissipation points. In the absence of dissipation and with a squeezed vacuum of power squeeze factor ~ 0.1 inserted into the output port so as to keep the circulating power down, the SQL can be beat by a factor 10 in power at all frequencies below some chosen $f_{\rm opt}\simeq 100$ Hz, with $W_{\rm circ}\simeq 800$ kW. Estimates are given of the amount by which vacuum leakage at dissipation points will debilitate this sensitivity; these losses are 10% or less over most of the frequency range of interest. 
  Using a novel approach, we work out the general relativistic effects on the quantum interference of de Broglie waves associated with thermal neutrons. The unified general formula is consistent with special relativistic results in the flat space limit. It is also shown that the exact geodesic equation contains in a natural way a gravitational analog of the Aharonov-Bohm effect. We work out two examples, one in general relativity and the other in heterotic string theory, in order to obtain the first order gravitational correction terms to the quantum fringe shift. Measurement of these terms is closely related to the validity of the equivalence principle at a quantum level. 
  According to quantum measurement theory, "speed meters" -- devices that measure the momentum, or speed, of free test masses -- are immune to the standard quantum limit (SQL). It is shown that a Sagnac-interferometer gravitational-wave detector is a speed meter and therefore in principle it can beat the SQL by large amounts over a wide band of frequencies. It is shown, further, that, when one ignores optical losses, a signal-recycled Sagnac interferometer with Fabry-Perot arm cavities has precisely the same performance, for the same circulating light power, as the Michelson speed-meter interferometer recently invented and studied by P. Purdue and the author. The influence of optical losses is not studied, but it is plausible that they be fairly unimportant for the Sagnac, as for other speed meters. With squeezed vacuum (squeeze factor $e^{-2R} = 0.1$) injected into its dark port, the recycled Sagnac can beat the SQL by a factor $ \sqrt{10} \simeq 3$ over the frequency band $10 {\rm Hz} \alt f \alt 150 {\rm Hz}$ using the same circulating power $I_c\sim 820$ kW as is used by the (quantum limited) second-generation Advanced LIGO interferometers -- if other noise sources are made sufficiently small. It is concluded that the Sagnac optical configuration, with signal recycling and squeezed-vacuum injection, is an attractive candidate for third-generation interferometric gravitational-wave detectors (LIGO-III and EURO). 
  Based on the equivalence between a gauge theory for the translation group and general relativity, a teleparallel version of the non-abelian Kaluza-Klein theory is constructed. In this theory, only the fiber-space turns out to be higher-dimensional, spacetime being kept always four-dimensional. The resulting model is a gauge theory that unifies, in the Kaluza-Klein sense, gravitational and gauge fields. In contrast to the ordinary Kaluza-Klein models, this theory defines a natural length-scale for the compact sub-manifold of the fiber space, which is shown to be of the order of the Planck length. 
  We obtain classes of two dimensional static Lorentzian manifolds, which through the supersymmetric formalism of quantum mechanics admit the exact solvability of Dirac equation on these curved backgrounds. Specially in the case of a modified supersymmetric harmonic oscillator the wave function and energy spectrum of Dirac equation is given explicitly. 
  The classical theory of gravity is formulated as a gauge theory on a frame bundle with spontaneous symmetry breaking caused by the existence of Dirac fermionic fields. The pseudo-Riemannian metric (tetrad field) is the corresponding Higgs field. We consider two variants of this theory. In the first variant, gravity is represented by the pseudo-Riemannian metric as in general relativity theory; in the second variant, it is represented by the effective metric as in the Logunov relativistic theory of gravity. The configuration space, Dirac operator, and Lagrangians are constructed for both variants. 
  In this article we discuss a few aspects of the space-time description of fields and particles. In sectionn II and III we demonstrate that fields are as fundamental as particles. In section IV we discuss non-equivalence of the Schwarzschild coordinates and the Kruskal-Szekeres coordinates. In section V we discuss that it is not possible to define causal structure in discrete space-time manifolds. In App.B we show that a line is not just a collection of points and we will have to introduce one-dimensional line-intervals as fundamental geometric elements. Similar discussions are valid for area and volume-elements. In App.C and App.D we make a comparative study of Quantum Field Theory and Quantum Mechanics and contradictions associated with probabilistics interpretation of these theories with space-time dimensional analysis. In App.E and App.F we discuss the geometry of Robertson-Walker model and electrostatic behavior of dielectrics respectively. In Sup.I we discuss the regularity of Spin-Spherical harmonics and also derive an energy-spectrum which is free of back-reaction problem. In Sup.II we discuss that in general the integral version of Gauss's divergence law in Electrodynamics is not valid and rederive Gauss's law and Ampere's law. We also show that under duality transformation magnetic charge conservation law do not remain time reversal symmetric. In Sup.III we derive the complete equation for viscous compressible fluids and make a few comments regarding some cotradictions associated with boundary conditions for fluid dynamics. In Sup.IV we discuss a few aspects on double slit interference experiments. We conclude this article with a few questions in Sup.V. 
  Since 1983 the meter is defined to be the "length of the path travelled by light in vacuum during a time interval of 1/299792458 of a second". If there was exactly one single consistent method of synchronizing clocks, or if all corresponding methods were equivalent, one could infer from the validity of special relativity theory on a definite value of the one-way speed of light c in inertial frames.   It is true that sufficiently slowly separated clocks always show middle in time reflection when sending and receiving light signals. But a simple consideration proves that the one-way speed of light is not a constant in rotating systems, in principle detectable with only one clock.   On basis of a new internal synchronization method, this also affects all local inertial frames on rotating Earth, too, violating an absolute constancy of the one-way speed of light. This is not a contradiction to Einstein's original theory of relativity but to the present definition of the meter. Based on the constant local average value c of light going there and back, however, a modification for the definition of the meter is suggested, which would not depend on synchronization of any distant clocks.   Appendix: FitzGerald-Lorentz contraction and time dilation, according to Einstein's generally accepted understanding, should be purely kinematic effects and would not need any dynamic explanation. But an analysis of Ehrenfest's paradox of the rotating disk shows that it is not possible to separate exactly relativistic kinematics from dynamics. The necessity of such a fundamental restriction is well known for a long time - though only from quantum mechanics till now. 
  Motivated by Wheeler's bottom up pregeometry, we introduce a pregeometric approach that does not assume Wheeler's probability amplitudes for establishing spacetime neighborhoods. Rather, a non-trivial metric is produced via the concept of a uniformity base, which is generated with discrete topological groups over some arbitrary fundamental, denumerable set. We show how the concept of entourage multiplication for the elements of our uniformity base mirrors the underlying group structure. This fact is then exploited to create entourage sequences of maximal length, whence a fine metric structure. The resulting metric structure is, for certain group structures, consistent with E4-embeddable graphs. Examples over Z2 x Z4, D4, Z6, D3, Z8, and Z5 are provided. Euclidean embeddability over Z7 and Q8 is discussed. Unlike the statistical approaches typical of graph theory, this method generates dimensionality over low-order sets. Possible applications to the pregeometric modeling of quantum stochasticity and non-locality/non-separability, wave function collapse, and the M4 structure of spacetime are provided in the context of Z2 x Z2 x Z4. 
  I study ultralight particle creation which becomes from the Yukawa interaction between the inflaton and the thermal bath during fresh inflation. Particle creation is important in the first stages of fresh inflation, when the nonequilibrium thermal effects are important. I find that the number density of the ultralight created particles is more important as the scale factor growth rate is more large (i.e., for $p$ large -- $a \sim t^p$). Ultralight boson fields created during fresh inflation could be an alternative mechanism to cosmological constant to explain the discrepancy between the observed $\Omega_m \simeq 0.2$ and $\Omega_{tot} \simeq 1$, predicted by inflationary models. 
  The 5d dilatonic gravity action with surface counterterms, motivated by the AdS/CFT correspondence and with contributions of brane quantum CFTs is considered with an AdS-like bulk. The role of quantum brane CFT consists in inducing complicated brane dilatonic gravity. For exponential bulk potentials, a number of AdS-like bulk spaces is found in an analytical form. The corresponding flat or curved (de Sitter or hyperbolic) dilatonic two-branes are created. 
  The metric perturbation induced by a particle in the Schwarzschild background is usually calculated in the Regge-Wheeler (RW) gauge, whereas the gravitational self-force is known to be given by the tail part of the metric perturbation in the harmonic gauge. Thus, to identify the gravitational self-force correctly in a specified gauge, it is necessary to find out a gauge transformation that connects these two gauges. This is called the gauge problem. As a direct approach to solve the gauge problem, we formulate a method to calculate the metric perturbation in the harmonic gauge on the Schwarzshild backgound. We apply the Fourier-harmonic expansion to the metric perturbation and reduce the problem to the gauge transformation of the Fourier-harmonic coefficients (radial functions) from the RW gauge to the harmonic gauge. We derive a set of decoupled radial equations for the gauge transformation. These equations are found to have a simple second-order form for the odd parity part and the forms of spin $s=0$ and 1 Teukolsky equations for the even parity part. As a by-product, we correct typos in Zerilli's paper and present a set of corrected equations in Appendix. 
  We consider a generalized scalar-tensor theory, where we let the coupling function $\omega(\phi)$ and the effective cosmological constants $\Lambda(\phi)$ undetermined. We obtain general expressions for $\omega(\phi)$ and $\Lambda(\phi)$ in terms of the scalar field and the scale factor, and show that $\omega(\phi)$ depends on the scalar field and the scale factor in a complicated way. In order to study the conditions for an accelerated expansion at the present time and a decelerated expansion in the past, we assume a power law evolution for the scalar field and the scale factor. We analyse the required conditions that allow our model to satisfy the weak field limits on $\omega(\phi)$, and at the same time, to obtain the correct values of cosmological parameters, as the energy density $ \Omega_{m0}$ and cosmological constant $\Lambda(t_0)$. We also study the conditions for a decelerated expansion at an early time dominated by radiation. We find values for $\omega(\phi)$ and $\Lambda(\phi)$ consistent with the expectations of a theory where the cosmological constant decreases with the time and the coupling function increases until the values accepted today. 
  In summer 1999 an experiment at ILL, Grenoble was conducted. So-called ultra-cold neutrons (UCN) were trapped in the vertical direction between the Fermi-potential of a smooth mirror below and the gravitational potential of the earth above [Ne00, Ru00]. If quantum mechanics turns out to be a sufficiently correct description of the phenomena in the regime of classical, weak gravitation, one should observe the forming of quantized bound states in the vertical direction above a mirror. Already in a simplified view, the data of the experiment provides strong evidence for the existence of such gravitationally bound quantized states. A successful quantum-mechanical description would then provide a convincing argument, that the socalled first quantization can be used for gravitation as an interaction potential, as this is widely expected. Furthermore, looking at the characteristic length scales of about 10 mikron of such bound states formed by UCN, one sees, that a complete quantum mechanical description of this experiment additionally would enable one to check for possible modifications of Newtonian gravitation on distance scales being one order of magnitude below currently available tests [Ad00]. The work presented here deals mainly with the development of a quantum mechanical description of the experiment. 
  The scalar invariant, I, constructed from the "square" of the first covariant derivative of the curvature tensor is used to probe the local geometry of static spacetimes which are also Einstein spaces. We obtain an explicit form of this invariant, exploiting the local warp-product structure of a 4-dimensional static spacetime, $~^{(3)}\Sigma \times_{f} \reals$, where $^{(3)}\Sigma $ is the Riemannian hypersurface orthogonal to a timelike Killing vector field with norm given by a positive function, $f$ on $^{(3)}\Sigma $. For a static spacetime which is an Einstein space, it is shown that the locally measurable scalar, I, contains a term which vanishes if and only if $^{(3)}\Sigma$ is conformally flat; also, the vanishing of this term implies (a) $~^{(3)}\Sigma$ is locally foliated by level surfaces of $f$, $^{(2)}S$, which are totally umbilic spaces of constant curvature, and (b) $^{(3)}\Sigma$ is locally a warp-product space. Futhermore, if $^{(3)}\Sigma$ is conformally flat it follows that every non-trivial static solution of the vacuum Einstein equation with a cosmological constant, is either Nariai-type or Kottler-type - the classes of spacetimes relevant to quantum aspects of gravity. 
  We take the $G = 0$ limit of the NUT space which yields a non flat space and show that source of its curvature is electromagnetic field generated by the NUT parameter defining the NUT symmetry. This is a very curious electrovac NUT space. Further it is also possible to superpose on it a global monopole. 
  A generalized scalar-tensor theory is investigated whose cosmological term depends on both a scalar field and its time derivative. A correspondence with solutions of five-dimensional Space-Time-Matter theory is noted. Analytic solutions are found for the scale factor, scalar field and cosmological term. Models with free parameters of order unity are consistent with recent observational data and could be relevant to both the dark-matter and cosmological-"constant" problems. 
  We generalize a result of Vollick constraining the possible behaviors of the renormalized expected stress-energy tensor of a free massless scalar field in two dimensional spacetimes that are globally conformal to Minkowski spacetime. Vollick derived a lower bound for the energy density measured by a static observer in a static spacetime, averaged with respect to the observers proper time by integrating against a smearing function. Here we extend the result to arbitrary curves in non-static spacetimes. The proof, like Vollick's proof, is based on conformal transformations and the use of our earlier optimal bound in flat Minkowski spacetime. The existence of such a quantum inequality was previously established by Fewster. 
  It is shown in the covariant phase space formalism that the Noether charges with respect to the diffeomorphism generated by vector fields and their horizontal variations in general relativity form a diffeomorphism algebra. It is also shown with the help of the null tetrad which is well defined everywhere that the central term of the reduced diffeomorphism algebra on the Killing horizon for a large class of vector fields vanishes. 
  Recently there has been a lot of intersest in the superluminal phenomena, and time varying velocity of light cosmological models. More than two decades ago at Einstein centenary symposium, Nagpur I had put forward space-time interaction hypothesis. One of its predictions was that velocity of light decreased with the age of Universe. In view of the profoundness of the hypothesis the original paper is reproduced here. We also mention that in a paper "Quasars, Tachyons and the early universe" proc. Einstein found. Intnl. 2(3), 1985 pp 69-75, it was suggested that, "The boundary region of the universe being the source of radiation, the slowing down of the speed of radiation from $4.4 \times 10^{10}$ cm/sec at $t_A = 10^{10}$ yrs. to the present value of $3 \times 10^{10}$ cm/sec should show up as a large red shifts. The successive boundary region radiations at various epoch will give rise to a wide range of red shifts. Therefore, defining quasars as the objects having two properties: large red shifts and powerful emission of radiations, we would like to identify these objects as the boundary regions of the universe". 
  We study the problem of existence of static spherically symmetric wormholes supported by the kink-like configuration of a scalar field. With this aim we consider a self-consistent, real, nonlinear, nonminimally coupled scalar field $\phi$ in general relativity with the symmetry-breaking potential $V(\phi)$ possessing two minima. We classify all possible field configurations ruling out those of them for which wormhole solutions are impossible. Field configurations admitting wormholes are investigated numerically. Such the configurations represent a spherical domain wall localized near the wormhole throat. 
  It is shown that the accelerated expansion of the universe in the framework of the relativistic theory of gravitation can be achieved by the introduction of the quintessential term in the energy-momentum tensor. The value of the minimum scaling factor and the modern observational data for the density and state parameters of the matter give the rough estimations for the maximum graviton mass and the maximum scaling factor. The former can be very low in the case of the primordial inflation and the latter can be extremely large for the scalar field model of the quintessence. In any case, the massive graviton stops the second inflation and provide the closed cosmological scenario in the agreement with the causality principle inherent to the theory. 
  The motion of classical test spinning particles in Godel universe in the realm of Einstein's General Relativity (GR) is investigated by making use of Killing conserved currents. We consider three distinct cases of motion of spinning particles polarized along the three distinct axes of the anisotropic metric. It is shown that in the case the spin is polarised along the y-direction the minimum energy of the motion is attained for only for spinless particles while the other two directions the minimum energy is obtained for spinning particles. The continuos energy spectrum is also computed. 
  The dynamics of pseudo-classical spinning particles in spacetime of gravitational plane waves of general polarization and harmonic profile is studied. The resulting equations of motion are solved exactly and the results are compared with those of the other approaches. The relative accelerations of nearby particles is also calculated. 
  A relativistic Hamiltonian mechanical system is seen as a conservative Dirac constraint system on the cotangent bundle of a pseudo-Riemannian manifold. We provide geometric quantization of this cotangent bundle where the quantum constraint serves as a relativistic quantum equation. 
  We discuss the Newtonian limit of boost-rotation symmetric spacetimes by means of the Ehler's frame theory. Conditions for the existence of such a limit are given and, in particular, we show that asymptotic flatness is an essential requirement for the existence of such a limit. Consequently, generalized boost-rotation symmetric spacetimes describing particles moving in uniform fields will not possess a Newtonian limit. In the cases where the boost-rotation symmetric spacetime is asymptotically flat and its Newtonian limit exists, then it is non-zero only for the instant of time symmetry and its value is given by a Poisson integral. The relation of this result with the (Newtonian) gravitational potential suggested by the weak field approximation is discussed. We illustrate our analysis through some examples: the two monopoles solution, the Curzon-Chazy particle solution, the generalized Bonnor-Swaminarayan solution, and the C metric. 
  We provide an analytic method to discriminate among different types of black holes on the ground of their strong field gravitational lensing properties. We expand the deflection angle of the photon in the neighbourhood of complete capture, defining a strong field limit, in opposition to the standard weak field limit. This expansion is worked out for a completely generic spherically symmetric spacetime, without any reference to the field equations and just assuming that the light ray follows the geodesics equation. We prove that the deflection angle always diverges logarithmically when the minimum impact parameter is reached. We apply this general formalism to Schwarzschild, Reissner-Nordstrom and Janis-Newman-Winicour black holes. We then compare the coefficients characterizing these metrics and find that different collapsed objects are characterized by different strong field limits. The strong field limit coefficients are directly connected to the observables, such as the position and the magnification of the relativistic images. As a concrete example, we consider the black hole at the centre of our galaxy and estimate the optical resolution needed to investigate its strong field behaviour through its relativistic images. 
  The paper addresses the quantization of minisuperspace cosmological models by studying a possible solution to the problem of time and time asymmetries in quantum cosmology. Since General Relativity does not have a privileged time variable of the newtonian type, it is necessary, in order to have a dynamical evolution, to select a physical clock. This choice yields, in the proposed approach, to the breaking of the so called clock-reversal invariance of the theory which is clearly distinguished from the well known motion-reversal invariance of both classical and quantum mechanics. In the light of this new perspective, the problem of imposing proper boundary conditions on the space of solutions of the Wheeler-DeWitt equation is reformulated. The symmetry-breaking formalism of previous papers is analyzed and a clarification of it is proposed in order to satisfy the requirements of the new interpretation. 
  A global definition of time-asymmetry is presented. Schulman's two arrows of time model is criticized. 
  We present a class of general relativistic soliton-like solutions composed of multiple minimally coupled, massive, real scalar fields which interact only through the gravitational field. We describe a two-parameter family of solutions we call ``phase-shifted boson stars'' (parameterized by central density rho_0 and phase delta), which are obtained by solving the ordinary differential equations associated with boson stars and then altering the phase between the real and imaginary parts of the field. These solutions are similar to boson stars as well as the oscillating soliton stars found by Seidel and Suen [E. Seidel and W.M. Suen, Phys. Rev. Lett. 66, 1659 (1991)]; in particular, long-time numerical evolutions suggest that phase-shifted boson stars are stable. Our results indicate that scalar soliton-like solutions are perhaps more generic than has been previously thought. 
  This paper surveys aspects of the convergence and degeneration of Riemannian metrics on a given manifold M - the Cheeger-Gromov theory - and extensions thereof to Ricci curvature in place of full curvature. This theory is then applied to study a collection of different issues in mathematical aapects of General Relativity. 
  Numerical solutions for: the integral curves of the velocity field (streamlines), the density contours, and the accretion rate of a steady-state flow of an ideal fluid with p=K n^(gamma) equation of state orbiting in a core-dipole-shell system are presented. For 1 < gamma < 2, we found that the non-linear contribution appearing in the partial differential equation for the velocity potential has little effect in the form of the streamlines and density contour lines, but can be noticed in the density values. The study of several cases indicates that this appears to be the general situation. The accretion rate was found to increase when the constant gamma decreases. 
  It has been widely believed that variability of the fine-structure constant alpha would imply detectable violations of the weak equivalence principle. This belief is not justified in general. It is put to rest here in the context of the general framework for alpha variability [J. D. Bekenstein, Phys. Rev. D 25, 1527 (1982)] in which the exponent of a scalar field plays the role of the permittivity and inverse permeability of the vacuum. The coupling of particles to the scalar field is necessarily such that the anomalous force acting on a charged particle by virtue of its mass's dependence on the scalar field is cancelled by terms modifying the usual Coulomb force. As a consequence a particle's acceleration in external fields depends only on its charge to mass ratio, in accordance with the principle. And the center of mass acceleration of a composite object can be proved to be independent of the object's internal constitution, as the weak equivalence principle requires. Likewise the widely employed assumption that the Coulomb energy of matter is the principal source of the scalar field proves wrong; Coulomb energy effectively cancels out in the continuum description of the scalar field's dynamics. This cancellation resolves a cosmological conundrum: with Coulomb energy as source of the scalar field, the framework would predict a decrease of alpha with cosmological expansion, whereas an increase is claimed to be observed. Because of the said cancellation, magnetic energy of cosmological baryonic matter is the main source of the scalar field. Consequently the expansion is accompanied by an increase in alpha; for reasonable values of the framework's sole parameter, this occurs at a rate consistent with the observers' claims. 
  Rigorous results on solutions of the Einstein-Vlasov system are surveyed. After an introduction to this system of equations and the reasons for studying it, a general discussion of various classes of solutions is given. The emphasis is on presenting important conceptual ideas, while avoiding entering into technical details. Topics covered include spatially homogenous models, static solutions, spherically symmetric collapse and isotropic singularities. 
  In a previous paper we have set up the Wheeler-DeWitt equation which describes the quantum general relativistic collapse of a spherical dust cloud. In the present paper we specialize this equation to the case of matter perturbations around a black hole, and show that in the WKB approximation, the wave-functional describes an eternal black hole in equilibrium with a thermal bath at Hawking temperature. 
  In this paper we present the results of our calculations of the Einsteinian strengths S_E(d) and numbers dynamical degrees of freedom N_{DF}(d) for alternative gravity theories in d >= 4 dimensions. In the first part we consider the numbers S_E(d) and N_{DF}(d) for metric-compatible and quadratic in curvature (or quadratic in curvature and in torsion) gravity. We show that in the entire set of the metric-compatible quadratic gravity in d >= 4 dimensions the 2-nd order Einstein-Gauss-Bonnet theory has the smallest numbers S_E(d) and N_{DF}(d), i.e., this quadratic theory of gravity has the strongest field equations. From the physical point of view this theory is the best one quadratic and metric-compatible theory of gravity in d >= 4 dimensions. 
  This work shows that the gravitational field is rather an unusual field and cannot be quantized due to the absence of a fermion charge carrier. When its existence is assumed quite strange results are obtained for its mass. And this means that the graviton does not exist either since bosons act between fermion states. 
  A consisten quantization with a clear notion of time and evolution is given for the anisotropic Kantowski-SDachs cosmological model. It is shown that a suitable coordinate choice allows to obtain a solution of the Wheeler-DeWitt equation in the form of definite energy states, and that the results can be associated to two disjoint equivalent theories, one for each sheet of the constraint surface. 
  A suite of three evolution systems is presented in the framework of the 3+1 formalism. The first one is of second order in space derivatives and has the same causal structure of the Baumgarte-Shapiro-Shibata-Nakamura (BSSN) system for a suitable choice of parameters. The second one is the standard first order version of the first one and has the same causal structure of the Bona-Masso system for a given parameter choice. The third one is obtained from the second one by reducing the space of variables in such a way that the only modes that propagate with zero characteristic speed are the trivial ones. This last system has the same structure of the ones recently presented by Kidder, Scheel and Teukolski: the correspondence between both sets of parameters is explicitly given. The fact that the suite started with a system in which all the dynamical variables behave as tensors (contrary to what happens with BSSN system) allows one to keep the same parametrization when passing from one system to the next in the suite. The direct relationship between each parameter and a particular characteristic speed, which is quite evident in the second and the third systems, is a direct consequence of the manifest 3+1 covariance of the approach. 
  We present a model for introducing dynamics into a space-time geometry. This space-time structure is constructed from a C*-algebra defined in terms of the generators of an irreducible unitary representation of a finite-dimensional Lie algebra G. This algebra is included as a subalgebra in a bigger algebra F, the generators of which mix the representations of G in a way that relates different space-times and creates the dynamics. This construction can be considered eventually as a model for 2-D quantum gravity. 
  We make a comparison between results from numerically generated, quasi-equilibrium configurations of compact binary systems of black holes in close orbits, and results from the post-Newtonian approximation. The post-Newtonian results are accurate through third PN order (O(v/c)^6 beyond Newtonian gravity), and include rotational and spin-orbit effects, but are generalized to permit orbits of non-zero eccentricity. Both treatments ignore gravitational radiation reaction. The energy E and angular momentum J of a given configuration are compared between the two methods as a function of the orbital angular frequency \Omega. For small \Omega, corresponding to orbital separations a factor of two larger than that of the innermost stable orbit, we find that, if the orbit is permitted to be slightly eccentric, with e ranging from \approx 0.03 to \approx 0.05, and with the two objects initially located at the orbital apocenter (maximum separation), our PN formulae give much better fits to the numerically generated data than do any circular-orbit PN methods, including various ``effective one-body'' resummation techniques. We speculate that the approximations made in solving the initial value equations of general relativity numerically may introduce a spurious eccentricity into the orbits. 
  We analyze the dynamical stability of black hole solutions in self-gravitating nonlinear electrodynamics with respect to arbitrary linear fluctuations of the metric and the electromagnetic field. In particular, we derive simple conditions on the electromagnetic Lagrangian which imply linear stability in the domain of outer communication. We show that these conditions hold for several of the regular black hole solutions found by Ayon-Beato and Garcia. 
  In order to investigate the effect of inhomogeneities on the volume expansion of the universe, we study modified Swiss-Cheese universe model. Since this model is an exact solution of Einstein equations, we can get an insight into non-linear dynamics of inhomogeneous universe from it. We find that inhomogeneities make the volume expansion slower than that of the background Einstein-de Sitter universe when those can be regarded as small fluctuations in the background universe. This result is consistent with the previous studies based on the second order perturbation analysis. On the other hand, if the inhomogeneities can not be treated as small perturbations, the volume expansion of the universe depends on the type of fluctuations. Although the volume expansion rate approaches to the background value asymptotically, the volume itself can be finally arbitrarily smaller than the background one and can be larger than that of the background but there is an upper bound on it. 
  The null splitting theorem (proved in math.DG/9909158) is discussed. As an application, a uniqueness theorem for Minkowski space and for de Sitter space associated with the occurrence of null lines (inextendible globally achronal null geodesics) is presented. 
  A simple group theoretic derivation is given of the family of space-time metrics with isometry group SO(2,1) X SO(2) X R first described by Godel, of which the Godel stationary cosmological solution is the member with a perfect-fluid stress-energy tensor. Other members of the family are shown to be interpretable as cosmological solutions with a electrically charged perfect fluid and a magnetic field. 
  We argue that a particular spacetime, a spherically symmetric spacetime with hyper-surface orthogonal, radial, homothetic Killing vector, is a physically meaningful spacetime that describes the problem of spherical gravitational collapse in its full "physical" generality. 
  A particular type of coupling of the dilaton field to the metric is shown to admit a simple geometric interpretation in terms of a volume element density independent from the metric. For dimension n = 4 two families of either magnetically or electrically charged static spherically symmetric solutions to the Maxwell-Einstein-Dilaton field equations are derived. Whereas the metrics of the "magnetic" spacetimes are smooth, geodesically complete and have the topology of a wormhole, the "electric" metrics behave similarly as the singular and geodesically incomplete classical Reissner-Nordstroem metrics. At the price of losing the simple geometric interpretation, a closely related "alternative" dilaton coupling can nevertheless be defined, admitting as solutions smooth "electric" metrics. 
  We numerically investigate the formation of D-dimensional black holes in high-energy particle collision with the impact parameter and evaluate the total cross section of the black hole production. We find that the formation of an apparent horizon occurs when the distance between the colliding particles is less than 1.5 times the effective gravitational radius of each particles. Our numerical result indicates that although both the one-dimensional hoop and the (D-3)-dimensional volume corresponding to the typical scale of the system give a fairly good condition for the horizon formation in the higher-dimensional gravity, the (D-3)-dimensional volume provide a better condition to judge the existence of the horizon. 
  The effects, upon the Klein--Gordon field, of nonconformal stochastic metric fluctuations, are analyzed. It will be shown that these fluctuations allow us to consider an effective mass, i.e., the mass detected in a laboratory is not the parameter appearing in the Klein--Gordon equation, but a function of this parameter and of the fluctuations of the metric. In other words, in analogy to the case of a nonrelativistic electron in interaction with a quantized electromagnetic field, we may speak of a bare mass, where the observed mass shows a dependence upon the stochastic terms included in the metric. Afterwards, we prove, resorting to the influence functional, that the energy--momentum tensor of the Klein--Gordon field inherites this stochastic behavior, and that this feature provokes decoherence upon a particle immersed in the region where this tensor is present. 
  We try to solve the problem about how to probe the inside of a black hole. We show that in the framework of revised quantum dynamics, which may naturally result from the combination of quantum mechanics and general relativity, the information inside a black hole can be gotten out in principle when using the conscious being as quantum measuring device. 
  Theoretical considerations of fundamental physics, as well as certain cosmological observations, persistently point out to permissibility, and maybe necessity, of macroscopic modifications of the Einstein general relativity. The field-theoretical formulation of general relativity helped us to identify the phenomenological seeds of such modifications. They take place in the form of very specific mass-terms, which appear in addition to the field-theoretical analog of the usual Hilbert-Einstein Lagrangian. We interpret the added terms as masses of the spin-2 and spin-0 gravitons. The arising finite-range gravity is a fully consistent theory, which smoothly approaches general relativity in the massless limit, that is, when both masses tend to zero and the range of gravity tends to infinity. We show that all local weak-field predictions of the theory are in perfect agreement with the available experimental data. However, some other conclusions of the non-linear massive theory are in a striking contrast with those of general relativity. We show in detail how the arbitrarily small mass-terms eliminate the black hole event horizon and replace a permanent power-law expansion of a homogeneous isotropic universe with an oscillatory behaviour. One variant of the theory allows the cosmological scale factor to exhibit an `accelerated expansion'instead of slowing down to a regular maximum of expansion. We show in detail why the traditional, Fierz-Pauli, massive gravity is in conflict not only with the static-field experiments but also with the available indirect gravitational-wave observations. At the same time, we demonstrate the incorrectness of the widely held belief that the non-Fierz-Pauli theories possess `negative energies' and `instabilities'. 
  The expectation value of the energy-momentum tensor and the Hawking flux of a scalar field on a Schwarzschild spacetime is calculated using the zeta-function regularisation of the heat kernel. In particular, massless particles are considered in a spherically reduced dilaton model. The effective action is thereby obtained by the covariant perturbation theory and the boundary conditions are fixed by means of the energy-momentum conservation equation. In contrast to previous approaches the expectation values are calculated directly from the effective action which itself is derived in a straightforward manner. 
  This is a summary of the lectures presented at the Xth Brazilian school on cosmology and gravitation. The style of the text is that of a lightly written descriptive summary of ideas with almost no formulas, with pointers to the literature. We hope this style can encourage new people to take a look into these results. We discuss the variables that Ashtekar introduced 18 years ago that gave rise to new momentum in this field, the loop representation, spin networks, measures in the space of connections modulo gauge transformations, the Hamiltonian constraint, application to cosmology and the connection with potentially observable effects in gamma-ray bursts and conclude with a discussion of consistent discretizations of general relativity on the lattice. 
  We have studied the problem of all sky search in reference to continuous gravitational wave particularly for such sources whose wave-form are known in advance. We have made an analysis of the number of templates required for matched filter analysis as applicable to these sources. We have employed the concept of {\it fitting factor} {\it (FF)}; treating the source location as the parameters of the signal manifold and have studied the matching of the signal with templates corresponding to different source locations. We have investigated the variation of FF with source location and have noticed a symmetry in template parameters, $\theta_T$ and $\phi_T$. It has been found that the two different template values in source location, each in $\theta_T$ and $\phi_T$, have same {\it FF}. We have also computed the number of templates required assuming the noise power spectral density $S_n(f)$ to be flat. It is observed that higher {\it FF} requires exponentially increasing large number of templates. 
  The nonequilibrium dynamics of quantum fields is studied in inflationary cosmology, with particular emphasis on applications to the problem of post-inflation reheating. The Schwinger-Keldysh closed-time-path (CTP) formalism is utilized along with the two-particle-irreducible (2PI) effective action in order to obtain coupled, nonperturbative equations for the mean field and variance in a general curved background spacetime. For a model consisting of a quartically self-interacting O(N) field theory (with unbroken symmetry) in spatially flat FRW spacetime, the dynamics of the mean field is studied numerically, at leading order in the large-N expansion. The time evolution of the scale factor is determined self-consistently using the semiclassical Einstein equation. It is found that cosmic expansion can dramatically affect the efficiency of parametric resonance-induced particle production. The production of fermions due to the oscillating inflaton mean field is studied for the case of a scalar inflaton coupled to a fermion field via a Yukawa coupling $f$. The dissipation and noise kernels appearing at $O(f^2)$ in the one-loop CTP effective action are shown to satisfy a zero-temperature fluctuation-dissipation relation (FDR). The effective stochastic equation obeyed by the inflaton zero mode at $O(f^4)$ contains multiplicative noise. It is shown that stochasticity becomes important to the dynamics of the inflaton zero mode before the end of reheating. The thermalization problem is discussed, and a strategy is presented for obtaining time-local equations for equal-time correlation functions which goes beyond the Hartree-Fock approximation. 
  The orbital motion of the Laser Interferometer Space Antenna (LISA) introduces modulations into the observed gravitational wave signal. These modulations can be used to determine the location and orientation of a gravitational wave source. The complete LISA response to an arbitary gravitational wave is derived using a coordinate free approach in the transverse-traceless gauge. The general response function reduces to that found by Cutler (PRD 57, 7089 1998) for low frequency, monochromatic plane waves. Estimates of the noise in the detector are found to be complicated by the time variation of the interferometer arm lengths. 
  Gravitational waves generated by the final merger of compact binary systems depend on the structure of the binary's members. If the binary contains neutron stars, measuring such waves can teach us about the properties of matter at extreme densities. Unfortunately, these waves are typically at high frequency where the sensitivity of broad-band detectors is not good. Learning about dense matter from these waves will require networks of broad-band detectors combined with narrow-band detectors that have good sensitivity at high frequencies. This paper presents an algorithm by which a network can be ``tuned'', in accordance with the best available information, in order to most effectively measure merger waves. The algorithm is presented in the context of a toy model that captures the qualitative features of narrow-band detectors and of certain binary neutron star merger wave models. By using what is learned from a sequence of merger measurements, the network can be gradually tuned in order to accurately measure the waves. The number of measurements needed to reach this stage depends upon the waves' signal strength, the number of narrow-band detectors available for the measurement, and the detailed characteristics of the waves that carry the merger information. Future studies will go beyond this toy model, encompassing a more realistic description of both the detectors and the gravitational waves. 
  In the Randall-Sundrum brane-world scenario and other non-compact Kaluza-Klein theories, the motion of test particles is higher-dimensional in nature. In other words, all test particles travel on five-dimensional geodesics but observers, who are bounded to spacetime, have access only to the 4D part of the trajectory. Conventionally, the dynamics of test particles as observed in 4D is discussed on the basis of the splitting of the geodesic equation in 5D. However, this procedure is {\em not} unique and therefore leads to some problems. The most serious one is the ambiguity in the definition of rest mass in 4D, which is crucial for the discussion of the dynamics. We propose the Hamilton-Jacobi formalism, instead of the geodesic one, to study the dynamics in 4D. On the basis of this formalism we provide an unambiguous expression for the rest mass and its variation along the motion as observed in 4D. It is independent of the coordinates and any parameterization used along the motion. Also, we are able to show a comprehensive picture of the various physical scenarios allowed in 4D, without having to deal with the subtle details of the splitting formalism. Moreover we study the extra non-gravitational forces perceived by an observer in 4D who describes the geodesic motion of a bulk test particle in 5D. Firstly, we show that the so-called fifth force fails to account for the variation of rest mass along the particle's worldline. Secondly, we offer here a new definition that correctly takes into account the change of mass observed in 4D. 
  Tentative observations and theoretical considerations have recently led to renewed interest in models of fundamental physics in which certain ``constants'' vary in time. Assuming fixed black hole mass and the standard form of the Bekenstein-Hawking entropy, Davies, Davis and Lineweaver have argued that the laws of black hole thermodynamics disfavor models in which the fundamental electric charge $e$ changes. I show that with these assumptions, similar considerations severely constrain ``varying speed of light'' models, unless we are prepared to abandon cherished assumptions about quantum gravity. Relaxation of these assumptions permits sensible theories of quantum gravity with ``varying constants,'' but also eliminates the thermodynamic constraints, though the black hole mass spectrum may still provide some restrictions on the range of allowable models. 
  An incoming gravity wave being a stress wave is a surface with intrinsic curvature. When a light beam is parallel transported on this non-Euclidian surface it acquires an excess phase which accumulates with each curcuit. We calculate the separate contributions to excess phase from the wave geometry as well as the dynamic response of mirrors in a Michelson interferometer. Using these results and a combination of analogue and digital signal processing techniques we show how a compact interferometer can be made sensitive to gravity waves of amplitude density 10^-23/&#8730;(Hz) within a frequency range 10^-Hz to 10^4Hz. As an example we describe a 10cm Michelson interferometer designed to measure gravity waves from sources as far as the Virgo cluster.   An incoming gravity wave being a stress wave is a surface with intrinsic curvature. When a light beam is parallel transported on this non-Euclidian surface it acquires an excess phase which accumulates with each curcuit. We calculate the separate contributions to excess phase from the wave geometry as well as the dynamic response of mirrors in a Michelson interferometer. Using these results and a combination of analogue and digital signal processing techniques we show how a compact interferometer can be made sensitive to gravity waves of amplitude density 10^-23/&#8730;(Hz) within a frequency range 10^-Hz to 10^4Hz. As an example we describe a 10cm Michelson interferometer designed to measure gravity waves from sources as far as the Virgo cluster. 
  A possible connection between the electromagnetic quantum vacuum and inertia was first published by Haisch, Rueda and Puthoff (1994). If correct, this would imply that mass may be an electromagnetic phenomenon and thus in principle subject to modification, with possible technological implications for propulsion. A multiyear NASA-funded study at the Lockheed Martin Advanced Technology Center further developed this concept, resulting in an independent theoretical validation of the fundamental approach (Rueda and Haisch, 1998ab). Distortion of the quantum vacuum in accelerated reference frames results in a force that appears to account for inertia. We have now shown that the same effect occurs in a region of curved spacetime, thus elucidating the origin of the principle of equivalence (Rueda, Haisch and Tung, 2001). A further connection with general relativity has been drawn by Nickisch and Mollere (2002): zero-point fluctuations give rise to spacetime micro-curvature effects yielding a complementary perspective on the origin of inertia. Numerical simulations of this effect demonstrate the manner in which a massless fundamental particle, e.g. an electron, acquires inertial properties; this also shows the apparent origin of particle spin along lines originally proposed by Schroedinger. Finally, we suggest that the heavier leptons (muon and tau) may be explainable as spatial-harmonic resonances of the (fundamental) electron. They would carry the same overall charge, but with the charge now having spatially lobed structure, each lobe of which would respond to higher frequency components of the electromagnetic quantum vacuum, thereby increasing the inertia and thus manifesting a heavier mass. 
  In the Cauchy problem of general relativity one considers initial data that satisfies certain constraints. The evolution equations guarantee that the evolved variables will satisfy the constraints at later instants of time. This is only true within the domain of dependence of the initial data. If one wishes to consider situations where the evolutions are studied for longer intervals than the size of the domain of dependence, as is usually the case in three dimensional numerical relativity, one needs to give boundary data. The boundary data should be specified in such a way that the constraints are satisfied everywhere, at all times. In this paper we address this problem for the case of general relativity linearized around Minkowski space using the generalized Einstein-Christoffel symmetric hyperbolic system of evolution equations. We study the evolution equations for the constraints, specify boundary conditions for them that make them well posed and further choose these boundary conditions in such a way that the evolution equations for the metric variables are also well posed. We also consider the case of a manifold with a non-smooth boundary, as is the usual case of the cubic boxes commonly used in numerical relativity. The techniques discussed should be applicable to more general cases, as linearizations around more complicated backgrounds, and may be used to establish well posedness in the full non-linear case. 
  We discuss various properties of the conformal field equations and their consequences for the asymptotic structure of space-times. 
  This thesis is devoted to the investigations of gravitational wave (GW) data analysis from a continuous source e.g. a pulsar, a binary star system. The first Chapter is an introduction to gravitational wave and second Chapter is on the data analysis concept for the detection of GW. In third Chapter we developed the Fourier Transform (FT) of a continuous gravitational wave (CGW) for ground based laser interferometric detectors for the data set of one day observation time incorporating the effects arising due to rotational as well as orbital motion of the earth. The transform is applicable for arbitrary location of detector and source. In Chapter four we have generalized the FT for the data set for (i) one year observation time and (ii) arbitrary observation time. As an application of the transform we considered spin down and N-component signal analysis. In fifth Chapter we have made an analysis of the number of templates required for matched filter analysis as applicable to these sources. We have employed the concept of {\it Fitting Factor (FF)}; treating the source location as the parameters of the signal manifold and have studied the matching of the signal with templates corresponding to different source locations. We have investigated the variation of {\it FF} with source location and have noticed a symmetry in template parameters, $\theta_T$ and $\phi_T$. It has been found that the two different template values in source location, each in $\theta_T$ and $\phi_T$, have same {\it FF}. We have also computed the number of templates required assuming the noise power spectral density $S_n(f)$ to be flat. It is observed that higher {\it FF} requires exponentially increasing large number of templates. Appendix contains the source codes developed for the computation. 
  We study a collection of discrete Markov chains related to the causal set approach to modeling discrete theories of quantum gravity. The transition probabilities of these chains satisfy a general covariance principle, a causality principle, and a renormalizability condition. The corresponding dynamics are completely determined by a sequence of nonnegative real coupling constants. Using techniques related to the classical moment problem, we give a complete description of any such sequence of coupling constants. We prove a representation theorem: every discrete theory of quantum gravity arising from causal set dynamics satisfying covariance, causality and renormalizability corresponds to a unique probability distribution function on the nonnegative real numbers, with the coupling constants defining the theory given by the moments of the distribution. 
  We have argued that quantum mechanics and general relativity give a lower bound $\delta l \gtrsim l^{1/3} l_P^{2/3}$ on the measurement uncertainty of any distance $l$ much greater than the Planck length $l_P$. Recently Baez and Olson have claimed that one can go below this bound by attaching the measuring device to a massive elastic rod. Here we refute their claim. We also reiterate (and invite our critics to ponder on) the intimate relationship and consistency between black hole physics (including the holographic principle) and our bound on distance measurements. 
  We present a systematic exposition of the Lagrangian field theory for the massive spin-two field generated in higher-derivative gravity. It has been noticed by various authors that this nonlinear field overcomes the well known inconsistency of the theory for a linear massive spin-two field interacting with Einstein's gravity. Starting from a Lagrangian quadratically depending on the Ricci tensor of the metric, we explore the two possible second-order pictures usually called "(Helmholtz-)Jordan frame" and "Einstein frame". In spite of their mathematical equivalence, the two frames have different structural properties: in Einstein frame, the spin-two field is minimally coupled to gravity, while in the other frame it is necessarily coupled to the curvature, without a separate kinetic term. We prove that the theory admits a unique and linearly stable ground state solution, and that the equations of motion are consistent, showing that these results can be obtained independently in either frame. The full equations of motion and the energy-momentum tensor for the spin--two field in Einstein frame are given, and a simple but nontrivial exact solution to these equations is found. The comparison of the energy-momentum tensors for the spin-two field in the two frames suggests that the Einstein frame is physically more acceptable. We point out that the energy-momentum tensor generated by the Lagrangian of the linearized theory is unrelated to the corresponding tensor of the full theory. It is then argued that the ghost-like nature of the nonlinear spin-two field, found long ago in the linear approximation, may not be so harmful to classical stability issues, as has been expected. 
  The stationary phase technique is used to calculate asymptotic formulae for SO(4) Relativistic Spin Networks. For the tetrahedral spin network this gives the square of the Ponzano-Regge asymptotic formula for the SU(2) 6j symbol. For the 4-simplex (10j-symbol) the asymptotic formula is compared with numerical calculations of the Spin Network evaluation. Finally we discuss the asymptotics of the SO(3,1) 10j-symbol. 
  All Lorentzian spacetimes with vanishing invariants constructed from the Riemann tensor and its covariant derivatives are determined. A subclass of the Kundt spacetimes results and we display the corresponding metrics in local coordinates. Some potential applications of these spacetimes are discussed. 
  Relativistic rotation is considered in the limit of angular velocity approaching zero and radial distance approaching infinity, such that centrifugal acceleration is immeasurably small while tangent velocity remains close to the speed of light. For this case, the predictions of the traditional approach to relativistic rotation using local co-moving Lorentz frames are compared and contrasted with those of the differential geometry based non-time-orthogonal analysis approach. Different predictions by the two approaches imply that only the non-time-orthogonal approach is valid. 
  We study spherically symmetric solutions to the Jordan-Brans-Dicke field equations under the assumption that the space-time may possess an arbitrary number of spatial dimensions. Assuming a perfect fluid with the equation of state, we show that there are static interior nontrivial solutions in three dimensional Jordan-Brans-Dicke gravity theory. 
  In this paper we analyze quantitatively the concept of LAGEOS--type satellites in critical supplementary orbit configuration (CSOC) which has proven capable of yielding various observables for many tests of General Relativity in the terrestrial gravitational field, with particular emphasis on the measurement of the Lense--Thirring effect. 
  The apparent discrepancy between the bending of light predicted by the equivalence principle and its corresponding value in general relativity is resolved by evaluating the deflection of light with respect to a direction that is parallel transported along the ray trajectory in 3-space. In this way the bending predicted by the equivalence principle is fulfilled in general relativity and other alternative metric theories of gravity. 
  We define the renormalization group flow for a renormalizable interacting quantum field in curved spacetime via its behavior under scaling of the spacetime metric, $\g \to \lambda^2 \g$. We consider explicitly the case of a scalar field, $\phi$, with a self-interaction of the form $\kappa \phi^4$, although our results should generalize straightforwardly to other renormalizable theories. We construct the interacting field--as well as its Wick powers and their time-ordered-products--as formal power series in the algebra generated by the Wick powers and time-ordered-products of the free field, and we determine the changes in the interacting field observables resulting from changes in the renormalization prescription. Our main result is the proof that, for any fixed renormalization prescription, the interacting field algebra for the spacetime $(M, \lambda^2 \g)$ with coupling parameters $p$ is isomorphic to the interacting field algebra for the spacetime $(M, \g)$ but with different values, $p(\lambda)$, of the coupling parameters. The map $p \to p(\lambda)$ yields the renormalization group flow. The notion of essential and inessential coupling parameters is defined, and we define the notion of a fixed point as a point, $p$, in the parameter space for which there is no change in essential parameters under renormalization group flow. 
  We consider a hydrogen atom in the background spacetimes generated by an infinitely thin cosmic string and by a point-like global monopole. In both cases, we find the solutions of the corresponding Dirac equations and we determine the energy levels of the atom. We investigate how the geometric and topological features of these spacetimes leads to shifts in the energy levels as compared with the flat Minkowski spacetime. 
  Electromagnetic wavelets are a family of 3x3 matrix fields W_z(x') parameterized by complex spacetime points z=x+iy with y timelike. They are translates of a \sl basic \rm wavelet W(z) holomorphic in the future-oriented union T of the forward and backward tubes. Applied to a complex polarization vector p (representing electric and magnetic dipole moments), W(z) gives an anti-selfdual solution W(z)p of Maxwell's equations derived from a selfdual Hertz potential Z(z)=-iS(z)p, where S is the \sl Synge function \rm acting as a Whittaker-like scalar Hertz potential. Resolutions of unity exist giving representations of sourceless electromagnetic fields as superpositions of wavelets. With the choice of a branch cut, S(z) splits into a difference of retarded and advanced \sl pulsed beams \rm whose limits as y\to 0 give the propagators of the wave equation. This yields a similar splitting of the wavelets and leads to their complete physical interpretation as EM pulsed beams absorbed and emitted by a \sl disk source \rm D(y) representing the branch cut. The choice of y determines the beam's orientation, collimation and duration, giving beams as sharp and pulses as short as desired. The sources are computed as spacetime distributions of electric and magnetic dipoles supported on D(y). The wavelet representation of sourceless electromagnetic fields now splits into representations with advanced and retarded sources. These representations are the electromagnetic counterpart of relativistic coherent-state representations previously derived for massive Klein-Gordon and Dirac particles. 
  The gravitational field of two identical rotating and counter-moving dust beams is found in full generality. The solution depends on an arbitrary function and a parameter. Some of its properties are studied. Previous particular solutions are derived as subcases. 
  The equations which determine the response of a charged particle moving in a magnetic field to an incident gravitational wave(GW) are derived in the linearized approximation to general relativity. We briefly discuss several astrophysical applications of the derived formulae taking into account the resonance between the wave and the particle's motion which occurs at $\omega_g=2\Omega$, whenever the GW is parallel to the constant magnetic field. In the case where the GW is perpendicular to the constant magnetic field, magnetic resonances appear at $\omega_g=\Omega$ and $\omega_g=2\Omega$. Such resonant mechanism may be useful to build models of GW driven cyclotron emitters. 
  We show that the spin-2 equations on Minkowski space in the gauge of the `regular finite initial value problem at space-like infinity' imply estimates which, together with the transport equations on the cylinder at space-like infinity, allow us to obtain for a certain class of initial data information on the behaviour of the solution near space-like and null infinity of any desired precision. 
  We present a class of solutions for a heat conducting fluid sphere, which radiates energy during collapse without the appearance of horizon at the boundary at any stage of the collapse. A simple model shows that there is no accumulation of energy due to collapse since it radiates out at the same rate as it is being generated. 
  For the quantised, massless, minimally coupled real scalar field in four-dimensional Minkowski space, we show (by an explicit construction) that weighted averages of the null-contracted stress-energy tensor along null geodesics are unbounded from below on the class of Hadamard states. Thus there are no quantum inequalities along null geodesics in four-dimensional Minkowski spacetime. This is in contrast to the case for two-dimensional flat spacetime, where such inequalities do exist. We discuss in detail the properties of the quantum states used in our analysis, and also show that the renormalized expectation value of the stress energy tensor evaluated in these states satisfies the averaged null energy condition (as expected), despite the nonexistence of a null-averaged quantum inequality. However, we also show that in any globally hyperbolic spacetime the null-contracted stress energy averaged over a timelike worldline does satisfy a quantum inequality bound (for both massive and massless fields). We comment briefly on the implications of our results for singularity theorems. 
  We argue that quantum theory in curved spacetime should be invariant under the continuous spacetime symmetries thaat are connected with the identity. For typical warped-product spacetimes, we prove that such invariance can be actually implemented, at least at the level of first quantization. Our approach rests on a mode decomposition which is special to the context of warped spacetimes. 
  It is well-known \cite{mtbh} that {\em all} black hole solutions of General Relativity are of Petrov-type D. It may thus be expected that the spacetime of {\em physically realizable} spherical gravitational collapse is also of Petrov-type D. We show that a radially homothetic spacetime, {\em ie}, a spherically symmetric spacetime with hyper-surface orthogonal, radial, homothetic Killing vector, is of Petrov-type D. As has been argued in \cite{prl1}, it is a spacetime of {\em physically realizable} spherical collapse. 
  The multiple Doppler readouts available on the Laser Interferometer Space Antenna (LISA) permit simultaneous formation of several interferometric observables. All these observables are independent of laser frequency fluctuations and have different couplings to gravitational waves and to the various LISA instrumental noises. Within the functional space of interferometric combinations LISA will be able to synthesize, we have identified a triplet of interferometric combinations that show optimally combined sensitivity. As an application of the method, we computed the sensitivity improvement for sinusoidal sources in the nominal, equal-arm LISA configuration. In the part of the Fourier band where the period of the wave is longer than the typical light travel-time across LISA, the sensitivity gain over a single Michelson interferometer is equal to $\sqrt{2}$. In the mid-band region, where the LISA Michelson combination has its best sensitivity, the improvement over the Michelson sensitivity is slightly better than $\sqrt{2}$, and the frequency band of best sensitivity is broadened. For frequencies greater than the reciprocal of the light travel-time, the sensitivity improvement is oscillatory and $\sim \sqrt{3}$, but can be greater than $\sqrt{3}$ near frequencies that are integer multiples of the inverse of the one-way light travel-time in the LISA arm. 
  An approach is developed which enables one to analyze gravitational effects without usage of any concrete model of geometry (or a class of models of geometry) of space-time and even without any coordinate system. Instead, the formalism of the group of paths is used. An element of this group (a class of curves in Minkowski space) is associated with each curve in the curved space-time and called its `flat model'. The analysis of observational data in terms of flat models of closed curves is interpreted as a formalization of the analysis which a `naive observer' (knowing nothing about the space-time being curved) applies to his observations. 
  The stationary gravitational field of two identical counter-moving beams of pure radiation is found in full generality. The solution depends on an arbitrary function and a parameter which sets the scale of the energy density. Some of its properties are studied. Previous particular solutions are derived as subcases. 
  The automorphisms of all 4-dimensional, real Lie Algebras are presented in a comprehensive way. Their action on the space of $4\times 4$, real, symmetric and positive definite, matrices, defines equivalence classes which are used for the invariant characterization of the 4-dimensional homogeneous spaces which possess an invariant basis. 
  We propose a method to determine the topological structure of an event horizon far in the future of a spacetime from the geometrical information of its future null infinity. In the present article, we mainly consider spacetimes with two black holes. Although, in most of cases, the black holes coalesce and their event horizon is topologically a single sphere far in the future, there are several possibilities that the black holes do not coalesce eternally and such exact solutions. In our formulation, the geometrical structure of future null infinity is related to the topological structure of the upper end of the future null infinity through the Poincare\'-Hopf's theorem. Since the upper end of the future null infinity determines the event horizon far in the future under the conformal embedding, the topology of event horizon far in the future will be affected by the geometrical structure of the future null infinity. Our method is not only for the case of black hole coalescence. Also we can consider more than two black holes or a black hole with non-trivial topology. 
  The Dirac eigenvalues form a subset of observables of the Euclidean gravity. The symplectic two-form in the covariant phase space could be expressed, in principle, in terms of the Dirac eigenvalues. We discuss the existence of the formal solution of the equations defining the components of the symplectic form in this framework. 
  A non-linear gravitational model with a multidimensional geometry and quadratic scalar curvature is considered. For certain parameter ranges, the extra dimensions are stabilized if the internal spaces have negative curvature. As a consequence, the 4-dimensional effective cosmological constant as well as the bulk cosmological constant become negative. The homogeneous and isotropic external space is asymptotically AdS. The connection between the D-dimensional and the 4-dimensional fundamental mass scales sets an additional restriction on the parameters of the considered non-linear models. 
  What is the quantum state of the universe? That is the central question of quantum cosmology. This essay describes the place of that quantum state in a final theory governing the regularities exhibited universally by all physical systems in the universe. It is possible that this final theory consists of two parts: (1) a dynamical theory such as superstring theory, and (2) a state of the universe such as Hawking's no-boundary wave function. Both are necessary because prediction in quantum mechanics requires both a Hamiltonian and a state. Complete ignorance of the state leads to predictions inconsistent with observation. The simplicity observed in the early universe gives hope that there is a simple, discoverable quantum state of the universe. It may be that, like the dynamical theory, the predictions of the quantum state for late time, low energy observations can be summarized by an effective cosmological theory. That should not obscure the need to provide a fundamental basis for such an effective theory which gives a a unified explanation of its features and is applicable without restrictive assumptions. It could be that there is one principle that determines both the dynamical theory and the quantum state. That would be a truly unified final theory. (talk given The Future of Theoretical Physics and Cosmology: Stephen Hawking 60th Birthday Symposium) 
  If a cat, a cannonball, and an economics textbook are all dropped from the same height, they fall to the ground with exactly the same acceleration under the influence of gravity. This equality of gravitational accelerations of different things is one of the most accurately tested laws of physics. That law, however, tells us little about cats, cannonballs, or economics. This lecture expands on this theme to address the question of what features of our world are predicted by a fundamental ``theory of everything'' governing the regularities exhibited universally by all physical systems. This may consist of two parts: a dynamical law governing regularities in time (e.g superstring theory) and a law of cosmological initial condition governing mostly regularities in space (e.g. Hawking's no-boundary initial condition). The lecture concludes that: (1) ``A theory of everything'' is not a theory of everything in a quantum mechanical universe. (2) If the laws are short enough to be discoverable then they are probably too short to predict everything. (3) The regularities of human history, economics, biology, geology, etc are consistent with the fundamental laws of physics but do not follow from them. (Public lecture given at The Future of Theoretical Physics and Cosmology: Stephen Hawking 60th Birthday Symposium.) 
  We continue recent work and formulate the gravitational vacuum Einstein equations over a locally finite spacetime by using the basic axiomatics, techniques, ideas and working philosophy of Abstract Differential Geometry. The whole construction is `fully covariant', `inherently quantum' (both expressions are analytically explained in the paper) and genuinely smooth background spacetime independent. 
  It is shown that in d=11 supergravity, under a very reasonable ansatz, the nearly flat spacetime in which we are living must be 4-dimensional without appealing to the Anthropic Principle. Can we dispel the Anthropic Principle completely from cosmology? 
  Absolute parallelism geometry is frequently used for physical applications. It has two main defects, from the point of view of applications. The first is the identical vanishing of its curvature tensor. The second is that its autoparallel paths do not represent physical trajectories. The present work shows how these defects were treated in the course of development of the geometry. The new version of this geometry contains simultaneous non-vanishing torsion and curvatures. Also, the new paths discovered in this geometry do represent physical trajectories. Advantages and disadvantages of this geometry are given for each stage of its development. Physical applications are just mentioned without giving any details. 
  We present a new covariant and gauge-invariant perturbation formalism for dealing with spacetimes having spherical symmetry (or some preferred spatial direction) in the background, and apply it to the case of gravitational wave propagation in a Schwarzschild black hole spacetime. The 1+3 covariant approach is extended to a `1+1+2 covariant sheet' formalism by introducing a radial unit vector in addition to the timelike congruence, and decomposing all covariant quantities with respect to this. The background Schwarzschild solution is discussed and a covariant characterisation is given. We give the full first-order system of linearised 1+1+2 covariant equations, and we show how, by introducing (time and spherical) harmonic functions, these may be reduced to a system of first-order ordinary differential equations and algebraic constraints for the 1+1+2 variables which may be solved straightforwardly. We show how both the odd and even parity perturbations may be unified by the discovery of a covariant, frame- and gauge-invariant, transverse-traceless tensor describing gravitational waves, which satisfies a covariant wave equation equivalent to the Regge-Wheeler equation for both even and odd parity perturbations. We show how the Zerilli equation may be derived from this tensor, and derive a similar transverse traceless tensor equivalent to this equation. The so-called `special' quasinormal modes with purely imaginary frequency emerge naturally. The significance of the degrees of freedom in the choice of the two frame vectors is discussed, and we demonstrate that, for a certain frame choice, the underlying dynamics is governed purely by the Regge-Wheeler tensor. The two transverse-traceless Weyl tensors which carry the curvature of gravitational waves are discussed. 
  This talk describes some recent results [16] regarding the problem of uniqueness in the large (also known as strong cosmic censorship) for the initial value problem in general relativity. In order to isolate the essential analytic features of the problem from the complicated setting of gravitational collapse in which it arises, some familiarity with conformal properties of certain celebrated special solutions of the theory of relativity will have to be developed. This talk is an attempt to present precisely these features to an audience of non-specialists, in a way which will hopefully fully motivate a certain characteristic initial value problem for the spherically-symmetric Einstein-Maxwell-Scalar Field system. The considerations outlined here leading to this particular initial value problem are well known in the physics relativity community, where the problem of uniqueness has been studied heuristically [1, 22] and numerically [2, 3]. In [16], the global behavior of solutions to this IVP, in particular, the issue of uniqueness, is mathematically completely understood. A statement of the relevant Theorems is included in Section 9. Only a sketch of the ideas of the proof is provided here, but the readers may refer to [16] for details. 
  We investigate a static solution with spherical symmetry of a recently proposed field theory of gravitation. In this so-called NDL theory, matter interacts with gravity in accordance with the Weak Equivalence Principle, while gravitons have a nonlinear self-interaction. It is shown that the predictions of NDL agree with those of General Relativity in the three classic tests. However, there are potential differences in the strong-field limit, which we illustrate by proving that this theory does not allow the existence of static and spherically symmetric black holes. 
  We derive a ``generic'' inhomogeneous ``bridge'' solution for a cosmological model in the presence of a real self-interacting scalar field. This solution connects a Kasner-like regime to an inflationary stage of evolution and therefore provides a dynamical mechanism for the quasi-isotropization of the universe. In the framework of a standard Arnowitt-Deser-Misner Hamiltonian formulation of the dynamics and by adopting Misner-Chitr\`e-like variables, we integrate the Einstein-Hamilton-Jacobi equation corresponding to a ``generic'' inhomogeneous cosmological model whose evolution is influenced by the coupling with a bosonic field, expected to be responsible for a spontaneous symmetry breaking configuration. The dependence of the detailed evolution of the universe on the initial conditions is then appropriately characterized. 
  In Paper II [N. G. Phillips and B. L. Hu, previous abstract] we presented the details for the regularization of the noise kernel of a quantum scalar field in optical spacetimes by the modified point separation scheme, and a Gaussian approximation for the Green function. We worked out the regularized noise kernel for two examples: hot flat space and optical Schwarzschild metric. In this paper we consider noise kernels for a scalar field in the Schwarzschild black hole. Much of the work in the point separation approach is to determine how the divergent piece conformally transforms. For the Schwarzschild metric we find that the fluctuations of the stress tensor of the Hawking flux in the far field region checks with the analytic results given by Campos and Hu earlier [A. Campos and B. L. Hu, Phys. Rev. D {\bf 58} (1998) 125021; Int. J. Theor. Phys. {\bf 38} (1999) 1253]. We also verify Page's result [D. N. Page, Phys. Rev. {\bf D25}, 1499 (1982)] for the stress tensor, which, though used often, still lacks a rigorous proof, as in his original work the direct use of the conformal transformation was circumvented. However, as in the optical case, we show that the Gaussian approximation applied to the Green function produces significant error in the noise kernel on the Schwarzschild horizon. As before we identify the failure as occurring at the fourth covariant derivative order. 
  Continuing our investigation of the regularization of the noise kernel in curved spacetimes [N. G. Phillips and B. L. Hu, Phys. Rev. D {\bf 63}, 104001 (2001)] we adopt the modified point separation scheme for the class of optical spacetimes using the Gaussian approximation for the Green functions a la Bekenstein-Parker-Page. In the first example we derive the regularized noise kernel for a thermal field in flat space. It is useful for black hole nucleation considerations. In the second example of an optical Schwarzschild spacetime we obtain a finite expression for the noise kernel at the horizon and recover the hot flat space result at infinity. Knowledge of the noise kernel is essential for studying issues related to black hole horizon fluctuations and Hawking radiation backreaction. We show that the Gaussian approximated Green function which works surprisingly well for the stress tensor at the Schwarzschild horizon produces significant error in the noise kernel there. We identify the failure as occurring at the fourth covariant derivative order. 
  We study the correlations between the particles emitted by a moving mirror. To this end, we first analyze $< T_{\mu\nu}(x) T_{\alpha\beta}(x') >$, the two-point function of the stress tensor of the radiation field. In this we generalize the work undertaken by Carlitz and Willey. To further analyze how the vacuum correlations on $I^-$ are scattered by the mirror and redistributed among the produced pairs of particles, we use a more powerful approach based on the value of $T_{\mu\nu}$ which is conditional to the detection of a given particle on $I^+$. We apply both methods to the fluxes emitted by a uniformly accelerated mirror. This case is particularly interesting because of its strong interferences which lead to a vanishing flux, and because of its divergences which are due to the infinite blue shift effects associated with the horizons. Using the conditional value of $T_{\mu\nu}$, we reveal the existence of correlations between created particles and their partners in a domain where the mean fluxes and the two-point function vanish. This demonstrates that the scattering by an accelerated mirror leads to a steady conversion of vacuum fluctuations into pairs of quanta. Finally, we study the scattering by two uniformly accelerated mirrors which follow symmetrical trajectories (i.e. which possess the same horizons). When using the Davies-Fulling model, the Bogoliubov coefficients encoding pair creation vanish because of perfectly destructive interferences. When using regularized amplitudes, these interferences are inevitably lost thereby giving rise to pair creation. 
  Exploiting a multiply warped product manifold scheme, we study the interior solutions of the Banados-Teitelboim-Zanelli black holes and the exterior solutions of the de Sitter black holes in the (2+1) dimensions. 
  In this work we study in detail new kinds of motions of the metric tensor. The work is divided into two main parts. In the first part we study the general existence of Kerr-Schild motions --a recently introduced metric motion. We show that generically, Kerr-Schild motions give rise to finite dimensional Lie algebras and are isometrizable, i.e., they are in a one-to-one correspondence with a subset of isometries of a (usually different) spacetime. This is similar to conformal motions. There are however some exceptions that yield infinite dimensional algebras in any dimension of the manifold. We also show that Kerr-Schild motions may be interpreted as some kind of metric symmetries in the sense of having associated some geometrical invariants. In the second part, we suggest a scheme able to cope with other new candidates of metric motions from a geometrical viewpoint. We solve a set of new candidates which may be interpreted as the seeds of further developments and relate them with known methods of finding new solutions to Einstein's field equations. The results are similar to those of Kerr-Schild motions, yet a richer algebraical structure appears. In conclusion, even though several points still remain open, the wealth of results shows that the proposed concept of generalized metric motions is meaningful and likely to have a spin-off in gravitational physics.We end by listing and analyzing some of those open points. 
  The present work deals with the search of useful physical applications of some generalized groups of metric transformations. We put forward different proposals and focus our attention on the implementation of one of them. Particularly, the results show how one can control very efficiently the kind of spacetimes related by a Generalized Kerr-Schild (GKS) Ansatz through Kerr-Schild groups. Finally a preliminar study regarding other generalized groups of metric transformations is undertaken which is aimed at giving some hints in new Ans\"atze to finding useful solutions to Einstein's equations. 
  A computer which has access to a closed timelike curve, and can thereby send the results of calculations into its own past, can exploit this to solve difficult computational problems efficiently. I give a specific demonstration of this for the problem of factoring large numbers, and argue that a similar approach can solve NP-complete and PSPACE-complete problems. I discuss the potential impact of quantum effects on this result. 
  We study optical activity induced by curvature. The optical activity model we present has two phenomenological gyration parameters, within which we analyze three model cases, namely, an exactly integrable model, the Landau-Lifshitz model and the Fedorov model, these latter two are solved in the short wavelength approximation. The model background is a gravitational pp-wave. The solutions show that the optical activity induced by curvature leads to Faraday rotation. 
  A class of spherically symmetric spacetimes invariantly defined by a zero flux condition is examined first from a purely geometrical point of view and then physically by way of Einstein's equations for a general fluid decomposition of the energy-momentum tensor. The approach, which allows a formal inversion of Einstein's equations, explains, for example, why spherically symmetric perfect fluids with spatially homogeneous energy density must be shearfree. 
  We present the analysis of a nonlinear effect of parametric oscillatory instability in power recycled LIGO interferometer with the Fabry-Perot (FP) cavities in the arms. The basis for this effect is the excitation of the additional (Stokes) optical mode and the mirror elastic mode, when the optical energy stored in the main FP cavity main mode exceeds the certain threshold and the frequencies are related so that sum of frequencies of Stokes and elastic modes are approximately equal to frequencyof main mode. The presence of anti-Stokes modes (with frequency approximately equal to sum of frequencies of main and elastic modes) can depress parametric instability. However, it is very likely that the anti-Stokes modes will not compensate the parametric instability completely. 
  We consider the quantum birth of a hot FRW universe from a vacuum-dominated quantum fluctuation with admixture of radiation and strings, which correspond to quantum tunnelling from a discrete energy level with a non-zero temperature. The presence of strings with the equation of state $p=-\epsilon/3$ mimics a positive curvature term which makes it possible, in the case of a negative deficit angle, the quantum birth of an open and flat universe. In the pre-de-Sitter domain radiation energy levels are quantized. We calculate the temperature spectrum and estimate the range of the model parameters restricting temperature fluctuations by the observational constraint on the CMB anisotropy. For the GUT scale of initial de Sitter vacuum the lower limit on the temperature at the start of classical evolution is close to the values as predicted by reheating theories, while the upper limit is far from the threshold for a monopole rest mass. 
  We experiment with modifications of the BSSN form of the Einstein field equations (a reformulation of the ADM equations) and demonstrate how these modifications affect the stability of numerical black hole evolution calculations. We use excision to evolve both non-rotating and rotating Kerr-Schild black holes in octant and equatorial symmetry, and without any symmetry assumptions, and obtain accurate and stable simulations for specific angular momenta J/M of up to about 0.9M. 
  A class of solutions to Einstein field equations is studied, which represents gravitational collapse of thick spherical shells made of self-similar and shear-free fluid with heat flow. It is shown that such shells satisfy all the energy conditions, and the corresponding collapse always forms naked singularities. 
  A complete family of non-expanding impulsive waves in spacetimes which are the direct product of two 2-spaces of constant curvature is presented. In addition to previously investigated impulses in Minkowski, (anti-)Nariai and Bertotti-Robinson universes, a new explicit class of impulsive waves which propagate in the exceptional electrovac Plebanski-Hacyan spacetimes with a cosmological constant Lambda is constructed. In particular, pure gravitational waves generated by null particles with an arbitrary multipole structure are described. The metrics are impulsive members of a more general family of the Kundt spacetimes of type II. The well-known pp-waves are recovered for Lambda=0. 
  In this study a rotationally and translationally invariant metric in Finsler space is investigated. We choose to rewrite the metric in Riemanian space by increasing the dimension of space-time and introducing additional coordinates such that for specific values of these coordinates, the geodesics of the four dimensional Finslerian space-time and six dimensional Riemanian space-time are identical. Cosmological solutions described by this metric give rise to an equation of state corresponding to a space dominated by domain walls and an internal space dominated by strings. 
  We discuss the gravitationally interacting system of a thick domain wall and a black hole. We numerically solve the scalar field equation in the Schwarzschild spacetime and obtain a sequence of static axi-symmetric solutions representing thick domain walls. We find that, for the walls near the horizon, the Nambu--Goto approximation is no longer valid. 
  Isotropic cosmological singularities are singularities which can be removed by rescaling the metric. In some cases already studied (gr-qc/9903008, gr-qc/9903009, gr-qc/9903018) existence and uniqueness of cosmological models with data at the singularity has been established. These were cosmologies with, as source, either perfect fluids with linear equations of state or massless, collisionless particles. In this article we consider how to extend these results to a variety of other matter models. These are scalar fields, massive collisionless matter, the Yang-Mills plasma of Choquet-Bruhat, or matter satisfying the Einstein-Boltzmann equation. 
  We obtain all ``regularization parameters'' (RP) needed for calculating the gravitational and electromagnetic self forces for an arbitrary geodesic orbit around a Schwarzschild black hole. These RP values are required for implementing the previously introduced mode-sum method, which allows a practical calculation of the self force by summing over contributions from individual multipole modes of the particle's field. In the gravitational case, we provide here full details of the analytic method and results briefly reported in a recent Letter [Phys. Rev. Lett. {\bf 88}, 091101 (2002)]. In the electromagnetic case, the RP are obtained here for the first time. 
  Let (M, g) be an (n+1) dimensional space-time, with bounded curvature with respect to a bounded framing. If (M, g) is vacuum or satisfies a mild condition on the stress-energy tensor, then we show that (M, g) locally admits coordinate systems in which the Lorentz metric is well-controlled in the (space-time) Sobolev space L^{2,p}, for any finite p. 
  We formulate the nonlinear isovector model in a curved background, and calculate the spherically symmetric solutions for weak and strong coupling regimes. The usual belief that gravity does not have appreciable effects on the structure of solitons will be examined, in the framework of the calculated solutions, by comparing the flat-space and curved-space solutions. It turns out that in the strong coupling regime, gravity has essential effects on the solutions. Masses of the self-gravitating solitons are calculated numerically using the Tolman expression, and its behavior as a function of the coupling constant of the model is studied. 
  A quantitative test for the validity of the semi-classical approximation in gravity is given. The criterion proposed is that solutions to the semi-classical Einstein equations should be stable to linearized perturbations, in the sense that no gauge invariant perturbation should become unbounded in time. A self-consistent linear response analysis of these perturbations, based upon an invariant effective action principle, necessarily involves metric fluctuations about the mean semi-classical geometry, and brings in the two-point correlation function of the quantum energy-momentum tensor in a natural way. This linear response equation contains no state dependent divergences and requires no new renormalization counterterms beyond those required in the leading order semi-classical approximation. The general linear response criterion is applied to the specific example of a scalar field with arbitrary mass and curvature coupling in the vacuum state of Minkowski spacetime. The spectral representation of the vacuum polarization function is computed in n dimensional Minkowski spacetime, and used to show that the flat space solution to the semi-classical Einstein equations for n=4 is stable to all perturbations on distance scales much larger than the Planck length. 
  Nonlinear wave interaction of low amplitude gravitational waves in flat space-time is considered. Analogy with optics is established. It is shown that the flat metric space-time is equivalent to a centro-symmetric optical medium, with no second order susceptibility. The lowest order nonlinear effects are those due to the third order nonlinearity and include self-phase modulation and high harmonic generation. These processes lead to an efficient energy dilution of the gravitational wave energy over an increasingly large spectral range. 
  The perfect dilaton-spin fluid (as a model of the dilaton matter, the particles of which are endowed with intrinsic spin and dilaton charge) is considered as the source of the gravitational field in a Weyl-Cartan spacetime. The variational theory of such fluid is constructed and the dilaton-spin fluid energy-momentum tensor is obtained. The variational formalism of the gravitational field in a Weyl-Cartan spacetime ia developed in the exterior form language. The homogeneous and isotropic Universe filled with the dilaton matter as the dark matter is considered and one of the field equations is represented as Einstein-like equation which leads to the modified Friedmann-Lemaitre equation. From this equation the absence of the initial singularity in the cosmological solution follows. Also the existence of two points of inflection of the scale factor function is established, the first of which corresponds to the early stage of the Universe and the second one corresponds to the modern era when the expansion with deceleration is replaced by the expansion with acceleration. The possible equations of state for the self-interacting cold dark matter are found on the basis of the modern observational data. The inflation-like solution is obtained. 
  Testing of the gravitation equations, proposed by one of the authors earlier, by a binary pulsar is considered. It has been shown that the formulas for the gravitation radiation of the system resulting from the equations do not contradict the available observations data. 
  The helicity-rotation coupling and its current empirical basis are examined. The modification of the Doppler effect due to the coupling of photon spin with the rotation of the observer is considered in detail in connection with its applications in the Doppler tracking of spacecraft. Further implications of this coupling and the possibility of searching for it in the intensity response of a rotating detector are briefly discussed. 
  We review some of the history and properties of theories for the variation of the gravitation and fine structure 'constants'. We highlight some general features of the cosmological models that exist in these theories with reference to recent quasar data that is consistent with time-variation in alpha since a redshift of 3.5. The behaviour of a simple class of varying-alpha cosmologies is outlined in the light of all the observational constraints. We discuss the key role played by non-zero vacuum energy and curvature in turning off the variation of constants in these theories and the issue of comparing extra-galactic and local observational data. We also show why black hole thermodynamics does not enable us to distinguish between time variations of different constants. 
  We argue that space-time properties are not absolute with respect to the used frame of reference as is to be expected according to ideas of relativity of space and time properties by Berkley - Leibnitz - Mach- Poincar\'{e}. From this point of view gravitation may manifests itself both as a field in Minkowski space-time and as space-time curvature. If the motion of test particles is described by the Thirring Lagrangian, then in the inertial frames of reference, where space-time is pseudo-Euclidean, gravitation manifests itself as a field. In reference frames, whose reference body is formed by point masses moving under the effect of the field, it appears as Riemannian curvature which in these frames is other than zero. For realization of this idea the author bimetric gravitation equations are considered. The spherically - symmetric solution of the equations in Minkowski space-time does not lead to the physical singularity in the center. The energy of the gravitational field of a point mass is finite. It follows from the properties of the gravitational force that there can exist stable compact supermassive configurations of Fermi-gas without an events horizon. 
  A general covariant extension of Einstein's field equations is considered with a view to Numerical Relativity applications. The basic variables are taken to be the metric tensor and an additional four-vector. The extended field equations, when supplemented by suitable coordinate conditions, determine the time evolution of all these variables without any constraint. Einstein's solutions are recovered when the additional four-vector vanishes, so that the energy and momentum constraints hold true. The extended system is well posed when using the natural extension of either harmonic coordinates or the harmonic slicing condition in normal coordinates 
  We study massless Duffin-Kemmer-Petiau (DKP) fields in the context of Einstein-Cartan gravitation theory, interacting via minimal coupling procedure. In the case of an identically vanishing torsion (Riemannian space-times) we show that there exists local gauge symmetries which reproduce the usual gauge symmetries for the massless scalar and electromagnetic fields. On the other hand, similarly to what happens with the Maxwell theory, a non-vanishing torsion, in general, breaks the usual U(1) local gauge symmetry of the electromagnetic field or, in a different point of view, impose conditions on the torsion. 
  The singularitiy inside a spherical charged black hole, coupled to a spherical, massless scalar field is studied numerically. The profile of the characteristic scalar field was taken to be a power of advanced time with an exponent $\alpha>0$. A critical exponent $\alpha_{\rm crit}$ exists. For exponents below the critical one ($\alpha<\alpha_{\rm crit}$) the singularity is a union of spacelike and null sectors, as is also the case for data with compact support. For exponents greater than the critical one ($\alpha>\alpha_{\rm crit}$) an all-encompassing, spacelike singularity evolves, which completely blocks the ``tunnel'' inside the black hole, preventing the use of the black hole as a portal for hyperspace travel. 
  Contents:  Community news:   Einstein prize update, by Clifford Will   World year of physics, by Richard H. Price   We hear that... by Jorge Pullin  Research briefs:   R-mode epitaph? by John Friedman and Nils Andersson   Gravitational waves from bumpy neutron stars, by Ben Owen   LIGO science operations begin!, by Gary Sanders  Conference reports:   Fourth international LISA symposium, by Peter Bender   Initial Data for Binary Systems, by Gregory Cook   Report on Joint LSC/Source Modeling Meeting, by Patrick Brady   Greek Relativity Conference, NEB-X by Kostas Kokkotas and Nick Stergioulas   Gravity, Astrophysics, and Strings @ the Black Sea, by Plamen Fiziev   Quantum field theory at ESI, by Robert Wald   School on quantum gravity in Chile, by Don Marolf   Apples with apples workshop, by Miguel Alcubierre   Radiation reaction focus session and 5th Capra meeting, by Eanna Flanagan   Numerical Relativity Workshop at IMA, by Manuel Tiglio 
  The cosmological perturbation theory is revisited from the holographic point of view. In the case of the single brane model, it turns out that the AdS/CFT correspondence plays an important role. In the case of the two-brane model, it is shown that the effective equations of motion becomes the quasi-scalar-tensor gravity. It is also demonstrated that the radion anisotropy gives the CMB fluctuations through the Sachs-Wolfe effect. 
  The low energy effective theory for the Randall-Sundrum two-brane system is investigated with an emphasis on the role of the non-linear radion in the brane world. It is shown that the gravity on the brane world is described by a quasi-scalar-tensor theory with a specific coupling function. 
  The Einstein-Hilbert action (and thus the dynamics of gravity) can be obtained by combining the principle of equivalence, special relativity and quantum theory in the Rindler frame and postulating that the horizon area must be proportional to the entropy. This approach uses the local Rindler frame as a natural extension of the local inertial frame, and leads to the interpretation that the gravitational action represents the free energy of the spacetime geometry. As an aside, one obtains an insight into the peculiar structure of Einstein-Hilbert action and a natural explanation to the questions:(i) Why does the covariant action for gravity contain second derivatives of the metric tensor? (ii) Why is the gravitational coupling constant is positive ? Some geometrical features of gravitational action are clarified. 
  The equations of motion of compact binary systems and their associated Lagrangian formulation have been derived in previous works at the third post-Newtonian (3PN) approximation of general relativity in harmonic coordinates. In the present work we investigate the binary's relative dynamics in the center-of-mass frame (center of mass located at the origin of the coordinates). We obtain the 3PN-accurate expressions of the center-of-mass positions and equations of the relative binary motion. We show that the equations derive from a Lagrangian (neglecting the radiation reaction), from which we deduce the conserved center-of-mass energy and angular momentum at the 3PN order. The harmonic-coordinates center-of-mass Lagrangian is equivalent, {\it via} a contact transformation of the particles' variables, to the center-of-mass Hamiltonian in ADM coordinates that is known from the post-Newtonian ADM-Hamiltonian formalism. As an application we investigate the dynamical stability of circular binary orbits at the 3PN order. 
  We compute the spectrum of normalizable fermion bound states in a Schwarzschild black hole background. The eigenstates have complex energies. The real part of the energies, for small couplings, closely follow a hydrogen-like spectrum. The imaginary parts give decay times for the various states, due to the absorption properties of the hole, with states closer to the hole having shorter half-lives. As the coupling increases, the spectrum departs from that of the hydrogen atom, as states close to the horizon become unfavourable. Beyond a certain coupling the 1S1/2 state is no longer the ground state, which shifts to the 2P3/2 state, and then to states of successively greater angular momentum. For each positive energy state a negative energy counterpart exists, with opposite sign of its real energy, and the same decay factor. It follows that the Dirac sea of negative energy states is decaying, which may provide a physical contribution to Hawking radiation. 
  The mathematical notion of foliated cobordism is presented, and its relationship to both the motion of extended particles and wave motion is detailed. The fact that wave motion, when represented in such a manner on a four-dimensional spacetime, leads to a reduction of the bundle of linear frames to an SO(2)-principle bundle is demonstrated. Invariants of foliated cobordism are discussed as they relate to the aforementioned cases of motion. 
  We classify all spherically symmetric and homothetic spacetimes that are allowed kinematically by constructing them from a small number of building blocks. We then restrict attention to a particular dynamics, namely perfect fluid matter with the scale-free barotropic equation of state p = alpha mu where 0<alpha<1 is a constant. We assign conformal diagrams to all solutions in the complete classification of Carr and Coley, and so establish which of the kinematic possibilities are realized for these dynamics. We pay particular attention to those solutions which arise as critical solutions during gravitational collapse. 
  We present constraints from various experimental data that limit any spatial anisotropy of the Gravitational constant to less than a part per billion or even smaller. This rules out with a wide margin the recently reported claim of a spatial anisotropy of G with a diurnal temporal signature. 
  Two solutions of the coupled Einstein-Maxwell field equations are found by means of the Horsky-Mitskievitch generating conjecture. The vacuum limit of those obtained classes of spacetimes is the seed gamma-metric and each of the generated solutions is connected with one Killing vector of the seed spacetime. Some of the limiting cases of our solutions are identified with already known metrics, the relations among various limits are illustrated through a limiting diagram. We also verify our calculation through the Ernst potentials. The existence of circular geodesics is briefly discussed in the Appendix. 
  In induced gravity theory the solution of the dynamics equations for the test particle on null path leads to additional force in four-dimensional space-time. We find such force from five-dimensional geodesic line equations and try to apply this approach to analysis of additional acceleration of Pioneer 10/11, using properties of the asymmetrically warped space-time. 
  The article presents computer algebra procedures and routines applied to the study of the Dirac field on curved spacetimes. The main part of the procedures is devoted to the construction of Pauli and Dirac matrices algebra on an anholonomic orthonormal reference frame. Then these procedures are used to compute the Dirac equation on curved spacetimes in a sequence of special dedicated routines. A comparative review of such procedures obtained for two computer algebra platforms (REDUCE + EXCALC and MAPLE + GRTensorII) is carried out. Applications for the calculus of Dirac equation on specific examples of spacetimes with or without torsion are pointed out. 
  There are several definitions of the notion of angular momentum in general relativity. However non of them can be said to capture the physical notion of intrinsic angular momentum of the sources in the presence of gravitational radiation. We present a definition which is appropriate for the description of intrinsic angular momentum in radiative spacetimes. This notion is required in calculations involving radiation of angular momentum, as for example is expected in binary coalescence of black holes. 
  The abstract boundary has, in recent years, proved a general and flexible way to define the singularities of space-time. In this approach an essential singularity is a non-regular boundary point of an embedding which is accessible by a chosen family of curves within finite parameter distance. Ashley and Scott proved the first theorem relating essential singularities in strongly causal space-times to causal geodesic incompleteness. Linking this with the work of Beem on the $C^{r}$-stability of geodesic incompleteness allows proof of the stability of these singularities. Here I present this result stating the conditions under which essential singularities are $C^{1}$-stable against perturbations of the metric. 
  We present an exact solution of the Bargmann-Michel-Telegdi (BMT) equations for the dynamics of a spin particle in external electromagnetic and gravitational pp-wave fields. We demonstrate that an anomalous magnetic moment gives rise to an additional spin rotation which is modulated both by the electromagnetic and by the gravitational wave periodicities. 
  We first describe a class of spinor-curvature identities (SCI) which have gravitational applications. Then we sketch the topic of gravitational energy-momentum, its connection with Hamiltonian boundary terms and the issues of positivity and (quasi)localization. Using certain SCIs several spinor expressions for the Hamiltonian have been constructed. One SCI leads to the celebrated Witten positive energy proof and the Dougan-Mason quasilocalization. We found two other SCIs which give alternate positive energy proofs and quasilocalizations. In each case the spinor field has a different role. These neat expressions for gravitational energy-momentum have much appeal. However it seems that such spinor formulations just have no room for angular momentum; which leads us to doubt that spinor formulations can really correctly capture the elusive gravitational energy-momentum. 
  We extend and improve earlier estimates of the ability of the proposed LISA (Laser Interferometer Space Antenna) gravitational wave detector to place upper bounds on the graviton mass, m_g, by comparing the arrival times of gravitational and electromagnetic signals from binary star systems. We show that the best possible limit on m_g obtainable this way is ~ 50 times better than the current limit set by Solar System measurements. Among currently known, well-understood binaries, 4U1820-30 is the best for this purpose; LISA observations of 4U1820-30 should yield a limit ~ 3-4 times better than the present Solar System bound. AM CVn-type binaries offer the prospect of improving the limit by a factor of 10, if such systems can be better understood by the time of the LISA mission. We briefly discuss the likelihood that radio and optical searches during the next decade will yield binaries that more closely approach the best possible case. 
  We solve Einstein's field equations coupled to relativistic hydrodynamics in full 3+1 general relativity to evolve astrophysical systems characterized by strong gravitational fields. We model rotating, collapsing and binary stars by idealized polytropic equations of state, with neutron stars as the main application. Our scheme is based on the BSSN formulation of the field equations. We assume adiabatic flow, but allow for the formation of shocks. We determine the appearance of black holes by means of an apparent horizon finder. We introduce several new techniques for integrating the coupled Einstein-hydrodynamics system. For example, we choose our fluid variables so that they can be evolved without employing an artificial atmosphere. We also demonstrate the utility of working in a rotating coordinate system for some problems. We use rotating stars to experiment with several gauge choices for the lapse function and shift vector, and find some choices to be superior to others. We demonstrate the ability of our code to follow a rotating star that collapses from large radius to a black hole. Finally, we exploit rotating coordinates to evolve a corotating binary neutron star system in a quasi-equilibrium circular orbit for more than two orbital periods. 
  We generalize a previously obtained result, for the case of a few other static hyperbolic universes with manifolds of nontrivial topology as spatial sections. 
  An algorithm based on the choice of a single monotone function (subject to boundary conditions) is presented which generates all regular static spherically symmetric perfect fluid solutions of Einstein's equations. For physically relevant solutions the generating functions must be restricted by non-trivial integral-differential inequalities. Nonetheless, the algorithm is demonstrated here by the construction of an infinite number of previously unknown physically interesting exact solutions. 
  Covariant loop gravity comes out of the canonical analysis of the Palatini action and the use of the Dirac brackets arising from dealing with the second class constraints (``simplicity'' constraints). Within this framework, we underline a quantization ambiguity due to the existence of a family of possible Lorentz connections. We show the existence of a Lorentz connection generalizing the Ashtekar-Barbero connection and we loop-quantize the theory showing that it leads to the usual SU(2) Loop Quantum Gravity and to the area spectrum given by the SU(2) Casimir. This covariant point of view allows to analyze closely the drawbacks of the SU(2) formalism: the quantization based on the (generalized) Ashtekar-Barbero connection breaks time diffeomorphisms and physical outputs depend non-trivially on the embedding of the canonical hypersurface into the space-time manifold. On the other hand, there exists a true space-time connection, transforming properly under all diffeomorphisms. We argue that it is this connection that should be used in the definition of loop variables. However, we are still not able to complete the quantization program for this connection giving a full solution of the second class constraints at the Hilbert space level. Nevertheless, we show how a canonical quantization of the Dirac brackets at a finite number of points leads to the kinematical setting of the Barrett-Crane model, with simple spin networks and an area spectrum given by the SL(2,C) Casimir. 
  In order to obtain stable and accurate general relativistic simulations, re-formulations of the Einstein equations are necessary. In a series of our works, we have proposed using eigenvalue analysis of constraint propagation equations for evaluating violation behavior of constraints. In this article, we classify asymptotical behaviors of constraint-violation into three types (asymptotically constrained, asymptotically bounded, and diverge), and give their necessary and sufficient conditions. We find that degeneracy of eigenvalues sometimes leads constraint evolution to diverge (even if its real-part is not positive), and conclude that it is quite useful to check the diagonalizability of constraint propagation matrices. The discussion is general and can be applied to any numerical treatments of constrained dynamics. 
  We study numerically the late-time tails of linearized fields with any spin $s$ in the background of a spinning black hole. Our code is based on the ingoing Kerr coordinates, which allow us to penetrate through the event horizon. The late time tails are dominated by the mode with the least multipole moment $\ell$ which is consistent with the equatorial symmetry of the initial data and is equal to or greater than the least radiative mode with $s$ and the azimuthal number $m$. 
  We extend previous analyses of the violation of Lorentz invariance induced in a non-critical string model of quantum space-time foam, discussing the propagation of low-energy particles through a distribution of non-relativistic D-particles.We argue that nuclear and atomic physics experiments do not constitute sensitive probes of this approach to quantum gravity due to a difference in the dispersion relations for massive probes as compared to those for massless ones, predicted by the model. 
  After an introduction on phenomena due to spin and mass-energy currents on clocks and photons, we review the 1995-2001 measurements of the gravitomagnetic field of Earth and Lense-Thirring effect obtained by analyzing the orbits of the two laser-ranged satellites LAGEOS and LAGEOS II; this method has provided a direct measurement of Earth's gravitomagnetism with accuracy of the order of 20 %. A future accurate measurement of the Lense-Thirring effect, at the level of 1 % accuracy, may include the LARES experiment that will also provide other basic tests of general relativity and gravitation. Finally, we report the latest measurement of the Lense-Thirring effect, obtained in 2002 with the LAGEOS satellites over nearly 8 years of data. This 2002 result fully confirms and improves our previous measurements of the Earth frame-dragging: the Lense-Thirring effect exists and its experimental value is within ~ 20 % of what is predicted by Einstein's theory of general relativity. 
  A seemingly obvious extension of the weak equivalence principle, in which all matter must respond to Post-Newtonian gravitational fields, such as Lense-Thirring and radiation fields, in a composition-independent way, is considered in light of the Kramers-Kronig dispersion relations for the linear response of any material medium to these fields. It is argued that known observational facts lead to violations of this extended form of the equivalence principle. (PACS numbers: 04.80.Cc, 04.80.Nn, 03.65.Ud, 67.40.Bz) 
  We review recent efforts to re-formulate the Einstein equations for fully relativistic numerical simulations. The so-called numerical relativity (computational simulations in general relativity) is a promising research field matching with ongoing astrophysical observations such as gravitational wave astronomy. Many trials for longterm stable and accurate simulations of binary compact objects have revealed that mathematically equivalent sets of evolution equations show different numerical stability in free evolution schemes. In this article, we first review the efforts of the community, categorizing them into the following three directions: (1) modifications of the standard Arnowitt-Deser-Misner equations initiated by the Kyoto group, (2) rewriting of the evolution equations in hyperbolic form, and (3) construction of an "asymptotically constrained" system. We next introduce our idea for explaining these evolution behaviors in a unified way using eigenvalue analysis of the constraint propagation equations. The modifications of (or adjustments to) the evolution equations change the character of constraint propagation, and several particular adjustments using constraints are expected to diminish the constraint-violating modes. We propose several new adjusted evolution equations, and include some numerical demonstrations. We conclude by discussing some directions for future research. 
  We present a systematic study of collisions of homogeneous, plane--fronted, impulsive light--like signals which do not interact after head--on collision. For the head--on collision of two such signals, six real parameters are involved, three from each of the incoming signals. We find two necessary conditions to be satisfied by these six parameters for the signals to be non--interacting after collision. We then solve the collision problem in general when these necessary conditions hold. After collision the two signals focus each other at Weyl curvature singularities on each others signal front. Our family of solutions contains some known collision solutions as special cases. 
  We compute the semi-classical quantum amplitude to go from an initial spherically symmetric bosonic matter and gravitational field configuration to a final radiation configuration, corresponding to the relic Hawking radiation from a non-rotating, chargeless black hole which evaporates completely. This is obtained via the classical action integral which is solely a boundary term. On discretising the classical action, the quantum amplitude can be expressed in terms of generalised coherent states of the harmonic oscillator. A squeezed-state representation is obtained by complexifying the proper time separation T at spatial infinity between the initial and final space-like hypersurfaces. Such a procedure is deemed necessary as the two-surface problem for Dirichlet boundary data and wave-like perturbations is not well posed. We find that infinitesimal rotation into the lower complex T plane is equivalent to a highly-squeezed final state for the relic radiation, similar to the relic gravitational-wave background in cosmology. This final state is a pure state, and so the unpredictability associated with the final momentarily-naked singularity is avoided. The cosmological analogy is the tunnelling from an initial smooth Euclidean or timeless state to a classical universe. The high-squeezing limit corresponds to a final state of the Hawking flux which is indistinguishable from a stochastic collection of standing waves. The phases conjugate to the field amplitudes are squeezed to discrete values. We also discuss the entropy of the final radiation in the high-squeezing limit. 
  In an extended, new form of general relativity, which is a teleparallel theory of gravity, we examine the energy-momentum and angular momentum carried by gravitational wave radiated from Newtonian point masses in a weak-field approximation. The resulting wave form is identical to the corresponding wave form in general relativity, which is consistent with previous results in teleparallel theory. The expression for the dynamical energy-momentum density is identical to that for the canonical energy-momentum density in general relativity up to leading order terms on the boundary of a large sphere including the gravitational source, and the loss of dynamical energy-momentum, which is the generator of \emph{internal} translations, is the same as that of the canonical energy-momentum in general relativity. Under certain asymptotic conditions for a non-dynamical Higgs-type field $\psi^{k}$, the loss of ``spin'' angular momentum, which is the generator of \emph{internal} $SL(2,C)$ transformations, is the same as that of angular momentum in general relativity, and the losses of canonical energy-momentum and orbital angular momentum, which constitute the generator of Poincar\'{e} \emph{coordinate} transformations, are vanishing. The results indicate that our definitions of the dynamical energy-momentum and angular momentum densities in this extended new general relativity work well for gravitational wave radiations, and the extended new general relativity accounts for the Hulse-Taylor measurement of the pulsar PSR1913+16. 
  Three dimensional (3D) numerical evolutions of static black holes with excision are presented. These evolutions extend to about 8000M, where M is the mass of the black hole. This degree of stability is achieved by using growth-rate estimates to guide the fine tuning of the parameters in a multi-parameter family of symmetric hyperbolic representations of the Einstein evolution equations. These evolutions were performed using a fixed gauge in order to separate the intrinsic stability of the evolution equations from the effects of stability-enhancing gauge choices. 
  This paper reconsider the problem of a Proca field in the exterior of a static black hole. The original Bekenstein's demonstration on the vanishing of this field, based on an integral identity, is improved by using more natural arguments at the event horizon. In particular, the use of the so-called standard integration measure in the horizon is fully justified. Accordingly, the horizon contribution to the Bekenstein integral identity is more involved and its vanishing can be only established using the related Einstein equations. With the new reasoning the ``no-hair'' theorem for the Proca field now rest on better founded grounds. 
  We construct an explicit representation of the algebra of local diffeomorphisms of a manifold with realistic dimensions. This is achieved in the setting of a general approach to the (quantum) dynamics of a physical system which is characterized by the fundamental role assigned to a basic underlying symmetry. The developed mathematical formalism makes contact with the relevant gravitational notions by means of the addition of some extra structure. The specific manners in which this is accomplished, together with their corresponding physical interpretation, lead to different gravitational models. Distinct strategies are in fact briefly outlined, showing the versatility of the present conceptual framework. 
  Parallel transport around closed circular orbits in the equatorial plane of the Taub-NUT spacetime is analyzed to reveal the effect of the gravitomagnetic monopole parameter on circular holonomy transformations. Investigating the boost/rotation decomposition of the connection 1-form matrix evaluated along these orbits, one finds a situation that reflects the behavior of the general orthogonally transitive stationary axisymmetric case and indeed along Killing trajectories in general. 
  We have studied the time dependence of w for an expanding universe in the generalised B-D theory and have obtained its explicit dependence on the nature of matter contained in the universe,in different era.Lastly we discuss how the observed accelerated expansion of the present universe can be accomodated in the formalism. 
  In the framework of the special theory of relativity, the relativistic theory of gravitation (RTG) is constructed. The energy-momentum tensor density of all the matter fields (including gravitational one) is treated as a source of the gravitational field. The energy-momentum and the angular momentum conservation laws are fulfilled in this theory. Such an approach permits us to unambiguously construct the gravitional field theory as a gauge theory. According to the RTG, the homogeneous and isotropic Universe is to be ``flat''. It evolves cyclewise from some maximal density to the minimal one, etc.   The book is designed for scientific workers, post-graduates and upper-year students majoring in theoretical physics. 
  The article presents some aspects concerning the construction of a new thorn for the Cactus code, a complete 3-dimensional machinery for numerical relativity. This thorn is completely dedicated to numerical simulations in cosmology, that means it can provide evolutions of different cosmological models, mainly based on Friedman-Robertson-Walker metric. Some numerical results are presented, testing the convergence, stability and the applicability of the code. 
  We study geodesic motion in expanding spherical impulsive gravitational waves propagating in a Minkowski background. Employing the continuous form of the metric we find and examine a large family of geometrically preferred geodesics. For the special class of axially symmetric spacetimes with the spherical impulse generated by a snapping cosmic string we give a detailed physical interpretation of the motion of test particles. 
  We investigate the class of quadratic detectors (i.e., the statistic is a bilinear function of the data) for the detection of poorly modeled gravitational transients of short duration. We point out that all such detection methods are equivalent to passing the signal through a filter bank and linearly combine the output energy. Existing methods for the choice of the filter bank and of the weight parameters rely essentially on the two following ideas: (i) the use of the likelihood function based on a (possibly non-informative) statistical model of the signal and the noise, (ii) the use of Monte-Carlo simulations for the tuning of parametric filters to get the best detection probability keeping fixed the false alarm rate. We propose a third approach according to which the filter bank is "learned" from a set of training data. By-products of this viewpoint are that, contrarily to previous methods, (i) there is no requirement of an explicit description of the probability density function of the data when the signal is present and (ii) the filters we use are non-parametric. The learning procedure may be described as a two step process: first, estimate the mean and covariance of the signal with the training data; second, find the filters which maximize a contrast criterion referred to as deflection between the "noise only" and "signal+noise" hypothesis. The deflection is homogeneous to the signal-to-noise ratio and it uses the quantities estimated at the first step. We apply this original method to the problem of the detection of supernovae core collapses. We use the catalog of waveforms provided recently by Dimmelmeier et al. to train our algorithm. We expect such detector to have better performances on this particular problem provided that the reference signals are reliable. 
  I study fresh inflation from a five-dimensional vacuum state, where the fifth dimension is constant. In this framework, the universe can be seen as inflating in a four-dimensional FRW metric embedding in a five-dimensional metric. Finally, the experimental data $n_s=1$ are consistent with $(p+\rho_t)/\rho_t \simeq1/3$ in the fresh inflationary scenario. 
  In the frame of the Kerr-Schild approach, we obtain a generalization of the Kerr solution to a nonstationary case corresponding to a rotating source moving with arbitrary acceleration. Similar to the Kerr solution, the solutions obtained have the geodesic and shear free principal null congruence. The current parameters of the solutions are determined by a complex retarded-time construction via a given complex worldline of source. The real part of the complex worldline defines the values of the boost and acceleration while the imaginary part controls the rotation. The acceleration of the source is accompanied by a lightlike radiation along the principal null congruence. The solutions obtained generalize to the rotating case the known Kinnersley class of the "photon rocket" solutions. 
  We describe a formalism for studying spherically symmetric collapse of the massless scalar field in any spacetime dimension, and for any value of the cosmological constant $\Lambda$. The formalism is used for numerical simulations of gravitational collapse in four spacetime dimensions with negative $\Lambda$. We observe critical behaviour at the onset of black hole formation, and find that the critical exponent is independent of $\Lambda$. 
  The merger of binary neutron stars is likely to lead to differentially rotating remnants. In this paper we numerically construct models of differentially rotating neutron stars in general relativity and determine their maximum allowed mass. We model the stars adopting a polytropic equation of state and tabulate maximum allowed masses as a function of differential rotation and stiffness of the equation of state. We also provide a crude argument that yields a qualitative estimate of the effect of stiffness and differential rotation on the maximum allowed mass. 
  We present a framework for analyzing black hole backreaction from the point of view of quantum open systems using influence functional formalism. We focus on the model of a black hole described by a radially perturbed quasi-static metric and Hawking radiation by a conformally coupled massless quantum scalar field. It is shown that the closed-time-path (CTP) effective action yields a non-local dissipation term as well as a stochastic noise term in the equation of motion, the Einstein-Langevin equation. Once the thermal Green's function in a Schwarzschild background becomes available to the required accuracy the strategy described here can be applied to obtain concrete results on backreaction. We also present an alternative derivation of the CTP effective action in terms of the Bogolyubov coefficients, thus making a connection with the interpretation of the noise term as measuring the difference in particle production in alternative histories. 
  It has been shown in the past, that the six Doppler data streams obtained LISA configuration can be combined by appropriately delaying the data streams for cancelling the laser frequency noise. Raw laser noise is several orders of magnitude above the other noises and thus it is essential to bring it down to the level of shot, acceleration noises. A rigorous and systematic formalism using the techniques of computational commutative algebra was developed which generates all the data combinations cancelling the laser frequency noise. The relevant data combinations form a first module of syzygies. In this paper we use this formalism for optimisation of the LISA sensitivity by analysing the noise and signal covariance matrices. The signal covariance matrix, averaged over polarisations and directions, is calculated for binaries whose frequency changes at most adiabatically. We then present the extremal SNR curves for all the data combinations in the module. They correspond to the eigenvectors of the noise and signal covariance matrices. We construct LISA `network' SNR by combining the outputs of the eigenvectors which improves the LISA sensitivity substantially. The maximum SNR curve can yield an improvement upto 70 % over the Michelson, mainly at high frequencies, while the improvement using the network SNR ranges from 40 % to over 100 %. Finally, we describe a simple toy model, in which LISA rotates in a plane. In this analysis, we estimate the improvement in the LISA sensitivity, if one switches from one data combination to another as it rotates. Here the improvement in sensitivity, if one switches optimally over three cyclic data combinations of the eigenvector is about 55 % on an average over the LISA band-width. The corresponding SNR improvement is 60 %, if one maximises over the module. 
  We study the gravitational time delay in ray propagation due to rotating masses in the linear approximation of general relativity. Simple expressions are given for the gravitomagnetic time delay that occurs when rays of radiation cross a slowly rotating shell and propagate in the field of a distant rotating source. Moreover, we calculate the local gravitational time delay in the Goedel universe. The observational consequences of these results in the case of weak gravitational lensing are discussed. 
  We reexamine the possibility of the detection of the cosmic topology in nearly flat hyperbolic Friedmann-Lemaitre-Robertson-Walker (FLRW) universes by using patterns repetition. We update and extend our recent results in two important ways: by employing recent observational constraints on the cosmological density parameters as well as the recent mathematical results concerning small hyperbolic 3-manifolds. This produces new bounds with consequences for the detectability of the cosmic topology. In addition to obtaining new bounds, we also give a concrete example of the sensitive dependence of detectability of cosmic topology on the uncertainties in the observational values of the density parameters. 
  Generalized definitions for angular and linear momentum are given and shown to reduce to the ADM (at spatial infinity) definitions and the definitions at null infinity in the appropriate limit. These definitions are used to express angular momentum in terms of linear momentum. The formalism allows one to see the connection with the classical and special relativitistic notions of momenta. Further, the techniques elucidate, for the first time, the geometric nature of these conserved quantities. The boosted Schwarzschild solution is used to illustrate some aspects. The definitions are useful and give insight in the region far from all masses where gravity waves are detected. 
  We present general relativistic hydrodynamics simulations of constant specific angular momentum tori orbiting a Schwarzschild black hole. These tori are expected to form as a result of stellar gravitational collapse, binary neutron star merger or disruption, can reach very high rest-mass densities and behave effectively as neutron stars but with a toroidal topology (i.e. ``toroidal neutron stars''). Our attention is here focussed on the dynamical response of these objects to axisymmetric perturbations. We show that, upon the introduction of perturbations, these systems either become unstable to the runaway instability or exhibit a regular oscillatory behaviour resulting in a quasi-periodic variation of the accretion rate as well as of the mass quadrupole. The latter, in particular, is responsible for the emission of intense gravitational radiation whose signal-to-noise ratio at the detector is comparable or larger than the typical one expected in stellar-core collapse, making these new sources of gravitational waves potentially detectable. We discuss a systematic investigation of the parameter space both in the linear and nonlinear regimes, providing estimates of how the gravitational radiation emitted depends on the mass of the torus and on the strength of the perturbation. 
  This letter describes a scalar curvature invariant for general relativity with a certain, distinctive feature. While many such invariants exist, this one vanishes in regions of space-time which can be said unambiguously to contain no gravitational radiation. In more general regions which incontrovertibly support non-trivial radiation fields, it can be used to extract local, coordinate-independent information partially characterizing that radiation. While a clear, physical interpretation is possible only in such radiation zones, a simple algorithm can be given to extend the definition smoothly to generic regions of space-time. 
  We show that, in the framework of Carath\'eodory's approach to thermodynamics, one can implement black hole thermodynamics by realizing that there exixts a quasi-homogeneity symmetry of the Pfaffian form $\deq$ representing the infinitesimal heat exchanged reversibly by a Kerr-Newman black hole; this allow us to calculate readily an integrating factor, and, as a consequence, a foliation of the thermodynamic manifold can be recovered. 
  We propose a generalized thermodynamics in which quasi-homogeneity of the thermodynamic potentials plays a fundamental role. This thermodynamic formalism arises from a generalization of the approach presented in paper [1], and it is based on the requirement that quasi-homogeneity is a non-trivial symmetry for the Pfaffian form $\delta Q_{rev}$. It is shown that quasi-homogeneous thermodynamics fits the thermodynamic features of at least some self-gravitating systems. We analyze how quasi-homogeneous thermodynamics is suggested by black hole thermodynamics. Then, some existing results involving self-gravitating systems are also shortly discussed in the light of this thermodynamic framework. The consequences of the lack of extensivity are also recalled. We show that generalized Gibbs-Duhem equations arise as a consequence of quasi-homogeneity of the thermodynamic potentials. An heuristic link between this generalized thermodynamic formalism and the thermodynamic limit is also discussed. 
  Gowdy's model of cosmological spacetimes is a much investigated subject in classical and quantum gravity. Depending on spatial topology recollapsing as well as expanding models are known. Several analytic tools were used in order to clarify singular behaviour in this class of spacetimes. Here we investigate the structure of a certain subclass, the polarized Gowdy models with spatial T^3-topology, in the large. The asymptotics for general solutions of the dynamical equation for one of the gravitational degrees of freedom plays a key role while the asymptotic behaviour of the remaining metric function is a result of solving the Hamiltonian constraint equation. Using both we are able to prove (future) geodesic completeness in all spacetimes of this type. 
  We consider classical and quantum dynamics of a free particle in de Sitter's space-times with different topologies to see what happens to space-time singularities of removable type in quantum theory. We find analytic solution of the classical dynamics. The quantum dynamics is solved by finding an essentially self-adjoint representation of the algebra of observables integrable to the unitary representations of the symmetry group of each considered gravitational system. The dynamics of a massless particle is obtained in the zero-mass limit of the massive case. Our results indicate that taking account of global properties of space-time enables quantization of particle dynamics in all considered cases. 
  The algebraic approach to black hole quantization requires the horizon area eigenvalues to be equally spaced. As shown previously, for a neutral non-rotating black hole, such eigenvalues must be $2^{n}$-fold degenerate if one constructs the black hole stationary states by means of a pair of creation operators subject to a specific algebra. We show that the algebra of these two building blocks exhibits $U(2)\equiv U(1)\times SU(2)$ symmetry, where the area operator generates the U(1) symmetry. The three generators of the SU(2) symmetry represent a {\it global} quantum number (hyperspin) of the black hole, and we show that this hyperspin must be zero. As a result, the degeneracy of the $n$-th area eigenvalue is reduced to $2^{n}/n^{3/2}$ for large $n$, and therefore, the logarithmic correction term $-3/2\log A$ should be added to the Bekenstein-Hawking entropy. We also provide a heuristic approach explaining this result, and an evidence for the existence of {\it two} building blocks. 
  This paper explores ``black hole'' solutions of various Einstein-wave matter systems admitting an isometry of their domain of outer communications taking every point to its future. In the first two parts, it is shown that such solutions, assuming in addition that they are spherically symmetric and the matter has a certain structure, must be Schwarzschild or Reissner-Nordstrom. Non-trivial examples of matter for which the result applies are a wave map and a massive charged scalar field interacting with an electromagnetic field. The results thus generalize work of Bekenstein [1] and Heusler [12] from the static to the periodic case. In the third part, which is independent of the first two, it is shown that Dirac fields preserved by an isometry of a spherically symmetric domain of outer communications of the type described above must vanish. It can be applied in particular to the Einstein-Dirac-Maxwell equations or the Einstein-Dirac-Yang/Mills equations, generalizing work of Finster, Smoller, and Yau [9], [7], [8], and also [6]. 
  We discuss in the framework of black hole thermodynamics some aspects relative to the third law in the case of black holes of the Kerr-Newman family. In the light of the standard proof of the equivalence between the unattainability of the zero temperature and the entropic version of the third law it is remarked that the unattainability has a special character in black hole thermodynamics. Also the zero temperature limit which obtained in the case of very massive black holes is discussed and it is shown that a violation of the entropic version in the charged case occurs. The violation of the Bekenstein-Hawking law in favour of zero entropy S_E=0 in the case of extremal black holes is suggested as a natural solution for a possible violation of the second law of thermodynamics. Thermostatic arguments in support of the unattainability are explored, and $S_E=0$ for extremal black holes is shown to be again a viable solution. The third law of black hole dynamics by W.Israel is then interpreted as a further strong corroboration to the picture of a discontinuity between extremal states and non-extremal ones. 
  We study through numerical simulation the spherical collapse of isothermal gas in Newtonian gravity. We observe a critical behavior which occurs at the threshold of gravitational instability leading to core formation. For a given initial density profile, we find a critical temperature, which is of the same order as the virial temperature of the initial configuration. For the exact critical temperature, the collapse converges to a self-similar form, the first member in Hunter's family of self-similar solutions. For a temperature close to the critical value, the collapse first approaches this critical solution. Later on, in the supercritical case, the collapse converges to another self-similar solution, which is called the Larson-Penston solution. In the subcritical case, the gas bounces and disperses to infinity. We find two scaling laws: one for the collapsed mass in the supercritical case and the other for the maximum density reached before dispersal in the subcritical case. The value of the critical exponent is measured to be $\simeq 0.11$ in the supercritical case, which agrees well with the predicted value $\simeq 0.10567$. These critical properties are quite similar to those observed in the collapse of a radiation fluid in general relativity. We study the response of the system to temperature fluctuation and discuss astrophysical implications for the insterstellar medium structure and for the star formation process. Newtonian critical behavior is important not only because it provides a simple model for general relativity but also because it is relevant for astrophysical systems such as molecular clouds. 
  Response of an interferometer becomes complicated for gravitational wave shorter than the arm-length of the detector, as nature of wave appears strongly. We have studied how parameter estimation for merging massive black hole binaries are affected by this complicated effect in the case of LISA. It is shown that three dimensional positions of some binaries might be determined much better than the past estimations that use the long wave approximation. For equal mass binaries this improvement is most prominent at $\sim 10^5\sol$. 
  We have measured the photothermal effect in a single cross-polarized interferometer at audio frequencies (5 Hz - 4 kHz). In a Fabry-Perot interferometer, light in one polarization is chopped to periodically heat the interferometer mirrors, while light in the orthogonal polarization measures the mirror length changes. Tests of a polished solid metal mirror show good agreement with relevant proposed theories by Braginsky et al. ["Thermodynamical fluctuations and photo-thermal shot noise in gravitational wave antennae," Physics Letters A 264, 1-10 (1999)] and Cerdonio et al. ["Thermoelastic effects at low temperatures and quantum limits in displacement measurements," Physical Review D 63 082003 (2001)] describing uncoated optics. 
  In this paper the effect of the post-Newtonian gravitomagnetic force on the mean longitudes $l$ of a pair of counter-rotating Earth artificial satellites following almost identical circular equatorial orbits is investigated. The possibility of measuring it is examined. The observable is the difference of the times required to $l$ in passing from 0 to 2$\pi$ for both senses of motion. Such gravitomagnetic time shift, which is independent of the orbital parameters of the satellites, amounts to 5$\times 10^{-7}$ s for Earth; it is cumulative and should be measured after a sufficiently high number of revolutions. The major limiting factors are the unavoidable imperfect cancellation of the Keplerian periods, which yields a constraint of 10$^{-2}$ cm in knowing the difference between the semimajor axes $a$ of the satellites, and the difference $I$ of the inclinations $i$ of the orbital planes which, for $i\sim 0.01^\circ$, should be less than $0.006^\circ$. A pair of spacecrafts endowed with a sophisticated intersatellite tracking apparatus and drag-free control down to 10$^{-9}$ cm s$^{-2}$ Hz$^{-{1/2}}$ level might allow to meet the stringent requirements posed by such a mission. 
  In this paper we give, for the first time, a complete description of the late-time evolution of non-tilted spatially homogeneous cosmologies of Bianchi type VIII. The source is assumed to be a perfect fluid with equation of state $p = (\gamma - 1)\mu$, where $\gamma$ is a constant which satisfies $1 \leq \gamma \leq 2$. Using the orthonormal frame formalism and Hubble-normalized variables, we rigorously establish the limiting behaviour of the models at late times, and give asymptotic expansions for the key physical variables. The main result is that asymptotic self-similarity breaking occurs, and is accompanied by the phenomenon of `Weyl curvature dominance', characterized by the divergence of the Hubble-normalized Weyl curvature at late times. 
  We address the issue of finding an optimal detection method for a discontinuous or intermittent gravitational wave stochastic background. Such a signal might sound something like popcorn popping. We derive an appropriate version of the maximum likelihood detection statistic, and compare its performance to that of the standard cross-correlation statistic both analytically and with Monte Carlo simulations. The maximum likelihood statistic performs better than the cross-correlation statistic when the background is sufficiently non-Gaussian. For both ground and space based detectors, this results in a gain factor, ranging roughly from 1 to 3, in the minimum gravitational-wave energy density necessary for detection, depending on the duty cycle of the background. Our analysis is exploratory, as we assume that the time structure of the events cannot be resolved, and we assume white, Gaussian noise in two collocated, aligned detectors. Before this detection method can be used in practice with real detector data, further work is required to generalize our analysis to accommodate separated, misaligned detectors with realistic, colored, non-Gaussian noise. 
  The Bogoliubov procedure in quantum field theory is used to describe a relativistic almost ideal Bose gas at zero temperature. Special attention is given to the study of a vortex. The radius of the vortex in the field description is compared to that obtained in the relativistic fluid approximation. The Kelvin waves are studied and, for long wavelengths, the dispersion relation is obtained by an asymptotic matching method and compared with the non relativistic result. 
  An expanding closed universe filled with radiation can either recollapse or tunnel to the regime of unbounded expansion, if the cosmological constant is nonzero. We re-examine the question of particle creation during tunneling, with the purpose of resolving a long-standing controversy. Using a perturbative superspace model with a conformally coupled massless scalar field, which is known to give no particle production, we explicitly show that the breakdown of the semiclassical approximation and the ``catastrophic particle production'' claimed earlier in the literature are due to an inappropriate choice of the initial quantum state prior to the tunneling. 
  The stability of the Cauchy horizon in spherically symmetric self-similar collapse is studied by determining the flux of scalar radiation impinging on the horizon. This flux is found to be finite. 
  The set N of all null geodesics of a globally hyperbolic (d+1)-dimensional spacetime (M,g) is naturally a smooth (2d-1)-dimensional contact manifold. The sky of an event is the subset of N defined by all null geodesics through that event, and is an embedded Legendrian submanifold of N diffeomorphic to a (d-1)-dimensional sphere. It was conjectured by Low that for d=2 two events are causally related iff their skies are linked (in an appropriate sense). We use the contact structure and knot polynomial calculations to prove this conjecture in certain particular cases, and suggest that for d=3 smooth linking should be replaced with Legendrian linking. 
  Averaged inhomogeneous cosmologies lie at the forefront of interest, since cosmological parameters like the rate of expansion or the mass density are to be considered as volume-averaged quantities and only these can be compared with observations. For this reason the relevant parameters are intrinsically scale-dependent and one wishes to control this dependence without restricting the cosmological model by unphysical assumptions. In the latter respect we contrast our way to approach the averaging problem in relativistic cosmology with shortcomings of averaged Newtonian models. Explicitly, we investigate the scale-dependence of Eulerian volume averages of scalar functions on Riemannian three-manifolds. We propose a complementary view of a Lagrangian smoothing of (tensorial) variables as opposed to their Eulerian averaging on spatial domains. This program is realized with the help of a global Ricci deformation flow for the metric. We explain rigorously the origin of the Ricci flow which, on heuristic grounds, has already been suggested as a possible candidate for smoothing the initial data set for cosmological spacetimes. The smoothing of geometry implies a renormalization of averaged spatial variables. We discuss the results in terms of effective cosmological parameters that would be assigned to the smoothed cosmological spacetime. 
  We review the arguments supporting the existence of a maximal acceleration for a massive particle and show that different values of this upper limit can be predicted in different physical situations. 
  We review the definition of (maximally supersymmetric) vacuum in supergravity theories, the currently known vacua in arbitrary dimensions and how the associated supersymmetry algebras can be found. (Invited talk at the Spanish Relativity Meeting (``EREs'') 2002, Mao, Menorca, September 21-23 2002.) 
  We present new static spherically-symmetric exact solutions of Einstein equations with the quintessential matter surrounding a black hole charged or not as well as for the case without the black hole. A condition of additivity and linearity in the energy-momentum tensor is introduced, which allows one to get correct limits to the known solutions for the electromagnetic static field implying the relativistic relation between the energy density and pressure, as well as for the extraordinary case of cosmological constant, i.e. de Sitter space. We classify the horizons, which evidently reveal themselves in the static coordinates, and derive the Gibbons-Hawking temperatures. An example of quintessence with the state parameter w=-2/3 is discussed in detail. 
  Misprints corrected, two references added. To appear in the Phys. Rev. D. 
  We investigate the dynamics of relativistic spinning test particles in the spacetime of a rotating black hole using the Papapetrou equations. We use the method of Lyapunov exponents to determine whether the orbits exhibit sensitive dependence on initial conditions, a signature of chaos. In the case of maximally spinning equal-mass binaries (a limiting case that violates the test-particle approximation) we find unambiguous positive Lyapunov exponents that come in pairs +/- lambda, a characteristic of Hamiltonian dynamical systems. We find no evidence for nonvanishing Lyapunov exponents for physically realistic spin parameters, which suggests that chaos may not manifest itself in the gravitational radiation of extreme mass-ratio binary black-hole inspirals (as detectable, for example, by LISA, the Laser Interferometer Space Antenna). 
  An efficient algorithm is presented for the identification of short bursts of gravitational radiation in the data from broad-band interferometric detectors. The algorithm consists of three steps: pixels of the time-frequency representation of the data that have power above a fixed threshold are first identified. Clusters of such pixels that conform to a set of rules on their size and their proximity to other clusters are formed, and a final threshold is applied on the power integrated over all pixels in such clusters. Formal arguments are given to support the conjecture that this algorithm is very efficient for a wide class of signals. A precise model for the false alarm rate of this algorithm is presented, and it is shown using a number of representative numerical simulations to be accurate at the 1% level for most values of the parameters, with maximal error around 10%. 
  A scalar field generalization of Xanthopoulos's cylindrically symmetric solutions of the vacuum Einstein equation is obtained. The obtained solution preserves the properties of the Xanthopoulos solution, which are regular on the axis, asymptotically flat and free from the curvature singularities. The solution describes stable, infinite length of rotating cosmic string interacting with gravitational and scalar waves. 
  In the context of the averaging problem in relativistic cosmology, we provide a key to the interpretation of cosmological parameters by taking into account the actual inhomogeneous geometry of the Universe. We discuss the relation between `bare' cosmological parameters determining the cosmological model, and the parameters interpreted by observers with a ``Friedmannian bias'', which are `dressed' by the smoothed-out geometrical inhomogeneities of the surveyed spatial region. 
  By employing an exact back-reaction geometry, Helliwell-Konkowski stability conjecture is shown to fail. This happens when a test null dust is inserted to the interaction region of cross-polarized Bell-Szekeres spacetime. 
  The early time behaviour of brane-world models is analysed in the presence of anisotropic stresses. It is shown that that the initial singularity cannot be isotropic, unless there is also an isotropic fluid stiffer than radiation present. Also, a magnetic Bianchi type I brane-world is analysed in detail. It is known that the Einstein equations for the magnetic Bianchi type I models are in general oscillatory and are believed to be chaotic, but in the brane-world model this chaotic behaviour does not seem to be possible. 
  The existence of time machines, understood as spacetime constructions exhibiting physically realised closed timelike curves (CTCs), would raise fundamental problems with causality and challenge our current understanding of classical and quantum theories of gravity. In this paper, we investigate three proposals for time machines which share some common features: cosmic strings in relative motion, where the conical spacetime appears to allow CTCs; colliding gravitational shock waves, which in Aichelburg-Sexl coordinates imply discontinuous geodesics; and the superluminal propagation of light in gravitational radiation metrics in a modified electrodynamics featuring violations of the strong equivalence principle. While we show that ultimately none of these constructions creates a working time machine, their study illustrates the subtle levels at which causal self-consistency imposes itself, and we consider what intuition can be drawn from these examples for future theories. 
  The frequencies of a cryogenic sapphire oscillator and a hydrogen maser are compared to set new constraints on a possible violation of Lorentz invariance. We determine the variation of the oscillator frequency as a function of its orientation (Michelson-Morley test) and of its velocity (Kennedy-Thorndike test) with respect to a preferred frame candidate. We constrain the corresponding parameters of the Mansouri and Sexl test theory to $\delta - \beta + 1/2 = (1.5\pm 4.2) \times 10^{-9}$ and $\beta - \alpha - 1 = (-3.1\pm 6.9) \times 10^{-7}$ which is equivalent to the best previous result for the former and represents a 30 fold improvement for the latter. 
  I study the Bona-Masso family of hyperbolic slicing conditions, considering in particular its properties when approaching two different types of singularities: focusing singularities and gauge shocks. For focusing singularities, I extend the original analysis of Bona et. al and show that both marginal and strong singularity avoidance can be obtained for certain types of behavior of the slicing condition as the lapse approaches zero. For the case of gauge shocks, I re-derive a condition found previously that eliminates them. Unfortunately, such a condition limits considerably the type of slicings allowed. However, useful slicing conditions can still be found if one asks for this condition to be satisfied only approximately. Such less restrictive conditions include a particular member of the 1+log family, which in the past has been found empirically to be extremely robust for both Brill wave and black hole simulations. 
  We construct a spin foam model of Yang-Mills theory coupled to gravity by using a discretized path integral of the BF theory with polynomial interactions and the Barret-Crane ansatz. In the Euclidian gravity case we obtain a vertex amplitude which is determined by a vertex operator acting on a simple spin network function. The Euclidian gravity results can be straightforwardly extended to the Lorentzian case, so that we propose a Lorentzian spin foam model of Yang-Mills theory coupled to gravity. 
  We study a simple analytic solution to Einstein's field equations describing a thin spherical shell consisting of collisionless particles in circular orbit. We then apply two independent criteria for the identification of circular orbits, which have recently been used in the numerical construction of binary black hole solutions, and find that both yield equivalent results. Our calculation illustrates these two criteria in a particularly transparent framework and provides further evidence that the deviations found in those numerical binary black hole solutions are not caused by the different criteria for circular orbits. 
  We report the result from a search for bursts of gravitational waves using data collected by the cryogenic resonant detectors EXPLORER and NAUTILUS during the year 2001, for a total measuring time of 90 days. With these data we repeated the coincidence search performed on the 1998 data (which showed a small coincidence excess) applying data analysis algorithms based on known physical characteristics of the detectors. With the 2001 data a new interesting coincidence excess is found when the detectors are favorably oriented with respect to the Galactic Disk. 
  Quantization is performed of a Friedmann-Robertson-Walker universe filled with a conformally invariant scalar field and a perfect fluid with equation of state $p=\alpha \rho$. A well-known discrete set of static quantum wormholes is shown to exist for radiation ($\alpha =1/3$), and a novel continuous set is found for cosmic strings ($\alpha = -1/3$), the latter states having throat radii of any size. In both cases wave-packet solutions to the Wheeler-DeWitt equation are obtained with all the properties of evolving quantum wormholes. In the case of a radiation fluid, a detailed analysis of the quantum dynamics is made in the context of the Bohm-de Broglie interpretation. It is shown that a repulsive quantum force inversely proportional to the cube of the scale factor prevents singularities in the quantum domain. For the states considered, there are no particle horizons either. 
  The correspondence between black holes and colliding waves extends to cover the near horizon geometry of rotating black holes and colliding waves with cross polarization. Extreme Kerr and Kerr-Newman geometries are given as examples. 
  Inspired by the discrete evolution implied by the recent work on loop quantum cosmology, we obtain a discrete time description of usual quantum mechanics viewing it as a constrained system. This description, obtained without any approximation or explicit discretization, mimics features of the discrete time evolution of loop quantum cosmology. We discuss the continuum limit, physical inner product and matrix elements of physical observables to bring out various issues regarding viability of a discrete evolution. We also point out how a continuous time could emerge without appealing to any continuum limit. 
  We reconsider the problem of modelling static spherically symmetric perfect fluid configurations with an equation of state from a point of view of that requires the use of the concept of principal transform of a 3-dimensional Riemannian metric. We discuss from this new point of view the meaning of those familiar quantities that we call density, pressure and geometry in a relativistic context. This is not simple semantics. To prove it we apply the new ideas to recalculate the maximum mass that a massive neutron core can have. This limit is found to be of the order of 3.8 $M_\odot$ substantially larger than the Oppenheimer and Volkoff limit. 
  We show that the graviton acquires a mass in a de Sitter background given by $m_{g}^{2}=-{2/3}\Lambda.$ This is precisely the fine-tuning value required for the perturbed gravitational field to mantain its two degrees of freedom. 
  In string theory the coupling ``constants'' appearing in the low-energy effective Lagrangian are determined by the vacuum expectation values of some (a priori) massless scalar fields (dilaton, moduli). This naturally leads one to expect a correlated variation of all the coupling constants, and an associated violation of the equivalence principle. We review some string-inspired theoretical models which incorporate such a spacetime variation of coupling constants while remaining naturally compatible both with phenomenological constraints coming from geochemical data (Oklo; Rhenium decay) and with present equivalence principle tests. Barring a very unnatural fine-tuning of parameters, a variation of the fine-structure constant as large as that recently ``observed'' by Webb et al. in quasar absorption spectra appears to be incompatible with these phenomenological constraints. Independently of any model, it is emphasized that the best experimental probe of varying constants are high-precision tests of the universality of free fall, such as MICROSCOPE and STEP. 
  The geometric calculus based on Clifford algebra is a very useful tool for geometry and physics. It describes a geometric structure which is much richer than the ordinary geometry of spacetime. A Clifford manifold (C-space) consists not only of points, but also of 1-loops, 2-loops, etc.. They are associated with multivectors which are the wedge product of the basis vectors, the generators of Clifford algebra. Within C-space we can perform the so called polydimensional rotations which reshuffle the multivectors, e.g., a bivector into a vector, etc.. A consequence of such a polydimensional rotation is that the signature can change: it is relative to a chosen set of basis vectors. Another important consequence is that the well known unconstrained Stueckelberg theory is embedded within the constrained theory based on C-space. The essence of the Stueckelberg theory is the existence of an evolution parameter which is invariant under the Lorentz transformations. The latter parameter is interpreted as being the true time - associated with our perception of the passage of time. 
  The ``generic'' family of classical sequential growth dynamics for causal sets provides cosmological models of causal sets which are a testing ground for ideas about the, as yet unknown, quantum theory. In particular we can investigate how general covariance manifests itself and address the problem of identifying and interpreting covariant ``observables'' in quantum gravity. The problem becomes, in this setting, that of identifying measurable covariant collections of causal sets, to each of which corresponds the question: ``Does the causal set that occurs belong to this collection?'' It has for answer the probability measure of the collection. Answerable covariant questions, then, correspond to measurable collections of causal sets which are independent of the labelings of the causal sets. However, what the transition probabilities of the classical sequential growth dynamics provide directly is a measure on the space of {\it labeled} causal sets and the physical interpretation of the covariant measurable collections is consequently obscured. We show that there is a physically meaningful characterisation of the class of measurable covariant sets as unions and differences of ``stem sets''. 
  Qualitative approach to homogeneous anisotropic Bianchi class A models in terms of dynamical systems reveals a hierarchy of invariant manifolds. By calculating the Kovalevski Exponents according to Adler - van Moerbecke method we discuss how algebraic integrability property is distributed in this class of models. In particular we find that algebraic nonintegrability of vacuum Bianchi VII_0 model is inherited by more general Bianchi VIII and Bianchi IX vacuum types. Matter terms (cosmological constant, dust and radiation) in the Einstein equations typically generate irrational or complex Kovalevski exponents in class A homogeneous models thus introducing an element of nonintegrability even though the respective vacuum models are integrable. 
  I examine the results obtained so far in exploring the recent proposal of theories of the relativistic transformations between inertial observers that involve both an observer-independent velocity scale and an observer-independent length/momentum scale. I also discuss what appear to be the key open issues for this research line. 
  We analyse the classical and quantum geometry of the Barrett-Crane spin foam model for four dimensional quantum gravity, explaining why it has to be considering as a covariant realization of the projector operator onto physical quantum gravity states. We discuss how causality requirements can be consistently implemented in this framework, and construct causal transiton amplitudes between quantum gravity states, i.e. realising in the spin foam context the Feynman propagator between states. The resulting causal spin foam model can be seen as a path integral quantization of Lorentzian first order Regge calculus, and represents a link between several approaches to quantum gravity as canonical loop quantum gravity, sum-over-histories formulations, dynamical triangulations and causal sets. In particular, we show how the resulting model can be rephrased within the framework of quantum causal sets (or histories). 
  In this paper the most recent developments in testing General Relativity in the gravitational field of the Earth with the technique of Satellite Laser Ranging are presented. In particular, we concentrate our attention on some gravitoelectric and gravitomagnetic post--Newtonian orbital effects on the motion of a test body in the external field of a central mass. 
  Recent observations of the fine structure of spectral lines in the early universe have been interpreted as a variation of the fine structure constant.   From the assumed validity of Maxwell equations in general relativity and well known experimental facts, it is proved that $e$ and $\hbar$ are absolute constants. On the other hand, the speed of light need not be constant. 
  4-dimensional homogeneous isotropic cosmological models obtained from solutions of vacuum 5-dimensional Einstein equations are considered. It is assumed, that the G(55)-component of the 5-d metric simulates matter in the comoving frame of reference. Observable 4-d metric is defined up to conformal transformations of the metric of 4-d section \tilde{g}, with a conformal factor as a function of the component G(55). It is demonstrated, that the form of this function determines the matter equation of state. Possible equations of state are analyzed separately for flat, open and close models. 
  In this paper we develop a formalism which models all massive particles as travelling at the speed of light(c). This is done by completing the 3-velocity v of a test particle to the speed of light by adding an auxiliary 3-velocity component z for the particle. According to the observations and laws of physics defined in our spacetime these vectors are generalized to domains and then two methods are developed to define c domain in terms of our spacetime(v domain). By using these methods the formalism is applied on relativistic quantum theory and general theory of relativity. From these, the relation between the formalism and Mach's principle is investigated. The ideas and formulae developed from application of the formalism on general relativity are compared with the characteristics of anomalous accelerations detected on Pioneer 10/11, Ulysses and Galileo spacecrafts and an explanation according to the formalism is suggested. Possible relationships between Mach's principle and the nondeterministic nature of the universe are also explored. In this study Mach's principle, on which current debate still continues, is expressed from an unconventional point of view, as a naturally arising consequence of the formalism, and the approaches are simplified accordingly. 
  We investigate the robustness of some recent results obtained for homogeneous and isotropic cosmological models with conformally coupled scalar fields. For this purpose, we investigate anisotropic homogeneous solutions of the models described by the action $$ S=\int d^4x \sqrt{-g}\left\{F(\phi)R - \partial_a\phi\partial^a\phi -2V(\phi) \right\}, $$ with general $F(\phi)$ and $V(\phi)$. We show that such a class of models leads generically to geometrical singularities if for some value of $\phi$, $F(\phi)=0$, rendering previous cosmological results obtained for the conformal coupling case highly unstable. We show that stable models can be obtained for suitable choices of $F(\phi)$ and $V(\phi)$. Implications for other recent results are also discussed. 
  Employing the approximate effective action constructed from the coincidence limit of the Hadamard-Minakshisundaram-DeWitt (HaMiDeW) coefficient a_{3}, the renormalized stress-energy tensor of the quantized massive scalar field in the spacetime of a static and electrically charged dilatonic black hole is calculated. Special attention is paid to the minimally and conformally coupled fields propagating in geometries with a=1, and to the power expansion of the general stress-energy tensor for small values of charge. A compact expression for the trace of the stress-energy tensor is presented. 
  We derive a new first-order formulation for Einstein's equations which involves fewer unknowns than other first-order formulations that have been proposed. The new formulation is based on the 3+1 decomposition with arbitrary lapse and shift. In the reduction to first order form only 8 particular combinations of the 18 first derivatives of the spatial metric are introduced. In the case of linearization about Minkowski space, the new formulation consists of symmetric hyperbolic system in 14 unknowns, namely the components of the extrinsic curvature perturbation and the 8 new variables, from whose solution the metric perturbation can be computed by integration. 
  The Multiverse is collection of parallel universes. In this article a formal theory and a topos-theoretic models of the multiverse are given. For this the Lawvere-Kock Synthetic Differential Geometry and topos models for smooth infinitesimal analysis are used. Physical properties of multi-variant and many-dimensional parallel universes are discussed. The source of multiplicity of physical objects is set of physical constants. 
  We consider the dynamics of a Bianchi type I spacetime in the presence of dilaton and magnetic fields. The general solution of the Einstein-Maxwell dilaton field equations can be obtained in an exact parametric form. Depending on the numerical values of the parameters of the model there are three distinct classes of solutions. The time evolution of the mean anisotropy, shear and deceleration parameter is considered in detail and it is shown that a magnetic-dilaton anisotropic Bianchi type I geometry does not isotropize, the initial anisotropy being present in the universe for all times. 
  We construct new axially symmetric solutions of SU(2) Yang-Mills theory in a four dimensional anti-de Sitter spacetime. Possessing nonvanishing nonabelian charges, these regular configurations have also a nonzero angular momentum. Numerical arguments are also presented for the existence of rotating solutions of Einstein-Yang-Mills equations in an asymptotically anti-de Sitter spacetime. 
  Some examples from the mathematics of shape are presented that question some of the almost hidden assumptions behind results on limiting behaviour of finitary approximations to space-time. These are presented so as to focus attention on the observational problem of refinement and suggest the necessity for an alternative theory of `fractafolds' against which the limiting theory of $C^\infty$-differential manifolds usually underlying (quantum) general relativity can be measured. 
  We study the effect of retardation of gravity in binary pulsars. It appears in pulsar timing formula as a periodic excess time delay to the Shapiro effect. The retardation of gravity effect can be large enough for observation in binary pulsars with the nearly edgewise orbits and relatively large ratio of the projected semimajor axis to the orbital period of the pulsar. If one succeeds in measuring the retardation of gravity it will give further experimental evidence in favor of General Relativity. 
  If there is a shortest length in nature, for example at the Planck scale of 10^-35m, then the cosmic expansion should continually create new comoving modes. A priori, each of the new modes comes with its own vacuum energy, which could contribute to the cosmological constant. I discuss possible mathematical models for a shortest length and an explicit model for a corresponding mode-generating mechanism. 
  Motivated by the recent interest in dynamical properties of topologically nontrivial spacetimes, we study linear perturbations of spatially closed Bianchi III vacuum spacetimes, whose spatial topology is the direct product of a higher genus surface and the circle. We first develop necessary mode functions, vectors, and tensors, and then perform separations of (perturbation) variables. The perturbation equations decouple in a way that is similar to but a generalization of those of the Regge--Wheeler spherically symmetric case. We further achieve a decoupling of each set of perturbation equations into gauge-dependent and independent parts, by which we obtain wave equations for the gauge-invariant variables. We then discuss choices of gauge and stability properties. Details of the compactification of Bianchi III manifolds and spacetimes are presented in an appendix. In the other appendices we study scalar field and electromagnetic equations on the same background to compare asymptotic properties. 
  The possibility that the dark energy may be described by the Chaplygin gas is discussed. Some observational constraints are established. These observational constraints indicate that a unified model for dark energy and dark matter through the employement of the Chaplygin gas is favored. 
  In this paper we derive homogeneous vacuum plane-wave solutions to Einstein's field equations in 4+1 dimensions. The solutions come in five different types of which three generalise the vacuum plane-wave solutions in 3+1 dimensions to the 4+1 dimensional case. By doing a Kaluza-Klein reduction we obtain solutions to the Einstein-Maxwell equations in 3+1 dimensions. The solutions generalise the vacuum plane-wave spacetimes of Bianchi class B to the non-vacuum case and describe spatially homogeneous spacetimes containing an extremely tilted fluid. Also, using a similar reduction we obtain 3+1 dimensional solutions to the Einstein equations with a scalar field. 
  We study the local behaviour of static solutions of a general 1+1 dimensional dilaton gravity theory coupled to scalar fields and Abelian gauge fields near horizons. This type of model includes in particular reductions of higher dimensional theories invariant under a sufficiently large isometry group. The solution near the horizon can in general be obtained solving a system of integral equations or in favourable cases in the form of a convergent series in the dilaton field. 
  It is shown that (except for two well defined cases), the necessary and sufficient condition for any spherically symmetric distribution of fluid to leave the state of equilibrium (or quasi-equilibrium), is that the Weyl tensor changes with respect to its value in the state of equilibrium (or quasi-equilibrium). 
  The standard procedure for detection of gravitational wave coalescing binaries signals is based on Wiener filtering with an appropriate bank of template filters. This is the optimal procedure in the hypothesis of addictive Gaussian and stationary noise. We study the possibility of improving the detection efficiency with a class of adaptive spectral identification techniques, analyzing their effect in presence of non stationarities and undetected non linearities in the noise 
  In the case of crossing thin dust shells the momentum conservation law is found. For two crossing isotropic shells it coincides with the 't Hooft-Dray formula. The system of one isotropic and one time-like shell is considered. In this case we found a very simple formula which relate velocities of dust shell before and after crossing. 
  A coordinate system is set up for a general accelerating observer and is used to determine the particle content of the Dirac vacuum for that observer. Equations are obtained for the spatial distribution and total number of particles for massless fermions as seen by this observer, generalising previous work. 
  Suppose the usual description of spacetime as a 4-dimensional manifold with a Lorentzian metric breaks down at Planck energies. Can we still construct sensible theoretical models of the universe? Are they testable? Do they lead to a consistent quantum cosmology? Is this cosmology different than the standard one? The answer is yes, to all these questions, assuming that quantum theory is still valid at this scale. I describe the basic features of such models, mainly quantum causal histories and spin foams. They are given by a partition function, similar to spin systems and lattice gauge theories. This suggests that we should treat this approach to quantum gravity as a problem in statistical physics, but with significant complications: there is no background and, in particular, no external time. Gravity and the familiar 3+1 manifold spacetime are to be derived as the low-energy continuum approximation to these models. 
  Three solutions of the Brans-Dicke theory with a self-interacting quartic potential and perfect fluid distribution are presented for a spatially flat geometry. The physical behavior is consistent with the recent cosmological scenario favored by type Ia supernova observations, indicating an accelerated expansion of the Universe. 
  We show a global existence theorem for Einstein-matter equations of $T^{3}$-Gowdy symmetric spacetimes with stringy matter. The areal time coordinate is used. It is shown that this spacetime has a crushing singularity into the past. From these results we can show that the spacetime is foliated by compact hypersurfaces of constant mean curvature. 
  Einstein's field equations in FRW space-times are coupled to the BGK equation in order to derive the stress energy tensor including dissipative effects up to second order in the thermodynamical forces. The space-time is assumed to be matter-dominated, but in a low density regime for which a second order (Burnett) coefficient becomes relevant. Cosmological implications of the solutions, as well as the physical meaning of transport coefficients in an isotropic homogeneous universe are discussed. 
  We discuss the implication of the introduction of an extra field to the dynamics of a scalar field conformally coupled to gravitation in a homogeneous isotropic spatially flat universe. We show that for some reasonable parameter values the dynamical effects are similar to those of our previous model with a single scalar field. Nevertheless for other parameter values new dynamical effects are obtained. 
  We argue that storage rings can be used for the detection of low-frequency gravitational-wave background. Proceeding from the measurements by Schin Date and Noritaka Kumagai (Nucl. Instrum. Meth. A421, 417 (1999)) and Masaru Takao and Taihei Shimada (Proceedings of EPAC 2000, Vienna, 2000, p.1572) of variations of the machine circumference of the SPring-8 storage ring we explain the systematic shrinkage of the machine circumference by the influence of the relic gravitational-wave background. We give arguments against a possibility to explain the observed shrinkage of the machine circumference of the SPring-8 storage ring by diastrophic tectonic forces. We show that the forces, related to the stiffness of the physical structures, governing the path of the beam, can be neglected for the analysis of the shrinkage of the machine circumference caused by the relic gravitational-wave background. We show the shrinkage of the machine circumference can be explained by a relic gravitational-wave background even if it is treated as a stochastic system incoming on the plane of the machine circumference from all quarters of the Universe. We show that the rate of the shrinkage of the machine circumference does not depend on the radius of the storage ring and it should be universal for storage rings with any radii. 
  We review recent results by the author, in collaboration with Erwann Delay, Olivier Lengard, and Rafe Mazzeo, on existence and properties of space-times with controlled asymptotic behavior at null infinity. 
  Internal friction effects are responsible for line widening of the resonance frequencies in mechanical oscillators and result in damped oscillations of its eigenmodes with a decay time Q/\omega. We study the solutions to the equations of motion for the case of spherical oscillators, to be used as next generation of acoustic gravitational wave detectors, based on various different assumptions about the material's constituent equations. Quality factor dependence on mode frequency is determined in each case, and a discussion of its applicability to actual gravitational wave detectors is made on the basis of available experimental evidence. 
  Quantum General Relativity (QGR), sometimes called Loop Quantum Gravity, has matured over the past fifteen years to a mathematically rigorous candidate quantum field theory of the gravitational field. The features that distinguish it from other quantum gravity theories are 1) background independence and 2) minimality of structures. Background independence means that this is a non-perturbative approach in which one does not perturb around a given, distinguished, classical background metric, rather arbitrary fluctuations are allowed, thus precisely encoding the quantum version of Einstein's radical perception that gravity is geometry. Minimality here means that one explores the logical consequences of bringing together the two fundamental principles of modern physics, namely general covariance and quantum theory, without adding any experimentally unverified additional structures. The approach is purposely conservative in order to systematically derive which basic principles of physics have to be given up and must be replaced by more fundamental ones. QGR unifies all presently known interactions in a new sense by quantum mechanically implementing their common symmetry group, the four-dimensional diffeomorphism group, which is almost completely broken in perturbative approaches. These lectures offer a problem -- supported introduction to the subject. 
  The Klein-Gordon and Dirac equations in a semi-infinite lab ($x > 0$), in the background metric $\ds^2 = u^2(x) (-\dt^2 + \dx^2) + \dy^2 + \dz^2$, are investigated. The resulting equations are studied for the special case $ u(x) = 1 + g x$. It is shown that in the case of zero transverse-momentum, the square of the energy eigenvalues of the spin-1/2 particles are less than the squares of the corresponding eigenvalues of spin-0 particles with same masses, by an amount of $mg\hbar c$. Finally, for nonzero transverse-momentum, the energy eigenvalues corresponding to large quantum numbers are obtained, and the results for spin-0 and spin-1/2 particles are compared to each other. 
  The issue of the vacuum energy of quantum fields is briefly reviewed. It is argued that this energy is normally either much too large or much too small to account for the dark energy, However, there are a few proposals in which it would be of the order needed to effect the dynamics of the present day universe. Backreaction models are reviewed, and the question of whether quantum effects can react against a cosmological constant is discussed. 
  A linear relationship between the Hubble expansion parameter and the time derivative of the scalar field is assumed in order to derive exact analytic cosmological solutions to Einstein's gravity with two fluids: a barotropic perfect fluid of ordinary matter, together with a self-interacting scalar field fluid accounting for the dark energy in the universe. A priori assumptions about the functional form of the self-interaction potential or about the scale factor behavior are not neccessary. These are obtained as outputs of the assumed linear relationship between the Hubble expansion parameter and the time derivative of the scalar field. As a consequence only a class of exponential potentials and their combinations can be treated. The relevance of the solutions found for the description of the cosmic evolution are discussed in some detail. The possibility to have superaccelerated expansion within the context of normal quintessence models is also discussed. 
  Filters developed in order to detect short bursts of gravitational waves in interferometric detector outputs are compared according to three main points. Conventional Receiver Operating Characteristics (ROC) are first built for all the considered filters and for three typical burst signals. Optimized ROC are shown for a simple pulse signal in order to estimate the best detection efficiency of the filters in the ideal case, while realistic ones obtained with filters working with several ``templates'' show how detection efficiencies can be degraded in a practical implementation. Secondly, estimations of biases and statistical errors on the reconstruction of the time of arrival of pulse-like signals are then given for each filter. Such results are crucial for future coincidence studies between Gravitational Wave detectors but also with neutrino or optical detectors. As most of the filters require a pre-whitening of the detector noise, the sensitivity to a non perfect noise whitening procedure is finally analysed. For this purpose lines of various frequencies and amplitudes are added to a Gaussian white noise and the outputs of the filters are studied in order to monitor the excess of false alarms induced by the lines. The comparison of the performances of the different filters finally show that they are complementary rather than competitive. 
  We study gravitational waves propagating through an anisotropic Bianchi I dust-filled universe (containing the Einstein-de-Sitter universe as a special case). The waves are modeled as small perturbations of this background cosmological model and we choose a family of null hypersurfaces in this space-time to act as the histories of the wavefronts of the radiation. We find that the perturbations we generate can describe pure gravitational radiation if and only if the null hypersurfaces are shear-free. We calculate the gauge-invariant small perturbations explicitly in this case. How these differ from the corresponding perturbations when the background space-time is isotropic is clearly exhibited. 
  We investigate spacetimes with their singular boundaries as noncommutative spaces. Such a space is defined by a noncommutative algebra on a transformation groupoid $\Gamma = E \times G$, where $E$ is the total space of the frame bundle over spacetime with its singular boundary, and $G$ its structural group. There is a bijective correspondence between unitary representations of the groupoid $\Gamma $ and the systems of imprimitivity of the group $G$. This allows us to apply the Mackey theorem, and deduce from it some information concerning singular fibres of the groupoid. A subgroup $K$ of $G$, from which -- according to the Mackey theorem -- the representation is induced to the whole of $G$, can be regarded as measuring the "richness" of the singularity structure. 
  In general relativity black holes can be formed from regular initial data that do not contain a black hole already. The space of regular initial data for general relativity therefore splits naturally into two halves: data that form a black hole in the evolution and data that do not. The spacetimes that are evolved from initial data near the black hole threshold have many properties that are mathematically analogous to a critical phase transition in statistical mechanics.   Solutions near the black hole threshold go through an intermediate attractor, called the critical solution. The critical solution is either time-independent (static) or scale-independent (self-similar). In the latter case, the final black hole mass scales as $(p-p_*)^\gamma$ along any one-parameter family of data with a regular parameter $p$ such that $p=p_*$ is the black hole threshold in that family. The critical solution and the critical exponent $\gamma$ are universal near the black hole threshold for a given type of matter.   We show how the essence of these phenomena can be understood using dynamical systems theory and dimensional analysis. We then review separately the analogy with critical phase transitions in statistical mechanics, and aspects specific to general relativity, such as spacetime singularities. We examine the evidence that critical phenomena in gravitational collapse are generic, and give an overview of their rich phenomenology. 
  We study the inertial modes of slowly rotating, fully relativistic compact stars. The equations that govern perturbations of both barotropic and non-barotropic models are discussed, but we present numerical results only for the barotropic case. For barotropic stars all inertial modes are a hybrid mixture of axial and polar perturbations. We use a spectral method to solve for such modes of various polytropic models. Our main attention is on modes that can be driven unstable by the emission of gravitational waves. Hence, we calculate the gravitational-wave growth timescale for these unstable modes and compare the results to previous estimates obtained in Newtonian gravity (i.e. using post-Newtonian radiation formulas). We find that the inertial modes are slightly stabilized by relativistic effects, but that previous conclusions concerning eg. the unstable r-modes remain essentially unaltered when the problem is studied in full general relativity. 
  We analyze the horizon structure of families of space times obtained by evolving initial data sets containing apparent horizons with several connected components. We show that under certain smallness conditions the outermost apparent horizons will also have several connected components. We further show that, again under a smallness condition, the maximal globally hyperbolic development of the many black hole initial data constructed by Chrusciel and Delay, or of hyperboloidal data of Isenberg, Mazzeo and Pollack, will have an event horizon, the intersection of which with the initial data hypersurface is not connected. This justifies the "many black hole" character of those space-times. 
  We study the optical paths of the light rays propagating inside a nonlinear moving dielectric media. For the rapidly moving dielectrics we show the existence of a distinguished surface which resembles, as far as the light propagation is concerned, the event horizon of a black hole. Our analysis clarifies the physical conditions under which electromagnetic analogues of the gravitational black holes can eventually be obtained in laboratory. 
  We compute characteristic (quasinormal) frequencies corresponding to decay of a massive charged scalar field in a Reissner-Nordstrom black hole background. It proves that, contrary to the behavior at very late times, at the stage of quasinormal ringing the neutral perturbations will damp slower than the charged ones. In the limit of the extremal black hole the damping rate of charged and neutral perturbations coincides. Possible connection of this with the critical collapse in a massive scalar electrodynamics is discussed. 
  The persistent, second order, anomalous signal found in the Brillet and Hall experiment is derived by applying 4D differential geometry in the rotating earth frame. By incorporating the off diagonal time-space components of the rotating frame metric directly into the analysis, rather than arbitrarily transforming them away, one finds a signal dependence on the surface speed of the earth due to rotation about its axis. This leads to a Brillet-Hall signal prediction in remarkably close agreement with experiment. No signal is predicted from the speed of the earth in solar or galactic orbit, as the associated metric for gravitational orbit has no off diagonal component. To corroborate this result, a repetition by other experimentalists of the Brillet-Hall experiment, in which the test apparatus turns with respect to the earth surface, is urged. 
  Neutrino asymmetry in general relativistic radiative spacetime exterior to spinning stars is investigating by making use of Newmann-Penrose (NP) spin coefficient formalism. It is shown that neutrino current depends on the direction of rotation of the star. The solution is obtained in test field approximation where the neutrinos do not generate gravitational fields. 
  Some approaches to quantization of the horizon area of black holes are discussed. The maximum entropy of a quantized surface is demonstrated to be proportional to the surface area in the classical limit. This result is valid for a rather general class of approaches to surface quantization. In the case of rotating black holes no satisfactory solution for the quantization problem has been found up to now. 
  Spherically symmetric black holes produce, by strong field lensing, two infinite series of relativistic images, formed by light rays winding around the black hole at distances comparable to the gravitational radius. In this paper, we address the relevance of the black hole spin for the strong field lensing phenomenology, focusing on trajectories close to the equatorial plane for simplicity. In this approximation, we derive a two-dimensional lens equation and formulae for the position and the magnification of the relativistic images in the strong field limit. The most outstanding effect is the generation of a non trivial caustic structure. Caustics drift away from the optical axis and acquire finite extension. For a high enough black hole spin, depending on the source extension, we can practically observe only one image rather than two infinite series of relativistic images. In this regime, additional non equatorial images may play an important role in the phenomenology. 
  We compute the production of particles from the gravitational field of an expanding mass shell. Contrary to the situation of Hawking radiation and the production of cosmological perturbations during cosmological inflation, the example of an expanding mass shell has no horizon and no singularity. We apply the method of `ray-tracing', first introduced by Hawking, and calculate the energy spectrum of the produced particles. The result depends on three parameters: the expansion velocity of the mass shell, its radius, and its mass. Contrary to the situation of a collapsing mass shell, the energy spectrum is non-thermal. Invoking time reversal we reproduce Hawking's thermal spectrum in a certain limit. 
  We present a self-contained formalism for analyzing scale invariant differential equations. We first cast the scale invariant model into its equidimensional and autonomous forms, find its fixed points, and then obtain power-law background solutions. After linearizing about these fixed points, we find a second linearized solution, which provides a distinct collection of power laws characterizing the deviations from the fixed point. We prove that generically there will be a region surrounding the fixed point in which the complete general solution can be represented as a generalized Frobenius-like power series with exponents that are integer multiples of the exponents arising in the linearized problem. This Frobenius-like series can be viewed as a variant of Liapunov's expansion theorem. As specific examples we apply these ideas to Newtonian and relativistic isothermal stars and demonstrate (both numerically and analytically) that the solution exhibits oscillatory power-law behaviour as the star approaches the point of collapse. These series solutions extend classical results. (Lane, Emden, and Chandrasekhar in the Newtonian case; Harrison, Thorne, Wakano, and Wheeler in the relativistic case.) We also indicate how to extend these ideas to situations where fixed points may not exist -- either due to ``monotone'' flow or due to the presence of limit cycles. Monotone flow generically leads to logarithmic deviations from scaling, while limit cycles generally lead to discrete self-similar solutions. 
  A plane-symmetric non-static cosmological model representing a bulk viscous fluid distribution has been obtained which is inhomogeneous and anisotropic and a particular case of which is gravitationally radiative. Without assuming any {\it adhoc} law, we obtain a cosmological constant as a decreasing function of time. The physical and geometric features of the models are also discussed. 
  In this talk we entertain the possibility that the synthesis of general covariance and quantum mechanics requires an extension of the basic kinematical setup of quantum mechanics. According to the holographic principle, regions of spacetime bounded by a finite area carry finite entropy. When we in addition assume that the origin of the entropy is a finite dimensional Hilbert space, and apply this to cosmological solutions using a suitable notion of complementarity, we find as a consequence that gravitational effects can lead to dynamical variation in the dimensionality of such Hilbert spaces. This happens generally in cosmological settings like our own universe. 
  It has been suggested in the literature that, given a black hole spacetime, a relativistic membrane can provide an effective description of the horizon dynamics. In this paper, we explore such a framework in the context of a 2+1-dimensional BTZ black hole. Following this membrane prescription, we are able to translate the horizon dynamics (now described by a string) into the convenient form of a 1+1-dimensional Klein-Gordon equation. We proceed to quantize the solutions and construct a thermodynamic partition function. Ultimately, we are able to extract the quantum-corrected entropy, which is shown to comply with the BTZ form of the Bekenstein-Hawking area law. We also substantiate that the leading-order correction is proportional to the logarithm of the area. 
  In this article we compute the density of Dirac particles created by a cosmological anisotropic Bianchi I universe in the presence of a constant electric field. We show that the particle distribution becomes thermal when one neglects the electric interaction. 
  We construct a large class of new singularity-free static Lorentzian four-dimensional solutions of the vacuum Einstein equations with a negative cosmological constant. The new families of metrics contain space-times with, or without, black hole regions. Two uniqueness results are also established. 
  Quantum effects lead to the annihilation of AdS Universe when dilaton is absent. We consider here the role of the form for the dilatonic potential in the quantum creation of a dilatonic AdS Universe and its stabilization. Using the conformal anomaly for dilaton coupled scalar, the anomaly induced action and the equations of motion are obtained. Using numerical methods the solutions of the full theory which correspond to quantum-corrected AdS Universe are given for a particular case of dilatonic potential. 
  Conformally flat spherically symmetric cosmological models representing a charged perfect fluid as well as a bulk viscous fluid distribution have been obtained. The cosmological constant \Lambda is found positive and is a decreasing function of time which is consistent with the recent supernovae observations. The physical and geometrical properties of the models are discussed. 
  A relativistic theory constructed on Riemann-Cartan manifold with a derived totally antisymmetric torsion is proposed. It follows the coincidence of the autoparallel curve and metric geodesic. The totally antisymmetric torsion naturally appears in the theory without any ad hoc imposed on. 
  Given a null geodesic $\gamma_0(\lambda)$ with a point $r$ in $(p,q)$ conjugate to $p$ along $\gamma_0(\lambda)$, there will be a variation of $\gamma_0(\lambda)$ which will give a time-like curve from $p$ to $q$. This is a well-known theory proved in the famous book\cite{2}. In the paper we prove that the time-like curves coming from the above-mentioned variation have a proper acceleration which approaches infinity as the time-like curve approaches the null geodesic. This means no observer can be infinitesimally near the light and begin at the same point with the light and finally catch the light. Only separated from the light path finitely, does the observer can begin at the same point with the light and finally catch the light. 
  A quantum mechanical upper limit on the value of particle accelerations, or maximal acceleration (MA), is applied to compact stars. A few MA fermions are at most present in canonical white dwarfs and neutron stars. They drastically alter a star's stability conditions. 
  In loop quantum gravity, matter fields can have support only on the `polymer-like' excitations of quantum geometry, and their algebras of observables and Hilbert spaces of states can not refer to a classical, background geometry. Therefore, to adequately handle the matter sector, one has to address two issues already at the kinematic level. First, one has to construct the appropriate background independent operator algebras and Hilbert spaces. Second, to make contact with low energy physics, one has to relate this `polymer description' of matter fields to the standard Fock description in Minkowski space. While this task has been completed for gauge fields, important gaps remained in the treatment of scalar fields. The purpose of this letter is to fill these gaps. 
  In this paper we wish to preliminary investigate if it would be possible to use the orbital data from the proposed OPTIS mission together with those from the existing geodetic passive SLR LAGEOS and LAGEOS II satellites in order to perform precise measurements of some general relativistic gravitoelectromagnetic effects, with particular emphasis on the Lense-Thirring effect. 
  Via a straightforward integration of the Einstein equations with cosmological constant, all static circularly symmetric perfect fluid 2+1 solutions are derived. The structural functions of the metric depend on the energy density, which remains in general arbitrary. Spacetimes for fluids fulfilling linear and polytropic state equations are explicitly derived; they describe, among others, stiff matter, monatomic and diatomic ideal gases, nonrelativistic degenerate fermions, incoherent and pure radiation. As a by--product, we demonstrate the uniqueness of the constant energy density perfect fluid within the studied class of metrics. A full similarity of the perfect fluid solutions with constant energy density of the 2+1 and 3+1 gravities is established. 
  The hypothesis that cold dark matter consists of primordial superheavy particles, the decay of short lifetime component of which led to the observable mass of matter while long living component survived up to modern times manifesting its presence in high energetic cosmic rays particles is investigated. 
  We investigate the inflation of Universe in a model of four dimensional dilatonic gravity with a massive dilaton field $\Phi$. The dilaton plays simultaneously the roles of an inflation field and a quintessence field. It yields a sequential {\em hyper}-inflation with a graceful exit to asymptotic de Sitter space-time, which is an attractor, and is approached as $\exp(-\sqrt{3\Lambda^{obs}} ct/2)$. The time duration of the inflation is reciprocal to the the mass of the dilaton: $\Delta t_{infl}\sim m_{{}_\Phi}^{-1}$. The typical number of e-folds in the simplest model of this type is shown to be realistic without fine tuning. 
  A general class of Lorentzian metrics, $M_0 x R^2$, $ds^2 = <.,.> + 2 du dv + H(x,u) du^2$, with $(M_0, <.,.>$ any Riemannian manifold, is introduced in order to generalize classical exact plane fronted waves. Here, we start a systematic study of their main geodesic properties: geodesic completeness, geodesic connectedness and multiplicity, causal character of connecting geodesics. These results are independent of the possibility of a full integration of geodesic equations. Variational and geometrical techniques are applied systematically.   In particular, we prove that the asymptotic behavior of $H(x,u)$ with $x$ at infinity determines many properties of geodesics. Essentially, a subquadratic growth of $H$ ensures geodesic completeness and connectedness, while the critical situation appears when $H(x,u)$ behaves in some direction as $|x|^2$, as in the classical model of exact gravitational waves 
  The capture of a straight, infinitely long cosmic string by a rotating black hole with rotation parameter $a$ is considered. We assume that a string is moving with velocity $v$ and that initially the string is parallel to the axis of rotation of the black hole and has the impact parameter $b$. The string can be either scattered or captured by the black hole. We demonstrate that there exists a critical value of the impact parameter $b_c(v,a)$ which separates these two regimes. Using numerical simulations we obtain the critical impact parameter curve for different values of the rotation parameter $a$. We show that for the prograde motion of the string this curve lies below the curve for the retrograde motion. Moreover, for ultrarelativistic strings moving in the prograde direction and nearly extremal black holes the critical impact parameter curve is found to be a multiply valued function of $v$. We obtain real time profiles of the scattered strings in the regime close to the critical. We also study critical scattering and capture of strings by the rotating black hole in the relativistic and ultrarelativistic regime and especially such relativistic effects as coil formation and wrapping effect. 
  We study gravitational collapse of the general spherically symmetric null strange quark fluid having the equation of state, $p = (\rho - 4B)/n$, where $B$ is the bag constant. An interesting feature that emerges is that the initial data set giving rise to naked singularity in the Vaidya collapse of null fluid gets covered due to the presence of strange quark matter component. Its implication to the Cosmic Censorship Conjecture is discussed. 
  In this paper we study the cosmological dynamics of Randall-Sundrum braneworld type scenarios in which the five-dimensional Weyl tensor has a non-vanishing projection onto the three-brane where matter fields are confined. Using dynamical systems techniques, we study how the state space of Friedmann-Lemaitre-Robertson-Walker (FLRW) and Bianchi type I scalar field models with an exponential potential is affected by the bulk Weyl tensor, focusing on the differences that appear with respect to standard general relativity and also Randall-Sundrum cosmological scenarios without the Weyl tensor contribution. 
  A new and universal method for implementing scale invariance, called best matching, is presented. It extends to scaling the method introduced by Bertotti and the author to create a fully relational dynamics that satisfies Mach's principle. The method is illustrated here in the context of non-relativistic gravitational particle dynamics.. It leads to far stronger predictions than general Newtonian dynamics. The energy and angular momentum of an `island universe' must be exactly zero and its size, measured by its moment of inertia, cannot change. This constancy is enforced because the scale invariance requires all potentials to be homogeneous of degree -2. It is remarkable that one can nevertheless exactly recover the standard observed Newtonian laws and forces, which are merely accompanied by an extremally weak universal force like the one due to Einstein's cosmological constant. In contrast to Newtonian and Einsteinian dynamics, both the gravitational constant G and the strength of the cosmological force are uniquely determined by the matter distribution of the universe. Estimates of their values in agreement with observations are obtained. Best matching implements a dynamics of pure shape for which the action is a dimensionless number. If the universe obeys such scale invariant law, steadily increasing inhomogeneity, not expansion of the universe, causes the Hubble red shift. The application of best matching to geometrodynamics is treated in a companion paper. 
  We present a scale-invariant theory, conformal gravity, which closely resembles the geometrodynamical formulation of general relativity (GR). While previous attempts to create scale-invariant theories of gravity have been based on Weyl's idea of a compensating field, our direct approach dispenses with this and is built by extension of the method of best matching w.r.t scaling developed in the parallel particle dynamics paper by one of the authors. In spatially-compact GR, there is an infinity of degrees of freedom that describe the shape of 3-space which interact with a single volume degree of freedom. In conformal gravity, the shape degrees of freedom remain, but the volume is no longer a dynamical variable. Further theories and formulations related to GR and conformal gravity are presented.   Conformal gravity is successfully coupled to scalars and the gauge fields of nature. It should describe the solar system observations as well as GR does, but its cosmology and quantization will be completely different. 
  Ori and Thorne have discussed the duration and observability (with LISA) of the transition from circular, equatorial inspiral to plunge for stellar-mass objects into supermassive ($10^{5}-10^{8}M_{\odot}$) Kerr black holes. We extend their computation to eccentric Kerr equatorial orbits. Even with orbital parameters near-exactly determined, we find that there is no universal length for the transition; rather, the length of the transition depends sensitively -- essentially randomly -- on initial conditions. Still, Ori and Thorne's zero-eccentricity results are essentially an upper bound on the length of eccentric transitions involving similar bodies (e.g., $a$ fixed). Hence the implications for observations are no better: if the massive body is $M=10^{6}M_{\odot}$, the captured body has mass $m$, and the process occurs at distance $d$ from LISA, then $S/N \lesssim (m/10 M_{\odot})(1\text{Gpc}/d)\times O(1)$, with the precise constant depending on the black hole spin. For low-mass bodies ($m \lesssim 7 M_\odot$) for which the event rate is at least vaguely understood, we expect little chance (probably [much] less than 10%, depending strongly on the astrophysical assumptions) of LISA detecting a transition event with $S/N>5$ during its run; however, even a small infusion of higher-mass bodies or a slight improvement in LISA's noise curve could potentially produce $S/N>5$ transition events during LISA's lifetime. 
  Expansions of the gravitational field arising from the development of asymptotically Euclidean, time symmetric, conformally flat initial data are calculated in a neighbourhood of spatial and null infinities up to order 6. To this ends a certain representation of spatial infinity as a cylinder is used. This set up is based on the properties of conformal geodesics. It is found that these expansions suggest that null infinity has to be non-smooth unless the Newman-Penrose constants of the spacetime, and some other higher order quantities of the spacetime vanish. As a consequence of these results it is conjectured that similar conditions occur if one were to take the expansions to even higher orders. Furthermore, the smoothness conditions obtained suggest that if a time symmetric initial data which is conformally flat in a neighbourhood of spatial infinity yields a smooth null infinity, then the initial data must in fact be Schwarzschildean around spatial infinity. 
  It is shown that there are no vacuum space-times (with or without cosmological constant) for which the Weyl-tensor is purely gravito-magnetic with respect to a normal and timelike congruence of observers. 
  The aim of this paper is twofold. First, we set up the theory of elastic matter sources within the framework of general relativity in a self-contained manner. The discussion is primarily based on the presentation of Carter and Quintana but also includes new methods and results as well as some modifications that in our opinion make the theory more modern and transparent. For instance, the equations of motion for the matter are shown to take a neat form when expressed in terms of the relativistic Hadamard elasticity tensor. Using this formulation we obtain simple formulae for the speeds of elastic wave propagation along eigendirections of the pressure tensor. Secondly, we apply the theory to static spherically symmetric configurations using a specific equation of state and consider models either having an elastic crust or core. 
  The radiation reaction in compact spinning binaries on eccentric orbits due to the quadrupole-monopole interaction is studied. This contribution is of second post-Newtonian order. As result of the precession of spins the magnitude $L$ of the orbital angular momentum is not conserved. Therefore a proper characterization of the perturbed radial motion is provided by the energy $E$ and angular average $\bar{L}$. As powerful computing tools, the generalized true and eccentric anomaly parametrizations are introduced. Then the secular losses in energy and magnitude of orbital angular momentum together with the secular evolution of the relative orientations of the orbital angular momentum and spins are found for eccentric orbits by use of the residue theorem. The circular orbit limit of the energy loss agrees with Poisson's earlier result. 
  Numerical relativity is the most promising tool for theoretically modeling the inspiral and coalescence of neutron star and black hole binaries, which, in turn, are among the most promising sources of gravitational radiation for future detection by gravitational wave observatories. In this article we review numerical relativity approaches to modeling compact binaries. Starting with a brief introduction to the 3+1 decomposition of Einstein's equations, we discuss important components of numerical relativity, including the initial data problem, reformulations of Einstein's equations, coordinate conditions, and strategies for locating and handling black holes on numerical grids. We focus on those approaches which currently seem most relevant for the compact binary problem. We then outline how these methods are used to model binary neutron stars and black holes, and review the current status of inspiral and coalescence simulations. 
  We construct a kinematical analogue of superluminal travel in the ``warped'' space-times curved by gravitation, in the form of ``super-phononic'' travel in the effective space-times of perfect nonrelativistic fluids. These warp-field space-times are most easily generated by considering a solid object that is placed as an obstruction in an otherwise uniform flow. No violation of any condition on the positivity of energy is necessary, because the effective curved space-times for the phonons are ruled by the Euler and continuity equations, and not by the Einstein field equations. 
  The following three geometrical structures on a manifold are studied in detail: (1) Leibnizian: a non-vanishing 1-form $\Omega$ plus a Riemannian metric $\h$ on its annhilator vector bundle. In particular, the possible dimensions of the automorphism group of a Leibnizian G-structure are characterized. (2) Galilean: Leibnizian structure endowed with an affine connection $\nabla$ (gauge field) which parallelizes $\Omega$ and $\h$. Fixed any vector field of observers Z ($\Omega (Z) = 1$), an explicit Koszul--type formula which reconstruct bijectively all the possible $\nabla$'s from the gravitational ${\cal G} = \nabla_Z Z$ and vorticity $\omega = rot Z/2$ fields (plus eventually the torsion) is provided. (3) Newtonian: Galilean structure with $\h$ flat and a field of observers Z which is inertial (its flow preserves the Leibnizian structure and $\omega = 0$). Classical concepts in Newtonian theory are revisited and discussed. 
  In order to illustrate a recently derived covariant formalism for computing asymptotic symmetries and asymptotically conserved superpotentials in gauge theories, the case of gravity with minimally coupled scalar fields is considered and the matter contribution to the asymptotically conserved superpotentials is computed. 
  Through the subsequent discussion we consider a certain particular sort of (topological) algebras, which may substitute the `` structure sheaf algebras'' in many--in point of fact, in all--the situations of a geometrical character that occur, thus far, in several mathematical disciplines, as for instance, differential and/or algebraic geometry, complex (geometric) analysis etc. It is proved that at the basis of this type of algebras lies the sheaf-theoretic notion of (functional) localization, which, in the particular case of a given topological algebra, refers to the respective ``Gel'fand transform algebra'' over the spectrum of the initial algebra. As a result, one further considers the so-called ``geometric topological algebras'', having special cohomological properties, in terms of their ``Gel'fand sheaves'', being also of a particular significance for (abstract) differential-geometric applications; yet, the same class of algebras is still ``closed'', with respect to appropriate inductive limits, a fact which thus considerably broadens the sort of the topological algebras involved, hence, as we shall see, their potential applications as well. 
  We study the evolution of spherically symmetric radiating fluid distributions using the effective variables method, implemented {\it ab initio} in Schwarzschild coordinates. To illustrate the procedure and to establish some comparison with the original method, we integrate numerically the set of equations at the surface for two different models. The first model is derived from the Schwarzschild interior solution. The second model is inspired in the Tolman VI solution. 
  The effect of induced Riemann geometry in nonlinear electrodynamics is considered. The possibility for description of real gravitation by this effect is discussed. 
  We prove that the thermodynamic properties of a Schwarzschild black hole are unaffected by an external magnetic field passing through it. Apart from the background substraction prescription, this result is obtained also by using a counterterm method. 
  A pair of wave equations for the electromagnetic and gravitational perturbations of the charged Kerr black hole are derived. The perturbed Einstein-Maxwell equations in a new gauge are employed in the derivation. The wave equations refer to the perturbed Maxwell spinor $\Phi_0$ and to the shear $\sigma$ of a principal null direction of the Weyl curvature. The whole construction rests on the tripod of three distinct derivatives of the first curvature $\kappa$ of a principal null direction. 
  The extended constraint equations arise as a special case of the conformal constraint equations that are satisfied by an initial data hypersurface $Z$ in an asymptotically simple spacetime satisfying the vacuum conformal Einstein equations developed by H. Friedrich. The extended constraint equations consist of a quasi-linear system of partial differential equations for the induced metric, the second fundamental form and two other tensorial quantities defined on $Z$, and are equivalent to the usual constraint equations that $Z$ satisfies as a spacelike hypersurface in a spacetime satisfying Einstein's vacuum equation. This article develops a method for finding perturbative, asymptotically flat solutions of the extended constraint equations in a neighbourhood of the flat solution on Euclidean space. This method is fundamentally different from the `classical' method of Lichnerowicz and York that is used to solve the usual constraint equations. 
  We have studied the Generalised Brans-Dicke theory and obtained exact solutions of a(t),phi(t),and omega(t) for different epochs of the cosmic evolution .We discuss how inflation,decceleration,cosmic acceleration can result from this solution.The time variation of G(t) is also examined. 
  Starting from an important application of Conformal Yano--Killing tensors for the existence of global charges in gravity, some new observations at $\scri^+$ are given. They allow to define asymptotic charges (at future null infinity) in terms of the Weyl tensor together with their fluxes through $\scri^+$. It occurs that some of them play a role of obstructions for the existence of angular momentum.  Moreover, new relations between solutions of the Maxwell equations and the spin-2 field are given. They are used in the construction of new conserved quantities which are quadratic in terms of the Weyl tensor. The obtained formulae are similar to the functionals obtained from the  Bel--Robinson tensor. 
  In this Letter we investigate uniformly rotating, homogeneous and axisymmetric relativistic fluid bodies with a toroidal shape. The corresponding field equations are solved by means of a multi-domain spectral method, which yields highly accurate numerical solutions. For a prescribed, sufficiently large ratio of inner to outer coordinate radius, the toroids exhibit a continuous transition to the extreme Kerr black hole. Otherwise, the most relativistic configuration rotates at the mass-shedding limit. For a given mass-density, there seems to be no bound to the gravitational mass as one approaches the black-hole limit and a radius ratio of unity. 
  We extend the description of gravitational waves emitted by binary black holes during the final stages of inspiral and merger by introducing in the third post-Newtonian (3PN) effective-one-body (EOB) templates seven new ``flexibility'' parameters that affect the two-body dynamics and gravitational radiation emission. The plausible ranges of these flexibility parameters, notably the parameter characterising the fourth post-Newtonian effects in the dynamics, are estimated. Using these estimates, we show that the currently available standard 3PN bank of EOB templates does ``span'' the space of signals opened up by all the flexibility parameters, in that their maximized mutual overlaps are larger than 96.5%. This confirms the effectualness of 3PN EOB templates for the detection of binary black holes in gravitational-wave data from interferometric detectors. The possibility to drastically reduce the number of EOB templates using a few ``universal'' phasing functions is suggested. 
  A discussion is given of the uncertainty principle in view of the introduction of a Gravitational Planck Constant. The need for such a gravitational constant is shown first. A reduced electromagnetic Planck constant and the analogous reduced gravitational Planck constant are defined as h/e^2 and H/m^2 respectively. An attempt is made to reconcile the quantum uncertainty concepts with a deterministic view of the physical world. This conclusion is achieved trough the detailed analysis of the measurement procedures of physical quantities. 
  This short note compares different methods to prove that Einstein-Dirac systems have no static, spherically symmetric solutions. 
  I study a fresh inflationary model with an increasing F-cosmological parameter. The model provides sufficiently e-folds to solve the flatness/horizon problem and the density fluctuations agree with experimental values. The temperature increases during fresh inflation and reach its maximum value when inflation ends. I find that entropy perturbations always remain below $10^{-4}$ during fresh inflation and become negligible when fresh inflation ends. Hence, the adiabatic fluctuations dominate the primordial spectrum at the end of fresh inflation. 
  In a static spacetime, the Killing time can be used to measure the time required for signals or objects to propagate between two of its orbits. By further restricting to spherically symmetric cases, one obtains a natural association between these orbits and timelike lines in Minkowski space. We prove a simple theorem to the effect that in any spacetime satisfying the weak energy condition the above signaling time is, in this sense, no faster than that for a corresponding signal in Minkowski space. The theorem uses a ormalization of Killing time appropriate to an observer at infinity. We then begin an investigation of certain related but more local questions by studying particular families of spacetimes in detail. Here we are also interested in restrictions imposed by the dominant energy condition. Our examples suggest that signaling in spacetimes satisfying this stronger energy condition may be significantly slower than the fastest spacetimes satisfying only the weak energy condition. 
  In this paper we propose, in a preliminary way, a new Earth-based laboratory experiment aimed to the detection of the gravitomagnetic field of the Earth. It consists of the measurement of the difference of the circular frequencies of two rotators moving along identical circular paths, but in opposite directions, on a horizontal friction-free plane in a vacuum chamber placed at South Pole. The accuracy of our knowledge of the Earth's rotation from VLBI and the possibility of measuring the rotators'periods over many revolutions should allow for the feasibility of the proposed experiment. 
  I describe how gravitational entropy is intimately connected with the concept of gravitational heat, expressed as the difference between the total and free energies of a given gravitational system. From this perspective one can compute these thermodyanmic quantities in settings that go considerably beyond Bekenstein's original insight that the area of a black hole event horizon can be identified with thermodynamic entropy. The settings include the outsides of cosmological horizons and spacetimes with NUT charge. However the interpretation of gravitational entropy in these broader contexts remains to be understood. 
  A covariant scheme for matter coupling with a GL(3,R) gauge formulation of gravity is studied. We revisit a known Yang-Mills type construction, where quadratical power of cosmological constant have to be considered in consistence with vacuum Einstein's gravity. Then, matter coupling with gravity is introduced and some constraints on fields and background appear. Finally we elucidate that introduction of auxiliary fields decreases the number of these constraints. 
  A family of spherically symmetric solutions in the model with 1-component anisotropic fluid is considered. The metric of the solution depends on a parameter q > 0 relating radial pressure and the density and contains n -1 parameters corresponding to Ricci-flat ``internal space'' metrics. For q = 1 and certain equations of state the metric coincides with the metric of black brane solutions in the model with antisymmetric form. A family of black hole solutions corresponding to natural numbers q = 1,2, ... is singled out. Certain examples of solutions (e.g. containing for q =1 Reissner-Nordstr\"{o}m, M2 and M5 black brane metrics) are considered. The post-Newtonian parameters beta and gamma corresponding to the 4-dimensional section of the metric are calculated. 
  Einstein's general theory of relativity predicts that an initially plane wave-front will curve because of gravity. This effect can now be measured using Very Long Baseline Interferometry (VLBI). A wave-front from a distant point source will curve as it passes the gravitational field of the Sun. We propose an experiment to directly measure this curvature, using four VLBI stations on earth, separated by intercontinental distances. Expressed as a time delay, the size of the effect is a few hundred picoseconds and may be measureable with present technology. 
  Many solutions of Einstein's field equations contain closed timelike curves (CTC). Some of these solutions refer to ordinary materials in situations which might occur in the laboratory, or in astrophysics. It is argued that, in default of a reasonable interpretation of CTC, general relativity does not give a satisfactory account of all phenomena within its terms of reference. 
  It is shown that the same phenomenological Newtonian model recently proposed by the author to explain the cosmological evolution of the fine structure constant suggests furthermore an explanation of the unmodelled acceleration $a_P\simeq 8.5\times 10^{-10}m/s^2$ of the Pioneer 10/11 spaceships reported by Anderson {\em et al} in 1998. In the view presented here, it is argued that the permittivity and permeability of empty space are decreasing adiabatically, and the light is accelerating therefore, as a consequence of the progressive attenuation of the quantum vacuum due to the combined effect of its gravitational interaction with all the expanding universe and the fourth Heisenberg relation. It is suggested that the spaceships do not have any extra acceleration (but follow the unchanged Newton laws), the observed effect being due to an adiabatic acceleration of the light equal to $a_P$, which has the same observational radio signature as the anomalous acceleration of the Pioneers. 
  We revisit the recently found equivalence for the response of a static scalar source interacting with a {\em massless} Klein-Gordon field when the source is (i) static in Schwarzschild spacetime, in the Unruh vacuum associated with the Hawking radiation and (ii) uniformly accelerated in Minkowski spacetime, in the inertial vacuum, provided that the source's proper acceleration is the same in both cases. It is shown that this equivalence is broken when the massless Klein-Gordon field is replaced by a {\em massive} one. 
  Relativistic elasticity on an arbitrary spacetime is formulated as a Lagrangian field theory which is covariant under spacetime diffeomorphisms. This theory is the relativistic version of classical elasticity in the hyperelastic, materially frame-indifferent case and, on Minkowski space, reduces to the latter in the non-relativistic limit . The field equations are cast into a first -- order symmetric hyperbolic system. As a consequence one obtains local--in--time existence and uniqueness theorems under various circumstances. 
  We study a massless scalar field propagating in the background of a five-dimensional rotating black hole. We showed that in the Myers-Perry metric describing such a black hole the massless field equation allows the separation of variables. The obtained angular equation is a generalization of the equation for spheroidal functions. The radial equation is similar to the radial Teukolsky equation for the 4-dimensional Kerr metric. We use these results to quantize the massless scalar field in the space-time of the 5-dimensional rotating black hole and to derive expressions for energy and angular momentum fluxes from such a black hole. 
  We evaluate the decay rate of the uniformly accelerated proton. We obtain an analytic expression for inverse beta decay process caused by the acceleration. We evaluate the decay rate both from the inertial frame and from the accelerated frame where we should consider thermal radiation by Unruh effect. We explicitly check that the decay rates obtained in both frame coincide with each other. 
  A cosmological model with perfect fluid and self-interacting quintessence field is considered in the framework of the spatially flat Friedmann-Robertson-Walker (FRW) geometry. By assuming that all physical quantities depend on the volume scale factor of the Universe, the general solution of the gravitational field equations can be expressed in an exact parametric form. The quintessence field is a free parameter. With an appropriate choice of the scalar field a class of exact solutions is obtained, with an exponential type scalar field potential fixed via the gravitational field equations. The general physical behavior of the model is consistent with the recent cosmological scenario favored by supernova Type Ia observations, indicating an accelerated expansion of the Universe. 
  An expression is derived where the mass is connected to an integral over the pressure of gravitating matter in the frame work of five dimensional(5D) space-time. 
  Charged, rotating black hole solutions of Einstein's gravitational equations are investigated in the presence of a cosmological constant. A pair of wave equations governing the electromagnetic and gravitational perturbations are derived. 
  We study master variables in the Regge-Wheeler-Zerilli formalism. We show that a specific choice of new variables is suitable for studying perturbation theory from the viewpoint of radiation reaction calculations. With explicit definition of the improved master variables in terms of components of metric perturbations, we present the master equations, with source terms, and metric reconstruction formulas. In the scheme using these new variables, we do not need any time and radial integrations except for solving the master equation. We also show that the master variable for even parity modes which satisfies the same homogeneous equation as the odd parity case, obtained via  Chandrasekhar transformation, does not have the good property in this sense. 
  There are non-radial null geodesics emanating from the shell focusing singularity formed at the symmetric center in a spherically symmetric dust collapse. In this article, assuming the self-similarity in the region filled with the dust fluid, we study these singular null geodesics in detail. We see the time evolution of the angular diameter of the central naked singularity and show that it might be bounded above by the value corresponding to the circular null geodesic in the Schwarzschild spacetime. We also investigate the angular frequency of a physical field which propagates along the singular null geodesic and find that it depends on the impact parameter. Further, we comment on the non-uniformity of the topology of the central naked singularity. 
  The diagonalization of the metrical and canonical Hamilton operators of a scalar field with an arbitrary coupling, with a curvature in N-dimensional homogeneous isotropic space is considered in this paper. The energy spectrum of the corresponding quasiparticles is obtained and then the modified energy-momentum tensor is constructed; the latter coincides with the metrical energy-momentum tensor for conformal scalar field. Under the diagonalization of corresponding Hamilton operator the energies of relevant particles of a nonconformal field are equal to the oscillator frequencies, and the density of such particles created in a nonstationary metric is finite. It is shown that the modified Hamilton operator can be constructed as a canonical Hamilton operator under the special choice of variables. 
  The global structure of solutions of the Einstein equations coupled to the Vlasov equation is investigated in the presence of a two-dimensional symmetry group. It is shown that there exist global CMC and areal time foliations. The proof is based on long-time existence theorems for the partial differential equations resulting from the Einstein-Vlasov system when conformal or areal coordinates are introduced. 
  Searching for a signal depending on unknown parameters in a noisy background with matched filtering techniques always requires an analysis of the data with several templates in parallel in order to ensure a proper match between the filter and the real waveform. The key feature of such an implementation is the design of the filter bank which must be small to limit the computational cost while keeping the detection efficiency as high as possible. This paper presents a geometrical method which allows one to cover the corresponding physical parameter space by a set of ellipses, each of them being associated to a given template. After the description of the main characteristics of the algorithm, the method is applied in the field of gravitational wave (GW) data analysis, for the search of damped sine signals. Such waveforms are expected to be produced during the de-excitation phase of black holes -- the so-called 'ringdown' signals -- and are also encountered in some numerically computed supernova signals. 
  Madelung's hydrodynamical forms of the Schrodinger equation and the Klein-Gordon equation are presented. The physical nature of the quantum potential is explored. It is demonstrated that the geometrical origin of the quantum potential is in the scalar curvature of the of the metric that defines the kinetic energy density for an extended particle and that the quantization of circulation (Bohr-Sommerfeld) is a consequence of associating an SO(2)-reduction of the Lorentz frame bundle with wave motion. The Madelung equations are then cast in a basis-free form in terms of exterior differential forms in such a way that they represent the equations for a timelike solution to the conventional wave equations whose rest mass density satisfies a differential equation of the "Klein-Gordon minus nonlinear term" type. The role of non-zero vorticity is briefly examined. 
  Some standard definitions and results concerning foliations of dimension one and codimension one are introduced. A proper time foliation of Minkowski space is defined and contrasted with the foliation that is defined by the time coordinate. The extent to which a Lorentz structure on a manifold defines foliations, and the issues concerning the extension of the proper time foliation of Minkowski space to a Lorentz manifold are discussed, such as proper time sections of a geodesic flow. 
  The exact 1+3 covariant dynamical fluid equations for a multi-component plasma, together with Maxwell's equations are presented in such a way as to make them suitable for a gauge-invariant analysis of linear density and velocity perturbations of the Friedmann-Robertson-Walker model. In the case where the matter is described by a two component plasma where thermal effects are neglected, a mode representing high-frequency plasma oscillations is found in addition to the standard growing and decaying gravitational instability picture. Further applications of these equations are also discussed. 
  The flatness of an accelerating universe model (characterized by a dark energy scalar field $\chi$) is mimicked from a curved model that is filled with, apart from the cold dark matter component, a quintessencelike scalar field $Q$. In this process, we characterize the original scalar potential $V(Q)$ and the mimicked scalar potential $V(\chi)$ associated to the scalar fields $Q$ and $\chi$, respectively. The parameters of the original model are fixed through the mimicked quantities that we relate to the present astronomical data, such that the equation state parameter $w_{_{\chi}}$ and the dark energy density parameter $\Omega_{\chi}$. 
  Artificial black holes, such as sonic holes in Bose-Einstein condensates, may give insights into the role of the physics at the event horizon beyond the Planck scale. We show that sonic white holes give rise to a discrete spectrum of instabilities that is insensitive to the analogue of trans-Planckian physics for Bose-Einstein condensates. 
  A new framework for analysing the gravitational fields in a stationary, axisymmetric configuration is introduced. The method is used to construct a complete set of field equations for the vacuum region outside a rotating source. These equations are under-determined. Restricting the Weyl tensor to type D produces a set of equations which can be solved, and a range of new techniques are introduced to simplify the problem. Imposing the further condition that the solution is asymptotically flat yields the Kerr solution uniquely. The implications of this result for the no-hair theorem are discussed. The techniques developed here have many other applications, which are described in the conclusions. 
  In this paper we give, for the first time, a qualitative description of the asymptotic dynamics of a class of non-tilted spatially homogeneous (SH) cosmologies, the so-called exceptional Bianchi cosmologies, which are of Bianchi type VI$_{-1/9}$. This class is of interest for two reasons. Firstly, it is generic within the class of non-tilted SH cosmologies, being of the same generality as the models of Bianchi types VIII and IX. Secondly, it is the SH limit of a generic class of spatially inhomogeneous $G_{2}$ cosmologies.   Using the orthonormal frame formalism and Hubble-normalized variables, we show that the exceptional Bianchi cosmologies differ from the non-exceptional Bianchi cosmologies of type VI$_{h}$ in two significant ways. Firstly, the models exhibit an oscillatory approach to the initial singularity and hence are not asymptotically self-similar. Secondly, at late times, although the models are asymptotically self-similar, the future attractor for the vacuum-dominated models is the so-called Robinson-Trautman SH model instead of the vacuum SH plane wave models. 
  A Spin-polarised cylindrically symmetric exact class of solutions endowed with magnetic fields in Einstein-Cartan-Maxwell gravity is obtained. Application of matching conditions to this interior solution having an exterior as Einstein's vacuum solution shows that for this class of metrics the Riemann-Cartan (RC) rotation vanishes which makes the solution static. Therefore we end up with a magnetized static spin polarised cylinder where the pressure along the symmetry axis is negative. 
  Regular generalizations of spherically and axially symmetric metrics and their properties are considered. Newton gravity law generalizations are reduced for null geodesic. 
  We contrast features of simple varying speed of light (VSL) cosmologies with inflationary universe models. We present new features of VSL cosmologies and show that they face problems explaining the cosmological isotropy problem. We also find that if c falls fast enough to solve the flatness and horizon problems then the quantum wavelengths of massive particle states and the radii of primordial black holes can grow to exceed the scale of the particle horizon. This may provide VSL cosmologies with a self-reproduction property. The constraint of entropy increase is also discussed. The new problems described in the this letter provide a set of bench tests for more sophisticated VSL theories to pass. 
  In a recent investigation of the effects of precession on the anticipated detection of gravitational-wave inspiral signals from compact object binaries with moderate total masses, we found that (i) if precession is ignored, the inspiral detection rate can decrease by almost a factor of 10, and (ii) previously proposed ``mimic'' templates cannot improve the detection rate significantly (by more than a factor of 2). In this paper we propose a new family of templates that can improve the detection rate by factors of 5--6 in cases where precession is most important. Our proposed method for these new ``mimic'' templates involves a hierarchical scheme of efficient, two-parameter template searches that can account for a sequence of spikes that appear in the residual inspiral phase, after one corrects for the any oscillatory modification in the phase. We present our results for two cases of compact object masses (10 and 1.4 solar masses and 7 and 3 solar masses) as a function of spin properties. Although further work is needed to fully assess the computational efficiency of this newly proposed template family, we conclude that these ``spiky templates'' are good candidates for a family of precession templates used in realistic searches, that can improve detection rates of inspiral events. 
  The results of canonical quantum gravity concerning geometric operators and black hole entropy are beset by an ambiguity labelled by the Immirzi parameter. We use a result from classical gravity concerning the quasinormal mode spectrum of a black hole to fix this parameter in a new way. As a result we arrive at the Bekenstein - Hawking expression of $A/4 l_P^2$ for the entropy of a black hole and in addition see an indication that the appropriate gauge group of quantum gravity is SO(3) and not its covering group SU(2). 
  Proceeding from a homogeneous and isotropic Friedmann universe a conceptional problem concerning light propagation in an expanding universe is brought up. As a possible solution of this problem it is suggested that light waves do not scale with R(t). With the aid of a Generalized Equivalence Principle a cosmologic model with variable "constants" c, H, and G is constructed. It is shown that with an appropriate variation of the Boltzmann "constant" k the thermal evolution of the universe is similar to the standard model. It is further shown that this model explains the cosmological redshift as well as certain problems of the standard model (horizon, flatness, accelerated expansion of the universe).   PACS numbers: 98.80.Bp, 98.80.Hw, 04.20.Cv.   Keywords: cosmology, velocity of light, expansion. 
  One of the conceptual tensions between quantum mechanics (QM) and general relativity (GR) arises from the clash between the spatial nonseparability} of entangled states in QM, and the complete spatial separability of all physical systems in GR, i.e., between the nonlocality implied by the superposition principle, and the locality implied by the equivalence principle. Possible experimental consequences of this conceptual tension will be discussed for macroscopically entangled, coherent quantum fluids, such as superconductors, superfluids, atomic Bose-Einstein condensates, and quantum Hall fluids, interacting with tidal and gravitational radiation fields. A minimal-coupling rule, which arises from the electron spin coupled to curved spacetime, leads to an interaction between electromagnetic (EM) and gravitational (GR) radiation fields mediated by a quantum Hall fluid. This suggests the possibility of a quantum transducer action, in which EM waves are convertible to GR waves, and vice versa. 
  Special Relativity (SR) kinematics is derived from very intuitive assumptions. Contrary to standard Einstein's derivation, no light signal is used in the construction nor it is assumed to exist. Instead we postulate the existence of two equivalence classes of physical objects: proportional clocks and proportional rulers. Simple considerations lead to Lorentz kinematics as one of three generic cases. The Lorentz case is characterized by the maximal relative speed of physical objects. The two others are the Galilean and the Euclidean cases. 
  An underlying fundamental assumption in relativistic perturbation theory is the existence of a parametric family of spacetimes that can be Taylor expanded around a background. Since the choice of the latter is crucial, sometimes it is convenient to have a perturbative formalism based on two (or more) parameters. A good example is the study of rotating stars, where generic perturbations are constructed on top of an axisymmetric configuration built by using the slow rotation approximation. Here, we discuss the gauge dependence of non-linear perturbations depending on two parameters and how to derive explicit higher order gauge transformation rules. 
  We show that the spacetimes of domain wall solutions to the coupled Einstein-scalar field equations with a given scalar field potential fall into two classes, depending on whether or not reflection symmetry on the wall is imposed. Solutions with reflection symmetry are dynamic, while the asymmetric ones are static. Asymmetric walls are asymptotically flat on one side and reduce to the Taub spacetime on the other. Examples of asymmetric thick walls in D-dimensional spacetimes are given, and results on the thin-wall limit of the dynamic, symmetric walls are extended to the asymmetric case. The particular case of symmetric, static spacetimes is considered and a new family of solutions, including previously known BPS walls, is presented. 
  Observations indicate that the universe is effectively flat, but they do not rule out a closed universe. The role of positive curvature is negligible at late times, but can be crucial in the early universe. In particular, positive curvature allows for cosmologies that originate as Einstein static universes, and then inflate and later reheat to a hot big bang era. These cosmologies have no singularity, no "beginning of time", and no horizon problem. If the initial radius is chosen to be above the Planck scale, then they also have no quantum gravity era, and are described by classical general relativity throughout their history. 
  In this paper Quantum Mechanics with Fundamental Length is built as a deformation of Quantum Mechanics. To this aim an approach is used which does not take into account commutator deformation as usually it is done, but density matrix deformation. The corresponding deformed density matrix, which is called density pro-matrix is given explicitly. It properties have been investigated as well as some dynamical aspects of the theory. In particular, the deformation of Liouville equation is analyzed in detail. It was shown that Liouville equation in Quantum Mechanics appears as a low energy limit of deformed Liouville equation in Quantum Mechanics with Fundamental Length. Some implications of obtained results are presented as well as their application to the calculation of black hole entropy. 
  This Resource Letter provides a guide to the literature on the physics and astrophysics of gravitational waves. Journals, books, reports, archives, and websites are provided as basic resources and for current research frontiers in detectors, data analysis, and astrophysical source modeling. 
  A new theory is considered according to which extended objects in $n$-dimensional space are described in terms of multivector coordinates which are interpreted as generalizing the concept of centre of mass coordinates. While the usual centre of mass is a point, by generalizing the latter concept, we associate with every extended object a set of $r$-loops, $r=0,1,..., n-1$, enclosing oriented $(r+1)$-dimensional surfaces represented by Clifford numbers called $(r+1)$-vectors or multivectors. Superpositions of multivectors are called polyvectors or Clifford aggregates and they are elements of Clifford algebra. The set of all possible polyvectors forms a manifold, called $C$-space. We assume that the arena in which physics takes place is in fact not Minkowski space, but $C$-space. This has many far reaching physical implications, some of which are discussed in this paper. The most notable is the finding that although we start from the constrained relativity in $C$-space we arrive at the unconstrained Stueckelberg relativistic dynamics in Minkowski space which is a subspace of $C$-space. 
  Let $M = M_0 \times \R^2$ be a pp--wave type spacetime endowed with the metric $<\cdot,\cdot>_z = <\cdot,\cdot>_x + 2 du dv + H(x,u) du^2$, where $(M_0, <\cdot,\cdot>_x) $ is any Riemannian manifold and $H(x,u)$ an arbitrary function. We show that the behaviour of $H(x,u)$ at spatial infinity determines the causality of $M$, say: (a) if $-H(x,u)$ behaves subquadratically (i.e, essentially $-H(x,u) \leq R_1(u) |x|^{2-\epsilon} $ for some $\epsilon >0$ and large distance $|x|$ to a fixed point) and the spatial part $(M_0, <\cdot,\cdot>_x) $ is complete, then the spacetime $M$ is globally hyperbolic, (b) if $-H(x,u)$ grows at most quadratically (i.e, $-H(x,u) \leq R_1(u) |x|^{2}$ for large $|x|$) then it is strongly causal and (c) $M$ is always causal, but there are non-distinguishing examples (and thus, non-strongly causal), even when $-H(x,u) \leq R_1(u) |x|^{2+\epsilon} $, for small $\epsilon >0$.   Therefore, the classical model $M_0 = \R^2$, $H(x,u) = \sum_{i,j} h_{ij}(u) x_i x_j (\not\equiv 0)$, which is known to be strongly causal but not globally hyperbolic, lies in the critical quadratic situation with complete $M_0$. This must be taken into account for realistic applications. In fact, we argue that $-H$ will be subquadratic (and the spacetime globally hyperbolic) if $M$ is asymptotically flat. The relation of these results with the notion of astigmatic conjugacy and the existence of conjugate points is also discussed. 
  Black-hole (BH) binaries with single-BH masses m=5--20 Msun, moving on quasicircular orbits, are among the most promising sources for first-generation ground-based gravitational-wave (GW) detectors. Until now, the development of data-analysis techniques to detect GWs from these sources has been focused mostly on nonspinning BHs. The data-analysis problem for the spinning case is complicated by the necessity to model the precession-induced modulations of the GW signal, and by the large number of parameters needed to characterize the system, including the initial directions of the spins, and the position and orientation of the binary with respect to the GW detector. In this paper we consider binaries of maximally spinning BHs, and we work in the adiabatic-inspiral regime to build families of modulated detection templates that (i) are functions of very few physical and phenomenological parameters, (ii) model remarkably well the dynamical and precessional effects on the GW signal, with fitting factors on average >~ 0.97, but (iii) might require increasing the detection thresholds, offsetting at least partially the gains in the fitting factors. Our detection-template families are quite promising also for the case of neutron-star--black-hole binaries, with fitting factors on average ~ 0.93. For these binaries we also suggest (but do not test) a further template family, which would produce essentially exact waveforms written directly in terms of the physical spin parameters. 
  A new topology of laser interferometric gravitational-wave antenna is considered. It is based on two schemes: {\em quantum speedmeter} and {\em zero-area Sagnac interferometer} and allows to obtain sensitivity better than the Standard Quantum Limit in wide band without any large-scale modifications of the standard topology of the laser interferometric antennae. 
  It has been proposed by Bekenstein and others that the horizon area of a black hole conforms, upon quantization, to a discrete and uniformly spaced spectrum. In this paper, we consider the area spectrum for the highly non-trivial case of a rotating (Kerr) black hole solution. Following a prior work by Barvinsky, Das and Kunstatter, we are able to express the area spectrum in terms of an integer-valued quantum number and an angular-momentum operator. Moreover, by using an analogy between the Kerr black hole and a quantum rotator, we are able to quantize the angular-momentum sector. We find the area spectrum to be $A_{n,J_{cl}}=8\pi\hbar(n+J_{cl}+1/2)$, where $n$ and $J_{cl}$ are both integers. The quantum number $J_{cl}$ is related to but distinct from the eigenvalue $j$ of the angular momentum of the black hole. Actually, it represents the ``classical'' angular momentum and, for $J_{cl}\gg 1$, $J_{cl}\approx j$. 
  Stochastic semiclassical gravity of the 90's is a theory naturally evolved from semiclassical gravity of the 70's and 80's. It improves on the semiclassical Einstein equation with source given by the expectation value of the stress-energy tensor of quantum matter fields in curved spacetimes by incorporating an additional source due to their fluctuations. In stochastic semiclassical gravity the main object of interest is the noise kernel, the vacuum expectation value of the (operator-valued) stress-energy bi-tensor, and the centerpiece is the (stochastic) Einstein-Langevin equation. We describe this new theory via two approaches: the axiomatic and the functional. The axiomatic approach is useful to see the structure of the theory from the framework of semiclassical gravity. The functional approach uses the Feynman-Vernon influence functional and the Schwinger-Keldysh close-time-path effective action methods which are convenient for computations. It also brings out the open systems concepts and the statistical and stochastic contents of the theory such as dissipation, fluctuations, noise and decoherence. We then describe the application of stochastic gravity to the backreaction problems in cosmology and black hole physics. Intended as a first introduction to this subject, this article places more emphasis on pedagogy than completeness. 
  This paper gives two complete and elementary proofs that if the speed of light over closed paths has a universal value $c$, then it is possible to synchronize clocks in such a way that the one-way speed of light is c. The first proof is an elementary version of a recent proof. The second provides high precision experimental evidence that it is possible to synchronize clocks in such a way that the one-way speed of light has a universal value. We also discuss an old incomplete proof by Weyl which is important from an historical perspective. 
  Hamilton-Jacobi formalism is used to study 2D-gravity and its SL(2, R) hidden symmetry. If the contribution of the surface term is considered the obtained results coincide with those given by the Dirac and Faddeev-Jackiw approaches. 
  We present a new class of exact homogeneous cosmological solutions with a radiation fluid for all scalar-tensor theories. The solutions belong to Bianchi type $VI_{h}$ cosmologies. Explicit examples of nonsingular homogeneous scalar-tensor cosmologies are also given. 
  In this paper, we consider the gravitational radiation generated by the collision of highly relativistic particles with rotating Kerr black holes. We use the Sasaki-Nakamura formalism to compute the waveform, energy spectra and total energy radiated during this process. We show that the gravitational spectrum for high-energy collisions has definite characteristic universal features, which are independent of the spin of the colliding objects. We also discuss possible connections between these results and the black hole-black hole collision at the speed of light process. With these results at hand, we predict that during the high speed collision of a non-rotating hole with a rotating one, about 35% of the total energy can get converted into gravitational waves. Thus, if one is able to produce black holes at the Large Hadron Collider, as much as 35% of the partons' energy should be emitted during the so called balding phase. This energy will be missing, since we don't have gravitational wave detectors able to measure such amplitudes. The collision at the speed of light between one rotating black hole and a non-rotating one or two rotating black holes turns out to be the most efficient gravitational wave generator in the Universe. 
  We consider the effects of adding a scale invariant $R^{2}$ term to the action of the scale invariant model (SIM) studied previously by one of us (E.I.G., Mod. Phys. Lett. A14, 1043 (1999)). The SIM belongs to the general class of theories, where an integration measure independent of the metric is introduced. To implement scale invariance (S.I.), a dilaton field is introduced. The integration of the equations of motion associated with the new measure gives rise to the spontaneous symmetry breaking (S.S.B) of S.I.. After S.S.B. of S.I. in the model with the $R^{2}$ term, it is found that a non trivial potential for the dilaton is generated. This potential contains two flat regions: one associated with the Planck scale and with an inflationary phase, while the other flat region is associated to a very small vacuum energy (V.E.) and is associated to the present slowly accelerated phase of the universe (S.A.PH). The smallness of the V.E. in the S.A.PH. is understood through the see saw mechanism introduced in S.I.M. 
  Singularity-free cosmological solutions may be obtained from the string action at tree level if the dimension of the space-time is greater than 10 and if brane configurations are taken into account. The behaviour of the dilaton field in this case is also regular. Asymptotically a radiative phase is attained indicating a smooth transition to the standard cosmological model. 
  Neutrino oscillation is visualized as coupled vibrations. Unlike existing models describing neutrino oscillations, our model involving a single fundamental mass parameter $m$ and first discussed in the context of a possible boson oscillation demonstrates that for a self consistent theory there should be three types of bosons paired with maximal mixing. The argument is easily extended to neutrinos (fermions) with the exception that in the case of boson oscillation means periodic variation of particle number, whereas in the case of fermions the particle number does not change, only the mass oscillates. The fermionic vacuum is not really empty but filled with zero energy fermions. The oscillation length calculated perturbatively conforms to the experimental findings. Also there is no fermionic number violation in our model. 
  We try to give hereafter an answer to some open questions about the definition of conserved quantities in Chern-Simons theory, with particular reference to Chern-Simons AdS_3 Gravity. Our attention is focused on the problem of global covariance and gauge invariance of the variation of Noether charges. A theory which satisfies the principle of covariance on each step of its construction is developed, starting from a gauge invariant Chern-Simons Lagrangian and using a recipe developed in gr-qc/0110104 and gr-qc/0107074 to calculate the variation of conserved quantities. The problem to give a mathematical well-defined expression for the infinitesimal generators of symmetries is pointed out and it is shown that the generalized Kosmann lift of spacetime vector fields leads to the expected numerical values for the conserved quantities when the solution corresponds to the BTZ black hole. The fist law of black holes mechanics for the BTZ solution is then proved and the transition between the variation of conserved quantities in Chern-Simons AdS_3 Gravity theory and the variation of conserved quantities in General Relativity is analysed in detail. 
  Spherically symmetric inhomogeneous dust collapse has been studied in higher dimensional space-time and the factors responsible for the appearance of a naked singularity are analyzed in the region close to the centre for the marginally bound case. It is clearly demonstrated that in the former case naked singularities do not appear in the space-time having more than five dimension, which appears to a strong result. The non-marginally bound collapse is also examined in five dimensions and the role of shear in developing naked singularities in this space-time is discussed in details. The five dimensional space-time is chosen in the later case because we have exact solution in closed form only in five dimension and not in any other case. 
  We extend the classical Avez-Seifert theorem to trajectories of charged test particles with fixed charge-to-mass ratio. In particular, given two events x_{0} and x_{1}, with x_{1} in the chronological future of x_{0}, we find an interval I=]-R,R[ such that for any q/m in I there is a timelike connecting solution of the Lorentz force equation. Moreover, under the assumption that there is no null geodesic connecting x_0 and x_1, we prove that to any value of |q/m| there correspond at least two connecting timelike solutions which coincide only if they are geodesics. 
  We show that existing low energy experiments, searching for the breaking of local Lorentz invariance, set bounds upon string theory inspired quantum gravity models that induce corrections to the propagation of fields. In the D-particle recoil model we find M > 1.2 x 10^5 M_P and v < 2 x 10^{-27}c for the mass and recoil speed of the D-particle, respectively. These bounds are \~10^8 times stronger than the latest astrophysical bounds. These results indicate that the stringy scenario for modified dispersion relations is as vulnerable to these types of tests as the loop quantum gravity schemes. 
  Spherically symmetric inhomogeneous dust collapse has been studied in higher dimensional space-time and appearance of naked singularity has been analyzed both for non-marginal and marginally bound cases. It has been shown that naked singularity is possible for any arbitrary dimension in non-marginally bound case. For marginally bound case we have examined the radial null geodesics from the singularity and found that naked singularity is possible upto five dimension. 
  This paper has been withdrawn. 
  It was suggested earlier that the gravitational redshift formula can be invalid when the effect of the backscattering is strong. It is demonstrated here numerically, for an exemplary electromagnetic pulse that is: i) initially located very close to the horizon of a Schwarzschild black hole and ii) strongly backscattered, that a mean frequency does not obey the standard redshift formula. Redshifts appear to depend on the frequency and there manifests a backscatter-induced blueshift in the outgoing radiation. 
  We describe recent progress in the numerical study of the structure of rapidly rotating superfluid neutron star models in full general relativity. The superfluid neutron star is described by a model of two interpenetrating and interacting fluids, one representing the superfluid neutrons and the second consisting of the remaining charged particles (protons, electrons, muons). We consider general stationary configurations where the two fluids can have different rotation rates around a common rotation axis. The previously discovered existence of configurations with one fluid in a prolate shape is confirmed. 
  Let G be a compact Lie group. Using suitable normalization conventions, we show that the evaluation of GxG-symmetric spin networks is non-negative whenever the edges are labeled by representations of the form V\otimes V^* where V is a representation of G, and the intertwiners are generalizations of the Barrett--Crane intertwiner. This includes in particular the relativistic spin networks with symmetry group Spin(4) or SO(4). We also present a counterexample, using the finite group S_3, to the stronger conjecture that all spin network evaluations are non-negative as long as they can be written using only group integrations and index contractions. This counterexample applies in particular to the product of five 6j-symbols which appears in the spin foam model of the S_3-symmetric BF-theory on the two-complex dual to a triangulation of the sphere S^3 using five tetrahedra. We show that this product is negative real for a particular assignment of representations to the edges. 
  A generalisation of a known theorem concerning the computation of the conformal algebra in 1+(n-1) decomposable spaces is presented. It is shown that the general form of Conformal Vector Fields (CVF) is the sum of a gradient CVF and a Killing or Homothetic (n-1)-vector. A simple criterion is established which enables one to check if a 1+(n-1) decomposable spacetime admits proper CVF. As an example, the complete conformal algebra of a G\"odel-type spacetime is computed. 
  The general relativistic gravitomagnetic clock effect involves a coupling between the orbital motion of a test particle and the rotation of the central mass and results in a difference in the proper periods of two counter-revolving satellites. It is shown that at O(c^-2) this effect has a simple analogue in the electromagnetic case. Moreover, in view of a possible measurement of the clock effect in the gravitational field of the Earth, we investigate the influence of some classical perturbing forces of the terrestrial space environment on the orbital motion of test bodies along opposite trajectories. 
  We study the quantum deformation of the Barrett-Crane Lorentzian spin foam model which is conjectured to be the discretization of Lorentzian Plebanski model with positive cosmological constant and includes therefore as a particular sector quantum gravity in de-Sitter space. This spin foam model is constructed using harmonic analysis on the quantum Lorentz group. The evaluation of simple spin networks are shown to be non commutative integrals over the quantum hyperboloid defined as a pile of fuzzy spheres. We show that the introduction of the cosmological constant removes all the infrared divergences: for any fixed triangulation, the integration over the area variables is finite for a large class of normalization of the amplitude of the edges and of the faces. 
  We study the action of diffeomorphisms on spin foam models. We prove that in 3 dimensions, there is a residual action of the diffeomorphisms that explains the naive divergences of state sum models. We present the gauge fixing of this symmetry and show that it explains the original renormalization of Ponzano-Regge model. We discuss the implication this action of diffeomorphisms has on higher dimensional spin foam models and especially the finite ones. 
  In the quantum-gravity literature there has been interest in the possibility that quantum properties of spacetime might affect the energy/momentum dispersion relation. The most used test theory for data analysis is based on a modification of the laws of propagation proposed in astro-ph/9712103 [Nature 393,763], and the present best limit on the quantum-gravity scale was obtained in gr-qc/9810044 [Phys.Rev.Lett.83,2108]. I derive an improved limit using recent experimental information on absorption by the infrared diffuse extragalactic background of $\gamma$-rays emitted by blazars. Foreseeable more accurate determinations of the absorption levels could achieve Planck-scale sensitivity. As a corollary I also show that, contrary to the recent claim of astro-ph/0208507v3, the test theory here considered does not allow decays of photons into electron-positron pairs, and I expose the limitations of phenomenological proposals, such as the one reported in astro-ph/0212190, in which one attempts to infer limits on the kinematic theory here considered through the ad hoc introduction of a dynamical framework. 
  Quantum vacua are characterized by the topological structure of their fermion zero modes. The vacua are distributed into universality classes protected by topology in momentum space. The vacua whose manifold of fermion zero modes has co-dimension 3 are of special interest because in the low-energy corner the fermionic excitations become the Weyl relativistic chiral fermions, while the dynamical bosonic collective modes of the fermionic vacuum interact with the chiral fermions as the effective gravity and gauge fields. The relativistic invariance, the chirality of fermions, the gauge and gravity fields, the relativistic spin, etc., are the emergent low-energy properties of the quantum vacuum with such fermion zero modes. The vacuum of the Standard Model and the vacuum of superfluid 3He-A belong to this universality class and thus they are described by similar effective theories. This allows us to use this quantum liquid for the theoretical and experimental simulations of many problems related to the quantum vacuum, such as the chiral anomaly and the cosmological constant problems. 
  Keeping the relativistic laws of motion a non-conventional Pioneer effect would prove an increase of the scale rate of atomic clocks in comparison with planetary ones. Together with a slowly decreasing amount of about 60% due to anisotropic radiation this would be a thinkable explanation for an apparent anomalous acceleration of the Pioneer 10/11 probes. Such a difference between atomic time and ephemeris time, however, (coincidentally corresponding to that of atomic time and cosmic time as derived from any cosmological model of general relativity, where the cosmic 'coordinate' speed of light is fixed to be c* = c) is ruled out by solar system's observational facts. Thus a non-conventional Pioneer effect would inevitably contradict relativity theory. 
  In this letter we reexamine the evaluation of \zone in some proposed tests of relativistic gravitomagnetism with existing and proposed laser--ranged LAGEOS--like satellites in the gravitational field of the Earth. A more conservative and realistic approach is followed by using the diagonal part only of the covariance matrix of the EGM96 Earth's gravity model up to degree l=20. It turns out that, within this context and according to the present level of knowledge of the terrestrial gravitational field, the best choice would be the use of a recently proposed combination using the nodes \Omega of LAGEOS, LAGEOS II and LARES and the perigees \omega of LAGEOS II and LARES. Indeed, it turns out to be insensitive both to the even zonal harmonics of degree higher than l=20 and to the correlation among them 
  We derive a new parametric class of exact cosmological solutions to Brans-Dicke theory of gravity with a self-interacting scalar field and a barotropic perfect fluid of ordinary matter, by assuming a linear relationship between the Hubble expansion parameter and the time derivative of the scalar field. As a consequence only a class of exponential potentials and their combinations can be treated. The relevance of the solutions found for the description of the cosmic evolution are discussed in detail. We focus our discussion mainly on the possibility to have superquintessence behavior. 
  We consider the five-dimensional bulk spacetime with negative Lambda described by the Nariai metric (which is not conformally flat) and match it with a vacuum brane satisfying the proper boundary conditions. It is shown that the brane metric corresponds to a cloud of string dust of constant energy density. 
  Test particle geodesic motion is analysed in detail for the background spacetimes of the degenerate Ferrari-Ibanez colliding gravitational wave solutions. Killing vectors have been used to reduce the equations of motion to a first order system of differential equations which have been integrated numerically. The associated constants of the motion have also been used to match the geodesics as they cross over the boundary between the single plane wave and interaction zones. 
  Interpretation of the cosmological red shift as light retardation with the amount of (d/dt)c = -Hc yields a photon rest mass hH/c^2. A system of natural units is introduced, in which the Planck mass is the geometric average of the photon rest mass and the universal mass. It is shown that the equivalence of expansion and light retardation results in a Cosmologic Uncertainty Principle (CUP), which determines the Heisenberg Principle. Within the Retarded Light Mode (RLM) the Dirac "coincidences" are found to be systematic, and an explanation is provided. 
  We propose a monitoring indicator of the normality of the output of a gravitational wave detector. This indicator is based on the estimation of the kurtosis (i.e., the 4th order statistical moment normalized by the variance squared) of the data selected in a time sliding window. We show how a low cost (because recursive) implementation of such estimation is possible and we illustrate the validity of the presented approach with a few examples using simulated random noises. 
  In spherically symmetric, static spacetime, we show that only j=1/2 fermions can satisfy both Einstein's field equation and Dirac's equation. It is also shown that neutrinos are able to have effective masses and cluster in the galactic halo when they are coupled to a global monopole situated at the galactic core. Astronomical implications of the results are discussed. 
  Several attempts to construct theories of gravity with variable mass are considered. The theoretical impacts of allowing the rest mass to vary with respect to time or an appropriate curve parameter are examined in the framework of Newtonian and Einsteinian gravity theories. In further steps, scalar-tensor theories are examined with respect to their relation to the variation of the mass and in an ultimate step, an additional coordinate is introduced and its possible relation to the mass is examined, yielding a five dimensional space-time-matter theory. 
  A simple model of spacetime foam, made by spherically symmetric wormholes, with or without a cosmological term is proposed. The black hole area quantization and its consequences are examined in this context. We open the possibility of probing Lorentz symmetry in this picture. 
  Starting from recent observations\cite{hod,dreyer1} about quasi-normal modes, we use semi-classical arguments to derive the Bekenstein-Hawking entropy spectrum for $d$-dimensional spherically symmetric black holes. We find that the entropy spectrum is equally spaced: $S_{BH}=k \ln(m_0)n$, where $m_0$ is a fixed integer that must be derived from the microscopic theory. As shown in \cite{dreyer1},4-$d$ loop quantum gravity yields precisely such a spectrum with $m_0=3$ providing the Immirzi parameter is chosen appropriately. For $d$-dimensional black holes of radius $R_H(M)$, our analysis requires the existence of a unique quasinormal mode frequency in the large damping limit $\omega^{(d)}(M) = \alpha^{(d)}c/ R_H(M)$ with coefficient $\alpha^{(d)} = {(d-3)/over 4\pi} \ln(m_0)$, where $m_0$ is an integer and $\Gamma^{(d-2)}$ is the volume of the unit $d-2$ sphere. 
  We examine the gravitational collapse of sphaleron type configurations in Einstein--Yang--Mills--Higgs theory. Working in spherical symmetry, we investigate the critical behavior in this model. We provide evidence that for various initial configurations, there can be three different critical transitions between possible endstates with different critical solutions sitting on the threshold between these outcomes. In addition, we show that within the dispersive and black hole regimes, there are new possible endstates, namely a stable, regular sphaleron and a stable, hairy black hole. 
  Some years ago Koutras presented a method of constructing a conformal Killing tensor from a pair of orthogonal conformal Killing vectors. When the vector associated with the conformal Killing tensor is a gradient, a Killing tensor (in general irreducible) can then be constructed. In this paper it is shown that the severe restriction of orthogonality is unnecessary and thus it is possible that many more Killing tensors can be constructed in this way. We also extend, and in one case correct, some results on Killing tensors constructed from a single conformal Killing vector. Weir's result that, for flat space, there are 84 independent conformal Killing tensors, all of which are reducible, is extended to conformally flat spacetimes. In conformally flat spacetimes it is thus possible to construct all the conformal Killing tensors and in particular all the Killing tensors (which in general will not be reducible) from conformal Killing vectors. 
  We start with a brief account of the latest analysis of the Oklo phenomenon providing the still most stringent constraint on time-variability of the fine- structure constant $\alpha$. Comparing this with the recent result from the measurement of distant QSO's appears to indicate a non-uniform time-dependence, which we argue to be related to another recent finding of the accelerating universe. This view is implemented in terms of the scalar-tensor theory, applied specifically to the small but nonzero cosmological constant. Our detailed calculation shows that these two phenomena can be understood in terms of a common origin, a particular behavior of the scalar field, dilaton. We also sketch how this theoretical approach makes it appropriate to revisit non- Newtonian gravity featuring small violation of Weak Equivalence Principle at medium distances. 
  We find the most general spherically symmetric non singular black hole solution in a special class of teleparallel theory of gravitation. If $r$ is large enough, the general solution coincides with the Schwarzschild solution. Whereas, if $r$ is small, the general solution behaves in a manner similar to that of de Sitter solution. Otherwise it describes a spherically symmetric black hole singularity free everywhere. Moreover, the energy associated with the general solution is calculated using the superpotential given by M{\o}ller 1978. 
  The reported anomalous acceleration acting on the Pioneers spacecrafts could be seen as a consequence of the existence of some local curvature in light geodesics when using the coordinate speed of light in an expanding space-time. The effect is related with the non synchronous character of the underlying metric and therefore, planets closed orbits can not reveal it. It is shown that the cosmic expansion rate -the Hubble parameter H- has been indeed detected. Additionally, a relation for an existing annual term is obtained which depends on the cosine of the ecliptic latitude of the spacecraft, suggestingan heuristic analogy between the effect and Foucault's experiment - light rays playing a similar role in the expanding space than Foucault's Pendulum does while determining Earth's rotation. This statement could be seen as a benchmark for future experiments. 
  Much work has been done after the possibility of a fine structure constant being time-varying. It has been taken as an indication of a time-varying speed of light. Here we prove that this is not the case. We prove that the speed of light may or may not vary with time, independently of the fine structure constant being constant or not. Time variations of the speed of light, if present, have to be derived by some other means and not from the fine structure constant. No implications based on the possible variations of the fine structure constant can be imposed on the speed of light. 
  In this paper, we consider the quantum area spectrum for a rotating and charged (Kerr-Newman) black hole. Generalizing a recent study on Kerr black holes (which was inspired by the static-black hole formalism of Barvinsky, Das and Kunstatter), we show that the quantized area operator can be expressed in terms of three quantum numbers (roughly related to the mass, charge and spin sectors). More precisely, we find that $A=8\pi\hbar[n+{1\over 2}+{p_1\over 2}+p_2]$, where $n$, $p_1$ and $p_2$ are strictly non-negative integers. In this way, we are able to confirm a uniformly spaced spectrum even for a fully general Kerr-Newman black hole. Along the way, we derive certain selection rules and use these to demonstrate that, in spite of appearances, the charge and spin spectra are not completely independent. 
  We study the 5-dimensional Einstein-Yang-Mills system with a cosmological constant. Assuming a spherically symmetric spacetime, we find a new analytic black hole solution, which approaches asymptotically "quasi-Minkowski", "quasi anti-de Sitter", or "quasi de Sitter" spacetime depending on the sign of a cosmological constant. Since there is no singularity except for the origin which is covered by an event horizon, we regard it as a localized object. This solution corresponds to a magnetically charged black hole.   We also present a singularity-free particle-like solution and a non-trivial black hole solution numerically. Those solutions correspond to the Bartnik-McKinnon solution and a colored black hole with a cosmological constant in the 4-dimensions. We analyze their asymptotic behaviors, spacetime structures and thermodynamical properties. We show that there is a set of stable solutions if a cosmological constant is negative. 
  The problem of defining and constructing representations of the Canonical Commutation Relations can be systematically approached via the technique of {\it algebraic quantization}. In particular, when the phase space of the system is linear and finite dimensional, the `vertical polarization' provides an unambiguous quantization. For infinite dimensional field theory systems, where the Stone-von Neumann theorem fails to be valid, even the simplest representation, the Schroedinger functional picture has some non-trivial subtleties. In this letter we consider the quantization of a real free scalar field --where the Fock quantization is well understood-- on an arbitrary background and show that the representation coming from the most natural application of the algebraic quantization approach is not, in general, unitary equivalent to the corresponding Schroedinger-Fock quantization. We comment on the possible implications of this result for field quantization. 
  Gravitational fields invariant for a 2-dimensional Lie algebra of Killing fields [ X,Y] =Y, with Y of light type, are analyzed. The conditions for them to represent gravitational waves are verified and the definition of energy and polarization is addressed; realistic generating sources are described. 
  Given a local quantum field theory net A on the de Sitter spacetime dS^d, where geodesic observers are thermalized at Gibbons-Hawking temperature, we look for observers that feel to be in a ground state, i.e. particle evolutions with positive generator, providing a sort of converse to the Hawking-Unruh effect. Such positive energy evolutions always exist as noncommutative flows, but have only a partial geometric meaning, yet they map localized observables into localized observables.   We characterize the local conformal nets on dS^d. Only in this case our positive energy evolutions have a complete geometrical meaning. We show that each net has a unique maximal expected conformal subnet, where our evolutions are thus geometrical.   In the two-dimensional case, we construct a holographic one-to-one correspondence between local nets A on dS^2 and local conformal non-isotonic families (pseudonets) B on S^1. The pseudonet B gives rise to two local conformal nets B(+/-) on S^1, that correspond to the H(+/-)-horizon components of A, and to the chiral components of the maximal conformal subnet of A. In particular, A is holographically reconstructed by a single horizon component, namely the pseudonet is a net, iff the translations on H(+/-) have positive energy and the translations on H(-/+) are trivial. This is the case iff the one-parameter unitary group implementing rotations on dS^2 has positive/negative generator. 
  We show that the torsion of a Cartan geometry can be associated to two spin-2 fields. This structure allows a new approach to deal with the proposal of geometrization of spin-2 fields besides the traditional one dealt with in General Relativity. We use the associated Hilbert-Einstein Lagrangian $R$ for generating a dynamics for the fields. 
  We develop an algebraic procedure to rotate a general Newman-Penrose tetrad in a Petrov type I spacetime into a frame with Weyl scalars $\Psi_{1}$ and $\Psi_{3}$ equal to zero, assuming that initially all the Weyl scalars are non vanishing. The new frame highlights the physical properties of the spacetime. In particular, in a Petrov Type I spacetime, setting $\Psi_{1}$ and $\Psi_{3}$ to zero makes apparent the superposition of a Coulomb-type effect $\Psi_{2}$ with transverse degrees of freedom $\Psi_{0}$ and $\Psi_{4}$. 
  We consider the operator product expansion for quantum field theories on general analytic 4-dimensional curved spacetimes within an axiomatic framework. We prove under certain general, model-independent assumptions that such an expansion necessarily has to be invariant under a simultaneous reversal of parity, time, and charge (PCT) in the following sense: The coefficients in the expansion of a product of fields on a curved spacetime with a given choice of time and space orientation are equal (modulo complex conjugation) to the coefficients for the product of the corresponding charge conjugate fields on the spacetime with the opposite time and space orientation. We propose that this result should be viewed as a replacement of the usual PCT theorem in Minkowski spacetime, at least in as far as the algebraic structure of the quantum fields at short distances is concerned. 
  In this work, focused on the production of exact inflationary solutions using dimensional analysis, it is shown how to explain inflation from a pragmatic and basic point of view, in a step-by-step process, starting from the one-dimensional harmonic oscillator. 
  One might raise a question if the gravitational scalar field (dilaton) mediates a finite-range force between local objects still behaving globally as being massless to implement the scenario of a decaying cosmological constant. We offer a non-negative reply by a detailed analysis of the field-theoretical quantization procedure in relation to the observationally required suppression of the vacuum-energy. 
  We compute the physical graviton two-point function in de Sitter spacetime with three-sphere spatial sections. We demonstrate that the large-distance growth present in the corresponding two-point function in spatially flat de Sitter spacetime is absent. We verify that our two-point function agrees with that in Minkowski spacetime in the zero cosmological constant limit. 
  Using a gauge-invariant formalism we derive and solve the perturbed cosmological equations for the BSBM theory of varying fine structure 'constant'. We calculate the time evolution of inhomogeneous perturbations of the fine structure constant, $\frac{\delta \alpha}{\alpha}$ on small and large scales with respect to the Hubble radius. In a radiation-dominated universe small inhomogeneities in $\alpha $ will decrease on large scales but on scales smaller than the Hubble radius they will undergo stable oscillations. In a dust-dominated universe small inhomogeneous perturbations in $\alpha $ will become constant on large scales and on small scales they will increase as $t^{2/3}$, and $\frac{\delta \alpha}{\alpha}$ will track $% \frac{\delta \rho _{m}}{\rho_{m}}$ . If the expansion accelerates, as in the case of a $\Lambda $ or quintessence-dominated phase, inhomogeneities in $\alpha $ will decrease on both large and small scales. The amplitude of perturbations in $\alpha $ will be much smaller than that of matter or radiation perturbations. We also present a numerical study of the non-linear evolution of spherical inhomogeneities in radiation and dust universes by means of a comparison between the evolution of flat and closed Friedmann models with time-varying $\alpha .$Various limitations of these simple models are also discussed. 
  We consider the application of the consistent lattice quantum gravity approach we introduced recently to the situation of a Friedmann cosmology and also to Bianchi cosmological models. This allows us to work out in detail the computations involved in the determination of the Lagrange multipliers that impose consistency, and the implications of this determination. It also allows us to study the removal of the Big Bang singularity. Different discretizations can be achieved depending on the version of the classical theory chosen as a starting point and their relationships studied. We analyze in some detail how the continuum limit arises in various models. In particular we notice how remnants of the symmetries of the continuum theory are embodied in constants of the motion of the consistent discrete theory. The unconstrained nature of the discrete theory allows the consistent introduction of a relational time in quantum cosmology, free from the usual conceptual problems. 
  A precise formulation of the strong Equivalence Principle is essential to the understanding of the relationship between gravitation and quantum mechanics. The relevant aspects are reviewed in a context including General Relativity, but allowing for the presence of torsion. For the sake of brevity, a concise statement is proposed for the Principle: "An ideal observer immersed in a gravitational field can choose a reference frame in which gravitation goes unnoticed". This statement is given a clear mathematical meaning through an accurate discussion of its terms. It holds for ideal observers (time-like smooth non-intersecting curves), but not for real, spatially extended observers. Analogous results hold for gauge fields. The difference between gravitation and the other fundamental interactions comes from their distinct roles in the equation of force. 
  The massless scalar field in the higher-dimensional Kerr black hole (Myers- Perry solution with a single rotation axis) has been investigated. It has been shown that the field equation is separable in arbitrary dimensions. The quasi-normal modes of the scalar field have been searched in five dimensions using the continued fraction method. The numerical result shows the evidence for the stability of the scalar perturbation of the five-dimensional Kerr black holes. The time scale of the resonant oscillation in the rapidly rotating black hole, in which case the horizon radius becomes small, is characterized by (black hole mass)^{1/2}(Planck mass)^{-3/2} rather than the light-crossing time of the horizon. 
  We consider the evolution of a 4D-universe embedded in a five-dimensional (bulk) world with a large extra dimension and a cosmological constant. The cosmology in 5D possesses "wave-like" character in the sense that the metric coefficients in the bulk are functions of the extra coordinate and time in a way similar to a pulse or traveling wave propagating along the fifth dimension. This assumption is motivated by some recent work presenting the big-bang as a higher dimensional shock wave. We show that this assumption, together with an equation of state for the effective matter quantities in 4D, allows Einstein's equations to be fully integrated. We then recover the familiar FLRW universes, on the four-dimensional hypersurfaces orthogonal to the extra dimension. Regarding the extra dimension we find that it is {\em growing} in size if the universe is speeding up its expansion. We also get an estimate for the relative change of the extra dimension over time. This estimate could have important observational implications, notably for the time variation of rest mass, electric charge and the gravitational "constant". Our results extend previous ones in the literature. 
  It is shown that the theory of dark matter can be derived from the first principles. Particles representing a new form of matter gravitate but do not interact electromagnetically, strongly and weakly with the known elementary particles. Physics of these particles is defined by the Planck scales. 
  The notions of centrifugal (centripetal) and Coriols' velocities and accelerations are introduced and considered in spaces with affine connections and metrics as velocities and accelerations of flows of mass elements (particles) moving in space-time. It is shown that these types of velocities and accelerations are generated by the relative motions between the mass elements. They are closely related to the kinematic characteristics of the relative velocity and the relative acceleration. The null (isotropic) vector fields are considered and their relations with the centrifugal (centripetal) velocity are established. The centrifugal (centripetal) velocity is found to be in connection with the Hubble law and the generalized Doppler effect in spaces with affine connections and metrics. The centrifugal (centripetal) acceleration could be interpreted as gravitational acceleration as it has been done in the Einstein theory of gravitation. This fact could be used as a basis for working out of new gravitational theories in spaces with affine connections and metrics. 
  I describe the construction of initial data for the Einstein vacuum equations that can represent a collision of two black holes. I stress in the main physical ideas. 
  We summarize results on the Penrose inequality bounding the ADM-mass or the Bondi mass in terms of the area of an outermost apparent horizon for asymptotically flat initial data of Einstein's equations. We first recall the proof, due to Geroch and to Jang and Wald, of monotonicity of the Geroch-Hawking mass under a smooth inverse mean curvature flow for data with non-negative Ricci scalar, which leads to a Penrose inequality if the apparent horizon is a minimal surface.We then sketch a proof of the Penrose inequality of Malec, Mars and Simon which holds for general horizons and for data satisfying the dominant energy condition, but imposes (in addition to smooth inverse mean curvature flow) suitable restrictions on the data on a spacelike surface. These conditions can, however, at least locally be fulfilled by a suitable choice of the initial surface in a given spacetime. Remarkably, they are also (formally) identical to ones employed earlier by Hayward in order to define a 2+1 foliation on null surfaces, with respect to which the Hawking mass is again monotonic. 
  Classical anti-commuting spinor fields and their dynamics are derived from the geometry of the Clifford bundle over spacetime via the BRST formulation. In conjunction with Kaluza-Klein theory, this results in a geometric description of all the fields and dynamics of the standard model coupled to gravity and provides the starting point for a new approach to quantum gravity. 
  A set of exact quasi-local conservation equations is obtained in the (1+1)-dimensional description of the Einstein's equations of (3+1)-dimensional spacetimes. These equations are interpreted as quasi-local energy, linear momentum, and angular momentum conservation equations. In the asymptotic region of asymptotically flat spacetimes, it is shown that these quasi-local conservation equations reduce to the conservation equations of Bondi energy, linear momentum, and angular momentum, respectively. When restricted to the quasi-local horizon of a generic spacetime, which is defined without referring to the infinity, the quasi-local conservation equations coincide with the conservation equations on the stretched horizon studied by Price and Thorne. All of these quasi-local quantities are expressed as invariant two-surface integrals, and geometrical interpretations in terms of the area of a given two-surface and a pair of null vector fields orthogonal to that surface are given. 
  The Sun's relativistic gravitational gradient accelerations of Earth and Moon, dependent on the motions of the latter bodies, act upon the system's internal angular momentum. This spin-orbit force (which plays a part in determining the gravity wave signal templates for astrophysical sources) slightly accelerates the Earth-Moon system as a whole, but it more robustly perturbs that system's internal dynamics with a 5 cm, synodically oscillating range contribution which is presently measured to 4 mm precision by more than three decades of lunar laser ranging. 
  Location dependence of physical parameters such as the electromagnetic fine structure constant and Newton's G produce body accelerations which violate universality of free fall rates testable with laboratory and space experiments. Theoretically related cosmological time variation of these same parameters are also constrained by experiments such as lunar laser ranging, and these time variations produce accelerations of bodies relative to a preferred cosmological frame. 
  In this paper we examine the Cosmic No-Hair Conjecture (CNHC) in brane world scenarios. For the validity of this conjecture, in addition to the strong and weak energy conditions for the matter field, a similar type of assumption is to be made on the quadratic correction term and there is a restriction on the non-local term. It is shown by examples with realistic fluid models that strong and weak energy conditions are sufficient for CNHC in brane world. 
  It is shown that a present acceleration with a past deceleration is a possible solution of the Friedmann equation by considering the Universe as a mixture of a scalar with a matter field and by including a non-equilibrium pressure term in the energy-momentum tensor. The dark energy density decays more slowly with respect to the time than the matter energy density does. The inclusion of the non-equilibrium pressure leads to a less pronounced decay of the matter field with a shorter period of past deceleration. 
  Conventional renormalization methods in statistical physics and lattice quantum field theory assume a flat metric background. We outline here a generalization of such methods to models on discretized spaces without metric background. Cellular decompositions play the role of discretizations. The group of scale transformations is replaced by the groupoid of changes of cellular decompositions. We introduce cellular moves which generate this groupoid and allow to define a renormalization groupoid flow.   We proceed to test our approach on several models. Quantum BF theory is the simplest example as it is almost topological and the renormalization almost trivial. More interesting is generalized lattice gauge theory for which a qualitative picture of the renormalization groupoid flow can be given. This is confirmed by the exact renormalization in dimension two.   A main motivation for our approach are discrete models of quantum gravity. We investigate both the Reisenberger and the Barrett-Crane spin foam model in view of their amenability to a renormalization treatment. In the second case a lack of tunable local parameters prompts us to introduce a new model. For the Reisenberger and the new model we discuss qualitative aspects of the renormalization groupoid flow. In both cases quantum BF theory is the UV fixed point. 
  In the frame of the Kerr-Schild approach, we consider the complex structure of Kerr geometry which is determined by a complex world line of a complex source. The real Kerr geometry is represented as a real slice of this complex structure. The Kerr geometry is generalized to the nonstationary case when the current geometry is determined by a retarded time and is defined by a retarded-time construction via a given complex world line of source. A general exact solution corresponding to arbitrary motion of a spinning source is obtained. The acceleration of the source is accompanied by a lightlike radiation along the principal null congruence. It generalizes to the rotating case the known Kinnersley class of "photon rocket" solutions. 
  We discuss various features and details of two versions of the Barrett-Crane spin foam model of quantum gravity, first of the Spin(4)-symmetric Riemannian model and second of the SL(2,C)-symmetric Lorentzian version in which all tetrahedra are space-like. Recently, Livine and Oriti proposed to introduce a causal structure into the Lorentzian Barrett--Crane model from which one can construct a path integral that corresponds to the causal (Feynman) propagator. We show how to obtain convergent integrals for the 10j-symbols and how a dimensionless constant can be introduced into the model. We propose a `Wick rotation' which turns the rapidly oscillating complex amplitudes of the Feynman path integral into positive real and bounded weights. This construction does not yet have the status of a theorem, but it can be used as an alternative definition of the propagator and makes the causal model accessible by standard numerical simulation algorithms. In addition, we identify the local symmetries of the models and show how their four-simplex amplitudes can be re-expressed in terms of the ordinary relativistic 10j-symbols. Finally, motivated by possible numerical simulations, we express the matrix elements that are defined by the model, in terms of the continuous connection variables and determine the most general observable in the connection picture. Everything is done on a fixed two-complex. 
  Self-gravitating scalar fields with nonminimal coupling to gravity and having a quartic self-interaction are considered in the domain of outer communications of a static black hole. It is shown that there is no value of the nonminimal coupling parameter $\zeta$ for which nontrivial static black hole solutions exist. This result establishes the correctness of Bekenstein ``no-scalar-hair'' conjecture for quartic self-interactions. 
  A detailed study of the Counter-Rotating Model (CRM) for generic electrostatic (magnetostatic) axially symmetric thin disks without radial pressure is presented. We find a general constraint over the counter-rotating tangential velocities needed to cast the surface energy-momentum tensor of the disk as the superposition of two counter-rotating charged dust fluids. We then show that this constraint is satisfied if we take the two counter-rotating streams as circulating along electrogeodesics with equal and opposite tangential velocities. We also find explicit expressions for the energy densities, electrostatic (magnetostatic) charge densities and velocities of the counter-rotating fluids. Three specific examples are considered where we obtain some CRM well behaved based in simple solutions to the Einstein-Maxwell equations. The considered solutions are Reissner-Nordstrom in the electrostatic case, its magnetostatic counterpart and two solutions obtained from Taub-NUT and Kerr solutions. 
  Rotating disks with nonzero radial pressure and finite radius are studied. The models are based in the Taub-NUT metric and constructed using the well-known ``displace, cut and reflect'' method. We find that the disks are made of perfect fluids with constant energy density and pressure. The energy density is negative, but the effective Newotnian density is possitive as the strong energy condition requires. We also find that the disks are not stable under radial perturbations and that there are regions of the disks where the particles move with superluminal velocities. 
  On incorporating special relativity theory into an extended equivalence principle, post-Newtonian gravitational phenomena beyond that originally predicted by Einstein are predicted (required), such as geodetic and gravitomagnetic precessions of local inertial frames, and precession of Mercury's orbital perihelion. Why were not these phenomena predicted in the years 1907-1911? The unique 1/c^2 order dynamical equations for clock rates and motion of both bodies and light in local gravity are derived which guarantee fulfillment of the special relativistic equivalence principle. 
  The key problem of the theory of exterior differential systems (EDS) is to decide whether or not a system is in involution. The special case of EDSs generated by one-forms (Pfaffian systems) can be adequately illustrated by a 2-dimensional example. In 4 dimensions two such problems arise in a natural way, namely, the Riemann-Lanczos and the Weyl-Lanczos problems. It is known from the work of Bampi and Caviglia that the Weyl-Lanczos problem is always in involution in both 4 and 5 dimensions but that the Riemann-Lanczos problem fails to be in involution even for 4 dimensions. However, singular solutions of it can be found. We give examples of singular solutions for the Goedel, Kasner and Debever-Hubaut spacetimes. It is even possible that the singular solution can inherit the spacetime symmetries as in the Debever-Hubaut case. We comment on the Riemann-Lanczos problem in 5 dimensions which is neither in involution nor does it admit a 5-dimensional involution of Vessiot vector fields in the generic case. 
  The Riemann-Lanczos problem for 4-dimensional manifolds was discussed by Bampi and Caviglia. Using exterior differential systems they showed that it was not an involutory differential system until a suitable prolongation was made. Here, we introduce the alternative Janet-Riquier theory and use it to consider the Riemann-Lanczos problem in 2 and 3 dimensions. We find that in 2 dimensions, the Riemann-Lanczos problem is a differential system in involution. It depends on one arbitrary function of 2 independent variables when no differential gauge condition is imposed but on 2 arbitrary functions of one independent variable when the differential gauge condition is imposed. For each of the two possible signatures we give the general solution in both instances to show that the occurrence of characteristic coordinates need not affect the result. In 3 dimensions, the Riemann-Lanczos problem is not in involution as a identity occurs. This does not prevent the existence of singular solutions. A prolongation of this problem, where an integrability condition is added, leads to an involutory prolonged system and thereby generates non-singular solutions of the prolonged Riemann-Lanczos problem. We give a singular solution for the unprolonged Riemann-Lanczos problem for the 3-dimensional reduced Goedel spacetime. 
  Using the work by Bampi and Caviglia, we write the Weyl-Lanczos equations as an exterior differential system. Using Janet-Riquier theory, we compute the Cartan characters for all spacetimes with a diagonal metric and for the plane wave spacetime since all spacetimes have a plane wave limit. We write the Lanczos wave equation as an exterior differential system and, with assistance from Janet-Riquier theory, we find that it forms a system in involution. This result can be derived from the scalar wave equation itself. We compute its Cartan characters and compare them with those of the Weyl-Lanczos equations. 
  Dilaton black hole solutions which are neither asymptotically flat nor (anti)-de Sitter but reduce to asymptotically flat solutions in some special limits have been known for a Liouville type dilatonic potential. It is shown how, by solving a pair of coupled differential equations, infinitesimally small angular momentum can be added to these static solutions to produce rotating black hole solutions. 
  We consider brane-world models embedded in a five-dimensional bulk spacetime with a large extra dimension and a cosmological constant. The cosmology in $5D$ possesses "wave-like" character in the sense that the metric coefficients in the bulk are assumed to have the form of plane waves propagating in the fifth dimension. We model the brane as the "plane" of collision of waves propagating in opposite directions along the extra dimension. This plane is a jump discontinuity which presents the usual ${\bf Z}_2$ symmetry of brane models. The model reproduces the {\em generalized} Friedmann equation for the evolution on the brane, regardless of the specific details in $5D$. Model solutions with spacelike extra coordinate show the usual {\em big-bang} behavior, while those with timelike extra dimension present a {\em big bounce}. This bounce is an genuine effect of a timelike extra dimension. We argue that, based on our current knowledge, models having a large timelike extra dimension cannot be dismissed as mathematical curiosities in non-physical solutions. The size of the extra dimension is small today, but it is {\em increasing} if the universe is expanding with acceleration. Also, the expansion rate of the fifth dimension can be expressed in a simple way through the four-dimensional "deceleration" and Hubble parameters as $- q H$. These predictions could have important observational implications, notably for the time variation of rest mass, electric charge and the gravitational "constant". They hold for the three $(k = 0, + 1, - 1)$ models with arbitrary cosmological constant, and are independent of the signature of the extra dimension. 
  The topological structure of Schwarzschild's space-time and its maximal analytic extension are investigated in context of brane-worlds. Using the embedding coordinates, these geometries are seen as different states of the evolution of a single brane-world. Comparing the topologies and the embeddings it is shown that this evolution must be followed by a signature change in the bulk. 
  We consider the thermodynamics of minimally coupled massive scalar field in 3+1 dimensional constant curvature black hole background. The brick wall model of 't Hooft is used. When Schwarzschild like coordinates are used it is found that two radial brick wall cut-off parameters are required to regularize the solution. Free energy of the scalar field is obtained through counting of states using the WKB approximation. It is found that the free energy and the entropy are divergent in both the cut-off parameters. 
  We investigate the occurrence and nature of a naked singularity in the gravitational collapse of an inhomogeneous dust cloud described by higher dimensional Tolman-Bondi space-time. The naked singularities are found to be gravitationally strong. 
  We investigate the end state of the gravitational collapse of an inhomogeneous dust cloud in higher dimensional space-time. The naked singularities are shown to be developing as the final outcome of non-marginally bound collapse. The naked singularities are found to be gravitationally strong in the sense of Tipler . 
  We shall investigate $D$-dimensional Lorentzian spacetimes in which all of the scalar invariants constructed from the Riemann tensor and its covariant derivatives are zero. These spacetimes are higher-dimensional generalizations of $D$-dimensional pp-wave spacetimes, which have been of interest recently in the context of string theory in curved backgrounds in higher dimensions. 
  The Causal Set approach to quantum gravity asserts that spacetime, at its smallest length scale, has a discrete structure. This discrete structure takes the form of a locally finite order relation, where the order, corresponding with the macroscopic notion of spacetime causality, is taken to be a fundamental aspect of nature.   After an introduction to the Causal Set approach, this thesis considers a simple toy dynamics for causal sets. Numerical simulations of the model provide evidence for the existence of a continuum limit. While studying this toy dynamics, a picture arises of how the dynamics can be generalized in such a way that the theory could hope to produce more physically realistic causal sets. By thinking in terms of a stochastic growth process, and positing some fundamental principles, we are led almost uniquely to a family of dynamical laws (stochastic processes) parameterized by a countable sequence of coupling constants. This result is quite promising in that we now know how to speak of dynamics for a theory with discrete time. In addition, these dynamics can be expressed in terms of state models of Ising spins living on the relations of the causal set, which indicates how non-gravitational matter may arise from the theory without having to be built in at the fundamental level.   These results are encouraging in that there exists a natural way to transform this classical theory, which is expressed in terms of a probability measure, to a quantum theory, expressed in terms of a quantum measure. A sketch as to how one might proceed in doing this is provided. Thus there is good reason to expect that Causal Sets are close to providing a background independent theory of quantum gravity. 
  Global properties of static, spherically symmetric configurations of scalar fields of sigma-model type with arbitrary potentials are studied in $D$ dimensions, including space-times containing multiple internal factor spaces. The latter are assumed to be Einstein spaces, not necessarily Ricci-flat, and the potential $V$ includes contributions from their curvatures. The following results generalize those known in four dimensions: (A) a no-hair theorem: in case $V\geq 0$, an asymptotically flat black hole cannot have varying scalar fields or moduli fields outside the event horizon; (B) nonexistence of particlelike solutions in models with $V\geq 0$; (C) nonexistence of wormholes under very general conditions; (D) a restriction on possible global causal structures (represented by Carter-Penrose diagrams). The list of structures in all models under consideration is the same as is known for vacuum with a cosmological constant in general relativity: Minkowski (or AdS), Schwarzschild, de Sitter and Schwarzschild--de Sitter, and horizons which bound a static region are always simple. The results are applicable to a wide range of Kaluza-Klein, supergravity and stringy models with multiple dilaton and moduli fields. 
  New methods are presented which enables one to analyze the thermodynamics of systems with long-range interactions. Generically, such systems have entropies which are non-extensive, (do not scale with the size of the system). We show how to calculate the degree of non-extensivity for such a system. We find that a system interacting with a heat reservoir is in a probability distribution of canonical ensembles. The system still possesses a parameter akin to a global temperature, which is constant throughout the substance. There is also a useful quantity which acts like a {\it local temperatures} and it varies throughout the substance. These quantities are closely related to counterparts found in general relativity. A lattice model with long-range spin-spin coupling is studied. This is compared with systems such as those encountered in general relativity, and gravitating systems with Newtonian-type interactions. A long-range lattice model is presented which can be seen as a black-hole analog. One finds that the analog's temperature and entropy have many properties which are found in black-holes. Finally, the entropy scaling behavior of a gravitating perfect fluid of constant density is calculated. For weak interactions, the entropy scales like the volume of the system. As the interactions become stronger, the entropy becomes higher near the surface of the system, and becomes more area-scaling. 
  We investigate the occurrence and nature of a naked singularity in the gravitational collapse of an inhomogeneous dust cloud described by a self-similar higher dimensional Tolman-Bondi space-time. Bound, marginally bound and unbound space-times are analyzed. The degree of inhomogeneity of the collapsing matter necessary to form a naked singularity is given. 
  Methods of dynamical systems have been used to study homogeneous and isotropic cosmological models with a varying speed of light (VSL). We propose two methods of reduction of dynamics to the form of planar Hamiltonian dynamical systems for models with a time dependent equation of state. The solutions are analyzed on two-dimensional phase space in the variables $(x, \dot{x})$ where $x$ is a function of a scale factor $a$. Then we show how the horizon problem may be solved on some evolutional paths. It is shown that the models with negative curvature overcome the horizon and flatness problems. The presented method of reduction can be adopted to the analysis of dynamics of the universe with the general form of the equation of state $p=\gamma(a)\epsilon$. This is demonstrated using as an example the dynamics of VSL models filled with a non-interacting fluid. We demonstrate a new type of evolution near the initial singularity caused by a varying speed of light. The singularity-free oscillating universes are also admitted for positive cosmological constant. We consider a quantum VSL FRW closed model with radiation and show that the highest tunnelling rate occurs for a constant velocity of light if $c(a) \propto a^n$ and $-1 < n \le 0$. It is also proved that the considered class of models is structurally unstable for the case of $n < 0$. 
  We carry out a covariant calculation of the measurable relativistic effects in an orbiting gyroscope experiment. The experiment, currently known as Gravity Probe B, compares the spin directions of an array of spinning gyroscopes with the optical axis of a telescope, all housed in a spacecraft that rolls about the optical axis. The spacecraft is steered so that the telescope always points toward a known guide star. We calculate the variation in the spin directions relative to readout loops rigidly fixed in the spacecraft, and express the variations in terms of quantities that can be measured, to sufficient accuracy, using an Earth-centered coordinate system. The measurable effects include the aberration of starlight, the geodetic precession caused by space curvature, the frame-dragging effect caused by the rotation of the Earth and the deflection of light by the Sun. 
  The basic concepts and equations of classical fluid mechanics are presented in the form necessary for the formulation of Newtonian cosmology and for derivation and analysis of a system of the averaged Navier-Stokes-Poisson equations. A special attention is paid to the analytic formulation of the definitions and equations of moving fluids and to their physical content. 
  The basic concepts and equations of Newtonian Cosmology are presented in the form necessary for the derivation and analysis of the averaged Navier-Stokes-Poisson equations. A particular attention is paid to the physical and cosmological hypotheses about the structure of Newtonian universes. The system of the Navier-Stokes-Poisson equations governing the cosmological dynamics of Newtonian universes is presented and discussed. A reformulation of the Navier-Stokes-Poisson equations in terms of the fluid kinematic quantities is given and the structure of this system of equations is analyzed. 
  The basic concepts and hypotheses of Newtonian Cosmology necessary for a consistent treatment of the averaged cosmological dynamics are formulated and discussed in details. The space-time, space, time and ensemble averages for the cosmological fluid fields are defined and analyzed with a special attention paid to their analytic properties. It is shown that all averaging procedures require an arrangement for a standard measurement device with the same measurement time interval and the same space region determined by the measurement device resolution to be prescribed to each position and each moment of time throughout a cosmological fluid configuration. The formulae for averaging out the partial derivatives of the averaged cosmological fluid fields and the main formula for averaging out the material derivatives have been proved. The full system of the averaged Navier-Stokes-Poisson equations in terms of the fluid kinematic quantities is derived. 
  A particular yet large class of non-diverging solutions which admits a cosmological constant, electromagnetic field, pure radiation and/or general non-null matter component is explicitly presented. These spacetimes represent exact gravitational waves of arbitrary profiles which propagate in background universes such as Minkowski, conformally flat (anti-)de Sitter, Edgar-Ludwig, Bertotti-Robinson, and type D (anti-)Nariai or Plebanski-Hacyan spaces, and their generalizations. All possibilities are discussed and are interpreted using a unifying simple metric form. Sandwich and impulsive waves propagating in the above background spaces with different geometries and matter content can easily be constructed. New solutions are identified, e.g. type D pure radiation or explicit type II electrovacuum waves in (anti-)Nariai universe. It is also shown that, in general, there are no conformally flat Einstein-Maxwell fields with a non-vanishing cosmological constant. 
  We study the Unruh effect for an observer with a finite lifetime, using the thermal time hypothesis. The thermal time hypothesis maintains that: (i) time is the physical quantity determined by the flow defined by a state over an observable algebra, and (ii) when this flow is proportional to a geometric flow in spacetime, temperature is the ratio between flow parameter and proper time. An eternal accelerated Unruh observer has access to the local algebra associated to a Rindler wedge. The flow defined by the Minkowski vacuum of a field theory over this algebra is proportional to a flow in spacetime and the associated temperature is the Unruh temperature. An observer with a finite lifetime has access to the local observable algebra associated to a finite spacetime region called a "diamond". The flow defined by the Minkowski vacuum of a (four dimensional, conformally invariant) quantum field theory over this algebra is also proportional to a flow in spacetime. The associated temperature generalizes the Unruh temperature to finite lifetime observers.   Furthermore, this temperature does not vanish even in the limit in which the acceleration is zero. The temperature associated to an inertial observer with lifetime T, which we denote as "diamond's temperature", is 2hbar/(pi k_b T).This temperature is related to the fact that a finite lifetime observer does not have access to all the degrees of freedom of the quantum field theory. 
  We consider the dynamics of a viscous cosmological fluid in the generalized Randall-Sundrum model for an anisotropic, Bianchi type I brane. To describe the dissipative effects we use the Israel-Hiscock-Stewart full causal thermodynamic theory. By assuming that the matter on the brane obeys a linear barotropic equation of state, and the bulk viscous pressure has a power law dependence on the energy density, the general solution of the field equations can be obtained in an exact parametric form. The obtained solutions describe generally a non-inflationary brane world. In the large time limit the brane Universe isotropizes, ending in an isotropic and homogeneous state. The evolution of the temperature and of the comoving entropy of the Universe is also considered, and it is shown that due to the viscous dissipative processes a large amount of entropy is created in the early stages of evolution of the brane world. 
  Different approaches are compared to formulation of quantum mechanics of a particle on the curved spaces. At first, the canonical, quasi-classical and path integration formalisms are considered for quantization of geodesic motion on the Rimannian configurational spaces. A unique rule of ordering of operators in the canonical formalism and a unique definition of the path integral are established and, thus, a part of ambiguities in the quantum counterpart of geodesic motion is removed. A geometric interpretation is proposed for non-invariance of the quantum mechanics on coordinate transformations. An approach alternative to the quantization of geodesic motion is surveyed, which starts with the quantum theory of a neutral scalar field. Consequences of this alternative approach and the three formalisms of quantization are compared. In particular, the field theoretical approach generates a deformation of the canonical commutation relations between coordinates and momenta of a prticle. A possible cosmological consequence of the deformation is presented in short. Key words: quantum mechanics, Riemannian space, geodesic motion, deformation. 
  We study the spectrum of the length and area operators in Lorentzian loop quantum gravity, in 2+1 spacetime dimensions. We find that the spectrum of spacelike intervals is continuous, whereas the spectrum of timelike intervals is discrete. This result contradicts the expectation that spacelike intervals are always discrete. On the other hand, it is consistent with the results of the spin foam quantization of the same theory. 
  A new approach to the description of spin-2 particle in flat and curved spacetime is developed on the basis of the teleparallel gravity theory. We show that such an approach is in fact a true and natural framework for the Fierz representation proposed recently by Novello and Neves. More specifically, we demonstrate how the teleparallel theory fixes uniquely the structure of the Fierz tensor, discover the transparent origin of the gauge symmetry of the spin 2 model, and derive the linearized Einstein operator from the fundamental identity of the teleparallel gravity. In order to cope with the consistency problem on the curved spacetime, similarly to the usual Riemannian approach, one needs to include the non-minimal (torsion dependent) coupling terms. 
  Assuming certain asymptotic conditions, we prove a general theorem on the non-existence of static regular (i.e., nondegenerate) black holes in spacetimes with a negative cosmological constant, given that the fundamental group of space is infinite. We use this to rule out the existence of regular negative mass AdS black holes with Ricci flat scri. For any mass, we also rule out a class of conformally compactifiable static black holes whose conformal infinity has positive scalar curvature and infinite fundamental group, subject to our asymptotic conditions. In a limited, but important, special case our result adds new support to the AdS/CFT inspired positive mass conjecture of Horowitz and Myers. 
  The teleparallel gravity theory, treated physically as a gauge theory of translations, naturally represents a particular case of the most general gauge-theoretic model based on the general affine group of spacetime. On the other hand, geometrically, the Weitzenboeck spacetime of distant parallelism is a particular case of the general metric-affine spacetime manifold. These physical and geometrical facts offer a new approach to the teleparallelism. We present a systematic treatment of the teleparallel gravity within the framework of the metric-affine theory. The symmetries, conservation laws and the field equations are consistently derived, and the physical consequences are discussed in detail. We demonstrate that the so-called teleparallel GR-equivalent model has a number of attractive features which distinguishes it among the general teleparallel theories, although it has a consistency problem when dealing with spinning matter sources. 
  We show that quantum noise in very sensitive interferometric measurements such as gravitational-waves detectors can be drastically modified by quantum feedback. We present a new scheme based on active control to lock the motion of a mirror to a reference mirror at the quantum level. This simple technique allows to reduce quantum effects of radiation pressure and to greatly enhance the sensitivity of the detection. 
  A theorem, giving necessary and sufficient condition for naked singularity formation in spherically symmetric non static spacetimes under hypotheses of physical acceptability, is formulated and proved. The theorem relates existence of singular null geodesics to existence of regular curves which are super-solutions of the radial null geodesic equation, and allows us to treat all the known examples of naked singularities from a unified viewpoint. New examples are also found using this approach, and perspectives are discussed. 
  General relativistic superfluid neutron stars have a significantly more intricate dynamics than their ordinary fluid counterparts. Superfluidity allows different superfluid (and superconducting) species of particles to have independent fluid flows, a consequence of which is that the fluid equations of motion contain as many fluid element velocities as superfluid species. Whenever the particles of one superfluid interact with those of another, the momentum of each superfluid will be a linear combination of both superfluid velocities. This leads to the so-called entrainment effect whereby the motion of one superfluid will induce a momentum in the other superfluid. We have constructed a fully relativistic model for entrainment between superfluid neutrons and superconducting protons using a relativistic $\sigma - \omega$ mean field model for the nucleons and their interactions. In this context there are two notions of ``relativistic'': relativistic motion of the individual nucleons with respect to a local region of the star (i.e. a fluid element containing, say, an Avogadro's number of particles), and the motion of fluid elements with respect to the rest of the star. While it is the case that the fluid elements will typically maintain average speeds at a fraction of that of light, the supranuclear densities in the core of a neutron star can make the nucleons themselves have quite high average speeds within each fluid element. The formalism is applied to the problem of slowly-rotating superfluid neutron star configurations, a distinguishing characteristic being that the neutrons can rotate at a rate different from that of the protons. 
  We consider the response of a uniformly accelerated monopole detector that is coupled to a superposition of an odd and an even power of a quantized, massless scalar field in flat spacetime in arbitrary dimensions. We show that, when the field is assumed to be in the Minkowski vacuum, the response of the detector is characterized by a Bose-Einstein factor in even spacetime dimensions, whereas a Bose-Einstein as well as a Fermi-Dirac factor appear in the detector response when the dimension of spacetime is odd. Moreover, we find that, it is possible to interpolate between the Bose-Einstein and the Fermi-Dirac distributions in odd spacetime dimensions by suitably adjusting the relative strengths of the detector's coupling to the odd and the even powers of the scalar field. We point out that the response of the detector is always thermal and we, finally, close by stressing the apparent nature of the appearance of the Fermi-Dirac factor in the detector response. 
  The Rainich problem for the Killing-Yano tensors posed by Collinson \cite{col} is solved. In intermediate steps, we first obtain the necessary and sufficient conditions for a 2+2 almost-product structure to determine the principal 2--planes of a skew-symmetric Killing-Yano tensor and then we give the additional conditions on a symmetric Killing tensor for it to be the square of a Killing-Yano tensor.We also analyze a similar problem for the conformal Killing-Yano and the conformal Killing tensors. Our results show that, in both cases, the principal 2--planes define a maxwellian structure. The associated Maxwell fields are obtained and we outline how this approach is of interest in studying the spacetimes that admit these kind of first integrals of the geodesic equation. 
  We give a classification of the type D spacetimes based on the invariant differential properties of the Weyl principal structure. Our classification is established using tensorial invariants of the Weyl tensor and, consequently, besides its intrinsic nature, it is valid for the whole set of the type D metrics and it applies on both, vacuum and non-vacuum solutions. We consider the Cotton-zero type D metrics and we study the classes that are compatible with this condition. The subfamily of spacetimes with constant argument of the Weyl eigenvalue is analyzed in more detail by offering a canonical expression for the metric tensor and by giving a generalization of some results about the non-existence of purely magnetic solutions. The usefulness of these results is illustrated in characterizing and classifying a family of Einstein-Maxwell solutions. Our approach permits us to give intrinsic and explicit conditions that label every metric, obtaining in this way an operational algorithm to detect them. In particular a characterization of the Reissner-Nordstr\"{o}m metric is accomplished. 
  It is shown that, for systems in which the entropy is an extensive function of the energy and volume, the Bekenstein and the holographic entropy bounds predict new results. More explicitly, the Bekenstein entropy bound leads to the entropy of thermal radiation (the Unruh-Wald bound) and the spherical entropy bound implies the "causal entropy bound". Surprisingly, the first bound shows a close relationship between black hole physics and the Stephan-Boltzmann law (for the energy and entropy flux densities of the radiation emitted by a hot blackbody). Furthermore, we find that the number of different species of massless fields is bounded by $\sim 10^{4}$. 
  Operating resonant mass detectors set interesting bounds on diffused backgrounds of gravitational radiation and in the next five years the wide-band interferometers will also look for stochastic sources. In this lecture the interplay among relic GW backgrounds and large scale magnetic fields will be discussed. Magnetic fields may significantly affect the thermal history of the Universe in particular at the epoch electroweak symmetry breaking and shortly after. A review of some old an new results on the spectral properties of stochastic GW backgrounds will be presented. The possible r\^ole of primordial magnetic fields as a source of gravitational radiation will be outlined. It will be shown that the usual bound on stochastic GW backgrounds coming from the standard big bang nucleosynthesis (BBN) scenario can be significantly relaxed. 
  We examine the gravitational collapse of an infinite cylindrical distribution of time like dust. In order to simplify the calculation we make an assumption that the axial and azimuthal metric functions are equal. It is shown that the resulting solution describes homogeneous collapse. We show that the interior metric can be matched to a time dependent exterior. We also discuss the nature of the singularity in the matter region and show that it is covered. 
  One may ask whether an extended group of invariance can naturally be attributed to the space of associative commutative Quadrahyperbolic Numbers? To search for a rigorous and positive answer to the question, we shall focus on the method of derivation of the respective invariance. The outcome that there exist 3--parametric nonlinear transformations which leave invariant the scalar product chosen appropriately for The Quadrahyperbolic Numbers, is the main result of the present publication. 
  We consider the gravitational properties of a global monopole on the basis of the simplest Higgs scalar triplet model in general relativity. We begin with establishing some common features of hedgehog-type solutions with a regular center, independent of the choice of the symmetry-breaking potential. There are six types of qualitative behavior of the solutions; we show, in particular, that the metric can contain at most one simple horizon. For the standard Mexican hat potential, the previously known properties of the solutions are confirmed and some new results are obtained. Thus, we show analytically that solutions with monotonically growing Higgs field and finite energy in the static region exist only in the interval $1<\gamma <3$, $\gamma $ being the squared energy of spontaneous symmetry breaking in Planck units. The cosmological properties of these globally regular solutions apparently favor the idea that the standard Big Bang might be replaced with a nonsingular static core and a horizon appearing as a result of some symmetry-breaking phase transition on the Planck energy scale. In addition to the monotonic solutions, we present and analyze a sequence of families of new solutions with oscillating Higgs field. These families are parametrized by $n$, the number of knots of the Higgs field, and exist for $\gamma < \gamma_n = 6/[(2n+1) (n+2)]$; all such solutions possess a horizon and a singularity beyond it. 
  We investigated dynamics of the test particle in the gravitational field of the charged black hole with dipoles in this paper. At first we have studied the gravitational potential, by the numerical simulations, we found, for appropriate parameters, that there are two different cases in the potential curve, one is a well case with a stable critical point, and the other is three wells case with three stable critical points and two unstable critical points. As consequence, the chaotic motion will rise. We have performed the evolution of the orbits of the test particle in phase space, we found that the orbits of the test particle randomly oscillate without any periods, even sensitively depend on the initial conditions and parameters. By performing Poincar\'{e} sections for different values of the parameters and initial condition, we have found regular motion and chaotic motion. By comparing these Poincar\'{e} sections, we further conformed that the chaotic motion of the test particle mainly origins from the dipoles of the black hole. 
  By using the complex angular momentum approach, we prove that the quasinormal mode complex frequencies of the Schwarzschild black hole are Breit-Wigner type resonances generated by a family of surface waves propagating close to the unstable circular photon (graviton) orbit at $r=3M$. Furthermore, because each surface wave is associated with a given Regge pole of the $S$-matrix, we can construct the spectrum of the quasinormal-mode complex frequencies from Regge trajectories. The notion of surface wave orbiting around black holes thus appears as a fundamental concept which could be profitably introduced in various areas of black hole physics in connection with the complex angular momentum approach. 
  We shall investigate the asymptotic properties of the Bianchi type IX cosmological model in the Brane-world scenario. The matter content is assumed to be a combination of a perfect fluid and a minimimally coupled scalar field that is restricted to the Brane. A detailed qualitative analysis of the Bianchi type IX braneworld containing a scalar field having an exponential potential is undertaken. It is found that the Brane-Robertson Walker solution is a local source for the expanding Bianchi IX models, and if $k^2<2$ the expanding Bianchi IX models asymptote to the power-law inflationary solution. The only other local sink is the contracting Brane-Robertson Walker solution. An analysis of the Bianchi IX models with a scalar field with a general potential is discussed, and it is shown that in the case of expanding models, for physical scalar field potentials close to the initial singularity, the scalar field is effectively massless, and the solution is approximated by the Brane-Robertson Walker model. 
  The asymptotic properties of the Bianchi type II cosmological model in the Brane-world scenario are investigated. The matter content is assumed to be a combination of a perfect fluid and a minimimally coupled scalar field that is restricted to the Brane. The isotropic braneworld solution is determined to represent the initial singularity in all brane-world cosmologies. Additionally, it is shown that it is the kinetic energy of the scalar field which dominates the initial dynamics in these brane-world cosmologies. It is important to note that, the dynamics of these brane-world cosmologies is not necessarily asymptotic to general relativistic cosmologies to the future in the case of a zero four-dimensional cosmological constant. 
  Recently it has been proposed that a strange logarithmic expression for the so-called Barbero-Immirzi parameter, which is one of the ingredients that are necessary for Loop Quantum Gravity (LQG) to predict the correct black hole entropy, is not another sign of the inconsistency of this approach to quantization of General Relativity, but is rather a meaningful number that can be independently justified in classical GR. The alternative justification involves the knowledge of the real part of the frequencies of black hole quasinormal states whose imaginary part blows up. In this paper we present an analytical derivation of the states with frequencies approaching a large imaginary number plus ln 3 / 8 pi M; this constant has been only known numerically so far. We discuss the structure of the quasinormal states for perturbations of various spin. Possible implications of these states for thermal physics of black holes and quantum gravity are mentioned and interpreted in a new way. A general conjecture about the asymptotic states is stated. Although our main result lends some credibility to LQG, we also review some of its claims in a critical fashion and speculate about its possible future relevance for Quantum Gravity. 
  We consider here the question if it is possible to recover cosmic censorship when a transition is made to higher dimensional spacetimes, by studying the spherically symmetric dust collapse in an arbitrary higher spacetime dimension. It is pointed out that if only black holes are to result as end state of a continual gravitational collapse, several conditions must be imposed on the collapsing configuration, some of which may appear to be restrictive, and we need to study carefully if these can be suitably motivated physically in a realistic collapse scenario. It would appear that in a generic higher dimensional dust collapse, both black holes and naked singularities would develop as end states as indicated by the results here. The mathematical approach developed here generalizes and unifies the earlier available results on higher dimensional dust collapse as we point out. Further, the dependence of black hole or naked singularity end states as collapse outcomes, on the nature of the initial data from which the collapse develops, is brought out explicitly and in a transparent manner as we show here. Our method also allows us to consider here in some detail the genericity and stability aspects related to the occurrence of naked singularities in gravitational collapse. 
  A generation procedure, based on the 5-dimensional covariance of the Kaluza-Klein theory, is developed. The procedure allows one to obtain exact solutions of the 4-dimensional Einstein equations with electromagnetic and scalar fields from vacuum 5-dimensional solutions using special 5-dimensional coordinate transformations. Relations between the physical properties of the resulting solutions and invariant geometrical properties of the generating Killing vectors are found out. 
  In some black hole solutions, these do not exist the same energy-momentum complexes associated with using definition of Einstein and M{\o}ller in given coordinates. Here, we consider the difference of energy between the Einstein and M{\o}ller prescription, and compare it with the energy density of those black hole solutions. We found out a special relation between the difference of energy between the Einstein and M{\o}ller prescription and the energy density for considered black hole solutions. 
  In this paper we try to answer the main question: what is a quantum black hole? 
  In a space-time $M$ with a Killing vector field $\xi^a$ which is either everywhere timelike or everywhere spacelike, the collection of all trajectories of $\xi^a$ gives a 3-dimension space $S$. Besides the symmetry-reduced action from that of Einstein-Hilbert, an alternative action of the fields on $S$ is also proposed, which gives the same fields equations as those reduced from the vacuum Einstein equation on $M$. 
  Applications of the Path Group (consisting of classes of continuous curves in Minkowski space-time) to gauge theory and gravity are reviewed. Covariant derivatives are interpreted as generators of an induced representation of Path Group. Non-Abelian generalization of Stokes theorem is naturally formulated and proved in terms of paths. Quantum analogue of Equivalence Principle is formulated in terms of Path Group and Feynman path integrals. 
  We present a practical method for calculating the gravitational self-force, as well as the electromagnetic and scalar self forces, for a particle in a generic orbit around a Kerr black hole. In particular, we provide the values of all the regularization parameters needed for implementing the (previously introduced) {\it mode-sum regularization} method. We also address the gauge-regularization problem, as well as a few other issues involved in the calculation of gravitational radiation-reaction in Kerr spacetime. 
  We investigate the dynamics of a flat isotropic brane Universe with two-component matter source: perfect fluid with the equation of state $p=(\gamma-1) \rho$ and a scalar field with a power-law potential $V \sim \phi^{\alpha}$. The index $\alpha$ can be either positive or negative. We describe solutions for which the scalar field energy density scales as a power-law of the scale factor (so called scaling solutions). In the nonstandard brane regime when the brane is driven by energy density square term these solutions are rather different from their analogs in the standard cosmology. A particular attention is paid to the inverse square potential. Its dynamical properties in the nonstandard brane regime are in some sense analogous to those of the exponential potential in the standard cosmology. Stability analysis of the scaling solutions are provided. We also describe solutions existing in regions of the parameter space where the scaling solutions are unstable or do not exist. 
  Recent developments on the rotational instabilities of relativistic stars are reviewed. The article provides an account of the theory of stellar instabilities with emphasis on the rotational ones. Special attention is being paid to the study of these instabilities in the general relativistic regime. Issues such as the existence relativistic r-modes, the existence of a continuous spectrum and the CFS instability of the w-modes are discussed in the second half of the article. 
  The dynamical behavior of traversable wormholes and black holes under impulsive radiation is studied in an exactly soluble dilaton gravity model. Simple solutions are presented where a traversable wormhole is constructed from a black hole, or the throat of a wormhole is stably enlarged or reduced. These solutions illustrate the basic operating principles needed to construct similar analytic solutions in full Einstein gravity. 
  We study integrability by quadrature of a spatially flat Friedmann model containing both a minimally coupled scalar field $\phi$ with an exponential potential $V(\phi)\sim\exp[-\sqrt{6}\sigma\kappa\phi]$, $\kappa=\sqrt{8\pi G_N}$, of arbitrary sign and a perfect fluid with barotropic equation of state $p=(1-h)\rho$. From the mathematical view point the model is pseudo-Euclidean Toda-like system with 2 degrees of freedom. We apply the methods developed in our previous papers, based on the Minkowsky-like geometry for 2 characteristic vectors depending on the parameters $\sigma$ and $h$. In general case the problem is reduced to integrability of a second order ordinary differential equation known as the generalized Emden-Fowler equation, which was investigated by discrete-group methods. We present 4 classes of general solutions for the parameters obeying the following relations: {\bf A}. $\sigma$ is arbitrary, $h=0$; {\bf B}. $\sigma=1-h/2$, $0<h<2$; {\bf C1}. $\sigma=1-h/4$, $0<h\leq 2$; {\bf C2}. $\sigma=|1-h|$, $0<h\leq 2$, $h\neq 1,4/3$. We discuss the properties of the exact solutions near the initial singularity and at the final stage of evolution. 
  Tilted Bianchi type I cosmological models filled with disordered radiation in presence of a bulk viscous fluid and heat flow are investigated. The coefficient of bulk viscosity is assumed to be a power function of mass density. Some physical and geometric properties of the models are also discussed. 
  Bianchi type I magnetized cosmological models in the presence of a bulk viscous fluid are investigated. The source of the magnetic field is due to an electric current produced along x-axis. The distribution consists of an electrically neutral viscous fluid with an infinite electrical conductivity. The coefficient of bulk viscosity is assumed to be a power function of mass density. The cosmological constant $\Lambda$ is found to be positive and is a decreasing function of time which is supported by results from recent supernovae observations. The behaviour of the models in presence and absence of magnetic field are also discussed. 
  The Regge calculus generalised to independent area tensor variables is considered. The continuous time limit is found and formal Feynman path integral measure corresponding to the canonical quantisation is written out. The quantum measure in the completely discrete theory is found which possesses the property to lead to the Feynman path integral in the continuous time limit whatever coordinate is chosen as time. This measure can be well defined by passing to the integration over imaginary field variables (area tensors). Averaging with the help of this measure gives finite expectation values for areas. 
  There are, at present, several gravitational and cosmological anomalies; the dark energy problem, the lambda problem, accelerating cosmological expansion, the anomalous Pioneer spacecraft acceleration, a spin-up of the Earth and an apparent variation of G observed from analysis of the evolution of planetary longitudes. These conundrums may be resolved in the theory of Self Creation Cosmology, in which the Principle of Mutual Interaction subsumes both Mach's Principle and the Local Conservation of Energy. The theory is conformally equivalent to General Relativity in vacuo with the consequence that predictions of the theory are identical with General Relativity in the standard solar system experiments. Other observable local and cosmological consequences offer an explanation for the anomalies above. The SCC universe expands linearly in its Einstein Frame and it is static in its Jordan Frame; hence, as there are no density, smoothness or horizon problems, there is no requirement for Inflation. The theory determines the total density parameter to be one third, and the cold dark matter density parameter to be two ninths, yet in the Jordan frame the universe is similar to Einstein's original static cylindrical model and spatially flat. Therefore there is no need for a 'Dark Energy' hypothesis. As the field equations determine the false vacuum energy density to be a specific, and feasibly small, value there is no 'Lambda Problem'. Finally certain observations in SCC would detect cosmic acceleration. 
  The condition R=0, where R is the four-dimensional scalar curvature, is used for obtaining a large class (with an arbitrary function of r) of static, spherically symmetric Lorentzian wormhole metrics. The wormholes are globally regular and traversable, can have throats of arbitrary size and can be both symmetric and asymmetric. These metrics may be treated as possible wormhole solutions in a brane world since they satisfy the vacuum Einstein equations on the brane where effective stress-energy is induced by interaction with the bulk gravitational field. Some particular examples are discussed. 
  Beginning with the Pauli-Fierz theory, we construct a model for multi-graviton theory. Couplings between gravitons belonging to nearest-neighbor ``theory spaces'' lead to a discrete mass spectrum. Our model coincides with the Kaluza-Klein theory whose fifth dimension is latticized.   We evaluate one-loop vacuum energy in models with a circular latticized extra dimension as well as with compact continuous dimensions. We find that the vacuum energy can take a positive value, if the dimension of the continuous space time is $6, 10,...$. Moreover, since the amount of the vacuum energy can be an arbitrary small value according to the choice of parameters in the model, our models is useful to explain the small positive dark energy in the present universe. 
  In the context of the braneworld inflation driven by a bulk scalar field, we study the energy dissipation from the bulk scalar field into the matter on the brane in order to understand the reheating after inflation. Deriving the late-time behavior of the bulk field with dissipation by using the Green's function method, we give a rigorous justification of the statement that the standard reheating process is reproduced in this bulk inflaton model as long as the Hubble parameter on the brane and the mass of the bulk scalar field are much smaller than the 5-dimensional inverse curvature scale. Our result supports the idea that the brane inflation model caused by a bulk scalar field is expected to be a viable alternative scenario of the early universe. 
  We first review the various definition of the total energy in the gravitational system. The naive definition has some defects, and we review how to modify the definition of the total energy. Then we explicitly demonstrate how to calculate the total energy of the system. Our example is the total energy of a black hole in the expanding closed de Sitter universe in (2+1) dimension. In general, we find that the contribution to the total energy comes only from the singularity. Then we can calculate the total energy by evaluating the contribution around the singularity. 
  Self-consistent solutions to the spinor, scalar and BI gravitational field equations are obtained. The problems of initial singularity and asymptotically isotropization process of the initially anisotropic space-time are studied. It is also shown that the introduction of the Cosmological constant ($\Lambda$-term) in the Lagrangian generates oscillations of the BI model, which is not the case in absence of $\Lambda$ term. 
  Loop Quantum Gravity is the major candidate of quantum gravity. It is interesting to consider its continuum limit, which corresponds to the classical limit. We consider the Gaussian weave state, which describes a semi-classical picture. We calculate the expectation value of metric operator with respect to this state. 
  2+1 dimensional anti-de Sitter space has been the subject of much recent investigation. Studies of the behaviour of point particles in this space have given us a greater understanding of the BTZ black hole solutions produced by topological identification of adS isometries. In this paper, we present a new configuration of two orbiting massive point particles that leads to an ``eternal'' time machine, where closed timelike curves fill the entire space. In contrast to previous solutions, this configuration has no event or chronology horizons. Another interesting feature is that there is no lower bound on the relative velocities of the point masses used to construct the time machine; as long as the particles exceed a certain mass threshold, an eternal time machine will be produced. 
  We consider a static spherically symmetric charged anisotropic fluid source of finite physical radius (\sim 10^{-16} cm) by introducing a scalar variable \Lambda dependent on the radial coordinate r under general relativity. From the solution sets a possible role of the cosmological constant is investigated which indicates the dependency of energy density of electron on the variable \Lambda. 
  Experimental result regarding the maximum limit of the radius of the electron \sim 10^{-16} cm and a few of the theoretical works suggest that the gravitational mass which is a priori a positive quantity in Newtonian mechanics may become negative in general theory of relativity. It is argued that such a negative gravitational mass and hence negative energy density also can be obtained with a better physical interpretation in the framework of Einstein-Cartan theory. 
  Gravitational physics of VLBI experiment conducted on September 8, 2002 and dedicated to measure the speed of gravity (a fundamental constant in the Einstein equations) is treated in the first post-Newtonian approximation. Explicit speed-of-gravity parameterization is introduced to the Einstein equations to single out the retardation effect caused by the finite speed of gravity in the relativistic time delay of light, passing through the variable gravitational field of the solar system. The speed-of-gravity 1.5 post-Newtonian correction to the Shapiro time delay is derived and compared with our previous result obtained by making use of the post-Minkowskian approximation. We confirm that the 1.5 post-Newtonian correction to the Shapiro delay depends on the speed of gravity $c_g$ that is a directly measurable parameter in the VLBI experiment. 
  Using media with extremely low group velocities one can create an optical analog of a curved space-time. Leonhardt and Piwnicki have proposed that a vortex flow will act as an optical black hole. We show that although the Leonhardt - Piwnicki flow has an orbit of no return and an infinite red-shift surface, it is not a true black hole since it lacks a null hypersurface. However a radial flow will produce a true optical black hole that has a Hawking temperature and obeys the first law of black hole mechanics. By combining the Leonhardt - Piwnicki flow with a radial flow we obtain the analog of the Kerr black hole. 
  The back reaction effect of the neutrino field at finite temperature in the background of the static Einstein universe is investigated. A relationship between the temperature of the universe and its radius is found. As in the previously studied cases of the massless scalar field and the photon field, this relation exhibit a minimum radius below which no self-consistent solution for the Einstein field equation can be found. A maximum temperature marks the transition from a vacuum dominated state to the radiation dominated state universe. In the light of the results obtained for the scalar, neutrino and photon fields the role of the back reaction of quantum fields in controling the value of the cosmological constant is briefly discussed. 
  We consider the dynamics of timelike spherical thin matter shells in vacuum. A general formalism for thin shells matching two arbitrary spherical spacetimes is derived, and subsequently specialized to the vacuum case. We first examine the relative motion of two dust shells by focusing on the dynamics of the exterior shell, whereby the problem is reduced to that of a single shell with different active Schwarzschild masses on each side. We then examine the dynamics of shells with non-vanishing tangential pressure $p$, and show that there are no stable--stationary, or otherwise--solutions for configurations with a strictly linear barotropic equation of state, $p=\alpha\sigma$, where $\sigma$ is the proper surface energy density and $\alpha\in(-1,1)$. For {\em arbitrary} equations of state, we show that, provided the weak energy condition holds, the strong energy condition is necessary and sufficient for stability. We examine in detail the formation of trapped surfaces, and show explicitly that a thin boundary layer causes the apparent horizon to evolve discontinuously. Finally, we derive an analytical (necessary and sufficient) condition for neighboring shells to cross, and compare the discrete shell model with the well-known continuous Lema\^{\i}tre-Tolman-Bondi dust case. 
  We consider the most general vacuum cylindrical spacetimes, which are defined by two global, spacelike, commuting, non-hypersurface-orthogonal Killing vector fields. The cylindrical waves in such spacetimes contain both + and $\times$ polarizations, and are thus said to be unpolarized. We show that there are no trapped cylinders in the spacetime, and present a formal derivation of Thorne's C-energy, based on a Hamiltonian reduction approach. Using the Brown-York quasilocal energy prescription, we compute the actual physical energy (per unit Killing length) of the system, which corresponds to the value of the Hamiltonian that generates unit proper-time translations orthogonal to a given fixed spatial boundary. The C-energy turns out to be a monotonic non-polynomial function of the Brown-York quasilocal energy. Finally, we show that the Brown-York energy at spatial infinity is related to an asymptotic deficit angle in exactly the same manner as the specific mass of a straight cosmic string is to the former. 
  Loop quantum gravity can account for the Bekenstein-Hawking entropy of a black hole provided a free parameter is chosen appropriately. Recently, it was proposed that a new choice of the Immirzi parameter could predict both black hole entropy and the frequencies of quasinormal modes in the large $n$ limit, but at the price of changing the gauge group of the theory. In this note we use a simple physical argument within loop quantum gravity to arrive at the same value of the parameter. The argument uses strongly the necessity of having fermions satisfying basic symmetry and conservation principles, and therefore supports SU(2) as the relevant gauge group of the theory. 
  The natural extension of Schwarzschild metric to the case of nonzero cosmological constant $\Lambda$ known as the Kottler metric is considered and it is discussed under what circumstances the given metric could describe the Schwarzschild black hole immersed in a medium with nonzero energy density. Under the latter situation such an object might carry topologically inequivalent configurations of various fields. The given possibility is analysed for complex scalar field and it is shown that the mentioned configurations might be tied with natural presence of Dirac monopoles on black hole under consideration. In turn, this could markedly modify the Hawking radiation process. 
  We argue that existing doubly special relativities may not be operationally distinguishable from the special relativity. In the process we point out that some of the phenomenologically motivated modifications of dispersion relations, and arrived conclusions, must be reconsidered. Finally, we reflect on the possible conceptual issues that arise in quest for a theory of spacetime with two invariant scales. 
  We investigate the matching of continuous gravitational wave (CGW) signals in an all sky search with reference to Earth based laser interferometric detectors. We consider the source location as the parameters of the signal manifold and templates corresponding to different source locations. It has been found that the matching of signals from locations in the sky that differ in their co-latitude and longitude by $\pi$ radians decreases with source frequency. We have also made an analysis with the other parameters affecting the symmetries. We observe that it may not be relevant to take care of the symmetries in the sky locations for the search of CGW from the output of LIGO-I, GEO600 and TAMA detectors. 
  We present an action principle formulation for the study of motion of an extended body in General Relativity in the limit of weak gravitational field. This gives the classical equations of motion for multipole moments of arbitrary order coupling to the gravitational field. In particular, a new force due to the octupole moment is obtained. The action also yields the gravitationally induced phase shifts in quantum interference experiments due to the coupling of all multipole moments. 
  In this talk we discuss the cosmic singularity. We motivate the need to correct general relativity in the study of singularities, and mention the kind of corrections provided by string theory. We review how string theory resolves time-like singularities with two examples. Then, a simple toy model with lightlike singularities is presented, and studied in classical string theory. It turns out that classical string theory cannot resolve these singularities, and therefore better understanding of the full quantum theory is needed. The implications of this result for the Ekpyrotic/Cyclic Model are discussed. We end by mentioning the known suggestions for explaining the cosmological singularity. 
  Propagation of fermions in curved space-time generates gravitational interaction due to coupling of its spin with space-time curvature connection. This gravitational interaction, which is an axial-four-vector multiplied by a four gravitational vector potential, appears as CPT violating term in the Lagrangian which generates an opposite sign and thus asymmetry between the left-handed and right handed partners under CPT transformation. In the case of neutrinos this property can generate neutrino asymmetry in the Universe. If the background metric is of rotating black hole, Kerr geometry, this interaction for neutrino is non-zero. Therefore the dispersion energy relation for neutrino and its anti-neutrino are different which gives rise to the difference in their number densities and neutrino asymmetry in the Universe in addition to the known relic asymmetry. 
  Traversable wormholes necessarily require violations of the averaged null energy condition; this being the definition of ``exotic matter''. However, the theorems which guarantee the energy condition violation are remarkably silent when it comes to making quantitative statements regarding the ``total amount'' of energy condition violating matter in the spacetime. We develop a suitable measure for quantifying this notion, and demonstrate the existence of spacetime geometries containing traversable wormholes that are supported by arbitrarily small quantities of ``exotic matter''. 
  We study the duality of quasilocal energy and charges with non-orthogonal boundaries in the (2+1)-dimensional low-energy string theory. Quasilocal quantities shown in the previous work and some new variables arisen from considering the non-orthogonal boundaries as well are presented, and the boost relations between those quantities are discussed. Moreover, we show that the dual properties of quasilocal variables such as quasilocal energy density, momentum densities, surface stress densities, dilaton pressure densities, and Neuve-Schwarz(NS) charge density, are still valid in the moving observer's frame. 
  A unified general approach is presented for construction of solutions of the characteristic initial value problems for various integrable hyperbolic reductions of Einstein's equations for space-times with two commuting isometries in General Relativity and in some string theory induced gravity models. In all cases the associated linear systems of similar structures are used, and their fundamental solutions admit an alternative representations by two ``scattering'' matrices of a simple analytical structures on the spectral plane. The condition of equivalence of these representations leads to the linear ``integral evolution equations'' whose scalar kernels and right hand sides are determined completely by the initial data for the fields specified on the two initial characteristics. If the initial data for the fields are given, all field components of the corresponding solution can be expressed in quadratures in terms of a unique solution of these quasi - Fredholm integral evolution equations. 
  We present a new numerical code designed to solve the Einstein field equations for axisymmetric spacetimes. The long term goal of this project is to construct a code that will be capable of studying many problems of interest in axisymmetry, including gravitational collapse, critical phenomena, investigations of cosmic censorship, and head-on black hole collisions. Our objective here is to detail the (2+1)+1 formalism we use to arrive at the corresponding system of equations and the numerical methods we use to solve them. We are able to obtain stable evolution, despite the singular nature of the coordinate system on the axis, by enforcing appropriate regularity conditions on all variables and by adding numerical dissipation to hyperbolic equations. 
  For an $SU(2)\times U(1)$-invariant $S^3$ boundary the classical Dirichlet problem of Riemannian quantum gravity is studied for positive-definite regular solutions of the Einstein equations with a negative cosmological constant within biaxial Bianchi-IX metrics containing bolts, i.e., within the family of Taub-Bolt-anti-de Sitter (Taub-Bolt-AdS) metrics. Such metrics are obtained from the two-parameter Taub-NUT-anti-de Sitter family. The condition of regularity requires them to have only one free parameter ($L$) and constrains $L$ to take values within a narrow range; the other parameter is determined as a double-valued function of $L$ and hence there is a bifurcation within the family. We found that {\it{any}} axially symmetric $S^3$-boundary can be filled in with at least one solution coming from each of these two branches despite the severe limit on the permissible values of $L$. The number of infilling solutions can be one, three or five and they appear or disappear catastrophically in pairs as the values of the two radii of $S^3$ are varied. The solutions occur simultaneously in both branches and hence the total number of independent infillings is two, six or ten. We further showed that when the two radii are of the same order and large the number of solutions is two. In the isotropic limit this holds for small radii as well. These results are to be contrasted with the one-parameter self-dual Taub-NUT-AdS infilling solutions of the same boundary-value problem studied previously. 
  The singularity of the black hole solutions obtained before in M{\o}ller's theory are studied. It is found that although the two solutions reproduce the same associated metric the asymptotic behavior of the scalars of torsion tensor and basic vector are quite different. The stability of the associated metric of those solutions which is spherically symmetric non singular black hole is studied using the equations of geodesic deviation. The condition for the stability is obtained. From this condition the stability of the Schwarzschild solution and di Sitter solution can be obtained. 
  We study the back reaction effect of massless minimally coupled scalar field at finite temperatures in the background of Einstein universe. Substituting for the vacuum expectation value of the components of the energy-momentum tensor on the RHS of the Einstein equation, we deduce a relationship between the radius of the universe and its temperature. This relationship exhibit a maximum temperature, below the Planck scale, at which the system changes its behaviour drastically. The results are compared with the case of a conformally coupled field. An investigation into the values of the cosmological constant exhibit a remarkable difference between the conformally coupled case and the minimally coupled one. 
  We investigate the dynamics of a bulk scalar field with various decay channels in the Randall-Sundrum infinite braneworld scenario. A bulk scalar field in this scenario has a quasi-localized mode which dominates the late-time behavior near the brane. As for this mode, an interesting point is the presence of dissipation caused by the escape of the energy in the direction away from the brane, even if the bulk scalar field does not have the interaction with the other bulk fields in the bulk and fields on the brane. We can interpret that this lost energy is transfered to the dark radiation. We show that such an effective 4-dimensional description for a bulk scalar field is valid including the various processes of energy dissipation. 
  It has been conjectured that in head-on collisions of neutron stars (NSs), the merged object would not collapse promptly even if the total mass is higher than the maximum stable mass of a cold NS. In this paper, we show that the reverse is true: even if the total mass is {\it less} than the maximum stable mass, the merged object can collapse promptly. We demonstrate this for the case of NSs with a realistic equation of state (the Lattimer-Swesty EOS) in head-on {\it and} near head-on collisions. We propose a ``Prompt Collapse Conjecture'' for a generic NS EOS for head on and near head-on collisions. 
  Embedding diagrams have been used extensively to visualize the properties of curved space in Relativity. We introduce a new kind of embedding diagram based on the {\it extrinsic} curvature (instead of the intrinsic curvature). Such an extrinsic curvature embedding diagram, when used together with the usual kind of intrinsic curvature embedding diagram, carries the information of how a surface is {\it embedded} in the higher dimensional curved space. Simple examples are given to illustrate the idea. 
  The Gross-Pitaevski (GP) equation describing helium superfluids is extended to non-Riemannian spacetime background where torsion is shown to induce the splitting in the potential energy of the flow. A cylindrically symmetric solution for Minkowski background with constant torsion is obtained which shows that torsion induces a damping on the superfluid flow velocity. The Sagnac phase shift is computed from the superfluid flow velocity obtained from the solution of GP equations. 
  We study the dynamical evolution of a large amplitude r-mode by numerical simulations. R-modes in neutron stars are unstable growing modes, driven by gravitational radiation reaction. In these simulations, r-modes of amplitude unity or above are destroyed by a catastrophic decay: A large amplitude r-mode gradually leaks energy into other fluid modes, which in turn act nonlinearly with the r-mode, leading to the onset of the rapid decay. As a result the r-mode suddenly breaks down into a differentially rotating configuration. The catastrophic decay does not appear to be related to shock waves at the star's surface. The limit it imposes on the r-mode amplitude is significantly smaller than that suggested by previous fully nonlinear numerical simulations. 
  This paper has being withdrawn by the authors due to an error in the conclusion. 
  We study motion of particles and light in a space-time of a 5-dimensional rotating black hole. We demonstrate that the Myers-Perry metric describing such a black hole in addition to three Killing vectors possesses also a Killing tensor. As a result, the Hamilton-Jacobi equations of motion allow a separation of variables. Using first integrals we present the equations of motion in the first-order form. We describe different types of motion of particles and light and study some interesting special cases. We proved that there are no stable circular orbits in equatorial planes in the background of this metric. 
  We give the construction modulo normalization of a new state sum model for lorentzian quantum general relativity, using the construction of Dirac's expansors to include quantum operators corresponding to edge lengths as well as the quantum bivectors of the Barrett-Crane model, and discuss the problem of its normalization. The new model gives rise to a new picture of quantum geometry in which lengths come in a discrete spectrum, while areas have a continuum of values. 
  Besides the well-known quasinormal modes, the gravitational spectrum of a Schwarzschild black hole also has a continuum part on the negative imaginary frequency axis. The latter is studied numerically for quadrupole waves. The results show unexpected striking behavior near the algebraically special frequency $\Omega=-4i$. This reveals a pair of unconventional damped modes very near $\Omega$, confirmed analytically. 
  A generalization of the notion of ellipsoids to curved Riemannian spaces is given and the possibility to use it in describing the shapes of rotating bodies in general relativity is examined. As an illustrative example, stationary, axisymmetric perfect-fluid spacetimes with a so-called confocal inside ellipsoidal symmetry are investigated in detail under the assumption that the 4-velocity of the fluid is parallel to a time-like Killing vector field. A class of perfect-fluid metrics representing interior NUT-spacetimes is obtained along with a vacuum solution with a non-zero cosmological constant. 
  The solutions of vacuum Einstein's field equations, for the class of Riemannian metrics admitting a non Abelian bidimensional Lie algebra of Killing fields, are explicitly described. They are parametrized either by solutions of a transcendental equation (the tortoise equation), or by solutions of a linear second order differential equation in two independent variables. Metrics, corresponding to solutions of the tortoise equation, are characterized as those that admit a 3-dimensional Lie algebra of Killing fields with bidimensional leaves. 
  A formalism (zeta-complex analysis), allowing one to construct global Einstein metrics by matching together local ones described in the papers Phys. Lett. B 513(2001)142-146; Diff. Geom. Appl. 16(2002)95-120, is developed. With this formalism the singularities of the obtained metrics are described naturally as well. 
  An exact solution of the Einstein equations for a Bianchi -I universe in the presence of dust, stiff matter and cosmological constant, generalising the well-known Heckmann-Schucking solution is presented. PACS: 04.20-q; 04.20.Dw Keywords: Exact cosmological solutions 
  The definition of entropy obtained for stationary black holes is extended in this paper to the case of non-stationary black holes. Entropy is defined as a macroscopical thermodynamical quantity which satisfies the first principle of thermodynamics. In the non-stationary case a volume term appears since the solution does not admit a Killing vector. 
  More than 30 years of lunar laser ranging has produced several key tests of gravitational theory, including confirmation that bodies fall in external gravity at rates independent of their internal gravitational binding energy, and that Newton's G is constant to a part in 10^12 per year precision. The fitting of LLR data depends on the entire scope of 1/c^2 order features of the gravitational equation of motion, including non-linearity, gravitomagnetism, and inductive inertial forces. 
  The first scientific runs of kilometer scale laser interferometric detectors like LIGO are underway. Data from these detectors will be used to look for signatures of gravitational waves (GW) from astrophysical objects like inspiraling neutron star/blackhole binaries using matched filtering. The computational resources required for online flat-search implementation of the matched filtering are large if searches are carried out for small total mass. In this paper we report an improved implementation of the hierarchical search, wherein we extend the domain of hierarchy to an extra dimension - namely the time of arrival of the signal in the bandwidth of the interferometer. This is accomplished by lowering the Nyquist sampling rate of the signal in the trigger stage. We show that this leads to further improvement in the efficiency of data analysis and speeds up the online computation by a factor of $\sim 65 - 70$ over the flat search. We also take into account and discuss issues related to template placement, trigger thresholds and other peculiar problems that do not arise in earlier implementation schemes of the hierarchical search. We present simulation results for 2PN waveforms embedded in the noise expected for initial LIGO detectors. 
  The holographic principle in a radiation dominated universe is extended to incorporate the case of a bulk-viscous cosmic fluid. This corresponds to a nonconformally invariant theory. Generalization of the Cardy-Verlinde entropy formula to the viscous case appears to be possible from a formal point of view, although we question on physical grounds the manner in which the Casimir energy is evaluated in this case. Also, we consider an observation recently made by Youm, namely that the entropy of the universe is no longer expressible in the conventional Cardy-Verlinde form if one relaxes the radiation dominance equation of state and instead merely assumes that the pressure is proportional to the energy density. We show that Youm's generalized entropy formula remains valid when the cosmic fluid is no longer ideal, but endowed with a constant bulk viscosity. 
  See hep-ph/0304045 
  We derive global weak solutions of Einstein's equations for spherically symmetric dust-filled space-times which admit shell-crossing singularities. In the marginally bound case, the solutions are weak solutions of a conservation law. In the non-marginally bound case, the equations are solved in a generalized sense involving metric functions of bounded variation. The solutions are not unique to the future of the shell-crossing singularity, which is replaced by a shock wave in the present treatment; the metric is bounded but not continuous. 
  The influence of higher order (stringly inspired) curvature corrections to the classical General Relativity spherically symmetric solution is studied. In string gravity these curvature corrections have a special form and can provide a singular contribution to the field equations because they generate higher derivatives of metric functions multiplied by a small parameter. Analytically and numerically it is shown that sometimes in 4D string gravity the Schwarzschild solution is not recovered when the string coupling constant vanishes and limited number of higher order curvature corrections is considered. 
  We derive the delay in travel time of photons due to the spin of a body both inside a rotating shell and outside a rotating body. We then show that this time delay by the spin of an astrophysical object might be detected in different images of the same source by gravitational lensing; it might be relevant in the determination of the Hubble constant using accurate measurements of the time delay between the images of some gravitational lens systems. The measurement of the spin-time-delay might also provide a further observable to estimate the dark matter content in galaxies, clusters, or super-clusters of galaxies. 
  We investigate the point-particle limit of the equations of motion valid for a system of extended bodies in a scalar alternative theory of gravitation: the size of one of the bodies being a small parameter xi, we calculate the limit, as xi tends towards 0, of the post-Newtonian (PN) acceleration of this small body. We use the asymptotic scheme of PN approximation, that expands all fields. We find that the PN acceleration A of the small body keeps a structure-dependent part at this limit. In particular, if the only massive body is static and spherical, then A differs from the PN acceleration of a test particle in a Schwarzschild field only by this structure-dependent part. The presence of the latter is due to the fact that the PN metric depends on the first spatial derivatives of the Newtonian potential. Since just the same form of PN metric is valid for the standard form of Schwarzschild's solution, the acceleration of a small body might keep a structure-dependent part at the point limit in general relativity also, depending on the gauge. The magnitude of the structure-dependent acceleration is already challenging on Earth. For the Pioneer spacecrafts, it is likely to discard the current version of the scalar theory. A modified version has been outlined in a quoted reference.   Keywords: Weak equivalence principle violation, Asymptotic post-Newtonian scheme 
  We discuss the expected amplitude of a cosmic background of massive, non-relativistic dilatons, and we report recent results about its possible detection. This paper is a contracted version of a talk given at the 15th SIGRAV Conference on "General Relativity and Gravitational Physics" (Villa Mondragone, Roma, September 2002). 
  Starting from a static spherically symmetric solution of the Einstein's field equations in the second approximation in the perfect fluid scheme, in the exterior of the source, the corresponding metrics depend on a dimensionless parameter $\alpha $ . Taking the Sun for the source, the influence of $\alpha $ on the classical Solar System tests of the general relativity is estimated, the only $\alpha $-dependent one being the Shapiro radar-echo delay experiment. Performing such a test in given conditions within a precision better than $10^{-2}$, it is possible to obtain an experimental value for $\alpha $ . If $\alpha =1$, the Einstein's equations in the weak field approximation take the D'Alembert form, which attests the existence of gravitational waves 
  The immensely fruitful concept of Grothendieck topology or covering issued from the efforts of algebraic geometers to study "sheaf-like" objects defined on categories more general than the lattice of open sets on a topological space. In the present paper the covering concept - here called a cover scheme -is presented and developed in the simple case when the underlying category is a preordered set. The relationship between cover schemes, frames (complete Heyting algebras), Kripke models, and frame-valued set theory is discussed. Finally cover schemes and frame-valued set theory are applied in the context of Markopoulou's 1999 account of discrete spacetime as sets "evolving" over a causal set. 
  Since the Randall-Sundrum 1999 papers, braneworlds have been a favourite playground to test string inspired cosmological models. The subject has developped into two main directions : elaborating more complex models in order to strenghten the connection with string theories, and trying to confront them with observations, in particular the Cosmic Microwave Background anisotropies. We review here the latter and see that, even in the simple, "paradigmatic", case of a single expanding brane in a 5D anti-de Sitter bulk, there is still a missing link between the "view from the brane" and the "view from the bulk" which prevents definite predictions. 
  We review here how Newton's law can be approximately recovered in the simple, "paradigmatic", case of a flat $Z_2$-symmetric brane in a 5-D anti-de Sitter bulk. We then comment on the difficulties encountered so far in extending this analysis to cosmological perturbations on a Robertson-Walker brane. 
  After a brief review of the historical development and CLASSICAL properties of the BLACK HOLES, we discuss how our present knowledge of some of their QUANTUM properties shed light on the very concept of ELEMENTARY PARTICLE. As an illustration, we discuss in this context the decay of accelerated protons, which may be also relevant to astrophysics. 
  In this article we compute the black hole entropy by finding a classical central charge of the Virasoro algebra of a Liouville theory using the Cardy formula. This is done by performing a dimensional reduction of the Einstein Hilbert action with the ansatz of spherical symmetry and writing the metric in conformally flat form. We obtain two coupled field equations. Using the near horizon approximation the field equation for the conformal factor decouples. The one concerning the conformal factor is a Liouville equation, it posses the symmetry induced by a Virasoro algebra. We argue that it describes the microstates of the black hole, namely the generators of this symmetry do not change the thermodynamical properties of the black hole. 
  The expansion of the closed two-component universe has been considered. The potential barrier of the expansion has been investigated and its overcoming condition has been obtained. The restrictions on the Friedmann integrals, cosmological constant and density parameters have been analyzed. The phase-space has been considered and the phase curves of eternally expanding closed universes have been plotted. A questionable coincidence of the our Universe Friedmann integrals has been discussed. 
  (Abridged Abstract) This paper deals with a number of technical achievements that are instrumental for a dis-solution of the so-called {\it Hole Argument} in general relativity. The work is carried through in metric gravity for the class of Christoudoulou-Klainermann space-times, in which the temporal evolution is ruled by the {\it weak} ADM energy. The main results of the investigation are the following: 1) A re-interpretation of {\it active} diffeomorphisms as {\it passive and metric-dependent} dynamical symmetries of Einstein's equations, a re-interpretation that discloses their (up to now unknown) connection to gauge transformations on-shell; understanding such connection also enlightens the real content of the {\it Hole Argument}. 2) The utilization of Bergmann-Komar {\it intrinsic coordinates} for a peculiar gauge-fixing to the superhamiltonian and supermomentum constraints which embodies on shell a {\it physical individuation} of the mathematical points of $M^4$ as point-events in terms of the non-local intrinsic degrees of freedom of the gravitational field (Dirac observables). 3) A clarification of the notion of {\it Bergmann observable} that leads to a main conjecture asserting the existence of i) special Dirac observables which are also Bergmann observables; ii) tensorial (scalar) gauge variables. 
  The spherically symmetric static spacetimes are classified according to their matter collineations. These are studied when the energy-momentum tensor is degenerate and also when it is non-degenerate. We have found a case where the energy-momentum tensor is degenerate but the group of matter collineations is finite. For the non-degenerate case, we obtain either {\it four}, {\it five}, {\it six} or {\it ten} independent matter collineations in which four are isometries and the rest are proper. We conclude that the matter collineations coincide with the Ricci collineations but the constraint equations are different which on solving can provide physically interesting cosmological solutions. 
  We develop our recent suggestion that inflation may be made past eternal, so that there is no initial cosmological singularity or "beginning of time". Inflation with multiple vacua generically approaches a steady-state statistical distribution of regions at these vacua, and our model follows directly from making this distribution hold at all times. We find that this corresponds (at the semi-classical level) to particularly simple cosmological boundary conditions on an infinite null surface near which the spacetime looks de Sitter. The model admits an interesting arrow of time that is well-defined and consistent for all physical observers that can communicate, even while the statistical description of the entire universe admits a symmetry that includes time-reversal. Our model suggests, but does not require, the identification of antipodal points on the manifold. The resulting "elliptic" de Sitter spacetime has interesting classical and quantum properties. The proposal may be generalized to other inflationary potentials, or to boundary conditions that give semi-eternal but non-singular cosmologies. 
  Using quantum liquids one can simulate the behavior of the quantum vacuum in the presence of the event horizon. The condensed matter analogs demonstrate that in most cases the quantum vacuum resists to formation of the horizon, and even if the horizon is formed different types of the vacuum instability develop, which are faster than the process of Hawking radiation. Nevertheless, it is possible to create the horizon on the quantum-liquid analog of the brane, where the vacuum life-time is long enough to consider the horizon as the quasistationary object. Using this analogy we calculate the Bekenstein entropy of the nearly extremal and extremal black holes, which comes from the fermionic microstates in the region of the horizon -- the fermion zero modes. We also discuss how the cancellation of the large cosmological constant follows from the thermodynamics of the vacuum. 
  In homogeneous universes the propagation of quantum fields gives rise to pair creation of quanta with opposite momenta. When computing expectation values of operators, the correlations between these quanta are averaged out and no space-time structure is obtained. In this article, by an appropriate use of wave packets, we reveal the space-time structure of these correlations. We show that every pair emerges from vacuum configurations which are torn apart so as to give rise to two semi-classical currents: that carried by the particle and that of its `partner'. The partner's current lives behind the Hubble horizon centered around the particle. Hence any measurement performed within a Hubble patch would correspond to an uncorrelated density matrix, as for Hawking radiation. However, when inflation stops, the Hubble radius grows and eventually encompasses the partner. When this is realized the coherence is recovered within a patch. Our analysis applies to rare pair creation events as well as to cases leading to arbitrary high occupation numbers. Hence it might be applied to primordial gravitational waves which evolve into highly squeezed states. 
  The singularity theorems of classical general relativity are briefly reviewed. The extent to which their conclusions might still apply when quantum theory is taken into account is discussed. There are two distinct quantum loopholes: quantum violation of the classical energy conditions, and the presence of quantum fluctuations of the spacetime geometry. The possible significance of each is discussed. 
  The flux tube solutions in 5D Kaluza-Klein theory can be considered as a string-like object - $\Delta-$string. The initial 5D metric can be reduced to some inner degrees of freedom living on the $\Delta-$string. The propagation of electromagnetic waves through the $\Delta-$string is considered. It is shown that the difference between $\Delta$ and ordinary strings are connected with the fact that for the $\Delta-$string such limitations as critical dimensions are missing. 
  We show that the path integral for the three-dimensional SU(2) BF theory with a Wilson loop or a spin network function inserted can be understood as the Rovelli-Smolin loop transform of a wavefunction in the Ashtekar connection representation, where the wavefunction satisfies the constraints of quantum general relativity with zero cosmological constant. This wavefunction is given as a product of the delta functions of the SU(2) field strength and therefore it can be naturally associated to a flat connection spacetime. The loop transform can be defined rigorously via the quantum SU(2) group, as a spin foam state sum model, so that one obtains invariants of spin networks embedded in a three-manifold. These invariants define a flat connection vacuum state in the q-deformed spin network basis. We then propose a modification of this construction in order to obtain a vacuum state corresponding to the flat metric spacetime. 
  We study the sensitivity limits of a broadband gravitational-waves detector based on dual resonators such as nested spheres. We determine both the thermal and back-action noises when the resonators displacements are read-out with an optomechanical sensor. We analyze the contributions of all mechanical modes, using a new method to deal with the force-displacement transfer functions in the intermediate frequency domain between the two gravitational-waves sensitive modes associated with each resonator. This method gives an accurate estimate of the mechanical response, together with an evaluation of the estimate error. We show that very high sensitivities can be reached on a wide frequency band for realistic parameters in the case of a dual-sphere detector. 
  The spin modulated gravitational wave signals, which we shall call smirches, emitted by stellar mass black holes tumbling and inspiralling into massive black holes have extremely complicated shapes. Tracking these signals with the aid of pattern matching techniques, such as Wiener filtering, is likely to be computationally an impossible exercise. In this article we propose using a mixture of optimal and non-optimal methods to create a search hierarchy to ease the computational burden. Furthermore, by employing the method of principal components (also known as singular value decomposition) we explicitly demonstrate that the effective dimensionality of the search parameter space of smirches is likely to be just three or four, much smaller than what has hitherto been thought to be about nine or ten. This result, based on a limited study of the parameter space, should be confirmed by a more exhaustive study over the parameter space as well as Monte-Carlo simulations to test the predictions made in this paper. 
  The paper discusses the problem of the Lorentz contraction in accelerated systems, in the context of the special theory of relativity. Equal proper accelerations along different world lines are considered, showing the differences arising when the world lines correspond to physically connected or disconnected objects. In all cases the special theory of relativity proves to be completely self-consistent 
  We study the nature of asymptotic symmetries in topological 3d gravity with torsion. After introducing the concept of asymptotically anti-de Sitter configuration, we find that the canonical realization of the asymptotic symmetry is characterized by the Virasoro algebra with classical central charge, the value of which is the same as in general relativity: c=3l/2G. 
  We study scalar, electromagnetic and gravitational perturbations of a Reissner-Nordstr\"om-anti-de Sitter (RN-AdS) spacetime, and compute its quasinormal modes (QNM's). We confirm and extend results previously found for Schwarzschild-anti-de Sitter (S-AdS) black holes. For ``large'' black holes, whose horizon is much larger than the AdS radius, different classes of perturbations are almost exactly {\it isospectral}; this isospectrality is broken when the black hole's horizon radius is comparable to the AdS radius. We provide very accurate fitting formulas for the QNM's, which are valid for black holes of any size and charge $Q<Q_{ext}/3$.   Electromagnetic and axial perturbations of large black holes are characterized by the existence of pure-imaginary (purely damped) modes. The damping of these modes tends to infinity as the black hole charge approaches the extremal value; if the corresponding mode amplitude does not tend to zero in the same limit, this implies that {\it extremally charged RN-AdS black holes are marginally unstable}. This result is relevant in view of the AdS/CFT conjecture, since, according to it, the AdS QNM's give the timescales for approach to equilibrium in the corresponding conformal field theory. 
  We study the weak-field limit of string-dilaton gravity and derive corrections to the Newtonian potential which strength directly depends on the self interaction potential and the nonminimal coupling of the dilaton scalar field. We discuss also possible astrophysical applications of the results, in particular the flat rotation curves of spiral galaxies. 
  General relativity can be cast as a gauge theory by introducing a tetrad field and a spin-connection. This formalism was extended by replacing the tetrad field with a mixed tensor field independent of the metric tensor in order to develop a mechanism of adjustment of the vacuum energy density that takes advantage of Weinberg's no-go theorem. With no anthropic considerations, it was shown that the vacuum energy density is bounded and the gravitational and cosmological constants are proportional to a tiny dimensionless parameter determined by the coupling constants of the model. 
  We study the motion of test particles and electromagnetic waves in the Kerr-Newman-Taub-NUT spacetime in order to elucidate some of the effects associated with the gravitomagnetic monopole moment of the source. In particular, we determine in the linear approximation the contribution of this monopole to the gravitational time delay and the rotation of the plane of the polarization of electromagnetic waves. Moreover, we consider "spherical" orbits of uncharged test particles in the Kerr-Taub-NUT spacetime and discuss the modification of the Wilkins orbits due to the presence of the gravitomagnetic monopole. 
  General relativistic spin-orbit interaction leads to the quasiresonant oscillation of the gyroscope mass center along the orbital normal. The beating amplitude does not include the speed of light and equals the ratio of the intrinsic momentum of the gyroscope to its orbital momentum. The modulation frequency equals the angular velocity of the geodetic precession that prevents the oscillation from resonance. The oscillation represents the precession of the gyroscope orbital momentum. Within an acceptable time the oscillation amplitude reaches the values that are amenable to being analyzed experimentally. Taking into account the source oblateness decreases the beating amplitude and increases the modulation frequency by the factor that is equal to the ratio of the quadrupole precession velocity to the geodetic precession velocity. The period of the quadrupole precession turns out to be a quite sufficient time to form a measurable amplitude of the oscillation. 
  We discuss the influence of the cosmological constant on the gravitomagnetic clock effect and the gravitational time delay of electromagnetic rays. Moreover, we consider the relative motion of a binary system to linear order in the cosmological constant $\Lambda$. The general expression for the effect of $\Lambda$ on pericenter precession is given for arbitrary orbital eccentricity. 
  Bounds on the scale parameter {\cal L} arising in loop quantum gravity theory are derived in the framework of Cerenkov's effect and neutrino oscillations. Assuming that {\cal L} is an universal constant, we infer {\cal L}> 10^{-18}eV^{-1}, a bound compatible with ones inferred in different physical context. 
  Koutras has proposed some methods to construct reducible proper conformal Killing tensors and Killing tensors (which are, in general, irreducible) when a pair of orthogonal conformal Killing vectors exist in a given space. We give the completely general result demonstrating that this severe restriction of orthogonality is unnecessary. In addition we correct and extend some results concerning Killing tensors constructed from a single conformal Killing vector. A number of examples demonstrate how it is possible to construct a much larger class of reducible proper conformal Killing tensors and Killing tensors than permitted by the Koutras algorithms. In particular, by showing that all conformal Killing tensors are reducible in conformally flat spaces, we have a method of constructing all conformal Killing tensors (including all the Killing tensors which will in general be irreducible) of conformally flat spaces using their conformal Killing vectors. 
  We present a method for computing the evolution of a spacetime containing a massive particle and a black hole. The essential idea is that the gravitational field is evolved using full numerical relativity, with the particle generating a non-zero source term in the Einstein equations. The matter fields are not evolved by hydrodynamic equations. Instead the particle is treated as a rigid body whose center follows a geodesic. The necessary theoretical framework is developed and then implemented in a computer code that uses the null-cone, or characteristic, formulation of numerical relativity. The performance of the code is illustrated in test runs, including a complete orbit (near r=9M) of a Schwarzschild black hole. 
  Can the spatial distance between two identical particles be explained in terms of the extent that one can be distinguished from the other? Is the geometry of space a macroscopic manifestation of an underlying microscopic statistical structure? Is geometrodynamics derivable from general principles of inductive inference? Tentative answers are suggested by a model of geometrodynamics based on the statistical concepts of entropy, information geometry, and entropic dynamics. 
  A numerical analysis shows that a class of scalar-tensor theories of gravity with a scalar field minimally and nonminimally coupled to the curvature allows static and spherically symmetric black hole solutions with scalar-field hair in asymptotically flat spacetimes. In the limit when the horizon radius of the black hole tends to zero, regular scalar solitons are found. The asymptotically flat solutions are obtained provided that the scalar potential $V(\phi)$ of the theory is not positive semidefinite and such that its local minimum is also a zero of the potential, the scalar field settling asymptotically at that minimum. The configurations for the minimal coupling case, although unstable under spherically symmetric linear perturbations, are regular and thus can serve as counterexamples to the no-scalar-hair conjecture. For the nonminimal coupling case, the stability will be analyzed in a forthcoming paper. 
  The theory of inverse spectra of $T_0$ Alexandroff topological spaces is used to construct a model of $T_0$-discrete four-dimensional spacetime. The universe evolution is interpreted in terms of a sequence of topology changes in the set of $T_0$-discrete spaces realized as nerves of the canonical partitions of three-dimensional compact manifolds. The cosmological time arrow arises being connected with the refinement of the canonical partitions, and it is defined by the action of homomorphisms in the proper inverse spectrum of three-dimensional $T_0$-discrete spaces. A new causal order relation in this spectrum is postulated having the basic properties of the causal order in the pseudo-Riemannian spacetime however also bearing certain quasi-quantum features. An attempt is made to describe topological changes between compact manifolds in terms of bifurcations of proper inverse spectra; this led us to the concept of bispectrum. As a generalization of this concept, inverse multispectra and superspectrum are introduced. The last one enables us to introduce the discrete superspace, a discrete counterpart of the Wheeler--DeWitt superspace. 
  The generalized Second Law of thermodynamics and the Holographic Principle are combined to obtain the maximum mass of black holes formed inside a static spherical box of size $R$ filled with radiation at initial temperature $T_{i}$. The final temperature after the formation of black holes is evaluated, and we show that a critical threshold exists for the radiation to be fully consumed by the process. We next argue that if some form of Holographic Principle holds, upper bounds to the mass density of PBHs formed in the early universe may be obtained. The limits are worked out for inflationary and non-inflationary cosmological models. This method is independent of the known limits based on the background fluxes (from cosmic rays, radiation and other forms of energy) and applies to potentially important epochs of PBH formation, resulting in quite strong constraints to $\Omega_{pbh}$. 
  The standard relativistic theory of accelerated reference frames in Minkowski spacetime is described. The measurements of accelerated observers are considered and the limitations of the standard theory, based on the hypothesis of locality, are pointed out. The physical principles of the nonlocal theory of accelerated observers are presented. The implications of the nonlocal theory are briefly discussed. 
  In gravitation theory with a background metric, a gravitational field is described by a (1,1)-tensor field. The energy-momentum conservation law imposes a gauge condition on this field. 
  This essay discusses some geometric effects associated with gravitomagnetic fields and gravitomagnetic charge as well as the gravity theory of the latter. Gravitomagnetic charge is the duality of gravitoelectric charge (mass) and is therefore also termed the dual mass which represents the topological property of gravitation. The field equation of gravitomagnetic matter is suggested and a static spherically symmetric solution of this equation is offered. A possible explanation of the anomalous acceleration acting on Pioneer spacecrafts are briefly proposed. 
  We show that back-action noise in interferometric measurements such as gravitational-waves detectors can be completely suppressed by a local control of mirrors motion. An optomechanical sensor with an optimized measurement strategy is used to monitor mirror displacements. A feedback loop then eliminates radiation-pressure effects without adding noise. This very efficient technique leads to an increased sensitivity for the interferometric measurement, which becomes only limited by phase noise. Back-action cancellation is furthermore insensitive to losses in the interferometer. 
  For a two-surface B tending to an infinite--radius round sphere at spatial infinity, we consider the Brown--York boundary integral H_B belonging to the energy sector of the gravitational Hamiltonian. Assuming that the lapse function behaves as N \sim 1 in the limit, we find agreement between H_B and the total Arnowitt--Deser--Misner energy, an agreement first noted by Braden, Brown, Whiting, and York. However, we argue that the Arnowitt--Deser--Misner mass--aspect differs from a gauge invariant mass--aspect by a pure divergence on the unit sphere. We also examine the boundary integral H_B corresponding to the Hamiltonian generator of an asymptotic boost, in which case the lapse N \sim x^k grows like one of the asymptotically Cartesian coordinate functions. Such an integral defines the kth component of the center of mass for a Cauchy surface \Sigma bounded by B. In the large--radius limit, we find agreement between H_B and an integral introduced by Beig and O'Murchadha. Although both H_B and the Beig--O'Murchadha integral are naively divergent, they are in fact finite modulo the Hamiltonian constraint. Furthermore, we examine the relationship between H_B and a certain two--surface integral linear in the spacetime Riemann curvature tensor. Similar integrals featuring the curvature appear in works by Ashtekar and Hansen, Penrose, Goldberg, and Hayward. Within the canonical 3+1 formalism, we define gravitational energy and center--of--mass as certain moments of Riemann curvature. 
  The Kalam Cosmological Argument is perhaps the most solid and widly discussed argument for a caused creation of the universe. The usual objections to the argument mainly focus on the second premise. In this paper we discuss the dependency of the first premise on the topological structure of the space-time manifold adopted for the underlying cosmological model. It is shown that in chronology-violating space-times the first premise is also violated. The chronology-violation, in turn, requires a massive violation of the so-called energy conditions which could have observational effects that are briefly discussed here. Hence, astronomical observations could be relevant for the validity of the metaphysical argument. In this sense, it is possible to talk of "observational theology". 
  Given asymptotically flat initial data on M^3 for the vacuum Einstein field equation, and given a bounded domain in M, we construct solutions of the vacuum constraint equations which agree with the original data inside the given domain, and are identical to that of a suitable Kerr slice (or identical to a member of some other admissible family of solutions) outside a large ball in a given end. The data for which this construction works is shown to be dense in an appropriate topology on the space of asymptotically flat solutions of the vacuum constraints. This construction generalizes work of the first author, where the time-symmetric case was studied. 
  The tetrad-based equations for vacuum gravity published by Estabrook, Robinson, and Wahlquist are simplified and adapted for numerical relativity. We show that the evolution equations as partial differential equations for the Ricci rotation coefficients constitute a rather simple first-order symmetrizable hyperbolic system, not only for the Nester gauge condition on the acceleration and angular velocity of the tetrad frames considered by Estabrook et al., but also for the Lorentz gauge condition of van Putten and Eardley, and for a fixed gauge condition. We introduce a lapse function and a shift vector to allow general coordinate evolution relative to the timelike congruence defined by the tetrad vector field. 
  Generalising an analysis of Corvino and Schoen, we study surjectivity properties of the constraint map in general relativity in a large class of weighted Sobolev spaces. As a corollary we prove several perturbation, gluing, and extension results: we show existence of non-trivial, singularity-free, vacuum space-times which are stationary in a neighborhood of $i^0$; for small perturbations of parity-covariant initial data sufficiently close to those for Minkowski space-time this leads to space-times with a smooth global Scri; we prove existence of initial data for many black holes which are exactly Kerr -- or exactly Schwarzschild -- both near infinity and near each of the connected components of the apparent horizon; under appropriate conditions we obtain existence of vacuum extensions of vacuum initial data across compact boundaries; we show that for generic metrics the deformations in the Isenberg-Mazzeo-Pollack gluings can be localised, so that the initial data on the connected sum manifold coincide with the original ones except for a small neighborhood of the gluing region; we prove existence of asymptotically flat solutions which are static or stationary up to $r^{-m}$ terms, for any fixed $m$, and with multipole moments freely prescribable within certain ranges. 
  A non-minimal photon-torsion axial coupling in the quantum electrodynamics (QED) framework is considered. The geometrical optics in Riemannian-Cartan spacetime is considering and a plane wave expansion of the electromagnetic vector potential is considered leading to a set of the equations for the ray congruence. Since we are interested mainly on the torsion effects in this first report we just consider the Riemann-flat case composed of the Minkowskian spacetime with torsion. It is also shown that in torsionic de Sitter background the vacuum polarisation does alter the propagation of individual photons, an effect which is absent in Riemannian spaces. 
  We study a $(4+D)$-dimensional Kaluza-Klein cosmology with a Robertson-Walker type metric having two scale factors $a$ and $R$, corresponding to $D$-dimensional internal space and 4-dimensional universe, respectively. By introducing an exotic matter as the space-time part of the higher dimensional energy-momentum tensor, a 4-dimensional decaying cosmological term is appeared as $\Lambda \sim R^{-2}$, playing the role of an evolving dark energy in the universe. The resulting field equations yield the exponential solutions for the scale factors. These exponential behaviors may account for the dynamical compactification of extra dimensions and the accelerating expansion of the 4-dimensional universe in terms of the Hubble parameter. The acceleration of universe may be explained by the negative pressure of the exotic matter. It is shown that the rate of compactification of higher dimensions depends on the dimension, $D$. We then obtain the Wheeler-DeWitt equation and find the general exact solutions in $D$-dimensions. A good correspondence between the classical solutions and the Wheeler-DeWitt solutions, in any dimension $D$, is obtained. 
  We discuss the application of computer algebra to problems commonly arising in numerical relativity, such as the derivation of 3+1-splits, manipulation of evolution equations and automatic code generation. Particular emphasis is put on working with abstract index tensor quantities as much as possible. 
  A gravitational field can be seen as the anholonomy of the tetrad fields. This is more explicit in the teleparallel approach, in which the gravitational field-strength is the torsion of the ensuing Weitzenboeck connection. In a tetrad frame, that torsion is just the anholonomy of that frame. The infinitely many tetrad fields taking the Lorentz metric into a given Riemannian metric differ by point-dependent Lorentz transformations. Inertial frames constitute a smaller infinity of them, differing by fixed-point Lorentz transformations. Holonomic tetrads take the Lorentz metric into itself, and correspond to Minkowski flat spacetime. An accelerated frame is necessarily anholonomic and sees the electromagnetic field strength with an additional term. 
  We present an exact expression for the quasinormal modes of scalar, electromagnetic and gravitational perturbations of a near extremal Scwarzschild-de Sitter black hole and we show why a previous approximation holds exactly in this near extremal regime. In particular, our results give the asymptotic behavior of the quasinormal frequencies for highly damped modes, which as recently attracted much attention due to the proposed identification of its real part with the Barbero-Immirzi parameter. 
  We apply the Hamiltonian formulation of teleparallel theories of gravity in 2+1 dimensions to a circularly symmetric geometry. We find a family of one-parameter black hole solutions. The BTZ solution fixes the unique free parameter of the theory. The resulting field equations coincide with the teleparallel equivalent of Einstein's three-dimensional equations. We calculate the gravitational energy of the black holes by means of the simple expression that arises in the Hamiltonian formulation and conclude that the resulting value is identical to that calculated by means of the Brown-York method. 
  A single master equation is given describing spin $s\le2$ test fields that are gauge- and tetrad-invariant perturbations of the Kerr-Taub-NUT spacetime representing a source with mass $M$, gravitomagnetic monopole moment $-\ell$ and gravitomagnetic dipole moment (angular momentum) per unit mass $a$. This equation can be separated into its radial and angular parts. The behavior of the radial functions at infinity and near the horizon is studied and used to examine the influence of $\ell$ on the phenomenon of superradiance, while the angular equation leads to spin-weighted spheroidal harmonic solutions generalizing those of the Kerr spacetime. Finally the coupling between the spin of the perturbing field and the gravitomagnetic monopole moment is discussed. 
  Motivated by improving the understanding of the quantum-to-classical transition we use a simple model of classical discrete interactions for studying the discrete-to-continuous transition in the classical harmonic oscillator. A parallel is traced with gravity for stressing the relevance of such discrete interaction models. 
  This paper was withdrawn by the authors because it has been supplanted by gr-qc/0311007 and gr-qc/0311038. 
  Non-vacuum exact gravitational waves invariant for a non Abelian two-dimensional Lie algebra generated by two Killing fields whose commutator is of light type, are described. The polarization of these waves, already known from previous works, is related to the sources. Non vacuum exact gravitational waves admitting only one Killing field of light type are also discussed. 
  We consider a spherically symmetric global monopole in general relativity in $(D=d+2)$-dimensional spacetime. The monopole is shown to be asymptotically flat up to a solid angle defect in case $\gamma < d-1$, where $\gamma$ is a parameter characterizing the gravitational field strength. In the range $d-1< \gamma < 2d(d+1)/(d+2)$ the monopole space-time contains a cosmological horizon. Outside the horizon the metric corresponds to a cosmological model of Kantowski-Sachs type, where spatial sections have the topology ${\R\times \S}^d$. In the important case when the horizon is far from the monopole core, the temporal evolution of the Kantowski-Sachs metric is described analytically. The Kantowski-Sachs space-time contains a subspace with a $(d+1)$-dimensional Friedmann-Robertson-Walker metric, and its possible cosmological application is discussed. Some numerical estimations in case $d=3$ are made showing that this class of nonsingular cosmologies can be viable. Other results, generalizing those known in the 4-dimensional space-time, are derived, in particular, the existence of a large class of singular solutions with multiple zeros of the Higgs field magnitude. 
  See hep-ph/0304045 
  See hep-ph/0304045 
  See hep-ph/0304045 
  See hep-ph/0304045 
  No realistic black holes localized on a 3-brane in the Randall-Sundrum infinite braneworld have been found so far. The problem of finding a static black hole solution is reduced to a boundary value problem. We solve it by means of a numerical method, and show numerical examples of a localized black hole whose horizon radius is small compared to the bulk curvature scale. The sequence of small localized black holes exhibits a smooth transition from a five-dimensional Schwarzschild black hole, which is a solution in the limit of small horizon radius. The localized black hole tends to flatten as its horizon radius increases. However, it becomes difficult to find black hole solutions as its horizon radius increases. 
  In the light of recent interest in quantum gravity in de Sitter space, we investigate semi-classical aspects of 4-dimensional Schwarzschild-de Sitter space-time using the method of complex paths. The standard semi-classical techniques (such as Bogoliubov coefficients and Euclidean field theory) have been useful to study quantum effects in space-times with single horizons; however, none of these approaches seem to work for Schwarzschild-de Sitter or, in general, for space-times with multiple horizons. We extend the method of complex paths to space-times with multiple horizons and obtain the spectrum of particles produced in these space-times. We show that the temperature of radiation in these space-times is proportional to the effective surface gravity -- inverse harmonic sum of surface gravity of each horizon. For the Schwarzschild-de Sitter, we apply the method of complex paths to three different coordinate systems -- spherically symmetric, Painleve and Lemaitre. We show that the equilibrium temperature in Schwarzschild-de Sitter is the harmonic mean of cosmological and event horizon temperatures. We obtain Bogoliubov coefficients for space-times with multiple horizons by analyzing the mode functions of the quantum fields near the horizons. We propose a new definition of entropy for space-times with multiple horizons analogous to the entropic definition for space-times with a single horizon. We define entropy for these space-times to be inversely proportional to the square of the effective surface gravity. We show that this definition of entropy for Schwarzschild-de Sitter satisfies the D-bound conjecture. 
  In the first part of this thesis the relativistic viscous fluid equations describing the outflow of high temperature matter created via Hawking radiation from microscopic black holes are solved numerically for a realistic equation of state. We focus on black holes with initial temperatures greater than 100 GeV and lifetimes less than 6 days. The spectra of direct photons and photons from neutral pion decay are calculated for energies greater than 1 GeV. We calculate the diffuse gamma ray spectrum from black holes distributed in our galactic halo. However, the most promising route for their observation is to search for point sources emitting gamma rays of ever-increasing energy. We also calculate the spectra of all three flavors of neutrinos arising from direct emission from the fluid at the neutrino-sphere and from the decay of pions and muons from their decoupling at much larger radii and smaller temperatures for neutrino energies between 1 GeV and the Planck energy. The results for neutrino spectra may be applicable for the last few hours and minutes of the lifetime of a microscopic black hole. In the second part of this thesis the combined field equations of gravity and a scalar field are studied. When a potential for a scalar field has two local minima there arise spherical shell-type solutions of the classical field equations due to gravitational attraction. We establish such solutions numerically in a space which is asymptotically de Sitter. It generically arises when the energy scale characterizing the scalar field potential is much less than the Planck scale. It is shown that the mirror image of the shell appears in the other half of the Penrose diagram. The configuration is smooth everywhere with no physical singularity. 
  A recent report on gravitational wave detector data from the NAUTILUS and EXPLORER detector groups claims a statistically significant excess of coincident events when the detectors are oriented in a way that maximizes their sensitivity to gravitational wave sources in the galactic plane. While not claiming a detection of gravitational waves, they do strongly suggest that the origin of the excess is of gravitational wave origin. In this note we show that the statistical analysis that led them to the conclusion that there is a statistical excess is flawed and that the reported observation is entirely consistent with the normal Poisson statistics of the reported detector background. 
  See hep-ph/0304045 
  See hep-ph/0304045 
  We investigate a relativistic self-interacting gas in the field of an external {\it pp} gravitational wave. Based on symmetry considerations we ask for those forces which are able to compensate the imprint of the gravitational wave on the macroscopic 4-acceleration of the gaseous fluid. We establish an exactly solvable toy model according to which the stationary states which characterize such a situation have negative entropy production and are accompanied by instabilities of the microscopic particle motion. These features are similar to those which one encounters in phenomena of self-organization in many-particle systems. 
  The goal of this paper is to express the Bach tensor of a four dimensional conformal geometry of an arbitrary signature by the Cartan normal conformal (CNC) connection. We show that the Bach tensor can be identified with the Yang-Mills current of the connection. It follows from that result that a conformal geometry whose CNC connection is reducible in an appropriate way has a degenerate Bach tensor. As an example we study the case of a CNC connection which admits a twisting covariantly constant twistor field. This class of conformal geometries of this property is known as given by the Fefferman metric tensors. We use our result to calculate the Bach tensor of an arbitrary Fefferman metric and show it is proportional to the tensorial square of the four-fold eigenvector of the Weyl tensor. Finally, we solve the Yang-Mills equations imposed on the CNC connection for all the homogeneous Fefferman metrics. The only solution is the Nurowski-Plebanski metric. 
  I provide a very brief sketch of some of Dirac's interests and work in gravity, particularly his Hamiltonian formulation of Einstein's theory and its relation to his earlier research. 
  Geometric structure of spherically-symmetric space-time in metric-affine gauge theory of gravity is studied. Restrictions on curvature tensor and Bianchi identities are obtained. By using certain simple gravitational Lagrangian the solution of gravitational equations for vacuum spherically-symmetric gravitational field is obtained. 
  We consider the problem of three body motion for a relativistic one-dimensional self-gravitating system. After describing the canonical decomposition of the action, we find an exact expression for the 3-body Hamiltonian, implicitly determined in terms of the four coordinate and momentum degrees of freedom in the system. Non-relativistically these degrees of freedom can be rewritten in terms of a single particle moving in a two-dimensional hexagonal well. We find the exact relativistic generalization of this potential, along with its post-Newtonian approximation. We then specialize to the equal mass case and numerically solve the equations of motion that follow from the Hamiltonian. Working in hexagonal-well coordinates, we obtaining orbits in both the hexagonal and 3-body representations of the system, and plot the Poincare sections as a function of the relativistic energy parameter $\eta $. We find two broad categories of periodic and quasi-periodic motions that we refer to as the annulus and pretzel patterns, as well as a set of chaotic motions that appear in the region of phase-space between these two types. Despite the high degree of non-linearity in the relativistic system, we find that the the global structure of its phase space remains qualitatively the same as its non-relativisitic counterpart for all values of $\eta $ that we could study. However the relativistic system has a weaker symmetry and so its Poincare section develops an asymmetric distortion that increases with increasing $\eta $. For the post-Newtonian system we find that it experiences a KAM breakdown for $\eta \simeq 0.26$: above which the near integrable regions degenerate into chaos. 
  The physically interesting gravitational analogue of magnetic monopole in electrodynamics is considered in the present paper. The author investigates the field equation of gravitomagnetic matter, and the exact static cylindrically symmetric solution of field equation as well as the motion of gravitomagnetic charge in gravitational fields. Use is made of the mechanism of gravitational Meissner effect, a potential interpretation of anomalous, constant, acceleration acting on the Pioneer 10/11, Galileo and Ulysses spacecrafts is also suggested. 
  Bianchi VIII vacuum solutions to Einstein's equations are causally geodesically complete to the future, given an appropriate time orientation, and the objective of this article is to analyze the asymptotic behaviour of solutions in this time direction. For the Bianchi class A spacetimes, there is a formulation of the field equations that was presented in an article by Wainwright and Hsu, and in a previous article we analyzed the asymptotic behaviour of solutions in these variables. One objective of this paper is to give an asymptotic expansion for the metric. Furthermore, we relate this expansion to the topology of the compactified spatial hypersurfaces of homogeneity. The compactified spatial hypersurfaces have the topology of Seifert fibred spaces and we prove that in the case of NUT Bianchi VIII spacetimes, the length of a circle fibre converges to a positive constant but that in the case of general Bianchi VIII solutions, the length tends to infinity at a rate we determine. 
  We investigate the interaction of the gravitational field with a quantum particle. We derive the wave equation in the curved galilean spacetime from the very broad Quantum mechanical assumptions and from covariance under the Milne group. The inertial and gravitational masses are equal in that equation. So, we give the proof of the equality for the non-relativistic quantum particle, without applying the equivalence principle to the Schr\"odinger equation and witout imposing any relation to the classical equations of motion. This result constitutes a substantial strengthening of the previous result obtained by Herdegen and the author. 
  We report a new implementation for axisymmetric simulation in full general relativity. In this implementation, the Einstein equations are solved using the Nakamura-Shibata formulation with the so-called cartoon method to impose an axisymmetric boundary condition, and the general relativistic hydrodynamic equations are solved using a high-resolution shock-capturing scheme based on an approximate Riemann solver. As tests, we performed the following simulations: (i) long-term evolution of non-rotating and rapidly rotating neutron stars, (ii) long-term evolution of neutron stars of a high-amplitude damping oscillation accompanied with shock formation, (iii) collapse of unstable neutron stars to black holes, and (iv) stellar collapses to neutron stars. The tests (i)--(iii) were carried out with the $\Gamma$-law equation of state, and the test (iv) with a more realistic parametric equation of state for high-density matter. We found that this new implementation works very well: It is possible to perform the simulations for stable neutron stars for more than 10 dynamical time scales, to capture strong shocks formed at stellar core collapses, and to accurately compute the mass of black holes formed after the collapse and subsequent accretion. In conclusion, this implementation is robust enough to apply to astrophysical problems such as stellar core collapse of massive stars to a neutron star and black hole, phase transition of a neutron star to a high-density star, and accretion-induced collapse of a neutron star to a black hole. The result for the first simulation of stellar core collapse to a neutron star started from a realistic initial condition is also presented. 
  We study the scalar and spinor perturbation, namely the Klein-Gordan and Dirac equations, in the Kerr-NUT space-time. The metric is invariant under the duality transformation involving the exchange of mass and NUT parameters on one hand and radial and angle coordinates on the other. We show that this invariance is also shared by the scalar and spinor perturbation equations. Further, by the duality transformation, one can go from the Kerr to the dual Kerr solution, and vice versa, and the same applies to the perturbation equations. In particular, it turns out that the potential barriers felt by the incoming scalar and spinor fields are higher for the dual Kerr than that for the Kerr. We also comment on existence of horizon and singularity. 
  We present an exhaustive analysis of the numerical evolution of the Einstein-Klein-Gordon equations for the case of a real scalar field endowed with a quadratic self-interaction potential. The self-gravitating equilibrium configurations are called oscillatons and are close relatives of boson stars, their complex counterparts. Unlike boson stars, for which the oscillations of the two components of the complex scalar field are such that the spacetime geometry remains static, oscillatons give rise to a geometry that is time-dependent and oscillatory in nature. However, they can still be classified into stable (S-branch) and unstable (U-branch) cases. We have found that S-oscillatons are indeed stable configurations under small perturbations and typically migrate to other S-profiles when perturbed strongly. On the other hand, U-oscillatons are intrinsically unstable: they migrate to the S-branch if their mass is decreased and collapse to black holes if their mass is increased even by a small amount. The S-oscillatons can also be made to collapse to black holes if enough mass is added to them, but such collapse can be efficiently prevented by the gravitational cooling mechanism in the case of diluted oscillatons. 
  In the brane-world framework, we consider static, spherically symmetric configurations of a scalar field with the Lagrangian $(\d\phi)^2/2 - V(\phi)$, confined on the brane. We use the 4D Einstein equations on the brane obtained by Shiromizu et al., containing the usual stress tensor $T\mN$, the tensor $\Pi\mN$, quadratic in $T\mN$, and $E\mN$ describing interaction with the bulk. For models under study, the tensor $\Pi\mN$ has zero divergence, so we can consider a "minimally coupled" brane with $E\mN = 0$, whose 4D gravity is decoupled from the bulk geometry. Assuming $E\mN =0$, we try to extend to brane worlds some theorems valid for scalar fields in general relativity (GR). Thus, the list of possible global causal structures in all models under consideration is shown to be the same as is known for vacuum with a $Lambda$ term in GR: Minkowski, Schwarzschild, (A)dS and Schwarzschild-(A)dS. A no-hair theorem, saying that, given a potential $V\geq 0$, asymptotically flat black holes cannot have nontrivial external scalar fields, is proved under certain restrictions. Some objects, forbidden in GR, are allowed on the brane, e.g, traversable wormholes supported by a scalar field, but only at the expense of enormous matter densities in the strong field region. 
  Dust configurations play an important role in astrophysics and are the simplest models for rotating bodies. The physical properties of the general--relativistic global solution for the rigidly rotating disk of dust, which has been found recently as the solution of a boundary value problem, are discussed. 
  In the formulation of (2+1)-dimensional gravity as a Chern-Simons gauge theory, the phase space is the moduli space of flat Poincar\'e group connections. Using the combinatorial approach developed by Fock and Rosly, we give an explicit description of the phase space and its Poisson structure for the general case of a genus g oriented surface with punctures representing particles and a boundary playing the role of spatial infinity. We give a physical interpretation and explain how the degrees of freedom associated with each handle and each particle can be decoupled. The symmetry group of the theory combines an action of the mapping class group with asymptotic Poincar\'e transformations in a non-trivial fashion. We derive the conserved quantities associated to the latter and show that the mapping class group of the surface acts on the phase space via Poisson isomorphisms. 
  We study the reparametrization invariant system of a classical relativistic particle moving in (5+1) dimensions, of which two internal ones are compactified to form a torus. A discrete physical time is constructed based on a quasi-local invariant observable. Due to ergodicity, it is simply related to the proper time on average. The external motion in Minkowski space can then be described as a unitary quantum mechanical evolution. 
  Radiative properties of gravitational and electromagnetic fields generated by uniformly accelerated charged black holes in asymptotically de Sitter spacetime are studied by analyzing the C-metric exact solution of the Einstein-Maxwell equations with a positive cosmological constant. Its global structure and physical properties are thoroughly discussed. We explicitly find and describe the specific pattern of radiation which exhibits the dependence of the fields on a null direction along which the (spacelike) conformal infinity is approached. This directional characteristic of radiation supplements the peeling behavior of the fields near the infinity. The interpretation of the solution is achieved by means of various coordinate systems, and suitable tetrads. Relation to the Robinson-Trautman framework is also presented. 
  We present a singularity excision algorithm appropriate for numerical simulations of black holes moving throughout the computational domain. The method is an extension of the excision procedure previously used to obtain stable simulations of single, non-moving black holes. The excision procedure also shares elements used in recent work to study the dynamics of a scalarfield in the background of a single, boosted black hole. The robustness of our excision method is tested with single black-hole evolutions using a coordinate system in which the coordinate location of the black hole, and thus the excision boundary, moves throughout the computational domain. 
  An approach to general relativity based on conformal flatness and quasiequilibrium (CFQE) assumptions has played an important role in the study of the inspiral dynamics and in providing initial data for fully general relativistic numerical simulations of coalescing compact binaries. However, the regime of validity of the approach has never been established. To this end, we develop an analysis that determines the violation of the CFQE approximation in the evolution of the binary described by the full Einstein theory. With this analysis, we show that the CFQE assumption is significantly violated even at relatively large orbital separations in the case of corotational neutron star binaries. We also demonstrate that the innermost stable circular orbit (ISCO) determined in the CFQE approach for corotating neutron star binaries may have no astrophysical significance. 
  In this article we review the present status of the spin foam formulation of non-perturbative (background independent) quantum gravity. The article is divided in two parts. In the first part we present a general introduction to the main ideas emphasizing their motivations from various perspectives. Riemannian 3-dimensional gravity is used as a simple example to illustrate conceptual issues and the main goals of the approach. The main features of the various existing models for 4-dimensional gravity are also presented here. We conclude with a discussion of important questions to be addressed in four dimensions (gauge invariance, discretization independence, etc.).   In the second part we concentrate on the definition of the Barrett-Crane model. We present the main results obtained in this framework from a critical perspective. Finally we review the combinatorial formulation of spin foam models based on the dual group field theory technology. We present the Barrett-Crane model in this framework and review the finiteness results obtained for both its Riemannian as well as its Lorentzian variants. 
  A new experiment to test for the time independence of the fine structure constant, alpha, is proposed. The experiment utilizes orthogonally polarized Transverse Electric and Transverse Magnetic Whispering Gallery Modes in a single sapphire resonator tuned to similar frequencies. When configured as a dual mode sapphire clock, we show that the anisotropy of sapphire makes it is possible to undertake a sensitive measurement from the beat frequency between the two modes. At infrared frequencies this is possible due to the different effect of the lowest phonon frequency on the two orthogonally polarized modes. At microwave frequencies we show that the phonon effect is too small. We show that the Electron Spin Resonance of paramagnetic impurities (such as Cr3+) in the lattice effects only one polarization with an alpha^6 dependence. This enables an enhancement of the sensitivity to temporal changes in a at microwave frequencies. 
  See hep-ph/0304045 
  See hep-ph/0304045 
  We discuss the origin of causal set structure and the emergence of classical space and time in the universe. Given that the universe is a closed self-referential quantum automaton with a quantum register consisting of a vast number of elementary quantum subregisters, we find two distinct but intimately related causal sets. One of these is associated with the factorization and entanglement properties of states of the universe and encodes phenomena such as quantum correlations and violations of Bell-type inqualities. The concepts of separations and entanglements of states are used to show how state reduction dynamics generates the familial relationships which gives this causal set structure. The other causal set structure is generated by the factorization properties of the observables (the Hermitian operators) over the quantum register. The concept of skeleton sets of operators is used to show how the factorization properties of these operators could generate the classical causal set structures associated with Einstein locality. 
  The properties of 5D gravitational flux tubes are considered. With the cross section and 5th dimension in the Planck region such tubes can be considered as string-like objects, namely $\Delta-$strings. A model of attachment of $\Delta-$string to a spacetime is offered. It is shown that the attachment point is a model of an electric charge for an observer living in the spacetime. Magnetic charges are forbidden in this model. 
  The quasi-isotropic inhomogeneous solution of the Einstein equations near a cosmological singularity in the form of a series expansion in the synchronous system of reference, first found by Lifshitz and Khalatnikov in 1960, is generalized to the case of a two-fluid cosmological model. This solution describes non-decreasing modes of adiabatic and isocurvature scalar perturbations and gravitational waves in the regime when deviations of a space-time metric from the homogeneous isotropic Friedmann-Robertson-Walker (FRW) background are large while locally measurable quantities like Riemann tensor components are still close to their FRW values. The general structure of the perturbation series is presented and the first coefficients of the series expansion for the metric tensor and the fluid energy densities and velocities are calculated explicitly. 
  The Einstein evolution equations have been written in a number of symmetric hyperbolic forms when the gauge fields--the densitized lapse and the shift--are taken to be fixed functions of the coordinates. Extended systems of evolution equations are constructed here by adding the gauge degrees of freedom to the set of dynamical fields, thus forming symmetric hyperbolic systems for the combined evolution of the gravitational and the gauge fields. The associated characteristic speeds can be made causal (i.e. less than or equal to the speed of light) by adjusting 14 free parameters in these new systems. And 21 additional free parameters are available, for example to optimize the stability of numerical evolutions. The gauge evolution equations in these systems are generalizations of the ``K-driver'' and ``Gamma-driver'' conditions that have been used with some success in numerical black hole evolutions. 
  The quantum evolution of a model of the universe with account of two scalar fields ({\it dilaton} and {\it inflaton}) is considered. For this case, the closed and flat models has been examined. It is shown that in both cases the realization of conditions necessary for inflation is strongly depends on coupling constant for dilaton and inflaton fields. 
  Black-hole quasinormal modes (QNM) have been the subject of much recent attention, with the hope that these oscillation frequencies may shed some light on the elusive theory of quantum gravity. We compare numerical results for the QNM spectrum of the (rotating) Kerr black hole with an {\it exact} formula Re$\omega \to T_{BH}\ln 3+\Omega m$, which is based on Bohr's correspondence principle. We find a close agreement between the two. Possible implications of this result to the area spectrum of quantum black holes are discussed. 
  We construct a Hamiltonian formulation of quasilocal general relativity using an extended phase space that includes boundary coordinates as configuration variables. This allows us to use Hamiltonian methods to derive an expression for the energy of a non-isolated region of space-time that interacts with its neighbourhood. This expression is found to be very similar to the Brown-York quasilocal energy that was originally derived by Hamilton-Jacobi methods. We examine the connection between the two formalisms and find that when the boundary conditions for the two are harmonized, the resulting quasilocal energies are identical. 
  We consider brane Kantowski-Sachs Universe when bulk space is five-dimensional Anti-deSitter space. The corresponding cosmological equations with perfect fluid are written. For several specific choices of relation between energy and pressure it is found the behavior of scale factors at early time. In particulary, for $\gamma=3/2$ Kantowski-Sachs brane cosmology is modified to become the isotropic one, while for $\gamma=1$ it remains the anisotropic cosmology in the process of evolution. 
  We develop the general formalism to study the low energy regime of the brane world. We apply our formalism to the single brane model where the AdS/CFT correspondence will take an important role. We also consider the two-brane system and show the system is described by the quasi-scalar tensor gravity. Our result provides a basis for predicting CMB fluctuations in the braneworld models. 
  We propose a cosmological braneworld scenario in which two branes collide and emerge as reborn branes with signs of tensions opposite to the original tensions of respective branes. In this scenario, the branes are assumed to be inflating. However, the whole dynamics is different from the usual inflation due to the non-trivial dynamics of the radion field. Transforming the conformal frame to the Einstein frame, this born-again scenario resembles the pre-big-bang scenario. Thus our scenario has features of both inflation and pre-big-bang scenarios. In particular, the gravitational waves produced from vacuum fluctuations will have a very blue spectrum, while the inflaton field will give rise to a standard scale-invariant spectrum. 
  We present results from axisymmetric stellar core collapse simulations in general relativity. Our hydrodynamics code has proved robust and accurate enough to allow for a detailed analysis of the global dynamics of the collapse. Contrary to traditional approaches based on the 3+1 formulation of the gravitational field equations, our framework uses a foliation based on a family of outgoing light cones, emanating from a regular center, and terminating at future null infinity. Such a coordinate system is well adapted to the study of interesting dynamical spacetimes in relativistic astrophysics such as stellar core collapse and neutron star formation. Perhaps most importantly this procedure allows for the unambiguous extraction of gravitational waves at future null infinity without any approximation, along with the commonly used quadrupole formalism for the gravitational wave extraction. Our results concerning the gravitational wave signals show noticeable disagreement when those are extracted by computing the Bondi news at future null infinity on the one hand and by using the quadrupole formula on the other hand. We have strong indication that for our setup the quadrupole formula on the null cone does not lead to physical gravitational wave signals. The Bondi gravitational wave signals extracted at infinity show typical oscillation frequencies of about 0.5 kHz. 
  Recent suggestion, that the emission of a quantum of energy corresponding to the asymptotic value of quasinormal modes of a Schwarzschild black hole should be associated with the loss of spin one punctures from the black hole horizon, fixes the Immirzi parameter to a definite value. We show that saturating the horizon with spin one punctures reproduces the earlier formula for the black hole entropy, including the $ln (area)$ correction with definite coefficient (- 3/2) for large area. 
  We propose a new definition for the mass and angular momentum of neutral or electrically charged black holes in 2+1 gravity with two Killing vectors. These finite conserved quantities, associated with the SL(2,R) invariance of the reduced mechanical system, are shown to be identical to the quasilocal conserved quantities for an improved gravitational action corresponding to mixed boundary conditions. They obey a general Smarr-like formula and, in all cases investigated, are consistent with the first law of black hole thermodynamics. Our framework is applied to the computation of the mass and angular momentum of black hole solutions to several field-theoretical models. 
  We consider quasi-extreme Kerr and quasi-extreme Schwarzschild-de Sitter black holes. From the known analytical expressions obtained for their quasi-normal modes frequencies, we suggest an area quantization prescription for those objects. 
  We review the classical and quantum mechanics of Stueckelberg, and introduce the compensation fields necessary for the gauge covariance of the Stueckelberg- Schr\"odinger equation. To achieve this, one must introduce a fifth, Lorentz scalar, compensation field, in addition to the four vector fields which compensate the action of the space-time derivatives. A generalized Lorentz force can be derived from the classical Hamilton equations associated with this evolution function. We show that the fifth (scalar) field can be eliminated through the introduction of a conformal metric on the spacetime manifold. The geodesic equation associated with this metric coincides with the Lorentz force, and is therefore dynamically equivalent. Since the generalized Maxwell equations for the five dimensional fields provide an equation relating the fifth field with the spacetime density of events, one can derive the spacetime event density associated with the Friedmann-Robertson-Walker solution of the Einstein equations. The resulting density, in the conformal coordinate space, is isotropic and homogeneous, decreasing as the square of the Robertson-Walker scale factor. Using the Einstein equations, one sees that both for the static and matter dominated models, the conformal time slice in which the events which generate the world lines are contained becomes progressively thinner as the inverse square of the scale factor, establishing a simple correspondence between the configurations predicted by the underlying Friedmann-Robertson-Walker dynamical model and the configurations in the conformal coordinates. 
  We calculated the evolution of the Newton gravitational in a scalar tensor theory, using parameters that holds for the present Universe. We analised the evolution from one billion of years ago. 
  The use of adaptive mesh refinement (AMR) techniques is crucial for accurate and efficient simulation of higher dimensional spacetimes. In this work we develop an adaptive algorithm tailored to the integration of finite difference discretizations of wave-like equations using characteristic coordinates. We demonstrate the algorithm by constructing a code implementing the Einstein-Klein-Gordon system of equations in spherical symmetry. We discuss how the algorithm can trivially be generalized to higher dimensional systems, and suggest a method that can be used to parallelize a characteristic code. 
  General relativity as well as Newtonian gravity admits self-similar solutions due to its scale-invariance. This is a review on these self-similar solutions and their relevance to gravitational collapse. In particular, our attention is mainly paid on the crucial role of self-similar solutions in the critical behavior and attraction in gravitational collapse. 
  We solve the Einstein equations for the 2+1 dimensions with and without scalar fields. We calculate the entropy, Hawking temperature and the emission probabilities for these cases. We also compute the Newman-Penrose coefficients for different solutions and compare them. 
  We derive formulae for the time variation of the gravitational ``constant'' and of the fine structure ``constant'' in various models with extra dimensions and analyze their consistency with the observational data. 
  We determine transformations between coordinate systems which are mutually in linear accelerated motion. In case of the symmetrical linear mutual acceleration, we immediately get the maximal acceleration limit which was derived by Caianiello from quantum mechanics. The derived results can play crucial role in modern particle physics. 
  We obtain an expression for the active gravitational mass (Tolman) of a source of the $\gamma$ metric, just after its departure from hydrostatic equilibrium, on a time scale of the order of (or smaller than) the hydrostatic time scale. It is shown that for very compact sources, even arbitrarily small departures from sphericity, produce significant decreasing (increasing) in the values of active gravitational mass of collapsing (expanding) spheres, with respect to its value in equilibrium, enhancing thereby the stability of the system. 
  We model a black hole spacetime as a causal set and count, with a certain definition, the number of causal links crossing the horizon in proximity to a spacelike or null hypersurface $\Sigma$. We find that this number is proportional to the horizon's area on $\Sigma $, thus supporting the interpretation of the links as the ``horizon atoms'' that account for its entropy. The cases studied include not only equilibrium black holes but ones far from equilibrium. 
  Cosmic strings are considered in two types of gauged sigma models, which generalize the gravitating Abelian Higgs model. The two models differ by whether the U(1) kinetic term is of the Maxwell or Chern-Simons form. We obtain the self-duality conditions for a general two-dimensional target space defined in terms of field dependent "dielectric functions". In particular, we analyze analytically and numerically the equations for the case of O(3) models (two-sphere as target space), and find cosmic string solutions of several kinds as well as gravitating vortices. We classify the solutions by their flux and topological charge. We note an interesting connection between the Maxwell and Chern-Simons type models, which is responsible for simple relations between the self-dual solutions of both types. There is however a significant difference between the two systems, in that only the Chern-Simons type sigma model gives rise to spinning cosmic vortices. 
  A simple method is presented which enables us to construct scalar field solutions from any given Einstein-Maxwell solution in colliding plane waves. As an application we give scalar field extensions of the solution found by Hogan, Barrabes and Bressange. 
  We present the concept of selective readout for broadband resonant mass gravitational wave detectors. This detection scheme is capable of specifically selecting the signal from the contributions of the vibrational modes sensitive to the gravitational waves, and efficiently rejecting the contribution from non gravitationally sensitive modes. Moreover this readout, applied to a dual detector, is capable to give an effective reduction of the back-action noise within the frequency band of interest. The overall effect is a significant enhancement in the predicted sensitivity, evaluated at the standard quantum limit for a dual torus detector. A molybdenum detector, 1 m in diameter and equipped with a wide area selective readout, would reach spectral strain sensitivities 2x10^{-23}/sqrt{Hz} between 2-6 kHz. 
  We estimate the upper frequency cutoff of the galactic white dwarf binaries gravitational wave background that will be observable by the LISA detector. This is done by including the modulation of the gravitational wave signal due the motion of the detector around the Sun. We find this frequency cutoff to be equal to $10^{-3.0}$Hz, a factor of 2 smaller than the values previously derived. This implies an increase in the number of resolvable signals in the LISA band by a factor of about 4. Our theoretical derivation is complemented by a numerical simulation, which shows that by using the maximum likelihood estimation technique it is possible to accurately estimate the parameters of the resolvable signals and then remove them from the LISA data. 
  There have been conflicting points of view concerning the Riemann--Lanczos problem in 3 and 4 dimensions. Using direct differentiation on the defining partial differential equations, Massa and Pagani (in 4 dimensions) and Edgar (in dimensions n > 2) have argued that there are effective constraints so that not all Riemann tensors can have Lanczos potentials; using Cartan's criteria of integrability of ideals of differential forms Bampi and Caviglia have argued that there are no such constraints in dimensions n < 5, and that, in these dimensions, all Riemann tensors can have Lanczos potentials. In this paper we give a simple direct derivation of a constraint equation, confirm explicitly that known exact solutions of the Riemann-Lanczos problem satisfy it, and argue that the Bampi and Caviglia conclusion must therefore be flawed. In support of this, we refer to the recent work of Dolan and Gerber on the three dimensional problem; by a method closely related to that of Bampi and Caviglia, they have found an 'internal identity' which we demonstrate is precisely the three dimensional version of the effective constraint originally found by Massa and Pagani, and Edgar. 
  Stated succinctly, the original version of the Campbell-Magaard theorem says that it is always possible to locally embed any solution of 4-dimensional general relativity in a 5-dimensional Ricci-flat manifold. We discuss the proof of this theorem (and its variants) in n dimensions, and its application to current theories that postulate that our universe is a 4-dimensional hypersurface Sigma_0 within a 5-dimensional manifold, such as Space-Time-Matter (STM) theory and the Randall & Sundrum (RS) braneworld scenario. In particular, we determine whether or not arbitrary spacetimes may be embedded in such theories, and demonstrate how these seemingly disparate models are interconnected. Special attention is given to the motion of test observers in 5 dimensions, and the circumstances under which they are confined to Sigma_0. For each 5-dimensional scenario considered, the requirement that observers be confined to the embedded spacetime places restrictions on the 4-geometry. For example, we find that observers in the thin braneworld scenario can be localized around the brane if its total stress-energy tensor obeys the 5-dimensional strong energy condition. As a concrete example of some of our technical results, we discuss a Z_2 symmetric embedding of the standard radiation-dominated cosmology in a 5-dimensional vacuum. 
  We show that every 2nd order ODE defines a 4-parameter family of projective connections on its 2-dimensional solution space. In a special case of ODEs, for which a certain point transformation invariant vanishes, we find that this family of connections always has a preferred representative. This preferred representative turns out to be identical to the projective connection described in Cartan's classic paper "Sur les Varietes a Connection Projective". 
  We consider the end state of collapsing null radiation with a string fluid. It is shown that, if diffusive transport is assumed for the string, that a naked singularity can form (at least locally). The model has the advantage of not being asymptotically flat. We also analyse the case of a radiation-string two-fluid and show that a locally naked singularity can result in the collapse of such matter. We contrast this model with that of strange quark matter. 
  We describe how the Barrett-Crane spin foam model defines transition amplitudes for quantum gravity states and how causality can be consistently implemented in it. 
  The Einstein-Gordon equations for Friedmann-Robertson-Walker (FRW) geometries in feedback reaction with the quartically self-interacting physical field, arisen from the "inner parity" spontaneous breaking, are explicitly formulated. The Hamiltonian density non-positive extrema would classically forbid both spatially closed and flat homogeneous and isotropic worlds if these were to allow the physical field to (repeatedly) go through and to (finally) settle down in a ground state. In this respect, the fixed point exact solutions of the spontaneous Z_2-symmetry breaking Einstein-Gordon equations (mandatory) describe (k=-1)-FRW manifolds which actually are either Milne or anti-de Sitter Universes. Setting the Z_2-invariance breaking scale at the one of the electroweak symmetry, we speculate on the cosmological implications of the Higgs-anti-de Sitter bubbles and derive a set of particular closed-form solutions to the S_2-cobordism with a spatially flat FRW Universe. 
  The problem of energy and its localization in general relativity is critically re-examined. The Tolman energy integral for the Eddington spinning rod is analyzed in detail and evaluated apart from a single term. It is shown that a higher order iteration is required to find its value. Details of techniques to solve mathematically challenging problems of motion with powerful computing resources are provided. The next phase of following a system from static to dynamic to final quasi-static state is described. 
  We prove global completeness in the expanding direction of spacetimes satisfying the vacuum Einstein equations on a manifold of the form $\Sigma \times S^{1}\times R$ where $\Sigma $ is a compact surface of genus $G>1.$ The Cauchy data are supposed to be invariant with respect to the group $S^{1}$ and sufficiently small, but we do not impose a restrictive hypothesis made in gr-qc 0112049 on the lowest eigenvalue of a relevant Laplacian. The total energy decay still holds, but its rate depends of the asymptotic value of this eigenvalue. 
  A massless spinor particle is considered in the background gravitational field due to a rotating body. In the weak field approximation it is shown that the solution of the Weyl equations depend on the angular momentum of the rotating body, which does not affect the curvature in this approximation. This result may be looked upon as a generalization of the gravitational Aharonov-Bohm effect. 
  We study the behaviour of a non-relativistic quantum particle interacting with different potentials, in the background spacetime generated by a cosmic string. We find the energy spectra for the quantum systems under consideration and discuss how they differ from their flat Minkowski spacetime values. 
  For mechanical Weber gravitational wave antennae, it is thought that gravity waves are weakly converted into acoustic vibrations. Acoustic vibrations in metals (such as Aluminum) are experimentally known to be attenuated by the creation of electron-hole pairs described via the electronic viscosity. These final state electronic excitations give rise to gravitational wave absorption cross sections which are considerably larger (by four orders of magnitude) than those in previous theories which have not explicitly considered electronic excitations. 
  These lectures aim at providing an introduction to the properties of gravitational waves and in particular to those gravitational waves that are expected as a consequence of perturbations of black holes and neutron stars. Imprinted in the gravitational radiation emitted by these objects is, in fact, a wealth of physical information. In the case of black holes, a detailed knowledge of the gravitational radiation emitted as a response to perturbations will reveal us important details about their mass and spin, but also about the fundamental properties of the event horizon. In the case of neutron stars, on the other hand, this information can provide a detailed map of their internal structure and tell us about the equation of state of matter at very high density, thus filling-in a gap in energies and densities that cannot be investigated by experiments in terrestrial laboratories. 
  The new theory of Self Creation Cosmology has been shown to yield a concordant cosmological solution that does not require inflation, exotic non-baryonic Dark matter or Dark Energy to fit observational constraints. In vacuo there is a conformal equivalence between this theory and canonical General Relativity and as a consequence an experimental degeneracy exists as the two theories predict identical results in the standard tests. However, there are three definitive experiments that are able to resolve this degeneracy and distinguish between the two theories. Here these standard tests and definitive experiments are described. One of the definitive predictions, that of the geodetic precession of a gyroscope, has just been measured on the Gravity Probe B satellite, which is at the present time of writing in the data processing stage. This is the first opportunity to falsify Self Creation Cosmology. The theory predicts a 'frame-dragging' result equal to GR but a geodetic precession of only 2/3 the GR value. When applied to the Gravity Probe B satellite, Self Creation Cosmology predicts an E-W gravitomagnetic/frame-dragging precession, equal to that of GR, of 40.9 milliarcsec/yr but a -S gyroscope (geodetic + Thomas) precession of just 4.4096 arcsec/yr. 
  We describe exact cosmological solutions with rotation and expansion in the low-energy effective string theory. These models are spatially homogeneous (closed Bianchi type IX) and they belong to the family of shear-free metrics which are causal (no closed timelike curves are allowed), admit no parallax effects and do not disturb the isotropy of the background radiation. The dilaton and the axion fields are nontrivial, in general, and we consider both cases with and without the central charge (effective cosmological constant). 
  A possible resolution of the incompatibility of quantum mechanics and general relativity is that the relativity principle is emergent. I show that the central paradox of black holes also occurs at a liquid-vapor critical surface of a bose condensate but is resolved there by the phenomenon of quantum criticality. I propose that real black holes are actually phase boundaries of the vacuum analogous to this, and that the Einstein field equations simply fail at the event horizon the way quantum hydrodynamics fails at a critical surface. This can occur without violating classical general relativity anywhere experimentally accessible to external observers. Since the low-energy effects that occur at critical points are universal, it is possible to make concrete experimental predictions about such surfaces without knowing much, if anything about the true underlying equations. Many of these predictions are different from accepted views about black holes - in particular the absence of Hawking radiation and the possible transparency of cosmological black hole surfaces. [To appear in the C. N Yang Festschrift (World Sci., Singapore, 2003).] 
  We present nonsingular cosmological models with a variable cosmological term described by the second-rank symmetric tensor $\Lambda_{mn}$ evolving from $\Lambda g_{mn}$ to $\lambda g_{mn}$ with $\lambda < \Lambda$. All $\Lambda_{mn}$ dominated cosmologies belong to Lemaitre type models for an anisotropic perfect fluid. The expansion starts from a nonsingular nonsimultaneous de Sitter bang, with $\Lambda$ on the scale responsible for the earliest accelerated expansion, which is followed by an anisotropic Kasner type stage. For a certain class of observers these models can be also identified as Kantowski-Sachs models with regular R regions. For Kantowski-Sachs observers the cosmological evolution starts from horizons with a highly anisotropic ``null bang'' where the volume of the spatial section vanishes. We study in detail the spherically symmetric case and consider the general features of cosmologies with planar and pseudospherical symmetries. Nonsingular $\Lambda_{mn}$ dominated cosmologies are Bianchi type I in the planar case and hyperbolic analogs of the Kantowski-Sachs models in the pseudospherical case. At late times all models approach a de Sitter asymptotic with small $\lambda$. 
  The emergence of time in the matter-gravity system is addressed within the context of the inflationary paradigm. A quantum minisuperspace-homogeneous minimally coupled inflaton system is studied with suitable initial conditions leading to inflation and the system is approximately solved in the limit for large scale factor. Subsequently normal matter (either non homogeneous inflaton modes or lighter matter) is introduced as a perturbation and it is seen that its presence requires the coarse averaging of a gravitational wave function (which oscillates at trans-Planckian frequencies) having suitable initial conditions. Such a wave function, which is common for all types of normal matter, is associated with a ``time density'' in the sense that its modulus is related to the amount of time spent in a given interval (or the rate of flow of time). One is then finally led to an effective evolution equation (Schroedinger Schwinger-Tomonaga) for ``normal'' matter. An analogy with the emergence of a temperature in statistical mechanics is also pointed out. 
  In this communication I analyze the problem of complete exceptionality of wave propagation in a class of spin 2 field theories. I show that, under the imposition of the good weak-field limit, only two Lagrangians are completely exceptional. These are the linear Fierz Lagrangian, and a Born-Infeld-like Lagrangian. As a byproduct, I reobtain the result that in a nonlinear theory, spin 2 particles follow an effective metric that depends on the nonlinearities of the Lagrangian. 
  In regards to the initial-boundary value problem of the Einstein equations, we argue that the projection of the Einstein equations along the normal to the boundary yields necessary and appropriate boundary conditions for a wide class of equivalent formulations. We explicitly show that this is so for the Einstein-Christoffel formulation of the Einstein equations in the case of spherical symmetry. 
  The analysis of vacuum general relativity by R. Beig and N. O Murchadha (Ann. Phys. vol 174, 463 (1987)) is extended in numerous ways. The weakest possible power-type fall-off conditions for the energy-momentum tensor, the metric, the extrinsic curvature, the lapse and the shift are determined, which, together with the parity conditions, are preserved by the energy-momentum conservation and the evolution equations. The algebra of the asymptotic Killing vectors, defined with respect to a foliation of the spacetime, is shown to be the Lorentz Lie algebra for slow fall-off of the metric, but it is the Poincare algebra for 1/r or faster fall-off. It is shown that the applicability of the symplectic formalism already requires the 1/r (or faster) fall-off of the metric. The connection between the Poisson algebra of the Beig-O Murchadha Hamiltonians and the asymptotic Killing vectors is clarified. The value H[K^a] of their Hamiltonian is shown to be conserved in time if K^a is an asymptotic Killing vector defined with respect to the constant time slices. The angular momentum and centre-of-mass, defined by the value of H[K^a] for asymptotic rotation-boost Killing vectors K^a, are shown to be finite only for 1/r or faster fall-off of the metric. Our center-of-mass expression is the difference of that of Beig and O Murchadha and the spatial momentum times the coordinate time. The spatial angular momentum and this centre-of-mass form a Lorentz tensor, which transforms in the correct way under Poincare transformations. 
  Rotating relativistic stars have been studied extensively in recent years, both theoretically and observationally, because of the information one could obtain about the equation of state of matter at extremely high densities and because they are considered to be promising sources of gravitational waves. The latest theoretical understanding of rotating stars in relativity is reviewed in this updated article. The sections on the equilibrium properties and on the nonaxisymmetric instabilities in f-modes and r-modes have been updated and several new sections have been added on analytic solutions for the exterior spacetime, rotating stars in LMXBs, rotating strange stars, and on rotating stars in numerical relativity. 
  As motivated in the full abstract, this paper further investigates Barbour, Foster and O Murchadha (BFO)'s 3-space formulation of GR. This is based on best-matched lapse-eliminated actions and gives rise to several theories including GR and a conformal gravity theory. We study the simplicity postulates assumed in BFO's work and how to weaken them, so as to permit the inclusion of the full set of matter fields known to occur in nature.   We study the configuration spaces of gravity-matter systems upon which BFO's formulation leans. In further developments the lapse-eliminated actions used by BFO become impractical and require generalization. We circumvent many of these problems by the equivalent use of lapse-uneliminated actions, which furthermore permit us to interpret BFO's formulation within Kuchar's generally covariant hypersurface framework. This viewpoint provides alternative reasons to BFO's as to why the inclusion of bosonic fields in the 3-space approach gives rise to minimally-coupled scalar fields, electromagnetism and Yang--Mills theory. This viewpoint also permits us to quickly exhibit further GR-matter theories admitted by the 3-space formulation. In particular, we show that the spin-1/2 fermions of the theories of Dirac, Maxwell--Dirac and Yang--Mills--Dirac, all coupled to GR, are admitted by the generalized 3-space formulation we present. Thus all the known fundamental matter fields can be accommodated. This corresponds to being able to pick actions for all these theories which have less kinematics than suggested by the generally covariant hypersurface framework. For all these theories, Wheeler's thin sandwich conjecture may be posed, rendering them timeless in Barbour's sense. 
  A model is proposed to describe a transition from a Schwarzschild black hole of mass $M_{0}$ to a Schwarzschild black hole of mass $M_{1}$ $\leq M_{0}$. The basic equations are derived from the non-vacuum Einstein field equations taking a source representing a null scalar field with a nonvanishing trace anomaly. It is shown that the nonvanishing trace anomaly of the scalar field prevents a complete evaporation. 
  The creation of brane universes induced by a totally antisymmetric tensor living in a fixed background spacetime is presented, where a term involving the intrinsic curvature of the brane is considered. A canonical quantum mechanical approach employing Wheeler-DeWitt equation is done. The probability nucleation for the brane is calculated taking into account both an instanton method and a WKB approximation. Some cosmological implications arose from the model are presented. 
  A thermodynamical description for the quasi-static collapse of radiating, self-gravitating spherical shells of matter in anti-de Sitter space-time is obtained. It is shown that the specific heat at constant area and other thermodynamical quantities may diverge before a black hole has eventually formed. This suggests the possibility of a phase transition occurring along the collapse process. The differences with respect to the asymptotically flat case are also highlighted. 
  In magnetized plasmas gravitational and electromagnetic waves may interact coherently and exchange energy between themselves and with plasma flows. We derive the wave interaction equations for these processes in the case of waves propagating perpendicular or parallel to the plasma background magnetic field. In the latter case, the electromagnetic waves are taken to be circularly polarized waves of arbitrary amplitude. We allow for a background drift flow of the plasma components which increases the number of possible evolution scenarios. The interaction equations are solved analytically and the characteristic time scales for conversion between gravitational and electromagnetic waves are found. In particular, it is shown that in the presence of a drift flow there are explosive instabilities resulting in the generation of gravitational and electromagnetic waves. Conversely, we show that energetic waves can interact to accelerate particles and thereby \emph{produce} a drift flow. The relevance of these results for astrophysical and cosmological plasmas is discussed. 
  In these lectures we review the basic structure of Poincare gauge theory of gravity, with emphasis on its fundamental principles and geometric interpretation. A specific limit of this theory, defined by the teleparallel geometry of spacetime, is discussed as a viable alternative for the description of macroscopic gravitational phenomena. 
  We analyze and discuss the quantum noise in signal-recycled laser interferometer gravitational-wave detectors, such as Advanced LIGO, using a heterodyne readout scheme and taking into account the optomechanical dynamics. Contrary to homodyne detection, a heterodyne readout scheme can simultaneously measure more than one quadrature of the output field, providing an additional way of optimizing the interferometer sensitivity, but at the price of additional noise. Our analysis provides the framework needed to evaluate whether a homodyne or heterodyne readout scheme is more optimal for second generation interferometers from an astrophysical point of view. As a more theoretical outcome of our analysis, we show that as a consequence of the Heisenberg uncertainty principle the heterodyne scheme cannot convert conventional interferometers into (broadband) quantum non-demolition interferometers. 
  Analytical solution of Weyl neutrino wave equation in Kerr geometry is presented by making use of the two-spinor component spin-coefficient Newman-Penrose (NP) calculus. So far only asymptotic or approximate solutions have been found for the Weyl equation in this background. It is shown that neutrino current asymmetry is also present in this solution. 
  See hep-ph/0304045 
  We compare different treatments of the constraints in canonical quantum gravity. The standard approach on the superspace of 3--geometries treats the constraints as the sole carriers of the dynamic content of the theory, thus rendering the traditional dynamical equations obsolete. Quantization of the constraints in both the Dirac and ADM square root Hamiltonian approaches leads to the well known problems of time evolution. These problems of time are of both an interpretational and technical nature. In contrast, the geometrodynamic quantization procedure on the superspace of the true dynamical variables separates the issues of quantization from the enforcement of the constraints. The resulting theory takes into account states that are off-shell with respect to the constraints, and thus avoids the problems of time. We develop, for the first time, the geometrodynamic quantization formalism in a general setting and show that it retains all essential features previously illustrated in the context of homogeneous cosmologies. 
  The coupling of gravity to matter is explored in the linearized gravity limit. The usual derivation of gravity-matter couplings within the quantum-field-theoretic framework is reviewed. A number of inconsistencies between this derivation of the couplings, and the known results of tidal effects on test particles according to classical general relativity are pointed out. As a step towards resolving these inconsistencies, a General Laboratory Frame fixed on the worldline of an observer is constructed. In this frame, the dynamics of nonrelativistic test particles in the linearized gravity limit is studied, and their Hamiltonian dynamics is derived. It is shown that for stationary metrics this Hamiltonian reduces to the usual Hamiltonian for nonrelativistic particles undergoing geodesic motion. For nonstationary metrics with long-wavelength gravitational waves (GWs) present, it reduces to the Hamiltonian for a nonrelativistic particle undergoing geodesic \textit{deviation} motion. Arbitrary-wavelength GWs couple to the test particle through a vector-potential-like field $N_a$, the net result of the tidal forces that the GW induces in the system, namely, a local velocity field on the system induced by tidal effects as seen by an observer in the general laboratory frame. Effective electric and magnetic fields, which are related to the electric and magnetic parts of the Weyl tensor, are constructed from $N_a$ that obey equations of the same form as Maxwell's equations . A gedankin gravitational Aharonov-Bohm-type experiment using $N_a$ to measure the interference of quantum test particles is presented. 
  Einstein's equations for a Robertson-Walker fluid source endowed with rotation Einstein's equations for a Robertson-Walker fluid source endowed with rotation are presented upto and including quadratic terms in angular velocity parameter. A family of analytic solutions are obtained for the case in which the source angular velocity is purely time-dependent. A subclass of solutions is presented which merge smoothly to homogeneous rotating and non-rotating central sources. The particular solution for dust endowed with rotation is presented. In all cases explicit expressions, depending sinusoidally on polar angle, are given for the density and internal supporting pressure of the rotating source. In addition to the non-zero axial velocity of the fluid particles it is shown that there is also a radial component of velocity which vanishes only at the poles. The velocity four-vector has a zero component between poles. 
  We calculate the energy distribution of a charged black hole solution in heterotic string theory in the M{\o}ller prescription. 
  Using precession of orbits due to non-Newtonian interaction between two celestial bodies and modern tracking data of satellites, planets and a pulsar we obtain new more precise limits on possible Yukawa-type deviations from the Newton law in planets (satellites)radii ranges. 
  First, the ideas introduced in the wormhole research field since the work of Morris and Thorne are briefly reviewed, namely, the issues of energy conditions, wormhole construction, stability, time machines and astrophysical signatures. Then, spherically symmetric and static traversable Morris-Thorne wormholes in the presence of a generic cosmological constant are analyzed. A matching of an interior solution to the unique exterior vacuum solution is done using directly the Einstein equations. The structure as well as several physical properties and characteristics of traversable wormholes due to the effects of the cosmological term are studied. Interesting equations appear in the process of matching. For instance, one finds that for asymptotically flat and anti-de Sitter spacetimes the surface tangential pressure of the thin-shell, at the boundary of the interior and exterior solutions, is always strictly positive, whereas for de Sitter spacetime it can take either sign as one could expect, being negative (tension) for relatively high cosmological constant and high wormhole radius, positive for relatively high mass and small wormhole radius, and zero in-between. Finally, some specific solutions with generic cosmological constant, based on the Morris-Thorne solutions, are provided. 
  We discuss the potential for detection of gravitational waves from a rapidly spinning neutron star produced by supernova 1987A taking the parameters claimed by Middleditch et al. (2000) at face value. Asssuming that the dominant mechanism for spin down is gravitational waves emitted by a freely precessing neutron star, it is possible to constrain the wobble angle, the effective moment of inertai of the precessing crust and the crust cracking stress limit. Our analysis, suggests that, if the interpretation of the Middleditch data is correct, the compact remnant of SN 1987A may well provide a reliable and predictable source of gravitational waves well within the capability of LIGO II. 
  Standard techniques of canonical gravity quantization on the superspace of 3--metrics are known to cause insurmountable difficulties in the description of time evolution. We forward a new quantization procedure on the superspace of true dynamic variables -- geometrodynamic quantization. This procedure takes into account the states that are ``off-shell'' with respect to the constraints and thus circumvents the notorious problems of time. In this approach quantum geometrodynamics, general covariance, and the interpretation of time emerge together as parts of the solution to the total problem of geometrodynamic evolution. 
  We introduce the concept of effective geometry by studying several systems in which it arises naturally. As an example of the power and conciseness of the method, it is shown that a flowing dielectric medium with a linear response to an external electric field can be used to generate an analog geometry that has many of the formal properties of a \Sch black hole for light rays, in spite of birefringence. The surface gravity of this analog black hole has a contribution that depends only on the dielectric properties of the fluid (in addition to the usual term dependent on the acceleration). This term may be give a hint to a new mechanism to increase the temperature of Hawking radiation. 
  Loop quantum gravity theory incorporates a new scale length ${\cal L}$ which induces a Lorentz invariance breakdown. This scale can be either an universal constant or can be fixed by the momentum of particles (${\cal L}\sim p^{-1}$). Effects of the scale parameter ${\cal L}$ and helicity terms occurring in the dispersion relation of fermions are reviewed in the framework of spin-flip conversion of neutrino flavors. 
  We review a viable alternative scenario of the inflationary universe in the context of the Randall-Sundrum (RS) braneworld. In this scenario, the dynamics of a 5-dimensional scalar field, which we call a bulk scalar field, plays the central role. Focusing on the second (single-brane) RS model, we discuss braneworld inflation driven by a bulk scalar field without introducing an inflaton on the brane. As a toy model, for the bulk scalar field, we consider a minimally coupled massive scalar field in the 5-dimensional spacetime, and look for a perturbative solution of the field equation in the anti-de Sitter background with an inflating brane. For a suitable range of the model parameters, we find a solution that realizes slow-roll inflation on the brane. When the Hubble parameter on the brane and the mass of a bulk scalar field are much smaller than a typical 5-dimensional mass scale, it is found that this proposed inflation scenario reproduces the standard inflation scenario in the 4-dimensional theory. 
  One of the possible sources of gamma-ray bursts are merging, compact neutronstar binaries. More than 90% of the binding energy of such a binary is released in the form of gravitational waves (GWs) in the last few seconds of the spiral-in phase before the formation of a black hole. In this article we investigate whether a fraction of this GW-energy is transferred to magnetohydrodynamic waves in the magnetized plasma wind around the binary. Using the 3+1 orthonormal tedrad formalism, we study the propagation of a monochromatic, plane fronted, linearly polarized GW perpendicular to the ambient magnetic field in an ultra-relativistic wind, first in the comoving and then in the observer frame. A closed set of general relativistic magnetohydrodynamic equations is derived in the form of conservation laws for electric charge, matter energy, momentum and magnetic energy densities. We linearize these equations under the action of a monochromatic GW, which acts as a driver and find that fast magneto-acoustic waves grow, with amplitudes proportional to the GW amplitude and frequency and the strength of the background magnetic field. 
  The object of this contribution is twofold. On one hand, it rises some general questions concerning the definition of the electromagnetic field and its intrinsic properties, and it proposes concepts and ways to answer them. On the other hand, and as an illustration of this analysis, a set of quadratic equations for the electromagnetic field is presented, richer in pure radiation solutions than the usual Maxwell equations, and showing a striking property relating geometrical optics to all the other Maxwell solutions. 
  Comparison of the oscillatory behavior of a gravitating infinite Nambu-Goto string and a test string is investigated using the general relativistic gauge invariant perturbation technique with two infinitesimal parameters on a flat spacetime background. Due to the existence of the pp-wave exact solution, we see that the conclusion that the dynamical degree of freedom of an infinite Nambu-Goto string is completely determined by that of gravitational waves, which was reached in our previous works [K. Nakamura, A. Ishibashi and H. Ishihara, Phys. Rev. D{\bf 62} (2002), 101502(R); K. Nakamura and H. Ishihara, Phys. Rev. D{\bf 63} (2001), 127501.], do not contradict to the dynamics of a test string. We also briefly discuss the implication of this result. 
  The initial data of gravity for a cylindrical matter distribution confined to a brane are studied in the framework of the single-brane Randall-Sundrum scenario. In this scenario, the 5-dimensional nature of gravity appears in the short-range gravitational interaction. We find that a sufficiently thin configuration of matter leads to the formation of a marginal surface, even if the configuration is infinitely long. This implies that the hoop conjecture proposed by Thorne does not hold on the brane: Even if a mass $M$ does not become compacted into a region whose circumference ${\cal C}$ in every direction satisfies ${\cal C}> 4\pi GM$, black holes with horizons can form in the Randall-Sundrum scenario. 
  Recently, Sahlmann proposed a new, algebraic point of view on the loop quantization. He brought up the issue of a star-algebra underlying that framework, studied the algebra consisting of the fluxes and holonomies and characterized its representations. We define the diffeomorphism covariance of a representation of the Sahlmann algebra and study the diffeomorphism covariant representations. We prove they are all given by Sahlmann's decomposition into the cyclic representations of the sub-algebra of the holonomies by using a single state only. The state corresponds to the natural measure defined on the space of the generalized connections. This result is a generalization of Sahlmann's result concerning the U(1) case. 
  In a recent paper we presented analytic expressions for the axis potential, the disk metric, and the surface mass density of the global solution to Einstein's field equations describing a rigidly rotating disk of dust. Here we add the complete solution in terms of ultraelliptic functions and quadratures. 
  In a previous paper, a binary black hole four-metric was presented in a post-Newtonian corotating coordinate system valid only up to the holes' apparent horizons. In this paper, I define an ingoing coordinate transformation that extends this corotating coordinate system through the holes' horizons and into their interiors. The motivation for using ingoing coordinates is that numerical simulations of black holes require the computational grid to extend inside the horizons. The coordinate transformation presented here makes the binary black hole four-metric suitable as a source of initial data for numerical simulations. 
  A new class of exact solutions to the axisymmetric and stationary vacuum Einstein equations containing n arbitrary complex parameters and one arbitrary real solution of the axisymmetric three-dimensional Laplace equation is presented. The solutions are related to Jacobi's inversion problem for hyperelliptic Abelian integrals. 
  In the minisuperspace models of quantum cosmology, the absence of time in the Wheeler-DeWitt (constraint) equation, is the main point leading to the generally accepted conclusion that in the quantum cosmology there is no possibility to describe the evolution of the universe procceding in the cosmic time (the time usually used in classical cosmology). We show that in spite of the constraint, under the specific circumstances, the averaging of some of the Heisenberg equations can give nontrivial additional information about explicit time dependence of the expectation values of certain dynamical variables in quantum cosmology. This idea is realized explicitly in a higher dimensional model with a negative cosmological constant and dust as the sources of gravity. When there is an anisotropy in the evolution of the universe, the above phenomenon (i.e. explicit cosmic time dependence of certain expectation values) appears and we find the new quantum effect which consists in "quantum inflationary phase" for some dimensions and simultaneous "quantum deflationary contraction" for the remaining dimensions. The expectation value of the "volume" of the universe remains constant during this quantum "inflation-deflation" process. 
  The recently introduced consistent discrete lattice formulation of canonical general relativity produces a discrete theory that is constraint-free. This immediately allows to overcome several of the traditional obstacles posed by the ``problem of time'' in totally constrained systems and quantum gravity and cosmology. In particular, one can implement the Page--Wootters relational quantization. This brief paper discusses this idea in the context of a simple model system --the parameterized particle-- that is usually considered one of the crucial tests for any proposal for solution to the problem of time in quantum gravity. 
  We introduce an exactly solvable example of timelike geodesic motion and geodesic deviation in the background geometry of a well-known two-dimensional black hole spacetime. The effective potential for geodesic motion turns out to be either a harmonic oscillator or an inverted harmonic oscillator or a linear function of the spatial variable, corresponding to the three different domains of a constant of the motion. The geodesic deviation equation also is exactly solvable. The corresponding deviation vector is obtained and the nature of the deviation is briefly discussed by highlighting a specific case. 
  In the present paper we consider, using our earlier results, the process of quantum gravitational collapse and argue that there exists the final quantum state when the collapse stops. This state, which can be called the ``no-memory state'', reminds the final ``no-hair state'' of the classical gravitational collapse. Translating the ``no-memory state'' into classical language we construct the classical analogue of quantum black hole and show that such a model has a topological temperature which equals exactly the Hawking's temperature. Assuming for the entropy the Bekenstein-Hawking value we develop the local thermodynamics for our model and show that the entropy is naturally quantized with the equidistant spectrum S + gamma_0*N. Our model allows, in principle, to calculate the value of gamma_0. In the simplest case, considered here, we obtain gamma_0 = ln(2). 
  The information contained in galactic rotation curves is examined under a minimal set of assumptions. If emission occurs from stable circular geodesic orbits of a static spherically symmetric field, with information propagated to us along null geodesics, observed rotation curves determine galactic potentials without specific reference to any metric theory of gravity. Given the potential, the gravitational mass can be obtained by way of an anisotropy function of this field. The gravitational mass and anisotropy function can be solved for simultaneously in a Newtonian limit without specifying any specific source. This procedure, based on a minimal set of assumptions, puts very strong constraints on any model of the "dark matter". 
  We discuss inhomogeneous cosmological models which satisfy the Copernican principle. We construct some inhomogeneous cosmological models starting from the ansatz that the all the observers in the models view an isotropic cosmic microwave background. We discuss multi-fluid models, and illustrate how more general inhomogeneous models may be derived, both in General Relativity and in scalar-tensor theories of gravity. Thus we illustrate that the cosmological principle, the assumption that the Universe we live in is spatially homogeneous, does not necessarily follow from the Copernican principle and the high isotropy of the cosmic microwave background. 
  In contrast to the phenomenon of nullification of the cosmological constant in the equilibrium vacuum, which is the general property of any quantum vacuum, there are many options in modifying the Einstein equation to allow the cosmological constant to evolve in a non-equilibrium vacuum. An attempt is made to extend the Einstein equation in the direction suggested by the condensed-matter analogy of the quantum vacuum. Different scenarios are found depending on the behavior of and the relation between the relaxation parameters involved, some of these scenarios having been discussed in the literature. One of them reproduces the scenario in which the effective cosmological constant emerges as a constant of integration. The second one describes the situation, when after the cosmological phase transition the cosmological constant drops from zero to the negative value; this scenario describes the relaxation from this big negative value back to zero and then to a small positive value. In the third example the relaxation time is not a constant but depends on matter; this scenario demonstrates that the vacuum energy (or its fraction) can play the role of the cold dark matter. 
  Several relativistic quantum gravitational effects such as spin-rotation coupling, gravitomagnetic charge and gravitational Meissner effect are investigated in the present letter. The field equation of gravitomagnetic matter is suggested and a static spherically symmetric solution of this equation is offered. With foreseeable improvements in detecting and measuring technology, it is possible for us to investigate quantum mechanics in weak-gravitational fields. The potential implications of these gravitational effects (or phenomena) to some problems are briefly discussed. 
  In the 3+1 framework of the Einstein equations for the case of vanishing shift vector and arbitrary lapse, we calculate explicitly the four boundary equations arising from the vanishing of the projection of the Einstein tensor along the normal to the boundary surface of the initial-boundary value problem. Such conditions take the form of evolution equations along (as opposed to across) the boundary for certain components of the extrinsic curvature and for certain space-derivatives of the intrinsic metric. We argue that, in general, such boundary conditions do not follow necessarily from the evolution equations and the initial data, but need to be imposed on the boundary values of the fundamental variables. Using the Einstein-Christoffel formulation, which is strongly hyperbolic, we show how three of the boundary equations should be used to prescribe the values of some incoming characteristic fields. Additionally, we show that the fourth one imposes conditions on some outgoing fields. 
  We use rigorous techniques from numerical analysis of hyperbolic equations in bounded domains to construct stable finite-difference schemes for Numerical Relativity, in particular for their use in black hole excision. As an application, we present 3D simulations of a scalar field propagating in a Schwarzschild black hole background. 
  We use the analytic continuation procedure proposed in our earlier works to study the thermodynamics of black holes in 2+1 dimensions. A general black hole in 2+1 dimensions has g handles hidden behind h horizons. The result of the analytic continuation is a hyperbolic 3-manifold having the topology of a handlebody. The boundary of this handlebody is a compact Riemann surface of genus G=2g+h-1. Conformal moduli of this surface encode in a simple way the physical characteristics of the black hole. The moduli space of black holes of a given type (g,h) is then the Schottky space at genus G. The (logarithm of the) thermodynamic partition function of the hole is the Kaehler potential for the Weil-Peterson metric on the Schottky space. Bekenstein bound on the black hole entropy leads us to conjecture a new strong bound on this Kaehler potential. 
  We investigate three-dimensional perfect fluid stars with polytropic equation of state, matched to the exterior three-dimensional black hole geometry of Banados, Teitelboim and Zanelli. A new class of exact solutions for a generic polytropic index is found and analysed. 
  A charge-free, point particle of infinitesimal mass orbiting a Kerr black hole is known to move along a geodesic. When the particle has a finite mass or charge, it emits radiation which carries away orbital energy and angular momentum, and the orbit deviates from a geodesic.   In this paper we assume that the deviation is small and show that the half-advanced minus half-retarded field surprisingly provides the correct radiation reaction force, in a time-averaged sense, and determines the orbit of the particle. 
  We show that generic anisotropic universes arbitrarily close to the open Friedmann universe allow information processing to continue into the infinite future if there is no cosmological constant or stable gravitationally repulsive stress, and the spatial topology is non-compact. An infinite amount of information can be processed by ``civilisations'' who harness the temperature gradients created by gravitational tidal energy. These gradients are driven by the gravitational waves that sustain the expansion shear and three-curvature anisotropy. 
  We present new analytical self-similar solutions describing a collapse of a massless scalar field in scalar-tensor theories. The solutions exhibit a type of critical behavior. The black hole mass for the near critical evolution is analytically obtained for several scalar-tensor theories and the critical exponent is calculated. Within the framework of the analytical models we consider it is found that the black hole mass law for some scalar-tensor theories is of the form $M_{BH}=f(p-p_{cr})(p-p_{cr})^\gamma$ which is slightly different from the general relativistic law $M_{BH}=const (p-p_{cr})^\gamma$. 
  We show that the square of the Weyl tensor can be negative by giving an example. 
  We analyze the thermodynamical behavior of black holes in closed finite boxes. First the black hole mass evolution is analyzed in an initially empty box. Using the conservation of the energy and the Hawking evaporation flux, we deduce a minimal volume above which one black hole can loss all of its mass to the box, a result which agrees with the previous analysis made by Page. We then obtain analogous results using a box initially containing radiation, allowed to be absorbed by the black hole. The equilibrium times and masses are evaluated and their behavior discussed to highlight some interesting features arising. These results are generalized to $N$ black holes + thermal radiation. Using physically simple arguments, we prove that these black holes achieve the same equilibrium masses (even that the initial masses were different). The entropy of the system is used to obtain the dependence of the equilibrium mass on the box volume, number of black holes and the initial radiation. The equilibrium mass is shown to be proportional to a {\it positive} power law of the effective volume (contrary to naive expectations), a result explained in terms of the detailed features of the system. The effect of the reflection of the radiation on the box walls which comes back into the black hole is explicitly considered. All these results (some of them counter-intuitive) may be useful to formulate alternative problems in thermodynamic courses for graduate and advanced undergraduate students. A handful of them are suggested in the Appendix. 
  Necessary and sufficient conditions for a space-time to be conformal to an Einstein space-time are interpreted in terms of curvature restrictions for the corresponding Cartan conformal connection. 
  We classify all five-dimensional Einstein manifolds that are static, have an SO(3) isometry group and have Petrov type 22. We use this classification to show that the localized black hole in the Randall-Sundrum scenario necessarily has Petrov type 4. 
  The relation between a recently proposed path integral for minisuperspaces and different canonical quantizations is established. The step of the procedure where a choice between non equivalent theories is made is identified. Coordinates avoiding such a choice are found for a class of homogeneous cosmologies. 
  A general covariant extension of Einstein\'{}s field equations is considered with a view to Numerical Relativity applications. The basic variables are taken to be the metric tensor and an additional four-vector $Z_\mu$. Einstein's solutions are recovered when the additional four-vector vanishes, so that the energy and momentum constraints amount to the covariant algebraic condition $Z_\mu=0$. The extended field equations can be supplemented by suitable coordinate conditions in order to provide symmetric hyperbolic evolution systems: this is actually the case for either harmonic coordinates or normal coordinates with harmonic slicing. 
  In the framework of the rest-frame instant form of tetrad gravity, where the Hamiltonian is the weak ADM energy ${\hat E}_{ADM}$, we define a special completely fixed 3-orthogonal Hamiltonian gauge, corresponding to a choice of {\it non-harmonic} 4-coordinates, in which the independent degrees of freedom of the gravitational field are described by two pairs of canonically conjugate Dirac observables (DO) $r_{\bar a}(\tau ,\vec \sigma)$, $\pi_{\bar a}(\tau ,\vec \sigma)$, $\bar a = 1,2$. We define a Hamiltonian linearization of the theory, i.e. gravitational waves, {\it without introducing any background 4-metric}, by retaining only the linear terms in the DO's in the super-hamiltonian constraint (the Lichnerowicz equation for the conformal factor of the 3-metric) and the quadratic terms in the DO's in ${\hat E}_{ADM}$. {\it We solve all the constraints} of the linearized theory: this amounts to work in a well defined post-Minkowskian Christodoulou-Klainermann space-time. The Hamilton equations imply the wave equation for the DO's $r_{\bar a}(\tau ,\vec \sigma)$, which replace the two polarizations of the TT harmonic gauge, and that {\it linearized Einstein's equations are satisfied} . Finally we study the geodesic equation, both for time-like and null geodesics, and the geodesic deviation equation. 
  A Hamiltonian linearization of the rest-frame instant form of tetrad gravity (gr-qc/0302084), where the Hamiltonian is the weak ADM energy ${\hat E}_{ADM}$, in a completely fixed (non harmonic) 3-orthogonal Hamiltonian gauge is defined. For the first time this allows to find an explicit solution of all the Hamiltonian constraints and an associated linearized solution of Einstein's equations. It corresponds to background-independent gravitational waves in a well defined post-Minkowskian Christodoulou-Klainermann space-time. 
  Effects of space time geometry fluctuations on fermionic fields have recently been looked for in nuclear physics experiments, and were found to be much lower than predicted, at a phenomenological level, by loop quantum gravity. We show that possible corrections to the canonical structure in the semi classical regime may introduce important changes in the outcome of the theory, and may explain the observed mismatch with experiments. 
  The Kerr vacuum has two independent invariants derivable from the Riemann tensor without differentiation. Both of these invariants must be examined in order to avoid an erroneous conclusion that the ring singularity of this spacetime is "directional". 
  It has been shown that the new Self Creation Cosmology theory predicts a universe with a total density parameter of one third yet spatially flat, which would appear to accelerate in its expansion. Although requiring a moderate amount of 'cold dark matter' the theory does not have to invoke the hypotheses of inflation, 'dark energy', 'quintessence' or a cosmological constant (dynamical or otherwise) to explain observed cosmological features. The theory also offers an explanation for the observed anomalous Pioneer spacecraft acceleration, an observed spin-up of the Earth and an problematic variation of G observed from analysis of the evolution of planetary longitudes. It predicts identical results as General Relativity in standard experimental tests but three definitive experiments do exist to falsify the theory. In order to match the predictions of General Relativity, and observations in the standard tests, the new theory requires the Brans Dicke omega parameter that couples the scalar field to matter to be -3/2 . Here it is shown how this value for the coupling parameter is determined by the theory's basic assumptions and therefore it is an inherent property of the principles upon which the theory is based. 
  The individuation of point-events and the Hamiltonian way of distinguishing gravitational from inertial effects in general relativity are discussed. 
  Much of the work in loop quantum gravity and quantum geometry rests on a mathematically rigorous integration theory on spaces of distributional connections. Most notably, a diffeomorphism invariant representation of the algebra of basic observables of the theory, the Ashtekar-Lewandowski representation, has been constructed. This representation is singled out by its mathematical elegance, and up to now, no other diffeomorphism invariant representation has been constructed. This raises the question whether it is unique in a precise sense.   In the present article we take steps towards answering this question. Our main result is that upon imposing relatively mild additional assumptions, the AL-representation is indeed unique.   As an important tool which is also interesting in its own right, we introduce a C*-algebra which is very similar to the Weyl algebra used in the canonical quantization of free quantum field theories. 
  Adiabatic perturbations propagate in the expanding universe like scalar massless fields in some effective Robertson-Walker space-time. 
  It is shown that the extra coordinate of 5D induced-matter and membrane theory is related in certain gauges to the inertial rest mass of a test particle. This implies that the Weak Equivalence Principle is a geometric symmetry, valid only in the limit in which the test mass is negligible compared to the source mass. Exact solutions illustrate this, and show the way to possible resolutions of the cosmological-constant and hierarchy problems. 
  Current interferometric gravitational wave detectors use test masses with mirror coatings formed from multiple layers of dielectric materials, most commonly alternating layers of SiO2 (silica) and Ta2O5 (tantala). However, mechanical loss in the Ta2O5/SiO2 coatings may limit the design sensitivity for advanced detectors. We have investigated sources of mechanical loss in the Ta2O5/SiO2 coatings, including loss associated with the coating-substrate interface, with the coating-layer interfaces, and with the bulk material. Our results indicate that the loss is associated with the bulk coating materials and that the loss of Ta2O5 is substantially larger than that of SiO2. 
  We show using covariant techniques that the Einstein static universe containing a perfect fluid is always neutrally stable against small inhomogeneous vector and tensor perturbations and neutrally stable against adiabatic scalar density inhomogeneities so long as c_{s}^2>1/5, and unstable otherwise. We also show that the stability is not significantly changed by the presence of a self-interacting scalar field source, but we find that spatially homogeneous Bianchi type IX modes destabilise an Einstein static universe. The implications of these results for the initial state of the universe and its pre-inflationary evolution are also discussed. 
  We discuss the Kretschmann, Chern-Pontryagin and Euler invariants among the second order scalar invariants of the Riemann tensor in any spacetime in the Newman-Penrose formalism and in the framework of gravitoelectromagnetism, using the Kerr-Newman geometry as an example. An analogy with electromagnetic invariants leads to the definition of regions of gravitoelectric or gravitomagnetic dominance. 
  Some of the possible consequences of a generalized uncertainty principle (which emerges in the context of string theory and quantum gravity models as a consequence of fluctuations of the background metric) are analyzed considering the case of a quantum particle immersed in a homogeneous gravitational field. It will be shown that the expectation value of the momentum operator depends in a novel way on the mass of the involved particle. This kind of physical characteristics could be, in principle, detected. In other words, one way of confronting against the experiment some of the models around quantum gravity is given by the detection of the dependence upon the mass parameter of the expectation value of the momentum operator. 
  In (1+2)-dimensional Poincar\'e gauge gravity, we start from a Lagrangian depending on torsion and curvature which includes additionally {\em translational} and {\em Lorentzian} Chern-Simons terms. Limiting ourselves to to a specific subcase, the Mielke-Baekler (MB) model, we derive the corresponding field equations (of Einstein-Cartan-Chern-Simons type) and find the general vacuum solution. We determine the properties of this solution, in particular its mass and its angular momentum. For vanishing torsion, we recover the BTZ-solution. We also derive the general conformally flat vacuum solution with torsion. In this framework, we discuss {\em Cartan's} (3-dimensional) {\em spiral staircase} and find that it is not only a special case of our new vacuum solution, but can alternatively be understood as a solution of the 3-dimensional Einstein-Cartan theory with matter of constant pressure and constant torque. 
  Much of the published work regarding the Isotropic Singularity is performed under the assumption that the matter source for the cosmological model is a barotropic perfect fluid, or even a perfect fluid with a $\gamma$-law equation of state. There are, however, some general properties of cosmological models which admit an Isotropic Singularity, irrespective of the matter source. In particular, we show that the Isotropic Singularity is a point-like singularity and that vacuum space-times cannot admit an Isotropic Singularity. The relationships between the Isotropic Singularity, and the energy conditions, and the Hubble parameter is explored. A review of work by the authors, regarding the Isotropic Singularity, is presented. 
  Although the laws of thermodynamics are well established for black hole horizons, much less has been said in the literature to support the extension of these laws to more general settings such as an asymptotic de Sitter horizon or a Rindler horizon (the event horizon of an asymptotic uniformly accelerated observer). In the present paper we review the results that have been previously established and argue that the laws of black hole thermodynamics, as well as their underlying statistical mechanical content, extend quite generally to what we call here "causal horizons". The root of this generalization is the local notion of horizon entropy density. 
  In light of recent study on the dark energy models that manifest an equation of state $w<-1$, we investigate the cosmological evolution of phantom field in a specific potential, exponential potential in this paper. The phase plane analysis show that the there is a late time attractor solution in this model, which address the similar issues as that of fine tuning problems in conventional quintessence models. The equation of state $w$ is determined by the attractor solution which is dependent on the $\lambda$ parameter in the potential. We also show that this model is stable for our present observable universe. 
  We exploit once again the analogy between the energy-momentum tensor and the so-called ``superenergy'' tensors in order to build conserved currents in the presence of Killing vectors. First of all, we derive the divergence-free property of the gravitational superenergy currents under very general circumstances, even if the superenergy tensor is not divergence-free itself. The associated conserved quantities are explicitly computed for the Reissner-Nordstrom and Schwarzschild solutions. The remaining cases, when the above currents are not conserved, lead to the possibility of an interchange of some superenergy quantities between the gravitational and other physical fields in such a manner that the total, mixed, current may be conserved. Actually, this possibility has been recently proved to hold for the Einstein-Klein-Gordon system of field equations. By using an adequate family of known exact solutions, we present explicit and completely non-obvious examples of such mixed conserved currents. 
  We investigate spontaneous symmetry breaking in a conformally invariant gravitational model. In particular, we use a conformally invariant scalar tensor theory as the vacuum sector of a gravitational model to examine the idea that gravitational coupling may be the result of a spontaneous symmetry breaking. In this model matter is taken to be coupled with a metric which is different but conformally related to the metric appearing explicitly in the vacuum sector. We show that after the spontaneous symmetry breaking the resulting theory is consistent with Mach's principle in the sense that inertial masses of particles have variable configurations in a cosmological context. Moreover, our analysis allows to construct a mechanism in which the resulting large vacuum energy density relaxes during evolution of the universe. 
  We investigate the dynamics of the Papapetrou equations in Kerr spacetime. These equations provide a model for the motion of a relativistic spinning test particle orbiting a rotating (Kerr) black hole. We perform a thorough parameter space search for signs of chaotic dynamics by calculating the Lyapunov exponents for a large variety of initial conditions. We find that the Papapetrou equations admit many chaotic solutions, with the strongest chaos occurring in the case of eccentric orbits with pericenters close to the limit of stability against plunge into a maximally spinning Kerr black hole. Despite the presence of these chaotic solutions, we show that physically realistic solutions to the Papapetrou equations are not chaotic; in all cases, the chaotic solutions either do not correspond to realistic astrophysical systems, or involve a breakdown of the test-particle approximation leading to the Papapetrou equations (or both). As a result, the gravitational radiation from bodies spiraling into much more massive black holes (as detectable, for example, by LISA, the Laser Interferometer Space Antenna) should not exhibit any signs of chaos. 
  We present a class of exact solutions of Einstein's gravitational field equations describing spherically symmetric and static anisotropic stellar type configurations. The solutions are obtained by assuming a particular form of the anisotropy factor. The energy density and both radial and tangential pressures are finite and positive inside the anisotropic star. Numerical results show that the basic physical parameters (mass and radius) of the model can describe realistic astrophysical objects like neutron stars. 
  On the basis of hypotheses, that a density of weakly interacting particles in the Universe has an order of nuclear matter density or more the Lagrangian is offered, through which one can be obtained a propagator of a vector boson with a non-zero rest-mass. The dependence of vector bosons masses on the time allows to explain the availability of the hot stage of the Universe evolution, not using to the hypothesis of the Universe expansion. 
  A formulation is developed for general relativistic ideal magnetohydrodynamics in stationary axisymmetric spacetimes. We reduce basic equations to a single second-order partial differential equation, the so-called Grad-Shafranov (GS) equation. Our formulation is most general in the sense that it is applicable even when a stationary axisymmetric spacetime is noncircular, that is, even when it is impossible to foliate a spacetime with two orthogonal families of two-surfaces. The GS equation for noncircular spacetimes is crucial for the study of relativistic stars with a toroidal magnetic field or meridional flow, such as magnetars, since the existence of a toroidal field or meridional flow violates the circularity of a spacetime. We also derive the wind equation in noncircular spacetimes, and discuss various limits of the GS equation. 
  To predict the outcome of (almost) any experiment we have to assume that our spacetime is globally hyperbolic. The wormholes, if they exist, cast doubt on the validity of this assumption. At the same time, no evidence has been found so far (either observational, or theoretical) that the possibility of their existence can be safely neglected. 
  See hep-ph/0304045 
  See hep-ph/0304045 
  See hep-ph/0304045 
  This is an introduction into the problem of how to set up black hole initial-data for the matter-free field equations of General Relativity. The approach is semi-pedagogical and addresses a more general audience of astrophysicists and students with no specialized training in General Relativity beyond that of an introductory lecture. 
  We study the quantized Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW) model minimally coupled to a free massless scalar field. In a previous paper, \cite{fab2}, solutions of this model were constructed as gaussian superpositions of negative and positive modes solutions of the Wheeler-DeWitt equation, and quantum bohmian trajectories were obtained in the framework of the Bohm-de Broglie (BdB) interpretation of quantum cosmology. In the present work, we analyze the quantum bohmian trajectories of a different class of gaussian packets. We are able to show that this new class generates bohmian trajectories which begin classical (with decelerated expansion), undergo an accelerated expansion in the middle of its evolution due to the presence of quantum cosmological effects in this period, and return to its classical decelerated expansion in the far future. We also show that the relation between luminosity distance and redshift in the quantum cosmological model can be made close to the corresponding relation coming from the classical model suplemented by a cosmological constant, for $z<1$. These results suggest the posibility of interpreting the present observations of high redshift supernovae as the manifestation of a quantum cosmological effect. 
  We discuss the quantum scattering process in the moduli space consisting of two maximally charged dilaton black holes. The black hole moduli space geometry has different structures for arbitrary dimensions and various values of dilaton coupling. We study the quantum effects of the different moduli space geometries with scattering process. Then, it is found that there is a resonance state on certain moduli spaces. 
  It is shown that the class of asymptotically flat solutions to the axisymmetric and stationary vacuum Einstein equations with reflection symmetry of the metric is uniquely characterized by a simple relation for the Ernst potential on the upper part of the symmetry axis. This result generalizes a well-known fact from potential theory. 
  Differential rotation of r-modes is investigated within the nonlinear theory up to second order in the mode amplitude in the case of a slowly-rotating, Newtonian, barotropic, perfect-fluid star. We find a nonlinear extension of the linear r-mode, which represents differential rotation that produces large scale drifts of fluid elements along stellar latitudes. This solution includes a piece induced by first-order quantities and another one which is a pure second-order effect. Since the latter is stratified on cylinders, it cannot cancel differential rotation induced by first-order quantities, which is not stratified on cylinders. It is shown that, unlikely the situation in the linearized theory, r-modes do not preserve vorticity of fluid elements at second-order. It is also shown that the physical angular momentum and energy of the perturbation are, in general, different from the corresponding canonical quantities. 
  The causal properties of curved spacetime, which underpin our sense of time in gravitational theories, are defined by the null cones of the spacetime metric. In classical general relativity, it is assumed that these coincide with the light cones determined by the physical propagation of light rays. However, the quantum vacuum acts as a dispersive medium for the propagation of light, since vacuum polarisation in QED induces interactions which effectively violate the strong equivalence principle (SEP). For low frequencies the phenomenon of gravitational birefringence occurs and indeed, for some metrics and polarisations, photons may acquire {\it superluminal} phase velocities. In this article, we review some of the remarkable features of SEP violating superluminal propagation in curved spacetime and discuss recent progress on the issue of dispersion, explaining why it is the high-frequency limit of the phase velocity that determines the characteristics of the effective wave equation and thus the physical causal structure. 
  It was shown in a previous work that, for systems in which the entropy is an extensive function of the energy and volume, the Bekenstein and the holographic entropy bounds predict new results. In this paper, we go further and derive improved upper bounds to the entropy of {\it extensive} charged and rotating systems. Furthermore, it is shown that for charged and rotating systems (including non-extensive ones), the total energy that appear in both the Bekenstein entropy bound (BEB) and the causal entropy bound (CEB) can be replaced by the {\it internal} energy of the system. In addition, we propose possible corrections to the BEB and the CEB. 
  Lyapunov exponents (LEs) are key indicators of chaos in dynamical systems. In general relativity the classical definition of LE meets difficulty because it is not coordinate invariant and spacetime coordinates lose their physical meaning as in Newtonian dynamics. We propose a new definition of relativistic LE and give its algorithm in any coordinate system, which represents the observed changing law of the space separation between two neighboring particles (an 'observer' and a 'neighbor'), and is truly coordinate invariant in a curved spacetime. 
  In this paper Quantum Mechanics with Fundamental Length is chosen as the theory for describing the early Universe. This is possible due to the presence in the theory of General Uncertainty Relations from which unavoidable it follows that in nature a fundamental length exits. Here Quantum  Mechanics with Fundamental Length is obtained as a deformation of Quantum Mechanics. The distinguishing feature of the proposed in this paper approach in comparison with previous ones, lies on the fact that here density matrix subjects to deformation as well as so far commutators had been deformed. The deformed density matrix mentioned above, is named throughout this paper density pro-matrix. Within our approach two main features of Quantum Mechanics are conserved: the probabilistic interpretation of the theory and exact predefined measuring procedure corresponding to that interpretation. The proposed here approach allows to describe dynamics. In particular, the explicit form of deformed Liouville's equation and the deformed Shr\"odinger's picture are given. Some implications of obtained results are discussed. In particular, the problem of singularity, the hypothesis of cosmic censorship, a possible improvement of the statistical entropy definition and the problem of information loss in black holes are considered. 
  We examine the dynamical behavior of matter coupled to gravity in the context of a linear Klein-Gordon equation coupled to a Friedman-Robertson-Walker metric. The resulting ordinary differential equations can be decoupled, the effect of gravity being traced in rendering the equation for the scalar field nonlinear. We obtain regular (in the massless case) and asymptotic (in the massive case) solutions for the resulting matter field and discuss their ensuing finite time blowup in the light of earlier findings. Finally, some potentially interesting connections of these blowups with features of focusing in the theory of nonlinear partial differential equations are outlined, suggesting the potential relevance of a nonlinear theory of quantum cosmology. 
  A metric with signature (-+++) can be constructed from a metric with signature (++++) and a double-sided vector field called the line element field. Some of the classical and quantum properties of this vector field are studied. 
  We address several criticisms by Amelino-Camelia of our recent analyses of two observational constraints on Lorentz violation at order E/M_{Planck}. In particular, we emphasize the role of effective field theory in our analysis of synchrotron radiation, and we strengthen the justification for the constraint coming from photon annihilation. 
  The principle of equivalence in gravitational physics and its mathematical base are reviewed. It is demonstrated how this principle can be realized in classical electrodynamis. In general, it is valid at any given single point or along a path without selfintersections unless the field considered satisfies some conditions. 
  We study the behavior of spatially homogeneous brane-worlds close to the initial singularity in the presence of both local and nonlocal stresses. It is found that the singularity in these brane-worlds can be locally either isotropic or anisotropic. We then investigate the Weyl curvature conjecture, according to which some measure of the Weyl curvature is related to a gravitational entropy. In particular, we study the Weyl curvature conjecture on the brane with respect to the dimensionless ratio of the Weyl invariant and the Ricci square and the measure proposed by Gr{\o}n and Hervik. We also argue that the Weyl curvature conjecture should be formulated on brane  (i.e., in the four-dimensional context). 
  We propose a way to construct manifestly gauge independent quantities out of the gauge dependent quantities occurring in the linearized Einstein equations. Thereupon, we show that these gauge-invariant combinations can be identified with measurable perturbations to the particle and energy densities.    In the radiation-dominated era we find, for small-scale perturbations, acoustic waves with an increasing amplitude, while standard treatments predict acoustic waves with a decaying amplitude. For large-scale perturbations we find exactly the same growth rates as in the standard literature.    When considering the non-relativistic limit of the linearized Einstein equations we find the Poisson equation.    It is shown, using the linearized Einstein equations, that the usual Newtonian treatment of density perturbations does not describe the evolution of density perturbations. 
  In this Brief Report we give the proof that the solution of any static test charge distribution in Schwarzschild space is unique. In order to give the proof we derive the first Green's identity written with p-forms on (pseudo) Riemannian manifolds. Moreover, the proof of uniqueness can be shown for either any purely electric or purely magnetic field configuration. The spacetime geometry is not crucial for the proof. 
  A semi-classical reasoning leads to the non-commutativity of the space and time coordinates near the horizon of Schwarzschild black hole. This non-commutativity in turn provides a mechanism to interpret the brick wall thickness hypothesis in 't Hooft's brick wall model as well as the boundary condition imposed for the field considered. For concreteness, we consider a noncommutative scalar field model near the horizon and derive the effective metric via the equation of motion of noncommutative scalar field. This metric displays a new horizon in addition to the original one associated with the Schwarzschild black hole. The infinite red-shifting of the scalar field on the new horizon determines the range of the noncommutativ space and explains the relevant boundary condition for the field. This range enables us to calculate the entropy of black hole as proportional to the area of its original horizon along the same line as in 't Hooft's model, and the thickness of the brick wall is found to be proportional to the thermal average of the noncommutative space-time range. The Hawking temperature has been derived in this formalism. The study here represents an attempt to reveal some physics beyond the brick wall model. 
  See hep-ph/0304045 
  Simple ideas that shed new light on the physics of rotation as it concerns two famous experiments: The Wilson and Wilson, and the Michelson and Morley experiments. 
  We utilize Moller's and Einstein's energy-momentum complexes in order to explicitly evaluate the energy distributions associated with the two-dimensional "Schwarzschild" and "Reissner-Nordstrom" black hole backgrounds. While Moller's prescription provides meaningful physical results, Einstein's prescription fails to do so in the aforementioned gravitational backgrounds. These results hold for all two-dimensional static black hole geometries. The results obtained within this context are exploited in order Seifert's hypothesis to be investigated. 
  We present here the general transformation that leaves unchanged the form of the field equations for perfect fluid Friedmann--Robertson--Walker and Bianchi V cosmologies. The symmetries found can be used as algorithms for generating new cosmological models from existing ones. A particular case of the general transformation is used to illustrate the crucial role played by the number of scalar fields in the occurrence of inflation. Related to this, we also study the existence and stability of Bianchi V power law solutions. 
  We compute the quasinormal frequencies of rotating black holes using the continued fraction method first proposed by Leaver. The main difference with former works, is that our results are obtained by a new numerical technique which avoids the use of two dimensional root-finding routines. The technique is applied to evaluate the angular eigenvalues of Teukolsky's angular equation. This method allow us to calculate both the slowly and the rapidly damped quasinormal frequencies with excellent accuracy. 
  We investigate superresonant scattering of acoustic disturbances from a rotating acoustic black hole in the low frequency range. We derive an expression for the reflection coefficient, exhibiting its frequency dependence in this regime. 
  We discuss the baseline optical configuration for the Laser Interferometer Space Antenna (LISA) mission, in which the lasers are not free-running, but rather one of them is used as the main frequency reference generator (the {\it master}) and the remaining five as {\it slaves}, these being phase-locked to the master (the {\it master-slave configuration}). Under the condition that the frequency fluctuations due to the optical transponders can be made negligible with respect to the secondary LISA noise sources (mainly proof-mass and shot noises), we show that the entire space of interferometric combinations LISA can generate when operated with six independent lasers (the {\it one-way method}) can also be constructed with the {\it master-slave} system design. The corresponding hardware trade-off analysis for these two optical designs is presented, which indicates that the two sets of systems needed for implementing the {\it one-way method}, and the {\it master-slave configuration}, are essentially identical. Either operational mode could therefore be implemented without major implications on the hardware configuration. We then....... 
  We study the late time evolution of flat and negatively curved FRW models with a perfect fluid matter source and a scalar field having an arbitrary non-negative potential function $V(\phi) .$ We prove using a dynamical systems approach four general results for a large class of non-negative potentials which show that almost always the universe ever expands. In particular, for potentials having a local zero minimum, flat and negatively curved FRW models are ever expanding and the energy density asymptotically approaches zero. We investigate the conditions under which the scalar field asymptotically approaches the minimum of the potential. We discuss the question of whether a closed FRW with ordinary matter can avoid recollapse due to the presence of a scalar field with a non-negative potential. 
  The theory of perfect fluids is reconsidered from the point of view of a covariant Lagrangian theory. It has been shown that the Euler-Lagrange equations for a perfect fluid could be found in spaces with affine connections and metrics from an unconstrained variational principle by the use of the method of Lagrangians with covariant derivatives (MLCD) and additional conditions for reparametrizations of the proper time of the mass elements (particles) of the perfect fluid. The last conditions are not related to the variational principle and are not considered as constraints used in the process of variations. The application of the whole structure of a Lagrangian theory with an appropriate choice of Lagrangian invariant as the pressure of the fluid shows that the Euler-Lagrange equations with their corresponding energy-momentum tensors lead to Navier-Stokes' equation identical with the Euler equation for a perfect fluid in a space with one affine connection and metrics. The Navier-Stokes equations appear as higher order equations with respect to the Euler-Lagrange equations. 
  In continuing our earlier research, we find the formulae needed to determine the arbitrary functions in the Lemaitre-Tolman model when the evolution proceeds from a given initial velocity distribution to a final state that is determined either by a density distribution or by a velocity distribution. In each case the initial and final distributions uniquely determine the L-T model that evolves between them, and the sign of the energy-function is determined by a simple inequality. We also show how the final density profile can be more accurately fitted to observational data than was done in the previous paper. We work out new numerical examples of the evolution: the creation of a galaxy cluster out of different velocity distributions, reflecting the current data on temperature anisotropies of CMB, the creation of the same out of different density distributions, and the creation of a void. The void in its present state is surrounded by a nonsingular wall of high density. 
  We study the behavior of spiky features in Gowdy spacetimes. Spikes with velocity initially high are, generally, driven to low velocity. Let n be any integer greater than or equal to 1. If the initial velocity of an upward pointing spike is between 4n-3 and 4n-1 the spike persists with final velocity between 1 and 2, while if the initial velocity is between 4n-1 and 4n+1, the spiky feature eventually disappears. For downward pointing spikes the analogous rule is that spikes with initial velocity between 4n-4 and 4n-2 persist with final velocity between 0 and 1, while spikes with initial velocity between 4n-2 and 4n eventually disappear. 
  Recent results from linear perturbation theory suggest that first-order cosmological quark-hadron phase transitions occurring as deflagrations may be ``borderline'' unstable, and those occurring as detonations may give rise to growing modes behind the interface boundary. However, since nonlinear effects can play important roles in the development of perturbations, unstable behavior cannot be asserted entirely by linear analysis, and the uncertainty of these recent studies is compounded further by nonlinearities in the hydrodynamics and self-interaction fields. In this paper we investigate the growth of perturbations and the stability of quark-hadron phase transitions in the early Universe by solving numerically the fully nonlinear relativistic hydrodynamics equations coupled to a scalar field with a quartic self-interaction potential regulating the transitions. We consider single, perturbed, phase transitions propagating either by detonation or deflagration, as well as multiple phase and shock front interactions in 1+2 dimensional spacetimes. 
  Motivated by the important work of Brown adn York on quasilocal energy, we propose definitions of quasilocal energy and momentum surface energy of a spacelike 2-surface with positive intrinsic curvature in a spacetime. We show that the quasilocal energy of the boundary of a compact spacelike hypersurface which satisfies the local energy condition is strictly positive unless the spacetime is flat along the spacelike hypersurface. 
  Based on the recent work \cite{PII} we put forward a new type of transformation for Lorentzian manifolds characterized by mapping every causal future-directed vector onto a causal future-directed vector. The set of all such transformations, which we call causal symmetries, has the structure of a submonoid which contains as its maximal subgroup the set of conformal transformations. We find the necessary and sufficient conditions for a vector field $\xiv$ to be the infinitesimal generator of a one-parameter submonoid of pure causal symmetries. We speculate about possible applications to gravitation theory by means of some relevant examples. 
  We consider the problem of finding the gravitational radiation output, or news, within the context of a numerical simulation of a spacetime by means of the null-cone, or characteristic, approach to numerical relativity. We develop a method for computing the news that uses an explicit coordinate transformation to a coordinate system that satisfies the Bondi conditions. The method has been implemented computationally. We present results of applying the method to certain test problems, demonstrating second order convergence of the news to the analytic value. 
  Problem of cosmological singularity of general relativity theory is discussed. The possible resolution of this problem in the framework of inflationary cosmology is proposed. Physical conditions leading to bouncing inflationary solutions in the frame of general relativity theory and gauge theories of gravitation are compared. It is shown that gauge theories of gravitation allow to build regular inflationary cosmological models of closed, open and flat type with dominating ultrarelativistic matter at a bounce. 
  The maximum entropy that can be stored in a bounded region of space is in dispute: it goes as volume, implies (non-gravitational) microphysics; it goes as the surface area, asserts the "holographic principle." Here I show how the holographic bound can be derived from elementary flat-spacetime quantum field theory when the total energy of Fock states is constrained gravitationally. This energy constraint makes the Fock space dimension (whose logarithm is the maximum entropy) finite for both Bosons and Fermions. Despite the elementary nature of my analysis, it results in an upper limit on entropy in remarkable agreement with the holographic bound. 
  In this paper we study a type of one-field model for open inflationary universe models in the context of the Jordan-Brans-Dicke theory. In the scenario of a one-bubble universe model we determine and characterize the existence of the Coleman-De Lucia instanton, together with the period of inflation after tunnelling has occurred. Our results are analogous to those found in the Einstein General Relativity models. 
  We discuss the asymptotic structure of spacetimes, presenting a new construction of ideal points at infinity and introducing useful topologies on the completed space. Our construction is based on structures introduced by Geroch, Kronheimer, and Penrose and has much in common with the modifications introduced by Budic and Sachs as well as those introduced by Szabados. However, these earlier constructions defined ideal points as equivalence classes of certain past and future sets, effectively defining the completed space as a quotient. Our approach is fundamentally different as it identifies ideal points directly as appropriate pairs consisting of a (perhaps empty) future set and a (perhaps empty) past set. These future and past sets are just the future and past of the ideal point within the original spacetime. This provides our construction with useful causal properties and leads to more satisfactory results in a number of examples. We are also able to endow the completion with a topology. In fact, we introduce two topologies, which illustrate different features of the causal approach. In both topologies, the completion is the original spacetime together with endpoints for all timelike curves. We explore this procedure in several examples, and in particular for plane wave solutions, with satisfactory results. 
  The most common spin foam models of gravity are widely believed to be discrete path integral quantizations of the Plebanski action. However, their derivation in present formulations is incomplete and lower dimensional simplex amplitudes are left open to choice. Since the large-spin behavior of these amplitudes determines the convergence properties of the state-sum, this gap has to be closed before any reliable conclusion about finiteness can be reached. It is shown that these amplitudes are directly related to the path integral measure and can in principle be derived from it. This requires a detailed knowledge of the constraint algebra and the corresponding gauge fixing of its first class part which in the case of gravity generates space-time diffeomorphisms. It has been suggested that the discretization of space-time in a spin foam model breaks the diffeomorphism gauge without introducing an explicit gauge fixing. Here we show that minimal requirements of background independence -which are reminiscent of cylindrical consistency in loop quantum gravity- provide non trivial restrictions on the form of an anomaly free measure. Many models in the literature do not satisfy these requirements. Moreover, we show that an anomaly free model will necessarily contain divergent amplitudes that could be interpreted as due to infinite contributions of gauge equivalent configurations. Exploring these issues we come across a simple model satisfying the above consistency requirements which can be thought of as a spin foam quantization of the Husain-Kuchar model. 
  Contents:  Community news:   GGR activities, by Richard Price   We hear that..., by Jorge Pullin   Institute of Physics Gravitational Physics Group, by Elizabeth Winstanley   Center for gravitational wave astronomy, by Mario Diaz  Research briefs:   LIGO's first preliminary science run, by Gary Sanders   Quantization of area: the plot thickens, by John Baez   Convergence of G measurements -Mysteries remain, by Riley Newman  Conference reports:   Brane world gravity, by Elizabeth Winstanley   Massive black holes focus session, by Steinn Sigurdsson   GWDAW 2002, by Peter Saulson   Source simulation focus session, by Pablo Laguna   RRI workshop on loop quantum gravity, by Fernando Barbero   Lazarus/Kudu Meeting, by Warren G. Anderson 
  Although slow light (electromagnetically induced transparency) would seem an ideal medium in which to institute a ``dumb hole'' (black hole analog), it suffers from a number of problems. We show that the high phase velocity in the slow light regime ensures that the system cannot be used as an analog displaying Hawking radiation. Even though an appropriately designed slow-light set-up may simulate classical features of black holes -- such as horizon, mode mixing, Bogoliubov coefficients, etc. -- it does not reproduce the related quantum effects. PACS: 04.70.Dy, 04.80.-y, 42.50.Gy, 04.60.-m. 
  The hypothesis of locality, its origin and consequences are discussed. This supposition is necessary for establishing the local spacetime frame of accelerated observers; in this connection, the measurement of length in a rotating system is considered in detail. Various limitations of the hypothesis of locality are examined. 
  The relation between logarithmic corrections to the area law for black hole entropy, due to thermal fluctuations around an equilibrium canonical ensemble, and those originating from quantum spacetime fluctuations within a microcanonical framework, is explored for three and four dimensional asymptotically anti-de Sitter black holes. For the BTZ black hole, the two logarithmic corrections are seen to precisely cancel each other, while for four dimensional adS-Schwarzschild black holes a partial cancellation is obtained. We discuss the possibility of extending the analysis to asymptotically flat black holes. 
  On the base of an exact solution for the static spherically symmetric Einstein equations with the quintessential dark matter, we explain the asymptotic behavior of rotation curves in spiral galaxies. The parameter of the quintessence state, i.e. the ratio of its pressure to the density is tending to -1/3. We present an opportunity to imitate the relevant quintessence by appropriate scalar fields in the space-time with extra 2 dimensions. 
  We present a new exact solution in Brans-Dicke theory. The solution describes inhomogeneous plane-symmetric perfect fluid cosmological model with an equation of state $p=\gamma \rho$. Some main properties of the solution are discussed. 
  This paper studies nonlinear deformations of the linear gauge theory of any number of spin-2 and spin-3/2 fields with general formal multiplication rules in place of standard Grassmann rules for manipulating the fields, in four spacetime dimensions. General possibilities for multiplication rules and coupling constants are simultaneously accommodated by regarding the set of fields equivalently as a single algebra-valued spin-2 field and single algebra-valued spin-3/2 field, where the underlying algebra is factorized into a field-coupling part and an internal multiplication part. The condition that there exist a gauge invariant Lagrangian (to within a divergence) for these algebra-valued fields is used to derive determining equations whose solutions give all allowed deformation terms, yielding nonlinear field equations and nonabelian gauge symmetries, together with all allowed formal multiplication rules as needed in the Lagrangian for demonstration of invariance under the gauge symmetries and for derivation of the field equations. In the case of spin-2 fields alone, the main result of this analysis is that all deformations (without any higher derivatives than appear in the linear theory) are equivalent to an algebra-valued Einstein gravity theory. By a systematic examination of factorizations of the algebra, a novel type of nonlinear gauge theory of two or more spin-2 fields is found, where the coupling for the fields is based on structure constants of an anticommutative, anti-associative algebra, and with formal multiplication rules that make the fields anticommuting (while products obey anti-associativity). Supersymmetric extensions of these results are obtained in the more general case when spin-3/2 fields are included. 
  The teleparallel versions of the Einstein and the Landau-Lifshitz energy-momentum complexes of the gravitational field are obtained. By using these complexes, the total energy of the universe, which includes the energy of both the matter and the gravitational fields, is then obtained. It is shown that the total energy vanishes independently of both the curvature parameter and the three dimensionless coupling constants of teleparallel gravity. 
  We consider a minimum uncertainty vacuum choice at a fixed energy scale Lambda as an effective description of trans-Planckian physics, and discuss its implications for the linear perturbations of a massless scalar field in power-law inflationary models. We find possible effects with a magnitude of order H/\Lambda in the power spectrum, in analogy with previous results for de-Sitter space-time. 
  The Chevreton superenergy tensor was introduced in 1964 as a counterpart, for electromagnetic fields, of the well-known Bel-Robinson tensor of the gravitational field. We here prove the unnoticed facts that, in the absence of electromagnetic currents, Chevreton's tensor (i) is completely symmetric, and (ii) has a trace-free divergence if Einstein-Maxwell equations hold. It follows that the trace of the Chevreton tensor is a rank-2, symmetric, trace-free, {\em conserved} tensor, which is different from the energy-momentum tensor, and nonetheless can be constructed for any test Maxwell field, or any Einstein-Maxwell spacetime. 
  A mathematically well-defined, manifestly covariant theory of classical and quantum field is given, based on Euclidean Poisson algebras and a generalization of the Ehrenfest equation, which implies the stationary action principle. The theory opens a constructive spectral approach to finding physical states both in relativistic quantum field theories and for flexible phenomenological few-particle approximations.   In particular, we obtain a Lorentz-covariant phenomenological multiparticle quantum dynamics for electromagnetic and gravitational interaction which provides a representation of the Poincare group without negative energy states. The dynamics reduces in the nonrelativistic limit to the traditional Hamiltonian multiparticle description with standard Newton and Coulomb forces.   The key that allows us to overcome the traditional problems in canonical quantization is the fact that we use the algebra of linear operators on a space of wave functions slightly bigger than traditional Fock spaces. 
  Penrose limits of inhomogeneous cosmologies admitting two abelian Killing vectors and their abelian T-duals are found in general. The wave profiles of the resulting plane waves are given for particular solutions. Abelian and non-abelian T-duality are used as solution generating techniques. Furthermore, it is found that unlike in the case of abelian T-duality, non-abelian T-duality and taking the Penrose limit are not commutative procedures. 
  We present the effective equations to describe the four-dimensional gravity of a brane world, assuming that a five-dimensional bulk spacetime satisfies the Einstein equations and gravity is confined on the $Z_2$ symmetric brane. Applying this formalism, we study the induced-gravity brane model first proposed by Dvali, Gabadadze and Porrati. In a generalization of their model, we show that an effective cosmological constant on the brane can be extremely reduced in contrast to the case of the Randall-Sundrum model even if a bulk cosmological constant and a brane tension are not fine-tuned. 
  A persistent challenge in numerical relativity is the correct specification of boundary conditions. In this work we consider a many parameter family of symmetric hyperbolic initial-boundary value formulations for the linearized Einstein equations and analyze its well posedness using the Laplace-Fourier technique. By using this technique ill posed modes can be detected and thus a necessary condition for well posedness is provided. We focus on the following types of boundary conditions: i) Boundary conditions that have been shown to preserve the constraints, ii) boundary conditions that result from setting the ingoing constraint characteristic fields to zero and iii) boundary conditions that result from considering the projection of Einstein's equations along the normal to the boundary surface. While we show that in case i) there are no ill posed modes, our analysis reveals that, unless the parameters in the formulation are chosen with care, there exist ill posed constraint violating modes in the remaining cases. 
  Using the first-order approximating solutions to the Einstein-Maxwell-Klein-Gordon system of equations for a complex scalar field minimally coupled to a spherically symmetric spacetime, we study the feedback of gravity and electric field on the charged scalar source. Within a perturbative approach, we compute, in the radiation zone, the transition amplitudes and the coherent source-field regeneration rate. 
  We show that an analytical continuation of the Vuorio solution to three-dimensional topologically massive gravity leads to a two-parameter family of black hole solutions, which are geodesically complete and causally regular within a certain parameter range. No observers can remain static in these spacetimes. We discuss their global structure, and evaluate their mass, angular momentum, and entropy, which satisfy a slightly modified form of the first law of thermodynamics. 
  The gravitational collapse of spherical, barotropic perfect fluids is analyzed here. For the first time, the final state of these systems is studied without resorting to simplifying assumptions - such as self-similarity - using a new approach based on non-linear o.d.e. techniques, and formation of naked singularities is shown to occur for solutions such that the mass function is analytic in a neighborhood of the spacetime singularity. 
  We investigate the dynamics of particles moving in a spacetime augmented by one extra dimension in the context of the induced matter theory of gravity. We examine the appearance of a fifth force as an effect caused by the extra dimension and discuss two different approaches to the fifth force formalism. We then give two examples of application of both approaches by considering the case of a Ricci-flat warped-product manifold and a generalized Randall-Sundrum space. 
  In this paper we prove a global existence theorem, in the direction of cosmological expansion, for sufficiently small perturbations of a family of spatially compact variants of the $k=-1$ Friedmann--Robertson--Walker vacuum spacetime. We use a special gauge defined by constant mean curvature slicing and a spatial harmonic coordinate condition, and develop energy estimates through the use of the Bel-Robinson energy and its higher order generalizations. In addition to the smallness condition on the data, we need a topological constraint on the spatial manifold to exclude the possibility of a non--trivial moduli space of flat spacetime perturbations, since the latter could not be controlled by curvature--based energies such as those of Bel--Robinson type. Our results also demonstrate causal geodesic completeness of the perturbed spacetimes (in the expanding direction) and establish precise rates of decay towards the background solution which serves as an attractor asymptotically. 
  This paper is a continuation of the paper gr-qc/0203045 and is devoted to the problem of the arrow of time. A deterministic past-directed dynamics is constructed, which results in the retrodictive universe. A future-directed dynamics of the latter is indeterministic and reproduces standard probabilistic quantum dynamics. The arrow of time is inherent in the retrodictive universe as well as a future-directed increase of informational entropy. 
  In this paper a hypothesis is considered, in which neutrinos and other weakly interacting particles play a fundamental role in Universe. In addition the Newton gravitational constant $G_N$ and the Hubble constant $H$ are interpreted as parameters, characterizing the neutrinos background of Universe. 
  A six dimensional braneworld scenario based on a model describing the interaction of gravity, gauge fields and 3+1 branes in a conformally invariant way is described. The action of the model is defined using a measure of integration built of degrees of freedom independent of the metric. There is no need to fine tune any bulk cosmological constant or the tension of the two (in the scenario described here) parallel branes to obtain zero cosmological constant, the only solutions are those with zero 4-D cosmological constant. The two extra dimensions are compactified in a "football" fashion and the branes lie on the two opposite poles of the compact "football-shaped" sphere. 
  We show that there are no vacuum solutions with a purely magnetic Weyl tensor with respect to an observer submitted to kinematic restrictions involving first order differential scalars. This result generalizes previous ones for the vorticity-free and shear-free cases. We use a covariant approach which makes evident that only the Bianchi identities are used and, consequently, the results are also valid for non vacuum solutions with vanishing Cotton tensor. 
  We show, that classical Kaluza-Klein theory possesses hidden nematic dynamics. It appears as a consequence of 1+4-decomposition procedure, involving 4D observers 1-form \lambda. After extracting of boundary terms the, so called, "effective matter" part of 5D geometrical action becomes proportional to square of anholonomicity 3-form \lambda\wedge d\lambda. It can be interpreted as twist nematic elastic energy, responsible for elastic reaction of 5D space-time on presence of anholonomic 4D submanifold, defined by \lambda. We derive both 5D covariant and 1+4 forms of 5D nematic equilibrium equations, consider simple examples and discuss some 4D physical aspects of generic 5D nematic topological defects. 
  We consider Gowdy spacetimes under the assumption that the spatial hypersurfaces are diffeomorphic to the torus. The relevant equations are then wave map equations with the hyperbolic space as a target. In an article by Grubisic and Moncrief, a formal expansion of solutions in the direction toward the singularity was proposed. Later, Kichenassamy and Rendall constructed a family of real analytic solutions with the maximum number of free functions and the desired asymptotics at the singularity. The condition of real analyticity was subsequently removed by Rendall. In an article by the author, it was shown that one can put a condition on initial data that leads to asymptotic expansions. In this article, we show the following. By fixing a point in hyperbolic space, we can consider the hyperbolic distance from this point to the solution at a given spacetime point. If we fix a spatial point for the solution, it is enough to put conditions on the rate at which the hyperbolic distance tends to infinity as time tends to the singularity in order to conclude that there are smooth expansions in a neighbourhood of the given spatial point. 
  We study characteristic (quasinormal) modes of a $D$-dimensional Schwarzshild black hole. It proves out that the real parts of the complex quasinormal modes, representing the real oscillation frequencies, are proportional to the product of the number of dimensions and inverse horizon radius $\sim$ $D$ $r_{0}^{-1}$. The asymptotic formula for large multipole number $l$ and arbitrary $D$ is derived. In addition the WKB formula for computing QN modes, developed to the 3th order beyond the eikonal approximation, is extended to the 6th order here. This gives us an accurate and economic way to compute quasinormal frequencies. 
  Propagation of fermion in curved space-time generates gravitational interaction due to the coupling between spin of the fermion and space-time curvature. This gravitational interaction, which is an axial-vector appears as CPT violating term in the Lagrangian. It is seen that this space-time interaction can generate neutrino asymmetry in Universe. If the back-ground metric is spherically asymmetric, say, of a rotating black hole, this interaction is non-zero, thus the net difference to the number density of the neutrino and anti-neutrino is nonzero. 
  Treating the Teukolsky perturbation equation numerically as a 2+1 PDE and smearing the singularities in the particle source term by the use of narrow Gaussian distributions, we have been able to reproduce earlier results for equatorial circular orbits that were computed using the frequency domain formalism. A time domain prescription for a more general evolution of nearly geodesic orbits under the effects of radiation reaction is presented. This approach can be useful when tackling the more realistic problem of a stellar-mass black hole moving on a generic orbit around a supermassive black hole under the influence of radiation reaction forces. 
  We describe light-like boosts of the Kerr gravitational field transverse and parallel to the symmetry axis. In the transverse case the boosted field is that of an impulsive gravitational wave having a line singularity displaced relative to its position if the rotation of the source were removed. The parallel boost is insensitive to the rotation of the source. The literature contains a number of diverse results for light-like boosts of the Kerr gravitational field. Our conclusions confirm the correctness of the limits calculated by Balasin and Nachbagauer [Class.and Quantum Grav.13(1996),731]. To avoid any ambiguity our approach is centered on evaluating the light-like boost of the Riemann tensor for the Kerr space-time with the metric playing a secondary role. 
  It is shown that there are no vacuum space-times (with or without cosmological constant) for which the Weyl-tensor is purely gravito-magnetic with respect to a congruence of freely falling observers. 
  I present results of two-dimensional general relativistic hydrodynamical simulations of constant specific angular momentum tori orbiting around a Schwarzschild black hole. After introducing axisymmetric perturbations, these objects either become unstable to the runaway instability or respond with regular oscillations. The latter, in particular, are responsible for quasi-periodic bursts of accretion onto the black hole as well as for the emission of intense gravitational radiation, with signal-to-noise ratios at the detector which are comparable or even larger than the typical ones expected in stellar-core collapse. 
  In many braneworld models, gravity is largely modified at the electro-weak scale ~ 1TeV. In such models, primordial black holes (PBHs) with lunar mass M ~ 10^{-7}M_sun might have been produced when the temperature of the universe was at ~ 1TeV. If a significant fraction of the dark halo of our galaxy consists of these lunar mass PBHs, a huge number of BH binaries will exist in our neighborhood. Third generation detectors such as EURO can detect gravitational waves from these binaries, and can also determine their chirp mass. With a new detector designed to be sensitive at high frequency bands greater than 1 kHz, the existence of extradimensions could be confirmed. 
  We study (3+1) Morris-Thorne wormhole to investigate its higher dimensional embedding structures and thermodynamic properties. It is shown that the wormhole is embedded in (5+2) global embedding Minkowski space. This embedding enables us to construct the wormhole entropy and Hawking temperature by exploiting Unruh effects. We also propose a possibility of negative temperature originated from exotic matter distribution of the wormhole. 
  A new approach is suggested to the problem of quantising causal sets, or topologies, or other such models for space-time (or space). The starting point is the observation that entities of this type can be regarded as objects in a category whose arrows are structure-preserving maps. This motivates investigating the general problem of quantising a system whose `configuration space' (or history-theory analogue) can be regarded as the set of objects in a category. In this first of a series of papers, we study this question in general and develop a scheme based on constructing an analogue of the group that is used in the canonical quantisation of a system whose configuration space is a manifold $Q\simeq G/H$ where G and H are Lie groups. In particular, we choose as the analogue of G the monoid of `arrow fields' on the category. Physically, this means that an arrow between two objects in the category is viewed as some sort of analogue of momentum. After finding the `category quantisation monoid', we show how suitable representations can be constructed using a bundle of Hilbert spaces over the set of objects. 
  In the present work the role that a generalized uncertainty principle could play in the quantization of the electromagnetic field is analyzed. It will be shown that we may speak of a Fock space, a result that implies that the concept of photon is properly defined. Nevertheless, in this new context the creation and annihilation operators become a function of the new term that modifies the Heisenberg algebra, and hence the Hamiltonian is not anymore diagonal in the occupation number representation. Additionally, we show the changes that the energy expectation value suffers as result of the presence of an extra term in the uncertainty principle. The existence of a deformed dispersion relation is also proved. 
  This paper has been withdrawn by the author. The paper has been accepted for publication in Communications on Pure and Applied Mathematics. 
  We study a scalar hyperbolic partial differential equation with non-linear terms similar to those of the equations of general relativity. The equation has a number of non-trivial analytical solutions whose existence rely on a delicate balance between linear and non-linear terms. We formulate two classes of second-order accurate central-difference schemes, CFLN and MOL, for numerical integration of this equation. Solutions produced by the schemes converge to exact solutions at any fixed time $t$ when numerical resolution is increased. However, in certain cases integration becomes asymptotically unstable when $t$ is increased and resolution is kept fixed. This behavior is caused by subtle changes in the balance between linear and non-linear terms when the equation is discretized. Changes in the balance occur without violating second-order accuracy of discretization. We thus demonstrate that a second-order accuracy and convergence at finite $t$ do not guarantee a correct asymptotic behavior and long-term numerical stability.  Accuracy and stability of integration are greatly improved by an exponential transformation of the unknown variable. 
  Using ODE techniques we prove the existence of large classes of initial data satisfying the constraints for the spherically symmetric Einstein-Vlasov-Maxwell system. These include data for which the ratio of total charge to total mass is arbitrarily large. 
  We study a tensorial exponential transformation of a three-dimensional metric of space-like hypersurfaces embedded in a four-dimensional space-time, $\gamma_{ij} = e^{\epsilon_{ikm}\theta_m} e^{\phi_k} e^{-\epsilon_{jkn}\theta_n}$, where $\phi_k$ are logarithms of the eigenvalues of $\gamma_{ij}$, $\theta_k$ are rotation angles, and $\epsilon_{ijk}$ is a fully anti-symmetric symbol. Evolution part of Einstein's equations, formulated in terms of $\phi_k$ and $\theta_k$, describes time evolution of the metric at every point of a hyper-surface as a continuous stretch and rotation of a local coordinate system in a tangential space. The exponential stretch-rotation (ESR) transformation generalizes particular exponential transformations used previously in cases of spatial symmetry. The ESR 3+1 formulation of Einstein's equations may have certain advantages for long-term stable integration of these equations. 
  We theoretically analyze the quantum noise of signal-recycled laser interferometric gravitational-wave detectors with additional input and output optics, namely frequency-dependent squeezing of the vacuum entering the dark port and frequency-dependent homodyne detection. We combine the work of Buonanno and Chen on the quantum noise of signal-recycled interferometers with ordinary input-output optics, and the work of Kimble el al. on frequency-dependent input-output optics with conventional interferometers. Analytical formulas for the optimal input and output frequency dependencies are obtained. It is shown that injecting squeezed light with the optimal frequency-dependent squeezing angle into the dark port yields an improvement on the noise spectral density by a factor of exp(-2r) (in power) over the entire squeezing bandwidth, where r is the squeezing parameter. It is further shown that frequency-dependent (variational) homodyne read-out leads to an additional increase in sensitivity which is significant in the wings of the doubly resonant structure. The optimal variational input squeezing in case of an ordinary output homodyne detection is shown to be realizable by applying two optical filters on a frequency-independent squeezed vacuum. Throughout this paper, we take as example the signal-recycled topology currently being completed at the GEO600 site. However, theoretical results obtained here are also applicable to the proposed topology of Advanced LIGO. 
  We propose two methods for obtaining the dual of non-linear relativity as previously formulated in momentum space. In the first we allow for the (dual) position space to acquire a non-linear representation of the Lorentz group independently of the chosen representation in momentum space. This requires a non-linear definition for the invariant contraction between momentum and position spaces. The second approach, instead, respects the linearity of the invariant contraction. This fully fixes the dual of momentum space and dictates a set of energy-dependent space-time Lorentz transformations. We discuss a variety of physical implications that would distinguish these two strategies. We also show how they point to two rather distinct formulations of theories of gravity with an invariant energy and/or length scale. 
  The Jackiw-Teitelboim gravity with non-vanishing cosmological constant coupled to Liouville theory is considered as a non-critical string on d dimensional flat space-time. It is discussed how the presence of cosmological constant leads to consider additional constraints on the parameters of the theory, even though the conformal anomaly is independent of the cosmological constant. The constraints agree with the necessary conditions required to ensure that the tachyon field turns out to be a primary prelogarithmic operator within the context of the world-sheet conformal field theory. Thus, the linearized tachyon field equation allows to impose the diagonal condition for the interaction term.   We analyze the neutralization of the Liouville mode induced by the coupling to the Jackiw-Teitelboim lagrangian. The standard free field prescription leads to obtain explicit expressions for three-point functions for the case of vanishing cosmological constant in terms of a product of Shapiro-Virasoro integrals; this fact is a consequence of the mentioned neutralization effect. 
  We study the properties of a modified version of the Bona-Masso family of hyperbolic slicing conditions. This modified slicing condition has two very important features: In the first place, it guarantees that if a spacetime is static or stationary, and one starts the evolution in a coordinate system in which the metric coefficients are already time independent, then they will remain time independent during the subsequent evolution, {\em i.e.} the lapse will not evolve and will therefore not drive the time lines away from the Killing direction. Second, the modified condition is naturally adapted to the use of a densitized lapse as a fundamental variable, which in turn makes it a good candidate for a dynamic slicing condition that can be used in conjunction with some recently proposed hyperbolic reformulations of the Einstein evolution equations. 
  Anthropic reasoning often begins with the premise that we should expect to find ourselves typical among all intelligent observers. However, in the infinite universe predicted by inflation, there are some civilizations which have spread across their galaxies and contain huge numbers of individuals. Unless the proportion of such large civilizations is unreasonably tiny, most observers belong to them. Thus anthropic reasoning predicts that we should find ourselves in such a large civilization, while in fact we do not. There must be an important flaw in our understanding of the structure of the universe and the range of development of civilizations, or in the process of anthropic reasoning. 
  Fuchsian methods and their applications to the study of the structure of spacetime singularities are surveyed. The existence question for spacetimes with compact Cauchy horizons is discussed. After some basic facts concerning Fuchsian equations have been recalled, various ways in which these equations have been applied in general relativity are described. Possible future applications are indicated. 
  Loop quantum cosmology of the closed isotropic model is studied with a special emphasis on a comparison with traditional results obtained in the Wheeler-DeWitt approach. This includes the relation of the dynamical initial conditions with boundary conditions such as the no-boundary or the tunneling proposal and a discussion of inflation from quantum cosmology. 
  Loop quantum cosmological methods are extended to homogeneous models in diagonalized form. It is shown that the diagonalization leads to a simplification of the volume operator such that its spectrum can be determined explicitly. This allows the calculation of composite operators, most importantly the Hamiltonian constraint. As an application the dynamics of the Bianchi I model is studied and it is shown that its loop quantization is free of singularities. 
  Much of the work in loop quantum gravity and quantum geometry rests on a mathematically rigorous integration theory on spaces of distributional connections. Most notably, a diffeomorphism invariant representation of the algebra of basic observables of the theory, the Ashtekar-Isham-Lewandowski representation, has been constructed. Recently, several uniqueness results for this representation have been worked out. In the present article, we contribute to these efforts by showing that the AIL-representation is irreducible, provided it is viewed as the representation of a certain C*-algebra which is very similar to the Weyl algebra used in the canonical quantization of free quantum field theories. 
  The separated radial part of a massive complex scalar wave equation in the Kerr- Sen geometry is shown to satisfy the generalized spheroidal wave equation which is, in fact, a confluent Heun equation up to a multiplier. The Hawking evaporation of scalar particles in the Kerr-Sen black hole background is investigated by the Damour-Ruffini-Sannan method. It is shown that quantum thermal effect of the Kerr-Sen black hole has the same character as that of the Kerr-Newman black hole. 
  The Newman-Penrose formalism is used to derive the Teukolsky master equations controlling massless scalar, neutrino, electromagnetic, gravitino, and gravitational field perturbations of the Kerr-de Sitter spacetime. Then the quantum entropy of a non-extreme Kerr-de Sitter black hole due to arbitrary spin fields is calculated by the improved thin-layer brick wall model. It is shown that the subleading order contribution to the entropy is dependent on the square of the spins of particles and that of the specific angular momentum of black holes as well as the cosmological constant. The logarithmic correction of the spins of particles to the entropy relies on the rotation of the black hole and the effect of the cosmological constant. 
  We investigate the stability of self-gravitating spherically symmetric anisotropic spheres under radial perturbations. We consider both the Newtonian and the full general-relativistic perturbation treatment. In the general-relativistic case, we extend the variational formalism for spheres with isotropic pressure developed by Chandrasekhar. We find that, in general, when the tangential pressure is greater than the radial pressure, the stability of the anisotropic sphere is enhanced when compared to isotropic configurations. In particular, anisotropic spheres are found to be stable for smaller values of the adiabatic index $\gamma$. 
  We evaluate both the massless and the massive Dirac quasi-normal mode frequencies in the Schwarzschild black hole spacetime using the WKB approximation. For the massless case, we find that, similar to those for the integral spin fields, the real parts of the frequencies increase with the angular momentum number $\kappa$, while the imaginary parts or the dampings increase with the mode number $n$ for fixed $\kappa$. For the massive case, the oscillation frequencies increase with the mass $m$ of the field, while the dampings decrease. Fields with higher masses will therefore decay more slowly. 
  This paper is withdrawn by the authors 
  The contribution of gravitational neutrino oscillations to the solar neutrino problem is studied by constructing the Dirac Hamiltonian and calculating the corresponding dynamical phase in the vicinity of the Sun in a non-Riemann background Kerr space-time with torsion and non-metricity. We show that certain components of non-metricity and the axial as well as non-axial components of torsion may contribute to neutrino oscillations. We also note that the rotation of the Sun may cause a suppression of transitions among neutrinos. However, the observed solar neutrino deficit could not be explained by any of these effects because they are of the order of Planck scales. 
  We consider an Einstein-Maxwell action modified by the addition of three terms coupling the electromagnetic strength to the curvature tensor. The corresponding generalized Maxwell equations imply a variation of the speed of light in a vacuum. We determine this variation in Friedmann-Robertson-Walker spacetimes. We show that light propagates at a speed greater than $c$ when a simple condition is satisfied. 
  We derive the gravitational Hamiltonian starting from the Gauss-Bonnet action, keeping track of all surface terms. This is done using the language of orthonormal frames and forms to keep things as tidy as possible. The surface terms in the Hamiltonian give a remarkably simple expression for the total energy of a spacetime. This expression is consistent with energy expressions found in hep-th/0212292. However, we can apply our results whatever the choice of background and whatever the symmetries of the spacetime. 
  In a recent paper Abramowicz and Klu{\'z}niak have discussed the problem of epicyclic oscillations in Newton's and Einstein's dynamics and have shown that Newton's dynamics in a properly curved three-dimensional space is identical to test-body dynamics in the three-dimensional optical geometry of Schwarzschild space-time. One of the main results of this paper was the proof that different behaviour of radial epicyclic frequency and Keplerian frequency in Newtonian and General Relativistic regimes had purely geometric origin contrary to claims that nonlinearity of Einstein's theory was responsible for this effect.   In this paper we obtain the same result from another perspective: by representing these two distinct problems (Newtonian and Einstein's test body motion in central gravitational field) in a uniform way -- as a geodesic motion. The solution of geodesic deviation equation reproduces the well known results concerning epicyclic frequencies and clearly demonstrates geometric origin of the difference between Newtonian and Einstein's problems. 
  A stochastic background of gravitational waves can be generated during a cosmological first order phase transition, at least by two distinct mechanisms: collisions of true vacuum bubbles and turbulence in the cosmic fluid. I compare these two contributions, analyzing their relative importance for a generic phase transition. In particular, a first order electroweak phase transition is expected to generate a gravitational wave signal peaked at a frequency which today falls just within the band of the planned space interferometer LISA. For this transition, I find constraints for the relevant parameters in order to produce a signal within the reach of the sensitivity of LISA. The result is that the transition must be strongly first order, alpha > 0.2. In this regime the signal coming from turbulence dominates over that from colliding bubbles. 
  These lectures discuss how the direct detection of gravitational waves can be used to probe the very early Universe. We review the main cosmological mechanisms which could have produced relic gravitational waves, and compare theoretical predictions with capabilities and time scales of current and upcoming experiments. 
  We show that the Kidder-Scheel-Teukolsky family of hyperbolic formulations of the 3+1 evolution equations of general relativity remains hyperbolic when coupled to a recently proposed modified version of the Bona-Masso slicing condition. 
  We analyze the Teleparallel Equivalent of General Relativity (TEGR) from the point of view of Hamilton-Jacobi approach for singular systems 
  The theory of gravitation field within the special theory of relativity is analyzed. 
  Measurements of the tunneling time are briefly reviewed. Next, time and matter in general relativity and quantum mechanics is examined. In particular, the question arises: How does gravitational radiation interact with a coherent quantum many-body system (a ``quantum fluid'')? A minimal coupling rule for the coupling of the electron spin to curved spacetime in general relativity implies the possibility of a coupling between electromagnetic (EM) and gravitational (GR) radiation mediated by a quantum Hall fluid. This suggests that quantum transducers between these two kinds of radiation fields might exist. We report here on a first attempt at a Hertz-type experiment, in which a high-$\rm{T_c}$ superconductor (YBCO) was the material used as a quantum transducer to convert EM into GR microwaves, and a second piece of YBCO in a separate apparatus was used to back-convert GR into EM microwaves. An upper limit on the conversion efficiency of YBCO was measured to be $1.6\times10^{-5}$. 
  The procedure to find gauge invariant variables for two-parameter nonlinear perturbations in general relativity is considered. For each order metric perturbation, we define the variable which is defined by the appropriate combination with lower order metric perturbations. Under the gauge transformation, this variable is transformed in the manner similar to the gauge transformation of the linear order metric perturbation. We confirm this up to third order. This implies that gauge invariant variables for higher order metric perturbations can be found by using a procedure similar to that for linear order metric perturbations. We also derive gauge invariant combinations for the perturbation of an arbitrary physical variable, other than the spacetime metric, up to third order. 
  The classical Bondi-Penrose approach to the gravitational radiation theory in asymptotically flat spacetimes is recalled and recent advances in the proofs of the existence of such spacetimes are briefly reviewed. We then mention the unique role of the boost-rotation symmetric spacetimes, representing uniformly accelerated objects, as the only explicit radiative solutions known which are asymptotically flat; they are used as test beds in numerical relativity and approximation methods.   The main part of the review is devoted to the examples of radiative fields in the vacuum spacetimes with positive cosmological constant. Type N solutions are analyzed by using the equation of geodesic deviation. Both these and Robinson-Trautman solutions of type II are shown to approach de Sitter universe asymptotically. Recent work on the the radiative fields due to uniformly accelerated charges in de Sitter spacetime ("cosmological Born's solutions") is reviewed and the properties of these fields are discussed with a perspective to characterize general features of radiative fields near a de Sitter-like infinity. 
  We analyze the effect of pressure on the evolution of perturbations of an Einstein-de Sitter Universe in the matter dominated epoch assuming an ideal gas equation of state. For the sake of simplicity the temperature is considered uniform. The goal of the paper is to examine the validity of the linear approximation. With this purpose the evolution equations are developed including quadratic terms in the derivatives of the metric perturbations and using coordinate conditions that, in the linear case, reduce to the longitudinal gauge. We obtain the general solution, in the coordinate space, of the evolution equation for the scalar mode, and, in the case of spherical symmetry, we express this solution in terms of unidimensional integrals of the initial conditions: the initial values of the Newtonian potential and its first time derivative. We find that the contribution of the initial first time derivative, which has been systematically forgotten, allows to form inhomogeneities similar to a cluster of galaxies starting with very small density contrast. Finally, we obtain the first non linear correction to the linearized solution due to the quadratic terms in the evolution equations. Here we find that a non null pressure plays a crucial role in constraining the non linear corrections. It is shown, by means of examples, that reasonable thermal velocities at the present epoch (non bigger than $10^{-6}$) make the ratio between the first non linear correction and the linear solution of the order of $10^{-2}$ for a galaxy cluster inhomogeneity. 
  The linearized stability of charged thin shell wormholes under spherically symmetric perturbations is analized. It is shown that the presence of a large value of charge provides stabilization to the system, in the sense that the constrains onto the equation of state are less severe than for non-charged wormholes. 
  We investigate the structure of the $\delta=2$ Tomimatsu-Sato spacetime. We show that this spacetime has degenerate horizons with two components, in contrast to the general belief that the Tomimatsu-Sato solutions with even $\delta$ do not have horizons. 
  The Regge-Wheeler equation for black-hole gravitational waves is analyzed for large negative imaginary frequencies, leading to a calculation of the cut strength for waves outgoing to infinity. In the--limited--region of overlap, the results agree well with numerical findings [Class. Quantum Grav._20_, L217 (2003)]. Requiring these waves to be outgoing into the horizon as well subsequently yields an analytic formula for the highly damped Schwarzschild quasinormal modes,_including_ the leading correction. Just as in the WKB quantization of, e.g., the harmonic oscillator, solutions in different regions of space have to be joined through a connection formula, valid near the boundary between them where WKB breaks down. For the oscillator, this boundary is given by the classical turning points; fascinatingly, the connection here involves an expansion around the black-hole singularity r=0. 
  After a more general assumption on the influence of the bulk on the brane, we extend some conclusions by Maartens et al. and Santos et al. on the asymptotic behavior of Bianchi I brane worlds. As a consequence of the nonlocal anisotropic stresses induced by the bulk, in most of our models, the brane does not isotropize and the nonlocal energy does not vanish in the limit in which the mean radius goes to infinity. We have also found the intriguing possibility that the inflation due to the cosmological constant might be prevented by the interaction with the bulk. We show that the problem for the mean radius can be completely solved in our models, which include as particular cases those in the references above. 
  This work extends to six dimensions the idea first proposed by Klein regarding a closed space in the context of a fifth dimension and its link to quantum theory. The main result is a formula that expresses the value of the characteristic length of the sixth dimension in terms of the strength of a magnetic monopole $g$. It is shown that in the case of Dirac's monopole, the ratio of the characteristic lengths of the fifth and sixth dimension corresponds to twice the fine structure constant $\alpha$. Possible consequences of the idea are discussed. 
  Riemann tensor irreducible part $E_{iklm} = {1/2} (g_{il}S_{km} + g_{km}S_{il} - g_{im}S_{kl} - g_{kl}S_{im})$ constructed from metric tensor $g_{ik}$ and traceless part of Ricci tensor $S_{ik} = R_{ik} - {1/4} g_{ik} R$ is expanded into bilinear combinations of bivectorial fields being eigenfunctions of $E$. Field equations for the bivectors induced by Bianchi identities are studied and it is shown that in general case it will be 3-parametric local symmetry group Yang-Mills field. 
  This work addresses and solves the problem of generically tracking black hole event horizons in computational simulation of black hole interactions. Solutions of the hyperbolic eikonal equation, solved on a curved spacetime manifold containing black hole sources, are employed in development of a robust tracking method capable of continuously monitoring arbitrary changes of topology in the event horizon, as well as arbitrary numbers of gravitational sources. The method makes use of continuous families of level set viscosity solutions of the eikonal equation with identification of the black hole event horizon obtained by the signature feature of discontinuity formation in the eikonal's solution. The method is employed in the analysis of the event horizon for the asymmetric merger in a binary black hole system. In this first such three dimensional analysis, we establish both qualitative and quantitative physics for the asymmetric collision; including: 1. Bounds on the topology of the throat connecting the holes following merger, 2. Time of merger, and 3. Continuous accounting for the surface of section areas of the black hole sources. 
  One of the conceptual tensions between quantum mechanics (QM) and general relativity (GR) arises from the clash between the spatial nonseparability of entangled states in QM, and the complete spatial separability of all physical systems in GR, i.e., between the nonlocality implied by the superposition principle, and the locality implied by the equivalence principle. Possible experimental consequences of this conceptual tension will be discussed for macroscopically entangled, coherent quantum fluids, such as superconductors, superfluids, atomic Bose-Einstein condensates, and quantum Hall fluids, interacting with tidal and gravitational radiation fields. A minimal-coupling rule, which arises from the electron spin coupled to curved spacetime, leads to an interaction between electromagnetic (EM) and gravitational (GR) radiation fields mediated by a quantum Hall fluid. This suggests the possibility of a quantum transducer action, in which EM waves are convertible to GR waves, and vice versa. 
  We performed a careful numerical analysis of the late tail behaviour of waves propagating in the Schwarzschild spacetime. Specifically the scalar monopole, the electromagnetic dipole and the gravitational axial quadrupole waves have been investigated. The obtained results agree with a falloff $1/t^{2l+3}$ for the general initial data and $1/t^{2l+4}$ for the initially static data. 
  We present an exact solution that describes collision of electromagnetic shock waves coupled with axion plane waves. The axion has a rather special coupling to the cross polarization term of the metric. The initial data on the null surfaces is well-defined and collision results in a singularity free interaction region. Our solution is a generalization of the Bell-Szekeres solution in the presence of an axion field. 
  The evolution of a Universe modelled as a mixture of a Chaplygin gas and radiation is determined by taking into account irreversible processes. This mixture could interpolate periods of a radiation dominated, a matter dominated and a cosmological constant dominated Universe. The results of a Universe modelled by this mixture are compared with the results of a mixture whose constituents are radiation and quintessence. Among other results it is shown that: (a) for both models there exists a period of a past deceleration with a present acceleration; (b) the slope of the acceleration of the Universe modelled as a mixture of a Chaplygin gas with radiation is more pronounced than that modelled as a mixture of quintessence and radiation; (c) the energy density of the Chaplygin gas tends to a constant value at earlier times than the energy density of quintessence does; (d) the energy density of radiation for both mixtures coincide and decay more rapidly than the energy densities of the Chaplygin gas and of quintessence. 
  We prescribe a choice of 18 variables in all that casts the equations of the fully nonlinear characteristic formulation of general relativity in first--order quasi-linear canonical form. At the analytical level, a formulation of this type allows us to make concrete statements about existence of solutions. In addition, it offers concrete advantages for numerical applications as it now becomes possible to incorporate advanced numerical techniques for first order systems, which had thus far not been applicable to the characteristic problem of the Einstein equations, as well as in providing a framework for a unified treatment of the vacuum and matter problems. This is of relevance to the accurate simulation of gravitational waves emitted in astrophysical scenarios such as stellar core collapse. 
  In the present work we approximate an ultrarelativistic jet by a homogeneous beam of null matter with finite width. Then, we study the influence of this beam over the space-time metric in the framework of higher-derivative gravity. We find an exact shock wave solution of the quadratic gravity field equations and compare it with the solution to Einstein's gravity. We show that the effect of higher-curvature gravity becomes negligible at large distances from the beam axis. We also observe that only the Ricci-squared term contribute to modify the Einstein's gravity prediction. Furthermore, we note that this higher-curvature term contribute to regularize the discontinuities associated to the solution to Einstein's general relativity. 
  Quantum weak energy inequalities (QWEI) provide state-independent lower bounds on averages of the renormalised energy density of a quantum field. We derive QWEIs for the electromagnetic and massive spin-one fields in globally hyperbolic spacetimes whose Cauchy surfaces are compact and have trivial first homology group. These inequalities provide lower bounds on weighted averages of the renormalized energy density as ``measured'' along an arbitrary timelike trajectory, and are valid for arbitrary Hadamard states of the spin-one fields. The QWEI bound takes a particularly simple form for averaging along static trajectories in ultrastatic spacetimes; as specific examples we consider Minkowski space [in which case the topological restrictions may be dispensed with] and the static Einstein universe.   A significant part of the paper is devoted to the definition and properties of Hadamard states of spin-one fields in curved spacetimes, particularly with regard to their microlocal behaviour. 
  In quasi-metric relativity it is necessary to separate between 2 different versions of the electromagnetic field tensor (EMFT): (1) The active EMFT determining the electromagnetic contribution to the active stress-energy tensor, and (2) The passive EMFT entering the equations of motion. The passive EMFT may be found from the usual Maxwell's equations in curved space-time, and local conservation laws for passive electromagnetism ensure that photons move on null geodesics in quasi-metric space-time. However, by construction the norm of the passive EMFT decreases secularly, defining a global cosmic attenuation (not noticeable locally) of the electromagnetic field. As a simple example the gravitational and electric fields outside a spherically symmetric, metrically static, charged source are calculated. It is found that the global cosmic expansion affects the electric and the gravitational fields differently, i.e., unlike the gravitational field the electric field does not expand. On the other hand, if radiative effects can be neglected it is found that electromagnetically bound, classical systems will participate in the cosmic expansion. But since quantum-mechanical states should not be affected by the expansion, there is no reason to believe that this calculation should apply to quantum-mechanical systems such as atoms. Finally it is shown that the main results of geometric optics hold in quasi-metric space-time. 
  We consider a braneworld inflation model driven by the dynamics of a scalar field living in the 5-dimensional bulk, the so-called ``bulk inflaton model'', and investigate the geometry in the bulk and large scale cosmological perturbations on the brane. The bulk gravitational effects on the brane are described by a projection of the 5-dimensional Weyl tensor, which we denote by $E_{\mu\nu}$. Focusing on a tachionic potential model, we take a perturbative approach in the anti-de Sitter (AdS$_5$) background with a single de Sitter brane. We first formulate the evolution equations for $E_{\mu\nu}$ in the bulk. Next, applying them to the case of a spatially homogeneous brane, we obtain two different integral expressions for $E_{\mu\nu}$. One of them reduces to the expression obtained previously when evaluated on the brane. The other is a new expression that may be useful for analyzing the bulk geometry. Then we consider superhorizon scale cosmological perturbations and evaluate the bulk effects onto the brane. In the limit $H^2\ell^2\ll1$, where $H$ is the Hubble parameter on the brane and $\ell$ is the bulk curvature radius, we find that the effective theory on the brane is identical to the 4-dimensional Einstein-scalar theory with a simple rescaling of the potential even under the presence of inhomogeneities. % atleast on super-Hubble horizon scales. In particular, it is found that the anticipated non-trivial bulk effect due to the spatially anisotropic part of $E_{\mu\nu}$ may appear only at %second order in the low energy expansion, i.e., at $O(H^4\ell^4)$. 
  This work establishes critical phenomena in the topological transition of black hole coalescence. We describe and validate a computational front tracking event horizon solver, developed for generic studies of the black hole coalescence problem. We then apply this to the Kastor - Traschen axisymmetric analytic solution of the extremal Maxwell - Einstein black hole merger with cosmological constant. The surprising result of this computational analysis is a power law scaling of the minimal throat proportional to time. The minimal throat connecting the two holes obeys this power law during a short time immediately at the beginning of merger. We also confirm the behavior analytically. Thus, at least in one axisymmetric situation a critical phenomenon exists. We give arguments for a broader universality class than the restricted requirements of the Kastor - Traschen solution. 
  We present an exact solution of Einstein's field equations in toroidal coordinates. The solution has three regions: an interior with a string equation of state; an Israel boundary layer; an exterior with constant isotropic pressure and constant density, locally isometric to anti-de Sitter spacetime. The exterior can be a cosmological vacuum with negative cosmological constant. The size and mass of the toroidal loop depend on the size of the cosmological constant. 
  The classical Avez-Seifert theorem is generalized to the case of the Lorentz force equation for charged test particles with fixed charge-to-mass ratio. Given two events x_{0} and x_{1}, with x_{1} in the chronological future of x_{0}, and a ratio q/m, it is proved that a timelike connecting solution of the Lorentz force equation exists provided there is no null connecting geodesic and the spacetime is globally hyperbolic. As a result, the theorem answers affirmatively to the existence of timelike connecting solutions for the particular case of Minkowski spacetime. Moreover, it is proved that there is at least one C^{1} connecting curve that maximizes the functional I[\gamma]=\int_{\gamma} ds+q/(mc^2) \omega over the set of C^{1} future-directed non-spacelike connecting curves. 
  We explore a theory of large-scale gravitational quantization, using the general relativistic Hamilton-Jacobi equation to create quantization conditions via a new scalar wave equation dependent upon the total mass and the total vector angular momentum only. Instead of h-bar, a local invariant quantity proportional to the total angular momentum dictates the quantization conditions. In the Schwarzschild metric the theory predicts eigenstates with quantized energy per mass and angular momentum per mass. We find excellent agreement to the orbital spacings of the satellites of the Jovian planets and to the planet spacings in the Solar System. For galaxies we derive the baryonic Tully-Fisher relation and the MOND acceleration, so galaxy velocity curves are explained without requiring 'dark matter'. For the universe, we derive a new Hubble relation that accounts for the accelerated expansion with a matter density at about 5% of the critical matter/energy density, with the remainder being large-scale quantization zero-point energy. A possible laboratory test is proposed. 
  The equivalence between the path integrals for first order gravity and the standard torsion-free, metric gravity in 3+1 dimensions is analyzed. Starting with the path integral for first order gravity, the correct measure for the path integral of the metric theory is obtained. 
  Quasinormal modes for scalar field perturbations of a Schwarzschild-de Sitter (SdS) black hole are investigated. An analytical approximation is proposed for the problem. The quasinormal modes are evaluated for this approximate model in the limit when black hole mass is much smaller than the radius of curvature of the spacetime. The model mirrors some striking features observed in numerical studies of time behaviour of scalar perturbations of the SdS black hole. In particular, it shows the presence of two sets of modes relevant at two different time scales, proportional to the surface gravities of the black hole and cosmological horizons respectively. These quasinormal modes are not complete - another feature observed in numerical studies. Refinements of this model to yield more accurate quantitative agreement with numerical studies are discussed. Further investigations of this model are outlined, which would provide a valuable insight into time behaviour of perturbations in the SdS spacetime. 
  Motivated by the Randall-Sundrum brane-world scenario, we discuss the classical and quantum dynamics of a (d+1)-dimensional boundary wall between a pair of (d+2)-dimensional topological Schwarzschild-AdS black holes. We assume there are quite general -- but not completely arbitrary -- matter fields living on the boundary ``brane universe'' and its geometry is that of an Friedmann-Lemaitre-Robertson-Walker (FLRW) model. The effective action governing the model in the mini-superspace approximation is derived. We find that the presence of black hole horizons in the bulk gives rise to a complex action for certain classically allowed brane configurations, but that the imaginary contribution plays no role in the equations of motion. Classical and instanton brane trajectories are examined in general and for special cases, and we find a subset of configuration space that is not allowed at the classical or semi-classical level; these correspond to spacelike branes carrying tachyonic matter. The Hamiltonization and Dirac quantization of the model is then performed for the general case; the latter involves the manipulation of the Hamiltonian constraint before it is transformed into an operator that annihilates physical state vectors. The ensuing covariant Wheeler-DeWitt equation is examined at the semi-classical level, and we consider the possible localization of the brane universe's wavefunction away from the cosmological singularity. This is easier to achieve for branes with low density and/or spherical spatial sections. 
  From the latest experimental readouts in this context an intriguing discrepancy has been elicited. Indeed, theory and experiment dissent by one per cent, and though this fact could be a consequence of the mounting of the experimental device, it might also embody a difference between the way in which gravity behaves in classical and quantum mechanics. In this work the effects, upon the interference pattern, of space--time torsion will be analyzed heeding its coupling with the spin of the neutron beam. It will be proved that, even with this contribution, there is enough leeway for a further discussion of the validity of the equivalence principle in nonrelativistic quantum mechanics. 
  At the time of this conference, in June 2002, The LIGO Science Collaboration was getting ready to perform its first Science Run, where data will be taken with all three LIGO detectors. We describe here the status of the LIGO detectors as of February 2003, their performance during the ``Engineering Run'' E7 (Dec 28'01-Jan 14'02) and subsequent Science Runs in 2002/3. We also describe ongoing efforts on data analysis for setting upper limits of different gravitational wave sources. 
  The quantization of gravity coupled to barotropic perfect fluid as matter field and cosmological constant is made and the wave function can be determined for any $\kappa$ in the FRW minisuperspace model. The meaning of the existence of the classical solution is discussed in the WKB semiclassical approximation 
  We present a general framework for analyzing spatially inhomogeneous cosmological dynamics. It employs Hubble-normalized scale-invariant variables which are defined within the orthonormal frame formalism, and leads to the formulation of Einstein's field equations with a perfect fluid matter source as an autonomous system of evolution equations and constraints. This framework incorporates spatially homogeneous dynamics in a natural way as a special case, thereby placing earlier work on spatially homogeneous cosmology in a broader context, and allows us to draw on experience gained in that field using dynamical systems methods. One of our goals is to provide a precise formulation of the approach to the spacelike initial singularity in cosmological models, described heuristically by Belinski\v{\i}, Khalatnikov and Lifshitz. Specifically, we construct an invariant set which we conjecture forms the local past attractor for the evolution equations. We anticipate that this new formulation will provide the basis for proving rigorous theorems concerning the asymptotic behavior of spatially inhomogeneous cosmological models. 
  We describe our present understanding of the relations between the behaviour of asymptotically flat Cauchy data for Einstein's vacuum field equations near space-like infinity and the asymptotic behaviour of their evolution in time at null infinity. 
  The recently published analysis of the coincidences between the EXPLORER and NAUTILUS gravitational wave detectors in the year 2001 (Astone et al. 2002) has drawn some criticism (Finn 2003). We do not hold with these objections, even if we agree that no claim can be made with our data. The paper we published reports data of unprecedented quality and sets a new procedure for the coincidence search, which can be repeated again by us and by other groups in order to search for signature of possible signals. About the reported coincidence excess, we remark that it is not destined to remain an intriguing observation for long: it will be confirmed or denied soon by interferometers and bars operating at their expected sensitivity. 
  We show that the coupling of a Dirac spinor field with the gravitational field in the teleparallel equivalent of general relativity is consistent. For an arbitrary SO(3,1) connection there are two possibilities for the coupling of the spinor field with the gravitational field. The problems of consistency raised by Y. N. Obukhov and J. G. Pereira in the paper {\it Metric-affine approach to teleparallel gravity} [gr-qc/0212080] take place only in the framework of one particular coupling. By adopting an alternative coupling the consistency problem disappears. 
  Regge calculus is considered as a particular case of the more general system where the linklengths of any two neighbouring 4-tetrahedra do not necessarily coincide on their common face. This system is treated as that one described by metric discontinuous on the faces. In the superspace of all discontinuous metrics the Regge calculus metrics form some hypersurface. Quantum theory of the discontinuous metric system is assumed to be fixed somehow in the form of quantum measure on (the space of functionals on) the superspace. The problem of reducing this measure to the Regge hypersurface is addressed. The quantum Regge calculus measure is defined from a discontinuous metric measure by inserting the $\delta$-function-like phase factor. The requirement that this reduction would respect natural physical properties (positivity, well-defined continuum limit, absence of lattice artefacts) put rather severe restrictions and allows to define practically uniquely this phase factor. 
  We investigate the general relativistic collapse of spherically symmetric, massless spin-1/2 fields at the threshold of black hole formation. A spherically symmetric system is constructed from two spin-1/2 fields by forming a spin singlet with no net spin-angular momentum. We study the system numerically and find strong evidence for a Type II critical solution at the threshold between dispersal and black hole formation, with an associated mass scaling exponent $\gamma ~ 0.26$. Although the critical solution is characterized by a continuously self-similar (CSS) geometry, the matter fields exhibit discrete self-similarity with an echoing exponent $\Delta ~ 1.34$. We then adopt a CSS ansatz and reduce the equations of motion to a set of ODEs. We find a solution of the ODEs that is analytic throughout the solution domain, and show that it corresponds to the critical solution found via dynamical evolutions. 
  We obtain the perturbed components of affine connection and Ricci tensor using algebraic computation. Naturally, the perturbed Einstein field equations for the vacuum can be written. The method can be used to obtain perturbed equations of the superior order. 
  In this paper, the real scalar field equation in Schwarzschild-de Sitter spacetime is solved numerically with high precision. A method called polynomial approximation is introduced to derive the relation between the tortoise coordinate x and the radius r. This method is di&#64256;erent from the tangent approximation [1] and leads to more accurate result. The Nariai black hole is then discussed in details. We find that the wave function is harmonic only near the horizons as I. Brevik and B. Simonsen [1] found. Howerver the wave function is not harmonic in the region of the potential peak, with amplitude increasing instead. Furthermore, we also find that, when cosmological constant decreases, the potential peak increases, and the maximum wave amplitude increases. 
  Under certain conditions imposed on the energy-momentum tensor, a theorem that characterizes a two-parameter family of static and spherically symmetric solutions to Einstein's field equations (black holes), is proved. A discussion on the asymptotics, regularity, and the energy conditions is provided. Examples that include the best known exact solutions within these symmetries are considered. A trivial extension of the theorem includes the cosmological constant {\it ab-initio}, providing then a three-parameter family of solutions. 
  In this paper we use, in a preliminary way, the recently released EIGEN2 Earth gravity model, which is based on six months of data of CHAMP only, in order to reassess the systematic error due to the mismodelling in the even zonal harmonics of geopotential in the LAGEOS-LAGEOS II Lense-Thirring experiment involving the nodes of both the LAGEOS satellites and the perigee of LAGEOS II. The first results from the GGM01C Earth gravity model including the first GRACE data are very promising. 
  We investigate relativistic spherically symmetric static perfect fluid models in the framework of the theory of dynamical systems. The field equations are recast into a regular dynamical system on a 3-dimensional compact state space, thereby avoiding the non-regularity problems associated with the Tolman-Oppenheimer-Volkoff equation. The global picture of the solution space thus obtained is used to derive qualitative features and to prove theorems about mass-radius properties. The perfect fluids we discuss are described by barotropic equations of state that are asymptotically polytropic at low pressures and, for certain applications, asymptotically linear at high pressures. We employ dimensionless variables that are asymptotically homology invariant in the low pressure regime, and thus we generalize standard work on Newtonian polytropes to a relativistic setting and to a much larger class of equations of state. Our dynamical systems framework is particularly suited for numerical computations, as illustrated by several numerical examples, e.g., the ideal neutron gas and examples that involve phase transitions. 
  In the context of threshold investigations of Lorentz violation, we discuss the fundamental principle of coordinate invariance, the role of an effective dynamical framework, and the conditions of positivity and causality. Our analysis excludes a variety of previously considered Lorentz-breaking parameters and opens an avenue for viable dispersion-relation investigations of Lorentz violation. 
  We review some recent works on the post-Newtonian theory of slowly-moving (post-Newtonian) sources, and its application to the problems of dynamics and gravitational radiation from compact binary systems. Our current knowledge is 3PN on the center-of-mass energy and 3.5PN on the gravitational-wave flux of inspiralling compact binaries. We compute the innermost circular orbit (ICO) of binary black-hole systems and find a very good agreement with the result of numerical relativity. We argue that the gravitational dynamics of two bodies of comparable masses in general relativity does not resemble that of a test particle on a Schwarzschild background. This leads us to question the validity of some ``Schwarzschild-like'' templates for binary inspiral which are constructed from post-Newtonian resummation techniques. 
  The Hessian of the entropy function can be thought of as a metric tensor on the state space. In the context of thermodynamical fluctuation theory Ruppeiner has argued that the Riemannian geometry of this metric gives insight into the underlying statistical mechanical system; the claim is supported by numerous examples. We study this geometry for some families of black holes. It is flat for the BTZ and Reissner-Nordstrom black holes, while curvature singularities occur for the Reissner-Nordstrom-anti-de Sitter and Kerr black holes. 
  The notion of G-structure is defined and various geometrical and topological aspects of such structures are discussed. A particular chain of subgroups in the affine group for Minkowski space is chosen and the canonical geometrical and topological objects that are asssociated with each reduction of the bundle of affine frames on a four-dimensional manifold are detailed. Their physical significance is discussed in the language of topological defects in ordered media. Particular attention is paid to how one topologically characterizes the wave phase of the spacetime vacuum manifold. 
  A cosmology is considered driven by a stress-energy tensor consisting of a perfect fluid, an inhomogeneous pressure term (which we call a ``tachyonic dust'' for reasons which will become apparent) and a cosmological constant. The inflationary, radiation dominated and matter dominated eras are investigated in detail. In all three eras, the tachyonic pressure decreases with increasing radius of the universe and is thus minimal in the matter dominated era. The gravitational effects of the dust, however, may still strongly affect the universe at present time. In case the tachyonic pressure is positive, it enhances the overall matter {\em density} and is a candidate for dark matter. In the case where the tachyonic pressure is negative, the recent acceleration of the universe can be understood without the need for a cosmological constant. The ordinary matter, however, has positive energy density at all times. In a later section, the extension to a variable cosmological term is investigated and a specific model is put forward such that recent acceleration and future re-collapse is possible. 
  This study looks at motion of particles using mathematical methods of chronometric invariants (physical observable values in General Relativity). It is shown that aside for mass-bearing particles and light-like particles "zero-particles" can exist in fully degenerated space-time (zero-space). For a regular observer zero-particles move instantly, thus transferring long-range action. Further we show existence of two separate areas in unhomogeneous space-time, where observable time flows into future and into past, while this duality is not found in homogeneous space-time. These areas are referred to as our world, where time flows into future and as the mirror Universe, where time flows in past. The areas are separated with a space-time membrane, referred to as zero-space, where observable time stops. 
  We obtain a global existence result for the Einstein equations. We show that in the maximal Cauchy development of vacuum $T^2$ symmetric initial data with nonvanishing twist constant, except for the special case of flat Kasner initial data, the area of the $T^2$ group orbits takes on all positive values. This result shows that the areal time coordinate $R$ which covers these spacetimes runs from zero to infinity, with the singularity occurring at R=0. 
  One of the greatest challenges facing gravitational wave astronomy in the low frequency band is the confusion noise generated by the vast numbers of unresolved galactic and extra galactic binary systems. Estimates of the binary confusion noise suffer from several sources of astrophysical uncertainty, such as the form of the initial mass function and the star formation rate. There is also considerable uncertainty about what defines the confusion limit. Various ad-hoc rules have been proposed, such as the one source per bin rule, and the one source per three bin rule. Here information theoretic methods are used to derive a more realistic estimate for the confusion limit. It is found that the gravitational wave background becomes unresolvable when there is, on average, more than one source per eight frequency bins. This raises the best estimate for the frequency at which galactic binaries become a source of noise from 1.45 mHz to 2.54 mHz. 
  We calculate momentum imparted by colliding gravitational waves in a closed Friedmann Robertson-Walker background and also by gravitational waves with toroidal wavefronts using an operational procedure. The results obtained for toroidal wavefronts are well behaved and reduce to the spherical wavefronts for a special choice. 
  In this work we estimate the performance of a method for the detection of burst events in the data produced by interferometric gravitational wave detectors. We compute the receiver operating characteristics in the specific case of a simulated noise having the spectral density expected for Virgo, using test signals taken from a library of possible waveforms emitted during the collapse of the core of Type II Supernovae. 
  We study neutron stars in a varying speed of light (VSL) theory of gravity in which the local speed of light depends upon the value of a scalar field $\phi$. We find that the masses and radii of the stars are strongly dependent on the strength of the coupling between $\phi$ and the matter field and that for certain choices of coupling parameters, the maximum neutron star mass can be arbitrarily small. We also discuss the phenomenon of cosmological evolution of VSL stars (analogous to the gravitational evolution in scalar-tensor theories) and we derive a relation showing how the fractional change in the energy of a star is related to the change in the cosmological value of the scalar field. 
  There is an abundance of empirical evidence in the numerical relativity literature that the form in which the Einstein evolution equations are written plays a significant role in the lifetime of numerical simulations. This paper attempts to present a consistent framework for modifying any system of evolution equations by adding terms that push the evolution toward the constraint hypersurface. The method is, in principle, applicable to any system of partial differential equations which can be divided into evolution equations and constraints, although it is only demonstrated here through an application to the Maxwell equations. 
  We discuss the gravitomagnetic time delay and the Lense-Thirring effect in the context of Brans-Dicke theory of gravity. We compare the theoretical results obtained with those predicted by general relativity. We show that within the accuracy of experiments designed to measure these effects both theories predict essentially the same result. 
  A minimal coupling rule for the coupling of the electron spin to curved spacetime in general relativity suggests the possibility of a coupling between electromagnetic and gravitational radiation mediated by means of a quantum fluid. Thus quantum transducers between these two kinds of radiation fields might exist. We report here on the first attempt at a Hertz-type experiment, in which a high-$\rm{T_c}$ superconductor (YBCO) was the sample material used as a possible quantum transducer to convert EM into GR microwaves, and a second piece of YBCO in a separate apparatus was used to back-convert GR into EM microwaves. An upper limit on the conversion efficiency of YBCO was measured to be $1.6\times10^{-5}$ at liquid nitrogen temperature. 
  An experiment in Low Earth Orbit (LEO) is proposed to measure components of the Riemann curvature tensor using atom interferometry. We show that the difference in the quantum phase $\Delta\phi$ of an atom that can travel along two intersecting geodesics is given by $mR_{0i0j}/\hbar$ times the spacetime volume contained within the geodesics. Our expression for $\Delta\phi$ also holds for gravitational waves in the long wavelength limit. 
  We consider spacetime to be a connected real 4-manifold equipped with a Lorentzian metric and an affine connection. The 10 independent components of the (symmetric) metric tensor and the 64 connection coefficients are the unknowns of our theory. We introduce an action which is quadratic in curvature and study the resulting system of Euler-Lagrange equations. In the first part of the paper we look for Riemannian solutions, i.e. solutions whose connection is Levi-Civita. We find two classes of Riemannian solutions: 1) Einstein spaces, and 2) spacetimes with metric of a pp-wave and parallel Ricci curvature. We prove that for a generic quadratic action these are the only Riemannian solutions. In the second part of the paper we look for non-Riemannian solutions. We define the notion of a "Weyl pseudoinstanton" (metric compatible spacetime whose curvature is purely Weyl) and prove that a Weyl pseudoinstanton is a solution of our field equations. Using the pseudoinstanton approach we construct explicitly a non-Riemannian solution which is a wave of torsion in Minkowski space. We discuss the possibility of using this non-Riemannian solution as a mathematical model for the graviton or the neutrino. 
  We investigate the transformation laws of coordinates in generalizations of special relativity with two observer-independent scales. The request of covariance leads to simple formulas if one assumes noncanonical Poisson brackets, corresponding to noncommuting spacetime coordinates. 
  We discuss simple integration methods for the calculation of rotating black hole scattering resonances both in the complex frequency plane (quasinormal modes) and the complex angular momentum plane (Regge poles). Our numerical schemes are based on variations of "phase-amplitude" methods. In particular, we discuss the Pruefer transformation, where the original (frequency domain) Teukolsky wave equation is replaced by a pair of first-order non-linear equations governing the introduced phase functions. Numerical integration of these equations, performed along the real coordinate axis, or along rotated contours in the complex coordinate plane, provides the required S-matrix element (the ratio of amplitudes of the outgoing and ingoing waves at infinity). Mueller's algorithm is then employed to conduct searches in the complex plane for the poles of this quantity (which are, by definition, the desired resonances). We have tested this method by verifying known results for Schwarzschild quasinormal modes and Regge poles, and provide new results for the Kerr black hole problem. We also describe a new method for estimating the "excitation coefficients" for quasinormal modes. The method is applied to scalar waves moving in the Kerr geometry, and the obtained results shed light on the long-lived quasinormal modes that exist for black holes rotating near the extreme Kerr limit. 
  We show that a particular set of global modes for the massive de Sitter scalar field (the de Sitter waves) allows to manage the group representations and the Fourier transform in the flat (Minkowskian) limit. This is in opposition to the usual acceptance based on a previous result, suggesting the appearance of negative energy in the limit process. This method also confirms that the Euclidean vacuum, in de Sitter spacetime, has to be preferred as far as one wishes to recover ordinary QFT in the flat limit. 
  We assume that space-time at the Planck scale is discrete, quantised in Planck units and "qubitsed" (each pixel of Planck area encodes one qubit), that is, quantum space-time can be viewed as a quantum computer. Within this model, one finds that quantum space-time itself is entangled, and can quantum-evaluate Boolean functions which are the laws of Physics in their discrete and fundamental form. 
  Using the one-parameter internal symmetry group in the Bianchi type-I spacetime for cosmological models with a perfect fluid, we show that a system of coordinates exists in the associated internal space where two scale factors become equal. We find the general solution for an anisotropic model containing a perfect fluid with constant baryotropic index and investigate the asymptotic regimes. We obtain exact solutions for a set of anisotropic fluids which includes an anisotropic stiff fluid. 
  Using the generalized Langevin equations involving the stress tensor approach, we study the dynamics of a perfectly reflecting mirror which is exposed to the electromagnetic radiation pressure by a laser beam in a fluid at finite temperature. Based on the fluctuation-dissipation theorem, the minimum uncertainty of the mirror's position measurement from both quantum and thermal noises effects including the photon counting error in the laser interferometer is obtained in the small time limit as compared with the "standard quantum limit".  The result of the large time behavior of fluctuations of the mirror's velocity in a dissipative environment can be applied to the laser interferometer of the ground-based gravitational wave detector. 
  Perfect fluid with kinematic self-similarity is studied in 2+1 dimensional spacetimes with circular symmetry, and various exact solutions to the Einstein field equations are given. In particular, these include all the solutions of dust and stiff perfect fluid with self-similarity of the first kind, and all the solutions of perfect fluid with a linear equation of state and self-similarity of the zeroth or second kind. It is found that some of these solutions represent gravitational collapse, and the final state of the collapse can be either black holes or naked singularities. 
  We discuss stability of spherically symmetric static solutions in Newtonian limit of Jordan, Brans-Dicke field equations. The behavior of the stable equilibrium solutions for the spherically symmetric configurations considered here, it emerges that the more compact a model is, the more stable it is. Moreover, linear stability analysis shows the existence of stable configurations for any polytropic index. 
  The Noether-charge realization and the Hamiltonian realization for the $\diff({\cal M})$ algebra in diffeomorphism invariant gravitational theories are studied in a covariant formalism. For the Killing vector fields, the Nother-charge realization leads to the mass formula as an entire vanishing Noether charge for the vacuum black hole spacetimes in general relativity and the corresponding first law of the black hole mechanics. It is analyzed in which sense the Hamiltonian functionals form the $\diff({\cal M})$ algebra under the Poisson bracket and shown how the Noether charges with respect to the diffeomorphism generated by vector fields and their variations in general relativity form this algebra. The asymptotic behaviors of vector fields generating diffeomorphism of the manifold with boundaries are discussed. In order to get more precise estimation for the "central extension" of the algebra, it is analyzed in the Newman-Penrose formalism and shown that the "central extension" for a large class of vector fields is always zero on the Killing horizon. It is also checked whether the Virasoro algebra may be picked up by choosing the vector fields near the horizon. The conclusion is unfortunately negative. 
  Copernicus realised we were not at the centre of the universe. A universe made finite by topological identifications introduces a new Copernican consideration: while we may not be at the geometric centre of the universe, some galaxy could be. A finite universe also picks out a preferred frame: the frame in which the universe is smallest. Although we are not likely to be at the centre of the universe, we must live in the preferred frame (if we are at rest with respect to the cosmological expansion). We show that the preferred topological frame must also be the comoving frame in a homogeneous and isotropic cosmological spacetime. Some implications of topologically identifying time are also discussed. 
  The existence of closed trapped surfaces need not imply a cosmological singularity when the spatial hypersurfaces are compact. This is illustrated by a variety of examples, in particular de Sitter spacetime admits many closed trapped surfaces and obeys the null convergence condition but is non-singular in the k=+1 frame. 
  The extended conformal algebra (so)(2,3) of global, quantum, constants of motion in 2+1 dimensional gravity with topology R x T^2 and negative cosmological constant is reviewed. It is shown that the 10 global constants form a complete set by expressing them in terms of two commuting spinors and the Dirac gamma matrices. The spinor components are the globally constant holonomy parameters, and their respective spinor norms are their quantum commutators. 
  We show that, contrary to the claim in hep-th/0304050, the propagation of a spin-2 field in an electromagnetic background is {\em causal}. 
  The prediction that black holes radiate due to quantum effects is often considered one of the most secure in quantum field theory in curved space-time. Yet this prediction rests on two dubious assumptions: that ordinary physics may be applied to vacuum fluctuations at energy scales increasing exponentially without bound; and that quantum-gravitational effects may be neglected. Various suggestions have been put forward to address these issues: that they might be explained away by lessons from sonic black hole models; that the prediction is indeed successfully reproduced by quantum gravity; that the success of the link provided by the prediction between black holes and thermodynamics justifies the prediction.   This paper explains the nature of the difficulties, and reviews the proposals that have been put forward to deal with them. None of the proposals put forward can so far be considered to be really successful, and simple dimensional arguments show that quantum-gravitational effects might well alter the evaporation process outlined by Hawking. Thus a definitive theoretical treatment will require an understanding of quantum gravity in at least some regimes. Until then, no compelling theoretical case for or against radiation by black holes is likely to be made.   The possibility that non-radiating "mini" black holes exist should be taken seriously; such holes could be part of the dark matter in the Universe. Attempts to place observational limits on the number of "mini" black holes (independent of the assumption that they radiate) would be most welcome. 
  A united approach of the large-scale structure of a closed universe and the local spherically symmetric gravitational field is given by supposing an appropriate boundary condition. The general feature of the model obtained are the following. The universe is approximately homogeneous and isotropic on the average on large scale and is expanding at present, as described by the standard model; while locally, the small exterior region of a star started long ago to contract, as expected by the gravitational collapse theory. 
  We give a classification of spherically symmetric kinematic self-similar solutions. This classification is complementary to that given in a previous work by the present authors [Prog. Theor. Phys. 108, 819 (2002)]. Dust solutions of the second, zeroth and infinite kinds, perfect-fluid solutions and vacuum solutions of the first kind are treated. The kinematic self-similarity vector is either parallel or orthogonal to the fluid flow in the perfect-fluid and vacuum cases, while the `tilted' case, i.e., neither parallel nor orthogonal case, is also treated in the dust case. In the parallel case, there are no dust solutions of the second (except when the self-similarity index $\alpha$ is 3/2), zeroth and infinite kinds, and in the orthogonal case, there are no dust solutions of the second and infinite kinds. Except in these cases, the governing equations can be integrated to give exact solutions. It is found that the dust solutions in the tilted case belong to a subclass of the Lema{\^ i}tre-Tolman-Bondi family of solutions for the marginally bound case. The flat Friedmann-Robertson-Walker (FRW) solution is the only dust solution of the second kind with $\alpha=3/2$ in the tilted and parallel cases and of the zeroth kind in the orthogonal case. The flat, open and closed FRW solutions with $p=-\mu/3$, where $p$ and $\mu$ are the pressure and energy density, respectively, are the only perfect-fluid first-kind self-similar solutions in the parallel case, while a new exact solution with $p=\mu$, which we call the ``singular stiff-fluid solution'', is the only such solution in the orthogonal case. The Minkowski solution is the only vacuum first-kind self-similar solution both in the parallel and orthogonal cases. Some important corrections and complements to the authors' previous work are also presented. 
  Line singularities including cosmic strings may be screened by photonic shells until they appear as a planar wall. 
  We study Bianchi I type brane cosmologies with scalar matter self-interacting through combinations of exponential potentials. Such models correspond in some cases to inflationary universes. We discuss the conditions for accelerated expansion to occur, and pay particular attention to the influence of extra dimensions and anisotropy. Our results show that the associated effects evolve in such a way that they become negligible in the late time limit, those related to the anisotropy disappearing earlier. This study focuses mainly on single field models, but we also consider a generalization yielding models with multiple non-interacting fields and examine its features briefly. We conclude that in the brane scenario, as happens in general relativity, an increase in the number of fields assists inflation. 
  We study several issues related to the different choices of time available for the classical and quantum treatment of linearly polarized cylindrical gravitational waves. We pay especial attention to the time evolution of creation and annihilation operators and the definition of Fock spaces for the different choices of time involved. We discuss also the issue of microcausality and the use of field commutators to extract information about the causal properties of quantum spacetime. 
  Present models describing the interaction of quantum Maxwell and gravitational fields predict a breakdown of Lorentz invariance and a non standard dispersion relation in the semiclassical approximation. Comparison with observational data however, does not support their predictions. In this work we introduce a different set of ab initio assumptions in the canonical approach, namely that the homogeneous Maxwell equations are valid in the semiclassical approximation, and find that the resulting field equations are Lorentz invariant in the semiclassical limit. We also include a phenomenological analysis of possible effects on the propagation of light, and their dependence on energy, in a cosmological context. 
  The big bounce singularity of a simple 5D cosmological model is studied. Contrary to the standard big bang space-time singularity, this big bounce singularity is found to be an event horizon at which the scale factor and the mass density of the universe are finite, while the pressure undergoes a sudden transition from negative infinity to positive infinity. By using coordinate transformation it is also shown that before the bounce the universe contracts deflationary, and the universe has been existed, according to the proper-time, for an infinitely long time. 
  An asymptotic stability analysis of spatially homogeneous models of Bianchi type containing tilted perfect fluids is performed. Using the known attractors for the non-tilted Bianchi type universes, we check whether they are stable against perturbations with respect to tilted perfect fluids. We perform the analysis for all Bianchi class B models and the Bianchi type VI_0 model. In particular, we find that none of the non-tilted equilibrium points are stable against tilted perfect fluids stiffer than radiation. We also indicate parts of the phase space where new tilted exact solutions might be found. 
  We show that Loop Quantum Gravity provides new mechanisms through which observed matter-antimatter asymmetry in the Universe can naturally arise at temperatures less than GUT scale. This is enabled through the introduction of a new length scale ${\cal L}$, much greater than Planck length ($l_P$), to obtain semi-classical weave states in the theory. This scale which depends on the momentum of the particle modifies the dispersion relation for different helicities of fermions and leads to lepton asymmetry. 
  At the 20-th Texas Symposium on Relativistic Astrophysics there was a plenary talk devoted to the recent developments in classical Relativity. In that talk the problems of gravitational collapse, collisions of black holes, and of black holes as celestial bodies were discussed. But probably the problems of the internal structure of black holes are a real great challenge. In my talk I want to outline the recent achievements in our understanding of the nature of the singularity (and beyond!) inside a realistic rotating black hole. This presentation also addresses the following questions: Can we see what happens inside a black hole? Can a falling observer cross the singularity without being crushed? An answer to these questions is probably "yes". 
  We derive an expression for the quasinormal modes of scalar perturbations in near extreme d-dimensional Schwarzschild-de Sitter and Reissner-Nordstrom-de Sitter black holes. We show that, in the near extreme limit, the dynamics of the scalar field is characterized by a Poschl-Teller effective potential. The results are qualitatively independent of the spacetime dimension and field mass. 
  We give a general geometric definition of asymptotic flatness at null infinity in $d$-dimensional general relativity ($d$ even) within the framework of conformal infinity. Our definition is arrived at via an analysis of linear perturbations near null infinity and shown to be stable under such perturbations. The detailed fall off properties of the perturbations, as well as the gauge conditions that need to be imposed to make the perturbations regular at infinity, are qualitatively different in higher dimensions; in particular, the decay rate of a radiating solution at null infinity differs from that of a static solution in higher dimensions. The definition of asymptotic flatness in higher dimensions consequently also differs qualitatively from that in $d=4$.   We then derive an expression for the generator conjugate to an asymptotic time translation symmetry for asymptotically flat spacetimes in $d$-dimensional general relativity ($d$ even) within the Hamiltonian framework, making use especially of a formalism developed by Wald and Zoupas. This generator is given by an integral over a cross section at null infinity of a certain local expression and is taken to be the definition of the Bondi energy in $d$ dimensions. Our definition yields a manifestly positive flux of radiated energy.   Our definitions and constructions fail in odd spacetime dimensions, essentially because the regularity properties of the metric at null infinity seem to be insufficient in that case. We also find that there is no direct analog of the well-known infinite set of angle dependent translational symmetries in more than 4 dimensions. 
  We analyze changes of the Immirzi parameter in loop quantum gravity and compare their consequences with those of Lorentz boosts and constant conformal transformations in black hole physics. We show that the effective value deduced for the Planck length in local measurements of vacuum black holes by an asymptotic observer may depend on its conformal or Lorentz frame. This introduces an apparent ambiguity in the expression of the black hole entropy which is analogous to that produced by the Immirzi parameter. For quantities involving a notion of energy, the similarity between the implications of the Immirzi ambiguity and a conformal scaling disappears, but the parallelism with boosts is maintained. 
  Black holes binaries support unstable orbits at very close separations. In the simplest case of geodesics around a Schwarzschild black hole the orbits, though unstable, are regular. Under perturbation the unstable orbits can become the locus of chaos. All unstable orbits, whether regular or chaotic, can be quantified by their Lyapunov exponents. The exponents are observationally relevant since the phase of gravitational waves can decohere in a Lyapunov time. If the timescale for dissipation due to gravitational waves is shorter than the Lyapunov time, chaos will be damped and essentially unobservable. We find the timescales can be comparable. We emphasize that the Lyapunov exponents must only be used cautiously for several reasons: they are relative and depend on the coordinate system used, they vary from orbit to orbit, and finally they can be deceptively diluted by transient behaviour for orbits which pass in and out of unstable regions. 
  We give a short update of our research program on nonequilibrium statistical field theory applied to quantum processes in the early universe and black holes, as well as the development of stochastic gravity theory as an extension of semiclassical gravity and an intermediary in the 'bottom-up' approach to quantum gravity. 
  An approximately scale invariant spectrum generating the seeds of structure formation is derived from a bimetric gravity theory. By requiring that the amplitude of the CMB fluctuations from the model matches the observed value, we determine the fundamental length scale in the model to be a factor of 10^5 times larger than the Planck length, which results in a scalar mode spectral index: n_s\approx 0.97, and its running: \alpha_s\approx -5\times 10^{-4}. This is accomplished in the variable speed of light (VSL) metric frame, in which the dynamics of perturbations of the bimetric scalar field are determined by a minimally-coupled Klein-Gordon equation, and it is assumed that modes are born in a ground state at a scale given by the fundamental length scale appearing in the bimetric structure. We show that while this is taking place for scales of interest, the background (primordial) radiation energy density is strongly suppressed as a result of the bimetric structure of the model. Nevertheless, the enlarged lightcone of matter fields ensures that the horizon and flatness problems are solved. 
  In quantum theory, the curved spacetime of Einstein's general theory of relativity acts as a dispersive optical medium for the propagation of light. Gravitational rainbows and birefringence replace the classical picture of light rays mapping out the null geodesics of curved spacetime. Even more remarkably, {\it superluminal} propagation becomes a real possibility, raising the question of whether it is possible to send signals into the past. In this article, we review recent developments in the quantum theory of light propagation in general relativity and discuss whether superluminal light is compatible with causality. 
  The ``optical bars''/``optical lever'' topologies of gravitational-wave antennae allow to obtain sensitivity better that the Standard Quantum Limit while keeping the optical pumping energy in the antenna relatively low. Element of the crucial importance in these schemes is the local meter which monitors the local test mirror position. Using cross-correlation of this meter back-action noise and its measurement noise it is possible to further decrease the optical pumping energy. In this case the pumping energy minimal value will be limited by the internal losses in the antenna only. Estimates show that for values of parameters available for contemporary and planned gravitational-wave antennae, sensitivity about one order of magnitude better than the Standard Quantum Limit can be obtained using the pumping energy about one order of magnitude smaller energy than is required in the traditional topology in order to obtain the the Standard Quantum Limit level of sensitivity. 
  The cosmological constant is not an absolute constant. The gravitating part of the vacuum energy is adjusted to the energy density of matter and to other types of the perturbations of the vacuum. We discuss how the vacuum energy responds (i) to the curvature of space in the Einstein closed Universe; (ii) to the expansion rate in the de Sitter Universe; and (iii) to the rotation in the Goedel Universe. In all these steady state Universes, the gravitating vacuum energy is zero in the absence of the perturbation, and is proportional to the energy density of perturbation. This is in a full agreement with the thermodynamic Gibbs-Duhem relation applicable to any quantum vacuum. It demonstrates that (i) the cosmological constant is not huge, since according to the Gibbs-Duhem relation the contribution of zero point fluctuations to the vacuum energy is cancelled by the trans-Planckian degrees of freedom; (ii) the cosmological constant is non-zero, since the perturbations of the vacuum state induce the non-zero vacuum energy; and (iii) the gravitating vacuum energy is on the order of the energy density of matter and/or of other perturbations. We also consider the vacuum response to the non-steady-state perturbations. In this case the Einstein equations are modified to include the non-covariant corrections, which are responsible for the relaxation of the cosmological constant. The connection to the quintessence is demonstrated. The problem of the energy-momentum tensor for the gravitational field is discussed in terms of effective gravity. The difference between the momentum and pseudo-momentum of gravitational waves in general relativity is similar to that for sound waves in hydrodynamics. 
  I present a new, simple method to dynamically control the growth of the discretized constraints during a free evolution of Einstein's equations. During an evolution, any given family of formulations is adjusted off the constraints surface in a way such that, for any chosen numerical method and arbitrary but fixed resolution, the constraints growth can be minimized with respect to the freedom allowed by the formulation. In particular, provided there is enough freedom, the discretized constraints can be maintained close to its initial truncation value for all times, or decay from it.   No a priori knowledge of the solution is needed, and the method can be applied to any formulation of Einstein's equations without affecting hyperbolicity. This method is independent of the numerical algorithm and accounts for constraint violating modes introduced both by continuum instabilities of the formulation and by the numerical method. 
  1) A wave equation is derived from the kinetic equations governing media with rotational as well as translational degrees of freedom. In this wave the fluctuating quantity is a vector, the bulk spin. The transmission is similar to compressive waves but propagation is possible even in the limit of incompressibility, where such disturbances could become dominant. 2) In this context a kinetic theory of space-time is introduced, in which hypothetical constituents of the space-time manifold possess such a rotational degree of freedom (spin). Physical fields (i.e. electromagnetic or gravitational) in such a theory are represented as moments of a statistical distribution of these constituents. The spin wave equation from 1) is treated as a candidate for governing light and metric. Such a theory duplicates to first order Maxwell's equations of electromagnetism, Schrodinger's equation for the electron, and the Lorentz transformations of special relativity. Slight deviations from the classical approach are predicted and should be experimentally verifiable. 
  We investigate the structure of the ZVW (Zipoy-Voorhees-Weyl) spacetime, which is a Weyl solution described by the Zipoy-Voorhees metric, and the delta=2 Tomimatsu-Sato spacetime. We show that the singularity of the ZVW spacetime, which is represented by a segment rho=0, -sigma<z<sigma in the Weyl coordinates, is geometrically point-like for delta<0, string-like for 0<delta<1 and ring-like for delta>1. These singularities are always naked and have positive Komar masses for delta>0. Thus, they provide a non-trivial example of naked singularities with positive mass. We further show that the ZVW spacetime has a degenerate Killing horizon with a ring singularity at the equatorial plane for delta=2,3 and delta>=4. We also show that the delta=2 Tomimatsu-Sato spacetime has a degenerate horizon with two components, in contrast to the general belief that the Tomimatsu-Sato solutions with even delta do not have horizons. 
  We show that a flow (timelike congruence) in any type $B_{1}$ warped product spacetime is uniquely and algorithmically determined by the condition of zero flux. (Though restricted, these spaces include many cases of interest.) The flow is written out explicitly for canonical representations of the spacetimes. With the flow determined, we explore an inverse approach to Einstein's equations where a phenomenological fluid interpretation of a spacetime follows directly from the metric irrespective of the choice of coordinates. This approach is pursued for fluids with anisotropic pressure and shear viscosity. In certain degenerate cases this interpretation is shown to be generically not unique. The framework developed allows the study of exact solutions in any frame without transformations. We provide a number of examples, in various coordinates, including spacetimes with and without unique interpretations. The results and algorithmic procedure developed are implemented as a computer algebra program called GRSource. 
  Explicit Riemannian metrics with Euclidean signature and anti-self dual curvature that do not admit any Killing vectors are presented. The metric and the Riemann curvature scalars are homogenous functions of degree zero in a single real potential and its derivatives. The solution for the potential is a sum of exponential functions which suggests that for the choice of a suitable domain of coordinates and parameters it can be the metric on a compact manifold. Then, by the theorem of Hitchin, it could be a class of metrics on $K3$, or on surfaces whose universal covering is $K3$. 
  We investigate the effects on cosmological density perturbations of dark radiation in a Randall-Sundrum 2 type brane-world. Dark radiation in the background is limited by observational constraints to be a small fraction of the radiation energy density, but it has an interesting qualitative effect in the radiation era. On large scales, it serves to slightly suppress the radiation density perturbations at late times, while boosting the perturbations in dark radiation. In a kinetic (stiff) era, the suppression is much stronger, and drives the density perturbations to zero. 
  We use the general solution to the trace of the 4-dimensional Einstein equations for static, spherically symmetric configurations as a basis for finding a general class of black hole (BH) metrics, containing one arbitrary function $g_{tt} = A(r)$ which vanishes at some $r = r_h > 0$, the horizon radius. Under certain reasonable restrictions, BH metrics are found with or without matter and, depending on the boundary conditions, can be asymptotically flat or have any other prescribed large $r$ behaviour. It is shown that this procedure generically leads to families of solutions unifying non-extremal globally regular BHs with a Kerr-like global structure, extremal BHs and symmetric wormholes. Horizons in space-times with zero scalar curvature are shown to be either simple or double. The same is generically true for horizons inside a matter distribution, but in special cases there can be horizons of any order. A few simple examples are discussed. A natural application of the above results is the brane world concept, in which the trace of the 4D gravity equations is the only unambiguous equation for the 4D metric, and its solutions can be continued into the 5D bulk according to the embedding theorems. 
  We discuss the electromagnetic measurements of rotating observers and study the propagation of electromagnetic waves in a uniformly rotating frame of reference. The phenomenon of helicity-rotation coupling is elucidated and some of the observational consequences of the coupling of the spin of a particle with the rotation of a gravitational source are briefly examined. 
  At the threshold of black hole formation in the gravitational collapse of a scalar field a naked singularity is formed through a universal critical solution that is discretely self-similar. We study the global spacetime structure of this solution. It is spherically symmetric, discretely self-similar, regular at the center to the past of the singularity, and regular at the past lightcone of the singularity. At the future lightcone of the singularity, which is also a Cauchy horizon, the curvature is finite and continuous but not differentiable. To the future of the Cauchy horizon the solution is not unique, but depends on a free function (the null data coming out of the naked singularity). There is a unique continuation with a regular center (which is self-similar). All other self-similar continuations have a central timelike singularity with negative mass. 
  We study tensorial perturbations (gravitational waves) in a universe with particle production (OSC). The background of gravitational waves produces a perturbation in the redshift observed from distant sources. The modes for the perturbation in the redshift (induced redshift) are calculated in a universe with particle production. 
  In this paper we present a class of exact inhomogeneous solutions to Einstein's equations for higher dimensional Szekeres metric with perfect fluid and a cosmological constant. We also show particular solutions depending on the choices of various parameters involved and for dust case. Finally, we examine the asymptotic behaviour of some of these solutions. 
  We discuss the modal properties of the $r$-modes of relativistic superfluid neutron stars, taking account of the entrainment effects between superfluids. In this paper, the neutron stars are assumed to be filled with neutron and proton superfluids and the strength of the entrainment effects between the superfluids are represented by a single parameter $\eta$. We find that the basic properties of the $r$-modes in a relativistic superfluid star are very similar to those found for a Newtonian superfluid star. The $r$-modes of a relativistic superfluid star are split into two families, ordinary fluid-like $r$-modes ($r^o$-mode) and superfluid-like $r$-modes ($r^s$-mode). The two superfluids counter-move for the $r^s$-modes, while they co-move for the $r^o$-modes. For the $r^o$-modes, the quantity $\kappa\equiv\sigma/\Omega+m$ is almost independent of the entrainment parameter $\eta$, where $m$ and $\sigma$ are the azimuthal wave number and the oscillation frequency observed by an inertial observer at spatial infinity, respectively. For the $r^s$-modes, on the other hand, $\kappa$ almost linearly increases with increasing $\eta$. It is also found that the radiation driven instability due to the $r^s$-modes is much weaker than that of the $r^o$-modes because the matter current associated with the axial parity perturbations almost completely vanishes. 
  Applications of Riemannian quantum geometry to cosmology have had notable successes. In particular, the fundamental discreteness underlying quantum geometry has led to a natural resolution of the big bang singularity. However, the precise mathematical structure underlying loop quantum cosmology and the sense in which it implements the full quantization program in a symmetry reduced model has not been made explicit. The purpose of this paper is to address these issues, thereby providing a firmer mathematical and conceptual foundation to the subject. 
  We investigate the occurrence and nature of naked singularities in the gravitational collapse of an adiabatic perfect fluid in self-similar higher dimensional space-times. It is shown that strong curvature naked singularities could occur if the weak energy condition holds. Its implication for cosmic censorship conjecture is discussed. Known results of analogous studies in four dimensions can be recovered. 
  We construct approximate solutions that will describe the last stage of cylindrically symmetric gravitational collapse of dust fluid. Just before the spacetime singularity formation, the speed of the dust fluid might be almost equal to the speed of light by gravitational acceleration. Therefore the analytic solution describing the dynamics of cylindrical null dust might be the crudest approximate solution of the last stage of the gravitational collapse. In this paper, we regard this null dust solution as a background and perform `high-speed approximation' to know the gravitational collapse of ordinary timelike dust fluid; the `deviation of the timelike  4-velocity vector field from null' is treated as a perturbation. In contrast with the null dust approximation, our approximation scheme can describe the generation of gravitational waves in the course of the cylindrically symmetric dust collapse. 
  In a recent paper by the author, a new approach was suggested for quantising space-time, or space. This involved developing a procedure for quantising a system whose configuration space--or history-theory analogue--is the set of objects in a (small) category $\Q$. In the present paper, we show how this theory can be applied to the special case when $\Q$ is a category of sets. This includes the physically important examples where $\Q$ is a category of causal sets, or topological spaces. 
  The anisotropic Bianchi I cosmological model coupled with perfect fluid is quantized in the minisuperspace. The perfect fluid is described by using the Schutz formalism which allows to attribute dynamical degrees of freedom to matter. A Schr\"odinger-type equation is obtained where the matter variables play the role of time. However, the signature of the kinetic term is hyperbolic. This Schr\"odinger-like equation is solved and a wave packet is constructed. The norm of the resulting wave function comes out to be time dependent, indicating the loss of unitarity in this model. The loss of unitarity is due to the fact that the effective Hamiltonian is hermitian but not self-adjoint. The expectation value and the bohmian trajectories are evaluated leading to different cosmological scenarios, what is a consequence of the absence of a unitary quantum structure. The consistency of this quantum model is discussed as well as the generality of the absence of unitarity in anisotropic quantum models. 
  Magueijo and Smolin have introduced a modification of the Lorentz group for the momentum-space transformations in Doubly-Special Relativity. As presented the group is non-linear, but we show that it is a group of fraction-linear transformations in 4 dimensional real projective space. We pass to the associated 5 dimensional linear space and identify the subgroup as a conjugate of the ordinary Lorentz group, giving the conjugating matrix. Taking the dual of the 5-space, we identify a subgroup as the Lorentz transformations about a point different to the origin. 
  We look for physically realistic initial data in numerical relativity which are in agreement with post-Newtonian approximations. We propose a particular solution of the time-symmetric constraint equation, appropriate to two momentarily static black holes, in the form of a conformal decomposition of the spatial metric. This solution is isometric to the post-Newtonian metric up to the 2PN order. It represents a non-linear deformation of the solution of Brill and Lindquist, i.e. an asymptotically flat region is connected to two asymptotically flat (in a certain weak sense) sheets, that are the images of the two singularities through appropriate inversion transformations. The total ADM mass M as well as the individual masses m_1 and m_2 (when they exist) are computed by surface integrals performed at infinity. Using second order perturbation theory on the Brill-Lindquist background, we prove that the binary's interacting mass-energy M-m_1-m_2 is well-defined at the 2PN order and in agreement with the known post-Newtonian result. 
  The event horizon of a charged black hole is, according to Ruffini\cite{Ruffini} and Preparata \emph{et al.}\cite{PreparataEtAl}, surrounded by a special region called the \emph{dyadosphere} where the electromagnetic field exceeds the Euler-Heisenberg critical value for electron-positron pair production. We obtain the energy distribution in the dyadosphere region for a Reissner-Nordstr\"{o}m black hole. We find that the energy-momentum prescriptions of Einstein, Landau-Lifshitz, Papapetrou, and Weinberg give the same and acceptable energy distribution. 
  Covariant generalizations of well-known wave equations predict the existence of inertial-gravitational effects for a variety of quantum systems that range from Bose-Einstein condensates to particles in accelerators. Additional effects arise in models that incorporate Born reciprocity principle and the notion of a maximal acceleration. Some specific examples are discussed in detail. 
  We study the interaction of an n-dimensional topological defect (n-brane) described by the Nambu-Goto action with a higher-dimensional Schwarzschild black hole moving in the bulk spacetime. We derive the general form of the perturbation equations for an n-brane in the weak field approximation and solve them analytically in the most interesting cases. We specially analyze applications to brane world models. We calculate the induced geometry on the brane generated by a moving black hole. From the point of view of a brane observer, this geometry can be obtained by solving (n+1)-dimensional Einstein's equations with a non-vanishing right hand side. We calculate the effective stress-energy tensor corresponding to this `shadow-matter'. We explicitly show that there exist regions on the brane where a brane observer sees an apparent violation of energy conditions. We also study the deflection of light propagating in the region of influence of this `shadow matter'. 
  The gravitational effects in the relativistic quantum mechanics are investigated in a relativistically derived version of Heaviside's speculative Gravity (in flat space-time) named here as Maxwellian Gravity. The standard Dirac's approach to the intrinsic spin in the fields of Maxwellian Gravity yields the gravitomagnetic moment of a Dirac (spin 1/2) particle exactly equals to its intrinsic spin. Violation of The Equivalence Principle (both at classical and quantum mechanical level) in the relativistic domain has also been reported in this work. 
  Black strings, one class of higher dimensional analogues of black holes, were shown to be unstable to long wavelength perturbations by Gregory and Laflamme in 1992, via a linear analysis. We revisit the problem through numerical solution of the full equations of motion, and focus on trying to determine the end-state of a perturbed, unstable black string. Our preliminary results show that such a spacetime tends towards a solution resembling a sequence of spherical black holes connected by thin black strings, at least at intermediate times. However, our code fails then, primarily due to large gradients that develop in metric functions, as the coordinate system we use is not well adapted to the nature of the unfolding solution. We are thus unable to determine how close the solution we see is to the final end-state, though we do observe rich dynamical behavior of the system in the intermediate stages. 
  The increasing interest in compact astrophysical objects (neutron stars, binaries, galactic black holes) has stimulated the search for rigorous methods, which allow a systematic general relativistic description of such objects. This paper is meant to demonstrate the use of the inverse scattering method, which allows, in particular cases, the treatment of rotating body problems. The idea is to replace the investigation of the matter region of a rotating body by the formulation of boundary values along the surface of the body. In this way we construct solutions describing rotating black holes and disks of dust ("galaxies"). Physical properties of the solutions and consequences of the approach are discussed. Among other things, the balance problem for two black holes can be tackled. 
  From the original papers an interesting and surprising connection is established between LaPlace and his early discovery that black holes may occur in nature, and a poem by Schiller. 
  Quantum area tensor Regge calculus is considered, some properties are discussed. The path integral quantisation is defined for the usual length-based Regge calculus considered as a particular case (a kind of a state) of the area tensor Regge calculus. Under natural physical assumptions the quantisation of interest is practically unique up to an additional one-parametric local factor of the type of a power of $\det\|g_{\lambda\mu}\|$ in the measure. In particular, this factor can be adjusted so that in the continuum limit we would have any of the measures usually discussed in the continuum quantum gravity, namely, Misner, DeWitt or Leutwyler measure. It is the latter two cases when the discrete measure turns out to be well-defined at small lengths and lead to finite expectation values of the lengths. 
  The standard cosmological model, based on general relativity with an inflationary era, is very effective in accounting for a broad range of observed features of the universe. However, the ongoing puzzles about the nature of dark matter and dark energy, together with the problem of a fundamental theoretical framework for inflation, indicate that cosmology may be probing the limits of validity of general relativity. The early universe provides a testing ground for theories of gravity, since gravitational dynamics can lead to characteristic imprints on the CMB and other cosmological observations. Precision cosmology is in principle a means to constrain and possibly falsify candidate quantum gravity theories like M theory. Generalized Randall-Sundrum brane-worlds provide a phenomenological means to test aspects of M theory. I outline the 1+3-covariant approach to cosmological perturbations in these brane-worlds, and its application to CMB anisotropies. 
  A near-field analysis based on Maxwells equations is presented which indicates that the fields generated by both an electric and a magnetic dipole or quadrapole, and also the gravitational waves generated by a quadrapole mass source propagate superluminally in the nearfield of the source and reduce to the speed of light as the waves propagate into the farfield. Both the phase speed and the group speed are shown to be superluminal in the nearfield of these systems. Although the information speed is shown to differ from group speed in the nearfield of these systems, provided the noise of the signal is small and the modulation method is known, the information can be extracted in a time period much smaller than the wave propagation time, thereby making the information speed only slightly less than the superluminal group speed. It is shown that relativity theory indicates that these superluminal signals can be reflected off of a moving frame causing the information to arrive before the signal was transmitted (i.e. backward in time). It is unknown if these signals can be used to change the past. 
  We find a gravitational wave solution to the linearized version of quadratic gravity by adding successive perturbations to the Einstein's linearized field equations. We show that only the Ricci squared quadratic invariant contributes to give a different solution of those found in Einstein's general relativity. The perturbative solution is written as a power series in the $\beta$ parameter, the coefficient of the Ricci squared term in the quadratic gravitational action. We also show that, for monochromatic waves of a given angular frequency $\omega$, the perturbative solution can be summed out to give an exact solution to linearized version of quadratic gravity, for $0<\omega<c/\mid\beta\mid^{1/2}$.   This result may lead to implications to the predictions for gravitational wave backgrounds of cosmological origin. 
  The quasinormal-mode spectrum of the Schwarzschild-de Sitter black hole is studied in the limit of near-equal black-hole and cosmological radii. It is found that the mode_frequencies_ agree with the P"oschl-Teller approximation to one more order than previously realized, even though the effective_potential_ does not. Whether the spectrum approaches the limiting one uniformly in the mode index is seen to depend on the chosen units (to the order investigated). A perturbation framework is set up, in which these issues can be studied to higher order in future. 
  Recent observations of Type Ia supernovae provide evidence for the acceleration of our universe, which leads to the possibility that the universe is entering an inflationary epoch. We simulate it under a ``big bounce'' model, which contains a time variable cosmological ``constant'' that is derived from a higher dimension and manifests itself in 4D spacetime as dark energy. By properly choosing the two arbitrary functions contained in the model, we obtain a simple exact solution in which the evolution of the universe is divided into several stages. Before the big bounce, the universe contracts from a $\Lambda $-dominated vacuum, and after the bounce, the universe expands. In the early time after the bounce, the expansion of the universe is decelerating. In the late time after the bounce, dark energy (i.e., the variable cosmological ``constant'') overtakes dark matter and baryons, and the expansion enters an accelerating stage. When time tends to infinity, the contribution of dark energy tends to two third of the total energy density of the universe, qualitatively in agreement with observations. 
  The first law of black hole mechanics is derived from the Einstein-Maxwell (EM) Lagrangian by comparing two infinitesimally nearby stationary black holes. With similar arguments, the first law of black hole mechanics in Einstein-Yang-Mills (EYM) theory is also derived. 
  We consider Bel-Robinson-like higher derivative conserved two-index tensors $H_\mn$ in simple matter models, following a recently suggested Maxwell field version. In flat space, we show that they are essentially equivalent to the true stress-tensors. In curved Ricci-flat backgrounds it is possible to redefine $H_\mn$ so as to overcome non-commutativity of covariant derivatives, and maintain conservation, but they become model- and dimension- dependent, and generally lose their simple "BR" form. 
  Bayesian reasoning is applied to the data by the ROG Collaboration, in which gravitational wave (g.w.) signals are searched for in a coincidence experiment between Explorer and Nautilus. The use of Bayesian reasoning allows, under well defined hypotheses, even tiny pieces of evidence in favor of each model to be extracted from the data. The combination of the data of several experiments can therefore be performed in an optimal and efficient way. Some models for Galactic sources are considered and, within each model, the experimental result is summarized with the likelihood rescaled to the insensitivity limit value (``${\cal R}$ function''). The model comparison result is given in in terms of Bayes factors, which quantify how the ratio of beliefs about two alternative models are modified by the experimental observation 
  Spherically symmetric static fluid sources are endowed with rotation and embedded in Kerr empty space-time up to an including quadratic terms in an angular velocity parameter using Darmois junction conditions. Einstein's equation's for the system are developed in terms of linear ordinary differential equations. The boundary of the rotating source is expressed explicitly in terms of sinusoidal functions of the polar angle which differ somewhat according to whether an equation of state exists between internal density and supporting pressure. 
  The results on local existence and continuation criteria obtained by G. Rein in [4] are extended to the case with a non-zero cosmological constant. It is also shown that for the spherically symmetric case and a positive cosmological constant there is a large class of initial data with global existence and inflationary asymptotics in the future as in the case of plane or hyperbolic symmetry treated in [7]. Furthermore we analyze the behaviour of the energy-momentum tensor at late times. 
  It is known that some results for spinors, and in particular for superenergy spinors, are much less transparent and require a lot more effort to establish, when considered from the tensor viewpoint. In this paper we demonstrate how the use of dimensionally dependent tensor identities enables us to derive a number of 4-dimensional identities by straightforward tensor methods in a signature independent manner. In particular, we consider the quadratic identity for the Bel-Robinson tensor ${\cal T}_{abcx}{\cal T}^{abcy} = \delta_x^y {\cal T}_{abcd}{\cal T}^{abcd}/4$ and also the new conservation laws for the Chevreton tensor, both of which have been obtained by spinor means; both of these results are rederived by {\it tensor} means for 4-dimensional spaces of any signature, using dimensionally dependent identities, and also we are able to conclude that there are no {\it direct} higher dimensional analogues. In addition we demonstrate a simple way to show non-existense of such identities via counter examples; in particular we show that there is no non-trivial Bel tensor analogue of this simple Bel-Robinson tensor quadratic identity. On the other hand, as a sample of the power of generalising dimensionally dependent tensor identities from four to higher dimensions, we show that the symmetry structure, trace-free and divergence-free nature of the four dimensional Bel-Robinson tensor does have an analogue for a class of tensors in higher dimensions. 
  The results of measurement of optical mirror coating are presented. These results indicate that Standard Quantum Limit of sensitivity can be reached in the second stage of LIGO project if it is limited by thermoelastic noise in the coating only. 
  A discussion is given of the quantisation of a physical system with finite degrees of freedom subject to a Hamiltonian constraint by treating time as a constrained classical variable interacting with an unconstrained quantum state. This leads to a quantisation scheme that yields a Schrodinger-type equation which is in general nonlinear in evolution. Nevertheless it is compatible with a probabilistic interpretation of quantum mechanics and in particular the construction of a Hilbert space with a Euclidean norm is possible. The new scheme is applied to the quantisation of a Friedmann Universe with a massive scalar field whose dynamical behaviour is investigated numerically. 
  Black holes emit thermal radiation (Hawking effect). If after black-hole evaporation nothing else were left, an arbitrary initial state would evolve into a thermal state (`information-loss problem'). Here it is argued that the whole evolution is unitary and that the thermal nature of Hawking radiation emerges solely through decoherence -- the irreversible interaction with further degrees of freedom. For this purpose a detailed comparison with an analogous case in cosmology (entropy of primordial fluctuations) is presented. Some remarks on the possible origin of black-hole entropy due to interaction with other degrees of freedom are added. This might concern the interaction with quasi-normal modes or with background fields in string theory. 
  The problem of cosmological constant and vacuum energy is usually thought of as the subject of general relativity. However, the vacuum energy is important for the Universe even in the absence of gravity, i.e. in the case when the Newton constant G is exactly zero, G=0. We discuss the response of the vacuum energy to the perturbations of the quantum vacuum in special relativity, and find that as in general relativity the vacuum energy density is on the order of the energy density of matter. In general relativity, the dependence of the vacuum energy on the equation of state of matter does not contain G, and thus is valid in the limit when G tends to zero. However, the result obtained for the vacuum energy in the world without gravity, i.e. when G=0 exactly, is different. 
  The theory of General Relativity explaines the advance of Mercury perihelion using space curvature and the Schwartzschild metric. We demonstrate that this phenomena can also be interpreted due to the cogravitational field produced by the apparent motion of the Sun around Mercury giving exactly the same estimate as derived from the Schwartzschild metric in general relativity theory. This is a surprising and new result because the estimate from both theoretical approaches match exactly the measured value. The discussion and implications of this result is out of the scope of the present work. 
  We show that a recent letter claiming to present exact cosmological solutions in Brans-Dicke theory actually uses a flawed set of equations as the starting point for their analysis. The results presented in the letter are therefore not valid. 
  In the general relativistic description of gravitation, geometry replaces the concept of force. This is possible because of the universal character of free fall, and would break down in its absence. On the other hand, the teleparallel version of general relativity is a gauge theory for the translation group and, as such, describes the gravitational interaction by a force similar to the Lorentz force of electromagnetism, a non-universal interaction. Relying on this analogy it is shown that, although the geometric description of general relativity necessarily requires the existence of the equivalence principle, the teleparallel gauge approach remains a consistent theory for gravitation in its absence. 
  We present results of the all-sky search for gravitational-wave signals from spinning neutron stars in the data of the EXPLORER resonant bar detector. Our data analysis technique was based on the maximum likelihood detection method. We briefly describe the theoretical methods that we used in our search. The main result of our analysis is an upper limit of ${\bf 2\times10^{-23}}$ for the dimensionless amplitude of the continuous gravitational-wave signals coming from any direction in the sky and in the narrow frequency band from 921.00 Hz to 921.76 Hz. 
  It is expected that interferometric gravitational wave detectors such as LIGO \cite{Barish99} will be eventually limited by fundamental noise sources like shot noise and Brownian motion, as well as by seismic noise. In the commissioning process, other technical noise sources (electronics noise, alignment fluctuations) limit the sensitivity and are eliminated one by one. We propose here a way to correlate the noise in the output of the gravitational wave detector with other detector and environmental signals not through their linear transfer functions (often unknown), but through their statistical properties. This could prove useful for identifying the frequency bands dominated by different noise sources in the final configuration, and also to help the commissioning process. 
  We study the massless scalar wave propagation in the time-dependent Schwarzschild black hole background. We find that the Kruskal coordinate is an appropriate framework to investigate the time-dependent spacetime. A time-dependent scattering potential is derived by considering dynamical black hole with parameters changing with time. It is shown that in the quasinormal ringing both the decay time-scale and oscillation are modified in the time-dependent background. 
  The requirements are formulated which lead to the existence of the class of globally regular solutions to the minimally coupled GR equations which are asymptotically de Sitter at the center. The brief review of the resulting geometry is presented. The source term, invariant under radial boots, is classified as spherically symmetric vacuum with variable density and pressure, associated with an r-dependent cosmological term, whose asymptotic in the origin, dictated by the weak energy condition, is the Einstein cosmological term. For this class of metrics the ADM mass is related to both de Sitter vacuum trapped in the origin and to breaking of space-time symmetry. In the case of the flat asymptotic, space-time symmetry changes smoothly from the de Sitter group at the center to the Lorentz group at infinity. Dependently on mass, de Sitter-Schwarzschild geometry describes a vacuum nonsingular black hole, or G-lump - a vacuum selfgravitating particlelike structure without horizons. In the case of de Sitter asymptotic at infinity, geometry is asymptotically de Sitter at both origin and infinity and describes, dependently on parameters and choice of coordinates, a vacuum nonsingular cosmological black hole, selfgravitating particlelike structure at the de Sitter background and regular cosmological models with smoothly evolving vacuum energy density. 
  A method is presented to generalize the power detectors for short bursts of gravitational waves that have been developed for single interferometers so that they can optimally process data from a network of interferometers. The performances of this method for the estimation of the position of the source are studied using numerical simulations. 
  We consider three methods by which one can generate new cosmological models. Two of these are based on the Lorentzian structure of spacetime. In a Lorentzian manifold there can exist horizons that separate regions of spacetime that can be interpreted as cosmological models from others that have the character of ``black holes.'' A number of well known solutions of this type can be used to generate both known cosmological models and others that do not seem to have been recognized. Another method based on the Lorentzian character of spacetime is to simply interchange some space variable with time and try to restructure the metric to make a viable cosmology.   A more broad-ranging method is the use of modern solution-generating techniques to construct new models. This method has been widely used to generate black hole solutions, but seems not to have been so widely used in cosmology. We will discuss examples of all three methods. 
  The Helmholtz equation for symmetric, traceless, second-rank tensor fields in three-dimensional flat space is solved in spherical and cylindrical coordinates by separation of variables making use of the corresponding spin-weighted harmonics. It is shown that any symmetric, traceless, divergenceless second-rank tensor field that satisfies the Helmholtz equation can be expressed in terms of two scalar potentials that satisfy the Helmholtz equation. Two such expressions are given, which are adapted to the spherical or cylindrical coordinates. The application to the linearized Einstein theory is discussed. 
  We present results from a numerical study of critical gravitational collapse of axisymmetric distributions of massless scalar field energy. We find threshold behavior that can be described by the spherically symmetric critical solution with axisymmetric perturbations. However, we see indications of a growing, non-spherical mode about the spherically symmetric critical solution. The effect of this instability is that the small asymmetry present in what would otherwise be a spherically symmetric self-similar solution grows. This growth continues until a bifurcation occurs and two distinct regions form on the axis, each resembling the spherically symmetric self-similar solution. The existence of a non-spherical unstable mode is in conflict with previous perturbative results, and we therefore discuss whether such a mode exists in the continuum limit, or whether we are instead seeing a marginally stable mode that is rendered unstable by numerical approximation. 
  We review some properties of the Einstein-"Gauss-Bonnet" equations for gravity--also called the Einstein-Lanczos equations in five and six dimensions, and the Lovelock equations in higher dimensions. We illustrate, by means of simple Kaluza-Klein and brane cosmological models, some consequences of the quasi-linearity of these equations on the Cauchy problem (a point first studied by Yvonne Choquet-Bruhat), as well as on "junction conditions". 
  The shear free condition is studied for dissipative relativistic self-gravitating fluids in the quasi-static approximation. It is shown that, in the Newtonian limit, such condition implies the linear homology law for the velocity of a fluid element, only if homology conditions are further imposed on the temperature and the emission rate. It is also shown that the shear-free plus the homogeneous expansion rate conditions are equivalent (in the Newtonian limit) to the homology conditions. Deviations from homology and their prospective applications to some astrophysical scenarios are discussed, and a model is worked out. 
  We study primordial gravitational waves from inflation in Randall-Sundrum braneworld model. The effect of small change of the Hubble parameter during inflation is investigated using a toy model given by connecting two de Sitter branes. We analyze the power spectrum of final zero-mode gravitons, which is generated from the vacuum fluctuations of both initial Kaluza-Klein modes and zero-mode. The amplitude of fluctuations is confirmed to agree with the four-dimensional one at low energies, whereas it is enhanced due to the normalization factor of zero-mode at high energies. We show that the five-dimensional spectrum can be well approximated by applying a simple mapping to the four-dimensional fluctuation amplitude. 
  We calculate the time delay between different relativistic images formed by black hole gravitational lensing in the strong field limit. For spherically symmetric black holes, it turns out that the time delay between the first two images is proportional to the minimum impact angle. Their ratio gives a very interesting and precise measure of the distance of the black hole. Moreover, using also the separation between the images and their luminosity ratio, it is possible to extract the mass of the black hole. The time delay for the black hole at the center of our Galaxy is just few minutes, but for supermassive black holes with M=10^8 - 10^9 solar masses in the neighbourhood of the Local Group the time delay amounts to few days, thus being measurable with a good accuracy. 
  We investigate the dynamics of Einstein equations in the vicinity of the two recently described types of singularity of anisotropic and homogeneous cosmological models described by the action $$ S=\int d^4x \sqrt{-g}{F(\phi)R - \partial_a\phi\partial^a\phi -2V(\phi)}, $$ with general $F(\phi)$ and $V(\phi)$. The dynamical nature of each singularity is elucidated, and we show that both are, in general, dynamically unavoidable, reinforcing the unstable character of previous isotropic and homogeneous cosmological results obtained for the conformal coupling case. 
  We show that the action of Einstein's gravity with a scalar field coupled in a generic way to spacetime curvature is invariant under a particular set of conformal transformations. These transformations relate dual theories for which the effective couplings of the theory are scaled uniformly. In the simplest case, this class of dualities reduce to the S-duality of low-energy effective action of string theory. 
  We present a definition of angular momentum for radiative spacetimes which does not suffer from any ambiguity of supertranslations. We succeed in providing an appropriate notion of {\it intrinsic} angular momentum; and at the same time a definition of center of mass frame at future null infinity.   We use the center of mass frame to present the asymptotic structure equations for vacuum spacetimes. 
  We consider here the dynamics of some homogeneous and isotropic cosmological models with $N$ interacting classical scalar fields non-minimally coupled to the spacetime curvature, as an attempt to generalize some recent results obtained for one and two scalar fields. We show that a Lyapunov function can be constructed under certain conditions for a large class of models, suggesting that chaotic behavior is ruled out for them. Typical solutions tend generically to the empty de Sitter (or Minkowski) fixed points, and the previous asymptotic results obtained for the one field model remain valid. In particular, we confirm that, for large times and a vanishing cosmological constant, even in the presence of the extra scalar fields, the universe tends to an infinite diluted matter dominated era. 
  It was previously shown by one of us that in any static, non-globally-hyperbolic, spacetime it is always possible to define a sensible dynamics for a Klein-Gordon scalar field. The prescription proposed for doing so involved viewing the spatial derivative part, $A$, of the wave operator as an operator on a certain $L^2$ Hilbert space $\mathcal H$ and then defining a positive, self-adjoint operator on $\mathcal H$ by taking the Friedrichs extension (or other positive extension) of $A$. However, this analysis left open the possibility that there could be other inequivalent prescriptions of a completely different nature that might also yield satisfactory definitions of the dynamics of a scalar field. We show here that this is not the case. Specifically, we show that if the dynamics agrees locally with the dynamics defined by the wave equation, if it admits a suitable conserved energy, and if it satisfies certain other specified conditions, then it must correspond to the dynamics defined by choosing some positive, self-adjoint extension of $A$ on $\mathcal H$. Thus, subject to our requirements, the previously given prescription is the only possible way of defining the dynamics of a scalar field in a static, non-globally-hyperbolic, spacetime. In a subsequent paper, this result will be applied to the analysis of scalar, electromagnetic, and gravitational perturbations of anti-de Sitter spacetime. By doing so, we will determine all possible choices of boundary conditions at infinity in anti-de Sitter spacetime that give rise to sensible dynamics. 
  We reformulate the Shiromizu, Maeda and Sasaki (SMS) braneworlds within the framework of the five-dimensional Einstein equations. In many applications of the braneworld Einstein field equations, the Weyl term is attributed to the bulk, thus splitting the non-Einsteinian terms into `bulk' and `brane' terms. Here by employing standard geometrical identities, we show that such a split is non-unique, since these terms get mixed up in different formulations. An important consequence of this non-uniqueness is that even though the full brane-bulk systems in all such formulations are completely equivalent, important differences can arise were one to truncate different formulations by throwing away the associated `bulk' terms. This is particularly likely to be the case in more general anisotropic/inhomogeneous settings with non-AdS bulks, in which the usual truncation of the SMS (which throws away the Weyl term) would not coincide with the full system. We emphasize that rather than providing support for any truncation, these differences show clearly the dangers of using any truncated equations and provide a strong argument in favour of studying the full brane-bulk system. The different formulations we provide also permit different ways of approaching the full brane-bulk system which may greatly facilitate its study. An example of this is the second-order nature of the formulations given here as opposed to the SMS's formulation which is third-order. 
  We perform numerical simulations of the critical gravitational collapse of a massive vector field. The result is that there are two critical solutions. One is equivalent to the Choptuik critical solution for a massless scalar field. The other is periodic. 
  A deformation of special relativity based on a dispersion relation with an energy independent speed of light and a symmetry between positive and negative energy states is proposed. The deformed Lorentz transformations, generators and algebra are derived and some consequences are discussed. 
  We show that the classification of Kantowski-Sachs, Bianchi Types I and III spacetimes admitting Matter Collineations (MCs) presented in a recent paper by Camci et al. [Camci, U., and Sharif, M. {Matter Collineations in Kantowski-Sachs, Bianchi Types I and III Spacetimes}, 2003 Gen. Relativ. Grav. vol. 35, 97-109] is incomplete. Furthermore for these spacetimes and when the Einstein tensor is non-degenerate, we give the complete Lie Algebra of MCs and the algebraic constraints on the spatial components of the Einstein tensor. 
  It is shown that in transitively self-similar spatially homogeneous tilted perfect fluid models the symmetry vector is not normal to the surfaces of spatial homogeneity. A direct consequence of this result is that there are no self-similar Bianchi VIII and IX tilted perfect fluid models. Furthermore the most general Bianchi VIII and IX spacetime which admit a four dimensional group of homotheties is given. 
  In this paper, we consider the effect of thermal fluctuations on the entropy of both neutral and charged black holes. We emphasize the distinction between fixed and fluctuating charge systems; using a canonical ensemble to describe the former and a grand canonical ensemble to study the latter. Our novel approach is based on the philosophy that the black hole quantum spectrum is an essential component in any such calculation. For definiteness, we employ a uniformly spaced area spectrum, which has been advocated by Bekenstein and others in the literature. The generic results are applied to some specific models; in particular, various limiting cases of an (arbitrary-dimensional) AdS-Reissner-Nordstrom black hole. We find that the leading-order quantum correction to the entropy can consistently be expressed as the logarithm of the classical quantity. For a small AdS curvature parameter and zero net charge, it is shown that, independent of the dimension, the logarithmic prefactor is +1/2 when the charge is fixed but +1 when the charge is fluctuating.We also demonstrate that, in the grand canonical framework, the fluctuations in the charge are large, $\Delta Q\sim\Delta A\sim S_{BH}^{1/2}$, even when $<Q> =0$. A further implication of this framework is that an asymptotically flat, non-extremal black hole can never achieve a state of thermal equilibrium. 
  We review a few topics in Planck-scale physics, with emphasis on possible manifestations in relatively low energy. The selected topics include quantum fluctuations of spacetime, their cumulative effects, uncertainties in energy-momentum measurements, and low energy quantum-gravity phenomenology. The focus is on quantum-gravity-induced uncertainties in some observable quantities. We consider four possible ways to probe Planck-scale physics experimentally: 1. looking for energy-dependent spreads in the arrival time of photons of the same energy from GRBs; 2. examining spacetime fluctuation-induced phase incoherence of light from extragalactic sources; 3. detecting spacetime foam with laser-based interferometry techniques; 4. understanding the threshold anomalies in high energy cosmic ray and gamma ray events. Some other experiments are briefly discussed. We show how some physics behind black holes, simple clocks, simple computers, and the holographic principle is related to Planck-scale physics. We also discuss a formulation of the Dirac equation as a difference equation on a discrete Planck-scale spacetime lattice, and a possible interplay between Planck-scale and Hubble-scale physics encoded in the cosmological constant (dark energy). 
  The noninvariance of Lyapunov exponents in general relativity has led to the conclusion that chaos depends on the choice of the space-time coordinates. Strikingly, we uncover the transformation laws of Lyapunov exponents under general space-time transformations and we find that chaos, as characterized by positive Lyapunov exponents, is coordinate invariant. As a result, the previous conclusion regarding the noninvariance of chaos in cosmology, a major claim about chaos in general relativity, necessarily involves the violation of hypotheses required for a proper definition of the Lyapunov exponents. 
  Quantum Gravity has been so elusive because we have tried to approach it by two paths which can never meet: standard quantum field theory and general relativity. The gateway is covariance under the complexified Clifford algebra of our space-time manifold M, and its spinor representations, which Sachs dubbed the Einstein group, E. On the microscopic scale, quantum gravity appears as the statistical mechanics of the null zig-zag rays of spinor fields in imaginary time T. Our unified field/particle action L_g also contains new couplings of gravitomagnetic fields to strong fields and weak potentials. These predict new physical phenomena: Axial jets of nuclear decay products emitted with left helicity along the axis of a massive, spinning body. 
  We apply the Generalized Uncertainty Principle (GUP) to the problem of maximum entropy and evaporation/absorption of energy of black holes near the Planck scale. We find within this general approach corrections to the maximum entropy, and indications for quenching of the evaporation because not only the evaporation term goes to a finite limit, but also because absorption of quanta seems to help the balance for black holes in a thermal bath. Then, residual masses around the Planck scale may be the final outcome of primordial black hole evaporation. 
  In recent years, many different numerical evolution schemes for Einstein's equations have been proposed to address stability and accuracy problems that have plagued the numerical relativity community for decades. Some of these approaches have been tested on different spacetimes, and conclusions have been drawn based on these tests. However, differences in results originate from many sources, including not only formulations of the equations, but also gauges, boundary conditions, numerical methods, and so on. We propose to build up a suite of standardized testbeds for comparing approaches to the numerical evolution of Einstein's equations that are designed to both probe their strengths and weaknesses and to separate out different effects, and their causes, seen in the results. We discuss general design principles of suitable testbeds, and we present an initial round of simple tests with periodic boundary conditions. This is a pivotal first step toward building a suite of testbeds to serve the numerical relativists and researchers from related fields who wish to assess the capabilities of numerical relativity codes. We present some examples of how these tests can be quite effective in revealing various limitations of different approaches, and illustrating their differences. The tests are presently limited to vacuum spacetimes, can be run on modest computational resources, and can be used with many different approaches used in the relativity community. 
  One often-used approximation in the study of binary compact objects (i.e., black holes and neutron stars) in general relativity is the instantaneously circular orbit assumption. This approximation has been used extensively, from the calculation of innermost circular orbits to the construction of initial data for numerical relativity calculations. While this assumption is inconsistent with generic general relativistic astrophysical inspiral phenomena where the dissipative effects of gravitational radiation cause the separation of the compact objects to decrease in time, it is usually argued that the timescale of this dissipation is much longer than the orbital timescale so that the approximation of circular orbits is valid. Here, we quantitatively analyze this approximation using a post-Newtonian approach that includes terms up to order ({Gm/(rc^2)})^{9/2} for non-spinning particles. By calculating the evolution of equal mass black hole / black hole binary systems starting with circular orbit configurations and comparing them to the more astrophysically relevant quasicircular solutions, we show that a minimum initial separation corresponding to at least 6 (3.5) orbits before plunge is required in order to bound the detection event loss rate in gravitational wave detectors to < 5% (20%). In addition, we show that the detection event loss rate is > 95% for a range of initial separations that include all modern calculations of the innermost circular orbit (ICO). 
  We examine the particle production during tunneling in quantum cosmology. We consider a minisuperspace model with a massive, conformally coupled scalar field and a uniform radiation background. In this model, we construct a semiclassical wave function describing a small recollapsing universe and a nucleated inflating universe (``tunneling from something''). We find that the quantum states of the scalar field in both the initial and the nucleated universe are close to the adiabatic vacuum, the number of created particles is small, and their backreaction on the metric is negligible. We show that the use of the semiclassical approximation is justified for this wave function. Our results imply that the creation of the universe from nothing can be understood as a limit of tunneling from a small recollapsing universe. 
  Nonlinear covariant parity-violating deformations of free spin-two gauge theory are studied in n>2 spacetime dimensions, using a linearized frame and spin-connection formalism, for a set of massless spin-two fields. It is shown that the only such deformations yielding field equations with a second order quasilinear form are the novel algebra-valued types in n=3 and n=5 dimensions already found in some recent related work concentrating on lowest order deformations. The complete form of the deformation to all orders in n=5 dimensions is worked out here and some features of the resulting new algebra-valued spin-two gauge theory are discussed. In particular, the internal algebra underlying this theory on 5-dimensional Minkowski space is shown to cause the energy for the spin-two fields to be of indefinite sign. Finally, a Kaluza-Klein reduction to n=4 dimensions is derived, giving a parity-violating nonlinear gauge theory of a coupled set of spin-two, spin-one, and spin-zero massless fields. 
  We investigate the behavior of a dynamical scalar field on a fixed Kerr background in Kerr-Schild coordinates using a 3+1 dimensional spectral evolution code, and we measure the power-law tail decay that occurs at late times. We compare evolutions of initial data proportional to f(r) Y_lm(theta,phi) where Y_lm is a spherical harmonic and (r,theta,phi) are Kerr-Schild coordinates, to that of initial data proportional to f(r_BL) Y_lm(theta_BL,phi), where (r_BL,theta_BL) are Boyer-Lindquist coordinates. We find that although these two cases are initially almost identical, the evolution can be quite different at intermediate times; however, at late times the power-law decay rates are equal. 
  The notion of null (isotropic) vector field is considered in spaces with affine connections and metrics as models of space or space-time. On its basis the propagation of signals in space-time is considered. The Doppler effect is generalized for these types of spaces. The notions of standard (longitudinal) Doppler effect and transversal Doppler effect are introduced. On their grounds, the Hubble effect and the aberration effect appear as Doppler effects with explicit forms of the centrifugal (centripetal) and Coriolis velocity vector fields in spaces with affine connections and metrics. The upper limit of the value of the general observed shift parameter z, generated by both the effects, based on the Doppler effects, is found to be z = 1.41. Doppler's effects, Hubble's effect, and aberration's effect could be used in mechanics of continuous media and in other classical field theories in the same way as the standard Doppler effect is used in classical and relativistic mechanics. PACS numbers: 04.20.Cv; 04.50.+h; 04.40.b; 04.90.+e; 83.10.Bb 
  We study the behavior near the singularity t=0 of Gowdy metrics. We prove existence of an open dense set of boundary points near which the solution is smoothly "asymptotically velocity term dominated" (AVTD). We show that the set of AVTD solutions satisfying a uniformity condition is open in the set of all solutions. We analyse in detail the asymptotic behavior of "power law" solutions at the (hitherto unchartered) points at which the asymptotic velocity equals zero or one. Several other related results are established. 
  It is shown that linearized gravitational radiation confined in a cavity can achieve thermal equilibrium if the mean density of the radiation and the size of the cavity satisfy certain constraints. 
  We analyze the weak gravity in the braneworld model proposed by Dvali-Gabadadze-Porrati, in which the unperturbed background spacetime is given by five dimensional Minkowski bulk with a brane which has the induced Einstein Hilbert term. This model has a critical length scale $r_c$. Naively, we expect that the four dimensional general relativity (4D GR) is approximately recovered at the scale below $r_c$. However, the simple linear perturbation does not work in this regime. Only recently the mechanism to recover 4D GR was clarified under the restriction to spherically symmetric configurations, and the leading correction to 4D GR was derived. Here, we develop an alternative formulation which can handle more general perturbations. We also generalize the model by adding bulk cosmological constant and the brane tension. 
  Some shortcomings in regard to our lack of conceptual understanding of string theory are displayed and prescription to untangle them is proposed. String theory should be a fundamental dynamics of four dimensional symmetric space-times. Properties of the two dimensional equivalent action are studied, in the hydrodynamic approximation. In the pressureless regime it is conformal invariant. Correlation of our proposal to 't Hooft work on quantization of black holes[7] and work on 2D black hole solutions established by Witten [14] are pointed out as perspectives of the present work. 
  Assuming equation of state for quintessential matter: $p=w(z)\rho$, we analyse dynamical behaviour of the scale factor in FRW cosmologies. It is shown that its dynamics is formally equivalent to that of a classical particle under the action of 1D potential $V(a)$. It is shown that Hamiltonian method can be easily implemented to obtain a classification of all cosmological solutions in the phase space as well as in the configurational space. Examples taken from modern cosmology illustrate the effectiveness of the presented approach. Advantages of representing dynamics as a 1D Hamiltonian flow, in the analysis of acceleration and horizon problems, are presented. The inverse problem of reconstructing the Hamiltonian dynamics (i.e. potential function) from the luminosity distance function $d_{L}(z)$ for supernovae is also considered. 
  We present a simple technique for generating new solutions of Einstein's equations using such function transformations that leave the field equations in the Ernst form. In this context we recover all the known covariant transformations of Ernst equations and we find the role of the analytic ones. Finally we obtain a new asymptotically flat solution starting from the Kerr solutions. 
  We present a simple novel derivation, ab initio, of the equations appropriate for stationary axisymmetric spacetimes using the Papapetrou form of the metric (Papapetrou gauge). It is shown that using coordinates which preserve the Papapetrou gauge three separated solutions of the Ernst equations appear in the case of Kerr metric. In this context a parameter arises which represents topological defects induced by an infinite static string along the z axis. Finally, we discuss a simple solution that may be derived from the Kerr ansatz. 
  We study the definition of perturbations in the presence of a submanifold, like e.g. a brane. In the standard theory of cosmological perturbations, one compares quantities at the same coordinate points in the non-perturbed and the perturbed manifolds, identified via a (non-unique) mapping between the two manifolds. In the presence of a physical submanifold one needs to modify this definition in order to evaluate perturbations of quantities at the submanifold location. As an application, we compute the perturbed metric and the extrinsic curvature tensors at the brane position in a general gauge. 
  We present a thorough analysis for the quasinormal (QN) behavior, associated with the decay of scalar, electromagnetic and gravitational perturbations, of Schwarzschild-anti-de Sitter black holes. As it is known the anti-de Sitter (AdS) QN spectrum crucially depends on the relative size of the black hole to the AdS radius. There are three different types of behavior depending on whether the black hole is large, intermediate, or small. The results of previous works, concerning lower overtones for large black holes, are completed here by obtaining higher overtones for all the three black hole regimes. There are two major conclusions that one can draw from this work: First, asymptotically for high overtones, all the modes are evenly spaced, and this holds for all three types of regime, large, intermediate and small black holes, independently of l, where l is the quantum number characterizing the angular distribution; Second, the spacing between modes is apparently universal, in that it does not depend on the field, i.e., scalar, electromagnetic and gravitational QN modes all have the same spacing for high overtones. We are also able to prove why scalar and gravitational perturbations are isospectral, asymptotically for high overtones, by introducing appropriate superpartner potentials. 
  This letter describes a novel derivation of general relativity by considering the (non)self-consistency of theories whose Hamiltonians are constraints. The constraints, from Hamilton's equations, generate the evolution, while the evolution, in turn, must preserve the constraints. This closure requirement can be used as a selection mechanism for general relativity starting from a very simple set of assumptions. The configuration space is chosen to be a family of $3 \times 3$ positive definite symmetric matrices on some bare 3-manifold. A general Hamiltonian is constructed on this space of matrices which consists of a single constraint per space point. It is assumed that this constraint looks like an energy balance relationship. It will be the sum of a `kinetic' term which is quadratic and undifferentiated in the momenta, and a `potential' term, which is any function of the configuration variables. Further, the constraint must be a scalar under the linear group, the natural symmetry group of the configuration space. This inexorably leads to the ADM Hamiltonian for general relativity. Both the space of Riemannian geometries (Wheeler's superspace), and spacetime are emergent quantities in this analysis. 
  I present a new general purpose event horizon finder for full 3D numerical spacetimes. It works by evolving a complete null surface backwards in time. The null surface is described as the zero level set of a scalar function, that in principle is defined everywhere. This description of the surface allows the surface, trivially, to change topology, making this event horizon finder able to handle numerical spacetimes, where two (or more) black holes merge into a single final black hole. 
  The conceptual basis for the nonlocality of accelerated systems is presented. The nonlocal theory of accelerated observers and its consequences are briefly described. Nonlocal field equations are developed for the case of the electrodynamics of linearly accelerated systems. 
  We investigate the cosmological consequences of a brane-world theory which incorporates time variations in the gravitational coupling G and the cosmological term Lambda. We analyze in detail the model where (dG/dt)/G ~ H and Lambda ~ H^2, which seems to be favored by observations. We show that these conditions single out models with flat space sections. We determine the behavior of the expansion scale factor, as well as, the variation of G, Lambda and H for different possible scenarios where the bulk cosmological constant can be zero, positive or negative. We demonstrate that the universe must recollapse, if it is embedded in an Anti-de Sitter five-dimensional bulk, which is the usual case in brane models. We evaluate the cosmological parameters, using some observational data, and show that we are nowhere near the time of recollapse. We conclude that the models with zero and negative bulk cosmological constant agree with the observed accelerating universe, while fitting simultaneously the observational data for the density and deceleration parameters. The age of the universe, even in the recollapsing case, is much larger than in the FRW universe. 
  Canonical quantization of the Brane-World effective action presented by Kanno and Soda containing higher order curvature invariant terms, has been performed. It requires introduction of an auxiliary variable. As observed in a series of publications by Sanyal and Modak, here again we infer that properly chosen auxiliary variable leads to a Schrodinger like equation where the kinetic part of a canonical variable disentangles from the rest of the variables giving a natural quantum mechanical flavour of time. Further, the effective Hamiltonian turns out to be hermitian, leading to the continuity ewuation. Thus, a quantum mechanical probability inter pretation is plausible. Finally, the extremization of the effective potential leads to Einstein's equation and a well behaved classical solution, which is a desirable feature of the gravitational action containing higher order curvature invariant terms. 
  In this paper a quantum N = 4 super Yang-Mills theory perturbed by dilaton-coupled scalars, is considered. The induced effective action for such a theory is calculated on a dilaton-gravitational background using the conformal anomaly found via AdS/CFT correspondence. Considering such an effective action (using the large N method) as a quantum correction to the classical gravity action with cosmological constant we study the effect from dilaton to the scale factor (which corresponds to the inflationary universe without dilaton). It is shown that, depending on the initial conditions for the dilaton, the dilaton may slow down, or accelerate, the inflation process. At late times, the dilaton is decaying exponentially. At the end of this work, we consider the question how the perturbation of the solution for the scale factor affects the stability of the solution for the equations of motion and therefore the stability of the Inflationary Universe, which could be eternal. 
  The isolated horizon framework is extended to include non-minimally coupled scalar fields. As expected from the analysis based on Killing horizons, entropy is no longer given just by (a quarter of) the horizon area but also depends on the scalar field. In a subsequent paper these results will serve as a point of departure for a statistical mechanical derivation of entropy using quantum geometry. 
  The stress-energy tensor for the massless spin 1/2 field is numerically computed outside and on the event horizons of both charged and uncharged static non-rotating black holes, corresponding to the Schwarzschild, Reissner-Nordstrom and extreme Reissner-Nordstr\"om solutions of Einstein's equations. The field is assumed to be in a thermal state at the black hole temperature. Comparison is made between the numerical results and previous analytic approximations for the stress-energy tensor in these spacetimes. For the Schwarzschild (charge zero) solution, it is shown that the stress-energy differs even in sign from the analytic approximation. For the Reissner-Nordstrom and extreme Reissner-Nordstrom solutions, divergences predicted by the analytic approximations are shown not to exist. 
  The phase shift due to the Sagnac Effect, for relativistic matter beams counter-propagating in a rotating interferometer, is deduced on the bases of a a formal analogy with the the Aharonov-Bohm effect. A procedure outlined by Sakurai, in which non relativistic quantum mechanics and newtonian physics appear together with some intrinsically relativistic elements, is generalized to a fully relativistic context, using the Cattaneo's splitting technique. This approach leads to an exact derivation, in a self-consistently relativistic way, of the Sagnac effect. Sakurai's result is recovered in the first order approximation. 
  We present an alternative field theoretical approach to the definition of conserved quantities, based directly on the field equations content of a Lagrangian theory (in the standard framework of the Calculus of Variations in jet bundles). The contraction of the Euler-Lagrange equations with Lie derivatives of the dynamical fields allows one to derive a variational Lagrangian for any given set of Lagrangian equations. A two steps algorithmical procedure can be thence applied to the variational Lagrangian in order to produce a general expression for the variation of all quantities which are (covariantly) conserved along the given dynamics. As a concrete example we test this new formalism on Einstein's equations: well known and widely accepted formulae for the variation of the Hamiltonian and the variation of Energy for General Relativity are recovered. We also consider the Einstein-Cartan (Sciama-Kibble) theory in tetrad formalism and as a by-product we gain some new insight on the Kosmann lift in gauge natural theories, which arises when trying to restore naturality in a gauge natural variational Lagrangian. 
  We specialize the radiation-reaction part of the Arnowitt-Deser-Misner (ADM) Hamiltonian for many non-spinning point-like bodies, calculated by Jaranowski and Schaefer [1], to third-and-a-half post-Newtonian approximation to general relativity, to binary systems. This Hamiltonian is used for the computation of the instantaneous gravitational energy loss of a binary to 1PN reactive order. We also derive the equations of motion, which include PN reactive terms via Hamiltonian and Euler-Lagrangian approaches. The results are consistent with the expressions for reactive acceleration provided by Iyer-Will formalism in Ref. [2] in a general class of gauges. 
  In conventional Maxwell--Lorentz electrodynamics, the propagation of light is influenced by the metric, not, however, by the possible presence of a torsion T. Still the light can feel torsion if the latter is coupled nonminimally to the electromagnetic field F by means of a supplementary Lagrangian of the type l^2 T^2 F^2 (l = coupling constant). Recently Preuss suggested a specific nonminimal term of this nature. We evaluate the spacetime relation of Preuss in the background of a general O(3)-symmetric torsion field and prove by specifying the optical metric of spacetime that this can yield birefringence in vacuum. Moreover, we show that the nonminimally coupled homogeneous and isotropic torsion field in a Friedmann cosmos affects the speed of light. 
  This paper investigates the possible cosmological implications of the presence of an antisymmetric tensor field related to a lack of commutatitivity of spacetime coordinates at the Planck era. For this purpose, such a field is promoted to a dynamical variable, inspired by tensor formalism. By working to quadratic order in the antisymmetric tensor, we study the field equations in a Bianchi I universe in two models: an antisymmetric tensor plus scalar field coupled to gravity, or a cosmological constant and a free massless antisymmetric tensor. In the first scenario, numerical integration shows that, in the very early universe, the effects of the antisymmetric tensor can prevail on the scalar field, while at late times the former approaches zero and the latter drives the isotropization of the universe. In the second model, an approximate solution is obtained of a nonlinear ordinary differential equation which shows how the mean Hubble parameter and the difference between longitudinal and orthogonal Hubble parameter evolve in the early universe. 
  The recently assembled laser-beam detectors of gravitational waves are approaching the planned level of sensitivity. In the coming 1 - 2 years, we may be observing the rare but powerful events of inspiral and merger of binary stellar-mass black holes. More likely, we will have to wait for a few years longer, until the advanced detectors become operational. Their sensitivity will be sufficient to meet the most cautious evaluations of the strength and event rates of astrophysical sources of gravitational waves. The experimental and theoretical work related to the space-based laser-beam detectors is also actively pursued. The current gravitational wave research is broad and interesting. Experimental innovations, source modelling, methods of data analysis, theoretical issues of principle are being studied and developed at the same time. The race for direct detection of relatively high-frequency waves is accompanied by vigorous efforts to discover the very low-frequency relic gravitational waves through the measurements of the cosmic microwave background radiation. In this update, we will touch upon each of these directions of research, including the recent data from the Wilkinson Microwave Anisotropy Probe (WMAP). 
  In this essay it will be shown that the introduction of a modification to Heisenberg algebra (here this feature means the existence of a minimal obserlvable length), as a fundamental part of the quantization process of the electrodynamical field, renders states in which the uncertainties in the two quadrature components violate the usual Heisenberg uncertainty relation. Hence in this context it may be asserted that any physically realistic generalization of the uncertainty principle must include, not only a minimal observable length, but also a minimal observable momentum. 
  New four-dimensional black hole solutions of Brans-Dicke equations with a negative constant $\omega$, coupled to a massless scalar field, are presented. The temperature of these black holes is zero and the horizon area is infinite. An astrophysical application is also discussed. 
  To confront the transplanckian problem encountered in the backward extrapolation of the cosmological expansion of the momenta of the modes of quantum field theory, it is proposed that there is a reservoir, depository of transplanckian degrees of freedom. These are solicited by the cisplanckian modes so as to keep their density fixed and the total energy density of vacuum at a minimum.   The mechanism is due to mode - reservoir interaction, whereupon virtual quantum processes give rise to an effective mode-mode attraction. A BCS condensate results. It has a massless and massy collective excitation, the latter identified with the inflaton. For an effective non dimensional mode-reservoir coupling constant, g approx 0.3, the order of magnitude of its mass is what is required to account for cosmological fluctuations i.e. O(10^-6 -> 10^-5)m_Planck. 
  Non-linear special relativity (or doubly special relativity) is a simple framework for encoding properties of flat quantum space-time. In this paper we show how this formalism may be generalized to incorporate curvature (leading to what might be called ``doubly general relativity''). We first propose a dual to non-linear realizations of relativity in momentum space, and show that for such a dual the space-time invariant is an energy-dependent metric. This leads to an energy-dependent connection and curvature, and a simple modification to Einstein's equations. We then examine solutions to these equations. We find the counterpart to the cosmological metric, and show how cosmologies based upon our theory of gravity may solve the ``horizon problem''. We discuss the Schwarzchild solution, examining the conditions for which the horizon is energy dependent. We finally find the weak field limit. 
  In this paper we adopt a global and non-entropic approach to the problem of the arrow of time, according to which the arrow of time is an intrinsic geometrical property of spacetime. Our main aim is to show the double role played by the energy-momentum tensor in the context of our approach. On the one hand, the energy-momentum tensor is the intermediate step that permits to turn the geometrical time-asymmetry of the universe into a local arrow of time manifested as a time-asymmetric energy flow. On the other hand, the energy-momentum tensor supplies the basis for deducing the time-asymmetry of quantum field theory, posed as an axiom in this theory. 
  Over the last few years the study of possible Planck-scale departures from classical Lorentz symmetry has been one of the most active areas of quantum-gravity research. We now have a satisfactory description of the fate of Lorentz symmetry in the most popular noncommutative spacetimes and several studies have been devoted to the fate of Lorentz symmetry in loop quantum gravity. Remarkably there are planned experiments with enough sensitivity to reveal these quantum-spacetime effects, if their magnitude is only linearly suppressed by the Planck length. Unfortunately, in some quantum-gravity scenarios even the strongest quantum-spacetime effects are suppressed by at least two powers of the Planck length, and many authors have argued that it would be impossible to test these quadratically-suppressed effects. I here observe that advanced cosmic-ray observatories and neutrino observatories can provide the first elements of an experimental programme testing the possibility of departures from Lorentz symmetry that are quadratically Planck-length suppressed. 
  Nonspherical stellar collapse to a black hole is one of the most promising gravitational wave sources for gravitational wave detectors. We numerically study gravitational waves from a slightly nonspherical stellar collapse to a black hole in linearized Einstein theory. We adopt a spherically collapsing star as the zeroth-order solution and gravitational waves are computed using perturbation theory on the spherical background. In this paper we focus on the perturbation of odd-parity modes. Using the polytropic equations of state with polytropic indices $n_p=1$ and 3, we qualitatively study gravitational waves emitted during the collapse of neutron stars and supermassive stars to black holes from a marginally stable equilibrium configuration. Since the matter perturbation profiles can be chosen arbitrarily, we provide a few types for them. For $n_p=1$, the gravitational waveforms are mainly characterized by a black hole quasinormal mode ringing, irrespective of perturbation profiles given initially. However, for $n_p=3$, the waveforms depend strongly on the initial perturbation profiles. In other words, the gravitational waveforms strongly depend on the stellar configuration and, in turn, on the ad hoc choice of the functional form of the perturbation in the case of supermassive stars. 
  The behaviour of expanding cosmological models with collisionless matter and a positive cosmological constant is analysed. It is shown that under the assumption of plane or hyperbolic symmetry the area radius goes to infinity, the spacetimes are future geodesically complete, and the expansion becomes isotropic and exponential at late times. This proves a form of the cosmic no hair theorem in this class of spacetimes. 
  I prove the existence of vacuum $S^{1}$ symmetric Einsteinian, unpolarized, space times which are complete in the direction of the expansion, for small initial data, without supposing that the $S^{1}$ orbits are orthogonal to the 3-manifolds, as was done in previous work in collaboration with V. Moncrief. 
  It is by now well known that various condensed matter systems may be used to mimic many of the kinematic aspects of general relativity, and in particular of curved-spacetime quantum field theory. In this essay we will take a look at what would be needed to mimic a cosmological spacetime -- to be precise a spatially flat FRW cosmology -- in one of these analogue models. In order to do this one needs to build and control suitable time dependent systems. We discuss here two quite different ways to achieve this goal. One might rely on an explosion, physically mimicking the big bang by an outflow of whatever medium is being used to carry the excitations of the analogue model, but this idea appears to encounter dynamical problems in practice. More subtly, one can avoid the need for any actual physical motion (and avoid the dynamical problems) by instead adjusting the propagation speed of the excitations of the analogue model. We shall focus on this more promising route and discuss its practicality. 
  We study the late time evolution of positively curved FRW models with a scalar field which arises in the conformal frame of the $R+\alpha R^{2}$ theory. The resulted three-dimensional dynamical system has two equilibrium solutions corresponding to a de Sitter space and an ever expanding closed universe. We analyze the structure of the first equilibrium with the methods of the center manifold theory and, for the second equilibrium we apply the normal form theory to obtain a simplified system, which we analyze with special phase plane methods. It is shown that an initially expanding closed FRW spacetime avoids recollapse. 
  We present, for the first time, an action principle that gives the equations of motion of an extended body possessing multipole moments in an external gravitational field, in the weak field limit. From the action, the experimentally observable quantum phase shifts in the wavefunction of an extended object due to the coupling of its multipole moments with the gravitational field are obtained. Also, since the theory may be quantized using the action, the present approach is useful in the interface between general relativity and quantum mechanics. 
  In this paper, we have studied Brans-Dicke Cosmology in anisotropic Kantowski-Sachs space-time model; considering variation of the velocity of light. We have addressed the flatness problem considering both the cases namely, (i) when the Brans-Dicke scale field $\phi$ is constant (ii) when $\phi$ varies, specially for radiation dominated era perturbatively and non-perturbatively and asymptotic behaviour have been studied. 
  It is shown that minimally coupled scalar field in Brans-Dicke theory with varying speed of light can solve the quintessence problem and it is possible to have a non-decelerated expansion of the present universe with BD-theory for anisotropic models without any matter. 
  We give a brief review of recent developments in five-dimensional theories of spacetime and highlight their geometrical structure mainly in connection with the Campbell-Magaard theorem. 
  A sketch of the affine quantum gravity program illustrates a different perspective on several difficult issues of principle: metric positivity; quantum anomalies; and nonrenormalizability. 
  Some issues of the cosmological constant or dark energy are briefly reviewed. There are an increasing number of observations that constrain the equation of state of dark energy more stringently and favor the time-independent cosmological constant. Then a plausible model of dark energy would be a theory with degenerate perturbative vacua in which its origin is explained by a nonperturbative effect so that, unlike quintessence, k-essence etc., it is separable from the perturbative problem why its amplitude is smaller than the Planckian density by a factor of $\order (10^{-120})$. 
  In physical theories, boundary or initial conditions play the role of selecting special situations which can be described by a theory with its general laws. Cosmology has long been suspected to be different in that its fundamental theory should explain the fact that we can observe only one particular realization. This is not realized, however, in the classical formulation and in its conventional quantization; the situation is even worse due to the singularity problem. In recent years, a new formulation of quantum cosmology has been developed which is based on quantum geometry, a candidate for a theory of quantum gravity. Here, the dynamical law and initial conditions turn out to be linked intimately, in combination with a solution of the singularity problem. 
  A very brief and popular account of the time machine problem. 
  An orbiting black hole binary will generate strong gravitational radiation signatures, making these binaries important candidates for detection in gravitational wave observatories. The gravitational radiation is characterized by the orbital parameters, including the frequency and separation at the inner-most stable circular orbit (ISCO). One approach to estimating these parameters relies on a sequence of initial data slices that attempt to capture the physics of the inspiral. Using calculations of the binding energy, several authors have estimated the ISCO parameters using initial data constructed with various algorithms. In this paper we examine this problem using conformally Kerr-Schild initial data. We present convergence results for our initial data solutions, and give data from numerical solutions of the constraint equations representing a range of physical configurations. In a first attempt to understand the physical content of the initial data, we find that the Newtonian binding energy is contained in the superposed Kerr-Schild background before the constraints are solved. We examine some deficiencies with the initial data approach to orbiting binaries, especially touching on the effects of prior motion and spin-orbital coupling of the angular momenta. Making rough estimates of these effects, we find that they are not insignificant compared to the binding energy, leaving some doubt of the utility of using initial data to predict ISCO parameters. In computations of specific initial-data configurations we find spin-specific effects that are consistent with analytical estimates. 
  Deconstruction provides a novel way of dealing with the notoriously difficult ultraviolet problems of four-dimensional gravity. This approach also naturally leads to a new perspective on the holographic principle, tying it to the fundamental requirements of unitarity and diffeomorphism invariance, as well as to a new viewpoint on the cosmological constant problem. The numerical smallness of the cosmological constant is implied by a unique combination of holography and supersymmetry, opening a new window into the fundamental physics of the vacuum. 
  Recent attempts to resolve the ambiguity in the loop quantum gravity description of the quantization of area has led to the idea that j=1 edges of spin-networks dominate in their contribution to black hole areas as opposed to j=1/2 which would naively be expected. This suggests that the true gauge group involved might be SO(3) rather than SU(2) with attendant difficulties. We argue that the assumption that a version of the Pauli principle is present in loop quantum gravity allows one to maintain SU(2) as the gauge group while still naturally achieving the desired suppression of spin-1/2 punctures. Areas come from j=1 punctures rather than j=1/2 punctures for much the same reason that photons lead to macroscopic classically observable fields while electrons do not. 
  In this letter, we discuss how thermal fluctuations can effect the stability of (generally) charged black holes when close to certain critical points. Our novel treatment utilizes the black hole area spectrum (which is, for definiteness, taken to be evenly spaced) and makes an important distinction between fixed and fluctuating charge systems (with these being modeled, respectively, as a canonical and grand canonical ensemble.) The discussion begins with a summary of a recent technical paper [gr-qc/0305018]. We then go on to consider the issue of stability when the system approaches the critical points of interest. These include the $d$-dimensional analogue of the Hawking-Page phase transition, a phase transition that is relevant to Reissner-Nordstrom black holes and various extremal-limiting cases. 
  We study the occurrence and nature of naked singularities for a dust model with non-zero cosmological constant in ($n+2$)-dimensional Szekeres space-times (which possess no Killing vectors) for $n\geq 2$. We find that central shell-focusing singularities may be locally naked in higher dimensions but depend sensitively on the choice of initial data. In fact, the nature of the initial density determines the possibility of naked singularity in space-times with more than five dimensions. The results are similar to the collapse in spherically symmetric Tolman-Bondi-Lema\^{\i}tre space-times. 
  In this paper, it is shown, using a geometrical approach, the isotropy of the velocity of light measured in a rotating frame in Minkowski space-time, and it is verified that this result is compatible with the Sagnac effect. Furthermore, we find that this problem can be reduced to the solution of geodesic triangles in a Minkowskian cylinder. A relationship between the problems established on the cylinder and on the Minkowskian plane is obtained through a local isometry. 
  We show that the cosmological large number coincidence can be interpreted as giving the filling factor in a Landau problem. The analogy with the Landau problem leads naturally to the noncommutativity of the gravitational and matter degrees of freedom. We present a toy model that supports this view. We discuss some of the physical consequences this noncommutativity implies, like a different insight into the semiclassical approximation of quantum gravity and a different tackling of the cosmological constant problem. 
  A new method of post-Newtonian approximation (PNA) for weak gravitational fields is presented together with its application to test an alternative, scalar theory of gravitation. The new method consists in defining a one-parameter family of systems, by applying a Newtonian similarity transformation to the initial data that defines the system of interest. This method is rigorous. Its difference with the standard PNA is emphasized. In particular, the new method predicts that the internal structure of the bodies does have an influence on the motion of the mass centers. The translational equations of motion obtained with this method in the scalar theory are adjusted in the solar system, and compared with an ephemeris based on the standard PNA of GR. 
  It is commonly accepted that the study of 2+1 dimensional quantum gravity could teach us something about the 3+1 dimensional case. The non-perturbative methods developed in this case share, as basic ingredient, a reformulation of gravity as a gauge field theory. However, these methods suffer many problems. Firstly, this perspective abandon the non-degeneracy of the metric and causality as fundamental principles, hoping to recover them in a certain low-energy limit. Then, it is not clear how these combinatorial techniques could be used in the case where matter fields are added, which are however the essential ingredients in order to produce non trivial observables in a generally covariant approach. Endly, considering the status of the observer in these approaches, it is not clear at all if they really could produce a completely covariant description of quantum gravity. We propose to re-analyse carefully these points. This study leads us to a really covariant description of a set of self-gravitating point masses in a closed universe. This approach is based on a set of observables associated to the measurements accessible to a participant-observer, they manage to capture the whole dynamic in Chern-Simons gravity as well as in true gravity. The Dirac algebra of these observables can be explicitely computed, and exhibits interesting algebraic features related to Poisson-Lie groupoids theory. 
  The Hamiltonian constraint remains the major unsolved problem in Loop Quantum Gravity (LQG). Seven years ago a mathematically consistent candidate Hamiltonian constraint has been proposed but there are still several unsettled questions which concern the algebra of commutators among smeared Hamiltonian constraints which must be faced in order to make progress. In this paper we propose a solution to this set of problems based on the so-called {\bf Master Constraint} which combines the smeared Hamiltonian constraints for all smearing functions into a single constraint. If certain mathematical conditions, which still have to be proved, hold, then not only the problems with the commutator algebra could disappear, also chances are good that one can control the solution space and the (quantum) Dirac observables of LQG. Even a decision on whether the theory has the correct classical limit and a connection with the path integral (or spin foam) formulation could be in reach. While these are exciting possibilities, we should warn the reader from the outset that, since the proposal is, to the best of our knowledge, completely new and has been barely tested in solvable models, there might be caveats which we are presently unaware of and render the whole {\bf Master Constraint Programme} obsolete. Thus, this paper should really be viewed as a proposal only, rather than a presentation of hard results, which however we intend to supply in future submissions. 
  It is generally accepted that the entropy of an asymptotically de Sitter universe is bounded by the area, in Planck units, of the de Sitter horizon. Based on an analysis of the entropy associated to the vacuum quantum fluctuations, we suggest that the existence of such a holographic bound constitutes a possible explanation for the observed value of the cosmological constant, theoretically justifying a relation proposed 35 years ago by Zel'dovich. 
  The black hole entropy calculation for type I isolated horizons, based on loop quantum gravity, is extended to include non-minimally coupled scalar fields. Although the non-minimal coupling significantly modifies quantum geometry, the highly non-trivial consistency checks for the emergence of a coherent description of the quantum horizon continue to be met. The resulting expression of black hole entropy now depends also on the scalar field precisely in the fashion predicted by the first law in the classical theory (with the same value of the Barbero-Immirzi parameter as in the case of minimal coupling). 
  The Einstein equivalence principle is certainly a key element in the development of new enhanced theories of gravity. Although being an important building block in Einstein's general relativity, theoretically predicted violations of its validity are a main feature in alternative, nonmetric gravitation theories if they are able to incorporate quantum mechanical principles.   We have investigated the metric-affine gauge theory of gravity which predict that a gravitational field singles out an orthogonal pair of polarization states of light that propagate with different phase velocities. This gravity-induced birefringence implies that propagation through a gravitational field can alter the polarization of light and, so, violates the Einstein equivalence principle. We use solar and white dwarf polarimetric data to constrain birefringence induced by the gravitational field of these objects and set limits on coupling constants required by such theories. 
  The phase shift due to the Sagnac Effect, for relativistic matter and electromagnetic beams, counter-propagating in a rotating interferometer, is deduced using two different approaches. From one hand, we show that the relativistic law of velocity addition leads to the well known Sagnac time difference, which is the same independently of the physical nature of the interfering beams, evidencing in this way the universality of the effect. Another derivation is based on a formal analogy with the phase shift induced by the magnetic potential for charged particles travelling in a region where a constant vector potential is present: this is the so called Aharonov-Bohm effect. Both derivations are carried out in a fully relativistic context, using a suitable 1+3 splitting that allows us to recognize and define the space where electromagnetic and matter waves propagate: this is an extended 3-space, which we call "relative space". It is recognized as the only space having an actual physical meaning from an operational point of view, and it is identified as the 'physical space of the rotating platform': the geometry of this space turns out to be non Euclidean, according to Einstein's early intuition. 
  A reference frame consists of: a reference space, a time scale and a spatial metric. The geometric structure induced by these objects in spacetime is developed. The existence of a class of spatial metrics that are rigid, have free mobility and can be derived as a slight deformation of the radar metric, is shown. 
  The generalized Chaplygin gas (GCG) model allows for an unified description of the recent accelerated expansion of the Universe and the evolution of energy density perturbations. This dark energy - dark matter unification is achieved through an exotic background fluid whose equation of state is given by $p = - A/\rho^{\alpha}$, where $A$ is a positive constant and $0 < \alpha \le 1$. Stringent constraints on the model parameters can be obtained from recent WMAP and BOOMERanG bounds on the locations of the first few peaks and troughs of the Cosmic Microwave Background Radiation (CMBR) power spectrum as well as SNe Ia data. 
  In this paper, I study the dynamics of the Kaluza-Klein internal manifold using its ``shape'' and the ``size'' as collective coordinates. The essential motivation is to be able to explain symmetry breaking in gauge theories through a transition of the internal manifold from a symmetrical metric space to a less symmetric one. We find that this may be possible depending on the values of certain group theoretical parameters. Further, this transition resembles a ``quintessential inflation'' scenario, the feasibility of which however, needs to be further studied in details. 
  We study the spherically symmetric collapse of a fluid with non-vanishing radial pressure in higher dimensional space-time. We obtain the general exact solution in the closed form for the equation of state ($P_r = \gamma \rho$) which leads to the explicit construction of the root equation governing the nature (black hole versus naked singularity) of the central singularity. A remarkable feature of the root equation is its invariance for the three cases: (${D+1}, {\gamma = -1}$),   (${D}, {\gamma = 0}$) and (${D - 1}, {\gamma = 1}$) where $D$ is the dimension of space-time. That is, for the ultimate end result of the collapse, $D$-dimensional dust, ${D+1}$ - AdS (anti de Sitter)-like and ${D-1}$ - dS-like are absolutely equivalent. 
  The usual form of the C-metric has the structure function G(\xi)=1-\xi^2-2mA\xi^3, whose cubic nature can make calculations cumbersome, especially when explicit expressions for its roots are required. In this paper, we propose a new form of the C-metric, with the explicitly factorisable structure function G(\xi)=(1-\xi^2)(1+2mA\xi). Although this form is related to the usual one by a coordinate transformation, it has the advantage that its roots are now trivial to write down. We show that this leads to potential simplifications, for example, when casting the C-metric in Weyl coordinates. These results also extend to the charged C-metric, whose structure function can be written in the new form G(\xi)=(1-\xi^2)(1+r_{+}A\xi)(1+r_{-}A\xi), where r_{\pm} are the usual locations of the horizons in the Reissner-Nordstrom solution. As a by-product, we explicitly cast the extremally charged C-metric in Weyl coordinates. 
  We construct numerical solutions for non-topological solitons in three-dimensional U(1)-gauged ${\cal N}=2$ supergravity. We find the region of the solutions showing with the BTZ mass, the angular momentum and the magnetic flux and discuss the relation among the physical parameters for various values of a cosmological constant. 
  Garcia and Campuzano claim to have found a previously overlooked family of stationary and axisymmetric conformally flat spacetimes, contradicting an old theorem of Collinson. In both these papers it is tacitly assumed that the isometry group is orthogonally transitive. Under the same assumption, we point out here that Collinson's result still holds if one demands the existence of an axis of symmetry on which the axial Killing vector vanishes. On the other hand if the assumption of orthogonal transitivity is dropped, a wider class of metrics is allowed and it is possible to find explicit counterexamples to Collinson's result. 
  In the standard approach to defining a Planck scale where gravity is brought into the quantum domain, the Schwarzschild gravitational radius is set equal to the Compton wavelength. However, ignored thereby are the charge and spin, the fundamental quantized aspects of matter. The gravitational and null-surface radii of the Kerr-Newman metric are used to introduce spin and charge into a new extended Planck scale. The fine structure constant appears in the extended Planck mass and the recent discovery of the $\alpha$ variation with the evolution of the universe adds further significance. An extended Planck charge and Planck spin are derived. There is an intriguing suggestion of a connection with the $\alpha$ value governing high-energy radiation in Z-boson production and decay. 
  The asymmetry in the time delay for light rays propagating on opposite sides of a spinning body is analyzed. A frequency shift in the perceived signals is found. A practical procedure is proposed for evidencing the asymmetry, allowing for a measurement of the specific angular momentum of the rotating mass. Orders of magnitude are considered and discussed. 
  Some physically interesting weak-gravitational effects and phenomena are reviewed and briefly discussed: particle geometric phases due to the time-dependent spin-rotation couplings, non-inertial gravitational wave in rotating reference of frame, hyperbolical geometric quantum phases and topological dual mass as well. 
  Recently Gambini and Pullin proposed a new consistent discrete approach to quantum gravity and applied it to cosmological models. One remarkable result of this approach is that the cosmological singularity can be avoided in a general fashion. However, whether the continuum limit of such discretized theories exists is model dependent. In the case of massless scalar field coupled to gravity with $\Lambda=0$, the continuum limit can only be achieved by fine tuning the recurrence constant. We regard this failure as the implication that cosmological constant should vary with time. For this reason we replace the massless scalar field by Chaplygin gas which may contribute an effective cosmological constant term with the evolution of the universe. It turns out that the continuum limit can be reached in this case indeed. 
  We propose a method to probe the equation of state of the early universe and its evolution, using the stochastic gravitational wave background from inflation. A small deviation from purely radiation dominated universe ($w= 1/3$) would be clearly imprinted on the gravitational wave spectrum  $\Omega_{GW}(f)$ due to the nearly scale invariant nature of inflationary generated waves. 
  The history of the particle concept is briefly reviewed, with particular emphasis on the `foliation dependence' of many particle creation models, and the possible connection between our notion of particle and our notion of simultaneity. It is argued that the concept of `radar time' (originally introduced by Sir Hermann Bondi in his work on k-calculus) provides a satisfactory concept of `simultaneity' for observers in curved spacetimes. This is used to propose an observer-dependent particle interpretation, applicable to an arbitrary observer, depending solely on that observers motion and not on a choice of coordinates or gauge. This definition is illustrated with application to non-inertial observers and simple cosmologies, demonstrating its generality and its consistency with known cases. 
  We show that a recent proposal for the quantization of gravity based on discrete space-time implies a modification of standard quantum mechanics that naturally leads to a loss of coherence in quantum states of the type discussed by Milburn. The proposal overcomes the energy conservation problem of previously proposed decoherence mechanisms stemming from quantum gravity. Mesoscopic quantum systems (as Bose--Einstein condensates) appear as the most promising testing grounds for an experimental verification of the mechanism. 
  Theories proposing a varying speed of light have recently been widely promoted under the claim that they offer an alternative way of solving the standard cosmological problems. Recent observational hints that the fine structure constant may have varied during over cosmological scales also has given impetus to these models. In theoretical physics the speed of light, $c$, is hidden in almost all equations but with different facets that we try to distinguish. Together with a reminder on scalar-tensor theories of gravity, this sheds some light on these proposed varying speed of light theories. 
  For slowly spinning matter the rate of energy loss via radiation of gravitational waves is estimated in General Relativity (GR) within a generally covariant superenergy approach. This estimation differs from Einstein's Quadrupole Formula (EQF) by a suppression factor ($\Pi\ll1$). For a symmetric two-body-like distribution of scalar matter $\Pi$ is estimated to be $\ll(v/c)^2(r/R)^2$, where $v$ is orbital velocity of the bodies, $c$ - velocity of light, $r$ - radius of each body, and $R$ -- the inter-body distance. This contradiction with EQF is briefly discussed. 
  The gravitational radiation-reaction force acting on perfect fluids at 3.5 post-Newtonian order is cast into a form which is directly applicable to numerical simulations. Extensive use is made of metric-coefficient changes induced by functional coordinate transformations, of the continuity equation, as well as of the equations of motion. We also present an expression appropriate for numerical simulations of the radiation field causing the worked out reaction force. 
  We show that it is possible to embed the 1+1 dimensional reduction of certain spherically symmetric black hole spacetimes into 2+1 Minkowski space. The spacetimes of interest (Schwarzschild de-Sitter, Schwarzschild anti de-Sitter, and Reissner-Nordstrom near the outer horizon) represent a class of metrics whose geometries allow for such embeddings. The embedding diagrams have a dynamic character which allows one to represent the motion of test particles. We also analyze various features of the embedding construction, deriving the general conditions under which our procedure provides a smooth embedding. These conditions also yeild an embedding constant related to the surface gravity of the relevant horizon. 
  A quintessence scalar field in self-interacting Brans-Dicke theory is shown to give rise to a non-decelerated expansion of the present universe for open, flat and closed models. Along with providing a non-decelerating solution, it can potentially solve the flatness problem too. 
  The quantum cosmological wavefunction for a quadratic gravity theory derived from the heterotic string effective action is obtained near the inflationary epoch and during the initial Planck era. Neglecting derivatives with respect to the scalar field, the wavefunction would satisfy a third-order differential equation near the inflationary epoch which has a solution that is singular in the scale factor limit $a(t)\to 0$. When scalar field derivatives are included, a sixth-order differential equation is obtained for the wavefunction and the solution by Mellin transform is regular in the $a\to 0$ limit. It follows that inclusion of the scalar field in the quadratic gravity action is necessary for consistency of the quantum cosmology of the theory at very early times. 
  The collision of pure electromagnetic plane waves with collinear polarization in N-dimensional (N=2+n) Einstein-Maxwell theory is considered. A class of exact solutions for the higher dimensional Bell-Szekeres metric is obtained and its singularity structure is examined. 
  We investigate and solve in the context of General Relativity the apparent paradox which appears when bodies floating in a background fluid are set in relativistic motion. Suppose some macroscopic body, say, a submarine designed to lie just in equilibrium when it rests (totally) immersed in a certain background fluid. The puzzle arises when different observers are asked to describe what is expected to happen when the submarine is given some high velocity parallel to the direction of the fluid surface. On the one hand, according to observers at rest with the fluid, the submarine would contract and, thus, sink as a consequence of the density increase. On the other hand, mariners at rest with the submarine using an analogous reasoning for the fluid elements would reach the opposite conclusion. The general relativistic extension of the Arquimedes law for moving bodies shows that the submarine sinks. 
  A vacuum solution of the Einstein gravitational field equation is given that follows from a general ansatz but fails to follow from it if a certain symmetric matrix is assumed to be in diagonal form from the beginning. 
  The problem of describing isolated rotating bodies in equilibrium in General Relativity has so far been treated under the assumption of the circularity condition in the interior of the body. For a fluid without energy flux, this condition implies that the fluid flow moves only along the angular direction, i.e. there is no convection. Using this simplification, some recent studies have provided us with uniqueness and existence results for asymptotically flat vacuum exterior fields given the interior sources. Here, the generalisation of the problem to include general sources is studied. It is proven that the convective motions have no direct influence on the exterior field, and hence, that the aforementioned results on uniqueness and existence of exterior fields apply equally in the general case. 
  In the framework of spacetime with torsion and without curvature, the Dirac particle spin precession in the rotational system is studied. We write out the equivalent tetrad of rotating frame, in the polar coordinate system, through considering the relativistic factor, and the resultant equivalent metric is a flat Minkowski one. The obtained rotation-spin coupling formula can be applied to the high speed rotating case, which is consistent with the expectation. 
  We summarize the consequences of the twin assumptions of (discrete) self-similarity and spherical symmetry for the global structure of a spacetime. All such spacetimes can be constructed from two building blocks, the "fan" and "splash". Each building block contains one radial null geodesic that is invariant under the self-similarity (self-similarity horizon). 
  We study the semiclassical evolution of a self-gravitating thick shell in Anti-de Sitter space-time. We treat the matter on the shell as made of quantized bosons and evaluate the back-reaction of the loss of gravitational energy which is radiated away as a non-adiabatic effect. A peculiar feature of anti-de Sitter is that such an emission also occurs for large shell radius, contrary to the asymptotically flat case. 
  We formulate the quantum mechanics of the solutions of a Klein-Gordon-type field equation: (\partial_t^2+D)\psi(t)=0, where D is a positive-definite operator acting in a Hilbert space \tilde H. We determine all the positive-definite inner products on the space H of the solutions of such an equation and establish their physical equivalence. We use a simple realization of the unique Hilbert space structure on H to construct the observables of the theory explicitly. In general, there are infinitely many choices for the Hamiltonian each leading to a different notion of time-evolution in H. Among these is a particular choice that generates t-translations in H and identifies t with time whenever D is t-independent. For a t-dependent D, we show that t-translations never correspond to unitary evolutions in H, and t cannot be identified with time. We apply these ideas to develop a formulation of quantum cosmology based on the Wheeler-DeWitt equation for a Friedman-Robertson-Walker model coupled to a real scalar field. We solve the Hilbert space problem, construct the observables, introduce a new set of the wave functions of the universe, reformulate the quantum theory in terms of the latter, reduce the problem of time to the determination of a Hamiltonian operator acting in L^2(\R)\oplus L^2(\R), show that the factor-ordering problem is irrelevant for the kinematics of the quantum theory, and propose a formulation of the dynamics. Our method allows for a genuine probabilistic interpretation and a unitary Schreodinger time-evolution. 
  We show explicitly that traversable wormholes requiring exotic matter in 4-dimensions nevertheless have acceptable stress-tensors obeying reasonable energy conditions in higher dimensions if the wormholes are regarded as being embedded in higher dimensional space-times satisfying Einstein's field equations. From the 4-dimensional perspective, the existence of higher dimensions may thus facilitate wormhole and time-machine constructions through access to "exotic matter". 
  We analyse the concept of active gravitational mass for Reissner-Nordstrom spacetime in terms of scalar polynomial invariants and the Karlhede classification. We show that while the Kretschmann scalar does not produce the expected expression for the active gravitational mass, both scalar polynomial invariants formed from the Weyl tensor, and the Cartan scalars, do. 
  Apparent horizons are structures of spacelike hypersurfaces that can be determined locally in time. Closed surfaces of constant expansion (CE surfaces) are a generalisation of apparent horizons. I present an efficient method for locating CE surfaces. This method uses an explicit representation of the surface, allowing for arbitrary resolutions and, in principle, shapes. The CE surface equation is then solved as a nonlinear elliptic equation.   It is reasonable to assume that CE surfaces foliate a spacelike hypersurface outside of some interior region, thus defining an invariant (but still slicing-dependent) radial coordinate. This can be used to determine gauge modes and to compare time evolutions with different gauge conditions. CE surfaces also provide an efficient way to find new apparent horizons as they appear e.g. in binary black hole simulations. 
  I review basic principles of the quantum mechanical measurement process in view of their implications for a quantum theory of general relativity. It turns out that a clock as an external classical device associated with the observer plays an essential role. This leads me to postulate a ``principle of the integrity of the observer''. It essentially requires the observer to be part of a classical domain connected throughout the measurement process. Mathematically this naturally leads to a formulation of quantum mechanics as a kind of topological quantum field theory. Significantly, quantities with a direct interpretation in terms of a measurement process are associated only with amplitudes for connected boundaries of compact regions of space-time. I discuss some implications of my proposal such as in-out duality for states, delocalization of the ``collapse of the wave function'' and locality of the description. Differences to existing approaches to quantum gravity are also highlighted. 
  After a brief introduction to classical and quantum gravity we discuss applications of loop quantum gravity in the cosmological realm. This includes the basic formalism and recent results of loop quantum cosmology, and a computation of modified dispersion relations for quantum gravity phenomenology. The presentation is held at a level which does not require much background knowledge in general relativity or mathematical techniques such as functional analysis, so as to make the article accessible to graduate students and researchers from other fields. 
  A new solution to the Einstein-Maxwell field equations is presented describing a cylindrically symmetric homogeneous cosmology. The solution is conformally flat, it possesses seven Killing vectors of which the timelike one is rotating and one of the spacelike, pseudorotating. Our solution also admits a Kerr-Schild form. It is alternatively produced by different electromagnetic sources some of which represent constant null electromagnetic fields, while the others, a circularly polarized plane electromagnetic wave (seemingly, a unique situation in general relativity). The concrete electromagnetic four-potentials are found from the assumption that they are proportional to the Killing covectors. The general solution is obtained for timelike and null geodesics. Finally, we find that this space-time admits closed timelike non-geodesic lines. 
  We numerically calculate equilibrium configurations of uniformly rotating and charged neutron stars, in the case of insulating material and neglecting the electromagnetic forces acting on the equilibrium of the fluid. This allows us to study the behaviour of the gyromagnetic ratio for those objects, when varying rotation rate and equation of state for the matter. Under the assumption of low charge and incompressible fluid, we find that the gyromagnetic ratio is directly proportional to the compaction parameter M/R of the star, and very little dependent on its angular velocity. Nevertheless, it seems impossible to have g=2 for these models with low charge-to-mass ratio, where matter consists of a perfect fluid and where the collapse limit is never reached. 
  In this paper, we extend the first-order post-Newtonian scheme in multiple systems presented by Damour-Soffel-Xu to the second-order contribution to light propagation without changing the virtueof the scheme on the linear partial differential equations of the potential and vector potential. The spatial components of the metric are extended to second order level both in a global coordinates ($q_{ij}/ c^4$) and a local coordinates ($Q_{ab}/ c^4$). The equations of $q_{ij}$ (or $Q_{ab}$) are obtained from the field equations.The relationship between $q_{ij}$ and $Q_{ab}$ are presented in this paper also. In special case of the solar system (isotropic condition is applied ($q_{ij} = \delta_{ij} q $)), we obtain the solution of $q$. Finally, a further extension of the second-order contributions in the parametrized post-Newtonian formalism is discussed. 
  We present a method for generating exact diagonal $G_2$-cosmological solutions in dilaton gravity coupled to a radiation perfect fluid and with a cosmological potential of a special type. The method is based on the symmetry group of the system of $G_2$-field equations. Several new classes of explicit exact inhomogeneous perfect fluid scalar-tensor cosmologies are presented. 
  The renewed serious interest to possible practical applications of gravitational waves is encouraging. Building on previous work, I am arguing that the strong variable electromagnetic fields are appropriate systems for the generation and detection of high-frequency gravitational waves (HFGW). The advantages of electromagnetic systems are clearly seen in the proposed complete laboratory experiment, where one has to ensure the efficiency of, both, the process of generation and the process of detection of HFGW. Within the family of electromagnetic systems, one still has a great variety of possible geometrical configurations, classical and quantum states of the electromagnetic field, detection strategies, etc. According to evaluations performed 30 years ago, the gap between the HFGW laboratory signal and its level of detectability is at least 4 orders of magnitude. Hopefully, new technologies of today can remove this gap and can make the laboratory experiment feasible. The laboratory experiment is bound to be expensive, but one should remember that a part of the cost is likely to be reimbursed from the Nobel prize money ! Electromagnetic systems seem also appropriate for the detection of high-frequency end of the spectrum of relic gravitational waves. Although the current effort to observe the stochastic background of relic gravitational waves is focused on the opposite, very low-frequency, end of the spectrum, it would be extremely valuable for fundamental science to detect, or put sensible upper limits on, the high-frequency relic gravitational waves. I will briefly discuss the origin of relic gravitational waves, the expected level of their high-frequency signal, and the existing estimates of its detectability. 
  In this paper, the dynamical equations and junction conditions at the interface between adjacent layers of different elastic properties for an elastic deformable astronomical body in the first post-Newtonian approximation of Einstein theory of gravity are discussed in both rotating Cartesian coordinates and rotating spherical coordinates. The unperturbed rotating body (the ground state) is described as uniformly rotating, stationary and axisymmetric configuration in an asymptotically flat space-time manifold. Deviations from the equilibrium configuration are described by means of a displacement field. In terms of the formalism of relativistic celestial mechanics developed by Damour, Soffel and Xu, and the framework established by Carter and Quintana the post Newtonian equations of the displacement field and the symmetric trace-free shear tensor are obtained. Corresponding post-Newtonian junction conditions at interfaces also the outer surface boundary conditions are presented. The PN junction condition is an extension of Wahr's one which is a Newtonian junction conditions without rotating. 
  In this paper, it is the first time to construct a complete post-Newtonian (PN) model of a rigid body by means of a new constraint on the mass current density and mass density. In our PN rigid body model most of relations, such as spin vector proportional to the angular velocity, the definition on the moment of inertia tensor, the key relation between the mass quadrupole moment and the moment of inertia tensor, rigid rotating formulae of mass quadrupole moment and the moment of inertia tensor, are just the extension of the main relations in Newtonian rigid body model. When all of $1/c^2$ terms are neglected, the PN rigid body model and the corresponding formulae reduce to Newtonian version. The key relation is obtained in this paper for the first time, which might be very useful in the future application to problems in geodynamics and astronomy. 
  We study gravitational lensing by the Gibbons-Maeda-Garfinkle-Horowitz-Strominger (GMGHS) charged black hole of heterotic string theory and obtain the angular position and magnification of the relativistic images. Modeling the supermassive central object of the galaxy as a GMGHS black hole, we estimate the numerical values of different strong-lensing parameters. We find that there is no significant string effect present in the lensing observables in the strong-gravity scenario. 
  In the simplest form of the Randall-Sundrum model, we consider the metric generated by a static, spherically symmetric distribution of matter on the physical brane. The solution to the five-dimensional Einstein equations, obtained numerically, describes a wormhole geometry. 
  Scalar-tensor theories are the best motivated alternatives to general relativity and provide a mathematically consistent framework to test the various observable predictions. They can involve three functions of the scalar field: (i) a potential (as in "quintessence" models), (ii) a matter-scalar coupling function (as in "extended quintessence", where it may also be rewritten as a nonminimal coupling of the scalar field to the scalar curvature), and (iii) a coupling function of the scalar field to the Gauss-Bonnet topological invariant. We recall the main experimental constraints on this class of theories, and underline that solar-system, binary-pulsar, and cosmological observations give qualitatively different tests. We finally show that the combination of these data is necessary to constrain the existence of a scalar-Gauss-Bonnet coupling. 
  We consider the implications for laser interferometry of the quantum-gravity-motivated modifications in the laws of particle propagation, which are presently being considered in attempts to explain puzzling observations of ultra-high-energy cosmic rays. We show that there are interferometric setups in which the Planck-scale effect on propagation leads to a characteristic signature. A naive estimate is encouraging with respect to the possibility of achieving Planck-scale sensitivity, but we also point out some severe technological challenges which would have to be overcome in order to achieve this sensitivity. 
  We show that puncture data for quasicircular binary black hole orbits allow a special gauge choice that realizes some of the necessary conditions for the existence of an approximate helical Killing vector field. Introducing free parameters for the lapse at the punctures we can satisfy the condition that the Komar and ADM mass agree at spatial infinity. Several other conditions for an approximate Killing vector are then automatically satisfied, and the 3-metric evolves on a timescale smaller than the orbital timescale. The time derivative of the extrinsic curvature however remains significant. Nevertheless, quasicircular puncture data are not as far from possessing a helical Killing vector as one might have expected. 
  We study interactions of electro-magnetic fields with the curvature tensor of the form $\lambda R_{\mu \nu \alpha \beta}F^{\mu \nu}F^{\alpha \beta}$. Such coupling terms though are invariant under general coordinate transformation and CPT, however violate the Einstein equivalence principle. These couplings do not cause any energy dependent dispersion of photons but they exhibit birefringence. We put constraints on the coupling constant $\lambda$ using results from solar system radar ranging experiments and millisecond-pulsar observations. We find that the most stringent constraint comes from pulsar observations and is given by $ \lambda < 10^{11} cm^2 $ obtained from the timing of binary pulsar PSR B1534+12. 
  This article is motivated by the observation, that calculations of the Unruh effect based on idealized particle detectors are usually made in a way that involves integrations along the {\em entire} detector trajectory up to the infinitely remote {\em future}. We derive an expression which allows time-dependence of the detector response in the case of a non-stationary trajectory and conforms more explicitely to the principle of causality, namely that the response at a given instant of time depends only on the detectors {\em past} movements. On trying to reproduce the thermal Unruh spectrum we are led to an unphysical result, which we trace down to the use of the standard regularization $t\to t-i\eps$ of the correlation function. By consistently employing a rigid detector of finite extension, we are led to a different regularization which works fine with our causal response function. 
  We review some string-inspired theoretical models which incorporate a correlated spacetime variation of coupling constants while remaining naturally compatible both with phenomenological constraints coming from geochemical data (Oklo; Rhenium decay) and with present equivalence principle tests. Barring unnatural fine-tunings of parameters, a variation of the fine-structure constant as large as that recently ``observed'' by Webb et al. in quasar absorption spectra appears to be incompatible with these phenomenological constraints. Independently of any model, it is emphasized that the best experimental probe of varying constants are high-precision tests of the universality of free fall, such as MICROSCOPE and STEP. Recent claims by Bekenstein that fine-structure-constant variability does not imply detectable violations of the equivalence principle are shown to be untenable. 
  The modification of the Doppler effect due to the coupling of the helicity of the radiation with the rotation of the source/receiver is considered in the case of the Pioneer 10/11 spacecraft. We explain why the Pioneer anomaly is not influenced by the helicity-rotation coupling. 
  A scalar theory with a preferred reference frame is summarized. To test that theory in celestial mechanics, an "asymptotic" post-Newtonian (PN) scheme has been developed. This associates a conceptual family of self-gravitating systems with the given system, in order to have a true small parameter available. The resulting equations for a weakly-self-gravitating system of extended bodies include internal-structure effects. The internal-structure influence subsists at the point-particle limit--a violation of the weak equivalence principle. If one could develop an "asymptotic" approximation scheme in general relativity also, this could plausibly be found there also, in a gauge where the PN space metric would not be "conformally Euclidean". 
  It is well-known that the Kerr-metric (rotating black hole in four dimensions) has Petrov type D. We prove a similar property in five dimensions. The Myers-Perry metric (rotating black hole in five dimensions) with one non-zero angular momentum has Petrov type \underline{22}. Conversely, we show that the Myers-Perry solution is unique within a certain restricted class of metrics of Petrov type \underline{22}. 
  We introduce N-parameter perturbation theory as a new tool for the study of non-linear relativistic phenomena. The main ingredient in this formulation is the use of the Baker-Campbell-Hausdorff formula. The associated machinery allows us to prove the main results concerning the consistency of the scheme to any perturbative order. Gauge transformations and conditions for gauge invariance at any required order can then be derived from a generating exponential formula via a simple Taylor expansion. We outline the relation between our novel formulation and previous developments. 
  We consider combining two important methods for constructing quasi-equilibrium initial data for binary black holes: the conformal thin-sandwich formalism and the puncture method. The former seeks to enforce stationarity in the conformal three-metric and the latter attempts to avoid internal boundaries, like minimal surfaces or apparent horizons. We show that these two methods make partially conflicting requirements on the boundary conditions that determine the time slices. In particular, it does not seem possible to construct slices that are quasi-stationary and avoid physical singularities and simultaneously are connected by an everywhere positive lapse function, a condition which must obtain if internal boundaries are to be avoided. Some relaxation of these conflicting requirements may yield a soluble system, but some of the advantages that were sought in combining these approaches will be lost. 
  The Einstein-Cartan theory of gravitation and the classical theory of defects in an elastic medium are presented and compared. The former is an extension of general relativity and refers to four-dimensional space-time, while we introduce the latter as a description of the equilibrium state of a three-dimensional continuum. Despite these important differences, an analogy is built on their common geometrical foundations, and it is shown that a space-time with curvature and torsion can be considered as a state of a four-dimensional continuum containing defects. This formal analogy is useful for illustrating the geometrical concept of torsion by applying it to concrete physical problems. Moreover, the presentation of these theories using a common geometrical basis allows a deeper understanding of their foundations. 
  The surface gravity for the extreme Reissner-Nordstr\"om black hole is zero suggesting that it has a zero temperature. However, the direct evaluation of the Bogolubov's coefficients, using the standard semi-classical analysis, indicates that the temperature of the extreme black hole is ill definite: the Bogolubov's coefficients obtained by performing the usual analysis of a collapsing model of a thin shell, and employing the geometrical optical approximation, do not obey the normalization conditions. We argue that the failure of the employement of semi-classical analysis for the extreme black hole is due to the absence of orthonormal quantum modes in the vicinity of the event horizon in this particular case. 
  A diagram for Bianchi spaces with vanishing vector of structure constants (type A in the Ellis-MacCallum classification) illustrates the relations among their different types under similarity transformations. The Ricci coefficients and the Ricci tensor are related by a Cremona transformation. 
  It is known that Newton's law of gravity holds asymptotically on a flat "brane" embedded in an anti-de Sitter "bulk" ; this was shown not only when gravity in the bulk is described by Einstein's theory but also in Einstein "Lanczos Lovelock Gauss-Bonnet"'s theory. We give here the expressions for the corrections to Newton's potential in both theories, in analytic form and valid for all distances. We find that in Einstein's theory the transition from the 1/r behaviour at small r to the 1/r^2 one at large r is quite slow. In the Einstein Gauss-Bonnet case on the other hand, we find that the correction to Newton's potential can be small for all r. Hence, Einstein Gauss-Bonnet equations in the bulk (rather than simply Einstein's) induce on the brane a better approximation to Newton's law. 
  These Lecture notes give an introduction to Regge calculus as a discrete model of General Relativity. 
  The problem of time in canonical quantum gravity is related to the fact that the canonical description is based on the prior choice of a spacelike foliation, hence making a reference to a spacetime metric. However, the metric is expected to be a dynamical, fluctuating quantity in quantum gravity. We show how this problem can be solved in the histories formulation of general relativity. We implement the 3+1 decomposition using metric-dependent foliations which remain spacelike with respect to all possible Lorentzian metrics. This allows us to find an explicit relation of covariant and canonical quantities which preserves the spacetime character of the canonical description. In this new construction, we also have a coexistence of the spacetime diffeomorphisms group, and the Dirac algebra of constraints. 
  Relying on known results of the Noether theory of symmetries extended to constrained systems, it is shown that there exists an obstruction that prevents certain tangent-space diffeomorphisms to be projectable to phase-space, for generally covariant theories. This main result throws new light on the old fact that the algebra of gauge generators in the phase space of General Relativity, or other generally covariant theories, only closes as a soft algebra and not a a Lie algebra.   The deep relationship between these two issues is clarified. In particular, we see that the second one may be understood as a side effect of the procedure to solve the first. It is explicitly shown how the adoption of specific metric-dependent diffeomorphisms, as a way to achieve projectability, causes the algebra of gauge generators (constraints) in phase space not to be a Lie algebra --with structure constants-- but a soft algebra --with structure {\it functions}. 
  We show in detail how the histories description of general relativity carries representations of both the spacetime diffeomorphisms group and the Dirac algebra of constraints. We show that the introduction of metric-dependent equivariant foliations leads to the crucial result that the canonical constraints are invariant under the action of spacetime diffeomorphisms. Furthermore, there exists a representation of the group of generalised spacetime mappings that are functionals of the four-metric: this is a spacetime analogue of the group originally defined by Bergmann and Komar in the context of the canonical formulation of general relativity. Finally, we discuss the possible directions for the quantization of gravity in histories theory. 
  Motivated by a nice result shown by E. Linder, a detailed discussion on the choice of the expansion parameters of the Maclaurin series for the equation of states of a perfect fluid is presented in this paper. We show that their nice recent result is in fact a linear approximation to the full Maclaurin series as a power series of the parameter $y=z/(1+z)$. The power series for the energy density function, the Hubble parameter and related physical quantities of interest are also shown in this paper. The method presented here will have significant application in the precision distance-redshift observations aiming to map out the recent expansion history of the universe. In addition, a complete analysis of all known advantageous parameterizations for the equation of states to high redshift is also presented. 
  It is a famous result of relativistic stellar structure that (under mild technical conditions) a static fluid sphere satisfies the Buchdahl--Bondi bound 2M/R <= 8/9; the surprise here being that the bound is not 2M/R <= 1. In this article we provide further generalizations of this bound by placing a number of constraints on the interior geometry (the metric components), on the local acceleration due to gravity, on various combinations of the internal density and pressure profiles, and on the internal compactness 2m(r)/r of static fluid spheres. We do this by adapting the standard tool of comparing the generic fluid sphere with a Schwarzschild interior geometry of the same mass and radius -- in particular we obtain several results for the pressure profile (not merely the central pressure) that are considerably more subtle than might first be expected. 
  Some cylindrically symmetric inhomogeneous viscous fluid cosmological models with electro-magnetic field are obtained. To get a solution a supplementary condition between metric potentials is used. The viscosity coefficient of bulk viscous fluid is assumed to be a power function of mass density. Without assumin g any {\it ad hoc} law, we obtain a cosmological constant as a decreasing function of time. The behaviour of the electro-magnetic field tensor together with some p hysical aspects of the model are also discussed. 
  It has been shown that Brans-Dicke (BD) theory in anisotropic cosmological model can alone solve the quintessence problem and we have accelerated expanding universe without any quintessence matter. Also the flatness problem has been discussed in this context. 
  Recently there was proposeda hypothesis about existence of the two large extradimensions. This hypothesis demands, e.g., modification of Newton law at submilimeter scale. In this brief report we show that this hypothesis cannot be correct in presen formulation. 
  Given, in an arbitrary spacetime, a 2-dimensional timelike submanifold (worldsheet) and an observer field on this worldsheet, we assign gravitational, centrifugal, Coriolis and Euler forces to every particle worldline in the worldsheet with respect to the observer field. We prove that centrifugal and Coriolis forces vanish, for all particle worldlines with respect to any observer field, if and only if the worldsheet is a photon 2-surface, i.e., generated by two families of lightlike geodesics. We further demonstrate that a photon 2-surface can be characterized in terms of gyroscope transport and we give several mathematical criteria for the existence of photon 2-surfaces. Finally, examples of photon 2-surfaces in conformally flat spacetimes, in Schwarzschild and Reissner-Nordstroem spacetimes, and in Goedel spacetime are worked out. 
  The project SYPOR wishes to use the global navigation satellite system GALILEO as an autonomous relativistic positioning system for the Earth. Motivations and a sketch of the basic concepts underlying the project are presented. For non geodetic (perturbed) satellites, a two-dimensional example describes how the dynamics of the constellation of satellites and that of the users may be deduced from the knowledge of the dynamics of only one of the satellites during a partial interval. 
  The Hawking radiation of Weyl neutrinos in an arbitrarily accelerating Kinnersley black hole is investigated by using a method of the generalized tortoise coordinate transformation. Both the location and temperature of the event horizon depend on the time and on the angles. They coincide with previous results, but the thermal radiation spectrum of massless spinor particles displays a kind of spin-acceleration coupling effect. 
  We construct an exact quantum gravitational state describing the collapse of an inhomogeneous spherical dust cloud using a lattice regularization of the Wheeler-DeWitt equation. In the semiclassical approximation around a black hole, this state describes Hawking radiation. We show that the leading quantum gravitational correction to Hawking radiation renders the spectrum non-thermal. 
  The results of our study of the motion of a three particle, self-gravitating system in general relativistic lineal gravity is presented for an arbitrary ratio of the particle masses. We derive a canonical expression for the Hamiltonian of the system and discuss the numerical solution of the resulting equations of motion. This solution is compared to the corresponding non-relativistic and post-Newtonian approximation solutions so that the dynamics of the fully relativistic system can be interpretted as a correction to the one-dimensional Newtonian self-gravitating system. We find that the structure of the phase space of each of these systems yields a large variety of interesting dynamics that can be divided into three distinct regions: annulus, pretzel, and chaotic; the first two being regions of quasi-periodicity while the latter is a region of chaos. By changing the relative masses of the three particles we find that the relative sizes of these three phase space regions changes and that this deformation can be interpreted physically in terms of the gravitational interactions of the particles. Furthermore, we find that many of the interesting characteristics found in the case where all of the particles share the same mass also appears in our more general study. We find that there are additional regions of chaos in the unequal mass system which are not present in the equal mass case. We compare these results to those found in similar systems. 
  The frequencies of a cryogenic sapphire oscillator and a hydrogen maser are compared to set new constraints on a possible violation of Lorentz invariance. We determine the variation of the oscillator frequency as a function of its orientation (Michelson-Morley test) and of its velocity (Kennedy-Thorndike test) with respect to a preferred frame candidate. We constrain the corresponding parameters of the Mansouri and Sexl test theory to $\delta - \beta + 1/2 \leq 3.4 \times 10^{-9}$ and $\beta - \alpha - 1 \leq 4.1 \times 10^{-7}$ which is of the same order as the best previous result for the former and represents a 50 fold improvement for the latter. These results correspond to an improvement of our previously published limits [Wolf P. et al., Phys. Rev. Lett. {\bf 90}, 6, 060402, (2003)] by about a factor 2. We describe the changes of the experiment, and show the new data that lead to that improvement. 
  I outline a series of results obtained in collaboration with A. Waldron on the properties of massive higher (s>1) spin fields in cosmological, constant curvature, backgrounds and the resulting unexpected qualitative effects on their degrees of freedom and unitarity properties. The dimensional parameter $\L$ extends the flat space m-line to a $(m^2,\L)$ "phase" plane in which these novel phenomena unfold. In this light, I discuss a possible partial resurrection of deSitter supergravity. I will also exhibit the well-known causality problems of coupling these systems to gravity and, for complex fields, to electromagnetism, systematizing some of the occasionally misunderstood obstacles to interactions, particularly for s = 3/2 and 2. 
  We consider random topologies of surfaces generated by cubic interactions. Such surfaces arise in various contexts in 2-dimensional quantum gravity and as world-sheets in string theory. Our results are most conveniently expressed in terms of a parameter h = n/2 + \chi, where n is the number of interaction vertices and \chi is the Euler characteristic of the surface. Simulations and results for similar models suggest that Ex[h] = log (3n) + \gamma + O(1/n) and Var[h] = log (3n) + \gamma - \pi^2/6 + O(1/n). We prove rigourously that Ex[h] = log n + O(1) and Var[h] = O(log n). We also derive results concerning a number of other characteristics of the topology of these random surfaces. 
  It was shown in a previous work that the data combinations canceling laser frequency noise constitute a module - the module of syzygies. The cancellation of laser frequency noise is crucial for obtaining the requisite sensitivity for LISA. In this work we show how the sensitivity of LISA can be optimised for a monochromatic source - a compact binary - whose direction is known, by using appropriate data combinations in the module. A stationary source in the barycentric frame appears to move in the LISA frame and our strategy consists of "coherently tracking" the source by appropriately "switching" the data combinations so that they remain optimal at all times. Assuming that the polarisation of the source is not known, we average the signal over the polarisations. We find that the best statistic is the `network' statistic, in which case LISA can be construed of as two independent detectors. We compare our results with the Michelson combination, which has been used for obtaining the standard sensitivity curve for LISA, and with the observable obtained by optimally switching the three Michelson combinations. We find that for sources lying in the ecliptic plane the improvement in SNR increases from 34% at low frequencies to nearly 90% at around 20 mHz. Finally we present the signal-to-noise ratios for some known binaries in our galaxy. We also show that, if at low frequencies SNRs of both polarisations can be measured, the inclination angle of the plane of the orbit of the binary can be estimated. 
  Space-time wormholes were introduced in Wheeler's idea of space-time foam. Traversible wormholes as defined by Morris & Thorne became popular as potential short cuts across the universe and even time machines. More recently, the author proposed a general theory of wormhole dynamics, unified with black-hole dynamics. This article gives a brief review of the above ideas and summarizes progress on wormhole dynamics in the last year. Firstly, a numerical study of dynamical perturbations of the first Morris-Thorne wormhole showed it to be unstable, either collapsing to a black hole or exploding to an inflationary universe. This provides a mechanism for inflating a wormhole from space-time foam to usable size. Intriguing critical behaviour was also discovered. Secondly, a wormhole solution supported by pure radiation was discovered and used to find analytic examples of dynamic wormhole processes which were also recently found in a two-dimensional dilaton gravity model: the construction of a traversible wormhole from a Schwarzschild black hole and vice versa, and the enlargement or reduction of the wormhole. 
  This review is concerned with the motion of a point scalar charge, a point electric charge, and a point mass in a specified background spacetime. In each of the three cases the particle produces a field that behaves as outgoing radiation in the wave zone, and therefore removes energy from the particle. In the near zone the field acts on the particle and gives rise to a self-force that prevents the particle from moving on a geodesic of the background spacetime. The field's action on the particle is difficult to calculate because of its singular nature: the field diverges at the position of the particle. But it is possible to isolate the field's singular part and show that it exerts no force on the particle -- its only effect is to contribute to the particle's inertia. What remains after subtraction is a smooth field that is fully responsible for the self-force. Because this field satisfies a homogeneous wave equation, it can be thought of as a free (radiative) field that interacts with the particle; it is this interaction that gives rise to the self-force. The mathematical tools required to derive the equations of motion of a point scalar charge, a point electric charge, and a point mass in a specified background spacetime are developed here from scratch. The review begins with a discussion of the basic theory of bitensors (part I). It then applies the theory to the construction of convenient coordinate systems to chart a neighbourhood of the particle's word line (part II). It continues with a thorough discussion of Green's functions in curved spacetime (part III). The review concludes with a detailed derivation of each of the three equations of motion (part IV). 
  Dual recycling is the combination of signal recycling and power recycling; both optical techniques improve the shot-noise-limited sensitivity of interferometric gravitational-wave detectors. In addition, signal recycling can reduce the loss of light power due to imperfect interference and allows, in principle, to beat the standard quantum limit. The interferometric gravitational-wave detector GEO600 is the first detector to use signal recycling. We have recently equipped the detector with a signal-recycling mirror with a transmittance of 1%. In this paper, we present details of the detector commissioning and the first locks of the dual- recycled interferometer. 
  We investigate the thermodynamics of a four-dimensional charged black hole in a finite cavity in asymptotically flat and asymptotically de Sitter space. In each case, we find a Hawking-Page-like phase transition between a black hole and a thermal gas very much like the known transition in asymptotically anti-de Sitter space. For a ``supercooled'' black hole--a thermodynamically unstable black hole below the critical temperature for the Hawking-Page phase transition--the phase diagram has a line of first-order phase transitions that terminates in a second order point. For the asymptotically flat case, we calculate the critical exponents at the second order phase transition and find that they exactly match the known results for a charged black hole in anti-de Sitter space. We find strong evidence for similar phase transitions for the de Sitter black hole as well. Thus many of the thermodynamic features of charged anti-de Sitter black holes do not really depend on asymptotically anti-de Sitter boundary conditions; the thermodynamics of charged black holes is surprisingly universal. 
  We present an overview of quantum noise in gravitational wave interferometers. Gravitational wave detectors are extensively modified variants of a Michelson interferometer and the quantum noise couplings are strongly influenced by the interferometer configuration. We describe recent developments in the treatment of quantum noise in the complex interferometer configurations of present-day and future gravitational-wave detectors. In addition, we explore prospects for the use of squeezed light in future interferometers, including consideration of the effects of losses, and the choice of optimal readout schemes. 
  In 3+1 numerical simulations of dynamic black hole spacetimes, it's useful to be able to find the apparent horizon(s) (AH) in each slice of a time evolution. A number of AH finders are available, but they often take many minutes to run, so they're too slow to be practically usable at each time step. Here I present a new AH finder,_AHFinderDirect_, which is very fast and accurate: at typical resolutions it takes only a few seconds to find an AH to $\sim 10^{-5} m$ accuracy on a GHz-class processor.   I assume that an AH to be searched for is a Strahlk\"orper (star-shaped region) with respect to some local origin, and so parameterize the AH shape by $r = h(angle)$ for some single-valued function $h: S^2 \to \Re^+$. The AH equation then becomes a nonlinear elliptic PDE in $h$ on $S^2$, whose coefficients are algebraic functions of $g_{ij}$, $K_{ij}$, and the Cartesian-coordinate spatial derivatives of $g_{ij}$. I discretize $S^2$ using 6 angular patches (one each in the neighborhood of the $\pm x$, $\pm y$, and $\pm z$ axes) to avoid coordinate singularities, and finite difference the AH equation in the angular coordinates using 4th order finite differencing. I solve the resulting system of nonlinear algebraic equations (for $h$ at the angular grid points) by Newton's method, using a "symbolic differentiation" technique to compute the Jacobian matrix._AHFinderDirect_ is implemented as a thorn in the_Cactus_ computational toolkit, and is freely available by anonymous CVS checkout. 
  The sensitivity of LISA depends on the suppression of several noise sources; dominant one is laser frequency noise. It has been shown that the six Doppler data streams obtained from three space-crafts can be appropriately time delayed and optimally combined to cancel this laser frequency noise. We show that the optimal data combinations when operated in a network mode improves the sensitivity over Michelson ranging from 40 % to 100 %. In this article, we summarize these results. We further show that the residual laser noise in the optimal data combination due to typical arm-length inaccuracy of 200 m is much below the level of optical path and the proof mass noises. 
  We consider stationary rotating black holes in SU(2) Einstein-Yang-Mills theory, coupled to a dilaton. The black holes possess non-trivial non-Abelian electric and magnetic fields outside their regular event horizon. While generic solutions carry no non-Abelian magnetic charge, but non-Abelian electric charge, the presence of the dilaton field allows also for rotating solutions with no non-Abelian charge at all. As a consequence, these special solutions do not exhibit the generic asymptotic non-integer power fall-off of the non-Abelian gauge field functions. The rotating black hole solutions form sequences, characterized by the winding number $n$ and the node number $k$ of their gauge field functions, tending to embedded Abelian black holes. The stationary non-Abelian black hole solutions satisfy a mass formula, similar to the Smarr formula, where the dilaton charge enters instead of the magnetic charge. Introducing a topological charge, we conjecture, that black hole solutions in SU(2) Einstein-Yang-Mills-dilaton theory are uniquely characterized by their mass, their angular momentum, their dilaton charge, their non-Abelian electric charge, and their topological charge. 
  We study formulation and probabilistic interpretation of a simple general-relativistic hamiltonian quantum system. The system has no unitary evolution in background time. The quantum theory yields transition probabilities between measurable quantities (partial observables). These converge to the classical predictions in the $\hbar\to 0$ limit. Our main tool is the kernel of the projector on the solutions of Wheeler-deWitt equation, which we analyze in detail. It is a real quantity, which can be seen as a propagator that propagates "forward" as well as "backward" in a local parameter time. Individual quantum states, on the other hand, may contain only "forward propagating" components. The analysis sheds some light on the interpretation of background independent transition amplitudes in quantum gravity. 
  Introductory Notes in Bosonic String Theory and its Canonical Quantization. 
  We study a model that the entropy per particle in the universe is constant. The sources for the entropy are the particle creation and a lambda decaying term. We find exact solutions for the Einstein field equations and show the compatibilty of the model with respect to the age and the acceleration of the universe. 
  We discuss the motion of extended objects in a spacetime by considering a gravitational field created by these objects. We define multipole moments of the objects as a classification by Lie group SO(3). Then, we construct an energy-momentum tensor for the objects and derive equations of motion from it. As a result, we reproduce the Papapetrou equations for a spinning particle. Furthermore, we will show that we can obtain more simple equations than the Papapetrou equations by changing the center-of-mass. 
  A new class of solutions to Einstein's classical field equations of general relativity is presented. The solutions describe a non-rotating, spherically symmetric, compact self gravitating object, residing in a static electro-vacuum space time.   The solutions generally have an interior non-zero matter-distribution. A wide class of interior solutions can be constructed for any specific set of exterior parameters (mass, charge). The original Schwarzschild and Reissner-Nordstroem solutions constitute special cases within the variety of interior solutions.   An outstanding feature of the new solutions is a non-continuous boundary of the matter-distribution, accompanied by a two dimensional membrane at the boundary. The membrane consists of an infinitesimally thin spherical shell of tangential pressure (surface tension/stress). The interior matter state generally has a locally anisotropic pressure.   A general procedure for generating the new solutions is given. A few solutions are derived and discussed briefly. In order to identify the physically most promising solutions, a selection principle is formulated, based on the holographic principle. One solution of particular interest emerges. It is characterized by the property, that the "stress-energy content" of the membrane is equal to the gravitating mass of the object. 
  In two recent papers by the author, a new approach was suggested for quantising space-time, or space. This involved developing a procedure for quantising a system whose configuration space--or history-theory analogue--is the set of objects, $\Ob\Q$, in a (small) category $\Q$. The quantum states in this approach are cross-sections of a bundle $A\leadsto\K[A]$ of Hilbert spaces over $\Ob\Q$. The Hilbert spaces $\K[A]$, $A\in\Ob\Q$, depend strongly on the object $A$, and have to be chosen so as to get an irreducible, faithful, representation of the basic `category quantisation monoid'. In the present paper, we develop a different approach in which the state vectors are complex-valued functions on the set of {\em arrows} in $\Q$. This throws a new light on the Hilbert bundle scheme: in particular, we recover the results of that approach in the, physically important, example when $\Q$ is a small category of finite sets. 
  We investigated the cosmology in a higher-curvature gravity where the dimensionality of spacetime gives rise to only quantitative difference, contrary to Einstein gravity. We found exponential type solutions for flat isotropic and homogeneous vacuum universe for the case in which the higher-curvature term in the Lagrangian density is quadratic in the scalar curvature, $\xi R^2$. The solutions are classified according to the sign of the cosmological constant, $\Lambda$, and the magnitude of $\Lambda\xi$. For these solutions 3-dimensional space has a specific feature in that the solutions are independent of the higher curvature term. For the universe filled with perfect fluid, numerical solutions are investigated for various values of the parameter $\xi$. Evolutions of the universes in different dimensionality of spacetime are compared. 
  The holostar is an exact spherically symmetric solution to the field equations of general relativity with anisotropic interior pressure. Its properties are similar to a black hole. It has an internal temperature inverse proportional to the square root of the radial coordinate value, from which the Hawking temperature law follows. The number of particles within any concentric region of the holostar's interior is proportional to the proper area of its boundary. The holostar-metric is static throughout the whole space-time. There are no trapped surfaces, no singularity and no event horizon. Information is not lost. The weak and strong energy conditions are fulfilled everywhere, except for a Planck-size region at the center.   Geodesic motion of massive particles in a large holostar is similar to what is observed in the universe today: A material observer moving geodesically experiences an isotropic outward directed Hubble-flow of massive particles. The total matter-density decreases over proper time by an inverse square law. The local Hubble radius increases linearly over time. Geodesic motion of photons preserves the Planck-distribution. The local radiation temperature decreases over time by an inverse square law. The current radial position r of an observer can be determined by measurements of the total local mass-density, the local radiation temperature or the local Hubble-flow. The values of r determined from the CBMR-temperature, the Hubble constant and the total mass-density of the universe are equal within an error of 15 percent to the radius of the observable universe.   The holographic solution also admits microscopic self-gravitating objects with a surface area of roughly the Planck-area and zero gravitating mass. 
  The holostar is an exact solution of the Einstein field equations with a singularity free interior matter-density rho = 1 / (8 pi r^2) and a boundary membrane consisting out of tangential pressure. Although the interior matter has on overall string equation of state, part of the matter can be interpreted in terms of particles. A simple thermodynamic model is presented, treating the matter as an ideal gas of (ultrarelativistic) fermions and bosons.  The number of ultra-relativistic particles within a holostar is proportional its surface-area, indicating that the holographic principle is valid in classical GR for self gravitating objects of any size. Using the grand canonical formalism we show, that the interior temperature is given by T \propto / \sqrt{r}. With a surface redshift z \propto \sqrt{r} the holostar's temperature at infinity is equal to the Hawking result, up to a constant factor. The factor depends on the number of particle degrees of freedom at the Planck energy, which is estimated as f ~ 7000. The holostar's total thermodynamic entropy is proportional to the area of its boundary membrane.  The ultra-relativistic fermions in the interior space-time must acquire a non-zero chemical potential, which acts as a natural source for a profound matter-antimatter asymmetry at high temperatures.  The local values of the interior temperature and matter-density are related to the holostar's temperature at infinity, enabling a "measurement" of the Hawking temperature from the interior space-time. Using the experimental values for the CMBR-temperature and the total matter-density of the universe determined by WMAP the Hawking result is verified to an accuracy of 1%.ior particles. Some properties expected from a rotating holostar are discussed briefly. 
  A charged holostar is an exact solution of the Einstein field equations. Its interior matter distribution rho = 1 / (8 pi r^2) is singularity free with an overall string equation of state. It has a boundary membrane of tangential pressure (but no mass-energy) situated roughly a Planck coordinate distance outside of the outer horizon of the RN-solution with the same mass and charge.   The geometric mass Mg = M + r0/2 of a charged holostar is always larger than its charge. r0 is a Planck size correction to the gravitational mass M with r0  2 r_Pl. For a large holostar this condition is practically identical to the classical condition M >= Q. Whereas RN solutions with M < Q are possible, a charged holostar with Mg > Q doesn't exist.   The total charge Q is derived by the proper integral over the interior charge density, which is attributed to the charged massive particles. The interior energy density splits into an electromagnetic and a "matter" contribution. Both contributions are proportional to 1/r^2. The ratio of electro-magnetic to total energy density rho_em / rho = 4 pi Q^2/A is constant throughout the whole interior. It is related to the dimensionless ratio of the exterior conserved quantities Q^2/A (or alternatively Q/M_g). An extremely charged holostar has a surface area A = 4 pi Q^2, so that its interior energy density consists entirely out of electromagnetic energy.   A large holostar can be regarded as the classical analogue of a loop quantum gravity (LQG) spin-network state. The Immirzi parameter is determined: g = s /(pi \sqrt{3}), where s is the mean entropy per particle. g is larger by a factor of ~4.8 than the LQG-result. An explanation for the discrepancy is given. 
  We consider axially symmetric static metrics in arbitrary dimension, both with and without a cosmological constant. The most obvious such solutions have an SO(n) group of Killing vectors representing the axial symmetry, although one can also consider abelian groups which represent a flat `internal space'. We relate such metrics to lower dimensional dilatonic cosmological metrics with a Liouville potential. We also develop a duality relation between vacuum solutions with internal curvature and those with zero internal curvature but a cosmological constant. This duality relation gives a solution generating technique permitting the mapping of different spacetimes. We give a large class of solutions to the vacuum or cosmological constant spacetimes. We comment on the extension of the C-metric to higher dimensions and provide a novel solution for a braneworld black hole. 
  We study the nature of boundary dynamics in the teleparallel 3D gravity. The asymptotic field equations with anti-de Sitter boundary conditions yield only two non-trivial boundary modes, related to a conformal field theory with classical central charge. After showing that the teleparallel gravity can be formulated as a Chern-Simons theory, we identify dynamical structure at the boundary as the Liouville theory. 
  We apply TDI, unfolding the general triangular configuration, to the special case of a linear array of three spacecraft. We show that such an array ("SyZyGy") has, compared with an equilateral triangle GW detector of the same scale, degraded (but non-zero) sensitivity at low-frequencies (f<<c/(arrany size)) but similar peak and high-frequency sensitivities to GWs. Sensitivity curves are presented for SyZyGys having various arm-lengths. A number of technical simplifications result from the linear configuration. These include only one faceted (e.g., cubical) proof mass per spacecraft, intra-spacecraft laser metrology needed only at the central spacecraft, placement in a single appropriate orbit can reduce Doppler drifts so that no laser beam modulation is required for ultra-stable oscillator noise calibration, and little or no time-dependent articulation of the telescopes to maintain pointing. Because SyZyGy's sensitivity falls off more sharply at low frequency than that of an equilateral triangular array, it may be more useful for GW observations in the band between those of ground-based interferometers (10-2000 Hz) and LISA (.1 mHz-.1 Hz). A SyZyGy with ~1 light- second scale could, for the same instrumental assumptions as LISA, make obseervations in this intermediate frequency GW band with 5 sigma sensitivity to sinusoidal waves of ~2.5 x 10^-23 in a year's integration. 
  In this paper we study thick-shell braneworld models in the presence of a Gauss-Bonnet term. We discuss the peculiarities of the attainment of the thin-shell limit in this case and compare them with the same situation in Einstein gravity. We describe the two simplest families of thick-brane models (parametrized by the shell thickness) one can think of. In the thin-shell limit, one family is characterized by the constancy of its internal density profile (a simple structure for the matter sector) and the other by the constancy of its internal curvature scalar (a simple structure for the geometric sector). We find that these two families are actually equivalent in Einstein gravity and that the presence of the Gauss-Bonnet term breaks this equivalence. In the second case, a shell will always keep some non-trivial internal structure, either on the matter or on the geometric sectors, even in the thin-shell limit. 
  Explicit regular coordinates that cover all of the Tangherlini solutions (Schwarzschild black holes of dimension $D>4$) are given. The coordinates reduce to Israel coordinates for D=4. 
  It was pointed out by Couch and Torrence that the extreme Reissner-Nordstrom solution possesses a discrete conformal isometry. Using results of Romans, it is shown that such a symmetry also exists when a non-zero cosmological constant is allowed. 
  We introduce a new numerical scheme for solving the initial value problem for quasiequilibrium binary neutron stars allowing for arbitrary spins. The coupled Einstein field equations and equations of relativistic hydrodynamics are solved in the Wilson-Mathews conformal thin sandwich formalism. We construct sequences of circular-orbit binaries of varying separation, keeping the rest mass and circulation constant along each sequence. Solutions are presented for configurations obeying an n=1 polytropic equation of state and spinning parallel and antiparallel to the orbital angular momentum. We treat stars with moderate compaction ((m/R) = 0.14) and high compaction ((m/R) = 0.19). For all but the highest circulation sequences, the spins of the neutron stars increase as the binary separation decreases. Our zero-circulation cases approximate irrotational sequences, for which the spin angular frequencies of the stars increases by 13% (11%) of the orbital frequency for (m/R) = 0.14 ((m/R) = 0.19) by the time the innermost circular orbit is reached. In addition to leaving an imprint on the inspiral gravitational waveform, this spin effect is measurable in the electromagnetic signal if one of the stars is a pulsar visible from Earth. 
  The relativity of Global Positioning System (GPS) pseudorange measurements is explored within the geometrical optics approximation in the curved space-time near Earth. A space-time grid for navigation is created by the discontinuities introduced in the electromagnetic field amplitude by the P-code broadcast by the GPS satellites. We compute the world function of space-time near Earth, and we use it to define a scalar phase function that describes the space-time grid. We use this scalar phase function to define the measured pseudorange, which turns out to be a two-point space-time scalar under generalized coordinate transformations. Though the measured pseudorange is an invariant, it depends on the world lines of the receiver and satellite. While two colocated receivers measure two different pseudoranges to the same satellite, they obtain correct position and time, independent of their velocity. We relate the measured pseudorange to the geometry of space-time and find corrections to the conventional model of pseudorange that are on the order of the gravitational radius of the Earth. 
  We report on the analysis and prototype-characterization of a dual-electrode electro-optic modulator that can generate both amplitude and phase modulations with a selectable relative phase, termed a universally tunable modulator (UTM). All modulation states can be reached by tuning only the electrical inputs, facilitating real-time tuning, and the device is shown to have good suppression and stability properties. A mathematical analysis is presented, including the development of a geometric phase representation for modulation. The experimental characterization of the device shows that relative suppressions of 38 dB, 39 dB and 30 dB for phase, single-sideband and carrier-suppressed modulations, respectively, can be obtained, as well as showing the device is well-behaved when scanning continuously through the parameter space of modulations. Uses for the device are discussed, including the tuning of lock points in optical locking schemes, single sideband applications, modulation fast-switching applications, and applications requiring combined modulations. 
  We explore numerically the evolution of a collapsing spherical shell of charged, massless scalar field. We obtain an external \RN space-time, and an inner space-time that is bounded by a singularity on the Cauchy Horizon. We compare these results with previous analysis and discuss some of the numerical problems encountered. 
  We give an overview of the current issues in early universe cosmology and consider the potential resolution of these issues in an as yet nascent spin foam cosmology. The model is the Barrett-Crane Model for quantum gravity along with a generalization of manifold complexes to complexes including conical singularities. 
  In this paper we analyze in the Wilson loop context the parallel transport of vectors and spinors around a closed loop in the background space-time of a rotating black string in order to classify its global properties. We also examine particular closed orbits in this space-time and verify the Mandelstam relations. 
  In this paper Quantum Mechanics with Fundamental Length is chosen as Quantum Mechanics at Planck's scale. This is possible due to the presence in the theory of General Uncertainty Relations. Here Quantum Mechanics with Fundamental Length is obtained as a deformation of Quantum Mechanics. The distinguishing feature of the proposed approach in comparison with previous ones, lies on the fact that here density matrix subjects to deformation whereas so far commutators have been deformed. The density matrix obtained by deformation of quantum-mechanical density one is named throughout this paper density pro-matrix. Within our approach two main features of Quantum Mechanics are conserved: the probabilistic interpretation of the theory and the well-known measuring procedure corresponding to that interpretation. The proposed approach allows to describe dynamics. In particular, the explicit form of deformed Liouville's equation and the deformed Shr\"odinger's picture are given. Some implications of obtained results are discussed. In particular, the problem of singularity, the hypothesis of cosmic censorship, a possible improvement of the definition of statistical entropy and the problem of information loss in black holes are considered. It is shown that obtained results allow to deduce in a simple and natural way the Bekenstein-Hawking's formula for black hole entropy in semiclassical approximation. 
  We develop a search method for gravitational ringing of black holes. The gravitational ringing is due to complex frequency modes called the quasi-normal modes that are excited when a black hole geometry is perturbed. The detection of it will be a direct confirmation of the existence of a black hole. Assuming that the ringdown waves are dominated by the fundamental mode with least imaginary part, we consider matched filtering and develop an optimal method to search for the ringdown waves that have damped sinusoidal wave forms.   When we use the matched filtering method, the data analysis with a lot of templates required. Here we have to ensure a proper match between the filter as a template and the real wave. It is necessary to keep the detection efficiency as high as possible under limited computational costs.   First, we consider the white noise case for which the matched filtering can be studied analytically. We construct an efficient method for tiling the template space. Then, using a fitting curve of the TAMA300 DT6 noise spectrum, we numerically consider the case of colored noise. We find our tiling method developed for the white noise case is still valid even if the noise is colored. 
  Yang-Mills theory in four dimensions formally admits an exact Chern-Simons wavefunction. It is an eigenfunction of the quantum Hamiltonian with zero energy. It is known to be unphysical for a variety of reasons, but it is still interesting to understand what it describes. We show that in expanding around this state, positive helicity gauge bosons have positive energy and negative helicity ones have negative energy. Some of the negative energy states would have negative norm. We also show that the Chern-Simons state is the supersymmetric partner of the naive fermion vacuum in which one does not fill the fermi sea. Finally, we give a sort of explanation of ``why'' this state exists. Similar properties can be expected for the analogous Kodama wavefunction of gravity. 
  We investigate the effect of quantum metric fluctuations on qubits that are gravitationally coupled to a background spacetime. In our first example, we study the propagation of a qubit in flat spacetime whose metric is subject to flat quantum fluctuations with a Gaussian spectrum. We find that these fluctuations cause two changes in the state of the qubit: they lead to a phase drift, as well as the expected exponential suppression (decoherence) of the off-diagonal terms in the density matrix. Secondly, we calculate the decoherence of a qubit in a circular orbit around a Schwarzschild black hole. The no-hair theorems suggest a quantum state for the metric in which the black hole's mass fluctuates with a thermal spectrum at the Hawking temperature. Again, we find that the orbiting qubit undergoes decoherence and a phase drift that both depend on the temperature of the black hole. Thirdly, we study the interaction of coherent and squeezed gravitational waves with a qubit in uniform motion. Finally, we investigate the decoherence of an accelerating qubit in Minkowski spacetime due to the Unruh effect. In this case decoherence is not due to fluctuations in the metric, but instead is caused by coupling (which we model with a standard Hamiltonian) between the qubit and the thermal cloud of Unruh particles bathing it. When the accelerating qubit is entangled with a stationary partner, the decoherence should induce a corresponding loss in teleportation fidelity. 
  Black hole perturbation theory, or more generally, perturbation theory on a Schwarzschild bockground, has been applied in several contexts, but usually under the simplifying assumption that the ADM momentum vanishes, namely, that the evolution is carried out and observed in the ``center of momentum frame''. In this paper we consider some consequences of the inclusion of a non vanishing ADM momentum in the initial data. We first provide a justification for the validity of the transformation of the initial data to the ``center of momentum frame'', and then analyze the effect of this transformation on the gravitational wave amplitude. The most significant result is the possibility of a type of gravitational memory effect that appears to have no simple relation with the well known Christodoulou effect. 
  It is well known that carriers of astrophysical information are massless spinning particles. These carriers are photons, neutrinos and, expectedly, gravitons. All these particles are emitted during supernova events. Information carried by these particles characterize their sources, but such information are affected by the trajectories of the carriers. Recently, it is shown that these trajectories are spin dependent. Knowing these trajectories and the arrival times of such particles to the detectors, a spin dependent model is constructed and compared with the conventional spin independent model. 
  Methods and properties regarding the linear perturbations are discussed for some spatially closed (vacuum) solutions of Einstein's equation. The main focus is on two kinds of spatially locally homogeneous solution; one is the Bianchi III (Thurston's H^2 x R) type, while the other is the Bianchi II (Thurston's Nil) type. With a brief summary of previous results on the Bianchi III perturbations, asymptotic solutions for the gauge-invariant variables for the Bianchi III are shown, with which (in)stability of the background solution is also examined. The issue of linear stability for a Bianchi II solution is still an open problem. To approach it, appropriate eigenfunctions are presented for an explicitly compactified Bianchi II manifold and based on that, some field equations on the Bianchi II background spacetime are studied. Differences between perturbation analyses for Bianchi class B (to which Bianchi III belongs) and class A (to which Bianchi II belongs) are stressed for an intention to be helpful for applications to other models. 
  Utilizing various gauges of the radial coordinate we give a description of static spherically symmetric space-times with point singularity at the center and vacuum outside the singularity. We show that in general relativity (GR) there exist a two-parameters family of such solutions to the Einstein equations which are physically distinguishable but only some of them describe the gravitational field of a single massive point particle with nonzero bare mass $M_0$. In particular, we show that the widespread Hilbert's form of Schwarzschild solution, which depends only on the Keplerian mass $M<M_0$, does not solve the Einstein equations with a massive point particle's stress-energy tensor as a source. Novel normal coordinates for the field and a new physical class of gauges are proposed, in this way achieving a correct description of a point mass source in GR. We also introduce a gravitational mass defect of a point particle and determine the dependence of the solutions on this mass defect. The result can be described as a change of the Newton potential $\phi_{{}_N}=-G_{{}_N}M/r$ to a modified one: $\phi_{{}_G}=-G_{{}_N}M/ (r+G_{{}_N} M/c^2\ln{{M_0}\over M})$ and a corresponding modification of the four-interval. In addition we give invariant characteristics of the physically and geometrically different classes of spherically symmetric static space-times created by one point mass. These space-times are analytic manifolds with a definite singularity at the place of the matter particle. 
  Surface plasmons at metal interfaces are collective excitations of the conduction electrons and the electromagnetic field. They exist in "curved three-dimensional space-times" defined by the shape of the metal surface and the spatial distribution of the dielectric constant near the surface. Here we show that surface plasmon toy models of many non-trivial space-time metrics, such as wormholes and black holes, can be easily built and studied in experiments. For example, a droplet of dielectric on the metal surface behaves as a black hole for surface plasmons within a substantial frequency range. On the other hand, a nanohole in a thin metal membrane may be treated as a wormhole connecting two "flat" surface plasmon worlds located on the opposite interfaces of the membrane. 
  A self consistent solution to Dirac equation in a Kerr Newman space-time with $M^2 > a^2 + Q^2$ is presented for the case when the Dirac particle is the source of the curvature and the electromagnetic field. The solution is localised, continuous everywhere and valid only for a special choice of the parameters appearing in the Dirac equation. 
  We define the concept of a Maximally symmetric osculating space-time at any event of any given Robertson-Walker model. We use this definition in two circumstances: i) to approximate any given cosmological model by a simpler one sharing the same observational parameters, i.e, the speed of light, the Hubble constant and the deceleration parameter at the time of tangency, and ii) to shed some light on the problem of considering an eventual influence of the overall behaviour of the Universe on localized systems at smaller scales, or viceversa. 
  Maximal signal and peak of high-frequency relic gravitational waves (GW's), recently expected by quintessential inflationary models, may be firmly localized in the GHz region, the energy density of the relic gravitons in critical units (i.e., $ h_0^2 \Omega_{GW}$) is of the order $10^{-6}$, roughly eight orders of magnitude larger than in ordinary inflationary models. This is just right best frequency band of the electromagnetic (EM) response to the high-frequency GW's in smaller EM detecting systems. We consider the EM response of a Gaussian beam passing through a static magnetic field to a high-frequency relic GW. It is found that under the synchroresonance condition, the first-order perturbative EM power fluxes will contain "left circular wave" and "right circular wave" around the symmetrical axis of the Gaussian beam, but the perturbative effects produced by the states of + polarization and $\times$ polarization of the relic GW have different properties, and the perturbations on behavior are obviously different from that of the background EM fields in the local regions. For the high-frequency relic GW with the typical parameters $ \nu_g = 10^{10}Hz$, $ h = 10^{- 30} $ in the quintessential inflationary models, the corresponding perturbative photon flux passing through the region $ 10^{- 2} m^{2} $ would be expected to be $ 10^{3}s^{-1} $. This is largest perturbative photon flux we recently analyzed and estimated using the typical laboratory parameters. In addition, we also discuss geometrical phase shift generated by the high-frequency relic GW in the Gaussian beam and estimate possible physical effects. 
  Three presumably unrelated open questions concerning gravity and the structure of the Universe are here discussed: 1) To which extent is Lorentz invariance an exact symmetry ? 2) What is the equation of state of the Universe ? 3) What is the origin of the so-called Pioneer anomaly ? 
  A coordinate system is constructed for a general accelerating observer in 1+1 dimensions, and is used to determine the particle density of the massless Dirac vacuum for that observer. Equations are obtained for the spatial distribution and frequency distribution of massless fermions seen by this observer, in terms of the rapidity function of the observer's worldline. Examples that are considered include the uniformly accelerating observer as a limiting case, but do not always involve particle horizons. Only the low frequency limit depends on the possible presence of particle horizons. The rest of the spectrum is `almost thermal' whenever the observer's acceleration is `almost uniform'. 
  Smolin has put forward the proposal that the universe fine tunes the values of its physical constants through a Darwinian selection process. Every time a black hole forms, a new universe is developed inside it that has different values for its physical constants from the ones in its progenitor. The most likely universe is the one which maximizes the number of black holes. Here we present a concrete quantum gravity calculation based on a recently proposed consistent discretization of the Einstein equations that shows that fundamental physical constants change in a random fashion when tunneling through a singularity. 
  In an effort to eliminate laser phase noise in laser interferometer spaceborne gravitational wave detectors, several combinations of signals have been found that allow the laser noise to be canceled out while gravitational wave signals remain. This process is called time delay interferometry (TDI). In the papers that defined the TDI variables, their performance was evaluated in the limit that the gravitational wave detector is fixed in space. However, the performance depends on certain symmetries in the armlengths that are available if the detector is fixed in space, but that will be broken in the actual rotating and flexing configuration produced by the LISA orbits. In this paper we investigate the performance of these TDI variables for the real LISA orbits. First, addressing the effects of rotation, we verify Daniel Shaddock's result that the Sagnac variables will not cancel out the laser phase noise, and we also find the same result for the symmetric Sagnac variable. The loss of the latter variable would be particularly unfortunate since this variable also cancels out gravitational wave signal, allowing instrument noise in the detector to be isolated and measured. Fortunately, we have found a set of more complicated TDI variables, which we call Delta-Sagnac variables, one of which accomplishes the same goal as the symmetric Sagnac variable to good accuracy. Finally, however, as we investigate the effects of the flexing of the detector arms due to non-circular orbital motion, we show that all variables, including the interferometer variables, which survive the rotation-induced loss of direction symmetry, will not completely cancel laser phase noise when the armlengths are changing with time. This unavoidable problem will place a stringent requirement on laser stability of 5 Hz per root Hz. 
  We review the work going on in black-hole physics during the last ten years, called the Choptuik Phenomenon. 
  We study the properties of the outgoing gravitational wave produced when a non-spinning black hole is excited by an ingoing gravitational wave. Simulations using a numerical code for solving Einstein's equations allow the study to be extended from the linearized approximation, where the system is treated as a perturbed Schwarzschild black hole, to the fully nonlinear regime. Several nonlinear features are found which bear importance to the data analysis of gravitational waves. When compared to the results obtained in the linearized approximation, we observe large phase shifts, a stronger than linear generation of gravitational wave output and considerable generation of radiation in polarization states which are not found in the linearized approximation. In terms of a spherical harmonic decomposition, the nonlinear properties of the harmonic amplitudes have simple scaling properties which offer an economical way to catalog the details of the waves produced in such black hole processes. 
  It is shown that the recently geometric formulation of quantum mechanics implies the use of Weyl geometry. It is discussed that the natural framework for both gravity and quantum is Weyl geometry. At the end a Weyl invariant theory is built, and it is shown that both gravity and quantum are present at the level of equations of motion. 
  In this paper we have applied Bohmian quantum theory to the linear field approximation of gravity and present a self--consistent quantum gravity theory in the linear field approximation. The theory is then applied to some specific problems, the Newtonian limit, and the static spherically symmetric solution. Some observable effects of the theory are investigated. 
  The energy distribution associated with a stringy charged black hole is studied using M{\o}ller's energy-momentum complex. Our result is reasonable and it differs from that known in literature using Einstein's energy-momentum complex. 
  The Leibniz bracket of an operator on a (graded) algebra is defined and some of its properties are studied. A basic theorem relating the Leibniz bracket of the commutator of two operators to the Leibniz bracket of them, is obtained. Under some natural conditions, the Leibniz bracket gives rise to a (graded) Lie algebra structure. In particular, those algebras generated by the Leibniz bracket of the divergence and the Laplacian operators on the exterior algebra are considered, and the expression of the Laplacian for the product of two functions is generalized for arbitrary exterior forms. 
  A universal scheme for describing gravitational thermal effects is developed as a generalization of Unruh effect. Quasi-Rindler (QR) coordinates are constructed in an arbitrary curved space-time in such a way that the imaginary QR time be periodical. The observer at rest in QR coordinates should experience a thermal effect. Application to de Sitter space-time is considered. 
  The often-asked question whether space-time is discrete or continuous may not be the right question to ask: Mathematically, it is possible that space-time possesses the differentiability properties of manifolds as well as the ultraviolet finiteness properties of lattices. Namely, physical fields in space-time could possess a finite density of degrees of freedom in the following sense: if a field's amplitudes are given on a sufficiently dense set of discrete points then the field's amplitudes at all other points of the manifold are fully determined and calculable. Which lattice of sampling points is chosen should not matter, as long as the lattices' spacings are tight enough, for example, not exceeding the Planck distance. This type of mathematical structure is known within information theory, as sampling theory, and it plays a central role in all of digital signal processing. 
  The recent, precise Michelson-Morley experiment performed by Muller et al. suggests a tiny anisotropy of the speed of light. I propose a quantitative explanation of the observed effect based on the interpretation of gravity as a density fluctuation of the Higgs condensate. 
  We study the stability of five-dimensional Myers-Perry black holes with a single angular momentum under linear perturbations, and we compute the quasinormal modes (QNM's) of the black hole metric projected on the brane, using Leaver's continued fraction method. In our numerical search we do not find unstable modes. The damping time of modes having l=m=2 and l=m=1 tends to infinity as the black hole spin tends to the extremal value, showing a behaviour reminiscent of the one observed for ordinary 4-dimensional Kerr black holes. 
  It seems to be expected, that a horizon of a quasi-local type, like a Killing or an isolated horizon, by analogy with a globally defined event horizon, should be unique in some open neighborhood in the spacetime, provided the vacuum Einstein or the Einstein-Maxwell equations are satisfied. The aim of our paper is to verify whether that intuition is correct. If one can extend a so called Kundt metric, in such a way that its null, shear-free surfaces have spherical spacetime sections, the resulting spacetime is foliated by so called non-expanding horizons. The obstacle is Kundt's constraint induced at the surfaces by the Einstein or the Einstein-Maxwell equations, and the requirement that a solution be globally defined on the sphere. We derived a transformation (reflection) that creates a solution to Kundt's constraint out of data defining an extremal isolated horizon. Using that transformation, we derived a class of exact solutions to the Einstein or Einstein-Maxwell equations of very special properties. Each spacetime we construct is foliated by a family of the Killing horizons. Moreover, it admits another, transversal Killing horizon. The intrinsic and extrinsic geometry of the transversal Killing horizon coincides with the one defined on the event horizon of the extremal Kerr-Newman solution. However, the Killing horizon in our example admits yet another Killing vector tangent to and null at it. The geometries of the leaves are given by the reflection. 
  Given a globally hyperbolic spacetime $M$, we show the existence of a {\em smooth spacelike} Cauchy hypersurface $S$ and, thus, a global diffeomorphism between $M$ and $\R \times S$. 
  Perfect fluid spheres, both Newtonian and relativistic, have attracted considerable attention as the first step in developing realistic stellar models (or models for fluid planets). Whereas there have been some early hints on how one might find general solutions to the perfect fluid constraint in the absence of a specific equation of state, explicit and fully general solutions of the perfect fluid constraint have only very recently been developed. In this article we present a version of Lake's algorithm [Phys. Rev. D 67 (2003) 104015; gr-qc/0209104] wherein: (1) we re-cast the algorithm in terms of variables with a clear physical meaning -- the average density and the locally measured acceleration due to gravity, (2) we present explicit and fully general formulae for the mass profile and pressure profile, and (3) we present an explicit closed-form expression for the central pressure. Furthermore we can then use the formalism to easily understand the pattern of inter-relationships among many of the previously known exact solutions, and generate several new exact solutions. 
  It is proposed that the Schrodinger equation for a free point particle has non-linear corrections which depend on the mass of the particle. It is assumed that the corrections become extremely small when the mass is much smaller or much larger than a critical value (the critical value being related to but smaller than Planck mass). The corrections become significant when the mass is close to this critical value and could play a role in explaining wave-function collapse. It appears that such corrections are not ruled out by present day experimental tests of the Schrodinger equation. Corrections to the energy levels of a harmonic oscillator are calculated. 
  We investigate the possibility of obtaining non-singular black-hole solutions in the brane world model by solving the effective field equations for the induced metric on the brane. The Reissner-Nordstrom solution on the brane was obtained by Dadhich etal by imposing the null energy condition on the 3-brane for a bulk having non zero Weyl curvature. In this work, we relax the condition of vanishing scalar curvature $R$, however, retaining the null condition. We have shown that it is possible to obtain class of static non-singular spherically symmetric brane space-times admitting horizon. We obtain one such class of solution which is a regular version of the Reissner-Nordstrom solution in the standard general relativity. 
  In this short note, we verify explicitly in static coordinates that the non trivial asymptotic Killing vectors at spatial infinity for anti-de Sitter space-times correspond one to one to the conformal Killing vectors of the conformally flat metric induced on the boundary. The fall-off conditions for the metric perturbations that guarantee finiteness of the associated conserved charges are derived. 
  We examine radiative corrections arising from Lorentz violating dimension five operators presumably associated with Planck scale physics as recently considered by Myers and Pospelov. We find that observational data result in bounds on the dimensionless parameters of the order $10^{-15}$. These represent the most stringent bounds on Lorentz violation to date. 
  We apply the Weyl method, as sanctioned by Palais' symmetric criticality theorems, to obtain those -highly symmetric -geometries amenable to explicit solution, in generic gravitational models and dimension. The technique consists of judiciously violating the rules of variational principles by inserting highly symmetric, and seemingly gauge fixed, metrics into the action, then varying it directly to arrive at a small number of transparent, indexless, field equations. Illustrations include spherically and axially symmetric solutions in a wide range of models beyond D=4 Einstein theory; already at D=4, novel results emerge such as exclusion of Schwarzschild solutions in cubic curvature models and restrictions on ``independent'' integration parameters in quadratic ones. Another application of Weyl's method is an easy derivation of Birkhoff's theorem in systems with only tensor modes. Other uses are also suggested. 
  A summary of how black holes grow in full, non-linear general relativity is presented. Specifically, a notion of "dynamical horizons" is introduced and expressions of fluxes of energy and angular momentum carried by gravitational waves across these horizons are obtained. Fluxes are local and the energy flux is positive. Change in the horizon area is related to these fluxes. The flux formulae also give rise to balance laws analogous to the ones obtained by Bondi and Sachs at null infinity and provide generalizations of the first and second laws of black hole mechanics. 
  Using "smooth brane" solutions of the field equations, we give an alternative derivation of the junction conditions for a "brane" in a five dimensional "bulk", when gravity is governed by the Einstein Lanczos (Gauss-Bonnet) equations. 
  Einstein equations are addressed with the energy-momentum tensor that appears if the equations under discussion are required to possess conformal invariance. It is proved that thus derived equations (equations of conformally invariant geometrodynamics) can have not only smooth solutions, but also solutions with discontinuities on space-like hypersurfaces. The solutions obtained are similar to the well-known discontinuous Einstein equation solutions like shock-wave solutions, extended-source solutions, etc.   For the centrally symmetric stationary solution discussed in the paper, the discontinuity surface removes the singularity. The degree of generality of this solution regularization mechanism is discussed. The issue of the mechanism that forces any smooth solution in the conformally invariant geometrodynamics to be rearranged into the discontinuous one when certain conditions are met is also discussed. The conditions can be: (1) sound speed becoming to be higher than light speed; (2) the solution becoming intolerant to smaller and smaller-scale perturbation modes. 
  In this paper we investigate asymptotic isotropization. We derive the asymptotic dynamics of spatially inhomogeneous cosmological models with a perfect fluid matter source and a positive cosmological constant near the de Sitter equilibrium state at late times, and near the flat FL equilibrium state at early times. Our results show that there exists an open set of solutions approaching the de Sitter state at late times, consistent with the cosmic no-hair conjecture. On the other hand, solutions that approach the flat FL state at early times are special and admit a so-called isotropic initial singularity. For both classes of models the asymptotic expansion of the line element contains an arbitrary spatial metric at leading order, indicating asymptotic spatial inhomogeneity. We show, however, that in the asymptotic regimes this spatial inhomogeneity is significant only at super-horizon scales. 
  We present a study of Bianchi class A tilted cosmological models admitting a proper homothetic vector field together with the restrictions, both at the geometrical and dynamical level, imposed by the existence of the simply transitive similarity group. The general solution of the symmetry equations and the form of the homothetic vector field are given in terms of a set of arbitrary integration constants. We apply the geometrical results for tilted perfect fluids sources and give the general Bianchi II self-similar solution and the form of the similarity vector field. In addition we show that self-similar perfect fluid Bianchi VII$_0$ models and irrotational Bianchi VI$_0$ models do not exist. 
  We review analytic methods to perform the post-Newtonian expansion of gravitational waves induced by a particle orbiting a massive compact body, based on the black hole perturbation theory. There exist two different methods of the post-Newtonian expansion. Both are based the Teukolsky equation. In one method, the Teukolsky equation is transformed into a Regge-Wheeler type equation that reduces to the standard Klein-Gordon equation in the flat space limit, while in the other method, which were introduced by Mano, Suzuki and Takasugi relatively recently, the Teukolsky equation is directly used in its original form. The former has an advantage that it is intuitively easy to understand how various curved space effects come into play. However, it becomes increasingly complicated when one goes on to higher and higher post-Newtonian orders. In contrast, the latter has an advantage that a systematic calculation to higher post-Newtonian orders is relatively easily implementable, but otherwise so mathematical that it is hard to understand the interplay of higher order terms. In this paper, we review both methods so that their pros and cons may be clearly seen. We also review some results of calculations of gravitational radiation emitted by a particle orbiting a black hole. 
  Hoyle and Narlikar's $C$-field cosmology is extended in the framework of higher dimensional spacetime and a class of exact solutions is obtained. Adjusting the arbitrary constants of integration one can show that our model is amenable to the desirable property of dimensional reduction so that the universe ends up in an effective 4D one.Further with matter creation from the $C$-field the mass density steadies with time and the usual bigbang singularity is avoided. An alternative mechanism is also suggested which seems to provide matter creation in the 4D spacetime although total matter in the 5D world remains conserved. Quintessence phenomenon and energy conditions are also discussed and it is found that in line with the physical requirements our model admits a solution with a decelerating phase in the early era followed by an accelerated expansion later. Moreover, as the contribution from the $C$-field is made negligible a class of our solutions reduces to the previously known higher dimensional models in the framework of Einstein's theory. 
  As part of our development of a computer code to perform 3D `constrained evolution' of Einstein's equations in 3+1 form, we discuss issues regarding the efficient solution of elliptic equations on domains containing holes (i.e., excised regions), via the multigrid method. We consider as a test case the Poisson equation with a nonlinear term added, as a means of illustrating the principles involved, and move to a "real world" 3-dimensional problem which is the solution of the conformally flat Hamiltonian constraint with Dirichlet and Robin boundary conditions. Using our vertex-centered multigrid code, we demonstrate globally second-order-accurate solutions of elliptic equations over domains containing holes, in two and three spatial dimensions. Keys to the success of this method are the choice of the restriction operator near the holes and definition of the location of the inner boundary. In some cases (e.g. two holes in two dimensions), more and more smoothing may be required as the mesh spacing decreases to zero; however for the resolutions currently of interest to many numerical relativists, it is feasible to maintain second order convergence by concentrating smoothing (spatially) where it is needed most. This paper, and our publicly available source code, are intended to serve as semi-pedagogical guides for those who may wish to implement similar schemes. 
  By efforts of several authors, it is recently established that the dynamical behavior of the cosmological perturbation on superhorizon scales is well approximated in terms of that in the long wavelength limit, and the latter can be constructed from the evolution of corresponding exactly homogeneous universe. Using these facts, we investigate the evolution of the cosmological perturbation on superhorizon scales in the universe dominated by oscillating multiple scalar fields which are generally interacting with each other, and the ratio of whose masses is incommensurable. Since the scalar fields oscillate rapidly around the local minimum of the potential, we use the action angle variables. We found that this problem can be formulated as the canonical perturbation theory in which the perturbed part appearing as the result of the expansion of the universe and the interaction of the scalar fields is bounded by the negative power ot time. We show that by constructing the canonical transformations properly, the transformed hamiltonian becomes simple enough to be solved. As the result of the invetigation using the long wavelength limit and the canonical perturbation theory, under the sufficiently general conditions, we prove that for the adiabatic growing mode the Bardeen parameter stays constant and that for all the other modes the Bardeen parameter decays.   From the viewpoint of the ergodic theory, it is discussed that as for the Bardeen parameter, the sigularities appear probabilistically. This analysis serves the understanding of the evolution of the cosmological perturbations on superhorizon scales during reheating. 
  The lowest 37000 eigenvalues of the area operator in loop quantum gravity is calculated and studied numerically. We obtain an asymptotical formula for the eigenvalues as a function of their sequential number. The multiplicity of the lowest few hundred eigenvalues is also determined and the smoothed spectral density is calculated. The spectral density is presented for various number of vertices, edges and SU(2) representations. A scaling form of spectral density is found, being a power law for one vertex, while following an exponential for several vertices. The latter case is explained on the basis of the one vertex spectral density. 
  A phase-locking configuration for LISA is proposed that provides a significantly simpler mode of operation. The scheme provides one Sagnac signal readout inherently insensitive to laser frequency noise and optical bench motion for a non-rotating LISA array. This Sagnac output is also insensitive to clock noise, requires no time shifting of data, nor absolute arm length knowledge. As all measurements are made at one spacecraft, neither clock synchronization nor exchange of phase information between spacecraft is required. The phase-locking configuration provides these advantages for only one Sagnac variable yet retains compatibility with the baseline approach for obtaining the other TDI variables. The orbital motion of the LISA constellation is shown to produce a 14 km path length difference between the counter-propagating beams in the Sagnac interferometer. With this length difference a laser frequency noise spectral density of 1 Hz/$\sqrt{\rm Hz}$ would consume the entire optical path noise budget of the Sagnac variables. A significant improvement of laser frequency stability (currently at 30 Hz/$\sqrt{\rm Hz}$) would be needed for full-sensitivity LISA operation in the Sagnac mode. Alternatively, an additional level of time-delay processing could be applied to remove the laser frequency noise. The new time-delayed combinations of the phase measurements are presented. 
  We study the Klein-Gordon and Dirac equations in the presence of a background metric ds^2 = -dt^2 + dx^2 + e^{-2gx}(dy^2 + dz^2) in a semi-infinite lab (x>0). This metric has a constant scalar curvature R=6g^2 and is produced by a perfect fluid with equation of state p=-\rho /3. The eigenfunctions of spin-0 and spin-1/2 particles are obtained exactly, and the quantized energy eigenvalues are compared. It is shown that both of these particles must have nonzero transverse momentum in this background. We show that there is a minimum energy E^2_{min}=m^2c^4 + g^2c^2\hbar^2$ for bosons E_{KG} > E_{min}, while the fermions have no specific ground state E_{Dirac}>mc^2. 
  It is shown that the free Dirac equation in spherically symmetric static backgrounds of any dimensions can be put in a simple form using a special version of Cartesian gauge in Cartesian coordinates. This is manifestly covariant under the transformations of the isometry group so that the generalized spherical coordinates can be separated in terms of angular spinors like in the flat case, obtaining a pair of radial equations. In this approach the equation of the free field Dirac in $AdS_{d+1}$ backgrounds is analytically solved obtaining the formula of the energy levels and the corresponding normalized eigenspinors. 
  The Sagnac time delay and the corresponding Sagnac phase shift, for relativistic matter and electromagnetic beams counter-propagating in a rotating interferometer, are deduced on the ground of relativistic kinematics. This purely kinematical approach allows to explain the ''universality'' of the effect, namely the fact that the Sagnac time difference does not depend on the physical nature of the interfering beams. The only prime requirement is that the counter-propagating beams have the same velocity with respect to any Einstein synchronized local co-moving inertial frame. 
  The spacetime homogeneous G\"odel-type spacetimes which have four classes of metrics are studied according to their matter collineations. The obtained results are compared with Killing vectors and Ricci collineations. It is found that these spacetimes have infinite number of matter collineations in degenerate case, i.e. det$(T_{ab}) = 0$, and do not admit proper matter collineations in non-degenerate case, i.e. det$(T_{ab}) \ne 0$. The degenerate case has the new constraints on the parameters $m$ and $w$ which characterize the causality features of the G\"odel-type spacetimes. 
  Many alternative formulations of Einstein's evolution have lately been examined, in an effort to discover one which yields slow growth of constraint-violating errors. In this paper, rather than directly search for well-behaved formulations, we instead develop analytic tools to discover which formulations are particularly ill-behaved. Specifically, we examine the growth of approximate (geometric-optics) solutions, studied only in the future domain of dependence of the initial data slice (e.g. we study transients). By evaluating the amplification of transients a given formulation will produce, we may therefore eliminate from consideration the most pathological formulations (e.g. those with numerically-unacceptable amplification). This technique has the potential to provide surprisingly tight constraints on the set of formulations one can safely apply. To illustrate the application of these techniques to practical examples, we apply our technique to the 2-parameter family of evolution equations proposed by Kidder, Scheel, and Teukolsky, focusing in particular on flat space (in Rindler coordinates) and Schwarzchild (in Painleve-Gullstrand coordinates). 
  We discuss a method of coincidence analysis to search for gravitational waves from inspiraling compact binaries using the data of two laser interferometer gravitational wave detectors. We examine the allowed difference of the wave's parameters estimated by each detector to obtain good detection efficiency. We also discuss a method to set an upper limit to the event rate from the results of the coincidence analysis. For the purpose to test above methods, we performed a coincidence analysis by applying these methods to the real data of TAMA300 and LISM detectors taken during 2001. We show that the fake event rate is reduced significantly by the coincidence analysis without losing real events very much. Results of the test analysis are also given. 
  A condition of supersymmetric cosmological solutions of simple (N=1) supergravity is formulated in the classical case. As an application we prove that supersymmetry is spontaneously broken in Friedmann-Robertson-Walker type cosmologies as well as in the Kasner universe, except for the Minkowski space. 
  Transforming Penrose's intuitive picture of a strong cosmic censorship principle, that generically forbids the appearance of locally naked space-time singularities, into a formal mathematical proof, remains at present, one of the most outstanding unsolved mathematical problems from the theory of gravitational collapse. Part of the difficulty lies in the fact that we do not possess yet a clear-cut understanding of the hypothesis needed for the establishment of some sort of strong cosmic censorship theorem. What we have is a selected list of solutions, which at first sight seem to go against cosmic censorship, but at the end they fail in some way. However, the space of solutions of Einstein's field equations is vast. In this article, we plan to increase one's intuition by establishing a link between certain inequalities for Cauchy-horizon stability and a set of generic conditions, such as a reasonable equation of state--which determines whether the space-time is asymptotically flat or not, an energy condition, and an hypothesis over the class of metrics on which Einstein's field equations ought to be solved to ensure strong cosmic censorship inside black-holes. With these tools in hand we examine the Cauchy-horizon stability of the theory created by Born and Infeld--whose action principle has been used as a prototype in superstring theory, and the singularity-free Bardeen's black-hole model. 
  The long time behavior of an evaporating Schwarzschild black hole is studied exploiting that it can be described by an effective theory in 2D, a particular dilaton gravity model.   A crucial technical ingredient is Izawa's result on consistent deformations of 2D BF theory, while the most relevant physical assumption is boundedness of the asymptotic matter flux during the whole evaporation process.   An attractor solution, the endpoint of the evaporation process, is found. Its metric is flat. However, the behavior of the dilaton field is nontrivial: it is argued that during the final flicker a first order phase transition occurs from a linear to a constant dilaton vacuum, thereby emitting a shock wave with a total energy of a fraction of the Planck mass. Another fraction of the Planck mass may reside in a cold remnant. [Note: More detailed abstract in the paper] 
  We generalize Israel's formalism to cover singular shells embedded in a non-vacuum Universe. That is, we deduce the relativistic equation of motion for a thin shell embedded in a Schwarzschild/Friedmann-Lemaitre-Robertson-Walker spacetime. Also, we review the embedding of a Schwarzschild mass into a cosmological model using "curvature" coordinates and give solutions with (Sch/FLRW) and without the embedded mass (FLRW). 
  The Einstein equations have proven surprisingly difficult to solve numerically. A standard diagnostic of the problems which plague the field is the failure of computational schemes to satisfy the constraints, which are known to be mathematically conserved by the evolution equations. We describe a new approach to rewriting the constraints as first-order evolution equations, thereby guaranteeing that they are satisfied to a chosen accuracy by any discretization scheme. This introduces a set of four subsidiary constraints which are far simpler than the standard constraint equations, and which should be more easily conserved in computational applications. We explore the manner in which the momentum constraints are already incorporated in several existing formulations of the Einstein equations, and demonstrate the ease with which our new constraint-conserving approach can be incorporated into these schemes. 
  The low-frequency resolution of space-based gravitational wave observatories such as LISA (Laser Interferometry Space Antenna) hinges on the orbital purity of a free-falling reference test mass inside a satellite shield. We present here a torsion pendulum study of the forces that will disturb an orbiting test mass inside a LISA capacitive position sensor. The pendulum, with a measured torque noise floor below 10 fNm/sqrt{Hz} from 0.6 to 10 mHz, has allowed placement of an upper limit on sensor force noise contributions, measurement of the sensor electrostatic stiffness at the 5% level, and detection and compensation of stray DC electrostatic biases at the mV level. 
  Einstein's spherically symmetric interior gravitational equations are investigated. Following Synge's procedure, the most general solution of the equations is furnished in case $T^{1}_{1}$ and $T^{4}_{4}$ are prescribed. The existence of a total mass function, $M(r,t)$, is rigorously proved. Under suitable restrictions on the total mass function, the Schwarzschild mass $M(r,t)=m$, implicitly defines the boundary of the spherical body as $r=B(t)$. Both Synge's junction conditions as well as the continuity of the second fundamental form are examined and solved in a general manner. The weak energy conditions for an \emph{arbitrary boost} are also considered. The most general solution of the spherically symmetric anisotropic fluid model satisfying both junction conditions is furnished. In the final section, various exotic solutions are explored using the developed scheme including gravitational instantons, interior $T$-domains and $D$-dimensional generalizations. 
  Aguirregabiria et al showed that Einstein, Landau and Lifshitz, Papapetrou, and Weinberg energy-momentum complexes coincide for all Kerr-Schild metric. Bringely used their general expression of the Kerr-Schild class and found energy and momentum densities for the Bonnor metric. We obtain these results without using Aguirregabiria et al results and verify that Bringley's results are correct. This also supports Aguirregabiria et al results as well as Cooperstock hypothesis. Further, we obtain the energy distribution of the space-time under consideration. 
  By proper co-ordinates of non-inertial observers (shortly - proper non-inertial co-ordinates) we understand the proper co-ordinates of an arbitrarily moving local observer. After a brief review of the theory of proper non-inertial co-ordinates, we apply these co-ordinates to discuss the relativistic effects seen by observers at different positions on a rotating ring. Although there is no relative motion among observers at different positions, they belong to different proper non-inertial frames. The relativistic length seen by an observer depends only on his instantaneous velocity, not on his acceleration or rotation. For any observer the velocity of light is isotropic and equal to $c$, provided that it is measured by propagating a light beam in a small neighbourhood of the observer. 
  The low-laying frequencies of characteristic quasi-normal modes (QNM) of Schwarzschild-de Sitter (SdS) black holes have been calculated for fields of different spin using the 6th-order WKB approximation and the approximation by the P\"{o}shl-Teller potential. The well-known asymptotic formula for large $l$ is generalized here on a case of the Schwarzchild-de Sitter black hole. In the limit of the near extreme $\L$ term the results given by both methods are in a very good agreement, and in this limit fields of different spin decay with the same rate. 
  We consider a spherically symmetric characteristic initial value problem for the Einstein-Maxwell-scalar field equations. On the initial outgoing characteristic, the data is assumed to satisfy the Price law decay widely believed to hold on an event horizon arising from the collapse of an asymptotically flat Cauchy surface. We establish that the heuristic mass inflation scenario put forth by Israel and Poisson is mathematically correct in the context of this initial value problem. In particular, the maximal domain of development has a future boundary, over which the spacetime is extendible as a continuous metric, but along which the Hawking mass blows up identically; thus, the spacetime is inextendible as a differentiable metric. In view of recent results of the author in collaboration with I. Rodnianski (gr-qc/0309115), which rigorously establish the validity of Price's law as an upper bound for the decay of scalar field hair, the continuous extendibility result applies to the collapse of complete asymptotically flat spacelike data where the scalar field is compactly supported on the initial hypersurface. This shows that under Christodoulou's C^0 formulation, the strong cosmic censorship conjecture is false for this system. 
  We study classical Hamiltonian systems in which the intrinsic proper time evolution parameter is related through a probability distribution to the physical time, which is assumed to be discrete. - This is motivated by the ``timeless'' reparametrization invariant model of a relativistic particle with two compactified extradimensions. In this example, discrete physical time is constructed based on quasi-local observables. - Generally, employing the path-integral formulation of classical mechanics developed by Gozzi et al., we show that these deterministic classical systems can be naturally described as unitary quantum mechanical models. The emergent quantum Hamiltonian is derived from the underlying classical one. It is closely related to the Liouville operator. We demonstrate in several examples the necessity of regularization, in order to arrive at quantum models with bounded spectrum and stable groundstate. 
  We present long-term stable and second-order convergent evolutions of an excised wobbling black hole. Our results clearly demonstrate that the use of a densitized lapse function extends the lifetime of simulations dramatically. We also show the improvement in the stability of single static black holes when an algebraic densitized lapse condition is applied. In addition, we introduce a computationally inexpensive approach for tracking the location of the singularity suitable for mildly distorted black holes. The method is based on investigating the fall-off behavior and asymmetry of appropriate grid variables. This simple tracking method allows one to adjust the location of the excision region to follow the coordinate motion of the singularity. 
  A linear relationship between the Hubble expansion parameter and the time derivative of the scalar field is explored in order to derive exact cosmological, attractor-like solutions, both in Einstein's theory and in Brans-Dicke gravity with two fluids: a background fluid of ordinary matter, together with a self-interacting scalar field accounting for the dark energy in the universe. A priori assumptions about the functional form of the self-interaction potential or about the scale factor behavior are not necessary. These are obtained as outputs of the assumed relationship between the Hubble parameter and the time derivative of the scalar field. A parametric class of scaling quintessence models given by a self-interaction potential of a peculiar form: a combination of exponentials with dependence on the barotropic index of the background fluid, arises. Both normal quintessence described by a self-interacting scalar field minimally coupled to gravity and Brans-Dicke quintessence given by a non-minimally coupled scalar field are then analyzed and the relevance of these models for the description of the cosmic evolution are discussed in some detail. The stability of these solutions is also briefly commented on. 
  We consider the head-on collision of two opposite-charged point particles moving at the speed of light. Starting from the field of a single charge we derive in a first step the field generated by uniformly accelerated charge in the limit of infinite acceleration. From this we then calculate explicitly the burst of radiation emitted from the head-on collision of two charges and discuss its distributional structure. The motivation for our investigation comes from the corresponding gravitational situation where the head-on collision of two ultrarelativistic particles (black holes) has recently aroused renewed interest. 
  It is done by introducing of an additional term proportional to the interior energy into the standard thermodynamic uncertainty relation that leads to existence of the lower limit of inverse temperature 
  We present a continuity equation for the gravitational energy-momentum, which is obtained in the framework of the teleparallel equivalent of general relativity. From this equation it follows a general definition for the gravitational energy-momentum flux. This definition is investigated in the context of plane waves and of cylindrical Einstein-Rosen waves. We obtain the well known value for the energy flux of plane gravitational waves, and conclude that the latter exhibit features similar to plane electromagnetic waves. 
  A conjectured connection to quantum gravity has led to a renewed interest in highly damped black hole quasinormal modes (QNMs). In this paper we present simple derivations (based on the WKB approximation) of conditions that determine the asymptotic QNMs for both Schwarzschild and Reissner-Nordstrom black holes. This confirms recent results obtained by  Motl and Neitzke, but our analysis fills several gaps left by their discussion. We study the Reissner-Nordstrom results in some detail, and show that, in contrast to the asymptotic QNMs of a Schwarzschild black hole, the Reissner-Nordstrom QNMs are typically not periodic in the imaginary part of the frequency. This leads to the charged black hole having peculiar properties which complicate an interpretation of the results. 
  A technique is given to generate coupled scalar field solutions in colliding Einstein - Maxwell (EM) waves. By employing the Bell - Szekeres solution as seed and depending on the chosen scalar field it is possible to construct nonsingular solutions. If the original EM solution is already singular addition of scalar fields does not make the physics any better. In particular, scalar field solution that is transformable to spherical symmetry is plagued with singularities. 
  We study the propagation of classical electromagnetic waves on the simplest four-dimensional spherically symmetric metric with a dilaton background field. Solutions to the relevant equations are obtained perturbatively in a parameter which measures the strength of the dilaton field (hence parameterizes the departure from Schwarzschild geometry). The loss of energy from outgoing modes is estimated as a back-scattering process against the dilaton background, which would affect the luminosity of stars with a dilaton field. The radiation emitted by a freely falling point-like source on such a background is also studied by analytical and numerical methods. 
  Within the framework of an exact general relativistic formulation of gluing manifolds, we consider the problem of matching an inhomogeneous overdense region to a Friedmann-Robertson-Walker background universe in the general spherical symmetric case of pressure-free models. It is shown that, in general, the matching is only possible through a thin shell, a fact ignored in the literature. In addition to this, in subhorizon cases where the matching is possible, an intermediate underdense region will necessarily arise. 
  Linearized perturbations of a Schwarzschild black hole are described, for each angular momentum $\ell$, by the well-studied discrete quasinormal modes (QNMs), and in addition a continuum. The latter is characterized by a cut strength $q(\gamma>0)$ for frequencies $\omega = -i\gamma$. We show that: (a) $q(\gamma\downarrow0) \propto \gamma$, (b) $q(\Gamma) = 0$ at $\Gamma = (\ell+2)!/[6(\ell-2)!]$, and (c) $q(\gamma)$ oscillates with period $\sim 1$ ($2M\equiv1$). For $\ell=2$, a pair of QNMs are found beyond the cut on the unphysical sheet very close to $\Gamma$, leading to a large dipole in the Green's function_near_ $\Gamma$. For a source near the horizon and a distant observer, the continuum contribution relative to that of the QNMs is small. 
  We consider a spacetime foam model of the Schwarzschild horizon, where the horizon consists of Planck size black holes. According to our model the entropy of the Schwarzschild black hole is proportional to the area of its event horizon. It is possible to express geometrical arguments to the effect that the constant of proportionality is, in natural units, equal to one quarter. 
  Cosmological equations for homogeneous isotropic models filled by scalar fields and ultrarelativistic matter are investigated in the framework of gauge theories of gravity. Regular inflationary cosmological models of flat, closed and open type with dominating ultrarelativistic matter at a bounce are discussed. It is shown that essential part of inflationary cosmological models has bouncing character. 
  We construct a sequence of binary black hole puncture data derived under the assumptions (i) that the ADM mass of each puncture as measured in the asymptotically flat space at the puncture stays constant along the sequence, and (ii) that the orbits along the sequence are quasi-circular in the sense that several necessary conditions for the existence of a helical Killing vector are satisfied. These conditions are equality of ADM and Komar mass at infinity and equality of the ADM and a rescaled Komar mass at each puncture. In this paper we explicitly give results for the case of an equal mass black hole binary without spin, but our approach can also be applied in the general case. We find that up to numerical accuracy the apparent horizon mass also remains constant along the sequence and that the prediction for the innermost stable circular orbit is similar to what has been found with the effective potential method. 
  A new approach to space-time asymptotics is presented, refining Penrose's idea of conformal transformations with infinity represented by the conformal boundary of space-time. Generalizing examples such as flat and Schwarzschild space-times, it is proposed that the Penrose conformal factor be a product of advanced and retarded conformal factors, which asymptotically relate physical and conformal null (light-like) coordinates and vanish at future and past null infinity respectively, with both vanishing at spatial infinity. A correspondingly refined definition of asymptotic flatness at both spatial and null infinity is given, including that the conformal boundary is locally a light cone, with spatial infinity as the vertex. It is shown how to choose the conformal factors so that this asymptotic light cone is locally a metric light cone. The theory is implemented in the spin-coefficient (or null-tetrad) formalism by a simple joint transformation of the spin-metric and spin-basis (or metric and tetrad). The advanced and retarded conformal factors may be used as expansion parameters near the respective null infinity, together with a dependent expansion parameter for both spatial and null infinity, essentially inverse radius. Asymptotic regularity conditions on the spin-coefficients are proposed, based on the conformal boundary locally being a smoothly embedded metric light cone. These conditions ensure that the Bondi-Sachs energy-flux integrals of ingoing and outgoing gravitational radiation decay at spatial infinity such that the total radiated energy is finite, and that the Bondi-Sachs energy-momentum has a unique limit at spatial infinity, coinciding with the uniquely rendered ADM energy-momentum. 
  Static, spherically symmetric solutions with regular origin are investigated of the Einstein-Yang-Mills theory with a negative cosmological constant $\Lambda$. A combination of numerical and analytical methods leads to a clear picture of the `moduli space' of the solutions. Some issues discussed in the existing literature on the subject are reconsidered and clarified. In particular the stability of the asymptotically AdS solutions is studied. Like for the Bartnik-McKinnon (BK) solutions obtained for $\Lambda=0$ there are two different types of instabilities -- `topological' and `gravitational'. Regions with any number of these instabilities are identified in the moduli space. While for BK solutions there is always a non-vanishing equal number of instabilities of both types, this degeneracy is lifted and there exist stable solutions, genuine sphalerons with exactly one unstable mode and so on. The boundaries of these regions are determined. 
  After a preliminary discussion of the relevance of the field nature of gravitation interaction, both for the fundamental interaction of particles and the topology of space time, a method is proposed to produce and detect a dynamical gravitational field, allowing the determination of the order of magnitude of its propagation velocity. 
  Self-phase modulation of spherical gravitational wavepackets propagating in a flat space-time in the presence of a tenuous distribution of matter is considered. Analogies with respect to similar effects in nonlinear optics are explored. Self phase modulation of waves emitted from a single source can eventually lead to an efficient energy dilution of the gravitational wave energy over an increasingly large spectral range. An explicit criterium for the occurrence of a significant spectral energy dilution is established. 
  Whereas semiclassical gravity is based on the semiclassical Einstein equation with sources given by the expectation value of the stress-energy tensor of quantum fields, stochastic semiclassical gravity is based on the Einstein-Langevin equation, which has in addition sources due to the noise kernel.In the first part, we describe the fundamentals of this new theory via two approaches: the axiomatic and the functional. In the second part, we describe three applications of stochastic gravity theory. First, we consider metric perturbations in a Minkowski spacetime: we compute the two-point correlation functions for the linearized Einstein tensor and for the metric perturbations. Second, we discuss structure formation from the stochastic gravity viewpoint. Third, we discuss the backreaction of Hawking radiation in the gravitational background of a quasi-static black hole. 
  We attribute the gravitational interaction between sources of curvature to the world being a crystal which has undergone a quantum phase transition to a nematic phase by a condensation of dislocations.   The model explains why spacetime has no observable torsion and predicts the existence of curvature sources in the form of world sheets, albeit with different high-energy properties than those of string models. 
  We study traversable Lorentzian wormholes in the three-dimensional low energy string theory by adding some matter source involving a dilaton field. It will be shown that there are two-different types of wormhole solutions such as BTZ and black string wormholes depending on the dilaton backgrounds, respectively. We finally obtain the desirable solutions which confine exotic matter near the throat of wormhole by adjusting NS charge. 
  Recently, it has been reported that a candidate for a quark star may have been observed. In this article, we pay attention to quark stars with radiation radii in the reported range. We calculate nonradial oscillations of $f$-, $w$- and $w_{\rm II}$-modes. Then, we find that the dependence of the $f$-mode quasi-normal frequency on the bag constant and stellar radiation radius is very strong and different from that of the lowest $w_{\rm II}$-mode quasi-normal frequency. Furthermore we deduce a new empirical formula between the $f$-mode frequency of gravitational waves and the parameter of the equation of state for quark stars. The observation of gravitational waves both of the $f$-mode and of the lowest $w_{\rm II}$-mode would provide a powerful probe for the equation of state of quark matter and the properties of quark stars. 
  We prove by explicit construction that there exists a maximal slicing of the Schwarzschild spacetime such that the lapse has zero gradient at the puncture. This boundary condition has been observed to hold in numerical evolutions, but in the past it was not clear whether the numerically obtained maximal slices exist analytically. We show that our analytical result agrees with numerical simulation. Given the analytical form for the lapse, we can derive that at late times the value of the lapse at the event horizon approaches the value ${3/16}\sqrt{3} \approx 0.3248$, justifying the numerical estimate of 0.3 that has been used for black hole excision in numerical simulations. We present our results for the non-extremal Reissner-Nordstr\"om metric, generalizing previous constructions of maximal slices. 
  A dynamical correspondence is established between positively curved, isotropic, perfect fluid cosmologies and quasi-two-dimensional, harmonically trapped Bose-Einstein condensates by mapping the equations of motion for both systems onto the one-dimensional Ermakov system. Parameters that characterize the physical properties of the condensate wavepacket, such as its width, momentum and energy, may be identified with the scale factor, Hubble expansion parameter and energy density of the universe, respectively. Different forms of cosmic matter correspond to different choices for the time-dependent trapping frequency of the condensate. The trapping frequency that mimics a radiation-dominated universe is determined. 
  We have investigated an LRS Bianchi Type I models with bulk viscosity in the cosmological theory based on Lyra's geometry. A new class of exact solutions have been obtained by considering a time-dependent displacement field for a constant value of the deceleration parameter and viscosity coefficient of bulk viscous fluid is assumed to be a power function of mass density. The physical behaviour of the models is also discussed. 
  Some Bianchi type IX viscous fluid cosmological models are investigated. To get a solution a supplementary condition between metric potentials is used. The viscosity coefficient of bulk viscous fluid is assumed to be a power function of mass density whereas the coefficient of shear viscosity is considered as proportional to scale of expansion in the model. The cosmological constant $\Lambda$ is found to be positive and is a decreasing function of time which is supported by results from recent supernova observations. Some physical and geometric properties of the models are also discussed. 
  It is shown that quantum particle detectors are not reliable probes of spacetime structure. In particular, they fail to distinguish between inertial and non-inertial motion in a general spacetime. To prove this, we consider detectors undergoing circular motion in an arbitrary static spherically symmetric spacetime, and give a necessary and sufficient condition for the response function to vanish when the field is in the static vacuum state. By examining two particular cases, we show that there is no relation, in general, between the vanishing of the response function and the fact that the detector motion is, or is not, geodesic. In static asymptotically flat spacetimes, however, all rotating detectors are excited in the static vacuum. Thus, in this particular case the static vacuum appears to be associated with a non-rotating frame. The implications of these results for the equivalence principle are considered. In particular, we discuss how to properly formulate the principle for particle detectors, and show that it is satisfied. 
  We investigate five dimensional Einstein spaces in warped geometries from the point of view of the four dimensional physically relevant Robertson-Walker-Friedman cosmological metric and the Schwarzschild metric. We show that a four-dimensional cosmology with a closed spacelike section and a cosmological constant can be imbedded into five-dimensional flat space-time. 
  An anomalous constant acceleration of (8.7 \pm 1.3) 10^-8 cm.s^-2 directed toward the Sun has been discovered by Anderson et al. in the motion of the Pioneer 10/11 and Galileo spacecrafts. In parallel, the WMAP results have definitively established the existence of a cosmological constant \Lambda=1/ L_U^2, and therefore of an invariant cosmic length-scale L_U=2.85 \pm 0.25 Mpc. We show that the existence of this invariant scale definitively implements Mach's principle in Einstein's theory of general relativity. Then we demonstrate, in the framework of an exact cosmological solution of Einstein's field equations which is valid both locally and globally, that the definition of inertial systems ultimately depends on this length-scale. As a consequence, usual local coordinates are not inertial, so that the motion of a free body is expected to contain an additional constant acceleration a_P = c^2/\sqrt{3}L_U = (5.9 \pm 0.5) 10^-8 cm.s^-2. Such an effect represents a major contribution to the Pioneer acceleration. 
  We treat the gravitational effects of quantum stress tensor fluctuations. An operational approach is adopted in which these fluctuations produce fluctuations in the focusing of a bundle of geodesics. This can be calculated explicitly using the Raychaudhuri equation as a Langevin equation. The physical manifestation of these fluctuations are angular blurring and luminosity fluctuations of the images of distant sources. We give explicit results for the case of a scalar field on a flat background in a thermal state. 
  In this work we study the existence of mechanisms of transition to global chaos in a closed Friedmann-Robertson-Walker universe with a massive conformally coupled scalar field. We propose a complexification of the radius of the universe so that the global dynamics can be understood. We show numerically the existence of heteroclinic connections of the unstable and stable manifolds to periodic orbits associated to the saddle-center equilibrium points. We find two bifurcations which are crucial in creating non-collapsing universes both in the real and imaginary version of the models. The techniques presented here can be employed in any cosmological model. 
  Conformally flat tilted Bianchi type V cosmological models in presence of a bulk viscous fluid and heat flow are investigated. The coefficient of bulk viscosity is assumed to be a power function of mass density. Some physical and geometric aspects of the models are also discussed. 
  We give a detailed description of the constant mean curvature foliations in the Schwarzschild solution; show that the lapse collapses exponentially, and compute the exponent. 
  An explicit CMC Schwarzschildean line element is derived near the critical point of the foliation, the lapse is shown to decay exponentially, and the coefficient of the exponent is calculated. 
  In this paper we find a solution for a quasi-isotropic inflationary Universe which allows to introduce in the problem a certain degree of inhomogeneity. We consider a model which generalize the (flat) FRW one by introducing a first order inhomogeneous term, whose dynamics is induced by an effective cosmological constant. The 3-metric tensor is constituted by a dominant term, corresponding to an isotropic-like component, while the amplitude of the first order one is controlled by a ``small'' function $\eta(t)$.   In a Universe filled with ultrarelativistic matter and a real self-interacting scalar field, we discuss the resulting dynamics, up to first order in $\eta$, when the scalar field performs a slow roll on a plateau of a symmetry breaking configuration and induces an effective cosmological constant.   We show how the spatial distribution of the ultrarelativistic matter and of the scalar field admits an arbitrary form but nevertheless, due to the required inflationary e-folding, it cannot play a serious dynamical role in tracing the process of structures formation (via the Harrison--Zeldovic spectrum). As a consequence, this paper reinforces the idea that the inflationary scenario is incompatible with a classical origin of the large scale structures. 
  Equatorial motion of test particles in the Kerr-de Sitter spacetimes is considered. Circular orbits are determined, their properties are discussed for both the black-hole and naked-singularity spacetimes, and their relevance for thin accretion discs is established. 
  Given a spherical spacelike three-geometry, there exists a very simple algebraic condition which tells us whether, and in which, Schwarzschild solution this geometry can be smoothly embedded. One can use this result to show that any given Schwarzschild solution covers a significant subset of spherical superspace and these subsets form a sequence of nested domains as the Schwarzschild mass increases. This also demonstrates that spherical data offer an immediate counter example to the thick sandwich `theorem'. 
  When applied to a dipole source subjected to acceleration which is violent and long lasting (``extreme acceleration''), Maxwell's equations predict radiative power which augments Larmor's classical radiation formula by a nontrivial amount. The physical assumptions behind this result are made possible by the kinematics of a system of geometrical clocks whose tickings are controlled by cavities which are expanding inertially. For the purpose of measuring the radiation from such a source we take advantage of the physical validity of a spacetime coordinate framework (``inertially expanding frame'') based on such clocks. They are compatible and commensurable with the accelerated clocks of the accelerated source. By contrast, a common Lorentz frame with its mutually static clocks won't do: it lacks that commensurability. Inertially expanding clocks give a physicist a window into the frame of a source with extreme acceleration. He thus can locate that source and measure radiation from it without being subjected to such acceleration himself. The conclusion is that inertially expanding reference frames reveal qualitatively distinct aspects of nature which would not be accessible if static inertial frames were the only admissible frames. 
  I reconsider the problem of the Newtonian limit in nonlinear gravity models in the light of recently proposed models with inverse powers of the Ricci scalar.   Expansion around a maximally symmetric local background with positive curvature scalar R_0 gives the correct Newtonian limit on length scales << R_0^{-1/2} if the gravitational Lagrangian f(R) satisfies |f(R_0)f''(R_0)|<< 1. I propose two models with f''(R_0)=0. 
  We obtain a new self-similar solution to the Einstein's equations in four-dimensions, representing the collapse of a spherically symmetric, minimally coupled, massless, scalar field. Depending on the value of certain parameters, this solution represents the formation of naked singularities and black holes. Since the black holes are identified as the Schwarzschild ones, one may naturally see how these black holes are produced as remnants of the scalar field collapse. 
  A method to evaluate spin networks for (2+1)-dimensional quantum gravity is given. We analyse the evaluation of spin networks for Lorentzian, Euclidean and a new limiting case of Newtonian quantum gravity. Particular attention is paid to the tetrahedron and to the study of its asymptotics. Moreover, we propose that all this technique can be extended to spin networks for quantum gravity in any dimension. 
  Solving the 4-d Einstein equations as evolution in time requires solving equations of two types: the four elliptic initial data (constraint) equations, followed by the six second order evolution equations. Analytically the constraint equations remain solved under the action of the evolution, and one approach is to simply monitor them ({\it unconstrained} evolution).   The problem of the 3-d computational simulation of even a single isolated vacuum black hole has proven to be remarkably difficult. Recently, we have become aware of two publications that describe very long term evolution, at least for single isolated black holes. An essential feature in each of these results is {\it constraint subtraction}. Additionally, each of these approaches is based on what we call "modern," hyperbolic formulations of the Einstein equations. It is generally assumed, based on computational experience, that the use of such modern formulations is essential for long-term black hole stability. We report here on comparable lifetime results based on the much simpler ("traditional") $\dot g$ - $\dot K$ formulation.   We have also carried out a series of {\it constrained} 3-d evolutions of single isolated black holes. We find that constraint solution can produce substantially stabilized long-term single hole evolutions. However, we have found that for large domains, neither constraint-subtracted nor constrained $\dot g$ - $\dot K$ evolutions carried out in Cartesian coordinates admit arbitrarily long-lived simulations. The failure appears to arise from features at the inner excision boundary; the behavior does generally improve with resolution. 
  We consider embedding diagrams for the Reissner-Nordstr\"om spacetime. We embed the $(r-t)$ and $(r-\phi)$ planes into 3-Minkowski/Euclidean space and discuss the relation between the diagrams and the corresponding curvature scalar of the 2-metrics 
  In this work the Quantum and Statistical Mechanics of the Early Universe, i.e. at Planck scale, is considered as a deformation of the well-known theories. In so doing the primary object under deformation in both cases is the density matrix. It is demonstrated that in construction of the deformed quantum mechanical and statistical density matrices referred to as density pro-matrices there is a complete analog. The principal difference lies in a nature of the deformation parameter that is associated with the fundamental length in the first case and with a maximum temperature in the second case. Consideration is also given to some direct consequences, specifically the use of the explicitly specified exponential ansatz in the derivation from the basic principles of a semiclassical Bekenstein-Hawking formula for the black hole entropy and of the high-temperature complement to the canonical Gibbs distribution. 
  The behaviour of magnetic field in anisotropic Bianchi type I cosmological model for bulk viscous distribution is investigated. The distribution consists of an electrically neutral viscous fluid with an infinite electrical conductivity. It is assumed that the component $\sigma^{1}_{1}$ of shear tensor $\sigma^{j}_{i}$ is proportional to expansion ($\theta$) and the coefficient of bulk viscosity is assumed to be a power function of mass density. Some physical and geometrical aspects of the models are also discussed in presence and also in absence of the magnetic field. 
  In this paper we consider the Long-Wavelength Approximation Scheme (LWAS) in the framework of the covariant fluid approach to general relativistic dynamics, specializing to the particular case of irrotational dust matter. We discuss the dynamics of these models during the approach to any spacelike singularity where a BKL-type evolution is expected, studying the validity of this approximation scheme and the role of the magnetic part of the Weyl tensor, $H_{ab}$. Our analytic results confirm a previous numerical analysis: it is $H_{ab}$ that destroys the pure Kasner-like approach to the singularity and eventually produces the bounce to another Kasner phase. Expanding regions evolve as separate universes where inhomogeneities and anisotropies die away. 
  Motivated by novel results in the theory of black-hole quantization, we study {\it analytically} the quasinormal modes (QNM) of ({\it rotating}) Kerr black holes. The black-hole oscillation frequencies tend to the asymptotic value $\omega_n=m\Omega+i2\pi T_{BH}n$ in the $n \to \infty$ limit. This simple formula is in agreement with Bohr's correspondence principle. Possible implications of this result to the area spectrum of quantum black holes are discussed. 
  In a 5-dimensional spacetime ($M,g_{ab}$) with a Killing vector field $\xi ^a$ which is either everywhere timelike or everywhere spacelike, the collection of all trajectories of $\xi ^a$ gives a 4-dimensional space $S$. The reduction of ($M,g_{ab}$) is studied in the geometric language, which is a generalization of Geroch's method for the reduction of 4-dimensional spacetime. A 4-dimensional gravity coupled to a vector field and a scalar field on $S$ is obtained by the reduction of vacuum Einstein's equations on $M$, which gives also an alternative description of the 5-dimensional Kaluza-Klein theory. Besides the symmetry-reduced action from the Hilbert action on $M$, an alternative action of the fields on $S$ is also obtained, the variations of which lead to the same fields equations as those reduced from the vacuum Einstein equation on $M$. 
  We consider global properties of gravitomagnetism by investigating the gravitomagnetic field of a rotating cosmic string. We show that although the gravitomagnetic field produced by such a configuration of matter vanishes locally, it can be detected globally. In this context we discuss the gravitational analogue of the Aharonov-Bohm effect. 
  We consider a possible (parity conserving) interaction between the electromagnetic field $F$ and a torsion field $T^\alpha$ of spacetime. For generic elementary torsion, gauge invariant coupling terms of lowest order fall into two classes that are both nonminimal and {\it quadratic} in torsion. These two classes are displayed explicitly. The first class of the type $\sim F T^2$ yields (undesirable) modifications of the Maxwell equations. The second class of the type $\sim F^2 T^2$ doesn't touch the Maxwell equations but rather modifies the constitutive tensor of spacetime. Such a modification can be completely described in the framework of metricfree electrodynamics. We recognize three physical effects generated by the torsion: (i) An axion field that induces an {\em optical activity} into spacetime, (ii) a modification of the light cone structure that yields {\em birefringence} of the vacuum, and (iii) a torsion dependence of the {\em velocity of light.} We study these effects in the background of a Friedmann universe with torsion. {\it File tor17.tex, 02 August 2003} 
  The gravitational collapse of a high-density null charged matter fluid, satisfying the Hagedorn equation of state, is considered in the framework of the Vaidya geometry. The general solution of the gravitational field equations can be obtained in an exact parametric form. The conditions for the formation of a naked singularity, as a result of the collapse of the compact object, are also investigated. For an appropriate choice of the arbitrary integration functions the null radial outgoing geodesic, originating from the shell focussing central singularity, admits one or more positive roots. Hence a collapsing Hagedorn fluid could end either as a black hole, or as a naked singularity. A possible astrophysical application of the model, to describe the energy source of gamma-ray bursts, is also considered. 
  We prove, for the relativistic Boltzmann equation in the homogeneous case, on the Minkowski space-time, a global in time existence and uniqueness theorem. The method we develop extends to the cases of some curved space-times such as the flat Robertson-Walker space-time and some Bianchi type I space-times. 
  We show that the families of effective actions considered by Jacobson et al. to study Lorentz invariance violations contain a class of models that represent pure General Relativity with Euclidean signature. We also point out that some members of this family of actions preserve Lorentz invariance in a generalized sense. 
  The general-covariant Z4 formalism is further analyzed. The gauge conditions are generalized with a view to Numerical Relativity applications and the conditions for obtaining strongly hyperbolic evolution systems are given both at the first and the second order levels. A symmetry-breaking mechanism is proposed that allows one, when applied in a partial way, to recover previously proposed strongly hyperbolic formalisms, like the BSSN and the Bona-Mass\'o ones. When applied in its full form, the symmetry breaking mechanism allows one to recover the full five-parameter family of first order KST systems. Numerical codes based in the proposed formalisms are tested. A robust stability test is provided by evolving random noise data around Minkowski space-time. A strong field test is provided by the collapse of a periodic background of plane gravitational waves, as described by the Gowdy metric. 
  We investigate the dependence of the gravitational wave spectrum from quintessential inflation on the reheating process. We consider two extreme reheating processes. One is the gravitational reheating by particle creation in the expanding universe in which the beginning of the radiation dominated epoch is delayed due to the presence of the epoch of domination of the kinetic energy of the inflaton (kination). The other is the instant preheating considered by Felder et al. in which the Universe becomes radiation dominated soon after the end of inflation. We find that the spectrum of the gravitational waves at $\sim 100$ MHz is quite sensitive to the reheating process. This result is not limited to quintessential inflation but applicable to various inflation models. Conversely, the detection or non-detection of primordial gravitational waves at $\sim$100 MHz would provide useful information regarding the reheating process in inflation. 
  In a recent work we presented a reformulation of the canonical quantum gravity, based on adding the so-called kinematical term to the gravity-matter action; this revised approach leads to a self-consistent canonical quantization of the 3-geometries, referred to the external time as provided via the added term.   Here, we show how the kinematical term can be interpreted in terms of a non relativistic dust fluid which plies the role of a ``real clock' for the quantum gravity theory, and, in the WKB limit of a cosmological problem, makes account for a dark matter component which, at present time, could play a dynamical role. 
  We study covariant entropy bounds in dynamical spacetimes with naked singularities. Specifically we study a spherically symmetric massless scalar field solution. The solution is an inhomogeneous cosmology with an initial spacelike singularity, and a naked timelike singularity at the origin. We construct the entropy flux 4-vector for the scalar field, and show by explicit computation that the generalized covariant bound $S_{L(B,B')}\le (A(B)-A(B'))/4 $ is violated for light sheets $L(B,B')$ in the neighbourhood of the (evolving) apparent horizon. We find no violations of the Bousso bound (for which $A(B')=0$), even though certain sufficient conditions for this bound do not hold. This result therefore shows that these conditions are not necessary. 
  Four-dimensional cylindrically symmetric spacetimes with homothetic self-similarity are studied in the context of Einstein's Theory of Gravity, and a class of exact solutions to the Einstein-massless scalar field equations is found. Their local and global properties are investigated and found that they represent gravitational collapse of a massless scalar field. In some cases the collapse forms black holes with cylindrical symmetry, while in the other cases it does not. The linear perturbations of these solutions are also studied and given in closed form. From the spectra of the unstable eigen-modes, it is found that there exists one solution that has precisely one unstable mode, which may represent a critical solution, sitting on a boundary that separates two different basins of attraction in the phase space. 
  Lensing in a spherically symmetric and static spacetime is considered, based on the lightlike geodesic equation without approximations. After fixing two radius values r_O and r_S, lensing for an observation event somewhere at r_O and static light sources distributed at r_S is coded in a lens equation that is explicitly given in terms of integrals over the metric coefficients. The lens equation relates two angle variables and can be easily plotted if the metric coefficients have been specified; this allows to visualize in a convenient way all relevant lensing properties, giving image positions, apparent brightnesses, image distortions, etc. Two examples are treated: Lensing by a Barriola-Vilenkin monopole and lensing by an Ellis wormhole. 
  We define conserved gravitational charges in -cosmologically extended- topologically massive gravity, exhibit them in surface integral form about their de-Sitter or flat vacua and verify their correctness in terms of two basic types of solution. 
  We study the stability of a spherically symmetric black hole with a global monopole hair. Asymptotically the spacetime is flat but has a deficit solid angle which depends on the vacuum expectation value of the scalar field. When the vacuum expectation value is larger than a certain critical value, this spacetime has a cosmological event horizon. We investigate the stability of these solutions against the spherical and polar perturbations and confirm that the global monopole hair is stable in both cases. Although we consider some particular modes in the polar case, our analysis suggests the conservation of the "topological charge" in the presence of the event horizons and violation of black hole no-hair conjecture in asymptotically non-flat spacetime. 
  We investigate gravitating lumps with a false vacuum core surrounded by the true vacuum in a scalar field potential. Such configurations become possible in the Einstein gravity in the presence of fermions at the core. Gravitational interactions as well as Yukawa interactions are essential for such lumps to exist. The mass and size of gravitating lumps sensitively depend on the scale characterizing the scalar field potential and the density of fermions. These objects can exist in the universe at various scales. 
  We study the head-on collision of linearly polarized, high frequency plane gravitational waves and their electromagnetic counterparts in the Einstein-Maxwell theory. The post-collision space-times are obtained by solving the vacuum Einstein-Maxwell field equations in the geometrical optics approximation. The head-on collisions of all possible pairs of these systems of waves is described and the results are then generalised to non-linearly polarized waves which exhibit the maximum two degrees of freedom of polarization. 
  A model for the static weak-field macroscopic medium is analyzed and the equation for the macroscopic gravitational potential is derived. This is a biharmonic equation which is a non-trivial generalization of the Poisson equation of Newtonian gravity. In case of the strong gravitational quadrupole polarization it essentially holds inside a macroscopic matter source. Outside the source the gravitational potential fades away exponentially. The equation is equivalent to a system of the Poisson equation and the nonhomogeneous modified Helmholtz equations. The general solution to this system is obtained by using Green's function method and it does not have a limit to Newtonian gravity. In case of the insignificant gravitational quadrupole polarization the equation for macroscopic gravitational potential becomes the Poisson equation with the matter density renormalized by the factor including the value of the quadrupole gravitational polarization of the source. The general solution to this equation obtained by using Green's function method has a limit to Newtonian gravity. 
  We show that topological 3D gravity with torsion can be formulated as a Chern-Simons gauge theory, provided a specific parameter, known as the effective cosmological constant, is negative. In that case, the boundary dynamics of the theory corresponding to anti-de Sitter boundary conditions is described by a conformal field theory with two different central charges. 
  The classical unified theory of Weyl is revisited. The possibility of stable extended electron model in the Einstein-Weyl space is suggested. 
  LISA is an array of three spacecraft in an approximately equilateral triangle configuration which will be used as a low-frequency gravitational wave detector. We present here new generalizations of the Michelson- and Sagnac-type time-delay interferometry data combinations. These combinations cancel laser phase noise in the presence of different up and down propagation delays in each arm of the array, and slowly varying systematic motion of the spacecraft. The gravitational wave sensitivities of these generalized combinations are the same as previously computed for the stationary cases, although the combinations are now more complicated. We introduce a diagrammatic representation to illustrate that these combinations are actually synthesized equal-arm interferometers. 
  We examine the global structure of scalar field critical collapse spacetimes using a characteristic double-null code. It can integrate past the horizon without any coordinate problems, due to the careful choice of constraint equations used in the evolution. The limiting sequence of sub- and supercritical spacetimes presents an apparent paradox in the expected Penrose diagrams, which we address in this paper. We argue that the limiting spacetime converges pointwise to a unique limit for all r>0, but not uniformly. The r=0 line is different in the two limits. We interpret that the two different Penrose diagrams differ by a discontinuous gauge transformation. We conclude that the limiting spacetime possesses a singular event, with a future removable naked singularity. 
  We propose a new formulation for 3+1 numerical relativity, based on a constrained scheme and a generalization of Dirac gauge to spherical coordinates. This is made possible thanks to the introduction of a flat 3-metric on the spatial hypersurfaces t=const, which corresponds to the asymptotic structure of the physical 3-metric induced by the spacetime metric. Thanks to the joint use of Dirac gauge, maximal slicing and spherical components of tensor fields, the ten Einstein equations are reduced to a system of five quasi-linear elliptic equations (including the Hamiltonian and momentum constraints) coupled to two quasi-linear scalar wave equations. The remaining three degrees of freedom are fixed by the Dirac gauge. Indeed this gauge allows a direct computation of the spherical components of the conformal metric from the two scalar potentials which obey the wave equations. We present some numerical evolution of 3-D gravitational wave spacetimes which demonstrates the stability of the proposed scheme. 
  The dynamics of physical theories is usually described by differential equations. Difference equations then appear mainly as an approximation which can be used for a numerical analysis. As such, they have to fulfill certain conditions to ensure that the numerical solutions can reliably be used as approximations to solutions of the differential equation. There are, however, also systems where a difference equation is deemed to be fundamental, mainly in the context of quantum gravity. Since difference equations in general are harder to solve analytically than differential equations, it can be helpful to introduce an approximating differential equation as a continuum approximation. In this paper implications of this change in view point are analyzed to derive the conditions that the difference equation should satisfy. The difference equation in such a situation cannot be chosen freely but must be derived from a fundamental theory. Thus, the conditions for a discrete formulation can be translated into conditions for acceptable quantizations. In the main example, loop quantum cosmology, we show that the conditions are restrictive and serve as a selection criterion among possible quantization choices. 
  The notion of standard positive probability distribution function (tomogram) which describes the quantum state of universe alternatively to wave function or to density matrix is introduced. Connection of the tomographic probability distribution with the Wigner function of the universe and with the star-product (deformation) quantization procedure is established.   Using the Radon transform the Wheeler-De Witt generic equation for the probability function is written in tomographic form. Some examples of the Wheeler-DeWitt equation in the minisuperspace are elaborated explicitly for a homogeneous isotropic cosmological models. Some interpretational aspects of the probability description of the quantum state are discussed. 
  Electromagnetic methods recently proposed for detecting gravitational waves modify the Michelson phase shift analysis (historically employed for special relativity). We suggest that a frequency modulation analysis is more suited to general relativity. An incident photon in the presence of a very long wavelength gravitational wave will have a finite probability of being returned as a final photon with a frequency shift whose magnitude is equal to the gravitational wave frequency. The effect is due to the non-linear coupling between electromagnetic and gravitational waves. The frequency modulation is derived directly from the Maxwell-Einstein equations. 
  The occurrence of a big smash singularity which ends the universe in a finite time in the future is investigated in the context of superquintessence, i.e. dark energy with effective equation of state parameter w<-1. The simplest model of superquintessence based on a single nonminimally coupled scalar field exhibits big smash solutions which are attractors in phase space. 
  We study the mechanics of Hayward's trapping horizons, taking isolated horizons as equilibrium states. Zeroth and second laws of dynamic horizon mechanics come from the isolated and trapping horizon formalisms respectively. We derive a dynamical first law by introducing a new perturbative formulation for dynamic horizons in which "slowly evolving" trapping horizons may be viewed as perturbatively non-isolated. 
  We analyse exhaustively the structure of \emph{non-degenerate} Cauchy horizons in Gowdy space-times, and we establish existence of a large class of non-polarized Gowdy space-times with such horizons.   Added in proof: Our results here, together with deep new results of H. Ringstr\"om (talk at the Miami Waves conference, January 2004), establish strong cosmic censorship in (toroidal) Gowdy space-times. 
  The spectrum of multiple level transitions of the quantum black hole is considered, and the line widths calculated. Initial evidence is found for these higher order transitions in the spectrum of quasinormal modes for Schwarzschild and Kerr black holes, further bolstering the idea that there exists a correspondence principle between quantum transitions and classical ``ringing modes''. Several puzzles are noted, including a fine-tuning problem between the line width and the level degeneracy. A more general explanation is provided for why setting the Immirzi parameter of loop quantum gravity from the black hole spectrum necessarily gives the correct value for the black hole entropy. 
  Two popular attempts to understand the quantum physics of gravitation are critically assessed. The talk on which this paper is based was intended for a general particle-physics audience. 
  The configuration space of general relativity is superspace - the space of all Riemannian 3-metrics modulo diffeomorphisms. However, it has been argued that the configuration space for gravity should be conformal superspace - the space of all Riemannian 3-metrics modulo diffeomorphisms and conformal transformations. Recently a manifestly 3-dimensional theory was constructed with conformal superspace as the configuration space. Here a fully 4-dimensional action is constructed so as to be invariant under conformal transformations of the 4-metric using general relativity as a guide. This action is then decomposed to a (3+1)-dimensional form and from this to its Jacobi form. The surprising thing is that the new theory turns out to be precisely the original 3-dimensional theory. The physical data is identified and used to find the physical representation of the theory. In this representation the theory is extremely similar to general relativity. The clarity of the 4-dimensional picture should prove very useful for comparing the theory with those aspects of general relativity which are usually treated in the 4-dimensional framework. 
  Many authors claimed that a large initial inhomogeneity prevents the onset of inflation and therefore inflation takes place only if the scalar field is homogeneous or appropriately chosen over the horizon scale. We show that those arguments do not apply to topological inflation. The core of a defect starts inflation even if it has much smaller size than the horizon and much larger gradient energy than the potential, as long as the vacuum expectation value is large enough ($\gtrsim0.3\mpl$) and the core is not contracting initially. This is due to stability of false vacuum. 
  It is shown that screening the background of super-strong interacting gravitons ensures the Newtonian attraction, if a part of single gravitons is pairing and graviton pairs are destructed by collisions with a body. If the considered quantum mechanism of classical gravity is realized in the nature, than an existence of black holes contradicts to the equivalence principle. In such the model, Newton's constant is proportional to $H^{2}/T^{4},$ where $H$ is the Hubble constant, $T$ is an equivalent temperature of the graviton background. The estimate of the Hubble constant is obtained for the Newtonian limit: $H=3.026 \cdot 10^{-18}  s^{-1}$ (or $94.576  km \cdot s^{-1} \cdot Mpc^{-1}$). 
  Starting from an anisotropic flat cosmological model(Bianchi type $I$), we show that conditions leading to isotropisation fall into 3 classes, respectively 1, 2, 3. We look for necessary conditions such that a Bianchi type $I$ model reaches a stable isotropic state due to the presence of several massive scalar fields minimally coupled to the metric with a perfect fluid for class 1 isotropisation. The conditions are written in terms of some functions $\ell$ of the scalar fields. Two types of theories are studied. The first one deals with scalar tensor theories resulting from extra-dimensions compactification, where the Brans-Dicke coupling functions only depend on their associated scalar fields. The second one is related to the presence of complex scalar fields. We give the metric and potential asymptotical behaviours originating from class 1 isotropisation. The results depend on the domination of the scalar field potential compared to the perfect fluid energy density. We give explicit examples showing that some hybrid inflation theories do not lead to isotropy contrary to some high-order theories, whereas the most common forms of complex scalar fields undergo a class 3 isotropisation, characterised by strong oscillations of the $\ell$ functions. 
  The Einstein-Maxwell equations in D-dimensions admitting (D-3) commuting Killing vector fields have been investigated. The existence of the electric, magnetic and twist potentials have been proved. The system is formulated as the harmonic map coupled to gravity on three-dimensional base space generalizing the Ernst system in the four-dimensional stationary Einstein-Maxwell theory. Some classes of the new exact solutions have been provided, which include the electro-magnetic generalization of the Myers-Perry solution, which describes the rotating black hole immersed in a magnetic universe, and the static charged black ring solution. 
  Quantum Mechanics at Planck scale is considered as a deformation of the conventional Quantum Mechanics. Similar to the earlier works of the author, the main object of deformation is the density matrix. On this basis a notion of the entropy density is introduced that is a matrix value used for a detail study of the Information Problem at the Universe, and in particular, for the Information Paradox Problem. 
  The massless scalar wave propagation in the time-dependent BTZ black hole background has been studied. It is shown that in the quasi-normal ringing both the decay and oscillation time-scales are modified in the time-dependent background. 
  Quantum Weak Energy Inequalities (QWEIs) have been established for a variety of quantum field theories in both flat and curved spacetimes. Dirac fields are known (by a result of Fewster and Verch) to satisfy QWEIs under very general circumstances. However this result does not provide an explicit formula for the QWEI bound, so its magnitude has not previously been determined. In this paper we present a new and explicit QWEI bound for Dirac fields of arbitrary mass in four-dimensional Minkowski space. We follow the methods employed by Fewster and Eveson for the scalar field, modified to take account of anticommutation relations. A key ingredient is an identity for Fourier transforms established by Fewster and Verch. We also compare our QWEI with those previously obtained for scalar and spin-1 fields. 
  We investigate the solutions of Einstein equations such that a hedgehog solution is matched to different exterior or interior solutions via a spherical shell. In the case where both the exterior and the interior regions are hedgehog solutions or one of them is flat, the resulting spherical shell becomes a stringy shell. We also consider more general matchings and see that in this case the shell deviates from its stringy character. 
  Network data analysis methods are the only way to properly separate real gravitational wave (GW) transient events from detector noise. They can be divided into two generic classes: the coincidence method and the coherent analysis. The former uses lists of selected events provided by each interferometer belonging to the network and tries to correlate them in time to identify a physical signal. Instead of this binary treatment of detector outputs (signal present or absent), the latter method involves first the merging of the interferometer data and looks for a common pattern, consistent with an assumed GW waveform and a given source location in the sky. The thresholds are only applied later, to validate or not the hypothesis made. As coherent algorithms use a more complete information than coincidence methods, they are expected to provide better detection performances, but at a higher computational cost. An efficient filter must yield a good compromise between a low false alarm rate (hence triggering on data at a manageable rate) and a high detection efficiency. Therefore, the comparison of the two approaches is achieved using so-called Receiving Operating Characteristics (ROC), giving the relationship between the false alarm rate and the detection efficiency for a given method. This paper investigates this question via Monte-Carlo simulations, using the network model developed in a previous article. 
  Trying to detect the gravitational wave (GW) signal emitted by a type II supernova is a main challenge for the GW community. Indeed, the corresponding waveform is not accurately modeled as the supernova physics is very complex; in addition, all the existing numerical simulations agree on the weakness of the GW emission, thus restraining the number of sources potentially detectable. Consequently, triggering the GW signal with a confidence level high enough to conclude directly to a detection is very difficult, even with the use of a network of interferometric detectors. On the other hand, one can hope to take benefit from the neutrino and optical emissions associated to the supernova explosion, in order to discover and study GW radiation in an event already detected independently. This article aims at presenting some realistic scenarios for the search of the supernova GW bursts, based on the present knowledge of the emitted signals and on the results of network data analysis simulations. Both the direct search and the confirmation of the supernova event are considered. In addition, some physical studies following the discovery of a supernova GW emission are also mentioned: from the absolute neutrino mass to the supernova physics or the black hole signature, the potential spectrum of discoveries is wide. 
  We consider (four dimensional) gravity coupled to a scalar field with potential V(\phi). The potential satisfies the positive energy theorem for solutions that asymptotically tend to a negative local minimum. We show that for a large class of such potentials, there is an open set of smooth initial data that evolve to naked singularities. Hence cosmic censorship does not hold for certain reasonable matter theories in asymptotically anti de Sitter spacetimes. The asymptotically flat case is more subtle. We suspect that potentials with a local Minkowski minimum may similarly lead to violations of cosmic censorship in asymptotically flat spacetimes, but we do not have definite results. 
  We present new classes of exact solutions with noncommutative symmetries constructed in vacuum Einstein gravity (in general, with nonzero cosmological constant), five dimensional (5D) gravity and (anti) de Sitter gauge gravity. Such solutions are generated by anholonomic frame transforms and parametrized by generic off-diagonal metrics. For certain particular cases, the new classes of metrics have explicit limits with Killing symmetries but, in general, they may be characterized by certain anholonomic noncommutative matrix geometries. We argue that different classes of noncommutative symmetries can be induced by exact solutions of the field equations in 'commutative' gravity modeled by a corresponding moving real and complex frame geometry. We analyze two classes of black ellipsoid solutions (in the vacuum case and with cosmological constant) in 4D gravity and construct the analytic extensions of metrics for certain classes of associated frames with complex valued coefficients. The third class of solutions describes 5D wormholes which can be extended to complex metrics in complex gravity models defined by noncommutative geometric structures. The anholonomic noncommutative symmetries of such objects are analyzed. We also present a descriptive account how the Einstein gravity can be related to gauge models of gravity and their noncommutative extensions and discuss such constructions in relation to the Seiberg-Witten map for the gauge gravity. Finally, we consider a formalism of vielbeins deformations subjected to noncommutative symmetries in order to generate solutions for noncommutative gravity models with Moyal (star) product. 
  We investigate the electromagnetic radiation released during the high energy collision of a charged point particle with a four-dimensional Schwarzschild black hole. We show that the spectra is flat, and well described by a classical calculation. We also compare the total electromagnetic and gravitational energies emitted, and find that the former is supressed in relation to the latter for very high energies. These results could apply to the astrophysical world in the case charged stars and small charged black holes are out there colliding into large black holes, and to a very high energy collision experiment in a four-dimensional world. In this latter scenario the calculation is to be used for the moments just after the black hole formation, when the collision of charged debris with the newly formed black hole is certainly expected. Since the calculation is four-dimensional, it does not directly apply to Tev-scale gravity black holes, as these inhabit a world of six to eleven dimensions, although our results should qualitatively hold when extrapolated with some care to higher dimensions. 
  Brief review of principal ideas, estimates and schemes proposed by Russian research groups in respect of the gravitational radiation generated and detected in the laboratory condition is presented. Analysis leads to conclusion that the more promising variant of the laboratory GW-Hertz experiment might be associated with power electromagnetic and acoustical impulsive or shock waves travelling and interacting in nonlinear optic-acoustical medium 
  Thermal fluctuations in the early universe plasma and in very hot astrophysical objects are an unavoidable source of gravitational waves (GW). Differently from previous studies on the subject, we approach this problem using methods based on field theory at finite temperature. Such an approach allows to probe the infrared region of the spectrum where dissipative effects are dominant. Incidentally, this region is the most interesting from the point of view of the detectability perspectives. We find significant deviations from a Planck spectrum. 
  In this paper we compute the holonomies along curves in the gravitational field of a slowly rotating massive body. We use our results to study the gravitational analogue of Aharanov-Bohm effect in this space-time. We also investigate the behaviour of a scalar quantum particle in this space-time and determine Berry's quantum phase acquired by this particle when transported along a closed curve surrounding the body. 
  We study gravitational and electromagnetic radiation generated by uniformly accelerated charged black holes in anti-de Sitter spacetime. This is described by the C-metric exact solution of the Einstein-Maxwell equations with a negative cosmological constant Lambda. We explicitly find and interpret the pattern of radiation that characterizes the dependence of the fields on a null direction from which the (timelike) conformal infinity is approached. This directional pattern exhibits specific properties which are more complicated if compared with recent analogous results obtained for asymptotic behavior of fields near a de Sitter-like infinity. In particular, for large acceleration the anti-de Sitter-like infinity is divided by Killing horizons into several distinct domains with a different structure of principal null directions, in which the patterns of radiation differ. 
  We give a definition of mass for conformally compactifiable initial data sets. The asymptotic conditions are compatible with existence of gravitational radiation, and the compactifications are allowed to be polyhomogeneous. We show that the resulting mass is a geometric invariant, and we prove positivity thereof in the case of a spherical conformal infinity. When R(g) - or, equivalently, the trace of the extrinsic curvature tensor - tends to a negative constant to order one at infinity, the definition is expressed purely in terms of three-dimensional or two-dimensional objects. 
  We improved the approximation of the model of the wormhole generated physical universe constructed by Choudhury and Pendharkar. We show here that the negative pressure of the unphysical wormhole can generate the right physical condition to explain the correct sign of the deacceleration parameter $q_0$. The negative sign ofsign of $q_0$ implies an accelerating universe. 
  We illustrate how form-invariance transformations can be used for constructing phantom cosmologies from standard scalar field universes. First, we discuss how to relate two flat Friedmann-Robertson-Walker cosmologies with different barotropic indexes $\gamma$ and $\bar \gamma$. Then, we consider the particular case $\bar \gamma=-\gamma$, and we show that if the matter content is interpreted in terms of self-interacting scalar fields, then the corresponding transformation provides the link between a standard and a phantom cosmology. After that, we illustrate the method by considering models with exponential potentials. Finally, we also show that the mentioned duality persists even if the typical braneworld modifications to the Friedmann equation are considered. 
  We provide a realisation of a singularity-free inflationary universe in the form of a simple cosmological model dominated at early times by a single minimally coupled scalar field with a physically based potential. The universe starts asymptotically from an initial Einstein static state, which may be large enough to avoid the quantum gravity regime. It enters an expanding phase that leads to inflation followed by reheating and a standard hot Big Bang evolution. We discuss the basic characteristics of this Emergent model and show that none is at odds with current observations. 
  We present a method for the introduction of small-scale structure into strings constructed from products of rotation matrices. We use this method to illustrate a range of possibilities for the shape of cusps that depends on the properties of the small-scale structure. We further argue that the presence of structure at cusps under most circumstances leads to the formation of loops at the size of the smallest scales. On the other hand we show that the gravitational waveform of a cusp remains generally unchanged; the primary effect of small-scale structure is to smooth out the sharp waveform emitted in the direction of cusp motion. 
  We study a uniform and isotropic cosmology with a decaying vacuum energy density, in the realm of a model with a time varying gravitational "constant". We show that, for late times, such a cosmology is in accordance with the observed values of the cosmological parameters. In particular, we can obtain the observed ratio between the matter density and the total energy density, with no necessity of any fine tuning. 
  It has been known that a B=2 skyrmion is axially symmetric. We consider the Skyrme model coupled to gravity and obtain static axially symmetric black hole solutions numerically. The black hole skyrmion no longer has integer baryonic charge but has fractional charge outside the horizon as in the spherically symmetric case. Therefore, the solution represents a black hole partially swallowing a deuteron. Recent studies of theories with large extra dimensions suggest an effective Planck scale of order a TeV and thus the deuteron black hole may be produced in the Linear Hadron Collider (LHC) in future. 
  Matter collineations of spherically Symmetric Lorentzian Manifolds are considered. These are investigated when the energy-momentum tensor is non-degenerate and also when it is degenerate. We have classified spacetimes admitting higher symmetries and spacetimes admitting SO(3) as the maximal isometry group. For the non-degenerate case, we obtain either {\it four}, {\it six}, {\it seven} or {\it ten} independent matter collineations in which {\it four} are isometries and the rest are proper. The results of the previous paper [1] are recovered as a special case. It is worth noting that we have also obtained two cases where the energy-momentum tensor is degenerate but the group of matter collineations is finite-dimensional, i.e. {\it four} or {\it ten}. 
  We construct asymptotically Euclidean solutions of the vacuum Einstein constraint equations with an apparent horizon boundary condition. Specifically, we give sufficient conditions for the constant mean curvature conformal method to generate such solutions. The method of proof is based on the barrier method used by Isenberg for compact manifolds without boundary, suitably extended to accommodate semilinear boundary conditions and low regularity metrics. As a consequence of our results for manifolds with boundary, we also obtain improvements to the theory of the constraint equations on asymptotically Euclidean manifolds without boundary. 
  We consider a local formalism in quantum field theory, in which no reference is made to infinitely extended spacial surfaces, infinite past or infinite future. This can be obtained in terms of a functional W[f,S] of the field f on a closed 3d surface S that bounds a finite region R of Minkowski spacetime. The dependence of W on S is governed by a local covariant generalization of the Schroedinger equation. Particles' scattering amplitudes that describe experiments conducted in the finite region R --the lab during a finite time-- can be expressed in terms of W. The dependence of W on the geometry of S expresses the dependence of the transition amplitudes on the relative location of the particle detectors. In a gravitational theory, background independence implies that W is independent from S. However, the detectors' relative location is still coded in the argument of W, because the geometry of the boundary surface is determined by the boundary value f of the gravitational field. This observation clarifies the physical meaning of the functional W defined by non perturbative formulations of quantum gravity, such as the spinfoam formalism. In particular, it suggests a way to derive particles' scattering amplitudes from a spinfoam model. In particular, we discuss the notion of vacuum in a generally covariant context. We distinguish the nonperturbative vacuum |0_S>, which codes the dynamics, from the Minkowski vacuum |0_M>, which is the state with no particles and is recovered by taking appropriate large values of the boundary metric. We derive a relation between the two vacuum states. We propose an explicit expression for computing the Minkowski vacuum from a spinfoam model. 
  Testing the effects predicted by the General Theory of Relativity, in its linearized weak field and slow motion approximation, in the Solar System is difficult because they are very small. Among them the post-Newtonian gravitomagnetic Lense-Thirring effect, or dragging of the inertial frames, on the orbital motion of a test particle is very interesting and, up to now, there is not yet an undisputable experimental direct test of it. Here we illustrate how it could be possible to measure it with an accuracy of the order of 1%, together with other tests of Special Relativity and post-Newtonian gravity, with a joint space based OPTIS/LARES mission in the gravitational field of Earth. Up to now, the data analysis of the orbits of the existing geodetic LAGEOS and LAGEOS II satellites has yielded a test of the Lense-Thirring effect with a claimed accuracy of 20%-30%. 
  Resonant gravitational wave detectors with an observation bandwidth of tens of hertz are a reality: the antenna Explorer, operated at CERN by the ROG collaboration, has been upgraded with a new read-out. In this new configuration, it exhibits an unprecedented useful bandwidth: in over 55 Hz about its frequency of operation of 919 Hz the spectral sensitivity is better than 10^{-20} /sqrt(Hz) . We describe the detector and its sensitivity and discuss the foreseable upgrades to even larger bandwidths. 
  The Newman-Penrose equations for spacetimes having one spacelike Killing vector are reduced -- in a geometrically defined "canonical frame'' -- to a minimal set, and its differential structure is studied. Expressions for the frame vectors in an arbitrary coordinate basis are given, and coordinate-independent choices of the metric functions are suggested which make the components of the Ricci tensor in the direction of the Killing vector vanish. 
  We report on the effects of an electrical charge on mechanical loss of a fused silica disk. A degradation of Q was seen that correlated with charge on the surface of the sample. We examine a number of models for charge damping, including eddy current damping and loss due to polarization. We conclude that rubbing friction between the sample and a piece of dust attracted by the charged sample is the most likely explanation for the observed loss. 
  We consider static, cylindrically symmetric configurations in general relativity coupled to nonlinear electrodynamics (NED) with an arbitrary gauge-invariant Lagrangian of the form $L_{em}= \Phi(F)$, $F =F_{mn}F^{mn}$. We study electric and magnetic fields with three possible orientations: radial (R), longitudinal (L) and azimuthal (A), and try to find solitonic stringlike solutions, having a regular axis and a flat metric at large $r$, with a possible angular defect. Assuming the function $\Phi(F)$ to be regular at small $F$, it is shown that a regular axis is impossible in R-fields if there is a nonzero effective electric charge and in A-fields if there is a nonzero effective electric current along the axis. Solitonic solutions are only possible for purely magnetic R-fields and purely electric A-fields, in cases when $\Phi(F)$ tends to a finite limit at large $F$. For both R- and A-fields, the desired large $r$ asymptotic is only possible with a non- Maxwell behaviour of $\Phi(F)$ at small $F$. For L-fields, solutions with a regular axis are easily obtained (and can be found by quadratures) whereas a desired large $r$ asymptotic is only possible in an exceptional solution; the latter gives rise to solitonic configurations in case $\Phi(F) = \const \cdot \sqrt{F}$. We give an explicit example of such a solution. 
  We consider the evolution of the orbit of a spinning compact object in a quasi-circular, planar orbit around a Schwarzschild black hole in the extreme mass ratio limit. We compare the contributions to the orbital evolution of both spin-orbit coupling and the local self force. Making assumptions on the behavior of the forces, we suggest that the decay of the orbit is dominated by radiation reaction, and that the conservative effect is typically dominated by the spin force. We propose that a reasonable approximation for the gravitational waveform can be obtained by ignoring the local self force, for adjusted values of the parameters of the system. We argue that this approximation will only introduce small errors in the astronomical determination of these parameters. 
  We present a characterization of general gravitational and electromagnetic fields near de Sitter-like conformal infinity which supplements the standard peeling behavior. This is based on an explicit evaluation of the dependence of the radiative component of the fields on the null direction from which infinity is approached. It is shown that the directional pattern of radiation has a universal character that is determined by the algebraic (Petrov) type of the spacetime. Specifically, the radiation field vanishes along directions opposite to principal null directions. 
  Kruskal's extension solves the problem of the arrow of time of the ``Schwarzschild solution'' through combining two Hilbert manifolds by a singular coordinate transformation. We discuss the implications for the singularity problem and the definition of the mass point.   The analogy set by Rindler between the Kruskal metric and the Minkowski spacetime is investigated anew. The question is answered, whether this analogy is limited to a similarity of the chosen "Bildr\"aume'', or can be given a deeper, intrinsic meaning. The conclusion is reached by observing a usually neglected difference: the left and right quadrants of Kruskal's metric are endowed with worldlines of absolute rest, uniquely defined through each event by the manifold itself, while such worldlines obviously do not exist in the Minkowski spacetime. 
  We present a general formalism for describing singular hypersurfaces in the Einstein theory of gravitation with a Gauss--Bonnet term. The junction conditions are given in a form which is valid for the most general embedding and matter content and for coordinates chosen independently on each side of the hypersurface. The theory is applied to both a time--like and a light--like hypersurface in brane--cosmology. 
  We discuss finite difference techniques for hyperbolic equations in non-trivial domains, as those that arise when simulating black hole spacetimes. In particular, we construct dissipative and difference operators that satisfy the {\it summation by parts} property in domains with excised multiple cubic regions. This property can be used to derive semi-discrete energy estimates for the associated initial-boundary value problem which in turn can be used to prove numerical stability. 
  It is expected that the realization of a convergent and long-term stable numerical code for the simulation of a black hole inspiral collision will depend greatly upon the construction of stable algorithms capable of handling smooth and, most likely, time dependent boundaries. After deriving single grid, energy conserving discretizations for axisymmetric systems containing the axis of symmetry, we present a new excision method for moving black holes using multiple overlapping coordinate patches, such that each boundary is fixed with respect to at least one coordinate system. This multiple coordinate structure eliminates all need for extrapolation, a commonly used procedure for moving boundaries in numerical relativity.   We demonstrate this excision method by evolving a massless Klein-Gordon scalar field around a boosted Schwarzschild black hole in axisymmetry. The excision boundary is defined by a spherical coordinate system co-moving with the black hole. Our numerical experiments indicate that arbitrarily high boost velocities can be used without observing any sign of instability. 
  Trapped surfaces are studied as inner boundary for the Einstein vacuum constraint equations. The trapped surface condition can be written as a non linear boundary condition for these equations. Under appropriate assumptions, we prove existence and uniqueness of solutions in the exterior region for this boundary value problem. We also discuss the relevance of this result for the study of black holes collisions. 
  This paper was withdrawn at the recommendation of the Cassini Radio Science Team. 
  We study constrained Hamiltonian systems by utilizing general forms of time discretization. We show that for explicit discretizations, the requirement of preserving the canonical Poisson bracket under discrete evolution imposes strong conditions on both allowable discretizations and Hamiltonians. These conditions permit time discretizations for a limited class of Hamiltonians, which does not include homogeneous cosmological models. We also present two general classes of implicit discretizations which preserve Poisson brackets for any Hamiltonian. Both types of discretizations generically do not preserve first class constraint algebras. Using this observation, we show that time discretization provides a complicated time gauge fixing for quantum gravity models, which may be compared with the alternative procedure of gauge fixing before discretization. 
  We investigate here how the shearing effects present within a collapsing matter cloud influence the outcome of gravitational collapse in terms of formation of either a black hole or a naked singularity as the final end state. For collapse of practically all physically reasonable matter fields, we prove that it would always end up in a black hole if it is either shear-free or has homogeneous density. Thus it follows that whenever a naked singularity forms as end product, the collapsing cloud must necessarily be shearing with inhomogeneous density. Our consideration brings out the physical forces at work which could cause a naked singularity to result as collapse end state, rather than a black hole. 
  The Hamiltonian dynamics of two-component spherically symmetric null dust is studied with regard to the quantum theory of gravitational collapse. The components--the ingoing and outgoing dusts--are assumed to interact only through gravitation. Different kinds of singularities, naked or "clothed", that can form during collapse processes are described. The general canonical formulation of the one-component null-dust dynamics by Bicak and Kuchar is restricted to the spherically symmetric case and used to construct an action for the two components. The transformation from a metric variable to the quasilocal mass is shown to simplify the mathematics. The action is reduced by a choice of gauge and the corresponding true Hamiltonian is written down. Asymptotic coordinates and energy densities of dust shells are shown to form a complete set of Dirac observables. The action of the asymptotic time translation on the observables is defined but it has been calculated explicitly only in the case of one-component dust (Vaidya metric). 
  We present a construction of the Hubble operator for the spatially flat isotropic loop quantum cosmology. This operator is a Dirac observable on a subspace of the space of physical solutions. This subspace gets selected dynamically, requiring that its action be invariant on the physical solution space. As a simple illustrative application of the expectation value of the operator, we do find a generic phase of (super)inflation, a feature shown by Bojowald from the analysis of effective Friedmann equation of loop quantum cosmology. 
  We study and look for similarities between the response rates $R^{\rm dS}(a_0, \Lambda)$ and $R^{\rm SdS}(a_0, \Lambda, M)$ of a static scalar source with constant proper acceleration $a_0$ interacting with a massless, conformally coupled Klein-Gordon field in (i) deSitter spacetime, in the Euclidean vacuum, which describes a thermal flux of radiation emanating from the deSitter cosmological horizon, and in (ii) Schwarzschild-deSitter spacetime, in the Gibbons-Hawking vacuum, which describes thermal fluxes of radiation emanating from both the hole and the cosmological horizons, respectively, where $\Lambda$ is the cosmological constant and $M$ is the black hole mass. After performing the field quantization in each of the above spacetimes, we obtain the response rates at the tree level in terms of an infinite sum of zero-energy field modes possessing all possible angular momentum quantum numbers. In the case of deSitter spacetime, this formula is worked out and a closed, analytical form is obtained. In the case of Schwarzschild-deSitter spacetime such a closed formula could not be obtained, and a numerical analysis is performed. We conclude, in particular, that $R^{\rm dS}(a_0, \Lambda)$ and $R^{\rm SdS}(a_0, \Lambda, M)$ do not coincide in general, but tend to each other when $\Lambda \to 0$ or $a_0 \to \infty$. Our results are also contrasted and shown to agree (in the proper limits) with related ones in the literature. 
  Gamma-ray bursts are believed to originate in core-collapse of massive stars.   This produces an active nucleus containing a rapidly rotating Kerr black hole surrounded by a uniformly magnetized torus represented by two counter-oriented current rings. We quantify black hole spin-interactions with the torus and charged particles along open magnetic flux-tubes subtended by the event horizon. A major output of Egw=4e53 erg is radiated in gravitational waves of frequency fgw=500 Hz by a quadrupole mass-moment in the torus. Consistent with GRB-SNe, we find (i) Ts=90s (tens of s, Kouveliotou et al. 1993), (ii) aspherical SNe of kinetic energy Esn=2e51 erg (2e51 erg in SN1998bw, Hoeflich et al. 1999) and (iii) GRB-energies Egamma=2e50 erg (3e50erg in Frail et al. 2001). GRB-SNe occur perhaps about once a year within D=100Mpc. Correlating LIGO/Virgo detectors enables searches for nearby events and their spectral closure density 6e-9 around 250Hz in the stochastic background radiation in gravitational waves. At current sensitivity, LIGO-Hanford may place an upper bound around 150MSolar in GRB030329. Detection of Egw thus provides a method for identifying Kerr black holes by calorimetry. 
  Analysis of radio-metric tracking data from the Pioneer 10/11 spacecraft at distances between 20 - 70 astronomical units (AU) from the Sun has consistently indicated the presence of an anomalous, small, constant Doppler frequency drift. The drift can be interpreted as being due to a constant acceleration of a_P= (8.74 \pm 1.33) x 10^{-8} cm/s^2 directed towards the Sun. Although it is suspected that there is a systematic origin to the effect, none has been found. As a result, the nature of this anomaly has become of growing interest. Here we present a concept for a deep-space experiment that will reveal the origin of the discovered anomaly and also will characterize its properties to an accuracy of at least two orders of magnitude below the anomaly's size. The proposed mission will not only provide a significant accuracy improvement in the search for small anomalous accelerations, it will also determine if the anomaly is due to some internal systematic or has an external origin. A number of critical requirements and design considerations for the mission are outlined and addressed. If only already existing technologies were used, the mission could be flown as early as 2010. 
  Considering the physical 3-space t = constant of the spacetime metrics as spheroidal and pseudo spheroidal, cosmological models which are generalizations of Robertson-Walker models are obtained. Specific forms of these general models as solutions of Einstein's field equations are also discussed in the radiation- and the matter-dominated eras of the universe. 
  Starting from the SO(2,2n) Chern-Simons form in (2n+1) dimensions we calculate the variation of conserved quantities in Lovelock gravity and Lovelock-Maxwell gravity through the covariant formalism developed in gr-qc/0305047. Despite the technical complexity of the Lovelock Lagrangian we obtain a remarkably simple expression for the variation of the charges ensuing from the diffeomorphism covariance of the theory. The viability of the result is tested in specific applications and the formal expression for the entropy of Lovelock black holes is recovered. 
  We extend the previously found accelerated Kerr-Schild metrics for Einstein-Maxwell-null dust and Einstein-Born-Infeld-null dust equations to the cases including the cosmological constant. This way we obtain the generalization of the charged de Sitter metrics in static space-times. We also give a generalization of the zero acceleration limit of our previous Einstein-Maxwell and Einstein-Born-Infeld solutions. 
  Quantum decoherence can arise due to classical fluctuations in the parameters which define the dynamics of the system. In this case decoherence, and complementary noise, is manifest when data from repeated measurement trials are combined. Recently a number of authors have suggested that fluctuations in the space-time metric arising from quantum gravity effects would correspond to a source of intrinsic noise, which would necessarily be accompanied by intrinsic decoherence. This work extends a previous heuristic modification of Schr\"{o}dinger dynamics based on discrete time intervals with an intrinsic uncertainty. The extension uses unital semigroup representations of space and time translations rather than the more usual unitary representation, and does the least violence to physically important invariance principles. Physical consequences include a modification of the uncertainty principle and a modification of field dispersion relations, in a way consistent with other modifications suggested by quantum gravity and string theory . 
  We examine how the new forthcoming Earth gravity models from the CHAMP and, especially, GRACE missions could improve the measurement of the general relativistic Lense-Thirring effect according to the various kinds of observables which could be adopted. In a very preliminary way, we use the recently released EIGEN2 CHAMP-only and GRACE01S GRACE-only Earth gravity models in order to assess the impact of the mismodelling in the even zonal harmonic coefficients of geopotential which represents one of the major sources of systematic errors in this kind of measurement. 
  We consider a Kerr black hole acting as a gravitational deflector within the geometrical optics, and point source approximations. The Kerr black hole gravitational lens geometry consisting of an observer and a source located far away and placed at arbitrary inclinations with respect to the black hole's equatorial plane is studied in the strong field regime. For this geometry the null geodesics equations of our interest can go around the black hole several times before reaching the observer. Such photon trajectories are written in terms of the angular positions in the observer's sky and therefore become "lens equations". As a consequence, we found for any image a simple classification scheme based in two integers numbers: the number of turning points in the polar coordinate $\theta$, and the number of windings around the black hole's rotation axis. As an application, and to make contact with the literature, we consider a supermassive Kerr black hole at the Galactic center as a gravitational deflector. In this case, we show that our proposed computational scheme works successfully by computing the positions and magnifications of the relativistic images for different source-observer geometries. In fact, it is shown that our general procedure and results for the positions and magnifications of the images off the black hole's equatorial plane, reduce and agree with well known cases found in the literature. 
  High power in narrow frequency bands, spectral lines, are a feature of an interferometric gravitational wave detector's output. Some lines are coherent between interferometers, in particular, the 2 km and 4 km LIGO Hanford instruments. This is of concern to data analysis techniques, such as the stochastic background search, that use correlations between instruments to detect gravitational radiation. Several techniques of `line removal' have been proposed. Where a line is attributable to a measurable environmental disturbance, a simple linear model may be fitted to predict, and subsequently subtract away, that line. This technique has been implemented (as the command oelslr) in the LIGO Data Analysis System (LDAS). We demonstrate its application to LIGO S1 data. 
  We consider a spherical thick shell immersed in two different spherically symmetric space-times. Using the fact that the boundaries of the thick shell with two embedding space-times must be nonsingular hypersurfaces, we develop a scheme to obtain the underlying equation of motion for the thick shell in general. As a simple example, the equation of motion of a spherical dustlike shell in vacuum is obtained. To compare our formalism with the thin shell one, the dynamical equation of motion of the thick shell is then expanded to the first order of its thickness. It is easily seen that the thin shell limit of our dynamical equation is exactly that given in the literature for the dynamics of a thin shell. It turns out that the effect of thickness is to speed up the collapse of the shell. 
  A generalization of the recently formulated nonlinear quantization of a parameterized theory is presented in the context of quantum gravity. The parametric quantization of a Friedmann universe with a massless scalar field is then considered in terms of analytic solutions of the resulting evolution equations. 
  We discuss the gravitational self-force on a particle in a black hole space-time. For a point particle, the full (bare) self-force diverges. It is known that the metric perturbation induced by a particle can be divided into two parts, the direct part (or the S part) and the tail part (or the R part), in the harmonic gauge, and the regularized self-force is derived from the R part which is regular and satisfies the source-free perturbed Einstein equations. In this paper, we consider a gauge transformation from the harmonic gauge to the Regge-Wheeler gauge in which the full metric perturbation can be calculated, and present a method to derive the regularized self-force for a particle in circular orbit around a Schwarzschild black hole in the Regge-Wheeler gauge. As a first application of this method, we then calculate the self-force to first post-Newtonian order. We find the correction to the total mass of the system due to the presence of the particle is correctly reproduced in the force at the Newtonian order. 
  It is argued that the `problem of time' in quantum gravity necessitates a refinement of the local inertial structure of the world, demanding a replacement of the usual Minkowski line element by a 4+2n dimensional pseudo-Euclidean line element, with the extra 2n being the number of internal phase space dimensions of the observed system. In the refined structure, the inverse of the Planck time takes over the role of observer-independent conversion factor usually played by the speed of light, which now emerges as an invariant but derivative quantity. In the relativistic theory based on the refined structure, energies and momenta turn out to be invariantly bounded from above, and lengths and durations similarly bounded from below, by their respective Planck scale values. Along the external timelike world-lines, the theory naturally captures the `flow of time' as a genuinely structural attribute of the world. The theory also predicts expected deviations--suppressed quadratically by the Planck energy--from the dispersion relations for free fields in the vacuum. The deviations from the special relativistic Doppler shifts predicted by the theory are also suppressed quadratically by the Planck energy. Nonetheless, in order to estimate the precision required to distinguish the theory from special relativity, an experiment with a binary pulsar emitting TeV range gamma-rays is considered in the context of the predicted deviations from the second-order shifts. 
  For a FRW-spacetime coupled to an arbitrary real scalar field, we endow the solution space of the associated Wheeler-DeWitt equation with a Hilbert-space structure, construct the observables, and introduce the physical wave functions of the universe that admit a genuine probabilistic interpretation. We also discuss a proposal for the formulation of the dynamics. The approach to quantum cosmology outlined in this article is based on the results obtained within the theory of pseudo-Hermitian operators. 
  We present quantization of particle dynamics on one-sheet hyperboloid embedded in three dimensional Minkowski space. Taking account of all global symmetries enables unique quantization. Making use of topology of canonical variables not only simplifies calculations but also gives proper framework for analysis. 
  Using the well-known ``displace, cut and reflect'' method used to generate disks from given solutions of Einstein field equations, we construct static disks made of perfect fluid based on vacuum Schwarzschild's solution in isotropic coordinates. The same method is applied to different exactsolutions to the Einstein'sequations that represent static spheres of perfect fluids. We construct several models of disks with axially symmetric perfect fluid halos.   All disks have some common features: surface energy density and pressures decrease monotonically and rapidly with radius. As the ``cut'' parameter $a$ decreases, the disks become more relativistic, with surface energy density and pressure more concentrated near the center. Also regions of unstable circular orbits are more likely to appear for high relativistic disks. Parameters can be chosen so that the sound velocity in the fluid and the tangential velocity of test particles in circular motion are less then the velocity of light. This tangential velocity first increases with radius and reaches a maximum. 
  The model of cylindrical gravitational waves is employed to work out and check a recent proposal in Ref. [11] how a diffeomorphism-invariant Hamiltonian dynamics is to be constructed. The starting point is the action by Ashtekar and Pierri because it contains the boundary term that makes it differentiable for non-trivial variations at infinity. With the help of parametrization at infinity, the notion of gauge transformation is clearly separated from that of asymptotic symmetry. The symplectic geometry of asymptotic symmetries and asymptotic time is described and the role of the asymptotic structures in defining a zero-motion frame for the Hamiltonian dynamics of Dirac observables is explained. Complete sets of Dirac observables associated with the asymptotic fields are found and the action of the asymptotic symmetries on them is calculated. The construction of the corresponding quantum theory is sketched: the Fock space, operators of asymptotic fields, the Hamiltonian and the scattering matrix are determined. 
  A detailed description of how black holes grow in full, non-linear general relativity is presented. The starting point is the notion of dynamical horizons. Expressions of fluxes of energy and angular momentum carried by gravitational waves across these horizons are obtained. Fluxes are local and the energy flux is positive. Change in the horizon area is related to these fluxes. A notion of angular momentum and energy is associated with cross-sections of the horizon and balance equations, analogous to those obtained by Bondi and Sachs at null infinity, are derived. These in turn lead to generalizations of the first and second laws of black hole mechanics. The relation between dynamical horizons and their asymptotic states --the isolated horizons-- is discussed briefly. The framework has potential applications to numerical, mathematical, astrophysical and quantum general relativity. 
  An analytic method is given for deriving the part of the retarded Green's function v(x,x') that contributes to the tail term in the radiation reaction force felt by a particle coupled to a massless minimally coupled scalar field. The method gives an expansion of v(x,x') for small separations of the points x, x' valid for an arbitrary static spherically symmetric spacetime. It is obtained by using a WKB approximation for the Euclidean Green's function for the massless minimally coupled scalar field and is equivalent to the DeWitt-Schwinger expansion for v(x,x'). The first few terms in this expansion are displayed here for the case of Schwarzschild spacetime. 
  We study locally spatially homogeneous solutions of the Einstein-Vlasov system with a positive cosmological constant. First the global existence of solutions of this system and the casual geodesic completeness are shown. Then the asymptotic behaviour of solutions in the future time is investigated in various aspects. 
  In this contribution we present two new proposals for measuring the general relativistic gravitomagnetic component of the gravitational field of the Earth. One proposal consists of the measurement of the difference of the rates of the perigee $\psi$ from the analysis of the laser--ranged data of two identical Earth'artificial satellites placed in equal orbits with supplementary inclinations. In this way the impact of the aliasing classical secular precessions due to the even zonal harmonics of the geopotential would be canceled out, although the non--gravitational perturbations, to which the perigees of LAGEOS--type satellites are particularly sensitive, should be a limiting factor in the obtainable accuracy. With a suitable choice of the inclinations of the orbital planes it would be possible to reduce the periods of such insidious perturbations so to use not too long observational time spans. However, the use of a pair of drag--free satellites would greatly reduce this problem, provided that the time span of the data analysis does not excess the lifetime of the drag--free apparatus. In the other proposal the difference of the rotational periods of two counter-revolving particles placed on a friction-free plane in a vacuum chamber at the South Pole should be measured in order to extract the relativistic gravitomagnetic signal. Among other very challenging practical implications, the Earth's angular velocity $\omega_{\oplus}$ should be known at a $10^{-15}$ rad s$^{-1}$ level from VLBI and the friction force of the plane should be less than $2\times 10^{-9}$ dyne. 
  We study the gravitational collapse in ($n+2$)-D quasi-spherical Szekeres space-time (which possess no killing vectors) with dust as the matter distribution. Instead of choosing the radial coordinate `$r$' as the initial value for the scale factor $R$, we consider a power function of $r$ as the initial scale for the radius $R$. We examine the influence of initial data on the formation of singularity in gravitational collapse. 
  The norms associated with the gradients of the two non-differential invariants of the Kerr vacuum are examined. Whereas both locally single out the horizons, their global behavior is more interesting. Both reflect the background angular momentum as the volume of space allowing a timelike gradient decreases with increasing angular momentum becoming zero in the degenerate and naked cases. These results extend directly to the Kerr-Newman geometry. 
  We develop a theoretical frame for the study of classical and quantum gravitational waves based on the properties of a nonlinear ordinary differential equation for a function $\sigma(\eta)$ of the conformal time $\eta$, called the auxiliary field equation. At the classical level, $\sigma(\eta)$ can be expressed by means of two independent solutions of the ''master equation'' to which the perturbed Einstein equations for the gravitational waves can be reduced. At the quantum level, all the significant physical quantities can be formulated using Bogolubov transformations and the operator quadratic Hamiltonian corresponding to the classical version of a damped parametrically excited oscillator where the varying mass is replaced by the square cosmological scale factor $a^{2}(\eta)$. A quantum approach to the generation of gravitational waves is proposed on the grounds of the previous $\eta-$dependent Hamiltonian. An estimate in terms of $\sigma(\eta)$ and $a(\eta)$ of the destruction of quantum coherence due to the gravitational evolution and an exact expression for the phase of a gravitational wave corresponding to any value of $\eta$ are also obtained. We conclude by discussing a few applications to quasi-de Sitter and standard de Sitter scenarios. 
  Scalar density cosmological perturbations, spectral indices and reheating in a chaotic inflationary universe model, in which a higher derivative term is added, are investigated. This term is supposed to play an important role in the early evolution of the Universe, specifically at times closer to the Planck era. 
  Gravitational radiation from the galactic population of white dwarf binaries is expected to produce a background signal in the LISA frequency band. At frequencies below 1 mHz, this signal is expected to be confusion-limited and has been approximated as gaussian noise. At frequencies above about 5 mHz, the signal will consist of separable individual sources. We have produced a simulation of the LISA data stream from a population of 90k galactic binaries in the frequency range between 1 - 5 mHz. This signal is compared with the simulated signal from globular cluster populations of binaries. Notable features of the simulation as well as potential data analysis schemes for extracting information are presented. 
  The Mathisson-Papapetrou-Dixon equations for a massive spinning test particle in plane gravitational waves are analysed and explicit solutions constructed in terms of solutions of certain linear ordinary differential equations. For harmonic waves this system reduces to a single equation of Mathieu-Hill type. In this case spinning particles may exhibit parametric excitation by gravitational fields. For a spinning test particle scattered by a gravitational wave pulse, the final energy-momentum of the particle may be related to the width, height, polarisation of the wave and spin orientation of the particle. 
  For 17 days in August and September 2002, the LIGO and GEO interferometer gravitational wave detectors were operated in coincidence to produce their first data for scientific analysis. Although the detectors were still far from their design sensitivity levels, the data can be used to place better upper limits on the flux of gravitational waves incident on the earth than previous direct measurements. This paper describes the instruments and the data in some detail, as a companion to analysis papers based on the first data. 
  If the situation of quantum gravity nowadays is nearly the same as that of the quantum mechanics in it's early time of Bohr and Sommerfeld, then a first step study of the quantum gravity from Sommerfeld's quantum condition of action might be helpful. In this short paper the spectra of Schwarzschild black hole(SBH) in quasi-classical approach of quantum mechanics is given. We find the quantum of area, the quantum of entropy and the Hawking evaporation will cease as the black hole reaches its ground state. 
  We describe the role of correlation measurements between the LIGO interferometer in Livingston, LA, and the ALLEGRO resonant bar detector in Baton Rouge, LA, in searches for a stochastic background of gravitational waves. Such measurements provide a valuable complement to correlations between interferometers at the two LIGO sites, since they are sensitive in a different, higher, frequency band. Additionally, the variable orientation of the ALLEGRO detector provides a means to distinguish gravitational wave correlations from correlated environmental noise. We describe the analysis underway to set a limit on the strength of a stochastic background at frequencies near 900 Hz using ALLEGRO data and data from LIGO's E7 Engineering Run. 
  The Friedmann equations for a brane with induced gravity are analyzed and compared with the standard general relativity and Randall-Sundrum cases. Randall-Sundrum gravity modified the early universe dynamics, whereas induced gravity changes the late universe evolution. The early and late time limits are investigated. Induced gravity effects can contribute to late-universe acceleration. This conditions for this are found. Qualitative analysis is given for a range of scalar field potentials. 
  A particle in the vicinity of a Schwarzschild black hole is known to trace a geodesic of the Schwarzschild background, to a first approximation. If the interaction of the particle with its own field (scalar, electromagnetic or gravitational) is taken into account, the path is no longer a background geodesic and the self-force that the particle experiences needs to be taken into consideration.  In this dissertation, a recently proposed method for the calculation of the self-force is implemented. According to this method the self-force comes from the interaction of the particle with the Regular-Remainder scalar field, electromagnetic potential or metric perturbation. That Regular-Remainder is obtained by subtracting the Singular part (which exerts no force) from the retarded scalar field, electromagnetic potential of metric perturbation generated by the moving particle. First, the Singular scalar fields, electromagnetic potentials and metric perturbations are calculated for different sources moving in a Schwarzschild background. For that, the Thorne-Hartle-Zhang coordinates in the vicinity of the moving source are used. Then a mode-sum regularization method initially proposed for the direct scalar field is followed, and the regularization parameters for the singular part of the scalar field and for the first radial derivative of the singular part of the self-force are calculated. Also, the numerical calculation of the retarded scalar field for a particle moving on a circular geodesic in a Schwarzschild spacetime is presented. Finally, the self-force for a scalar particle moving on a circular Schwarzschild orbit is calculated and some results about the effects of the self-force on the orbital frequency of the circular orbit are presented. 
  These notes introduce the subject of quantum field theory in curved spacetime and some of its applications and the questions they raise. Topics include particle creation in time-dependent metrics, quantum origin of primordial perturbations, Hawking effect, the trans-Planckian question, and Hawking radiation on a lattice. 
  There are two major alternatives for violating the (usual) Lorentz invariance at large (Planckian) energies or momenta - either not all inertial frames (in the Planck regime) are equivalent (e.g., there is an effectively preferred frame) or the transformations from one frame to another are (non-linearly) deformed (``doubly special relativity''). We demonstrate that the natural (and reasonable) assumption of an energy-dependent speed of light in the latter method goes along with violations of locality/separability (and even translational invariance) on macroscopic scales.   PACS: 03.30.+p, 11.30.Cp, 04.60.-m, 04.50.+h. 
  Data collected by the GEO600 and LIGO interferometric gravitational wave detectors during their first observational science run were searched for continuous gravitational waves from the pulsar J1939+2134 at twice its rotation frequency. Two independent analysis methods were used and are demonstrated in this paper: a frequency domain method and a time domain method. Both achieve consistent null results, placing new upper limits on the strength of the pulsar's gravitational wave emission. A model emission mechanism is used to interpret the limits as a constraint on the pulsar's equatorial ellipticity. 
  We study the gravitational radiation reaction in compact binary systems composed of neutron stars with spin and huge magnetic dipole moments (magnetars). The magnetic dipole moments undergo a precessional motion about the respective spins. At sufficiently high values of the magnetic dipole moments, their interaction generates second post-Newtonian order contributions both to the equations of motion and to the gravitational radiation escaping the system. We parametrize the radial motion and average over a radial period in order to find the secular contributions to the energy and magnitude of the orbital angular momentum losses, in the generic case of \textit{eccentric} orbits. Similarly as for the spin-orbit, spin-spin, quadrupole-monopole interactions, here too we deduce the secular evolution of the relative orientations of the orbital angular momentum and spins. These equations, supplemented by the evolution equations for the angles characterizing the orientation of the dipole moments form a first order differential system, which is closed. The circular orbit limit of the energy loss agrees with Ioka and Taniguchi's earlier result. 
  We consider D-particles coupled to the CGHS dilaton gravity and obtain the exact wormhole geometry and trajectories of D-particles by introducing the exotic matter. The initial static wormhole background is not stable after infalling D-particles due to the classical backreaction of the geometry so that the additional exotic matter source should be introduced for the stability. Then, the traversable wormhole geometry naturally appears and the D-particles can travel through it safely. Finally, we discuss the dynamical evolution of the wormhole throat and the massless limit of D-particles. 
  In this paper we investigate the feasibility of a recently proposed space-based experiment aimed to the detection of the effect of the Earth gravitomagnetic field in spaceborne semiconductors carrying radial electric currents and following identical circular equatorial orbits along opposite directions. It turns out that the deviations from this idealized situation due to the unavoidable orbital injection errors would make impossible the measurement of the gravitomagnetic voltage of interest. 
  We present three reasons for the formation of gravitational bound states of primordial black holes,called holeums,in the early universe.Using Newtonian gravity and nonrelativistic quantum mechanics we find a purely quantum mechanical mass-dependant exclusion property for the nonoverlap of the constituent black holes in a holeum.This ensures that the holeum occupies space just like ordinary matter.A holeum emits only gravitational radiation whose spectrum is an exact analogue of that of a hydrogen atom. A part of this spectrum lies in the region accessible to the detectors being built.The holeums would form haloes around the galaxies and would be an important component of the dark matter in the universe today.They may also be the constituents of the invisible domain walls in the universe. 
  This paper treats the global existence question for a collection of general relativistic collisionless particles, all having the same mass. The spacetimes considered are globally hyperbolic, with Cauchy surface a 3-torus. Furthermore, the spacetimes considered are isometrically invariant under a two-dimensional group action, the orbits of which are spacelike 2-tori. It is known from previous work that the area of the group orbits serves as a global time coordinate. In the present work it is shown that the area takes on all positive values in the maximal Cauchy development. 
  We investigate the embedding of four-dimensional branes in five-dimensional spaces. We firstly consider the case when the embedding space is a vacuum bulk whose energy-momentum tensor consists of a Dirac delta function with support in the brane. We then consider the embedding in the context of Randall-Sundrum-type models, taking into account $Z_{2}$ symmetry and a cosmological constant. We employ the Campbell-Magaard theorem to construct the embeddings and are led to the conclusion that the content of energy-matter of the brane does not necessarily determine its curvature. Finally, as an application to illustrate our results, we construct the embedding of Minkowski spacetime filled with dust. 
  This diploma thesis analyses static, spherically symmetric perfect fluid solutions to Einstein's field equations with cosmological constant. Constant density solutions are derived for different values of the cosmological constant. Eleven types of solutions are found, with an overview given at page 41. Furthermore the existence of a global solution is proved for a cosmological constant smaller than 4 Pi the boundary density, which is given by the equation of state. 
  The presence of a bulk viscosity for the cosmic fluid on a single Randall-Sundrum brane is considered. The spatial curvature is assumed to be zero. The five-dimensional Friedmann equation is derived, together with the energy conservation equation for the viscous fluid. These governing equations are solved for some special cases: (i) in the low-energy limit when the matter energy density is small compared with brane tension; (ii) for a matter-dominated universe, and (iii) for a radiation-dominated universe. Rough numerical estimates, for the extreme case when the universe is at its Planck time, indicate that the viscous effect can be significant. 
  An analysis is presented of the Bianchi type I cosmological models with a bulk viscosity when the universe is filled with the stiff fluid $p = \epsilon$ while the viscosity is a power function of the energy density, such as $\eta = \alpha |\epsilon|^n$. Although the exact solutions are obtainable only when the $2n$ is an integer, the characteristics of evolution can be clarified for the models with arbitrary value of $n$. It is shown that, except for the $n = 0$ model that has solutions with infinite energy density at initial state, the anisotropic solutions that evolve to positive Hubble functions in the later stage will begin with Kasner-type curvature singularity and zero energy density at finite past for the $n> 1$ models, and with finite Hubble functions and finite negative energy density at infinite past for the $n < 1$ models. In the course of evolution, matters are created and the anisotropies of the universe are smoothed out. At the final stage, cosmologies are driven to infinite expansion state, de Sitter space-time, or Friedman universe asymptotically. However, the de Sitter space-time is the only attractor state for the $n <1/2 $ models. The solutions that are free of cosmological singularity for any finite proper time are singled out. The extension to the higher-dimensional models is also discussed. 
  Bianchi type I cosmological models are studied that contain a stiff fluid with a shear viscosity that is a power function of the energy density, such as $\zeta = \alpha \epsilon^n$. These models are analyzed by describing the cosmological evolutions as the trajectories in the phase plane of Hubble functions. The simple and exact equations that determine these flows are obtained when $n$ is an integer. In particular, it is proved that there is no Einstein initial singularity in the models of $0\leq n < 1$. Cosmologies are found to begin with zero energy density and in the course of evolution the gravitational field will create matter. At the final stage, cosmologies are driven to the isotropic Fnedmann universe. It is also pointed out that although the anisotropy will always be smoothed out asymptotically, there are solutions that simultaneously possess non-positive and non-negative Hubble functions for all time. This means that the cosmological dimensional reduction can work even if the matter fluid having shear viscosity. These characteristics can also be found in any-dimensional models. 
  Self-consistent system of spinor, scalar and BI gravitational fields is considered. Exact solutions to the field equations in terms of volume scale of the BI metric are obtained. Einstein field equations in account of the cosmological constant $\Lambda$ and perfect fluid are studied. Oscillatory mode of expansion of the universe is obtained. It is shown that for the interaction term being a power function of the invariants of bilinear spinor forms and $\Lambda > 0$ and given other parameters, e.g., coupling constant, spinor mass etc., there exists a finite range of integration constant which generates oscillatory mode of evolution. 
  Searches for gravitational wave signals which do not have a precise model describing the shape of their waveforms are often performed using power detectors based on a quadratic form of the data. A new, optimal method of generalizing these power detectors so that they operate coherently over a network of interferometers is presented. Such a mode of operation is useful in obtaining better detection efficiencies, and better estimates of the position of the source of the gravitational wave signal. Numerical simulations based on a realistic, computationally efficient hierarchical implementation of the method are used to characterize its efficiency, for detection and for position estimation. The method is shown to be more efficient at detecting signals than an incoherent approach based on coincidences between lists of events. It is also shown to be capable of locating the position of the source. 
  Recent observational results suggest that our universe is nearly flat and well modelled within a $\Lambda$CDM framework. The observed values of $\Omega_{m}$ and $\Omega_{\Lambda}$ inevitably involve uncertainties. Motivated by this, we make a systematic study of the necessary and sufficient conditions for undetectability as well as detectability (in principle) of cosmic topology (using pattern repetition) in presence of such uncertainties. We do this by developing two complementary methods to determine detectability for nearly flat universes. Using the first method we derive analytical conditions for undetectability for infinite redshift, the accuracy of which is then confirmed by the second method. Estimates based on WMAP data together with other measurements of the density parameters are used to illustrate both methods, which are shown to provide very similar results for high redshifts. 
  Sijie Gao has recently investigated Hawking radiation from spherically symmetrical gravitational collapse to an extremal R-N black hole for a real scalar field. Especially he estimated the upper bound for the expected number of particles in any wave packet belonging to $\mathcal{H}_{out}$ spontaneously produced from the state $|0>_{in}$, which confirms the traditional belief that extremal black holes do not radiate particles. Making some modifications, we demonstrate that the analysis can go through for a charged scalar field. 
  Quantum fluctuations of lightcone are examined in a 4-dimensional spacetime with two parallel planes. Both the Dirichlet and the Neumann boundary conditions are considered. In all the cases we have studied, quantum lightcone fluctuations are greater where the Neumann boundary conditions are imposed, suggesting that quantum lightcone fluctuations depend not only on the geometry and topology of the spacetime as has been argued elsewhere but also on boundary conditions. Our results also show that quantum lightcone fluctuations are larger here than that in the case of a single plane. Therefore, the confinement of gravitons in a smaller region by the presence of a second plane reinforces the quantum fluctuations and this can be understood as a consequence of the uncertainty principle. 
  Due to the growing interest in embeddings of space-time in higher-dimensional spaces we consider a specific type of embedding. After proving an inequality between intrinsically defined curvature invariants and the squared mean curvature, we extend the notion of ideal embeddings from Riemannian geometry to the indefinite case. Ideal embeddings are such that the embedded manifold receives the least amount of tension from the surrounding space. Then it is shown that the de Sitter spaces, a Robertson-Walker space-time and some anisotropic perfect fluid metrics can be ideally embedded in a five-dimensional pseudo-Euclidean space. 
  We use covariant techniques to describe the properties of the Godel universe and then consider its linear response to a variety of perturbations. Against matter aggregations, we find that the stability of the Godel model depends primarily upon the presence of gradients in the centrifugal energy, and secondarily on the equation of state of the fluid. The latter dictates the behaviour of the model when dealing with homogeneous perturbations. The vorticity of the perturbed Godel model is found to evolve as in almost-FRW spacetimes, with some additional directional effects due to shape distortions. We also consider gravitational-wave perturbations by investigating the evolution of the magnetic Weyl component. This tensor obeys a simple plane-wave equation, which argues for the neutral stability of the Godel model against linear gravity-wave distortions. The implications of the background rotation for scalar-field Godel cosmologies are also discussed. 
  We formulate a new analytical method for regularizing the self-force acting on a particle of small mass $\mu$ orbiting a black hole of mass $M$, where $\mu\ll M$. At first order in $\mu$, the geometry is perturbed and the motion of the particle is affected by its self-force. The self-force, however, diverges at the location of the particle, and hence should be regularized. It is known that the properly regularized self-force is given by the tail part (or the $R$-part) of the self-field, obtained by subtracting the direct part (or the $S$-part) from the full self-field. The most successful method of regularization proposed so far relies on the spherical harmonic decomposition of the self-force, the so-called mode-sum regularization or mode decomposition regularization. However, except for some special orbits, no systematic analytical method for computing the regularized self-force has been given. In this paper, utilizing a new decomposition of the retarded Green function in the frequency domain, we formulate a systematic method for the computation of the self-force. Our method relies on the post-Newtonian (PN) expansion but the order of the expansion can be arbitrarily high. To demonstrate the essence of our method, in this paper, we focus on a scalar charged particle on the Schwarzschild background. The generalization to the gravitational case is straightforward, except for some subtle issues related with the choice of gauge (which exists irrespective of regularization methods). 
  We report on a search for gravitational waves from coalescing compact binary systems in the Milky Way and the Magellanic Clouds. The analysis uses data taken by two of the three LIGO interferometers during the first LIGO science run and illustrates a method of setting upper limits on inspiral event rates using interferometer data. The analysis pipeline is described with particular attention to data selection and coincidence between the two interferometers. We establish an observational upper limit of $\mathcal{R}<$1.7 \times 10^{2}$ per year per Milky Way Equivalent Galaxy (MWEG), with 90% confidence, on the coalescence rate of binary systems in which each component has a mass in the range 1--3 $M_\odot$. 
  A general ansatz for gravitational entropy can be provided using the criterion that, any patch of area which acts as a horizon for a suitably defined accelerated observer, must have an entropy proportional to its area. After providing a brief justification for this ansatz, several consequences are derived: (i) In any static spacetime with a horizon and associated temperature $\beta^{-1}$, this entropy satisfies the relation $S=(1/2)\beta E$ where $E$ is the energy source for gravitational acceleration, obtained as an integral of $(T_{ab}-(1/2)Tg_{ab})u^au^b$. (ii) With this ansatz of $S$, the minimization of Einstein-Hilbert action is equivalent to minimizing the free energy $F$ with $\beta F=\beta U-S$ where $U$ is the integral of $T_{ab}u^au^b$. We discuss the conditions under which these results imply $S\propto E^2$ and/or $S\propto U^2$ thereby generalizing the results known for black holes. This approach links with several other known results, especially the holographic views of spacetime. 
  General relativity is extended by promoting the three-dimensional gravitational Chern-Simons term to four dimensions. This entails choosing an embedding coordinate v_\mu -- an external quantity, which we fix to be a non-vanishing constant in its time component. The theory is identical to one in which the embedding coordinate is itself a dynamical variable, rather than a fixed, external quantity. Consequently diffeomorphism symmetry breaking is hidden in the modified theory: the Schwarzschild metric is a solution; gravitational waves possess two polarizations, each traveling at the velocity of light; a conserved energy-momentum (pseudo-) tensor can be constructed. The modification is visible in the intensity of gravitational radiation: the two polarizations of a gravity wave carry intensities that are suppressed/enchanced by the extension. 
  We prove that for a large class of generalized Randall-Sundrum II type models the characterization of brane-gravity sector by the effective Einstein equation, Codazzi equation and the twice-contracted Gauss equation is equivalent with the bulk Einstein equation. We give the complete set of equations in the generic case of non-$Z_{2}$-symmetric bulk and arbitrary energy-momentum tensors both on the brane and in the bulk. Among these, the effective Einstein equation contains a varying cosmological constant and two new source terms. The first of these represents the deviation from $Z_{2}$ symmetry, while the second arises from the bulk energy-momentum tensor. We apply the formalism for the case of perfect fluid on a Friedmann brane embedded in a generic bulk. The generalized Friedmann and Raychaudhuri equations are given in a form independent of both the embedding and the bulk matter. They contain two new functions obeying a first order differential system, both depending on the bulk matter and the embedding. Then we focus on Friedmann branes separating two non-identical (inner or outer) regions of Reissner-Nordstr\"{o}m-Anti de Sitter bulk space-times, generalizing previous non-$Z_{2}$-symmetric treatments. Finally the analysis is repeated for the Vaidya-Anti de Sitter bulk space-time, allowing for both ingoing and outgoing radiation in each region. 
  A new concept of internal time (viewed as a scalar temporal field) is introduced which allows one to solve the energy problem in General Relativity. The law of energy conservation means that the total energy density of the full system of interacting fields (including gravitational field) does not vary with time, thus being the first integral of the system. It is demonstrated that direct introduction of the temporal field permits to derive the general covariant four dimensional Maxwell equations for the electric and magnetic fields from the equations of electromagnetic fields considering in General Relativity. It means that the fundamental physical laws are in full correspondence with the essence of time. Theory of time presented here predicts the existence of matter outside the time. 
  This paper is the first of three in which I study the moduli space of isometry classes of (compact) globally hyperbolic spacetimes (with boundary). I introduce a notion of Gromov-Hausdorff distance which makes this moduli space into a metric space. Further properties of this metric space are studied in the next papers. The importance of the work can be situated in fields such as cosmology, quantum gravity and - for the mathematicians - global Lorentzian geometry. 
  In this second paper, I construct a limit space of a Cauchy sequence of globally hyperbolic spacetimes. In the second section, I work gradually towards a construction of the limit space. I prove the limit space is unique up to isometry. I als show that, in general, the limit space has quite complicated causal behaviour. This work prepares the final paper in which I shall study in more detail properties of the limit space and the moduli space of (compact) globally hyperbolic spacetimes (cobordisms). As a fait divers, I give in this paper a suitable definition of dimension of a Lorentz space in agreement with the one given by Gromov in the Riemannian case. 
  In induced matter Kaluza-Klein gravity theory the solution of the dynamics equations for the test particle on null path leads to additional force in four-dimensional space-time. We find such force from five-dimensional geodesic line equations and apply this approach to analysis of the asymmetrically warped space-time. 
  We calculate high-order quasinormal modes with large imaginary frequencies for electromagnetic and gravitational perturbations in nearly extremal Schwarzschild-de Sitter spacetimes. Our results show that for low-order quasinormal modes, the analytical approximation formula in the extremal limit derived by Cardoso and Lemos is a quite good approximation for the quasinormal frequencies as long as the model parameter $r_1\kappa_1$ is small enough, where $r_1$ and $\kappa_1$ are the black hole horizon radius and the surface gravity, respectively. For high-order quasinormal modes, to which corresponds quasinormal frequencies with large imaginary parts, on the other hand, this formula becomes inaccurate even for small values of $r_1\kappa_1$. We also find that the real parts of the quasinormal frequencies have oscillating behaviors in the limit of highly damped modes, which are similar to those observed in the case of a Reissner-Nordstr{\" o}m black hole. The amplitude of oscillating ${\rm Re(\omega)}$ as a function of ${\rm Im}(\omega)$ approaches a non-zero constant value for gravitational perturbations and zero for electromagnetic perturbations in the limit of highly damped modes, where $\omega$ denotes the quasinormal frequency. This means that for gravitational perturbations, the real part of quasinormal modes of the nearly extremal Schwarzschild-de Sitter spacetime appears not to approach any constant value in the limit of highly damped modes. On the other hand, for electromagnetic perturbations, the real part of frequency seems to go to zero in the limit. 
  We obtain the Schwarzschild solution based on teleparallel gravity (TG) theory formulated in a space-time with torsion only. The starting point is the Poincar\UNICODE{0xe9} gauge theory (PGT).The general structure of TG and its connection with general relativity (GR) are presented and the Schwarzschild solution is obtained by solving the field equations of TG. Most of calculations are performed using the GRTensorII package, running on the MapleV platform. 
  We consider the electromagnetic (EM) response of a Gaussian beam passing through a static magnetic field to be the high-frequency gravitational waves (HFGW) as generated by several devices discussed at this conference. It is found that under the synchroresonance condition, the first-order perturbative EM power fluxes will contain a ''left circular wave'' and a ''right circular wave'' around the symmetrical axis of the Gaussian beam. However, the perturbative effects produced by the states of + polarization and \times polarization of the GW have a different physical behavior. For the HFGW of $\nu_{g}=3GHz$, $h=10^{-30}$ (which corresponds to the power flux density $~ 10^{-6} W m^{-2}$) to $\nu_{g}=1.3THz$, $ h=10^{-28}$ (which corresponds to the power flux density $~10^{3} W m^{-2}$) expected by the HFGW generators described at this conference, the corresponding perturbative photon fluxes passing through a surface region of $10^{-2} m^{2}$ would be expected to be $10^{3} s^{-1} - 10^{4} s^{-1}$. They are the orders of magnitude of the perturbative photon flux we estimated using typical laboratory parameters that could lead to the development of sensitive HFGW receivers. Moreover, we will also discuss the relative background noise problems and the possibility of displaying the HFGW. A laboratory test bed for juxtaposed HFGW generators and our detecting scheme is explored and discussed. 
  Using squeezed vacuum state formalism of quantum optics, an approximate solution to the semiclassical Einstein equation is obtained in Bianchi type-I universe. The phenomena of nonclassical particle creation is also examined in the anisotropic background cosmology. 
  The derivation of the recently proposed nonlinear quantum evolution of gravity from an action principle is considered in this brief note. It is shown to be possible if a set of consistency conditions are satisfied that are analogous to the Dirac relations for the super-Hamiltonian and momenta in classical canonical gravity. 
  A simple thought experiment suggests that, contrary to assertions in an earlier Letter, constancy across materials of the ratio of active to passive gravitational mass does not rule out that electrons (and other leptons) could have active gravitational mass zero, thus might not generate gravity. If they do not, then widely held assumptions about the gravitational effects of various forms of energy cannot be sustained. 
  With the aid of a Fermi-Walker chart associated with an orthonormal frame attached to a time-like curve in spacetime, a discussion is given of relativistic balance laws that may be used to construct models of massive particles with spin, electric charge and a magnetic moment,interacting with background electromagnetic fields and gravitation described by non-Riemannian geometries. A natural generalisation to relativistic Cosserat media is immediate. 
  Considering the vacuum as characterized by the presence of only the gravitational field, we show that the vacuum energy density of the de Sitter space, in the realm of the teleparallel equivalent of general relativity, can acquire arbitrarily high values. This feature is expected to hold in the consideration of realistic cosmological models, and may possibly provide a simple explanation to the cosmological constant problem. 
  We consider the general problem of estimating the inflight LISA noise power spectra and cross-spectra, which are needed for detecting and estimating the gravitational wave signals present in the LISA data. For the LISA baseline design and in the long wavelength limit, we bound the error on all spectrum estimators that rely on the use of the fully symmetric Sagnac combination ($\zeta$). This procedure avoids biases in the estimation that would otherwise be introduced by the presence of a strong galactic background in the LISA data. We specialize our discussion to the detection and study of the galactic white dwarf-white dwarf binary stochastic signal. 
  We exactly calculate the particle number $N$ of scalar fields which are created from an initial vacuum in certain higher-dimensional cosmological models. The spacetimes in these models are the four-dimensional Chitre-Hartle or radiation-dominated universe with extra spaces which are static or power-law contracting. Except for some models in which no particles could be produced, the distribution of created particles shows a thermal behavior, at least in the limit of high three-dimensional "momentum" $k$. In some models, $N$ does not depend on the magnitude of the extra-dimensional "momentum" $k_c$ if $k_c$ is nonvanishing. A cutoff momentum $k_c$ may emerge in some models, and particles with $k\le k_c$ could not be produced. We also discuss these results. 
  According to Mach's principle inertia has its reason in the presence of all masses in the universe. Despite there is a lot of sympathy for this plausible idea, only a few quantitative frameworks have been proposed to test it. In this paper a tentative theory is given which is based on Mach's critisism on Newton's rotating bucket. Taking this criticism seriously, one is led to the hypothesis that the rotation of our galaxy is the reason for gravitation. Concretely, a functional dependence of the gravitational constant on the size, mass and angular momentum of the milky way is proposed that leads to a spatial, but not to a temporal variation of G. Since Newton's inverse-square law is modified, flat rotation curves of galaxies can be explained that usually need the postulate of dark matter. While the consequences for stellar evolution are discussed briefly, a couple of further observational coincidences are noted and possible experimental tests are proposed. 
  Within the scope of Bianchi type VI (BVI) model the self-consistent system of nonlinear spinor and gravitational fields is considered. Exact self-consistent solutions to the spinor and gravitational field equations are obtained for some special choice of spatial inhomogeneity and nonlinear spinor term. The role of inhomogeneity in the evolution of spinor and gravitational field is studied. Oscillatory mode of expansion of the BVI universe is obtained for some special choice of spinor field nonlinearity. 
  We report on parallel observations in two seemingly unrelated areas of dynamical network research. The one is the so-called small world phenomenon and/or the observation of scale freeness in certain types of large (empirical) networks and their theoretical analysis. The other is a discrete cellular network approach to quantum space-time physics on the Planck scale we developed in the recent past. In this context we formulated a kind of geometric renormalisation group or coarse graining process in order to construct some fixed point which can be associated to our macroscopic space-time (physics). Such a fixed point can however only emerge if the network on the Planck scale has very peculiar critical geometric properties which strongly resemble the phenomena observed in the above mentioned networks. A particularly noteworthy phenomenon is the appearance of translocal bridges or short cuts connecting widely separated regions of ordinary space-time and which we expect to become relevant in various of the notorious quantum riddles. 
  The interferometers being planned for second generation LIGO promise and order of magnitude increase in broadband strain sensitivity--with the corresponding cubic increase in detection volume--and an extension of the observation band to lower frequencies. In addition, one of the interferometers may be designed for narrowband performance, giving further improved sensitivity over roughly an octave band above a few hundred Hertz. This article discusses the physics and technology of these new interferometer designs, and presents their projected sensitivity spectra. 
  We fully develop the concept of causal symmetry introduced in Class. Quant. Grav. 20 (2003) L139. A causal symmetry is a transformation of a Lorentzian manifold (V,g) which maps every future-directed vector onto a future-directed vector. We prove that the set of all causal symmetries is not a group under the usual composition operation but a submonoid of the diffeomorphism group of V. Therefore, the infinitesimal generating vector fields of causal symmetries --causal motions-- are associated to local one-parameter groups of transformations which are causal symmetries only for positive values of the parameter --one-parameter submonoids of causal symmetries--. The pull-back of the metric under each causal symmetry results in a new rank-2 future tensor, and we prove that there is always a set of null directions canonical to the causal symmetry. As a result of this it makes sense to classify causal symmetries according to the number of independent canonical null directions. This classification is maintained at the infinitesimal level where we find the necessary and sufficient conditions for a vector field to be a causal motion. They involve the Lie derivatives of the metric tensor and of the canonical null directions. In addition, we prove a stability property of these equations under the repeated application of the Lie operator. Monotonicity properties, constants of motion and conserved currents can be defined or built using casual motions. Some illustrative examples are presented. 
  We construct manifold structures on various sets of solutions of the general relativistic initial data sets. 
  Topological charged black holes coupled with a cosmological constant in $R^{2}\times X^{D-2}$ spacetimes are studied, where $X^{D-2}$ is an Einstein space of the form ${}^{(D-2)}R_{AB} = k(D-3) h_{AB}$. The global structure for the four-dimensional spacetimes with $k = 0$ is investigated systematically. The most general solutions that represent a Type $II$ fluid in such a high dimensional spacetime are found, and showed that topological charged black holes can be formed from the gravitational collapse of such a fluid. When the spacetime is (asymptotically) self-similar, the collapse always forms black holes for $k = 0, -1$, in contrast to the case $k = 1$, where it can form either balck holes or naked singularities. 
  Recently, the gravitational collapse of an infinite cylindrical thin shell of matter in an otherwise empty spacetime with two hypersurface orthogonal Killing vectors was studied by Gon\c{c}alves [Phys. Rev. {\bf D65}, 084045 (2002).]. By using three "alternative" criteria for trapped surfaces, the author claimed to have shown that {\em they can never form either outside or on the shell, regardingless of the matter content for the shell, except at asymptotical future null infinite}.   Following Penrose's original idea, we first define trapped surfaces in cylindrical spacetimes in terms of the expansions of null directions orthogonal to the surfaces, and then show that the first criterion used by Gon\c{c}alves is incorrect. We also show that his analysis of non-existence of trapped surfaces in vacuum is incomplete. To confirm our claim, we present an example that is a solution to the vacuum Einstein field equations and satisfies all the regular conditions imposed by Gon\c{c}alves. After extending the solution to the whole spacetime, we show explicitly that trapped surfaces exist in the extended region. 
  Mach's "fixed stars" are actually not fixed at all. The distant clusters of galaxies are not only receding from each observer but they are also accelerating since the rate of cosmological expansion is not constant. If the distant cosmic masses in someway constitute the frame of inertial reference then an additional force should be generated among local bodies in reaction to the apparent cosmological accelerations of the distant galaxies. 
  Four-dimensional Riemannian spacetimes with two commuting spacelike Killing vectors are studied in Einstein's theory of gravity, and found that no outer apparent horizons exist, provided that the dominant energy condition holds. 
  We clarify the status of two known solutions to the 5-dimensional vacuum Einstein field equations derived by Liu, Mashhoon & Wesson (LMW) and Fukui, Seahra & Wesson (FSW), respectively. Both 5-metrics explicitly embed 4-dimensional Friedman-Lemaitre-Robertson-Walker cosmologies with a wide range of characteristics. We show that both metrics are also equivalent to 5-dimensional topological black hole (TBH) solutions, which is demonstrated by finding explicit coordinate transformations from the TBH to LMW and FSW line elements. We argue that the equivalence is a direct consequence of Birkhoff's theorem generalized to 5 dimensions. Finally, for a special choice of parameters we plot constant coordinate surfaces of the LMW patch in a Penrose-Carter diagram. This shows that the LMW coordinates are regular across the black and/or white hole horizons. 
  Assuming that general relativity is the correct theory of gravity in the strong field limit, can gravitational wave observations distinguish between black hole and other compact object sources? Alternatively, can gravitational wave observations provide a test of one of the fundamental predictions of general relativity? Here we describe a definitive test of the hypothesis that observations of damped, sinusoidal gravitational waves originated from a black hole or, alternatively, that nature respects the general relativistic no-hair theorem. For astrophysical black holes, which have a negligible charge-to-mass ratio, the black hole quasi-normal mode spectrum is characterized entirely by the black hole mass and angular momentum and is unique to black holes. In a different theory of gravity, or if the observed radiation arises from a different source (e.g., a neutron star, strange matter or boson star), the spectrum will be inconsistent with that predicted for general relativistic black holes. We give a statistical characterization of the consistency between the noisy observation and the theoretical predictions of general relativity, together with a numerical example. 
  Recently, the study of three-dimensional spaces is becoming of great interest. In these dimensions the Cotton tensor is prominent as the substitute for the Weyl tensor. It is conformally invariant and its vanishing is equivalent to conformal flatness. However, the Cotton tensor arises in the context of the Bianchi identities and is present in any dimension. We present a systematic derivation of the Cotton tensor. We perform its irreducible decomposition and determine its number of independent components for the first time. Subsequently, we exhibit its characteristic properties and perform a classification of the Cotton tensor in three dimensions. We investigate some solutions of Einstein's field equations in three dimensions and of the topologically massive gravity model of Deser, Jackiw, and Templeton. For each class examples are given. Finally we investigate the relation between the Cotton tensor and the energy-momentum in Einstein's theory and derive a conformally flat perfect fluid solution of Einstein's field equations in three dimensions. 
  These are lecture notes on causal set theory prepared in Jan. 2002 for a Summer School in Valdivia, Chile. In some places, they are more complete, in others much less so, regrettably. An extensive set of references and a glossary of terms can be found at the end of the notes. 
  We clarify the correspondence between the first order formalism and the second order formalism in the generalized gravity where the Lagrangian density is given by a function of the scalar curvature f(R). 
  ASTROD (Astrodynamical Space Test of Relativity using Optical Devices) is a mission concept with three spacecraft -- one near L1/L2 point, one with an inner solar orbit and one with an outer solar orbit, ranging coherently with one another using lasers to test relativistic gravity, to measure the solar system and to detect gravitational waves. ASTROD I with one spacecraft ranging optically with ground stations is the first step toward the ASTROD mission. In this paper, we present the ASTROD I payload and accelerometer requirements, discuss the gravitational-wave sensitivities for ASTROD and ASTROD I, and compare them with LISA and radio-wave PDoppler-tracking of spacecraft. 
  FINESSE is a software simulation that allows to compute the optical properties of laser interferometers as they are used by the interferometric gravitational-wave detectors today. It provides a fast and versatile tool which has proven to be very useful during the design and the commissioning of gravitational-wave detectors. The basic algorithm of FINESSE numerically computes the light amplitudes inside an interferometer using Hermite-Gauss modes in the frequency domain. In addition, FINESSE provides a number of commands to easily generate and plot the most common signals like, for example, power enhancement, error or control signals, transfer functions and shot-noise-limited sensitivities.   Among the various simulation tools available to the gravitational wave community today, FINESSE is the most advanced general optical simulation that uses the frequency domain. It has been designed to allow general analysis of user defined optical setups while being easy to install and easy to use. 
  Very recently, a ``Comment'' by Wang [gr-qc/0309003] on a paper by Gon\c{c}alves [Phys. Rev. D {\bf 65}, 084045 (2002)] appeared, claiming that Gon\c{c}alves' analysis of trapped surfaces in certain kinds of cylindrical spacetimes was incomplete. Specifically, Wang claims to have found a coordinate extension of the spacetime used by Gon\c{c}alves (the Einstein-Rosen spacetime) which contains trapped surfaces; in addition, Wang also claims that some such trapped surfaces are apparent horizons. Here, I comment on Wang's ``Comment'', and argue that, while Wang's spacetime extension appears to exist and contain trapped surfaces, it does not render Gon\c{c}alves' results incomplete in the sense Wang claims. I also show that, contrary to Wang's claim, his spacetime extension does {\em not} contain apparent horizons, i.e., it does not contain outer marginally trapped surfaces which are the outer boundary of a trapped region. Further peripheral comments by Wang are also commented on. 
  We consider (1+4) generalization of classical electrodynamics including gravitation field. With this approach it is assumed a presence of an extra component of extended field stress tensor, whose physical interpretation is based on necessity to obtain Newton gravity law as particular case. In model analyzed in present work mass of having in 4D a rest mass particle moves in five-dimensional space-time along fifth coordinate with light velocity and its electric charge is a stationary value in additional dimension. Generalized gravity law that is obtained from extended Maxwell equations confirmed by existence of variable component of Pioneer 10 additional acceleration, whose analyze is made in frame of this model. 
  We generalize the known equivalence between higher order gravity theories and scalar tensor theories to a new class of theories. Specifically, in the context of a first order or Palatini variational principle where the metric and connection are treated as independent variables, we consider theories for which the Lagrangian density is a function f of (i) the Ricci scalar computed from the metric, and (ii) a second Ricci scalar computed from the connection. We show that such theories can be written as tensor-multi-scalar theories with two scalar fields with the following features: (i) the two dimensional sigma-model metric that defines the kinetic energy terms for the scalar fields has constant, negative curvature; (ii) the coupling function determining the coupling to matter of the scalar fields is universal, independent of the choice of function f; and (iii) if both mass eigenstates are long ranged, then the Eddington post-Newtonian parameter has value 1/2. Therefore in order to be compatible with solar system experiments at least one of the mass eigenstates must be short ranged. 
  A representation of spatial infinity based in the properties of conformal geodesics is used to obtain asymptotic expansions of the gravitational field near the region where null infinity touches spatial infinity. These expansions show that generic time symmetric initial data with an analytic conformal metric at spatial infinity will give rise to developments with a certain type of logarithmic singularities at the points where null infinity and spatial infinity meet. These logarithmic singularities produce a non-smooth null infinity. The sources of the logarithmic singularities are traced back down to the initial data. It is shown that is the parts of the initial data responsible for the non-regular behaviour of the solutions are not present, then the initial data is static to a certain order. On the basis of these results it is conjectured that the only time symmetric data sets with developments having a smooth null infinity are those which are static in a neighbourhood of infinity. This conjecture generalises a previous conjecture regarding time symmetric, conformally flat data. The relation of these conjectures to Penrose's proposal for the description of the asymptotic gravitational field of isolated bodies is discussed. 
  In this paper, we develop an iterative scheme to enable the explicit calculation of an arbitrary post-Newtonian order for a relativistic body that reduces to the Maclaurin spheroid in the appropriate limit. This scheme allows for an analysis of the structure of the solution in the vicinity of bifurcation points along the Maclaurin sequence. The post-Newtonian expansion is solved explicitly to the fourth order and its accuracy and convergence are studied by comparing it to highly accurate numerical results. 
  One ambiguity in loop quantum gravity is the appearance of a free parameter which is called Immirzi parameter. Recently Dreyer has argued that this parameter may be fixed by considering the quasinormal mode spectrum of black holes, while at the price of changing the gauge group to SO(3) rather than the original one SU(2). Physically such a replacement is not quite natural or desirable. In this paper we study the relationship between the black hole entropy and the quasi normal mode spectrum in the loop quantization of N=1 supergravity. We find that a single value of the Immirzi parameter agrees with the semiclassical expectations as well. But in this case the lowest supersymmetric representation dominates, fitting well with the result based on statistical consideration. This suggests that, so long as fermions are included in the theory, supersymemtry may be favored for the consistency of the low energy limit of loop quantum gravity. 
  The expression of a time-dependent cosmological constant $\lambda \propto 1/t^2$ is interpreted as the energy density of a special type of the quaternionic field. The Lorenz-like force acting on the moving body in the presence of this quaternionic field is determined. The astronomical and terrestrial effects of this field are presented, and the ways how it can be observably detected is discussed. Finally, a new mechanism of the particle creation and an alternative cosmological scenario in the presence of the cosmic quatertionic field is suggested. 
  We introduce here the concept of relative space, an extended 3-space which is recognized as the only space having an operational meaning in the study of the space geometry of a rotating disk. Accordingly, we illustrate how space measurements are performed in the relative space, and we show that an old-aged puzzling problem, that is the Ehrenfest's paradox, is explained in this purely relativistic context. Furthermore, we illustrate the kinematical origin of the tangential dilation which is responsible for the solution of the Ehrenfest's paradox. 
  The scattering problems arising when considering the contribution of the topologically inequivalent configurations of the spinors on Reissner-Nordstr\"{o}m black holes to the Hawking radiation are correctly stated. The corresponding $S$-matrices are described and presented in the form convenient to numerical computations. 
  The extreme Reissner-Nordstr\"om black holes have zero surface gravity. However, a semi-classical analysis seems to be ill-definite for these objects and apparently no notion of temperature exists for them. It is argued here that these properties are shared for all kind of black holes whose surface gravity is zero. Two examples are worked out explicitely: the scalar-tensor cold black holes and extreme black holes resulting from a gravity system coupled to a generalized Maxwell field in higher dimensions. The reasons for this anomolous behaviour are discussed as well as its thermodynamics implications. 
  Implementing Poincar\'e's `geometric conventionalism' a scalar Lorentz-covariant gravity model is obtained based on gravitationally modified Lorentz transformations (or GMLT). The modification essentially consists of an appropriate space-time and momentum-energy scaling ("normalization") relative to a nondynamical flat background geometry according to an isotropic, nonsingular gravitational `affecting' function Phi(r). Elimination of the gravitationally `unaffected' S_0 perspective by local composition of space-time GMLT recovers the local Minkowskian metric and thus preserves the invariance of the locally observed velocity of light. The associated energy-momentum GMLT provides a covariant Hamiltonian description for test particles and photons which, in a static gravitational field configuration, endorses the four `basic' experiments for testing General Relativity Theory: gravitational i) deflection of light, ii) precession of perihelia, iii) delay of radar echo, iv) shift of spectral lines. The model recovers the Lagrangian of the Lorentz-Poincar\'e gravity model by Torgny Sj\"odin and integrates elements of the precursor gravitational theories, with spatially Variable Speed of Light (VSL) by Einstein and Abraham, and gravitationally variable mass by Nordstr\"om. 
  As a rule in General Relativity the spacetime metric fixes the Einstein tensor and through the Field Equations (FE) the energy-momentum tensor. However one cannot write the FE explicitly until a class of observers has been considered. Every class of observers defines a decomposition of the energy-momentum tensor in terms of the dynamical variables energy density ($\mu$), the isotropic pressure ($p$), the heat flux $q^a$ and the traceless anisotropic pressure tensor $\pi_{ab}$. The solution of the FE requires additional assumptions among the dynamical variables known with the generic name equations of state. These imply that the properties of the matter for a given class of observers depends not only on the energy-momentum tensor but on extra a priori assumptions which are relevant to that particular class of observers. This makes difficult the comparison of the Physics observed by different classes of observers for the {\it same} spacetime metric. One way to overcome this unsatisfactory situation is to define the extra condition required among the dynamical variables by a geometric condition, which will be based on the metric and not to the observers. Among the possible and multiple conditions one could use the consideration of collineations. We examine this possibility for the Friedmann-Lema\^{i}tre-Robertson-Walker models admitting matter and Ricci collineations and determine the equations of state for the comoving observers. We find linear and non-linear equations of state, which lead to solutions satisfying the energy conditions, therefore describing physically viable cosmological models. 
  A broader perspective is suggested for the study of higher dimensional cosmological models. [1986: Considerations involving the Einstein constraints and the Ricci form of the evolution equations for spatially homogeneous spacetimes in 4 or more dimensions in the Taub time gauge (constant densitized lapse) are used to show how many of the special exact solutions found at random fit together into a larger picture in which the Taub spacetime solution plays an instructive role. Concludes with some remarks on chaos.] 
  Non-rotating black holes in three and four dimensions are shown to possess a canonical entropy obeying the Bekenstein-Hawking area law together with a leading correction (for large horizon areas) given by the logarithm of the area with a {\it universal} finite negative coefficient, provided one assumes that the quantum black hole mass spectrum has a power law relation with the quantum area spectrum found in Non-perturbative Canonical Quantum General Relativity. The thermal instability associated with asymptotically flat black holes appears in the appropriate domain for the index characterising this power law relation, where the canonical entropy (free energy) is seen to turn complex. 
  A global picture is drawn tying together most exact cosmological solutions of gravitational theories in four or more spacetime dimensions. 
  I review the formalism of loop quantum gravity, in both its real and complex formulations, and spin foam theory which is its path integral counterpart. Spin networks for non-compact groups are introduced (following hep-th/0205268) to deal with gauge invariant structures based on the Lorentz group. The whole formalism is studied in details in three dimensions in both its canonical formulation (loop gravity) and its spin foam formulation. The main output (following gr-qc/0212077) is the discreteness of timelike intervals and the continuous character of spacelike distances even at the quantum level. Then it is explained how to extend these considerations to the 4-dimensional case. I review the Barrett-Crane model, its geometrical interpretation, its link with general relativity and the role of causality. It is shown to be the history formulation of a covariant canonical formulation of loop gravity (following gr-qc/0209105), whose link with standard loop quantum gravity is discussed. Similarly to the 3d case, spacelike areas turn out continuous. Finally, ways of extracting informations from the non-perturbative spin foam structures are discussed. 
  We show that four-dimensional Lorentzian metrics admitting a global spacelike Lie group of isometries, $G_{1}={\mathbb R}$, which obey the Einstein equations for vacuum and certain types of matter, cannot contain apparent horizons. The assumed global isometry allows for the dimensional reduction of the (3+1) system to a (2+1) picture, wherein the four-dimensional metric fields act formally as matter fields. A theorem by Ida allows one to check for the absence of apparent horizons in the dimensionally reduced spacetime, with the four-dimensional results following from the topological product nature of the corresponding manifold. We argue that the absence of apparent horizons in spacetimes with translational symmetry constitutes strong evidence for the validity of the hoop conjecture, and also hints at possible (albeit arguably unlikely) generic violations of strong cosmic censorship. 
  We consider the brane-world generalisation of the Godel universe and analyse its dynamical interaction with the bulk. The exact homogeneity of the standard Godel spacetime no longer holds, unless the bulk is also static. We show how the anisotropy of the Godel-type brane is dictated by that of the bulk and find that the converse is also true. This determines the precise evolution of the nonlocal anisotropic stresses, without any phenomenological assumptions, and leads to a self-consistent closed set of equations for the evolution of the Godel brane. We also examine the causality of the Godel brane and show that the presence of the bulk cannot prevent the appearance of closed timelike curves. 
  As we showed in a preceding arXiv:gr-qc Einstein equations, conveniently written, provide the more orthodox and simple description of cosmological models with a time dependent speed of light $c$. We derive here the concomitant dependence of the electric permittivity $\epsilon$, the magnetic permeability $\mu$, the unit of charge $e$, Plank's constant $h$, under the assumption of the constancy of the fine structure constant $\alpha$, and the masses of elementary particles $m$. As a consequence of these concomitant dependences on time they remain constant their ratios $e/m$ as well as their Compton wave length $\lambda_c$ and their classical radius $r_0$. 
  Motivation is given for trying a theory of gravity with a preferred reference frame (``ether'' for short). One such theory is summarized, that is a scalar bimetric theory. Dynamics is governed by an extension of Newton's second law. In the static case, geodesic motion is recovered together with Newton's attraction field. In the static spherical case, Schwarzschild's metric is got. An asymptotic scheme of post-Minkowskian (PM) approximation is built by associating a conceptual family of systems with the given weakly-gravitating system. It is more general than the post-Newtonian scheme in that the velocity may be comparable with $c$. This allows to justify why the 0PM approximation of the energy rate may be equated to the rate of the Newtonian energy, as is usually done. At the 0PM approximation of this theory, an isolated system loses energy by quadrupole radiation, without any monopole or dipole term. It seems plausible that the observations on binary pulsars (the pulse data) could be nicely fitted with a timing model based on this theory. 
  The Rayleigh criterion is used to study the stability of circular orbits of particles moving around static black holes surrounded by different axially symmetric structures with reflection symmetry, like disks, rings and halos. We consider three models of disks one of infinite extension and two finite, and one model of rings. The halos are represented by external quadrupole moments (either oblate or prolate). Internal quadrupole perturbation (oblate and prolate) are also considered. For this class of disks the counter-rotation hypothesis implies that the stability of the disks is equivalent to the stability of test particles. The stability of Newtonian systems is also considered and compared with the equivalent relativistic situation. We find that the general relativistic dynamics favors the formation of rings. 
  A new method is presented for the determination of Ricci Collineations (RC) and Matter Collineations (MC) of a given spacetime, in the cases where the Ricci tensor and the energy momentum tensor are non-degenerate and have a similar form with the metric. This method reduces the problem of finding the RCs and the MCs to that of determining the KVs whereas at the same time uses already known results on the motions of the metric. We employ this method to determine all hypersurface homogeneous locally rotationally symmetric spacetimes, which admit proper RCs and MCs. We also give the corresponding collineation vectors. 
  The Lorentzian Hamiltonian constraint is solved for isotropic loop quantum cosmology coupled to a massless scalar field. As in the Euclidean case, the discreteness of quantum geometry removes the classical singularity from the quantum Friedmann models. In spite of the absence of the classical singularity, a modified DeWitt initial condition is incompatible with a late-time smooth behavior. Further, the smooth behavior is recovered only for positive or negatives times but not both. An important feature, which is shared with the Euclidean case, is a minimal initial energy of the order of the Planck energy required for the system to evolve dynamically. By forming wave packets of the matter field an explicit evolution in terms of an internal time is obtained. 
  In this note I discuss the problem of cosmological singularities within gauge theories of gravitation. Solutions of cosmological equations with the scalar field are considered. 
  This paper has been withdrawn, because the result turned out to be well known. 
  Exact solutions, in terms of special functions, of all wave equations $% u_{xx} - u_{tt} = V(x) u(t,x)$, characterised by eight inequivalent time independent potentials and by variable separation, have been found. The real valueness of the solutions from computer algebra programs is not always manifest and in this work we provide ready to use solutions. We discussed especially the potential $\cosh^{-2}x (m_1 + m_2 \sinh x)$. Such potential approximates the Schwarzschild black hole potential for even parity. 
  This work deals with the motion of a radially falling star in Schwarzschild geometry and correctly identifies radiation reaction terms by the perturbative method. The results are: i) identification of all terms up to first order in perturbations, second in trajectory deviation, and mixed terms including lowest order radiation reaction terms; ii) renormalisation of all divergent terms by the $\zeta$ Riemann and Hurwitz functions. The work implements a method previously identified by one of the authors and corrects some current misconceptions and results. 
  We have found that the hierarchial problems appearing in cosmology is a manifestation of the quantum nature of the universe. The universe is still described by the same formulae that once hold at Planck's time. The universe is found to be governed by the Machian equation, $G M=Rc^2$, where $M$ and $R$ are mass and radius of the universe. A Planck's constant for different cosmic scales is provided. The status of the universe at different stages is shown to be described in terms of the fundamental constants (eg. \textcolor{blue}{$c, \hbar, G, \Lambda, H$}) only. The concept of maximal (minimal) acceleration, power, temperature, etc., is introduced and justified. The electromagnetic interactions are shown to be active at a cosmic level. Their contribution would exclude the inclusion of dark energy in cosmology. 
  The quantum gauge general relativity is proposed in the framework of quantum gauge theory of gravity. It is formulated based on gauge principle which states that the correct symmetry for gravitational interactions should be gravitational gauge symmetry. The gravitational gauge group is studied in the paper. Then gravitational gauge interactions of pure gravitational gauge field is studied. It is found that the field equation of gravitational gauge field is just the Einstein's field equation. After that, the gravitational interactions of scalar field, Dirac field and vector fields are studied, and unifications of fundamental interactions are discussed. Path integral quantization of the theory is studied in the paper. The quantum gauge general relativity discussed in this paper is a perturbatively renormalizable quantum gravity, which is one of the most important advantage of the quantum gauge general relativity proposed in this paper. A strict proof on the renormalizability of the theory is also given in this paper. Another important advantage of the quantum gauge general relativity is that it can explain both classical tests of gravity and quantum effects of gravitational interactions, such as gravitational phase effects found in COW experiments and gravitational shielding effects found in Podkletnov experiments. For all classical effects of gravitational interactions, such as classical tests of gravity and cosmological model, quantum gauge general relativity gives out the same theoretical predictions as that of the Einstein's general relaitvity. 
  The perfect fluid cosmology in the 1+d+D dimensional Kaluza-Klein spacetimes for an arbitrary barotropic equation of state $p= n \rho$ is quantized by using the Schutz's variational formalism. We make efforts in the mathematics to solve the problems in two cases. For the first case of the stiff fluid $n=1$ we exactly solve the Wheeler-DeWitt equation when the $d$ space is flat. After the superposition of the solutions we analyze the Bohmian trajectories of the final-stage wave-packet functions and show that the flat $d$ spaces and the compact $D$ spaces will eventually evolve into finite scale functions. For the second case of $n \approx 1$, we use the approximated wavefunction in the Wheeler-DeWitt equation to find the analytic forms of the final-stage wave-packet functions. After analyzing the Bohmian trajectories we show that the flat $d$ spaces will be expanding forever while the scale function of the contracting $D$ spaces would not become zero within finite time. Our investigations indicate that the quantum effect in the quantum perfect-fluid cosmology could prevent the extra compact $D$ spaces in the Kaluza-Klein theory from collapsing into a singularity or that the "crack-of-doom" singularity of the extra compact dimensions is made to occur at $t=\infty$. 
  Motivated from recent string theoretic results, a tachyonic potential is constructed for a spatially homogeneous and anisotropic background cosmology. 
  The principle of nuclear democracy is invoked to prove the formation of stable quantized gravitational bound states of primordial black holes called Holeums. The latter come in four varieties: ordinary Holeums H, Black Holeums BH, Hyper Holeums HH and the massless Lux Holeums LH.These Holeums are invisible because the gravitational radiation emitted by their quantum transitions is undetectable now. The copiously produced Holeums form an important component of the dark matter and the Lux Holeums an important component of the dark energy in the universe. A segregation property puts the Holeums mainly in the galactic haloes (GH) and the domain walls (DW) explaining the latters' invisibility now. Cosmic rays (CR) are produced by two exploding black holes created in a pressure-ionization of a stable Holeum. Our prediction that more CRs will be emitted by the haloes than by the discs of galaxies already has a strong empirical support. The concentration of the Hs and the HHs in the GHs and the DWs lead to the formation of Holeum-stars emitting the CRs and the gravitational waves(GW).Innumerable explosions of BHs at the time of decoupling of gravity from the other interactions lead to inflation and baryon asymmetry. A substantial cosmic back ground of matter and GWs and an infra-quantum gravity (infra-QG) band and an ultra-QG band of GWs and their emission frequencies are predicted. A unique quantum system containing matter-energy oscillations is found. 
  Quasiequilibrium sequences of binary neutron stars are numerically calculated in the framework of the Isenberg-Wilson-Mathews (IWM) approximation of general relativity. The results are presented for both rotation states of synchronized spins and irrotational motion, the latter being considered as the realistic one for binary neutron stars just prior to the merger. We assume a polytropic equation of state and compute several evolutionary sequences of binary systems composed of different-mass stars as well as identical-mass stars with adiabatic indices gamma=2.5, 2.25, 2, and 1.8. From our results, we propose as a conjecture that if the turning point of binding energy (and total angular momentum) locating the innermost stable circular orbit (ISCO) is found in Newtonian gravity for some value of the adiabatic index gamma_0, that of the ADM mass (and total angular momentum) should exist in the IWM approximation of general relativity for the same value of the adiabatic index. 
  The dimensional structure of space-time is investigated according to physical and mathematical methods. We show that ther are various empirical and theoretical restrictions on the number of independent dimensions of space-time, consequently there is no physical and mathematical evidence for a space-time with four independent dimensions. 
  It is assumed that a black hole can be disturbed in such a way that a ringdown gravitational wave would be generated. This ringdown waveform is well understood and is modelled as an exponentially damped sinusoid. In this work we use this kind of waveform to study the performance of the SCHENBERG gravitational wave detector. This first realistic simulation will help us to develop strategies for the signal analysis of this Brazilian detector. We calculated the signal-to-noise ratio as a function of frequency for the simulated signals and obtained results that show that SCHENBERG is expected to be sensitive enough to detect this kind of signal up to a distance of $\sim 20\mathrm{kpc}$. 
  We show that for generic sliced spacetimes global hyperbolicity is equivalent to space completeness under the assumption that the lapse, shift and spatial metric are uniformly bounded. This leads us to the conclusion that simple sliced spaces are timelike and null geodesically complete if and only if space is a complete Riemannian manifold. 
  We establish exactly solvable models for the motion of neutral particles, electrically charged point and spin particles (U(1) symmetry), isospin particles (SU(2) symmetry), and particles with color charges (SU(3) symmetry) in a gravitational wave background. Special attention is devoted to parametric effects induced by the gravitational field. In particular, we discuss parametric instabilities of the particle motion and parametric oscillations of the vectors of spin, isospin, and color charge. 
  The propagation of signals in space-time is considered on the basis of the notion of null (isotropic) vector field in spaces with affine connections and metrics as models of space or of space-time. The Doppler effect is generalized for these types of spaces. The notions of aberration, standard (longitudinal) Doppler effect, and transversal Doppler effect are considered. On their grounds, the Hubble effect appears as Doppler effect with explicit forms of the centrifugal (centripetal) and Coriolis velocities and accelerations. Doppler effect, Hubble effect, and aberration could be used in mechanics of continuous media and in other classical field theories in the same way as the standard Doppler effect is used in classical and relativistic mechanics. 
  The notions of centrifugal (centripetal) and Coriolis velocities and accelerations are introduced and considered in spaces with affine connections and metrics used as models of space or of space-time. It is shown that these types of velocities and accelerations are generated by the relative motions between mass elements in a continuous media or of particles. The velocities and accelerations are closely related to the kinematic characteristics of the relative velocity and of the relative acceleration. The relation between the centrifugal (centripetal) velocity and the Hubble law is found. The centrifugal (centripetal) acceleration could be interpreted as gravitational acceleration as it has been done in the Einstein theory of gravitation. This fact could be used as a basis for working out of new gravitational theories in spaces with affine connections and metrics. 
  A recently presented method for the study of evolving self-gravitating relativistic spheres is applied to the description of the evolution of relativistic polytropes. Two different definitions of relativistic polytrope, yielding the same Newtonian limit, are considered. Some examples are explicitly worked out, describing collapsing polytropes and bringing out the differences between both types of polytropes. 
  In any space-time, it is possible to have a family of observers who have access to only part of the space-time manifold, because of the existence of a horizon. We demand that \emph{physical theories in a given coordinate system must be formulated entirely in terms of variables that an observer using that coordinate system can access}. In the coordinate frame in which these observers are at rest, the horizon manifests itself as a (coordinate) singularity in the metric tensor. Regularization of this singularity removes the inaccessible region, and leads to the following consequences: (a) The non-trivial topological structure for the effective manifold allows one to obtain the standard results of quantum field theory in curved space-time. (b) In case of gravity, this principle requires that the effect of the unobserved degrees of freedom should reduce to a boundary contribution $A_{\rm boundary}$ to the gravitational action. When the boundary is a horizon, $A_{\rm boundary}$ reduces to a single, well-defined term proportional to the area of the horizon. Using the form of this boundary term, it is possible to obtain the full gravitational action in the semiclassical limit. (c) This boundary term must have a quantized spectrum with uniform spacing, $\Delta A_{boundary}=2\pi\hbar$, in the semiclassical limit. This, in turn, yields the following results for semiclassical gravity: (i) The area of any one-way membrane is quantized. (ii) The information hidden by a one-way membrane amounts to an entropy, which is always one-fourth of the area of the membrane in the leading order. (iii) In static space-times, the action for gravity can be given a purely thermodynamic interpretation and the Einstein equations have a formal similarity to laws of thermodynamics. 
  Each approach to the quantum-gravity problem originates from expertise in one or another area of theoretical physics. The particle-physics perspective encourages one to attempt to reproduce in quantum gravity as much as possible of the successes of the Standard Model of particle physics, and therefore, as done in String Theory, the core features of quantum gravity are described in terms of graviton-like exchange in a background classical spacetime. From the general-relativity perspective it is natural to renounce to any reference to a background spacetime, and to describe spacetime in a way that takes into account the in-principle limitations of measurements. The Loop Quantum Gravity approach and the approaches based on noncommutative geometry originate from this general-relativity perspective. The condensed-matter perspective, which has been adopted in a few recent quantum-gravity proposals, naturally leads to scenarios in which some familiar properties of spacetime are only emergent, just like, for example, some emergent collective degrees of freedom are relevant to the description of certain physical systems only near a critical point. Both from the general-relativity perspective and from the condensed-matter perspective it is natural to explore the possibility that quantum gravity might have significant implications for the fate of Lorentz symmetry in the Planckian regime. From the particle-physics perspective there is instead no obvious reason to renounce to exact Lorentz symmetry, although (``spontaneous'') Lorentz symmetry breaking is of course possible. A fast-growing phenomenological programme looking for Planck-scale departures from Lorentz symmetry can contribute to this ongoing debate. 
  We prove that strictly stationary spacetimes cannot contain closed trapped nor marginally trapped surfaces. The result is purely geometric and holds in arbitrary dimension. Other results concerning the interplay between (generalized) symmetries and trapped submanifolds are also presented. 
  We study radial perturbations of general relativistic stars with elastic matter sources. We find that these perturbations are governed by a second order differential equation which, along with the boundary conditions, defines a Sturm-Liouville type problem that determines the eigenfrequencies. Although some complications arise compared to the perfect fluid case, leading us to consider a generalisation of the standard form of the Sturm-Liouville equation, the main results of Sturm-Liouville theory remain unaltered. As an important consequence we conclude that the mass-radius curve for a one-parameter sequence of regular equilibrium models belonging to some particular equation of state can be used in the same well-known way as in the perfect fluid case, at least if the energy density and the tangential pressure of the background solutions are continuous. In particular we find that the fundamental mode frequency has a zero for the maximum mass stars of the models with solid crusts considered in Paper I of this series. 
  In this paper, using the Newman-Penrose formalism, we find the Maxwell equations in NUT space and after separation into angular and radial components solve them analytically. All the angular equations are solved in terms of Jaccobi polynomials. The radial equations are transformed into Hypergeometric and Heun's equations with the right hand sides including terms of different order in the frequency of the perturbation which allow solutions in the expansion of this parameter. 
  Gravitational waves bring about the relative motion of free test masses. The detailed knowledge of this motion is important conceptually and practically, because the mirrors of laser interferometric detectors of gravitational waves are essentially free test masses. There exists an analogy between the motion of free masses in the field of a gravitational wave and the motion of free charges in the field of an electromagnetic wave. In particular, a gravitational wave drives the masses in the plane of the wave-front and also, to a smaller extent, back and forth in the direction of the wave's propagation. To describe this motion, we introduce the notion of `electric' and `magnetic' components of the gravitational force. This analogy is not perfect, but it reflects some important features of the phenomenon. Using different methods, we demonstrate the presence and importance of what we call the `magnetic' component of motion of free masses. It contributes to the variation of distance between a pair of particles. We explicitely derive the full response function of a 2-arm laser interferometer to a gravitational wave of arbitrary polarization. We give a convenient description of the response function in terms of the spin-weighted spherical harmonics. We show that the previously ignored `magnetic' component may provide a correction of up to 10 %, or so, to the usual `electric' component of the response function. The `magnetic' contribution must be taken into account in the data analysis, if the parameters of the radiating system are not to be mis-estimated. 
  A homogeneous, Kantowski-Sachs type, bouncing brane-world universe is presented. The bulk has a positive cosmological constant and the Killing algebra $so(1,3)\oplus so(3)$. The totality of the source terms of the effective Einstein equation combine to a solid with different radial and tangential pressures. 
  The main results of papers gr-qc/0307026 and gr-qc/0312068 are formulated. These results are opposite to conclusions of paper astro-ph/0305039 and comments gr-qc/0309036. 
  We present positive energy theorems in asymptotically translationally invariant spacetimes which can be applicable to black strings and charged branes. We also address the bound property of the tension and charge of branes. 
  Self-consistent system of spinor, scalar and BI gravitational fields in presence of magneto-fluid and $\Lambda$-term is considered. Assuming that the expansion of the BI universe is proportional to the $\sigma_1^1$ component of the shear tensor, exact solutions for the metric functions, as well as for scalar and spinor fields are obtained. For a non-positive $\Lambda$ the initially anisotropic space-time becomes isotropic one in the process of expansion, whereas, for $\Lambda > 0$ an oscillatory mode of expansion of the BI model occurs. 
  We explore some fundational issues regarding the splitting of D-dimensional EFE's w.r.t timelike and spacelike (D-1)-dimensional hypersurfaces, first without and then with thin matter sheets such as branes. We begin to implement methodology, that is well-established for the GR CP and IVP, in the new field of GR-based braneworlds, identifying and comparing many different choices of procedure. We abridge fragmentary parts of the literature of embeddings, putting the Campbell--Magaard theorem into context. We recollect and refine arguments why York and not elimination methods are used for the GR IVP. We compile a list of numerous mathematical and physical impasses to using timelike splits, whereas spacelike splits are known to be well-behaved. We however pursue both options to make contact with the current braneworld literature which is almost entirely based on timelike splits. We look at the Shiromizu-Maeda -Sasaki braneworld by means of reformulations which emphasize different aspects from the original formulation. We show that what remains of the York method in the timelike case generalizes heuristic bulk construction schemes. We formulate timelike (brane) versions of the thin sandwich conjecture. We discuss whether it is plausible to remove singularities by timelike embeddings. We point out how the braneworld geodesic postulates lead to further difficulties with the notion of singularities than in GR where these postulates are simpler. Having argued for the use of the spacelike split, we study how to progress to the construction of more general data sets for spaces partially bounded by branes. Boundary conditions are found and algorithms provided. Working with (finitely) thick branes would appear to facilitate such a study. 
  We discuss the Einstein energy-momentum complex and the Bergmann-Thomson angular momentum complex in general relativity and calculate them for space-time homogeneous Goedel universes. The calculations are performed for a dust acausal model and for a scalar-field causal model. It is shown that the Einstein pseudotensor is traceless, not symmetric, the gravitational energy is "density" is negative and the gravitational Poynting vector vanishes. Significantly, the total (gravitational and matter) energy "density" fro the acausal model is zero while for the casual model it is negative.The Bergmann-Thomson angular momentum complex does not vanish for both G\"odel models. 
  Contents:  Community news:    We hear that... by Jorge Pullin  Research briefs:    Update on a busy year for LIGO, by Stan Whitcomb    First year results from WMAP, by Rachel Bean    Short range searches for non-Newtonian gravity, by M. Varney and J. Long  Conference reports:    Xth Brazilian school of cosmology and gravitation, by Mario Novello    6th East coast gravity meeting, by David Fiske    5th Edoardo Amaldi meeting, by Alain Brillet    Pacific coast gravity meeting, by Charles Torre    Astrophysics of gravitational wave sources, by Joan Centrella    Gravitational interaction of compact objects, by M. Choptuik, E. Flanagan, L. Lehner    PriceFest, by John Whelan    Gravitation: a decennial perspective, by Jorge Pullin 
  The temperature correction to the free energy of the gravitational field is considered which does not depend on the Planck energy physics. The leading correction may be interpreted in terms of the temperature dependent effective gravitational constant G_{eff}. The temperature correction to G_{eff}^{-1} appears to be valid for all temperatures T<< E_{Planck}. It is universal since it is determined only by the number of fermionic and bosonic fields with masses m<< T, does not contain the Planck energy scale E_{Planck} which determines the gravitational constant at T=0, and does not depend on whether or not the gravitational field obeys the Einstein equations. That is why this universal modification of the free energy for gravitational field can be used to study thermodynamics of quantum systems in condensed matter (such as quantum liquids superfluid 3He and 4He), where the effective gravity emerging for fermionic and/or bosonic quasiparticles in the low-energy corner is quite different from the Einstein gravity. 
  DC electric fields can combine with test mass charging and thermal dielectric voltage noise to create significant force noise acting on the drag-free test masses in the LISA (Laser Interferometer Space Antenna) gravitational wave mission. This paper proposes a simple technique to measure and compensate average stray DC potentials at the mV level, yielding substantial reduction in this source of force noise. We discuss the attainable resolution for both flight and ground based experiments. 
  We investigate the properties of global monopoles with an event horizon. We find that there is an unstable circular orbit even if a particle does not have an angular momentum when the core mass is negative. We also obtain the asymptotic form of solutions when the event horizon is much larger than the core radius of the monopole, and discuss if they could be a model of galactic halos. 
  An exact analytical solution describing the interior of a charged strange quark star is found under the assumption of spherical symmetry and the existence of a one-parameter group of conformal motions. The solution describes a unique static charged configuration of quark matter with radius $R=9.46$ km and total mass $M=2.86M_{\odot}$. 
  The abstract boundary construction of Scott and Szekeres is a general and flexible way to define singularities in General Relativity. The abstract boundary construction also proves of great utility when applied to questions about more general boundary features of space-time. Within this construction an essential singularity is a non-regular boundary point which is accessible by a curve of interest (e.g. a geodesic) within finite (affine) parameter distance and is not removable. Ashley and Scott proved the first theorem linking abstract boundary essential singularities with the notion of causal geodesic incompleteness for strongly causal, maximally extended space-times. The relationship between this result and the classical singularity theorems of Penrose and Hawking has enabled us to obtain abstract boundary singularity theorems. This paper describes essential singularity results for maximally extended space-times and presents our recent efforts to establish a relationship between the strong curvature singularities of Tipler and Krolak and abstract boundary essential singularities. 
  To find what influence the charge of the black hole $Q$ will bring to the evolution of the quasinormal modes, we calculate the quasinormal frequencies of the neutrino field (charge $e=0$) perturbations and those of the massless Dirac field ($e\neq 0$) perturbations in the RN metric. The influences of $Q$, $e$, the momentum quantum number $l$, and the mode number $n$ are discussed. Among the conclusions, the most important one is that, at the stage of quasinormal ringing, the larger when the black hole and the field have the same kind of charge ($eQ>0$), the quasinormal modes of the massless charged Dirac field decay faster than those of the neutral ones, and when $eQ<0$, the massless charged Dirac field decays slower. 
  In this article I present a simple Newtonian heuristic for deriving a weak-field approximation for the spacetime geometry of a point particle. The heuristic is based on Newtonian gravity, the notion of local inertial frames [the Einstein equivalence principle], plus the use of Galilean coordinate transformations to connect the freely falling local inertial frames back to the ``fixed stars''. Because of the heuristic and quasi-Newtonian manner in which the spacetime geometry is obtained, we are at best justified in expecting it to be a weak-field approximation to the true spacetime geometry. However, in the case of a spherically symmetric point mass the result is coincidentally an exact solution of the full vacuum Einstein field equations -- it is the Schwarzschild geometry in Painleve--Gullstrand coordinates.   This result is much stronger than the well-known result of Michell and Laplace whereby a Newtonian argument correctly estimates the value of the Schwarzschild radius -- using the heuristic presented in this article one obtains the entire Schwarzschild geometry. The heuristic also gives sensible results -- a Riemann flat geometry -- when applied to a constant gravitational field. Furthermore, a subtle extension of the heuristic correctly reproduces the Reissner--Nordstrom geometry and even the de Sitter geometry. Unfortunately the heuristic construction is not truly generic. For instance, it is incapable of generating the Kerr geometry or anti-de Sitter space.   Despite this limitation, the heuristic does have useful pedagogical value in that it provides a simple and direct plausibility argument for the Schwarzschild geometry. 
  For general relativistic equilibrium stellar models (stationary axisymmetric asymptotically flat and convection-free) with differential rotation, it is shown that for a wide class of rotation laws the distribution of angular velocity of the fluid has a sign, say "positive", and then both the dragging rate and the angular momentum density are positive. In addition, the "mean value" (with respect to an intrinsic density) of the dragging rate is shown to be less than the mean value of the fluid angular velocity (in full general, without having to restrict the rotation law, nor the uniformity in sign of the fluid angular velocity); this inequality yields the positivity and an upper bound of the total rotational energy. 
  There is proven a theorem, to the effect that a material body in general relativity, in a certain limit of sufficiently small size and mass, moves along a geodesic. 
  We consider a particular solution to Slavnov-Taylor identity in four-dimensional supergravity. The consideration is performed for pure supergravity, no matter superfields are included. The solution is obtained by inserting dressing functions into ghost part of the classical action for supergravity.As a consequence, physical part of the effective action is local invariant with respect to diffeomorphism and structure groups of transformation for dressed effective superfields of vielbein and spin connection. 
  An approximation method to study the properties of a small black hole located on the TeV brane in the Randall-Sundrum type I scenario is presented. The method enables us to find the form of the metric close to the matter distribution when its asymptotic form is given. The short range solution is found as an expansion in the ratio between the Schwarzschild radius of the black hole and the curvature length of the bulk. Long range properties are introduced using the linearized gravity solution as an asymptotic boundary condition. The solution is found up to first order. It is valid in the region close to the horizon but is not valid on the horizon. The regularity of the horizon is still under study. 
  We show that some of the recent results reported in gr-qc/0308049 are based on assumptions which are in contrast with general properties of ``Doubly Special Relativity'' and/or with Planck-scale physics models. 
  We present an exhaustive analysis of scalar, electromagnetic and gravitational perturbations in the background of a Schwarzchild-de Sitter spacetime. The field propagation is considered by means of a semi-analytical (WKB) approach and two numerical schemes: the characteristic and general initial value integrations. The results are compared near the extreme regime, and a unifying picture is established for the dynamics of different spin fields. Although some of the results just confirm usual expectations, a few surprises turn out to appear, as the dependence on the non-characteristic initial conditions of the non-vanishing asymptotic value for l=0 mode scalar fields. 
  We present an exhaustive analysis of scalar, electromagnetic and gravitational perturbations in the background of Schwarzchild-de Sitter and Reissner-Nordstrom-de Sitter spacetimes. The field propagation is considered by means of a semi-analytical (WKB) approach and two numerical schemes: the characteristic and general initial value integrations. The results are compared near the extreme cosmological constant regime, where analytical results are presented. A unifying picture is established for the dynamics of different spin fields. 
  If the observable universe is a braneworld of Randall-Sundrum type, then particle interactions at high energies will produce 5-dimensional gravitons that escape into the bulk. As a result, the Weyl energy density on the brane does not behave like radiation in the early universe, but does so only later, in the low energy regime. Recently a simple model was proposed to describe this modification of the Randall-Sundrum cosmology. We investigate the dynamics of this model, and find the exact solution of the field equations. We use a dynamical systems approach to analyze global features of the phase space of solutions. 
  In the context of canonical quantum gravity, we study an alternative real quantisation scheme, which is arising by relating simpler Riemannian quantum theory to the more complicated physical Lorentzian theory - the generalised Wick transform. On the symmetry reduced models, homogenous Bianchi cosmology and 2+1 gravity, we investigate its generalised construction principle, demonstrate that the emerging quantum theory is equivalent to the one obtained from standard quantisation and how to obtain physical states in Lorentzian gravity from Wick transforming solutions of Riemannian quantum theory. 
  We show that rotating dyonic black holes with static and counterrotating horizon exist in Einstein-Maxwell-dilaton theory when the dilaton coupling constant exceeds the Kaluza-Klein value. The black holes with static horizon bifurcate from the static black holes. Their mass decreases with increasing angular momentum, their horizons are prolate. 
  We consider the asymptotic dynamics of the Einstein-Maxwell field equations for the class of non-tilted Bianchi cosmologies with a barotropic perfect fluid and a pure homogeneous source-free magnetic field, with emphasis on models of Bianchi type VII_{0}, which have not been previously studied. Using the orthonormal frame formalism and Hubble-normalized variables, we show that, as is the case for the previously studied class A magnetic Bianchi models, the magnetic Bianchi VII_{0} cosmologies also exhibit an oscillatory approach to the initial singularity. However, in contrast to the other magnetic Bianchi models, we rigorously establish that typical magnetic Bianchi VII_{0} cosmologies exhibit the phenomena of asymptotic self-similarity breaking and Weyl curvature dominance in the late-time regime. 
  A cosmological scenario with two branes (A and B) moving in a 5-dimensional bulk is considered. As in the case of ecpyrotic and born-again braneworld models it is possible that the branes collide. The energy-momentum tensor is taken to describe a perfect barotropic fluid on the A-brane and a phenomenological time-dependent "cosmological constant" on the B-brane. The A-brane is identified with our Universe and its cosmological evolution in the approximation of a homogeneous and isotropic brane is analysed. The dynamics of the radion (a scalar field on the brane) contains information about the proper distance between the branes. It is demonstrated that the deSitter type solutions are obtained for late time evolution of the braneworld and accelerative behaviour is anticipated at the present time. 
  We study the generalized scalar tensor theory with a potential in the Bianchi type I model by using the ADM formalism. We examine the conditions for the Universe to be in expansion, isotropic and with a positive potential at late time in the Brans-Dicke and Einstein frames. In particular, we analyse the two important cases where metric functions tend, in an asymptotic way, toward power or exponential laws in the Einstein frame. 
  The dynamics of the hyperextended scalar-tensor theory in the empty Bianchi type I model is investigated. We describe a method giving the sign of the first and second derivatives of the metric functions whatever the coupling function. Hence, we can predict if a theory gives birth to expanding, contracting, bouncing or inflationary cosmology. The dynamics of a string inspired theory without antisymetric field strength is analysed. Some exact solutions are found. 
  We look for sufficient conditions such that the scalar curvature, Ricci and Kretchmann scalars be bounded in Hyperextended Scalar Tensor theory for Bianchi models. We find classes of gravitation functions and Brans-Dicke coupling functions such that the theories thus defined avoid the singularity. We compare our results with these found by Rama in the framework of the Generalised Scalar Tensor theory for the FLRW models. 
  We study in which conditions the Hyperextended Scalar Tensor theory in an FLRW background admits a Noether symmetry and derive the vectors field generating it. 
  A new approach to the dynamics of the universe based on work by O Murchadha, Foster, Anderson and the author is presented. The only kinematics presupposed is the spatial geometry needed to define configuration spaces in purely relational terms. A new formulation of the relativity principle based on Poincare's analysis of the problem of absolute and relative motion (Mach's principle) is given. The enire dynamics is based on shape and nothing else. It leads to much stronger predictions than standard Newtonian theory. For the dynamics of Riemannian 3-geometries on which matter fields also evolve, implementation of the new relativity principle establishes unexpected links between special relativity, general relativity and the gauge principle. They all emerge together as a self-consistent complex from a unified and completely relational approach to dynamics. A connection between time and scale invariance is established. In particular, the representation of general relativity as evolution of the shape of space leads to unique definition of simultaneity. This opens up the prospect of a solution of the problem of time in quantum gravity on the basis of a fundamental dynamical principle. 
  Following the monodromy technique performed by Motl and Neitzke, we consider the analytic determination of the highly damped (asymptotic) quasi-normal modes of small Schwarzschild-de Sitter (SdS) black holes. We comment the result as compared to the recent numerical data of Konoplya and Zhidenko. 
  We show that the absence of equilibrium states of two uncharged spinning particles located on the symmetry axis, revealed in an approximate approach recently employed by Bonnor, can be explained by a non-general character of his approximation scheme which lacks an important arbitrary parameter representing a strut. The absence of this parameter introduces an artificial restriction on the particles' angular momenta, making it impossible to find a physical solution to the balance equations. 
  Among the effects predicted by the General Theory of Relativity for the orbital motion of a test particle, the post-Newtonian gravitomagnetic Lense-Thirring effect is very interesting and, up to now, there is not yet an undisputable experimental direct test of it. To date, the data analysis of the orbits of the existing geodetic LAGEOS and LAGEOS II satellites has yielded a test of the Lense-Thirring effect with a claimed accuracy of 20%-30%. According to some scientists such estimates could be optimistic. Here we wish to discuss the improvements obtainable in this measurement, in terms of reliability of the evaluation of the systematic error and reduction of its magnitude, due to the new CHAMP and GRACE Earth gravity models. 
  The Hyperextended Scalar Tensor theory with a potential is defined by three free functions: the gravitational function, the Brans-Dicke coupling function and the potential. Starting from the expression of the 3-volume and the potential as function of the proper time, we determine the exact solutions of this theory. We study two important cases corresponding to power and exponential laws for the 3-volume and the potential. 
  In this paper we study the isotropisation of a Generalized Scalar-Tensor theory with a massive scalar field. We find it depends on a condition on the Brans-Dicke coupling function and the potential and show that asymptotically the metric functions always tend toward a power or exponential law of the proper time. These results generalise and unify these of De Sitter in the case of a cosmological constant and of Cooley and Kitada in the case of an exponential potential. 
  We look for necessary conditions such that minimally coupled scalar-tensor theory with a massive scalar field and a perfect fluid in the Bianchi type I model isotropises. Then we derive the dynamical asymptotical properties of the Universe. 
  We look for necessary isotropisation conditions of Bianchi class $A$ models with curvature in presence of a massive and minimally coupled scalar field when a function $\ell$ of the scalar field tends to a constant, diverges monotonically or with sufficiently small oscillations. Isotropisation leads the metric functions to tend to a power or exponential law of the proper time $t$ and the potential respectively to vanish as $t^{-2}$ or to a constant. Moreover, isotropisation always requires late time accelerated expansion and flatness of the Universe. 
  We study when a cosmological constant is a natural issue if it is mimicked by the potential of a massive Hyperextended Scalar Tensor theory with a perfect fluid for Bianchi type I and V models. We then deduce a reciprocal Wald theorem giving the conditions such that the potential tends to a non vanishing constant when the gravitational function varies. We also get the conditions allowing the potentiel to vanish or diverge. 
  In this paper we explore the impact of an era -right after reheating- dominated by mini black holes and radiation on the spectrum of the relic gravitational waves. This era may lower the spectrum several orders of magnitude. 
  We show that the Wyman's solution may be obtained from the four-dimensional Einstein's equations for a spherically symmetric, minimally coupled, massless scalar field by using the continuous self-similarity of those equations. The Wyman's solution depends on two parameters, the mass $M$ and the scalar charge $\Sigma$. If one fixes $M$ to a positive value, say $M_0$, and let $\Sigma^2$ take values along the real line we show that this solution exhibits critical behaviour. For $\Sigma^2 >0$ the space-times have eternal naked singularities, for $\Sigma^2 =0$ one has a Schwarzschild black hole of mass $M_0$ and finally for $-M_0^2 \leq \Sigma^2 < 0$ one has eternal bouncing solutions. 
  The cosmological constant combined with Planck's constant and the speed of light implies a quantum of mass of approximately 2 x 10^{-65}g. This follows either from a generic dimensional analysis, or from a specific analysis where the cosmological constant appears in 4D spacetime as the result of a dimensional reduction from higher dimensional relativity (such as 5D induced-matter and membrane theory). In the latter type of theory, all the particles in the universe can be in higher-dimensional contact. 
  The field equations associated with the Born-Infeld-Einstein action are derived using the Palatini variational technique. In this approach the metric and connection are varied independently and the Ricci tensor is generally not symmetric. For sufficiently small curvatures the resulting field equations can be divided into two sets. One set, involving the antisymmetric part of the Ricci tensor $R_{\stackrel{\mu\nu}{\vee}}$, consists of the field equation for a massive vector field. The other set consists of the Einstein field equations with an energy momentum tensor for the vector field plus additional corrections. In a vacuum with $R_{\stackrel{\mu\nu}{\vee}}=0$ the field equations are shown to be the usual Einstein vacuum equations. This extends the universality of the vacuum Einstein equations, discussed by Ferraris et al. \cite{Fe1,Fe2}, to the Born-Infeld-Einstein action. In the simplest version of the theory there is a single coupling constant and by requiring that the Einstein field equations hold to a good approximation in neutron stars it is shown that mass of the vector field exceeds the lower bound on the mass of the photon. Thus, in this case the vector field cannot represent the electromagnetic field and would describe a new geometrical field. In a more general version in which the symmetric and antisymmetric parts of the Ricci tensor have different coupling constants it is possible to satisfy all of the observational constraints if the antisymmetric coupling is much larger than the symmetric coupling. In this case the antisymmetric part of the Ricci tensor can describe the electromagnetic field, although gauge invariance will be broken. 
  We present here the extended-object approach for the explanation and calculation of the self-force phenomenon. In this approach, one considers a charged extended object of a finite size $\epsilon$ that accelerates in a nontrivial manner, and calculates the total force exerted on it by the electromagnetic field (whose source is the charged object itself). We show that at the limit $\epsilon \to 0$ this overall electromagnetic field yields a universal result, independent on the object's shape, which agrees with the standard expression for the self force acting on a point-like charge. This approach has already been considered by many authors, but previous analyses ended up with expressions for the total electromagnetic force that include $O(1/\epsilon)$ terms which do not have the form required by mass-renormalization. (In the special case of a spherical charge distribution, this $\propto 1/\epsilon $ term was found to be 4/3 times larger than the desired quantity.) We show here that this problem was originated from a too naive definition of the notion of ''total electromagnetic force'' used in previous analyses. Based on energy-momentum conservation combined with proper relativistic kinematics, we derive here the correct notion of total electromagnetic force. This completely cures the problematic $O(1/\epsilon)$ term, for any object's shape, and yields the correct self force at the limit $\epsilon \to 0$. In particular, for a spherical charge distribution, the above ''4/3 problem'' is resolved. 
  We derive a new regularization method for the calculation of the (massless) scalar self force in curved spacetime. In this method, the scalar self force is expressed in terms of the difference between two retarded scalar fields: the massless scalar field, and an auxiliary massive scalar field. This field difference combined with a certain limiting process gives the expression for the scalar self-force. This expression provides a new self force calculation method. 
  An explicit fluid flow simulation of electromagnetic wave propagation in the gravitational field of a Schwarzschild black hole is given. The fluid has a constant refractive index and a spherically symmetric inward directed flow. The resulting form of the metric leads to a new coordinate system in which the Schwarzschild vacuum is written in Gordon's form. It is shown that a closely related coordinate system interpolates between the Kerr-Schild and Painleve-Gullstrand coordinates. 
  In this paper we examine the possibility of testing the equivalence principle, in its weak form, by analyzing the orbital motion of a pair of artificial satellites of different composition moving along orbits of identical shape and size in the gravitational field of Earth. It turns out that the obtainable level of accuracy is, realistically, of the order of 10^-10 or slightly better. It is limited mainly by the fact that, due to the unavoidable orbital injection errors, it would not be possible to insert the satellites in orbits with exactly the same radius and that such difference could be known only with a finite precision. The present-day level of accuracy, obtained with torsion balance Earth-based measurements and the analysis of Earth-Moon motion in the gravitational field of Sun with the Lunar Laser Ranging technique, is of the order of 10^-13. The proposed space-based missions STEP, \muSCOPE, GG and SEE aim to reach a 10^-15-10^-18 precision level. 
  If a physical significance should be attributed to the cosmological large number relationship obtained from Sciama's formulation of Mach's Principle, then a number of interesting physical conclusions may be drawn. The Planck length is naturally obtained as the amplitude of waves in a medium whose properties are implied by the relationship. The relativistic internal energy associated with a rest mass is explicitly related to the gravitational potential energy of the Universe, and consistency with the Einstein photon energy is demonstrated. Broader cosmological consequences of this formulation are addressed. 
  We extend the treatment of the problem of the gravitational waves produced by a particle of negligible mass orbiting a Kerr black hole using black hole perturbation theory in the time domain, to elliptic and inclined orbits. We model the particle by smearing the singularities in the source term using narrow Gaussian distributions. We compare results (energy and angular momentum fluxes) for such orbits with those computed using the frequency domain formalism. 
  We calculate the time delay between different relativistic images formed by the gravitational lensing produced by the Gibbons-Maeda-Garfinkle-Horowitz-Stromiger (GMGHS) charged black hole of heterotic string theory. Modeling the supermassive central objects of some galaxies as GMGHS black holes, numerical values of the time delays are estimated and compared with the correspondient Reissner-Nordstrom black holes . The time difference amounts to hours, thus being measurable and permiting to distinguish between General Relativity and String Theory charged black holes. 
  Taylor expanding the cosmological equation of state around the current epoch is the simplest model one can consider that does not make any a priori restrictions on the nature of the cosmological fluid. Most popular cosmological models attempt to be ``predictive'', in the sense that once somea priori equation of state is chosen the Friedmann equations are used to determine the evolution of the FRW scale factor a(t). In contrast, a retrodictive approach might usefully take observational dataconcerning the scale factor, and use the Friedmann equations to infer an observed cosmological equation of state. In particular, the value and derivatives of the scale factor determined at the current epoch place constraints on the value and derivatives of the cosmological equation of state at the current epoch. Determining the first three Taylor coefficients of the equation of state at the current epoch requires a measurement of the deceleration, jerk, and snap -- the second, third, and fourth derivatives of the scale factor with respect to time. Higher-order Taylor coefficients in the equation of state are related to higher-order time derivatives of the scale factor. Since the jerk and snap are rather difficult to measure, being related to the third and fourth terms in the Taylor series expansion of the Hubble law, it becomes clear why direct observational constraints on the cosmological equation of state are so relatively weak; and are likely to remain weak for the foreseeable future. 
  We give an example of a spacetime having an infinite thin rotating cylindrical shell constituted by a charged perfect fluid as a source. As the interior of the shell the Bonnor--Melvin universe is considered, while its exterior is represented by Datta--Raychaudhuri spacetime. We discuss the energy conditions and we show that our spacetime contains closed timelike curves. Trajectories of charged test particles both inside and outside the cylinder are also examined. Expression for the angular velocity of a circular motion inside the cylinder is given. 
  The Universe is modeled as a binary mixture whose constituents are described by a van der Waals fluid and by a dark energy density. The dark energy density is considered either as the quintessence or as the Chaplygin gas. The irreversible processes concerning the energy transfer between the van der Waals fluid and the gravitational field are taken into account. This model can simulate: (a) an inflationary period where the acceleration grows exponentially and the van der Waals fluid behaves like an inflaton; (b) an inflationary period where the acceleration is positive but it decreases and tends to zero whereas the energy density of the van der Waals fluid decays; (c) a decelerated period which corresponds to a matter dominated period with a non-negative pressure; and (d) a present accelerated period where the dark energy density outweighs the energy density of the van der Waals fluid. 
  We make a thorough investigation of the asymptotic quasinormal modes of the four and five-dimensional Schwarzschild black hole for scalar, electromagnetic and gravitational perturbations. Our numerical results give full support to all the analytical predictions by Motl and Neitzke, for the leading term. We also compute the first order corrections analytically, by extending to higher dimensions, previous work of Musiri and Siopsis, and find excellent agreement with the numerical results. For generic spacetime dimension number D the first-order corrections go as $\frac{1}{n^{(D-3)/(D-2)}}$. This means that there is a more rapid convergence to the asymptotic value for the five dimensional case than for the four dimensional case, as we also show numerically. 
  This paper concerns the generation and evolution of the cosmological (large-scale $\sim Mpc$) magnetic fields in an inflationary universe. The universe during inflation is represented by de Sitter space-time. We started with the Maxwell equations in spatially flat Friedmann-Robertson-Walker (FRW) Cosmologies. Then we calculated the wave equations of the magnetic field and electric field for the evolution. We consider the input current that was produced from a massless charged scalar complex field. This field minimally coupled to both gravity and the electromagnetic fields. The Lagrangian for massless scalar electrodynamics is then $L=\sqrt{-g}(D_\mu\phi(D^\mu\phi)^*-{1/4}F_{\mu\nu}F^{\mu\nu})$ . The complex scalar field couples to electromagnetism through the usual gauge covariant derivative $D_\mu=\partial_\mu-ieA_\mu$ . After the quantum field theoretical deduction for the current, we put it back into the wave equation of the magnetic field.   After solving this wave equation, our result is $a^2B\sim \frac{eH}{\sqrt{2}k^2}\mid\sin\sqrt{2}k\eta\mid$ . At the time $\eta_{RH}$ we have $B_{RH}=\frac{e}{k_{phys}}$. This may imply that the breaking of the conformal invariance due to the minimal coupling of a massless charged scalar complex field to both gravitational and electromagnetic fields is not sufficient for the production of seed galactic magnetic fields during inflation. But since we are interested in the large-scale cosmological magnetic field, this could be still a candidate, because of the $1/k$ factor. 
  The curved geometry of a spacetime manifold arises as a solution of Einstein's gravitational field equation. We show that the metric of a spherically symmetric gravitational field configuration can be viewed as an optical metric created by the moving material fluid with nontrivial dielectric and magnetic properties. Such a "hydrodynamical" approach provides a simple physical interpretation of a horizon. 
  A well-known open problem in general relativity, dating back to 1972, has been to prove Price's law for an appropriate model of gravitational collapse. This law postulates inverse-power decay rates for the gravitational radiation flux on the event horizon and null infinity with respect to appropriately normalized advanced and retarded time coordinates. It is intimately related both to astrophysical observations of black holes and to the fate of observers who dare cross the event horizon. In this paper, we prove a well-defined (upper bound) formulation of Price's law for the collapse of a self-gravitating scalar field with spherically symmetric initial data. We also allow the presence of an additional gravitationally coupled Maxwell field. Our results are obtained by a new mathematical technique for understanding the long-time behavior of large data solutions to the resulting coupled non-linear hyperbolic system of p.d.e.'s in 2 independent variables. The technique is based on the interaction of the conformal geometry, the celebrated red-shift effect, and local energy conservation; we feel it may be relevant for the problem of non-linear stability of the Kerr solution. When combined with previous work of the first author (gr-qc/0307013) concerning the internal structure of charged black holes, which assumed the validity of Price's law, our results can be applied to the strong cosmic censorship conjecture for the Einstein-Maxwell-real scalar field system with complete spacelike asymptotically flat spherically symmetric initial data. Under Christodoulou's C^0 formulation, the conjecture is proven to be false. 
  In the context of the recently proposed formulation of quantum geometrodynamics in extended phase space we discuss the problem how the behavior of the Universe, initially managed by quantum laws, has become classical. In this version of quantum geometrodynamics we quantize gauge gravitational degrees of freedom on an equal basis with physical degrees of freedom. As a consequence of this approach, a wave function of the Universe depends not only on physical fields but also on gauge degrees of freedom. From this viewpoint, one should regard the physical Universe as a subsystem whose properties are formed in interaction with the subsystem of gauge degrees of freedom. We argue that the subsystem of gauge degrees of freedom may play the role of environment, which, being taken into account, causes the density matrix to be diagonal. We show that under physically reasonable fixing of gauge condition the density matrix describing the physical subsystem of the Universe may have a Gaussian peak in some variable, but it could take the Gaussian form only within a spacetime region where a certain gauge condition is imposed. If spacetime manifold consists of regions covered by different coordinate charts the Universe cannot behave in a classical manner nearby borders of these regions. Moreover, in this case the Universe could not stay in the same quantum state, but its state would change in some irreversible way. 
  A sum-over-histories generalized quantum theory is developed for homogeneous minisuperspace type A Bianchi cosmological models, focussing on the particular example of the classically recollapsing Bianchi IX universe. The decoherence functional for such universes is exhibited. We show how the probabilities of decoherent sets of alternative, coarse-grained histories of these model universes can be calculated. We consider in particular the probabilities for classical evolution defined by a suitable coarse-graining. For a restricted class of initial conditions and coarse grainings we exhibit the approximate decoherence of alternative histories in which the universe behaves classically and those in which it does not. For these situations we show that the probability is near unity for the universe to recontract classically if it expands classically. We also determine the relative probabilities of quasi-classical trajectories for initial states of WKB form, recovering for such states a precise form of the familiar heuristic "J d\Sigma" rule of quantum cosmology, as well as a generalization of this rule to generic initial states. 
  For relativistic stars rotating slowly and differentially with a positive angular velocity, some properties in relation to the positiveness of the rate of rotational dragging and of the angular momentum density are derived. Also, a new proof for the bounds on the rotational mass-energy is given. 
  We construct two models of the formation a galaxy with a central black hole, starting from a small initial fluctuation at recombination. This is an application of previously developed methods to find a Lemaitre-Tolman model that evolves from a given initial density or velocity profile to a given final density profile. We show that the black hole itself could be either a collapsed object, or a non-vacuum generalisation of a full Schwarzschild-Kruskal-Szekeres wormhole. Particular attention is paid to the black hole's apparent and event horizons. 
  We consider the homogeneous and isotropic cosmological fluid dynamics which is compatible with a homothetic, timelike motion, equivalent to an equation of state $\rho + 3P = 0$. By splitting the total pressure $P$ into the sum of an equilibrium part $p$ and a non-equilibrium part $\Pi$, we find that on thermodynamical grounds this split is necessarily given by $p = \rho$ and $\Pi = - (4/3)\rho$, corresponding to a dissipative stiff (Zel'dovich) fluid. 
  The present work is a study of the unitarity problem for Quantum Mechanics at Planck Scale considered as Quantum Mechanics with Fundamental Length (QMFL).In the process QMFL is described as deformation of a well-known Quantum Mechanics (QM). Similar to previous works of the author, the basic approach is based on deformation of the density matrix (density pro-matrix) with concurrent development of the wave function deformation in the respective Schrodinger picture. It is demonstrated that the existence of black holes in the suggested approach in the end twice results in nonunitary transitions (first after the Big Bang of QMFL to QM, and then when on trapping of the matter into the black hole the situation is just the opposite - from QM to QMFL)and hence in recovery of the unitarity. In parallel this problem is considered in the deformation terms of Heisenberg algebra, showing the identity of the basic results. From this an explicit solution for Hawking's informaion paradox has been derived 
  A massless scalar field is quantized in the background of a spinning string with cosmic dislocation. By increasing the spin density toward the dislocation parameter, a region containing closed timelike curves (CTCs) eventually forms around the defect. Correspondingly, the propagator tends to the ordinary cosmic string propagator, leading therefore to a mean-square field fluctuation, which remains well behaved throughout the process, unlike the vacuum expectation value of the energy-momentum tensor, which diverges due to a subtle mechanism. These results suggest that back reaction leads to the formation of a "horizon" that protects from the appearance of CTCs. 
  It is widely believed that the large redshifts for distant supernovae are explained by the vacuum energy dominance, or, in other words, by the cosmological constant in Einstein's equations, which is responsible for the anti-gravitation effect. A tacit assumption is that particles move along a geodesic for the background metric. This is in the same spirit as the consensus regarding the uniform Galilean motion of a free electron. However, there is a runaway solution to the Lorentz--Dirac equation governing the behavior of a radiating electron, in addition to the Galilean solution. Likewise, a runaway solution to the entire system of equations, both gravitation and matter equations of motion including, may provide an alternative explanation for the accelerated expansion of the Universe, without recourse to the hypothetic cosmological constant. 
  We analyze type I vacuum solutions admitting an isometry whose Killing 2--form is aligned with a principal bivector of the Weyl tensor, and we show that these solutions belong to a family of type I metrics which admit a group $G_3$ of isometries. We give a classification of this family and we study the Bianchi type for each class. The classes compatible with an aligned Killing 2--form are also determined. The Szekeres-Brans theorem is extended to non vacuum spacetimes with vanishing Cotton tensor. 
  The graviton is pictured as a bound state of a fermion and anti-fermion with the spacetime metric assumed to be a composite object of spinor fields, based on a globally Lorentz invariant action proposed by Hebecker and Wetterich. The additional degrees of freedom beyond those of the graviton are described by Goldstone boson gravitational degrees of freedom. If we assume that the fermion is a light neutrino with mass $m_nu\sim 10^{-3}$ eV, then we obtain the effective vacuum density ${\bar\rho}_\lambda\sim (10^{-3} eV)^4$, which agrees with the estimates for the cosmological constant from WMAP and SNIa data. 
  If perturbations beyond the horizon have the velocities prescribed everywhere then the dragging of inertial frames near the origin is suppressed by an exponential factor. However if perturbations are prescribed in terms of their angular momenta there is no such suppression.   We resolve this paradox and in doing so give new explicit results on the dragging of inertial frames in closed, flat and open universe with and without a cosmological constant. 
  Explicit expressions are found for the axisymmetric metric perturbations of the closed, flat and open FRW universes caused by toroidal motions of the cosmic fluid. The perturbations are decomposed in vector spherical harmonics on 2-spheres, but the radial dependence is left general. Solutions for general odd-parity $l$-pole perturbations are given for either angular velocities or angular momenta prescribed. In particular, in case of closed universes the solutions require a special treatment of the Legendre equation. 
  We write the equation of geodesic deviations in the spacetime of $pp$-waves in terms of the Newman-Penrose scalars and apply it to study gravitational waves in quadratic curvature gravity. We show that quadratic curvature gravity $pp$-waves can have a transverse helicity-0 polarization mode and two transverse helicity-2 general relativity-like wave polarizations. A concrete example is given in which we analyze the wave polarizations of an exact impulsive gravitational wave solution to quadratic curvature gravity. 
  With the help of Nordtvedt's scalar tensor theory an exact analytical model of a non-minimally coupled scalar field cosmology in which the gravitational coupling $G$ and the Hubble factor $H$ oscillate during the radiation era is presented. A key feature is that the oscillations are confined to the early stages of the radiation dominated era with $G$ approaching its present constant value while $H$ becoming a monotonically decreasing function of time. The Brans Dicke parameter $\omega$ is chosen to be a function of Brans Dicke scalar field so that no conflict with observational constraints regarding its present value arises. 
  The energy distributions of four 2+1 dimensional black hole solutions were obtained by using the Einstein and M{\o}ller energy-momentum complexes. while $r \to \infty$, the energy distributions of these four solutions become divergence. 
  We consider boson star solutions in a $D$-dimensional, asymptotically anti-de Sitter spacetime and investigate the influence of the cosmological term on their properties. We find that for $D>4$ the boson star properties are close to those in four dimensions with a vanishing cosmological constant. A different behavior is noticed for the solutions in the three dimensional case. We establish also the non-existence of static, spherically symmetric black holes with a harmonically time-dependent complex scalar field in any dimension greater than two. 
  This work discusses the apriori possible asymptotic behavior to the future, for (vacuum) space-times which are geodesically complete to the future and which admit a foliation by compact constant mean curvature Cauchy surfaces. 
  A quantum model of universe is constructed in which values of dimensionless coupling constants of the fundamental interactions (including the cosmological constant) are determined via certain topological invariants of manifolds forming finite ensembles of 3D Seifert fibrations. The characteristic values of the coupling constants are explicitly calculated as the set of rational numbers (up to the factor $2\pi$) on the basis of a hypothesis that these values are proportional to the mean relative fluctuations of discrete volumes of manifolds in these ensembles. The discrete volumes are calculated using the standard Alexandroff procedure of constructing $T_0$-discrete spaces realized as nerves corresponding to characteristic canonical triangulations which are compatible with the Milnor representation of Seifert fibered homology spheres being the building material of all used 3D manifolds. Moreover, the determination of all involved homology spheres is based on the first nine prime numbers ($p_1=2, >..., p_9=23$). The obtained hierarchy of coupling constants at the present evolution stage of universe well reproduces the actual hierarchy of the experimentally observed dimensionless low-energy coupling constants. 
  Heisenberg's uncertainty relation is commonly regarded as defining a level of unpredictability that is fundamentally incompatible with the deterministic laws embodied in classical field theories such as Einstein's general relativity. We here show that this is not necessarily the case. Using 5D as an example of dimensionally-extended relativity, we employ a novel metric to derive the standard quantum rule for the action and a form of Heisenberg's relation that applies to real and virtual particles. The philosophical implications of these technical results are somewhat profound. 
  Observationally, the universe appears virtually critical. Yet, there is no simple explanation for this state. In this article we advance and explore the premise that the dynamics of the universe always seeks equilibrium conditions. Vacuum-induced cosmic accelerations lead to creation of matter-energy modes at the expense of vacuum energy. Because they gravitate, such modes constitute inertia against cosmic acceleration. On the other extreme, the would-be ultimate phase of local gravitational collapse is checked by a phase transition in the collapsing matter fields leading to a de Sitter-like fluid deep inside the black hole horizon, and at the expense of the collapsing matter fields. As a result, the universe succumbs to neither vacuum-induced run-away accelerations nor to gravitationally induced spacetime curvature singularities. Cosmic dynamics is self-regulating. We discuss the physical basis for these constraints and the implications, pointing out how the framework relates and helps resolve standing puzzles such as "why did cosmic inflation end?", "why is Lambda small now?" and "why does the universe appear persistently critical?". The approach does, on the one hand, suggest a future course for cosmic dynamics, while on the other hand it provides some insight into the physics inside black hole horizons. The interplay between the background vacuum and matter fields suggests an underlying symmetry that links spacetime acceleration with spacetime collapse and global (cosmic) dynamics with local (black hole) dynamics. 
  The periodic standing-wave method for binary inspiral computes the exact numerical solution for periodic binary motion with standing gravitational waves, and uses it as an approximation to slow binary inspiral with outgoing waves. Important features of this method presented here are: (i) the mathematical nature of the ``mixed'' partial differential equations to be solved, (ii) the meaning of standing waves in the method, (iii) computational difficulties, and (iv) the ``effective linearity'' that ultimately justifies the approximation. The method is applied to three dimensional nonlinear scalar model problems, and the numerical results are used to demonstrate extraction of the outgoing solution from the standing-wave solution, and the role of effective linearity. 
  Recent developments in ``Einstein Dehn filling'' allow the construction of infinitely many Einstein manifolds that have different topologies but are geometrically close to each other. Using these results, we show that for many spatial topologies, the Hartle-Hawking wave function for a spacetime with a negative cosmological constant develops sharp peaks at certain calculable geometries. The peaks we find are all centered on spatial metrics of constant negative curvature, suggesting a new mechanism for obtaining local homogeneity in quantum cosmology. 
  We try to show that the energy density of the cosmic quaternionic field might be a possible candidate for the black energy. 
  We study the dynamics of a spherically symmetric dust shell separating two spacetime domains, the 'interior' one being a part of the de Sitter spacetime and the exterior one having the 'extremal' Reissner-Nordstroem geometry. Extending the ideas of previous works on the subject, we show that the it is possible to determine the (metastable) WKB quantum states of this gravitational system. 
  We review the integrable systems which arise as symmetry reductions of Plebanski's heavenly equations, and their generalisations. We also show that all four-dimensional null Kahler-Einstein (or type N hyper-heavenly) metrics with symmetry can be found from solutions to a variable coefficient generalisation of the dispersionless Kadomtsev-Petviashvili equation. 
  The oscillating gravitational field of an oscillaton of finite mass M causes it to lose energy by emitting classical scalar field waves, but at a rate that is non-perturbatively tiny for small GMm, where m is the scalar field mass: d(GM)/dt ~ -3797437.776333015 e^[-39.433795197160163/(GMm)]/(GMm)^2. Oscillatons also decay by the quantum process of the annihilation of scalarons into gravitons, which is only perturbatively small in GMm, giving by itself d(GM)/dt ~ - 0.008513223934732692 G m^2 (GMm)^5. Thus the quantum decay is faster than the classical one for Gmm < 39.4338/[ln(1/Gm^2)}-7ln(GMm)+19.9160]. The time for an oscillaton to decay away completely into free scalarons and gravitons is ~ 2/(G^5 m^11) ~ 10^324 yr (1 meV/m)^11. Oscillatons of more than one real scalar field of the same mass generically asymptotically approach a static-geometry U(1) boson star configuration with GMm = GM_0 m, at the rate d(GM/c^3)/dt ~ [(C/(GMm)^4)e^{-alpha/(GMm)}+Q(m/m_{Pl})^2(GMm)^3] [(GMm)^2-(GM_0 m)^2], with GM_0 m depending on the magnitudes and relative phases of the oscillating fields, and with the same constants C, alpha, and Q given numerically above for the single-field case that is equivalent to GM_0 m = 0. 
  We analyze both the feasibility and reasonableness of a classical Euclidean Theory of Everything (TOE), which we understand as a TOE based on an Euclidean space and an absolute time over which deterministic models of particles and forces are built. The possible axiomatic complexity of a TOE in such a framework is considered and compared to the complexity of the assumptions underlying the Standard Model. Current approaches to relevant (for our purposes) reformulations of Special Relativity, General Relativity, inertia models and Quantum Theory are summarized, and links between some of these reformulations are exposed. A qualitative framework is suggested for a research program on a classical Euclidean TOE. Within this framework an underlying basis is suggested, in particular, for the Principle of Relativity and Principle of Equivalence. A model for gravity as an inertial phenomenon is proposed. Also, a basis for quantum indeterminacy and wave function collapse is suggested in the framework. 
  Black holes are presumed to have an ideal ability to absorb and keep matter. Whatever comes close to the event horizon, a boundary separating the inside region of a black hole from the outside world, inevitably goes in and remains inside forever. This work shows, however, that quantum corrections make possible a surprising process, reflection: a particle can bounce back from the event horizon. For low energy particles this process is efficient, black holes behave not as holes, but as mirrors, which changes our perception of their physical nature. Possible ways for observations of the reflection and its relation to the Hawking radiation process are outlined. 
  In this paper, we investigate the asymptotic nature of the quasinormal modes for "dirty" black holes -- generic static and spherically symmetric spacetimes for which a central black hole is surrounded by arbitrary "matter" fields. We demonstrate that, to the leading asymptotic order, the [imaginary] spacing between modes is precisely equal to the surface gravity, independent of the specifics of the black hole system.   Our analytical method is based on locating the complex poles in the first Born approximation for the scattering amplitude. We first verify that our formalism agrees, asymptotically, with previous studies on the Schwarzschild black hole. The analysis is then generalized to more exotic black hole geometries. We also extend considerations to spacetimes with two horizons and briefly discuss the degenerate-horizon scenario. 
  The motion of light and a neutral test particle around the charged D-star has been studied. The difference of the deficit angle of light from the case in asymptotically flat spacetime is in a factor $(1-\epsilon^2)$. The motion of a test particle is affected by the deficit angle and the charge. Through the phase analysis, we prove the existence of the periodic solution to the equation of motion and the effect of the deficit angle and the charge to the critical point and its type. We also give the conditions under which the critical point is a stable center and an unstable saddle point. 
  We present an alternative approach to setting initial data in general relativity. We do not use a conformal decomposition, but instead express the 3-metric in terms of a given unit vector field and one unknown scalar field. In the case of axisymmetry, we have written a program to solve the resulting nonlinear elliptic equation. We have obtained solutions, both numerically and from a linearized analytic method, for a general perturbation of Schwarzschild. 
  A thermal squeezed state representation of inflaton is constructed for a flat Friedmann-Robertson-Walker background metric and the phenomenon of particle creation is examined during the oscillatory phase of inflaton, in the semiclassical theory of gravity. An approximate solution to the semiclassical Einstein equation is obtained in thermal squeezed state formalism by perturbatively and is found obey the same power-law expansion as that of classical Einstein equation. In addition to that the solution shows oscillatory in nature except on a particular condition. It is also noted that, the coherently oscillating nonclassical inflaton, in thermal squeezed vacuum state, thermal squeezed state and thermal coherent state, suffer particle production and the created particles exhibit oscillatory behavior. The present study can account for the post inflation particle creation due to thermal and quantum effects of inflaton in a flat FRW universe. 
  We consider the stability of spatially homogeneous plane-wave spacetimes. We carry out a full analysis for plane-wave spacetimes in (4+1) dimensions, and find there are two cases to consider; what we call non-exceptional and exceptional. In the non-exceptional case the plane waves are stable to (spatially homogeneous) vacuum perturbations as well as a restricted set of matter perturbations. In the exceptional case we always find an instability. Also we consider the Milne universe in arbitrary dimensions and find it is also stable provided the strong energy condition is satisfied. This implies that there exists an open set of stable plane-wave solutions in arbitrary dimensions. 
  It is shown that the field equations derived from an effective interaction hamiltonian for Maxwell and gravitational fields in the semiclassical approximation of loop quantum gravity using rotational invariant states (such as weave states) are Lorentz invariant. To derive this result, which is in agreement with the observational evidence, we use the geometrical properties of the electromagnetic field. 
  A cosmological event horizon develops in a vacuum-dominated Friedmann universe. The Schwarschild radius of the vacuum energy within the horizon equals the horizon radius. Black hole thermodynamics and the holographic conjecture indicate a finite number of degrees of freedom within the horizon. The average energy per degree of freedom equals the energy of a massless quantum with wavelength of the horizon circumference. This suggests identifying the degrees of freedom with the presence or absence, in each Planck area on one horizon quadrant, of a 0S2 vibrational mode of the horizon with the z axis passing through that area. Pressure waves on the horizon (the superposition of 0S2 vibrational modes) can be envisioned to propagate into the observable universe within the horizon at the speed of light. So, the vacuum energy and pressure throughout the observable universe could (in principle) be determined from the vacuum equation of state. 
  Classically, the dynamics in a non-globally hyperbolic spacetime is ill posed. Previously, a prescription was given for defining dynamics in static spacetimes in terms of a second order operator acting on a Hilbert space defined on static slices. The present work extends this result by giving a similar prescription for defining dynamics in stationary spacetimes obeying certain mild assumptions. The prescription is defined in terms of a first order operator acting on a different Hilbert space from the one used in the static prescription. It preserves the important properties of the earlier one: the formal solution agrees with the Cauchy evolution within the domain of dependence, and smooth data of compact support always give rise to smooth solutions. In the static case, the first order formalism agrees with second order formalism (using specifically the Friedrichs extension). Applications to field quantization are also discussed. 
  Space-borne interferometric gravitational wave detectors, sensitive in the low-frequency (millihertz) band, will fly in the next decade. In these detectors the spacecraft-to-spacecraft light-travel-times will necessarily be unequal, time-varying, and (due to aberration) have different time delays on up- and down-links. Reduction of data from moving interferometric laser arrays in solar orbit will in fact encounter non-symmetric up- and downlink light time differences that are about 100 times larger than has previously been recognized. The time-delay interferometry (TDI) technique uses knowledge of these delays to cancel the otherwise dominant laser phase noise and yields a variety of data combinations sensitive to gravitational waves. Under the assumption that the (different) up- and downlink time delays are constant, we derive the TDI expressions for those combinations that rely only on four inter-spacecraft phase measurements. We then turn to the general problem that encompasses time-dependence of the light-travel times along the laser links. By introducing a set of non-commuting time-delay operators, we show that there exists a quite general procedure for deriving generalized TDI combinations that account for the effects of time-dependence of the arms. By applying our approach we are able to re-derive the ``flex-free'' expression for the unequal-arm Michelson combinations $X_1$, first presented in \cite{STEA}, and obtain the generalized expressions for the TDI combinations called Relay, Beacon, Monitor, and Symmetric Sagnac. 
  Using Einstein and Papapetrou energy-momentum complexes, we explicitly calculate the energy and momentum distribution associated with spacetime homogeneous G$\ddot{o}$del-type metrics. We obtain that the two definitions of energy-momentum complexes do not provide the same result for these type of metrics. However, it is shown that the results obtained are reduced to the energy-momentum densities of G$\ddot{o}$del metric already available in the literature 
  In this paper we classify static plane symmetric spacetimes according to their matter collineations. These have been studied for both cases when the energy-momentum tensor is non-degenerate and also when it is degenerate. It turns out that the non-degenerate case yields either {\it four}, {\it five}, {\it six}, {\it seven} or {\it ten} independent matter collineations in which {\it four} are isometries and the rest are proper. There exists three interesting cases where the energy-momentum tensor is degenerate but the group of matter collineations is finite-dimensional. The matter collineations in these cases are either {\it four}, {\it six} or {\it ten 
  We consider a few thought experiments of radial motion of massive particles in the gravitational fields outside and inside various celestial bodies: Earth, Sun, black hole. All other interactions except gravity are disregarded. For the outside motion there exists a critical value of coordinate velocity ${\rm v}_c = c/\sqrt 3$: particles with ${\rm v} < {\rm v}_c$ are accelerated by the field, like Newtonian apples, particles with ${\rm v} > {\rm v}_c$ are decelerated like photons. Particles moving inside a body with constant density have no critical velocity; they are always accelerated. We consider also the motion of a ball inside a tower, when it is thrown from the top (bottom) of the tower and after classically bouncing at the bottom (top) comes back to the original point. The total time of flight is the same in these two cases if the initial proper velocity $v_0$ is equal to $c/\sqrt 2$. 
  We consider a nearly free falling Earth satellite where atomic wave interferometers are tied to a telescope pointing towards a faraway star. They measure the acceleration and the rotation relatively to the local inertial frame.   We calculate the rotation of the telescope due to the aberrations and the deflection of the light in the gravitational field of the Earth. We show that the deflection due to the quadrupolar momentum of the gravity is not negligible if one wants to observe the Lense-Thirring effect of the Earth.   We consider some perturbation to the ideal device and we discuss the orders of magnitude of the phase shifts due to the residual tidal gravitational field in the satellite and we exhibit the terms which must be taken into account to calculate and interpret the full signal.   Within the framework of a geometric model, we calculate the various periodic components of the signal which must be analyzed to detect the Lense-Tirring effect. We discuss the results which support a reasonable optimism.   As a conclusion we put forward the necessity of a more complete, realistic and powerful model in order to obtain a final conclusion on the theoretical feasibility of the experiment as far as the observation of the Lense-Thirring effect is involved. 
  The properties of principal null directions of a perturbed black hole are investigated. It shown that principal null directions are directly observable quantities characterizing the space-time. A definition of a perturbed space-time, generalizing that given by Stewart and Walker is proposed. This more general framework allows one to include descriptions of a given space-time other than by a pair $(M,g)$ where $M$ is a four-dimensional differential manifold and $g$ a Lorentz metric. Examples of alternative characterizations are the curvature representation of Karlhede and others, the Newman-Penrose representation or observable quantities involving principal null directions. The conditions are studied under which the various alternative choices of observables provide equivalent descriptions of the space-time. 
  In this paper we have investigated an LRS Bianchi I anisotropic cosmological model of the universe by taking time varying $G$ and $\Lambda$ in the presence of bulk viscous fluid source described by full causal non-equilibrium thermodynamics. We obtain a cosmological constant as a decreasing function of time and for $m, n > 0$, the value of cosmological ``constant'' for this model is found to be small and positive which is supported by the results from recent supernovae observations. 
  We consider the scalar wave equation in the Kerr geometry for Cauchy data which is smooth and compactly supported outside the event horizon. We derive an integral representation which expresses the solution as a superposition of solutions of the radial and angular ODEs which arise in the separation of variables. In particular, we prove completeness of the solutions of the separated ODEs.   This integral representation is a suitable starting point for a detailed analysis of the long-time dynamics of scalar waves in the Kerr geometry. 
  Five-dimensional cosmological models with two 3-branes and with a buck cosmological constant are studied. It is found that for all the three cases ($\Lambda =0$, $\Lambda >0$, and $\Lambda <0$), the conventional space-time singularity ``big bang'' could be replaced by a matter singularity ``big bounce'', at which the ``size'' of the universe and the energy density are finite while the pressure diverges, and across which the universe evolves from a pre-existing contracting phase to the present expanding phase. It is also found that for the $\Lambda >0$ case the brane solutions could give an oscillating universe model in which the universe oscillates with each cosmic cycle begins from a ``big bounce'' and ends to a ``big crunch'', with a distinctive characteristic that in each subsequent cycle the universe expands to a larger size and then contracts to a smaller (but non-zero) size. By studying the gravitational force acted on a test particle in the bulk, a gravitational stability condition is derived and then is used to analyze those brane models. It predicts that if dark energy takes over ordinary matter, particles on the brane may become unstable in the sense that they may escape from our 4D-world and dissolve in the bulk due to the repulsive force of dark energy. 
  We study frequency dependent (FD) input-output schemes for signal-recycling interferometers, the baseline design of Advanced LIGO and the current configuration of GEO 600. Complementary to a recent proposal by Harms et al. to use FD input squeezing and ordinary homodyne detection, we explore a scheme which uses ordinary squeezed vacuum, but FD readout. Both schemes, which are sub-optimal among all possible input-output schemes, provide a global noise suppression by the power squeeze factor, while being realizable by using detuned Fabry-Perot cavities as input/output filters. At high frequencies, the two schemes are shown to be equivalent, while at low frequencies our scheme gives better performance than that of Harms et al., and is nearly fully optimal. We then study the sensitivity improvement achievable by these schemes in Advanced LIGO era (with 30-m filter cavities and current estimates of filter-mirror losses and thermal noise), for neutron star binary inspirals, and for narrowband GW sources such as low-mass X-ray binaries and known radio pulsars. Optical losses are shown to be a major obstacle for the actual implementation of these techniques in Advanced LIGO. On time scales of third-generation interferometers, like EURO/LIGO-III (~2012), with kilometer-scale filter cavities, a signal-recycling interferometer with the FD readout scheme explored in this paper can have performances comparable to existing proposals. [abridged] 
  It is known that the imaginary parts of the frequencies of the quasi normal modes of the Schwarzschild black hole are equally spaced, with the level spacing dependent only on the surface gravity. We generalize this result to a wider class of spacetimes and provide a simple derivation of the imaginary parts of the frequencies. The analysis shows that the result is closely linked to the thermal nature of horizons and arises from the exponential redshift of the wave modes close to the horizon. 
  A third post-Newtonian (3 PN) equation of motion for an inspiralling binary consisting of two spherical compact stars with strong internal gravity is derived under harmonic coordinate condition using the strong field point particle limit. The equation of motion is complete in a sense that it is Lorentz invariant in the post-Newtonian perturbative sense, admits conserved energy of the orbital motion, and is unambiguous, that is, with no undetermined coefficient. In this paper, we show explicit expressions of the 3 PN equation of motion and an energy of the binary orbital motion in case of the circular orbit (neglecting the 2.5 PN radiation reaction effect) and in the center of the mass frame. It is argued that the 3 PN equation of motion we obtained is physically unambiguous. Full details will be reported elsewhere. 
  An equation of motion for relativistic compact binaries is derived through the third post-Newtonian (3 PN) approximation of general relativity. The strong field point particle limit and multipole expansion of the stars are used to solve iteratively the harmonically relaxed Einstein equations. We take into account the Lorentz contraction on the multipole moments defined in our previous works. We then derive a 3 PN acceleration of the binary orbital motion of the two spherical compact stars based on a surface integral approach which is a direct consequence of local energy momentum conservation. Our resulting equation of motion admits a conserved energy (neglecting the 2.5 PN radiation reaction effect), is Lorentz invariant and is unambiguous: there exist no undetermined parameter reported in the previous works. We shall show that our 3 PN equation of motion agrees physically with the Blanchet and Faye 3 PN equation of motion if $\lambda = - 1987/3080$, where $\lambda$ is the parameter which is undetermined within their framework. This value of $\lambda$ is consistent with the result of Damour, Jaranowski, and Sch\"afer who first completed a 3 PN iteration of the ADM Hamiltonian in the ADMTT gauge using the dimensional regularization. 
  We present results of three dimensional numerical simulations of the merger of unequal-mass binary neutron stars in full general relativity. A $\Gamma$-law equation of state $P=(\Gamma-1)\rho\epsilon$ is adopted, where $P$, $\rho$, $\varep$, and $\Gamma$ are the pressure, rest mass density, specific internal energy, and the adiabatic constant, respectively. We take $\Gamma=2$ and the baryon rest-mass ratio $Q_M$ to be in the range 0.85--1. The typical grid size is $(633,633,317)$ for $(x,y,z)$ . We improve several implementations since the latest work. In the present code, the radiation reaction of gravitational waves is taken into account with a good accuracy. This fact enables us to follow the coalescence all the way from the late inspiral phase through the merger phase for which the transition is triggered by the radiation reaction. It is found that if the total rest-mass of the system is more than $\sim 1.7$ times of the maximum allowed rest-mass of spherical neutron stars, a black hole is formed after the merger irrespective of the mass ratios. The gravitational waveforms and outcomes in the merger of unequal-mass binaries are compared with those in equal-mass binaries. It is found that the disk mass around the so formed black holes increases with decreasing rest-mass ratios and decreases with increasing compactness of neutron stars. The merger process and the gravitational waveforms also depend strongly on the rest-mass ratios even for the range $Q_M= 0.85$--1. 
  $\Lambda^{\mu}_{\nu}$-geometry is a geometry with a variable cosmological term described by a second-rank symmetric tensor $\Lambda^{\mu}_{\nu}$ whose asymptotics are Einstein cosmological term $\Lambda \delta ^{\mu}_{\nu}$ at the origin and $\lambda \delta ^{\mu}_{\nu}$ at infinity (with $\lambda < \Lambda$).  It corresponds to extension of the algebraic structure of the Einstein cosmological term $\Lambda \delta ^{\mu}_{\nu}$ in such a way that a scalar $\Lambda$ describing vacuum energy density as $\rho_{vac}=8\pi G \Lambda$ (with $\rho_{vac}$=const by virtue of the Bianchi identities), becomes explicite related to the appropriate component, $\Lambda^0_0$, of an appropriate stress-energy tensor, $T^{\mu}_{\nu}=8\pi G\Lambda^{\mu}_{\nu}$ whose vacuum properties follow from its symmetry, $T_0^0=T_1^1$, and whose variability follows from the contracted Bianchi identities. In the spherically symmetric case existence of such geometries in frame of GR follows from imposing on Einstein equations requirements of finiteness of the ADM mass $m$, and of regularity of density and pressures. Dependently on parameters $m$ and $q=\sqrt{\Lambda /\lambda}$, $\Lambda^{\mu}_{\nu}$ geometry describes five types of configurations. We summarize here the results which tell us how these configurations look from the point of view of different observers: a static observer, a Lemaitre co-moving observer, and a Kantowski-Sachs observer. 
  The gravitational instability of Yang-Mills cosmologies is numerically studied with the hamiltonian formulation of the spherically symmetric Einstein-Yang-Mills equations with SU(2) gauge group. On the short term, the expansion dilutes the energy densities of the Yang-Mills fluctuations due to their conformal invariance. In this early regime, the gauge potentials appear oscillating quietly in an interaction potential quite similar to the one of the homogeneous case. However, on the long term, the expansion finally becomes significantly inhomogeneous and no more mimics a conformal transformation of the metric. Thereafter, the Yang-Mills fluctuations enter a complex non-linear regime, accompanied by diffusion, while their associated energy contrasts grow. 
  We show that the tilted perfect fluid Bianchi VI$_0$ family of self-similar models found by Rosquist and Jantzen [K. Rosquist and R. T. Jantzen, \emph{% Exact power law solutions of the Einstein equations}, 1985 Phys. Lett. \textbf{107}A 29-32] is the most general class of tilted self-similar models but the state parameter $\gamma $ lies in the interval $(\frac 65,\frac 32) $. The model has a four dimensional stable manifold indicating the possibility that it may be future attractor, at least for the subclass of tilted Bianchi VI$_0$ models satisfying $n_\alpha ^\alpha =0$ in which it belongs. In addition the angle of tilt is asymptotically significant at late times suggesting that for the above subclasses of models the tilt is asymptotically extreme. 
  The detection of the gravitational waves (GWs) emitted by precessing binaries of spinning compact objects is complicated by the large number of parameters (such as the magnitudes and initial directions of the spins, and the position and orientation of the binary with respect to the detector) that are required to model accurately the precession-induced modulations of the GW signal. In this paper we describe a fast matched-filtering search scheme for precessing binaries, and we adopt the physical template family proposed by Buonanno, Chen, and Vallisneri [Phys.Rev.D 67, 104025 (2003)] for ground-based interferometers. This family provides essentially exact waveforms, written directly in terms of the physical parameters, for binaries with a single significant spin, and for which the observed GW signal is emitted during the phase of adiabatic inspiral (for LIGO-I and VIRGO, this corresponds to a total mass M < 15Msun). We show how the detection statistic can be maximized automatically over all the parameters (including the position and orientation of the binary with respect to the detector), except four (the two masses, the magnitude of the single spin, and the opening angle between the spin and the orbital angular momentum), so the template bank used in the search is only four-dimensional; this technique is relevant also to the searches for GW from extreme--mass-ratio inspirals and supermassive blackhole inspirals to be performed using the space-borne detector LISA. Using the LIGO-I design sensitivity, we compute the detection threshold (~10) required for a false-alarm probability of 10^(-3)/year, and the number of templates (~76,000) required for a minimum match of 0.97, for the mass range (m1,m2)=[7,12]Msun*[1,3]Msun. 
  In information theory, the link between continuous information and discrete information is established through well-known sampling theorems. Sampling theory explains, for example, how frequency-filtered music signals are reconstructible perfectly from discrete samples. In this Letter, sampling theory is generalized to pseudo-Riemannian manifolds. This provides a new set of mathematical tools for the study of space-time at the Planck scale: theories formulated on a differentiable space-time manifold can be completely equivalent to lattice theories. There is a close connection to generalized uncertainty relations which have appeared in string theory and other studies of quantum gravity. 
  Two homodyne Michelson interferferometers aboard the LISA Pathfinder spacecraft will measure the the positions of two free-floating test masses, as part of the NASA ST7 mission. The interferometer is required to measure the separation between the test masses with sensitivity of 30 pm/sqrt(Hz) at 10 mHz. The readout scheme is described, error sources are analyzed, and experimental results are presented. 
  In this comment we point out some problems with the approach used in the paper "Newtonian limit of String-Dilaton Gravity" in the attempt to explain the properties of a weak field approximation for gravity theories. 
  We continue our studies of spherically symmetric self-similar solutions in the SU(2) sigma model coupled to gravity. Using mixed numerical and analytical methods we show existence of an unstable periodic solution lying at the boundary between the basins of two generic attractors. 
  We review some of our recent work on the conformal geometry corresponding to the triangulated surfaces used in 2-dimensional simplicial quantum gravity. In particular, we discuss the regularized Liouville action associated with random Regge triangulations and its connection with Hodge-Deligne theory. 
  An open problem in general relativity has been to construct an asymptotically flat solution to a reasonable Einstein-matter system containing a black hole in the future and yet past-causally geodesically complete, in particular, containing no white holes. We give such an example in this paper--in fact, a family of such examples, stable in a suitable sense--for the case of a self-gravitating scalar field. 
  An approximation to Einstein's field equations in Arnowitt-Deser-Misner (ADM) canonical formalism is presented which corresponds to the magneto-hydrodynamics (MHD) approximation in electrodynamics. It results in coupled elliptic equations which represent the maximum of elliptic-type structure of Einstein's theory and naturally generalizes previous conformal-flat truncations of the theory. The Hamiltonian, in this approximation, is identical with the non-dissipative part of the Einsteinian one through the third post-Newtonian order. The proposed scheme, where stationary spacetimes are exactly reproduced, should be useful to construct {\em realistic} initial data for general relativistic simulations as well as to model astrophysical scenarios, where gravitational radiation reaction can be neglected. 
  We present results of 3D numerical simulations using a finite difference code featuring fixed mesh refinement (FMR), in which a subset of the computational domain is refined in space and time. We apply this code to a series of test cases including a robust stability test, a nonlinear gauge wave and an excised Schwarzschild black hole in an evolving gauge. We find that the mesh refinement results are comparable in accuracy, stability and convergence to unigrid simulations with the same effective resolution. At the same time, the use of FMR reduces the computational resources needed to obtain a given accuracy. Particular care must be taken at the interfaces between coarse and fine grids to avoid a loss of convergence at higher resolutions, and we introduce the use of "buffer zones" as one resolution of this issue. We also introduce a new method for initial data generation, which enables higher-order interpolation in time even from the initial time slice. This FMR system, "Carpet", is a driver module in the freely available Cactus computational infrastructure, and is able to endow generic existing Cactus simulation modules ("thorns") with FMR with little or no extra effort. 
  A family of spherically symmetric solutions in the model with m-component anisotropic fluid is considered. The metric of the solution depends on parameters q_s, s = 1,...,m, relating radial pressures and the densities and contains (n -1)m parameters corresponding to Ricci-flat "internal space" metrics and obeying certain m(m-1)/2 ("orthogonality") relations. For q_s = 1 (for all s) and certian equations of state (p_i^s = \pm \rho^s) the metric coincides with the metric of intersecting black brane solution in the model with antisymmetric forms. A family of solutions with (regular) horizon corresponding to natural numbers q_s = 1,2,... is singled out. Certain examples of "generalized simulation" of intersecting M-branes in D=11 supergravity are considered. The post-Newtonian parameters \beta and \gamma corresponding to the 4-dimensional section of the metric are calculated. 
  A generally covariant extension of general relativity (GR) in which a dynamical unit timelike vector field is coupled to the metric is studied in the asymptotic weak field limit of spherically symmetric static solutions. The two post-Newtonian parameters known as the Eddington-Robertson-Schiff parameters are found to be identical to those in the case of pure GR, except for some non-generic values of the coefficients in the Lagrangian. 
  We consider the effects of a cosmological constant on the dynamics of constant angular momentum discs orbiting Schwarzschild-de Sitter black holes. The motivation behind this study is to investigate whether the presence of a radial force contrasting the black hole's gravitational attraction can influence the occurrence of the runaway instability, a robust feature of the dynamics of constant angular momentum tori in Schwarzschild and Kerr spacetimes. In addition to the inner cusp near the black hole horizon through which matter can accrete onto the black hole, in fact, a positive cosmological constant introduces also an outer cusp through which matter can leave the torus without accreting onto the black hole. To assess the impact of this outflow on the development of the instability we have performed time-dependent and axisymmetric hydrodynamical simulations of equilibrium initial configurations in a sequence of background spacetimes of Schwarzschild-de Sitter black holes with increasing masses. The simulations have been performed with an unrealistic value for the cosmological constant which, however, yields sufficiently small discs to be resolved accurately on numerical grids and thus provides a first qualitative picture of the dynamics. The calculations, carried out for a wide range of initial conditions, show that the mass-loss from the outer cusp can have a considerable impact on the instability, with the latter being rapidly suppressed if the outflow is large enough. 
  We study global flat embeddings, three accelerations and Hawking temperatures of the BTZ black holes in the framework of two-time physics scheme associated with Sp(2) local symmetry, to construct their corresponding SO(3,2) global symmetry invariant Lagrangians both inside and outside event horizons. Moreover, the Sp(2) local symmetry is discussed in terms of the metric time-independence. 
  We construct a chiral theory of gravity in 7 and 8 dimensions, which are equivalent to Einstein-Cartan theory using less variables. In these dimensions, we can construct such higher dimensional chiral gravity because of the existence of gravitational instanton. The octonionic-valued variables in the theory represent the deviation from the gravitational instanton, and from their non-associativity, prevents the theory to be SO(n) gauge invariant. Still the chiral gravity holds G_2 (7-D), and Spin(7) (8-D) gauge symmetry. 
  It is proved that a stationary solutions to the vacuum Einstein field equations with non-vanishing angular momentum have no Cauchy slice that is maximal, conformally flat, and non-boosted. The proof is based on results coming from a certain type of asymptotic expansions near null and spatial infinity --which also show that the developments of Bowen-York type of data cannot have a development admitting a smooth null infinity--, and from the fact that stationary solutions do admit a smooth null infinity. 
  We describe the search for gravitational waves from inspiraling neutron star binary systems, using data from the first Scientific Run of the LIGO Science Collaboration. 
  We continue our studies of spherically symmetric self-similar solutions in the SU(2) sigma model coupled to gravity. For some values of the coupling constant we present numerical evidence for the chaotic solution and the fractal threshold behavior. We explain this phenomenon in terms of horseshoe-like dynamics and heteroclinic intersections. 
  Black holes are capable of reflection: there is a finite probability for any particle that approaches the event horizon to bounce back. The albedo of the black hole depends on its temperature and the energy of the incoming particle. The reflection shares its physical origins with the Hawking process of radiation, both of them arise as consequences of the mixing of the incoming and outgoing waves that takes place on the event horizon. 
  We present a systematic treatment of the initial conditions and evolution of cosmological perturbations in a universe containing photons, baryons, neutrinos, cold dark matter, and a scalar quintessence field. By formulating the evolution in terms of a differential equation involving a matrix acting on a vector comprised of the perturbation variables, we can use the familiar language of eigenvalues and eigenvectors. As the largest eigenvalue of the evolution matrix is fourfold degenerate, it follows that there are four dominant modes with non-diverging gravitational potential at early times, corresponding to adiabatic, cold dark matter isocurvature, baryon isocurvature and neutrino isocurvature perturbations. We conclude that quintessence does not lead to an additional independent mode. 
  A scalar field can be inserted in Maxwell and/or Einstein theory to effect symmetry breaking. Consequences of such a modification are discussed. Possible dynamics for the scalar field are presented. 
  A subclassification of stationary spacetimes, endowed with one timelike and one spacelike Killing vectors, i.e., Petrov $G{_2}I$ on $T_2$ spaces, is proposed. Special attention deserves the Collison's theorem [1] and the branch of metrics circularly cyclicly (axially) symmetric possessing additionally the conformal flatness property reported by Garcia and Campuzano [2]. 
  The mass spectrum of a model constructed in a theory space is expressed by eigenvalues of the Laplacian on the graph structure of the theory space. The nature of the one-loop UV divergence in the vacuum energy is then controlled only by the degree matrix of the graph. Using these facts, we can construct models of induced gravity which do not suffer from divergences at the one-loop level. 
  We study asymptotic dynamics of photons propagating in the polarized vacuum of a locally de Sitter Universe. The origin of the vacuum polarization is fluctuations of a massless, minimally coupled, scalar, which we model by the one-loop vacuum polarization tensor of scalar electrodynamics. We show that late time dynamics of the electric field on superhorizon scales approaches that of an Airy oscillator. The magentic field amplitude, on the other hand, asymptotically approaches a nonvanishing constant (plus an exponentially small oscillatory component), which is suppressed with respect to the initial (vacuum) amplitude. This implies that the asymptotic photon dynamics is more intricate than that of a massive photon obeying the local Proca equation. 
  Some aspects of Cosmology with primordial black holes are briefly reviewed 
  In the field equations of Einstein-Cartan theory with cosmological constant a static spherically symmetric perfect fluid with spin density satisfying the Weyssenhoff restriction is considered. This serves as a rough model of space filled with (fermionic) dark matter. From this the Einstein static universe with constant torsion is constructed, generalising the Einstein Cosmos to Einstein-Cartan theory.   The interplay between torsion and the cosmological constant is discussed. A possible way out of the cosmological constant's sign problem is suggested. 
  Recent measurements of the propagation of the quasar's radio signal past Jupiter are directly sensitive to the time-dependent effect from the geometric sector of general relativity which is proportional to the speed of propagation of gravity but not the speed of light. It provides a first confirmative measurement of the fundamental speed of the Einstein general principle of relativity for gravitational field. 
  Solutions for scalar fields superdense gravitating systems of flat, open and closed type obtained in the frame of gauge theories of gravitation are discussed. Properties of these systems in dependence on parameter $\beta$ and initial conditions are analyzed. 
  We investigate the properties of a closed-form analytic solution recently found by Manko et al. (2000) for the exterior spacetime of rapidly rotating neutron stars. For selected equations of state we numerically solve the full Einstein equations to determine the neutron star spacetime along constant rest mass sequences. The analytic solution is then matched to the numerical solutions by imposing the condition that the quadrupole moment of the numerical and analytic spacetimes be the same. For the analytic solution we consider, such a matching condition can be satisfied only for very rapidly rotating stars. When solutions to the matching condition exist, they belong to one of two branches. For one branch the current octupole moment of the analytic solution is very close to the current octupole moment of the numerical spacetime; the other branch is more similar to the Kerr solution. We present an extensive comparison of the radii of innermost stable circular orbits (ISCOs) obtained with a) the analytic solution, b) the Kerr metric, c) an analytic series expansion derived by Shibata and Sasaki (1998) and d) a highly accurate numerical code. In most cases where a corotating ISCO exists, the analytic solution has an accuracy consistently better than the Shibata-Sasaki expansion. The numerical code is used for tabulating the mass-quadrupole and current-octupole moments for several sequences of constant rest mass. 
  A scalar bimetric theory of gravity with a preferred reference frame is summarized. Dynamics is governed by an extension of Newton's second law. In the static case, geodesic motion is recovered together with Newton's attraction field. In the static spherical case, Schwarzschild's metric is found. Asymptotic schemes of post-Newtonian (PN) and post-Minkowskian (PM) approximation are built, each based on associating a conceptual family of systems with the given system. At the 1PN approximation, there is no preferred-frame effect for photons, hence the standard predictions of GR for photons are got. At the 0PM approximation, an isolated system loses energy by quadrupole radiation, without any monopole or dipole term. Inserting this loss into the Newtonian 2-body problem gives the Peters-Mathews coefficients of the theory. 
  We study gravitational lensing by a Reissner-Nordstrom (RN) black hole in the weak field limit. We obtain the basic equations for the deflection angle and time delay and find analytical expressions for the positions and amplifications of the primary and secondary images. Due to a net positive charge, the separation between images increases, but no change in the total magnification occurs. 
  We show how the use of the normal projection of the Einstein tensor as a set of boundary conditions relates to the propagation of the constraints, for two representations of the Einstein equations with vanishing shift vector: the ADM formulation, which is ill posed, and the Einstein-Christoffel formulation, which is symmetric hyperbolic. Essentially, the components of the normal projection of the Einstein tensor that act as non-trivial boundary conditions are linear combinations of the evolution equations with the constraints that are not preserved at the boundary, in both cases. In the process, the relationship of the normal projection of the Einstein tensor to the recently introduced ``constraint-preserving'' boundary conditions becomes apparent. 
  Recent papers by Samuel declared that the linearized post-Newtonian v/c effects are too small to have been measured in the recent experiment involving Jupiter and quasar J0842+1845 that was used to measure the ultimate speed of gravity defined as a fundamental constant entering in front of each time derivative of the metric tensor in the Einstein gravity field equations. We describe our Lorentz-invariant formulation of the Jovian deflection experiment and confirm that v/c effects are do observed, as contrasted to the erroneous claim by Samuel, and that they vanish if and only if the speed of gravity is infinite. 
  Odd-type spin 2 perturbations of Einstein's equation can be reduced to the scalar Regge-Wheeler equation. We show that the weighted norms of solutions are in L^2 of time and space. This result uses commutator methods and applies uniformly to all relevant spherical harmonics. 
  We study a five-dimensional cosmological model, which suggests that the universe bagan as a discontinuity in a (Higgs-type) scalar field, or alternatively as a conventional four-dimensional phase transition. 
  The application of quantum theory to gravity is beset with many technical and conceptual problems. After a short tour d'horizon of recent attempts to master those problems by the introduction of new approaches, we show that the aim, a background independent quantum theory of gravity, can be reached in a particular area, 2d dilaton quantum gravity, without any assumptions beyond standard quantum field theory. 
  We study gravitational theory in 1+2 spacetime dimensions which is determined by the Lagrangian constructed as a sum of the Einstein-Hilbert term plus the two (translational and rotational) gravitational Chern-Simons terms. When the corresponding coupling constants vanish, we are left with the purely Einstein theory of gravity. We obtain new exact solutions for the gravitational field equations with the nontrivial material sources. Special attention is paid to plane-fronted gravitational waves (in case of the Maxwell field source) and to the circularly symmetric as well as the anisotropic cosmological solutions which arise for the ideal fluid matter source. 
  We show that Petrov type I vacuum solutions admitting a Killing vector whose Papapetrou field is aligned with a principal bivector of the Weyl tensor are the Kasner and Taub metrics, their counterpart with timelike orbits and their associated windmill-like solutions, as well as the Petrov homogeneous vacuum solution. We recover all these metrics by using an integration method based on an invariant classification which allows us to characterize every solution. In this way we obtain an intrinsic and explicit algorithm to identify them. 
  Suppose a spacetime $M$ is a quotient of a spacetime $V$ by a discrete group of isometries. It is shown how causality conditions in the two spacetimes are related, and how can one learn about the future causal boundary on $M$ by studying structures in $V$. The relations between the two are particularly simple (the boundary of the quotient is the quotient of the boundary) if both $V$ and $M$ have spacelike future boundaries and if it is known that the quotient of the future completion of $V$ is past-distinguishing. (That last assumption is automatic in the case of $M$ being multi-warped.) 
  For an island-like distribution of matter the gravitational energy-momentum tensor is defined according to Weinberg as a source of metric. If this source is formed by self-interactions of gravitons, so that nonphysical degrees of freedom are excluded, then this source is a reasonable candidate for the energy-momentum tensor of gravitational field. The disastrous influence of the nonphysical degrees of freedom is demonstrated by comparing the gravitational energy-momentum tensors in the harmonic, isotropic and standard frames for the Schwarzschild solution. The harmonic frame is clearly preferable for defining the gravitational energy-momentum tensor. 
  Sequences of initial-data sets representing binary black holes in quasi-circular orbits have been used to calculate what may be interpreted as the innermost stable circular orbit. These sequences have been computed with two approaches. One method is based on the traditional conformal-transverse-traceless decomposition and locates quasi-circular orbits from the turning points in an effective potential. The second method uses a conformal-thin-sandwich decomposition and determines quasi-circular orbits by requiring the existence of an approximate helical Killing vector. Although the parameters defining the innermost stable circular orbit obtained from these two methods differ significantly, both approaches yield approximately the same initial data, as the separation of the binary system increases. To help understanding this agreement between data sets, we consider the case of initial data representing a single boosted or spinning black hole puncture of the Bowen-York type and show that the conformal-transverse-traceless and conformal-thin-sandwich methods yield identical data, both satisfying the conditions for the existence of an approximate Killing vector. 
  This paper gives a theoretical discussion of the orbits and isotropies which arise in a space-time which admits a Lie algebra of Killing vector fields. The submanifold structure of the orbits is explored together with their induced Killing vector structure. A general decomposition of a space-time in terms of the nature and dimension of its orbits is given and the concept of stability and instability for orbits introduced. A general relation is shown linking the dimensions of the Killing algebra, the orbits and the isotropies. The well-behaved nature of "stable" orbits and the possible miss-behaviour of the "unstable" ones is pointed out and, in particular, the fact that independent Killing vector fields in space-time may not induce independent such vector fields on unstable orbits. Several examples are presented to exhibit these features. Finally, an appendix is given which revisits and attempts to clarify the well-known theorem of Fubini on the dimension of Killing orbits. 
  The causal boundary construction of Geroch, Kronheimer, and Penrose has some universal properties of importance for general studies of spacetimes, particularly when equipped with a topology derived from the causal structure. Properties of the causal boundary are detailed for spacetimes with spacelike boundaries, for multi-warped spacetimes, for static spacetimes, and for spacetimes with group actions. 
  A study is made of the possible holonomy group types of a space-time for which the energy-momentum tensor corresponds to a null or non-null electromagnetic field, a perfect fluid or a massive scalar field. The case of an Einstein space is also included. The techniques developed are also applied to vacuum and conformally flat space-times and contrasted with already known results in these two cases. Examples are given. 
  We present a new reformulation of the canonical quantum geometrodynamics, which allows to overcome the fundamental problem of the frozen formalism and, therefore, to construct an appropriate Hilbert space associate to the solution of the restated dynamics. More precisely, to remove the ambiguity contained in the Wheeler-DeWitt approach, with respect to the possibility of a (3 + 1)-splitting when the space-time is in a quantum regime, we fix the reference frame (i.e. the lapse function and the shift vector) by introducing the so-called kinematical action; as a consequence the new super-Hamiltonian constraint becomes a parabolic one and we arrive to a Schroedinger-like approach for the quantum dynamics. In the semiclassical limit our theory provides General Relativity in the presence of an additional energy-momentum density contribution coming from no longer zero eigenvalues of the Hamiltonian constraints; the interpretation of these new contributions comes out in natural way as soon as it is recognized that the kinematical action can be recasted in such a way it describes a pressureless, but, in general, non geodesic perfect fluid. 
  We investigate the question of how an observer in 4D perceives the five-dimensional geodesic motion. We consider the interpretation of null and non-null bulk geodesics in the context of brane theory, space-time-matter theory (STM) and other non-compact approaches. We develop a "frame-invariant" formalism that allows the computation of the rest mass and its variation as observed in 4D. We find the appropriate expression for the four-acceleration and thus obtain the extra force observed in 4D. Our formulae extend and generalize all previous results in the literature. An important result here is that the extra force in brane-world models with ${\bf Z}_{2}$-symmetry is continuous and well defined across the brane. This is because the momentum component along the extra dimension is discontinuous across the brane, which effectively compensates the discontinuity of the extrinsic curvature. We show that brane theory and STM produce identical interpretation of the bulk geodesic motion. This holds for null and non-null bulk geodesics. Thus, experiments with test particles are unable to distinguish whether our universe is described by the brane world scenario or by STM. However, they do discriminate between the brane/STM scenario and other non-compact approaches. Among them the canonical and embedding approaches, which we examine in detail here. 
  We consider the possibilities for obtaining information about the equation of state for quark matter by using future direct observational data on gravitational waves. We study the nonradial oscillations of both fluid and spacetime modes of pure quark stars. If we observe the $f$ and the lowest $w_{\rm II}$ modes from quark stars, by using the simultaneously obtained radiation radius we can constrain the bag constant $B$ with reasonable accuracy, independently of the $s$ quark mass. 
  A model is proposed of a collapsing quasi-spherical radiating star with matter content as shear-free isotropic fluid undergoing radial heat-flow with outgoing radiation. To describe the radiation of the system, we have considered both plane symmetric and spherical Vaidya solutions. Physical conditions and thermodynamical relations are studied using local conservation of momentum and surface red-shift. We have found that for existence of radiation on the boundary, pressure on the boundary is not necessary. 
  After commenting on the early search for a mechanism explaining the Newtonian action-at-a-distance gravitational law we review non-Newtonian effects occurring in certain ansatzes for shielding, screening and absorption effects in pre-relativistic theories of gravity. Mainly under the aspect of absorption and suppression (or amplification), we then consider some implications of these ansatzes for relativistic theories of gravity and discuss successes and problems in establishing a general framework for a comparison of alternative relativistic theories of gravity. We examine relativistic representatives of theories with absorption and suppression (or amplification) effects, such as fourth-order theories, tetrad theories and the Einstein-Cartan-Kibble-Sciama theory. 
  In discussing fundamentals of general-relativistic irreversible continuum thermodynamics, this theory is shown to be characterized by the feature that no thermodynamical degrees of freedom are ascribed to gravitation. However, accepting that black hole thermodynamics seems to oppose this harmlessness of gravitation one is called on consider other approaches. Therefore, in brief some gravitational and thermodynamical alternatives are reviewed. 
  We present the complete family of space-times with a non-expanding, shear-free, twist-free, geodesic principal null congruence (Kundt waves) that are of algebraic type III and for which the cosmological constant ($\Lambda_c$) is non-zero. The possible presence of an aligned pure radiation field is also assumed. These space-times generalise the known vacuum solutions of type N with arbitrary $\Lambda_c$ and type III with $\Lambda_c=0$. It is shown that there are two, one and three distinct classes of solutions when $\Lambda_c$ is respectively zero, positive and negative. The wave surfaces are plane, spherical or hyperboloidal in Minkowski, de Sitter or anti-de Sitter backgrounds respectively, and the structure of the family of wave surfaces in the background space-time is described. The weak singularities which occur in these space-times are interpreted in terms of envelopes of the wave surfaces. 
  A formalism is introduced which may describe both standard linearized waves and gravitational waves in Isaacson's high-frequency limit. After emphasizing main differences between the two approximation techniques we generalize the Isaacson method to non-vacuum spacetimes. Then we present three large explicit classes of solutions for high-frequency gravitational waves in particular backgrounds. These involve non-expanding (plane, spherical or hyperboloidal), cylindrical, and expanding (spherical) waves propagating in various universes which may contain a cosmological constant and electromagnetic field. Relations of high-frequency gravitational perturbations of these types to corresponding exact radiative spacetimes are described. 
  We show, using a covariant and gauge-invariant charged multifluid perturbation scheme, that velocity perturbations of the matter-dominated dust Friedmann-Lemaitre-Robertson-Walker (FLRW) model can lead to the generation of cosmic magnetic fields. Moreover, using cosmic microwave background (CMB) constraints, it is argued that these fields can reach strengths of between 10^{-28} and 10^{-29} G at the time the dynamo mechanism sets in, making them plausible seed field candidates. 
  We study the renormalized energy-momentum tensor (EMT) of cosmological scalar fluctuations during the slow-rollover regime for chaotic inflation with a quadratic potential and find that it is characterized by a negative energy density which grows during slow-rollover. We also approach the back-reaction problem as a second-order calculation in perturbation theory finding no evidence that the back-reaction of cosmological fluctuations is a gauge artifact. In agreement with the results on the EMT, the average expansion rate is decreased by the back-reaction of cosmological fluctuations. 
  We analyze the compatibility between the geometrodynamics and thermodynamics of a binary mixture of perfect fluids which describe inhomogeneous cosmological models. We generalize the thermodynamic scheme of general relativity to include the chemical potential of the fluid mixture with non-vanishing entropy production. This formalism is then applied to the case of Szekeres and Stephani families of cosmological models. The compatibility conditions turn out to impose symmetry conditions on the cosmological models in such a way that only the limiting case of the Friedmann-Robertson-Walker model remains compatible. This result is an additional indication of the incompatibility between thermodynamics and relativity. 
  A conventional explanation of the correlation between the Pioneer 10/11 anomalous acceleration and spin-rate change is given. First, the rotational Doppler shift analysis is improved. Finally, a relation between the radio beam reaction force and the spin-rate change is established. Computations are found in good agreement with observational data. The relevance of our result to the main Pioneer 10/11 anomalous acceleration is emphasized. Our analysis leads us to conclude that the latter may not be merely artificial. 
  We analyze asymptotic structure of general gravitational and electromagnetic fields near an anti-de Sitter-like conformal infinity. Dependence of the radiative component of the fields on a null direction along which the infinity is approached is obtained. The directional pattern of outgoing and ingoing radiation, which supplements standard peeling property, is determined by the algebraic (Petrov) type of the fields and also by orientation of principal null directions with respect to the timelike infinity. The dependence on the orientation is a new feature if compared to spacelike infinity. 
  A black hole, surrounded by a reflecting shell, acts as an effective star-like object with respect to the outer region that leads to vacuum polarization outside, where the quantum fields are in the Boulware state. We find the quantum correction to the Hawking temperature, taking into account this circumstance. It is proportional to the integral of the trace of the total quantum stress-energy tensor over the whole space from the horizon to infinity. For the shell, sufficiently close to the horizon, the leading term comes from the boundary contribution of the Boulware state. 
  The semilinear wave equation on the (outer) Schwarzschild manifold is studied. We prove local decay estimates for general (non-radial) data, deriving a-priori Morawetz type estimates. 
  New geometries were obtained by adding a suitable surface term involving the components of the angular momentum to the corresponding free Lagrangians. Killing vectors, Killing-Yano and Killing tensors of the obtained manifolds were investigated. 
  Quadratic Lagrangians are introduced adding surface terms to a free particle Lagrangian. Geodesic equations are used in the context of the Hamilton-Jacobi formulation of constrained sysytem. Manifold structure induced by the quadratic Lagrangian is investigated. 
  Using Hamilton-Jacobi formalism we investigated the massive Yang-Mills theory on both extended and reduced phase-space. The integrability conditions were discussed and the actions were calculated. 
  The conventional group of four-dimensional diffeomorphisms is not realizeable as a canonical transformation group in phase space. Yet there is a larger field-dependent symmetry transformation group which does faithfully reproduce 4-D diffeomorphism symmetries. Some properties of this group were first explored by Bergmann and Komar. More recently the group has been analyzed from the perspective of projectability under the Legendre map. Time translation is not a realizeable symmetry, and is therefore distinct from diffeomorphism-induced symmetries. This issue is explored further in this paper. It is shown that time is not "frozen". Indeed, time-like diffeomorphism invariants must be time-dependent. Intrinsic coordinates of the type proposed by Bergmann and Komar are used to construct invariants. Lapse and shift variables are retained as canonical variables in this approach, and therefore will be subject to quantum fluctuations in an eventual quantum theory. Concepts and constructions are illustrated using the relativistic classical and quantum free particle. In this example concrete time-dependent invariants are displayed and fluctuation in proper time is manifest. 
  Any candidate theory of quantum gravity must address the breakdown of the classical smooth manifold picture of space-time at distances comparable to the Planck length. String theory, in contrast, is formulated on conventional space-time. However, we show that in the low energy limit, the dynamics of generally curved Dirichlet p-branes possess an extended local isometry group, which can be absorbed into the brane geometry as an almost product structure. The induced kinematics encode two invariant scales, namely a minimal length and a maximal speed, without breaking general covariance. Quantum gravity effects on D-branes at low energy are then seen to manifest themselves by the kinematical effects of a maximal acceleration. Experimental and theoretical implications of such new kinematics are easily derived. We comment on consequences for brane world phenomenology. 
  We consider the quasinormal modes for a class of black hole spacetimes that, informally speaking, contain a closely ``squeezed'' pair of horizons. (This scenario, where the relevant observer is presumed to be ``trapped'' between the horizons, is operationally distinct from near-extremal black holes with an external observer.) It is shown, by analytical means, that the spacing of the quasinormal frequencies equals the surface gravity at the squeezed horizons. Moreover, we can calculate the real part of these frequencies provided that the horizons are sufficiently close together (but not necessarily degenerate or even ``nearly degenerate''). The novelty of our analysis (which extends a model-specific treatment by Cardoso and Lemos) is that we consider ``dirty'' black holes; that is, the observable portion of the (static and spherically symmetric) spacetime is allowed to contain an arbitrary distribution of matter. 
  Using Noether's identities, we define a superpotential with respect to a background for the Einstein Gauss-Bonnet theory of gravity. As an example, we show that its associated conserved charge yields the mass-energy of a D-dimensional Gauss-Bonnet black hole in an anti-de Sitter spacetime. 
  We investigate black hole solutions in the Einstein-Born-Infeld system. We clarify the role played by derivative corrections to the Born-Infeld (BI) action. The qulitative differences from the case without derivative corrections are: (i) there is no particlelike solution. (ii) the existence of the inner horizon is restricted to the near extreme solutions. (iii) contribution of the BI parameter $b$ to the gravitational mass and the Hawking temperature works in the opposite direction. 
  We explore the numerical stability properties of an evolution system suggested by Alekseenko and Arnold. We examine its behavior on a set of standardized testbeds, and we evolve a single black hole with different gauges. Based on a comparison with two other evolution systems with well-known properties, we discuss some of the strengths and limitations of such simple tests in predicting numerical stability in general. 
  We examine the Pioneer anomaly - a reported anomalous acceleration affecting the Pioneer 10/11, Galileo and Ulysses spacecrafts - in the context of a braneworld scenario. We show that effects due to the radion field cannot account for the anomaly, but that a scalar field with an appropriate potential is able to explain the phenomena. Implications and features of our solution are analyzed. 
  We show that the conclusion that matter stress-energy tensor satisfies the usual covariant continuity law, and the cosmological constant is still a constant of integration arrived at by Finkelstein et al (42, 340, 2001) is not valid. 
  Two flat Randall - Sundrum three-branes are analyzed, at fixed mutual distance, in the case where each brane contains an ideal isotropic fluid. Both fluids are to begin with assumed to obey the equation of state p=(\gamma -1)\rho, where \gamma is a constant. Thereafter, we impose the condition that there is zero energy flux from the branes into the bulk, and assume that the tension on either brane is zero. It then follows that constant values of the fluid energies at the branes are obtained only if the value of \gamma is equal to zero (i.e., a `vacuum' fluid). The fluids on the branes are related: if one brane is a dS_4 brane (the effective four-dimensional constant being positive), then the other brane is dS_4 also, and if the fluid energy density on one brane is positive, the energy density on the other brane is larger in magnitude but negative. This is a non-acceptable result, which sheds some light on how far it is possible to give a physical interpretation of the two-brane scenario. Also, we discuss the graviton localization problem in the two-brane setting, generalizing prior works. 
  We show global existence theorems for Gowdy symmetric spacetimes with type IIB stringy matter. The areal and constant mean curvature time coordinates are used. Before coming to that, it is shown that a wave map describes the evolution of this system. 
  Using Komar's definition, we give expressions for the mass and angular momentum of a rotating acoustic black hole. We show that the mass and angular momentum so defined, obey the equilibrium version of the first law of Black Hole thermodynamics. We also show that when a phonon passes by a vortex with a sink, its trajectory is bent. The angle of bending of the sound wave to leading order is quadratic in $A/cb$ and $B/cb$, where $b$ is the impact parameter and $A$ and $B$ are the parameters in the velocity of the fluid flow. The time delay in the propagation of sound wave which to first order depends only on $B/c^2$ and is independent of $A$. 
  Considering gravitational collapse of Type I matter fields, we prove that, given an arbitrary $C^{2}$- mass function $\textit{M}(r,v)$ and a $C^{1}$- function $h(r,v)$ (through the corresponding $C^{1}$- metric function $\nu(t,r)$), there exist infinitely many choices of energy distribution function $b(r)$ such that the `true' initial data ($\textit{M},h(r,v)$) leads the collapse to the formation of naked singularity. We further prove that the occurrence of such a naked singularity is stable with respect to small changes in the initial data. We remark that though the initial data leading to both black hole and naked singularity form a "big" subset of the true initial data set, their occurrence is not generic. The terms `stability' and `genericity' are appropriately defined following the theory of dynamical systems. The particular case of radial pressure $p_{r}(r)$ has been illustrated in details to get clear picture of how naked singularity is formed and how, it is stable with respect to initial data. 
  The "gravastar" picture developed by Mazur and Mottola is one of a very small number of serious challenges to our usual conception of a "black hole". In the gravastar picture there is effectively a phase transition at/ near where the event horizon would have been expected to form, and the interior of what would have been the black hole is replaced by a segment of de Sitter space. While Mazur and Mottola were able to argue for the thermodynamic stability of their configuration, the question of dynamic stability against spherically symmetric perturbations of the matter or gravity fields remains somewhat obscure. In this article we construct a model that shares the key features of the Mazur-Mottola scenario, and which is sufficiently simple for a full dynamical analysis. We find that there are some physically reasonable equations of state for the transition layer that lead to stability. 
  "There are several interpretations and approaches to relativity. All of them are characterized by the fact that none of them is accepted by physicists without doubts, even the Einsteinian General Relativity! Only those theories can get into the spotlight that predicts something that is different from predictions of other concurrent theories. From this point of view the theory of Janossy [1] is not an excellent idea, as he tried to show that his materialistic approach also corresponds to the general principles and equations of relativity. As his program was basically successful there was not any additional result, except the philosophical part."   Since the death of Janossy, his work has almost been forgotten. Both what he achieved and what he was not succeeded in. He was one of the founders of KFKI (Central Physics Research Institute of the Hungarian Scientific Academy) but his effort has not been carried over even there, however his collegues are still remember his name and his work. Luckily his books are still available in the Hungarian libraries. Additionally the most informed etherists in the world who are lucky enough to know his work are consider his work as a No1 reference.   This paper is designated to refresh the idea of the ether based gravitation theory of Janossy and introduce a well founded way to adjust it to be equivalent with the experiences and General Relativity. 
  Recently a scale invariant theory of gravity was constructed by imposing a conformal symmetry on general relativity. The imposition of this symmetry changed the configuration space from superspace - the space of all Riemannian 3-metrics modulo diffeomorphisms - to conformal superspace - the space of all Riemannian 3-metrics modulo diffeomorphisms and conformal transformations. However, despite numerous attractive features, the theory suffers from at least one major problem: the volume of the universe is no longer a dynamical variable. In attempting to resolve this problem a new theory is found which has several surprising and atractive features from both quantisation and cosmological perspectives. Furthermore, it is an extremely restrictive theory and thus may provide testable predictions quickly and easily. One particularly interesting feature of the theory is the resolution of the cosmological constant problem. 
  A nonintegrable phase-factor global approach to gravitation is developed by using the similarity of teleparallel gravity with electromagnetism. The phase shifts of both the COW and the gravitational Aharonov-Bohm effects are obtained. It is then shown, by considering a simple slit experiment, that in the classical limit the global approach yields the same result as the gravitational Lorentz force equation of teleparallel gravity. It represents, therefore, the quantum mechanical version of the classical description provided by the gravitational Lorentz force equation. As teleparallel gravity can be formulated independently of the equivalence principle, it will consequently require no generalization of this principle at the quantum level. 
  The Einsteinian Theory of Gravitation ("General Theory of Relativity") is founded essentially; on the reception that the geometrical properties of the 4-dimensional space-time continuum are defined from the matter in it. Contrary to this, in the Newtonian Mechanics space and time obey a absolute, matter-independent meaning. This thesis offers a compromise between the two conceptions: The spontaneous splitting of space-time in a "universal time" and a "absolute space" in the sense of Newton is accepted, but this (1+3)-splitting will receive the status of a dynamical object in the sense of Einstein. Herein, the (large scale) dynamics of the (1+3) splitting is coupled only weakly to the (local) fluctuation of the matter density with the help of the Einsteinian equations, so that these will keep their validity, regarding the gravitational phenomena, in a bounded domain of space (planets, stars, galaxies). However, at a cosmic scale, the properties of the universe as whole will be determined essentially from the selfdynamics of the space-time splitting. The geometry of the (1+3)-splitting and the expansion of the empty universe will be examined in detail. Contrary to the Einsteinian theory, the resulting new theory of gravitation contains a new degree of freedom, that comes into question as a carrier for the energy-momentum tensor of the the gravitational interaction. The "ground state" of the empty universe (-> maximal symmetry)is described by means of a deSitter-geometry, where only the expanding universe renders stable. 
  We review some results concerning the properties of static, spherically symmetric solutions of multidimensional theories of gravity: various scalar-tensor theories and a generalized string-motivated model with multiple scalar fields and fields of antisymmetric forms associated with p-branes. A Kaluza-Klein type framework is used: there is no dependence on internal coordinates but multiple internal factor spaces are admitted. We discuss the causal structure and the existence of black holes, wormholes and particle-like configurations in the case of scalar vacuum with arbitrary potentials as well as some observational predictions for exactly solvable systems with p-branes: post-Newtonian coefficients, Coulomb law violation and black hole temperatures. Particular attention is paid to conformal frames in which the theory is initially formulated and which are used for its comparison with observations; it is stressed that, in general, these two kinds of frames do not coincide. 
  It is postulated in general relativity that the matter energy-momentum tensor (named the stress tensor) vanishes if and only if all the matter fields vanish. In classical lagrangian field theory the stres tensor is the variational (symmetric) one and a priori it might occur that for some systems the tensor is identically zero for all field configurations whereas evolution of the system is subject to deterministic equations of motion. Such a system would not generate its own gravitational field. To check if suchsystems may exist we find a relationship between the stress tensor and the Euler operator. We prove that if a system of n interacting scalar fields (n cannot exceed the spacetime dimension d) or a single vector field (if d is even) has the stress tensor such that its divergence is identically zero ("on and off shell"), then the Lagrange equations of motion hold identically too. These systems are unphysical as having no propagation equations at all. Thus nontrivial field equations imply the nontrivial stress tensor. The theorem breaks down if the of the field components n is greater than d. We show that for n>d matter systems without energy and their own gravity (and yet detectable) are in principle admissible. Their equations of motion are degenerate. We also show for which matter systems their stress tensors cannot vanish for all solutions of the field equations. 
  This short review deals with a multidimensional gravitational model containing dilatonic scalar fields and antisymmetric forms. The manifold is chosen in the product form. The sigma-model approach and exact solutions are reviewed. 
  When 4-dimensional general relativity is extended by a 3-dimensional gravitational Chern-Simons term an apparent violation of diffeormorphism invariance is extinguished by the dynamical equations of motion for the modified theory. The physical predictions of this recently proposed model show little evidence of symmetry breaking, but require the vanishing of the Pontryagin density. 
  We investigate the dynamics of self-gravitating, spherically-symmetric distributions of fluid through numerical means. In particular, systems involving neutron star models driven far from equilibrium in the strong-field regime of general relativity are studied. Hydrostatic solutions of Einstein's equations using a stiff, polytropic equation of state are used for the stellar models. Many of the scenarios we examine involve highly-relativistic flows that require improvements upon previously published numerical methods to simulate. Here our particular focus is on the physical behavior of the coupled fluid-gravitational system at the threshold of black hole formation--so-called black hole critical phenomena. To investigate such phenomena starting from conditions representing stable stars, we must drive the star far from its initial stable configuration. We use one of two different mechanisms to do this: setting the initial velocity profile of the star to be in-going, or collapsing a shell of massless scalar field onto the star. Both of these approaches give rise to a large range of dynamical scenarios that the star may follow. These scenarios have been extensively surveyed by using different initial star solutions, and by varying either the magnitude of the velocity profile or the amplitude of the scalar field pulse. In addition to illuminating the critical phenomena associated with the fluid collapse, the resulting phase diagram of possible outcomes provides an approximate picture of the stability of neutron stars to large, external perturbations that may occur in nature. 
  The problem of constructing a model of an extended charged particle within the context of general relativity has a long and distinguished history. The distinctive feature of these models is that, in some way or another, they require the presence of negative mass in order to maintain stability against Coulomb's repulsion. Typically, the particle contains a core of $negative$ mass surrounded by a positive-mass outer layer, which emerges from the Reissner-Nordstr\"{o}m field.   In this work we show how the Einstein-Maxwell field equations can be used to construct an extended model where the mass is positive everywhere. This requires the principal pressures to be unequal inside the particle. The model is obtained by setting the "effective" matter density, rather than the rest matter density, equal to zero. The Schwarzschild mass of the particle arises from the electrical and gravitational field (Weyl tensor) energy. The model satisfies the energy conditions of Hawking and Ellis. A particular solution that illustrates the results is presented. 
  We study global flat embeddings inside and outside of event horizons of black holes such as Schwarzschild and Reissner-Nordstr\"{o}m black holes, and of de Sitter space. On these overall patches of the curved manifolds we investigate four accelerations and Hawking temperatures by introducing relevant Killing vectors. 
  We have developed a torsion pendulum facility for LISA gravitational reference sensor ground testing that allows us to put significant upper limits on residual stray forces exerted by LISA-like position sensors on a representative test mass and to characterize specific sources of disturbances for LISA. We present here the details of the facility, the experimental procedures used to maximize its sensitivity, and the techniques used to characterize the pendulum itself that allowed us to reach a torque sensitivity below 20 fNm /sqrt{Hz} from 0.3 to 10 mHz. We also discuss the implications of the obtained results for LISA. 
  In an n dimensional vector space, any tensor which is antisymmetric in k>n arguments must vanish; this is a trivial consequence of the limited number of dimensions. However, when other possible properties of tensors, for example trace-freeness, are taken into account, such identities may be heavily disguised. Tensor identities of this kind were first considered by Lovelock, and later by Edgar and Hoeglund.   In this paper we continue their work. We obtain dimensionally dependent identities for highly structured expressions of products of (2,2)-forms. For tensors possessing more symmetries, such as block symmetry W_{abcd} = W_{cdab}, or the first Bianchi identity W_{a[bcd]} = 0, we derive identities for less structured expressions.   These identities are important tools when studying super-energy tensors, and, in turn, deriving identities for them. As an application we are able to show that the Bel-Robinson tensor, the super-energy tensor for the Weyl tensor, satisfies the equation T_{abcy}T^{abcx} = 1/4g^{x}_{y}T_{abcd}T^{abcd} in four dimensions, irrespective of the signature of the space. 
  We study the gravitational waves in spacetimes of arbitrary dimension. They generalize the pp-waves and the Kundt waves, obtained earlier in four dimensions. Explicit solutions of the Einstein and Einstein-Maxwell equations are derived for an arbitrary cosmological constant. 
  We construct here a special class of perfect fluid collapse models which generalizes the homogeneous dust collapse solution in order to include non-zero pressures and inhomogeneities into evolution. It is shown that a black hole is necessarily generated as end product of continued gravitational collapse, rather than a naked singularity. We examine the nature of the central singularity forming as a result of endless collapse and it is shown that no non-spacelike trajectories can escape from the central singularity. Our results provide some insights into how the dynamical collapse works, and into the possible formulations of the cosmic censorship hypothesis, which is as yet a major unsolved problem in black hole physics. 
  This article has been replaced by gr-qc/0412011 
  The Einstein-Schrodinger theory is modified by defining the metric and electromagnetic field differently than in previous work, and by adding a cosmological constant contribution caused by zero-point fluctuations. This ``extrinsic'' cosmological constant which multiplies the symmetric metric is assumed to be nearly cancelled by Schrodinger's ``bare'' cosmological constant which multiplies the nonsymmetric fundamental tensor, such that the total cosmological constant is consistent with measurement. This modified Einstein-Schrodinger theory is shown to approximate ordinary general relativity and electromagnetism so closely that differences between the two theories may be too small to detect by experiment. The modified theory correctly predicts the equation of motion for charged particles, and avoids problems with negative energy particles. An exact solution to the modified field equations is derived which closely approximates the Reissner-Nordstrom solution for a non-rotating charged mass. The divergence of the Einstein equations vanishes exactly, allowing external fields to be accounted for with an energy-momentum tensor. Some ideas are presented concerning the uniqueness of the theory. 
  Captures of stellar-mass compact objects (COs) by massive ($\sim 10^6 M_\odot$) black holes (MBHs) are potentially an important source for LISA, the proposed space-based gravitational-wave (GW) detector. The orbits of the inspiraling COs are highly complicated; they can remain rather eccentric up until the final plunge, and display extreme versions of relativistic perihelion precession and Lense-Thirring precession of the orbital plane. The strongest capture signals will be ~10 times weaker than LISA's instrumental noise, but in principle (with sufficient computing power) they can be disentangled from the noise by matched filtering. The associated template waveforms are not yet in hand, but theorists will very likely be able to provide them before LISA launches. Here we introduce a family of approximate (post-Newtonian) capture waveforms, given in (nearly) analytic form, for use in advancing LISA studies until more accurate versions are available. Our model waveforms include most of the key qualitative features of true waveforms, and cover the full space of capture-event parameters (including orbital eccentricity and the MBH's spin). Here we use our approximate waveforms to (i) estimate the relative contributions of different harmonics (of the orbital frequency) to the total signal-to-noise ratio, and (ii) estimate the accuracy with which LISA will be able to extract the physical parameters of the capture event from the measured waveform. For a typical source (a $10 M_\odot$ CO captured by a $10^6 M_\odot$ MBH at a signal-to-noise ratio of 30), we find that LISA can determine the MBH and CO masses to within a fractional error of $\sim 10^{-4}$, measure $S/M^2$ (where $S$ and $M$ are the MBH's mass and spin) to within $\sim 10^{-4}$, and determine the sky location of the source to within $\sim 10^{-3}$ stradians. 
  We review some recent results concerning the properties of a spherically symmetric global monopole in $(D=d+2)$-dimensional general relativity. Some common features of monopole solutions are found independently of the choice of the symmetry-breaking potential. Thus, the solutions show six types of qualitative behavior and can contain at most one simple horizon. For the standard Mexican hat potential, we analytically find the $D$-dependent range of $\gamma$ (the gravitational field strength parameter) in which there exist globally regular solutions with a monotonically growing Higgs field, containing a horizon and a Kantowski-Sachs (KS) cosmology outside it, where the topology of spatial sections is $\R\times \S^d$. Their cosmological properties favor the idea that the standard Big Bang might be replaced with a nonsingular static core and a horizon appearing as a result of some symmetry-breaking phase transition on the Planck energy scale. We have also found families of new solutions with an oscillating Higgs field, parametrized by the number of its knots. All such solutions describe space-times of finite size, possessing a regular center, a horizon and a singularity beyond it. 
  The dynamics of Gowdy vacuum spacetimes is considered in terms of Hubble-normalized scale-invariant variables, using the timelike area temporal gauge. The resulting state space formulation provides for a simple mechanism for the formation of ``false'' and ``true spikes'' in the approach to the singularity, and a geometrical formulation for the local attractor. 
  We consider an Anti-de Sitter Universe filled by quantum CFT with classical phantom matter and perfect fluid. The model represents the combination of a trace-anomaly annihilated and a phantom driven Anti-de Sitter Universes. The influence exerted by the quantum effects and phantom matter on the AdS space is discussed. Different energy conditions in this type of Universe are investigated and compared with those for the corresponding model in a de Sitter Universe. 
  By parametrizing the action integral for the standard Schrodinger equation we present a derivation of the recently proposed method for quantizing a parametrized theory. The reformulation suggests a natural extension from conventional to nonlinear quantum mechanics. This generalization enables a unitary description of the quantum evolution for a broad class of constrained Hamiltonian systems with a nonlinear kinematic structure. In particular, the new theory is applicable to the quantization of cosmological models where a chosen gravitational degree of freedom acts as geometric time. This is demonstrated explicitly using three cosmological models: the Friedmann universe with a massless scalar field and Bianchi type I and IX models. Based on these investigations, the prospect of further developing the proposed quantization scheme in the context of quantum gravity is discussed. 
  We discuss a semiclassical treatment to inflationary models from Kaluza-Klein theory without the cylinder condition. We conclude that the evolution of the early universe could be described by a geodesic trayectory of a cosmological 5D metric here proposed, so that the effective 4D FRW background metric should be a hypersurface on a constant fifth dimension. 
  We study the gravitomagnetic effect in the context of absolute parallelism with the use of a modified geodesic equation via a free parameter b. We calculate the time difference in two atomic clocks orbiting the Earth in opposite directions and find a small correction due to the coupling between the torsion of the spacetime and the internal structure of atomic clocks measured by the free parameter. 
  In this article we extend to higher dimensional space-times a recent theorem proved by Salgado which characterizes a three-parameter family of static and spherically symmetric solutions to the Einstein Field Equations. As it happens in four dimensions, it is shown that the Schwarzschild, Reissner-Nordstrom and global monopole solutions in higher dimensions are particular cases from this family. 
  Static spherically symmetric anisotropic source has been studied for the Einstein-Maxwell field equations assuming the erstwhile cosmological constant $ \Lambda $ to be a space-variable scalar, viz., $ \Lambda = \Lambda(r) $. Two cases have been examined out of which one reduces to isotropic sphere. The solutions thus obtained are shown to be electromagnetic in origin as a particular case. It is also shown that the generally used pure charge condition, viz., $ \rho + p_r = 0 $ is not always required for constructing electromagnetic mass models. 
  The quantum corrections make the black hole capable of reflection: any particle that approaches the event horizon can bounce back in the outside world. The albedo of the black hole depends on its temperature. The reflection shares physical origins with the phenomenon of Hawking radiation; both effects are explained as consequences of the singular nature that the event horizon exhibits on the quantum level. 
  We consider rotating boson star solutions in a three-dimensional anti-de Sitter spacetime and investigate the influence of the rotation on their properties. The mass and angular momentum of these configurations are computed by using the counterterm method. No regular solution is found in the limit of vanishing cosmological constant. 
  Non commutative geometry is creating new possibilities for physics. Quantum spacetime geometry and post inflationary models of the universe with matter creation have an enormous range of scales of time, distance and energy in between. There is a variety of physics possible till the nucleosynthesis epoch is reached. The use of topology and non commutative geometry in cosmology is a recent approach. This paper considers the possibility of topological solutions of a vortex kind given by non commutative structures. These are interpreted as dark matter, with the grand unified Yang-Mills field theory energy scale used to describe its properties. The relation of the model with other existing theories is discussed. 
  Diffeomorphism covariant theories with dynamical background metric, like gravity coupled to matter fields in the way expressed by Einstein-Hilbert's action or relativistic strings described by Polyakov's action, have `on-shell' vanishing energy-momentum tensor $t_{\mu\nu}$ because $t_{\mu\nu}$ is, essentially, the Eulerian derivative associated with the dynamical background metric and thus $t_{\mu\nu}$ vanishes `on-shell.' Therefore, the equations of motion for the dynamical background metric play a double role: as equations of motion themselves and as a reflection of the fact that $t_{\mu\nu}=0$. Alternatively, the vanishing property of $t_{\mu\nu}$ can be seen as a reflection of the so-called `problem of time' present in diffeomorphism covariant theories in the sense that $t_{\mu\nu}$ are written as linear combinations of first class constraints only. 
  Spherically symmetric thin-shell wormholes in the presence of a cosmological constant are constructed applying the cut-and-paste technique implemented by Visser. Using the Darmois-Israel formalism the surface stresses, which are concentrated at the wormhole throat, are determined. This construction allows one to apply a dynamical analysis to the throat, considering linearized radial perturbations around static solutions. For a large positive cosmological constant, i.e., for the Schwarzschild-de Sitter solution, the region of stability is significantly increased, relatively to the null cosmological constant case, analyzed by Poisson and Visser. With a negative cosmological constant, i.e., the Schwarzschild-anti de Sitter solution, the region of stability is decreased. In particular, considering static solutions with a generic cosmological constant, the weak and dominant energy conditions are violated, while for $a_0 \leq 3M$ the null and strong energy conditions are satisfied. The surface pressure of the static solution is strictly positive for the Schwarzschild and Schwarzschild-anti de Sitter spacetimes, but takes negative values, assuming a surface tension in the Schwarzschild-de Sitter solution, for high values of the cosmological constant and the wormhole throat radius. 
  In classical general relativity, the generic approach to the initial singularity is very complicated as exemplified by the chaos of the Bianchi IX model which displays the generic local evolution close to a singularity. Quantum gravity effects can potentially change the behavior and lead to a simpler initial state. This is verified here in the context of loop quantum gravity, using methods of loop quantum cosmology: the chaotic behavior stops once quantum effects become important. This is consistent with the discrete structure of space predicted by loop quantum gravity. 
  Homogeneous cosmological models with non-vanishing intrinsic curvature require a special treatment when they are quantized with loop quantum cosmological methods. Guidance from the full theory which is lost in this context can be replaced by two criteria for an acceptable quantization, admissibility of a continuum approximation and local stability. A quantization of the corresponding Hamiltonian constraints is presented and shown to lead to a locally stable, non-singular evolution compatible with almost classical behavior at large volume. As an application, the Bianchi IX model and its modified behavior close to its classical singularity is explored. 
  A unifying definition of trapped submanifold for arbitrary codimension by means of its mean curvature vector is presented. Then, the interplay between (generalized) symmetries and trapped submanifolds is studied, proving in particular that (i) stationary spacetimes cannot contain closed trapped nor marginally trapped submanifolds S of any codimension; (ii) S can be within the subset where there is a null Killing vector only if it is marginally trapped with mean curvature vector parallel to the null Killing; (iii) any submanifold orthogonal to a timelike or null Killing vector has a mean curvature vector orthogonal to it. All results are purely geometric, hold in arbitrary dimension, and can be appropriately generalized to many non-Killing vector fields, such as conformal Killing vectors and the like. A simple criterion to ascertain the trapping or not of a family of codimension-2 submanifolds is given. A path allowing to generalize the singularity theorems is conjectured as feasible and discussed. 
  With the aid of a simple family of examples, we show that the quasi-local mass defined by Kijowski and Liu and Yau, and shown by Liu and Yau to be positive, may be strictly positive for space-like, topologically spherical 2-surfaces in flat space-time. 
  Geometric (Clifford) algebra provides an efficient mathematical language for describing physical problems. We formulate general relativity in this language. The resulting formalism combines the efficiency of differential forms with the straightforwardness of coordinate methods. We focus our attention on orthonormal frames and the associated connection bivector, using them to find the Schwarzschild and Kerr solutions, along with a detailed exposition of the Petrov types for the Weyl tensor. 
  Recently, an approximated solution of the Einstein equations for a rotating body whose mass effects are negligible with respect to the rotational ones has been derived by Tartaglia. At first sight, it seems to be interesting because both external and internal metric tensors have been consistently found, together an appropriate source tensor; moreover, it may suggest possible experimental checks since the conditions of validity of the considered metric are well satisfied at Earth laboratory scales. However, it should be pointed out that reasonable doubts exist if it is physically meaningful because it is not clear if the source tensor related to the adopted metric can be realized by any real extended body. Here we derive the geodesic equations of the metric and analyze the allowed motions in order to disclose possible unphysical features which may help in shedding further light on the real nature of such approximated solution of the Einstein equations. 
  We consider three possible approaches to formulating coordinate transformations on position space associated with non-linear Lorentz transformations on momentum space. The first approach uses the definition of velocity and gives the standard Lorentz transformation. In the second method, we translate the behavior in momentum space into position space by means of Fourier transform. Under certain conditions, it also gives the standard Lorentz transformation on position space. The third approach investigates the covariance of the modified Klein-Gordon equation obtained from the dispersion relation. 
  A novel approach to the calculation of the deflection of highly relativistic test particles in gravitational fields is described. We make use of the light-like boosts of the gravitational fields of the sources. Examples are given of the deflection of highly relativistic particles in the Schwarzschild and Kerr gravitational fields, in the field of a static, axially symmetric, multipole source and in the field of a cosmic string. The deflection of spinning particles is also discussed. 
  Due to its large number of symmetries the Schwarzschild Black Hole can be described by a specific two-dimensional dilaton gravity model. After reviewing classical, semi-classical and quantum properties and a brief discussion of virtual black holes deformations are studied: the first part is devoted to deformations of the Lorentz-symmetry, the second part to dynamical deformations and its role for the long time evaporation of the Schwarzschild Black Hole. 
  Recent progress in quantum gravity and string theory has raised interest among scientists to whether or not nature behaves discretely at the Planck scale. There are two attitudes twoards this discretenes i.e. top-down and bottom-up approach. We have followed up the bottom-up approach. Here we have tried to describe how macroscopic space-time or its underlying mesoscopic substratum emerges from a more fundamental concept. The very concept of space-time, causality may not be valid beyond Planck scale. We have introduced the concept of generalised time within the framework of Sheaf Cohomology where the physical time emrges around and above Planck scale. The possible physical amd metaphysical implications are discussed. 
  We present a complete treatment in the strong field limit of gravitational retro-lensing by a static spherically symmetric compact object having a photon sphere. The results are compared with those corresponding to ordinary lensing in similar strong field situations. As examples of application of the formalism, a supermassive black hole at the galactic center and a stellar mass black hole in the galactic halo are studied as retro-lenses, in both cases using the Schwarzschild and Reissner-Nordstrom geometries. 
  A stability criterion is derived in general relativity for self-similar solutions with a scalar field and those with a stiff fluid, which is a perfect fluid with the equation of state $P=\rho$. A wide class of self-similar solutions turn out to be unstable against kink mode perturbation. According to the criterion, the Evans-Coleman stiff-fluid solution is unstable and cannot be a critical solution for the spherical collapse of a stiff fluid if we allow sufficiently small discontinuity in the density gradient field in the initial data sets. The self-similar scalar-field solution, which was recently found numerically by Brady {\it et al.} (2002 {\it Class. Quantum. Grav.} {\bf 19} 6359), is also unstable. Both the flat Friedmann universe with a scalar field and that with a stiff fluid suffer from kink instability at the particle horizon scale. 
  A free test particle in 5-dimensional Kaluza-Klein spacetime will show its electricity in the reduced 4-dimensional spacetime when it moves along the fifth dimension. In the light of this observation, we study the coupling of a 5-dimensional dust field with the Kaluza-Klein gravity. It turns out that the dust field can curve the 5-dimensional spacetime in such a way that it provides exactly the source of the electromagnetic field in the 4-dimensional spacetime after the dimensional reduction. 
  In this thesis, we explore three phenomenological alternatives to the current paradigm of the standard inflationary big bang scenario. The three alternative themes are spin torsion (or Einstein-Cartan-Kibble-Sciama) theories, extra dimensions (braneworld cosmology) and changing global symmetry. In the spin torsion theories, we found new cosmological solutions with a cosmological constant as alternative to the standard scalar field driven inflationary scenario and we conclude that these toy models do not exhibit an inflationary phase. In the theme of extra dimensions, we discuss the dynamics of linearized scalar and tensor perturbations in an almost Friedmann-Robertson-Walker braneworld cosmology of Randall-Sundrum type II using the 1+3 covariant approach. We derive a complete set of frame-independent equations for the total matter variables, and a partial set of equations for the non-local variables, which arise from the projection of the Weyl tensor in the bulk. The latter equations are incomplete since there is no propagation equation for the non-local anisotropic stress. In the simplest approximation, we show the braneworld imprint as a correction to the power spectra for standard temperature and polarization anisotropies and similarly show that the tensor anisotropies are also insensitive to the high energy effects. Finally in the theme of changing global symmetry, we constructed a bounded isothermal solution embedded in an expanding Einstein de Sitter universe and showed that there is a possible phase transition in the far future. 
  We numerically solve the inhomogeneous Zerilli-Moncrief and Regge-Wheeler equations in the time domain. We obtain the gravitational waveforms produced by a point-particle of mass $\mu$ traveling around a Schwarzschild black hole of mass M on arbitrary bound and unbound orbits. Fluxes of energy and angular momentum at infinity and the event horizon are also calculated. Results for circular orbits, selected cases of eccentric orbits, and parabolic orbits are presented. The numerical results from the time-domain code indicate that, for all three types of orbital motion, black hole absorption contributes less than 1% of the total flux, so long as the orbital radius r_p(t) satisfies r_p(t)> 5M at all times. 
  An approximate analytical and non-linear solution of the Einstein field equations is derived for a system of multiple non-rotating black holes. The associated space-time has the same asymptotic structure as the Brill-Lindquist initial data solution for multiple black holes. The system admits an Arnowitt-Deser-Misner (ADM) Hamiltonian that can particularly evolve the Brill-Lindquist solution over finite time intervals. The gravitational field of this model may properly be referred to as a skeleton approximate solution of the Einstein field equations. The approximation is based on a conformally flat truncation, which excludes gravitational radiation, as well as a removal of some additional gravitational field energy. After these two simplifications, only source terms proportional to Dirac delta distributions remain in the constraint equations. The skeleton Hamiltonian is exact in the test-body limit, it leads to the Einsteinian dynamics up to the first post-Newtonian approximation, and in the time-symmetric limit it gives the energy of the Brill-Lindquist solution exactly. The skeleton model for binary systems may be regarded as a kind of analytical counterpart to the numerical treatment of orbiting Misner-Lindquist binary black holes proposed by Gourgoulhon, Grandclement, and Bonazzola, even if they actually treat the corotating case. Along circular orbits, the two-black-hole skeleton solution is quasi-stationary and it fulfills the important property of equality of Komar and ADM masses. Explicit calculations for the determination of the last stable circular orbit of the binary system are performed up to the tenth post-Newtonian order within the skeleton model. 
  General Relativity has so far passed almost all the ground-based and solar-system experiments. Any reasonable extended gravity models should consistently reduce to it at least in the weak field approximation. In this work we derive the gravitational potential for the Palatini formulation of the modified gravity of the L(R) type which admits a de Sitter vacuum solution. We conclude that the Newtonian limit is always obtained in those class of models and the deviations from General Relativity is very small for a slowly moving source. 
  This paper discusses new fundamental physics experiment that will test relativistic gravity at the accuracy better than the effects of the second order in the gravitational field strength, $\propto G^2$. The Laser Astrometric Test Of Relativity (LATOR) mission uses laser interferometry between two micro-spacecraft whose lines of sight pass close by the Sun to accurately measure deflection of light in the solar gravity. The key element of the experimental design is a redundant geometry optical truss provided by a long-baseline (100 m) multi-channel stellar optical interferometer placed on the International Space Station (ISS). The spatial interferometer is used for measuring the angles between the two spacecraft and for orbit determination purposes. The geometric redundancy enables LATOR to measure the departure from Euclidean geometry caused by the solar gravity field to a very high accuracy. LATOR will not only improve the value of the parameterized post-Newtonian (PPN) $\gamma$ to unprecedented levels of accuracy of 1 part in 10$^{8}$, it will also reach ability to measure effects of the next post-Newtonian order ($c^{-4}$) of light deflection resulting from gravity's intrinsic non-linearity. The solar quadrupole moment parameter, $J_2$, will be measured with high precision, as well as a variety of other relativistic effects including Lense-Thirring precession. LATOR will lead to very robust advances in the tests of Fundamental physics: this mission could discover a violation or extension of general relativity, or reveal the presence of an additional long range interaction in the physical law. There are no analogs to the LATOR experiment; it is unique and is a natural culmination of solar system gravity experiments. 
  Accurate analysis of precision ranges to the Moon has provided several tests of gravitational theory including the Equivalence Principle, geodetic precession, parameterized post-Newtonian (PPN) parameters $\gamma$ and $\beta$, and the constancy of the gravitational constant {\it G}. Since the beginning of the experiment in 1969, the uncertainties of these tests have decreased considerably as data accuracies have improved and data time span has lengthened. We are exploring the modeling improvements necessary to proceed from cm to mm range accuracies enabled by the new Apache Point Observatory Lunar Laser-ranging Operation (APOLLO) currently under development in New Mexico. This facility will be able to make a significant contribution to the solar system tests of fundamental and gravitational physics. In particular, the Weak and Strong Equivalence Principle tests would have a sensitivity approaching 10$^{-14}$, yielding sensitivity for the SEP violation parameter $\eta$ of $\sim 3\times 10^{-5}$, $v^2/c^2$ general relativistic effects would be tested to better than 0.1%, and measurements of the relative change in the gravitational constant, $\dot{G}/G$, would be $\sim0.1$% the inverse age of the universe. Having this expected accuracy in mind, we discusses the current techniques, methods and existing physical models used to process the LLR data. We also identify the challenges for modeling and data analysis that the LLR community faces today in order to take full advantage of the new APOLLO ranging station. 
  We argue that whether the universe is infinite or finite is less crucial than usually supposed. Paradoxes of repeating behaviour in the infinite, or eternal inflationary, universe can be alleviated by a realistic definition of differing lives. We also critically question the notion that our universe could simply be a simulation in somebody else's computer. 
  The first science run of the LIGO and GEO gravitational wave detectors presented the opportunity to test methods of searching for gravitational waves from known pulsars. Here we present new direct upper limits on the strength of waves from the pulsar PSR J1939+2134 using two independent analysis methods, one in the frequency domain using frequentist statistics and one in the time domain using Bayesian inference. Both methods show that the strain amplitude at Earth from this pulsar is less than a few times $10^{-22}$. 
  The paper contains a discussion of the properties of the gravito-magnetic interaction in non stationary conditions. A direct deduction of the equivalent of Faraday-Henry law is given. A comparison is made between the gravito-magnetic and the electro-magnetic induction, and it is shown that there is no Meissner-like effect for superfluids in the field of massive spinning bodies. The impossibility of stationary motions in directions not along the lines of the gravito-magnetic field is found. Finally the results are discussed in relation with the behavior of superconductors. 
  The present work considers (4+1)-dimensional spatially homogeneous vacuum cosmological models. Exact solutions -- some already existing in the literature, and others believed to be new -- are exhibited. Some of them are the most general for the corresponding Lie group with which each homogeneous slice is endowed, and some others are quite general. The characterization ``general'' is given based on the counting of the essential constants, the line-element of each model must contain; indeed, this is the basic contribution of the work. We give two different ways of calculating the number of essential constants for the simply transitive spatially homogeneous (4+1)-dimensional models. The first uses the initial value theorem; the second uses, through Peano's theorem, the so-called time-dependent automorphism inducing diffeomorphisms 
  In the first part of this article I present a system of retarded coordinates based at an arbitrary world line of an arbitrary curved spacetime. The retarded-time coordinate labels forward light cones that are centered on the world line, the radial coordinate is an affine parameter on the null generators of these light cones, and the angular coordinates are constant on each of these generators. The spacetime metric in the retarded coordinates is displayed as an expansion in powers of the radial coordinate and expressed in terms of the world line's acceleration vector and the spacetime's Riemann tensor evaluated at the world line. The formalism is illustrated in two examples, the first involving a comoving world line of a spatially-flat cosmology, the other featuring an observer in circular motion in the Schwarzschild spacetime. The main application of the formalism is presented in the second part of the article, in which I consider the motion of a small black hole in an empty external universe. I use the retarded coordinates to construct the metric of the small black hole perturbed by the tidal field of the external universe, and the metric of the external universe perturbed by the presence of the black hole. Matching these metrics produces the MiSaTaQuWa equations of motion for the small black hole. 
  Classical solutions of the spherically symmetric Nordstr\"{o}m-Vlasov system are shown to exist globally in time. The main motivation for investigating the mathematical properties of the Nordstr\"{o}m-Vlasov system is its relation to the Einstein-Vlasov system. The former is not a physically correct model, but it is expected to capture some of the typical features of the latter, which constitutes a physically satisfactory, relativistic model but is mathematically much more complex. We show that classical solutions of the spherically symmetric Nordstr\"{o}m-Vlasov system exist globally in time for compactly supported initial data under the additional condition that there is a lower bound on the modulus of the angular momentum of the initial particle system. We emphasize that this is not a smallness condition and that our result holds for arbitrary large initial data satisfying this hypothesis. 
  In this essay, I wish to share a novel perspective based on the principle of universalization in arriving at the relativistic and quantum world from the classical world. I also delve on some insightful discussion on going ``beyond''. 
  We give a general survey of the solution of the Einstein constraints by the conformal method on n dimensional compact manifolds. We prove some new results about solutions with low regularity (solutions in $H_{2}$ when n=3), and solutions with unscaled sources. 
  The main theoretical aspects of gravitoelectromagnetism ("GEM") are presented. Two basic approaches to this subject are described and the role of the gravitational Larmor theorem is emphasized. Some of the consequences of GEM are briefly mentioned. 
  In (V. Galdi et al., Phys. Rev. E57, 6470, 1998) a thorough characterization in terms of receiver operating characteristics (ROCs) of stochastic-resonance (SR) detectors of weak harmonic signals of known frequency in additive gaussian noise was given. It was shown that strobed sign-counting based strategies can be used to achieve a nice trade-off between performance and cost, by comparison with non-coherent correlators. Here we discuss the more realistic case where besides the sought signal (whose frequency is assumed known) further unwanted spectrally nearby signals with comparable amplitude are present. Rejection properties are discussed in terms of suitably defined false-alarm and false-dismissal probabilities for various values of interfering signal(s) strength and spectral separation. 
  Considering the Barrett-Crane spin foam model for quantum gravity with (positive) cosmological constant, we show that speeds must be quantized and we investigate the physical implications of this effect such as the emergence of an effective deformed Poincare symmetry. 
  A recently found (gr-qc/0303036) 2-index, symmetric, trace-free, divergence-free tensor is introduced for arbitrary source-free electromagnetic fields. The tensor can be constructed for any test Maxwell field in Einstein spaces (including proper vacuum), and more importantly for any Einstein-Maxwell spacetime. The tensor is explicitly given and analyzed in some special situations, such as general null electromagnetic fields, Reissner-Nordstr\"om solution, or classical electrodynamics. We present an explicit example where the conserved currents derived from the energy-momentum tensor using symmetries are trivial, but those derived from the new tensor are not. 
  The configuration space of general relativity is superspace - the space of all Riemannian 3-metrics modulo diffeomorphisms. However, it has been argued that the configuration space for gravity should be conformal superspace - the space of all Riemannian 3-metrics modulo diffeomorphisms and conformal transformations. Taking this conformal nature seriously leads to a new theory of gravity which although very similar to general relativity has some very different features particularly in cosmology and quantisation. It should reproduce the standard tests of general relativity. The cosmology is studied in some detail. The theory is incredibly restrictive and as a result admits an extremely limited number of possible solutions. The problems of the standard cosmology are addressed and most remarkably the cosmological constant problem is resolved in a natural way. The theory also has several attractive features with regard to quantisation particularly regarding the problem of time. 
  New electromagnetic conservation laws have recently been proposed: in the absence of electromagnetic currents, the trace of the Chevreton superenergy tensor, $H_{ab}$ is divergence-free in four-dimensional (a) Einstein spacetimes for test fields, (b) Einstein-Maxwell spacetimes. Subsequently it has been pointed out, in analogy with flat spaces, that for Einstein spacetimes the trace of the Chevreton superenergy tensor $H_{ab}$ can be rearranged in the form of a generalised wave operator $\square_L$ acting on the energy momentum tensor $T_{ab}$ of the test fields, i.e., $H_{ab}=\square_LT_{ab}/2$. In this letter we show, for Einstein-Maxwell spacetimes in the full non-linear theory, that, although, the trace of the Chevreton superenergy tensor $H_{ab}$ can again be rearranged in the form of a generalised wave operator $\square_G$ acting on the electromagnetic energy momentum tensor, in this case the result is also crucially dependent on Einstein's equations; hence we argue that the divergence-free property of the tensor $H_{ab}=\square_GT_{ab}/2$ has significant independent content beyond that of the divergence-free property of $T_{ab}$. 
  Spacetimes with horizons show a resemblance to thermodynamic systems and it is possible to associate the notions of temperature and entropy with them. Several aspects of this connection are reviewed in a manner appropriate for broad readership. The approach uses two essential principles: (a) the physical theories must be formulated for each observer entirely in terms of variables any given observer can access and (b) consistent formulation of quantum field theory requires analytic continuation to the complex plane. These two principles, when used together in spacetimes with horizons, are powerful enough to provide several results in a unified manner. Since spacetimes with horizons have a generic behaviour under analytic continuation, standard results of quantum field theory in curved spacetimes with horizons can be obtained directly (Sections III to VII). The requirements (a) and (b) also put strong constraints on the action principle describing the gravity and, in fact, one can obtain the Einstein-Hilbert action from the thermodynamic considerations. The latter part of the review (Sections VIII to X) investigates this deeper connection between gravity, spacetime microstructure and thermodynamics of horizons. This approach leads to several interesting results in the semiclassical limit of quantum gravity, which are described. 
  We present the preliminary results of the analysis to search for inspiraling compact binaries using TAMA300 DT8 data which was taken during 2003. We compare the quality and the stability of the data with that taken during DT6 in 2001. We find that the DT8 data has better quality and stability than the DT6 data. 
  Since Schwarzshild discovered the point-mass solution to Einstein's equations that bears his name, many equivalent forms of the metric have been catalogued. Using an elementary coordinate transformation, we derive the most general form for the stationary, spherically-symmetric vacuum metric, which contains one free function. Different choices for the function correspond to common expressions for the line element. From the general metric, we obtain particle and photon trajectories, and use them to specify several time coordinates adapted to physical situations. The most general form of the metric is only slightly more complicated than the Schwarzschild form, which argues effectively for teaching the general line element in place of the diagonal metric. 
  This paper addresses the motivation, technology and recent results in the tests of the general theory of relativity (GR) in the solar system. We specifically discuss Lunar Laser Ranging (LLR), the only technique available to test the Strong Equivalence Principle (SEP) and presently the most accurate method to test for the constancy of the gravitational constant, G. The new Apache Point Observatory Lunar Laser-ranging Operation (APOLLO) will enable tests of the Weak and Strong Equivalence Principles with a sensitivity approaching 10e-14, translating to a test of the SEP violation parameter, \eta, to a precision of ~3x10e-5. In addition, the (v/c)^2 general relativistic effects would be tested to better than 0.1%, and measurements of the relative change in the gravitational constant, \dot{G}/G, would be ~0.1% the inverse age of the universe.   We also discuss the Laser Astrometric Test Of Relativity (LATOR) mission that will be able to improve the value of the PPN parameter \gamma to accuracy of 1 part in 10e8 and will measure effects of the next post-Newtonian order (1/c^4) of light deflection resulting from gravity's intrinsic non-linearity, as well as measure a variety of other relativistic effects. LATOR will lead to very robust advances in the tests of fundamental physics: this mission could discover a violation or extension of GR, or reveal the presence of an additional long range interaction in the physical law. There are no analogs to the LATOR experiment; it is unique and is a natural culmination of solar system gravity experiments. 
  The influence of the shape of scalar field potential on the outcome of vacuum decay in de Sitter universe is studied. Sufficient condition for vacuum decay via bubble formation, described by Coleman - de Luccia instanton, is revisited and necessary condition is found. Both conditions require that the curvature of the potential is greater than 4 $\times$ (Hubble constant)$^2$, but while the sufficient condition states that this inequality is to be valid at the top of the barrier, the necessary condition requires that it holds at least somewhere throughout the barrier. The conditions leave a 'grey zone' in parameter space, which however seems to be forbidden for quartic potential as well as for quadratic potential with a narrow peak proposed by Linde in the context of open inflation. 
  In this paper a quantum N = 4 super Yang-Mills theory perturbed by dilaton-coupled scalars and spinor fields is considered. The induced effective action for such a theory is calculated on a dilaton-gravitational background using the conformal anomaly found via the AdS/CFT correspondence. Considering such an effective action (using the large N method) as a quantum correction to the classical gravity action with cosmological constant, we study the effect from the dilaton on the scale factor (this corresponds to an inflationary universe without dilaton). It is shown that, depending on the initial conditions for the dilaton, the dilaton may slow down or accelerate the inflation process. At late times the dilaton is decaying exponentially. Different possible cases corresponding to a dilatonic dS Universe are analyzed with respect to the equations of motion. 
  We investigate the conformal invariance of massless Duffin-Kemmer-Petiau theory coupled to riemannian space-times. We show that, as usual, in the minimal coupling procedure only the spin 1 sector of the theory -which corresponds to the electromagnetic field- is conformally invariant. We show also that the conformal invariance of the spin 0 sector can be naturally achieved by introducing a compensating term in the lagrangian. Such a procedure -besides not modifying the spin 1 sector- leads to the well-known conformal coupling between the scalar curvature and the massless Klein-Gordon-Fock field. Going beyond the riemannian spacetimes, we briefly discuss the effects of a nonvanishing torsion in the scalar case. 
  We use the Gu-Huang model for a special case when the universe is seven dimensional. In the core of extra dimension we place a modified Gidding-Strominger wormhole. This wormhole is separated by a thin wall from the extra dimensional space. The wormhole content is assumed to satisfy the adiabatic gas law. The wormhole pressure penetrates into the extra dimension. We then solve the Einstein equation assuming that the real universe and the extra dimension contains only inextendable fluid with negligible local pressure. We show the physical universe expands because the time dependent Hubble parameter is positive. Under certain condition the deacceleration parameter, $q_0$, is also positive. But the most significant outcome of our investigation is the fact that $q_0$ fluctuates. If we can detect by observation that the acceleration fluctuates, our model will be an alternative explanation without the help of dark energy. We could however link dark energy with work done by wormhole pressure. 
  Superthin and superlong solutions in 5D Kaluza-Klein gravity are considered. It is shown that they can be cosidered as a hybrid between Einstein's and string paradigmes. 
  We consider a system of interacting spinor and scalar fields in a gravitational field given by a Bianchi type-I cosmological model filled with perfect fluid. The interacting term in the Lagrangian is chosen in the form of derivative coupling, i.e., ${\cal L}_{\rm int} = \frac{\lambda}{2} \vf_{,\alpha}\vf^{,\alpha} F$, with $F$ being a function of the invariants $I$ an $J$ constructed from bilinear spinor forms $S$ and $P$. We consider the cases when $F$ is the power or trigonometric functions of its arguments. Self-consistent solutions to the spinor, scalar and BI gravitational field equations are obtained. The problems of initial singularity and asymptotically isotropization process of the initially anisotropic space-time are studied. It is also shown that the introduction of the Cosmological constant ($\Lambda$-term) in the Lagrangian generates oscillations of the BI model, which is not the case in absence of $\Lambda$ term. Unlike the case when spinor field nonlinearity is induced by self-action, in the case in question, wehere nonlinearity is induced by the scalar field, there exist regular solutions even without broken dominant energy condition. 
  We compute analytically the tidal field and polarizations of an exact gravitational wave generated by a cylindrical beam of null matter of finite width and length in quadratic curvature gravity. We propose that this wave can represent the gravitational wave that keep up with the high energy photons produced in a gamma ray burst (GRB) source. 
  Corrections to Newton's gravitational law inspired by extra dimensional physics and by the exchange of light and massless elementary particles between the atoms of two macrobodies are considered. These corrections can be described by the potentials of Yukawa-type and by the power-type potentials with different powers. The strongest up to date constraints on the corrections to Newton's gravitational law are reviewed following from the E\"{o}tvos- and Cavendish-type experiments and from the measurements of the Casimir and van der Waals force. We show that the recent measurements of the Casimir force gave the possibility to strengthen the previously known constraints on the constants of hypothetical interactions up to several thousand times in a wide interaction range. Further strengthening is expected in near future that makes Casimir force measurements a prospective test for the predictions of fundamental physical theories. 
  The idea of the quantum state of the Universe described by some density matrix, i.e mixture of at least two vacua, the trivial symmetric and the nontrivial one with spontaneously broken symmetry is discussed. Nonzero cosmological constant necessarily arises for such a state and has the observable value if one takes the axion mass for the vacuum expectation value. The Higgs model, Nambu's model and discrete symmetry breaking are considered. Human observers can observe only the world on the nonsymmetric vacuum, the world on the other vacuum is some dark matter. Gravity is due to action of two worlds. Tachyons nonobservable for visible matter can be present in the dark matter, leading to some effects of nonlocality in the space of the Universe. 
  This paper discusses new Fundamental physics experiment that will test relativistic gravity at the accuracy better than the effects of the second order in the gravitational field strength, ~G^2. The Laser Astrometric Test Of Relativity (LATOR) mission uses laser interferometry between two micro-spacecraft whose lines of sight pass close by the Sun to accurately measure deflection of light in the solar gravity. The key element of the experimental design is a redundant geometry optical truss provided by a long-baseline (100 m) multi-channel stellar optical interferometer placed on the International Space Station (ISS). The spatial interferometer is used for measuring the angles between the two spacecraft and for orbit determination purposes. LATOR will not only improve the value of the parameterized post-Newtonian (PPN) $\gamma$ to unprecedented levels of accuracy of 1 part in 10e8, it will also reach ability to measure effects of the next post-Newtonian order (1/c^4) of light deflection resulting from gravity's intrinsic non-linearity. The solar quadrupole moment parameter, J2, will be measured with high precision, as well as a variety of other relativistic effects including Lense-Thirring precession. LATOR will lead to very robust advances in the tests of Fundamental physics: this mission could discover a violation or extension of general relativity, or reveal the presence of an additional long range interaction in the physical law. There are no analogs to the LATOR experiment; it is unique and is a natural culmination of solar system gravity experiments. 
  The general solution of the gravitational field equations for a full causal bulk viscous stiff cosmological fluid, with bulk viscosity coefficient proportional to the energy density to the power 1/4, is obtained in the flat Friedmann-Robertson-Walker geometry. The solution describes a non-inflationary Universe, which starts its evolution from a singular state. The time variation of the scale factor, deceleration parameter, viscous pressure, viscous pressure-thermodynamic pressure ratio, comoving entropy and Ricci and Kretschmann invariants is considered in detail. 
  One can construct families of static solutions that can be viewed as interpolating between nonsingular spacetimes and those containing black holes. Although everywhere nonsingular, these solutions come arbitrarily close to having a horizon. To an observer in the exterior region, it becomes increasingly difficulty to distinguish these from a true black hole as the critical limiting solution is approached. In this paper we use the Majumdar-Papapetrou formalism to construct such quasi-black hole solutions from extremal charged dust. We study the gravitational properties of these solutions, comparing them with the the quasi-black hole solutions based on magnetic monopoles. As in the latter case, we find that solutions can be constructed with or without hair. 
  Dimensional regularization is used to derive the equations of motion of two point masses in harmonic coordinates. At the third post-Newtonian (3PN) approximation, it is found that the dimensionally regularized equations of motion contain a pole part [proportional to 1/(d-3)] which diverges as the space dimension d tends to 3. It is proven that the pole part can be renormalized away by introducing suitable shifts of the two world-lines representing the point masses, and that the same shifts renormalize away the pole part of the "bulk" metric tensor g_munu(x). The ensuing, finite renormalized equations of motion are then found to belong to the general parametric equations of motion derived by an extended Hadamard regularization method, and to uniquely determine the heretofore unknown 3PN parameter lambda to be: lambda = - 1987/3080. This value is fully consistent with the recent determination of the equivalent 3PN static ambiguity parameter, omega_s = 0, by a dimensional-regularization derivation of the Hamiltonian in Arnowitt-Deser-Misner coordinates. Our work provides a new, powerful check of the consistency of the dimensional regularization method within the context of the classical gravitational interaction of point particles. 
  We construct a model unifying general relativity and quantum mechanics in a broader structure of noncommutative geometry. The geometry in question is that of a transformation groupoid given by the action of a finite group G on a space E. We define the algebra of smooth complex valued functions on the groupoid, with convolution as multiplication, in terms of which the groupoid geometry is developed. Owing to the fact that the group G is finite the model can be computed in full details. We show that by suitable averaging of noncommutative geometric quantities one recovers the standard space-time geometry. The quantum sector of the model is explored in terms of the regular representation of the groupoid algebra, and its correspondence with the standard quantum mechanics is established. 
  We review the implications of modern higher-dimensional theories of gravity for astrophysics and cosmology. In particular, we discuss the latest developments of space-time-matter theory in connection with dark matter, particle dynamics and the cosmological constant, as well as related aspects of quantum theory. There are also more immediate tests of extra dimensions, notably involving perturbations of the cosmic 3K microwave background and the precession of a supercooled gyroscope in Earth orbit. We also outline some general features of embeddings, and include pictures of the big bang as viewed from a higher dimension. 
  Contrary to what is often stated, a fundamental spacetime discreteness need not contradict Lorentz invariance. A causal set's discreteness is in fact locally Lorentz invariant, and we recall the reasons why. For illustration, we introduce a phenomenological model of massive particles propagating in a Minkowski spacetime which arises from an underlying causal set. The particles undergo a Lorentz invariant diffusion in phase space, and we speculate on whether this could have any bearing on the origin of high energy cosmic rays. 
  We discuss the entropy change due to fragmentation for black hole solutions in various dimensions. We find three different types of behavior. The entropy may decrease, increase or have a mixed behavior, characterized by the presence of a threshold mass. For two-dimensional (2D) black holes we give a complete characterization of the entropy behavior under fragmentation, in the form of sufficient conditions imposed on the function J, which defines the 2D gravitational model. We compare the behavior of the gravitational solutions with that of free field theories in d dimensions. This excludes the possibility of finding a gravity/field theory realization of the holographic principle for a broad class of solutions, including asymptotically flat black holes. We find that the most natural candidates for holographic duals of the black hole solutions with mixed behavior are field theories with a mass gap. We also discuss the possibility of formulating entropy bounds that make reference only to the energy of a system. 
  The conformal formulation provides a method for constructing and parametrizing solutions of the Einstein constraint equations by mapping freely chosen sets of conformal data to solutions, provided a certain set of coupled, elliptic determined PDEs (whose expression depends on the chosen conformal data) admit a unique solution. For constant mean curvature (CMC) data, it is known in almost all cases which sets of conformal data allow these PDEs to have solutions, and which do not. For non CMC data, much less is known. Here we exhibit the first class of non CMC data for which we can prove that no solutions exist. 
  After a review of the 1+3 point of view on non-inertial observers and of the problems of rotating reference frames, we underline that was is lacking in their treatment is a good global notion of simultaneity due to the restricted validity of the existing 4-coordinates associated to an accelerated observer (like the Fermi normal ones). We show that the relativistic Hamiltonian 3+1 point of view, based on a 3+1 splitting of Minkowski space-time with a foliation whose space-like leaves are both simultaneity and Cauchy surfaces, allows to find a solution to such problems, if we take into account M$\o$ller's definition of allowed 4-coordinate transformations extended to radar 4-coordinates. Rigidly rotating relativistic reference frames are shown not to exist. We give explicit foliations, with simultaneity surfaces (also space-like hyper-planes) non orthogonal to the non-inertial observer world-line, which correspond to a good notion of simultaneity for suitable (mutually balancing) translational and rotational accelerations. This allows to evaluate the simultaneity-dependent one-way speed of light and to give the 3+1 description of both the rotating disk and the Sagnac effect. It is also shown how a GPS system of spacecrafts may establish a grid of admissible radar 4-coordinates and how the ACES mission can employ a variant of this method adapted to Earth's rotation in the evaluation of the one-way time transfer to detect the deviation from Einstein convention on the synchronization of inertial clocks implied by such a notion of simultaneity. We show that in parametrized Minkowski theories all the admissible notions of simultaneity are gauge equivalent ({\it conventionality of simultaneity}) and, as an example, we describe Maxwell theory in non-inertial systems with any admissible notion of simultaneity. 
  We provide a mechanism by which, from a background independent model with no quantum mechanics, quantum theory arises in the same limit in which spatial properties appear. Starting with an arbitrary abstract graph as the microscopic model of spacetime, our ansatz is that the microscopic dynamics can be chosen so that 1) the model has a low low energy limit which reproduces the non-relativistic classical dynamics of a system of N particles in flat spacetime, 2) there is a minimum length, and 3) some of the particles are in a thermal bath or otherwise evolve stochastically. We then construct simple functions of the degrees of freedom of the theory and show that their probability distributions evolve according to the Schroedinger equation. The non-local hidden variables required to satisfy the conditions of Bell's theorem are the links in the fundamental graph that connect nodes adjacent in the graph but distant in the approximate metric of the low energy limit. In the presence of these links, distant stochastic fluctuations are transferred into universal quantum fluctuations. 
  The recent observations of type Ia supernovae strongly support that the universe is accelerating now and decelerated in the recent past. This may be the evidence of the breakdown of the standard Friemann equation. We consider a general modified Friedmann equation. Three different models are analyzed in detail. The current supernovae data and the Wilkinson microwave anisotropy probe data are used to constrain these models. A detailed analysis of the transition from the deceleration phase to the acceleration phase is also performed. 
  We show that the quantum locking scheme recently proposed by Courty {\it et al.} [Phys. Rev. Lett. {\bf 90}, 083601 (2003)] for the reduction of back action noise is able to significantly improve the sensitivity of the next generation of gravitational wave interferometers. 
  Experimental and theoretical studies of linear and nonlinear optics of surface plasmon toy wormholes and black holes have been performed. These models are based on dielectric microdroplets on the metal surfaces and on nanoholes drilled in thin metal films. Toy surface plasmon black holes and wormholes are shown to exhibit strongly enhanced nonlinear optical behavior in the frequency range near the surface plasmon resonance of a metal-liquid interface. Various possibilities to emulate such nontrivial gravitation theory effects as Hawking radiation and Cauchy horizons are discussed. 
  The classical problem of self-energy divergence was studied in the framework of Lagrangian formulation of Relativistic Mechanics. The conclusion was made that a revision of mass-energy concept is needed for the development of singularity-free gravitational and electromagnetic field theory. Perspectives of the development of unified field theory are discussed. 
  It is known that the imaginary parts of the quasi normal mode (QNM) frequencies for the Schwarzschild black hole are evenly spaced with a spacing that depends only on the surface gravity. On the other hand, for massless minimally coupled scalar fields, there exist no QNMs in the pure DeSitter spacetime. It is not clear what the structure of the QNMs would be for the Schwarzschild-DeSitter (SDS) spacetime, which is characterized by two different surface gravities. We provide a simple derivation of the imaginary parts of the QNM frequencies for the SDS spacetime by calculating the scattering amplitude in the first Born approximation and determining its poles. We find that, for the usual set of boundary conditions in which the incident wave is scattered off the black hole horizon, the imaginary parts of the QNM frequencies have a equally spaced structure with the level spacing depending on the surface gravity of the black hole. Several conceptual issues related to the QNM are discussed in the light of this result and comparison with previous work is presented. 
  It is well known that radiative corrections evaluated in nontrivial backgrounds lead to effective dispersion relations which are not Lorentz invariant. Since gravitational interactions increase with energy, gravity-induced radiative corrections could be relevant for the trans-Planckian problem. As a first step to explore this possibility, we compute the one-loop radiative corrections to the self-energy of a scalar particle propagating in a thermal bath of gravitons in Minkowski spacetime. We obtain terms which originate from the thermal bath and which indeed break the Lorentz invariance that possessed the propagator in the vacuum. Rather unexpectedly, however, the terms which break Lorentz invariance vanish in the high three-momentum limit. We also found that the imaginary part, which gives the rate of approach to thermal equilibrium, vanishes at one loop. 
  Spin foam models are a new approach to a formulation of quantum gravity which is fully background independent, non-perturbative, and covariant, in the spirit of path integral formulations of quantum field theory. In this thesis we describe in details the general ideas and formalism of spin foam models, and review many of the results obtained recently in this approach. We concentrate, for the case of 3-dimensional quantum gravity, on the Turaev-Viro model, and, in the 4-dimensional case, which is our main concern, on the Barrett-Crane model. In particular, for the Barrett-Crane model: we describe the general ideas behind its construction, and review what has been achieved up to date, discuss in details its links with the classical formulations of gravity as constrained topological field theory; we show a derivation of the model from a lattice gauge theory perspective, in the general case of manifold with boundaries, presenting also a few possible variations of the procedure used, discussing the problems they present; we analyse in details the classical and quantum geometry; we also describe how, from the same perspective, a spin foam model that couples quantum gravity to any gauge theory may be constructed; finally, we describe a general scheme for causal spin foam models, how the Barrett-Crane model can be modified to implement causality and to fit in such a scheme, and the resulting link with the quantum causal set approach to quantum gravity. 
  We argue that the recently introduced "statefinder parameters" (Sahni et al., JETP Lett. 77, 201 (2003)), that include the third derivative of the cosmic scale factor, are useful tools to characterize interacting quitessence models. We specify the statefinder parameters for two classes of models that solve, or at least alleviate, the coincidence problem. 
  Recent developments in string theory suggest that there might exist extra spatial dimensions, which are not small nor compact. The framework of most brane cosmological models is that in which the matter fields are confined on a brane-world embedded in five dimensions (the bulk). Motivated by this we reexamine the classification of the second order symmetric tensors in 5--D, and prove two theorems which collect together some basic results on the algebraic structure of these tensors in 5-dimensional space-times. We also briefly indicate how one can obtain, by induction, the classification of symmetric two-tensors (and the corresponding canonical forms) on n-dimensional spaces from the classification on 4-dimensional spaces. This is important in the context of 11--D supergravity and 10--D superstrings. 
  Planning is underway for several space-borne gravitational wave observatories to be built in the next ten to twenty years. Realistic and efficient forward modeling will play a key role in the design and operation of these observatories. Space-borne interferometric gravitational wave detectors operate very differently from their ground based counterparts. Complex orbital motion, virtual interferometry, and finite size effects complicate the description of space-based systems, while nonlinear control systems complicate the description of ground based systems. Here we explore the forward modeling of space-based gravitational wave detectors and introduce an adiabatic approximation to the detector response that significantly extends the range of the standard low frequency approximation. The adiabatic approximation will aid in the development of data analysis techniques, and improve the modeling of astrophysical parameter extraction. 
  We consider light waves propagating clockwise and other light waves propagating counterclockwise around a closed path in a plane (theoretically with the help of stationary mirrors). The time difference between the two light propagating path orientations constitutes the Sagnac effect. The general relativistic expression for the Sagnac effect is discussed. It is shown that a gravitational wave incident to the light beams at an arbitrary angle will not induce a Sagnac effect so long as the wave length of the weak gravitational wave is long on the length scale of the closed light beam paths. The gravitational wave induced Sagnac effect is thereby null. 
  As was recently pointed out by Cadoni, a certain class of two-dimensional gravitational theories will exhibit (black hole) thermodynamic behavior that is reminiscent of a free field theory. In the current letter, a direct correspondence is established between these two-dimensional models and the strongly curved regime of (arbitrary-dimensional) anti-de Sitter gravity. On this basis, we go on to speculatively argue that two-dimensional gravity may ultimatley be utilized for identifying and perhaps even understanding holographic dualities. 
  According to electrodynamical equations in curved spacetime we consider the coupling of a linearized weak gravitational wave (GW) to a Gaussian beam passing through a static magnetic field. It is found that unlike the properties of the "left-circular" and "right-circular" waves of the tangential perturbative photon fluxes in the cylindrical polar coordinates, the resultant effect of the tangential and radial perturbations can produce the unique nonvanishing photon flux propagating along the direction of the electric field of the Gaussian beam. This result might provide a larger detecting space for the high-frequency GWs in GHz band. Moreover, we also discuss the relevant noise issues. 
  It is generally argued that the combined effect of Heisenberg principle and general relativity leads to a minimum time uncertainty. Most of the analyses supporting this conclusion are based on a perturbative approach to quantization. We consider a simple family of gravitational models, including the Einstein-Rosen waves, in which the (non-linearized) inclusion of gravity changes the normalization of time translations by a monotonic energy-dependent factor. In these circumstances, it is shown that a maximum time resolution emerges non-perturbatively only if the total energy is bounded. Perturbatively, however, there always exists a minimum uncertainty in the physical time. 
  In the Randall-Sundrum two-brane model (RS1), a Kerr black hole on the brane can be naturally identified with a section of rotating black string. To estimate Kaluza-Klein (KK) corrections on gravitational waves emitted by perturbed rotating black strings, we give the effective Teukolsky equation on the brane which is separable equation and hence numerically manageable. In this process, we derive the master equation for the electric part of the Weyl tensor $E_{\mu\nu}$ which would be also useful to discuss the transition from black strings to localized black holes triggered by Gregory-Laflamme instability. 
  We find new Melvin-like solutions in Einstein-Maxwell-dilaton gravity with a Liouville-type dilaton potential. The properties of the corresponding solution in Freedman-Schwarz gauged supergravity model are extensively studied. We show that this configuration is regular and geodesically complete but do not preserve any supersymmetry. An exact solution describing travelling waves in this Melvin-type background is also presented. 
  It is shown that introducing the quantum effects using deBroglie--Bohm theory in the canonical formulation of gravity would change the constraints algebra. The new algebra is derived and shown that it is the clear projection of general coordinate transformations to the spatial and temporal diffeomorphisms. The quantum Einstein's equations are derived and it is shown that they are manifestly covariant under the above diffeomorphisms, as it would be. 
  A method of solving perfect fluid Einstein equations with two commuting spacelike Killing vectors is presented. Given a spacelike 2-dimensional surface in the 3-dimensional nonphysical Minkowski space the field equations reduce to a single nonlinear differential equation. An example is discussed. 
  A method to construct exact general relativistic thick disks that is a simple generalization of the ``displace, cut and reflect'' method commonly used in Newtonian, as well as, in Einstein theory of gravitation is presented. This generalization consists in the addition of a new step in the above mentioned method. The new method can be pictured as a ``displace, cut, {\it fill} and reflect'' method. In the Newtonian case, the method is illustrated in some detail with the Kuzmin-Toomre disk. We obtain a thick disk with acceptable physical properties. In the relativistic case two solutions of the Weyl equations, the Weyl gamma metric (also known as Zipoy-Voorhees metric) and the Chazy-Curzon metric are used to construct thick disks. Also the Schwarzschild metric in isotropic coordinates is employed to construct another family of thick disks. In all the considered cases we have non trivial ranges of the involved parameter that yield thick disks in which all the energy conditions are satisfied. 
  We construct perfect fluid metrics corresponding to spacelike surfaces invariant under a 1-dimensional group of isometries in 3-dimensional Minkowski space. Under additional assumptions we obtain new cosmological solutions of Bianchi type II, VI_0 and VII_0. The solutions depend on an arbitrary function of time, which can be specified in order to satisfy an equation of state. 
  The bounded orbital motion of a massive spinless test particle in the background of a Kerr Brans-Dicke geometry is analysed in terms of worldlines that are auto-parallels of different metric compatible spacetime connections. In one case the connection is that of Levi-Civita with zero-torsion. In the second case the connection has torsion determined by the gradient of the Brans-Dicke background scalar field. The calculations permit in principle to discriminate between these possibilities. 
  Using the iterative Scheme we prove the local existence and uniqueness of solutions of the spherically symmetric Einstein-Vlasov-Maxwell system with small initial data. We prove a continuation criterion to global in-time solutions. 
  This article is meant as a summary and introduction to the ideas of effective field theory as applied to gravitational systems.  Contents:   1. Introduction   2. Effective Field Theories   3. Low-Energy Quantum Gravity   4. Explicit Quantum Calculations   5. Conclusions 
  After some remarks about the history and the mystery of the vacuum energy I shall review the current evidence for a cosmologically significant nearly homogeneous exotic energy density with negative pressure (`Dark Energy'). Special emphasis will be put on the recent polarization measurements by WMAP and their implications. I shall conclude by addressing the question: Do the current observations really imply the existence of a dominant dark energy component? 
  It is shown that 5D Kaluza-Klein theory stabilized by an external bulk scalar field may solve the discrepant laboratory G measurements. This is achieved by an effective coupling between gravitation and the geomagnetic field. Experimental considerations are also addressed. 
  The gravitational collapse of a magnetised medium is investigated by studying qualitatively the convergence of a timelike family of non-geodesic worldlines in the presence of a magnetic field. Focusing on the field's tension we illustrate how the winding of the magnetic forcelines due to the fluid's rotation assists the collapse, while shear-like distortions in the distribution of the field's gradients resist contraction. We also show that the relativistic coupling between magnetism and geometry, together with the tension properties of the field, lead to a magneto-curvature stress that opposes the collapse. This tension stress grows stronger with increasing curvature distortion, which means that it could potentially dominate over the gravitational pull of the matter. If this happens, a converging family of non-geodesic lines can be prevented from focusing without violating the standard energy conditions. 
  There is an apparent discrepancy in the literature with regard to the quasinormal mode frequencies of Schwarzschild-de Sitter black holes in the degenerate-horizon limit. On the one hand, a Poschl-Teller-inspired method predicts that the real part of the frequencies will depend strongly on the orbital angular momentum of the perturbation field whereas, on the other hand, the degenerate limit of a monodromy-based calculation suggests there should be no such dependence (at least, for the highly damped modes). In the current paper, we provide a possible resolution by critically re-assessing the limiting procedure used in the monodromy analysis. 
  Measurements of the mass or angular momentum of a black hole are onerous, particularly if they have to be frequently repeated, as when one is required to transform a black hole to prescribed parameters. Irradiating a black hole of the Kerr-Newman family with scalar or electromagnetic waves provides a way to drive it to prescribed values of its mass, charge and angular momentum without the need to repeatedly measure mass or angular momentum throughout the process. I describe the mechanism, which is based on Zel'dovich-Misner superradiance and its analog for charged black holes. It represents a possible step in the development of preparation procedures for quantum black holes. 
  We investigate the ultrarelativistic boost of a Schwarzschild black hole immersed in an external electromagnetic field, described by an exact solution of the Einstein-Maxwell equations found by Ernst (the ``Schwarzschild-Melvin'' metric). Following the classical method of Aichelburg and Sexl, the gravitational field generated by a black hole moving ``with the speed of light'' and the transformed electromagnetic field are determined. The corresponding exact solution describes an impulsive gravitational wave propagating in the static, cylindrically symmetric, electrovac universe of Melvin, and for a vanishing electromagnetic field it reduces to the well known Aichelburg-Sexl pp-wave. In the boosting process, the original Petrov type I of the Schwarzschild-Melvin solution simplifies to the type II on the impulse, and to the type D elsewhere. The geometry of the wave front is studied, in particular its non-constant Gauss curvature. In addition, a more general class of impulsive waves in the Melvin universe is constructed by means of a six-dimensional embedding formalism adapted to the background. A coordinate system is also presented in which all the impulsive metrics take a continuous form. Finally, it is shown that these solutions are a limiting case of a family of exact gravitational waves with an arbitrary profile. This family is identified with a solution previously found by Garfinkle and Melvin. We thus complement their analysis, in particular demonstrating that such spacetimes are of type II and belong to the Kundt class. 
  We analyze the spacetimes admitting a direction for which the relative electric and magnetic Weyl fields are aligned. We give an invariant characterization of these metrics and study the properties of its Debever null vectors. The directions 'observing' aligned electric and magnetic Weyl fields are obtained for every Petrov type. The results on the no existence of purely magnetic solutions are extended to the wider class having homothetic electric and magnetic Weyl fields. 
  This article has been replaced by gr-qc/0412011 
  There is a number of completely integrable gravity theories in two dimensions. We study the metric-affine approach on a 2-dimensional spacetime and display a new integrable model. Its properties are described and compared with the known results of Poincare gauge gravity. 
  Working within the HPO (History Projection Operator) Consistent Histories formalism, we follow the work of Savvidou on (scalar) field theory and that of Savvidou and Anastopolous on (first-class) constrained systems to write a histories theory (both classical and quantum) of Electromagnetism. We focus particularly on the foliation-dependence of the histories phase space/Hilbert space and the action thereon of the two Poincare groups that arise in histories field theory. We quantise in the spirit of the Dirac scheme for constrained systems. 
  In this paper the static, spherically symmetric and electrically charged black hole solutions in Einstein-Born-Infeld gravity with massive dilaton are investigated numerically.   The Continuous Analog of Newton Method (CANM) is used to solve the corresponding nonlinear multipoint boundary value problems (BVPs). The linearized BVPs are solved numerically by means of collocation scheme of fourth order.   A special class of solutions are the extremal ones. We show that the extremal horizons within the framework of the model satisfy some nonlinear system of algebraic equations. Depending on the charge $q$ and dilaton mass $\gamma$, the black holes can have no more than three horizons. This allows us to construct some Hermite polynomial of third order. Its real roots describe the number, the type and other characteristics of the horizons. 
  Spin foam models for gravity or BF theory can be constructed by path integral formulation of the classical discrete models formulated on simplicial manifolds. Using this, we discuss the rigorous construction of Lorentzian spin foam models for gravity and BF theory based on the Gelfand-Naimark theory of the representations of SL(2,C). First we construct the simplex amplitude for the BF SL(2,C) model. Next we discuss the implementation of the Barrett-Crane constraints on this model to derive the spin foam model for gravity. The non-trivial constraints are the cross simplicity constraints which state that the sum of the bivectors associated to any two triangles of a quantum tetrahedron is simple. We do not complete the construction of the model, but ultimately we derive an equation corresponding to the cross simplicity constraints that the Lorentzian spin foam model of gravity has to satisfy. In the appendix we give a simple derivation of the Clebsch-Gordan coefficients for SL(2,C). 
  We present three-dimensional simulations of Einstein equations implementing a symmetric hyperbolic system of equations with dynamical lapse. The numerical implementation makes use of techniques that guarantee linear numerical stability for the associated initial-boundary value problem. The code is first tested with a gauge wave solution, where rather larger amplitudes and for significantly longer times are obtained with respect to other state of the art implementations. Additionally, by minimizing a suitably defined energy for the constraints in terms of free constraint-functions in the formulation one can dynamically single out preferred values of these functions for the problem at hand. We apply the technique to fully three-dimensional simulations of a stationary black hole spacetime with excision of the singularity, considerably extending the lifetime of the simulations. 
  Gravitation lensing calculations, which are generally done for light ray, are extended to that for a massive particle. Many interesting results were observed. We discuss the scattering cross section along-with many consequential quantities here. In particular, the case of Schrwarzschild metric was taken as illustration, though the analysis applies to a wide range of cases, such as extended black holes. 
  We investigate timelike junctions (with surface layer) between spherically symmetric solutions of the Einstein-field equation. In contrast to previous investigations this is done in a coordinate system in which the junction surface motion is absorbed in the metric, while all coordinates are continuous at the junction surface.   The evolution equations for all relevant quantities are derived. We discuss the no-surface layer case (boundary surface) and study the behaviour for small surface energies. It is shown that one should expect cases in which the speed of light is reached within a finite proper time.   We carefully discuss necessary and sufficient conditions for a possible matching of spherically symmetric sections.   For timelike junctions between spherically symmetric space-time sections we show explicitly that the time component of the Lanczos equation always reduces to an identity (independently of the surface equation of state).   The results are applied to the matching of FLRW models. We discuss `vacuum bubbles' and closed-open junctions in detail. As illustrations several numerical integration results are presented, some of them indicate that the junction surface can reach the speed of light within a finite time. 
  A cellular automata approach using a Directed Cyclic Graph is used to model interrelationships of fluctuating time, state and space. This model predicts phenomena including a constant and maximum speed at which any moving entity can travel, time dilation effects in accordance with special relativity, calculation for the Doppler effect, propagation in three spatial dimensions, an explanation for the non-local feature of collapse and a speculation on the origin of gravitation. The approach has proven amenable to computer modelling. 
  An analysis is made of a moving disturbance using a directed cyclic graph.   A statistical approach is used to calculate the alternative positions in space and state of the disturbance with a defined observed time. The probability for a freely moving entity interacting in a particular spatial position is calculated and a formulation is derived for the minimum locus of uncertainty in position and momentum. This is found to accord with calculations for quantum mechanics. The model has proven amenable to computer modelling; a copy of the "SimulTime" program is available on request. 
  A field theory is proposed where the regular fermionic matter and the dark fermionic matter are different states of the same "primordial" fermion fields. In regime of the fermion densities typical for normal particle physics, the primordial fermions split into three families identified with regular fermions. When fermion energy density becomes comparable with dark energy density, the theory allows new type of states. The possibility of such Cosmo-Low Energy Physics (CLEP) states is demonstrated by means of solutions of the field theory equations describing FRW universe filled by homogeneous scalar field and uniformly distributed nonrelativistic neutrinos. Neutrinos in CLEP state are drawn into cosmological expansion by means of dynamically changing their own parameters. One of the features of the fermions in CLEP state is that in the late time universe their masses increase as a^{3/2}. The energy density of the cold dark matter consisting of neutrinos in CLEP state scales as a sort of dark energy; this cold dark matter possesses negative pressure and for the late time universe its equation of state approaches that of the cosmological constant. The total energy density of such universe is less than it would be in the universe free of fermionic matter at all. The (quintessence) scalar field is coupled to dark matter but its coupling to regular fermionic matter appears to be extremely strongly suppressed. The key role in obtaining these results belongs to a fundamental constraint (which is consequence of the action principle) that plays the role of a new law of nature. 
  The varying speed of light theories have been recently proposed to solve the standard model problems and anomalies in the ultra high energy cosmic rays. These theories try to formulate a new relativity with no assumptions about the constancy of the light speed. In this regard, we study two theories and want to show that these theories are not the new theories of relativity, but only re-descriptions of Einstein's special relativity. 
  Teleparallel gravity can be seen as a gauge theory for the translation group. As such, its fundamental field is neither the tetrad nor the metric, but a gauge potential assuming values in the Lie algebra of the translation group. This gauge character makes of teleparallel gravity, despite its equivalence to general relativity, a rather peculiar theory. A first important point is that it does not rely on the universality of free fall, and consequently does not require the equivalence principle to describe the gravitational interaction. Another peculiarity is its similarity with Maxwell's theory, which allows an Abelian nonintegrable phase factor approach, and consequently a global formulation for gravitation. Application of these concepts to the motion of spinless particles, as well as to the COW and gravitational Aharonov-Bohm effects are presented and discussed. 
  In this paper we analyze the cosmological dynamics of phantom field in a variety of potentials unbounded from above. We demonstrate that the nature of future evolution generically depends upon the steepness of the phantom potential and discuss the fate of Universe accordingly. 
  We calculate the unregularized monopole and dipole contributions to the self-force acting on a particle of small mass in a circular orbit around a Schwarzschild black hole. From a self-force point of view, these non-radiating modes are as important as the radiating modes with l greater than 2. In fact, we demonstrate how the dipole self-force contributes to the dynamics even at the Newtonian level. The self-acceleration of a particle is an inherently gauge-dependent concept, but the Lorenz gauge is often preferred because of its hyperbolic wave operator. Our results are in the Lorenz gauge and are also obtained in closed form, except for the even-parity dipole case where we formulate and implement a numerical approach. 
  Sagnac interferometry has been employed in the context of gravity as a proposal for the detection of the so called gravitomagnetic effect. In the present work we explore the possibilities that this experimental device could open up in the realm of non--Newtonian gravity. It will be shown that this experimental approach allows us to explore an interval of values of the range of the new force that up to now remains unexplored, namely, $\lambda\geq 10^{14}$ m. 
  We discuss the stability of (charged) static black holes in higher-dimensional spacetimes with and without cosmological constant by using gauge-invariant master equations of the Schroedinger equation type for black hole perturbations derived by the authors recently. In particular, we show that the stability of higher-dimensional Schwarzschild black holes can be proved with the help of a technique called S-deformation of the master equations. We also point out that higher-dimensional static black holes might be unstable only against scalar-type perturbations in the neutral case and in the charged case with spherically symmetric or flat horizons. 
  We calculate the harmonic-gauge even l=1 mode of the linear metric perturbation (MP) produced by a particle in a weak-field circular orbit around a Schwarzschild black hole (BH). We focus on the Newtonian limit, i.e. the limit in which the mass M of the central BH approaches zero (while fixing the orbital radius and the small-object mass), and obtain explicit expressions for the MP in this limit. We find that the MP are anomalous in this limit, namely, they do not approach their standard, Coulomb-like, flat-space values. Instead, the MP grows on approaching the BH, and this growth becomes worse as M decreases. This anomalous behavior leads to some pathologies which we briefly discuss. We also derive here the next-order correction (in the orbital frequency $\Omega $) to the MP. 
  We investigate refined algebraic quantisation with group averaging in a finite-dimensional constrained Hamiltonian system that provides a simplified model of general relativity. The classical theory has gauge group SL(2,R) and a distinguished o(p,q) observable algebra. The gauge group of the quantum theory is the double cover of SL(2,R), and its representation on the auxiliary Hilbert space is isomorphic to the (p,q) oscillator representation. When p>1, q>1 and p+q == 0 (mod 2), we obtain a physical Hilbert space with a nontrivial representation of the o(p,q) quantum observable algebra. For p=q=1, the system provides the first example known to us where group averaging converges to an indefinite sesquilinear form. 
  We study here the spherical gravitational collapse assuming initial data to be necessarily smooth, as motivated by the requirements based on physical reasonableness. A tangential pressure model is constructed and analyzed in order to understand the final fate of collapse explicitly in terms of the density and pressure parameters at the initial epoch from which the collapsedevelops. It is seen that both black holes and naked singularities are produced as collapse end states even when the initial data is smooth. We show that the outcome is decided entirely in terms of the initial data, as given by density, pressure and velocity profiles at the initial epoch, from which the collapse evolves. 
  The possible design of QND gravitational-wave detector based on speed meter principle is considered with respect to optical losses. The detailed analysis of speed meter interferometer is performed and the ultimate sensitivity that can be achieved is calculated. It is shown that unlike the position meter signal-recycling can hardly be implemented in speed meter topology to replace the arm cavities as it is done in signal-recycled detectors, such as GEO 600. It is also shown that speed meter can beat the Standard Quantum Limit (SQL) by the factor of $\sim 3$ in relatively wide frequency band, and by the factor of $\sim 10$ in narrow band. For wide band detection speed meter requires quite reasonable amount of circulating power $\sim 1$ MW. The advantage of the considered scheme is that it can be implemented with minimal changes in the current optical layout of LIGO interferometer. 
  A time-varying cosmological "constant" Lambda is consistent with Einstein's equation, provided matter and/or radiation is created or destroyed to compensate for it. Supposing an empty primordial universe endowed with a very large cosmological term, matter will emerge gradually as Lambda decays. Provided only radiation or ultrarelativistic matter is initially created, the universe starts in a nearly de Sitter phase, which evolves towards a FRW regime as expansion proceeds. If, at some cosmological time, the cosmological term begins increasing again, as presently observed, expansion will accelerate and matter and/or radiation will be transformed back into dark energy. It is shown that such accelerated expansion is a route towards a new kind of gravitational singular state, characterized by an empty, conformally transitive spacetime in which all energy is dark. 
  We calculate the probability of creation of a universe with space topology $S^{1}\times T_{g}$, where $S^{1}$ is the circle and $T_{g}$ is a compact hyperbolic surface of genus $g\geq 2$. We use the method of path integrals as applied to quantum cosmology. 
  We present a new exact perfect fluid interior solution for a particular scalar-tensor theory. The solution is regular everywhere and has a well defined boundary where the fluid pressure vanishes. The metric and the dilaton field match continuously the external solution. 
  A positive cosmological constant simplifies the asymptotics of forever expanding cosmological solutions of the Einstein equations. In this paper a general mathematical analysis on the level of formal power series is carried out for vacuum spacetimes of any dimension and perfect fluid spacetimes with linear equation of state in spacetime dimension four. For equations of state stiffer than radiation evidence for development of large gradients, analogous to spikes in Gowdy spacetimes, is found. It is shown that any vacuum solution satisfying minimal asymptotic conditions has a full asymptotic expansion given by the formal series. In four spacetime dimensions, and for spatially homogeneous spacetimes of any dimension, these minimal conditions can be derived for appropriate initial data. Using Fuchsian methods the existence of vacuum spacetimes with the given formal asymptotics depending on the maximal number of free functions is shown without symmetry assumptions. 
  We point out that the Myers-Perry metric in five dimensions is algebraically special. It has Petrov type \underline{22}, which is the Petrov type of the five-dimensional Schwarzschild metric. 
  We investigate harmonic maps in the context of isometric embeddings when the target space is Ricci-flat and has codimension one. With the help of the Campbell-Magaard theorem we show that any $n$-dimensional ($n\geqslant 3$) Lorentzian manifold can be isometrically and harmonically embedded in a (n+1)-dimensional semi-Riemannian Ricci-flat space. We then extend our analysis to the case when the target space is an Einstein space. Finally, as an example, we work out the harmonic and isometric embedding of a Friedmann-Robertson-Walker spacetime in a five-dimensional Ricci-flat space and proceed to obtain a general scheme to minimally embed any vacuum solution of general relativity in Ricci-flat spaces with codimension one. 
  It has been suggested recently that the microcanonical entropy of a system may be accurately reproduced by including a logarithmic correction to the canonical entropy. In this paper we test this claim both analytically and numerically by considering three simple thermodynamic models whose energy spectrum may be defined in terms of one quantum number only, as in a non-rotating black hole. The first two pertain to collections of noninteracting bosons, with logarithmic and power-law spectra. The last is an area ensemble for a black hole with equi-spaced area spectrum. In this case, the many-body degeneracy factor can be obtained analytically in a closed form. We also show that in this model, the leading term in the entropy is proportional to the horizon area A, and the next term is ln A with a negative coefficient. 
  According to the theory of unimodular relativity developed by Anderson and Finkelstein, the equations of general relativity with a cosmological constant are composed of two independent equations, one which determines the null-cone structure of space-time, another which determines the measure structure of space-time. The field equations that follow from the restricted variational principle of this version of general relativity only determine the null-cone structure and are globally scale-invariant and scale-free. We show that the electromagnetic field may be viewed as a compensating gauge field that guarantees local scale invariance of these field equations. In this way, Weyl's geometry is revived. However, the two principle objections to Weyl's theory do not apply to the present formulation: the Lagrangian remains first order in the curvature scalar and the non-integrability of length only applies to the null-cone structure. 
  A criterion is presented and discussed to detect when a divergence-free perfect fluid energy tensor in the space-time describes an evolution in local thermal equilibrium.   This criterion is applied to the class II Szafron-Szekeres perfect fluid space-times solutions, giving a very simple characterization of those that describe such thermal evolutions. For all of them, the significant thermodynamic variables are explicitly obtained.   Also, the specific condition is given under which the divergence-free perfect fluid energy tensors may be interpreted as an ideal gas. 
  The Stephani Universes that can be interpreted as an ideal gas evolving in local thermal equilibrium are determined, and the method to obtain the associated thermodynamic schemes is given 
  Einstein's field equations with cosmological constant are analysed for a static, spherically symmetric perfect fluid having constant density. Five new global solutions are described.   One of these solutions has the Nariai solution joined on as an exterior field. Another solution describes a decreasing pressure model with exterior Schwarzschild-de Sitter spacetime having decreasing group orbits at the boundary. Two further types generalise the Einstein static universe.   The other new solution is unphysical, it is an increasing pressure model with a geometric singularity. 
  This article deals with the gravitational lensing (GL) of gravitational waves (GW). We compute the increase in the number of detected GW events due to GL. First, we check that geometrical optics is valid for the GW frequency range on which Earth-based detectors are sensitive, and that this is also partially true for what concerns the future space-based interferometer LISA. To infer this result, both the diffraction parameter and a cut-off frequency are computed. Then, the variation in the number of GW signals is estimated in the general case, and applied to some lens models: point mass lens and singular isothermal sphere (SIS profile). An estimation of the magnification factor has also been done for the softened isothermal sphere and for the King profile. The results appear to be strongly model-dependent, but in all cases the increase in the number of detected GW signals is negligible. The use of time delays among images is also investigated. 
  We employ WKB approximation up to the third order to determine the low-lying quasinormal modes for Weyl neutrino field in R-N black holes, which are the most relevant to the evolution of the field around a black hole in the intermediate stage. It is showed that the quasinormal mode frequencies for Weyl neutrino field in R-N black holes are different from those in Schwartzchild black holes owning to the charge-induced additional gravitation, and the variations of the quasinormal mode frequencies for Weyl neutrino field are similar to those for integral spin fields in R-N black holes. 
  This paper reports on our effort in modeling realistic astrophysical neutron star binaries in general relativity. We analyze under what conditions the conformally flat quasiequilibrium (CFQE) approach can generate ``astrophysically relevant'' initial data, by developing an analysis that determines the violation of the CFQE approximation in the evolution of the binary described by the full Einstein theory. We show that the CFQE assumptions significantly violate the Einstein field equations for corotating neutron stars at orbital separations nearly double that of the innermost stable circular orbit (ISCO) separation, thus calling into question the astrophysical relevance of the ISCO determined in the CFQE approach. With the need to start numerical simulations at large orbital separation in mind, we push for stable and long term integrations of the full Einstein equations for the binary neutron star system. We demonstrate the stability of our numerical treatment and analyze the stringent requirements on resolution and size of the computational domain for an accurate simulation of the system. 
  Injection of simulated binary inspiral signals into detector hardware provides an excellent test of the inspiral detection pipeline. By recovering the physical parameters of an injected signal, we test our understanding of both instrumental calibration and the data analysis pipeline. We describe an inspiral search code and results from hardware injection tests and demonstrate that injected signals can be recovered by the data analysis pipeline. The parameters of the recovered signals match those of the injected signals. 
  Generalizing the Lie derivative of smooth tensor fields to distribution-valued tensors, we examine the Killing symmetries and the collineations of the curvature tensors of some distributional domain wall geometries. The chosen geometries are rigorously the distributional thin wall limit of self gravitating scalar field configurations representing thick domain walls and the permanence and/or the rising of symmetries in the limit process is studied. We show that, for all the thin wall spacetimes considered, the symmetries of the distributional curvature tensors turns out to be the Killing symmetries of the pullback of the metric tensor to the surface where the singular part of these tensors is supported. Remarkably enough, for the non-reflection symmetric domain wall studied, these Killing symmetries are not necessarily symmetries of the ambient spacetime on both sides of the wall. 
  Lattice universes are spatially closed space-times of spherical topology in the large, containing masses or black holes arranged in the symmetry of a regular polygon or polytope. Exact solutions for such spacetimes are found in 2+1 dimensions for Einstein gravity with a non-positive cosmological constant. By means of a mapping that preserves the essential nature of geodesics we establish analogies between the flat and the negative curvature cases. This map also allows treatment of point particles and black holes on a similar footing. 
  Space-time coordinates in DSR theories with two invariant scales based on a dispersion relation with an energy independent speed of light are introduced by the demand, that boost and rotation generators are invariant under a transformation from SR to DSR variables. This turns out to be equivalent to a recent suggestion postulating the existence of plane wave solutions in DSR theories. The momentum space representation of coordinates is derived, yielding a noncommutative space-time and the deformed algebra. 
  In this work we present a mathematical model for the mechanical response of the Brazilian Mario SCHENBERG gravitational wave (GW) detector to such waves. We found the physical parameters that are involved in this response assuming a linear elastic theory. Adopting this approach we determined the system's resonance frequencies for the case when six $i$-mode mechanical resonators are coupled to the antenna surface according to the arrangement suggested by Johnson and Merkowitz: the truncated icosahedron configuration. This configuration presents special symmetries that allow for the derivation of an analytic expression for the mode channels, which can be experimentally monitored and which are directly related to the tensorial components of the GW. Using this model we simulated how the system behaves under a gravitational sinewave quadrupolar force and found the relative amplitudes that result from this excitation. The mechanical resonators made the signal $\approx 5340$ times stronger. We found $i+1$ degenerate triplets plus $i$ non-degenerate system mode resonances within a band around ${3.17-3.24\rm{kHz}}$ that are sensitive to signals higher than ${\tilde h\sim 10^{-22}\rm{Hz}^{-1/2}}$ when we considerate the effects of thermal noise only. 
  We determine the innermost stable circular orbit (ISCO) of binary neutron stars (BNSs) by performing dynamical simulations in full general relativity. Evolving quasiequilibrium (QE) binaries that begin at different separations, we bracket the location of the ISCO by distinguishing stable circular orbits from unstable plunges. We study Gamma=2 polytropes of varying compactions in both corotational and irrotational equal-mass binaries. For corotatonal binaries we find an ISCO orbital angular frequency somewhat smaller than that determined by applying turning-point methods to QE initial data. For the irrotational binaries the initial data sequences terminate before reaching a turning point, but we find that the ISCO frequency is reached prior to the termination point. Our findings suggest that the ISCO frequency varies with compaction but does not depend strongly on the stellar spin. Since the observed gravitational wave signal undergoes a transition from a nearly periodic ``chirp'' to a burst at roughly twice the ISCO frequency, the measurement of its value by laser interferometers (e.g LIGO) will be important for determining some of the physical properites of the underlying stars 
  GR and other theories have been obtained from 3-space rather than spacetime principles. I explore consequences of this as regards the Problem of Time. 
  It is shown that in the standard vacuum 5D Kaluza-Klein gravity there is wormhole-like solutions which look like strings attached to two D-branes. The asymptotical behaviour of the corresponding metric is investigated. 
  Sources of high frequency gravitational waves are reviewed. Gravitational collapse, rotational instabilities and oscillations of the remnant compact objects are potentially important sources of gravitational waves. Significant and unique information for the various stages of the collapse, the evolution of protoneutron stars and the details of the equations of state of such objects can be drawn from careful study of the gravitational wave signal. 
  A modified version of the Ozer and Taha nonsingular cosmological model is presented on the assumption that the universe's radius is complex if it is regarded as empty, but it contains matter when the radius is real. It also predicts the values: Omega_M =rho_M /rho_C \approx 4/3, Omega_V = rho_V /rho_C \approx 2/3, and Omega_ = rho_ /rho_C << 1 in the present nonrelativistic era, where rho_M = matter density, rho_V = vacuum density, rho_= negative energy density and rho_{C} = critical density. 
  Recently Flanagan [astro-ph/0308111] has argued that the Palatini form of 1/R gravity is ruled out by experiments such as electron-electron scattering. His argument involves adding minimally coupled fermions in the Jordan frame and transforming to the Einstein frame. This produces additional terms that are ruled out experimentally.   Here I argue that this conclusion is false. It is well known that conformally related theories are mathematically equivalent but not physically equivalent. As discussed by Magnano and Sokolowski [2] one must decide, in the vacuum theory, which frame is the physical frame and add the minimally coupled Lagrangian in this frame. If this procedure is followed the resulting theory is not ruled out experimentally. The discussions in this paper also show that the equivalence between the generalized gravitational theories and scalar tensor theories discussed by Flanagan [gr-qc/0309015] is only mathematical, not physical. 
  A simple method is presented for removing the amplitude, frequency and phase modulations from the Laser Interferometer Space Antenna (LISA) data stream for sources at any sky location. When combined with an excess power trigger or the fast chirp transform, the total demodulation procedure allows the majority of LISA sources to be identified without recourse to matched filtering. 
  We construct explicitly a (12g-12)-dimensional space P of unconstrained and independent initial data for 't Hooft's polygon model of (2+1) gravity for vacuum spacetimes with compact genus-g spacelike slices, for any g >= 2. Our method relies on interpreting the boost parameters of the gluing data between flat Minkowskian patches as the lengths of certain geodesic curves of an associated smooth Riemann surface of the same genus. The appearance of an initial big-bang or a final big-crunch singularity (but never both) is verified for all configurations. Points in P correspond to spacetimes which admit a one-polygon tessellation, and we conjecture that P is already the complete physical phase space of the polygon model. Our results open the way for numerical investigations of pure (2+1) gravity. 
  The interactions of different particle species with the foamy space-time fluctuations expected in quantum gravity theories may not be universal, in which case different types of energetic particles may violate Lorentz invariance by varying amounts, violating the equivalence principle. We illustrate this possibility in two different models of space-time foam based on D-particle fluctuations in either flat Minkowski space or a stack of intersecting D-branes. Both models suggest that Lorentz invariance could be violated for energetic particles that do not carry conserved charges, such as photons, whereas charged particles such electrons would propagate in a Lorentz-inavariant way. The D-brane model further suggests that gluon propagation might violate Lorentz invariance, but not neutrinos. We argue that these conclusions hold at both the tree (lowest-genus) and loop (higher-genus) levels, and discuss their implications for the phenomenology of quantum gravity. 
  We compare the recent loop quantum cosmology approach of Bojowald and co-workers with earlier quantum cosmological schemes. Because the weak-energy condition can now be violated at short distances, and not necessarily with a high energy density, a number of possible instabilities are suggested: flat space unstable to expansion or baby universe production. Or else a Machian type principle is required to prevent such behaviour. Allowing a bounce to prevent an approaching singularity seems incompatible with other standard notions concerning the arrow of time and unitarity.  Preventing rapid oscillations in the wavefunction appears in conflict with more general scalar-tensor gravity.   Other approaches such as ``creation from nothing'' or from some quiescent state, static or time machine, are also assessed on grounds of naturalness and fine tuning. 
  Another connection of harmonic maps to gravity is presented. Using 1-soliton and anti-soliton solutions of the sine-Gordon equation, we construct a pair of harmonic maps that we express in terms of a particular dilaton field in Jackiw-Teitelboim gravity. This field satisfies a linearized sine-Gordon equation. We use it also to construct an explicit transformation that relates the corresponding solitonic metric to a two dimensional black hole metric. 
  In 1973, R. Penrose presented an argument that the total mass of a space-time which contains black holes with event horizons of total area $A$ should be at least $\sqrt{A/16\pi}$. An important special case of this physical statement translates into a very beautiful mathematical inequality in Riemannian geometry known as the Riemannian Penrose inequality. This inequality was first established by G. Huisken and T. Ilmanen in 1997 for a single black hole and then by one of the authors (H.B.) in 1999 for any number of black holes. The two approaches use two different geometric flow techniques and are described here. We further present some background material concerning the problem at hand, discuss some applications of Penrose-type inequalities, as well as the open questions remaining. 
  The various phase spaces involved in the dynamics of parametrized nonrelativistic Hamiltonian systems are displayed by using Crnkovic and Witten's covariant canonical formalism. It is also pointed out that in Dirac's canonical formalism there exists a freedom in the choice of the symplectic structure on the extended phase space and in the choice of the equations that define the constraint surface with the only restriction that these two choices combine in such a way that any pair (of these two choices) generates the same gauge transformation. The consequence of this freedom on the algebra of observables is also discussed. 
  The technique of dimensional reduction was used in a recent paper (Zanchin et al, Phys. Rev. D66, 064022,(2002)) where a three-dimensional (3D) Einstein-Maxwell-Dilaton theory was built from the usual four-dimensional (4D) Einstein-Maxwell-Hilbert action for general relativity. Starting from a class of 4D toroidal black holes in asymptotically anti-de Sitter (AdS) spacetimes several 3D black holes were obtained and studied in such a context. In the present work we choose a particular case of the 3D action which presents Maxwell field, dilaton field and an extra scalar field, besides gravity field and a negative cosmological constant, and obtain new 3D static black hole solutions whose horizons may have spherical or planar topology. We show that there is a 3D static spherically symmetric solution analogous to the 4D Reissner-Nordstr\"om-AdS black hole, and obtain other new 3D black holes with planar topology. From the static spherical solutions, new rotating 3D black holes are also obtained and analyzed in some detail. 
  Some time ago Mbelek and Lachieze-Rey proposed that the discrepancy between the results of the various measurements of Newton's constant could be explained by introducing a gravielectric coupling between the Earth's gravitational and magnetic fields mediated by two scalar fields. A critical assessment of this model is performed. By calculating the static field configuration of the relevant scalar around a nucleus in the linearised theory and then folding this result with the mass density of the nucleus its effective gravitational mass is determined. Considering test bodies of different materials one finds violations of the weak equivalence principle for torsion-balance experiments which are four orders of magnitude beyond the current experimental limit, thus rendering the model non-viable. The method presented can be applied to generic theories with gravielectric coupling and seems to rule out in general the explanation of the discrepant measurements of Newton's constant by such couplings. 
  In this paper we consider nonlinear interaction between gravitational and electromagnetic waves in a strongly magnetized plasma. More specifically, we investigate the propagation of gravitational waves with the direction of propagation perpendicular to a background magnetic field, and the coupling to compressional Alfv\'{e}n waves. The gravitational waves are considered in the high frequency limit and the plasma is modelled by a multifluid description. We make a self-consistent, weakly nonlinear analysis of the Einstein-Maxwell system and derive a wave equation for the coupled gravitational and electromagnetic wave modes. A WKB-approximation is then applied and as a result we obtain the nonlinear Schr\"{o}dinger equation for the slowly varying wave amplitudes. The analysis is extended to 3D wave pulses, and we discuss the applications to radiation generated from pulsar binary mergers. It turns out that the electromagnetic radiation from a binary merger should experience a focusing effect, that in principle could be detected. 
  We study the conformal symmetry and the energy-momentum conservation of scalar field interacting with a curved background at D=2. We avoid to incorporate the metric determinant into the measure of the scalar field to explain the conformal anomaly and the consequent energy-momentum conservation. Contrarily, we split the scalar field in two other fields, in such a way that just one of them can be quantized. We show that the same usual geometric quantities of the anomaly are obtained, which are accompanied by terms containing the new field of the theory. 
  Modern formulation of Finsler geometry of a manifold M utilizes the equivalence between this geometry and the Riemannian geometry of VTM, the vertical bundle over the tangent bundle of M, treating TM as the base space. We argue that this approach is unsatisfactory when there is an indefinite metric on M because the corresponding Finsler fundamental function would not be differentiable over TM (even without its zero section) and therefore TM cannot serve as the base space. We then make the simple observation that for any differentiable Lorentzian metric on a smooth space-time, the corresponding Finsler fundamental function is differentiable exactly on a proper subbundle of TM. This subbundle is then used, in place of TM, to provide a satisfactory basis for modern Finsler geometry of manifolds with Lorentzian metrics. Interestingly, this Finslerian property of Lorentzian metrics does not seem to exist for general indefinite Finsler metrics and thus, Lorentzian metrics appear to be of special relevance to Finsler geometry. We note that, in contrast to the traditional formulation of Finsler geometry, having a Lorentzian metric in the modern setting does not imply reduction to pseudo- Riemannian geometry because metric and connection are entirely disentangled in the modern formulation and there is a new indispensable non-linear connection, necessary for construction of Finsler tensor bundles. It is concluded that general relativity--without any modification--has a close bearing on Finsler geometry and a modern Finsler formulation of the theory is an appealing idea... 
  We propose a model describing Einstein gravity coupled to a scalar field with an exponential potential. We show that the weak-field limit of the model has static solutions given by a gravitational potential behaving for large distances as \ln r . The Newtonian term GM/r appears only as subleading. Our model can be used to give a phenomenological explanation of the rotation curves of the galaxies without postulating the presence of dark matter. This can be achieved only by giving up at galactic scales Einstein equivalence principle. 
  The inflationary mechanism of mode amplification predicts that the state of each mode with a given wave vector is correlated to that of its partner mode with the opposite vector. This implies nonlocal correlations which leave their imprint on temperature anisotropies in the cosmic microwave background. Their spatial properties are best revealed by using local wave packets. This analysis shows that all density fluctuations giving rise the large scale structures originate in pairs which are born near the reheating. In fact each local density fluctuation is paired with an oppositely moving partner with opposite amplitude. To obtain these results we first apply a ``wave packet transformation'' with respect to one argument of the two point correlation function. A finer understanding of the correlations is then reached by making use of coherent states. The knowledge of the velocity field is required to extract the contribution of a single pair of wave packets. Otherwise, there is a two-folded degeneracy which gives three aligned wave packets arising from two pairs. The applicability of these methods to observational data is briefly discussed. 
  We report on a search for gravitational wave bursts using data from the first science run of the LIGO detectors. Our search focuses on bursts with durations ranging from 4 ms to 100 ms, and with significant power in the LIGO sensitivity band of 150 to 3000 Hz. We bound the rate for such detected bursts at less than 1.6 events per day at 90% confidence level. This result is interpreted in terms of the detection efficiency for ad hoc waveforms (Gaussians and sine-Gaussians) as a function of their root-sum-square strain h_{rss}; typical sensitivities lie in the range h_{rss} ~ 10^{-19} - 10^{-17} strain/rtHz, depending on waveform. We discuss improvements in the search method that will be applied to future science data from LIGO and other gravitational wave detectors. 
  Of the current known pulsars, the Crab pulsar (B0531+21) is one of the most promising sources of gravitational waves. The relatively large timing noise of the Crab causes its phase evolution to depart from a simple spin-down model. This effect needs to be taken in to account when performing time domain searches for the Crab pulsar in order to avoid severely degrading the search efficiency. The Jodrell Bank Crab pulsar ephemeris is examined to see if it can be used for tracking the phase evolution of any gravitational wave signal from the pulsar, and we present a method of heterodyning the data that takes account of the phase wander. The possibility of obtaining physical information about the pulsar from comparisons of the electromagnetically and a gravitationally observed timing noise is discussed. Finally, additional problems caused by pulsar glitches are discussed. 
  It is shown that inhomogeneous Szekeres and Stephani universes exist corresponding to non-dissipative binary mixtures of perfect fluids in local thermal equilibrium. This result contradicts a recent statement by Z\'arate and Quevedo (2004 Class. Quantum Grav. {\bf 21} 197, {\it Preprint} gr-qc/0310087), which affirms that the only Szekeres and Stephani universes compatible with these fluids are the homogeneous Friedmann-Robertson-Walker models. Thus, contrarily to their conclusion, their thermodynamic scheme do not gives new indications of incompatibility between thermodynamics and relativity. Two of the points that have generated this error are commented. 
  The observable universe could be a 1+3-surface (the "brane") embedded in a 1+3+d-dimensional spacetime (the "bulk"), with standard-model particles and fields trapped on the brane while gravity is free to access the bulk. At least one of the d extra spatial dimensions could be very large relative to the Planck scale, which lowers the fundamental gravity scale, possibly even down to the electroweak (~ TeV) level. This revolutionary picture arises in the framework of recent developments in M theory. The 1+10-dimensional M theory encompasses the known 1+9-dimensional superstring theories, and is widely considered to be a promising potential route to quantum gravity. General relativity cannot describe gravity at high enough energies and must be replaced by a quantum gravity theory, picking up significant corrections as the fundamental energy scale is approached. At low energies, gravity is localized at the brane and general relativity is recovered, but at high energies gravity "leaks" into the bulk, behaving in a truly 1+3+d-dimensional way. This introduces significant changes to gravitational dynamics and perturbations, with interesting testable implications for high-energy astrophysics, black holes and cosmology. Brane-world models offer a phenomenological way to test some of the novel predictions and corrections to general relativity that are implied by M theory. This review discusses the geometry, dynamics and perturbations of simple brane-world models for cosmology and astrophysics, mainly focusing on warped 5-dimensional brane-worlds based on the Randall-Sundrum models. 
  If the diffeomorphism symmetry of general relativity is fully implemented into a path integral quantum theory, the path integral leads to a partition function which is an invariant of smooth manifolds. We comment on the physical implications of results on the classification of smooth and piecewise-linear 4-manifolds which show that the partition function can already be computed from a triangulation of space-time. Such a triangulation characterizes the topology and the differentiable structure, but is completely unrelated to any physical cut-off. It can be arbitrarily refined without affecting the physical predictions and without increasing the number of degrees of freedom proportionally to the volume. Only refinements at the boundary have a physical significance as long as the experimenters who observe through this boundary, can increase the resolution of their measurements. All these are consequences of the symmetries. The Planck scale cut-off expected in quantum gravity is rather a dynamical effect. 
  We consider a three dimensional family of filters based on broken power law spectra to search for gravitational wave stochastic backgrounds in the data from Earth-based laser interferometers. We show that such templates produce the necessary fitting factor for a wide class of cosmological backgrounds and astrophysical foregrounds and that the total number of filters required to search for those signals in the data from first generation laser interferometers operating at the design sensitivity is fairly small 
  We pursue the view that quantum theory may be an emergent structure related to large space-time scales. In particular, we consider classical Hamiltonian systems in which the intrinsic proper time evolution parameter is related through a probability distribution to the discrete physical time. This is motivated by studies of ``timeless'' reparametrization invariant models, where discrete physical time has recently been constructed based on coarse-graining local observables. Describing such deterministic classical systems with the help of path-integrals, primordial states can naturally be introduced which follow unitary quantum mechanical evolution in suitable limits. 
  We replace the usual Hamiltonian constraint of quantum gravity H|psi>=0 by a weaker one <psi|H|psi>=0. This allows |psi> to satisfy the time-dependent functional Schrodinger equation. In general, only the phase of the wave function appears to be time independent. The resulting quantum theory has the correct classical limit and thus provides a viable theory of quantum gravity that solves the problem of time without introducing additional nongravitational degrees of freedom. 
  Recent developments in string theory suggest that there might exist extra spatial dimensions, which are not small nor compact. The framework of a great number of brane cosmological models is that in which the matter fields are confined on a brane-world embedded in five dimensions (the bulk). Motivated by this we review the main results on the algebraic classification of second order symmetric tensors in 5-dimensional space-times. All possible Segre types for a symmetric two-tensor are found, and a set of canonical forms for each Segre type is obtained. A limiting diagram for the Segre types of these symmetric tensors in 5-D is built. Two theorems which collect together some basic results on the algebraic structure of second order symmetric tensors in 5-D are presented. We also show how one can obtain, by induction, the classification and the canonical forms of a symmetric two-tensor on n-dimensional (n > 5) spaces from its classification in 5-D spaces, present the Segre types in n-D and the corresponding canonical forms. This classification of symmetric two-tensors in any n-D spaces and their canonical forms are important in the context of n-dimensional brane-worlds context and also in the framework of 11-D supergravity and 10-D superstrings. 
  The absorption cross section for scalar particle impact on a Schwarzschild black hole is found. The process is dominated by two physical phenomena. One of them is the well-known greybody factor that arises from the energy-dependent potential barrier outside the horizon that filters the incoming and outgoing waves. The other is related to the reflection of particles on the horizon (Kuchiev 2003). This latter effect strongly diminishes the cross section for low energies, forcing it to vanish in the infrared limit. It is argued that this is a general property, the absorption cross section vanishes in the infrared limit for scattering of particles of arbitrary spin. 
  W discuss the evolution of the fluctuations in a symmetric $\phi_c$-exponential potential which provides a power-law expansion during inflation using both, the gauge invariant field $\Phi$ and the Sasaki-Mukhanov field. 
  The variational principle for a spherical configuration consisting of a thin spherical dust shell in gravitational field is constructed. The principle is consistent with the boundary-value problem of the corresponding Euler-Lagrange equations, and leads to ``natural boundary conditions''. These conditions and the field equations following from the variational principle are used for performing of the reduction of this system. The equations of motion for the shell follow from the obtained reduced action. The transformation of the variational formula for the reduced action leads to two natural variants of the effective action. One of them describes the shell from a stationary interior observer's point of view, another from the exterior one. The conditions of isometry of the exterior and interior faces of the shell lead to the momentum and Hamiltonian constraints. 
  Problem of cosmological singularity is discussed in the framework of gauge theories of gravitation. Generalizing cosmological Friedmann equations (GCFE) for homogeneous isotropic models including scalar fields and usual gravitating matter are introduced. It is shown that by certain restrictions on equation of state of gravitating matter and indefinite parameter of GCFE generic feature of inflationary cosmological models of flat, open and closed type is their regular bouncing character. 
  If certain conditions are met, a propagating phase boundary can be a sonic horizon. Sonic Hawking radiation from such a phase boundary is expected in the quantum theory. The Hawking temperature for typical values of system parameters can be as large as $\sim 0.04$ K. Since the setup does not require the physical transport of material, it evades the seemingly insurmountable difficulties of the usual proposals to create a sonic horizon in which fluid is required to flow at supersonic speeds. Issues that are likely to present difficulties that are particular to this setup are discussed. Hawking evaporation of the sonic horizon is also expected and is predicted to lead to a deceleration of the phase boundary. 
  The Hartle-Thorne metric is an exact solution of vacuum Einstein field equations that describes the exterior of any slowly and rigidly rotating, stationary and axially symmetric body. The metric is given with accuracy up to the second order terms in the body's angular momentum, and first order in its quadrupole moment. We give, with the same accuracy, analytic formulae for circular geodesics in the Hartle-Thorne metrics. They describe angular velocity, angular momentum, energy, epicyclic frequencies, shear, vorticity and Fermi-Walker precession. These quantities are relevant to several astrophysical phenomena, in particular to the observed high frequency, kilohertz Quasi Periodic Oscillations (kHz QPOs) in the X-ray luminosity from black hole and neutron star sources. It is believed that kHz QPO data may be used to test the strong field regime of Einstein's general relativity, and the physics of super-dense matter of which neutron stars are made of. 
  Using the methods developed by Fewster and colleagues, we derive a quantum inequality for the free massive spin-${3\over 2}$ Rarita-Schwinger fields in the four dimensional Minkowski spacetime. Our quantum inequality bound for the Rarita-Schwinger fields is weaker, by a factor of 2, than that for the spin-${1\over 2}$ Dirac fields. This fact along with other quantum inequalities obtained by various other authors for the fields of integer spin (bosonic fields) using similar methods lead us to conjecture that, in the flat spacetime, separately for bosonic and fermionic fields, the quantum inequality bound gets weaker as the the number of degrees of freedom of the field increases. A plausible physical reason might be that the more the number of field degrees of freedom, the more freedom one has to create negative energy, therefore, the weaker the quantum inequality bound. 
  The production of gravitational vacuum defects and their contribution in energy density of the Universe are discussed. These topological microstructures could be produced as the result of defect creation of the Universe from "nothing" as well as the result of the first relativistic phase transition. They must be isotropically distributed on background of the expanding Universe. After Universe inflation these microdefects smoothed, stretched and broke up. Parts of them have survived and now they are perceived as the structures of Lambda-term (quintessence) and unclustered dark matter. It is shown that for phenomenological description of vacuum topological defects of different dimensions (worm-holes, micromembranes, microstrings and monopoles) the parametrizational noninvariant members of Wheeler -DeWitt equation can be used. The mathematical illustration of these processes may be the spontaneous breaking of local Lorentz-invariance of quasi-classical equations of gravity. In addition, 3-dimensional topological defects revalues Lambda-term. 
  It is argued that the initial cosmological singularity is isotropic in spatially inhomogeneous brane-world models. This implies that brane cosmology may naturally give rise to a set of initial data that provide the conditions for inflation to subsequently take place, consequently offering a plausible solution to the initial conditions problem in cosmology. 
  Quantum--mechanical operators corresponding to canonical momentum and position of a point--like particle, which follow from the quantum field theory in the general Riemannian space-time, satisfy generally to a deformation of the canonical commutation relations with $c^{-2}$ as the parameter of deformation. For operators of the quasi-Cartesian coordinates in the closed and open Friedman--Robertson--Walker models, the deformation reproduces the spatial part of the well--known Snyder formula for quantization of the Minkowsky space-time. The spatially-flat models are distinguished apart by that the deformation is reduced exactly to the standard canonical commutation relations, which correlates remarkably with the fact of the observed flatness of the Universe. Conditions are briefly discussed for which the deformation could have cosmological manifestations. Key words: quantum mechanics, cosmology, quantized space. 
  We discuss the asymptotic dynamical evolution of spatially inhomogeneous brane-world cosmological models close to the initial singularity. By introducing suitable scale-invariant dependent variables and a suitable gauge, we write the evolution equations of the spatially inhomogeneous $G_{2}$ brane cosmological models with one spatial degree of freedom as a system of autonomous first-order partial differential equations. We study the system numerically, and we find that there always exists an initial singularity, which is characterized by the fact that spatial derivatives are dynamically negligible. More importantly, from the numerical analysis we conclude that there is an initial isotropic singularity in all of these spatially inhomogeneous brane cosmologies for a range of parameter values which include the physically important cases of radiation and a scalar field source. The numerical results are supported by a qualitative dynamical analysis and a calculation of the past asymptotic decay rates. Although the analysis is local in nature, the numerics indicates that the singularity is isotropic for all relevant initial conditions. Therefore this analysis, and a preliminary investigation of general inhomogeneous ($G_0$) models, indicates that it is plausible that the initial singularity is isotropic in spatially inhomogeneous brane-world cosmological models and consequently that brane cosmology naturally gives rise to a set of initial data that provide the conditions for inflation to subsequently take place. 
  We have developed a formalism to study an inflationary scenario driven by a bulk inflaton in the two-brane system. The 4-dimensional low energy effective action is obtained using the gradient expansion method. It is also found that the dark radiation and the dark scalar source are expressed by the radion. In the single-brane limit, we find these dark components disappear. Therefore, it turns out that the inflation due to the bulk inflaton successfully takes place. Kaluza-Klein corrections are also taken into account in this case. 
  In this work we present a discussion of the existing links between the procedures of endowing the quantum gravity with a real time and of including in the theory a physical reference frame. More precisely, as first step, we develop the canonical quantum dynamics, starting from the Einstein equations in presence of a dust fluid and arrive to a Schroedinger evolution. Then, by fixing the lapse function in the path integral of gravity, we get a Schroedinger quantum dynamics, of which eigenvalues problem provides the appearance of a dust fluid in the classical limit. The main issue of our analysis is to claim that a theory, in which the time displacement invariance, on a quantum level, is broken, is indistinguishable from a theory for which this symmetry holds, but a real reference fluid is included. 
  By using the 't Hooft's "brick wall" model and the Pauli-Villars regularization scheme we calculate the statistical-mechanical entropies arising from the quantum scalar field in different coordinate settings, such as the Painlev\'{e} and Lemaitre coordinates. At first glance, it seems that the entropies would be different from that in the standard Schwarzschild coordinate since the metrics in both the Painlev\'{e} and Lemaitre coordinates do not possess the singularity at the event horizon as that in the Schwarzschild-like coordinate. However, after an exact calculation we find that, up to the subleading correction, the statistical-mechanical entropies in these coordinates are equivalent to that in the Schwarzschild-like coordinate. The result is not only valid for black holes and de Sitter spaces, but also for the case that the quantum field exerts back reaction on the gravitational field provided that the back reaction does not alter the symmetry of the spacetime. 
  The quasinormal modes of the Reissner-Nordstr\"om de Sitter black hole for the massless Dirac fields are studied using the P\"oshl-Teller potential approximation. We find that the magnitude of the imaginary part of the quasinormal frequencies decreases as the cosmological constant or the orbital angular momentum increases, but it increases as the charge or the overtone number increases. An interesting feature is that the imaginary part is almost linearly related to the real part as the cosmological constant changes for fixed charge, and the linearity becomes better as the orbital angular momentum increases. We also prove exactly that the Dirac quasinormal frequencies are the same for opposite chirality. 
  Generic spherically symmetric self-similar collapse results in strong naked-singularity formation. In this paper we are concerned with particle creation during a naked-singularity formation in spherically symmetric self-similar collapse without specifying the collapsing matter. In the generic case, the power of particle emission is found to be proportional to the inverse square of the remaining time to the Cauchy horizon (CH). The constant of proportion can be arbitrarily large in the limit to marginally naked singularity. Therefore, the unbounded power is especially striking in the case that an event horizon is very close to the CH because the emitted energy can be arbitrarily large in spite of a cutoff expected from quantum gravity. Above results suggest the instability of the CH in spherically symmetric self-similar spacetime from quantum field theory and seem to support the existence of a semiclassical cosmic censor. The divergence of redshifts and blueshifts of emitted particles is found to cause the divergence of power to positive or negative infinity, depending on the coupling manner of scalar fields to gravity. On the other hand, it is found that there is a special class of self-similar spacetimes in which the semiclassical instability of the CH is not efficient. The analyses in this paper are based on the geometric optics approximation, which is justified in two dimensions but needs justification in four dimensions. 
  We present an approach to quantum gravity based on the general boundary formulation of quantum mechanics, path integral quantization, spin foam models and renormalization. 
  Using equations of motion accurate to the third post-Newtonian (3PN) order (O(v/c)^6 beyond Newtonian gravity), we derive expressions for the total energy E and angular momentum J of the orbits of compact binary systems (black holes or neutron stars) for arbitrary orbital eccentricity. We also incorporate finite-size contributions such as spin-orbit and spin-spin coupling, and rotational and tidal distortions, calculated to the lowest order of approximation, but we exclude the effects of gravitational radiation damping. We describe how these formulae may be used as an accurate diagnostic of the physical content of quasi-equilibrium configurations of compact binary systems of black holes and neutron stars generated using numerical relativity. As an example, we show that quasi-equilibrium configurations of corotating neutron stars recently reported by Miller et al. can be fit by our diagnostic to better than one per cent with a circular orbit and with physically reasonable tidal coefficients. 
  Using the analogy with stationary axisymmetric solutions, we present a method to generate new analytic cosmological solutions of Einstein's equation belonging to the class of $T^3$ Gowdy cosmological models. We show that the solutions can be generated from their data at the initial singularity and present the formal general solution for arbitrary initial data. We exemplify the method by constructing the Kantowski-Sachs cosmological model and a generalization of it that corresponds to an unpolarized $T^3$ Gowdy model. 
  Relativistic spin-orbit and spin-spin couplings has been shown to modify the gravitational waveforms expected from inspiraling binaries with a black hole and a neutron star. As a result inspiral signals may be missed due to significant losses in signal-to-noise ratio, if precession effects are ignored in gravitational-wave searches. We examine the sensitivity of the anticipated loss of signal-to-noise ratio on two factors: the accuracy of the precessing waveforms adopted as the true signals and the expected distributions of spin-orbit tilt angles, given the current understanding of their physical origin. We find that the results obtained using signals generated by approximate techniques are in good agreement with the ones obtained by integrating the 2PN equations. This shows that a complete account of all high-order post-Newtonian effects is usually not necessary for the determination of detection efficiencies. Based on our current astrophysical expectations, large tilt angles are not favored and as a result the decrease in detection rate varies rather slowly with respect to the black hole spin magnitude and is within 20--30% of the maximum possible values. 
  David Albert claims that classical electromagnetic theory is not time reversal invariant. He acknowledges that all physics books say that it is, but claims they are "simply wrong" because they rely on an incorrect account of how the time reversal operator acts on magnetic fields. On that account, electric fields are left intact by the operator, but magnetic fields are inverted. Albert sees no reason for the asymmetric treatment, and insists that neither field should be inverted. I argue, to the contrary, that the inversion of magnetic fields makes good sense and is, in fact, forced by elementary geometric considerations. I also suggest a way of thinking about the time reversal invariance of classical electromagnetic theory -- one that makes use of the invariant four-dimensional formulation of the theory -- that makes no reference to magnetic fields at all. It is my hope that it will be of interest in its own right, Albert aside. It has the advantage that it allows for arbitrary curvature in the background spacetime structure, and is therefore suitable for the framework of general relativity. The only assumption one needs is temporal orientability. 
  The ultrarelativistic limit of twodimensional dilaton gravity is presented and its associated (anti-)selfdual energy momentum tensor is derived. It is localized on a null line, although the line element remains twice differentiable. Relations to the Aichelburg-Sexl spacetime and constant dilaton vacua are pointed out. Geodesics are found to be smooth for minimally coupled test particles but non-smooth -- with a finite jump in the acceleration -- for test particles coupled non-minimally to the dilaton. Quantization on boosted backgrounds is discussed; no anomalous trace of the energy momentum tensor arises and the 1-loop flux component can be adjusted to be equal to the classical flux of the shock wave. 
  We analyze the geometry of a rotating disk with a tangential acceleration in the framework of the Special Theory of Relativity, using the kinematic linear differential system that verifies the relative position vector of time-like curves in a Fermi reference. A numerical integration of these equations for a generic initial value problem is made up and the results are compared with those obtained in other works. 
  We present the analysis of between 50 and 100 hrs of coincident interferometric strain data used to search for and establish an upper limit on a stochastic background of gravitational radiation. These data come from the first LIGO science run, during which all three LIGO interferometers were operated over a 2-week period spanning August and September of 2002. The method of cross-correlating the outputs of two interferometers is used for analysis. We describe in detail practical signal processing issues that arise when working with real data, and we establish an observational upper limit on a f^{-3} power spectrum of gravitational waves. Our 90% confidence limit is Omega_0 h_{100}^2 < 23 in the frequency band 40 to 314 Hz, where h_{100} is the Hubble constant in units of 100 km/sec/Mpc and Omega_0 is the gravitational wave energy density per logarithmic frequency interval in units of the closure density. This limit is approximately 10^4 times better than the previous, broadband direct limit using interferometric detectors, and nearly 3 times better than the best narrow-band bar detector limit. As LIGO and other worldwide detectors improve in sensitivity and attain their design goals, the analysis procedures described here should lead to stochastic background sensitivity levels of astrophysical interest. 
  Two theories of special relativity with an additional invariant scale, "doubly special relativity" (DSR), are tested with calculations of particle process kinematics. Using the Judes-Visser modified conservation laws, thresholds are studied in both theories. In contrast to some linear approximations, which allow for particle processes forbidden in special relativity, both the Amelino-Camelia and Maguejo-Smolin frameworks allow no additional processes. To first order, the Amelino-Camelia framework thresholds are lowered and the Maguejo-Smolin framework thresholds may be raised or lowered. 
  We present preliminar results and tests of a new general relativistic code to simulate the hydrodynamic collapse of a 21 solar masses star. We have assumed spherical symmetry and used the formalism of Misner and Sharp to construct a finite-difference scheme to solve the Einstein's equations, energy-momentum conservation equations and baryonic conservation equation. The code is similar to the one originally developed by May and White (1967). Here we discuss the capabilities of the code that make it well suited for numerical relativity on a personal computer and some caveats based on the experiments we have made with it. 
  Recently A. Rathke has argued that the KK$\psi$ model explanation of the discrepant measurements of Newton's constant is already ruled out due to E\"otv\"os experiments by several orders of magnitude. The structure of the action of the KK$\psi$ model is even qualified as inconsistent in the sense that it would yield a negative energy of the electromagnetic field. Here, I refute both claims and emphasize the possibility still open to reconcile the experimental bounds on the test of the weak equivalence principle (WEP) with scalar-tensor theories in general by some compensating mechanism. 
  The classical observational cosmological tests (Hubble diagram, count of sources, etc.) are considered for a homogeneous and isotropic model of the Universe in the framework of the five-dimensional Projective Unified Field Theory in which gravitation is described by both space-time curvature and some hypothetical scalar field (sigma-field). It is shown that the presence of the sigma-field can essentially affect conclusions obtained from the cosmological tests. The surface brightness-redshift relation can be used as a critical test for sigma-field effects. It seems reasonable to say that the available experimental data testify that the sigma-field decreases with time. It is concluded that the spatial curvature is positive or negative depending on whether the mass density is larger or smaller than some critical parameter which is smaller than the critical density and can even take negative values. It is shown that the increase in the number of the observational cosmological parameters as compared to the standard Friedmann model can essentially facilitate coordination of the existing observational data. 
  By using a nonholonomic moving frame version of the general covariance principle, an active version of the equivalence principle, an analysis of the gravitational coupling prescription of teleparallel gravity is made. It is shown that the coupling prescription determined by this principle is always equivalent with the corresponding prescription of general relativity, even in the presence of fermions. An application to the case of a Dirac spinor is made. 
  We examine the singularity resolution issue in quantum gravity by studying a new quantization of standard Friedmann-Robertson-Walker geometrodynamics. The quantization procedure is inspired by the loop quantum gravity programme, and is based on an alternative to the Schr\"odinger representation normally used in metric variable quantum cosmology. We show that in this representation for quantum geometrodynamics there exists a densely defined inverse scale factor operator, and that the Hamiltonian constraint acts as a difference operator on the basis states. We find that the cosmological singularity is avoided in the quantum dynamics. We discuss these results with a view to identifying the criteria that constitute "singularity resolution" in quantum gravity. 
  Heuristic arguments and numerical simulations support the Belinskii et al (BKL) claim that the approach to the singularity in generic gravitational collapse is characterized by local Mixmaster dynamics (LMD). Here, one way to identify LMD in collapsing spatially inhomogeneous cosmologies is explored. By writing the metric of one spacetime in the standard variables of another, signatures for LMD may be found. Such signatures for the dynamics of spatially homogeneous Mixmaster models in the variables of U(1)-symmetric cosmologies are reviewed. Similar constructions for U(1)-symmetric spacetimes in terms of the dynamics of generic $T^2$-symmetric spacetime are presented. 
  A dynamical, non-Euclidean spacetime geometry in general relativity theory implies the possibility of gravitational radiation. Here we explore novel methods of detecting such radiation from astrophysical sources by means of matter-wave interferometers (MIGOs), using atomic beams emanating from supersonic atomic sources that are further cooled and collimated by means of optical molasses. While the sensitivities of such MIGOs compare favorably with LIGO and LISA, the sizes of MIGOs can be orders of magnitude smaller, and their bandwidths wider. Using a pedagogical approach, we place this problem into the broader context of problems at the intersection of quantum mechanics with general relativity. 
  We present the first results from our new general relativistic, Lagrangian hydrodynamics code, which treats gravity in the conformally flat (CF) limit. The evolution of fluid configurations is described using smoothed particle hydrodynamics (SPH), and the elliptic field equations of the CF formalism are solved using spectral methodsin spherical coordinates. The code was tested on models for which the CF limit is exact, finding good agreement with the classical Oppenheimer-Volkov solution for a relativistic static spherical star as well as the exact semi-analytic solution for a collapsing spherical dust cloud. By computing the evolution of quasi-equilibrium neutron star binary configurations in the absence of gravitational radiation backreaction, we have confirmed that these configurations can remain dynamically stable all the way to the development of a cusp. With an approximate treatment of radiation reaction, we have calculated the complete merger of an irrotational binary configuration from the innermost point on an equilibrium sequence through merger and remnant formation and ringdown, finding good agreement withprevious relativistic calculations. In particular, we find that mass loss is highly suppressed by relativistic effects, but that, for a reasonably stiff neutron star equation of state, the remnant is initially stable against gravitational collapse because of its strong differential rotation. The gravity wave signal derived from our numerical calculation has an energy spectrum which matches extremely well with estimates based solely on quasi-equilibrium results, deviating from the Newtonian power-law form at frequencies below 1 kHz, i.e., within the reach of advanced interferometric detectors. 
  To estimate Kaluza-Klein (KK) corrections on gravitational waves emitted by perturbed rotating black strings, we give the effective Teukolsky equation on the brane which is separable equation and hence numerically manageable. 
  Gravitomagnetic charge that can also be referred to as the {\it dual mass} or {\it magnetic mass} is the topological charge in gravity theory. A gravitomagnetic monopole at rest can produce a stationary gravitomagnetic field. Due to the topological nature of gravitomagnetic charge, the metric of spacetime where the gravitomagnetic matter is present will be nonanalytic. In this paper both the dual curvature tensors (which can characterize the dynamics of gravitational charge/monopoles) and the antisymmetric gravitational field equation of gravitomagnetic matter are presented. We consider and discuss the mathematical formulation and physical properties of the dual curvature tensors and scalar, antisymmetric source tensors, dual spin connection (including the low-motion weak-field approximation), dual vierbein field as well as dual current densities of gravitomagnetic charge. It is shown that the dynamics of gravitomagnetic charge can be founded within the framework of the above dual quantities. In addition, the dual relationship between the dynamical theories of gravitomagnetic charge (dual mass) and gravitoelectric charge (mass) is also taken into account in the present paper. 
  It is shown that atom interferometry allows for the construction of MIGO, the Matter-wave Interferometric Gravitational-wave Observatory. MIGOs of the same sensitivity as LIGO or LISA are expected to be orders of magnitude smaller than either one. A design for MIGO using crystalline diffraction gratings is introduced, and its sensitivity is calculated. 
  We investigate the effects of cosmological expansion on the spectrum of small-scale structure on a cosmic string. We simulate the evolution of a string with two modes that differ in wavelength by one order of magnitude. Once the short mode is inside the horizon, we find that its physical amplitude remains unchanged, in spite of the fact that its comoving wavelength decreases as the longer mode enters the horizon. Thus the ratio of amplitude to wavelength for the short mode becomes larger than it would be in the absence of the long mode. 
  The equations which determine the response of a spinning charged particle moving in a uniform magnetic field to an incident gravitational wave are derived in the linearized approximation to general relativity. We verify that 1) the components of the 4-momentum, 4-velocity and the components of the spinning tensor, both electric and magnetic moments, exhibit resonances and 2) the co-existence of the uniform magnetic field and the GW are responsible for the resonances appearing in our equations. In the absence of the GW, the magnetic field and the components of the spin tensor decouple and the magnetic resonances disappear. 
  For many years, the most active area of quantum cosmology has been the issue of choosing boundary conditions for the wave function of a universe. Recently, loop quantum cosmology, which is obtained from loop quantum gravity, has shed new light on this question. In this case, boundary conditions are not chosen by hand with some particular physical intuition in mind, but they are part of the dynamical law. It is then natural to ask if there are any relations between these boundary conditions and the ones provided before. After discussing the technical foundation of loop quantum cosmology which leads to crucial differences to the Wheeler-DeWitt quantization, we compare the dynamical initial conditions of loop quantum cosmology with the tunneling and the no-boundary proposal and explain why they are closer to the no-boundary condition. We end with a discussion of recent developments and several open problems of loop quantum cosmology. 
  We study the stability of new neutral and electrically charged four-dimensional black hole solutions of Einstein's equations with a positive cosmological constant and conformally coupled scalar field. The neutral black holes are always unstable. The charged black holes are also shown analytically to be unstable for the vast majority of the parameter space of solutions, and we argue using numerical techniques that the configurations corresponding to the remainder of the parameter space are also unstable. 
  The behaviour of a massless Dirac field on a general spacetime background representing two colliding gravitational plane waves is discussed in the Newman-Penrose formalism. The geometrical properties of the neutrino current are analysed and explicit results are given for the special Ferrari-Ibanez solution. 
  Evaluating Kaluza-Klein (KK) corrections is indispensable to test the braneworld scenario. In this report, we propose a novel symmetry approach to an effective 4-dimensional action with KK corrections for the Randall-Sundrum two-brane system. 
  In the Randall-Sundrum two-brane model (RS1), a Kerr black hole on the brane can be naturally identified with a section of rotating black string. To estimate Kaluza-Klein (KK) corrections on gravitational waves emitted by perturbed rotating black strings, we give the effective Teukolsky equation on the brane which is separable equation and hence numerically manageable. In this process, we derive the master equation for the electric part of the Weyl tensor $E_{\mu\nu}$ which would be also useful to discuss the transition from black strings to localized black holes triggered by Gregory-Laflamme instability. 
  We investigate the possibility that the late acceleration observed in the rate of expansion of the universe is due to vacuum quantum effects arising in curved spacetime. The theoretical basis of the vacuum cold dark matter (VCDM), or vacuum metamorphosis, cosmological model of Parker and Raval is revisited and improved. We show, by means of a manifestly nonperturbative approach, how the infrared behavior of the propagator (related to the large-time asymptotic form of the heat kernel) of a free scalar field in curved spacetime causes the vacuum expectation value of its energy-momentum tensor to exhibit a resonance effect when the scalar curvature R of the spacetime reaches a particular value related to the mass of the field. we show that the back reaction caused by this resonance drives the universe through a transition to an accelerating expansion phase, very much in the same way as originally proposed by Parker and Raval. Our analysis includes higher derivatives that were neglected in the earlier analysis, and takes into account the possible runaway solutions that can follow from these higher-derivative terms. We find that the runaway solutions do not occur if the universe was described by the usual classical FRW solution prior to the growth of vacuum energy-density and negative pressure (i.e., vacuum metamorphosis) that causes the transition to an accelerating expansion of the universe in this theory. 
  In the context of the Randall-Sundrum (RS) single-brane scenario, we discuss the bulk geometry and dynamics of a cosmological brane in terms of the local energy conservation law which exists for the bulk that allows slicing with a maximally symmetric 3-space. This conservation law enables us to define a local mass in the bulk. We show that there is a unique generalization of the dark radiation on the brane, which is given by the local mass. We find there also exists a conserved current associated with the Weyl tensor, and the corresponding local charge, which we call the Weyl charge, is given by the sum of the local mass and a certain linear combination of the components of the bulk energy-momentum tensor. This expression of the Weyl charge relates the local mass with the projected Weyl tensor, $E_{\mu\nu}$, which plays a central role in the geometrical formalism of the RS braneworld. On the brane, in particular, this gives a decomposition of the projected Weyl tensor into the local mass and the bulk energy-momentum tensor. Then, as an application of these results, we consider a null dust model for the bulk energy-momentum tensor and discuss the black hole formation in the bulk. We investigate the causal structure by identifying the locus of the apparent horizon and clarify possible brane trajectories in the bulk. We find that the brane stays always outside the black hole as long as it is expanding. We also find an upper bound on the value of the Hubble parameter in terms of the matter energy density on the brane, irrespective of the energy flux emitted from the brane. 
  It is well known that a closed universe with a minimally coupled massive scalar field always collapses to a singularity unless the initial conditions are extremely fine tuned. We show that the corrections to the equations of motion for the massive scalar field, given by loop quantum gravity in high curvature regime, always lead to a bounce independently of the initial conditions. In contrast to the previous works in loop quantum cosmology, we note that the singularity can be avoided even at the semi-classical level of effective dynamical equations with non-perturbative quantum gravity modifications, without using a discrete quantum evolution. 
  We heuristically discuss the Ashtekar type canonical analysis of gravity using temporal foliations instead of spacial foliations and analyze its implications on classical and quantum gravity in general. The constraints and the constraint algebra are essentially the same as before. The gauge group of the real gravity theory is now SO(2,1) instead of SU(2) on the phase of canonical gravity restricted to real triads. We briefly discuss the classical and quantum aspects of the theory. We propose a generalisation of the Ashtekar formalism in which the type of the foliation is included in the initial conditions. The gauge group and the phase space of Ashtekar's canonical (real) gravity is clarified. The area spectrum is unchanged if we use only the finite dimensional representations of SL(2,C) to construct the quantum Hilbert space of the theory. If we use the infinite dimensional representations of SL(2,C) to construct the Hilbert space of Loop quantum gravity, we suggest the possibility that the Ashtekar formulation using spacial foliations is related to the Lorentzian Barrett-Crane model based on SL(2,C)/SU(2) and the Ashtekar formulation with temporal foliations is related to the Rovelli-Perez model based on SL(2,C)/SU(1,1)Z_2. This work has implications for loop quantum gravity. 
  We present numerical simulations of binary black hole systems which for the first time last for about one orbital period for close but still separate black holes as indicated by the absence of a common apparent horizon. An important part of the method is the construction of comoving coordinates, in which both the angular and radial motion is minimized through a dynamically adjusted shift condition. We use fixed mesh refinement for computational efficiency. 
  The fundamental features of the detection of non-stationary undulatory perturbations of metrics based on the interference effects are considered. The advantage of the Aharonov-Bohm effect in superconductors for these purposes in comparison with the ordinary optical interference is demonstrated. Some circuitries of the interferometric detectors in order to be used with SQUID are suggested. The possibilities of lowering the noise temperature of the ultraweak signals detectors based on the analogy between the processes of high-sensitive measurements and the reversible calculations are discussed. 
  The role that the auxiliary scalar field $\phi$ played in Brans-Dicke cosmology is discussed. If a constant vacuum energy is assumed to be the origin of dark energy, then the corresponding density parameter would be a quantity varying with $\phi$; and almost all of the fundamental components of our universe can be unified into the dynamical equation for $\phi$. As a generalization of Brans-Dicke theory, we propose a new gravity theory with a complex scalar field $\phi$ which is coupled to the cosmological curvature scalar. Through such a coupling, the Higgs mechanism is naturally incorporated into the evolution of the universe, and a running density of the field vacuum energy is obtained which may release the particle standard model from the rigorous cosmological constant problem in some sense. Our model predicts a running mass scale of the fundamental particles in which the gauge symmetry breaks spontaneously. The running speed of the mass scale in our case could survive all existing experiments. 
  The cosmological constant problem is reviewed and a possible quantum gravity resolution is proposed. A space satellite E\"otv\"os experiment for zero-point vacuum energy is proposed to see whether Casimir vacuum energy falls in a gravitational field at the same rate as ordinary matter. 
  This note is devoted to the detailed mathematical treatment of the coupling of graviton spin to gravitomagnetic fields. The expression (i.e., $\sim g_{0m}\dot{g}_{0n}(\partial_{m}g_{0n}-\partial_{n}g_{0m})$) for the graviton spin-gravitomagnetic (S-G) coupling in the Lagrangian/Hamiltonian density of the weak gravitational fields is presented in this note. 
  Numerical simulations of the approach to the singularity in vacuum spacetimes are presented here. The spacetimes examined have no symmetries and can be regarded as representing the general behavior of singularities. It is found that the singularity is spacelike and that as it is approached, the spacetime dynamics becomes local and oscillatory. 
  A recent refinement of Penrose's conformal framework for asymptotically flat space-times is summarized. The key idea concerns advanced and retarded conformal factors, which allow a rigid description of infinity as a locally metric light cone. In the new framework, the Bondi-Sachs energy-flux integrals of ingoing and outgoing gravitational radiation decay at spatial infinity such that the total radiated energy is finite, and the Bondi-Sachs energy-momentum has a unique limit at spatial infinity, coinciding with the uniquely rendered ADM energy-momentum. 
  We present current theories about the structure of space and time, where the building blocks are some fundamental entities (yes-no experiment, quantum processes, spin net-work, preparticles) that do not presuppose the existence of space and time. The relations among these objects are the base for a pregeometry of discrete character, the continuous limit of which gives rise to the physical properties of the space and time. 
  We have shown that DeWitt constraint H=0 on the physical states of the Universe does not prevent Heisenberg operators and its mean values to evolve with time. Mean value from observable, which is singular in classical theory, is also singular in a quantum case. 
  We show that (3+1) vacuum spacetimes admitting a global, spacelike, one-parameter Lie group of isometries of translational type cannot contain apparent horizons. The only assumption made is that of the existence of a global spacelike Killing vector field with infinite open orbits; the four-dimensional vacuum spacetime metric is otherwise arbitrary. This result may thus be viewed as a hoop conjecture theorem for vacuum gravity with one spacelike translational Killing symmetry. 
  We consider four-dimensional spacetimes $(M,{\mathbf g})$ which obey the Einstein equations ${\mathbf G}={\mathbf T}$, and admit a global spacelike $G_{1}={\mathbb R}$ isometry group. By means of dimensional reduction and local analyis on the reduced (2+1) spacetime, we obtain a sufficient condition on ${\mathbf T}$ which guarantees that $(M,{\mathbf g})$ cannot contain apparent horizons. Given any (3+1) spacetime with spacelike translational isometry, the no-horizon condition can be readily tested without the need for dimensional reduction. This provides thus a useful and encompassing apparent horizon test for $G_{1}$-symmetric spacetimes. We argue that this adds further evidence towards the validity of the hoop conjecture, and signals possible violations of strong cosmic censorship. 
  We consider spherically symmetric spacetimes with matter whose timelike flow is assumed to be shear-free. A number of results on the formation and visibility of spacetime singularities is proven, with the main one being that shear-free collapse cannot admit locally naked singularities (which implies absence of globally naked singularities). We conjecture that shear is a necessary condition for the occurrence of locally naked singularities in generic gravitational collapse. 
  Investigations of the possibility that some novel ``quantum" properties of spacetime might induce a modification dispersion relation focused at first on scenarios with Planck-scale violations of Lorentz symmetry. More recently several studies have considered the possibility of a ``doubly special relativity", in which the modification of the dispersion relation emerges from a framework with both the Planck scale and the speed-of-light scale as characteristic scales of a deformation of the Lorentz transformations. For the schemes with broken Lorentz symmetry at the Planck scale there is a large literature on the derivation of experimental limits. We provide here a corresponding analysis for the doubly-special-relativity framework. We find that the analyses of photon stability, synchrotron radiation, and threshold conditions for particle production in collision processes, the three contexts which are considered as most promising for constraining the broken-Lorentz-symmetry scenario, cannot provide significant constraints on doubly-special-relativity parameter space. However, certain types of analyses of gamma-ray bursts are sensitive to the symmetry deformation. A key element of our study is an observation that removes a possible sign ambiguity for the doubly-special-relativity framework. This result also allows us to characterize more sharply the differences between the doubly-special-relativity framework and the framework of k-Poincare Hopf algebras, two frameworks which are often confused with each other in the literature. 
  We review some modern theories about the structure of space and time, in particular those related to discrete space and time. Following an epistemological method we start from theories which discuss discrete space and time as a mathematical tool to solve physical models. Antother theories look for physical content of the discrete structure of space and time, based in relational theories of space and time which are derived from the relations of some fundamental entities. Finally we present some philosophical positions who try to find the ontological foundation of the relational theories os space and time. 
  We classify parallelizable noncommutative manifold structures on finite sets of small size in the general formalism of framed quantum manifolds and vielbeins introduced previously. The full moduli space is found for $\le 3$ points, and a restricted moduli space for 4 points. The topological part of the moduli space is found for $\le 9$ points based on the known atlas of regular graphs. We also discuss aspects of the quantum theory defined by functional integration. 
  For scalar, electromagnetic, or gravitational wave propagation on a fixed Schwarzschild blackhole background, we describe the exact nonlocal radiation outer boundary conditions (ROBC) appropriate for a spherical outer boundary of finite radius enclosing the blackhole. Derivation of the ROBC is based on Laplace and spherical-harmonic transformation of the Regge-Wheeler equation, the PDE governing the wave propagation, with the resulting radial ODE an incarnation of the confluent Heun equation. For a given angular index l the ROBC feature integral convolution between a time-domain radiation boundary kernel (TDRK) and each of the corresponding 2l+1 spherical-harmonic modes of the radiating wave. The TDRK is the inverse Laplace transform of a frequency-domain radiation kernel (FDRK) which is essentially the logarithmic derivative of the asymptotically outgoing solution to the radial ODE. We numerically implement the ROBC via a rapid algorithm involving approximation of the FDRK by a rational function. Such an approximation is tailored to have relative error \epsilon uniformly along the axis of imaginary Laplace frequency. Theoretically, \epsilon is also a long-time bound on the relative convolution error. Via study of one-dimensional radial evolutions, we demonstrate that the ROBC capture the phenomena of quasinormal ringing and decay tails. Moreover, carrying out a numerical experiment in which a wave packet strikes the boundary at an angle, we find that the ROBC yield accurate results in a three-dimensional setting. Our work is a partial generalization to Schwarzschild wave propagation and Heun functions of the methods developed for flatspace wave propagation and Bessel functions by Alpert, Greengard, and Hagstrom. 
  When the cosmological constant of spacetime is derived from the 5D induced-matter theory of gravity, we show that a simple gauge transformation changes it to a variable measure of the vacuum which is infinite at the big bang and decays to an astrophysically-acceptable value at late epochs. We outline implications of this for cosmology and galaxy formation. 
  Multidimensional cosmological models with factorizable geometry and their dimensional reduction to effective four-dimensional theories are analyzed on sensitivity to different scalings. It is shown that a non-correct gauging of the effective four-dimensional gravitational constant within the dimensional reduction results in a non-correct rescaling of the cosmological constant and the gravexciton/radion masses. The relationship between the effective gravitational constants of theories with different dimensions is discussed for setups where the lower dimensional theory results via dimensional reduction from the higher dimensional one and where the compactified space components vary dynamically. 
  The initial value problem is well-defined on a class of spacetimes broader than the globally hyperbolic geometries for which existence and uniqueness theorems are traditionally proved. Simple examples are the time-nonorientable spacetimes whose orientable double cover is globally hyperbolic. These spacetimes have generalized Cauchy surfaces on which smooth initial data sets yield unique solutions. A more difficult problem is to characterize the class of spacetimes with closed timelike curves that admit a well-posed initial value problem. Examples of spacetimes with closed timelike curves are given for which smooth initial data at past null infinity has been recently shown to yield solutions. These solutions appear to be unique, and uniquesness has been proved in particular cases. Other examples, however, show that confining closed timelike curves to compact regions is not sufficient to guarantee uniqueness. An approach to the characterization problem is suggested by the behavior of congruences of null rays. Interacting fields have not yet been studied, but particle models suggest that uniqueness (and possibly existence) is likely to be lost as the strength of the interaction increases. 
  The Sagnac effect for a uniformly moving observer is discussed. Starting from a recent measurement we show that the Sagnac effect is in fact not a consequence of the rotation of the observer, but simply of its, even inertial, motion with respect to the device resending light towards the (moving) source. 
  Motivated by recent studies on the uniqueness or non-uniqueness of higher dimensional black hole spacetime, we investigate the asymptotic structure of spatial infinity in n-dimensional spacetimes($n \geq 4$). It turns out that the geometry of spatial infinity does not have maximal symmetry due to the non-trivial Weyl tensor {}^{(n-1)}C_{abcd} in general. We also address static spacetime and its multipole moments P_{a_1 a_2 ... a_s}. Contrasting with four dimensions, we stress that the local structure of spacetimes cannot be unique under fixed a multipole moments in static vacuum spacetimes. For example, we will consider the generalized Schwarzschild spacetimes which are deformed black hole spacetimes with the same multipole moments as spherical Schwarzschild black holes. To specify the local structure of static vacuum solution we need some additional information, at least, the Weyl tensor {}^{(n-2)}C_{abcd} at spatial infinity. 
  In this paper, we study the perturbation problem of the scalar, electromagnetic, and gravitational waves under the traversable Lorentzian wormhole geometry. The unified form of the potential for the Schr\"odinger type one-dimensional wave equation is found. 
  We discuss the algebraic classification of the Weyl tensor in higher dimensional Lorentzian manifolds. This is done by characterizing algebraically special Weyl tensors by means of the existence of aligned null vectors of various orders of alignment. Further classification is obtained by specifying the alignment type and utilizing the notion of reducibility. For a complete classification it is then necessary to count aligned directions, the dimension of the alignment variety, and the multiplicity of principal directions. The present classification reduces to the classical Petrov classification in four dimensions. Some applications are briefly discussed. 
  Quantum singularities in general relativistic spacetimes are determined by the behavior of quantum test particles. A static spacetime is quantum mechanically singular if the spatial portion of the wave operator is not essentially self-adjoint. Here Weyl's limit point-limit circle criterion is used to determine whether a wave operator is essentially self-adjoint. This test is then applied to scalar wave packets in Levi-Civita spacetimes to help elucidate the physical properties of the spacetimes in terms of their metric parameters. 
  We develop a dimension-independent theory of alignment in Lorentzian geometry, and apply it to the tensor classification problem for the Weyl and Ricci tensors. First, we show that the alignment condition is equivalent to the PND equation. In 4D, this recovers the usual Petrov types. For higher dimensions, we prove that, in general, a Weyl tensor does not possess aligned directions. We then go on to describe a number of additional algebraic types for the various alignment configurations. For the case of second-order symmetric (Ricci) tensors, we perform the classification by considering the geometric properties of the corresponding alignment variety. 
  We obtain global space-time weighted-L^2 (Morawetz) and L^4 (Strichartz) estimates for a massless chargeless scalar field propagating on a super-extremal (overcharged) Reissner-Nordstrom background. We begin by discussing the question of well-posedness of the Cauchy problem for scalar fields on non-globally hyperbolic manifolds and the role played by the Friedrichs extension, go over the construction of the function spaces involved, show how to transform the problem to one about the wave equation on the Minkowski space with a singular potential, and finally prove that the potential thus obtained satisfies the various conditions needed in order for the estimates to hold. 
  Doppler effect and Hubble effect in different models of space-time in the case of auto-parallel motion of the observer are considered. The Doppler effect and shift frequency parameter are specialized for the case of auto-parallel motion of the observer. The Hubble effect and shift frequency parameter are considered for the same case. It is shown that by the use of the variation of the shift frequency parameter during a time perod, considered locally in the proper frame of reference of an observer, one can directly determine the centrifugal (centripetal) relative velocity and acceleration as well as the Coriolis relative velocity and acceleration of an astronomical object moving relatively to the observer. All results are obtained on purely kinematic basis without taking into account the dynamic reasons for the considered effect. PACS numbers: 98.80.Jk; 98.62.Py; 04.90.+e; 04.80.Cc 
  A higher dimensional frame formalism is developed in order to study implications of the Bianchi identities for the Weyl tensor in vacuum spacetimes of the algebraic types III and N in arbitrary dimension $n$. It follows that the principal null congruence is geodesic and expands isotropically in two dimensions and does not expand in $n-4$ spacelike dimensions or does not expand at all. It is shown that the existence of such principal geodesic null congruence in vacuum (together with an additional condition on twist) implies an algebraically special spacetime. We also use the Myers-Perry metric as an explicit example of a vacuum type D spacetime to show that principal geodesic null congruences in vacuum type D spacetimes do not share this property. 
  The detected anomalous frequency drift acceleration in Pioneer's radar data finds its explanation in a Berry phase that obtains the quantum state of a photon that propagates within an expanding space-time. The clock acceleration is just the adiabatic expansion rate and an analogy between the effect and Foucault's experiment is fully suggested. In this sense, light rays play a similar role in the expanding space than Foucault's Pendulum does while determining Earth's rotation. On the other hand, one could speculate about a suitable future experiment at "laboratory" scales able to measure the local cosmological expansion rate using the procedure outlined in this paper. 
  Quantum foam, also known as spacetime foam, has its origin in quantum fluctuations of spacetime. Its physics is intimately linked to that of black holes and computation. Arguably it is the source of the holographic principle which severely limits how densely information can be packed in space. Various proposals to detect the foam are briefly discussed. Its detection will provide us with a glimpse of the ultimate structure of space and time. 
  We formulate a premetric version of classical electrodynamics in terms of the excitation H and the field strength F. A local, linear, and symmetric spacetime relation between H and F is assumed. It yields, if electric/magnetic reciprocity is postulated, a Lorentzian metric of spacetime thereby excluding Euclidean signature (which is, nevertheless, discussed in some detail). Moreover, we determine the Dufay law (repulsion of like charges and attraction of opposite ones), the Lenz rule (the relative sign in Faraday's law), and the sign of the electromagnetic energy. In this way, we get a systematic understanding of the sign rules and the sign conventions in electrodynamics. The question in the title of the paper is answered affirmatively. 
  The frequencies of a cryogenic sapphire oscillator and a hydrogen maser are compared to set new constraints on a possible violation of Lorentz invariance. We give a detailed description of microwave resonators operating in Whispering Gallery modes and then apply it to derive explicit models for Lorentz violating effects in our experiment. Models are calculated in the theoretical framework of Robertson, Mansouri and Sexl and in the standard model extension (SME) of Kostelecky and co-workers. We constrain the parameters of the Mansouri and Sexl test theory to $1/2 - \beta_{MS} + \delta_{MS} = (1.2 \pm 2.2) \times 10^{-9}$ and $\beta_{MS} - \alpha_{MS} - 1 = (1.6 \pm 3.0) \times 10^{-7}$ which is of the same order as the best results from other experiments for the former and represents a 70 fold improvement for the latter. These results correspond to an improvement of our previously published limits [Wolf P. et al., Phys. Rev. Lett. {\bf 90}, 6, 060402, (2003)] by about a factor 2. 
  General aspects of the boundary value problem for the constraint equations and their application to black holes are discussed. 
  A class of dynamical shift conditions is shown to lead to a strongly hyperbolic evolution system, both in the Z4 and in the BSSN Numerical Relativity formalisms. This class generalizes the harmonic shift condition, where light speed is the only non-trivial characteristic speed, and it is contained into the multi-parameter family of minimal distortion shift conditions recently proposed by Lindblom and Scheel. The relationship with the analogous 'dynamical freezing' shift conditions used in black hole simulations discussed. 
  It has been known that a B=2 skyrmion is axially symmetric. We consider the Skyrme model coupled to gravity and obtain static axially symmetric regular and black hole solutions numerically. Computing the energy density of the skyrmion, we discuss the effect of gravity to the energy density and baryon density of the skyrmion. 
  Gravitation might make a preferred frame appear, and with it a clear space-time separation, which is needed by quantum mechanics (QM) in curved space-time. Several models of gravitation with an ether are discussed: they assume metrical effects in an heterogeneous ether and/or a Lorentz-symmetry breaking. One scalar model is detailed. It sees gravity as a pressure force and has been developed to a complete theory including continuum dynamics, cosmology, and links with electromagnetism and QM. To test the theory, an asymptotic scheme of post-Newtonian approximation has been built. It predicts an internal-structure effect, even at the point limit. The same should be true in general relativity (GR) if one could develop a similar scheme. Adjusting the equations of planetary motion on an ephemeris leaves a residual difference with it; one should adjust the equations using primary observations. The same effects on light rays are predicted as with GR, and a similar energy loss applies to binary pulsars. 
  Here we study some general properties of spherical shear-free collapse. Its general solution when imposing conformal flatness is reobtained [1,2] and matched to the outgoing Vaidya spacetime. We propose a simple model satisfying these conditions and study its physical consequences. Special attention deserve, the role played by relaxational processes and the conspicuous link betweeen dissipation and density inhomogeneity. 
  It was shown by Ford and Roman in 1996 that quantum field theory severely constrains wormhole geometries on a macroscopic scale. The first part of this paper discusses a wide class of wormhole solutions that meet these constraints. The type of shape function used is essentially generic. The constraints are then discussed in conjunction with various redshift functions. Violations of the weak energy condition and traversability criteria are also considered. The second part of the paper analyzes analogous time-dependent (dynamic) wormholes with the aid of differential forms. It is shown that a violation of the weak energy condition is not likely to be avoidable even temporarily. 
  We examine various well known exact solutions available in the literature to investigate the recent criterion obtained in ref. [20] which should be fulfilled by any static and spherically symmetric solution in the state of hydrostatic equilibrium. It is seen that this criterion is fulfilled only by (i) the regular solutions having a vanishing surface density together with the pressure, and (ii) the singular solutions corresponding to a non-vanishing density at the surface of the configuration . On the other hand, the regular solutions corresponding to a non-vanishing surface density do not fulfill this criterion. Based upon this investigation, we point out that the exterior Schwarzschild solution itself provides necessary conditions for the types of the density distributions to be considered inside the mass, in order to obtain exact solutions or equations of state compatible with the structure of general relativity. The regular solutions with finite centre and non-zero surface densities which do not fulfill the criterion [20], in fact, can not meet the requirement of the `actual mass' set up by exterior Schwarzschild solution. The only regular solution which could be possible in this regard is represented by uniform (homogeneous) density distribution.   The criterion [20] provides a necessary and sufficient condition for any static and spherical configuration (including core-envelope models) to be compatible with the structure of general relativity. Thus, it may find application to construct the appropriate core-envelope models of stellar objects like neutron stars and may be used to test various equations of state for dense nuclear matter and the models of relativistic stellar structures like star clusters. 
  In this article we discuss the analogy between superfluids and a spinning thick cosmic string. We use the geometrical approach to obtain the geometrical phases for a phonon in the presence of a vortex. We use loop variables for a geometric description of Aharonov-Bohm effect in these systems. We use holonomy transformations to characterize globally the "space-time" of a vortex and in this point of view we study the gravitational analog of the Aharonov-Bohm effect in this system. We demonstrate that in the general case the Aharonov-Bohm effect has a contribution both from the rotational and the translational holonomy. We study also Berrys quantum phase for phonons in this systems. 
  In braneworld models a variable vacuum energy may appear if the size of the extra dimension changes during the evolution of the universe. In this scenario the acceleration of the universe is related not only to the variation of the cosmological term, but also to the time evolution of $G$ and, possibly, to the variation of other fundamental "constants" as well. This is because the expansion rate of the extra dimension appears in different contexts, notably in expressions concerning the variation of rest mass and electric charge. We concentrate our attention on spatially-flat, homogeneous and isotropic, brane-universes where the matter density decreases as an inverse power of the scale factor, similar (but at different rate) to the power law in FRW-universes of general relativity.   We show that these braneworld cosmologies are consistent with the observed accelerating universe and other observational requirements. In particular, $G$ becomes constant and $\Lambda_{(4)} \approx const \times H^2$ asymptotically in time. Another important feature is that the models contain no "adjustable" parameters. All the quantities, even the five-dimensional ones, can be evaluated by means of measurements in 4D. We provide precise constrains on the cosmological parameters and demonstrate that the "effective" equation of state of the universe can, in principle, be determined by measurements of the deceleration parameter alone. We give an explicit expression relating the density parameters $\Omega_{\rho}$, $\Omega_{\Lambda}$ and the deceleration parameter $q$. These results constitute concrete predictions that may help in observations for an experimental/observational test of the model. 
  The gravitational force harbours a fundamental instability against collapse. In standard General Relativity without Quantum Mechanics, this implies the existence of black holes as natural, stable solutions of Einstein's equations. If one attempts to quantize the gravitational force, one should also consider the question how Quantum Mechanics affects the behaviour of black holes. In this lecture, we concentrate on the horizon. One would have expected that its properties could be derived from general coordinate transformations out of a vacuum state. In contrast, it appears that much new physics is needed. Much of that is still poorly understood, but one may speculate on the way information is organized at a horizon, and how refined versions of Quantum Theory may lead to answers. 
  This paper generalizes the static and spherically symmetric traversable wormhole geometry to a rotating axially symmetric one with a time-dependent angular velocity by means of an exact solution. It was found that the violation of the weak energy condition, although unavoidable, is considerably less severe than in the static spherically symmetric case. The radial tidal constraint is more easily met due to the rotation. Similar improvements are seen in one of the lateral tidal constraints. The magnitude of the angular velocity may have little effect on the weak energy condition violation for an axially symmetric wormhole. For a spherically symmetric one, however, the violation becomes less severe with increasing angular velocity. The time rate of change of the angular velocity, on the other hand, was found to have no effect at all. Finally, the angular velocity must depend only on the radial coordinate, confirming an earlier result. 
  An exact solution of the Einstein field equations is found under the assumption of spherically symmetry and the existence of one-parameter group of homothetic motions. This solution has a singularity at $r = 0$, and has non-vanishing expansion, acceleration and shear. Tidal forces in radial direction will not stretch an observer falling in this fluid. If the material is represented by perfect fluid, we can verify that the solution coincides with stiff matter case. In this case, the solution has zero expansion. Tidal forces in radial direction will not stretch an observer falling in this fluid and they not squeeze him in transverse directions. 
  In this work we study a charged particle in the presence of both a continuous distribution of disclinations and a continuous distribution of edge dislocations in the framework of the geometrical theory of defects. We obtain the self-energy for a single charge both in the internal and external regions of either distribution. For both distributions the result outside the defect distribution is the self-energy that a single charge experiments in the presence of a single defect. 
  We solve numerically equations of motion for real self-interacting scalar fields in the background of Reissner-Nordstrom black hole and obtained a sequence of static axisymmetric solutions representing thick domain walls charged black hole systems. In the case of extremal Reissner-Nordstrom black hole solution we find that there is a parameter depending on the black hole mass and the width of the domain wall which constitutes the upper limit for the expulsion to occur. 
  The Friedmann equation for a FRW-brane-world in a flat bulk is derived and applied to the accelerated expansion of the universe. 
  We show that the BMPV metric has Petrov type 22. This means that the BMPV metric is less algebraically special than the five-dimensional Schwarzschild metric, which has Petrov type \underline{22}. 
  We have directly observed broadband thermal noise in silica/tantala coatings in a high-sensitivity Fabry-Perot interferometer. Our result agrees well with the prediction based on indirect, ring-down measurements of coating mechanical loss, validating that method as a tool for the development of advanced interferometric gravitational-wave detectors. 
  The search for a quantum theory of gravity has followed two parallel but different paths. One aims at arriving at the final theory starting from a priori assumptions as to its form and building it from the ground up. The other tries to infer as much as possible about the unknown theory from the existing ones and use our current knowledge to constrain the possibilities for the quantum theory of gravity. Probably the biggest success of the second path has been the results of black hole thermodynamics. The subject of this essay is a new, highly promising such result, the application of quasinormal modes in quantum gravity. 
  In this paper, we study the rotational wormhole and scalar perturbation under the spacetime. We found the Schr\"odinger like equation and consider the asymptotic solutions for the special cases. 
  The one-loop effective potential for $\phi ^4$ theory on a Bianchi type-I universe is evaluated in the adiabatic approximation. It is used to see the quantum-field effects on symmetry breaking and restoration in anisotropic spacetimes. The results show that the fate of symmetry will not be changed in the cases of conformal coupling or a vanishing scale curvature, and only for some suitable values of scalar-gravitational coupling could the symmetry be radiatively broken or restored. 
  Levi-Civita spacetimes have both classical and quantum singularities. The relationship between the two is used here to study and clarify the physical aspects of the enigmatic Levi-Civita spacetimes. 
  We calculate the energy and momentum distributions associated with a G\"{o}del-type space-time, using the well-known energy-momentum complexes of Landau and Lifshitz and M{\o}ller. We show that the definitions of Landau and Lifshitz and M{\o}ller do not furnish a consistent result. 
  We will briefly review the definition and classification of classical and quantum singularities in general relativity. Examples of classically singular spacetimes that do not have quantum singularities will be given. We will present results on quantum singularities in quasiregular spacetimes. We will also show that a strong repulsive "potential" near the classical singularity can turn a classically singular spacetime into a quantum mechanically nonsingular spacetime. 
  We give three different spherically symmetric spacetimes for the coupled gravitational and electromagnetic fields with charged source in the tetrad theory of gravitation. One of these, contains an arbitrary function and generates the others. These spacetimes give the Reissner Nordstr$\ddot{o}$m metric black hole. We then, calculated the energy associated with these spacetimes using the superpotential method. We find that unless the time-space components of the tetrad field go to zero faster than ${1/\sqrt{r}}$ at infinity, one gets different results for the energy. 
  We use the $\zeta$-function regularization method to evaluate the finite temperature 1-loop effective potential for $\phi^4$ theory in the Godel spacetime. It is used to study the effects of temperature and curvature coupling on the cosmological phase transition in the rotational spacetime. From our results the critical temperature of symmetry restoration, which is a function of curvature coupling and magnitude of spacetime rotation, can be determined. 
  A common error, originally due to Ehrenfest, is corrected. 
  We study the helicity and chirality transitions of a high-energy neutrino propagating in a Schwarzschild space-time background. Using both traditional Schwarzschild and isotropic spherical co-ordinates, we derive an ultrarelativistic approximation of the Dirac Hamiltonian to first-order in the neutrino's rest mass, via a generalization of the Cini-Touschek transformation that incorporates non-inertial frame effects due to the noncommutative nature of the momentum states in curvilinear co-ordinates. Under general conditions, we show that neutrino's helicity is not a constant of the motion in the massless limit due to space-time curvature, while the chirality transition rate still retains an overall dependence on mass. We show that the chirality transition rate generally depends on the zeroth-order component of the neutrino's helicity transition rate under the Cini-Touschek transformation. It is also shown that the chiral current for high-energy neutrinos is altered by corrections due to curvature and frame-dependent effects, but should have no significant bearing on the chiral anomaly in curved space-time. We determine the upper bound for helicity and chirality transitions near the event horizon of a black hole. The special case of a weak-field approximation is also considered, which includes the gravitational analogue of Berry's phase first proposed by Cai and Papini. Finally, we propose a method for estimating the absolute neutrino mass and the number of right-handed chiral states from the expectation values of the helicity and chirality transition rates in the weak-field limit. 
  We study the radiation of energy and linear momentum emitted to infinity by the headon collision of binary black holes, starting from rest at a finite initial separation, in the extreme mass ratio limit. For these configurations we identify the radiation produced by the initially conformally flat choice of the three geometry. This identification suggests that the radiated energy and momentum of headon collisions will not be dominated by the details of the initial data for evolution of holes from initial proper separations $L_0\geq7M$. For non-headon orbits, where the amount of radiation is orders of magnitude larger, the conformally flat initial data may provide a relative even better approximation. 
  The one-loop renormalized effective potentials for the massive $\phi^4$ theory on the spatially homogeneous models of Bianchi type I and Kantowski-Sachs type are evaluated. It is used to see how the quantum field affects the cosmological phase transition in the anisotropic spacetimes. For reasons of the mathematical technique it is assumed that the spacetimes are slowly varying or have specially metric forms. We obtain the analytic results and present detailed discussions about the quantum field corrections to the symmetry breaking or symmetry restoration in the model spacetimes. 
  We study the collapse of a homogeneous braneworld dust cloud in the context of the various curvature correction scenarios, namely, the induced-gravity, the Gauss-Bonnet, and the combined induced-gravity and Gauss-Bonnet. In accordance to the Randall-Sundrum model, and contrary to four-dimensional general relativity, we show in all cases that the exterior spacetime on the brane is non-static. 
  Recent studies have shown that (a) quantum effects may be sufficient to support a wormhole throat and (b) the total amount of "exotic matter" can be made arbitrarily small. Unfortunately, using only small amounts of exotic matter may result in a wormhole that flares out too slowly to be traversable in a reasonable length of time. Combined with the Ford-Roman constraints, the wormhole may also come close to having an event horizon at the throat. This paper examines a model that overcomes these difficulties, while satisfying the usual traversability conditions. This model also confirms that the total amount of exotic matter can indeed be made arbitrarily small. 
  We consider some classes of solutions of the static, spherically symmetric gravitational field equations in the vacuum in the brane world scenario, in which our Universe is a three-brane embedded in a higher dimensional space-time. The vacuum field equations on the brane are reduced to a system of two ordinary differential equations, which describe all the geometric properties of the vacuum as functions of the dark pressure and dark radiation terms (the projections of the Weyl curvature of the bulk, generating non-local brane stresses). Several classes of exact solutions of the vacuum gravitational field equations on the brane are derived. In the particular case of a vanishing dark pressure the integration of the field equations can be reduced to the integration of an Abel type equation. A perturbative procedure, based on the iterative solution of an integral equation, is also developed for this case. Brane vacuums with particular symmetries are investigated by using Lie group techniques. In the case of a static vacuum brane admitting a one-parameter group of conformal motions the exact solution of the field equations can be found, with the functional form of the dark radiation and pressure terms uniquely fixed by the symmetry. The requirement of the invariance of the field equations with respect to the quasi-homologous group of transformations also imposes a unique, linear proportionality relation between the dark energy and dark pressure. A homology theorem for the static, spherically symmetric gravitational field equations in the vacuum on the brane is also proven. 
  In this letter we have considered the eigenvalues and eigenfunctions of relativistic massless scalar particle which conformally coupled to the background of Einstein universe. We found the eigenvalues and eigenfunctions exactly. 
  We consider a massless scalar field, conformally coupled to the Ricci scalar curvature, in the pre-inflation era of a closed FLRW Universe. The scalar field potential can be of the form of the Coleman-Weinberg one-loop potential, which is flat at the origin and drives the inflationary evolution. For positive values of the conformal parameter \xi, less than the critical value xi_c=(1/6), the model admits exact solutions with non-zero scale factor and zero initial Hubble parameter. Thus these solutions can be matched smoothly to the so called Pre-Big-Bang models. At the end of this pre-inflation era one can match inflationary solutions by specifying the form of the potential and the whole solution is of the class C^(1). 
  We compute for the first time very highly damped quasinormal modes of the (rotating) Kerr black hole. Our numerical technique is based on a decoupling of the radial and angular equations, performed using a large-frequency expansion for the angular separation constant_{s}A_{l m}. This allows us to go much further in overtone number than ever before. We find that the real part of the quasinormal frequencies approaches a non-zero constant value which does not depend on the spin s of the perturbing field and on the angular index l: \omega_R=m\varpi(a). We numerically compute \varpi(a). Leading-order corrections to the asymptotic frequency are likely to be of order 1/\omega_I. The imaginary part grows without bound, the spacing between consecutive modes being a monotonic function of a. 
  Regge calculus configuration superspace can be embedded into a more general superspace where the length of any edge is defined ambiguously depending on the 4-tetrahedron containing the edge. Moreover, the latter superspace can be extended further so that even edge lengths in each the 4-tetrahedron are not defined, only area tensors of the 2-faces in it are. We make use of our previous result concerning quantisation of the area tensor Regge calculus which gives finite expectation values for areas. Also our result is used showing that quantum measure in the Regge calculus can be uniquely fixed once we know quantum measure on (the space of the functionals on) the superspace of the theory with ambiguously defined edge lengths. We find that in this framework quantisation of the usual Regge calculus is defined up to a parameter. The theory may possess nonzero (of the order of Plank scale) or zero length expectation values depending on whether this parameter is larger or smaller than a certain value. Vanishing length expectation values means that the theory is becoming continuous, here {\it dynamically} in the originally discrete framework. 
  Working in the lagrangian framework, we develop a geometric theory in vacuum with propagating torsion; the antisymmetric and trace parts of the torsion tensor, considered as derived from local potential fields, are taken and, using the minimal action principle, their field equations are calculated. Actually these will show themselves to be just equations for propagating waves giving torsion a behavior similar to that of metric which, as known, propagates through gravitational waves. Then we establish a principle of minimal substitution to derive test particles equation of motion, obtaining, as result, that they move along autoparallels. We then calculate the analogous of the geodesic deviation for these trajectories and analyze their behavior in the nonrelativistic limit, showing that the torsion trace potential $\phi$ has a phenomenology which is indistinguishable from that of the gravitational newtonian field; in this way we also give a reason for why there have never been evidence for it. 
  We present a detailed discussion of the inflationary scenario in the context of inhomogeneous cosmologies. After a review of the fundamental features characterizing the inflationary model, as referred to a homogeneous and isotropic Universe, we develop a generalization in view of including small inhomogeneous corrections in the theory. A second step in our discussion is devoted to show that the inflationary scenario provides a valuable dynamical ``bridge'' between a generic Kasner-like regime and a homogeneous and isotropic Universe in the horizon scale. This result is achieved by solving the Hamilton-Jacobi equation for a Bianchi IX model in the presence of a cosmological space-dependent term. In this respect, we construct a quasi-isotropic inflationary solution based on the expansion of the Einstein equations up to first-two orders of approximation, in which the isotropy of the Universe is due to the dominance of the scalar field kinetic term; the first order of approximation corresponds to the inhomogeneous corrections and is driven by the matter evolution. We show how such a quasi-isotropic solution contains a certain freedom in fixing the space functions involved in the problem. The main physical issue of this analysis corresponds to outline the impossibility for the classical origin of density perturbations, due to the exponential decay of the matter term during the de Sitter phase. 
  This work is an extension of the study into statistical mechanics of the early Universe that has been the subject in prior works of the author, the principal approach being the density matrix deformation. In the work it is demonstrated that the previously derived exponential ansatz may be successfully applied to the derivation of the free and average energy deformation as well as entropy deformation. Based on the exponential ansatz, the derivation of a statistical-mechanical Liouville equation as a deformation of the quantum-mechanical counterpart is presented. It is shown that deformed Liouville equation will possess nontrivial components as compared to the normal equation in two cases: for the original singularity (i.e. early Universe) and for black hole, that is in complete agreement with the results obtained by the author with coworkers in earlier works devoted to the deformation in quantum mechanics at Planck scale. In conclusion some possible applications of the proposed methods are given, specifically for investigation into thermodynamics of black holes. 
  We introduce a canonical method for pair production by electromagnetic fields. The canonical method in the space-dependent gauge provides pair-production rate even for inhomogeneous fields. Further, the instanton action including all corrections leads to an accurate formula for the pair-production rate. We discuss various aspects of the canonical method and clarify terminology for pair production. We study pair production by charged black holes first by finding states of the field equation that describe pair production and then by applying the canonical method. 
  This paper is twofold. First of all a complete unified picture of $n$-dimensional quantum gravity is proposed in the following sense: In spin foam models of quantum gravity the evaluation of spin networks play a very important role. These evaluations correspond to amplitudes which contribute in a state sum model of quantum gravity. In \cite{fk}, the evaluation of spin networks as integrals over internal spaces was described. This evaluation was restricted to evaluations of spin networks in $n$-dimensional Euclidean quantum gravity. Here we propose that a similar method can be considered to include Lorentzian quantum gravity. We therefore describe the the evaluation of spin networks in the Lorentzian framework of spin foam models. We also include a limit of the Euclidean and Lorentzian spin foam models which we call Newtonian. This Newtonian limit was also considered in \cite{jm}.   Secondly, we propose an alternative formulation of spin foam models of quantum gravity with its corresponding evaluation of spin networks. This alternative formulation is a non-archimedean or $p$-adic spin foam model. The interest on this description is that it is based on a discrete space-time, which is the expected situation we might have at the Planck length; this description might lead us to an alternative regularisation of quantum gravity. Moreover a non-commutative formulation follows from the non-archimedean one. 
  A third post-Newtonian (3 PN) equation of motion for two spherical compact stars in a harmonic coordinate has been derived based on the surface integral approach and the strong field point particle limit. The strong field point particle limit enables us to incorporate a notion of a self-gravitating regular star into general relativity. The resulting 3 PN equation of motion is Lorentz invariant, unambiguous, and conserves an energy of the binary orbital motion. 
  The evolution of a Universe confined onto a 3-brane embedded in a five-dimensional space-time is investigated where the cosmological fluid on the brane is modeled by the van der Waals equation of state. It is shown that the Universe on the brane evolves in such a manner that three distinct periods concerning its acceleration field are attained: (a) an initial accelerated epoch where the van der Waals fluid behaves like a scalar field with a negative pressure; (b) a past decelerated period which has two contributions, one of them is related to the van der Waals fluid which behaves like a matter field with a positive pressure, whereas the other contribution comes from a term of the Friedmann equation on the brane which is inversely proportional to the scale factor to the fourth power and can be interpreted as a radiation field, and (c) a present accelerated phase due to a cosmological constant on the brane. 
  We investigate matter symmetries of cylindrically symmetric static spacetimes. These are classified for both cases when the energy-momentum tensor is non-degenerate and also when it is degenerate. It is found that the non-degenerate energy-momentum tensor gives either three, four, five, six, seven or ten independent matter collineations in which three are isometries and the rest are proper. The worth mentioning cases are those where we obtain the group of matter collineations finite-dimensional even the energy-momentum tensor is degenerate. These are either three, four, five or ten. Some examples are constructed satisfying the constraints on the energy-momentum tensor. 
  Intersecting hypersurfaces in classical Lovelock gravity were studied in [hep-th/0306220], exploiting the description of the Lovelock Lagrangian as a sum of dimensionally continued Euler densities. We wish to simplify and demystify the calculations, providing an interesting geometrical interpretation. This analysis allows us to deal most efficiently with the division of space-time into a honeycomb network of cells which one might expect from membranes of matter. We exploit a kind of duality between an intersection and a simplex or simplicial complex in the space of Homotopy parameters. This approach is valid for Euler (and Pontryagin) densities but also for a dimensionally continued Euler density. As an implication, in the nth order Lovelock gravity, surfaces up to co-dimension $n$ naturally carry localised matter. 
  This paper discusses new fundamental physics experiment to test relativistic gravity at the accuracy better than the effects of the 2nd order in the gravitational field strength. The Laser Astrometric Test Of Relativity (LATOR) mission uses laser interferometry between two micro-spacecraft whose lines of sight pass close by the Sun to accurately measure deflection of light in the solar gravity. The key element of the experimental design is a redundant geometry optical truss provided by a long-baseline (100 m) multi-channel stellar optical interferometer placed on the International Space Station. The geometric redundancy enables LATOR to measure the departure from Euclidean geometry caused by the solar gravity field to a very high accuracy. LATOR will not only improve the value of the parameterized post-Newtonian (PPN) parameter gamma to unprecedented levels of accuracy of 1 part in 1e8, it will also reach ability to measure effects of the next post-Newtonian order (1/c^4) of light deflection resulting from gravity's intrinsic non-linearity. The solar quadrupole moment parameter, J2, will be measured with high precision, as well as a variety of other relativistic. LATOR will lead to very robust advances in the tests of fundamental physics: this mission could discover a violation or extension of general relativity, or reveal the presence of an additional long range interaction in the physical law. There are no analogs to the LATOR experiment; it is unique and is a natural culmination of solar system gravity experiments. 
  We have solved cosmological gravitational Wave(GW)equation in the frame work of Generalised Brans-Dicke(GBD) theory for all epochs of the Universe.The solutions are expressed in terms of the present value of the Brans-Dicke coupling parameter $\omega(\phi)$.It is seen that the solutions represent travelling growing modes for negative values of $\omega_{0}$ for all epochs of the Universe. 
  The time equation associated to the Dirac Equation (DE) is studied for the radiation-dominated Friedmann-Robertson-Walker (FRW) universe. The results are analysed for small and large values of time. We also incorporate the corrections of the paper studied by Zecca [1] for the matter-dominated FRW universe. 
  Some mathematical aspects of using the translation group as an internal symmetry group in a gauge field theory are presented and discussed. The traditional manner in which gravitation can be accounted for by the introduction of a global frame field on a parallelizable spacetime is reviewed. It is then discussed in the more general context of a global frame field on the bundle of linear frames. In the process, the elements of variational field theory for physical fields defined on G-structures are set down. It is suggested that it is probably more proper to attribute gravitation to a reduction of the bundle of linear frames to {e} -- at least over a generic submanifold of spacetime -- than to a reduction to the Lorentz group since the Lorentz group is more intrinsic to electromagnetism and gravitation has the character of a "residual" symmetry of spacetime at the astrophysical level. 
  Nonstandard q-deformed algebras U'_q(so_n), proposed a decade ago for the needs of representation theory, essentially differ from the standard Drinfeld-Jimbo quantum deformation of the algebras U(so_n) and possess with regard to the latter a number of important advantages. We discuss possible application of the q-algebras U'_q(so_n), within two different contexts of quantum/q-deformed gravity: one concerns q-deforming of D-dimensional (D >= 3) euclidean gravity, the other applies to 2+1 anti-De Sitter quantum gravity (with space surface of genus g) in the approach of Nelson and Regge. 
  Spacetimes in which the electric part of the Weyl tensor vanishes (relative to some timelike unit vector field) are said to be purely magnetic. Examples of purely magnetic spacetimes are known and are relatively easy to construct, if no restrictions are placed on the energy-momentum tensor. However it has long been conjectured that purely magnetic vacuum spacetimes (with or without a cosmological constant) do not exist. The history of this conjecture is reviewed and some advances made in the last year are described briefly. A generalisation of this conjecture first suggested for type D vacuum spacetimes by Ferrando and Saez is stated and proved in a number of special cases. Finally an approach to a general proof of the conjecture is described using the Newman-Penrose formalism based on a canonical null tetrad of the Weyl tensor. 
  The behavior near the initial singular state of the anisotropy parameter of the arbitrary type, homogeneous and anisotropic Bianchi models is considered in the framework of the brane world cosmological models. The matter content on the brane is assumed to be an isotropic perfect cosmological fluid, obeying a barotropic equation of state. To obtain the value of the anisotropy parameter at an arbitrary moment an evolution equation is derived, describing the dynamics of the anisotropy as a function of the volume scale factor of the Universe. The general solution of this equation can be obtained in an exact analytical form for the Bianchi I and V types and in a closed form for all other homogeneous and anisotropic geometries. The study of the values of the anisotropy in the limit of small times shows that for all Bianchi type space-times filled with a non-zero pressure cosmological fluid, obeying a linear barotropic equation of state, the initial singular state on the brane is isotropic. This result is obtained by assuming that in the limit of small times the asymptotic behavior of the scale factors is of Kasner-type. For brane worlds filled with dust, the initial values of the anisotropy coincide in both brane world and standard four-dimensional general relativistic cosmologies. 
  Various approaches to black hole entropy yield the area law with logarithmic corrections, many involving a coefficient 1/2, and some involving 3/2. It is pointed out here that the standard quantum geometry formalism is not consistent with 3/2 and favours 1/2. 
  We study the vacuum C-metric and its physical interpretation in terms of the exterior spacetime of a uniformly accelerating spherically - symmetric gravitational source. Wave phenomena on the linearized C-metric background are investigated. It is shown that the scalar perturbations of the linearized C-metric correspond to the gravitational Stark effect. This effect is studied in connection with the Pioneer anomaly. 
  A new approach has been used to evaluate the momentum and angular momentum of the isotropic and homogeneous cosmological models. It is shown that the results obtained for momentum exactly coincide with those already available in the literature. However, the angular momentum expression coincides only for the closed Friedmann model. 
  We take an axisymmetric rotating universe model by crossing with a time dependent factor and evaluate its force and momentum in this evolving universe. It is concluded that it behaves exactly like a Friedmann model. We also extend this conclusion to the most general cosmological model. 
  Two models are given by crossing the Friedmann metrics with Schwarzschild and Kerr metrics. In these evolving universes with a gravitational source, the force four-vector and the corresponding potentials are evaluated. 
  We derive matter collineations for some static spherically symmetric spacetimes and compare the results with Killing, Ricci and Curvature symmetries. We conclude that matter and Ricci collineations are not, in general, the same. 
  We present a numerical code designed to study astrophysical phenomena involving dynamical spacetimes containing black holes in the presence of relativistic hydrodynamic matter. We present evolutions of the collapse of a fluid star from the onset of collapse to the settling of the resulting black hole to a final stationary state. In order to evolve stably after the black hole forms, we excise a region inside the hole before a singularity is encountered. This excision region is introduced after the appearance of an apparent horizon, but while a significant amount of matter remains outside the hole. We test our code by evolving accurately a vacuum Schwarzschild black hole, a relativistic Bondi accretion flow onto a black hole, Oppenheimer-Snyder dust collapse, and the collapse of nonrotating and rotating stars. These systems are tracked reliably for hundreds of M following excision, where M is the mass of the black hole. We perform these tests both in axisymmetry and in full 3+1 dimensions. We then apply our code to study the effect of the stellar spin parameter J/M^2 on the final outcome of gravitational collapse of rapidly rotating n = 1 polytropes. We find that a black hole forms only if J/M^2<1, in agreement with previous simulations. When J/M^2>1, the collapsing star forms a torus which fragments into nonaxisymmetric clumps, capable of generating appreciable ``splash'' gravitational radiation. 
  In this paper, we assume that the observer is fixed in a comoving frame of reference with $g_{00}=\frac{\lambda}{\Lambda}$, where $\lambda$ and $\Lambda$ denote the comoving parameter and the cosmological constant, respectively. By using the {\it comoving suppression mechanism} and {\it Mach's principle} (the latter of which is used to determine the comoving parameter $\lambda$), we calculate the vacuum energy density of quantum fluctuation field in the above-mentioned comoving frame of reference. It is shown that in such a comoving frame of reference, the cosmological constant will greatly decrease by many orders of magnitude (if Mach's principle is applied to this calculation, then it will be shown that $\Lambda$ is reduced by about 120 orders of magnitude). Additionally, we briefly discuss the related topics such as the varying observed speed of light ($\frac{{\rm d}c}{{\rm d}t}={\mathcal O}(10^{-9}{\rm m/s}^{2})$) and the mystery of anomalous acceleration ($\sim 10^{-9}{\rm m/s}^{2}$) acquired by the Pioneer 10/11, Galileo and Ulysses spacecrafts. 
  In an attempt to clarify what is the velocity of a particle in doubly special relativity, we solve Maxwell's equations invariant under the position-space nonlinear Lorentz transformation proposed by Kimberly, Magueijo, and Medeiros. We show that only the amplitude of the Maxwell wave, not the phase, is affected by the nonlinearity of the transformation. Thus, although the Maxwell wave appears to have infinitely large energy near the Planck time, the wave velocity is the same as the conventional light velocity. Surprisingly, the velocity of the Maxwell wave is not the same as the maximum signal velocity determined by the null geodesic condition, which is infinitely large near the Planck time and monotonically decreases in time to the conventional light velocity when time approaches infinity. This implies that, depending on the position of the particle in question, the light cone determined by Maxwell's equations may be inside or outside the null cone determined by the null geodesic equation, which may lead to the causality problem. 
  We investigate solutions of Einstein field equations for the non-static spherically symmetric perfect fluid case using different equations of state. The properties of an exact spherically symmetric perfect fluid solutions are obtained which contain shear. We obtain three different solutions out of these one turns out to be an incoherent dust solution and the other two are stiff matter solutions. 
  We consider radiation-dominated Friedmann universe and evaluate its force four-vector and momentum. We analyse and compare the results with the already evaluated for the matter-dominated Friedmann model. It turns out that the results are physically acceptable. 
  We show that the recent work of Lee [23] implies existence of a large class of new singularity-free strictly static Lorentzian vacuum solutions of the Einstein equations with a negative cosmological constant. This holds in all space-time dimensions greater than or equal to four, and leads both to strictly static solutions and to black hole solutions. The construction allows in principle for metrics (whether black hole or not) with Yang-Mills-dilaton fields interacting with gravity through a Kaluza-Klein coupling. 
  Quantum gravitational effects may induce stochastic fluctuations in the structure of space-time, to produce a characteristic foamy structure. It has been known for some time now that these fluctuations may have observable consequences for the propagation of cosmic ray particles over cosmological distances. While invoked as a possible explanation for the detection of the puzzling cosmic rays with energies in excess of the threshold for photopion production (the so-called super-GZK particles), we demonstrate here that lower energy observations may provide strong constraints on the role of a fluctuating space-time structure. We note also that the same fluctuations, if they exist, imply that some decay reactions normally forbidden by elementary conservation laws, become kinematically allowed, inducing the decay of particles that are seen to be stable in our universe. Due to the strength of the prediction, we are led to consider this finding as the most severe constraint on the classes of models that may describe the effects of gravity on the structure of space-time. We also propose and discuss several potential loopholes of our approach, that may affect our conclusions. In particular, we try to identify the situations in which despite a fluctuating energy-momentum of the particles, the reactions mentioned above may not take place. 
  Applying the Darmois-Israel thin shell formalism, we construct static and dynamic thin shells around traversable wormholes. Firstly, by applying the cut-and-paste technique we apply a linearized stability analysis to thin-shell wormholes in the presence of a generic cosmological constant. We find that for large positive values of the cosmological constant, i.e., the Schwarzschild-de Sitter solution, the regions of stability significantly increase relatively to the Schwarzschild case, analyzed by Poisson and Visser. Secondly, we construct static thin shell solutions by matching an interior wormhole solution to a vacuum exterior solution at a junction surface. In the spirit of minimizing the usage of exotic matter we analyze the domains in which the weak and null energy conditions are satisfied at the junction surface. The characteristics and several physical properties of the surface stresses are explored, namely, we determine regions where the sign of tangential surface pressure is positive and negative (surface tension). An equation governing the behavior of the radial pressure across the junction surface is deduced. Specific dimensions of the wormhole, namely, the throat radius and the junction interface radius, are found by taking into account the traversability conditions, and estimates for the traversal time and velocity are also determined. 
  The status of canonical reduction for metric and tetrad gravity in space-times of the Christodoulou-Klainermann type, where the ADM energy rules the time evolution, is reviewed. Since in these space-times there is an asymptotic Minkowski metric at spatial infinity, it is possible to define a Hamiltonian linearization in a completely fixed (non harmonic) 3-orthogonal gauge without introducing a background metric. Post-Minkowskian background-independent gravitational waves are obtained as solutions of the linearized Hamilton equations. 
  In $D$-dimentional gravity on arbitrary curved backgrounds using proven methods conserved currents, divergences of antisymmetrical tensor densities (superpotentials), are constructed. These superpotentials have two remarkable properties: they depend in an essential way on second derivatives in the Lagrangian and are independent on divergences added to it. The conserved currents are thus particulary well adapted to the case of perturbations in Gauss-Bonett cosmological brane theories. 
  We analyze the Bianchi IX dynamics (Mixmaster) in view of its stochastic properties; in the present paper we address either the original approach due to Belinski, Khalatnikov and Lifshitz (BKL) as well as a Hamiltonian one relying on the Arnowitt--Deser--Misner (ADM) reduction. We compare these two frameworks and show how the BKL map is related to the geodesic flow associated with the ADM dynamics. In particular, the link existing between the \textit{anisotropy parameters} and the \textit{Kasner indices} is outlined. 
  We give an introductory account to the renormalization of models without metric background. We sketch the application to certain discrete models of quantum gravity such as spin foam models. 
  We present a first attempt to apply the approach of deformation quantization to linearized Einstein's equations. We use the analogy with Maxwell equations to derive the field equations of linearized gravity from a modified Maxwell Lagrangian which allows the construction of a Hamiltonian in the standard way. The deformation quantization procedure for free fields is applied to this Hamiltonian. As a result we obtain the complete set of quantum states and its discrete spectrum. 
  This article is a continuation of a previous work that dealt with the topological obstructions to the reductions of the bundle of linear frames on a spacetime manifold for a particular chain of subgroups of GL(4). In this article, the corresponding geometrical information, such as connections, torsion, curvature, and automorphisms of the reductions will be discussed. The details are elaborated upon for a certain sequence of reductions of GL(M) when M is a four-dimensional spacetime manifold. 
  A theory has been presented previously in which the geometrical structure of a real four-dimensional space time manifold is expressed by a real orthonormal tetrad, and the group of diffeomorphisms is replaced by a larger group. The group enlargement was accomplished by including those transformations to anholonomic coordinates under which conservation laws are covariant statements. Field equations have been obtained from a variational principle which is invariant under the larger group. These field equations imply the validity of the Einstein equations of general relativity with a stress-energy tensor that is just what one expects for the electroweak field and associated currents. In this paper, as a first step toward quantization, a consistent Hamiltonian for the theory is obtained. Some concluding remarks are given concerning the need for further development of the theory. These remarks include discussion of a possible method for extending the theory to include the strong interaction. 
  The possibility of performing post-Newtonian gravitoelectromagnetic measurements with a joint LAGEOS-LAGEOS II-OPTIS space-based mission is investigated 
  In numerical computations of Einstein's equations for black hole spacetimes, it will be necessary to use approximate boundary conditions at a finite distance from the holes. We point out here that ``tails,'' the inverse power-law decrease of late-time fields, cannot be expected for such computations. We present computational demonstrations and discussions of features of late-time behavior in an evolution with a boundary condition. 
  In this paper we report some results on the expectation values of a set of observables introduced for 3-dimensional Riemannian quantum gravity with positive cosmological constant, that is, observables in the Turaev-Viro model. Instead of giving a formal description of the observables, we just formulate the paper by examples. This means that we just show how an idea works with particular cases and give a way to compute 'expectation values' in general by a topological procedure. 
  We study the scalar and spinor perturbation to Kerr-NUT space-time, that is, Klein-Gordan and Dirac equation therein. The equations are invariant under duality transformation between the gravitational electric (M) and magnetic (l) charge, radial and angular coordinate, and radial and angular component of the field. We solve the equations separating into radial and angular parts. Moreover, if sets of Klein-Gordan and Dirac equation and corresponding solutions are known for Kerr space-time, under duality transformation, those in dual Kerr space-time are shown to be achieved. A few examples of solution are shown. We comment about the horizon and singularity conditions. 
  Propagation of fermions in curved space-time generates a gravitational interaction due to the coupling between spin of the fermion and space-time curvature. This gravitational interaction, which is an axial-vector appears as the CPT violating term in the Lagrangian, can generate the neutrino asymmetry in Universe. If the back-ground metric is spherically asymmetric, say, of rotating black hole, the axisymmetric space-time of an expanding Universe, this interaction as well as the neutrino asymmetry is non-zero. 
  The asymmetry in the time delay for light rays propagating on opposite sides of a spinning body is analyzed. A frequency shift in the perceived signals is found. A practical procedure is proposed for evidencing the asymmetry, allowing for a measurement of the specific angular momentum of the rotating mass. Orders of magnitude are discussed. 
  Some kinematical speculations on the infinite curvature limit of the conjectured duality of Maldacena between ten-dimensional strings living in $AdS_5\times S_5$ and a ordinary four-dimensional quantum field theory, namely ${\cal{N}}=4$ super Yang-Mills with gauge group SU(N) are given. 
  We perform an analytic late time analysis for maximal slicing of the Reissner-Nordstr\"om black hole spacetime. In particular, we discuss the collapse of the lapse in terms of its late time behavior at the throat and at the event horizon for the even and the puncture lapse. In the latter case we also determine the value of the lapse at the puncture. Furthermore, in the limit of late times slice stretching effects are studied as they arise for maximal slicing of puncture evolutions. We perform numerical experiments for a Schwarzschild black hole with puncture lapse and find agreement with the analytical results. 
  Various interpretations of the Riemann Curvature Tensor, Ricci Tensor, and Scalar Curvature are described. Also, the physical meanings of the Einstein Tensor and Einstein's Equations are discussed. Finally a derivation of Newtonian Gravity from Einstein's Equations is given. 
  We study the boundary value problem for the stationary rotating black hole solutions to the five-dimensional vacuum Einstein equation. Assuming the two commuting rotational symmetry and the sphericity of the horizon topology, we show that the black hole is uniquely characterized by the mass, and a pair of the angular momenta. 
  The double torus provides a relativistic model for a closed 2D cosmos with topology of genus 2 and constant negative curvature. Its unfolding into an octagon extends to an octagonal tessellation of its universal covering, the hyperbolic space H^2. The tessellation is analysed with tools from hyperbolic crystallography. Actions on H^2 of groups/subgroups are identified for SU(1, 1), for a hyperbolic Coxeter group acting also on SU(1, 1), and for the homotopy group \Phi_2 whose extension is normal in the Coxeter group. Closed geodesics arise from links on H^2 between octagon centres. The direction and length of the shortest closed geodesics is computed. 
  This work concerns a new reformulation of quantum geometrodynamics, which allows to overcome a fundamental ambiguity contained in the canonical approach to quantum gravity: the possibility of performing a (3+1)-slicing of space-time, when the metric tensor is in a quantum regime. Our formulation provides also a procedure to solve the problems connected to the so called "frozen formalism". In particular we fix the reference frame (i.e. the lapse function and the shift vector) by introducing the so called "kinematical action"; as a consequence, the new hamiltonian constraints become parabolic, so arriving to evolutive (Schroedinger-like) equations for the quantum dynamics. The kinematical action can be interpreted as the action of a pressureless, but, in general, non geodesic perfect fluid, so in the semi classical limit our theory leads to the dynamics of the gravitational field coupled to a dust which represents the material reference frame we have introduced fixing the slicing. We also investigate the cosmological implications of the presence of the dust, which, in the WKB limit of a cosmological problem, makes account for a "dark matter" component and could play, at present time, a dynamical role. 
  It has been suggested that space-time might undergo fluctuations because of its intrinsic quantum nature. These fluctuations would pose a fundamental limit to the ability of measuring distances with arbitrary precision, beyond any limitations due to standard quantum mechanics. Laser interferometers have recently been proposed as being suited for a search for the existence of space-time fluctuations. Here we present results of a search for space-time fluctuations of very low fluctuation frequencies, in the range from 1 microHz to 0.5 Hz. Rigid optical interferometers made out of sapphire and operated at cryogenic temperature were used. We find an upper limit of 1.10^-24 Hz^-1 for the normalized distance noise spectral density at 6 microHz, and of 1.10^-28 Hz^-1 above 5 mHz, and establish an experimental limit for the parameter of a recently proposed random-walk hypothesis. 
  We present a new numerical scheme for solving the initial value problem for quasiequilibrium binary neutron stars allowing for arbitrary spins. We construct sequences of circular-orbit binaries of varying separation, keeping the rest mass and circulation constant along each sequence. The spin angular frequency of the stars is shown to vary along the sequence, a result that can be derived analytically in the PPN limit. This spin effect, in addition to leaving an imprint on the gravitational waveform emitted during binary inspiral, is measurable in the electromagnetic signal if one of the stars is a pulsar visible from Earth. 
  Taking a hint from Dirac's large number hypothesis, we note the existence of cosmic combined conservation laws that work to cosmologically long time. We thus modify or generalize Einstein's theory of general relativity with fixed gravitation constant $G$ to a theory for varying $G$, which can be applied to cosmology without inconsistency, where a tensor arising from the variation of G takes the place of the cosmological constant term. We then develop on this basis a systematic theory of evolving natural constants $m_{e},m_{p},e,\hslash ,k_{B}$ by finding out their cosmic combined counterparts involving factors of appropriate powers of $G$ that remain truly constant to cosmologically long time. As $G$ varies so little in recent centuries, so we take these natural constants to be constant. 
  Gravitational waves in isotropic cosmologies were recently studied using the gauge-invariant approach of Ellis-Bruni. We now construct the linearised metric perturbations of the background Robertson-Walker space-time which reproduce the results obtained in that study. The analysis carried out here also facilitates an easy comparison with Bardeen. 
  Spin networks are graphs derived from 3nj symbols of angular momentum. The surface embedding, the topology and dualization of these networks are considered. Embeddings into compact surfaces include the orientable sphere S^2 and the torus T, and the not orientable projective space P^2 and Klein's bottle K. Two families of 3nj graphs admit embeddings of minimal genus into S^2 and P^2. Their dual 2-skeletons are shown to be triangulations of these surfaces. 
  The LISA time-delay--interferometry responses to a gravitational wave signal are rewritten in a form that accounts for the motion of the LISA constellation around the Sun; the responses are given in closed analytic forms valid for any frequency in the band accessible to LISA. We present a complete procedure, based on the principle of maximum likelihood, to search for stellar-mass binary systems in the LISA data. We define the required optimal filters, the amplitude-maximized detection statistic (analog to the F statistic used in pulsar searches with ground-based interferometers), and discuss the false-alarm and detection probabilities. Last, we test the procedure in numerical simulations of gravitational-wave detection. 
  We present an interpretation of loop quantization in the framework of lattice gauge theory. Within this context the lack of appropriate notions of effective theories and renormalization group flow exhibit loop quantization as an incomplete framework. This interpretation includes a construction of embedded spin foam models which does not rely on the choice of any auxiliary structure (e.g. triangulation) and has the following straightforward consequences: (1) The values of the coupling constants need to be those of an UV-attractive fixed point (2) The kinematics of canonical loop quantization and embedded spin foam models are compatible (3) The weights assigned to embedded spin foams are independent of the 2-polyhedron used to regularize the path integral, $|J|_x = |J|_{x'}$ (4) An area spectrum with edge contributions proportional to $l_{\rm PL}^2 (j+1 / 2)$ is not compatible with embedded spin foam models and/or canonical loop quantization 
  We compare two area spectra proposed in loop quantum gravity in different approaches to compute the entropy of the Schwarzschild black hole. We describe the black hole in general microcanonical and canonical area ensembles for these spectra. We show that in the canonical ensemble, the results for all statistical quantities for any spectrum can be reproduced by a heuristic picture of Bekenstein up to second order. For one of these spectra - the equally-spaced spectrum - in light of a proposed connection of the black hole area spectrum to the quasinormal mode spectrum and following hep-th/0304135, we present explicit calculations to argue that this spectrum is completely consistent with this connection. This follows without requiring a change in the gauge group of the spin degrees of freedom in this formalism from SU(2) to SO(3). We also show that independent of the area spectrum, the degeneracy of the area observable is bounded by $C A\exp(A/4)$, where $A$ is measured in Planck units and $C$ is a constant of order unity. 
  New coordinates are given which describe non-degenerate Kerr black holes in dual-null foliations based on the outer (or inner) horizons, generalizing the Kruskal form for Schwarzschild black holes. The construction involves an area radius for the transverse surfaces and a generalization of the Regge-Wheeler radial function, both functions of the original radial coordinate only. 
  The folk questions in Lorentzian Geometry, which concerns the smoothness of time functions and slicings by Cauchy hypersurfaces, are solved by giving simple proofs of: (a) any globally hyperbolic spacetime $(M,g)$ admits a smooth time function $\tau$ whose levels are spacelike Cauchy hyperfurfaces and, thus, also a smooth global splitting $M= \R \times {\cal S}$, $g= - \beta(\tau,x) d\tau^2 + \bar g_\tau $, (b) if a spacetime $M$ admits a (continuous) time function $t$ (i.e., it is stably causal) then it admits a smooth (time) function $\tau$ with timelike gradient $\nabla \tau$ on all $M$. 
  The lack of regularity of geometric variables at the origin is often a source of serious problem for spherically symmetric evolution codes in numerical relativity. One usually deals with this by restricting the gauge and solving the hamiltonian constraint for the metric. Here we present a generic algorithm for dealing with the regularization of the origin that can be used directly on the evolution equations and that allows very general gauge choices. Our approach is similar in spirit to the one introduced by Arbona and Bona for the particular case of the Bona-Masso formulation. However, our algorithm is more general and can be used with a wide variety of evolution systems. 
  To every axi-symmetric isolated horizon we associate two sets of numbers, $M_n$ and $J_n$ with $n = 0, 1, 2, ...$, representing its mass and angular momentum multipoles. They provide a diffeomorphism invariant characterization of the horizon geometry. Physically, they can be thought of as the `source multipoles' of black holes in equilibrium. These structures have a variety of potential applications ranging from equations of motion of black holes and numerical relativity to quantum gravity. 
  We present an equation of state for elastic matter which allows for purely longitudinal elastic waves in all propagation directions, not just principal directions. The speed of these waves is equal to the speed of light whereas the transversal type speeds are also very high, comparable to but always strictly less than that of light. Clearly such an equation of state does not give a reasonable matter description for the crust of a neutron star, but it does provide a nice causal toy model for an extremely rigid phase in a neutron star core, should such a phase exist. Another reason for focusing on this particular equation of state is simply that it leads to a very simple recipe for finding stationary rigid motion exact solutions to the Einstein equations. In fact, we show that a very large class of stationary spacetimes with constant Ricci scalar can be interpreted as rigid motion solutions with this matter source. We use the recipe to derive a static spherically symmetric exact solution with constant energy density, regular centre and finite radius, having a nontrivial parameter that can be varied to yield a mass-radius curve from which stability can be read off. It turns out that the solution is stable down to a tenuity R/M slightly less than 3. The result of this static approach to stability is confirmed by a numerical determination of the fundamental radial oscillation mode frequency. We also present another solution with outwards decreasing energy density. Unfortunately, this solution only has a trivial scaling parameter and is found to be unstable. 
  Unruh effect states that the vacuum of a quantum field theory on Minkovski space-time looks like a thermal state for an eternal uniformly accelerated observer. Adaptation to the non eternal case causes a serious problem: if the thermalization of the vacuum depends on the lifetime of the observer, then in principle the latest is able to deduce its lifetime from the measurement of the temperature. This short note aims at underlining that time-energy uncertainty relation allows to adapt Unruh effect to non-eternal observers without breaking causality. In particular we show that our adaptation - the diamonds's temperature- of Bisognano-Wichman approach to Unruh effect is causally acceptable. This note is self-contained but it is fully meaningful as a complement to gr-qc/0212074 as well as a comment on gr-qc/0306022. 
  We use the idea of the symmetry between the spacetime coordinates x^\mu and the energy-momentum p^\mu in quantum theory to construct a momentum space quantum gravity geometry with a metric s_{\mu\nu} and a curvature P^\lambda_{\mu\nu\rho}. For a closed maximally symmetric momentum space with a constant 3-curvature, the volume of the p-space admits a cutoff with an invariant maximum momentum a. A Wheeler-DeWitt-type wave equation is obtained in the momentum space representation. The vacuum energy density and the self-energy of a charged particle are shown to be finite, and modifications of the electromagnetic radiation density and the entropy density of a system of particles occur for high frequencies. 
  We reduce Boyer-Finley equation to a family of compatible systems of hydrodynamic type, with characteristic speeds expressed in terms of spaces of rational functions. The systems of hydrodynamic type are then solved by the generalized hodograph method, providing solutions of the Boyer-Finley equation including functional parameters. 
  The quantum noise of the light field is a fundamental noise source in interferometric gravitational wave detectors. Injected squeezed light is capable of reducing the quantum noise contribution to the detector noise floor to values that surpass the so-called Standard-Quantum-Limit (SQL). In particular, squeezed light is useful for the detection of gravitational waves at high frequencies where interferometers are typically shot-noise limited, although the SQL might not be beaten in this case. We theoretically analyze the quantum noise of the signal-recycled laser interferometric gravitational-wave detector GEO600 with additional input and output optics, namely frequency-dependent squeezing of the vacuum state of light entering the dark port and frequency-dependent homodyne detection. We focus on the frequency range between 1 kHz and 10 kHz, where, although signal recycled, the detector is still shot-noise limited. It is found that the GEO600 detector with present design parameters will benefit from frequency dependent squeezed light. Assuming a squeezing strength of -6 dB in quantum noise variance, the interferometer will become thermal noise limited up to 4 kHz without further reduction of bandwidth. At higher frequencies the linear noise spectral density of GEO600 will still be dominated by shot-noise and improved by a factor of 10^{6dB/20dB}~2 according to the squeezing strength assumed. The interferometer might reach a strain sensitivity of 6x10^{-23} above 1 kHz (tunable) with a bandwidth of around 350 Hz. We propose a scheme to implement the desired frequency dependent squeezing by introducing an additional optical component to GEO600s signal-recycling cavity. 
  A special subclass of shear-free null congruences (SFC) is studied, with tangent vector field being a repeated principal null direction of the Weyl tensor. We demonstrate that this field is parallel with respect to an effective affine connection which contains the Weyl nonmetricity and the skew symmetric torsion. On the other hand, a Maxwell-like field can be directly associated with any special SFC, and the electric charge for bounded singularities of this field turns to be ``self-quantized''. Two invariant differential operators are introduced which can be thought of as spinor analogues of the Beltrami operators and both nullify the principal spinor of any special SFC. 
  Two aspects of the widely accepted heuristic picture of the final state of gravitational collapse are the so-called Price law tails, describing the asymptotics of the exterior region of the black hole that forms, and Israel-Poisson's mass inflation scenario, describing the internal structure of the black hole. (The latter scenario, if valid, would indicate that the maximal development of initial data is extendible as a C^0 metric, putting into question the validity of Penrose's strong cosmic censorship conjecture.) In this talk, I shall discuss a series of rigorous results proving both Price's law and the mass inflation scenario in an appropriate spherically symmetric setting. The proof of Price's law is joint work with I. Rodnianski. 
  Recent attempts to resolve the ambiguity in the loop quantum gravity description of the quantization of area has led to the idea that $j=1$ edges of spin-networks dominate in their contribution to black hole areas as opposed to $j=1/2$ which would naively be expected. This suggests that the true gauge group involved might be SO(3) rather than SU(2) with attendant difficulties. We argue that the assumption that a version of the Pauli principle is present in loop quantum gravity allows one to maintain SU(2) as the gauge group while still naturally achieving the desired suppression of spin-1/2 punctures. Areas come from $j=1$ punctures rather than $j=1/2$ punctures for much the same reason that photons lead to macroscopic classically observable fields while electrons do not. 
  For geodesic motion of a particle in a stationary spacetime the $U_0$ component of particle 4-velocity is constant and is considered to be a conserved mechanical energy. We show that this concept of a conserved mechanical energy can be extended to particles that move under the influence of a ``normal force,'' a force that, in the stationary frame, is orthogonal to the motion of the particle. We illustrate the potential usefulness of the concept with a simple example. 
  Stable gravitating lumps with a false vacuum core surrounded by the true vacuum in a scalar field potential exist in the presence of fermions at the core. These objects may exist in the universe at various scales. 
  We study the evolution of Bianchi-I space-times filled with a global unidirectional electromagnetic field $F_{mn}$ interacting with a massless scalar dilatonic field according to the law \Psi(\phi) F^{mn} F_{mn} where \Psi(\phi) > 0 is an arbitrary function. A qualitative study, among other results, shows that (i) the volume factor always evolves monotonically, (ii) there exist models becoming isotropic at late times and (iii) the expansion generically starts from a singularity but there can be special models starting from a Killing horizon preceded by a static stage. All these features are confirmed for exact solutions found for the usually considered case \Psi = e^{2\lambda\phi}, \lambda = const. In particular, isotropizing models are found for |\lambda| > 1/\sqrt{3}. In the special case |\lambda| = 1, which corresponds to models of string origin, the string metric behaviour is studied and shown to be qualitatively similar to that of the Einstein frame metric. 
  This thesis start by a review of different approaches to classical and quantum gravity. The main theme is Lorentzian Gromov Hausdorff theory which is an active diffeomorphism invariant theory on the space of Lorentz spaces (think about globally hyperbolic spacetimes). It is argued why such theory might be of significant importance for Lorentzian approaches to quantum gravity such as causal set theory and Lorentzian dynamical triangulations 
  We provide a discussion about the necessity to fix the reference frame before quantizing the gravitational field. Our presentation is based on stressing how the 3+1-slicing of the space time becomes an ambiguous procedure as referred to a quantum 4-metric. 
  A Friedman-Robertson-Walker cosmology arising from a five-dimensional Chern-Simons (CS) theory for the group S0(1,5) coupled to matter is considered as an alternative model for dark energy and matter. The four-dimensional reduction describes an accelerating universe having a time dependent Newton's coupling G and a positive cosmological constant. Five dimensional matter gives rise to what we interprete as four dimensional ordinary plus dark matter and a dark energy is provided by a cosmological constant term plus a fluid arising from the CS coupling. The case of five dust is studied in detail, leading to acceptable limits for most of the cosmological parameters considered, in the context of an open non-flat universe. Nevertheless, a value for ${\dot G}/G $ which is two orders of magnitude higher than recent bounds is predicted. 
  Are the classical singularities of general relativistic spacetimes, normally defined by the incompleteness of classical particle paths, still singular if quantum mechanical particles are used instead? This is the question we will attempt to answer for particles obeying the quantum mechanical wave equations for scalar, null vector and spinor particles. The analysis will be restricted to certain static general relativistic spacetimes that classically contain the mildest true classical singularities, quasiregular singularities. 
  The purpose of this article is to draw attention to some fundamental issues in General Relativity. It is argued that these deep issues cannot be resolved within the standard approach to general relativity that considers {\em every} solution of Einstein's field equations to be of relevance to some, hypothetical or not, physical situation. Hence, to resolve the considered problems of the standard approach to general relativity, one must go beyond it. A possible approach, a theory of everything, is outlined in the present article and will be developed in details subsequently. 
  The problem of quantization of general relativity is considered in the framework of noncommutative differential geometry. Operator analogues for interval, scalar curvature, values of the Einstein tensor are proposed. Quantum measurements of these observables lead to a paradox: different procedures of measurements can supply non equivalent geometrical pictures of space-time. A concrete example of such situation is provided. 
  Local Lorentz invariance violation can be realized by introducing extra tensor fields in the action that couple to matter. If the Lorentz violation is rotationally invariant in some frame, then it is characterized by an ``aether'', i.e. a unit timelike vector field. General covariance requires that the aether field be dynamical. In this paper we study the linearized theory of such an aether coupled to gravity and find the speeds and polarizations of all the wave modes in terms of the four constants appearing in the most general action at second order in derivatives. We find that in addition to the usual two transverse traceless metric modes, there are three coupled aether-metric modes. 
  Implications of noncommutative field theories with commutator of the coordinates of the form $[x^{\mu},x^{\nu}]=i \Lambda_{\quad \omega}^{\mu \nu}x^{\omega}$with nilpotent structure constants are investigated. It is shown that a free quantum field theory is not affected by noncommutativity, but that invariance under translations is broken and hence the energy-momentum conservation is not respected. The new energy-momentum law is expressed by a Poincar\'e-invariant equation and the resulting kinematics is developed and applied to the astrophysical puzzle related with the observed violation of the GZK cutoff. 
  This talk reviews the constraints imposed by binary-pulsar data on gravity theories, focusing on ``tensor-scalar'' ones which are the best motivated alternatives to general relativity. We recall that binary-pulsar tests are qualitatively different from solar-system experiments, because of nonperturbative strong-field effects which can occur in compact objects like neutron stars, and because one can observe the effect of gravitational radiation damping. Some theories which are strictly indistinguishable from general relativity in the solar system are ruled out by binary-pulsar observations. During the last months, several impressive new experimental data have been published. Today, the most constraining binary pulsar is no longer the celebrated (Hulse-Taylor) PSR B1913+16, but the neutron star-white dwarf system PSR J1141-6545. In particular, in a region of the ``theory space'', solar-system tests were known to give the tightest constraints; PSR J1141-6545 is now almost as powerful. We also comment on the possible scalar-field effects for the detection of gravitational waves with future interferometers. The presence of a scalar partner to the graviton might be detectable with the LISA space experiment, but we already know that it would have a negligible effect for LIGO and VIRGO, so that the general relativistic wave templates can be used securely for these ground interferometers. 
  A class of non-Markoffian nonunitary models for Newtonian gravity is characterized as following from some rather natural hypotheses. One of such models was previously obtained as the Newtonian limit of a classically stable version of higher derivative gravity. They give rise to a mass threshold around $10^{11}$ proton masses for gravity induced localization, to a breaking of linearity and to the possible identification of thermodynamic and von Neumann entropies. 
  I give a brief overview of some Quantum-Gravity-Phenomenology research lines, focusing on studies of cosmic rays and gamma-ray bursts that concern the fate of Lorentz symmetry in quantum spacetime. I also stress that the most valuable phenomenological analyses should not mix too many conjectured new features of quantum spacetime, and from this perspective it appears that it should be difficult to obtain reliable guidance on the quantum-gravity problem from the analysis of synchrotron radiation from the Crab nebula and from the analysis of phase coherence of light from extragalactic sources. Forthcoming observatories of ultra-high-energy neutrinos should provide several opportunities for clean tests of some simple hypothesis for the short-distance structure of spacetime. In particular, these neutrino studies, and some related cosmic-ray studies, should provide access to the regime $E>\sqrt{m E_p}$. 
  I argue that GR Cauchy and Initial Value Problem mathematics is a valuable source of ideas for braneworlds based on higher-dimensional Einstein field equations. 
  We investigate the corrections to the inflationary cosmological dynamics due to a $R^2$ term in the Palatini formulation which may arise as quantum corrections to the effective Lagrangian in early universe. We found that the standard Friedmann equation will not be changed when the scalar field is in the potential energy dominated era. However, in the kinetic energy dominated era, the standard Friedmann equation will be modified and in the case of closed and flat universe, the Modified Friedmann equation will automatically require that the initial kinetic energy density of the scalar field must be in sub-Planckian scale. 
  We present here the general transformation that leaves unchanged the form of the field equations for perfect fluid cosmologies in the DGP braneworld model. Specifically, a prescription for relating exact solutions with different equations of state is provided, and the symmetries found can be used as algorithms for generating new cosmological models from previously known ones. We also present, implicitly, the first known exact DGP perfect fluid spacetime. A particular case of the general transformation is used to illustrate the crucial role played both by the number of scalar fields and the extra dimensional effects in the occurrence of inflation. In particular, we see that assisted inflation does not proceed at all times for one of the two possible ways in which the brane can be embedded into the bulk. 
  We show that the non-perturbative vacuum structure associated with neutrino mixing leads to a non-zero contribution to the value of the cosmological constant. Such a contribution comes from the specific nature of the mixing phenomenon. Its origin is completely different from the one of the ordinary contribution of a massive spinor field. We estimate this neutrino mixing contribution by using the natural cut--off appearing in the quantum field theory formalism for neutrino mixing and oscillation. 
  The interaction of a charged particle with its own field results in the "self-force" on the particle, which includes but is more general than the radiation reaction force. In the vicinity of the particle in curved spacetime, one may follow Dirac and split the retarded field of the particle into two parts, (1) the singular source field, ~q/r, and (2) the regular remainder field. The singular source field exerts no force on the particle, and the self-force is entirely caused by the regular remainder. We describe an elementary multipole decomposition of the singular source field which allows for the calculation of the self-force on a scalar-charged particle orbiting a Schwarzschild black hole. 
  Neutron stars that are cold enough should have two or more superfluids/supercondutors in their inner crusts and cores. The implication of superfluidity/superconductivity for equilibrium and dynamical neutron star states is that each individual particle species that forms a condensate must have its own, independent number density current and equation of motion that determines that current. An important consequence of the quasiparticle nature of each condensate is the so-called entrainment effect, i.e. the momentum of a condensate is a linear combination of its own current and those of the other condensates. We present here the first fully relativistic modelling of slowly rotating superfluid neutron stars with entrainment that is accurate to the second-order in the rotation rates. The stars consist of superfluid neutrons, superconducting protons, and a highly degenerate, relativistic gas of electrons. We use a relativistic $\sigma$ - $\omega$ mean field model for the equation of state of the matter and the entrainment. We determine the effect of a relative rotation between the neutrons and protons on a star's total mass, shape, and Kepler, mass-shedding limit. 
  Flat space-time has not heretofore been thought a suitable locus in which to construct model universes because of the presumed necessity of incorporating gravitation in such models and because of the historical lack of a theory of gravitation in flat space-time. It is here shown that a Lorentz-invariant theory of gravitation can be formulated by incorporating in it the mass-energy relation. Such a theory correctly predicts the well-known relativistic effects (advance of perihelion, gravitational refraction, gravitational red shift, echo delay of sun-grazing radio signals, and others). The equations of motion, properly stated, are also seen to be identical to those of electromagnetism and lead to the correct prediction of gravitational radiation. Therefore Milne's kinematic model of the universe, mappable into his dynamical (or Newtonian) model, offers a unique alternative to the general relativistic models which are encumbered with both theoretically and observationally objectionable features. 
  A model is proposed to describe a transition from a charged black hole of mass $M$ and charge $Q$ to one of mass $\bar{M}$ and charge $\bar{Q}$. The basic equations are derived from the non-vacuum Einstein field equations sourced by the Coulomb field and by a null scalar field with a nonvanishing trace anomaly. It is shown that the nonvanishing trace of the energy-momentum tensor prevents the formation of a naked singularity. 
  We consider one-loop effects in general relativity which result in quantum long-range corrections to the Newton law, as well as to the gravitational spin-dependent and velocity-dependent interactions. Some contributions to these effects can be interpreted as quantum corrections to the Schwarzschild and Kerr metric. 
  The dynamics of the Mixmaster Universe is analized in a covariant picture via Misner--Chitre-like variables for an ADM Hamiltonian approach. The system outcomes as isomorphic to a billiard on the Lobachevsky plane and Lyapunov exponents are calculated explicitly. 
  The LTP (LISA Testflight Package), to be flown aboard the ESA / NASA LISA Pathfinder mission, aims to demonstrate drag-free control for LISA test masses with acceleration noise below 30 fm/s^2/Hz^1/2 from 1-30 mHz. This paper describes the LTP measurement of random, position independent forces acting on the test masses. In addition to putting an overall upper limit for all source of random force noise, LTP will measure the conversion of several key disturbances into acceleration noise and thus allow a more detailed characterization of the drag-free performance to be expected for LISA. 
  The magnetic field due to an axially symmetric, hot and highly conducting plasma, taken as an ideal magnetohydrodynamic fluid, surrounding a slow rotating compact gravitational object is studied within the context of Einstein-Maxwell field equations. It is assumed that whereas the plasma is effected by the background spacetime it does not effect the spacetime itself. The Einstein-Maxwell equations are then solved for the magnetic field in a comoving frame with the background spacetime described by the slow rotating Kerr black hole spacetime. It is found that the solutions are magnetic waves travelling along the azimuthal angle with velocity equal to the angular velocity of a free falling intertial frame. These general solutions, when applied to various particular cases of physical interest, show that for a fixed value of the azimuthal angle the magnetic field is completely induced by the dragging of the background spacetime. 
  In the general case, torsion couples to the spin current of the Dirac field. In General Relativity, the apparent torsion field to which the spin current of the Dirac field couples is a mere manifestation of the tetrad anholonomy. Seen from the tetrad frame itself, it has for components the anholonomy coefficients. The latter represent mechanical characteristics of the frame. In the teleparallel equivalent of General Relativity, this coefficient turns out to be the only torsion present. 
  We present a Master equation for description of fermions and bosons for special relativities with two invariant scales, SR2, (c and lambda_P). We introduce canonically-conjugate variables (chi^0, chi) to (epsilon, pi) of Judes-Visser. Together, they bring in a formal element of linearity and locality in an otherwise non-linear and non-local theory. Special relativities with two invariant scales provide all corrections, say, to the standard model of the high energy physics, in terms of one fundamental constant, lambda_P. It is emphasized that spacetime of special relativities with two invariant scales carries an intrinsic quantum-gravitational character. In an addenda, we also comment on the physical importance of a phase factor that the whole literature on the subject has missed and present a brief critique of SR2. In addition, we remark that the most natural and physically viable SR2 shall require momentum-space and spacetime to be non-commutative with the non-commutativity determined by the spin content and C, P, and T properties of the examined representation space. Therefore, in a physically successful SR2, the notion of spacetime is expected to be deeply intertwined with specific properties of the test particle. 
  It is introduced a hypothesis that the gravitational potential in the universe changes linearly with the time. This enables to explain the Hubble red shift and the anomalous acceleration of Pioneer 10 and 11. 
  I summarize some results obtained from a canonical quantization of gravitational collapse. The quantization is carried out in Kuchar variables on the LeMaitre-Tolman-Bondi family of spacetimes. I show how mass quantization, the black hole entropy and Hawking radiation may be understood in these models. Hawking radiation is obtained in the WKB approximation but the first order quantum gravity correction makes the near-horizon spectrum non-thermal, suggesting that unitarity is preserved. The quantization may be used to study quantum gravity effects in collapse leading to the formation of both covered and naked singularities. 
  In this work we study the magnitude-redshift relation of a non-standard cosmological model. The model under consideration was firstly investigated within a special case of metric-affine gravity (MAG) and was recently recovered via different approaches by two other groups. Apart from the usual cosmological parameters for pressure-less matter $\Omega_{\rm m}$, cosmological constant/dark energy $\Omega_{\lambda}$, and radiation $\Omega_{\rm r}$ a new density parameter $\Omega_\psi$ emerges. The field equations of the model reduce to a system which is effectively given by the usual Friedmann equations of general relativity, supplied by a correction to the energy density and pressure in form of $\Omega_\psi$, which is related to the non-Riemannian structure of the underlying spacetime. We search for the best-fit parameters by using recent SN Ia data sets and constrain the possible contribution of a new dark-energy like component at low redshifts, thereby we put an upper limit on the presence of non-Riemannian quantities in the late stages of the universe. In addition the impact of placing the data in redshift bins of variable size is studied. The numerical results of this work also apply to several anisotropic cosmological models which, on the level of the field equations, exhibit a similar scaling behavior of the density parameters like our non-Riemannian model. 
  Motivated by the need to control the exponential growth of constraint violations in numerical solutions of the Einstein evolution equations, two methods are studied here for controlling this growth in general hyperbolic evolution systems. The first method adjusts the evolution equations dynamically, by adding multiples of the constraints, in a way designed to minimize this growth. The second method imposes special constraint preserving boundary conditions on the incoming components of the dynamical fields. The efficacy of these methods is tested by using them to control the growth of constraints in fully dynamical 3D numerical solutions of a particular representation of the Maxwell equations that is subject to constraint violations. The constraint preserving boundary conditions are found to be much more effective than active constraint control in the case of this Maxwell system. 
  The existence of a new fundamental scale may lead to modified dispersion relations for particles at high energies. Such modifications seem to be realized with the Planck scale in certain descriptions of quantum gravity. We apply effective field theory to this problem and identify dimension 5 operators that would lead to cubic modifications of dispersion relations for Standard Model particles. We also discuss other issues related to this approach including various experimental bounds on the strength of these interactions. Further we sketch a scenario where mixing of these operators with dimensions 3 and 4 due to quantum effects is minimal. 
  We propose a criterion for the validity of semiclassical gravity (SCG) which is based on the stability of the solutions of SCG with respect to quantum metric fluctuations. We pay special attention to the two-point quantum correlation functions for the metric perturbations, which contain both intrinsic and induced fluctuations. These fluctuations can be described by the Einstein-Langevin equation obtained in the framework of stochastic gravity. Specifically, the Einstein-Langevin equation yields stochastic correlation functions for the metric perturbations which agree, to leading order in the large $N$ limit, with the quantum correlation functions of the theory of gravity interacting with $N$ matter fields. The homogeneous solutions of the Einstein-Langevin equation are equivalent to the solutions of the perturbed semiclassical equation, which describe the evolution of the expectation value of the quantum metric perturbations. The information on the intrinsic fluctuations, which are connected to the initial fluctuations of the metric perturbations, can also be retrieved entirely from the homogeneous solutions. However, the induced metric fluctuations proportional to the noise kernel can only be obtained from the Einstein-Langevin equation (the inhomogeneous term). These equations exhibit runaway solutions with exponential instabilities. A detailed discussion about different methods to deal with these instabilities is given. We illustrate our criterion by showing explicitly that flat space is stable and a description based on SCG is a valid approximation in that case. 
  A thick Z_2-symmetric domain wall supported by a scalar field with an arbitrary potential V(\phi) in 5D general relativity is considered as a candidate brane world. We show that, under the global regularity requirement, such a configuration (i) has always an AdS asymptotic far from the brane, (ii) is only possible if V(\phi) has an alternating sign and (iii) V(\phi) satisfies a certain fine-tuning type equality. The thin brane limit is well defined and conforms to the Randall-Sundrum (RS2) brane world model if the asymptotic value of V(\phi) (related to \Lambda, the effective cosmological constant) is kept thickness-independent. Universality of such a transition is demonstrated using as an example exact solutions for stepwise potentials of different shapes. Also, due to scale invariance of the Einstein-scalar equations, any given regular solution creates a one-parameter family of solutions with different potentials. In such families, a thin brane limit does not exist while the ratio \Lambda/(brane tension)^2 is thickness-independent and is in general different from its value in the RS2 model. 
  The recent type Ia supernova data suggest that the universe is accelerating now and decelerated in recent past. This may provide the evidence that the standard Friedmann equation needs to be modified. We analyze in detail a new model in the context of modified Friedmann equation using the supernova data published by the High-$z$ Supernova Search Team and the Supernova Cosmology Project. The new model explains recent acceleration and past deceleration. Furthermore, the new model also gives a decelerated universe in the future. 
  Based on the gauge invariant variables proposed in [K. Nakamura, Prog. Theor. Phys. vol.110 (2003), 723.], general framework of the second order gauge invariant perturbation theory on arbitrary background spacetime is considered. We derived formulae of the perturbative Einstein tensor of each order, which have the similar form to the definitions of gauge invariant variables for arbitrary perturbative fields. As a result, each order Einstein equation is necessarily given in terms of gauge invariant variables. 
  We discuss expansions for the Cotton-York tensor near infinity for arbitrary slices of stationary spacetimes. From these expansions it follows directly that a necessary condition for the existence of conformally flat slices in stationary solutions is the vanishing of a certain quantity of quadrupolar nature (obstruction). The obstruction is nonzero for the Kerr solution. Thus, the Kerr metric admits no conformally flat slices. An analysis of higher orders in the expansions of the Cotton-York tensor for solutions such that the obstruction vanishes suggests that the only stationary solution admitting conformally flat slices are the Schwarzschild family of solutions. 
  The displacement noise in the test mass mirrors of interferometric gravitational wave detectors is proportional to their elastic dissipation at the observation frequencies. In this paper, we analyze one fundamental source of dissipation in thin coatings, thermoelastic damping associated with the dissimilar thermal and elastic properties of the film and the substrate. We obtain expressions for the thermoelastic dissipation factor necessary to interpret resonant loss measurements, and for the spectral density of displacement noise imposed on a Gaussian beam reflected from the face of a coated mass. The predicted size of these effects is large enough to affect the interpretation of loss measurements, and to influence design choices in advanced gravitational wave detectors. 
  Cosmological models where spatial sections are the Poincar\'e dodecahedral space D have been recently invoked to give an account of the lower modes of the angular anisotropies of the cosmic microwave background. Further explorations of this possibility require the knowledge of the eigenmodes of the Laplacian of D. Only the first modes have been calculated numerically. Here we give an explicit form for these modes up to arbitrary order, in term of eigenvectors of a small rank matrix, very easy to calculate numerically. As an illustration we give the first modes, up to the eigenvalue $-k (k+2)$ for $k=62$. These results are obtained by application of a more general method (presented in a previous work) which allows to express the properties of any eigenfunction of the Laplacian of the three sphere under an arbitrary rotation of SO(4). 
  After a brief review of topological gravity, we present a superspace approach to this theory. This formulation allows us to recover in a natural manner various known results and to gain some insight into the precise relationship between different approaches to topological gravity. Though the main focus of our work is on the vielbein formalism, we also discuss the metric approach and its relationship with the former formalism. 
  We investigate the instability of charged massive scalar fields in Kerr-Newman spacetime. Due to the super-radiant effect of the background geometry, the bound state of the scalar field is unstable, and its amplitude grows in time. By solving the Klein-Gordon equation of the scalar field as an eigenvalue problem, we numerically obtain the growth rate of the amplitude of the scalar field. Although the dependence of the scalar field mass and the scalar field charge on this growth rate agrees with the result of the analytic approximation, the maximum value of the growth rate is three times larger than that of the analytic approximation. We also discuss the effect of the electric charge on the instability of the scalar field. 
  The Markov chain Monte Carlo methods offer practical procedures for detecting signals characterized by a large number of parameters and under conditions of low signal-to-noise ratio. We present a Metropolis-Hastings algorithm capable of inferring the spin and orientation parameters of a neutron star from its periodic gravitational wave signature seen by laser interferometric detectors 
  We study homogeneous and isotropic cosmologies in a Weyl spacetime. We show that for homogeneous and isotropic spacetimes, the field equations can be reduced to the Einstein equations with a two-fluid source. We write the equations as a two-dimensional dynamical system and analyze the qualitative, asymptotic behavior of the models. We examine the possibility that in certain theories the Weyl 1-form may give rise to a late accelerated expansion of the Universe and conclude that such behaviour is not met as a generic feature of the simplest cosmologies. 
  Axisymmetric numerical simulations of rotating stellar core collapse to a neutron star are performed in the framework of full general relativity. The so-called Cartoon method, in which the Einstein field equations are solved in the Cartesian coordinates and the axisymmetric condition is imposed around the $y=0$ plane, is adopted. The hydrodynamic equations are solved in the cylindrical coordinates (on the $y=0$ plane in the Cartesian coordinates) using a high-resolution shock-capturing scheme with the maximum grid size $(2500,2500)$. A parametric equation of state is adopted to model collapsing stellar cores and neutron stars following Dimmelmeier et al. It is found that the evolution of central density during the collapse, bounce, and formation of protoneutron stars agree well with those in the work of Dimmelmeier et al. in which an approximate general relativistic formulation is adopted. This indicates that such approximation is appropriate for following axisymmetric stellar core collapses and subsequent formation of protoneutron stars. Gravitational waves are computed using a quadrupole formula. It is found that the waveforms are qualitatively in good agreement with those by Dimmelmeier et al. However, quantitatively, two waveforms do not agree well. Possible reasons for the disagreement are discussed. 
  The covariant canonical formalism for four-dimensional BF theory is performed. The aim of the paper is to understand in the context of the covariant canonical formalism both the reducibility that some first class constraints have in Dirac's canonical analysis and also the role that topological terms play. The analysis includes also the cases when both a cosmological constant and the second Chern character are added to the pure BF action. In the case of the BF theory supplemented with the second Chern character, the presymplectic 3-form is different to the one of the BF theory in spite of the fact both theories have the same equations of motion while on the space of solutions they both agree to each other. Moreover, the analysis of the degenerate directions shows some differences between diffeomorphisms and internal gauge symmetries. 
  The Data Analysis programme of the Australian Consortium for Interferometric Gravitational Astronomy (ACIGA) was set up in 1998 by the first author to complement the then existing ACIGA programmes working on suspension systems, lasers and optics, and detector configurations. The ACIGA Data Analysis programme continues to contribute significantly in the field; we present an overview of our activities. 
  We focus on N = 3 chiral supergravity (SUGRA) which is the lowest N theory involving a spin-1/2 field, and derive the Ashtekar's canonical formulation of N = 3 SUGRA starting with the chiral Lagrangian constructed by closely following the standard SUGRA. The polynomiality of constraints in terms of canonical variables and the graded algebraic structure of constraints are discussed in the canonical formulation. In particular, we show the polynomiality of the {\it rescaled} right- and left-handed SUSY constraints by a nonpolynomial factor. And also we show the graded algebraic structure of Osp(3/2) in the constraint algebra by calculating the Poisson brackets of Gauss, SU(2) gauge and right-handed SUSY constraints, although the algebra among only those three types of constraints does not closed. 
  We investigate the entropy of black holes in Gauss-Bonnet and Lovelock gravity using the Noether charge approach, in which the entropy is given as the integral of a suitable (n-2) form charge over the event horizon. We compare the results to those obtained in other approaches. We also comment on the appearance of negative entropies in some cases, and show that there is an additive ambiguity in the definition of the entropy which can be appropriately chosen to avoid this problem. 
  It has been recently argued by Hertog, Horowitz and Maeda, that generic reasonable initial data in asymptotically anti deSitter, spherically symmetric, space-times within an Einstein-Higgs theory, will evolve toward a naked singularity, in clear violation of the cosmic censor conjecture. We will argue that there is a logical and physically plausible loophole in the argument and that the numerical evidence in a related problem suggests that this loophole is in fact employed by physics. 
  We give an exact solution of the $5D$ Einstein equations which in 4D can be interpreted as a spherically symmetric dissipative distribution of matter, with heat flux, whose effective density and pressure are nonstatic, nonuniform, and satisfy the equation of state of radiation. The matter satisfies the usual energy and thermodynamic conditions. The energy density and temperature are related by the Stefan-Boltzmann law. The solution admits a homothetic Killing vector in $5D$, which induces the existence of self-similar symmetry in 4D, where the line element as well as the dimensionless matter quantities are invariant under a simple "scaling" group. 
  The system consisting of a self gravitating perfect fluid and scalar field is considered in detail. The scalar fields considered are the quintessence and ``tachyonic'' forms which have important application in cosmology. Mathematical properties of the general system of equations are studied including the algebraic and differential identities as well as the eigenvalue structure. The Cauchy problem for both quintessence and the tachyon is presented. We discuss the initial constraint equations which must be satisfied by the initial data. A Cauchy evolution scheme is presented in the form of a Taylor series about the Cauchy surface. Finally, a simple numerical example is provided to illustrate this scheme. 
  We carefully analyse the contribution to the oscillations of a metallic gravitational antenna due to the interaction between the electrons of the bar and the incoming gravitational wave. To this end, we first derive the total microscopic Hamiltonian of the wave-antenna system and then compute the contribution to the attenuation factor due to the electron-graviton interaction. As compared to the ordinary damping factor, which is due to the electron viscosity, this term turns out to be totally negligible. This result confirms that the only relevant mechanism for the interaction of a gravitational wave with a metallic antenna is its direct coupling with the bar normal modes. 
  This is the last article in a series of three initiated by the second author. We elaborate on the concepts and theorems constructed in the previous articles. In particular, we prove that the GH and the GGH uniformities previously introduced on the moduli space of isometry classes of globally hyperbolic spacetimes are different, but the Cauchy sequences which give rise to well-defined limit spaces coincide. We then examine properties of the strong metric introduced earlier on each spacetime, and answer some questions concerning causality of limit spaces. Progress is made towards a general definition of causality, and it is proven that the GGH limit of a Cauchy sequence of $\mathcal{C}^{\pm}_{\alpha}$, path metric Lorentz spaces is again a $\mathcal{C}^{\pm}_{\alpha}$, path metric Lorentz space. Finally, we give a necessary and sufficient condition, similar to the one of Gromov for the Riemannian case, for a class of Lorentz spaces to be precompact. 
  Starting from the defining transformations of complex matrices for the SO(4) group, we construct the fundamental representation and the tensor and spinor representations of the group SO(4). Given the commutation relations for the corresponding algebra, the unitary representations of the group in terms of the generalized Euler angles are constructed. These mathematical results help us to a more complete description of the Barrett-Crane model in Quantum Gravity. In particular a complete realization of the weight function for the partition function is given and a new geometrical itnerpretation of the asymptotic limit for the Regge action is presented. 
  In this letter we investigate the nature of generic cosmological singularities using the framework developed by Uggla et al. We do so by studying the past asymptotic dynamics of general vacuum G2 cosmologies, models that are expected to capture the singular behavior of generic cosmologies with no symmetries at all. In particular, our results indicate that asymptotic silence holds, i.e., that particle horizons along all timelines shrink to zero for generic solutions. Moreover, we provide evidence that spatial derivatives become dynamically insignificant along generic timelines, and that the evolution into the past along such timelines is governed by an asymptotic dynamical system which is associated with an invariant set -- the silent boundary. We also identify an attracting subset on the silent boundary that organizes the oscillatory dynamics of generic timelines in the singular regime. In addition, we discuss the dynamics associated with recurring spike formation. 
  Geodesics are studied in one of the Weyl metrics, referred to as the M--Q solution. First, arguments are provided, supporting our belief that this space--time is the more suitable (among the known solutions of the Weyl family) for discussing the properties of strong quasi--spherical gravitational fields. Then, the behaviour of geodesics is compared with the spherically symmetric situation, bringing out the sensitivity of the trajectories to deviations from spherical symmetry. Particular attention deserves the change of sign in proper radial acceleration of test particles moving radially along symmetry axis, close to the $r=2M$ surface, and related to the quadrupole moment of the source. 
  Aspects of the full theory of loop quantum gravity can be studied in a simpler context by reducing to symmetric models like cosmological ones. This leads to several applications where loop effects play a significant role when one is sensitive to the quantum regime. As a consequence, the structure of and the approach to classical singularities are very different from general relativity: The quantum theory is free of singularities, and there are new phenomenological scenarios for the evolution of the very early universe including inflation. We give an overview of the main effects, focussing on recent results obtained by several different groups. 
  In this work the late-time evolution of Bianchi type $VIII$ models is discussed. These cosmological models exhibit a chaotic behaviour towards the initial singularity and our investigations show that towards the future, far from the initial singularity, these models have a non-chaotic evolution, even in the case of vacuum and without inflation. These space-time solutions turn out to exhibit a particular time asymmetry. On the other hand, investigations of the late-time behaviour of type $VIII$ models by another author have the result that chaos continues for ever in the far future and that these solutions have a time symmetric behaviour: this result was obtained using the approximation methods of Belinski, Khalatnikov and Lifshitz ($BKL$) and we try to find out a possible reason explaining why the different approaches lead to distinct outcomes. It will be shown that, at a heuristic level, the $BKL$ method gives a valid approximation of the late-time evolution of type $VIII$ models, agreeing with the result of our investigations. 
  The unification of quantum field theory and general relativity is a fundamental goal of modern physics. In many cases, theoretical efforts to achieve this goal introduce auxiliary gravitational fields, ones in addition to the familiar symmetric second-rank tensor potential of general relativity, and lead to nonmetric theories because of direct couplings between these auxiliary fields and matter. Here, we consider an example of a metric-affine gauge theory of gravity in which torsion couples nonminimally to the electromagnetic field. This coupling causes a phase difference to accumulate between different polarization states of light as they propagate through the metric-affine gravitational field. Solar spectropolarimetric observations are reported and used to set strong constraints on the relevant coupling constant k: k^2 < (2.5 km)^2. 
  We study and report on the class of vacuum Maxwell fields in Minkowski space that possess a non-degenerate, diverging, principle null vector field (null eigenvector field of the Maxwell tensor) that is tangent to a shear-free null geodesics congruence. These congruences can be either surface forming (the tangent vectors proportional to gradients) or not, i.e., the twisting congruences. In the non-twisting case, the associated Maxwell fields are precisely the Lienard-Wiechert fields, i.e., those Maxwell fields arising from an electric monopole moving on an arbitrary worldline. The null geodesic congruence is given by the generators of the light-cones with apex on the world-line. The twisting case is much richer, more interesting and far more complicated. In a twisting subcase, where our main interests lie, it can be given the following strange interpretation. If we allow the real Minkowski space to be complexified so that the real Minkowski coordinates x^a take complex values, i.e., x^a => z^a=x^a+iy^a with complex metric g=eta_abdz^adz^b, the real vacuum Maxwell equations can be extended into the complex and rewritten as curlW =iWdot, divW with W =E+iB. This subcase of Maxwell fields can then be extended into the complex so as to have as source, a complex analytic world-line, i.e., to now become complex Lienard-Wiechart fields. When viewed as real fields on the real Minkowski space, z^a=x^a, they possess a real principle null vector that is shear-free but twisting and diverging. The twist is a measure of how far the complex world-line is from the real 'slice'. Most Maxwell fields in this subcase are asymptotically flat with a time-varying set of electric and magnetic moments, all depending on the complex displacements and the complex velocities. 
  A family of quasilocal mass definitions that includes as special cases the Hawking mass and the Brown-York ``rest mass'' energy is derived for spacelike 2-surfaces in spacetime. The definitions involve an integral of powers of the norm of the spacetime mean curvature vector of the 2-surface, whose properties are connected with apparent horizons. In particular, for any spacelike 2-surface, the direction of mean curvature is orthogonal (dual in the normal space) to a unique normal direction in which the 2-surface has vanishing expansion in spacetime. The quasilocal mass definitions are obtained by an analysis of boundary terms arising in the gravitational ADM Hamiltonian on hypersurfaces with a spacelike 2-surface boundary, using a geometric time-flow chosen proportional to the dualized mean curvature vector field at the boundary surface. A similar analysis is made choosing a geometric rotational flow given in terms of the twist covector of the dual pair of mean curvature vector fields, which leads to a family of quasilocal angular momentum definitions involving the squared norm of the twist. The large sphere limit of these definitions is shown to yield the ADM mass and angular momentum in asymptotically flat spacetimes, while at apparent horizons a quasilocal version of the Gibbons-Penrose inequality is derived. Finally, some results concerning positivity are proved for the quasilocal masses, motivated by consideration of spacelike mean curvature flow of 2-surfaces in spacetime. 
  To analyze linear field equations on a locally homogeneous spacetime by means of separation of variables, it is necessary to set up appropriate harmonics according to its symmetry group. In this paper, the harmonics are presented for a spatially compactified Bianchi II cosmological model -- the nilgeometric model. Based on the group structure of the Bianchi II group (also known as the Heisenberg group) and the compactified spatial topology, the irreducible differential regular representations and the multiplicity of each irreducible representation, as well as the explicit form of the harmonics are all completely determined. They are also extended to vector harmonics. It is demonstrated that the Klein-Gordon and Maxwell equations actually reduce to systems of ODEs, with an asymptotic solution for a special case. 
  We examine in the context of general relativity the dynamics of a spatially flat Robertson-Walker universe filled with a classical minimally coupled scalar field \phi of exponential potential ~ e^{-\mu\phi} plus pressureless baryonic matter. This system is reduced to a first-order ordinary differential equation, providing direct evidence on the acceleration/deceleration properties of the system. As a consequence, for positive potentials, passage into acceleration not at late times is generically a feature of the system, even when the late-times attractors are decelerating. Furthermore, the structure formation bound, together with the constraints on the present values of \Omega_{m}, w_{\phi} provide, independently of initial conditions and other parameters, necessary conditions on \mu. Special solutions are found to possess intervals of acceleration. For the almost cosmological constant case w_{\phi} ~ -1, as well as, for the generic late-times evolution, the general relation \Omega_{\phi}(w_{\phi}) is obtained. 
  We study the structural stability of tachyonic inflation against changes in the shape of the potential. Following Lidsey (Gen. Rel. Grav. 25 (1993) 399), the concepts of rigidity and fragility are defined through a condition on the functional form of the Hubble factor. We find that the models are rigid in the sense that the attractor solutions never change as long as the conditions for inflation are met. 
  The thermodynamics of general relativistic systems with boundary, obeying a Hamiltonian constraint in the bulk, is argued to be determined solely by the boundary quantum dynamics, and hence by the area spectrum. Assuming, for large area of the boundary, (a) an area spectrum as determined by Non-perturbative Canonical Quantum General Relativity (NCQGR), (b) an energy spectrum that bears a power law relation to the area spectrum, (c) an area law for the leading order microcanonicai entropy, leading thermal fluctuation corrections to the canonical entropy are shown to be logarithmic in area with a universal coefficient. Since the microcanonical entropy also has univeral logarithmic corrections to the area law (from quantum spacetime fluctuations, as found earlier) the canonical entropy then has a universal form including logarithmic corrections to the area law. This form is shown to be independent of the index appearing in assumption (b). The index, however, is crucial in ascertaining the domain of validity of our approach based on thermal equilibrium. 
  This is a summary of the talk presented by JP at ICGC2004. It covered some developments in canonical quantum gravity occurred since ICGC2000, emphasizing the recently introduced consistent discretizations of general relativity. 
  Observations have established that extremely compact, massive objects are common in the universe. It is generally accepted that these objects are black holes. As observations improve, it becomes possible to test this hypothesis in ever greater detail. In particular, it is or will be possible to measure the properties of orbits deep in the strong field of a black hole candidate (using x-ray timing or with gravitational-waves) and to test whether they have the characteristics of black hole orbits in general relativity. Such measurements can be used to map the spacetime of a massive compact object, testing whether the object's multipoles satisfy the strict constraints of the black hole hypothesis. Such a test requires that we compare against objects with the ``wrong'' multipole structure. In this paper, we present tools for constructing bumpy black holes: objects that are almost black holes, but that have some multipoles with the wrong value. The spacetimes which we present are good deep into the strong field of the object -- we do not use a large r expansion, except to make contact with weak field intuition. Also, our spacetimes reduce to the black hole spacetimes of general relativity when the ``bumpiness'' is set to zero. We propose bumpy black holes as the foundation for a null experiment: if black hole candidates are the black holes of general relativity, their bumpiness should be zero. By comparing orbits in a bumpy spacetime with those of an astrophysical source, observations should be able to test this hypothesis, stringently testing whether they are the black holes of general relativity. (Abridged) 
  We examine and compare different area spectra that have been recently considered in Loop Quantum Gravity (LQG). In particular we focus our attention on a Equally Spaced (ES) spectrum operator introduced by Alekseev et. al. that has gained recent attention. We show that such operator is not well defined within the LQG framework, and comment on the issue regarding area spectra and QNM frequencies. 
  The initial quantum state during inflation may evolve to a highly squeezed quantum state due to the amplification of the time-dependent parameter, $\omega_{phys}(k/a)$, which may be the modified dispersion relation in trans-Planckian physics. This squeezed quantum state is a nonclassical state that has no counterpart in the classical theory. We have considered the nonclassical states such as squeezed, squeezed coherent, and squeezed thermal states, and calculated the power spectrum of the gravitational wave perturbation when the mode leaves the horizon. 
  We investigate the effects of cosmological constant on the characteristic peaks in the iron line profile in the black hole accretion disk X-ray spectrum. Our results show that for a fixed mass black hole, the peaks become less pronounced and closer together with increasing cosmological constant. This effect is mainly due to the slower rotational velocity of Keplerian orbits at large radii in the Schwarzschild-de Sitter spacetime as compared to those for the Schwarzschild spacetime. This change of the iron line profile is similar to that obtained from increasing the outer radius of the accretion disk size or reducing the emission intensity power law exponent in the accretion disk emissivity models. 
  We discuss brane wormhole solution when classical brane action contains 4d curvature. The equations of motion for the cases with R=0 and $R\ne 0$ are obtained. Their numerical solutions corresponding to wormhole are found for specific boundary conditions. 
  According to the AdS/CFT correspondence conjecture, the Randall-Sundrum infinite braneworld is equivalent to four dimensional Einstein gravity with ${\cal N}=4$ super Yang-Mills fields at low energies. Here we derive a four dimensional effective equation of motion for tensor-type perturbations in two different pictures, and demonstrate their equivalence. 
  We consider the spacetime geometry of a static but otherwise generic black hole (that is, the horizon geometry and topology are not necessarily spherically symmetric). It is demonstrated, by purely geometrical techniques, that the curvature tensors, and the Einstein tensor in particular, exhibit a very high degree of symmetry as the horizon is approached. Consequently, the stress-energy tensor will be highly constrained near any static Killing horizon. More specifically, it is shown that -- at the horizon -- the stress-energy tensor block-diagonalizes into ``transverse'' and ``parallel'' blocks, the transverse components of this tensor are proportional to the transverse metric, and these properties remain invariant under static conformal deformations. Moreover, we speculate that this geometric symmetry underlies Carlip's notion of an asymptotic near-horizon conformal symmetry controlling the entropy of a black hole. 
  A Hilbert manifold structure is described for the ADM phase space of asymptotically flat initial data $(g,\pi)$ with local $H^2\times H^1$ Sobolev regularity. Solutions of the constraint equations form a Hilbert submanifold. A regularized RT Hamiltonian is defined and smooth on the full phase space and generates the Einstein evolution for any lapse-shift asymptotic to a (time) translation at infinity. Critical points for the total (ADM) mass, considered as a function on the Hilbert manifold of constraint solutions, arise precisely at initial data generating stationary vacuum spacetimes. 
  In the framework of a flat FLRW model we derive an inflationary regime in which the scalar field, laying on the plateau of its potential, admits a linear time dependence and remains close to a constant value. The behaviour of inhomogeneous perturbations is determined on the background metric in agreement to the "slow-rolling" approximation. We show that the inhomogeneous scales which before inflation were not much greater then the physical horizon, conserve their spectrum (almost) unaltered after the de Sitter phase. 
  The chaoticity of the Mixmaster is discussed in the framework of Statistical Mechanics by using Misner--Chitre-like variables and an ADM reduction of its dynamics. We show that such a system is well described by a microcanonical ensemble whose invariant measure is induced by the corresponding Liouville one and is uniform. The covariance with respect to the choice of the temporal gauge of the obtained invariant measure is outlined. 
  We discuss the possibility to obtain, from a five-dimensional free spinor Lagrangian, the Quantum Electro-Dynamics (QED) coupling via a Kaluza-Klein reduction of the theory. This result is achieved taking a phase dependence of the spinor field on the extra-coordinate and modifying the corresponding connection. The five-dimensional spinor theory is covariant under the admissible coordinates transformations and its four-dimensional reduction provides the QED coupling term. 
  We have used the results of renormalization of a two-dimensional quantum stress tensor to develop a conformally invariant dynamical model. The model requires the consideration of those conformal frames in which there exists a correspondence between the trace anomaly and a cosmological constant. We apply this model to a two dimensional-Schwarzschild (-de Sitter) spacetime to show that in these conformal frames one may achieve Hawking radiation without recourse to the trace anomaly . 
  We studied the expectation value of the scale factor in radiation and dust quantum perfect fluid cosmology. We used Schutz variational formalism to describe perfect fluid and selected the conjugate coordinate of perfect fluid be dynamical variable. After quantization and solving the Wheeler-DeWitt equation can obtain the exact solution. By superposition of the exact solution, we obtained one wave packets and used it to compute the expectation value of the scale factor. We found that if one select different dynamical variable be the time variable in each of these two systems, the expectation value of the scale factor of these two systems can fit in with the prediction of General Relativity. Therefore we thought that the selection of reference time can be different for different quantum perfect fluid systems. 
  The effective evolution of an inhomogeneous cosmological model may be described in terms of spatially averaged variables. We point out that in this context, quite naturally, a measure arises which is identical to a fluid model of the `Kullback-Leibler Relative Information Entropy', expressing the distinguishability of the local inhomogeneous mass density field from its spatial average on arbitrary compact domains. We discuss the time-evolution of `effective information' and explore some implications. We conjecture that the information content of the Universe -- measured by Relative Information Entropy of a cosmological model containing dust matter -- is increasing. 
  It is shown that if two Reissner-Nordstr\"{o}m space-times, both with the same mass m and charge e, glued together in the singularities, then the light ray in black hole of the first space-time can go continuously through the singularity into black hole of the second. The behavior of tidal forces near the Reissner-Nordstr\"{o}m space-time singularity is examined by considering what happens between two particles falling freely towards the singularity. 
  We study particle production of coherently oscillating inflaton in the semiclassical theory of gravity by representing inflaton in coherent and squeezed state formalisms. A comparative study of the inflaton in classical gravity with coherent state inflaton in semiclassical gravity is also presented. 
  Motivated by the initial-boundary value problem for the Einstein equations, we propose a definition of symmetric hyperbolicity for systems of evolution equations that are first order in time but second order in space. This can be used to impose constraint-preserving boundary conditions. The general methods are illustrated in detail in the toy model of electromagnetism. 
  The variation of the energy for a gravitational system is directly defined from the Hamiltonian field equations of General Relativity. When the variation of the energy is written in a covariant form it splits into two (covariant) contributions: one of them is the Komar energy, while the other is the so-called covariant ADM correction term. When specific boundary conditions are analyzed one sees that the Komar energy is related to the gravitational heat while the ADM correction term plays the role of the Helmholtz free energy. These properties allow to establish, inside a classical geometric framework, a formal analogy between gravitation and the laws governing the evolution of a thermodynamic system. The analogy applies to stationary spacetimes admitting multiple causal horizons as well as to AdS Taub-bolt solutions. 
  Geometric inequalities of classical differential geometry are used to extend to higher dimensional spacetimes the Penrose-Gibbons isoperimetric inequalities and the hoop conjecture of general reltivity. 
  Using perturbative techniques, we investigate the existence and properties of a new static solution for the Einstein equation with a negative cosmological constant, which we call the deformed black hole. We derive a solution for a static and axisymmetric perturbation of the Schwarzschild-anti-de Sitter black hole that is regular in the range from the horizon to spacelike infinity. The key result is that this perturbation simultaneously deforms the two boundary surfaces--i.e., both the horizon and spacelike two-surface at infinity. Then we discuss the Abbott-Deser mass and the Ashtekar-Magnon one for the deformed black hole, and according to the Ashtekar-Magnon definition, we construct the thermodynamic first law of the deformed black hole. The first law has a correction term which can be interpreted as the work term that is necessary for the deformation of the boundary surfaces. Because the work term is negative, the horizon area of the deformed black hole becomes larger than that of the Schwarzschild-anti-de Sitter black hole, if compared under the same mass, indicating that the quasistatic deformation of the Schwarzschild-anti-de Sitter black hole may be compatible with the thermodynamic second law (i.e., the area theorem). 
  We investigate the response function of LISA and consider the adequacy of its commonly used approximation in the high-frequency range of the observational band. We concentrate on monochromatic binary systems, such as white dwarf binaries. We find that above a few mHz the approxmation starts becoming increasingly inaccurate. The transfer function introduces additional amplitude and phase modulations in the measured signal that influence parameter estmation and, if not properly accounted for, lead to losses of signal-to-noise ratio. 
  It is shown that the main contribution to the rotational curve of a spiral galaxy may be due essentially to the interaction, in the general relativistic spacetime, of the galactic matter with a very light long range scalar field which respects the weak equivalence principle. The comparison of the theoretical results with 23 spiral galaxy rotation curves shows a good agreement between our proposal and observations. 
  We study stability, dispersion and dissipation properties of four numerical schemes (Iterative Crank-Nicolson, 3'rd and 4'th order Runge-Kutta and Courant-Fredrichs-Levy Non-linear). By use of a Von Neumann analysis we study the schemes applied to a scalar linear wave equation as well as a scalar non-linear wave equation with a type of non-linearity present in GR-equations. Numerical testing is done to verify analytic results. We find that the method of lines (MOL) schemes are the most dispersive and dissipative schemes. The Courant-Fredrichs-Levy Non-linear (CFLN) scheme is most accurate and least dispersive and dissipative, but the absence of dissipation at Nyquist frequency, if fact, puts it at a disadvantage in numerical simulation. Overall, the 4'th order Runge-Kutta scheme, which has the least amount of dissipation among the MOL schemes, seems to be the most suitable compromise between the overall accuracy and damping at short wavelengths. 
  A relatively recent study by Mars and Senovilla provided us with a uniqueness result for the exterior vacuum gravitational field of global models describing finite isolated rotating bodies in equilibrium in General Relativity (GR). The generalisation to exterior electrovacuum gravitational fields, to include charged rotating objects, is presented here. 
  We show that static metrics solving vacuum Einstein equations (possibly with a cosmological constant) are one-sided analytic at non-degenerate Killing horizons. We further prove analyticity in a two-sided neighborhood of "bifurcate horizons".   It is a pleasure to dedicate this work to Prof. Staruszkiewicz, on the occasion of his 65th birthday. 
  In a previous work \cite{mbeleka}, we have modelled the rotation curves (RC) of spiral galaxies by including in the equation of motion dynamical terms from an external real self-interacting scalar field, $\phi$, minimally coupled to gravity and which respects the equivalence principle in the absence of electromagnetic fields. This model appears to have three free parameters : the turnover radius, $r_{0}$, the maximum rotational velocity, $v_{max} = v(r_{0})$, plus a strictly positive integer, $n$. Here, the coupling of the $\phi$-field to other kinds of matter is emphasized at the expense of its self-interaction. This reformulation presents the very advantageous possibility that the same potential may be used now for all galaxies. New correlations are established. 
  Solution for a stationary spherically symmetric accretion of the relativistic perfect fluid with an equation of state $p(\rho)$ onto the Schwarzschild black hole is presented. This solution is a generalization of Michel solution and applicable to the problem of dark energy accretion. It is shown that accretion of phantom energy is accompanied with the gradual decrease of the black hole mass. Masses of all black holes tend to zero in the phantom energy universe approaching to the Big Rip. 
  General relativity in the form where gravitational perturbations together with other physical fields propagate on an auxiliary background is considered. With using the Katz-Bi{\v{c}}\'ak-Lynden-Bell technique new conserved currents, divergences of antisymmetric tensor densities(superpotentials), in an arbitrary curved spacetime are constructed. 
  Recent attempts to resolve the ambiguity in the loop quantum gravity description of the quantization of area has led to the idea that j=1 edges of spin-networks dominate in their contribution to black hole areas as opposed to j=1/2 which would naively be expected. This suggests that the true gauge group involved might be SO(3) rather than SU(2). We argue that the idea that a version of the Pauli principle is present in loop quantum gravity allows one to maintain SU(2) as the gauge group while still naturally achieving the desired suppression of spin-1/2 punctures. Such an idea can be motivated by arguments from geometric quantization even though the SU(2) under consideration does not have the geometrical interpretation of rotations in 3-dimensional space, and its representation labels do not correspond to physical angular momenta. In this picture, it is natural that macroscopic areas come almost entirely from j=1 punctures rather than j=1/2 punctures, and this is for much the same reason that photons lead to macroscopic classically observable fields while electrons do not. 
  The idea of a role for DSR (doubly-special relativity) in quantum gravity finds some encouragement in a few scenarios, but in order to explore some key conceptual issues it is necessary to find a well-understood toy-quantum-gravity model that is fully compatible with the DSR principles. Perhaps the most significant source of encouragement comes from the recent proposal of a path for the emergence of DSR in Loop Quantum Gravity, which however relies on a few assumptions on the results of some computations that we are still unable to perform. Indications in favor of the possibility of using some elements of $\kappa$-Poincar\'e Hopf algebras (and of the related $\kappa$-Minkowski noncommutative spacetime) for the construction of a DSR theory have been discussed extensively, but a few stubborn open issues must still be resolved, especially in the two-particle sector. It has been recently observed that certain structures encountered in a formulation of 2+1-dimensional classical-gravity models would fit naturally in a DSR framework, but some key elements of these 2+1-dimensional models, including the description of observers, might be incompatible with the DSR principles. 
  The closed form solution for the geodesics of classical particles in SdS space are obtained in terms of hyperelliptic modular functions and multiple hypergeometric functions. The closed form solution for the five roots of the fifth degree polynomial are found giving the branch places on the genus two Riemann surface. The `Inversion Problem', for the genus two hyperelliptic integral is solved in a closed form. Thus a couple of mathematical problems that have been around for a couple of centuries are solved. The solution is important in astrophysical applications of measuring the cosmological constant. 
  It was originally thought that Bonnor's solution in Einstein-Maxwell theory describes a singular point-like magnetic dipole. Lately, however, it has been demonstrated that indeed it may describe a black {\it dihole}, i.e., a pair of static, oppositely-charged extremal black holes with regular horizons. Motivated particularly by this new interpretation, in the present work, the construction and extensive analysis of a solution in the context of the Brans-Dicke-Maxwell theory representing a black dihole are attempted. It has been known for some time that the solution-generating algorithm of Singh and Rai produces stationary, axisymmetric, charged solutions in Brans-Dicke-Maxwell theory from the known such solutions in Einstein-Maxwell theory. Thus this algorithm of Singh and Rai's is employed in order to construct a Bonnor-type magnetic black dihole solution in Brans-Dicke-Maxwell theory from the known Bonnor solution in Einstein-Maxwell theory. The peculiar feature of the new solution including internal infinity nature of the symmetry axis and its stability issue have been discussed in full detail. 
  We derive generic equations for a vector field driving the evolution of flat homogeneous isotropic universe and give a comparison with a scalar filed dynamics in the cosmology. Two exact solutions are shown as examples, which can serve to describe an inflation and a slow falling down of dynamical ``cosmological constant'' like it is given by the scalar quintessence. An attractive feature of vector field description is a generation of ``induced mass'' proportional to a Hubble constant, which results in a dynamical suppression of actual cosmological constant during the evolution. 
  We analyze cylindrical gravitational waves in vacuo with general polarization and develop a viewpoint complementary to that presented recently by Niedermaier showing that the auxiliary sigma model associated with this family of waves is not renormalizable in the standard perturbative sense. 
  The cross section for a gravitational wave antenna to absorb a graviton may be directly expressed in terms of the non-local viscous response function of the metallic crystal. Crystal viscosity is dominated by electronic processes which then also dominate the graviton absorption rate. To compute this rate from a microscopic Hamiltonian, one must include the full Coulomb interaction in the Maxwell electric field pressure and also allow for strongly non-adiabatic transitions in the electronic kinetic pressure. The view that the electrons and phonons constitute ideal gases with a weak electron phonon interaction is not sufficiently accurate for estimating the full strength of the electronic interaction with a gravitational wave. 
  Iterative solutions to fourth-order gravity describing static and electrically charged black holes are constructed. Obtained solutions are parametrized by two integration constants which are related to the electric charge and the exact location of the event horizon. Special emphasis is put on the extremal black holes. It is explicitly demonstrated that in the extremal limit, the exact location of the (degenerate) event horizon is given by $\rp = |e|.$ Similarly to the classical Reissner-Nordstr\"om solution, the near-horizon geometry of the charged black holes in quadratic gravity, when expanded into the whole manifold, is simply that of Bertotti and Robinson. Similar considerations have been carried out for the boundary conditions of second type which employ the electric charge and the mass of the system as seen by a distant observer. The relations between results obtained within the framework of each method are briefly discussed. 
  We construct solutions of plane symmetric wormholes in the presence of a negative cosmological constant by matching an interior spacetime to the exterior anti-de Sitter vacuum solution. The spatial topology of this plane symmetric wormhole can be planar, cylindrical and toroidal. As usual the null energy condition is necessarily violated at the throat. At the junction surface, the surface stresses are determined. By expressing the tangential surface pressure as a function of several parameters, namely, that of the matching radius, the radial derivative of the redshift function and of the surface energy density, the sign of the tangential surface pressure is analyzed. We then study four specific equations of state at the junction: zero surface energy density, constant redshift function, domain wall equation of state, and traceless surface stress-energy tensor. The equation governing the behavior of the radial pressure, in terms of the surface stresses and the extrinsic curvatures, is also displayed. Finally, we construct a model of a plane symmetric traversable wormhole which minimizes the usage of the exotic matter at the throat, i.e., the null energy condition is made arbitrarily small at the wormhole throat, while the surface stresses on the junction surface satisfy the weak energy condition, and consequently the null energy condition. The construction of these wormholes does not alter the topology of the background spacetime (i.e., spacetime is not multiply-connected), so that these solutions can instead be considered domain walls. Thus, in general, these wormhole solutions do not allow time travel. 
  A relativistic 3-brane can be given a conformally invariant, gauge-type, formulation provided the embedding space is six-dimensional. The implementation of conformal invariance requires the use of a modified measure, independent of the metric in the action. A brane-world scenario without the need of a cosmological constant in 6D can be constructed. Thus, no ``old'' cosmological constant problem appears at this level. 
  Higher dimensional space-time models provide us an alternative interpretation of nature, and give us different dynamical aspects than the traditional four-dimensional space-time models. Motivated by such recent interests, especially for future numerical research of higher-dimensional space-time, we study the dimensional dependence of constraint propagation behavior. The $N+1$ Arnowitt-Deser-Misner evolution equation has matter terms which depend on $N$, but the constraints and constraint propagation equations remain the same. This indicates that there would be problems with accuracy and stability when we directly apply the $N+1$ ADM formulation to numerical simulations as we have experienced in four-dimensional cases. However, we also conclude that previous efforts in re-formulating the Einstein equations can be applied if they are based on constraint propagation analysis. 
  By means of a highly accurate, multi-domain, pseudo-spectral method, we investigate the solution space of uniformly rotating, homogeneous and axisymmetric relativistic fluid bodies. It turns out that this space can be divided up into classes of solutions. In this paper, we present two new classes including relativistic core-ring and two-ring solutions. Combining our knowledge of the first four classes with post-Newtonian results and the Newtonian portion of the first ten classes, we present the qualitative behaviour of the entire relativistic solution space. The Newtonian disc limit can only be reached by going through infinitely many of the aforementioned classes. Only once this limiting process has been consummated, can one proceed again into the relativistic regime and arrive at the analytically known relativistic disc of dust. 
  The whole class of minimally coupled and massive scalar fields which may be responsible for flattening of galactic rotation curves is found. An interesting relation with a class of scalar-tensor theories able to isotropise anisotropic models of Universe is shown. The resulting metric is found and its stability and scalar field properties are tested with respect to the presence of a second scalar field or a small perturbation of the rotation velocity at galactic outer radii. 
  We look for the necessary conditions allowing the Universe isotropisation in presence of a minimally coupled and massive scalar field with a perfect fluid. We conclude that it arises only when the Universe is scalar field dominated, leading to flat spacelike sections and accelerated expansion, and examine the case of a SUGRA theory. 
  Ashtekar and Samuel have shown that Bianchi cosmological models with compact spatial sections must be of Bianchi class A. Motivated by general results on the symmetry reduction of variational principles, we show how to extend the Ashtekar-Samuel results to the setting of weakly locally homogeneous spaces as defined, e.g., by Singer and Thurston. In particular, it is shown that any m-dimensional homogeneous space G/K admitting a G-invariant volume form will allow a compact discrete quotient only if the Lie algebra cohomology of G relative to K is non-vanishing at degree m. 
  We show that positivity of energy for stationary, asymptotically flat, non-singular domains of outer communications is a simple corollary of the Lorentzian splitting theorem. 
  Some unusual relations between stress tensors, conservation and equations of motion are briefly reviewed. 
  Using a recently introduced formalism we discuss slow-roll inflaton from Kaluza-Klein theory without the cylinder condition. In particular, some examples corresponding to polynomic and hyperbolic $\phi$-potentials are studied. We find that the evolution of the fifth coordinate should be determinant for both, the evolution of the early inflationary universe and the quantum fluctuations. 
  We consider a gravitational wave of arbitrary frequency incident on a normal or a super-conductor. The gravitationally induced fields inside the conductor are derived. The outward propagating EM waves are calculated for a low frequency wave on a small sphere and for a high frequency wave incident on a large disk. We estimate for both targets the GW to EM conversion efficiencies and also the magnitude of the superconductor's phase perturbation. 
  In this paper, we address the problem of the dynamics in three dimensional loop quantum gravity with zero cosmological constant. We construct a rigorous definition of Rovelli's generalized projection operator from the kinematical Hilbert space--corresponding to the quantization of the infinite dimensional kinematical configuration space of the theory--to the physical Hilbert space. In particular, we provide the definition of the physical scalar product which can be represented in terms of a sum over (finite) spin-foam amplitudes. Therefore, we establish a clear-cut connection between the canonical quantization of three dimensional gravity and spin-foam models. We emphasize two main properties of the result: first that no cut-off in the kinematical degrees of freedom of the theory is introduced (in contrast to standard `lattice' methods), and second that no ill-defined sum over spins (`bubble' divergences) are present in the spin foam representation. 
  We consider the coupling between three dimensional gravity with zero cosmological constant and massive spinning point particles. First, we study the classical canonical analysis of the coupled system. Then, we go to the Hamiltonian quantization generalizing loop quantum gravity techniques. We give a complete description of the kinematical Hilbert space of the coupled system. Finally, we define the physical Hilbert space of the system of self-gravitating massive spinning point particles using Rovelli's generalized projection operator which can be represented as a sum over spin foam amplitudes. In addition we provide an explicit expression of the (physical) distance operator between two particles which is defined as a Dirac observable. 
  We present a rigorous regularization of Rovellis's generalized projection operator in canonical 2+1 gravity. This work establishes a clear-cut connection between loop quantum gravity and the spin foam approach in this simplified setting. The point of view adopted here provides a new perspective to tackle the problem of dynamics in the physically relevant  3+1 case. 
  We consider the coupling between massive and spinning particles and three dimensional gravity. This allows us to construct geometric operators (distances between particles) as Dirac observables. We quantize the system a la loop quantum gravity: we give a description of the kinematical Hilbert space and construct the associated spin-foam model. We construct the physical disctance operator and consider its quantization. 
  A deduction of a solution of the Einstein's equations, employing the Mitskievich's field theoretic description of perfect fluids, is presented. This solution describes a dust-space-time with a spherical-like symmetry and a NUT-like rotation. This solution is of Petrov type D, and has an isometry group G4. It also admits closed timelike geodesics. It has Minkowski space-time as a limit, when both dust and rotation disappear. 
  We study the cosmology of the Randall-Sundrum brane-world where the Einstein-Hilbert action is modified by curvature correction terms: a four-dimensional scalar curvature from induced gravity on the brane, and a five-dimensional Gauss-Bonnet curvature term. The combined effect of these curvature corrections to the action removes the infinite-density big bang singularity, although the curvature can still diverge for some parameter values. A radiation brane undergoes accelerated expansion near the minimal scale factor, for a range of parameters. This acceleration is driven by the geometric effects, without an inflaton field or negative pressures. At late times, conventional cosmology is recovered. 
  We re-formulate the notion of a Newton-Cartan manifold and clarify the compatibility conditions of a connection with torsion with the Newton-Cartan structure. 
  The anisotropic Bianchi I cosmological model coupled with perfect fluid is quantized in the minisuperspace. The perfect fluid is described by using the Schutz formalism which allows to attribute dynamical degrees of freedom to matter. It is shown that the resulting model is non-unitary. This breaks the equivalence between the many-worlds and dBB interpretations of quantum mechanics. 
  The use of a relational time in quantum mechanics is a framework in which one promotes to quantum operators all variables in a system, and later chooses one of the variables to operate like a ``clock''. Conditional probabilities are computed for variables of the system to take certain values when the ``clock'' specifies a certain time. This framework is attractive in contexts where the assumption of usual quantum mechanics of the existence of an external, perfectly classical clock, appears unnatural, as in quantum cosmology. Until recently, there were problems with such constructions in ordinary quantum mechanics with additional difficulties in the context of constrained theories like general relativity. A scheme we recently introduced to consistently discretize general relativity removed such obstacles. Since the clock is now an object subject to quantum fluctuations, the resulting evolution in the time is not exactly unitary and pure states decohere into mixed states. Here we work out in detail the type of decoherence generated, and we find it to be of Lindblad type. This is attractive since it implies that one can have loss of coherence without violating the conservation of energy. We apply the framework to a simple cosmological model to illustrate how a quantitative estimate of the effect could be computed. For most quantum systems it appears to be too small to be observed, although certain macroscopic quantum systems could in the future provide a testing ground for experimental observation. 
  In this paper a family of non-singular cylindrical perfect fluid cosmologies is derived. The equation of state corresponds to a stiff fluid. The family depends on two independent functions under very simple conditions. A sufficient condition for geodesic completeness is provided. 
  The situation with respect to the experiments is presented of a recently proposed model that gives an explanation of the Pioneer anomalous acceleration $a_{\rm P}$. The model is based on an idea already discovered by Einstein in 1907: the light speed depends on the gravitational potential $\Phi$, so that it is larger the higher if $\Phi$. The potential due to all the mass and energy in the universe increases in time because of its expansion, which has the consequence that light must be slowly accelerating. Moreover it turns out that the observational effects of a universal adiabatic acceleration of light $a_\ell =a_{\rm P}$ and of an extra acceleration towards the  Sun $a_{\rm P}$ of a spaceship would be the same: a blue shift increasing linearly in time, precisely what was observed. The phenomenon would be due to a cosmological acceleration of the proper time of bodies with respect to the coordinate time. It is shown that it agrees with the experimental tests of special relativity and the weak equivalence principle if the cosmological variation of the fine structure constant is zero or very small, as it seems now. 
  I withdraw the previous version of the paper since it contains conceptual and mathematical mistakes. I will soon replace it with a radically revised version. 
  An introduction into the fundamental quests addressed in space missions is given. These quests are the exploration of the relativistic gravitational field, the Universality of Free Fall, the Universality of the Gravitational Redshift, Local Lorentz Invariance, the validity of Einstein's field equations, etc. In each case, the correspoding missions take advantage of the space conditions which are essential for the improvement of the accuracy of the experiments as compared to experiments on ground. A list and a short description of past, current and planned projects is given. Also the key technologies employed in space missions are addressed. 
  BSSN-type evolution equations are discussed. The name refers to the Baumgarte, Shapiro, Shibata, and Nakamura version of the Einstein evolution equations, without introducing the conformal-traceless decomposition but keeping the three connection functions and including a densitized lapse. It is proved that a pseudo-differential first order reduction of these equations is strongly hyperbolic. In the same way, densitized Arnowitt-Deser-Misner evolution equations are found to be weakly hyperbolic. In both cases, the positive densitized lapse function and the spacelike shift vector are arbitrary given fields. This first order pseudodifferential reduction adds no extra equations to the system and so no extra constraints. 
  In this paper we analyse the possibility of constructing singularity-free inhomogeneous cosmological models with a pure radiation field as matter content. It is shown that the conditions for regularity are very easy to implement and therefore there is a huge number of such spacetimes. 
  We study non-degenerate (Petrov type I) silent universes in the presence of a non-vanishing cosmological constant L. In contrast to the L=0 case, for which the orthogonally spatially homogeneous Bianchi type I metrics most likely are the only admissible metrics, solutions are shown to exist when L is positive. The general solution is presented for the case where one of the eigenvalues of the expansion tensor is 0. 
  It was recently shown by Whinnett & Torres \cite{WT} that the phenomenon of spontaneous scalarization (SC) in compact objects (polytropes) was accompanied also by a {\it spontaneous violation of the weak energy condition} (WEC). Notably, by the encounter of negative-energy densities at several star places as measured by a static observer. Here we argue that the negativeness of such densities is not generic of scalar tensor theories of gravity (STT). We support this conclusion by numerical results within a class of STT and by using three realistic models of dense matter. However, we show that the ``angular parts'' of the additional conditions for the WEC to hold $\rho_{\rm eff} + T^{i ({\rm eff})}_{i} \geq 0$ tend to be ``slightly violated'' at the outskirts of the star. 
  The world is four-dimensional according to fundamental physics, governed by basic laws that operate in a spacetime that has no unique division into space and time. Yet our subjective experience is divided into present, past, and future. This paper discusses the origin of this division in terms of simple models of information gathering and utilizing systems (IGUSes). Past, present, and future are not properties of four-dimensional spacetime but notions describing how individual IGUSes process information. Their origin is to be found in how these IGUSes evolved or were constructed. The past, present, and future of an IGUS is consistent with the four-dimensional laws of physics and can be described in four-dimensional terms. The present, for instance, is not a moment of time in the sense of a spacelike surface in spacetime. Rather there is a localized notion of present at each point along an IGUS' world line. The common present of many localized IGUSes is an approximate notion appropriate when they are sufficiently close to each other and have relative velocities much less than that of light. But modes of organization that are different from present, past and future can be imagined that are consistent with the physical laws. We speculate why the present, past, and future organization might be favored by evolution and therefore a cognitive universal. 
  We study interaction of rotating higher dimensional black holes with a brane in space-times with large extra dimensions. We demonstrate that a rotating black hole attached to a brane can be stationary only if the null Killing vector generating the black hole horizon is tangent to the brane world-sheet. The characteristic time when a rotating black hole with the gravitational radius $r_0$ reaches this final stationary state is $T\sim r_0^{p-1}/(G\sigma)$, where $G$ is the higher dimensional gravitational coupling constant, $\sigma$ is the brane tension, and $p$ is the number of extra dimensions. 
  This article starts with the mathematical definition, concrete description, and physical meaning of Cartan's torsion. I proceed with the argumentation that torsion is required for the description of intrinsic spin. Moreover I argue that the duality between curvature and torsion is analogous to the duality between electricity and magnetism. I conclude this article by pointing out that the aligned rotation axes of the galaxies of the Perseus-Pisces supercluster may be interpreted as a topological defect generated by torsion. 
  We generalize the optical geometry to include, not only conformally static spacetimes, but any spacetime that admits a hypersurface orthogonal shearless congruence of worldlines. In the generalized optical geometry, which in general is time dependent, photons move with constant speed along spatial geodesics. Gyroscopes moving along spatial geodesics do not precess. We also classify inertial forces within this framework. For a Schwarzschild black hole the generalized optical geometry works from infinity, across the horizon and arbitrarily close to the singularity. 
  Most black holes are known to be unstable to emitting Hawking radiation (in asymptotically flat spacetime). If the black holes are non-extreme, they have positive temperature and emit thermally. If they are extremal rotating black holes, they still spontaneously emit particles like gravitons and photons. If they are extremal electrically charged black holes, they are unstable to emitting electrons or positrons. The only exception would be extreme magnetically charged black holes if there do not exist any magnetic monopoles for them to emit. However, here we show that even in this case, vacuum polarization causes all magnetic black holes to be unstable to emitting smaller magnetic black holes. 
  The full set of equations governing the evolution of self--gravitating spherically symmetric dissipative fluids with anisotropic stresses is deployed and used to carry out a general study on the behaviour of such systems, in the context of general relativity. Emphasis is given to the link between the Weyl tensor, the shear tensor, the anisotropy of the pressure and the density inhomogeneity. In particular we provide the general, necessary and sufficient, condition for the vanishing of the spatial gradients of energy density, which in turn suggests a possible definition of a gravitational arrow of time. Some solutions are also exhibited to illustrate the discussion. 
  We discuss several topics related to the notion of strong hyperbolicity which are of interest in general relativity. After introducing the concept and showing its relevance we provide some covariant definitions of strong hyperbolicity. We then prove that is a system is strongly hyperbolic with respect to a given hypersurface, then it is also strongly hyperbolic with respect to any near by one.   We then study for how much these hypersurfaces can be deformed and discuss then causality, namely what is the maximal propagation speed in any given direction.   To show that the concept is needed in the area of general relativity we discuss two results for which the theory of symmetric hyperbolic systems shows to be insufficient. The first deals with the hyperbolicity analysis of systems which are second order in space derivatives, they include certain versions of the ADM and the BSSN families of equations.   The second result deals with systems having constraints. The question posed is which hyperbolicity properties, if any, are inherited from the original evolution system by the subsidiary system satisfied by the constraint quantities. The answer is that, subject to some condition on the constraints, if the evolution system is strongly hyperbolic then the subsidiary system is also strongly hyperbolic and the causality properties of both are identical. 
  We obtain expressions for the vacuum expectations of the energy-momentum tensor of the scalar field with an arbitrary coupling to the curvature in an N-dimensional homogeneous isotropic space for the vacuum determined by diagonalization of the Hamiltonian. We generalize the n-wave procedure to N-dimensional homogeneous isotropic space-time. Using the dimensional regularization, we investigate the geometric structure of the terms subtracted from the vacuum energy-momentum tensor in accordance with the n-wave procedure. We show that the geometric structures of the first three subtractions in the n-wave procedure and in the effective action method coincide. We show that all the subtractions in the n-wave procedure in a four- and five-dimensional homogeneous isotropic spaces correspond to a renormalization of the coupling constants of the bare gravitational Lagrangian. 
  The quasigroup approach to the conservation laws (Phys. Rev. D56, R7498 (1997)) is completed by imposing new gauge conditions for asymptotic symmetries. Noether charge associated with an arbitrary element of the Poincar\'e quasialgebra is free from the supertranslational ambiquity and identically vanishes in a flat spacetime 
  Beginning with the stress-energy tensor of an elastic string this paper derives a relativistic string and its form in a parallel transported Fermi frame including its reduction in the Newtonian limit to a Cosserat string. In a Fermi frame gravitational curvature is seen to induce three dominant relative acceleration terms dependent on: position, velocity and position, strain and position, respectively. An example of a string arranged in an axially flowing ring (a lasso) is shown to have a set of natural frequencies that can be parametrically excited by a monochromatic plane gravitational wave. The lasso also exhibits, in common with spinning particles, oscillation about geodesic motion in proportion to spin magnitude and wave amplitude when the spin axis lies in the gravitational wave front. Coordinate free notation is used throughout including the development of the properties of the Fermi frame. 
  We analyse the Hawking(-Unruh) effect for a massive Dirac spinor on the Z_2 quotient of Kruskal spacetime known as the RP^3 geon. There are two distinct Hartle-Hawking-like vacua, depending on the choice of the spin structure, and suitable measurements in the static region (which on its own has only one spin structure) distinguish these two vacua. However, both vacua appear thermal in the usual Hawking temperature to certain types of restricted operators, including operators with support in the asymptotic future (or past). Similar results hold in a family of topologically analogous flat spacetimes, where we show the two vacua to be distinguished also by the shear stresses in the zero-mass limit. As a by-product, we display the explicit Bogolubov transformation between the Rindler-basis and the Minkowski-basis for massive Dirac fermions in four-dimensional Minkowski spacetime. 
  The global properties of static perfect-fluid cylinders and their external Levi-Civita fields are studied both analytically and numerically. The existence and uniqueness of global solutions is demonstrated for a fairly general equation of state of the fluid. In the case of a fluid admitting a non-vanishing density for zero pressure, it is shown that the cylinder's radius has to be finite. For incompressible fluid, the field equations are solved analytically for nearly Newtonian cylinders and numerically in fully relativistic situations. Various physical quantities such as proper and circumferential radii, external conicity parameter and masses per unit proper/coordinate length are exhibited graphically. 
  A model is presented in which the Pioneer anomaly is not related to the motion of the spaceship, but is a consequence of the acceleration of the cosmological proper time $\tau$ with respect to the coordinate parametric time $t$, what is an effect of the background gravitational potential of the entire universe. The light speed, while being constant if defined with respect to $\tau$ ({\it i. e.} as ${\rm d}\ell /{\rm d} \tau$), would suffer an adiabatic secular acceleration, $a_\ell={\rm d}c/{\rm d}t >0$, if defined in terms of $t$ ({\it i. e.} as ${\rm d}\ell /{\rm d} t$). Such an adiabatic acceleration of light, and a small acceleration of the Pioneer towards the Sun $a_{\rm P}$ could be mistaken the one for the other, because they do have the same fingerprint: a blue shift. However, this shift would be quite unrelated to any anomalous motion of the Pioneer, being just an observational effect of the acceleration of light with respect to time $t$. The Pioneer anomaly turns out then to be an interesting case of the dynamics of time, its explanation involving the interplay between the two times $\tau$ and $t$. 
  We study the anisotropies of the Galactic confusion noise background and its effects on LISA data analysis. LISA has two data streams of the gravitational waves signals relevant for low frequency regime. Due to the anisotropies of the background, the matrix for their confusion noises has off-diagonal components and depends strongly on the orientation of the detector plane. We find that the sky-averaged confusion noise level $\sqrt {S(f)}$ could change by a factor of 2 in three months, and would be minimum when the orbital position of LISA is either around the spring or autumn equinox. 
  In a quantum cosmological model consisting of a Euclidean region and a Lorentzian region, Hartle-Hawking's no-bounary wave function, and Linde's wave function and Vilenkin's tunneling wave function are briefly described and compared with each other. We put a particular emphasis on semiclassical gravity from quantum cosmology and compare it with the conventional quantum field theory in curved spacetimes. Finally, we discuss the recent debate on catastrophic particle production in the tunneling universe between Rubakov and Vilenkin within the semiclassical gravity. 
  A Dirac spinor is coupled to topologically massive gravity and the D=3 dimensional action is reduced to D=2 dimensions with a metric that includes both the electromagnetic potential 1-form A and a dilaton scalar \phi. The dimensionnaly reduced spinor is made a mass eigenstate with a (local) chiral rotation. The non-trivial interactions thus induced are discussed. 
  Scale invariance is considered in the context of a gravitational theory where the action, in the first order formalism, is of the form S = \int L_{1} \Phi d^4x + \int L_{2}\sqrt{-g}d^4x where \Phi is a density built out of degrees of freedom independent of the metric. For global scale invariance, a "dilaton" \phi has to be introduced, with non-trivial potentials V(\phi)=f_{1}e^{\alpha\phi} in L_1 and U(\phi) = f_{2}e^{2\alpha\phi} in L_2. In the effective Einstein frame, this leads to a non-trivial \phi potential (of the Morse type) which has a flat region with energy density f_{1}^{2}/4f_{2} as \phi\to\infty. The addition of an R^{2} term produces an effective potential with two connected flat regions: one of the Planck scale, that can be responsible for early inflation, and another for the description of the present universe. 
  The late-time tails of a massive scalar field in the spacetime of black holes are studied numerically. Previous analytical results for a Schwarzschild black hole are confirmed: The late-time behavior of the field as recorded by a static observer is given by $\psi(t)\sim t^{-5/6}\sin [\omega (t)\times t]$, where $\omega(t)$ depends weakly on time. This result is carried over to the case of a Kerr black hole. In particular, it is found that the power-law index of -5/6 depends on neither the multipole mode $\ell$ nor on the spin rate of the black hole $a/M$. In all black hole spacetimes, massive scalar fields have the same late-time behavior irrespective of their initial data (i.e., angular distribution). Their late-time behavior is universal. 
  We present two families of first-order in time and second-order in space formulations of the Einstein equations (variants of the Arnowitt-Deser-Misner formulation) that admit a complete set of characteristic variables and a conserved energy that can be expressed in terms of the characteristic variables. The associated constraint system is also symmetric hyperbolic in this sense, and all characteristic speeds are physical. We propose a family of constraint-preserving boundary conditions that is applicable if the boundary is smooth with tangential shift. We conjecture that the resulting initial-boundary value problem is well-posed. 
  The analysis of certain singularities in scalar-tensor gravity contained in a recent paper is completed, and situations are pointed out in which these singularities cannot occur. 
  Progress and plans are reported for a program of gravitational physics experiments using cryogenic torsion pendula undergoing large amplitude torsional oscillation. The program includes a UC Irvine project to measure the gravitational constant G and joint UC Irvine - U. Washington projects to test the gravitational inverse square law at a range of about 10 cm and to test the weak equivalence principle. 
  Modulo conventional scale factors, the Simon and Simon-Mars tensors are defined for stationary vacuum spacetimes so that their equality follows from the Bianchi identities of the second kind. In the nonvacuum case one can absorb additional source terms into a redefinition of the Simon tensor so that this equality is maintained. Among the electrovacuum class of solutions of the Einstein-Maxwell equations, the expression for the Simon tensor in the Kerr-Newman-Taub-NUT spacetime in terms of the Ernst potential is formally the same as in the vacuum case (modulo a scale factor), and its vanishing guarantees the simultaneous alignment of the principal null directions of the Weyl tensor, the Papapetrou field associated with the timelike Killing vector field, the electromagnetic field of the spacetime and even the Killing-Yano tensor. 
  Each spacecraft in the Laser Interferometer Space Antenna houses a proof mass which follows a geodesic through spacetime. Disturbances which change the proof mass position, momentum, and/or acceleration will appear in the LISA data stream as additive quadratic functions. These data disturbances inhibit signal extraction and must be removed. In this paper we discuss the identification and fitting of monochromatic signals in the data set in the presence of data disturbances. We also present a preliminary analysis of the extent of science result limitations with respect to the frequency of data disturbances. 
  We prove that a completely symmetric and trace-free rank-4 tensor is, up to sign, a Bel-Robinson type tensor, i.e., the superenergy tensor of a tensor with the same algebraic symmetries as the Weyl tensor, if and only if it satisfies a certain quadratic identity. This may be seen as the first Rainich theory result for rank-4 tensors. 
  Doppler effect and Hubble effect in different models of space-time related to the space-time velocity of an observer are considered. The Doppler effect and Doppler shift frequency parameter are connected with the kinematic characteristics of the relative velocity and the relative acceleration of the emitter with respect to the observer (detector). The Hubble effect and Hubble shift frequency parameter are considered in analogous way. It is shown that by the use of the variation of the shift frequency parameter during a time period, considered locally in the proper frame of reference of an observer, one can directly determine the radial (centrifugal, centripetal) relative velocity and acceleration as well as the tangential (Coriolis) relative velocity and acceleration of an astronomical object moving relatively to the observer. All results are obtained on purely kinematic basis without taking into account the dynamic reasons for the considered effect. PACS numbers: 98.80.Jk; 98.62.Py; 04.90.+e; 04.80.Cc 
  We establish that the Einstein tensor takes on a highly symmetric form near the Killing horizon of any stationary but non-static (and non-extremal) black hole spacetime. [This follows up on a recent article by the current authors, gr-qc/0402069, which considered static black holes.] Specifically, at any such Killing horizon -- irrespective of the horizon geometry -- the Einstein tensor block-diagonalizes into ``transverse'' and ``parallel'' blocks, and its transverse components are proportional to the transverse metric. Our findings are supported by two independent procedures; one based on the regularity of the on-horizon geometry and another that directly utilizes the elegant nature of a bifurcate Killing horizon. It is then argued that geometrical symmetries will severely constrain the matter near any Killing horizon. We also speculate on how this may be relevant to certain calculations of the black hole entropy. 
  We investigate in detail the special case of an infinitely thin static cylindrical shell composed of counter-rotating photons on circular geodetical paths separating two distinct parts of Minkowski spacetimes--one inside and the other outside the shell--and compare it to a static disk shell formed by null particles counter-rotating on circular geodesics within the shell located between two sections of flat spacetime. One might ask whether the two cases are not, in fact, merely one. 
  We explore the use of Fabry-P\'erot cavities as high-pass filters for squeezed light, and show that they can increase the sensitivity of interferometric gravitational-wave detectors without the need for long (kilometer scale) filter cavities. We derive the parameters for the filters, and analyze the performance of several possible cavity configurations in the context of a future gravitational-wave interferometer with squeezed light (vacuum) injected into the output port. 
  We present a new three-dimensional fully general-relativistic hydrodynamics code using high-resolution shock-capturing techniques and a conformal traceless formulation of the Einstein equations. Besides presenting a thorough set of tests which the code passes with very high accuracy, we discuss its application to the study of the gravitational collapse of uniformly rotating neutron stars to Kerr black holes. The initial stellar models are modelled as relativistic polytropes which are either secularly or dynamically unstable and with angular velocities which range from slow rotation to the mass-shedding limit. We investigate the gravitational collapse by carefully studying not only the dynamics of the matter, but also that of the trapped surfaces, i.e. of both the apparent and event horizons formed during the collapse. The use of these surfaces, together with the dynamical horizon framework, allows for a precise measurement of the black-hole mass and spin. The ability to successfully perform these simulations for sufficiently long times relies on excising a region of the computational domain which includes the singularity and is within the apparent horizon. The dynamics of the collapsing matter is strongly influenced by the initial amount of angular momentum in the progenitor star and, for initial models with sufficiently high angular velocities, the collapse can lead to the formation of an unstable disc in differential rotation. All the simulations performed with uniformly rotating initial data and a polytropic or ideal-fluid equation of state show no evidence of shocks or of the presence of matter on stable orbits outside the black hole. 
  We investigate the quasinormal mode frequencies for the massless Dirac field in static four dimensional $AdS$ space-time. The separation of the Dirac equation is achieved for the first time in $AdS$ space. Besides the relevance that this calculation can have in the framework of the $AdS/CFT$ correspondence between M-theory on $AdS_4\times S^7$ and SU(N) super Yang-Mills theory on $M_3$, it also serves to fill in a gap in the literature, which has only been concerned with particles of integral spin $0,1,2$. 
  In this paper I show that the Newman-Tamburino spherical metrics always admit a Killing vector, correcting a claim by Collinson and French, (1967 J. Math. Phys. 8 701) and also admit a homothety. A similar calculation is given for the limit of the Newman-Tamburino cylindrical metric. 
  This paper investigates the global properties of a class of spherically symmetric spacetimes. The class contains the maximal development of asymptotically flat spherically symmetric initial data for a wide variety of coupled Einstein-matter systems. For this class, it is proven here that the existence of a single trapped surface or marginally trapped surface implies the completeness of future null infinity and the formation of an event horizon whose area radius is bounded by twice the final Bondi mass. 
  We consider the problem of collapse of a self-gravitating Higgs field, with potential bounded below by a (possibly negative) constant. The behaviour at infinity may be either asymptotically flat or asymptotically AdS. This problem has received much attention as a source for possible violations of weak cosmic censorship in string theory. In this paper, we prove under spherical symmetry that ``first singularities'' arising in the non-trapped region must necessarily emanate from the centre. In particular, this excludes the formation of a certain type of naked singularity which was recently conjectured to occur. 
  In recent work of Allen at. al., heuristic and numerical arguments were put forth to suggest that boundary value problems for black hole evolution, where an appropriate Sommerfeld radiation condition is imposed, would fail to produce Price law tails. The interest in this issue lies in its possible implications for numerical relativity, where black hole evolution is typically studied in terms of such boundary formulations. In this note, it is shown rigorously that indeed, Price law tails do not arise in this case, i.e. that Sommerfeld (and more general) radiation conditions lead to decay faster than any polynomial power. Our setting is the collapse of a spherically symmetric self-gravitating scalar field. We allow an additional gravitationally coupled Maxwell field. The proof also applies to the easier problem of a spherically symmetric solution of the wave equation on a Schwarzschild or Reissner-Nordstrom background. The method relies on previous work of the authors. 
  Recently, static and spherically symmetric configurations of globally regular self-gravitating scalar solitons were found. These configurations are unstable with respect to radial linear perturbations. In this paper we study the dynamical evolution of such configurations and show that, depending on the sign of the initial perturbation, the solitons either collapse to a Schwarzschild black hole or else ``explode'' into an outward moving domain wall. 
  We study gravitational collapse of rapidly rotating relativistic polytropes of the adiabatic index $\Gamma = 1.5$ and 2, in which the spin parameter $q \equiv J/M^{2} > 1$ where $J$ and $M$ are total angular momentum and gravitational mass, in full general relativity.   First, analyzing initial distributions of the mass and the spin parameter inside stars, we predict the final outcome after the collapse. Then, we perform fully general relativistic simulations on assumption of axial and equatorial symmetries and confirm our predictions. As a result of simulations, we find that in contrast with the previous belief, even for stars with $q > 1$, the collapse proceeds to form a seed black hole at central region, and the seed black hole subsequently grows as the ambient fluids accrete onto it. We also find that growth of angular momentum and mass of the seed black hole can be approximately determined from the initial profiles of the density and the specific angular momentum. We define an effective spin parameter at the central region of the stars, $q_{c}$, and propose a new criterion for black hole formation as $q_{c} \alt 1$. Plausible reasons for the discrepancy between our and previous results are clarified. 
  An effect of geometrical phase shift is predicted for a light beam propagating in the field of a gravitational wave. Gravitational radiation detection experiments are proposed using this new effect, the corresponding estimates being given. 
  We study the structure of the phase space of generic models of deformed special relativity that gives rise to a definition of velocity consistent with the deformed Lorentz symmetry. As a byproduct we also determine the laws of transformation of spacetime coordinates. 
  The energy and momentum densities associated with the Weyl metric are calculated using M{\o}ller's energy-momentum complex. These results are compared with the results obtained by using the energy-momentum complexes of Einstein, Landau and Lifshitz, Papapetrou and Bergmann. We show that the aforementioned different prescriptions and that of M{\o}ller do not give the same energy density, while give the same momentum density. 
  We study the asymptotic behaviour of the Bianchi type VI$_0$ universes with a tilted $\gamma$-law perfect fluid. The late-time attractors are found for the full 7-dimensional state space and for several interesting invariant subspaces. In particular, it is found that for the particular value of the equation of state parameter, $\gamma=6/5$, there exists a bifurcation line which signals a transition of stability between a non-tilted equilibrium point to an extremely tilted equilibrium point. The initial singular regime is also discussed and we argue that the initial behaviour is chaotic for $\gamma<2$. 
  In this paper a mathematically precise global (i.e. not the usual local) approach is presented to the variational principles of general relativistic classical field theories.   Problems of the classic (usual) approaches are also discussed in comparison.   The aim of developing a global approach is to provide a possible tool for future efforts on proving global existence theorems of field theoretical solutions. 
  We prove that generic solutions of the vacuum constraint Einstein equations do not possess any global or local space-time Killing vectors, on an asymptotically flat Cauchy surface, or on a compact Cauchy surface with mean curvature close to a constant, or for CMC asymptotically hyperbolic initial data sets. More generally, we show that non-existence of global symmetries implies, generically, non-existence of local ones. As part of the argument, we prove that generic metrics do not possess any local or global conformal Killing vectors. 
  We show that a Maxwell-like system of equations for spatial gravitational fields $\bf g$ and $\bf B$ (latter being the analogy of a magnetic field), modified to include an extra term for the $\bf B$ field in the expression for force, leads to the correct values for the photon deflection angle and for the precession of the periastron. 
  A new quasigroup approach to conservation laws in general relativity is applied to study asymptotically flat at future null infinity spacetime. The infinite-parametric Newman-Unti group of asymptotic symmetries is reduced to the Poincar\'e quasigroup and the Noether charge associated with any element of the Poincar\'e quasialgebra is defined. The integral conserved quantities of energy-momentum and angular momentum are linear on generators of Poincar\'e quasigroup, free of the supertranslation ambiguity, posess the flux and identically equal to zero in Minkowski spacetime. 
  Using the known connection between Schroedinger-like equations and Dirac-like equations in the supersymmetric context, we discuss an extension of FRW barotropic cosmologies in which a Dirac mass-like parameter is introduced. New Hubble cosmological parameters H_K(eta) depending on the Dirac-like parameter are plotted and compared with the standard Hubble case H_0(eta). The new H_K(eta) are complex quantities. The imaginary part is a supersymmetric way of introducing dissipation and instabilities in the barotropic FRW hydrodynamics 
  The LIGO experiment aims to detect and study gravitational waves using ground based laser interferometry. A critical factor to the performance of the interferometers, and a major consideration in the design of possible future upgrades, is isolation of the interferometer optics from seismic noise. We present the results of a detailed program of measurements of the seismic environment surrounding the LIGO interferometers. We describe the experimental configuration used to collect the data, which was acquired over a 613 day period. The measurements focused on the frequency range 0.1-10 Hz, in which the secondary microseismic peak and noise due to human activity in the vicinity of the detectors was found to be particularly critical to interferometer performance. We compare the statistical distribution of the data sets from the two interferometer sites, construct amplitude spectral densities of seismic noise amplitude fluctuations with periods of up to 3 months, and analyze the data for any long term trends in the amplitude of seismic noise in this critical frequency range. 
  We study the separability of the state space of loop quantum gravity. In the standard construction, the kinematical Hilbert space of the diffeomorphism-invariant states is nonseparable. This is a consequence of the fact that the knot-space of the equivalence classes of graphs under diffeomorphisms is noncountable. However, the continuous moduli labeling these classes do not appear to affect the physics of the theory. We investigate the possibility that these moduli could be only the consequence of a poor choice in the fine-tuning of the mathematical setting. We show that by simply choosing a minor extension of the functional class of the classical fields and coordinates, the moduli disappear, the knot classes become countable, and the kinematical Hilbert space of loop quantum gravity becomes separable. 
  We present an algorithm for treating mesh refinement interfaces in numerical relativity. We detail the behavior of the solution near such interfaces located in the strong field regions of dynamical black hole spacetimes, with particular attention to the convergence properties of the simulations. In our applications of this technique to the evolution of puncture initial data with vanishing shift, we demonstrate that it is possible to simultaneously maintain second order convergence near the puncture and extend the outer boundary beyond 100M, thereby approaching the asymptotically flat region in which boundary condition problems are less difficult and wave extraction is meaningful. 
  One of the predictions of quantum gravity phenomenology is that, in situations where Planck-scale physics and the notion of a quantum spacetime are relevant, field propagation will be described by a modified set of laws. Descriptions of the underlying mechanism differ from model to model, but a general feature is that electromagnetic waves will have non-trivial dispersion relations. A physical phenomenon that offers the possibility of experimentally testing these ideas in the foreseeable future is the propagation of high-energy gamma rays from GRB's at cosmological distances. With the observation of non-standard dispersion relations within experimental reach, it is thus important to find out whether there are competing effects that could either mask or be mistaken for this one. In this letter, we consider possible effects from standard physics, due to electromagnetic interactions, classical as well as quantum, and coupling to classical geometry. Our results indicate that, for currently observed gamma-ray energies and estimates of cosmological parameter values, those effects are much smaller than the quantum gravity one if the latter is first-order in the energy; some corrections are comparable in magnitude with the second-order quantum gravity ones, but they have a very different energy dependence. 
  A covariant reformulation of General Relativity is briefly considered from three points of view: geometrodynamics, Lagrange-Euler field theory, and gauge field theory. From a geometrodynamics perspective, a definition of the reference frame as a differential manifold with an affine connection results in separation of the respective contributions of inertial and gravitational fields represented by the affine connection and the tensor of nonmetricity within the Levi-Civita connection of GR. Resulting decomposition of the Einstein curvature tensor into affine and nonmetric parts allows to recast Einstein's field equations in a form invariant with respect to the choice of a reference frame wherein the gravity is described by nonmetricity of space-time . A covariant Lagrangian is proposed leading to the same field equation. All three approaches ultimately lead to the same fully covariant theory of gravitation with a covariant tensor of energy-momentum of the gravitational field and differential and integral conservation laws. The role of the frames of reference, as distinguished from coordinate systems, is discussed. 
  GGR News:   We hear that... by Jorge Pullin  Research Briefs:   Too many coincidences?, by Laura Mersini   The Quest for a Realistic Cosmology in String Theory, by Andrew Chamblin   SFB/TR 7, by A. Gopakumar and D. Petroff   The mock LISA data archive, by John Baker  Conference reports:   Second Gravitational Wave Phenomenology Workshop, by Neil Cornish   Apples with apples II, by Miguel Alcubierre   3 Conferences for 30 years of Gravity at UNAM, by Alejandro Corichi   Building bridges, by Mario Diaz   Strings Meet Loops, by Martin Bojowald   ICGC -- 2004, by Ghanashyam Date   27th Spanish Relativity Meeting (ERE-2003), by J Miralles, J Font, J Pons   Mathematical Relativity: New Ideas and Developments, by Simonetta Frittelli 
  The Einstein-Schrodinger theory is modified to include a large cosmological constant caused by zero-point fluctuations. This ``extrinsic'' cosmological constant which multiplies the symmetric metric is assumed to be nearly cancelled by Schrodinger's ``bare'' cosmological constant which multiplies the nonsymmetric fundamental tensor, such that the total cosmological constant is consistent with measurement. This modified Einstein-Schrodinger theory is expressed in Newman-Penrose form, and tetrad methods are used to confirm that it closely approximates ordinary general relativity and electromagnetism. A solution for the connections in terms of the fundamental tensor is derived in the tetrad frame. The tetrad form of an exact electric monopole solution is shown to approximate the Reissner-Nordstrom solution and to be of Petrov type-D. 
  Trying to combine standard quantum field theories with gravity leads to a breakdown of the usual structure of space-time at around the Planck length, 1.6*10^{-35} m, with possible violations of Lorentz invariance. Calculations of preferred-frame effects in quantum gravity have further motivated high precision searches for Lorentz violation. Here, we explain that combining known elementary particle interactions with a Planck-scale preferred frame gives rise to Lorentz violation at the percent level, some 20 orders of magnitude higher than earlier estimates, unless the bare parameters of the theory are unnaturally strongly fine-tuned. Therefore an important task is not just the improvement of the precision of searches for violations of Lorentz invariance, but also the search for theoretical mechanisms for automatically preserving Lorentz invariance. 
  We study interaction of rotating higher dimensional black holes with a brane in space-times with large extra dimensions. We demonstrate that in a general case a rotating black hole attached to a brane can loose bulk components of its angular momenta. A stationary black hole can have only those components of the angular momenta which are connected with Killing vectors generating transformations preserving a position of the brane. In a final stationary state the null Killing vector generating the black hole horizon is tangent to the brane. We discuss first the interaction of a cosmic string and a domain wall with the 4D Kerr black hole. We then prove the general result for slowly rotating higher dimensional black holes interacting with branes. The characteristic time when a rotating black hole with the gravitational radius $r_0$ reaches this final stationary state is $T\sim r_0^{p-1}/(G\sigma)$, where $G$ is the higher dimensional gravitational coupling constant, $\sigma$ is the brane tension, and $p$ is the number of extra dimensions. 
  A set of world-line deviation equations is derived in the framework of Mathisson-Papapetrou-Dixon description of pseudo-classical spinning particles. They generalize the geodesic deviation equations. We examine the resulting equations for particles moving in the space-time of a plane gravitational wave. 
  It is shown that in spite of a generally accepted concept, there exist nondiffeomorphic solutions to the Cauchy problem in nonempty spacetime, which implies the necessity for canonical complementary conditions. It is nonlocal quantum jumps that provide a canonical global structure of spacetime manifold and, by the same token, the canonical complementary conditions. 
  Spacetime foam, also known as quantum foam, has its origin in quantum fluctuations of spacetime. Arguably it is the source of the holographic principle, which severely limits how densely information can be packed in space. Its physics is also intimately linked to that of black holes and computation. In particular, the same underlying physics is shown to govern the computational power of black hole quantum computers. 
  There is derived, for a conformally flat three-space, a family of linear second-order partial differential operators which send vectors into tracefree, symmetric two-tensors. These maps, which are parametrized by conformal Killing vectors on the three-space, are such that the divergence of the resulting tensor field depends only on the divergence of the original vector field. In particular these maps send source-free electric fields into TT-tensors. Moreover, if the original vector field is the Coulomb field on $\mathbb{R}^3\backslash \lbrace0\rbrace$, the resulting tensor fields on $\mathbb{R}^3\backslash \lbrace0\rbrace$ are nothing but the family of TT-tensors originally written down by Bowen and York. 
  We study a spatially flat Friedmann model containing a pressureless perfect fluid (dust) and a scalar field with an unbounded from below potential of the form $V(\fii)=W_0 - V_0\sinh(\sqrt{3/2}\kappa\fii)$, where the parameters $W_0$ and $V_0$ are arbitrary and $\kappa=\sqrt{8\pi G_N}=M_p^{-1}$. The model is integrable and all exact solutions describe the recollapsing universe. The behavior of the model near both initial and final points of evolution is analyzed. The model is consistent with the observational parameters. We single out the exact solution with the present-day values of acceleration parameter $q_0=0.5$ and dark matter density parameter $\Omega_{\rho 0}=0.3$ describing the evolution within the time approximately equal to $2H_0^{-1}$. 
  Fomalont and Kopeikin have recently succeeded in measuring the velocity-dependent component of the Shapiro time delay of light from a quasar passing behind Jupiter. While there is general agreement that this observation tests the gravitomagnetic properties of the gravitational field, a controversy has emerged over the question of whether the results depend on the speed of light, $c$, or the speed of gravity, $c_g$. By analyzing the Shapiro time delay in a set of ``preferred frame'' models, I demonstrate that this question is ill-posed: the distinction can only be made in the context of a class of theories in which $c\ne c_g$, and the answer then depends on the specific class of theories one chooses. It remains true, however, that for a large class of theories ``close enough'' to general relativity, the leading contribution to the time delay depends on $c$ and not $c_g$; within this class, observations are thus not yet accurate enough to measure the speed of gravity. 
  Dirac equation is solved in the near horizon limit geometry of an extreme Kerr black hole. We decouple equations first as usual, into an axial and angular part. The axial equation turns out to be independent of the mass and is solved exactly. The angular equation reduces, in the massless case, to a confluent Heun equation. In general for nonzero mass, the angular equation is expressible at best, as a set of coupled first order differential equations apt for numerical investigation. The axial potentials corresponding to the associated Schrodinger-type equations and their conserved currents are found. Finally, based on our solution, we verify in a similar way the absence of superradiance for Dirac particles in the near horizon, a result which is well-known within the context of general Kerr background. 
  We review the essential features of the Chaplygin gas cosmological models and provide some examples of appearance of the Chaplygin gas equation of state in modern physics. A possible theoretical basis for the Chaplygin gas in cosmology is discussed. The relation with scalar field and tachyon cosmological models is also considered. 
  It has frequently been claimed in the literature that the classical physical predictions of scalar tensor theories of gravity depend on the conformal frame in which the theory is formulated. We argue that this claim is false, and that all classical physical predictions are conformal-frame invariants. We also respond to criticisms by Vollick [gr-qc/0312041], in which this issue arises, of our recent analysis of the Palatini form of 1/R gravity. 
  Treating macro-black hole as quantum states, and using Brown-York quaselocal gravitational energy definition and Heisenberg uncertainty principle, we find out the classical horizon with singularity spreads into a quantum horizon in which the space-time is non-commutative and the spread range is determined dynamically. A Quantum Field Theory (QFT) model in curved space with quantum horizon is constructed. By using it, the black hole entropy and the Hawking temperature are calculated successfully. The $\phi-$field mode number is predicted and our quantum horizon model favors to support the Minimal Super-symmetric Standard Model. 
  In this paper we construct negatively curved Einstein spaces describing gravitational waves having a solvegeometry wave-front (i.e., the wave-fronts are solvable Lie groups equipped with a left-invariant metric). Using the Einstein solvmanifolds (i.e., solvable Lie groups considered as manifolds) constructed in a previous paper as a starting point, we show that there also exist solvegeometry gravitational waves. Some geometric aspects are discussed and examples of spacetimes having additional symmetries are given, for example, spacetimes generalising the Kaigorodov solution. The solvegeometry gravitational waves are also examples of spacetimes which are indistinguishable by considering the scalar curvature invariants alone. 
  We present a local gluing construction for general relativistic initial data sets. The method applies to generic initial data, in a sense which is made precise. In particular the trace of the extrinsic curvature is not assumed to be constant near the gluing points, which was the case for previous such constructions. No global conditions on the initial data sets such as compactness, completeness, or asymptotic conditions are imposed. As an application, we prove existence of spatially compact, maximal globally hyperbolic, vacuum space-times without any closed constant mean curvature spacelike hypersurface. 
  In the paper are studied the deformations of the planetary orbits caused by the time dependent gravitational potential in the universe. It is shown that the orbits are not axially symmetric and the time dependent potential does not cause perihelion precession. It is found a simple formula for the change of the orbit period caused by the time dependent gravitational potential and it is tested for two binary pulsars. 
  We discuss a covariant generalization of the parametrized post-Newtonian (PPN) formalism in a class of scalar-tensor theories of gravity. It includes an exact construction of a set of global and local (Fermi-like) references frames for an isolated N-body astronomical system as well as PPN multipolar decomposition of gravitational field in these frames. We derive PPN equations of translational and rotational motion of extended bodies taking into account all gravitational multipoles and analyze the body finite-size effects in relativistic dynamics that can be important at the latest stages of orbital evolution of coalescing binary systems. We also reconcile the IAU 2000 resolutions on the general relativistic reference frames in the solar system with the PPN equations of motion of the solar system bodies used in JPL ephemerides. 
  A black hole has characteristic quasi-normal modes that will be excited when it is formed or when the geometry is perturbed. The state of a black hole when the quasi-normal modes are excited is called the gravitational ringing, and detections of it will be a direct confirmation of the existence of black holes. To detect it, a method based on matched filtering needs to be developed. Generically, matched filtering requires a large number of templates, because one has to ensure a proper match of a real gravitational wave with one of template waveforms to keep the detection efficiency as high as possible. On the other hand, the number of templates must be kept as small as possible under limited computational costs. In our previous paper, assuming that the gravitational ringing is dominated by the least-damped (fundamental) mode with the least imaginary part of frequency, we constructed an efficient method for tiling the template space. However, the dependence of the template space metric on the initial phase of a wave was not taken into account. This dependence arises because of an unavoidable mismatch between the parameters of a signal waveform and those given discretely in the template space. In this paper, we properly take this dependence into account and present an improved, efficient search method for gravitational ringing of black holes. 
  In many cases a nonlinear scalar field with potential $V$ can lead to accelerated expansion in cosmological models. This paper contains mathematical results on this subject for homogeneous spacetimes. It is shown that, under the assumption that $V$ has a strictly positive minimum, Wald's theorem on spacetimes with positive cosmological constant can be generalized to a wide class of potentials. In some cases detailed information on late-time asymptotics is obtained. Results on the behaviour in the past time direction are also presented. 
  The Weyl gravity appears to be a very peculiar theory. The contribution of the Weyl linear parameter to the effective geodesic potential is opposite for massive and nonmassive geodesics. However, photon geodesics do not depend on the unknown conformal factor, unlike massive geodesics. Hence light deflection offers an interesting test of the Weyl theory.   In order to investigate light deflection in the setting of Weyl gravity, we first distinguish between a weak field and a strong field approximation. Indeed, the Weyl gravity does not turn off asymptotically and becomes even stronger at larger distances.   We then take full advantage of the conformal invariance of the photon effective potential to provide the key radial distances in Weyl gravity. According to those, we analyze the weak and strong field regime for light deflection. We further show some amazing features of the Weyl theory in the strong regime. 
  In this thesis, we study some aspects of a possible holographic correspondence in two different systems: three dimensional Chern-Simons theory and asymptotically flat space-times. In the former we use simplicial techniques to study CS/WZW correspondence and in particular we construct the discretized WZW partition function for SU(2) group at level 1. In the latter we outline the main characteristics of a field theory living at null infinity invariant under the action of the asymptotic symmetry group: the BMS group. In particular, using fibre bundle techniques, we derive the covariant wave equations for fields carrying BMS representations in order to investigate the nature of boundary degrees of freedom. 
  There is described a spacetime formulation of both nonrelativistic and relativistic elasticity. Specific attention is devoted to the causal structure of the theories and the availability of local existence theorems for the initial-value problem. Much of the presented material is based on joint work of B.G.Schmidt and the author (in Class.Quantum Grav.\textbf{20} (2003), 889-904). 
  According to the teleparallel equivalent of general relativity, curvature and torsion are two equivalent ways of describing the same gravitational field. Despite equivalent, however, they act differently: whereas curvature yields a geometric description, in which the concept of gravitational force is absent, torsion acts as a true gravitational force, quite similar to the Lorentz force of electrodynamics. As a consequence, the right-hand side of a spinless-particle equation of motion (which would represent a gravitational force) is always zero in the geometric description, but not in the teleparallel case. This means essentially that the gravitational coupling prescription can be minimal only in the geometric case. Relying on this property, a new gravitational coupling prescription in the presence of curvature and torsion is proposed. It is constructed in such a way to preserve the equivalence between curvature and torsion, and its basic property is to be equivalent with the usual coupling prescription of general relativity. According to this view, no new physics is connected with torsion, which appears as a mere alternative to curvature in the description of gravitation. An application of this formulation to the equations of motion of both a spinless and a spinning particle is made 
  We show that the solution published in Ref.1 is geodesically complete and singularity-free. We also prove that the solution satisfies the stronger energy and causality conditions, such as global hyperbolicity, causal symmetry and causal stability. A detailed discussion about which assumptions in the singularity theorems are not fulfilled is performed, and we show explicitly that the solution is in accordance with those theorems. A brief discussion of the results is given. 
  An adaptive mesh refinement (AMR) scheme is implemented in a distributed environment using Message Passing Interface (MPI) to find solutions to the nonlinear sigma model. Previous work studied behavior similar to black hole critical phenomena at the threshold for singularity formation in this flat space model. This work is a follow-up describing extensions to distribute the grid hierarchy and presenting tests showing the correctness of the model. 
  This paper reports results from numerical simulations of the gravitational radiation emitted from non--rotating compact objects(both neutron stars and Schwarzschild black holes) as a result of the accretion of matter. A hybrid procedure is adopted: we evolve, in axisymmetry, the linearized equations describing metric and fluid perturbations, coupled with a nonlinear hydrodynamics code that calculates the motion of the accreting matter. The initial matter distribution is shaped in the form of extended quadrupolar shells of dust or perfect fluid. Self--gravity and radiation reaction effects of the accreting fluid are neglected. This idealized setup is used to understand the qualitative features appearing in the energy spectrum of the gravitational wave emission from compact stars or black holes, subject to accretion processes involving extended objects. A comparison for the case of point--like particles falling radially onto black holes is also provided. Our results show that, when the central object is a black hole, the spectrum is far from having only one clear, monochromatic peak at the frequency of the fundamental quasi-normal mode, but it shows a complex pattern, with interference fringes produced by the interaction between the infalling matter and the underlying perturbed spacetime: most of the energy is emitted at frequencies lower than that of the fundamental mode of the black hole. Similar results are obtained for extended shells accreting onto neutron stars, but in this case the stellar fundamental mode is clearly excited. Our analysis shows that the gravitational wave signal driven by accretion is influenced more by the details and dynamics of the process, than by the quasi--normal mode structure of the central object. 
  A numerical simulation is presented here of the evolution of initial data of the kind that was conjectured by Hertog, Horowitz and Maeda to be a violation of cosmic censorship. That initial data is essentially a thick domain wall connecting two regions of anti-deSitter space. The initial data has a free parameter that is the initial size of the wall. The simulation shows no violation of cosmic censorship, but rather the formation of a small black hole. The simulation described here is for a moderate wall size and leaves open the possibility that cosmic censorship might be violated for larger walls. 
  We study topology change in (2+1)D gravity coupling with non-Abelian SO(2,1) Higgs field from the point of view of Morse theory. It is shown that the Higgs potential can be identified as a Morse function. The critical points of the latter ({\em i.e.} loci of change of the spacetime topology) coincide with zeros of the Higgs field. In these critical points two-dimensional metric becomes degenerate, but the curvature remains bounded. 
  In order to detect the rare astrophysical events that generate gravitational wave (GW) radiation, sufficient stability is required for GW antennas to allow long-term observation. In practice, seismic excitation is one of the most common disturbances effecting stable operation of suspended-mirror laser interferometers. A straightforward means to allow more stable operation is therefore to locate the antenna, the ``observatory'', at a ``quiet'' site. A laser interferometer gravitational wave antenna with a baseline length of 20m (LISM) was developed at a site 1000m underground, near Kamioka, Japan. This project was a unique demonstration of a prototype laser interferometer for gravitational wave observation located underground. The extremely stable environment is the prime motivation for going underground. In this paper, the demonstrated ultra-stable operation of the interferometer and a well-maintained antenna sensitivity are reported. 
  This is the first of a couple of papers in which, by exploiting the capabilities of the Hamiltonian approach to general relativity, we get a number of technical achievements that are instrumental both for a disclosure of \emph{new} results concerning specific issues, and for new insights about \emph{old} foundational problems of the theory. The first paper includes: 1) a critical analysis of the various concepts of symmetry related to the Einstein-Hilbert Lagrangian viewpoint on the one hand, and to the Hamiltonian viewpoint, on the other. This analysis leads, in particular, to a re-interpretation of {\it active} diffeomorphisms as {\it passive and metric-dependent} dynamical symmetries of Einstein's equations, a re-interpretation which enables to disclose the (nearly unknown) connection of a subgroup of them to Hamiltonian gauge transformations {\it on-shell}; 2) a re-visitation of the canonical reduction of the ADM formulation of general relativity, with particular emphasis on the geometro-dynamical effects of the gauge-fixing procedure, which amounts to the definition of a \emph{global (non-inertial) space-time laboratory}. This analysis discloses the peculiar \emph{dynamical nature} that the traditional definition of distant simultaneity and clock-synchronization assume in general relativity, as well as the {\it gauge relatedness} of the "conventions" which generalize the classical Einstein's convention. 
  Attention is drawn to the fact that the well-known expression for the red-shift of spectral lines due to a gravitational field may be derived with no recourse to the theory of general relativity. This raises grave doubts over the inclusion of the measurement of this gravitational red-shift in the list of crucial tests of the theory of general relativity. 
  We study global existence problems and asymptotic behavior of higher-dimensional inhomogeneous spacetimes with a compact Cauchy surface in the Einstein-Maxwell-dilaton (EMD) system. Spacelike $T^{D-2}$-symmetry is assumed, where $D\geq 4$ is spacetime dimension. The system of the evolution equations of the EMD equations in the areal time coordinate is reduced to a wave map system, and a global existence theorem for the system is shown. As a corollary of this theorem, a global existence theorem in the constant mean curvature time coordinate is obtained. Finally, for vacuum Einstein gravity in arbitrary dimension, we show existence theorems of asymptotically velocity-terms dominated singularities in the both cases which free functions are analytic and smooth. 
  We show that a singularity can occur at a finite future time in an expanding Friedmann universe even when the density is positive and the density plus the sum of the principal pressures is positive. Explicit examples are constructed and a simple condition is given which can be used to eliminate behaviour of this sort if it is judged to be unphysical. 
  One can view quantum mechanics as a generalization of classical probability theory that provides for pairwise interference among alternatives. Adopting this perspective, we ``quantize'' the classical random walk by finding, subject to a certain condition of ``strong positivity'', the most general Markovian, translationally invariant ``decoherence functional'' with nearest neighbor transitions. 
  The Casimir effect in compact hyperbolic Universes was numerically obtained in previous publications. In this talk, I expose these results. 
  This contribution is divided in two parts. The first part provides a text-book level introduction to gravitational radiation. The key concepts required for a discussion of gravitational-wave physics are introduced. In particular, the quadrupole formula is applied to the anticipated ``bread-and-butter'' source for detectors like LIGO, GEO600, EGO and TAMA300: inspiralling compact binaries. The second part provides a brief review of high frequency gravitational waves. In the frequency range above (say) 100Hz, gravitational collapse, rotational instabilities and oscillations of the remnant compact objects are potentially important sources of gravitational waves. Significant and unique information concerning the various stages of collapse, the evolution of protoneutron stars and the details of the supranuclear equation of state of such objects can be drawn from careful study of the gravitational-wave signal. As the amount of exciting physics one may be able to study via the detections of gravitational waves from these sources is truly inspiring, there is strong motivation for the development of future generations of ground based detectors sensitive in the range from hundreds of Hz to several kHz. 
  Japanese laser interferometric gravitational wave detectors, TAMA300 and LISM, performed a coincident observation during 2001. We perform a coincidence analysis to search for inspiraling compact binaries. The length of data used for the coincidence analysis is 275 hours when both TAMA300 and LISM detectors are operated simultaneously. TAMA300 and LISM data are analyzed by matched filtering, and candidates for gravitational wave events are obtained. If there is a true gravitational wave signal, it should appear in both data of detectors with consistent waveforms characterized by masses of stars, amplitude of the signal, the coalescence time and so on. We introduce a set of coincidence conditions of the parameters, and search for coincident events. This procedure reduces the number of fake events considerably, by a factor $\sim 10^{-4}$ compared with the number of fake events in single detector analysis. We find that the number of events after imposing the coincidence conditions is consistent with the number of accidental coincidences produced purely by noise. We thus find no evidence of gravitational wave signals. We obtain an upper limit of 0.046 /hours (CL $= 90 %$) to the Galactic event rate within 1kpc from the Earth. The method used in this paper can be applied straightforwardly to the case of coincidence observations with more than two detectors with arbitrary arm directions. 
  We present and describe an exact solution of Einstein's equations which represents a snapping cosmic string in a vacuum background with a cosmological constant $\Lambda$. The snapping of the string generates an impulsive spherical gravitational wave which is a particular member of a known family of such waves. The global solution for all values of $\Lambda$ is presented in various metric forms and interpreted geometrically. It is shown to represent the limit of a family of sandwich type N Robinson-Trautman waves. It is also derived as a limit of the C-metric with $\Lambda$, in which the acceleration of the pair of black holes becomes unbounded while their masses are scaled to zero. 
  We study the slow-roll inflationary dynamics in a self-gravitating induced gravity braneworld model with bulk cosmological constant. For E \gg M_{5}^{3}/M^{2}_{2} we find important corrections to the four-dimensional Friedmann equation which bring the standard chaotic inflationary scenario in closer agreement with recent observations. For \lambda/M^{3}_{5} \ll E \ll M^{3}_{5}/M^{2}_{4} we find five-dimensional corrections to the Friedmann equation, which give the known Randall-Sundrum results of the inflationary parameters. 
  We study the dynamics of a bounded gravitational collapsing configuration emitting gravitational waves, where the exterior spacetime is described by Robinson-Trautman geometries. The full nonlinear regime is examined by using the Galerkin method that allows us to reduce the equations governing the dynamics to a finite-dimensional dynamical system, after a proper truncation procedure. Amongst the obtained results of the nonlinear evolution, one of the most impressive is the fact that the distribution of the mass fraction extracted by gravitational wave emission satisfies the distribution law of nonextensive statistics and this result is independent of the initial configurations considered. 
  We study nonlinear gravity theories in both the metric and the Palatini (metric-affine) formalisms. The nonlinear character of the gravity lagrangian in the metric formalism causes the appearance of a scalar source of matter in Einstein's equations that can be interpreted as a quintessence field. However, in the Palatini case no new energy sources appear, though the equations of motion get modified in such a way that usual matter can lead to repulsive gravity at very low densities. Thus, the Palatini formalism could provide a mechanism to explain the recent acceleration of the universe without the necessity of dark energy sources. We also show that in contrast to the metric formalism where only the Einstein frame should be considered as physical, the Palatini formalism allows both the Einstein and the Jordan frames to be physically acceptable. 
  This article derives an optimal (i.e., unbiased, minimum variance) estimator for the pseudo-detector strain for a pair of co-located gravitational wave interferometers (such as the pair of LIGO interferometers at its Hanford Observatory), allowing for possible instrumental correlations between the two detectors. The technique is robust and does not involve any assumptions or approximations regarding the relative strength of gravitational wave signals in the detector pair with respect to other sources of correlated instrumental or environmental noise. An expression is given for the effective power spectral density of the combined noise in the pseudo-detector. This can then be introduced into the standard optimal Wiener filter used to cross-correlate detector data streams in order to obtain an optimal estimate of the stochastic gravitational wave background. In addition, a dual to the optimal estimate of strain is derived. This dual is constructed to contain no gravitational wave signature and can thus be used as on "off-source" measurement to test algorithms used in the "on-source" observation. 
  In suitably chosen domains of space-time, the world function may be a powerful tool for modelling the deflection of light and the time/frequency transfer. In this paper we work out a recursive procedure for expanding the world function into a perturbative series of ascending powers of the Newtonian gravitational constant G. We show rigorously that each perturbation term is given by a line integral taken along the unperturbed geodesic between two points. Once the world function is known, it becomes possible to determine the time transfer functions giving the propagation time of a photon between its emission and its reception. We establish that the direction of a light ray as measured in the 3-space relative to a given observer can be derived from these time transfer functions, even if the metric is not stationary. We show how to deduce these functions up to any given order in G from the perturbative expansion of the world function. To illustrate the method, we carry out the calculation of the world function and of the time transfer function outside a static, spherically symmetric body up to the order G^2, the metric containing three arbitrary parameters. 
  In this paper a new approach to investigation of Quantum and Statistical Mechanics of the Early Universe (Planck scale) - density matrix deformation - is proposed. The deformation is understood as an extension of a particular theory by inclusion of one or several additional parameters in such a way that the initial theory appears in the limiting transition... 
  We show that the analysis presented in a recent comment by Coll and Ferrando \cite{comment} (qr-qc/0312058) is based on the erroneous assumption that the chemical potential and fractional concentration of a {\it mixture} of perfect fluids are unknown variables. 
  We use the Einstein and Papapetrou energy-momentum complexes to calculate the energy and momentum densities of Weyl metric as well as Curzon metric. We show that these two different definitions of energy-momentum complexes do not provide the same energy density for Weyl metric, although they give the same momentum density. We show that, in the case of Curzon metric, these two definitions give the same energy only when $R \to \infty$. Furthermore, we compare these results with those obtained using Landau and Lifshitz, Bergmann and M{\o}ller. 
  In this paper, by applying Newman-Janis algorithm in spherical symmetric metrics, a class of embedded rotating solutions of field equations is presented. These solutions admit non-perfect fluids 
  The infinite cosmological "constant" limit of the de Sitter solutions to Einstein's equation is studied. The corresponding spacetime is a singular, four-dimensional cone-space, transitive under proper conformal transformations, which constitutes a new example of maximally-symmetric spacetime. Grounded on its geometric and thermodynamic properties, some speculations are made in connection with the primordial universe. 
  We investigate the possible bounds which could be placed on alternative theories of gravity using gravitational wave detection from inspiralling compact binaries with the proposed LISA space interferometer. Specifically, we estimate lower bounds on the coupling parameter \omega of scalar-tensor theories of the Brans-Dicke type and on the Compton wavelength of the graviton \lambda_g in hypothetical massive graviton theories. In these theories, modifications of the gravitational radiation damping formulae or of the propagation of the waves translate into a change in the phase evolution of the observed gravitational waveform. We obtain the bounds through the technique of matched filtering, employing the LISA Sensitivity Curve Generator (SCG), available online. For a neutron star inspiralling into a 10^3 M_sun black hole in the Virgo Cluster, in a two-year integration, we find a lower bound \omega > 3 * 10^5. For lower-mass black holes, the bound could be as large as 2 * 10^6. The bound is independent of LISA arm length, but is inversely proportional to the LISA position noise error. Lower bounds on the graviton Compton wavelength ranging from 10^15 km to 5 * 10^16 km can be obtained from one-year observations of massive binary black hole inspirals at cosmological distances (3 Gpc), for masses ranging from 10^4 to 10^7 M_sun. For the highest-mass systems (10^7 M_sun), the bound is proportional to (LISA arm length)^{1/2} and to (LISA acceleration noise)^{-1/2}. For the others, the bound is independent of these parameters because of the dominance of white-dwarf confusion noise in the relevant part of the frequency spectrum. These bounds improve and extend earlier work which used analytic formulae for the noise curves. 
  Asymptotically flat gravitating systems have 10 conserved quantities, which lack proper local densities. It has been hoped that the teleparallel equivalent of Einstein's GR (TEGR, aka GR${}_{||}$) could solve this gravitational energy-momentum localization problem. Meanwhile a new idea: quasilocal quantities, has come into favor. The earlier quasilocal investigations focused on energy-momentum. Recently we considered quasilocal angular momentum for the teleparallel theory and found that the popular expression (unlike our ``covariant-symplectic'' one) gives the correct result only in a certain frame. We now report that the center-of-mass moment, which has largely been neglected, gives an even stronger requirement. We found (independent of the frame gauge) that our ``covariant symplectic'' Hamiltonian-boundary-term quasilocal expression succeeds for all the quasilocal quantities, while the usual expression cannot give the desired center-of-mass moment. We also conclude, contrary to hopes, that the teleparallel formulation appears to have no advantage over GR with regard to localization. 
  A method for interpreting discontinuities of the twist potential of vacuum stationary axisymmetric solutions of Einstein's equations is introduced. Surface densities for the angular momentum of the source can be constructed after solving a linear partial differential equation with boundary conditions at infinity. This formalism is applied to the Kerr metric, obtaining a regularized version of the density calculated with other formalisms. The main result is that the integral defining the total angular momentum is finite for the Kerr metric. 
  Gravitating systems have no well-defined local energy-momentum density. Various quasilocal proposals have been made, however the center-of-mass moment (COM) has generally been overlooked. Asymptotically flat graviating systems have 10 total conserved quantities associated with the Poincar{\'e} symmetry at infinity. In addition to energy-momentum and angular momentum (associated with translations and rotations) there is the boost quantity: the COM. A complete quasilocal formulation should include this quantity. Getting good values for the COM is a fairly strict requirement, imposing the most restrictive fall off conditions on the variables. We take a covariant Hamiltonian approach, associating Hamiltonian boundary terms with quasilocal quantities and boundary conditions. Unlike several others, our {\it covariant symplectic} quasilocal expressions do have the proper asymptotic form for all 10 quantities. 
  A de-Sitter gauge theory of the gravitational field is developed using a spherical symmetric Minkowski space-time as base manifold. The gravitational field is described by gauge potentials and the mathematical structure of the underlying space-time is not affected by physical events. The field equations are written and their solutions without singularities are obtained by imposing some constraints on the invariants of the model. An example of such a solution is given and its dependence on the cosmological constant is studied. A comparison with results obtained in General Relativity theory is also presented.   Keywords: gauge theory, gravitation, singularity, computer algebra 
  We present the results of quality factor measurements for rod samples made of fused silica. To decrease the dissipation we annealed our samples. The highest quality factor that we observed was $Q=(2.03\pm0.01)\times10^8$ for a mode at 384 Hz. This is the highest published value of $Q$ in fused silica measured to date. 
  In loop quantum gravity, modifications to the geometrical density cause a self-interacting scalar field to accelerate away from a minimum of its potential. In principle, this mechanism can generate the conditions that subsequently lead to slow-roll inflation. The consequences for this mechanism of various quantization ambiguities arising within loop quantum cosmology are considered. For the case of a quadratic potential, it is found that some quantization procedures are more likely to generate a phase of slow--roll inflation. In general, however, loop quantum cosmology is robust to ambiguities in the quantization and extends the range of initial conditions for inflation. 
  The Energy Problem (EP) in General Relativity (GR) is analyzed in the context of GR's axiomatic inconsistencies. EP is classified according to its local and global aspects. The local aspects of the EP include noncovariance of the energy-momentum pseudotensor (EMPT) of the gravitational field, non-uniqueness of the EMPT, asymmetry of EMPT, and vanishing metric energy-momentum tensor. The global aspect of the EP relates to the lack of integral conservation laws due to the general difficulties in defining invariant integrals of tensors in non-Euclidean space. These difficulties are related to the lack of precise definition of a reference frame in the GR. A reference frame is defined here as a differential manifold with an affine connection. The resulting unique decomposition of the Levi-Civita connection into its affine and nonmetric parts allows for a covariant definition of the gravitational energy-momentum tensor. It is pointed out that the invariance of the Lagrangian (or action functional) is a necessary but not sufficient condition to secure the covariance of the Lagrange-Euler field theory. A rigorous definition of the Lagrange Field Structure (LFS) on differential manifolds is proposed. A covariant generalization of the first Noether theorem for LFS is obtained. Different approaches to the EP are discussed. 
  The basic motivation of this work is to attempt to explain the rapid primordial inflation and the observed slow late-time inflation by using the Brans-Dicke theory of gravity. We show that the ratio of these two inflation parameters is proportional to the square root of the Brans-Dicke parameter $% \omega$ $(\omega\gg1) $. We also calculate the Hubble parameter $% H$ and the time variation of the time dependent Newtonian gravitational constant $G$ for both regimes. The variation of the Hubble parameter predicted by Brans-Dicke cosmology is shown to be consistent with recent measurements: The value of $H$ in the late-time future is predicted as 0.86 times the present value of $H$. 
  The near horizon limit of the extreme nonlinear black hole is investigated. It is shown that resulting geometry belongs to the AdS2xS2 class with different modules of curvatures of subspaces and could be described in terms of the Lambert functions. It is demonstrated that the considered class of Lagrangians does not admit solutions of the Bertotti-Robinson type. 
  The equilibrium conditions for charges and currents, apparent in exact solutions of the field equations, lead one to regard the Hermitian theory of relativity as the theory of a field endowed with two sources: electromagnetic and colour four-currents. 
  We discuss the Sagnac effect in standard Minkowski coordinates and with an alternative synchronization convention. We find that both approaches lead to the same result without any contradictions. When applying standard coordinates to the complete rim of the rotating disk, a time-lag has to be taken into account which accounts for the global anisotropy. We propose a closed Minkowski space-time as an exact equivalent to the rim of a disk, both in the rotating and non-rotating case. In this way the Sagnac effect can be explained as being purely topological, neglecting the radial acceleration altogether. This proves that the rim of the disk can be treated as an inertial system. In the same context we discuss the twin paradox and find that the standard scenario is equivalent to an unaccelerated version in a closed space-time. The closed topology leads to preferred frame effects which can be detected only globally. The relation of synchronization conventions to the measurement of lengths is discussed in the context of Ehrenfest's paradox. This leads to a confirmation of the classical arguments by Ehrenfest and Einstein. 
  It is shown that space-time may possess the differentiability properties of manifolds as well as the ultraviolet finiteness properties of lattices. Namely, if a field's amplitudes are given on any sufficiently dense set of discrete points this could already determine the field's amplitudes at all other points of the manifold. The criterion for when samples are sufficiently densely spaced could be that they are apart on average not more than at a Planck distance. The underlying mathematics is that of classes of functions that can be reconstructed completely from discrete samples. The discipline is called sampling theory and is at the heart of information theory. Sampling theory establishes the link between continuous and discrete forms of information and is used in ubiquitous applications from scientific data taking to digital audio. 
  Zipoy used spheroidal coordinates to construct a family of static axisymmetric gravitational fields included in Weyl's class. We calculate the mass moment source for the dipole solution using techniques previously developed for magnetic and angular momentum densities and compare it with the Newtonian analog studied by Bonnor and Sackfield. 
  Presented is a summary of studies by the LIGO Scientific Collaboration's Inspiral Analysis Group on the development of possible vetoes to be used in evaluation of data from the first two LIGO science data runs. Numerous environmental monitor signals and interferometer control channels have been analyzed in order to characterize the interferometers' performance. The results of studies on selected data segments are provided in this paper. The vetoes used in the compact binary inspiral analyses of LIGO's S1 and S2 science data runs are presented and discussed. 
  Models of gravity with variable G and Lambda have acquired greater relevance after the recent evidence in favour of the Einstein theory being nonperturbatively renormalizable in the Weinberg sense. The present paper applies the Arnowitt-Deser-Misner (ADM) formalism to such a class of gravitational models. Are modified action functional is then built which reduces to the Einstein-Hilbert action when G is constant, and leads to a power-law growth of the scale factor for pure gravity and for a massless Phi**4 theory in a Universe with Robertson-Walker symmetry, in agreement with the recently developed fixed-point cosmology. Interestingly, the renormalization-group flow at the fixed point is found to be compatible with a Lagrangian description of the running quantities G and Lambda. 
  We use covariant techniques to examine the implications of the dynamical equivalence between geodesic motions and adiabatic hydrodynamic flows. Assuming that the metrics of a geodesically and a non-geodesically moving fluid are conformally related, we calculate and compare their mass densities. The density difference is then expressed in terms of the fundamental physical quantities of the fluid, such as its energy and isotropic pressure. Both the relativistic and the non-relativistic case are examined and their differences identified. Our analysis suggests that observational determinations of astrophysical masses based on purely Keplerian motions could underestimate the available amount of matter. 
  This brief paper investigates the consequences for the metric tensor of space-time when the Weyl tensor (in its conformally invariant form) and the energy-momentum tensor is specified. It is shown that, unless rather special conditions hold, the metric is uniquely determined up to a constant conformal factor. 
  A formulation for stationary axisymmetric electromagnetic fields in general relativity is derived by casting them into the form of an anisotropic fluid. Several simplifications of the formalism are carried out in order to analyze different features of the fields, such as the derivation of electromagnetic sources for the Maxwell field in the form of thin layers, construction of new solutions, and generation techniques. 
  The modified gravity with 1/R term (R being scalar curvature) and the Einstein-Hilbert term is studied by incorporating the phantom scalar field. A number of cosmological solutions are derived in the presence of the phantom field in the perfect fluid background. It is shown the current inflation obtained in the modified gravity is affected by the existence the phantom field. 
  We calculate the Hilbert action for the Bondi-Sachs metrics. It yields the Einstein vacuum equations in a closed form. Following the Dirac approach to constrained systems we investigate the related Hamiltonian formulation. 
  It is often argued that superluminal velocities and nontrivial spacetime topologies, allowed by the theory of relativity, may lead to causal paradoxes. By emphasizing that the notion of causality assumes the existence of a time arrow (TA) that points from the past to the future, the apparent paradoxes appear to be an artefact of the wrong tacit assumption that the relativistic coordinate TA coincides with the physical TA. The latter should be identified with the thermodynamic TA, which, by being absolute and irrotational, does not lead to paradoxes. 
  The equations of motion of two point masses in harmonic coordinates are derived through the third post-Newtonian (3PN) approximation. The problem of self-field regularization (necessary for removing the divergent self-field of point particles) is dealt with in two separate steps. In a first step the extended Hadamard regularization is applied, resulting in equations of motion which are complete at the 3PN order, except for the occurence of one and only one unknown parameter. In a second step the dimensional regularization (in d dimensions) is used as a powerful argument for fixing the value of this parameter, thereby completing the 3-dimensional Hadamard-regularization result. The complete equations of motion and associated energy at the 3PN order are given in the case of circular orbits. 
  The energy and momentum for different cosmological models using various prescriptions are evaluated. In particular, we have focused our attention on the energy and momentum for gravitational waves and discuss the results. It is concluded that there are methods which can provide physically acceptable results. 
  The concept "centre of mass" is analyzed in spaces with torsion free flat linear connection. It is shown that under sufficiently general conditions it is almost uniquely defined, the corresponding arbitrariness in its definition being explicitly described. 
  Backgrounds depending on time and on "angular" variable, namely polarized and unpolarized $S^1 \times S^2$ Gowdy models, are generated as the sector inside the horizons of the manifold corresponding to axisymmetric solutions. As is known, an analytical continuation of ordinary $D$-branes, $iD$-branes allows one to find $S$-brane solutions. Simple models have been constructed by means of analytic continuation of the Schwarzchild and the Kerr metrics. The possibility of studying the $i$-Gowdy models obtained here is outlined with an eye toward seeing if they could represent some kind of generalized $S$-branes depending not only on time but also on an ``angular'' variable. 
  We study coherent states for Bianchi type I cosmological models, as examples of semiclassical states for time-reparametrization invariant systems. This simple model allows us to study explicitly the relationship between exact semiclassical states in the kinematical Hilbert space and corresponding ones in the physical Hilbert space, which we construct here using the group averaging technique. We find that it is possible to construct good semiclassical physical states by such a procedure in this model; we also discuss the sense in which the original kinematical states may be a good approximation to the physical ones, and the situations in which this is the case. In addition, these models can be deparametrized in a natural way, and we study the effect of time evolution on an "intrinsic" coherent state in the reduced phase space, in order to estimate the time for this state to spread significantly. 
  There have been a number of claims in the literature about gravity shielding effects of superconductors and more recently on the weight reduction of superconductors passing through their critical temperature. We report several experiments to test the weight of superconductors under various conditions. First, we report tests on the weight of YBCO and BSCCO high temperature superconductors passing through their critical temperature. No anomaly was found within the equipment accuracy ruling out claimed anomalies by Rounds and Reiss. Our experiments extend the accuracy of previous measurements by two orders of magnitude. In addition, for the first time, the weight of a rotating YBCO superconductor was measured. Also in this case, no weight anomaly could be seen within the accuracy of the equipment used. In addition, also weight measurements of a BSCCO superconductor subjected to extremely-low-frequency (ELF) radiation have been done to test a claim of weight reduction under these conditions by De Aquino, and again, no unusual behavior was found. These measurements put new important boundaries on any inertial effect connected with superconductivity - and consequently on possible space related applications. 
  We give an explicit proof of the result that non trivial conserved n-2 forms for a spin 2 field on a background corresponding to a solution to Einstein's equation (with or without cosmological constant) are characterized uniquely by the Killing vector fields of the background. 
  We study the Einstein-Vlasov system coupled to a nonlinear scalar field with a nonnegative potential in locally spatially homogeneous spacetime, as an expanding cosmological model. It is shown that solutions of this system exist globally in time. When the potential of the scalar field is of an exponential form, the cosmological model corresponds to accelerated expansion. The Einstein-Vlasov system coupled to a nonlinear scalar field whose potential is of an exponential form shows the causal geodesic completeness of the spacetime towards the future. The asymptotic behaviour of solutions of this system in the future time is analysed in various aspects, which shows power-law expansion. 
  In this paper a new method is derived for constructing electromagnetic surface sources for stationary axisymmetric electrovac spacetimes endowed with non-smooth or even discontinuous  Ernst potentials. This can be viewed as a generalization of some classical potential theory results, since lack of continuity of the potential is related to dipole density and lack of smoothness, to monopole density. In particular this approach is useful for constructing the dipole source for the magnetic field. This formalism involves solving a linear elliptic differential equation with boundary conditions at infinity. As an example, two different models of surface densities for the Kerr-Newman electrovac spacetime are derived. 
  We consider charged black holes within dilaton gravity with exponential-linear dependence of action coefficients on dilaton and minimal coupling to quantum scalar fields. This includes, in particular, CGHS and RST black holes in the uncharged limit. For non-extremal configuration quantum correction to the total mass, Hawking temperature, electric potential and metric are found explicitly and shown to obey the first generalized law. We also demonstrate that quantum-corrected extremal black holes in these theories do exist and correspond to the classically forbidden region of parameters in the sense that the total mass $M_{tot}<Q$ ($Q$ is a charge). We show that in the limit $T_{H}\to 0$ (where $T_{H}$ is the Hawking temperature) the mass and geometry of non-extremal configuration go smoothly to those of the extremal one, except from the narrow near-horizon region. In the vicinity of the horizon the quantum-corrected geometry (however small quantum the coupling parameter $\kappa $ would be) of a non-extremal configuration tends to not the quantum-corrected extremal one but to the special branch of solutions with the constant dilaton (2D analog of the Bertotti-Robinson metric) instead. Meanwhile, if $\kappa =0$ exactly, the near-extremal configuration tends to the extremal one. We also consider the dilaton theory which corresponds classically to the spherically-symmetrical reduction from 4D case and show that for the quantum-corrected extremal black hole $M_{tot}>Q$. 
  A model of the Universe as a mixture of a scalar (inflaton or rolling tachyon from the string theory) and a matter field (classical particles) is analyzed. The particles are created at the expense of the gravitational energy through an irreversible process whereas the scalar field is supposed to interact only with itself and to be minimally coupled with the gravitational field. The irreversible processes of particle creation are related to the non-equilibrium pressure within the framework of the extended (causal or second-order) thermodynamic theory. The scalar field (inflaton or tachyon) is described by an exponential potential density added by a parameter which represents its asymptotic value and can be interpreted as the vacuum energy. This model can simulate three phases of the acceleration field of the Universe, namely,(a) an inflationary epoch with a positive acceleration followed by a decrease of the acceleration field towards zero, (b) a past decelerated period where the acceleration field decreases to a maximum negative value followed by an increase towards zero, and (c) a present accelerated epoch. For the energy densities there exist also three distinct epochs which begin with a scalar field dominated period followed by a matter field dominated epoch and coming back to a scalar field dominated phase. 
  A model for the Universe is proposed where it is considered as a mixture of scalar and matter fields. The particle production is due to an irreversible transfer of energy from the gravitational field to the matter field and represented by a non-equilibrium pressure. This model can simulate three distinct periods of the Universe: (a) an accelerated epoch where the energy density of the scalar field prevails over the matter field, (b) a past decelerated period where the energy density of the matter field becomes more predominant than the scalar energy density, and (c) a present acceleration phase where the scalar energy density overcomes the energy density of the matter field. 
  Oscillatons are solutions of the coupled Einstein-Klein-Gordon (EKG) equations that are globally regular and asymptotically flat. By means of a Legendre transformation we are able to visualize the behaviour of the corresponding objects in non-linear gravity where the scalar field has been absorbed by means of the conformal mapping. 
  Within the first order formalism static solutions of generic dilaton gravity in 2D with self-interacting (scalar) matter can be discussed with ease. The question of (non)existence of Killing horizons is addressed and the interplay with asymptotic conditions is investigated. Naturally, such an analysis has to be a global one. A central element in the discussion is the rank of the Jacobi matrix of the underlying dynamical system. With some (pathological) exceptions Killing horizons exist only if it equals to 3. For certain self-interactions asymptotically flat black holes with scalar hair do exist. An example relevant to general relativity is provided. Finally, generalizations are addressed including 2D type 0A string theory as a particular example. Additionally, in a pedagogical appendix the mass definition in dilaton gravity is briefly reviewed and a unique prescription to fix scaling and shift ambiguity is presented. 
  Using numerical techniques, we study the collapse of a scalar field configuration in the Newtonian limit of the spherically symmetric Einstein--Klein--Gordon (EKG) system, which results in the so called Schr\"odinger--Newton (SN) set of equations. We present the numerical code developed to evolve the SN system and topics related, like equilibrium configurations and boundary conditions. Also, we analyze the evolution of different initial configurations and the physical quantities associated to them. In particular, we readdress the issue of the gravitational cooling mechanism for Newtonian systems and find that all systems settle down onto a 0--node equilibrium configuration. 
  We investigate the following question: Consider a small mass, with $\epsilon$ (the ratio of the Schwarzschild radius and the bulk curvature length) much smaller than 1, that is confined to the TeV brane in the Randall-Sundrum I scenario. Does it form a black hole with a regular horizon, or a naked singularity? The metric is expanded in $\epsilon$ and the asymptotic form of the metric is given by the weak field approximation (linear in the mass). In first order of $\epsilon$ we show that the iteration of the weak field solution, which includes only integer powers of the mass, leads to a solution that has a singular horizon. We find a solution with a regular horizon but its asymptotic expansion in the mass also contains half integer powers. 
  We use perturbations in order to study the stability of the Cauchy Horizon in a Reissner-Nordstr\"om space-time. The perturbations are either scalar or gravitational, and indicate some strong instabilities. 
  A cylindrically symmetric perfect fluid spacetime with no curvature singularity is shown. The equation of state for the perfect fluid is that of a stiff fluid. The metric is diagonal and non-separable in comoving coordinates for the fluid. It is proven that the spacetime is geodesically complete and globally hyperbolic. 
  The goal of this article is to present an introduction to loop quantum gravity -a background independent, non-perturbative approach to the problem of unification of general relativity and quantum physics, based on a quantum theory of geometry. Our presentation is pedagogical. Thus, in addition to providing a bird's eye view of the present status of the subject, the article should also serve as a vehicle to enter the field and explore it in detail. To aid non-experts, very little is assumed beyond elements of general relativity, gauge theories and quantum field theory. While the article is essentially self-contained, the emphasis is on communicating the underlying ideas and the significance of results rather than on presenting systematic derivations and detailed proofs. (These can be found in the listed references.) The subject can be approached in different ways. We have chosen one which is deeply rooted in well established physics and also has sufficient mathematical precision to ensure that there are no hidden infinities. In order to keep the article to a reasonable size, and to avoid overwhelming non-experts, we have had to leave out several interesting topics, results and viewpoints; this is meant to be an introduction to the subject rather than an exhaustive review of it. 
  The general relativistic accretion onto a black hole is investigated in which the motion is steady and spherically symmetrical, the gas being at rest at infinity. Two models with different equations of state are compared. Numerical calculations show that the predictions of the models are similar in most aspects. In the ultrarelativistic regime the allowed band of the asymptotic speed of sound and the mass accretion rate can be markedly different. 
  We compute numerically eigenvalues and eigenfunctions of the Laplacian in a three-dimensional hyperbolic space. Applying the results to cosmology, we demonstrate that the methods learned in quantum chaos can be used in other fields of research. 
  We construct a state in the loop quantum gravity theory with zero cosmological constant, which should correspond to the flat spacetime vacuum solution. This is done by defining the loop transform coefficients of a flat connection wavefunction in the holomorphic representation which satisfies all the constraints of quantum General Relativity and it is peaked around the flat space triads. The loop transform coefficients are defined as spin foam state sum invariants of the spin networks embedded in the spatial manifold for the SU(2) quantum group. We also obtain an expression for the vacuum wavefunction in the triad represntation, by defining the corresponding spin networks functional integrals as SU(2) quantum group state sums. 
  In this diplom-arbeit I consider a specific class of "analogue models" of curved spacetime that are specifically based on the use of Bose-Einstein condensates.   As is usual in "analogue models", we are primarily interested in the kinematics of fields and quanta immersed in a curved-space background. We are not directly concerned with the Einstein equations of general relativity. Over the last few years numerous papers concerning "analogue models" have been published, the key result being that in many dynamical systems the perturbations have equations of motion that are governed by an "effective metric" that can often be interpreted in terms of an equivalent gravitational field.   After a brief introduction concerning Bose-Einstein condensates and general relativity, I explain the connection between these two fields. Several specific examples are then explored in a little more detail:   1) Sinks and acoustic black holes [dumb holes].   2) Ring-shaped Laval nozzles and acoustic horizons.   3) the de Sitter universe.   In particular, the de Sitter universe is modelled by a freely expanding condensate obtained by suddenly switching off the trap that normally holds the condensate in place. 
  We present a suite of Mathematica-based computer-algebra packages, termed "Kranc", which comprise a toolbox to convert (tensorial) systems of partial differential evolution equations to parallelized C or Fortran code. Kranc can be used as a "rapid prototyping" system for physicists or mathematicians handling very complicated systems of partial differential equations, but through integration into the Cactus computational toolkit we can also produce efficient parallelized production codes. Our work is motivated by the field of numerical relativity, where Kranc is used as a research tool by the authors. In this paper we describe the design and implementation of both the Mathematica packages and the resulting code, we discuss some example applications, and provide results on the performance of an example numerical code for the Einstein equations. 
  Free scalar field theory on a flat spacetime can be cast into a generally covariant form known as parametrised field theory in which the action is a functional of the scalar field as well as the embedding variables which describe arbitrary, in general curved, foliations of the flat spacetime.  We construct the path integral quantization of parametrised field theory in order to analyse issues at the interface of quantum field theory and general covariance in a path integral context. We show that the measure in the Lorentzian path integral is non-trivial and is the analog of the Fradkin- Vilkovisky measure for quantum gravity. We construct Euclidean functional integrals in the generally covariant setting of parametrised field theory using key ideas of Schleich and show that our constructions imply the existence of non-standard `Wick rotations' of the standard free scalar field 2 point function. We develop a framework to study the problem of time through computations of scalar field 2 point functions. We illustrate our ideas through explicit computation for a time independent 1+1 dimensional foliation. Although the problem of time seems to be absent in this simple example, the general case is still open. We discuss our results in the contexts of the path integral formulation of quantum gravity and the canonical quantization of parametrised field theory. 
  We present a Markov chain Monte Carlo technique for detecting gravitational radiation from a neutron star in laser interferometer data. The algorithm can estimate up to six unknown parameters of the target, including the rotation frequency and frequency derivative, using reparametrization, delayed rejection and simulated annealing. We highlight how a simple extension of the method, distributed over multiple computer processors, will allow for a search over a narrow frequency band. The ultimate goal of this research is to search for sources at a known locations, but uncertain spin parameters, such as may be found in SN1987A. 
  This is a brief summary with comments on selected contributions to the Cosmology and Gravitation section at the $24^{th}$ Brazilian Meeting on Particle and Fields (ENFPC XXIV), held at Caxambu, from September 30 to October 4, 2003. 
  We systematically measured and compared the mechanical losses of various kinds of bulk fused silica. Their quality factors ranged widely from 7x10^5 to 4x10^7, the latter being one of the highest reported among bulk fused silica. We observed frequency-dependent losses and a decrease in the losses upon annealing. 
  In this paper we compare the dimensional method with the Lie groups tactic in order to show the limitations and advantages of each technique. For this purpose we study in detail a perfect fluid cosmological model with time-varying "constants" by using dimensional analysis and the symmetry method. We revise our previous conclusion about the variation of the fine structure constant finding for example that in the radiation predominance era if $\alpha$ varies is only due to the variation of $e^{2}\epsilon_{0}^{-1}$ since $c\hbar=const$ in this era. 
  In this paper we propose a new strategy for gravitational waves detection from coalescing binaries, using IIR Adaptive Line Enhancer (ALE) filters. This strategy is a classical hierarchical strategy in which the ALE filters have the role of triggers, used to select data chunks which may contain gravitational events, to be further analyzed with more refined optimal techniques, like the the classical Matched Filter Technique. After a direct comparison of the performances of ALE filters with the Wiener-Komolgoroff optimum filters (matched filters), necessary to discuss their performance and to evaluate the statistical limitation in their use as triggers, we performed a series of tests, demonstrating that these filters are quite promising both for the relatively small computational power needed and for the robustness of the algorithms used. The performed tests have shown a weak point of ALE filters, that we fixed by introducing a further strategy, based on a dynamic bank of ALE filters, running simultaneously, but started after fixed delay times. The results of this global trigger strategy seems to be very promising, and can be already used in the present interferometers, since it has the great advantage of requiring a quite small computational power and can easily run in real-time, in parallel with other data analysis algorithms. 
  We discuss the inhomogeneous multidimensional mixmaster model in view of appearing, near the cosmological singularity, a scenario for the dimensional compactification in correspondence to an 11-dimensional space-time. Our analysis candidates such a collapsing picture toward the singularity to describe the actual expanding 3-dimensional Universe and an associated collapsed 7-dimensional space. To this end, a conformal factor is determined in front of the 4-dimensional metric to remove the 4-curvature divergences and the resulting Universe expands with a power-law.inflation. Thus we provide an additional peculiarity of the eleven space-time dimensions in view of implementing a geometrical theory of unification. 
  We provide an inhomogeneous solution concerning the dynamics of a real self interacting scalar field minimally coupled to gravity in a region of the configuration space where it performs a slow rolling on a plateau of its potential. During the inhomogeneous de Sitter phase the scalar field dominant term is a function of the spatial coordinates only. This solution specialized nearby the FLRW model allows a classical origin for the inhomogeneous perturbations spectrum. 
  We present exact expressions for the Sagnac effect of Goedel's Universe. For this purpose we first derive a formula for the Sagnac time delay along a circular path in the presence of an arbitrary stationary metric in cylindrical coordinates. We then apply this result to Goedel's metric for two different experimental situations: First, the light source and the detector are at rest relative to the matter generating the gravitational field. In this case we find an expression that is formally equivalent to the familiar nonrelativistic Sagnac time delay. Second, the light source and the detector are rotating relative to the matter. Here we show that for a special rotation rate of the detector the Sagnac time delay vanishes. Finally we propose a formulation of the Sagnac time delay in terms of invariant physical quantities. We show that this result is very close to the analogous formula of the Sagnac time delay of a rotating coordinate system in Minkowski spacetime. 
  Critical collapse of a self-gravitating scalar field in a (2+1)-dimensional spacetime with negative cosmological constant seems to be dominated by a continuously self-similar solution of the field equations without cosmological constant. However, previous studies of linear perturbations in this background were inconclusive. We extend the continuously self-similar solutions to solutions of the field equations with negative cosmological constant, and analyse their linear perturbations. The extended solutions are characterized by a continuous parameter. A suitable choice of this parameter seems to improve the agreement with the numerical results. We also study the dynamics of the apparent horizon in the extended background. 
  Quantum cosmology may permit to determine the initial conditions of the Universe. In particular, it may select a specific model between many possible classical models. In this work, we study a quantum cosmological model based on the string effective action coupled to matter. The Schutz's formalism is employed in the description of the fluid. A radiation fluid is considered. In this way, a time coordinate may be identified and the Wheeler-DeWitt equation reduces in the minisuperspace to a Schr\"odinger-like equation. It is shown that, under some quite natural assumptions, the expectation values indicate a null axionic field and a constant dilatonic field. At the same time the scale factor exhibits a bounce revealing a singularity-free cosmological model. In some cases, the mininum value of the scale factor can be related to the value of gravitational coupling. 
  We study stationary string configurations in a space-time of a higher-dimensional rotating black hole. We demonstrate that the Nambu-Goto equations for a stationary string in the 5D Myers-Perry metric allow a separation of variables. We present these equations in the first-order form and study their properties. We prove that the only stationary string configuration which crosses the infinite red-shift surface and remains regular there is a principal Killing string. A worldsheet of such a string is generated by a principal null geodesic and a timelike at infinity Killing vector field. We obtain principal Killing string solutions in the Myers-Perry metrics with an arbitrary number of dimensions. It is shown that due to the interaction of a string with a rotating black hole there is an angular momentum transfer from the black hole to the string. We calculate the rate of this transfer in a spacetime with an arbitrary number of dimensions. This effect slows down the rotation of the black hole. We discuss possible final stationary configurations of a rotating black hole interacting with a string. 
  Rapid growth of constraints is often observed in free evolutions of highly gravitating systems. To alleviate this problem we investigate the effect of adding spatial derivatives of the constraints to the right hand side of the evolution equations, and we look at how this affects the character of the system and the treatment of boundaries. We apply this technique to two formulations of Maxwell's equations, the so-called fat Maxwell and the Knapp-Walker-Baumgarte systems, and obtain mixed hyperbolic-parabolic problems in which high frequency constraint violations are damped. Constraint-preserving boundary conditions amount to imposing Dirichlet boundary conditions on constraint variables, which translate into Neumann-like boundary conditions for the main variables. The success of the numerical tests presented in this work suggests that this remedy may bring benefits to fully nonlinear simulations of General Relativity. 
  We consider a scalar field with a negative kinetic term minimally coupled to gravity. We obtain an exact non-static spherically symmetric solution which describes a wormhole in cosmological setting. The wormhole is shown to connect two homogeneous spatially flat universes expanding with acceleration. Depending on the wormhole's mass parameter $m$ the acceleration can be constant (the de Sitter case) or infinitely growing. 
  It is a common belief now that the explanation of the microscopic origin of the Bekenstein-Hawking entropy of black holes should be available in quantum gravity theory, whatever this theory will finally look like. Calculations of the entropy of certain black holes in string theory do support this point of view. In the last few years there also appeared a hope that an understanding of black hole entropy may be possible even without knowing the details of quantum gravity. The thermodynamics of black holes is a low energy phenomenon, so only a few general features of the fundamental theory may be really important. The aim of this review is to describe some of the proposals in this direction and the results obtained. 
  The Bianchi IX model has been used often to investigate the structure close to singularities of general relativity. Its classical chaos is expected to have, via the BKL scenario, implications even for the approach to general inhomogeneous singularities. Thus, it is a popular model to test consequences of modifications to general relativity suggested by quantum theories of gravity. This paper presents a detailed proof that modifications coming from loop quantum gravity lead to a non-chaotic effective behavior. The way this is realized, independently of quantization ambiguities, suggests a new look at initial and final singularities. 
  If the cosmological constant $\Lambda $ can not be neglected, we will show in this short report, that graviton should have a rest mass $m_g=\sqrt{2\Lambda }=\sqrt{2\Lambda}\hbar c^{-1}$ different from zero. 
  We study the Hamiltonian formulation of Plebanski theory in both the Euclidean and Lorentzian cases. A careful analysis of the constraints shows that the system is non regular, i.e. the rank of the Dirac matrix is non-constant on the non-reduced phase space. We identify the gravitational and topological sectors which are regular sub-spaces of the non-reduced phase space. The theory can be restricted to the regular subspace which contains the gravitational sector. We explicitly identify first and second class constraints in this case. We compute the determinant of the Dirac matrix and the natural measure for the path integral of the Plebanski theory (restricted to the gravitational sector). This measure is the analogue of the Leutwyler-Fradkin-Vilkovisky measure of quantum gravity. 
  We study equations of a non-gauge vector field in a spherically symmetric static metric. The constant vector field with a scale arrangement of components: the time component about the Planck mass m_{Pl} and the radial component about M suppressed with respect to the Planck mass, serves as a source of metric reproducing flat rotation curves in dark halos of spiral galaxies, so that the velocity of rotation v_0 is determined by the hierarchy of scales: \sqrt{2} v_0^2= M/m_{Pl}, and M\sim 10^{12} GeV. A natural estimate of Milgrom's acceleration about the Hubble rate is obtained. 
  In this work we compile a few differential equations (ODEs) that arise from the relativistic equations in cosmological models that consider the ``constants'' as scalars functions dependent on time and they are described as perfect as well as viscous fluids. The general idea of the paper is to show how to solve the equations of the models through dimensional techniques (self-similarity). The results are compared with those obtained by other authors and new solutions are introduced. 
  We present a flat (K=0) cosmological model, described by a perfect fluid with the ``constants'' $G,c$ and $\Lambda$ varying with cosmological time $t$. We introduce Planck\'s ``constant'' $\hbar$ in the field equations through the equation of state for the energy density of radiation. We then determine the behaviour of the ``constants'' by using the zero divergence of the second member of the modified Einstein\'s field equations i.e. $div(\frac{G}{c^{4}}T_{i}^{j}+\delta_{i}^{j}\Lambda)=0,$ together with the equation of state and the Einstein cosmological equations. Assuming realistic physical and mathematical conditions we obtain a consistent result with $\hbar c=constant$. In this way we obtain gauge invariance for the Schr\"{o}dinger equation and the behaviour of the remaining ``constants'' 
  We study the evolution of a flat Friedmann-Robertson-Walker Universe, filled with a bulk viscous cosmological fluid, in the presence of time varying ``constants''. The dimensional analysis of the model suggests a proportionality between the bulk viscous pressure of the dissipative fluid and the energy density. On using this assumption and with the choice of the standard equations of state for the bulk viscosity coefficient, temperature and relaxation time, the general solution of the field equations can be obtained, with all physical parameters having a power-law time dependence. The symmetry analysis of this model, performed by using Lie group techniques, confirms the unicity of the solution for this functional form of the bulk viscous pressure. In order to find another possible solution we relax the hypotheses assuming a concrete functional dependence for the ``constants''. 
  In this paper, we study in detail a perfect fluid cosmological model with time-varying "constants" using dimensional analysis and the symmetry method. We examine the case of variable "constants" in detail without considering the perfect fluid model as a limiting case of a model with a causal bulk viscous fluid as discussed in a recent paper. We obtain some new solutions through the Lie method and show that when matter creation is considered, these solutions are physically relevant. 
  The goal of this paper is to present the physics behind Brill initial data sets as an excellent tool for numerical experiments of axisymmetric spacetimes, data sets which are practical applications of HPC for numerical solutions of Einstein's vacuum equations. 
  A suspension-point interferometer (SPI) is an auxiliary interferometer for active vibration isolation, implemented at the suspension points of the mirrors of an interferometric gravitational wave detector. We constructed a prototype Fabry-Perot interferometer equipped with an SPI and observed vibration isolation in both the spectrum and transfer function. The noise spectrum of the main interferometer was reduced by 40 dB below 1 Hz. Transfer function measurements showed that the SPI also produced good vibration suppression above 1 Hz. These results indicate that SPI can improve both the sensitivity and the stability of the interferometer. 
  Recent measurements of the Chandra satellite have shown that a supermassive black hole of $M = 2.6 \times 10^{6} M_{\odot}$ is located in the Galactic Center; it seems probable that, from other observations, this fact is common in the majority of galaxies. On the other hand, GRB explosions are typical phenomenon linked to the galactic dynamics. In the present paper we discuss the possibility that GRBs are tidal disruption of stars by supermassive black holes located in the center of galaxies. This conjecture can be tested by a gravitational wave detector of the class of AURIGA. 
  A general formalism for the dynamics of non rotating cylindrical thin-shell wormholes is developed. The time evolution of the throat is explicitly obtained for thin-shell wormholes whose metric has the form associated to local cosmic strings. It is found that the throat collapses to zero radius, remains static or expands forever, depending only on the sign of its initial velocity. 
  Black holes arising in the context of scalar-tensor gravity theories, where the scalar field is non-minimally coupled to the curvature term, have zero surface gravity. Hence, it is generally stated that their Hawking temperature is zero, irrespectivelly of their gravitational and scalar charges. The proper analysis of the Hawking temperature requires to study the propagation of quantum fields in the space-time determined by these objects. We study scalar fields in the vicinity of the horizon of these black holes. It is shown that the scalar modes do not form an orthonormal set. Hence, the Hilbert space is ill-definite in this case, and no notion of temperature can be extracted for such objects. 
  A discretization of a continuum theory with constraints or conserved quantities is called mimetic if it mirrors the conserved laws or constraints of the continuum theory at the discrete level. Such discretizations have been found useful in continuum mechanics and in electromagnetism. We have recently introduced a new technique for discretizing constrained theories. The technique yields discretizations that are consistent, in the sense that the constraints and evolution equations can be solved simultaneously, but it cannot be considered mimetic since it achieves consistency by determining the Lagrange multipliers. In this paper we would like to show that when applied to general relativity linearized around a Minkowski background the technique yields a discretization that is mimetic in the traditional sense of the word. We show this using the traditional metric variables and also the Ashtekar new variables, but in the latter case we restrict ourselves to the Euclidean case. We also argue that there appear to exist conceptual difficulties to the construction of a mimetic formulation of the full Einstein equations, and suggest that the new discretization scheme can provide an alternative that is nevertheless close in spirit to the traditional mimetic formulations. 
  The consequences of taking the generalized Chaplygin gas as the dark energy constituent of the Universe on the gravitational waves are studied and the spectrum obtained from this model, for the flat case, is analyzed. Besides its importance for the study of the primordial Universe, the gravitational waves represent an additional perspective (besides the CMB temperature and polarization anisotropies) to evaluate the consistence of the different dark energy models and establish better constraints to their parameters. The analysis presented here takes this fact into consideration to open one more perspective of verification of the generalized Chapligin gas model applicability. Nine particular cases are compared: one where no dark energy is present; two that simulate the $\Lambda$-CDM model; two where the gas acts like the traditional Chaplygin gas; and four where the dark energy is the generalized Chaplygin gas. The different spectra permit to distinguish the $\Lambda$-CDM and the Chaplygin gas scenarios. 
  In a purely relational theory there exists a tension between the relational character of the theory and the existence of quantities like distance and duration. We review this issue in the context of the Leibniz-Clarke correspondence. We then address this conflict by showing that a purely relational definition of length and time can be given, provided the dynamics of the theory is known. We further show that in such a setting it is natural to expect Lorentz transformations to describe the mapping between different observers. We then comment on how these insights can be used to make progress in the search for a theory of quantum gravity. 
  We review an idea that uses details of the quasinormal mode spectrum of a black hole to obtain the Bekenstein-Hawking entropy of $A/4$ in Loop Quantum Gravity. We further comment on a recent proposal concerning the quasinormal mode spectrum of rotating black holes. We conclude by remarking on a recent proposal to include supersymmetry. 
  We calculate puncture initial data corresponding to both single and binary black hole solutions of the constraint equations by means of a pseudo-spectral method applied in a single spatial domain. Introducing appropriate coordinates, these methods exhibit rapid convergence of the conformal factor and lead to highly accurate solutions. As an application we investigate small mass ratios of binary black holes and compare these with the corresponding test mass limit that we obtain through a semi-analytical limiting procedure. In particular, we compare the binding energy of puncture data in this limit with that of a test particle in the Schwarzschild spacetime and find that it deviates by 50% from the Schwarzschild result at the innermost stable circular orbit of Schwarzschild, if the ADM mass at each puncture is used to define the local black hole masses. 
  The first terms of the general solution for an asymptotically flat stationary axisymmetric vacuum spacetime endowed with an equatorial symmetry plane are calculated from the corresponding Ernst potential up to seventh order in the radial pseudospherical coordinate. The metric is used to determine the influence of high order multipoles in the perihelion precession of an equatorial orbit and in the node line precession of a non-equatorial orbit with respect to a geodesic circle. Both results are written in terms of invariant quantities such as the Geroch-Hansen multipoles and the energy and angular momentum of the orbit. 
  We calculate the response functions of a freely falling Unruh detector in de Sitter space coupled to scalar fields of different coupling to the curvature, including the minimally coupled massless case. Although the responses differ strongly in the infrared as a consequence of the amplification of superhorizon modes, the energy levels of the detector are thermally populated. 
  When using black hole excision to numerically evolve a black hole spacetime with no continuous symmetries, most 3+1 finite differencing codes use a Cartesian grid. It's difficult to do excision on such a grid, because the natural $r = \text{constant}$ excision surface must be approximated either by a very different shape such as a contained cube, or by an irregular and non-smooth "LEGO(tm) sphere" which may introduce numerical instabilities into the evolution. In this paper I describe an alternate scheme, which uses multiple $\{r \times (\text{angular coordinates}) \}$ grid patches, each patch using a different (nonsingular) choice of angular coordinates. This allows excision on a smooth $r = \text{constant}$ 2-sphere.   I discuss the key design choices in such a multiple-patch scheme, including the choice of ghost-zone versus internal-boundary treatment of the interpatch boundaries, the number and shape of the patches, the details of how the ghost zones are "synchronized" by interpolation from neighboring patches, the tensor basis for the Einstein equations in each patch, and the handling of non-tensor field variables such as the BSSN $\tilde{\Gamma}^i$.   I present sample numerical results from a prototype implementation of this scheme. This code simulates the time evolution of the (asymptotically flat) spacetime around a single (excised) black hole, using 4th-order finite differencing in space and time. Using Kerr initial data with $J/m^2 = 0.6$, I present evolutions to $t \gtsim 1500m$. The lifetime of these evolutions appears to be limited only by outer boundary instabilities, not by any excision instabilities or by any problems inherent to the multiple-patch scheme. 
  In the paper is presented a new gravitational theory, which is in accordance of the experimental results which confirm the general relativity. Some new tests in the solar system are proposed. 
  We consider branes $N=I\times\so$, where $\so$ is an $n$\ndash dimensional space form, not necessarily compact, in a Schwarzschild-AdS_{(n+2)} bulk $\mc N$. The branes have a big crunch singularity. If a brane is an ARW space, then, under certain conditions, there exists a smooth natural transition flow through the singularity to a reflected brane $\hat N$, which has a big bang singularity and which can be viewed as a brane in a reflected Schwarzschild-AdS_{(n+2)} bulk $\hat{\mc N}$. The joint branes $N\uu \hat N$ can thus be naturally embedded in $R^2\times \so$, hence there exists a second possibility of defining a smooth transition from big crunch to big bang by requiring that $N\uu\hat N$ forms a $C^\infty$-hypersurface in $R^2\times \so$. This last notion of a smooth transition also applies to branes that are not ARW spaces, allowing a wide range of possible equations of state. 
  In this paper we discuss and compare a node-only LAGEOS-LAGEOS II combination and a node-only LAGEOS-LAGEOS II-Ajisai-Jason1 combination for the determination of the Lense-Thirring effect. The new combined EIGEN-CG01C Earth gravity model has been adopted. The second combination cancels the first three even zonal harmonics along with their secular variations but introduces the non-gravitational perturbations of Jason1. The first combination is less sensitive to the non-conservative forces but is sensitive to the secular variations of the uncancelled even zonal harmonics of low degree J4 and J6 whose impact grows linearly in time. 
  Over the last few years numerous papers concerning analog models for gravity have been published. It was shown that the dynamical equation of several systems (e.g. Bose-Einstein condensates with a sink or a vortex) have the same wave equation as light in a curved-space (e.g. black holes). In the last few months several papers were released which deal with simulations of the universe.   In this article the de-Sitter universe will be compared with a freely expanding three-dimensional spherical Bose-Einstein condensate. Initially the condensate is in a harmonic trap, which suddenly will be switched off. At the same time a small perturbation will be injected in the center of the condensate cloud.   The motion of the perturbation in the expanding condensate will be discussed, and after some transformations the similarity to an expanding universe will be shown. 
  Searches for binary inspiral signals in data collected by interferometric gravitational wave detectors utilize matched filtering techniques. Although matched filtering is optimal in the case of stationary Gaussian noise, data from real detectors often contains "glitches" and episodes of excess noise which cause filter outputs to ring strongly. We review the standard \chi^2 statistic which is used to test whether the filter output has appropriate contributions from several different frequency bands. We then propose a new type of waveform consistency test which is based on the time history of the filter output. We apply one such test to the data from the first LIGO science run and show that it cleanly distinguishes between true inspiral waveforms and large-amplitude false signals which managed to pass the standard \chi^2 test. 
  We analyze the action $\int d^4x \sqrt{\det||{\cal B} g_{\mu\nu}+ {\cal C}   R_{\mu\nu}}||$ as a possible alternative or addition to the Einstein gravity. Choosing a particular form of ${\cal B}(R)= \sqrt {R}$ we can restore the Einstein gravity and, if ${\cal B}=m^2$, we obtain the cosmological constant term. Taking ${\cal B} = m^2 + {\cal B}_1 R$ and expanding the action in $ 1/m^2$, we obtain as a leading term the Einstein Lagrangian with a cosmological constant proportional to $m^4$ and a series of higher order operators. In general case of non-vanishing ${\cal B}$ and ${\cal C}$ new cosmological solutions for the Robertson-Walker metric are obtained. 
  A scalar-tensor bimetric gravity model of early universe cosmology is reviewed. The metric frame with a variable speed of light (VSL) and a constant speed of gravitational waves is used to describe a Friedmann-Robertson-Walker universe. The Friedmann equations are solved for a radiation dominated equation of state and the power spectrum is predicted to be scale invariant with a scalar mode spectral index $n_s=0.97$. The scalar modes are born in a ground state superhorizon and the fluctuation modes are causally connected by the VSL mechanism. The cosmological constant is equated to zero and there is no significant dependence on the scalar field potential energy. A possible way of distinguishing the metric gravity model from standard inflationary models is discussed. 
  If quantum gravity violates Lorentz symmetry, the prospects for observational guidance in understanding quantum gravity improve considerably. This article briefly reviews previous work on Lorentz violation (LV) and discusses aspects of the effective field theory framework for parametrizing LV effects. Current observational constraints on LV are then summarized, focusing on effects in QED at order E/M_Planck. 
  We investigate geodesics in specific Kundt type N (or conformally flat) solutions to Einstein's equations. Components of the curvature tensor in parallelly transported tetrads are then explicitly evaluated and analyzed. This elucidates some interesting global properties of the spacetimes, such as an inherent rotation of the wave-propagation direction, or the character of singularities. In particular, we demonstrate that the characteristic envelope singularity of the rotated wave-fronts is a (non-scalar) curvature singularity, although all scalar invariants of the Riemann tensor vanish there. 
  We re-derive a formula relating the areal and luminosity distances, entirely in the framework of the classical Maxwell theory, assuming a geometric-optics type condition. 
  In relation to the BSSN formulation of the Einstein equations, we write down the boundary conditions that result from the vanishing of the projection of the Einstein tensor normally to a timelike hypersurface. Furthermore, by setting up a well-posed system of propagation equations for the constraints, we show explicitly that there are three constraints that are incoming at the boundary surface and that the boundary equations are linearly related to them. This indicates that such boundary conditions play a role in enforcing the propagation of the constraints in the region interior to the boundary. Additionally, we examine the related problem for a strongly hyperbolic first-order reduction of the BSSN equations and determine the characteristic fields that are prescribed by the three boundary conditions, as well as those that are left arbitrary. 
  Second-order formulations of the 3+1 Einstein equations obtained by eliminating the extrinsic curvature in terms of the time derivative of the metric are examined with the aim of establishing whether they are well posed, in cases of somewhat wide interest, such as ADM, BSSN and generalized Einstein-Christoffel. The criterion for well-posedness of second-order systems employed is due to Kreiss and Ortiz. By this criterion, none of the three cases are strongly hyperbolic, but some of them are weakly hyperbolic, which means that they may yet be well posed but only under very restrictive conditions for the terms of order lower than second in the equations (which are not studied here). As a result, intuitive transferences of the property of well-posedness from first-order reductions of the Einstein equations to their originating second-order versions are unwarranted if not false. 
  We explore the different geometric structures that can be constructed from the class of pairs of 2nd order PDE's that satisfy the condition of a vanishing generalized W\"{u}nschmann invariant. This condition arises naturally from the requirement of a vanishing torsion tensor. In particular, we find that from this class of PDE's we can obtain all four-dimensional conformal Lorentzian metrics as well as all Cartan normal conformal O(4,2) connections.   To conclude, we briefly discuss how the conformal Einstein equations can be imposed by further restricting our class of PDE's to those satisfying additional differential conditions. 
  In this work we apply the de Broglie-Bohm interpretation of quantum mechanics to the quantized spherically symmetric black-hole coupled to a massless scalar field. The wave-functional used was first obtained by Tomimatsu using the standard ADM quantization and a gauge that places the observer close to the black-hole horizon. Using the causal interpretation, we compute quantum trajectories determined by the initial conditions. We show that the quantum trajectories for the black-hole mass can either increase or decrease with time. The quantum trajectories that show increasing mass represent the usual black-hole behavior of continuous energy absorption. The mass-decreasing quantum trajectories are a purely quantum mechanical phenomena. They can be physically interpreted as describing a black-hole that evaporates. 
  We propose additional conditions (beyond those considered in our previous papers) that should be imposed on Wick products and time-ordered products of a free quantum scalar field in curved spacetime. These conditions arise from a simple ``Principle of Perturbative Agreement'': For interaction Lagrangians $L_1$ that are such that the interacting field theory can be constructed exactly--as occurs when $L_1$ is a ``pure divergence'' or when $L_1$ is at most quadratic in the field and contains no more than two derivatives--then time-ordered products must be defined so that the perturbative solution for interacting fields obtained from the Bogoliubov formula agrees with the exact solution. The conditions derived from this principle include a version of the Leibniz rule (or ``action Ward identity'') and a condition on time-ordered products that contain a factor of the free field $\phi$ or the free stress-energy tensor $T_{ab}$. The main results of our paper are (1) a proof that in spacetime dimensions greater than 2, our new conditions can be consistently imposed in addition to our previously considered conditions and (2) a proof that, if they are imposed, then for {\em any} polynomial interaction Lagrangian $L_1$ (with no restriction on the number of derivatives appearing in $L_1$), the stress-energy tensor $\Theta_{ab}$ of the interacting theory will be conserved. Our work thereby establishes (in the context of perturbation theory) the conservation of stress-energy for an arbitrary interacting scalar field in curved spacetimes of dimension greater than 2. Our approach requires us to view time-ordered products as maps taking classical field expressions into the quantum field algebra rather than as maps taking Wick polynomials of the quantum field into the quantum field algebra. 
  In this paper we consider vacuum Kasner spacetimes, focusing on those that can be parametrized as linear perturbations of the special Petrov type D case. For these quasi-D Kasner models we first investigate the modification to the principal null directions, then a Teukolsky Master Equation for fields of any spin, considering in particular the quasi-D models as curvature perturbations of the type D background. Considering the speciality index and the principal null directions and comparing the results for the exact solutions and those for the perturbative ones, this simple Kasner example allows us to clarify that perturbed spacetime do not retain in general the speciality character of the background. There are four distinct principal null directions, although they are not necessarily first order perturbations of the background principal null directions, as our example of the quasi-D Kasner models shows.   For the quasi-D Kasner models the use of a Teukolsky Master Equation, a classical tool for studying black hole perturbations, allows us to show, from a completely new point of view, the well known absence of gravitational waves in Kasner spacetimes. This result, used together with an explicit expression of the electric Weyl tensor in terms of Weyl scalars, provides an example of the fact that the presence of transverse curvature terms does not necessarily imply the presence of gravitational waves. 
  The nonsymmetric gravitational theory predicts an acceleration law that modifies the Newtonian law of attraction between particles. For weak fields a fit to the flat rotation curves of galaxies is obtained in terms of the mass (mass-to-light ratio M/L) of galaxies. The fits assume that the galaxies are not dominated by exotic dark matter. The equations of motion for test particles reduce for weak gravitational fields to the GR equations of motion and the predictions for the solar system and the binary pulsar PSR 1913+16 agree with the observations. The gravitational lensing of clusters of galaxies can be explained without exotic dark matter. 
  An energy conservation law is described, expressing the increase in mass-energy of a general black hole in terms of the energy densities of the infalling matter and gravitational radiation. For a growing black hole, this first law of black-hole dynamics is equivalent to an equation of Ashtekar & Krishnan, but the new integral and differential forms are regular in the limit where the black hole ceases to grow. An effective gravitational-radiation energy tensor is obtained, providing measures of both ingoing and outgoing, transverse and longitudinal gravitational radiation on and near a black hole. Corresponding energy-tensor forms of the first law involve a preferred time vector which plays the role for dynamical black holes which the stationary Killing vector plays for stationary black holes. Identifying an energy flux, vanishing if and only if the horizon is null, allows a division into energy-supply and work terms, as in the first law of thermodynamics. The energy supply can be expressed in terms of area increase and a newly defined surface gravity, yielding a Gibbs-like equation, with a similar form to the so-called first law for stationary black holes. 
  We study a toy model for phantom cosmology recently introduced in the literature and consisting of two oscillators, one of which carries negative kinetic energy. The results are compared with the exact phase space picture obtained for similar dynamical systems describing, respectively, a massive canonical scalar field conformally coupled to the spacetime curvature, and a conformally coupled massive phantom. Finally, the dynamical system describing exactly a minimally coupled phantom is studied and compared with the toy model. 
  The extreme weakness of the gravitational interaction has as one of its consequences that appreciable intensities of gravitational waves (GW) can only be generated in large size astrophysical and cosmological sources. Earth based detectors face unsurmountable problems to be sensitive to signals at frequencies below 10 Hz due to seismic vibrations. In order to see lower frequency signals, a space based detector is the natural solution. LISA (Laser Interferometer Space Antenna) is a joint ESA-NASA project aimed at detecting GWs in a range of frequencies between 10^{-4} Hz and 10^{-1} Hz, and consists in a constellation of three spacecraft in heliocentric orbit, whose GW-induced armlength variations are monitored by high precision interferometry. This article reviews the main features and scientific goals of the LISA mission, as well as a shorter description of its precursor tecnological mission LPF (LISA Pathfinder). 
  We prepare a general framework for analyzing the dynamics of a cylindrical shell in the spacetime with cylindrical symmetry. Based on the framework, we investigate a particular model of a cylindrical shell-collapse with rotational pressure, accompanying the radiation of gravitational waves and massless particles. The model has been introduced previously but has been awaiting for proper analysis. Here the analysis is put forward: It is proved that, as far as the weak energy condition is satisfied outside the shell, the collapsing shell bounces back at some point irrespective of the initial conditions, and escapes from the singularity formation.   The behavior after the bounce depends on the sign of the shell pressure in the z-direction. When the pressure is non-negative, the shell continues to expand without re-contraction. On the other hand, when the pressure is negative (i.e. it has a tension), the behavior after the bounce can be more complicated depending on the details of the model. However, even in this case, the shell never reaches the zero-radius configuration. 
  Kerr-Schild solutions to the vacuum Einstein equations are considered from the viewpoint of integral equations. We show that, for a class of Kerr-Schild fields, the stress-energy tensor can be regarded as a total divergence in Minkowski spacetime. If one assumes that Minkowski coordinates cover the entire manifold (no maximal extension), then Gauss' theorem can be used to reveal the nature of any sources present. For the Schwarzschild and Vaidya solutions the fields are shown to result from a delta-function point source. For the Reissner-Nordstrom solution we find that inclusion of the gravitational fields removes the divergent self-energy familiar from classical electromagnetism. For more general solutions a complex structure is seen to arise in a natural, geometric manner with the role of the unit imaginary fulfilled by the spacetime pseudoscalar. The Kerr solution is analysed leading to a novel picture of its global properties. Gauss' theorem reveals the presence of a disk of tension surrounded by the matter ring singularity. Remarkably, the tension profile over this disk has a simple classical interpretation. It is also shown that the matter in the ring follows a light-like path, as one expects for the endpoint of rotating, collapsing matter. Some implications of these results for physically-realistic black holes are discussed. 
  By applying the Palatini approach to the 1/R-gravity model it is possible to explain the present accelerated expansion of the Universe. Investigation of the late Universe limiting case shows that: (i) due to the curvature effects the energy-momentum tensor of the matter field is not covariantly conserved; (ii) however, it is possible to reinterpret the curvature corrections as sources of the gravitational field, by defining a modified energy-momentum tensor; (iii) with the adoption of this modified energy-momentum tensor the Einstein's field equations are recovered with two main modifications: the first one is the weakening of the gravitational effects of matter whereas the second is the emergence of an effective varying "cosmological constant"; (iv) there is a transition in the evolution of the cosmic scale factor from a power-law scaling $a\propto t^{11/18}$ to an asymptotically exponential scaling $a\propto \exp(t)$; (v) the energy density of the matter field scales as $\rho_m\propto (1/a)^{36/11}$; (vi) the present age of the Universe and the decelerated-accelerated transition redshift are smaller than the corresponding ones in the $\Lambda$CDM model. 
  An equidistant spectrum of the horizon area of a quantized black hole does not follow from the correspondence principle or from general statistical arguments. On the other hand, such a spectrum obtained in loop quantum gravity (LQG) either does not comply with the holographic bound, or demands a special choice of the Barbero-Immirzi parameter for the horizon surface, distinct from its value for other quantized surfaces. The problem of distinguishability of edges in LQG is discussed, with the following conclusion. Only under the assumption of partial distinguishability of the edges, the microcanonical entropy of a black hole can be made both proportional to the horizon area and satisfying the holographic bound. 
  Geroch's theorem about the splitting of globally hyperbolic spacetimes is a central result in global Lorentzian Geometry. Nevertheless, this result was obtained at a topological level, and the possibility to obtain a metric (or, at least, smooth) version has been controversial since its publication in 1970. In fact, this problem has remained open until a definitive proof, recently provided by the authors. Our purpose is to summarize the history of the problem, explain the smooth and metric splitting results (including smoothability of time functions in stably causal spacetimes), and sketch the ideas of the solution. 
  Using the multipolar post-Minkowskian and matching formalism we compute the gravitational waveform of inspiralling compact binaries moving in quasi-circular orbits at the second and a half post-Newtonian (2.5PN) approximation to general relativity. The inputs we use include notably the mass-type quadrupole at the 2.5PN order, the mass octupole and current quadrupole at the 2PN order, the mass $2^5$-pole and current $2^4$-pole at 1PN. The non-linear hereditary terms come from the monopole-quadrupole multipole interactions or tails, present at the 1.5PN, 2PN and 2.5PN orders, and the quadrupole-quadrupole interaction arising at the 2.5PN level. In particular, the specific effect of non-linear memory is computed using a simplified model of binary evolution in the past. The ``plus'' and ``cross'' wave polarisations at the 2.5PN order are obtained in ready-to-use form, extending the 2PN results calculated earlier by Blanchet, Iyer, Will and Wiseman. 
  Assuming a minimum value for area measurement, the emergence of quantum mechanics can be easily motivated from naive consideration of gravitational force. Here we provide some pedagogical examples and extensions.   At the same time, the role of Planck mass is shown to be of some theoretical influence even at low energies. 
  2D $R^2$ quantum gravity in infinitely large invariant volume is considered. In weak coupling limit the dynamics is reduced to quantum mechanics of a single degree of freedom. The correspondent two - point Green function is calculated explicitly in Gaussian approximation. 
  The gauge symmetry of classical general relativity under space-time diffeomorphisms implies that any path integral quantization which can be interpreted as a sum over space-time geometries, gives rise to a formal invariant of smooth manifolds. This is an opportunity to review results on the classification of smooth, piecewise-linear and topological manifolds. It turns out that differential topology distinguishes the space-time dimension d=3+1 from any other lower or higher dimension and relates the sought-after path integral quantization of general relativity in d=3+1 with an open problem in topology, namely to construct non-trivial invariants of smooth manifolds using their piecewise-linear structure. In any dimension d<=5+1, the classification results provide us with triangulations of space-time which are not merely approximations nor introduce any physical cut-off, but which rather capture the full information about smooth manifolds up to diffeomorphism. Conditions on refinements of these triangulations reveal what replaces block-spin renormalization group transformations in theories with dynamical geometry. The classification results finally suggest that it is space-time dimension rather than absence of gravitons that renders pure gravity in d=2+1 a `topological' theory. 
  The generalized Szekeres family of solution for quasi-spherical space-time of higher dimensions are obtained in the scalar tensor theory of gravitation. Brans-Dicke field equations expressed in Dicke's revised units are exhaustively solved for all the subfamilies of the said family. A particular group of solutions may also be interpreted as due to the presence of the so-called C-field of Hoyle and Narlikar and for a chosen sign of the coupling parameter. The models show either expansion from a big bang type of singularity or a collapse with the turning point at a lower bound. There is one particular case which starts from the big bang, reaches a maximum and collapses with the in course of time to a crunch. 
  We consider 2+1--dimensional analogues of the Bertotti-Robinson (BR) spacetimes in the sense that the coefficient at the angular part is a constant. We show that such BR-like solutions are either pure static or uncharged rotating. We trace the origin of the inconsistency between a charge and rotation, considering the BR-like spacetime as a result of the limiting transition of a non-extremal black hole to the extremal state. We also find that the quasilocal energy and angular momentum of such BR-like spacetimes calculated within the boundary $l=const$ ($l$ is the proper distance) are constants independent of the position of the boundary. 
  In case of spacetimes with single horizon, there exist several well-established procedures for relating the surface gravity of the horizon to a thermodynamic temperature. Such procedures, however, cannot be extended in a straightforward manner when a spacetime has multiple horizons. In particular, it is not clear whether there exists a notion of global temperature characterizing the multi-horizon spacetimes. We examine the conditions under which a global temperature can exist for a spacetime with two horizons using the example of Schwarzschild-De Sitter (SDS) spacetime. We systematically extend different procedures (like the expectation value of stress tensor, response of particle detectors, periodicity in the Euclidean time etc.) for identifying a temperature in the case of spacetimes with single horizon to the SDS spacetime. This analysis is facilitated by using a global coordinate chart which covers the entire SDS manifold. We find that all the procedures lead to a consistent picture characterized by the following features: (a) In general, SDS spacetime behaves like a non-equilibrium system characterized by two temperatures. (b) It is not possible to associate a global temperature with SDS spacetime except when the ratio of the two surface gravities is rational (c) Even when the ratio of the two surface gravities is rational, the thermal nature depends on the coordinate chart used. There exists a global coordinate chart in which there is global equilibrium temperature while there exist other charts in which SDS behaves as though it has two different temperatures. The coordinate dependence of the thermal nature is reminiscent of the flat spacetime in Minkowski and Rindler coordinate charts. The implications are discussed. 
  The main goal of numerical relativity is the long time simulation of highly nonlinear spacetimes that cannot be treated by perturbation theory. This involves analytic, computational and physical issues. At present, the major impasses to achieving global simulations of physical usefulness are of an analytic/computational nature. We present here some examples of how analytic insight can lend useful guidance for the improvement of numerical approaches. 
  Black holes play a fundamental role in modern physics. They have characteristic oscillation modes, called quasinormal modes. Past studies have shown that these modes are important to our understanding of the dynamics of astrophysical black holes. Recent studies indicate that they are important as a link between gravitation and quantum mechanics. Thus, the investigation of these modes is a timeliness topic.   Quasinormal modes dominate almost every process involving black holes, in particular gravitational wave emission during, for example, the collision between two black holes. It may be possible to create black holes at future accelerators, according to recent theories advocating the existence of extra dimensions in our universe. It is therefore important to study in depth the gravitational radiation emitted in high energy collision between two black holes in several dimensions, and also to make a theoretical study of gravitational waves in higher dimensions.   In this thesis we shall make a thorough study of the quasinormal modes of black holes in several kinds of background spacetimes. We shall investigate the gravitational radiation given away when highly energetic particles collide with black holes, and also when two black holes collide with each other. Finally, we shall study the properties of gravitational waves in higher dimensions, for instance, we generalize Einstein's quadrupole formula. 
  The relative classical motion of membranes is governed by an equation of the form D(hessian D separation)=riemann times separation times momentum. This is a generalization of the geodesic deviation equation and can be derived from a simple lagrangian. Quantum mechanically the picture is less clear. Some quantizations of the classical equations are attempted so that the question as to whether the Universe started with a quantum fluctuation can be addressed. 
  The holographic principle in a radiation dominated universe, as discussed first by Verlinde (2000), is extended so as to incorporate the case of a bulk-viscous cosmic fluid. This corresponds to a non-conformally invariant theory. Generalization of the Cardy-Verlinde entropy formula to the viscous case appears to be formally possible, although on physical grounds one may question some elements in this type of theory, especially the manner in which the Casimir energy is evaluated. Also, we consider the observation made by Youm (2002),namely that the entropy of the universe is no longer expressible in the conventional Cardy-Verlinde form if one relaxes the radiation dominance equation of state for the fluid and instead merely assumes that the pressure is proportional to the energy density. We show that Youm's generalized entropy formula remains valid when the cosmic fluid is no longer ideal, but endowed with a constant bulk viscosity. In the introductory part of this article, we take a general point of view and survey the essence of cosmological theory applied to a fluid containing both a constant shear viscosity and a constant bulk viscosity. 
  In a recent paper dealing with maximum likelihood detection of gravitational wave chirps from coalescing binaries with unknown parameters we introduced an accurate representation of the no-signal cumulative distribution of the supremum of the whole correlator bank. This result can be used to derive a refined estimate of the number of templates yielding the best tradeoff between detector's performance (in terms of lost signals among those potentially detectable) and computational burden. 
  This note suggests a generalization of the Born--Infeld action (1932) on case of electroweak and gravitational fields in four-dimensional spacetime. The action is constructed from Dirac matrices, $\gamma_a$, and dimensionless covariant derivatives, $\pi_{a} = - i\ell \nabla_{a}$, where $\ell$ is of order of magnitude of Planck's length. By a postulate, the action possesses additional symmetry with respect to global transformations of the Lorentz group imposed on pairs ($\gamma_{a}$, $\pi_{a}$). It's shown, that parameter of the Lorentz group is associated with a constant value of the electroweak potential at spatial infinity. It follows, that in linear and quadratic in $\ell^2$ approximation, action for gravitational field coincides with Einstein-Hilbert (EH) and Gauss-Bonnet (GB). 
  We show that the RST model describing the exactly soluble black hole model can have a dynamical wormhole solution along with an appropriate boundary condition. The necessary exotic matter which is usually negative energy density is remarkably produced by the quantization of the infalling matter fields. Then the asymptotic geometry in the past is two-dimensional anti-de Sitter(AdS$_2$), which implies the exotic matter is negative. As time goes on, the wormhole eventually evolves into the black hole and its Hawking radiation appears. The throat of the static RST wormhole is lower-bounded but in the presence of infalling matter it collapses to a black hole. 
  The only efficient and robust method of generating consistent initial data in general relativity is the conformal technique initiated by Lichnerowicz and perfected by York. In the spatially compact case, the complete scheme consists of the Arnowitt-Deser-Misner (ADM) Hamiltonian and momentum constraints, the ADM Euler-Lagrange equations, York's constant-mean-curvature (CMC) condition, and a lapse-fixing equation (LFE) that ensures propagation of the CMC condition by the Euler-Lagrange equations. The Hamiltonian constraint is rewritten as the Lichnerowicz-York equation for the conformal factor (psi) of the physical metric (psi)^4(g_{ij}) given an initial unphysical 3-metric (g_{ij}). The CMC condition and LFE introduce a distinguished foliation (definition of simultaneity) on spacetime, and separate scaling laws for the canonical momenta and their trace are used. In this article, we derive all these features in a single package by seeking a gauge theory of geometrodynamics (evolving 3-geometries) invariant under both three-dimensional diffeomorphisms and volume-preserving conformal transformations. 
  We investigate refined algebraic quantisation with group averaging in a constrained Hamiltonian system whose gauge group is the connected component of the lower triangular subgroup of SL(2,R). The unreduced phase space is T^*R^{p+q} with p>0 and q>0, and the system has a distinguished classical o(p,q) observable algebra. Group averaging with the geometric average of the right and left invariant measures, invariant under the group inverse, yields a Hilbert space that carries a maximally degenerate principal unitary series representation of O(p,q). The representation is nontrivial iff (p,q) is not (1,1), which is also the condition for the classical reduced phase space to be a symplectic manifold up to a singular subset of measure zero. We present a detailed comparison to an algebraic quantisation that imposes the constraints in the sense H_a Psi = 0 and postulates self-adjointness of the o(p,q) observables. Under certain technical assumptions that parallel those of the group averaging theory, this algebraic quantisation gives no quantum theory when (p,q) = (1,2) or (2,1), or when p>1, q>1 and p+q is odd. 
  We give a surface integral derivation of post-1-Newtonian translational equations of motion for a system of arbitrarily structured bodies, including the coupling to all the bodies' mass and current multipole moments. The derivation requires only that the post-1-Newtonian vacuum field equations are satisfied in weak-field regions between the bodies; the bodies' internal gravity can be arbitrarily strong. The derivation extends previous results of Damour, Soffel and Xu (DSX) for weakly self-gravitating bodies for which the post-1-Newtonian field equations are satisfied everywhere. The derivation consists of a number of steps: (i) The definition of each body's current and mass multipole moments and center-of-mass worldline in terms of the behavior of the metric in a weak-field region surrounding the body. (ii) The definition for each body of a set of gravitoelectric and gravitomagnetic tidal moments, again in terms of the behavior of the metric in a weak-field region surrounding the body. For the special case of weakly self-gravitating bodies, our definitions of these multipole and tidal moments agree with definitions given previously by DSX. (iii) The derivation of a formula, for any given body, of the second time derivative of its mass dipole moment in terms of its other multipole and tidal moments and their time derivatives. This formula was obtained previously by DSX for weakly self-gravitating bodies. (iv) A derivation of the relation between the tidal moments acting on each body and the multipole moments and center-of-mass worldlines of all the other bodies. A formalism to compute this relation was developed by DSX; we simplify their formalism and compute the relation explicitly. (v) The deduction from the previous steps of the explicit translational equations of motion, whose form has not been previously derived. 
  Understanding quantum theory in terms of a geometric picture sounds great. There are different approaches to this idea. Here we shall present a geometric picture of quantum theory using the de-Broglie--Bohm causal interpretation of quantum mechanics. We shall show that it is possible to understand the key character of de-Broglie--Bohm theory, the quantum potential, as the conformal degree of freedom of the space--time metric. In this way, gravity should give the causal structure of the space--time, while quantum phenomena determines the scale. Some toy models in terms of tensor and scalar--tensor theories will be presented. Then a few essential physical aspects of the idea including the effect on the black holes, the initial Big--Bang singularity and non locality are investigated. We shall formulate a quantum equivalence principle according to which gravitational effects can be removed by going to a freely falling frame while quantum effects can be eliminated by choosing an appropriate scale. And we shall see that the best framework for both quantum and gravity is Weyl geometry. Then we shall show how one can get the de-Broglie--Bohm quantum theory out of a Weyl covariant theory. Extension to the case of many particle systems and spinning particles is discussed at the end. 
  Quantum gravity effects of zeroth order in the Planck constant are investigated in the framework of the low-energy effective theory. A special emphasis is placed on establishing the correspondence between classical and quantum theories, for which purpose transformation properties of the \hbar^0-order radiative contributions to the effective gravitational field under deformations of a reference frame are determined. Using the Batalin-Vilkovisky formalism it is shown that the one-loop contributions violate the principle of general covariance, in the sense that the quantities which are classically invariant under such deformations take generally different values in different reference frames. In particular, variation of the scalar curvature under transitions between different reference frames is calculated explicitly. Furthermore, the long-range properties of the two-point correlation function of the gravitational field are examined. Using the Schwinger-Keldysh formalism it is proved that this function is finite in the coincidence limit outside the region of particle localization. In this limit, the leading term in the long-range expansion of the correlation function is calculated explicitly, and the relative value of the root mean square fluctuation of the Newton potential is found to be 1/\sqrt{2}. It shown also that in the case of a macroscopic gravitating body, the terms violating general covariance, and the field fluctuation are both suppressed by a factor 1/N, where N is the number of particles in the body. This leads naturally to a macroscopic formulation of the correspondence between classical and quantum theories of gravitation. As an application of the obtained results, the secular precession of a test particle orbit in the field of a black hole is determined. 
  We consider solutions with conformal symmetry of the static, spherically symmetric gravitational field equations in the vacuum in the brane world scenario. By assuming that the vector field generating the symmetry is non-static, the general solution of the field equations on the brane can be obtained in an exact parametric form, with the conformal factor taken as parameter. As a physical application of the obtained solutions we consider the behavior of the angular velocity of a test particle moving in a stable circular orbit. In this case the tangential velocity can be expressed as a function of the conformal factor and some integration constants only. For a specific range of the integration constants, the tangential velocity of the test particle tends, in the limit of large radial distances, to a constant value. This behavior is specific to the galactic rotation curves, and is explained usually by invoking the hypothesis of the dark matter. The limiting value of the angular velocity of the test particle can be obtained as a function of the baryonic mass and radius of the galaxy. The behavior of the dark radiation and dark pressure terms is also considered in detail, and it is shown that they can be expressed in terms of the rotational velocity of a test particle. Hence all the predictions of the present model can be tested observationally. Therefore the existence of the non-local effects, generated by the free gravitational field of the bulk in a conformally symmetric brane, may provide an explanation for the dynamics of the neutral hydrogen clouds at large distances from the galactic center. 
  We present a reformulation of the canonical quantization of gravity, as referred to the minisuperspace; the new approach is based on fixing a Gaussian (or synchronous) reference frame and then quantizing the system via the reconstruction of a suitable constraint; then the quantum dynamics is re-stated in a generic coordinates system and it becomes dependent on the lapse function. The analysis follows a parallelism with the case of the non-relativistic particle and leads to the minisuperspace implementation of the so-called {\em kinematical action} as proposed in \cite{M02} (here almost coinciding also with the approach presented in \cite{KT91}). The new constraint leads to a Schr\"odinger equation for the system. i.e. to non-vanishing eigenvalues for the super-Hamiltonian operator; the physical interpretation of this feature relies on the appearance of a ``dust fluid'' (non-positive definite) energy density, i.e. a kind of ``materialization'' of the reference frame. As an example of minisuperspace model, we consider a Bianchi type IX Universe, for which some dynamical implications of the revised canonical quantum gravity are discussed. We also show how, on the classical limit, the presence of the dust fluid can have relevant cosmological issues. Finally we upgrade our analysis by its extension to the generic cosmological solution, which is performed in the so-called long-wavelength approximation. In fact, near the Big-Bang, we can neglect the spatial gradients of the dynamical variables and arrive to implement, in each space point, the same minisuperspace paradigm valid for the Bianchi IX model. 
  Within a cosmological framework, we provide a Hamiltonian analysis of the Mixmaster Universe dynamics on the base of a standard Arnowitt-Deser-Misner approach, showing how the chaotic behavior characterizing the evolution of the system near the cosmological singularity can be obtained as the semiclassical limit of the canonical quantization of the model in the same dynamical representation. The relation between this intrinsic chaotic behavior and the indeterministic quantum dynamics is inferred through the coincidence between the microcanonical probability distribution and the semiclassical quantum one. 
  The possibility of analyzing the node Omega of the GP-B satellite in order to measure also the Lense-Thirring effect on its orbit is examined. This feature is induced by the general relativistic gravitomagnetic component of the Earth gravitational field. The GP-B mission has been launched in April 2004 and is aimed mainly to the measurement of the gravitomagnetic precession of four gyroscopes carried onboard at a claimed accuracy of 1% or better. The aliasing effect of the solid Earth and ocean components of the solar K_1 tidal perturbations would make the measurement of the Lense-Thirring effect on the orbit unfeasible. Indeed, the science period of the GP-B mission amounts to almost one year. During this time span the Lense-Thirring shift on the GP-B node would be 164 milliarcseconds (mas), while the tidal perturbations on its node would have a period of the order of 10^3 years and amplitudes of the order of 10^5 mas. 
  We obtain the energy and momentum densities of a general static axially symmetric vacuum space-time described by the Weyl metric, using Landau-Lifshitz and Bergmann-Thomson energy-momentum complexes. These two definitions of the energy-momentum complex do not provide the same energy density for the space-time under consideration, while give the same momentum density. We show that, in the case of Curzon metric which is a particular case of the Weyl metric, these two definitions give the same energy only when $R \to \infty$. Furthermore, we compare these results with those obtained using Einstein, Papapetrou and M{\o}ller energy momentum complexes. 
  It is known that the formation of apparent horizons with non-spherical topology is possible in higher-dimensional spacetimes. One of these is the black ring horizon with $S^1\times S^{D-3}$ topology where $D$ is the spacetime dimension number. In this paper, we investigate the black ring horizon formation in systems with $n$-particles. We analyze two kinds of system: the high-energy $n$-particle system and the momentarily-static $n$-black-hole initial data. In the high-energy particle system, we prove that the black ring horizon does not exist at the instant of collision for any $n$. But there remains a possibility that the black ring forms after the collision and this result is not sufficient. Because calculating the metric of this system after the collision is difficult, we consider the momentarily-static $n$-black-hole initial data that can be regarded as a simplified $n$-particle model and numerically solve the black ring horizon that surrounds all the particles. Our results show that there is the minimum particle number that is necessary for the black ring formation and this number depends on $D$. Although many particle number is required in five-dimensions, $n=4$ is sufficient for the black ring formation in the $D\ge 7$ cases. The black ring formation becomes easier for larger $D$. We provide a plausible physical interpretation of our results and discuss the validity of Ida and Nakao's conjecture for the horizon formation in higher-dimensions. Finally we briefly discuss the probable methods of producing the black rings in accelerators. 
  The theoretical and phenomenological status of negative energies is reviewed in Quantum Field theory leading to the conclusion that hopefully their rehabilitation might only be completed in a modified general relativistic model 
  Cosmologies with a Chaplygin gas have recently been explored with the objective of explaining the transition from a dust dominated epoch towards an accelerating expansion stage. We consider the hypothesis that the transition to the accelerated period involves a quantum mechanical process. Three physically admissible cases are possible. In particular, we identify a minisuperspace configuration with two Lorentzian sectors, separated by a classically forbidden region. The Hartle-Hawking and Vilenkin wave functions are computed, together with the transition amplitudes towards the accelerating epoch. Furthermore, it is found that for specific initial conditions, the parameters characterizing the generalized Chaplygin gas become related through an expression involving an integer $n$. We also introduce a phenomenological association between some brane-world scenarios and a FRW minisuperspace cosmology with a generalized Chaplygin gas. The aim is to promote a discussion and subsequent research on the quantum creation of brane cosmologies from such a perspective. Results suggest that the brane tension would become related with generalized Chaplygin gas parameters through another expression involving an integer. 
  We consider the inverse mean curvature flow in Robertson-Walker spacetimes that satisfy the Einstein equations and have a big crunch singularity and prove that under natural conditions the rescaled inverse mean curvature flow provides a smooth transition from big crunch to big bang. We also construct an example showing that in general the transition flow is only of class $C^3$. 
  Recent claims point out that possible violations of Lorentz symmetry appearing in some semiclassical models of extended matter dynamics motivated by loop quantum gravity can be removed by a different choice of canonically conjugated variables. In this note we show that such alternative is inconsistent with the choice of variables in the underlying quantum theory together with the semiclassical approximation, as long as the correspondence principle is maintained. A consistent choice will violate standard Lorentz invariance. Thus, to preserve a relativity principle in this framework, the linear realization of Lorentz symmetry should be extended or superseded. 
  Higher order curvature gravity has recently received a lot of attention due to the fact that it gives rise to cosmological models which seem capable of solving dark energy and quintessence issues without using "ad hoc" scalar fields. In this letter, a gravitational potential is obtained which differs from the Newtonian one because of a repulsive correction increasing with distance. We evaluate the rotation curve of our Galaxy and compare it with the observed data in order both to test the viability of these theories and to estimate the scalelength of the correction. It is remarkable that the Milky Way rotation curve is well fitted without the need of any dark matter halo and a similar result tentatively holds also for other galaxies. 
  Non Conventional treatments in modern cosmology, in both Steady State and Big Bang, are given. The motivation behind these treatments is to solve some of the problems of the conventional treatments in cosmology. For this aim, different geometric structures and alternative field theories, used to construct world models, are given. A brief review of Absolute parallelism (AP) geometry and its parameterized version(PAP), as a wider geometry than the Riemannian one, is presented. World models constructed using alternative field theories, constructed in the AP geometry, are discussed and compared. Some points about using topology in the field of cosmology are commented. A new path equation, admitted by the PAP geometry, is used to get the effect of spin-gravity interaction on the cosmological parameters. 
  We study scalar field and electromagnetic perturbations on Locally Rotationally Symmetric (LRS) class II spacetimes, exploiting a recently developed covariant and gauge-invariant perturbation formalism. From the Klein-Gordon equation and Maxwell's equations, respectively, we derive covariant and gauge-invariant wave equations for the perturbation variables and thereby find the generalised Regge-Wheeler equations for these LRS class II spacetime perturbations. As illustrative examples, the results are discussed in detail for the Schwarzschild and Vaidya spacetime, and briefly for some classes of dust Universes. 
  Recently we studied the effects of information carrying waves propagating through isotropic cosmologies. By information carrying we mean that the waves have an arbitrary dependence on a function. We found that the waves introduce shear and anisotropic stress into the universe. We then constructed explicit examples of pure gravity wave perturbations for which the presence of this anisotropic stress is essential and the null hypersurfaces playing the role of the histories of the wave-fronts in the background space-time are shear-free. Motivated by this result we now prove that these two properties are true for all information carrying waves in isotropic cosmologies. 
  Slice stretching effects such as slice sucking and slice wrapping arise when foliating the extended Schwarzschild spacetime with maximal slices. For arbitrary spatial coordinates these effects can be quantified in the context of boundary conditions where the lapse arises as a linear combination of odd and even lapse. Favorable boundary conditions are then derived which make the overall slice stretching occur late in numerical simulations. Allowing the lapse to become negative, this requirement leads to lapse functions which approach at late times the odd lapse corresponding to the static Schwarzschild metric. Demanding in addition that a numerically favorable lapse remains non-negative, as result the average of odd and even lapse is obtained. At late times the lapse with zero gradient at the puncture arising for the puncture evolution is precisely of this form. Finally, analytic arguments are given on how slice stretching effects can be avoided. Here the excision technique and the working mechanism of the shift function are studied in detail. 
  We provide a brief chronological guide to the literature on non-Riemannian cosmological models. Developments in this field are traced back to the early seventies and are given in table form. 
  Spherically symmetric solutions of generic gravitational models are optimally, and legitimately, obtained by expressing the action in terms of the two surviving metric components. This shortcut is not to be overdone, however: a one-function ansatz invalidates it, as illustrated by the incorrect solutions of [1]. 
  In this mostly expository article, elements of higher category theory essential to the construction of a class of four dimensional quantum geometric models are reviewed. These models improve current state sum models for Quantum Gravity, such as the Barrett-Crane model, in that they appear, for instance, to remove degeneracies which swamp the partition function. Much work remains to be done before a complete construction is reached, but the crucial categorical notion of internalisation already illuminates the idea that a full unified model may result from few, albeit as yet poorly understood, additional principles. In particular, a spacetime and matter duality principle is employed through an understanding of the role of pseudomonoidal objects in categorified cohomology. 
  Since the main open problem of contemporary physics is to find a unified description of the four interactions, we present a possible scenario which, till now only at the classical level, is able to englobe experiments ranging from experimental space gravitation to atomic and particle physics. After a reformulation of special relativistic physics in a form taking into account the non-dynamical chrono-geometrical structure of Minkowski space-time (parametrized Minkowski theories and rest-frame instant form) and in particular the conventionality of simultaneity (re-phrased as a gauge freedom), a model of canonical metric and tetrad gravity is proposed in a class of space-times where the deparametrization to Minkowski space-time is possible. In them it is possible to give a post-Minkowskian background-independent description of the gravitational field and of matter. The study of the dynamical chrono-geometrical structure of these space-times allows to face interpretational problems like the physical identification of point-events (the Hole Argument), the distinction between inertial (gauge) and tidal (Dirac observables) effects, the dynamical nature of simultaneity in general relativity and to find background-independent gravitational waves. These developments are possible at the Hamiltonian level due to a systematic use of Dirac-Bergmann theory of constraints. Finally there is a proposal for a new coordinate- and background-independent quantization scheme for gravity. 
  In the search for unmodeled gravitational wave bursts, there are a variety of methods that have been proposed to generate candidate events from time series data. Block Normal is a method of identifying candidate events by searching for places in the data stream where the characteristic statistics of the data change. These change-points divide the data into blocks in which the characteristics of the block are stationary. Blocks in which these characteristics are inconsistent with the long term characteristic statistics are marked as Event-Triggers which can then be investigated by a more computationally demanding multi-detector analysis. 
  To investigate how chaos affects gravitational waves, we study the gravitational waves from a spinning test particle moving around a Kerr black hole, which is a typical chaotic system. To compare the result with those in non-chaotic dynamical system, we also analyze a spinless test particle, which orbit can be complicated in the Kerr back ground although the system is integrable. We estimate the emitted gravitational waves by the multipole expansion of a gravitational field. We find a striking difference in the energy spectra of the gravitational waves. The spectrum for a chaotic orbit of a spinning particle, contains various frequencies, while some characteristic frequencies appear in the case of a spinless particle. 
  Some solutions of the Einstein equations for the eight-dimensional Riemann extension of the classical four-dimensional Schwarzschild metric are considered. 
  Friedmann-Lemaitre universes driven by a scalar field, spatially closed and bouncing, were recently studied by Martin and Peter in [1], with the conclusion that the spectrum of their large scale matter perturbations was generically modified when going through the bounce. In this Note we give the properties of the scalar field potentials which underly such models. 
  We model the system Earth-Moon-Sun from the point of view of a frame of reference co-moving with the Earth and we derive a detailed prediction of the outcome of future Kennedy-Thorndike's type experiments to be seen as light tides. 
  We provide a method for analytically constructing high-accuracy templates for the gravitational wave signals emitted by compact binaries moving in inspiralling eccentric orbits. By contrast to the simpler problem of modeling the gravitational wave signals emitted by inspiralling {\it circular} orbits, which contain only two different time scales, namely those associated with the orbital motion and the radiation reaction, the case of {\it inspiralling eccentric} orbits involves {\it three different time scales}: orbital period, periastron precession and radiation-reaction time scales. By using an improved `method of variation of constants', we show how to combine these three time scales, without making the usual approximation of treating the radiative time scale as an adiabatic process. We explicitly implement our method at the 2.5PN post-Newtonian accuracy. Our final results can be viewed as computing new `post-adiabatic' short period contributions to the orbital phasing, or equivalently, new short-period contributions to the gravitational wave polarizations, $h_{+,\times}$, that should be explicitly added to the `post-Newtonian' expansion for $h_{+,\times}$, if one treats radiative effects on the orbital phasing of the latter in the usual adiabatic approximation. Our results should be of importance both for the LIGO/VIRGO/GEO network of ground based interferometric gravitational wave detectors (especially if Kozai oscillations turn out to be significant in globular cluster triplets), and for the future space-based interferometer LISA. 
  The Laser Interferometer Space Antenna (LISA) will detect thousands of gravitational wave sources. Many of these sources will be overlapping in the sense that their signals will have a non-zero cross-correlation. Such overlaps lead to source confusion, which adversely affects how well we can extract information about the individual sources. Here we study how source confusion impacts parameter estimation for galactic compact binaries, with emphasis on the effects of the number of overlaping sources, the time of observation, the gravitational wave frequencies of the sources, and the degree of the signal correlations. Our main findings are that the parameter resolution decays exponentially with the number of overlapping sources, and super-exponentially with the degree of cross-correlation. We also find that an extended mission lifetime is key to disentangling the source confusion as the parameter resolution for overlapping sources improves much faster than the usual square root of the observation time. 
  Clock synchronization is the backbone of applications such as high-accuracy satellite navigation, geolocation, space-based interferometry, and cryptographic communication systems. The high accuracy of synchronization needed over satellite-to-ground and satellite-to-satellite distances requires the use of general relativistic concepts. The role of geometrical optics and antenna phase center approximations are discussed in high accuracy work. The clock synchronization problem is explored from a general relativistic point of view, with emphasis on the local measurement process and the use of the tetrad formalism as the correct model of relativistic measurements. The treatment makes use of J. L. Synge's world function of space-time as a basic coordinate independent geometric concept. A metric is used for space-time in the vicinity of the Earth, where coordinate time is proper time on the geoid. The problem of satellite clock syntonization is analyzed by numerically integrating the geodesic equations of motion for low-Earth orbit (LEO), geosynchronous orbit (GEO), and highly elliptical orbit (HEO) satellites. Proper time minus coordinate time is computed for satellites in these orbital regimes. The frequency shift as a function of time is computed for a signal observed on the Earth's geoid from a LEO, GEO, and HEO satellite. Finally, the problem of geolocation in curved space-time is briefly explored using the world function formalism. 
  Taking a hint from Dirac's large number hypothesis, we note the existence of cosmologically combined conservation laws that work to cosmologically long time. We thus modify Einstein's theory of general relativity with fixed gravitation constant $G$ to a theory for varying $G$, with a tensor term arising naturally from the derivatives of $G$ in place of the cosmological constant term usually introduced \textit{ad hoc}. The modified theory, when applied to cosmology, is consistent with Dirac's large number hypothesis, and gives a theoretical Hubble's relation not contradicting the observational data. For phenomena of duration and distance short compared with that of the universe, our theory reduces to Einstein's theory with $G$ being constant outside the gravitating matter, and thus also passes the crucial tests of Einstein's theory. 
  We reconsider the toy model studied in [1] of a spatially closed Friedmann-Lemaitre universe, driven by a massive scalar field, which deflates quasi-exponentially, bounces and then enters a period of standard inflation. We find that the equations for the matter density perturbations can be solved analytically, at least at lowest order in some "slow-roll" parameter. We can therefore give, in that limit, the explicit spectrum of the post-bounce perturbations in terms of their pre-bounce initial spectrum. Our result is twofold. If the pre-bounce growing and decaying modes are of the same order of magnitude at the bounce, then the spectrum of the pre-bounce growing modes is carried over to the post-bounce decaying modes ("mode inversion"). On the other hand, if, more likely, the pre-bounce growing modes dominate, then resolution at next order indicates that their spectrum is nicely carried over, with reduced amplitude, to the post-bounce growing modes. 
  We study Rainich-like conditions for symmetric and trace-free tensors T. For arbitrary even rank we find a necessary and sufficient differential condition for a tensor to satisfy the source free field equation. For rank 4, in a generic case, we combine these conditions with previously obtained algebraic conditions to obtain a complete set of algebraic and differential conditions on T for it to be a superenergy tensor of a Weyl candidate tensor satisfying the Bianchi vacuum equations. By a result of Bell and Szekeres this implies that in vacuum, generically, T must be the Bel-Robinson tensor of the spacetime. For the rank 3 case we derive a complete set of necessary algebraic and differential conditions for T to be the superenergy tensor of a massless spin 3/2 field satisfying the source free field equation. 
  The initial value problem is introduced after a thorough review of the essential geometry. The initial value equations are put into elliptic form using both conformal transformations and a treatment of the extrinsic curvature introduced recently. This use of the metric and the extrinsic curvature is manifestly equivalent to the author's conformal thin sandwich formulation. Therefore, the reformulation of the constraints as an elliptic system by use of conformal techniques is complete. 
  A generalized Tomonaga--Schwinger equation, holding on the entire boundary of a {\em finite} spacetime region, has recently been considered as a tool for studying particle scattering amplitudes in background-independent quantum field theory. The equation has been derived using lattice techniques under assumptions on the existence of the continuum limit. Here I show that in the context of continuous euclidean field theory the equation can be directly derived from the functional integral formalism, using a technique based on Hadamard's formula for the variation of the propagator. 
  This paper is a significantly expanded version of gr-qc/0306066. It discusses the geometric properties of the so called holographic solution, an exact, spherically symmetric solution to the Einstein field equations with zero cosmological constant. The holographic solution can be regarded as the simplest solution of the field equations including matter. Its interior matter-density follows an inverse square law: e = 1 / (8 pi r2). The interior principle pressures are P_r = - e in the radial direction and P_tan = 0 in the tangential direction. This is the equation of state of a radial arrangement of strings. The interior string type matter state is densely packed, each string occupying a transverse extension of exactly one Planck area, and bounded by a membrane consisting out of tangential pressure. The membrane's energy density is zero, as expected from string theory. Despite its simple structure, the results that can be derived from this solution are far from trivial. It is not possible to give a fair account of all relevant results in a 20-line abstract. The reader is referred to the abstract in the paper, which summarizes some of the more important results. The author appreciates any comments, critique or suggestions of new material. 
  In the spherically symmetric case the Einstein field equations take on their simplest form for a matter-density rho = 1 / (8 pi r^2), from which a radial metric coefficient g_{rr} \propto r follows. The boundary of an object with such an interior matter-density is situated slightly outside of its gravitational radius. Its surface-redshift scales with z \propto \sqrt{r}, so that any such large object is practically indistinguishable from a black hole, as seen from exterior space-time. The interior matter has a well defined temperature, T \propto 1 / \sqrt{r}. Under the assumption, that the interior matter can be described as an ultra-relativistic gas, the object's total entropy and its temperature at infinity can be calculated by microscopic statistical thermodynamics. They are equal to the Hawking result up to a possibly different constant factor. The simplest solution of the field equations with rho = 1 / (8 pi r^2) is the so called holographic solution, short "holostar". It has an interior string equation of state. The strings are densely packed, explaining why the solution does not collapse to a singularity. The holographic solution has been shown to be a very accurate model for the universe as we see it today in Ref[7]. The factor relating the holostar's temperature at infinity to the Hawking temperature can be expressed in terms the holostar's interior (local) radiation temperature and its (local) matter-density, allowing an experimental verification of the Hawking temperature law. Using the recent experimental data for the CMBR-temperature and the total matter-density in the universe measured by WMAP, the Hawking formula is verified to an accuracy of 1%. 
  We construct a sheaf theoretical representation of Quantum Observables Algebras over a base Category equipped with a Grothendieck topology, consisting of epimorphic families of commutative Observables Algebras, playing the role of local arithmetics in measurement situations. This construction makes possible the application of the methodology of Abstract Differential Geometry in a Category theoretical environment, and subsequently, the extension of the mechanism of differentials in the Quantum regime. 
  It is shown, that self-gravitating systems can be classified by a dimensionless constant positive number $\kappa = S T / E$, which can be determined from the (global) values for the entropy, temperature and (total) energy. The Kerr-Newman black hole family is characterized by $\kappa$ in the range $0-1/2$, depending on the dimensionless ratios of angular momentum and charge squared to the horizon area, $J/A$ and $Q^2/A$. By analyzing the most general case of an ultra-relativistic ideal gas with non-zero chemical potential it is shown, that $\kappa$ is an important parameter which determines the (local) thermodynamic properties of an ultra-relativistic gas. $\kappa$ only depends on the chemical potential per temperature $u = \mu / T$ and on the ratio of bosonic to fermionic degrees of freedom $r_F = f_B / f_F$. A gas with zero chemical potential has $\kappa = 4/3$. Whenever $\kappa < 4/3$ the gas must acquire a non-zero chemical potential. This non-zero chemical potential induces a natural matter-antimatter asymmetry, whenever microscopic statistical thermodynamics can be applied. The recently discovered holographic solution describes a compact self gravitating black hole type object with an interior, well defined matter state. One can associate a local - possibly observer-dependent - value of $\kappa$ to the interior matter, which lies in the range $2/3-1$ (for the uncharged case). This finding is used to construct an alternative scenario of baryogenesis in the context of the holographic solution, based on quasi-equilibrium thermodynamics. 
  The holographic solution is a new exact solution to the Einstein field equations. It describes a compact self-gravitating object with properties similar to a black hole. Its entropy and temperature at infinity are proportional to the Hawking result. Instead of an event horizon, the holographic solution has a real spherical boundary membrane, situated roughly two Planck distances outside of the object's gravitational radius. The interior matter-state is singularity free. It consists out of string type matter, which is densely packed. Each string has a transverse extension of exactly one Planck area. This dense package of strings might be the reason, why the solution does not collapse to a singularity. The local string tension is inverse proportional to the average string length. This purely classical result has its almost exact correspondence in a recent result in string theory. The holographic solution suggest, that string theory is relevant also on cosmological scales. The large scale phenomena in the universe can be explained naturally in a string context. Due to the zero active gravitational mass-density of the strings the Hubble constant in a string dominated universe is related to its age by H t = 1. The WMAP measurements have determined H t \approx 1.02 \pm 0.02 experimentally. The nearly unaccelerated expansion in a string dominated universe is compatible with the recent supernova measurements. Under the assumption, that the cold dark matter (CDM) consists out of strings, the ratio of CDM to baryonic matter is estimated as \Omega_CDM / \Omega_b \approx 6.45. Some arguments are given, which suggest that the universe might be constructed hierarchically out of its most basic building blocks: strings and membranes. 
  The general thermodynamic analysis of the quantum vacuum, which is based on our knowledge of the vacua in condensed-matter systems, is consistent with the Einstein earlier view on the cosmological constant. In the equilibrium Universes the value of the cosmological constant is regulated by matter. In the empty Universe, the vacuum energy is exactly zero, lambda=0. The huge contribution of the zero point motion of the quantum fields to the vacuum energy is exactly cancelled by the higher-energy degrees of freedom of the quantum vacuum. In the equilibrium Universes homogeneously filled by matter, the vacuum is disturbed, and density of the vacuum energy becomes proportional to the energy density of matter. This consideration applies to any equilibrium vacuum irrespective of whether the vacuum is false or true, and is valid both in the Einstein general theory of relativity and within the special theory of relativity, i.e. in the world without gravity. 
  In this paper we analyse Abelian diagonal orthogonally transitive spacetimes with spacelike orbits for which the matter content is a stiff perfect fluid. The Einstein equations are cast in a suitable form for determining their geodesic completeness. A sufficient condition on the metric of these spacetimes is obtained, that is fairly easy to check and to implement in exact solutions. These results confirm that non-singular spacetimes are abundant among stiff fluid cosmologies. 
  The necessary and sufficient condition for the false vacuum decay in a de Sitter universe via Coleman - de Luccia instanton is applied on the scalar field with quasi - exponential potential. 
  A scalar gravity model is developed according the 'geometric conventionalist' approach introduced by Poincare (Einstein 1921, Poincare 1905, Reichenbach 1957, Gruenbaum1973). In principle this approach allows an alternative interpretation and formulation of General Relativity Theory (GRT), with distinct i) physical congruence standard, and ii) gravitation dynamics according Hamilton-Lagrange mechanics, while iii) retaining empirical indistinguishability with GRT. In this scalar model the congruence standards have been expressed as gravitationally modified Lorentz Transformations (Broekaert 2002). The first type of these transformations relate quantities observed by gravitationally 'affected' (natural geometry) and 'unaffected' (coordinate geometry) observers and explicitly reveal a spatially variable speed of light (VSL). The second type shunts the unaffected perspective and relates affected observers, recovering i) the invariance of the locally observed velocity of light, and ii) the local Minkowski metric (Broekaert 2003). In the case of a static gravitation field the model retrieves the phenomenology implied by the Schwarzschild metric. The case with proper source kinematics is now described by introduction of a 'sweep velocity' field w: The model then provides a hamiltonian description for particles and photons in full accordance with the first Post-Newtonian approximation of GRT (Weinberg 1972, Will 1993). 
  Slice-stretching effects are discussed as they arise at the event horizon when geodesically slicing the extended Schwarzschild black-hole spacetime while using singularity excision. In particular, for Novikov and isotropic spatial coordinates the outward movement of the event horizon (``slice sucking'') and the unbounded growth there of the radial metric component (``slice wrapping'') are analyzed. For the overall slice stretching, very similar late time behavior is found when comparing with maximal slicing. Thus, the intuitive argument that attributes slice stretching to singularity avoidance is incorrect. 
  The detailed numerical and analytical approximate analysis of wormhole-like solutions in 5D Kaluza-Klein gravity is given. It is shown that some part of these solutions with $E \approx H, E>H$ relation between electric $E$ and magnetic $H$ fields can be considered as a superthin and superlong gravitational flux tube filled with electric and magnetic fields, namely $\Delta-$strings. The solution behaviour near hypersurface $ds^2=0$ and the model of electric charge on the basis of $\Delta-$string are discussed. The comparison of the properties of $\Delta-$string and ordinary string in string theory is carried out. Some arguments are given that fermionic degrees of freedom can be build in on the $\Delta-$string. These degrees of freedom are connected with quantum wormholes of spacetime foam. It is shown that the natural theory for these spinor fields is supergravity. 
  In this paper we study the Hawking evaporation of masses of variable-charged Reissner-Nordstrom and Kerr-Newman, black holes embedded into the de Sitter universe by considering the charge to be function of radial coordinate of the spherically symmetric metric. 
  In this paper we study the Hawking radiation in Reissner-Nordstrom and Kerr-Newman black holes by considering the charge to be the function of radial coordinate. 
  In this comment we discuss big rip singularities occurring in typical phantom models by violation of the weak energy condition. After that, we compare them with future late-time singularities arising in models where the scale factor ends in a constant value and there is no violation of the strong energy condition. In phantom models the equation of state is well defined along the whole evolution, even at the big rip. However, both the pressure and the energy density of the phantom field diverge. In contrast, in the second kind of model the equation of state is not defined at the big rip because the pressure bursts at a finite value of the energy density. 
  Soliton solutions are recovered as scale-invariant asymptotic states of vacuum inhomogeneous cosmologies using renormalization group method. The stability analysis of these states is also given. 
  Consider the configuration space Q for some physical system, and a continuous group of transformations G whose action on the configurations is declared to be physically irrelevant. Implement G indirectly by adjoining 1 auxiliary g per independent generator of G to Q, by writing the system's action in an arbitrary G-frame (G-AF), and then passing to the quotient Q/G thanks to the constraints encoded by g-variation. I show that this G-AF principle supercedes (and indeed leads to a derivation of) the Barbour--Bertotti (BB) best matching principle. My other consideration is that absolute external time is meaningless for the universe as a whole. For various choices of Q and G, these lead to BB's proposed absolute structure free replacement of Newtoninan mechanics, to Gauge Theory and to the 3-space approach (TSA) formulation of GR. For the latter with matter fields, I discuss the SR postulates, Principle of Equivalence (POE) and simplicity postulates. I explain how a full enough set of fundamental matter fields to describe nature can be accommodated in the TSA, and compare the TSA with the `split spacetime formulation' of Kuchar. I explain the emergence of broken and unbroken Gauge Theories as a consequence of the POE. I also consider as further examples of the G-AF principle the further quotienting out of conformal transformations (CT) or volume preserving CT. Depending on which choices are made, this leads to York's initial value formulation (IVF) of GR, new alternative foundations for it, or alternative theories of gravity built out of similar conformal mathematics which nevertheless admit no GR-like spacetime interpretation. 
  The general problem of computing the false-alarm rate vs. detection-threshold relationship for a bank of correlators is addressed, in the context of maximum-likelihood detection of gravitational waves, with specific reference to chirps from coalescing binary systems. Accurate (lower-bound) approximants for the cumulative distribution of the whole-bank supremum are deduced from a class of Bonferroni-type inequalities. The asymptotic properties of the cumulative distribution are obtained, in the limit where the number of correlators goes to infinity. The validity of numerical simulations made on small-size banks is extended to banks of any size, via a gaussian-correlation inequality. The result is used to estimate the optimum template density, yielding the best tradeoff between computational cost and detection efficiency, in terms of undetected potentially observable sources at a prescribed false-alarm level, for the simplest case of Newtonian chirps. 
  A formalization of the recently introduced formalism for inflation is developed from noncompact Kaluza-Klein theory. In particular, the case for a single scalar field inflationary model is studied. We obtain that the scalar potential, which assume different representations in different frames, has a geometrical origin. 
  We show how our theory of large-scale gravitational quantization explains the large angle gravitational lensing by galaxies without requiring "dark matter". A galaxy is treated as a collective system of billions of stars in each quantization state with each star experiencing an average gravitational environment analogous to that for nucleons in the atomic nucleus. Consequently, each star is in an approximate finite depth square well type of gravitational potential. The "effective potential" is shown to be about ten times greater than the Newtonian gravitational potential, so the gravitational lensing effects of a galaxy are about ten times greater also, in agreement with the measured gravitational lensing. 
  We study the spherically symmetric collapse of a perfect fluid using area-radial coordinates. We show that analytic mass functions describe a static regular centre in these coordinates. In this case, a central singularity can not be realized without an infinite discontinuity in the central density. We construct mass functions involving fluid dynamics at the centre and investigate the relationship between those and the nature of the singularities. 
  A reference frame in event space is a smooth field of orthonormal frames. Every reference frame is equipped by anholonomic coordinates. Using anholonomic coordinates allows to find out relative speed of two observers and appropriate Lorentz transformation.   Synchronization of a reference frame is an anholonomic time coordinate. Simple calculations show how synchronization influences time measurement in the vicinity of the Earth.   Measurement of Doppler shift from the star orbiting the black hole helps to determine mass of the black hole. According observations of Sgr A, if non orbiting observer estimates age of S2 about 10 Myr, this star is 0.297 Myr younger. 
  We call a manifold with torsion and nonmetricity the metric-affine manifold. The nonmetricity leads to a difference between the auto parallel line and the extreme line, and to a change in the expression of the Frenet transport and moving basis. The torsion leads to a change in the Killing equation. We also need to add a similar equation for the connection.   The analysis of the Frenet transport leads to the concept of the Cartan transport and an introduction of the connection compatible with the metric tensor. The dynamics of a particle follows to the Cartan transport. We need additional physical constraints to make a nonmetricity observable. 
  We consider the first laws of thermodynamics for a pair of systems made up of the two horizons of a Kerr-Newman black hole. These two systems are constructed in such a way that we only demand their ``horizon areas'' to be the sum and difference of that of the outer and inner horizons of their prototype. Remarkably, these two copies bear a striking resemblance to the right- and left-movers in string theory and D-brane physics. Our reformulation of the first law of black hole thermodynamics can be thought of as an analogy of thermodynamics of effective string or D-brane models. 
  Gunnar Nordstrom constructed the first relativistic theory of gravitation formulated in terms of interactions with a scalar field. It was an important precursor to Einstein's general theory of relativity a couple of years later. He was also the first to introduce an extra dimension to our spacetime so that gravitation would be just an aspect of electromagnetic interactions in five dimensions. His scalar theory and generalizations thereof are here presented in a bit more modern setting. Extra dimensions are of great interest in physics today and can give scalar gravitational interactions with similar properties as in Nordstrom's original theory. 
  Taking the triangle areas as independent variables in the theory of Regge calculus can lead to ambiguities in the edge lengths, which can be interpreted as discontinuities in the metric. We construct solutions to area Regge calculus using a triangulated lattice and find that on a spacelike hypersurface no such discontinuity can arise. On a null hypersurface however, we can have such a situation and the resulting metric can be interpreted as a so-called refractive wave. 
  Horizon solutions for the axial perturbations of the spherically symmetric metric are analyzed in the framework of the relativistic theory of gravitation. The gravitational perturbations can not be absorbed by the horizon that results in the excitation of the new type of the normal modes trapped by the Regge-Wheeler potential. The obtained results demonstrate testable differences between the collapsar and the black hole near-horizon physics. 
  A new gauge theory of gravity is presented. The theory is constructed in a flat background spacetime and employs gauge fields to ensure that all relations between physical quantities are independent of the positions and orientations of the matter fields. In this manner all properties of the background spacetime are removed from physics, and what remains are a set of `intrinsic' relations between physical fields. The properties of the gravitational gauge fields are derived from both classical and quantum viewpoints. Field equations are then derived from an action principle, and consistency with the minimal coupling procedure selects an action that is unique up to the possible inclusion of a cosmological constant. This in turn singles out a unique form of spin-torsion interaction. A new method for solving the field equations is outlined and applied to the case of a time-dependent, spherically-symmetric perfect fluid. A gauge is found which reduces the physics to a set of essentially Newtonian equations. These equations are then applied to the study of cosmology, and to the formation and properties of black holes. The existence of global solutions enables one to discuss the properties of field lines inside the horizon due to a point charge held outside it. The Dirac equation is studied in a black hole background and provides a quick derivation of the Hawking temperature. 
  We show that previously known non-asymptotically flat static black hole solutions of Einstein-Maxwell-dilaton theory may be obtained as near-horizon limits of asymptotically flat black holes. Specializing to the case of the dilaton coupling constant $\alpha^2 = 3$, we generate from the non-asymptotically flat magnetostatic or electrostatic black holes two classes of rotating dyonic black hole solutions. The rotating dyonic black holes of the ``magnetic'' class are dimensional reductions of the five-dimensional Myers-Perry black holes relative to one of the azimuthal angles, while those of the ``electric'' class are twisted dimensional reductions of rotating dyonic Rasheed black strings. We compute the quasi-local mass and angular momentum of our rotating dyonic black holes, and show that they satisfy the first law of black hole thermodynamics, as well as a generalized Smarr formula. We also discuss the construction of non-asymptotically flat multi-extreme black hole configurations. 
  Talk presented at the 2003 Coral Gables conference in honor and appreciation of the work of Professor Behram Kursunoglu, general relativist extraordinaire and founder of the Coral Gables series of conferences, whose untimely death occurred shortly before the 2003 conference. 
  The detection of gravitational waves from astrophysical sources of gravitational waves is a realistic goal for the current generation of interferometric gravitational-wave detectors. Short duration bursts of gravitational waves from core-collapse supernovae or mergers of binary black holes may bring a wealth of astronomical and astrophysical information. The weakness of the waves and the rarity of the events urges the development of optimal methods to detect the waves. The waves from these sources are not generally known well enough to use matched filtering however; this drives the need to develop new ways to exploit source simulation information in both detections and information extraction. We present an algorithmic approach to using catalogs of gravitational-wave signals developed through numerical simulation, or otherwise, to enhance our ability to detect these waves. As more detailed simulations become available, it is straightforward to incorporate the new information into the search method. This approach may also be useful when trying to extract information from a gravitational-wave observation by allowing direct comparison between the observation and simulations. 
  Realistic modelling of radiation transfer in and from variable accretion disks around black holes requires the solution of the problem: find the constants of motion and equation of motion of a light-like geodesic connecting two arbitrary points in space. Here we give the complete solution of this problem in the Schwarzschild space-time. 
  We suggest a scalar model of dark energy with the SO(1,1) symmetry. The model may be reformulated in terms of a real scalar field $\Phi$ and the scale factor $a$ so that the Lagrangian may be decomposed as that of the real quintessence model plus the negative coupling energy term of $\Phi$ to $a$. The existence of the coupling term $L^c$ leads to a wider range of $w_{\Phi}$ and overcomes the problem of negative kinetic energy in the phantom universe model. We propose a power-law expansion model of univese with time-dependent power, which can describe the phantom universe and the universe transition from ordinary acceleration to super acceleration. 
  We prove in the cases of spherical, plane and hyperbolic symmetry a local in time existence theorem and continuation criteria for cosmological solutions of the Einstein-Vlasov-scalar field system, with the sources generated by a distribution function and a scalar field, subject to the Vlasov and wave equations respectively. This system describes the evolution of self-gravitating collisionless matter and scalar waves within the context of general relativity. In the case where the only source is a scalar field it is shown that a global existence result can be deduced from the general theorem. 
  Corrected misprints and omissions. 
  We study the effects of shear and density inhomogeneities in the formation of naked singularities in spherically symmetric dust space--times. We find that in general neither of these physical features alone uniquely specifies the end state of the gravitational collapse. We do this by (i) showing that, for open sets of initial data, the same initial shear (or initial density contrast) can give rise to both naked and covered solutions. In particular this can happen for zero initial shear or zero initial density contrast; (ii) demonstrating that both shear and density contrast are invariant under a one parameter set of linear transformations acting on the initial data set and (iii) showing that asymptotically (near the singularities) one cannot in general establish a direct relationship between the rate of change of shear (or density contrast) and the nature of the singularities. However, one can uniquely determine the nature of the singularity if both the initial shear and initial density contrast are known.   These results are important in understanding the effects of the initial physical state and in particular the role of shear, in determiming the end state of the gravitational collapse. 
  We discuss the fundamental principles underlying the current physical theories and the prospects of further improving their knowledge through experiments in space. 
  We study generalisations of the Einstein--Straus model in cylindrically symmetric settings by considering the matching of a static space-time to a non-static spatially homogeneous space-time, preserving the symmetry. We find that such models possess severe restrictions, such as constancy of one of the metric coefficients in the non-static part. A consequence of this is that it is impossible to embed static locally cylindrically symmetric objects in reasonable spatially homogeneous cosmologies. 
  Searches for gravitational-wave bursts have often focused on the loudest event(s) in searching for detections and in determining upper limits on astrophysical populations. Typical upper limits have been reported on event rates and event amplitudes which can then be translated into constraints on astrophysical populations. We describe the mathematical construction of such upper limits. 
  Searches for known waveforms in gravitational wave detector data are often done using matched filtering. When used on real instrumental data, matched filtering often does not perform as well as might be expected, because non-stationary and non-Gaussian detector noise produces large spurious filter outputs (events). This paper describes a chi-squared time-frequency test which is one way to discriminate such spurious events from the events that would be produced by genuine signals. The method works well only for broad-band signals. The case where the filter template does not exactly match the signal waveform is also considered, and upper bounds are found for the expected value of chi-squared. 
  We discuss the physical nature of elementary singularities arising in the complexified Maxwell field extended into complex spacetime, i.e., in Lanczos-Newman electrodynamics. We show that the translation of the world-line of a bare electric-monopole singularity into imaginary space is adding a magnetic-dimonopole component to it, so that it may be interpreted as a pseudo-scalar pion-proton interaction current. On the other hand, the electric-monopole magnetic-dipole singularity characteristic of a Dirac electron is obtained by another operation on the world-line, which however does not seem to have a simple geometric interpretation. Nevertheless, both type of operations can be given a field-theoretical interpretation which can be generalized to yield the interaction currents of other elementary particles, and therefore to provide a link between elementary particle physics and general relativity theory. 
  We discuss the possibility that galactic gravitational wave sources might give burst signals at a rate of several events per year, detectable by state-of-the-art detectors. We are stimulated by the results of the data collected by the EXPLORER and NAUTILUS bar detectors in the 2001 run, which suggest an excess of coincidences between the two detectors, when the resonant bars are orthogonal to the galactic plane. Signals due to the coalescence of galactic compact binaries fulfill the energy requirements but are problematic for lack of known candidates with the necessary merging rate. We examine the limits imposed by galactic dynamics on the mass loss of the Galaxy due to GW emission, and we use them to put constraints also on the GW radiation from exotic objects, like binaries made of primordial black holes. We discuss the possibility that the events are due to GW bursts coming repeatedly from a single or a few compact sources. We examine different possible realizations of this idea, such as accreting neutron stars, strange quark stars, and the highly magnetized neutron stars (``magnetars'') introduced to explain Soft Gamma Repeaters. Various possibilities are excluded or appear very unlikely, while others at present cannot be excluded. 
  Based on Einstein's theory of gravitation, we discuss the influence of a spherically symmetric gravitational field on Maxwell's law of velocity distribution. We derive the equilibrium velocity distribution of low-energy particles in the spherically symmetric gravitational field and calculate the escape rate of low-energy particles in a container with a leak placed in the spherically symmetric gravitational field. They can serve as the tests of Einstein's theory of gravitation. 
  We show that the naked singularities arising in dust collapse from smooth initial data (which include those discovered by Eardley and Smarr, Christodoulou, and Newman) are removed when we make transition to higher dimensional spacetimes. The cosmic censorship is then restored for dust collapse which will always produce a black hole as the collapse end state for dimensions $D\ge6$, under conditions to be motivated physically such as the smoothness of initial data from which the collapse develops. 
  We reexamine our proposed counterexample (gr-qc/0307102) to cosmic censorship in anti de Sitter (AdS) space, and find a gap in the construction. We mention some possible ways to close the gap, but at present the question of whether cosmic censorship is violated in AdS remains open. 
  Three new classes (II-IV) of solutions of the vacuum low energy effective string theory in four dimensions are derived. Wormhole solutions are investigated in those solutions including the class I case both in the Einstein and in the Jordan (string) frame. It turns out that, of the eight classes of solutions investigated (four in the Einstein frame and four in the corresponding string frame), massive Lorentzian traversable wormholes exist in five classes. Nontrivial massless limit exists only in class I Einstein frame solution while none at all exists in the string frame. An investigation of test scalar charge motion in the class I solution in the two frames is carried out by using the Plebanski-Sawicki theorem. A curious consequence is that the motion around the extremal zero (Keplerian) mass configuration leads, as a result of scalar-scalar interaction, to a new hypothetical "mass" that confines test scalar charges in bound orbits, but does not interact with neutral test particles. 
  Static, spherically symmetric solutions of the Yang-Mills-Dilaton theory are studied. It is shown that these solutions fall into three different classes. The generic solutions are singular. Besides there is a discrete set of globally regular solutions further distinguished by the number of nodes of their Yang-Mills potential. The third class consists of oscillating solutions playing the role of limits of regular solutions, when the number of nodes tends to infinity. We show that all three sets of solutions are non-empty. Furthermore we give asymptotic formulae for the parameters of regular solutions and confront them with numerical results. 
  Here we explore the possibility of extending to the Solar System scenario the combined residual approach employed for the deterination of the Lense-Thirring effect in the Earth-LAGEOS satellites system. After calculating the secular gravitoelectric advance of the mean anomaly, we propose several linear combinations of the residuals of the nodes, the perihelia and the mean anomalies of some inner planets in order to measure the gravitoelectric advance, the Lense-Thirring effect and the solar quadrupole mass moment. The obtainable accuracies are comparable to those of new complex and expensive missions involving the launch of one or more spacecrafts. They are 10^-4 for the PPN parameter beta, 10^-5 for gamma, 10^-9 for the solar J2, 20-30% for the Lense-Thirring effect. In the case of the solar gravitomagnetic field, it would be the first attempt to measure its effect on the planetary motions. However, the improvements in the ephemerides of Mercury thanks to the Messenger and BepiColombo missions will map into an increasing accuracy of the proposed tests. 
  f(R)-type gravity in the first order formalism is interpreted as Einstein gravity with non-minimal coupling arising from the use of unphysical frame. Identification of the corresponding second order higher-curvature gravity in the physical frame is proposed by requiring that the action is the same. 
  It is shown that any spatially flat and isotropic universe undergoing accelerated expansion driven by a self-interacting scalar field can be directly related to a contracting, decelerating cosmology. The duality is made manifest by expressing the scale factor and Hubble parameter as functions of the scalar field and simultaneously interchanging these two quantities. The decelerating universe can be twinned with a cosmology sourced by a phantom scalar field by inverting the scale factor and leaving the Hubble parameter invariant. The accelerating model can be related to the same phantom universe by identifying the scale factor with the inverse of the Hubble parameter. The duality between accelerating and decelerating backgrounds can be extended to spatially curved cosmologies and models containing perfect fluids. A similar triality and associated scale factor duality is found in the Randall-Sundrum type II braneworld scenario. 
  The crease set of an event horizon or a Cauchy horizon is an important object which determines qualitative properties of the horizon. In particular, it determines the possible topologies of the spatial sections of the horizon. By Fermat's principle in geometric optics, we relate the crease set and the Maxwell set of a smooth function in the context of singularity theory. We thereby give a classification of generic topological structure of the Maxwell sets and the generic topologies of the spatial section of the horizon. 
  A framework which combines ideas from Connes' noncommutative geometry, or spectral geometry, with recent ideas on generally covariant quantum field theory, is proposed in the present work. A certain type of spectral geometries modelling (possibly noncommutative) globally hyperbolic spacetimes is introduced in terms of so-called globally hyperbolic spectral triples. The concept is further generalized to a category of globally hyperbolic spectral geometries whose morphisms describe the generalization of isometric embeddings. Then a local generally covariant quantum field theory is introduced as a covariant functor between such a category of globally hyperbolic spectral geometries and the category of involutive algebras (or *-algebras). Thus, a local covariant quantum field theory over spectral geometries assigns quantum fields not just to a single noncommutative geometry (or noncommutative spacetime), but simultaneously to ``all'' spectral geometries, while respecting the covariance principle demanding that quantum field theories over isomorphic spectral geometries should also be isomorphic. It is suggested that in a quantum theory of gravity a particular class of globally hyperbolic spectral geometries is selected through a dynamical coupling of geometry and matter compatible with the covariance principle. 
  We give a surface integral derivation of the leading-order evolution equations for the spin and energy of a relativistic body interacting with other bodies in the post-Newtonian expansion scheme. The bodies can be arbitrarily shaped and can be strongly self-gravitating. The effects of all mass and current multipoles are taken into account. As part of the computation one of the 2PN potentials parametrizing the metric is obtained. The formulae obtained here for spin and energy evolution coincide with those obtained by Damour, Soffel and Xu for the case of weakly self-gravitating bodies. By combining an Einstein-Infeld-Hoffman-type surface integral approach with multipolar expansions we extend the domain of validity of these evolution equations to a wide class of strongly self-gravitating bodies. This paper completes in a self-contained way a previous work by Racine and Flanagan on translational equations of motion for compact objects. 
  In this short note we comment about some criticisms - appeared in a recent paper by Iguchi et al - to our previous works on gravitational collapse of perfect fluids. We show that those criticisms are incorrect on their own. 
  The Volume Operator plays a crucial role in the definition of the quantum dynamics of Loop Quantum Gravity (LQG). Efficient calculations for dynamical problems of LQG can therefore be performed only if one has sufficient control over the volume spectrum. While closed formulas for the matrix elements are currently available in the literature, these are complicated polynomials in 6j symbols which in turn are given in terms of Racah's formula which is too complicated in order to perform even numerical calculations for the semiclassically important regime of large spins. Hence, so far not even numerically the spectrum could be accessed. In this article we demonstrate that by means of the Elliot -- Biedenharn identity one can get rid of all the 6j symbols for any valence of the gauge invariant vertex, thus immensely reducing the computational effort. We use the resulting compact formula to study numerically the spectrum of the gauge invariant 4 -- vertex. The techniques derived in this paper could be of use also for the analysis of spin -- spin interaction Hamiltonians of many -- particle problems in atomic and nuclear physics. 
  This is a written, expanded version of the summary talk given at the conclusion of the ICGC-2004 held at Cochin. Brief introductory remarks are included to provide a slightly wider context to the theme talks. 
  The concept of acoustic metric introduced previously by Unruh (PRL-1981) is extended to include Cartan torsion by analogy with the scalar wave equation in Riemann-Cartan (RC) spacetime. This equation describes irrotational perturbations in rotational non-relativistic fluids. This physical motivation allows us to show that the acoustic line element can be conformally mapped to the line element of a stationary torsion loop in non-Riemannian gravity. Two examples of such sonic analogues are given. The first is when we choose the static torsion loop in teleparallel gravity. In this case Cartan torsion vector in the far from the vortex approximation is shown to be proportional to the quantum vortex number of the superfluid. Also in this case the torsion vector is shown to be proportional to the superfluid vorticity in the presence of vortices. Torsion loops in RC spacetime does not favor the formation of superfluid vortices. It is suggested that the teleparallel model may help to find a model for superfluid neutron stars vortices based on non-Riemannian gravity. 
  We have developed a formalism to study non-adiabatic, non-radial oscillations of non-rotating compact stars in the frequency domain, including the effects of thermal diffusion in the framework of general relativistic perturbation theory. When a general equation of state depending on temperature is used, the perturbations of the fluid result in heat flux which is coupled with the spacetime geometry through the Einstein field equations. Our results show that the frequency of the first pressure (p) and gravity (g) oscillation modes is significantly affected by thermal diffusion, while that of the fundamental (f) mode is basically unaltered due to the global nature of that oscillation. The damping time of the oscillations is generally much smaller than in the adiabatic case (more than two orders of magnitude for the p- and g-modes) reflecting the effect of thermal dissipation. Both the isothermal and adiabatic limits are recovered in our treatment and we study in more detail the intermediate regime. Our formalism finds its natural astrophysical application in the study of the oscillation properties of newly born neutron stars, neutron stars with a deconfined quark core phase, or strange stars which are all promising sources of gravitational waves with frequencies in the band of the first generation and advanced ground-based interferometric detectors. 
  When sources are added at their right-hand sides, and g_{(ik)} is a priori assumed to be the metric, the equations of Einstein's Hermitian theory of relativity were shown to allow for an exact solution that describes the general electrostatic field of n point charges. Moreover, the injunction of spherical symmetry of g_{(ik)} in the infinitesimal neighbourhood of each of the charges was proved to yield the equilibrium conditions of the n charges in keeping with ordinary electrostatics. The tensor g_{(ik)}, however, cannot be the metric of the theory, since it enters neither the eikonal equation nor the equation of motion of uncharged test particles. A physically correct metric that rules both the behaviour of wave fronts and of uncharged matter is the one indicated by H\'ely. In the present paper it is shown how the electrostatic solution predicts the structure of the n charged particles and their mutual positions of electrostatic equilibrium when H\'ely's physically correct metric is adopted. 
  Static cylindrical shells composed of massive particles arising from matching of two different Levi-Civita space-times are studied for the shell satisfying either isotropic or anisotropic equation of state. We find that these solutions satisfy the energy conditions for certain ranges of the parameters. 
  We present a supersymmetric model of space-time foam with two stacks of eight D8-branes with equal string tensions, separated by a single bulk dimension containing D0-brane particles that represent quantum fluctuations in the space-time foam. The ground state configuration with static D-branes has zero vacuum energy. However, gravitons and other closed-string states propagating through the bulk may interact with the D0-particles, causing them to recoil and the vacuum energy to become non zero. This provides a possible origin of dark energy. Recoil also distorts the background metric felt by energetic massless string states, which travel at less than the usual (low-energy) velocity of light. On the other hand, the propagation of chiral matter anchored on the D8 branes is not affected by such space-time foam effects. 
  Gamma-Ray Bursts (GRBs) are now considered as relativistic jets. We analyze the gravitational waves from the acceleration stage of the GRB jets. We show that (i) the point mass approximation is not appropriate if the opening half-angle of the jet is larger than the inverse of the Lorentz factor of the jet, (ii) the gravitational waveform has many step function like jumps, and  (iii) the practical DECIGO and BBO may detect such an event if the GRBs occur in Local group of galaxy. We found that the light curve of GRBs and the gravitational waveform are anti-correlated so that the detection of the gravitational wave is indispensable to determine the structure of GRB jets. 
  The coupling of the electromagnetic field directly with gravitational gauge fields leads to new physical effects that can be tested using astronomical data. Here we consider a particular case for closer scrutiny, a specific nonminimal coupling of torsion to electromagnetism, which enters into a metric-affine geometry of space-time. We show that under the assumption of this nonminimal coupling, spacetime is birefringent in the presence of such a gravitational field. This leads to the depolarization of light emitted from extended astrophysical sources. We use polarimetric data of the magnetic white dwarf ${RE J0317-853}$ to set strong constraints on the essential coupling constant for this effect, giving $k^2 \lsim (19 {m})^2 $. 
  This paper is the continuation of a study into the information paradox problem started by the author in his earlier works. As previously, the key instrument is a deformed density matrix in quantum mechanics of the early universe. It is assumed that the latter represents quantum mechanics with fundamental length. It is demonstrated that the obtained results agree well with the canonical viewpoint that in the processes involving black holes pure states go to the mixed ones in the assumption that all measurements are performed by the observer in a well-known quantum mechanics. Also it is shown that high entropy for Planck remnants of black holes appearing in the assumption of the Generalized Uncertainty Relations may be explained within the scope of the density matrix entropy introduced by the author previously. It is noted that the suggested paradigm is consistent with the Holographic Principle. Because of this, a conjecture is made about the possibility for obtaining the Generalized Uncertainty Relations from the covariant entropy bound at high energies in the same way as R. Bousso has derived Heisenberg uncertainty principle for the flat space. 
  The conversion of the read-out from the anti-symmetric port of the LIGO interferometers into gravitational strain has thus far been performed in the frequency domain. Here we describe a conversion in the time domain which is based on the method developed by GEO. We illustrate the method using the Hanford 4km interferometer during the second LIGO science run (S2). 
  A physical metric is constructed as one that gives a coordinate independent result for the time delay in infinite order in the perturbation expansion in the gravitational constant. A compact form for the metric is obtained. One result is that the metric functions are positive definite. Another is an exact expression for the gravitational red shift. The metric can be used to calculate general relativity predictions in higher order for any process. A relationship between the spacetimes of the physical metric and the Schwarzschild metric is discussed. 
  In any spacetime, it is possible to have a family of observers following a congruence of timelike curves such that they do not have access to part of the spacetime. This lack of information suggests associating a (congruence dependent) notion of entropy with the horizon that blocks the information from these observers. While the blockage of information is absolute in classical physics, quantum mechanics will allow tunneling across the horizon. This process can be analysed in a simple, yet general, manner and we show that the probability for a system with energy $E$ to tunnel across the horizon is $P(E)\propto\exp[-(2\pi/\kappa)E)$ where $\kappa$ is the surface gravity of the horizon. If the surface gravity changes due to the leakage of energy through the horizon, then one can associate an entropy $S(M)$ with the horizon where $dS = [ 2\pi / \kappa (M) ] dM$ and $M$ is the active gravitational mass of the system. Using this result, we discuss the conditions under which, a small patch of area $\Delta A$ of the horizon contributes an entropy $(\Delta A/4L_P^2)$, where $L_P^2$ is the Planck area. 
  Hamiltonian description of gravitational field contained in a spacetime region with boundary $S$ being a null-like hypersurface (a wave front) is discussed. Complete generating formula for the Hamiltonian dynamics (with no surface integrals neglected) is presented. A quasi-local proof of the 1-st law of black holes thermodynamics is obtained as a consequence, in case when $S$ is a non-expanding horizon. The 0-th law and Penrose inequalities are discussed from this point of view. 
  It is shown that the extreme Kerr black hole is the only candidate for a black hole limit of rotating fluid bodies in equilibrium. 
  We give a detailed study of the asymptotic behavior of field commutators for linearly polarized, cylindrically symmetric gravitational waves in different physically relevant regimes. We also discuss the necessary mathematical tools to carry out our analysis. Field commutators are used here to analyze microcausality, in particular the smearing of light cones owing to quantum effects. We discuss in detail several issues related to the semiclassical limit of quantum gravity, in the simplified setting of the cylindrical symmetry reduction considered here. We show, for example, that the small G behavior is not uniform in the sense that its functional form depends on the causal relationship between spacetime points. We consider several physical issues relevant for this type of models such as the emergence of large gravitational effects. 
  The nonsymmetric gravitational theory leads to a modified acceleration law that can at intermediate distance ranges account for the anomalous acceleration experienced by the Pioneer 10 and 11 spacecraft. 
  We present a new numerical method for the construction of quasiequilibrium models of black hole-neutron star binaries. We solve the constraint equations of general relativity, decomposed in the conformal thin-sandwich formalism, together with the Euler equation for the neutron star matter. We take the system to be stationary in a corotating frame and thereby assume the presence of a helical Killing vector. We solve these coupled equations in the background metric of a Kerr-Schild black hole, which accounts for the neutron star's black hole companion. In this paper we adopt a polytropic equation of state for the neutron star matter and assume large black hole--to--neutron star mass ratios. These simplifications allow us to focus on the construction of quasiequilibrium neutron star models in the presence of strong-field, black hole companions. We summarize the results of several code tests, compare with Newtonian models, and locate the onset of tidal disruption in a fully relativistic framework. 
  Spacetime undergoes quantum fluctuations, giving rise to spacetime foam, a.k.a. quantum foam. We discuss some properties of spacetime foam, and point out the conceptual interconnections in the physics of quantum foam, black holes, and quantum computation. We also discuss the phenomenology of quantum foam, and conclude that it may be difficult, but by no means impossible, to detect its tiny effects in the not-too-distant future. 
  We review and discuss the original Kaluza-Klein theory in the framework of modern embedding theories of the spacetime, such as the recent induced matter approach. We show that in spite of their seeming similarity they constitute rather distinct proposals as far as their geometrical structure is concerned. 
  We derive two sets of explicit algebraic constraint preserving boundary conditions for the linearized BSSN system. The approach can be generalized to inhomogeneous differential and evolution conditions, the examples of which are given. The proposed conditions are justified by an energy estimate on the original BSSN variables. 
  The problem of dynamical generation of 4-D space-time signature at small scales and its stabilization towards Lorentzian signature at large scales is studied in the context of Higgs mechanism in a two-time scenario. It is also shown that Lorentz invariance at small scales can be violated but at large space-time scales is restored. 
  It is commonly assumed that quantum field theory arises by applying ordinary quantum mechanics to the low energy effective degrees of freedom of a more fundamental theory defined at ultra-high-energy/short-wavelength scales. We shall argue here that, even for free quantum fields, there are holistic aspects of quantum field theory that cannot be properly understood in this manner. Specifically, the ``subtractions'' needed to define nonlinear polynomial functions of a free quantum field in curved spacetime are quite simple and natural from the quantum field theoretic point of view, but are at best extremely ad hoc and unnatural if viewed as independent renormalizations of individual modes of the field. We illustrate this point by contrasting the analysis of the Casimir effect, the renormalization of the stress-energy tensor in time-dependent spacetimes, and anomalies from the point of quantum field theory and from the point of view of quantum mechanics applied to the independent low energy modes of the field. Some implications for the cosmological constant problem are discussed. 
  We discuss whether the future extrapolation of the present cosmological state may lead to a singularity even in case of "conventional" (negative) pressure of the dark energy field, namely $w=p/\rho \geq -1$. The discussion is based on an often neglected aspect of scalar-tensor models of gravity: the fact that different test particles may follow the geodesics of different metric frames, and the need for a frame-independent regularization of curvature singularities. 
  We explore a possible connection between two aspects of Loop Quantum Gravity which have been extensively studied in the recent literature: the black-hole area-entropy law and the energy-momentum dispersion relation. We observe that the original Bekenstein argument for the area-entropy law implicitly requires information on the energy-momentum dispersion relation. Recent results show that in first approximation black-hole entropy in Loop Quantum Gravity depends linearly on the area, with small correction terms which have logarithmic or inverse-power dependence on the area. Preliminary studies of the Loop-Quantum-Gravity dispersion relation reported some evidence of the presence of terms that depend linearly on the Planck length, but we observe that this possibility is excluded since it would require, for consistency, a contribution to black-hole entropy going like the square root of the area. 
  A common misconception is that Lorentz invariance is inconsistent with a discrete spacetime structure and a minimal length: under Lorentz contraction, a Planck length ruler would be seen as smaller by a boosted observer. We argue that in the context of quantum gravity, the distance between two points becomes an operator and show through a toy model, inspired by Loop Quantum Gravity, that the notion of a quantum of geometry and of discrete spectra of geometric operators, is not inconsistent with Lorentz invariance. The main feature of the model is that a state of definite length for a given observer turns into a superposition of eigenstates of the length operator when seen by a boosted observer. More generally, we discuss the issue of actually measuring distances taking into account the limitations imposed by quantum gravity considerations and we analyze the notion of distance and the phenomenon of Lorentz contraction in the framework of ``deformed (or doubly) special relativity'' (DSR), which tentatively provides an effective description of quantum gravity around a flat background. In order to do this we study the Hilbert space structure of DSR, and study various quantum geometric operators acting on it and analyze their spectral properties. We also discuss the notion of spacetime point in DSR in terms of coherent states. We show how the way Lorentz invariance is preserved in this context is analogous to that in the toy model. 
  We study the propagation of gravitational waves (GW) in a uniformly magnetized plasma at arbitrary angles to the magnetic field. No a priori assumptions are made about the temperature, and we consider both a plasma at rest and a plasma flowing out at ultra-relativistic velocities. In the 3+1 orthonormal tetrad description, we find that all three fundamental low-frequency plasma wave modes are excited by the GW. Alfven waves are excited by a x polarized GW, whereas the slow and fast magneto-acoustic modes couple to the + polarization. The slow mode, however, doesn't interact coherently with the GW. The most relevant wave mode is the fast magneto-acoustic mode which in a strongly magnetized plasma has a vanishingly small phase lag with respect to the GW allowing for coherent interaction over large length scales. When the background magnetic field is almost, but not entirely, parallel to the GW's direction of propagation even the Alfven waves grow to first order in the GW amplitude. Finally, we calculate the growth of the magneto-acoustic waves and the damping of the GW. 
  Understanding the end state of black hole evaporation, the microscopic origin of black hole entropy, the information loss paradox, and the nature of the singularity arising in gravitational collapse - these are outstanding challenges for any candidate quantum theory of gravity. Recently, a midisuperspace model of quantum gravitational collapse has been solved using a lattice regularization scheme. It is shown that the mass of an eternal black hole follows the Bekenstein spectrum, and a related argument provides a fairly accurate estimate of the entropy. The solution also describes a quantized mass-energy distribution around a central black hole, which in the WKB approximation, is precisely Hawking radiation. The leading quantum gravitational correction makes the spectrum non-thermal, thus providing a plausible resolution of the information loss problem. 
  We construct low regularity solutions of the vacuum Einstein constraint equations. In particular, on 3-manifolds we obtain solutions with metrics in $H^s\loc$ with $s>{3\over 2}$. The theory of maximal asymptotically Euclidean solutions of the constraint equations descends completely the low regularity setting. Moreover, every rough, maximal, asymptotically Euclidean solution can be approximated in an appropriate topology by smooth solutions. These results have application in an existence theorem for rough solutions of the Einstein evolution equations. 
  In brane-world cosmology gravitational waves can propagate in the higher dimensions (i.e., in the `bulk'). In some appropriate regimes the bulk gravitational waves may be approximated by plane waves. We systematically study five-dimensional gravitational waves that are algebraically special and of type N. In the most physically relevant case the projected non-local stress tensor on the brane is formally equivalent to the energy-momentum tensor of a null fluid. Some exact solutions are studied to illustrate the features of these branes; in particular, we show explicity that any plane wave brane can be embedded into a 5-dimensional Siklos spacetime. More importantly, it is possible that in some appropriate regime the bulk can be approximated by gravitational plane waves and thus may act as initial conditions for the gravitational field in the bulk (thereby enabling the field equations to be integrated on the brane). 
  The gravitational waveforms emitted during the adiabatic inspiral of precessing binaries with two spinning compact bodies of comparable masses, evaluated within the post-Newtonian approximation, can be reproduced rather accurately by the waveforms obtained by setting one of the two spins to zero, at least for the purpose of detection by ground-based gravitational-wave interferometers. Here we propose to use this quasi-physical family of single-spin templates to search for the signals emitted by double-spin precessing binaries, and we find that its signal-matching performance is satisfactory for source masses (m1,m2) in [3,15]Msun x [3,15]Msun. For this mass range, using the LIGO-I design sensitivity, we estimate that the number of templates required to yield a minimum match of 0.97 is ~320,000. We discuss also the accuracy to which the single-spin template family can be used to estimate the parameters of the original double-spin precessing binaries. 
  A possible deviation from the precession of the Gravity Probe-B gyroscope predicted by general relativity is obtained in the nonsymmetric gravity theory. The time delay of radio signals emitted by spacecraft at planetary distances from the Sun, in nonsymmetric gravity theory is the same as in general relativity. A correction to the precession of the gyroscope would provide a possible experimental signature for the Gravity Probe-B gyroscope experiment. The Lense-Thirring frame-dragging effect is predicted to be the same as in GR. 
  We review the properties of the constraint equations, from their geometric origin in hypersurface geometry through to their roles in the Cauchy problem and the Hamiltonian formulation of the Einstein equations. We then review properties of the space of solutions and construction techniques, including the conformal and conformal thin sandwich methods, the thin sandwich method, quasi-spherical and generalized QS methods, gluing techniques and the Corvino-Schoen projection. 
  We present a new idea that allows us to detect gravitational waves without being disturbed by any kind of displacement noise, based on the fact that gravitational waves and test-mass motions affect the propagations of light differently. We demonstrate this idea by analyzing a simple toy model consisting three equally-separated objects on a line. By taking a certain combination of light travel times between these objects, we construct an observable free from the displacement of each object which has a reasonable sensitivity to gravitational waves. 
  The premature acceptance of the standard cosmological model, the 'LambdaCDM' paradigm, is questioned; Self Creation Cosmology is offered as an alternative and shown to be as equally concordant with observed cosmological constraints and local observations including the EEP. The Brans Dicke theory is modified to enable the creation of matter and energy out of the self contained gravitational and scalar fields constrained by the local conservation of energy so that rest masses vary whereas the observed Newtonian Gravitation 'constant' does not. There is a conformal equivalence between self-creation and General Relativity in vacuo, which results in the predictions of the two theories being equal in the standard tests. In self-creation test particles in vacuo follow the geodesics of General Relativity. Nevertheless there are three types of experiment, including the LIGO apparatus & Gravity Probe B geodetic precession, which are able to distinguish between the two theories. Self-creation is as consistent with cosmological constraints as the standard paradigm, without the addition of the undiscovered physics of Inflation, dark non-baryonic matter, or dark energy. It does demand an exotic equation of state, which requires a false vacuum energy determined by the field equations. The two conformal frames conserve energy or energy-momentum and choose photons or atoms as the standard of measurement respectively. In the former frame the universe is stationary and eternal with exponentially shrinking rulers and accelerating atomic clocks, and in the latter frame the universe is freely coasting, expanding linearly from a Big Bang with rigid rulers and regular atomic clocks. A novel representation of space-time geometry is suggested. (Abridged) 
  The timelike geodesic equations resulting from the Kerr gravitational metric element are derived and solved exactly including the contribution from the cosmological constant. The geodesic equations are derived, by solving the Hamilton-Jacobi partial differential equation by separation of variables. The solutions can be applied in the investigation of the motion of a test particle in the Kerr and Kerr-(anti) de Sitter gravitational fields. In particular, we apply the exact solutions of the timelike geodesics i) to the precise calculation of dragging (Lense-Thirring effect) of a satellite's spherical polar orbit in the gravitational field of Earth assuming Kerr geometry, ii) assuming the galactic centre is a rotating black hole we calculate the precise dragging of a stellar polar orbit aroung the galactic centre for various values of the Kerr parameter including those supported by recent observations. The exact solution of non-spherical geodesics in Kerr geometry is obtained by using the transformation theory of elliptic functions. The exact solution of spherical polar geodesics with a nonzero cosmological constant can be expressed in terms of Abelian modular theta functions that solve the corresponding Jacobi's inversion problem. 
  The backreaction equations for the linearized quantum fluctuations in an acoustic black hole are given. The solution near the horizon, obtained within a dimensional reduction, indicates that acoustic black holes, unlike Schwarzschild ones, get cooler as they radiate phonons. They show remarkable analogies with near-extremal Reissner-Nordstrom black holes. 
  In this work we analyze the viability of use a particular models of scalar fields in the context of the galactic dark matter problem. These models are based on a single scalar field, minimally coupled to the gravity in a asymptotically flat or asymptotically deSitter spacetime. We discuss the opening possibility of constructing a unified model for both the cosmological and the galactic dark matter. 
  We investigate the backreaction equations for an acoustic black hole formed in a Laval nozzle under the assumption that the motion of the fluid is one-dimensional. The solution in the near-horizon region shows that as phonons are (thermally) radiated the sonic horizon shrinks and the temperature decreases. This contrasts with the behaviour of Schwarzschild black holes, and is similar to what happens in the evaporation of (near-extremal) Reissner-Nordstrom black holes (i.e. infinite evaporation time). Finally, by appropriate boundary conditions the solution is extended in both the asymptotic regions of the nozzle. 
  We use null spherical (observational) coordinates to describe a class of inhomogeneous cosmological models. The proposed cosmological construction is based on the observer past null cone. A known difficulty in using inhomogeneous models is that the null geodesic equation is not integrable in general. Our choice of null coordinates solves the radial ingoing null geodesic by construction. Furthermore, we use an approach where the velocity field is uniquely calculated from the metric rather than put in by hand. Conveniently, this allows us to explore models in a non-comoving frame of reference. In this frame, we find that the velocity field has shear, acceleration and expansion rate in general. We show that a comoving frame is not compatible with expanding perfect fluid models in the coordinates proposed and dust models are simply not possible. We describe the models in a non-comoving frame. We use the dust models in a non-comoving frame to outline a fitting procedure. 
  Nowadays, the Global Navigation Satellite Systems (GNSS), working like global positioning systems, are the GPS (NAVSTAR) and the GLONASS, which only are operative when several relativistic effects are corrected. In the next years the Galileo system will be constructed, copying the GPS System if there is no an alternative project. In this work, it will be exposed that there is one alternative to the mere copy by means of the SYPOR project, using relativistic concepts, and without utilize the Newtonian ideas that are in the basic conception, so much of the GPS as of the GLONASS. According to the SYPOR project, the Galileo system would be exact, with no need of corrections, and it would have additional technological advantages. 
  We report a new critical solution found at the threshold of axisymmetric gravitational collapse of a complex scalar field with angular momentum. To carry angular momentum the scalar field cannot be axisymmetric; however, its azimuthal dependence is defined so that the resulting stress energy tensor and spacetime metric are axisymmetric. The critical solution found is non-spherical, discretely self-similar with an echoing exponent of 0.42 (+- 4%), and exhibits a scaling exponent of 0.11 (+- 10%) in near critical collapse. Our simulations suggest that the solution is universal (within the imposed symmetry class), modulo a family-dependent constant phase in the complex plane. 
  We study the matching of LRS spatially homogeneous collapsing dust space-times with non-stationary vacuum exteriors in cylindrical symmetry. Given an interior with diagonal metric we prove existence and uniqueness results for the exterior. The matched solutions contain trapped surfaces, singularities and Cauchy horizons. The solutions cannot be asymptotically flat and we present evidence that they are singular on the Cauchy horizons. 
  The `theoretical' existence of traversable Lorentzian wormholes in the classical, macroscopic world is plagued by the violation of the well--known energy conditions of General Relativity. In this brief article we show : (i) how the extent of violation can be quantified using certain volume integrals (ii) whether this `amount of violation' can be minimised for some specific cut--and--paste geometric constructions. Examples and possibilities are also outlined. 
  A nonvanishing cosmological term in Einstein's equations implies a nonvanishing spacetime curvature even in absence of any kind of matter. It would, in consequence, affect many of the underlying kinematic tenets of physical theory. The usual commutative spacetime translations of the Poincare' group would be replaced by the mixed conformal translations of the de Sitter group, leading to obvious alterations in elementary concepts such as time, energy and momentum. Although negligible at small scales, such modifications may come to have important consequences both in the large and for the inflationary picture of the early Universe. A qualitative discussion is presented which suggests deep changes in Hamiltonian, Quantum and Statistical Mechanics. In the primeval universe as described by the standard cosmological model, in particular, the equations of state of the matter sources could be quite different from those usually introduced. 
  Starting from the assumption that general relativity might be an emergent phenomenon showing up at low-energies from an underlying microscopic structure, we re-analyze the stability of a static closed Universe filled with radiation. In this scenario, it is sensible to consider the effective general-relativistic configuration as in a thermal contact with an "environment" (the role of environment can be played, for example, by the higher-dimensional bulk or by the trans-Planckian degrees of freedom). We calculate the free energy at a fixed temperature of this radiation-filled static configuration. Then, by looking at the free energy we show that the static Einstein configuration is stable under the stated condition. 
  The structure of extended loop wave function is investigated in terms of the operator formalism. It is found that the extended loop wave function is characterized by the family number and classified by the partition of the family number. It is pointed out that the constraints to the extended loop function in quantum gravity exhibit a hierarchy structure. 
  General lectures on quantum gravity. 
  We review some basic natural geometric objects on null hypersurfaces. Gauss-Codazzi constraints are given in terms of the analog of canonical ADM momentum which is a well defined tensor density on the null surface. Bondi cones are analyzed with the help of this object. 
  This article--summarizing the authors' then novel formulation of General Relativity--appeared as Chapter 7 of an often cited compendium edited by L. Witten in 1962, which is now long out of print. Intentionally unretouched, this posting is intended to provide contemporary accessibility to the flavor of the original ideas. Some typographical corrections have been made: footnote and page numbering have changed--but not section nor equation numbering etc. The authors' current institutional affiliations are encoded in: arnowitt@physics.tamu.edu, deser@brandeis.edu, misner@physics.umd.edu . 
  The two-phase structure is imposed on the world continuum, with the graviton emerging as the tensor Goldstone boson during the spontaneous transition from the affinely connected phase to the metric one. The physics principle of metarelativity, extending the respective principle of special relativity, is postulated. The theory of metagravitation as the general nonlinear model GL(4,R)/SO(1,3) in the arbitrary background continuum is built. The concept of Metauniverse as the ensemble of the regions of the metric phase inside the affinely connected phase is introduced, and the possible bearing of the emerging multiple universes to the fine tuning of our Universe is conjectured. 
  Vacuum fluctuations and the Casimir effect are considered in a cosmological setting. It is suggested that the dark energy, which recent observations suggest make up 73% of our universe, is vacuum energy due to a causal boundary effect at the cosmological horizon. After a discussion of the similarities and differences between material boundaries in flat spacetime and causal horizons in general relativity, a simple model with a purely vacuum energy de Sitter interior and Schwarzschild exterior, separated by a thin boundary layer is outlined. The boundary layer is a quantum transition region which replaces the event horizons of the classical de Sitter and Schwarzschild solutions, through which the vacuum energy changes. 
  This essay presents a new asymmetry that arises from the interplay of charge conjugation and Lense-Thirring effect. When applied to Majorana neutrinos, the effects predicts nu_e <-> overline{nu}_e oscillations in gravitational environments with rotating sources. Parameters associated with astrophysical environments indicate that the presented effect is presently unobservable for solar neutrinos. But, it will play an important role in supernova explosions, and carries relevance for the observed matter-antimatter asymmetry in the universe. 
  The gravitational interaction is scale-free in both Newtonian gravity and general theory of relativity. The concept of self-similarity arises from this nature. Self-similar solutions reproduce themselves as the scale changes. This property results in great simplification of the governing partial differential equations. In addition, some self-similar solutions can describe the asymptotic behaviors of more general solutions. Newtonian gravity contains only one dimensional constant, the gravitational constant, while the general relativity contains another dimensional constant, the speed of light, besides the gravitational constant. Due to this crucial difference, incomplete similarity can be more interesting in general relativity than in Newtonian gravity. Kinematic self-similarity has been defined and studied as an example of incomplete similarity in general relativity, in an effort to pursue a wider application of self-similarity in general relativity. We review the mathematical and physical aspects of kinematic self-similar solutions in general relativity. 
  Examples are given of the creation of closed timelike curves by choices of coordinate identifications. Following G\"odel's prescription, it is seen that flat spacetime can produce closed timelike curves with structure similar to that of G\"odel. In this context, coordinate identifications rather than exotic gravitational effects of general relativity are shown to be the source of closed timelike curves. Removing the periodic time coordinate restriction, the modified G\"odel family of curves is expressed in a form that retains the timelike and spacelike character of the coordinates. With these coordinates, the nature of the timelike curves is clarified. A helicoidal surface unifies the families of timelike, spacelike and null curves. In all of these, it is seen that as in ordinary flat spacetime, periodicity in the spatial position does not naturally carry over into closure in time. Thus, the original source of serious scientific speculation regarding time machines is seen to be misconceived. 
  In this essay, I wish to share a novel perspective which envisions universalization as a guide from the classical world to relativistic and quantum world. It is the incorporation of zero mass particle in mechanics which leads to special relativity while its interaction with a universal field shared by all particles leads to general relativity. We also give a very simple classical argument to show that why the universal force has to be attractive. We try to envisage what sort of directions does this principle of universality point to for the world beyond general relativity? 
  It is known that for nonlinear electrodynamics the First Law of Black Hole Mechanics holds, however the Smarr's formula for the total mass does not. In this contribution we discuss the point and determine the corresponding expressions for the Bardeen black hole solution that represents a nonlinear magnetic monopole. The same is done for the regular black hole solution derived by Ayon-Beato and Garcia, showing that in the case that variations of the electric charge are involved, the Smarr's formula does not longer is valid. 
  We introduce a self-consistent stochastic coarse-graining method, which includes both metric and scalar field fluctuations, to investigate the back reaction of long wavelength perturbations in single-scalar driven inflation, up to the second (one loop) order. We demonstrate that, although back reaction cannot be significant during the last 70 e-foldings of inflation with a smooth potential, there exist non-smooth inflaton potentials which allow significant back reaction, and are also consistent with cosmological observations. Such non-smooth potentials may lead to the generation of massive primordial black holes, which could be further used to constrain/verify these models. 
  It is well-known that one of the most interesting and challenging problems of General Relativity is the energy and momentum localization. There are many attempts to evaluate the energy distribution in a general relativistic system. One of the methods used for the energy and momentum localization is the one which used the energy-momentum complexes. After the Einstein work, a large number of definitions for the energy distribution was given. We mention the expressions proposed by Landau and Lifshitz, Papapetrou, Bergmann, Weinberg and M{\o}ller. The Einstein, Landau and Lifshitz, Papapetrou, Bergmann and Weinberg energy-momentum complexes are restricted to calculate the energy distribution in quasi-Cartesian coordinates. The energy-momentum complex of M{\o}ller gives the possibility to make the calculations in any coordinate system. In this paper we calculate the energy distribution of three stringy black hole solutions in the M{\o}ller prescription. The M{\o}ller energy-momentum complex gives us a consistent result for these three situations.   Keywords: M{\o}ller energy-momentum complex, charged black hole solution in heterotic string theory PACS: 04. 20 Dw, 04. 70. Bw, 
  We continue an analysis of representations of cylindrical functions and fluxes which are commonly used as elementary variables of Loop Quantum Gravity. We consider an arbitrary principal bundle of a compact connected structure group and following Sahlmann's ideas define a holonomy-flux *-algebra whose elements correspond to the elementary variables. There exists a natural action of automorphisms of the bundle on the algebra; the action generalizes the action of analytic diffeomorphisms and gauge transformations on the algebra considered in earlier works. We define the automorphism covariance of a *-representation of the algebra on a Hilbert space and prove that the only Hilbert space admitting such a representation is a direct sum of spaces L^2 given by a unique measure on the space of generalized connections. This result is a generalization of our previous work (Class. Quantum. Grav. 20 (2003) 3543-3567, gr-qc/0302059) where we assumed that the principal bundle is trivial, and its base manifold is R^d. 
  D-dimensional cosmological model describing the evolution of a multicomponent perfect fluid with variable barotropic parameters in n Ricci-flat spaces is investigated. The equations of motion are integrated for the case, when each component possesses an isotropic pressure with respect to all spaces. Exact solutions are presented in the Kasner-like form. Some explicit examples are given: 4-dimensional model with an isotropic accelerated expansion at late times and (4+d)-dimensional model describing a compactification of extra dimensions. 
  We construct multi-time theory, in frame of which the cosmological acceleration is a natural phenomenon without cosmological constant or anything like that. The main point of this theory is each of the gravity interaction and electromagnetic interaction has its own time respectively. Also we give a concrete model of this theory which contents all the observations. Further more we present some predictions of this model which may be tested in the future. 
  The late-time tail behavior of massive Dirac fields is investigated in the Schwarzschild black-hole geometry and the result is compared with that of the massive scalar fields. It is shown that in the intermediate times there are three kinds of differences between the massive Dirac and scalar fields, (I) the asymptotic behavior of massive Dirac fields is dominated by a decaying tail without any oscillation, but the massive scalar field by a oscillatory inverse power-law decaying tail, (II) the dumping exponent for the massive Dirac field depends not only on the multiple number of the wave mode but also on the mass of the Dirac field, while that for the massive scalar field depends on the multiple number only, and (III) the decay of the massive Dirac field is slower than that of the massive scalar field. 
  A v_J/c correction to the Shapiro time delay seems verified by a 2002 Jovian observation by VLBI. In this Essay, this correction is interpreted as an effect of the aberration of light in an optically refractive medium which supplies an analog of Jupiter's gravity field rather than as a measurement of the speed of gravity, as it was first proposed by other authors. The variation of the index of refraction is induced by the Lorentz invariance of the weak gravitational field equations for Jupiter in a uniform translational slow motion with velocity v_J=13.5 km/s. The correction on time delay and deflection is due not to the Kerr (or Lense-Thirring) stationary gravitomagnetic field of Jupiter, but to its Schwarzschild gravitostatic field when measured from the barycenter of the solar system. 
  The variational theory of the perfect hypermomentum fluid is developed. The new type of the generalized Frenkel condition is considered. The Lagrangian density of such fluid is stated, and the equations of motion of the fluid and the Weyssenhoff-type evolution equation of the hypermomentum tensor are derived. The expressions of the matter currents of the fluid (the canonical energy-momentum 3-form, the metric stress-energy 4-form and the hypermomentum 3-form) are obtained. The Euler-type hydrodynamic equation of motion of the perfect hypermomentum fluid is derived. It is proved that the motion of the perfect fluid without hypermomentum in a metric-affine space coincides with the motion of this fluid in a Riemann space. 
  We study separability of the Hamilton-Jacobi and massive Klein-Gordon equations in the general Kerr-de Sitter spacetime in all dimensions. Complete separation of both equations is carried out in 2n+1 spacetime dimensions with all n rotation parameters equal, in which case the rotational symmetry group is enlarged from (U(1))^n to U(n). We explicitly construct the additional Killing vectors associated with the enlarged symmetry group which permit separation. We also derive first-order equations of motion for particles in these backgrounds and examine some of their properties. 
  Loop quantum cosmology leads to a difference equation for the wave function of a universe, which in general has solutions changing rapidly even when the volume changes only slightly. For a semiclassical regime such small-scale oscillations must be suppressed, by choosing the parameters of the solution appropriately. For anisotropic models this is not possible to do numerically by trial and error; instead, it is shown here for the Bianchi I LRS model how this can be done analytically, using generating function techniques. Those techniques can also be applied to more complicated models, and the results gained allow conclusions about initial value problems for other systems. 
  We cast the Reissner Nordstrom solution in a particular co-ordinate system which shows dynamical evolution from initial data. The initial data for the $E<M$ case is regular. This procedure enables us to treat the metric as a collapse to a singularity. It also implies that one may assume Wald axioms to be valid globally in the Cauchy development, especially when Hadamard states are chosen. We can thus compare the semiclassical behaviour with spherical dust case, looking upon the metric as well as state specific information as evolution from initial data. We first recover the divergence on the Cauchy horizon obtained earlier. We point out that the semiclassical domain extends right upto the Cauchy horizon. This is different from the spherical dust case where the quantum gravity domain sets in before. We also find that the backreaction is not negligible near the central singularity, unlike the dust case. Apart from these differences, the Reissner Nordstrom solution has a similarity with dust in that it is stable over a considerable period of time. The features appearing dust collapse mentioned above were suggested to be generally applicable within spherical symmetry. Reissner Nordstrom background (along with the quantum state) generated from initial data, is shown not to reproduce them. 
  We show that the behaviour of the outgoing radial null geodesic congruence on the boundary of the trapped region (suitably defined as a four dimensional region) is related to the property of nakedness in spherical dust collapse. The argument involves a conformal transformation which justifies the difference in the Penrose diagrams in the naked and covered dust collapse scenarios. 
  The late-time tail behavior of massive scalar fields is studied analytically in a stationary axisymmetric EMDA black hole geometry. It is shown that the asymptotic behavior of massive perturbations is dominated by the oscillatory inverse power-law decaying tail $ t^{-(l+3/2)}\sin(\mu t)$ at the intermediate late times, and by the asymptotic tail $ t^{-5/6}\sin(\mu t)$ at asymptotically late times. Our result seems to suggest that the intermediate tails $ t^{-(l+3/2)}\sin(\mu t)$ and the asymptotically tails $t^{-5/6} \sin(\mu t)$ may be quite general features for evolution of massive scalar fields in any four dimensional asymptotically flat rotating black hole backgrounds. 
  We identify conditions for the presence of negative specific heat in non-relativistic self-gravitating systems and similar systems of attracting particles.   The method used, is to analyse the Virial theorem and two soluble models of systems of attracting particles, and to map the sign of the specific heat for different combinations of the number of spatial dimensions of the system, $D$($\geq 2$), and the exponent, $\nu$($\neq 0$), in the force potential, $\phi=Cr^\nu$. Negative specific heat in such systems is found to be present exactly for $\nu=-1$, at least for $D \geq 3$. For many combinations of $D$ and $\nu$ representing long-range forces, the specific heat is positive or zero, for both models and the Virial theorem. Hence negative specific heat is not caused by long-range forces as such. We also find that negative specific heat appears when $\nu$ is negative, and there is no singular point in a certain density distribution. A possible mechanism behind this is suggested. 
  We analyze the canonical treatment of classical constrained mechanical systems formulated with a discrete time. We prove that under very general conditions, it is possible to introduce nonsingular canonical transformations that preserve the constraint surface and the Poisson or Dirac bracket structure. The conditions for the preservation of the constraints are more stringent than in the continuous case and as a consequence some of the continuum constraints become second class upon discretization and need to be solved by fixing their associated Lagrange multipliers. The gauge invariance of the discrete theory is encoded in a set of arbitrary functions that appear in the generating function of the evolution equations. The resulting scheme is general enough to accommodate the treatment of field theories on the lattice. This paper attempts to clarify and put on sounder footing a discretization technique that has already been used to treat a variety of systems, including Yang--Mills theories, BF-theory and general relativity on the lattice. 
  The notion that the geometry of our space-time is not only a static background but can be physically dynamic is well established in general relativity. Geometry can be described as shaped by the presence of matter, where such shaping manifests itself as gravitational force. We consider here probabilistic or atomistic models of such space-time, in which the active geometry emerges from a statistical distribution of 'atoms'. Such atoms are not to be confused with their chemical counterparts, however the shift of perspective obtained in analyzing a gas via its molecules rather than its bulk properties is analogous to this "second atomization". In this atomization, space-time itself (i.e. the meter and the second) is effectively atomized, so the atoms themselves must exist in a 'subspace'.      Here we build a simple model of such a space-time from the ground up, establishing a route for more complete theories, and enabling a review of recent work. We first introduce the motivation behind statistical interpretations and atomism, and look at applications to the realm of dynamic space-time theories. We then consider models of kinetic media in subspace compatible with our understanding of light. From the equations governing the propagation of light in subspace we can build a metric geometry, describing the dynamic and physical space-time of general relativity. Finally, implications of the theory on current frontiers of general relativity including cosmology, black holes, and quantum gravity are discussed. 
  A complete study of the structure of Ricci collineations for type B warped spacetimes is carried out. This study can be used as a method to obtain these symetries in such spacetimes. Special cases as 2+2 reducible spacetimes, and plane and spherical symmetric spacetimes are considered specifically. 
  It is unlikely that uniqueness theorem holds for stationary black holes in higher dimensional spacetimes. However, we will examine the possibility that the higher multipole moments classify vacuum solutions uniquely. Especially, we compute the potentials associated with rotational Killing vectors and look at the dependence on the total mass M and angular momentum J. Consequently, there is a potential $\sigma$ which we cannot write down in terms of integer power of M and J explicitly. This may be regarded as an evidence for the uniqueness using multipole moments generated by $\sigma$. 
  We created a model of several dimensional physical universe. The extra dimensions associated with the four dimensional physical universe is assumed to have modified Gidding-Strominger wormhole core. This core is separated by a flexible wall, but it allows the adiabatic pressure generated in the wormhole to penetrate in the extra dimensions. We assume that the extra dimensions are a contracting Robinson-Walker space. We show that the associated physical universe accelerates under certain restriction of the parameters introduced. The extra dimensional space is very large at the begining, however at present time this space will be very reduced. As a result the physical universe will appear to us four dimensional the way we observe it now. 
  Astronomical observations in the electromagnetic window - microwave, radio and optical - have revealed that most of the Universe is dark. The only reason we know that dark matter exists is because of its gravitational influence on luminous matter. It is plausible that a small fraction of that dark matter is clumped, and strongly gravitating. Such systems are potential sources of gravitational radiation that can be observed with a world-wide network of gravitational wave antennas. Electromagnetic astronomy has also revealed objects and phenomena - supernovae, neutron stars, black holes and the big bang - that are without doubt extremely strong emitters of the radiation targeted by the gravitational wave interferometric and resonant bar detectors. In this talk I will highlight why gravitational waves arise in Einstein's theory, how they interact with matter, what the chief astronomical sources of the radiation are, and in which way by observing them we can gain a better understanding of the dark and dense Universe. 
  One of the most exciting prospects for the LISA gravitational wave observatory is the detection of gravitational radiation from the inspiral of a compact object into a supermassive black hole. The large inspiral parameter space and low amplitude of the signal makes detection of these sources computationally challenging. We outline here a first cut data analysis scheme that assumes realistic computational resources. In the context of this scheme, we estimate the signal-to-noise ratio that a source requires to pass our thresholds and be detected. Combining this with an estimate of the population of sources in the Universe, we estimate the number of inspiral events that LISA could detect. The preliminary results are very encouraging -- with the baseline design, LISA can see inspirals out to a redshift z=1 and should detect over a thousand events during the mission lifetime. 
  In the framework of parallelism general relativity (PGR), the Dirac particle spin precession in the rotational gravitational field is studied. In terms of the equivalent tetrad of Kerr frame, we investigate the torsion axial-vector spin coupling in PGR. In the case of the weak field and slow rotation approximation, we obtain that the torsion axial-vector has the dipole-like structure, but different from the gravitomagnetic field, which indicates that the choice of the Kerr tetrad will influence on the physics interpretation of the axial-vector spin coupling. 
  Local simultaneity conventions are mathematically represented by connections on the bundle of timelike curves that defines the frame. Those simultaneity conventions having an holonomy proportional to the Riemann tensor have a special interest since they exhibit a good global behavior in the weak field limit. By requiring that the simultaneity convention depends on the the acceleration, vorticity vector and the angle between them we are able to restrict considerably the allowed simultaneity conventions. In particular, we focus our study on the simplest among the allowed conventions. It should be preferred over Einstein's in flat spacetime and in general if the tidal force between neighboring particles is weaker than the centrifugal force. It reduces to the Einstein convention if the vectorial product between the acceleration and the vorticity vector vanishes. We finally show how to use this convention in practice. 
  Asymptotic properties of electromagnetic waves are studied within the context of Friedmann-Robertson-Walker (FRW) cosmology. Electromagnetic fields are considered as small perturbations on the background spacetime and Maxwell's equations are solved for all three cases of flat, closed and open FRW universes. The asymptotic character of these solutions are investigated and their relevance to the problem of cosmological tails of electromagnetic waves is discussed. 
  We discuss a class of uniform and isotropic, spatially flat, decaying Lambda cosmologies, in the realm of a model where the gravitation constant G is a function of the cosmological time. Besides the usual de Sitter solution, the models at late times are characterized by a constant ratio between the matter and total energy densities. One of them presents a coasting expansion where the matter density parameter is equal to 1/3, and the age of the universe satisfies Ht = 1. From considerations in line with the holographic conjecture, it is argued that, among the non-decelerating solutions, the coasting expansion is the only acceptable from a thermodynamic viewpoint, and that the time varying cosmological term must be proportional to the square of the Hubble parameter, a result earlier obtained using different arguments. 
  A suitable derivative of Einstein's equations in the framework of the teleparallel equivalent of general relativity (TEGR) yields a continuity equation for the gravitational energy-momentum. In particular, the time derivative of the total gravitational energy is given by the sum of the total fluxes of gravitational and matter fields energy. We carry out a detailed analysis of the continuity equation in the context of Bondi and Vaidya's metrics. In the former space-time the flux of the gravitational energy is given by the well known expression in terms of the square of the news function. It is known that the energy definition in the realm of the TEGR yields the ADM (Arnowitt-Deser-Misner) energy for appropriate boundary conditions. Here we show that the same definition also describes the Bondi energy. The analysis of the continuity equation in Vaidya's space-time shows that the variation of the total gravitational energy is given by the energy flux of matter only. 
  The static Kottler metric is the Schwarzschild vacuum metric extended to include a cosmological constant. Angular momentum is added to the Kottler metric by using Newman and Janis' complexifying algorithm. The new metric is the Lambda generalization of the Kerr spacetime. It is stationary, axially symmetric, Petrov type II, and has Kerr-Schild form. 
  A general classical theorem is presented according to which all invariant relations among the space time metric scalars, when turned into functions on the Phase Space of full Pure Gravity (using the Canonical Equations of motion), become weakly vanishing functions of the Quadratic and Linear Constraints. The implication of this result is that (formal) Dirac consistency of the Quantum Operator Constraints (annihilating the wave Function) suffices to guarantee space time covariance f the ensuing quantum theory: An ordering for each invariant relation will always exist such that the emanating operator has an eigenvalue identical to the classical value. The example of 2+1 Quantum Cosmology is explicitly considered: The four possible ``Cosmological Solutions'' --two for pure Einstein's equations plus two more when a $\Lambda$ term is present- are exhibited and the corresponding models are quantized. The invariant relations describing the geometries are explicitly calculated and promoted to operators whose eigenvalues are their corresponding classical values. 
  We show that sudden variations in the composition and structure of an hybrid star can be triggered by its rapid spin-down, induced by r-mode instabilities. The discontinuity of this process is due to the surface tension between hadronic and quark matter and in particular to the overpressure needed to nucleate new structures of quark matter in the mixed phase. The consequent mini-collapses in the star can produce highly energetic gravitational wave bursts. The possible connection between the predictions of this model and the burst signal found by EXPLORER and NAUTILUS detectors during the year 2001 is also investigated. 
  We compare three different models of rotating neutron star spacetimes: the Hartle-Thorne (HT) slow-rotation approximation at second order in rotation, the exact analytic vacuum solution of Manko et al. and a numerical solution of the full Einstein equations. We integrate the HT structure equations for five representative equations of state. Then we match the HT models to numerical solutions of the Einstein equations, imposing that the mass and angular momentum of the models be the same. We estimate the limits of validity of the HT expansion computing relative errors in the spacetime's quadrupole moment Q and in the ISCO radii. We find that ISCO radii computed in the HT approximation are accurate to better than 1%, even for the fastest observed ms pulsar. At the same rotational rates the accuracy on Q is of order 20%. In the second part of the paper we focus on the exterior vacuum spacetimes. We introduce a physically motivated `quasi-Kinnersley' Newman-Penrose frame. In this frame we evaluate the speciality index S, a coordinate-independent quantity measuring the deviation of each model from Petrov Type D. On the equatorial plane this deviation is smaller than 5%, even for the fastest rotating models. Our main conclusion is that the HT approximation is very reliable for most astrophysical applications. 
  Time-Delay Interferometry (TDI) is the data processing technique needed for generating interferometric combinations of data measured by the multiple Doppler readouts available onboard the three LISA spacecraft. Within the space of all possible interferometric combinations TDI can generate, we have derived a specific combination that has zero-response to the gravitational wave signal, and called it the {\it Zero-Signal Solution} (ZSS). This is a two-parameter family of linear combinations of the generators of the TDI space, and its response to a gravitational wave becomes null when these two parameters coincide with the values of the angles of the source location in the sky. Remarkably, the ZSS does not rely on any assumptions about the gravitational waveform, and in fact it works for waveforms of any kind. Our approach is analogous to the data analysis method introduced by G\"ursel & Tinto in the context of networks of Earth-based, wide-band, interferometric gravitational wave detectors observing in coincidence a gravitational wave burst. The ZSS should be regarded as an application of the G\"ursel & Tinto method to the LISA data. 
  We show that recent, persistent discrepancies between theory and experiment can be interpreted as corrections to the gyro-gravitational ratio of the muon and lead to improved upper limits on the violation of discrete symmetries in rotational inertia. 
  In this paper, the relation between the modified Lorenz boosts, proposed in the doubly relativity theories and a linear combination of Conformal Group generators in $R^{1,d-1}$ is investigated. The introduction of a new generator is proposed in order to deform the Conformal Group to achieve the connection conjectured. The new generator is obtained trough a formal dimensional reduction from a free massless particle living in a $R^{2,d}$ space. Due this treatment it is possible to say that even DSR theories modify light cone structure in $R^{1,d-1}$, it could remains, in some cases, untouched in $R^{2,d}$. 
  We study the inhomogeneous cosmological evolution of the Newtonian gravitational 'constant' G in the framework of scalar-tensor theories. We investigate the differences that arise between the evolution of G in the background universes and in local inhomogeneities that have separated out from the global expansion. Exact inhomogeneous solutions are found which describe the effects of masses embedded in an expanding FRW Brans-Dicke universe. These are used to discuss possible spatial variations of G in different regions. We develop the technique of matching different scalar-tensor cosmologies of different spatial curvature at a boundary. This provides a model for the linear and non-linear evolution of spherical overdensities and inhomogeneities in G. This allows us to compare the evolution of G and \dot{G} that occurs inside a collapsing overdense cluster with that in the background universe. We develop a simple virialisation criterion and apply the method to a realistic lambda-CDM cosmology containing spherical overdensities. Typically, far slower evolution of \dot{G} will be found in the bound virialised cluster than in the cosmological background. We consider the behaviour that occurs in Brans-Dicke theory and in some other representative scalar-tensor theories. 
  On the bases of the Papapetrou equations with various supplementary conditions and other approaches a comparative analysis of the equations of motion of rotating bodies in general relativity is made. The motion of a body with vertical spin in a circular orbit is considered. An expression for the spin-orbit force in a post-Newtonian approximation is investigated. 
  We give a well posed initial value formulation of the Baumgarte-Shapiro-Shibata-Nakamura form of Einstein's equations with gauge conditions given by a Bona-Masso like slicing condition for the lapse and a frozen shift. This is achieved by introducing extra variables and recasting the evolution equations into a first order symmetric hyperbolic system. We also consider the presence of artificial boundaries and derive a set of boundary conditions that guarantee that the resulting initial-boundary value problem is well posed, though not necessarily compatible with the constraints. In the case of dynamical gauge conditions for the lapse and shift we obtain a class of evolution equations which are strongly hyperbolic and so yield well posed initial value formulations. 
  We review some material connecting gravity and the quantum potential and provide a few new observations. 
  We discuss from the condensed-matter point of view the recent idea that the Poisson fluctuations of cosmological constant about zero could be a source of the observed dark energy. We argue that the thermodynamic fluctuations of Lambda are much bigger. Since the amplitude of fluctuations is proportional to V^{-1/2}, where V is the volume of the Universe, the present constraint on the cosmological constant provides the lower limit for V, which is much bigger than the volume within the cosmological horizon. 
  Applying the Ginzburg-Landau theory including frame dragging effects to the case of a rotating superconductor, we were able to express the absolute value of the gravitomagnetic field involved to explain the Cooper pair mass anomaly previously reported by Tate. Although our analysis predicts large gravitomagnetic fields originated by superconductive gyroscopes, those should not affect the measurement of the Earth gravitomagnetic field by the Gravity Probe-B satellite. However, the hypothesis might be well suited to explain a mechanical momentum exchange phenomena reported for superfluid helium. As a possible explanation for those abnormally large gravitomagnetic fields in quantum materials, the reduced speed of light (and gravity) that was found in the case of Bose-Einstein condensates is analysed. 
  The well-known ``displace, cut and reflect'' method used to generate disks from given solutions of Einstein field equations is applied to the superposition of twoextreme Reissner-Nordstrom black holes to construct disks made of charged dust and alsonon-axisymmetric planar distributions of charged dust on the z=0 plane. They are symmetric with respect to twoor one coordinate axes, depending whether the black holes have equal or unequal masses, respectively.For these non-axisymmetric distributions of matter we also study the effective potential for geodesic motion of neutral test particles. 
  We comment on the presence of spurious observables and on a subtle violation of irreducibility in loop quantum cosmology. 
  The response of laser interferometers to gravitational waves has been calculated in a number of different ways, particularly in the transverse-traceless and the local Lorentz gauges. At first sight, it would appear that these calculations lead to different results when the separation between the test masses becomes comparable to the wavelength of the gravitational wave. In this paper this discrepancy is resolved. We describe the response of free test masses to plane gravitational waves in the coordinate frame of a local observer and show that it acquires contributions from three different effects: the displacement of the test masses, the apparent change in the photon velocity, and the variation in the clock speed of the local observer, all of which are induced by the gravitational wave. Only when taken together do these three effects represent a quantity which is translationally invariant. This translationally-invariant quantity is identical to the response function calculated in the transverse-traceless gauge. We thus resolve the well-known discrepancy between the two coordinates systems, and show that the results found in the coordinate frame of a local observer are valid for large separation between the masses. 
  Under certain conditions, a $(1+1)$-dimensional slice $\hat{g}$ of a spherically symmetric black hole spacetime can be equivariantly embedded in $(2+1)$-dimensional Minkowski space. The embedding depends on a real parameter that corresponds physically to the surface gravity $\kappa$ of the black hole horizon.   Under conditions that turn out to be closely related, a real surface that possesses rotational symmetry can be equivariantly embedded in 3-dimensional Euclidean space. The embedding does not obviously depend on a parameter. However, the Gaussian curvature is given by a simple formula: If the metric is written $g = \phi(r)^{-1} dr^2 + \phi(r) d\theta^2$, then $\K_g=-{1/2}\phi''(r)$.   This note shows that metrics $g$ and $\hat{g}$ occur in dual pairs, and that the embeddings described above are orthogonal facets of a single phenomenon. In particular, the metrics and their respective embeddings differ by a Wick rotation that preserves the ambient symmetry.   Consequently, the embedding of $g$ depends on a real parameter. The ambient space is not smooth, and $\kappa$ is inversely proportional to the cone angle at the axis of rotation. Further, the Gaussian curvature of $\hat{g}$ is given by a simple formula that seems not to be widely known. 
  In spite of a recent reply by Quevedo and Z\'arate (gr-qc/0403096), their assertion that their thermodynamic scheme for a perfect fluid binary mixture is incompatible with Szekeres and Stephani families of universes, except those of FRW ones, remains wrong. 
  The gravitational radiation from point particle binaries is computed at the third post-Newtonian (3PN) approximation of general relativity. Three previously introduced ambiguity parameters, coming from the Hadamard self-field regularization of the 3PN source-type mass quadrupole moment, are consistently determined by means of dimensional regularization, and proved to have the values xi = -9871/9240, kappa = 0 and zeta = -7/33. These results complete the derivation of the general relativistic prediction for compact binary inspiral up to 3.5PN order, and should be of use for searching and deciphering the signals in the current network of gravitational wave detectors. 
  All powered spacecraft experience residual systematic acceleration due to anisotropy of the thermal radiation pressure and fuel leakage. The residual acceleration limits the accuracy of any test of gravity that relies on the precise determination of the spacecraft trajectory. We describe a novel two-step laser ranging technique, which largely eliminates the effects of non-gravity acceleration sources and enables celestial mechanics checks with unprecedented precision. A passive proof mass is released from the mother spacecraft on a solar system exploration mission. Retro-reflectors attached to the proof mass allow its relative position to the spacecraft to be determined using optical ranging techniques. Meanwhile, the position of the spacecraft relative to the Earth is determined by ranging with a laser transponder. The vector sum of the two is the position, relative to the Earth, of the proof mass, the measurement of which is not affected by the residual accelerations of the mother spacecraft. We also describe the mission concept of the Dark Matter Explorers (DMX), which will demonstrate this technology and will use it to test the hypothesis that dark matter congregates around the sun. This hypothesis implies a small apparent deviation from the inverse square law of gravity, which can be detected by a sensitive experiment. We expect to achieve an acceleration resolution of $\sim 10^{-14} m/s^2$. DMX will also be sensitive to acceleration towards the galactic center, which has a value of $\sim 10^{-10} m/s^2$. Since dark matter dominates the galactic acceleration, DMX can also test whether dark matter obeys the equivalence principle to a level of 100 ppm by ranging to several proof masses of different composition from the mother spacecraft. 
  It is shown that the recently claimed two new Brans-Dicke wormhole solutions [F. He and S-W. Kim, Phys. Rev. D{\bf 65}, 084022 (2002)] are not really new solutions. They are just the well known Brans-Dicke solutions of Class I and II in a different conformal gauge. 
  In reply to the criticism made by Mielke in the pereceding Comment [Phys. Rev. D69 (2004) 128501] on our recent paper, we once again explicitly demonstrate the inconsistency of the coupling of a Dirac field to gravitation in the teleparallel equivalent of general relativity. Moreover, we stress that the mentioned inconsistency is generic for {\it all} sources with spin and is by no means restricted to the Dirac field. In this sense the $SL(4,R)$-covariant generalization of the spinor fields in the teleparallel gravity theory is irrelevant to the inconsistency problem. 
  The stationary cosmological model without closed timelike curves of G\"odel type is obtained for the ideal dust matter source within the framework of the teleparallel gravity. For a specific choice of the teleparallel gravity parameters, this solution reproduces the causality violating stationary G\"odel solution in general relativity, in accordance with the teleparallel equivalent of general relativity. The relation between the axial-vector torsion and the cosmic vorticity is clarified. 
  A semi-classical reasoning leads to the non-commutativity of space and time coordinates near the horizon of static non-extreme black hole, and renders the classical horizon spreading to {\it Quantum Horizon} . In terms of the background metric of the black hole with the {\it Quantum Horizon}, a quantum field theory in curved space without ultraviolet divergency near the horizon is formulated. In this formulism, the black hole thermodynamics is reproduced correctly without both ambiguity and additional hypothesis in the deriving the hole's Hawking radiations and entropies, and a new interesting prediction on the number of radiative field modes $N$ is provided. Specifically, the main results are follows: 1, Hawking radiations rightly emerge as an effect of quantum tunneling through the quantum horizon, and hence the ambiguities due to going across the singularity on the classical horizon were got rid of; 2, 't Hooft's brick wall thickness hypothesis and the boundary condition imposed for the field considered in his brick wall model were got rid of also, and related physics has been interpreted; 3, The present theory is parameter free. So, the theory has power to predict the multiplicity $N$ of radiative field modes according to the requirement of normalization of Hawking-Bekenstein entropy. It has been found that $N\simeq 162$, which is just in good agreement with one in the Minimal Super-symmetric Standard Model. The studies in this paper represent an attempt to reveal some physics near the horizon at Planck scale. This paper serves a brief review on the author's works on this subject. 
  Relativistic stars are endowed with intense electromagnetic fields but are also subject to oscillations of various types. We here investigate the impact that oscillations have on the electric and magnetic fields external to a relativistic star in vacuum. In particular, modelling the star as a relativistic polytrope with infinite conductivity, we consider the solution of the general relativistic Maxwell equations both in the vicinity of the stellar surface and far from it, once a perturbative velocity field is specified. In this first paper we present general analytic expressions that are not specialized to a particular magnetic field topology or velocity field. However, as a validating example and an astrophysically important application, we consider a dipolar magnetic field and the velocity field corresponding to the rotation of the misaligned dipole. Besides providing analytic expressions for the electromagnetic fields produced by this configuration, we calculate, for the first time, the general relativistic energy loss through dipolar electromagnetic radiation. We find that the widely used Newtonian expression under-estimates this loss by a factor of 2-6 depending on the stellar compactness. This correction could have important consequences in the study of the spin evolution of pulsars. 
  The problem of finding independent components of an indexed object (e.g., a tensor) with arbitrary number of indices and arbitrary linear symmetries is discussed. It is proved that the number of independent components $f(k)$ is a polynomial of degree not greater than the number of indices $n$, $k$ being the dimension of the space. Several algorithms to compute $f(k)$ for arbitrary $k$ are described and discussed. It is shown that in the worst case finding $f(k)$ for arbitrary $k$ requires solving at most P(n) systems of linear equations with at most $(n!)^2$ equations for at most of $n!$ unknowns, P(n) being the number of partitions of $n$. As a by-product, an efficient algorithm to parametrize all components of the object through its independent components is found and implemented in \Mathematica. 
  We present a formalism for constructing quasi-equilibrium binary black hole initial data suitable for numerical evolution. We construct quasi-equilibrium models by imposing an approximate helical Killing symmetry appropriate for quasi-circular orbits. We use the sum of two Kerr-Schild metrics as our background metric, thereby improving on conformally flat backgrounds that do not accommodate rotating black holes and providing a horizon-penetrating lapse, convenient for implementing black hole excision. We set inner boundary conditions at an excision radius well inside the apparent horizon and construct these boundary conditions to incorporate the quasi-equilibrium condition and recover the solution for isolated black holes in the limit of large separation. We use our formalism both to generate initial data for binary black hole evolutions and to construct a crude quasi-equilibrium, inspiral sequence for binary black holes of fixed irreducible mass. 
  Using an estimate, we prove that if solution of the spherically symmetric Einstein-Vlasov-Maxwell system develops a singularity at all time, then the first one has to appear at the center of symmetry. 
  Stochastic quantisation normally involves the introduction of a fictitious extra time parameter, which is taken to infinity so that the system evolves to an equilibrium state.In the case of a locally supersymmetric theory, an interesting new possibility arises due to the existence of a Nicolai map. In this case it turns out that no additional time parameter is required, as the existence of the Nicolai map ensures that the same job can be done by the existing time parameter after Euclideanisation. This provides the quantum theory with a natural probabilistic interpretation, without any reference to the concept of an inner product or a Hilbert space structure. 
  In this paper I argue for a reassessment of special relativity. The fundamental theory of relativity applicable in this Universe has to be consistent with the existence of the massive Universe, and with the effects of its gravitational interaction on local physics. A reanalysis of the situation suggests that all relativistic effects that are presently attributed to kinematics of relative motion in flat space-time are in fact gravitational effects of the nearly homogeneous and isotropic Universe. The correct theory of relativity is the one with a preferred cosmic rest frame. Yet, the theory preserves Lorentz invariance. I outline the new theory of Cosmic Relativity, and its implications to local physics, especially to physics of clocks and to quantum physics. This theory is generally applicable to inertial and noninertial motion. Most significanlty, experimental evidence support and favour Cosmic Relativity. There are observed effects that can be consistently explained only within Cosmic Relativity. The most amazing of these is the dependence of the time dilation of clocks on their `absolute' velocity relative to the cosmic rest frame. Important effects on quantum systems include the physical cause of the Thomas precession responsible for part of the spectral fine structure, and the phase changes responsible for the spin-statistics connection. At a deeper level it is conlcuded that relativity in flat space-time with matter reiterates Mach's principle. There will not be any relativistic effect in an empty Universe. 
  A proposal for the issue of time and observables in any parameterized theory such as general relativity is addressed. Introduction of a gauge potential 3-form A in the theory of relativity enables us to define a gauge-invariant quantity which can be used by observers as a clock to measure the passage of time. This dynamical variable increases monotonically and continuously along a world line. Then we define world line observables to be any covariantly defined quantity obtained from the field configurations on any such causal past with dynamical time T. 
  Plane symmetric self-similar solutions to Einstein's four-dimensional theory of gravity are studied and all such solutions are given analytically in closed form. The local and global properties of these solutions are investigated and it is shown that some of the solutions can be interpreted as representing gravitational collapse of the scalar field. During the collapse, trapped surfaces are never developed. As a result, no black hole is formed. Although the collapse always ends with spacetime singularities, it is found that these singularities are spacelike and not naked. 
  All the 2+1-dimensional circularly symmetric solutions with kinematic self-similarity of the second kind to the Einstein-massless-scalar field equations are found and their local and global properties are studied. It is found that some of them represent gravitational collapse of a massless scalar field, in which black holes are always formed. 
  We develop a new model for the universe based on two key axioms: first, the inertial energy of the universe is a constant, and second, the total energy of a particle, the inertial plus the gravitational potential energy produced by the other mass in the universe, is zero. This model allows the speed of light and the total mass of the universe to vary as functions of the cosmological time, where we assume the gravitational constant to be a constant. By means of these assumptions the relations between the scale factor and the other parameters are derived. The Einstein equation by making it compatible with varying-$c$ is used and the Friedmann equations in this model are obtained. Assuming the matter content of the universe to be perfect fluids, the model fixes $\gamma$ to be 2/3. That is, the whole universe always exhibits a negative pressure. Moreover, the behaviour of the scale factor is the same for any value of the curvature. It is also shown that the universe has begun from a big bang with zero initial mass and expands forever even with positive curvature, but it always decelerates. At the end, solutions to some famous problems, mainly of the standard big bang model, and an explanation for the observational data about the accelerating universe are provided. 
  Quantization of general relativity in terms of SL(2,C)-connections (i.e. in terms of the complex Ashtekar variables) is technically difficult because of the non-compactness of SL(2,C). The difficulties concern the construction of a diffeomorphism invariant Hilbert space structure on the space of cylindrical functions of the connections. We present here a 'toy' model of such a Hilbert space built over connections whose structure group is the group of real numbers. We show that in the case of any Hilbert space built analogously over connections with any non-compact structure group (this includes some models presented in the literature) there exists an obstacle which does not allow to define a *-representation of cylindrical functions on the Hilbert space by the multiplication map which is the only known way to define a diffeomorphism invariant representation of the functions. 
  In this paper, we are exploring some of the properties of the self-similar solutions of the first kind. In particular, we shall discuss the kinematic properties and also check the singularities of these solutions. We discuss these properties both in co-moving and also in non co-moving (only in the radial direction) coordinates. Some interesting features of these solutions turn up. 
  The resolution of the problem of cosmological singularity in the framework of gauge theories of gravitation is discussed. Generalized cosmological Friedmann equations for homogeneous isotropic models filled by interacting scalar fields and usual gravitating matter are deduced. It is shown that generic feature of cosmological models of flat, open and closed type is their regular bouncing character. 
  We consider a scenario where the interior spacetime,described by a heat conducting fluid sphere is matched to a Vaidya metric in higher dimensions.Interestingly we get a class of solutions, where following heat radiation the boundary surface collapses without the appearance of an event horizon at any stage and this happens with reasonable properties of matter field.The non-occurrence of a horizon is due to the fact that the rate of mass loss exactly counterbalanced by the fall of boundary radius.Evidently this poses a counter example to the so-called cosmic censorship hypothesis.Two explicit examples of this class of solutions are also given and it is observed that the rate of collapse is delayed with the introduction of extra dimensions.The work extends to higher dimensions our previous investigation in 4D. 
  We argue that every finite piecewise smooth spacelike two-surface of spacetime possesses an entropy which is, in natural units, one quarter of its area. However, the thermodynamical properties of a two-surface become apparent only to the observers having that two-surface as a horizon. Consequences of this result are discussed. 
  Exact cosmological solutions are obtained for a five dimensional inhomogeneous fluid distribution along with a Brans-Dicke type of scalar field. The set includes varied forms of matter field including $\rho+p=0$, where p is the 3D isotropic pressure. Depending on the signature of 4-space curvature our solutions admit of indefinite expansion in the usual 3-space and dimensional reduction of the fifth dimension. Due to the presence of the scalar field the case $p=-\rho$ does not yield an exponential expansion of the scale factor, which strikingly differs from our earlier investigations without scalar field.The \emph{effective} four dimensional values of entropy and matter are calculated and possible consequences of entropy and matter generation in the 4D world as a result of dimensional reduction of the extra space are also discussed. Encouraging to point out that aside from the well known big bang singularity our inhomogeneous cosmology is spatially regular everywhere. Further our model seems to suggest an alternative mechanism pointing to a smooth pass over from a primordial, inhomogeneous cosmological phase to a 4D homogeneous one. 
  The Conditional Probability Interpretation (CPI), first introduced by Page and Wootters, is reviewed and refined. It is argued that in it's refined form the CPI is capable of answering various past criticisms. In particular, questions involving more than one clock time are described in detail, resolving the problems raised in Kuchar's ``reduction ad absurdum''. In the case of Parametrized Particle Dynamics, conventional quantum mechanics is recovered in the ideal clock limit. When E=0 is among the continuous spectrum of the Hamiltonian, the induced inner product is used to construct the physical Hilbert space $\clh_{\rm ph}$ from the generalized eigenvectors in (the topological dual of) $\clh_{\rm aux}$. This allows the CPI to be applied to these `continuous-spectrum' cases in a more rigorous fashion than that described previously. The discrete spectrum case is also treated. 
  The measured values of the matter energy density and the vacuum energy density are obtained using an adiabatic black hole percolating model at the critical point. The percolation of black holes is related with an expanding isotropic universe filled with the most entropic fluid saturating the holographic bound. 
  We consider for the first time the solutions of Klein-Gordon equation in gravitational field of {\em a massive} point source in GR. We examine numerically the basic bounded quantum state and the next few states in the discrete spectrum for different values of the orbital momentum. A novel feature of the solutions under consideration is the essential dependence if their physical properties on the gravitational mass defect of the point source, even not introduced up to recently. It yields a repulsion or an attraction of the quantum levels up to their quasi-crossing. 
  Any canonical quantum theory can be understood to arise from the compatibility of the statistical geometry of distinguishable observations with the canonical Poisson structure of Hamiltonian dynamics. This geometric perspective offers a novel, background independent non-perturbative formulation of quantum gravity. We invoke a quantum version of the equivalence principle, which requires both the statistical and symplectic geometries of canonical quantum theory to be fully dynamical quantities. Our approach sheds new light on such basic issues of quantum gravity as the nature of observables, the problem of time, and the physics of the vacuum. In particular, the observed numerical smallness of the cosmological constant can be rationalized in this approach. 
  The standard post-Newtonian approximation to gravitational waveforms, called T-approximants, from non-spinning black hole binaries are known not to be sufficiently accurate close to the last stable orbit of the system. A new approximation, called P-approximants, is believed to improve the accuracy of the waveforms rendering them applicable up to the last stable orbit. In this study we apply P-approximants to the case of a test-particle in equatorial orbit around a Kerr black hole parameterized by a spin parameter q that takes values between -1 and 1. In order to assess the performance of the two approximants we measure their effectualness (i.e. larger overlaps with the exact signal), and faithfulness (i.e. smaller biases while measuring the parameters of the signal) with the exact (numerical) waveforms. We find that in the case of prograde orbits, that is orbits whose angular momentum is in the same sense as the spin angular momentum of the black hole, T-approximant templates obtain an effectualness of ~ 0.99 for spins q < 0.75. For 0.75 < q < 0.95, the effectualness drops to about 0.82. The P-approximants achieve effectualness of > 0.99 for all spins up to q = 0.95. The bias in the estimation of parameters is much lower in the case of P-approximants than T-approximants. We find that P-approximants are both effectual and faithful and should be more effective than T-approximants as a detection template family when q>0. For q<0 both T- and P-approximants perform equally well so that either of them could be used as a detection template family. 
  The Laser Interferometer Space Antenna (LISA) is expected to provide the largest observational sample of binary systems of faint sub-solar mass compact objects, in particular white-dwarfs, whose radiation is monochromatic over most of the LISA observational window. Current astrophysical estimates suggest that the instrument will be able to resolve about 10000 such systems, with a large fraction of them at frequencies above 3 mHz, where the wavelength of gravitational waves becomes comparable to or shorter than the LISA arm-length. This affects the structure of the so-called LISA transfer function which cannot be treated as constant in this frequency range: it introduces characteristic phase and amplitude modulations that depend on the source location in the sky and the emission frequency. Here we investigate the effect of the LISA transfer function on detection and parameter estimation for monochromatic sources. For signal detection we show that filters constructed by approximating the transfer function as a constant (long wavelength approximation) introduce a negligible loss of signal-to-noise ratio -- the fitting factor always exceeds 0.97 -- for f below 10mHz, therefore in a frequency range where one would actually expect the approximation to fail. For parameter estimation, we conclude that in the range 3mHz to 30mHz the errors associated with parameter measurements differ from about 5% up to a factor of 10 (depending on the actual source parameters and emission frequency) with respect to those computed using the long wavelength approximation. 
  Three weak gravitational effects associated with the gravitomagnetic fields are taken into account in this paper: (i) we discuss the background Lorentz transformation and gauge transformation in a linearized gravity theory, and obtain the expression for the spin of gravitational field by using the canonical procedure and Noether theorem; (ii) we point out that by using the coordinate transformation from the fixed frame to the rotating frame, it is found that the nature of Mashhoon's spin-rotation coupling is in fact an interaction between the gravitomagnetic moment of a spinning particle and the gravitomagnetic fields. The fact that the rotational angular velocity of a rotating frame can be viewed as a gravitomagnetic field is demonstrated; (iii) a purely gravitational generalization of Mashhoon's spin-rotation coupling, i.e., the interaction of the graviton spin with the gravitomagnetic fields is actually a self-interaction of the spacetime (gravitational fields). In the present paper, we will show that this self-interaction will also arise in a non-inertial frame of reference itself: specifically, a rotating frame that experiences a fluctuation of its rotational frequency (i.e., the change in the rotational angular frequency) will undergo a weak self-interaction. The self-interaction of the rotating frame, which can also be called the self-interaction of the spacetime of the rotating frame, is just the non-inertial generalization of the interaction of the graviton spin with the gravitomagnetic fields. 
  In this paper we first calculate the post-Newtonian gravitoelectric secular rate of the mean anomaly of a test particle freely orbiting a spherically symmetric central mass. Then, we propose a novel approach to suitably combine the presently available planetary ranging data to Mercury, Venus and Mars in order to determine, simultaneously and independently of each other, the Sun's quadrupole mass moment J_2 and the secular advances of the perihelion and the mean anomaly. This would also allow to obtain the PPN parameters gamma and beta independently. We propose to analyze the time series of three linear combinations of the experimental residuals of the rates of the nodes, the longitudes of perihelia and mean anomalies of Mercury, Venus and Mars built up in order to absorb the secular precessions induced by the solar oblateness and the post-Newtonian gravitoelectric forces. The values of the three investigated parameters can be obtained by fitting the expected linear trends with straight lines, determining their slopes in arcseconds per century and suitably normalizing them. According to the present-day EPM2000 ephemerides accuracy, the obtainable precision would be of the order of 10^-4-10^-5 for the PPN parameters and, more interestingly, of 10^-9 for J_2. The future BepiColombo mission should improve the Mercury's orbit by one order of magnitude. 
  Positively-curved, oscillatory universes are studied within the context of Loop Quantum Cosmology subject to a consistent semi-classical treatment. The semi-classical effects are reformulated in terms of an effective phantom fluid with a variable equation of state. In cosmologies sourced by a massless scalar field, these effects lead to a universe that undergoes ever-repeating cycles of expansion and contraction. The presence of a self-interaction potential for the field breaks the symmetry of the cycles and can enable the oscillations to establish the initial conditions for successful slow-roll inflation, even when the field is initially at the minimum of its potential with a small kinetic energy. The displacement of the field from its minimum is enhanced for lower and more natural values of the parameter that sets the effective quantum gravity scale. For sufficiently small values of this parameter, the universe can enter a stage of eternal self-reproduction. 
  I suggest that the Spin-Statistics connection is a consequence of the phase shifts on quantum scattering amplitudes due to the induced gravitomagnetic field of the whole Universe at critical density. This connection was recently brought out in the context of a new theory of relativity in flat space with matter, called Cosmic Relativity (gr-qc/0406023). This prediction of the correct gravitational phases is a consequence of any relativistic gravitation theory in the presence of the massive Universe. This can also be interpreted as related to the Mach's principle applied to quantum phenomena. Perhaps this is the simplest valid proof of the Spin-Statistics Theorem, and it finally identifies the physical origin of the connection. 
  There has been substantial interest, as of late, in the quantum-corrected form of the Bekenstein-Hawking black hole entropy. The consensus viewpoint is that the leading-order correction should be a logarithm of the horizon area; however, the value of the logarithmic prefactor remains a point of notable controversy. Very recently, Hod has employed statistical arguments that constrain this prefactor to be a non-negative integer. In the current paper, we invoke some independent considerations to argue that the "best guess" for the prefactor might simply be zero. Significantly, this value complies with the prior prediction and, moreover, seems suggestive of some fundamental symmetry. 
  In this paper we ask whether the phenomenon of timing noise long known in electromagnetic pulsar astronomy is likely to be important in gravitational wave (GW) observations of spinning-down neutron stars. We find that timing noise is strong enough to be of importance only in the young pulsars, which must have larger triaxialities than theory predicts for their GW emission to be detectable. However, assuming that their GW emission is detectable, we list the pulsars for which timing noise is important, either because it is strong enough that its neglect by the observer would render the source undetectable, or else because it is a measurable feature of the GW signal. We also find that timing noise places a limit on the observation duration of a coherent blind GW search, and suggest that hierarchical search techniques might be able to cope with this problem. Demonstration of the presence or absence of timing noise in the GW channel would give a new probe of neutron star physics. 
  The spectra of relic gravitational waves produced as a result of cosmological expansion of the inflationary models are derived in Brans-Dicke theory of gravity.The time dependence of the very early Hubble parameter and matter energy density are derived from frequency dependent spectrum of relic gravitational waves.Also it is found that Brans-Dicke scalar field contributes to the energy density of relic gravitons. 
  I describe the Einstein's gravitation of 3+1 dimensional spacetimes using the (2,2) formalism without assuming isometries. In this formalism, quasi-local energy, linear momentum, and angular momentum are identified from the four Einstein's equations of the divergence-type, and are expressed geometrically in terms of the area of a two-surface and a pair of null vector fields on that surface. The associated quasi-local balance equations are spelled out, and the corresponding fluxes are found to assume the canonical form of energy-momentum flux as in standard field theories. The remaining non-divergence-type Einstein's equations turn out to be the Hamilton's equations of motion, which are derivable from the {\it non-vanishing} Hamiltonian by the variational principle. The Hamilton's equations are the evolution equations along the out-going null geodesic whose {\it affine} parameter serves as the time function. In the asymptotic region of asymptotically flat spacetimes, it is shown that the quasi-local quantities reduce to the Bondi energy, linear momentum, and angular momentum, and the corresponding fluxes become the Bondi fluxes. The quasi-local angular momentum turns out to be zero for any two-surface in the flat Minkowski spacetime. I also present a candidate for quasi-local {\it rotational} energy which agrees with the Carter's constant in the asymptotic region of the Kerr spacetime. Finally, a simple relation between energy-flux and angular momentum-flux of a generic gravitational radiation is discussed, whose existence reflects the fact that energy-flux always accompanies angular momentum-flux unless the flux is an s-wave. 
  By calculating the Newman-Penrose Weyl tensor components of a perturbed spherically symmetric space-time with respect to invariantly defined classes of null tetrads, we give a physical interpretation, in terms of gravitational radiation, of odd parity gauge invariant metric perturbations. We point out how these gauge invariants may be used in setting boundary and/or initial conditions in perturbation theory. 
  The quantum measure in area tensor Regge calculus can be constructed in such the way that it reduces to the Feynman path integral describing canonical quantisation if the continuous limit along any of the coordinates is taken. This construction does not necessarily mean that Lorentzian (Euclidean) measure satisfies correspondence principle, that is, takes the form proportional to $e^{iS}$ ($e^{-S}$) where $S$ is the action. Requirement to fit this principle means some restriction on the action, or, in the context of representation of the Regge action in terms of independent rotation matrices (connections), restriction on such representation. We show that the representation based on separate treatment of the selfdual and antiselfdual rotations allows to modify the derivation and give sense to the conditionally convergent integrals to implement both the canonical quantisation and correspondence principles. If configurations are considered such that the measure is factorisable into the product of independent measures on the separate areas (thus far it was just the case in our analysis), the considered modification of the measure does not effect the vacuum expectation values. 
  A comparative analysis of the versions of quantum measure in the area tensor Regge calculus is performed on the simplest configurations of the system. The quantum measure is constructed in such the way that it reduces to the Feynman path integral describing canonical quantisation if the continuous limit along any of the coordinates is taken. As we have found earlier, it is possible to implement also the correspondence principle (proportionality of the Lorentzian (Euclidean) measure to $e^{iS}$ ($e^{-S}$), $S$ being the action). For that a certain kind of the connection representation of the Regge action should be used, namely, as a sum of independent contributions of selfdual and antiselfdual sectors (that is, effectively 3-dimensional ones). There are two such representations, the (anti)selfdual connections being SU(2) or SO(3) rotation matrices according to the two ways of decomposing full SO(4) group, as SU(2) $\times$ SU(2) or SO(3) $\times$ SO(3). The measure from SU(2) rotations although positive on physical surface violates positivity outside this surface in the general configuration space of arbitrary independent area tensors. The measure based on SO(3) rotations is expected to be positive in this general configuration space on condition that the scale of area tensors considered as parameters is bounded from above by the value of the order of Plank unit. 
  Using the canonical quantum theory of spherically symmetric pure gravitational systems, we present a direct correspondence between the Friedmann-Robertson-Walker (FRW) cosmological model in the interior of a Schwarzschild black hole and the nth energy eigenstate of a linear harmonic oscillator. Such type of universe has a quantized mass of the order of the Planck mass and harmonic oscillator wave functions 
  We investigate here the gravitational collapse end states for a spherically symmetric perfect fluid with an equation of state $p=k\rho$. It is shown that given a regular initial data in terms of the density and pressure profiles at the initial epoch from which the collapse develops, the black hole or naked singularity outcomes depend on the choice of rest of the free functions available, such as the velocities of the collapsing shells, and the dynamical evolutions as allowed by Einstein equations. This clarifies the role that equation of state and initial data play towards determining the final fate of gravitational collapse. 
  The Majumdar-Papapetrou system is the subset of the Einstein-Maxwell-charged dust matter theory, when the charge of each particle is equal to its mass. Solutions for this system are less difficult to find, in general one does not need even to impose any spatial symmetry a priori. For instance, any number of extreme Reissner-Nordstrom solutions (which in vacuum reduce to extreme Reissner-Nordstrom black holes) located at will is a solution. In matter one can also find solutions with some ease. Here we find an exact solution of the Majumdar-Papapetrou system, a spherically symmetric charged thick shell, with mass m, outer radius r_o, and inner radius r_i. This solution consists of three regions, an inner Minkowski region, a middle region with extreme charged dust matter, and an outer Reissner-Nordstrom region. The matching of the regions, obeying the usual junction conditions for boundary surfaces, is continuous. For vanishing inner radius, one obtains a Bonnor star, whereas for vanishing thickness, one obtains an infinitesimally thin shell. For sufficiently high mass of the thick shell or sufficiently small outer radius, it forms an extreme Reissner-Nordstrom quasi-black hole, i.e., a star whose gravitational properties are virtually indistinguishable from a true extreme black hole. This quasi-black hole has no hair and has a naked horizon, meaning that the Riemann tensor at the horizon on an infalling probe diverges. At the critical value, when the mass is equal to the outer radius, m=r_o, there is no smooth manifold. Above the critical value when m>r_o there is no solution, the shell collapses into a singularity. Systems with m< r_o are neutrally stable. Many of these properties are similar to those of gravitational monopoles. 
  It has earlier been argued that there should exist a formulation of quantum mechanics which does not refer to a background spacetime. In this paper we propose that, for a relativistic particle, such a formulation is provided by a noncommutative generalisation of the Hamilton-Jacobi equation. If a certain form for the metric in the noncommuting coordinate system is assumed, along with a correspondence rule for the commutation relations, it can be argued that this noncommutative Hamilton-Jacobi equation is equivalent to standard quantum mechanics. 
  We presented a model for unification of electricity and gravity. We have found a consistent description of all physical quantities pertaining to the system. We have provided limiting values for all physical values. These values are neither zero nor infinity. Our universe is described at all times by the four dimensional constants $c, \hbar, k, G$ only. The remnant of vacuum remains at all epochs with different values. The present cosmological puzzles are justified as due to the consequences of cosmic quantization developed in this work. 
  The Casimir energy density calculated for a spherical shell of radius equal to the size of the Universe projected back to the Planck time is almost equal to the present day critical density. Is it just a coincidence, or is it a solution to the `cosmic dark energy' and the `cosmic coincidence' problems? The correspondence is too close to be ignored as a coincidence, especially since this solution fits the conceptual and numerical ideas about the dark energy, and also answers why this energy is starting to dominate at the present era in the evolution of the Universe. 
  We consider a Volovik's analog model for description of a topological defects in a superfluid and we investigate the scattering of quasiparticles in this background. The analog of the gravitational Aharonov-Bohm in this system is found. An analysis of this problem employing loop variables is considered and corroborates for the existence of the Aharonov-Bohm effect in this system. The results presented here may be used to study the Aharonov-Bohm effect in superconductors. 
  The variation of the velocity of a periodic signal and its frequency along the world line of a standard emitter (at rest with an observer) are considered in a space with affine connections and metrics. It is shown that the frequency of the emitted periodic signal is depending on the kinematic characteristics of the motion of the emitter in space-time related to its shear and expansion velocities. The same conclusions are valid for a standard clock moving with an observer. 
  Dilatations by means of a constant factor can be seen in a double way: as a simple change of units length or as a conformal mapping of the starting spacetime into a ``stretched'' one with the same units length. The numerical value of the black hole entropy depends on the interpretations made for the stretched manifold. Further, we study the possibility to choose an unusual ``mass dependent'' normalization for the timelike Killing vector for a Kerr black hole with and without a cosmic string. 
  We present a simple method to obtain vacuum solutions of Einstein's equations in parabolic coordinates starting from ones with cylindrical symmetries. Furthermore, a generalization of the method to a more general situation is given together with a discussion of the possible relations between our method and the Belinsky-Zakharov soliton-generating solutions. 
  The massless field perturbations of the accelerating Minkowski and Schwarzschild spacetimes are studied. The results are extended to the propagation of the Proca field in Rindler spacetime. We examine critically the possibility of existence of a general spin-acceleration coupling in complete analogy with the well-known spin-rotation coupling. We argue that such a direct coupling between spin and linear acceleration does not exist. 
  In this web note, we reply to a recent paper, gr-qc/0404126, confirming a previous work of ours in which a cosmological bouncing phase was shown to have the ability of modifying the spectrum of primordial perturbations (PRD 68, 103517 2003), but challenging its physical conditions of validity. Explicitly, Ref. gr-qc/0404126, besides pretending our Taylor series expansion of the scale factor close to the bounce amounts to choosing a family of polynomial scale factors, also claims that the bounce affects the spectrum only if the mass scale of the scalar field driving the dynamics is of the order of the Planck mass. We show that these objections are either misleading or incorrect since the minimum size of the Universe a_0 (value of the scale factor at the bounce) is either not physically specified, as required in a closed Universe, or implicitly assumed to be the Planck mass. We calculate this mass and obtain that, unsurprisingly, for a reasonable value of a_0, i.e. much larger than the Planck length, the scalar field mass is smaller than the Planck mass. 
  In 4 dimensions, general relativity can be formulated as a constrained $BF$ theory; we show that the same is true in 2 dimensions. We describe a spinfoam quantization of this constrained BF-formulation of 2d riemannian general relativity, obtained using the Barrett-Crane technique of imposing the constraint as a restriction on the representations summed over. We obtain the expected partition function, thus providing support for the viability of the technique. The result requires the nontrivial topology of the bundle where the gravitational connection is defined, to be taken into account. For this purpose, we study the definition of a principal bundle over a simplicial base space. The model sheds light also on several other features of spinfoam quantum gravity: the reality of the partition function; the geometrical interpretation of the Newton constant, and the issue of possible finiteness of the partition function of quantum general relativity. 
  Discussions on the Langevin Twins 'paradox' are most often based on a "triangular" diagram which outlines the twins spacetime travels. It won't be our way, avoiding what we think to be a problem at the basis of numerous controversies. Our approach relies on a fundamentally different Equivalence Principle, namely the so-called "Punctual Equivalence Principle", from which we think that a very conformal aspect proceeds. We present a resolution of this paradox in the framework of the so-called "scale gravity". This resolution hinges on a clear determinism of the Twins proper times, in some definite situations, and a fundamental under-determinism in some other particular ones, the physical discrimination of which being achieved out of a precise mathematical description of conformal geometry. Moreover, we find that the time discrepancy between the twins, could somehow be at the root expression of the second fundamental law of thermodynamics. 
  Motivated by the critical remarks of several authors, we have re-analyzed the classical ether-drift experiments with the conclusion that the small observed deviations should not be neglected. In fact, within the framework of Lorentzian Relativity, they might indicate the existence of a preferred frame relatively to which the Earth is moving with a velocity v_earth\sim 200$ km/s (value projected in the plane of the interferometer). We have checked this idea by comparing with the modern ether-drift experiments, those where the observation of the fringe shifts is replaced by the difference \Delta \nu in the relative frequencies of two cavity-stabilized lasers, upon local rotations of the apparatus or under the Earth's rotation. It turns out that, even in this case, the most recent data are consistent with the same value of the Earth's velocity, once the vacuum within the cavities is considered a physical medium whose refractive index is fixed by General Relativity. We thus propose a sharp experimental test that can definitely resolve the issue. If the small deviations observed in the classical ether-drift experiments were not mere instrumental artifacts, by replacing the high vacuum in the resonating cavities with a dielectric gaseous medium (e.g. air), the typical measured \Delta\nu\sim 1 Hz should increase by orders of magnitude. This expectation is consistent with the characteristic modulation of a few kHz observed in the original experiment with He-Ne masers. However, if such enhancement would not be confirmed by new and more precise data, the existence of a preferred frame can be definitely ruled out. 
  The effects of compact extra dimensions upon quantum stress tensor fluctuations are discussed. It is argued that as the compactification volume decrease, these fluctuations increase in magnitude. In principle, this would have the potential to create observable effects, such as luminosity fluctuations or angular blurring of distant sources, and lead to constraints upon Kaluza-Klein theories. However, the dependence of the four-dimensional Newton's constant upon the compactification volume causes the gravitational effects of the stress tensor fluctuations to be finite in the limit of small volume. Consequently, no observational constraints upon Kaluza-Klein theories are obtained. 
  We reexamine here a problem considered in detail before by Waugh and Lake: the solutions of spherically symmetric Einstein's equations with a radial flow of unpolarized radiation (the Vaidya metric) in double-null coordinates. This problem is known to be not analytically solvable, the only known explicit solutions correspond to the constant mass case (Schwarzschild solution in Kruskal-Szekeres form) and the linear and exponential mass functions originally discovered by Waugh and Lake. We present here a semi-analytical approach that can be used to discuss some qualitative and quantitative aspects of the Vaidya metric in double-null coordinates for generic mass functions. We present also a new analytical solution corresponding to $(1/v)$-mass function. 
  Recent observations of Type Ia supernova at high redshifts establish that the dark energy component of the universe has (a probably constant) ratio between pressure and energy density $w=p/\rho=-1.02(^{+0.13}_{-0.19})$. The conventional quintessence models for dark energy are restricted to the range $-1\le w < 0$, with the cosmological constant corresponding to $w=-1$. Conformally coupled quintessence models are the simplest ones compatible with the marginally allowed superaccelerated regime ($w<-1$). However, they are known to be plagued with anisotropic singularities.   We argue here that the extension of the classical approach to the semiclassical one, with the inclusion of quantum counterterms necessary to ensure the renormalization, can eliminate the anisotropic singularities preserving the isotropic behavior of conformally coupled superquintessence models. Hence, besides of having other interesting properties, they are consistent candidates to describe the superaccelerated phases of the universe compatible with the present experimental data. 
  We show that the inclusion of a term $C_{abcd}C^{abcd}$ in the action can remove the recently described anisotropic singularity occurring on the hypersurface $F(\phi)=0$ of scalar-tensor theories of gravity of the type $$ S=\int d^4x \sqrt{-g} {F(\phi)R - \partial_a\phi\partial^a\phi -2V(\phi)}, $$ preserving, by construction, all of their isotropic solutions. We show that, in principle, a higher order term of this type can arise from considerations about the renormalizability of the semiclassical approach to the theory. Such result brings again into consideration the quintessential models recently proposed based in a conformally coupled scalar field ($F(\phi)=1-{1/6}\phi^2$) with potential $V(\phi)=\frac{m}{2}\phi^2 -\frac{\Omega}{4}\phi^4$, that have been discharged as unrealistic precisely by their anisotropic instabilities on the hypersurface $F(\phi)=0$. 
  We review the arguments and counter arguments about the recent proposal for generic censorship violation. In particular the argument made in gr-qc/0405050 against our proposal for a possible expanding domain wall that could encompass a large black hole, is shown to have a serious flaw. Other problems of the original idea are also discussed. 
  Recent researches show that the fluctuations of the dielectric mirrors coating thickness can introduce a substantial part of the future laser gravitational-wave antennae total noise budget. These fluctuations are especially large in the high-reflectivity end mirrors of the Fabry-Perot cavities which are being used in the laser gravitational-wave antennae.   We show here that the influence of these fluctuations can be substantially decreased by using additional short Fabry-Perot cavities, tuned in anti-resonance instead of the end mirrors. 
  To systematically analyze the dynamical implications of the matter content in cosmology, we generalize earlier dynamical systems approaches so that perfect fluids with a general barotropic equation of state can be treated. We focus on locally rotationally symmetric Bianchi type IX and Kantowski-Sachs orthogonal perfect fluid models, since such models exhibit a particularly rich dynamical structure and also illustrate typical features of more general cases. For these models, we recast Einstein's field equations into a regular system on a compact state space, which is the basis for our analysis. We prove that models expand from a singularity and recollapse to a singularity when the perfect fluid satisfies the strong energy condition. When the matter source admits Einstein's static model, we present a comprehensive dynamical description, which includes asymptotic behavior, of models in the neighborhood of the Einstein model; these results make earlier claims about ``homoclinic phenomena and chaos'' highly questionable. We also discuss aspects of the global asymptotic dynamics, in particular, we give criteria for the collapse to a singularity, and we describe when models expand forever to a state of infinite dilution; possible initial and final states are analyzed. Numerical investigations complement the analytical results. 
  In this paper, the general procedure to solve the General Relativistic Hydrodynamical(GRH) equations with Adaptive-Mesh Refinement (AMR) is presented. In order to achieve, the GRH equations are written in the conservation form to exploit their hyperbolic character. The numerical solutions of general relativistic hydrodynamic equations are done by High Resolution Shock Capturing schemes (HRSC), specifically designed to solve non-linear hyperbolic systems of conservation laws. These schemes depend on the characteristic information of the system. The Marquina fluxes with MUSCL left and right states are used to solve GRH equations. First, different test problems with uniform and AMR grids on the special relativistic hydrodynamics equations are carried out to verify the second order convergence of the code in 1D, 2D and 3D. Results from uniform and AMR grid are compared. It is found that adaptive grid does a better job when the number of resolution is increased. Second, the general relativistic hydrodynamical equations are tested using two different test problems which are Geodesic flow and Circular motion of particle In order to this, the flux part of GRH equations is coupled with source part using Strang splitting. The coupling of the GRH equations is carried out in a treatment which gives second order accurate solutions in space and time. 
  We address the question of radiation emitted from a collapsing star. We consider the simple model of a spherical star consisting of pressure-free dust and we derive the emission spectrum via a systematic asymptotic expansion of the complete Bogolubov amplitude. We present a review of standard features of the problem and we point out certain inconsistencies in previous derivations. 
  We recalculate the spectrum of radiation emitted from a collapsing star. We consider the simple model of a spherical star consisting of pressure-free dust and we derive the thermal spectrum via a systematic asymptotic expansion of the complete Bogolubov amplitude. Inconsistencies in previous derivations are pointed out. 
  For future configurations, we study the relation between the abatement of the noise sources and the Signal to Noise Ratio (SNR) for coalescing binaries. Our aim is not the proposition of a new design, but an indication of where in the bandwidth or for which noise source, a noise reduction would be most efficient. We take VIRGO as the reference for our considerations, solely applicable to the inspiralling phase of a coalescing binary. Thus, only neutron stars and small black holes of few solar masses are encompassed by our analysis. The contributions to the SNR given by final merge and quasi-normal ringing are neglected. It is identified that i) the reduction in the mirror thermal noise band provides the highest gain for the SNR, when the VIRGO bandwidth is divided according to the dominant noises; ii) it exists a specific frequency at which lies the potential largest increment in the SNR, and that the enlargement of the bandwidth, where the noise is reduced, produces a shift of such optimal frequency to higher values; iii) the abatement of the pendulum thermal noise provides the largest, but modest, gain, when noise sources are considered separately. Our recent astrophysical analysis on event rates for neutron stars leads to a detection rate of one every 148 or 125 years for VIRGO and LIGO, respectively, while a recently proposed and improved, but still conservative, VIRGO configuration would provide an increase to 1.5 events per year. Instead, a bi-monthly event rate, similar to advanced LIGO, requires a 16 times gain. We analyse the 3D (pendulum, mirror, shot noises) parameter space showing how such gain could be achieved. 
  We show that positivity of energy for stationary, or strongly uniformly Schwarzschildian, asymptotically flat, non-singular domains of outer communications can be proved using Galloway's null rigidity theorem. 
  It is shown that in 5D Kaluza-Klein theory there are everywhere regular wormhole-like solutions in which the magnetic field at the center is the origin of a rotation on the peripheral part of these solutions. The time on the peripheral part is topologically non-trivial and magnetic field is suppressed in comparison with the electric one. 
  A new approach to quantum gravity is presented based on a nonlinear quantization scheme for canonical field theories with an implicitly defined Hamiltonian. The constant mean curvature foliation is employed to eliminate the momentum constraints in canonical general relativity. It is, however, argued that the Hamiltonian constraint may be advantageously retained in the reduced classical system to be quantized. This permits the Hamiltonian constraint equation to be consistently turned into an expectation value equation on quantization that describes the scale factor on each spatial hypersurface characterized by a constant mean exterior curvature. This expectation value equation augments the dynamical quantum evolution of the unconstrained conformal three-geometry with a transverse traceless momentum tensor density. The resulting quantum theory is inherently nonlinear. Nonetheless, it is unitary and free from a nonlocal and implicit description of the Hamiltonian operator. Finally, by imposing additional homogeneity symmetries, a broad class of Bianchi cosmological models are analyzed as nonlinear quantum minisuperspaces in the context of the proposed theory. 
  The theoretical construction of a traversable wormhole from a Schwarzschild black hole is described, using analytic solutions in Einstein gravity. The matter model is pure phantom radiation (pure radiation with negative energy density) and the idealization of impulsive radiation is employed. 
  A new approach to the model of the universe based on work by Rippl, Romero, Tavakol is presented. We have used the scheme for relating the vacuum (D + 1) dimensional theories to D dimensional theories for setting up a correspondence between vacuum 4-dimensional Einstein theory with 3-dimensional gravity theory with temporal scalar field. These ideas we continued by using the 3-dimensional analog of Jordan, Brans-Dicke theory with temporal scalar field. As the result space and time are treated in completely different ways. For the case of a static spherically symmetric field new vacuum static solutions are found. 
  It is shown using a space-time curvature classification and decomposition that for certain holonomy types of a space-time, proper projective vector fields cannot exist. Existence is confirmed, by example, for the remaining holonomy types. In all except the most general holonomy type, a local uniqueness theorem for proper projective symmetry is established. 
  "Warp drive" spacetimes are useful as "gedanken-experiments" that force us to confront the foundations of general relativity, and among other things, to precisely formulate the notion of "superluminal" communication. We verify the non-perturbative violation of the classical energy conditions of the Alcubierre and Natario warp drive spacetimes and apply linearized gravity to the weak-field warp drive, testing the energy conditions to first and second order of the non-relativistic warp-bubble velocity. We are primarily interested in a secondary feature of the warp drive that has not previously been remarked upon, if it could be built, the warp drive would be an example of a "reaction-less drive". For both the Alcubierre and Natario warp drives we find that the occurrence of significant energy condition violations is not just a high-speed effect, but that the violations persist even at arbitrarily low speeds.   An interesting feature of this construction is that it is now meaningful to place a finite mass spaceship at the center of the warp bubble, and compare the warp field energy with the mass-energy of the spaceship. There is no hope of doing this in Alcubierre's original version of the warp-field, since by definition the point in the center of the warp bubble moves on a geodesic and is "massless". That is, in Alcubierre's original formalism and in the Natario formalism the spaceship is always treated as a test particle, while in the linearized theory we can treat the spaceship as a finite mass object. For both the Alcubierre and Natario warp drives we find that even at low speeds the net (negative) energy stored in the warp fields must be a significant fraction of the mass of the spaceship. 
  New geometric objects on null thin layers are introduced and their importance for crossing null-like shells are discussed. The Barrab\`es--Israel equations are represented in a new geometric form and they split into decoupled system of equations for two different geometric objects: tensor density ${\bf G}^a{_b}$ and vector field $I$. Continuity properties of these objects through a crossing sphere are proved. In the case of spherical symmetry Dray--t'Hooft--Redmount formula results from continuity property of the corresponding object. 
  Non-linear interactions among the inertial modes of a rotating fluid can be described by a network of coupled oscillators. We use such a description for an incompressible fluid to study the development of the r-mode instability of rotating neutron stars. A previous hydrodynamical simulation of the r-mode reported the catastrophic decay of large amplitude r-modes. We explain the dynamics and timescale of this decay analytically by means of a single three mode coupling. We argue that at realistic driving and damping rates such large amplitudes will never actually be reached. By numerically integrating a network of nearly 5000 coupled modes, we find that the linear growth of the r-mode ceases before it reaches an amplitude of around 10^(-4). The lowest parametric instability thresholds for the r-mode are calculated and it is found that the r-mode becomes unstable to modes with 13<n<15 if modes up to n=30 are included. Using the network of coupled oscillators, integration times of 10^6 rotational periods are attainable for realistic values of driving and damping rates. Complicated dynamics of the modal amplitudes are observed. The initial development is governed by the three mode coupling with the lowest parametric instability. Subsequently a large number of modes are excited, which greatly decreases the linear growth rate of the r-mode. 
  An effective action of ghost condensate with higher derivatives creates a source of gravity and mimics a dark matter in spiral galaxies. We present a spherically symmetric static solution of Einstein--Hilbert equations with the ghost condensate at large distances, where flat rotation curves are reproduced in leading order over small ratio of two energy scales characterizing constant temporal and spatial derivatives of ghost field: mu_*^2 and mu_\star^2, respectively, with a hierarchy mu_\star\ll \mu_*. We assume that a mechanism of hierarchy is provided by a global monopole in the center of galaxy. An estimate based on the solution and observed velocities of rotations in the asymptotic region of flatness, gives mu_* ~ 10^{19} GeV and the monopole scale in a GUT range mu_\star ~ 10^{16} GeV, while a velocity of rotation v_0 is determined by the ratio: sqrt{2} v_0^2= mu_\star^2/mu_*^2. A critical acceleration is introduced and naturally evaluated of the order of Hubble rate, that represents the Milgrom's acceleration. 
  We discuss the connection between the Fock space introduced by Ashtekar and Pierri for Einstein-Rosen waves and its perturbative counterpart based on the concept of particle that arises in linearized gravity with a de Donder gauge. We show that the gauge adopted by Ashtekar and Pierri is a generalization of the de Donder gauge to full (i.e. non-linearized) cylindrical gravity. This fact allows us to relate the two descriptions of the Einstein-Rosen waves analyzed here by means of a simple field redefinition. Employing this redefinition, we find the highly non-linear relation that exists between the annihilation and creation-like variables of the two approaches. We next represent the particle-like variables of the perturbative approach as regularized operators, introducing a cut-off. These can be expanded in powers of the annihilation and creation operators of the Ashtekar and Pierri quantization, each additional power being multiplied by an extra square-root of the three-dimensional gravitational constant, $\sqrt{G}$. In principle, the perturbative vacuum may be reached as the limit of a state annihilated by these regularized operators when the cut-off is removed. This state can be written as the vacuum of the Ashtekar and Pierri quantization corrected by a perturbative series in $\sqrt{G}$ with no contributions from particles with energies above the cut-off. We show that the first-order correction is in fact a state of infinite norm. This result is interpreted as indicating that the Fock quantizations in the two approaches are unitarily inequivalent and, in any case, proves that the perturbative vacuum is not analytic in the interaction constant. Therefore a standard perturbative quantum analysis fails. 
  Affine variational principle for General Relativity, proposed in 1978 by one of us (J.K.), is a good remedy for the non-universal properties of the standard, metric formulation, arising when the matter Lagrangian depends upon the metric derivatives. Affine version of the theory cures the standard drawback of the metric version, where the leading (second order) term of the field equations depends upon matter fields and its causal structure violates the light cone structure of the metric. Choosing the affine connection (and not the metric one) as the gravitational configuration, simplifies considerably the canonical structure of the theory and is more suitable for purposes of its quantization along the lines of Ashtekar and Lewandowski (see http://www.arxiv.org/gr-qc/0404018). We show how the affine formulation provides a simple method to handle boundary integrals in general relativity theory. 
  Astrophysical sources of high frequency gravitational radiation are considered in association with a new interest to very sensitive HFGW receivers required for the laboratory GW Hertz experiment. A special attention is paid to the phenomenon of primordial black holes evaporation. They act like black body to all kinds of radiation, including gravitons, and, therefore, emit an equilibrium spectrum of gravitons during its evaporation. Limit on the density of high frequency gravitons in the Universe is obtained, and possibilities of their detection are briefly discussed. 
  The content of this review is summarized here through the titles of its sections, as follows:   1. Introduction: Schwarzschild's original solution and the ``Schwarzschild solution''.   2. The wrong arrow of time of Hilbert's manifold is at the origin of the Kruskal extension.   3. An invariant, local, intrinsic quantity that diverges at the Schwarzschild surface.   4. The singularity at the Schwarzschild surface is both intrinsic and physical.   5. Conclusion. 
  In this article we investigate the principle and properties of a vertical passive seismic noise attenuator conceived for ground based gravitational wave interferometers. This mechanical attenuator based on a particular geometry of cantilever blades called monolithic geometric anti springs (MGAS) permits the design of mechanical harmonic oscillators with very low resonant frequency (below 100mHz).   Here we address the theoretical description of the mechanical device, focusing on the most important quantities for the low frequency regime, on the distribution of internal stresses, and on the thermal stability.   In order to obtain physical insight of the attenuator peculiarities, we devise some simplified models, rather than use the brute force of finite element analysis. Those models have been used to optimize the design of a seismic attenuation system prototype for LIGO advanced configurations and for the next generation of the TAMA interferometer. 
  We consider the backreaction of the vacuum polarization effect for a massive charged scalar field in the presence of a singular magnetic massless string on the background metric. Using semiclassical approach, we find the first-order (in $\hbar$ units) metric modifications and the corresponding gravitational potential and deficit angle. It is shown that, in certain region of values of coupling constant and magnetic flux, the gravitational potential and deficit angle can be positive as well as negative over all distances from the string and can even change its sign. Unlike the case of massless scalar field, the gravitational corrections were found to have short-range behavior. 
  The Efroimsky perturbation scheme for consistent treatment of gravitational waves and their influence on the background is summarized and compared with classical Isaacson's high-frequency approach. We demonstrate that the Efroimsky method in its present form is not compatible with the Isaacson limit of high-frequency gravitational waves, and we propose its natural generalization to resolve this drawback. 
  We study rotating black holes in Einstein-Yang-Mills-Higgs theory. These black holes emerge from static black holes with monopole hair when a finite horizon angular velocity is imposed. At critical values of the horizon angular velocity and the horizon radius, they bifurcate with embedded Kerr-Newman black holes. The non-Abelian black holes possess an electric dipole moment, but no electric charge is induced by the rotation. We deduce that gravitating regular monopoles possess a gyroelectric ratio g_el=2. 
  Matter collineations (MCs) are the vector fields along which the energy-momentum tensor remains invariant under the Lie transport. Invariance of the metric, the Ricci and the Riemann tensors have been studied extensively and the vectors along which these tensors remain invariant are called Killing vectors (KVs), Ricci collineations (RCs) and curvature collineations (CCs), respectively. In this paper plane symmetric static spacetimes have been studied for their MCs. Explicit form of MCs together with the Lie algebra admitted by them has been presented. Examples of spacetimes have been constructed for which MCs have been compared with their RCs and KVs. The comparison shows that neither of the sets of RCs and MCs contains the other, in general. 
  There now exists in the literature two different expressions for the phase shift of a matter-wave interferometer caused by the passage of a gravitation wave. The first, a commonly accepted expression that was first derived in the 1970s, is based on the traditional geodesic equation of motion (EOM) for a test particle. The second, a more recently derived expression, is based on the geodesic deviation EOM. The power-law dependence on the frequency of the gravitational wave for both expressions for the phase shift is different, which indicates fundamental differences in the physics on which these calculations are based. Here we compare the two approaches by presenting a series of side-by-side calculations of the phase shift for one specific matter-wave-interferometer configuration that uses atoms as the interfering particle. By looking at the low-frequency limit of the different expressions for the phase shift obtained, we find that the phase shift calculated via the geodesic deviation EOM is correct, and the ones calculated via the geodesic EOM are not. 
  A perturbed black hole has characteristic frequencies (quasi-normal modes). Here I apply a quantum measurement analysis of the quasi-normal mode frequency in the limit of high damping. It turns out that a measurement of this mode necessarily adds noise to it. For a Schwarzschild black hole, this corresponds exactly to the Hawking temperature. The situation for other black holes is briefly discussed. 
  We report on a revision of our previous computation of the renormalized expectation value of the stress-energy tensor of a massless, minimally coupled scalar with a quartic self-interaction on a locally de Sitter background. This model is important because it demonstrates that quantum effects can lead to violations of the weak energy condition on cosmological scales - on average, not just in fluctuations - although the effect in this particular model is far too small to be observed. The revision consists of modifying the propagator so that dimensional regularization can be used when the dimension of the renormalized theory is not four. Although the finite part of the stress-energy tensor does not change (in D=4) from our previous result, the counterterms do. We also speculate that a certain, finite and separately conserved part of the stress tensor can be subsumed into a natural correction of the initial state from free Bunch-Davies vacuum. 
  For a given asymptotically flat initial data set for Einstein equations a new geometric invariant is constructed. This invariant measure the departure of the data set from the stationary regime, it vanishes if and only if the data is stationary. In vacuum, it can be interpreted as a measure of the total amount of radiation contained in the data. 
  We argue that a (slightly) curved space-time probed with a finite resolution, equivalently a finite minimal length, is effectively described by a flat non-commutative space-time. More precisely, a small cosmological constant (so a constant curvature) leads the kappa-deformed Poincar\'e flat space-time of deformed special relativity (DSR) theories. This point of view eventually helps understanding some puzzling features of DSR. It also explains how DSR can be considered as an effective flat (low energy) limit of a (true) quantum gravity theory. This point of view leads us to consider a possible generalization of DSR to arbitrary curvature in momentum space and to speculate about a possible formulation of an effective quantum gravity model in these terms. It also leads us to suggest a {\it doubly deformed special relativity} framework for describing particle kinematics in an effective low energy description of quantum gravity. 
  In the Schwarzchild black hole spacetime, we show that chaotic motion can be triggered by the spin of a particle. Taking the spin of the particle as a perturbation and using the Melnikov method, we find that the perturbed stable and unstable orbits are entangled with each other and that illustrates the onset of chaotic behavior in the motion of the particle. 
  If Einstein's equations are to describe a field theory of gravity in Minkowski spacetime, then causality requires that the effective curved metric must respect the flat background metric's null cone. The kinematical problem is solved using a generalized eigenvector formalism based on the Segr\'{e} classification of symmetric rank 2 tensors with respect to a Lorentzian metric. Securing the correct relationship between the two null cones dynamically plausibly is achieved using the naive gauge freedom. New variables tied to the generalized eigenvector formalism reduce the configuration space to the causality-respecting part. In this smaller space, gauge transformations do not form a group, but only a groupoid. The flat metric removes the difficulty of defining equal-time commutation relations in quantum gravity and guarantees global hyperbolicity. 
  Recently the neglected issue of the causal structure in the flat spacetime approach to Einstein's theory of gravity has been substantially resolved. Consistency requires that the flat metric's null cone be respected by the null cone of the effective curved metric. While consistency is not automatic, thoughtful use of the naive gauge freedom resolves the problem. After briefly recapitulating how consistent causality is achieved, we consider the flat Robertson-Walker Big Bang model. The Big Bang singularity in the spatially flat Robertson-Walker cosmological model is banished to past infinity in Minkowski spacetime. A modified notion of singularity is proposed to fit the special relativistic approach, so that the Big Bang becomes nonsingular. 
  Prediction in quantum cosmology requires a specification of the universe's quantum dynamics and its quantum state. We expect only a few general features of the universe to be predicted with probabilities near unity conditioned on the dynamics and quantum state alone. Most useful predictions are of conditional probabilities that assume additional information beyond the dynamics and quantum state. Anthropic reasoning utilizes probabilities conditioned on `us'. This paper discusses the utility, limitations, and theoretical uncertainty involved in using such probabilities. The predictions resulting from various levels of ignorance of the quantum state are discussed including those related to uncertainty in the vacuum of string theory. Some obstacles to using anthropic reasoning to determine this vacuum are described. 
  In this work we study static perfect fluid stars in 2+1 dimensions with an exterior BTZ spacetime. We found the general expression for the metric coefficients as a function of the density and pressure of the fluid. We found the conditions to have regularity at the origin throughout the analysis of a set of linearly independent invariants. We also obtain an exact solution of the Einstein equations, with the corresponding equation of state $p=p(\rho)$, which is regular at the origin. 
  High-precision interpolation of LISA phase measurements allows signal reconstruction and formulation of Time-Delay Interferometry (TDI) combinations to be conducted in post-processing. The reconstruction is based on phase measurements made at approximately 10 Hz, at regular intervals independent of the TDI delay times. Interpolation introduces an error less than 1e-8 with continuous data segments as short as two seconds in duration. Potential simplifications in the design and operation of LISA are presented. 
  We reinvestigate the thermodynamics of black objects (holes and strings) in four-dimensional braneworld models that are originally constructed by Emparan, Horowitz and Myers based on the anti-de Sitter (AdS) C-metric. After proving the uniqueness of slicing the AdS C-metric, we derive thermodynamic quantities of the black objects by means of the Euclidean formulation and find that we have no necessity of requiring any regularization to calculate their classical action. We show that there exist the Bekenstein-Hawking law and the thermodynamic first law. The thermodynamic mass of the localized black hole on a flat brane is negative, and it differs from the one previously derived. We discuss the thermodynamic stabilities and show that the BTZ black string is more stable than the localized black holes in a canonical ensemble, except for an extreme case. We also find a braneworld analogue of the Hawking-Page transition between the BTZ black string and thermal AdS branes. The localized black holes on a de Sitter brane is discussed by considering Nariai instanton, comparing the study of "black cigar" in the five-dimensional braneworld model. 
  Recently the mechanism was found which allows avoidance of the cosmological singularity within the semi-classical formulation of Loop Quantum Gravity. Numerical studies show that the presence of self-interaction potential of the scalar field allows generation of initial conditions for successful slow-roll inflation. In this paper qualitative analysis of dynamical system, corresponding to cosmological equations of Loop Quantum Gravity is performed. The conclusion on singularity avoidance in positively curved cosmological models is confirmed. Two cases are considered, the massless (with flat potential) and massive scalar field. Explanation of initial conditions generation for inflation in models with massive scalar field is given. The bounce is discussed in models with zero spatial curvature and negative potentials. 
  The well-known Schwarzschild black hole was first obtained as a stationary, spherically symmetric solution of the Einstein's vacuum field equations. But until thirty years later, efforts were made for the analytic extension from the exterior area $(r>2GM)$ to the interior one $(r<2GM)$. As a contrast to its maximally extension in the Kruskal coordinates, we provide a comoving coordinate system from the view of the observers freely falling into the black hole in the radial direction. We find an interesting fact that the spatial part in this coordinate system is maximally symmetric $(E_3)$, i.e., along the world lines of these observers, the Schwarzschild black hole can be decomposed into a family of maximally symmetric subspaces. 
  The it Global Time Theory (GTT) is the further development of the General Relativity (GR). GTT significantly differs from GR in the general physical concepts, but retains 90% of the mathematical structure and main results. The dynamics equations are derived from Lagrangian, and the Hamiltonian of gravitation is nonzero. The quantum theory of gravitation can be built on the basis of the Schroedinger equation, as for other fields. The quantum model of the Big Bang is demonstrated. 
  A semi-Riemannian metric in a n-manifold has n(n-1)/2 degrees of freedom, i.e. as many as the number of components of a differential 2-form. We prove that any semi-Riemannian metric can be obtained as a deformation of a constant curvature metric, this deformation being parametrized by a 2-form 
  Introduced herein are the foundations of Global Time Theory (GTT) that further develop the General Relativity (GR). GTT is significantly different from GR in the general physical concepts, but retains 90% of the mathematical structure and main results.Meanwhile, description of the cosmos dynamics in GTT leads to significant modifications. The dynamics equations are derived from Lagrangian, and the Hamiltonian of gravitation is nonzero. Detailed solutions to the cosmic vortexes, that possess a weak principle of superposition and a huge energy, are derived. The virial theorem of space is formulated and proved. 
  Analytic solutions are presented which describe the construction of a traversable wormhole from a Schwarzschild black hole, and the enlargement of such a wormhole, in Einstein gravity. The matter model is pure radiation which may have negative energy density (phantom or ghost radiation) and the idealization of impulsive radiation (infinitesimally thin null shells) is employed. 
  We consider the response of an Unruh detector to scalar fields in an expanding space-time. When combining transition elements of the scalar field Hamiltonian with the interaction operator of detector and field, one finds at second order in time-dependent perturbation theory a transition amplitude, which actually dominates in the ultraviolet over the first order contribution. In particular, the detector response faithfully reproduces the particle number implied by the stress-energy of a minimally coupled scalar field, which is inversely proportional to the energy of a scalar mode. This finding disagrees with the contention that in de Sitter space, the response of the detector drops exponentially with particle energy and therefore indicates a thermal spectrum. 
  The evolution of the inflaton field fluctuations from gauge-invariant metric fluctuations is discused. In particular, the case of a symmetric $\phi_c$-exponential potential is analyzed. 
  The energy method can be used to identify well-posed initial boundary value problems for quasi-linear, symmetric hyperbolic partial differential equations with maximally dissipative boundary conditions. A similar analysis of the discrete system can be used to construct stable finite difference equations for these problems at the linear level. In this paper we apply these techniques to some test problems commonly used in numerical relativity and observe that while we obtain convergent schemes, fast growing modes, or ``artificial instabilities,'' contaminate the solution. We find that these growing modes can partially arise from the lack of a Leibnitz rule for discrete derivatives and discuss ways to limit this spurious growth. 
  In this note we give an alternative geometrical derivation of the results recently presented by Garcia-Godinez, Newman and Silva-Ortigoza in [1] on the class of all two-dimensional riemannian and lorentzian metrics from 2nd order ODEs which are in duality with the two dimensional Hamilton-Jacobi equation. We show that, as it happens in the Null Surface Formulation of General Relativity, the Wuschmann-like condition can be obtained as a requirement of a vanishing torsion tensor. Furthermore, from these second order ODEs we obtain the associated Cartan connections. 
  In the description of \emph{relative} motion in accelerated systems and gravitational fields, inertial and tidal accelerations must be taken into account, respectively. These involve a critical speed that in the first approximation can be simply illustrated in the case of motion in one dimension. For one-dimensional motion, such first-order accelerations are multiplied by $(1-V^2/V_c^2)$, where $V_c=c/\sqrt{2}$ is the critical speed. If the speed of relative motion exceeds $V_c$, there is a sign reversal with consequences that are contrary to Newtonian expectations. 
  We study models of axi-dilaton gravity in space-time geometries with torsion. We discuss conformal re-scaling rules in both Riemannian and non-Riemannian formulations. We give static, spherically symmetric solutions and examine their singularity structure. 
  We use a dynamical systems approach to study Bianchi type VI$_0$ cosmological models containing two tilted $\gamma$-law perfect fluids. The full state space is 11-dimensional, but the existence of a monotonic function simplifies the analysis considerably. We restrict attention to a particular, physically interesting, invariant subspace and find all equilibrium points that are future stable in the full 11-dimensional state space; these are consequently local attractors and serve as late-time asymptotes for an open set of tilted type VI$_0$ models containing two tilted fluids. We find that if one of the fluids has an equation of state parameter $\gamma<6/5$, the stiffest fluid will be dynamically insignificant at late times. For the value $\gamma=6/5$ there is a 2-dimensional bifurcation set, and if both fluids are stiffer than $\gamma=6/5$ both fluids will have extreme tilt asymptotically. We investigate the case in which one fluid is extremely tilting in detail. We also consider the case with one stiff fluid ($\gamma=2$) close to the initial singularity, and find that the chaotic behaviour which occurs in general Bianchi models with $\gamma<2$ is suppressed. 
  A generalised equivalence principle is put forward according to which space-time symmetries and internal quantum symmetries are indistinguishable before symmetry breaking. Based on this principle, a higher-dimensional extension of Minkowski space is proposed and its properties examined. In this scheme the structure of space-time is intrinsically quantum mechanical. It is shown that the causal geometry of such a quantum space-time possesses a rich hierarchical structure. The natural extension of the Poincare group to quantum space-time is investigated. In particular, we prove that the symmetry group of a quantum space-time is generated in general by a system of irreducible Killing tensors. When the symmetries of a quantum space-time are spontaneously broken, then the points of the quantum space-time can be interpreted as space-time valued operators. The generic point of a quantum space-time in the broken symmetry phase thus becomes a Minkowski space-time valued operator. Classical space-time emerges as a map from quantum space-time to Minkowski space. It is shown that the general such map satisfying appropriate causality-preserving conditions ensuring linearity and Poincare invariance is necessarily a density matrix. 
  The Stephani universes that can be interpreted as an ideal gas evolving in local thermal equilibrium are determined. Five classes of thermodynamic schemes are admissible, which give rise to five classes of regular models and three classes of singular models. No Stephani universes exist representing an exact solution to a classical ideal gas (one for which the internal energy is proportional to the temperature). But some Stephani universes may approximate a classical ideal gas at first order in the temperature: all of them are obtained. Finally, some features about the physical behavior of the models are pointed out. 
  The French-Italian interferometric gravitational wave detector VIRGO is currently being commissioned. Its principal instrument is a Michelson interferometer with 3 km long optical cavities in the arms and a power-recycling mirror. This paper gives an overview of the present status of the system. We report on the presently attained sensitivity and the system's performance during the recent commissioning runs.   After a sequence of intermediate stages, the interferometer is now being used in the so-called recombined configuration. The input laser beam is spatially filtered by a 144 m long input mode-cleaner before being injected to the main interferometer. The main optics are suspended from so-called \sa s, which provide an excellent seismic isolation. The two 3 km long Fabry-Perot arm cavities are kept in resonance with the laser light, and the Michelson interferometer is held on the dark fringe. An automatic mirror alignment system based on the Anderson technique has been implemented for the arm cavities. The light leaving the dark port contains the gravitational wave signal; this light is filtered by an output mode-cleaner before being detected by a photo detector. This setup is the last step on the way to the final configuration, which will include power recycling. 
  This paper describes an incoherent method to search for continuous gravitational waves based on the Hough transform, a well known technique used for detecting patterns in digital images. We apply the Hough transform to detect patterns in the time-frequency plane of the data produced by an earth-based gravitational wave detector. Two different flavors of searches will be considered, depending on the type of input to the Hough transform: either Fourier transforms of the detector data or the output of a coherent matched-filtering type search. We present the technical details for implementing the Hough transform algorithm for both kinds of searches, their statistical properties, and their sensitivities. 
  We exploit four-dimensional tensor identities to give a very simple proof of the existence of a Lanczos potential for a Weyl tensor in four dimensions with any signature, and to show that the potential satisfies a simple linear second order differential equation, e.g., a wave equation in Lorentz signature. Furthermore, we exploit higher dimensional tensor identities to obtain the analogous results for (m,m)-forms in 2m dimensions. 
  I wish to expound a novel perspective of probing universal character of gravity. To begin with, inclusion of zero mass particle in mechanics leads to special relativity while its interaction with a universal force shared by all particles leads to general relativity. The universal nature of force further suggests that it is intrinsically attractive, self interactive and higher dimensional. I argue that the principle of universality could serve as a good guide for future directions. 
  Parallel transport along circular orbits in orthogonally transitive stationary axisymmetric spacetimes is described explicitly relative to Lie transport in terms of the electric and magnetic parts of the induced connection. The influence of both the gravitoelectromagnetic fields associated with the zero angular momentum observers and of the Frenet-Serret parameters of these orbits as a function of their angular velocity is seen on the behavior of parallel transport through its representation as a parameter-dependent Lorentz transformation between these two inner-product preserving transports which is generated by the induced connection. This extends the analysis of parallel transport in the equatorial plane of the Kerr spacetime to the entire spacetime outside the black hole horizon, and helps give an intuitive picture of how competing "central attraction forces" and centripetal accelerations contribute with gravitomagnetic effects to explain the behavior of the 4-acceleration of circular orbits in that spacetime. 
  In these lectures I review, in as much pedagogical way as possible, various theoretical ideas and motivation for violation of CPT invariance in some models of Quantum Gravity, and discuss the relevant phenomenology. Since the subject is vast, I pay particular emphasis on the CPT Violating decoherence scenario for quantum gravity, due to space-time foam. In my opinion this seems to be the most likely scenario to be realised in Nature, should quantum gravity be responsible for the violation of this symmetry. In this context, I also discuss how the CPT Violating decoherence scenario can explain experimental ``anomalies'' in neutrino data, such as LSND results, in agreement with the rest of the presently available data, without enlarging the neutrino sector. 
  We show analytically that the vacuum electromagnetic stress-energy tensor outside a ball with constant dielectric constant and permeability always obeys the weak, null, dominant, and strong energy conditions. There are still no known examples in quantum field theory in which the averaged null energy condition in flat spacetime is violated. 
  (abridged)The achievements of the present work include: a) A clarification of the multiple definition given by Bergmann of the concept of {\it (Bergmann) observable. This clarification leads to the proposal of a {\it main conjecture} asserting the existence of i) special Dirac's observables which are also Bergmann's observables, ii) gauge variables that are coordinate independent (namely they behave like the tetradic scalar fields of the Newman-Penrose formalism). b) The analysis of the so-called {\it Hole} phenomenology in strict connection with the Hamiltonian treatment of the initial value problem in metric gravity for the class of Christoudoulou -Klainermann space-times, in which the temporal evolution is ruled by the {\it weak} ADM energy. It is crucial the re-interpretation of {\it active} diffeomorphisms as {\it passive and metric-dependent} dynamical symmetries of Einstein's equations, a re-interpretation which enables to disclose their (nearly unknown) connection to gauge transformations on-shell; this is expounded in the first paper (gr-qc/0403081). The use of the Bergmann-Komar {\it intrinsic pseudo-coordinates} allows to construct a {\it physical atlas} of 4-coordinate systems for the 4-dimensional {\it mathematical} manifold, in terms of the highly non-local degrees of freedom of the gravitational field (its four independent {\it Dirac observables}), and to realize the {\it physical individuation} of the points of space-time as {\it point-events} as a gauge-fixing problem, also associating a non-commutative structure to each 4-coordinate system. 
  A binary compact object early in its inspiral phase will be picked up by its nearly monochromatic gravitational radiation by LISA. But even this innocuous appearing candidate poses interesting detection challenges. The data that will be scanned for such sources will be a set of three functions of LISA's twelve data streams obtained through time-delay interferometry, which is necessary to cancel the noise contributions from laser-frequency fluctuations and optical-bench motions to these data streams. We call these three functions pseudo-detectors. The sensitivity of any pseudo-detector to a given sky position is a function of LISA's orbital position. Moreover, at a given point in LISA's orbit, each pseudo-detector has a different sensitivity to the same sky position. In this work, we obtain the optimal statistic for detecting gravitational wave signals, such as from compact binaries early in their inspiral stage, in LISA data. We also present how the sensitivity of LISA, defined by this optimal statistic, varies as a function of sky position and LISA's orbital location. Finally, we show how a real-time search for inspiral signals can be implemented on the LISA data by constructing a bank of templates in the sky positions. 
  This paper purposes to study quasi-normal modes due to massive scalar fields. We, in particular, investigate the dependence of QNM frequencies on the field mass. By this research, we find that there are quasi-normal modes with arbitrarily long life when the field mass has special values. It is also found that QNM can disappear when the field mass exceed these values. 
  We study the possibility of brane-world generalization of the Einstein-Straus Swiss-cheese cosmological model. We find that the modifications induced by the brane-world scenario are excessively restrictive. At a first glance only the motion of the boundary is modified and the fluid in the exterior region is allowed to have pressure. The general relativistic Einstein-Straus model emerges in the low density limit. However by imposing that the central mass in the Schwarzschild voids is constant, a combination of the junction conditions and modified cosmological evolution leads to the conclusion that the brane is flat. Thus no generic Swiss-cheese universe can exist on the brane. The conclusion is not altered by the introduction of a cosmological constant in the FLRW regions. This shows that although allowed in the low density limit, the Einstein-Straus universe cannot emerge from cosmological evolution in the brane-world scenario. 
  Techniques are developed for projecting the solutions of symmetric hyperbolic evolution systems onto the constraint submanifold (the constraint-satisfying subset of the dynamical field space). These optimal projections map a field configuration to the ``nearest'' configuration in the constraint submanifold, where distances between configurations are measured with the natural metric on the space of dynamical fields. The construction and use of these projections is illustrated for a new representation of the scalar field equation that exhibits both bulk and boundary generated constraint violations. Numerical simulations on a black-hole background show that bulk constraint violations cannot be controlled by constraint-preserving boundary conditions alone, but are effectively controlled by constraint projection. Simulations also show that constraint violations entering through boundaries cannot be controlled by constraint projection alone, but are controlled by constraint-preserving boundary conditions. Numerical solutions to the pathological scalar field system are shown to converge to solutions of a standard representation of the scalar field equation when constraint projection and constraint-preserving boundary conditions are used together. 
  The Teukolsky formalism of black hole perturbation theory describes weak gravitational radiation generated by a mildly dynamical hole near equilibrium. A particular null tetrad of the background Kerr geometry, due to Kinnersley, plays a singularly important role within this formalism. In order to apply the rich physical intuition of Teukolsky's approach to the results of fully non-linear numerical simulations, one must approximate this Kinnersley tetrad using raw numerical data, with no a priori knowledge of a background. This paper addresses this issue by identifying the directions of the tetrad fields in a quasi-Kinnersley frame. This frame provides a unique, analytic extension of Kinnersley's definition for the Kerr geometry to a much broader class of space-times including not only arbitrary perturbations, but also many examples which differ non-perturbatively from Kerr. This paper establishes concrete limits delineating this class and outlines a scheme to calculate the quasi-Kinnersley frame in numerical codes based on the initial-value formulation of geometrodynamics. 
  The Newman-Penrose formalism may be used in numerical relativity to extract coordinate-invariant information about gravitational radiation emitted in strong-field dynamical scenarios. The main challenge in doing so is to identify a null tetrad appropriately adapted to the simulated geometry such that Newman-Penrose quantities computed relative to it have an invariant physical meaning. In black hole perturbation theory, the Teukolsky formalism uses such adapted tetrads, those which differ only perturbatively from the background Kinnersley tetrad. At late times, numerical simulations of astrophysical processes producing isolated black holes ought to admit descriptions in the Teukolsky formalism. However, adapted tetrads in this context must be identified using only the numerically computed metric, since no background Kerr geometry is known a priori. To do this, this paper introduces the notion of a quasi-Kinnersley frame. This frame, when space-time is perturbatively close to Kerr, approximates the background Kinnersley frame. However, it remains calculable much more generally, in space-times non-perturbatively different from Kerr. We give an explicit solution for the tetrad transformation which is required in order to find this frame in a general space-time. 
  We show that for general relativity in odd spacetime dimensions greater than 4, all components of the unphysical Weyl tensor for arbitrary smooth, compact spatial support perturbations of Minkowski spacetime fail to be smooth at null infinity at leading nonvanishing order. This implies that for nearly flat radiating spacetimes, the non-smoothness of the unphysical metric at null infinity manifests itself at the same order as it describes deviations from flatness of the physical metric. Therefore, in odd spacetime dimensions, it does not appear that conformal null infinity can be in any way useful for describing radiation. 
  The behavior of spin-1/2 particle in a weak static gravitational field is considered. The Dirac Hamiltonian is diagonalized by the Foldy-Wouthuysen transformation providing also the simple form for the momentum and spin polarization operators. The operator equations of momentum and spin motion are derived for a first time. Their semiclassical limit is analyzed. The dipole spin-gravity coupling in the previously found (another) Hamiltonian does not lead to any observable effects. The general agreement between the quantum and classical analysis is established, contrary to several recent claims. The expression for gravitational Stern-Gerlach force is derived. The helicity evolution in the gravitational field and corresponding accelerated frame coincides, being the manifestation of the equivalence principle. 
  In this work we present an approximate solution of the Einstein equations describing a global model for the gravitational field generated by a bounded, self-gravitating stationary and axisymmetric body rotating rigidly with constant angular velocity. 
  The kinematical setting of spherically symmetric quantum geometry, derived from the full theory of loop quantum gravity, is developed. This extends previous studies of homogeneous models to inhomogeneous ones where interesting field theory aspects arise. A comparison between a reduced quantization and a derivation of the model from the full theory is presented in detail, with an emphasis on the resulting quantum representation. Similar concepts for Einstein-Rosen waves are discussed briefly. 
  The spherically symmetric volume operator is discussed and all its eigenstates and eigenvalues are computed. Even though the operator is more complicated than its homogeneous analog, the spectra are related in the sense that the larger spherically symmetric volume spectrum adds fine structure to the homogeneous spectrum. The formulas of this paper complete the derivation of an explicit calculus for spherically symmetric models which is needed for future physical investigations. 
  We have withdrawn this paper because we have presented the basic results in Brunini & Cionco (2005), Icarus 177, 264 
  We consider here a recently proposed geometrical criterion for local instability based on the geodesic deviation equation. Although such a criterion can be useful in some cases, we show here that, in general, it is neither necessary nor sufficient for the occurrence of chaos. To this purpose, we introduce a class of chaotic two-dimensional systems with Gaussian curvature everywhere positive and, hence, locally stable. We show explicitly that chaotic behavior arises from some trajectories that reach certain non convex parts of the boundary of the effective Riemannian manifold. Our result questions, once more, the viability of local, curvature-based criteria to predict chaotic behavior. 
  We obtain conditions for the existence and stability of de Sitter attractors in the phase space of spatially homogeneous and isotropic cosmology in generalized theories of gravity (including non-linear and scalar-tensor theories). These conditions are valid for any form of the coupling functions of the theory. Stability with respect to inhomogeneous perturbations is analyzed using a covariant and gauge-invariant formalism. The relevance for inflationary scenarios of the early universe and for quintessence models of the present era is discussed. 
  Using only the principle of relativity and Euclidean geometry we show in this pedagogical article that the square of proper time or length in a two-dimensional spacetime diagram is proportional to the Euclidean area of the corresponding causal domain. We use this relation to derive the Minkowski line element by two geometric proofs of the "spacetime Pythagoras theorem". 
  The anomalous time depending blueshift, the so-called "Pioneer anomaly", that was detected in the radio-metric data from Pioneer 10/11, Ulysses and Galileo spacecraft may not result from a real change of velocity. Rather, the Pioneer anomaly may be understood within the framework of general relativity as a time depending gravitational frequency shift accounting for the time dependence of the density of the dark energy when the latter is identified with quintessence. Thus, instead of being in conflict with Einstein equivalence principle, the main Pioneer anomaly appears merely as a new validation of general relativity in the weak field and low velocity limit. 
  A value of the cosmological constant in a toy model of the five-dimensional universe is calculated in such a manner that it remains in agreement with both astronomical observations and the quantum field theory concerning the zero-point fluctuations of the vacuum. The (negative) cosmological constant is equal to the inverse of the Planck length squared, which means that in the toy model the vanishing of the observed value of the cosmological constant is a consequence of the existence of an energy cutoff exactly at the level of the Planck scale. In turn, a model for both a virtual and a real particle-antiparticle pair is proposed which describes properly some energetic properties of both the vacuum fluctuations and created particles, as well as it allows one to calculate the discrete "bare" values of an elementary-particle mass, electric charge and intrinsic angular momentum (spin) at the energy cutoff. The relationships between the discussed model and some phenomena such as the Zitterbewegung and the Unruh-Davies effect are briefly analyzed, too. The proposed model also allows one to derive the Lorentz transformation and the Maxwell equations while considering the properties of the vacuum filled with the sea of virtual particles and their antiparticles. Finally, the existence of a finite value of the vacuum-energy density resulting from the toy model leads us to the formulation of dimensionless Einstein field equations which can be derived from the Lagrangian with a dimensionless (naively renormalized) coupling constant. 
  In this paper we describe the performance of the WaveBurst algorithm which was designed for detection of gravitational wave bursts in interferometric data. The performance of the algorithm was evaluated on the test data set collected during the second LIGO Scientific run. We have measured the false alarm rate of the algorithm as a function of the threshold and estimated its detection efficiency for simulated burst waveforms. 
  New formulae are obtained for the energy of K.N. b.h.'s that point out a gravitomagnetic energy effect. The results are valid for slowly or rapidly rotating black-holes. The expression of the energy density of Kerr-Newman back-holes in the slow rotation case, is obtained afterwards, and shown to be essentially positive. Subsequently,we show how to attain a "repulsive" gravitation (antigravitation) state identified with negative energy distribution contents in a limited region of space, without violating the Positive Energy Theorem. 
  We make use of possible high energy correction to the Friedmann equation to implement the bounce and study the behavior of massive scalar field before and after bounce semianalytically and numerically. We find that the slow-roll inflation can be preceded by the kinetic dominated contraction. During this process, the field initially in the bottom of its potential can be driven by the anti-frictional force resulted from the contraction and roll up its potential hill, and when it rolls down after the bounce, it can driven a period inflation. The required e-folds number during the inflation limits the energy scale of bounce. Further unlike that expected, the field during the contraction can not be driven to arbitrary large value, even though the bounce occurs at Planck scale. 
  We investigate the motion of a test particle in a d-dimensional, spherically symmetric and static space-time supported by a mass $M$ plus a $\Lambda$-term. The motion is strongly dependent on the sign of $\Lambda$. In Schwarzschild-de Sitter (SdS) space-time ($\Lambda > 0$), besides the physical singularity at $r=0$ there are cases with two horizons and two turning points, one horizon and one turning point and the complete absence of horizon and turning points. For Schwarzschild-Anti de Sitter (SAdS) space-time ($\Lambda < 0$) the horizon coordinate is associated to a unique turning point. 
  In this work we discuss the possibility of positive-acceleration regimes, and their transition to decelerated regimes, in two-dimensional (2D) cosmological models. We use general relativity and the thermodynamics in a 2D space-time, where the gas is seen as the sources of the gravitational field. An early-Universe model is analyzed where the state equation of van der Waals is used, replacing the usual barotropic equation. We show that this substitution permits the simulation of a period of inflation, followed by a negative-acceleration era. The dynamical behavior of the system follows from the solution of the Jackiw-Teitelboim equations (JT equations) and the energy-momentum conservation laws. In a second stage we focus the Callan-Giddings-Harvey-Strominger model (CGHS model); here the transition from the inflationary period to the decelerated period is also present between the solutions, although this result depend strongly on the initial conditions used for the dilaton field. The temporal evolution of the cosmic scale function, its acceleration, the energy density and the hydrostatic pressure are the physical quantities obtained in through the analysis. 
  We study separability of the Hamilton-Jacobi and massive Klein-Gordon equations in the general Myers-Perry black hole background in all dimensions. Complete separation of both equations is carried out in cases when there are two sets of equal black hole rotation parameters, which significantly enlarges the rotational symmetry group. We explicitly construct a nontrivial irreducible Killing tensor associated with the enlarged symmetry group which permits separation. We also derive first-order equations of motion for particles in these backgrounds and examine some of their properties. 
  The burst search in LIGO relies on the coincident detection of transient signals in multiple interferometers. As only minimal assumptions are made about the event waveform or duration, the analysis pipeline requires loose coincidence in time, frequency and amplitude. Confidence in the resulting events and their waveform consistency is established through a time-domain coherent analysis: the r-statistic test.   This paper presents a performance study of the r-statistic test for triple coincidence events in the second LIGO Science Run (S2), with emphasis on its ability to suppress the background false rate and its efficiency at detecting simulated bursts of different waveforms close to the S2 sensitivity curve. 
  A lower bound on the size of a Lorentzian wormhole can be obtained by semiclassically introducing the Planck cut-off on the magnitude of tidal forces (Horowitz-Ross constraint). Also, an upper bound is provided by the quantum field theoretic constraint in the form of the Ford-Roman Quantum Inequality for massless minimally coupled scalar fields. To date, however, exact static solutions belonging to this scalar field theory have not been worked out to verify these bounds. To fill this gap, we examine the wormhole features of two examples from the Einstein frame description of the vacuum low energy string theory in four dimensions which is the same as the minimally coupled scalar field theory. Analyses in this paper support the conclusion of Ford and Roman that wormholes in this theory can have sizes that are indeed only a few order of magnitudes larger than the Planck scale. It is shown that the two types of bounds are also compatible. In the process, we point out a "wormhole" analog of naked black holes. 
  A new prescription, in the framework of condensate models for space-times, for physical stationary gravitational fields is presented. We show that the spinning cosmic string metric describes the gravitational field associated with the single vortex in a superfluid condensate model for space-time outside the vortex core. This metric differs significantly from the usual acoustic metric for the Onsager-Feynman vortex. We also consider the question of what happens when many vortices are present, and show that on large scales a G\"{o}del-like metric emerges. In both the single and multiple vortex cases the failure of general relativity exemplified by the presence of closed time-like curves is attributed to the breakdown of superfluid rigidity. 
  It is still uncertain whether the cosmic censorship conjecture is true or not. To get a new insight into this issue, we propose the concept of the border of spacetime as a generalization of the spacetime singularity and discuss its visibility. The visible border, corresponding to the naked singularity, is not only relevant to mathematical completeness of general relativity but also a window into new physics in strongly curved spacetimes, which is in principle observable. 
  Axially symmetric gravitating multi-skyrmion configurations are obtained using the harmonic map ansatz introduced in [1]. In particular, the effect of gravity on the energy and baryon densities of the SU(3) non-gravitating multi-skyrmion configurations is studied in detail. 
  Two relations, the virial relation $M_{\rm ADM}=M_{\rm K}$ and the first law in the form $\delta M_{\rm ADM}=\Omega \delta J$, should be satisfied by a solution and a sequence of solutions describing binary compact objects in quasiequilibrium circular orbits. Here, $M_{\rm ADM}$, $M_{\rm K}$, $J$, and $\Omega$ are the ADM mass, Komar mass, angular momentum, and orbital angular velocity, respectively. $\delta$ denotes an Eulerian variation. These two conditions restrict the allowed formulations that we may adopt. First, we derive relations between $M_{\rm ADM}$ and $M_{\rm K}$ and between $\delta M_{\rm ADM}$ and $\Omega \delta J$ for general asymptotically flat spacetimes. Then, to obtain solutions that satisfy the virial relation and sequences of solutions that satisfy the first law at least approximately, we propose a formulation for computation of quasiequilibrium binary neutron stars in general relativity. In contrast to previous approaches in which a part of the Einstein equation is solved, in the new formulation, the full Einstein equation is solved with maximal slicing and in a transverse gauge for the conformal three-metric. Helical symmetry is imposed in the near zone, while in the distant zone, a waveless condition is assumed. We expect the solutions obtained in this formulation to be excellent quasiequilibria as well as initial data for numerical simulations of binary neutron star mergers. 
  A new, globally regular model describing a static, non spherical gravitating object in General Relativity is presented. The model is composed by a vacuum Weyl--Levi-Civita special field - the so called gamma metric - generated by a regular static distribution of mass-energy. Standard requirements of physical reasonableness such as, energy, matching and regularity conditions are satisfied. The model is used as a toy in investigating various issues related to the directional behavior of naked singularities in static spacetimes and the blackhole (Schwarschild) limit. 
  Following fresh attempts to resolve the problem of the energy density of the vacuum, we reconsider the case where the cosmological constant is derived from a higher-dimensional version of general relativity, and interpret the gauge-dependence of $\Lambda $ as a dynamical effect. This leads to a relation between the change in $\Lambda $ and the line element (action) which is independent of gauge choices and fundamental constants: $d\Lambda ds^{2}=-3$. This implies that the (classical) vacuum is unstable, with implications for particle production. 
  We study the possibility of using matter wave interferometry techniques to build a gravitational wave detector. We derive the response function and find that it contains a term proportional to the derivative of the gravitational wave, a point which has been disputed recently. We then study in detail the sensitivity that can be reached by such a detector and find that, if it is operated near resonance, it can reach potentially interesting values in the high frequency regime. The correlation between two or more of such devices can further improve the sensitivity for a stochastic signal. 
  We present the full set of evolution equations for the spatially homogeneous cosmologies of type VI$_h$ filled with a tilted perfect fluid and we provide the corresponding equilibrium points of the resulting dynamical state space. It is found that only when the group parameter satisfies $h>-1$ a self-similar solution exists. In particular we show that for $h>-{\frac 19}$ there exists a self-similar equilibrium point provided that $\gamma \in ({\frac{2(3+\sqrt{-h})}{5+3\sqrt{-h}}},{\frac 32}) $ whereas for $h<-{\frac 19}$ the state parameter belongs to the interval $\gamma \in (1,{\frac{2(3+\sqrt{-h})}{5+3\sqrt{-h}}}) $. This family of new exact self-similar solutions belongs to the subclass $n_\alpha ^\alpha =0$ having non-zero vorticity. In both cases the equilibrium points have a five dimensional stable manifold and may act as future attractors at least for the models satisfying $n_\alpha ^\alpha =0$. Also we give the exact form of the self-similar metrics in terms of the state and group parameters. As an illustrative example we provide the explicit form of the corresponding self-similar radiation model ($\gamma={\frac 43}$), parametrised by the group parameter $h$. Finally we show that there are no tilted self-similar models of type III and irrotational models of type VI$_h$. 
  We attempt to match the most general cylindrically symmetric vacuum space-time with a Robertson-Walker interior. The matching conditions show that the interior must be dust filled and that the boundary must be comoving. Further, we show that the vacuum region must be polarized. Imposing the condition that there are no trapped cylinders on an initial time slice, we can apply a result of Thorne's and show that trapped cylinders never evolve. This results in a simplified line element which we prove to be incompatible with the dust interior. This result demonstrates the impossibility of the existence of an isotropic cylindrically symmetric star (or even a star which has a cylindrically symmetric portion). We investigate the problem from a different perspective by looking at the expansion scalars of invariant null geodesic congruences and, applying to the cylindrical case, the result that the product of the signs of the expansion scalars must be continuous across the boundary. The result may also be understood in relation to recent results about the impossibility of the static axially symmetric analogue of the Einstein-Straus model. 
  Over the past three decades, black holes have played an important role in quantum gravity, mathematical physics, numerical relativity and gravitational wave phenomenology. However, conceptual settings and mathematical models used to discuss them have varied considerably from one area to another. Over the last five years a new, quasi-local framework was introduced to analyze diverse facets of black holes in a unified manner. In this framework, evolving black holes are modeled by dynamical horizons and black holes in equilibrium by isolated horizons. We review basic properties of these horizons and summarize applications to mathematical physics, numerical relativity and quantum gravity. This paradigm has led to significant generalizations of several results in black hole physics. Specifically, it has introduced a more physical setting for black hole thermodynamics and for black hole entropy calculations in quantum gravity; suggested a phenomenological model for hairy black holes; provided novel techniques to extract physics from numerical simulations; and led to new laws governing the dynamics of black holes in exact general relativity. 
  The problem of comparison of the stationary axisymmetric vacuum solutions obtained within the framework of exact and approximate approaches for the description of the same general relativistic systems is considered. We suggest two ways of carrying out such comparison: (i) through the calculation of the Ernst complex potential associated with the approximate solution whose form on the symmetry axis is subsequently used for the identification of the exact solution possessing the same multipole structure, and (ii) the generation of approximate solutions from exact ones by expanding the latter in series of powers of a small parameter. The central result of our paper is the derivation of the correct approximate analogues of the double-Kerr solution possessing the physically meaningful equilibrium configurations. We also show that the interpretation of an approximate solution originally attributed to it on the basis of some general physical suppositions may not coincide with its true nature established with the aid of a more accurate technique. 
  Erroneous statements on the mathematical and physical properties of the double-Kerr solution made in a recent paper (Class. Quantum Grav. 21 (2004) 2723) are commented. 
  The metric for a Reissner-Nordstr$\ddot{o}$m black hole in the background of the Friedman-Robertson-Walker universe is obtained. Then we verified it and discussed the influence of the evolution of the universe on the size of the black hole. To study the problem of the orbits of a planet in the expanding universe, we rewrote the metric in the Schwarzschild coordinates system and deduced the equation of motion for a planet. 
  In this work we find cosmological solutions in the brane-bulk system starting from a 5-D line element which is a simple extension, for cosmological applications, of the pioneering Randall-Sundrum line element. From the knowledge of the bulk metric, assumed to have the form of plane waves propagating in the fifth dimension, we solve the corresponding 4-D Einstein equations on the brane with a well defined energy-momentum tensor. 
  Here we explore a novel approach in order to try to measure the post-Newtonian 1/c^2 Lense-Thirring secular effect induced by the gravitomagnetic field of the Sun on the planetary orbital motion. Due to the relative smallness of the solar angular momentum J and the large values of the planetary semimajor axes a, the gravitomagnetic precessions, which affect the nodes Omega and the perihelia omega and are proportional to J/a^3, are of the order of 10^-3 arcseconds per century only for, e.g., Mercury. This value lies just at the edge of the present-day observational sensitivity in reconstructing the planetary orbits, although future missions to Mercury like Messenger and BepiColombo could allow to increase it. The major problems come from the main sources of systematic errors. They are the aliasing classical precessions induced by the multipolar expansion of the Sun's gravitational potential and the classical secular N-body precessions which are of the same order of magnitude or much larger than the Lense-Thirring precessions of interest. This definitely rules out the possibility of analyzing only one orbital element of, e.g., Mercury. In order to circumvent these problems, we propose a suitable linear combination of the orbital residuals of the nodes of Mercury, Venus and Mars which is, by construction, independent of such classical secular precessions. A 1-sigma reasonable estimate of the obtainable accuracy yields a 36% error. Since the major role in the proposed combination is played by the Mercury's node, it could happen that the new, more accurate ephemerides available in future thanks to the Messenger and BepiColombo missions will offer an opportunity to improve the present unfavorable situation. 
  It is argued that static electric or magnetic fields induce Weyl-Majumdar-Papapetrou solutions for the metric of spacetime. Their gravitational acceleration includes a term many orders of magnitude stronger than usual perturbative terms. It gives rise to a number of effects, which can be detected experimentally. Four electrostatic and four magnetostatic examples of physical set-ups with simple symmetries are proposed. The different ways in which mass sources enter and complicate the pure electromagnetic picture are described. 
  We present Keplerian-type parametrization for the solution of third post-Newtonian (3PN) accurate equations of motion for two non-spinning compact objects moving in an eccentric orbit. The orbital elements of the parametrization are explicitly given in terms of the 3PN accurate conserved orbital energy and angular momentum in both Arnowitt, Deser, and Misner-type and harmonic coordinates. Our representation will be required to construct post-Newtonian accurate `ready to use' search templates for the detection of gravitational waves from compact binaries in inspiralling eccentric orbits. Due to the presence of certain 3PN accurate gauge invariant orbital elements, the parametrization should be useful to analyze the compatibility of general relativistic numerical simulations involving compact binaries with the corresponding post-Newtonian descriptions. If required, the present parametrization will also be needed to compute post-Newtonian corrections to the currently employed `timing formula' for the radio observations of relativistic binary pulsars. 
  The first objective of this work is to obtain practical prescriptions to calculate the absorption of mass and angular momentum by a black hole when external processes produce gravitational radiation. These prescriptions are formulated in the time domain within the framework of black-hole perturbation theory. Two such prescriptions are presented. The first is based on the Teukolsky equation and it applies to general (rotating) black holes. The second is based on the Regge-Wheeler and Zerilli equations and it applies to nonrotating black holes. The second objective of this work is to apply the time-domain absorption formalisms to situations in which the black hole is either small or slowly moving. In the context of this small-hole/slow-motion approximation, the equations of black-hole perturbation theory can be solved analytically, and explicit expressions can be obtained for the absorption of mass and angular momentum. The changes in the black-hole parameters can then be understood in terms of an interaction between the tidal gravitational fields supplied by the external universe and the hole's tidally-induced mass and current quadrupole moments. For a nonrotating black hole the quadrupole moments are proportional to the rate of change of the tidal fields on the hole's world line. For a rotating black hole they are proportional to the tidal fields themselves. 
  Quantum Geometry (the modern Loop Quantum Gravity using graphs and spin-networks instead of the loops) provides microscopic degrees of freedom that account for the black-hole entropy. However, the procedure for state counting used in the literature contains an error and the number of the relevant horizon states is underestimated. In our paper a correct method of counting is presented. Our results lead to a revision of the literature of the subject. It turns out that the contribution of spins greater then 1/2 to the entropy is not negligible. Hence, the value of the Barbero-Immirzi parameter involved in the spectra of all the geometric and physical operators in this theory is different than previously derived. Also, the conjectured relation between Quantum Geometry and the black hole quasi-normal modes should be understood again. 
  We calculate the black hole entropy in Loop Quantum Gravity as a function of the horizon area and provide the exact formula for the leading and sub-leading terms. By comparison with the Bekenstein-Hawking formula we uniquely fix the value of the 'quantum of area' in the theory. 
  We present gravitoelectromagnetism and other decompositions of the Riemann tensor from the differential-geometrical point of view. 
  A code to evolve boson stars in 3D is presented as the starting point for the evolution of scalar field systems with arbitrary symmetries. It was possible to reproduce the known results related to perturbations discovered with 1D numerical codes in the past, which include evolution of stable and unstable equilibrium configurations. In addition, the apparent and event horizons masses of a collapsing boson star are shown for the first time. The present code is expected to be useful at evolving possible sources of gravitational waves related to scalar field objects and to handle toy models of systems perturbed with scalar fields in 3D. 
  I show that attempts to detect Hawking quanta would reduce the quantum state to one containing ultra-energetic incoming particles; couplings of these to other systems would extract ultra-high energies from the gravitational collapse. As the collapse proceeds, these energies grow exponentially, rapidly become trans-Planckian, and quantum-gravitational effects must enter. 
  The present work is devoted to massive gauge fields in special relativity with two fundamental constants-the velocity of light, and the Planck length, so called doubly special relativity (DSR). The two invariant scales are accounted for by properly modified boost parameters. Within above framework we construct the vector potential as the (1/2,0)x(0,1/2) direct product, build the associated field strength tensor together with the Dirac spinors and use them to calculate various observables as functions of the Planck length. 
  One of the brightest Gamma Ray Burst ever recorded, GRB030329, occurred during the second science run of the LIGO detectors. At that time, both interferometers at the Hanford, WA LIGO site were in lock and acquiring data. The data collected from the two Hanford detectors was analyzed for the presence of a gravitational wave signal associated with this GRB. This paper presents a detailed description of the search algorithm implemented in the current analysis. 
  The black-hole black-string system is known to exhibit critical dimensions and therefore it is interesting to vary the spacetime dimension $D$, treating it as a parameter of the system. We derive the large $D$ asymptotics of the critical, i.e. marginally stable, string following an earlier numerical analysis. For a background with an arbitrary compactification manifold we give an expression for the critical mass of a corresponding black brane. This expression is completely explicit for ${\bf T}^n$, the $n$ dimensional torus of an arbitrary shape. An indication is given that by employing a higher dimensional torus, rather than a single compact dimension, the total critical dimension above which the nature of the black-brane black-hole phase transition changes from sudden to smooth could be as low as $D\leq 11$. 
  We reveal an underlying flaw in Reginald T. Cahill's recently promoted quantum foam inflow theory of gravity. It appears to arise from a confusion of the idea of the Galilean invariance of the acceleration of an individual flow with what is obtained as an acceleration when a homogeneous flow is superposed with an inhomogeneous flow. We also point out that the General Relativistic covering theory he creates by substituting a generalized Painleve-Gullstrand metric into Einstein's field equations leads to absurd results. 
  A new geometric interpretation for General Relativity (GR) is proposed. We show that in the presence of an arbitrary affine connection, the gravitational field is described as nonmetricity of the affine connection. An affine connection can be interpreted as induced by a frame of reference (FR). Although the gravitational field equations are identical to Einstein's equations of GR, this formulation leads to a covariant tensor (instead of the pseudotensor) of energy-momentum of the gravitational field and covariant conservation laws. We further develop a geometric representation of FR as a metric-affine space, with transition between FR represented as affine deformation of the connection. Geodesic and autoparallel worldlines are considered. We show that the affine connection of a NIFR has curvature and may have torsion. We calculate the curvature for the uniformly accelerated FR. Finally, we show that GR is inadequate to describe the gravitational field in a NIFR. We propose a generalization of GR, which describes gravity as nonmetricity of the affine connection induced in a FR. This generalization contains GR as a special case of the inertial FR. 
  We study a closed model of the universe filled with viscous fluid and quintessence matter components in a Brans-Dicke type cosmological model. The dynamical equations imply that the universe may look like an accelerated flat Friedmann-Robertson-Walker universe at low redshift. We consider here dissipative processes which follow a causal thermodynamics. The theory is applied to viscous fluid inflation, where accepted values for the total entropy in the observable universe is obtained. 
  Results on the behaviour in the past time direction of cosmological models with collisionless matter and a cosmological constant $\Lambda$ are presented. It is shown that under the assumption of non-positive $\Lambda$ and spherical or plane symmetry the area radius goes to zero at the initial singularity. Under a smallness assumption on the initial data, these properties hold in the case of hyperbolic symmetry and negative $\Lambda$ as well as in the positive $\Lambda$ case. Furthermore in the latter cases past global existence of spatially homogeneous solutions is proved for generic initial data. The early-time asymptotics is shown to be Kasner-like for small data. 
  We present a set of boundary conditions for solving the elliptic equations in the Initial Data Problem for space-times containing a black hole, together with a number of constraints to be satisfied by the otherwise freely specifiable standard parameters of the Conformal Thin Sandwich formulation. These conditions altogether are sufficient for the construction of a horizon that is instantaneously in equilibrium in the sense of the Isolated Horizons formalism. We then investigate the application of these conditions to the Initial Data Problem of binary black holes and discuss the relation of our analysis with other proposals that exist in the literature. 
  Following the method of Hoenselaers and Perj\'{e}s we present a new corrected and dimensionally consistent set of multipole gravitational and electromagnetic moments for stationary axisymmetric spacetimes. Furthermore, we use our results to compute the multipole moments, both gravitational and electromagnetic, of a Kerr-Newman black hole. 
  We consider Maxwell fields associated with any shear-free null geodesic congruence on Minkowski or Riemannian background space-time. Bounded singular loci of these fields are treated as particle-like formations, possess "self-quantized" electric charge and undergo self-consistent time dynamics. Complicated singular solutions of Maxwell (as well as related complex Yang-Mills) equations can be obtained in a purely algebraic way using the Kerr theorem. 
  When the equatorial spin velocity, $v$, of a charged conducting sphere approaches $c$, the Lorentz force causes a remarkable rearrangement of the total charge $q$.   Charge of that sign is confined to a narrow equatorial belt at latitudes $b \leqslant \sqrt{3} (1 - v^2/c^2)^{{1/2}}$ while charge of the opposite sign occupies most of the sphere's surface. The change in field structure is shown to be a growing contribution of the `magic' electromagnetic field of the charged Kerr-Newman black hole with Newton's G set to zero. The total charge within the narrow equatorial belt grows as $(1-v^2/c^2)^{-{1/4}}$ and tends to infinity as $v$ approaches $c$. The electromagnetic field, Poynting vector, field angular momentum and field energy are calculated for these configurations.   Gyromagnetic ratio, g-factor and electromagnetic mass are illustrated in terms of a 19th Century electron model. Classical models with no spin had the small classical electron radius $e^2/mc^2\sim$ a hundredth of the Compton wavelength, but models with spin take that larger size but are so relativistically concentrated to the equator that most of their mass is electromagnetic.   The method of images at inverse points of the sphere is shown to extend to charges at points with imaginary co-ordinates. 
  Standard general relativity fails to take into account the changes in coordinates induced by the variation of metric in the Hilbert action principle. We propose to include such changes by introducing a fundamental compensating tensor field and modifying the usual variational procedure. 
  In this paper we revise a perfect fluid FRW model with time-varying constants \textquotedblleft but\textquotedblright taking into account the effects of a \textquotedblleft$c$-variable\textquotedblright into the curvature tensor. We study the model under the following assumptions, $div(T)=0$ and $div(T)\neq0,$ and in each case the flat and the non-flat cases are studied. Once we have outlined the new field equations, it is showed in the flat case i.e. K=0, that there is a non-trivial homothetic vector field i.e. that this case is self-similar. In this way, we find that there is only one symmetry, the scaling one, which induces the same solution that the obtained one in our previous model. At the same time we find that \textquotedblleft constants" $G$ and $c$ must verify, as integration condition of the field equations, the relationship $G/c^{2}=const.$ in spite of that both \textquotedblleft constants" vary. We also find that there is a narrow relationship between the equation of state and the behavior of the time functions $G,c$ and the sign of $\Lambda$ in such a way that these functions may be growing as well as decreasing functions on time $t,$ while $\Lambda$ may be a positive or negative decreasing function on time $t.$ In the non-flat case it will be showed that there is not any symmetry. For the case $div(T)\neq0,$ it will be studied again the flat and the non-flat cases. In order to solve this case it is necessary to make some assumptions on the behavior of the time functions $G,c$ and $\Lambda.$ We also find the flat case with $div(T)=0,$ is a particular solution of the general case $div(T)\neq0.$ 
  Non-perturbative corrections from loop quantum cosmology (LQC) to the scalar matter sector is already known to imply inflation. We prove that the LQC modified scalar field generates exponential inflation in the small scale factor regime, for all positive definite potentials, independent of initial conditions and independent of ambiguity parameters. For positive semi-definite potentials it is always possible to choose, without fine tuning, a value of one of the ambiguity parameters such that exponential inflation results, provided zeros of the potential are approached at most as a power law in the scale factor. In conjunction with generic occurrence of bounce at small volumes, particle horizon is absent thus eliminating the horizon problem of the standard Big Bang model. 
  This paper concerns the so-called cosmological constant problem. In order to solve it, we propose a toy model providing an extension of the dimensionality of spacetime, with an additional spatial dimension which is macroscopically unobservable. The toy model introduces no corrections to most predictions of the "standard" general relativity regarding, among others, the so-called "five tests of general relativity". However, it seems that the toy model could provide an explanation to the flatness of circular velocity curves of spiral galaxies without introducing any dark matter. The proposed model has quite important cosmological consequences. By introducing certain corrections to Friedmann's currently accepted model(s), the toy model allows one to solve problems related to the present density of matter in the Universe and, finally, it does not contain the initial singularity. 
  The inflationary production of magnetic field seeds for galaxies is discussed. The analysis is carried out by writing the wave equation of the electromagnetic field in curved spacetimes. The conformal invariance is broken by taking into account of the interaction of the electromagnetic field with the curvature tensor of the form $\lambda R_{\alpha\beta\gamma\delta} F^{\alpha\beta} F^{\gamma\delta}$. Such a term induces an amplification of the magnetic field during the reheating phase of the universe, but no growth of the magnetic field occurs in the de Sitter epoch. The resulting primordial magnetic field turns out to have strengths of astrophysical interest. 
  We address the question of existence of regular spherically symmetric electrically charged solutions in Nonlinear Electrodynamics coupled to General Relativity. Stress-energy tensor of the electromagnetic field has the algebraic structure $T_0^0=T_1^1$. In this case the Weak Energy Condition leads to the de Sitter asymptotic at approaching a regular center. In de Sitter center of an electrically charged NED structure, electric field, geometry and stress-energy tensor are regular without Maxwell limit which is replaced by de Sitter limit: energy density of a field is maximal and gives an effective cut-off on self-energy density, produced by NED coupled to gravity and related to cosmological constant $\Lambda$. Regular electric solutions satisfying WEC, suffer from one cusp in the Lagrangian ${\cal L}(F)$, which creates the problem in an effective geometry whose geodesics are world lines of NED photons. We investigate propagation of photons and show that their world lines never terminate which suggests absence of singularities in the effective geometry. To illustrate these results we present the new exact analytic spherically symmetric electric solution with the de Sitter center. 
  For a class of solutions of the fundamental difference equation of isotropic loop quantum cosmology, the difference equation can be replaced by a differential equation valid for {\em all} values of the triad variable. The differential equation admits a `unique' non-singular continuation through vanishing triad. A WKB approximation for the solutions leads to an effective continuum Hamiltonian. The effective dynamics is also non-singular (no big bang singularity) and approximates the classical dynamics for large volumes. The effective evolution is thus a more reliable model for further phenomenological implications of the small volume effects. 
  The absence of isotropic singularity in loop quantum cosmology can be understood in an effective classical description as the universe exhibiting a Big Bounce. We show that with scalar matter field, the big bounce is generic in the sense that it is independent of quantization ambiguities and details of scalar field dynamics. The volume of the universe at the bounce point is parametrized by a single parameter. It provides a minimum length scale which serves as a cut-off for computations of density perturbations thereby influencing their amplitudes. 
  A new final state of gravitational collapse is proposed. By extending the concept of Bose-Einstein condensation to gravitational systems, a cold, dark, compact object with an interior de Sitter condensate $p_{_V} = -\rho_{_V}$ and an exterior Schwarzschild geometry of arbitrary total mass $M$ is constructed. These are separated by a shell with a small but finite proper thickness $\ell$ of fluid with equation of state $p=+\rho$, replacing both the Schwarzschild and de Sitter classical horizons. The new solution has no singularities, no event horizons, and a global time. Its entropy is maximized under small fluctuations and is given by the standard hydrodynamic entropy of the thin shell, which is of order $k_{_B}\ell Mc/\hbar$, instead of the Bekenstein-Hawking entropy formula, $S_{_{BH}}= 4\pi k_{_B} G M^2/\hbar c$. Hence unlike black holes, the new solution is thermodynamically stable and has no information paradox. 
  We discuss in this Chapter a series of theoretical developments which motivate the introduction of a quantum evolution equation for which the eikonal approximation results in the geodesics of a four dimensional manifold. This geodesic motion can be put into correspondence with general relativity. One obtains in this way a quantum theory on a flat spacetime, obeying the rules of the standard quantum theory in Lorentz covariant form, with a spacetime dependent Lorentz tensor $g_{\mu\nu}$, somewhat analogous to a gauge field, coupling to the kinetic terms. Since the geodesics predicted by the eikonal approximation, with appropriate choice of $g_{\mu\nu}$, can be those of general relativity, this theory provides a quantum theory which could be underlying to classical gravitation, and coincides with it in this classical ray approximation. In order to understand the possible origin of the structure of this equation, we appeal to the approach of Nelson in constructing a Schroedinger equation from the properties of Brownian motion. Extending the notion of Browninan motion to spacetime in a covariant way, we show that such an equation follows from correlations between spacetime dimensions in the stochastic process. 
  Several recently found properties of the event horizon of black holes are discussed. One of them is the reflection of the incoming particles on the horizon. A particle approaching the black hole can bounce on the horizon back, into the outside world, which drastically reduces the absorption cross section in the infrared region. Another, though related phenomenon takes place for particles inside the horizon. A locked inside particle has, in fact, an opportunity to escape into the outside world. Thus, the confinement inside the horizon is not absolute. The escape from within the interior region of the horizon allows the transfer of information from this region into the outside world. This result may help resolve the information paradox for black holes. Both the reflection and escape phenomena happen due to pure quantum reasons, being impossible in the classical approximation. 
  We define and extensively test a set of boundary conditions that can be applied at black hole excision surfaces when the Hamiltonian and momentum constraints of general relativity are solved within the conformal thin-sandwich formalism. These boundary conditions have been designed to result in black holes that are in quasiequilibrium and are completely general in the sense that they can be applied with any conformal three-geometry and slicing condition. Furthermore, we show that they retain precisely the freedom to specify an arbitrary spin on each black hole. Interestingly, we have been unable to find a boundary condition on the lapse that can be derived from a quasiequilibrium condition. Rather, we find evidence that the lapse boundary condition is part of the initial temporal gauge choice. To test these boundary conditions, we have extensively explored the case of a single black hole and the case of a binary system of equal-mass black holes, including the computation of quasi-circular orbits and the determination of the inner-most stable circular orbit. Our tests show that the boundary conditions work well. 
  We answer an important question in general relativity about the volume integral theorem for exotic matter by suggesting an exact integral quantifier for matter violating Averaged Null Energy Condition (ANEC). It is checked against some well known static, spherically symmetric traversable wormhole solutions of general relativity with a sign reversed kinetic term minimally coupled scalar field. The improved quantifier is consistent with the principle that traversable wormholes can be supported by arbitrarily small quantities of exotic matter. 
  We consider the evolution of electromagnetic fields in curved spacetimes and calculate the exact wave equations of the associated electric and magnetic components. Our analysis is fully covariant, applies to a general spacetime and isolates all the sources that affect the propagation of these waves. Among others, we explicitly show how the different parts of the gravitational field act as driving sources of electromagnetic disturbances. When applied to perturbed FRW cosmologies, our results argue for a superadiabatic-type amplification of large-scale cosmological magnetic fields in Friedmann models with open spatial curvature. 
  With (non-barotropic) equations of state valid even when the neutron, proton and electron content of neutron star cores is not in beta equilibrium, we study inertial and composition gravity modes of relativistic rotating neutron stars. We solve the relativistic Euler equations in the time domain with a three dimensional numerical code based on spectral methods, in the slow rotation, relativistic Cowling and anelastic approximations. Principally, after a short description of the gravity modes due to smooth composition gradients, we focus our analysis on the question of how the inertial modes are affected by non-barotropicity of the nuclear matter. In our study, the deviation with respect to barotropicity results from the frozen composition of non-superfluid matter composed of neutrons, protons and electrons, when beta equilibrium is broken by millisecond oscillations. We show that already for moderatly fast rotating stars the increasing coupling between polar and axial modes makes those two cases less different than for very slowly rotating stars. In addition, as we directly solve the Euler equations, without coupling only a few number of spherical harmonics, we always found, for the models that we use, a discrete spectrum for the $l = m = 2$ inertial mode. Finally, we find that, for non-barotropic stars, the frequency of this mode, which is our main focus, decreases in a non-negligible way, whereas the time dependence of the energy transfer between polar and axial modes is substantially different due to the existence of low-frequencies gravity modes. 
  The metric by Carmeli accurately produces the Tully-Fisher type relation in spiral galaxies, a relation showing the fourth power of the rotation speed proportional to the mass of the galaxy. And therefore it is claimed that it is also no longer necessary to invoke dark matter to explain the anomalous dynamics in the arms of spiral galaxies. An analysis is presented here showing Carmeli?s 5 dimensional space-time-velocity metric can also indeed describe the rotation curves of spiral galaxies based on the properties of the metric alone. 
  Carmeli's 5D brane cosmology has been applied to the expanding accelerating universe and it has been found that the distance redshift relation will fit the data of the high-z supernova teams without the need for dark matter. Also the vacuum energy contribution to gravity indicates that the universe is asymptotically expanding towards a spatially flat state, where the total mass/energy density tends to unity. 
  A general principle of non-equivalence for bodies and observers in different G potentials (GP) was derived from correspondence of the Einstein's equivalence principle either with optical physics or with gravitational experiments in which bodies and observers are in different GP. According to it some relative physical changes occur to any well defined part of an object after a change of GP.. Such changes cannot be measured by observers travelling with the object because his instruments change in identical proportions. The same principle was derived from a new gravitational theory based on a particle model made up of photons in stationary states. Such model accounts for the inertial and gravitational properties of matter. This principle is not consistent with both, the classical hypotheses on the relative invariability of the bodies after a change of GP and with the G field energy hypothesis. The two kinds of errors are of the same magnitude and opposite signs. Such errors are cancelled but only when the two hypotheses are used. This accounts for the good predictions of general relativity for the classical gravitational tests. The new properties of the universe derived from the new principle, are radically different from the classical ones. They are more clearly consistent with the astronomical observations. 
  We study the quantum effects induced by bulk scalar fields in a model with a de Sitter (dS) brane in a flat bulk (the Vilenkin-Ipser-Sikivie model) in more than four dimensions. In ordinary dS space, it is well known that the stress tensor in the dS invariant vacuum for an effectively massless scalar ($m_\eff^2=m^2+\xi {\cal R}=0$ with ${\cal R}$ the Ricci scalar) is infrared divergent except for the minimally coupled case. The usual procedure to tame this divergence is to replace the dS invariant vacuum by the Allen Follaci (AF) vacuum. The resulting stress tensor breaks dS symmetry but is regular. Similarly, in the brane world context, we find that the dS invariant vacuum generates $\tmn$ divergent everywhere when the lowest lying mode becomes massless except for massless minimal coupling case. A simple extension of the AF vacuum to the present case avoids this global divergence, but $\tmn$ remains to be divergent along a timelike axis in the bulk. In this case, singularities also appear along the light cone emanating from the origin in the bulk, although they are so mild that $\tmn$ stays finite except for non-minimal coupling cases in four or six dimensions. We discuss implications of these results for bulk inflaton models. We also study the evolution of the field perturbations in dS brane world. We find that perturbations grow linearly with time on the brane, as in the case of ordinary dS space. In the bulk, they are asymptotically bounded. 
  We study the quantum-mechanical decay of a Schwarzschild-like black hole into almost-flat space and weak radiation at a very late time, evaluating quantum amplitudes (not just probabilities) for transitions from initial to final states. No information is lost. The model contains gravity and a massless scalar field. The quantum amplitude to go from given initial to final bosonic data in a slightly complexified time-interval $T={\tau}{\exp}(-i{\theta})$ at infinity is approximately $\exp(-I)$, where $I$ is the (complex) Euclidean action of the classical solution filling in between the boundary data. And in a locally supersymmetric (supergravity) theory, the amplitude const. exp(-I) is exact. Dirichlet boundary data for gravity and the scalar field are posed on an initial spacelike hypersurface extending to spatial infinity, just prior to collapse, and on a corresponding final spacelike surface, sufficiently far to the future of the initial surface to catch all the Hawking radiation. In an averaged sense this radiation has an approximately spherically-symmetric distribution. If the time-interval $T$ were exactly real, the resulting `hyperbolic Dirichlet boundary-value problem' would not be well posed. If instead (`Euclidean strategy'), one takes $T$ complex, as above ($0<\theta{\leq}{\pi}/2$), the field equations become strongly elliptic, with a unique solution to the classical boundary-value problem. Expanding the bosonic part of the action to quadratic order in perturbations about the classical solution gives the quantum amplitude for weak-field final configurations, up to normalization. Such amplitudes are calculated for weak final scalar fields. 
  Atomic wave interferometers are tied to a telescope pointing towards a faraway star in a nearly free falling satellite. Such a device is sensitive to the acceleration and the rotation relatively to the local inertial frame and to the tidal gravitational effects too. We calculate the rotation of the telescope due to the aberration and the deflection of the light in the gravitational field of a central mass (the Earth and Jupiter). Within the framework of a general parametrized description of the problem, we discuss the contributions which must be taken into account in order to observe the Lense-Thirring effect. Using a geometrical model, we consider some perturbations to the idealized device and we calculate the corresponding effect on the periodic components of the signal. Some improvements in the knowledge of the gravitational field are still necessary as well as an increase of the experimental capabilities; however our conclusions support a reasonable optimism for the future. Finally we put forward the necessity of a more complete, realistic and powerful model in order to obtain a definitive conclusion on the feasibility of the experiment as far as the observation of the Lense-Thirring effect is involved. 
  We show that Einstein equations are compatible with the presence of massive point particles and find corresponding two parameter family of their solutions which depends on the bare mechanical mass $M_0>0$ and the Keplerian mass $M<M_0$ of the point source of gravity. The global analytical properties of these solutions in the complex plane define a unique preferable radial variable of the problem. 
  We present a specific model for cosmological inflation driven by the Liouville field in a non-critical supersymmetric string framework, in which the departure from criticality is due to open strings stretched between the two moving Type-II 5-branes. We use WMAP and other data on fluctuations in the cosmic microwave background to fix parameters of the model, such as the relative separation and velocity of the 5-branes, respecting also the constraints imposed by data on light propagation from distant gamma-ray bursters. The model also suggests a small, relaxing component in the present vacuum energy that may accommodate the breaking of supersymmetry. 
  We show that black hole formation and evaporation in curved-space quantum field theory can be described in a unitary manner consistent with previous results that take scattering and stimulated emission into account. In particular, we show that the entropy accreted by a black hole when particles cross the event horizon is exactly balanced by a commensurate entropy increase of the rest of the universe, owing to the quantum entanglement between the black hole, Hawking radiation, and scattered radiation (including stimulated emission). Radiation emitted by a black hole in response to infalling matter is non-thermal, and contains all of the information of the initial state. 
  We investigate the (conservative) dynamics of binary black holes using the Hamiltonian formulation of the post-Newtonian (PN) equations of motion. The Hamiltonian we use includes spin-orbit coupling, spin-spin coupling, and mass monopole/spin-induced quadrupole interaction terms. In the case of both quasi-circular and eccentric orbits, we search for the presence of chaos (using the method of Lyapunov exponents) for a large variety of initial conditions. For quasi-circular orbits, we find no chaotic behavior for black holes with total mass 10 - 40 solar masses when initially at a separation corresponding to a Newtonian gravitational-wave frequency less than 150 Hz. Only for rather small initial radial distances, for which spin-spin induced oscillations in the radial separation are rather important, do we find chaotic solutions, and even then they are rare. Moreover, these chaotic quasi-circular orbits are of questionable astrophysical significance, since they originate from direct parametrization of the equations of motion rather than from widely separated binaries evolving to small separations under gravitational radiation reaction. In the case of highly eccentric orbits, which for ground-based interferometers are not astrophysically favored, we again find chaotic solutions, but only at pericenters so small that higher order PN corrections, especially higher spin PN corrections, should also be taken into account. 
  Generalization of an idea may lead to very interesting result. Learning how torsion influences on tidal force reveals similarity between tidal equation for geodesic and the Killing equation of second type.   The relationship between tidal acceleration, curvature and torsion gives an opportunity to measure torsion. 
  We prove that a globally hyperbolic spacetime with its causality relation is a bicontinuous poset whose interval topology is the manifold topology. This provides an abstract mathematical setting in which one can study causality independent of geometry and differentiable structure. 
  Beginning from only a countable dense set of events and the causality relation, it is possible to reconstruct a globally hyperbolic spacetime in a purely order theoretic manner. The ultimate reason for this is that globally hyperbolic spacetimes belong to a category that is equivalent to a special category of domains called interval domains. 
  We present mathematical details of several cosmological models, whereby the topological and the geometrical background will be emphasized. 
  A new varying-$c$ cosmological model constructed using two additional assumptions, which was introduced in our previous work, is briefly reviewed and the dynamic equation of the model is derived distinctly from a semi-Newtonian approach. The results of this model, using a $\Lambda$ term and an extra energy-momentum tensor, are considered separately. It is shown that the Universe began from a hot Big Bang and expands forever with a constant deceleration parameter regardless of its curvature. Finally, the age, the radius, and the energy content of the Universe are estimated and some discussion about the type of the geometry of the Universe is provided. 
  We apply techniques recently introduced in quantum cosmology to the Schwarzschild metric inside the horizon and near the black hole singularity at r = 0. In particular, we use the quantization introduced by Husain and Winkler, which is suggested by Loop Quantum Gravity and is based on an alternative to the Schrodinger representation introduced by Halvorson. Using this quantization procedure, we show that the black hole singularity disappears and spacetime can be dynamically extended beyond the classical singularity. 
  Deformed Special Relativity (DSR) is obtained by imposing a maximal energy to Special Relativity and deforming the Lorentz symmetry (more exactly the Poincar\'e symmetry) to accommodate this requirement. One can apply the same procedure deforming the Galilean symmetry in order to impose a maximal speed (the speed of light). This leads to a non-commutative space structure, to the expected deformations of composition of speed and conservation of energy-momentum. In doing so, one runs into most of the ambiguities that one stumbles onto in the DSR context. However, this time, Special Relativity is there to tell us what is the underlying physics, in such a way that we can understand and interpret these ambiguities. We use these insights to comment on the physics of DSR. 
  We consider the asymptoic behaviour of the Chern - Simons (CS) theory with matter in curved space - time. The asymptotics of effective couplings are discussed. 
  We discuss some global properties of cosmological spacetimes of de Sitter type, based on results with Lars Andersson obtained in hep-th/0202161. We relate the geometry and topology of conformal infinity to the occurrence of singularities in such spacetimes, thereby extending to a much broader context certain properties of some well-known cosmological models. 
  We study the renormalized energy-momentum tensor of gravitons in a de Sitter space-time. After canonically quantizing only the physical degrees of freedom, we adopt the standard adiabatic subtraction used for massless minimally coupled scalar fields as a regularization procedure and find that the energy density of gravitons in the E(3) invariant vacuum is proportional to H^4, where H is the Hubble parameter, but with a positive sign. According to this result the scalar expansion rate, which is gauge invariant in de Sitter space-time, is increased by the fluctuations. This implies that gravitons may then add to conformally coupled matter in driving the Starobinsky model of inflation. 
  We report on three numerical experiments on the implementation of Time-Delay Interferometry (TDI) for LISA, performed with Synthetic LISA, a C++/Python package that we developed to simulate the LISA science process at the level of scientific and technical requirements. Specifically, we study the laser-noise residuals left by first-generation TDI when the LISA armlengths have a realistic time dependence; we characterize the armlength-measurements accuracies that are needed to have effective laser-noise cancellation in both first- and second-generation TDI; and we estimate the quantization and telemetry bitdepth needed for the phase measurements. Synthetic LISA generates synthetic time series of the LISA fundamental noises, as filtered through all the TDI observables; it also provides a streamlined module to compute the TDI responses to gravitational waves according to a full model of TDI, including the motion of the LISA array and the temporal and directional dependence of the armlengths. We discuss the theoretical model that underlies the simulation, its implementation, and its use in future investigations on system characterization and data-analysis prototyping for LISA. 
  Motivated by the dark energy issue, the minisuperspace approach for general relativistic cosmological theories is outlined. 
  When constructing general relativity (GR), Einstein required 4D general covariance. In contrast, we derive GR (in the compact, without boundary case) as a theory of evolving 3-dimensional conformal Riemannian geometries obtained by imposing two general principles: 1) time is derived from change; 2) motion and size are relative. We write down an explicit action based on them. We obtain not only GR in the CMC gauge, in its Hamiltonian 3 + 1 reformulation but also all the equations used in York's conformal technique for solving the initial-value problem. This shows that the independent gravitational degrees of freedom obtained by York do not arise from a gauge fixing but from hitherto unrecognized fundamental symmetry principles. They can therefore be identified as the long-sought Hamiltonian physical gravitational degrees of freedom. 
  It is proven that, under mild physical assumptions, an isolated stationary relativistic perfect fluid consists of a finite number of cells fibred by invariant annuli or invariant tori. For axially symmetric circular flows it is shown that the fluid consists of cells fibred by rigidly rotating annuli or tori. 
  We are presenting a quantum traversable wormhole in an exactly soluble two-dimensional model. This is different from previous works since the exotic negative energy that supports the wormhole is generated from the quantization of classical energy-momentum tensors. This explicit illustration shows the quantum-mechanical energy can be used as a candidate for the exotic source. As for the traversability, after a particle travels through the wormhole, the static initial wormhole geometry gets a back reaction which spoils the wormhole structure. However, it may still maintain the initial structure along with the appropriate boundary condition. 
  The standard energy conditions of classical general relativity are applied to FLRW cosmologies containing sudden future singularities. Here we show, in a model independent way, that although such cosmologies can satisfy the null, weak and strong energy conditions, they always fail to satisfy the dominant energy condition. They require a divergent spacelike energy flux in all but the comoving frame. 
  Linear perturbation theory is appropriate to describe small oscillations of stars, while a mild non-linearity is still tractable perturbatively but requires to consider mode coupling. It is natural to start to look at this problem by considering the coupling between linear radial and non-radial modes. Radial pulsations of a spherical compact objects do not per se emit gravitational waves but, if the coupling is efficient in driving and possibly amplifying the non-radial modes, gravitational radiation could then be produced to a significant level.   In this paper we develop the relativistic formalism to study the coupling of radial and non-radial first order perturbations of a compact spherical star.   From a mathematical point of view, it is convenient to treat the two sets of perturbations as separately parametrized, using a 2-parameter perturbative expansion of the metric, the energy-momentum tensor and Einstein equations in which $\lambda$ is associated with the radial modes, $\epsilon$ with the non-radial perturbations, and the $\lambda\epsilon$ terms describe the coupling. This approach provides a well-defined framework to consider the gauge dependence of perturbations, allowing us to use $\epsilon$ order gauge-invariant non-radial variables on the static background and to define new second order $\lambda\epsilon$ gauge-invariant variables describing the non-linear coupling. We present the evolution and constraint equations for our variables outlining the setup for numerical computations, and briefly discuss the surface boundary conditions in terms of the second order $\lambda\epsilon$ Lagrangian pressure perturbation. 
  Gravitational collapse is one of the most striking phenomena in gravitational physics. The cosmic censorship conjecture has provided strong motivation for researches in this field. In the absence of general proof for the censorship, many examples have been proposed, in which naked singularity is the outcome of gravitational collapse. Recent development has revealed that there are examples of naked singularity formation in the collapse of physically reasonable matter fields, although the stability of these examples is still uncertain. We propose the concept of ``effective naked singularities'', which will be quite helpful because general relativity has the limitation of its application for high-energy end. The appearance of naked singularities is not detestable but can open a window for new physics of strongly curved spacetimes. 
  A new numerical scheme to solve the Einstein field equations based upon the generalized harmonic decomposition of the Ricci tensor is introduced. The source functions driving the wave equations that define generalized harmonic coordinates are treated as independent functions, and encode the coordinate freedom of solutions. Techniques are discussed to impose particular gauge conditions through a specification of the source functions. A 3D, free evolution, finite difference code implementing this system of equations with a scalar field matter source is described. The second-order-in-space-and-time partial differential equations are discretized directly without the use first order auxiliary terms, limiting the number of independent functions to fifteen--ten metric quantities, four source functions and the scalar field. This also limits the number of constraint equations, which can only be enforced to within truncation error in a numerical free evolution, to four. The coordinate system is compactified to spatial infinity in order to impose physically motivated, constraint-preserving outer boundary conditions. A variant of the Cartoon method for efficiently simulating axisymmetric spacetimes with a Cartesian code is described that does not use interpolation, and is easier to incorporate into existing adaptive mesh refinement packages. Preliminary test simulations of vacuum black hole evolution and black hole formation via scalar field collapse are described, suggesting that this method may be useful for studying many spacetimes of interest. 
  Contrary to the wide-spread belief, the correspondence principle does not dictate any relation between the asymptotics of quasinormal modes and the spectrum of quantized black holes. Moreover, this belief is in conflict with simple physical arguments. 
  Some Bianchi type I viscous fluid cosmological models with a variable cosmological constant are investigated in which the expansion is considered only in two direction i.e. one of the Hubble parameter $(H_{1} = \frac{A_{4}}{A})$ is zero. The viscosity coefficient of bulk viscous fluid is assumed to be a power function of mass density whereas the coefficient of shear viscosity is considered as constant in first case whereas in other case it is taken as proportional to scale of expansion in the model. The cosmological constant $\Lambda$ is found to be positive and is a decreasing function of time which is supported by results from recent supernovae Ia observations. Some physical and geometric properties of the models are also discussed. 
  As in other parts of physics, we advocate the interaction approach: experiments <--> phenomenology <--> low-energy effective (field) theory <--> microscopic theory to probe the microscopic origin of gravity. Using chi-g phenomenological framework, we discuss the tests of equivalence principles. The only experimentally unconstrained degree of freedom is the axion freedom. It has effects on the long-range astrophysical/cosmological propagation of electromagnetic waves and can be tested/measured using future generation of polarization measurement of cosmic background radiation. The verification or refutal of this axionic effect will be a crucial step for constructing effective theory and probing the microscopic origin of gravity. The interaction of spin with gravity is another important clue for probing microscopic origin of gravity. The interplay of experiments, phenomenology and effective theory is expounded. An ideal way to reveal the microscopic origin of gravity is to measure the gyrogravitational ratio of particles. Three potential experimental methods are considered. 
  Using the methods of loop quantum gravity, we derive a framework for describing an inflationary, homogeneous universe in a purely quantum theory. The classical model is formulated in terms of the Ashtekar-Sen connection variables for a general subclass of Bianchi class A spacetimes. This formulation then provides a means to develop a corresponding quantum theory. An exact solution is derived for the classical Bianchi type I model and the corresponding semi-classical quantum state is found to also be an exact solution to the quantum theory. The development of a normalizable quantum state from this exact solution is presented and the implications for the normalizability of the Kodama state discussed. We briefly consider some consequences of such a quantum cosmological framework and show that the quantum scale factor of the universe has a continuous spectrum in this model. This result may suggest further modifications that are required to build a more accurate theory. We close this study by suggesting a variety of directions in which to take the results presented. 
  The first-order gravity effects of Dirac wave functions are found from the inertial effects in the accelerated frames of reference. Derivations and discussions about Lense-Thirring effect and the gyrogravitational ratio for intrinsic spin are presented. We use coordinate transformations among reference frames to study and understand the Lense-Thirring effect of a scalar particle. For a Dirac particle, the wave-function transformation operator from an inertial frame to a moving accelerated frame is obtained. From this, the Dirac wave function is solved and its change of polarization gives the gyrogravitational ratio 1 for the first-order gravitational effects. The eikonal approach to this problem is presented in the end for ready extension to investigations involving curvature terms. 
  The paper summarizes the most important effects in Einsteinian gravitomagnetic fields related to propagating light rays, moving clocks and atoms, orbiting objects, and precessing spins. Emphasis is put onto the gravitational interaction of spinning objects. The gravitomagnetic field lines of a rotating or spinning object are given in analytic form. 
  We tackle the problem of the accelerating universe by reconsidering the most general form of the metric imposed by the cosmological principle when the mathematical device employed to describe the gravitational interaction is a 4-D manifold which is purely spatial in nature. A new varying speed of light (VSL) model naturally emerges from this approach. We show that a general coordinate transformation made with purpose of reverting to a constant g_{tt}, while possible because of diffeomorphism invariance, destroys the direct relation between the evolution of the scale factor and that of the wavelengths of the radiation propagating in the universe. In this model, the expansion rate and the acceleration of the universe turn out to be apparent effects. Solutions to field equations, when combined with observations, show that the universe is currently undergoing a contracting motion. A possible qualitative global picture is that of an eternally cycling universe. Only dust and radiation are put into the energy-momentum tensor. No sort of exotic (and so far unobserved) fluids, not even a cosmological constant, are needed to match our solutions with observations. A nice incidental and surprising result is the proof that a possible evolution of the Newton constant G(t) would not have any influence on the dynamics of the universe. 
  This paper has been withdrawn by the authors due to the inconsistency of the computations used for extra dimensions and the fractal dimensional approach of the paper.   An updated version of the paper will be published under a different title. 
  I argue against the widespread notion that manifest de Sitter invariance on the full de Sitter manifold is either useful or even attainable in gauge theories. Green's functions and propagators computed in a de Sitter invariant gauge are generally more complicated than in some noninvariant gauges. What is worse, solving the gauge-fixed field equations in a de Sitter invariant gauge generally leads to violations of the original, gauge invariant field equations. The most interesting free quantum field theories possess no normalizable, de Sitter invariant states. This precludes the existence of de Sitter invariant propagators. Even had such propagators existed, infrared divergent processes would still break de Sitter invariance. 
  Solving the 4-d Einstein equations as evolution in time requires solving equations of two types: the four elliptic initial data (constraint) equations, followed by the six second order evolution equations. Analytically the constraint equations remain solved under the action of the evolution, and one approach is to simply monitor them ({\it unconstrained} evolution). Since computational solution of differential equations introduces almost inevitable errors, it is clearly "more correct" to introduce a scheme which actively maintains the constraints by solution ({\it constrained} evolution). This has shown promise in computational settings, but the analysis of the resulting mixed elliptic hyperbolic method has not been completely carried out. We present such an analysis for one method of constrained evolution, applied to a simple vacuum system, linearized gravitational waves.   We begin with a study of the hyperbolicity of the unconstrained Einstein equations. (Because the study of hyperbolicity deals only with the highest derivative order in the equations, linearization loses no essential details.) We then give explicit analytical construction of the effect of initial data setting and constrained evolution for linearized gravitational waves. While this is clearly a toy model with regard to constrained evolution, certain interesting features are found which have relevance to the full nonlinear Einstein equations. 
  A model is constructed for the confinement of test particles moving on a brane. Within the classical framework of this theory, confining a test particle to the brane eliminates the effects of extra dimensions, rendering them undetectable. However, in the quantized version of the theory, the effects of the gauge fields and extrinsic curvature are pronounced and this might provide a hint for detecting them. As a consequence of confinement the mass of the test particle is shown to be quantized. The condition of stability against small perturbations along extra dimensions is also studied and its relation to dark matter is discussed. 
  The nature of Mashhoon's spin-rotation coupling is the interaction between a particle spin (gravitomagnetic moment) and a gravitomagnetic field. Here we will consider the coupling of graviton spin to the weak gravitomagnetic fields by analyzing the Lagrangian density of weak gravitational field, and hence study the purely gravitational generalization of Mashhoon's spin-rotation couplings. 
  The application of Regge calculus, a lattice formulation of general relativity, is reviewed in the context of numerical relativity. Particular emphasis is placed on problems of current computational interest, and the strengths and weaknesses of the lattice approach are highlighted. Several new and illustrative applications are presented, including initial data for the head on collision of two black holes, and the time evolution of vacuum axisymmetric Brill waves. 
  We outline the construction of differential invariants for higher--rank tensors. 
  The Ashtekar-Krishnan energy-balance law for dynamical horizons, expressing the increase in mass-energy of a general black hole in terms of the infalling matter and gravitational radiation, is expressed in terms of trapping horizons, allowing the inclusion of null (isolated) horizons as well as spatial (dynamical) horizons. This first law of black-hole dynamics is given in differential and integral forms, regular in the null limit. An effective gravitational-radiation energy tensor is obtained, providing measures of both ingoing and outgoing, transverse and longitudinal gravitational radiation on and near a black hole. Corresponding energy-tensor forms of the first law involve a preferred time vector which plays the role for dynamical black holes which the stationary Killing vector plays for stationary black holes. Identifying an energy flux, vanishing if and only if the horizon is null, allows a division into energy-supply and work terms, as in the first law of thermodynamics. The energy supply can be expressed in terms of area increase and a newly defined surface gravity, yielding a Gibbs-like equation, with a similar form to the so-called first law for stationary black holes. A Clausius-like relation suggests a definition of geometric entropy flux. Taking entropy as area/4 for dynamical black holes, it is shown that geometric entropy is conserved: the entropy of the black hole equals the geometric entropy supplied by the infalling matter and gravitational radiation. The area or entropy of a dynamical horizon increases by the so-called second law, not because entropy is produced, but because black holes classically are perfect absorbers. 
  Addressing the question of whether the Hawking effect depends on degrees of freedom at ultra-high (e.g., Planckian) energies/momenta, we propose three rather general conditions on these degrees of freedom under which the Hawking effect is reproduced to lowest order. As a generalization of Corley's results, we present a rather general model based on non-linear dispersion relations satisfying these conditions together with a derivation of the Hawking effect for that model. However, we also demonstrate counter-examples, which do not appear to be unphysical or artificial, displaying strong deviations from Hawking's result. Therefore, whether real black holes emit Hawking radiation remains an open question and could give non-trivial information about Planckian physics.   PACS: 04.70.Dy, 04.62.+v, 04.60.-m, 04.20.Cv. 
  We review the different aspects of the interaction of mesoscopic quantum systems with gravitational fields. We first discuss briefly the foundations of general relativity and quantum mechanics. Then, we consider the non-relativistic expansions of the Klein-Gordon and Dirac equations in the post-Newtonian approximation. After a short overview of classical gravitational waves, we discuss two proposed interaction mechanisms: (i) the use of quantum fluids as generator and/or detector of gravitational waves in the laboratory, and (ii) the inclusion of gravitomagnetic fields in the study of the properties of rotating superconductors. The foundations of the proposed experiments are explained and evaluated. 
  We consider fluctuations in a perfect irrotational fluid coupled to gravity in an Einstein static universe background. We show that the homogeneous linear perturbations of the scalar and metric fluctuations in the Einstein static universe must be present if the second order constraint equations are to be integrable. I.e., the 'linearization stability' constraint forces the presence of these homogeneous modes. Since these linear homogeneous scalar modes are well known to be exponentially unstable, the tactic of neglecting these modes to create a long-lived, almost Einstein universe does not work, even if all higher order (L $>$ 1) modes are dynamically stable. 
  We study the second order Coleman - de Luccia instanton which appears as the curvature of the effective potential reaches a sufficiently large value. We show how one can find the approximative formula for this instanton by perturbative expansion in the case when the second derivative of the effective potential divided by the Hubble parameter squared is close to -10, and we perform a numerical study of this instanton in the case of quasi-exponential potential. 
  Within classical general relativity, a particle cannot reach the horizon of a black hole during a finite time, in the reference frame of an external observer; a particle inside cannot escape from a black hole; and the horizon does not produce any reflection. We argue that these processes may possibly be allowed in the quantum world. It is known that quantum mechanics allows pair creation at the horizon (one particle inside, another particle outside) and Hawking radiation. One can extend this idea to propose other processes. Tunneling of an external particle inside black hole may be produced by the creation of a pair at the horizon, followed by the annihilation of one created particle with the initial particle outside, with the other created particle appearing inside. Escape of a particle from a black hole may result from the creation of a pair, followed by the annihilation of one created particle with the particle inside, with the other created particle appearing outside. The escape may allow the transfer of information to the outside.Finally, the reflection of an external particle from the horizon may be modelled by a combination of the two processes presented above. The relationship between these "pair creation-annihilation'' mechanisms and the "horizon tunneling" calculations [1-5] is discussed. 
  It is very common to ignore the non-local bulk effects in the study of brane-world cosmologies using the brane-world approach. However, we shall illustrate through the use of three different scenarios, that the non-local bulk-effect ${\cal P}_{\mu\nu}$ does indeed have significant impact on both the initial and future behaviour of brane-world cosmologies. 
  A numerical solution of Einstein field equations for a spherical symmetric and stationary system of identical and auto-gravitating particles in phase transition is presented. The fluid possess a perfect fluid energy momentum tensor, and the internal interactions of the system are represented by a van der Walls like equation of state able to describe a first order phase transition of the type gas-liquid. We find that the space-time curvature, the radial component of the metric, and the pressure and density show discontinuities in their radial derivatives in the phase coexistence region. This region is found to be a spherical surface concentric with the star and the system can be thought as a foliation of acronal, concentric and isobaric surfaces in which the coexistence of phases occurs in only one of these surfaces. This kind of system can be used to represent a star with a high energy density core and low energy density mantle in hydrodynamic equilibrium. 
  In the present work we found the geodesic structure of an AdS black hole. By means of a detailed analyze of the corresponding effective potentials for particles and photon, we found all the possible motions which are allowed by the energy levels. Radial and non radial trajectories were exactly evaluated for both geodesics. The founded orbits were plotted in order to have a direct visualization of the allowed motions. We show that the geodesic structure of this black hole presents new type of motions not allowed by the Schwarzschild spacetime. 
  We complete the historical overview about the geometry of a Schwarzschild black hole at its horizon by emphasizing the contribution made by J. L. Synge in 1950 to its clarification. 
  The approach to the singularity in Gowdy spacetimes consists of velocity term dominated behavior, except at a set of isolated points. At and near these points, spiky features grow. This paper reviews what is known about these spikes. 
  The nature of gravitational singularities, long mysterious, has now become clear through a combination of mathematical and numerical analysis. As the singularity is approached, the time derivative terms in the field equations dominate, and the singularity behaves locally like a homogeneous oscillatory spacetime. 
  In the lowest nonlinear approximation I compare two gravitational wave equations,- those of Weinberg and Papapetrou. The first one is simply a form of Einstein equation and the second is claimed to be yet another field theoretical form in which the energy-momentum tensor is obtained by Belinfante or Rosenfeld method. I show that for interacting gravitational field these methods lead to different energy-momentum tensors. Both these tensors need to be complemented "by hand" with some interaction energy-momentum tensors in order that the conservation laws of the total energy-momentum tensor give equation of motion for particles in agreement with general relativity. In approximation considered by Thirring, the Papapetrou wave equation must coincide with that of Thirring. But they differ because Thirring inserted the necessary interaction term. I show that Thirring wave equation is equivalent to Weinberg's one. Hence the Papapetrou equation is not yet another form of Einstein equation. 
  Tensor type perturbations in the expanding brane world of the Randall Sundrum type are investigated. We consider a model composed of slow-roll inflation phase and the succeeding radiation phase. The effect of the presence of an extra dimension through the transition to the radiation phase is studied, giving an analytic formula for leading order corrections. 
  The so-called ``analogue models of general relativity'' provide a number of specific physical systems, well outside the traditional realm of general relativity, that nevertheless are well-described by the differential geometry of curved spacetime. Specifically, the propagation of acoustic disturbances in moving fluids are described by ``effective metrics'' that carry with them notions of ``causal structure'' as determined by an exchange of sound signals. These acoustic causal structures serve as specific examples of what can be done in the presence of a Lorentzian metric without having recourse to the Einstein equations of general relativity. (After all, the underlying fluid mechanics is governed by the equations of traditional hydrodynamics, not by the Einstein equations.) In this article we take a careful look at what can be said about the causal structure of acoustic spacetimes, focusing on those containing sonic points or horizons, both with a view to seeing what is different from standard general relativity, and to seeing what the similarities might be. 
  Conventional explanations for observations of anomalous behaviour of mechanical systems during solar eclipses are critically reviewed. These observations include the work of Allais with paraconical pendula, those of Saxl and Allen with a torsion pendulum and measurements with gravimeters. Attempts of replications of these experiments and recent gravimeter results are discussed and unpublished data by Latham and by Saxl et al. is presented. Some of the data are summarized and re-analyzed. Especially, attention is paid to observations of tilt of the vertical, which seems to play an important role in this matter and recommendations for future research are given. It is concluded that all the proposed conventional explanations either qualitatively or quantitatively fail to explain the observations. 
  Light deflection offers an unbiased test of Weyl's gravity since no assumption on the conformal factor needs to be made. In this second paper of our series ``Light deflection in Weyl gravity'', we analyze the constraints imposed by light deflection experiments on the linear parameter of Weyl's theory. Regarding solar system experiments, the recent CASSINI Doppler measurements are used to infer an upper bound, $\sim 10^{-19} $m$^{-1}$, on the absolute value of the above Weyl parameter. In non-solar system experiments, a condition for unbound orbits together with gravitational mirage observations enable us to further constrain the allowed negative range of the Weyl parameter to $\sim -10^{-31} $m$^{-1}$. We show that the characteristics of the light curve in microlensing or gravitational mirages, deduced from the lens equation, cannot be recast into the General Relativistic predictions by a simple rescaling of the deflector mass or of the ring radius. However, the corrective factor, which depends on the Weyl parameter value and on the lensing configuration, is small, even perhaps negligible, owing to the upper bound inferred on the absolute value of a negative Weyl parameter. A statistical study on observed lensing systems is required to settle the question.   Our Weyl parameter range is more reliable than the single value derived by Mannheim and Kazanas from fits to galactic rotation curves, $\sim +10^{-26}\ $m$^{-1}$. Indeed, the latter, although consistent with our bounds, is biased by the choice of a specific conformal factor. 
  We review a recent proposal for the construction of a quantum theory of the gravitational field. The proposal is based on approximating the continuum theory by a discrete theory that has several attractive properties, among them, the fact that in its canonical formulation it is free of constraints. This allows to bypass many of the hard conceptual problems of traditional canonical quantum gravity. In particular the resulting theory implies a fundamental mechanism for decoherence and bypasses the black hole information paradox. 
  A field theory is proposed where the regular fermionic matter and the dark fermionic matter are different states of the same "primordial" fermion fields. In regime of the fermion densities typical for normal particle physics, each of the primordial fermions splits into three generations identified with regular fermions. In a simple model, this fermion families birth effect is accompanied with the right lepton numbers conservation laws. It is possible to fit the muon to electron mass ratio without fine tuning of the Yukawa coupling constants. When fermion energy density becomes comparable with dark energy density, the theory allows new type of states - Cosmo-Low Energy Physics (CLEP) states. Neutrinos in CLEP state can be both a good candidate for dark matter and responsible for a new type of dark energy. In the latter case the total energy density of the universe is less than it would be in the universe free of fermionic matter at all. The (quintessence) scalar field is coupled to dark matter but its coupling to regular fermionic matter appears to be extremely suppressed. 
  We intend to analyse the constraint structure of Teleparallelism employing the Hamilton-Jacobi formalism for singular systems. This study is conducted without using an ADM 3+1 decomposition and without fixing time gauge condition. It can be verified that the field equations constitute an integrable system. 
  We consider a quantized massless and minimally coupled scalar field on a circular closed string with a time-dependent radius $R(t)$, whose undisturbed dynamics is governed by the Nambu-Goto action. Within the semi-classical treatment, the back-reaction of the quantum field onto the string dynamics is taken into account in terms of the renormalized expectation value of the energy-momentum tensor including the trace anomaly. The results indicate that the back-reaction could prevent the collapse of the circle $R\downarrow0$ -- however, the semi-classical picture fails to describe the string dynamics at the turning point (i.e., possible bounce) at finite values of $R$ and $\dot R$. The fate of the closed string after that point (e.g., oscillation or eternal acceleration) cannot be determined within the semi-classical picture and thus probably requires the full quantum treatment. PACS: 04.62.+v, 03.70.+k, 11.15.Kc, 04.60.-m. 
  In the present work, it is shown that, the application of the Bazanski approach to Lagrangians, written in AP-geometry and including the basic vector of the space, gives rise to a new class of path equations. The general equation representing this class contains four extra terms, whose vanishing reduces this equation to the geodesic one. If the basic vector of the  AP-geometry is considered as playing the role of the electromagnetic potential, as done in a previous work, then the second term  (of the extra terms) will represent Lorentz force while the fourth term gives a direct effect of the electromagnetic potential on the motion of the charged particle. This last term may give rise to an effect similar to the Aharanov-Bohm effect. It is to be considered that all extra terms will vanish if the space-time used is torsion-less. 
  We consider scalar tensor theories in D-dimensional spacetime, D \ge 4. They consist of metric and a non minimally coupled scalar field, with its non minimal coupling characterised by a function. The probes couple minimally to the metric only. We obtain vacuum solutions - both cosmological and static spherically symmetric ones - and study their properties. We find that, as seen by the probes, there is no singularity in the cosmological solutions for a class of functions which obey certain constraints. It turns out that for the same class of functions, there are static spherically symmetric solutions which exhibit novel properties: {\em e.g.} near the ``horizon'', the gravitational force as seen by the probe becomes repulsive. 
  Up to now attempts to measure the general relativistic Lense-Thirring effect in the gravitational field of Earth have been performed by analyzing a suitable J_2-J_4-free combination of the nodes Omega of LAGEOS and LAGEOS II and the perigee omega of LAGEOS II with the Satellite Laser Ranging technique. The claimed total accuracy is of the order of 20-30%, but, according to some scientists, it could be an optimistic estimate. The main sources of systematic errors are the mismodelling in the even zonal harmonic coefficients J_l of the multipolar expansion of the gravitational potential of Earth and the non-gravitational perturbations which plague especially the perigee of LAGEOS II and whose impact on the proposed measurement is difficult to be reliably assessed. Here we present some evaluations of the accuracy which could be reached with a different J_2-free observable built up with the nodes of LAGEOS and LAGEOS II in view of the new preliminary 2nd-generation Earth gravity models from the GRACE mission. According to the GRACE-only based EIGEN-GRACE02S solution, a 1-sigma upper bound of 4% for the systematic error due to the even zonal harmonics can be obtained. In the near future it could be possible to perform a reliable measurement of the Lense-Thirring effect by means of the existing LAGEOS satellites with an accuracy of a few percent by adopting a time span of a few years. The choice of a not too long observational temporal interval would be helpful in reducing the impact of the secular variations of the uncancelled even zonal harmonics dot J_4 and dot J_6 whose impact is difficult to be reliably evaluated. 
  Using the black string between two branes as a model of a brane-world black hole, we compute the gravity wave perturbations and identify the features arising from the additional polarizations of the graviton. The standard four-dimensional gravitational wave signal acquires late-time oscillations due to massive modes of the graviton. The Fourier transform of these oscillations shows a series of spikes associated with the masses of the Kaluza-Klein modes, providing in principle a spectroscopic signature of extra dimensions. 
  We argue that counting black hole states in loop quantum gravity one should take into account only states with the minimal spin at the horizon. 
  The standard, scale-invariant, inflationary perturbation spectrum will be modified if the effects of trans-Planckian physics are incorporated into the dynamics of the matter field in a phenomenological manner, say, by the modification of the dispersion relation. The spectrum also changes if we retain the standard dynamics but modify the initial quantum state of the matter field. We show that, given {\it any} spectrum of perturbations, it is possible to choose a class of initial quantum states which can exactly reproduce this spectrum with the standard dynamics. We provide an explicit construction of the quantum state which will produce the given spectrum. We find that the various modified spectra that have been recently obtained from `trans-Planckian considerations' can be constructed from suitable squeezed states above the Bunch-Davies vacuum in the standard theory. Hence, the CMB observations can, at most, be useful in determining the initial state of the matter field in the standard theory, but it can {\it not} help us to discriminate between the various Planck scale models of matter fields. We study the problem in the Schrodinger picture, clarify various conceptual issues and determine the criterion for negligible back reaction due to modified initial conditions. 
  A choice of first-order variables for the characteristic problem of the linearized Einstein equations is found which casts the system into manifestly well-posed form. The concept of well-posedness for characteristic problems invoked is that there exists an \textit{a priori} estimate of the solution of the characteristic problem in terms of the data. The notion of manifest well-posedness consists of an algebraic criterion sufficient for the existence of the estimates, and is to characteristic problems as symmetric hyperbolicity is to Cauchy problems. Both notions have been made precise elsewere. 
  The definitions of classical and quantum singularities in general relativity are reviewed. The occurence of quantum mechanical singularities in certain spherically symmetric and cylindrically symmetric (including infinite line mass)spacetimes is considered. A strong repulsive ``potential'' near the classical singularity is shown to turn a classically singular spacetime into a quantum mechanically nonsingular spacetime. 
  Based on a number of experimentally verified physical observations, it is argued that the standard principles of quantum mechanics should be applied to the Universe as a whole. Thus, a paradigm is proposed in which the entire Universe is represented by a pure state wavefunction contained in a factorisable Hilbert space of enormous dimension, and where this statevector is developed by successive applications of operators that correspond to unitary rotations and Hermitian tests. Moreover, because by definition the Universe contains everything, it is argued that these operators must be chosen self-referentially; the overall dynamics of the system is envisaged to be analogous to a gigantic, self-governing, quantum computation. The issue of how the Universe could choose these operators without requiring or referring to a fictitious external observer is addressed.   The processes by which conventional physics might be recovered from this fundamental, mathematical and global description of reality are particularly investigated. Specifically, it is demonstrated that by considering the changing properties, separabilities and factorisations of both the state and the operators as the Universe proceeds though a sequence of discrete computations, familiar notions such as classical distinguishability, particle physics, space, time, special relativity and endo-physical experiments can all begin to emerge from the proposed picture. A pregeometric vision of cosmology is therefore discussed, with all of physics ultimately arising from the relationships occurring between the elements of the underlying mathematical structure. 
  I consider a power-law inflationary model taking into account back-reaction effects. The interesting result is that the spectrum for the scalar field fluctuations does not depends on the expansion rate of the universe $p$ and that it result to be scale invariant for cosmological scales. However, the amplitude for these fluctuations depends on $p$. 
  We derive the power spectrum P_\psi(k) of the gravitational waves produced during general classes of inflation with second order corrections. Using this result, we also derive the spectrum and the spectral index in the standard slow-roll approximation with new higher order corrections. 
  The speciality index, which has been mainly used in Numerical Relativity for studying gravitational waves phenomena as an indicator of the special or non-special Petrov type character of a spacetime, is applied here in the context of Mixmaster cosmology, using the Belinski-Khalatnikov-Lifshitz map. Possible applications for the associated chaotic dynamics are discussed. 
  This article reports results from numerical simulations of the gravitational radiation emitted from nonrotating relativistic stars as a result of the axisymmetric accretion of layers of perfect fluid matter, shaped in the form of quadrupolar shells. We adopt a {\em hybrid} procedure where we evolve numerically the polar nonspherical perturbations equations of the star coupled to a fully nonlinear hydrodynamics code that calculates the motion of the accreting matter. Self-gravity of the accreting fluid as well as radiation reaction effects are neglected. 
  There has been some recent speculation that a connection may exist between the quasinormal-mode spectra of highly damped black holes and the fundamental theory of quantum gravity. This notion follows from a conjecture by Hod that the real part of the highly damped mode frequencies can be used to calibrate the semi-classical level spacing in the black hole quantum area spectrum. However, even if the level spacing can be fixed in this manner, it still remains unclear whether this implies a physically significant "duality" or merely a numerical coincidence. This tapestry of ideas serves as the motivation for the current paper. We utilize the "monodromy approach" to calculate the quasinormal-mode spectra for a generic class of black holes in two-dimensional dilatonic gravity. Our results agree with the prior literature whenever a direct comparison is possible and provide the analysis of a much more diverse class of black hole models than previously considered. 
  Conformally flat tilted Bianchi type V cosmological models in presence of a bulk viscous fluid and heat flow are investigated. The coefficient of bulk viscosity is assumed to be a power function of mass density. The cosmological constant is found to be a decreasing function of time, which is supported by results from recent type Ia supernovae observations. Some physical and geometric aspects of the models are also discussed. 
  The Jang equation in the spherically symmetric case reduces to a first order equation. This permits an easy analysis of the role apparent horizons play in the (non)existence of solutions. We demonstrate that the proposed derivation of the Penrose inequality based on the Jang equation cannot work in the spherically symmetric case. Thus it is fruitless to apply this method, as it stands, to the general case. We show also that those analytic criteria for the formation of horizons that are based on the use of the Jang equation are of limited validity for the proof of the trapped surface conjecture. 
  A `resolution' of the interior singularity of the spherically symmetric Schwarzschild solution of the Einstein equations for the gravitational field of a point-particle is carried out entirely and solely by finitistic and algebraic means. To this end, the background differential spacetime manifold and Calculus-free purely algebraic (:sheaf-theoretic) conceptual and technical machinery of Abstract Differential Geometry (ADG) is employed via Sorkin's finitary (:locally finite) poset substitutes of continuous manifolds in their Gel'fand-dual picture in terms of discrete differential incidence algebras and the finitary spacetime sheaves thereof. It is shown that the Einstein equations hold not only at the finitary poset level of `discrete events', but also at a suitable `classical continuum limit' of the said finitary sheaves and the associated differential triads that they define ADG-theoretically. We infer that the law of gravity does not break down in any (differential geometric) sense in the vicinity of the locus of the point-mass as the usual manifold based analysis of spacetime singularities in General Relativity has hitherto maintained. Various possible implications that such a total evasion of smooth gravitational singularities, as well as some anticipations of the wider significance that the general ADG-framework, may have for certain current `hot' issues in both classical and quantum gravity research are briefly discussed at the end. 
  We discuss the existence, stability and classical thermodynamics of four-dimensional, spherically symmetric black hole solutions of the Einstein equations with a conformally coupled scalar field. We review the solutions existing in the literature with zero, positive and negative cosmological constant. We also outline new results on the thermodynamics of these black holes when the cosmological constant is non-zero. 
  In order to discuss the well-posed initial value formulation of the teleparallel gravity and apply it to numerical relativity a symmetric hyperbolic system in the self-dual teleparallel gravity which is equivalent to the Ashtekar formulation is posed. This system is different from the ones in other works by that the reality condition of the spatial metric is included in the symmetric hyperbolicity and then is no longer an independent condition. In addition the constraint equations of this system are rather simpler than the ones in other works. 
  A self-dual and anti-self-dual decomposition of the teleparallel gravity is carried out and the self-dual Lagrangian of the teleparallel gravity which is equivalent to the Ashtekar Lagrangian in vacuum is obtained. Its Hamiltonian formulation and the constraint analysis are developed. Starting from Witten's equation Nester's gauge condition is derived directly and a new expression of the boundary term is obtained. Using this expression and Witten's identity the proof of the positive energy theorem by Nester et al is extended to a case including momentum. 
  By means of Ernst complex potential formalism it is shown, that previously studied static axisymmetric Einstein-Maxwell fields obtained though the application of the Horsky-Mitskievitch generating conjecture represent a combination of Kinnersley's transformations [W. Kinnersley: J. Math. Phys. 14 (1973) 651]. New theoretical background for the conjecture is suggested and commented. 
  We have recently argued that if one introduces a relational time in quantum mechanics and quantum gravity, the resulting quantum theory is such that pure states evolve into mixed states. The rate at which states decohere depends on the energy of the states. There is therefore the question of how this can be reconciled with Galilean invariance. More generally, since the relational description is based on objects that are not Dirac observables, the issue of covariance is of importance in the formalism as a whole. In this note we work out an explicit example of a totally constrained, generally covariant system of non-relativistic particles that shows that the formula for the relational conditional probability is a Galilean scalar and therefore the decoherence rate is invariant. 
  It is very likely that the quantum description of spacetime is quite different from what we perceive at large scales, $l\gg (G\hbar/c^3)^{1/2}$. The long wave length description of spacetime, based on Einstein's equations, is similar to the description of a continuum solid made of a large number of microscopic degrees of freedom. This paradigm provides a novel interpretation of coordinate transformations as deformations of "spacetime solid" and allows one to obtain Einstein's equations as a consistency condition in the long wavelength limit. The entropy contributed by the microscopic degrees of freedom reduces to a pure surface contribution when Einstein's equations are satisfied. The horizons arises as "defects" in the "spacetime solid" (in the sense of well defined singular points) and contributes an entropy which is one quarter of the horizon area. Finally, the response of the microstructure to vacuum energy leads to a near cancellation of the cosmological constant, leaving behind a tiny fluctuation which matches with the observed value. 
  The present paper has the purpose to illustrate the importance of the ideas and constructions of the Non-Euclidean (Lobachevsky) Geometry, which can be applied even today for solving some conceptually important problems. We study the static and spherically symmetric solutions to the Einstein field equations under the assumption that the space-time may possess an arbitrary number of spatial dimensions. A new exact solution of a perfect fluid sphere of constant (homogeneous) energy-density which agrees with interior Lobachevsky geometry for 3D and 4D spaces are found. We discuss the property of temporal scalar field arise in lower-dimensional theories as the reduction of extra dimension. 
  An introduction to solutions of the Einstein equations defining cosmological models with accelerated expansion is given. Connections between mathematical and physical issues are explored. Theorems which have been proved for solutions with positive cosmological constant or nonlinear scalar fields are reviewed. Some remarks are made on more exotic models such as the Chaplygin gas, tachyons and $k$-essence. 
  Using the Hamilton-Jacobi formalism, we study extra force and extra mass in a recently introduced noncompact Kaluza-Klein cosmological model. We examine the inertial 4D mass $m_0$ of the inflaton field on a 4D FRW bulk in two examples. We find that $m_0$ has a geometrical origin and antigravitational effects on a non inertial 4D bulk should be a consequence of the motion of the fifth coordinate with respect to the 4D bulk. 
  We present detailed numerical simulations of a laser phase stabilization scheme for LISA, where both lasers emitting along one arm are locked to each other. Including the standard secondary noises and spacecraft motions that approximately mimic LISA's orbit, we verify that very stable laser phases can be obtained, and that time delay interferometry can be used to remove the laser phase noise from measurements of gravitational wave strains. Most importantly, we show that this locking scheme can provide significant simplifications over LISA's baseline design in the implementation of time delay interferometry. 
  In this communication, we analyze the case of 3+1 dimensional scalar field cosmologies in the presence, as well as in the absence of spatial curvature, in isotropic, as well as in anisotropic settings. Our results extend those of Hawkins and Lidsey [Phys. Rev. D {\bf 66}, 023523 (2002)], by including the non-flat case. The Ermakov-Pinney methodology is developed in a general form, allowing through the converse results presented herein to use it as a tool for constructing new solutions to the original equations. As an example of this type a special blowup solution recently obtained in Christodoulakis {\it et al.} [gr-qc/0302120] is retrieved. Additional solutions of the 3+1 dimensional gravity coupled with the scalar field are also obtained. To illustrate the generality of the approach, we extend it to the anisotropic case of Bianchi types I and V and present some related open problems. 
  Exact solution of the Einstein's equation describing a spherically symmetric cosmological model without a big bang or any other kind of singularity recently obtained by Dadhich and Patel (2000) is revisited. The matter content of the model is a shear-free perfect fluid with isotropic pressure and a radial heat flux. The cosmological constant $\Lambda$ is found to be positive and decreasing function of time which is supported by the results from recent type Ia supernovae observations. 
  By establishing that Palatini formulation of $L(R)$ gravity is equivalent to $\omega=-3/2$ Brans-Dicke theory, we show that energy-momentum tensor is covariantly conserved in this type of modified gravity theory. 
  This paper presents a method for computing two-dimensional constant mean curvature surfaces. The method in question uses the variational aspect of the problem to implement an efficient algorithm. In principle it is a flow like method in that it is linked to the gradient flow for the area functional, which gives reliable convergence properties. In the background a preconditioned conjugate gradient method works, that gives the speed of a direct elliptic multigrid method. 
  In this paper we obtain unstable even-parity eigenmodes to the static regular spherically symmetric solutions of the SU(2) Yang-Mills-dilaton coupled system of equations in 3+1 Minkowski space-time. The corresponding matrix Sturm-Liouville problem is solved numerically by means of the continuous analogue of Newton's method. The method, being the powerful tool for solving both boundary-value and Sturm-Liouville problems, is described in details. 
  The elliptic system of equations, which is general-covariant and locally SU(2)-covariant, is investigated. The new condition of the Dirichlet problem solvability and the condition of zeros absence for solutions are obtained for this system, which contains in particular case the Sen-Witten equation. On this basis it is proved the existence of the wide class of hypersurfaces, in all points of which there exists a correspondence between the Sen-Witten spinor field and three-frame, which generalizes the Nester orthoframe. The Nester special orthoframe also exists on a certain subclass containing not only the maximal hypersurfaces. 
  The Conformal Einstein equations and the representation of spatial infinity as a cylinder introduced by Friedrich are used to analyse the behaviour of the gravitational field near null and spatial infinity for the development of data which are asymptotically Euclidean, conformally flat and time asymmetric. Our analysis allows for initial data whose second fundamental form is more general than the one given by the standard Bowen-York Ansatz. The Conformal Einstein equations imply upon evaluation on the cylinder at spatial infinity a hierarchy of transport equations which can be used to calculate in a recursive way asymptotic expansions for the gravitational field. It is found that the the solutions to these transport equations develop logarithmic divergences at certain critical sets where null infinity meets spatial infinity. Associated to these, there is a series of quantities expressible in terms of the initial data (obstructions), which if zero, preclude the appearance of some of the logarithmic divergences. The obstructions are, in general, time asymmetric. That is, the obstructions at the intersection of future null infinity with spatial infinity are different, and do not generically imply those obtained at the intersection of past null infinity with spatial infinity. The latter allows for the possibility of having spacetimes where future and past null infinity have different degrees of smoothness. Finally, it is shown that if both sets of obstructions vanish up to a certain order, then the initial data has to be asymptotically Schwarzschildean to some degree. 
  Expression for the Witten-Nester 4-spinor 3-form of the Hamiltonian density of gravitational field in the asymptotically flat space-time in terms of the Sommers-Sen spinors, direct with a certain orthonormal three-frame connect, is obtained. A direct connection between the one and the ADM Hamiltonian density in the Sen-Witten frame is established on this basis. 
  A numerical and analytic treatment is presented here of the evolution of initial data of the kind that was conjectured by Hertog, Horowitz and Maeda to lead to a violation of cosmic censorship. That initial data is essentially a thick domain wall connecting two regions of anti de Sitter space. The evolution results in no violation of cosmic censorship, but rather the formation of a small black hole. 
  We look at a gas of dust and investigate how its entropy evolves with time under a spherically symmetric gravitational collapse. We treat the problem perturbatively and find that the classical thermodynamic entropy does actually increase to first order when one allows for gravitational potential energy to be transferred to thermal energy during the collapse. Thus, in this situation there is no need to resort to the introduction of an intrinsic gravitational entropy in order to satisfy the second law of thermodynamics. 
  A spherically symmetric spacetime is presented with an initial data set that is asymptotically flat, satisfies the dominant energy condition, and such that on this initial data $M<\sqrt{A/16\pi}$, where M is the total (ADM) mass and A is the area of the apparent horizon. This provides a counterexample to a commonly stated version of the Penrose inequality, though it does not contradict the ``true'' Penrose inequality. 
  We provide a simple derivation of the Schwarzschild solution in General Relativity, generalizing an early approach by Weyl, to include Birkhoff's theorem: constancy of the mass; its deeper, Hamiltonian, basis is also given. Our procedure is illustrated by a parallel derivation of the Coulomb field and constancy of electric charge, in electrodynamics. 
  We prove that the space of causal curves between compact subsets of a separable globally hyperbolic poset is itself compact in the Vietoris topology. Although this result implies the usual result in general relativity, its proof does not require the use of geometry or differentiable structure. 
  Recently D. Vollick [Phys. Rev. D68, 063510 (2003)] has shown that the inclusion of the 1/R curvature terms in the gravitational action and the use of the Palatini formalism offer an alternative explanation for cosmological acceleration. In this work we show not only that this model of Vollick does not have a good Newtonian limit, but also that any f(R) theory with a pole of order n in R=0 and its second derivative respect to R evaluated at Ro is not zero, where Ro is the scalar curvature of background, does not have a good Newtonian limit. 
  Bianchi type V perfect fluid cosmological models are investigated with cosmological term $\Lambda$ varying with time. Using a generation technique (Camci {\it et al.}, 2001), it is shown that the Einstein's field equations are solvable for any arbitrary cosmic scale function. Solutions for particular forms of cosmic scale functions are also obtained. The cosmological constant is found to be decreasing function of time, which is supported by results from recent type Ia supernovae observations. Some physical aspects of the models are also discussed. 
  In all dimensions and arbitrary signature, we demonstrate the existence of a new local potential -- a double (2,3)-form -- for the Weyl curvature tensor, and more generally for all tensors with the symmetry properties of the Weyl curvature tensor. The classical four-dimensional Lanczos potential for a Weyl tensor -- a double (2,1)-form -- is proven to be a particular case of the new potential: its double dual. 
  A new proof of Friedrich's theorem on the existence and stability of asymptotically de Sitter spaces in 3+1 dimensions is given, which extends to all even dimensions. In addition, we characterize the possible limits of spaces which are globally asymptotically de Sitter, to the past and future. 
  Linearized gravitational waves in Brans-Dicke and scalar-tensor theories carry negative energy. A gauge-invariant analysis shows that the background Minkowski space is stable at the classical level with respect to linear scalar and tensor inhomogeneous perturbations. 
  The use of techniques from loop quantum gravity for cosmological models may solve some difficult problems in quantum cosmology. The solutions under a number of circumstances have been well studied. We will analyse the behaviour of solutions in the closed model, focusing on the behaviour of a universe containing a massless scalar field. The asymptotic behaviour of the solutions is examined, and is used to determine requirements of the initial conditions. 
  Finite temperature effects in brane world cosmology are studied by considering the interaction between scalar field and bulk gravity. One-loop correction to zero-temperature potential is computed by taking into account, interaction of scalar field and bulk gravity. Phase transitions and high temperature symmetry restoration are examined. Critical temperature of phase transitions depends on the interaction constant of the scalar field and bulk gravity, and these constant is an order parameter. Present study can account for second order phase transition in early universe, in brane world cosmological scenario. 
  LISA is an array of three spacecraft flying in an approximately equilateral triangle configuration, which will be used as a low-frequency detector of gravitational waves. Recently a technique has been proposed for suppressing the phase noise of the onboard lasers by locking them to the LISA arms. In this paper we show that the delay-induced effects substantially modify the performance of this technique, making it different from the conventional locking of lasers to optical resonators. We analyze these delay-induced effects in both transient and steady-state regimes and discuss their implications for the implementation of this technique on LISA. 
  An L-pole perturbation in Schwarzschild spacetime generally falls off at late times t as t^{-2L-3}. It has recently been pointed out by Karkowski, Swierczynski and Malec, that for initial data that is of compact support, and is initially momentarily static, the late-time behavior is different, going as t^{-2L-4}. By considering the Laplace transforms of the fields, we show here why the momentarily stationary case is exceptional. We also explain, using a time-domain description, the special features of the time development in this exceptional case. 
  The exact five-dimensional charged black hole solution in Lovelock gravity coupled to Born-Infeld electrodynamics is presented. This solution interpolates between the Hoffmann black hole for the Einstein-Born-Infeld theory and other solutions in the Lovelock theory previously studied in the literature. The conical singularity of the metric around the origin can be removed by a proper choice of the black hole parameters. The thermodynamic properties of the solution are also analyzed and, in particular, it is shown that the behaviour of the specific heat indicates the existence of a stability transition point in the vacuum solutions. We discuss the similarities existing between this five-dimensional geometry and the three-dimensional black hole. Like BTZ black hole, the Lovelock black hole has an infinite lifetime. 
  We study how physical information can be extracted from a background independent quantum system. We use an extremely simple `minimalist' system that models a finite region of 3d euclidean quantum spacetime with a single equilateral tetrahedron. We show that the physical information can be expressed as a boundary amplitude. We illustrate how the notions of "evolution" in a boundary proper-time and "vacuum" can be extracted from the background independent dynamics. 
  We work in the locally de Sitter background of an inflating universe and consider a massless, minimally coupled scalar with a quartic self-interaction. We use dimensional regularization to compute the fully renormalized scalar self-mass-squared at one and two loop order for a state which is released in Bunch-Davies vacuum at t=0. Although the field strength and coupling constant renormalizations are identical to those of lfat space, the geometry induces a non-zero mass renormalization. The finite part also shows a sort of growing mass that competes with the classical force in eventually turning off this system's super-acceleration. 
  We show that spherical accretion onto astrophysical black holes can be considered as a natural example of analogue system. We provide, for the first time, an exact analytical scheme for calculating the analogue Hawking temperature and surface gravity for general relativistic accretion onto astrophysical black holes. Our calculation may bridge the gap between the theory of transonic astrophysical accretion and the theory of analogue Hawking radiation. We show that the domination of the analogue Hawking temperature over the actual Hawking temperature may be a real astrophysical phenomena. We also discuss the possibilities of the emergence of analogue white holes around astrophysical black holes. Our calculation is general enough to accommodate accreting black holes with any mass. 
  We describe an approach to the quantization of (2+1)--dimensional gravity with topology R x T^2 and negative cosmological constant, which uses two quantum holonomy matrices satisfying a q--commutation relation. Solutions of diagonal and upper--triangular form are constructed, which in the latter case exhibit additional, non--trivial internal relations for each holonomy matrix. This leads to the notion of quantum matrix pairs. These are pairs of matrices with non-commuting entries, which have the same pattern of internal relations, q-commute with each other under matrix multiplication, and are such that products of powers of the matrices obey the same pattern of internal relations as the original pair. This has implications for the classical moduli space, described by ordered pairs of commuting SL(2,R) matrices modulo simultaneous conjugation by SL(2,R) matrices. 
  The ADM Formalism is discussed in the context of 2+1--dimensional gravity, uniting two areas of relativity theory in which Stanley Deser has been particularly active. For spacetimes with topology R x T^2 the partially reduced and fully reduced ADM formalism are related and quantized, and the role of "large diffeomorphisms" (the modular group) in the quantum theory is illustrated. 
  At high energies on a cosmological brane of Randall-Sundrum type, particle interactions can produce gravitons that are emitted into the bulk and that can feed a bulk black hole. We generalize previous investigations of such radiating brane-worlds by allowing for a breaking of Z_2-symmetry, via different bulk cosmological constants and different initial black hole masses on either side of the brane. One of the notable features of asymmetry is a suppression of the asymptotic level of dark radiation, which means that nucleosynthesis constraints are easier to satisfy. There are also models where the radiation escapes to infinity on one or both sides, rather than falling into a black hole, but these models can have negative energy density on the brane. 
  We find a class of warp drive spacetimes possessing Newtonian limits, which we then determine. The same method is used to compute Newtonian limits of the Schwarzschild solution and spatially flat Friedmann-Robertson-Walker cosmological models. 
  We compute the energy and momentum of a regular black hole of type defined by Mars, Martin-Prats, and Senovilla using the Einstein and Papapetrou definitions for energy-momentum density. Some other definitions of energy-momentum density are shown to give mutually contradictory and less reasonable results. Results support the Cooperstock hypothesis. 
  An exact solution to the dynamic equations for a massive boson traveling in a pp-wave gravitational background under the influence of the force induced by curvature, is presented. We focus on the effect of anomalous polarization-curvature interaction and consider models in which the coupling constant of such an interaction is treated to be either a deterministic quantity or a random variable. 
  The computation of the radiation flux related to the Hawking temperature of a Schwarzschild Black Hole or another geometric background is still well-known to be fraught with a number of delicate problems. In spherical reduction, as shown by one of the present authors (W. K.) with D.V. Vassilevich, the correct black body radiation follows when two ``basic components'' (conformal anomaly and a ``dilaton'' anomaly) are used as input in the integrated energy-momentum conservation equation. The main new element in the present work is the use of a quite different method, the covariant perturbation theory of Barvinsky and Vilkovisky, to establish directly the full effective action which determines these basic components. In the derivation of W. K. and D.V. Vassilevich the computation of the dilaton anomaly implied one potentially doubtful intermediate step which can be avoided here. Moreover, the present approach also is sensitive to IR (renormalisation) effects. We realize that the effective action naturally leads to expectation values in the Boulware vacuum which, making use of the conservation equation, suffice for the computation of the Hawking flux in other quantum states, in particular for the relevant Unruh state. Thus, a rather comprehensive discussion of the effects of (UV and IR) renormalisation upon radiation flux and energy density is possible. 
  We solve for single distorted black hole initial data using the puncture method, where the Hamiltonian constraint is written as an elliptic equation in R^3 for the nonsingular part of the metric conformal factor. With this approach we can generate isometric and non--isometric black hole data. For the isometric case, our data are directly comparable to those obtained by Bernstein et al., who impose isometry boundary conditions at the black hole throat. Our numerical simulations are performed using a parallel multigrid elliptic equation solver with adaptive mesh refinement. Mesh refinement allows us to use high resolution around the black hole while keeping the grid boundaries far away in the asymptotic region. 
  The late-time evolution of the charged massive Dirac fields in the background of a Reissner-Norstr\"om (RN) black hole is studied. It is found that the intermediate late-time behavior is dominated by an inverse power-law decaying tail without any oscillation in which the dumping exponent depends not only on the multiple number of the wave mode but also on the field parameters. It is also found that, at very late times, the oscillatory tail has the decay rate of $t^{-5/6}$ and the oscillation of the tail has the period $2\pi/\mu$ which is modulated by two types of long-term phase shifts. 
  The existing high technology laser-beam detectors of gravitational waves may find very useful applications in an unexpected area - geophysics. To make possible the detection of weak gravitational waves in the region of high frequencies of astrophysical interest, ~ 30 - 10^3 Hz, control systems of laser interferometers must permanently monitor, record and compensate much larger external interventions that take place in the region of low frequencies of geophysical interest, ~ 10^{-5} - 3 X 10^{-3} Hz. Such phenomena as tidal perturbations of land and gravity, normal mode oscillations of Earth, oscillations of the inner core of Earth, etc. will inevitably affect the performance of the interferometers and, therefore, the information about them will be stored in the data of control systems. We specifically identify the low-frequency information contained in distances between the interferometer mirrors (deformation of Earth) and angles between the mirrors' suspensions (deviations of local gravity vectors and plumb lines). We show that the access to the angular information may require some modest amendments to the optical scheme of the interferometers, and we suggest the ways of doing that. The detailed evaluation of environmental and instrumental noises indicates that they will not prevent, even if only marginally, the detection of interesting geophysical phenomena. Gravitational-wave instruments seem to be capable of reaching, as a by-product of their continuous operation, very ambitious geophysical goals, such as observation of the Earth's inner core oscillations. 
  In a blind search for continuous gravitational wave signals scanning a wide frequency band one looks for candidate events with significantly large values of the detection statistic. Unfortunately, a noise line in the data may also produce a moderately large detection statistic.   In this paper, we describe how we can distinguish between noise line events and actual continuous wave (CW) signals, based on the shape of the detection statistic as a function of the signal's frequency. We will analyze the case of a particular detection statistic, the F statistic, proposed by Jaranowski, Krolak, and Schutz.   We will show that for a broad-band 10 hour search, with a false dismissal rate smaller than 1e-6, our method rejects about 70 % of the large candidate events found in a typical data set from the second science run of the Hanford LIGO interferometer. 
  We consider cosmological solutions and their stability with respect to homogeneous and isotropic perturbations in the braneworld model with the scalar-curvature term in the action for the brane. Part of the results are similar to those obtained by Campos and Sopuerta for the Randall-Sundrum braneworld model. Specifically, the expanding de Sitter solution is an attractor, while the expanding Friedmann solution is a repeller. In the braneworld theory with the scalar-curvature term in the action for the brane, static solutions with matter satisfying the strong energy condition exist not only with closed spatial geometry but also with open and flat ones even in the case where the dark-radiation contribution is absent. In a certain range of parameters, static solutions are stable with respect to homogeneous and isotropic perturbations. 
  The intuitive classical space-time picture breaks down in quantum gravity, which makes a comparison and the development of semiclassical techniques quite complicated. Using ingredients of the group averaging method to solve constraints one can nevertheless introduce a classical coordinate time into the quantum theory, and use it to investigate the way a semiclassical continuous description emerges from discrete quantum evolution. Applying this technique to test effective classical equations of loop cosmology and their implications for inflation and bounces, we show that the effective semiclassical theory is in good agreement with the quantum description even at short scales. 
  We show that there is a generic transport of energy between the scalar field generated by the conformal transformation of higher order gravity theories and the matter component. We give precise relations of this exchange and show that, unless we are in a stationary spacetime, slice energy is not generically conserved. These results translate into statements about the relative behaviour of ordinary matter, dark matter and dark energy in the context of higher order gravity. 
  By using a nonholonomous-frame formulation of the general covariance principle, seen as an active version of the strong equivalence principle, an analysis of the gravitational coupling prescription in the presence of curvature and torsion is made. The coupling prescription implied by this principle is found to be always equivalent with that of general relativity, a result that reinforces the completeness of this theory, as well as the teleparallel point of view according to which torsion does not represent additional degrees of freedom for gravity, but simply an alternative way of representing the gravitational field. 
  We study backreaction analytically using the parabolic Lemaitre-Tolman-Bondi universe as a toy model. We calculate the average expansion rate and energy density on two different hypersurfaces and compare the results. We also consider the Hubble law and find that backreaction slows down the expansion if measured with proper time, but speeds it up if measured with energy density. 
  The assumption that an ensemble of classical particles is subject to nonclassical momentum fluctuations, with the fluctuation uncertainty fully determined by the position uncertainty, has been shown to lead from the classical equations of motion to the Schroedinger equation. This 'exact uncertainty' approach may be generalised to ensembles of gravitational fields, where nonclassical fluctuations are added to the field momentum densities, of a magnitude determined by the uncertainty in the metric tensor components. In this way one obtains the Wheeler-DeWitt equation of quantum gravity, with the added bonus of a uniquely specified operator ordering. No a priori assumptions are required concerning the existence of wavefunctions, Hilbert spaces, Planck's constant, linear operators, etc. Thus this approach has greater transparency than the usual canonical approach, particularly in regard to the connections between quantum and classical ensembles. Conceptual foundations and advantages are emphasised. 
  Many properties of black holes can be studied using acoustic analogues in the laboratory through the propagation of sound waves. We investigate in detail sound wave propagation in a rotating acoustic (2+1)-dimensional black hole, which corresponds to the ``draining bathtub'' fluid flow. We compute the quasinormal mode frequencies of this system and discuss late-time power-law tails. Due to the presence of an ergoregion, waves in a rotating acoustic black hole can be superradiantly amplified. We compute superradiant reflection coefficients and instability timescales for the acoustic black hole bomb, the equivalent of the Press-Teukolsky black hole bomb. Finally we discuss quasinormal modes and late-time tails in a non-rotating canonical acoustic black hole, corresponding to an incompressible, spherically symmetric (3+1)-dimensional fluid flow. 
  We consider noncommutative quantum cosmology in the case of the low-energy string effective theory. Exacts solutions are found and compared with the commutative case.The Noncommutative quantum cosmology is considered in the case of the low-energy string effective theory. Exacts solutions are found and compared with the commutative case. 
  We show that duality transformations of linearized gravity in four dimensions, i.e., rotations of the linearized Riemann tensor and its dual into each other, can be extended to the dynamical fields of the theory so as to be symmetries of the action and not just symmetries of the equations of motion. Our approach relies on the introduction of two "superpotentials", one for the spatial components of the spin-2 field and the other for their canonically conjugate momenta. These superpotentials are two-index, symmetric tensors. They can be taken to be the basic dynamical fields and appear locally in the action. They are simply rotated into each other under duality. In terms of the superpotentials, the canonical generator of duality rotations is found to have a Chern-Simons like structure, as in the Maxwell case. 
  The behavior of the action of the instantons describing vacuum decay in a de Sitter is investigated. For a near-to-limit instanton (a Coleman-de Luccia instanton close to some Hawking-Moss instanton) we find approximate formulas for the Euclidean action by expanding the scalar field and the metric of the instanton in the powers of the scalar field amplitude. The order of the magnitude of the correction to the Hawking-Moss action depends on the order of the instanton (the number of crossings of the barrier by the scalar field): for instantons of odd and even orders the correction is of the fourth and third order in the scalar field amplitude, respectively. If a near-to-limit instanton of the first order exists in a potential with the curvature at the top of the barrier greater than 4 $\times$ (Hubble constant)$^2$, which is the case if the fourth derivative of the potential at the top of the barrier is greater than some negative limit value, the action of the instanton is less than the Hawking-Moss action and, consequently, the instanton determines the outcome of the vacuum decay if no other Coleman-de Luccia instanton is admitted by the potential. A numerical study shows that for the quartic potential the physical mode of the vacuum decay is given by the Coleman-de Luccia instanton of the first order also in the region of parameters in which the potential admits two instantons of the second order. 
  We show that a recent claim that matter wave interferometers have a much higher sensitivity than laser interferometers for a comparable physical setup is unfounded. We point out where the mistake in the earlier analysis is made. We also disprove the claim that only a description based on the geodesic deviation equation can produce the correct physical result. The equations for the quantum dynamics of non-relativistic massive particles in a linearly perturbed spacetime derived here are useful for treating a wider class of related physical problems. A general discussion on the use of atom interferometers for the detection of gravitational waves is also provided. 
  We study the low-energy dynamics of an acoustic black hole near the sonic horizon. For the experimental test of black hole evaporation in the laboratory, the decay rate (greybody factor) of the acoustic black hole (sonic hole) can be calculated by the usual low-energy perturbation method. As a consequence, we obtain the decay rate of the sonic horizon from the absorption and the reflection coefficients. Moreover, we show that the thermal emission from the sonic horizon is only proportional to a control parameter which describes the velocity of the fluid. 
  Astrophysical bounds on the cosmological constant are examined for spherically symmetric bodies. Similar limits emerge from hydrostatical and gravitational equilibrium and the validity of the Newtonian limit. It is argued that the bound coming from the analysis of exact solutions is not a feature of the relativistic theory since it also emerges from the Newtonian theory.   The methods in use seem to be disjoint from the basic principles, however they have the same implication regarding the upper bounds. Therefore we will compare different inequalities and comment on the possible relationship between them. 
  We analyze the tensor mode perturbations of static, spherically symmetric solutions of the Einstein equations with a quadratic Gauss-Bonnet term in dimension $D > 4$. We show that the evolution equations for this type of perturbations can be cast in a Regge-Wheeler-Zerilli form, and obtain the exact potential for the corresponding Schr\"odinger-like stability equation. As an immediate application we prove that for $D \neq 6$ and $\alpha >0$, the sign choice for the Gauss-Bonnet coefficient suggested by string theory, all positive mass black holes of this type are stable. In the exceptional case $D =6$, we find a range of parameters where positive mass asymptotically flat black holes, with regular horizon, are unstable. This feature is found also in general for $\alpha < 0$. 
  This paper is a contribution to the development of a framework, to be used in the context of semiclassical canonical quantum gravity, in which to frame questions about the correspondence between discrete spacetime structures at "quantum scales" and continuum, classical geometries at large scales. Such a correspondence can be meaningfully established when one has a "semiclassical" state in the underlying quantum gravity theory, and the uncertainties in the correspondence arise both from quantum fluctuations in this state and from the kinematical procedure of matching a smooth geometry to a discrete one. We focus on the latter type of uncertainty, and suggest the use of statistical geometry as a way to quantify it. With a cell complex as an example of discrete structure, we discuss how to construct quantities that define a smooth geometry, and how to estimate the associated uncertainties. We also comment briefly on how to combine our results with uncertainties in the underlying quantum state, and on their use when considering phenomenological aspects of quantum gravity. 
  A `quantum inequality' (a conjectured relation between the energy density of a free quantum field and the time during which this density is observed) has recently been used to rule out some of the macroscopic wormholes and warp drives. I discuss the possibility of generalizing that result to other similar spacetimes and first show that the problem amounts to verification of a slightly different inequality. This new inequality \emph{can} replace the original one, if an additional assumption (concerning homogeneity of the `exotic matter' distribution) is made, and \emph{must} replace it if the assumption is relaxed. Then by an explicit example I show that the `new' inequality breaks down even in a simplest case (a free field in a simply connected two dimensional space). Which suggests that there is no grounds today to consider such spacetimes `unphysical'. 
  We develop further the integration procedure in the generalised invariant formalism, and demonstrate its efficiency by obtaining a class of Petrov type N pure radiation metrics without any explicit integration, and with comparatively little detailed calculations. The method is similar to the one exploited by Edgar and Vickers when deriving the general conformally flat pure radiation metric. A major addition to the technique is the introduction of non-intrinsic elements in generalised invariant formalism, which can be exploited to keep calculations manageable. 
  We consider a scalar field with a Gauss-Bonnet-type coupling to the curvature in a curved space-time. For such a quadratic coupling to the curvature, the metric energy-momentum tensor does not contain derivatives of the metric of orders greater than two. We obtain the metric energy-momentum tensor and find the geometric structure of the first three counterterms to the vacuum averages of the energy-momentum tensor for an arbitrary background metric of an N-dimensional space-time. In a homogeneous isotropic space, we obtain the first three counterterms of the n-wave procedure, which allow calculating the renormalized values of the vacuum averages of the energy-momentum tensors in the dimensions N=4,5. Using dimensional regularization, we establish that the geometric structures of the counterterms in the $n$-wave procedure coincide with those in the effective action method. 
  Captures of compact objects (COs) by massive black holes (MBHs) in galactic nuclei will be an important source for LISA, the space-based gravitational-wave (GW) detector. However, a large fraction of captures will not be individually resolvable--either because they are too distant, have unfavorable orientation, or have too many years to go before final plunge--and so will constitute a source of ``confusion noise,'' obscuring other types of sources. Here we estimate the shape and overall magnitude of the spectrum of confusion noise from CO captures. The overall magnitude depends on the capture rates, which are rather uncertain, so we present results for a plausible range of rates. We show that the impact of capture confusion noise on the total LISA noise curve ranges from insignificant to modest, depending on these rates. Capture rates at the high end of estimated ranges would raise LISA's overall (effective) noise level by at most a factor \sim 2. While this slightly elevated noise level would somewhat decrease LISA's sensitivity to other classes of sources, overall, this would be a pleasant problem for LISA to have: It would also imply that detection rates for CO captures were at nearly their maximum possible levels (given LISA's baseline design). This paper includes several other results that should be useful in further studies of LISA capture sources, including (i) a calculation of the total GW energy output from generic inspirals into Kerr MBHs, and (ii) an approximate GW energy spectrum for a typical capture. 
  We study conformal gravity in d=2+1, where the Cotton tensor is equated to a--necessarily traceless--matter stress tensor, for us that of the improved scalar field. We first solve this system exactly in the $pp$ wave regime, then show it to be equivalent to topologically massive gravity. 
  We fully determine a uniformly valid asymptotic behaviour for large $a \omega$ and fixed $m$ of the angular solutions and eigenvalues of the spin-weighted spheroidal differential equation. We fully complement the analytic work with a numerical study. 
  The analytic forms of the asymptotic quasinormal frequencies of a coupled scalar field in the Garfinkle-Horowitz-Strominger dilaton spacetime is investigated by using the monodromy technique proposed by Motl and Neitzke. It is found that the asymptotic quasinormal frequencies depend not only on the structure parameters of the background spacetime, but also on the coupling between the scalar fields and gravitational field. Moreover, our results show that only in the minimal couple case, i.e., $\xi$ tends zero, the real parts of the asymptotic quasinormal frequencies agrees with the Hod's conjecture, $T_H\ln{3}$. 
  The spacetime geometry on the equatorial slice through a Kerr black hole is formally equivalent to the geometry felt by phonons entrained in a rotating fluid vortex. We analyse this situation in some detail: First, we find the most general ``acoustic geometry'' compatible with the fluid dynamic equations in a collapsing/expanding perfect-fluid line vortex. Second, we demonstrate that there is a suitable choice of coordinates on the equatorial slice through a Kerr black hole that puts it into this vortex form; though it is not possible to put the entire Kerr spacetime into perfect-fluid ``acoustic'' form. Finally, we briefly discuss the implications of this formal equivalence; both with respect to gaining insight into the Kerr spacetime and with respect to possible vortex-inspired experiments. 
  We re-examine the brick-wall model in the context of spacetime foam. In particular we consider a foam composed by wormholes of different sizes filling the black hole horizon. The contribution of such wormholes is computed via a scale invariant distribution. We obtain that the brick wall divergence appears to be logarithmic when the cutoff is sent to zero. 
  We regard the Wheeler-De Witt equation as a Sturm-Liouville problem with the cosmological constant considered as the associated eigenvalue. The used method to study such a problem is a variational approach with Gaussian trial wave functionals. We approximate the equation to one loop in a Schwarzschild background. A zeta function regularization is involved to handle with divergences. The regularization is closely related to the subtraction procedure appearing in the computation of Casimir energy in a curved background. A renormalization procedure is introduced to remove the infinities together with a renormalization group equation. 
  Fermi coordinates are the natural generalization of inertial Cartesian coordinates to accelerated systems and gravitational fields. We study the motion of ultrarelativistic particles and light rays in Fermi coordinates and investigate inertial and tidal effects beyond the critical speed c/sqrt(2). In particular, we discuss the black-hole tidal acceleration mechanism for ultrarelativistic particles in connection with a possible origin for high-energy cosmic rays. 
  We match an interior solution of a spherically symmetric traversable wormhole to a unique exterior vacuum solution, with a generic cosmological constant, at a junction interface, and the surface stresses on the thin shell are deduced. In the spirit of minimizing the usage of exotic matter we determine regions in which the weak and null energy conditions are satisfied on the junction surface. The characteristics and several physical properties of the surface stresses are explored, namely, regions where the sign of the tangential surface pressure is positive and negative (surface tension) are determined. This is done by expressing the tangential surface pressure as a function of several parameters, namely, that of the matching radius, the redshift parameter, the surface energy density and of the generic cosmological constant. An equation governing the behavior of the radial pressure across the junction surface is also deduced. 
  Exhaustive ghost solutions to Einstein-Weyl equations for two dimensional spacetimes are obtained, where the ghost neutrinos propagate in the background spacetime, but do not influence the background spacetime due to the vanishing stress-energy-momentum tensor for the ghost neutrinos. Especially, those non-trivial ghost solutions provide a counterexample to the traditional claim that the Einstein-Hilbert action has no meaningful two dimensional analogue. 
  We shall present here the causal interpretation of canonical quantum gravity in terms of new variables. Then we shall apply it to the minisuperspace of cosmology. A vacuum solution of quantum cosmology is obtained and the Bohmian trajectory is investigated. At the end a coherent state with matter is considered in the cosmological model. 
  The teleparallel coframe gravity may be viewed as a generalization of the standard GR. A coframe (a field of four independent 1-forms) is considered, in this approach, to be a basic dynamical variable. The metric tensor is treated as a secondary structure. The general Lagrangian, quadratic in the first order derivatives of the coframe field is not unique. It involves three dimensionless free parameters. We consider a weak field approximation of the general coframe teleparallel model. In the linear approximation, the field variable, the coframe, is covariantly reduced to the superposition of the symmetric and antisymmetric field. We require this reduction to be preserved on the levels of the Lagrangian, of the field equations and of the conserved currents. This occurs if and only if the pure Yang - Mills type term is removed from the Lagrangian. The absence of this term is known to be necessary and sufficient for the existence of the viable (Schwarzschild) spherical-symmetric solution. Moreover, the same condition guarantees the absence of ghosts and tachyons in particle content of the theory. The condition above is shown recently to be necessary for a well defined Hamiltonian formulation of the model. Here we derive the same condition in the Lagrangian formulation by means of the weak field reduction. 
  We show that globally and regularly hyperbolic future geodesically incomplete isotropic universes, except for the standard all-encompassing `big crunch', can accommodate singularities of only one kind, namely, those having a non-integrable Hubble parameter, $H$. We analyze several examples from recent literature which illustrate this result and show that such behaviour may arise in a number of different ways. We also discuss the existence of new types of lapse singularities in inhomogeneous models, impossible to meet in the isotropic ones. 
  In the braneworld scenario, the four dimensional effective Einstein equation has extra terms which arise from the embedding of the 3-brane in the bulk. We show that in this modified theory of gravity, it is possible to model observations of galaxy rotation curves and the X-ray profiles of clusters of galaxies, without the need for dark matter. In this scenario, a traceless tensor field which arises from the projection of the bulk Weyl tensor on the brane, provides the extra gravitational acceleration which is usually explained through dark matter. We also predict that gravitational lensing observations can possibly discriminate between the proposed higher dimensional effects and dark matter, the deflection angles predicted in the proposed scenario being around 75% to 80% of the usual predictions based on dark matter. 
  A new exactly solvable model for the evolution of relativistic kinetic system interacting with an internal stochastic reservoir under the influence of a gravitational background expansion is established. This model of self-interaction is based on the relativistic kinetic equation for the distribution function defined in the extended phase space. The supplementary degree of freedom is described by the scalar stochastic variable (Langevin source), which is considered to be the constructive element of the effective one-particle force. The expansion of the Universe is shown to be accelerated for the suitable choice of the non-minimal self-interaction force. 
  An electromagnetic analog of the Kerr-Newman solution in general relativity is derived, based on Minkowski's formulation for electromagnetic fields in moving media. The equivalent system is a distribution of charges and currents largely localized within a spinning disk of radius a. This occurs in a rotating medium of inhomogeneous index of refraction and "frame dragging" at exactly half the angular velocity of the electrical charge. 
  The axes of gyroscopes experimentally define local non-rotating frames, i.e. the time-evolution of axes of inertial frames. But what physical cause governs the time-evolution of gyroscope axes? Starting from an unperturbed FRW cosmology with k=0 we consider linear cosmological vorticity perturbations and ask: Will cosmological vorticity perturbations exactly drag the axes of gyroscopes relative to the directions of geodesics to galaxies in the asymptotic FRW space? Using Cartan's formalism with local orthonormal bases we cast the laws of gravitomagnetism into a form showing the close correspondence with the laws of ordinary magnetism. Our results, valid for any equation of state, are: 1) The dragging of a gyroscope axis by rotational perturbations beyond the H-dot radius (H=Hubble constant) is exponentially suppressed. 2) If the perturbation is a homogeneous rotation inside a perturbation radius, then exact dragging of the gyroscope axis by the rotational perturbation is reached exponentially fast, as the perturbation radius gets larger than the H-dot radius. 3) The time-evolution of a gyroscope axis exactly follows a specific average of the matter inside the H-dot radius. In this sense Mach's Principle is a consequence of cosmology with Einstein gravity. 
  In many numerical implementations of the Cauchy formulation of Einstein's field equations one encounters artificial boundaries which raises the issue of specifying boundary conditions. Such conditions have to be chosen carefully. In particular, they should be compatible with the constraints, yield a well posed initial-boundary value formulation and incorporate some physically desirable properties like, for instance, minimizing reflections of gravitational radiation.   Motivated by the problem in General Relativity, we analyze a model problem, consisting of a formulation of Maxwell's equations on a spatially compact region of spacetime with timelike boundaries. The form in which the equations are written is such that their structure is very similar to the Einstein-Christoffel symmetric hyperbolic formulations of Einstein's field equations. For this model problem, we specify a family of Sommerfeld-type constraint-preserving boundary conditions and show that the resulting initial-boundary value formulations are well posed. We expect that these results can be generalized to the Einstein-Christoffel formulations of General Relativity, at least in the case of linearizations about a stationary background. 
  We show how to prescribe the initial data of a characteristic problem satisfying the costraints, the smallness, the regularity and the asymptotic decay suitable to prove a global existence result. In this paper, the first of two, we show in detail the construction of the initial data and give a sketch of the existence result. This proof, which mimicks the analogous one for the non characteristic problem in [Kl-Ni], will be the content of a subsequent paper. 
  General Relativity with nonvanishing torsion has been investigated in the first order formalism of Poincare gauge field theory. In the presence of torsion, either side of the Einstein equation has the nonvanishing covariant divergence. This fact turned out to be self-consistent in the framework under consideration. By using Noether's procedure with the definition of the Lie derivative where the general coordinate transformation and the local Lorentz rotation are combined, the revised covariant divergence of the energy momentum is consistently obtained. Subsequently we have definitely derived the spin correction to the energy momentum tensor for the Dirac field and the Rarita-Schwinger field in the Einstein equation. We conjecture that the accelerated expansion of the universe possibly arises due to the spin correction in our framework. 
  Static and spherically symmetric perfect fluid solutions of Einstein's field equations with cosmological constant are analysed. After showing existence and uniqueness of a regular solution at the centre the extension of this solution is discussed. Then the existence of global solutions with given equation of state and cosmological constant bounded by 4 pi rho_b, where rho_b is the boundary density (given by the equation of state) of the perfect fluid ball, is proved. 
  An equidistant spectrum of the horizon area of a quantized black hole does not follow from the correspondence principle or from general statistical arguments. Such a spectrum obtained earlier in loop quantum gravity (LQG) does not comply with the holographic bound. This bound fixes the Barbero-Immirzi parameter of LQG, and thus leads to the unique result for the spectrum of horizon area. 
  We describe conformally flat initial data, with explicitly given analytic extrinsic curvature solving the vacuum momentum constraints. They follow from a solution of Dain and Friedrich discovered in 2001. The cylindrically symmetric subcase of the Bowen-York solution is a subclass of this general configuration. 
  A partially alternative derivation of the expression for the time dilation effect in a uniform static gravitational field is obtained by means of a thought experiment in which rates of clocks at rest at different heights are compared using as reference a clock bound to a free falling reference system (FFRS). Derivations along these lines have already been proposed, but generally introducing some shortcut in order to make the presentation elementary. The treatment is here exact: the clocks whose rates one wishes to compare are let to describe their world lines (Rindler's hyperbolae) with respect to the FFRS, and the result is obtained by comparing their lengths in space-time. Only at the end of the paper the corresponding GR metric is derived, to the purpose of making a comparison to the solutions of Einstein field equation. The exercise may nonetheless prove pedagogically instructive insofar as it shows that the exact result of General Relativity (GR) can be obtained in terms of physical and geometrical reasoning without having recourse to the general formalism. It also compels to deal with a few subtle points inherent in the very foundations of GR. 
  Equal-arm interferometric detectors of gravitational radiation allow phase measurements many orders of magnitude below the intrinsic phase stability of the laser injecting light into their arms. This is because the noise in the laser light is common to both arms, experiencing exactly the same delay, and thus cancels when it is differenced at the photo detector. In this situation, much lower level secondary noises then set overall performance. If, however, the two arms have different lengths (as will necessarily be the case with space-borne interferometers), the laser noise experiences different delays in the two arms and will hence not directly cancel at the detector. In order to solve this problem, a technique involving heterodyne interferometry with unequal arm lengths and independent phase-difference readouts has been proposed. It relies on properly time-shifting and linearly combining independent Doppler measurements, and for this reason it has been called Time-Delay Interferometry (or TDI). This article provides an overview of the theory and mathematical foundations of TDI as it will be implemented by the forthcoming space-based interferometers such as the Laser Interferometer Space Antenna (LISA) mission. We have purposely left out from this first version of our ``Living Review'' article on TDI all the results of more practical and experimental nature, as well as all the aspects of TDI that the data analysts will need to account for when analyzing the LISA TDI data combinations. Our forthcoming ``second edition'' of this review paper will include these topics. 
  de-Broglie--Bohm causal interpretation of canonical quantum gravity in terms of Ashtekar new variables is built. The Poisson brackets of (deBroglie--Bohm) constraints are derived and it is shown that the Poisson bracket of Hamiltonian with itself would change with respect to its classical counterpart. 
  We linearize extended ADM-gravity around the flat torus, and use the associated Fock vacuum to construct a state that could play the role of a free vacuum in loop quantum gravity. The state we obtain is an element of the gauge-invariant kinematic Hilbert space and restricted to a cutoff graph, as a natural consequence of the momentum cutoff of the original Fock state. It has the form of a Gaussian superposition of spin networks. We show that the peak of the Gaussian lies at weave-like states and derive a relation between the coloring of the weaves and the cutoff scale. Our analysis indicates that the peak weaves become independent of the cutoff length when the latter is much smaller than the Planck length. By the same method, we also construct multiple-graviton states. We discuss the possible use of these states for deriving a perturbation series in loop quantum gravity. 
  Previously we found that large amplitude $r$-modes could decay catastrophically due to nonlinear hydrodynamic effects. In this paper we found the particular coupling mechanism responsible for this catastrophic decay, and identified the fluid modes involved. We find that for a neutron star described by a polytropic equation of state with polytropic index $\Gamma=2$, the coupling strength of the particular three-mode interaction causing the decay is strong enough that the usual picture of the $r$-mode instability with a flow pattern dominated by that of an $r$-mode can only be valid for the dimensionless $r$-mode amplitude less than $O(10^{-2})$. 
  We study various configurations in which a domain wall (or cosmic string), described by the Nambu-Goto action, is embedded in a background space-time of a black hole in $(3+1)$ and higher dimensional models. We calculate energy fluxes through the black hole horizon. In the simplest case, when a static domain wall enters the horizon of a static black hole perperdicularly, the energy flux is zero. In more complicated situations, where parameters which describe the domain wall surface are time and position dependent, the flux is non-vanishing is principle. These results are of importance in various conventional cosmological models which accommodate the existence of domain walls and strings and also in brane world scenarios. 
  In three spacetime dimensions, general relativity drastically simplifies, becoming a ``topological'' theory with no propagating local degrees of freedom. Nevertheless, many of the difficult conceptual problems of quantizing gravity are still present. In this review, I summarize the rather large body of work that has gone towards quantizing (2+1)-dimensional vacuum gravity in the setting of a spatially closed universe. 
  In the framework of string-inspired dilatonic gravity theories (from 4 to $D$ space-time dimensions), we construct new non-asymptotically flat black hole or black brane solutions. For particular values of the dilatonic coupling constant, we generalize static solutions to rotating ones, using the target space isometry group. We compute their masses and their angular momentum using the modern approach to the computation of energy in General Relativity, the quasilocal formalism, and we check the agreement of these solutions with the first law of black hole thermodynamics. Finally, we study a new black hole family in the 2+1 dimensional theory of Topologically Massive Gravity. 
  Using the Newman-Penrose formalism, we obtain the explicit expressions for the polarization modes of weak, plane gravitational waves with a massive graviton. Our analysis is restricted for a specific bimetric theory whose term of mass, for the graviton, appears as an effective extra contribution to the stress-energy tensor. We obtain for such kind of theory that the extra states of polarization have amplitude several orders of magnitude smaller than the polarizations purely general relativity (GR), h_(+) and h_(x), in the VIRGO-LIGO frequency band. This result appears using the best limit to the graviton mass inferred from solar system observations and if we consider that all the components of the metric perturbation have the same amplitude h. However, if we consider low frequency gravitational waves (e.g., f_(GW) ~ 10^(-7) Hz), the extra polarization states produce similar Newman-Penrose amplitudes that the polarization states purely GR. This particular characteristic of the bimetric theory studied here could be used, for example, to directly impose limits on the mass of the graviton from future experiments that study the cosmic microwave background (CMB). 
  The gravitational collapse of cylindrically distributed perfect fluid is studied. We assume the collapsing speed of fluid is very large and investigate such a situation by recently proposed high-speed approximation scheme. We show that if the value of the pressure divided by the energy density is bounded below by some positive value, the high-speed collapse is necessarily halted. This suggests that the collapsing perfect fluid of realistic ideal gas experiences the pressure bounce. However even in the case of mono-atomic ideal gas, arbitrarily large tidal force for freely falling observers are realizable by setting the initial collapsing velocity exceedingly large. In order that the high-speed collapse of cylindrical perfect fluid forms spacetime singularity, the equation of state should be very soft. 
  In a recent preprint, Krasnikov has claimed that to show that quantum energy inequalities (QEIs) are violated in curved spacetime situations, by considering the example of a free massless scalar field in two-dimensional de Sitter space. We show that this claim is incorrect, and based on misunderstandings of the nature of QEIs. We also prove, in general two-dimensional spacetimes, that flat spacetime QEIs give a good approximation to the curved spacetime results on sampling timescales short in comparison with natural geometric scales. 
  The inflaton is highly likely to settle in a squeezed vacuum state after inflation. The relic inflaton after inflation and reheating undergoes a damped oscillatory motion and contributes to the effective cosmological constant. We interpret the renormalized energy density from the squeezed vacuum state as an effective cosmological constant. Using the recent observational data on the cosmological constant, we find the constraint on the squeeze parameter of the inflaton in the early universe. 
  We reconsider the unified model of gravitation and Yang--Mills interactions proposed by Chakraborty and Peld\'an, in the light of recent formal developments in loop quantum gravity. In particular, we show that one can promote the Hamiltonian constraint of the unified model to a well defined anomaly-free quantum operator using the techniques introduced by Thiemann, at least for the Euclidean theory. The Lorentzian version of the model can be consistently constructed, but at the moment appears to yield a correct weak field theory only under restrictive assumptions, and its quantization appears problematic. 
  Contents:  Community news:   Message from the Chair, by Jim Isenberg   We hear that..., by Jorge Pullin   THE TGG WYP Speakers Program, by Richard Price  Research Briefs:   Gravity Probe B is launched, by Bill Hamilton   Questions and progress in mathematical general relativity, by Jim Isenberg   Summary of recent preliminary LIGO results, by Alan Wiseman for the LSC   100 Years ago, by Jorge Pullin  Conference reports:   Einstein 125, by Abhay Ashtekar   The 7th Eastern Gravity Meeting, by Deirdre Shoemaker   2004 Aspen GWADW, by Syd Meshkov   Fifth LISA Symposium, by Curt Cutler   GR17, by Brien Nolan   Loops and Spinfoams, by Carlo Rovelli   20th Pacific coast gravity meeting, by Michele Vallisneri 
  We establish an optimal gluing construction for general relativistic initial data sets. The construction is optimal in two distinct ways. First, it applies to generic initial data sets and the required (generically satisfied) hypotheses are geometrically and physically natural. Secondly, the construction is completely local in the sense that the initial data is left unaltered on the complement of arbitrarily small neighborhoods of the points about which the gluing takes place. Using this construction we establish the existence of cosmological, maximal globally hyperbolic, vacuum space-times with no constant mean curvature spacelike Cauchy surfaces. 
  R-modes of a rotating neutron star are unstable because of the emission of gravitational radiation. We explore the saturation amplitudes of these modes determined by nonlinear mode-mode coupling. Modelling the star as incompressible allows the analytic computation of the coupling coefficients. All couplings up to n=30 are obtained, and analytic values for the shear damping and mode normalization are presented. In a subsequent paper we perform numerical simulations of a large set of coupled modes. 
  This is the first of series of papers in which we investigate stability of the spherically symmetric space-time with de Sitter center. Geometry, asymptotically Schwarzschild for large $r$ and asymptotically de Sitter as $r\to 0$, describes a vacuum nonsingular black hole for $m\geq m_{cr}$ and particle-like self-gravitating structure for $m < m_{cr}$ where a critical value $m_{cr}$ depends on the scale of the symmetry restoration to de Sitter group in the origin. In this paper we address the question of stability of a vacuum non-singular black hole with de Sitter center to external perturbations. We specify first two types of geometries with and without changes of topology. Then we derive the general equations for an arbitrary density profile and show that in the whole range of the mass parameter $m$ objects described by geometries with de Sitter center remain stable under axial perturbations. In the case of the polar perturbations we find criteria of stability and study in detail the case of the density profile $\rho(r)=\rho_0 e^{-r^3/r_0^2 r_g}$ where $\rho_0$ is the density of de Sitter vacuum at the center, $r_0$ is de Sitter radius and $r_g$ is the Schwarzschild radius. 
  Thermoelastic noise will be the most significant noise source in advanced-LIGO interferometers with sapphire test masses. The standard plan for advanced-LIGO has optimized the optics, within the framework of conventional mirrors, to reduce thermoelastic noise. Recently, we and our collaborators have proposed going beyond the bounds of traditional optics to increase the effective beam spot size and thus lower thermoelastic noise. One particular proposal for mirror shapes (``Mexican-hat mirrors'') yields the class of ``mesa'' beams.   In this paper, we outline a general procedure for analyzing light propagating in individual arm cavities, and the associated thermoelastic noise, in the presence of arbitrary optics. We apply these procedures to study the Mexican-hat proposal. Results obtained by the techniques of this paper were presented elsewhere, to demonstrate that the Mexican-hat proposal for advanced-LIGO both significantly lowers thermoelastic noise and does not significantly complicate the function of the interferometer. 
  We present a parametrization of $T^3$ and $S^1\times S^2$ Gowdy cosmological models which allows us to study both types of topologies simultaneously. We show that there exists a coordinate system in which the general solution of the linear polarized special case (with both topologies) has exactly the same functional dependence. This unified parametrization is used to investigate the existence of Cauchy horizons at the cosmological singularities, leading to a violation of the strong cosmic censorship conjecture. Our results indicate that the only acausal spacetimes are described by the Kantowski-Sachs and the Kerr-Gowdy metrics. 
  The construction of a radar coordinate system about the world line of an observer is discussed. Radar coordinates for a hyperbolic observer as well as a uniformly rotating observer are described in detail. The utility of the notion of radar distance and the admissibility of radar coordinates are investigated. Our results provide a critical assessment of the physical significance of radar coordinates. 
  The present article lies at the interface between gravity, a highly nonlinear phenomenon, and quantum field theory. The nonlinear field equations of Einstein permit the theoretical existence of classical wormholes. One of the fundamental questions relates to the practical viability of such wormholes. One way to answer this question is to assess if the total \emph{volume} of exotic matter needed to maintain the wormhole is finite. Using this value as the lower bound, we propose a modified semiclassical volume Averaged Null Energy Condition (ANEC) constraint as a method of discarding many solutions as being possible self-consistent wormhole solutions of semiclassical gravity. The proposed constraint is consistent with known results. It turns out that a class of Morris-Thorne wormholes can be ruled out on the basis of this constraint. 
  The notion of particle plays an essential role in quantum field theory (QFT). Some recent theoretical developments, however, have questioned this notion; for instance, QFT on curved spacetimes suggests that preferred particle states might not exist in general. Furthermore, a certain tension derives from the fact that QFT's particle states are intrinsically nonlocal, while experiments are localized. These considerations have lead some to suggest that in general QFT should not be interpreted in terms of particle states, but rather, say, in terms of eigenstates of local operators. On the other hand, it is not completely obvious how to reconcile this view with the empirically-observed ubiquitous particle-like behavior of quantum fields. We observe here that already in flat space there exist --strictly speaking-- two distinct notions of particles: globally defined n-particle Fock states and local particle states. The last describe the physical objects detected by the real finite-size particle detectors and are eigenstates of local field operators. In the limit in which the particle detectors are large compared, say, to the Compton wavelength, global and local particle states converge in a weak topology defined by physical measurements (but not in norm). This observation reconciles the two point of view and provides a local definition of particle state that remains well-defined even when the conventional global particle states are not defined. This definition could play an important role in quantum gravity, when asymptotic regions may not be available. 
  ``Eternal inflation'' is often confused with ``chaotic inflation''. Moreover, the term ``chaotic inflation'' is being used in three different meanings (all of them unrelated to eternal inflation). I make some suggestions in an effort to untangle this terminological mess. I also give a brief review of the origins of eternal inflation. 
  In Klein geometric model of space the mass is manifestation of the quantized charges oscillations in additional compactified dimension. We analyze model in which common in four-dimensional space-time for mass and electric charge of the particle trajectory is disintegrated in five dimensions on movement of the mass along null geodesic line and trajectory of the charge corresponding to the time-like interval in 5D volume. We find relation between five-velocity vector of electric charge and mass. This scheme is regarded to have concern with many worlds theory. Considered approach is applied to the model of rotating space having four-dimensional spherical symmetry. One proposed appearance additional force in included 4D space-time, which may be explanation of the Pioneer-effect. We analyze also possible part of this force in conservation of the substance in Galaxy area. 
  We apply the ``consistent discretization'' approach to general relativity leaving the spatial slices continuous. The resulting theory is free of the diffeomorphism and Hamiltonian constraints, but one can impose the diffeomorphism constraint to reduce its space of solutions and the constraint is preserved exactly under the discrete evolution. One ends up with a theory that has as physical space what is usually considered the kinematical space of loop quantum geometry, given by diffeomorphism invariant spin networks endowed with appropriate rigorously defined diffeomorphism invariant measures and inner products. The dynamics can be implemented as a unitary transformation and the problem of time explicitly solved or at least reduced to as a numerical problem. We exhibit the technique explicitly in 2+1 dimensional gravity. 
  We consider gravity from the quantum field theory point of view and introduce a natural way of coupling gravity to matter by following the gauge principle for particle interactions. The energy-momentum tensor for the matter fields is shown to be conserved and follows as a consequence of the dynamics in a spontaneously broken SO(3,2) gauge theory of gravity. All known interactions are described by the gauge principle at the microscopic level. 
  Let the reciprocal Newton 'constant' be an apparently non-dynamical Brans-Dicke scalar field damped oscillating towards its General Relativistic VEV. We show, without introducing additional matter fields or dust, that the corresponding cosmological evolution averagely resembles, in the Jordan frame, the familiar dark radiation -> dark matter -> dark energy domination sequence. The fingerprints of our theory are fine ripples, hopefully testable, in the FRW scale factor; they die away at the General Relativity limit. The possibility that the Brans-Dicke scalar also serves as the inflaton is favorably examined. 
  It is shown that superthin and superlong gravitational flux tube solutions in the 5D Kaluza-Klein gravity have the region $(\approx l_{Pl})$ where the metric signature changes from $\{+,-,-,-,- \}$ to $\{-,-,-,-,+ \}$. Such change is too quickly from one of the paradigms of quantum gravity which tells that the Planck length is the minimal length in the nature and consequently the physical quantities can not change very quickly in the course of this length. For avoiding such dynamic it is supposed that a pure quantum freezing of the dynamic of the $5^{th}$ dimension takes place. As the continuation of the flux tube metric in the longitudinal direction the Reissner-Nordstr\"om metric is proposed. In the consequence of such construction one can avoid the appearance of a point-like singularity in the extremal Reissner-Nordstr\"om solution. 
  These notes are a didactic overview of the non perturbative and background independent approach to a quantum theory of gravity known as loop quantum gravity. The definition of real connection variables for general relativity, used as a starting point in the program, is described in a simple manner. The main ideas leading to the definition of the quantum theory are naturally introduced and the basic mathematics involved is described. The main predictions of the theory such as the discovery of Planck scale discreteness of geometry and the computation of black hole entropy are reviewed. The quantization and solution of the constraints is explained by drawing analogies with simpler systems. Difficulties associated with the quantization of the scalar constraint are discussed.In a second part of the notes, the basic ideas behind the spin foam approach are presented in detail for the simple solvable case of 2+1 gravity. Some results and ideas for four dimensional spin foams are reviewed. 
  We present a general form for the solution of an expanding general-relativistic Friedmann universe that encounters a singularity at finite future time. The singularity occurs in the material pressure and acceleration whilst the scale factor, expansion rate and material density remain finite and the strong energy condition holds. We also show that the same phenomenon occurs, but under different conditions, for Friedmann universes in gravity theories arising from the variation of an action that is an arbitrary analytic function of the scalar curvature. 
  We continue our program of unifying general relativity and quantum mechanics in terms of a noncommutative algebra ${\cal A}$ on a transformation groupoid $\Gamma = E \times G$ where $E$ is the total space of a principal fibre bundle over spacetime, and $G$ a suitable group acting on $\Gamma $. We show that every $a \in {\cal A}$ defines a random operator, and we study the dynamics of such operators. In the noncommutative regime, there is no usual time but, on the strength of the Tomita-Takesaki theorem, there exists a one-parameter group of automorphisms of the algebra ${\cal A}$ which can be used to define a state dependent dynamics; i.e., the pair $({\cal A}, \phi)$, where $\phi $ is a state on ${\cal A}$, is a ``dynamic object''. Only if certain additional conditions are satisfied, the Connes-Nikodym-Radon theorem can be applied and the dependence on $\phi $ disappears. In these cases, the usual unitary quantum mechanical evolution is recovered. We also notice that the same pair $({\cal A}, \phi)$ defines the so-called free probability calculus, as developed by Voiculescu and others, with the state $\phi $ playing the role of the noncommutative probability measure. This shows that in the noncommutative regime dynamics and probability are unified. This also explains probabilistic properties of the usual quantum mechanics. 
  We apply a covariant and generic procedure to obtain explicit expressions of the transverse frames that a type I spacetime admits in terms of an arbitrary initial frame. We also present a simple and general algorithm to obtain the Weyl scalars $\Psi_2^T$, $\Psi_0^T$ and $\Psi_4^T$ associated with these transverse frames. In both cases it is only necessary to choose a particular root of a cubic expression. 
  We present a fourth order convergent (2+1) numerical code to solve the Teukolsky equation in the time domain. Our approach is to rewrite the Teukolsky equation as a system of first order differential equations. In this way we get a system that has the form of an advection equation. This is used in combination with a series expansion of the solution in powers of time. To obtain a fourth order scheme we kept terms up to fourth derivative in time and use the advection-like system of differential equations to substitute the temporal derivatives by spatial derivatives. A local stability study leads to a Courant factor of 1.5 for the nonrotating case. This scheme is used to evolve gravitational perturbations in Schwarzschild and Kerr backgrounds. Our numerical method proved to be fourth order convergent in r* and theta directions. The correct power-law tail, ~1/t^{2\ell+3}, for general initial data, and ~1/t^{2\ell+4}, for time symmetric data, was found in the simulations where the duration in time of the tail phase was long enough. We verified that it is crucial to resolve accurately the angular dependence of the mode at late times in order to obtain these values of the exponents in the power-law decay. In other cases, when the decay was too fast and round-off error was reached before a tail was developed, the quasinormal modes frequencies provided a test to determine the validity of our code. 
  We present an alternative way of tracing the existence of a scalar field based on the analysis of the gravitational wave spectrum of a vibrating neutron star. Scalar-tensor theories in strong-field gravity can potentially introduce much greater differences in the parameters of a neutron star than the uncertainties introduced by the various equations of state. The detection of gravitational waves from neutron stars can set constraints on the existence and the strength of scalar fields. We show that the oscillation spectrum is dramatically affected by the presence of a scalar field, and can provide unique confirmation of its existence. 
  The metric space-time is revised as a priori existing. It is substituted by the world continuum endowed only with the affine connection. The metric, accompanied by the tensor Goldstone boson, is to emerge during the spontaneous breaking of the global affine symmetry. Implications for gravity and the Universe are indicated. 
  In recent papers (astro-ph/0306630, gr-qc/0312041) I have argued that the observed cosmological acceleration can be accounted for by the inclusion of a 1/R term in the gravitational action in the Palatini formalism. Subsequently, Flanagan (astro-ph/0308111, gr-qc/0403063) argued that this theory is equivalent to a scalar-tensor theory which produces corrections to the standard model that are ruled out experimentally.   In this article I examine the Dirac field coupled to 1/R gravity. The Dirac action contains the connection which was taken to be the Christoffel symbol, not an independent quantity, in the papers by Flanagan. Since the metric and connection are taken to be independent in the Palatini approach it is natural to allow the connection that appears in the Dirac action to be an independent quantity. This is the approach that is taken in this paper. The resulting theory is very different and much more complicated than the one discussed in Flanagan's papers. 
  We argue that the model of a quantum computer with N qubits on a quantum space background, which is a fuzzy sphere with n=2^N elementary cells, can be viewed as the minimal model for Quantum Gravity. In fact, it is discrete, has no free parameters, is Lorentz invariant, naturally realizes the Holographic Principle, and defines a subset of punctures of spin networks' edges of Loop Quantum Gravity labelled by spins j=2^(N-1)-1/2. In this model, the discrete area spectrum of the cells, which is not equally spaced, is given in units of the minimal area of Loop Quantum Gravity (for j=1/2), and provides a discrete emission spectrum for quantum black holes. When the black hole emits one string of N bits encoded in one of the n cells, its horizon area decreases of an amount equal to the area of one cell. 
  The motion of a classical test particle moving on a 4-dimensional brane embedded in an $n$-dimensional bulk is studied in which the brane is allowed to fluctuate along the extra dimensions. It is shown that these fluctuations produce three different forces acting on the particle, all stemming from the effects of extra dimensions. Interpretations are offered to describe the origin of these forces and a relationship between the 4 and $n$-dimensional mass of the particle is obtained by introducing charges associated with large extra dimensions 
  The coframe field model is known as a viable model for gravity. The principle problem is an interpretation of six additionaldegrees of freedom. We construct a general family of connections which includes the connections of Levi-Civita and Weitzenb\"{o}ck as the limiting cases. We show that for a special choice of parameters, a subfamily of connections is invariant when the infinitesimal field of transformations (antisymmetric tensor) satisfies the pair of vacuum Maxwell equations -- one for torsion and one for non-metricity. Moreover, the vacuum Maxwell equations turn to be the necessary and sufficient conditions for invariance of the viable coframe action (alternative to GR). Consequently, for the viable models, the coframe field is proved to have the Maxwell-type behavior in addition to the known gravity sector. 
  We consider premetric electrodynamics with a local and linear constitutive law for the vacuum. Within this framework, we find quartic Fresnel wave surfaces for the propagation of light. If we require vanishing birefringence in vacuum, then a Riemannian light cone is implied. No proper Finslerian structure can occur. This is generalized to dynamical equations of any order. 
  This preliminary report proposes integrating the Maxwell equations in Minkowski spacetime using coordinates where the spacelike surfaces are hyperboloids asymptotic to null cones at spatial infinity. The space coordinates are chosen so that Scri+ occurs at a finite coordinate and a smooth extension beyond Scri+ is obtained. The question addressed is whether a Cauchy evolution numerical integration program can be easily modified to compute this evolution.   In the spirit of the von Neumann and Richtmyer artificial viscosity which thickens a shock by many orders of magnitude to facilitate numerical simulation, I propose artificial cosmology to thicken null infinity Scri+ to approximate it by a de Sitter cosmological horizon where, in conformally compactified presentation, it provides a shell of purely outgoing null cones where asymptotic waves can be read off as data on a spacelike pure outflow outer boundary. This should be simpler than finding Scri+ as an isolated null boundary or imposing outgoing wave conditions at a timelike boundary at finite radius. 
  It is demonstrated that dark energy, driven by tachyons having non-minimal coupling with curvature and self-interacting inverse cubic potential, decays to cold dark matter in the late universe. It is found that this penomenon yields a solution to `` cosmic coincidence problem''. 
  In the baseline design for advanced LIGO interferometers, the most serious noise source is tiny, dynamically fluctuating bumps and valleys on the faces of the arm-cavity mirrors, caused by random flow of heat in the mirrors' sapphire substrates: so-called "thermoelastic noise". We propose replacing the interferometers' baseline arm-cavity light beams, which have Gaussian-shaped intensity profiles, by beams with mesa-shaped profiles that are flat in their central ~7 cm of radius, and that then fall toward zero as quickly as is allowed by diffraction in LIGO's 4 km arms. The mesa beams average the bumps and valleys much more effectively than the Gaussian beams. As a result, if the beam radii are adjusted so diffraction losses per bounce are about 10 ppm, replacing Gaussian beams by mesa beams reduces the thermoelastic noise power by about a factor 3. If other thermal noises are kept negligible, this will permit advanced LIGO to beat the Standard Quantum Limit by about a factor 1.5 over a bandwidth about equal to frequency, and the event rate for inspiraling neutron star binaries will increase by about a factor 2.5. The desired mesa beams can be produced from input, Gaussian-profile laser light, by changing the shapes of the arm cavities' mirror faces from their baseline spherical shapes (with radii of curvature of order 60 km) to Mexican-hat (MH) shapes that have a shallow bump in the center but are otherwise much flatter in the central 10 cm than the spherical mirrors, and then flare upward strongly in the outer 6 cm, like a sombrero. We describe mesa beams and MH mirrors mathematically and we report the results of extensive modeling calculations, which show that mesa-beam interferometers are not much more sensitive than the Gaussian-beam interferometers to errors in mirror figures, positions, and orientations. 
  These lectures were addressed to nonspecialists willing to learn some basic facts, approaches, tools and observational evidence which conform modern cosmology. The aim is also to try to complement the many excellent treatises that exists on the subject (an exhaustive treatment being in any case impossible for lack of time, in the lectures, and of space here), instead of trying to cover everything in a telegraphic way. We start by recalling in the introduction a couple of philosophical questions that have always upset inquiring minds. We then present some original mathematical approaches to investigate a number of basic questions, as the comparison of two point distributions (each point corresponding to a galaxy or galaxy cluster), the use of non-standard statistics in the analysis of possible non-Gaussianities, and the use of zeta regularization in the study of the contributions of vacuum energy effects at the cosmological scale. And we also summarize a number of important issues which are both undoubtedly beautiful (from the physical viewpoint) and useful in present-day observational cosmology. To finish, the reader should be warned that, for the reasons already given and lack of space, some fundamental issues, as inflation, quantum gravity and string theoretical fundamental approaches to cosmology will not be dealt with here. A minimal treatment of any of them would consume more pages than the ones at disposal and, again, a number of excellent treatments of these subjects are available. 
  Implications of some proposed theories of quantum gravity for neutrino flavor oscillations are explored within the context of modified dispersion relations of special relativity. In particular, approximate expressions for Planck-scale-induced deviations from the standard oscillation length are obtained as functions of neutrino mass, energy, and propagation distance. Grounding on these expressions, it is pointed out that, in general, even those deviations that are suppressed by the second power of the Planck energy may be observable for ultra-high-energy neutrinos, provided they originate at cosmological distances. In fact, for neutrinos in the highest energy range of EeV to ZeV, deviations that are suppressed by as much as the seventh power of the Planck energy may become observable. Accordingly, realistic possibilities of experimentally verifying these deviations by means of the next generation neutrino detectors--such as IceCube and ANITA--are investigated. 
  The aim of this work is to analyze the dynamical behavior of relativistic infinite axial-symmetric shells with flat interior and a radiation filled curved exterior spacetimes. It will be proven, by the use of conservation equations of Israel, that the given configuration does not let expansion or collapse of the shell which was proposed before, but rather the shell stays at constant radius. The case of null-collapse will also be considered in this work and it will be shown that the shell collapses to zero radius, and moreover, if cylindrical flatness is imposed a boundary layer is obtained still contrary to previous works. 
  The beamplitter in high-power interferometers is subject to significant radiation-pressure fluctuations. As a consequence, the phase relations which appear in the beamsplitter coupling equations oscillate and phase modulation fields are generated which add to the reflected fields. In this paper, the transfer function of the various input fields impinging on the beamsplitter from all four ports onto the output field is presented including radiation-pressure effects. We apply the general solution of the coupling equations to evaluate the input-output relations of the dual-recycled laser-interferometer topology of the gravitational-wave detector GEO600 and the power-recycling, signal-extraction topology of advanced LIGO. We show that the input-output relation exhibits a bright-port dark-port coupling. This mechanism is responsible for bright-port contributions to the noise density of the output field and technical laser noise is expected to decrease the interferometer's sensitivity at low frequencies. It is shown quantitatively that the issue of technical laser noise is unimportant in this context if the interferometer contains arm cavities. 
  A generic-curved spacetime Dirac-like equation in 3D is constructed. It has, owing to the $\bar{SL}(n,R)$ group deunitarizing automorphism, a physically correct unitarity and flat spacetime particle properties. The construction is achieved by embedding $\bar{SL}(3,R)$ vector operator $X_{\mu}$, that plays a role of Dirac's $\gamma_{\mu}$ matrices, into $\bar{SL}(4,R)$. Decomposition of the unitary irreducible spinorial $\bar{SL}(4,R)$ representations gives rise to an explicit form of the infinite $X_{\mu}$ matrices. 
  The best motivated alternatives to general relativity are scalar-tensor theories, in which the gravitational interaction is mediated by one or several scalar fields together with the usual graviton. The analysis of their various experimental constraints allows us to understand better which features of the models have actually been tested, and to suggest new observations able to discriminate between them. This talk reviews three classes of constraints on such theories, which are qualitatively different from each other: (i) solar-system experiments; (ii) binary-pulsar tests and future detections of gravitational waves from inspiralling binaries; (iii) cosmological observations. While classes (i) and (ii) impose precise bounds respectively on the first and second derivatives of the matter-scalar coupling function, (iii) a priori allows us to reconstruct the full shapes of the functions of the scalar field defining the theory, but obviously with more uncertainties and/or more theoretical hypotheses needed. Simple arguments such as the absence of ghosts (to guarantee the stability of the field theory) nevertheless suffice to rule out a wide class of scalar-tensor models. Some of them can be probed only if one takes simultaneously into account solar-system and cosmological observations. 
  It is useful to study the space of all cosmological models from a dynamical systems perspective, that is, by formulating the Einstein field equations as a dynamical system using appropriately normalized variables. We will discuss various aspects of this work, the choices of normalization factor, multiple representations of models, the past attractor, nonlinear dynamics in close-to-Friedmann-Lemaitre models, Weyl curvature dominance, and numerical simulations. 
  Advanced LIGO's present baseline design uses arm cavities with Gaussian light beams supported by spherical mirrors. Because Gaussian beams have large intensity gradients in regions of high intensity, they average poorly over fluctuating bumps and valleys on the mirror surfaces, caused by random thermal fluctuations (thermoelastic noise). Flat-topped light beams (mesa beams) are being considered as an alternative because they average over the thermoelastic fluctuations much more effectively. However, the proposed mesa beams are supported by nearly flat mirrors, which experience a very serious tilt instability. In this paper we propose an alternative configuration in which mesa-shaped beams are supported by nearly concentric spheres, which experience only a weak tilt instability. The tilt instability is analyzed for these mirrors in a companion paper by Savov and Vyatchanin. We also propose a one-parameter family of light beams and mirrors in which, as the parameter alpha varies continuously from 0 to pi, the beams and supporting mirrors get deformed continuously from the nearly flat-mirrored mesa configuration ("FM") at alpha=0, to the nearly concentric-mirrored mesa configuration ("CM") at alpha=pi. The FM and CM configurations at the endpoints are close to optically unstable, and as alpha moves away from 0 or pi, the optical stability improves. 
  Sidles and Sigg have shown that advanced LIGO interferometers will encounter a serious tilt instability, in which symmetric tilts of the mirrors of an arm cavity cause the cavity's light beam to slide sideways, so its radiation pressure exerts a torque that increases the tilt. Sidles and Sigg showed that the strength T of this torque is 26.2 times greater for advanced LIGO's baseline cavities -- nearly flat spherical mirrors which support Gaussian beams (``FG'' cavities), than for nearly concentric spherical mirrors which support Gaussian beams with the same diffraction losses as the baseline case -- ``CG'' cavities: T^{FG}/T^{CG} = 26.2. This has motivated a proposal to change the baseline design to nearly concentric, spherical mirrors. In order to reduce thermoelastic noise in advanced LIGO, O'Shaughnessy and Thorne have proposed replacing the spherical mirrors and their Gaussian beams by ``Mexican-Hat'' (MH) shaped mirrors which support flat-topped, ``mesa'' shaped beams. In this paper we compute the tilt-instability torque for advanced-LIGO cavities with nearly flat MH mirrors and mesa beams (``FM'' cavities) and nearly concentric MH mirrors and mesa beams (``CM'' cavities), with the same diffraction losses as in the baseline FG case. We find that the relative sizes of the restoring torques are T^{CM}/T^{CG} = 0.91, T^{FM}/T^{CG} = 96, T^{FM}/T^{FG} = 3.67. Thus, the nearly concentric MH mirrors have a weaker tilt instability than any other configuration. Their thermoelastic noise is the same as for nearly flat MH mirrors, and is much lower than for spherical mirrors. 
  Acoustic analogs of static, spherically symmetric massive traversable Lorentzian wormholes are constructed as a {\em formal} extension of acoustic black holes. The method is straightforward but the idea is interesting in itself. The analysis leads to a new acoustic invariant for the massless counterpart of the Einstein-Rosen model of an elementary particle. It is shown that there is a marked, in a sense even counterintuitive, physical difference between the acoustic analogs of black holes and wormholes. The analogy allows us to also portray the nature of curvature singularity in the acoustic language. It is demonstrated that the light ray trajectories in an optical medium are the same as the sound trajectories in its acoustic analog. The implications of these analogies in the laboratory set up and in the different context of phantom energy accretion have been speculated. 
  Here, an accelerated phantom model for the late universe is explored, which is free from future singularity. It is interseting to see that this model exhibits strong curvature for all time in future, unlike models with `big-rip singularity' showing high curvature near singularity time only. So, quantum gravity effects grow dominant as time increases in the late universe too. More importantly, it is demonstrated that that quantum corrections to FRW equations lead to non-violation of `cosmic energy conditions' of general relativity, which are violated for accelerating universe without these corrections. 
  For a certain example of a "doubly special relativity theory" the modified space-time Lorentz transformations are obtained from momentum space transformations by using canonical methods. In the sequel an energy-momentum dependent space-time metric is constructed, which is essentially invariant under the modified Lorentz transformations. By associating such a metric to every Planck cell in space and the energy-momentum contained in it, a solution of the problem of macroscopic bodies in doubly special relativity is suggested. 
  Exploiting the existence of two "cosmological" constants, one associated with the classical Lovelock theorem and one with the vacuum energy density, we argue, in a model independent way, that in spatially closed FLRW cosmologies with a positive definite effective cosmological constant there exists a range in this constant that serves as a sufficient condition for the satisfaction of the null, weak, strong and dominant energy conditions at a bounce. The application of energy conditions is not unambiguous and we show how the bounce can be considered classically and how, we believe more reasonably, it can be considered a matter of quantum cosmology. 
  There is a general belief, reinforced by statements in standard textbooks, that: (i) one can obtain the full non-linear Einstein's theory of gravity by coupling a massless, spin-2 field $h_{ab}$ self-consistently to the total energy momentum tensor, including its own; (ii) this procedure is unique and leads to Einstein-Hilbert action and (iii) it only uses standard concepts in Lorentz invariant field theory and does not involve any geometrical assumptions. After providing several reasons why such beliefs are suspect -- and critically re-examining several previous attempts -- we provide a detailed analysis aimed at clarifying the situation. First, we prove that it is \textit{impossible} to obtain the Einstein-Hilbert (EH) action, starting from the standard action for gravitons in linear theory and iterating repeatedly. Second, we use the Taylor series expansion of the action for Einstein's theory, to identify the tensor $\mathcal{S}^{ab}$, to which the graviton field $h_{ab}$ couples to the lowest order. We show that the second rank tensor $\mathcal{S}^{ab}$ is {\it not} the conventional energy momentum tensor $T^{ab}$ of the graviton and provide an explanation for this feature. Third, we construct the full nonlinear Einstein's theory with the source being spin-0 field, spin-1 field or relativistic particles by explicitly coupling the spin-2 field to this second rank tensor $\mathcal{S}^{ab}$ order by order and summing up the infinite series. Finally, we construct the theory obtained by self consistently coupling $h_{ab}$ to the conventional energy momentum tensor $T^{ab}$ order by order and show that this does {\it not} lead to Einstein's theory. (condensed). 
  This essay reviews some of the recent progress in the area of energy conditions and wormholes. Most of the discussion centers on the subject of ``quantum inequality'' restrictions on negative energy. These are bounds on the magnitude and duration of negative energy which put rather severe constraints on its possible macroscopic effects. Such effects might include the construction of wormholes and warp drives for faster-than-light travel, and violations of the second law of thermodynamics. Open problems and future directions are also discussed. 
  This article examines the claim that the Brans-Dicke scalar field \phi \to \phi_{0} + O(1/\sqrt{\omega}) for large $\omega$ when the matter field is traceless. It is argued that such a claim can not be true in general. 
  A scalar theory of gravity with a preferred reference frame is presented. It is insisted on the dynamics, which involves a (non-trivial) extension of Newton's second law, and on the new version (v2) with isotropic space metric. We display the energy conservation equation obtained with v2. Then the principles of the asymptotic post-Newtonian approximation are discussed in some detail. The results of its application to the motion of a small extended body in a weakly-gravitating system are given and discussed: the weak equivalence principle was violated in v1, due to its anisotropic space metric (as the standard Schwarzschild metric), but is valid with v2. 
  We study the quantum stress tensor correlation function for a massless scalar field in a flat two-dimensional spacetime containing a moving mirror. We construct the correlation functions for right-moving and left-moving fluxes for an arbitrary trajectory, and then specialize them to the case of a mirror trajectory for which the expectation value of the stress tensor describes a pair of delta-function pulses, one of negative energy and one of positive energy. The flux correlation function describes the fluctuations around this mean stress tensor, and reveals subtle changes in the correlations between regions where the mean flux vanishes. 
  Continuing previous work on the 3PN-accurate gravitational wave generation from point particle binaries, we obtain the binary's 3PN mass-type quadrupole and dipole moments for general (not necessarily circular) orbits in harmonic coordinates. The final expressions are given in terms of their ``core'' parts, resulting from the application of the pure Hadamard-Schwartz (pHS) self-field regularization scheme, and augmented by an ``ambiguous'' part. In the case of the 3PN quadrupole we find three ambiguity parameters, xi, kappa and zeta, but only one for the 3PN dipole, in the form of the particular combination xi+kappa. Requiring that the dipole moment agree with the center-of-mass position deduced from the 3PN equations of motion in harmonic coordinates yields the relation xi+kappa=-9871/9240. Our results will form the basis of the complete calculation of the 3PN radiation field of compact binaries by means of dimensional regularization. 
  The paper addresses matrix spaces, whose properties and dynamics are determined by Dirac matrices in Riemannian spaces of different dimension and signature. Among all Dirac matrix systems there are such ones, which nontrivial scalar, vector or other tensors cannot be made up from. These Dirac matrix systems are associated with the vacuum state of the matrix space. The simplest vacuum system realization can be ensured using the orthonormal basis in the internal matrix space. This vacuum system realization is not however unique. The case of 7-dimensional Riemannian space of signature 7(-) is considered in detail. In this case two basically different vacuum system realizations are possible: (1) with using the orthonormal basis; (2) with using the oblique-angled basis, whose base vectors coincide with the simple roots of algebra E_{8}.   Considerations are presented, from which it follows that the least-dimension space bearing on physics is the Riemannian 11-dimensional space of signature 1(-)& 10(+). The considerations consist in the condition of maximum vacuum energy density and vacuum fluctuation energy density. 
  Dust configurations are the simplest models for astrophysical objects. Here we examine the gravitational collapse of an infinite cylinder of dust and give an analytic interior solution. Surprisingly, starting with a cylindrically symmetric ansatz one arrives at a 3-space with constant curvature, i.e. the resulting metric describes a piece of the Friedman interior of the Oppenheimer-Snyder collapse. Indeed, by introducing double polar coordinates, a 3-space of constant curvature can be interpreted as a cylindrically symmetric space as well. This result shows afresh that topology is not fixed by the Einstein equations. 
  The effects of thermal fluctuations of the mass (horizon area) and electric charge, on the entropy of non-rotating charged {\it macroscopic} black holes, are analyzed using a grand canonical ensemble. Restricting to Gaussian fluctuations around equilibrium, and assuming a power law type of relation between the black hole mass, charge and horizon area, characterized by two real positive indices, the grand canonical entropy is shown to acquire a logarithmic correction with a positive coefficient proportional to the sum of the indices. However, the root mean squared fluctuations of mass and charge relative to the mean values of these quantities turn out to be independent of the details of the assumed mass-area relation. We also comment on possible cancellation between log (area) corrections arising due to {\it fixed area} quantum spacetime fluctuations and that due to thermal fluctuations of the area and other quantities. 
  Chiral cosmic strings are naturally produced at the end of D-term inflation and they present very interesting cosmological consequences. In this work, we investigate the formation and evolution of wakes by a chiral string. We show that, for cold dark matter, the mechanism of forming wakes by a chiral string is similar to the mechanism by an ordinary string. 
  The original computation did not take into account the initial phase shift. A corrected computation, included as section II in the revised version of gr-qc/0407039, still gives a term proportional to the derivative of the signal, qualitatively confirming the original result. 
  We use a dynamical systems approach to analyse the tilting spatially homogeneous Bianchi models of solvable type (e.g., types VI$_h$ and VII$_h$) with a perfect fluid and a linear barotropic $\gamma$-law equation of state. In particular, we study the late-time behaviour of tilted Bianchi models, with an emphasis on the existence of equilibrium points and their stability properties. We briefly discuss the tilting Bianchi type V models and the late-time asymptotic behaviour of irrotational Bianchi VII$_0$ models. We prove the important result that for non-inflationary Bianchi type VII$_h$ models vacuum plane-wave solutions are the only future attracting equilibrium points in the Bianchi type VII$_h$ invariant set. We then investigate the dynamics close to the plane-wave solutions in more detail, and discover some new features that arise in the dynamical behaviour of Bianchi cosmologies with the inclusion of tilt. We point out that in a tiny open set of parameter space in the type IV model (the loophole) there exists closed curves which act as attracting limit cycles. More interestingly, in the Bianchi type VII$_h$ models there is a bifurcation in which a set of equilibrium points turn into closed orbits. There is a region in which both sets of closed curves coexist, and it appears that for the type VII$_h$ models in this region the solution curves approach a compact surface which is topologically a torus. 
  The recent discovery of a double-pulsar PSR J0737-3039A/B provides an opportunity of unequivocally observing, for the first time, spin effects in general relativity. Existing efforts involve detection of the precession of the spinning body itself. However, for a close binary system, spin effects on the orbit may also be discernable. Not only do they add to the advance of the periastron (by an amount which is small compared to the conventional contribution) but they also give rise to a precession of the orbit about the spin direction. The measurement of such an effect would also give information on the moment of inertia of pulsars. 
  The complete form of the constraints following from their conformal structure is extended so as to include constant mean curvature and other mean curvature foliations. This step is demonstrated using the momentum phase space approach. This approach yields equations of exactly the same form as the extended conformal thin sandwich approach. In solving the equations, it is never necessary actually to perform a tensor decomposition. 
  Static spherically symmetric distributions of electrically counterpoised dust (ECD) are used to construct solutions to Einstein-Maxwell equations in Majumdar--Papapetrou formalism. Unexpected bifurcating behaviour of solutions with regard to source strength is found for localized, as well as for the delta-function ECD distributions. Unified treatment of general ECD distributions is accomplished and it is shown that for certain source strengths one class of regular solutions approaches Minkowski spacetime, while the other comes arbitrarily close to black hole solutions. 
  We study the evolution of a homogeneous, anisotropic Universe given by a Bianchi type-I cosmological model filled with viscous fluid, in the presence of a cosmological constant $\Lambda$. The role of viscous fluid and $\Lambda$ term in the evolution the BI space-time is studied. Though the viscosity cannot remove the cosmological singularity, it plays a crucial part in the formation of a qualitatively new behavior of the solutions near singularity. It is shown that the introduction of the $\Lambda$ term can be handy in the elimination of the cosmological singularity. In particular, in case of a bulk viscosity, it provides an everlasting process of evolution ($\Lambda < 0$), whereas, for some positive values of $\Lambda$ and the bulk viscosity being inverse proportional to the expansion, the BI Universe admits a singularity-free oscillatory mode of expansion. In case of a constant bulk viscosity and share viscosity being proportional to expansion, the model allows oscillatory mode accompanied by an exponential growth even with a negative $\Lambda$. Space-time singularity in this case occurs at $t \to -\infty$. 
  The axiomatic bases of Special Relativity Theory (SRT) are thoroughly re-examined from an operational point of view, with particular emphasis on the status of Einstein synchronization in the light of the possibility of arbitrary synchronization procedures in inertial reference frames. Once correctly and explicitly phrased, the principles of SRT allow for a wide range of `theories' that differ from the standard SRT only for the difference in the chosen synchronization procedures, but are wholly equivalent to SRT in predicting empirical facts. This results in the introduction, in the full background of SRT, of a suitable synchronization gauge. A complete hierarchy of synchronization gauges is introduced and elucidated, ranging from the useful Selleri synchronization gauge (which should lead, according to Selleri, to a multiplicity of theories alternative to SRT) to the more general Mansouri-Sexl synchronization gauge and, finally, to the even more general Anderson-Vetharaniam-Stedman's synchronization gauge. It is showed that all these gauges do not challenge the SRT, as claimed by Selleri, but simply lead to a number of formalisms which leave the geometrical structure of Minkowski spacetime unchanged. Several aspects of fundamental and applied interest related to the conventional aspect of the synchronization choice are discussed, encompassing the issue of the one-way velocity of light on inertial and rotating reference frames, the GPS's working, and the recasting of Maxwell equations in generic synchronizations. Finally, it is showed how the gauge freedom introduced in SRT can be exploited in order to give a clear explanation of the Sagnac effect for counter-propagating matter beams. 
  Using dynamical systems theory and a detailed numerical analysis, the late-time behaviour of tilting perfect fluid Bianchi models of types IV and VII$_h$ are investigated. In particular, vacuum plane-wave spacetimes are studied and the important result that the only future attracting equilibrium points for non-inflationary fluids are the plane-wave solutions in Bianchi type VII$_h$ models is discussed. A tiny region of parameter space (the loophole) in the Bianchi type IV model is shown to contain a closed orbit which is found to act as an attractor (the Mussel attractor). From an extensive numerical analysis it is found that at late times the normalised energy-density tends to zero and the normalised variables 'freeze' into their asymptotic values. A detailed numerical analysis of the type VII$_h$ models then shows that there is an open set of parameter space in which solution curves approach a compact surface that is topologically a torus. 
  We generalize the work of Deser and Levin on the unified description of Hawking radiation and Unruh effect to general stationary motions in spherically symmetric black holes. We have also matched the chemical potential term of the thermal spectrum of the two sides for uncharged black holes. 
  Using the well-known ``displace, cut and reflect'' method used to generate disks from given solutions of Einstein field equations, we construct static charged disks made of perfect fluid based on the Reissner-Nordstr\"{o}m solution in isotropic coordinates. We also derive a simple stability condition for charged and non charged perfect fluid disks. As expected, we find that the presence of charge increases the regions of instability of the disks. 
  An exact but simple general relativistic model for the gravitational field of active galactic nuclei is constructed, based on the superposition in Weyl coordinates of a black hole, a Chazy-Curzon disk and two rods, which represent matter jets. The influence of the rods on the matter properties of the disk and on its stability is examined. We find that in general they contribute to destabilize the disk. Also the oscillation frequencies for perturbed circular geodesics on the disk are computed, and some geodesic orbits for the superposed metric are numerically calculated. 
  The stability of general relativistic thin disks is investigated under a general first order perturbation of the energy momentum tensor. In particular, we consider temporal, radial and azimuthal "test matter" perturbations of the quantities involved on the plane $z=0$. We study the thin disks generated by applying the "displace, cut and reflect" method, usually known as the image method, to the Schwarzschild metric in isotropic coordinates and to the Chazy-Curzon metric and the Zipoy-Voorhees metric ($\gamma$-metric) in Weyl coordinates. In the case of the isotropic Schwarzschild thin disk, where a radial pressure is present to support the gravitational attraction, the disk is stable and the perturbation favors the formation of rings. Also, we found the expected result that the thin disk models generated by the Chazy-Curzon and Zipoy-Voorhees metric with only azimuthal pressure are not stable under a general first order perturbation 
  This paper focuses on the mission design for the Laser Astrometric Test Of Relativity (LATOR). This mission uses laser interferometry between two micro-spacecraft whose lines of sight pass close by the Sun to accurately measure deflection of light in the solar gravity. The key element of the experimental design is a redundant geometry optical truss provided by a long-baseline (100 m) multi-channel stellar optical interferometer placed on the International Space Station (ISS). The spatial interferometer is used for measuring the angles between the two spacecraft and for orbit determination purposes. The geometric redundancy enables LATOR to measure the departure from Euclidean geometry caused by the solar gravity field to a very high accuracy. Such a design enables LATOR to improve the value of the parameterized post-Newtonian (PPN) parameter $\gamma$ to unprecedented levels of accuracy of 1 part in 10$^{8}$; the misison will also measure effects of the next post-Newtonian order ($\propto G^2$) of light deflection resulting from gravity's intrinsic non-linearity. The solar quadrupole moment parameter, $J_2$, will be measured with high precision, as well as a variety of other relativistic effects including Lense-Thirring precession. LATOR will lead to very robust advances in the tests of Fundamental physics: this mission could discover a violation or extension of general relativity, or reveal the presence of an additional long range interaction in the physical law. 
  Consider a bipartite entangled system half of which falls through the event horizon of an evaporating black hole, while the other half remains coherently accessible to experiments in the exterior region. Beyond complete evaporation, the evolution of the quantum state past the Cauchy horizon cannot remain unitary, raising the questions: How can this evolution be described as a quantum map, and how is causality preserved? What are the possible effects of such nonstandard quantum evolution maps on the behavior of the entangled laboratory partner? More generally, the laws of quantum evolution under extreme conditions in remote regions (not just in evaporating black-hole interiors, but possibly near other naked singularities and regions of extreme spacetime structure) remain untested by observation, and might conceivably be non-unitary or even nonlinear, raising the same questions about the evolution of entangled states. The answers to these questions are subtle, and are linked in unexpected ways to the fundamental laws of quantum mechanics. We show that terrestrial experiments can be designed to probe and constrain exactly how the laws of quantum evolution might be altered, either by black-hole evaporation, or by other extreme processes in remote regions possibly governed by unknown physics. 
  The recent observations on the far quasars absorption lines spectra and comparison of these lines with laboratory ones provide a framework for explantation of these observations by considering a varying fine structure constant, over the cosmological time-scale. Also, there seems to be an anomalous acceleration in the Pioneer spacecraft 10/11 about $ 10^{-10} {\rm m}/{\rm s^2}$. These matters lead Ranada to study the quantum vacuum to explain these problems by introducing a phenomenological model for the variation of $\alpha$. In this manuscript we want to show that this model is not a quantum model; it is a classical model that is only in accordance with mentioned observations by adjusting some parameters and is not based on a fundamental physical intuition. 
  The paper is the first of two parts of a work reviewing some approaches to the problem of time in quantum cosmology, which were put forward last decade, and which demonstrated their relation to the problems of reparametrization and gauge invariance of quantum gravity. In the present part we remind basic features of quantum geometrodynamics and minisuperspace cosmological models, and discuss fundamental problems of the Wheeler - DeWitt theory. Various attempts to find a solution to the problem of time are considered in the framework of the canonical approach. Possible solutions to the problem are investigated making use of minisuperspace models, that is, systems with a finite number of degrees of freedom. At the same time, in the last section of the paper we expand our consideration beyond the minisuperspace approximation and briefly review promising ideas by Brown and Kuchar, who propose that dust interacting only gravitationally can be used for time measuring, and the unitary approach by Barvinsky and collaborators. The latter approach admits both the canonical and path integral formulations and anticipates the consideration of recent developments in the path integral approach in the second part of our work. 
  The irrotational vortex geometry carachter of torsion loops is displayed by showing that torsion loops and nonradial flow acoustic metrics are conformally equivalent in $(1+1)$ dimensions while radial flow acoustic spacetime are conformally related in $(2+1)$ dimensional spacetime. The analysis of 2-dimensional space allows us to express the fluid density in terms of the parameters of torsion loop metric. These results lead us to conclude that the acoustic metric of vortex flows is the gravitational analog of torsion loop spacetime. Since no vorticity in the fluids is considered we do not make explicit use of non-Riemannian geometry of vortex acoustics in classical fluids. Acoustic nonradial flows are shown to exihibit a full analogy with torsion loop metric. 
  Acoustic torsion recently introduced in the literature (Garcia de Andrade,PRD(2004),7,64004) is extended to rotational incompressible viscous fluids represented by the generalised Navier-Stokes equation. The fluid background is compared with the Riemann-Cartan massless scalar wave equation, allowing for the generalization of Unruh acoustic metric in the form of acoustic torsion, expressed in terms of viscosity, velocity and vorticity of the fluid. In this work the background vorticity is nonvanishing but the perturbation of the flow is also rotational which avoids the problem of contamination of the irrotational perturbation by the background vorticity. The acoustic Lorentz invariance is shown to be broken due to the presence of acoustic torsion in strong analogy with the Riemann-Cartan gravitational case presented recently by Kostelecky (PRD 69,2004,105009). An example of analog gravity describing acoustic metric is given based on the teleparallel loop where the acoustic torsion is given by the Lense-Thirring rotation and the acoustic line element corresponds to the Lense-Thirring metric. 
  Analysis of the radio-metric tracking data from the Pioneer 10/11 spacecraft at distances between 20--70 astronomical units (AU) from the Sun has consistently indicated the presence of an anomalous, small, constant Doppler frequency drift. The drift is a blue-shift, uniformly changing with rate a_t = (2.92 +/- 0.44) x 10^(-18) s/s^2. It can also be interpreted as a constant acceleration of a_P = (8.74 +/- 1.33) x 10^(-8) cm/s^2 directed towards the Sun. Although it is suspected that there is a systematic origin to the effect, none has been found. As a result, the nature of this anomaly has become of growing interest. Here we discuss the details of our recent investigation focusing on the effects both external to and internal to the spacecraft, as well as those due to modeling and computational techniques. We review some of the mechanisms proposed to explain the anomaly and show their inability to account for the observed behavior of the anomaly. We also present lessons learned from this investigation for a potential deep-space experiment that will reveal the origin of the discovered anomaly and also will characterize its properties with an accuracy of at least two orders of magnitude below the anomaly's size. A number of critical requirements and design considerations for such a mission are outlined and addressed. 
  We describe the quasi-static collapse of a radiating, spherical shell of matter in de Sitter space-time using a thermodynamical formalism. It is found that the specific heat at constant area and other thermodynamical quantities exhibit singularities related to phase transitions during the collapse. 
  The paper is the second part of the work devoted to the problem of time in quantum cosmology. Here we consider in detail two approaches within the scope of Feynman path integration scheme: The first, by Simeone and collaborators, is gauge-invariant and lies within the unitary approach to a consistent quantization of gravity. It is essentially based on the idea of deparametrization (reduction to physical degrees of freedom) as a first step before quantization. The other approach by Savchenko, Shestakova and Vereshkov is rather radical. It is an attempt to take into account peculiarities of the Universe as a system without asymptotic states that leads to the conclusion that quantum geometrodynamics constructed for such a system is, in general, a gauge-noninvariant theory. However, this theory is shown to be mathematically consistent and the problem of time is solved in this theory in a natural way. 
  In previous papers [1,2], it was proved that a covariant quantization of the minimally coupled scalar field in de Sitter space is achieved through addition of the negative norm states. This causal approach which eliminates the infrared divergence, was generalized further to the calculation of the graviton propagator in de Sitter space [3] and one-loop effective action for scalar field in a general curved space-time [4]. This method gives a natural renormalization of the above problems. Pursuing this approach, in the present paper the tree-level scattering amplitudes of the scalar field, with one graviton exchange, has been calculated in de Sitter space. It is shown that the infrared divergence disappears and the theory automatically reaches a renormalized solution of the problem. 
  The generalized Stokes theorem (connecting integrals of dimensions 3 and 4) is formulated in a curved space-time in terms of paths in Minkowski space (forming Path Group). A covariant integral form of the conservation law for the energy-momentum of matter is then derived in General Relativity. It extends Einstein's equivalence principle on the energy conservation, since it formulates the conservation law for the energy-momentum of matter without explicit including the gravitational field in the formulation. 
  Given a particular prescription for the Einstein field equations (EFE's), it is important to have general protective theorems that lend support to it. The prescription of data on a timelike hypersurface for the (n + 1)-d EFE's arises in `noncompact Kaluza--Klein theory', and in certain kinds of braneworlds and low-energy string theory. The Campbell--Magaard (CM) theorem, which asserts local existence (and, with extra conditions, uniqueness) of analytic embeddings of completely general n-d manifolds into vacuum (n + 1)-d manifolds, has often recently been invoked as a protective theorem for such prescriptions. But in this paper I argue that there are problems with loosening the CM thoerem of restrictive meanings in its statement, which is worthwhile thing to do in pursuit of the proposed applications. While I remedy some problems by identifying the required topology, delineating what `local' can be taken to mean, and offering a new, more robust and covariant proof, other problems remain unsurmountable. The theorem lends only inadequate support, both because it offers no guarantee of continuous dependence on the data and because it disregards causality. Furthermore, the theorem is only for the analytic functions which renders it inappropriate for the study of the relativistic equations of modern physics. Unfortunately, there are no known general theorems that offer adequate protection to the proposed applications' prescription. I conclude by making some suggestions for more modest progress. 
  This thesis concerns the split of Einstein's field equations (EFE's) with respect to nowhere null hypersurfaces. Areas covered include A) the foundations of relativity, deriving geometrodynamics from relational first principles and showing that this form accommodates a sufficient set of fundamental matter fields to be classically realistic, alternative theories of gravity that arise from similar use of conformal mathematics. B) GR Initial value problem (IVP) methods, the badness of timelike splits of the EFE's and studying braneworlds under guidance from GR IVP and Cauchy problem methods. 
  We review the status of "Einstein-Aether theory", a generally covariant theory of gravity coupled to a dynamical, unit timelike vector field that breaks local Lorentz symmetry. Aspects of waves, stars, black holes, and cosmology are discussed, together with theoretical and observational constraints. Open questions are stressed. 
  In a previous paper, we showed that the traditional form of the charged C-metric can be transformed, by a change of coordinates, into one with an explicitly factorizable structure function. This new form of the C-metric has the advantage that its properties become much simpler to analyze. In this paper, we propose an analogous new form for the rotating charged C-metric, with structure function G(\xi)=(1-\xi^2)(1+r_{+}A\xi)(1+r_{-}A\xi), where r_\pm are the usual locations of the horizons in the Kerr-Newman black hole. Unlike the non-rotating case, this new form is not related to the traditional one by a coordinate transformation. We show that the physical distinction between these two forms of the rotating C-metric lies in the nature of the conical singularities causing the black holes to accelerate apart: the new form is free of torsion singularities and therefore does not contain any closed timelike curves. We claim that this new form should be considered the natural generalization of the C-metric with rotation. 
  We give here an interior metric which matches Gurses metric. The metric given here is a sphere of electrically counterpoised dust (ECD) of constant density. We use Lane Emden equation for obtaining the interior solution. 
  This paper is aimed to elaborate the problem of energy-momentum in General Relativity. In this connection, we use the prescriptions of Einstein, Landau-Lifshitz, Papapetrou and M\"{o}ller to compute the energy-momentum densities for two exact solutions of Einstein field equations. The spacetimes under consideration are the non-null Einstein-Maxwell solutions and the singularity-free cosmological model. The electromagnetic generalization of the G\"{o}del solution and the G\"{o}del metric become special cases of the non-null Einstein-Maxwell solutions. It turns out that these prescriptions do not provide consistent results for any of these spacetimes. These inconsistence results verify the well-known proposal that the idea of localization does not follow the lines of pseudo-tensorial construction but instead follows from the energy-momentum tensor itself. These differences can also be understood with the help of the Hamiltonian approach. 
  We consider the conversion of gravitons and photons as a four-wave mixing process. A nonlinear coupled systems of equations involving two gravitons and two photons is obtained, and the energy exchange between the different degrees of freedom is found. The scattering amplitudes are obtained, from which a crossection for incoherent processes can be found. An analytical example is given, and applications to the early Universe are discussed. 
  Global geometric properties of product manifolds ${\cal M}= M \times \R^2$, endowed with a metric type $<\cdot, \cdot > = < \cdot, \cdot >_R + 2 dudv + H(x,u) du^2$ (where $<\cdot, \cdot >_R$ is a Riemannian metric on $M$ and $H:M \times \R \to \R$ a function), which generalize classical plane waves, are revisited. Our study covers causality (causal ladder, inexistence of horizons), geodesic completeness, geodesic connectedness and existence of conjugate points. Appropiate mathematical tools for each problem are emphasized and the necessity to improve several Riemannian (positive definite) results is claimed. 
  We place direct upper limits on the amplitude of gravitational waves from 28 isolated radio pulsars by a coherent multi-detector analysis of the data collected during the second science run of the LIGO interferometric detectors. These are the first direct upper limits for 26 of the 28 pulsars. We use coordinated radio observations for the first time to build radio-guided phase templates for the expected gravitational wave signals. The unprecedented sensitivity of the detectors allow us to set strain upper limits as low as a few times $10^{-24}$. These strain limits translate into limits on the equatorial ellipticities of the pulsars, which are smaller than $10^{-5}$ for the four closest pulsars. 
  We discuss whether an appropriately defined dimensionless scalar function might be an acceptable candidate for the gravitational entropy, by explicitly considering Szekeres and Bianchi type VI$_{h}$ models that admit an isotropic singularity. We also briefly discuss other possible gravitational entropy functions, including an appropriate measure of the velocity dependent Bel-Robinson tensor. 
  The energy-momentum vacuum average of a conformally coupled massless scalar field vibrating around a cosmic dislocation (a cosmic string with a dislocation along its axis) is taken as source of the linearized semiclassical Einstein equations. The solution up to first order in the Planck constant is derived. Motion of a test particle is then discussed, showing that under certain circumstances a helical-like dragging effect, with no classical analogue around the cosmic dislocation, is induced by back reaction. 
  We further develop a noncommutative model unifying quantum mechanics and general relativity proposed in {\it Gen. Rel. Grav.} (2004) {\bf 36}, 111-126. Generalized symmetries of the model are defined by a groupoid $\Gamma $ given by the action of a finite group on a space $E$. The geometry of the model is constructed in terms of suitable (noncommutative) algebras on $\Gamma $. We investigate observables of the model, especially its position and momentum observables. This is not a trivial thing since the model is based on a noncommutative geometry and has strong nonlocal properties. We show that, in the position representation of the model, the position observable is a coderivation of a corresponding coalgebra, "coparallelly" to the well known fact that the momentum observable is a derivation of the algebra. We also study the momentum representation of the model. It turns out that, in the case of the algebra of smooth, quickly decreasing functions on $\Gamma $, the model in its "quantum sector" is nonlocal, i.e., there are no nontrivial coderivations of the corresponding coalgebra, whereas in its "gravity sector" such coderivations do exist. They are investigated. 
  For a particle of mass mu and scalar charge q, we compute the effects of the scalar field self-force upon circular orbits, upon slightly eccentric orbits and upon the innermost stable circular orbit of a Schwarzschild black hole of mass M. For circular orbits the self force is outward and causes the angular frequency at a given radius to decrease. For slightly eccentric orbits the self force decreases the rate of the precession of the orbit. The effect of the self force moves the radius of the innermost stable circular orbit inward by 0.122701 q^2/mu, and it increases the angular frequency of the ISCO by the fraction 0.0291657 q^2/mu M. 
  The gauge fixed polygon model of 2+1 gravity with zero cosmological constant and arbitrary number of spinless point particles is reconstructed from the first order formalism of the theory in terms of the triad and the spin connection. The induced symplectic structure is calculated and shown to agree with the canonical one in terms of the variables. 
  In the paper we consider the energy and its flux in the Bonnor spacetime. We construct some non-local expressions from the Einstein canonical energy-momentum pseudotensor of the gravitational field which show that the gravitational energy and its flux in this spacetime are different from zero and do not vanish even outside of the material source (a stationary beam of the null dust) of this spacetime. 
  The Misner and Sharp approach to the study of gravitational collapse is extended to the dissipative case in, both, the streaming out and the diffusion approximations. The role of different terms in the dynamical equation are analyzed in detail. The dynamical equation is then coupled to a causal transport equation in the context of Israel--Stewart theory. The decreasing of the inertial mass density of the fluid, by a factor which depends on its internal thermodynamics state, is reobtained, at any time scale. In accordance with the equivalence principle, the same decreasing factor is obtained for the gravitational force term. Prospective applications of this result to some astrophysical scenarios are discussed. 
  I investigate a new idea of perturbation theory in covariant canonical quantization. I present preliminary results for a toy model of a harmonic oscillator with a quartic perturbation, and show that this method reproduces the quantized spectrum of standard quantum theory. This result indicates that when the exact solutions to classical equations are not known, covariant canonical quantization via perturbation theory could be a viable approximation scheme for finding observables, and suggests a physically interesting way of extending the scope of covariant canonical quantization in quantum gravity 
  A method is presented to construct initial data for Einstein's equations as a superposition of a gravitational wave perturbation on an arbitrary stationary background spacetime. The method combines the conformal thin sandwich formalism with linear gravitational waves, and allows detailed control over characteristics of the superposed gravitational wave like shape, location and propagation direction. It is furthermore fully covariant with respect to spatial coordinate changes and allows for very large amplitude of the gravitational wave. 
  Beside diffeomorphism invariance also manifest SO(3,1) local Lorentz invariance is implemented in a formulation of Einstein Gravity (with or without cosmological term) in terms of initially completely independent vielbein and spin connection variables and auxiliary two-form fields. In the systematic study of all possible embeddings of Einstein gravity into that formulation with auxiliary fields, the introduction of a ``bi-complex'' algebra possesses crucial technical advantages. Certain components of the new two-form fields directly provide canonical momenta for spatial components of all Cartan variables, whereas the remaining ones act as Lagrange multipliers for a large number of constraints, some of which have been proposed already in different, less radical approaches. The time-like components of the Cartan variables play that role for the Lorentz constraints and others associated to the vierbein fields. Although also some ternary ones appear, we show that relations exist between these constraints, and how the Lagrange multipliers are to be determined to take care of second class ones. We believe that our formulation of standard Einstein gravity as a gauge theory with consistent local Poincare algebra is superior to earlier similar attempts. 
  We discuss a numerical method to compute the homogeneous solutions of the Teukolsky equation which is the basic equation of the black hole perturbation method. We use the formalism developed by Mano, Suzuki and Takasugi, in which the homogeneous solutions of the radial Teukolsky equation are expressed in terms of two kinds of series of special functions, and the formulas for the asymptotic amplitudes are derived explicitly.Although the application of this method was previously limited to the analytical evaluation of the homogeneous solutions, we find that it is also useful for numerical computation. We also find that so-called "renormalized angular momentum parameter", $\nu$, can be found only in the limited region of $\omega$ for each $l,m$ if we assume $\nu$ is real (here, $\omega$ is the angular frequency, and $l$ and $m$ are degree and order of the spin-weighted spheroidal harmonics respectively). We also compute the flux of the gravitational waves induced by a compact star in a circular orbit on the equatorial plane around a rotating black hole. We find that the relative error of the energy flux is about $10^{-14}$ which is much smaller than the one obtained by usual numerical integration methods. 
  We study the structural stability of a cosmic acceleration (inflation) in a class of k-essence cosmologies against changes in the shape of the potential. Those models may be viewed as generalized tachyon cosmologies and this analysis extends previous results on the structural stability of cosmic acceleration in tachyon cosmologies. The study considers both phantom and non-phantom cases. The concepts of rigidity and fragility are defined through a condition on the functional form of the Hubble factor. Given the known result of the existence of inflationary (non-phantom) and super-inflationary (phantom) attractors we formulate the question of their structural stability. We find that those attractors are rigid in the sense that they never change as long as the conditions for inflation or super-inflation are met. 
  We discuss the definition of velocity as dE/dp, where E,p are the energy and momentum of a particle, in Doubly Special Relativity (DSR). If this definition matches dx/dt appropriate for the space-time sector, then space-time can in principle be built consistently with the existence of an invariant length scale. We show that, within different possible velocity definitions, a space-time compatible with momentum-space DSR principles can not be derived. 
  New expressions for the multipole moments of an isolated post-Newtonian source, in the form of surface integrals in the outer near-zone, are derived. As an application we compute the ``source'' quadrupole moment of a Schwarzschild solution boosted to uniform velocity, at the third post-Newtonian (3PN) order. We show that the consideration of this boosted Schwarzschild solution (BSS) is enough to uniquely determine one of the ambiguity parameters in the recent computation of the gravitational wave generation by compact binaries at 3PN order: zeta=-7/33. We argue that this value is the only one for which the Poincar\'e invariance of the 3PN wave generation formalism is realized. As a check, we confirm the value of zeta by a different method, based on the far-zone expansion of the BSS at fixed retarded time, and a calculation of the relevant non-linear multipole interactions in the external metric at the 3PN order. 
  We use the recently developed massive field approach to calculate the scalar self-force on a static particle in a Schwarzschild spacetime. In this approach the scalar self-force is obtained from the difference between the (massless) scalar field, and an auxiliary massive scalar field combined with a certain limiting process. By applying this approach to a static particle in Schwarzschild we show that the scalar self-force vanishes in this case. This result conforms with a previous analysis by Wiseman . 
  We have developed a theoretical model and a numerical code for stationary rotating superfluid neutron stars in full general relativity. The underlying two-fluid model is based on Carter's covariant multi-fluid hydrodynamic formalism. The two fluids, representing the superfluid neutrons on one hand, and the protons and electrons on the other, are restricted to uniform rotation around a common axis, but are allowed to have different rotation rates. We have performed extensive tests of the numerical code, including quantitative comparisons to previous approximative results for these models. The results presented here are the first ``exact'' calculations of such models in the sense that no approximations (other than that inherent in a discretized numerical treatment) are used. Using this code we reconfirm the existence of prolate-oblate shaped configurations. We studied the dependency of the Kepler rotation limit and of the mass-density relation on the relative rotation rate. We further demonstrate how one can simulate a (albeit fluid) neutron-star ``crust'' by letting one fluid extend further outwards than the other, which results in interesting cases where the Kepler limit is actually determined by the outermost but slower fluid. 
  Based on the gauge invariant variables proposed in our previous paper [K. Nakamura, Prog. Theor. Phys. vol.110 (2003), 723.], some formulae of the perturbative curvatures of each order are derived. We follow the general framework of the second order gauge invariant perturbation theory on arbitrary background spacetime to derive these formulae. These perturbative curvatures do have the same form as the definitions of gauge invariant variables for arbitrary perturbative fields which are previously proposed. As a result, we explicitly see that any perturbative Einstein equations are given in terms of gauge invarinat form. We briefly discuss physical situations to which this framework should be applied. 
  We have studied the evolution of the massless scalar field propagating in time-dependent charged Vaidya black hole background. A generalized tortoise coordinate transformation were used to study the evolution of the massless scalar field. It is shown that, for the slowest damped quasinormal modes, the approximate formulae in stationary Reissner-Nordstr\"{o}m black hole turn out to be a reasonable prescription, showing that results from quasinormal mode analysis are rather robust. 
  We show how, in canonical quantum cosmology, the frame fixing induces a new energy density contribution having features compatible with the (actual) cold dark matter component of the Universe. First we quantize the closed Friedmann-Robertson-Walker (FRW) model in a sinchronous reference and determine the spectrum of the super-Hamiltonian in the presence of ultra-relativistic matter and a perfect gas contribution. Then we include in this model small inhomogeneous (spherical) perturbations in the spirit of the Lemaitre-Tolman cosmology. The main issue of our analysis consists in outlining that, in the classical limit, the non-zero eigenvalue of the super-Hamiltonian can make account for the present value of the dark matter critical parameter. Furthermore we obtain a direct correlation between the inhomogeneities in our dark matter candidate and those one appearing in the ultra-relativistic matter. 
  In this paper we establish dualities between inflationary, cyclic/ekpyrotic, and phantom cosmologies within the patch formalism approximating high-energy effects in scenarios with extra dimensions. The exact dualities relating the four-dimensional spectra are broken in favour of their braneworld counterparts; the dual solutions display new interesting features because of the modification of the effective Friedmann equation on the brane. We then address some qualitative issues about phantomlike cosmologies without phantom matter. 
  We report the first high-precision interferometer using large sapphire mirrors, and we present the first direct, broadband measurements of the fundamental thermal noise in these mirrors. Our results agree well with the thermoelastic-damping noise predictions of Braginsky, et al. [Phys. Lett. A 264, 1(1999)] and Cerdonio, et al.[Phys. Rev. D 63, 082003 (2001)], which have been used to predict the astrophysical reach of advanced interferometric gravitational wave detectors. 
  In this work, based on some mathematical results obtained by Yamabe, Osgood, Phillips and Sarnak, we demonstrate that in dimensions three and higher the famous Ginzburg-Landau equations used in theory of phase transitions can be obtained (without any approximations) by minimization of the Riemannian-type Hilbert-Einstein action functional for pure gravity in the presence of cosmological term. We use this observation in order to bring to completion the work by Lifshitz (done in 1941) on group-theoretical refinements of the Landau theory of phase transitions. In addition, this observation allows us to develop a systematic extension to higher dimensions of known string-theoretic path integral methods developed for calculation of observables in two dimensional conformal field theories. 
  We consider the gravity field of a Bose-Einstein condensate in a quantum superposition. The gravity field then is also in a quantum superposition which is in principle observable. Hence we have ``quantum gravity'' far away from the so-called Planck scale. 
  By linearly scaling the initial data set (mass and kinetic energy functions) together with the initial area radius of a collapsing dust sphere, we find a symmetry of the collapse dynamics. That is, the linear transformation defines an equivalence class of data sets which lead to the same end result as well as its evolution all through. In particular, the density and shear remain invariant initially as well as during the collapse. What the transformation is exhibiting is an interesting scaling relationship between mass, kinetic energy and the size of the collapsing dust sphere. 
  The main cosmological models on the brane are presented. A generic equation is given, from which the Friedmann equations of the Randall-Sundrum, induced gravity, Gauss-Bonnet and the combined induced gravity and Gauss-Bonnet cosmological models are obtained. We discuss the modifications they bring to the standard cosmology and the main features of their inflationary dynamics. 
  We find that sudden future singularities may also appear in spatially inhomogeneous Stephani models of the universe. They are temporal pressure singularities and may appear independently of the spatial finite density singularities already known to exist in these models. It is shown that the main advantage of the homogeneous sudden future singularities which is the fulfillment of the strong and weak energy conditions may not be the case for inhomogeneous models. 
  The issue of implementing the principle of general relativity in Einstein equations has been widely discussed, since Kretschmann's well-known criticism stated that general covariance of the Einstein equations is not suffice to express the principle of general relativity (the equivalence of all the coordinate systems). This failure is usually rooted in the fact that metric in Einstein equations is not univocally determined by the matter distribution. We show that the condition of univocal determination of the metric by the matter distribution is stronger than the requirement of equivalence of all coordinate systems. In order to separate the uniqueness problem in Einstein equations from the issue of the principle of general relativity, we define the "equivalence group" instead of the notion of covariance group which is empty of physical content. Moreover, we have complemented in a positive way Kretschmann's objection by supplementing Einstein equations with a sufficient condition for the equivalence of all the coordinate systems. 
  We consider two approaches to evading paradoxes in quantum mechanics with closed timelike curves (CTCs). In a model similar to Politzer's, assuming pure states and using path integrals, we show that the problems of paradoxes and of unitarity violation are related; preserving unitarity avoids paradoxes by modifying the time evolution so that improbable events bewcome certain. Deutsch has argued, using the density matrix, that paradoxes do not occur in the "many worlds interpretation". We find that in this approach account must be taken of the resolution time of the device that detects objects emerging from a wormhole or other time machine. When this is done one finds that this approach is viable only if macroscopic objects traversing a wormhole interact with it so strongly that they are broken into microscopic fragments. 
  Non-Riemannian geometry of acoustic non-relativistic turbulent flows is irrotationally perturbed generating a acoustic geometry model with acoustic metric and acoustic Cartan contortion. The contortion term is due to nonlinearities in the turbulent fluid. The acoustic curvature and acoustic contortion are given by Dirac delta distributions. Violation of Lorentz invariance due to turbulence is considered and analog gravity is suggested to be linked to planar acoustic domain walls. 
  Gravitational radiation from a slightly distorted black hole with ringdown waveform is well understood in general relativity. It provides a probe for direct observation of black holes and determination of their physical parameters, masses and angular momenta (Kerr parameters). For ringdown searches using data of gravitational wave detectors, matched filtering technique is useful. In this paper, we describe studies on problems in matched filtering analysis in realistic gravitational wave searches using observational data. Above all, we focus on template constructions, matches or signal-to-noise ratios (SNRs), detection probabilities for Galactic events, and accuracies in evaluation of waveform parameters or black hole hairs. We have performed matched filtering analysis for artificial ringdown signals which are generated with Monte-Carlo technique and injected into the TAMA300 observational data. It is shown that with TAMA300 sensitivity, the detection probability for Galactic ringdown events is about 50% for black holes of masses greater than $20 M_{\odot}$ with SNR $> 10$. The accuracies in waveform parameter estimations are found to be consistent with the template spacings, and resolutions for black hole masses and the Kerr parameters are evaluated as a few % and $\sim 40 %$, respectively. They can be improved up to $< 0.9 %$ and $< 24 %$ for events of ${\rm SNR} \ge 10$ by using fine-meshed template bank in the hierarchical search strategy. 
  This paper demonstrates a dynamical evolution model of the black hole (BH) horizon. The result indicates that a kinetic area-cells model of the BH's horizon can model the evolution of BH due to the Hawking radiation, and this area-cell system can be considered as an interacting geometrical particle system. Thus the evolution turns into a problem of statistical physics. In the present work, this problem is treated in the framework of non-equilibrium statistics. It is proposed that each area-cell possesses the energy like a microscopic black hole, and has the gravitational interaction with the other area-cells. We consider both a non-interaction ideal system, and a system with small nearest-neighbor interactions, and obtain an analytic expression of the expected value of the horizon area of a dynamical BH. We find that, after a long enough evolution, a dynamical BH with the Hawking radiation can be in equilibrium with a finite temperature radiation field. However, we also find that, the system has a critical point, and when the temperature of the radiation field surrounding the BH approaches the critical temperature of the BH, a critical slowing down phenomenon occurs. 
  Gravito-electromagnetism is somewhat ubiquitous in relativity. In fact, there are many situations where the effects of gravitation can be described by formally introducing "gravito-electric" and "gravito-magnetic" fields, starting from the corresponding potentials, in analogy with the electromagnetic theory (see also A. Tartaglia's contribution to these proceedings). The "many faces of gravito-electromagnetism" are related to rotation effects in both approximated and full theory approaches. Here we show that, by using a 1+3 splitting, relativistic dynamics can be described in terms of gravito-electromagnetic (GEM) fields in full theory. On the basis of this formalism, we introduce a "gravito-magnetic Aharonov-Bohm effect", which allows to interpret some rotation effects as gravito-magnetic effects. Finally, we suggest a way for measuring the angular momentum of celestial bodies by studying the gravito-magnetic effects on the propagation of electromagnetic signals. 
  We present a numerical study of general relativistic boson stars in both spherical symmetry and axisymmetry. We consider both time-independent problems, involving the solution of equilibrium equations for rotating boson stars, and time-dependent problems, focusing on black hole critical behaviour associated with boson stars.   The study of the critical phenomena that arise at the threshold of black hole formation has been a subject of intense interest among relativists and applied mathematicians over the past decade. Type I critical phenomena were previously observed in the dynamics of spherically symmetric boson stars by Hawley and Choptuik. We extend this work and show that, contrary to previous claims, the subcritical end-state is well described by a stable boson star executing a large amplitude oscillation with a frequency in good agreement with that predicted for the fundamental normal mode of the end-state star from linear perturbation theory.   We then extend our studies of critical phenomena to the axisymmetric case, studying two distinct classes of parametrized families of initial data whose evolution generates families of spacetimes that ``interpolate'' between those than contain a black hole and those that do not. In both cases we find strong evidence for a Type I transition at threshold, and are able to demonstrate scaling of the lifetime for near-critical configurations of the type expected for such a transition. This is the first time that Type I critical solutions have been simulated in axisymmetry. 
  We construct a class of spherically symmetric collapse models in which a naked singularity may develop as the end state of collapse. The matter distribution considered has negative radial and tangential pressures, but the weak energy condition is obeyed throughout. The singularity forms at the center of the collapsing cloud and continues to be visible for a finite time. The duration of visibility depends on the nature of energy distribution. Hence the causal structure of the resulting singularity depends on the nature of the mass function chosen for the cloud. We present a general model in which the naked singularity formed is timelike, neither pointlike nor null. Our work represents a step toward clarifying the necessary conditions for the validity of the Cosmic Censorship Conjecture. 
  In Einstein's general relativity, geometry replaces the concept of force in the description of the gravitation interaction. Such an approach rests on the universality of free-fall--the weak equivalence principle--and would break down without it. On the other hand, the teleparallel version of general relativity, a gauge theory for the translation group, describes the gravitational interaction by a force similar to the Lorentz force of electromagnetism, a non-universal interaction. It is shown that, similarly to the Maxwell's description of electromagnetism, the teleparallel gauge approach provides a consistent theory for gravitation even in the absence of the weak equivalence principle. 
  The space-time curvature carried by electromagnetic fields is discovered and a new unification of geometry and electromagnetism is found. Curvature is invariant under charge reversal symmetry. Electromagnetic field equations are examined with De Rham co homology theory. Radiative electromagnetic fields must be exact and co exact to preclude unobserved massless topological charges. Weyl's conformal tensor, here called ``the gravitational field'', is decomposed into a divergence-free non-local piece with support everywhere and a local piece with the same support as the matter. By tuning a local gravitational field to a Maxwell field the electromagnetic field's local gravitational field is discovered. This gravitational field carries the electromagnetic field's polarization or phase information, unlike Maxwell's stress-energy tensor. The unification assumes Einstein's equations and derives Maxwell's equations from curvature assumptions. Gravity forbids magnetic monopoles! This unification is stronger than the Einstein-Maxwell equations alone, as those equations must produce the electromagnetic field's local gravitational field and not just any conformal tensor. Charged black holes are examples. Curvature of radiative null electromagnetic fields is characterized. 
  This paper discusses experimental design for the Laser Astrometric Test Of Relativity (LATOR) mission. LATOR is designed to reach unprecedented accuracy of 1 part in 10^8 in measuring the curvature of the solar gravitational field as given by the value of the key Eddington post-Newtonian parameter \gamma. This mission will demonstrate the accuracy needed to measure effects of the next post-Newtonian order (~G^2) of light deflection resulting from gravity's intrinsic non-linearity. LATOR will provide the first precise measurement of the solar quadrupole moment parameter, J2, and will improve determination of a variety of relativistic effects including Lense-Thirring precession. The mission will benefit from the recent progress in the optical communication technologies -- the immediate and natural step above the standard radio-metric techniques. The key element of LATOR is a geometric redundancy provided by the laser ranging and long-baseline optical interferometry. We discuss the mission and optical designs, as well as the expected performance of this proposed mission. LATOR will lead to very robust advances in the tests of Fundamental physics: this mission could discover a violation or extension of general relativity, or reveal the presence of an additional long range interaction in the physical law. There are no analogs to the LATOR experiment; it is unique and is a natural culmination of solar system gravity experiments. 
  Membrane method is used to compute the entropy of the NUT-Kerr-Newman black holes. It is found that even though the Euler characteristic is greater than two, the Bekenstein-Hawking area law is still satisfied. The formula $S=\chi A/8$ relating the entropy and the Euler characteristic becomes inapplicable for non-extreme four dimensional NUT-Kerr-Newman black holes. 
  A detailed analysis of dynamics of cosmological models based on $R^{n}$ gravity is presented. We show that the cosmological equations can be written as a first order autonomous system and analyzed using the standard techniques of dynamical system theory. In absence of perfect fluid matter, we find exact solutions whose behavior and stability are analyzed in terms of the values of the parameter $n$. When matter is introduced, the nature of the (non-minimal) coupling between matter and higher order gravity induces restrictions on the allowed values of $n$. Selecting such intervals of values and following the same procedure used in the vacuum case, we present exact solutions and analyze their stability for a generic value of the parameter $n$. From this analysis emerges the result that for a large set of initial conditions an accelerated expansion is an attractor for the evolution of the $R^n$ cosmology. When matter is present a transient almost-Friedman phase can also be present before the transition to an accelerated expansion. 
  We consider the collisions of plane gravitational and electromagnetic waves with distinct wavefronts and of arbitrary polarizations in a Minkowski background. We first present a new, completely geometric formulation of the characteristic initial value problem for solutions in the wave interaction region for which initial data are those associated with the approaching waves. We present also a general approach to the solution of this problem which enables us in principle to construct solutions in terms of the specified initial data. This is achieved by re-formulating the nonlinear dynamical equations for waves in terms of an associated linear problem on the spectral plane. A system of linear integral ``evolution'' equations which solve this spectral problem for specified initial data is constructed. It is then demonstrated explicitly how various colliding plane wave space-times can be constructed from given characteristic initial data. 
  We apply a Harrison transformation to higher dimensional asymptotically flat black hole solutions, which puts them into an external magnetic field. First, we magnetize the Schwarzschild-Tangherlini metric in arbitrary spacetime dimension n>=4. The thus generated exact solution of the Einstein-Maxwell equations describes a static black hole immersed in a Melvin "fluxbrane", and generalizes previous results by Ernst for the case n=4. The magnetic field deforms the shape of the event horizon, but the total area (as a function of the mass) and the thermodynamics remain unaffected. The amount of flux through a one-dimensional loop on the horizon exhibits a maximum for a finite value of the magnetic field strength, and decreases for larger values. In the Aichelburg-Sexl ultrarelativistic limit, the magnetized black hole becomes an impulsive gravitational wave propagating in the Melvin background. Furthermore, we discuss possible applications of a similar Harrison transformation to rotating black objects. This enables us to magnetize the Myers-Perry hole and the (dipole) Emparan-Reall ring at least in the special case when the vector potential is parallel to a nonrotating Killing field. In particular, dipole rings may be held in equilibrium even when their spin vanishes, thus demonstrating (infinite) non-uniqueness of magnetized static uncharged black holes in five dimensions. Physical properties of such rings are discussed. 
  We give a brief overview of the nature of spacetime emerging from string theory. This is radically different from the familiar spacetime of Einstein's relativity. At a perturbative level, the spacetime metric appears as ``coupling constants" in a two dimensional quantum field theory. Nonperturbatively (with certain boundary conditions), spacetime is not fundamental but must be reconstructed from a holographic, dual theory. 
  We discuss the late-time property of universe and phantom field in the SO(1,1) dark energy model for the potential $V=V_0e^{-\beta\Phi^\alpha}$ with $\alpha$ and $\beta$ two positive constants. We assume in advance some conditions satisfied by the late-time field to simplify equations, which are confirmed to be correct from the eventual results. For $\alpha<2$, the filed falls exponentially off and the phantom equation of state rapidly approaches -1. When $\alpha=2$, the kinetic energy $\rho_k$ and the coupling energy $\rho_c$ become comparable but there is always $\rho_k<-\rho_c$ so that the phantom property of field proceeds to hold. The analysis on the perturbation to the late-time field $\Phi$ illustrates the square effective mass of the perturbation field is always positive and thus the phantom is stable. The universe considered currently may evade the future sudden singularity and will evolve to de Sitter expansion phase. 
  We discuss the optimal detection strategy for a stochastic background of gravitational waves in the case n detectors are available. In literature so far, only two cases have been considered: 2- and n-point correlators. We generalize these analysises to m-point correlators (with m<n) built out of the n detector signals, obtaining the result that the optimal choice is to combine 2-point correlators. Correlating n detectors in this optimal way will improve the (suitably defined) signal-to-noise ratio with respect to the n=2 case by a factor equal to the fourth root of n(n-1)/2. Finally we give an estimation of how this could improve the sensitivity for a network of multi-mode spherical antennas. 
  We present two kinds of acoustic models for the massless electric charge conceived by Einstein and Rosen in the form of a bridge (wormhole throat). It is found that the first kind of modelling requires a thin layer of exotic matter at the bridge. We also derive an acoustic equation that exclusively characterizes the model. Using a second kind of model, it is demonstrated that the Einstein-Rosen charge has a sonic Hawking-Unruh temperature proportional to +-1/$beta$, where $beta$ is the size of the charge. This suggests that (squealing!) wormholes can also be formally accommodated into Unruh's fluid model. 
  Inflation has the potential to seed the galactic magnetic fields observed today. However, there is an obstacle to the amplification of the quantum fluctuations of the electromagnetic field during inflation: namely the conformal invariance of electromagnetic theory on a conformally flat underlying geometry. As the existence of a preferred minimal length breaks the conformal invariance of the background geometry, it is plausible that this effect could generate some electromagnetic field amplification. We show that this scenario is equivalent to endowing the photon with a large negative mass during inflation. This effective mass is negligibly small in a radiation and matter dominated universe. Depending on the value of the free parameter of the theory, we show that the seed required by the dynamo mechanism can be generated. We also show that this mechanism can produce the requisite galactic magnetic field without resorting to a dynamo mechanism. 
  The goal of this article is to present a broad perspective on quantum gravity for \emph{non-experts}. After a historical introduction, key physical problems of quantum gravity are illustrated. While there are a number of interesting and insightful approaches to address these issues, over the past two decades sustained progress has primarily occurred in two programs: string theory and loop quantum gravity. The first program is described in Horowitz's contribution while my article will focus on the second. The emphasis is on underlying ideas, conceptual issues and overall status of the program rather than mathematical details and associated technical subtleties. 
  The parity, time reversal and space/time exchange invariant actions, equations and their conjugate metric solutions are obtained in the context of a general relativistic model modified in order to take into account discrete symmetries. The equations are not covariant and the PPN formalism breaksdown however the new Schwarzschild metric solution in vacuum only starts to differ from that of General Relativity at the Post-Post-Newtonian order for a source at rest relative to the CMB. Preferred frame gravitomagnetic effects well above expectations in the PPN formalism and within the accuracy of the Gravity Probe B experiment are predicted. No coordinate singularity (black hole) arises in the privileged frame where the energy of gravity is found to vanish. The context is very promising to help the cancelation of vacuum energies as gravitational sources. A flat universe accelerated expansion phase is obtained without resorting to inflation nor a cosmological constant and the Big-Bang singularity is avoided. The Pioneer anomalous blue-shift is a natural outcome. The context is also promising to help us elucidate several outstanding enigmas such as flat galactic rotation curves or the universe voids. A wave solution is obtained leading to the same binary pulsar decay rate as observed and predicted in GR. 
  The nature of cosmological solutions for a homogeneous, anisotropic Universe given by a Bianchi type-I (BI) model in the presence of a Cosmological constant $\Lambda$ is investigated by taking into account dissipative process due to viscosity. The system in question is thoroughly studied both analytically and numerically. It is shown the viscosity, as well as the $\Lambda$ term exhibit essential influence on the character of the solutions. In particular a negative $\Lambda$ gives rise to an ever-expanding Universe, whereas, a suitable choice of initial conditions plus a positive $\Lambda$ can result in a singularity-free oscillatory mode of expansion. For some special cases it is possible to obtain oscillations in the exponential mode of expansion of the BI model even with a negative $\Lambda$, where oscillations arise by virtue of viscosity. 
  Gravitational wave bursters are sources which emit repeatedly bursts of gravitational waves, and have been recently suggested as potentially interesting candidates for gravitational wave (GW) detectors. Mechanisms that could give rise to a GW burster can be found for instance in highly magnetized neutron stars (the magnetars which explain the phenomenon of soft gamma repeaters), in accreting neutron stars and in hybrid stars with a quark core. We point out that these sources have very distinctive experimental signatures. In particular, as already observed in the gamma-ray bursts from soft gamma repeaters, the energy spectrum of the events is a power-law, dN\sim E^{-\gamma}dE with \gamma\simeq 1.6, and they have a distribution of waiting times (the times between one outburst and the next) significantly different from the distribution of uncorrelated events. We discuss possible detection strategies that could be used to search for these events in existing gravitational wave detectors. 
  The indefinite sign of the Hamiltonian constraint means that solutions to Einstein's equations must achieve a delicate balance--often among numerically large terms that nearly cancel. If numerical errors cause a violation of the Hamiltonian constraint, the failure of the delicate balance could lead to qualitatively wrong behavior rather than just decreased accuracy. This issue is different from instabilities caused by constraint-violating modes. Examples of stable numerical simulations of collapsing cosmological spacetimes exhibiting local mixmaster dynamics with and without Hamiltonian constraint enforcement are presented. 
  We extend the problem of $(1+1)$ circular dilaton gravity to include charged particles. We examine the two (charged) particle case in detail and find an exact equilibrium solution. We then extend this to $N-$particles and obtain a solution for this case as well. This class of solutions corresponds to $N$-particles of the same mass, spaced evenly around the circle with charges chosen so that the electric field satisfies $E^{2}=$constant. We discuss the relation of these solutions to the previous uncharged equilibrium solutions and examine the behavior when the number of particles is large. We comment on the challenges in further generalizing the solutions we obtain. 
  We investigate the resonant interaction to the weak gravitational waves in a coupling electromagnetic system, which consists of a Gaussian beam with the double polarized transverse electric modes, a static magnetic field and the fractal membranes. We find that under the syncroresonance condition a high-frequency GW (HFGW) of h=10^-30,v_g=3GHz may produce the perturbative photon flux (PPF) of 2.15*10/s in a surface of 0.01m^2. The PPF can be pumped out from the background photon fluxes and one might obtain the amplified signal photon flux of 2.15*10^4s^-1 by cascade fractal membranes. It appears to be worthwhile to study this effect for the detection of the high-frequency relic GWs in quintessential inflationary models and the HFGWs expected by possible laboratory schemes. 
  Using appropriate harmonics, we study the future asymptotic behavior of massless scalar fields on a class of cosmological vacuum spacetimes. The spatial manifold is assumed to be a circle bundle over a higher genus surface with a locally homogeneous metric. Such a manifold corresponds to the SL(2,R)-geometry (Bianchi VIII type) or the H^2 x R-geometry (Bianchi III type). After a technical preparation including an introduction of suitable harmonics for the circle-fibered Bianchi VIII to separate variables, we derive systems of ordinary differential equations for the scalar field. We present future asymptotic solutions for these equations in a special case, and find that there is a close similarity with those on the circle-fibered Bianchi III spacetime. We discuss implications of this similarity, especially to (gravitational) linear perturbations. We also point out that this similarity can be explained by the "fiber term dominated behavior" of the two models. 
  Keeping Einstein's equations in second order form can be appealing for computational efficiency, because of the reduced number of variables and constraints. Stability issues emerge, however, which are not present in first order formulations. We show that a standard discretization of the second order ``shifted'' wave equation leads to an unstable semi-discrete scheme if the shift parameter is too large. This implies that discretizations obtained using integrators such as Runge-Kutta, Crank-Nicholson, leap-frog are unstable for any fixed value of the Courant factor. We argue that this situation arises in numerical relativity, particularly in simulations of spacetimes containing black holes, and discuss several ways of circumventing this problem. We find that the first order reduction in time based on ``ADM'' type variables is very effective. 
  The gauge-independent invariant approach to investigation of the linear scalar perturbations of inflaton and gravitational fields is developed in self-consistent way. This approach allows to compare various gauges used by other researchers and to find unambiguous selection criteria of physical and coordinate effects. We have shown that the so-called longitudinal gauge commonly used for studying the gravitational instability leads to overestimation of physical effects due to the presence of nonphysical proper time perturbations. Equation of invariant dynamics (EID) is derived. The general long-wave solution of EID for an arbitrary potential U(phi) has been obtained. We have also found analytical solutions for all wave lengths at all stages of the universe evolution in the framework of simplest potential U(phi)=m^2*phi^2/2. We have constructed the analytical expressions for the energy density perturbations spectrum Delta(k,t) at all possible k and t. Amplitude of the long-wave spectrum in the case of the transition from short waves to long ones occurs at the inflationary stage is almost flat, i.e. has the Harrison-Zeldovich form, for arbitrary potential U(phi). 
  We apply the variable speed of light into general relativity in order to solve the problems we met in the standard cosmology. We're surprised to find that, the results from the general relativity in cosmology are exactly the same as those we got from Newtonian dynamics. The relation between the Newtonian dynamics and the relativistic dynamics can be demonstrated with the variable speed of light. With this approach, some problems in the standard cosmology such as the flatness problem and the horizon problem doesn't arise any more. All the cosmological results and the physical results are reasonable and natural. There are no any difficulties in the standard cosmology. 
  In this paper we study the evolution of a LRS Bianchi I Universe, filled with a bulk viscous cosmological fluid in the presence of time varying constants "but" taking into account the effects of a c-variable into the curvature tensor. We find that the only physical models are those which ``constants'' $G$ and $c$ are growing functions on time $t$, while the cosmological constant $\Lambda$ is a negative decreasing function. In such solutions the energy density obeys the ultrastiff matter equation of state i.e. $\omega=1$. 
  We present a general formalism for studying an inflationary scenario in the two-brane system. We explain the gradient expansion method and obtain the 4-dimensional effective action. Based on this effective action, we give a basic equations for the background homogeneous universe and the cosmological perturbations. As an application, we propose a born-again braneworld scenario. 
  We study a codimension 2 braneworld in the Einstein-Gauss-Bonnet gravity. In the linear regime, we show the conventional Einstein gravity can be recovered on the brane. While, in the nonlinear regime, we find corrections due to the thickness and the bulk geometry. 
  A mathematical mistake due to an unjustified formal assumption in the variational principle of general relativity theory, which obviously has confused Einstein as well as Hilbert, is shown and cleared up here.   Thereby the 'amicable controversy' of spring 1915 with Levi-Civita - otherwise still confusing - seems finally dissolved.   In particular, it is proved that also Hilbert could not have found the correct Einstein tensor by a practicable calculation from the basic relations of his widely discussed "first note" (published 1916) straightforwardly. 
  It is usually accepted that General Relativity is the only consistent theory which can be obtained starting from the linear Fiertz-Pauli Lagrangian. It is the aim of the present paper to study whether, under certain requirements, a different and consistent theory can be found. These requirements will be the common ones encountered in flat field theory: removal of the non physical degrees of freedom and conservation of the energy momentum currents determined by Noether's Theorem. It will be shown that imposing certain constraint (related to the elimination of the undesired components of the reducible representation) on the field manifold, a consistent theory (at least to first order in nonlinearties) is achieved. The theory obtained proceeding this way is characterizd, to the lowest nonlinear order, by certain parameter epsilon. General Relativity's corresponding term is found to be a limit case when the parameter tends to zero. So, epsilon measures at this level the size of the breaking of the global symmetry appearing in General Relativity i.e. diff.invariance. It remains as open questions the matters of the theory's solvability to all orders and the appearence of it's quantized version. 
  We study manifolds with Lorentzian signature and prove that all scalar curvature invariants of all orders vanish in a higher-dimensional Lorentzian spacetime if and only if there exists an aligned non-expanding, non-twisting, geodesic null direction along which the Riemann tensor has negative boost order. 
  A scheme for calculating corrections to all orders to the entropy of any thermodynamic system due to statistical fluctuations around equilibrium has been developed. It is then applied to the BTZ black hole, AdS-Schwarzschild black Hole and Schwarzschild black Hole in a cavity. The scheme that we present is a model-independent scheme and hence universally applicable to all classical black holes with positive specific heat. It has been seen earlier that the microcanonical entropy of a system can be more accurately reproduced by considering a logarithmic correction to the canonical entropy function. The higher order corrections will be a step further in calculating the microcanonical entropy of a black hole. 
  Two mechanisms for nonlinear mode saturation of the r-mode in neutron stars have been suggested: the parametric instability mechanism involving a small number of modes and the formation of a nearly continuous Kolmogorov-type cascade. Using a network of oscillators constructed from the eigenmodes of a perfect fluid incompressible star, we investigate the transition between the two regimes numerically. Our network includes the 4995 inertial modes up to n<= 30 with 146,998 direct couplings to the r-mode and 1,306,999 couplings with detuning< 0.002 (out of a total of approximately 10^9 possible couplings).  The lowest parametric instability thresholds for a range of temperatures are calculated and it is found that the r-mode becomes unstable to modes with 13<n<15. In the undriven, undamped, Hamiltonian version of the network the rate to achieve equipartition is found to be amplitude dependent, reminiscent of the Fermi-Pasta-Ulam problem. More realistic models driven unstable by gravitational radiation and damped by shear viscosity are explored next. A range of damping rates, corresponding to temperatures 10^6K to 10^9K, is considered. Exponential growth of the r-mode is found to cease at small amplitudes, approximately 10^-4. For strongly damped, low temperature models, a few modes dominate the dynamics. The behavior of the r-mode is complicated, but its amplitude is still no larger than about 10^-4 on average. For high temperature, weakly damped models the r-mode feeds energy into a sea of oscillators that achieve approximate equipartition. In this case the r-mode amplitude settles to a value for which the rate to achieve equipartition is approximately the linear instability growth rate. 
  In the present work we show that planetary mean distances can be calculated through considering the Weyl geometry. We interpret the Weyl gauge field as a vector field associated with the hypercharge of the particles and apply the gauge concept of the Weyl geometry. The results obtained are shown to agree with the observed orbits of all the planets and of the asteroid belt in the solar system, with some empty states. 
  We present an exact expression for the quasinormal modes of acoustic disturbances in a rotating 2+1 dimensional sonic black hole (draining bathtub fluid flow) in the low frequency limit and evaluate the adiabatic invariant proposed by Kunstatter. We also compute,via Bohr-Sommerfeld quantization rule the equivalent area spectrum for this acoustic black hole, and we compute the superradiance phenomena for pure spinning 2+1 black holes. 
  The inclusion of a $\Lambda $ term, today, in the relativistic field equations is consequence of the evolution of an observational cosmology. In this work we argue the "ansatz" $\Lambda = \beta H^2 +\frac{\alpha}{R^2}$ is applicable to the total energy density of the universe, discuss some cosmological consequences and compare the growing mode of the density contrast with the corresponding mode of the standard model. 
  A scheme, in which gravitons are super-strong interacting, is described. The graviton background with the Planckian spectrum and a small effective temperature is considered as a reservoir of gravitons. A cross-section of interaction of a graviton with any particle is assumed to be a bilinear function of its energies. Any pair of bodies are attracting not due to an exchange with its own gravitons, but due to a pressure of external gravitons of this background. A graviton pairing is necessary to obtain classical gravity. Any divergencies are not possible in such the model because of natural smooth cut-offs of the graviton spectrum from both sides. Some cosmological consequences of this scheme are discussed, too. Also it is shown here that the main conjecture of this approach may be verified at present on the Earth. 
  This paper is a supplement of our earlier work JHEP 0410 (2004) 011[gr-qc/0409107].We map the vector potential of charged black holes into GEMS and find that its effect on the thermal spectrum is the same as that on the black hole side, i.e., it will induce a chemical potential in the thermal spectrum which is the same as that in the charged black holes.We also argue that the generalization of GEMS approach to non-stationary motions is not possible. 
  In a recent paper, Mallett found a solution of the Einstein equations in which closed timelike curves (CTC's) are present in the empty space outside an infinitely long cylinder of light moving in circular paths around an axis. Here we show that, for physically realistic energy densities, the CTC's occur at distances from the axis greater than the radius of the visible universe by an immense factor. We then show that Mallett's solution has a curvature singularity on the axis, even in the case where the intensity of the light vanishes. Thus it is not the solution one would get by starting with Minkowski space and establishing a cylinder of light. 
  The family of generalized-harmonic gauge conditions, which is currently used in Numerical Relativity for its singularity-avoidant behavior, is analyzed by looking for pathologies of the corresponding spacetime foliation. The appearance of genuine shocks, arising from the crossing of characteristic lines, is completely discarded. Runaway solutions, meaning that the lapse function can grow without bound at an accelerated rate, are instead predicted. Black Hole simulations are presented, showing spurious oscillations due to the well known slice stretching phenomenon. These oscillations are made to disappear by switching the numerical algorithm to a high-resolution shock-capturing one, of the kind currently used in Computational Fluid Dynamics. Even with these shock-capturing algorithms, runaway solutions are seen to appear and the resulting lapse blow-up is causing the simulations to crash. As a side result, a new method is proposed for obtaining regular initial data for Black Hole spacetimes, even inside the horizons. 
  The dynamics of relativistic thin shells is a recurrent topic in the literature about the classical theory of gravitating systems and the still ongoing attempts to obtain a coherent description of their quantum behavior. Certainly, a good reason to make this system a preferred one for many models is the clear, synthetic description given by Israel junction conditions. Using some results from an effective Lagrangian approach to the dynamics of spherically symmetric shells, we show a general way to obtain WKB states for the system; a simple example is also analyzed. 
  We introduce horizon pretracking as a method for analysing numerically generated spacetimes of merging black holes. Pretracking consists of following certain modified constant expansion surfaces during a simulation before a common apparent horizon has formed. The tracked surfaces exist at all times, and are defined so as to include the common apparent horizon if it exists. The method provides a way for finding this common apparent horizon in an efficient and reliable manner at the earliest possible time. We can distinguish inner and outer horizons by examining the distortion of the surface. Properties of the pretracking surface such as its expansion, location, shape, area, and angular momentum can also be used to predict when a common apparent horizon will appear, and its characteristics. The latter could also be used to feed back into the simulation by adapting e.g. boundary or gauge conditions even before the common apparent horizon has formed. 
  We study the behaviour of spinning test particles in the Schwarzschild spacetime. Using Mathisson-Papapetrou equations of motion we confine our attention to spatially circular orbits and search for observable effects which could eventually discriminate among the standard supplementary conditions namely the Corinaldesi-Papapetrou, Pirani and Tulczyjew. We find that if the world line chosen for the multipole reduction and whose unit tangent we denote as $U$ is a circular orbit then also the generalized momentum $P$ of the spinning test particle is tangent to a circular orbit even though $P$ and $U$ are not parallel four-vectors. These orbits are shown to exist because the spin induced tidal forces provide the required acceleration no matter what supplementary condition we select. Of course, in the limit of a small spin the particle's orbit is close of being a circular geodesic and the (small) deviation of the angular velocities from the geodesic values can be of an arbitrary sign, corresponding to the possible spin-up and spin-down alignment to the z-axis. When two spinning particles orbit around a gravitating source in opposite directions, they make one loop with respect to a given static observer with different arrival times. This difference is termed clock effect. We find that a nonzero gravitomagnetic clock effect appears for oppositely orbiting both spin-up or spin-down particles even in the Schwarzschild spacetime. This allows us to establish a formal analogy with the case of (spin-less) geodesics on the equatorial plane of the Kerr spacetime. This result can be verified experimentally. 
  We study the motion of spinning test particles in Kerr spacetime using the Mathisson-Papapetrou equations; we impose different supplementary conditions among the well known Corinaldesi-Papapetrou, Pirani and Tulczyjew's and analyze their physical implications in order to decide which is the most natural to use. We find that if the particle's center of mass world line, namely the one chosen for the multipole reduction, is a spatially circular orbit (sustained by the tidal forces due to the spin) then the generalized momentum $P$ of the test particle is also tangent to a spatially circular orbit intersecting the center of mass line at a point. There exists one such orbit for each point of the center of mass line where they intersect; although fictitious, these orbits are essential to define the properties of the spinning particle along its physical motion. In the small spin limit, the particle's orbit is almost a geodesic and the difference of its angular velocity with respect to the geodesic value can be of arbitrary sign, corresponding to the spin-up and spin-down possible alignment along the z-axis. We also find that the choice of the supplementary conditions leads to clock effects of substantially different magnitude. In fact, for co-rotating and counter-rotating particles having the same spin magnitude and orientation, the gravitomagnetic clock effect induced by the background metric can be magnified or inhibited and even suppressed by the contribution of the individual particle's spin. Quite surprisingly this contribution can be itself made vanishing leading to a clock effect undistiguishable from that of non spinning particles. The results of our analysis can be observationally tested. 
  This work proposes an explanation of the Pioneer anomaly, the unmodelled and as yet unexplained blueshift detected in the microwave signal of the Pioneer 10 and other spaceships by Anderson {\it et al} in 1998. What they observed is similar to the effect that would have either (i) an anomalous acceleration $a_{\rm P}$ of the ship towards the Sun or (ii) an acceleration of the clocks $a_{\rm t}=a_{\rm P}/c$. The second alternative is investigated here, with a phenomenological model in which the anomaly is an effect of the background gravitational potential $\Psi (t)$ that pervades all the universe and is increasing because of the expansion. It is shown that $2a_{\rm t}={\rm d}\Psi /{\rm d}t ={\rm d}^2\tau_{\rm clocks} /{\rm d}t^2$, evaluated at present time $t_0$, where $t$ and $\tau_{\rm clocks}$ are the coordinate time and the time measured by the clocks, respectively. The result of a simple estimate gives the value $a_{\rm t}\simeq 1.8\times 10^{-18}{s}^{-1}$, while Anderson {\it et al} suggested $a_{\rm t}= (2.9\pm 0.4)\times 10^{-18}{s}^{-1}$ on the basis of their observations. The calculation are performed near the Newtonian limit but in the frame of general relativity. 
  We show that the field equations of the Schwarzschild geometry are invariant under passive Lorentz transformations to a freely falling system. We decompose the field equations with respect to the accelerated system and find that the force of gravity is not transformed away but dynamically compensated. 
  It is commonplace, in the literature, to find that the Bekenstein-Hawking entropy has been endowed with having an explicit statistical interpretation. In the following essay, we discuss why such a viewpoint warrants a certain degree of caution. 
  Firstly, we review the pointwise and averaged energy conditions, the quantum inequality and the notion of the ``volume integral quantifier'', which provides a measure of the ``total amount'' of energy condition violating matter. Secondly, we present a specific metric of a spherically symmetric traversable wormhole in the presence of a generic cosmological constant, verifying that the null and the averaged null energy conditions are violated, as was to be expected. Thirdly, a pressureless dust shell is constructed around the interior wormhole spacetime by matching the latter geometry to a unique vacuum exterior solution. In order to further minimize the usage of exotic matter, we then find regions where the surface energy density is positive, thereby satisfying all of the energy conditions at the junction surface. An equation governing the behavior of the radial pressure across the junction surface is also deduced. Lastly, taking advantage of the construction, specific dimensions of the wormhole, namely, the throat radius and the junction interface radius, and estimates of the total traversal time and maximum velocity of an observer journeying through the wormhole, are also found by imposing the traversability conditions. 
  We consider wormhole geometries subject to a gravitational action consisting of non-linear powers of the Ricci scalar. Specifically, wormhole throats are studied in the case where Einstein gravity is supplemented with a Ricci-squared and inverse Ricci term. In this modified theory it is found that static wormhole throats respecting the weak energy condition can exist. The analysis is done locally in the vicinity of the throat, which eliminates certain restrictions on the models introduced by considering the global topology. 
  We present a method for constructing stationary, asymptotically flat, rotating solutions of Einstein's field equations. One of the spun-up solutions has quasilocal mass but no global mass. It has an ergosphere but no event horizon. The angular momentum is constant everywhere beyond the ergosphere. The energy-momentum content of this solution can be interpreted as a rotating string-fluid. 
  We study the Friedmann-Robertson-Walker cosmological model to investigate non-smooth curvatures associated with multiple discontinuities involved in the evolution of the universe, by exploiting Lorentzian warped product scheme. Introducing double discontinuities occurred at the radiation-matter and matter-lambda phase transitions, we also discuss non-smooth features of the spatially flat Friedmann-Robertson-Walker universe. 
  We construct new regular solutions in Einstein-Yang-Mills theory. They are static, axially symmetric and asymptotically flat. They are characterized by a pair of integers (k,n), where k is related to the polar angle and $n$ to the azimuthal angle. The known spherically and axially symmetric EYM solutions have k=1. For k>1 new solutions arise, which form two branches. They exist above a minimal value of n, that increases with k. The solutions on the lower mass branch are related to certain solutions of Einstein-Yang-Mills-Higgs theory, where the nodes of the Higgs field form rings. 
  Improved Wentzel-Kramers-Brillouin (WKB)-type approximations are presented in order to study cosmological perturbations beyond the lowest order. Our methods are based on functions which approximate the true perturbation modes over the complete range of the independent (Langer) variable, from sub-horizon to super-horizon scales, and include the region near the turning point. We employ both a perturbative Green's function technique and an adiabatic (or ``semiclassical'') expansion (for a linear turning point) in order to compute higher order corrections. Improved general expressions for the WKB scalar and tensor power spectra are derived for both techniques. We test our methods on the benchmark of power-law inflation, which allows comparison with exact expressions for the perturbations, and find that the next-to-leading order adiabatic expansion yields the amplitude of the power spectra with excellent accuracy, whereas the next-to-leading order with the perturbative Green's function method does not improve the leading order result significantly. However, in more general cases, either or both methods may be useful. 
  The joint NASA-ESA mission LISA relies crucially on the stability of the three spacecraft constellation. Each of the spacecraft is in heliocentric orbits forming a stable triangle. The principles of such a formation flight have been formulated long ago and analysis performed, but seldom presented if ever, even to LISA scientists. We nevertheless need these details in order to carry out theoretical studies on the optical links, simulators etc. In this article, we present in brief, a model of the LISA constellation, which we believe will be useful for the LISA community. 
  The eigenmodes of the Poincar\'e dodecahedral 3-manifold $M$ are constructed as eigenstates of a novel invariant operator. The topology of $M$ is characterized by the homotopy group $\pi_1(M)$, given by loop composition on $M$, and by the isomorphic group of deck transformations $deck(\tilde{M})$, acting on the universal cover $\tilde{M}$. ($\pi_1(M)$, $\tilde{M}$) are known to be the binary icosahedral group ${\cal H}_3$ and the sphere $S^3$ respectively.   Taking $S^3$ as the group manifold $SU(2,C)$ it is shown that $deck(\tilde{M}) \sim {\cal H}^r_3$ acts on $SU(2,C)$ by right multiplication. A semidirect product group is constructed from ${\cal H}^r_3$ as normal subgroup and from a second group ${\cal H}^c_3$ which provides the icosahedral symmetries of $M$. Based on F. Klein's fundamental icosahedral ${\cal H}_3$-invariant, we construct a novel hermitian ${\cal H}_3$-invariant polynomial (generalized Casimir) operator ${\cal K}$. Its eigenstates with eigenvalues $\kappa$ quantize a complete orthogonal basis on Poincar\'{e}'s dodecahedral 3-manifold. The eigenstates of lowest degree $\lambda=12$ are 12 partners of Klein's invariant polynomial. The analysis has applications in cosmic topology \cite{LA},\cite{LE}. If the Poincar\'{e} 3-manifold $M$ is assumed to model the space part of a cosmos, the observed temperature fluctuations of the cosmic microwave background must admit an expansion in eigenstates of ${\cal K}$. 
  Binary black hole spacetimes with a helical Killing vector, which are discussed as an approximation for the early stage of a binary system, are studied in a projection formalism. In this setting the four dimensional Einstein equations are equivalent to a three dimensional gravitational theory with a $SL(2,\mathbb{C})/SO(1,1)$ sigma model as the material source. The sigma model is determined by a complex Ernst equation. 2+1 decompositions of the 3-metric are used to establish the field equations on the orbit space of the Killing vector. The two Killing horizons of spherical topology which characterize the black holes, the cylinder of light where the Killing vector changes from timelike to spacelike, and infinity are singular points of the equations. The horizon and the light cylinder are shown to be regular singularities, i.e. the metric functions can be expanded in a formal power series in the vicinity. The behavior of the metric at spatial infinity is studied in terms of formal series solutions to the linearized Einstein equations. It is shown that the spacetime is not asymptotically flat in the strong sense to have a smooth null infinity under the assumption that the metric tends asymptotically to the Minkowski metric. In this case the metric functions have an oscillatory behavior in the radial coordinate in a non-axisymmetric setting, the asymptotic multipoles are not defined. The asymptotic behavior of the Weyl tensor near infinity shows that there is no smooth null infinity. 
  We find that the expansion of the universe is accelerating by analyzing the recent observation data of type $\textsc{I}a$ supernova(SN-Ia) .It indicates that the equation of state of the dark energy might be smaller than -1,which leads to the introduction of phantom models featured by its negative kinetic energy to account for the regime of equation of state parameter $w<-1$.In this paper the possibility of using a non-minimally coupled real scalar field as phantom to realize the equation of state parameter $w<-1$ is discussed.The main equations which govern the evolution of the universe are obtained.Then we rewrite them with the observable quantities. 
  The scalar-tensor theory of gravitation has been and still is one of the most widely discussed "alternative theories" to General Relativity (GR). Despite nearly half a century of its age, it continues to attract renewed interests of not only theorists but also experimentalists when we now face such issues like the accelerating universe and possible time-variability of the fine-structure constant, both viewed as something beyond the standard GR. It appears that the theory provides realistic results sometimes even beyond what is expected from the quintessence approach aimed primarily to be more phenomenological. It seems nevertheless as if some of the unique aspects of this theory are not fully understood, even leading to occasional confusions. I try in my lectures, partly using the contents of our book (Y.F. and K. Maeda, Scalar-tensor theory of gravitation, Cambridge University Press, 2003), to discuss some of the most crucial concepts starting from elementary introduction to the theory. Particular emphases will be placed on the unique features of the nonminimal coupling term, the roles of the conformal transformations together with the choice of a physical conformal frame and the value of the coupling strength to the matter. Readers are advised to refer to the references for more details. 
  We consider the static vacuum C metric that represents the gravitational field of a black hole of mass $m$ undergoing uniform translational acceleration $A$ such that $mA<1/(3\sqrt{3})$. The influence of the inertial acceleration on the exterior perturbations of this background are investigated. In particular, we find no evidence for a direct spin-acceleration coupling. 
  We consider gravitational self interaction in the lowest approximation and assume that graviton interacts with gravitational energy-momentum tensor in the same way as it interacts with particles. We show that, using gravitational vertex with a preferred gravitational energy-momentum tensor, it is possible to obtain a metric necessary for explaining perihelion precession. The preferred gravitational energy-momentum tensor gives positive gravitational energy density of Newtonian center. We show also that, employing "improvement" technique, any gravitational energy-momentum tensor can be made suitable for using in gravitational wave equation for obtaining metric which explains perihelion precession. Yet the "improvement" leads to negative gravitational energy density of the Newtonian center. 
  The divergence of the constraint quantities is a major problem in computational gravity today. Apparently, there are two sources for constraint violations. The use of boundary conditions which are not compatible with the constraint equations inadvertently leads to 'constraint violating modes' propagating into the computational domain from the boundary. The other source for constraint violation is intrinsic. It is already present in the initial value problem, i.e. even when no boundary conditions have to be specified. Its origin is due to the instability of the constraint surface in the phase space of initial conditions for the time evolution equations. In this paper, we present a technique to study in detail how this instability depends on gauge parameters. We demonstrate this for the influence of the choice of the time foliation in context of the Weyl system. This system is the essential hyperbolic part in various formulations of the Einstein equations. 
  We discuss the question, whether the Reissner-Nordstr\"{o}m RN) metric can be glued to another solutions of Einstein-Maxwell equations in such a way that (i) the singularity at r=0 typical of the RN metric is removed (ii), matching is smooth. Such a construction could be viewed as a classical model of an elementary particle balanced by its own forces without support by an external agent. One choice is the Minkowski interior that goes back to the old Vilenkin and Fomin's idea who claimed that in this case the bare delta-like stresses at the horizon vanish if the RN metric is extremal. However, the relevant entity here is the integral of these stresses over the proper distance which is infinite in the extremal case. As a result of the competition of these two factors, the Lanczos tensor does not vanish and the extremal RN cannot be glued to the Minkowski metric smoothly, so the elementary-particle model as a ball empty inside fails. We examine the alternative possibility for the extremal RN metric - gluing to the Bertotti-Robinson (BR) metric. For a surface placed outside the horizon there always exist bare stresses but their amplitude goes to zero as the radius of the shell approaches that of the horizon. This limit realizes the Wheeler idea of "mass without mass" and "charge without charge". We generalize the model to the extremal Kerr-Newman metric glued to the rotating analog of the BR metric. 
  The geometry around a rotating massive body, which carries charge and electrical currents, could be described by its multipole moments (mass moments, mass-current moments, electric moments, and magnetic moments). When a small body is orbiting this massive body, it will move on geodesics, at least for a time interval that is short with respect to the characteristic time of the binary due to gravitational radiation. By monitoring the waves emitted by the small body we are actually tracing the geometry of the central object, and hence, in principle, we can infer all its multipole moments. This paper is a generalization of previous similar results by Ryan. The fact that the electromagnetic moments of spacetime can be measured demonstrates that one can obtain information about the electromagnetic field purely from gravitational wave analysis. Additionally, these measurements could be used as a test of the no-hair theorem for black holes. 
  It has often been suggested (especially by Carlip) that spacetime symmetries in the neighborhood of a black hole horizon may be relevant to a statistical understanding of the Bekenstein-Hawking entropy. A prime candidate for this type of symmetry is that which is exhibited by the Einstein tensor. More precisely, it is now known that this tensor takes on a strongly constrained (block-diagonal) form as it approaches any stationary, non-extremal Killing horizon. Presently, exploiting the geometrical properties of such horizons, we provide a particularly elegant argument that substantiates this highly symmetric form for the Einstein tensor. It is, however, duly noted that, on account of a "loophole", the argument does fall just short of attaining the status of a rigorous proof. 
  We propose a structure called a causal site to use as a setting for quantum geometry, replacing the underlying point set. The structure has an interesting categorical form, and a natural "tangent 2-bundle," analogous to the tangent bundle of a smooth manifold. Examples with reasonable finiteness conditions have an intrinsic geometry, which can approximate classical solutions to general relativity. We propose an approach to quantization of causal sites as well. 
  We present a physically reasonable source for an static, axially--symmetric solution to the Einstein equations. Arguments are provided, supporting our belief that the exterior space--time produced by such source, describing a quadrupole correction to the Schwarzschild metric, is particularly suitable (among known solutions of the Weyl family) for discussing the properties of quasi--spherical gravitational fields. 
  Regarding long life particles produced during preheating after inflation as dark matter, we find that its back reaction on the field $\phi$ could lock $\phi$ in a false vacuum up to today. This false vacuum can drive the accelerated expansion of universe at late time and play the role of dark energy. When the number density of dark matter particles is dilute to some value, the field $\phi$ becomes tachyonic and rolls to its true minima rapidly, and the acceleration of universe ceases. We discuss the constraints on the parameters of model from the observations of dark energy and dark matter halos on subgalactic scale. 
  The study of the quasinormal modes (QNMs) of the 2+1 dimensional rotating draining bathtub acoustic black hole, the closest analogue found so far to the Kerr black hole, is performed. Both the real and imaginary parts of the quasinormal (QN) frequencies as a function of the rotation parameter B are found through a full non-linear numerical analysis. Since there is no change in sign in the imaginary part of the frequency as B is increased we conclude that the 2+1 dimensional rotating draining bathtub acoustic black hole is stable against small perturbations. 
  Both scalar and Dirac quasinormal modes in Garfinkle-Horowitz-Strominger black hole spacetime are studied by using the WKB approximation and the P\"{o}schl-Teller approximation. For scalar field, we find that the QNMs with higher dilatons decay more rapidly than that with lower ones. However, this is not the case for mass $m$. Fields with higher mass will decay more slowly. The similar behaviors appear in the case of Dirac field. We also find that QNM frequencies evaluated by using WKB approximation and P\"{o}schl-Teller approximation have a good agreement with each other when mode number $n$ is small. 
  A closed form analytic solution is found for the electromagnetic field of the charged uniformly rotating conducting disk for all values of the tip speed $v$ up to $c$. For $v=c$ it becomes the Magic field of the Kerr-Newman black hole with $G$ set to zero.   The field energy, field angular momentum and gyromagnetic ratio are calculated and compared with those of the electron.   A new mathematical expression that sums products of 3 Legendre functions each of a different argument, is demonstrated. 
  In this brief note some comments about the observable used in a recently published paper on the measurement of the general relativistic Lense-Thirring in the gravitational field of the Earth are presented. It turns out that, among other things, the authors might have yielded an optimistic evaluation of the error budget because of an underestimation of the impact of the secular variations of the even zonal harmonics of the geopotential. More realistic evaluations point towards a 15-45% error at 1-3sigma level, respectively. 
  Following past investigations, we explore the symmetries of the Hamilton-Jacobi cosmological equations in the generalized patch formalism describing braneworld and tachyon scenarios. Dualities between different patches are established and regular dual solutions, either contracting or phantom-like, are constructed. 
  The quantized Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW) model minimally coupled to a free massless scalar field is studied and interpreted in the Bohm-de Broglie framework. We analyze the quantum bohmian trajectories corresponding to a certain class of gaussian packets, solutions of the Wheeler-DeWitt equation. We show that these bohmian trajectories undergo an accelerated expansion in the middle of its evolution due to the presence of quantum cosmological effects in this period. It is shown that the luminosity-redshift relation in the quantum cosmological model can be made close to the corresponding relation coming from the classical model suplemented by a cosmological constant, for $z<1$. In this way we have the posibility of interpreting the present observations of high redshift supernovae as the consequence of a quantum cosmological effect. 
  In this article we develop a physical interpretation for the deformed (doubly) special relativity theories (DSRs), based on a modification of the theory of measurement in special relativity. We suggest that it is useful to regard the DSRs as reflecting the manner in which quantum gravity effects induce Planck-suppressed distortions in the measurement of the "true" energy and momentum. This interpretation provides a framework for the DSRs that is manifestly consistent, non-trivial, and in principle falsifiable. However, it does so at the cost of demoting such theories from the level of "fundamental" physics to the level of phenomenological models -- models that should in principle be derivable from whatever theory of quantum gravity one ultimately chooses to adopt. 
  Levi-Civita spacetimes have classical naked singularities. They also have quantum singularities. Quantum singularities in general relativistic spacetimes are determined by the behavior of quantum test particles. A static spacetime is said to be quantum mechanically singular if the spatial portion of the wave operator is not essentially self-adjoint on a $C_{0}^{\infty}$ domain in $L^{2}$, a Hilbert space of square integrable functions. Here we summarize how Weyl's limit point-limit circle criterion can be used to determine whether a wave operator is essentially self-adjoint and how this test can then be applied to scalar wave packets in Levi-Civita spacetimes with and without a cosmological constant to help elucidate the physical properties of these spacetimes. 
  In a previous paper, based on the black hole perturbation approach, we formulated a new analytical method for regularizing the self-force acting on a particle of small mass $\mu$ orbiting a Schwarzschild black hole of mass $M$, where $\mu\ll M$. In our method, we divide the self-force into the $\tilde S$-part and $\tilde R$-part. All the singular behaviors are contained in the $\tilde S$-part, and hence the $\tilde R$-part is guaranteed to be regular. In this paper, focusing on the case of a scalar-charged particle for simplicity, we investigate the precision of both the regularized $\tilde S$-part and the $\tilde R$-part required for the construction of sufficiently accurate waveforms for almost circular inspiral orbits. For the regularized $\tilde S$-part, we calculate it for circular orbits to 18 post-Newtonian (PN) order and investigate the convergence of the post-Newtonian expansion. We also study the convergence of the remaining $\tilde{R}$-part in the spherical harmonic expansion. We find that a sufficiently accurate Green function can be obtained by keeping the terms up to $\ell=13$. 
  We obtain the correct hamiltonian which describes the dynamics of classes of asymptotic open Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW) spacetimes, which includes Tolman geometries. We calculate the surface term that has to be added to the usual hamiltonian of General Relativity in order to obtain an improved hamiltonian with well defined functional derivatives. For asymptotic flat FLRW spaces, this surface term is zero, but for asymptotic negative curvature FLRW spaces it is not null in general. In the particular case of the Tolman geometries, they vanish. The surface term evaluated on a particular solution of Einstein's equations may be viewed as the ``energy'' of this solution with respect to the FLRW spacetime they approach asymptotically. Our results are obtained for a matter content described by a dust fluid, but they are valid for any perfect fluid, including the cosmological constant. 
  I make a review on the aplications of the Bohm-De Broglie interpretation of quantum mechanics to quantum cosmology. In the framework of minisuperspaces models, I show how quantum cosmological effects in Bohm's view can avoid the initial singularity, isotropize the Universe, and even be a cause for the present observed acceleration of the Universe. In the general case, we enumerate the possible structures of quantum space and time. 
  The post-Newtonian gravitoelectric secular rate of the mean anomaly M is worked out for a two-body system in the framework of the General Theory of Relativity. The possibility of using such an effect, which is different from the well known decrease of the orbital period due to gravitational wave emission, as a further post-Keplerian parameter in binary systems including one pulsar is examined. The resulting effect is almost three times larger than the periastron advance. E.g., for the recently discovered double pulsar system PSR J0737-3039 A+B it would amount to -47.79 deg yr^-1. This implies that it could be extracted from the linear part of a quadratic fit of the orbital phase because the uncertainties both in the linear drift due to the mean motion and in the quadratic shift due to the gravitational wave are smaller. The availability of such additional post-Keplerian parameter would be helpful in further constraining the General Theory of Relativity, especially for such systems in which some of the other post-Keplerian parameters can be measured with limited accuracy. Moreover, also certain pulsar-white dwarf binary systems, characterized by circular orbits like PSR B1855+09 and a limited number of measured post-Keplerian parameters, could be used for constraining competing theories of gravity. 
  Examples of nonsingular cosmological models are presented on the basis of exact solutions to multidimensional gravity equations. These examples involve pure imaginary scalar fields, or, in other terms, ``phantom'' fields with an unusual sign of the kinetic term in the Lagrangian. We show that, with such fields, hyperbolic nonsingular models with a cosmological bounce (unlike spherical and spatially flat models) emerge without special relations among the integration constants, i.e., without fine tuning. In such models, the extra-dimension scale factors as well as scalar fields evolve smoothly between different finite asymptotic values. Examples of theories which create phantom scalar fields are obtained from string-inspired multidimensional field models and from theories of gravity in integrable Weyl space-times. 
  The quantum behaviour of weak gravitational fields admits an adequate, albeit approximate, description by those graviton states in which the expectation values and fluctuations of the linearised gravitational field are small. Such states must approximate corresponding states in full quantum gravity. We analyse the nature of this approximation for the graviton vacuum state in the context of kinematical Loop Quantum Gravity (LQG) wherein the constraints are ignored. We identify the graviton vacuum state with kinematically non-normalizable, distributional states in LQG by demanding that relations between linearised operator actions on the former are mirrored by those of their non-linear counterparts on the latter. We define a semi- norm on the space of kinematical distributions and show that the identification is approximate upto distributions which are small in this semi-norm. We argue that our candidate states are annihilated by the linearised constraints (expressed as operators in the full theory) to leading order in the parameter characterising the approximation. This suggests the possibility, in a scheme such as ours, of solving the full constraints order by order in this parameter. The main drawback of our considerations is that they depend on certain auxilliary constructions which, though mathematically well defined, do not arise from physical insight. Our work is an attempt to implement an earlier proposal of Iwasaki and Rovelli. 
  The merger of two black holes is one of the most extraordinary events in the natural world. Made of pure gravity, the holes combine to form a single hole, emitting a strong burst of gravitational radiation. Ground-based detectors are currently searching for such bursts from holes formed in the evolution of binary stars, and indeed the very first gravitational wave event detected may well be a black-hole merger. The space-based LISA detector is being designed to search for such bursts from merging massive black holes in the centers of galaxies, events that would emit many thousands of solar masses of pure gravitational wave energy over a period of only a few minutes. To assist gravitational wave astronomers in their searches, and to be in a position to understand the details of what they see, numerical relativists are performing supercomputer simulations of these events. I review here the state of the art of these simulations, what we have learned from them so far, and what challenges remain before we have a full prediction of the waveforms to be expected from these events. 
  Space-borne interferometric gravitational-wave detectors, sensitive in the low-frequency (mHz) band, will fly in the next decade. In these detectors, the spacecraft-to-spacecraft light-travel times will necessarily be unequal and time-varying, and (because of aberration) will have different values on up- and down-links. In such unequal-armlength interferometers, laser phase noise will be canceled by taking linear combinations of the laser-phase observables measured between pairs of spacecraft, appropriately time-shifted by the light propagation times along the corresponding arms. This procedure, known as time-delay interferometry (TDI), requires an accurate knowledge of the light-time delays as functions of time. Here we propose a high-accuracy technique to estimate these time delays and study its use in the context of the Laser Interferometer Space Antenna (LISA) mission. We refer to this ranging technique, which relies on the TDI combinations themselves, as Time-Delay Interferometric Ranging (TDIR). For every TDI combination, we show that, by minimizing the rms power in that combination (averaged over integration times $\sim 10^4$ s) with respect to the time-delay parameters, we obtain estimates of the time delays accurate enough to cancel laser noise to a level well below the secondary noises. Thus TDIR allows the implementation of TDI without the use of dedicated inter-spacecraft ranging systems, with a potential simplification of the LISA design. In this paper we define the TDIR procedure formally, and we characterize its expected performance via simulations with the \textit{Synthetic LISA} software package. 
  The theory of symmetries of systems of coupled, ordinary differential equations (ODE's) is used to develop a concise algorithm for cartographing the space of solutions to vacuum Bianchi Einstein's Field Equations (EFE). The symmetries used are the well known automorphisms of the Lie algebra for the corresponding isometry group of each Bianchi Type, as well as the scaling and the time eparameterization symmetry. Application of the method to Type III results in: a) the recovery of all known solutions without prior assumption of any extra symmetry, b) the enclosure of the entire unknown part of the solution space into a single, second order ODE in terms of one dependent variable and c) a partial solution to this ODE. It is also worth-mentioning the fact that the solution space is seen to be naturally partitioned into three distinct, disconnected pieces: one consisting of the known Siklos (pp-wave) solution, another occupied by the Type III member of the known Ellis-MacCallum family and the third described by the aforementioned ODE in which an one parameter subfamily of the known Kinnersley geometries resides. Lastly, preliminary results reported show that the unknown part of the solution space for other Bianchi Types is described by a strikingly similar ODE, pointing to a natural operational unification as far as the problem of solving the cosmological EFE's is concerned. 
  In this paper we analyze the effect of recently proposed classes of sudden future singularities on causal geodesics of FLRW spacetimes. Geodesics are shown to be extendible and just the equations for geodesic deviation are singular, although tidal forces are not strong enough to produce a Big Rip. For the sake of completeness, we compare with the typical sudden future singularities of phantom cosmologies. 
  We study the classical and quantum theory of spherically symmetric spacetimes with scalar field coupling in general relativity. We utilise the canonical formalism of geometrodynamics adapted to the Painleve-Gullstrand coordinates, and present a new quantisation of the resulting field theory. We give an explicit construction of operators that capture curvature properties of the spacetime and use these to show that the black hole curvature singularity is avoided in the quantum theory. 
  In this thesis we investigate cosmological models more general than the isotropic and homogeneous Friedmann-Lemaitre models. We focus on cosmologies with one spatial degree of freedom, whose matter content consists of a perfect fluid and the cosmological constant. We formulate the Einstein field equations as a system of quasilinear first order partial differential equations, using scale-invariant variables.   The primary goal is to study the dynamics in the two asymptotic regimes, i.e. near the initial singularity and at late times. We highlight the role of spatially homogeneous dynamics as the background dynamics, and analyze the inhomogeneous aspect of the dynamics. We perform a variety of numerical simulations to support our analysis and to explore new phenomena. 
  The self-force describes the effect of a particle's own gravitational field on its motion. While the motion is geodesic in the test-mass limit, it is accelerated to first order in the particle's mass. In this contribution I review the foundations of the self-force, and show how the motion of a small black hole can be determined by matched asymptotic expansions of a perturbed metric. I next consider the case of a point mass, and show that while the retarded field is singular on the world line, it can be unambiguously decomposed into a singular piece that exerts no force, and a smooth remainder that is responsible for the acceleration. I also describe the recent efforts, by a number of workers, to compute the self-force in the case of a small body moving in the field of a much more massive black hole. The motivation for this work is provided in part by the Laser Interferometer Space Antenna, which will be sensitive to low-frequency gravitational waves. Among the sources for this detector is the motion of small compact objects around massive (galactic) black holes. To calculate the waves emitted by such systems requires a detailed understanding of the motion, beyond the test-mass approximation. 
  In this paper, a braneworld black hole is studied as a gravitational lens, using the strong field limit to obtain the positions and magnifications of the relativistic images. Standard lensing and retrolensing situations are analyzed in a unified setting, and the results are compared with those corresponding to the Schwarschild black hole lens. The possibility of observing the strong field images is discussed. 
  In brane-worlds, our universe is assumed to be a submanifold, or brane, embedded in a higher-dimensional bulk spacetime. Focusing on scenarios with a curved five-dimensional bulk spacetime, I discuss their gravitational and cosmological properties. 
  We present a brief review of brane-world models - models in which our observable Universe with its standard matter fields is assumed as localized on a domain wall (three-brane) in a higher dimensional surrounding (bulk) spacetime. Models of this type arise naturally in M-theory and have been intensively studied during the last years. We pay particular attention to the covariant projection approach, the Cardassian scenario, to induced gravity models, self-tuning models and the Ekpyrotic scenario. A brief discussion is given of their basic properties and their connection with conventional FRW cosmology. 
  We study the Robertson-Walker minisuperspace model in histories theory, motivated by the results that emerged from the histories approach to general relativity. We examine, in particular, the issue of time-reparameterisation in such systems. The model is quantised using an adaptation of reduced-state-space quantisation. We finally discuss the classical limit, the implementation of initial cosmological conditions and estimation of probabilities in the histories context. 
  Spinor gravity is a functional integral formulation of gravity based only on fundamental spinor fields. The vielbein and metric arise as composite objects. Due to the lack of local Lorentz-symmetry new invariants in the effective gravitational action lead to a modification of Einstein's equations. We discuss different geometrical viewpoints of spinor gravity with particular emphasis on nonlinear fields. The effective gravitational field equations arise as solutions to the lowest order Schwinger-Dyson equation for spinor gravity. 
  We calculate the angles of deflection of high speed particles projected in an arbitrary direction into the Kerr gravitational field. This is done by first calculating the light-like boost of the Kerr gravitational field in an arbitrary direction and then using this boosted gravitational field as an approximation to the gravitational field experienced by a high speed particle. In the rest frame of the Kerr source the angles of deflection experienced by the high speed test particle can then easily be evaluated. 
  We link the notion causality with the orientation of the 2-complex on which spin foam models are based. We show that all current spin foam models are orientation-independent, pointing out the mathematical structure behind this independence. Using the technology of evolution kernels for quantum fields/particles on Lie groups/homogeneous spaces, we construct a generalised version of spin foam models, introducing an extra proper time variable and prove that different ranges of integration for this variable lead to different classes of spin foam models: the usual ones, interpreted as the quantum gravity analogue of the Hadamard function of QFT or as a covariant definition of the inner product between quantum gravity states; and a new class of causal models, corresponding to the quantum gravity analogue of the Feynman propagator in QFT, non-trivial function of the orientation data, and implying a notion of ''timeless ordering''. 
  We show how to compute the mass of a Kerr-anti-de Sitter spacetime with respect to the anti-de Sitter background in any dimension, using a superpotential which has been derived from standard Noether identities. The calculation takes no account of the source of the curvature and confirms results obtained for black holes via the first law of thermodynamics. 
  In recent papers [1-3], we have discussed matter symmetries of non-static spherically symmetric spacetimes, static plane symmetric spacetimes and cylindrically symmetric static spacetimes. These have been classified for both cases when the energy-momentum tensor is non-degenerate and also when it is degenerate. Here we add up some consequences and the missing references about the Ricci tensor. 
  A novel but elementary geometric construction produces on the seven-dimensional manifold of rotated spheres in Euclidean three-space a finslerian geometry whose geodesics are interpreted as the paths of free, spinning, spherical particles moving through de Sitter's expanding universe. A particle of nonzero inertial rest mass typically follows a helical track and exhibits behavior remindful of the phenomenon of ``Zitterbewegung'' of spinning electrons first deduced by Schroedinger from Dirac's relativistic wave equation. Its velocity vector and its spin vector precess about the axial direction of the helix, with their projections onto that direction at all times parallel or at all times antiparallel. Particles of zero rest mass follow straight tracks at the speed of light with their spin vectors parallel or antiparallel to their velocity vectors, thereby replicating behavior of spinning photons predicted by the quantum theory of light. 
  We present a general method for constructing acoustic analogs of the black hole solutions of two-dimensional (2D) dilaton gravity. Because by dimensional reduction every spherically symmetric, four-dimensional (4D) black hole admits a 2D description, the method can be also used to construct analogue models of 4D black holes. We also show that after fixing the gauge degrees of freedom the 2D gravitational dynamics is equivalent to an one-dimensional fluid dynamics.  This enables us to find a natural definition of mass $M$, temperature $T$ and entropy $S$ of the acoustic black hole. In particular the first principle of thermodynamics $dM=TdS$ becomes a consequence of the fluid dynamics equations. We also discuss the general solutions of the fluid dynamics and two particular cases, the 2D Anti-de sitter black hole and the 4D Schwarzschild black hole. 
  It was shown recently that in four dimensions scalar sources with fixed proper acceleration minimally coupled to a massless Klein-Gordon field lead to the same responses when they are (i) uniformly accelerated in Minkowski spacetime (in the inertial vacuum) and (ii) static in the Schwarzschild spacetime (in the Unruh vacuum). Here we show that this equivalence is broken if the spacetime dimension is more than four. 
  The frequencies and damping times of neutron star (and quark star) oscillations have been computed using the most recent equations of state available in the literature. We find that some of the empirical relations that connect the frequencies and damping times of the modes to the mass and radius of the star, and that were previously derived in the literature need to be modified. 
  We provide a mathematical definition of the gauge fixed Ponzano-Regge model showing that it gives a measure on the space of flat connections whose volume is well defined. We then show that the Ponzano-Regge model can be equivalently expressed as Reshetikhin-Turaev evaluation of a colored chain mail link based on D(SU(2)): a non compact quantum group being the Drinfeld double of SU(2) and a deformation of the Poincare algebra. This proves the equivalence between spin foam quantization and Chern-Simons quantization of three dimensional gravity without cosmological constant. We extend this correspondence to the computation of expectation value of physical observables and insertion of particles. 
  In this paper we present an analysis to determine the existence of singularities in spatially homogeneous anisotropic universes filled with nonlinear electromagnetic radiation. These spaces are conformal to Bianchi spaces admitting a three parameter group of motions G$_3$. For these models we study geodesic completeness. It is shown that with nonlinear electromagnetic field some of the Bianchi spaces are geodesically complete, like G$_3$IX and G$_3$VIII; however, completeness depends on the curvature of the space. When certain topology is assumed, Bianchi G$_3$IX presents geodesics that are imprisoned. It is surprising that in the linear limit (Maxwell field) the spacetimes are singularity-free even if the curvature parameter is zero. 
  We develop a nonperturbative quantum scalar field formalism from a noncompact Kaluza-Klein (KK) theory using the induced-matter theory of gravity during inflation. We study the particular case of a de Sitter expansion for the universe. 
  We construct here a class of collapsing scalar field models with a non-zero potential, which result in a naked singularity as collapse end state. The weak energy condition is satisfied by the collapsing configuration. It is shown that physically it is the rate of collapse that governs either the black hole or naked singularity formation as the final state for the dynamical evolution. It is seen that the cosmic censorship is violated in dynamical scalar field collapse. 
  Space time is described as a continuum four-dimensional medium similar to ordinary elastic continua. Exploiting the analogy internal stress states are considered. The internal ''stress'' is originated by the presence of defects. The defects are described according to the typical Volterra process. The case of a point defect in an otherwise isotropic four-dimensional medium is discussed showing that the resulting metric tensor corresponds to an expanding (or contracting) universe filled up with a non-zero energy-momentum density. 
  With a help of a generalized Raychaudhuri equation non-expanding null surfaces are studied in arbitrarily dimensional case. The definition and basic properties of non-expanding and isolated horizons known in the literature in the 4 and 3 dimensional cases are generalized. A local description of horizon's geometry is provided. The Zeroth Law of black hole thermodynamics is derived. The constraints have a similar structure to that of the 4 dimensional spacetime case. The geometry of a vacuum isolated horizon is determined by the induced metric and the rotation 1-form potential, local generalizations of the area and the angular momentum typically used in the stationary black hole solutions case. 
  Isolated horizon conditions specialized to spherical symmetry can be imposed directly at the quantum level. This answers several questions concerning horizon degrees of freedom, which are seen to be related to orientation, and its fluctuations at the kinematical as well as dynamical level. In particular, in the absence of scalar or fermionic matter the horizon area is an approximate quantum observable. Including different kinds of matter fields allows to probe several aspects of the Hamiltonian constraint of quantum geometry that are important in inhomogeneous situations. 
  We build up a new phenomenological framework associated with a minimal generalization of Einsteinian gravitation theory. When linearity, stationarity and isotropy are assumed, tests in the solar system are characterized by two potentials which generalize respectively the Newton potential and the parameter $\gamma $ of parametrized post-Newtonian formalism. The new framework seems to have the capability to account for the Pioneer anomaly besides other gravity tests. 
  We show that the gravitating static soliton in the 2+1 dimensional O(3) $\sigma$ model does not exist in the presence of a negative cosmological constant. 
  I consider the initial-boundary-value-problem of nonlinear general relativistic vacuum spacetimes, which today cannot yet be evolved numerically in a satisfactory manner. Specifically, I look at gauge conditions, classifying them into gauge evolution conditions and gauge fixing conditions. In this terminology, a gauge fixing condition is a condition that removes all gauge degrees of freedom from a system, whereas a gauge evolution condition determines only the time evolution of the gauge condition, while the gauge condition itself remains unspecified. I find that most of today's gauge conditions are only gauge evolution conditions.   I present a system of evolution equations containing a gauge fixing condition, and describe an efficient numerical implementation using constrained evolution. I examine the numerical behaviour of this system for several test problems, such as linear gravitational waves or nonlinear gauge waves. I find that the system is robustly stable and second-order convergent. I then apply it to more realistic configurations, such as Brill waves or single black holes, where the system is also stable and accurate. 
  We express stress tensor correlators using the Schwinger-Keldysh formalism. The absence of off-diagonal counterterms in this formalism ensures that the +- and -+ correlators are free of primitive divergences. We use dimensional regularization in position space to explicitly check this at one loop order for a massless scalar on a flat space background. We use the same procedure to show that the ++ correlator contains the divergences first computed by `t Hooft and Veltman for the scalar contribution to the graviton self-energy. 
  A review on the main results concerning the algebraic and differential properties of the averaging and coordination operators and the properties of the space-time averages of macroscopic gravity is given. The algebraic and differential properties of the covariant space-time averaging procedure by means of using the parallel transportation averaging bivector operator are analyzed. The structure of the pseudo-Riemannian space-time manifolds of general relativity averaged by means of this procedure is discussed. A comparison of both procedures is given and the directions of further development of space-time averaging procedures of the physical classical fields are outlined. 
  It is investigated the behaviour of the ``constants'' $G,$ $c$ and $\Lambda $ in the framework of a perfect fluid LRS Bianchi I cosmological model. It has been taken into account the effects of a $c-$variable into the curvature tensor. Two exact cosmological solutions are investigated, arriving to the conclusion that if $q<0$ (deceleration parameter) then $G,$ $c$ are growing functions on time $t$ while $\Lambda $ is a negative decreasing function on time. 
  Classical black hole analogues (alternatively, the analogue systems) are fluid dynamical analogue of general relativistic black holes. Such analogue effects may be observed when acoustic perturbations (sound waves) propagate through a classical dissipation-less tran-sonic fluid. The acoustic horizon, which resembles the actual black hole event horizon in many ways, may be generated at the transonic point in the fluid flow. Acoustic horizon emits quasi thermal phonon spectra, which is analogous to the actual Hawking radiation, and possesses the temperature referred as the analogue Hawking temperature, or simply, the analogue temperature.   Transonic accretion onto astrophysical black holes is a very interesting example of classical analogue system found naturally in the Universe. An accreting black holes system as a classical analogue is unique in the sense that only for such a system, both kind of horizons, the electromagnetic and the acoustic (generated due to transonicity of accreting fluid) are simultaneously present in the same system. Hence an accreting astrophysical black hole is the ideal-most candidate to theoretically study and to compare the properties of these two different kind of horizons. Also such system is unique in the aspect that general relativistic spherical accretion onto the Schwarzschild black hole represents the only classical analogue system found in the nature so far, where the analogue Hawking temperature may be higher than the actual Hawking temperature. 
  In this paper by deriving the Modified Friedmann equation in the Palatini formulation of $R^2$ gravity, first we discuss the problem of whether in Palatini formulation an additional $R^2$ term in Einstein's General Relativity action can drive an inflation. We show that the Palatini formulation of $R^2$ gravity cannot lead to the gravity-driven inflation as in the metric formalism. If considering no zero radiation and matter energy densities, we obtain that only under rather restrictive assumption about the radiation and matter energy densities there will be a mild power-law inflation $a(t)\sim t^2$, which is obviously different from the original vacuum energy-like driven inflation. Then we demonstrate that in the Palatini formulation of a more generally modified gravity, i.e., the $1/R+R^2$ model that intends to explain both the current cosmic acceleration and early time inflation, accelerating cosmic expansion achieved at late Universe evolution times under the model parameters satisfying $\alpha\ll\beta$. 
  We review selected recent results concerning the global structure of solutions of the vacuum Einstein equations. The topics covered include quasi-local mass, strong cosmic censorship, non-linear stability, new constructions of solutions of the constraint equations, improved understanding of asymptotic properties of the solutions, existence of solutions with low regularity, and construction of initial data with trapped surfaces or apparent horizons.   This is an expanded version of a plenary lecture, sponsored by Classical and Quantum Gravity, held at the GR17 conference in Dublin in July 2004. 
  It was argued recently that there exists an unexpected phenomenon, the reflection of incoming particles on the event horizon of black holes (Kuchiev(2003)). This means that a particle approaching the black hole can bounce back into the outside world due to those events that take part strictly on the horizon. Previously the effect was discussed in relation to eternal black holes. The present work shows that the effect exists for collapsing black holes as well. 
  This article presents some of the more topical results of a study into the LISA phase measurement system. This system is responsible for measuring the phase of the heterodyne signal caused by the interference of the laser beams between the local and far spacecraft. Interactions with the LISA systems that surround the phase measurement system imply additional non-trivial requirements on the phase measurement system. 
  We investigate some cylindrically symmetric nonstationary and nonstatic solutions of Einstein field equations. We first study some physical properties of a solution which can be considered as Kasner generalization of static Levi-Civita vacuum solution. Then we generalize this metric to include a solution where a space-time filled with null dust or a stiff fluid. 
  It is widely believed that quantum field fluctuation in an inflating background creates the primeval seed perturbation which through subsequent evolution leads to the observed large scale structure of the universe. The standard inflationary scenario produces scale invariant power spectrum quite generically but it produces, unless fine tuned, too large amplitude for the primordial density perturbation than observed. Using similar techniques it is shown that loop quantum cosmology induced inflationary scenario can produce scale invariant power spectrum as well as small amplitude for the primordial density perturbation without fine tuning. Further its power spectrum has a qualitatively distinct feature which is in principle falsifiable by observation and can distinguish it from the standard inflationary scenario. 
  We will pick up the concepts of partial and complete observables introduced by Rovelli in order to construct Dirac observables in gauge systems. We will generalize these ideas to an arbitrary number of gauge degrees of freedom. Different methods to calculate such Dirac observables are developed. For background independent field theories we will show that partial and complete observables can be related to Kucha\v{r}'s Bubble Time Formalism. Moreover one can define a non-trivial gauge action on the space of complete observables and also state the Poisson brackets of these functions.   Additionally we will investigate, whether it is possible to calculate Dirac observables starting with partially invariant partial observables, for instance functions, which are invariant under the spatial diffeomorphism group. 
  We take a three dimensional Euclidian metric in toroidal coordinates and consider the corresponding Laplace equation. The simplest solution of this equation is taken. Based on this we build a Weyl space-time. This space-time is transformed to cylindrical coordinates. It is shown by using Mathematica that Weyl equations in cylindrical coordinates are satisfied. Geodesic motion is considered along the symmetric axis as well as along the radii of the singularity, which is the cause of the space time. 
  In this paper we have considered a model of modified Chaplygin gas and its role in accelerating phase of the universe. We have assumed that the equation of state of this modified model is valid from the radiation era to $\Lambda$CDM model. We have used recently developed statefinder parameters in characterizing different phase of the universe diagrammatically. 
  The Einstein-Schrodinger theory is extended to include spin-0 and spin-1/2 sources, and the theory is derived from a Lagrangian density which allows other fields to be easily added. The original theory is also modified by including a cosmological constant caused by zero-point fluctuations. This "extrinsic" cosmological constant which multiplies the symmetric metric is assumed to be nearly cancelled by Schrodinger's "bare" cosmological constant which multiplies the nonsymmetric fundamental tensor, such that the total cosmological constant is consistent with measurement. The resulting theory is shown to closely approximate ordinary one-particle quantum mechanics, electromagnetism and general relativity. In particular, the field equations match the ordinary Einstein and Maxwell equations except for additional terms which are $<10^{-16}$ of the usual terms for worst-case field strengths and rates-of-change accessible to measurement. The theory is also shown to predict the exact Klein-Gordon and Dirac equations, and the exact Lorentz force equation. Lastly, we discuss the merits of our Lagrangian density compared to the Einstein-Maxwell Lagrangian density. 
  We discuss the sensitivity to anisotropies of stochastic gravitational-wave backgrounds (GWBs) observed via space-based interferometer. In addition to the unresolved galactic binaries as the most promising GWB source of the planned Laser Interferometer Space Antenna (LISA), the extragalactic sources for GWBs might be detected in the future space missions. The anisotropies of the GWBs thus play a crucial role to discriminate various components of the GWBs. We study general features of antenna pattern sensitivity to the anisotropies of GWBs beyond the low-frequency approximation. We show that the sensitivity of space-based interferometer to GWBs is severely restricted by the data combinations and the symmetries of the detector configuration. The spherical harmonic analysis of the antenna pattern functions reveals that the angular power of the detector response increases with frequency and the detectable multipole moments with effective sensitivity h_{eff} \sim 10^{-20} Hz^{-1/2} may reach $\ell \sim$ 8-10 at $f \sim f_*=10$ mHz in the case of the single LISA detector. However, the cross correlation of optimal interferometric variables is blind to the monopole (\ell=0) intensity anisotropy, and also to the dipole (\ell=1) in some case, irrespective of the frequency band. Besides, all the self-correlated signals are shown to be blind to the odd multipole moments (\ell=odd), independently of the frequency band. 
  We reexamine in detail a canonical quantization method a la Gupta-Bleuler in which the Fock space is built over a so-called Krein space. This method has already been successfully applied to the massless minimally coupled scalar field in de Sitter spacetime for which it preserves covariance. Here, it is formulated in a more general context. An interesting feature of the theory is that, although the field is obtained by canonical quantization, it is independent of Bogoliubov transformations. Moreover no infinite term appears in the computation of $T^{\mu\nu}$ mean values and the vacuum energy of the free field vanishes: $<0|T^{00}|0>=0$. We also investigate the behaviour of the Krein quantization in Minkowski space for a theory with interaction. We show that one can recover the usual theory with the exception that the vacuum energy of the free theory is zero. 
  We present a hyperbolic formulation of the evolution equations describing non-radial perturbations of slowly rotating relativistic stars in the Regge--Wheeler gauge. We demonstrate the stability preperties of the new evolution set of equations and compute the polar w-modes for slowly rotating stars. 
  Launched more than thirty years ago and now drifting in space with no further contact, the Pioneer 10 and 11 spacecraft are currently at the center of a small but developing concern: are they under the influence of an anomalous acceleration that reflects new, yet unknown physics, or merely experiencing some unexpected technical problem ? The proposals vary from basic engineering principles to extra spatial dimensions, but most probably an answer can only be obtained after a dedicated mission is underway. 
  This is a progress report on our study of the coupling of first-order radial and non-radial relativistic perturbations of a static spherical star. Our goal is to investigate the effects of this coupling on the gravitational wave signal of neutron stars. In particular, we are looking for the existence of resonances and parametric amplifications, changes in the damping time of non-radial oscillations, etc. To that end, we have developed a formalism that introduces gauge invariant quantities to describe the coupling. Their equations have the same structure as the equations for first-order non-radial perturbations plus some source terms, which makes them very appealing for time domain studies. 
  In this talk I will briefly outline work in progress in two different contexts in astrophysical relativity, i.e. the study of rotating star spacetimes and the problem of reliably extracting gravitational wave templates in numerical relativity. In both cases the use of Weyl scalars and curvature invariants helps to clarify important issues. 
  A simple unified closed form derivation of the non-linearities of the Einstein, Yang-Mills and spinless (e.g., chiral) meson systems is given. For the first two, the non-linearities are required by locality and consistency; in all cases, they are determined by the conserved currents associated with the initial (linear) gauge invariance of the first kind. Use of first-order formalism leads uniformly to a simple cubic self-interaction. 
  In this paper we will show in detail that the performed attempts aimed at the detection of the general relativistic Lense-Thirring effect in the gravitational field of the Earth with the existing LAGEOS satellites are often presented in an optimistic and misleading way which is inadequate for such an important test of fundamental physics. E.g., in the latest reported measurement of the gravitomagnetic shift with the nodes of the LAGEOS satellites and the 2nd generation GRACE-only EIGEN-GRACE02S Earth gravity model over an observational time span of 11 years a 5-10% total accuracy is claimed at 1-3sigma, respectively. We will show that, instead, it might be 15-45% (1-3sigma) if the impact of the secular variations of the even zonal harmonics is considered as well. 
  This is a short review of the quasinormal mode spectrum of Schwarzschild, Reissner-Nordstrom and Kerr black holes. The summary includes previously unpublished calculations of i) the eigenvalues of spin-weighted spheroidal harmonics, and ii) quasinormal frequencies of extremal Reissner-Nordstrom black holes. 
  We show that the empirical signs of the fundamental {\it static} Coulomb/Newton forces are dictated by the seemingly unrelated requirement that the photons/gravitons in the respective underlying Maxwell/Einstein physics be stable. This linkage, which is imposed by special relativity, is manifested upon decomposing the corresponding fields and sources in a gauge-invariant way, and without appeal to static limits. The signs of these free field excitation energies determine those of the instantaneous forces between sources; opposite Coulomb/Newton signs are direct consequences of the Maxwell/Einstein free excitations' odd/even spins. 
  We present data-analysis schemes and results of observations with the TAMA300 gravitational-wave detector, targeting burst signals from stellar-core collapse events. In analyses for burst gravitational waves, the detection and fake-reduction schemes are different from well-investigated ones for a chirp-wave analysis, because precise waveform templates are not available. We used an excess-power filter for the extraction of gravitational-wave candidates, and developed two methods for the reduction of fake events caused by non-stationary noises of the detector. These analysis schemes were applied to real data from the TAMA300 interferometric gravitational wave detector. As a result, fake events were reduced by a factor of about 1000 in the best cases. The resultant event candidates were interpreted from an astronomical viewpoint. We set an upper limit of 2.2x10^3 events/sec on the burst gravitational-wave event rate in our Galaxy with a confidence level of 90%. This work sets a milestone and prospects on the search for burst gravitational waves, by establishing an analysis scheme for the observation data from an interferometric gravitational wave detector. 
  In the present work we analize the behavior of 5-dimensional gravitational waves propagating on a Kaluza-Klein background and we face separately the two cases in which respectively the waves are generated before and after the process of dimensional compactification. We show that if the waves are originated on a 5-d space-time which fulfills the principle of general relativity, then the process of compactification can not reduce the dynamics to the pure 4-dimensional scalar, vector and tensor degrees of freedom. In particular, while the electromagnetic waves evolve independently, the scalar and tensor fields couple to each other; this feature appears because, when the gauge conditions are splitted, the presence of the scalar ripple prevents that the 4-d gravitational waves are traceless. The phenomenological issue of this scheme consists of an anomalous relative amplitude of the two independent polarizations which characterize the 4-d gravitational waves. Such profile of polarization amplitudes, if detected, would outline the extra-dimension in a very reliable way, because a wave with non-zero trace can not arise from ordinary matter sources. We discuss the above mentioned phenomenon either in the case of a unit constant value of the background scalar component (when the geodesic deviation is treated with precise outputs), and assuming such background field as a dynamical degree (only qualitative conclusion are provided here, because the details of the polarization amplitudes depend on the choice of specific metric forms). Finally we perturb a real Kaluza-Klein theory showing that in this context, while the electromagnetic waves propagate independently, the 4-d gravitational waves preserve their ordinary structure, while the scalar plays for them the role of source. 
  The tetrad partial differential equations formulated by Buchman and Bardeen for vacuum gravity are shown to be well posed by calculation of the Cartan characters of an associated exterior differential system. Gauge specializations are discussed. A Cartan 4-form is found for this field theory, together with its intrinsic version the Lagrangian density. 
  Evidence for free precession has been observed in the radio signature of several pulsars. Freely precessing pulsars radiate gravitationally at frequencies near the rotation rate and twice the rotation rate, which for rotation frequencies greater than $\sim 10$ Hz is in the LIGO band. In older work, the gravitational wave spectrum of a precessing neutron star has been evaluated to first order in a small precession angle. Here we calculate the contributions to second order in the wobble angle, and we find that a new spectral line emerges. We show that for reasonable wobble angles, the second-order line may well be observable with the proposed advanced LIGO detector for precessing neutron stars as far away as the galactic center. Observation of the full second-order spectrum permits a direct measurement of the star's wobble angle, oblateness, and deviation from axisymmetry, with the potential to significantly increase our understanding of neutron star structure. 
  In her recent work, Dittrich generalized Rovelli's idea of partial observables to construct Dirac observables for constrained systems to the general case of an arbitrary first class constraint algebra with structure functions rather than structure constants. Here we use this framework and propose a new way for how to implement explicitly a reduced phase space quantization of a given system, at least in principle, without the need to compute the gauge equivalence classes. The degree of practicality of this programme depends on the choice of the partial observables involved. The (multi-fingered) time evolution was shown to correspond to an automorphism on the set of Dirac observables so generated and interesting representations of the latter will be those for which a suitable preferred subgroup is realized unitarily. We sketch how such a programme might look like for General Relativity. We also observe that the ideas by Dittrich can be used in order to generate constraints equivalent to those of the Hamiltonian constraints for General Relativity such that they are spatially diffeomorphism invariant. This has the important consequence that one can now quantize the new Hamiltonian constraints on the partially reduced Hilbert space of spatially diffeomorphism invariant states, just as for the recently proposed Master constraint programme. 
  We extend the ideas introduced in the previous work to a more general space-time. In particular we consider the Kantowski-Sachs space time with space section with topology $R \times S^2$. In this way we want to study a general space time that we think to be the space time inside the horizon of a black hole. In this case the phase space is four dimensional and we simply apply the quantization procedure suggested by Loop Quantum Gravity and based on an alternative to the Schroedinger representation introduced by H. Halvorson. Through this quantization procedure we show that the inverse of the volume density and the Schwarzschild curvature invariant are upper bounded and so the space time is singularity free. Also in this case we can extend dynamically the space time beyond the classical singularity. 
  The metric of arbitrary dimensional Schwarzschild black hole in the background of Friedman-Robertson-Walker universe is presented in the cosmic coordinates system. In particular, the arbitrary dimensional Schwarzschild-de Sitter metric is rewritten in the Schwarzschild coordinates system and basing on which the even more generalized higher dimensional Schwarzschild-de Sitter metric with another extra dimensions is found. The generalized solution shows that the cosmological constant may roots in the extra dimensions of space. 
  Gravitational lensing in a weak but otherwise arbitrary gravitational field can be described in terms of a 3 x 3 tensor, the "effective refractive index". If the sources generating the gravitational field all have small internal fluxes, stresses, and pressures, then this tensor is automatically isotropic and the "effective refractive index" is simply a scalar that can be determined in terms of a classic result involving the Newtonian gravitational potential. In contrast if anisotropic stresses are ever important then the gravitational field acts similarly to an anisotropic crystal. We derive simple formulae for the refractive index tensor, and indicate some situations in which this will be important. 
  A proper counting of states for black holes in the quantum geometry approach shows that the dominant configuration for spins are distributions that include spins exceeding one-half at the punctures. This raises the value of the Immirzi parameter and the black hole entropy. However, the coefficient of the logarithmic correction remains -1/2 as before. 
  We define the notion of an aligned null direction, a Lorentz-signature analogue of the eigenvector concept that is valid for arbitrary tensor types. The set of aligned null directions is described by a a system of alignment polynomials whose coefficients are derived from the components of the tensor. The algebraic properties of the alignment polynomials can be used to classify the corresponding tensors and to put them into normal form. The alignment classification paradigm is illustrated with a discussion of bivectors and of Weyl-type tensors. Note: an earlier version of this manuscript was published in the proceedings of SPT 2004. The present version has been expanded to include a discussion of complexified alignment. Section 4 also corrects errors contained in the earlier manuscript. 
  A new approximation scheme, designed to solve the covariant Maxwell equations inside a rotating hollow slender conducting cavity (modelling a ring-laser), is constructed. It is shown that for well-defined conditions there exist TE and TM modes with respect to the longitudinal axis of the cavity. A twisted mode spectrum is found to depend on the integrated Frenet torsion of the cavity and this in turn may affect the Sagnac beat frequency induced by a non-zero rotation of the cavity. The analysis is motivated by attempts to use ring-lasers to measure terrestrial gravito-magnetism or the Lense-Thirring effect produced by the rotation of the Earth. 
  We revisit the integral formulation (or Green's function approach) of Einstein's equations in the context of braneworlds. The integral formulation has been proposed independently by several authors in the past, based on the assumption that it is possible to give a reinterpretation of the local metric field in curved spacetimes as an integral expression involving sources and boundary conditions. This allows one to separate source-generated and source-free contributions to the metric field. As a consequence, an exact meaning to Mach's Principle can be achieved in the sense that only source-generated (matter fields) contributions to the metric are allowed for; universes which do not obey this condition would be non-Machian. In this paper, we revisit this idea concentrating on a Randall-Sundrum-type model with a non-trivial cosmology on the brane. We argue that the role of the surface term (the source-free contribution) in the braneworld scenario may be quite subtler than in the 4D formulation. This may pose, for instance, an interesting issue to the cosmological constant problem. 
  In this paper we present a model of Riemannian loop quantum cosmology with a self-adjoint quantum scalar constraint. The physical Hilbert space is constructed using refined algebraic quantization. When matter is included in the form of a cosmological constant, the model is exactly solvable and we show explicitly that the physical Hilbert space is separable consisting of a single physical state. We extend the model to the Lorentzian sector and discuss important implications for standard loop quantum cosmology. 
  By inspecting some known solutions to Einstein equations, we present the metric of higher dimensional Reissner-Nordstr$\ddot{o}$m black hole in the background of Friedman-Robertson-Walker universe. Then we verify the solution with a perfect fluid. The discussion of the event horizon of the black hole reveals that the scale of the black hole would increase with the expansion of the universe and decrease with the contraction of the universe. 
  We investigate a condensed matter ``black hole'' analogue, taking the Gross-Pitaevskii (GP) equation as a starting point. The linearized GP equation corresponds to a wave equation on a black hole background, giving quasinormal modes under some appropriate conditions. We suggest that we can know the detailed characters and corresponding geometrical information about the acoustic black hole by observing quasinormal ringdown waves in the low temperature condensed matters. 
  Thin shells in general relativity have been used in the past as keystones to obtain realistic models of cosmological and astrophysical situations. A crucial role for these developments was played by the compact description of their dynamics in terms of Israel's junction conditions. Starting from this geometrical formulation we present a problem related to the WKB regime of shell dynamics and suggest a possible solution. 
  We present a generalization to the N-dimensional case for the nucleation coefficient of a spherical p-brane, separating two (anti-)de Sitter spacetimes. We use a semiclassical approximation based on the analytical continuation to the Euclidean sector of a suitable effective action describing a p-brane in General Relativity. 
  We outline the covariant nature,with respect to the choice of a reference frame, of the chaos characterizing the generic cosmological solution near the initial singularity, i.e. the so-called inhomogeneous Mixmaster model. Our analysis is based on a "gauge" independent ADM-reduction of the dynamics to the physical degrees of freedom. The out coming picture shows how the inhomogeneous Mixmaster model is isomorphic point by point in space to a billiard on a Lobachevsky plane. Indeed, the existence of an asymptotic (energy-like) constant of the motion allows to construct the Jacobi metric associated to the geodesic flow and to calculate a non-zero Lyapunov exponent in each space point. The chaos covariance emerges from the independence of our scheme with respect to the form of the lapse function and the shift vector; the origin of this result relies on the dynamical decoupling of the space-points which takes place near the singularity, due to the asymptotic approach of the potential term to infinite walls. At the ground of the obtained dynamical scheme is the choice of Misner-Chitre' like variables which allows to fix the billiard potential walls. 
  We show the existence of an infinite family of finite-time singularities in isotropically expanding universes which obey the weak, strong, and dominant energy conditions. We show what new type of energy condition is needed to exclude them ab initio. We also determine the conditions under which finite-time future singularities can arise in a wide class of anisotropic cosmological models. New types of finite-time singularity are possible which are characterised by divergences in the time-rate of change of the anisotropic-pressure tensor. We investigate the conditions for the formation of finite-time singularities in a Bianchi type $VII_{0}$ universe with anisotropic pressures and construct specific examples of anisotropic sudden singularities in these universes. 
  We compute the vacuum polarization associated with quantum massless fields around stars with spherical symmetry. The nonlocal contribution to the vacuum polarization is dominant in the weak field limit, and induces quantum corrections to the exterior metric that depend on the inner structure of the star. It also violates the null energy conditions. We argue that similar results also hold in the low energy limit of quantum gravity. Previous calculations of the vacuum polarization in spherically symmetric spacetimes, based on local approximations, are not adequate for newtonian stars. 
  The aim of this paper is to study the triviality of $\lambda\phi^{4}$ theory in a classical gravitational model. Starting from a conformal invariant scalar tensor theory with a self-interaction term $\lambda\phi^{4}$, we investigate the effect of a conformal symmetry breaking emerging from the gravitational coupling of the large-scale distribution of matter in the universe. Taking in this cosmological symmetry breaking phase the infinite limit of the maximal length (the size of the universe) and the zero limit of the minimal length (the Planck length) implies triviality, i.e. a vanishing coupling constant $\lambda$. It suggests that the activity of the self-interaction term $\lambda\phi^{4}$ in the cosmological context implies that the universe is finite and a minimal fundamental length exists. 
  I discuss the range of validity of Detweiler's formula for the resonant frequencies of rapidly rotating Kerr black holes. While his formula is correct for extremal black holes, it has also been commonly accepted that it describes very well the resonant frequencies of near extremal black holes, and that therefore there is a large number of modes clustering on the real axis as the black hole becomes extremal. I will show that this last statement is not only incorrect, but that it also does not follow from Detweiler's formula, provided it is handled with due care. It turns out that only the first n << -log{(r_+-r_-)/r_+} modes are well described by that formula, which translates, for any astrophysical black hole, into one or two modes only. All existing numerical data gives further support to this claim. I also discuss some implications of this result for recent investigations on the late-time dynamics of rapidly rotating black holes. 
  We discuss a new torsion pendulum design for ground testing of prototype LISA (Laser Interferometer Space Antenna) displacement sensors. This new design is directly sensitive to net forces and therefore provides a more representative test of the noisy forces and parasitic stiffnesses acting on the test mass as compared to previous ground-based experiments. We also discuss a specific application to the measurement of thermal gradient effects. 
  Solutions for a class of wave equations with effective potentials are obtained by a method of a Laplace-transform. Quasinormal modes appear naturally in the solutions only in a spatially truncated form; their coefficients are uniquely determined by the initial data and are constant only in some region of spacetime -- in contrast to normal modes. This solves the problem of divergence of the usual expansion into spatially unbounded quasinormal modes and a contradiction with the causal propagation of signals. It also partially answers the question about the region of validity of the expansion. Results of numerical simulations are presented. They fully support the theoretical predictions. 
  For stationary vacuum spacetimes the Bianchi identities of the second kind equate the Simon tensor to the Simon-Mars tensor, the latter having a clear geometrical interpretation. The equivalence of these two tensors is broken in the nonvacuum case by additional source energy-momentum terms, but absorbing these source terms into a redefinition of the Simon tensor restores the equality. Explicit examples are discussed for electrovacuum and rigidly rotating matter fields. 
  The classical electromagnetic and gravitomagnetic fields in the vacuum, in (3+2) dimensions, described by the Maxwell-Nordstrom equations, are quantized. These equations are rederived from the field tensor which follows from a five-dimensional form of the Dirac equation. The electromagnetic field depends on the customary time t, and the hypothetical gravitomagnetic field depends on the second time variable u. The total field energy is identified with the component T44 of the five-dimensional energy-stress tensor of the electromagnetic and gravitomagnetic fields. In the ground state, the electromagnetic field and the gravitomagnetic field energies cancel out. The quanta of the gravitomagnetic field have spin 1. 
  This article explores the overall geometric manner in which human beings make sense of the world around them by means of their physical theories; in particular, in what are nowadays called pregeometric pictures of Nature. In these, the pseudo-Riemannian manifold of general relativity is considered a flawed description of spacetime and it is attempted to replace it by theoretical constructs of a different character, ontologically prior to it. However, despite its claims to the contrary, pregeometry is found to surreptitiously and unavoidably fall prey to the very mode of description it endeavours to evade, as evidenced in its all-pervading geometric understanding of the world. The question remains as to the deeper reasons for this human, geometric predilection--present, as a matter of fact, in all of physics--and as to whether it might need to be superseded in order to achieve the goals that frontier theoretical physics sets itself at the dawn of a new century: a sounder comprehension of the physical meaning of empty spacetime. 
  Affine quantum gravity, which differs notably from either string theory or loop quantum gravity, is briefly reviewed. Emphasis in this article is placed on the use of affine coherent states in this program. 
  All attempts to quantize gravity face several difficult problems. Among these problems are: (i) metric positivity (positivity of the spatial distance between distinct points), (ii) the presence of anomalies (partial second-class nature of the quantum constraints), and (iii) perturbative nonrenormalizability (the need for infinitely many distinct counterterms). In this report, a relatively nontechnical discussion is presented about how the program of affine quantum gravity proposes to deal with these problems. 
  Anisotropy of the cosmic microwave background radiation (CMB) originates from both tensor and scalar perturbations. To study the characteristics of each of these two kinds of perturbations, one has to determine the contribution of each to the anisotropy of CMB. For example, the ratio of the power spectra of tensor/scalar perturbations can be used to tighten bounds on the scalar spectral index. We investigate here the implications for the tensor/scalar ratio of the recent discovery (noted in astro-ph/0410139) that the introduction of a minimal length cutoff in the structure of spacetime does not leave boundary terms invariant. Such a cutoff introduces an ambiguity in the choice of action for tensor and scalar perturbations, which in turn can affect this ratio. We numerically solve for both tensor and scalar mode equations in a near-de-sitter background and explicitly find the cutoff dependence of the tensor/scalar ratio during inflation. 
  The conservative dynamics of two point masses given in harmonic coordinates up to the third post-Newtonian (3pN) order is treated within the framework of constrained canonical dynamics. A representation of the approximate Poincar\'e algebra is constructed with the aid of Dirac brackets. Uniqueness of the generators of the Poincar\'e group resp. the integrals of motion is achieved by imposing their action on the point mass coordinates to be identical with that of the usual infinitesimal Poincar\'e transformations. The second post-Coulombian approximation to the dynamics of two point charges as predicted by Feynman-Wheeler electrodynamics in Lorentz gauge is treated similarly. 
  We briefly review the arising of an inflationary phase in the Universe evolution in order to discuss an inhomogeneous cosmological solution in presence of a real self interacting scalar field minimally coupled to gravity in the region of a slow rolling potential plateau. During the inhomogeneous de Sitter phase the scalar field dominant term is a function of the spatial coordinates only. We apply this generic solution to the Coleman-Weinberg potential and to the Lemaitre-Tolman metric. This framework specialized nearby the FLRW model allows a classical origin for the inhomogeneous perturbations spectrum. 
  The decay of massive scalar field in the Schwarzschild black hole background is investigated here by consideration its quasinormal spectrum. It has been proved that the so-called $quasi-resonant$ modes, which are arbitrary long living (purely real) modes, can exist only if the effective potential is not zero at least at one of the boundaries of the $R$-region. We have observed that the quasinormal spectrum exists for all field masses and proved both analytically and numerically that when $n \to \infty$ the real part of the frequencies approaches the same asymptotical value ($\ln3/(8\pi M)$) as in the case of the massless field. 
  This paper presents an under-appreciated way to conceptualize stationary black holes, which we call the river model. The river model is mathematically sound, yet simple enough that the basic picture can be understood by non-experts. %that can by understood by non-experts. In the river model, space itself flows like a river through a flat background, while objects move through the river according to the rules of special relativity. In a spherical black hole, the river of space falls into the black hole at the Newtonian escape velocity, hitting the speed of light at the horizon. Inside the horizon, the river flows inward faster than light, carrying everything with it. We show that the river model works also for rotating (Kerr-Newman) black holes, though with a surprising twist. As in the spherical case, the river of space can be regarded as moving through a flat background. However, the river does not spiral inward, as one might have anticipated, but rather falls inward with no azimuthal swirl at all. Instead, the river has at each point not only a velocity but also a rotation, or twist. That is, the river has a Lorentz structure, characterized by six numbers (velocity and rotation), not just three (velocity). As an object moves through the river, it changes its velocity and rotation in response to tidal changes in the velocity and twist of the river along its path. An explicit expression is given for the river field, a six-component bivector field that encodes the velocity and twist of the river at each point, and that encapsulates all the properties of a stationary rotating black hole. 
  An extensive investigation is made of the interior structure of self-similar accreting charged black holes. In this, the first of two papers, the black hole is assumed to accrete a charged, electrically conducting, relativistic baryonic fluid. The mass and charge of the black hole are generated self-consistently by the accreted material. The accreted baryonic fluid undergoes one of two possible fates: either it plunges directly to the spacelike singularity at zero radius, or else it drops through the Cauchy horizon. The baryons fall directly to the singularity if the conductivity either exceeds a certain continuum threshold kappa_oo, or else equals one of an infinite spectrum kappa_n of discrete values. Between the discrete values kappa_n, the solution is characterized by the number of times that the baryonic fluid cycles between ingoing and outgoing. If the conductivity is at the continuum threshold kappa_oo, then the solution cycles repeatedly between ingoing and outgoing, displaying a discrete self-similarity reminiscent of that observed in critical collapse. Below the continuum threshold kappa_oo, and except at the discrete values kappa_n, the baryonic fluid drops through the Cauchy horizon, and in this case undergoes a shock, downstream of which the solution terminates at an irregular sonic point where the proper acceleration diverges, and there is no consistent self-similar continuation to zero radius. As far as the solution can be followed inside the Cauchy horizon, the radial direction is timelike. If the radial direction remains timelike to zero radius (which cannot be confirmed because the self-similar solutions terminate), then there is presumably a spacelike singularity at zero radius inside the Cauchy horizon, which is distinctly different from the vacuum solution for a charged black hole. 
  (Abridged) This is the second of two companion papers on the interior structure of self-similar accreting charged black holes. In the first paper, the black hole was allowed to accrete only a single fluid of charged baryons. In this second paper, the black hole is allowed to accrete in addition a neutral fluid of almost non-interacting dark matter. Relativistic streaming between outgoing baryons and ingoing dark matter leads to mass inflation near the inner horizon. When enough dark matter has been accreted that the center of mass frame near the inner horizon is ingoing, then mass inflation ceases and the fluid collapses to a central singularity. A null singularity does not form on the Cauchy horizon. Although the simultaneous presence of ingoing and outgoing fluids near the inner horizon is essential to mass inflation, reducing one or other of the ingoing dark matter or outgoing baryonic streams to a trace relative to the other stream makes mass inflation more extreme, not the other way round as one might naively have expected. Consequently, if the dark matter has a finite cross-section for being absorbed into the baryonic fluid, then the reduction of the amount of ingoing dark matter merely makes inflation more extreme, the interior mass exponentiating more rapidly and to a larger value before mass inflation ceases. However, if the dark matter absorption cross-section is effectively infinite at high collision energy, so that the ingoing dark matter stream disappears completely, then the outgoing baryonic fluid can drop through the Cauchy horizon. In all cases, the solutions do not depend on what happens in the infinite past or future. We discuss in some detail the physical mechanism that drives mass inflation. 
  We study the stability of static, spherically symmetric, traversable wormholes existing due to conformal continuations in a class of scalar-tensor theories with zero scalar field potential (so that Fisher's well-known scalar-vacuum solution holds in the Einstein conformal frame). Specific examples of such wormholes are those with nonminimally (e.g., conformally) coupled scalar fields. All boundary conditions for scalar and metric perturbations are taken into account. All such wormholes are shown to be unstable under spherically symmetric perturbations. The instability is proved analytically with the aid of the theory of self-adjoint operators in Hilbert space and is confirmed by a numerical computation. 
  I discuss possible implications a symmetry relating gravity with antigravity might have for smoothing out of the cosmological constant puzzle. For this purpose, a very simple model with spontaneous symmetry breaking is explored, that is based on Einstein-Hilbert gravity with two self-interacting scalar fields. The second (exotic) scalar particle with negative energy density, could be interpreted, alternatively, as an antigravitating particle with positive energy. 
  This letter contains a brief discussion on the leading-order canonical correction to the Bekenstein-Hawking (black hole) entropy. In particular, we address some recent criticism directed at an earlier commentary. 
  We consider a 5-dimensional scalar-tensor theory which is a direct generalization of the original 4-dimensional Brans-Dicke theory to 5-dimensions. By assuming that there is a hypersurface-orthogonal spacelike Killing vector field in the underlying 5-dimensional spacetime, the theory is reduced to a 4-dimensional theory where the 4-metric is coupled with two scalar fields. The cosmological implication of this reduced theory is then studied in the Robertson-Walker model. It turns out that the two scalar fields may account naturally for the present accelerated expansion of our universe. The observational restriction of the reduced cosmological model is also analyzed. 
  In this review, I discuss briefly how the presence of a cosmological constant in the Universe may imply a decoherent evolution of quantum matter in it, and as a consequence a fundamental irreversibility of time unrelated in principle to CP properties (Cosmological CPT Violation).In this context, I also discuss recently suggested novel possible contributions of massive neutrinos to the cosmological constant, which are not due to the standard loop expansion in quantum field theory, but rather due to unconventional properties of (some version of) the quantum theory underlying flavour mixing. It is also argued that quantum space time foam may be responsible for the neutrino mass differences, observed today, and through the above considerations, for the (majority of the) dark energy of the Universe in the present era. In the above context, I also present a fit of all the currently available neutrino oscillation data, including the LSND ``anomalous'' experimental results, based on such a CPT Violating decoherent neutrino model. The key feature is to use different decoherent parameters between neutrinos and antineutrinos, due to the above mentioned CPT violation. This points to the necessity of future experiments, concentrating on the antineutrino sector, in order to falsify the model. 
  In a recent paper [1] we have constructed the spin and tensor representations of SO(4) from which the invariant weight can be derived for the Barrett-Crane model in quantum gravity. By analogy with the SO(4) group, we present the complexified Clebsch-Gordan coefficients in order to construct the Biedenharn-Dolginov function for the SO(3,1) group and the spherical function as the Lorentz invariant weight of the model. 
  Minkowski diagrams in 1+1 dimensional flat space-time are given a strictly geometric derivation, directly from two gedanken experiments incorporating the principle of the constancy of the velocity of light and the principle of (special) relativity. Rectangles of photon trajectories play a central role in determining the simultaneity convention and in establishing the invariance of the interval. 
  We study the vacuum, plane-wave Bianchi $VII{}_{h}$ spacetimes described by the Lukash metric. Combining covariant with orthonormal frame techniques, we describe these models in terms of their irreducible kinematical and geometrical quantities. This covariant description is used to study analytically the response of the Lukash spacetime to linear perturbations. We find that the stability of the vacuum solution depends crucially on the background shear anisotropy. The stronger the deviation from the Hubble expansion, the more likely the overall linear instability of the model. Our analysis addresses rotational, shear and Weyl curvature perturbations and identifies conditions sufficient for the linear growth of these distortions. 
  Here we describe the mission design for SMART-2/LISA Pathfinder. The best trade-off between the requirements of a low-disturbance environment and communications distance is found to be a free-insertion Lissajous orbit around the first co-linear Lagrange point of the Sun-Earth system L1, 1.5x 10^6 km from Earth. In order to transfer SMART-2/LISA Pathfinder from a low Earth orbit, where it will be placed by a small launcher, the spacecraft carries out a number of apogee-raise manoeuvres, which ultimatively place it to a parabolic escape trajectory towards L1. The challenges of the design of a small mission are met, fulfilling the very demanding technology demonstration requirements without creating excessive requirements on the launch system or the ground segment. 
  Recent work has shown that differential rotation, producing large scale drifts of fluid elements along stellar latitudes, is an unavoidable feature of r-modes in the nonlinear theory. We investigate the role of this differential rotation in the evolution of the l=2 r-mode instability of a newly born, hot, rapidly rotating neutron star. It is shown that the amplitude of the r-mode saturates a few hundred seconds after the mode instability sets in. The saturation amplitude depends on the amount of differential rotation at the time the instability becomes active and can take values much smaller than unity. It is also shown that, independently of the saturation amplitude of the mode, the star spins down to rotation rates that are comparable to the inferred initial rotation rates of the fastest pulsars associated with supernova remnants. Finally, it is shown that, when the drift of fluid elements at the time the instability sets in is significant, most of the initial angular momentum of the star is transferred to the r-mode and, consequently, almost none is carried away by gravitational radiation. 
  The careful analysis of the duality properties of Riemann's curvature tensor points to possibility of extension of Einstein's General Relativity to the nonabelian Yang-Mills theory. The motion equations of the theory are Yang-Mills' equations for the curvature tensor. Einstein's equations (with cosmological term to appear as an integration constant) are contained in the theory proposed. New is that now gravitational field is not exceptionally determined by matter energy-momentum but can possess its own non-Einsteinian dynamics(vacuum fluctuations, self-interaction)which is generally an attribute of nonabelian gauge field. The gravitational equations proper due to either matter energy-momentum or vacuum fluctuations are side conditions imposed on the Riemann tensor, like self-duality conditions. 
  This paper is a review of a recently introduced cosmological model from a noncompact Kaluza-Klein theory for a single scalar field minimally coupled to gravity. We obtain that the 4D scalar potential has a geometrical origin and assume different representations in different frames. It should be responsible for the expansion of the universe. In this framework we explain the (neutral scalar field governed) evolution of the universe from an initially inflationary expansion that has a change of phase towards a decelerated expansion and thereinafter evolves towards the present day observed accelerated (quintessential) expansion. Finally, using the Hamilton-Jacobi formalism, we study extra force and extra mass from this 5D cosmological model. 
  We prove that "first singularities" in the non-trapped region of the maximal development of spherically symmetric asymptotically flat data for the Einstein-Vlasov system must necessarily emanate from the center. The notion of "first" depends only on the causal structure and can be described in the language of terminal indecomposable pasts (TIPs). This result suggests a local approach to proving weak cosmic censorship for this system. It can also be used to give the first proof of the formation of black holes by the collapse of collisionless matter from regular initial configurations. 
  A one-parameter deformation of Einstein?Hilbert gravity with an inverse Riemann curvature term is derived as the classical limit of quantum gravity compatible with an accelerating universe. This result is based on the investigation of semi-classical theories with sectional curvature bounds which are shown not to admit static spherically symmetric black holes if otherwise of phenomenological interest. We discuss the impact on the canonical quantization of gravity, and observe that worldsheet string theory is not affected. 
  The radio-metric Doppler tracking data from the Pioneer 10/11 spacecraft, from between 20-70 AU, yields an unambiguous and independently confirmed anomalous blue shift drift of a_t = (2.92 \pm 0.44)\times 10^{-18} s/s^2. It can be interpreted as being due to a constant acceleration of a_P = (8.74 \pm 1.33) \times 10^{-8} cm/s^2 directed towards the Sun. No systematic effect has been able to explain the anomaly, even though such an origin is an obvious candidate. We discuss what has been learned (and what might still be learned) from the data about the anomaly, its origin, and the mission design characteristics that would be needed to test it. Future mission options are proposed. 
  We discuss critical gravitational collapse on the threshold of apparent horizon formation as a model both for the discussion of global aspects of critical collapse and for numerical studies in a compactified context. For our matter model we choose a self-gravitating massless scalar field in spherical symmetry, which has been studied extensively in the critical collapse literature. Our evolution system is based on Bondi coordinates, the mass function is used as an evolution variable to ensure regularity at null infinity. We compute radiation quantities like the Bondi mass and news function and find that they reflect the DSS behavior. Surprisingly, the period of radiation at null infinity is related to the formal result for the leading quasi-normal mode of a black hole with rapidly decreasing mass. Furthermore, our investigations shed some light on global versus local issues in critical collapse, and the validity and usefulness of the concept of null infinity when predicting detector signals. 
  The gravity water wave black (GWBH) hole analog discovered by Schutzhold and Unruh (SU) is extended to allow for the presence of turbulent shear flow. The Riemannian geometry of turbulent black holes (BH) analogs in water waves is computed in the case of laminar tirbulent shear flow. The Riemann curvature is constant and the geodesic deviation equation shows that the curvature acts locally as a diverging lens and the stream lines on opposite sides of the analog black hole flow apart from each other. In this case it is shown that the curvature quantities can be expressed in terms of the Newtonian gravitational constant in the ergoregion. The dispersion relation is obtained for the case of constant flow injection. 
  We consider a self-consistent system of Bianchi type-I (BI) gravitational field and a binary mixture of perfect fluid and dark energy given by a cosmological constant. The perfect fluid is chosen to be the one obeying either the usual equation of state, i.e., $p = \zeta \ve$, with $\zeta \in [0, 1]$ or a van der Waals equation of state. Role of the $\Lambda$ term in the evolution of the BI Universe has been studied. 
  In this article I will review some basic results on elliptic boundary value problems with applications to General Relativity. 
  Current and future optical technologies will aid exploration of the Moon and Mars while advancing fundamental physics research in the solar system. Technologies and possible improvements in the laser-enabled tests of various physical phenomena are considered along with a space architecture that could be the cornerstone for robotic and human exploration of the solar system. In particular, accurate ranging to the Moon and Mars would not only lead to construction of a new space communication infrastructure enabling an improved navigational accuracy, but will also provide a significant improvement in several tests of gravitational theory: the equivalence principle, geodetic precession, PPN parameters $\beta$ and $\gamma$, and possible variation of the gravitational constant $G$. Other tests would become possible with an optical architecture that would allow proceeding from meter to centimeter to millimeter range accuracies on interplanetary distances. This paper discusses the current state and the future improvements in the tests of relativistic gravity with Lunar Laser Ranging (LLR). We also consider precision gravitational tests with the future laser ranging to Mars and discuss optical design of the proposed Laser Astrometric Test of Relativity (LATOR) mission. We emphasize that already existing capabilities can offer significant improvements not only in the tests of fundamental physics, but may also establish the infrastructure for space exploration in the near future. Looking to future exploration, what characteristics are desired for the next generation of ranging devices, what is the optimal architecture that would benefit both space exploration and fundamental physics, and what fundamental questions can be investigated? We try to answer these questions. 
  A brief sketch of the present status of gravitational wave experiments is given. Attention is concentrated to recent observations with the gravitational detector network. The project OGRAN for a combined optic-interferometrical and acoustical gravitation wave antenna planned for installation into underground facilities of the Baksan Neutrino Observatory is presented. We describe general principles of the apparatus, expected sensitivity and current characteristics of the antenna prototype; some ways for sensitivity improvement are also discussed. 
  With quantitative numerical simulations we show that the secular variations of the even zonal harmonics of the geopotential do affect the measurement of the Lense-Thirring effect with the nodes of LAGEOS and LAGEOS II at a \sim 14% level. Thus, the 1-sigma total error in the performed test with the 2nd generation CHAMP-only EIGEN2 Earth gravity model amounts to 51%, contrary to 18% claimed by Ciufolini. 
  Recently a Hamiltonian formulation for the evolution of the universe dominated by multiple oscillatory scalar fields was developed by the present author and was applied to the investigation of the evolution of cosmological perturbations on superhorizon scales in the case that the scalar fields have incommensurable masses. In the present paper, the analysis is extended to the case that the masses of the scalar fields satisfy resonance relations approximately. In this case, the action-angle variables for the system can be classified into fast changing variables and slowly changing variables. We show that after an appropriate canonical transformation, the part of the Hamiltonian that depends on fast changing angle variables can be made negligibley small, so that the dynamics of the system can be effectively determined by a truncated Hamiltonian that describes a closed dynamics of the slowly changing varables. Utilizing this formulation, we show that the system is unstable if this truncated Hamiltonian system has hyperbolic fixed point and as a consequence, the Bardeen parameter for a perturbation of the system grows. 
  We extend the derivation of the Hawking temperature of a Schwarzschild black hole via the Heisenberg uncertainty principle to the de Sitter and anti-de Sitter spacetimes. The thermodynamics of the Schwarzschild-(anti-)de Sitter black holes is obtained from the generalized uncertainty principle of string theory and non-commutative geometry. This may explain why the thermodynamics of (anti-)de Sitter-like black holes admits a holographic description in terms of a dual quantum conformal field theory, whereas the thermodynamics of Schwarzschild-like black holes does not. 
  Integral equations for the spin-weighted spheroidal wave functions is given. For the prolate spheroidal wave function with m=0, there exists the integral equation whose kernel is(sin x)/x, and the sinc function kernel (sin x)/x is of great mathematical significance. In the paper, we also extend the similar sinc function kernel (sin x)/x to the case m and s both are not zero, which interestingly turn out as some kind of Hankel transformation. 
  Here, non-minimally coupled tachyon to gravity is considered as a source of "dark energy". It is demonstrated that with expansion of the universe, tachyon dark energy decays to "dark matter" providing a solution to "cosmic coincidence problem".Moreover, it is found that universe undergoes accelerated expansion simultaneously. 
  The existence of a minimum time uncertainty is usually argued to be a consequence of the combination of quantum mechanics and general relativity. Most of the studies that point to this result are nonetheless based on perturbative quantization approaches, in which the effect of matter on the geometry is regarded as a correction to a classical background. In this paper, we consider rainbow spacetimes constructed from doubly special relativity by using a modification of the proposals of Magueijo and Smolin. In these models, gravitational effects are incorporated (at least to a certain extent) in the definition of the energy-momentum of particles without adhering to a perturbative treatment of the back reaction. In this context, we derive and compare the expressions of the time uncertainty in quantizations that use as evolution parameter either the background or the rainbow time coordinates. These two possibilities can be regarded as corresponding to perturbative and non-perturbative quantization schemes, respectively. We show that, while a non-vanishing time uncertainty is generically unavoidable in a perturbative framework, an infinite time resolution can in fact be achieved in a non-perturbative quantization for the whole family of doubly special relativity theories with unbounded physical energy. 
  We analysis the radial equations for massive Dirac fields in Schwarzschild black hole spacetimes. Different approximation formulae under the WKB scheme are developed for the transmission probability ${\cal T}$ of the radial wavefunction with $E^{2}\gg V_{m}$, $E^{2}\approx V_{m}$, and $E^{2}\ll V_{m}$, where $E$ is the energy of the field and $V_{m}$ is the maximum value of the effective potential. Explicit results of ${\cal T}$ in these approximations are given for various values of $E$, the mass $m$, and the angular momentum parameter $\kappa$ of the fields. We also discuss the dependence of ${\cal T}$ on these parameters. 
  Newton's theory predicts that the velocity $V$ of free test particles on circular orbits around a spherical gravity center is a decreasing function of the orbital radius $r$, $dV/dr < 0$. Only very recently, Aschenbach (A&A 425, p. 1075 (2004)) has shown that, unexpectedly, the same is not true for particles orbiting black holes: for Kerr black holes with the spin parameter $a>0.9953$, the velocity has a positive radial gradient for geodesic, stable, circular orbits in a small radial range close to the black hole horizon. We show here that the {\em Aschenbach effect} occurs also for non-geodesic circular orbits with constant specific angular momentum $\ell = \ell_0 = const$. In Newton's theory it is $V = \ell_0/R$, with $R$ being the cylindrical radius. The equivelocity surfaces coincide with the $R = const$ surfaces which, of course, are just co-axial cylinders. It was previously known that in the black hole case this simple topology changes because one of the ``cylinders'' self-crosses. We show here that the Aschenbach effect is connected to a second topology change that for the $\ell = const$ tori occurs only for very highly spinning black holes, $a>0.99979$. 
  After a short introduction to the characteristic geometry underlying weakly hyperbolic systems of partial differential equations we review the notion of symmetric hyperbolicity of first-order systems and that of regular hyperbolicity of second-order systems. Numerous examples are provided, mainly taken from nonrelativistic and relativistic continuum mechanics. 
  We have found that proposals addressing the old cosmological constant problem come in various categories. The aim of this paper is to identify as many different, credible mechanisms as possible and to provide them with a code for future reference. We find that they all can be classified into five different schemes of which we indicate the advantages and drawbacks.   Besides, we add a new approach based on a symmetry principle mapping real to imaginary spacetime. 
  We study how different types of blow-ups can occur in systems of hyperbolic evolution equations of the type found in general relativity. In particular, we discuss two independent criteria that can be used to determine when such blow-ups can be expected. One criteria is related with the so-called geometric blow-up leading to gradient catastrophes, while the other is based upon the ODE-mechanism leading to blow-ups within finite time. We show how both mechanisms work in the case of a simple one-dimensional wave equation with a dynamic wave speed and sources, and later explore how those blow-ups can appear in one-dimensional numerical relativity. In the latter case we recover the well known ``gauge shocks'' associated with Bona-Masso type slicing conditions. However, a crucial result of this study has been the identification of a second family of blow-ups associated with the way in which the constraints have been used to construct a hyperbolic formulation. We call these blow-ups ``constraint shocks'' and show that they are formulation specific, and that choices can be made to eliminate them or at least make them less severe. 
  Analysis of Lunar Laser Ranging (LLR) data provides science results: gravitational physics and ephemeris information from the orbit, lunar science from rotation and solid-body tides, and Earth science. Sensitive tests of gravitational physics include the Equivalence Principle, limits on the time variation of the gravitational constant G, and geodetic precession. The equivalence principle test is used for an accurate determination of the parametrized post-Newtonian (PPN) parameter \beta. Lunar ephemerides are a product of the LLR analysis used by current and future spacecraft missions. The analysis is sensitive to astronomical parameters such as orbit, masses and obliquity. The dissipation-caused semimajor axis rate is 37.9 mm/yr and the associated acceleration in orbital longitude is -25.7 ''/cent^2, dominated by tides on Earth with a 1% lunar contribution. Lunar rotational variation has sensitivity to interior structure, physical properties, and energy dissipation. The second-degree lunar Love numbers are detected; k_2 has an accuracy of 11%. Lunar tidal dissipation is strong and its Q has a weak dependence on tidal frequency. A fluid core of about 20% the Moon's radius is indicated by the dissipation data. Evidence for the oblateness of the lunar fluid-core/solid-mantle boundary is getting stronger. This would be independent evidence for a fluid lunar core. Moon-centered coordinates of four retroreflectors are determined. Station positions and motion, Earth rotation variations, nutation, and precession are determined from analyses. Extending the data span and improving range accuracy will yield improved and new scientific results. Adding either new retroreflectors or precise active transponders on the Moon would improve the accuracy of the science results. 
  The warp drive envisioned by Alcubierre that can move a spaceship faster than light can, with modification, levitate it as if it were lighter than light, even allow it to go below a black hole's horizon and return unscathed. Wormhole-like versions of the author's `drainhole' (1973) might provide the drive, in the form of a by-pass of the spaceship composed of a multitude of tiny topological tunnels. The by-pass would divert the gravitational `ether' into a sink covering part of the spaceship's hull, connected by the tunnels to a source covering the remainder of the hull, to produce an ether flow like that of a river that disappears underground only to spring forth at a point downstream. This diversion would effectively shield the spaceship from external gravity. 
  The Randall-Sundrum scenario, with a 1+3-dimensional brane in a 5-dimensional bulk spacetime, can be generalized in various ways. We consider the case where the Z2-symmetry at the brane is relaxed, and in addition the gravitational action is generalized to include an induced gravity term on the brane. We derive the complete set of equations governing the gravitational dynamics for a general brane and bulk, and identify how the asymmetry and the induced gravity act as effective source terms in the projected field equations on the brane. For a Friedmann brane in an anti de Sitter bulk, the solution of the Friedmann equation is given by the solution of a quartic equation. We find the perturbative solutions for small asymmetry, which has an effect at late times. 
  The motion of a spinning test particle given by the Mathisson-Papapetrou equations is studied on an exterior vacuum C metric background spacetime describing the accelerated motion of a spherically symmetric gravitational source. We consider circular orbits of the particle around the direction of acceleration of the source. The symmetries of this configuration lead to the reduction of the differential equations of motion to algebraic relations. The spin supplementary conditions as well as the coupling between the spin of the particle and the acceleration of the source are discussed. 
  Using Cartan's equivalence method for point transformations we obtain from first principles the conformal geometry associated with third order ODEs and a special class of PDEs in two dimensions. We explicitly construct the null tetrads of a family of Lorentzian metrics, the conformal group in three and four dimensions and the so called normal metric connection. A special feature of this connection is that the non vanishing components of its torsion depend on one relative invariant, the (generalized) W\"unschmann Invariant. We show that the above mentioned construction naturally contains the Null Surface Formulation of General Relativity. 
  There are many solutions to the Einstein field equations that demonstrate naked singularity (NS) formation after regular evolution. It is possible, however, that such a quantum effect as particle creation prevents NSs from forming. We investigate the relation between the curvature strength and the quantum effects of NSs in a very wide class of spherical dust collapse. Through a perturbative calculation, we find that if the NS is very strong, the quantum particle creation diverges as the Cauchy horizon is approached, while if the NS is very weak, the creation should be finite. In the context of cosmic censorship, strong NSs will be subjected to the backreaction of quantum effects and may disappear or be hidden behind horizons, while weak NSs will not. 
  A simple model is constructed which allows to compute modified dispersion relations with effects from loop quantum gravity. Different quantization choices can be realized and their effects on the order of corrections studied explicitly. A comparison with more involved semiclassical techniques shows that there is agreement even at a quantitative level.   Furthermore, by contrasting Hamiltonian and Lagrangian descriptions we show that possible Lorentz symmetry violations may be blurred as an artifact of the approximation scheme. Whether this is the case in a purely Hamiltonian analysis can be resolved by an improvement in the effective semiclassical analysis. 
  In a companion article (referred hearafter as paper I) a detailed study of the simply transitive Spatially Homogeneous (SH) models of class A concerning the existence of a simply transitive similarity group has been given. The present work (paper II) continues and completes the above study by considering the remaining set of class B models. Following the procedure of paper I we find all SH models of class B subjected only to the minimal geometric assumption to admit a proper Homothetic Vector Field (HVF). The physical implications of the obtained geometric results are studied by specialising our considerations to the case of vacuum and $\gamma -$law perfect fluid models. As a result we regain all the known exact solutions regarding vacuum and non-tilted perfect fluid models. In the case of tilted fluids we find the \emph{general }self-similar solution for the exceptional type VI$_{-1/9}$ model and we identify it as equilibrium point in the corresponding dynamical state space. It is found that this \emph{new} exact solution belongs to the subclass of models $n_\alpha ^\alpha =0$, is defined for $\gamma \in (\frac 43,\frac 32)$ and although has a five dimensional stable manifold there exist always two unstable modes in the restricted state space. Furthermore the analysis of the remaining types, guarantees that tilted perfect fluid models of types III, IV, V and VII$_h$ cannot admit a proper HVF strongly suggesting that these models either may not be asymptotically self-similar (type V) or may be extreme tilted at late times. Finally for each Bianchi type, we give the extreme tilted equilibrium points of their state space. 
  A non-Riemannian geometrical approach to the investigation of an acoustic black hole in irrotational mean flows, based on the Lighthill vortex sound theory is given. This additional example of analog gravity based on classical fluids is used to investigate the acoustic Lorentz violation. An example is given where the contortion vector is distributed along a ring inside the fluid which can be gravitational analog of the torsion thick string spacetime defect. It is found that the linear background flow velocity approximation, acoustic Lorentz symmetry is breaking by the acoustic Cartan contortion in analogy to the spontaneous gravitational Lorentz breaking in Riemann-Cartan spacetime discovered recently by Kostelecky. We also show that although the acoustic torsion contributes to the fiducial observer acceleration, it is not present in Hawking radiation since is not presnt in the surface gravity of the acoustic black hole. 
  In a previous work (Mbelek 2001), we modelled the rotation curves (RC) of spiral galaxies by including in the equation of motion of the stars the dynamical terms from an external real self-interacting scalar field, $\psi$, minimally coupled to gravity and which respects the equivalence principle in the weak fields and low velocity approximation. This model appeared to have three free parameters : the turnover radius, $r_{0}$, the maximum tangential velocity, $v_{\theta max} = v_{\theta}(r_{0})$, plus a strictly positive integer, $n$. Here, we propose a new improved version where the coupling of the $\psi$-field to dark matter is emphasized at the expense of its self-interaction. This reformulation presents the very advantageous possibility that the same potential is used for all galaxies. Using at the same time a quasi-isothermal dark matter density and the scalar field helps to better fit the RC of spiral galaxies. In addition, new correlations are established. 
  Theoretical and observational arguments are listed in favor of a new principle of relativity of units of measurements as the basis of a conformal-invariant unification of General Relativity and Standard Model by replacement of all masses with a scalar (dilaton) field. The relative units mean conformal observables: the coordinate distance, conformal time, running masses, and constant temperature. They reveal to us a motion of a universe along its hypersurface in the field space of events like a motion of a relativistic particle in the Minkowski space, where the postulate of the vacuum as a state with minimal energy leads to arrow of the geometric time. In relative units, the unified theory describes the Cold Universe Scenario, where the role of the conformal dark energy is played by a free minimal coupling scalar field in agreement with the most recent distance-redshift data from type Ia supernovae. In this Scenario, the evolution of the Universe begins with the effect of intensive creation of primordial W-Z-bosons explaining the value of CMBR temperature, baryon asymmetry, tremendous deficit of the luminosity masses in the COMA-type superclusters and large-scale structure of the Universe. 
  We construct monopole-antimonopole chain and vortex solutions in Yang-Mills-Higgs theory coupled to Einstein gravity. The solutions are static, axially symmetric and asymptotically flat. They are characterized by two integers (m,n) where m is related to the polar angle and n to the azimuthal angle. Solutions with n=1 and n=2 correspond to chains of m monopoles and antimonopoles. Here the Higgs field vanishes at m isolated points along the symmetry axis. Larger values of n give rise to vortex solutions, where the Higgs field vanishes on one or more rings, centered around the symmetry axis. When gravity is coupled to the flat space solutions, a branch of gravitating monopole-antimonopole chain or vortex solutions arises, and merges at a maximal value of the coupling constant with a second branch of solutions. This upper branch has no flat space limit. Instead in the limit of vanishing coupling constant it either connects to a Bartnik-McKinnon or generalized Bartnik-McKinnon solution, or, for m>4, n>4, it connects to a new Einstein-Yang-Mills solution. In this latter case further branches of solutions appear. For small values of the coupling constant on the upper branches, the solutions correspond to composite systems, consisting of a scaled inner Einstein-Yang-Mills solution and an outer Yang-Mills-Higgs solution. 
  We present the whole set of equations with regularity and matching conditions required for the description of physically meaningful static cylindrically symmmetric distributions of matter, smoothly matched to Levi-Civita vacuum spacetime. It is shown that the conformally flat solution with equal principal stresses represents an incompressible fluid. It is also proved that any conformally flat cylindrically symmetric static source cannot be matched through Darmois conditions to the Levi-Civita spacetime. Further evidence is given that when the Newtonian mass per unit length reaches 1/2 the spacetime has plane symmetry. 
  Continuing previous work reported in an earlier paper [L.M. Burko, A.I. Harte, and E. Poisson, Phys. Rev. D 65, 124006 (2002)] we calculate the self-force acting on a point scalar charge in a wide class of cosmological spacetimes. The self-force produces two types of effect. The first is a time-changing inertial mass, and this is calculated exactly for a particle at rest relative to the cosmological fluid. We show that for certain cosmological models, the mass decreases and then increases back to its original value. For all other models except de Sitter spacetime, the mass is restored only to a fraction of its original value. For de Sitter spacetime the mass steadily decreases. The second effect is a deviation relative to geodesic motion, and we calculate this for a charge that moves slowly relative to the dust in a matter-dominated cosmology. We show that the net effect of the self-force is to push on the particle. We show that this is not an artifact of the scalar theory: The electromagnetic self-force acting on an electrically charged particle also pushes on the particle. The paper concludes with a demonstration that the pushing effect can also occur in the context of slow-motion electrodynamics in flat spacetime. 
  Recent attempts to calculate the black-hole entropy in loop quantum gravity are demonstrated to be erroneous. The correct solution of the problem is pointed out. 
  The constraint-preserving approach, which aim is to provide consistent boundary conditions for Numerical Relativity simulations, is discussed in parallel with other recent developments. The case of the Z4 system is considered, and constraint-preserving boundary conditions of the Sommerfeld type are provided. A necessary condition for the stability of the proposed boundary conditions is obtained, which amounts to the requirement of a symmetric ordering of space derivatives. This requirement is numerically seen to be also sufficient in the absence of corners and edges. Maximally dissipative boundary conditions are also implemented. In this case, a less restrictive stability condition is obtained, which is shown numerically to be also sufficient even in the presence of corners and edges. 
  In this work, the cosmological implications of brane world scenario are investigated when the gravitational coupling $G$ and the cosmological term $\Lambda$ are not constant but rather there are time variation of them. From observational point of view, these time variations are taken in the form $\frac{\dot{G}}{G}\sim H$ and $\Lambda \sim H^{2}$. The behavior of scale factors and different kinematical parameters are investigated for different possible scenarios where the bulk cosmological constant $\Lambda_{5}$ can be zero, positive or negative. 
  In this paper we describe in detail the computational algorithm used by our parallel multigrid elliptic equation solver with adaptive mesh refinement. Our code uses truncation error estimates to adaptively refine the grid as part of the solution process. The presentation includes a discussion of the orders of accuracy that we use for prolongation and restriction operators to ensure second order accurate results and to minimize computational work. Code tests are presented that confirm the overall second order accuracy and demonstrate the savings in computational resources provided by adaptive mesh refinement. 
  Analyses of laser ranges to the Moon provide increasingly stringent limits on any violation of the Equivalence Principle (EP); they also enable several very accurate tests of relativistic gravity. We report the results of our recent analysis of Lunar Laser Ranging (LLR) data giving an EP test of \Delta (M_G/M_I)_{EP} =(-1.0 +/- 1.4) x 10^{-13}. This result yields a Strong Equivalence Principle (SEP) test of \Delta (M_G/M_I)_{SEP} =(-2.0 +/- 2.0) x 10^{-13}. Also, the corresponding SEP violation parameter \eta is (4.4 +/- 4.5) x 10^{-4}, where \eta=4\beta-\gamma-3 and both \beta and \gamma are parametrized post-Newtonian (PPN) parameters. Using the recent Cassini result for the parameter \gamma, PPN parameter \beta is determined to be \beta-1=(1.2 +/- 1.1) x 10^{-4}. The geodetic precession test, expressed as a relative deviation from general relativity, is K_{gp}=-0.0019 +/- 0.0064. The search for a time variation in the gravitational constant results in \dot G/G=(4 +/- 9) x 10^{-13} yr^{-1}, consequently there is no evidence for local (~1AU) scale expansion of the solar system. 
  Quantum energy inequalities (QEIs) were established by Flanagan for the massless scalar field on two-dimensional Lorentzian spacetimes globally conformal to Minkowski space. We extend his result to all two-dimensional globally hyperbolic Lorentzian spacetimes and use it to show that flat spacetime QEIs give a good approximation to the curved spacetime results on sampling timescales short in comparison with natural geometric scales. This is relevant to the application of QEIs to constrain exotic spacetime metrics. 
  The asymptotic behaviour of the components of the Weyl tensor and of the energy-momentum tensor in the Penrose limit is determined. In both cases a peeling-off property is found. Examples of different types of matter are provided. The expansion and shear of the congruence of null geodesics along which the Penrose limit is taken are determined. Finally, the approach to the singularity in the Penrose limit of cosmological space-times is discussed. 
  The French-Italian interferometric gravitational wave detector VIRGO is currently being commissioned. Its principal instrument is a Michelson laser interferometer with 3 km long optical cavities in the arms and a power-recycling mirror. The interferometer resides in an ultra-high vacuum system and the mirrors are suspended from multistage pendulums for seismic isolation.   This type of laser interferometer reaches its maximum sensitivity only when the optical setup is held actively very accurately at a defined operating point: control systems using the precise interferometer signals stabilise the longitudinal and angular positions of the optical component. This paper gives an overview of the control system for the angular degrees of freedom; we present the current status of the system and report the first experimental demonstration of the Anderson technique on a large-scale interferometer. 
  We consider a topological field theory derived from the Chern - Simons action in (2+1) dimensions with the I(ISO(2,1)) group,and we investigate in detail the canonical structure of this theory.Originally developed as a topological theory of Einstein gravity minimally coupled to topological matter fields in (2+1) dimensions, it admits a BTZ black-hole solutions, and can be generalized to arbitrary dimensions.In this paper, we further study the canonical structure of the theory in (2+1) dimensions, by identifying all the distinct gauge equivalence classes of solutions as they result from holonomy considerations. The equivalence classes are discussed in detail, and examples of solutions representative of each class are constructed or identified. 
  Algebraically special fields with no gravitational radiation are described. Kerr-Schild fields, which include as a concrete case the Kinnersley photon rocket, form an important subclass of them. 
  We will address the question of the consistency of teleparallel theories in presence of spinning matter which has been a controversial subject of discussion over the last twenty years. We argue that the origin of the problem is not simply the symmetry or asymmetry of the stress-energy tensor of the matter fields, which has been recently analyzed by several authors, but arises at a more fundamental level, namely from the invariance of the field equatins under a frame change, a problem that has been discussed long time ago by Kopczynski in the framework of the teleparallel equivalent of general relativity. More importantly, we show that the problem is not only confined to the purely teleparallel theory but arises actually in every Poincare gauge theory that admits a teleparallel geometry in the absence of spinning sources, i.e. in its classical limit. 
  In this work, we deal with the chiral string model for which the world-sheet current is null in the framework of a scalar-tensor gravity. Our main goal is to analyse the impact of such a current on the gravitational macroscopic effects. For the purpose of this analysis, we first study the gravitational properties of the spacetime generated by this string in the presence of a dilaton field. Then, we carry out an investigation of the mechanism of formation and evolution of wakes in this framework, showing the explicit contribution of the chirality to this effect. 
  The smooth gravitational singularities of the differential spacetime manifold based General Relativity (GR) are viewed from the perspective of the background manifold independent and, in extenso, Calculus-free Abstract Differential Geometry (ADG). In particular, the inner Schwarzschild singularity is being `resolved' ADG-theoretically in two different ways. A plethora of important mathematical, physical and philosophical issues in current classical and quantum gravity research are addressed and tackled. 
  For there is always a wrong sign in the mass of graviton in the so-called perturbation expansion approximation of both Minkowski and de Sitter spacetimes, the existence of gravitational wave from the metric perturbation of de Sitter spacetime is doubtful. We try another way to start from the assumption that the gravitational wave equation should be both general covariant and conformal invariant and find that graviton is no longer a part of metric field, it has an effective mass of $m_g=\sqrt{R/6}=% \sqrt{2\Lambda/3}$ with correct sign in de Sitter spacetime, though it's intrinsic mass remains zero. 
  We describe a method by which gravitational wave observations of eccentric binary systems could be used to test General Relativity's prediction that gravitational waves are dispersionless. We present our results in terms of the graviton having a non-zero rest mass, or equivalently a non-infinite Compton wavelength. We make a rough estimate of the bounds that might be obtained following gravitational wave detections by the space-based LISA interferometer. The bounds we find are comparable to those obtainable from a method proposed by Will, and several orders of magnitude stronger than other dynamic (i.e. gravitational wave based) tests that have been proposed. The method described here has the advantage over those proposed previously of being simple to apply, as it does not require the inspiral to be in the strong field regime nor correlation with electromagnetic signals. We compare our results with those obtained from static (i.e. non-gravitational wave based) tests. 
  Within the framework of loop quantum cosmology, there exists a semi-classical regime where spacetime may be approximated in terms of a continuous manifold, but where the standard Friedmann equations of classical Einstein gravity receive non-perturbative quantum corrections. An approximate, analytical approach to studying cosmic dynamics in this regime is developed for both spatially flat and positively-curved isotropic universes sourced by a self-interacting scalar field. In the former case, a direct correspondence between the classical and semi-classical field equations can be established together with a scale factor duality that directly relates different expanding and contracting universes. Some examples of non-singular, bouncing cosmologies are presented together with a scaling, power-law solution. 
  We study oscillatory universes within the context of Loop Quantum Cosmology. We make a comparative study of flat and positively curved universes sourced by scalar fields with either positive or negative potentials. We investigate how oscillating universes can set the initial conditions for successful slow-roll inflation, while ensuring that the semi-classical bounds are satisfied. We observe rich oscillatory dynamics with negative potentials, although it is difficult to respect the semi-classical bounds in models of this type. 
  We develop models of void formation starting from a small initial fluctuation at recombination and growing to a realistic present day density profile in agreement with observations of voids. The model construction is an extension of previously developed algorithms for finding a Lemaitre-Tolman metric that evolves between two profiles of either density or velocity specified at two times. Of the 4 profiles of concern -- those of density and velocity at recombination and at the present day -- two can be specified and the other two follow from the derived model. We find that, in order to reproduce the present-day void density profiles, the initial velocity profile is more important than the initial density profile. Extrapolation of current CMB observations to the scales relevant to proto-voids is very uncertain. Even so, we find that it is very difficult to make both the initial density and velocity fluctuation amplitudes small enough, and still obtain a realistic void by today. 
  We present the first relativistic calculations of the final phase of inspiral of a binary system consisting of two stars built predominantely of strange quark matter (strange quark stars). We study the precoalescing stage within the Isenberg-Wilson-Mathews approximation of general relativity using a multidomain spectral method. A hydrodynamical treatment is performed under the assumption that the flow is either rigidly rotating or irrotational, taking into account the finite density at the stellar surface -- a distinctive feature with respect to the neutron star case. The gravitational-radiation driven evolution of the binary system is approximated by a sequence of quasi-equilibrium configurations at fixed baryon number and decreasing separation. We find that the innermost stable circular orbit (ISCO) is given by an orbital instability both for synchronized and irrotational systems. This constrasts with neutron stars for which the ISCO is given by the mass-shedding limit in the irrotational case. The gravitational wave frequency at the ISCO, which marks the end of the inspiral phase, is found to be 1400 Hz for two irrotational 1.35 Msol strange stars and for the MIT bag model of strange matter with massless quarks and a bag constant B=60 MeV/fm^3. Detailed comparisons with binary neutrons star models, as well as with third order Post-Newtonian point-mass binaries are given. 
  Gravitational waves (GWs) propagating through a uniformly magnetized plasma interact directly with the magnetic field and excite magnetohydrodynamic (MHD) waves with both electromagnetic and matter components. We study this process for arbitrary geometry in the MHD approximation and find that all three fundamental MHD modes -- slow and fast magnetosonic, and Alfven -- are excited depending on both the polarization of the GW and the orientation of the ambient magnetic field. The latter two modes can interact coherently with the GW resulting in damping of the GW and linear growth of the plasma waves. 
  We investigate the effect of spin-orbit and spin-spin couplings on the estimation of parameters for inspiralling compact binaries of massive black holes, and for neutron stars inspiralling into intermediate-mass black holes, using hypothetical data from the proposed Laser Interferometer Space Antenna (LISA). We work both in Einstein's theory and in alternative theories of gravity of the scalar-tensor and massive-graviton types. We restrict the analysis to non-precessing spinning binaries, i.e. to cases where the spins are aligned normal to the orbital plane. We find that the accuracy with which intrinsic binary parameters such as chirp mass and reduced mass can be estimated within general relativity is degraded by between one and two orders of magnitude. We find that the bound on the coupling parameter omega_BD of scalar-tensor gravity is significantly reduced by the presence of spin couplings, while the reduction in the graviton-mass bound is milder. Using fast Monte-Carlo simulations of 10^4 binaries, we show that inclusion of spin terms in massive black-hole binaries has little effect on the angular resolution or on distance determination accuracy. For stellar mass inspirals into intermediate-mass black holes, the angular resolution and the distance are determined only poorly, in all cases considered. We also show that, if LISA's low-frequency noise sensitivity can be extrapolated from 10^-4 Hz to as low as 10^-5 Hz, the accuracy of determining both extrinsic parameters (distance, sky location) and intrinsic parameters (chirp mass, reduced mass) of massive binaries may be greatly improved. 
  We propose in this paper an alternative method for the quantisation of systems with first-class constraints. This method is a combination of the coherent-state-path-integral quantisation developed by Klauder, with the ideas of reduced state space quantisation. The key idea is that the physical Hilbert space may be defined by a coherent-state path-integral on the reduced state space and that the metric on the reduced state space that is necessary for the regularisation of the path-integral may be computed from the geometry of the classical reduction procedure. We provide a number of examples--notably the relativistic particle. Finally we discuss the quantisation of systems, whose reduced state space has orbifold-like singularities. 
  How much of modern cosmology is really cosmography? How much of modern cosmology is independent of the Einstein equations? (Independent of the Friedmann equations?) These questions are becoming increasingly germane -- as the models cosmologists use for the stress-energy content of the universe become increasingly baroque, it behoves us to step back a little and carefully disentangle cosmological kinematics from cosmological dynamics. The use of basic symmetry principles (such as the cosmological principle) permits us to do a considerable amount, without ever having to address the vexatious issues of just how much "dark energy", "dark matter", "quintessence", and/or "phantom matter" is needed in order to satisfy the Einstein equations. This is the sub-sector of cosmology that Weinberg refers to as "cosmography", and in this article I will explore the extent to which cosmography is sufficient for analyzing the Hubble law and so describing many of the features of the universe around us. 
  We examine static charged perfect fluid configurations in the presence of a dilaton field. A method for construction of interior solutions is given. An explicit example of an interior solution which matches continuously the external Gibbons-Maeda-Garfinkle-Horowitz-Strominger solution is presented. Extremely charged perfect fluid configurations with a dilaton are also examined. We show that there are two types of extreme configurations. For each type the field equations are reduced to a single nonlinear equation on a space of a constant curvature. In the particular case of a perfect fluid with a linear equation of state, the field equations of the first type configurations are reduced to a Helmlotz equation on a space with a constant curvature. An explicit example of an extreme configuration is given and discussed. 
  We revisit the classical and quantum cosmology of a universe in which a self interacting scalar field is coupled to gravity with a flat FRW type metric undergoing continuous signature transition. We arrange for quantum cosmologically allowed discontinuity in the classical solutions at the signature changing hypersurface, provided these solutions be dual in some respects. This may be of some importance in the study of early universe within the signature changing scenarios. 
  The effects of quantized conformally invariant massless fields on the evolution of cosmological models containing a ``Big Rip'' future expansion singularity are examined. Quantized scalar, spinor, and vector fields are found to strengthen the accelerating expansion of such models as they approach the expansion singularity. 
  We compute the mass and angular momenta of rotating anti-de Sitter spacetimes in Einstein-Gauss-Bonnet theory of gravity using a superpotential derived from standard Noether identities. The calculation relies on the fact that the Einstein and Einstein-Gauss-Bonnet vacuum equations are the same when linearized on maximally symmetric backgrounds and uses the recently discovered D-dimensional Kerr-anti-de Sitter solutions to Einstein's equations. 
  We study the static black hole solutions of generalized two-dimensional dilaton-gravity theories generated by pointlike mass sources, in the hypothesis that the matter is conformally coupled. We also discuss the motion of test particles. Due to conformal coupling, these follow the geodesics of a metric obtained by rescaling the canonical metric with the dilaton. 
  We perform both distorted black hole evolutions and binary black hole head on collisions and compare the results of using a full grid to results obtained by excising the black hole interiors. In both cases the evolutions are found to run essentially indefinitely, and produce the same, convergent waveforms. Further, since both the distorted black holes and the head-on collision of puncture initial data can be carried out without excision, they provide an excellent dynamical test-bed for excision codes. This provides a strong numerical demonstration of the validity of the excision idea, namely the event horizon can be made to "protect" the spacetime from the excision boundary and allow an accurate exterior evolution. 
  Recently the Master Constraint Programme for Loop Quantum Gravity (LQG) was proposed as a classically equivalent way to impose the infinite number of Wheeler -- DeWitt constraint equations in terms of a single Master Equation. While the proposal has some promising abstract features, it was until now barely tested in known models. In this series of five papers we fill this gap, thereby adding confidence to the proposal. We consider a wide range of models with increasingly more complicated constraint algebras, beginning with a finite dimensional, Abelean algebra of constraint operators which are linear in the momenta and ending with an infinite dimensional, non-Abelean algebra of constraint operators which closes with structure functions only and which are not even polynomial in the momenta. In all these models we apply the Master Constraint Programme successfully, however, the full flexibility of the method must be exploited in order to complete our task. This shows that the Master Constraint Programme has a wide range of applicability but that there are many, physically interesting subtleties that must be taken care of in doing so. In this first paper we prepare the analysis of our test models by outlining the general framework of the Master Constraint Programme. The models themselves will be studied in the remaining four papers. As a side result we develop the Direct Integral Decomposition (DID) for solving quantum constraints as an alternative to Refined Algebraic Quantization (RAQ). 
  This is the second paper in our series of five in which we test the Master Constraint Programme for solving the Hamiltonian constraint in Loop Quantum Gravity. In this work we begin with the simplest examples: Finite dimensional models with a finite number of first or second class constraints, Abelean or non -- Abelean, with or without structure functions. 
  This is the third paper in our series of five in which we test the Master Constraint Programme for solving the Hamiltonian constraint in Loop Quantum Gravity. In this work we analyze models which, despite the fact that the phase space is finite dimensional, are much more complicated than in the second paper: These are systems with an $SL(2,\Rl)$ gauge symmetry and the complications arise because non -- compact semisimple Lie groups are not amenable (have no finite translation invariant measure). This leads to severe obstacles in the refined algebraic quantization programme (group averaging) and we see a trace of that in the fact that the spectrum of the Master Constraint does not contain the point zero. However, the minimum of the spectrum is of order $\hbar^2$ which can be interpreted as a normal ordering constant arising from first class constraints (while second class systems lead to $\hbar$ normal ordering constants). The physical Hilbert space can then be be obtained after subtracting this normal ordering correction. 
  This is the fourth paper in our series of five in which we test the Master Constraint Programme for solving the Hamiltonian constraint in Loop Quantum Gravity. We now move on to free field theories with constraints, namely Maxwell theory and linearized gravity. Since the Master constraint involves squares of constraint operator valued distributions, one has to be very careful in doing that and we will see that the full flexibility of the Master Constraint Programme must be exploited in order to arrive at sensible results. 
  This is the final fifth paper in our series of five in which we test the Master Constraint Programme for solving the Hamiltonian constraint in Loop Quantum Gravity. Here we consider interacting quantum field theories, specificlly we consider the non -- Abelean Gauss constraints of Einstein -- Yang -- Mills theory and 2+1 gravity. Interestingly, while Yang -- Mills theory in 4D is not yet rigorously defined as an ordinary (Wightman) quantum field theory on Minkowski space, in background independent quantum field theories such as Loop Quantum Gravity (LQG) this might become possible by working in a new, background independent representation. 
  After the heroic epoch of Causality Theory, problems concerning the smoothability of time functions and Cauchy hypersurfaces remained as unanswered folk questions. Just recently solved, our aim is to discuss the state of the art on this topic, including self-contained proofs for questions on causally continuous, stably causal and globally hyperbolic spacetimes. 
  We canonically quantize the dynamics of the brane universe embedded into the five-dimensional Schwarzschild-anti-deSitter bulk space-time. We show that in the brane-world settings the formulation of the quantum cosmology, including the problem of initial conditions, is conceptually more simple than in the 3+1-dimensional case. The Wheeler-deWitt equation is a finite-difference equation. It is exactly solvable in the case of a flat universe and we find the ground state of the system. The closed brane universe can be created as a result of decay of the bulk black hole. 
  We consider elastic bodies in rigid rotation, both nonrelativistically and in special relativity. Assuming a body to be in its natural state in the absence of rotation, we prove the existence of solutions to the elastic field equations for small angular velocity. 
  (Abridged) We revisit the problem of parameter estimation of gravitational-wave chirp signals from inspiralling non-spinning compact binaries in the light of the recent extension of the post-Newtonian (PN) phasing formula to order $(v/c)^7$ beyond the leading Newtonian order. We study in detail the implications of higher post-Newtonian orders from 1PN up to 3.5PN in steps of 0.5PN ($\sim v/c$), and examine their convergence. In both initial and advanced detectors the estimation of the chirp mass (${\cal M}$) and symmetric mass ratio ($\eta$) improve at higher PN orders but oscillate with every half-a-PN order. We compare parameter estimation in different detectors and assess their relative performance in two different ways: at a {\it fixed SNR,} with the aim of understanding how the bandwidth improves parameter estimation, and for a {\it fixed source}, to gauge the importance of sensitivity. Errors in parameter estimation at a fixed SNR are smaller for VIRGO than for both initial and advanced LIGO. However, for sources at a fixed distance it is advanced LIGO that achieves the lowest errors owing to its greater sensitivity. Finally, we compute the amplitude corrections due to the `frequency-sweep' in the Fourier domain representation of the waveform within the stationary phase approximation and discuss its implication on parameter estimation. 
  Inspiraling compact binaries are expected to be the strongest sources of gravitational waves for VIRGO, LIGO and other laser interferometers. We present the first computations of quasi-equilibrium sequences of compact binaries containing two strange quark stars (which are currently considered as a possible alternative to neutron stars). We study a precoalescing stage in the conformal flatness approximation of general relativity using a multidomain spectral method. A hydrodynamical treatment is performed under the assumption that the flow is irrotational. 
  The 2-surface characterization of special classical radiative Higgs-, Yang-Mills and linear zero-rest-mass (l.z.r.m) fields with any spin is investigated. We determine all the zero quasi-local mass Higgs- and Yang-Mills field configurations with compact semisimple gauge groups, and show that they are plane waves (provided the Higgs field is massless and linear) and appropriate generalizations of plane waves (`Yang-Mills pp-waves'), respectively. A tensor field (generalizing the energy-momentum tensor for the Maxwell field and of the Bel-Robinson tensor for the linearized gravitational field) is found by means of which the pp-wave nature of the solutions of the l.z.r.m. field equations with any spin can be characterized equivalently. It is shown that these radiative Yang-Mills and l.z.r.m. fields, given on a finite globally hyperbolic domain D, are determined completely by certain unconstrained data set on a closed spacelike 2-surface, the `edge of D'. These pure radiative solutions are shown to determine a dense subset in the set of solutions of various (Yang-Mills and l.z.r.m.) field equations. Thus for these field configurations some `classical quasi-local holography' holds. 
  We study the fully nonlinear dynamical evolution of binary black hole data, whose orbital parameters are specified via the effective potential method for determining quasi-circular orbits. The cases studied range from the Cook-Baumgarte innermost stable circular orbit (ISCO) to significantly beyond that separation. In all cases we find the black holes to coalesce (as determined by the appearance of a common apparent horizon) in less than half an orbital period. The results of the numerical simulations indicate that the initial holes are not actually in quasi-circular orbits, but that they are in fact nearly plunging together. The dynamics of the final horizon are studied to determine physical parameters of the final black hole, such as its spin, mass, and oscillation frequency, revealing information about the inspiral process. We show that considerable resolution is required to extract accurate physical information from the final black hole formed in the merger process, and that the quasi-normal modes of the final hole are strongly excited in the merger process. For the ISCO case, by comparing physical measurements of the final black hole formed to the initial data, we estimate that less than 3% of the total energy is radiated in the merger process. 
  We discuss the gravitational self-force on a particle in a black hole space-time. For a point particle, the full (bare) self-force diverges. The metric perturbation induced by a particle can be divided into two parts, the direct part (or the S part) and the tail part (or the R part), in the harmonic gauge, and the regularized self-force is derived from the R part which is regular and satisfies the source-free perturbed Einstein equations. But this formulation is abstract, so when we apply to black hole-particle systems, there are many problems to be overcome in order to derive a concrete self-force. These problems are roughly divided into two parts. They are the problem of regularizing the divergent self-force, i.e., ``subtraction problem'' and the problem of the singularity in gauge transformation, i.e., ``gauge problem''. In this paper, we discuss these problems in the Schwarzschild background and report some recent progress. 
  After a brief introduction on the scientific objectives of the LARES/WEBER-SAT satellite we present the recent measurement of the Lense-Thirring effect using the nodes of the LAGEOS and LAGEOS 2 satellites and using the Earth gravity model EIGENGRACE02S obtained by the GRACE space mission, we also include some determination of the rate of change of the lowest order Earth's even zonal harmonics. Finally, we describe an interesting possibility of testing the Brane-World unified theory of fundamental interactions by the use of a specially designed LARES/WEBER-SAT satellite. 
  The conformal method for constructing initial data for Einstein's equations is presented in both the Hamiltonian and Lagrangian picture (extrinsic curvature decomposition and conformal thin sandwich formalism, respectively), and advantages due to the recent introduction of a weight-function in the extrinsic curvature decomposition are discussed. I then describe recent progress in numerical techniques to solve the resulting elliptic equations, and explore innovative approaches toward the construction of astrophysically realistic initial data for binary black hole simulations. 
  Equilibrium states of black holes can be modelled by isolated horizons. If the intrinsic geometry is spherical, they are called type I while if it is axi-symmetric, they are called type II. The detailed theory of geometry of \emph{quantum} type I horizons and the calculation of their entropy can be generalized to type II, thereby including arbitrary distortions and rotations. The leading term in entropy of large horizons is again given by 1/4th of the horizon area for the \emph{same} value of the Barbero-Immirzi parameter as in the type I case. Ideas and constructions underlying this extension are summarized. 
  In many different ways, Deformed Special Relativity (DSR) has been argued to provide an effective limit of quantum gravity in almost-flat regime. Some experiments will soon be able to test some low energy effects of quantum gravity, and DSR is a very promising candidate to describe these latter. Unfortunately DSR is up to now plagued by many conceptual problems (in particular how it describes macroscopic objects) which forbids a definitive physical interpretation and clear predictions. Here we propose a consistent framework to interpret DSR. We extend the principle of relativity: the same way that Special Relativity showed us that the definition of a reference frame requires to specify its speed, we show that DSR implies that we must also take into account its mass. We further advocate a 5-dimensional point of view on DSR physics and the extension of the kinematical symmetry from the Poincare group to the Poincare-de Sitter group (ISO(4,1)). This leads us to introduce the concept of a pentamomentum and to take into account the renormalization of the DSR deformation parameter kappa. This allows the resolution of the "soccer ball problem" (definition of many-particle-states) and provides a physical interpretation of the non-commutativity and non-associativity of the addition the relativistic quadrimomentum. In particular, the coproduct of the kappa-Poincare algebra is interpreted as defining the law of change of reference frames and not the law of scattering. This point of view places DSR as a theory, half-way between Special Relativity and General Relativity, effectively implementing the Schwarzschild mass bound in a flat relativistic context. 
  Braneworld theory provides a natural setting to treat, at a classical level, the cosmological effects of vacuum energy. Non-static extra dimensions can generally lead to a variable vacuum energy, which in turn may explain the present accelerated cosmic expansion. We concentrate our attention in models where the vacuum energy decreases as an inverse power law of the scale factor. These models agree with the observed accelerating universe, while fitting simultaneously the observational data for the density and deceleration parameter. The redshift at which the vacuum energy can start to dominate depends on the mass density of ordinary matter. For Omega = 0.3, the transition from decelerated to accelerated cosmic expansion occurs at z approx 0.48 +/- 0.20, which is compatible with SNe data. We set a lower bound on the deceleration parameter today, namely q > - 1 + 3 Omega/2, i.e., q > - 0.55 for Omega = 0.3. The future evolution of the universe crucially depends on the time when vacuum starts to dominate over ordinary matter. If it dominates only recently, at an epoch z < 0.64, then the universe is accelerating today and will continue that way forever. If vacuum dominates earlier, at z > 0.64, then the deceleration comes back and the universe recollapses at some point in the distant future. In the first case, quintessence and Cardassian expansion can be formally interpreted as the low energy limit of our model, although they are entirely different in philosophy. In the second case there is no correspondence between these models and ours. 
  We linearize the Einstein equations when the metric is Bondi-Sachs, when the background is Schwarzschild or Minkowski, and when there is a matter source in the form of a thin shell whose density varies with time and angular position. By performing an eigenfunction decomposition, we reduce the problem to a system of linear ordinary differential equations which we are able to solve. The solutions are relevant to the characteristic formulation of numerical relativity: (a) as exact solutions against which computations of gravitational radiation can be compared; and (b) in formulating boundary conditions on the $r=2M$ Schwarzschild horizon. 
  Teleparallel gravity models, in which the curvature and the nonmetricity of spacetime are both set zero, are widely studied in the literature. We work a different teleparallel theory, in which the curvature and the torsion of spacetime are both constrained to zero, but the nonmetricity is nonzero. After reformulating the general relativity in this spacetime we find a solution and investigate its singularity structure. 
  We study dynamics of a scalar field in the near-horizon region described by a static Klein-Gordon operator which is the Hamiltonian of the system. The explicite construction of a time operator near-horizon is given and its self-adjointness discussed. 
  The gravitational self-force on a point particle moving in a vacuum background spacetime can be expressed as an integral over the past worldline of the particle, the so-called tail term. In this paper, we consider that piece of the self-force obtained by integrating over a portion of the past worldline that extends a proper time ${\Delta}{\tau}$ into the past, provided that ${\Delta}{\tau}$ does not extend beyond the normal neighborhood of the particle. We express this ``quasi-local'' piece as a power series in the proper time interval ${\Delta}{\tau}$. We argue from symmetries and dimensional considerations that the $O({\Delta}{\tau}^0)$ and $O({\Delta}{\tau})$ terms in this power series must vanish, and compute the first two non-vanishing terms which occur at $O({\Delta}{\tau}^2)$ and $O({\Delta}{\tau}^3)$. The coefficients in the expansion depend only on the particle's four velocity and on the Weyl tensor and its derivatives at the particle's location. The result may be useful as a foundation for a practical computational method for gravitational self-forces in the Kerr spacetime, in which the portion of the tail integral in the distant past is computed numerically from a mode sum decomposition. 
  Coalescing compact star binaries are expected to be among the strongest sources of gravitational radiation to be seen by laser interferometers. We present calculations of the final phase of inspiral of equal mass irrotational neutron star binaries and strange quark star binaries. Six types of equations of state at zero temperature are used - three realistic nuclear equations of state of various softness and three different MIT bag models of strange quark matter. We study the precoalescing stage within the Isenberg-Wilson-Mathews approximation of general relativity using a multidomain spectral method. The gravitational-radiation driven evolution of the binary system is approximated by a sequence of quasi-equilibrium configurations at fixed baryon number and decreasing separation. We find that the innermost stable circular orbit (ISCO) is given by an orbital instability for binary strange quark stars and by the mass-shedding limit for neutron star binaries. The gravitational wave frequency at the ISCO, which marks the end of the inspiral phase, is found to be around 1100-1460 Hz for two 1.35 solar masses irrotational strange stars described by the MIT bag model and between 800 Hz and 1230 Hz for neutron stars. 
  In this article we lay foundations for a formal relationship of spin foam models of gravity and BF theory to their continuum canonical formulations. First the derivation of the spin foam model of the BF theory from the discrete BF theory action in n dimensions is reviewed briefly. By foliating the underlying n dimensional simplicial manifold using n-1 dimensional simplicial hypersurfaces, the spin foam model is reformulated. Then it is shown that spin network functionals arise naturally on the foliations. The graphs of these spin network functionals are dual to the triangulations of the foliating hypersurfaces. Quantum Transition amplitudes are defined. I calculate the transition amplitudes related to 2D BF theory explicitly and show that these amplitudes are triangulation independent. The application to the spin foam models of gravity is discussed briefly. 
  We construct new geon-type black holes in D>3 dimensions for Einstein's theory coupled to gauge fields. A static nondegenerate vacuum black hole has a geon quotient provided the spatial section admits a suitable discrete isometry, and an antisymmetric tensor field of rank 2 or D-2 with a pure F^2 action can be included by an appropriate (and in most cases nontrivial) choice of the field strength bundle. We find rotating geons as quotients of the Myers-Perry(-AdS) solution when D is odd and not equal to 7. For other D we show that such rotating geons, if they exist at all, cannot be continuously deformed to zero angular momentum. With a negative cosmological constant, we construct geons with angular momenta on a torus at the infinity. As an example of a nonabelian gauge field, we show that the D=4 spherically symmetric SU(2) black hole admits a geon version with a trivial gauge bundle. Various generalisations, including both black-brane geons and Yang-Mills theories with Chern-Simons terms, are briefly discussed. 
  This paper studies false vacuum lumps surrounded by the true vacuum in a real scalar field potential in flat spacetime. Fermions reside in the core of the lump, which are coupled with the scalar field via Yukawa interaction. Such lumps are stable against spherical collapse and deformation from spherical shape based on energetics considerations. The fermions inside the lump are treated as a uniform Fermi gas. We consider the Fermi gas in both ultrarelativistic and nonrelativistic limits. The mass and size of these lumps depend on the scale characterizing the scalar field potential as well as the mass density of the fermions. 
  We use the numerical solution describing the evolution of a perturbed black string presented in Choptuik et al. (2003) to elucidate the intrinsic behavior of the horizon. It is found that by the end of the simulation, the affine parameter on the horizon has become very large and the expansion and shear of the horizon in turn very small. This suggests the possibility that the horizon might pinch off in infinite affine parameter. 
  We define an Isometry germ at any given event $x$ of space-time as a vector field $\xi$ defined in a neighborhood of $x$ such that the Lie derivative of both the metric and the Riemannian connection are zero at this event. Two isometry germs can be said to be equivalent if their values and the values of their first derivatives coincide at $x$. The corresponding quotient space can be endowed with a structure of a bracket algebra which is a deformation of de Sitter's Lie algebra. Each isometry germ defines also a local stationary frame of reference, the consideration of the family of adapted coordinate transformations between any two of them leading to a local novel structure that generalizes the Lorentz group. 
  We describe conditions assuring that the Kerr--Schild type solutions of Einstein equations with pure radiation fields are asymptotically flat at future null infinity. Such metrics cannot describe ``true'' gravitational radiation from bounded sources--it is shown that the Bondi news function vanishes identically. We obtain formulae for the total energy and angular momentum at scri+. As an example we consider non-stationary generalization of the Kerr metric given by Vaidya and Patel. Angular momentum and total energy are expressed in closed form as functions of retarded time. 
  This text consists on a series of introductory lectures on cosmology for mathematicians and physicists who are not specialized on the subject. 
  We compute the radiation reaction force on the orbital motion of compact binaries to the 3.5 post-Newtonian (3.5PN) approximation, i.e. one PN order beyond the dominant effect. The method is based on a direct PN iteration of the near-zone metric and equations of motion of an extended isolated system, using appropriate ``asymptotically matched'' flat-space-time retarded potentials. The formalism is subsequently applied to binary systems of point particles, with the help of the Hadamard self-field regularisation. Our result is the 3.5PN acceleration term in a general harmonic coordinate frame. Restricting the expression to the centre-of-mass frame, we find perfect agreement with the result derived in a class of coordinate systems by Iyer and Will using the energy and angular momentum balance equations. 
  Numerical relativity has come a long way in the last three decades and is now reaching a state of maturity. We are gaining a deeper understanding of the fundamental theoretical issues related to the field, from the well posedness of the Cauchy problem, to better gauge conditions, improved boundary treatment, and more realistic initial data. There has also been important work both in numerical methods and software engineering. All these developments have come together to allow the construction of several advanced fully three-dimensional codes capable of dealing with both matter and black holes. In this manuscript I make a brief review the current status of the field. 
  We show that a set of conformally invariant equations derived from the Fefferman-Graham tensor can be used to construct global solutions of the vacuum Einstein equations, in all even dimensions. This gives, in particular, a new, simple proof of Friedrich's result on the future hyperboloidal stability of Minkowski space-time, and extends its validity to even dimensions. 
  A path deviation equation in the Parameterized Absolute Parallelism (PAP) geometry is derived. This equation includes curvature and torsion terms. These terms are found to be naturally quantized. The equation represents the deviation from a general path equation, in the PAP-geometry, derived by the author in a previous work. It is shown that, as the effect of the torsion, on the deviation, increases, the effect of the curvature decreases. It is also shown that the general path deviation equation can be reduced to the geodesic deviation equation if PAP-geometry becomes Riemannian. The equation can be used to study the deviation from the trajectories of spinning elementary particles. 
  I describe analysis of correlations in the outputs of the three LIGO interferometers from LIGO's first science run, held over 17 days in August and September of 2002, and the resulting upper limit set on a stochastic background of gravitational waves. By searching for cross-correlations between the LIGO detectors in Livingston, LA and Hanford, WA, we are able to set a 90% confidence level upper limit of h_{100}^2 Omega_0 < 23 +/- 4.6. 
  We study the Klein-Gordon equation for a massive scalar field in Kerr spacetime in the time-domain. We demonstrate that under conditions of super-radiance, the scalar field becomes unstable and its amplitude grows without bound. We also estimate the growth rate of this instability. 
  We study the universality in gravitational waves emitted from non-rotating neutron stars characterized by different equations of state (EOS). We find that the quasi-normal mode frequencies of such waves, including the $w$-modes and the $f$-mode, display similar universal scaling behaviours that hold for most EOS. Such behaviours are shown to stem from the mathematical structure of the axial and the polar gravitational wave equations, and the fact that the mass distribution function can be approximated by a cubic-quintic polynomial in radius. As a benchmark for other realistic neutron stars, a simple model of neutron stars is adopted here to reproduce the pulsation frequencies and the generic scaling behaviours mentioned above with good accuracy. 
  Recently it has been suggested to use the perigee of the proposed LARES/WEBER-SAT satellite in order to measure the secular precession which would be induced on such Keplerian orbital element by a weak-field modification of gravity occurring in some Brane-World scenarios put forth by Dvali, Gabadadze and Porrati. This precession, derived for the first time by Lue and Starkman, amounts to 4\times 10-3 milliarcseconds per year for an Earth orbiting satellite. In this paper we show that, according to the recently released EIGEN-CG01C Earth gravity model the quite larger systematic errors due to the Newtonian part of the terrestrial gravitational potential would vanish any attempts to detect a so small effect. The situation is much more favorable in the Solar System scenario. Indeed, the non-Newtonian perihelion advance of Mars, which is the currently best tracked planet, has recently been measured with an accuracy of 1\times 10^-4 arcseconds per century; the Dvali precession is 4\times 10^-4 arcseconds per century. A suitable combination of the perihelia and the nodes of Mars and Mercury, which disentangles the Dvali effect from the competing larger Newtonian and general relativistic precessions, might allow to reach a 1-sigma 47% level of accuracy. 
  We contrast the initial condition requirements of various contemporary cosmological models, including inflationary and bouncing cosmologies. Various proposals such as Hartle-Hawking's no boundary, or Tunnelling boundary conditions are assessed on grounds of naturalness and fine tuning. Alternatively a quiescent or ``time machine'' state is considered. Extensions to brane models are also addressed. Further ideas about universe creation from a meta-universe are outlined.   We compare the recent loop quantum cosmology of Bojowald and coworkers with these earlier proposals. A number of possible difficulties and limitations are outlined. 
  Causality principle is a powerful criterion that allows us to discriminate between what is possible or not. In this paper we study the transition from decelerated to accelerated expansion in the context of Cardassian and dark energy models. We distinguish two important events during the transition. The first one is the end of the matter-dominated phase, which occurs at some time $t_{eq}$. The second one is the actual crossover from deceleration to acceleration, which occurs at some $t_{T}$. Causality requires $t_{T} \geq t_{eq}$. We demonstrate that dark energy models, with constant $w$, and Cardassian expansion, are compatible with causality only if $(\Omega_{M} - \bar{q}) \leq 1/2$. However, observational data indicate that the most probable option is $(\Omega_{M} - \bar{q}) > 1/2$. Consequently, the transition from deceleration to acceleration in dark energy and Cardassian models occurs before the matter-dominated epoch comes to an end, i.e., $t_{eq} > t_{T}$. Which contradicts causality principle. 
  We discuss the asymptotic behavior of regulated field commutators for linearly polarized, cylindrically symmetric gravitational waves and the mathematical techniques needed for this analysis. We concentrate our attention on the effects brought about by the introduction of a physical cut-off in the study of the microcausality of the model and describe how the different physically relevant regimes are affected by its presence. Specifically we discuss how genuine quantum gravity effects can be disentangled from those originating in the introduction of a regulator. 
  By using suitably improved surface integrals, we give a unified geometric derivation of the generalized Smarr relation for higher dimensional Kerr black holes which is valid both in flat and in anti-de Sitter backgrounds. The improvement of the surface integrals, which allows one to use them simultaneously at infinity and on the horizon, consists in integrating them along a path in solution space. Path independence of the improved charges is discussed and explicitly proved for the higher dimensional Kerr AdS black holes. It is also shown that the charges for these black holes can be correctly computed from the standard Hamiltonian or Lagrangian surface integrals. 
  Within the framework of a Kaluza-Klein theory, we provide the geometrization of a generic (Abelian and non-Abelian) gauge coupling, which comes out by choosing a suitable matter fields dependence on the extra-coordinates.   We start by the extension of the Nother theorem to a multidimensional spacetime being the direct sum of a 4-dimensional Minkowski space and of a compact homogeneous manifold (whose isometries reflect the gauge symmetry); we show, how on such a ``vacuum'' configuration, the extra-dimensional components of the field momentum correspond to the gauge charges. Then we analyze the structure of a Dirac algebra as referred to a spacetime with the Kaluza-Klein restrictions and, by splitting the corresponding free-field Lagrangian, we show how the gauge coupling terms outcome. 
  Dynamics of a null thin shell immersed in a generic spherically symmetric spacetime is obtained within the distributional formalism. It has been shown that the distributional formalism leads to the same result as in the conventional formalism. 
  The Palatini formulation is used to develop a genuine connection theory for general relativity, in which the gravitational field is represented by a Lorentz-valued spin connection. The existence of a tetrad field, given by the Fock-Ivanenko covariant derivative of the tangent-space coordinates, implies a coupling between the spin connection and the coordinate vector-field, which turns out to be the responsible for the onset of curvature. This connection-coordinate coupling can thus be considered as the very foundation of the gravitational interaction. The peculiar form of the tetrad field is shown to reduce both Bianchi identities of general relativity to a single one, which brings this theory closer to the gauge theories describing the other fundamental interactions of Nature. Some further properties of this approach are also examined. 
  (Abridged): The standard adiabatic approximation to phasing of gravitational waves from inspiralling compact binaries uses the post-Newtonian expansions of the binding energy and gravitational wave flux both truncated at the same relative post-Newtonian order. Motivated by the eventual need to go beyond the adiabatic approximation we must view the problem as the dynamics of the binary under conservative post-Newtonian forces and gravitational radiation damping.   From the viewpoint of the dynamics of the binary, the standard approximation at leading order is equivalent to retaining the 0PN and 2.5PN terms in the acceleration and neglecting the intervening 1PN and 2PN terms. A complete mathematically consistent treatment of the acceleration at leading order should include all PN terms up to 2.5PN without any gaps. These define the 'standard' and 'complete' non-adiabatic approximants respectively. We propose a new and simple complete adiabatic approximant constructed from the energy and flux functions. At the leading order it uses the 2PN energy function rather than the 0PN one in the standard approximation so that in spirit it corresponds to the dynamics where there are no missing post-Newtonian terms in the acceleration. We compare the overlaps of the standard and complete adiabatic approximants with the exact waveforms for a test-particle orbiting a Schwarzschild black hole. The complete adiabatic approximants lead to a remarkable improvement in the effectualness at lower PN (< 3PN) orders. However, standard adiabatic approximants of order $\geq$ 3PN are nearly as good as the complete adiabatic approximants for the construction of effectual templates. Standard and complete approximants beyond the adiabatic approximation are next studied using the Lagrangian models of Buonanno, Chen and Vallisneri. 
  The general covariance principle, seen as an active version of the principle of equivalence, is used to study the gravitational coupling prescription in the presence of curvature and torsion. It is concluded that the coupling prescription determined by this principle is always equivalent with the corresponding prescription of general relativity. An application to the case of a Dirac spinor is made. 
  We discuss the notion of causality in Quantum Gravity in the context of sum-over-histories approaches, in the absence therefore of any background time parameter. In the spin foam formulation of Quantum Gravity, we identify the appropriate causal structure in the orientation of the spin foam 2-complex and the data that characterize it; we construct a generalised version of spin foam models introducing an extra variable with the interpretation of proper time and show that different ranges of integration for this proper time give two separate classes of spin foam models: one corresponds to the spin foam models currently studied, that are independent of the underlying orientation/causal structure and are therefore interpreted as a-causal transition amplitudes; the second corresponds to a general definition of causal or orientation dependent spin foam models, interpreted as causal transition amplitudes or as the Quantum Gravity analogue of the Feynman propagator of field theory, implying a notion of ''timeless ordering''. 
  The temperature gradient of microwave background radiation (CMBR) is calculated in the Self Consistent Model. An expected values for Hubble parameter have been presented in two different cases. In the first case the temperature is treated as a function of time only, while in the other one the temperature depends on relaxation of isotropy condition in the self-consistent model and the assumption that the universe expands adiabatically. The COBE's or WMAP's fluctuations in temperature of CMBR may be used to predict a value for Hubble parameter. 
  A new tetrad is introduced within the framework of geometrodynamics for non-null electromagnetic fields. This tetrad diagonalizes the electromagnetic stress-energy tensor and allows for maximum simplification of the expression of the electromagnetic field. The Einstein-Maxwell equations will also be simplified. 
  We present the history of fourth order metric theories of gravitation from its beginning in 1918 until 1988. 
  Using a recently developed quantization of spherically symmetric gravity coupled to a scalar field, we give a construction of null expansion operators that allow a definition of general, fully dynamical quantum black holes. These operators capture the intuitive idea that classical black holes are defined by the presence of trapped surfaces, that is surfaces from which light cannot escape outward. They thus provide a mechanism for classifying quantum states of the system into those that describe quantum black holes and those that do not. We find that quantum horizons fluctuate, confirming long-held heuristic expectations. We also give explicit examples of quantum black hole states. The work sets a framework for addressing the puzzles of black hole physics in a fully quantized dynamical setting. 
  The evolution of weak discontinuity is investigated in the flat FRW universe with a single scalar field and with multiple scalar fields. We consider both massless scalar fields and scalar fields with exponential potentials. Then we find that a new type of instability, i.e. kink instability develops in the flat FRW universe with massless scalar fields. The kink instability develops with scalar fields with considerably steep exponential potentials, while less steep exponential potentials do not suffer from kink instability. In particular, assisted inflation with multiple scalar fields does not suffer from kink instability. The stability of general spherically symmetric self-similar solutions is also discussed. 
  We discuss the quantum creation scenario of a Kerr-de Sitter black hole in all dimensions. We show that its relative creation probability is the exponential to the entropy of the black hole, using a topological argument. The action of the regular Euclidean instanton can be calculated in the same way. 
  Hamiltonian dynamics of gravitational field contained in a spacetime region with boundary $S$ being a null-like hypersurface (a wave front) is discussed. Complete Hamiltonian formula for the dynamics (with no surface integrals neglected) is derived. A quasi-local proof of the first law of black holes thermodynamics is obtained as a consequence, in case when $S$ is a non-expanding horizon. The zeroth law and Penrose inequalities are discussed from this point of view. 
  Recent astronomical observations verify the new scenario resulting from new conservation laws and a new relativity principle fixed either by dual properties of light or by new gravitational (G) tests and the Einstein's equivalence principle. This scenario is radically different from the classical one. During a free fall, the relative masses of free bodies, with respect to the observer at rest in a G field, remain constants. The energy released during the stops in different G potentials (GP) comes not from the G field but from the bodies. The relative properties of the bodies at rest with respect to the observer depend on the differences of GP between the bodies and the observer. The increase of GP due to universe expansion expands bodies in identical proportion. The universe age may be infinite. Its entropy is conserved because the new black hole, without singularity, after absorbing radiation, explodes regenerating new gas that transforms dark galaxies into luminous ones. Galaxies evolve, indefinitely in closed cycles with luminous and dark periods. All of their phases are found anywhere in the universe. They solve fundamental dilemmas like dark matter and radiation backgrounds 
  In this paper we critically analyze the so far performed and proposed tests for measuring the general relativistic Lense-Thirring effect in the gravitational field of the Earth with some of the existing accurately tracked artificial satellites. The impact of the 2nd generation GRACE-only EIGEN-GRACE02S Earth gravity model and of the 1st CHAMP+GRACE+terrestrial gravity combined EIGEN-CG01C Earth gravity model is discussed. The role of the proposed LARES is discussed as well. 
  This article uses the conformal Einstein equations and the conformal representation of spatial infinity introduced by Friedrich to analyse the behaviour of the gravitational field near null and spatial infinity for the development of initial data which are, in principle, non-conformally flat and time asymmetric. This article is the continuation of the investigation started in Class. Quantum Grav. 21 (2004) 5457-5492, where only conformally flat initial data sets were considered. For the purposes of this investigation, the conformal metric of the initial hypersurface is assumed to have a very particular type of non-smoothness at infinity in order to allow for the presence of non-Schwarzschildean initial data sets in the class under study. The calculation of asymptotic expansions of the development of these initial data sets reveals --as in the conformally flat case-- the existence of a hierarchy of obstructions to the smoothness of null infinity which are expressible in terms of the initial data. This allows for the possibility of having spacetimes where future and past null infinity have different degrees of smoothness. A conjecture regarding the general structure of the hierarchy of obstructions is presented. 
  We consider the possible modification of the Friedman equation due to operator ordering parameter entering the Wheeler-DeWitt equation. 
  Computing spherical harmonic decompositions is a ubiquitous technique that arises in a wide variety of disciplines and a large number of scientific codes. Because spherical harmonics are defined by integrals over spheres, however, one must perform some sort of interpolation in order to compute them when data is stored on a cubic lattice. Misner (2004, Class. Quant. Grav., 21, S243) presented a novel algorithm for computing the spherical harmonic components of data represented on a cubic grid, which has been found in real applications to be both efficient and robust to the presence of mesh refinement boundaries. At the same time, however, practical applications of the algorithm require knowledge of how the truncation errors of the algorithm depend on the various parameters in the algorithm. Based on analytic arguments and experience using the algorithm in real numerical simulations, I explore these dependencies and provide a rule of thumb for choosing the parameters based on the truncation errors of the underlying data. I also demonstrate that symmetries in the spherical harmonics themselves allow for an even more efficient implementation of the algorithm than was suggested by Misner in his original paper. 
  The intimate link between complex geometry and the problem of the pre-metric formulation of electromagnetism is explored. In particular, the relationship between 3+1 decompositions of R4 and the decompositions of the vector space of bivectors over R4 into real and imaginary subspaces relative to a choice of complex structure is emphasized. The role of the various scalar products on the space of bivectors that are defined in terms of a volume element on R4 and a complex structure on the space of bivectors that makes it C-linear isomorphic to C3 is discussed in the context of formulation of a theory of electromagnetism in which the Lorentzian metric on spacetime follows as a consequence of the existence of electromagnetic waves, not a prior assumption. 
  Laser pulses fired at retroreflectors on the Moon provide very accurate ranges. Analysis yields information on Earth, Moon, and orbit. The highly accurate retroreflector positions have uncertainties less than a meter. Tides on the Moon show strong dissipation, with Q=33 \pm 4 at a month and a weak dependence on period. Lunar rotation depends on interior properties; a fluid core is indicated with radius ~20% that of the Moon. Tests of relativistic gravity verify the equivalence principle to \pm 1.4 x 10^{-13}, limit deviations from Einstein's general relativity, and show no rate for the gravitational constant $\dot{G}/G$ with uncertainty 9 x 10^{-13} 1/yr. 
  We construct simple and useful approximation for the relativistic gas of massive particles. The equation of state is given by an elementary function and admits analytic solution of the Friedmann equation, including more complex cases when the relativistic gas of massive particles is considered together with radiation or with dominating cosmological constant. The model of relativistic gas may be interesting for the description of primordial Universe, especially as a candidate for the role of a Dark Matter. 
  Some new five dimensional minimal scalar-Einstein exact solutions are presented. These new solutions are tested against various criteria used to measure interaction with the fifth dimension. 
  We study multidimensional cosmological models with a higher-dimensional product manifold, that consists of spherical and flat spaces, in the presence of a minimal free scalar field. Dynamical behaviour of the model is analyzed both in Einstein and Brans-Dicke conformal frames. For a number of particular cases, it is shown that external space-time undergoes an accelerated expansion 
  In the context of star capture by a black hole, a new noticeable difference between Brans-Dicke theory and general relativity gravitational radiation is pointed out. This feature stems from the non-stationarity of the black hole state, barring Hawking's theorem. 
  We show that the gravitational collapse of a black-hole terminates in the birth of a white-hole, due to repulsive gravitation (antigravitation); in particular, the infinite energy density singularity does NOT occur. 
  We evaluate the energy-momentum of the gravitational field of a Schwarzschild black hole of mass M in the frame of a moving observer that asymptotically undergoes a Lorentz boost. The analysis is carried out in the framework of the teleparallel equivalent of general relativity (TEGR). We find that the total expression for the energy-momentum of the gravitational field is similar to the usual relativistic expression for the energy-momentum four-vector of a particle of inertial mass M under a Lorentz boost in flat space-time. Moreover we conclude that if the observer accelerates with respect to the black hole he will experience gravitational energy radiation, in similarity to the expected radiation of an accelerated charged particle in electrodynamics. We show that the increase of the mass of the black hole by the usual factor gamma as observed in the moving frame, which is a typical feature of special relativity if the black hole is considered asymptotically as a body of mass M, is due to gravitational radiation. 
  The approximate stress-energy tensor of the conformally invariant massless spin-1/2 field in the Hartle-Hawking state in the Schwarzschild spacetime is constructed. It is shown that by solving the conservation equation in conformal space and utilizing the regularity conditions in a physical metric one obtains the stress-energy tensor that is in a good agreement with the numerical calculations. The back reaction of the quantized field upon the spacetime metric is briefly discussed. 
  We discuss the impact of the present-day uncertainties in the recently released CHAMP and/or GRACE Earth gravity models on the measurement of the Lense-Thirring effect with the nodes of the LAGEOS satellites. Also the role of the secular variations Jdots of the even zonal harmonics is quantitatively assessed via numerical simulations and tests. While the systematic error due to the static part of the geopotential ranges from 4% (EIGEN-GRACE02S) to 9% (GGM02S), the impact of the Jdots amounts to 13% over 11 years. This yields a 19% 1-sigma total error in the test recently performed with EIGEN-GRACE02S. 
  We argue for black holes do not represent a strict consequence of general relativity. 
  The canonical description is based on the prior choice of a spacelike foliation, hence making a reference to a spacetime metric. However, the metric is expected to be a dynamical, fluctuating quantity in quantum gravity. After presenting the developments in the History Projection Operator histories theory in the last seven years--giving special emphasis on the novel temporal structure of the formalism--we show how this problem can be solved in the histories formulation of general relativity. We implement the 3+1 decomposition using metric-dependent foliations which remain spacelike with respect to all possible Lorentzian metrics. This allows us to find an explicit relation of covariant and canonical quantities which preserves the spacetime character of the canonical description. In this new construction we have a coexistence of the spacetime diffeomorphisms group Diff(M) and the Dirac algebra of constraints. 
  Studying Yang-Mills field and gravitational field in class A Bianchi spacetimes, we find that chaotic behavior appears in the late phase (the asymptotic future). In this phase, the Yang-Mills field behaves as that in Minkowski spacetime, in which we can understand it by a potential picture, except for the types VIII and IX. At the same time, in the initial phase (near the initial singularity), we numerically find that the behavior seems to approach the Kasner solution. However, we show that the Kasner circle is unstable and the Kasner solution is not an attractor. From an analysis of stability and numerical simulation, we find a Mixmaster-like behavior in Bianchi I spacetime. Although this result may provide a counterexample to the BKL (Belinskii, Khalatnikov and Lifshitz) conjecture, we show that the BKL conjecture is still valid in Bianchi IX spacetime. We also analyze a multiplicative effect of two types of chaos, that is, chaos with the Yang-Mills field and that in vacuum Bianchi IX spacetime. Two types of chaos seem to coexist in the initial phase. However, the effect due to the Yang-Mills field is much smaller than that of the curvature term. 
  We present a general construction of initial data for Einstein's equations containing an arbitrary number of black holes, each of which is instantaneously in equilibrium. Each black hole is taken to be a marginally trapped surface and plays the role of the inner boundary of the Cauchy surface. The black hole is taken to be instantaneously isolated if its outgoing null rays are shear-free. Starting from the choice of a conformal metric and the freely specifiable part of the extrinsic curvature in the bulk, we give a prescription for choosing the shape of the inner boundaries and the boundary conditions that must be imposed there. We show rigorously that with these choices, the resulting non-linear elliptic system always admits solutions. 
  Combining deeper insight of Einstein's equations with sophisticated numerical techniques promises the ability to construct accurate numerical implementations of these equations. We illustrate this in two examples, the numerical evolution of ``bubble'' and single black hole spacetimes. The former is chosen to demonstrate how accurate numerical solutions can answer open questions and even reveal unexpected phenomena. The latter illustrates some of the difficulties encountered in three-dimensional black hole simulations, and presents some possible remedies. 
  Results are presented from general relativistic numerical computations of primordial black-hole formation during the radiation-dominated era of the universe. Growing-mode perturbations are specified within the linear regime and their subsequent evolution is followed as they become nonlinear. We use a spherically symmetric Lagrangian code and study both super-critical perturbations, which go on to produce black holes, and sub-critical perturbations, for which the overdensity eventually disperses into the background medium. For super-critical perturbations, we confirm the results of previous work concerning scaling-laws but note that the threshold amplitude for a perturbation to lead to black-hole formation is substantially reduced when the initial conditions are taken to represent purely growing modes. For sub-critical cases, where an initial collapse is followed by a subsequent re-expansion, strong compressions and rarefactions are seen for perturbation amplitudes near to the threshold. We have also investigated the effect of including a significant component of vacuum energy and have calculated the resulting changes in the threshold and in the slope of the scaling law. 
  It is shown that Einstein gravity tends to modify the electric and magnetic fields appreciably at distances of the order of the Compton wavelength. At that distance the gravitational field becomes spin dominated rather than mass dominated. The gravitational field couples to the electromagnetic field via the Einstein-Maxwell equations which in the simplest model causes the electrostatic field of charged spinning particles to acquire an oblate structure relative to the spin direction. For electrons and protons, a pure Coulomb field is therefore likely to be incompatible with general relativity at the Compton scale. In the simplest model, the magnetic dipole corresponds to the Dirac g-factor, g=2. Also, it follows from the form of the electric field that the electric dipole moment vanishes, in agreement with current experimental limits for the electron. Quantitatively, the classical Einstein-Maxwell theory predicts the magnetic and electric dipoles of the electron to an accuracy of about one part in 10^{-3} or better. Going to the next multipole order, one finds that the first non-vanishing higher multipole is the electric quadrupole moment which is predicted to be -124 barn for the electron. Any non-zero value of the electric quadrupole moment for the electron or the proton would be a clear sign of curvature due to the implied violation of rotation invariance. There is also a possible spherical modification of the Coulomb force proportional to r^{-4}. However, the size of this effect is well below current experimental limits. The corrections to the hydrogen spectrum are expected to be small but possibly detectable. 
  ''Warp drive'' spacetimes are useful as ''gedanken-experiments'' and as a theoretician's probe of the foundations of general relativity. Applying linearized gravity to the weak-field warp drive, i.e., for non-relativistic warp-bubble velocities, we find that the occurrence of energy condition violations in this class of spacetimes is generic to the form of the geometry under consideration and is not simply a side-effect of the ''superluminal'' properties. Using the linearized construction it is now possible to compare the warp field energy with the mass-energy of the spaceship, and applying the ''volume integral quantifier'', extremely stringent conditions on the warp drive spacetime are found. 
  We incorporate a massless scalar field into a 3-dimensional code for the characteristic evolution of the gravitational field. The extended 3-dimensional code for the Einstein--Klein--Gordon system is calibrated to be second order convergent. It provides an accurate calculation of the gravitational and scalar radiation at infinity. As an application, we simulate the fully nonlinear evolution of an asymmetric scalar pulse of ingoing radiation propagating toward an interior Schwarzschild black hole and compute the backscattered scalar and gravitational outgoing radiation patterns. The amplitudes of the scalar and gravitational outgoing radiation modes exhibit the predicted power law scaling with respect to the amplitude of the initial data. For the scattering of an axisymmetric scalar field, the final ring down matches the complex frequency calculated perturbatively for the $\ell=2$ quasinormal mode. 
  We give a complete classification of all static, spherically symmetric solutions of the SU(2) Einstein-Yang-Mills theory with a positive cosmological constant. Our classification proceeds in two steps. We first extend solutions of the radial field equations to their maximal interval of existence. In a second step we determine the Carter-Penrose diagrams of all 4-dimensional space-times constructible from such radial pieces. Based on numerical studies we sketch a complete phase space picture of all solutions with a regular origin. 
  Certain peculiar features of Einstein-Hilbert (EH) action provide clues towards a holographic approach to gravity which is independent of the detailed microstructure of spacetime. These features of the EH action include: (a) the existence of second derivatives of dynamical variables; (b) a non trivial relation between the surface term and the bulk term; (c) the fact that surface term is non analytic in the coupling constant, when gravity is treated as a spin-2 perturbation around flat spacetime and (d) the form of the variation of the surface term under infinitesimal coordinate transformations. The surface term can be derived directly from very general considerations and using (d) one can obtain Einstein's equations {\it just from the surface term of the action}. Further one can relate the bulk term to the surface term and derive the full EH action based on purely thermodynamic considerations. The features (a), (b) and (c) above emerge in a natural fashion in this approach. It is shown that action $A_{grav}$ splits into two terms $-S+\beta E$ in a natural manner \textit{in any stationary spacetime with horizon}, where $E$ is essentially an integral over ADM energy density and $S$ arises from the integral of the surface gravity over the horizon. This analysis shows that the true degrees of freedom of gravity reside in the surface term of the action, making gravity intrinsically holographic. It also provides a close connection between gravity and gauge theories, and highlights the subtle role of the singular coordinate transformations. 
  We show how the quantum to classical transition of the cosmological fluctuations produced during inflation can be described by means of the influence functional and the master equation. We split the inflaton field into the system-field (long-wavelength modes), and the environment, represented by its own short-wavelength modes. We compute the decoherence times for the system-field modes and compare them with the other time scales of the model. 
  We have studied the scalar field perturbations on six-dimensional ultra-spinning black holes. We have numerically calculated the quasinormal modes of rotating black holes. Our results suggest that such perturbations are stable. 
  Deriving the gravitational field of the high-energy objects is an important theme because the black holes might be produced in particle collisions in the brane world scenario. In this paper, we develop a method for boosting the metric of the stationary objects and taking the lightlike limit in higher-dimensional spacetimes, using the Kerr black hole as one example. We analytically construct the metric of lightlike Kerr black holes and find that this method contains no ambiguity in contrast to the four-dimensional case. We discuss the possible implications of our results in the brane world context. 
  Using the canonical formalism, we study the asymptotic symmetries of the topological 3-dimensional gravity with torsion. In the anti-de Sitter sector, the symmetries are realized by two independent Virasoro algebras with classical central charges. In the simple case of the teleparallel vacuum geometry, the central charges are equal to each other and have the same value as in general relativity, while in the general Riemann-Cartan geometry, they become different. 
  We study the decay of charged scalar and spinor fields around Reissner-Nordstrom black holes in de Sitter spacetime through calculations of quasinormal frequencies of the fields. The influence of the parameters of the black hole (charge, mass), of the decaying fields (charge, spin), and of the spacetime (cosmological constant) on the decay is analyzed. The analytic formula for calculation quasinormal frequencies for a large multipole number (eikonal approximation) is derived both for the spinor and scalar fieldss. 
  In a globally hyperbolic spacetime any pair of chronologically related events admits a connecting geodesic. We present two theorems which prove that, more generally, under weak assumptions, given a charge-to-mass ratio there is always a connecting solution of the Lorentz force equation having that ratio. A geometrical interpretation of the charged-particle action is given which shows that the constructed solutions are maximizing. 
  The main purpose of this work is to obtain the metric of a Charge Tempered Cosmological Model, a slightly modified Standard Cosmological Model by a small excess of charge density, distributed uniformly in accordance with the Cosmological Principle, the global Coulomb interaction incorporated in this metric. The particularity of this model is that the commoving observer referential where the metric belongs is non inertial, which consequence is that clocks at different position can not be synchronized. The new metric is constrained to k goint to 0, with dependence on a charge parameter, and related to a modified Friedmann equation, but it is constrained to a positive deceleration parameter and the hyperbolic solution . Nevertheless, there are corrections to do, valid just for a long range distances. For example, the red shift has, now, dependences on the gravitational potential together the recessional motion. In any way, this model accepts as well the cosmological constant and its physical counterpart, the dark energy. 
  We consider the issue of computability at the most fundamental level of physical reality: the Planck scale. To this aim, we consider the theoretical model of a quantum computer on a non commutative space background, which is a computational model for quantum gravity. In this domain, all computable functions are the laws of physics in their most primordial form, and non computable mathematics finds no room in the physical world. Moreover, we show that a theorem that classically was considered true but non computable, at the Planck scale becomes computable but non decidable. This fact is due to the change of logic for observers in a quantum-computing universe: from standard quantum logic and classical logic, to paraconsistent logic. 
  In a Robertson-Walker space-time a spinning particle model is investigated and we show that in a stationary case, there exists a class of new structures called f-symbols which can generate reducible Killing tensors and supersymmetry algebras. 
  We consider a self-consistent system of Bianchi type-I (BI) gravitational field and a binary mixture of perfect fluid and dark energy. The perfect fluid is taken to be the one obeying the usual equation of state, i.e., $p = \zeta \ve$, with $\zeta \in [0, 1]$ whereas, the dark energy density is considered to be either the quintessence or the Chaplygin gas. Exact solutions to the corresponding Einstein equations are obtained. 
  In many different ways, Deformed Special Relativity (DSR) has been argued to provide an effective limit of quantum gravity in almost-flat regime. Unfortunately DSR is up to now plagued by many conceptual problems (in particular how it describes macroscopic objects) which forbids a definitive physical interpretation and clear predictions. Here we propose a consistent framework to interpret DSR. We extend the principle of relativity: the same way that Special Relativity showed us that the definition of a reference frame requires to specify its speed, we show that DSR implies that we must also take into account its mass. We further advocate a 5-dimensional point of view on DSR physics and the extension of the kinematical symmetry from the Poincare group to the Poincare-de Sitter group (ISO(4,1)). This leads us to introduce the concept of a pentamomentum and to take into account the renormalization of the DSR deformation parameter kappa. This allows the resolution of the "soccer ball problem" (definition of many-particle-states) and provides a physical interpretation of the non-commutativity and non-associativity of the addition the relativistic quadrimomentum. 
  We take further steps in the development of the characteristic approach to enable handling the physical problem of a compact self-gravitating object, such as a neutron star, in close orbit around a black hole. We examine different options for setting the initial data for this problem and, in order to shed light on their physical relevance, we carry out short time evolution of this data. To this end we express the matter part of the characteristic gravity code so that the hydrodynamics are in conservation form. The resulting gravity plus matter relativity code provides a starting point for more refined future efforts at longer term evolution. In the present work we find that, independently of the details of the initial gravitational data, the system quickly flushes out spurious gravitational radiation and relaxes to a quasi-equilibrium state with an approximate helical symmetry corresponding to the circular orbit of the star. 
  Observing the list of compatible second order equations of Absolute Parallelism (AP) found by Einstein and Mayer (for D=4), we have indicated the one-parameter class of equations taking on 3-linear form (when contra-frame density of some weight is in use).   Spherically symmetric solutions to these equations are considered, and we try not to add any \delta-sources (ie, $\delta(r)$-sources of unknown nature) during integrations concerned with high symmetry. Using two different ways to fix radius and time, we have found that only non-static solutions (except for trivial one, of course) are possible. If D=5, such solutions, looking like a single wave (or a file of single waves) moving along the radius, could serve as (a base of) an expanding cosmological model.   It is interesting that the one coordinate choice used leads to the first order system, which is closely similar to gas-dynamic equations for Chaplygin gas, with solutions being free of gradient catastrophe. On the contrary, the other coordinates used show the possibility of solutions with arising ("weak") singularities (it is still not clear whether the measure of such solutions is not zero). 
  Recent developments on the final state of a gravitationally collapsing massive matter cloud are summarized and reviewed here. After a brief background on the problem, we point out how the black hole and naked singularity end states arise naturally in spherical collapse. We see that it is the geometry of trapped surfaces that governs this phenomena. 
  A brief review of main features of the new approach named ``quantum geometrodynamics in extended phase space'' is given and its possible prospects are discussed. Gauge degrees of freedom are treated as a subsystem of the Universe which affects the evolution of the physical subsystem. Three points can be singled out when the gauge subsystem shows itself as a real constituent of the Universe: a chosen gauge condition determines the form of equation for the physical part of wave function, the form of density matrix and the measure in physical subspace. An example is considered when a physically relevant choice of gauge condition leads to almost diagonal density matrix. The analogy between a transition to another reference frame (another basis in physical subspace) and a transition to accelerating reference frame in Rindler space is suggested. 
  What is the influence of cosmology (the expansion law and its acceleration, the cosmological constant...) on the dynamics and optics of a local system like the solar system, a galaxy, a cluster, a supercluster...? The answer requires the solution of Einstein equation with the local source, which tends towards the cosmological model at large distance. There is, in general, no analytic expression for the corresponding metric, but we calculate here an expansion in a small parameter, which allows to answer the question. First, we derive a static expression for the pure cosmological (Friedmann-Lema\^itre) metric, whose validity, although local, extends in a very large neighborhood of the observer. This expression appears as the metric of an osculating de Sitter model. Then we propose an expansion of the cosmological metric with a local source, which is valid in a very large neighborhood of the local system. This allows to calculate exactly the (tiny) influence of cosmology on the dynamics of the solar system: it results that, contrary to some claims, cosmological effects fail to account for the unexplained acceleration of the Pioneer probe by several order of magnitudes. Our expression provide estimations of the cosmological influence in the calculations of rotation or dispersion velocity curves in galaxies, clusters, and any type of cosmic structure, necessary for precise evaluations of dark matter and/or cosmic flows. The same metric can also be used to estimate the influence of cosmology on gravitational optics in the vicinity of such systems. 
  We consider a preferred-frame bimetric theory in which the scalar gravitational field both influences the metric and has direct dynamical effects. A modified version ("v2") is built, by assuming now a locally-isotropic dilation of physically measured distances, as compared with distances evaluated with the Euclidean space metric. The dynamical equations stay unchanged: they are based on a consistent formulation of Newton's second law in a curved space-time. To obtain a local conservation equation for energy with the new metric, the equation for the scalar field is modified: now its l.h.s. is the flat wave operator. Fluid dynamics is formulated and the asymptotic scheme of post-Newtonian approximation is adapted to v2. The latter also explains the gravitational effects on light rays, as did the former version (v1). The violation of the weak equivalence principle found for gravitationally-active bodies at the point-particle limit, which discarded v1, is proved to not exist in v2. Thus that violation was indeed due to the anisotropy of the space metric assumed in v1. 
  OBITUARY The Article and we have been friends for more than half a year. With it, we shared many experiences, both in planetary dynamics and field theory. This research is something I shall always remember with a smile on my face, and a pain in my heart.   Today is a day of sadness and mourning for the loss of our ill-born Article. After more thorough medical examination we came to the conclusion that the precession of planetary orbits cannot be used to bound anything except human fantasy.   Yet it can also be viewed as a day of celebration! Why? Because without it, we would never have been touched by the shared experiences on physics's eventful journey. Whilst it will now forever be absent in Arxive, it will be with us always in spirit. As it might have said, "I have a long journey to take, and must bid the company farewell" (Sir Walter Raleigh). I say three things:   Gone? - Yes!   Forgotten? - Never!   Remembered? - Always! by K.G.Z. 
  We discuss generalizations of the recent theorem by Dafermos (hep-th/0403033) forbidding a certain class of naked singularities in the spherical collapse of a scalar field. Employing techniques similar to the ones Dafermos used, we consider extending the theorem (1) to higher dimensions, (2) by including more general matter represented by a stress-energy tensor satisfying certain assumptions, and (3) by replacing the spherical geometry by a toroidal or higher genus (locally hyperbolic) one. We show that the extension to higher dimensions and a more general topology is straightforward; on the other hand, replacing the scalar field by a more general matter content forces us to shrink the class of naked singularities we are able to exclude. We then show that the most common matter theories (scalar field interacting with a non-abelian gauge field and a perfect fluid satisfying certain conditions) obey the assumptions of our weaker theorem, and we end by commenting on the applicability of our results to the five-dimensional AdS scenarii considered recently in the literature. 
  Newtonian limit of Extended Theories of Gravity (in particular, higher--order and scalar--tensor theories) is theoretically discussed taking into account recent observational and experimental results. 
  We study the quantum cosmology of an empty (4+1)-dimensional Kaluza-Klein cosmology with a negative cosmological constant and a FRW type metric with two scale factors, one for 4-D universe and the other for one compact extra dimension. By assuming the noncommutativity in the corresponding mini-superspace we suggest a solution for the Hierarchy problem, at the level of Wheeler-DeWitt equation. 
  Causal radial geodesics with a positive interval in the Schwarzschild metric include a subset of trajectories completely confined under a horizon, which compose a thermal statistical ensemble with the Hawking-Gibbons temperature. The Bekenstein--Hawking entropy is given by an action at corresponding geodesics of particles with a summed mass equal to that of black hole in the limit of large mass. 
  The probability representation for quantum states of the universe in which the states are described by a fair probability distribution instead of wave function (or density matrix) is developed to consider cosmological dynamics. The evolution of the universe state is described by standard positive transition probability (tomographic transition probability) instead of the complex transition probability amplitude (Feynman path integral) of the standard approach. The latter one is expressed in terms of the tomographic transition probability. Examples of minisuperspaces in the framework of the suggested approach are presented. Possibility of observational applications of the universe tomographs are discussed. 
  In this paper, we study the equation of state admissible for a flat FRW models filled with a bulk viscous fluid by using the Lie group method. It is founded that the model admits scaling symmetries iff the bulk viscous parameter $\gamma =1/2$. In this case, it is found that the main quantities follow a power law solution and in particular the bulk viscous pressure $\Pi $ has the same order of magnitude as the energy density $\rho ,$ in such a way that it is possible to formulate the equation of state $\Pi =\varkappa \rho ,$ where $\varkappa \in \mathbb{R}^{-}$ (i.e. is a negative numerical constant)$.$ If we assume such relationship we find again that the model is scale invariant iff $\gamma =1/2.$ We conclude that the model accepts a scaling symmetry iff $\gamma =1/2$ and that for this value of the viscous parameter, $\Pi =\varkappa \rho ,$ but the hypothesis $\Pi =\varkappa \rho $ does not imply $\gamma =1/2,$ and that the model is scale invariant. 
  Achieving the low frequency LISA sensitivity requires that the test masses acting as the interferometer end mirrors are free-falling with an unprecedented small degree of deviation. Magnetic disturbances, originating in the interaction of the test mass with the environmental magnetic field, can significantly deteriorate the LISA performance and can be parameterized through the test mass remnant dipole moment $\vec{m}_r$ and the magnetic susceptibility $\chi$. While the LISA test flight precursor LTP will investigate these effects during the preliminary phases of the mission, the very stringent requirements on the test mass magnetic cleanliness make ground-based characterization of its magnetic proprieties paramount. We propose a torsion pendulum technique to accurately measure on ground the magnetic proprieties of the LISA/LTP test masses. 
  Given a planar graph derived from a spherical, euclidean or hyperbolic tessellation, one can define a discrete curvature by combinatorial properties, which after embedding the graph in a compact 2d-manifold, becomes the Gaussian curvature. 
  Photon-graviton mixing in an electromagnetic field is a process of potential interest for cosmology and astrophysics. At the tree level it has been studied by many authors. We consider the one-loop contribution to this amplitude involving a charged spin 0 or spin 1/2 particle in the loop and an arbitrary constant field. In the first part of this article, the worldline formalism is used to obtain a compact two-parameter integral representation for this amplitude, valid for arbitrary photon energies and background field strengths. The calculation is manifestly covariant througout. 
  A general concept of potential field is introduced. The potential field that one puts in correspondence with dark matter, has fundamental geometrical interpretation (parallel transport) and has intrinsically inherent in local symmetry. The equations of dark matter field are derived that are invariant with respect to the local transformations. It is shown how to reduce these equations to the Maxwell equations. Thus, the dark matter field may be considered as generalized electromagnetic field and a simple solution is given of the old problem to connect electromagnetic field with geometrical properties of the physical manifold itself. It is shown that gauge fixing renders generalized electromagnetic field effectively massive while the Maxwell electromagnetic field remains massless. To learn more about interactions between matter and dark matter on the microscopical level (and to recognize the fundamental role of internal symmetry) the general covariant Dirac equation is derived in the Minkowski space--time which describe the interactions of spinor field with dark matter field. 
  The entropy of charged black hole is calculated by using the partition function evaluated at radial geodesics confined under horizons. We establish two quantum phase states inside the black hole and a transition between them. 
  Spherical reduction of generic four-dimensional theories is revisited. Three different notions of "spherical symmetry" are defined. The following sectors are investigated: Einstein-Cartan theory, spinors, (non-)abelian gauge fields and scalar fields. In each sector a different formalism seems to be most convenient: the Cartan formulation of gravity works best in the purely gravitational sector, the Einstein formulation is convenient for the Yang-Mills sector and for reducing scalar fields, and the Newman-Penrose formalism seems to be the most transparent one in the fermionic sector. Combining them the spherically reduced Standard Model of particle physics together with the usually omitted gravity part can be presented as a two-dimensional (dilaton gravity) theory. 
  In this paper we investigate a class of basic super-energy tensors, namely those constructed from Killing-Yano tensors, and give a generalization of super-energy tensors for cases when we start not with a single tensor, but with a pair of tensors. 
  Formulation of the Penrose inequality becomes ambiguous when the past and future apparent horizons do cross. We test numerically several natural possibilities of stating the inequality in punctured and boosted single- and double- black holes, in a Dain-Friedrich class of initial data and in conformally flat spheroidal data.The Penrose inequality holds true in vacuum configurations for the outermost element amongst the set of disjoint future and past apparent horizons (as expected)and (unexpectedly) for each of the outermost past and future apparent horizons, whenever these two bifurcate from an outermost minimal surface, regardless of whether they intersect or remain disjoint. In systems with matter the conjecture breaks down only if matter does not obey the dominant energy condition. 
  We analyze the excision strategy for simulating black holes. The problem is modeled by the propagation of quasi-linear waves in a 1-dimensional spatial region with timelike outer boundary, spacelike inner boundary and a horizon in between. Proofs of well-posed evolution and boundary algorithms for a second differential order treatment of the system are given for the separate pieces underlying the finite difference problem. These are implemented in a numerical code which gives accurate long term simulations of the quasi-linear excision problem. Excitation of long wavelength exponential modes, which are latent in the problem, are suppressed using conservation laws for the discretized system. The techniques are designed to apply directly to recent codes for the Einstein equations based upon the harmonic formulation. 
  The answer to the question, what physical meaning should be attributed to the so-called boost-rotation symmetric exact solutions to the field equations of general relativity, is provided within the general interpretation scheme for the ``theories of relativity'', based on group theoretical arguments, and set forth by Erich Kretschmann already in the year 1917. 
  A torsion pendulum allows ground-based investigation of the purity of free-fall for the LISA test masses inside their capacitive position sensor. This paper presents recent improvements in our torsion pendulum facility that have both increased the pendulum sensitivity and allowed detailed characterization of several important sources of acceleration noise for the LISA test masses. We discuss here an improved upper limit on random force noise originating in the sensor. Additionally, we present new measurement techniques and preliminary results for characterizing the forces caused by the sensor's residual electrostatic fields, dielectric losses, residual spring-like coupling, and temperature gradients. 
  A known realization of the Lorentz group Racah coefficients is given by an integral of a product of 6 ``propagators'' over 4 copies of the hyperbolic space. These are ``bulk-to-bulk'' propagators in that they are functions of two points in the hyperbolic space. It is known that the bulk-to-bulk propagator can be constructed out of two bulk-to-boundary ones. We point out that there is another way to obtain the same object. Namely, one can use two bulk-to-boundary and one boundary-to-boundary propagator. Starting from this construction and carrying out the bulk integrals we obtain a realization of the Racah coefficients that is ``holographic'' in the sense that it only involves boundary objects. This holographic realization admits a geometric interpretation in terms of an ``extended'' tetrahedron. 
  The structure of the general, inhomogeneous solution of (bosonic) Einstein-matter systems in the vicinity of a cosmological singularity is considered. We review the proof (based on ideas of Belinskii-Khalatnikov-Lifshitz and technically simplified by the use of the Arnowitt-Deser-Misner Hamiltonian formalism) that the asymptotic behaviour, as one approaches the singularity, of the general solution is describable, at each (generic) spatial point, as a billiard motion in an auxiliary Lorentzian space. For certain Einstein-matter systems, notably for pure Einstein gravity in any spacetime dimension D and for the particular Einstein-matter systems arising in String theory, the billiard tables describing asymptotic cosmological behaviour are found to be identical to the Weyl chambers of some Lorentzian Kac-Moody algebras. In the case of the bosonic sector of supergravity in 11 dimensional spacetime the underlying Lorentzian algebra is that of the hyperbolic Kac-Moody group E_{10}, and there exists some evidence of a correspondence between the general solution of the Einstein-three-form system and a null geodesic in the infinite dimensional coset space E_{10} / K (E_{10}), where K (E_{10}) is the maximal compact subgroup of E_{10}. 
  We consider an Anti-de Sitter universe filled by quantum conformal matter with the contribution from the usual tachyon and a perfect fluid. The model represents the combination of a trace-anomaly annihilated and a tachyon driven Anti-de Sitter universe. The influence exerted by the quantum effects and by the tachyon on the AdS space is studied. The radius corresponding to this universe is calculated and the effect of the tachyon potential is discussed, in particular, concerning to the possibility to get an accelerated scale factor for the proposed model (implying an accelerated expansion of the AdS type of universe). Fulfillment of the cosmological energy conditions in the model is also investigated 
  This paper explores the idea that within the framework of three-dimensional quantum gravity one can extend the notion of Feynman diagram to include the coupling of the particles in the diagram with quantum gravity. The paper concentrates on the non-trivial part of the gravitational response, which is to the large momenta propagating around a closed loop. By taking a limiting case one can give a simple geometric description of this gravitational response. This is calculated in detail for the example of a closed Feynman loop in the form of a trefoil knot. The results show that when the magnitude of the momentum passes a certain threshold value, non-trivial gravitational configurations of the knot play an important role.   The calculations also provide some new information about a limit of the coloured Jones polynomial which may be of independent mathematical interest. 
  Recently a multidimensional generalization of Isolated Horizon framework has been proposed by Lewandowski and Pawlowski (gr-qc/0410146). Therein the geometric description was easily generalized to higher dimensions and the structure of the constraints induced by the Einstein equations was analyzed. In particular, the geometric version of the zeroth law of the black hole thermodynamics was proved. In this work we show how the IH mechanics can be formulated in a dimension--independent fashion and derive the first law of BH thermodynamics for arbitrary dimensional IH. We also propose a definition of energy for non--rotating horizons. 
  We use the overlapping grids method to construct a fourth order accurate discretization of a first order reduction of the Klein-Gordon scalar field equation on a boosted spinning black hole blackground in axisymmetry. This method allows us to use a spherical outer boundary and excise the singularity from the domain with a spheroidal inner boundary which is moving with respect to the main grid. We discuss the use of higher order accurate energy conserving schemes to handle the axis of symmetry and compare it with a simpler technique based on regularity conditions. We also compare the single grid long term stability property of this formulation of the wave equation with that of a different first order reduction. 
  We specify an angular motion on geodesics to reduce the problem to the case of radial motion elaborated in previous chapters. An appropriate value of entropy for a charged and rotating black hole is obtained by calculating the partition function on thermal geodesics confined under horizons. The quantum aggregation is classified in a similar way to the Reissner--Nordstrom black hole. 
  We apply the Keski-Vakkuri, Kraus and Wilczek (KKW) generalized analysis to a magnetic stringy black hole solution to compute its temperature and entropy. The solution that we choose in the Einstein-dilaton-Maxwell theory is the dual solution known as the magnetic black hole solution. Our results show that the expressions of the temperature and entropy of this non-Schwarzschild-type black hole are not the Hawking temperature and the Bekenstein-Hawking entropy, respectively. In addition, the extremal magnetic stringy black hole is not frozen because it has a constant non-zero temperature. 
  We examine and compare the behaviour of the scalar field slice energy in different classes of theories of gravity, in particular higher-order and scalar-tensor theories. We find a universal formula for the energy and compare the resulting conservation laws with those known in general relativity. This leads to a comparison between the inflaton, the dilaton and other forms of scalar fields present in these generalized theories. It also shows that all such conformally-related, generalized theories of gravitation allow for the energy on a slice to be invariably defined and its fundamental properties be insensitive to conformal transformations. 
  Local conformal transformations are known as a useful tool in various applications of the gravitational theory, especially in cosmology. We describe some new aspects of these transformations, in particular using them for derivation of Einstein equations for the cosmological and Schwarzschild metrics. Furthermore, the conformal transformation is applied for the dimensional reduction of the Gauss-Bonnet topological invariant in $d=4$ to the spaces of lower dimensions. 
  We investigate the cosmological perturbations of the brane-induced (Dvali-Gabadadze-Porrati) model which exhibits a van Dam-Veltman-Zakharov (vDVZ) discontinuity when linearized over a Minkowski background. We show that the linear brane scalar cosmological perturbations over an arbitrary Friedmann-Lemaitre-Robertson-Walker (FLRW) space-time have a well defined limit when the radius of transition between 4D and 5D gravity is sent to infinity with respect to the background Hubble radius. This radius of transition plays for the brane-induced gravity model a role equivalent to the Compton wavelength of the graviton in a Pauli-Fierz theory, as far as the vDVZ discontinuity is concerned. This well defined limit is shown to obey the linearized 4D Einstein's equations whenever the Hubble factor is non vanishing. This shows the disappearance of the vDVZ discontinuity for general FLRW background, and extends the previously know result for maximally-symmetric space-times of non vanishing curvature. Our reasoning is valid for matter with simple equation of state such as a scalar field, or a perfect fluid with adiabatic perturbations, and involves to distinguish between space-times with a vanishing scalar curvature and space-times with a non vanishing one. We also discuss the validity of the linear perturbation theory, in particular for those FLRW space-times where the Ricci scalar is vanishing only on a set of zero measure. In those cases, we argue that the linear perturbation theory breaks down when the Ricci scalar vanishes (and the radius of transition is sent to infinity), in a way similar to what has been found to occur around sources on a Minkowski background. 
  Outer boundary conditions for strongly and symmetric hyperbolic formulations of 3D Einstein's field equations with a live gauge condition are discussed. The boundary conditions have the property that they ensure constraint propagation and control in a sense made precise in this article the physical degrees of freedom at the boundary. We use Fourier-Laplace transformation techniques to find necessary conditions for the well posedness of the resulting initial-boundary value problem and integrate the resulting three-dimensional nonlinear equations using a finite-differencing code. We obtain a set of constraint-preserving boundary conditions which pass a robust numerical stability test. We explicitly compare these new boundary conditions to standard, maximally dissipative ones through Brill wave evolutions. Our numerical results explicitly show that in the latter case the constraint variables, describing the violation of the constraints, do not converge to zero when resolution is increased while for the new boundary conditions, the constraint variables do decrease as resolution is increased. As an application, we inject pulses of ``gravitational radiation'' through the boundaries of an initially flat spacetime domain, with enough amplitude to generate strong fields and induce large curvature scalars, showing that our boundary conditions are robust enough to handle nonlinear dynamics.   We expect our boundary conditions to be useful for improving the accuracy and stability of current binary black hole and binary neutron star simulations, for a successful implementation of characteristic or perturbative matching techniques, and other applications. We also discuss limitations of our approach and possible future directions. 
  New boundary conditions are constructed and tested numerically for a general first-order form of the Einstein evolution system. These conditions prevent constraint violations from entering the computational domain through timelike boundaries, allow the simulation of isolated systems by preventing physical gravitational waves from entering the computational domain, and are designed to be compatible with the fixed-gauge evolutions used here. These new boundary conditions are shown to be effective in limiting the growth of constraints in 3D non-linear numerical evolutions of dynamical black-hole spacetimes. 
  Using thermal quantization of geodesics confined under horizons and reasonable conjecture on massless modes, we evaluate quantum spectrum of Kerr black hole masses compatible with superstring symmetries. 
  In this paper we derive a class of rotating embedded black holes. Then we study Hawking's radiation effects on these embedded black holes. The surface gravity, entropy and angular velocity are given for each of these black holes. 
  We present two search algorithms that implement logarithmic tiling of the time-frequency plane in order to efficiently detect astrophysically unmodeled bursts of gravitational radiation. The first is a straightforward application of the dyadic wavelet transform. The second is a modification of the windowed Fourier transform which tiles the time-frequency plane for a specific Q. In addition, we also demonstrate adaptive whitening by linear prediction, which greatly simplifies our statistical analysis. This is a methodology paper that aims to describe the techniques for identifying significant events as well as the necessary pre-processing that is required in order to improve their performance. For this reason we use simulated LIGO noise in order to illustrate the methods and to present their preliminary performance. 
  We use Moeller's energy-momentum complex in order to explicitly evaluate the energy and momentum density distributions associated with the three-dimensional magnetic solution to the Einstein-Maxwell equations. The magnetic spacetime under consideration is a one-parametric solution describing the distribution of a radial magnetic field in a three-dimensional AdS background, and representing the superposition of the magnetic field with a 2+1 Einstein static gravitational field. 
  The maximum entropy of a quantized surface is demonstrated to be proportional to the surface area in the classical limit. The general structure of the horizon spectrum and the value of the Barbero-Immirzi parameter are found. The discrete spectrum of thermal radiation of a black hole fits naturally the Wien profile. The natural widths of the lines are very small as compared to the distances between them. The total intensity of the thermal radiation is calculated. 
  It is shown that the universal property of gravitational field to slow down the rate of time leads in the field theory to a fundamental property -- generation of effective forces of repulsion. 
  We describe the plans for a joint search for unmodelled gravitational wave bursts being carried out by the LIGO and TAMA collaborations using data collected during February-April 2003. We take a conservative approach to detection, requiring candidate gravitational wave bursts to be seen in coincidence by all four interferometers. We focus on some of the complications of performing this coincidence analysis, in particular the effects of the different alignments and noise spectra of the interferometers. 
  In this work, we discuss observable characteristics of the radiation emitted from a surface of a collapsing object. We study a simplified model in which a radiation of massless particles has a sharp in time profile and it happens at the surface at the same moment of comoving time. Since the radiating surface has finite size the observed radiation will occur during some finite time. Its redshift and bending angle are affected by the strong gravitational field. We obtain a simple expression for the observed flux of the radiation as a function of time. To find an explicit expression for the flux we develop an analytical approximation for the bending angle and time delay for null rays emitted by a collapsing surface. In the case of the bending angle this approximation is an improved version of the earlier proposed Beloborodov-Leahy-approximation. For rays emitted at $R > 2R_g$ the accuracy of the proposed improved approximations for the bending angle and time delay is of order (or less) than 2-3%. By using this approximation we obtain an approximate analytical expression for the observed flux and study its properties. 
  In dilaton gravity theories, we consider a string-like topological defect formed during U(1) gauge symmetry-breaking phase transition in the early Universe, and far from the cosmic string we have vacuum solutions of the generalized Einstein equation. We discuss how they can be related to the flatness of galactic rotation curves. 
  We present cosmological perturbation theory based on generalized gravity theories including string theory correction terms and a tachyonic complication. The classical evolution as well as the quantum generation processes in these variety of gravity theories are presented in unified forms. These apply both to the scalar- and tensor-type perturbations. Analyses are made based on the curvature variable in two different gauge conditions often used in the literature in Einstein's gravity; these are the curvature variables in the comoving (or uniform-field) gauge and the zero-shear gauge. Applications to generalized slow-roll inflations and its consequent power spectra are derived in unified forms which include wide range of inflationary scenarios based on Einstein's gravity and others. 
  It is well known that couplings occur among the scalar-, vector-, and tensor-type perturbations of Friedmann world model in the second perturbational order. Here, we prove that, except for the gravitational wave contribution, the relativistic zero-pressure irrotational fluid perturbed to second order in a flat Friedmann background coincides exactly with the Newtonian result. Since we include the cosmological constant, our results are relevant to currently favoured cosmology. As we prove that the Newtonian hydrodynamic equations are valid in all cosmological scales to the second order, our result has an important practical implication that one can now use the large-scale Newtonian numerical simulation more reliably even as the simulation scale approaches and even goes beyond the horizon. That is, our discovery shows that, in the zero-pressure case, except for the gravitational wave contribution, there are no relativistic correction terms even near and beyond the horizon to the second-order perturbation. 
  The dynamic world model and its linear perturbations were first studied in Einstein's gravity. In the system without pressure the relativistic equations coincide exactly with the later known ones in Newton's gravity. Here we prove that, except for the gravitational wave contribution, even to the second-order perturbations, equations for the relativistic irrotational zero-pressure fluid in a flat Friedmann background coincide exactly with the previously known Newtonian equations. Thus, to the second order, we correctly identify the relativistic density and velocity perturbation variables, and we expand the range of applicability of the Newtonian medium without pressure to all cosmological scales including the super-horizon scale. In the relativistic analyses, however, we do not have a relativistic variable which corresponds to the Newtonian potential to the second order. Mixed usage of different gauge conditions is useful to make such proofs and to examine the result with perspective. We also present the gravitational wave equation to the second order. Since our correspondence includes the cosmological constant, our results are relevant to currently favoured cosmology. Our result has an important practical implication that one can use the large-scale Newtonian numerical simulation more reliably even as the simulation scale approaches near horizon. 
  We consider a general relativistic zero-pressure irrotational cosmological medium perturbed to the third order. We assume a flat Friedmann background but include the cosmological constant. We ignore the rotational perturbation which decays in expanding phase. In our previous studies we discovered that, to the second-order perturbation, except for the gravitational wave contributions, the relativistic equations coincide exactly with the previously known Newtonian ones. Since the Newtonian second-order equations are fully nonlinear, any nonvanishing third and higher order terms in the relativistic analyses are supposed to be pure relativistic corrections. In this work we derive such correction terms appearing in the third order. Continuing our success in the second-order perturbations we take the comoving gauge. We discover that the third-order correction terms are of $\phi_v$-order higher than the second-order terms where $\phi_v$ is a gauge-invariant combination related to the three-space curvature perturbation in the comoving gauge; compared with the Newtonian potential we have $\delta \Phi \sim {3 \over 5} \phi_v$ to the linear order. Therefore, the pure general relativistic effects are of $varphi_v$-order higher than the Newtonian ones. The corrections terms are independent of the horizon scale and depend only on the linear order gravitational potential perturbation strength. From the temperature anisotropy of cosmic microwave background we have ${\delta T \over T} \sim {1 \over 3} \delta \Phi \sim {1 \over 5} \phi_v \sim 10^{-5}$. Therefore, our present result reinforces our previous important practical implication that near current era one can use the large-scale Newtonian numerical simulation more reliably even as the simulation scale approaches near the horizon. 
  PhD thesis and author's Summary (in Russian). Absolute parallelism has many remarkable features - large symmetry group, irreducibility (with respect to this group), and variety of differential covariants.  In large list of compatible second order equations of AP, there is the unique variant -- unique or optimal equation (non-Lagrangian) -- whose solutions of general position seem to be free of arising singularities (only in D=5).  Being non-lagrangian, the unique equation leads to energy-momentum tensor where key role belongs to second order differential covariant f_{\mu\nu} (looks like EM-field, but no gradient symmetry). We note that a wave-packet of energy transferring f-component (there are solutions with f=0) should move along usual riemannian geodesics - as in GR; however, spin or polarization evolution should depend also on rank three skew-symmetric tensor S, which is certainly absent in GR. (Neglecting extra dimension and using unique equation, one may introduce pseudoscalar phi : \phi_{,\mu}=h\epsilon_{\mu\nu\a\b}S^{\nu\a\b}. The presence of dipole-like \phi-field near rotating Earth could be of interest in view of forthcoming results of GP-B mission - relating to some unknown forces.)  In absence of singularities (degenerated h^a_\mu-matrices are inaccessible and should be eliminated from field set), AP becomes topological theory. Besides topological charge, we introduce the concept of topological quasi-charge (QC) for field configurations having some symmetry. QC-groups and their morphisms induced by "embedding of symmetries" are summarized (for D=5) in the table.  Further comments on quasi-particle phenomenology (configurations carrying QC) on expanding cosmological O_4-background (superposition principle, path integral -- due to huge, undeveloped extra dimension) are added to Summary. 
  Utilizing various gauges of the radial coordinate we give a description of static spherically symmetric space-times with point singularity at the center and vacuum outside the singularity. We show that in general relativity (GR) there exist a two-parameters family of such solutions to the Einstein equations which are physically distinguishable but only some of them describe the gravitational field of a single massive point particle with nonzero bare mass $M_0$. In particular, the widespread Hilbert's form of Schwarzschild solution, which depends only on the Keplerian mass $M<M_0$, does not solve the Einstein equations with a massive point particle's stress-energy tensor as a source. Novel normal coordinates for the field and a new physical class of gauges are proposed, in this way achieving a correct description of a point mass source in GR. We also introduce a gravitational mass defect of a point particle and determine the dependence of the solutions on this mass defect. The result can be described as a change of the Newton potential $\phi_{{}_N}=-G_{{}_N}M/r$ to a modified one: $\phi_{{}_G}=-G_{{}_N}M/ (r+G_{{}_N} M/c^2\ln{{M_0}\over M})$ and a corresponding modification of the four-interval. In addition we give invariant characteristics of the physically and geometrically different classes of spherically symmetric static space-times created by one point mass. These space-times are analytic manifolds with a definite singularity at the place of the matter particle. 
  We derive a class of non-stationary embedded rotating black holes and study the Hawking's radiation effects on these embedded black holes. The surface gravity, entropy and angular velocity, which are three important properties of black holes, are presented for each of these embedded black holes. 
  The vacuum expectation value of the stress-energy tensor is calculated for spin $1\over 2$ massive fields in several multiply connected flat spacetimes. We examine the physical effects of topology on manifolds such as $R^3 \times S^1$, $R^2\times T^2$, $R^1 \times T^3$, the Mobius strip and the Klein bottle. We find that the spinor vacuum stress tensor has the opposite sign to, and twice the magnitude of, the scalar tensor in orientable manifolds. Extending the above considerations to the case of Misner spacetime, we calculate the vacuum expectation value of spinor stress-energy tensor in this space and discuss its implications for the chronology protection conjecture. 
  We study the canonical structure of the topological 3D gravity with torsion, assuming the anti-de Sitter asymptotic conditions. It is shown that the Poisson bracket algebra of the canonical generators has the form of two independent Virasoro algebras with classical central charges. In contrast to the case of general relativity with a cosmological constant, the values of the central charges are different from each other. 
  We extend the theory of Kimberly and Magueijo for the spacetime variation of the electroweak couplings in the unified Glashow-Salam-Weinberg model of the electroweak interaction to include quantum corrections. We derive the effective quantum-corrected dilaton evolution equations in the presence of a background cosmological matter density that is composed of weakly interacting and non-weakly-interacting non-relativistic dark-matter components. 
  After a brief review of the first phase of development of Quantum-Gravity Phenomenology, I argue that this research line is now ready to enter a more advanced phase: while at first it was legitimate to resort to heuristic order-of-magnitude estimates, which were sufficient to establish that sensitivity to Planck-scale effects can be achieved, we should now rely on detailed analyses of some reference test theories. I illustrate this point in the specific example of studies of Planck-scale modifications of the energy/momentum dispersion relation, for which I consider two test theories. Both the photon-stability analyses and the Crab-nebula synchrotron-radiation analyses, which had raised high hopes of ``beyond-Plankian'' experimental bounds, turn out to be rather ineffective in constraining the two test theories. Examples of analyses which can provide constraints of rather wide applicability are the so-called ``time-of-flight analyses'', in the context of observations of gamma-ray bursts, and the analyses of the cosmic-ray spectrum near the GZK scale. 
  The same but different: That might describe two metrics. On the surface CLASSI may show two metrics are locally equivalent, but buried beneath one may be a wealth of further structure. This was beautifully described in a paper by M.A.H. MacCallum in 1998. Here I will illustrate the effect with two flat metrics -- one describing ordinary Minkowski spacetime and the other describing a three-parameter family of Gal'tsov-Letelier-Tod spacetimes. I will dig out the beautiful hidden classical singularity structure of the latter (a structure first noticed by Tod in 1994) and then show how quantum considerations can illuminate the riches. I will then discuss how quantum structure can help us understand classical singularities and metric parameters in a variety of exact solutions mined from the Exact Solutions book. 
  Fock representations are constructed for a free scalar field in the closed and quasi-Euclidean isotropic cosmological models. Invariance of their cyclic vector (vacuum) under isometries and the correspondence principle single out a class of unitarily equivalent representations. 
  We consider black-string-type solutions in five-dimensional Einstein-Gauss-Bonnet gravity. Numerically constructed solutions under static, axially symmetric and translationally invariant metric ansatz are presented. The solutions are specified by two asymptotic charges: mass of a black string and a scalar charge associated with the radion part of the metric. Regular black string solutions are found if and only if the two charges satisfy a fine-tuned relation, and otherwise the spacetime develops a singular event horizon or a naked singularity. We can also generate bubble solutions from the black strings by using a double Wick rotation. 
  Recent review article by S. Samuel "On the speed of gravity and the Jupiter/Quasar measurement" published in the International Journal of Modern Physics D13, 1753 (2004) provides the reader with a misleading "theory" of the relativistic time delay in general theory of relativity. Furthermore, it misquotes original publications by Kopeikin and Fomalont & Kopeikin related to the measurement of the speed of gravity by VLBI. We summarize the general relativistic principles of the Lorentz-invariant theory of propagation of light in time-dependent gravitational field, derive Lorentz-invariant expression for the relativistic time delay, and finally explain why Samuel's "theory" is conceptually incorrect and confuses the speed of gravity with the speed of light. 
  Charged axially symmetric solution of the coupled gravitational and electromagnetic fields in the tetrad theory of gravitation is derived. The metric associated with this solution is an axially symmetric metric which is characterized by three parameters ``$ $the gravitational mass $M$, the charge parameter $Q$ and the rotation parameter $a$". The parallel vector fields and the electromagnetic vector potential are axially symmetric. We calculate the total exterior energy. The energy-momentum complex given by M{\o}ller in the framework of the Weitzenb$\ddot{o}$ck geometry ``$ ${\it characterized by vanishing the curvature tensor constructed from the connection of this geometry}" has been used. This energy-momentum complex is considered as a better definition for calculation of energy and momentum than those of general relativity theory. The energy contained in a sphere is found to be consistent with pervious results which is shared by its interior and exterior. Switching off the charge parameter, one finds that no energy is shared by the exterior of the charged axially symmetric solution. The components of the momentum density are also calculated and used to evaluate the angular momentum distribution. We found no angular momentum contributes to the exterior of the charged axially symmetric solution if zero charge parameter is used. 
  Necessary conditions for various algebraic types of the Weyl tensor are determined. These conditions are then used to find Weyl aligned null directions for the black ring solution. It is shown that the black ring solution is algebraically special, of type I_i, while locally on the horizon the type is II. One exceptional subclass - the Myers-Perry solution - is of type D. 
  A point particle of mass $\mu$ moving on a geodesic creates a perturbation $h_{ab}$, of the spacetime metric $g_{ab}$, that diverges at the particle. Simple expressions are given for the singular $\mu/r$ part of $h_{ab}$ and its distortion caused by the spacetime. This singular part $h^\SS_{ab}$ is described in different coordinate systems and in different gauges. Subtracting $h^\SS_{ab}$ from $h_{ab}$ leaves a regular remainder $h^\R_{ab}$. The self-force on the particle from its own gravitational field adjusts the world line at $\Or(\mu)$ to be a geodesic of $g_{ab}+h^\R_{ab}$; this adjustment includes all of the effects of radiation reaction. For the case that the particle is a small non-rotating black hole, we give a uniformly valid approximation to a solution of the Einstein equations, with a remainder of $\Or(\mu^2)$ as $\mu\to0$.   An example presents the actual steps involved in a self-force calculation. Gauge freedom introduces ambiguity in perturbation analysis. However, physically interesting problems avoid this ambiguity. 
  We investigate the canonical quantization of the electromagnetic field on the Kerr background. We give new expressions for the expectation value of the electromagnetic stress-energy tensor in various vacua states and give a physical interpretation of the separate terms appearing in them. We numerically calculate the luminosity in these states. We also study the form of the renormalized stress-energy tensor close to the horizon when the electromagnetic field is in the past Boulware state. 
  The trajectories of test particles moving in the gravitational field of a non-spherically symmetric mass distribution become affected by the presence of multipole moments. In the case of hyperbolic trajectories, the quadrupole moment of an oblate mass induces a displacement of the trajectory towards the mass source, an effect that can be interpreted as an additional acceleration directed towards the source. Although this additional acceleration is not constant, we perform a general relativistic analysis in order to evaluate the possibility of explaining Pioneer's anomalous acceleration by means of the observed Solar quadrupole moment, within the range of accuracy of the observed anomalous acceleration. We conclude that the Solar quadrupole moment generates an acceleration which is of the same order of magnitude of Pioneer's constant acceleration only at distances of a few astronomical units. 
  Experiments aimed at searching for gravitational waves from astrophysical sources have been under development for the last 40 years, but only now are sensitivities reaching the level where there is a real possibility of detections being made within the next five years. In this article a history of detector development will be followed by a description of current detectors such as LIGO, VIRGO, GEO 600, TAMA 300, Nautilus and Auriga. Preliminary results from these detectors will be discussed and related to predicted detection rates for some types of sources. Experimental challenges for detector design are introduced and discussed in the context of detector developments for the future. 
  We implement a code to find the gravitational news at future null infinity by using data from a Cauchy code as boundary data for a characteristic code. This technique of Cauchy-characteristic Extraction (CCE) allows for the unambiguous extraction of gravitational waves from numerical simulations. 
  We present the complete solution of the first order metric and density perturbation equations in a spatially flat (K=0), Friedmann-Robertson-Walker (FRW) universe filled with pressureless ideal fluid, in the presence of cosmological constant. We use covariant linear perturbation formalism and the comoving gauge condition to obtain the field and conservation equations. The solution contains all modes of the perturbations, i.e. scalar, vector and tensor modes, and we show that our results are in agreement with the Sachs & Wolfe metric perturbation formalism. 
  It is confirmed rigorously that the Killing-Cauchy horizons, which sometimes occur in space-times representing the collision and subsequent interaction of plane gravitational waves in a Minkowski background, are unstable with respect to bounded perturbations of the initial waves, at least for the case in which the initial waves have constant aligned polarizations. 
  We derive Keplerian-type parametrization for the solution of post-Newtonian (PN) accurate conservative dynamics of spinning compact binaries moving in eccentric orbits. The PN accurate dynamics that we consider consists of the third post-Newtonian accurate conservative orbital dynamics influenced by the leading order spin effects, namely the leading order spin-orbit interactions. The orbital elements of the representation are explicitly given in terms of the conserved orbital energy, angular momentum and a quantity that characterizes the leading order spin-orbit interactions in Arnowitt, Deser, and Misner-type coordinates. Our parametric solution is applicable in the following two distinct cases: (i) the binary consists of equal mass compact objects, having two arbitrary spins, and (ii) the binary consists of compact objects of arbitrary mass, where only one of them is spinning with an arbitrary spin. As an application of our parametrization, we present gravitational wave polarizations, whose amplitudes are restricted to the leading quadrupolar order, suitable to describe gravitational radiation from spinning compact binaries moving in eccentric orbits. The present parametrization will be required to construct `ready to use' reference templates for gravitational waves from spinning compact binaries in inspiralling eccentric orbits. Our parametric solution for the post-Newtonian accurate conservative dynamics of spinning compact binaries clearly indicates, for the cases considered, the absence of chaos in these systems. Finally, we note that our parametrization provides the first step in deriving a fully second post-Newtonian accurate `timing formula', that may be useful for the radio observations of relativistic binary pulsars like J0737-3039. 
  It is considered an equation for the Lyapunov exponent $% \gamma $ in a random metric for a scalar propagating wave field. At first order in frequency this equation is solved explicitly. The localization length $L_{c}$ (reciprocal of Re($\gamma $)) is obtained as function of the metric-fluctuation-distance $\Delta R$ (function of disorder) and the frequency $\omega $ of the wave. Explicitly, low-frequencies propagate longer than high, that is $L_{c}\omega ^{2}=C^{te}$. Direct applications with cosmological quantities like background radiation microwave ($\lambda \sim 1/2\times 10^{-3}$ [m]) and the Universe-length (`localization length' $L_{c}\sim 1.6\times 10^{25}$ [m]) permits to evaluate the metric-fluctuations-distance as $\Delta R\sim 10^{-35}$ [m], a number at order of the Planck's length. 
  Conditions under which gravity coupled to self interacting scalar field determines singularity formation are found and discussed. It is shown that, under a suitable matching with an external space, the boundary, if collapses completely, may give rise to a naked singularity. Issues related to the strength of the singularity are discussed. 
  A static non-singular 10-dimensional closed Friedmann universe of Planck size, filled with a perfect fluid with an equation of state with w = -2/3, can arise spontaneously by a quantum fluctuation from nothing in 11-dimensional spacetime. A quantum transition from this state can initiate the inflationary quantum cosmology outlined in Ref. 2 [General Relativity and Gravitation 33, 1415, 2001 - gr-qc/0103021]. With no fine-tuning, that cosmology predicts about 60 e-folds of inflation and a vacuum energy density depending only on the number of extra space dimensions (seven), G, h, c and the ratio between the strength of gravity and the strength of the strong force. The fraction of the total energy in the universe represented by this vacuum energy depends on the Hubble constant. Hubble constant estimates from WMAP, SDSS, the Hubble Key Project and Sunyaev-Zeldovich and X-ray flux measurements range from 60 to 72 km/(Mpc sec). With a mid-range Hubble constant of 65 km/(Mpc sec), the model in Ref. 2 predicts Omega-sub-Lambda = 0.7 
  We analyse the physics of nonlinear gravitational processes inside a spherical charged black hole perturbed by a self-gravitating massless scalar field. For this purpose we created an appropriate numerical code. Throughout the paper, in addition to investigation of the properties of the mathematical singularities where some curvature scalars are equal to infinity, we analyse the properties of the physical singularities where the Kretschmann curvature scalar is equal to the planckian value. Using a homogeneous approximation we analyse the properties of the spacetime near a spacelike singularity in spacetimes influenced by different matter contents namely a scalar field, pressureless dust and matter with ultrarelativistic isotropic pressure. We also carry out full nonlinear analyses of the scalar field and geometry of spacetime inside black holes by means of an appropriate numerical code with adaptive mesh refinement capabilities. We use this code to investigate the nonlinear effects of gravitational focusing, mass inflation, matter squeeze, and these effects dependence on the initial boundary conditions. It is demonstrated that the position of the physical singularity inside a black hole is quite different from the positions of the mathematical singularities. In the case of the existence of a strong outgoing flux of the scalar field inside a black hole it is possible to have the existence of two null singularities and one central $r=0$ singularity simultaneously. 
  Loop quantum cosmology, the symmetry reduction of quantum geometry for the study of various cosmological situations, leads to a difference equation for its quantum evolution equation. To ensure that solutions of this equation act in the expected classical manner far from singularities, additional restrictions are imposed on the solution. In this paper, we consider the Bianchi I model, both the vacuum case and the addition of a cosmological constant, and show using generating function techniques that only the zero solution satisfies these constraints. This implies either that there are technical difficulties with the current method of quantizing the evolution equation, or else loop quantum gravity imposes strong restrictions on the physically allowed solutions. 
  The role played by torsion in gravitation is critically reviewed. After a description of the problems and controversies involving the physics of torsion, a comprehensive presentation of the teleparallel equivalent of general relativity is made. According to this theory, curvature and torsion are alternative ways of describing the gravitational field, and consequently related to the same degrees of freedom of gravity. However, more general gravity theories, like for example Einstein-Cartan and gauge theories for the Poincare and the affine groups, consider curvature and torsion as representing independent degrees of freedom. By using an active version of the strong equivalence principle, a possible solution to this conceptual question is reviewed. This solution favors ultimately the teleparallel point of view, and consequently the completeness of general relativity. A discussion of the consequences for gravitation is presented. 
  By utilizing non-standard slicings of 5-dimensional Schwarzschild and Schwarzschild-AdS manifolds based on isotropic coordinates, we generate static and spherically symmetric braneworld spacetimes containing shell-like naked null singularities. For planar slicings, we find that the brane-matter sourcing the solution is a perfect fluid with an exotic equation of state and a pressure singularity where the brane crosses the bulk horizon. From a relativistic point of view, such a singularity is required to maintain matter infinitesimally above the surface of a black hole. From the point of view of the AdS/CFT conjecture, the singular horizon can be seen as one possible quantum correction to a classical black hole geometry. Various generalizations of planar slicings are also considered for a Ricci-flat bulk, and we find that singular horizons and exotic matter distributions are common features. 
  We study a full causal bulk viscous cosmological model with flat FRW symmetries and where the ``constants'' $G,c$ and $\Lambda $ vary. We take into account the possible effects of a $c-$variable into the curvature tensor in order to outline the field equations. Using the Lie method we find the possible forms of the ``constants'' $G$ and $c$ that make integrable the field equations as well as the equation of state for the viscous parameter. It is found that $G,c$ and $\Lambda $ follow a power law solution verifying the relationship $G/c^{2}=\kappa .$ Once these possible forms have been obtained we calculate the thermodynamical quantities of the model in order to determine the possible values of the parameters that govern the quantities, finding that only a growing $G$ and $c$ are possible while $% \Lambda $ behaves as a negative decreasing function. 
  There is constructed, for each member of a one-parameter family of cosmological models, which is obtained from the Kottler-Schwarzschild-de Sitter spacetime by identification under discrete isometries, a slicing by spherically symmetric Cauchy hypersurfaces of constant mean curvature. These slicings are unique up to the action of the static Killing vector. Analytical and numerical results are found as to when different leaves of these slicings do not intersect, i.e. when the slicings form foliations. 
  Finding signatures of quantum gravity in cosmological observations is now actively pursued both from the theoretical and the experimental side. Recent work has concentrated on finding signatures of light-cone fluctuations in the CMB. Because in inflationary scenarios a Gravitational Wave Background (GWB) is always emitted much before the CMB, we can ask, in the hypothesis where this GWB could be observed, what is the imprint of light cone fluctuations on this GWB. We show that due to the flat nature of the GWB spectrum, the effect of lightcone fluctuations are negligible. 
  We give a brief account of the description of the standard model in noncommutative geometry as well as the thermal time hypothesis, questioning their relevance for quantum gravity. 
  The kind of flat-earth gravity used in introductory physics appears in an accelerated reference system in special relativity. From this viewpoint, we work out the special relativistic description of a ballistic projectile and a simple pendulum, two examples of simple motion driven by earth-surface gravity. The analysis uses only the basic mathematical tools of special relativity typical of a first-year university course. 
  The standard geometrodynamics is transformed into a theory of conformal geometrodynamics by extending the ADM phase space for canonical general relativity to that consisting of York's mean exterior curvature time, conformal three-metric and their momenta. Accordingly, an additional constraint is introduced, called the conformal constraint. In terms of the new canonical variables, a diffeomorphism constraint is derived from the original momentum constraint. The Hamiltonian constraint then takes a new form. It turns out to be the sum of an expression that previously appeared in the literature and extra terms quadratic in the conformal constraint. The complete set of the conformal, diffeomorphism and Hamiltonian constraints are shown to be of first class through the explicit construction of their Poisson brackets. The extended algebra of constraints has as subalgebras the Dirac algebra for the deformations and Lie algebra for the conformorphism transformations of the spatial hypersurface. This is followed by a discussion of potential implications of the presented theory on the Dirac constraint quantization of general relativity. An argument is made to support the use of the York time in formulating the unitary functional evolution of quantum gravity. Finally, the prospect of future work is briefly outlined. 
  This paper is intended to give a review of the recent developments on black holes with Skyrme hair. The Einstein-Skyrme system is known to possess black hole solutions with Skyrme hair. The spherically symmetric black hole skyrmion with B=1 was the first discovered counter example of the no-hair conjecture for black holes. Recently we found the B=2 axially symmetric black hole skyrmion. In this system, the black hole at the center of the skyrmion absorbs the baryon number partially, leaving fractional charge outside the horizon. Therefore the baryon number is no longer conserved. We examine the B=1, 2 black hole solutions in detail in this paper. The model has a natural extension to the gauged version which can describe monopole black hole skyrmions. Callan and Witten discussed the monopole catalysis of proton decay within the Skyrme model. We apply the idea to the Einstein-Maxwell-Skyrme system and obtain monopole black hole skyrmions. Remarkably there exist multi-black hole skyrmion solutions in which the gravitational, electromagnetic, and strong forces between the monopoles are all in balance. The solutions turn out to be stable under spherically symmetric linear perturbations. 
  An origin and necessity of so called conformal (or,Penrose-Chernikov-Tagirov) coupling of scalar field to the metric of n-dimensional Riemannian space-time is discussed in brief. The corresponding general-relativistic field equation implies a one-particle (quantum mechanical) Schrodinger Hamiltonian which depends on the space-time dimensionality n, contrary to the Hamiltonian constructed by quantization of geodesic motion, which is the same for any value of n. In general, the Hamiltonians can coincide only for n = 4, the dimensionality of the ordinarily observed Universe. In view of the fundamental role of a scalar field in various cosmological models, this fact may be of interest for models of brane worlds where n > 4 . 
  A recently introduced discrete formalism allows to solve the problem of time in quantum gravity in a relational manner. Quantum mechanics formulated with a relational time is not exactly unitary and implies a fundamental mechanism for decoherence of quantum states. The mechanism is strong enough to render the black hole information puzzle unobservable. 
  Riemannian effective spacetime description of Hawking radiation in $^{3}He-A$ superfluids is extended to non-Riemannian effective spacetime. An example is given of non-Riemannian effective geometry of the rotational motion of the superfluid vacuum around the vortex where the effective spacetime Cartan torsion can be associated to the Hawking giving rise to a physical interpretation of effective torsion recently introduced in the literature in the form of an acoustic torsion in superfluid $^{4}He$ (PRD-70(2004),064004). Curvature and torsion singularities of this $^{3}He-A$ fermionic superfluid are investigated. This Lense-Thirring effective metric, representing the superfluid vacuum in rotational motion, is shown not support Hawking radiation when the isotropic $^{4}He$ is restored at far distances from the vortex axis. Hawking radiation can be expressed also in topological solitons (moving domain walls) in fermionic superfluids in non-Riemannian (teleparallel) $(1+1)$ dimensional effective spacetime. A teleparallel solution is proposed where the quasiparticle speed is determined from the teleparallel geometry. 
  We present a stochastic theory for the nonequilibrium dynamics of charges moving in a quantum scalar field based on the worldline influence functional and the close-time-path (CTP or in-in) coarse-grained effective action method. We summarize (1) the steps leading to a derivation of a modified Abraham-Lorentz-Dirac equation whose solutions describe a causal semiclassical theory free of runaway solutions and without pre-acceleration patholigies, and (2) the transformation to a stochastic effective action which generates Abraham-Lorentz-Dirac-Langevin equations depicting the fluctuations of a particle's worldline around its semiclassical trajectory. We point out the misconceptions in trying to directly relate radiation reaction to vacuum fluctuations, and discuss how, in the framework that we have developed, an array of phenomena, from classical radiation and radiation reaction to the Unruh effect, are interrelated to each other as manifestations at the classical, stochastic and quantum levels. Using this method we give a derivation of the Unruh effect for the spacetime worldline coordinates of an accelerating charge. Our stochastic particle-field model, which was inspired by earlier work in cosmological backreaction, can be used as an analog to the black hole backreaction problem describing the stochastic dynamics of a black hole event horizon. 
  The quantization of the gravitational field is discussed within the exact uncertainty approach. The method may be described as a Hamilton-Jacobi quantization of gravity. It differs from previous approaches that take the classical Hamilton-Jacobi equation as their starting point in that it incorporates some new elements, in particular the use of a formalism of ensembles in configuration space and the postulate of an exact uncertainty relation. These provide the fundamental elements needed for the transition from the classical theory to the quantum theory. 
  We show that with a small modification, the formulation of the Einstein equations of Uggla et al, which uses tetrad variables normalised by the expansion, is a mixed symmetric hyperbolic/parabolic system. Well-posedness of the Cauchy problem follows from a standard theorem. 
  The metric of a tidally distorted, nonrotating black hole is presented in a light-cone coordinate system that penetrates the event horizon and possesses a clear geometrical meaning. The metric is expressed as an expansion in powers of r/R << 1, where r is a measure of distance from the black hole and R is the local radius of curvature of the external spacetime; this is assumed to be much larger than M, the mass of the black hole. The metric is calculated up to a remainder of order (r/R)^4, and it depends on a family of tidal gravitational fields which characterize the hole's local environment. The coordinate system allows an easy identification of the event horizon, and expressions are derived for its surface gravity and the rates at which the tidal interaction transfers mass and angular momentum to the black hole. 
  In asymptotically anti-de Sitter gravity, diffeomorphisms that change the conformal boundary data can be promoted to genuine physical degrees of freedom. I show that in 2+1 dimensions, the dynamics of these degrees of freedom is described by a Liouville action, with the correct central charge to reproduce the entropy of the BTZ black hole. 
  In this paper we show how the student can be led to an understanding of the connection between special relativity and general relativity by considering the time dilation effect of clocks placed on the surface of the Earth. This paper is written as a Socratic dialog between a lecturer Sam and a student Kim. 
  In this paper, the role of anisotropy and inhomogeneity has been studied in quasi-spherical gravitational collapse. Also the role of initial data has been investigated in characterizing the final state of collapse. Finally, a linear transformation on the initial data set has been presented and its impact has been discussed. 
  We formulate in an intuitive manner several conceptual aspects of the field-to-particle transition problem which intends to extract physical properties of elementary particles from specific field configurations. We discuss the possibility of using the conceptual basis of the holographic principle and the mathematical fundaments of nonlinear sigma models for the field-to-particle transition. It is shown that certain classical gravitational configurations in vacuum may contain physical parameters with discrete values, and that they behave under rotations as particle-like objects. 
  By generalizing the Hodge dual operator to the case of soldered bundles, and working in the context of the teleparallel equivalent of general relativity, an analysis of the duality symmetry in gravitation is performed. Although the basic conclusion is that, at least in the general case, gravitation is not dual symmetric, there is a particular theory in which this symmetry shows up. It is a self dual (or anti-self dual) teleparallel gravity in which, due to the fact that it does not contribute to the interaction of fermions with gravitation, the purely tensor part of torsion is assumed to vanish. The ensuing fermionic gravitational interaction is found to be chiral. Since duality is intimately related to renormalizability, this theory may eventually be more amenable to renormalization than teleparallel gravity or general relativity. 
  Experimental tests of Newton law put stringent constraints on potential deviations from standard theory with ranges from the millimeter to the size of planetary orbits. Windows however remain open for short range deviations, below the millimeter, as well as long range ones, of the order of or larger than the size of the solar system. We discuss here the relation between long range tests of the Newton law and the anomaly recorded on Pioneer 10/11 probes. 
  The generalized second law of gravitational thermodynamics is applied to the present era of accelerated expansion of the Universe. In spite of the fact that the entropy of matter and relic gravitational waves inside the event horizon diminish, the mentioned law is fulfilled provided that the expression for the entropy density of the gravitational waves satisfies a certain condition. 
  We investigate the evolution of infinite strings as a part of a complete cosmic string network in flat space. We perform a simulation of the network which uses functional forms for the string position and thus is exact to the limits of computer arithmetic. Our results confirm that the wiggles on the strings obey a scaling law described by universal power spectrum. The average distance between long strings also scales accurately with the time. These results suggest that small-scale structure will also scale in expanding universe, even in the absence of gravitational damping. 
  Einstein's special theory of relativity revolutionized physics by teaching us that space and time are not separate entities, but join as ``spacetime''. His general theory of relativity further taught us that spacetime is not just a stage on which dynamics takes place, but is a participant: The field equation of general relativity connects matter dynamics to the curvature of spacetime. Curvature is responsible for gravity, carrying us beyond the Newtonian conception of gravity that had been in place for the previous two and a half centuries. Much research in gravitation since then has explored and clarified the consequences of this revolution; the notion of dynamical spacetime is now firmly established in the toolkit of modern physics. Indeed, this notion is so well established that we may now contemplate using spacetime as a tool for other science. One aspect of dynamical spacetime -- its radiative character, ``gravitational radiation'' -- will inaugurate entirely new techniques for observing violent astrophysical processes. Over the next one hundred years, much of this subject's excitement will come from learning how to exploit spacetime as a tool for astronomy. This article is intended as a tutorial in the basics of gravitational radiation physics. 
  The evolution of weak discontinuity is investigated on horizons in the $n$-dimensional static solutions in the Einstein-Maxwell-scalar-$\Lambda$ system, including the Reissner-Nordstr\"om-(anti) de Sitter black hole. The analysis is essentially local and nonlinear. We find that the Cauchy horizon is unstable, whereas both the black-hole event horizon and the cosmological event horizon are stable. This new instability, the so-called kink instability, of the Cauchy horizon is completely different from the well-known ``infinite-blueshift'' instability. The kink instability makes the analytic continuation beyond the Cauchy horizon unstable. 
  Due to the complexity of the required numerical codes, many of the new formulations for the evolution of the gravitational fields in numerical relativity are not tested on binary evolutions. We introduce in this paper a new testing ground for numerical methods based on the simulation of binary neutron stars. This numerical setup is used to develop a new technique, the Hamiltonian relaxation (HR), that is benchmarked against the currently most stable simulations based on the BSSN method. We show that, while the length of the HR run is somewhat shorter than the equivalent BSSN simulation, the HR technique improves the overall quality of the simulation, not only regarding the satisfaction of the Hamiltonian constraint, but also the behavior of the total angular momentum of the binary. The latest quantity agrees well with post-Newtonian estimations for point-mass binaries in circular orbits. 
  In this paper, we study the steady solutions of Euler-Poisson equations in bounded domains with prescribed angular velocity. This models a rotating Newtonian star consisting of a compressible perfect fluid with given equation of state $P=e^S\rho^{\gamma}$. When the domain is a ball and the angular velocity is constant, we obtain both existence and non-existence theorems, depending on the adiabatic gas constant $\gamma$. In addition we obtain some interesting properties of the solutions; e.g., monotonicity of the radius of the star with both angular velocity and central density. We also prove that the radius of a rotating spherically symmetric star, with given constant angular velocity and constant entropy, is uniformly bounded independent of the central density . This is physically striking and in sharp contrast to the case of the nonrotating star. For general domains and variable angular velocities, both an existence result for the isentropic equations of state and non-existence result for the non-isentropic equation of state are also obtained. 
  We show how the quantum potential arises in various ways and trace its connection to quantum fluctuations and Fisher information along with its realization in terms of Weyl curvature. It is a quantization factor for certain classical systems as well as an expression for quantum matter in gravity theories of Weyl-Dirac type. We extract theories and examples from the literature, providing connections and interpretations, and make a few new observations. 
  Accurate calculation of the motion of a compact object in a background spacetime induced by a supermassive black hole is required for the future detection of such binary systems by the gravitational-wave detector LISA. Reaching the desired accuracy requires calculation of the second-order gravitational perturbations produced by the compact object. At the point particle limit the second-order gravitational perturbation equations turn out to have highly singular source terms, for which the standard retarded solutions diverge. Here we study a simplified scalar toy-model in which a point particle induces a nonlinear scalar field in a given curved spacetime. The corresponding second-order scalar perturbation equation in this model is found to have a similar singular source term, and therefore its standard retarded solutions diverge. We develop a regularization method for constructing well-defined causal solutions for this equation. Notably these solutions differ from the standard retarded solutions, which are ill-defined in this case. 
  The braneworld cosmological model was constructed by embedding a 3-brane into a higher dimensional bulk background geometry and the usual matter of our universe was assumed to be confined in the brane while the gravity can propagate in the bulk. By considering that the bulk geometry is anisotropic, after reviewed the separable solution for the bulk metric and the result that the anisotropic property of the bulk can support the perfect fluid kind of the matter in the brane, we develop a formalism of the cosmological perturbation for this kind of anisotropic braneworld model. As in isotropic case, we can also decompose the perturbation into scalar, vector and tensor modes, we find that the formalism for the anisotropic braneworld cosmological perturbation are very different from the isotropic case. The anisotropic effect can be reflected in the tensor modes which dominated the cosmological gravitation waves. Finally, we also discussed the perturbed Einstein equations governed the dynamics of the bulk geometry and the brane with the junction conditions. 
  All quantum gravity approaches lead to small modifications in the standard laws of physics which lead to violations of Lorentz invariance. One particular example is the extended standard model (SME). Here, a general phenomenological approach for extensions of the Maxwell equations is presented which turns out to be more general than the SME and which covers charge non--conservation (CNC), too. The new Lorentz invariance violating terms cannot be probed by optical experiments but need, instead, the exploration of the electromagnetic field created by a point charge or a magnetic dipole. Some scalar--tensor theories and higher dimensional brane theories predict CNC in four dimensions and some models violating Special Relativity have been shown to be connected with CNC and its relation to the Einstein Equivalence Principle has been discussed. Due to this upcoming interest, the experimental status of electric charge conservation is reviewed. Up to now there seem to exist no unique tests of charge conservation. CNC is related to the precession of polarization, to a modification of the $1/r$--Coulomb potential, and to a time-dependence of the fine structure constant. This gives the opportunity to describe a dedicated search for CNC. 
  In this paper, the two-dimensional Reissner-Nordstr\"{o}m black hole is considered as a system of the Casimir type. In this background the Casimir effect for the massless Dirac field is discussed. The massless Dirac field is confined between two ``parallel plates'' separated by a distance $L$ and there is no particle current drilling through the boundaries. The vacuum expectation values of the stress tensor of the massless Dirac field at infinity are calculated separately in the Boulware state, the Hartle-Hawking state and the Unruh state. 
  Riemannian and teleparallel geometrical approaches to the investigation of Maxwell electrodynamics shown that a unified field theory of gravitation and electromagnetism a la Einstein can be obtained from a stationary metric. This idea contrasts with the recently proposed pre-metric electrodynamics by Hehl and Obukhov. In the teleparallel case the definition of the electric field is obtained straightforward from the spacetime metric and the orthonormal basis frame of teleparallelism. In this case the only nonvanishing component of Cartan torsion is defined as the effective electric field. In this approach the gravitational potentials or metric coefficients are expressed in terms of the effective or analogous electric and magnetic potentials. Thefore the Maxwell equations in vacuum can be obtained by derivation of this electric field definition as usual. In the Riemannian case we consider an electrostatic spacetime where the Einstein equations in vacuum in the approximation of linear fields. The constraint of Einstein equations in vacuum are shown to lead or to the Coulomb equation or to a singular behaviour on the metric which would represent a kind of effective electrodynamic black hole event horizon. 
  Anisotropic cosmological models such as the G\"{o}del universe and its extensions - G\"{o}del type solutions, are embedded on a visible 3-brane in the Randall-Sundrum 1 model. The size of the extra dimension is stabilized by tuning the rotation parameter to a very small value so that hierarchy problem can be solved. A limiting case also yields the Randall-Sundrum 2 model. The rotation parameter on the visible brane turns out to be of order $10^{-32}$, which implies that visible brane essentially lacks rotation. 
  We extend to the Einstein Maxwell Higgs system results first obtained previously in collaboration with V. Moncrief for Einstein equations in vacuum. 
  We give an overview of ongoing searches for effects motivated by the study of the quantum-gravity problem. We describe in greater detail approaches which have not been covered in recent ``Quantum Gravity Phenomenology'' reviews. In particular, we outline a new framework for describing Lorentz invariance violation in the Maxwell sector. We also discuss the general strategy on the experimental side as well as on the theoretical side for a search for quantum gravity effects. The role of test theories, kinematical and dymamical, in this general context is emphasized. The present status of controlled laboratory experiments is described, and we also summarize some key results obtained on the basis of astrophysical observations. 
  We present a simple method for applying excision boundary conditions for the relativistic Euler equations. This method depends on the use of Reconstruction-Evolution methods, a standard class of HRSC methods. We test three different reconstruction schemes, namely TVD, PPM and ENO. The method does not require that the coordinate system is adapted to the excision boundary. We demonstrate the effectiveness of our method using tests containing discontinuites, static test-fluid solutions with black holes, and full dynamical collapse of a neutron star to a black hole. A modified PPM scheme is introduced because of problems arisen when matching excision with the original PPM reconstruction scheme. 
  As shown by the development of Special Relativity the simultaneity concept should be related to that of reference frame. Poincare' proposed to define the simultaneity of two events by means of light signals following what is nowadays known as the Einstein simultaneity convention. The need of a simultaneity definition is present also in general relativity and in curved spacetimes in order to provide the observers with a coordinate time.   It is recognized that the old Einstein simultaneity convention is nothing but a connection on a suitable trivial bundle that defines the reference frame. Unfortunately, it has a non vanishing holonomy in curved and even in flat spacetimes a fact that makes it almost useless. We point out the advantage of local simultaneity conventions showing that they are represented by local simultaneity connections. Among them there is one, uniquely determined by the reference frame, which is particularly well-behaved globally. 
  Brane model of universe is considered for zero-mass particle. Equation of Wheeler - de Witt type is obtained using variation principle from the well-known conservation laws inside the brane. This equation includes term accounting the variation of brane topology. Solutions are obtained analytically at some simplifications and the dispersion relations are derived for frequency of wave associated with the particle. 
  We construct a new class of exact solutions describing spacetimes possessing Lie algebroid symmetry. They are described by generic off-diagaonal 5D metrics embedded in bosonic string gravity and possess nontrivial limits to the Einstein gravity. While we focus on nonholonomic vielbein transforms of the Schwarzschild metrics to 5D ansatz with solitonic backgrounds, much of the analysis continues to hold for more general configurations with nontrivial Lie algebroid structure and nonlinear connections. We carefully investigate some examples when the anchor structure is related to 3D solitonic interactions. The approach defines a general geometric method of constructing exact solutions with various type of symmetries and new developments and applications of the Lie aglebroid theory. 
  A model for noncommutative scalar fields coupled to gravity based on the generalization of the Moyal product is proposed. Solutions compatible with homogeneous and isotropic flat Robertson-Walker spaces to first non-trivial order in the perturbation of the star-product are presented. It is shown that in the context of a typical chaotic inflationary scenario, at least in the slow-roll regime, noncommutativity yields no observable effect. 
  The aim of this paper is to study the time delay on electromagnetic signals propagating across a binary stellar system. We focus on the antisymmetric gravitomagnetic contribution due to the angular momentum of one of the stars of the pair. Considering a pulsar as the source of the signals, the effect would be manifest both in the arrival times of the pulses and in the frequency shift of their Fourier spectra. We derive the appropriate formulas and we discuss the influence of different configurations on the observability of gravitomagnetic effects. We argue that the recently discovered PSR J0737-3039 binary system does not permit the detection of the effects because of the large size of the eclipsed region. 
  We study a supersymmetric model of space-time foam with two stacks each of eight D8-branes with equal string tensions, separated by a single bulk dimension containing D0-brane particles that represent quantum fluctuations. The ground-state configuration with static D-branes has zero vacuum energy, but, when they move, the interactions among the D-branes and D-particles due to the exchanges of strings result in a non-trivial, positive vacuum energy. We calculate its explicit form in the limits of small velocities and large or small separations between the D-branes and/or the D-particles. This non-trivial vacuum energy appears as a central charge deficit in the non-critical stringy $\sigma$ model describing perturbative string excitations on a moving D-brane. These calculations enable us to characterise the ground state of the D-brane/D-particle system, and provide a framework for discussing brany inflation and the possibility of residual Dark Energy in the present-day Universe. 
  We study here what it means for the Universe to be nearly flat, as opposed to exactly flat. We give three definitions of nearly flat, based on density, geometry and dynamics; all three definitions are equivalent and depend on a single constant flatness parameter epsilon that quantifies the notion of nearly flat. Observations can only place an upper limit on epsilon, and always allow the possibility that the Universe is infinite with k=-1 or finite with k=1. We use current observational data to obtain a numerical upper limit on the flatness parameter and discuss its implications, in particular the "naturalness" of the nearly flat Universe. 
  We prove in the cases of plane and hyperbolic symmetries a global in time existence result in the future for comological solutions of the Einstein-Vlasov-scalar field system, with the sources generated by a distribution function and a scalar field, subject to the Vlasov and wave equations respectively. The spacetime is future geodesically complete in the special case of plane symmetry with only a scalar field. Causal geodesics are also shown to be future complete for homogeneous solutions of the Einstein-Vlasov-scalar field system with plane and hyperbolic symmetry. 
  A similarity variable is introduced for the Einstein - Maxwell equations with one Killing vector that reduces the partial differential equations in three independent variables to ordinary differential equations. These equations are then solved, providing new solutions of the Einstein - Maxwell equations. 
  The structure of the general, inhomogeneous solution of (bosonic) Einstein-matter systems in the vicinity of a cosmological singularity is considered. We review the proof (based on ideas of Belinskii-Khalatnikov-Lifshitz and technically simplified by the use of the Arnowitt-Deser-Misner Hamiltonian formalism) that the asymptotic behaviour, as one approaches the singularity, of the general solution is describable, at each (generic) spatial point, as a billiard motion in an auxiliary Lorentzian space. For certain Einstein-matter systems, notably for pure Einstein gravity in any spacetime dimension D and for the particular Einstein-matter systems arising in String theory, the billiard tables describing asymptotic cosmological behaviour are found to be identical to the Weyl chambers of some Lorentzian Kac-Moody algebras. In the case of the bosonic sector of supergravity in 11 dimensional spacetime the underlying Lorentzian algebra is that of the hyperbolic Kac-Moody group E(10), and there exists some evidence of a correspondence between the general solution of the Einstein-three-form system and a null geodesic in the infinite dimensional coset space E(10)/K (E(10)), where K (E(10)) is the maximal compact subgroup of E(10). 
  A simple method to deal with four dimensional Hamilton-Jacobi equation for null hypersurfaces is introduced. This method allows to find simple geometrical conditions which give rise to the failure of the WKB approximation on curved spacetimes. The relation between such failure, extreme blackholes and the Cosmic Censor hypothesis is briefly discussed. 
  We have carried out numerical simulations of strongly gravitating systems based on the Einstein equations coupled to the relativistic hydrodynamic equations using adaptive mesh refinement (AMR) techniques. We show AMR simulations of NS binary inspiral and coalescence carried out on a workstation having an accuracy equivalent to that of a $1025^3$ regular unigrid simulation, which is, to the best of our knowledge, larger than all previous simulations of similar NS systems on supercomputers. We believe the capability opens new possibilities in general relativistic simulations. 
  We consider a self-consistent system of Bianchi type-I (BI) gravitational field and a binary mixture of perfect fluid and dark energy. The perfect fluid is taken to be the one obeying the usual equation of state, i.e., $p = \zeta \ve$, with $\zeta \in [0, 1]$ whereas, the dark energy is considered to be obeying a quintessence-like equation of state. Exact solutions to the corresponding Einstein equations are obtained. The model in consideration gives rise to a Universe which is spatially finite. Depending on the choice of problem parameters the Universe is either close with a space-time singularity, or an open one which is oscillatory, regular and infinite in time. 
  We have performed a search for bursts of gravitational waves associated with the very bright Gamma Ray Burst GRB030329, using the two detectors at the LIGO Hanford Observatory. Our search covered the most sensitive frequency range of the LIGO detectors (approximately 80-2048 Hz), and we specifically targeted signals shorter than 150 ms. Our search algorithm looks for excess correlated power between the two interferometers and thus makes minimal assumptions about the gravitational waveform. We observed no candidates with gravitational wave signal strength larger than a pre-determined threshold. We report frequency dependent upper limits on the strength of the gravitational waves associated with GRB030329. Near the most sensitive frequency region, around 250 Hz, our root-sum-square (RSS) gravitational wave strain sensitivity for optimally polarized bursts was better than h_RSS = 6E-21 Hz^{-1/2}. Our result is comparable to the best published results searching for association between gravitational waves and GRBs. 
  We give an up-to-date perspective with a general overview of the theory of causal properties, the derived causal structures, their classification and applications, and the definition and construction of causal boundaries and of causal symmetries, mostly for Lorentzian manifolds but also in more abstract settings. 
  We develop a relativistic velocity space called \emph{rapidity space} from the single assumption of Lorentz invariance, and use it to visualize and calculate effects resulting from the successive application of non-colinear Lorentz boosts. In particular, we show how rapidity space provides a geometric approach to Wigner rotation and Thomas precession in the same way that spacetime provides a geometrical approach to kinematic effects in special relativity. 
  A binary system composed of an oscillating and rotating coplanar dusty disk and a point mass is considered. The conservative dynamics is treated on the Newtonian level. The effects of gravitational radiation reaction and wave emission are studied to leading quadrupole order. The related waveforms are given. The dynamical evolution of the system is determined semi-analytically exploiting the Hamiltonian equations of motion which comprise the effects both of the Newtonian tidal interaction and the radiation reaction on the motion of the binary system in elliptic orbits. Tidal resonance effects between orbital and oscillatory motions are considered in the presence of radiation damping. 
  It is shown that spatially homogeneous solutions of the Einstein equations coupled to a nonlinear scalar field and other matter exhibit accelerated expansion at late times for a wide variety of potentials $V$. These potentials are strictly positive but tend to zero at infinity. They satisfy restrictions on $V'/V$ and $V''/V'$ related to the slow-roll approximation. These results generalize Wald's theorem for spacetimes with positive cosmological constant to those with accelerated expansion driven by potentials belonging to a large class. 
  A new parameterization of unconstrained degrees of freedom for gravitational field, used in Classical and Quantum Gravity {\bf 11}, 1055-1068 (1994), has been generalized to one-parameter family of such parameterizations, depending on a real parameter $\beta \in [0,2]$. The description introduced in CQG 11, 1055 corresponds to the special choice $\beta=0$. The method is closely related to the proof of the positivity of the energy presented in Phys. Rev. D {\bf 36}, 1041-1044 (1987) where $\beta$-foliations have been introduced (see also applications to black holes dynamics in Classical and Quantum Gravity {\bf 6}, 1535-1539 (1989) and Acta Physica Polonica B {\bf 25}, 1413-17 (1994)). Spherically symmetric initial data corresponding to trivial degrees of freedom is analyzed along these lines. In particular, the quasi-local energy content of the Schwarzschild initial data is analyzed for different choices of the $\beta$-gauge. 
  The studies in general relativity of rotating finite objects in equilibrium have usually focused on the case when they are truly isolated, this is, the models to describe finite objects are embedded in an asymptotically flat exterior vacuum. Known results ensure the uniqueness of the vacuum exterior field by using the boundary data for the exterior field given at the surface of the object plus the decay of the exterior field at infinity. The final aim of the present work is to study the consequences on the interior models by changing the boundary condition at infinity to one accounting for the embedding of the object in a cosmological background. Considering first the FLRW standard cosmological backgrounds, we are studying the general matching of FLRW with stationary axisymmetric spacetimes in order to find the new boundary condition for the vacuum region. Here we present the first results. 
  Loop Quantum Gravity defines the quantum states of space geometry as spin networks and describes their evolution in time. We reformulate spin networks in terms of harmonic oscillators and show how the holographic degrees of freedom of the theory are described as matrix models. This allow us to make a link with non-commutative geometry and to look at the issue of the semi-classical limit of LQG from a new perspective. This work is thought as part of a bigger project of describing quantum geometry in quantum information terms. 
  One of the most important issues in quantum gravity is to identify its semi-classical regime. First the issue is to define for we mean by a semi-classical theory of quantum gravity, then we would like to use it to extract physical predictions. Writing an effective theory on a flat background is a way to address this problem and I explain how the non-commutative spacetime of deformed special relativity is the natural arena for such considerations. On the other hand, I discuss how the definition of the semi-classical regime can be formulated in a background independent fashion in terms of quantum information and renormalisation of geometry. 
  The covariant Dirac equation in Robertson-Walker space-time is studied under the comoving coordinates. The exact forms of the spatial factor of wave function are respectively acquired in closed, spatially flat, and open universes. 
  We study the renormalisation group running of the cosmological and the Newton constant, where the renormalisation scale is given by the inverse of the radius of the cosmological event horizon. In this framework, we discuss the future evolution of the universe, where we find stable de Sitter solutions, but also "big crunch"-like and "big rip"-like events, depending on the choice of the parameters in the model. 
  In this note we discuss the possibility to define a space-time with a DSR based approach. We show that the strategy of defining a non linear realization of the Lorentz symmetry with a consistent vector composition law cannot be reconciled with the extra request of an invariant length (time) scale. The latter request forces to abandon the group structure of the translations and leaves a space-time structure where points with relative distances smaller or equal to the invariant scale can not be unambiguously defined. 
  We investigate new paths to black hole formation by considering the general relativistic evolution of a differentially rotating polytrope with toroidal shape. We find that this polytrope is unstable to nonaxisymmetric modes, which leads to a fragmentation into self-gravitating, collapsing components. In the case of one such fragment, we apply a simplified adaptive mesh refinement technique to follow the evolution to the formation of an apparent horizon centered on the fragment. This is the first study of the one-armed instability in full general relativity. 
  The operational meaning of spacetime fluctuations is discussed. Classical spacetime geometry can be viewed as encoding the relations between the motions of test particles in the geometry. By analogy, quantum fluctuations of spacetime geometry can be interpreted in terms of the fluctuations of these motions. Thus one can give meaning to spacetime fluctuations in terms of observables which describe the Brownian motion of test particles. We will first discuss some electromagnetic analogies, where quantum fluctuations of the electromagnetic field induce Brownian motion of test particles. We next discuss several explicit examples of Brownian motion caused by a fluctuating gravitational field. These examples include lightcone fluctuations, variations in the flight times of photons through the fluctuating geometry, and fluctuations in the expansion parameter given by a Langevin version of the Raychaudhuri equation. The fluctuations in this parameter lead to variations in the luminosity of sources. Other phenomena which can be linked to spacetime fluctuations are spectral line broadening and angular blurring of distant sources. 
  A discrete theory of gravity locally invariant under the Poincar\'e group is considered as in a companion paper. We define a first order theory, in the sense of Palatini, on the metric-dual Voronoi complex of a simplicial complex. We follow the same spirit of the continuum theory of General Relativity in the Cartan formalism. The field equations are carefully derived taking in account the constraints of the theory. They look very similar to first order Einstein equations in the Cartan formalism. It is shown that in the limit of {\it small deficit angles} these equations have Regge Calculus, locally, as the only solution. A quantum measure is easly defined which does not suffer the ambiguities of Regge Calculus, and a coupling with fermionic matter is easily introduced 
  We extend the conformal gluing construction of Isenberg-Mazzeo-Pollack [18] by establishing an analogous gluing result for field theories obtained by minimally coupling Einstein's gravitational theory with matter fields. We treat classical fields such as perfect fluids and the Yang-Mills equations as well as the Einstein-Vlasov system, which is an important example coming from kinetic theory. In carrying out these extensions, we extend the conformal gluing technique to higher dimensions and codify it in such a way as to make more transparent where it can, and can not, be applied. In particular, we show exactly what criteria need to be met in order to apply the construction, in its present form, to any other non-vacuum field theory. 
  We derive a tidal potential for a self-gravitating fluid star orbiting Kerr black hole along a timelike geodesic extending previous works by Fishbone and Marck. In this paper, the tidal potential is calculated up to the third and fourth-order terms in $R/r$, where $R$ is the stellar radius and $r$ the orbital separation, in the Fermi-normal coordinate system following the framework developed by Manasse and Misner. The new formulation is applied for determining the tidal disruption limit (Roche limit) of corotating Newtonian stars in circular orbits moving on the equatorial plane of Kerr black holes. It is demonstrated that the third and fourth-order terms quantitatively play an important role in the Roche limit for close orbits with $R/r \agt 0.1$. It is also indicated that the Roche limit of neutron stars orbiting a stellar-mass black hole near the innermost stable circular orbit may depend sensitively on the equation of state of the neutron star. 
  We state a condition for an observer to be comoving with another observer in general relativity, based on the concept of lightlike simultaneity. Taking into account this condition, we study relative velocities, Doppler effect and light aberration. We obtain that comoving observers observe the same light ray with the same frequency and direction, and so gravitational redshift effect is a particular case of Doppler effect. We also define a distance between an observer and the events that it observes, that coincides with the known affine distance. We show that affine distance is a particular case of radar distance in the Minkowski space-time and generalizes the proper radial distance in the Schwarzschild space-time. Finally, we show that affine distance gives us a new concept of distance in Robertson-Walker space-times, according to Hubble law. 
  We discuss the effect of Kaluza-Klein (KK) modes of bulk metric perturbations on the second Randall-Sundrum (RS II) type brane cosmology, taking the possible backreaction in the bulk and on the brane into account. KK gravitons may be produced via quantum fluctuations during a de Sitter (dS) inflating phase of our brane universe. In an effective 4-dimensional theory in which one integrates out the extra-dimensional dependence in the action, KK gravitons are equivalent to massive gravitons on the brane with masses $m>3H/2$, where $H$ represents the expansion rate of a dS brane. Thus production of even a tiny amount of KK gravitons may eventually have a significant impact on the late-time brane cosmology. As a first step to quantify the effect of KK gravitons on the brane, we calculate the effective energy density and pressure for a single KK mode. Surprisingly, we find that a KK mode behaves as cosmic dust with a negative energy density on the brane. We note that the bulk energy density of a KK mode is positive definite and there occurs no singular phenomenon in the bulk. 
  We study the phase space of the spherically symmetric solutions of Einstein Gauss-Bonnet system nonminimally coupled to a scalar field and show that in four dimensions the only regular black hole solutions are asymptotically flat 
  Perturbation theory of rotating black holes is described in terms of the Weyl scalars $\psi_4$ and $\psi_0$; each satisfying the Teukolsky's complex master wave equation with spin $s=\mp2$, and respectively representing outgoing and ingoing radiation. We explicitly construct the metric perturbations out of these Weyl scalars in the Regge-Wheeler gauge in the nonrotating limit. We propose a generalization of the Regge-Wheeler gauge for Kerr background in the Newman-Penrose language, and discuss the approach for building up the perturbed spacetime of a rotating black hole. We also provide both-way relationships between waveforms defined in the metric and curvature approaches in the time domain, also known as the (inverse-) Chandrasekhar transformations, generalized to include matter. 
  Laser interferometer gravitational wave detectors can be operated at their free spectral range frequency. We show that in this case and when the interferometer is well understood one could detect a stochastic background using a single detector. 
  All existing 4-coordinate systems centered on the world-line of an accelerated observer are only locally defined like it happens for Fermi coordinates both in special and general relativity. As a consequence, it is not known how non-inertial observers can build {\it equal-time surfaces} which a) correspond to a conventional observer-dependent definition of synchronization of distant clocks; b) are good Cauchy surfaces for Maxwell equations. Another type of coordinate singularities are those connected to the relativistic rotating coordinate systems (the rotating disk).We show that the use of Hamiltonian methods based on 3+1 splittings of space-time allows to define as many observer-dependent globally defined radar 4-coordinate systems as nice foliations of space-time with space-like hyper-surfaces admissible according to M$\o$ller (for instance only differentially rotating relativistic coordinate system are allowed). All these conventional notions of an {\it instantaneous 3-space} for an arbitrary observer can be empirically defined by introducing generalizations of Einstein ${1\over 2}$ convention for clock synchronization in inertial frames. Each admissible 3+1 splitting corresponds to a non-rigid non-inertial frame centered on the observer. When there is a Lagrangian description of an isolated relativistic system, its reformulation as a parametrized Minkowski theory allows to show that all the admissible synchronization conventions are {\it gauge equivalent}, as it also happens in canonical metric and tetrad gravity, where, however, the chrono-geometrical structure of space-time is dynamically determined. 
  We give a brief and a critical review of the Barret-Crane spin foam models of quantum gravity. Then we describe two new spin foam models which are obtained by direct quantization of General Relativity and do not have some of the drawbacks of the Barret-Crane models. These are the model of spin foam invariants for the embedded spin networks in loop quantum gravity and the spin foam model based on the integration of the tetrads in the path integral for the Palatini action. 
  Einstein's theory of general relativity is written in terms of the variables obtained from a conformal--traceless decomposition of the spatial metric and extrinsic curvature. The determinant of the conformal metric is not restricted, so the action functional and equations of motion are invariant under conformal transformations. With this approach the conformal--traceless variables remain free of density weights. The conformal invariance of the equations of motion can be broken by imposing an evolution equation for the determinant of the conformal metric g. Two conditions are considered, one in which g is constant in time and one in which g is constant along the unit normal to the spacelike hypersurfaces. This approach is used to write the Baumgarte--Shapiro--Shibata--Nakamura system of evolution equations in conformally invariant form. The presentation includes a discussion of the conformal thin sandwich construction of gravitational initial data, and the conformal flatness condition as an approximation to the evolution equations. 
  We construct a canonical formulation of general relativity for the case of a timelike foliation of spacetime. The formulation possesses explicit covariance with respect to Lorentz transformations in the tangent space. Applying the loop approach to quantize the theory we derive the spectrum of the area operator of a two-dimensional surface. Its different branches are naturally associated to spacelike and timelike surfaces. The results are compared with the predictions of Lorentzian spin foam models. A restriction of the representations labeling spin networks leads to perfect agreement between the states as well as the area spectra in the two approaches. 
  The existence of a maximal acceleration for deterministic finslerian models at the Planck scale is argued. Possible phenomenological relativity groups O(1,6), O(1,7) and O(2,6) are introduced from the perspective of finslerian deterministic models. The effect of maximal acceleration delaying the speed of light is studied. Finally, a mechanism generating maximal acceleration in connection with bosonic string theory is discussed. 
  Light deflection in the post-linear gravitational field of two bounded point-like masses is treated. Both the light source and the observer are assumed to be located at infinity in an asymptotically flat space. The equations of light propagation are explicitly integrated to the second order in $G/c^2$. Some of the integrals are evaluated by making use of an expansion in powers of the ratio of the relative separation distance to the impact parameter $(r_{12}/\xi)$. A discussion of which orders must be retained to be consistent with the expansion in terms of $G/c^2$ is given. It is shown that the expression obtained in this paper for the angle of light deflection is fully equivalent to the expression obtained by Kopeikin and Sch\"afer up to the order given there. The deflection angle takes a particularly simple form for a light ray originally propagating orthogonal to the orbital plane of a binary with equal masses. Application of the formulae for the deflection angle to the double pulsar PSR J0737-3039 for an impact parameter five times greater than the relative separation distance of the binary's components shows that the corrections to the Epstein-Shapiro light deflection angle of about $10^{-6}$ arcsec lie between $10^{-7}$ and $10^{-8}$ arcsec. 
  We investigate the possibility of dressing a four-dimensional black hole with classical scalar field hair which is non-minimally coupled to the space-time curvature. Our model includes a cosmological constant but no self-interaction potential for the scalar field. We are able to rule out black hole hair except when the cosmological constant is negative and the constant governing the coupling to the Ricci scalar curvature is positive. In this case, non-trivial hairy black hole solutions exist, at least some of which are linearly stable. However, when the coupling constant becomes too large, the black hole hair becomes unstable. 
  We investigate the scattering of phase oscillation of Bose-Einstein Condensate by a 'draining of bathtub' type fluid motion. We derive a relation between the reflection and transmission coefficients which exhibits existence of analog of 'Superradiance effect' in BEC vortex with sink. 
  The Newman-Penrose formalism is used to deal with the massless scalar, neutrino, electromagnetic, gravitino and gravitational quasinormal modes (QNMs) in Schwarzschild black holes in a united form. The quasinormal mode frequencies evaluated by using the 3rd-order WKB potential approximation show that the boson perturbations and the fermion perturbations behave in a contrary way for the variation of the oscillation frequencies with spin, while this is no longer true for the damping's, which variate with $s$ in a same way both for boson and fermion perturbations. 
  The aim of this paper is to study foliations that remain invariant by parallel transports along the integral curves of vector fields of another foliations. According to this idea, we define a new concept of stability between foliations. A particular case of stability (called regular stability) is studied, giving a useful characterization in terms of the Riemann curvature tensor. This characterization allows us to prove that there are no regularly self-stable foliations of dimension greater than 1 in Schwarzschild and Robertson-Walker space-times. Finally, we study the existence of regularly self-stable foliations in other space-times, like $pp$-wave space-times. 
  We introduce new concepts and properties of lightlike distributions and foliations (of dimension and co-dimension 1) in a space-time manifold of dimension $n$, from a purely geometric point of view. Given an observer and a lightlike distribution $\Omega $ of dimension or co-dimension 1, its lightlike direction is broken down into two vector fields: a timelike vector field $U$ representing the observer and a spacelike vector field $S$ representing the relative direction of propagation of $\Omega $ for this observer. A new distribution $\Omega_U^-$ is defined, with the opposite relative direction of propagation for the observer $U$. If both distributions $\Omega $ and $\Omega _U^-$ are integrable, the pair \Omega ,\Omega_U^- $ represents the wave fronts of a stationary wave for the observer $U$. However, we show in an example that the integrability of $\Omega $ does not imply the integrability of $\Omega_U^-$. 
  We consider hydrodynamics with non conserved number of particles and show that it can be modeled with effective fluid Lagrangians which explicitly depend on the velocity potentials. For such theories, the {}``shift symetry'' $\phi\to\phi+$const. leading to the conserved number of fluid particles in conventional hydrodynamics is globaly broken and, as a result, the non conservation of particle number appears as a source term in the continuity equation. The particle number non-conservation is balanced by the entropy change, with both the entropy and the source term expresed in terms of the fluid velocity potential. Equations of hydrodynamics are derived using a modified version of Schutz's variational principle method. Examples of fluids described by such Lagrangians (tachyon condensate, k-essence) in spatially flat isotropic universe are briefly discussed. 
  One of the qualitatively distinct and robust implication of Loop Quantum Gravity (LQG) is the underlying discrete structure. In the cosmological context elucidated by Loop Quantum Cosmology (LQC), this is manifested by the Hamiltonian constraint equation being a (partial) difference equation. One obtains an effective Hamiltonian framework by making the continuum approximation followed by a WKB approximation. In the large volume regime, these lead to the usual classical Einstein equation which is independent of both the Barbero-Immirzi parameter $\gamma$ as well as $\hbar$. In this work we present an alternative derivation of the effective Hamiltonian by-passing the continuum approximation step. As a result, the effective Hamiltonian is obtained as a close form expression in $\gamma$. These corrections to the Einstein equation can be thought of as corrections due to the underlying discrete (spatial) geometry with $\gamma$ controlling the size of these corrections. These corrections imply a bound on the rate of change of the volume of the isotropic universe. In most cases these are perturbative in nature but for cosmological constant dominated isotropic universe, there are significant deviations. 
  New families of exact general relativistic thick disks are constructed using the ``displace, cut, fill and reflect'' method. A class of functions used to ``fill'' the disks is derived imposing conditions on the first and second derivatives to generate physically acceptable disks. The analysis of the function's curvature further restrict the ranges of the free parameters that allow phisically acceptable disks. Then this class of functions together with the Schwarzschild metric is employed to construct thick disks in isotropic, Weyl and Schwarzschild canonical coordinates. In these last coordinates an additional function must be added to one of the metric coefficients to generate exact disks. Disks in isotropic and Weyl coordinates satisfy all energy conditions, but those in Schwarzschild canonical coordinates do not satisfy the dominant energy condition. 
  We review recent work and present new examples about the character of singularities in globally and regularly hyperbolic, isotropic universes. These include recent singular relativistic models, tachyonic and phantom universes as well as inflationary cosmologies. 
  We compute the graviton one loop contribution to a classical energy in a traversable wormhole background. Such a contribution is evaluated by means of a variational approach with Gaussian trial wave functionals. A zeta function regularization is involved to handle with divergences. A renormalization procedure is introduced and the finite one loop energy is considered as a self-consistent source for the traversable wormhole. 
  We consider the extension of the Majumdar-type class of static solutions for the Einstein-Maxwell equations, proposed by Ida to include charged perfect fluid sources. We impose the equation of state $\rho+3p=0$ and discuss spherically symmetric solutions for the linear potential equation satisfied by the metric. In this particular case the fluid charge density vanishes and we locate the arising neutral perfect fluid in the intermediate region defined by two thin shells with respective charges $Q$ and $-Q$. With its innermost flat and external (Schwarzschild) asymptotically flat spacetime regions, the resultant condenser-like geometries resemble solutions discussed by Cohen and Cohen in a different context. We explore this relationship and point out an exotic gravitational property of our neutral perfect fluid. We mention possible continuations of this study to embrace non-spherically symmetric situations and higher dimensional spacetimes. 
  This paper has been withdrawn by the author, See J.Krishna Rao, J. Phys. A: Gen. Phys., 4, 17 (1971) for radiating Levi-Civita metric. 
  This paper was withdrawn by arXiv admin due to authors' misrepresentation of identity/affiliation. 
  The matched filtering technique is used to search for gravitational wave signals of a known form in the data taken by ground-based detectors. However, the analyzed data contains a number of artifacts arising from various broad-band transients (glitches) of instrumental or environmental origin which can appear with high signal-to-noise ratio on the matched filtering output. This paper describes several techniques to discriminate genuine events from the false ones, based on our knowledge of the signals we look for. Starting with the $\chi^2$ discriminator, we show how it may be optimized for free parameters. We then introduce several alternative vetoing statistics and discuss their performance using data from the GEO600 detector. 
  We show how to use the quasi-Maxwell formalism to obtain solutions of Einstein's field equations corresponding to homogeneous cosmologies - namely Einstein's universe, Godel's universe and the Ozsvath-Farnsworth-Kerr class I solutions - written in frames for which the associated observers are stationary. 
  A mathematical structure is presented that allows one to define a physical process independent of any background. That is, it is possible, for a set of objects, to choose an object from that set through a choice process that is defined solely in terms of the objects in the set itself. It is conjectured that this background free structure is a necessary ingredient for a self-consistent description of physical processes and that these same physical processes are determined by the absence of any background. The properties of the mathematical structure, denoted Q, are equivalent to the three-dimensional topological manifold 2T^3 + 3(S^1 x S^2), two three-tori plus three handles, embedded in four dimensions. The topology of Q reproduces the basic properties of QED and Einstein gravity. 
  Gravitational gauge theories with de Sitter, Poincare and affine symmetry group are investigated under the aspect of the breakdown of the initial symmetry group down to the Lorentz subgroup. We review the theory of spontaneously broken de Sitter gravity by Stelle and West and apply a similar approach to the case of the Poincare and affine groups. Especially, we find that the groundstate of the metric affine theory leads to the determination of the Lorentzian signature of the metric in the groundstate. We show that the Higgs field remains in its groundstate, i.e., that the metric will have Lorentzian signature, unless we introduce matter fields that explicitely couple to the symmetric part of the connection. We also show that some features, like the necessity of the introduction of a dilaton field, that seem artificial in the context of the affine theory, appear most natural if the gauge group is taken to be the special linear group in five dimensions. Finally, we present an alternative model which is based on the spinor representation of the Lorentz group and is especially adopted to the description of spinor fields in a general linear covariant way, without the use of the infinite dimensional representations which are usually considered to be unavoidable. 
  We study the evolution of relic D3-branes after the D3/\bd-brane inflation in string warped compactification. The motion of D3-branes can be frozen under certain condition during the radiation/matter domination. These D3-branes can not be released until the D3/\bd-branes potential energy becomes dominated at late time. Subsequently they will move towards to \bd-branes, which play the role of uplifting AdS minimum to dS minimum, near the apex of throats. The annihilation of D3/\bd-branes leads to the disappearance of dS vacua. This process may be regarded as a rapid decay channel of present dS vacua. We discuss the parameter spaces required by this process and calculate the decay time. 
  The general relativistic treatment of gravitation can be extended by preserving the geometrical nature of the theory but modifying the form of the coupling between curvature and stress tensors. The gravitation constant is thus replaced by two running coupling constants which depend on scale and differ in the sectors of traceless and traced tensors. When calculated in the solar system in a linearized approximation, the metric is described by two gravitation potentials. This extends the parametrized post-Newtonian (PPN) phenomenological framework while allowing one to preserve compatibility with gravity tests performed in the solar system. Consequences of this extension are drawn here for phenomena correctly treated in the linear approximation. We obtain a Pioneer-like anomaly for probes with an eccentric motion as well as a range dependence of Eddington parameter $\gamma$ to be seen in light deflection experiments. 
  We review recent work on the use of the slice energy concept in generalized theories of gravitation. We focus on two special features in these theories, namely, the energy exchange between the matter component and the scalar field generated by the conformal transformation to the Einstein frame of such theories and the issue of the physical equivalence of different conformal frame representations. We show that all such conformally-related, generalized theories of gravitation allow for the slice energy to be invariably defined and its fundamental properties be insensitive to conformal transformations. 
  We study the matching conditions for a collapsing anisotropic cylindrical perfect fluid, and we show that its radial pressure is non zero on the surface of the cylinder and proportional to the time dependent part of the field produced by the collapsing fluid. This result resembles the one that arises for the radiation - though non-gravitational - in the spherically symmetric collapsing dissipative fluid, in the diffusion approximation. 
  The quasinormal modes (QNMs) associated with the decay of Dirac field perturbation around a Schwarzschild-anti-de Sitter (SAdS) black hole is investigated by using Horowitz-Hubeny approach. We find that both the real and the imaginary parts of the fundamental quasinormal frequencies for large black holes are linear functions of the Hawking temperature, and the real part of the fundamental quasinormal frequencies for intermediate and small black holes approximates a temperature curve and the corresponding imaginary part is almost a linear function of the black hole radius. The angular quantum number has the surprising effect of increasing the damping time scale and decreasing the oscillation time scale for intermediate and small black holes. We also find that the rates between quasinormal frequencies and black hole radii for large, intermediate and small black holes are linear functions of the overtone number, i. e., all the modes are evenly spaced. 
  The quasinormal modes (QNMs) of Dirac field perturbations of a Reissner-Nordstr\"om black hole in an asymptotically Anti-de Sitter spacetime are investigated. We find that both the real and imaginary parts of the fundamental quasinormal frequencies for large black holes are the linear functions of the Hawking temperature, and the slope of the lines for the real parts decreases while that for the magnitude of the imaginary parts increases as the black hole charge increases. According to the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence, the fact shows that different charge presents different time scale in three-dimensional CFT. Another interesting result is that the quasinormal frequencies become evenly spaced for high overtone number, and in the spacing expressions the real part decreases while the magnitude of the imaginary part increases as the charge increases. We also study the relation between quasinormal frequencies and angular quantum number and find that the real part increases while the magnitude of the imaginary part decreases as the angular quantum number increases. 
  In this article we develop a method of finding the static axisymmetric space-time corresponding to any given set of multipole moments. In addition to an implicit algebraic form for the general solution, we also give a power series expression for all finite sets of multipole moments. As conjectured by Geroch we prove in the special case of axisymmetry, that there is a static space-time for any given set of multipole moments subject to a (specified) convergence criterion. We also use this method to confirm a conjecture of Hernandez-Pastora and Martin concerning the monopole-quadropole solution. 
  An infinite 3D plate of homogeneous incompressible fluid is considered, with finite thickness, together with a 2D infinite homogeneous mass in its centre. Einstein equations are exactly solved, in the interior of the 3D mass. The solution is joined to the exterior vacuum metric of Taub. Every value for the 2D mass, positive or negative, allows a perfect junction. Also the joining to a vacuum metric of Rindler is shown; if an imperfect joining is allowed, then again every value of the 2D mass is possible. Some of our results contradict assertions foun in the literature. The text is also available in English by e-mail, ask the author.   --   Estu nefinhava 3D-a plato de unuforma nepremebla fluido, kun finhava diko, kune kun nefinhava 2D-a plano de unuforma maso en /gia mezo. Ekvacioj de Einstein estas ekzakte solvataj, ene la 3D-a maso. La solvo estas kunigata al la malena metriko de vakuo, de Taub. /Ciu valoro por la 2D-a maso, pozitiva a/u negativa, permesas perfektan kunigon. Anka/u la kunigo al la metriko de vakuo de Rindler estas montrata; se neperfekta kunigo estas akceptata, denove /ciu valoro de la 2D-a maso estas ebla. Kelke da niaj rezultoj kontra/udiras asertojn findatajn en la literaturo. 
  Contrary to a recent claim by Anderson ["The Mathematical Theory of Cosmic Strings", I.O.P. Publishing, Bristol 2003], we show that the Letelier-Gal'tsov metric does represent a system of crossed straight infinite cosmic strings moving with arbitrary constant velocities. 
  We study the phase space of spatially homogeneous and isotropic cosmology in general scalar-tensor theories. A reduction to a two-dimensional phase space is performed when possible-in these situations the phase space is usually a two-dimensional curved surface embedded in a three-dimensional space and composed of two sheets attached to each other, possibly with complicated topology. The results obtained are independent of the choice of the coupling function of the theory and, in certain situations, also of the potential. 
  Although most fundamental laws are invariant under time reversal, experience exhibits the presence of irreversible phenomena -- the arrows of time. Their origin lies in cosmology, and I argue that only quantum cosmology can provide the appropriate formal framework. After briefly reviewing the formalism, I discuss how a simple and natural boundary condition can lead to the observed arrows of time. This yields at the same time interesting consequences for black holes. 
  We construct an approximate solution which describes the gravitational emission from a naked singularity formed by the gravitational collapse of a cylindrical thick shell composed of dust. The assumed situation is that the collapsing speed of the dust is very large. In this situation, the metric variables are obtained approximately by a kind of linear perturbation analysis in the background Morgan solution which describes the motion of cylindrical null dust. The most important problem in this study is what boundary conditions for metric and matter variables should be imposed at the naked singularity. We find a boundary condition that all the metric and matter variables are everywhere finite at least up to the first order approximation. This implies that the spacetime singularity formed by this high-speed dust collapse is very similar to that formed by the null dust and thus the gravitational emission from a naked singularity formed by the cylindrical dust collapse can be gentle. 
  Entangled states in the universe may change interpretation of observations and even revise the concept of dark energy. 
  Quantization of spinor and vector free fields in 4-dimensional de Sitter space-time, in the ambient space notation, has been studied in the previous works. Various two-points functions for the above fields are presented in this paper. The interaction between the spinor field and the vector field is then studied by the abelian gauge theory. The U(1) gauge invariant spinor field equation is obtained in a coordinate independent way notation and their corresponding conserved currents are computed. The solution of the field equation is obtained by use of the perturbation method in terms of the Green's function. The null curvature limit is discussed in the final stage. 
  The possibility of measuring the post-Newtonian gravitoelectric correction to the orbital period of a test particle freely orbiting a spherically symmetric mass in the Solar System is analyzed. It should be possible, in principle, to detect it for Mercury at a precision level of 10^-4. This level is mainly set by the unavoidable systematic errors due to the mismodelling in the Keplerian period which could not be reduced by accumulating a large number of orbital revolutions. Future missions like Messenger and BepiColombo should allow to improve it by increasing our knowledge of the Mercury's orbital parameters. The observational accuracy is estimated to be 10^-4 from the knowledge of the International Celestial Reference Frame (ICRF) axes. It could be improved by observing as many planetary transits as possible. It is not possible to measure such an effect in the gravitational field of the Earth by analyzing the motion of artificial satellites or the Moon because of the unavoidable systematic errors related to the uncertainties in the Keplerian periods. In the case of some recently discovered exoplanets the problems come from the observational errors which are larger than the relativistic effect. 
  In flat spacetime, as a simple 4-vector, a particle's 4-velocity cannot be changed by translation. Parallel translation then produces constant velocity, motion without force. Here we consider a richer, but less well-known, representation of the Poincare group of symmetries of flat spacetime in which translation adds a non-linear connection term to the 4-vector. Then 4-vectors can be meaningfully parallel translated, curved geodesics can be developed, and the curvature for the connection can be found. By adding an assumption, that the arc length is calculated with a position dependent metric, it is shown that a geodesic has a 4-acceleration which is the sum of a Christoffel connection term and a term that is the scalar product of 4-velocity and an antisymmetric tensor. This is just like the force law obeyed by a charged, massive particle in general relativity. Thus the dynamical laws of electrodynamics and general relativity can be deduced as geodesic equations from the way particle proper time depends on position and the way 4-vectors can change upon translation in flat spacetime. 
  Discrete symmetries, parity, time reversal, antipodal, and charge conjugation transformations for spinor field in de Sitter space, are presented in the ambient space notation, i.e. in a coordinate independent way. The PT and PCT transformations are also discussed in this notation. The five-current density is studied and their transformation under the discrete symmetries is discussed. 
  The quasinormal modes (QNMs) associated with the decay of Dirac field perturbation around a Schwarzschild black hole is investigated by using continued fraction and Hill-determinant approaches. It is shown that the fundamental quasinormal frequencies become evenly spaced for large angular quantum number and the spacing is given by $\omega_{\lambda+1}- \omega_{\lambda}=0.38490-0.00000i$. The angular quantum number has the surprising effect of increasing real part of the quasinormal frequencies, but it almost does not affect imaginary part, especially for low overtones. In addition, the quasinormal frequencies also become evenly spaced for large overtone number and the spacing for imaginary part is $Im(\omega_{n+1})-Im(\omega_n)\approx -i/4M$ which is same as that of the scalar, electromagnetic, and gravitational perturbations. 
  We present exact spherically symmetric dyonic black hole solutions in four-dimensional Einstein-Maxwell-dilaton gravity with Liouville-type potentials for the dilaton field. These solutions have unusual asymptotics--they are neither asymptotically flat nor asymptotically (anti-) de Sitter. The solutions have one or two horizons hiding a curvature singularity at the origin. A class of topological dyonic black holes with topology of a torus is also presented. Some basic properties of the black holes are discussed. 
  It is believed that a primary principle of the theory of quantum gravity is the Holographic Principle according to which a physical system can be described only by degrees of freedom living on its boundary. The generalized covariant formulation of the principle considers the entropy content on truncated light-sheets: light-like hypersurfaces of non-positive expansion orthogonally generated from a boundary. When we construct the truncated light-sheets for cosmological observers we find a general expression for the minimum cosmic time interval from the apparent horizon to verify the holographic prescription; this minimum time is related to quantum effects involved in the entropy computation. Finally, we arrive to the uncertainty relation from the Holographic Principle, which suggests a deep connection between general covariance, entropy bounds and quantum mechanics. 
  Brane-World models suggest modification of Newton's law of gravity on the 3-brane at submillimeter scales. The brane-world induced corrections are in higher powers of inverse distance and appear as additional terms with the Newtonian potential. The average inter-particle distance in white dwarf and neutron stars are $10^{-10} cms$ and $10^{-13} cms$ respectively, and therefore, the effect of submillimeter corrections needs to be investigated. We show, by carrying out simple manybody calculations, that the mass and mass-radius relationship of the white dwarf and neutron stars are not effected by submillimeter corrections. However, our analysis shows that the correction terms in the effective theory give rise to force akin to surface tension in normal liquids. 
  Using a nonperturbative Kaluza-Klein (KK) quantum scalar field formalism recently introduced where the fifth dimension is noncompact, we study the dynamics for this field in different epochs of the evolution of the universe. 
  We present a formalism to study the metric perturbations of the Schwarzschild spacetime. The formalism is gauge invariant, and it is also covariant under two-dimensional coordinate transformations that leave the angular coordinates unchanged. The formalism is applied to the typical problem of calculating the gravitational waves produced by material sources moving in the Schwarzschild spacetime. We examine the radiation escaping to future null infinity as well as the radiation crossing the event horizon. The waveforms, the energy radiated, and the angular-momentum radiated can all be expressed in terms of two gauge-invariant scalar functions that satisfy one-dimensional wave equations. The first is the Zerilli-Moncrief function, which satisfies the Zerilli equation, and which represents the even-parity sector of the perturbation. The second is the Cunningham-Price-Moncrief function, which satisfies the Regge-Wheeler equation, and which represents the odd-parity sector of the perturbation. The covariant forms of these wave equations are presented here, complete with covariant source terms that are derived from the stress-energy tensor of the matter responsible for the perturbation. Our presentation of the formalism is concluded with a separate examination of the monopole and dipole components of the metric perturbation. 
  F. Rohrlich has recently published two papers, including the paper under review, advocating a particular delay-differential equation as an approximate equation of motion for classical charged particles, which he characterizes as providing a "fully acceptable classical electrodynamics". This Comment notes some mathematical and physical problems with this equation. It points out that most of the claims of these papers are unproved, while some appear to be false as stated. 
  Recent interest in brane world models motivates the investigation of generic first order dilaton gravity actions, with potentials having some non-smoothness. We consider two different types of \delta-like contributions in the action and analyse their effects on the solutions. Furthermore a second source of non-smoothness arises due to the remaining ambiguities in the solutions in the separated smooth patches, after fixing all other constants by matching and asymptotic conditions. If moreover staticity is assumed, we explicitly construct exact solutions.   With the methods described, for example models with point like sources or brane world models (where the second source of non-smoothness becomes crucial), can now be treated as non-smooth dilaton gravity theories. 
  Using the Einstein, Bergmann-Thomson and Landau-Lifshitz's energy-momentum definitions and theirs teleparallel gravity versions, we evaluate the energy and momentum distribution (due to matter and fields including gravitation) of the inhomogeneous cosmological models and show that energy-momentum complexes give same results in both of gravitation theories for a given space-time. Although results obtained by using Einstein and Bergmann-Thomson prescriptions agree with each other, Landau-Lifshitz's definition gives different result. However, in the special limit, we find the same energy distributions from these three prescriptions. 
  Salti and Havare showed that the massless Duffin-Kemmer-Petiau equation and the free-space Maxwell equations are equivalent in the Shuwer. In this study, we consider nine different well-known space-time models and discuss the results obtained in this paper. To find out the exact solution, we solve the second order differential equation which was given in the previous paper. We compute the oscillatory behavior in the region and frequency spectrum of the photon for a given geometry. The counterpart of the Maxwell equations in the general relativistic quantum mechanics, we obtained as the zero-mass limit of the Duffin-Kemmer-Petiau equation with appropriate identification of the components of the Duffin-Kemmer-Petiau spinor with electromagnetic field strength. 
  To solve the problem of exact integration of the field equations or equations of motion of matter in curved spacetimes one can use a class of Riemannian metrics for which the simplest equations of motion can be integrated by the complete separation of variables method. Here, we consider the particular case of the class of Stackel metrics. These metrics admit integration of the Hamilton-Jacobi equation for test particle by the complete separation of variables method. It appears that the other important equations of motion (Klein-Gordon-Fock, Dirac, Weyl) in curved spacetimes can be integrated by complete separation of variables method only for the metrics, belonging to the class of Stackel spaces. 
  The periodic standing wave (PSW) method for the binary inspiral of black holes and neutron stars computes exact numerical solutions for periodic standing wave spacetimes and then extracts approximate solutions of the physical problem, with outgoing waves. The method requires solution of a boundary value problem with a mixed (hyperbolic and elliptic) character.   We present here a new numerical method for such problems, based on three innovations: (i) a coordinate system adapted to the geometry of the problem, (ii) an expansion in multipole moments of these coordinates and a filtering out of higher moments, and (iii) the replacement of the continuum multipole moments with their analogs for a discrete grid. We illustrate the efficiency and accuracy of this method with nonlinear scalar model problems. Finally, we take advantage of the ability of this method to handle highly nonlinear models to demonstrate that the outgoing approximations extracted from the standing wave solutions are highly accurate even in the presence of strong nonlinearities. 
  A conformally flat accelerated charge metric is found in an arbitrary dimension $D$. It is a solution of the Einstein-Maxwell-null fluid with a cosmological constant in $D \ge 4$ dimensions. When the acceleration is zero our solution reduces to the Levi-Civita-Bertotti-Robinson metric. We show that the charge loses its energy, for all dimensions, due to the acceleration. 
  We devise a technique for defining and computing n-point functions in the context of a background-independent gravitational quantum field theory. We construct a tentative implementation of this technique in a perturbatively-finite loop/spinfoam model. 
  This paper is concerned exclusively with axisymmetric spacetimes. We want to develop reductions of Einstein's equations which are suitable for numerical evolutions. We first make a Kaluza-Klein type dimensional reduction followed by an ADM reduction on the Lorentzian 3-space, the (2+1)+1 formalism. We include also the Z4 extension of Einstein's equations adapted to this formalism. Our gauge choice is based on a generalized harmonic gauge condition. We consider vacuum and perfect fluid sources.   We use these ingredients to construct a strongly hyperbolic first-order evolution system and exhibit its characteristic structure. This enables us to construct constraint-preserving stable outer boundary conditions. We use cylindrical polar coordinates and so we provide a careful discussion of the coordinate singularity on axis. By choosing our dependent variables appropriately we are able to produce an evolution system in which each and every term is manifestly regular on axis. 
  We show that the set of states of the Ashtekar-Isham-Lewandowski holonomy algebra defined by elements of the Ashtekar-Lewandowski Hilbert space is dense in the space of all states. We consider weak convergence properties of a modified version of the cut-off procedure currently in use in loop quantum gravity. This version is adapted to vector states rather than to general distributions. 
  Using post-Newtonian equations of motion for fluid bodies that include radiation-reaction terms at 2.5 and 3.5 post-Newtonian (PN) order (O[(v/c)^5] and O[(v/c)^7] beyond Newtonian order), we derive the equations of motion for binary systems with spinning bodies. In particular we determine the effects of radiation-reaction coupled to spin-orbit effects on the two-body equations of motion, and on the evolution of the spins. For a suitable definition of spin, we reproduce the standard equations of motion and spin-precession at the first post-Newtonian order. At 3.5PN order, we determine the spin-orbit induced reaction effects on the orbital motion, but we find that radiation damping has no effect on either the magnitude or the direction of the spins. Using the equations of motion, we find that the loss of total energy and total angular momentum induced by spin-orbit effects precisely balances the radiative flux of those quantities calculated by Kidder et al. The equations of motion may be useful for evolving inspiraling orbits of compact spinning binaries. 
  Most general relativity textbooks devote considerable space to the simplest example of a black hole containing a singularity, the Schwarzschild geometry. However only a few discuss the dynamical process of gravitational collapse, by which black holes and singularities form. We present here two types of analytic models for this process, which we believe are the simplest available; the first involves collapsing spherical shells of light, analyzed mainly in Eddington-Finkelstein coordinates; the second involves collapsing spheres filled with a perfect fluid, analyzed mainly in Painleve-Gullstrand coordinates. Our main goal is pedagogical simplicity and algebraic completeness, but we also present some results that we believe are new, such as the collapse of a light shell in Kruskal-Szekeres coordinates. 
  We review the theory of stationary black hole solutions of vacuum Einstein equations. 
  In the present work, considering the teleparallel gravity versions of the Einstein and Landau-Lifshitz energy-momentum prescriptions, energy and momentum distributions (due to matter and fields including gravitation) of the universe based on the Bianchi-I type metrics are found to be zero and these results are the same as those obtained by Banerjee-Sen, Xulu and Aydogdu-Salt{\i}. Another point is that our study agree with the previous works of Cooperstock-Israelit, Rosen, Johri {\it et al.} Furthermore, the result that the total energy of the universe based Bianchi-I type spacetime is supports the viewpoints of Albrow and Tryon and valid not only in the teleparallel equivalent of general relativity, but also in any teleparallel model 
  Considering the Moller, Weinberg and Qadir-Sharif's definitions in general relativity, we find the momentum four-vector of the closed universe based on the Bianchi-type metrics. The momentum four-vector(due to matter plus fields) is found to be zero. This result supports the viewpoints of Albrow and Tryon and extends the previous works by Cooperstock-Israelit, Rosen, Johri et al., Banerjee-Sen and Vargas who investigated the problem of the energy in Friedmann-Robertson-Walker universe and Salti-Havare who studied the problem of the energy-momentum of the viscous Kasner-type space-times. 
  The Heisenberg picture of the minisuperspace model is considered. The suggested quantization scheme interprets all the observables including the Universe scale factor as the (quasi)Heisenberg operators. The operators arise as a result of the re-quantization of the Heisenberg operators that is required to obtain the hermitian theory. It is shown that the DeWitt constraint H=0 on the physical states of the Universe does not prevent a time-evolution of the (quasi)Heisenberg operators and their mean values. Mean value of an observable, which is singular in a classical theory, is also singular in a quantum case. The (quasi)Heisenberg operator equations are solved in an analytical form in a first order on the interaction constant for the quadratic inflationary potential. Operator solutions are used to evaluate the observables mean values and dispersions. A late stage of the inflation is considered numerically in the framework of the Wigner-Weyl phase-space formalism. It is found that the dispersions of the observables do not vanish at the inflation end. 
  Quantum gravity is made more difficult in part by its constraint structure. The constraints are classically first-class; however, upon quantization they become partially second-class. To study such behavior, we focus on a simple problem with finitely many degrees of freedom and demonstrate how the projection operator formalism for dealing with quantum constraints is well suited to this type of example. 
  We investigate the quasinormal modes for gravitational perturbations of rotating black holes in four dimensional Anti-de Sitter (AdS) spacetime. The study of the quasinormal frequencies related to these modes is relevant to the AdS/CFT correspondence. Although results have been obtained for Schwarzschild and Reissner-Nordstrom AdS black holes, quasinormal frequencies of Kerr-AdS black holes are computed for the first time. We solve the Teukolsky equations in AdS spacetime, providing a second order and a Pade approximation for the angular eigenvalues associated to the Teukolsky angular equation. The transformation theory and the Regge-Wheeler-Zerilli equations for Kerr-AdS are obtained. 
  It is argued that static electromagnetic sources induce Weyl-Majumdar-Papapetrou solutions for the spacetime metric. The acceleration in such fields has a term many orders of magnitude stronger than usual perturbative terms. Two electrostatic and two magnetostatic examples are given. 
  A framework for quantum field theory coupled to three-dimensional quantum gravity is proposed. The coupling with quantum gravity regulates the Feynman diagrams. One recovers the usual Feynman amplitudes in the limit as the cosmological constant tends to zero. 
  New exact solutions of the Einstein-Maxwell field equations that describe $pp$-waves are presented. 
  An ensemble of cosmological models based on generalized BF-theory is constructed where the role of vacuum (zero-level) coupling constants is played by topologically invariant rational intersection forms (cosmological-constant matrices) of 4-dimensional plumbed V-cobordisms which are interpreted as Euclidean spacetime regions. For these regions describing topology changes, the rational and integer intersection matrices are calculated. A relation is found between the hierarchy of certain elements of these matrices and the hierarchy of coupling constants of the universal (low-energy) interactions.   PACS numbers: 0420G, 0240, 0460 
  We revisit a theorem, somehow neglected at present, due to E. Beltrami, through which the integration of the geodesic equations of a curved manifold is obtained by means of a method which, even if inspired by Hamilton-Jacobi method, is merely geometric. The application of this theorem to the Schwarzschild and Kerr metrics allows us to obtain in a straightforward and general way the solution of the geodesic equations. This way of dealing with the problem is, in our opinion, well in accordance with the geometric spirit of the Theory of General Relativity. On the contrary, the usually applied methods carry out the integration of the geodesic equations by translating back the geometrical problem into a mechanical one. 
  Captures of compact objects (COs) by massive black holes in galactic nuclei (aka ``extreme-mass-ratio inspirals'') will be an important source for LISA. However, a large fraction of captures will not be individually resolvable, and so will constitute a source of ``confusion noise,'' obscuring other types of sources. Here we estimate the shape and overall magnitude of the spectrum of confusion noise from CO captures. The overall magnitude depends on the capture rates, which are rather uncertain, so we present results for a plausible range of rates. We show that the impact of capture confusion noise on the total LISA noise curve ranges from insignificant to modest, depending on these rates. Capture rates at the high end of estimated ranges would raise LISA's overall (effective) noise level by at most a factor $\sim 2$. While this would somewhat decrease LISA's sensitivity to other classes of sources, it would be a pleasant problem for LISA to have, overall, as it would also imply that detection rates for CO captures were at nearly their maximum possible levels (given LISA's baseline design). 
  We argue that the Einstein-Yang-Mills-Higgs theory presents nontrivial solutions with a NUT charge. These solutions approach asymptotically the Taub-NUT spacetime and generalize the known dyon black hole configurations. The main properties of the solutions and the differences with respect to the asymptotically flat case are discussed. We find that a nonabelian magnetic monopole placed in the field of gravitational dyon necessarily acquires an electric field, while the magnetic charge may take arbitrary values. 
  In this report the theoretical and experimental activities for the development of superconducting microwave cavities for the detection of gravitational waves are presented. 
  James York, in a major extension of Andr\'e Lichnerowicz's work, showed how to construct solutions to the constraint equations of general relativity. The York method consists of choosing a 3-metric on a given manifold; a divergence-free, tracefree (TT) symmetric 2-tensor wrt this metric; and a single number, the trace of the extrinsic curvature. One then obtains a quasi-linear elliptic equation for a scalar function, the Lichnerowicz-York (L-Y) equation. The solution of this equation is used as a conformal factor to transform the data into a set that satisfies the constraints. If the manifold is compact and without boundary, one quantity that emerges is the volume of the physical space. This article reinterprets the L-Y equation as an eigenvalue equation so as to get a set of data with a preset physical volume. One chooses the conformal metric, the TT tensor, and the physical volume, while regarding the trace of the extrinsic curvature as a free parameter. The resulting equation has extremely nice uniqueness and existence properties. A even more radical approach would be to fix the base (conformal) metric, the physical volume, and the trace. One also selects a TT tensor, but one is free to multiply it by a constant(unspecified). One then solves the L-Y equation as an eigenvalue equation for this constant. A third choice would be to fix the TT tensor and and multiply the base metric by a constant. Each of these three formulations has good uniqueness and existence properties. 
  Solutions of the semilinear wave equation are found numerically in three spatial dimensions with no assumed symmetry using distributed adaptive mesh refinement. The threshold of singularity formation is studied for the two cases in which the exponent of the nonlinear term is either $p=5$ or $p=7$. Near the threshold of singularity formation, numerical solutions suggest an approach to self-similarity for the $p=7$ case and an approach to a scale evolving static solution for $p=5$. 
  In this study, using Moller and Tolman prescriptions we calculate energy and momentum densities for the general cylindrically symmetric spacetime metric. We find that results are finite and well defined in these complexes. We also give the results for some cylindrically symmetric spacetime models. 
  Using general relativity analogs of Bergmann-Thomson, Papapetrou, Landau-Lifshitz and Einstein energy and momentum definitions, we find the energy distribution (due to matter plus fields) in the LTB Space-time. The energy distribution is found well defined and the same in all of these energy-momentum complexes. 
  In this paper, a general relativistic wave equation is written to deal with electromagnetic waves in the background of the Shuwer. We obtain the exact form of this equation in a second order form. On the other hand, by using spinor form of the Maxwell equations the propagation problem is reduced to the solution of the second order differential equation of complex combination of the electric and magnetic fields. For these two different approach, we obtain the spinors in terms of field strength tensor. We show that the Maxwell equations are the equivalence with the mDKP equation in the Shuwer. 
  Using the Bergmann-Thomson energy momentum complex and its tele-parallel gravity version, we obtain the energy and momentum of the universe in viscous Kasner-type cosmological models. The energy and momentum components (due to matter plus field) are found to be zero and this agree with a previous work of Rosen and Johri et al. who investigated the problem of the energy in Friedmann-Robertson-Walker universe. The result that the total energy and momentum components of the universe in these models is zero supports the viewpoint of Tryon. Rosen found that the energy of the Friedmann-Robertson-Walker space-time is zero, which agrees with the studies of Tryon. 
  We considered the massless Duffin-Kemmer-Petiau equation for the general rotating space-times, then found its second order form for a given geometry. Using this second order differential equation for two well-known cosmological model, the exact solutions of the massless Duffin-Kemmer-Petiau equation were obtained. On the other hand, by using spinor form of the Maxwell equations the propagation problem is reduced to the solution of the second order differential equation of complex combination of the electric and magnetic fields. For these two different approach we obtain the spinors in terms of field strength tensor. 
  The periastron position advance for geodesic motion in axially symmetric solutions of the Einstein field equations belonging to the Weyl class of vacuum solutions is investigated. Explicit examples corresponding to either static solutions (single Chazy-Curzon, Schwarzschild and a pair of them), or stationary solution (single rotating Chazy-Curzon and Kerr black hole) are discussed. The results are then applied to the case of S2-SgrA$^*$ binary system of which the periastron position advance will be soon measured with a great accuracy. 
  We study the exterior vacuum problem for first and second order stationary and axially symmetric perturbations of static bodies. The boundary conditions and their compatibility for the existence of an asymptotically flat exterior solution are discussed. 
  The theory of gauge-invariant non-spherical metric perturbations of Schwarzschild black hole spacetimes is now well established. Yet, as different notations and conventions have been used throughout the years, the literature on the subject is often confusing and sometimes confused. The purpose of this paper is to review and collect the relevant expressions related to the Regge-Wheeler and Zerilli equations for the odd and even-parity perturbations of a Schwarzschild spacetime. Special attention is paid to the form they assume in the presence of matter-sources and, for the two most popular conventions in the literature, to the asymptotic expressions and gravitational-wave amplitudes. Besides pointing out some inconsistencies in the literature, the expressions collected here could serve as a quick reference for the calculation of the perturbations of Schwarzschild black hole spacetimes driven by generic sources and for those approaches in which gravitational waves are extracted from numerically generated spacetimes. 
  We compute numerically the quasinormal modes of Kerr-Newman black holes in the scalar case, for which the perturbation equations are separable. Then we study different approximations to decouple electromagnetic and gravitational perturbations of the Kerr-Newman metric, computing the corresponding quasinormal modes. Our results suggest that the Teukolsky-like equation derived by Dudley and Finley gives a good approximation to the dynamics of a rotating charged black hole for Q<M/2. Though insufficient to deal with Kerr-Newman based models of elementary particles, the Dudley-Finley equation should be adequate for astrophysical applications. 
  `Einstein-Aether' theory, in which gravity couples to a dynamical, time-like, unit-norm vector field, provides a means for studying Lorentz violation in a generally covariant setting. Demonstrated here is the effect of a redefinition of the metric and vector field in terms of the original fields and two free parameters. The net effect is a change of the coupling constants appearing in the action. Using such a redefinition, one of the coupling constants can be set to zero, simplifying studies of solutions of the theory. 
  We apply the puncture approach to conformal thin-sandwich black-hole initial data. We solve numerically the conformal thin-sandwich puncture (CTSP) equations for a single black hole with non-zero linear momentum. We show that conformally flat solutions for a boosted black hole have the same maximum gravitational radiation content as the corresponding Bowen-York solution in the conformal transverse-traceless decomposition. We find that the physical properties of these data are independent of the free slicing parameter. 
  In this paper we investigate the opportunities offered by the new Earth gravity models from the dedicated CHAMP and, especially, GRACE missions to the project of measuring the general relativistic Lense-Thirring effect with a new Earth's artificial satellite. It turns out that it would be possible to abandon the stringent, and expensive, requirements on the orbital geometry of the originally prosed LARES mission (same semimajor axis a=12270 km of the existing LAGEOS and inclination i=70 deg) by inserting the new spacecraft in a relatively low, and cheaper, orbit (a=7500-8000 km, i\sim 70 deg) and suitably combining its node Omega with those of LAGEOS and LAGEOS II in order to cancel out the first even zonal harmonic coefficients of the multipolar expansion of the terrestrial gravitational potential J_2, J_4 along with their temporal variations. The total systematic error due to the mismodelling in the remaining even zonal harmonics would amount to \sim 1% and would be insensitive to departures of the inclination from the originally proposed value of many degrees. No semisecular long-period perturbations would be introduced because the period of the node, which is also the period of the solar K_1 tidal perturbation, would amount to \sim 10^2 days. Since the coefficient of the node of the new satellite would be smaller than 0.1 for such low altitudes, the impact of the non-gravitational perturbations of it on the proposed combination would be negligible. Then, a particular financial and technological effort for suitably building the satellite in order to minimize the non-conservative accelerations would be unnecessary. 
  Five-dimensional black holes are studied in Lovelock gravity coupled to Hoffmann-Infeld non-linear electrodynamics. It is shown that some of these solutions present a double peak behavior of the temperature as a function of the horizon radius. This feature implies that the evaporation process, though drastic for a period, leads to an eternal black hole remnant. Moreover, the form of the caloric curve corresponds to the existence of a plateau in the evaporation rate, which implies that black holes of intermediate scales turn out to be unstable. The geometrical aspects, such as the absence of conical singularity, the structure of horizons, etc. are also discussed. In particular, solutions that are asymptotically AdS arise for special choices of the parameters, corresponding to charged solutions of five-dimensional Chern-Simons gravity. 
  We show that generalized gravity theories involving the curvature invariants of the Ricci tensor and the Riemann tensor as well as the Ricci scalar are equivalent to multi- scalar-tensor gravities with four derivatives terms. By expanding the action around a vacuum spacetime, the action is reduced to that of the Einstein gravity with four derivative terms, and consequently there appears a massive spin-2 ghost in such generalized gravity theories in addition to a massive spin-0 field. 
  We argue that if the velocity of electromagnetic waves in vacuum, \cem, is different from the limiting velocity for massive particles, \cst, then, a proton moving with velocity $\cem < v < \cst$ would radiate Cherenkov radiation. Because we have seen protons with energy $\sim 10^{19}$ eV coming from a distance of order more than 1 Mpc, we can put a constraint on $\Delta := (\cst - \cem)/\cst$: $\Delta < 10^{-27}$. 
  We briefly review how to compute the mass and angular momenta of rotating, asymptotically anti-de Sitter spacetimes in Einstein-Gauss-Bonnet theory of gravity using superpotentials derived from standard Noether identities. The calculations depend on the asymptotic form of the metrics only and hence take no account of the source of the curvature. When the source of curvature is a black hole, the results can be checked using the first law of black hole thermodynamics. 
  In this work we construct charged thin-shell Lorentzian wormholes in dilaton gravity. The exotic matter required for the construction is localized in the shell and the energy conditions are satisfied outside the shell. The total amount of exotic matter is calculated and its dependence with the parameters of the model is analysed. 
  We show that in order to avoid a breakdown of general covariance at the quantum level the total flux in each outgoing partial wave of a quantum field in a black hole background must be equal to that of a (1+1)-dimensional blackbody at the Hawking temperature. 
  The matching between two 4-dimensional PP-waves is discussed by using Israel's matching conditions. Physical consequences on the dynamics of (cosmic) strings are analyzed. The extension to space-time of arbitrary dimension is discussed and some interesting features related to the brane world scenario, BPS states in gravity and Dirac-like quantization conditions are briefly described. 
  We study the classical and quantum cosmology of a universe in which the matter source is a massive Dirac spinor field and consider cases where such fields are either free or self-interacting. We focus attention on the spatially flat Robertson-Walker cosmology and classify the solutions of the Einstein-Dirac system in the case of zero, negative and positive cosmological constant $\Lambda$. For $\Lambda<0$, these solutions exhibit signature transitions from a Euclidean to a Lorentzian domain. In the case of massless spinor fields it is found that signature changing solutions do not exist when the field is free while in the case of a self-interacting spinor field such solutions may exist. The resulting quantum cosmology and the corresponding Wheeler-DeWitt equation are also studied for both free and self interacting spinor fields and closed form expressions for the wavefunction of the universe are presented. These solutions suggest a quantization rule for the energy. 
  For the inverse linear potential, the SO(1,1) field behaves as phantom for late time and the Big Rip will occur. The field approaches zero as time approaches the Big Rip, here. For this potential the phantom equation of state takes the late-time minimum $w_\Phi=-3$. We give some discussions that the Big Rip in the SO(1,1) model may be treated as either the transition point of universe from expansion to extract phase or the final state. In the latter picture of the universe, the field has the $T$ symmetry and the scale factor possesses the $CT$ symmetry, for which the SO(1,1) charge $\bar{Q}$ plays a crucial role. 
  A one-to-one correspondence is established between linearized space-time metrics of general relativity and the wave equations of quantum mechanics. Also, the key role of boundary conditions in distinguishing quantum mechanics from classical mechanics will emerge naturally from the procedure. Finally, we will find that the methodology will enable us to introduce not only test charges but also test masses by means of gauges. 
  A detailed simulation of Advanced LIGO test mass optical cavities shows that parametric instabilities will excite acoustic modes in the test masses in the frequency range 28-35 kHz and 64-72 kHz. Using nominal Advanced LIGO optical cavity parameters with fused silica test masses, parametric instability excites 7 acoustic modes in each test mass, with parametric gain R up to 7. For the alternative sapphire test masses only 1 acoustic mode is excited in each test mass with R ~ 2. Fine tuning of the test mass radii of curvature cause the instabilities to sweep through various modes with R as high as ~2000. Sapphire test mass cavities can be tuned to completely eliminate instabilities using thermal g-factor tuning with negligible degradation of the noise performance. In the case of fused silica test mass, instabilities can be minimized but not eliminated. 
  Generalizing earlier results on dust collapse in higher dimensions, we show here that cosmic censorship can be restored in gravitational collapse with tangential pressure present if we take the spacetime dimension to be $N\ge6$. This is under conditions to be motivated physically, such as the smoothness of initial data from which the collapse develops. The models considered here incorporating a non-zero tangential pressure include the Einstein cluster spacetime. 
  We study here the gravitational collapse of a matter cloud with a non-vanishing tangential pressure in the presence of a non-zero cosmological term. Conditions for bounce and singularity formation are derived for the model. It is also shown that when the tangential pressures vanish, the bounce and singularity conditions reduce to that of the dust case studied earlier. The collapsing interior is matched with an exterior which is asymptotically de Sitter or anti de Sitter, depending on the sign of cosmological constant. The junction conditions for matching the cloud to exterior are specified. The effect of the cosmological term on apparent horizons is studied in some detail, and the nature of central singularity is analyzed. We also discuss here the visibility of the singularity and implications for the cosmic censorship conjecture. 
  In this paper we construct the Hamiltonian constraint operator of loop quantum cosmology using holonomies defined for arbitrary irreducible SU(2) representations labeled by spin J. We show that modifications to the effective semi-classical equations of motion arise both in the gravitational part of the constraint as well as matter terms. The modifications are important for phenomenological investigations of the cosmological imprints of loop quantum cosmology. We discuss the implications for the early universe evolution. 
  Using the P\"{o}shl-Teller approximation, we evaluate the neutrino quasinormal modes (QNMs) of a Kerr-Newman-de Sitter black hole. The result shows that for a Kerr-Newman-de Sitter black hole, massless neutrino perturbation of large $\Lambda$, positive $m$ and small value of $n$ will decay slowly. 
  We extend the notion of phantom energy--which is generally accepted for homogeneously distributed matter with $w<-1$ in the universe--on inhomogeneous spherically symmetric spacetime configurations. A spherically symmetric distribution of phantom energy is shown to be able to support the existence of static wormholes. We find an exact solution describing a static spherically symmetric wormhole with phantom energy and show that a spatial distribution of the phantom energy is mainly restricted by the vicinity of the wormhole's throat. The maximal size of the spherical region, surrounding the throat and containing the most part of the phantom energy, depends on the equation-of-state parameter $w$ and cannot exceed some upper limit. 
  Misner initial data are a standard example of time-symmetric initial data with two apparent horizons. Compact formulae describing such data are presented in the cases of equal or non-equal masses (i.e. isometric or non-isometric horizons). The interaction energy in the "Schwarzschild + test particle" limit of the Misner data is analyzed. 
  Modification to the behavior of geometrical density at short scales is a key result of loop quantum cosmology, responsible for an interesting phenomenology in the very early universe. We demonstrate the way matter with arbitrary scale factor dependence in Hamiltonian incorporates this change in its effective dynamics in the loop modified phase. For generic matter, the equation of state starts varying near a critical scale factor, becomes negative below it and violates strong energy condition. This opens a new avenue to generalize various phenomenological applications in loop quantum cosmology. We show that different ways to define energy density may yield radically different results, especially for the case corresponding to classical dust. We also discuss implications for frequency dispersion induced by modification to geometric density at small scales. 
  I discuss the accuracy requirements on numerical relativity calculations of inspiraling compact object binaries whose extracted gravitational waveforms are to be used as templates for matched filtering signal extraction and physical parameter estimation in modern interferometric gravitational wave detectors. Using a post-Newtonian point particle model for the pre-merger phase of the binary inspiral, I calculate the maximum allowable errors for the mass and relative velocity and positions of the binary during numerical simulations of the binary inspiral. These maximum allowable errors are compared to the errors of state-of-the-art numerical simulations of multiple-orbit binary neutron star calculations in full general relativity, and are found to be smaller by several orders of magnitude. A post-Newtonian model for the error of these numerical simulations suggests that adaptive mesh refinement coupled with second order accurate finite difference codes will {\it not} be able to robustly obtain the accuracy required for reliable gravitational wave extraction on Terabyte-scale computers. I conclude that higher order methods (higher order finite difference methods and/or spectral methods) combined with adaptive mesh refinement and/or multipatch technology will be needed for robustly accurate gravitational wave extraction from numerical relativity calculations of binary coalescence scenarios. 
  Newtonian gravity and special relativity combine to produce a gravitomagnetic precession of an orbiting gyroscope that is one fourth as large as predicted by General Relativity. The geodetic effect is the same in both cases. 
  A theory of gravity in higher dimensions is considered. The usual Einstein-Hilbert action is supplemented with Lovelock terms, of higher order in the curvature tensor. These terms are important for the low energy action of string theories.   The intersection of hypersurfaces is studied in the Lovelock theory. The study is restricted to hypersurfaces of co-dimension 1, $(d-1)$-dimensional submanifolds in a $d$-dimensional space-time. It is found that exact thin shells of matter are admissible, with a mild form of curvature singularity: the first derivative of the metric is discontinuous across the surface. Also, with only this mild kind of curvature singularity, there is a possibility of matter localised on the intersections.   This gives a classical analogue of the intersecting brane-worlds in high energy String phenomenology. Such a possibility does not arise in the pure Einstein-Hilbert case. 
  Recently the structure of the Letelier-Gal'tsov spacetime has become a matter of some controversy. I show that the metric proposed in \cite{letgal} is defined only on a dense subset of the whole manifold. In the case when it can be defined on the remainder by continuity, the resulting spacetime corresponds to a system of parallel cosmic strings at rest w.r.t. each other. 
  The main purpose of this article is to provide a guide to theorems on global properties of solutions to the Einstein-Vlasov system. This system couples Einstein's equations to a kinetic matter model. Kinetic theory has been an important field of research during several decades in which the main focus has been on nonrelativistic and special relativistic physics, {\it i.e.} to model the dynamics of neutral gases, plasmas, and Newtonian self-gravitating systems. In 1990, Rendall and Rein initiated a mathematical study of the Einstein-Vlasov system. Since then many theorems on global properties of solutions to this system have been established. The Vlasov equation describes matter phenomenologically and it should be stressed that most of the theorems presented in this article are not presently known for other such matter models ({\it i.e.} fluid models). This paper gives introductions to kinetic theory in non-curved spacetimes and then the Einstein-Vlasov system is introduced. We believe that a good understanding of kinetic theory in non-curved spacetimes is fundamental to good comprehension of kinetic theory in general relativity. 
  We study five dimensional(5D) spherically symmetric self-similar perfect fluid space-time with adiabatic equation of state, considering all the families of future directed non-spacelike geodesics. The space-time admits globally strong curvature naked singularities in the sense of Tipler and thus violates the cosmic censorship conjecture provided a certain algebraic equation has real positive roots. We further show that it is the weak energy condition (WEC) that is necessary for visibility of singularities for a finite period of time and for singularities to be gravitationally strong. We, also, match the solution to 5D Schwarzschild solution using the junction conditions. 
  Much research has been done in the latter years on the subject of Lorentz violation induced by Quantum Gravity effects. On the theoretical side it has been shown that both Loop Quantum Gravity and String Theory predict that Lorentz violation can be induced at an energy near to the Planck scale. On the other hand, most of the experimental results in the latter years, have confirmed that the laws of physics are Lorentz invariant at low energy with very high accuracy.   The inclusion of one- and two-loop contributions from a Lorentz violating Lagrangian dramatically change the above picture: the loop momenta run into the Planck scale and above and from the "divergent" terms finite Lorentz violating contributions of order one arise. These can be suppressed through suitable counterterms in the Lagrangian, originating a strong fine tuning problem.   A brief discussion of these issues and their possible influence in future research follows. 
  The main issue of the present letter is to fix specific features (which turn out being independent of extradimension size) of gravitational waves generated before a dimensional compactification process. Valuable is the possibility to detect our prediction from gravitational wave experiment without high energy laboratory investigation. In particular we show how gravitational waves can bring information on the number of Universe dimensions. Within the framework of Kaluza-Klein hypotheses, a different morphology arises between waves generated before than the compactification process settled down and ordinary 4-dimensional waves. In the former case the scalar and tensor degrees of freedom can not be resolved. As a consequence if were detected gravitational waves having the feature here predicted (anomalous polarization amplitudes), then they would be reliable markers for the existence of an extra dimension. 
  We analyze the directional properties of general gravitational, electromagnetic, and spin-s fields near conformal infinity I. The fields are evaluated in normalized tetrads which are parallelly propagated along null geodesics which approach a point P of I. The standard peeling-off property is recovered and its meaning is discussed and refined. When the (local) character of the conformal infinity is null, such as in asymptotically flat spacetimes, the dominant term which is identified with radiation is unique. However, for spacetimes with a non-vanishing cosmological constant the conformal infinity is spacelike (for Lambda>0) or timelike (for Lambda<0), and the radiative component of each field depends substantially on the null direction along which P is approached. The directional dependence of asymptotic fields near such de Sitter-like or anti-de Sitter-like I is explicitly found and described. We demonstrate that the corresponding directional structure of radiation has a universal character that is determined by the algebraic (Petrov) type of the field. In particular, when Lambda>0 the radiation vanishes only along directions which are opposite to principal null directions. For Lambda<0 the directional dependence is more complicated because it is necessary to distinguish outgoing and ingoing radiation. Near such anti-de Sitter-like conformal infinity the corresponding directional structures differ, depending not only on the number and degeneracy of the principal null directions at P but also on their specific orientation with respect to I. The directional structure of radiation near (anti-)de Sitter-like infinities supplements the standard peeling-off property of spin-s fields. This offers a better understanding of the asymptotic behaviour of the fields near conformal infinity. 
  The directional behavior of dominant components of algebraically special spin-s fields near a spacelike, timelike or null conformal infinity is studied. By extending our previous general investigations we concentrate on fields which admit a pair of equivalent algebraically special null directions, such as the Petrov type D gravitational fields or algebraically general electromagnetic fields. We introduce and discuss a canonical choice of the reference tetrad near infinity in all possible situations, and we present the corresponding asymptotic directional structures using the most natural parametrizations. 
  Motivated by ideas about quantum gravity, a tremendous amount of effort over the past decade has gone into testing Lorentz invariance in various regimes. This review summarizes both the theoretical frameworks for tests of Lorentz invariance and experimental advances that have made new high precision tests possible. The current constraints on Lorentz violating effects from both terrestrial experiments and astrophysical observations are presented. 
  We discuss the propagation of neutrino wave packets in a Lense-Thirring space-time using a gravitational phase approach. We show that the neutrino oscillation length is altered by gravitational corrections and that neutrinos are subject to helicity flip induced by stellar rotation. For the case of a rapidly rotating neutron star, we show that absolute neutrino masses can be derived, in principle, from rotational contributions to the mass-induced energy shift, without recourse to mass generation models presently discussed in the literature. 
  It has been suggested that a possible candidate for the present accelerated expansion of the Universe is ''phantom energy''. The latter possesses an equation of state of the form $\omega\equiv p/\rho<-1$, consequently violating the null energy condition. As this is the fundamental ingredient to sustain traversable wormholes, this cosmic fluid presents us with a natural scenario for the existence of these exotic geometries. Due to the fact of the accelerating Universe, macroscopic wormholes could naturally be grown from the submicroscopic constructions that originally pervaded the quantum foam. One could also imagine an advanced civilization mining the cosmic fluid for phantom energy necessary to construct and sustain a traversable wormhole.   In this context, we investigate the physical properties and characteristics of traversable wormholes constructed using the equation of state $p=\omega \rho$, with $\omega<-1$. We analyze specific wormhole geometries, considering asymptotically flat spacetimes and imposing an isotropic pressure. We also construct a thin shell around the interior wormhole solution, by imposing the phantom energy equation of state on the surface stresses. Using the ''volume integral quantifier'' we verify that it is theoretically possible to construct these geometries with vanishing amounts of averaged null energy condition violating phantom energy. Specific wormhole dimensions and the traversal velocity and time are also deduced from the traversability conditions for a particular wormhole geometry. These phantom energy traversable wormholes have far-reaching physical and cosmological implications. For instance, an advanced civilization may use these geometries to induce closed timelike curves, consequently violating causality. 
  The inspirals of stellar-mass compact objects into supermassive black holes are some of the most important sources for LISA. Detection techniques based on fully coherent matched filtering have been shown to be computationally intractable. We describe an efficient and robust detection method that utilizes the time-frequency evolution of such systems. We show that a typical extreme mass ratio inspiral (EMRI) source could possibly be detected at distances of up to ~2 Gpc, which would mean ~10s of EMRI sources can be detected per year using this technique. We discuss the feasibility of using this method as a first step in a hierarchical search. 
  We have implemented a novel scheme of signal readout for resonant gravitational wave detectors. For the first time, a capacitive resonant transducer has been matched to the signal amplifier by means of a tuned high Q electrical resonator. The resulting 3-mode detection scheme widens significantly the bandwidth of the detector. We present here the results achieved by this signal readout equipped with a two-stage SQUID amplifier. Once installed on the AURIGA detector, the one-sided spectral sensitivity obtained with the detector operated at 4.5 K is better than 10^-20 Hz^-1/2 over 110 Hz and in good agreement with the expectations. 
  The aim of the present work is twofold: first, we show how all the $n$-dimensional Riemannian and Lorentzian metrics can be constructed from a certain class of systems of second-order PDE's which are in duality to the Hamilton-Jacobi equation and second we impose the Einstein equations to these PDE's. 
  In this work we investigate the quantum dynamics of an electric dipole in a $(2+1)$-dimensional conical spacetime. For specific conditions, the Schr\"odinger equation is solved and bound states are found with the energy spectrum and eigenfunctions determined. We find that the bound states spectrum extends from minus infinity to zero with a point of accumulation at zero. This unphysical result is fixed when a finite radius for the defect is introduced. 
  We use Fuchsian methods to show that, for any two dimensional manifold $\Sigma^2$, there is a large family of U(1) symmetric solutions of the vacuum Einstein equations on the manifold $\Sigma \times S^1 \times \mathbb{R}$, each of which has AVTD behavior in the neighborhood of its singularity. 
  Taking quantum physics as well as large scale astronomical observations into account, a spacetime metric is introduced, such that the nonlinear part of the Einstein tensor contains effects of the order of Planck's constant. 
  The necessity of a newly proposed (PRD 70 (2004) 64004) non-Riemannian acoustic spacetime structure called acoustic torsion of sound wave equation in fluids with vorticity are discussed. It is shown that this structure, although not always necessary is present in fluids with vorticity even when the perturbation is rotational. This can be done by solving the Bergliaffa et al (Physica D (2004)) gauge invariant equations for sound, superposed to a general background flow, needs to support a non-Riemannian acoustic geometry in effective spacetime. Bergliaffa et al have previously shown that a Riemannian structure cannot be associated to this gauge invariant general system. 
  We study spherically symmetric black hole solutions with Skyrme hair in the Einstein-Skyrme theory with a negative cosmological constant. The dependence of the skyrmion field configuration on the cosmological constant is examined. The stability is investigated in detail by solving the linearly perturbed equation numerically. It is shown that there exist linearly stable solutions in the branch which represents unstable configuration in the asymptotically flat spacetime. 
  This paper gives a new, simple and concise derivation of brane actions and brane dynamics in general relativity and in Einstein-Gauss-Bonnet gravity. We present a unified treatment, applicable to timelike surface layers and spacelike transition layers, and including consideration of the more difficult lightlike case. 
  In this paper we consider a cosmological Friedmann Robertson Walker brane world imbedded in a 5-dimensional anti-de Sitter Schwarzschild bulk. We show, using potential diagrams, that for an anti-de Sitter bulk the null geodesics never return to the brane. Null geodesics do however return for k=+1 when we include the Schwarzschild like mass and the condition of return is obtained from the corresponding effective potential. We next obtain the condition for a gravitational signal to be emitted from a FRW brane with isotropically distributed matter. We use these results to investigate the conditions under which shortcuts through the bulk are possible. 
  These notes summarize a set of lectures on phenomenological quantum gravity which one of us delivered and the other attended with great diligence. They cover an assortment of topics on the border between theoretical quantum gravity and observational anomalies. Specifically, we review non-linear relativity in its relation to loop quantum gravity and high energy cosmic rays. Although we follow a pedagogic approach we include an open section on unsolved problems, presented as exercises for the student. We also review varying constant models: the Brans-Dicke theory, the Bekenstein varying $\alpha$ model, and several more radical ideas. We show how they make contact with strange high-redshift data, and perhaps other cosmological puzzles. We conclude with a few remaining observational puzzles which have failed to make contact with quantum gravity, but who knows... We would like to thank Mario Novello for organizing an excellent school in Mangaratiba, in direct competition with a very fine beach indeed. 
  We consider dynamics of a quantum scalar field, minimally coupled to classical gravity, in the near-horizon region of a Schwarzschild black-hole. It is described by a static Klein-Gordon operator which in the near-horizon region reduces to a scale invariant Hamiltonian of the system. This Hamiltonian is not essentially self-adjoint, but it admits a one-parameter family of self-adjoint extension. The time-energy uncertainty relation, which can be related to the thermal black-hole mass fluctuations, requires explicit construction of a time operator near-horizon. We present its derivation in terms of generators of the affine group. Matrix elements involving the time operator should be evaluated in the affine coherent state representation. 
  We construct a general relativistic analogy of an infinite solenoid, i.e., of an infinite cylinder with zero electric charge and non-zero electric current in the direction tangential to the cylinder and perpendicular to its axis. We further show that the solution has a good weak-field limit. 
  This paper discusses the Laser Astrometric Test Of Relativity (LATOR) mission. By using a combination of independent time-series of highly accurate gravitational deflection of light in the immediate proximity to the Sun along with measurements of the Shapiro time delay on the interplanetary scales (to a precision respectively better than $10^{-13}$ radians and 1 cm), LATOR will significantly improve our knowledge of relativistic gravity. The primary mission objective is to i) measure the key post-Newtonian Eddington parameter $\gamma$ with accuracy of a part in 10$^9$. $(1-\gamma)$ is a direct measure for presence of a new interaction in gravitational theory, and, in its search, LATOR goes a factor 30,000 beyond the present best result, Cassini's 2003 test. Other mission objectives include: ii) first measurement of gravity's non-linear effects on light to $\sim$0.01% accuracy; including both the traditional Eddington $\beta$ parameter and also the spatial metric's 2nd order potential contribution (never been measured before); iii) direct measurement of the solar quadrupole moment $J_2$ (currently unavailable) to accuracy of a part in 200 of its expected size; iv) direct measurement of the ``frame-dragging'' effect on light by the Sun's rotational gravitomagnetic field to one percent accuracy. LATOR's primary measurement pushes to unprecedented accuracy the search for cosmologically relevant scalar-tensor theories of gravity by looking for a remnant scalar field in today's solar system. The key element of LATOR is a geometric redundancy provided by the laser ranging and long-baseline optical interferometry. We discuss the mission and optical designs of this proposed experiment. 
  In this work we examine the Cauchy convergence of both post-Newtonian (T-approximant) and re-summed post-Newtonian (P-approximant) templates for the case of a test-mass orbiting a Kerr black hole along a circular equatorial orbit. The Cauchy criterion demands that the inner product between the $n$ and $n+1$ order approximation approaches unity, as we increase the order of approximation. In previous works, it has been shown that we achieve greater fitting factors and better parameter estimation using the P-approximant templates for both Schwarzschild and Kerr black holes. In this work, we show that the P-approximant templates also display a faster Cauchy convergence making them a superior template to the standard post-Newtonian templates. 
  We examine backreaction of quantum massive fields on multiply-degenerate (ultraextremal) horizons. It is shown that, under influence of the quantum backreaction, the horizon of such a kind moves to a new position, near which the metric does not change its asymptotics, so the ultraextremal black holes and cosmological spacetimes do exist as self-consistent solutions of the semiclassical field equations. 
  We construct a solution satisfying initial conditions for accelerating cosmologies from string/M-theory. Gowdy symmetric spacetimes with a positive potential are considered. Also, a global existence theorem for the spacetimes is shown. 
  A new way to implement the causality condition on the event horizon of black holes is discovered. The metric of a black hole is shown to be a function of the complex-valued gravitational radius r_g => r_g + i0. The relation between this modification of the metric and the causality condition is established using the analyticity of the S-matrix, which describes scattering of probing particles on a back hole. The found property of the metric has strong manifestations in scattering and related phenomena. One of them is the unexpected effect of reflection of incoming particles on the event horizon, which strongly reduces the absorption cross section. 
  In this work we revisit the growth of small primordial black holes (PBHs) immersed in a quintessential field and/or radiation to the supermassive black hole (SMBHs) scale. We show the difficulties of scenarios in which such huge growth is possible. For that purpose we evaluated analytical solutions of the differential equations (describing mass evolution) and point out the strong fine tuning for that conclusions. The timescale for growth in a model with a constant quintessence flux is calculated and we show that it is much bigger than the Hubble time.The fractional gain of the mass is further evaluated in other forms, including quintessence and/or radiation. We calculate the cosmological density $\Omega$ due to quintessence necessary to grow BHs to the supermassive range and show it to be much bigger than one. We also describe the set of complete equations analyzing the evolution of the BH+quintessence universe, showing some interesting effects such the quenching of the BH mass growth due to the evolution of the background energy. Additional constraints obtained by using the Holographic Bound are also described. The general equilibrium conditions for evaporating/accreting black holes evolving in a quintessence/radiation universe are discussed in the Appendix. 
  Liouville string theory is a natural framework for discussing the non-equilibrium evolution of the Universe. It enables non-critical strings to be treated in mathematically consistent manner, in which target time is identified with a world-sheet renormalization-group scale parameter, preserving target-space general coordinate invariance and the existence of an S-matrix. We review our proposals for a unified treatment of inflation and the current acceleration of the Universe. We link the current acceleration of the Universe with the value of the string coupling. In such a scenario, the dilaton plays an essential background role, driving the acceleration of the Universe during the present era after decoupling as a constant during inflation. 
  Space-time singularities, viz. Big bang, Big crunch and black holes have been shown to follow from the singularity theorems of General relativity. Whether the entropy at such infinite proper-time objects can be other than zero has also been a longstanding subject of research. Currently the property most commonly chosen to calculate their entropy is a multiple of the surface area of the event horizon and usually gives non-zero entropy values. Though popular, this choice still leaves some substantial questions unanswered hence the motivation for alternative methods for entropy derivation. Here, we use a different property, the proper-time at singularities based on the General relativity predicted behavior of clocks, to derive their entropy. We find, firstly within statistical and thermodynamic principles, secondly when this property is taken into account in the Bekenstein-Hawking formula and thirdly illustrating with a natural analogue, that the entropy of black holes and all other gravitational singularities cannot be other than zero as had been earlier classically speculated. 
  In calculations of the inspiral of binary black holes an intermediate approximation is needed that can bridge the post-Newtonian methods of the early inspiral and the numerical relativity computations of the final plunge. We describe here the periodic standing wave approximation: A numerical solution is found to the problem of a periodic rotating binary with helically symmetric standing wave fields, and from this solution an approximation is extracted for the physically relevant problem of inspiral with outgoing waves. The approximation underlying this approach has been recently confirmed with innovative numerical methods applied to nonlinear model problems. 
  Recently, a relativistic gravitation theory has been proposed [J. D. Bekenstein, Phys. Rev. D {\bf 70}, 083509 (2004)] that gives the Modified Newtonian Dynamics (or MOND) in the weak acceleration regime. The theory is based on three dynamic gravitational fields and succeeds in explaining a large part of extragalactic and gravitational lensing phenomenology without invoking dark matter. In this work we consider the strong gravity regime of TeVeS. We study spherically symmetric, static and vacuum spacetimes relevant for a non-rotating black hole or the exterior of a star. Two branches of solutions are identified: in the first the vector field is aligned with the time direction while in the second the vector field has a non-vanishing radial component. We show that in the first branch of solutions the \beta and \gamma PPN coefficients in TeVeS are identical to these of general relativity (GR) while in the second the \beta PPN coefficient differs from unity violating observational determinations of it (for the choice of the free function $F$ of the theory made in Bekenstein's paper). For the first branch of solutions, we derive analytic expressions for the physical metric and discuss their implications. Applying these solutions to the case of black holes, it is shown that they violate causality (since they allow for superluminal propagation of metric, vector and scalar waves) in the vicinity of the event horizon and/or that they are characterized by negative energy density carried by the fields. 
  We numerically calculate the dissipative part of the self-force on a scalar charge moving on a circular, geodesic, equatorial orbit in Kerr spacetime. The solution to the scalar field equation is computed by separating variables and is expressed as a mode sum over radial and angular modes. The force is then computed in two ways: a direct, instantaneous force calculation which uses the half-retarded-minus-half-advanced field, and an indirect method which uses the energy and angular momentum flux at the horizon and at infinity to infer the force. We are able to show numerically and analytically that the force-per-mode is the same for both methods. To enforce the boundary conditions (ingoing radiation at the horizon and outgoing radiation at infinity for the retarded solution) numerical solutions to the radial equation are matched to asymptotic expansions for the fields at the boundaries. Recursion relations for the coefficients in the asymptotic expansions are given in an appendix. 
  We obtain a fourth order accurate numerical algorithm to integrate the Zerilli and Regge-Wheeler wave equations, describing perturbations of nonrotating black holes, with source terms due to an orbiting particle. Those source terms contain the Dirac's delta and its first derivative. We also re-derive the source of the Zerilli and Regge-Wheeler equations for more convenient definitions of the waveforms, that allow direct metric reconstruction (in the Regge-Wheeler gauge). 
  Tensor perturbations in an expanding braneworld of the Randall Sundrum type are investigated. We consider a model composed of a slow-roll inflation phase and the succeeding radiation phase. The effect of the presence of an extra dimension through the transition to the radiation phase is studied, giving an analytic formula for leading order corrections. 
  What happens to the entropy increase principle as the Universe evolve to form the big-crunch singularity? What happens to the uncertainity relations along the process of gravitational collapses? What is the quantum mechanical description of a Radon atom in a rigid box when the distance of consecutive nodes and antinodes of $\psi$ is equal to or less than the diameter of the atom? What is the position-space wave function of two finite volume massive bosons if we take contact interaction into account? How a photon produce electron-positron pair with finite volume concentrate rest masses? What are the charges of the electron-positron pairs forming loops in the vacuum? How two particles with three-momentums $k_1, k_2$ $(k_1 \neq k_2)$ produced to form a loop at a space-time point always arrive at another spacial point simultaneously? What is the microscopic explanation in terms of particle exchanges of the force in the Casimir effect? What is the mechanism of the collapse of the momentum-space wave function of a particle knocking out an elctron from an atom? What is meant by $|\Psi> = {c_1 (t)}|\Psi_{U^{238}}> + {c_2 (t)}|\Psi_{Th^{234}}> ~?$ Quantum mechanically the region between the rigid walls (which is equiprobable in classical mechanics) is non-homogeneous for a particle in a rigid box ! The large scale structure of the Universe is homogeneous. 
  It is shown that among the different classes of claimed static wormhole solutions of the vacuum Brans-Dicke theory only Brans Class I solution with coupling constant $\omega$ less than -1.5 (excluding the point $\omega =2$) gives rise to physically viable traversable wormhole geometry. Usability of this wormhole geometry for interstellar travel has been examined. 
  The time evolution of a wormhole in a Friedmann universe approaching the Big Rip is studied. The wormhole is modeled by a thin spherical shell accreting the superquintessence fluid - two different models are presented. Contrary to recent claims that the wormhole overtakes the expansion of the universe and engulfs it before the Big Rip is reached, it is found that the wormhole becomes asymptotically comoving with the cosmic fluid and the future evolution of the universe is fully causal. 
  Using second-order gauge-invariant perturbation theory, a self-consistent framework describing the non-linear coupling between gravitational waves and a large-scale homogeneous magnetic field is presented. It is shown how this coupling may be used to amplify seed magnetic fields to strengths needed to support the galactic dynamo. In situations where the gravitational wave background is described by an `almost' Friedmann-Lema{\^i}tre-Robertson-Walker (FLRW) cosmology we find that the magnitude of the original magnetic field is amplified by an amount proportional to the magnitude of the gravitational wave induced shear anisotropy and the square of the field's initial co-moving scale. We apply this mechanism to the case where the seed field and gravitational wave background are produced during inflation and find that the magnitude of the gravitational boost depends significantly on the manner in which the estimate of the shear anisotropy at the end of inflation is calculated. Assuming a seed field of $10^{-34}$ $\rm{G}$ spanning a comoving scale of about $10 \rm{kpc}$ today, the shear anisotropy at the end of inflation must be at least as large as $10^{-40}$ in order to obtain a generated magnetic field of the same order of magnitude as the original seed. Moreover, contrasting the weak field approximation to our gauge-invariant approach, we find that while both methods agree in the limit of high conductivity, their corresponding solutions are otherwise only compatible in the limit of infinitely long-wavelength gravitational waves. 
  Ever since Karl Schwarzschild's 1916 discovery of the spacetime geometry describing the interior of a particular idealized general relativistic star -- a static spherically symmetric blob of fluid with position-independent density -- the general relativity community has continued to devote considerable time and energy to understanding the general-relativistic static perfect fluid sphere. Over the last 90 years a tangle of specific perfect fluid spheres has been discovered, with most of these specific examples seemingly independent from each other. To bring some order to this collection, in this article we develop several new transformation theorems that map perfect fluid spheres into perfect fluid spheres. These transformation theorems sometimes lead to unexpected connections between previously known perfect fluid spheres, sometimes lead to new previously unknown perfect fluid spheres, and in general can be used to develop a systematic way of classifying the set of all perfect fluid spheres. 
  We study the condition of black hole formation in five-dimensional space-time. We analytically solve the constraint equations of five-dimensional Einstein equations for momentarily static and conformally flat initial data of a spheroidal mass. We numerically search for an apparent horizon in various initial hypersurfaces and find both necessary and sufficient conditions for the horizon formation in terms of inequalities relating a geometric quantity and a mass defined in an appropriate manner. In the case of infinitely thin spheroid, our results suggest a possibility of naked singularity formation by the spindle gravitational collapse in five-dimensional space-time. 
  The effective field equations on a 3-brane are established considering the massless bosonic sector of the type IIB string compactified on S^5. The covariant embedding formalism in a space endowed with Z_2-symmetry is applied. Recently the derivation of effective equations on the 3-brane, where only gravity penetrates in the bulk has been performed by Shiromizu, Maeda, and Sasaki. We extend this analysis to the situation when the bulk contains a set of fields given by the type IIB string. The notion of the Einstein-Cartan space is considered in order to avoid extra suppositions about the embedding of these fields. The interactions between the brane and the bulk fields are understood in a purely geometric way, which fixes the form of these interactions. Finally, we present the dynamically equivalent effective equations have expressed completely in Riemannian terms and make conclusions. 
  We consider charged rotating black holes in 5-dimensional Einstein-Maxwell theory. These black holes are asymptotically flat, they possess a regular horizon of spherical topology and two independent angular momenta associated with two distinct planes of rotation. We discuss their global and horizon properties, and derive a generalized Smarr formula. We construct these black holes numerically, focussing on black holes with a single angular momentum, and with two equal-magnitude angular momenta. 
  We present an approximate metric for a binary black hole spacetime to construct initial data for numerical relativity. This metric is obtained by asymptotically matching a post-Newtonian metric for a binary system to a perturbed Schwarzschild metric for each hole. In the inner zone near each hole, the metric is given by the Schwarzschild solution plus a quadrupolar perturbation corresponding to an external tidal gravitational field. In the near zone, well outside each black hole but less than a reduced wavelength from the center of mass of the binary, the metric is given by a post-Newtonian expansion including the lowest-order deviations from flat spacetime. When the near zone overlaps each inner zone in a buffer zone, the post-Newtonian and perturbed Schwarzschild metrics can be asymptotically matched to each other. By demanding matching (over a 4-volume in the buffer zone) rather than patching (choosing a particular 2-surface in the buffer zone), we guarantee that the errors are small in all zones. The resulting piecewise metric is made formally $C^\infty$ with smooth transition functions so as to obtain the finite extrinsic curvature of a 3-slice. In addition to the metric and extrinsic curvature, we present explicit results for the lapse and the shift, which can be used as initial data for numerical simulations. This initial data is not accurate all the way to the asymptotically flat ends inside each hole, and therefore must be used with evolution codes which employ black hole excision rather than puncture methods. This paper lays the foundations of a method that can be sraightforwardly iterated to obtain initial data to higher perturbative order. 
  The Brazilian spherical antenna (Schenberg) is planned to detect high frequency gravitational waves (GWs) ranging from 3.0 kHz to 3.4 kHz. There is a host of astrophysical sources capable of being detected by the Brazilian antenna, namely: core collapse in supernova events; (proto)neutron stars undergoing hydrodynamical instability; f-mode unstable neutron stars, caused by quakes and oscillations; excitation of the first quadrupole normal mode of 4-9 solar mass black holes; coalescence of neutron stars and/or black holes; exotic sources such as bosonic or strange matter stars rotating at 1.6 kHz; and inspiralling of mini black hole binaries. We here address our study in particular to the neutron stars, which could well become f-mode unstable producing therefore GWs. We estimate, for this particular source of GWs, the event rates that in principle can be detected by Schenberg and by the Dutch Mini-Grail antenna. 
  Diffeomorphism-induced symmetry transformations and time evolution are distinct operations in generally covariant theories formulated in phase space. Time is not frozen. Diffeomorphism invariants are consequently not necessarily constants of the motion. Time-dependent invariants arise through the choice of an intrinsic time, or equivalently through the imposition of time-dependent gauge fixation conditions. One example of such a time-dependent gauge fixing is the Komar-Bergmann use of Weyl curvature scalars in general relativity. An analogous gauge fixing is also imposed for the relativistic free particle and the resulting complete set time-dependent invariants for this exactly solvable model are displayed. In contrast with the free particle case, we show that gauge invariants that are simultaneously constants of motion cannot exist in general relativity. They vary with intrinsic time. 
  Intrinsic time-dependent invariants are constructed for classical, flat, homogeneous, anisotropic cosmology with a massless scalar material source. Invariance under the time reparameterization-induced canonical symmetry group is displayed explicitly. 
  Employing the Quark Mass Denisity- and temperature- dependent model and the Hartle's method, We have studied the slowly rotating strange star with uniform angular velocity. The mass-radius relation, the moment of inertia and the frame dragging for different frequencies are given. We found that we cannot use the strange star to solve the challenges of Stella and Vietri for the horizontal branch oscillations and the moment of inertia $I_{45}/(M/M_ s)>2.3$. Furthermore, we extended the Hartle's method to study the differential rotating strange star and found that the differential rotation is an effective way to get massive strange star. 
  We present the first calculation of gravitational wave emission produced in the gravitational collapse of uniformly rotating neutron stars to black holes in fully three-dimensional simulations. The initial stellar models are relativistic polytropes which are dynamically unstable and with angular velocities ranging from slow rotation to the mass-shedding limit. An essential aspect of these simulations is the use of progressive mesh-refinement techniques which allow to move the outer boundaries of the computational domain to regions where gravitational radiation attains its asymptotic form. The waveforms have been extracted using a gauge-invariant approach in which the numerical spacetime is matched with the non-spherical perturbations of a Schwarzschild spacetime. Overall, the results indicate that the waveforms have features related to the properties of the initial stellar models (in terms of their w-mode oscillations) and of the newly produced rotating black holes (in terms of their quasi-normal modes). While our waveforms are in good qualitative agreement with those computed by Stark and Piran in two-dimensional simulations, our amplitudes are about one order of magnitude smaller and this difference is mostly likely due to our less severe pressure reduction. For a neutron star rotating uniformly near mass-shedding and collapsing at 10 kpc, the signal-to-noise ratio computed uniquely from the burst is S/N ~ 0.25, but this grows to be S/N <~ 4 in the case of LIGO II. 
  I examine the standard formalism of calculating curvature perturbations in inflation at horizon crossing, and derive a general relation which must be satisfied for the horizon crossing formalism to be valid. This relation is satisfied for the usual cases of power-law and slow roll inflation. I then consider a model for which the relation is strongly violated, and the curvature perturbation evolves rapidly on superhorizon scales. This model has Hubble slow roll parameter $\eta = 3$, but predicts a scale-invariant spectrum of density perturbations. I consider the case of hybrid inflation with large $\eta$, and show that such solutions do not solve the ``$\eta$ problem'' in supergravity. These solutions correspond to field evolution which has not yet relaxed to the inflationary attractor solution, and may make possible new, more natural models on the string landscape. 
  Harmonic coordinate conditions in stationary asymptotically flat spacetimes with matter sources have more than one solution. The solutions depend on the degree of smoothness of the metric and its first derivatives, which we wish to impose across the material boundary, and on the conditions at infinity and at a suitable point inside the matter. This is illustrated in detail by simple fully solvable examples of static spherically symmetric spacetimes in global harmonic coordinates. Examples of stationary electrovacuum spacetimes described simply in harmonic coordinates are also given. They can represent the exterior fields of material discs.   The use of an appropriate background metric considerably simplifies the calculations. 
  We study the absorption of massive spin-half particles by a small Schwarzschild black hole by numerically solving the single-particle Dirac equation in Painleve-Gullstrand coordinates. We calculate the absorption cross section for a range of gravitational couplings Mm/m_P^2 and incident particle energies E. At high couplings, where the Schwarzschild radius R_S is much greater than the wavelength lambda, we find that the cross section approaches the classical result for a point particle. At intermediate couplings we find oscillations around the classical limit whose precise form depends on the particle mass. These oscillations give quantum violations of the equivalence principle. At high energies the cross section converges on the geometric-optics value of 27 \pi R_S^2/4, and at low energies we find agreement with an approximation derived by Unruh. When the hole is much smaller than the particle wavelength we confirm that the minimum possible cross section approaches \pi R_S^2/2. 
  Loop quantum cosmology applies techniques derived for a background independent quantization of general relativity to cosmological situations and draws conclusions for the very early universe. Direct implications for the singularity problem as well as phenomenology in the context of inflation or bouncing universes result, which will be reviewed here. The discussion focuses on recent new results for structure formation and generalizations of the methods. 
  The Pioneer 10 and 11 spacecraft yielded the most precise navigation in deep space to date. However, while at heliocentric distance of $\sim$ 20--70 AU, the accuracies of their orbit reconstructions were limited by a small, anomalous, Doppler frequency drift. This drift can be interpreted as a sunward constant acceleration of $a_P = (8.74 \pm 1.33)\times 10^{-8}$ cm/s$^2$ which is now commonly known as the Pioneer anomaly. Here we discuss the Pioneer anomaly and present the next steps towards understanding of its origin. They are: 1) Analysis of the entire set of existing Pioneer 10 and 11 data, obtained from launch to the last telemetry received from Pioneer 10, on 27 April 2002, when it was at a heliocentric distance of 80 AU. This data could yield critical new information about the anomaly. If the anomaly is confirmed, 2) Development of an instrumental package to be operated on a deep space mission to provide an independent confirmation on the anomaly. If further confirmed, 3) Development of a deep-space experiment to explore the Pioneer anomaly in a dedicated mission with an accuracy for acceleration resolution at the level of $10^{-10}$ cm/s$^2$ in the extremely low frequency range. In Appendices we give a summary of the Pioneer anomaly's characteristics, outline in more detail the steps needed to perform an analysis of the entire Pioneer data set, and also discuss the possibility of extracting some useful information from the Cassini mission cruise data. 
  In three spacetime dimensions, general relativity becomes a topological field theory, whose dynamics can be largely described holographically by a two-dimensional conformal field theory at the ``boundary'' of spacetime. I review what is known about this reduction--mainly within the context of pure (2+1)-dimensional gravity--and discuss its implications for our understanding of the statistical mechanics and quantum mechanics of black holes. 
  The Einstein field equations can be derived in $n$ dimensions ($n>2$) by the variations of the Palatini action. The Killing reduction of 5-dimensional Palatini action is studied on the assumption that pentads and Lorentz connections are preserved by the Killing vector field. A Palatini formalism of 4-dimensional action for gravity coupled to a vector field and a scalar field is obtained, which gives exactly the same fields equations in Kaluza-Klein theory. 
  We consider the Palatini formalism of gravity with cosmological constant $\Lambda$ coupled to a scalar field $\phi$ in $n$-dimensions. The $n$-dimensional Einstein equations with $\Lambda$ can be derived by the variation of the coupled Palatini action provided $n>2$. The Hamiltonian analysis of the coupled action is carried out by a $1+(n-1)$ decomposition of the spacetime. It turns out that both Palatini action and Hilbert action lead to the same geometric dynamics in the presence of $\Lambda$ and $\phi$. While, the $n$-dimensional Palatini action could not give a connection dynamics formalism directly. 
  We consider static, spherically symmetric vacuum solutions to the equations of a theory of gravity with the Lagrangian f(R) where R is the scalar curvature and f is an arbitrary function. Using a well-known conformal transformation, the equations of f(R) theory are reduced to the ``Einstein picture'', i.e., to the equations of general relativity with a source in the form of a scalar field with a potential. We have obtained necessary and sufficient conditions for the existence of solutions admitting conformal continuations. The latter means that a central singularity that exists in the Einstein picture is mapped, in the Jordan picture (i.e., in the manifold corresponding to the original formulation of the theory), to a certain regular sphere, and the solution to the field equations may be smoothly continued beyond it. The value of the curvature R on the transition sphere corresponds to an extremum of the function f(R). Specific examples are considered. 
  We investigate the ultrarelativistic boost of the five-dimensional Emparan-Reall non-rotating black ring. Following the classical method of Aichelburg and Sexl, we determine the gravitational field generated by a black ring moving ``with the speed of light'' in an arbitrary direction. In particular, we study in detail two different boosts along axes orthogonal and parallel to the plane of the ring circle, respectively. In both cases, after the limit one obtains a five-dimensional impulsive pp-wave propagating in Minkowski spacetime. The curvature singularity of the original static spacetime becomes a singular source within the wave front, in the shape of a ring or a rod according to the direction of the boost. In the case of an orthogonal boost, the wave front contains also a remnant of the original disk-shaped membrane as a component of the Ricci tensor (which is everywhere else vanishing). We also analyze the asymptotic properties of the boosted black ring at large spatial distances from the singularity, and its behaviour near the sources. In the limit when the singularity shrinks to a point, one recovers the well known five-dimensional analogue of the Aichelburg-Sexl ``monopole'' solution. 
  We study the propagation of neutrinos in gravitational fields using wave functions that are exact to first order in the metric deviation. For illustrative purposes, the geometrical background is represented by the Lense-Thirring metric. We derive explicit expressions for neutrino deflection, helicity transitions, flavor oscillations and oscillation Hamiltonian. 
  The purpose of this work is to review, clarify, and critically analyse modern mathematical cosmology. The emphasis is upon mathematical objects and structures, rather than numerical computations. This paper concentrates on general relativistic cosmology. The opening section reviews and clarifies the Friedmann-Robertson-Walker models of general relativistic cosmology, while Section 2 deals with the spatially homogeneous models. Particular attention is paid in these opening sections to the topological and geometrical aspects of cosmological models. Section 3 explains how the mathematical formalism can be linked with astronomical observation. In particular, the informal, observational notion of the celestial sphere is given a rigorous mathematical implementation. Part II of this work will concentrate on inflationary cosmology and quantum cosmology. 
  The purpose of the paper, of which this is part II, is to review, clarify, and critically analyse modern mathematical cosmology. The emphasis is upon mathematical objects and structures, rather than numerical computations. Part II provides a critical analysis of inflationary cosmology and quantum cosmology, with particular attention to the claims made that these theories can explain the creation of the universe. 
  The existence of gauge pathologies associated with the Bona-Masso family of generalized harmonic slicing conditions is proven for the case of simple 1+1 relativity. It is shown that these gauge pathologies are true shocks in the sense that the characteristic lines associated with the propagation of the gauge cross, which implies that the name ``gauge shock'' usually given to such pathologies is indeed correct. These gauge shocks are associated with places where the spatial hypersurfaces that determine the foliation of spacetime become non-smooth. 
  We give a Hamiltonian analysis of the asymptotically flat spherically symmetric system of gravity coupled to a scalar field. This 1+1 dimensional field theory may be viewed as the "standard model" for studying black hole physics. Our analysis is adapted to the flat slice Painleve-Gullstrand coordinates. We give a Hamiltonian action principle for this system, which yields an asymptotic mass formula. We then perform a time gauge fixing that gives a Hamiltonian as the integral of a local density. The Hamiltonian takes a relatively simple form compared to earlier work in Schwarzschild gauge, and therefore provides a setting amenable to full quantisation. 
  Using K-Causal relation introduced by Sorkin and Woolgar,we generalize results of Garcia-Parrado and Senovilla on causal maps.We also introduce new concepts like K-future sets,K-reflecting and K-future distinguishing space-times and prove some of their properties.This approach is simpler and more general as compared to the traditional causal approach and it has been used by Penrose et.al in giving a new proof of positivity of mass theorem. 
  We study the one-loop effective action for gravity in a cosmological setup to determine possible cosmological effects of quantum corrections to Einstein theory. By considering the effect of the universal non-local terms in a toy model, we show that they can play an important role in the very early universe. We find that during inflation, the non-local terms are significant, leading to deviations from the standard inflationary expansion. 
  We analyze two types of relativistic simultaneity associated to an observer: the spacelike simultaneity, given by Landau submanifolds, and the lightlike simultaneity (also known as observed simultaneity), given by past-pointing horismos submanifolds. We study some geometrical conditions to ensure that Landau submanifolds are spacelike and we prove that horismos submanifolds are always lightlike. Finally, we establish some conditions to guarantee the existence of foliations in the space-time whose leaves are these submanifolds of simultaneity generated by an observer. 
  A conjecture stated by Raychaudhuri which claims that the only physical perfect fluid non-rotating non-singular cosmological models are comprised in the Ruiz-Senovilla and Fernandez-Jambrina families is shown to be incorrect. An explicit counterexample is provided and the failure of the argument leading to the result is explicitly pointed out. 
  In terms of two-spinors a chiral formulation of general relativity with the Ashtekar Lagrangian and its Hamiltonian formalism in which the basic dynamic variables are the dyad spinors are presented. The extended Witten identities are derived. A new expression of the Hamiltonian boundary term is obtained. Using this expression and the extended Witten identities the proof of the positive energy theorem is extended to a case including momentum and angular momentum. 
  Among the several proposals to solve the incompatibility between the observed small value of the cosmological constant and the huge value obtained by quantum field theories, we can find the idea of a decaying vacuum energy density, leading from high values at early times of universe evolution to the small value observed nowadays. In this paper we consider a variation law for the vacuum density recently proposed by Schutzhold on the basis of quantum field estimations in the curved, expanding background, characterized by a vacuum density proportional to the Hubble parameter. We show that, in the context of an isotropic and homogeneous, spatially flat model, the corresponding solutions retain the well established features of the standard cosmology, and, in addition, are in accordance with the observed cosmological parameters. Our scenario presents an initial phase dominated by radiation, followed by a dust era long enough to permit structure formation, and by an epoch dominated by the cosmological term, which tends asymptotically to a de Sitter universe. Taking the matter density equals to half of the vacuum energy density, as suggested by observation, we obtain a universe age given by Ht = 1.1, and a decelerating parameter equals to -1/2. 
  In the present work we analyze the possibility of detecting some deformed dispersion relations, emerging in some quantum--gravity models, resorting to the so--called Hanbury--Brown--Twiss effect. It will be proved that in some scenarios the possibilities are not pessimistic. Forsooth, for some values of the corresponding parameters the aforementioned effect could render interesting outcomes. 
  In quantum gauge theory of gravity, the gravitational field is represented by gravitational gauge field. The field strength of gravitational gauge field has both gravitational electric component and gravitational magnetic component. In classical level, gauge theory of gravity gives out classical Newtonian gravitational interactions in a relativistic form. Besides, it gives out gravitational Lorentz force which is the gravitational force on a moving object in gravitational magnetic field. The direction of gravitational Lorentz force does not along that of classical gravitational Newtonian force. Effects of gravitational Lorentz force should be detectable, and these effects can be used to discriminate gravitational magnetic field from ordinary electromagnetic magnetic field. 
  We investigate Lorentzian spacetimes where all zeroth and first order curvature invariants vanish and discuss how this class differs from the one where all curvature invariants vanish (VSI). We show that for VSI spacetimes all components of the Riemann tensor and its derivatives up to some fixed order can be made arbitrarily small. We discuss this in more detail by way of examples. 
  Quantum gravity is expected to remove the classical singularity that arises as the end-state of gravitational collapse. To investigate this, we work with a toy model of a collapsing homogeneous scalar field. We show that non-perturbative semi-classical effects of Loop Quantum Gravity cause a bounce and remove the black hole singularity. Furthermore, we find a critical threshold scale, below which no horizon forms -- quantum gravity may exclude very small astrophysical black holes. 
  We consider the amplification of cosmological magnetic fields by gravitational waves as it was recently presented in [gr-qc/0503006]. That study confined to infinitely conductive environments, arguing that on spatially flat Friedmann backgrounds the gravito-magnetic interaction proceeds always as if the universe were a perfect conductor. We explain why this claim is not correct and then re-examine the Maxwell-Weyl coupling at the limit of ideal magnetohydrodynamics. We find that the scales of the main results of [gr-qc/0503006] were not properly assessed and that the incorrect scale assessment has compromised both the physical and the numerical results of the paper. This comment aims to clarify these issues on the one hand, while on the other it takes a closer look at the gauge-invariance and the nonlinearity of [gr-qc/0503006]. 
  For the fields depending on two of the four space-time coordinates only, the spaces of local solutions of various integrable reductions of Einstein's field equations are shown to be the subspaces of the spaces of local solutions of the ``null-curvature'' equations constricted by a requirement of a universal (i.e. solution independent) structures of the canonical Jordan forms of the unknown matrix variables. These spaces of solutions of the ``null-curvature'' equations can be parametrized by a finite sets of free functional parameters -- arbitrary holomorphic (in some local domains) functions of the spectral parameter which can be interpreted as the monodromy data on the spectral plane of the fundamental solutions of associated linear systems. Direct and inverse problems of such mapping (``monodromy transform''), i.e. the problem of finding of the monodromy data for any local solution of the ``null-curvature'' equations with given canonical forms, as well as the existence and uniqueness of such solution for arbitrarily chosen monodromy data are shown to be solvable unambiguously. The linear singular integral equations solving the inverse problems and the explicit forms of the monodromy data corresponding to the spaces of solutions of the symmetry reduced Einstein's field equations are derived. 
  Dimensional regularization is applied to the computation of the gravitational wave field generated by compact binaries at the third post-Newtonian (3PN) approximation. We generalize the wave generation formalism from isolated post-Newtonian matter systems to d spatial dimensions, and apply it to point masses (without spins), modelled by delta-function singularities. We find that the quadrupole moment of point-particle binaries in harmonic coordinates contains a pole when epsilon = d-3 -> 0 at the 3PN order. It is proved that the pole can be renormalized away by means of the same shifts of the particle world-lines as in our recent derivation of the 3PN equations of motion. The resulting renormalized (finite when epsilon -> 0) quadrupole moment leads to unique values for the ambiguity parameters xi, kappa and zeta, which were introduced in previous computations using Hadamard's regularization. Several checks of these values are presented. These results complete the derivation of the gravitational waves emitted by inspiralling compact binaries up to the 3.5PN level of accuracy which is needed for detection and analysis of the signals in the gravitational-wave antennas LIGO/VIRGO and LISA. 
  We study a stochastic formalism for a nonperturbative treatment of the inflaton field in the framework of a noncompact Kaluza-Klein (KK) theory during an inflationary (de Sitter) expansion, without the slow-roll approximation. 
  The possible description of the vacuum of quantum gravity through the so called kappa--Poincare group is analyzed considering some of the consequences of this symmetry in the path integral formulation of nonrelativistic quantum theory. This study is carried out with two cases, firstly, a free particle, and finally, the situation of a particle immersed in a homogeneous gravitational field. It will be shown that the kappa--Poincare group implies the loss of some of the basic properties associated to Feynman's path integral. For instance, loss of the group characteristic related to the time dependence of the evolution operator, or the breakdown of the composition law for amplitudes of events occurring successively in time. Additionally some similarities between the present idea and the so called restricted path integral formalism will be underlined. These analogies advocate the claim that if the kappa--Poincare group contains some of the physical information of the quantum gravity vacuum, then this vacuum could entail decoherence. This last result will also allow us to consider the possibility of analyzing the continuous measurement problem of quantum theory from a group--theoretical point of view, but now taking into account the kappa--Poincare symmetries. 
  Relativistic irreversible thermodynamics is reformulated following the conventional approach proposed by Meixner in the non-relativistic case. Clear separation between mechanical and non-mechanical energy fluxes is made. The resulting equations for the entropy production and the local internal energy have the same structure as the non-relativistic ones. Assuming linear constitutive laws, it is shown that consistency is obtained both with the laws of thermodynamics and causality. 
  We prove, for the relativistic Boltzmann equation on a Bianchi type I space-time, a global existence and uniqueness theorem, for arbitrarily large initial data. 
  We find the group of symmetry transformations generated by interacting fluids in spatially flat Friedmann-Robertson-Walker (FRW) spacetime which links cosmologies with the same scale factor {\it (identity)} or with scale factors $a$ and $a^{-1}$ {\it (duality)}. There exists a duality between contracting and superaccelerated expanding scenarios associated with {\it (phantom)} cosmologies. We investigate the action of this symmetry group on self-interacting minimally(conformally) coupled quintessence and $k$-essence cosmologies. 
  We consider an electric charge, minimally coupled to the Maxwell field, rotating around a Schwarzschild black hole. We investigate how much of the radiation emitted from the swirling charge is absorbed by the black hole and show that most of the photons escape to infinity. For this purpose we use the Gupta-Bleuler quantization of the electromagnetic field in the modified Feynman gauge developed in the context of quantum field theory in Schwarzschild spacetime. We obtain that the two photon polarizations contribute quite differently to the emitted power. In addition, we discuss the accurateness of the results obtained in a full general relativistic approach in comparison with the ones obtained when the electric charge is assumed to be orbiting a massive object due to a Newtonian force. 
  Einstein's equations are derived from a linear theory in flat spacetime using free field gauge invariance and universal coupling. The gravitational potential can be either covariant or contravariant and of almost any density weight. These results are adapted to yield universally coupled massive variants of Einstein's equations, yielding two one-parameter families of distinct theories with spin 2 and spin 0. The Freund-Maheshwari-Schonberg theory is thus not the unique universally coupled massive generalization of Einstein's theory, though it is privileged in some respects. The theories derived are a subset of those found by V. I. Ogievetsky and I. V. Polubarinov by other means. The question of positive energy remains open, but might be addressed by numerical relativists in the context of spherical symmetry. A few remarks are made on the issue of causality with two observable metrics and the possible need for gauge freedom. Some criticisms by Padmanabhan of field derivations of Einstein-like equations are addressed. 
  I reconsider Hawking's analysis of the effects of gravitational collapse on quantum fields, taking into account interactions between the fields. The ultra-high energy vacuum fluctuations, which had been considered to be an awkward peripheral feature of the analysis, are shown to play a key role. By interactions, they can scatter particles to, or create pairs of particle at, ultra-high energies. The energies rapidly become so great that quantum gravity must play a dominant role. Thus the vicinities of black holes are essentially quantum-gravitational regimes. 
  Black holes are extreme manifestations of general relativity, so one might hope that exotic quantum effects would be amplified in their vicinities, perhaps providing clues to quantum gravity. The commonly accepted treatment of quantum corrections to the physics around the holes, however, has provided only limited encouragement of this hope. The predicted corrections have been minor (for macroscopic holes): weak fluxes of low-energy thermal radiation which hardly disturb the classical structures of the holes. Here, I argue that this accepted treatment must be substantially revised. I show that when interactions among fields are taken into account (they were largely neglected in the earlier work) the picture that is drawn is very different. Not only low-energy radiation but also ultra-energetic quanta are produced in the gravitationally collapsing region. The energies of these quanta grow exponentially quickly, so that by the time the hole can be said to have formed, they have passed the Planck scale, at which quantum gravity must become dominant. The vicinities of black holes are windows on quantum gravity. 
  It is shown that the equations of motion of a test point particle with spin in a given gravitational field, so called Mathisson - Papapetrou equations, can be derived from Euler - Lagrange equations of the relativistic pseudomechanics -- relativistic mechanics, which side by side uses the conventional (commuting) and Grassmannian (anticommuting) variables. In this approach the known difficulties of the Mathisson - Papapetrou equations, namely, the problem of the choice of supplementary conditions and the problem of higher derivatives are not appear. 
  We study thick brane world models as Z_2-symmetric domain walls supported by a scalar field with an arbitrary potential V(\phi) in 5D general relativity. Under the global regularity requirement, such configurations (i) have always an AdS asymptotic far from the brane, (ii) are only possible if V(\phi) has an alternating sign and (iii) V(\phi) should satisfy a certain fine-tuning type equality. Thus a thick brane with any admissible V(\phi) is a regularized version of the RS2 brane immersed in the AdS_5 bulk. The thin brane limit is realized in a universal manner by including an arbitrary thick brane model in a one-parameter family, where the parameter "a" is associated with brane thickness; the asymptotic value of V(\phi) (related to \Lambda_5, the effective cosmological constant) remains a-independent. The problem of ordinary matter confinement on the brane is discussed for a test scalar field. Its stress-energy tensor is found to diverge at the AdS horizon for both thin and thick branes, making a serious problem for this class of brane world models. 
  We extend the notion of numerical stability of finite difference approximations to include hyperbolic systems that are first order in time and second order in space, such as those that appear in Numerical Relativity. By analyzing the symbol of the second order system, we obtain necessary and sufficient conditions for stability in a discrete norm containing one-sided difference operators. We prove stability for certain toy models and the linearized Nagy-Ortiz-Reula formulation of Einstein's equations.   We also find that, unlike in the fully first order case, standard discretizations of some well-posed problems lead to unstable schemes and that the Courant limits are not always simply related to the characteristic speeds of the continuum problem. Finally, we propose methods for testing stability for second order in space hyperbolic systems. 
  I argue that the true quantum gravity scale cannot be much larger than the Planck length, because if it were then the quantum gravity-induced fluctuations in $\Lam$ would be insufficient to produce the observed cosmic ``dark energy''. If one accepts this argument, it rules out scenarios of the ``large extra dimensions'' type. I also point out that the relation between the lower and higher dimensional gravitational constants in a Kaluza-Klein theory is precisely what is needed in order that a black hole's entropy admit a consistent higher dimensional interpretation in terms of an underlying spatio-temporal discreteness. 
  The brane world scenario is a new approach to resolve the problem on how to compactify the higher dimensional spacetime to our 4-dimensional world. One of the remarkable features of this scenario is the higher dimensional effects in classical gravitational interactions at short distances. Due to this feature, there are black string solutions in our 4-dimensional world. In this paper, assuming the simplest model of complex minimally coupled scalar field with the local U(1) symmetry, we show a possibility of black-string formation by merging processes of type I long cosmic strings in our 4-dimensional world. No fine tuning for the parameters in the model might be necessary. 
  We derive two-dimensional (2D) solutions of a generic dilaton gravity model coupled with matter, which describe D-dimensional static black holes with pointlike sources. The equality between the mass M of the D-dimensional gravitational solution and the mass m of the source can be preserved also at the level of the 2D gravity model. 
  We compare the respective efficiencies of three quantization methods (group theoretical, coherent state and geometric) by quantizing the dynamics of a free massive particle in two-dimensional de Sitter space. For each case we consider the realization of the principal series representation of $SO_0(1,2) $ group and its two-fold covering SU(1,1). We demonstrate that standard technique for finding an irreducible representation within the geometric quantization scheme fails. For consistency we recall our earlier results concerning the other two methods, make some improvements and generalizations. 
  Generic models of cosmological inflation and the recently proposed scenarios of a recycling universe and the string theory landscape predict spacetimes whose global geometry is a stochastic, self-similar fractal. To visualize the complicated causal structure of such a universe, one usually draws a conformal (Carter-Penrose) diagram. I develop a new method for drawing conformal diagrams, applicable to arbitrary 1+1-dimensional spacetimes. This method is based on a qualitative analysis of intersecting lightrays and thus avoids the need for explicit transformations of the spacetime metric. To demonstrate the power and simplicity of this method, I present derivations of diagrams for spacetimes of varying complication. I then apply the lightray method to three different models of an eternally inflating universe (scalar-field inflation, recycling universe, and string theory landscape) involving the nucleation of nested asymptotically flat, de Sitter and/or anti-de Sitter bubbles. I show that the resulting diagrams contain a characteristic fractal arrangement of lines. 
  Diffeomorphisms not connected to the identity can act nontrivially on the quantum state space for gravity. However, in stark contrast to the case of nonabelian Yang-Mills field theories, for which the quantum state space is always in 1 dimensional representation of the large gauge transformations, the quantum state space for gravity can have higher dimensional representations. In particular, the Kodama state will have 2 dimensional representations, that is sectors with spin 1/2, for many topologies that admit positive scalar curvature. The existence of these spin 1/2 states are used to point out a possible answer to certain criticisms raised recently in the literature. 
  First, for each case to be tested, a specific target inspiral signal is selected for parameter extraction. In a future real analysis, the target signal would be a real signal actually observed by a gravitational wave detector such as LISA. In this study, however, the target signals are themselves simulations. Some cases were selected to resemble sources likely to be detected by LISA when it flies; others were selected to facilitate comparison with previous work using Fisher matrix techniques. Then, for each target inspiral signal, a grid search of the input parameter space is conducted to determine the set of input parameters that produce a simulated inspiral output signal compatible with the target. In this study, we consider four parameters: the two masses, the spin of the larger black hole, and the eccentricity of the orbit. Searching through this four dimensional parameter space requires that hundreds of possible input source parameter combinations be simulated for each target signal analyzed. For each input parameter combination, the detailed time history of the phase of the resulting inspiral is simulated and compared with the phase history of the target signal. The simulation, comparison, and grid search technique used in this study requires more work than the Fisher matrix technique used in most previous studies of this topic. However, this method yields a detailed map of the acceptable region of input parameter space, in contrast to the multidimensional ellipsoids of the Fisher matrix method. Nevertheless, the final results are in general agreement with those obtained previously by the Fisher matrix method, providing a partly independent confirmation of both results. 
  We find numerical solutions of Einstein equations and scalar field equation for a global defect in higher dimensional spacetimes ($\geq 6$). We examine in detail the relation among the expansion rate $H$ and the symmetry-breaking scale $\eta$ and the number of extra dimensions $n$ for these solutions. We find that even if the extra dimensions do not have a cigar geometry, the expansion rate $H$ grows as $\eta$ increases, which is opposite to what is needed for the recently proposed mechanism for solving the cosmological constant problem. We also find that the expansion rate $H$ decreases as $n$ increases. 
  In isotropic loop quantum cosmology, non-perturbatively modified dynamics of a minimally coupled scalar field violates weak, strong and dominant energy conditions when they are stated in terms of equation of state parameter. The violation of strong energy condition helps to have non-singular evolution by evading singularity theorems thus leading to a generic inflationary phase. However, the violation of weak and dominant energy conditions raises concern, as in general relativity these conditions ensure causality of the system and stability of vacuum via Hawking-Ellis conservation theorem. It is shown here that the non-perturbatively modified kinetic term contributes negative pressure but positive energy density. This crucial feature leads to violation of energy conditions but ensures positivity of energy density, as scalar matter Hamiltonian remains bounded from below. It is also shown that the modified dynamics restricts group velocity for inhomogeneous modes to remain sub-luminal thus ensuring causal propagation across spatial distances. 
  We treat the problem of a Michelson interferometer in the field of a plane gravitational wave in the framework of general relativity. The arms of the interferometer are regarded as the world lines of the light beams, whose motion is determined by the Hamilton-Jacobi equation for a massless particle. In the case of a weak monochromatic wave we find that the formula for the delay of a light beam agrees with the result obtained by solving the linearized coupled Einstein-Maxwell equations. We also calculate this delay in the next (quadratic) approximation. 
  We explore further the proposal that general relativity is the hydrodynamic limit of some fundamental theories of the microscopic structure of spacetime and matter, i.e., spacetime described by a differentiable manifold is an emergent entity and the metric or connection forms are collective variables valid only at the low energy, long wavelength limit of such micro-theories. In this view it is more relevant to find ways to deduce the microscopic ingredients of spacetime and matter from their macroscopic attributes than to find ways to quantize general relativity because it would only give us the equivalent of phonon physics, not the equivalents of atoms or quantum electrodyanmics. It may turn out that spacetime is merely a representation of collective state of matter in some limiting regime of interactions, which is the view expressed by Sakharov. In this talk, working within the conceptual framework of geometro-hydrodynamics, we suggest a new way to look at the nature of spacetime inspired by Bose-Einstein Condensate (BEC) physics. We ask the question whether spacetime could be a condensate, even without the knowledge of what the `atom of spacetime' is. We begin with a summary of the main themes for this new interpretation of cosmology and spacetime physics, and the `bottom-up' approach to quantum gravity. We then describe the `Bosenova' experiment of controlled collapse of a BEC and our cosmology-inspired interpretation of its results. We discuss the meaning of a condensate in different context. We explore how far this idea can sustain, its advantages and pitfalls, and its implications on the basic tenets of physics and existing programs of quantum gravity. 
  We describe the possible scenarios for the evolution of a thin spherically symmetric self-gravitating phantom shell around the Schwarzschild black hole. The general equations describing the motion of the shell with a general form of equation of state are derived and analyzed. The different types of space-time R- and T-regions and shell motion are classified depending on the parameters of the problem. It is shown that in the case of a positive shell mass there exist three scenarios for the shell evolution with an infinite motion and two distinctive types of collapse. Analogous scenarios were classified for the case of a negative shell mass. In particular this classification shows that it is impossible for the physical observer to detect the fantom energy flow. We shortly discuss the importance of our results for astrophysical applications. 
  "The last remnant of physical objectivity of space-time" is disclosed, beyond the Leibniz equivalence, in the case of a continuous family of spatially non-compact models of general relativity. The {\it physical individuation} of point-events is furnished by the intrinsic degrees of freedom of the gravitational field, (viz, the {\it Dirac observables}) that represent - as it were - the {\it ontic} part of the metric field. The physical role of the {\it epistemic} part (viz. the {\it gauge} variables) is likewise clarified. At the end, a peculiar four-dimensional {\it holistic and structuralist} view of space-time emerges which includes elements common to the tradition of both {\it substantivalism} and {\it relationism}. The observables of our models undergo real {\it temporal change} and thereby provide a counter-example to the thesis of the {\it frozen-time} picture of evolution.   Invited Contribution to the ESF 2004 Oxford Conference on Space-Time 
  We consider the twin paradox of special relativity in a universe with a compact spatial dimension. Such topology allows two twin observers to remain inertial yet meet periodically. The paradox is resolved by considering the relationship of each twin to a preferred inertial reference frame which exists in such a universe because global Lorentz invariance is broken. The twins can perform "global" experiments to determine their velocities with respect to the preferred reference frame (by sending light signals around the cylinder, for instance). Here we discuss the possibility of doing so with local experiments. Since one spatial dimension is compact, the electrostatic field of a point charge deviates from $1/r^2$. We show that although the functional form of the force law is the same for all inertial observers, as required by local Lorentz invariance, the deviation from 1/r2 is observer-dependent. In particular, the preferred observer measures the largest field strength for fixed distance from the charge. 
  We present long-term-stable and convergent evolutions of head-on black hole collisions and extraction of gravitational waves generated during the merger and subsequent ring-down. The new ingredients in this work are the use of fixed mesh-refinement and dynamical singularity excision techniques. We are able to carry out head-on collisions with large initial separations and demonstrate that our excision infrastructure is capable of accommodating the motion of the individual black holes across the computational domain as well as their their merger. We extract gravitational waves from these simulations using the Zerilli-Moncrief formalism and find the ring-down radiation to be, as expected, dominated by the l=2, m=0 quasi-normal mode. The total radiated energy is about 0.1 % of the total ADM mass of the system. 
  A class of exact solutions of the gravitational field equations in the vacuum on the brane are obtained by assuming the existence of a conformal Killing vector field, with non-static and non-central symmetry. In this case the general solution of the field equations can be obtained in a parametric form in terms of the Bessel functions. The behavior of the basic physical parameters describing the non-local effects generated by the gravitational field of the bulk (dark radiation and dark pressure) is also considered in detail, and the equation of state satisfied at infinity by these quantities is derived. As a physical application of the obtained solutions we consider the behavior of the angular velocity of a test particle moving in a stable circular orbit. The tangential velocity of the particle is a monotonically increasing function of the radial distance and, in the limit of large values of the radial coordinate, tends to a constant value, which is independent on the parameters describing the model. Therefore a brane geometry admitting a one-parameter group of conformal motions may provide an explanation for the dynamics of the neutral hydrogen clouds at large distances from the galactic center, which is usually explained by postulating the existence of the dark matter. 
  It is likely that the holographic principle will be a consequence of the would be theory of quantum gravity. Thus, it is interesting to try to go in the opposite direction: can the holographic principle fix the gravitational interaction? It is shown that the classical gravitational interaction is well inside the set of potentials allowed by the holographic principle. Computations clarify which role such a principle could have in lowering the value of the cosmological constant computed in QFT to the observed one. 
  We propose a mechanism to make gravitational waves (GWs) visible in the electromagnetic domain. Gravitational waves that propagate through a strongly magnetized plasma interact with the plasma through its anisotropic stress-energy tensor and excite magnetohydrodynamic (MHD) wave modes. In catastrophic events such as the merger of a double neutron star binary, a large fraction of the total binding energy of the system is released in the form of GWs observable by LIGO, and the amount of energy transferred to the MHD waves is substantial. These modes, however, are excited at the same frequency as the GW and are not directly observable. In this paper we investigate radiation processes that operate in the presence of the gravitationally excited MHD waves and radiate in the radio regime accessible to LOFAR. We present order of magnitude estimates for the spectral flux of a merger detectable by a LOFAR. 
  In the continuation of a preceding work, we derive a new expression for the metric in the near zone of an isolated matter system in post-Newtonian approximations of general relativity. The post-Newtonian metric, a solution of the field equations in harmonic coordinates, is formally valid up to any order, and is cast in the form of a particular solution of the wave equation, plus a specific homogeneous solution which ensures the asymptotic matching to the multipolar expansion of the gravitational field in the exterior of the system. The new form provides some insights on the structure of the post-Newtonian expansion in general relativity and the gravitational radiation reaction terms therein. 
  We establish a new self-consistent system of equations for the gravitational and electromagnetic fields. The procedure is based on a non-minimal non-linear extension of the standard Einstein-Hilbert-Maxwell action. General properties of a three-parameter family of non-minimal linear models are discussed. In addition, we show explicitly, that a static spherically symmetric charged object can be described by a non-minimal model, second order in the derivatives of the metric, when the susceptibility tensor is proportional to the double-dual Riemann tensor 
  We present a class of curved-spacetime vacuum solutions which develope closed timelike curves at some particular moment. We then use these vacuum solutions to construct a time-machine model. The causality violation occurs inside an empty torus, which constitutes the time-machine core. The matter field surrounding this empty torus satisfies the weak, dominant, and strong energy conditions. The model is regular, asymptotically-flat, and topologically-trivial. Stability remains the main open question. 
  The idea that quantum gravity manifestations would be associated with a violation of Lorentz invariance is very strongly bounded and faces serious theoretical challenges. This leads us to consider an alternative line of thought for such phenomenological search. We discuss the underlying viewpoint and briefly mention its possible connections with current theoretical ideas. We also outline the challenges that the experimental search of the effects would seem to entail. 
  Exploiting a rotating Schwarzschild black hole metric, we study hydrodynamic properties of perfect fluid whirling inward toward the black holes along a conical surface. On the equatorial plane of the rotating Schwarzschild black hole, we derive radial equations of motion with effective potentials and the Euler equation for steady state axisymmetric fuid. Moreover, numerical analysis is performed to figure out effective potentials of particles on the rotating Schwarzschild manifolds in terms of angular velocity, total energy and angular momentum per unit rest mass. Higher dimensional global embeddings are also constructed inside and outside the event horizons of the rotating Schwarzschild black holes. 
  We study the singularity formation of smooth solutions of the relativistic Euler equations in $(3+1)$-dimensional spacetime for both finite initial energy and infinite initial energy. For the finite initial energy case, we prove that any smooth solution, with compactly supported non-trivial initial data, blows up in finite time. For the case of infinite initial energy, we first prove the existence, uniqueness and stability of a smooth solution if the initial data is in the subluminal region away from the vacuum. By further assuming the initial data is a smooth compactly supported perturbation around a non-vacuum constant background, we prove the property of finite propagation speed of such a perturbation. The smooth solution is shown to blow up in finite time provided that the radial component of the initial "generalized" momentum is sufficiently large. 
  Many theories which unify gravity with the other known forces of nature predict the existence of an intermediate-range ``fifth force'' similar to gravity. Such a force could be manifest as a deviation from the gravitational inverse-square law. Currently, at distances near $10^{-1}$m, the inverse-square law is known to be correct to about one part per thousand. I present the design of an experiment that will improve this limit by two orders of magnitude. This is accomplished by constructing a torsion pendulum and source mass apparatus that are particularly insensitive to Newtonian gravity and, simultaneously, maximally sensitive to violations of the same. 
  The description of a point mass in general relativity (GR) is given in the framework of the field formulation of GR where all the dynamical fields, including the gravitational field, are considered in a fixed background spacetime. With the use of stationary (not static) coordinates non-singular at the horizon, the Schwarzschild solution is presented as a point-like field configuration in a whole background Minkowski space. The requirement of a stable $\eta$-causality stated recently in [J.B.Pitts and W.C.Schieve, Found. Phys., v. 34, 211 (2004)] is used essentially as a criterion for testing configurations. 
  In the present article I propose a non-linear relativistic 4-d field model originated by the internal dynamics in CP(N-1). There is no initially distinction between `particle' and `field', and the space-time manifold is derivable. The main idea is to base the theory on the relative amplitudes solely. Quantum measurements will be described in terms of the parallel transport of the local dynamical variables and a specific gauge reduction of the full state vector to the Qubit coherent state. I will discuss here field equations of quantum particle arising in the dynamical space-time. 
  The "measurability" of the non-minimal coupling is discussed by considering the correction to the Newtonian static potential in the semi-classical approach. The coefficient of the "gravitational Darwin term" (GDT) gets redefined by the non-minimal torsion-scalar couplings. Based on a similar analysis of the GDT in the effective field theory approach to non-minimal scalar we conclude that for reasonable values of the couplings the correction is very small. 
  We present an extension to a previous work to study the collapse of a radiating, slow-rotating self-gravitating relativistic configuration. In order to simulate dissipation effects due to the transfer of photons and/or neutrinos within the matter configuration, we introduce the flux factor, the variable Eddington factor and a closure relation between them. Rotation in General Relativity is considered in the slow rotation approximation, i.e. tangential velocity of every fluid element is much less than the speed of light and the centrifugal forces are little compared with the gravitational ones. Solutions are properly matched, up to the first order in the Kerr parameter, to the exterior Kerr-Vaidya metric and the evolution of the physical variables are obtained inside the matter configuration. To illustrate the method we work out three models with different equations of state and several closure relations. We have found that, for the closure relations considered, the matching conditions implies that a total diffusion regime can not be attained at the surface of the configuration. It has also been obtained that the eccentricity at the surface of radiating configurations is greater for models near the diffusion approximation than for those in the free streaming out limit. At least for the static ``seed'' equations of state considered, the simulation we performed show that these models have differential rotation and that the more diffusive the model is, the slower it rotates. 
 GGR News:   Message from the Chair, by Jim Isenberg   Einstein@Home, by Bernard Schutz   We hear that..., by Jorge Pullin   100 Years ago, by Jorge Pullin  Research Briefs:   What's new in LIGO, by David Shoemaker   Frame-dragging in the news in 2004, by Cliff Will   Cosmic (super)strings and LIGO, by Xavier Siemens  Conference reports:   The first gulf coast gravity conference, by Richard Price   Imagining the future, by Shane Larson   VI Mexican School, by Alejandro Corichi 
  Energy transport mechanisms can be generated by imposing relations between null tetrad Ricci components. Several kinds of mass and density transport generated by these relations are studied for the generalized Vaidya system. 
  Relativistic Riemannian superfluid hydrodynamics used in general relativity to investigate superfluids in pulsars is extended to non-Riemannian background spacetime endowed with Cartan torsion. From the Gross-Pitaeviskii (GP) it is shown that in the weak field Cartan torsion approximation, the torsion vector is orthogonal to the superfluid plane wave velocity. Torsion vector is also shown to be aligned along the vortex direction in the superfluid. The background torsion is shown to induce rotation on the fluid as happens with the acoustic torsion in the analogue non-Riemannian non-relativistic superfluid models. The torsion part of the current would be connected to the normal part of the superfluid velocity while the Riemannian part of the velocity would be connected to the superfluid velocity itself. Magnus effect and the rotation of the superfluid are analysed. Since the Kalb-Ramond field is easily associated with torsion our method seems to be equivalent to the vortex-cosmic string relativistic superfluid method developed by Carter and Langlois to investigate rotating neutron stars. 
  The scattering of sound wave perturbations from vortex excitations of Bose-Einstein condensates(BEC) is investigated by numerical integration of the associated Klein-Gordon equation. It is found that, at sufficiently high angular speeds, sound wave-packets can extract a sizeable fraction of the vortex energy through a mechanism of superradiant scattering. It is conjectured that this superradiant regime may be detectable in BEC experiments. 
  Our Universe may be a domain separated by physical phase boundaries from other domain-Universes with different vacuum energy density and matter content. The coexistence of different quantum vacua is perhaps regulated by the exchange of global fermionic charges or by fermion zero modes on the phase boundary. An example would be a static de-Sitter Universe embedded in an asymptotically flat spacetime. 
  The concept of the space-time as emerging in the world phase transition, vs. a priori exiting, is put forward. The theory of gravity with two basic symmetries, the global affine one and the general covariance, is developed. Implications for the Universe are indicated. 
  A partially alternative derivation of the expression for the time dilation effect in a uniform static gravitational field is obtained by means of a thought experiment in which rates of clocks at rest at different heights are compared using as reference a clock bound to a free falling reference system (FFRS). Derivations along these lines have already been proposed, but generally introducing some shortcut in order to make the presentation elementary. The treatment is here exact: the clocks whose rates one wishes to compare are let to describe their world lines (Rindler's hyperbolae) with respect to the FFRS, and the result is obtained by comparing their lengths in space-time. The exercise may nonetheless prove pedagogically instructive insofar as it shows that the some results of General Relativity (GR) can be obtained in terms of physical and geometrical reasoning without having recourse to the general formalism. The corresponding GR metric is derived, to the purpose of making a comparison with solutions of Einstein field equation and with other metrics. For this reasons this paper also compels to deal with a few subtle points inherent in the very foundations of GR. 
  In this paper it is argued that even invoking the dominant energy condition, the Einstein's field equations admit nonglobally hyperbolic, asymptotically de Sitter spacetimes with locally naked singularities and with Cauchy horizons failing to be surfaces of infinite gravitational blueshift; what is more, it is shown the existence of space-time solutions containing Cauchy horizons expected to be stable against small linear nonstationary axisymmetric perturbations in an open set of the physical parameter space where the positive cosmological constant is nearly zero. The price to pay is a violation at ultrahigh densities of Le Chatelier's principle. Given an equation of state, the line of thought introduces black-hole Cauchy horizon instability criteria in terms of the associated adiabatic index, revealing a connection between this subject and the stability theory of relativistic stars. This is used to put forward a tentative, limited formulation of the strong cosmic censorship principle inside black holes (once called dark stars). 
  Quantum non-cloning theorem and a thought experiment are discussed for charged black holes whose global structure exhibits an event and a Cauchy horizon. We take Reissner-Norstr\"{o}m black holes and two-dimensional dilaton black holes as concrete examples. The results show that the quantum non-cloning theorem and the black hole complementarity are far from consistent inside the inner horizon. The relevance of this work to non-local measurements is briefly discussed. 
  We present arguments for the existence of both globally regular and black hole solutions of the Einstein equations with a conformally coupled scalar field, in the presence of a negative cosmological constant, for space-time dimensions greater than or equal to four. These configurations approach asymptotically anti-de Sitter spacetime and are indexed by the central value of the scalar field. We also study the stability of these solutions, and show that, at least for all the solutions studied numerically, they are linearly stable. 
  In the present work the problem of distinguishing between essential and spurious (i.e., absorbable) constants contained in a metric tensor field in a Riemannian geometry is considered. The contribution of the study is the presentation of a sufficient and necessary criterion, in terms of a covariant statement, which enables one to determine whether a constant is essential or not. It turns out that the problem of characterization is reduced to that of solving a system of partial differential equations of the first order. In any case, the metric tensor field is assumed to be smooth with respect to the constant to be tested. It should be stressed that the entire analysis is purely of local character. 
  Wormholes have been advanced as both a method for circumventing the limitations of the speed of light as well as a means for building a time machine (to travel to the past). Thus it is argued that General Relativity may allow both of these possibilities. In this note I argue that traversable wormholes connecting otherwise causally disconnected regions, violate two of the most fundamental principles physics, namely local energy conservation and the energy-time uncertainty principle. 
  For the cylindrically symmetric ''asymptotically flat'' Einstein equations in the case of electro-vacuum it is known that solutions exist globally and also that this class of spacetimes is causally geodesically complete. Hence strong cosmic censorship holds for this class. An interesting question is whether these results can be generalized to include spacetimes with phenomenological matter, e.g. collisionless matter described by the Vlasov equation. Spherically symmetric asymptotically flat solutions of the Einstein-Vlasov system with small initial data are known to be causally geodesically complete. For arbitrary (in size) data it has been shown that if a singularity occurs, the first one occurs at the center of symmetry. In this paper we begin to study the question of global existence for the cylindrically symmetric Einstein-Vlasov system with general (in size) data and we show that if a singularity occurs at all, the first one occurs at the axis of symmetry. 
  A new model of the observed universe, using solutions to the full Einstein equations, is developed from the hypothesis that our observable universe is an underdense bubble, with an internally inhomogeneous fractal bubble distribution of bound matter systems, in a spatially flat bulk universe. It is argued on the basis of primordial inflation and resulting structure formation, that the clocks of the isotropic observers in average galaxies coincide with clocks defined by the true surfaces of matter homogeneity of the bulk universe, rather than the comoving clocks at average spatial positions in the underdense bubble geometry, which are in voids. This understanding requires a systematic reanalysis of all observed quantities in cosmology. I begin such a reanalysis by giving a model of the average geometry of the universe, which depends on two measured parameters: the present matter density parameter, Omega_m, and the Hubble constant, H_0. The observable universe is not accelerating. Nonetheless, inferred luminosity distances are larger than naively expected, in accord with the evidence of distant type Ia supernovae. The predicted age of the universe is 15.3 +/-0.7 Gyr. The expansion age is larger than in competing models, and may account for observed structure formation at large redshifts. 
  We present convergent gravitational waveforms extracted from three-dimensional, numerical simulations in the wave zone and with causally disconnected boundaries. These waveforms last for multiple periods and are very accurate, showing a peak error to peak amplitude ratio of 2% or better. Our approach includes defining the Weyl scalar Psi_4 in terms of a three-plus-one decomposition of the Einstein equations; applying, for the first time, a novel algorithm due to Misner for computing spherical harmonic components of our wave data; and using fixed mesh refinement to focus resolution on non-linear sources while simultaneously resolving the wave zone and maintaining a causally disconnected computational boundary. We apply our techniques to a (linear) Teukolsky wave, and then to an equal mass, head-on collision of two black holes. We argue both for the quality of our results and for the value of these problems as standard test cases for wave extraction techniques. 
  In this work a generic set of boundary conditions for $\mathcal{N}=1$ SUGRA is proposed. This conditions defines that Hamiltonian charges equals Noether ones, including supercharge. 
  We construct the first examples of deformed non-abelian black strings in a 5-dimensional Einstein-Yang-Mills model. Assuming all fields to be independent of the extra coordinate, we construct deformed black strings, which in the 4-dimensional picture correspond to axially symmetric non-abelian black holes in gravity-dilaton theory. These solutions thus have deformed S^2 x R horizon topology. We study fundamental properties of the black strings and find that for all choices of the gravitational coupling two branches of solutions exist. The limiting behaviour of the second branch of solutions however depends strongly on the choice of the gravitational coupling. 
  We discuss how embeddings in connection with the Campbell-Magaard (CM) theorem can have a physical interpretation. We show that any embedding whose local existence is guaranteed by the CM theorem can be viewed as a result of the dynamical evolution of initial data given in a four-dimensional spacelike hypersurface. By using the CM theorem, we establish that for any analytic spacetime, there exist appropriate initial data whose Cauchy development is a five-dimensional vacuum space into which the spacetime is locally embedded. We shall see also that the spacetime embedded is Cauchy stable with respect these the initial data. 
  Dynamical evolution of test fields in background geometry with a naked singularity is an important problem relevant to the Cauchy horizon instability and the observational signatures different from black hole formation. In this paper we study electromagnetic perturbations generated by a given current distribution in collapsing matter under a spherically symmetric self-similar background. Using the Green's function method, we construct the formula to evaluate the outgoing energy flux observed at the future null infinity. The contributions from "quasi-normal" modes of the self-similar system as well as "high-frequency" waves are clarified. We find a characteristic power-law time evolution of the outgoing energy flux which appears just before naked singularity formation, and give the criteria as to whether or not the outgoing energy flux diverges at the future Cauchy horizon. 
  The entropy increases enormously when a star collapses into a black hole. This entropy increase is interpreted as the decrease of the temperature due to the increase of the volume. Through these investigation the microscopic states of the black hole entropy is naturally understood as the statistical states of the enlarged phase space. 
  Scalar particles--i.e., scalar-field excitations--in de Sitter space exhibit behavior unlike either classical particles in expanding space or quantum particles in flat spacetime. Their energies oscillate forever, and their interactions are spread out in energy. Here it is shown that these features characterize not only normal-mode excitations spread out over all space, but localized particles or wave packets as well. Both one-particle and coherent states of a massive, minimally coupled scalar field in de Sitter space, associated with classical wave packets, are constructed explicitly. Their energy expectation values and corresponding Unruh-DeWitt detector response functions are calculated. Numerical evaluation of these quantities for a simple set of classical wave packets clearly displays these novel features. Hence, given the observed accelerating expansion of the Universe, it is possible that observation of an ultralow-mass scalar particle could yield direct confirmation of distinct predictions of quantum field theory in curved spacetime. 
  Last couple of decades have been the golden age for cosmology. High quality data confirmed the broad paradigm of standard cosmology but have thrusted upon us a preposterous composition for the universe which defies any simple explanation, thereby posing probably the greatest challenge theoretical physics has ever faced. Several aspects of these developments are critically reviewed, concentrating on conceptual issues and open questions. [Topics discussed include: Cosmological Paradigm, Growth of structures in the universe, Inflation and generation of initial perturbations, Temperature anisotropies of the CMBR, Dark energy, Cosmological Constant, Deeper issues in cosmology.] 
  We examine the possibility of a constraint-free quantization of linearized gravity, based on the Teukolsky equation for black hole perturbations. We exhibit a simple quadratic (but complex) Lagrangian for the Teukolsky equation, leading to the interpretation that the elementary excitations (gravitons bound to the Kerr black hole) are unstable. 
  We first show that the intrinsic, geometrical structure of a dynamical horizon is unique. A number of physically interesting constraints are then established on the location of trapped and marginally trapped surfaces in the vicinity of any dynamical horizon. These restrictions are used to prove several uniqueness theorems for dynamical horizons. Ramifications of some of these results to numerical simulations of black hole spacetimes are discussed. Finally several expectations on the interplay between isometries and dynamical horizons are shown to be borne out. 
  The main pairs of leader operators of the quantum models of relativistic rotating oscillators in arbitrary dimensions are derived. To this end one exploits the fact that these models generate P\"{o}schl-Teller radial problems with remarkable properties of supersymmetry and shape invariance. 
  We investigate the backreaction of cosmological long wavelength perturbations on the evolution of the Universe. By applying the renormalization group method to a Friedmann-Robertson-Walker universe with long wavelength fluctuations, we demonstrate that the renormalized solution with the backreaction effect is equivalent to that of the separate universe. Then, using the effective Friedmann equation, we show that only non-adiabatic mode of long wavelength fluctuations affects the expansion law of the spatially averaged universe. 
  Present knowledge about the nature of spacetime singularities in the context of classical general relativity is surveyed. The status of the BKL picture of cosmological singularities and its relevance to the cosmic censorship hypothesis are discussed. It is shown how insights on cosmic censorship also arise in connection with the idea of weak null singularities inside black holes. Other topics covered include matter singularities and critical collapse. Remarks are made on possible future directions in research on spacetime singularities. 
  The isolated horizon formalism recently introduced by Ashtekar et al. aims at providing a quasi-local concept of a black hole in equilibrium in an otherwise possibly dynamical spacetime. In this formalism, a hierarchy of geometrical structures is constructed on a null hypersurface. On the other side, the 3+1 formulation of general relativity provides a powerful setting for studying the spacetime dynamics, in particular gravitational radiation from black hole systems. Here we revisit the kinematics and dynamics of null hypersurfaces by making use of some 3+1 slicing of spacetime. In particular, the additional structures induced on null hypersurfaces by the 3+1 slicing permit a natural extension to the full spacetime of geometrical quantities defined on the null hypersurface. This 4-dimensional point of view facilitates the link between the null and spatial geometries. We proceed by reformulating the isolated horizon structure in this framework. We also reformulate previous works, such as Damour's black hole mechanics, and make the link with a previous 3+1 approach of black hole horizon, namely the membrane paradigm. We explicit all geometrical objects in terms of 3+1 quantities, putting a special emphasis on the conformal 3+1 formulation. This is in particular relevant for the initial data problem of black hole spacetimes for numerical relativity. Illustrative examples are provided by considering various slicings of Schwarzschild and Kerr spacetimes. 
  The possibilities that, in the realm of the detection of the so--called deformed dispersion relation, a light source with a continuous distribution of frequencies offers is discussed. It will be proved that the presence of finite coherence length entails the emergence of a new term in the interference pattern. This is a novel trait, which renders a new possibility in the quest for bounds associated with these deformed dispersion relations. 
  Special relativity is generalized to extra dimensions and quantized energy levels of particles are obtained. By calculating the probability of particles' motion in extra dimensions at high temperature of the early universe, it is proposed that the branes may have not existed since the very beginning of the universe, but formed later. Meanwhile, before the formation, particles of the universe may have filled in the whole bulk, not just on the branes. This scenario differs from that in the standard big bang cosmology in which all particles are assumed to be in the 4D spacetime. So, in brane models, whether our universe began from a 4D big bang singularity is questionable. A cosmological constraint on the number of extra dimensions is also given which favors $N\geq 7$. 
  If there exists a formulation of quantum mechanics which does not refer to a background classical spacetime manifold, it then follows as a consequence, (upon making one plausible assumption), that a quantum description of gravity should be necessarily non-linear. This is true independent of the mathematical structure used for describing such a formulation of quantum mechanics. A specific model which exhibits this non-linearity is constructed, using the language of noncommutative geometry. We derive a non-linear Schrodinger equation for the quantum dynamics of a particle; this equation reduces to the standard linear Schrodinger equation when the mass of the particle is much smaller than Planck mass. It turns out that the non-linear equation found by us is very similar to a non-linear Schrodinger equation found by Doebner and Goldin in 1992 from considerations of unitary representaions of the infinite-dimensional group of diffeomorphisms in three spatial dimensions. Our analysis suggests that the diffusion constant introduced by Doebner and Goldin depends on the mass of the particle, and that this constant tends to zero in the limit in which the particle mass is much smaller than Planck mass, so that in this limit the non-linear theory reduces to standard linear quantum mechanics. A similar effective non-linear Schrodinger equation was also found for the quantum dynamics of a system of D0-branes, by Mavromatos and Szabo. 
  We study the stability under linear perturbations of a class of static solutions of Einstein-Gauss-Bonnet gravity in $D=n+2$ dimensions with spatial slices of the form $\Sigma_{\k}^n \times {\mathbb R}^+$, $\Sigma_{\k}^n$ an $n-$manifold of constant curvature $\k$. Linear perturbations for this class of space-times can be generally classified into tensor, vector and scalar types. The analysis in this paper is restricted to tensor perturbations. 
  The M{\o}ller energy complex of Schwarzschild black hole solution in several coodinates are evaluated. Our results show that the M{\o}ller energy complex is independent of not only the purely spatial transformation, but also the shift of time coordinate. So, we could conclude that a shift of time coordinates will not change the energy which is obtained by using the definition of M{\o}ller energy complex. 
  We present numerical results of three-dimensional simulations for the merger of binary neutron stars (BNSs) in full general relativity. Hybrid equations of state (EOSs) are adopted to mimic realistic nuclear EOSs. In this approach, we divide the EOSs into two parts, i.e., the thermal part and the cold part. For the cold part, we assign a fitting formula for realistic EOSs of cold nuclear matter slightly modifying the formula developed by Haensel and Potekhin. We adopt the SLy and FPS EOSs for which the maximum allowed ADM mass of cold and spherical neutron stars (NSs) is ~ 2.04Mo and 1.80Mo, respectively. Simulations are performed for BNSs of the total ADM mass in the range between 2.4Mo and 2.8Mo with the rest-mass ratio Q_M to be in the range 0.9 < Q_M < 1. It is found that if the total ADM mass of the system is larger than a threshold M_{thr}, a black hole (BH) is promptly formed in the merger irrespective of the mass ratios. In the other case, the outcome is a hypermassive NS of a large ellipticity, which results from the large adiabatic index of the realistic EOSs adopted. The value of M_{thr} depends on the EOS: ~ 2.7Mo and ~ 2.5Mo for the SLy and FPS EOSs, respectively. Gravitational waves are computed in terms of a gauge-invariant wave extraction technique. In the formation of the hypermassive NS, quasiperiodic gravitational waves of a large amplitude and of frequency between 3 and 4 kHz are emitted. The estimated emission time scale is < 100 ms, after which the hypermassive NS collapses to a BH. Because of the long emission time, the effective amplitude may be large enough to be detected by advanced laser interferometric gravitational wave detectors if the distance to the source is smaller than ~ 100 Mpc. 
  Generic cosmological models in non-critical string theory have a time-dependent dilaton background at a late epoch. The cosmological deceleration parameter Q_0 is given by the square of the string coupling, g_s^2, up to a negative sign. Hence the expansion of the Universe must accelerate eventually, and the observed value of Q_0 coresponds to g_s^2 ~ 0.6. In this scenario, the string coupling is asymptotically free at large times, but its present rate of change is imperceptibly small. 
  The Laser Interferometer Space Antenna (LISA) is expected to detect gravitational radiation from a large number of compact binary systems. We present a method by which these signals can be identified and have their parameters estimated. Our approach uses Bayesian inference, specifically the application of a Markov chain Monte Carlo method. The simulation study that we present here considers a large number of sinusoidal signals in noise, and our method estimates the number of periodic signals present in the data, the parameters for these signals and the noise level. The method is significantly better than classical spectral techniques at performing these tasks and does not use stopping criteria for estimating the number of signals present. 
  According to the Campbell-Magaard theorem, any analytical spacetime can be locally and analytically embedded into a five-dimensional pseudo-Riemannian Ricci-flat manifold. We find explicitly this embedding for Godel's universe. The embedding space is Ricci-flat and has a non-Lorentzian signature of type (++--). We also show that the embedding found is global. 
  We study spherically symmetric regular and black hole solutions in the Einstein-Skyrme theory with a negative cosmological constant. The Skyrme field configuration depends on the value of the cosmological constant in a similar manner to effectively varying the gravitational constant. We find the maximum value of the cosmological constant above which there exists no solution. The properties of the solutions are discussed in comparison with the asymptotically flat solutions. The stability is investigated in detail by solving the linearly perturbed equation numerically. We show that there exists a critical value of the cosmological constant above which the solution in the branch representing unstable configuration in the asymptotically flat spacetime turns to be linearly stable. 
  Post-Newtonian expansions of the binding energy and gravitational wave flux truncated at the {\it same relative} post-Newtonian order form the basis of the {\it standard adiabatic} approximation to the phasing of gravitational waves from inspiralling compact binaries. Viewed in terms of the dynamics of the binary, the standard approximation is equivalent to neglecting certain conservative post-Newtonian terms in the acceleration. In an earlier work, we had proposed a new {\it complete adiabatic} approximant constructed from the energy and flux functions. At the leading order it employs the 2PN energy function rather than the 0PN one in the standard approximation, so that, effectively the approximation corresponds to the dynamics where there are no missing post-Newtonian terms in the acceleration. In this paper, we compare the overlaps of the standard and complete adiabatic templates with the exact waveform in the adiabatic approximation of a test-mass motion in the Schwarzschild spacetime, for the VIRGO and the Advanced LIGO noise spectra. It is found that the complete adiabatic approximants lead to a remarkable improvement in the {\it effectualness} at lower PN ($<$ 3PN) orders, while standard approximants of order $\geq$ 3PN provide a good lower-bound to the complete approximants for the construction of effectual templates. {\it Faithfulness} of complete approximants is better than that of standard approximants except for a few post-Newtonian orders. Standard and complete approximants beyond the adiabatic approximation are also studied using the Lagrangian templates of Buonanno, Chen and Vallisneri. 
  Singularities in the dark energy universe are discussed, assuming that there is a bulk viscosity in the cosmic fluid. In particular, it is shown how the physically natural assumption of letting the bulk viscosity be proportional to the scalar expansion in a spatially flat FRW universe can drive the fluid into the phantom region (w < -1), even if lies in the quintessence region (w > -1) in the non-viscous case. 
  A set of sufficient conditions for the generalized covariant entropy bound given by Strominger and Thompson is as follows: Suppose that the entropy of matter can be described by an entropy current $s^a$. Let $k^a$ be any null vector along $L$ and $s\equiv -k^a s_a$. Then the generalized bound can be derived from the following conditions: (i) $s'\leq 2\pi T_{ab}k^ak^b$, where $s'=k^a\grad_a s$ and $T_{ab}$ is the stress energy tensor; (ii) on the initial 2-surface $B$, $s(0)\leq -{1/4}\theta(0)$, where $\theta$ is the expansion of $k^a$. We prove that condition (ii) alone can be used to divide a spacetime into two regions: The generalized entropy bound holds for all light sheets residing in the region where $s<-{1/4}\theta$ and fails for those in the region where $s>-{1/4}\theta$. We check the validity of these conditions in FRW flat universe and a scalar field spacetime. Some apparent violations of the entropy bounds in the two spacetimes are discussed. These holographic bounds are important in the formulation of the holographic principle. 
  A theorem providing a characterisation of Schwarzschildean initial data sets on slices with an asymptotically Euclidean end is proved. This characterisation is based on the proportionality of the Weyl tensor and its D'Alambertian that holds for some vacuum Petrov Type D spacetimes (e.g. the Schwarzschild spacetime, the C-metric, but not the Kerr solution). The 3+1 decomposition of this proportionality condition renders necessary conditions for an initial data set to be a Schwarzschildean initial set. These conditions can be written as quadratic expressions of the electric and magnetic parts of the Weyl tensor --and thus, involve only the freely specifiable data. In order to complete our characterisation, a study of which vacuum static Petrov type D spacetimes admit asymptotically Euclidean slices is undertaken. Furthermore, a discussion of the ADM 4-momentum for boost-rotation symmetric spacetimes is given. Finally, a generalisation of our characterisation, valid for Schwarzschildean hyperboloidal initial data sets is put forward. 
  An intriguing question related to black hole thermodynamics is that the entropy of a region shall scale as the area rather than the volume. In this essay we propose that the microscopical degrees of freedom contained in a given region of space, are statistically related in such a way that obey a non-standard statistics, in which case an holographic hypothesis would be not needed. This could provide us with some insight about the nature of degrees of freedom of the geometry and/or the way in which gravitation plays a role in the statistic correlation between the degrees of freedom of a system. 
  We show that the Einstein-aether theory of Jacobson and Mattingly (J&M) can be understood in the framework of the metric-affine (gauge theory of) gravity (MAG). We achieve this by relating the aether vector field of J&M to certain post-Riemannian nonmetricity pieces contained in an independent linear connection of spacetime. Then, for the aether, a corresponding geometrical curvature-square Lagrangian with a massive piece can be formulated straightforwardly. We find an exact spherically symmetric solution of our model. 
  The space of phase-parameters (sky-position, frequency, spindowns) of a coherent matched-filtering search for continuous gravitational waves from isolated neutron stars shows strong global correlations (``circles in the sky''). In the local limit this can be analysed in terms of a parameter-space metric, but the global properties are less well studied. In this work we report on our recent progress in understanding these global correlations analytically for short to intermediate (less than a month, say) observation times and neglecting spindowns. The location of these correlation-circles in parameter-space is found to be determined mostly by the orbital velocity of the earth, while the spin-motion of the detector and the antenna-patterns only contribute significantly to the amplitude of the detection statistic along these circles. 
  We show that a differential variant of the Heisenberg uncertainty relations emerges naturally from induced matter theory, as a sum of line elements in both momentum and Minkowski spaces. 
  It is shown that formally regular solutions in 5D Kaluza-Klein gravity have singularities. This phenomenon is connected with the existence of a minimal length in the nature. The calculation of the derivative of the $G_{55}$ metric component leads to the appearance of the Dirac's $\delta-$function. In this case the Ricci scalar becomes singular since there is a square of this derivative. 
  Because of lunar librations, the retroreflectors left on the moon do not, in general, face directly at the Earth. Usually this is regarded as a disadvantage. It results in a spread of arrival times, because each cube that comprises the retroreflector is at a slightly different distance from the earth. However, we can turn this same effect into an advantage. Using pulses and detectors somewhat faster than those currently used for lunar ranging, we can resolve at least some of the structure of a retroreflector, at least when the libration angles are large. This additional structure in the transfer function means that a unique mm level fit can be obtained with many fewer photons. Fitting to the expected reflector transfer function in general requires fewer photons than straight averaging, and smoothly reduces to averaging in the cases where no structure can be resolved. The gains from resolving the reflectors are largest at large libration angles, exactly the case where averaging is most inefficient. In these cases the number of photons needed can be reduced by an order of magnitude or more. Analysis shows that angles for which the gain is very high happen several times each month, with the details depending on the exact librations. Experimental validation of this technique should be possible with existing SLR stations and mockups of the lunar reflectors. 
  The Kodama State for Lorentzian gravity presupposes a particular value for the Immirzi-parameter, namely $\beta=-i$. However, the derivation of black hole entropy in Loop Quantum Gravity suggests that the Immirzi parameter is a fixed value whose magnitude is on the order of unity but larger than one. Since the Kodama state has de-Sitter spacetime as its classical limit, to get the proper radiation temperature, the Kodama state should be extended to incorporate a more physical value for $\beta$. Thus, we present an extension of the Kodama state for arbitrary values of the Immirzi parameter, $\beta$, that reduces to the ordinary Chern-Simons state for the particular value $\beta=-i$. The state for real values of $\beta$ is free of several of the outstanding problems that cast doubts on the original Kodama state as a ground state for quantum general relativity. We show that for real values of $\beta$, the state is invariant under large gauge transformations, it is CPT invariant (but not CP invariant), and it is expected to be delta-function normalizable with respect to the kinematical inner product. To aid in the construction, we first present a general method for solving the Hamiltonian constraint for imaginary values of $\beta$ that allows one to use the simpler self-dual and anti-self-dual forms of the constraint as an intermediate step. 
  The F-statistic, derived by Jaranowski, Krolak & Schutz (1998), is the optimal (frequentist) statistic for the detection of nearly periodic gravitational waves from known neutron stars, in the presence of stationary, Gaussian detector noise. The F-statistic was originally derived for the case of a single detector, whose noise spectral density was assumed constant in time, and for a single known neutron star. Here we show how the F-statistic can be straightforwardly generalized to the cases of 1) a network of detectors with time-varying noise curves, and 2) a population of known sources. Fortunately, all the important ingredients that go into our generalized F-statistics are already calculated in the single-source/single-detector searches that are currently implemented, e.g., in the LIGO Software Library, so implementation of optimal multi-detector, multi-source searches should require negligible additional cost in computational power or software development. 
  One of the main sources of gravitational waves for the LISA space-borne interferometer are galactic binary systems. The waveforms for these sources are represented by eight parameters, of which four are extrinsic, and four are intrinsic to the system. Geometrically, these signals exist in an 8-d parameter space. By calculating the metric tensor on this space, we calculate the number of templates needed to search for such sources. We show in this study that below a particular monochromatic frequency, we can ignore one of the intrinsic parameters and search over a 7-d space. Beyond this frequency, we have a sudden change in dimensionality of the parameter space from 7 to 8 dimensions, which results in a change in the scaling of the growth of template number as a function of monochromatic frequency. 
  We argue that in the context of string theory, the usual restriction to globally hyperbolic spacetimes should be considerably relaxed. We exhibit an example of a spacetime which only satisfies the causal condition, and so is arbitrarily close to admitting closed causal curves, but which has a well-behaved dual description, free of paradoxes. 
  In Gen. Rel. Grav. (36, 111-126 (2004); in press, gr-qc/0410010) we have proposed a model unifying general relativity and quantum mechanics based on a noncommutative geometry. This geometry was developed in terms of a noncommutative algebra A defined on a transformation groupoid given by the action of a group G on a space E. Owing to the fact that G was assumed to be finite it was possible to compute the model in full details. In the present paper we develop the model in the case when G is a noncompact group. It turns out that also in this case the model works well. The case is important since to obtain physical effects predicted by the model we should assume that G is a Lorentz group or some of its representations. We show that the generalized Einstein equation of the model has the form of the eigenvalue equation for the generalized Ricci operator, and all relevant operators in the quantum sector of the model are random operators; we study their dynamics. We also show that the model correctly reproduces general relativity and the usual quantum mechanics. It is interesting that the latter is recovered by performing the measurement of any observable. In the act of such a measurement the model ``collapses'' to the usual quantum mechanics. 
  We describe progress evolving an important limit of binary orbits in general relativity, that of a stellar mass compact object gradually spiraling into a much larger, massive black hole. These systems are of great interest for gravitational wave observations. We have developed tools to compute for the first time the radiated fluxes of energy and angular momentum, as well as instantaneous snapshot waveforms, for generic geodesic orbits. For special classes of orbits, we compute the orbital evolution and waveforms for the complete inspiral by imposing global conservation of energy and angular momentum. For fully generic orbits, inspirals and waveforms can be obtained by augmenting our approach with a prescription for the self force in the adiabatic limit derived by Mino. The resulting waveforms should be sufficiently accurate to be used in future gravitational-wave searches. 
  The asymptotic scheme of post-Newtonian approximation defined for general relativity (GR) in the harmonic gauge by Futamase & Schutz (1983) is based on a family of initial data for the matter fields of a perfect fluid and for the initial metric, defining a family of weakly self-gravitating systems. We show that Weinberg's (1972) expansion of the metric and his general expansion of the energy-momentum tensor ${\bf T}$, as well as his expanded equations for the gravitational field and his general form of the expanded dynamical equations, apply naturally to this family. Then, following the asymptotic scheme, we derive the explicit form of the expansion of ${\bf T}$ for a perfect fluid, and the expanded fluid-dynamical equations. (These differ from those written by Weinberg.) By integrating these equations in the domain occupied by a body, we obtain a general form of the translational equations of motion for a 1PN perfect-fluid system in GR. To put them into a tractable form, we use an asymptotic framework for the separation parameter $\eta $, by defining a family of well-separated 1PN systems. We calculate all terms in the equations of motion up to the order $\eta ^3$ included. To calculate the 1PN correction part, we assume that the Newtonian motion of each body is a rigid one, and that the family is quasi-spherical, in the sense that in all bodies the inertia tensor comes close to being spherical as $\eta \to 0$. Apart from corrections that cancel for exact spherical symmetry, there is in the final equations of motion one additional term, as compared with the Lorentz-Droste (Einstein-Infeld-Hoffmann) acceleration. This term depends on the spin of the body and on its internal structure. 
  Observations of binary inspirals with LISA will allow us to place bounds on alternative theories of gravity and to study the merger history of massive black holes (MBH). These possibilities rely on LISA's parameter estimation accuracy. We update previous studies of parameter estimation including non-precessional spin effects. We work both in Einstein's theory and in alternative theories of gravity of the scalar-tensor and massive-graviton types. Inclusion of non-precessional spin terms in MBH binaries has little effect on the angular resolution or on distance determination accuracy, but it degrades the estimation of the chirp mass and reduced mass by between one and two orders of magnitude. The bound on the coupling parameter of scalar-tensor gravity is significantly reduced by the presence of spin couplings, while the reduction in the graviton-mass bound is milder. LISA will measure the luminosity distance of MBHs to better than ~10% out to z~4 for a (10^6+10^6) Msun binary, and out to z~2 for a (10^7+10^7) Msun binary. The chirp mass of a MBH binary can always be determined with excellent accuracy. Ignoring spin effects, the reduced mass can be measured within ~1% out to z=10 and beyond for a (10^6+10^6) Msun binary, but only out to z~2 for a (10^7+10^7) Msun binary. Present-day MBH coalescence rate calculations indicate that most detectable events should originate at z~2-6: at these redshifts LISA can be used to measure the two black hole masses and their luminosity distance with sufficient accuracy to probe the merger history of MBHs. If the low-frequency LISA noise can only be trusted down to 10^-4 Hz, parameter estimation for MBHs (and LISA's ability to perform reliable cosmological observations) will be significantly degraded. 
  This is an expanded version of my talk given at the international conference "Zeldovich-90". I start with a brief recollection of interactions with Zeldovich in the context of the study of relic gravitational waves. I then summarise the principles and early results on the quantum-mechanical generation of cosmological perturbations. The expected amplitudes of relic gravitational waves are different in different frequency windows, and therefore the techniques and prospects of their detection are different. One section of the paper describes the present state of efforts in direct detection of relic gravitational waves. Another section is devoted to indirect detection via the anisotropy and polarisation measurements of the cosmic microwave background radiation (CMB). It is emphasized throughout the paper that the conclusions on the existence and expected amount of relic gravitational waves are based on a solid theoretical foundation and the best available cosmological observations. I also explain in great detail what went wrong with the so-called `inflationary gravitational waves', whose amount is predicted by inflationary theorists to be negligibly small, thus depriving them of any observational significance. 
  The occurrence of a spacetime singularity indicates the breakdown of Einstein gravitation theory in these extreme regimes. We consider here the singularity issue and various black hole paradoxes at classical and quantum levels. It is pointed out that a possible resolution to these problems could be arrived at by avoiding the formation of trapped surfaces during a continual gravitational collapse. A class of perfect fluid collapse models is constructed which realizes such a possibility. While the pressure could be negative in the interior of the cloud, the weak energy condition is satisfied. The collapsing star radiates away most of its matter as the process of gravitational collapse evolves, so as to avoid the formation of trapped surfaces and the spacetime singularity. The collapsing interior is matched to an exterior which is a generalized Vaidya spacetime to complete the model. 
  Accurate to the first order in the uniform angular velocity, the general relativity frame dragging effect of the moments of inertia and radii of gyration of two kinds of neutron stars are calculated in a relativistic $\sigma-\omega$ model. The calculation shows that the dragging effect will diminish the moments of inertia and radii of gyration. 
  In Spacetime-Matter theory we assume that the 4D induced matter of the $5D $ Ricci-flat bouncing cosmological solutions contains a perfect fluid as well as an induced scalar field. Then we show that the conventional 4D quintessence and phantom models of dark energy could be recovered from the $5D$ cosmological solutions. By using the phase-plane analysis to study the stability of evolution of the $5D$ models, we find that the conventional 4D late-time attractor solution is also recovered. This attractor solution shows that the scale factors of the phantom dominated universes in both the 4D and $5D$ theories will reach infinity in a finite time and the universes will be ended at a new kind of spacetime singularity at which everything will be annihilated. We also find that the repulsive force of the phantom may provide us with a mechanics to explain the bounce. 
  We show that certain structures defined on the complex four dimensional space known as H-Space have considerable relevance for its closely associated asymptotically flat real physical space-time. More specifically for every complex analytic curve on the H-space there is an asymptotically shear-free null geodesic congruence in the physical space-time. There are specific geometric structures that allow this world-line to be chosen in a unique canonical fashion giving it physical meaning and significance. 
  We investigate the analogue of the Randall-Sundrum brane-world in the case when the bulk contains a black hole. Instead of the static vacuum Minkowski brane of the RS model, we have an Einstein static vacuum brane. We find that the presence of the bulk black hole has a dramatic effect on the gravity that is felt by brane observers. In the RS model, the 5D graviton has a stable localized zero-mode that reproduces 4D gravity on the brane at low energies. With a bulk black hole, there is no such solution -- gravity is delocalized by the 5D horizon. However, the brane does support a discrete spectrum of metastable massive bound states, or quasinormal modes, as was recently shown to be the case in the RS scenario. These states should dominate the high frequency component of the bulk gravity wave spectrum on a cosmological brane. We expect our results to generalize to any bulk spacetime containing a Killing horizon. 
  Based on unified theory of electromagnetic interactions and gravitational interactions, the non-relativistic limit of the equation of motion of a charged Dirac particle in gravitational field is studied. From the Schrodinger equation obtained from this non-relativistic limit, we could see that the classical Newtonian gravitational potential appears as a part of the potential in the Schrodinger equation, which can explain the gravitational phase effects found in COW experiments. And because of this Newtonian gravitational potential, a quantum particle in earth's gravitational field may form a gravitationally bound quantized state, which had already been detected in experiments. Three different kinds of phase effects related to gravitational interactions are discussed in this paper, and these phase effects should be observable in some astrophysical processes. Besides, there exists direct coupling between gravitomagnetic field and quantum spin, radiation caused by this coupling can be used to directly determine the gravitomagnetic field on the surface of a star. 
  We consider the equatorial circular motion of a test particle of specific charge q/m << 1 in the Kerr-Newman geometry of a rotating charged black hole. We find the particle's conserved energy and conserved projection of the angular momentum on the black hole's axis of rotation as corrections, in leading order of q/m, to the corresponding energy and angular momentum of a neutral particle. We determine the centripetal force acting on the test particle and, consequently, we find a classical pseudo-Newtonian potential with which one can mimic this general relativistic problem. 
  LISA (Laser Interferometer Space Antenna) is a proposed space mission, which will use coherent laser beams exchanged between three remote spacecraft to detect and study low-frequency cosmic gravitational radiation. In the low-part of its frequency band, the LISA strain sensitivity will be dominated by the incoherent superposition of hundreds of millions of gravitational wave signals radiated by inspiraling white-dwarf binaries present in our own galaxy. In order to estimate the magnitude of the LISA response to this background, we have simulated a synthesized population that recently appeared in the literature. We find the amplitude of the galactic white-dwarf binary background in the LISA data to be modulated in time, reaching a minimum equal to about twice that of the LISA noise for a period of about two months around the time when the Sun-LISA direction is roughly oriented towards the Autumn equinox. Since the galactic white-dwarfs background will be observed by LISA not as a stationary but rather as a cyclostationary random process with a period of one year, we summarize the theory of cyclostationary random processes and present the corresponding generalized spectral method needed to characterize such process. We find that, by measuring the generalized spectral components of the white-dwarf background, LISA will be able to infer properties of the distribution of the white-dwarfs binary systems present in our Galaxy. 
  We consider a Vaidya-type radiating spacetime in Einstein gravity with the Gauss-Bonnet combination of quadratic curvature terms. Simply generalizing the known static black hole solutions in Einstein-Gauss-Bonnet gravity, we present an exact solution in arbitrary dimensions with the energy-momentum tensor given by a null fluid form. As an application, we derive an evolution equation for the ``dark radiation'' in the Gauss-Bonnet braneworld. 
  We obtain a general spherically symmetric solution of a null dust fluid in $n (\geq 4)$-dimensions in Gauss-Bonnet gravity. This solution is a generalization of the $n$-dimensional Vaidya-(anti)de Sitter solution in general relativity. For $n=4$, the Gauss-Bonnet term in the action does not contribute to the field equations, so that the solution coincides with the Vaidya-(anti)de Sitter solution. Using the solution for $n \ge 5$ with a specific form of the mass function, we present a model for a gravitational collapse in which a null dust fluid radially injects into an initially flat and empty region. It is found that a naked singularity is inevitably formed and its properties are quite different between $n=5$ and $n \ge 6$. In the $n \ge 6$ case, a massless ingoing null naked singularity is formed, while in the $n=5$ case, a massive timelike naked singularity is formed, which does not appear in the general relativistic case. The strength of the naked singularities is weaker than that in the general relativistic case. These naked singularities can be globally naked when the null dust fluid is turned off after a finite time and the field settles into the empty asymptotically flat spacetime. 
  A paradigm describing black hole evaporation in non-perturbative quantum gravity is developed by combining two sets of detailed results: i) resolution of the Schwarzschild singularity using quantum geometry methods; and ii) time-evolution of black holes in the trapping and dynamical horizon frameworks. Quantum geometry effects introduce a major modification in the traditional space-time diagram of black hole evaporation, providing a possible mechanism for recovery of information that is classically lost in the process of black hole formation. The paradigm is developed directly in the Lorentzian regime and necessary conditions for its viability are discussed. If these conditions are met, much of the tension between expectations based on space-time geometry and structure of quantum theory would be resolved. 
  The quantum nature of a black hole is revealed using the simplest terms that one learns in undergraduate and beginning graduate courses. The exposition demonstrates -- vividly -- the importance and power of the quantum oscillator in contemporary research in theoretical physics. 
  According to general relativity, the present analysis shows on geometrical grounds that the cosmological constant problem is an artifact due to the unfounded link of this fundamental constant to vacuum energy density of quantum fluctuations. 
  Recent observations from type Ia Supernovae and from cosmic microwave background (CMB) anisotropies have revealed that most of the matter of the Universe interacts in a repulsive manner, composing the so-called dark energy constituent of the Universe. The analysis of cosmic gravitational waves (GW) represents, besides the CMB temperature and polarization anisotropies, an additional approach in the determination of parameters that may constrain the dark energy models and their consistence. In recent work, a generalized Chaplygin gas model was considered in a flat universe and the corresponding spectrum of gravitational waves was obtained. The present work adds a massless gas component to that model and the new spectrum is compared to the previous one. The Chaplygin gas is also used to simulate a $\Lambda$-CDM model by means of a particular combination of parameters so that the Chaplygin gas and the $\Lambda$-CDM models can be easily distinguished in the theoretical scenarios here established. The lack of direct observational data is partialy solved when the signature of the GW on the CMB spectra is determined. 
  In the context of effective Friedmann equation we classify the cosmologies in multi-scalar models with an arbitrary scalar potential $V$ according to their geometric properties. It is shown that all flat cosmologies are geodesics with respect to a conformally rescaled metric on the `augmented' target space. Non-flat cosmologies with V=0 are also investigated. It is shown that geodesics in a `doubly-augmented' target space yield cosmological trajectories for any curvature $k$ when projected onto a given hypersurface. 
  We study the gravitational collapse of Type I matter field in $N$ dimensional spacetime with radial pressure $p_{r}$ as a function of $r $. We find that for a given smooth initial data set satisfying physical requirements, naked singularities exist for spacetime dimensions N=4 and 5 while for $N \geq 6$ Cosmic Censorship Conjecture withholds its ground. We, also, study the collapse with linear equation of state and find that, similar to dust collapse with appropriate choice of initial data naked singularities occur in all dimensions. 
  The light-like limit of the Kerr gravitational field relative to a distant observer moving rectilinearly in an arbitrary direction is an impulsive plane gravitational wave with a singular point on its wave front. By colliding particles with this wave we show that they have the same focussing properties as high speed particles scattered by the original black hole. By colliding photons with the gravitational wave we show that there is a circular disk, centered on the singular point on the wave front, having the property that photons colliding with the wave within this disk are reflected back and travel with the wave. This result is approximate in the sense that there are observers who can see a dim (as opposed to opaque) circular disk on their sky. By colliding plane electromagnetic waves with the gravitational wave we show that the reflected electromagnetic waves are the high frequency waves. 
  We provide an infinity of spacetimes which contain part of both the Schwarzschild vacuum and de Sitter space. The transition, which occurs below the Schwarzschild event horizon, involves only boundary surfaces (no surface layers). An explicit example is given in which the weak and strong energy conditions are satisfied everywhere (except in the de Sitter section) and the dominant energy condition is violated only in the vicinity of the boundary to the Schwarzschild section. The singularity is avoided by way of a change in topology in accord with a theorem due to Borde.. 
  Basic properties of black holes are explained in terms of trapping horizons. It is shown that matter and information will escape from an evaporating black hole. A general scenario is outlined whereby a black hole evaporates completely without singularity, event horizon or loss of energy or information. 
  The supposed information paradox for black holes is based on the fundamental misunderstanding that black holes are usefully defined by event horizons. Understood in terms of locally defined trapping horizons, the paradox disappears: information will escape from an evaporating black hole. According to classical properties of trapping horizons, a general scenario is outlined whereby a black hole evaporates completely without singularity, event horizon or loss of energy or information. 
  In a recent paper on wormholes (gr-qc/0503097), the author of that paper demonstrated that he didn't know what he was talking about. In this paper I correct the author's naive erroneous misconceptions. 
  The thermal time hypothesis proposed by Rovelli [1] regards the physical basis for the flow of time as thermodynamical and provides a definition of the temperature for some special cases. We verify this hypothesis in the case of de Sitter spacetime by relating the uniformly accelerated observer in de Sitter spacetime to the diamond in Minkowski spacetime. Then, as an application of it, we investigate the thermal effect for the uniformly accelerated observer with a finite lifetime in dS spacetime, which generalizes the corresponding result for the case of Minkowski spacetime [2].   Furthermore, noticing that a uniformly accelerated dS observer with a finite lifetime corresponds to a Rindler observer with a finite lifetime in the embedding Minkowski spacetime, we show that the global-embedding-Minkowski-spacetime (GEMS) picture of spacetime thermodynamics is valid in this case. This is a rather nontrivial and unexpected generalization of the GEMS picture, as well as a further verification of both the thermal time hypothesis and the GEMS picture. 
  For every mapping of a perturbed spacetime onto a background and with any vector field $\xi$ we construct a conserved covariant vector density $I(\xi)$, which is the divergence of a covariant antisymmetric tensor density, a "superpotential". $ I(\xi)$ is linear in the energy-momentum tensor perturbations of matter, which may be large; $I(\xi)$ does not contain the second order derivatives of the perturbed metric. The superpotential is identically zero when perturbations are absent.   By integrating conserved vectors over a part $\Si$ of a hypersurface $S$ of the background, which spans a two-surface $\di\Si$, we obtain integral relations between, on the one hand, initial data of the perturbed metric components and the energy-momentum perturbations on $\Si$ and, on the other hand, the boundary values on $\di\Si$. We show that there are as many such integral relations as there are different mappings, $\xi$'s, $\Si$'s and $\di\Si$'s. For given boundary values on $\di\Si$, the integral relations may be interpreted as integral constraints (e.g., those of Traschen) on local initial data including the energy-momentum perturbations. Conservation laws expressed in terms of Killing fields $\Bar\xi$ of the background become "physical" conservation laws.   In cosmology, to each mapping of the time axis of a Robertson-Walker space on a de Sitter space with the same spatial topology there correspond ten conservation laws. The conformal mapping leads to a straightforward generalization of conservation laws in flat spacetimes. Other mappings are also considered. ... 
  It is computationally expensive to search the large parameter space associated with a gravitational wave signal of uncertain frequency, such as might be expected from the possible pulsar generated by SN1987A. To address this difficulty we have developed a Markov Chain Monte Carlo method that performs a time-domain Bayesian search for a signal over a 4 Hz frequency band and a spindown of magnitude of up to $1\ee{-9}$ Hz/s. We use Monte Carlo simulations to set upper limits on signal amplitude with this technique, which we intend to apply to a gravitational wave search. 
  We apply the recent results in Loop Quantum Cosmology and in the resolution of Black Hole singularity to the gravitational collapse of a star. We study the dynamic of the space time in the interior of the Schwarzschild radius. In particular in our simple model we obtain the evolution of the matter inside the star and of the gravity outside the region where the matter is present. The boundary condition identify an unique time inside and outside the region where the matter is present. We consider a star during the collapse in the particular case in which inside the collapsing star we take null pressure, homogeneity and isotropy. The space-time outside the matter is homogeneous and anisotropic. We show that the space time is singularity free and that we can extend dynamically the space-time beyond the classical singularity. 
  Using developed earlier our methods for multidimensional models \cite{M1,M2,M3} a family of cosmological-type solutions in D-dimensional model with two sets of scalar fields \vec{\phi} and \vec{\psi} and exponential potential depending upon \vec{\phi} is considered. The solutions are defined on a product of n Ricci-flat spaces. The fields from \vec{\phi} have positive kinetic terms and \vec{\psi} are "phantom" fields with negative kinetic terms. For vector coupling constant obeying 0< \vec{\lambda}^2 < (D-1)/(D-2) a subclass of non-singular solutions is singled out. The solutions from this subclass are regular for all values of synchronous "time" \tau \in (- \infty, + \infty). For \vec{\lambda}^2 < 1/(D-2) we get an asymptotically accelerated and isotropic expansion for large values of \tau. 
  The Tolman-Bondi and Vaidya solutions are two solutions to Einstein equations which describe dust particles and null fluid, respectively. We show that it is possible to match the two solutions in one single spacetime, the Tolman-Bondi--Vaidya spacetime. The new spacetime is divided by a null surface with Tolman-Bondi dust on one side and Vaidya fluid on the other side. The differentiability of the spacetime is discussed. By constructing a specific solution, we show that the metric across the null surface can be at least $C^1$ and the stress-energy tensor is continuous. 
  In the search for binary systems inspiral signal in interferometric gravitational waves detectors, one needs the generation and placement of a grid of templates. We present an original technique for the placement in the associated parameter space, that makes use of the variation of size of the isomatch ellipses in order to reduce the number of templates necessary to cover the parameter space. This technique avoids the potentially expensive computation of the metric at every point, at the cost of having a small number of ``holes'' in the coverage, representing a few percent of the surface of the parameter space, where the match is slightly lower than specified. A study of the covering efficiency, as well as a comparison with a very simple regular tiling using a single ellipse is made. Simulations show an improvement varying between 6% and 30% for the computing cost in this comparison. 
  We consider the Cauchy problem for the scalar wave equation in the Kerr geometry for smooth initial data supported outside the event horizon. We prove that the solutions decay in time in L^\infty_loc. The proof is based on a representation of the solution as an infinite sum over the angular momentum modes, each of which is an integral of the energy variable on the real line. This integral representation involves solutions of the radial and angular ODEs which arise in the separation of variables. 
  The scattering process of a dynamic perturbation impinging on a draining-tub model of an acoustic black hole is numerically solved in the time domain. Analogies with real black holes of General Relativity are explored by using recently developed mathematical tools involving finite elements methods, excision techniques, and constrained evolution schemes for strongly hyperbolic systems. In particular it is shown that superradiant scattering of a quasi-monochromatic wavepacket can produce strong amplification of the signal, offering the possibility of a significant extraction of rotational energy at suitable values of the angular frequency of the vortex and of the central frequency of the wavepacket. The results show that theoretical tools recently developed for gravitational waves can be brought to fruition in the study of other problems in which strong anisotropies are present. 
  When testing multiple hypothesis in a survey --e.g. many different source locations, template waveforms, and so on-- the final result consists in a set of confidence intervals, each one at a desired confidence level. But the probability that at least one of these intervals does not cover the true value increases with the number of trials. With a sufficiently large array of confidence intervals, one can be sure that at least one is missing the true value. In particular, the probability of false claim of detection becomes not negligible. In order to compensate for this, one should increase the confidence level, at the price of a reduced detection power. False discovery rate control is a relatively new statistical procedure that bounds the number of mistakes made when performing multiple hypothesis tests. We shall review this method, discussing exercise applications to the field of gravitational wave surveys. 
  This article reports on a project that is the first step the LIGO Scientific Collaboration and the Virgo Collaboration have taken to prepare for the mutual search for inspiral signals. The project involved comparing the analysis pipelines of the two collaborations on data sets prepared by both sides, containing simulated noise and injected events. The ability of the pipelines to detect the injected events was checked, and a first comparison of how the parameters of the events were recovered has been completed. 
  The material tensor of linear response in electrodynamics is constructed out of products of two symmetric second rank tensor fields which in the approximation of geometrical optics and for uniaxial symmetry reduce to "optical" metrics, describing the phenomenon of birefringence. This representation is interpreted in the context of an underlying internal geometrical structure according to which the symmetric tensor fields are vectorial elements of an associated two-dimensional space. 
  The notion of semi-classical states is first sharpened by clarifying two issues that appear to have been overlooked in the literature. Systems with linear and quadratic constraints are then considered and the group averaging procedure is applied to kinematical coherent states to obtain physical semi-classical states. In the specific examples considered, the technique turns out to be surprisingly efficient, suggesting that it may well be possible to use kinematical structures to analyze the semi-classical behavior of physical states of an interesting class of constrained systems. 
  In this paper we explicitly work out the secular perturbations induced on all the Keplerian orbital elements of a test body to order O(e^2) in the eccentricity e by the weak-field long-range modifications of the usual Newton-Einstein gravity due to the Dvali-Gabadadze-Porrati (DGP) braneworld model. The Gauss perturbative scheme is used. It turns out that the argument of pericentre and the mean anomaly are affected by secular rates which are independent of the semimajor axis of the orbit of the test particle. The first nonvaishing eccentricity-dependent corrections are of order O(e^2). For circular orbits the Lue-Starkman (LS) effect on the pericentre is obtained. Some observational consequences are discussed for the Solar System planetary mean longitudes lambda which would undergo a 1.2\cdot 10^-3 arcseconds per century braneworld secular precession. According to recent data analysis over 92 years for the EPM2004 ephemerides, the 1-sigma formal accuracy in determining the Martian mean longitude amounts to 3\cdot 10^-3 milliarcseconds, while the braneworld effect over the same time span would be 1.159 milliarcseconds. The major limiting factor is the 2.6\cdot 10^-3 arcseconds per century systematic error due to the mismodelling in the Keplerian mean motion of Mars. A suitable linear combination of the mean longitudes of Mars and Venus may overcome this problem. The formal, 1-sigma obtainable observational accuracy would be \sim 7%. The systematic error due to the present-day uncertainties in the solar quadrupole mass moment, the Keplerian mean motions, the general relativistic Schwarzschild field and the asteroid ring would amount to some tens of percent. 
  The functional formulation and one-loop effective action for scalar self-interacting theory non-linearly coupled with some power of the curvature are studied. After the explicit one-loop renormalization at weak curvature, we investigated numerically the phase structure for such unusual phi^4 theory. It is demonstrated the possibility of curvature-induced phase transitions for positive values of the curvature power, while for negative values the radiative symmetry breaking does not take place. The dynamical mechanism for the explanation of the current smallness of the cosmological constant is presented for several models from the class of theories under consideration. 
  The stability of the Schwarzschild black hole is studied. Regge and Wheeler treated the problem first at 1957 and obtained the dynamical equations for the small perturbation. There are two kinds of perturbations: odd one and even one. Using the Painlev\'{e} coordinate, we reconsider the odd perturbation and find that: the white-hole-connected universe(r>2m, see text) is unstable. Because the odd perturbation may be regarded as the angular perturbation, therefore, the physical mean to it may be that the white-hole-connected universe is unstable with respect to the rotating perturbation. 
  This paper has been withdrawn by the author. The metrics presented are known since 1969 (and are referred to as C-metrics) 
  The current accelerated universe could be produced by modified gravitational dynamics as it can be seen in particular in its Palatini formulation. We analyze here a specific non-linear gravity-scalar system in the first order Palatini formalism which leads to a FRW cosmology different from the purely metric one. It is shown that the emerging FRW cosmology may lead either to an effective quintessence phase (cosmic speed-up) or to an effective phantom phase. Moreover, the already known gravity assisted dark energy dominance occurs also in the first order formalism. Finally, it is shown that a dynamical theory able to resolve the cosmological constant problem exists also in this formalism, in close parallel with the standard metric formulation. 
  Perturbed equations for an arbitrary metric theory of gravity in $D$ dimensions are constructed in the vacuum of this theory. The nonlinear part together with matter fields are a source for the linear part and are treated as a total energy-momentum tensor. A generalized family of conserved currents expressed through divergences of anti-symmetrical tensor densities (superpotentials) linear in perturbations is constructed. The new family generalizes the Deser and Tekin currents and superpotentials in quadratic curvature gravity theories generating Killing charges in dS and AdS vacua. As an example, the mass of the $D$-dimensional Schwarzschild black hole in an effective AdS spacetime (a solution in the Einstein-Gauss-Bonnet theory) is examined. 
  We review the dual transformation from pure lattice gauge theory to spin foam models with an emphasis on a geometric viewpoint. This allows us to give a simple dual formulation of SU(N) Yang-Mills theory, where spin foam surfaces are weighted with the exponentiated area. In the case of gravity, we introduce a symmetry condition which demands that the amplitude of an individual spin foam depends only on its geometric properties and not on the lattice on which it is defined. For models that have this property, we define a new sum over abstract spin foams that is independent of any choice of lattice or triangulation. We show that a version of the Barrett-Crane model satisfies our symmetry requirement. 
  We present a comparative study of 6 search methods for gravitational wave bursts using simulated LIGO and Virgo noise data. The data's spectra were chosen to follow the design sensitivity of the two 4km LIGO interferometers and the 3km Virgo interferometer. The searches were applied on replicas of the data sets to which 8 different signals were injected. Three figures of merit were employed in this analysis: (a) Receiver Operator Characteristic curves, (b) necessary signal to noise ratios for the searches to achieve 50 percent and 90 percent efficiencies, and (c) variance and bias for the estimation of the arrival time of a gravitational wave burst. 
  In previous work it has been shown that the electromagnetic quantum vacuum, or electromagnetic zero-point field, makes a contribution to the inertial reaction force on an accelerated object. We show that the result for inertial mass can be extended to passive gravitational mass. As a consequence the weak equivalence principle, which equates inertial to passive gravitational mass, appears to be explainable. This in turn leads to a straightforward derivation of the classical Newtonian gravitational force. We call the inertia and gravitation connection with the vacuum fields the quantum vacuum inertia hypothesis. To date only the electromagnetic field has been considered. It remains to extend the hypothesis to the effects of the vacuum fields of the other interactions. We propose an idealized experiment involving a cavity resonator which, in principle, would test the hypothesis for the simple case in which only electromagnetic interactions are involved. This test also suggests a basis for the free parameter $\eta(\nu)$ which we have previously defined to parametrize the interaction between charge and the electromagnetic zero-point field contributing to the inertial mass of a particle or object. 
  We report on the development of the LISA Technology Package (LTP) experiment that will fly on board the LISA Pathfinder mission of the European Space Agency in 2008. We first summarize the science rationale of the experiment aimed at showing the operational feasibility of the so called Transverse-Traceless coordinate frame within the accuracy needed for LISA. We then show briefly the basic features of the instrument and we finally discuss its projected sensitivity and the extrapolation of its results to LISA. 
  To study asymptotic structures, we regularize Einstein's field equations by means of conformal transformations. The conformal factor is chosen so that it carries a dimensional scale that captures crucial asymptotic features. By choosing a conformal orthonormal frame we obtain a coupled system of differential equations for a set of dimensionless variables, associated with the conformal dimensionless metric, where the variables describe ratios with respect to the chosen asymptotic scale structure. As examples, we describe some explicit choices of conformal factors and coordinates appropriate for the situation of a timelike congruence approaching a singularity. One choice is shown to just slightly modify the so-called Hubble-normalized approach, and one leads to dimensionless first order symmetric hyperbolic equations. We also discuss differences and similarities with other conformal approaches in the literature, as regards, e.g., isotropic singularities. 
  It is shown that a non-rotating macroscopic black hole with very large horizon area can remain in stable thermal equilibrium with Hawking radiation provided {\it its mass, as a function of horizon area, exceeds its microcanonical entropy, i.e., its entropy when isolated, without thermal radiation or accretion, and having a constant horizon area} (in appropriate units). The analysis does not use properties of specific classical spacetimes, but depends only on the plausible assumption that the mass is a function of the horizon area for large areas. 
  The LIGO Scientific Collaboration is currently engaged in the first search for binary black hole inspiral signals in real data. We are using the data from the second LIGO science run and we focus on inspiral signals coming from binary systems with component masses between 3 and 20 solar masses. We describe the analysis methods used and report on preliminary estimates for the sensitivities of the LIGO instruments during the second science run. 
  A classical precursor to a full quantum dynamics for causal sets has been forumlated in terms of a stochastic sequential growth process in which the elements of the causal set arise in a sort of accretion process. The transition probabilities of the Markov growth process satisfy certain physical requirements of causality and general covariance, and the generic solution with all transition probabilities non-zero has been found. Here we remove the assumption of non-zero probabilities, define a reasonable extension of the physical requirements to cover the case of vanishing probabilities, and find the completely general solution to these physical conditions. The resulting family of growth processes has an interesting structure reminiscent of an ``infinite tower of turtles'' cosmology. 
  LIGO recently conducted its third scientific data run, S3. Here we summarize the veto and data quality studies conducted by the LIGO Scientific Collaboration in connection with the search for binary inspiral signals in the S3 data. LIGO's interferometer channels and physical environmental monitors were monitored, and events in these channels coincident with inspiral triggers were examined. 
  We construct models of rotating stars using the perturbative approach introduced by J. Hartle in 1967, and a set of equations of state proposed to model hadronic interactions in the inner core of neutron stars. We integrate the equations of stellar structure to third order in the angular velocity and show, comparing our results to those obtained with fully non linear codes, to what extent third order corrections are needed to accurately reproduce the moment of inertia of a star which rotates at rates comparable to that of the fastest isolated pulsars. 
  Classical sequential growth models for causal sets provide an important step towards the formulation of a quantum causal set dynamics. The covariant observables in a class of these models known as generalised percolation have been completely characterised in terms of physically well-defined ``stem sets'' and yield an insight into the nature of observables in quantum causal set cosmology. We discuss a recent extension of generalised percolation and show that the characterisation of covariant observables in terms of stem sets is also complete in this extension. 
  The dynamics of the dust-shell model of universe is exactly solved for the modified Schwarzschild solution. This solution is used to derive the cosmology corresponding to the modified gravity. 
  We present a Monte Carlo simulation for the response of the Laser Interferometer Space Antenna (LISA) to the galactic gravitational wave background. The simulated data streams are used to estimate the number and type of binary systems that will be individually resolved in a 1-year power spectrum. We find that the background is highly non-Gaussian due to the presence of individual bright sources, but once these sources are identified and removed, the remaining signal is Gaussian. We also present a new estimate of the confusion noise caused by unresolved sources that improves on earlier estimates. 
  It is well known that, for pressureless matter, Newtonian and relativistic cosmologies are equivalent. We show that this equivalence breaks down in the quantum level. In addition, we find some cases for which quantum Newtonian cosmology can be related to quantum cosmology in (2+1) dimensions. Two exact solutions for the wave function of the Newtonian universe are also obtained. 
  Modifying the standard hot big bang model of cosmology with an inflationary event has been very successful in resolving most of the outstanding cosmological problems. The various inflationary mechanisms proposed depend on the production of expansion from exotic phenomena, false vacuum, scalar fields or other exotic particle behaviour within an environment of astronomically high energy, making such proposals relatively inaccessible for verification by experimental tests. Though descriptive of how space has been expanding, the models do not give a complete and consistent mathematical or physical explanation that compels space appearance and propels its expansion. Here we describe another mechanism for achieving exponential inflation based substantially on already tested physics and equations, particularly the thermodynamic equation, dS = dE/T and relate this to the creation event. 
  We carry out the Hamiltonian formulation of the three- dimensional gravitational teleparallelism without imposing the time gauge condition, by rigorously performing the Legendre transform. Definition of the gravitational angular momentum arises by suitably interpreting the integral form of the constraint equation Gama^ik=0 as an angular momentum equation. The gravitational angular momentum is evaluated for the gravitational field of a rotating BTZ black hole. 
  The equations describing the adiabatic, small radial oscillations of general relativistic stars are generalized to include the effects of a cosmological constant. The generalized eigenvalue equation for the normal modes is used to study the changes in the stability of the homogeneous sphere induced by the presence of the cosmological constant. The variation of the critical adiabatic index as a function of the central pressure is studied numerically for different trial functions. The presence of a large cosmological constant significantly increases the value of the critical adiabatic index. The dynamical stability condition of the homogeneous star in the Schwarzschild-de Sitter geometry is obtained and several bounds on the maximum allowable value for a cosmological constant are derived from stability considerations. 
  It is known that, through inflation, Planck scale phenomena should have left an imprint in the cosmic microwave background. The magnitude of this imprint is expected to be suppressed by a factor $\sigma^n$ where $\sigma\approx 10^{-5}$ is the ratio of the Planck length to the Hubble length during inflation. While there is no consensus about the value of $n$, it is generally thought that $n$ will determine whether the imprint is observable. Here, we suggest that the magnitude of the imprint may not be suppressed by any power of $\sigma$ and that, instead, $\sigma$ may merely quantify the amount of fine tuning required to achieve an imprint of order one. To this end, we show that the UV/IR scale separation, $\sigma$, in the analogous case of the Casimir effect plays exactly this role. 
  In the framework of the teleparallel equivalent of general relativity it is possible to establish the energy-momentum tensor of the gravitational field. This tensor has the following essential features: (1) it is identified directly in Einstein's field equations; (2) it is conserved and traceless; (3) it yields expressions for the energy and momentum of the gravitational field; (4)it is free of second (and highest) derivatives of the field variables; (5) the gravitational and matter energy-momentum tensors take place in the field equations on the same footing; (6) it is unique. However, it is not symmetric. We show that the spatial components of this tensor yield a consistent definition of the gravitational pressure. 
  For gravitational deflections of massless particles of given helicity from a classical rotating body, we describe the general relativity corrections to the geometric optics approximation. We compute the corresponding scattering cross sections for neutrinos, photons and gravitons to lowest order in the gravitational coupling constant. We find that the helicity coupling to spacetime geometry modifies the ray deflection formula of the geometric optics, so that rays of different helicity are deflected by different amounts. We also discuss the validity range of the Born approximation. 
  We compute the angular pattern and the overlap reduction functions for the geodesic and non-geodesic response of the common mode of two interferometers interacting with a stochastic, massive scalar background. We also discuss the possible overlap between common and differential modes. We find that the cross-correlated response of two common modes to a non-relativistic background may be higher than the response of two differential modes to the same background. 
  A new method to obtain thick domain wall solutions to the coupled Einstein scalar field system is presented. The procedure allows the construction of irregular walls from well known ones, such that the spacetime associated to them are physically different. As consequence of the approach, we obtain two irregular geometries corresponding to thick domain walls with $dS$ expansion and topological double kink embedded in $AdS$ spacetime. In particular, the double brane can be derived from a fake superpotential. 
  Let (M, g) be an asymptotically flat static vacuum initial data set with non-empty compact boundary. We prove that (M, g) is isometric to a spacelike slice of a Schwarzschild spacetime under the mere assumption that the boundary of (M, g) has zero mean curvature, hence generalizing a classic result of Bunting and Masood-ul-Alam. In the case that the boundary has constant positive mean curvature and satisfies a stability condition, we derive an upper bound of the ADM mass of (M, g) in terms of the area and mean curvature of the boundary. Our discussion is motivated by Bartnik's quasi-local mass definition. 
  Latest general relativistic simulations for merger of binary neutron stars with realistic equations of state (EOSs) show that a hypermassive neutron star of an ellipsoidal figure is formed after the merger if the total mass is smaller than a threshold value which depends on the EOSs. The effective amplitude of quasiperiodic gravitational waves from such hypermassive neutron stars is $\sim 6$--$7 \times 10^{-21}$ at a distance of 50 Mpc, which may be large enough for detection by advanced laser interferometric gravitational wave detectors although the frequency is high $\sim 3$ kHz. We point out that the detection of such signal may lead to constraining the EOSs for neutron stars. 
  What happens to the entropy increase principle as the Universe evolve to form the big-crunch singularity? What happens to the uncertainity relations along the process of gravitational collapses? What is the quantum mechanical description of a Radon atom in a rigid box when the distance of consecutive nodes and antinodes of $\psi$ is equal to or less than the diameter of the atom? What is the position-space wave function of two finite volume massive bosons if we take contact interaction into account? How a photon produce electron-positron pair with finite volume concentrate rest masses? What are the charges of the electron-positron pairs forming loops in the vacuum? How two particles with three-momentums $k_1, k_2$ $(k_1 \neq k_2)$ produced to form a loop at a space-time point always arrive at another spacial point simultaneously? What is the microscopic explanation in terms of particle exchanges of the force in the Casimir effect? What is the mechanism of the collapse of the momentum-space wave function of a particle knocking out an elctron from an atom? What is meant by $|\Psi> = {c_1 (t)}|\Psi_{U^{238}}> + {c_2 (t)}|\Psi_{Th^{234}}> ~?$ Quantum mechanically the region between the rigid walls (which is equiprobable in classical mechanics) is non-homogeneous for a particle in a rigid box. A photon can not reproduce Maxwell's equations apart from moving with velocity $c$. How can a process involving only a few photons be described starting from the Maxwell's equations? The large scale structure of the Universe is homogeneous.   What is the screen in our brains to view objects, as they are, of sizes larger than our brains? 
  I address some recently raised issues regarding the time-parametrization dependence in stochastic descriptions of eternal inflation. To clarify the role of the choice of the time gauge, I show examples of gauge-dependent as well as gauge-independent statements about physical observables in eternally inflating spacetimes. In particular, the relative abundance of thermalized and inflating regions is highly gauge-dependent. The unbounded growth of the 3-volume of the inflating regions is found in certain time gauges, such as the proper time or the scale factor gauge. Yet in the same spacetimes there exist time foliations with a finite and monotonically decreasing 3-volume, which I demonstrate by an explicit construction. I also show that there exists no "correct" choice of the time gauge that would yield an unbiased stationary probability distribution for observables in thermalized regions. 
  This article is a partly pedagogical, partly historical and partly technical review of special relativity and its experimental foundations, in honor of the centenary of Einstein's annus mirabilis. 
  We review the experimental evidence for Einstein's special and general relativity. A variety of high precision null experiments verify the weak equivalence principle and local Lorentz invariance, while gravitational redshift and other clock experiments support local position invariance. Together these results confirm the Einstein Equivalence Principle which underlies the concept that gravitation is synonymous with spacetime geometry, and must be described by a metric theory. Solar system experiments that test the weak-field, post-Newtonian limit of metric theories strongly favor general relativity. The Binary Pulsar provides tests of gravitational-wave damping and of strong-field general relativity. Recently discovered binary pulsar systems may provide additional tests. Future and ongoing experiments, such as the Gravity Probe B Gyroscope Experiment, satellite tests of the Equivalence principle, and tests of gravity at short distance to look for extra spatial dimensions could constrain extensions of general relativity. Laser interferometric gravitational-wave observatories on Earth and in space may provide new tests of gravitational theory via detailed measurements of the properties of gravitational waves. 
  We study the thermodynamics of de Sitter black holes with a conformally coupled scalar field. The geometry is that of the ``lukewarm'' Reissner-Nordstrom-de Sitter black holes, with the event and cosmological horizons at the same temperature. This means that the region between the event and cosmological horizons can form a regular Euclidean instanton. The entropy is modified by the non-minimal coupling of the scalar field to the geometry, but can still be derived from the Euclidean action, provided suitable modifications are made to deal with the electrically charged case. We use the first law as derived from the isolated horizons formalism to compute the local horizon energies for the event and cosmological horizons. 
  Perturbations in a Chaplygin gas, characterized by an equation of state $p = -A/\rho$, may acquire non-adiabatic contributions if spatial variations of the parameter $A$ are admitted. This feature is shown to be related to a specific internal structure of the Chaplygin gas. We investigate how perturbations of this type modify the adiabatic sound speed and influence the time dependence of the gravitational potential which gives rise to the Integrated Sachs-Wolfe effect in the anisotropy spectrum of the cosmic microwave background. 
  Prompted by the recent more precise determination of the basic cosmological parameters and growing evidence that the matter-energy content of the universe is now dominated by dark energy and dark matter we present the general solution of the equation that describes the evolution of density perturbations in the linear approximation. It turns out that as in the standard CDM model the density perturbations grow very slowly during the radiation dominated epoch and their amplitude increases by a factor of about 4000 in the matter and later dark energy dominated epoch of expansion of the universe. 
  In this paper dynamics and quantum mechanical coherent states of a simple harmonic oscillator are considered in the framework of Generalized Uncertainty Principle(GUP). Equations of motion for simple harmonic oscillator are derived and some of their new implications are discussed. Then coherent states of harmonic oscillator in the case of GUP are compared with relative situation in ordinary quantum mechanics. It is shown that in the framework of GUP there is no considerable difference in definition of coherent states relative to ordinary quantum mechanics. But, considering expectation values and variances of some operators, based on quantum gravitational arguments one concludes that although it is possible to have complete coherency and vanishing broadening in usual quantum mechanics, gravitational induced uncertainty destroys complete coherency in quantum gravity and it is not possible to have a monochromatic ray in principle. 
  A classical foundation for an idea of reality condition in the context of spin foams (Barrett-Crane models) is developed. I extract classical real general relativity (all signatures) from complex general relativity by imposing the area metric reality constraint; the area metric is real iff a non-degenerate metric is real or imaginary. First I review the Plebanski theory of complex general relativity starting from a complex vectorial action. Then I modify the theory by adding a Lagrange multiplier to impose the area metric reality condition and derive classical real general relativity. I investigate two types of action: Complex and Real. All the non-trivial solutions of the field equations of the theory with the complex action correspond to real general relativity. Half the non-trivial solutions of the field equations of the theory with the real action correspond to real general relativity. Discretization of the area metric reality constraint in the context of Barrett-Crane theory is discussed. In the context of Barrett-Crane theory the area metric reality condition is equivalent to the condition that the scalar products of the bivectors associated to the triangles of a four simplex be real. The Plebanski formalism for the degenerate case and Palatini formalism are also briefly discussed by including the area metric reality condition. 
  The Barrett-Crane model for the SO(4,C) general relativity is systematically derived. This procedure makes rigorous the calculation of the Barrett-Crane intertwiners from the Barrett-Crane constraints of both real and complex Riemannian general relativity. The reality of the scalar products of the complex bivectors associated with the triangles of a flat four simplex is equivalent to the reality of the associated flat geometry. Spin foam models in 4D for the real and complex orthogonal gauge groups are discussed in a unified manner from the point of view of the bivector scalar product reality constraints. Many relevant issues are discussed and generalizations of the ideas are introduced. The asymptotic limit of the SO(4,C) general relativity is discussed. The asymptotic limit is controlled by the SO(4,C) Regge calculus which unifies the Regge calculus theories for all the real general relativity cases. The spin network functionals for the 3+1 formulation of the spin foams are discussed. The field theory over group formulation for the Barrett-Crane models is discussed briefly. I introduce the idea of a mixed Lorentzian Barrett-Crane model which mixes the intertwiners for the Lorentzian Barrett-Crane models. A mixed propagator is calculated. I also introduce a multi-signature spin foam model for real general relativity which is made by splicing together the four simplex amplitudes for the various signatures. Further research that is to be done is listed and discussed. 
  We show that for asymptotically vanishing Maxwell fields in Minkowski space with non-vanishing total charge, one can find a unique geometric structure, a null direction field, at null infinity. From this structure a unique complex analytic world-line in complex Minkowski space that can be found and then identified as the complex center of charge. By ''sitting'' - in an imaginary sense, on this world-line both the (intrinsic) electric and magnetic dipole moments vanish. The (intrinsic) magnetic dipole moment is (in some sense) obtained from the `distance' the complex the world line is from the real space (times the charge). This point of view unifies the asymptotic treatment of the dipole moments For electromagnetic fields with vanishing magnetic dipole moments the world line is real and defines the real (ordinary center of charge). We illustrate these ideas with the Lienard-Wiechert Maxwell field. In the conclusion we discuss its generalization to general relativity where the complex center of charge world-line has its analogue in a complex center of mass allowing a definition of the spin and orbital angular momentum - the analogues of the magnetic and electric dipole moments. 
  Asymptotics of solutions of a perfect fluid when coupled with a cosmological constant in four-dimensional spacetime with toroidal symmetry are studied. In particular, it is found that the problem of self-similar solutions of the first kind for a fluid with the equation of state, $p = k \rho$, can be reduced to solving a master equation of the form, $$ 2 F(q, k)\frac{q''(\xi)}{q'(\xi)} - G(q,k) q'(\xi) = \frac{4}{\xi}. $$ For $k = 0$ and $k = -1/3$ the general solutions are obtained and their main local and global properties are studied in detail. 
  In this article we consider nonholonomic deformations of disk solutions in general relativity to generic off-diagonal metrics defining knew classes of exact solutions in 4D and 5D gravity. These solutions possess Lie algebroid symmetries and local anisotropy and define certain generalizations of manifolds with Killing and/ or Lie algebra symmetries. For Lie algebroids, there are structures functions depending on variables on a base submanifold and it is possible to work with singular structures defined by the 'anchor' map. This results in a number of new physical implications comparing with the usual manifolds possessing Lie algebra symmetries defined by structure constants. The spacetimes investigated here have two physically distinct properties: First, they can give rise to disk type configurations with angular/ time/ extra dimension gravitational polarizations and running constants. Second, they define static, stationary or moving disks in nontrivial solitonic backgrounds, with possible warped factors, additional spinor and/or noncommutative symmetries. Such metrics may have nontrivial limits to 4D gravity with vanishing, or nonzero torsion. The work develops the results of Ref. gr-qc/0005025 and emphasizes the solutions with Lie algebroid symmetries following similar constructions for solutions with noncommutative symmetries gr-qc/0307103. 
  This article will summarize selected aspects of the semiclassical theory of gravity, which involves a classical gravitational field coupled to quantum matter fields. Among the issues which will be discussed are the role of quantum effects in black hole physics and in cosmology, the effects of quantum violations of the classical energy conditions, and inequalities which constrain the extent of such violations. We will also examine the first steps beyond semiclassical gravity, when the effects of spacetime geometry fluctuations start to appear. 
  Singularities in the metric of the classical solutions to the Einstein equations (Schwarzschild, Kerr, Reissner -- Nordstr\"om and Kerr -- Newman solutions) lead to appearance of generalized functions in the Einstein tensor that are not usually taken into consideration. The generalized functions can be of a more complex nature than the Dirac $\d$-function. To study them, a technique has been used based on a limiting solution sequence. The solutions are shown to satisfy the Einstein equations everywhere, if the energy-momentum tensor has a relevant singular addition of non-electromagnetic origin. When the addition is included, the total energy proves finite and equal to $mc^2$, while for the Kerr and Kerr--Newman solutions the angular momentum is $mc {\bf a}$. As the Reissner--Nordstr\"om and Kerr--Newman solutions correspond to the point charge in the classical electrodynamics, the result obtained allows us to view the point charge self-energy divergence problem in a new fashion. 
  New exact interior solutions to the Einstein field equations for anisotropic spheres are found. We utilise a procedure that necessitates a choice for the energy density and the radial pressure. This class contains the constant density model of Maharaj and Maartens (Gen. Rel. Grav., Vol 21, 899-905, 1989) and the variable density model of Gokhroo and Mehra (Gen. Rel. Grav., Vol 26, 75-84, 1994) as special cases. These anisotropic spheres match smoothly to the Schwarzschild exterior and gravitational potentials are well behaved in the interior. A graphical analysis of the matter variables is performed which points to a physically reasonable matter distribution. 
  Bekenstein-Hawking Black hole thermodynamics should be corrected to incorporate quantum gravitational effects. Generalized Uncertainty Principle(GUP) provides a perturbational framework to perform such modifications. In this paper we consider the most general form of GUP to find black holes thermodynamics in microcanonical ensemble. Our calculation shows that there is no logarithmic pre-factor in perturbational expansion of entropy. This feature will solve part of controversies in literatures regarding existence or vanishing of this pre-factor. 
  In loop quantum cosmology, a difference equation for the wave function describes the evolution of a universe model. This is different from the differential equations that arise in Wheeler-DeWitt quantizations, and some aspects of general properties of solutions can appear differently. Properties of particular interest are boundedness and the presence of small-scale oscillations. Continued fraction techniques are used to show in different matter models the presence of special initial conditions leading to bounded solutions, and an explicit expression for these initial values is derived. 
  This paper is devoted to the investigation of the consequences of timelike and spacelike matter inheritance vectors in specific forms of energy-momentum tensor, i.e., for string cosmology (string cloud and string fluid) and perfect fluid. Necessary and sufficient conditions are developed for a spacetime with string cosmology and perfect fluid to admit a timelike matter inheritance vector, parallel to $u^a$ and spacelike matter inheritance vector, parallel to $x^a$. We compare the outcome with the conditions of conformal Killing vectors. This comparison provides us the conditions for the existence of matter inheritance vector when it is also a conformal Killing vector. Finally, we discuss these results for the existence of matter inheritance vector in the special cases of the above mentioned spacetimes. 
  This paper is devoted to discuss some of the features of self-similar solutions of the first kind. We consider the cylindrically symmetric solutions with different homotheties. We are interested in evaluating the quantities acceleration, rotation, expansion, shear, shear invariant and expansion rate. These kinematical quantities are discussed both in co-moving as well as in non-co-moving coordinates (only in radial direction). Finally, we would discuss the singularity feature of these solutions. It is expected that these properties would help in exploring some interesting features of the self-similar solutions. 
  This paper is devoted to find out cylindrically symmetric kinematic self-similar perfect fluid and dust solutions. We study the cylindrically symmetric solutions which admit kinematic self-similar vectors of second, zeroth and infinite kinds, not only for the tilted fluid case but also for the parallel and orthogonal cases. It is found that the parallel case gives contradiction both in perfect fluid and dust cases. The orthogonal perfect fluid case yields a vacuum solution while the orthogonal dust case gives contradiction. It is worth mentioning that the tilted case provides solution both for the perfect as well as dust cases. 
  We seek exact solutions to the Einstein field equations which arise when two spacetime geometries are conformally related. Whilst this is a simple method to generate new solutions to the field equations, very few such examples have been found in practice. We use the method of Lie analysis of differential equations to obtain new group invariant solutions to conformally related Petrov type D spacetimes. Four cases arise depending on the nature of the Lie symmetry generator. In three cases we are in a position to solve the master field equation in terms of elementary functions. In the fourth case special solutions in terms of Bessel functions are obtained. These solutions contain known models as special cases. 
  We treat a very crude model of an exploding star, in the weak field approximation of the Brans-Dicke theory, in a scenario that resembles some characteristics data of a Type Ia Supernova. The most noticeable feature, in the electromagnetic component, is the relationship between the absolute magnitude at maximum brightness of the star and the decline rate in one magnitude from that maximum. This characteristic has become one of the most accurate method to measure luminosity distances to objects at cosmological distances. An interesting result is that the active mass associated with the scalar field is totally radiated to infinity, representing a mass loss in the ratio of the "tensor" component to the scalar component of 1 to $(2 \omega + 3)$ ($\omega$ is the Brans-Dicke parameter), in agreement with a general result of Hawking. Then, this model shows explicitly, in a dynamical case, the mechanism of radiation of scalar field, which is necessary to understand the Hawking result. 
  The LIGO detectors collected about 4 months of data in 2003-2004 during two science runs, S2 and S3. Several environmental and auxiliary channels that monitor the instruments' physical environment and overall interferometric operation were analyzed in order to establish the quality of the data as well as the presence of transients of non-astrophysical origin. This analysis allowed better understanding of the noise character of the instruments and the establishment of correlations between transients in these channels and the one recording the gravitational wave strain. In this way vetoes for spurious burst were identified. We present the methodology we followed in this analysis and the results from the S2 and S3 veto analysis within the context of the search for gravitational wave bursts. 
  The anomalous acceleration of Pioneer 10 and 11 spacecraft of (8.74 \pm 1.33) \times 10^{-8} cm. s^{-2} fits with a theoretical prediction of a minimal acceleration in nature of about 7.61 \times 10^{-8} cm. s^{-2} 
  Motivated by a recently-invented scheme of displacement-noise-free gravitational-wave detection, we demonstrate the existence of gravitational-wave detection schemes insusceptible to both displacement and timing (laser) noises, and are thus realizable by shot-noise-limited laser interferometry. This is possible due to two reasons: first, gravitational waves and displacement disturbances contribute to light propagation times in different manners; second, for an N-detector system, the number of signal channels is of the order O(N^2), while the total number of timing- and displacement-noise channels is of the order O(N). 
  A test of Lorentz invariance for electromagnetic waves was performed by comparing the resonance frequencies of two optical resonators as a function of orientation in space. In terms of the Robertson-Mansouri-Sexl theory, we obtain $\beta-\delta-1/2=(+0.5\pm 3\pm 0.7) E-10$, a ten-fold improvement compared to the previous best results. We also set a first upper limit for a so far unknown parameter of the Standard Model Extension test theory, $|(\tilde{\kappa}_{e-})^{ZZ}| < 2\cdot E-14$. 
  We study the propagation of axial gravitational waves in Friedman universes. The evolution equation is obtained in the Regge-Wheeler gauge. The gravitational waves obey the Huygens principle in the radiation dominated era, but in the matter dominated universe their propagation depends on their wavelengths, with the scale fixed essentially by the Hubble radius. Short waves practically satisfy the Huygens principle while long waves can backscatter off the curvature of a spacetime. 
  We present a formulation for potential-density pairs to describe axisymmetric galaxies in the Newtonian limit of scalar-tensor theories of gravity. The scalar field is described by a modified Helmholtz equation with a source that is coupled to the standard Poisson equation of Newtonian gravity. The net gravitational force is given by two contributions: the standard Newtonian potential plus a term stemming from massive scalar fields. General solutions have been found for axisymmetric systems and the multipole expansion of the Yukawa potential is given. In particular, we have computed potential-density pairs of galactic disks for an exponential profile and their rotation curves. 
  LISA (Laser Interferometer Space Antenna) is a proposed space mission, which will use coherent laser beams exchanged between three remote spacecraft to detect and study low-frequency cosmic gravitational radiation. In the low-part of its frequency band, the LISA strain sensitivity will be dominated by the incoherent superposition of hundreds of millions of gravitational wave signals radiated by inspiraling white-dwarf binaries present in our own galaxy. In order to estimate the magnitude of the LISA response to this background, we have simulated a synthesized population that recently appeared in the literature. We find the amplitude of the galactic white-dwarf binary background in the LISA data to be modulated in time, reaching a minimum equal to about twice that of the LISA noise for a period of about two months around the time when the Sun-LISA direction is roughly oriented towards the Autumn equinox. Since the galactic white-dwarfs background will be observed by LISA not as a stationary but rather as a cyclostationary random process with a period of one year, we summarize the theory of cyclostationary random processes, present the corresponding generalized spectral method needed to characterize such process, and make a comparison between our analytic results and those obtained by applying our method to the simulated data. We find that, by measuring the generalized spectral components of the white-dwarf background, LISA will be able to infer properties of the distribution of the white-dwarfs binary systems present in our Galaxy. 
  Recently, Angheben et.al. [hep-th/0503081] have presented a refined method for calculating the (tree-level) black hole temperature by way of the tunneling paradigm. In the current letter, we demonstrate how their formalism can be suitably adapted to accommodate the (higher-order) effects of the gravitational back-reaction. 
  We show that by adding suitable lower-order terms to the Z4 formulation of the Einstein equations, all constraint violations except constant modes are damped. This makes the Z4 formulation a particularly simple example of a lambda-system as suggested by Brodbeck et al. We also show that the Einstein equations in harmonic coordinates can be obtained from the Z4 formulation by a change of variables that leaves the implied constraint evolution system unchanged. Therefore the same method can be used to damp all constraints in the Einstein equations in harmonic gauge. 
  In the macroscopic gravity approach to the averaging problem in cosmology, the Einstein field equations on cosmological scales are modified by appropriate gravitational correlation terms. We present exact cosmological solutions to the equations of macroscopic gravity for a spatially homogeneous and isotropic macroscopic space-time and find that the correlation tensor is of the form of a spatial curvature term. We briefly discuss the physical consequences of these results. 
  In 1859, Le Verrier discovered the mercury perihelion advance anomaly. This anomaly turned out to be the first relativistic-gravity effect observed. During the 141 years to 2000, the precisions of laboratory and space experiments, and astrophysical and cosmological observations on relativistic gravity have been improved by 3 orders of magnitude. In 1999, we envisaged a 3-6 order improvement in the next 30 years in all directions of tests of relativistic gravity. In 2000, the interferometric gravitational wave detectors began their runs to accumulate data. In 2003, the measurement of relativistic Shapiro time-delay of the Cassini spacecraft determined the relativistic-gravity parameter gamma&#947; with a 1.5-order improvement. In October 2004, Ciufolini and Pavlis reported a measurement of the Lense-Thirring effect on the LAGEOS and LAGEOS2 satellites to 10 percent of the value predicted by general relativity. In April 2004, Gravity Probe B was launched and has been accumulating science data for more than 170 days now. MICROSCOPE is on its way for a 2007 launch to test Galileo equivalence principle to 10-15. STEP (Satellite Test of Equivalence Principle), and ASTROD (Astrodynamical Space Test of Relativity using Optical Devices) are in the good planning stage. Various astrophysical tests and cosmological tests of relativistic gravity will reach precision and ultra-precision stages. Clock tests and atomic interferometry tests of relativistic gravity will reach an ever-increasing precision. These will give revived interest and development both in experimental and theoretical aspects of gravity, and may lead to answers to some profound questions of gravity and the cosmos. 
  We consider the Weyl-Dirac theory which in the weak field approximation leads to a gravitational potential differing from that of the Newtonian by a repulsive correction term increasing with distance. The scale of the correction term appears to be determined by the time variation rate of the gravitational coupling. It is shown that if the time variation rate of gravitational coupling is chosen from its observational bound, the theory can explain the flat rotation curves of typical spiral galaxies, without the dark matter hypothesis. In this case the intergalactic effects and the gravitational lensing of clusters of galaxies are also obtained. 
  We study the foundation of space-time theory in the framework of first-order logic (FOL). Since the foundation of mathematics has been successfully carried through (via set theory) in FOL, it is not entirely impossible to do the same for space-time theory (or relativity). First we recall a simple and streamlined FOL-axiomatization SpecRel of special relativity from the literature. SpecRel is complete with respect to questions about inertial motion. Then we ask ourselves whether we can prove usual relativistic properties of accelerated motion (e.g., clocks in acceleration) in SpecRel. As it turns out, this is practically equivalent to asking whether SpecRel is strong enough to "handle" (or treat) accelerated observers. We show that there is a mathematical principle called induction (IND) coming from real analysis which needs to be added to SpecRel in order to handle situations involving relativistic acceleration. We present an extended version AccRel of SpecRel which is strong enough to handle accelerated motion, in particular, accelerated observers. Among others, we show that the Twin Paradox becomes provable in AccRel, but it is not provable without IND. 
  A late time asymptotic perturbative analysis of curvature coupled complex scalar field models with accelerated cosmological expansion is carried out on the level of formal power series expansions. For this, algebraic analogues of the Einstein scalar field equations in Gaussian coordinates for space-time dimensions greater than two are postulated and formal solutions are constructed inductively and shown to be unique. The results obtained this way are found to be consistent with already known facts on the asymptotics of such models. In addition, the algebraic expansions are used to provide a prospect of the large time behaviour that might be expected of the considered models. 
  Interactions between outgoing Hawking particles and ingoing matter are determined by gravitational forces and Standard Model interactions. In particular the gravitational interactions are responsible for the unitarity of the scattering against the horizon, as dictated by the holographic principle, but the Standard Model interactions also contribute, and understanding their effects is an important first step towards a complete understanding of the horizon's dynamics. The relation between in- and outgoing states is described in terms of an operator algebra. In this paper, the first of a series, we describe the algebra induced on the horizon by U(1) vector fields and scalar fields, including the case of an Englert-Brout-Higgs mechanism, and a more careful consideration of the transverse vector field components. 
  We calculate the expectation values of the energy-momentum tensor T_{{\mu}{\nu}} for massive scalar and spinor fields, in the Minkowski-like vacuum states on the two flat spaces which are quotients of Minkowski space under the discrete isometries (t,x,y,z)\mapsto(t,x,y,z+2a) and (t,x,y,z)\mapsto(t,-x,-y,z+a).   The results on the first space confirm the literature.   The results on the second space are new. We note some qualitative differences between the massless and massive fields in the limits of large a and large x^2+y^2. 
  The scope of this talk is to present some preliminary results on an effort, currently in progress, to generate an exact solution of Einstein's equation, suitable for describing spacetime around a rotating compact object. Specifically, the form of the Ernst potential on the symmetry axis and its connection with the multipole moments is discussed thoroughly. The way to calculate the multipole moments of spacetime directly from the value of the Ernst potential on the symmetry axis is presented. Finally, a mixed ansatz is formed for the Ernst potential including parameters additional to the ones dictated by Sibgatullin. Thus, we believe that this talk can also serve as a comment on choosing the appropriate ansatz for the Ernst potential. 
  A direct pathway from Hilbert's ``Foundation of Physics'' to Quantum Gravity is established through Dirac's Hamiltonian reduction of General Relativity and Bogoliubov's transformation by analogy with a similar pathway passed by QFT in 20th century. The cosmological scale factor appears on this pathway as a zero mode of the momentum constraints treated as a global excitation of the Landau superfluid liquid type. This approach would be considered as the foundation of the well--known Lifshitz cosmological perturbation theory, if it did not contain the double counting of the scale factor as an obstruction to the Dirac Hamiltonian method. After avoiding this ``double counting'' the Hamiltonian cosmological perturbation theory does not contain the time derivatives of gravitational potentials that are responsible for the CMB ``primordial power spectrum'' in the inflationary model.   The Hilbert -- Dirac -- Bogoliubov Quantum Gravity gives us another possibility to explain this ``spectrum'' and other topical problems of cosmology by the cosmological creation of both universes and particles from Bogoliubov's vacuum. We listed the set of theoretical and observational arguments in favor of that the CMB radiation can be a final product of primordial vector W-, Z- bosons cosmologically created from the vacuum when their Compton length coincides with the universe horizon. The equations describing longitudinal vector bosons in SM, in this case, are close to the equations of the inflationary model used for description of the ``power primordial spectrum'' of the CMB radiation. 
  The pattern of acoustic peaks in the sub-horizon power spectrum of primordial density anisotropies at recombination can be naturally understood in the framework of standard Friedmann-Robertson-Walker cosmology (without inflation) as a consequence of the boundary conditions imposed by the causal horizon on the statistical two-points correlation functions: the sub-horizon spectrum is discrete (harmonic), with comoving modes located at $k_n = n \frac{\pi}{H^{-1}_{eq}}$, $n = 1,2,...$, because the causally connected patch of the universe at recombination is compact, with comoving radius $H^{-1}_{eq}$. The results presented in this paper complement those presented in [1], where it was shown that the scale invariance of the primordial density anisotropies over comoving scales of cosmological size is also a consequence of the boundary conditions imposed by causality. Together these results lay an appealing theoretical alternative to the inflationary paradigm as the ultimate answer for the origin of cosmological structures in standard cosmology. 
  The leading quantum correction to Einstein-Hilbert Hamiltonian coming from large volume vacuum isotropic loop quantum cosmology, is independent of quantization ambiguity parameters. It is shown here that this correction can be viewed as finite volume gravitational Casimir energy due to one-loop `graviton' contributions. In vacuum case sub-leading quantum corrections and in non-vacuum case even leading quantum correction depend on ambiguity parameters. It may be recalled that these are in fact analogous features of perturbative quantum gravity where it is well-known that pure gravity (on-shell) is one-loop finite whereas higher-loops contributions are not even renormalizable. These features of the quantum corrections coming from non-perturbative quantization, sheds a new light on a major open issue; how to communicate between non-perturbative and perturbative approaches of quantum gravity. 
  The Isolated Horizons (IH) formalism, together with a simple phenomenological model for colored black holes has been used to predict non-trivial formulae that relate the ADM mass of the solitons and hairy Black Holes of Gravity-Matter system on the one hand, and several horizon properties of the black holes in the other. In this article, the IH formalism is tested numerically for spherically symmetric solutions to an Einstein-Higgs system where hairy black holes were recently found to exist. It is shown that the mass formulae still hold and that, by appropriately extending the current model, one can account for the behavior of the horizon properties of these new solutions. An empirical formula that approximates the ADM mass of hairy solutions is put forward, and some of its properties are analyzed. 
  We study ``draining bathtub'' as an acoustic analogue of a three-dimensional rotating black hole. Rotating fluid near the sonic horizon necessarily gives rise to the superradiant modes, which are partially responsible for the thermodynamic quantities in this rotating vortex-like hole. Using the recently suggested thin-layer method overcoming some difficulties from the well-known brick-wall method, we explicitly calculate the free energy of the system by treating the superradiance carefully and obtain the desirable entropy formula. 
  We present the status of the joint search for gravitational waves from inspiraling neutron star binaries in the LIGO Science Run 2 and TAMA300 Data Taking Run 8 data, which was taken from February 14 to April 14, 2003, by the LIGO and TAMA collaborations. In this paper we discuss what has been learned from an analysis of a subset of the data sample reserved as a ``playground''. We determine the coincidence conditions for parameters such as the coalescence time and chirp mass by injecting simulated Galactic binary neutron star signals into the data stream. We select coincidence conditions so as to maximize our efficiency of detecting simulated signals. We obtain an efficiency for our coincident search of 78 %, and show that we are missing primarily very distant signals for TAMA300. We perform a time slide analysis to estimate the background due to accidental coincidence of noise triggers. We find that the background triggers have a very different character from the triggers of simulated signals. 
  In hyperbolic reductions of the Einstein equations the evolution of gauge conditions or constraint quantities is controlled by subsidiary systems. We point out a class of non-linearities in these systems which may have the potential of generating catastrophic growth of gauge resp. constraint violations in numerical calculations. 
  The general class of Robinson-Trautman metrics that describe gravitational radiation in the exterior of bounded sources in four space-time dimensions is shown to admit zero curvature formulation in terms of appropriately chosen two-dimensional gauge connections. The result, which is valid for either type II or III metrics, implies that the gravitational analogue of the Lienard-Wiechert fields of Maxwell equations form a new integrable sector of Einstein equations for any value of the cosmological constant. The method of investigation is similar to that used for integrating the Ricci flow in two dimensions. The zero modes of the gauge symmetry (factored by the center) generate Kac's K_2 simple Lie algebra with infinite growth. 
  We propose a spin foam model of four-dimensional quantum gravity which is based on the integration of the tetrads in the path integral for the Palatini action of General Relativity. In the Euclidian gravity case we show that the model can be understood as a modification of the Barrett-Crane spin foam model. Fermionic matter can be coupled by using the path integral with sources for the tetrads and the spin connection, and the corresponding state sum is based on a spin foam where both the edges and the faces are colored independently with the irreducible representations of the spacetime rotations group. 
  Gravitational and hydrodynamical perturbations are analysed in a relativistic plasma containing a mixture of interacting fluids characterized by a non-negligible bulk viscosity coefficient. The energy-momentum transfer between the cosmological fluids, as well as the fluctuations of the bulk viscosity coefficients, are analyzed simultaneously with the aim of deriving a generalized set of evolution equations for the entropy and curvature fluctuations. For typical length scales larger than the Hubble radius, the fluctuations of the bulk viscosity coefficients and of the decay rate provide source terms for the evolution of both the curvature and the entropy fluctuations. According to the functional dependence of the bulk viscosity coefficient on the energy densities of the fluids composing the system, the mixing of entropy and curvature perturbations is scrutinized both analytically and numerically. 
  The vacuum decay in a de Sitter universe is studied within semiclassical approximation for the class of effective inflaton potentials whose curvature at the top is close to a critical value. By comparing the actions of the Hawking - Moss instanton and the Coleman - de Luccia instanton(s) the mode of vacuum decay is determined. The case when the fourth derivative of the effective potential at its top is less than a critical value is discussed. 
  The thermo-mechanical properties of silicon make it of significant interest as a possible material for mirror substrates and suspension elements for future long-baseline gravitational wave detectors. The mechanical dissipation in 92um thick <110> single-crystal silicon cantilevers has been observed over the temperature range 85 K to 300 K, with dissipation approaching levels down to phi = 4.4E-7. 
  One of the firm predictions of inflationary cosmology is the consistency relation between scalar and tensor spectra. It has been argued that such a relation -if experimentally confirmed- would offer strong support for the idea of inflation. We examine the possibility that trans-Planckian physics violates the consistency relation in the framework of inflation with a cut-off proposed in astro-ph/0009209. We find that despite the ambiguity that exists in choosing the action, Planck scale physics modifies the consistency relation considerably. It also leads to the running of the spectral index. For modes that are larger than our current horizon, the tensor spectral index is positive. For a window of k values with amplitudes of the same order of the modes which are the precursor to structure formation, the behavior of tensor spectral index is oscillatory about the standard Quantum Field theory result, taking both positive and negative values. There is a hope that in the light of future experiments, one can verify this scenario of short distance physics. 
  We present a matrix method for obtaining new classes of exact solutions for Einstein's equations representing static perfect fluid spheres. By means of a matrix transformation, we reduce Einstein's equations to two independent Riccati type differential equations for which three classes of solutions are obtained. One class of the solutions corresponding to the linear barotropic type fluid with an equation of state $p=\gamma \rho $ is discussed in detail. 
  We shall in the framework of Bohmian quantum gravity show that it is possible to find a {\it pure} quantum state which leads to the static Einstein universe whose classical counterpart is flat space--time. We obtain the solution not only in the long--wavelength approximation but also exactly. 
  In this paper we have investigated the classical limit in Bohmian quantum cosmology. It is observed that in the quantum regime where the quantum potential is greater than the classical one, one has an expansion in terms of negative powers of the Planck constant. But in the classical limit there are regions having positive powers of the Planck constant, and regions having negative powers and also regions having both. The conclusion is that the Bohmian classical limit cannot be obtained by letting the Planck constant goes to zero. 
  We investigate the response of the traversable wormholes to the external perturbations through finding their characteristic frequencies and time-domain profiles. The considered solution describes traversable wormholes between the branes in the two brane Randall-Sundrum model and was previously found within Einstein gravity with a conformally coupled scalar field. The evolution of perturbations of a wormhole is similar to that of a black hole and represents damped oscillations (ringing) at intermediately late times, which are suppressed by power law tails (proportional to t^{-2} for monopole perturbations) at asymptotically late times. 
  A world-wide network of interferometric gravitational wave detectors is currently operational. The detectors in the network are still in their commissioning phase and are expected to achieve their design sensitivity over the next year or so. Each detector is a complex instrument involving many subsystems and each subsystem is a source of noise at the output of the detector. Therefore, in addition to recording the main gravitational wave data channel at the output of the interferometer, the state of each detector subsystem is monitored and recorded. This subsidiary data is both large in volume as well as complex in nature. We require an online monitoring and analysis tool which can process all the data channels for various noise artefacts and summarize the results of the analysis in a manner that can be accessed and interpreted conveniently.    In this paper we describe the GEO600 Online Detector Characterization System (GODCS), which is the tool that is being used to monitor the output of the GEO600 gravitational wave detector situated near Hannover in Germany. We describe the various algorithms that we use and how the results of several algorithms can be combined to make meaningful statements about the state of the detector. This paper will be useful to researchers in the area of gravitational wave astronomy as a record of the various analyses and checks carried out to ensure the quality and reliability of the data before searching the data for the presence of gravitational waves. 
  Motion of classical point-like impurities in the homogeneous Einstein condensate of bosons is studied in the framework of second quantization method. A toy model is proposed and its general solution within the Bogoliubov approximation is obtained. The effective Minkowski space-time structure arises naturally in this non-relativistic quantum many-body system in the low energy regime. This is shown to be true in this model. Several examples are discussed in order to illustrate our model. The homogeneous condensate produces an effective Yukawa type attractive force between impurities sitting in condensate. Landau's criterion is naturally derived in a case of linear motion of impurity. The analytic expressions for spectra of Bogoliubov excitations produced by the accelerated motions of impurities are obtained. A quick look at the analytic expression reveals that the spectrum of gapless excitations emitted by the linearly accelerated impurity {\it is not thermal}. If the homogeneous condensate is the physically correct model for Minkowski space-time then it follows that the apparent thermal response of the simple linearly accelerated detector models may be the result of improper regularization. 
  The conformal thin sandwich (CTS) equations are a set of four of the Einstein equations, which generalize the Laplace-Poisson equation of Newton's theory. We examine numerically solutions of the CTS equations describing perturbed Minkowski space, and find only one solution. However, we find {\em two} distinct solutions, one even containing a black hole, when the lapse is determined by a fifth elliptic equation through specification of the mean curvature. While the relationship of the two systems and their solutions is a fundamental property of general relativity, this fairly simple example of an elliptic system with non-unique solutions is also of broader interest. 
  We search for acoustic analogues of a spherical symmetric black hole with a pointlike source. We show that the gravitational system has a dynamical counterpart in the constrained, steady motion of a fluid with a planar source. The equations governing the dynamics of the gravitational system can be exactly mapped in those governing the motion of the fluid. The different meaning that singularities and sources have in fluid dynamics and in general relativity is also discussed. Whereas in the latter a pointlike source is always associated with a (curvature) singularity in the former the presence of sources does not necessarily imply divergences of the fields. 
  We study the consequences of the existence of timelike and spacelike Ricci collineation vectors (RCVs) for string fluid in the context of general relativity. Necessary and sufficient conditions are derived for a space-time with string fluid to admit a timelike RCV, parallel to $u^a$, and a spacelike RCV, parallel to $n^a$. In these cases, some results obtained are discussed. 
  The space-based gravitational-wave observatory LISA, a NASA-ESA mission to be launched after 2012, will achieve its optimal sensitivity using Time Delay Interferometry (TDI), a LISA-specific technique needed to cancel the otherwise overwhelming laser noise in the inter-spacecraft phase measurements. The TDI observables of the Michelson and Sagnac types have been interpreted physically as the virtual measurements of a synthesized interferometer. In this paper, I present Geometric TDI, a new and intuitive approach to extend this interpretation to all TDI observables. Unlike the standard algebraic formalism, Geometric TDI provides a combinatorial algorithm to explore exhaustively the space of second-generation TDI observables (i.e., those that cancel laser noise in LISA-like interferometers with time-dependent armlengths). Using this algorithm, I survey the space of second-generation TDI observables of length (i.e., number of component phase measurements) up to 24, and I identify alternative, improved forms of the standard second-generation TDI observables. The alternative forms have improved high-frequency gravitational-wave sensitivity in realistic noise conditions (because they have fewer nulls in the gravitational-wave and noise response functions), and are less susceptible to instrumental gaps and glitches (because their component phase measurements span shorter time periods). 
  The relative motion of a classical relativistic spinning test particle is studied with respect to a nearby free test particle in the gravitational field of a rotating source. The effects of the spin-curvature coupling force are elucidated and the implications of the results for the motion of rotating plasma clumps in astrophysical jets are discussed. 
  Loop quantum gravity is an approach to quantum gravity that starts from the Hamiltonian formulation in terms of a connection and its canonical conjugate. Quantization proceeds in the spirit of Dirac: First one defines an algebra of basic kinematical observables and represents it through operators on a suitable Hilbert space. In a second step, one implements the constraints. The main result of the paper concerns the representation theory of the kinematical algebra: We show that there is only one cyclic representation invariant under spatial diffeomorphisms.   While this result is particularly important for loop quantum gravity, we are rather general: The precise definition of the abstract *-algebra of the basic kinematical observables we give could be used for any theory in which the configuration variable is a connection with a compact structure group. The variables are constructed from the holonomy map and from the fluxes of the momentum conjugate to the connection. The uniqueness result is relevant for any such theory invariant under spatial diffeomorphisms or being a part of a diffeomorphism invariant theory. 
  In this work the definition of a quasilocal energy for four dimensional first order gravity is developed. Using this an action principle which is adequate for the canonical ensemble is obtained. The microcanonical action principle is obtained as well. 
  We show that the well-known NUT solution can be correctly interpreted as describing the exterior field of two counter-rotating semi-infinite sources possessing negative masses and infinite angular momenta which are attached to the poles of a static finite rod of positive mass. 
  In classical general relativity, the generic approach to the initial singularity is usually understood in terms of the BKL scenario. In this scenario, along with the Bianchi IX model, the exact, singular, Kasner solution of vacuum Bianchi I model also plays a pivotal role. Using an effective classical Hamiltonian obtained from loop quantization of vacuum Bianchi I model, exact solution is obtained which is non-singular due to a discreteness parameter. The solution is parameterized in exactly the same manner as the usual Kasner solution and reduces to the Kasner solution as discreteness parameter is taken to zero. At the effective Hamiltonian level, the avoidance of Kasner singularity uses a mechanism distinct from the `inverse volume' modifications characteristic of loop quantum cosmology. 
  Black hole solutions in higher dimensional Einstein and Einstein-Maxwell gravity have been discussed by Tangherlini as well as Myers and Perry a long time ago. These solutions are the generalizations of the familiar Schwarzschild, Reissner-Nordstrom and Kerr solutions of four-dimensional general relativity. However, higher dimensional generalization of the Kerr-Newman solution in four dimensions has not been found yet. As a first step in this direction I shall report on a new solution of the Einstein-Maxwell system of equations that describes an electrically charged and slowly rotating black hole in five dimensions. 
  The crease set of an event horizon is studied in a spacetime with discrete or continuous symmetry. It determines possible topologies on spatial sections of an event horizon. We thereby investigate the classification of stable topological structure of the crease sets in a spacetime with any symmetry. In practice, we show the classification of the crease set in axially symmetric spacetime. By that we realize the topological structure of axially symmetric event horizons. We will finc that many new topological structures become stable which is not stable without symmetry. 
  We produce an analog of the Davies-Unruh effect for superfluid helium four. There are two temperatures which result--one is associated with ordinary, or first, sound and the other with second sound. 
  Exact solutions to static and non-static Einstein-Maxwell equations in the presence of extremely charged dust embedded on thin shells are constructed. Singularities of multi-black hole Majumdar-Papapetrou and Kastor-Trashen solutions are removed by placing the matter on thin shells. Double spherical thin shell solution is given as an illustration and the matter densitiies on the shells are derived. 
  New solutions to the static, spherically symmetric Einstein-Yang-Mills-Higgs equations with the Higgs field in the triplet resp. doublet representation are presented. They form continuous families parametrized by $\alpha=M_W/M_Pl$ ($M_W$ resp. $M_Pl$ denoting the W-boson resp. the Planck mass). The corresponding spacetimes are regular and have spatially compact sections. A particularly interesting class with the Yang-Mills amplitude being nodeless is exhibited and is shown to be linearly stable with respect to spherically symmetric perturbations. For some solutions with nodes of the Yang-Mills amplitude a new stabilization phenomenon is found, according to which their unstable modes disappear as $\alpha$ increases (for the triplet) or decreases (for the doublet). 
  We propose an extension of the original thought experiment proposed by Geroch, which sparked much of the actual debate and interest on black hole thermodynamics, and show that the generalized second law of thermodynamics is in compliance with it. 
  The results on cosmic rays detected by the gravitational antenna NAUTILUS have motivated an experiment (RAP) based on a suspended cylindrical bar, which is made of the same aluminum alloy as NAUTILUS and is exposed to a high energy electron beam. Mechanical vibrations originate from the local thermal expansion caused by warming up due to the energy lost by particles crossing the material. The aim of the experiment is to measure the amplitude of the fundamental longitudinal vibration at different temperatures. We report on the results obtained down to a temperature of about 4 K, which agree at the level of about 10% with the predictions of the model describing the underlying physical process. 
  We study the consequences of the existence of spacelike Ricci inheritance vectors (SpRIVs) parallel to $x^a$ for model of string cloud and string fluid stress tensor in the context of general relativity. Necessary and sufficient conditions are derived for a spacetime with a model of string cloud and string fluid stress tensor to admit a SpRIV and a SpRIV which is also a spacelike conformal Killing vector (SpCKV). Also, some results are obtained. 
  The quadrupole formula in four-dimensional Einstein gravity is a useful tool to describe gravitational wave radiation. We derive the quadrupole formula for the Kaluza-Klein (KK) modes in the Randall-Sundrum braneworld model. The quadrupole formula provides transparent representation of the exterior weak gravitational field induced by localized sources. We find that a general isolated dynamical source gives rise to the 1/r^2 correction to the leading 1/r gravitational field. We apply the formula to an evaluation of the effective energy carried by the KK modes from the viewpoint of an observer on the brane. Contrary to the ordinary gravitational waves (zero mode), the flux of the induced KK modes by the non-spherical part of the quadrupole moment vanishes at infinity and only the spherical part contributes to the flux. Since the effect of the KK modes appears in the linear order of the metric perturbations, the effective energy flux observed on the brane is not always positive, but can become negative depending on the motion of the localized sources. 
  The scale transformation laws produce, on the motion equations of gravitating bodies and under some peculiar assumptions, effects which are anologous to those of a "macroscopic quantum mechanics". When we consider time and space scales such that the description of the trajectories of these bodies (planetesimals in the case of planetary system formation, interstellar gas and dust in the case of star formation, etc...) is in the shape of non-differentiable curves, we obtain fractal curves of fractal dimension 2. Continuity and non-differentiability yield a fractal space and a symmetry breaking of the differential time element which gives a doubling of the velocity fields. The application of a geodesics principle leads to motion equations of Schrodinger-type. When we add an outside gravitational field, we obtain a Schrodinger-Poisson system. We give here the derivation of the Schrodinger equation for chaotic systems, i.e., with time scales much longer than their Lyapounov chaos-time. 
  We investigate the canonical quantization of supergravity N=1 in the case of a midisuperspace described by Gowdy $T^3$ cosmological models. The quantum constraints are analyzed and the wave function of the universe is derived explicitly. Unlike the minisuperspace case, we show the existence of physical states in midisuperspace models. The analysis of the wave function of the universe leads to the conclusion that the classical curvature singularity present in the evolution of Gowdy models is removed at the quantum level due to the presence of the Rarita-Schwinger field. 
  The transport equations for polarized radiation transfer in non-Riemannian, Weyl-Cartan type space-times are derived, with the effects of both torsion and non-metricity included. To obtain the basic propagation equations we use the tangent bundle approach. The equations describing the time evolution of the Stokes parameters, of the photon distribution function and of the total polarization degree can be formulated as a system of coupled first order partial differential equations. As an application of our results we consider the propagation of the cosmological gamma ray bursts in spatially homogeneous and isotropic spaces with torsion and non-metricity. For this case the exact general solution of the equation for the polarization degree is obtained, with the effects of the torsion and non-metricity included. The presence of a non-Riemannian geometrical background in which the electromagnetic fields couple to torsion and/or non-metricity affect the polarization of photon beams. Consequently, we suggest that the observed polarization of prompt cosmological gamma ray bursts and of their optical afterglows may have a propagation effect component, due to a torsion/non-metricity induced birefringence of the vacuum. A cosmological redshift and frequency dependence of the polarization degree of gamma ray bursts also follows from the model, thus providing a clear observational signature of the torsional/non-metric effects. On the other hand, observations of the polarization of the gamma ray bursts can impose strong constraints on the torsion and non-metricity and discriminate between different theoretical models. 
  There has been recent speculation that the tunneling paradigm for Hawking radiation could -- after quantum-gravitational effects have suitably been incorporated -- provide a means for resolving the (black hole) information loss paradox. A prospective quantum-gravitational effect is the logarithmic-order correction to the Bekenstein-Hawking entropy/area law. In this letter, it is demonstrated that, even with the inclusion of the logarithmic correction (or, indeed, the quantum correction up to any perturbative order), the tunneling formalism is still unable to resolve the stated paradox. Moreover, we go on to show that the tunneling framework effectively constrains the coefficient of this logarithmic term to be non-negative. Significantly, the latter observation implies the necessity for including the canonical corrections in the quantum formulation of the black hole entropy. 
  We show that the generic solutions of the Lovelock equations with spherical, planar or hyperbolic symmetry are locally isometric to the corresponding static Lovelock black hole. As a consequence, these solutions are locally static: they admit an additional Killing vector that can either be space-like or time-like, depending on the position. This result also holds in the presence of an abelian gauge field, in which case the solutions are locally isometric to a charged static black hole. 
  Coherent or semiclassical states in canonical quantum gravity describe the classical Schwarzschild space-time. By tracing over the coherent state wavefunction inside the horizon, a density matrix is derived. Bekenstein-Hawking entropy is obtained from the density matrix, modulo the Immirzi parameter. The expectation value of the area and curvature operator is evaluated in these states. The behaviour near the singularity of the curvature operator shows that the singularity is resolved. We then generalise the results to space-times with spherically symmetric apparent horizons. 
  The null infinity limit of the gravitational energy-momentum and energy flux determined by the covariant Hamiltonian quasi-local expressions is evaluated using the NP spin coefficients. The reference contribution is considered by three different embedding approaches. All of them give the expected Bondi energy and energy flux. 
  Starting from a critical analysis of recently reported surprisingly large uncertainties in length and position measurements deduced within the framework of quantum gravity, we embark on an investigation both of the correlation structure of Planck scale fluctuations and the role the holographic hypothesis is possibly playing in this context. While we prove the logical independence of the fluctuation results and the holographic hypothesis (in contrast to some recent statements in that direction) we show that by combining these two topics one can draw quite strong and interesting conclusions about the fluctuation structure and the microscopic dynamics on the Planck scale. We further argue that these findings point to a possibly new and generalized form of quantum statistical mechanics of strongly (anti)correlated systems of degrees of freedom in this fundamental regime. 
  In this Letter we demonstrate that any interaction of pressureless dark matter with holographic dark energy, whose infrared cutoff is set by the Hubble scale, implies a constant ratio of the energy densities of both components thus solving the coincidence problem. The equation of state parameter is obtained as a function of the interaction strength. For a variable degree of saturation of the holographic bound the energy density ratio becomes time dependent which is compatible with a transition from decelerated to accelerated expansion. 
  We present a simple method to derive the semiclassical equations of motion for a spinning particle in a gravitational field. We investigate the cases of classical, rotating particles (pole-dipole particles), as well as particles with intrinsic spin. We show that, starting with a simple Lagrangian, one can derive equations for the spin evolution and momentum propagation in the framework of metric theories of gravity and in theories based on a Riemann-Cartan geometry (Poincare gauge theory), without explicitly referring to matter current densities (spin and energy-momentum). Our results agree with those derived from the multipole expansion of the current densities by the conventional Papapetrou method and from the WKB analysis for elementary particles. 
  Korn's inequality plays an important role in linear elasticity theory. This inequality bounds the norm of the derivatives of the displacement vector by the norm of the linearized strain tensor. The kernel of the linearized strain tensor are the infinitesimal rigid-body translations and rotations (Killing vectors). We generalize this inequality by replacing the linearized strain tensor by its trace free part. That is, we obtain a stronger inequality in which the kernel of the relevant operator are the conformal Killing vectors. The new inequality has applications in General Relativity. 
  We review recent efforts to construct gravitational theories on discrete space-times, usually referred to as the ``consistent discretization'' approach. The resulting theories are free of constraints at the canonical level and therefore allow to tackle many problems that cannot be currently addressed in continuum quantum gravity. In particular the theories imply a natural method for resolving the big bang (and other types) of singularities and predict a fundamental mechanism for decoherence of quantum states that might be relevant to the black hole information paradox. At a classical level, the theories may provide an attractive new path for the exploration of issues in numerical relativity. Finally, the theories can make direct contact with several kinematical results of continuum loop quantum gravity. We review in broad terms several of these results and present in detail as an illustration the classical treatment with this technique of the simple yet conceptually challenging model of two oscillators with constant energy sum. 
  We present an exact solution to the Einstein field equations which is Ricci and Riemann flat in five dimensions, but in four dimensions is a good model for the early vacuum-dominated universe. 
  We develop a symmetric teleparallel gravity model in a space-time with only the non-metricity is nonzero, in terms of a Lagrangian quadratic in the non-metricity tensor. We present a detailed discussion of the variations that may be used for any gravitational formulation. We seek Schwarzschild-type solutions because of its observational significance and obtain a class of solutions that includes Schwarzschild-type, Schwarzschild-de Sitter-type and Reissner-Nordstr\"{o}m-type solutions for certain values of the parameters. We also discuss the physical relevance of these solutions. 
  We demonstrate the ``peeling property'' of the Weyl tensor in higher dimensions in the case of even dimensions (and with some additional assumptions), thereby providing a first step towards understanding of the general peeling behaviour of the Weyl tensor, and the asymptotic structure at null infinity, in higher dimensions. 
  We show that in multidimensional gravity vector fields completely determine the structure and properties of singularity. It turns out that in the presence of a vector field the oscillatory regime exists in all spatial dimensions and for all homogeneous models. By analyzing the Hamiltonian equations we derive the Poincar\'e return map associated to the Kasner indexes and fix the rules according to which the Kasner vectors rotate. In correspondence to a 4-dimensional space time, the oscillatory regime here constructed overlap the usual Belinski-Khalatnikov-Liftshitz one. 
  Beetle and Burko recently introduced a background--independent scalar curvature invariant for general relativity that carries information only about the gravitational radiation in generic spacetimes, in cases where such radiation is incontrovertibly defined. In this paper we adopt a formalism that only uses spatial data as they are used in numerical relativity and compute the Beetle--Burko radiation scalar for a number of analytical examples, specifically linearized Einstein--Rosen cylindrical waves, linearized quadrupole waves, the Kerr spacetime, Bowen--York initial data, and the Kasner spacetime. These examples illustrate how the Beetle--Burko radiation scalar can be used to examine the gravitational wave content of numerically generated spacetimes, and how it may provide a useful diagnostic for initial data sets. 
  We perform a search for gravitational wave bursts using data from the second science run of the LIGO detectors, using a method based on a wavelet time-frequency decomposition. This search is sensitive to bursts of duration much less than a second and with frequency content in the 100-1100Hz range. It features significant improvements in the instrument sensitivity and in the analysis pipeline with respect to the burst search previously reported by LIGO. Improvements in the search method allow exploring weaker signals, relative to the detector noise floor, while maintaining a low false alarm rate, O(0.1) microHz. The sensitivity in terms of the root-sum-square (rss) strain amplitude lies in the range of hrss~10^{-20} - 10^{-19}/sqrt(Hz) No gravitational wave signals were detected in 9.98 days of analyzed data. We interpret the search result in terms of a frequentist upper limit on the rate of detectable gravitational wave bursts at the level of 0.26 events per day at 90% confidence level. We combine this limit with measurements of the detection efficiency for given waveform morphologies in order to yield rate versus strength exclusion curves as well as to establish order-of-magnitude distance sensitivity to certain modeled astrophysical sources. Both the rate upper limit and its applicability to signal strengths improve our previously reported limits and reflect the most sensitive broad-band search for untriggered and unmodeled gravitational wave bursts to date. 
  Loop quantization of diagonalized Bianchi class A models, leads to a partial difference equation as the Hamiltonian constraint at the quantum level. A criterion for testing a viable semiclassical limit has been formulated in terms of existence of the so-called pre-classical solutions. We demonstrate the existence of pre-classical solutions of the quantum equation for the vacuum Bianchi I model. All these solutions avoid the classical singularity at vanishing volume. 
  The violation of the general covariance is proposed as a resource of the gravitational dark matter. The minimal violation of the covariance to the unimodular one is associated with the massive scalar graviton as the simplest representative of such a matter. The Lagrangian formalism for the continuous medium, the perfect fluid in particular, in the scalar graviton environment is developed. The implications for cosmology are shortly indicated. 
  Loop Quantum Cosmology (LQC), mainly due to Bojowald, is not the cosmological sector of Loop Quantum Gravity (LQG). Rather, LQC consists of a truncation of the phase space of classical General Relativity to spatially homogeneous situations which is then quantized by the methods of LQG. Thus, LQC is a quantum mechanical toy model (finite number of degrees of freedom) for LQG(a genuine QFT with an infinite number of degrees of freedom) which provides important consistency checks. However, it is a non trivial question whether the predictions of LQC are robust after switching on the inhomogeneous fluctuations present in full LQG. Two of the most spectacular findings of LQC are that 1. the inverse scale factor is bounded from above on zero volume eigenstates which hints at the avoidance of the local curvature singularity and 2. that the Quantum Einstein Equations are non -- singular which hints at the avoidance of the global initial singularity. We display the result of a calculation for LQG which proves that the (analogon of the) inverse scale factor, while densely defined, is {\it not} bounded from above on zero volume eigenstates. Thus, in full LQG, if curvature singularity avoidance is realized, then not in this simple way. In fact, it turns out that the boundedness of the inverse scale factor is neither necessary nor sufficient for curvature singularity avoidance and that non -- singular evolution equations are neither necessary nor sufficient for initial singularity avoidance because none of these criteria are formulated in terms of observable quantities.After outlining what would be required, we present the results of a calculation for LQG which could be a first indication that our criteria at least for curvature singularity avoidance are satisfied in LQG. 
  In this paper we deliver the proofs for the claims, made in a companion paper, concerning the avoidance of cosmological curvature singularities in in full Loop Quantum Gravity (LQG). 
  We present exact solutions of the gravitational field equations in the generalized Randall-Sundrum model for an anisotropic brane with Bianchi type I geometry, with a generalized Chaplygin gas as matter source. The generalized Chaplygin gas, which interpolates between a high density relativistic era and a non-relativistic matter phase, is a popular dark energy candidate. For a Bianchi type I space-time brane filled with a cosmological fluid obeying the generalized Chaplygin equation of state the general solution of the gravitational field equations can be expressed in an exact parametric form, with the comoving volume taken as parameter. In the limiting cases of a stiff cosmological fluid, with pressure equal to the energy density, and for a pressureless fluid, the solution of the field equations can be expressed in an exact analytical form. The evolution of the scalar field associated to the Chaplygin fluid is also considered and the corresponding potential is obtained. The behavior of the observationally important parameters like shear, anisotropy and deceleration parameter is considered in detail. 
  By using a linearized non-vacuum late time solution in Brans-Dicke cosmology we account for the seventy five percent dark energy contribution but not for approximately twenty-three percent dark matter contribution to the present day energy density of the universe. 
  We study even parity metric and matter perturbations of all angular modes in self-similar Vaidya space-time. We focus on the case where the background contains a naked singularity. Initial conditions are imposed describing a finite perturbation emerging from the portion of flat space-time preceding the matter-filled region of space-time. The most general perturbation satisfying the initial conditions is allowed impinge upon the Cauchy horizon (CH), whereat the perturbation remains finite: there is no ``blue-sheet'' instability. However when the perturbation evolves through the CH and onto the second future similarity horizon of the naked singularity, divergence necessarily occurs: this surface is found to be unstable. The analysis is based on the study of individual modes following a Mellin transform of the perturbation. We present an argument that the full perturbation remains finite after resummation of the (possibly infinite number of) modes. 
  If our universe underwent inflation, its entropy during the inflationary phase was substantially lower than it is today. Because a low-entropy state is less likely to be chosen randomly than a high-entropy one, inflation is unlikely to arise through randomly-chosen initial conditions. To resolve this puzzle, we examine the notion of a natural state for the universe, and argue that it is a nearly-empty spacetime. If empty space has a small vacuum energy, however, inflation can begin spontaneously in this background. This scenario explains why a universe like ours is likely to have begun via a period of inflation, and also provides an origin for the cosmological arrow of time. 
  The holonomy-flux *-algebra was recently proposed as an algebra of basic kinematical observables for loop quantum gravity. We show the conventional GNS construction breaks down when the the holonomy-flux *-algebra is allowed to be a Jordan algebra of observables. To remedy this, we give a Jordan GNS construction for the holonomy-flux *-algebra that is based on trace. This is accomplished by assuming the holonomy-flux *-algebra is an algebra of observables that is also a Banach algebra, hence a JB algebra. We show the Jordan GNS construction produces a state that is invariant under all inner derivations of the holonomy-flux *-algebra. Implications for the corresponding Jordan-Schrodinger equation are also discussed. 
  We construct all independent local scalar monomials in the Riemann tensor at arbitrary dimension, for the special regime of static, spherically symmetric geometries. Compared to general spaces, their number is significantly reduced: the extreme example is the collapse of all invariants ~ Weyl^k, to a single term at each k. The latter is equivalent to the Lovelock invariant L_k. Depopulation is less extreme for invariants involving rising numbers of Ricci tensors, and also depends on the dimension. The corresponding local gravitational actions and their solution spaces are discussed. 
  After a historical introduction to Poisson's equation for Newtonian gravity, its analog for static gravitational fields in Einstein's theory is reviewed. It appears that the pressure contribution to the active mass density in Einstein's theory might also be noticeable at the Newtonian level. A form of its surprising appearance, first noticed by Richard Chase Tolman, was discussed half a century ago in the Hamburg Relativity Seminar and is resolved here. 
  We use 373 hours ($\approx$ 15 days) of data from the second science run of the LIGO gravitational-wave detectors to search for signals from binary neutron star coalescences within a maximum distance of about 1.5 Mpc, a volume of space which includes the Andromeda Galaxy and other galaxies of the Local Group of galaxies. This analysis requires a signal to be found in data from detectors at the two LIGO sites, according to a set of coincidence criteria. The background (accidental coincidence rate) is determined from the data and is used to judge the significance of event candidates. No inspiral gravitational wave events were identified in our search. Using a population model which includes the Local Group, we establish an upper limit of less than 47 inspiral events per year per Milky Way equivalent galaxy with 90% confidence for non-spinning binary neutron star systems with component masses between 1 and 3 $M_\odot$. 
  We use data from the second science run of the LIGO gravitational-wave detectors to search for the gravitational waves from primordial black hole (PBH) binary coalescence with component masses in the range 0.2--$1.0 M_\odot$. The analysis requires a signal to be found in the data from both LIGO observatories, according to a set of coincidence criteria. No inspiral signals were found. Assuming a spherical halo with core radius 5 kpc extending to 50 kpc containing non-spinning black holes with masses in the range 0.2--$1.0 M_\odot$, we place an observational upper limit on the rate of PBH coalescence of 63 per year per Milky Way halo (MWH) with 90% confidence. 
  We apply the consistent discretization scheme to general relativity particularized to the Gowdy space-times. This is the first time the framework has been applied in detail in a non-linear generally-covariant gravitational situation with local degrees of freedom. We show that the scheme can be correctly used to numerically evolve the space-times. We show that the resulting numerical schemes are convergent and preserve approximately the constraints as expected. 
  We calculate analytically the highly damped quasinormal mode spectra of generic single-horizon black holes using the rigorous WKB techniques of Andersson and Howls\cite{Andersson}. We thereby provide a firm foundation for previous analysis, and point out some of their possible limitations. The numerical coefficient in the real part of the highly damped frequency is generically determined by the behavior of coupling of the perturbation to the gravitational field near the origin, as expressed in tortoise coordinates. This fact makes it difficult to understand how the famous $ln(3)$ could be related to the quantum gravitational microstates near the horizon. 
  The 2+1-dimensional geodesic circularly symmetric solutions of Einstein-massless-scalar field equations with negative cosmological constant are found and their local and global properties are studied. It is found that one of them represents gravitational collapse where black holes are always formed. 
  We present in this work the study of the linear perturbations of the 2+1-dimensional circularly symmetric solution, obtained in a previous work, with kinematic self-similarity of the second kind. We have obtained an exact solution for the perturbation equations and the possible perturbation modes. We have shown that the background solution is a stable solution. 
  It seems to be not well known that the metrics of general relativity (GR) can be obtained without integrating Einstein equations. To that, we need only define a unit for GR-interval $\Delta s$, and observe 10 geodesics (out of which at least one must be nonnull). Even without using any unit, we can have $\kappa g_{\mu\nu}(x^\rho)$, where $\kappa=$const. Our notes attempt to simplify the articles of E. Kretschmann (1917) and of H.A. Lorentz (1923) about this last subject. The text of this article in English will soon be available, in LaTeX. Please ask the author.   -----   /Sajne estas malmulte konata ke la metrikoj de /generala relativeco (/GR) povas esti havataj sen integri Einstein-ajn ekvaciojn. Por tio, ni bezonas difini nur unuon por /GR-tempo $\Delta s$, kaj observi 10 geodezajn (el kiuj, almena/u unu devas esti nenulan). E/c sen uzi iun unuon, ni povas havi $\kappa g_{\mu\nu}(x^\rho)$, kie $\kappa$=konst. Niaj notoj tentas simpligi la artikolojn de E. Kretschmann (1917) kaj de H.A. Lorentz (1923) pri tiu lasta afero. 
  We consider the familiar junction conditions described by Israel for thin timelike walls in Einstein-Hilbert gravity. One such condition requires the induced metric to be continuous across the wall. Now, there are many spacetimes with sources confined to a thin wall for which this condition is violated and the Israel formalism does not apply. However, we explore the conjecture that the induced metric is in fact continuous for any thin wall which models spacetimes containing only positive energy matter. Thus, the usual junction conditions would hold for all positive energy spacetimes. This conjecture is proven in various special cases, including the case of static spacetimes with spherical or planar symmetry as well as settings without symmetry which may be sufficiently well approximated by smooth spacetimes with well-behaved null geodesic congruences. 
  Classical black holes and event horizons are highly non-local objects, defined in terms of the causal past of future null infinity. Alternative, (quasi)local definitions are often used in mathematical, quantum, and numerical relativity. These include apparent, trapping, isolated, and dynamical horizons, all of which are closely associated to two-surfaces of zero outward null expansion. In this paper we show that three-surfaces which can be foliated with such two-surfaces are suitable boundaries in both a quasilocal action and a phase space formulation of general relativity. The resulting formalism provides expressions for the quasilocal energy and angular momentum associated with the horizon. The values of the energy and angular momentum are in agreement with those derived from the isolated and dynamical horizon frameworks. 
  Several results of black holes thermodynamics can be considered as firmly founded and formulated in a very general manner. From this starting point we analyse in which way these results may give us the opportunity to gain a better understanding in the thermodynamics of ordinary systems for which a pre-relativistic description is sufficient. First, we investigated the possibility to introduce an alternative definition of the entropy basically related to a local definition of the order in a spacetime model rather than a counting of microstates. We show that such an alternative approach exists and leads to the traditional results provided an equilibrium condition is assumed. This condition introduces a relation between a time interval and the reverse of the temperature. We show that such a relation extensively used in the black hole theory, mainly as a mathematical trick, has a very general and physical meaning here; in particular its derivation is not related to the existence of a canonical density matrix. Our dynamical approach of thermodynamic equilibrium allows us to establish a relation between action and entropy and we show that an identical relation exists in the case of black holes. The derivation of such a relation seems impossible in the Gibbs ensemble approach of statistical thermodynamics. From these results we suggest that the definition of entropy in terms of order in spacetime should be more general that the Boltzmann one based on a counting of microstates. Finally we point out that these results are obtained by reversing the traditional route going from the Schr\"{o}dinger equation to statistical thermodynamics. 
  A non-linear relativistic 4D field model of a quantum particle which emerges from the internal dynamics in the quantum phase space $CP(N-1)$ is proposed. In this model there is no distinction between `particle' and its `surrounding field', and the space-time manifold emerges from the description of the quantum state. The quantum observables of the `quantum particle field' are described in terms of the affine parallel transport of the local dynamical variables in $CP(N-1)$. 
  We argue that recent developments in discretizations of classical and quantum gravity imply a new paradigm for doing research in these areas. The paradigm consists in discretizing the theory in such a way that the resulting discrete theory has no constraints. This solves many of the hard conceptual problems of quantum gravity. It also appears as a useful tool in some numerical simulations of interest in classical relativity. We outline some of the salient aspects and results of this new framework. 
  We use the Einstein energy-momentum complex to calculate the energy distribution of static plane-symmetric solutions of the Einstein-Maxwell equations in 3+1 dimensions with asymptotic anti-de Sitter behavior. This solution is expressed in terms of three parameters: the mass, electric charge and cosmological constant. We compare the energy distribution to that of the Reissner-Nordstrom-anti-de Sitter solution, pointing to qualitative differences between the models. Finally, we examine these results within the context of the Cooperstock hypothesis. 
  Natural wormholes and its astrophysical signatures have been sugested in various oportunities. By applying the strong field limit of gravitational lensing theory, we calculate the deflection angle produced by Morris-Thorne wormholes in asimptotically flat space-times. The results show that wormholes act like convergent lenses. Therefore it is hard to distinguish them from black holes using the gravitational lens effect, in contrast with the results reported by Cramer et.al. and Safanova et.al. 
  We present techniques for successfully performing numerical relativity simulations of binary black holes with fourth-order accuracy. Our simulations are based on a new coding framework which currently supports higher order finite differencing for the BSSN formulation of Einstein's equations, but which is designed to be readily applicable to a broad class of formulations. We apply our techniques to a standard set of numerical relativity test problems, demonstrating the fourth-order accuracy of the solutions. Finally we apply our approach to binary black hole head-on collisions, calculating the waveforms of gravitational radiation generated and demonstrating significant improvements in waveform accuracy over second-order methods with typically achievable numerical resolution. 
  Interactions within the cosmic medium modify its equation of state. We discuss implications of interacting dark energy models both for the spatially homogenous background and for the perturbation dynamics. 
  The expansion of our universe, when followed backward in time, implies that it emerged from a phase of huge density, the big bang. These stages are so extreme that classical general relativity combined with matter theories is not able to describe them properly, and one has to refer to quantum gravity. A complete quantization of gravity has not yet been developed, but there are many results about key properties to be expected. When applied to cosmology, a consistent picture of the early universe arises which is free of the classical pathologies and has implications for the generation of structure which are potentially observable in the near future. 
  It is a review paper. General relativity (GR) is presented in the field theoretical form, where gravitational field (metric perturbations) together with other physical fields are propagated in an auxiliary arbitrary curved background spacetime. Conserved currents are constructed and expressed through divergences of antisymmetrical tensor densities (superpotentials). This permits to connect local properties of perturbations with the quasi-local nature of the conserved quantities in GR. The problem of the non-localization of energy in GR is presented in exact mathematical expressions. A modification of GR developed recently by Babak and Grishchuk on the basis of the field formulation of GR is described. Their theory includes massive of spin-2 and spin-0 gravitons. All its local weak-field predictions are in agreement with experimental data. The exact equations of the massive theory eliminate the black hole event horizons and give an oscillator behavior for the homogeneous isotropic universe. 
  In this paper a new double-domain spectral method to compute binary black hole excision initial data is presented. The method solves a system of elliptic partial differential equations in the exterior of two excised spheres. At the surface of these spheres, boundary conditions need to be imposed. As such, the method can be used to construct arbitrary initial data corresponding to binary black holes with specific boundary conditions at their apparent horizons. We give representative examples corresponding to initial data that fulfill the requirements of the quasi-stationary framework, which combines the thin sandwich formulation of the constraint equations with the isolated horizon conditions for black holes in quasi-equilibrium. For all examples considered, numerical solutions with extremely high accuracy were obtained with moderate computational effort. Moreover, the method proves to be applicable even when tending toward limiting cases such as large radius ratios for the black holes. 
  Highly accurate numerical solutions to the problem of Black Holes surrounded by uniformly rotating rings in axially symmetric, stationary spacetimes are presented. The numerical methods developed to handle the problem are discussed in some detail. Related Newtonian problems are described and numerical results provided, which show that configurations can reach an inner mass-shedding limit as the mass of the central object increases. Exemplary results for the full relativistic problem for rings of constant density are given and the deformation of the event horizon due to the presence of the ring is demonstrated. Finally, we provide an example of a system for which the angular momentum of the central Black Hole divided by the square of its mass exceeds one. 
  It is shown that the general belief that the frequency and the absolute value of the velocity of periodic signals sent by a standard emitter do not change on the world line of the emitter needs to be revised and new conditions for the existence of a calibrted standard emitter should be taken into account. The notions of a standard clock and of a calibrated standard clock are introduced in a space with affine connections and metrics. The variation of the velocity and of the frequency of a standard clock could be compared with the constant velocity and the constant frequency of a calibrated standard clock along the world line of the observer. This calibrated standard clock is transported by meand of a generalized Fermi-Walker transport along the same world line of the observer. Some remarks about the synchronization of standard clocks in spaces with affine connections and metrics are given. PACS numbers: 95.30.Sf; 04.90.+h; 04.20.Cv; 04.90.+e 
  Linear perturbations of homothetic self-similar stiff fluid solutions, $S[n]$, with circular symmetry in 2+1 gravity are studied. It is found that, except for those with $n = 1$ and $n = 3$, none of them is stable and all have more than one unstable mode. Hence, {\em none of these solutions can be critical}. 
  In this work we investigate the behavior of two-dimensional (2D) cosmological models, starting with the Jackiw-Teitelboim (JT) theory of gravitation. A geometrical term, non-linear in the scalar curvature $R$, is added to the JT dynamics to test if it could play the role of dark energy in a 2D expanding universe. This formulation makes possible, first, the description of an early (inflationary) 2D universe, when the van der Waals (vdW) equation of state is used to construct the energy-momentum tensor of the gravitational sources. Second, it is found that for later times the non-linear term in $R$ can generate an old 2D universe in accelerated expansion, where an ordinary matter dominated era evolves into a decelerated/accelerated transition, giving to the dark energy effects a geometrical origin. The results emerge through numerical analysis, following the evolution in time of the scale factor, its acceleration, and the energy densities of constituents. 
  We present results from a covariance study for the proposed Laser Astrometric Test of Relativity (LATOR) mission. This mission would send two laser-transmitter spacecraft behind the Sun and measure the relative gravitational light bending of their signals using a hundred-meter-baseline optical interferometer to be constructed on the International Space Station. We assume that each spacecraft is equipped with a $ < 1.9 \times 10^{-13} \mathrm{m} \mathrm{s}^2 \mathrm{Hz}^{-1/2} $ drag-free system and assume approximately one year of data. We conclude that the observations allow a simultaneous determination of the orbit parameters of the spacecraft and of the Parametrized Post-Newtonian (PPN) parameter $\gamma$ with an uncertainty of $2.4 \times 10^{-9}$. We also find a $6 \times 10^{-9}$ determination of the solar quadrupole moment, $J_2$, as well as the first measurement of the second-order post-PPN parameter $\delta$ to an accuracy of about $10^{-3}$. 
  Analogue models of (and for) gravity have a long and distinguished history dating back to the earliest years of general relativity. In this review article we will discuss the history, aims, results, and future prospects for the various analogue models. We start the discussion by presenting a particularly simple example of an analogue model, before exploring the rich history and complex tapestry of models discussed in the literature. The last decade in particular has seen a remarkable and sustained development of analogue gravity ideas, leading to some hundreds of published articles, a workshop, two books, and this review article. Future prospects for the analogue gravity programme also look promising, both on the experimental front (where technology is rapidly advancing) and on the theoretical front (where variants of analogue models can be used as a springboard for radical attacks on the problem of quantum gravity). 
  I point out that according to the Copernican principle our universe is not unique. The way to make sense out of this statement is for us to construct a gravitational instanton that will tunnel out of our vacuum into another, to form a universe other than our Hubble bubble. 
  The covering of the affine symmetry group, a semidirect product of translations and special linear transformations, in $D \geq 3$ dimensional spacetime is considered. Infinite dimensional spinorial representations on states and fields are presented. A Dirac-like affine equation, with infinite matrices generalizing the $\gamma$ matrices, is constructed. 
  Entropy plays a crucial role in characterization of information and entanglement, but it is not a scalar quantity and for many systems it is different for different relativistic observers. Loop quantum gravity predicts the Bekenstein-Hawking term for black hole entropy and logarithmic correction to it. The latter originates in the entanglement between the pieces of spin networks that describe black hole horizon. Entanglement between gravity and matter may restore the unitarity in the black hole evaporation process. If the collapsing matter is assumed to be initially in a pure state, then entropy of the Hawking radiation is exactly the created entanglement between matter and gravity. 
  We link observational parameters such as the deceleration parameter, the jerk, the kerk (snap) and higher-order derivatives of the scale factor, called statefinders, to the conditions which allow to develop sudden future singularities of pressure with finite energy density. In this context, and within the framework of Friedmann cosmology, we also propose higher-order energy conditions which relate time derivatives of the energy density and pressure which may be useful in general relativity. 
  It is shown that the slowing down of the rate of time referencing to the inertial time leads in the field theory of gravitation to arising of repulsive forces which remove the cosmological singularity in the evolution of a homogeneous and isotropic universe and stop the collapse of large masses. 
  We propose a description of open universes in the Chern-Simons formulation of (2+1)-dimensional gravity where spatial infinity is implemented as a puncture. At this puncture, additional variables are introduced which lie in the cotangent bundle of the Poincar\'e group, and coupled minimally to the Chern-Simons gauge field. We apply this description of spatial infinity to open universes of general genus and with an arbitrary number of massive spinning particles. Using results of [9] we give a finite dimensional description of the phase space and determine its symplectic structure. In the special case of a genus zero universe with spinless particles, we compare our result to the symplectic structure computed by Matschull in the metric formulation of (2+1)-dimensional gravity. We comment on the quantisation of the phase space and derive a quantisation condition for the total mass and spin of an open universe. 
  We investigated the relation between the behavior of gravitational wave at late time and the limit structure of future null infinity tangent which will determine the topology of the event horizon far in the future. In the present article, we mainly consider a spacetime with two black holes. Although in most of cases, the black holes coalesce and its event horizon is topologically a single sphere far in the future, there are several possibilities that the black holes never coalesce and such exact solutions as examples. In our formulation, the tangent vector of future null infinity is, under conformal embedding, related to the number of black holes far in the future through the Poincar\'e-Hopf's theorem. Under the conformal embedding, the topology of event horizon far in the future will be affected by the geometrical structure of the future null infinity. In this article, we related the behavior of Weyl curvature to this limit behavior of the generator vector of the future null infinity. We show if Weyl curvature decays sufficiently slowly at late time in the neighborhood of future null infinity, two black holes never coalesce. 
  Considering a planar gravitating thick domain wall of the $\lambda \phi^4$ theory, we demonstrate how the Darmois junction conditions written on the boundaries of the thick wall with the embedding spacetimes reproduce the Israel junction condition across the wall when one takes its thin wall limit. 
  Based on recently reported results, we present arguments that sign changes in proper acceleration of test particles on the symmetry axis and close to the r=2M surface of quasi-spherical objects - related to the quadrupole moment of the source - might be at the origin of relativistic jets of quasars and micro-quasars. 
  A key source for LISA will be the inspiral of compact objects into supermassive black holes. Recently Mino has shown that in the adiabatic limit, gravitational waveforms for these sources can be computed using for the radiation reaction force the gradient of one half the difference between the retarded and advanced metric perturbations. Using post-Newtonian expansions, we argue that the resulting waveforms should be sufficiently accurate for signal detection with LISA. Data-analysis templates will require higher accuracy, going beyond adiabaticity; this remains a significant challenge. We describe an explicit computational procedure for obtaining waveforms based on Mino's result, for the case of a point particle coupled to a scalar field. We derive an expression for the time-averaged time derivative of the Carter constant, and verify that the expression correctly predicts that circular orbits remain circular while evolving via radiation reaction. The derivation uses detailed properties of mode expansions, Green's functions and bound geodesic orbits in the Kerr spacetime, which we review in detail. This paper is about three quarters review and one quarter new material. The intent is to give a complete and self-contained treatment of scalar radiation reaction in the Kerr spacetime, in a single unified notation, starting with the Kerr metric and ending with formulae for the time evolution of all three constants of the motion that are sufficiently explicit to be used immediately in a numerical code. 
  We present upper limits on the amplitude of gravitational waves from 28 isolated pulsars using data from the second science run of LIGO. The results are also expressed as a constraint on the pulsars' equatorial ellipticities. We discuss a new way of presenting such ellipticity upper limits that takes account of the uncertainties of the pulsar moment of inertia. We also extend our previous method to search for known pulsars in binary systems, of which there are about 80 in the sensitive frequency range of LIGO and GEO 600. 
  In addition to its long-term constancy, the Pioneer (spacecraft) anomaly appears to only exist for bodies whose mass is less than that of: planets, moons, comets, and heavy asteroids of known mass. Assuming the observational evidence is reliable, and not the result of an unknown systematic effect, a violation of the Weak Principle of Equivalence is implied. To propose an additional force fails to satisfy this constraint. This paper presents a new hypothesis involving additional field energy in the form of: a finite number of lunar sourced constant amplitude (Lorentz invariant) wave-like undulations upon the gravitational field. Although apparently a futile suggestion, the author's model overcomes concerns regarding wave dissipation, wave generation, and the apparent constancy of the anomaly. A shortfall in motion arises because a tiny proportion of spacecraft kinetic energy is directed into a superposition of non-translational longitudinal oscillatory components. The restriction of this effect to low mass bodies is also addressed. Additionally, the annual residual of the Pioneer anomaly may be attributed to a real 356 day Callisto-Titan wave resonance. This hypothesis may also be readily applied to other solar system anomalies including: the Earth flyby anomaly, an apparent absence of small comets, an apparent paucity of smaller bodies in the Main Belt of asteroids, and residual doubts concerning the migrating planets hypothesis that addresses the too rapid formation of the ice giants Uranus and Neptune. 
  Considering the Moller energy definition in both Einstein's theory of general relativity and tele-parallel theory of gravity, we find the energy of the universe based on viscous Kasner-type metrics. The energy distribution which includes both the matter and gravitational field is found to be zero in both of these different gravitation theories and this result agrees with previous works of Cooperstock and Israelit, Rosen, Johri et al., Banerjee-Sen, Vargas who investigated the problem of the energy in Friedmann-Robertson-Walker universe in Einstein's theory of general relativity and Aydogdu-Salti who considered the same problem in tele-parallel gravity. In all of these works, they found that the energy of the Friedmann-Robertson-Walker space-time is zero. Our result is the same as obtained in the studies of Salti and Havare. They used the viscous Kasner-type metric and found total energy and momentum by using Bergmann-Thomson energy-momentum formulation in both general relativity and tele-parallel gravity. The result that the total energy and momentum components of the universe is zero supports the viewpoints of Albrow and Tryon. 
  In this paper, using the energy definition in M{\o}ller's tetrad theory of gravity we calculate the total energy of the universe in Bianchi-type I cosmological models which includes both the matter and gravitational fields. The total energy is found to be zero and this result agrees with a previous works of Banerjee-Sen who investigated this problem using the general relativity version of the Einstein energy-momentum complex and Xulu who investigated same problem using the general relativity versions of the Landau-lifshitz, Papapetrou and Weinberg's energy-momentum complexes. The result that total energy of the universe in Bianchi-type I universes is zero supports the viewpoint of Tryon. 
  An effective energy tensor for gravitational radiation is identified for uniformly expanding flows of the Hawking mass-energy. It appears in an energy conservation law expressing the change in mass due to the energy densities of matter and gravitational radiation, with respect to a Killing-like vector encoding a preferred flow of time outside a black hole. In a spin-coefficient formulation, the components of the effective energy tensor can be understood as the energy densities of ingoing and outgoing, transverse and longitudinal gravitational radiation. By anchoring the flow to the trapping horizon of a black hole in a given sequence of spatial hypersurfaces, there is a locally unique flow and a measure of gravitational radiation in the strong-field regime. 
  The Immirzi parameter is a constant appearing in the general relativity action used as a starting point for the loop quantization of gravity. The parameter is commonly believed not to show up in the equations of motion, because it appears in front of a term in the action that vanishes on shell. We show that in the presence of fermions, instead, the Immirzi term in the action does not vanish on shell, and the Immirzi parameter does appear in the equations of motion. It determines the coupling constant of a four-fermion interaction. Therefore the Immirzi parameter leads to effects that are observable in principle, even independently from nonperturbative quantum gravity. 
  We formulate and optimize a computational search strategy for detecting gravitational waves from isolated, previously-unknown neutron stars (that is, neutron stars with unknown sky positions, spin frequencies, and spin-down parameters). It is well known that fully coherent searches over the relevant parameter-space volumes are not computationally feasible, and so more computationally efficient methods are called for. The first step in this direction was taken by Brady & Creighton (2000), who proposed and optimized a two-stage, stack-slide search algorithm. We generalize and otherwise improve upon the Brady-Creighton scheme in several ways. Like Brady & Creighton, we consider a stack-slide scheme, but here with an arbitrary number of semi-coherent stages and with a coherent follow-up stage at the end. We find that searches with three semi-coherent stages are significantly more efficient than two-stage searches (requiring about 2-5 times less computational power for the same sensitivity) and are only slightly less efficient than searches with four or more stages. We calculate the signal-to-noise ratio required for detection, as a function of computing power and neutron star spin-down-age, using our optimized searches. 
  The multifractal spectrum of various three-dimensional representations of Packed Swiss Cheese cosmologies in open, closed, and flat spaces are measured, and it is determined that the curvature of the space does not alter the associated fractal structure. These results are compared to observational data and simulated models of large scale galaxy clustering, to assess the viability of the PSC as a candidate for such structure formation. It is found that the PSC dimension spectra do not match those of observation, and possible solutions to this discrepancy are offered, including accounting for potential luminosity biasing effects. Various random and uniform sets are also analyzed to provide insight into the meaning of the multifractal spectrum as it relates to the observed scaling behaviors. 
  In this paper we study a type of model for closed inflationary universe models using the Jordan-Brans-Dicke theory. Herein we determine and characterize the existence of $\Omega>$1, together with the period of inflation. We have found that our model, which takes into account a Jordan-Brans-Dicke type of theory, is less restrictive than the one used in Einstein's theory of general relativity. Our results are compared to those found in Einstein's theory of Relativity. 
  We give a quantum field theoretical derivation of the scalar Abraham-Lorentz-Dirac (ALD) equation and the self-force for a scalar charged particle interacting with a quantum scalar field in curved spacetime. We regularize the causal Green's function using a quasi-local expansion in the spirit of effective field theory and obtain a regular expression for the self-force. The scalar ALD equation obtained in this way for the classical motion of the particle checks with the equation obtained by Quinn earlier \cite{Quinn}. We further derive a scalar ALD-Langevin equation with a classical stochastic force accounting for the effect of quantum fluctuations in the field, which causes small fluctuations on the particle trajectory. This equation will be useful for the study of stochastic motion of charges under the influence of both quantum and classical noise sources, derived either self-consistently (as done here) or put in by hand (with warnings). We show the possibility of secular effects from such stochastic influences on the trajectory that may impact on the present calculations of gravitational waveform templates. 
  The role of the Taub time gauge in cosmology is linked to the use of the densitized lapse function instead of the lapse function in the variational principle approach to the Einstein equations. The spatial metric variational equations then become the Ricci evolution equations, which are then supplemented by the Einstein constraints which result from the variation with respect to the densitized lapse and the usual shift vector field. In those spatially homogeneous cases where the least disconnect occurs between the general theory and the restricted symmetry scenario, the recent adjustment of the conformal approach to solving the initial value problem resulting from densitized lapse considerations is seen to be inherent in the use of symmetry-adapted metric variables. The minimal distortion shift vector field is a natural vector potential for the new York thin sandwich initial data approach to the constraints, which in this case corresponds to the diagonal spatial metric gauge. For generic spacetimes, the new approach suggests defining a new minimal distortion shift gauge which agrees with the old gauge in the Taub time gauge, but which also makes its defining differential equation agree with the vector potential equation for solving the supermomentum constraint in any time gauge. 
  Simple scalar field cosmological models are considered describing gravity assisted crossing of the phantom divide line. This crossing or (de)-phantomization characterized by the change of the sign of the kinetic term of the scalar field is smooth and driven dynamically by the Einstein equations. Different cosmological scenarios, including the phantom phase of matter are sketched. 
  Generalizations of gravitational Born-Infeld type lagrangians are investigated. Phenomenological constraints (reduction to Einstein-Hilbert action for small curvature, spin two ghost freedom and absence of Coulomb like Schwarschild singularity) select one effective lagrangian whose dynamics is dictated by the tensors g_{\mu\nu} and R_{\mu\nu\rho\sigma}(not R_{\mu\nu} or the scalar R). 
  When a black hole is put in an "empty" space (zero temperature space) on which there is no matter except the matter of the Hawking radiation (Hawking field), then an outgoing energy flow from the black hole into the empty space exists. By the way, an equilibrium between two arbitrary systems can not allow the existence of an energy (heat) flow from one system to another. Consequently, in the case of a black hole evaporation in the empty space, the Hawking field should be in a nonequilibrium state. Hence the total behaviour of the evaporation, for example the time evolution of the total entropy, should be analysed with a nonequilibrium thermodynamics for the Hawking field. This manuscript explains briefly the way of constructing a nonequilibrium thermodynamic theory for a radiation field, and apply it to a simplified model of a black hole evaporation to calculate the time evolution of the total entropy. 
  In the experimental tests of gravity, there have been considerable interests in the possibility of intermediate-range gravity. In this paper, we use the earth-satellite measurement of earth gravity, the lunar orbiter measurement of lunar gravity, and lunar laser ranging measurement to constrain the intermediate-range gravity from lambda=1.2*10^{7}m - 3.8*10^{8}m. The limits for this range are alpha=10^{-8}-5*10^{-8}, which improve previous limits by about one order of magnitude in the range lambda=1.2*10^{7}m-3.8*10^{8}m. 
  A scenario is proposed in which the matter-antimatter asymmetry behaves like a seed for the inflationary phase of the universe. The mechanism which makes this scenario plausible is the holographic principle: this scheme is supported by a good prediction of the number of e-folds. It seems that such a mechanism can only work in the presence of a Hagedorn-like phase transition. The issue of the "graceful exit" can also be naturally accounted for. 
  Two Lagrangian functions are used to construct geometric field theories. One of these Lagrangians depends on the conventional curvature of space, while the other depends on curvature and torsion. It is shown that the theory constructed from the first Lagrangian gives rise to pure gravity, while the theory constructed using the second Lagrangian gives rise to both gravity and electromagnetism. The two theories are constructed in a version of absolute parallelism geometry in which both curvature and torsion are, simultaneously, non-vanishing. One single geometric object, {\it the non-conventional curvature tensor}, reflecting the properties of curvature and torsion, is defined in this version and is used to construct the second theory. The main conclusion is that a necessary condition for geometric representation of electromagnetism is the presence of a non-vanishing torsion in the geometry used. 
  The study of the matching of stationary and axisymmetric spacetimes with Friedmann-Lemaitre-Robertson-Walker spacetimes preserving the axial symmetry is presented. We show, in particular, that any orthogonally transitive stationary and axisymmetric region in FLRW must be static, irrespective of the matter content. Therefore, previous results on static regions in FLRW cosmologies apply. As a result, the only stationary and axisymmetric vacuum region that can be matched to a (non-static) FLRW spacetime is a spherically symmetric region of Schwarzschild. This constitutes another uniqueness result for the Einstein-Straus model (as well as its Oppenheimer-Snyder counterpart), and hence another indication of its unsuitability as an answer to the influence of the cosmic expansion on local physics. 
  We obtain plane fronted gravitational waves (PFGWs) in arbitrary dimension in Lovelock gravity, to any order in the Riemann tensor. We exhibit pure gravity as well as Lovelock-Yang-Mills PFGWs. Lovelock-Maxwell and $pp$ waves arise as particular cases. The electrovac solutions trivially satisfy the Lovelock-Born-Infeld field equations. The peculiarities that arise in degenerate Lovelock theories are also analyzed. 
  We investigate the dynamics of an homogenous distribution of galaxies moving under the cosmological expansion through Euler-Poisson equations system. The solutions are interpreted with the aim of understanding the cosmic velocity fields in the Local Super Cluster, and in particular the presence of a bulk flow. Among several solutions, we shows a planar kinematics with constant (eternal) and rotational distortion, the velocity field is not potential. 
  In this paper we show that by considering a universe dominated by two interacting components a superaccelerated expansion can be obtained from a decelerated one by applying a dual transformation that leaves the Einstein's field equations invariant. 
  We investigate refined algebraic quantisation of the constrained Hamiltonian system introduced by Boulware as a simplified version of the Ashtekar-Horowitz model. The dimension of the physical Hilbert space is finite and asymptotes in the semiclassical limit to 1/(2\pi\hbar) times the volume of the reduced phase space. The representation of the physical observable algebra is irreducible for generic potentials but decomposes into irreducible subrepresentations for certain special potentials. The superselection sectors are related to singularities in the reduced phase space and to the rate of divergence in the formal group averaging integral. There is no tunnelling into the classically forbidden region of the unreduced configuration space, but there can be tunnelling between disconnected components of the classically allowed region. 
  Within the weak-field approximation of general relativity, new exact solutions are derived for the gravitational field of a mass moving with arbitrary velocity and acceleration. Owing to an inertial-pushing effect, a mass having a constant velocity greater than 3^-1/2 times the speed of light gravitationally repels other masses at rest within a narrow cone. At high Lorentz factors (gamma >> 1), the force of repulsion in the forward direction is about -8(gamma^5) times the Newtonian force, offering opportunities for laboratory tests of gravity at extreme velocities. 
  The Schwarzschild solution is used to find the exact relativistic motion of a payload in the gravitational field of a mass moving with constant velocity. At radial approach or recession speeds faster than 3^-1/2 times the speed of light, even a small mass gravitationally repels a payload. At relativistic speeds, a suitable mass can quickly propel a heavy payload from rest nearly to the speed of light with negligible stresses on the payload. 
  Full interpretation of data from gravitational wave observations will require accurate numerical simulations of source systems, particularly binary black hole mergers. A leading approach to improving accuracy in numerical relativity simulations of black hole systems is through fixed or adaptive mesh refinement techniques. We describe a manifestation of numerical interface truncation error which appears as slowly converging, artificial reflections from refinement boundaries in a broad class of mesh refinement implementations, potentially compromising the effectiveness of mesh refinement techniques for some numerical relativity applications if left untreated. We elucidate this numerical effect by presenting a model problem which exhibits the phenomenon, but which is simple enough that its numerical error can be understood analytically. Our analysis shows that the effect is caused by variations in finite differencing error generated across low and high resolution regions, and that its slow convergence is caused by the presence of dramatic speed differences among propagation modes typical of 3+1 relativity. Lastly, we resolve the problem, presenting a class of finite differencing stencil modifications, termed mesh-adapted differencing (MAD), which eliminate this pathology in both our model problem and in numerical relativity examples. 
  In this work we show that the gravity lagrangian f(R) at relatively low curvatures in both metric and Palatini formalisms is a bounded function that can only depart from the linearity within the limits defined by well known functions. We obtain those functions by analysing a set of inequalities that any f(R) theory must satisfy in order to be compatible with laboratory and solar system observational constraints. This result implies that the recently suggested f(R) gravity theories with nonlinear terms that dominate at low curvatures are incompatible with observations and, therefore, cannot represent a valid mechanism to justify the cosmic speed-up. 
  The INSPIRAL program is the LIGO Scientific Collaboration's computational engine for the search for gravitational waves from binary neutron stars and sub-solar mass black holes. We describe how this program, which makes use of the FINDCHIRP algorithm (discussed in a companion paper), is integrated into a sophisticated data analysis pipeline that was used in the search for low-mass binary inspirals in data taken during the second LIGO science run. 
  We present an emergent universe scenario making use of a new solution of the Starobinsky model. The solution belongs to a one parameter family of solutions, where the parameter is determined by the number and the species (spin-values) of primordial fields. The general features of the model have also been studied. 
  We compare quantum hydrodynamics and quantum gravity. They share many common features. In particular, both have quadratic divergences, and both lead to the problem of the vacuum energy, which in the quantum gravity transforms to the cosmological constant problem. We show that in quantum liquids the vacuum energy density is not determined by the quantum zero-point energy of the phonon modes. The energy density of the vacuum is much smaller and is determined by the classical macroscopic parameters of the liquid including the radius of the liquid droplet. In the same manner the cosmological constant is not determined by the zero-point energy of quantum fields. It is much smaller and is determined by the classical macroscopic parameters of the Universe dynamics: the Hubble radius, the Newton constant and the energy density of matter. The same may hold for the Higgs mass problem: the quadratically divergent quantum correction to the Higgs potential mass term is also cancelled by the microscopic (trans-Planckian) degrees of freedom due to thermodynamic stability of the whole quantum vacuum. 
  We show that short-range interactions between the fundamental particles in the universe can drive a period of accelerated expansion. This description fits the early universe. In the present day universe, if one postulates short-range interactions or a sort of "shielded gravity", the picture may repeat. 
  The impact of the latest combined CHAMP/GRACE/terrestrial measurements Earth gravity model EIGEN-CG03C on the measurement of the Lense-Thirring effect with some linear combinations of the nodes of some of the existing Earth's artificial satellites is presented. The 1-sigma upper bound of the systematic error in the node-node LAGEOS-LAGEOS II combination is 3.9% (4% with EIGEN-GRACE02S, \sim 6% with EIGEN-CG01C and \sim 9% with GGM02S), while it is 1$% for the node-only LAGEOS-LAGEOS II-Ajisai-Jason-1 combination (2% with EIGEN-GRACE02S, 1.6% with EIGEN-CG01C and 2.7% with GGM02S). 
  In this paper we investigate the possibility of measuring the general relativistic gravitoelectric contribution P^(GE) to the orbital period P of the transiting exoplanet HD 209458b 'Osiris'. It turns out that the predicted magnitude of such an effect is \sim 0.1 s, while the most recent determinations of the orbital period of HD 209458b with the photometric transit method are accurate to \sim 0.01 s. The present analysis shows that the major limiting factor is the \sim 1 m s^-1 sensitivity in the measurement of the projected semiamplitude of the star's radial velocity K. Indeed, it affects the determination of the mass m of the planet which, in turn, induces a systematic error in the Keplerian period P^(0) of \sim 8 s. It is of crucial importance because P^(0) should be subtracted from the measured period in order to extract the relativistic correction. The present-day uncertainty in $m$ does not yet make necessary the inclusion of relativistic corrections in the data-reduction process of the determination of the system's parameters. The present situation could change only if improvements of one-two orders of magnitude in the ground-based Doppler spectroscopy technique occurred. 
  Recently it was shown that the exact cosmological solutions known as the vacuum plane-wave solutions are late-time attractors for an open set of the spatially homogeneous Bianchi universes containing a non-inflationary $\gamma$-law perfect fluid. In this paper we study inhomogeneous perturbations of these plane-wave spacetimes. By using expansion-normalised scale-invariant variables we show that these solutions are unstable to generic inhomogeneous perturbations. The crucial observation for establishing this result is a divergence of the expansion-normalised frame variables which ultimately leads to unstable modes. 
  The causal interpretation of quantum mechanics is applied to a homogeneous and isotropic quantum universe, whose matter content is composed by non interacting dust and radiation. For wave functions which are eigenstates of the total dust mass operator, we find some bouncing quantum universes which reachs the classical limit for scale factors much larger than its minimum size. However these wave functions do not have unitary evolution. For wave functions which are not eigenstates of the dust total mass operator but do have unitary evolution, we show that, for flat spatial sections, matter can be created as a quantum effect in such a way that the universe can undergo a transition from an exotic matter dominated era to a matter dominated one. 
  The presence of gravity implies corrections to the Einstein-Planck formula $E=h \nu$. This gives hope that the divergent blueshift in frequency, associated to the presence of a black hole horizon, could be smoothed out for the energy. Using simple arguments based on Einstein's equivalence principle we show that this is only possible if a black hole emits, in first approximation, not just a single particle, but thermal radiation. 
  Black hole thermodynamics suggests that the maximum entropy that can be contained in a region of space is proportional to the area enclosing it rather than its volume. I argue that this follows naturally from loop quantum gravity and a result of Kolmogorov and Bardzin' on the the realizability of networks in three dimensions. This represents an alternative to other approaches in which some sort of correlation between field configurations helps limit the degrees of freedom within a region. It also provides an approach to thinking about black hole entropy in terms of states inside rather than on its surface. Intuitively, a spin network complicated enough to imbue a region with volume only lets that volume grow as quickly as the area bounding it. 
  We study analytically the quasinormal mode spectrum of near-extremal (rotating) Kerr black holes. We find an analytic expression for these black-hole resonances in terms of the black-hole physical parameters: its Bekenstein-Hawking temperature T_{BH} and its horizon's angular velocity \Omega, which is valid in the intermediate asymptotic regime 1<<\omega<<1/T_{BH}. 
  The universality observed in gravitational wave spectra of non-rotating neutron stars is analyzed here. We show that the universality in the axial oscillation mode can be reproduced with a simple stellar model, namely the centrifugal barrier approximation (CBA), which captures the essence of the Tolman VII model of compact stars. Through the establishment of scaled co-ordinate logarithmic perturbation theory (SCLPT), we are able to explain and quantitatively predict such universal behavior. In addition, quasi-normal modes of individual neutron stars characterized by different equations of state can be obtained from those of CBA with SCLPT. 
  The spherically symmetric Einstein-Vlasov system in Schwarzschild coordinates (i.e. polar slicing and areal radial coordinate) is considered. An improved continuation criterion for global existence of classical solutions is given. Two other types of criteria which prevent finite time blow-up are also given. 
  During inflation explicit perturbative computations of quantum field theories which contain massless, non-conformal fields exhibit secular effects that grow as powers of the logarithm of the inflationary scale factor. Starobinskii's technique of stochastic inflation not only reproduces the leading infrared logarithms at each order in perturbation theory, it can sometimes be summed to reveal what happens when inflation has proceeded so long that the large logarithms overwhelm even very small coupling constants. It is thus a cosmological analogue of what the renormalization group does for the ultraviolet logarithms of quantum field theory, and generalizing this technique to quantum gravity is a problem of great importance. There are two significant differences between gravity and the scalar models for which stochastic formulations have so far been given: derivative interactions and the presence of constrained fields. We use explicit perturbative computations in two simple scalar models to infer a set of rules for stochastically formulating theories with these features. 
  In this paper we investigate the structure of the Mirror Universes. The two universes are coupled with transformation t to -t. It is shown that for Planck scale the oscillations of temperature of the universes are observed. For the Mirror Universes the temperature fields are shifted in phase.   Key words: Gravity; Universe temperature; Oscillation of temperature. 
  In our paper Phys. Rev. Lett. 92, 151801 (2004), the oscillations of strongly deformed Bogomolny-Prasad-Sommerfield magnetic monopoles have been studied on a fixed Minkowski background. The purpose of the present article is to provide a more detailed account on the results yielded by our numerical simulations. In particular, an analysis on the dependence on the strength of the initial excitation is carried out in order to distinguish features which are already present for small perturbations from those which are consequences of the nonlinearity of the system. 
  We look for the global in time solution of the Cauchy problem corresponding to the asymptotically flat spherically symmetric EVM system with small initial data. Using an estimate, we also prove that if solution of the system stated above develops a singularity at all time, then the first one has to appear at the center of symmetry. 
  The information loss paradox for Schwarzschild black holes is examined, using the ADS/CFT correspondence extended to the $M_6 (4,2)$ bulk. It is found that the only option compatible with the preservation of the quantum unitarity is when a regular remnant region of the black hole survives to the black hole evaporation process, where information can be stored and eventually retrieved. 
  I construct initial data for equal-mass irrotational binary black holes using the conformal thin-sandwich puncture (CTSP) approach. I locate quasi-circular orbits using the effective-potential method, and estimate the location of the innermost stable circular orbit (ISCO). The ISCO prediction is consistent with results for conformal thin-sandwich data produced using excision techniques. These results also show that the ISCOs predicted by the effective-potential and ADM-Komar mass-comparison methods agree for conformal thin-sandwich data, just as they did for Bowen-York data. 
  Among the suggested solutions to the cosmological constant problem, we find the idea of a dynamic vacuum, with an energy density decaying with the universe expansion. We investigate the possibility of a variation in the gravitational constant as well, induced, at the cosmological scale, by the vacuum decay. We consider an effective Brans-Dicke theory in the spatially flat FLRW spacetime, finding late time solutions characterized by a constant ratio between the matter and vacuum energy densities. By using the observed limits for the universe age, we fix the only free parameter of our solutions, obtaining a relative matter density in the range 0.25-0.4. In particular, for Ht = 1 we obtain a relative matter density equals to 1/3. This constitutes a possible explanation for another problem related to the cosmological term, the cosmic coincidence problem. 
  Some properties of eight-dimensional Riemann extension of Minkowsky space-time metric in rotating coordinate system are studied. 
  We study a nonperturbative single field (inflaton) governed cosmological model from a 5D Noncompact Kaluza-Klein (NKK) theory of gravity. The inflaton field fluctuations are estimated for different epochs of the evolution of the universe. We conclude that the inflaton field has been sliding down its (quadratic) potential hill along all the evolution of the universe and a mass of the order of the Hubble parameter. In the model here developed the only free parameter is the Hubble parameter, which could be reconstructed in future from Super Nova Acceleration Probe (SNAP) data. 
  The Ellis wormhole is known to be an exact solution of the Einstein--scalar system where the scalar field has negative kinetic energy (phantom). We show here that the same geometry (in 3+1 dimensions) can also be obtained with `tachyon matter' as a source term in the field equations and a positive cosmological constant. The kink--like tachyon field profile and the nature of the tachyon potential are analyzed. It is shown that the field configuration and the spacetime are stable against linear scalar perturbations. Finally, we comment on extensions of the 3+1 dimensional Ellis wormhole (with tachyon matter source) in diverse dimensions (d=2+1 and d>4). 
  We discuss inflationary solutions of the coupled Einstein-Klein-Gordon equations for a complex field in a five dimensional spacetime with a compact $x^5$-dimension. As a new feature, the scalar field contains a dependence on the extra dimension of the form $\exp(i m x^5) $, corresponding to Kaluza-Klein excited modes. In a four dimensional picture, a nonzero $m$ implies the presence of a new term in the scalar field potential. An interesting feature of these solutions is the possible existence of several periods of oscillation of the scalar field around the equilibrium value at the minimum of the potential. These oscillations lead to cosmological periods of accelerated expansion of the universe. 
  We study the Friedmann-Robertson-Walker model with dynamical dark energy modelled in terms of the equation of state $p_{x}=w_{x}(a(z)) \rho_{x}$ in which the coefficient $w_{x}$ is parameterized by the scale factor $a$ or redshift $z$. We use methods of qualitative analysis of differential equations to investigate the space of all admissible solutions for all initial conditions on the two-dimensional phase plane. We show advantages of representing this dynamics as a motion of a particle in the one-dimensional potential $V(a)$. One of the features of this reduction is the possibility of investigating how typical are big rip singularities in the future evolution of the model. The properties of potential function $V$ can serve as a tool for qualitative classification of all evolution paths. Some important features like resolution of the acceleration problem can be simply visualized as domains on the phase plane. Then one is able to see how large is the class of solutions (labelled by the inset of the initial conditions) leading to the desired property. 
  We obtain a new kind of exact solution to vaccum Einstein field equations that contain both Minkowsian world and a special 5D curved spacetime with particularly free structure. This special world is defined by an arbitrary function and a space of three parameters. We suggest that this solution could correct (in principle) certain aspects of the physics in flat spacetime. 
  An explicit proof of the vanishing of the covariant divergence of the energy-momentum tensor in modified theories of gravity is presented. The gravitational action is written in arbitrary dimensions and allowed to depend nonlinearly on the curvature scalar and its couplings with a scalar field. Also the case of a function of the curvature scalar multiplying a matter Lagrangian is considered. The proof is given both in the metric and in the first-order formalism, i.e. under the Palatini variational principle. It is found that the covariant conservation of energy-momentum is built-in to the field equations. This crucial result, called the generalized Bianchi identity, can also be deduced directly from the covariance of the extended gravitational action. Furthermore, we demonstrate that in all of these cases, the freely falling world lines are determined by the field equations alone and turn out to be the geodesics associated with the metric compatible connection. The independent connection in the Palatini formulation of these generalized theories does not have a similar direct physical interpretation. However, in the conformal Einstein frame a certain bi-metricity emerges into the structure of these theories. In the light of our interpretation of the independent connection as an auxiliary variable we can also reconsider some criticisms of the Palatini formulation originally raised by Buchdahl. 
  Using the brick wall regularization of 't Hooft, the entropy of non-extreme and extreme black holes is investigated in a general static, spherically symmetric spacetime. We classify the singularity in the entropy by introducing a {\it new} index $\delta $ with respect to the brick wall cut-off $\epsilon $. The leading contribution to entropy for non-extreme case $(\delta \neq 0)$ is shown to satisfy the area law with quadratic divergence with respect to the invariant cut-off $\epsilon_{{\rm inv}}$ while the extreme case $(\delta =0)$ exhibits logarithmic divergence or constant value with respect to $\epsilon $. The general formula is applied to Reissner-Nordstr\"{o}m, dilaton and brane-world black holes and we obtain consistent results. 
  We obtain an exact solution for the Einstein's equations with cosmological constant coupled to a scalar, static particle in static, "spherically" symmetric background in 2+1 dimensions. 
  In this paper, we investigate cosmological particle production using quantum field theory (QFT). We will consider how production of scalar particles can occur in an expanding universe. By introducing a time-dependent energy parameter representing the time evolution of the universe, the initial vacuum state will be excited. Consequently, creation of particles is present. Here, our focus is mainly creation of minimally coupled scalar particles in Minkowski spacetime. 
  A direct and more general calculation of the limb effect connected with red-shift observations is obtained. Result agrees completely, in its general form, with that obtained from observations of the solar spectra but with different value for the maximum effect. 
  This article is a guide to theorems on existence and global dynamics of solutions of the Einstein equations. It draws attention to open questions in the field. The local-in-time Cauchy problem, which is relatively well understood, is surveyed. Global results for solutions with various types of symmetry are discussed. A selection of results from Newtonian theory and special relativity that offer useful comparisons is presented. Treatments of global results in the case of small data and results on constructing spacetimes with prescribed singularity structure or late-time asymptotics are given. A conjectural picture of the asymptotic behaviour of general cosmological solutions of the Einstein equations is built up. Some miscellaneous topics connected with the main theme are collected in a separate section. 
  We show that the strong self-interaction of the scalar polarization of a massive graviton can be understood in terms of the propagation of an extra ghost-like degree of freedom, thus relating strong coupling to the sixth degree of freedom discussed by Boulware and Deser in their Hamiltonian analysis of massive gravity. This enables one to understand the Vainshtein recovery of solutions of massless gravity as being due to the effect of the exchange of this ghost which gets frozen at distances larger than the Vainshtein radius. Inside this region, we can trust the two-field Lagrangian perturbatively, while at larger distances one can use the higher derivative formulation. We also compare massive gravity with other models, namely deconstructed theories of gravity, as well as DGP model. In the latter case we argue that the Vainshtein recovery process is of different nature, not involving a ghost degree of freedom. 
  We compute the complete post-Newtonian limit of the metric form of f(R) gravities using a scalar-tensor representation. By comparing the predictions of these theories with laboratory and solar system experiments, we find a set of inequalities that any lagrangian f(R) must satisfy. The constraints imposed by those inequalities allow us to find explicit bounds to the possible nonlinear terms of the lagrangian. We conclude that the lagrangian f(R) must be almost linear in R and that corrections that grow at low curvatures are incompatible with observations. This result shows that modifications of gravity at very low cosmic densities cannot be responsible for the observed cosmic speed-up. 
  We compute the complete post-Newtonian limit of the Palatini form of f(R) gravities using a scalar-tensor representation. By comparing the predictions of these theories with laboratory and solar system experiments, we find a set of inequalities that any lagrangian f(R) must satisfy. The constraints imposed by those inequalities allow us to find explicit functions that bound from above and from below the possible nonlinear terms of the lagrangian. We conclude that the lagrangian f(R) must be almost linear in R and that nonlinear corrections that grow at low curvatures are incompatible with observations. This result shows that modifications of gravity at very low cosmic densities cannot be responsible for the observed cosmic speed-up. 
  One of the very small number of serious alternatives to the usual concept of an astrophysical black hole is the "gravastar" model developed by Mazur and Mottola; and a related phase-transition model due to Laughlin et al. We consider a generalized class of similar models that exhibit continuous pressure -- without the presence of infinitesimally thin shells. By considering the usual TOV equation for static solutions with negative central pressure, we find that gravastars cannot be perfect fluids -- anisotropic pressures in the "crust" of a gravastar-like object are unavoidable. The anisotropic TOV equation can then be used to bound the pressure anisotropy. The transverse stresses that support a gravastar permit a higher compactness than is given by the Buchdahl--Bondi bound for perfect fluid stars. Finally we comment on the qualitative features of the equation of state that gravastar material must have if it is to do the desired job of preventing horizon formation. 
  In this paper we review the hole argument for the space-time points and elementary particles and generalize the hole argument to include all geometric object fields and diffeomorphisms; and, by application of forgetful functors to abstract from differentiability and even continuity, the hole argument is applied to a much wider class of mathematical objects. We discuss the problem concerning the individuation of the objects in more general settings such that fibered manifolds, fibered sets and n-ary relations. 
  We study the closed universe recollapse conjecture for positively curved FRW models with a perfect fluid matter source and a scalar field which arises in the conformal frame of the $R+\alpha R^{2}$ theory. By including ordinary matter, we extend the analysis of a previous work. We analyze the structure of the resulted four-dimensional dynamical system with the methods of the center manifold theory and the normal form theory. It is shown that an initially expanding closed FRW universe, starting close to the Minkowski spacetime, cannot avoid recollapse. We discuss the posibility that potentials with a positive minimum may prevent the recollapse of closed universes. 
  Effects of the vacuum polarization leading to change of the effective gravitational constant and particle creation in the early Universe are discussed. Gauss-Bonnet type coupling to the curvature is considered. Renormalization methods are generalized for such coupling and the N-dimensional space-time. Calculations of creation of entropy and visible matter of the Universe by the gravity of dark matter are made. 
  It is shown that among the four classes of the static spherically symmetric solution of the vacuum Brans-Dicke theory of gravity only two are really independent. Further by matching exterior and interior (due to physically reasonable spherically symmetric matter source) scalar fields it is found that only Brans class I solution with certain restriction on solution parameters may represent exterior metric for a nonsingular massive object. The physical viability of the black hole nature of the solution is investigated. It is concluded that no physical black hole solution different from the Schwarzschild black hole is available in the Brans-Dicke theory. 
  The Newtonian theory of gravitation and electrostatics admit equilibrium configurations of charged fluids where the charge density can be equal to the mass density, in appropriate units. The general relativistic analog for charged dust stars was discovered by Majumdar and by Papapetrou. In the present work we consider Einstein-Maxwell solutions in d-dimensional spacetimes and show that there are Majumdar-Papapetrou type solutions for all ${\rm d} \geq 4$. It is verified that the equilibrium is independent of the shape of the distribution of the charged matter. It is also showed that for perfect fluid solutions satisfying the Majumdar-Papapetrou condition with a boundary where the pressure is zero, the pressure vanishes everywhere, and that the $({\rm d}-1)$-dimensional spatial section of the spacetime is conformal to a Ricci-flat space. The Weyl d-dimensional axisymmetric solutions are generalized to include electric field and charged matter. 
  We consider axially symmetric, rotating boson stars. Their flat space limits represent spinning Q-balls. We discuss their properties and determine their domain of existence. Q-balls and boson stars are stationary solutions and exist only in a limited frequency range. The coupling to gravity gives rise to a spiral-like frequency dependence of the boson stars. We address the flat space limit and the limit of strong gravitational coupling. For comparison we also determine the properties of spherically symmetric Q-balls and boson stars. 
  We calculate the maximum mass of the class of compact stars described by Vaidya-Tikekar \cite{VT01} model. The model permits a simple method of systematically fixing bounds on the maximum possible mass of cold compact stars with a given value of radius or central density or surface density. The relevant equations of state are also determined. Although simple, the model is capable of describing the general features of the recently observed very compact stars. For the calculation, no prior knowledge of the equation of state (EOS) is required. This is in contrast to the earlier calculations for maximum mass which were done by choosing first the relevant EOSs and using those to solve the TOV equation with appropriate boundary conditions. The bounds obtained by us are comparable and, in some cases, more restrictive than the earlier results. 
  We investigate in detail gravitational waves in an Schwarzschild-anti-de Sitter bulk spacetime surrounded by an Einstein static brane with generic matter content. Such a model provides a useful analogy to braneworld cosmology at various stages of its evolution, and generalizes our previous work [gr-qc/0504023] on pure tension Einstein-static branes. We find that the behaviour of tensor-mode perturbations is completely dominated by quasi-normal modes, and we use a variety of numeric and analytic techniques to find the frequencies and lifetimes of these excitations. The parameter space governing the model yields a rich variety of resonant phenomena, which we thoroughly explore. We find that certain configurations can support a number of lightly damped `quasi-bound states'. A zero-mode which reproduces 4-dimensional general relativity is recovered on infinitely large branes. We also examine the problem in the time domain using Green's function techniques in addition to direct numeric integration. We conclude by discussing how the quasi-normal resonances we find here can impact on braneworld cosmology. 
  We reexamine here the issue of consistency of minimal action formulation with the minimal coupling procedure (MCP) in spaces with torsion. In Riemann-Cartan spaces, it is known that a proper use of the MCP requires that the trace of the torsion tensor be a gradient, $T_\mu=\partial_\mu\theta$, and that the modified volume element $\tau_\theta = e^\theta \sqrt{g} dx^1\wedge...\wedge dx^n $ be used in the action formulation of a physical model. We rederive this result here under considerably weaker assumptions, reinforcing some recent results about the inadequacy of propagating torsion theories of gravity to explain the available observational data. The results presented here also open the door to possible applications of the modified volume element in the geometric theory of crystalline defects. 
  It is shown that an infinite gravitational flux tube solution in 5D Kaluza-Klein gravity with the cross section in the Planck region after $5D \to 4D$ reduction and isometrical embedding in a Minkowski spacetime can be considered as a moving infinite string-like object. Such object has a flux of the electric and magnetic fields. The 4D gravitational waves on the tube are considered. 
  In this paper,we discussed the particle creation in de Sitter spacetime. When discussing the particle creation in a very short span, in used of the temperature of de Sitter space, We found that we must quantize the time. 
  The most promising way to compute the gravitational waves emitted by binary black holes (BBHs) in their last dozen orbits, where post-Newtonian techniques fail, is a quasistationary approximation introduced by Detweiler and being pursued by Price and others. In this approximation the outgoing gravitational waves at infinity and downgoing gravitational waves at the holes' horizons are replaced by standing waves so as to guarantee that the spacetime has a helical Killing vector field. Because the horizon generators will not, in general, be tidally locked to the holes' orbital motion, the standing waves will destroy the horizons, converting the black holes into naked singularities that resemble black holes down to near the horizon radius. This paper uses a spherically symmetric, scalar-field model problem to explore in detail the following BBH issues: (i) The destruction of a horizon by the standing waves. (ii) The accuracy with which the resulting naked singularity resembles a black hole. (iii) The conversion of the standing-wave spacetime (with a destroyed horizon) into a spacetime with downgoing waves by the addition of a ``radiation-reaction field''. (iv) The accuracy with which the resulting downgoing waves agree with the downgoing waves of a true black-hole spacetime (with horizon). The model problem used to study these issues consists of a Schwarzschild black hole endowed with spherical standing waves of a scalar field. It is found that the spacetime metric of the singular, standing-wave spacetime, and its radiation-reaction-field-constructed downgoing waves are quite close to those for a Schwarzschild black hole with downgoing waves -- sufficiently close to make the BBH quasistationary approximation look promising for non-tidally-locked black holes. 
  The purpose of this paper is to survey the possible topologies of branching space-times, and, in particular, to refute the popular notion in the literature that a branching space-time requires a non-Hausdorff topology. 
  There exist at least a few different kind of averaging of the differences of the energy-momentum and angular momentum in normal coordinates {\bf NC(P)} which give tensorial quantities. The obtained averaged quantities are equivalent mathematically because they differ only by constant scalar dimensional factors. One of these averaging was used in our papers [1-8] giving the {\it canonical superenergy and angular supermomentum tensors}.   In this paper we present one other averaging of the energy-momentum and angular momentum differences which gives tensorial quantities with proper dimensions of the energy-momentum and angular momentum densities. But these averaged energy-momentum and angular momentum tensors, closely related to the canonical superenergy and angular supermomentum tensors, {\it depend on some fundamental length L}.   The averaged energy-momentum and angular momentum tensors of the gravitational field obtained in the paper can be applied, like the canonical superenergy and angular supermomentum tensors, to coordinate independent local (and also global) analysis of this field. 
  I point out a symmetry, between equations of state for polytropic fluids, in the equation of motion of a spherically symmetric singular shell embedded in 4-d and 5-d vacuum spacetimes. In particular the equation of motion of a shell consisting of radiation has the same form as for a vacuum shell or domain wall. 
  Radiating black holes pose a number of puzzles for semiclassical and quantum gravity. These include the transplanckian problem -- the nearly infinite energies of Hawking particles created near the horizon, and the final state of evaporation. A definitive resolution of these questions likely requires robust inputs from quantum gravity. We argue that one such input is a quantum bound on curvature. We show how this leads to an upper limit on the redshift of a Hawking emitted particle, to a maximum temperature for a black hole, and to the prediction of a Planck scale remnant. 
  We point out that $\Lambda$CDM cosmology has an ignored assumption. That is, the $\Lambda$ component of the universe moves synchronously with ordinary matters on Hubble scales. If cosmological constant is vacuum energy, this assumption may be very difficult to be understood.   We then propose a new mechanism which can explain the accelerating recession of super-novaes. That is, considering the pressures originating from the random moving (including Hubble recession) of galaxy clusters and galaxies. We provide an new analytical solution of Einstein equation which may describe a universe whose pressures originating from the random moving of galaxy clusters and galaxies are considered. 
  FRW cosmologies with conformally coupled scalar fields are investigated in a geometrical way by the means of geodesics of the Jacobi metric. In this model of dynamics, trajectories in the configuration space are represented by geodesics. Because of the singular nature of the Jacobi metric on the boundary set $\partial\mathcal{D}$ of the domain of admissible motion, the geodesics change the cone sectors several times (or an infinite number of times) in the neighborhood of the singular set $\partial\mathcal{D}$. We show that this singular set contains interesting information about the dynamical complexity of the model. Firstly, this set can be used as a Poincar{\'e} surface for construction of Poincar{\'e} sections, and the trajectories then have the recurrence property. We also investigate the distribution of the intersection points. Secondly, the full classification of periodic orbits in the configuration space is performed and existence of UPO is demonstrated. Our general conclusion is that, although the presented model leads to several complications, like divergence of curvature invariants as a measure of sensitive dependence on initial conditions, some global results can be obtained and some additional physical insight is gained from using the conformal Jacobi metric. We also study the complex behavior of trajectories in terms of symbolic dynamics. 
  We describe the possibility of using LISA's gravitational-wave observations to study, with high precision, the response of a massive central body to the tidal gravitational pull of an orbiting, compact, small-mass object. Motivated by this application, we use first-order perturbation theory to study tidal coupling for an idealized case: a massive Schwarzschild black hole, tidally perturbed by a much less massive moon in a distant, circular orbit. We investigate the details of how the tidal deformation of the hole gives rise to an induced quadrupole moment in the hole's external gravitational field at large radii. In the limit that the moon is static, we find, in Schwarzschild coordinates and Regge-Wheeler gauge, the surprising result that there is no induced quadrupole moment. We show that this conclusion is gauge dependent and that the static, induced quadrupole moment for a black hole is inherently ambiguous. For the orbiting moon and the central Schwarzschild hole, we find (in agreement with a recent result of Poisson) a time-varying induced quadrupole moment that is proportional to the time derivative of the moon's tidal field. As a partial analog of a result derived long ago by Hartle for a spinning hole and a stationary distant companion, we show that the orbiting moon's tidal field induces a tidal bulge on the hole's horizon, and that the rate of change of the horizon shape leads the perturbing tidal field at the horizon by a small angle. 
  A classical pp-wave is a 4-dimensional Lorentzian spacetime which admits a nonvanishing parallel spinor field; here the connection is assumed to be Levi-Civita. We generalise this definition to metric compatible spacetimes with torsion and describe basic properties of such spacetimes. We use our generalised pp-waves for constructing new explicit vacuum solutions of quadratic metric-affine gravity. 
  We develop a semiclassical approximation scheme for the constraint equations of supersymmetric canonical quantum gravity. This is achieved by a Born-Oppenheimer type of expansion, in analogy to the case of the usual Wheeler-DeWitt equation. The formalism is only consistent if the states at each order depend on the gravitino field. We recover at consecutive orders the Hamilton-Jacobi equation, the functional Schrodinger equation, and quantum gravitational correction terms to this Schrodinger equation. In particular, the following consequences are found:   (i) the Hamilton-Jacobi equation and therefore the background spacetime must involve the gravitino, (ii) a (many fingered) local time parameter has to be present on $SuperRiem \Sigma$ (the space of all possible tetrad and gravitino fields), (iii) quantum supersymmetric gravitational corrections affect the evolution of the very early universe. The physical meaning of these equations and results, in particular the similarities to and differences from the pure bosonic case, are discussed. 
  We construct two distinct classes of exact type III solutions of the D=4 Einstein-Yang-Mills system. The solutions are embeddings of the non-abelian plane waves in spacetimes in Kundt's class. Reduction of the solutions to type N leads to generalized $pp$ and Kundt waves. The geodesic equations are briefly discussed. 
  The gravitational back-reaction on a certain type of rigidly-rotating cosmic string loop, first discovered by Allen, Casper and Ottewill, is studied at the level of the weak-field approximation. The near-field metric perturbations are calculated and used to construct the self-acceleration vector of the loop. Although the acceleration vector is divergent at the two kink points on the loop, its net effect on the trajectory over a single oscillation period turns out to be finite. The net back-reaction on the loop over a single period is calculated using a method due to Quashnock and Spergel, and is shown to induce a uniform shrinkage of the loop while preserving its original shape. The loop therefore evolves by self-similar evaporation. 
  The Newman-Penrose formalism is used to deal with the quasinormal modes(QNM's) of Rarita-Schwinger perturbations outside a Reissner-Nordstr\"{o}m black hole. We obtain four kinds of possible expressions of effective potentials, which are proved to be of the same spectra of quasinormal mode frequencies. The quasinormal mode frequencies evaluated by the WKB potential approximation show that, similar to those for Dirac perturbations, the real parts of the frequencies increase with the charge $Q$ and decrease with the mode number $n$, while the dampings almost keep unchanged as the charge increases. 
  The influence of spin on a photon's motion in a Schwarzschild and FRW spacetimes is studied. The first order correction to the geodesic motion is found. It is shown that unlike the world-lines of spinless particles, the photons world-lines do not lie in a plane. 
  We present the results of an analysis of superradiant energy flow due to scalar fields incident on an acoustic black hole. In addition to providing independent confirmation of the recent results in [5], we determine in detail the profile of energy flow everywhere outside the horizon. We confirm explicitly that in a suitable frame the energy flow is inward at the horizon and outward at infinity, as expected on physical grounds. 
  It has recently been shown that traversable wormholes may be supported by phantom energy. In this work phantom wormhole geometries are modelled by matching an interior traversable wormhole solution, governed by the equation of state $p=\omega \rho$ with $\omega<-1$, to an exterior vacuum spacetime at a finite junction interface. The stability analysis of these phantom wormholes to linearized spherically symmetric perturbations about static equilibrium solutions is carried out. A master equation dictating the stability regions is deduced, and by separating the cases of a positive and a negative surface energy density, it is found that the respective stable equilibrium configurations may be increased by strategically varying the wormhole throat radius. The first model considered, in the absence of a thin shell, is that of an asymptotically flat phantom wormhole spacetime. The second model constructed is that of an isotropic pressure phantom wormhole, which is of particular interest, as the notion of phantom energy is that of a spatially homogeneous cosmic fluid, although it may be extended to inhomogeneous spherically symmetric spacetimes. 
  We review a recent theoretical progress in the so-called self-force problem of a general relativistic two-body system. Although a two-body system in Newtonian gravity is a very simple problem, some fundamental issues are involved in relativistic gravity. Besides, because of recent projects for gravitational wave detection, it comes to be possible to see those phenomena directly via gravitational waves, and the self-force problem becomes one of urgent and highly-motivated problems in general relativity. Roughly speaking, there are two approaches to investigate this problem; the so-called post-Newtonian approximation, and a black hole perturbation.   In this paper, we review a theoretical progress in the self-force problem using a black hole perturbation. Although the self-force problem seems to be just a problem to calculate a self-force, we discuss that the real problem is to define a gauge invariant concept of a motion in a gauge dependent metric perturbation. 
  We propose a new metric perturbation scheme under a possible constraint of the gauge conditions in which we obtain a physically expected prediction of the orbital evolution caused by the MiSaTaQuWa self-force. In this new scheme of a metric perturbation, an adiabatic approximation is applied to both the metric perturbation and the orbit. As a result, we are able to predict the gravitational evolution of the system in the so-called radiation reaction time scale, which is longer than the dephasing time scale. However, for gravitational wave detection by LISA, this may still be insufficient. We further consider a gauge transformation in this new metric perturbation scheme, and find a special gauge condition with which we can calculate the gravitational waveform of a time scale long enough for gravitational wave detection by LISA. 
  The general metric for conformally flat stationary cyclic symmetric noncircular spacetimes is explicitly given. In spite of the complexity introduced by the inclusion of noncircular contributions, the related metric is derived via the full integration of the conformal flatness constraints. It is also shown that the conditions for the existence of a rotation axis (axisymmetry) are the same ones which restrict the above class of spacetimes to be static. As a consequence, a known theorem by Collinson is just part of a more general result: any conformally flat stationary cyclic symmetric spacetime, even a noncircular one, is additionally axisymmetric if and only if it is also static. Since recent astrophysical motivations point in the direction of considering noncircular configurations to describe magnetized neutron stars, the above results seem to be relevant in this context. 
  The present paper proposes a new explanation for the 3-dimensional Einstein general theory of relativity which is free of contradictions and consistent with usual 4-dimensional physics. We discuss the property of the new gravity theory with temporal scalar field arise in lower-dimensional theories as the reduction of timelike extra dimension. These ideas we continued by using the 3-dimensional analog of Jordan, Brans-Dicke theory with temporal scalar field where space and time are treated in different ways. 
  We establish a state of stopping the Hawking radiation by quantum Schwarzschild black hole in the framework of quasi-classical thermal quantization for particles behind the horizon. The mechanism of absorption and radiation by the black hole is presented. 
  Based on Lorentz and Levi-Civita's conservation laws, it can be shown that the energy of the matter field in the universe might originate from the gravitational field as a result of the latter field's energy decrease and the total entropy increase followed by cosmic expansion. By exploring this possibility and by using some new evidences discovered from recent astronomical observations, we establish an alternative theory of cosmology, which gives a new interpretation about the evolution of the cosmos and a number of new explanations regarding dark energy and dark matter. 
  For a successful detection of gravitational waves by LISA, it is essential to construct theoretical waveforms in a reliable manner. We discuss gravitational waves from an extreme mass ratio binary system which is expected to be a promising target of the LISA project.   The extreme mass ratio binary is a binary system of a supermassive black hole and a stellar mass compact object. As the supermassive black hole dominates the gravitational field of the system, we suppose that the system might be well approximated by a metric perturbation of a Kerr black hole. We discuss a recent theoretical progress in calculating the waveforms from such a system. 
  Using a collapsing matter model at the center of an expanding universe as described by Weinberg we assume a special type of generated pressure. This pressure transmits into the surrounding expanding universe. Under certain restriction the ensuing hubble parameter is positive. The deacceleration parameter fluctuates with time, indicating that the universe accelerates for certain time and decelerates for other time intervals. 
  The kink stability of self-similar solutions of a massless scalar field with circular symmetry in 2+1 gravity is studied, and found that such solutions are unstable against the kink perturbations along the sonic line (self-similar horizon). However, when perturbations outside the sonic line are considered, and taking the ones along the sonic line as their boundary conditions, we find that non-trivial perturbations do not exist. In other words, the consideration of perturbations outside the sonic line limits the unstable mode of the perturbations found along the sonic line. As a result, the critical solution for the scalar collapse remains critical even after the kink perturbations are taken into account. 
  We describe solutions of the Klein-Gordon equation which are spherically symmetric and localized, and may be regarded as massive particles without charge or spin. The proposed model, which is based on the action for a complex scalar field minimally coupled to the electromagnetic and gravitational fields, contains no adjustable parameters and predicts five particle species with masses of the order of the Planck mass. These particles appear to be candidates for dark matter. 
  ASTRODynamical Space Test of Relativity using Optical Devices I (ASTROD I) mainly aims at testing relativistic gravity and measuring the solar-system parameters with high precision, by carrying out laser ranging between a spacecraft in a solar orbit and ground stations. In order to achieve these goals, the magnitude of the total acceleration disturbance of the proof mass has to be less than 10&#8722;13 m s&#8722;2 Hz&#8722;1/2 at 0.1 m Hz. In this paper, we give a preliminary overview of the sources and magnitude of acceleration disturbances that could arise in the ASTROD I proof mass. Based on the estimates of the acceleration disturbances and by assuming a simple controlloop model, we infer requirements for ASTROD I. Our estimates show that most of the requirements for ASTROD I can be relaxed in comparison with Laser Interferometer Space Antenna (LISA). 
  Given a spacelike foliation of a spacetime and a marginally outer trapped surface S on some initial leaf, we prove that under a suitable stability condition S is contained in a ``horizon'', i.e. a smooth 3-surface foliated by marginally outer trapped slices which lie in the leaves of the given foliation. We also show that under rather weak energy conditions this horizon must be either achronal or spacelike everywhere. Furthermore, we discuss the relation between ``bounding'' and ``stability'' properties of marginally outer trapped surfaces. 
  We show succinctly that all metric theories with second order field equations obey Birkhoff's theorem: their spherically symmetric solutions are static. 
  The Advanced Laser Interferometer Antenna (ALIA) and the Big Bang Observer (BBO) have been proposed as follow on missions to the Laser Interferometer Space Antenna (LISA). Here we study the capabilities of these observatories, and how they relate to the science goals of the missions. We find that the Advanced Laser Interferometer Antenna in Stereo (ALIAS), our proposed extension to the ALIA mission, will go considerably further toward meeting ALIA's main scientific goal of studying intermediate mass black holes. We also compare the capabilities of LISA to a related extension of the LISA mission, the Laser Interferometer Space Antenna in Stereo (LISAS). Additionally, we find that the initial deployment phase of the BBO would be sufficient to address the BBO's key scientific goal of detecting the Gravitational Wave Background, while still providing detailed information about foreground sources. 
  We analyze a class of exact type II solutions of the Robinson-Trautman family which contain pure radiation and (possibly) a cosmological constant. It is shown that these spacetimes exist for any sufficiently smooth initial data, and that they approach the spherically symmetric Vaidya-(anti-)de Sitter metric. We also investigate extensions of the metric, and we demonstrate that their order of smoothness is in general only finite. Some applications of the results are outlined. 
  We study the behavior of nonzero rest mass spinning test particles moving along circular orbits in the Schwarzschild spacetime in the case in which the components of the spin tensor are allowed to vary along the orbit, generalizing some previous work. 
  We define the cohomogeneity one string, string with continuous symmetries, as its world surface is tangent to a Killing vector field of a target space. We classify the Killing vector fields by an equivalence relation using isometries of the target space. We find that the equivalence classes of Killing vectors in Minkowski spacetime are partitioned into seven families. It is clarified that there exist seven types of strings with spacelike symmetries and four types of strings with timelike symmetries, stationary strings. 
  Topological classification of the 4-manifolds bridges computation theory and physics. A proof of the undecidability of the homeomorphy problem for 4-manifolds is outlined here in a clarifying way. It is shown that an arbitrary Turing machine with an arbitrary input can be encoded into the topology of a 4-manifold, such that the 4-manifold is homeomorphic to a certain other 4-manifold if and only if the corresponding Turing machine halts on the associated input. Physical implications are briefly discussed. 
  Cosmic horizons arise in general relativity in the context of black holes and in certain cosmologies. Classically, regions beyond a horizon are inaccessible to causal observers. However, quantum mechanical correlations may exist across horizons that may influence local observations. For the case of de Sitter space, we show how a single particle excitation behind the horizon changes the density matrix governing local observables. As compared to the vacuum state, we calculate the change in the average energy and entropy per unit volume. This illustrates what may be a generic property allowing some features of spacetime beyond a horizon to be inferred. 
  A covariant scalar-tensor-vector gravity theory is developed which allows the gravitational constant $G$, a vector field coupling $\omega$ and the vector field mass $\mu$ to vary with space and time. The equations of motion for a test particle lead to a modified gravitational acceleration law that can fit galaxy rotation curves and cluster data without non-baryonic dark matter. The theory is consistent with solar system observational tests. The linear evolutions of the metric, vector field and scalar field perturbations and their consequences for the observations of the cosmic microwave background are investigated. 
  We provide a generating functional for the gravitational field, associated to the relaxation of the primary constraints as extended to the quantum sector. This requirement of the theory, relies on the assumption that a suitable time variable exist, when taking the T-products of the dynamical variables. More precisely, we start from the gravitational field equations written in the Hamiltonian formalism and expressed via Misner-like variables; hence we construct the equation to which the T-products of the dynamical variables obey and transform this paradigm in terms of the generating functional, as taken on the theory phase-space. We show how the relaxation of the primary constraints (which correspond to break down the invariance of the quantum theory under the 4-diffeomorphisms) is summarized by a free functional taken on the Lagrangian multipliers, accounting for such constraints in the classical theory. The issue of our analysis is equivalent to a Gupta-Bleuler approach on the quantum implementation of all the gravitational constraints; in fact, in the limit of small $\hbar$, the quantum dynamics is described by a Schr\"odinger equation, as soon as the mean values of the momenta, associated to the lapse function and the shift vector, are not vanishing. Finally we show how, in the classical limit, the evolutionary quantum gravity reduces to General Relativity in the presence of an Eckart fluid, which corresponds to the classical counterpart of the physical clock, introduced in the quantum theory. 
  A semiclassical calculation of entropy of a scalar field in the background of a class of brane world black holes (BWBH) is carried out in the presence of a brick wall cutoff when the 5D-bulk induced \textquotedblleft tidal charge" has generic or extreme values. 
  In this paper, we discuss the properties of one-parameter sequences that arise when solving the Hamiltonian constraint in Bianchi I loop quantum cosmology using a separation of variables method. In particular, we focus on finding an expression for the sequence for all real values of the parameter, and discuss the pre-classicality of this function. We find that the behavior of these preclassical sequences imply time asymmetry on either side of the classical singularity in Bianchi I cosmology. 
  We describe the cross-correlation measurements being carried out on data from the LIGO Livingston Observatory and the ALLEGRO resonant bar detector. The LIGO data are sampled at 16384 Hz while the ALLEGRO data are base-banded, i.e., heterodyned at 899 Hz and then sampled at 250 Hz. We handle these different sampling parameters by working in the Fourier domain, and demonstrate the approximate equivalence of this measurement to a hypothetical time-domain method in which both data streams are upsampled. 
  We study the fluctuations of the stress tensor for a massless scalar field in two and four-dimensional Minkowski spacetime in the vacuum state. Covariant expressions for the stress tensor correlation function are obtained as sums of derivatives of a scalar function. These expressions allow one to express spacetime averages of the correlation function as finite integrals. We also study the correlation between measurements of the energy density along a worldline. We find that these measurements may be either positively correlated or anticorrelated. The anticorrelated measurements can be interpreted as telling us that if one measurement yields one sign for the averaged energy density, a successive measurement with a suitable time delay is likely to yield a result with the opposite sign. 

  The false vacuum decay in a brane world model is studied in this work. We investigate the vacuum decay via the Coleman-de Luccia instanton, derive explicit approximative expressions for the Coleman-de Luccia instanton which is close to a Hawking-Moss instanton and compare the results with those already obtained within Einstein's theory of relativity. 
  We extend the "analogue spacetime" programme by investigating a condensed-matter system that is in principle capable of simulating the massive Klein--Gordon equation in curved spacetime. Since many elementary particles have mass, this is an essential step in building realistic analogue models, and a first step towards simulating quantum gravity phenomenology. Specifically, we consider the class of two-component BECs subject to laser-induced transitions between the components. This system exhibits a complicated spectrum of normal mode excitations, which can be viewed as two interacting phonon modes that exhibit the phenomenon of "refringence". We study the conditions required to make these two phonon modes decouple. Once decoupled, the two distinct phonons generically couple to distinct effective spacetimes, representing a bi-metric model, with one of the modes acquiring a mass. In the eikonal limit the massive mode exhibits the dispersion relation of a massive relativistic particle: omega = sqrt[omega_0^2 + c^2 k^2], plus curved-space modifications. Furthermore, it is possible to tune the system so that both modes can be arranged to travel at the same speed, in which case the two phonon excitations couple to the same effective metric. From the analogue spacetime perspective this situation corresponds to the Einstein equivalence principle being satisfied. 
  We obtain the electrostatic energy of two opposite charges near the horizon of stationary black-holes in the massive Schwinger model. Besides the confining aspects of the model, we discuss the Bekenstein entropy upper bound of a charged object using the generalized second law. We show that despite the massless case, in the massive Schwinger model the entropy of the black hole and consequently the Bekenstein bound are altered by the vacuum polarization. 
  The purpose of this article is to highlight the fascinating, but only very incompletely understood relation between Einstein's theory and its generalizations on the one hand, and the theory of indefinite, and in particular hyperbolic, Kac Moody algebras on the other. The elucidation of this link could lead to yet another revolution in our understanding of Einstein's theory and attempts to quantize it. 
  Given two observers, we define the ``relative velocity'' of one observer with respect to the other in four different ways. All four definitions are given intrinsically, i.e. independently of any coordinate system. Two of them are given in the framework of spacelike simultaneity and, analogously, the other two are given in the framework of observed (lightlike) simultaneity. Properties and physical interpretations are discussed. Finally, we study relations between them in special relativity, and we give some examples in Schwarzschild and Robertson-Walker spacetimes. 
  In a recent letter by H. Davoudiasl, R. Kitano, T. Li and H. Murayama ``The new Minimal Standard Model'' (NMSM) was constructed that incorporates new physics beyond the Minimal Standard Model (MSM) of particle physics. The authors follow the principle of minimal particle content and therefore adopt the viewpoint of particle physicists. It is shown that a generalisation of the geometric structure of spacetime can also be used to explain physics beyond the MSM. It is explicitly shown that for example inflation, i.e. an exponentially expanding universe, can easily be explained within the framework of Einstein-Cartan theory. 
  In the context of a nonlinear gauge theory of the Poincar\'e group, we show that covariant derivatives of Dirac fields include a coupling to the translational connections, manifesting itself in the matter action as a universal background mass contribution to fermions. 
  We take a step toward a nonperturbative gravitational path integral for black-hole geometries by deriving an expression for the expansion rate of null geodesic congruences in the approach of causal dynamical triangulations. We propose to use the integrated expansion rate in building a quantum horizon finder in the sum over spacetime geometries. It takes the form of a counting formula for various types of discrete building blocks which differ in how they focus and defocus light rays. In the course of the derivation, we introduce the concept of a Lorentzian dynamical triangulation of product type, whose applicability goes beyond that of describing black-hole configurations. 
  We study a Hamiltonian realization of the phase space of kappa-Poincare algebra that yields a definition of velocity consistent with the deformed Lorentz symmetry. We are also able to determine the laws of transformation of spacetime coordinates and to define an invariant spacetime metric, and discuss some possible experimental consequences. 
  A possible definition of strong/symmetric hyperbolicity for a second-order system of evolution equations is that it admits a reduction to first order which is strongly/symmetric hyperbolic. We investigate the general system that admits a reduction to first order and give necessary and sufficient criteria for strong/symmetric hyperbolicity of the reduction in terms of the principal part of the original second-order system. An alternative definition of strong hyperbolicity is based on the existence of a complete set of characteristic variables, and an alternative definition of symmetric hyperbolicity is based on the existence of a conserved (up to lower order terms) energy. Both these definitions are made without any explicit reduction. Finally, strong hyperbolicity can be defined through a pseudo-differential reduction to first order. We prove that both definitions of symmetric hyperbolicity are equivalent and that all three definitions of strong hyperbolicity are equivalent (in three space dimensions). We show how to impose maximally dissipative boundary conditions on any symmetric hyperbolic second order system. We prove that if the second-order system is strongly hyperbolic, any closed constraint evolution system associated with it is also strongly hyperbolic, and that the characteristic variables of the constraint system are derivatives of a subset of the characteristic variables of the main system, with the same speeds. 
  We discuss the local Lorentz invariance in the context of N=1 supergravity and show that a previous attempt to find explicit solutions to the Lorentz constraint in terms of $\gamma-$matrices is not correct. We improve that solution by using a different representation of the Lorentz operators in terms of the generators of the rotation group, and show its compatibility with the matrix representation of the fermionic field. We find the most general wave functional that satisfies the Lorentz constraint in this representation. 
  In ``Decoherence of macroscopic closed systems within Newtonian quantum gravity'' (Kay B S 1998 Class. Quantum Grav. 15 L89-L98) it was argued that, given a many-body Schroedinger wave function \psi(x_1,...,x_N) for the centre-of-mass degrees of freedom of a closed system of N identical uniform-mass balls of mass M and radius R, taking account of quantum gravitational effects and then tracing over the gravitational field amounts to multiplying the position-space density matrix \rho(x_1,...,x_N; x_1',...,x_N')= \psi(x_1,...,x_N)\psi*(x_1',...,x_N') by a multiplicative factor, which, if the positions {x_1,...,x_N; x_1',...,x_N'} are all much further away from one another than R, is well-approximated by the product from 1 to N over I, J, K (I<J) of ((|x_K-x_K'|/R)(|x_I'-x_J||x_I-x_J'|/|x_I-x_J||x_I'-x_J'|))^{-24M^2}. Here we show that if each uniform-mass ball is replaced by a grainy ball or more general-shaped lump of similar size consisting of a number, n, of well-spaced small balls of mass m and radius r and, in the above formula, R is replaced by r, M by m and the products are taken over all Nn positions of all the small balls, then the result is well-approximated by replacing R in the original formula by a new value R_eff. This suggests that the original formula will apply in general to physically realistic lumps -- be they macroscopic lumps of ordinary matter with the grains atomic nuclei etc. or be they atomic nuclei themselves with their own (quantum) grainy substructure -- provided R is chosen suitably. In the case of a cubical lump consisting of n=(2L+1)^3 small balls (L > 0) of radius r with centres at the vertices of a cubic lattice of spacing a (assumed to be very much bigger than 2r) and side 2La we establish the bound e^{-1/3}(r/a)^{1/n}La < R_eff < 2\sqrt 3(r/a)^{1/n} La. 
  We present a full investigation of scalar perturbations in a rather generic model for a regular bouncing universe, where the bounce is triggered by an effective perfect fluid with negative energy density. Long before and after the bounce the universe is dominated by a source with positive energy density, which may be a perfect fluid, a scalar field, or any other source with an intrinsic isocurvature perturbation. Within this framework, we present an analytical method to accurately estimate the spectrum of large-scale scalar perturbations until their reentry, long after the bounce. We also propose a simple way to identify non-singular gauge-invariant variables through the bounce and present the results of extensive numerical tests in several possible realizations of the scenario. In no case do we find that the spectrum of the pre-bounce growing mode of the Bardeen potential can be transferred to a post-bounce constant mode. 
  Using canonical quantization of a flat FRW cosmological model containing a real scalar field $\phi$ endowed with a scalar potential $V(\phi)$, we are able to obtain exact and semiclassical solutions of the so called Wheeler-DeWitt equation for a particular family of scalar potentials. Some features of the solutions and their classical limit are discussed. 
  The Maxwell equations expressed in terms of the excitation $\H=({\cal H}, {\cal D})$ and the field strength $F=(E,B)$ are metric-free and require an additional constitutive law in order to represent a complete set of field equations. In vacuum, we call this law the ``spacetime relation''. We assume it to be local and linear. Then $\H=\H(F)$ encompasses 36 permittivity/permeability functions characterizing the electromagnetic properties of the vacuum. These 36 functions can be grouped into 20+15+1 functions. Thereof, 20 functions finally yield the dilaton field and the metric of spacetime, 1 function represents the axion field, and 15 functions the (traceless) skewon field $\notS_i{}^j$ (S slash), with $i,j=0,1,2,3$. The hypothesis of the existence of $\notS_i{}^j$ was proposed by three of us in 2002. In this paper we discuss some of the properties of the skewon field, like its electromagnetic energy density, its possible coupling to Einstein-Cartan gravity, and its corresponding gravitational energy. 
  We study the WKB approximation beyond leading order for cosmological perturbations during inflation. To first order in the slow-roll parameters, we show that an improved WKB approximation leads to analytical results agreeing to within 0.1% with the standard slow-roll results. Moreover, the leading WKB approximation to second order in the slow-roll parameters leads to analytical predictions in qualitative agreement with those obtained by the Green's function method. 
  We present a discussion of the effects induced by the bulk viscosity on the very early Universe stability. The matter filling the cosmological (isotropic and homogeneous) background is described by a viscous fluid having an ultrarelativistic equation of state and whose viscosity coefficient is related to the energy density via a power-law of the form $\zeta=\zeta_0 \rho^\nu$. The analytic expression of the density contrast (obtained for $\nu=1/2$) shows that, for small values of the constant $\zeta_0$, its behavior is not significantly different from the non-viscous one derived by E.M. Lifshitz. But as soon as $\zeta_0$ overcomes a critical value, the growth of the density contrast is suppressed forward in time by the viscosity and the stability of the Universe is favored in the expanding picture. On the other hand, in such a regime, the asymptotic approach to the initial singularity (taken at $t=0$) is deeply modified by the apparency of significant viscosity in the primordial thermal bath i.e. the isotropic and homogeneous Universe admits an unstable collapsing picture. In our model this feature regards also scalar perturbations while in the non-viscous case it appears only for tensor modes. 
  We show that the Letelier-Gal'tsov (LG) metric describing multiple crossed strings in relative motion does solve the Einstein equations, in spite of the discontinuity uncovered recently by Krasnikov [gr-qc/0502090] provided the strings are straight and moving with constant velocities. 
  The purpose of this work is to return, with a new observation and rather unconventional point of view, to the study of asymptotically flat solutions of Einstein equations. The essential observation is that from a given asymptotically flat space-time with a given Bondi shear, one can find (by integrating a partial differential equation) a class of asymptotically shear-free (but, in general, twistiing) null geodesic congruences. The class is uniquely given up to the arbitrary choice of a complex analytic world-line in a four-parameter complex space. Surprisingly this parameter space turns out to be the H-space that is associated with the real physical space-time under consideration. The main development in this work is the demonstration of how this complex world-line can be made both unique and also given a physical meaning. More specifically by forcing or requiring a certain term in the asymptotic Weyl tensor to vanish, the world-line is uniquely determined and becomes (by several arguments) identified as the `complex center-of-mass'. Roughly, its imaginary part becomes identified with the intrinsic spin-angular momentum while the real part yields the orbital angular momentum. 
  Stellar oscillation under the combined influences of incident gravitational wave and radiation loss is studied in a simple toy model. The star is approximated as a uniform density ellipsoid in the Newtonian gravity including radiation damping through quadrupole formula. The time evolution of the oscillation is significantly controlled by the incident wave amplitude $h$, frequency $\nu$ and damping time $\tau$. If a combination $ h \nu \tau $ exceeds a threshold value, which depends on the resonance mode, the resonant growth is realized. 
  We study quasi-spherical Szekeres space-time (which possess no killing vectors) for perfect fluid, matter with tangential stress only and matter with anisotropic pressure respectively. In the first two cases cosmological solutions have been obtained and their asymptotic behaviour have been examined while for anisotropic pressure, gravitational collapse has been studied and the role of the pressure has been discussed. 
  Matter collineations of locally rotationally symmetric spacetimes are considered. These are investigated when the energy-momentum tensor is degenerate. We know that the degenerate case provides infinite dimensional matter collineations in most of the cases. However, an interesting case arises where we obtain proper matter collineations. We also solve the constraint equations for a particular case to obtain some cosmological models. 
  We study the consequences of nonzero neutrino masses for black holes evaporating by the emission of Hawking radiation. We find that the evolution of small, hot, black holes may be unaffected (if neutrinos are Majorana particles), or may show an increase in neutrino luminosity and a decrease in lifetime by up to a factor of 1.85 (if neutrinos are Dirac particles). However, for sufficiently large (e.g., stellar mass) black holes, neutrino emission is largely or entirely suppressed, resulting in a decrease in emitted power and an increase in lifetime by up to a factor of 7.5. 
  We show how the quantum to classical transition of the cosmological fluctuations produced during inflation can be described by means of the influence functional and the master equation. We split the inflaton field into the system-field (long-wavelength modes), and the environment, represented by its own short-wavelength modes. We compute the decoherence times for the system-field modes and compare them with the other time scales of the model. We present the renormalized stochastic Langevin equation for an homogeneous system-field and then we analyze the influence of the environment on the power spectrum for some modes in the system. 
  Binary black hole coalescence produces a finite burst of gravitational radiation which propagates towards quiescent infinity. These spacetimes are analytic about infinity and contain a dimensionless coupling constant $M/s$, where $M$ denotes the total mass-energy and $s$ an imaginary distance. This introduces globally convergent Fourier series on a complex radial coordinate, allowing spectral representation of black hole spacetimes in all three dimensions. We illustrate this representation theory on a Fourier-Legendre expansion of Boyer-Lindquist initial data and a scalar wave equation with signal recovery by Cauchy's integral formula. 
  In the framework of thermal quantization of radial geodesics completely confined behind the horizons we calculate the entropy of BTZ black hole in agreement with the Bekenstein--Hawking relation. Particles in the BTZ black hole occupy the only quantum ground level. The quantization allow us to find a linear dependence of black hole mass versus its orbital momentum. 
  We ask if the conventional variable separation techniques in the studying of standard cosmology and the collapsing of extremely dense stars introduce Newton's absolute space-time concepts. If this is the case, then a completely relative cosmology is needed. We build the basic frame-works for such a cosmology and illustrate that, the observed luminosity-distance v.s. red-shift relations of supernovaes can be explained naturally even without any conception of dark energies. 
  One of the greatest data analysis challenges for the Laser Interferometer Space Antenna (LISA) is the need to account for a large number of gravitational wave signals from compact binary systems expected to be present in the data. We introduce the basis of a Bayesian method that we believe can address this challenge, and demonstrate its effectiveness on a simplified problem involving one hundred synthetic sinusoidal signals in noise. We use a reversible jump Markov chain Monte Carlo technique to infer simultaneously the number of signals present, the parameters of each identified signal, and the noise level. Our approach therefore tackles the detection and parameter estimation problems simultaneously, without the need to evaluate formal model selection criteria, such as the Akaike Information Criterion or explicit Bayes factors. The method does not require a stopping criterion to determine the number of signals, and produces results which compare very favorably with classical spectral techniques. 
  We use dimensional regularization to compute the 1PI 1-point function of quantum gravity at one loop order in a locally de Sitter background. As with other computations, the result is a finite constant at this order. It corresponds to a small positive renormalization of the cosmological constant. 
  We compute the energy and angular momenta of recent D-dimensional Kerr-AdS solutions to cosmological Einstein gravity, as well as of the BTZ metric, using our invariant charge definitions. 
  In this paper we present new estimates of the coalescence rate of neutron star binaries in the local universe and we discuss its consequences for the first generations of ground based interferometers. Our approach based on both evolutionary and statistical methods gives a galactic merging rate of 1.7 10$^{-5}$ yr$^{-1}$, in the range of previous estimates 10$^{-6}$ - 10$^{-4}$ yr$^{-1}$. The local rate which includes the contribution of elliptical galaxies is two times higher, in the order of 3.4 10$^{-5}$ yr$^{-1}$. We predict one detection every 148 and 125 years with initial VIRGO and LIGO, and up to 6 events per year with their advanced configuration. Our recent detection rate estimates from investigations on VIRGO future improvements are quoted. 
  The Laser Interferometer Space Antenna (LISA) is expected to simultaneously detect many thousands of low frequency gravitational wave signals. This presents a data analysis challenge that is very different to the one encountered in ground based gravitational wave astronomy. LISA data analysis requires the identification of individual signals from a data stream containing an unknown number of overlapping signals. Because of the signal overlaps, a global fit to all the signals has to be performed in order to avoid biasing the solution. However, performing such a global fit requires the exploration of an enormous parameter space with a dimension upwards of 50,000. Markov Chain Monte Carlo (MCMC) methods offer a very promising solution to the LISA data analysis problem. MCMC algorithms are able to efficiently explore large parameter spaces, simultaneously providing parameter estimates, error analyses and even model selection. Here we present the first application of MCMC methods to simulated LISA data and demonstrate the great potential of the MCMC approach. Our implementation uses a generalized F-statistic to evaluate the likelihoods, and simulated annealing to speed convergence of the Markov chains. As a final step we super-cool the chains to extract maximum likelihood estimates, and estimates of the Bayes factors for competing models. We find that the MCMC approach is able to correctly identify the number of signals present, extract the source parameters, and return error estimates consistent with Fisher information matrix predictions. 
  We derive the perturbation equations for relativistic stars in scalar-tensor theories of gravity and study the corresponding oscillation spectrum. We show that the frequency of the emitted gravitational waves is shifted proportionally to the scalar field strength. Scalar waves which might be produced from such oscillations can be a unique probe for the theory, but their detectability is questionable if the radiated energy is small. However we show that there is no need for a direct observation of scalar waves: the shift in the gravitational wave spectrum could unambiguously signal the presence of a scalar field. 
  This study is purposed to elaborate the problem of energy and momentum distribution of the viscous Kasner-type universe in General theory of relativity. In this connection, we use the energy-momentum definitions of Einstein, Papapetrou and Landau-Lifshitz and obtained that the energy-momentum distributions (due to matter plus field) of the closed universes which is based on the viscous Kasner-type metric are vanishing everywhere. This results are exactly the same as obtained by Salti et al. and agree with a previous work of Rosen and Johri et al. who investigated the problem of the energy in Friedmann-Robertson-Walker universe. The result that the total energy-momentum of the universe in these models are zero support the viewpoint of Tryon. 
  In this paper we study solution of the photon equation (the Massless Duffin-Kemmer-Petiau equation (mDKP)) in anisotropic expanding the Bianchi-I type spacetime using the Fourier analyze method. The harmonic oscillator behavior of the solutions is found. It is shown that Maxwell equations are equivalent to the photon equation. 
  Why are there no fundamental scalar fields actually observed in physics today? Scalars are the simplest fields, but once we go beyond Galilean-Newtonian physics they appear only in speculations, as possible determinants of the gravitational constants in the so-called Scalar-Tensor theories in non-quantum physics, and as Higgs particles, dilatons, etc., in quantum physics. Actually, scalar fields have had a long and controversial life in gravity theories, with a history of deaths and resurrections. This paper presents a brief overview of this history. 
  We study the D=5 Emparan-Reall spinning black ring under an ultrarelativistic boost along an arbitrary direction. We analytically determine the resulting shock pp-wave, in particular for boosts along axes orthogonal and parallel to the plane of rotation. The solution becomes physically more interesting and simpler if one enforces equilibrium between the forces on the ring. We also comment on the ultrarelativistic limit of recently found supersymmetric black rings with two independent angular momenta. Essential distinct features with respect to the boosted Myers-Perry black holes are pointed out. 
  This survey paper is divided into two parts. In the first (section 2), I give a brief account of the structure of classical relativity theory. In the second (section 3), I discuss three special topics: (i) the status of the relative simultaneity relation in the context of Minkowski spacetime; (ii) the "geometrized" version of Newtonian gravitation theory (also known as Newton-Cartan theory); and (iii) the possibility of recovering the global geometric structure of spacetime from its "causal structure". 
  In a previous work, we used a polarization condition to show that there is a family of U(1) symmetric solutions of the vacuum Einstein equations such that each exhibits AVTD (Asymptotic Velocity Term Dominated) behavior in the neighborhood of its singularity. Here we consider the general case of U(1) bundles and determine a condition, called the half polarization condition, necessary and sufficient in our context, for AVTD behavior near the singularity. 
  We present a new group field theory model, generalising the Boulatov model, which incorporates both 3-dimensional gravity and matter coupled to gravity. We show that the Feynman diagram amplitudes of this model are given by Riemannian quantum gravity spin foam amplitudes coupled to a scalar matter field. We briefly discuss the features of this model and its possible generalisations. 
  This article deals with empty spacetime and the question of its physical reality. By "empty spacetime" we mean a collection of bare spacetime points, the remains of ridding spacetime of all matter and fields. We ask whether these geometric objects--themselves intrinsic to the concept of field--might be observable through some physical test. By taking quantum-mechanical notions into account, we challenge the negative conclusion drawn from the diffeomorphism invariance postulate of general relativity, and we propose new foundational ideas regarding the possible observation--as well as conceptual overthrow--of this geometric ether. 
  Scalar QFT on the boundary $\Im^+$ at null infinity of a general asymptotically flat 4D spacetime is constructed using the algebraic approach based on Weyl algebra associated to a BMS-invariant symplectic form. The constructed theory is invariant under a suitable unitary representation of the BMS group with manifest meaning when the fields are interpreted as suitable extensions to $\Im^+$ of massless minimally coupled fields propagating in the bulk. The analysis of the found unitary BMS representation proves that such a field on $\Im^+$ coincides with the natural wave function constructed out of the unitary BMS irreducible representation induced from the little group $\Delta$, the semidirect product between SO(2) and the two dimensional translational group. The result proposes a natural criterion to solve the long standing problem of the topology of BMS group. Indeed the found natural correspondence of quantum field theories holds only if the BMS group is equipped with the nuclear topology rejecting instead the Hilbert one. Eventually some theorems towards a holographic description on $\Im^+$ of QFT in the bulk are established at level of $C^*$ algebras of fields for strongly asymptotically predictable spacetimes. It is proved that preservation of a certain symplectic form implies the existence of an injective $*$-homomorphism from the Weyl algebra of fields of the bulk into that associated with the boundary $\Im^+$. Those results are, in particular, applied to 4D Minkowski spacetime where a nice interplay between Poincar\'e invariance in the bulk and BMS invariance on the boundary at $\Im^+$ is established at level of QFT. It arises that the $*$-homomorphism admits unitary implementation and Minkowski vacuum is mapped into the BMS invariant vacuum on $\Im^+$. 
  Quasi-periodic oscillations of high density thick accretion disks orbiting a Schwarzschild black hole have been recently addressed as interesting sources of gravitational waves. The aim of this paper is to compare the gravitational waveforms emitted from these sources when computed using (variations of) the standard quadrupole formula and gauge-invariant metric perturbation theory. To this goal we evolve representative disk models using an existing general relativistic hydrodynamics code which has been previously employed in investigations of such astrophysical systems. Two are the main results of this work: First, for stable and marginally stable disks, no excitation of the black hole quasi-normal modes is found. Secondly, we provide a simple, relativistic modification of the Newtonian quadrupole formula which, in certain regimes, yields excellent agreement with the perturbative approach. This holds true as long as back-scattering of GWs is negligible. Otherwise, any functional form of the quadrupole formula yields systematic errors of the order of 10%. 
  An approach to the discrete quantum gravity based on the Regge calculus is discussed which was developed in a number of our papers. Regge calculus is general relativity for the subclass of general Riemannian manifolds called piecewise flat ones. Regge calculus deals with the discrete set of variables, triangulation lengths, and contains continuous general relativity as a particular limiting case when the lengths tend to zero. In our approach the quantum length expectations are nonzero and of the order of Plank scale $10^{-33}cm$. This means the discrete spacetime structure on these scales. 
  Wick rotation in area tensor Regge calculus is considered. The heuristical expectation is confirmed that the Lorentzian quantum measure on a spacelike area should coincide with the Euclidean measure at the same argument. The consequence is validity of probabilistic interpretation of the Lorentzian measure as well (on the real, i.e. spacelike areas). 
  We construct and investigate a compactified version of the four-dimensional Reissner-Nordstrom-NUT solution, generalizing the compactified Schwarzschild black hole that has been previously studied by several workers. Our approach to compactification is based on dimensional reduction with respect to the stationary Killing vector, resulting in three-dimensional gravity coupled to a nonlinear sigma model. Using that the original non-compactified solution corresponds to a target space geodesic, the problem can be linearized much in the same way as in the case of no electric nor NUT charge. An interesting feature of the solution family is that for nonzero electric charge but vanishing NUT charge, the solution has a curvature singularity on a torus that surrounds the event horizon, but this singularity is removed when the NUT charge is switched on. We also treat the Schwarzschild case in a more complete way than has been done previously. In particular, the asymptotic solution (the Levi-Civita solution with the height coordinate made periodic) has to our knowledge only been calculated up to a determination of the mass parameter. The periodic Levi-Civita solution contains three essential parameters, however, and the remaining two are explicitly calculated here. 
  We show that the 4+1 dimensional vacuum Einstein equations admit gravitational waves with radial symmetry. The dynamical degrees of freedom correspond to deformations of the three-sphere orthogonal to the $(t,r)$ plane. Gravitational collapse of such waves is studied numerically and shown to exhibit discretely self-similar Type II critical behavior at the threshold of black hole formation. 
  Einstein's general relativity is increasingly important in contemporary physics on the frontiers of both the very largest distance scales (astrophysics and cosmology) and the very smallest(elementary particle physics). This paper makes the case for a `physics first' approach to introducing general relativity to undergraduate physics majors. 
  We study q-stars in Brans-Dicke gravitational theory. We find that when the Brans-Dicke constant, $\omega_{\textrm{BD}}$, tends to infinity, the results of General Relativity are reproduced. For other values of $\omega_{\textrm{BD}}$, the particle number, mass and radius of the star and the absolute value of the matter field are a few percent larger than in the case of General Relativity. We also investigate the general scalar-tensor gravitational theory and find that the star parameters are a few percent larger than in the case of General Relativity. 
  A metric representing the Kerr geometry has been obtained by Pretorius and Israel. We make a coordinate transformation on this metric, thereby bringing it into Bondi-Sachs form. In order to validate the metric, we evaluate it numerically on a regular grid of the new coordinates. The Ricci tensor is then computed, for different discretizations, and found to be convergent to zero. We also investigate the behaviour of the metric near the axis of symmetry and confirm regularity. Finally we investigate a Bondi-Sachs representation of the Kerr geometry reported by Fletcher and Lun; we confirm numerically that their metric is Ricci flat, but find that it has an irregular behaviour at the pole. 
  This article reviews the current status of black hole astrophysics, focusing on topics of interest to a physics audience. Astronomers have discovered dozens of compact objects with masses greater than 3 solar masses, the likely maximum mass of a neutron star. These objects are identified as black hole candidates. Some of the candidates have masses of 5 to 20 solar masses and are found in X-ray binaries, while the rest have masses from a million to a billion solar masses and are found in galactic nuclei. A variety of methods are being tried to estimate the spin parameters of the candidate black holes. There is strong circumstantial evidence that many of the objects have event horizons. Recent MHD simulations of magnetized plasma accreting on rotating black holes seem to hint that relativistic jets may be produced by a magnetic analog of the Penrose process. 
  In cosmology one labels the time t since the Big Bang in terms of the redshift of light emitted at t, as we see it now. In this Note we derive a formula that relates t to z which is valid for all redshifts. One can go back in time as far as one wishes, but not to the Big Bang at which the redshift tends to infinity. 
  Changing the set of independent variables of Poincare gauge theory and considering, in a manner similar to the second order formalism of general relativity, the Riemannian part of the Lorentz connection as function of the tetrad field, we construct theories that do not contain second or higher order derivatives in the field variables, possess a full general relativity limit in the absence of spinning matter fields, and allow for propagating torsion fields in the general case. A concrete model is discussed and the field equations are reduced by means of a Yasskin type ansatz to a conventional Einstein-Proca system. Approximate solutions describing the exterior of a spin polarized neutron star are prsented and the possibility of an experimental detection of the torsion fields is briefly discussed. 
  In the present work, it is shown that the geometerization philosophy has not been exhausted. Some quantum roots are already built in non-symmetric geometries. Path equations in such geometries give rise to spin-gravity interaction. Some experimental evidences (the results of the COW-experiment) indicate the existence of this interaction. It is shown that the new quantum path equations could account for the results of the COW-experiment. Large scale applications, of the new path equations, admitted by such geometries, give rise to tests for the existence of this interaction on the astrophysical and cosmological scales. As a byproduct, it is shown that the quantum roots appeared explicitly, in the path equations, can be diffused in the whole geometry using a parameterization scheme. 
  Under quite natural general assumptions, the following results are obtained. The maximum entropy of a quantized surface is demonstrated to be proportional to the surface area in the classical limit. The general structure of the horizon spectrum is found. The discrete spectrum of thermal radiation of a black hole Under quite natural general assumptions, the following results are obtained. The maximum entropy of a quantized surface is demonstrated to be proportional to the surface area in the classical limit. The general structure of the horizon spectrum is found. The discrete spectrum of thermal radiation of a black hole fits the Wien profile. The natural widths of the lines are much smaller than the distances between them. The total intensity of the thermal radiation is estimated.   In the special case of loop quantum gravity, the value of the Barbero -- Immirzi parameter is found. Different values for this parameter, obtained under additional assumption that the horizon is described by a U(1) Chern -- Simons theory, are demonstrated to be in conflict with the firmly established holographic bound. 
  We review sources of high-frequency gravitational waves, summarizing our current understanding of emission mechanisms, expected amplitudes and event rates. The most promising sources are gravitational collapse (formation of black holes or neutron stars) and subsequent ringing of the compact star, secular or dynamical rotational instabilities and high-mass compact objects formed through the merger of binary neutron stars. Significant and unique information for the various stages of the collapse, the structure of protoneutron stars and the high density equation of state of compact objects can be drawn from careful study of gravitational wave signals. 
  Doubly Special Relativity (DSR) is a theory with two observer-independent scales, of velocity and mass, which is expected to replace Special Relativity at ultra-high energies. In these notes we first discuss the postulates of DSR, and then turn to presenting arguments supporting the hypothesis that DSR can be regarded as a flat space, semiclassical limit of gravity. The notes are based on the talk presented at the conference ``Special Relativity -- Will it Survive the Next 100 Years?'' 
  We construct low regularity solutions of the vacuum Einstein constraint equations on compact manifolds. On 3-manifolds we obtain solutions with metrics in $H^s$ where $s>3/2$. The constant mean curvature (CMC) conformal method leads to a construction of all CMC initial data with this level of regularity. These results extend a construction from Ma04 that treated the asymptotically Euclidean case. 
  In this article we study multipole moments of axisymmetric stationary asymptotically flat spacetimes. We show how the tensorial recursion of Geroch and Hansen can be reduced to a recursion of scalar functions. We also demonstrate how a careful choice of conformal factor collects all moments into one complex valued function on R, where the moments appear as the derivatives at 0. As an application, we calculate the moments of the Kerr solution. We also discuss the freedom in choosing the potential for the moments. 
  We consider an explicit example of a process, where the entropy carried by radiation through an accelerating two-plane is proportional to the decrease in the area of that two-plane even when the two-plane is not a part of any horizon of spacetime. Our results seem to support the view that entropy proportional to area is possessed not only by horizons but by all spacelike two-surfaces of spacetime. 
  The tomographic histories approach is presented. As an inverse problem, we recover in an operational way the effective topology of the extended configuration space of a system. This means that from a series of experiments we get a set of points corresponding to events. The difference between effective and actual topology is drawn. We deduce the topology of the extended configuration space of a non-relativistic system, using certain concepts from the consistent histories approach to Quantum Mechanics, such as the notion of a record. A few remarks about the case of a relativistic system, preparing the ground for a forthcoming paper sequel to this, are made in the end. 
  We propose quantifying the quantum gravitational back-reaction on inflation with an invariant measure of the local acceleration rather than the expansion rate. Our observable is suitable for models in which there is no scalar inflaton to provide a preferred velocity field with which to define the expansion. As an example, we use stochastic techniques to evaluate the local acceleration at one loop order for Lambda-driven inflation in pure quantum gravity. 
  We illustrate how the group of symmetry transformations, which preserve the form of the n--dimensional flat Friedmann--Robertson--Walker cosmologies satisfying Einstein equations, acts in any dimension. This group relates the energy density and the isotropic pressure of the cosmic fluid to the expansion rate. The freedom associated with the dimension of the space time yields assisted inflation even when the energy density of the fluid is a dimensional invariant and enriches the set of duality transformations leading to phantom cosmologies. 
  The field equations associated with the Born-Infeld-Einstein action including matter are derived using a Palatini variational principle. Scalar, electromagnetic, and Dirac fields are considered. It is shown that an action can be chosen for the scalar field that produces field equations identical to the usual Einstein field equations minimally coupled to a scalar field. In the electromagnetic and Dirac cases the field equations reproduce the standard equations only to lowest order. The spherically symmetric electrovac equations are studied in detail. It is shown that the resulting Einstein equations correspond to gravity coupled to a modified Born-Infeld theory. It is also shown that point charges are not allowed. All particles must have a finite size. Mass terms for the fields are also considered. 
  Geodesic motion of a point particle in Kerr geometry has three constants of motion, energy $E$, azimuthal angular momentum $L$, and Carter constant $Q$. Under the adiabatic approximation, radiation reaction effect is characterized by the time evolution of these constants. In this letter we show that the scheme to evaluate them can be dramatically simplified. 
  We show in this letter that gravity coupled to a massless scalar field with full cylindrical symmetry can be exactly quantized by an extension of the techniques used in the quantization of Einstein-Rosen waves. This system provides a useful testbed to discuss a number of issues in quantum general relativity such as the emergence of the classical metric, microcausality, and large quantum gravity effects. It may also provide an appropriate framework to study gravitational critical phenomena from a quantum point of view, issues related to black hole evaporation, and the consistent definition of test fields and particles in quantum gravity. 
  We consider the embedding of 3+1 dimensional cosmology in 4+1 dimensional Jordan-Brans-Dicke theory. We show that exponentially growing and power law scale factors are implied. Whereas the 4+1 dimensional scalar field is approximately constant for each, the effective 3+1 dimensional scalar field is constant for exponentially growing scale factor and time dependent for power law scale factor. 
  We present a unified treatment of the phase space of a spatially flat homogeneous and isotropic universe dominated by a phantom field. Results on the dynamics and the late time attractors (Big Rip, de Sitter, etc.) are derived without specifying the form of the phantom potential, using only general assumptions on its shape. Many results found in the literature are quickly recovered and predictions are made for new scenarios. 
  We consider the Newtonian limit of modified theories of gravity that include inverse powers of the curvature in the action in order to explain the cosmic acceleration. It has been shown that the simplest models of this kind are in conflict with observations at the solar system level. In this letter we point out that when one adds to the action inverse powers of curvature invariants that do not vanish for the Schwarzschild geometry one generically recovers an acceptable Newtonian limit at small distances. Gravity is however modified at large distances. We compute the first correction to the Newtonian potential in a quite general class of models. The characteristic distance entering in these modifications is of the order of 10pc for the Sun and of the order of 10^2 kpc for a galaxy. 
  We calculate analytically the past asymptotic decay rates close to an initial singularity in general G_0 spatially inhomogeneous perfect fluid models with an effective equation of state which is stiff or ultra-stiff (i.e., $\gamma \ge 2$). These results are then supported by numerical simulations in a special class of G_2 cosmological models. Our analysis confirms and extends the BKL conjectures and lends support to recent isotropization results in cosmological models of current interest (with $\gamma > 2$). 
  The Dirac quasinormal modes (QNMs) of the Kerr-Newman black hole are investigated using continued fraction approach. It is shown that the quasinormal frequencies in the complex $\omega$ plane move counterclockwise as the charge or angular momentum per unit mass of the black hole increases. They get a spiral-like shape, moving out of their Schwarzschild or Reissner-Nordstr\"om values and "looping in" towards some limiting frequencies as the charge and angular momentum per unit mass tend to their extremal values. The number of the spirals increases as the overtone number increases but decreases as the angular quantum number increases. It is also found that both the real and imaginary parts are oscillatory functions of the angular momentum per unit mass, and the oscillation becomes faster as the overtone number increases but slower as the angular quantum number increases. 
  Recalling the universal covering group of de Sitter, the transformation properties of the spinor fields $\psi(x)$ and ${\bar\psi}(x)$, in the ambient space notation, are presented in this paper. The charge conjugation symmetry of the de Sitter spinor field is then discussed in the above notation. Using this spinor field and charge conjugation, de Sitter supersymmetry algebra in the ambient space notation has been established. It is shown that a novel dS-superalgebra can be attained by the use of spinor field and charge conjugation in the ambient space notation. 
  We examine the gravitational collapse of spherically symmetric inhomogeneous dust in (2+1) dimensions, with cosmological constant. We obtain the analytical expressions for the interior metric. We match the solution to a vacuum exterior. We discuss the nature of the singularity formed by analyzing the outgoing radial null geodesics. We examine the formation of trapped surfaces during the collapse. 
  We use the conformal method to obtain solutions of the Einstein-scalar field gravitational constraint equations. Handling scalar fields is a bit more challenging than handling matter fields such as fluids, Maxwell fields or Yang-Mills fields, because the scalar field introduces three extra terms into the Lichnerowicz equation, rather than just one. Our proofs are constructive and allow for arbitrary dimension (>2) as well as low regularity initial data. 
  James L. Anderson analyzed the novelty of Einstein's theory of gravity as its lack of "absolute objects." Michael Friedman's related work has been criticized by Roger Jones and Robert Geroch for implausibly admitting as absolute the timelike 4-velocity field of dust in cosmological models in Einstein's theory. Using the Rosen-Sorkin Lagrange multiplier trick, I complete Anna Maidens's argument that the problem is not solved by prohibiting variation of absolute objects in an action principle. Recalling Anderson's proscription of "irrelevant" variables, I generalize that proscription to locally irrelevant variables that do no work in some places in some models. This move vindicates Friedman's intuitions and removes the Jones-Geroch counterexample: some regions of some models of gravity with dust are dust-free and so naturally lack a timelike 4-velocity, so diffeomorphic equivalence to (1,0,0,0) is spoiled. Torretti's example involving constant curvature spaces is shown to have an absolute object on Anderson's analysis, viz., the conformal spatial metric density. The previously neglected threat of an absolute object from an orthonormal tetrad used for coupling spinors to gravity appears resolvable by eliminating irrelevant fields. However, given Anderson's definition, GTR itself has an absolute object (as Robert Geroch has observed recently): a change of variables to a conformal metric density and a scalar density shows that the latter is absolute. 
  We study the thermodynamic and gravitational stability of Kerr anti-de Sitter black holes in five and higher dimensions. We show, in the case of equal rotation parameters, $a_i=a$, that the Kerr-AdS background metrics become stable, both thermodynamically and gravitationally, when the rotation parameters $a_i$ take values comparable to the AdS curvature radius. In turn, a Kerr-AdS black hole can be in thermal equilibrium with the thermal radiation around it only when the rotation parameters become not significantly smaller than the AdS curvature radius. We also find with equal rotation parameters that a Kerr-AdS black hole is thermodynamically favored against the existence of a thermal AdS space, while the opposite behavior is observed in the case of a single non-zero rotation parameter. The five dimensional case is however different and also special in that there is no high temperature thermal AdS phase regardless of the choice of rotation parameters. We also verify that at fixed entropy, the temperature of a rotating black hole is always bounded above by that of a non-rotating black hole, in four and five dimensions, but not in six and more dimensions (especially, when the entropy approaches zero or the minimum of entropy does not correspond to the minimum of temperature). In this last context, the six dimensional case is marginal. 
  The Laser Astrometric Test Of Relativity (LATOR) is a joint European-U.S. Michelson-Morley-type experiment designed to test the pure tensor metric nature of gravitation - a fundamental postulate of Einstein's theory of general relativity. By using a combination of independent time-series of highly accurate gravitational deflection of light in the immediate proximity to the Sun, along with measurements of the Shapiro time delay on interplanetary scales (to a precision respectively better than 0.1 picoradians and 1 cm), LATOR will significantly improve our knowledge of relativistic gravity. The primary mission objective is to i) measure the key post-Newtonian Eddington parameter \gamma with accuracy of a part in 10^9. (1-\gamma) is a direct measure for presence of a new interaction in gravitational theory, and, in its search, LATOR goes a factor 30,000 beyond the present best result, Cassini's 2003 test. The mission will also provide: ii) first measurement of gravity's non-linear effects on light to ~0.01% accuracy; including both the Eddington \beta parameter and also the spatial metric's 2nd order potential contribution (never measured before); iii) direct measurement of the solar quadrupole moment J2 (currently unavailable) to accuracy of a part in 200 of its expected size; iv) direct measurement of the "frame-dragging" effect on light by the Sun's gravitomagnetic field, to 1% accuracy. LATOR's primary measurement pushes to unprecedented accuracy the search for cosmologically relevant scalar-tensor theories of gravity by looking for a remnant scalar field in today's solar system. We discuss the mission design of this proposed experiment. 
  In this paper, we discuss general relativistic, self-gravitating and uniformly rotating perfect fluid bodies with a toroidal topology (without central object). For the equations of state describing the fluid matter we consider polytropic as well as completely degenerate, perfect Fermi gas models. We find that the corresponding configurations possess similar properties to the homogeneous relativistic Dyson rings. On the one hand, there exists no limit to the mass for a given maximal mass-density inside the body. On the other hand, each model permits a quasistationary transition to the extreme Kerr black hole. 
  A large-scale smoothed-out model of the universe ignores small-scale inhomogeneities, but the averaged effects of those inhomogeneities may alter both observational and dynamical relations at the larger scale. This article discusses these effects, and comments briefly on the relation to gravitational entropy. 
  Numerical simulations of the approach to the singularity in spacetimes with stiff fluid matter are presented here. The spacetimes examined have no symmetries and can be regarded as representing the general behavior of singularities in the presence of such matter. It is found that the singularity is spacelike and that as it is approached, the spacetime dynamics becomes local and non-oscillatory. 
  Dynamical wave function collapse models entail the continuous liberation of a specified rate of energy arising from the interaction of a fluctuating scalar field with the matter wave function. We consider the wave function collapse process for the constituents of dark matter in our universe. Beginning from a particular early era of the universe chosen from physical considerations, the rate of the associated energy liberation is integrated to yield the requisite magnitude of dark energy at the time of galaxy formation. Further, the equation of state for the liberated energy approaches $w \to -1$ asymptotically, providing a mechanism to generate the present acceleration of the universe. 
  We use the Hough transform to analyze data from the second science run of the LIGO interferometers, to look for gravitational waves from isolated pulsars. We search over the whole sky and over a large range of frequencies and spin-down parameters. Our search method is based on the Hough transform, which is a semi-coherent, computationally efficient, and robust pattern recognition technique. We also present a validation of the search pipeline using hardware signal injections. 
  In several approaches to the quantum-gravity problem evidence has emerged of the validity of a "GUP" (a Generalized position-momentum Uncertainty Principle) and/or a "MDR" (a modification of the energy-momentum dispersion relation), but very little is known about the implications of GUPs and MDRs for black-hole thermodynamics, another key topic for quantum-gravity research. We investigate an apparent link, already suggested in an earlier exploratory study involving two of us, between the possibility of a GUP and/or a MDR and the possibility of a log term in the area-entropy black-hole formula. We then obtain, from that same perspective, a modified relation between the mass of a black hole and its temperature, and we examine the validity of the "Generalized Second Law of black-hole thermodynamics" in theories with a GUP and/or a MDR. After an analysis of GUP- and MDR-modifications of the black-body radiation spectrum, we conclude the study with a description of the black-hole evaporation process. 
  In this paper we present a non-singular black hole model as a possible end-product of gravitational collapse. The depicted spacetime which is type [II,(II)], by Petrov classification, is an exact solution of the Einstein equations and contains two horizons. The equation of state in the radial direction, is a well-behaved function of the density and smoothly reproduces vacuum-like behavior near r=0 while tending to a polytrope at larger r, low density, values. The final equilibrium configuration comprises of a de Sitter-like inner core surrounded by a family of 2-surfaces of matter fields with variable equation of state. The fields are all concentrated in the vicinity of the radial center r=0. The solution depicts a spacetime that is asymptotically Schwarzschild at large r, while it becomes de Sitter-like for vanishing r. Possible physical interpretations of the macro-state of the black hole interior in the model are offered. We find that the possible state admits two equally viable interpretations, namely either a quintessential intermediary region or a phase transition in which a two-fluid system is in both dynamic and thermodynamic equilibrium. We estimate the ratio of pure matter present to the total energy and in both (interpretations) cases find it to be virtually the same, being 0.83. Finally, the well-behaved dependence of the density and pressure on the radial coordinate provides some insight on dealing with the information loss paradox. 
  Implications of cosmological model with a cosmological term of the form $\Lambda = \beta \frac{\ddot {a}}{a}$, where $\beta$ is a constant, are analyzed in multidimensional space time. The proper distance, the luminosity distance-redshift, the angular diameter distance-redshift, and look back time-redshift for the model are presented. It has been shown that such models are found to be compatible with the recent observations. This work has thus generalized to higher dimensions the well-know result in four dimensional space time. It is found that there may be significant difference in principle at least,from the analogous situation in four dimensional space time. 
  The behavior of magnetic field in plane symmetric inhomogeneous cosmological models for bulk viscous distribution is investigated. The coefficient of bulk viscosity is assumed to be a power function of mass density $(\xi =\xi_{0}\rho^{n})$. The values of cosmological constant for these models are found to be small and positive which are supported by the results from recent supernovae Ia observations. Some physical and geometric aspects of the models are also discussed. 
  A new approach is suggested for the study of geometric symmetries in general relativity, leading to an invariant characterization of the evolutionary behaviour for a class of Spatially Homogeneous (SH) vacuum and orthogonal $\gamma -$law perfect fluid models. Exploiting the 1+3 orthonormal frame formalism, we express the kinematical quantities of a generic symmetry using expansion-normalized variables. In this way, a specific symmetry assumption lead to geometric constraints that are combined with the associated integrability conditions, coming from the existence of the symmetry and the induced expansion-normalized form of the Einstein's Field Equations (EFE), to give a close set of compatibility equations. By specializing to the case of a \emph{Kinematic Conformal Symmetry} (KCS), which is regarded as the direct generalization of the concept of self-similarity, we give the complete set of consistency equations for the whole SH dynamical state space. An interesting aspect of the analysis of the consistency equations is that, \emph{at least} for class A models which are Locally Rotationally Symmetric or lying within the invariant subset satisfying $N_{\alpha}^{\alpha}=0 $, a proper KCS \emph{always exists} and reduces to a self-similarity of the first or second kind at the asymptotic regimes, providing a way for the ``geometrization'' of the intermediate epoch of SH models. 
  In 1+1 dimensional case, Einstein equation cannot give us any information on the evolution of the universe because the Einstein tensor of the system is identically zero. We study such a 1+1 dimensional cosmology and find the metric of it according to cosmological principle and special relativity, but the results contradict the usual expression of cosmological principle of standard cosmology. So we doubt in 1+3 dimensional case, cosmological principle is expressed faithfully by standard cosmology. 
  The inspirals of stellar-mass compact objects into supermassive black holes constitute some of the most important sources for LISA. Detection of these sources using fully coherent matched filtering is computationally intractable, so alternative approaches are required. In a previous paper (Wen and Gair 2005, gr-qc/0502100), we outlined a detection method based on looking for excess power in a time-frequency spectrogram of the LISA data. The performance of the algorithm was assessed using a single `typical' trial waveform and approximations to the noise statistics. In this paper we present results of Monte Carlo simulations of the search noise statistics and examine its performance in detecting a wider range of trial waveforms. We show that typical extreme mass ratio inspirals (EMRIs) can be detected at distances of up to 1--3 Gpc, depending on the source parameters. We also discuss some remaining issues with the technique and possible ways in which the algorithm can be improved. 
  "Doubly-special relativity" (DSR), the idea of a Planck-scale Minkowski limit that is still a relativistic theory, but with both the Planck scale and the speed-of-light scale as nontrivial relativistic invariants, was proposed (gr-qc/0012051) as a physics intuition for several scenarios which may arise in the study of the quantum-gravity problem, but most DSR studies focused exclusively on the search of formalisms for the description of a specific example of such a Minkowski limit. A novel contribution to the DSR physics intuition came from a recent paper by Smolin (hep-th/0501091) suggesting that the emergence of the Planck scale as a second nontrivial relativistic invariant might be inevitable in quantum gravity, relying only on some rather robust expectations concerning the semiclassical approximation of quantum gravity. I here attempt to strengthen Smolin's argument by observing that an analysis of some independently-proposed Planck-scale particle-localization limits, such as the "Generalized Uncertainty Principle" often attributed to string theory in the literature, also suggests that the emergence of a DSR Minkowski limit might be inevitable. I discuss a possible link between this observation and recent results on logarithmic corrections to the entropy-area black-hole formula, and I observe that both the analysis here reported and Smolin's analysis appear to suggest that the examples of DSR Minkowski limits for which a formalism has been sought in the literature might not be sufficiently general. I also stress that, as we now contemplate the hypothesis of a DSR Minkowski limit, there is an additional challenge for those in the quantum-gravity community attributing to the Planck length the role of "fundamental length scale". 
  We consider the Bianchi identity as a field equation for the distortion of the shapes of images produced by weak gravitational lensing. Using the spin coefficient formalism of Newman and Penrose [1962], we show that certain complex components of the Weyl and Ricci curvature tensors are directly related to fundamental observables in weak gravitational lensing. In the case of weak gravitational fields, we then show that the Bianchi identity provides a field equation for the Ricci tensor assuming a known Weyl tensor. From the Bianchi identity, we derive the integral equation for weak lensing presented by Miralda-Escude [1996], thus making the Bianchi identity a first principles equation of weak gravitational lensing. This equation is integrated in the important case of an axially symmetric lens and explicitly demonstrated in the case of a point lens and a SIS model. 
  We investigate the generic behaviour of marginally trapped tubes (roughly time-evolved apparent horizons) using simple, spherically symmetric examples of dust and scalar field collapse/accretion onto pre-existing black holes. We find that given appropriate physical conditions the evolution of the marginally trapped tube may be either null, timelike, or spacelike and further that the marginally trapped two-sphere cross-sections may either expand or contract in area. Spacelike expansions occur when the matter falling into a black hole satisfies $\rho - P \leq 1/A$, where $A$ is the area of the horizon while $\rho$ and $P$ are respectively the density and pressure of the matter. Timelike evolutions occur when $(\rho - P)$ is greater than this cut-off and so would be expected to be more common for large black holes. Physically they correspond to horizon "jumps" as extreme conditions force the formation of new horizons outside of the old. 
  Motivated by the existence of black holes with various topologies in four-dimensional spacetimes with a negative cosmological constant, we study axisymmetric static solutions describing any large distortions of Schwarzschild-anti-de Sitter black holes parametrized by the mass $m$. Under the approximation such that $m$ is much larger than the anti-de Sitter radius, it is found that a cylindrically symmetric black string is obtained as a special limit of distorted spherical black holes. Such a prolonged distortion of the event horizon connecting a Schwarzschild-anti-de Sitter black hole to a black string is allowed without violating both the usual black hole thermodynamics and the hoop conjecture for the horizon circumference. 
  We comment on the paper [1] by Albert Einstein from 1918 to Willem De Sitter's solution [2] of the Einstein field equation from today's point of view. To this end, we start by describing the geometry of the De Sitter space-time and present its importance for the inflationary cosmological model. 
  We study the spectrum of metric fluctuation in $\kappa$-deformed inflationary universe. We write the theory of scalar metric fluctuations in the $\kappa-$deformed Robertson-Walker space, which is represented as a non-local theory in the conventional Robertson-Walker space. One important consequence of the deformation is that the mode generation time is naturally determined by the structure of the $\kappa-$deformation.   We expand the non-local action in $H^2/\kappa^2$, with $H$ being the Hubble parameter and $\kappa$ the deformation parameter, and then compute the power spectra of scalar metric fluctuations both for the cases of exponential and power law inflations up to the first order in $H^2/\kappa^2$. We show that the power spectra of the metric fluctuation have non-trivial corrections on the time dependence and on the momentum dependence compared to the commutative space results. Especially for the power law inflation case, the power spectrum for UV modes is weakly blue shifted early in the inflation and its strength decreases in time. The power spectrum of far-IR modes has cutoff proportional to $k^3$ which may explain the low CMB quadrupole moment. 
  Alternative theories of gravity have been recently studied in connection with their cosmological applications, both in the Palatini and in the metric formalism. The aim of this paper is to propose a theoretical framework (in the Palatini formalism) to test these theories at the solar system level and possibly at the galactic scales. We exactly solve field equations in vacuum and find the corresponding corrections to the standard general relativistic gravitational field. On the other hand, approximate solutions are found in matter cases starting from a Lagrangian which depends on a phenomenological parameter. Both in the vacuum case and in the matter case the deviations from General Relativity are controlled by parameters that provide the Post-Newtonian corrections which prove to be in good agreement with solar system experiments. 
  Laser frequency stabilization is notably one of the major challenges on the way to a space-borne gravitational wave observatory. The proposed Laser Interferometer Space Antenna (LISA) is presently under development in an ESA, NASA collaboration. We present a novel method for active laser stabilization and phase noise suppression in such a gravitational wave detector. The proposed approach is a further evolution of the "arm locking" method, which in essence consists of using an interferometer arm as an optical cavity, exploiting the extreme long-run stability of the cavity size in the frequency band of interest. We extend this method by using the natural interferometer arm length differences and existing interferometer signals as additional information sources for the reconstruction and active suppression of the quasi-periodic laser frequency noise, enhancing the resolution power of space-borne gravitational wave detectors. 
  It is a brief review of the physical theories embodying the idea of extra dimensions, starting from the pre-historic times to the present day. Here we have classified the developments into three eras, such as Pre-Einstein, Einstein and Kaluza-Klein. Here the views and flow of thoughts are emphasized rather rigorous mathematical details. Majour developments in Quantum field theory and Particle physics are outlined. Some well known higher dimensional approaches to unification are discussed. This is concluded with some examples for visualizing extra dimensions and a short discussion on the cosmological implications and possible existence of the same. 
  Regular (non-singular) space-times are given which describe the formation of a (locally defined) black hole from an initial vacuum region, its quiescence as a static region, and its subsequent evaporation to a vacuum region. The static region is Bardeen-like, supported by finite density and pressures, vanishing rapidly at large radius and behaving as a cosmological constant at small radius. The dynamic regions are Vaidya-like, with ingoing radiation of positive energy flux during collapse and negative energy flux during evaporation, the latter balanced by outgoing radiation of positive energy flux and a surface pressure at a pair creation surface. The black hole consists of a compact space-time region of trapped surfaces, with inner and outer boundaries which join circularly as a single smooth trapping horizon. 
  We present some basic facts concerning simultaneity in both special and general relativity. We discuss Weyl's proof of the consistence of Einstein's synchronization convention and consider the general relativistic problem of assigning a time function to a congruence of timelike curves. 
  Spherically symmetric space-times provide many examples for interesting black hole solutions, which classically are all singular. Following a general program, space-like singularities in spherically symmetric quantum geometry, as well as other inhomogeneous models, are shown to be absent. Moreover, one sees how the classical reduction from infinitely many kinematical degrees of freedom to only one physical one, the mass, can arise, where aspects of quantum cosmology such as the problem of initial conditions play a role. 
  We investigate here quantum effects in gravitational collapse of a scalar field model which classically leads to a naked singularity. We show that non-perturbative semi-classical modifications near the singularity, based on loop quantum gravity, give rise to a strong outward flux of energy. This leads to the dissolution of the collapsing cloud before the singularity can form. Quantum gravitational effects thus censor naked singularities by avoiding their formation. Further, quantum gravity induced mass flux has a distinct feature which may lead to a novel observable signature in astrophysical bursts. 
  Recently, it was shown that the extreme Kerr black hole is the only candidate for a (Kerr) black hole limit of stationary and axisymmetric, uniformly rotating perfect fluid bodies with a zero temperature equation of state. In this paper, necessary and sufficient conditions for reaching the black hole limit are presented. 
  Global existence results in the past time direction of cosmological models with collisionless matter and a massless scalar field are presented. It is shown that the singularity is crushing and that the Kretschmann scalar diverges uniformly as the singularity is approached. In the case without Vlasov matter, the singularity is velocity dominated and the generalized Kasner exponents converge at each spatial point as the singularity is approached. 
  The scalar field degree of freedom in Einstein's plus Matter field equations is decoupled for Bianchi type I and V general cosmological models. The source, apart from the minimally coupled scalar field with arbitrary potential V(Phi), is provided by a perfect fluid obeying a general equation of state p =p(rho). The resulting ODE is, by an appropriate choice of final time gauge affiliated to the scalar field, reduced to 1st order, and then the system is completely integrated for arbitrary choices of the potential and the equation of state. 
  Within the causal set approach to quantum gravity, a discrete analog of a spacelike region is a set of unrelated elements, or an antichain. In the continuum approximation of the theory, a moment-of-time hypersurface is well represented by an inextendible antichain. We construct a richer structure corresponding to a thickening of this antichain containing non-trivial geometric and topological information. We find that covariant observables can be associated with such thickened antichains and transitions between them, in classical stochastic growth models of causal sets. This construction highlights the difference between the covariant measure on causal set cosmology and the standard sum-over-histories approach: the measure is assigned to completed histories rather than to histories on a restricted spacetime region. The resulting re-phrasing of the sum-over-histories may be fruitful in other approaches to quantum gravity. 
  We consider models of accelerated cosmological expansion described by the Einstein equations coupled to a nonlinear scalar field with a suitable exponential potential. We show that homogeneous and isotropic solutions are stable under small nonlinear perturbations without any symmetry assumptions. Our proof is based on results on the nonlinear stability of de Sitter spacetime and Kaluza-Klein reduction techniques. 
  In this work we analyze the particle creation process of spin-1 particles which have no mass by considering Bianchi-type I cosmological models. We choose two models and solve massless Duffin-Kemmer-Petiau (mDKP) (Photon) equation for these models, and use the Bogoliubov coefficients to relate vacuum state in the asymptotic regions of gravitational background. By using the Bogoliubov transformation technique we calculate the density of created particles. 
  We discuss a practical method to compute the self-force on a particle moving through a curved spacetime. This method involves two expansions to calculate the self-force, one arising from the particle's immediate past and the other from the more distant past. The expansion in the immediate past is a covariant Taylor series and can be carried out for all geometries. The more distant expansion is a mode sum, and may be carried out in those cases where the wave equation for the field mediating the self-force admits a mode expansion of the solution. In particular, this method can be used to calculate the gravitational self-force for a particle of mass mu orbiting a black hole of mass M to order mu^2, provided mu/M << 1. We discuss how to use these two expansions to construct a full self-force, and in particular investigate criteria for matching the two expansions. As with all methods of computing self-forces for particles moving in black hole spacetimes, one encounters considerable technical difficulty in applying this method; nevertheless, it appears that the convergence of each series is good enough that a practical implementation may be plausible. 
  The possibility that the strength of gravitational interactions might slowly increase with distance, is explored by formulating a set of effective field equations, which incorporate the gravitational, vacuum-polarization induced, running of Newton's constant $G$. The resulting long distance (or large time) behaviour depends on only one adjustable parameter $\xi$, and the implications for the Robertson-Walker universe are calculated, predicting an accelerated power-law expansion at later times $t \sim \xi \sim 1/H$. 
  We derive the metric of an expanding universe with zero accelerations by pure kinematic method. By doing so we expatiate physics related with co-moving coordinate system in details. The most important discovery or our study is, in an expanding universe with zero accelerations, the red-shift of photons from distance galaxies is determined by the co-moving coordinate of the source galaxy instead of the scale factor's time dependence.  Our discovery is consistent with the current observed super-novaes's luminosity-distance v.s. red-shift relations. 
  The Pioneer 10 and 11 spacecraft yielded the most precise navigation in deep space to date. These spacecraft had exceptional acceleration sensitivity. However, analysis of their radio-metric tracking data has consistently indicated that at heliocentric distances of $\sim 20-70$ astronomical units, the orbit determinations indicated the presence of a small, anomalous, Doppler frequency drift. The drift is a blue-shift, uniformly changing with a rate of $\sim(5.99 \pm 0.01)\times 10^{-9}$ Hz/s, which can be interpreted as a constant sunward acceleration of each particular spacecraft of $a_P = (8.74 \pm 1.33)\times 10^{-10} {\rm m/s^2}$. This signal has become known as the Pioneer anomaly. The inability to explain the anomalous behavior of the Pioneers with conventional physics has contributed to growing discussion about its origin. There is now an increasing number of proposals that attempt to explain the anomaly outside conventional physics. This progress emphasizes the need for a new experiment to explore the detected signal. Furthermore, the recent extensive efforts led to the conclusion that only a dedicated experiment could ultimately determine the nature of the found signal. We discuss the Pioneer anomaly and present the next steps towards an understanding of its origin. We specifically focus on the development of a mission to explore the Pioneer Anomaly in a dedicated experiment conducted in deep space. 
  Field theoretical scheme of regular Big Bang in 4-dimensional physical space-time, built in the framework of gauge approach to gravitation, is discussed. Regular bouncing character of homogeneous isotropic cosmological models is ensured by gravitational repulsion effect at extreme conditions without quantum gravitational corrections. The most general properties of regular inflationary cosmological models are examined. Developing theory is valid, if energy density of gravitating matter is positive and energy dominance condition is fulfilled. 
  Experimental discovery of the gravitomagnetic fields generated by translational and/or rotational currents of matter is one of primary goals of modern gravitational physics. The rotational (intrinsic) gravitomagnetic field of the Earth is currently measured by the Gravity Probe B. The present paper makes use of a parametrized post-Newtonian (PN) expansion of the Einstein equations to demonstrate how the extrinsic gravitomagnetic field generated by the translational current of matter can be measured by observing the relativistic time delay caused by a moving gravitational lens. We prove that measuring the extrinsic gravitomagnetic field is equivalent to testing relativistic effect of the aberration of gravity caused by the Lorentz transformation of the gravitational field. We unfold that the recent Jovian deflection experiment is a null-type experiment testing the Lorentz invariance of the gravitational field (aberration of gravity), thus, confirming existence of the extrinsic gravitomagnetic field associated with orbital motion of Jupiter with accuracy 20%. We comment on erroneous interpretations of the Jovian deflection experiment given by a number of researchers who are not familiar with modern VLBI technique and subtleties of JPL ephemeris. We propose to measure the aberration of gravity effect more accurately by observing gravitational deflection of light by the Sun and processing VLBI observations in the geocentric frame with respect to which the Sun is moving with velocity 30 km/s. 
  The most general form of the deviation equations in spaces with linear connection with arbitrary torsion is derived. 
  We study the conserved charges of supersymmetric solutions in the topologically massive gravity theory for both asymptotically flat and constant curvature geometries. 
  The need to smoothly cover a computational domain of interest generically requires the adoption of several grids. To solve the problem of interest under this grid-structure one must ensure the suitable transfer of information among the different grids involved. In this work we discuss a technique that allows one to construct finite difference schemes of arbitrary high order which are guaranteed to satisfy linear numerical and strict stability. The technique relies on the use of difference operators satisfying summation by parts and {\it penalty techniques} to transfer information between the grids. This allows the derivation of semidiscrete energy estimates for problems admitting such estimates at the continuum. We analyze several aspects of this technique when used in conjuction with high order schemes and illustrate its use in one, two and three dimensional numerical relativity model problems with non-trivial topologies, including truly spherical black hole excision. 
  In this paper we find the first and second order perturbations of the induced metric and the extrinsic curvature of a non-degenerate hypersurface $\Sigma$ in a spacetime $(M,g)$, when the metric $g$ is perturbed arbitrarily to second order and the hypersurface itself is allowed to change perturbatively (i.e. to move within spacetime) also to second order. The results are fully general and hold in arbitrary dimensions and signature. An application of these results for the perturbed matching theory between spacetimes is presented. 
  This work develops the dynamics of perfectly elastic solid model for application to the outer crust of a magnetised neutron star. Particular attention is given to the Noether identities responsible for energy-momentum conservation, using a formulation that is fully covariant, not only (as is usual) in a fully relativistic treatment but also (sacrificing accuracy and elegance for economy of degrees of gravitational freedom) in the technically more complicated case of the Newtonian limit. The results are used to obtain explicit (relativistic and Newtonian) formulae for the propagation speeds of generalised (Alfven type) magneto-elastic perturbation modes. 
  We propose a generalization of the condition for harmonic spatial coordinates analogous to the generalization of the harmonic time slices introduced by Bona et al., and closely related to dynamic shift conditions recently proposed by Lindblom and Scheel, and Bona and Palenzuela. These generalized harmonic spatial coordinates imply a condition for the shift vector that has the form of an evolution equation for the shift components. We find that in order to decouple the slicing condition from the evolution equation for the shift it is necessary to use a rescaled shift vector. The initial form of the generalized harmonic shift condition is not spatially covariant, but we propose a simple way to make it fully covariant so that it can be used in coordinate systems other than Cartesian. We also analyze the effect of the shift condition proposed here on the hyperbolicity of the evolution equations of general relativity in 1+1 dimensions and 3+1 spherical symmetry, and study the possible development of blow-ups. Finally, we perform a series of numerical experiments to illustrate the behavior of this shift condition. 
  This paper discusses traversable wormholes that differ slightly but significantly from those of the Morris-Thorne type under the assumption of cylindrical symmetry. The throat is a piecewise smooth cylindrical surface resulting in a shape function that is not differentiable at some value. It is proposed that the regular derivative be replaced by a one-sided derivative at this value. The resulting wormhole geometry satisfies the weak energy condition. 
  The main part of the thesis deals with continuously and discretely self-similar solutions and type II critical phenomena in a family of self-gravitating non-linear sigma-models. The phenomena strongly depend on the dimensionless coupling constant. For small couplings we numerically construct continuously self-similar (CSS) solutions and analyze their stability properties. For large couplings we construct a discretely self-similar (DSS) solution with one unstable mode. We argue that at some critical coupling the DSS solution bifurcates from the first CSS excitation in a heteroclinic loop bifurcation. We study critical phenomena between dispersal and singularity formation (at very small couplings) respectively black hole formation (for larger couplings). We give numerical evidence that for very small couplings the generic end state of ``intermediately strong'' data is the stable CSS ground state. For small couplings the critical solution is the first CSS excitation whereas for strong couplings the threshold of black hole formation is governed by the DSS solution. We describe the phenomena occurring at intermediate couplings where the critical solution changes from CSS to DSS.   Content and references are at the state of October 2001. 
  We study cosmological application of the holographic energy density in the Brans-Dicke theory. Considering the holographic energy density as a dynamical cosmological constant, it is more natural to study it in the Brans-Dicke theory than in general relativity. Solving the Friedmann and Brans-Dicke field equations numerically, we clarify the role of Brans-Dicke field during evolution of the universe. When the Hubble horizon is taken as the IR cutoff, the equation of state ($w_{\Lmd}$) for the holographic energy density is determined to be 5/3 when the Brans-Dicke parameter $\omg$ goes infinity. This means that the Brans-Dicke field plays a crucial role in determining the equation of state. For the particle horizon IR cutoff, the Brans-Dicke scalar mediates a transition from $w_{\Lmd} = -1/3$ (past) to $w_{\Lmd} = 1/3$ (future). If a dust matter is present, it determines future equation of state. In the case of future event horizon cutoff, the role of the Brans-Dicke scalar and dust matter are turned out to be trivial, whereas the holographic energy density plays an important role as a dark energy candidate with $w_{\Lmd} =-1$. 
  Inspiraling compact binaries are promising sources of gravitational waves for ground and space-based laser interferometric detectors. The time-dependent signature of these sources in the detectors is a well-characterized function of a relatively small number of parameters; thus, the favored analysis technique makes use of matched filtering and maximum likelihood methods. Current analysis methodology samples the matched filter output at parameter values chosen so that the correlation between successive samples is 97% for which the filtered output is closely correlated. Here we describe a straightforward and practical way of using interpolation to take advantage of the correlation between the matched filter output associated with nearby points in the parameter space to significantly reduce the number of matched filter evaluations without sacrificing the efficiency with which real signals are recognized. Because the computational cost of the analysis is driven almost exclusively by the matched filter evaluations, this translates directly into an increase in computational efficiency, which in turn, translates into an increase in the size of the parameter space that can be analyzed and, thus, the science that can be accomplished with the data. As a demonstration we compare the present "dense sampling" analysis methodology with our proposed "interpolation" methodology, restricted to one dimension of the multi-dimensional analysis problem. We find that the interpolated search reduces by 25% the number of filter evaluations required by the dense search with 97% correlation to achieve the same efficiency of detection for an expected false alarm probability. Generalized to higher dimensional space of a generic binary including spins suggests an order of magnitude increase in computational efficiency. 
  In this article we investigate exact cylindrically symmetric solutions to the modified Einstein field equations in the brane world gravity scenarios. It is shown that for the special choice of the equation of state $2U+P=0$ for the dark energy and dark pressure, the solutions found could be considered formally as solutions of the Einstein-Maxwell equations in 4-D general relativity. 
  Recently several models of traversable wormholes have been proposed which require only arbitrarily small amounts of negative energy to hold them open against self-collapse. If the exotic matter is assumed to be provided by quantum fields, then quantum inequalities can be used to place constraints on the negative energy densities required. In this paper, we introduce an alternative method for obtaining constraints on wormhole geometries, using a recently derived quantum inequality bound on the null-contracted stress-energy averaged over a timelike worldline. The bound allows us to perform a simplified analysis of general wormhole models, not just those with small quantities of exotic matter. We then use it to study, in particular, the models of Visser, Kar, and Dadhich (VKD) and the models of Kuhfittig. The VKD models are constrained to be either submicroscopic or to have a large discrepancy between throat size and curvature radius. A recent model of Kuhfittig is shown to be non-traversable. This is due to the fact that the throat of his wormhole flares outward so slowly that light rays and particles, starting from outside the throat, require an infinite lapse of affine parameter to reach the throat. 
  We describe early success in the evolution of binary black hole spacetimes with a numerical code based on a generalization of harmonic coordinates. Indications are that with sufficient resolution this scheme is capable of evolving binary systems for enough time to extract information about the orbit, merger and gravitational waves emitted during the event. As an example we show results from the evolution of a binary composed of two equal mass, non-spinning black holes, through a single plunge-orbit, merger and ring down. The resultant black hole is estimated to be a Kerr black hole with angular momentum parameter a~0.70. At present, lack of resolution far from the binary prevents an accurate estimate of the energy emitted, though a rough calculation suggests on the order of 5% of the initial rest mass of the system is radiated as gravitational waves during the final orbit and ringdown. 
  In the framework of black hole spectroscopy, we extend the results obtained for a charged black hole in an asymptotically flat spacetime to the scenario with non vanishing negative cosmological constant. In particular, exploiting Hamiltonian techniques, we construct the area spectrum for an AdS Reissner-Nordstrom black hole. 
  It has been shown that t00 component of the energy-momentum pseudotensor in the case of cylindrically symmetrical static gravitational field cannot be interpreted as energy density of the gravitation field. An approach has been suggested allows one to express the energy density of the cylindrically or spherically symmetrical static gravitation field in terms of the metric tensor components. The approach based on the consideration of the process of isothermal compression of a cylinder consisted of incoherent matter. 
  A quantum theory of spherically symmetric thin shells of null dust and their gravitational field is studied. In Nucl. Phys. 603 (2001) 515 (hep-th/0007005), it has been shown how superpositions of quantum states with different geometries can lead to a solution of the singularity problem and black hole information paradox: the shells bounce and re-expand and the evolution is unitary. The corresponding scattering times will be defined in the present paper. To this aim, a spherical mirror of radius R_m is introduced. The classical formula for scattering times of the shell reflected from the mirror is extended to quantum theory. The scattering times and their spreads are calculated. They have a regular limit for R_m\to 0 and they reveal a resonance at E_m = c^4R_m/2G. Except for the resonance, they are roughly of the order of the time the light needs to cross the flat space distance between the observer and the mirror. Some ideas are discussed of how the construction of the quantum theory could be changed so that the scattering times become considerably longer. 
  This paper has been withdrawn because the new one gr-qc/0512095 includes all its results (as well as those in gr-qc/0511016) in a clearer way. 
  It has been suggested that the highly damped quasinormal modes of black holes provide information about the microscopic quantum gravitational states underlying black hole entropy. This interpretation requires the form of the highly damped quasinormal mode frequency to be universally of the form: $\hbar\omega_R = \ln(l)kT_{BH}$, where $l$ is an integer, and $T_{BH}$ is the black hole temperature. We summarize the results of an analysis of the highly damped quasinormal modes for a large class of single horizon, asymptotically flat black holes. 
  We present the concomitant decomposition of an (s+2)-dimensional spacetime both with respect to a timelike and a spacelike direction. The formalism we develop is suited for the study of the initial value problem and for canonical gravitational dynamics in brane-world scenarios. The bulk metric is replaced by two sets of variables. The first set consist of one tensorial (the induced metric $g_{ij}$), one vectorial ($M^{i}$) and one scalar ($M$) dynamical quantity, all defined on the s-space. Their time evolutions are related to the second fundamental form (the extrinsic curvature $K_{ij}$), the normal fundamental form ($\mathcal{K}^{i}$) and normal fundamental scalar ($\mathcal{K}$), respectively. The non-dynamical set of variables is given by the lapse function and the shift vector, which however has one component less. The missing component is due to the externally imposed constraint, which states that physical trajectories are confined to the (s+1)-dimensional brane. The pair of dynamical variables ($g_{ij}$, $K_{ij}$), well-known from the ADM-decomposition is supplemented by the pairs ($M^{i}$, $\mathcal{K}^{i}$) and ($M$, $\mathcal{K}$) due to the bulk curvature. We give all projections of the junction condition across the brane and prove that for a perfect fluid brane neither of the dynamical variables has jump across the brane. Finally we complete the set of equations needed for gravitational dynamics by deriving the evolution equations of $K_{ij}$, $\mathcal{K}^{i}$ and $\mathcal{K}$ on a brane with arbitrary matter. 
  An exact solution of Einstein's equations which represents a pair of accelerating and rotating black holes (a generalised form of the spinning C-metric) is presented. The starting point is a form of the Plebanski-Demianski metric which, in addition to the usual parameters, explicitly includes parameters which describe the acceleration and angular velocity of the sources. This is transformed to a form which explicitly contains the known special cases for either rotating or accelerating black holes. Electromagnetic charges and a NUT parameter are included, the relation between the NUT parameter $l$ and the Plebanski-Demianski parameter $n$ is given, and the physical meaning of all parameters is clarified. The possibility of finding an accelerating NUT solution is also discussed. 
  We show that if the total internal energy of a black hole is constructed as the sum of $N$ photons all having a fixed wavelength chosen to scale with the Schwarzschild radius as $\lambda=\alpha R_{s}$, then $N$ will scale with $R_{s}^{2}$. A statistical mechanical calculation of the configuration proposed yields (\alpha = 4 \pi^2 / \ln(2)) and a total entropy of the system $S=k_{B}N \ln(2)$, in agreement with the Bekenstein entropy of a black hole . It is shown that the critical temperature for Bose-Einstein condensation for relativistic particles of $\lambda=\alpha R_{s}$ is always well below the Hawking temperature of a black hole, in support of the proposed internal configuration. We then examine our results from the point of view of recent loop quantum gravity ideas and find that a natural consistency of both approaches appears.  We show that the Jeans criterion for gravitational instability can be generalised to the special and general relativistic regimes and holds for any type of mass--energy distribution. 
  Ultrarelativistic circular orbits of spinning particles in a Schwarzschild field described by the Mathisson-Papapetrou equations are considered. The preliminary estimates of the possible synchrotron electromagnetic radiation of highly relativistic protons and electrons on these orbits in the gravitational field of a black hole are presented 
  It is shown that the gravitational ultrarelativistic spin-orbit interaction violates the weak equivalence principle in the traditional sense. This fact is a direct consequence of the Mathisson-Papapetrou equations in the frame of reference comoving with a spinning test particle. The widely held assumption that the deviation of a spinning test body from a geodesic trajectory is caused by tidal forces is not correct 
  The response of the ultrarelativistic particle with spin in a Schwarzschild field to the gravitomagnetic components as measured by the comoving observer is investigated. The dependence of the particle's spin-orbit acceleration on the Lorentz \gamma - factor and the spin orientation is studied. The concrete circular ultrarelativistic orbit of radius r=3m is considered as a partial solution of the Mathisson-Papapetrou equations and as the corresponding high-energy quantum state of the Dirac particle. Numerical estimates for protons and electrons near black holes are given. A tendency of gravitational and electromagnetic interactions to approach in quantitative terms at ultrarelativistic velocities is discussed 
  Configuration space of general relativity is extended by inclusion of the determinant of the metric as a new independent variable. As the consequence the Hilbert-Einstein action takes a polynomial form. 
  The present paper reconsiders the Newtonian limit of models of modified gravity including higher order terms in the scalar curvature in the gravitational action. This was studied using the Palatini variational principle in [Meng X. and Wang P.: Gen. Rel. Grav. {\bf 36}, 1947 (2004)] and [Dom\'inguez A. E. and Barraco D. E.: Phys. Rev. D {\bf 70}, 043505 (2004)] with contradicting results. Here a different approach is used, and problems in the previous attempts are pointed out. It is shown that models with negative powers of the scalar curvature, like the ones used to explain the present accelerated expansion, as well as their generalization which include positive powers, can give the correct Newtonian limit, as long as the coefficients of these powers are reasonably small. Some consequences of the performed analysis seem to raise doubts for the way the Newtonian limit was derived in the purely metric approach of fourth order gravity [Dick R.: Gen. Rel. Grav. {\bf 36}, 217 (2004)]. Finally, we comment on a recent paper [Olmo G. J.: Phys. Rev. D {\bf 72}, 083505 (2005)] in which the problem of the Newtonian limit of both the purely metric and the Palatini formalism is discussed, using the equivalent Brans--Dicke theory, and with which our results partly disagree. 
  A system of effective Einstein equations for spatially averaged scalar variables of inhomogeneous cosmological models can be solved by providing a `cosmic equation of state'. Recent efforts to explain Dark Energy focus on `backreaction effects' of inhomogeneities on the effective evolution of cosmological parameters in our Hubble volume, avoiding a cosmological constant in the equation of state. In this Letter it is argued that, if kinematical backreaction effects are indeed of the order of the averaged density (or larger as needed for an accelerating domain of the Universe), then the state of our regional Hubble volume would have to be in the vicinity of a far-from-equilibrium state that balances kinematical backreaction and average density. This property, if interpreted globally, is shared by a stationary cosmos with effective equation of state $p_{\rm eff} = -1/3 \rho_{\rm eff}$. It is concluded that a confirmed explanation of Dark Energy by kinematical backreaction may imply a paradigmatic change of cosmology. 
  We construct physical semi-classical states annihilated by the Hamiltonian constraint operator in the framework of loop quantum cosmology as a method of systematically determining the regime and validity of the semi-classical limit of the quantum theory. Our results indicate that the evolution can be effectively described using continuous classical equations of motion with non-perturbative corrections down to near the Planck scale below which the universe can only be described by the discrete quantum constraint. These results, for the first time, provide concrete evidence of the emergence of classicality in loop quantum cosmology and also clearly demarcate the domain of validity of different effective theories. We prove the validity of modified Friedmann dynamics incorporating discrete quanum geometry effects which can lead to various new phenomenological applications. Furthermore the understanding of semi-classical states allows for a framework for interpreting the quantum wavefunctions and understanding questions of a semi-classical nature within the quantum theory of loop quantum cosmology. 
  We investigate gravitational waves from a dust disk around a Schwarzschild black hole to focus on whether we can extract any of its physical properties from a direct detection of gravitational waves. We adopt a black hole perturbation approach in a time domain, which is a satisfactory approximation to illustrate a dust disk in a supermassive black hole. We find that we can determine the radius of the disk by using the power spectrum of gravitational waves and that our method to extract the radius works for a disk of arbitrary density distribution. Therefore we believe a possibility exists for determining the radius of the disk from a direct observation of gravitational waves detected by the Laser Interferometer Space Antenna. 
  Maximally extended, explicit and regular coverings of the Schwarzschild - de Sitter family of vacua are given, first in spacetime (generalizing a result due to Israel) and then for all dimensions $D$ (assuming a $D-2$ sphere). It is shown that these coordinates offer important advantages over the well known Kruskal - Szekeres procedure. 
  Multidimensional configurations with Minkowski external space-time and a spherical global monopole in extra dimensions are discussed in the context of the brane world concept. The monopole is formed with a hedgehog-like set of scalar fields \phi^i with a symmetry-breaking potential V depending on the magnitude \phi^2 = \phi^i \phi^i. All possible kinds of globally regular configurations are singled out without specifying the shape of V(\phi). These variants are governed by the maximum value \phi_m of the scalar field, characterizing the energy scale of symmetry breaking. If \phi_m < \phi_cr (where \phi_cr is a critical value of \phi related to the multidimensional Planck scale), the monopole reaches infinite radii while in the ``strong field regime'', when \phi_m\geq \phi_cr, the monopole may end with a cylinder of finite radius or possess two regular centers. The warp factors of monopoles with both infinite and finite radii may either exponentially grow or tend to finite constant values far from the center. All such configurations are shown to be able to trap test scalar matter, in striking contrast to RS2 type 5D models. The monopole structures obtained analytically are also found numerically for the Mexican hat potential with an additional parameter acting as a cosmological constant. 
  We explore some of the gravitational features of a uniform ring both in the Newtonian potential theory and in General Relativity. We use a spacetime associated to a Weyl static solution of the vacuum Einstein's equations with ring like singularity. The Newtonian motion for a test particle in the gravitational field of the ring is studied and compared with the corresponding geodesic motion in the given spacetime. We have found a relativistic peculiar attraction: free falling particle geodesics are lead to the inner rim but never hit the ring. 
  We present an observation about the proposal that four-dimensional modification of general relativity may explain the observed cosmic acceleration today. Assuming that the thermodynamical nature of gravity theory continues to hold in modified gravity theories, we derive the modified horizon entropy formula from the modified Friedmann equation. We argue that our results imply that there are conceptual problems in some models of four-dimensional modification of general relativity. 
  We prove a global result in time for the initial value problem for the relativistic Boltzmann equation on the flat Robertson-Walker sapace time, in the functional framework appropriate to the coupling with Einstein's equations. We had nowhere to restrict the size of the initial data which can hence be taken arbitrarily large. 
  The volume operator plays a pivotal role for the quantum dynamics of Loop Quantum Gravity (LQG). It is essential in order to construct Triad operators that enter the Hamiltonian constraint and which become densely defined operators on the full Hilbert space even though in the classical theory the triad becomes singular when classical GR breaks down. The expression for the volume and triad operators derives from the quantisation of the fundamental electric flux operator of LQG by a complicated regularisation procedure. In fact, there are two inequivalent volume operators available in the literature and, moreover, both operators are unique only up to a finite, multiplicative constant which should be viewed as a regularisation ambiguity. Now on the one hand, classical volumes and triads can be expressed directly in terms of fluxes and this fact was used to construct the corresponding volume and triad operators. On the other hand, fluxes can be expressed in terms of triads and therefore one can also view the volume operator as fundamental and consider the flux operator as a derived operator. In this paper we examine whether the volume, triad and flux quantisations are consistent with each other. The results of this consistency analysis are rather surprising. Among other findings we show: 1. The regularisation constant can be uniquely fixed. 2. One of the volume operators can be ruled out as inconsistent. 3. Factor ordering ambiguities in the definition of triad operators are immaterial for the classical limit of the derived flux operator. The results of this paper show that within full LQG triad operators are consistently quantized. In this paper we present ideas and results of the consistency check. In a companion paper we supply detailed proofs. 
  In this paper we provide the techniques and proofs for the resuls presented in our companion paper concerning the consistency check on volume and triad operator quantisation in Loop Quantum Gravity. 
  This is the written version of a lecture given at the ``VI Mexican School of Gravitation and Mathematical Physics" (Nov 21-27, 2004, Playa del Carmen, Mexico), introducing the basics of Loop Quantum Geometry. The purpose of the written contribution is to provide a Primer version, that is, a first entry into Loop Quantum Gravity and to present at the same time a friendly guide to the existing pedagogical literature on the subject. This account is geared towards graduate students and non-experts interested in learning the basics of the subject. 
  We study the metric solutions for the gravitational equations in Modified Gravity Models (MGMs). In models with negative powers of the scalar curvature, we show that the Newtonian Limit (NL) is well defined as a limit at intermediate energies, in contrast with the usual low energy interpretation. Indeed, we show that the gravitational interaction is modified at low densities or low curvatures. 
  We consider anew some puzzling aspects of the equivalence of the quantum field theoretical description of Bremsstrahlung from the inertial and accelerated observer's perspectives. More concretely, we focus on the seemingly paradoxical situation that arises when noting that the radiating source is in thermal equilibrium with the thermal state of the quantum field in the wedge in which it is located, and thus its presence does not change there the state of the field, while it clearly does not affect the state of the field on the opposite wedge. How then is the state of the quantum field on the future wedge changed, as it must in order to account for the changed energy momentum tensor there? This and related issues are carefully discussed. 
  The post-Newtonian general relativistic gravitomagnetic Lense-Thirring precessions of the perihelia of the inner planets of the Solar System amount to 10^-3 arcseconds per century. Up to now they were always retained too small to be detected. Recent improvements in the planetary ephemerides determination yield the first observational evidence of such a tiny effect. Indeed, extra-corrections to the known perihelion advances of -0.0036+-0.0050, -0.0002+-0.0004 and 0.0001+-0.0005 arcseconds per century were recently determined by E.V. Pitjeva for Mercury, the Earth and Mars, respectively. They were based on the EPM2004 ephemerides and a set of more than 317 000 observations of various kinds. The predicted relativistic Thirring-Lense precessions for these planets are -0.0020, -0.0001 and -3 10^-5 arcseconds per century, respectively and are compatible with the measured perihelia corrections, although the experimental errors are still large. The data from the forthcoming BepiColombo mission to Mercury will improve our knowledge of the orbital motion of this planet and, consequently, the precision of the measurement of the Thirring-Lense effect. As a by-product of the present analysis, it is also possible to constrain the strength of a Yukawa-like fifth force to a 10^-12-10^-13 level at scales of about one Astronomical Unit. 
  We prove a global in time existence theorem for the initial values problem for the Einstein-Boltzmann system with cosmological constant and arbitrarily large initial data, in the spatially homogeneous case, in a Robertson-Walker space-time. 
  We study the Casimir effect for free massless scalar fields propagating on a two-dimensional cylinder with a metric that admits a change of signature from Lorentzian to Euclidean. We obtain a nonzero pressure, on the hypersurfaces of signature change, which destabilizes the signature changing region and so alters the energy spectrum of scalar fields. The modified region and spectrum, themselves, back react on the pressure. Moreover, the central term of diffeomorphism algebra of corresponding infinite conserved charges changes correspondingly. 
  We present a parameter-free gauge formulation of general relativity in terms of a new set of real spin connection variables. The theory is constructed by extending the phase space of the recently formulated conformal geometrodynamics for canonical gravity to accommodate a spin gauge description. This leads to a further enlarged set of first class gravitational constraints consisting of a reduced Hamiltonian constraint and the canonical generators for spin gauge and conformorphism transformations. Owing to the incorporated conformal symmetry, the new theory is shown to be free from an ambiguity of the Barbero-Immirzi type. 
  A theory of classical metric and matter fields in spacetime is {\it locally causal} if the probability distribution for the fields in any region is determined solely by physical data in its past, i.e. it is independent of events at space-like separated points. This is the case according to general relativity, and it is natural to hypothesise that it should also hold true in any theory in which the fundamental description of space-time is classical and geometric -- for instance, some hypothetical theory which stochastically couples a classical spacetime geometry to a quantum field theory of matter.   On the other hand, a quantum theory of gravity should allow the creation of spacetimes which macroscopically violate local causality. This paper describes a feasible experiment to test the local causality of spacetime, and hence to test whether gravity is better described, in this respect, by general relativity or by quantum theory. The experiment will either identify a definite limit to the domain of validity of quantum theory or else produce significant evidence for the hypothesis that gravity is described by a quantum theory. 
  The Titius-Bode law for planetary distances is reviewed. A model describing the basic features of this law in the "quantum-like" language of a wave equation is proposed. Some considerations about the 't Hooft idea on the quantum behaviour of deterministic systems with dissipation are discussed. 
  We construct dyons, and electrically charged monopole-antimonopole pairs and vortex rings in Yang-Mills-Higgs theory coupled to Einstein gravity. The solutions are stationary, axially symmetric and asymptotically flat. The dyons with magnetic charge $n\ge 2$ represent non-static solutions with vanishing angular momentum. The electrically charged monopole-antimonopole pairs and vortex rings, in contrast, possess vanishing magnetic charge, but finite angular momentum, equaling $n$ times their electric charge. 
  Models with extra space-time dimensions produce, tipically, a 4D effective theory whose vacuum is not exactly Lorentz invariant but can be considered a physical medium whose refractive index is determined by the gravitational field. This leads to a version of relativity with a preferred frame and to look for experimental tests with the new generation of ether-drift experiments using rotating cryogenic optical resonators. Considering various types of cosmic motion, we formulate precise predictions for the modulations of the signal induced by the Earth's rotation and its orbital revolution around the Sun. We also compare with recent experimental results that might represent the first modern experimental evidence for a preferred frame. 
  Using the energy momentum complex given by M{\o}ller in 1978 based on the absolute parallelism, the energy distribution in Kerr spacetime is evaluated. The energy with this spacetime is found to be the same as it was earlier evaluated using different definitions mainly based on the metric tensor. 
  Exploiting results recently proved in a technical paper (and some of them are reviewed herein in the language of theoretical physicists) we focus on quantization of the metric of a black hole restricted to the Killing horizon with universal radius r_0. The metric is represented in a suitable manner after imposing spherical symmetry and, after restriction to the Killing horizon, it is quantized employing chiral currents. Two ''components of the metric'' are in fact quantized: one behaves as an affine scalar fields under changes of coordinates and the other is a proper scalar field. The symplectic group acts on both fields as subgroup of diffeomorphisms of the horizon and this action, in some cases depending on the choice of the vacuum state, can be implemented by means of a unitary group. If the reference state of the scalar field is not a vacuum state but a coherent state, spontaneous breaking of conformal symmetry arises and the state contains a Bose-Einstein condensate. In this case the order parameter fixes the actual size of the black hole with respect to r_0. This state together with that associated with the affine scalar when restricted in a half horizon (the future boundary of the external region of the black hole) is recognized to be thermal (KMS) with respect to Schwarschild Killing time restricted to the horizon. The value of the order parameter individuates Hawking temperature as well. As a result it is found that the densities, energy and entropy of this state scales like the mass and the entropy of the black hole and they coincide with them provided the universal parameter r_0 is fixed appropriately not depending on the size of the actual black hole. 
  The existence of spacetime singularities is irrelevant for the irreversible appearance of black holes. However, confirmation of the latter's unitary dynamics would require the preparation of a coherent superposition of a tremendous number of appropriate ``Everett worlds''. 
  Analysis of the radio tracking data from the Pioneer 10/11 spacecraft at distances between about 20 - 70 AU from the Sun has consistently indicated the presence of an unmodeled, small, constant, Doppler blue shift drift of order 6 \times 10^{-9} Hz/s. After accounting for systematics, this drift can be interpreted as a constant acceleration of a_P= (8.74 \pm 1.33) \times 10^{-8} cm/s^2 directed towards the Sun, or perhaps as a time acceleration of a_t = (2.92 \pm 0.44)\times 10^{-18} s/s^2. Although it is suspected that there is a systematic origin to this anomaly, none has been unambiguously demonstrated. We review the current status of the anomaly, and then point out how the analysis of early data, which was never analyzed in detail, could allow a more clear understanding of the origin of the anomaly, be it a systematic or a manifestation of unsuspected physics. 
  A theory in which 4-dimensional spacetime is generalized to a larger space, namely a 16-dimensional Clifford space (C-space) is investigated. Curved Clifford space can provide a realization of Kaluza-Klein theory. A covariant Dirac equation in curved C-space is explored. The generalized Dirac field is assumed to be a polyvector-valued object (a Clifford number) which can be written as a superposition of four independent spinors, each spanning a different left ideal of Clifford algebra. The general transformations of a polyvector can act from the left and/or from the right, and form a large gauge group which may contain the group U(1)xSU(2)xSU(3) of the standard model. The generalized spin connection in C-space has the properties of Yang-Mills gauge fields. It contains the ordinary spin connection related to gravity (with torsion), and extra parts describing additional interactions, including those described by the antisymmetric Kalb-Ramond fields. 
  In this paper we analyze the interaction of a uniformly accelerated detector with a quantum field in (3+1)D spacetime, aiming at the issue of how kinematics can render vacuum fluctuations the appearance of thermal radiance in the detector (Unruh effect) and how they engender flux of radiation for observers afar. Two basic questions are addressed in this study: a) How are vacuum fluctuations related to the emitted radiation? b) Is there emitted radiation with energy flux in the Unruh effect? We adopt a method which places the detector and the field on an equal footing and derive the two-point correlation functions of the detector and of the field separately with full account of their interplay. From the exact solutions, we are able to study the complete process from the initial transient to the final steady state, keeping track of all activities they engage in and the physical effects manifested. We derive a quantum radiation formula for a Minkowski observer. We find that there does exist a positive radiated power of quantum nature emitted by the detector, with a hint of certain features of the Unruh effect. We further verify that the total energy of the dressed detector and a part of the radiated energy from the detector is conserved. However, this part of the radiation ceases in steady state. So the hint of the Unruh effect in radiated power is actually not directly from the energy flux that the detector experiences in Unruh effect. Since all the relevant quantum and statistical information about the detector (atom) and the field can be obtained from the results presented here, they are expected to be useful, when appropriately generalized, for addressing issues of quantum information processing in atomic and optical systems, such as quantum decoherence, entanglement and teleportation. 
  In gauge theory of gravity, there is direct coupling between the spin of a particle and gravitomagnetic field. In the surface of a neutron star or near black hole, the coupling energy between spin and gravitomagnetic field can be large and detectable. Precise measurement of the position of spectrum lines of the corresponding emission or absorption can help us to determine the gravitomagnetic field and electromagnetic field simultaneously. The ratio $\frac{\Delta E_e}{\Delta E_p}$ can be served as a quantitative criteria of black hole. In GRBs or X-ray pulsar, absorption spectral lines of electron were observed. If the absorption spectral lines of electron and proton can be observed simultaneously, using the method given in this paper, we can determine the gravitomagnetic field in the surface of the star, and discriminate black hole from neutron star. 
  The equations of general relativity in the form of timelike and null geodesics that describe motion of test particles and photons in Kerr spacetime are solved exactly including the contribution from the cosmological constant. We then perform a systematic application of the exact solutions obtained to the following cases. The exact solutions derived for null, spherical, polar and non-polar orbits are applied for the calculation of frame dragging (Lense-Thirring effect) for the orbit of a photon around the galactic centre, assuming that the latter is a Kerr black hole for various values of the Kerr parameter including those supported by recent observations. Unbound null polar orbits are investigated, and an analytical expression for the deviation angle of a polar photon orbit from the gravitational Kerr field is derived. In addition, we present the exact solution for timelike and null equatorial orbits. In the former case, we derive an analytical expression for the precession of the point of closest approach (perihelion, periastron) for the orbit of a test particle around a rotating mass whose surrounding curved spacetime geometry is described by the Kerr field. In the latter case, we calculate an exact expression for the deflection angle for a light ray in the gravitational field of a rotating mass (the Kerr field). We apply this calculation for the bending of light from the gravitational field of the galactic centre for various values of the Kerr parameter and the impact factor. 
  We present a model of an inhomogeneous universe that leads to accelerated expansion after taking spatial averaging. The model universe is the Tolman-Bondi solution of the Einstein equation and contains both a region with positive spatial curvature and a region with negative spatial curvature. We find that after the region with positive spatial curvature begins to re-collapse, the deceleration parameter of the spatially averaged universe becomes negative and the averaged universe starts accelerated expansion. We also discuss the generality of the condition for accelerated expansion of the spatially averaged universe. 
  The Poincare Gauge Theory of gravitation with a Lagrangian quadratic in the field strengths is applied to a classical cosmological model. It predicts a constant value of the non-riemannian curvature scalar, which acts as a cosmological constant. As the value of the scalar depends on the context, vacuum solutions may differ from the predictions based on Einstein's constant. The corresponding deviations from General Relativity are discussed on the basis of exact solutions for the field of a mass point. 
  We investigate the energy of a theory with a unit vector field (the "aether") coupled to gravity. Both the Weinberg and Einstein type energy-momentum pseudotensors are employed. In the linearized theory we find expressions for the energy density of the 5 wave modes. The requirement that the modes have positive energy is then used to constrain the theory. In the fully non-linear theory we compute the total energy of an asymptotically flat spacetime. The resulting energy expression is modified by the presence of the aether due to the non-zero value of the unit vector at infinity and its 1/r falloff. The question of non-linear energy positivity is also discussed, but not resolved. 
  A canonical formalism of f(R)-type gravity is proposed, resolving the problem in the formalism of Buchbinder and Lyakhovich(BL). The new coordinates corresponding to the time derivatives of the metric are taken to be its Lie derivatives which is the same as in BL. The momenta canonically conjugate to them and Hamiltonian density are defined similarly to the formalism of Ostrogradski. It is shown that our method surely resolves the problem of BL. 
  In the Bondi formulation of the axisymmetric vacuum Einstein equations, we argue that the ``surface area'' coordinate condition determining the ``radial'' coordinate can be considered as part of the initial data and should be chosen in a way that gives information about the physical problem whose solution is sought. For the two-body problem, we choose this coordinate by imposing a condition that allows it to be interpreted, near infinity, as the (inverse of the) Newtonian potential. In this way, two quantities that specify the problem -- the separation of the two particles and their mass ratio -- enter the equations from the very beginning. The asymptotic solution (near infinity) is obtained and a natural identification of the Bondi "news function" in terms of the source parameters is suggested, leading to an expression for the radiated energy that differs from the standard quadrupole formula but agrees with recent non-linear calculations. When the free function of time describing the separation of the two particles is chosen so as to make the new expression agree with the classical result, closed-form analytic expressions are obtained, the resulting metric approaching the Schwarzschild solution with time. As all physical quantities are defined with respect to the flat metric at infinity, the physical interpretation of this solution depends strongly on how these definitions are extended to the near-zone and, in particular, how the "time" function in the near-zone is related to Bondi's null coordinate. 
  A relatively recent study by Mars and Senovilla provided us with a uniqueness result for the exterior vacuum gravitational field generated by an isolated distribution of matter in axial rotation in equilibrium in General Relativity. The generalisation to exterior electrovacuum gravitational fields, to include charged rotating objects, is presented here. 
  We analyze the stability of generic spherically symmetric thin shells to linearized perturbations around static solutions. We include the momentum flux term in the conservation identity, deduced from the ''ADM'' constraint and the Lanczos equations. Following the Ishak-Lake analysis, we deduce a master equation which dictates the stable equilibrium configurations. Considering the transparency condition, we study the stability of thin shells around black holes, showing that our analysis is in agreement with previous results. Applying the analysis to traversable wormhole geometries, by considering specific choices for the form function, we deduce stability regions, and find that the latter may be significantly increased by considering appropriate choices for the redshift function. 
  We attempt to see how closely we can formally obtain the planetary and light path equations of General Relativity by employing certain operations on the familiar Newtonian equation. This article is intended neither as an alternative to nor as a tool for grasping Einstein's General Relativity. Though the exercise is understandable by readers at large, it is especially recommended to the teachers of Relativity for an appreciative understanding of its peculiarity as well as its pedagogical value in the teaching of differential equations. 
  It has been shown that in the context of General Relativity (GR) enriched with a new set of discrete symmetry reversal conjugate metrics, negative energy states can be rehabilitated while avoiding the well-known instability issues. We review here some cosmological implications of the model and confront them with the supernovae and CMB data. The predicted flat universe constantly accelerated expansion phase is found to be in rather good agreement with the most recent cosmological data. 
  We discuss certain general features of type B warped spacetimes which have important consequences on the material content they may admit and its associated dynamics. We show that, for Warped B spacetimes, if shear and anisotropy are nonvanishing, they have to be proportional. We also study some of the physics related to the warping factor and of the underlying decomposable metric. Finally we explore the only possible cases compatible with a type B Warped geometry which satisfy the dominant energy conditions. As an example of the above mentioned consequences we consider a radiating fluid and two non-spherically symmetric metrics which depend upon an arbitrary parameter, such that if the parameter vanishes the spherical symmetry is recovered. 
  We present an analytic method based on the Hadamard-WKB expansion to calculate the self-force for a particle with scalar charge that undergoes radial infall in a Schwarzschild spacetime after being held at rest until a time t = 0. Our result is valid in the case of short duration from the start. It is possible to use the Hadamard-WKB expansion in this case because the value of the integral of the retarded Green's function over the particle's entire past trajectory can be expressed in terms of two integrals over the time period that the particle has been falling. This analytic result is expected to be useful as a check for numerical prescriptions including those involving mode sum regularization and for any other analytical approximations to self-force calculations. 
  We present a numerical study of the evolution of a non-linearly disturbed black hole described by the Bondi--Sachs metric, for which the outgoing gravitational waves can readily be found using the news function. We compare the gravitational wave output obtained with the use of the news function in the Bondi--Sachs framework, with that obtained from the Weyl scalars, where the latter are evaluated in a quasi-Kinnersley tetrad. The latter method has the advantage of being applicable to any formulation of Einstein's equations---including the ADM formulation and its various descendants---in addition to being robust. Using the non-linearly disturbed Bondi--Sachs black hole as a test-bed, we show that the two approaches give wave-extraction results which are in very good agreement. When wave extraction through the Weyl scalars is done in a non quasi-Kinnersley tetrad, the results are markedly different from those obtained using the news function. 
  In this paper, we evaluate energy and momentum density distributions for the Weyl metric by using the well-known prescriptions of Einstein, Landau-Lifshitz, Papaterou and M$\ddot{o}$ller. The metric under consideration is the static axisymmetric vacuum solution to the Einstein field equations and one of the field equations represents the Laplace equation. Curzon metric is the special case of this spacetime. We find that the energy density is different for each prescription. However, momentum turns out to be constant in each case. 
  Quantum power corrections to the gravitational spin-orbit and spin-spin interactions, as well as to the Lense-Thirring effect, were found for particles of spin 1/2. These corrections arise from diagrams of second order in Newton gravitational constant G with two massless particles in the unitary cut in the t-channel. The corrections obtained differ from the previous calculation of the corrections to spin effects for rotating compound bodies with spinless constituents. 
  Gauge theories of gravity provide an elegant and promising extension of general relativity. In this paper we show that the Poincar\'e gauge theory exhibits gravity-induced birefringence under the assumption of a specific gauge invariant nonminimal coupling between torsion and Maxwell's field. Furthermore we give for the first time an explicit expression for the induced phaseshift between two orthogonal polarization modes within the Poincar\'e framework. Since such a phaseshift can lead to a depolarization of light emitted from an extended source this effect is, in principle, observable. We use white dwarf polarimetric data to constrain the essential coupling constant responsible for this effect. 
  We show that the method used in the Schwarzschild black hole for finding the elementary solution of the electrostatic equation in closed form cannot extend in higher dimensions. By contrast, we prove the existence of static, spherically symmetric geometries with a non-degenerated horizon in which the static scalar equation can be solved in closed form. We give the explicit results in 6 dimensions. We determine moreover the expressions of the electrostatic potential and of the static scalar field for a point source in the extremal Reissner-Nordstrom black holes in higher dimensions. 
  In semiclassical gravity the back-reaction of the classical gravitational field interacting with quantum matter fields is described by the semiclassical Einstein equations. A criterion for the validity of semiclassical gravity based on the stability of the solutions of the semiclassical Einstein equations with respect to quantum metric perturbations is discussed. The two-point quantum correlation functions for the metric perturbations can be described by the Einstein-Langevin equation obtained in the framework of stochastic gravity. These correlation functions agree, to leading order in the large $N$ limit, with the quantum correlation functions of the theory of gravity interacting with $N$ matter fields. The Einstein-Langevin equations exhibit runaway solutions and methods to deal with these solutions are discussed. The validity criterion is used to show that flat spacetime as a solution of semiclassical gravity is stable and, consequently, a description based on semiclassical gravity is a valid approximation in that case. 
  Dynamics of a self-gravitating shell of matter is derived from the Hilbert variational principle and then described as an (infinite dimensional, constrained) Hamiltonian system. A method used here enables us to define singular Riemann tensor of a non-continuous connection {\em via} standard formulae of differential geometry, with derivatives understood in the sense of distributions. Bianchi identities for the singular curvature are proved. They match the conservation laws for the singular energy-momentum tensor of matter. Rosenfed-Belinfante and Noether theorems are proved to be still valid in case of these singular objects. Assumption about continuity of the four-dimensional spacetime metric is widely discussed. 
  We construct a static solution for 4+1 dimensional bulk such that the 3+1 dimensional world has a linear warp factor and describes the Schwarzschild-dS_{4} black hole. For m=0 this four dimensional universe and Friedmann Robertson Walker universe are related with an explicit coordinate transformation. We emphasize that for linear warp factors the effect of bulk on the brane world shows up as the dS_{4} background which is favored by the big bang cosmology. 
  We study the gravitational collapse in five-dimensional de Sitter (dS) spacetime and discuss the existence of the conformal boundaries at future timelike infinity from the perspective of the dS/CFT correspondence. We investigate the motion of a spherical dust shell and the black hole area bounds. The latter includes the analysis of the trapping horizon and the initial data with spindle-shaped matter distribution. In all above analyses we find the evidences that guarantee the existence of the conformal boundaries at future timelike infinity which are essential to apply the dS/CFT correspondence. 
  We present classical and quantum dynamics of a test particle in the compactified Milne space. Background spacetime includes one compact space dimension undergoing contraction to a point followed by expansion. Quantization consists in finding a self-adjoint representation of the algebra of particle observables. Our model offers some insight into the nature of the cosmic singularity. 
  After reviewing various interpretations of structural realism, I adopt here a definition that allows both relations between things that are already individuated (which I call ``relations between things'') and relations that individuate previously un-individuated entities ("things between relations"). Since both space-time points in general relativity and elementary particles in quantum theory fall into the latter category, I propose a principle of maximal permutability as a criterion for the fundamental entities of any future theory of ``quantum gravity''; i.e., a theory yielding both general relativity and quantum field theory in appropriate limits. Then I review of a number of current candidates for such a theory. First I look at the effective field theory and asymptotic quantization approaches to general relativity, and then at string theory. Then a discussion of some issues common to all approaches to quantum gravity based on the full general theory of relativity argues that processes, rather than states should be taken as fundamental in any such theory. A brief discussion of the canonical approach is followed by a survey of causal set theory, and a new approach to the question of which space-time structures should be quantized ends the paper. 
  Suppose, the Universe comes into existence (as classical spacetime) already with an empty spherically symmetric macroscopic wormhole present in it. Classically the wormhole would evolve into a part of the Schwarzschild space and thus would not allow any signal to traverse it. I consider semiclassical corrections to that picture and build a model of an evaporating wormhole. The model is based on the assumption that the vacuum polarization and its backreaction on the geometry of the wormhole are weak. The lack of information about the era preceding the emergence of the wormhole results in appearance of three parameters which -- along with the initial mass -- determine the evolution of the wormhole. For some values of these parameters the wormhole turns out to be long-lived enough to be traversed and to transform into a time machine. 
  A brief introduction is given to rotating black holes in more than four spacetime dimensions. 
  We report on the first joint search for gravitational waves by the TAMA and LIGO collaborations. We looked for millisecond-duration unmodelled gravitational-wave bursts in 473 hr of coincident data collected during early 2003. No candidate signals were found. We set an upper limit of 0.12 events per day on the rate of detectable gravitational-wave bursts, at 90% confidence level. From simulations, we estimate that our detector network was sensitive to bursts with root-sum-square strain amplitude above approximately 1-3x10^{-19} Hz^{-1/2} in the frequency band 700-2000 Hz. We describe the details of this collaborative search, with particular emphasis on its advantages and disadvantages compared to searches by LIGO and TAMA separately using the same data. Benefits include a lower background and longer observation time, at some cost in sensitivity and bandwidth. We also demonstrate techniques for performing coincidence searches with a heterogeneous network of detectors with different noise spectra and orientations. These techniques include using coordinated signal injections to estimate the network sensitivity, and tuning the analysis to maximize the sensitivity and the livetime, subject to constraints on the background. 
  It is argued that in Weyl electrovacuum solutions the linear term in the metric cannot be eliminated just on grounds of gauge invariance. Its importance is stressed. 
  A primary objective of the Lunar Laser Ranging (LLR) experiment is to provide precise observations of the lunar orbit that contribute to a wide range of science investigations. Time series of the highly accurate measurements of the distance between the Earth and Moon provide unique information used to determine whether, in accordance with the Equivalence Principle (EP), both of these celestial bodies are falling towards the Sun at the same rate, despite their different masses, compositions, and gravitational self-energies. Current LLR solutions give $(-1.0 \pm 1.4) \times 10^{-13}$ for any possible inequality in the ratios of the gravitational and inertial masses for the Earth and Moon, $\Delta(M_G/M_I)$. This result, in combination with laboratory experiments on the weak equivalence principle, yields a strong equivalence principle (SEP) test of $\Delta(M_G/M_I)_{\tt SEP} = (-2.0 \pm 2.0) \times 10^{-13}$. Such an accurate result allows other tests of gravitational theories. The result of the SEP test translates into a value for the corresponding SEP violation parameter $\eta$ of $(4.4 \pm 4.5)\times10^{-4}$, where $\eta = 4\beta -\gamma -3$ and both $\gamma$ and $\beta$ are parametrized post-Newtonian (PPN) parameters. The PPN parameter $\beta$ is determined to be $\beta - 1 = (1.2 \pm 1.1) \times 10^{-4}$. Focusing on the tests of the EP, we discuss the existing data, and characterize the modeling and data analysis techniques. The robustness of the LLR solutions is demonstrated with several different approaches that are presented in the text. We emphasize that near-term improvements in the LLR ranging accuracy will further advance the research of relativistic gravity in the solar system, and, most notably, will continue to provide highly accurate tests of the Equivalence Principle. 
  It is shown in this work that all free physical fields should have a nonzero rest mass according to the field theory of gravitation. 
  Dirac's constraint analysis and the symplectic structure of geodesic equations are obtained for the general cylindrically symmetric stationary spacetime. For this metric, using the obtained first order Lagrangian, the geodesic equations of motion are integrated, and found some solutions for Lewis, Levi-Civita, and Van Stockum spacetimes. 
  We study the equilibria of a self-gravitating scalar field in the region outside a reflecting barrier. By introducing a potential difference between the barrier and infinity, we create a kink which cannot decay to a zero energy state. In the realm of small amplitude, the kink decays to a known static solution of the Einstein-Klein-Gordon equation. However, for larger kinks the static equilibria are degenerate, forming a system with two energy levels. The upper level is unstable and, under small perturbations, decays to the lower energy stable equilibrium. Under large perturbations, the unstable upper level undergoes collapse to a black hole. The equilibrium of the system provides a remarkably simple and beautiful illustration of a turning point instability. 
  Starting from generic bilinear Hamiltonians, constructed by covariant vector, bivector or tensor fields, it is possible to derive a general symplectic structure which leads to holonomic and anholonomic formulations of Hamilton equations of motion directly related to a hydrodynamic picture. This feature is gauge free and it seems a deep link common to all interactions, electromagnetism and gravity included. This scheme could lead toward a full canonical quantization. 
  We study the critical behaviour of spherically symmetric scalar field collapse to black holes in spacetime dimensions other than four. We obtain reliable values for the scaling exponent in the supercritical region for dimensions in the range $3.5\leq D\leq 14$. The critical exponent increases monotonically to an asymptotic value at large $D$ of $\gamma\sim0.466$. The data is well fit by a simple exponential of the form: $\gamma \sim 0.466(1-e^{-0.408 D})$. 
  In this paper we investigate quantum fields propagating on given, static, spherically symmetric spacetimes, which are isometric to a part of the Schwarzschild spacetime. Without specifying the internal geometry we show, that there exist bounds on the energy densities of ground states of a quantum scalar field on such spacetimes. The bounds (from above and below) come from the so-called Quantum Energy Inequalities, and are centered around the energy density of the Boulware state (the ground state for Schwarzschild spacetime). The specific value of the bound from below depends critically on the distance $\ell$ from the horizon, where the spacetimes of compact objects cease to be isometric to the Schwarzschild spacetime. In the limit of small $\ell$ we prove, that the energy densities of ground states cannot be below the Boulware level. 
  The maximal acceleration (MA) problem associated with the position-dependent rest mass concept is considered. New arguments in favor of the mass-dependent maximal acceleration (MDMA) are put forward. The hypothesis that there exists a maximal force with the numerical value equal to the inverse Einstein's gravitation constant is advanced. The Lagrangian and Hamiltonian classical dynamics of a point-like particle with the coordinate-dependent mass is given. The effective Lagrangian for the pure gravitational interaction of a test particle is proposed. Within the scope of this model the typical spiral galaxy rotation is described. It is shown that by this model the peculiar form of the corresponding rotation curve is as a whole reproduced without recourse to the dark matter concept. Also, it is demonstrated that the canonical quantization of this model leads directly to the Dirac oscillator model for a particle with Plank's mass. 
  In this work two different boundary conditions for first order gravity, corresponding to a null and a negative cosmological constant respectively, are studied. Both boundary conditions allows to obtain the standard black hole thermodynamics. Furthermore both boundary conditions define a canonical ensemble. Additionally the quasilocal energy definition is obtained for the null cosmological constant case. 
  We study the integrability of geodesic flow in the background of some recently discovered charged rotating solutions of supergravity in four and five dimensions. Specifically, we work with the gauged multicharge Taub-NUT-Kerr-(Anti) de Sitter metric in four dimensions, and the $U(1)^3$ gauged charged-Kerr-(Anti) de Sitter black hole solution of N = 2 supergravity in five dimensions. We explicitly construct the Killing tensors that permit separation of the Hamilton-Jacobi equation in these spacetimes. These results prove integrability for a large class of previously known supergravity solutions, including several BPS solitonic states. We also derive first-order equations of motion for particles in these backgrounds and examine some of their properties. Finally, we also examine the Klein-Gordon equation for a scalar field in these spacetimes and demonstrate separability. 
  Under a Lorentz-transformation, Mie's 1912 gravitational mass behaves identical as de Broglie's 1923 clock-like frequency. The same goes for Mie's inertial mass and de Broglie's wave-like frequency. This allows the interpretation of de Broglie's "Harmony of the Phases" as a "Principle of Equivalence" for Quantum Gravity. Thus, the particle-wave duality can be given a realist interpretation. The "Mie-de Broglie" interpretation suggests a correction of Hamilton's variational principle in the quantum domain. The equivalence of the masses can be seen as the classical "limit" of the quantum equivalence of the phases. 
  The existence of a thermodynamic arrow of time in the present universe implies that the initial state of the observable portion of our universe at (or near) the ``big bang'' must have been very ``special''. We argue that it is not plausible that these special initial conditions have a dynamical origin. 
  Poincare gauge theories that, in the absence of spinning matter, reduce to the one-parameter teleparallel theory are investigated with respect to their mathematical consistency and experimental viability. It is argued that the theories can be consistently coupled to the known standard model particles. Moreover, we establish the result that in the classical limit, such theories share a large class of solutions with general relativity, containing, among others, the four classical black hole solutions (Schwarzschild, Reisner-Nordstrom, Kerr and Kerr-Newman), as well as the complete class of Friedman-Robertson-Walker cosmological solutions, thereby extending older viability results that were restricted to the correct Newtonian limit and to the existence of the Schwarzschild solution. 
  We study the Hamilton-Jacobi and massive Klein-Gordon equations in the general Kerr-(Anti) de Sitter black hole background in all dimensions. Complete separation of both equations is carried out in cases when there are two sets of equal black hole rotation parameters. We analyze explicitly the symmetry properties of these backgrounds that allow for this Liouville integrability and construct a nontrivial irreducible Killing tensor associated with the enlarged symmetry group which permits separation. We also derive first-order equations of motion for particles in these backgrounds and examine some of their properties. This work greatly generalizes previously known results for both the Myers-Perry metrics, and the Kerr-(Anti) de Sitter metrics in higher dimensions. 
  We have compiled measurements of the mechanical loss in fused silica from samples spanning a wide range of geometries and resonant frequency in order to model the known variation of the loss with frequency and surface-to-volume ratio. This improved understanding of the mechanical loss has contributed significantly to the design of advanced interferometric gravitational wave detectors, which require ultra-low loss materials for their test mass mirrors. 
  It is commonly accepted that the combination of quantum mechanics and general relativity gives rise to the emergence of a minimum uncertainty both in space and time. The arguments that support this conclusion are mainly based on perturbative approaches to the quantization, in which the gravitational interactions of the matter content are described as corrections to a classical background. In a recent paper, we analyzed the existence of a minimum time uncertainty in the framework of doubly special relativity. In this framework, the standard definition of the energy-momentum of particles is modified appealing to possible quantum gravitational effects, which are not necessarily perturbative. Demanding that this modification be completed into a canonical transformation determines the implementation of doubly special relativity in position space and leads to spacetime coordinates that depend on the energy-momentum of the particle. In the present work, we extend our analysis to the quantum length uncertainty. We show that, in generic cases, there actually exists a limit in the spatial resolution, both when the quantum evolution is described in terms of the auxiliary time corresponding to the Minkowski background or in terms of the physical time. These two kinds of evolutions can be understood as corresponding to perturbative and non-perturbative descriptions, respectively. This result contrasts with that found for the time uncertainty, which can be made to vanish in all models with unbounded physical energy if one adheres to a non-perturbative quantization. 
  A recent paper by Lucas-Serrano et al. indicates that a high-resolution central (HRC) scheme is robust enough to yield accurate hydrodynamical simulations of special relativistic flows in the presence of ultrarelativistic speeds and strong shock waves. In this paper we apply this scheme in full general relativity (involving {\it dynamical} spacetimes), and assess its suitability by performing test simulations for oscillations of rapidly rotating neutron stars and merger of binary neutron stars. It is demonstrated that this HRC scheme can yield results as accurate as those by the so-called high-resolution shock-capturing (HRSC) schemes based upon Riemann solvers. Furthermore, the adopted HRC scheme has increased computational efficiency as it avoids the costly solution of Riemann problems and has practical advantages in the modeling of neutron star spacetimes. Namely, it allows simulations with stiff equations of state by successfully dealing with very low-density unphysical atmospheres. These facts not only suggest that such a HRC scheme may be a desirable tool for hydrodynamical simulations in general relativity, but also open the possibility to perform accurate magnetohydrodynamical simulations in curved dynamic spacetimes. 
  Previous work on applications of Abstract Differential Geometry (ADG) to discrete Lorentzian quantum gravity is brought to its categorical climax by organizing the curved finitary spacetime sheaves of quantum causal sets involved therein, on which a finitary (:locally finite), singularity-free, background manifold independent and geometrically prequantized version of the gravitational vacuum Einstein field equations were seen to hold, into a topos structure. This topos is seen to be a finitary instance of both an elementary and a Grothendieck topos, generalizing in a differential geometric setting, as befits ADG, Sorkin's finitary substitutes of continuous spacetime topologies. The paper closes with a thorough discussion of four future routes we could take in order to further develop our topos-theoretic perspective on ADG-gravity along certain categorical trends in current quantum gravity research. 
  We give a review of classical, thermodynamic and quantum properties of black holes relevant to fundamental physics. 
  We discuss the procedure for the exact solution of the Riemann problem in special relativistic magnetohydrodynamics (MHD). We consider both initial states leading to a set of only three waves analogous to the ones in relativistic hydrodynamics, as well as generic initial states leading to the full set of seven MHD waves. Because of its generality, the solution presented here could serve as an important test for those numerical codes solving the MHD equations in relativistic regimes. 
  Foundations of the Poincar\'{e}-gauge theory of gravity are developed. It is shown that the Poincar\'{e}-gauge field consists of two components: the translational gauge field ($t$-field), which is generated by the energy-momentum current of external fields, and the rotational gauge field ($r$-field), which is generated by the sum of the angular and spin momentum currents of external fields. Therefore, a physical field generated by the angular momentum of a rotating mass should exist. 
  It is proposed to consider dark matter as a perfect dilaton-spin fluid (with particles endowed with intrinsic spin and dilaton charge) in the framework of a gravitational theory with a Weyl-Cartan geometrical structure. The modified Friedmann-Lema\^{\i}tre equation (with a cosmological term) is obtained for the homogeneous and isotropic Universe filled with the dilaton-spin dark matter. On the basis of this equation, we develop a nonsingular cosmological model starting from an inflation-like stage (for super-stiff equation of state), passing radiation-dominated and matter-dominated decelerating stages and turning into a post-Friedmann accelerating era. 
  The joint NASA-ESA mission LISA relies crucially on the stability of the three spacecraft constellation. All three spacecraft are on heliocentric and weakly eccentric orbits forming a stable triangle. It has been shown that for certain spacecraft orbits, the arms keep constant distances to the first order in the eccentricities. However, exact orbitography shows the so-called `breathing modes' of the arms where the arms slowly change their lengths over the time-scale of a year. In this paper we analyse the breathing modes (the flexing of the arms) with the help of the geodesic deviation equations to octupole order which are shown to be equivalent to higher order Clohessy-Wiltshire equations. We show that the flexing of the arms of LISA as given by the `exact' solution of Keplerian orbits, which gives constant armlengths to the first order in eccentricity and whose maximum flexing amplitude is $\sim 115,000$ km, can be improved, by tilting the plane of the LISA triangle slightly from the proposed orientation of $60^\circ$ with the ecliptic to obtain a maximum flexing amplitude of $\sim 48,000$ km, reducing it by a factor of $\sim 2.4$. The reduction factor is even larger if we consider the corresponding Doppler shifts, for which the reduction factor reaches almost a factor of 6. We solve the second order equations and obtain the general solution. We then use the general solution to establish the optimality of the solutions that we have found. 
  In this work we will consider the concepts of partial and complete observables for canonical general relativity. These concepts provide a method to calculate Dirac observables. The central result of this work is that one can compute Dirac observables for general relativity by dealing with just one constraint. For this we have to introduce spatial diffeomorphism invariant Hamiltonian constraints. It will turn out that these can be made to be Abelian. Furthermore the methods outlined here provide a connection between observables in the space--time picture, i.e. quantities invariant under space--time diffeomorphisms, and Dirac observables in the canonical picture. 
  The Campbell-Magaard theorem is widely seen as a way of embedding Einstein's 4D theory of general relativity in a 5D theory of the Kaluza-Klein type. We give a brief history of theorem, present a short account of it, and show that it provides a geometrical frame for new physics related to the unification of the forces. Anderson's recent vituperative attack on Campbell's theorem (gr-qc/0409122)is errant. 
  By considering the Einstein vacuum field equations linearized about the Minkowski metric, the evolution equations for the gauge-invariant quantities characterizing the gravitational field are written in a Hamiltonian form by using a conserved functional as Hamiltonian; this Hamiltonian is not the analog of the energy of the field. A Poisson bracket between functionals of the field, compatible with the constraints satisfied by the field variables, is obtained. The generator of spatial translations associated with such bracket is also obtained. 
  We analyse the flat behaviour of the rotational curves in some galaxies in the framework of a dilatonic, current-carrying string. We determine the expression of the tangential velocity of test objects following a stable circular equatorial orbit in this spacetime. 
  We construct new black hole solutions in Einstein-Yang-Mills theory. They are static, axially symmetric and asymptotically flat. They are characterized by their horizon radius and a pair of integers (k,n), where k is related to the polar angle and n to the azimuthal angle. The known spherically and axially symmetric EYM black holes have k=1. For k>1, pairs of new black hole solutions appear above a minimal value of n, that increases with k. Emerging from globally regular solutions, they form two branches, which merge and end at a maximal value of the horizon radius. The difference of their mass and their horizon mass equals the mass of the corresponding regular solution, as expected from the isolated horizon framework. 
  The theory of general relativity, which is extremely well verified by classic tests in the solar system as well as by the radiation of the binary pulsar, is one of the fundamental tools of nowadays astrophysics. It permits the computation of the gravitational wave form emitted during the inspiral phase of binary systems of neutron stars and black holes. Based on the so-called post-Newtonian approximation (developped to high order), the prediction of general relativity is used as a "template" for searching and analysing the signals in the network of gravitational-wave detectors VIRGO/LIGO. 
  Extreme mass ratio binary systems, binaries involving stellar mass objects orbiting massive black holes, are considered to be a primary source of gravitational radiation to be detected by the space-based interferometer LISA. The numerical modelling of these binary systems is extremely challenging because the scales involved expand over several orders of magnitude. One needs to handle large wavelength scales comparable to the size of the massive black hole and, at the same time, to resolve the scales in the vicinity of the small companion where radiation reaction effects play a crucial role. Adaptive finite element methods, in which quantitative control of errors is achieved automatically by finite element mesh adaptivity based on posteriori error estimation, are a natural choice that has great potential for achieving the high level of adaptivity required in these simulations. To demonstrate this, we present the results of simulations of a toy model, consisting of a point-like source orbiting a black hole under the action of a scalar gravitational field. 
  The basic principles of Affine Quantum Gravity are presented in a pedagogical style with a limited number of equations. 
  We present a perturbative reconstruction method to make a skymap of gravitational-wave backgrounds (GWBs) observed via space-based interferometer. In the presence of anisotropies in GWBs, the cross-correlated signals of observed GWBs are inherently time-dependent due to the non-stationarity of the gravitational-wave detector. Since the cross-correlated signal is obtained through an all-sky integral of primary signals convolving with the antenna pattern function of gravitational-wave detectors, the non-stationarity of cross-correlated signals, together with full knowledge of antenna pattern functions, can be used to reconstruct an intensity map of the GWBs. Here, we give two simple methods to reconstruct a skymap of GWBs based on the perturbative expansion in low-frequency regime. The first one is based on harmonic-Fourier representation of data streams and the second is based on "direct" time-series data. The latter method enables us to create a skymap in a direct manner. The reconstruction technique is demonstrated in the case of the Galactic gravitational wave background observed via planned space interferometer, LISA. Although the angular resolution of low-frequency skymap is rather restricted, the methodology presented here would be helpful in discriminating the GWBs of galactic origins by those of the extragalactic and/or cosmological origins. 
  In this paper it is shown that the Brans - Dicke scalar field itself can serve the purpose of providing an early deceleration and a late time acceleration of the universe without any need of quintessence field if one considers an interaction, i.e, transfer of energy between the dark matter and the Brans - Dicke scalar field. 
  We show that the pathology which afflicts the Hartle-Hawking vacuum on the Kerr black hole space-time can be regarded as due to rigid rotation of the state with the horizon in the sense that when the region outside the speed-of-light surface is removed by introducing a mirror, there is a state with the defining features of the Hartle-Hawking vacuum. In addition, we show that when the field is in this state, the expectation value of the energy-momentum stress tensor measured by an observer close to the horizon and rigidly rotating with it corresponds to that of a thermal distribution at the Hawking temperature rigidly rotating with the horizon. 
  Black holes were predicted by Einstein General Relativity (GR). Because of unusual properties of these objects their existence is almost unbelievable. There are gravitation theories which do not predict the black hole appearance. By now, astronomers discovered a few hundreds of massive and highly compact objects with observational features which are very similar to the properties of black holes predicted by GR. To confirm existence of black holes in the Universe, a number of groundbased and space experiments are planned to be held in the coming decade. 
  Recently considered a very attracting possibility to detect retro-MACHOs, i.e. retro-images of the Sun by a Schwarzschild black hole. In this paper we discuss glories (mirages) formed near rapidly rotating Kerr black hole horizons and propose a procedure to measure masses and rotation parameters analyzing these forms of mirages. In some sense that is a manifestation of gravitational lens effect in the strong gravitational field near black hole horizon and a generalization of the retro-gravitational lens phenomenon. We analyze the case of a Kerr black hole rotating at arbitrary speed for some selected positions of a distant observer with respect to the equatorial plane of a Kerr black hole. We discuss glories (mirages) formed near rapidly rotating Kerr black hole horizons and propose a procedure to measure masses and rotation parameters analyzing these forms of mirages. Some time ago suggested to search shadows at the Galactic Center. In this paper we present the boundaries for shadows calculated numerically. We also propose to use future radio interferometer RADIOASTRON facilities to measure shapes of mirages (glories) and to evaluate the black hole spin as a function of the position angle of a distant observer. 
  Solution for a stationary spherically symmetric accretion of the relativistic perfect fluid with an equation of state p(rho) onto the Schwarzschild black hole is presented. This solution is a generalization of Michel solution and applicable to the problem of dark energy accretion. It is shown that accretion of phantom energy is accompanied with the gradual decrease of the black hole mass. Masses of all black holes tend to zero in the phantom energy universe approaching to the Big Rip. 
  After a preliminary discussion of the relevance of the field nature of gravitation interaction, both for the fundamental interaction of particles and the topology of space time, a method is proposed to produce and detect a dynamical gravitational field, allowing the determination of the order of magnitude of its propagation velocity. 
  Today, the Global Navigation Satellite Systems, used as global positioning systems, are the GPS and the GLONASS. They are based on a Newtonian model and hence they are only operative when several relativistic effects are taken into account. The most important relativistic effects (to order 1/c^2) are: the Einstein gravitational blue shift effect of the satellite clock frequency (Equivalence Principle of General Relativity) and the Doppler red shift of second order, due to the motion of the satellite (Special Relativity).   On the other hand, in a few years the Galileo system will be built, copying the GPS system unless an alternative project is designed. In this work, it will be also shown that the SYPOR project, using fully relativistic concepts, is an alternative to a mere copy of the GPS system. According to this project, the Galileo system would be exact and there would be no need for relativistic corrections. 
  The expression of the gravitational energy-momentum defined in the context of the teleparallel equivalent of general relativity is extended to an arbitrary set of real-valued tetrad fields, by adding a suitable reference space subtraction term. The characterization of tetrad fields as reference frames is addressed in the context of the Kerr space-time. It is also pointed out that Einstein's version of the principle of equivalence does not preclude the existence of a definition for the gravitational energy-momentum density. 
  Stringy and disklike sources of the rotating compact astrophysical objects are considered on the base of the Kerr geometry. It is argued that analyticity of the Kerr solutions may result the appearance of singular strings, which may be the source of two important astrophysical effects: the jets and QPOs phenomena. 
  We discuss the most interesting approaches to derivation of the Bekenstein-Hawking entropy formula from a statistical theory. 
  The origin of self-similar (according to Y.Kulakov) structure of the Universe is discussed from a position of the theory of dynamic systems (DS). A probable nature of the isomorphism of DS configurations of different levels is revealed. Nucleon DS configuration like black hole (BH) might be acquired by the last as a result of Hawking radiation of initial BH and serve further as a genome of the Universe development. 
  An effective action of ghost condensate with higher derivatives creates a source of gravity and mimics a dark matter in spiral galaxies. We present a spherically symmetric static solution of Einstein--Hilbert equations with the ghost condensate at large distances, where flat rotation curves are reproduced in leading order over small ratio of two energy scales characterizing constant temporal and spatial derivatives of ghost field: $\mu_*^2$ and $\mu_\star^2$, respectively, with a hierarchy $\mu_\star\ll \mu_*$. We assume that a mechanism of hierarchy is provided by a global monopole in the center of galaxy. An estimate based on the solution and observed velocities of rotations in the asymptotic region of flatness, gives $\mu_*\sim 10^{19}$ GeV and the monopole scale in a GUT range $\mu_\star\sim 10^{16}$ GeV, while a velocity of rotation $v_0$ is determined by the ratio: $ \sqrt{2} v_0^2= \mu_\star^2/\mu_*^2$. A critical acceleration is introduced and naturally evaluated of the order of Hubble rate, that represents the Milgrom's acceleration. 
  In this work a new asymptotically flat solution of the coupled Einstein-Born-Infeld equations for a static spherically symmetric space-time is obtained. When the intrinsic mass is zero the resulting spacetime is regular everywhere, in the sense given by B. Hoffmann and L. Infeld in 1937, and the Einstein-Born-Infeld theory leads to the identification of the gravitational with the electromagnetic mass. 
  The concept of the space-time as emerging in the world phase transition, vs. a priori existing, is put forward. The theory of gravity with two basic symmetries, the global affine one and the general covariance, is developed. Implications for the Universe are indicated. 
  In this talk the description of gauge theories associated with internal symmetries is extended to the case in which the symmetry group is the space-time translation group (recovering Einstein's theory) using the standard jet-bundle formalism. We also reformulate these theories introducing the idea of jet-gauge and jet-diffeomorphism groups. Finally, we attempt to a simple, yet non-trivial, mixing of gravity and electromagnetism (or more general internal interaction) by turning to gauge symmetry a central extension of the Poincare group. 
  If Newtonian gravitation is modified to use surface-to-surface separation between particles, it can have the strength of nuclear force between nucleons. This may be justified by possible existence of quantum wormholes in particles. All gravitational interactions would be between coupled wormholes, emitting graviton flux in proportional to particle size, allowing for the point-like treatment above. When the wormholes are 1 Planck length apart, the resultant force is 10^40 times the normal gravitational strength for nucleons. 
  In the framework of the field theory it is shown that a time (viewed as a scalar temporal field) is an internal property of the physical system, which defines its causal structure and evolution. A new concept of internal time allows to solve the energy problem in General Relativity and predicts the existence of matter outside the time. It is demonstrated that introduction of the temporal field permits to derive the physical laws of the electromagnetic field(the general covariant four dimensional Maxwell equations for the electric and magnetic fields) from the geometrical equations of this field. It means that the fundamental physical laws are in full correspondence with the essence of time. On this ground, from the geometrical laws of the gravitational field the physical evolution equations of this field are derived. Two characteristic solutions of these equations are obtained (including the Schwarzschild solution). 
  This paper has been withdrawn by the authors because the authors excluded the paper from electronic proceedings of the conference. 
  The problems concerning a possible discovery of the mini block holes at earthly accelerators are discussed. 
  In this paper we consider $(n+1)$-dimensional cosmological model with scalar field and antisymmetric $(p+2)$-form. Using an electric composite $Sp$-brane ansatz the field equations for the original system reduce to the equations for a Toda-like system with $n(n-1)/2$ quadratic constraints on the charge densities. For certain odd dimensions ($D = 4m+1 = 5, 9, 13, ...$) and $(p+2)$-forms ($p = 2m-1 = 1, 3, 5, ...$) these algebraic constraints can be satisfied with the maximal number of charged branes (i.e.} all the branes have non-zero charge densities). These solutions are characterized by self-dual or anti-self-dual charge density forms $Q$ (of rank $2m$). For these algebraic solutions with the particular $D$, $p$, $Q$ and non-exceptional dilatonic coupling constant $\lambda$ we obtain general cosmological solutions to the field equations and some properties of these solutions are examined. In particuilar Kasner-like behavior, the existence of attractor solutions. 
  Application of the noncommutative geometry to several physical models is considered. 
  Some highlights of the priority in the discovery of the gravitational field equations are given. 
  In this paper we present an analytical treatment of gravitational lensing by Kerr black holes in the limit of very large deflection angles, restricting to observers in the equatorial plane. We accomplish our objective starting from the Schwarzschild black hole and adding corrections up to second order in the black hole spin. This is sufficient to provide a full description of all caustics and the inversion of lens mapping for sources near them. On the basis of these formulae we argue that relativistic images of Low Mass X-ray Binaries around Sgr A* are very likely to be seen by future X-ray interferometry missions. 
  BLACK HOLES ON EARTH AND IN SPACE 
  It has been pointed out that it is impossible to obtain a unitary implementation of the dynamics for the polarized Gowdy $T^{3}$ cosmologies in an otherwise satisfactory, nonperturbative canonical quantization proposed for these spacetimes. By introducing suitable techniques to deal with deparametrized models in cosmology that possess an explicit time dependence (as it is the case for the toroidal Gowdy model), we present in this paper a detailed analysis about the roots of this failure of unitarity. We investigate the impediments to a unitary implementation of the evolution by considering modifications to the dynamics. These modifications may be regarded as perturbations. We show in a precise manner why and where unitary implementability fails in our system, and prove that the obstructions are extremely sensitive to modifications in the Hamiltonian that dictates the time evolution of the symmetry-reduced model. We are able to characterize to a certain extent how far the model is from unitarity. Moreover, we demonstrate that the dynamics can actually be approximated as much as one wants by means of unitary transformations. 
  Exact outer boundary conditions for gravitational perturbations of the Schwarzschild metric feature integral convolution between a time-domain boundary kernel and each radiative mode of the perturbation. For both axial (Regge-Wheeler) and polar (Zerilli) perturbations, we study the Laplace transform of such kernels as an analytic function of (dimensionless) Laplace frequency. We present numerical evidence indicating that each such frequency-domain boundary kernel admits a "sum-of-poles" representation. Our work has been inspired by Alpert, Greengard, and Hagstrom's analysis of nonreflecting boundary conditions for the ordinary scalar wave equation. 
  Many advances in physics have in common that some idea which was previously accepted as fundamental, general, and inescapable was subsequently seen to be consequent, special, and dispensable. The idea was not truly a general feature of the world, but only perceived to be general because of our special place in the universe and the limited range of our experience. It was excess baggage which had to be jettisoned to reach a more a more general perspective. This article discusses excess baggage from the perspective of quantum cosmology which aims at a theory of the universe's quantum initial state. We seek to answer the question `Which features of our current theoretical framework are fundamental and which reflect our special position in the universe or its special initial condition?' Past instances of cosmological excess baggage are reviewed such as the idea that the Earth was at the center of the universe or that the second law of thermodynamics was fundamental. Examples of excess baggage in our current understanding of quantum cosmology are the notion that measurement is central to formulating quantum mechanics, a fundamental quantum mechanical arrow of time, and the idea that a preferred time is needed to formulate quantum theory. We speculate on candidates for future excess baggage. (This article appeared in 1991 but is posted here for accessibility and possible current interest.) 
  A study of Bianchi type I space-times according to its proper affine vector field is given by using holonomy and decomposability, the rank of the Rieman matrix and direct integration techinques. It is shown that the special class of the above space-times admits proper affine vector fields. 
  An identity is derived from Einstein equation for any hypersurface H which can be foliated by spacelike two-dimensional surfaces. In the case where the hypersurface is null, this identity coincides with the two-dimensional Navier-Stokes-like equation obtained by Damour in the membrane approach to a black hole event horizon. In the case where H is spacelike or null and the 2-surfaces are marginally trapped, this identity applies to Hayward's trapping horizons and to the related dynamical horizons recently introduced by Ashtekar and Krishnan. The identity involves a normal fundamental form (normal connection 1-form) of the 2-surface, which can be viewed as a generalization to non-null hypersurfaces of the Hajicek 1-form used by Damour. This 1-form is also used to define the angular momentum of the horizon. The generalized Damour-Navier-Stokes equation leads then to a simple evolution equation for the angular momentum. 
  We relate the geometrical construction of (2+1)-spacetimes via grafting to phase space and Poisson structure in the Chern-Simons formulation of (2+1)-dimensional gravity with vanishing cosmological constant on manifolds of topology $R\times S_g$, where $S_g$ is an orientable two-surface of genus $g>1$. We show how grafting along simple closed geodesics \lambda is implemented in the Chern-Simons formalism and derive explicit expressions for its action on the holonomies of general closed curves on S_g. We prove that this action is generated via the Poisson bracket by a gauge invariant observable associated to the holonomy of $\lambda$. We deduce a symmetry relation between the Poisson brackets of observables associated to the Lorentz and translational components of the holonomies of general closed curves on S_g and discuss its physical interpretation. Finally, we relate the action of grafting on the phase space to the action of Dehn twists and show that grafting can be viewed as a Dehn twist with a formal parameter $\theta$ satisfying $\theta^2=0$. 
  We use Moeller's energy-momentum complex in order to explicitly compute the energy and momentum density distributions for an exact solution of Einstein's field equations with a negative cosmological constant minimally coupled to a static massless scalar field in a static, spherically symmetric background in (2+1)-dimensions. 
  It is shown that some regular solutions in 5D Kaluza-Klein gravity may have interesting properties if one from the parameters is in the Planck region. In this case the Kretschman metric invariant runs up to a maximal reachable value in nature, i.e. practically the metric becomes singular. This observation allows us to suppose that in this situation the problems with such soft singularity will be much easier resolved in the future quantum gravity then by the situation with the ordinary hard singularity (Reissner-Nordstr\"om singularity, for example). It is supposed that the analogous consideration can be applied for the avoiding the hard singularities connected with the gauge charges. 
  In various background independent approaches, quantum gravity is defined in terms of a field propagation kernel: a sum over paths interpreted as a transition amplitude between 3-geometries, expected to project quantum states of the geometry on the solutions of the Wheeler-DeWitt equation. We study the relation between this formalism and conventional quantum field theory methods. We consider the propagation kernel of 4d Lorentzian general relativity in the temporal gauge, defined by a conventional formal Feynman path integral, gauge fixed a\' la Fadeev--Popov. If space is compact, this turns out to depend only on the initial and final 3--geometries, while in the asymptotically flat case it depends also on the asymptotic proper time. We compute the explicit form of this kernel at first order around flat space, and show that it projects on the solutions of all quantum constraints, including the Wheeler-DeWitt equation, and yields the correct vacuum and n-graviton states. We also illustrate how the Newtonian interaction is coded into the propagation kernel, a key open issue in the spinfoam approach. 
  We give an explicit characterization of all functions on the phase space for the polarized Gowdy 3-torus spacetimes which have weakly vanishing Poisson brackets with the Hamiltonian and momentum constraint functions. 
  A systematic method is developed to study classical motion of a mass point in gravitational gauge field. First, the formulation of gauge theory of gravity in arbitrary curvilinear coordinates is given. Then in spherical coordinates system, a spherical symmetric solution of the field equation of gravitational gauge field is obtained, which is just the Schwarzschild solution. In gauge theory of gravity, the equation of motion of a classical mass point in gravitational gauge field is given by Newton's second law of motion. A relativistic form of the gravitational force on a mass point is deduced in this paper. Based on the spherical symmetric solution of the field equation and Newton's second law of motion, we can discuss classical tests of gauge theory of gravity, including the deflection of light by the sun, the precession of the perihelia of the orbits of the inner planets and the time delay of radar echoes passing the sun. It is found that the theoretical predictions of these classical tests given by gauge theory of gravity are completely the same as those given by general relativity. From the study in this paper, an important qualitative conclusion on the nature of gravity is that gravity can be treated as a kind of physical interactions in flat Minkowski space-time, and the equation of motion of mass point in gravitational field can be given by Newton's second law of motion. 
  We discuss the stability of semiclassical gravity solutions with respect to small quantum corrections by considering the quantum fluctuations of the metric perturbations around the semiclassical solution. We call the attention to the role played by the symmetrized two-point quantum correlation function for the metric perturbations, which can be naturally decomposed into two separate contributions: intrinsic and induced fluctuations. We show that traditional criteria on the stability of semiclassical gravity are incomplete because these criteria based on the linearized semiclassical Einstein equation can only provide information on the expectation value and the intrinsic fluctuations of the metric perturbations. By contrast, the framework of stochastic semiclassical gravity provides a more complete and accurate criterion because it contains information on the induced fluctuations as well. The Einstein-Langevin equation therein contains a stochastic source characterized by the noise kernel (the symmetrized two-point quantum correlation function of the stress tensor operator) and yields stochastic correlation functions for the metric perturbations which agree, to leading order in the large N limit, with the quantum correlation functions of the theory of gravity interacting with N matter fields. These points are illustrated with the example of Minkowski spacetime as a solution to the semiclassical Einstein equation, which is found to be stable under both intrinsic and induced fluctuations. 
  We construct a gedanken experiment, in which a weak wave packet of the complex massive scalar field interacts with a four-parameter (mass, angular momentum, electric and magnetic charges) extreme Kerr-Newman black hole. We show that the resulting black hole does not violate the cosmic censorship conjecture for any black hole parameters and wave packet configuration. 
  In the context of the currently ongoing efforts to improve the accuracy and reliability of the measurement of the Lense-Thirring effect in the gravitational field of the Earth it has recently been proposed to use the data from the existing spacecraft endowed with some active mechanisms of compensation of the non-gravitational accelerations like GRACE. In this paper we critically discuss this interesting possibility. Unfortunately, it turns out to be unpracticable because of the impact of the uncancelled even zonal harmonic coefficients of the multipolar expansion of the terrestrial gravitational potential and of some time-dependent tidal perturbations which would resemble as superimposed linear trends over the necessarily limited observational time span of the analysis. 
  The possibility of a symmetry between gravitating and anti-gravitating particles is examined. The properties of the anti-gravitating fields are defined by their behavior under general diffeomorphisms. The equations of motion and the conserved canonical currents are derived, and it is shown that the kinetic energy remains positive whereas the new fields can make a negative contribution to the source term of Einstein's field equations. The interaction between the two types of fields is naturally suppressed by the Planck scale. 
  Einstein's system of equations in the ADM decomposition involves two subsystems of equations: evolution equations and constraint equations. For numerical relativity, one typically solves the constraint equations only on the initial time slice, and then uses the evolution equations to advance the solution in time. Our interest is in the case when the spatial domain is bounded and appropriate boundary conditions are imposed. A key difficulty, which we address in this thesis, is what boundary conditions to place at the artificial boundary that lead to long time stable numerical solutions. We develop an effective technique for finding well-posed constraint preserving boundary conditions for constrained first order symmetric hyperbolic systems. By using this technique, we study the preservation of constraints by some first order symmetric hyperbolic formulations of Einstein's equations derived from the ADM decomposition linearized around Minkowski spacetime with arbitrary lapse and shift perturbations, and the closely related question of their equivalence with the linearized ADM system. Our main result is the finding of well-posed maximal nonnegative constraint preserving boundary conditions for each of the first order symmetric hyperbolic formulations under investigation, for which the unique solution of the corresponding initial boundary value problem provides a solution to the linearized ADM system on polyhedral domains. 
  We compute the one loop self-mass of a charged massless, minimally coupled scalar in a locally de Sitter background geometry. The computation is done in two different gauges: the noninvariant generalization of Feynman gauge which gives the simplest expression for the photon propagator and the de Sitter invariant gauge of Allen and Jacobson. In each case dimensional regularization is employed and fully renormalized results are obtained. By using our result in the linearized, effective field equations one can infer how the scalar responds to the dielectric medium produced by inflationary particle production. We also work out the result for a conformally coupled scalar. Although the conformally coupled case is of no great physical interest the fact that we obtain a manifestly de Sitter invariant form for its self-mass-squared establishes that our noninvariant gauge introduces no physical breaking of de Sitter invariance at one loop order. 
  The content of Einstein's theory of gravitation is encoded in the properties of the solutions to his field equations. There has been obtained a wealth of information about these solutions in the ninety years the theory has been around. It led to the prediction and the observation of physical phenomena which confirm the important role of general relativity in physics. The understanding of the domain of highly dynamical, strong field configurations is, however, still quite limited. The gravitational wave experiments are likely to provide soon observational data on phenomena which are not accessible by other means. Further theoretical progress will require, however, new methods for the analysis and the numerical calculation of the solutions to Einstein's field equations on large scales and under general assumptions. We discuss some of the problems involved, describe the status of the field and recent results, and point out some open problems. 
  In this work, we give a general class of solutions of the spinning cosmic string in Einstein's theory of gravity. After treating same problem in Einstein Cartan (EC) theory of gravity, the exact solution satisfying both exterior and interior space-times representing a spin fluid moving along the symmetry axis is presented in the EC theory. The existence of closed timelike curves in this spacetime are also examined. 
  Using the teleparallel gravity versions of the Einstein and Landau-Lifshitz's energy and/or momentum complexes, I obtain the energy and momentum of the universe in viscous Kasner-type cosmological models. The energy and momentum components (due to matter plus field) are found to be zero and this agree with a previous work of Rosen and Johri it et al. who investigated the problem of the energy in Friedmann-Robertson-Walker universe. The result that the total energy and momentum components of the universe in these models is zero same as Bergmann-Thomson's energy-momentum and props the viewpoint of Tryon. Rosen found that the energy of the Friedmann-Robertson-Walker space-time is zero, which agrees with the studies of Tryon. 
  It is proposed that after the macroscopic fluctuation of energy density that is responsible for inflation dies away, a class of microscopic fluctuations, always present, survives to give the present day dark energy. This latter is simply a reinterpretation of the causet mechanism of Ahmed, Dodelson, Green and Sorkin, wherein the emergence of space is dropped but only energy considerations are maintained. At postinflation times, energy is exchanged between the "cisplanckian" cosmos and an unknown foam-like transplanckian reservoir. Whereas during inflation, the energy flows only from the latter to the former after inflation it fluctuates in sign thereby accounting for the tiny effective cosmological constant that seems to account for dark energy. 
  Einstein's field equations for a spherically symmetric metric coupled to a massless scalar field are reduced to a system effectively of second order in time, in terms of the variables $\mu=m/r$ and $y=(\alpha/ra)$, where $a$, $\alpha$, $r$ and $m$ are as in [W.M. Choptuik, ``Universality and Scaling in Gravitational Collapse of Massless Scalar Field", \textit{Physical Review Letters} {\bf{70}} (1993), 9-12]. Solutions for which $\mu $ and $y$ are time independent may arise either from scalar fields with $\phi_t=0$ or with $\phi_s=0$ but $\phi$ linear in $t$, called respectively the positive and negative branches having the Schwarzschild solution characterized by $\phi=0 $ and $\mu_s+\mu=0$ in common. For the positive branch we obtain an exact solution which have been in fact obtained first in [I.Z. Fisher,``Scalar mesostatic field with regard for gravitational effects", \textit{Zh. Eksp. Teor. Fiz.} {\bf{18}} (1948), 636-640, gr-qc/9911008] and rediscovered many times (see D. Grumiller, ``Quantum dilaton gravity in two dimensions with matter", PhD thesis, \textit{Technische Universit$\ddot{a}$t, Wien} (2001), gr-qc/0105078) and we prove that the trivial solution $\mu=0$ is a global attractor for the region $\mu_s+\mu>0 $, $\mu<1/2$. For the negative branch discussed first in [M. Wyman, ``Static spherically symmetric scalar fields in general relativity", \textit{Physical Review D} {\bf{24}} (1981), 839-841] perturbatively, we prove that $\mu=0$ is a saddle point for the linearized system, but the non-vacuum solution $\mu=1/4$ is a stable focus and a global attractor for the region $\mu_s+\mu>0$, $\mu<1/2$. 
  We consider a new approach for gravity theory coupled to Chaplygin matter in which the {\it{relativistic}} formulation of the latter is of crucial importance. We obtain a novel form of matter with dust like density $(\sim (volume)^{-1})$ and negative pressure. We explicitly show that our results are compatible with a relativistic generalization of the energy conservation principle, derived here. 
  The QND intracavity topologies of gravitational-wave detectors proposed several years ago allow, in principle, to obtain sensitivity significantly better than the Standard Quantum Limit using relatively small anount of optical pumping power. In this article we consider an improved more ``practical'' version of the optical lever intracavity scheme. It differs from the original version by the symmetry which allows to suppress influence of the input light amplitude fluctuation. In addition, it provides the means to inject optical pumping inside the scheme without increase of optical losses.   We consider also sensitivity limitations imposed by the local meter which is the key element of the intracavity topologies. Two variants of the local meter are analyzed, which are based on the spectral variation measurement and on the Discrete Sampling Variation Measurement, correspondingly. The former one, while can not be considered as a candidate for a practical implementation, allows, in principle, to obtain the best sensitivity and thus can be considered as an ideal ``asymptotic case'' for all other schemes. The DSVM-based local meter can be considered as a realistic scheme but its sensitivity, unfortunately, is by far not so good just due to a couple of peculiar numeric factors specific for this scheme.   From our point of view search of new methods of mechanical QND measurements probably based on improved DSVM scheme or which combine the local meter with the pondermotive squeezing technique, is necessary. 
  The book contains a collection of works on Riemann-Cartan and metric-affine manifolds provided with nonlinear connection structure and on generalized Finsler-Lagrange and Cartan-Hamilton geometries and Clifford structures modelled on such manifolds. The choice of material presented has evolved from various applications in modern gravity and geometric mechanics and certain generalizations to noncommutative Riemann-Finsler geometry.   The authors develop and use the method of anholonomic frames with associated nonlinear connection structure and apply it to a number of concrete problems: constructing of generic off-diagonal exact solutions, in general, with nontrivial torsion and nonmetricity, possessing noncommutative symmetries and describing black ellipsoid/torus configurations, locally anisotropic wormholes, gravitational solitons and warped factors and investigation of stability of such solutions; classification of Lagrange/ Finsler -- affine spaces; definition of nonholonomic Dirac operators and their applications in commutative and noncommutative Finsler geometry. 
  We consider spacetime to be a 4-dimensional differentiable manifold that can be split locally into time and space. No metric, no linear connection are assumed. Matter is described by classical fields/fluids. We distinguish electrically charged from neutral matter. Electric charge and magnetic flux are postulated to be conserved. As a consequence, the inhomogeneous and the homogeneous Maxwell equations emerge expressed in terms of the excitation H and the field strength F, respectively. H and F are assumed to fulfill a local and linear "spacetime relation" with 36 constitutive functions. The propagation of electromagnetic waves is considered under such circumstances in the geometric optics limit. We forbid birefringence in vacuum and find the light cone including its Lorentzian signature. Thus the conformally invariant part of the metric is recovered. If one sets a scale, one finds the pseudo-Riemannian metric of spacetime. 
  We study the problem of motion of a relativistic, ideal elastic solid with free surface boundary by casting the equations in material form ("Lagrangian coordinates"). By applying a basic theorem due to Koch, we prove short-time existence and uniqueness for solutions close to a trivial solution. This trivial, or natural, solution corresponds to a stress-free body in rigid motion. 
  The Hamiltonian for a gravitating region includes a boundary term which determines not only the quasi-local values but also, via the boundary variation principle, the boundary conditions. Using our covariant Hamiltonian formalism, we found four particular quasi-local energy-momentum boundary term expressions; each corresponds to a physically distinct and geometrically clear boundary condition. Here, from a consideration of the asymptotics, we show how a fundamental Hamiltonian identity naturally leads to the associated quasi-local energy flux expressions. For electromagnetism one of the four is distinguished: the only one which is gauge invariant; it gives the familiar energy density and Poynting flux. For Einstein's general relativity two different boundary condition choices correspond to quasi-local expressions which asymptotically give the ADM energy, the Trautman-Bondi energy and, moreover, an associated energy flux (both outgoing and incoming). Again there is a distinguished expression: the one which is covariant. 
  We study regular, static, spherically symmetric solutions of Yang-Mills theories employing higher order invariants of the field strength coupled to gravity in $d$ dimensions. We consider models with only two such invariants characterised by integers $p$ and $q$. These models depend on one dimensionless parameter $\alpha$ leading to one-parameter families of regular solutions, obtainable by numerical solution of the corresponding boundary value problem. Much emphasis is put on an analytical understanding of the numerical results. 
  We show how to define and go from the spin-s spherical harmonics to the tensorial spin-s harmonics. These quantities, which are functions on the sphere taking values as Euclidean tensors, turn out to be extremely useful for many calculations in General Relativity. In the calculations, products of these functions, with their needed decompositions which are given here, often arise naturally. 
  It is shown that, on a manifold with a Finsler metric of Lorentzian signature, the lightlike geodesics satisfy the following variational principle. Among all lightlike curves from a point (emission event) to a timelike curve (worldline of receiver), the lightlike geodesics make the arrival time stationary. Here ``arrival time'' refers to a parametrization of the timelike curve. This variational principle can be applied (i) to the vacuum light rays in an alternative spacetime theory, based on Finsler geometry, and (ii) to light rays in an anisotropic non-dispersive medium with a general-relativistic spacetime as background. 
  A model is suggested to unify the Einstein GR and Dirac Cosmology. There is one adjusted parameter $b_2$ in our model. After adjusting the parameter $b_2$ in the model by using the supernova data, we have calculated the gravitational constant $\bar G$ and the physical quantities of $a(t)$, $q(t)$ and $\rho_r(t)/ \rho_b(t)$ by using the present day quantities as the initial conditions and found that the equation of state parameter $w_{\theta}$ equals to -0.83, the ratio of the density of the addition creation $\Omega_{\Lambda}=0.8$ and the ratio of the density of the matter including multiplication creation, radiation and normal matter $\Omega_m =0.2$ at present. The results are self-consistent and in good agreement with present knowledge in cosmology. These results suggest that the addition creation and multiplication creation in Dirac cosmology play the role of the dark energy and dark matter. 
  We consider the evolution of a flat Friedmann-Roberstson-Walker Universe in a higher derivative theories, including $\alpha R^{2}$ terms to the Einstein-Hilbert action in the presence of a variable gravitational and cosmological constants. We study here the evolution of the gravitational and cosmological constants in the presence of radiation and matter domination era of the universe. We present here new cosmological solutions which are physically interesting for model building. 
  In this study we apply post-Newtonian (T-approximants) and resummed post-Newtonian (P-approximants) to the case of a test-particle in equatorial orbit around a Kerr black hole. We compare the two approximants by measuring their effectualness (i.e. larger overlaps with the exact signal), and faithfulness (i.e. smaller biases while measuring the parameters of the signal) with the exact (numerical) waveforms. We find that in the case of prograde orbits, T-approximant templates obtain an effectualness of ~0.99 for spins q < 0.75. For 0.75 < q < 0.95, the effectualness drops to about 0.82. The P-approximants achieve effectualness of > 0.99 for all spins up to q = 0.95. The bias in the estimation of parameters is much lower in the case of P-approximants than T-approximants. We find that P-approximants are both effectual and faithful and should be more effective than T-approximants as a detection template family when q > 0. For q < 0 both T- and P-approximants perform equally well so that either of them could be used as a detection template family. However, for parameter estimation, the P-approximant templates still outperforms the T-approximants. 
  We consider globally regular and black holes solutions for the Einstein-Yang-Mills system with negative cosmological constant in $d-$spacetime dimensions. We find that the ADM mass of the spherically symmetric solutions generically diverges for $d>4$. Solutions with finite mass are found by considering corrections to the YM Lagrangean consisting in higher therm of the Yang--Mills hierarchy. Such systems can occur in the low energy effective action of string theory. A discussion of the main properties of the solutions and the differences with respect to the four dimensional case is presented. The mass of these configurations is computed by using a counterterm method. 
  In this paper, it is shown that the relative entropy is an increasing function of time in both the linear regime and the non-linear regime during the large scale structure formation in cosmology. 
  The equations of pre-metric electromagnetism are formulated as an exterior differential system on the bundle of exterior differential 2-forms over the spacetime manifold. The general form for the symmetry equations of the system is computed and then specialized to various possible forms for an electromagnetic constitutive law, namely, uniform linear, non-uniform linear, and uniform nonlinear. It is shown that in the uniform linear case, one has four possible ways of prolonging the symmetry Lie algebra, including prolongation to a Lie algebra of infinitesimal projective transformations of a real four-dimensional projective space. In the most general non-uniform linear case, th effect of non-uniformity on symmetry seems inconclusive in the absence of further specifics, and in the uniform nonlinear case, the overall difference from the uniform linear case amounts to a deformation of the electromagnetic constitutive tensor by the electromagnetic fields strengths, which induces a corresponding deformation of the symmetry Lie algebra that was obtained in the uniform linear case. 
  Using the Ehlers transformation along with the gravitoelectromagnetic approach to stationary spacetimes we start from the Morgan-Morgan disk spacetime (without radial pressure) as the seed metric and find its corresponding stationary spacetime. As expected from the Ehlers transformation the stationary spacetime obtained suffers from a NUT-type singularity and the new parameter introduced in the stationary case could be interpreted as the gravitomagnetic monopole charge (or the NUT factor). As a consequence of this singularity there are closed timelike curves (CTCs) in the singular region of the spacetime. Some of the properties of this spacetime including its particle velocity distribution, gravitational redshift, stability and energy conditions are discussed. 
  The energy levels of the left and the right handed neutrinos is split in the background of gravitational waves generated during inflation which in presence of lepton number violating interactions gives rise to a net lepton asymmetry at equilibrium. Lepton number violation is achieved by the same dimension five operator which gives rise to neutrino masses after electro-weak symmetry breaking. A net baryon asymmetry of the same magnitude can be generated from this lepton asymmetry by electroweak sphaleron processes. 
  We discuss a modified form of gravity implying that the action contains a power \alpha of the scalar curvature. Coupling with the cosmic fluid is assumed. As equation of state for the fluid, we take the simplest version where the pressure is proportional to the density. Based upon a natural ansatz for the time variation of the scale factor, we show that the equations of motion are satisfied for general \alpha. Also the condition of conservation of energy and momentum is satisfied. Moreover, we investigate the case where the fluid is allowed to possess a bulk viscosity, and find the noteworthy fact that consistency of the formalism requires the bulk viscosity to be proportional to the power (2\alpha -1) of the scalar expansion. In Einstein's gravity, where \alpha=1, this means that the bulk viscosity is proportional to the scalar expansion. This mathematical result is of physical interest; as discussed recently by the present authors, there exists in principle a viscosity-driven transition of the fluid from the quintessence region into the phantom region, implying a future Big Rip singularity. 
  Self-similar models are important in general relativity and other fundamental theories. In this paper we shall discuss the ``similarity hypothesis'', which asserts that under a variety of physical circumstances solutions of these theories will naturally evolve to a self-similar form. We will find there is good evidence for this in the context of both spatially homogenous and inhomogeneous cosmological models, although in some cases the self-similar model is only an intermediate attractor. There are also a wide variety of situations, including critical pheneomena, in which spherically symmetric models tend towards self-similarity. However, this does not happen in all cases and it is it is important to understand the prerequisites for the conjecture. 
  We have studied the sound perturbation of Unruh's acoustic geometry and we present an exact expression for the quasinormal modes of this geometry. We are obtain that the quasinormal frequencies are pure-imaginary, that give a purely damped modes. 
  An exact energy expression for a physical black hole is derived by considering the escape of a photon from the black hole. The mass of the black hole within its horizon is found to be twice its mass as observed at infinity. This result is important in understanding gravitational waves in black hole collisions. 
  A network of gravitational wave detectors is called redundant if, given the direction to a source, the strain induced by a gravitational wave in one or more of the detectors can be fully expressed in terms of the strain induced in others in the network. Because gravitational waves have only two polarizations, any network of three or more differently oriented interferometers with similar observing bands is redundant. For LISA, this ``null'' output is known as the Sagnac mode, and its use in discriminating between detector noise and a cosmological gravitational wave background is well understood [1][2]. But the usefulness of the null veto for ground-based detector networks has been ignored until now. We show that it should make it possible to discriminate in a model-independent way between real gravitational waves and accidentally coincident non-Gaussian noise ``events'' in redundant networks of two or more broadband detectors (e.g, three LIGO detectors and GEO). It has been shown that with three detectors, the null output can be used to locate the direction to the source, and then two other linear combinations of detector outputs give the optimal ``coherent'' reconstruction of the two polarization components of the signal. We discuss briefly the implementation of such a detection strategy in realistic networks. 
  A unified field theory for the description of matter in a curved space is discussed. The description is based on a standard Lagrangianian formalism in a pseudo-Euclidian 4D continuum using a 3-index tensor as independent variables. The theory defines gravitational matter as a vector field and contains the Einstein-Schwarzschild solution for the metric tensor in the case of stationary spherical symmetry. Maxwell's equations, generalized for curved space, offer an alternative to the expanding Universe explanation for the red shift phenomenon. The theory contains a covariant law for conservation of matter and strictly upports the Mach principle. A perturbation method allows a natural transition to the special relativity, providing an explanation and the scale of elementary particles parameters. 
  In this paper we study gauge-invariant metric fluctuations from a Noncompact Kaluza-Klein (NKK) theory of gravity in a de Sitter expansion. We recover the well known result $\delta\rho/\rho \simeq 2\Phi$, obtained from the standard 4D semiclassical approach to inflation. The spectrum for these fluctuations should be dependent of the fifth (spatial-like) coordinate. 
  The physically relevant singularities occurring in FRW cosmologies had traditionally been thought to be limited to the "big bang", and possibly a "big crunch". However, over the last few years, the zoo of cosmological singularities considered in the literature has become considerably more extensive, with "big rips" and "sudden singularities" added to the mix, as well as renewed interest in non-singular cosmological events such as "bounces" and "turnarounds".   In this article we present a complete catalogue of such cosmological milestones, both at the kinematical and dynamical level. First, using generalized power series, purely kinematical definitions of these cosmological events are provided in terms of the behaviour of the scale factor a(t). The notion of a "scale-factor singularity'" is defined, and its relation to curvature singularities (polynomial and differential) is explored. Second, dynamical information is extracted by using the Friedmann equations (without assuming even the existence of any equation of state) to place constraints on whether or not the classical energy conditions are satisfied at the cosmological milestones. We use these considerations to derive necessary and sufficient conditions for the existence of cosmological milestones such as bangs, bounces, crunches, rips, sudden singularities, and extremality events. Since the classification is extremely general, the corresponding results are to a high degree model-independent: In particular, we provide a complete characterization of the class of bangs, crunches, and sudden singularities for which the dominant energy condition is satisfied. 
  Some reasonings are presented that the problem of a singularity in general relativity with the problem of freezing of the $5^{th}$ dimension can be connected. It is shown that some solutions in the 5D Kaluza-Klein gravity with the cross section in the Planck region have a region $(\approx l_{Pl})$ where the metric signature changes from $\{+,-,-,-,- \}$ to $\{-,-,-,-,+ \}$. The idea is discussed that such switching can not occur following some general point of view that the Planck length is the minimal length in the nature and consequently the physical quantities can not change very quickly in the course of this length. In this case the dynamic of the $G_{55}$ metric component should be frozen. 
  An unexpected secular increase of the Astronomical Unit, the length scale of the Solar System, has recently been reported by three different research groups (Krasinsky and Brumberg, Pitjeva, Standish). The latest JPL measurements amount to 7+-2 m cy^-1. At present, there are no explanations able to accommodate such an observed phenomenon, neither in the realm of classical physics nor in the usual four-dimensional framework of the Einsteinian General Relativity. The Dvali-Gabadadze-Porrati braneworld scenario, which is a multi-dimensional model of gravity aimed to the explanation of the observed cosmic acceleration without dark energy, predicts, among other things, a perihelion secular shift, due to Lue and Starkman, of 5 10^-4 arcsec cy^-1 for all the planets of the Solar System. It yields a variation of about 6 m cy^-1 for the Earth-Sun distance which is compatible at 1-sigma level with the observed rate of the Astronomical Unit. The recently measured corrections to the secular motions of the perihelia of the inner planets of the Solar System are in agreement, at 1-sigma level, with the predicted value of the Lue-Starkman effect for Mercury and Mars and at 2-sigma level for the Earth. 
  This paper studies the interpretation of physics near a Schwarzschild black hole. A scenario for creation and growth is proposed that avoids the conundrum of information loss. In this picture the horizon recedes as it is approached and has no physical reality. Radiation is likely to occur, but it cannot be predicted. 
  The capture of compact bodies by black holes in galactic nuclei is an important prospective source for low frequency gravitational wave detectors, such as the planned Laser Interferometer Space Antenna. This paper calculates, using a semirelativistic approximation, the total energy and angular momentum lost to gravitational radiation by compact bodies on very high eccentricity orbits passing close to a supermassive, nonspinning black hole; these quantities determine the characteristics of the orbital evolution necessary to estimate the capture rate. The semirelativistic approximation improves upon treatments which use orbits at Newtonian-order and quadrupolar radiation emission, and matches well onto accurate Teukolsky simulations for low eccentricity orbits. Formulae are presented for the semirelativistic energy and angular momentum fluxes as a function of general orbital parameters. 
  The equations for the second-order gravitational perturbations produced by a compact-object have highly singular source terms at the point particle limit. At this limit the standard retarded solutions to these equations are ill-defined. Here we construct well-defined and physically meaningful solutions to these equations. These solutions are important for practical calculations: the planned gravitational-wave detector LISA requires preparation of waveform templates for the potential gravitational-waves. Construction of templates with desired accuracy for extreme mass ratio binaries, in which a compact-object inspirals towards a supermassive black-hole, requires calculation of the second-order gravitational perturbations produced by the compact-object. 
  We present a new two-parameter family of solutions of Einstein gravity with negative cosmological constant in 2+1 dimensions. These solutions are obtained by squashing the anti-de Sitter geometry along one direction and posses four Killing vectors. Global properties as well as the four dimensional generalization are discussed, followed by the investigation of the geodesic motion. A simple global embedding of these spaces as the intersection of four quadratic surfaces in a seven dimensional space is obtained. We argue also that these geometries describe the boundary of a four dimensional nutty-bubble solution and are relevant in the context of AdS/CFT correspondence. 
  The expansion of the universe is often viewed as a uniform stretching of space that would affect compact objects, atoms and stars, as well as the separation of galaxies. One usually hears that bound systems do not take part in the general expansion, but a much more subtle question is whether bound systems expand partially. In this paper, a very definitive answer is given for a very simple system: a classical ``atom'' bound by electrical attraction. With a mathemical description appropriate for undergraduate physics majors, we show that this bound system either completely follows the cosmological expansion, or -- after initial transients -- completely ignores it. This ``all or nothing'' behavior can be understood with techniques of junior-level mechanics. Lastly, the simple description is shown to be a justifiable approximation of the relativistically correct formulation of the problem. 
  We consider the possibility of observing the onset of the late time acceleration of our patch of the Universe. The Hubble size criterion and the event horizon criterion are applied to several dark energy models to discuss the problem of future inflation of the Universe. We find that the acceleration has not lasted long enough to be confirmed by present observations for the dark energy model with constant equation of state, the holographic dark energy model and the generalized Chaplygin gas (GCG) model. For the flat $\Lambda$CDM model with $\Omega_{m0}=0.3$, we find that if we use the Hubble size criterion, we need to wait until the scale factor $a_v$ reaches 3.59 times of the scale factor $a_T$ when the Universe started acceleration, to confirm the onset of acceleration, and we need to wait until $a_v=2.3 a_T$ to confirm the onset of acceleration if we use the event horizon criterion. For the flat holographic dark energy model with $d=1$, we find that $a_v=3.46 a_T$ and $a_v=2.34 a_T$, respectively. For the flat GCG model with the best supernova fitting parameter $\alpha=1.2$, we find that $a_v=5.50 a_T$ and $a_v=2.08 a_T$, respectively. 
  We study the effect of transport processes (diffusion and free--streaming) on a collapsing spherically symmetric distribution of matter in a self--similar space--time. A very simple solution shows interesting features when it is matched with the Vaidya exterior solution. In the mixed case (diffusion and free--streaming), we find a barotropic equation of state in the stationary regime. In the diffusion approximation the gravitational potential at the surface is always constant; if we perturb the stationary state, the system is very stable, recovering the barotropic equation of state as time progresses. In the free--streaming case the self--similar evolution is stationary but with a non--barotropic equation of state. 
  We study spherical, charged and self--similar distributions of matter in the diffusion approximation. We propose a simple, dynamic but physically meaningful solution. For such a solution we obtain a model in which the distribution becomes static and changes to dust. The collapse is halted with damped mass oscillations about the absolute value of the total charge. 
  We study the evolution of radiating and viscous fluid spheres assuming an additional homothetic symmetry on the spherically simmetric space--time. We match a very simple solution to the symmetry equations with the exterior one (Vaidya). We then obtain a system of two ordinary differential equations which rule the dynamics, and find a self--similar collapse which is shear--free and with a barotropic equation of state. Considering a huge set of initial self--similar dynamics states, we work out a model with an acceptable physical behavior. 
  We have found a simple exact solution of spherically-symmetrical Einstein equations describing a wormhole for an inhomogeneous distribution of the phantom energy. The equation of state is linear but highly anisotropic: while the radial pressure is negative, the transversal one is positive. At infinity the spacetime is not asymptotically flat and possesses on each side of the bridge a regular cosmological Killing horizon with an infinite area, impenetrable for any particles. This horizon does not arise if the wormhole region is glued to the Schwarzschild region. In doing so, the wormhole can enclose an arbitrary amount of the phantom energy. The configuration under discussion has a limit in which the phantom energy turns into the string dust, the areal radius tends to the constant. In this limit, the strong gravitational mass defect is realized in that the gravitational active mass is finite and constant while the proper mass integrated over the total manifold is infinite. 
  Fewster and Mistry have given an explicit, non-optimal quantum weak energy inequality that constrains the smeared energy density of Dirac fields in Minkowski spacetime. Here, their argument is adapted to the case of flat, two-dimensional spacetime. The non-optimal bound thereby obtained has the same order of magnitude, in the limit of zero mass, as the optimal bound of Vollick. In contrast with Vollick's bound, the bound presented here holds for all (non-negative) values of the field mass. 
  An exact expression for the gravitational field strength in a self-gravitating dust continuum is derived within the Lagrangian picture of continuum mechanics. From the Euler-Newton system a transport equation for the gravitational field strength is formulated and then integrated along trajectories of continuum elements. The resulting integral solves one of the Lagrangian equations of the corresponding Lagrange-Newton system in general. Relations to known exact solutions without symmetry in Newtonian gravity are discussed. The presented integral may be employed to access the non-perturbative regime of structure formation in Newtonian cosmology, and to apply iterative Lagrangian schemes to solve the Lagrange-Newton system. 
  It was pointed out by Fewster and Roman that some of the wormhole models discussed by Kuhfittig suffer from the failure to distinguish proper from coordinate distances. One of the advantages of "designer wormholes" is that models can be altered. The purpose of this note is to show that by adjusting the metric coefficients, some of these problems can be corrected. By doing so, the basic idea can be retained: wormholes containing only small amounts of exotic matter can still be traversable. 
  Stationary, axisymmetric, vacuum, solutions of Einstein's equations are obtained as critical points of the total mass among all axisymmetric and $(t,\phi)$ symmetric initial data with fixed angular momentum. In this variational principle the mass is written as a positive definite integral over a spacelike hypersurface. It is also proved that if absolute minimum exists then it is equal to the absolute minimum of the mass among all maximal, axisymmetric, vacuum, initial data with fixed angular momentum. Arguments are given to support the conjecture that this minimum exists and is the extreme Kerr initial data. 
  In this paper we investigate the critical collapse of an ultrarelativistic perfect fluid with the equation of state $P=(\Gamma-1)\rho$ in the limit of $\Gamma\to 1$. We calculate the limiting continuously self similar (CSS) solution and the limiting scaling exponent by exploiting self-similarity of the solution. We also solve the complete set of equations governing the gravitational collapse numerically for $(\Gamma-1) = 10^{-2},...,10^{-6}$ and compare them with the CSS solutions. We also investigate the supercritical regime and discuss the hypothesis of naked singularity formation in a generic gravitational collapse. The numerical calculations make use of advanced methods such as high resolution shock capturing evolution scheme for the matter evolution, adaptive mesh refinement, and quadruple precision arithmetic. The treatment of vacuum is also non standard. We were able to tune the critical parameter up to 30 significant digits and to calculate the scaling exponents accurately. The numerical results agree very well with those calculated using the CSS ansatz. The analysis of the collapse in the supercritical regime supports the hypothesis of the existence of naked singularities formed during a generic gravitational collapse. 
  Motivated by the TeV-scale gravity scenarios, we study gravitational radiation in the head-on collision of two black holes in higher dimensional spacetimes using a close-limit approximation. We prepare time-symmetric initial data sets for two black holes (the so-called Brill-Lindquist initial data) and numerically evolve the spacetime in terms of a gauge invariant formulation for the perturbation around the higher-dimensional Schwarzschild black holes. The waveform and radiated energy of gravitational waves emitted in the head-on collision are clarified. Also, the complex frequencies of fundamental quasinormal modes of higher-dimensional Schwarzschild black holes, which have not been accurately derived so far, are determined. 
  Detection template families (DTFs) are built to capture the essential features of true gravitational waveforms using a small set of phenomenological waveform parameters. Buonanno, Chen, and Vallisneri [Phys. Rev. D 67, 104025 (2003)] proposed the ``BCV2'' DTF to perform computationally efficient searches for signals from precessing binaries of compact stellar objects. Here we test the signal-matching performance of the BCV2 DTF for asymmetric--mass-ratio binaries, and specifically for double--black-hole binaries with component masses (m1,m2): (6~12Msun, 1~3Msun), and for black-hole--neutron-star binaries with component masses (m1,m2) = (10Msun, 1.4Msun); we take all black holes to be maximally spinning. We find a satisfactory signal-matching performance, with fitting factors averaging between 0.94 and 0.98. We also scope out the region of BCV2 parameters needed for a template-based search, we evaluate the template match metric, we discuss a template-placement strategy, and we estimate the number of templates needed for searches at the LIGO design sensitivity. In addition, after gaining more insight in the dynamics of spin--orbit precession, we propose a modification of the BCV2 DTF that is parametrized by physical (rather than phenomenological) parameters. We test this modified ``BCV2P'' DTF for the (10Msun, 1.4Msun) black-hole--neutron-star system, finding a signal-matching performance comparable to the BCV2 DTF, and a reliable parameter-estimation capability for target-binary quantities such as the chirp mass and the opening angle (the angle between the black-hole spin and the orbital angular momentum). 
  We perform a wide parameter space search for continuous gravitational waves over the whole sky and over a large range of values of the frequency and the first spin-down parameter. Our search method is based on the Hough transform, which is a semi-coherent, computationally efficient, and robust pattern recognition technique. We apply this technique to data from the second science run of the LIGO detectors and our final results are all-sky upper limits on the strength of gravitational waves emitted by unknown isolated spinning neutron stars on a set of narrow frequency bands in the range 200-$400 $Hz. The best upper limit on the gravitational wave strain amplitude that we obtain in this frequency range is $4.43\times 10^{-23}$. 
  The spin axes of gyroscopes experimentally define local non-rotating frames. But what physical cause governs the time-evolution of gyroscope axes? We consider linear perturbations of Friedmann-Robertson-Walker cosmologies with k=0. We ask: Will cosmological vorticity perturbations exactly drag the spin axes of gyroscopes relative to the directions of geodesics to quasars in the asymptotic unperturbed FRW space? Using Cartan's formalism with local orthonormal bases we cast the laws of linear cosmological gravitomagnetism into a form showing the close correspondence with the laws of ordinary magnetism. Our results, valid for any equation of state for cosmological matter, are: 1) The dragging of a gyroscope axis by rotational perturbations of matter beyond the Hubble-dot radius from the gyroscope is exponentially suppressed, where dot is the derivative with respect to cosmic time. 2) If the perturbation of matter is a homogeneous rotation inside some radius around a gyroscope, then exact dragging of the gyroscope axis by the rotational perturbation is reached exponentially fast as the rotation radius grows beyond the H-dot radius. 3) For the most general linear cosmological perturbations the time-evolution of all gyroscope spin axes exactly follow a weighted average of the energy currents of cosmological matter. The weight function is the same as in Ampere's law except that the inverse square law is replaced by the Yukawa force with the Hubble-dot cutoff. Our results demonstrate (in first order perturbation theory for FRW cosmologies with k = 0) the validity of Mach's hypothesis that axes of local non-rotating frames precisely follow an average of the motion of cosmic matter. 
  We investigate the non-adiabatic dynamics of spinning black hole binaries by using an analytical Hamiltonian completed with a radiation-reaction force, containing spin couplings, which matches the known rates of energy and angular momentum losses on quasi-circular orbits. We consider both a straightforward post-Newtonian-expanded Hamiltonian (including spin-dependent terms), and a version of the resummed post-Newtonian Hamiltonian defined by the Effective One-Body approach. We focus on the influence of spin terms onto the dynamics and waveforms. We evaluate the energy and angular momentum released during the final stage of inspiral and plunge. For an equal-mass binary the energy released between 40Hz and the frequency beyond which our analytical treatment becomes unreliable is found to be, when using the more reliable Effective One-Body dynamics: 0.6% M for anti-aligned maximally spinning black holes, 5% M for aligned maximally spinning black hole, and 1.8% M for non-spinning configurations. In confirmation of previous results, we find that, for all binaries considered, the dimensionless rotation parameter J/E^2 is always smaller than unity at the end of the inspiral, so that a Kerr black hole can form right after the inspiral phase. By matching a quasi-normal mode ringdown to the last reliable stages of the plunge, we construct complete waveforms approximately describing the gravitational wave signal emitted by the entire process of coalescence of precessing binaries of spinning black holes. 
  We propose a coherent method for the detection and reconstruction of gravitational wave signals for a network of interferometric detectors. The method is derived using the likelihood functional for unknown signal waveforms. In the standard approach, the global maximum of the likelihood over the space of waveforms is used as the detection statistic. We identify a problem with this approach. In the case of an aligned pair of detectors, the detection statistic depends on the cross-correlation between the detectors as expected, but this dependence dissappears even for infinitesimally small misalignments. We solve the problem by applying constraints on thelikelihood functional and obtain a new class of statistics. The resulting method can be applied to the data from a network consisting of any number of detectors with arbitrary detector orientations. The method allows us reconstruction of the source coordinates and the waveforms of two polarization components of a gravitational wave. We study the performance of the method with numerical simulation and find the reconstruction of the source coordinates to be more accurate than in the standard approach. 
  In this paper, we wish to point out the possibility of using a complex scalar field to account for the inflationary stage and the current acceleration. By the analysis of the dynamical system and numerical work, we show that the amplitude of the complex scalar field plays the role of the inflaton whereas the phase is the quintessence field. The numerical solutions describe heteroclinic orbits, which interpolate between an unstable critical point and a late-time de Sitter attractor. Therefore, this model is more natural to explain the two stages of acceleration. 
  For two-black-hole time-symmetric initial data we consider the Corvino construction of gluing an exact Schwarzschild end. We propose to do this by using Brill waves. We address the question of whether this method can be used to reduce the overall energy, which seems to relate to the question of whether it can reduce the amount of `spurious' gravitational radiation. We find a positive answer at first order in the inverse gluing radius. 
  To ask a question about a black hole in quantum gravity, one must restrict initial or boundary data to ensure that a black hole is actually present. For two-dimensional dilaton gravity, and probably a much wider class of theories as well, the imposition of a "stretched horizon" constraint alters the algebra of symmetries at the horizon, introducing a central term. Standard conformal field theory techniques can then then be used to obtain the asymptotic density of states, reproducing the Bekenstein-Hawking entropy. The microscopic states responsible for black hole entropy can thus be viewed as "would-be pure gauge" states that become physical because the symmetry is altered by the requirement that a horizon exist. 
  In the Hartle-Hawking ``no boundary'' approach to quantum cosmology, a real tunneling geometry is a configuration that represents a transition from a compact Riemannian spacetime to a Lorentzian universe. I complete an earlier proof that in three spacetime dimensions, such a transition is ``probable,'' in the sense that the required Riemannian geometry yields a genuine maximum of the semiclassical wave function. 
  The high precision attained by cosmological data in the last few years has increased the interest in exact solutions. Analytic expressions for solutions in the Standard Model are presented here for all combinations of $\Lambda = 0$, $\Lambda \ne 0$, $\kappa = 0$ and $\kappa \ne 0$, in the presence and absence of radiation and nonrelativistic matter. The most complete case (here called the $\Lambda \gamma CDM$ Model) has $\Lambda \ne 0, \kappa \ne 0$, and supposes the presence of radiation and dust. It exhibits clearly the recent onset of acceleration. The treatment includes particular models of interest such as the $\Lambda$CDM Model (which includes the cosmological constant plus cold dark matter as source constituents). 
  A static Einstein metric that generalizes the Schwarzschild metric is considered. The event horizon is not necessarily a sphere and the term $dt\sp2$ is a function on such horizon. That the metric is Einstein establishes a relation between its terms. One demonstrates that the scalar curvature of the horizon is constant, and that the term $dt\sp2$ gives rise to (i) the metric of the horizon being Einstein, or (ii) the scalar curvature of the horizon being proportional to an eigenvalue of the Laplace operator. 
  The probability for quantum creation of an inflationary universe with a pair of black holes in 1/R - gravitational theory has been studied. Considering a gravitational action which includes a cosmological constant ($\Lambda$) in addition to $ \delta R^{- 1} $ term, the probability has been evaluated in a semiclassical approximation with Hartle-Hawking boundary condition. We obtain instanton solutions determined by the parameters $\delta$ and $\Lambda$ satisfying the constraint $ \delta \leq \frac{4 \Lambda^{2}}{3}$. However, we note that two different classes of instanton solutions exists in the region $0 < \delta < \frac{4 \Lambda^{2}}{3}$. The probabilities of creation of such configurations are evaluated. It is found that the probability of creation of a universe with a pair of black holes is strongly suppressed with a positive cosmological constant except in one case when $0 < \delta < \Lambda^{2}$. It is also found that gravitational instanton solution is permitted even with $\Lambda = 0$ but one has to consider $\delta < 0$. However, in the later case a universe with a pair of black holes is less probable. 
  In this paper, we are going to put in a single consistent framework apparently unrelated pieces of information, i.e. zero-point length, extra-dimensions, string T-duality. More in details we are going to introduce a modified Kaluza-Klein theory interpolating between (high-energy) string theory and (low-energy) quantum field theory. In our model zero-point length is a four dimensional ``virtual memory'' of compact extra-dimensions length scale. Such a scale turns out to be determined by T-duality inherited from the underlying fundamental string theory. From a low energy perspective short distance infinities are cut off by a minimal length which is proportional to the square root of the string slope, i.e. \sqrt{\alpha^\prime}. Thus, we provide a ``bridge'' between the ultra-relativistic string domain and the low energy arena of point-particle quantum field theory. 
  Progress is reported on the development of a mathematical model based on the relationist principle that the position of an object can only be defined respect to other matter. Formal propositions are presented, obeying quantum logic and describing hypothetical measurement results. A connection, called a teleconnection, is defined between initial and final states using teleparallelism. Quantum covariance is defined, to take into account that a change of apparatus is implicit in Lorentz transformation. A metric is described in terms of particle interactions, using a generalisation of Bondi's k-calculus. Schwarzschild is derived for a system containing a single elementary particle in an eigenstate of position. The implications to the structure of black holes and to the horizon problem are discussed. The model makes empirical predictions about cosmological redshift which are distinct from the standard model. These have been discussed by Francis (2006), and shown to be consistent for a universe of just over critical mass. 
  The Einstein-Vlasov-Maxwell (EVM) system can be viewed as a relativistic generalization of the Vlasov-Poisson (VP) system. As it is proved below, one of nice property obeys by the first system is that the strong energy condition holds and this allows to conclude that the above system is physically viable. We show in this paper that in the context of spherical symmetry, solutions of the perturbed (EVM) system by $\gamma := 1/c^{2}$, $c$ being the speed of light, exist and converge uniformly in $L^{\infty}$-norm, as $c$ goes to infinity on compact time intervals to solutions of the non-relativistic (VP) system. 
  We find the most general Self-dual Lorentzian Wormholes in a special class of teleparallel theory of gravitation. The spacetime of these wormholes is a static and it includes the Schwarzschild black hole, a family of naked singularity and a disjoint family of Lorentzian wormholes all of which have a vanishing scalar curvature R{}. The stability is studied using the equations of geodesic deviation. The condition of stability is obtained from which the stability of Schwarzschild solution can be obtained. 
  We consider a nonlinear sigma model coupled to the metric of a conic space. We obtain restrictions for a nonlinear sigma model to be a source of the conic space. We then study nonlinear sigma model in the conic space background. We find coordinate transformations which reduce the chiral fields equations in the conic space background to field equations in Minkowski spacetime. This enables us to apply the same methods for obtaining exact solutions in Minkowski spacetime to the case of a conic spacetime. In the case the solutions depend on two spatial coordinates we employ Ivanov's geometrical ansatz. We give a general analysis and also present classes of solutions in which there is dependence on three and four coordinates. We discuss with special attention the intermediate instanton and meron solutions and their analogous in the conic space. We find differences in the total actions and topological charges of these solutions and discuss the role of the deficit angle. 
  Ten conservation laws in useful polynomial form are derived from a Cartan form and Exterior Differential System (EDS) for the tetrad equations of vacuum relativity. The Noether construction of conservation laws for well posed EDS is introduced first, and an illustration given, deriving 15 conservation laws of the free field Maxwell Equations from symmetries of its EDS. The Maxwell EDS and tetrad gravity EDS have parallel structures, with their numbers of dependent variables, numbers of generating 2-forms and generating 3-forms, and Cartan character tables all in the ratio of 1 to 4. They have 10 corresponding symmetries with the same Lorentz algebra, and 10 corresponding conservation laws. 
  A suitable splitting of tachyon field equation is able to disclose non trivial properties of Born-Infeld (some known, some unexpected) and Polyakov actions; the tachyon equation also can be analyzed in some detail. The analysis displays an intriguing connection between sine-Gordon theory and some crucial issues such as the emergence of perturbative string states when a D-Brane and an anti-D-Brane annihilate and the confinement of charged D-Branes. 
  Homogeneous cosmology in the braneworld can be studied without solving bulk equations of motion explicitly. The reason is simply because the symmetry of the spacetime restricts possible corrections in the 4-dimensional effective equations of motion. It would be great if we could analyze cosmological perturbations without solving the bulk. For this purpose, we combine the geometrical approach and the low energy gradient expansion method to derive the 4-dimensional effective action. Given our effective action, the standard procedure to obtain the cosmological perturbation theory can be utilized and the temperature anisotropy of the cosmic background radiation can be computed without solving the bulk equations of motion explicitly. 
  We present a general formalism for black string perturbations in Randall-Sundrum 1 model (RS1). First, we derive the master equation for the electric part of the Weyl tensor $E_{\mu\nu}$. Solving the master equation using the gradient expansion method, we give the effective Teukolsky equation on the brane at low energy. It is useful to estimate gravitational waves emitted by perturbed rotating black strings. We also argue the effect of the Gregory-Laflamme instability on the brane using our formalism. 
  We are interested in black holes in Loop Quantum Gravity (LQG). We study the simple model of static black holes: the horizon is made of a given number of identical elementary surfaces and these small surfaces all behaves as a spin-s system accordingly to LQG. The chosen spin-s defines the area unit or area resolution, which the observer uses to probe the space(time) geometry. For s=1/2, we are actually dealing with the qubit model, where the horizon is made of a certain number of qubits. In this context, we compute the black hole entropy and show that the factor in front of the logarithmic correction to the entropy formula is independent of the unit s. We also compute the entanglement between parts of the horizon. We show that these correlations between parts of the horizon are directly responsible for the asymptotic logarithmic corrections. This leads us to speculate on a relation between the evaporation process and the entanglement between a pair of qubits and the rest of the horizon. Finally, we introduce a concept of renormalisation of areas in LQG. 
  We quantize three Friedmann-Robertson-Walker models in the presence of a negative cosmological constant and radiation. The models differ from each other by the constant curvature of the spatial sections, which may be positive, negative or zero. They give rise to Wheeler-DeWitt equations for the scale factor which have the form of the Schroedinger equation for the quartic anharmonic oscillator. We find their eigenvalues and eigenfunctions by using a method first developed by Chhajlany and Malnev, and use the eigenfunctions in order to construct wave packets for each case and evaluate the time-dependent expected value of the scale factors. We find for all of them that the expected values of the scale factors oscillate between maximum and minimum values. Since the expectation values of the scale factors never vanish, we conclude that these models do not have singularities. 
  Some bulk viscous general solutions are found for domain walls in Lyra geometry in the plane symmetric inhomogeneous spacetime. Expressions for the energy density and pressure of domain walls are derived in both cases of uniform and time varying displacement field $\beta$. The viscosity coefficient of bulk viscous fluid is assumed to be a power function of mass density. Some physical consequences of the models are also given. Finally, the geodesic equations and acceleration of the test particle are discussed. 
  We describe here some new results concerning the Lorentzian Barrett-Crane model, a well-known spin foam formulation of quantum gravity. Generalizing an existing finiteness result, we provide a concise proof of finiteness of the partition function associated to all non-degenerate triangulations of 4-manifolds and for a class of degenerate triangulations not previously shown. This is accomplished by a suitable re-factoring and re-ordering of integration, through which a large set of variables can be eliminated. The resulting formulation can be interpreted as a ``dual variables'' model that uses hyperboloid variables associated to spin foam edges in place of representation variables associated to faces. We outline how this method may also be useful for numerical computations, which have so far proven to be very challenging for Lorentzian spin foam models. 
  In this paper we have analyzed the Kaluza-Klein type Robertson Walker (RW) cosmological models by considering three different forms of variable $\Lambda$: $\Lambda\sim(\frac{\dot{a}}{a})^2$,$\Lambda\sim(\frac{\ddot{a}} {a})$ and $\Lambda \sim \rho$. It is found that, the connecting free parameters of the models with cosmic matter and vacuum energy density parameters are equivalent, in the context of higher dimensional space time. The expression for the look back time, luminosity distance and angular diameter distance are also derived. This work has thus generalized to higher dimensions the well-known results in four dimensional space time. It is found that there may be significant difference in principle at least, from the analogous situation in four dimensional space time. 
  We study the classical and quantum evolution of a universe in which the matter source is a massive Dirac spinor field and the universe is described by a Bianchi type I metric. We focus attention on those classical solutions that admit a degenerate metric in which the scale factors have smooth behavior in transition from a Euclidean to a Lorentzian domain and show that this transition happens when the cosmological constant, $\Lambda$, is negative. The resulting quantum cosmology and the corresponding Wheeler-DeWitt equation are also studied and closed form expressions for the wave functions of the universe is presented. We have shown that there is a close relationship between the quantum states and signature changing classical solutions, suggesting a mechanism for creation of a Lorentzian universe from a Euclidean region by a continuous change of signature. The quantum solutions also represent a quantization rule for the mass of the spinor field. 
  We are concerned with the issue of quantization of a scalar field in a diffeomorphism invariant manner. We apply the method used in Loop Quantum Gravity. It relies on the specific choice of scalar field variables referred to as the polymer variables. The quantization, in our formulation, amounts to introducing the `quantum' polymer *-star algebra and looking for positive linear functionals, called states. The assumed in our paper homeomorphism invariance allows to determine a complete class of the states. Except one, all of them are new. In this letter we outline the main steps and conclusions, and present the results: the GNS representations, characterization of those states which lead to essentially self adjoint momentum operators (unbounded), identification of the equivalence classes of the representations as well as of the irreducible ones. The algebra and topology of the problem, the derivation, all the technical details and more are contained in the paper-part II. 
  The \emph{silent universe conjecture} (Sopuerta 1997, van Elst et al. 1997) states that the only algebraically general silent universes are the orthogonally spatially homogeneous Bianchi I models. In the same paper by Sopuerta this was confirmed for the subcase where the spacetime also admits a group G3 of isometries. However the proof contains a conceptual mistake. We recover the result in a different way. 
  This paper examines the classical dynamics of false vacuum regions embedded in surrounding regions of true vacuum, in the thin-wall limit. The dynamics of all generally relativistically allowed solutions -- most but not all of which have been previously studied -- are derived, enumerated, and interpreted. We comment on the relation of these solutions to possible mechanisms whereby inflating regions may be spawned from non-inflating ones. We then calculate the dynamics of first order deviations from spherical symmetry, finding that many solutions are unstable to such aspherical perturbations. The parameter space in which the perturbations on bound solutions inevitably become nonlinear is mapped. This instability has consequences for the Farhi-Guth-Guven mechanism for baby universe production via quantum tunneling. 
  There are now several analyses reporting quantized differences in the redshifts between pairs of galaxies. In the simplest cases, these differential redshifts are found to be harmonics of fundamental periods of approximately 72 km/s and 37.5 km/s. In this paper a wave equation is derived based on cosmological general relativity, which is a space-velocity theory of the expanding Universe. The wave equation is approximated to first order and comparisons are made between the quantized solutions and the reported observations. 
  Despite of over thirty years of research of the black hole thermodynamics our understanding of the possible role played by the inner horizons of Reissner-Nordstr\"om and Kerr-Newman black holes in black hole thermodynamics is still somewhat incomplete: There are derivations which imply that the temperature of the inner horizon is negative and it is not quite clear what this means. Motivated by this problem we perform a detailed analysis of the radiation emitted by the inner horizon of the Reissner-Nordstr\"om black hole. As a result we find that in a maximally extended Reissner-Nordstr\"om spacetime virtual particle-antiparticle pairs are created at the inner horizon of the Reissner-Nordstr\"om black hole such that real particles with positive energy and temperature are emitted towards the singularity from the inner horizon and, as a consequence, antiparticles with negative energy are radiated away from the singularity through the inner horizon. We show that these antiparticles will come out from the white hole horizon in the maximally extended Reissner-Nordstr\"om spacetime, at least when the hole is near extremality. The energy spectrum of the antiparticles leads to a positive temperature for the white hole horizon. In other words, our analysis predicts that in addition to the radiation effects of black hole horizons, also the white hole horizon radiates. The black hole radiation is caused by the quantum effects at the outer horizon, whereas the white hole radiation is caused by the quantum effects at the inner horizon of the Reissner-Nordstr\"om black hole. 
  We present a method of searching for, and parameterizing, signals from known radio pulsars in data from interferometric gravitational wave detectors. This method has been applied to data from the LIGO and GEO 600 detectors to set upper limits on the gravitational wave emission from several radio pulsars. Here we discuss the nature of the signal and the performance of the technique on simulated data. We show how to perform a coherent multiple detector analysis and give some insight in the covariance between the signal parameters. 
  I review the development of numerical evolution codes for general relativity based upon the characteristic initial value problem. Progress is traced from the early stage of 1D feasibility studies to 2D axisymmetric codes that accurately simulate the oscillations and gravitational collapse of relativistic stars and to current 3D codes that provide pieces of a binary black spacetime. A prime application of characteristic evolution is to compute waveforms via Cauchy-characteristic matching, which is also reviewed. 
  The software tool GRworkbench is an ongoing project in visual, numerical General Relativity at The Australian National University. Recently, GRworkbench has been significantly extended to facilitate numerical experimentation in analytically-defined space-times. The numerical differential geometric engine has been rewritten using functional programming techniques, enabling objects which are normally defined as functions in the formalism of differential geometry and General Relativity to be directly represented as function variables in the C++ code of GRworkbench. The new functional differential geometric engine allows for more accurate and efficient visualisation of objects in space-times and makes new, efficient computational techniques available. Motivated by the desire to investigate a recent scientific claim using GRworkbench, new tools for numerical experimentation have been implemented, allowing for the simulation of complex physical situations. 
  In this paper the observational consequence of the cosmological models and the expression for the neoclassical tests, luminosity distance, angular diameter distance and look back time are analyzed in the framework of Lyra geometry. It is interesting to note that the space time of the universe is not only free of Big Bang singularity but also exhibits acceleration during its evolution. 
  The current understanding of the quantum origin of cosmic structure is discussed critically. We point out that in the existing treatments a transition from a symmetric quantum state to an (essentially classical) non-symmetric state is implicitly assumed, but not specified or analyzed in any detail. In facing the issue we are led to conclude that new physics is required to explain the apparent predictive power of the usual schemes. Furthermore we show that the novel way of looking at the relevant issues opens new windows from where relevant information might be extracted regarding cosmological issues and perhaps even clues about aspects of quantum gravity. 
  We demonstrate the existence of equilibrium states in the limiting cases of the double-Kerr solution when one of the constituents is an extreme object. In the `extreme-subextreme' case the negative mass of one of the constituents is required for the balance, whereas in the `extreme-superextreme' equilibrium configurations both Kerr particles may have positive masses. We also show that the well-known relation |J|=M^2 between the mass and angular momentum in the extreme single Kerr solution ceases to be a characteristic property of the extreme Kerr particle in a binary system. 
  We study the effects of frequency-dependent squeeze amplitude attenuation and squeeze angle rotation by electromagnetically induced transparency (EIT) on gravitational wave (GW) interferometers. We propose the use of low-pass, band-pass, and high-pass EIT filters, an S-shaped EIT filter, and an intra-cavity EIT filter to generate frequency-dependent squeezing for injection into the antisymmetric port of GW interferometers. We find that the EIT filters have several advantages over the previous filter designs with regard to optical losses, compactness, and the tunability of the filter linewidth. 
  We obtain a general covariant conservation law of energy-momentum for Randall-Sundrum models by the general displacement transform. The energy-momentum currents have also a superpotential and are therefore identically conserved. It is shown that for Randall-Sundrum solution, the momentum vanishes and most of the bulk energy is located near the Planck brane. When the radius r of the extra dimension is taken the limit r->0, the extra dimension disappears and the energy vanishes too. 
  This paper can be seen as an exercise in how to adapt quantum mechanics from a strict relativistic perspective while being respectful and critical towards the experimental achievements of the contemporary theory. The result is a fully observer independent relativistic quantum mechanics for N particle systems without tachyonic solutions. A remaining worry for the moment is Bell's theorem. 
  We discuss cosmology in the context of Liouville strings, characterized by a central-charge deficit Q^2, in which target time is identified with (the world-sheet zero mode of the) Liouville field: Q-Cosmology. We use a specific example of colliding brane worlds to illustrate the phase diagram of this cosmological framework. The collision provides the necessary initial cosmological instability, expressed as a departure from conformal invariance in the underlying string model. The brane motion provides a way of breaking target-space supersymmetry, and leads to various phases of the brane and bulk Universes. Specifically, we find a hot metastable phase for the bulk string Universe soon after the brane collision in which supersymmetry is broken, which we describe by means of a subcritical world-sheet sigma model dressed by a space-like Liouville field, representing finite temperature (Euclidean time). This phase is followed by an inflationary phase for the brane Universe, in which the bulk string excitations are cold. This is described by a super-critical Liouville string with a time-like Liouville mode, whose zero mode is identified with the Minkowski target time. Finally, we speculate on possible ways of exiting the inflationary phase, either by means of subsequent collisions or by deceleration of the brane Universe due to closed-string radiation from the brane to the bulk. While phase transitions from hot to cold configurations occur in the bulk string universe, stringy excitations attached to the brane world remain thermalized throughout, at a temperature which can be relatively high. The late-time behaviour of the model results in dilaton-dominated dark energy and present-day acceleration of the expansion of the Universe, asymptoting eventually to zero. 
  The idea of treating general relativistic theories in a perturbative expansion around a topological theory has been recently put forward in the quantum gravity literature. Here we investigate the viability of this idea, by applying it to conventional Yang--Mills theory on flat spacetime. We find that the expansion around the topological theory coincides with the usual expansion around the abelian theory, though the equivalence is non-trivial. In this context, the technique appears therefore to be viable, but not to bring particularly new insights. Some implications for gravity are discussed. 
  Classical black holes and event horizons are highly non-local objects, defined in relation to the causal past of future null infinity. Alternative, quasilocal characterizations of black holes are often used in mathematical, quantum, and numerical relativity. These include apparent, killing, trapping, isolated, dynamical, and slowly evolving horizons. All of these are closely associated with two-surfaces of zero outward null expansion. This paper reviews the traditional definition of black holes and provides an overview of some of the more recent work on alternative horizons. 
  Is the geometry of space a macroscopic manifestation of an underlying microscopic statistical structure? Is geometrodynamics - the theory of gravity - derivable from general principles of inductive inference? Tentative answers are suggested by a model of geometrodynamics based on the statistical concepts of entropy, information geometry, and entropic dynamics. The model shows remarkable similarities with the 3+1 formulation of general relativity. For example, the dynamical degrees of freedom are those that specify the conformal geometry of space; there is a gauge symmetry under 3d diffeomorphisms; there is no reference to an external time; and the theory is time reversible. There is, in adition, a gauge symmetry under scale transformations. I conjecture that under a suitable choice of gauge one can recover the usual notion of a relativistic space-time. 
  The causal set approach to quantum gravity embodies the concepts of causality and discreteness. This article explores some foundational and conceptual issues within causal set theory. 
  We present a modification to the Berger and Oliger adaptive mesh refinement algorithm designed to solve systems of coupled, non-linear, hyperbolic and elliptic partial differential equations. Such systems typically arise during constrained evolution of the field equations of general relativity. The novel aspect of this algorithm is a technique of "extrapolation and delayed solution" used to deal with the non-local nature of the solution of the elliptic equations, driven by dynamical sources, within the usual Berger and Oliger time-stepping framework. We show empirical results demonstrating the effectiveness of this technique in axisymmetric gravitational collapse simulations. We also describe several other details of the code, including truncation error estimation using a self-shadow hierarchy, and the refinement-boundary interpolation operators that are used to help suppress spurious high-frequency solution components ("noise"). 
  A first order symmetric hyperbolic tetrad formulation of the Einstein equations developed by Estabrook and Wahlquist and put into a form suitable for numerical relativity by Buchman and Bardeen (the WEBB formulation) is adapted to explicit spherical symmetry and tested for accuracy and stability in the evolution of spherically symmetric black holes (the Schwarzschild geometry). The lapse and shift which specify the evolution of the coordinates relative to the tetrad congruence are reset at frequent time intervals to keep the constant-time hypersurfaces nearly orthogonal to the tetrad congruence and the spatial coordinate satisfying a kind of minimal rate of strain condition. By arranging through initial conditions that the constant-time hypersurfaces are asymptotically hyperbolic, we simplify the boundary value problem and improve stability of the evolution. Results are obtained for both tetrad gauges (``Nester'' and ``Lorentz'') of the WEBB formalism using finite difference numerical methods. We are able to obtain stable unconstrained evolution with the Nester gauge for certain initial conditions, but not with the Lorentz gauge. 
  In this paper we investigate the possibility of measuring the post-Newtonian general relativistic gravitomagnetic Lense-Thirring effect in the Jovian system of its Galilean satellites Io, Europa, Ganymede and Callisto in view of recent developments in processing and modelling their optical observations spanning a large time interval (125 years). The present day best observations have an accuracy between several kilometers to few tens of kilometers, which is just the order of magnitude of the Lense-Thirring shifts of the orbits of the Galilean satellites over almost a century. From a comparison between analytical development and numerical integration it turns out that, unfortunately, most of the secular component of the gravitomagnetic signature is removed in the process of fitting the initial conditions. Indeed, an estimation of the magnitude of the Lense-Thirring effect in the ephemerides residuals is given; the resulting residuals have a maximum magnitude of 20 meters only (over 125 years). 
  In Randall-Sundrum models, by the use of general Noether theorem, the covariant angular momentum conservation law is obtained with the respect to the local Lorentz transformations. The angular momentum current has also superpotential and is therefore identically conserved. The space-like components $J_{ij}$ of the angular momentum for Randall-Sundrum models are zero. But the component $J_{04}$ is infinite. 
  We give a short personally-biased review on the recent progress in our understanding of gravitational radiation reaction acting on a point particle orbiting a black hole. The main motivation of this study is to obtain sufficiently precise gravitational waveforms from inspiraling binary compact stars with a large mass ratio. For this purpose, various new concepts and techniques have been developed to compute the orbital evolution taking into account the gravitational self-force. Combining these ideas with a few supplementary new ideas, we try to outline a path to our goal here. 
  The gravastar picture is an alternative model to the concept of a black hole, where there is an effective phase transition at or near where the event horizon is expected to form, and the interior is replaced by a de Sitter condensate. In this work, a generalization of the gravastar picture is explored, by considering a matching of an interior solution governed by the dark energy equation of state, $\omega\equiv p/ \rho<-1/3$, to an exterior Schwarzschild vacuum solution at a junction interface. The motivation for implementing this generalization arises from the fact that recent observations have confirmed an accelerated cosmic expansion, for which dark energy is a possible candidate. Several relativistic dark energy stellar configurations are analyzed by imposing specific choices for the mass function. The first case considered is that of a constant energy density, and the second choice, that of a monotonic decreasing energy density in the star's interior. The dynamical stability of the transition layer of these dark energy stars to linearized spherically symmetric radial perturbations about static equilibrium solutions is also explored. It is found that large stability regions exist that are sufficiently close to where the event horizon is expected to form, so that it would be difficult to distinguish the exterior geometry of the dark energy stars, analyzed in this work, from an astrophysical black hole. 
  Brane-world models offer the possibility of explaining the late acceleration of the universe via infra-red modifications to General Relativity, rather than a dark energy field. However, one also expects ultra-violet modifications to General Relativity, when high-energy stringy effects in the early universe begin to grow. We generalize the DGP brane-world model via an ultra-violet modification, in the form of a Gauss-Bonnet term in the bulk action. The combination of infra-red and ultra-violet modifications produces an intriguing cosmology. The DGP feature of late-time acceleration without dark energy is preserved, but there is an entirely new feature - there is no hot big bang in the early universe. The universe starts with finite density and pressure, from a "sudden" curvature singularity. 
  We derive an exact solution to the Einstein's equations with a stress-energy tensor corresponding to an opposite-sign scalar field, and show that such a solution describes the internal region of a rotating wormhole. We also derive an static wormhole asymptotically flat solution and match them on both regions, thus obtaining an analytic solution for the complete space-time. We explore some of the features of these solutions. 
  Degenerate geometrical configurations in quantum gravity are important to understand if the fate of classical singularities is to be revealed. However, not all degenerate configurations arise on an equal footing, and one must take into account dynamical aspects when interpreting results: While there are many degenerate spatial metrics, not all of them are approached along the dynamical evolution of general relativity or a candidate theory for quantum gravity. For loop quantum gravity, relevant properties and steps in an analysis are summarized and evaluated critically with the currently available information, also elucidating the role of degrees of freedom captured in the sector provided by loop quantum cosmology. This allows an outlook on how singularity removal might be analyzed in a general setting and also in the full theory. The general mechanism of loop quantum cosmology will be shown to be insensitive to recently observed unbounded behavior of inverse volume in the full theory. Moreover, significant features of this unboundedness are not a consequence of inhomogeneities but of non-Abelian effects which can also be included in homogeneous models. 
  We investigate the evolution equation of linear density perturbations in the Friedmann-Robertson-Walker universe with matter, radiation and the cosmological constant. The concept of solvability by quadratures is defined and used to prove that there are no "closed form" solutions except for the known Chernin, Heath, Meszaros and simple degenerate ones. The analysis is performed applying Kovacic's algorithm. The possibility of the existence of other, more general solutions involving special functions is also investigated. 
  I briefly review the current status of quantum gravity. After giving some general motivations for the need of such a theory, I discuss the main approaches in quantizing general relativity: Covariant approaches (perturbation theory, effective theory, and path integrals) and canonical approaches (quantum geometrodynamics, loop quantum gravity). I then address quantum gravitational aspects of string theory. This is followed by a discussion of black holes and quantum cosmology. I end with some remarks on the observational status of quantum gravity. 
  Due to quantum fluctuations, spacetime is probably ``foamy'' on very small scales. We propose to detect this texture of spacetime foam by looking for core-halo structures in the images of distant quasars. We find that the Very Large Telescope interferometer will be on the verge of being able to probe the fabric of spacetime when it reaches its design performance. Our method also allows us to use spacetime foam physics and physics of computation to infer the existence of dark energy/matter, independent of the evidence from recent cosmological observations. 
  This talk discusses the relation between spacetime-dependent scalars, such as couplings or fields, and the violation of Lorentz symmetry. A specific cosmological supergravity model demonstrates how scalar fields can acquire time-dependent expectation values. Within this cosmological background, excitations of these scalars are governed by a Lorentz-breaking dispersion relation. The model also contains couplings of the scalars to the electrodynamics sector leading to the time dependence of both the fine-structure parameter alpha and the theta angle. Through these couplings, the variation of the scalars is also associated with Lorentz- and CPT-violating effects in electromagnetism. 
  In this paper, we study the bulk motion of a classical extended charge in flat spacetime. A formalism developed by W. G. Dixon is used to determine how the details of such a particle's internal structure influence its equations of motion. We place essentially no restrictions (other than boundedness) on the shape of the charge, and allow for inhomogeneity, internal currents, elasticity, and spin. Even if the angular momentum remains small, many such systems are found to be affected by large self-interaction effects beyond the standard Lorentz-Dirac force. These are particularly significant if the particle's charge density fails to be much greater than its 3-current density (or vice versa) in the center-of-mass frame. Additional terms also arise in the equations of motion if the dipole moment is too large, and when the `center-of-electromagnetic mass' is far from the `center-of-bare mass' (roughly speaking). These conditions are often quite restrictive. General equations of motion were also derived under the assumption that the particle can only interact with the radiative component of its self-field. These are much simpler than the equations derived using the full retarded self-field; as are the conditions required to recover the Lorentz-Dirac equation. 
  We study the graviton propagator in euclidean loop quantum gravity, using the spinfoam formalism. We use boundary-amplitude and group-field-theory techniques, and compute one component of the propagator to first order, under a number of approximations, obtaining the correct spacetime dependence. In the large distance limit, the only term of the vertex amplitude that contributes is the exponential of the Regge action: the other terms, that have raised doubts on the physical viability of the model, are suppressed by the phase of the vacuum state, which is determined by the extrinsic geometry of the boundary. 
  Courses in introductory special and general relativity have increasingly become part of the curriculum for upper-level undergraduate physics majors and master's degree candidates. One of the topics rarely discussed is symmetry, particularly in the theory of general relativity. The principal tool for its study is the Killing vector. We provide an elementary introduction to the concept of a Killing vector field, its properties, and as an example of its utility apply these ideas to the rigorous determination of gravitational and cosmological redshifts. 
  We present a geometrical unification theory in a Kaluza-Klein approach that achieve the geometrization of a generic gauge theory bosonic component.   We show how it is possible to derive the gauge charge conservation from the invariance of the model under extra-dimensional translations and to geometrize gauge connections for spinors, thus we can introduce the matter just by free spinorial fields. Then, we present the applications to i)a pentadimensional manifold $V^{4}\otimes S^{1}$, so reproducing the original Kaluza-Klein theory, unless some extensions related to the rule of the scalar field contained in the metric and the introduction of matter by spinors with a phase dependence from the fifth coordinate, ii)a seven-dimensional manifold $V^{4}\otimes S^{1}\otimes S^{2}$, in which we geometrize the electro-weak model by introducing two spinors for any leptonic family and quark generation and a scalar field with two components with opposite hypercharge, responsible of spontaneous symmetry breaking. 
  The 3+1 formulation of scalar-tensor theories of gravity (STT) is obtained in the physical (Jordan) frame departing from the 4+0 covariant field equations. Contrary to the common belief (folklore), the new system of ADM-like equations shows that the Cauchy problem of STT is well formulated (in the sense that the whole system of evolution equations is of first order in the time-derivative). This is the first step towards a full first order (in time and space) formulation from which a subsequent hyperbolicity analysis (a well-posedness determination) can be performed. Several gauge (lapse and shift) conditions are considered and implemented for STT. In particular, a generalization of the harmonic gauge for STT allows us to prove the well posedness of the STT using a second order analysis which is very similar to the one used in general relativity. Some spacetimes of astrophysical and cosmological interest are considered as specific applications. Several appendices complement the ideas of the main part of the paper. 
  We study a model of power-law inflationary inflation using the Space-Time-Matter (STM) theory of gravity for a five dimensional (5D) canonical metric that describes an apparent vacuum. In this approach the expansion is governed by a single scalar (neutral) quantum field. In particular, we study the case where the power of expansion of the universe is $p \gg 1$. This kind of model is more successful than others in accounting for galaxy formation. 
  Addition of matter and/or radiation to a de Sitter Universe breaks the symmetry generated by four of its Killing fields. Detailed examination of the Killing equations in Robertson--Walker coordinates shows that this symmetry-breaking is rather peculiar: the Killing fields timelike components simply vanish in the Friedmann--Robertson--Walker (FRW) case. It is of common usage to relate the generators responsible for homogeneity to the space--section itself which is homogeneous: the latter is identified with the very space of translation parameters. In that case, the present-day Universe (the space--section of cosmological spacetime) is {\it not} a simple deformation or modification of the pure de Sitter space. The picture of a de Sitter Universe with cosmological constant $\Lambda $ which, by insertion of matter, would ``reduce'' to the $\Lambda + $FRW model fails, because the point sets are not the same. That picture of natural transition would be suggested by the Generalized Robertson Transformation here constructed step-by-step and which converts the de Sitter static interval into the standard FRW line element. 
  We present a new approach to describing transformations between two uniformly accelerated systems, which is based only on the general principle of relativity and the symmetry following from it. By assuming the Clock Hypothesis, we derive explicit form of the space-time transformations from an inertial system to a system uniformly accelerated to the inertial system.   Without assuming the Clock Hypothesis, we are able to derive transformation between accelerated comoving systems. For this transformation we use the proper velocity-time description of events instead of the space-time description. We derive linear Lorentz-type transformation which depend on the relative acceleration and some constant. The transformations become Galilean if this constant equal zero, which is equivalent to validity of the Clock Hypothesis. Otherwise, the proper velocity-time transformations are Lorentz type transformations. In this case, the existence of an invariant maximal acceleration is predicted. 
  We describe the effects to be expected of unwanted or voluntary deviations from the vertical of the axis of the active rotation of modern high precision experiments of the Michelson-Morley type. The theoretical description that we use is a particular implementation of the Principle of free mobility. 
  A version of the cosmological perturbation theory in general relativity (GR) is developed, where the cosmological scale factor is identified with spatial averaging of the metric determinant logarithm and the cosmic evolution acquires the pattern of a superfluid motion: the absence of "friction-type" interaction, the London-type wave function, and the Bogoliubov condensation of quantum universes. This identification keeps the number of variables of GR and leads to a new type of potential perturbations. A set of arguments is given in favor of that this "superfluid" version of GR is in agreement with the observational data. 
  A test particle falling into a classical black hole crosses the event horizon and ends up in the singularity within finite eigentime. In the `more realistic' case of a `classical' evaporating black hole, an observer falling onto a black hole observes a sudden evaporation of the hole. This illustrates the fact that the discussion of the classical process commonly found in the literature may become obsolete when the black hole has a finite lifetime. The situation is basically the same for more complex cases, e.g. where a particle collides with two merging black holes. It should be pointed out that the model used in this paper is mainly of academic interest, since the description of the physics near a black hole horizon still presents a difficult problem which is not yet fully understood, but our model provides a valuable possibility for students to enter the interesting field of black hole physics and to perform numerical calculations of their own which are not very involved from the computational point of view. 
  Within the context of modified gravity and dark energy scenarios of the accelerating universe, we study the stability of de Sitter space with respect to inhomogeneous perturbations using a gauge-independent formalism. In modified gravity the stability condition is exactly the same that one obtains from a homogeneous perturbation analysis, while the stability condition in scalar-tensor gravity is more restrictive. 
  The nonanalytic property of metric resulting from the presence of gravitomagnetic monopoles is considered. The curvature tensors, dual curvature tensors, dual Einstein tensor (and hence the gravitational field equation of gravitomagnetic matter) expressed in terms of nonanalytic metric are analyzed. It is shown that the spinor gravitomagnetic monopole may be one of the potential origins of the cosmological constant. An alternative approach to the cosmological constant problem is thus proposed based on the concept of gravitomagnetic monopole. 
  We develop a theory in which there are couplings amongst Dirac spinor, dilaton and non-Riemannian gravity and explore the nature of connection-induced dilaton couplings to gravity and Dirac spinor when the theory is reformulated in terms of the Levi-Civita connection. After presenting some exact solutions without spinors, we investigate the minimal spinor couplings to the model and in conclusion we can not find any nontrivial dilaton couplings to spinor. 
  We consider (4+D)-dimensional Kaluza-Klein cosmological model with two scaling factors, as real, p-adic and adelic quantum mechanical one. One of the scaling factors corresponds to the D-dimensional internal space, and second one to the 4-dimensional universe. In standard quantum cosmology, i.e. over the field of real numbers R, it leads to dynamical compactification of additional dimensions and to the accelerating evolution of 4-dimensional universe. We construct corresponding $p$-adic quantum model and explore existence of its p-adic ground state. In addition, we explore evolution of this model and a possibility for its adelic generalization. It is necessary for the further investigation of space-time discreteness at very short distances, i.e. in a very early universe 
  We compute the dimensionless relativistic periastron advance parameter $k$, which is measurable from the timing of relativistic binary pulsars. We employ for the computation the recently derived Keplerian-type parametric solution to the post-Newtonian (PN) accurate conservative dynamics of spinning compact binaries moving in eccentric orbits. The parametric solution and hence the parameter $k$ are applicable for the cases of \emph{simple precession}, namely, case (i), the binary consists of equal mass compact objects, having two arbitrary spins, and case (ii), the binary consists of compact objects of arbitrary mass, where only one of them is spinning with an arbitrary spin. Our expression, for the cases considered, is in agreement with a more general formula for the 2PN accurate $k$, relevant for the relativistic double pulsar PSR J0737--3039, derived by Damour and Sch\"afer many years ago, using a different procedure. 
  The decoherent histories approach to quantum theory is applied to a class of reparametrization invariant models, which includes systems described by the Klein-Gordon equation, and by a minisuperspace Wheeler-DeWitt equation. A key step in this approach is the construction of class operators characterizing the questions of physical interest, such as the probability of the system entering a given region of configuration space without regard to time. In non-relativistic quantum mechanics these class operators are given by time-ordered products of projection operators. But in reparametrization invariant models, where there is no time, the construction of the class operators is more complicated, the main difficulty being to find operators which commute with the Hamiltonian constraint (and so respect the invariance of the theory). Here, inspired by classical considerations, we put forward a proposal for the construction of such class operators for a class of reparametrization-invariant systems. They consist of continuous infinite temporal products of Heisenberg picture projection operators. We investigate the consequences of this proposal in a number of simple models and also compare with the evolving constants method. 
  Motivated by conventional gauge theories, we consider a theory of gravity in which the Einstein-Hilbert action is replaced by a term that is quadratic in the Riemann tensor. We focus on cosmological solutions to the field equations in flat, open and closed universes. The gravitational action is scale invariant, so the only matter source considered is radiation. The theory can also accommodate isotropic torsion and this generically removes singularities from the evolution equations. For general initial conditions the Hubble parameter H(t) is driven in a seemingly chaotic fashion by torsion to produce irregularly occuring inflationary regions. In the absence of torsion, the theory reproduces the standard cosmological solutions of a simple big bang model. A satisfying feature is that a cosmological constant arises naturally as a constant of integration, and does not have to be put into the Lagrangian by hand. 
  An almost-stationary gauge condition is proposed with a view to Numerical Relativity applications. The time lines are defined as the integral curves of the timelike solutions of the harmonic almost-Killing equation. This vector equation is derived by a variational principle, by minimizing the deviations from isometry. The corresponding almost-stationary gauge condition allows one to put the field equations in hyperbolic form, both in the free-evolution ADM and in the Z4 formalisms. 
  We study thermodynamics of (1+1) dimensional dilatonic black holes in global embedding Minkowski space scheme. Exploiting geometrical entropy correction we construct consistent entropy for the charged dilatonic black hole. Moreover, (1+1) dilatonic black holes with higher order terms are shown to possess (3+2) global flat embedding structures regardless of the details of the lapse function, and to yield a generic entropy. 
  Einstein-Infeld-Hoffmann method is used to solve the problem of motion of two bodies when the equations of general relativity are of the generalized form: they have been reduced to a form invariant under conformal transformations. It is proved that not only metric degrees of freedom, but also derivatives of vector $A_{\alpha}$ appearing in the generalized equations can exert influence on the motion of bodies in a certain space-time domain. This influence can account for the recently observed anomalous acceleration of spacecrafts Pioneer 10, Pioneer 11. The impact of vector $A_{\alpha}$ on the motion of bodies is interpreted as a consequence of viscosity in geometrodynamic continuum. 
  An overview of the searches for gravitational waves from radio pulsars with LIGO and GEO is given. We give a brief description of the algorithm used in these targeted searches and provide end-to-end validation of the technique through hardware injections. We report on some aspects of the recent S3/S4 LIGO and GEO search for signals from several pulsars. The gaussianity of narrow frequency bands of S3/S4 LIGO data, where pulsar signals are expected, is assessed with Kolmogorov-Smirnov tests. Preliminary results from the S3 run with a network of four detectors are given for pulsar J1939+2134. 
  Lunar Laser Ranging (LLR), which has been carried out for more than 35 years, is used to determine many parameters within the Earth-Moon system. This includes coordinates of terrestrial ranging stations and that of lunar retro-reflectors, as well as lunar orbit, gravity field, and its tidal acceleration. LLR data analysis also performs a number of gravitational physics experiments such as test of the equivalence principle, search for time variation of the gravitational constant, and determines value of several metric gravity parameters. These gravitational physics parameters cause both secular and periodic effects on the lunar orbit that are detectable with LLR. Furthermore, LLR contributes to the determination of Earth orientation parameters (EOP) such as nutation, precession (including relativistic precession), polar motion, and UT1. The corresponding LLR EOP series is three decades long. LLR can be used for the realization of both the terrestrial and selenocentric reference frames. The realization of a dynamically defined inertial reference frame, in contrast to the kinematically realized frame of VLBI, offers new possibilities for mutual cross-checking and confirmation. Finally, LLR also investigates the processes related to the Moon's interior dynamics. Here, we review the LLR technique focusing on its impact on Geodesy and Relativity. We discuss the modern observational accuracy and the level of existing LLR modeling. We present the near-term objectives and emphasize improvements needed to fully utilize the scientific potential of LLR. 
  We study a locally anisotropic model of General Relativity in the framework of a more general geometrical structure than the Riemannian one. In this model the observable anisotropy of the CMBR (WMAP) is represented by a tensor of anisotropy and it is included in the metric structure of space-time. As well, some interesting special cases of spaces are considered. 
  The Doppler tracking data from two deep-space spacecraft, Pioneer 10 and 11, show an anomalous blueshift, which has been dubbed the "Pioneer anomaly". The effect is most commonly interpreted as a real deceleration of the spacecraft - an interpretation that faces serious challenges from planetary ephemerides. The Pioneer anomaly could as well indicate an unknown effect on the radio signal itself. Several authors have made suggestions how such a blueshift could be related to cosmology. We consider this interpretation of the Pioneer anomaly and study the impact of an anomalous blueshift on the Laser Interferometer Space Antenna (LISA), a planned joint ESA-NASA mission aiming at the detection of gravitational waves. The relative frequency shift (proportional to the light travel time) for the LISA arm length is estimated to 10E-16, which is much bigger than the expected amplitude of gravitational waves. The anomalous blueshift enters the LISA signal in two ways, as a small term folded with the gravitational wave signal, and as larger term at low frequencies. A detail analysis shows that both contributions remain undetectable and do not impair the gravitational-wave detection. This suggests that the Pioneer anomaly will have to be tested in the outer Solar system regardless if the effect is caused by an anomalous blueshift or by a real force. 
  In this paper, we utilize the teleparallel gravity analogs of the energy and momentum definitions of Bergmann-Thomson and Landau-Lifshitz in order to explicitly evaluate the energy distribution(due to matter and fields including gravity) based on the Bonnor space-time. it is shown that for a stationary beam of light, these energy-momentum definitions give the same result. Furthermore, this result supports the viewpoint of Cooperstock and also agree with the previous works by Bringley and Gad. 
  The paper is purposed to elaborate the problem of gravitational energy localization in de Sitter(dS) C-space-time (the C space-time in a background with a cosmological constant $\Lambda$). In this connection, using the energy-momentum definition of Einstein, we find the same energy in both general relativity and tele-parallel gravity. 
  In this review article I attempt to summarise past and present-ongoing-work on the problem of the inspiral of a small body in the gravitational field of a much more massive Kerr black hole. Such extreme mass ratio systems, expected to occur in galactic nuclei, will constitute prime sources of gravitational radiation for the future LISA gravitational radiation detector. The article's main goal is to provide a survey of basic celestial mechanics in Kerr spacetime and calculations of gravitational waveforms and backreaction on the small body's orbital motion, based on the traditional `flux-balance' method and the Teukolsky black hole perturbation formalism. 
  The measurement of spin effects in general relativity has recently taken centre stage with the successfully launched Gravity Probe B experiment coming toward an end, coupled with recently reported measurements using laser ranging. Many accounts of these experiments have been in terms of frame-dragging. We point out that this terminology has given rise to much confusion and that a better description is in terms of spin-orbit and spin-spin effects. In particular, we point out that the de Sitter precession (which has been mesured to a high accuracy) is also a frame-dragging effect and provides an accurate benchmark measurement of spin-orbit effects which GPB needs to emulate. 
  We consider the perturbations of the massive vector field around Schwarzschild black hole, (generally, with non-vanishing $\Lambda$ - term). The monopole massive vector perturbation equations can be reduced to a single wave-like equation. We have proved the stability against these perturbations and investigated the quasinormal spectrum. The quasinormal behaviour for Schwarzschild black hole is quite unexpected: the fundamental mode and all higher overtones shows totally different dependence on the mass of the field $m$: as $m$ is increasing, the damping rate of the fundamental mode is decreasing, what results in appearing of the infinitely long living modes, while, on contrary, damping rate of all higher overtones are increasing, and their real oscillation frequencies gradually go to tiny values. Thereby, for all higher overtones, almost non-oscillatory, damping modes can exist. In the limit of asymptotically high damping, $Re \omega$ goes to $ln3/(8 \pi M)$, while imaginary part shows equidistant behaviour with spacing $Im \omega_{n+1}- Im \omega_{n}=i/4M$. In addition, we have found quasinormal spectrum of massive vector field for Schwarzschild-anti-de Sitter black hole. 
  Bi-metricity and Hawking radiation are exhibit in non-relativistic moving magnetohydrodynamics (MHD) plasma medium generating two Riemannian effective spacetimes. The first metric is a flat metric although the speed of "light" is given by a time dependent signal where no Hawking radiation or effective black holes are displayed. This metric comes from a wave equation which the scalar function comes from the scalar potential of the background velocity of the fluid and depends on the perturbation of the magnetic background field. The second metric is an effective spacetime metric which comes from the perturbation of the background MHD fluid. This Riemann metric exhibits a horizon and Hawking radiation which can be expressed in terms of the background constant magnetic field. The effective velocity is given Alfven wave velocity of plasma physics. The effective black hole found here is analogous to the optical black hole in moving dielectrics found by De Lorenci et al [Phys. Rev. D (2003)] where bi-metricity and Hawking radiation in terms of the electric field are found. 
  Using quadratic spinor techniques we demonstrate that the Immirzi parameter can be expressed as ratio between scalar and pseudo-scalar contributions in the theory and can be interpreted as a measure of how Einstein gravity differs from a generally constructed covariant theory for gravity. This interpretation is independent of how gravity is quantized. One of the important advantage of deriving the Immirzi parameter using the quadratic spinor techniques is to allow the introduction of renormalization scale associated with the Immirzi parameter through the expectation value of the spinor field upon quantization. 
  A modified model of gravity with additional positive and negative powers of the scalar curvature, $R$, in the gravitational action is studied. This is done using the Palatini variational principle. It is demonstrated that using such a model might prove useful to explain both the early time inflation and the late time cosmic acceleration without the need for any form of dark energy. 
  Reconsideration of the Regge-Wheeler equation is processed by using the Painlev\'{e} coordinate and "good" timelier to define the initial time. We find that: the Regge-Wheeler equation could has positive imaginary frequency. Because the Regge-Wheeler equation is the odd (angular) perturbation to the Schwarzschild black hole, the conclusion is that the Schwarzschild black hole is unstable with respect to the rotating perturbation. 
  We prove a global in time existence theorem, for the initial value problem for the Einstein-Boltzmann system, with arbitrarily large initial data, in the homogeneous case, in a Bianchi type I space-time 
  We use expansion-normalised variables to investigate the Bianchi type VII$_0$ model with a tilted $\gamma$-law perfect fluid. We emphasize the late-time asymptotic dynamical behaviour of the models and determine their asymptotic states. Unlike the other Bianchi models of solvable type, the type VII$_0$ state space is unbounded. Consequently we show that, for a general non-inflationary perfect fluid, one of the curvature variables diverges at late times, which implies that the type VII$_0$ model is not asymptotically self-similar to the future. Regarding the tilt velocity, we show that for fluids with $\gamma<4/3$ (which includes the important case of dust, $\gamma=1$) the tilt velocity tends to zero at late times, while for a radiation fluid, $\gamma=4/3$, the fluid is tilted and its vorticity is dynamically significant at late times. For fluids stiffer than radiation ($\gamma>4/3$), the future asymptotic state is an extremely tilted spacetime with vorticity. 
  Equilibrium states of black holes can be modelled by isolated horizons. If the intrinsic geometry is spherical, they are called type I while if it is axi-symmetric, they are called type II. The detailed theory of geometry of quantum type I horizons and the calculation of their entropy can be generalized to type II, thereby including arbitrary distortions and rotations. The leading term in entropy of large horizons is again given by 1/4th of the horizon area for the same value of the Barbero-Immirzi parameter as in the type I case. Ideas and constructions underlying this extension are summarized. 
  Analogue black holes in non-Riemannian effective spacetime of moving vortical plasmas described by moving magnetohydrodynamic (MHD) flows. This example is an extension of acoustic torsion recently introduced in the literature (Garcia de Andrade,PRD(2004),7,64004), where now the presence of artificial black holes in moving plasmas is obtained by the presence of an horizon in the non-Riemannian spacetime. Hawking radiation is computed in terms of the background magnetic field and the magnetic permeability. The metric is singular although Cartan analogue torsion is not necessarily singular. The effective Lorentz invariance is shown to be broken due to the presence of effective torsion in strong analogy with the Riemann-Cartan gravitational case presented recently by Kostelecky (PRD 69,2004,105009). 
  We analyze a variety of Weyl invariant dynamical problems in three dimensions. 
  We perform in this paper a statefinder diagnostic to a dark energy model with two scalar fields, called "quintom", where one of the scalar fields has a canonical kinetic energy term and the other has a negative one. Several kinds of potentials are discussed. Our results show that the statefinder diagnostic can differentiate quintom model with other dark energy models. 
  Models of Lorentz/diffeomorphism violation frequently make use of a time-dependent scalar field. We investigate space-times produced by such a field. 
  The black hole as the thermodynamical system in equilibrium possesses the periodicity of motion in imaginary time, that allows us to formulate the quasi-classical rule of quantization. The rule yields the equidistant spectrum for the entropy of non-rotating black holes as well as for the appropriately scaled entropy in the case of rotation. We clarify and discuss a role of quasi-normal modes. 
  This bibliography attempts to give a comprehensive overview of all the literature related to what is known as the Ashtekar-Sen connection and the Rovelli-Smolin loop variables, from which the program currently known as Loop Quantum Gravity emerged. The original version was compiled by Peter Huebner in 1989, and it has been subsequently updated by Gabriela Gonzalez, Bernd Bruegmann, Monica Pierri, Troy Schilling, Christopher Beetle, Alejandro Corichi and Alberto Hauser. The criteria for inclusion in this list are the following: A paper in the classical theory is included if it deals with connection variables for gravity. If the paper is in the quantum domain, it is included when it is related directly with gravity using connection/loop variables, with mathematical aspects of connections, or when it introduces techniques that might be useful for the construction of the (loop) quantum theory of gravity. Information about additional literature, new preprints, and especially corrections are always welcome. 
  We investigate a model of the interacting holographic dark energy with cold dark matter (CDM). If the holographic energy density decays into CDM, we find two types of the effective equation of state. In this case we have to use the effective equations of state ($\omega^{\rm eff}_{\rm \Lambda}$) instead of the equation of state ($\omega_{\rm \Lambda})$. For a fixed ratio of two energy densities, their effective equations of state are given by the same negative constant. Actually, the cosmic anti-friction arisen from the vacuum decay process may induce the acceleration with $\omega^{\rm eff}_{\rm \Lambda}<-1/3$. For a variable ratio, their effective equations of state are slightly different, but they approach the same negative constant in the far future. Consequently, we show that such an interacting holographic energy model cannot accommodate a transition from the dark energy with $\omega^{\rm eff}_{\rm \Lambda}\ge-1$ to the phantom regime with $\omega^{\rm eff}_{\rm \Lambda}<-1$. 
  We examine the benefits of performing a joint LIGO--Virgo search for transient signals. We do this by adding burst and inspiral signals to 24 hours of simulated detector data. We find significant advantages to performing a joint coincidence analysis, above either a LIGO only or Virgo only search. These include an increased detection efficiency, at a fixed false alarm rate, to both burst and inspiral events and an ability to reconstruct the sky location of a signal. 
  We have studied the closed universe model with the variable cosmological term, which is presented as a sum of two terms: Lambda=Lambda_0 -k R. First term Lambda_0 is a constant and it is describing a sum of quantum field's zero oscillations. In the geometrical sense it determines an own constant curvature of the space-time. The second term is variable, it is proportional to curvature of the space-time and can be interpretable as response of vacuum to curvature of the space-time. In the framework of such model we have investigated the time dynamic of Lambda-term and try to explain a mechanism which to give rise to a large value of Lambda hence leading to the "cosmological constant problem". It was studied models for different dark matter equation of state: p=0 and p=varepsilon. It was shown that the presence of a variable Lambda-term alters the evolution of the scale factor a in the very early epoch. In the context of this model, a severe decrease of the module of vacuum energy density at the primitive stage of Universe's development is explained. The numerical results for the value of the Lambda-term agree with theoretical predictions for the early epoch and with recent observation data. 
  In the Tolman model there exist two quite different branches of solutions - generic Lemaitre-Tolman-Bondi (LTB) ones and T-spheres as a special case. We show that, nonetheless, T-spheres can be obtained as a limit of the class of LTB solutions having no origin and extending to infinity with the areal radius approaching constant. It is shown that all singularities of T-models are inherited from those of corresponding LBT solutions. In doing so, the disc type singularity of a T-sphere is the analog of shell-crossing. 
  In this work we investigate the behavior of three-dimensional (3D) cosmological models. The simulation of inflationary and dark-energy-dominated eras are among the possible results in these 3D formulations; taking as starting point the results obtained by Cornish and Frankel.   Motivated by those results, we investigate, first, the inflationary case where we consider a two-constituent cosmological fluid: the scalar field represents the hypothetical inflaton which is in gravitational interaction with a matter/radiation contribution. For the description of an old universe, it is possible to simulate its evolution starting with a matter dominated universe that faces a decelerated/accelerated transition due to the presence of the additional constituent (simulated by the scalar field or ruled by an exotic equation of state) that plays the role of dark energy. We obtain, through numerical analysis, the evolution in time of the scale factor, the acceleration, the energy densities, and the hydrostatic pressure of the constituents. The alternative scalar cosmology proposed by Cornish and Frankel is also under investigation in this work. In this case an inflationary model can be constructed when another non-polytropic equation of state (the van der Waals equation) is used to simulate the behavior of an early 3D universe. 
  A nonlinear scalar field theory from which an effective metric can be deduced is considered. This metric is shown to be compatible with requirements of general relativity. It is demonstrated that there is a class of solutions which fulfill both the nonlinear field equation as the Einstein equations for this metric. 
  Based on the Mathisson-Papapetrou-Dixon (MPD) equations and the Vaidya metric, the motion of a spinning point particle orbiting a non-rotating star while undergoing radiation-induced gravitational collapse is studied in detail. A comprehensive analysis of the orbital dynamics is performed assuming distinct central mass functions which satisfy the weak energy condition, in order to determine a correspondence between the choice of mass function and the spinning particle's orbital response, as reflected in the gravitational waves emitted by the particle. The analysis presented here is likely most beneficial for the observation of rotating solar mass black holes or neutron stars in orbit around intermediate-sized Schwarzschild black holes undergoing radiation collapse. The possibility of detecting the effects of realistic mass accretion based on this approach is considered. While it seems unlikely to observe such effects based on present technology, they may perhaps become observable with the advent of future detectors. 
  In this paper, using Einstein and Landau and Lifshitz's energy-momentum complexes in both general relativity and teleparallel gravity, we calculate the total energy distribution(due to matter and fields including gravitation) associated with Locally Rotationally Symmetric(LRS) Bianchi type II cosmological models. We show that energy density in these different gravitation theories is the same, so agree with each other. We obtain that the total energy is zero. This result agrees with previous works of Cooperstock and Israelit, Rosen, Johri et al., Banerjee and Sen, Vargas, Aydogdu and Salti. Moreover, our result supports the viewpoints of Albrow and Tryon. 
  We consider how the mass of the black hole decreases by the Hawking radiation in the Vaidya spacetime, using the concept of dynamical horizon equation, proposed by Ashtekar and Krishnan. Using the formula for the change of the dynamical horizon, we derive an equation for the mass incorporating the Hawking radiation. It is shown that final state is the Minkowski spacetime in our particular model. 
  We present a simple method to calculate certain sums of the eigenvalues of the volume operator in loop quantum gravity. We derive the asymptotic distribution of the eigenvalues in the classical limit of very large spins which turns out to be of a very simple form. The results can be useful for example in the statistical approach to quantum gravity. 
  The fundamental physical object of the Global Time Theory is a three-dimensional curved space dynamically developing in global time. The equations of its dynamics are derived from the Lagrangian, and the Hamiltonian of ravitation turns out to be nonzero. The General Relativity solutions are shown to be a subset of the GTT solutions with zero energy density. In Global time Theory, the quantum theory of gravitation can be built on the basis of the Schredinger equation, as for other fields. The quantum model of the Big Bang is presented in some detailes. 
  General relativity and quantum mechanics are conflicting theories. The seeds of discord are the fundamental principles on which these theories are grounded. General relativity, on one hand, is based on the equivalence principle, whose strong version establishes the local equivalence between gravitation and inertia. Quantum mechanics, on the other hand, is fundamentally based on the uncertainty principle, which is essentially nonlocal in the sense that a particle does not follow one trajectory, but infinitely many trajectories, each one with a different probability. This difference precludes the existence of a quantum version of the strong equivalence principle, and consequently of a quantum version of general relativity. Furthermore, there are compelling experimental evidences that a quantum object in the presence of a gravitational field violates the weak equivalence principle. Now it so happens that, in addition to general relativity, gravitation has an alternative, though equivalent description, given by teleparallel gravity, a gauge theory for the translation group. In this theory torsion, instead of curvature, is assumed to represent the gravitational field. These two descriptions lead to the same classical results, but are conceptually different. In general relativity, curvature geometrizes the interaction, while torsion in teleparallel gravity acts as a force, similar to the Lorentz force of electrodynamics. Because of this peculiar property, teleparallel gravity describes the gravitational interaction without requiring any of the equivalence principles. The replacement of general relativity by teleparallel gravity may, in consequence, lead to a conceptual reconciliation of gravitation with quantum mechanics. 
  We represent and discuss a theory of gravitational holography in which all the involved waves; subject, reference and illuminator are gravitational waves (GW). Although these waves are so weak that no terrestrial experimental set-ups, even the large LIGO, VIRGO, GEO and TAMA facilities, were able up to now to directly detect them they are, nevertheless, known under certain conditions (such as very small wavelengths) to be almost indistinguishable (see P. 962 in Ref. 18) from their analogue electromagnetic waves (EMW). We, therefore theoretically, show, using the known methods of optical holography and taking into account the very peculiar nature of GW, that it is also possible to reconstruct subject gravitational waves. 
  We study a two-level atom in interaction with a real massless scalar quantum field in a spacetime with a reflecting boundary. The presence of the boundary modifies the quantum fluctuations of the scalar field, which in turn modifies the radiative properties of atoms. We calculate the rate of change of the mean atomic energy of the atom for both inertial motion and uniform acceleration. It is found that the modifications induced by the presence of a boundary make the spontaneous radiation rate of an excited inertial atom to oscillate near the boundary and this oscillatory behavior may offer a possible opportunity for experimental tests for geometrical (boundary) effects in flat spacetime. While for accelerated atoms, the transitions from ground states to excited states are found to be possible even in vacuum due to changes in the vacuum fluctuations induced by both the presence of the boundary and the acceleration of atoms, and this can be regarded as an actual physical process underlying the Unruh effect. 
  It is argued that substantial portions of both Newtonian particle mechanics and general relativity can be viewed as relational (rather than absolute) theories. I furthermore use the relational particle models as toy models to investigate the problem of time in closed-universe canonical quantum general relativity. I consider thus in particular the internal time, semiclassical and records tentative resolutions of the problem of time. 
  The geodesics of a spacetime seldom coincide with those of an embedded submanifold of codimension one. We investigate this issue for higher-dimensional general relativity-like models, firstly in the simpler case without branes to isolate which features are already present, and then in the more complicated case with branes. The framework in which we consider branes is general enough to include asymmetric braneworlds but not thick branes. We apply our results on geodesics to study both the equivalence principle and cosmological singularities. Among the models we study these considerations favour $Z_2$ symmetric braneworlds with a negative bulk cosmological constant. 
  It is shown that the internal solution of the Schwarzschild type in the Relativistic Theory of Gravitation does not lead to an {infinite pressure} inside a body as it holds in the General Theory of Relativity. This happens due to the graviton rest mass, because of the stopping of the time slowing down. 
  Transonic accretion onto astrophysical objects is a unique example of analogue black hole realized in nature. In the framework of acoustic geometry we study axially symmetric accretion and wind of a rotating astrophysical black hole or of a neutron star assuming isentropic flow of a fluid described by a polytropic equation of state. In particular we analyze the causal structure of multitransonic configurations with two sonic points and a shock. Retarded and advanced null curves clearly demonstrate the presence of the acoustic black hole at regular sonic points and of the white hole at the shock. We calculate the analogue surface gravity and the Hawking temperature for the inner and the outer acoustic horizons. 
  Thermodynamical fluctuations of temperature in mirrors may produce surface fluctuations not only through thermal expansion in mirror body but also through thermal expansion in mirror coating. We analyze the last "surface" effect which can be larger than the first "volume" one due to larger thermal expansion coefficient of coating material and smaller effective volume. In particular, these fluctuations may be important in laser interferometric gravitational antennae. 
  We study the cosmological and weak-field properties of theories of gravity derived by extending general relativity by means of a Lagrangian proportional to $R^{1+\delta}$. This scale-free extension reduces to general relativity when $\delta \to 0$. In order to constrain generalisations of general relativity of this power class we analyse the behaviour of the perfect-fluid Friedmann universes and isolate the physically relevant models of zero curvature. A stable matter-dominated period of evolution requires $\delta >0$ or $\delta <-1/4$. The stable attractors of the evolution are found. By considering the synthesis of light elements (helium-4, deuterium and lithium-7) we obtain the bound $-0.017<\delta <0.0012.$ We evaluate the effect on the power spectrum of clustering via the shift in the epoch of matter-radiation equality. The horizon size at matter--radiation equality will be shifted by $\sim 1%$ for a value of $\delta \sim 0.0005.$ We study the stable extensions of the Schwarzschild solution in these theories and calculate the timelike and null geodesics. No significant bounds arise from null geodesic effects but the perihelion precession observations lead to the strong bound $\delta =2.7\pm 4.5\times 10^{-19}$ assuming that Mercury follows a timelike geodesic. The combination of these observational constraints leads to the overall bound $0\leq \delta <7.2\times 10^{-19}$ on theories of this type. 
  We prove that the number of odd parity instabilities of the n-th SU(2) Einstein-Yang-Mills-Dilaton soliton and black hole equals n. 
  The problem of finding the momentum 4-vector(due to matter and fields including gravitation) associated with static Bianchi-type space-times is studied in tele-parallel gravity. 
  We study the stability of static, spherically symmetric, traversable wormholes with or without an electric charge, existing due to conformal continuations in a class of scalar-tensor theories with zero scalar field potential (so that Penney's or Fisher's well-known solutions hold in the Einstein conformal frame). Specific examples of such wormholes are those with nonminimally (e.g., conformally) coupled scalar fields. All boundary conditions for scalar and metric perturbations are taken into account. All such wormholes with zero or small electric charge are shown to be unstable under spherically symmetric perturbations. The instability is proved analytically with the aid of the theory of self-adjoint operators in Hilbert space and is confirmed by numerical computations. 
  A new theorem for black holes is found. It is called the horizon mass theorem. The horizon mass is the mass which cannot escape from the horizon of a black hole. For all black holes: neutral, charged or rotating, the horizon mass is always twice the irreducible mass observed at infinity. Previous theorems on black holes are: 1. the singularity theorem, 2. the area theorem, 3. the uniqueness theorem, 4. the positive energy theorem. The horizon mass theorem is possibly the last general theorem for classical black holes. It is crucial for understanding Hawking radiation and for investigating processes occurring near the horizon. 
  In this review, the fundamental structure of loop quantum gravity is presented pedagogically. Our main aim is to help non-experts to understand the motivations, basic structures, as well as general results. We will focus on the theoretical framework itself, rather than its applications, and do our best to write it in modern and precise langauge while keeping the presentation accessible for beginners. After reviewing the classical connection dynamical formalism of general relativity, as a foundation, the construction of kinematical Ashtekar-Isham-Lewandowski representation is introduced in the content of quantum kinematics. In the content of quantum dynamics, we mainly introduce the construction of a Hamiltonian constraint operator and the master constraint project. It should be noted that this strategy of quantizing gravity can also be extended to obtain other background independent quantum gauge theories. There is no divergence within this background independent and diffeomorphism invariant quantization programme of matter coupled to gravity. 
  In certain string inspired higher dimensional cosmological models it has been conjectured that there is generic, chaotic oscillating behavior near the initial singularity -- the Kasner parameters which characterize the asymptotic form of the metric "jump" between different, locally constant values and exhibit a never-ending oscillation as one approaches the singularity. In this paper we investigate a class of cosmological solutions with form fields and diagonal metrics which have a "maximal" number of composite electric S-branes. We look at two explicit examples in D=4 and D=5 dimensions that do not have chaotic oscillating behavior near the singularity. When the composite branes are replaced by non-composite branes chaotic oscillating 
  By comparing with the most recent experimental results, we point out the model dependence of the present bounds on the anisotropy of the speed of light. In fact, by replacing the CMB with a class of preferred frames that can better account for the experimental data, one obtains values of the RMS anisotropy parameter (1/2 -beta + delta) that are one order of magnitude larger than the presently quoted ones. The resulting non-zero anisotropy can be understood starting from the observation that the speed of light in the Earth's gravitational field is not the basic parameter c=1 entering Lorentz transformations. In this sense, light can propagate isotropically only in one `preferred' frame. 
  This paper considers 4-dimensional manifolds upon which there is a Lorentz metric, h, and a symmetric connection and which are originally assumed unrelated. It then derives sufficient conditions on the metric and connection (expressed through the curvature tensor) for the connection to be the Levi-Civita connection of some (local) Lorentz metric, g, and calculates the relationship between g and h. Some examples are provided which help to assess the strength of the sufficient conditions derived. 
  We study the isotropisation of the homogeneous but anisotropic Bianchi class A models in presence of a minimally coupled and massive scalar field with or without a perfect fluid. To this end, we use the Hamiltonian formalism of Arnowitt, Deser and Misner(ADM) and the dynamical systems analysis methods. Our results allow to define three kinds of isotropisation called class 1, 2 and 3. We have specifically studied the class 1 and obtained some general constraints on scalar-tensor theories which are necessary conditions for isotropisation. The asymptotical behaviors of the metric functions and potential in the neighborhood of isotropy have also been determined when the isotropic state is reached sufficiently quickly. We show that the scalar field responsible for isotropisation may be quintessent and that the presence of curvature favor a late times acceleration and quintessence. Some applications are made with the well known exponential law potential by using our theoretical results but also by help of numerical analysis. The isotropisation process with a power law potential is also explored. We think this work represents a framework able to guide some future researches on the isotropisation of homogeneous models in scalar-tensor theories and we argue by discussing briefly about some recent results we have obtained in presence of a non minimally coupled scalar field or several scalar fields. 
  We study the homogeneous but anisotropic cosmological models of Bianchi in presence of a massive scalar field using the ADM Hamiltonian formalism. We begin to describe the main steps to find the ADM Hamiltonian of the General Relativity with a massive scalar field and then we study the dynamics of the flat Bianchi type $I$ anisotropic Universe according to initial and final values of this Hamiltonian and sign of the potential. After a brief recall of the conditions necessary to isotropise an anisotropic Bianchi class A model with such a field, we extend them to a non minimally coupled scalar field by using a conformal transformation of the metric which casts the General Relativity with a scalar field into a scalar-tensor theory. The new line element then corresponds to the so-called Brans-Dicke frame, the former one being the Einstein frame. Then, we study the isotropisation process of the Bianchi class A model when we consider the low energy form of the string theory without its antisymmetric tensor and the Brans-Dicke theory with some exponential or power laws of the scalar field for the potential. Finally, assuming an isotropic Universe such as all the metric functions behave as some power or exponential laws of the proper time, we find the conditions such that the gravitation function and the potential of the scalar field are bounded as it is observed today, and compare them with the necessary conditions for isotropy. 
  In previous works, we studied the isotropisation of Bianchi class A models with a minimally coupled scalar field. In this paper, we extend these results to the case of a non minimally coupled one. We first make the calculations in the Einstein frame where the scalar field is minimally coupled to the curvature but non minimally coupled to the perfect fluid. Then, we use a conformal transformation to generalise our results to a scalar field non minimally coupled to the curvature. Universe isotropisation for the Brans-Dicke and low energy string theories are studied. 
  We use numerical integrations to study the asymptotical behaviour of a homogeneous but anisotropic Bianchi type IX model in General Relativity with a massive scalar field. As it is well known, for a Brans-Dicke theory, the asymptotical behaviour of the metric functions is ruled only by the Brans-Dicke coupling constant with respect to the value -3/2. In this paper we examine if such a condition still exists with a massive scalar field. We also show that, contrary to what occurs for a massless scalar field, the singularity oscillatory approach may exist in presence of a massive scalar field having a positive energy density. 
  We extend the Gross-Perry-Yaffe approach of hot flat space instability to Minkowski space. This is done by a saddle point approximation of the partition function in a Schwarzschild wormhole background which is coincident with an eternal black hole. The appearance of an instability in the whole manifold is here interpreted as a black hole pair creation. 

  In this paper we have considered the multidimensional cosmological implications of a decay law for $\Lambda$ term that is proportional to $\beta \frac{\ddot {a}}{a}$, where $\beta$ is a constant and $a$ is the scale factor of RW-space time. We discuss the cosmological consequences of a model for the vanishing pressure for the case $k=0$. It has been observed that such models are compatible with the result of recent observations and cosmological term $\Lambda$ gradually reduces as the universe expands. In this model $\Lambda$ varies as the inverse square of time, which matches its natural units. The proper distance, the luminosity distance-redshift, the angular diameter distance-redshift, and look back time-redshift for the model are presented in the frame work of higher dimensional space time. The model of the Freese {\it et al.} ({\it Nucl. Phys. B} {\bf 287}, 797 (1987)) for $n=2$ is retrieved for the particular choice of $A_{0}$ and also Einstein-de Sitter model is obtained for $A_{0} = {2/3}$. This work has thus generalized to higher dimensions the well-know result in four dimensional space time. It is found that there may be significant difference in principle at least, from the analogous situation in four dimensional space time. 
  In homogeneous cosmologies, quantum geometry effects lead to a resolution of the classical singularity without having to invoke special boundary conditions at the singularity or introduce ad-hoc elements such as unphysical matter. The same effects are shown to lead to a resolution of the Schwarzschild singularity. The resulting quantum extension of space-time is likely to have significant implications to the black hole evaporation process. Similarities and differences with the situation in quantum geometrodynamics are pointed out. 
  We study the Lorentzian analogues of the squashed 3-sphere, namely 2+1 dimensional anti-de Sitter space, squashed or stretched along fibres that are either spacelike or timelike. The causal structure, and the property of being an Einstein--Weyl space, depend critically on whether we squash or stretch. We argue that squashing, and stretching, completely destroys the conformal boundary of the unsquashed spacetime. As a physical application we observe that the near horizon geometry of the extremal Kerr black hole, at constant Boyer--Lindquist latitude, is anti-de Sitter space squashed along compactified spacelike fibres. 
  It is shown that the definition for the volume of stationary black holes advocated in hep-th/0508108 readily generalizes to the case of dilaton gravity in D=2. The dilaton field is included as part of the measure. A feature observed in D=3 and 4 has been the impossibility to obtain infinite volume while retaining finite area without encountering some kind of pathology. It is demonstrated that this also holds in D=2. Consistency with spherically reduced gravity is shown. For the Witten black hole it is found that the area is proportional to the volume. 
  In this paper we consider the Kantowski-Sachs space-time in Ashtekar variables and the quantization of this space-time starting from the complete loop quantum gravity theory. The Kanthowski-Sachs space-time coincides with the Schwarzschild black hole solution inside the horizon. By studying this model we can obtain information about the black hole singularity and about the dynamics across the point r=0. We studied this space-time in ADM variables in two previous papers where we showed that the classical black hole singularity disappears in quantum theory. In this work we study the same model in Ashtekar variables and we obtain a regular space-time inside the horizon region and that the dynamics can be extend further the classical singularity. 
  The spherically symmetric steady accretion of polytropic perfect fluids onto a black hole is the simplest flow model that can demonstrate the effects of backreaction. The analytic and numerical investigation reveals that backreaction keeps intact most of the characteristics of the sonic point. For any such system, with the only free parameter being the relative abundance of the fluid, the mass accretion rate achieves maximal value when the mass of the fluid is universally 1/3 of the total mass. 
  The Barrett-Crane models of Lorentzian quantum gravity are a family of spin foam models based on the Lorentz group. We show that for various choices of edge and face amplitudes, including the Perez-Rovelli normalization, the amplitude for every triangulated closed 4-manifold is a non-negative real number. Roughly speaking, this means that if one sums over triangulations, there is no interference between the different triangulations. We prove non-negativity by transforming the model into a ``dual variables'' formulation in which the amplitude for a given triangulation is expressed as an integral over three copies of hyperbolic space for each tetrahedron. Then we prove that, expressed in this way, the integrand is non-negative. In addition to implying that the amplitude is non-negative, the non-negativity of the integrand is highly significant from the point of view of numerical computations, as it allows statistical methods such as the Metropolis algorithm to be used for efficient computation of expectation values of observables. 
  In this paper we introduce a numerical approximation technique to obtain pre-classical solutions to models of loop quantum gravity. In particular, we apply the technique to vacuum Bianchi I cosmological models and recover known solutions. We also present a pre-classical solution to the Bianchi I LRS model with cosmological constant, which has not appeared elsewhere. 
  A gravity-driven mechanism (``objective reduction'') proposed to explain quantum state reduction is analyzed in light of the possible existence of large extra dimensions in the ADD scenario. By calculating order-of-magnitude estimates for nucleon superpositions, it is shown that if the mechanism at question is correct, constraints may be placed on the number and size of extra dimensions. Hence, measurement of superposition collapse times ({\it e.g.} through diffraction or reflection experiments) could represent a new probe of extra dimensions. The influence of a time-dependent gravitational constant on the gravity-driven collapse scheme with and without the presence of extra dimensions is also discussed. 
  We analyze the observational and theoretical constraints on ``Einstein-aether theory", a generally covariant theory of gravity coupled to a dynamical, unit, timelike vector field that breaks local Lorentz symmetry. The results of a computation of the remaining post-Newtonian parameters are reported. These are combined with other results to determine the joint post-Newtonian, vacuum-Cerenkov, nucleosynthesis, stability, and positive-energy constraints. All of these constraints are satisfied by parameters in a large two-dimensional region in the four-dimensional parameter space defining the theory. 
  We examine the gyration motion of a charged particle, viewed from a reference observer falling along the Z axis into a Schwarzschild black hole. It is assumed that the magnetic field is constant and uniform along the Z axis, and that the particle has a circular orbit in the X-Y plane far from the gravitational source. When the particle as well as the reference observer approaches the black hole, its orbit is disrupted by the tidal force. The final plunging velocity increases in the non-relativistic case, but decreases if the initial circular velocity exceeds a critical value, which is approximately 0.7c. This toy model suggests that disruption of a rapidly rotating star due to a velocity-dependent tidal force may be quite different from that of a non-relativistic star. The model also suggested that collapse of the orbit after the disruption is slow in general, so that the particle subsequently escapes outside the valid Fermi coordinates. 
  We have found new anisotropic vacuum solutions for the scale-invariant gravity theories which generalise Einstein's general relativity to a theory derived from the Lagrangian $R^{1+\delta}$. These solutions are expanding universes of Kasner form with an initial spacetime singularity at $t=0 $ and exist for $-1/2<\delta <1/4$ but possess different Kasner index relations to the classic Kasner solution of general relativity to which they reduce when $\delta =0$. These solutions are unperturbed by the introduction of non-comoving perfect-fluid matter motions if $p<\rho $ on approach to the singularity and should not exhibit an infinite sequence of chaotic Mixmaster oscillations when $\delta >0$. 
  The purpose of the present work is to extend the earlier results for asymptotically flat vacuum space-times to asymptotically flat solutions of the Einstein-Maxwell equations. Once again, in this case, we get a class of asymptotically shear-free null geodesic congruences depending on a complex world-line in the same four-dimensional complex space. However in this case there will be, in general, two distinct but uniquely chosen world-lines. One of which can be assigned as the complex center-of- charge while the other could be called the complex center of mass. Rather than investigating the situation where there are two distinct complex world-lines, we study instead the special degenerate case where the two world-lines coincide, i.e., where there is a single unique world-line. This mimics the case of algebraically special Einstein-Maxwell fields where the degenerate principle null vector of the Weyl tensor coincides with a Maxwell principle null vector. Again we obtain equations of motion for this world-line - but explicitly found here only in an approximation. Though there are ambiguities in assigning physical meaning to different terms it appears as if reliance on the Kerr and charged Kerr metrics and classical electromagnetic radiation theory helps considerably in this identification. In addition, the resulting equations of motion appear to have many of the properties of a particle with intrinsic spin and an intrinsic magnetic dipole moment. At first order there is even the classical radiation-reaction term 2/3{q^{2}}{c^{-3}}ddot{v}, now obtained without any use of the Lorentz force law but obtained directly from the asymptotic fields themselves. One even sees the possible suppression, via the Bondi mass loss, of the classical runaway solutions due to the radiation reaction force. 
  New spherically symmetric gravastar solutions, stable to radial perturbations, are found by utilising the construction of Visser and Wiltshire. The solutions possess an anti--de Sitter or de Sitter interior and a Schwarzschild--(anti)--de Sitter or Reissner--Nordstr\"{o}m exterior. We find a wide range of parameters which allow stable gravastar solutions, and present the different qualitative behaviours of the equation of state for these parameters. 
  We study the effects of pressure anisotropy and heat dissipation in a spherically symmetric radiating star undergoing gravitational collapse. An exact solution of the Einstein field equations is presented in which the model has a Friedmann-like limit when the heat flux vanishes. The behaviour of the temperature profile of the evolving star is investigated within the framework of causal thermodynamics. In particular, we show that there are significant differences between the relaxation time for the heat flux and the relaxation time for the shear stress. 
  This is a two-part, `2-in-1' paper. In Part I, the introductory talk at `Glafka--2004: Iconoclastic Approaches to Quantum Gravity' international theoretical physics conference is presented in paper form (without references). In Part II, the more technical talk, originally titled ``Abstract Differential Geometric Excursion to Classical and Quantum Gravity'', is presented in paper form (with citations). The two parts are closely entwined, as Part I makes general motivating remarks for Part II. 
  The formulation of General Relativity in which the 4-dimensional space-time is embedded in a flat host space of higher dimension is reconsidered. New classes of embeddings (modeled after Nash's classical free embeddings) are introduced. They present the important advantage of being deformable and therefore physically realistic. Explicit examples of embeddings whose deformations DO describe gravitational waves around their respective backgrounds are given for several space-times, including the Schwarzschild black hole. New variational principles which give back Einstein's General Relativity are proposed. In this framework, the 4-D space-time is a membrane moving in a flat host space. 
  We consider the propagation of light in a anisotropic medium with a topological line defect in the realm of geometrical optics. It is shown that the effective geometry perceived by light propagating in such medium is that of a spacial section of the cosmic string spacetime. 
  We study the implications of adopting hyperbolic driver coordinate conditions motivated by geometrical considerations. In particular, conditions that minimize the rate of change of the metric variables. We analyze the properties of the resulting system of equations and their effect when implementing excision techniques. We find that commonly used coordinate conditions lead to a characteristic structure at the excision surface where some modes are not of outflow-type with respect to any excision boundary chosen inside the horizon. Thus, boundary conditions are required for these modes. Unfortunately, the specification of these conditions is a delicate issue as the outflow modes involve both gauge and main variables. As an alternative to these driver equations, we examine conditions derived from extremizing a scalar constructed from Killing's equation and present specific numerical examples. 
  The (four componenet) vector graviton contained in metric, the scalar component incorporated, is attributed to the violation of the general covariance to the residual isoharmonic one. In addition to the previously studied (singlet) scalar graviton, the vector graviton may constitute one more fraction of the gravitational dark matter. The gravity interactions of the vector graviton, as well as its impact on the continuous medium are studied. 
  Numerical relativity is an essential tool for solving Einstein's equations of general relativity for dynamical systems characterized by high velocities and strong gravitational fields. The implementation of new algorithms that can solve these nonlinear equations in 3+1 dimensions has enabled us to tackle many long-standing problems of astrophysical interest, leading to an explosion of important new results. Numerical relativity has been used to simulate the evolution of a diverse array of physical systems, including coalescing black hole and neutron star binaries, rotating and collapsing compact objects (stars, collisionless clusters, and scalar fields), and magnetic and viscous stars, to name a few. Numerical relativity has been exploited to address fundamental points of principle, including critical phenomena and cosmic censorship. It holds great promise as a guide for interpreting observations of gravitational waves and gamma-ray bursts and identifying the sources of such radiation. Highlights of a few recent developments in numerical relativity are sketched in this brief overview. 
  Experimental projects using spherical antennas to detect gravitational waves are nowdays a concrete reality. The main purpose of this paper is to give a possible way of interpreting output data from such a system. Responses of the five fundamental quadrupole modes and of the six resonators in TIGA collocations are shown as a function of the incoming direction of the incident wave. Then, for a source lying in the galactic plane, sidereal time and galactic longitude dependence is given. Thus, once a candidate source of gravitational waves is considered, we can exactly predict the resonators' response as a function of time. 
  The Dirac field is studied in a Lyra space-time background by means of the classical Schwinger Variational Principle. We obtain the equations of motion, establish the conservation laws, and get a scale relation relating the energy-momentum and spin tensors. Such scale relation is an intrinsic property for matter fields in Lyra background. 
  We study the co-evolution of Yang-Mills fields and perfect fluids in Bianchi type I universes. We investigate numerically the evolution of the universe and the Yang-Mills fields during the radiation and dust eras of a universe that is almost isotropic. The Yang-Mills field undergoes small amplitude chaotic oscillations, which are also displayed by the expansion scale factors of the universe. The results of the numerical simulations are interpreted analytically and compared with past studies of the cosmological evolution of magnetic fields in radiation and dust universes. We find that, whereas magnetic universes are strongly constrained by the microwave background anisotropy, Yang-Mills universes are principally constrained by primordial nucleosynthesis and the bound is comparatively weak, and Omega_YM < 0.105 Omega_rad. 
  We study the pulsar timing, focusing on the time delay induced by the gravitational field of the binary systems. In particular, we study the gravito-magnetic correction to the Shapiro time delay in terms of Keplerian and post-Keplerian parameters, and we introduce a new post-Keplerian parameter which is related to the intrinsic angular momentum of the stars. Furthermore, we evaluate the magnitude of these effects for the binary pulsar systems known so far. The expected magnitude is indeed small, but the effect is important per se. 
  We discuss the possibility to solve Modern Numerical Relativity problems using finite element methods (FEM). Adopting a "user friendly" software for handling totally general systems of nonlinear partial differential equations, FEMLAB, we model and numerically solve in a short time a Gowdy vacuum spacetime, representing an inhomogeneous cosmology. Results agree perfectly with existing simulations in the recent literature based not of FEMs but on finite differences methods. Possible applications for non relativistic Astrophysics, General Relativity, elementary particle physics and more general theories of gravitation like EMDA and branes are discussed. 
  In this article we observe that the self-adjoint extension of the Hamiltonian of a particle moving around a shielded cosmic string gives rise to a gravitational analogue of the bound state Aharonov-Bohm effect. 
  Using black hole perturbation theory, we calculate the gravitational waves produced by test particles moving on bound geodesic orbits about rotating black holes. The orbits we consider are generic - simultaneously eccentric and inclined. The waves can be described as having radial, polar, and azimuthal "voices", each of which can be made to dominate by varying eccentricity and inclination. Although each voice is generally apparent in the waveform, the radial voice is prone to overpowering the others. We also compute the radiative fluxes of energy and axial angular momentum at infinity and through the event horizon. These fluxes, coupled to a prescription for the radiative evolution of the Carter constant, will be used in future work to adiabatically evolve through a sequence of generic orbits. This will enable the calculation of inspiral waveforms that, while lacking certain important features, will approximate those expected from astrophysical extreme mass ratio captures sufficiently well to aid development of measurement algorithms on a relatively short timescale. 
  We investigate the thermodynamics of static black objects such as black holes, black strings and their generalizations to D dimensions (`black branes') in a gravitational theory containing the four dimensional Gauss-Bonnet term in the action, when D-4 of the dimensions are compactified on a torus. The entropies of black holes and black branes are compared to obtain information on the stability of these objects and to find their phase diagrams. We demonstrate the existence of a critical mass, which depends on the scale of the compactified dimensions, below which the black hole entropy dominates over the entropy of the black membrane. 
  A construction of conformal infinity in null and spatial directions is constructed for the Rainbow-flat space-time corresponding to doubly special relativity. From this construction a definition of asymptotic DSRness is put forward which is compatible with the correspondence principle of Rainbow gravity. Furthermore a result equating asymptotically flat space-times with asymptotically DSR spacetimes is presented. 
  The existence of a simple spherically symmetric and static solution of the Einstein equations in the presence of a cosmological constant vanishing outside a definite value of the radial distance is investigated. A particular succession of field configurations, which are solutions of the Einstein equations in the presence of the considered cosmological term and auxiliary external sources, is constructed. Then, it is shown that the associated succession of external sources tend to zero in the sense of the generalized functions. The type of weak solution that is found becomes the deSitter homogeneous space-time for the interior region, and the Schwartzschild space in the outside zone. 
  Gravity theory is the basis of modern cosmological models. Thirring-Feynman's tensor field approach to gravitation is an alternative to General Relativity (GR). Though Field Gravity (FG) approach is still developing subject, it opens new understanding of gravitational interaction, stimulates novel experiments on the nature of gravity and gives possibility to construct new cosmological models in Minkowski space. According to FG, the universal gravity force is caused by exchange of gravitons - the quanta of gravity field. Energy of this field is well-defined and excludes the singularity. All classical relativistic effects are the same as in GR, though there are new effects, such as free fall of rotating bodies, scalar gravitational radiation, surface of relativistic compact bodies, which may be tested experimentally. The intrinsic scalar (spin 0) part of gravity field corresponds to "antigravity" and only together with the pure tensor (spin 2) part gives the usual Newtonian force. Laboratory and astrophysical experiments for testing new predictions of FG, will be performed in near future. In particular observations with bar and interferometric detectors, like Explorer, Nautilus, LIGO and VIRGO, will check the predicted scalar gravitational waves from supernova explosions. 
  A new criterion for inextendibility of expanding cosmological models with symmetry is presented. It is applied to derive a number of new results and to simplify the proofs of existing ones. In particular it shows that the solutions of the Einstein-Vlasov system with $T^2$ symmetry, including the vacuum solutions, are inextendible in the future. The technique introduced adds a qualitatively new element to the available tool-kit for studying strong cosmic censorship. 
  Hawking's theorem on the topology of black holes asserts that cross sections of the event horizon in 4-dimensional asymptotically flat stationary black hole spacetimes obeying the dominant energy condition are topologically 2-spheres. This conclusion extends to outer apparent horizons in spacetimes that are not necessarily stationary. In this paper we obtain a natural generalization of Hawking's results to higher dimensions by showing that cross sections of the event horizon (in the stationary case) and outer apparent horizons (in the general case) are of positive Yamabe type, i.e., admit metrics of positive scalar curvature. This implies many well-known restrictions on the topology, and is consistent with recent examples of five dimensional stationary black hole spacetimes with horizon topology $S^2 \times S^1$. The proof is inspired by previous work of Schoen and Yau on the existence of solutions to the Jang equation (but does not make direct use of that equation). 
  No. It is simply not plausible that cosmic acceleration could arise within the context of general relativity from a back-reaction effect of inhomogeneities in our universe, without the presence of a cosmological constant or ``dark energy.'' We point out that our universe appears to be described very accurately on all scales by a Newtonianly perturbed FLRW metric. (This assertion is entirely consistent with the fact that we commonly encounter $\delta \rho/\rho > 10^{30}$.) If the universe is accurately described by a Newtonianly perturbed FLRW metric, then the back-reaction of inhomogeneities on the dynamics of the universe is negligible. If not, then it is the burden of an alternative model to account for the observed properties of our universe. We emphasize with concrete examples that it is {\it not} adequate to attempt to justify a model by merely showing that some spatially averaged quantities behave the same way as in FLRW models with acceleration. A quantity representing the ``scale factor'' may ``accelerate'' without there being any physically observable consequences of this acceleration. It also is {\it not} adequate to calculate the second-order stress energy tensor and show that it has a form similar to that of a cosmological constant of the appropriate magnitude. The second-order stress energy tensor is gauge dependent, and if it were large, contributions of higher perturbative order could not be neglected. We attempt to clear up the apparent confusion between the second-order stress energy tensor arising in perturbation theory and the ``effective stress energy tensor'' arising in the ``shortwave approximation.'' 
  The response of a cross-correlation measurement to an isotropic stochastic gravitational-wave background depends on the observing geometry via the overlap reduction function. If one of the detectors being correlated is a resonant bar whose orientation can be changed, the response to stochastic gravitational waves can be modulated. I derive the general form of this modulation as a function of azimuth, both in the zero-frequency limit and at arbitrary frequencies. Comparisons are made between pairs of nearby detectors, such as LIGO Livingston-ALLEGRO, Virgo-AURIGA, Virgo-NAUTILUS, and EXPLORER-AURIGA, with which stochastic cross-correlation measurements are currently being performed, planned, or considered. 
  We show that in the framework of the classical general relativity the presence of a positive cosmological constant implies the existence of a minimal mass and of a minimal density in nature. These results rigorously follow from the generalized Buchdahl inequality in the presence of a cosmological constant. 
  The geodesic deviation equation is generalized to worldline deviation equations describing the relative accelerations of charged spinning particles in the framework of Dixon-Souriau equations of motion. 
  We present a simple higher dimensional FRW type of model where the acceleration is apparently caused by the presence of the extra dimensions. Assuming an ansatz in the form of the deceleration parameter we get a class of solutions some of which shows the desirable feature of dimensional reduction as well as reasonably good physical properties of matter. Interestingly we do not have to invoke an extraneous scalar field or a cosmological constant to account for this acceleration. One argues that the terms containing the higher dimensional metric coefficients produces an extra negative pressure that apparently drives the inflation of the 4D space with an accelerating phase. It is further found that in line with the physical requirements our model admits of a decelerating phase in the early era along with an accelerating phase at present.Further the models asymptotically mimic a steady state type of universe although it starts from a big type of singularity. Correspondence to Wesson's induced matter theory is also briefly discussed and in line with it it is argued that the terms containing the higher dimensional metric coefficients apparently creates a negative pressure which drives the inflation of the 3-space with an accelerating phase. 
  We study Lorentzian spacetimes for which all scalar invariants constructed from the Riemann tensor and its covariant derivatives are constant ($CSI$ spacetimes). We obtain a number of general results in arbitrary dimensions. We study and construct warped product $CSI$ spacetimes and higher-dimensional Kundt $CSI$ spacetimes. We show how these spacetimes can be constructed from locally homogeneous spaces and $VSI$ spacetimes. The results suggest a number of conjectures. In particular, it is plausible that for $CSI$ spacetimes that are not locally homogeneous the Weyl type is $II$, $III$, $N$ or $O$, with any boost weight zero components being constant. We then consider the four-dimensional $CSI$ spacetimes in more detail. We show that there are severe constraints on these spacetimes, and we argue that it is plausible that they are either locally homogeneous or that the spacetime necessarily belongs to the Kundt class of $CSI$ spacetimes, all of which are constructed. The four-dimensional results lend support to the conjectures in higher dimensions. 
  Lunar laser ranging (LLR) is used to conduct high-precision measurements of ranges between an observatory on Earth and a laser retro-reflector on the lunar surface. Over the years, LLR has benefited from a number of improvements both in observing technology and data modeling, which led to the current accuracy of post-fit residuals of ~2 cm. Today LLR is a primary technique to study the dynamics of the Earth-Moon system and is especially important for gravitational physics, geodesy and studies of the lunar interior. LLR is used to perform high-accuracy tests of the equivalence principle, to search for a time-variation in the gravitational constant, and to test predictions of various alternative theories of gravity. On the geodesy front, LLR contributes to the determination of Earth orientation parameters, such as nutation, precession (including relativistic precession), polar motion, and UT1, i.e. especially to the long-term variation of these effects. LLR contributes to the realization of both the terrestrial and selenocentric reference frames. The realization of a dynamically defined inertial reference frame, in contrast to the kinematically realized frame of VLBI, offers new possibilities for mutual cross-checking and confirmation. Finally, LLR also investigates the processes related to the Moon's interior dynamics. Here, we review the LLR technique focusing on its impact on relativity and give an outlook to further applications, e.g. in geodesy. We present results of our dedicated studies to investigate the sensitivity of LLR data with respect to the relativistic quantities. We discuss the current observational situation and the level of LLR modeling implemented to date. We also address improvements needed to fully utilize the scientific potential of LLR. 
  A linear second order wave equation is presented based on cosmological general relativity, which is a space-velocity theory of the expanding Universe. The wave equation is shown to be exactly solvable, based on the Gaussian hypergeometric function. 
  Searches for gravitational waves from binary neutron stars or sub-solar mass black holes by the LIGO Scientific Collaboration use the findchirp algorithm: an implementation of standard matched filter techniques with innovations to improve performance on detector data that has non-stationary and non-Gaussian artifacts. We provide details on the methods used in the findchirp algorithm and describe some future improvements. 
  Massless scalar and vector fields are coupled to Lyra geometry by means of Duffin-Kemmer-Petiau (DKP) theory. Using Schwinger Variational Principle, equations of motion, conservation laws and gauge symmetry are implemented. We find that the scalar field couples to the anholonomic part of the torsion tensor, and the gauge symmetry of the electromagnetic field is not breaking by the coupling with torsion. 
  One of the main achievements of LQG is the consistent quantization of the Wheeler-DeWitt equation which is free of UV problems. However, ambiguities associated to the intermediate regularization procedure lead to an apparently infinite set of possible theories. The absence of an UV problem is intimately linked with the ambiguities arising in the quantum theory. Among these ambiguities there is the one associated to the SU(2) unitary rep. used in the diffeomorphism covariant pointsplitting regularization of nonlinear funct. of the connection. This ambiguity is labelled by a halfinteger m and, here, it is referred to as the m-ambiguity. The aim of this paper is to investigate the important implications of this ambiguity./ We first study 2+1 gravity quantized in canonical LQG. Only when the regularization of the quantum constraints is performed in terms of the fundamental rep. of the gauge group one obtains the usual TQFT. In all other cases unphysical local degrees of freedom arise at the level of the regulated theory that conspire against the existence of the continuum limit. This shows that there is a clear cut choice in the quantization of the constraints in 2+1 LQG./ We then analyze the effects of the ambiguity in 3+1 gravity exhibiting the existence of spurious solutions for higher unit. rep. quantizations of the Hamiltonian constraint. Although the analysis is not complete in D=3+1--due to the difficulties associated to the definition of the physical inner product--it provides evidence supporting the definitions quantum dynamics of loop quantum gravity in terms of the fundamental representation of the gauge group as the only consistent possibilities. If the gauge group is SO(3) we find physical solutions associated to spin-two local excitations. 
  We present strongly stable semi-discrete finite difference approximations to the quarter space problem (x>0, t>0) for the first order in time, second order in space wave equation with a shift term. We consider space-like (pure outflow) and time-like boundaries, with either second or fourth order accuracy. These discrete boundary conditions suggest a general prescription for boundary conditions in finite difference codes approximating first order in time, second order in space hyperbolic problems, such as those that appear in numerical relativity. As an example we construct boundary conditions for the Nagy-Ortiz-Reula formulation of the Einstein equations coupled to a scalar field in spherical symmetry. 
  Quantum theory is a probabilistic theory with fixed causal structure. General relativity is a deterministic theory but where the causal structure is dynamic. It is reasonable to expect that quantum gravity will be a probabilistic theory with dynamic causal structure. The purpose of this paper is to present a framework for such a probability calculus. We define an operational notion of space-time, this being composed of elementary regions. Central to this formalism is an object we call the causaloid. This object captures information about causal structure implicit in the data by quantifying the way in which the number of measurements required to establish a state for a composite region is reduced when there is a causal connection between the component regions. This formalism puts all elementary regions on an equal footing. It does not require that we impose fixed causal structure. In particular, it is not necessary to assume the existence of a background time. Remarkably, given the causaloid, we can calculate all relevant probabilities and so the causaloid is sufficient to specify the predictive aspect of a physical theory. We show how certain causaloids can be represented by suggestive diagrams and we show how to represent both classical probability theory and quantum theory by a causaloid. We do not give a causaloid formulation for general relativity though we speculate that this is possible. The work presented here suggests a research program aimed at finding a theory of quantum gravity. The idea is to use the causaloid formalism along with principles taken from the two theories to marry the dynamic causal structure of general relativity with the probabilistic structure of quantum theory. 
  The Noether charge method for defining the Hamiltonian of a diffeomorphism-invariant field theory is applied to "Einstein-aether" theory, in which gravity couples to a dynamical, timelike, unit-norm vector field. Using the method, expressions are obtained for the total energy, momentum, and angular momentum of an Einstein-aether space-time. The method is also used to discuss the mechanics of Einstein-aether black holes. The derivation of Wald, and Iyer and Wald, of the first law of black hole thermodynamics fails for this theory, because the unit vector is necessarily singular at the bifurcation surface of the Killing horizon. A general identity relating variations of energy and angular momentum to a surface integral at the horizon is obtained, but a thermodynamic interpretation, including a definitive expression for the black hole entropy, is not found. 
  A small body moving in the field of a much larger black hole and subjected to its own gravity moves on an accelerated world line in the background spacetime of the large black hole. The acceleration is produced by the body's gravitational self-force, which is constructed from the body's retarded gravitational field. The adiabatic approximation to the gravitational self-force is obtained instead from the half-retarded minus half-advanced field; it is known to produce the same dissipative effects as the true self-force. We argue that the adiabatic approximation is limited, because it discards important conservative terms which lead to the secular evolution of some orbital elements. We argue further that this secular evolution has measurable consequences; in particular, it affects the phasing of the orbit and the phasing of the associated gravitational wave. Our argument rests on a simple toy model involving a point electric charge moving slowly in the weak gravitational field of a central mass; the charge is also subjected to its electromagnetic self-force. In this simple context the true self-force is known explicitly and it can cleanly be separated into conservative and radiation-reaction pieces. We observe that the conservative part of the self-force produces a secular regression of the orbit's periapsis. We explain how the conclusions reached on the basis of the toy model can be extended to the gravitational self-force, and to fast motions and strong fields. While the limitations of the adiabatic approximation are quite severe in a post-Newtonian context in which the motion is slow and the gravitational field weak, they appear to be less so for rapid motions and strong fields. 
  The well-known Regge-Wheeler equation describes the axial perturbations of Schwarzschild metric in the linear approximation. From a mathematical point of view it presents a particular case of the confluent Heun equation and can be solved exactly, due to recent mathematical developments. We present the basic properties of its general solution. A novel analytical approach and numerical techniques for study the boundary problems which correspond to quasi-normal modes of black holes and other simple models of compact objects are developed. 
  In the framework of spatially averaged inhomogeneous cosmologies in classical General Relativity, effective Einstein equations govern the regional and the global dynamics of averaged scalar variables of cosmological models. A particular solution may be characterized by a cosmic equation of state. In this paper it is pointed out that a globally static averaged dust model is conceivable without employing a compensating cosmological constant. Much in the spirit of Einstein's original model we discuss consequences for the global, but also for the regional properties of this cosmology. We then consider the wider class of globally stationary cosmologies that are conceivable in the presented framework. All these models are based on exact solutions of the averaged Einstein equations and provide examples of cosmologies in an out-of-equilibrium state, which we characterize by an information-theoretical measure. It is shown that such cosmologies preserve high-magnitude kinematical fluctuations and so tend to maintain their global properties. The same is true for a $\Lambda-$driven cosmos in such a state despite of exponential expansion. We outline relations to inflationary scenarios, and put the Dark Energy problem into perspective. Here, it is argued, on the grounds of the discussed cosmologies, that a classical explanation of Dark Energy through backreaction effects is theoretically conceivable, if the matter-dominated Universe emerged from a non-perturbative state in the vicinity of the stationary solution. We also discuss a number of caveats that furnish strong counter arguments in the framework of structure formation in a perturbed Friedmannian model. 
  Cosmological models of Bianchi type V and I containing a perfect fluid with a linear equation of state plus cosmological constant are investigated using a tetrad approach where our variables are the Riemann tensor, the Ricci rotation coefficients and a subset of the tetrad vector components. This set, in the following called S, describes a spacetime when its elements are constrained by certain integrability conditions and due to a theorem by Cartan this set gives a complete local description of the spacetime. The system obtained by imposing the integrability conditions and Einstein's equations can be reduced to an integrable system of five coupled first order ordinary differential equations. The general solution is tilted and describes a fluid with expansion, shear and vorticity. With the help of standard bases for Bianchi V and I the full line element is found in terms of the elements in S. We then construct the solutions to the linearized equations around the open Friedmann model. The full system is also studied numerically and the perturbative solutions agree well with the numerical ones in the appropriate domains. We also give some examples of numerical solutions in the non-perturbative regime. 
  The circular noise is important in connection to Mach's principle, and also as a possible probe of the Unruh effect. In this letter the power spectrum of the detector following the Trocheries-Takeno motion in the Minkowski vacuum is analytically obtained in the form of an infinite series. A mean distribution function and corresponding energy density are obtained for this particular detected noise. The analogous of a non constant temperature distribution is obtained. And in the end, a brief discussion about the equilibrium configuration is given. 
  Within the framework of Geodesic Brane Gravity, the deviation from General Relativity is parameterized by the conserved bulk energy. The corresponding geodesic evolution/nucleation of a de-Sitter brane is shown to be exclusively driven by a double-well Higgs potential, rather than by a plain cosmological constant. The (hairy) horizon serves then as the locus of unbroken $Z_{2}$ symmetry. The quartic structure of the scalar potential, singled out on finiteness grounds of the total (including the dark component) energy density, chooses the Hartle-Hawking no-boundary proposal. 
  Construction of astrophysically realistic initial data remains a central problem when modelling the merger and eventual coalescence of binary black holes in numerical relativity. The objective of this paper is to provide astrophysically realistic freely specifiable initial data for binary black hole systems in numerical relativity, which are in agreement with post-Newtonian results. Following the approach taken by Blanchet, we propose a particular solution to the time-asymmetric constraint equations, which represent a system of two moving black holes, in the form of the standard conformal decomposition of the spatial metric and the extrinsic curvature. The solution for the spatial metric is given in symmetric tracefree form, as well as in Dirac coordinates. We show that the solution differs from the usual post-Newtonian metric up to the 2PN order by a coordinate transformation. In addition, the solutions, defined at every point of space, differ at second post-Newtonian order from the exact, conformally flat, Bowen-York solution of the constraints. 
  We report on a search for gravitational waves from binary black hole inspirals in the data from the second science run of the LIGO interferometers. The search focused on binary systems with component masses between 3 and 20 solar masses. Optimally oriented binaries with distances up to 1 Mpc could be detected with efficiency of at least 90%. We found no events that could be identified as gravitational waves in the 385.6 hours of data that we searched. 
  Particle production by slow-changing gravitational fields is usually described using quantum field theory in curved spacetime. Calculations require a definition of the vacuum state, which can be given using the adiabatic (WKB) approximation. I investigate the best attainable precision of the resulting approximate definition of the particle number. The standard WKB ansatz yields a divergent asymptotic series in the adiabatic parameter. I derive a novel formula for the optimal number of terms in that series and demonstrate that the error of the optimally truncated WKB series is exponentially small. This precision is still insufficient to describe particle production from vacuum, which is typically also exponentially small. An adequately precise approximation can be found by improving the WKB ansatz through perturbation theory. I show quantitatively that the fundamentally unavoidable imprecision in the definition of particle number in a time-dependent background is equal to the particle production expected to occur during that epoch. The results are illustrated by analytic and numerical examples. 
  Two Fokker actions and corresponding equations of motion are obtained for two point particles in a post-Minkowski framework, in which the field of each particle is given by the half-retarded + half-advanced solution to the linearized Einstein equations. The first action is parametrization invariant, the second a generalization of the affinely parametrized quadratic action for a relativistic particle. Expressions for a conserved 4-momentum and angular momentum tensor are obtained in terms of the particles' trajectories in this post-Minkowski approximation. A formal solution to the equations of motion is found for a binary system with circular orbits. For a bound system of this kind, the post-Minkowski solution is a toy model that omits nonlinear terms of relevant post-Newtonian order; and we also obtain a Fokker action that is accurate to first post-Newtonian order, by adding to the post-Minkowski action a term cubic in the particle masses. Curiously, the conserved energy and angular momentum associated with the Fokker action are each finite and well-defined for this bound 2-particle system despite the fact that the total energy and angular momentum of the radiation field diverge. Corresponding solutions and conserved quantities are found for two scalar charges (for electromagnetic charges we exhibit the solution found by Schild). For a broad class of parametrization-invariant Fokker actions and for the affinely parametrized action, binary systems with circular orbits satisfy the relation $dE = \Omega dL$ (a form of the first law of thermodynamics), relating the energy, angular velocity and angular momentum of nearby equilibrium configurations. 
  The possibility of a Newtonian gravitomagnetic field is considered here with its immediate and far-reaching implications for the interpretation of 2004 LAGEOS experimental results confirming the general relativistic prediction of Lense-Thirring effect. 
  Barotropic FRW cosmologies are presented from the standpoint of nonrelativistic supersymmetry. First, we reduce the barotropic FRW system of differential equations to simple harmonic oscillator differential equations. Employing the factorization procedure, the solutions of the latter equations are divided into the two classes of bosonic (nonsingular) and fermionic (singular) cosmological solutions. We next introduce a coupling parameter denoted by K between the two classes of solutions and obtain barotropic cosmologies with dissipative features acting on the scale factors and spatial curvature of the universe. The K-extended FRW equations in comoving time are presented in explicit form in the low coupling regime. The standard barotropic FRW cosmologies correspond to the dissipationless limit K =0 
  Solutions are found to field equations constructed from the Pauli, Bach and Gauss-Bonnet quadratic tensors to the Kasner and Kasner brane spacetimes in up to five dimensions. 
  We study the energy problem in the $R0X$ spacetimes in 5D. We found that for the Einstein, Landau-Lifschitz and the Moller complexes are null. 
  We show that inflation and current cosmic acceleration can be generated by a metric-affine f(R) gravity formulated in the Einstein conformal frame, if the gravitational Lagrangian L(R) contains both positive and negative powers of the curvature scalar R. In this frame, we give the equations for the expansion of the homogeneous and isotropic matter-dominated universe in the case L(R)=R+{R^3}/{\beta^2}-{\alpha^2}/{3R}, where \alpha and \beta are constants. We also show that gravitational effects of matter in such a universe at very late stages of its expansion are weakened by a factor that tends to 3/4, and the energy density of matter \epsilon scales the same way as in the \Lambda-CDM model only when \kappa*\epsilon<<\alpha. 
  We confront the predictions of S. Hod, Phys. Rev. D 60, 104053 (1999) for the late-time decay rate of black hole perturbations with numerical data. Specifically, we ask two questions: First, are corrections to the Price tail dominated by logarithmic terms, as predicted by Hod? Second, if there were logarithmic correction terms, do they take the specific form predicted in Hod's paper? The answer to both questions is ``no.'' 
  The spectrum of Hawking radiation by quantum fields in the curved spacetime is continuous, so the explanation of Hawking radiation using quasinormal modes can be suspected to be impossible. We find that quasinormal modes do not explain the relation between the state observed in a region far away from a black hole and the short distance behavior of the state on the horizon. 
  It is well-known that energy-momentum is the source of gravitational field. For a long time, it is generally believed that only stars with huge masses can generate strong gravitational field. Based on the unified theory of gravitational interactions and electromagnetic interactions, a new mechanism of the generation of gravitational field is studied. According to this mechanism, in some special conditions, electromagnetic energy can be directly converted into gravitational energy, and strong gravitational field can be generated without massive stars. Gravity impulse found in experiments is generated by this mechanism. 
  Recently the Master Constraint Programme (MCP) for Loop Quantum Gravity (LQG) was launched which replaces the infinite number of Hamiltonian constraints by a single Master constraint. The MCP is designed to overcome the complications associated with the non -- Lie -- algebra structure of the Dirac algebra of Hamiltonian constraints and was successfully tested in various field theory models. For the case of 3+1 gravity itself, so far only a positive quadratic form for the Master Constraint Operator was derived. In this paper we close this gap and prove that the quadratic form is closable and thus stems from a unique self -- adjoint Master Constraint Operator. The proof rests on a simple feature of the general pattern according to which Hamiltonian constraints in LQG are constructed and thus extends to arbitrary matter coupling and holds for any metric signature. With this result the existence of a physical Hilbert space for LQG is established by standard spectral analysis. 
  Scalar field contribution to entropy is studied in arbitrary D dimensional one parameter rotating spacetime by semiclassical method. By introducing the zenithal angle dependent cutoff parameter, the generalized area law is derived. The non-rotating limit can be taken smoothly and it yields known results. The derived area law is then applied to the Banados-Teitelboim-Zanelli (BTZ) black hole in (2+1) dimension and the Kerr-Newman black hole in (3+1) dimension. The generalized area law is reconfirmed by the Euclidean path integral method for the quantized scalar field. The scalar field mass contribution is discussed briefly. 
  We investigate the causal temperature profiles in a recent model of a radiating star undergoing dissipative gravitational collapse without the formation of an horizon. It is shown that this simple exact model provides a physically reasonable behaviour for the temperature profile within the framework of extended irreversible thermodynamics. 
  We introduce a master constraint operator $\hat{\mathbf{M}}$ densely defined in the diffeomorphism invariant Hilbert space in loop quantum gravity, which corresponds classically to the master constraint in the programme. It is shown that $\hat{\mathbf{M}}$ is positive and symmetric, and hence has its Friedrichs self-adjoint extension. The same conclusion is tenable for an alternative master operator $\hat{\mathbf{M'}}$, whose quadratic form coincides with the one proposed by Thiemann. So the master constraint programme for loop quantum gravity can be carried out in principle by employing either of the two operators. 
  The Einstein-Hilbert action has a bulk term and a surface term (which arises from integrating a four divergence). I show that one can obtain Einstein's equations from the surface term alone. This leads to: (i) a novel, completely self contained, perspective on gravity and (ii) a concrete mathematical framework in which the description of spacetime dynamics by Einstein's equations is similar to the description of a continuum solid in the thermodynamic limit. 
  We discuss the initial value problem of general relativity in its recently unified Lagrangian and Hamiltonian pictures and present a multi-domain pseudo-spectral collocation method to solve the resulting coupled nonlinear partial differential equations. Using this code, we explore several approaches to construct initial data sets containing one or two black holes: We compute quasi-circular orbits for spinning equal mass black holes and unequal mass (nonspinning) black holes using the effective potential method with Bowen-York extrinsic curvature. We compare initial data sets resulting from different decompositions, and from different choices of the conformal metric with each other. Furthermore, we use the quasi-equilibrium method to construct initial data for single black holes and for binary black holes in quasi-circular orbits. We investigate these binary black hole data sets and examine the limits of large mass-ratio and wide separation. Finally, we propose a new method for constructing spacetimes with superposed gravitational waves of possibly very large amplitude. 
  The nature of 'time', 'space' and 'reality' are to large extent dependent on our interpretation of Special (SRT) and General Relativity Theory (GRT). In SRT essentially two distinct interpretations exist; the "geometrical" interpretation by Einstein based on the Principle of Relativity and the Invariance of the velocity of light and, the "physical" Lorentz-Poincar\'e interpretation with underpinning by rod contractions, clock slowing and light synchronization, see e.g. (Bohm 1965, Bell 1987). It can be questioned whether the "Lorentz-Poincar\'e"-interpretation of SRT can be continued into GRT. We have shown that till first Post-Newtonian order this is indeed possible (Broekaert 2004). This requires the introduction of gravitationally modified Lorentz transformations, with an intrinsical spatially-variable speed of light $c(r)$, a scalar scaling field $\Phi$ and induced velocity field $w$. Still the invariance of the locally observed velocity of light is maintained (Broekaert 2005). The Hamiltonian description of particles and photons recovers the 1-PN approximation of GRT. At present we show the model does obey the Weak Equivalence Principle from a fixed perspective, and that the implied acceleration transformations are equivalent with those of GRT. 
  As an alternative view to the standard big bang cosmology the quasi-steady state cosmology(QSSC) argues that the universe was not created in a single great explosion; it neither had a beginning nor will it ever come to an end. The creation of new matter in the universe is a regular feature occurring through finite explosive events. Each creation event is called a mini-bang or, a mini creation event(MCE). Gravitational waves are expected to be generated due to any anisotropy present in this process of creation. Mini creation event ejecting matter in two oppositely directed jets is thus a source of gravitational waves which can in principle be detected by laser interferometric detectors. In the present work we consider the gravitational waveforms propagated by linear jets and then estimate the response of laser interferometric detectors like LIGO and LISA. 
  We reformulate the theory of Schwarzschild black hole perturbations in terms of the metric perturbation in the Lorenz gauge. In this formulation, each tensor-harmonic mode of the perturbation is constructed algebraically from 10 scalar functions, satisfying a set of 10 wavelike equations, which are decoupled at their principal parts. We solve these equations using numerical evolution in the time domain, for the case of a pointlike test particle set in a circular geodesic orbit around the black hole. Our code uses characteristic coordinates, and incorporates a constraint damping scheme. The axially-symmetric, odd-parity modes of the perturbation are obtained analytically. The approach developed here is especially advantageous in applications requiring knowledge of the local metric perturbation near a point particle; in particular, it offers a useful framework for calculations of the gravitational self force. 
  We investigate the dynamic stability of inspiraling neutron stars by performing multiple-orbit numerical relativity simulations of the binary neutron star inspiral process. By introducing eccentricities in the orbits of the neutron stars, significant changes in orbital separation are obtained within orbital timescales. We find that as the binary system evolves from apastron to periastron (as the binary separation decreases), the central rest mass density of each star decreases, thus stabilizing the stars against individual prompt collapse. As the binary system evolves from periastron to apastron, the central rest mass density increases; the neutron stars re-compress as the binary separation increases. 
 GGR News:   The WYP speakers program, by Richard Price   We hear that..., by Jorge Pullin   100 Years ago, by Jorge Pullin  Research Briefs:   What's new in LIGO, by David Shoemaker   Recent developments in the information loss paradox, by Eanna Flanagan   Gravity Probe B mission ends, by Bob Kahn  Conference reports:   6th Edoardo Amaldi Meeting, by Matthew Benacquista   Workshop on Numerical Relativity, BIRS, by Carsten Gundlach   8th Capra Meeting on Radiation Reaction, by Leor Barack   Theory and experiment in quantum gravity, by Elizabeth Winstanley 
  Braneworld scenarios are motivated by string/M-theory and can be characterized by the way in which they modify the conventional Friedmann equations of Einstein gravity. An alternative approach to quantum gravity, however, is the loop quantum cosmology program. In the semi-classical limit, the cosmic dynamics in this scenario can also be described by a set of modified Friedmann equations. We demonstrate that a dynamical correspondence can be established between these two paradigms at the level of the effective field equations. This allows qualitatively similar features between the two approaches to be compared and contrasted as well as providing a framework for viewing braneworld scenarios in terms of constrained Hamiltonian systems. As concrete examples of this correspondence, we illustrate the relationships between different cosmological backgrounds representing scaling solutions. 
  The coordinate transformation which maps the Kerr metric written in standard Boyer-Lindquist coordinates to its corresponding form adapted to the natural local coordinates of an observer at rest at a fixed position in the equatorial plane, i.e., Fermi coordinates for the neighborhood of a static observer world line, is derived and discussed in a way which extends to any uniformly circularly orbiting observer there. 
  This is a review of the chrono-geometrical structure of special and general relativity with a special emphasis on the role of non-inertial frames and of the conventions for the synchronization of distant clocks. ADM canonical metric and tetrad gravity are analyzed in a class of space-times suitable to incorporate particle physics by using Dirac theory of constraints, which allows to arrive at a separation of the genuine degrees of freedom of the gravitational field, the Dirac observables describing generalized tidal effects, from its gauge variables, describing generalized inertial effects. A background-independent formulation (the rest-frame instant form of tetrad gravity) emerges, since the chosen boundary conditions at spatial infinity imply the existence of an asymptotic flat metric. By switching off the Newton constant in presence of matter this description deparametrizes to the rest-frame instant form for such matter in the framework of parametrized Minkowski theories. The problem of the objectivity of the space-time point-events, implied by Einstein's Hole Argument, is analyzed. 
  We investigate the possibility for a flat Bianchi I brane Universe to recollaps due to the presence of a negative "dark radiation" and an anisotropic stress in the form of a homogeneous magnetic field, localized on the brane. 
  We summarize a recent work done on the title's subject. First, we present the asymptotic scheme of post-Newtonian (PN) approximation for general relativity in the harmonic gauge. Then, we discuss the definition of the mass centers and the derivation of equations for their motion, following that scheme. Finally, we briefly analyze the reason why a new term has thus been found in the equations of motion. 
  The familiar approach to quantum radiation following collapse to a black hole proceeds via Bogoliubov transformations, and yields probabilities for final outcomes. In our (complex) approach, we find quantum amplitudes, not just probabilities, by following Feynman's $+i\epsilon$ prescription. Initial and final data for Einstein gravity and (say) a massless scalar field are specified on a pair of asymptotically-flat space-like hypersurfaces $\Sigma_I$ and $\Sigma_F$; both are diffeomorphic to ${\Bbb R}^3$. Denote by $T$ the (real) Lorentzian proper-time interval between the surfaces, as measured at spatial infinity. Then rotate: $T\to{\mid}T{\mid}\exp(-i\theta),0<\theta\leq \pi/2$. The {\it classical} boundary-value problem is expected to be well-posed on a region of topology $I\times{\Bbb R}^3$, where $I$ is a closed interval. For a locally-supersymmetric theory, the quantum amplitude should be dominated by the semi-classical expression $\exp(iS_{\rm class})$, where $S_{\rm class}$ is the classical action. One finds the Lorentzian quantum amplitude from the limit $\theta\to 0_+$. In the usual approach, the only possible such final surfaces are in the strong-field region shortly before the curvature singularity. In our approach one can put arbitrary smooth gravitational data on $\Sigma_F$, provided that it has the correct mass $M$ -- the singularity is by-passed in the analytic continuation. Here, we consider Bogoliubov transformations and their possible relation to the probability distribution and density matrix in the traditional approach. We find that our probability distribution for configurations of the final scalar field cannot be expressed in terms of the diagonal elements of some non-trivial density-matrix distribution. 
  Here we examine the quantum-mechanical decay of a Schwarzschild-like black hole, formed by gravitational collapse, into almost-flat space-time and weak radiation at a very late time, in order to evaluate quantum amplitudes (not just probabilities) for final states. No information is lost in collapse to a black hole. Boundary data are specified on initial and final hypersurfaces $\Sigma_{I, F}$, separated by a Lorentzian proper-time interval $T$, as measured at spatial infinity. For simplicity, consider Einstein gravity coupled minimally to a massless scalar field $\phi$. In Lorentzian signature, the classical Dirichlet boundary-value problem, corresponding to specification of the intrinsic spatial metric $h_{ij} (i,j =1,2,3)$ and $\phi$ on the bounding surfaces, is badly posed, being a boundary-value problem for a wave-like (hyperbolic) set of equations. Following Feynman's $+i\epsilon$ prescription, the problem is made well-posed by rotating the asymptotic time interval $T$ into the complex: $T\to{\mid} T{\mid}\exp(-i\theta)$, with $0<\theta\leq\pi/2$. After calculating the amplitude for $\theta>0$, one takes the 'Lorentzian limit' $\theta\to 0_+$ to obtain the Lorentzian quantum amplitude. 
  This work on spin-0 amplitudes in black-hole evaporation is based on the underlying results and methods outlined in our first paper, "I. Complex Approach". The main result here, and the model calculation for work on all higher spins, as described in several further papers, is the computation of the quantum amplitude (rather than merely the probability) for a given slightly anisotropic configuration of a scalar field $\phi$ on a space-like hypersurface $\Sigma_F$ at a very late time $T$. For simplicity, one may take the initial data for gravity and the massless scalar field at an initial surface $\Sigma_I$ to be spherically symmetric. This applies to perturbations of spherically-symmetric collapse to a black hole, starting from a diffuse, nearly-stationary configuration, where the bosonic part of the Lagrangian consists of Einstein gravity and the massless scalar field. As in Paper I, Feynman's $+i\epsilon$ approach is taken; this involves a rotation into the complex: $T\to {\mid}T{\mid} \exp (-i\theta)$, with $0<\theta\leq\pi/2$. A complex solution of the classical boundary-value problem is expected to exist, provided $\theta>0$; although for $\theta =0$ (Lorentzian time-separation), the classical boundary-value problem is badly posed. Once the amplitude is found for $\theta>0$, one can take the limit $\theta\to 0_+$ to find the Lorentzian amplitude. The paper also includes a discussion of adiabatic solutions of the scalar wave equation, needed for the spin-0 calculation. 
  Our earlier work on the quantum amplitude for a scalar field in black-hole evaporation, following gravitational collapse, is here extended to Maxwell theory. Boundary data are specified on initial and final space-like hypersurfaces $\Sigma_{I,F}$, separated by a large Lorentzian proper-time interval $T$, as measured at spatial infinity. The initial boundary data may be chosen (say) to be spherically symmetric, corresponding to a nearly-spherical configuration prior to gravitational collapse. The final data include the intrinsic 3-metric and scalar field, restricted to $\Sigma_F$, in addition to spin-1 data, naturally taken to be the magnetic field $B_i$ on $\Sigma_{I,F}  (i=1,2,3)$. For a locally-supersymmetric theory, the quantum amplitude should be proportional to $\exp(iS_{\rm class})$, apart from corrections which are very small when the frequencies in the boundary data are small compared to the Planck scale. Here, $S_{\rm class}$ is the action of the classical solution. The Lorentzian amplitude is found by taking the limit $\theta\to 0_+$. By a method similar to that used in the spin-0 case, one obtains the quantum amplitude for photon data on $\Sigma_F$. The magnetic boundary conditions are related by supersymmetry to the natural spin-2 (gravitational-wave) boundary conditions, which involve fixing the magnetic part of the Weyl tensor. 
  From more critical tests for gravitational (G) hypotheses it has been proved that the relative properties of nonlocal (NL) bodies at rest with respect to an observer depend on the difference of G potential between bodies and observer. The G energy comes not from the field but from a fraction of the mass-energy of the bodies. These results are in opposition with two traditional hypotheses used in Physics. In the classical tests of general relativity their errors are compensated because they have the same absolute value and opposite signs. The general theory based on such tests, called non-local relativity, is consistent with quantum mechanics and with all of the traditional G tests. The new cosmological scenario, which is radically different from the standard one, has also been verified from astronomical observations. 
  In this paper we deduce a quite general formula which allows the relation of clock rates at two different space time points to be discussed. In the case of a perturbed Robertson-Walker metric, our analysis leads to an equation for the comparison of clock rates at different cosmic space time points, which includes the Hubble redshift, the Doppler effect, the gravitational redshift and the Rees-Sciama effects. In the case of the solar system, when the 2PN metric is substituted into the general formula, the comparison of the clock rates on both the earth and a space station could be made. It might be useful for the discussion on the precise measurements on future ACES and ASTROD. 
  Since Wilson's work on lattice gauge theory in the 1970s, discrete versions of field theories have played a vital role in fundamental physics. But there is recent interest in certain higher dimensional analogues of gauge theory, such as p-form electromagnetism, including the Kalb-Ramond field in string theory, and its nonabelian generalizations. It is desirable to discretize such `higher gauge theories' in a way analogous to lattice gauge theory, but with the fundamental geometric structures in the discretization boosted in dimension. As a step toward studying discrete versions of more general higher gauge theories, we consider the case of p-form electromagnetism. We show that discrete p-form electromagnetism admits a simple algebraic description in terms of chain complexes of abelian groups. Moreover, the model allows discrete spacetimes with quite general geometry, in contrast to the regular cubical lattices usually associated with lattice gauge theory. After constructing a suitable model of discrete spacetime for p-form electromagnetism, we quantize the theory using the Euclidean path integral formalism. The main result is a description of p-form electromagnetism as a `chain field theory' -- a theory analogous to topological quantum field theory, but with chain complexes replacing manifolds. This, in particular, gives a notion of time evolution from one `spacelike slice' of discrete spacetime to another. 
  In our previous paper, based on the Carter & Quintana framework and the Damour-Soffel-Xu scheme, we deduced a complete and closed set of post-Newtonian dynamical equations for elastically deformable astronomical bodies. In this paper, we expand the general relativistic perturbation equations of elastic deformable bodies (field equations, stress-strain relation, Euler equation) in terms of Generalized Spherical Harmonics. This turns the set of complicated partial differential equations into a set of ordinary differential equations. This will be useful for numerical applications that mainly deal with the global dynamics of the Earth. 
  Quantum amplitudes for $s=2$ gravitational-wave perturbations of Einstein/scalar collapse to a black hole are treated by analogy with $s=1$ Maxwell perturbations. The spin-2 perturbations split into parts with odd and even parity. We use the Regge-Wheeler gauge; at a certain point we make a gauge transformation to an asymptotically-flat gauge, such that the metric perturbations have the expected falloff behaviour at large radii. By analogy with $s=1$, for $s=2$ natural 'coordinate' variables are given by the magnetic part $H_{ij} (i,j=1,2,3)$ of the Weyl tensor, which can be taken as boundary data on a final space-like hypersurface $\Sigma_F$. For simplicity, we take the data on the initial surface $\Sigma_I$ to be exactly spherically-symmetric. The (large) Lorentzian proper-time interval between $\Sigma_I$ and $\Sigma_F$, measured at spatial infinity, is denoted by $T$. We follow Feynman's $+i\epsilon$ prescription and rotate $T$ into the complex: $T\to{\mid}T{\mid} \exp(-i\theta)$, for $0<\theta\leq\pi/2$. The corresponding complexified {\it classical} boundary-value problem is expected to be well-posed. The Lorentzian quantum amplitude is recovered by taking the limit as $\theta\to 0_+$. For boundary data well below the Planck scale, and for a locally supersymmetric theory, this involves only the semi-classical amplitude $\exp(iS^{(2)}_{\rm class}$, where $S^{(2)}_{\rm class}$ denotes the second-variation classical action. The relations between the $s=1$ and $s=2$ natural boundary data, involving supersymmetry, are investigated using 2-component spinor language in terms of the Maxwell field strength $\phi_{AB}=\phi_{(AB)}$ and the Weyl spinor $\Psi_{ABCD}=\Psi_{(ABCD)}$. 
  We extend to the fermionic spin-1/2 case earlier work on quantum amplitudes arising from gravitational collapse to a black hole. Boundary data are specified on initial and final asymptotically-flat space-like hypersurfaces $\Sigma_{I,F}$, separated by a Lorentzian proper-time interval $T$, measured at spatial infinity. Following Feynman's $+i\epsilon$ prescription, one makes the problem well-posed by rotating $T$ into the complex: $T\to{\mid}T{\mid} \exp(-i\theta)$, with $0<\theta\leq\pi/2$. After calculating the amplitude for $\theta>0$, one takes the 'Lorentzian limit' $\theta\to 0_+$. In this paper, we treat quantum amplitudes for the case of fermionic massless spin-1/2 (neutrino) final boundary data; working in the holomorphic representation, we take these boundary data to be odd elements of a Grassmann algebra. Making use of boundary conditions originally developed for local supersymmetry, we find that this fermionic case can be treated in a way which parallels the bosonic case. With these boundary conditions, for $\theta > 0$, one obtains a unique fermionic classical solution, and we calculate its classical action as a functional of the fermionic data on the late-time surface $\Sigma_F$; the quantum amplitude follows straightforwardly from this. 
  In the present paper, using the Fourier analyze method, we study the behavior of the massless spin-1 particles in the teleparallel gravity, and show that the free-space Maxwell equations in the background of the Bianchi-I type space-time give the same results as obtained in teleparallel gravity. Also, using our results, we find the harmonic oscillator behavior of the solutions. 
  We study the phenomena of creation of massless spin-1 particles in the universe based on a Godel-type space-time model. First, we solve the massless Duffin-Kemmer-Petiau equation. Next, using the exact solution, we calculate, via the Bogoliubov transformations technique, the density number of massless spin-1 particles created. Furthermore, we also compute the oscillating region and the frequency spectrum of these particles for the background considered. In appendix, we show that the Maxwell equations give the same solution as the massless Duffin-Kemmer-Petiau equation. 
  We calculate QNMs of the scalar hairy black hole in the AdS background using Horowitz-Hubeny method for the potential that is not known in analytical form. For some black hole parameters we found pure imaginary frequencies. Increasing of the scalar field mass does not cause the imaginary part to vanish, it reaches some minimum and then increases, thus in the case under consideration the infinitely long living modes (quasi-resonances) do not appear. 
  Recently we have studied, using a boundary-value approach, quantum amplitudes resulting from gravitational collapse to a black hole. Suitable boundary data for all fields present are posed on initial and final space-like asymptotically flat hypersurfaces $\Sigma_{I,F}$. The Lorentzian proper-time separation between the surfaces, as measured at spatial infinity, is denoted by $T$. Following Feynman's $+i\epsilon$ approach, we rotate $T$ into the complex: $T\to {\mid}T{\mid} \exp(-i\theta)$, where $0<\theta\leq\pi/2$. The corresponding {\it classical} complex boundary-value problem is expected to be well-posed for $\theta > 0$. The Lorentzian amplitude is found by taking the limit $\theta \to 0_+$ of the quantum amplitude, itself closely approximated by the semi-classical expression $\exp(iS_{\rm class})$, where $S_{\rm class}$ is the classical action. For given weak anisotropic spin-0 and spin-2 boundary data on $\Sigma_F$, one can compute an effective classical energy-momentum tensor in the interior, which has been averaged over several wave-lengths of the radiation. This averaged extra contribution will be spherically symmetric, equivalent to a null fluid, and describing the radial outward streaming of the radiation (of quantum origin). The corresponding space-time metric, in this region containing radially-outgoing radiation, is of the Vaidya form. This, in turn, justifies the treatment of the adiabatic radial mode equations, for spins $s=0$ and $s=2$, which is used throughout this larger project. 
  The active mass density in Einstein's theory of gravitation in the analog of Poisson's equation in a local inertial system is proportional to $\rho+3p/c^2$. Here $\rho$ is the density of energy and $p$ its pressure for a perfect fluid. By using exact solutions of Einstein's field equations in the static case we study whether the pressure term contributes towards the mass. 
  Quantum mechanics in its presently known formulation requires an external classical time for its description. A classical spacetime manifold and a classical spacetime metric are produced by classical matter fields. In the absence of such classical matter fields, quantum mechanics should be formulated without reference to a classical time. If such a new formulation exists, it follows as a consequence that standard linear quantum mechanics is a limiting case of an underlying non-linear quantum theory. A possible approach to the new formulation is through the use of noncommuting spacetime coordinates in noncommutative differential geometry. Here, the non-linear theory is described by a non-linear Schrodinger equation which belongs to the Doebner-Goldin class of equations, discovered some years ago. This mass-dependent non-linearity is significant when particle masses are comparable to Planck mass, and negligible otherwise. Such a non-linearity is in principle detectable through experimental tests of quantum mechanics for mesoscopic systems, and is a valuable empirical probe of theories of quantum gravity. We also briefly remark on the possible connection our approach could have with loop quantum gravity and string theory. 
  Our boundary-value approach to quantum processes in the gravitational collapse to a black hole leads to quantum amplitudes (not just probabilities) for transitions between data posed on initial and final hypersurfaces $\Sigma_{I,F}$, separated by a Lorentzian proper-time interval $T$, measured at spatial infinity. Following Feynman's $+i\epsilon$ approach, we rotate: $T\to {\mid}T{\mid} \exp(-i\theta)$, for $0<\theta\leq\pi/2$. The {\it classical} complexified boundary-value problem is expected to be well-posed for $0<\theta\leq\pi/2$, with classical action $S_{\rm class}$. For a locally supersymmetric Lagrangian, containing supergravity, possibly coupled to supermatter, the resulting quantum amplitude will be proportional to $\exp(iS_{\rm class})$, apart from possible loop corrections which are negligible for boundary data with frequencies below the Planck scale. The Lorentzian quantum amplitude is recovered by taking the limit as $\theta\to 0_+$ of this amplitude. In the present paper, a connection is made between this boundary value approach and the original approach to quantum evaporation in gravitational collapse to a black hole, {\it via} Bogoliubov coefficients. This connection is developed through consideration of the radial equation obeyed by the (adiabatic) non-spherical classical perturbations. When one studies the resulting final probability distribution, based on our quantum amplitudes above, one finds that this distribution can also be interpreted in terms of the Wigner quasi-probability distribution for harmonic oscillators. 
  It is dealt with the question, under which circumstances the canonical Noether stress-energy tensor is equivalent to the gravitational (Hilbert) tensor for general matter fields under the influence of gravity. In the framework of general relativity, the full equivalence is established for matter fields that do not couple to the metric derivatives. Spinor fields are included into our analysis by reformulating general relativity in terms of tetrad fields, and the case of Poincare gauge theory, with an additional, independent Lorentz connection, is also investigated. Special attention is given to the flat limit, focusing on the expressions for the matter field energy (Hamiltonian). The Dirac-Maxwell system is investigated in detail, with special care given to the separation of free (kinetic) and interaction (or potential) energy. Moreover, the stress-energy tensor of the gravitational field itself is briefly discussed. 
  The Astrodynamical Space Test of Relativity using Optical Devices (ASTROD) mission consists of three spacecraft in separate solar orbits and carries out laser interferometric ranging. ASTROD aims at testing relativistic gravity, measuring the solar system and detecting gravitational waves. Because of the larger arm length, the sensitivity of ASTROD to gravitational waves is estimated to be about 30 times better than Laser Interferometer Space Antenna (LISA) in the frequency range lower than about 0.1 mHz. ASTROD I is a simple version of ASTROD, employing one spacecraft in a solar orbit. It is the first step for ASTROD and serves as a technology demonstration mission for ASTROD. In addition, several scientific results are expected in the ASTROD I experiment. The required acceleration noise level of ASTROD I is 10^-13 m s^-2 Hz^{-1/2} at the frequency of 0.1 mHz. In this paper, we focus on local gravity gradient noise that could be one of the largest acceleration disturbances in the ASTROD I experiment. We have carried out gravitational modelling for the current test-mass design and simplified configurations of ASTROD I by using an analytical method and the Monte Carlo method. Our analyses can be applied to figure out the optimal designs of the test mass and the constructing materials of the spacecraft, and the configuration of compensation mass to reduce local gravity gradients. 
  In earlier papers, it is found that the Ricci scalar behaves in dual manner (i) like a matter field and (ii) like a geometrical field. Using dual role of the Ricci scalar, inhomogeneous cosmological models are derived. The essential features of these models is capability of these to exhibit gravitational effect of compact objects also in an expanding universe. Here, production of spinless and spin-1/2 particles are demonstrated in these models. 
  In the context of the natural splitting, the standard relative dynamics can be expressed in terms of gravito-electromagnetic fields, which allow to formally introduce a gravito-magnetic Aharonov-Bohm effect. We showed elsewhere that this formal analogy can be used to derive the Sagnac effect in flat space-time as a gravito-magnetic Aharonov-Bohm effect. Here, we generalize those results to study the General Relativistic corrections to the Sagnac effect in some stationary and axially symmetric geometries, such as the space-time around a weakly gravitating and rotating source, Kerr space-time, G\"{odel} universe and Schwarzschild space-time. 
  In a recent paper published in Classical and Quantum Gravity, 2004, vol. 21, p. 3803 Carlip used a vector-tensor theory of gravity to calculate the Shapiro time delay by a moving gravitational lens. He claimed that the relativistic correction of the order of v/c beyond the static part of the Shapiro delay depends on the speed of light c and, hence, the Fomalont-Kopeikin experiment is not sensitive to the speed of gravity c_g. In this letter we analyze Carlip's calculation and demonstrate that it implies a gravitodynamic (non-metric) system of units based on the principle of the constancy of the speed of gravity but it is disconnected from the practical method of measurement of astronomical distances based on the principle of the constancy of the speed of light and the SI metric (electrodynamic) system of units. Re-adjustment of theoretically-admissible but practically unmeasurable Carlip's coordinates to the SI metric system of units used in JPL ephemeris, reveals that the velocity-dependent correction to the static part of the Shapiro time delay does depend on the speed of gravity c_g as shown by Kopeikin in Classical and Quantum Gravity, 2004, vol. 21, p. 1. This analysis elucidates the importance of employing the metric system of units for physically meaningful interpretation of gravitational experiments. 
  We investigate particle detector responses in some topologically non-trivial spacetimes. We extend a recently proposed regularization of the massless scalar field Wightman function in 4-dimensional Minkowski space to arbitrary dimension, to the massive scalar field, to quotients of Minkowski space under discrete isometry groups and to the massless Dirac field. We investigate in detail the transition rate of inertial and uniformly accelerated detectors on the quotient spaces under groups generated by $(t,x,y,z)\mapsto(t,x,y,z+2a)$, $(t,x,y,z)\mapsto(t,-x,y,z)$, $(t,x,y,z)\mapsto(t,-x,-y,z)$, $(t,x,y,z)\mapsto(t,-x,-y,z+a)$ and some higher dimensional generalizations. For motions in at constant $y$ and $z$ on the latter three spaces the response is time dependent. We also discuss the response of static detectors on the RP^3 geon and inertial detectors on RP^3 de Sitter space via their associated global embedding Minkowski spaces (GEMS). The response on RP^3 de Sitter space, found both directly and in its GEMS, provides support for the validity of applying the GEMS procedure to detector responses and to quotient spaces such as RP^3 de Sitter space and the RP^3 geon where the embedding spaces are Minkowski spaces with suitable identifications. 
  We show the equivalence of the Lorentz-covariant canonical formulation considered for the Immirzi parameter $\beta=i$ to the selfdual Ashtekar gravity. We also propose to deal with the reality conditions in terms of Dirac brackets derived from the covariant formulation and defined on an extended phase space which involves, besides the selfdual variables, also their anti-selfdual counterparts. 
  In this paper, we prove that the 5-dimensional Schwarzschild-Tangherlini solution of the Einstein vacuum equations is orbitally stable (in the fully non-linear theory) with respect to vacuum perturbations of initial data preserving triaxial Bianchi IX symmetry. More generally, we prove that 5-dimensional vacuum spacetimes developing from suitable asymptotically flat triaxial Bianchi IX symmetric data and containing a trapped or marginally trapped homogeneous 3-surface possess a complete null infinity whose past is bounded to the future by a regular event horizon, whose cross-sectional volume in turn satisfies a Penrose inequality, relating it to the final Bondi mass. In particular, the results of this paper give the first examples of vacuum black holes which are not stationary exact solutions. 
  We investigate the possibility that a background independent quantum theory of gravity is not a theory of quantum geometry. We provide a way for global spacetime symmetries to emerge from a background independent theory without geometry. In this, we use a quantum information theoretic formulation of quantum gravity and the method of noiseless subsystems in quantum error correction. This is also a method that can extract particles from a quantum geometric theory such as a spin foam model. 
  As an inverse problem, we recover the topology of the effective spacetime that a system lies in, in an operational way. This means that from a series of experiments we get a set of points corresponding to events. This continues the previous work done by the authors. Here we use the existence of upper bound in the speed of transfer of matter and information to induce a partial order on the set of events. While the actual partial order is not known in our operational set up, the grouping of events to (unordered) subsets corresponding to possible histories, is given. From this we recover the partial order up to certain ambiguities that are then classified. Finally two different ways to recover the topology are sketched and their interpretation is discussed. 
  Wisdom has recently unveiled a new relativistic effect, called ``spacetime swimming'', where quasi-rigid free bodies in curved spacetimes can "speed up", "slow down" or "deviate" their falls by performing "local" cyclic shape deformations. We show here that for fast enough cycles this effect dominates over a non-relativistic related one, named here ``space swinging'', where the fall is altered through "nonlocal" cyclic deformations in Newtonian gravitational fields. We expect, therefore, to clarify the distinction between both effects leaving no room to controversy. Moreover, the leading contribution to the swimming effect predicted by Wisdom is enriched with a higher order term and the whole result is generalized to be applicable in cases where the tripod is in large red-shift regions. 
  By the inclusion of an additional term, non-linear in the scalar curvature $R$, it is tested if dark energy could rise as a geometrical effect in 3D gravitational formulations. We investigate a cosmological fluid obeying a non-polytropic equation of state (the van der Waals equation) that is used to construct the energy-momentum tensor of the sources, representing the hypothetical inflaton in gravitational interaction with a matter contribution.  Following the evolution in time of the scale factor, its acceleration, and the energy densities of constituents it is possible to construct the description of an inflationary 3D universe, followed by a matter dominated era. For later times it is verified that, under certain conditions, the non-linear term in $R$ can generate the old 3D universe in accelerated expansion, where the ordinary matter is represented by the barotropic limit of the van der Waals constituent. 
  To determine whether the Shapiro time delay of light passing near a moving object depends on the ``speed of gravity'' or the ``speed of light,'' one must analyze observations in a bimetric framework in which these two speeds can be different. In a recent comment (gr-qc/0510048), Kopeikin has argued that such a computation -- described in gr-qc/0403060 -- missed a hidden dependence on the speed of gravity. By analyzing the observables in the relevant bimetric model, I show that this claim is incorrect, and that the conclusions of gr-qc/0403060 stand. 
  The future LISA detector will constitute the prime instrument for high-precision gravitational wave observations.LISA is expected to provide information for the properties of spacetime in the vicinity of massive black holes which reside in galactic nuclei.Such black holes can capture stellar-mass compact objects, which afterwards slowly inspiral,radiating gravitational waves.The body's orbital motion and the associated waveform carry information about the spacetime metric of the massive black hole,and it is possible to extract this information and experimentally identify (or not!) a Kerr black hole.In this paper we lay the foundations for a practical `spacetime-mapping' framework. Our work is based on the assumption that the massive body is not necessarily a Kerr black hole, and that the vacuum exterior spacetime is stationary axisymmetric,described by a metric which deviates slightly from the Kerr metric. We first provide a simple recipe for building such a `quasi-Kerr' metric by adding to the Kerr metric the deviation in the value of the quadrupole moment. We then study geodesic motion in this metric,focusing on equatorial orbits. We proceed by computing `kludge' waveforms which we compare with their Kerr counterparts. We find that a modest deviation from the Kerr metric is sufficient for producing a significant mismatch between the waveforms, provided we fix the orbital parameters. This result suggests that an attempt to use Kerr waveform templates for studying EMRIs around a non-Kerr object might result in serious loss of signal-to-noise ratio and total number of detected events. The waveform comparisons also unveil a `confusion' problem, that is the possibility of matching a true non-Kerr waveform with a Kerr template of different orbital parameters. 
  Properties of (skew-symmetric) conformal Yano--Killing tensors are reviewed. Explicit forms of three symmetric conformal Killing tensors in Kerr spacetime are obtained from the Yano--Killing tensor. The relation between spin-2 fields and solutions to the Maxwell equations is used in the construction of a new conserved quantity which is quadratic in terms of the Weyl tensor. The formula obtained is similar to the functional obtained from the Bel--Robinson tensor and is examined in Kerr spacetime. A new interpretation of the conserved quantity obtained is proposed. 
  In addition to the pericentre \omega, the mean anomaly M and, thus, the mean longitude \lambda, also the orbital period Pb and the mean motion $n$ of a test particle are modified by the Dvali-Gabadadze-Porrati gravity. While the correction to Pb depends on the mass of the central body and on the geometrical features of the orbital motion around it, the correction to $n$ is independent of them, up to terms of second order in the eccentricity $e$. The latter one amounts to about 2\times 10^-3 arcseconds per century. The present-day accuracy in determining the mean motions of the inner planets of the Solar System from radar ranging and differential Very Long Baseline Interferometry is 10^-2-5\times 10^-3 arcseconds per century, but it should be improved in the near future when the data from the spacecraft to Mercury and Venus will be available. 
  It is shown that there exists, in the field theory of gravitation, contrary to the General Theory of Relativity (GTR), a bound for admissible time slowing down by the gravitational field which excludes a possibility of unbounded compression of matter by the gravity forces. 
  We discuss the case of massive gravitons and their relation with the cosmological constant, considered as an eigenvalue of a Sturm-Liouville problem. A variational approach with Gaussian trial wave functionals is used as a method to study such a problem. We approximate the equation to one loop in a Schwarzschild background and a zeta function regularization is involved to handle with divergences. The regularization is closely related to the subtraction procedure appearing in the computation of Casimir energy in a curved background. A renormalization procedure is introduced to remove the infinities together with a renormalization group equation. 
  The cosmological constant appearing in the Wheeler-De Witt equation is considered as an eigenvalue of the associated Sturm-Liouville problem. A variational approach with Gaussian trial wave functionals is used as a method to study such a problem. We approximate the equation to one loop in a Schwarzschild background and a zeta function regularization is involved to handle with divergences. The regularization is closely related to the subtraction procedure appearing in the computation of Casimir energy in a curved background. A renormalization procedure is introduced to remove the infinities together with a renormalization group equation. The case of massive gravitons is discussed. 
  Consider a distant source of light such that the photons emitted from it have the same constant 4-momentum in the spacetime region under consideration. Every inertial observer can set up a semitransparent screen perpendicular to the direction of light. An opaque object projects a shadow that the observers can see in their respective screens. We point out that the Poincar\'e group of 1+3 Minkowski spacetime has a 1+2 Galilean quotient subgroup and that the `translational' generators $W_1$, $W_2$, of the little group for massless particles generate Galilean boosts. Using this fact we show that the shadow has the same shape in all the screens and that the transformation that relates two screens of different inertial observers is a Galilean transformation of 1+2 spacetime enlarged to include time dilations. If the two observers measure the same frequency then the transformation is a Galilean transformation. As a consequence, a Minkowskian 1+3 physical theory admitting a symmetry in a null direction is equivalent to a 1+2 Galilean theory. 
  We algebraically classify some higher dimensional spacetimes, including a number of vacuum solutions of the Einstein field equations which can represent higher dimensional black holes. We discuss some consequences of this work. 
  In this paper the currently held view that the endpoint of gravitational collapse is a singularity is refuted. A quantum mechanical calculation is done showing that spin 0 and spin 1/2 particles inside a black hole's schwarzschild radius aren't confined to an infinitesimal point but form bound state orbits. As with the case of electric collapse, if an electron cannot spiral into a nucleus, then neither does this happen in plasma consisting of many electrons and nuclei. Showing that after undergoing gravitational collapse, like plasma, matter will not contract to an infinitesimal point. That quantum effects prevent singularity formation. 
  We study in detail the quasinormal modes of linear gravitational perturbations of plane-symmetric anti-de Sitter black holes. The wave equations are obtained by means of the Newman-Penrose formalism and the Chandrasekhar transformation theory. We show that oscillatory modes decay exponentially with time such that these black holes are stable against gravitational perturbations. Our numerical results show that in the large (small) black hole regime the frequencies of the ordinary quasinormal modes are proportional to the horizon radius $r_{+}$ (wave number $k$). The frequency of the purely damped mode is very close to the algebraically special frequency in the small horizon limit, and goes as $ik^{2}/3r_{+}$ in the opposite limit. This result is confirmed by an analytical method based on the power series expansion of the frequency in terms of the horizon radius. The same procedure applied to the Schwarzschild anti-de Sitter spacetime proves that the purely damped frequency goes as $i(l-1)(l+2)/3r_{+}$, where $l$ is the quantum number characterizing the angular distribution. Finally, we study the limit of high overtones and find that the frequencies become evenly spaced in this regime. The spacing of the frequency per unit horizon radius seems to be a universal quantity, in the sense that it is independent of the wave number, perturbation parity and black hole size. 
  We discuss prospects for direct measurement of stochastic gravitational wave background around 0.1-1Hz with future space missions. It is assumed to use correlation analysis technique with the optimal TDI variables for two sets of LISA-type interferometers. The signal to noise for detection of the background and the estimation errors for its basic parameters (amplitude, spectral index) are evaluated for proposed missions. 
  Einstein gravitation theory can be extended by preserving its geometrical nature but changing the relation between curvature and energy-momentum tensors. This change accounts for radiative corrections, replacing the Newton gravitation constant by two running couplings which depend on scale and differ in the two sectors of traceless and traced tensors. The metric and curvature tensors in the field of the Sun, which were obtained in previous papers within a linearized approximation, are then calculated without this restriction. Modifications of gravitational effects on geodesics are then studied, allowing one to explore phenomenological consequences of extensions lying in the vicinity of general relativity. Some of these extended theories are able to account for the Pioneer anomaly while remaining compatible with tests involving the motion of planets. The PPN Ansatz corresponds to peculiar extensions of general relativity which do not have the ability to meet this compatibility challenge. 
  We study the stability under linear perturbations of a class of static solutions of Einstein-Gauss-Bonnet gravity in $D=n+2$ dimensions with spatial slices of the form $\Sigma_{\k}^n \times {\mathbb R}^+$, $\Sigma_{\k}^n$ an $n-$manifold of constant curvature $\k$. Linear perturbations for this class of space-times can be generally classified into tensor, vector and scalar types. In a previous paper, tensor perturbations were analyzed. In this paper we study vector and scalar perturbations. We show that vector perturbations can be analyzed in general using an S-deformation approach and do not introduce instabilities. On the other hand, we show by analyzing an explicit example that, contrary to what happens in Einstein gravity, scalar perturbations may lead to instabilities in black holes with spherical horizons when the Gauss-Bonnet string corrections are taken into account. 
  We find an exact solution in dimensionally continued gravity in arbitrary dimensions which describes the gravitational collapse of a null dust fluid. Considering the situation that a null dust fluid injects into the initially anti-de Sitter spacetime, we show that a naked singularity can be formed. In even dimensions, a massless ingoing null naked singularity emerges. In odd dimensions, meanwhile, a massive timelike naked singularity forms. These naked singularities can be globally naked if the ingoing null dust fluid is switched off at a finite time; the resulting spacetime is static and asymptotically anti-de Sitter spacetime. The curvature strength of the massive timelike naked singularity in odd dimensions is independent of the spacetime dimensions or the power of the mass function. This is a characteristic feature in Lovelock gravity. 
  We study the scalar perturbations of rotating black holes in framework of extra dimensions type Randall-Sundrum(RS). 
  The status of experimental tests of general relativity and of theoretical frameworks for analysing them is reviewed. Einstein's equivalence principle (EEP) is well supported by experiments such as the Eotvos experiment, tests of special relativity, and the gravitational redshift experiment. Future tests of EEP and of the inverse square law are searching for new interactions arising from unification or quantum gravity. Tests of general relativity at the post-Newtonian level have reached high precision, including the light deflection, the Shapiro time delay, the perihelion advance of Mercury, and the Nordtvedt effect in lunar motion. Gravitational-wave damping has been detected in an amount that agrees with general relativity to better than half a percent using the Hulse-Taylor binary pulsar, and other binary pulsar systems have yielded other tests, especially of strong-field effects. When direct observation of gravitational radiation from astrophysical sources begins, new tests of general relativity will be possible. 
  We establish an algorithm that produces a new solution to the Einstein field equations, with an anisotropic matter distribution, from a given seed isotropic solution. The new solution is expressed in terms of integrals of known functions, and the integration can be completed in principle. The applicability of this technique is demonstrated by generating anisotropic isothermal spheres and anisotropic constant density Schwarzschild spheres. Both of these solutions are expressed in closed form in terms of elementary functions, and this facilitates physical analysis. 
  The first order post Newtonian scheme in multiple systems presented by Damour-Soffel-Xu is extended to the second order one for light propagation without changing the advantage of the scheme on the linear partial differential equations of potential and vector potential. The spatial components of the metric tensor are extended to the second order level both in the global coordinates ($q_{ij}/c^4$ term) and in a local coordinates ($Q_{ab}/c^4$ term). The equations of $q_{ij}$ (or $Q_{ab}$) are deduced from Einstein field equations. The linear relationship between $q_{ij}$ and $Q_{ab}$ are presented also. The 2PN equations of light ray based on the extended scheme are deduced by means of the iterative method. We also use parametrized second post Newtonian metric tensor to substitute into the null geodetic equations to obtain the parametrized second order equations of light ray which might be useful in the observation and measurement in the future space missions. 
  We study which geometric structure can be constructed from the vierbein (frame/coframe) variables and which field models can be related to this geometry. The coframe field models, alternative to GR, are known as viable models for gravity, since they have the Schwarzschild solution. Since the local Lorentz invariance is violated, a physical interpretation of additional six degrees of freedom is required. The geometry of such models is usually given by two different connections -- the Levi-Civita symmetric and metric-compatible connection and the Weitzenbock flat connection. We construct a general family of linear connections of the same type, which includes two connections above as special limiting cases. We show that for dynamical propagation of six additional degrees of freedom it is necessary for the gauge field of infinitesimal transformations (antisymmetric tensor) to satisfy the system of two first order differential equations. This system is similar to the vacuum Maxwell system and even coincides with it on a flat manifold. The corresponding ``Maxwell-compatible connections'' are derived. Alternatively, we derive the same Maxwell-type system as a symmetry conditions of the viable models Lagrangian. Consequently we derive a nontrivial decomposition of the coframe field to the pure metric field plus a dynamical field of infinitesimal Lorentz rotations. Exact spherical symmetric solution for our dynamical field is derived. It is bounded near the Schwarzschild radius. Further off, the solution is close to the Coulomb field. 
  We explicitly construct the metric of a Kerr black hole that is tidally perturbed by the external universe in the slow-motion approximation. This approximation assumes that the external universe changes slowly relative to the rotation rate of the hole, thus allowing the parameterization of the Newman-Penrose scalar $\psi_0$ by time-dependent electric and magnetic tidal tensors. This approximation, however, does not constrain how big the spin of the background hole can be and, in principle, the perturbed metric can model rapidly spinning holes. We first generate a potential by acting with a differential operator on $\psi_0$. From this potential we arrive at the metric perturbation by use of the Chrzanowski procedure in the ingoing radiation gauge. We provide explicit analytic formulae for this metric perturbation in spherical Kerr-Schild coordinates, where the perturbation is finite at the horizon. This perturbation is parametrized by the mass and Kerr spin parameter of the background hole together with the electric and magnetic tidal tensors that describe the time evolution of the perturbation produced by the external universe. In order to take the metric accurate far away from the hole, these tidal tensors should be determined by asymptotically matching this metric to another one valid far from the hole. The tidally perturbed metric constructed here could be useful in initial data constructions to describe the metric near the horizons of a binary system of spinning holes. This perturbed metric could also be used to construct waveforms and study the absorption of mass and angular momentum by a Kerr black hole when external processes generate gravitational radiation. 
  Experimental verification of the existence of gravitomagnetic fields generated by currents of matter is important for a complete understanding and formulation of gravitational physics. Although the rotational (intrinsic) gravitomagnetic field has been extensively studied and is now being measured by the Gravity Probe B, the extrinsic gravitomagnetic field generated by the translational current of matter is less well studied. The present paper uses the post-Newtonian parametrized Einstein and light geodesics equations to show that the extrinsic gravitomagnetic field generated by the translational current of matter can be measured by observing the relativistic time delay and/or light deflection caused by the moving mass. We prove that the extrinsic gravitomagnetic field is generated by the relativistic effect of the aberration of the gravity force caused by the Lorentz transformation of the metric tensor and the Levi-Civita connection. We show that the Lorentz transformation of the gravity field variables is equivalent to the technique of the retarded Lienard-Wiechert gravitational potentials predicting that a light particle is deflected by gravitational field of a moving body from its retarded position so that both general-relativistic phenomena -- the aberration and the retardation of gravity -- are tightly connected and prove that gravity has a causal nature. We explain in this framework the 2002 deflection experiment of a quasar by Jupiter where the aberration of gravity from its orbital motion was measured with accuracy 20%. We describe a theory of VLBI experiments to measure the gravitational deflection of radio waves from a quasar by the Sun, as viewed by a moving observer from the geocentric frame, to improve the measurement accuracy of the aberration of gravity to a few percent. 
  We examine the importance of second order corrections to linearized cosmological perturbation theory in an inflationary background, taken to be a spatially flat FRW spacetime. The full second order problem is solved in the sense that we evaluate the effect of the superhorizon second order corrections on the inhomogeneous and homogeneous modes of the linearized flucuations. These second order corrections enter in the form of a {\it cumulative} contribution from {\it all} of their Fourier modes. In order to quantify their physical significance we study their effective equation of state by looking at the perturbed energy density and isotropic pressure to second order. We define the energy density (isotropic pressure) in terms of the (averaged) eigenvalues associated with timelike (spacelike) eigenvectors of a total stress energy for the metric and matter fluctuations. Our work suggests that that for many parameters of slow-roll inflation, the second order contributions to these energy density and pressures may dominate over the first order effects for the case of super-Hubble evolution. These results hold in our choice of first and second order coordinate conditions however we also argue that other `reasonable` coordinate conditions do not alter the relative importance of the second order terms. We find that these second order contributions approximately take the form of a cosmological constant in this coordinate gauge, as found by others using effective methods. 
  A quantum inequality bound on the expectation value of the null-contracted stress tensor in an arbitrary Hadamard state is used to obtain constraints on the geometries of traversable wormholes. Particular attention is given to the wormhole models of Visser, Kar, and Dadhich (VKD) and to those of Kuhfittig. These are models which use arbitrarily small amounts of exotic matter for wormhole maintenance. It is shown that macroscopic VKD models are either ruled out or severely constrained, while a recent model of Kuhfittig is shown to be, in fact, non-traversable. 
  The present work has a double aim. On the one hand we call attention on the relationship existing between the Ashtekar formalism and other gauge-theoretical approaches to gravity, in particular the Poincar\'e Gauge Theory. On the other hand we study two kinds of solutions for the constraints of General Relativity, consisting of two mutually independent parts, namely a general three-metric-dependent contribution to the extrinsic curvature $K_{ab}$ in terms of the Cotton-York tensor, and besides it further metric independent contributions, which we analyze in particular in the presence of isotropic three-metrics. 
  The Pioneer 10/11 spacecraft yielded the most precise navigation in deep space to date. However, their radio-metric tracking data has consistently indicated the presence of a small, anomalous, Doppler frequency drift. The drift is a blue-shift, uniformly changing with a rate of ~6 x 10^{-9} Hz/s and can be interpreted as a constant sunward acceleration of each particular spacecraft of a_P =(8.74 +/- 1.33) x 10^{-10} m/s^2. The nature of this anomaly remains unexplained. Here we summarize our current knowledge of the discovered effect and review some of the mechanisms proposed for its explanation. Currently we are preparing for the analysis of the entire set of the available Pioneer 10/11 Doppler data which may shed a new light on the origin of the anomaly. We present a preliminary assessment of such an intriguing possibility. 
  Since the proposal of the AdS/CFT correspondence, made by Maldacena and Witten, there has been some controversy about the definition of conserved Noether charges associated to asymptotic isometries in asymptotically AdS spacetimes, namely, whether they form an anomalous (i.e., a nontrivial central extension) representation of the Lie algebra of the conformal group in odd bulk dimensions or not. In the present work, we shall review the derivation of these charges by using covariant phase space techniques, emphasizing the principle of locality underlying it. We shall also comment on how these issues manifest themselves in the quantum setting. 
  We consider a simple physical model for an evolving horizon that is strongly interacting with its environment, exchanging arbitrarily large quantities of matter with its environment in the form of both infalling material and outgoing Hawking radiation. We permit fluxes of both lightlike and timelike particles to cross the horizon, and ask how the horizon grows and shrinks in response to such flows. We place a premium on providing a clear and straightforward exposition with simple formulae.   To be able to handle such a highly dynamical situation in a simple manner we make one significant physical restriction, that of spherical symmetry, and two technical mathematical restrictions: (1) We choose to slice the spacetime in such a way that the space-time foliations (and hence the horizons) are always spherically symmetric. (2) Furthermore we adopt Painleve-Gullstrand coordinates (which are well suited to the problem because they are nonsingular at the horizon) in order to simplify the relevant calculations.   We find particularly simple forms for surface gravity, and for the first and second law of black hole thermodynamics, in this general evolving horizon situation. Furthermore we relate our results to Hawking's apparent horizon, Ashtekar et al's isolated and dynamical horizons, and Hayward's trapping horizons. The evolving black hole model discussed here will be of interest, both from an astrophysical viewpoint in terms of discussing growing black holes, and from a purely theoretical viewpoint in discussing black hole evaporation via Hawking radiation. 
  The extremely high precision of current observations demands a much better theoretical treatment of relativistic effects in the propagation of electromagnetic signals through variable gravitational fields of isolated astronomical systems emitted gravitational waves. A consistent approach giving a complete and exhaustive solution to this problem in the first post-Minkowskian approximation of General Relativity is presented in this paper. 
  The gravitational field exterior respectively interior to a metrically stationary, axially symmetric, isolated spinning source made of perfect fluid is examined within the quasi-metric framework. (A metrically stationary system is defined as a system which is stationary except for the direct effects of the global cosmic expansion on the space-time geometry.) Field equations are set up and solved approximately for the exterior part. The gravitomagnetic part of the found metric family corresponds with the Kerr metric in the metric approximation. On the other hand, the gravitoelectric part of the found metric family includes a tidal term describing the effect of source deformation due to the rotation. This tidal term has a counterpart in Newtonian gravitation, but not in the Kerr metric. Finally the predicted geodesic effect for a gyroscope in a circular orbit is calculated. There is a correction term, unfortunately barely too small to be detectable by Gravity Probe B, to the standard result. 
  Ricci scalar is the key ingredient of non-Newtonian theory of gravity, where space-time geometry has a crucial role. Normally, it is supposed to be a geometrical field, but interestingly it also behaves like a physical field. Thus it plays dual role in the arena of gravitation. This article is an overview of the work related to dual roles of the Ricci scalar. A scalar is a mathematical concept representing a spinless particle.Here, particle concept, manifesting the physical aspect of the Ricci scalar, is termed as riccion.It is a scalar particle with (mass)$^2$ inversely proportional to the gravitational constant. Many intereseting consequences of dual role of the Ricci scalar are discussed here. It causes inflationary scenario in the early universe without taking another scalar like ``inflaton''. It is found that a riccion behaves like an instanton also. This feature inspires``primordial inflation''.It is interesting to see that a riccion, obtained from higher-dimensional space-time, decouples into fermion and anti-fermion pair, if parity is violated. One-loop renormalization of riccion indicates fractal geometry at high energy.Homogeneous and inhomogeneous models of the early universe are derived using dual role of the Ricci scalar. Production of spinless and spin-1/2 particles, due to riccion coupling, is discussed here. Contribution of riccion to the cosmic dark energy is obtained here through one-loop renormalization and it is shown that dark energy decays to dark matter during expansion of the universe. It inspires a new cosmological scenario consistent with observational evidences. 
  We analyze the effects induced by the bulk viscosity on the dynamics associated to the extreme gravitational collapse. Aim of the work is to investigate whether the presence of viscous corrections to the evolution of a collapsing gas cloud influence the fragmentation process. To this end we study the dynamics of a uniform and spherically symmetric cloud with corrections due to the negative pressure contribution associated to the bulk viscosity phenomenology. Within the framework of a Newtonian approach (whose range of validity is outlined), we extend to the viscous case either the Lagrangian, either the Eulerian motion of the system and we treat the asymptotic evolution in correspondence to a viscosity coefficient of the form $\zeta=\zeta_0 \rho^{nu}$ ($\rho$ being the cloud density and $\zeta_0=const.$). We show how, in the adiabatic-like behavior of the gas (i.e. when the politropic index takes values $4/3<\gamma\leq5/3$), density contrasts acquire, asymptotically, a vanishing behavior which prevents the formation of sub-structures. We can conclude that in the adiabatic-like collapse the top down mechanism of structures formation is suppressed as soon as enough strong viscous effects are taken into account. Such a feature is not present in the isothermal-like (i.e. $1\leq\gamma<4/3$) collapse because the sub-structures formation is yet present and outlines the same behavior as in the non-viscous case. We emphasize that in the adiabatic-like collapse the bulk viscosity is also responsible for the appearance of a threshold scale beyond which perturbations begin to increase. 
  According to general relativity, the present analysis shows on geometrical grounds that the cosmological constant problem is an artifact due to the unfounded link of this fundamental constant to vacuum energy density of quantum fluctuations. 
  Nonlinear realizations of spacetime groups are presented as a versatile mathematical tool providing a common foundation for quite different formulations of gauge theories of gravity. We apply nonlinear realizations in particular to both the Poincar\'e and the affine group in order to develop Poincar\'e gauge theory (PGT) and metric-affine gravity (MAG) respectively. Regarding PGT, two alternative nonlinear treatments of the Poincar\'e group are developed, one of them being suitable to deal with the Lagrangian and the other one with the Hamiltonian version of the same gauge theory. We argue that our Hamiltonian approach to PGT is closely related to Ashtekar's approach to gravity. On the other hand, a brief survey on MAG clarifies the role played by the metric--affine metric tensor as a Goldsone field. All gravitational quantities in fact --the metric as much as the coframes and connections-- are shown to acquire a simple gauge--theoretical interpretation in the nonlinear framework. 
  A panoramic view, preceded by a short background of Newtonian mechanics and Maxwellian electrodynamics, is offered on the extent of how Einstein's space-time geometry, believed to be central to an understanding of the structure of the universe, is overshadowed by several hitherto unheard of features like dark matter and dark energy, that seem to be necessary, but by no means sufficient, for a more complete picture. 
  In the perspective of unifying quantum field theories with general relativity,the equations of the internal dynamics of the vacuum and mass structures of a set of interacting particles are proved to be in one-to-one correspondence with the equations of general relativity. This leads us to envisage a high value for the cosmological constant,as expected theoretically. 
  Observers at rest in a stationary spacetime flat at infinity can measure small amounts of rest-mass+internal energies+kinetic energies+pressure energy in a small volume of fluid attached to a local inertial frame. The sum of these small amounts is the total "matter energy" for those observers. The total mass-energy minus the matter energy is the binding gravitational energy.   Misner, Thorne and Wheeler evaluated the gravitational energy of a spherically symmetric static spacetime. Here we show how to calculate gravitational energy in any static and stationary spacetime for isolated sources with a set of observers at rest.   The result of MTW is recovered and we find that electromagnetic and gravitational 3-covariant energy densities in conformastatic spacetimes are of opposite signs. Various examples suggest that gravitational energy is negative in spacetimes with special symmetries or when the energy-momentum tensor satisfies usual energy conditions. 
  We describe several new ways of specifying the behaviour of Lemaitre-Tolman (LT) models, in each case presenting the method for obtaining the LT arbitrary functions from the given data, and the conditions for existence of such solutions. In addition to our previously considered `boundary conditions', the new ones include: a simultaneous big bang, a homogeneous density or velocity distribution in the asymptotic future, a simultaneous big crunch, a simultaneous time of maximal expansion, a chosen density or velocity distribution in the asymptotic future, only growing or only decaying fluctuations. Since these conditions are combined in pairs to specify a particular model, this considerably increases the possible ways of designing LT models with desired properties. 
  We study the motion of a pseudo-classical charged particle with spin in the space-time of a gravitational pp wave in the presence of a uniform magnetic field. 
  We consider generic static spacetimes with Killing horizons and study properties of curvature tensors in the horizon limit. It is determined that the Weyl, Ricci, Riemann and Einstein tensors are algebraically special and mutually aligned on the horizon. It is also pointed out that results obtained in the tetrad adjusted to a static observer in general differ from those obtained in a free-falling frame. This is connected to the fact that a static observer becomes null on the horizon.   It is also shown that finiteness of the Kretschmann scalar on the horizon is compatible with the divergence of the Weyl component $\Psi_{3}$ or $\Psi_{4}$ in the freely falling frame. Furthermore finiteness of $\Psi_{4}$ is compatible with divergence of curvature invariants constructed from second derivatives of the Riemann tensor.   We call the objects with finite Krestschmann scalar but infinite $\Psi_{4}$ ``truly naked black holes''. In the (ultra)extremal versions of these objects the structure of the Einstein tensor on the horizon changes due to extra terms as compared to the usual horizons, the null energy condition being violated at some portions of the horizon surface. The demand to rule out such divergencies leads to the constancy of the factor that governs the leading term in the asymptotics of the lapse function and in this sense represents a formal analog of the zeroth law of mechanics of non-extremal black holes. In doing so, all extra terms in the Einstein tensor automatically vanish. 
  The LIGO Scientific Collaboration recently reported a new upper limit on an isotropic stochastic background of gravitational waves obtained based on the data from the 3rd LIGO science Run (S3). Now I present a new method for obtaining directional upper limits that the LIGO Scientific Collaboration intends to use for future LIGO science runs and that essentially implements a gravitational wave radiometer. 
  This course introduces the use of semigroup methods in the solution of linear and nonlinear (quasi-linear) hyperbolic partial differential equations, with particular application to wave equations and Hermitian hyperbolic systems. Throughout the course applications to problems from current relativistic (hyperbolic) physics are provided, which display the potential of semigroup methods in the solution of current research problems in physics. 
  While it is generally accepted, in the framework of Poincare gauge theory, that the Lorentz connection couples minimally to spinor fields, there is no general agreement on the coupling of the translational gauge field to fermions. We will show that the assumption that spinors carry a full Poincare representation leads to inconsistencies, whose origins will be traced back by considering the Poincare group both as the contraction of the de Sitter group, and as a subgroup of the conformal group. As a result, the translational fields do not minimally couple to fermions, and consequently, fermions do not possess an intrinsic momentum. 
  Lie groups involving potential symmetries are applied in connection with the system of magnetohydrodynamic equations for incompressible matter with Ohm's law for finite resistivity and Hall current in cylindrical geometry. Some simplifications allow to obtain a Fokker-Planck type equation. Invariant solutions are obtained involving the effects of time-dependent flow and the Hall-current. Some interesting side results of this approach are new exact solutions that do not seem to have been reported in the literature. 
  The vacuum expectation value of the stress energy tensor for a massive scalar field with arbitrary coupling in flat spaces with non-trivial topology is discussed. We calculate the Casimir energy in these spaces employing the recently proposed {\it optical approach} based on closed classical paths. The evaluation of the Casimir energy consists in an expansion in terms of the lengths of these paths. We will show how different paths with corresponding weight factors contribute in the calculation. The optical approach is also used to find the mass and temperature dependence of the Casimir energy in a cavity and it is shown that the massive fields cannot be neglected in high and low temperature regimes. The same approach is applied to twisted as well as spinor fields and the results are compared with those in the literature. 
  The C-metric is one of few known exact solutions of Einstein's field equations which describes the gravitational field of moving sources. For a vanishing or positive cosmological constant, the C-metric represents two accelerated black holes in asymptotically flat or de Sitter spacetime. For a negative cosmological constant the structure of the spacetime is more complicated. Depending on the value of the acceleration, it can represent one black hole or a sequence of pairs of accelerated black holes in the spacetime with an anti-de Sitter-like infinity. The global structure of this spacetime is analyzed and compared with an empty anti-de Sitter universe. It is illustrated by 3D conformal-like diagrams. 
  The motivation of this communication is to investigate global geometric properties of the phase space in general relativity in the CMC gauge, for manifolds with sigma constant less or equal than zero, under some assumptions on the Bel-Robinson energies Q_{0} and Q_{1} and the reduced hamiltonian H. In particular we prove that small data states (g,K) (i.e. with Q_{0}+(H-H_{infimum})< e for e sufficiently small) are close to a strong geometrization. As an application we establish the stability of the Flat Cone solution under no restriction in the three dimensional hyperbolic geometry, providing in general a proof that had been partially obtained by L.Andersson and V.Moncrief for rigid hyperbolic manifolds. 
  We extend the WKB method for the computation of cosmological perturbations during inflation beyond leading order and provide the power spectra of scalar and tensor perturbations to second order in the slow-roll parameters. Our method does not require that the slow-roll parameters be constant. Although leading and next-to-leading results in the slow-roll parameters depend on the approximation technique used in the computation, we find that the inflationary theoretical predictions obtained may reach the accuracy required by planned observations. In two technical appendices, we compare our techniques and results with previous findings. 
  We examine a class of braneworld models in which the expanding universe encounters a "quiescent" future singularity. At a quiescent singularity, the energy density and pressure of the cosmic fluid as well as the Hubble parameter remain finite while all derivatives of the Hubble parameter diverge (i.e., ${\dot H}$, ${\ddot H}$, etc. $\to \infty$). Since the Kretschmann invariant diverges ($R_{iklm}R^{iklm} \to \infty$) at the singularity, one expects quantum effects to play an important role as the quiescent singularity is approached. We explore the effects of vacuum polarization due to massless conformally coupled fields near the singularity and show that these can either cause the universe to recollapse or, else, lead to a softer singularity at which $H$, ${\dot H}$, and ${\ddot H}$ remain finite while ${\dddot H}$ and higher derivatives of the Hubble parameter diverge. An important aspect of the quiescent singularity is that it is encountered in regions of low density, which has obvious implications for a universe consisting of a cosmic web of high and low density regions -- superclusters and voids. In addition to vacuum polarization, the effects of quantum particle production of non-conformal fields are also likely to be important. A preliminary examination shows that intense particle production can lead to an accelerating universe whose Hubble parameter shows oscillations about a constant value. 
  The multidimensional braneworld gravity model by Dvali, Gabadadze and Porrati was primarily put forth to explain the observed acceleration of the expansion of the Universe without resorting to dark energy. One of the most intriguing features of such a model is that it also predicts small effects on the orbital motion of test particles which could be tested in such a way that local measurements at Solar System scales would allow to get information on the global properties of the Universe. Lue and Starkman derived a secular extra-perihelion \omega precession of 5\times 10^-4 arcseconds per century, while Iorio showed that the mean longitude \lambda is affected by a secular precession of about 10^-3 arcseconds per century. Such effects depend only on the eccentricities e of the orbits via second-order terms: they are, instead, independent of their semimajor axes a. Up to now, the observational efforts focused on the dynamics of the inner planets of the Solar System whose orbits are the best known via radar ranging. Since the competing Newtonian and Einsteinian effects like the precessions due to the solar quadrupole mass moment J2, the gravitoelectric and gravitomagnetic part of the equations of motion reduce with increasing distances, it would be possible to argue that an analysis of the orbital dynamics of the outer planets of the Solar System, with particular emphasis on Saturn because of the ongoing Cassini mission with its precision ranging instrumentation, could be helpful in evidencing the predicted new features of motion. In this note we investigate this possibility in view of the latest results in the planetary ephemeris field. Unfortunately, the current level of accuracy rules out this appealing possibility and it appears unlikely that Cassini and GAIA will ameliorate the situation. 
  Charged stars have the potential of becoming charged black holes or even naked singularities. It is presented a set of numerical solutions of the Tolman-Oppenheimer-Volkov equations that represents spherical charged compact stars in hydrostatic equilibrium. The stellar models obtained are evolved forward in time integrating the Einstein-Maxwell field equations. It is assumed an equation of state of a neutron gas at zero temperature. The charge distribution is taken as been proportional to the rest mass density distribution. The set of solutions present an unstable branch, even with charge to mass ratios arbitrarily close to the extremum case. It is performed a direct check of the stability of the solutions under strong perturbations, and for different values of the charge to mass ratio. The stars that are in the stable branch oscillates and do not collapse, while models in the unstable branch collapse directly to form black holes. Stars with a charge greater or equal than the extreme value explode. When a charged star is suddenly discharged, it don't necessarily collapse to form a black hole. A non-linear effect that gives rise to the formation of an external shell of matter (see Ghezzi and Letelier 2005), is negligible in the present simulations. The results are in agreement with the third law of black hole thermodynamics and with the cosmic censorship conjecture. 
  We show that the problem of stabilization of extra dimensions in Kaluza-Klein type cosmology may be solved in a theory of gravity involving high-order curvature invariants. The method suggested (employing a slow-change approximation) can work with rather a general form of the gravitational action. As examples, we consider pure gravity with Lagrangians quadratic and cubic in the scalar curvature and some more complex ones in a simple Kaluza-Klein framework. After a transition to the 4D Einstein conformal frame, this results in effective scalar field theories with certain effective potentials, which in many cases possess positive minima providing stable small-size extra dimensions. Estimates made in the original (Jordan) conformal frame show that the problem of a small value of the cosmological constant in the present Universe is softened in this framework but is not solved completely.} 
  New static regular axially symmetric solutions of SU(2) Yang-Mills-Higgs theory are constructed. They are asymptotically flat and represent gravitating monopole-monopole pairs. The solutions form two branches linked to the second Bartnik-McKinnon solution on upper mass branch and to the unstable monopole-monopole configuration in flat space on the lower branch, respectively. 
  Recent results on the non-unitary character of quantum time evolution in the family of Gowdy T**3 spacetimes bring the question of whether one should renounce in cosmology to the most sacred principle of unitary evolution. In this work we show that the answer is in the negative. We put forward a full nonperturbative canonical quantization of the polarized Gowdy T**3 model that implements the dynamics while preserving unitarity. We discuss possible implications of this result. 
  Using a formalism recently introduced we study the decaying of the cosmological parameter during the early evolution of an universe, whose evolution is governed by a vacuum equation of state. We use a stochastic approach in a nonperturbative treatment of the inflaton field from a Noncompact Kaluza-Klein (NKK) theory, to study the evolution of energy density fluctuations in the early universe. 
  Efforts to achieve better accuracy in numerical relativity have so far focused either on implementing second order accurate adaptive mesh refinement or on defining higher order accurate differences and update schemes. Here, we argue for the combination, that is a higher order accurate adaptive scheme. This combines the power that adaptive gridding techniques provide to resolve fine scales (in addition to a more efficient use of resources) together with the higher accuracy furnished by higher order schemes when the solution is adequately resolved. To define a convenient higher order adaptive mesh refinement scheme, we discuss a few different modifications of the standard, second order accurate approach of Berger and Oliger. Applying each of these methods to a simple model problem, we find these options have unstable modes. However, a novel approach to dealing with the grid boundaries introduced by the adaptivity appears stable and quite promising for the use of high order operators within an adaptive framework. 
  We investigate the behavior of a noncommutative radiating Schwarzschild black hole. It is shown that coordinate noncommutativity cures usual problems encountered in the description of the terminal phase of black hole evaporation. More in detail, we find that: the evaporation end-point is a zero temperature extremal black hole even in the case of electrically neutral, non-rotating, objects; there exists a finite maximum temperature that the black hole can reach before cooling down to absolute zero; there is no curvature singularity at the origin, rather we obtain a regular DeSitter core at short distance. 
  A formulation of Continuum Mechanics within the context of General Relativity is presented that allows for the incorporation of certain types of anelastic material behaviour, such as viscoelasticity and plasticity. The approach is based on the concept of a four-dimensional body-time complex structured as a principal bundle. The anelastic response is regarded as the result of a continuous distribution of inhomogeneities, whose evolution is dictated by a suggested relativistic version of the Eshelby tensor. The role played by various groups is emphasized throughout the presentation and illustrated by means of the example of an anelastic fluid. 
  There exist at least a few different kind of averaging of the differences of the energy-momentum and angular momentum in normal coordinates {\bf NC(P)} which give tensorial quantities. The obtained averaged quantities are equivalent mathematically because they differ only by constant scalar dimensional factors. One of these averaging was used in our papers [1-8] giving the {\it canonical superenergy and angular supermomentum tensors}.   In this paper we present another averaging of the differences of the energy-momentum and angular momentum which gives tensorial quantities with proper dimensions of the energy-momentum and angular momentum densities. But these averaged relative energy-momentum and angular momentum tensors, closely related to the canonical superenergy and angular supermomentum tensors, {\it depend on some fundamental length $L>0$}.   The averaged relative energy-momentum and angular momentum tensors of the gravitational field obtained in the paper can be applied, like the canonical superenergy and angular supermomentum tensors, to {\it coordinate independent} analysis (local and in special cases also global) of this field.   We have applied the averaged relative energy-momentum tensors to analyze vacuum gravitational energy and momentum and to analyze energy and momentum of the Friedman (and also more general) universes. The obtained results are very interesting, e.g., the averaged relative energy density is {\it positive definite} for the all Friedman universes. 
  We study the evaporation of black holes in non-commutative space-times. We do this by calculating the correction to the detector's response function for a moving mirror in terms of the noncommutativity parameter $\Theta$ and then extracting the number density as modified by this parameter. We find that allowing space and time to be non-commutative increases the decay rate of a black hole. 
  The property and gravitational field of global monopole of tachyon are investigated in a four dimensional static space-time. We give an exact solution of the tachyon field in the flat space-time background. Using the linearized approximation of gravity, we get the approximate solution of the metric. We also solve analytically the coupled Einstein and tachyon field equations which is beyond the linearized approximation to determine the gravitational properties of the monopole solution. We find that the metric of tachyon monopole represents an asymptotically AdS space-time with a small effective mass at the origin. We show that this relatively tiny mass is actually negative, as it is in the case of ordinary scalar field. 
  We present some cosmological solutions of Brans-Dicke theory, characterized by a decaying vacuum energy density and by a constant relative matter density. With these features, they shed light on the cosmological constant problems, leading to a presently small vacuum term, and to a constant ratio between the vacuum and matter energy densities. By fixing the only free parameter of our solutions, we obtain cosmological parameters in accordance with observations of the relative matter density, the universe age and redshift-distance relations. 
  Causal sets are particular partially ordered sets which have been proposed as a basic model for discrete space-time in quantum gravity. We show that the class C of all countable past-finite causal sets contains a unique causal set (U,<) which is universal (i.e., any member of C can be embedded into (U,<)) and homogeneous (i.e., (U,<) has maximal degree of symmetry). Moreover, (U,<) can be constructed both probabilistically and explicitly. In contrast, the larger class of all countable causal sets does not contain a universal object. 
  Using the Penner--Fock parameterization for Teichmuller spaces of Riemann surfaces with holes, we construct the string-like free-field representation of the Poisson and quantum algebras of geodesic functions in the continuous-genus limit. The mapping class group acts naturally in the obtained representation. 
  We argue that the energy levels of an Unruh detector experience an effect similar to the Lamb shift in Quantum Electrodynamics. As a consequence, the spectrum of energy levels in a curved background is different from that in flat space. As examples, we consider a detector in an expanding Universe and in Rindler space, and for the latter case we suggest a new expression for the local virtual energy density seen by an accelerated observer. In the ultraviolet domain, that is when the space between the energy levels is larger than the Hubble rate or the acceleration of the detector, the Lamb shift quantitatively dominates over the thermal response rate. 
  We introduce a new template for the detection of gravitational waves from compact binary systems which is based on Chebyshev polynomials of the first kind. As well as having excellent convergence properties, these polynomials are also very closely related to the elusive minimax polynomial. In this study we have limited ourselves to the test-mass regime, where we model a test particle in a circular equatorial orbit around a Schwarzschild black hole. Our objective is to model the numerical gravitational wave flux function, starting with the post-Newtonian expansion from Black Hole Perturbation Theory. We introduce a new Chebyshev approximation to the flux function, which due to a process called Chebyshev economization gives a better model than either post-Newtonian or Pade based methods. A graphical examination of the new flux function shows that it gives an excellent fit to the numerical flux, but more importantly we find that at the last stable orbit the error is reduced, < 1.8%, at all orders of approximation. We also find that the templates constructed using the Chebyshev approximation give better fitting factors, in general > 0.99, and smaller errors, < 1/10%, in the estimation of the Chirp mass when compared to a fiducial exact waveform, constructed using the numerical flux and the exact expression for the orbital energy function, again at all orders of approximation. We also show that in the test-mass case, the new Chebyshev template is superior to both PN and Pade approximant templates, especially at lower orders of approximation. 
  The Lazarus project was designed to make the most of limited 3D binary black-hole simulations, through the identification of perturbations at late times, and subsequent evolution of the Weyl scalar $\Psi_4$ via the Teukolsky formulation. Here we report on new developments, employing the concept of the ``quasi-Kinnersley'' (transverse) frame, valid in the full nonlinear regime, to analyze late-time numerical spacetimes that should differ only slightly from Kerr. This allows us to extract the essential information about the background Kerr solution, and through this, to identify the radiation present. We explicitly test this procedure with full numerical evolutions of Bowen-York data for single spinning black holes, head-on and orbiting black holes near the ISCO regime. These techniques can be compared with previous Lazarus results, providing a measure of the numerical-tetrad errors intrinsic to the method, and give as a by-product a more robust wave extraction method for numerical relativity. 
  We calculate the total energy (due to matter plus fields) of the universe considering Bergmann-Thomson's energy momentum formulation in both Einstein's theory of general relativity and tele-parallel gravity on two different space-times; namely Reboucas-Tiomno-Korotkii-Obukhov and Godel-type metrics. We also compute some kinematical quantities for these space-times and find that these space-times have shear-free expansion and non-vanishing four-acceleration and vorticity. Different approximations of the Bergmann-Thomson energy-momentum formulation in these different gravitation theories give the same energy density and agree with each other. The results advocate the importance of energy-momentum definitions. 
  In general relativity, gravitational waves propagate at the speed of light, and so gravitons are massless. The masslessness can be traced to symmetry under diffeomorphisms. However, another elegant possibility exists: masslessness can instead arise from spontaneous violation of local Lorentz invariance. We construct the corresponding theory of gravity. It reproduces the Einstein-Hilbert action of general relativity at low energies and temperatures. Detectable signals occur for sensitive experiments, and potentially profound implications emerge for our theoretical understanding of gravity. 
  We present an analogue spacetime model that reproduces the salient features of the most common ansatz for quantum gravity phenomenology. We do this by investigating a system of two coupled Bose-Einstein condensates. This system can be tuned to have two "phonon" modes (one massive, one massless) which share the same limiting speed in the hydrodynamic approximation [Phys. Rev. D72 (2005) 044020, gr-qc/0506029; cond-mat/0409639]. The system nevertheless possesses (possibly non-universal) Lorentz violating terms at very high energies where "quantum pressure" becomes important. We investigate the physical interpretation of the relevant fine-tuning conditions, and discuss the possible lessons and hints that this analogue spacetime could provide for the phenomenology of real physical quantum gravity. In particular we show that the effective field theory of quasi-particles in such an emergent spacetime does not exhibit the so called "naturalness problem". 
  `How do our ideas about quantum mechanics affect our understanding of spacetime?' This familiar question leads to quantum gravity. The complementary question is also important: `How do our ideas about spacetime affect our understanding of quantum mechanics?' This short abstract of a talk given at the Gafka2004 conference contains a very brief summary of some of the author's papers on generalizations of quantum mechanics needed for quantum gravity. The need for generalization is motivated. The generalized quantum theory framework for such generalizations is described and illustrated for usual quantum mechanics and a number of examples to which it does not apply. These include spacetime alternatives extended over time, time-neutral quantum theory, quantum field theory in fixed background spacetime not foliable by spacelike surfaces, and systems with histories that move both forward and backward in time. A fully four-dimensional, sum-over-histories generalized quantum theory of cosmological geometries is briefly described. The usual formulation of quantum theory in terms of states evolving unitarily through spacelike surfaces is an approximation to this more general framework that is appropriate in the late universe for coarse-grained descriptions of geometry in which spacetime behaves classically. This abstract is unlikely to be clear on its own, but references are provided to the author's works where the ideas can be followed up. 
  The Unruh and Hawking effects are investigated on certain families of topologically non-trivial spacetimes using a variety of techniques. First we present the Bogolubov transformation between Rindler and Minkowski quantizations on two flat spacetimes with topology ${\R}^3\times{S^1}$ (M_0 and M_-) for massive Dirac spinors. The two inequivalent spin structures on each are considered. Results show modifications to the Minkowski space Unruh effect. This provides a flat space model for the Hawking effect on Kruskal and RP^3 geon black hole spacetimes which is the subject of the rest of this part.   Secondley we present the expectation values of the stress tensor for massive scalar and spinor fields on $M_0$ and $M_-$, and for massive scalar fields on Minkowski space with a single infinite plane boundary, in the Minkowski-like vacua.   Finally we investigate particle detector models. We investigate Schlicht's regularization of the Wightman function and extend it to an arbitrary spacetime dimension, to quotient spaces of Minkowski space, to non-linear couplings, to a massless Dirac field, and to conformally flat spacetimes. Secondly we present some detector responses, including the time dependent responses of inertial and uniformly accelerated detectors on $M_-$ and $M$ with boundary with motion perpendicular to the boundary. Responses are also considered for static observers in the exterior of the RP^3 geon and comoving observers in RP^3 de Sitter space, via those in the associated GEMS. 
  An analysis of the solutions for the field equations of generalized scalar-tensor theories of gravitation is performed through the study of the geometry of the phase space and the stability of the solutions, with special interest in the Brans-Dicke model. Particularly, we believe to be possible to find suitable forms of the Brans-Dicke parameter omega and potential V of the scalar field, using the dynamical systems approach, in such a way that they can be fitted in the present observed scenario of the Universe. 
  We present an improved version of the approximate scheme for generating inspirals of test-bodies into a Kerr black hole recently developed by Glampedakis, Hughes and Kennefick. Their original "hybrid" scheme was based on combining exact relativistic expressions for the evolution of the orbital elements (the semi-latus rectum p and eccentricity e) with approximate, weak-field, formula for the energy and angular momentum fluxes, amended by the assumption of constant inclination angle, iota, during the inspiral. Despite the fact that the resulting inspirals were overall well-behaved, certain pathologies remained for orbits in the strong field regime and for orbits which are nearly circular and/or nearly polar. In this paper we eliminate these problems by incorporating an array of improvements in the approximate fluxes. Firstly, we add certain corrections which ensure the correct behaviour of the fluxes in the limit of vanishing eccentricity and/or 90 degrees inclination. Secondly, we use higher order post-Newtonian formulae, adapted for generic orbits. Thirdly, we drop the assumption of constant inclination. Instead, we first evolve the Carter constant by means of an approximate post-Newtonian expression and subsequently extract the evolution of iota. Finally, we improve the evolution of circular orbits by using fits to the angular momentum and inclination evolution determined by Teukolsky based calculations. As an application of the improved scheme we provide a sample of generic Kerr inspirals and for the specific case of nearly circular orbits we locate the critical radius where orbits begin to decircularise under radiation reaction. These easy-to-generate inspirals should become a useful tool for exploring LISA data analysis issues and may ultimately play a role in source detection. 
  We investigate the conditions for a bounce to occur in Friedmann-Robertson-Walker cosmologies for the class of fourth order gravity theories. The general bounce criterion is determined and constraints on the parameters of three specific models are given in order to obtain bounces solutions. It is found that unlike the case of General Relativity, a bounce appears to be possible in open and flat cosmologies. 
  We propose an experiment to extract ponderomotive squeezing from an interferometer with high circulating power and low mass mirrors. In this interferometer, optical resonances of the arm cavities are detuned from the laser frequency, creating a mechanical rigidity that dramatically suppresses displacement noise. After taking into account imperfection of optical elements, laser noise, and other technical noise consistent with existing laser and optical technologies and typical laboratory environments, we expect the output light from the interferometer to have measurable squeezing of ~5 dB, with a frequency-independent squeeze angle for frequencies below 1 kHz. This squeeze source is well suited for injection into a gravitational-wave interferometer, leading to improved sensitivity from reduction in the quantum noise. Furthermore, this design provides an experimental test of quantum-limited radiation pressure effects, which have not previously been tested. 
  The Barrett-Crane intertwiner for the Riemannian general relativity is systematically derived by solving the quantum Barrett-Crane constraints corresponding to a tetrahedron (except for the non-degeneracy condition). It was shown by Reisenberger that the Barrett-Crane intertwiner is the unique solution. The systematic derivation can be considered as an alternative proof of the uniqueness. The new element in the derivation is the rigorous imposition of the cross-simplicity constraint. 
  The generalized Chaplygin gas (GCG) is a candidate for the unification of dark energy and dark matter, and is parametrized by an exotic equation of state given by $p_{ch}=-A/\rho_{ch}^{\alpha}$, where $A$ is a positive constant and $0<\alpha \leq 1$. In this paper, exact solutions of spherically symmetric traversable wormholes supported by the GCG are found, possibly arising from a density fluctuation in the GCG cosmological background. To be a solution of a wormhole, the GCG equation of state imposes the following generic restriction $A<(8\pi r_0^2)^{-(1+\alpha)}$, where $r_0$ is the wormhole throat radius, consequently violating the null energy condition. The spatial distribution of the exotic GCG is restricted to the throat neighborhood, and the physical properties and characteristics of these Chaplygin wormholes are further analyzed. Four specific solutions are explored in some detail, namely, that of a constant redshift function, a specific choice for the form function, a constant energy density, and finally, isotropic pressure Chaplygin wormhole geometries. 
  Black holes acting as gravitational lenses produce, besides the primary and secondary weak field images, two infinite sets of relativistic images. These images can be studied using the strong field limit, an analytic method based on a logarithmic asymptotic approximation of the deflection angle. In this work, braneworld black holes are analyzed as gravitational lenses in the strong field limit and the feasibility of observation of the images is discussed. 
  This paper constructs a kinematic basis for spin networks with planar or cylindrical symmetry, by exploiting the fact that the basis elements are representations of an O(3) subgroup of O(4). The action of the volume operator on this basis gives a difference equation for the eigenvalues and eigenvectors of the volume operator. For basis elements of low spin, the difference equation can be solved readily on a computer. For higher spins, I solve for the eigenvalues using a WKBJ method. This paper considers only the case where the gravitational wave can have both polarizations. The single polarization case is considered in a spearate paper. 
  A previous paper constructed a kinematic basis for spin networks with planar or cylindrical symmetry and arbitrary polarization. This paper imposes a constraint which limits the gravitational wave to a single polarization. The spectrum of the constraint contains a physically reasonable number of zero eigenvalues, and the zero eigenvectors can be constructed explicitly. Commutation of the constraint with the Hamiltonian is expected to lead to a further constraint. This new constraint is not investigated in this paper, but I argue it will be non-local, relating states at two or more neighboring vertices. 
  In this paper we review a model based on loop quantum cosmology that arises from a symmetry reduction of the self dual Plebanski action. In this formulation the symmetry reduction leads to a very simple Hamiltonian constraint that can be quantized explicitly in the framework of loop quantum cosmology. We investigate the phenomenological implications of this model in the semi-classical regime and compare those with the known results of the standard Loop Quantum Cosmology. 
  The ``optical springs'' regime of the signal-recycled configuration of laser interferometric gravitational-wave detectors is analyzed taking in account optical losses in the interferometer arm cavities. This regime allows to obtain sensitivity better than the Standard Quantum Limits both for a free test mass and for a conventional harmonic oscillator. The optical losses restrict the gain in sensitivity and achievable signal-to-noise ratio. Nevertheless, for parameters values planned for the Advanced LIGO gravitational-wave detector, this restriction is insignificant. 
  We formally show that the conservative second post-Newtonian (PN) accurate dynamics of spinning compact binaries moving in eccentric orbits, when spin effects are restricted to the leading order spin-orbit interaction cannot be chaotic for the following two distinct cases: (i) the binary consists of compact objects of arbitrary mass, where only one of them is spinning with an arbitrary spin and (ii) the binary consists of equal mass compact objects, having two arbitrary spins. We rest our arguments on the recent determination of PN accurate Keplerian-type parametric solutions to the above cases, indicating that the PN accurate dynamics is integrable in these two situations. We compare predictions of our case (i) with those from a numerical investigation of an equivalent scenario that observed chaos in the associated dynamics. We also present possible reasons for the discrepancies. 
  Einstein-Maxwell field equations correspoding to higher dimensional description of static spherically symmetric space-time have been solved under two specific set of conditions, viz., (i) $\rho \ne 0$, $\nu^\prime= 0$ and (ii) $\rho=0$, $ \nu^\prime\ne 0$ where $\rho$ and $\nu$ represent the mass density and metric potential. The solution sets thus obtained satisfy the criteria of being electromagnetic mass model such that the gravitational mass vanishes for the vanishing charge density $\sigma$ and also the space-time becomes flat. Physical features related to other parameters also have been discussed. 
  We show that for hypersurface orthogonal Killing vectors, the corresponding Chevreton superenergy currents will be conserved and proportional to the Killing vectors. This holds for four-dimensional Einstein-Maxwell spacetimes with an electromagnetic field that is sourcefree and inherits the symmetry of the spacetime. A similar result also holds for the trace of the Chevreton tensor. The corresponding Bel currents have previously been proven to be conserved and our result can be seen as giving further support to the concept of conserved mixed superenergy currents. The analogous case for a scalar field has also previously been proven to give conserved currents and we show, for completeness, that these currents also are proportional to the Killing vectors. 
  Liko and Wesson have recently introduced a new 5-dimensional induced matter solution of the Einstein equations, a negative curvature Robertson-Walker space embedded in a Riemann flat 5-dimensional manifold. We show that this solution is a special case of a more general theorem prescribing the structure of certain N+1-dimensional Riemann flat spaces which are all solutions of the Einstein equations. These solutions encapsulate N-dimensional curved manifolds. Such spaces are said to "induce matter" in the sub-manifolds by virtue of their geometric structure alone. We prove that the N-manifold can be any maximally symmetric space. 
  It is shown that coupling system between fractal membranes and a Gaussian beam passing through a static magnetic field has strong selection capability for the stochastic relic gravitational wave background. The relic GW components propagating along the positive direction of the symmetrical axis of the Gaussian beam might generate an optimal electromagnetic perturbation while the perturbation produced by the relic GW components propagating along the negative and perpendicular directions to the symmetrical axis will be much less than the former.The influence of the random fluctuation of the relic GWs to such effect can be neglected and the influence of the random fluctuation of the relic GWs to such effect can be neglected. 
  An idea of reality conditions in the context of spin foams (Barrett-Crane models) is developed. The square of areas are the most elementary observables in the case of spin foams. This observation implies that simplest reality conditions in the context of the Barrett-Crane models is that the all possible scalar products of the bivectors associated to the triangles of a four simplex be real. The continuum generalization of this is the area metric reality constraint: the area metric is real iff a non-degenerate metric is real or imaginary. Classical real general relativity (all signatures) can be extracted from complex general relativity by imposing the area metric reality constraint. The Plebanski theory can be modified by adding a Lagrange multiplier to impose the area metric reality condition to derive classical real general relativity. I discuss the SO(4,C) BF model and SO(4,C) Barrett-Crane model. It appears that the spin foam models in 4D for all the signatures are the projections of the SO(4,C) spin foam model using the reality constraints on the bivectors. 
  The structure of a light cone in the Goedel universe is studied. We derive the intrinsic cone metric, calculate the rotation coefficients of the ray congruence forming the cone, determine local differential invariants up to second order, describe the crossover (keel) singularities and give a first discussion of its focal points. Contrary to many rotation coefficients, some inner differential invariants attain simple finite standard values at focal singularities. 
  This paper has been withdrawn because the new one gr-qc/0512095 includes all its results (as well as those in gr-qc/0507018), in a clearer way. 
  Marginally trapped surfaces (MTSs) are commonly used in numerical relativity to locate black holes. For dynamical black holes, it is not known generally if this procedure is sufficiently reliable. Even for Schwarzschild black holes, Wald and Iyer constructed foliations which come arbitrarily close to the singularity but do not contain any MTSs. In this paper, we review the Wald-Iyer construction, discuss some implications for numerical relativity, and generalize to the well known Vaidya spacetime describing spherically symmetric collapse of null dust. In the Vaidya spacetime, we numerically locate non-spherically symmetric trapped surfaces which extend outside the standard spherically symmetric trapping horizon. This shows that MTSs are common in this spacetime and that the event horizon is the most likely candidate for the boundary of the trapped region. 
  Numerical algorithms based on variational and symplectic integrators exhibit special features that make them promising candidates for application to general relativity and other constrained Hamiltonian systems. This paper lays part of the foundation for such applications. The midpoint rule for Hamilton's equations is examined from the perspectives of variational and symplectic integrators. It is shown that the midpoint rule preserves the symplectic form, conserves Noether charges, and exhibits excellent long--term energy behavior. The energy behavior is explained by the result, shown here, that the midpoint rule exactly conserves a phase space function that is close to the Hamiltonian. The presentation includes several examples. 
  We are developing a general, unified, and rigorous analytical framework for using gravitational lensing by compact objects to test different theories of gravity beyond the weak-deflection limit. In this paper we present the formalism for computing corrections to lensing observables for static, spherically symmetric gravity theories in which the corrections to the weak-deflection limit can be expanded as a Taylor series in one parameter, namely the gravitational radius of the lens object. We take care to derive coordinate-independent expressions and compute quantities that are directly observable. We compute first- and second-order corrections to the image positions, magnifications, and time delays. Interestingly, we find that the first-order corrections to the total magnification and centroid position vanish in all gravity theories that agree with general relativity in the weak-deflection limit, but they can remain nonzero in modified theories that disagree with GR in the weak-deflection limit. For the Reissner-Nordstrom metric and a related metric from heterotic string theory, our formalism reveals an intriguing connection between lensing observables and the condition for having a naked singularity, which could provide an observational method for testing the existence of such objects. We apply our formalism to the Galactic black hole and predict that the corrections to the image positions are at the level of 10 micro-arcseconds, while the correction to the time delay is a few hundredths of a second. These corrections would be measurable today if a pulsar were found to be lensed by the Galactic black hole; and they should be readily detectable with planned missions like MAXIM. 
  Experimental tests of gravity performed in the solar system show a good agreement with general relativity. The latter is however challenged by the Pioneer anomaly which might be pointing at some modification of gravity law at ranges of the order of the size of the solar system. We introduce a metric extension of general relativity which, while preserving the equivalence principle, modifies the coupling between curvature and stress tensors and, therefore, the metric solution in the solar system. The ``post-Einsteinian extension'' replaces Newton gravitation constant by two running coupling constants, which depend on the scale and differ in the sectors of traceless and traced tensors, so that the metric solution is characterized by two gravitation potentials. The extended theory has the capability to preserve compatibility with gravity tests while accounting for the Pioneer anomaly. It can also be tested by new experiments or, maybe, by having a new look at data of already performed experiments. 
  An inspection of the DSS and 2MASS images of selected Milky Way regions has led to the discovery of 66 stellar groupings whose morphologies, color-magnitude diagrams, and stellar density distributions suggest that these objects are possible open clusters that do not yet appear to be listed in any catalogue. For 24 of these groupings, which we consider to be the most likely to be candidates, we provide extensive descriptions on the basis of 2MASS photometry and their visual impression on DSS and 2MASS. Of these cluster candidates, 9 have fundamental parameters determined by fitting the color-magnitude diagrams with solar metallicity Padova isochrones. An additional 10 cluster candidates have distance moduli and reddenings derived from K magnitudes and (J-K) color indices of helium-burning red clump stars. As an addendum, we also provide a list of a number of apparently unknown galactic and extragalactic objects that were also discovered during the survey. 
  We report on experimental observation of radiation-pressure induced effects in a high-power optical cavity. These effects play an important role in next generation gravitational wave (GW) detectors, as well as in quantum non-demolition (QND) interferometers. We measure the properties of an optical spring, created by coupling of an intense laser field to the pendulum mode of a suspended mirror; and also the parametric instability (PI) that arises from the nonlinear coupling between acoustic modes of the cavity mirrors and the cavity optical mode. Specifically, we measure an optical rigidity of $K = 3 \times 10^4$ N/m, and PI value $R = 3$. 
  Path and path deviation equations for charged, spinning and spinning charged objects in different versions of Kaluza-Klein (KK) theory using a modified Bazanski Lagrangian have been derived. The significance of motion in five dimensions, especially for a charged spinning object, has been examined. We have also extended the modified Bazanski approach to derive the path and path deviation equations of a test particle in a version of non-symmetric KK theory. 
  The field equations of a special class of teleparallel theory of gravitation and electromagnetic fields have been applied to tetrad space having cylindrical symmetry with four unknown functions of radial coordinate $r$ and azimuth angle $\theta$. The vacuum stress-energy momentum tensor with one assumption concerning its specific form generates one non-trivial exact analytic solution. This solution is characterized by a constant magnetic field parameter $B_0$. If $B_0=0$ then, the solution will reduces to the flat spacetime. The energy content is calculated using the superpotential given in the framework of teleparallel geometry. The energy contained in a sphere is found to be different from the pervious results. 
  Recently the asymptotic limit of the Barrett-Crane models has been studied by Barrett and Steele. Here by a direct study, I show that we can extract the bivectors which satisfy the essential Barrett-Crane constraints from the asymptotic limit. Because of this the Schlaffi identity is implied by the asymptotic limit, rather than to be imposed as a constraint. 
  A fully relativistic modified gravitational theory including a fifth force skew symmetric field is fitted to the Pioneer 10/11 anomalous acceleration. The theory allows for a variation with distance scales of the gravitational constant G, the fifth force skew symmetric field coupling strength omega and the mass of the skew symmetric field mu=1/lambda. A fit to the available anomalous acceleration data for the Pioneer 10/11 spacecraft is obtained for a phenomenological representation of the "running" constants and values of the associated parameters are shown to exist that are consistent with fifth force experimental bounds. The fit to the acceleration data is consistent with all current satellite, laser ranging and observations for the inner planets. 
  The Spin Foam Model for the SO(4,C) BF theory is discussed. The Barrett-Crane intertwiner for the SO(4,C) general relativity is systematically derived. The SO(4,C) Barret-Crane interwiner is unique. The propagators of the SO(4,C) Barrett-Crane model are discussed. The asymptotic limit of the SO(4,C) general relativity is discussed. The asymptotic limit is controlled by the SO(4,C) Regge calculus. 
  In this note, we study the integrability of geodesic flow in the background of a very general class of spacetimes with NUT-charge(s) in higher dimensions. This broad set encompasses multiply NUT-charged solutions, electrically and magnetically charged solutions, solutions with a cosmological constant, and time dependant bubble-like solutions. We also derive first-order equations of motion for particles in these backgrounds. Separability turns out to be possible due to the existence of non-trivial irreducible Killing tensors. Finally, we also examine the Klein-Gordon equation for a scalar field in these spacetimes and demonstrate complete separability. 
  Global monochromatic solutions of the scalar wave equation are obtained in flat wormholes of dimensions 2+1 and 3+1. The solutions are in the form of infinite series involving cylindirical and spherical wave functions and they are elucidated by the multiple scattering method. Explicit solutions for some limiting cases are illustrated as well. The results presented in this work constitute instances of solutions of the scalar wave equation in a spacetime admitting closed timelike curves. 
  The Moller energy (due to matter and fields including gravity) distribution of the traversable Lorentzian wormhole space-time by the scalar field or electric charged is studied in two different approaches of gravity such as general relativity and tele-parallel gravity. The results are found exactly the same in these different approximations. The energy found in tele-parallel gravity is also independent of the tele-parallel dimensionless coupling constant, which means that it is valid in any tele-parallel model. Our results sustains that (a) the importance of the energy-momentum definitions in the evaluation of the energy distribution of a given space-time and (b) the viewpoint of Lassner that the Moller energy-momentum complex is a powerful concept of energy and momentum. 
  In this article we elaborate on a recently proposed interpretation of DSR as an effective measurement theory in the presence of non-negligible (albeit small) quantum gravitational fluctuations. We provide several heuristic arguments to explain how such a new theory can emerge and discuss the possible observational consequences of this framework. 
  In this article, dedicated to one of the best specialist of the FOSH systems, we couple the Bianchi equations with the equations satisfied by the dynamical acceleration of a charged fluid and the derivatives of the associated Maxwell field. 
  An approximate strategy for studying the evolution of binary systems of extended objects is introduced. The stars are assumed to be polytropic ellipsoids. The surfaces of constant density maintain their ellipsoidal shape during the time evolution. The equations of hydrodynamics then reduce to a system of ordinary differential equations for the internal velocities, the principal axes of the stars and the orbital parameters. The equations of motion are given within Lagrangian and Hamiltonian formalism. The special case when both stars are axially symmetric fluid configurations is considered. Leading order gravitational radiation reaction is incorporated, where the quasi-static approximation is applied to the internal degrees of freedom of the stars. The influence of the stellar parameters, in particular the influence of the polytropic index $n$, on the leading order gravitational waveforms is studied. 
  Recently the 1/R gravity has been proposed in order to explain the accelerated expansion of the universe. However, it was argued that the 1/R gravity conflicts with solar system tests. While this statement is true if one views the 1/R gravity as an effective theory, we find that this difficulty might be avoided if one treats the 1/R term as a correction to the scalar curvature term in the high curvature limit $R \gg \mu^2$. 
  In this paper we consider a model in which the gravitational ``constant'' $G$ is determined by a tachyon-like field whose Lagrangian is of the form $\mathcal{L}_{\phi} = -V(\phi)\sqrt{1 - \partial_{\mu}\phi\partial^{\mu}\phi}$ and the potential is of the form $ V(\phi) = \lambda \phi ^{-4}$. We study the cosmological consequence of this theory in the matter dominated era and show that this leads to a transition from an initial decelerated expansion to an accelerated expansion phase at the present epoch. 
  When a spacetime takes Bondi radiating metric, and is vacuum and asymptotically flat at spatial infinity which ensures the positive mass theorem, we prove that the standard ADM energy-momentum is the past limit of the Bondi energy-momentum. We also derive a formula relating the ADM energy-momentum of any asymptotically flat spacelike hypersurface to the Bondi energy-momentum of any null hypersurface. The formula indicates that the Bondi mass is always less than the ADM total energy if the system has {\it news}.   The assumed asymptotic flatness precludes gravitational radiation. We therefore study further the relation between the ADM total energy and the Bondi mass when gravitational radiation emits. We find that in this case the ADM total energy is no longer the past limit of the Bondi mass. They differ by certain quantity relating to the {\it news} of the system. 
  In a vacuum spacetime equips with the Bondi's radiating metric which is asymptotically flat at spatial infinity including gravitational radiation ({\bf Condition D}), we establish the relation between the ADM total linear momentum and the Bondi momentum. The relation between the ADM total energy and the Bondi mass in this case was established earlier in [12]. 
  Ernst Mach (1838-1916) suggested that the origin of gravitational interaction could depend on the presence of all masses in the universe. A corresponding hypothesis of Sciama (1953) on the gravitational constant, $c^2/G = \sum m_i/r_i$, is linked to Dicke's (1957) proposal of an electromagnetic origin of gravitation, a precursor of scalar-tensor-theories. In this an equivalent description in terms of a variable speed of light (VSL) is given, and the agreement with all GR tests regarding time and length scales is shown. Moreover, VSL opens the possibility to write the total energy of a particle as $E=mc^2$; this necessarily leads to the proportionality of inertial and gravitating mass, the equivalence principle. Furthermore, a formula for c depending on the mass distribution is given that reproduces Newton's law of gravitation. This mass distribution allows to calculate a slightly variable term that corresponds to the `constant' G. The present proposal may also supply an alternative explanation to the flatness problem and the horizon problem in cosmology. 
  The model where the universe is considered as an expanding spherical 3-brane allows us to explain its expansion rate without a dark energy component. In this scenario the computed redshift that corresponds to the transition from cosmic deceleration to acceleration is in a good agreement with observations. 
  Under an appropriate change of the perturbation variable Lifshitz-Khalatnikov propagation equations for the scalar perturbation reduce to d'Alembert equation. The change of variables is based on the Darboux transform. 
  The concept of rigid reference frame and of constricted spatial metric, given in the previous work [\emph{Class. Quantum Grav.} {\bf 21}, 3067,(2004)] are here applied to some specific space-times: In particular, the rigid rotating disc with constant angular velocity in Minkowski space-time is analyzed, a new approach to the Ehrenfest paradox is given as well as a new explanation of the Sagnac effect. Finally the anisotropy of the speed of light and its measurable consequences in a reference frame co-moving with the Earth are discussed. 
  The canonical approach to Riemannian quantum gravity is reviewed with reference to local supersymmetry, to the classical boundary-value problem arising from the Hartle-Hawking quantum state, and particularly for (anti-)self-dual geometries. Two examples of the boundary-value problem for the Einstein equations, possibly with a cosmological constant \Lambda, are treated, both of Bianchi-IX type. These close smoothly in the interior with a NUT or a BOLT. The Hamiltonian approach to general relativity is described using Ashtekar variables; for non-zero \Lambda and anti-self-dual Weyl tensor, the classical solution corresponds, with the most naive choice of boundary data, to the Chern-Simons functional of the boundary data, the classical action being I_{CS}. Hence, one is led to the corresponding quantum states exp(\pm I_{CS}). Apparently, the classical solutions have the undesirable feature that, in general, the resulting Riemannian classical geometry, arising from the Hamilton-Jacobi equation, does not close smoothly in the interior. The canonical quantum theory of supergravity is also described, and may lead to very streamlined (finite) calculations of loop amplitudes for N=1 supergravity with gauged supermatter. If one uses Ashtekar/Jacobson variables for canonical supergravity, then again (for \Lambda\neq 0) one arrives at a (supersymmetric) Chern-Simons action and quantum state in the (anti-)self-dual case. 
  In this letter we work out the secular precession of the spin of a gyroscope in geodesic motion around a central mass in the framework of the Dvali-Gabadadze-Porrati multidimensional gravity model. Such an effect, which depends on the mass of the central body and on the orbit radius of the gyroscope, contrary to the precessions of the orbital elements of the orbit of a test body, is far too small to be detected. 
  In this paper, we study the gravitational collapse of null dust in the cylindrically symmetric spacetime. The naked singularity necessarily forms at the symmetry axis. We consider the situation in which null dust is emitted again from the naked singularity formed by the collapsed null dust and investigate the back-reaction by this emission for the naked singularity. We show a very peculiar but physically important case in which the same amount of null dust as that of the collapsed one is emitted from the naked singularity as soon as the ingoing null dust hits the symmetry axis and forms the naked singularity. In this case, although this naked singularity satisfies the strong curvature condition by Kr\'{o}lak (limiting focusing condition), geodesics which hit the singularity can be extended uniquely across the singularity. Therefore we may say that the collapsing null dust passes through the singularity formed by itself and then leaves for infinity. Finally the singularity completely disappears and the flat spacetime remains. 
  In this paper we study long distance modifications of gravity obtained by considering actions that are singular in the limit of vanishing curvature. In particular, we showed in a previous publication that models that include inverse powers of curvature invariants that diverge for r->0 in the Schwarzschild geometry, recover an acceptable weak field limit at short distances from sources. We study then the linearisation of generic actions of the form L=F[R,P,Q] where P=R_{ab}R^{ab} and Q=R_{abcd}R^{abcd}. We show that for the case in which F[R,P,Q]=F[R,Q-4P], the theory is ghost free. Assuming this is the case, in the models that can explain the acceleration of the Universe without recourse to Dark Energy there is still an extra scalar field in the spectrum besides the massless spin two graviton. The mass of this extra excitation is of the order of the Hubble scale in vacuum. We nevertheless recover Einstein gravity at short distances because the mass of this scalar field depends on the background in such a way that it effectively decouples when one gets close to any source. Remarkably, for the values of the parameters necessary to explain the cosmic acceleration the induced modifications of gravity are suppressed at the Solar System level but can be important for systems like a galaxy. 
  Finite expressions for the mean value of the stress tensor corresponding to a scalar field with a generalized dispersion relation in a Friedman--Robertson--Walker universe are obtained using adiabatic renormalization. Formally divergent integrals are evaluated by means of dimensional regularization. The renormalization procedure is shown to be equivalent to a redefinition of the cosmological constant and the Newton constant in the semiclassical Einstein equations. 
  Black-hole quasinormal modes have been the subject of much recent attention, with the hope that these oscillation frequencies may shed some light on the elusive theory of quantum gravity. We study {\it analytically} the asymptotic quasinormal spectrum of a {\it charged} scalar field in the (charged) Reissner-Nordstr\"om spacetime. We find an analytic expression for these black-hole resonances in terms of the black-hole physical parameters: its Bekenstein-Hawking temperature $T_{BH}$, and its electric potential $\Phi$. We discuss the applicability of the results in the context of black-hole quantization. In particular, we show that according to Bohr's correspondence principle, the asymptotic resonance corresponds to a fundamental area unit $\Delta A=4\hbar\ln2$. 
  We present a new algorithm for evolving orbiting black-hole binaries that does not require excision or a corotating shift. Our algorithm is based on a novel technique to handle the singular puncture conformal factor. This system, based on the BSSN formulation of Einstein's equations, when used with a `pre-collapsed' initial lapse, is non-singular at the start of the evolution, and remains non-singular and stable provided that a good choice is made for the gauge. As a test case, we use this technique to fully evolve orbiting black-hole binaries from near the Innermost Stable Circular Orbit (ISCO) regime. We show fourth order convergence of waveforms and compute the radiated gravitational energy and angular momentum from the plunge. These results are in good agreement with those predicted by the Lazarus approach. 
  A single Israel layer can be created when two metrics adjoin with no continuous metric derivative across the boundary. The properties of the layer depend only on the two metrics it separates. By using a fractional derivative match, a family of Israel layers can be created between the same two metrics. The family is indexed by the order of the fractional derivative. The method is applied to Tolman IV and V interiors and a Schwarzschild vacuum exterior. The method creates new ranges of modeling parameters for fluid spheres. A thin shell analysis clarifies pressure/tension in the family of boundary layers. 
  The formalism of electric - magnetic duality, first pioneered by Reinich and Wheeler, extends General Relativity to encompass non-Abelian fields. Several energy Tensors T^uv with non-vanishing trace matter are developed solely as a function of the field strength tensor F^uv, including the Euler tensor, and tensors for matter in flux, pressure in flux, and stationary pressure. The spacetime metric g_uv is not only a solution to the second-order Einstein equation based on T^uv, but is also constrained by a third-order equation involving the Bianchi identity together with the gravitational energy components kappa_u for each T^uv. The common appearance of F^uv in all of the T^uv and kappa_v makes it possible to obtain quantum solutions for the spacetime metric, thereby geometrizing quantum physics as a non-linear theory. 
  We investigate the laws of thermodynamics in an accelerating universe driven by dark energy with a time-dependent equation of state. In the case we consider that the physically relevant part of the Universe is that envelopped by the dynamical apparent horizon, we have shown that both the first law and second law of thermodynamics are satisfied. On the other hand, if the boundary of the Universe is considered to be the cosmological event horizon the thermodynamical description based on the definitions of boundary entropy and temperature breaks down. No parameter redefinition can rescue the thermodynamics laws from such a fate, rendering the cosmological event horizon unphysical from the point of view of the laws of thermodynamics. 
  It is shown that by making use of the Kodama vector field, as a preferred time evolution vector field, in spherically symmetric dynamical systems unexpected simplifications arise. In particular, the evolution equations relevant for the case of a massless scalar field minimally coupled to gravity are investigated. The simplest form of these equations in the 'canonical gauge' are known to possess the character of a mixed first order elliptic-hyperbolic system. The advantages related to the use of the Kodama vector field are two-folded although they show up simultaneously. First, it is found that the true degrees of freedom separate. Second, a subset of the field equations possessing the form of a first order symmetric hyperbolic system for these preferred degrees of freedom is singled out. It is also demonstrated, in the appendix, that the above results generalise straightforwardly to the case of a generic self-interacting scalar field. 
  We discuss a gravitationally induced nonlinearity in hierarchic systems. We consider the generation of extremely low-frequency radio waves with a frequency of the periodic gravitational radiation; the generation is due to an induced nonlinear self-action of electromagnetic radiation in the vicinity of the gravitational-radiation source. These radio waves are a fundamentally new type of response of an electrodynamic system to gravitational radiation. That is why we here use an unconventional term: radio-wave messengers of periodic gravitational radiation. 
  Based on the micro-black hole \emph{gedanken} experiment as well as on general considerations of quantum mechanics and gravity the generalized uncertainty principle (GUP) is analyzed by using the running Newton constant. The result is used to decide between the GUP and quantum gravitational effects as a possible mechanism leading to the black hole remnants of about Planck mass. 
  The dynamic structure factor of a simple relativistic fluid is calculated. The coupling of acceleration with the heat flux present in Eckart's version of irreversible relativistic thermodynamics is examined using the Rayleigh-Brillouin spectrum of the fluid. A modification of the width of the Rayleigh peak associated to Eckart's picture of the relativistic nature of heat is predicted and estimated. 
  The leading term of the asymptotic of quasinormal modes in the Schwarzschild background, omega_n = - i n/2, is obtained in two straightforward analytical ways for arbitrary spins. One of these approaches requires almost no calculations. As simply we demonstrate that for any odd integer spin, described by the Teukolsky equation, the first correction to the leading term vanishes. Then, this correction for half-integer spins is obtained in a slightly more intricate way. At last, we derive analytically the general expression for the first correction for all spins, described by the Teukolsky equation. 
  The cosmological constant sets certain scales important in cosmology. We show that \Lambda in conjunction with other parameters like the Schwarzschild radius leads to scales relevant not only for cosmological but also for astrophysical applications. Of special interest is the extension of orbits and velocity of test particles traveling over Mpc distances. We will show that there exists a lower and an upper cut-off on the possible velocities of test particles. For a test body moving in a central gravitational field \Lambda enforces a maximal value of the angular momentum if we insist on bound orbits of the test body which move at a distance larger than the Schwarzschild radius. 
  The relation between an isotropic and an anisotropic model in loop quantum cosmology is discussed in detail, comparing the strict symmetry reduction with a perturbative implementation of symmetry. While the latter cannot be done in a canonical manner, it allows to consider the dynamics including the role of small non-symmetric degrees of freedom for the symmetric evolution. This serves as a model for the general situation of perturbative degrees of freedom in a background independent quantization such as loop quantum gravity, and for the more complicated addition of perturbative inhomogeneities. While being crucial for cosmological phenomenology, it is shown that perturbative non-symmetric degrees of freedom do not allow definitive conclusions for the singularity issue and in such a situation could even lead to wrong claims. 
  The boundary conditions for canonical vacuum general relativity is investigated at the quasi-local level. It is shown that fixing the area element on the 2- surface S (rather than the induced 2-metric) is enough to have a well defined constraint algebra, and a well defined Poisson algebra of basic Hamiltonians parameterized by shifts that are tangent to and divergence-free on $. The evolution equations preserve these boundary conditions and the value of the basic Hamiltonian gives 2+2 covariant, gauge-invariant 2-surface observables. The meaning of these observables is also discussed. 
  It is shown by a straightforward argument that the Hamiltonian generating the time evolution of the Dirac wave function in relativistic quantum mechanics is not hermitian with respect to the covariantly defined inner product whenever the background metric is time dependent. An alternative, hermitian, Hamiltonian is found and is shown to be directly related to the canonical field Hamiltonian used in quantum field theory. 
  In this paper we include spin and multipole moment effects in the formalism used to describe the motion of extended objects recently introduced in hep-th/0409156. A suitable description for spinning bodies is developed and spin-orbit, spin-spin and quadrupole-spin Hamiltonians are found at leading order. The existence of tidal, as well as self induced finite size effects is shown, and the contribution to the Hamiltonian is calculated in the latter. It is shown that tidal deformations start formally at O(v^6) and O(v^10) for maximally rotating general and compact objects respectively, whereas self induced effects can show up at leading order. Agreement is found for the cases where the results are known. 
  In his recent work on a tilt instability for advanced LIGO interferometers, P. Savov discovered numerically a unique duality relation between the eigenspectra of paraxial optical cavities with non-spherical mirrors: he found a one-to-one mapping between eigenstates and eigenvalues of cavities deviating from flat mirrors by h(r) and cavities deviating from concentric mirrors by -h(r), where h need not be a small perturbation. In this paper, we analytically prove and generalize this remarkable result. We then illustrate its application to interferometric gravitational-wave detectors; in particular, we employ it to confirm the numerical results of Savov and Vyatchanin for the impact of optical-pressure torques on LIGO's Fabry-Perot arm cavities (i.e. the tilt instability), when the mirrors are designed to support beams with rather flat intensity profiles over the mirror surfaces. This unique mapping might be very useful in future studies of alternative optical designs for LIGO interferometers, when an important feature is the intensity distribution on the cavity optics. 
  We present a new type of gravitational mass defect in which an infinite amount of matter may be bounded in a zero ADM mass. This interpolates between effects typical of closed worlds and T-spheres. We consider the Tolman model of dust distribution and show that this phenomenon reveals itself for a solution that has no origin on one side but is closed on the other side. The second class of examples corresponds to smooth gluing T-spheres to the portion of the Friedmann-Robertson-Walker solution. The procedure is generalized to combinations of smoothly connected T-spheres, FRW and Schwarzschild metrics. In particular, in this approach a finite T-sphere is obtained that looks for observers in two R-regions as the Schwarzschild metric with two different masses one of which may vanish. 
  We consider the vacuum gravitational collapse for cohomogeneity-two solutions of the nine dimensional Einstein equations. Using combined numerical and analytical methods we give evidence that within this model the Schwarzschild-Tangherlini black hole is asymptotically stable. In addition, we briefly discuss the critical behavior at the threshold of black hole formation. 
  In this paper, charged black holes in general relativity coupled to Born-Infeld electrodynamics are studied as gravitational lenses. The positions and magnifications of the relativistic images are obtained using the strong deflection limit, and the results are compared with those corresponding to a Reissner-Nordstrom black hole with the same mass and charge. As numerical examples, the model is applied to the supermassive Galactic center black hole and to a small size black hole situated in the Galactic halo. 
  We revisit the problem of finding the entanglement entropy of a scalar field on a lattice by tracing over its degrees of freedom inside a sphere. It is known that this entropy satisfies the area law -- entropy proportional to the area of the sphere -- when the field is assumed to be in its ground state. We show that the area law continues to hold when the scalar field degrees of freedom are in generic coherent states and a class of squeezed states. However, when excited states are considered, the entropy scales as a lower power of the area. This suggests that for large horizons, the ground state entropy dominates, whereas entropy due to excited states gives power law corrections. We discuss possible implications of this result to black hole entropy. 
  In braneworld models, Space-Time-Matter and other Kaluza-Klein theories, our spacetime is devised as a four-dimensional hypersurface {\it orthogonal} to the extra dimension in a five-dimensional bulk. We show that the FRW line element can be "reinvented" on a dynamical four-dimensional hypersurface, which is {\it not} orthogonal to the extra dimension, without any internal contradiction. This hypersurface is selected by the requirement of continuity of the metric and depends explicitly on the evolution of the extra dimension. The main difference between the "conventional" FRW, on an orthogonal hypersurface, and the new one is that the later contains higher-dimensional modifications to the regular matter density and pressure in 4D. We compare the evolution of the spacetime in these two interpretations. We find that a wealth of "new" physics can be derived from a five-dimensional metric if it is interpreted on a dynamical (non-orthogonal) 4D hypersurface. In particular, in the context of a well-known cosmological metric in $5D$, we construct a FRW model which is consistent with the late accelerated expansion of the universe, while fitting simultaneously the observational data for the deceleration parameter. The model predicts an effective equation of state for the universe, which is consistent with observations. 
  This paper concerns the absolute versus relative motion debate. The Barbour and Bertotti 1982 work may be viewed as an indirectly set up relational formulation of a portion of Newtonian mechanics. I consider further direct formulations of this and argue that the portion in question -- universes with zero total angular momentum, that are conservative and with kinetic terms that are (homogeneous) quadratic in their velocities -- is capable of accommodating a wide range of classical physics phenomena. Furthermore, as I develop in Paper II, this relational particle model is a useful toy model for canonical general relativity.   I consider what happens if one quantizes relational rather than absolute mechanics, indeed whether the latter is misleading. By exploiting Jacobi coordinates, I show how to access many examples of quantized relational particle models and then interpret these from a relational perspective. By these means, previous suggestions of bad semiclassicality for such models can be eluded. I show how small (particle number) universe relational particle model examples display eigenspectrum truncation, gaps, energy interlocking and counterbalanced total angular momentum. These features mean that these small universe models make interesting toy models for some aspects of closed universe quantum cosmology. While, these features do not compromise the recovery of reality as regards the practicalities of experimentation in a large universe such as our own. 
  Relational particle models are employed as toy models for the study of the Problem of Time in quantum geometrodynamics. These models' analogue of the thin sandwich is resolved. It is argued that the relative configuration space and shape space of these models are close analogues from various perspectives of superspace and conformal superspace respectively. The geometry of these spaces and quantization thereupon is presented. A quantity that is frozen in the scale invariant relational particle model is demonstrated to be an internal time in a certain portion of the relational particle reformulation of Newtonian mechanics. The semiclassical approach for these models is studied as an emergent time resolution for these models, as are consistent records approaches. 
  The conventional spacetime formulation of general relativity may be recast as a dynamics of spatial 3-geometries (geometrodynamics). Furthermore, geometrodynamics can be derived from first principles. I investigate two distinct sets of these: (i) Hojman, Kucha\v{r} and Teitelboim's, which presuppose that the spatial 3-geometries are embedded in spacetime. (ii) The 3-space approach of Barbour, Foster, \'{O} Murchadha and Anderson in which the spatial 3-geometries are presupposed but spacetime is not. I consider how the constituent postulates of the conventional approach to relativity emerge or are to be built into these formulations. I argue that the 3-space approach is a viable description of classical physics (fundamental matter fields included), and one which affords considerable philosophical insight because of its `relationalist' character. From these assumptions of less structure, it is also interesting that conventional relativity can be recovered (albeit as one of several options).   However, contrary to speculation in the earlier 3-space approach papers, I also argue that this approach is not selective over which sorts of fundamental matter physics it admits. In particular, it does not imply the equivalence principle. 
  We study the present, flat isotropic universe in 1/R-modified gravity. We use the Palatini (metric-affine) variational principle and the Einstein (metric-compatible connected) conformal frame. We show that the energy density scaling deviates from the usual scaling for nonrelativistic matter, and the largest deviation occurs in the present epoch. We find that the current deceleration parameter derived from the apparent matter density parameter is consistent with observations. There is also a small overlap between the predicted and observed values for the redshift derivative of the deceleration parameter. The predicted redshift of the deceleration-to-acceleration transition agrees with that in the \Lambda-CDM model but it is larger than the value estimated from SNIa observations. 
  The most precise measurements are done at present by timing of radiopulsars in binary systems with two neutron stars. The timing measurements of the Taylor-Hulse pulsar B1913+16 gave the most precise results on testing of general relativity (GR), finding implicit proof of existence of gravitational waves. We show that available results of existing measurements, obtained to the year 1993, in combination with the results of the Mariner 10 in (1992), give the boundaries for the variation of the gravitational constant ${\dot G}/{G}$ inside the limits $(-0.6 \div +2)\cdot 10^{-12}$ year{$^{-1}$}. 
  This Resource Letter provides some guidance on issues that arise in teaching general relativity at both the undergraduate and graduate levels. Particular emphasis is placed on strategies for presenting the mathematical material needed for the formulation of general relativity. 
  We describe the general exact solution of Einstein's equation corresponding to a homogenous gravitational field. We study the geodesics in it, and we find that this simple spacetime exhibits very nice properties. In particular, it has a repelling boundary and all geodesics bounce off it. 
  We present an analysis of well-posedness of constrained evolution of 3+1 formulations of GR. In this analysis we explicitly take into account the energy and momentum constraints as well as possible algebraic constraints on the evolution of high-frequency perturbations of solutions of Einstein's equations. In this respect, our approach is principally different from standard analyses of well-posedness of free evolution in general relativity. Our study reveals the existence of subsets of the linearized Einstein's equations that control the well-posedness of constrained evolution. It is demonstrated that the well-posedness of ADM, BSSN and other 3+1 formulations derived from ADM by adding combinations of constraints to the right-hand-side of ADM and/or by linear transformation of the dynamical ADM variables depends entirely on the properties of the gauge. For certain classes of gauges we formulate conditions for well-posedness of constrained evolution. This provides a new basis for constructing stable numerical integration schemes for a classical Arnowitt--Deser--Misner (ADM) and many other 3+1 formulations of general relativity. 
  We determine the general conditions for the existence of Godel, Einstein static, and de Sitter universes in gravity theories derived from a Lagrangian that is an arbitrary function of the scalar curvature and Ricci and Riemann curvature invariants. Explicit expressions for the solutions are found in terms of the parameters defining the Lagrangian. We also determine the conditions on the Lagrangian of the theory under which time-travel is allowed in the Godel universes. 
  It is known that the Einstein-Hilbert action with a positive cosmological constant can be represented as a perturbation of the SO(4,1) BF theory by a symmetry-breaking term quadratic in the B field. Introducing fermionic matter generates additional terms in the action which are polynomial in the tetrads and the spin connection. We describe how to construct the generating functional in the spin foam formalism for a generic BF theory when the sources for the B and the gauge field are present. This functional can be used to obtain a path integral for General Relativity with matter as a perturbative series whose the lowest order term is a path integral for a topological gravity coupled to matter. 
  A generalization of the notion of surfaces of revolution in the spaces of General Relativity is presented. We apply this definition to the case of Carter's family [A] of solutions and we study the Kerr's metric with respect the above mentioned foliation. 
  We consider brane cosmologies within the context of five-dimensional actions with O(a') higher curvature corrections. The actions are compatible with bulk string amplitude calculations from heterotic string theory. We find wrapped solutions that satisfy the field equations in an approximate but acceptable manner given their complexity, where the internal four-dimensional scale factor is naturally inflating, having an exponential De-Sitter form. The temporal dependence of the metric components is non-trivial so that this metric cannot be factored as in a conformally flat case. The effective Planck mass is finite and the brane solutions localize four-dimensional gravity, while the four-dimensional gravitational constant varies with time. The Hubble constant can be freely specified through the initial value of the scalar field, to conform with recent data. 
  We describe a class of spin foam models of four-dimensional quantum gravity which is based on the integration of the tetrad one-forms in the path integral for the Palatini action of General Relativity. In the Euclidian gravity case this class of models can be understood as a modification of the Barrett-Crane spin foam model. Fermionic matter can be coupled by using the path integral with sources for the tetrads and the spin connection, and the corresponding state sum is based on a spin foam where both the edges and the faces are colored independently with the irreducible representations of the spacetime rotations group. 
  We review the results of a model of how nucleation of a new universe occurs, assuming a di quark identification for soliton-anti soliton constituent parts of a scalar field. Initially, we employ a false vacuum potential system; however ,when cosmological expansion is dominated by the Einstein cosmological constant at the end of chaotic inflation, the initial di quark scalar field is not consistent with respect to a semi classical consistency conditions we analyze as the potential changes to the chaotic inflationary potential utilized by Guth. We use Scherrer's derivation of a sound speed being zero during initial inflationary cosmology, and obtain a sound speed approaching unity as the slope of the scalar field moves away from a thin wall approximation. All this is to aid in a data reconstruction problem of how to account for the initial origins of CMB due to dark matter since effective field theories as presently constructed require a cut off value for applicability of their potential structure. This is often at the cost of, especially in early universe theoretical models, of clearly defined baryogenesis, and of a well defined mechanism of phase transitions. 
  Cooperstock and Tieu have proposed a model to account for galactic rotation curves without invoking dark matter. I argue that no model of this type can work. 
  The detection of gravitational radiation raises some subtle issues having to do with the coordinate invariance of general relativity. This paper explains these issues and their resolution by using an analogy with the Aharonov-Bohm effect of quantum mechanics. 
  We argue that the effective gauge group for {\it pure} four-dimensional loop quantum gravity(LQG) is SO(3) (or $SO(3,C)$) instead of SU(2) (or $SL(2,C)$). As a result, links with half-integer spins in spin network states are not realized for {\it pure} LQG, implying a modification of the spectra of area and volume operators. Our observations imply a new value of $\gamma \approx 0.170$ for the Immirzi parameter which is obtained from matching the Bekenstein-Hawking entropy to the number of states from LQG calculations. Moreover, even if the dominant contribution to the entropy is not assumed to come from configurations with the minimum spins, the results of both pure LQG and the supersymmetric extension of LQG can be made compatible when only integer spins are realized for the former, while the latter also contains half-integer spins, together with an Immirzi parameter for the supersymmetric case which is twice the value of the SO(3) theory. We also verify that the $-{1/2}$ coefficient of logarithmic correction to the Bekenstein-Hawking entropy formula is robust, independent of whether only integer, or also half-integer spins, are realized. 
  Using the third-order WKB approximation, we evaluate the quasinormal frequencies of massless scalar field perturbation around the black hole which is surrounded by the static and spherically symmetric quintessence. Our result shows that due to the presence of quintessence, the scalar field damps more rapidly. Moreover, we also note that the quintessential state parameter $\epsilon$ (the ratio of pressure $p_q$ to the energy density $\rho_q$) play an important role for the quasinormal frequencies. As the state parameter $\epsilon$ increases the real part increases and the absolute value of the imaginary part decreases. This means that the scalar field decays more slowly in the larger $\epsilon$ quintessence case. 
  A design of a configuration for violation of the averaged null energy condition (ANEC) and consequently other classic energy conditions (CECs), is presented. The methods of producing effective exotic matter (EM) for a traversable wormhole (TW) are discussed. Also, the approaches of less necessity of TWs to EM are considered. The result is, TW and similar structures; i.e., warp drive (WD) and Krasnikov tube are not just theoretical subjects for teaching general relativity (GR) or objects only an advanced civilization would be able to manufacture anymore, but a quite reachable challenge for current technology. Besides, a new compound metric is introduced as a choice for testing in the lab. 
  We prove that for any vacuum, maximal, asymptotically flat, axisymmetric initial data for Einstein equations close to extreme Kerr data, the inequality $\sqrt{J} \leq m$ is satisfied, where $m$ and $J$ are the total mass and angular momentum of the data. The proof consists in showing that extreme Kerr is a local minimum of the mass. 
  I study the canonical formulation and quantization of some simple parametrized systems, including the non-relativistic parametrized particle and the relativistic parametrized particle. Using Dirac's formalism I construct for each case the classical reduced phase space and study the dependence on the gauge fixing used. Two separate features of these systems can make this construction difficult: the actions are not invariant at the boundaries, and the constraints may have disconnected solution spaces. The relativistic particle is affected by both, while the non-relativistic particle displays only by the first. Analyzing the role of canonical transformations in the reduced phase space, I show that a change of gauge fixing is equivalent to a canonical transformation. In the relativistic case, quantization of one branch of the constraint at the time is applied and I analyze the electromagenetic backgrounds in which it is possible to quantize simultaneously both branches and still obtain a covariant unitary quantum theory. To preserve unitarity and space-time covariance, second quantization is needed unless there is no electric field. I motivate a definition of the inner product in all these cases and derive the Klein-Gordon inner product for the relativistic case. I construct phase space path integral representations for amplitudes for the BFV and the Faddeev path integrals, from which the path integrals in coordinate space (Faddeev-Popov and geometric path integrals) are derived. 
  The usual quantization of a classical space-time field does not touch the non-geometrical character of quantum mechanics. We believe that the deep problems of unification of general relativity and quantum mechanics are rooted in this poor understanding of the geometrical character of quantum mechanics. In Einstein's theory gravitation is expressed by geometry of space-time, and the solutions of the field equation are invariant w.r.t. a certain equivalence class of reference frames. This class can be characterized by the differential structure of space-time. We will show that matter is the transition between reference frames that belong to different differential structures, that the set of transitions of the differential structure is given by a Temperley-Lieb algebra which is extensible to a $C^{*}$-algebra comprising the field operator algebra of quantum mechanics and that the state space of quantum mechanics is the linear space of the differential structures. Furthermore we are able to explain the appearance of the complex numbers in quantum theory. The strong relation to Loop Quantum Gravity is discussed in conclusion. 
  The non-abelian Einstein-Born-Infeld-Dilaton theory, which rules the dynamics of tensor-scalar gravitation coupled to a $su(2)$-valued gauge field ruled by Born-Infeld lagrangian, is studied in a cosmological framework. The microscopic energy exchange between the gauge field and the dilaton which results from a non-universality of the coupling to gravity modifies the usual behaviour of tensor-scalar theories coupled to matter fluids. General cosmological evolutions are derived for different couplings to gravitation and a comparison to universal coupling is highlighted. Evidences of cosmic acceleration are presented when the evolution is interpreted in the Jordan physical frame of a matter respecting the weak equivalence principle. The importance for the mechanism of cosmic acceleration of the dynamics of the Born-Infeld gauge field, the attraction role of the matter fluid and the non-universality of the gravitational couplings is briefly outlined. 
  The Plebanski-Demianski metric, and those that can be obtained from it by taking coordinate transformations in certain limits, include the complete family of space-times of type D with an aligned electromagnetic field and a possibly non-zero cosmological constant. Starting with a new form of the line element which is better suited both for physical interpretation and for identifying different subfamilies, we review this entire family of solutions. Our metric for the expanding case explicitly includes two parameters which represent the acceleration of the sources and the twist of the repeated principal null congruences, the twist being directly related to both the angular velocity of the sources and their NUT-like properties. The non-expanding type D solutions are also identified. All special cases are derived in a simple and transparent way. 
  The Big Bang Observer (BBO) is a proposed space-based gravitational-wave (GW) mission designed primarily to search for an inflation-generated GW background in the frequency range 0.1-1 Hz. The major astrophysical foreground in this range is gravitational radiation from inspiraling compact binaries. This foreground is expected to be much larger than the inflation-generated background, so to accomplish its main goal, BBO must be sensitive enough to identify and subtract out practically all such binaries in the observable universe. It is somewhat subtle to decide whether BBO's current baseline design is sufficiently sensitive for this task, since, at least initially, the dominant noise source impeding identification of any one binary is confusion noise from all the others. Here we present a self-consistent scheme for deciding whether BBO's baseline design is indeed adequate for subtracting out the binary foreground. We conclude that the current baseline should be sufficient. However if BBO's instrumental sensitivity were degraded by a factor 2-4, it could no longer perform its main mission. It is impossible to perfectly subtract out each of the binary inspiral waveforms, so an important question is how to deal with the "residual" errors in the post-subtraction data stream. We sketch a strategy of "projecting out" these residual errors, at the cost of some effective bandwidth. We also provide estimates of the sizes of various post-Newtonian effects in the inspiral waveforms that must be accounted for in the BBO analysis. 
  Approximate equations are derived for the motion of a gyroscope on the earth's gravitational field using the Einstein, Infeld, Hoffmann surface integral method. This method does not require a knowledge of the energy-momentum-stress tensor associated with the gyroscope and uses only its exterior field for its characterization. The resulting equations of motion differ from those of previous derivations. 
  Conditions for the existence and stability of de Sitter space in modified gravity are derived by considering inhomogeneous perturbations in a gauge-invariant formalism. The stability condition coincides with the corresponding condition for stability with respect to homogeneous perturbations, while this is not the case in scalar-tensor gravity. The stability criterion is applied to various modified gravity models of the early and the present universe. 
  Using the Moller, Einstein, Bergmann-Thomson and Landau-Lifshitz energy-momentum definitions both in general relativity and teleparallel gravity, we find the energy-momentum of the closed universe based on the generalized Bianchi-type I metric. 
  We apply the ``consistent discretization'' technique to the Regge action for (Euclidean and Lorentzian) general relativity in arbitrary number of dimensions. The result is a well defined canonical theory that is free of constraints and where the dynamics is implemented as a canonical transformation. This provides a framework for the discussion of topology change in canonical quantum gravity. In the Lorentzian case, the framework appears to be naturally free of the ``spikes'' that plague traditional formulations. It also provides a well defined recipe for determining the measure of the path integral. 
  We study the gravitational vacuum star (gravastar) configuration as proposed by other authors in a model where the interior de Sitter spacetime segment is continuously extended to the exterior Schwarzschild spacetime. The multilayered structure in previous papers is replaced by a continuous stress-energy tensor at the price of introducing anisotropy in the (fluid) model of the gravastar. Either with an ansatz for the equation of state connecting the radial $p_r$ and tangential $p_t$ pressure or with a calculated equation of state with non-homogeneous energy/fluid density, solutions are obtained which in all aspects satisfy the conditions expected for an anisotropic gravastar. Certain energy conditions have been shown to be obeyed and a polytropic equation of state has been derived. Stability of the solution with respect to possible axial perturbation is shown to hold. 
  The late-time tail behavior of a coupled scalar field perturbation is investigated in a Schwarzschild black hole with a global monopole background. We find that the decay rate for the massless scalar field is described by $t^{-2\rho-2}$ and for the massive case at intermediate late-time is $t^{-\rho-1}\cos{[\mu t-(\rho+1)\pi/2]}$, where $\rho$ is the function of the coupling constant $\xi$ and the symmetry breaking scale $\eta_0$, namely, $\rho=\sqrt{\frac{l(l+1)+16\xi \pi G\eta^2_0}{1-8\pi G\eta^2_0}+{1/4}}$. Our result shows that the larger $\xi$ and $\eta_0$, the faster the decay of the scalar fields. However, the decay rate for the asymptotically late-time tail of the massive scalar field still is $t^{-5/6}$, and not affected by the couple between the scalar and gravitational fields. 
  In this work it is investigated if fermionic sources could be responsible for accelerated periods during the evolution of a universe where a matter field would answer for the decelerated period. The self-interaction potential of the fermionic field is considered as a function of the scalar and pseudo-scalar invariants. Irreversible processes of energy transfer between the matter and gravitational fields are also considered. It is shown that the fermionic field could behave like an inflaton field in the early universe and as dark energy for an old universe. 
  In this work we deal with the extension of the Kaluza-Klein approach to a non-Abelian gauge theory; we show how we need to consider the link between the n-dimensional model and a four-dimensional observer physics, in order to reproduce fields equations and gauge transformations in the four-dimensional picture. More precisely, in fields equations any dependence on extra-coordinates is canceled out by an integration, as consequence of the unobservability of extra-dimensions. Thus, by virtue of this extra-dimensions unobservability, we are able to recast the multidimensional Einstein equations into the four-dimensional Einstein-Yang-Mills ones, as well as all the right gauge transformations of fields are induced. The same analysis is performed for the Dirac equation describing the dynamics of the matter fields and, again, the gauge coupling with Yang-Mills fields are inferred from the multidimensional free fields theory, together with the proper spinors transformations. 
  The inequality $\sqrt{J}\leq m$ is proved for vacuum, asymptotically flat, maximal and axisymmetric data close to extreme Kerr data. The physical significance of this inequality and its relation to the standard picture of the gravitational collapse are discussed. 
  A highly accurate computer program is used to study axially symmetric and stationary spacetimes containing a Black Hole surrounded by a ring of matter. It is shown that the matter ring affects the properties of the Black Hole drastically. In particular, the absolute value of the ratio of the Black Hole's angular momentum to the square of its mass not only exceeds one, but can be greater than ten thousand (|J|/M^2 > 10^4). Indeed, the numerical evidence suggests that this quantity is unbounded. 
  We present new techniqes for evolving binary black hole systems which allow the accurate determination of gravitational waveforms directly from the wave zone region of the numerical simulations. Rather than excising the black hole interiors, our approach follows the "puncture" treatment of black holes, but utilzing a new gauge condition which allows the black holes to move successfully through the computational domain. We apply these techniques to an inspiraling binary, modeling the radiation generated during the final plunge and ringdown. We demonstrate convergence of the waveforms and good conservation of mass-energy, with just over 3% of the system's mass converted to gravitional radiation. 
  We investigate the connection between the entanglement system in Minkowski spacetime and the black hole using the scaling analysis. Here we show that the entanglement system satisfies the Bekenstein entropy bound. Even though the entropies of two systems are the same form, the entanglement energy is different from the black hole energy. Introducing the Casimir energy of the vacuum energy fluctuations rather than the entanglement energy, it shows a feature of the black hole energy. Hence the Casimir energy is more close to the black hole than the entanglement energy. Finally, we find that the entanglement system behaves like the black hole if the gravitational effects are included properly. 
  So called "analogue models" use condensed matter systems (typically hydrodynamic) to set up an "effective metric" and to model curved-space quantum field theory in a physical system where all the microscopic degrees of freedom are well understood. Known analogue models typically lead to massless minimally coupled scalar fields. We present an extended "analogue space-time" programme by investigating a condensed-matter system - in and beyond the hydrodynamic limit - that is in principle capable of simulating the massive Klein-Gordon equation in curved spacetime. Since many elementary particles have mass, this is an essential step in building realistic analogue models, and an essential first step towards simulating quantum gravity phenomenology. Specifically, we consider the class of two-component BECs subject to laser-induced transitions between the components, and we show that this model is an example for Lorentz invariance violation due to ultraviolet physics. Furthermore our model suggests constraints on quantum gravity phenomenology in terms of the "naturalness problem" and "universality issue". 
  Adopting the monodromy technique devised by Motl and Neitzke, we investigate analytically the asymptotic quasinormal frequencies of a coupled scalar field in the Gibbons-Maeda dilaton spacetime. We find that it is described by $ e^{\beta \omega}=-[1+2\cos{(\frac{\sqrt{2\xi+1}}{2} \pi)}]-e^{-\beta_I \omega}[2+2\cos{(\frac{\sqrt{2\xi+1}}{2}\pi)}]$, which depends on the structure parameters of the background spacetime and on the coupling between the scalar and gravitational fields. As the parameters $\xi$ and $\beta_I$ tend to zero, the real parts of the asymptotic quasinormal frequencies becomes $T_H\ln{3}$, which is consistent with Hod's conjecture. When $\xi={91/18} $, the formula becomes that of the Reissner-Nordstr\"{o}m spacetime. 
  The relation between symmetry reduction before and after quantization of a field theory is discussed using a toy model: the axisymmetric Klein-Gordon field. We consider three possible notions of symmetry at the quantum level: invariance under the group action, and two notions derived from imposing symmetry as a system of constraints a la Dirac, reformulated as a first class system. One of the latter two turns out to be the most appropriate notion of symmetry in the sense that it satisfies a number of physical criteria, including the commutativity of quantization and symmetry reduction. Somewhat surprisingly, the requirement of invariance under the symmetry group action is not appropriate for this purpose. A generalization of the physically selected notion of symmetry to loop quantum gravity is presented and briefly discussed. 
  Variables adapted to the quantum dynamics of spherically symmetric models are introduced, which further simplify the spherically symmetric volume operator and allow an explicit computation of all matrix elements of the Euclidean and Lorentzian Hamiltonian constraints. The construction fits completely into the general scheme available in loop quantum gravity for the quantization of the full theory as well as symmetric models. This then presents a further consistency check of the whole scheme in inhomogeneous situations, lending further credence to the physical results obtained so far mainly in homogeneous models. New applications in particular of the spherically symmetric model in the context of black hole physics are discussed. 
  For self-gravitating, static, spherically symmetric, minimally coupled scalar fields with arbitrary potentials and negative kinetic energy (favored by the cosmological observations), we give a classification of possible regular solutions to the field equations with flat, de Sitter and AdS asymptotic behavior. Among the 16 presented classes of regular rsolutions are traversable wormholes, Kantowski-Sachs (KS) cosmologies beginning and ending with de Sitter stages, and asymptotically flat black holes (BHs). The Penrose diagram of a regular BH is Schwarzschild-like, but the singularity at $r=0$ is replaced by a de Sitter infinity, which gives a hypothetic BH explorer a chance to survive. Such solutions also lead to the idea that our Universe could be created from a phantom-dominated collapse in another universe, with KS expansion and isotropization after crossing the horizon. Explicit examples of regular solutions are built and discussed. Possible generalizations include $k$-essence type scalar fields (with a potential) and scalar-tensor theories of gravity. 
  Bekenstein-Hawking formalism of black hole thermodynamics should be modified to incorporate quantum gravitational effects. Generalized Uncertainty Principle(GUP) provides a suitable framework to perform such modifications. In this paper, we consider a general form of GUP to find black hole thermodynamics in a model universe with large extra dimensions. We will show that black holes radiate mainly in the four-dimensional brane. Existence of black holes remnants as a possible candidate for dark matter is discussed. 
  Spin-weighted spheroidal harmonics are useful in a variety of physical situations, including light scattering, nuclear modeling, signal processing, electromagnetic wave propagation, black hole perturbation theory in four and higher dimensions, quantum field theory in curved space-time and studies of D-branes. We first review analytic and numerical calculations of their eigenvalues and eigenfunctions in four dimensions, filling gaps in the existing literature when necessary. Then we compute the angular dependence of the spin-weighted spheroidal harmonics corresponding to slowly-damped quasinormal mode frequencies of the Kerr black hole, providing numerical tables and approximate formulas for their scalar products. Finally we present an exhaustive analytic and numerical study of scalar spheroidal harmonics in (n+4) dimensions. 
  The electric and the magnetic part of the Weyl tensor, as well as the invariants obtained from them, are calculated for the Bondi vacuum metric. One of the invariants vanishes identically and the other only exhibits contributions from terms of the Weyl tensor containing the static part of the field. It is shown that the necessary and sufficient condition for the spacetime to be purely electric is that such spacetime be static. It is also shown that the vanishing of the electric part implies Minkowski spacetime. Unlike the electric part, the magnetic part does not contain contributions from the static field. Finally a speculation about the link between the vorticity of world lines of observers at rest in a Bondi frame, and gravitational radiation, is presented. 
  This article analyzes the present anomalies of cosmology from the point of view of integrable Weyl geometry. It uses P.A.M. Dirac's proposal for a weak extension of general relativity, with some small adaptations. Simple models with interesting geometrical and physical properties, not belonging to the Friedmann-Lema\^{\i}tre class, are studied in this frame. Those with positive spatial curvature (Einstein-Weyl universes) go well together with observed mass density $\Omega_m$, CMB, supernovae Ia data, and quasar frequencies. They suggest a physical role for an equilibrium state of the Maxwell field proposed by I.E. Segal in the 1980s (Segal background) and for a time invariant balancing condition of vacuum energy density. The latter leads to a surprising agreement with the BF-theoretical calculation proposed by C. Castro (2002). 
  In this paper we consider magnetized black holes and black rings in the higher dimensional dilaton gravity. Our study is based on exact solutions generated by applying a Harrison transformation to known asymptotically flat black hole and black ring solutions in higher dimensional spacetimes. The explicit solutions include the magnetized version of the higher dimensional Schwarzschild-Tangherlini black holes, Myers-Perry black holes and five dimensional (dipole) black rings. The basic physical quantities of the magnetized objects are calculated. We also discuss some properties of the solutions and their thermodynamics. The ultrarelativistic limits of the magnetized solutions are briefly discussed and an explicit example is given for the $D$-dimensional magnetized Schwarzschild-Tangherlini black holes. 
  Having in mind applications to gravitational wave theory (in connection with the radiation reaction problem), stochastic semiclassical gravity (in connection with the regularization of the noise kernel) and quantum field theory in higher-dimensional curved spacetime (in connection with the Hadamard regularization of the stress-energy tensor), we improve the DeWitt-Schwinger and Hadamard representations of the Feynman propagator of a massive scalar field theory defined on an arbitrary gravitational background by deriving higher-order terms for the covariant Taylor series expansions of the geometrical coefficients -- i.e., the DeWitt and Hadamard coefficients -- that define them. 
  Every relativistic particle has 4-speed equal to $c$, since $g_{\mu \nu} \frac{dx^\mu}{d\tau} \frac{dx^\nu}{d\tau} = c^2$. With the choice of $k = +1$ in the FRW metric, the cosmological scale factor $a(t)$ has the natural interpretation of the radius of the sphere $S^3_a = \{x \in \mathbb{R}^4 : (x, x) = a^2\}$. Thus, a particle at rest in the cosmological frame has 4-speed equal to $\frac{da}{dt}$. This leads us to infer that $\dot a = c$, which respresents a simple kinematic constraint linking the speed of light to the cosmological scale factor. This drastically changes the $k=+1$ picture from a closed deaccelerating universe to an open accelerating universe, settles the horizon problem, and provides for a new cosmological model more appealing to our natural intuition. In this paper we shall consider ramifications of this model. 
  We present a general description of the propagation properties of quantum gravity modified electrodynamics characterized by constitutive relations up to second order in the correction parameter. The effective description corresponds to an electrodynamics in a dispersive and absorptive non-local medium, where the Green functions and the refraction indices can be explicitly calculated. The reality of the electromagnetic field together with the requirement of causal propagation in a given referrence frame leads to restrictions in the form of such refraction indices. In particular, absorption must be present in all cases and, contrary to the usual assumption, it is the dominant aspect in those effective models which exhibit linear effects in the correction parameter not related to birefringence. In such a situation absorption is linear while propagation is quadratical in the correction parameter. 
  This paper is devoted to the analysis of charged superselection sectors in the framework of the locally covariant quantum field theories. We shall analize sharply localizable charges, and use net-cohomology of J.E. Roberts as a main tool. We show that to any 4-dimensional globally hyperbolic spacetime it is attached a unique, up to equivalence, symmetric tensor $\Crm^*-$category with conjugates (in case of finite statistics); to any embedding between different spacetimes, the corresponding categories can be embedded, contravariantly, in such a way that all the charged quantum numbers of sectors are preserved. This entails that to any spacetime is associated a unique gauge group, up to isomorphisms, and that to any embedding between two spacetimes there corresponds a group morphism between the related gauge groups. This form of covariance between sectors also brings to light the issue whether local and global sectors are the same. We conjecture this holds that at least on simply connected spacetimes. It is argued that the possible failure might be related to the presence of topological charges. Our analysis seems to describe theories which have a well defined short-distance asymptotic behaviour. 
  We discuss the unique existence, arising by analogy to that in algebraically special space-times, of a CR structure realized on null infinity for any asymptotically flat Einstein or Einstein-Maxwell space-time. 
  The structure and dynamics of the standard model and gravity are described by a Clifford valued connection and its curvature. 
  We use the Kruskal time coordinate T to define the initial time. By this way, it naturally divides the stable study into one connected with the two regions: the white-hole-connected region and the black-hole-connected region. The union of the two regions covers the Schwarzschild space-time (r>2m). We also obtain the very reasonable conclusion: the white-hole-connected region is instable; whereas the black-hole-connected region is stable. If we take the instability with caution and seriousness, it might be not unreasonable to regard that the Schwarzschild black hole might be instable too. 
  The complete family of exact solutions representing accelerating and rotating black holes with possible electromagnetic charges and a NUT parameter is known in terms of a modified Plebanski-Demianski metric. This demonstrates the singularity and horizon structure of the sources but not that the complete space-time describes two causally separated black holes. To demonstrate this property, the metric is first cast in the Weyl-Lewis-Papapetrou form. After extending this up to the acceleration horizon, it is then transformed to the boost-rotation-symmetric form in which the global properties of the solution are manifest. The physical interpretation of these solutions is thus clarified. 
  As a homage to A K Raychaudhuri, I derive in a straightforward way his famous equation and also indicate the problems he was last engaged in. 
  A theory in which 16-dimensional curved Clifford space (C-space) provides a realization of Kaluza-Klein theory is investigated. No extra dimensions of spacetime are needed: "extra dimensions" are in C-space. We explore the spin gauge theory in C-space and show that the generalized spin connection contains the usual 4-dimensional gravity and Yang-Mills fields of the U(1)xSU(2)xSU(3) gauge group. The representation space for the latter group is provided by 16-component generalized spinors composed of four usual 4-component spinors, defined geometrically as the members of four independent minimal left ideals of Clifford algebra. 
  In this paper we discuss matter inheritance collineations by giving a complete classification of spherically symmetric static spacetimes by their matter inheritance symmetries. It is shown that when the energy-momentum tensor is degenerate, most of the cases yield infinite dimensional matter inheriting symmetries. It is worth mentioning here that two cases provide finite dimensional matter inheriting vectors even for the degenerate case. The non-degenerate case provides finite dimensional matter inheriting symmetries. We obtain different constraints on the energy-momentum tensor in each case. It is interesting to note that if the inheriting factor vanishes, matter inheriting collineations reduce to be matter collineations already available in the literature. This idea of matter inheritance collineations turn out to be the same as homotheties and conformal Killing vectors are for the metric tensor. 
  We discuss the twin paradox or the clock paradox under the small velocity approximation of special relativity. In this paper the traveller twin of the standard twin parable sets out with a non-relativistic speed for the trip leaving behind the stay-at-home one on earth and continues up to a distance and finally returns home with the same speed when the siblings can compare their ages or their initially synchronized wrist watches. The common knowledge that at small velocities the length contraction and time dilation effects of special relativity can be ignored so that the world becomes essentially classical, tends to lead to a paradox in connection with the twin problem, which a "clever" student eventually discovers. After discussing and resolving the issue, some more related paradoxes have been presented. The resolution of all these problems provides some additional insight into the century old paradox. 
  We show that in theories of gravity that add quadratic curvature invariants to the Einstein-Hilbert action there exist expanding vacuum cosmologies with positive cosmological constant which do not approach the de Sitter universe. Exact solutions are found which inflate anisotropically. This behaviour is driven by the Ricci curvature invariant and has no counterpart in the general relativistic limit. These examples show that the cosmic no-hair theorem does not hold in these higher-order extensions of general relativity and raises new questions about the ubiquity of inflation in the very early universe and the thermodynamics of gravitational fields. 
  The curvaton reheating in a tachyonic inflationary universe model with an exponential potential is studied. We have found that the energy density in the kinetic epoch, has a complicated dependencies of the scale factor. For different scenarios, the temperature of reheating is computed. These temperature result to be analogous to those obtained in the standard case of the curvaton scenario. 
  Fermi coordinates are directly constructed in de Sitter and Goedel spacetimes and the corresponding exact coordinate transformations are given explicitly. The quasi-inertial Fermi coordinates are then employed to discuss the dynamics of a free test particle in these spacetimes and the results are compared to the corresponding generalized Jacobi equations that contain only the lowest-order tidal terms. The domain of validity of the generalized Jacobi equation is thus examined in these cases. Furthermore, the difficulty of constructing explicit Fermi coordinates in black-hole spacetimes is demonstrated. 
  This paper discusses some of the physical properties of plane symmetric self-similar solutions of the first kind (i.e., homothetic solutions). We are interested in calculating the expansion, the acceleration, the rotation, the shear tensor, the shear invariant, and the expansion rate (given by Raychaudhuri's equation). We check these properties both in co-moving and non-co-moving coordinates (only in the radial direction). Further, the singularity structure of such solutions will be explored. This analysis provides some interesting features of self-similar solutions. 
  Kerr-Schild solutions of the Einstein-Maxwell field equations, containing semi-infinite axial singular lines, are investigated.   It is shown that axial singularities break up the black hole, forming holes in the horizon. As a result, a tube-like region appears which allows matter to escape from the interior without crossing the horizon. It is argued that axial singularities of this kind, leading to very narrow beams, can be created in black holes by external electromagnetic or gravitational excitations and may be at the origin of astrophysically observable effects such as jet formation. 
  We take arbitrary gravitational perturbations of a 5d spacetime and reduce it to the form an axially symmetric warped braneworld. Then, we write the filed equations for the linearized gravity perturbations. We obtain the equations that describes the graviton, gravivector and the graviscalar fluctuations and analyse the effects of the Schr\"odinger potentials that appear in these equations. 
  Perturbations around black holes have been an intriguing topic in the last few decades. They are particularly important today, since they relate to the gravitational wave observations which may provide the unique fingerprint of black holes' existence. Besides the astrophysical interest, theoretically perturbations around black holes can be used as testing grounds to examine the proposed AdS/CFT and dS/CFT correspondence. 
  We present new exact solutions of the Einstein-Yang-Mills system. The solutions are described by a null Yang-Mills field in a Kundt spacetime. They generalize a previously known solution for a metric of $pp$ wave type. The solutions are formally of Petrov type III. 
  Consider, in the domain of outer communication of a Kerr-Newman black hole, a point (observation event) and a timelike curve (worldline of light source). Assume that the worldline of the source (i) has no past end-point, (ii) does not intersect the caustic of the past light-cone of the observation event, and (iii) goes neither to the horizon nor to infinity in the past. We prove that then for infinitely many positive integers k there is a past-pointing lightlike geodesic of (Morse) index k from the observation event to the worldline of the source, hence an observer at the observation event sees infinitely many images of the source. Moreover, we demonstrate that all lightlike geodesics from an event to a timelike curve in the domain of outer communication are confined to a certain spherical shell. Our characterization of this spherical shell shows that in the Kerr-Newman spacetime the occurrence of infinitely many images is intimately related to the occurrence of centrifugal-plus-Coriolis force reversal. 
  Equilibria of binary neutron stars in close circular orbits are computed numerically in a waveless formulation: The full Einstein-relativistic-Euler system is solved on an initial hypersurface to obtain an asymptotically flat form of the 4-metric and an extrinsic curvature whose time derivative vanishes in a comoving frame. Two independent numerical codes are developed, and solution sequences that model inspiraling binary neutron stars during the final several orbits are successfully computed. The binding energy of the system near its final orbit deviates from earlier results of third post-Newtonian and of spatially conformally flat calculations. The new solutions may serve as initial data for merger simulations and as members of quasiequilibrium sequences to generate gravitational wave templates, and may improve estimates of the gravitational-wave cutoff frequency set by the last inspiral orbit. 
  In this note we show that the latest determinations of the residual Mercury's perihelion advance, obtained by accounting for almost all known Newtonian and post-Newtonian orbital effects, yields only very broad constraints on the cosmological constant. Indeed, from \delta\dot\omega=-0.0036 + - 0.0050 arcseconds per century one gets -2 10^-34 km^-2 < Lambda < 4 10^-35 km^-2. The currently accepted value for Lambda, obtained from many independent cosmological and large-scale measurements, amounts to almost 10^-46 km^-2. 
  According to the braneworld model of gravity by Dvali, Gabadadze and Porrati, our Universe is a four-dimensional space-time brane embedded in a larger, infinite five-dimensional bulk space. Contrary to the other forces constrained to remain on the brane, gravity is able to explore the entire bulk getting substantially modified at large distances. This model has not only cosmological consequences allowing to explain the observed acceleration of the expansion of our Universe without resorting to the concept of dark energy, but makes also testable predictions at small scales. Interestingly, such local effects can yield information on the global properties of the Universe and on the kind of expansion currently ongoing. Indeed, among such predictions there are extra precessions of the perihelia and the mean longitudes of the planetary orbits which are affected by a twofold degeneration sign: one sign refers to a Friedmann-Lemaitre-Robertson-Walker phase while the opposite sign is for a self-sccelerated phase. In this paper we report on recent observations of planetary motions in the Solar System which are compatible with the existence of a fifth dimension as predicted in the Dvali-Gabadadze-Porrati model with a self-accelerated cosmological phase, although the errors are still large. The Friedmann-Lemaitre-Robertson-Walker phase is, instead, ruled out. 
  We discuss three complementary aspects of scalar curvature singularities: asymptotic causal properties, asymptotic Ricci and Weyl curvature, and asymptotic spatial properties. We divide scalar curvature singularities into two classes: so-called asymptotically silent singularities and non-generic singularities that break asymptotic silence. The emphasis in this paper is on the latter class which have not been previously discussed. We illustrate the above aspects and concepts by describing the singularities of a number of representative explicit perfect fluid solutions. 
  We compute the one loop fermion self-energy for massless Dirac + Einstein in the presence of a locally de Sitter background. We employ dimensional regularization and obtain a fully renormalized result by absorbing all divergences with BPHZ counterterms. An interesting technical aspect of this computation is the need for a noninvariant counterterm owing to the breaking of de Sitter invariance by our gauge condition. Our result can be used in the quantum-corrected Dirac equation to search for inflation-enhanced quantum effects from gravitons, analogous to those which have been found for massless, minimally coupled scalars. 
  We consider the coupling of scalar topological matter to (2+1)-dimensional gravity. The matter fields consist of a 0-form scalar field and a 2-form tensor field. We carry out a canonical analysis of the classical theory, investigating its sectors and solutions. We show that the model admits both BTZ-like black-hole solutions and homogeneous/inhomogeneous FRW cosmological solutions.We also investigate the global charges associated with the model and show that the algebra of charges is the extension of the Kac-Moody algebra for the field-rigid gauge charges, and the Virasoro algebrafor the diffeomorphism charges. Finally, we show that the model can be written as a generalized Chern-Simons theory, opening the perspective for its formulation as a generalized higher gauge theory. 
  The Keski-Vakkuri, Kraus and Wilczek (KKW) analysis is used to compute the temperature and entropy in the dyadosphere of a charged black hole solution. For our purpose we choose the dyadosphere region of the Reissner-Nordstrom black hole solution. Our results show that the expressions of the temperature and entropy in the dyadosphere of this charged black hole are not the Hawking temperature and the Bekenstein-Hawking entropy, respectively. 
  Full relativistic simulations in three dimensions invariably develop runaway modes that grow exponentially and are accompanied by violations of the Hamiltonian and momentum constraints. Recently, we introduced a numerical method (Hamiltonian relaxation) that greatly reduces the Hamiltonian constraint violation and helps improve the quality of the numerical model. We present here a method that controls the violation of the momentum constraint. The method is based on the addition of a longitudinal component to the traceless extrinsic curvature generated by a vector potential w_i, as outlined by York. The components of w_i are relaxed to solve approximately the momentum constraint equations, pushing slowly the evolution toward the space of solutions of the constraint equations. We test this method with simulations of binary neutron stars in circular orbits and show that effectively controls the growth of the aforementioned violations. We also show that a full numerical enforcement of the constraints, as opposed to the gentle correction of the momentum relaxation scheme, results in the development of instabilities that stop the runs shortly. 
  We investigate the gravitational implosion of magnetized matter by studying the inhomogeneous collapse of a weakly magnetized Tolman-Bondi spacetime. The role of the field is analyzed by looking at the convergence of neighboring particle worldlines. In particular, we identify the magnetically related stresses in the Raychaudhuri equation and use the Tolman-Bondi metric to evaluate their impact on the collapsing dust. We find that, despite the low energy level of the field, the Lorentz force dominates the advanced stages of the collapse, leading to a strongly anisotropic contraction. In addition, of all the magnetic stresses, those that resist the collapse are found to grow faster. 
  Future missions of gravitational-wave astronomy will be operated by space-based interferometers, covering very wide range of frequency. Search for stochastic gravitational-wave backgrounds (GWBs) is one of the main targets for such missions, and we here discuss the prospects for direct measurement of isotropic and anisotropic components of (primordial) GWBs around the frequency 0.1-10 Hz. After extending the theoretical basis for correlation analysis, we evaluate the sensitivity and the signal-to-noise ratio for the proposed future space interferometer missions, like Big-Bang Observer (BBO), Deci-Hertz Interferometer Gravitational-wave Observer (DECIGO) and recently proposed Fabry-Perot type DECIGO. The astrophysical foregrounds which are expected at low frequency may be a big obstacle and significantly reduce the signal-to-noise ratio of GWBs. As a result, minimum detectable amplitude may reach h^2 \ogw = 10^{-15} \sim 10^{-16}, as long as foreground point sources are properly subtracted. Based on correlation analysis, we also discuss measurement of anisotropies of GWBs. As an example, the sensitivity level required for detecting the dipole moment of GWB induced by the proper motion of our local system is closely examined. 
  We report on a search for gravitational wave bursts in data from the three LIGO interferometric detectors during their third science run. The search targets subsecond bursts in the frequency range 100-1100 Hz for which no waveform model is assumed, and has a sensitivity in terms of the root-sum-square (rss) strain amplitude of hrss ~ 10^{-20} / sqrt(Hz). No gravitational wave signals were detected in the 8 days of analyzed data. 
  We present a simple model in which a brane with anisotropic metric is embedded in an AdS bulk. We discuss the localization of the massless mode and the amplification of both the massless and the massive modes on the branes, paying particular attention to the normalization of the perturbed action and to the evaluation of the effective coupling constant that controls the amplitude of the spectrum. In the model under investigation there is no mass gap between massless and massive modes, and the massive modes can be amplified, with mass-dependent amplitudes. 
  In what respect does terrestrial physics reflect the two unique features of the global gravitational field: its infinite range and its equivalence to spacetime curvature? We quote the evidence that true irreversibility, i.e the growth of the Boltzmann entropy of any finite system, is the consequence of the global state of the gravitation dominated and expanding universe. Moreover, as another example, we calculate the effect of global expansion and of the gravitational potential observed in our Local Group on spacetime metric in terms of curvature. Surprisingly, we find an energy density which is in numerical agreement with the purely quantum theoretical result for the Casimir energy density containing Planck's constant. 
  The Wheeler-DeWitt equation of Friedmann models with a massless quantum field is formulated with arbitrary factor ordering of the Hamiltonian constraint operator. A scalar product of wave functions is constructed, giving rise to a probability interpretation and making comparison with the classical solution possible. In general the bahaviour of the wave function of the model depends on a critical energy of the matter field, which, in turn, depends on the chosen factor ordering. By certain choices of the ordering the critical energy can be pushed down to zero. 
  We consider the scenario where our observable universe is devised as a dynamical four-dimensional hypersurface embedded in a five-dimensional bulk spacetime, with a large extra dimension, which is the {\it generalization of the flat FRW cosmological metric to five dimensions}. This scenario generates a simple analytical model where different stages of the evolution of the universe are approximated by distinct parameterizations of the {\it same} spacetime. In this model the evolution from decelerated to accelerated expansion can be interpreted as a "first-order" phase transition between two successive stages. The dominant energy condition allows different parts of the universe to evolve, from deceleration to acceleration, at different redshifts within a narrow era. This picture corresponds to the creation of bubbles of new phase, in the middle of the old one, typical of first-order phase transitions. Taking $\Omega_{m} = 0.3$ today, we find that the cross-over from deceleration to acceleration occurs at $z \sim 1-1.5 $, regardless of the equation of state in the very early universe. In the case of primordial radiation, the model predicts that the deceleration parameter "jumps" from $q \sim + 1.5$ to $q \sim - 0.4$ at $z \sim 1.17$. At the present time $q = - 0.55$ and the equation of state of the universe is $w = p/\rho \sim - 0.7 $, in agreement with observations and some theoretical predictions. 
  We investigate the adiabatic orbital evolution of a point particle in the Kerr spacetime due to the emission of gravitational waves. In the case that the timescale of the orbital evolution is enough smaller than the typical timescale of orbits, the evolution of orbits is characterized by the change rates of three constants of motion, the energy $E$, the azimuthal angular momentum $L$, and the Carter constant $Q$. For $E$ and $L$, we can evaluate their change rates from the fluxes of the energy and the angular momentum at infinity and on the event horizon according to the balance argument. On the other hand, for the Carter constant, we cannot use the balance argument because we do not know the conserved current associated with it. %and the corresponding conservation law. Recently, Mino proposed a new method of evaluating the averaged change rate of the Carter constant by using the radiative field. In our previous paper we developed a simplified scheme for practical evaluation of the evolution of the Carter constant based on the Mino's proposal. In this paper we describe our scheme in more detail, and derive explicit analytic formulae for the change rates of the energy, the angular momentum and the Carter constant. 
  An overview of the searches for continuous gravitational wave signals in LIGO and GEO 600 performed on different recent science runs and results are presented. This includes both searching for gravitational waves from known pulsars as well as blind searches over a wide parameter space. 
  An argument is made to show that the singularity in the General Theory of Relativity (GTR) is the expression of a non-Machian feature. It can be avoided with a scale-invariant dynamical theory, a property lacking in GTR. It is further argued that the global non-conservation of energy in GTR also results from the lack of scale-invariance and the field formulation presented by several authors can only resolve the problem in part. A truly scale-invariant theory is required to avoid these two problems in a more consistent approach 
  Computational methods are essential to provide waveforms from coalescing black holes, which are expected to produce strong signals for the gravitational wave observatories being developed. Although partial simulations of the coalescence have been reported, scientifically useful waveforms have so far not been delivered. The goal of the AppleswithApples (AwA) Alliance is to design, coordinate and document standardized code tests for comparing numerical relativity codes. The first round of AwA tests have now being completed and the results are being analyzed. These initial tests are based upon periodic boundary conditions designed to isolate performance of the main evolution code. Here we describe and carry out an additional test with periodic boundary conditions which deals with an essential feature of the black hole excision problem, namely a non-vanishing shift. The test is a shifted version of the existing AwA gauge wave test. We show how a shift introduces an exponentially growing instability which violates the constraints of a standard harmonic formulation of Einstein's equations. We analyze the Cauchy problem in a harmonic gauge and discuss particular options for suppressing instabilities in the gauge wave tests. We implement these techniques in a finite difference evolution algorithm and present test results. Although our application here is limited to a model problem, the techniques should benefit the simulation of black holes using harmonic evolution codes. 
  In a previous paper (hep-th/0407256) local scalar QFT (in Weyl algebraic approach) has been constructed on degenerate semi-Riemannian manifolds $\bS^1\times \Sigma$ corresponding to the extension of Killing horizons by adding points at infinity to the null geodesic forming the horizon. It has been proved that the theory admits a natural representation of $PSL(2,\bR)$ in terms of $*$-automorphisms and this representation is unitarily implementable if referring to a certain invariant state $\lambda$. Among other results it has been proved that the theory admits a class of inequivalent algebraic (coherent) states $\{\lambda_\zeta\}$, with $\zeta\in L^2(\Sigma)$, which break part of the symmetry, in the sense that each of them is not invariant under the full group $PSL(2,\bR)$ and so there is no unitary representation of whole group $PSL(2,\bR)$ which leaves fixed the cyclic GNS vector. These states, if restricted to suitable portions of $\bM$ are invariant and extremal KMS states with respect a surviving one-parameter group symmetry. In this paper we clarify the nature of symmetry breakdown. We show that, in fact, {\em spontaneous} symmetry breaking occurs in the natural sense of algebraic quantum field theory: if $\zeta \neq 0$, there is no unitary representation of whole group $PSL(2,\bR)$ which implements the $*$-automorphism representation of $PSL(2,\bR)$ itself in the GNS representation of $\lambda_\zeta$ (leaving fixed or not the state). 
  Here we study an anisotropic model of the universe with constant energy per particle. A decaying cosmological constant and particle production in an adiabatic process are considered as the sources for the entropy. The statefinder parameters $\{r,s\}$ are defined and their behaviour are analyzed graphically in some cases. 
  The joint ESA/NASA LISA mission consists in three spacecraft on heliocentric orbits, flying in a triangular formation of 5 Mkm each side, linked by infrared optical beams. The aim of the mission is to detect gravitational waves in a low frequency band. For properly processing the science data, the propagation delays between spacecraft must be accurately known. We thus analyse the propagation of light between spacecraft in order to systematically derive the relativistic effects due to the static curvature of the Schwarzschild spacetime in which the spacecraft are orbiting with time-varying light-distances. In particular, our analysis allows to evaluate rigorously the Sagnac effect, and the gravitational (Einstein) redshift. 
  There are a number of mathematical theorems in the literature on the dynamics of cosmological models with accelerated expansion driven by a positive cosmological constant $\Lambda$ or a nonlinear scalar field with potential $V$ (quintessence) which do not assume homogeneity and isotropy from the beginning. The aim of this paper is to generalize these results to the case of $k$-essence models which are defined by a Lagrangian having a nonlinear dependence on the kinetic energy. In particular, Lagrangians are included where late time acceleration is driven by the kinetic energy, an effect which is qualitatively different from anything seen in quintessence models. A general criterion for isotropization is derived and used to strengthen known results in the case of quintessence. 
  We study the spectrum of loops as a part of a complete network of cosmic strings in flat spacetime. After a long transient regime, characterized by production of small loops at the scale of the initial conditions, it appears that a true scaling regime takes over. In this final regime the characteristic length of loops scales as $0.1 t$, in contrast to earlier simulations which found tiny loops. We expect the expanding-universe behavior to be qualitatively similar. The large loop sizes have important cosmological implications. In particular, the nucleosynthesis bound becomes $G\mu \lesssim 10^{-7}$, much tighter than before. 
  We present a fully model-independent analysis of the extensive observations reported by a recent ether-drift experiment in Berlin. No a priori assumption is made on the nature of a hypothetical preferred frame. We find a remarkable consistency with an Earth's cosmic motion exhibiting an average declination angle |\gamma|\sim 43^o and with values of the RMS anisotropy parameter (1/2-\beta+\delta) that are one order of magnitude larger than the presently quoted ones. This might represent the first modern indication for a preferred frame and for a non-zero anisotropy of the speed of light. 
  The spin network simulator model represents a bridge between (generalized) circuit schemes for standard quantum computation and approaches based on notions from Topological Quantum Field Theories (TQFT). More precisely, when working with purely discrete unitary gates, the simulator is naturally modelled as families of quantum automata which in turn represent discrete versions of topological quantum computation models. Such a quantum combinatorial scheme, which essentially encodes SU(2) Racah--Wigner algebra and its braided counterpart, is particularly suitable to address problems in topology and group theory and we discuss here a finite states--quantum automaton able to accept the language of braid group in view of applications to the problem of estimating link polynomials in Chern--Simons field theory. 
  We discuss some new developments in three-dimensional gravity with torsion, based on Riemann-Cartan geometry. Using the canonical approach, we study the structure of asymptotic symmetry, clarify its fundamental role in defining the gravitational conserved charges, and explore the influence of the asymptotic structure on the black hole entropy. 
  During the last few years progress has been made on several fronts making it possible to revisit Cauchy-perturbative matching (CPM) in numerical relativity in a more robust and accurate way. This paper is the first in a series where we plan to analyze CPM in the light of these new results.   Here we start by testing high-order summation-by-parts operators, penalty boundaries and contraint-preserving boundary conditions applied to CPM in a setting that is simple enough to study all the ingredients in great detail: Einstein's equations in spherical symmetry, describing a black hole coupled to a massless scalar field. We show that with the techniques described above, the errors introduced by Cauchy-perturbative matching are very small, and that very long term and accurate CPM evolutions can be achieved. Our tests include the accretion and ring-down phase of a Schwarzschild black hole with CPM, where we find that the discrete evolution introduces, with a low spatial resolution of \Delta r = M/10, an error of 0.3% after an evolution time of 1,000,000 M. For a black hole of solar mass, this corresponds to approximately 5 s, and is therefore at the lower end of timescales discussed e.g. in the collapsar model of gamma-ray burst engines.   (abridged) 
  We observe critical phenomena in spherical gravitational collapse of a matter model with vanishing radial pressure and non-zero tangential pressure. We show analytically that the collapsing cloud either forms a black hole or completely disperses depending on values of initial parameters and velocity profile of the cloud. Near the threshold of black hole formation, a scaling relation is seen for the mass of black hole and same critical exponent is obtained for different initial parameters. 
  Some properties of the G\"odel space-time metric and its Riemann extension are studied 
  We construct new, efficient, and accurate high-order finite differencing operators which satisfy summation by parts. Since these operators are not uniquely defined, we consider several optimization criteria: minimizing the bandwidth, the truncation error on the boundary points, the spectral radius, or a combination of these. We examine in detail a set of operators that are up to tenth order accurate in the interior, and we surprisingly find that a combination of these optimizations can improve the operators' spectral radius and accuracy by orders of magnitude in certain cases. We also construct high-order dissipation operators that are compatible with these new finite difference operators and which are semi-definite with respect to the appropriate summation by parts scalar product. We test the stability and accuracy of these new difference and dissipation operators by evolving a three-dimensional scalar wave equation on a spherical domain consisting of seven blocks, each discretized with a structured grid, and connected through penalty boundary conditions. 
  We study the issue of coupling among 4-simplices in the context of spin foam models obtained from a group field theory formalism. We construct a generalisation of the Barrett-Crane model in which an additional coupling between the normals to tetrahedra, as defined in different 4-simplices that share them, is present. This is realised through an extension of the usual field over the group manifold to a five argument one. We define a specific model in which this coupling is parametrised by an additional real parameter that allows to tune the degree of locality of the resulting model, interpolating between the usual Barrett-Crane model and a flat BF-type one. Moreover, we define a further extension of the group field theory formalism in which the coupling parameter enters as a new variable of the field, and the action presents derivative terms that lead to modified classical equations of motion. Finally, we discuss the issue of renormalisation of spin foam models, and how the new coupled model can be of help regarding this. 
  Here we analyze in detail some aspects of the proposed use of Ajisai and Jason-1, together with the LAGEOS satellites, to measure the general relativistic Lense-Thirring effect in the gravitational field of the Earth. A linear combination of the nodes of such satellites is the proposed observable. The systematic error due to the mismodelling in the uncancelled even zonal harmonics would be \sim 1% according to the latest present-day CHAMP/GRACE-based Earth gravity models. In regard to the non-gravitational perturbations especially affecting Jason-1, only relatively high-frequency harmonic perturbations should occur: neither semisecular nor secular bias of non-gravitational origin should affect the proposed combination: their maximum impact is evaluated to \sim 4% over 2 years. Our estimation of the root-sum-square total error is about 4-5% over at least 3 years of data analysis required to average out the uncancelled tidal perturbations. 
  We give a short and simple proof that the Lorentzian 10j symbol, which forms a key part of the Barrett-Crane model of Lorentzian quantum gravity, is finite. The argument is very general, and applies to other integrals. For example, we show that the Lorentzian and Riemannian causal 10j symbols are finite, despite their singularities. Moreover, we show that integrals that arise in Cherrington's work are finite. Cherrington has shown that this implies that the Lorentzian partition function for a single triangulation is finite, even for degenerate triangulations. Finally, we also show how to use these methods to prove finiteness of integrals based on other graphs and other homogeneous domains. 
  After explaining the physical origin of the quasinormal modes of perturbations in the background geometry of a black hole, I critically review the recent proposal for the quantization of the black-hole area based on the real part of quasinormal modes. As instantons due to the barriers of black-hole potentials lie at the root of a discrete set of complex quasinormal modes frequencies, it is likely that the physics of quasinormal modes can be learned from quantum theory. I propose a connection of a system of quasinormal modes of black holes with a dissipative open system, in particular, the Feshbach-Tikochinsky oscillator. This argument is supported in part by the fact that these two systems have the same group structure SU(1,1) and the same group representation of Hamiltonians; thereby, their quantum states exhibit the same behavior. 
  In a recent preprint, gr-qc/0511123, Dadhich has given a brief yet beautiful exposition on some of the research works by Prof. A.K. Raychaudhuri. Here Dadhich highlights the fact that the apparently ``self-evident'' assumption of occurrence of ``trapped surfaces'' may not be realized atleast in some specific cosmological models though no general proof for non-occurrence of trapped surfaces exists in the cosmological context. However, Dadhich added, without sufficient justification, that trapped surfaces should occur for collapse of isolated bodies. We point out that actually trapped surfaces do not occur even for collapse of spherically symmetric isolated bodies. Further unlike the cosmological case, for isolated bodies, an exact proof for generic non-occurrence of trapped surfaces is available. Thus for isolated bodies, the above referred apparently ``self-evident'' assumption fails much more acutely than in cosmology. Many recent astrophysical observations tend to corroborate the fact trapped surfaces do not occur for isolated bodies. Two recent specific papers (PRD) are cited to show that when radiative non-diispative collapse can prevent formation of trapped surfaces. 
  We study the expansion of the universe at late times in the case that the cosmological constant obeys certain scaling laws motivated by renormalisation group running in quantum theories. The renormalisation scale is identified with the Hubble scale and the inverse radii of the event and particle horizon, respectively. We find de Sitter solutions, power-law expansion and super-exponential expansion in addition to future singularities of the Big Rip and Big Crunch type. 
  In this work, numerical simulations were used to investigate the gravitational stochastic background produced by coalescences occurring up to $z \sim 5$ of double neutron star systems. The cosmic coalescence rate was derived from Monte Carlo methods using the probability distributions for forming a massive binary and to occur a coalescence in a given redshift. A truly continuous background is produced by events located only beyond the critical redshift $z_* = 0.23$. Events occurring in the redshift interval $0.027<z<0.23$ give origin to a "popcorn" noise, while those arising closer than $z = 0.027$ produce a shot noise. The gravitational density parameter $\Omega_{gw}$ for the continuous background reaches a maximum around 670 Hz with an amplitude of $1.1\times 10^{-9}$, while the "popcorn" noise has an amplitude about one order of magnitude higher and the maximum occurs around a frequency of 1.2 kHz. The signal is below the sensitivity of the first generation of detectors but could be detectable by the future generation of ground based interferometers. Correlating two coincident advanced-LIGO detectors or two EGO interferometers, the expected S/N ratio are respectively 0.5 and 10. 
  The Lorentz transformations are represented by Einstein velocity addition on the ball of relativistically admissible velocities. This representation is by projective maps. The Lie algebra of this representation defines the relativistic dynamic equation. If we introduce a new dynamic variable, called symmetric velocity, the above representation becomes a representation by conformal, instead of projective maps. In this variable, the relativistic dynamic equation for systems with an invariant plane, becomes a non-linear analytic equation in one complex variable. We obtain explicit solutions for the motion of a charge in uniform, mutually perpendicular electric and magnetic fields. By assuming the Clock Hypothesis and using these solutions, we are able to describe the space-time transformations between two uniformly accelerated and rotating systems. 
  From the point of view of an uncompromising field theorist quantum gravity is beset with serious technical and, above all, conceptual problems with regard especially to the meaning of genuine "physical" observables. This situation is not really improved by the appearance of recent attempts to reformulate gravity within some novel framework.However, the original aim, a background-independent quantum theory of gravity, can be achieved in a particular area, namely 2d dilaton quantum gravity without any assumptions beyond standard quantum field theory. Some important by-products of the research of the "Vienna School" include the introduction of the concept of Poisson-Sigma models, a verification of the "virtual Black Hole" and the extensions to N = (1,1) and N = (2,2) 2d-supergravity, for which complete solutions of some old problems have been possible which are relevant for superstring theory 
  A Lorentz-invariant cosmological model is constructed within the framework of five-dimensional gravity. The five-dimensional theorem which is analogical to the generalized Birkhoff theorem is proved, that corresponds to the Kaluza's ``cylinder condition''. The five-dimensional vacuum Einstein equations have an integral of motion corresponding to this symmetry, the integral of motion is similar to the mass function in general relativity (GR). Space closure with respect to the extra dimensionality follows from the requirement of the absence of a conical singularity. Thus, the Kaluza-Klein (KK) model is realized dynamically as a Lorentz-invariant mode of five-dimensional general relativity. After the dimensional reduction and conformal mapping the model is reduced to the GR configuration. It contains a scalar field with a vanishing conformally invariant energy-momentum tensor on the flat space-time background. This zero mode can be interpreted as a vacuum configuration in GR. As a result the vacuum-like configuration in GR can be considered as a manifestation of the Lorentz-invariant empty five-dimensional space. 
  It is shown by explicit construction of new metrics, that General Relativity can solve the exact Poinc$\acute{a}$re recurrence problem. In these solutions, the light cone, flips periodically between past and future, due to a periodically alternating arrow of the proper time. The geodesics in these universes show periodic Loschmidt's velocity reversion $v \to -v$, at critical points, which leads to recurrence. However, the matter tensors of some of these solutions exhibit unusual properties - such as, periodic variations in density and pressure. While this is to be expected in periodic models, the physical basis for such a variation is not clear. Present paper therefore can be regarded as an extension of Tipler's "no go theorem for recurrence in an expanding universe", to other space-time geometries. 
  The idea that quantum gravity manifestations would be associated with a violation of Lorentz invariance is very strongly bounded and faces serious theoretical challenges. Other related ideas seem to be drowning in interpretational quagmires. This leads us to consider alternative lines of thought for such phenomenological search. We discuss the underlying viewpoints and briefly mention their possible connections with other current theoretical ideas. 
  The space based gravitational wave detector LISA is expected to observe a large population of Galactic white dwarf binaries whose collective signal is likely to dominate instrumental noise at observational frequencies in the range 10^{-4} to 10^{-3} Hz. The motion of LISA modulates the signal of each binary in both frequency and amplitude, the exact modulation depending on the source direction and frequency. Starting with the observed response of one LISA interferometer and assuming only doppler modulation due to the orbital motion of LISA, we show how the distribution of the entire binary population in frequency and sky position can be reconstructed using a tomographic approach. The method is linear and the reconstruction of a delta function distribution, corresponding to an isolated binary, yields a point spread function (psf). An arbitrary distribution and its reconstruction are related via smoothing with this psf. Exploratory results are reported demonstrating the recovery of binary sources, in the presence of white Gaussian noise. 
  The neutrino quasinormal modes of the Reissner-Nordstr\"om (RN) black hole are investigated using continued fraction approach. We find, for large angular quantum number, that the quasinormal frequencies become evenly spaced and the spacing of the real part depends on the charge of the black hole and that of the imaginary part is zero. We then find that the quasinormal frequencies in the complex $\omega$ plane move counterclockwise as the charge increases. They get a spiral-like shape, moving out of their Schwarzschild value and ``looping in" towards some limiting frequency as the charge tends to the extremal value. The number of the spirals increases as the overtone number increases but it decreases as the angular quantum number increases. We also find that both the real and imaginary parts are oscillatory functions of the charge, and the oscillation becomes faster as the overtone number increases but it becomes slower as the angular quantum number increases. 
  Deterministic dynamical models are discussed which can be described in quantum mechanical terms. -- In particular, a local quantum field theory is presented which is a supersymmetric classical model. The Hilbert space approach of Koopman and von Neumann is used to study the classical evolution of an ensemble of such systems. Its Liouville operator is decomposed into two contributions, with positive and negative spectrum, respectively. The unstable negative part is eliminated by a constraint on physical states, which is invariant under the Hamiltonian flow. Thus, choosing suitable variables, the classical Liouville equation becomes a functional Schroedinger equation of a genuine quantum field theory. -- We briefly mention an U(1) gauge theory with ``varying alpha'' or dilaton coupling where a corresponding quantized theory emerges in the phase space approach. It is energy-parity symmetric and, therefore, a prototype of a model in which the cosmological constant is protected by a symmetry. 
  Although several models of $f(R)$ theories of gravity within the Palatini approach have been studied already, the interest was concentrated on those that have an effect on the late-time evolution of the universe, by the inclusion for example of terms inversely proportional to the scalar curvature in the gravitational action. However, additional positive powers of the curvature also provide interesting early-time phenomenology, like inflation, and the presence of such terms in the action is equally, if not more, probable. In the present paper models with both additional positive and negative powers of the scalar curvature are studied. Their effect on the evolution of the universe is investigated for all cosmological eras, and various constraints are put on the extra terms in the actions. Additionally, we examine the extent to which the new terms in positive powers affect the late-time evolution of the universe and the related observables, which also determines our ability to probe their presence in the gravitational action. 
  Starting from a perfect cosmological fluid represented by the energy momentum tensor T_uv, one class of frequency metrics that satisfies both Einstein's general relativistic equation and the perfect fluid condition is: g_uv = e^iwt N_uv. Mathematically, such metrics indicate spacetime behaves locally like a simple harmonic oscillator. In the cosmological model presented here these small spacetime oscillations compress vacuum energy into a standing wave inside a dynamic Casimir cavity. At peak compression a phase shift occurs and the standing wave forms into a particle having relativistic mass-energy equal to the compressive work required to produce it. At this point the newly formed particle does isobaric work to expand the volume against the external pressure given T_ii. Equilibrium is achieved when the collision rate on the volume's internal and external surfaces equalizes. By treating spacetime as a classical thermodynamic problem and oscillator, such quantities as the mass of the compressed particle--that of an axion, the radii of the initial and final volume of compression, and the angular frequency of compression can be determined. During axion collision the photon frequency of the particle is calculated to be in the microwave range and inversely equal to that of the frequency of the spacetime compression that produced the particle. This suggests axion production is a source for the 2.7K cosmic background radiation and dark matter that pervades spacetime. 
  Time dilation $\frac{1}{\sqrt{1-v^2}}$ and relative velocity $v$ are observationally indistinguishable in the special theory of relativity, a duality that carries over into the general theory under Fermi coordinates along a curve (in coordinate-independent language, in the tangent Minkowski space along the curve). For example, on a clock stationary at radius $r$, a distant observer sees time dilation of $\frac{1}{\sqrt{1-v^2}}=\frac{1}{\sqrt{1-2M/r}}$ under the Schwarzschild metric and sees the clock receding with a relative velocity of $v=\sqrt{2M/r}$ under the Painlev{\'e}-Gullstrand free fall metric. Duality implies that during gravitational collapse, the intensifying time dilation observed at the star's center from a fixed radius $r>0$ is indistinguishable (along a curve) from an increasing relative velocity at which the center recedes as seen from any direction, implying a local inflation. 
  We present the static spherically symmetric vacuum solutions of the Jordan, Brans-Dicke field equations. The new solutions are obtained by considering a polar Gaussian, isothermal and radial hyperbolic metrics. 
  The hole argument was developed by Einstein in 1913 while he was searching for a relativistic theory of gravitation. Einstein used the language of coordinate systems and coordinate invariance, rather than the language of manifolds and diffeomorphism invariance. He formulated the hole argument against covariant field equations and later found a way to avoid it using coordinate language.   In this paper we shall use the invariant language of categories, manifolds and natural objects to give a coordinate-free description of the hole argument and a way of avoiding it. Finally we shall point out some important implications of further extensions of the hole argument to sets and relations for the problem of quantum gravity. 
  We apply the method of matched asymptotic expansions to analyse whether cosmological variations in physical `constants' and scalar fields are detectable, locally, on the surface of local gravitationally bound systems such as planets and stars, or inside virialised systems like galaxies and clusters. We assume spherical symmetry and derive a sufficient condition for the local time variation of the scalar fields that drive varying constants to track the cosmological one. We calculate a number of specific examples in detail by matching the Schwarzschild spacetime to spherically symmetric inhomogeneous Tolman-Bondi metrics in an intermediate region by rigorously construction matched asymptotic expansions on cosmological and local astronomical scales which overlap in an intermediate domain. We conclude that, independent of the details of the scalar-field theory describing the varying `constant', the condition for cosmological variations to be measured locally is almost always satisfied in physically realistic situations. The proof of this statement provides a rigorous justification for using terrestrial experiments and solar system observations to constrain or detect any cosmological time variations in the traditional `constants' of Nature. 
  A discussion is given of recent developments in canonical gravity that assimilates the conformal analysis of gravitational degrees of freedom. The work is motivated by the problem of time in quantum gravity and is carried out at the metric and the triad levels. At the metric level, it is shown that by extending the Arnowitt-Deser-Misner (ADM) phase space of general relativity (GR), a conformal form of geometrodynamics can be constructed. In addition to the Hamiltonian and diffeomorphism constraints, an extra first class constraint is introduced to generate conformal transformations. This phase space consists of York's mean extrinsic curvature time, conformal three-metric and their momenta. At the triad level, the phase space of GR is further enlarged by incorporating spin-gauge as well as conformal symmetries. This leads to a canonical formulation of GR using a new set of real spin connection variables. The resulting gravitational constraints are first class, consisting of the Hamiltonian constraint and the canonical generators for spin-gauge and conformorphism transformations. The formulation has a remarkable feature of being parameter-free. Indeed, it is shown that a conformal parameter of the Barbero-Immirzi type can be absorbed by the conformal symmetry of the extended phase space. This gives rise to an alternative approach to loop quantum gravity that addresses both the conceptual problem of time and the technical problem of functional calculus in quantum gravity. 
  The static potential for a massless scalar field shares the essential features of the scalar gravitational mode in a tensorial perturbation analysis about the background solution. Using the fluxbrane construction of [8] we calculate the lowest order of the static potential of a massless scalar field on a thin brane using series solutions to the scalar field's Klein Gordon equation and we find that it has the same form as Newton's Law of Gravity. We claim our method will in general provide a quick and useful check that one may use to see if their model will recover Newton's Law to lowest order on the brane. 
  The super-Hamiltonian of four-dimensional gravity as simplified by Ashtekar through the use of gauge potential and densitized triad variables can furthermore be succinctly expressed as a Poisson bracket between fundamental invariants. Even when a cosmological constant is present, the constraint is equivalent to the vanishing of the Poisson Bracket between the volume element and a combination of the integral of the trace of the extrinsic curvature and the Chern-Simons functional. This observation naturally suggests a reformulation of non-perturbative quantum gravity wherein the Wheeler-DeWitt Equation is reduced to the requirement of the vanishing of the expectation value of the corresponding commutator. Remarkably, this formulation singles out spin network states as explicit realizations of the physical states. Moreover, by requiring physical states to be simultaneous eigenstates of the commuting operators, the formulation also yields a Schrodinger Equation with "intrinsic-time development". 
  We assume a flat brane located at y=0, surrounded by an AdS space, and consider the 5D Einstein equations when the energy flux component of the energy-momentum tensor is related to the Hubble parameter through a constant Q. We calculate the metric tensor, as well as the Hubble parameter on the brane, when Q is small. As a special case, if the brane is tensionless, the influence from Q on the Hubble parameter is absent. We also consider the emission of gravitons from the brane, by means of the Boltzmann equation. Comparing the energy conservation equation derived herefrom with the energy conservation equation for a viscous fluid on the brane, we find that the entropy change for the fluid in the emission process has to be negative. This peculiar effect is related to the fluid on the brane being a non-closed thermodynamic system. The negative entropy property for non-closed systems is encountered in other areas in physics also, in particular, in connection with the Casimir effect at finite temperature. 
  Recent papers by Fewster and Roman have emphasized that wormholes supported by arbitrarily small amounts of exotic matter will have to be incredibly fine-tuned if they are to be traversable. This paper discusses a wormhole model that strikes a balance between two conflicting requirements, reducing the amount of exotic matter and fine-tuning the metric coefficients, ultimately resulting in an engineering challenge: one requirement can only be met at the expense of the other. The wormhole model is macroscopic and satisfies various traversability criteria. 
  The description of extreme-mass-ratio binary systems in the inspiral phase is a challenging problem in gravitational wave physics with significant relevance for the space interferometer LISA. The main difficulty lies in the evaluation of the effects of the small body's gravitational field on itself. To that end, an accurate computation of the perturbations produced by the small body with respect the background geometry of the large object, a massive black hole, is required. In this paper we present a new computational approach based on Finite Element Methods to solve the master equations describing perturbations of non-rotating black holes due to an orbiting point-like object. The numerical computations are carried out in the time domain by using evolution algorithms for wave-type equations. We show the accuracy of the method by comparing our calculations with previous results in the literature. Finally, we discuss the relevance of this method for achieving accurate descriptions of extreme-mass-ratio binaries. 
  We study five dimensional thin-shell wormholes in Einstein-Maxwell theory with a Gauss-Bonnet term. The linearized stability under radial perturbations and the amount of exotic matter are analyzed as a function of the parameters of the model. We find that the inclusion of the quadratic correction substantially widens the range of possible stable configurations, and besides it allows for a reduction of the exotic matter required to construct the wormholes. 
  The experimental evidence that the equation of state (EOS) of the dark energy (DE) could be evolving with time/redshift (including the possibility that it might behave phantom-like near our time) suggests that there might be dynamical DE fields that could explain this behavior. We propose, instead, that a variable cosmological term (including perhaps a variable Newton's gravitational coupling too) may account in a natural way for all these features. 
  We investigate the dynamics of spatially homogeneous solutions of the Einstein-Vlasov equations with Bianchi type I symmetry by using dynamical systems methods. All models are forever expanding and isotropize toward the future; toward the past there exists a singularity. We identify and describe all possible past asymptotic states; in particular, on the past attractor set we establish the existence of a heteroclinic network, which is a new type of feature in general relativity. This illustrates among other things that Vlasov matter can lead to quite different dynamics of cosmological models as compared to perfect fluids. 
  We present a numerical model of a collapsing radiating sphere, whose boundary surface undergoes bouncing due to a decreasing of its inertial mass density (and, as expected from the equivalence principle, also of the ``gravitational'' force term) produced by the ``inertial'' term of the transport equation. This model exhibits for the first time the consequences of such an effect, and shows that under physically reasonable conditions this decreasing of the gravitational term in the dynamic equation may be large enough as to revert the collapse and produce a bouncing of the boundary surface of the sphere. 
  We discuss the hyperboloidal evolution problem in general relativity from a numerical perspective, and present some new results. Families of initial data which are the hyperboloidal analogue of Brill waves are constructed numerically, and a systematic search for apparent horizons is performed. Schwarzschild-Kruskal spacetime is discussed as a first application of Friedrich's general conformal field equations in spherical symmetry, and the Maxwell equations are discussed on a nontrivial background as a toy model for continuum instabilities. 
  We investigate the formation via tunneling of inflating (false-vacuum) bubbles in a true-vacuum background, and the reverse process. Using effective potentials from the junction condition formalism, all true- and false-vacuum bubble solutions with positive interior and exterior cosmological constant, and arbitrary mass are catalogued. We find that tunneling through the same effective potential appears to describe two distinct processes: one in which the initial and final states are separated by a wormhole (the Farhi-Guth-Guven mechanism), and one in which they are either in the same hubble volume or separated by a cosmological horizon. In the zero-mass limit, the first process corresponds to the creation of an inhomogenous universe from nothing, while the second mechanism is equivalent to the nucleation of true- or false-vacuum Coleman-De Luccia bubbles. We compute the probabilities of both mechanisms in the WKB approximation using semi-classical Hamiltonian methods, and find that -- assuming both process are allowed -- neither mechanism dominates in all regimes. 
  Based on the observations that there exists an analogy between the Reissner-Nordstr\"om-anti-de Sitter (RN-AdS) black holes and the van der Waals-Maxwell liquid-gas system, in which a correspondence of variables is $(\phi, q) \leftrightarrow (V,P)$, we study the Ruppeiner geometry, defined as Hessian matrix of black hole entropy with respect to the internal energy (not the mass) of black hole and electric potential (angular velocity), for the RN, Kerr and RN-AdS black holes. It is found that the geometry is curved and the scalar curvature goes to negative infinity at the Davies' phase transition point for the RN and Kerr black holes.  Our result for the RN-AdS black holes is also in good agreement with the one about phase transition and its critical behavior in the literature. 
  In this paper, it is shown completely analytically that a spintessence model can very well serve the purpose of providing an early deceleration and the present day acceleration. 
  It is shown explicitly that when the characteristic vector field that defines a Godel-type metric is also a Killing vector, there always exist closed timelike or null curves in spacetimes described by such a metric. For these geometries, the geodesic curves are also shown to be characterized by a lower dimensional Lorentz force equation for a charged point particle in the relevant Riemannian background. Moreover, two explicit examples are given for which timelike and null geodesics can never be closed. 
  Inspired by the generalization of quantum theory for the case of non-Hermitian Hamiltonians with CPT symmetry, we construct a simple classical cosmological scalar field based model describing a smooth transition from ordinary dark energy to the phantom one. 
  The detection of the Cosmic Microwave Background Radiation (CMB) was one of the most important cosmological discoveries of the last century. With the development of interferometric gravitational wave detectors, we may be in a position to detect the gravitational equivalent of the CMB in this century. The Cosmic Gravitational Background (CGB) is likely to be isotropic and stochastic, making it difficult to distinguish from instrument noise. The contribution from the CGB can be isolated by cross-correlating the signals from two or more independent detectors. Here we extend previous studies that considered the cross-correlation of two Michelson channels by calculating the optimal signal to noise ratio that can be achieved by combining the full set of interferometry variables that are available with a six link triangular interferometer. In contrast to the two channel case, we find that the relative orientation of a pair of coplanar detectors does not affect the signal to noise ratio. We apply our results to the detector design described in the Big Bang Observer (BBO) mission concept study and find that BBO could detect a background with $\Omega_{gw} > 2.2 \times 10^{-17}$. 
  The canonical quantum theory of gravity -- Quantum Geometrodynamics (QG) is applied to the homogeneous Bianchi type IX cosmological model. As a result, the framework for the quantum theory of homogeneous cosmologies is developed. We show that the theory is internally consistent, and prove that it possesses the correct classical limit (the theory of general relativity). To emphasize the special role that the constraints play in this new theory we, compare it to the traditional ADM square-root and Wheeler-DeWitt quantization schemes. We show that, unlike the traditional approaches, QG leads to a well-defined Schrodinger equation for the wave-function of the universe that is inherently coupled to the expectation value of the constraint equations. This coupling to the constraints is responsible for the appearance of a coherent spacetime picture. Thus, the physical meaning of the constraints of the theory is quite different from Dirac's interpretation. In light of this distinctive feature of the theory, we readdress the question of the dark energy effects in the Bianchi IX cosmological model for highly non-classical quantum states. We show that, at least for this model, for any choice of the initial wave function, the quantum corrections will not produce the accelerated expansion of the universe. 
  We present a simple proof of the non-existence of degenerate components of the event horizon in static, vacuum, regular, four-dimensional black hole spacetimes. We discuss the generalisation to higher dimensions and the inclusion of a cosmological constant. 
  We show that the borderline cases in the proof of the positive energy theorem for initial data sets, on spin manifolds, in dimensions $n\ge 3$, are only possible for initial data arising from embeddings in Minkowski space-time. 
  We show that static electro--vacuum black hole space--times containing an asymptotically flat spacelike hypersurface with compact interior and with both degenerate and non--degenerate components of the event horizon do not exist. This is done by a careful study of the near-horizon geometry of degenerate horizons, which allows us to eliminate the last restriction of the static electro-vacuum no-hair theory. 
  Based on the general considerations of quantum mechanics and gravity the generalized uncertainty principle (GUP) is determined in higher-dimensional case and on the brane, respectively. The result is used to evaluate the effect of GUP on the dynamics of evaporation and lifetime of mini black holes in the brane-world models. 
  We address the initial conditions for an expanding cosmology using the holographic principle. For the case of a closed model, the old prescription of Fishler and Susskind, that uses the particle horizon to encode the bulk degrees of freedom, can be implemented for accelerated models with enough acceleration. As a bonus we have singularity free bouncing models. The bound is saturated for co-dimension one branes dominated universes. 
  Starting from an interpretation of the classical-quantum correspondence, we derive the Dirac equation by factorizing the algebraic relation satisfied by the classical Hamiltonian, before applying the correspondence. This derivation applies in the same form to a free particle, to one in an electromagnetic field, and to one subjected to geodesic motion in a static metric, and leads to the same, usual form of the Dirac equation--in special coordinates. To use the equation in the static-gravitational case, we need to rewrite it in more general coordinates. This can be done only if the usual, spinor transformation of the wave function is replaced by the 4-vector transformation. We show that the latter also makes the flat-space-time Dirac equation Lorentz-covariant, although the Dirac matrices are not invariant. Because the equation itself is left unchanged in the flat case, the 4-vector transformation does not alter the main physical consequences of that equation in that case. However, the equation derived in the static-gravitational case is not equivalent to the standard (Fock-Weyl) gravitational extension of the Dirac equation. 
  We extend the classical and quantum treatment of the Lemaitre-Tolman-Bondi (LTB) model to the non-marginal case (defined by the fact that the shells of the dust cloud start with a non-vanishing velocity at infinity). We present the classical canonical formalism and address with particular care the boundary terms in the action. We give the general relation between dust time and Killing time. Employing a lattice regularization, we then derive and discuss for particular factor orderings exact solutions to all quantum constraints. 
  We give a very brief introduction to the group field theory approach to quantum gravity, a generalisation of matrix models for 2-dimensional quantum gravity to higher dimension, that has emerged recently from research in spin foam models. 
  Scalar BMS-invariant QFT defined on the causal boundary $\scri$ of an asymptotically flat spacetime is discussed. (a)(i) It is noticed that the natural $BMS$ invariant pure quasifree state $\lambda$ on $\cW(\scri)$, recently introduced by Dappiaggi, Moretti an Pinamonti, enjoys positivity of the self-adjoint generator of $u$-translations with respect to {\em every} Bondi coordinate frame $(u,\z,\bz)$ on $\scri$, $u\in \bR$ being the affine parameter of the null geodesics forming $\scri$. This fact may be interpreted as a remnant of spectral condition inherited from Minkowski spacetime. (ii) It is proved cluster property under $u$-displacements holds for $u$-invariant pure state on $\cW(\scri)$. (iii) It is proved that there is a unique algebraic pure quasifree state invariant under $u$-displacements (of a fixed Bondi frame) having positive self-adjoint generator of $u$-displacements. It coincides with the GNS-invariant state $\lambda$.(iv) It is showed that in the folium of a pure $u$-invariant state $\omega$ (not necessarily quasifree) on $\cW(\scri)$, $\omega$ is the only state invariant under $u$-displacement. (b) It is proved that the theory can formulated for spacetimes asymptotically flat at null infinity which admit future time completion. In this case a $*$-isomorphism $\imath$ exists which identifies the (Weyl) algebra of observables of linear fields in the bulk with a sub algebra of $\cW(\scri)$. A preferred state on the field algebra in the bulk is induced by the $BMS$-invariant state $\lambda$. 
  A self-consistent system of interaction nonlinear spinor and scalar fields within the scope of a BI cosmological model filled with perfect fluid is considered. The role of spinor field in the evolution of the Universe is studied. It is shown that the spinor field nonlinearity can generate a negative effective pressure, which can be seen as an alternative source for late time acceleration of the Universe. 
  Within the framework of Bianchi type-I space-time we study the Bel-Robinson tensor and its impact on the evolution of the Universe. We use different definitions of the Bel-Robinson tensor existing in the literature and compare the results. Finally we investigate the so called "dominant super-energy property" for the Bel-Robinson tensor as a generalization of the usual dominant energy condition for energy momentum tensors.   Keywords: Bianchi type I model, super-energy tensors   Pacs: 03.65.Pm and 04.20.Ha 
  We consider the asymptotic quasinormal frequencies of various spin fields in Schwarzschild and Reissner-Nordstr\"om black holes. In the Schwarzschild case, the real part of the asymptotic frequency is ln3 for the spin 0 and the spin 2 fields, while for the spin 1/2, the spin 1, and the spin 3/2 fields it is zero. For the non-extreme charged black holes, the spin 3/2 Rarita-Schwinger field has the same asymptotic frequency as that of the integral spin fields. However, the asymptotic frequency of the Dirac field is different, and its real part is zero. For the extremal case, which is relevant to the supersymmetric consideration, all the spin fields have the same asymptotic frequency, the real part of which is zero. For the imaginary parts of the asymptotic frequencies, it is interesting to see that it has a universal spacing of $1/4M$ for all the spin fields in the single-horizon cases of the Schwarzschild and the extreme Reissner-Nordstr\"om black holes. The implications of these results to the universality of the asymptotic quasinormal frequencies are discussed. 
  A summary of some lines of ideas leading to model-independent frameworks of relativistic quantum field theory is given. It is followed by a discussion of the Reeh-Schlieder theorem and geometric modular action of Tomita-Takesaki modular objects associated with the quantum field vacuum state and certain algebras of observables. The distillability concept, which is significant in specifying useful entanglement in quantum information theory, is discussed within the setting of general relativistic quantum field theory. 
  Novel solutions for the static spherically symmetric extremally charged dust in the Majumdar-Papapetrou system have been found. For a certain amount of the allocated mass/charge, the solutions show existence of singularity of a type which could render them physically unacceptable, since the corresponding physically relevant quantities are singular as well. These solutions, with a number of zero-nodes in the metric tensor, can be regularized by a simple redefinition of the charge/density distribution. The bifurcating behaviour of regular solutions is in these singular solutions lost, but quantized-like behaviour in the total mass is observed. 
  We give analytic expressions for the gravitational inner spherical multipole moments, q_{lm} with l <= 5, for 11 elementary solid shapes. These moments, in conjunction with their known rotational and translational properties, can be used to calculate precisely the moments of complex objects that may be assembled from the elementary shapes. We also give an analytic expression for the gravitational force between two rectangular solids at arbitrary separations. These expressions are useful for computing the gravitational properties of complex instruments, such as those used in equivalence principle tests, and in the gravitational balancing of drag-free spacecraft. 
  It is possible that the expansion of the universe began with an inflationary phase, in which the inflaton driving the process also was a Higgs field capable of stabilizing magnetic monopoles in a grand-unified gauge theory. If so, then the smallness of intensity fluctuations observed in the cosmic microwave background radiation implies that the self-coupling of the inflaton-Higgs field was exceedingly weak. It is argued here that the resulting broad, flat maximum in the Higgs potential makes the presence or absence of a topological zero in the field insignificant for inflation. There may be monopoles present in the universe, but the universe itself is not in the inflating core of a giant magnetic monopole. 
  We analyze the dynamics of models of warm inflation with general dissipative effects. We consider phenomenological terms both for the inflaton decay rate and for viscous effects within matter. We provide a classification of the asymptotic behavior of these models and show that the existence of a late-time scaling regime depends not only on an asymptotic behavior of the scalar field potential, but also on an appropriate asymptotic behavior of the inflaton decay rate. There are scaling solutions whenever the latter evolves to become proportional to the Hubble rate of expansion regardless of the steepness of the scalar field exponential potential. We show from thermodynamic arguments that the scaling regime is associated to a power-law dependence of the matter-radiation temperature on the scale factor, which allows a mild variation of the temperature of the matter/radiation fluid. We also show that the late time contribution of the dissipative terms alleviates the depletion of matter, and increases the duration of inflation. 
  In the earlier works on quantum geometrodynamics in extended phase space it has been argued that a wave function of the Universe should satisfy a Schrodinger equation. Its form, as well as a measure in Schrodinger scalar product, depends on a gauge condition (a chosen reference frame). It is known that the geometry of an appropriate Hilbert space is determined by introducing the scalar product, so the Hilbert space structure turns out to be in a large degree depending on a chosen gauge condition. In the present work we analyse this issue from the viewpoint of the path integral approach. We consider how the gauge condition changes as a result of gauge transformations. In this respect, three kinds of gauge transformations can be singled out: Firstly, there are residual gauge transformations, which do not change the gauge condition. The second kind is the transformations whose parameters can be related by homotopy. Then the change of gauge condition could be described by smoothly changing function. In particular, in this context time dependent gauges could be discussed. We also suggest that this kind of gauge transformations leads to a smooth changing of solutions to the Schrodinger equation. The third kind of the transformations includes those whose parameters belong to different homotopy classes. They are of the most interest from the viewpoint of changing the Hilbert space structure. In this case the gauge condition and the very form of the Schrodinger equation would change in discrete steps when we pass from a spacetime region with one gauge condition to another region with another gauge condition. In conclusion we discuss the relation between quantum gravity and fundamental problems of ordinary quantum mechanics. 
  Irving Ezra Segal (1918-1998) has proposed some axioms for mathematical cosmology. These are here re-examined and Segal's redshift formula and energy conservation in the Einstein universe are established in full on their basis. A detailed table of contents appears at the end. 
  The 2D time dependent solution of thin accretion disk in a close binary system have been presented on the equatorial plane around the Schwarzschild black hole. To do that, the special part of the General Relativistic Hydrodynamical(GRH) equations are solved using High Resolution Shock Capturing (HRSC) schemes. The spiral shock waves on the accretion disk are modeled using perfect fluid equation of state with adiabatic indices $\gamma = 1.05, 1.2$ and 5/3. The results show that the spiral shock waves are created for gammas except the case $\gamma=5/3$. These results consistent with results from Newtonian hydrodynamic code except close to black hole. Newtonian approximation does not give good solution while matter closes to black hole. Our simulations illustrate that the spiral shock waves are created close to black hole and the location of inner radius of spiral shock wave is around $10M$ and it depends on the specific heat rates. We also find that the smaller $\gamma$ is the more tightly the spiral winds. 
  Spherical accretion flows are simple enough for analytical study, by solution of the corresponding fluid dynamic equations. The solutions of stationary spherical flow are due to Bondi. The questions of the choice of a physical solution and of stability have been widely discussed. The answer to these questions is very dependent on the problem of boundary conditions, which vary according to whether the accretor is a compact object or a black hole. We introduce a particular, simple form of stationary spherical flow, namely, self-similar Bondi flow, as a case with physical interest in which analytic solutions for perturbations can be found. With suitable no matter-flux-perturbation boundary conditions, we will show that acoustic modes are stable in time and have no spatial instability at r=0. Furthermore, their evolution eventually becomes ergodic-like and shows no trace of instability or of acquiring any remarkable pattern. 
  Modified Newtonian dynamics, a successful alternative to the cosmic dark matter model, proposes that gravitational field deviates from the Newtonian law when the field strength $g$ is weaker than a critical value $g_0$. We will show that the dynamics of MOND can be derived from an induced gravity model. New dynamics is shown to be compatible with the spatial deformation of scalar fields coupled to the system. Approximate solutions are shown explicitly for a simple toy model. 
  An overview of the main points related to data analysis in resonant-mass gravitational wave detectors will be presented. Recent developments on the data analysis system for the Brazilian detector SCHENBERG will be emphasized. 
  The detection of gravitational waves is a very active research field at the moment. In Brazil the gravitational wave detector is called Mario SCHENBERG. Due to its high sensitivity it is necessary to model mathematically all known noise sources so that digital filters can be developed that maximize the signal-to-noise ratio. One of the noise sources that must be considered are the disturbances caused by electromagnetic pulses due to lightning close to the experiment. Such disturbances may influence the vibrations of the antenna's normal modes and mask possible gravitational wave signals. In this work we model the interaction between lightning and SCHENBERG antenna and calculate the intensity of the noise due to a close lightning stroke in the detected signal. We find that the noise generated does not disturb the experiment significantly. 
  We present a brief description of the ``consistent discretization'' approach to classical and quantum general relativity. We exhibit a classical simple example to illustrate the approach and summarize current classical and quantum applications. We also discuss the implications for the construction of a well defined quantum theory and in particular how to construct a quantum continuum limit. 
  The notion of being totally umbilic is considered for non-degenerate and degenerate submanifolds of semi-Riemanian manifolds. After some remarks on the general case, timelike and lightlike totally umbilic submanifolds of Lorentzian manifolds are discussed, along with their physical interpretation in view of general relativity. In particular, the mathematical notion of totally umbilic submanifolds is linked to the notions of photon surfaces and of null strings which have been used in the physics literature. 
  We point out that the field equations in 5D, with spatial spherical symmetry, possess an extra symmetry that leaves them invariant. This symmetry corresponds to certain simultaneous interchange of coordinates and metric coefficients. As a consequence a single solution in 5D can generate very different scenarios in 4D, ranging from static configurations to cosmological situations. A new perspective emanates from our work. Namely, that different astrophysical and cosmological scenarios in 4D might correspond to the same physics in 5D. We present explicit examples that illustrate this point of view. 
  It is known that the electromagnetic constitutive tensor can be algebraically decomposed in three parts: the so-called principal part, the axion part and the skewon part (see gr-qc/0506042). The aim of this paper is to provide a deeper decomposition of the principal part. 
  We construct a generalised formalism for group field theories, in which the domain of the field is extended to include additional proper time variables, as well as their conjugate mass variables. This formalism allows for different types of quantum gravity transition amplitudes in perturbative expansion, and we show how both causal spin foam models and the usual a-causal ones can be derived from it, within a sum over triangulations of all topologies. We also highlight the relation of the so-derived causal transition amplitudes with simplicial gravity actions. 
  We use the dynamical systems approach to investigate the Bianchi type VIII models with a tilted $\gamma$-law perfect fluid. We introduce expansion-normalised variables and investigate the late-time asymptotic behaviour of the models and determine the late-time asymptotic states. For the Bianchi type VIII models the state space is unbounded and consequently, for all non-inflationary perfect fluids, one of the curvature variables grows without bound. Moreover, we show that for fluids stiffer than dust ($1<\gamma<2$), the fluid will in general tend towards a state of extreme tilt. For dust ($\gamma=1$), or for fluids less stiff than dust ($0<\gamma< 1$), we show that the fluid will in the future be asymptotically non-tilted. Furthermore, we show that for all $\gamma\geq 1$ the universe evolves towards a vacuum state but does so rather slowly, $\rho/H^2\propto 1/\ln t$. 
  Quantum-gravity corrections to the probability of emission of a particle from a black hole in the Parikh-Wilczek tunneling framework are studied. We consider the effects of zero-point quantum fluctuations of the metric on the emission probability for a tunneling shell. Quantum properties of the geometry are responsible for the formation of a "quantum egosphere" whose effects on the emission probability can be related to the emergence of a logarithmic correction to the Bekenstein-Hawking entropy-area formula. 
  A coarse-graining of spin networks is expressed in terms of partial tracing, thus allowing to use tools of quantum information theory. This is illustrated by the analysis of a simple black hole model, where the logarithmic correction of the Bekenstein-Hawking entropy is shown to be equal to the total amount of correlations on the horizon. Finally other applications of entanglement to quantum gravity are briefly discussed. 
  This paper presents the results of a computational study related to the path-geodesic correspondence in causal sets. For intervals in flat spacetimes, and in selected curved spacetimes, we present evidence that the longest maximal chains (the longest paths) in the corresponding causal set intervals statistically approach the geodesic for that interval in the appropriate continuum limit. 
  We investigate the evolution of the non-linear long wavelength fluctuations during preheating after inflation. By using the separate universe approach, the temporal evolution of the power spectrum of the scalar fields and the curvature variable is obtained numerically. We found that the amplitude of the large scale fluctuations is suppressed after non-linear evolution during preheating. 
  A new solution has been presented for the spherically symmetric space time describing wormholes with Phantom Energy. The model suggests that the existence of wormhole is supported by arbitrarily small quantity of Phantom Energy. 
  I discuss group averaging as a method for quantising constrained systems whose gauge group is a noncompact Lie group. Focussing on three case studies, I address the convergence of the averaging, possible indefiniteness of the prospective physical inner product and the emergence of superselection sectors. 
  The "conformal mass prescriptions" were used recently to calculate the mass of spacetimes in higher dimensional and higher curvature theories of gravity. These definitions are closely related to Komar integrals for spacetimes that are conformally flat at great distances from the sources. We derive these relations without using the conformal infinity formalism. 
  We search for coincident gravitational wave signals from inspiralling neutron star binaries using LIGO and TAMA300 data taken during early 2003. Using a simple trigger exchange method, we perform an inter-collaboration coincidence search during times when TAMA300 and only one of the LIGO sites were operational. We find no evidence of any gravitational wave signals. We place an observational upper limit on the rate of binary neutron star coalescence with component masses between 1 and 3 M_sun of 49 per year per Milky Way equivalent galaxy at a 90% confidence level. The methods developed during this search will find application in future network inspiral analyses. 
  We first review asymptotic twistor theory with its real subspace of null asymptotic twistors. This is followed by a description of an asymptotic version of the Kerr theorem that produces regular asymptotically shear free null geodesic congruences in arbitrary asymptotically flat Einstein or Einstein-Maxwell spacetimes. 
  The energy (due to matter and fields including gravitation) of the Schwarzschild-de Sitter spacetime is investigated by using the Moller energy-momentum definition in both general relativity and teleparallel gravity. We found the same energy distribution for a given metric in both of these different gravitation theories. It is also independent of the teleparallel dimensionless coupling constant, which means that it is valid in any teleparallel model. Our results sustain that (a) the importance of the energy-momentum definitions in the evaluation of the energy distribution of a given spacetime and (b) the viewpoint of Lessner that the Moller energy-momentum complex is a powerful concept of energy and momentum. 
  The emission spectrum from a simple accretion disk model around a compact object is compared for the cases of a black hole (BH) and a boson star (BS) playing the role of the central object. It was found in the past that such a spectrum presents a hardening at high frequencies; however, here it is shown that the self-interaction and compactness of the BS have the effect of softening the spectrum, the less compact the star is, the softer the emission spectrum at high frequencies. Because the mass of the boson fixes the mass of the star and the self-interaction the compactness of the star, we find that, for certain values of the BS parameters, it is possible to produce similar spectra to those generated when the central object is a BH. This result presents two important implications: (i) using this simple accretion model, a BS can supplant a BH in the role of compact object accreting matter, and (ii) within the assumptions of the present accretion disk model we do not find a prediction that could help distinguish a BH from a BS with appropriate parameters of mass and self-interaction. 
  We present an analysis of the behaviour of the electromagnetic self-force for charged particles in a conformally static spacetime, interpreting the results with the help of optical geometry. Some conditions for the vanishing of the local terms in the self-force are derived and discussed. 
  A modified formulation of energy-momentum relation is proposed in the context of doubly special relativity. We investigate its impact on black hole physics. It turns out that such modification will give corrections to both the temperature and the entropy of black holes. In particular this modified dispersion relation also changes the picture of Hawking radiation greatly when the size of black holes approaching the Planck scale. It can prevent black holes from total evaporation, as a result providing a plausible mechanism to treat the remnant of black holes as a candidate for dark matter. 
  We study the thermodynamics of modified black holes proposed in the context of gravity's rainbow. A notion of intrinsic temperature and entropy for these black holes is introduced. In particular for a specific class of modified Schwarzschild solutions, their temperature and entropy are obtained and compared with those previously obtained from modified dispersion relations in deformed special relativity. It turns out that the results of these two different strategies coincide, and this may be viewed as a support for the proposal of deformed equivalence principle. 
  We show that in tilting perfect fluid cosmological models with an ultra-radiative equation of state, generically the tilt becomes extreme at late times and, as the tilt instability sets in, observers moving with the tilting fluid will experience singular behaviour in which infinite expansion is reached within a finite proper time, similar to that of phantom cosmology (but without the need for exotic forms of matter). 
  A study of conformally flat but non flat Bianchi type I and cylindrically symmetric static space-times according to proper projective symmetry is given by using some algebraic and direct integration techniques. It is shown that the special class of the above space-times admit proper projective vector fields. 
  We briefly overview the Petrov classification in four dimensions and its generalization to higher dimensions. 
  We obtain the effects of time dilation and length contraction starting with the law of universal gravitation and the influence of gravitational field on the light. Covariant theory of gravity is derived from the law of universal gravitation and special relativity. We calculate the standard tests of general relativity and find that the results basically agree with that of general relativity. However, our theory differs from general relativity in the predictions for GP-B test, which expects the geodetic effect to be zero and frame-dragging effect 1/4 of the result in general relativity. The anomalous precession of DI Herculis and origin of quasars are explained. The decelerated expansion of the universe is discussed in the paper. 
  The general exact solution of the Einstein-matter field equations describing spherically symmetric shells satisfying an equation of state in closed form is discussed under general assumptions of physical reasonableness. The solutions split into two classes: a class of "astro-physically interesting" solutions describing "ordinary" matter with positive density and pressure, and a class of "phantom-like" solutions with positive density but negative active gravitational mass, which can also be of interest in several ``very strong fields" regimes. Known results on linear-barotropic equations of state are recovered as particular cases. 
  We investigate the cosmological dynamics of a brane Universe when quantum corrections from vacuum polarization are taken into account. New vacuum de Sitter points existing on Randall-Sundrum brane are described. We show also that quantum correction can destroy the DGP de Sitter solution on induced gravity brane. 
  We determine sufficient and necessary conditions for a spherically symmetric initial data set to satisfy the dynamical horizon conditions in the spacetime development. The constraint equations reduce to a single second order linear master equation, which leads to a systematic construction of all spherically symmetric dynamical horizons (SSDH) satisfying certain boundedness conditions. We also find necessary and sufficient conditions for a given spherically symmetric spacetime to contain a SSDH. 
  Proceedings of the international Workshop on ``Dynamics and Thermodynamics of Blackholes and Naked Singularities``, that took place at the Department of Mathematics of the Politecnico of Milano from 13 to 15 May 2004. 
  A new representation of the Einstein evolution equations is presented that is first order, linearly degenerate, and symmetric hyperbolic. This new system uses the generalized harmonic method to specify the coordinates, and exponentially suppresses all small short-wavelength constraint violations. Physical and constraint-preserving boundary conditions are derived for this system, and numerical tests that demonstrate the effectiveness of the constraint suppression properties and the constraint-preserving boundary conditions are presented. 
  We review a number of results recently obtained in the area of constructing rotating solitons in a four dimensional asymptotically flat spacetime. Various models are examined, special attention being paid to the monopole-antimonopole and gauged skyrmion configurations, which have a nonvanishing total angular momentum. For all known examples of rotating solitons, the angular momentum is fixed by some conserved charge of the matter fields. 
  Recently, folk questions on the smoothability of Cauchy hypersurfaces and time functions of a globally hyperbolic spacetime M, have been solved. Here we give further results, applicable to several problems:   (1) Any compact spacelike acausal submanifold H with boundary can be extended to a spacelike Cauchy hypersurface S. If H were only achronal, counterexamples to the smooth extension exist, but a continuous extension (in fact, valid for any compact achronal subset K) is still possible.   (2) Given any spacelike Cauchy hypersurface S, a Cauchy temporal function T (i.e., a smooth function with past-directed timelike gradient everywhere, and Cauchy hypersurfaces as levels) with S equal to one of the levels, is constructed -thus, the spacetime splits orthogonally as $R \times S$ in a canonical way. Even more, accurate versions of this result are obtained if the Cauchy hypersurface S were non-spacelike (including non-smooth, or achronal but non-acausal). 
  A new form of the dynamical equations of vacuum general relativity is proposed. This form involves a new Hamiltonian structure and non canonical variables. The new field variables are the electric field E and the magnetic field of A from the Ashtekar representation of the (complex) gravitational phase space. The Poisson brackets between functionals of the field, which emerge from this framework, are compatible with the Gauss constraint. The quantization is briefly outlined. 
  Conditions are given which, subject to a genericity condition on the Ricci tensor, are both necessary and sufficient for a 3-metric to arise from a static spacetime metric. 
  Hawking radiation can usefully be viewed as a semi-classical tunnelling process that originates at the black hole horizon. The conservation of energy implies the effect of self-gravitation. For a static black hole, a generalized Painleve coordinate system is introduced, and Hawking radiation as tunnelling under the effect of self-gravitation is investigated. The corrected radiation is consistent with the underlying unitary theory. 
  Hawking radiation can usefully be viewed as a semi-classical tunnelling process that originates at the black hole horizon. For the stationary axisymmetric BTZ black hole, a generalized Painleve coordinate system (Painleve-BTZ coordinates) is introduced, and Hawking radiation as tunnelling under the effect of self-gravitation is investigated. The corrected radiation is obtained which is not precise thermal spectrum. The result is consistent with the underlying unitary theory. Moreover, Bekenstein-Hawking entropy of BTZ black hole is not necessarily corrected when we choose appropriate coordinate system to study the tunnelling effect. 
  In this letter we complete a previously introduced formalism to study the gauge-invariant metric fluctuations from a noncompact Kaluza-Klein theory of gravity, to study the evolution of the early universe. The evolution of both, metric and inflaton field fluctuations are reciprocally related. We obtain that <\delta\rho/\rho_b$ depends on the coupling of $\Phi$ with $\delta\phi$ and the spectral index of its spectrum is: $0.9483 < n_1 < 1$. 
  The stability of the Schwarzschild black hole is studied. Using the Painlev\'{e} coordinate, our region can be defined as the black-hole-connected region(r>2m, see text) of the Schwarzschild black hole or the white-hole-connected region(r>2m, see text) of the Schwarzschild black hole. We study the stable problems of the black-hole-connected region. The conclusions are: (1) in the black-hole-connected region, the initially regular perturbation fields must have real frequency or complex frequency whose imaginary must not be greater than -1/4m, so the black-hole-connected regionis stable in physicist' viewpoint; (2) On the contrary, in the mathematicians' viewpoint, the existence of the real frequencies means that the stable problem is unsolved by the linear perturbation method in the black-hole-connected region. 
  Recently, a proposal has appeared for the extraction of the 2-point function of linearised quantum gravity, within the spinfoam formalism. This relies on the use of a boundary state, which introduces a semi-classical flat geometry on the boundary. In this paper, we investigate this proposal considering a toy model in the (Riemannian) 3d case, where the semi-classical limit is better understood. We show that in this limit the propagation kernel of the model is the one for the harmonic oscillator. This is at the origin of the expected 1/L behaviour of the 2-point function. Furthermore, we numerically study the short scales regime, where deviations from this behaviour occur. 
  This is an introduction to the group field theory approach to quantum gravity, with emphasis on motivations and basic formalism, more than on recent results; we elaborate on the various ingredients, both conceptual and formal, of the approach, giving some examples, and we discuss some perspectives of future developments. 
  In this paper, we have solved 1D special relativistic hydrodynamical equations using different numerical method in computational gas dynamics. The numerical solutions of these equations for smooth wave cases give better solution when we use $Non-TVD$(Total Variable Diminishing) but solution of discontinuity wave produces some oscillation behind the shock. On the other hand, $TVD$ type schemes give good approximation at discontinuity cases. Because $TVD$ schemes completely remove the oscillations, they reduce locally the accuracy of the solution around the extrema. 
  Conformally invariant wave equations in de Sitter space, for scalar and vector fields, are introduced in the present paper. Solutions of their wave equations and the related two-point functions, in the ambient space notation, have been calculated. The ``Hilbert'' space structure and the field operator, in terms of coordinate independent de Sitter plane waves, have been defined. The construction of the paper is based on the analyticity in the complexified pseudo-Riemanian manifold, presented first by Bros et al.. Minkowskian limits of these functions are analyzed. The relation between the ambient space notation and the intrinsic coordinates is then studied in the final stage. 
  We derive integral and sup-estimates for the curvature of stably marginally outer trapped surfaces in a sliced space-time. The estimates bound the shear of a marginally outer trapped surface in terms of the intrinsic and extrinsic curvature of a slice containing the surface. These estimates are well adapted to situations of physical insterest, such as dynamical horizons. 
  The quasi-Heisenberg picture of minisuperspace model is considered. The The quasi-Heisenberg picture of minisuperspace model is considered. The suggested scheme consists in quantizing of the equation of motion and interprets all observables including the Universe scale factor as the time-dependent (quasi-Heisenbeg) operators acting in the space of solutions of the Wheeler--DeWitt equation. The Klein-Gordon normalization of the wave function and corresponding to it quantization rules for the equation of motion allow a time-evolution of the mean values of operators even under constraint H=0 on the physical states of Universe. Besides, the constraint H=0 appears as the relation connecting initial values of the quasi-Heisenbeg operators at $t=0$. A stage of the inflation is considered numerically in the framework of the Wigner--Weyl phase-space formalism. For an inflationary model of the ``chaotic inflation'' type it is found that a dispersion of the Universe scale factor grows during inflation, and thus, does not vanish at the inflation end. It was found also, that the ``by hand'' introduced dependence of the cosmological constant from the scale factor in the model with a massless scalar field leads to the decrease of dispersion of the Universe scale factor. The measurement and interpretation problems arising in the framework of our approach are considered, as well. 
  We present a detailed analysis of binary black hole evolutions in the last orbit, and demonstrate consistent and convergent results for the trajectories of the individual bodies. The gauge choice can significantly affect the overall accuracy of the evolution. It is possible to reconcile certain gauge dependent discrepancies by examining the convergence limit. We illustrate these results using an initial data set recently evolved by Bruegmann (Phys. Rev. Lett. 92, 211101). For our highest resolution and most accurate gauge, we estimate the duration of this data set's last orbit to be approximately $59 M_{ADM}$. 
  We propose a class of actions for the spacetime metric that introduce corrections to the Einstein-Hilbert Lagrangian depending on the logarithm of some curvature scalars. We show that for some choices of these invariants the models are ghost free and modify Newtonian gravity below a characteristic acceleration scale given by a_0 = c\mu, where c is the speed of light and \mu is a parameter of the model that also determines the late-time Hubble constant: H_0 \sim \mu. In these models, besides the massless spin two graviton, there is a scalar excitation of the spacetime metric whose mass depends on the background curvature. This dependence is such that this scalar, although almost massless in vacuum, becomes massive and effectively decouples when one gets close to any source and we recover an acceptable weak field limit at short distances. There is also a (classical) ``running'' of Newton's constant with the distance to the sources and gravity is easily enhanced at large distances by a large ratio. We comment on the possibility of building a model with a MOND-like Newtonian limit that could explain the rotation curves of galaxies without introducing Dark Matter using this kind of actions. We also explore briefly the characteristic gravitational phenomenology that these models imply: besides a long distance modification of gravity they also predict deviations from Newton's law at short distances. This short distance scale depends on the local background curvature of spacetime, and we find that for experiments on the Earth surface it is of order \sim 0.1mm, while this distance would be bigger in space where the local curvature is significantly lower. 
  We construct infinite dimensional families of non-singular stationary space times, solutions of the vacuum Einstein equations with a negative cosmological constant. 
  We offer a novel derivation of the electromagnetic self-force acting on a charged particle moving in an arbitrary curved spacetime. Our derivation is based on a generalization from flat spacetime to curved spacetime of the extended-body approach of Ori and Rosenthal. In this approach the charged particle is first modeled as a body of finite extension s, the net force acting on the extended body is computed, and the limit s -> 0 is taken at the end of the calculation. Concretely our extended body is a dumbbell that consists of two point charges that are maintained at a constant spacelike separation s. The net force acting on the dumbbell includes contributions from the mutual forces exerted on each charge by the field created by the other charge, the individual self-forces exerted on each charge by its own field, and the external force which is mostly responsible for the dumbbell's acceleration. These contributions are added up, in a way that respects the curved nature of the spacetime, and all diverging terms in the net force are shown to be removable by mass renormalization. Our end result, in the limit s -> 0, is the standard expression for the electromagnetic self-force in curved spacetime. 
  In this paper we compute the square root of the generalized squared total angular momentum operator $J$ for a Dirac particle in the Kerr-Newman metric. The separation constant $\lambda$ arising from the Chandrasekahr separation ansatz turns out to be the eigenvalue of $J$. After proving that $J$ is a symmetry operator, we show the completeness of Chandrasekhar Ansatz for the Dirac equation in oblate spheroidal coordinates and derive an explicit formula for the propagator $e^{-itH}$. 
  It is well known that a C-field, generated by a certain source equation leads to interesting changes in the cosmological solutions of Einstein's equations. In this article we present and analyze a simple Lorentzian vacuum wormhole in the presence of C-field. 
  We study the gravitational properties of a global monopole in $(D = d + 2)$ dimensional space-time in presence of electromagnetic field. 
  This is the Preface to the special issue of 'International Journal of Geometric Methods in Modern Physics', v.3, N.1 (2006) dedicated to the 50th aniversary of gauge gravitation theory. It addresses the geometry underlying gauge gravitation theories, their higher-dimensional, supergauge and non-commutatuve extensions. 
  We show, under certain conditions, that regular Israel-Wilson-Perj\'es black holes necessarily belong to the Majumdar-Papapetrou family. 
  We describe a rigorous construction, using matched asymptotic expansions, which establishes under very general conditions that local terrestrial and solar-system experiments will measure the effects of varying `constants' of Nature occurring on cosmological scales to computable precision. In particular, `constants' driven by scalar fields will still be found to evolve in time when observed within virialised structures like clusters, galaxies, and planetary systems. This provides a justification for combining cosmological and terrestrial constraints on the possible time variation of many assumed `constants' of Nature, including the fine structure constant and Newton's gravitation constant. 
  We develop the Hadamard renormalization of the stress-energy tensor for a massive scalar field theory defined on a general spacetime of arbitrary dimension. For spacetime dimension up to six, we explicitly describe this procedure. For spacetime dimension from seven to eleven, we provide the framework permitting the interested reader to perform this procedure explicitly in a given spacetime. Our formalism represents an improvement and a generalization of the usual methods and will be helpful in treating some aspects of the quantum physics of extra spatial dimensions. 
  We consider solutions to the linear wave equation on a (maximally extended) Schwarzschild spacetime, assuming only that the solution decays suitably at spatial infinity on a complete Cauchy hypersurface. (In particular, we allow the support of the solution to contain the bifurcate event horizon.) We prove uniform decay bounds for the solution in the exterior regions, including the uniform bound Cv_+^{-1}, where v_+ denotes max{v,1} and v denotes Eddington-Finkelstein advanced time. We also prove uniform decay bounds for the flux of energy through the event horizon and null infinity. The estimates near the event horizon exploit an integral energy identity normalized to local observers. This estimate can be thought to quantify the celebrated red-shift effect. The results in particular give an independent proof of the classical uniform boundedness theorem of Kay and Wald, without recourse to the discrete isometries of spacetime. 
  Gravitational collapse of FRW brane world embedded in a conformaly flat bulk is considered for matter cloud consists of dark matter and dark energy with equation of state $p=\epsilon \rho$ $(\epsilon<-{1/3})$. The effect of dark matter and dark energy is being considered first separately and then a combination of them both with and without interaction. In some cases the collapse leads to black hole in some other cases naked singularity appears. 
  Radiometric tracking data from Pioneer 10 and 11 spacecraft has consistently indicated the presence of a small, anomalous, Doppler frequency drift, uniformly changing with a rate of ~6 x 10^{-9} Hz/s; the drift can be interpreted as a constant sunward acceleration of each particular spacecraft of a_P = (8.74 \pm 1.33) x 10^{-10} m/s^2. This signal is known as the Pioneer anomaly; the nature of this anomaly remains unexplained. We discuss the efforts to retrieve the entire data sets of the Pioneer 10/11 radiometric Doppler data. We also report on the recently recovered telemetry files that may be used to reconstruct the engineering history of both spacecraft using original project documentation and newly developed software tools. We discuss possible ways to further investigate the discovered effect using these telemetry files in conjunction with the analysis of the much extended Doppler data. We present the main objectives of new upcoming study of the Pioneer anomaly, namely i) analysis of the early data that could yield the direction of the anomaly, ii) analysis of planetary encounters, that should tell more about the onset of the anomaly, iii) analysis of the entire dataset, to better determine the anomaly's temporal behavior, iv) comparative analysis of individual anomalous accelerations for the two Pioneers, v) the detailed study of on-board systematics, and vi) development of a thermal-electric-dynamical model using on-board telemetry. The outlined strategy may allow for a higher accuracy solution for a_P and, possibly, will lead to an unambiguous determination of the origin of the Pioneer anomaly. 
  We study a set of static solutions of the Einstein equations in presence of a massless scalar field and establish their connection to the Kantowski-Sachs cosmological solutions based on some kind of duality transformations. The physical properties of the limiting case of an empty hyperbolic spacetime (pseudo-Schwarzschild geometry) are analyzed in some detail. 
  Homogeneous isotropic gravitating models are discussed in the framework of gauge approach to gravitation. Generalized cosmological Friedmann equations without specific solutions are deduced for models filled by scalar fields and usual gravitating matter. Extreme conditions, by which gravitational repulsion effect takes place, are analyzed. 
  We study relativistic gyratons which carry an electric charge. The Einstein-Maxwell equations in arbitrary dimensions are solved exactly in the case of a charged gyraton propagating in an asymptotically flat metric. 
  Some concepts of real and complex projective geometry are applied to the fundamental physical notions that relate to Minkowski space and the Lorentz group. In particular, it is shown that the transition from an infinite speed of propagation for light waves to a finite one entails the replacement of a hyperplane at infinity with a light cone and the replacement of an affine hyperplane - or rest space - with a proper time hyperboloid. The transition from the metric theory of electromagnetism to the pre-metric theory is discussed in the context of complex projective geometry, and ultimately it is proposed that the geometrical issues are more general than electromagnetism, namely, they pertain to the transition from point mechanics to wave mechanics. 
  The equations of pre-metric electromagnetism are summarized and then formulated as an exterior differential system on the total space of the bundle of 2-forms over the spacetime manifold. The Harrison-Estabrook method of computing the symmetries of the system is then applied, with the result that of the four possible formal algebras of infinitesimal symmetries, the most physically compelling one is the Lie algebra of infinitesimal projective transformations of real four-dimensional projective space. 
  Astrophysical tests of Planck-suppressed Lorentz violations had been extensively studied in recent years and very stringent constraints have been obtained within the framework of effective field theory. There are however still some unresolved theoretical issues, in particular regarding the so called "naturalness problem" - which arises when postulating that Planck-suppressed Lorentz violations arise only from operators with mass dimension greater than four in the Lagrangian. In the work presented here we shall try to address this problem by looking at a condensed-matter analogue of the Lorentz violations considered in quantum gravity phenomenology. Specifically, we investigate the class of two-component BECs subject to laser-induced transitions between the two components, and we show that this model is an example for Lorentz invariance violation due to ultraviolet physics. We shall show that such a model can be considered to be an explicit example high-energy Lorentz violations where the ``naturalness problem'' does not arise. 
  The basic concepts of the formulation of Maxwellian electromagnetism in the absence of a Minkowski scalar product on spacetime are summarized, with particular emphasis on the way that the electromagnetic constitutive law on the space of bivectors over spacetime supplants the role of the Minkowski scalar product on spacetime itself. The complex geometry of the space of bivectors is summarized, with the intent of showing how an isomorphic copy of the Lorentz group appears in that context. The use of complex 3-spinors to represent electromagnetic fields is then discussed, as well as the expansion of scope that the more general complex projective geometry of the space of bivectors suggests. 
  The Laser Astrometric Test of Relativity (LATOR) is a Michelson-Morley-type experiment designed to test the Einstein's general theory of relativity in the most intense gravitational environment available in the solar system -- the close proximity to the Sun. By using independent time-series of highly accurate measurements of the Shapiro time-delay (laser ranging accurate to 1 cm) and interferometric astrometry (accurate to 0.1 picoradian), LATOR will measure gravitational deflection of light by the solar gravity with accuracy of 1 part in a billion, a factor ~30,000 better than currently available. LATOR will perform series of highly-accurate tests of gravitation and cosmology in its search for cosmological remnants of scalar field in the solar system. We present science, technology and mission design for the LATOR mission. 
  Inflationary homogeneous isotropic cosmological models filled by scalar fields and ultrarelativistic matter are examined in the framework of gauge theories of gravitation. By using quadratic scalar field potential numerical analysis of flat, open and closed models is curried out. Properties of cosmological models are investigated in dependence on indefinite parameter of cosmological equations and initial conditions at a bounce. Fulfilled analysis demonstrates regular character of all cosmological models. 
  We study the linear magnetohydrodynamic (MHD) equations, both in the Newtonian and the general-relativistic limit, as regards a viscous magnetized fluid of finite conductivity and discuss instability criteria. In addition, we explore the excitation of cosmological perturbations in anisotropic spacetimes, in the presence of an ambient magnetic field. Acoustic, electromagnetic (e/m) and fast-magnetosonic modes, propagating normal to the magnetic field, can be excited, resulting in several implications of cosmological significance. 
  We study the phase space of the spherically symmetric solutions of Einstein-Maxwell-Gauss-Bonnet system nonminimally coupled to a scalar field and prove the existence of solutions with unusual asymptotics in addition to asymptotically flat ones. We also find new dyonic solutions of dilatonic Einstein-Maxwell theory. 
  We use a Harrison transformation on solutions to the stationary axisymmetric Einstein equations to generate solutions of the Einstein-Maxwell equations. The case of hyperelliptic solutions to the Ernst equation is studied in detail. Analytic expressions for the metric and the multipole moments are obtained. As an example we consider the transformation of a family of counter-rotating dust disks. The resulting solutions can be interpreted as disks with currents and matter with a purely azimuthal pressure or as two streams of freely moving charged particles. We discuss interesting limiting cases as the extreme limit where the charge becomes identical to the mass, and the ultrarelativistic limit where the central redshift diverges. 
  We review explicit solutions to the stationary axisymmetric Einstein-Maxwell equations which can be interpreted as disks of charged dust. The disks of finite or infinite extension are infinitesimally thin and constitute a surface layer at the boundary of an electro-vacuum. The Einstein-Maxwell equations in the presence of one Killing vector are obtained by using a projection formalism. The SU(2,1) invariance of the stationary Einstein-Maxwell equations can be used to construct solutions for the electro-vacuum from solutions to the pure vacuum case via a so-called Harrison transformation. It is shown that the corresponding solutions will always have a non-vanishing total charge and a gyromagnetic ratio of 2. Since the vacuum and the electro-vacuum equations in the stationary axisymmetric case are completely integrable, large classes of solutions can be constructed with techniques from the theory of solitons. The richest class of physically interesting solutions to the pure vacuum case due to Korotkin is given in terms of hyperelliptic theta functions. The Harrison transformed hyperelliptic solutions are discussed. 
  It is possible to generate an accelerated period of expansion from reasonable potentials acting between the universe particle constituents. The pressure of primordial nucleons interacting via a simple nuclear potential is obtained via Mayer's cluster expansion technique. The attractive part of the potential engenders a negative pressure and may therefore be responsible for the cosmic acceleration. 
  Building on previous work on the critical behavior in gravitational collapse of the self-gravitating SU(2) $\sigma$-field and using high precision numerical methods we uncover a fine structure hidden in a narrow window of parameter space. We argue that this numerical finding has a natural explanation within a dynamical system framework of critical collapse. 
  The list of putative sources of gravitational waves possibly detected by the ongoing worldwide network of large scale interferometers has been continuously growing in the last years. For some of them, the detection is made difficult by the lack of a complete information about the expected signal. We concentrate on the case where the expected GW is a quasi-periodic frequency modulated signal i.e., a chirp. In this article, we address the question of detecting an a priori unknown GW chirp. We introduce a general chirp model and claim that it includes all physically realistic GW chirps. We produce a finite grid of template waveforms which samples the resulting set of possible chirps. If we follow the classical approach (used for the detection of inspiralling binary chirps, for instance), we would build a bank of quadrature matched filters comparing the data to each of the templates of this grid. The detection would then be achieved by thresholding the output, the maximum giving the individual which best fits the data. In the present case, this exhaustive search is not tractable because of the very large number of templates in the grid. We show that the exhaustive search can be reformulated (using approximations) as a pattern search in the time-frequency plane. This motivates an approximate but feasible alternative solution which is clearly linked to the optimal one. [abridged version of the abstract] 
  Global topological defects described by real scalar field in (3,1) dimensions coupled to gravity are analyzed. We consider a class of scalar potentials with explicit dependence with distance, evading Derrick's theorem and leading to defects with spherical symmetry. The analysis shows that the defects have finite energy on flat space, contrary to the observed for the global monopole. With the aim to study the gravitational field produced by such defects, after an {\it Ansatz} for the static metric with spherical symmetry, we obtain the coupled system of Einstein and field equations. On the Newtonian approximation, we numerically find that the defects have a repulsive gravitational field. This field is like one generated by a negative mass distributed on a spherical shell. In the weak gravity regime a relation between the Newtonian potential and one of the metric coefficients is obtained. The numerical analysis in this regime leads to a spacetime with a deficit solid angle on the core of the defect. 
  Effective field theories (EFTs) have been widely used as a framework in order to place constraints on the Planck suppressed Lorentz violations predicted by various models of quantum gravity. There are however technical problems in the EFT framework when it comes to ensuring that small Lorentz violations remain small -- this is the essence of the "naturalness" problem. Herein we present an "emergent" space-time model, based on the "analogue gravity'' programme, by investigating a specific condensed-matter system that is in principle capable of simulating the salient features of an EFT framework with Lorentz violations. Specifically, we consider the class of two-component BECs subject to laser-induced transitions between the components, and we show that this model is an example for Lorentz invariance violation due to ultraviolet physics. Furthermore our model explicitly avoids the "naturalness problem", and makes specific suggestions regarding how to construct a physically reasonable quantum gravity phenomenology. 
  We study the conditions of validity of generalized second law in phantom dominated era. 
  It is shown that in the 4d Euclidean space there are two causal structures defined by the temporal field. One of them is well-known Minkowski spacetime. In this case the gravitational potential (the positive definite Riemann metric) and temporal field satisfy the Einstein equations with trivial energy-momentum tensor. However, in the case of the second causal structure the gravitational potential and temporal field should be connected with some nontrivial energy-momentum tensor. We consider the simplest case with energy-momentum tensor of the real scalar field and derive exact solution of the field equations. It can be viewed as the ground to consider the second causal structure on the equal footing with the Minkowski spacetime, i.e., as an object interesting from the physical point of view, especially in the framework of the field theory. 
  We present a paralell approach to discrete geometry: the first one introduces Voronoi cell complexes from statistical tessellations in order to know the mean scalar curvature in term of the mean number of edges of a cell. The second one gives the restriction of a graph from a regular tessellation in order to calculate the curvature from pure combinatorial properties of the graph.   Our proposal is based in some epistemological pressupositions: the macroscopic continuous geometry is only a fiction, very usefull for describing phenomena at certain sacales, but it is only an approximation to the true geometry. In the discrete geometry one starts from a set of elements and the relation among them without presuposing space and time as a background. 
  We present static cylindrically symmetric electrovac solutions in the framework of the Brans-Dicke theory and show that our solution yields some of the well-known solutions for special values of the parameters of the resulting metric functions. 
  This is a review paper about one of the approaches to unify Quantum Mechanics and the theory of General Relativity. Starting from the pioneer work of Regge and Penrose other scientists have constructed state sum models, as Feymann path integrals, that are topological invariant on the triangulated Riemannian surfaces, and that in the continuous limit become the Hilbert-Einstein action. 
  This article discusses methods of geometric analysis in general relativity, with special focus on the role of "critical surfaces" such as minimal surfaces, marginal surface, maximal surfaces and null surfaces. 
  We show a kind of converse to some results of Hall and Reginatto on exact uncertainty related to the Schroedinger and Wheeler-deWitt equations. Some survey material on statistical geometrodynamics is also sketched. 
  A new algorithm for solving the general relativistic MHD equations is described in this paper. We design our scheme to incorporate black hole excision with smooth boundaries, and to simplify solving the combined Einstein and MHD equations with AMR. The fluid equations are solved using a finite difference Convex ENO method. Excision is implemented using overlapping grids. Elliptic and hyperbolic divergence cleaning techniques allow for maximum flexibility in choosing coordinate systems, and we compare both methods for a standard problem. Numerical results of standard test problems are presented in two-dimensional flat space using excision, overlapping grids, and elliptic and hyperbolic divergence cleaning. 
  My research work can be classified into two parts namely, (i) Cosmological phenomena with varying speed of light and (ii) Gravitational collapse and black holes.   We have investigated several cosmological phenomena when velocity of light varies with time. We have considered the variation of light as power-law in time and have studied the Brans-Dicke cosmology, Quintessence problem and others. We have also examined whether some cosmological problems can be solved using it for anisotropic space-time model.   In brane scenario we have studied some cosmological implications namely, the Cosmic No-Hair Conjecture (CNHC) and variable gravity theory. In CNHC we have shown that in realistic situation no more extra conditions are needed in brane scenario.   In gravitational collapse, we have mainly concentrated to spherical collapse for dust distribution. We have considered both four and higher dimensional space-time and have examined the possibility for formation of black hole or naked singularity. In marginally bound case we have found naked singularity is possible only for four and five dimension, while for non-marginally bound case the possibility of black hole increases with the increase in the dimension of the space-time.   In quasi-spherical gravitational collapse for dust distribution, we have (n+2) dimensional Szekeres space-time and have examined the possibility for formation of black hole or naked singularity. In marginally bound case, we have found that possibility of naked singularity depends on dimension of the space-time. 
  We investigate the use of asymptotically null slices combined with stretching or compactification of the radial coordinate for the numerical simulation of asymptotically flat spacetimes. We consider a 1-parameter family of coordinates characterised by the asymptotic relation $r\sim R^{1-n}$ between the physical radius $R$ and coordinate radius $r$, and the asymptotic relation $K\sim R^{n/2-1}$ for the extrinsic curvature of the slices. These slices are asymptotically null in the sense that their Lorentz factor relative to stationary observers diverges as $\Gamma\sim R^{n/2}$. While $1<n\le 2$ slices intersect $\scri$, $0< n\le 1$ slices end at $i^0$. We carry out numerical tests with the spherical wave equation on Minkowski and Schwarzschild spacetime. Simulations using our coordinates with $0<n\le 2$ achieve higher accuracy at lower computational cost in following outgoing waves to very large radius than using standard $n=0$ slices without compactification. Power-law tails in Schwarzschild are also correctly represented. 
  In this paper, we find some new exact solutions to the Einstein-Gauss-Bonnet equations. First, we prove a theorem which allows us to find a large family of solutions to the Einstein-Gauss-Bonnet gravity in $n$-dimensions. This family of solutions represents dynamic black holes and contains, as particular cases, not only the recently found Vaidya-Einstein-Gauss-Bonnet black hole, but also other physical solutions that we think are new, such as, the Gauss-Bonnet versions of the Bonnor-Vaidya(de Sitter/anti-de Sitter) solution, a global monopole and the Husain black holes. We also present a more general version of this theorem in which less restrictive conditions on the energy-momentum tensor are imposed. As an application of this theorem, we present the exact solution describing a black hole radiating a charged null fluid in a Born-Infeld nonlinear electrodynamics. 
  We develop the gauge approach based on the Lorentz group to the gravity with torsion. With a Lagrangian quadratic in curvature we show that the Einstein-Hilbert action can be induced from a simple gauge model due to quantum corrections of torsion via formation of a gravito-magnetic condensate. An effective theory of cosmic knots at Planckian scale is proposed. 
  Within the framework of brane-world models it is possible to account for the cosmological constant by assuming supersymmetry is broken on the 3-brane but preserved in the bulk. An effective Casimir energy is induced on the brane due to the boundary conditions imposed on the compactified extra dimensions. It will be demonstrated that modification of these boundary conditions allows a spacecraft to travel at any desired speed due to a local adjustment of the cosmological constant which effectively contracts/expands space-time in the front/rear of the ship resulting in motion potentially faster than the speed of light as seen by observers outside the disturbance. 
  In this letter, Parikh-Wilczek tunnelling framework, which treats Hawking radiation as a tunnelling process, is extended, and the emission rate of a charged particle tunnelling from the Kerr-Newman black hole is calculated. The emission spectrum takes the same functional form as that of uncharged particles and consists with an underlying unitary theory but deviates from the pure thermal spectrum. Moreover, our calculation indicates that the emission process is treated as a reversible process in the Parikh-Wilczek tunnelling framework, and the information conservation is a natural result of the first law of black hole thermodynamics. 
  A lower bound on the light neutrino mass $m_\nu$ is derived in the framework of a geometrical interpretation of quantum mechanics. Using this model and the time of flight delay data for neutrinos coming from SN1987A, we find that the neutrino masses are bounded from below by $m_\nu\gtrsim 10^{-4}-10^{-3}$eV, in agreement with the upper bound $m_\nu\lesssim$ $({\cal O}(0.1) - {\cal O} (1))$ eV currently available. When the model is applied to photons with effective mass, we obtain a lower limit on the electron density in intergalactic space that is compatible with recent baryon density measurements. 
  A new class of exact solutions of Einstein's field equations with bulk viscous fluid for an LRS Bianchi type-I spacetime is obtained by using deceleration parameter as variable. The value of Hubble's constant $H_{0}$ is found to be less than one for non-flat model and is equal to 1.5 for flat model which are of the physical interest. Some physical and geometric properties of the models are also discussed. 
  An exact solution has an axial symmetry is obtained in the teleparallel theory of gravitation. The associated metric has the structure function G(xi)=1-xi^2-2mA(xi)^3. The cubic nature of the structure function can make calculations cumbersome. Using a coordinate transformation we get a tetrad that its associated metric has the structure function in a factorisable form. This new form has the advantage that its roots are now trivial to write down. The singularities of the obtained tetrad are studied. Using another coordinate transformation we get a tetrad that its associated metric gives the Schwarzschild spacetime. Calculate the energy content of this tetrad we get a meaningless result! 
  We study light propagation and gravitational lensing in scalar-tensor theories of gravity by using a static, axisymmetric exterior solution. The solution has asymptotic flatness properties and is reduced to Voorhees's one in the case of a constant scalar field. Our studies are done by using a technique of the conformal transformation such that their results are independent of details of scalar-tensor theories. For some specific cases, we analytically obtain a deflection angle of the light path and find that it can become negative. The appearance of a negative deflection angle indicates ``reflection'' of a light path, and we investigate under which conditions the light reflection occurs. As for the optical scalars, the Weyl source-term shows significantly different properties compared with that in the Schwarzschild spacetime. We therefore classify a space of the model parameters into four distinct regions on the basis of the qualitative properties of the Weyl source-term and find a close relationship between this classification and the occurrence of the light reflection. We finally solve the null geodesic equations and the optical scalar equations numerically. We find that a picture of the thin lens is applicable and give a simple analytic model for the optical scalars. As for the properties of gravitational lensing, the deflection angle and the image distortion rate are obtained as functions of the impact parameter. Again, we find a close relationship between their qualitative properties and the classification above. 
  In this work we study to which extent the knowledge of spatial topology may place constraints on the parameters of the generalized Chaplygin gas (GCG) model for unification of dark energy and dark matter. By using both the Poincar\'e dodecahedral and binary octahedral spaces as the observable spatial topologies, we examine the current type Ia supernovae (SNe Ia) constraints on the GCG model parameters. We show that the knowledge of spatial topology does provide additional constraints on the $A_s$ parameter of the GCG model but does not lift the degeneracy of the $\alpha$ parameter. 
  Exact solutions of Einstein field equations invariant for a non-Abelian 2-dimensional Lie algebra of Killing fields are described. Physical properties of these gravitational fields are studied, their wave character is checked by making use of covariant criteria and the observable effects of such waves are outlined. The possibility of detection of these waves with modern detectors, spherical resonant antennas in particular, is sketched. 
  Newly formed black holes are expected to emit characteristic radiation in the form of quasi-normal modes, called ringdown waves, with discrete frequencies. LISA should be able to detect the ringdown waves emitted by oscillating supermassive black holes throughout the observable Universe. We develop a multi-mode formalism, applicable to any interferometric detectors, for detecting ringdown signals, for estimating black hole parameters from those signals, and for testing the no-hair theorem of general relativity. Focusing on LISA, we use current models of its sensitivity to compute the expected signal-to-noise ratio for ringdown events, the relative parameter estimation accuracy, and the resolvability of different modes. We also discuss the extent to which uncertainties on physical parameters, such as the black hole spin and the energy emitted in each mode, will affect our ability to do black hole spectroscopy. 
  The possibility that the universe may have a fundamental and positive cosmological constant has motivated an interesting cosmological model, in which initially the universe is in a cosmological constant sea, then the local quantum fluctuations violating the null energy condition create some islands with matter and radiation, which under certain conditions might corresponds to our observable universe. We in this note study the perturbation spectra of scalar fields not affecting the evolution of background during the fluctuation. We will examine whether they can be interesting, and responsible for the structure formation of observable universe. 
  One of the most important challenges of contemporary physics is to find experimental signatures of quantum gravity. It is expected that quantum gravitational effects lead to proton decay but on time scales way beyond what is of any relevance to experiments. At non-zero temperatures there are reasons to believe that the situation is much more favourable. We will argue that at the temperatures and densities reached at present and future fusion facilities there is a realistic possibility that proton decay could be detectable. 
  The infrared problem of the effective action in 2D is discussed in the framework of the Covariant Perturbation Theory. The divergences are regularised by a mass and the leading term is evaluated up to the third order of perturbation theory. A summation scheme is proposed which isolates the divergences from the finite part of the series and results in a single term. The latter turns out to be equivalent to the coupling to a certain classical external field. This suggests a renormalisation by factorisation. 
  We consider a cosmological model with a variable gravitational constant, G, based on a scalar-tensor theory. Using the recent observational data for the Hubble diagram of type Ia supernovae (SNeIa) we find a phenomenological expression describing the variation of G. The corresponding variation of the fine structure constant \alpha within multidimensional theories is also computed and is shown not to support known constraints on \Delta \alpha / \alpha. 
  We study {\it analytically} the asymptotic quasinormal spectrum of fermionic fields in the Kerr spacetime. We find an analytic expression for these black-hole resonances in terms of the black-hole physical parameters: its Bekenstein-Hawking temperature $T_{BH}$, and its horizon's angular velocity $\Omega$, which is valid in the asymptotic limit $1 \ll \omega_I \ll \omega_R$. It is shown that according to Bohr's correspondence principle, the emission of a Rarita-Schwinger quantum ($s=3/2$) corresponds to a fundamental black-hole area change $\Delta A=4\hbar \ln 2$, while the emission of a Weyl neutrino field ($s=1/2$) corresponds to an adiabatic quantum transition with $\Delta A=0$. 
  Lorentz invariant supersymmetric deformations of superspaces based on Moyal star product parametrized by Majorana spinor $\lambda_{a}$ and Ramond grassmannian vector $\psi_{m}=-{1\over 2}(\bar\theta\gamma_{m}\lambda)$ in the spinor realization \cite{VZ} are proposed. The map of supergravity background into composite supercoordinates: $(B^{-1}_{mn}, \Psi^{a}_{m}, C_{ab}) \leftrightarrow (i\psi_{m}\psi_{n}, \psi_{m}\lambda^{a}, \lambda_{a}\lambda_{b})$ valid up to the second order corrections in deformation parameter $h$ and transforming the background dependent Lorentz noninvariant (anti)commutators of supercoordinates into their invariant Moyal brackets is revealed. We found one of the deformations to depend on the axial vector $\psi_{1m}={1/2}(\bar\theta\gamma_{m}\gamma_{5}\lambda)$ and to vanish for the $\theta$ components with the same chiralities. The deformations in the (super)twistor picture are discussed. 
  This is a study of the behavior of wave equations in conformally compactified spacetimes suited to the use of computational boundaries beyond Scri+. There light cones may be adjusted for computational convenience and/or Scri+ may be approximated by a "proto-Scri" spacelike hypersurface just outside a de Sitter horizon. One expects a numerical implementation to excise the physically unnecessary universe somewhat beyond the outer horizon. As an entry level example I study forms of the Maxwell equations and causal relations for an outer boundary in that example. 
  Relationship between the speed of gravity c_g and the speed of light c_e in the bi-metric theory of gravity is discussed. We reveal that the speed of light is a function of the speed of gravity which is a primary fundamental constant. Thus, experimental measurement of relativistic bending of light propagating in time-dependent gravitational field directly compares the speed of gravity versus the speed of light and tests if there is any aether associated with the gravitational field considered as a transparent `medium' with the constant refraction index. 
  Event and apparent horizons are key diagnostics for the presence and properties of black holes. In this article I review numerical algorithms and codes for finding event and apparent horizons in numerically-computed spacetimes, focusing on calculations done using the 3+1 ADM formalism.   There are 3 basic algorithms for finding event horizons, based respectively on integrating null geodesics \emph{forwards} in time, integrating null geodesics \emph{backwards} in time, and integrating null \emph{surfaces} backwards in time. The last of these is generally the most efficient and accurate.   There are a large number of apparent-horizon finding algorithms, with differing trade-offs between speed, robustness, accuracy, and ease of programming. In axisymmetry, shooting algorithms work well and are fairly easy to program. In slices with no continuous symmetries, Nakamura et al.'s algorithm and elliptic-PDE algorithms are fast and accurate, but require good initial guesses to converge. In many cases Schnetter's "pretracking" algorithm can greatly improve an elliptic-PDE algorithm's robustness. Flow algorithms are generally quite slow, but can be very robust in their convergence. 
  We investigate the non-linear coupling between radial and non-radial oscillations of static spherically symmetric neutron stars as a possible mechanism for the generation of gravitational waves that may lead to observable signatures. In this paper we concentrate on the axial sector of the non-radial perturbations. By using a multi-parameter perturbative framework we introduce a complete description of the non-linear coupling between radial and axial non-radial oscillations; we study the gauge invariant character of the associated perturbative variables and develop a computational scheme to evolve the non-linear coupling perturbations in the time domain. We present results of simulations corresponding to different physical situations and discuss the dynamical behaviour of this non-linear coupling. Of particular interest is the occurrence of signal amplifications in the form of resonance phenomena when a frequency associated with the radial pulsations is close to a frequency associated with one of the axial w-modes of the star. Finally, we mention possible extensions of this work and improvements towards more astrophysically motivated scenarios. 
  We discuss some aspects of the differential geometry of curves in Minkowski space. We establish the Serret-Frenet equations in Minkowski space and use them to give a very simple proof of the fundamental theorem of curves in Minkowski space. We also state and prove two other theorems which represent Minkowskian versions of a very known theorem of the differential geometry of curves in tridimensional Euclidean space. We discuss the general solution for torsionless paths in Minkowki space. We then apply the four-dimensional Serret-Frenet equations to describe the motion of a charged test particle in a constant and uniform electromagnetic field and show how the curvature and the torsions of the four-dimensional path of the particle contain information on the electromagnetic field acting on the particle. 
  In the recent times a lot of effort has been devoted to improve our knowledge about the space of string theory vacua (``the landscape'') to find statistical grounds to justify how and why the theory selects its vacuum. Particularly interesting are those vacua that preserve some supersymmetry, which are always supersymmetric solutions of some supergravity theory. After an general introduction to how the pursuit of unification has lead to the vacuum selection problem, we are going to review some recent results on the problem of finding all the supersymmetric solutions of a supergravity theory applied to the N=4,d=4 supergravity case. 
  We present a formulation of gravity in terms of a theory based on complex SU(2) gauge fields with a general coordinate invariant action functional quadratic in the field strength. Self-duality or anti-self-duality of the field strength emerges as a constraint from the equations of motion of this theory. This in turn leads to Einstein gravity equations for a dilaton and an axion conformally coupled to gravity for the self-dual constraint. The analysis has also been extended to N=1 and 2 super Yang-Mills theory of complex SU(2) gauge fields. This leads, besides other equations of motion, to self-duality/anti-self-duality of generalized supercovariant field-strengths. The self-dual case is then shown to yield as its solutions $N = 1, 2$ supergravity equations respectively. 
  We investigate a local cosmic string with a phenomenological energy momentum tensor as prescribed by Vilenkin, in presence of C-field . The solutions of full nonlinear Einstein's equations for exterior and interior regions of such a string are presented. 
  The role of torsion in quantum three-dimensional gravity is investigated by studying the partition function of the Euclidean theory in Riemann-Cartan spacetime. The entropy of the black hole with torsion is found to differ from the standard Bekenstein-Hawking result, but its form is in complete agreement with the first law of black hole thermodynamics. 
  We study spatial variations of the fine-structure constant in the presence of static straight cosmic strings in the weak-field approximation in Einstein gravity. We work in the context of a generic Bekenstein-type model and consider a gauge kinetic function linear in the scalar field. We determine an analytical form for the scalar field and the string metric at large distances from the core. We show that the gravitational effects of $\alpha$-varying strings can be seen as a combination of the gravitational effects of global and local strings. We also verify that at large distances to the core the space-time metric is similar to that of a global string. We study the motion of test particles approaching from infinity and show that photons are scattered to infinity while massive particles are trapped in bounded trajectories. We also calculate an overall limit on the magnitude of the variation of $\alpha$ for a GUT string, by considering suitable cosmological constraints coming from the Equivalence Principle. 
  An encyclopedia article on mathematical aspects of quantum field theory in curved spacetime. Section titles are: Introduction and preliminaries; Construction of *-algebra for a real linear scalar field on globally hyperbolic spacetimes and some general theorems; Particle creation and the limitations of the particle concept; Theory of the stress-energy tensor; Hawking and Unruh effects; Non-globally hyperbolic spacetimes and the time-machine question; Other related topics and some warnings. 
  We prove a global in time existence theorem for the initial value problem for the Einstein-Boltzmann system, with positive cosmological constant and arbitrarily large initial data, in the spatially homogeneous case, in a Robertson-Walker space-time. 
  A pedagogical introduction to the Dirac equation for massive particles in Rindler space is presented. The spin connection coefficients are explicitly derived using techniques from general relativity. We then apply the Lagrange-Green identity to greatly simplify calculation of the inner products needed to normalize the states. Finally, the Bogolubov coefficients relating the Rindler and Minkowski modes are derived in an intuitive manner. These derivations are useful for students interested in learning about quantum field theory in a curved space-time. 
  We discuss cosmological models in six-dimensional spacetime. For codimension-1 branes, we consider a (4+1) braneworld model and discuss its cosmological evolution. For codimension-2 branes, we consider an infinitely thin conical braneword model in the presence of an induced gravity term on the brane and a Gauss-Bonnet term in the bulk. We discuss the cosmological evolution of isotropic and anisotropic matter on the brane. We also briefly discuss cosmological models in six-dimensional supergravity. 
  A cosmological constant, Lambda, is the most natural candidate to explain the origin of the dark energy (DE) component in the Universe. However, due to experimental evidence that the equation of state (EOS) of the DE could be evolving with time/redshift (including the possibility that it might behave phantom-like near our time) has led theorists to emphasize that there might be a dynamical field (or some suitable combination of them) that could explain the behavior of the DE. While this is of course one possibility, here we show that there is no imperative need to invoke such dynamical fields and that a variable cosmological constant (including perhaps a variable Newton's constant too) may account in a natural way for all these features. 
  The Ashtekar-Barbero-Immirzi formulation of General Relativity is extended to include spinor matter fields. Our formulation applies to generic values of the Immirzi parameter and reduces to the Ashtekar-Romano-Tate approach when the Immirzi parameter is taken equal to the imaginary unit. The dynamics of the gravity-fermions coupled system is described by the Holst plus Dirac action with a non-minimal coupling term. The non-minimal interaction together with the Holst modification to the Hilbert-Palatini action reconstruct the Nieh-Yan invariant, so that the effective action coming out is the one of Einstein-Cartan theory with a typical Fermi-like interaction term: in spite of the presence of spinor matter fields, the Immirzi parameter plays no role in the classical effective dynamics and results to be only a multiplicative factor in front of a total divergence.   We reduce the total action of the theory to the sum of dynamically independent Ashtekar-Romano-Tate actions for self and anti-self dual connections, with different weights depending on the Immirzi parameter. This allows to calculate the constraints of the complete theory in a simple way, it is only necessary to realize that the Barbero-Immirzi connection is a weighted sum of the self and anti-self dual Ashtekar connections. Finally the obtained constraints for the separated action result to be polynomial in terms of the self and anti-self dual connections, this could have implications in the inclusion of spinor matter in the framework of non-perturbative quantum gravity. 
  A class of time dependent solutions to $(3+1)$ Einstein--Maxwell-dilaton theory with attractive electric force is found from Einstein--Weyl structures in (2+1) dimensions corresponding to dispersionless Kadomtsev--Petviashvili and $SU(\infty)$ Toda equations. These solutions are obtained from time--like Kaluza--Klein reductions of $(3+2)$ solitons. 
  In this paper, we provide a detailed description of our recent analysis and determination of the frame-dragging effect obtained using the nodes of the satellites LAGEOS and LAGEOS 2, in reply to the paper "On the reliability of the so-far performed tests for measuring the Lense-Thirring effect with the LAGEOS satellites" by L. Iorio 
  A modified gravitational action is considered which involves the quantity $F_{\mu\nu}=\partial_{\mu}\Gamma_{\nu}-\partial_{\nu}\Gamma_{\mu}$, where $\Gamma_{\mu}=\Gamma^{\alpha}_{\mu\alpha}$. Since $\Gamma_{\mu}$ transforms like a U(1) gauge field under coordinate transformations terms such as $F^{\mu\nu}F_{\mu\nu}$ are invariant under coordinate transformations. If such a term is added to the usual gravitational action the resulting field equations, obtained from a Palatini variation, are the Einstein-Proca equations. The vector field can be coupled to point charges or to a complex scalar density of weight $ie$, where $e$ is the charge of the field. If this scalar density is taken to be $g^{-ie/2}$ and the overall factor of the scalar density Lagrangian takes on a particular value the resulting field equations are the Einstein-Maxwell equations. 
  We consider a way to avoid black hole singularities by gluing a black hole exterior to an interior with a tube-like geometry consisting of a direct product of two-dimensional AdS, dS, or Rindler spacetime with a two-sphere of constant radius. As a result we obtain a spacetime with either "cosmological" or "acceleration" (event) horizons but without an apparent horizon. The inner region is everywhere regular and supported by matter with the vacuum-like equation of state $p_{r}+\rho =0$ where $p_{r}=T_{r}^{r}$ is the longitudinal pressure, $\rho =-T_{0}^{0}$ is the energy density, $T_{\mu}^{\nu}$ is the stress-energy tensor. When the surface of gluing approaches the horizon, surface stresses vanish, while $p_{r}$ may acquire a finite jump on the boundary. Such composite spacetimes accumulate an infinitely large amount of matter inside the horizon but reveal themselves for an external observer as a sphere of a finite ADM mass and size. If the throat of the inner region is glued to two black hole exteriors, one obtains a wormhole of an arbitrarily large length. Wormholes under discussion are static but not traversable, so the null energy condition is not violated. In particular, they include the case with an infinite proper distance to the throat. We construct also gravastars with an infinite tube as a core and traversable wormholes connected by a finite tube-like region. 
  In Parikh and Wilczek's original works, the laws of black hole thermodynamics are not referred and it seems that there is no relation between Hawking radiation via tunnelling and the laws of black hole thermodynamics in their works. However, taking examples for the R-N black hole and the Kerr black hole, we find that they are correlated and even consistent if the tunnelling process is a reversible process. 
  We examine the adiabatic approximation in the study of a relativistic two-body problem with the gravitational radiation reaction. We recently pointed out that the usual metric perturbation scheme using a perturbation of the stress-energy tensor may not be appropriate for study of the dissipative dynamics of the bodies due to the radiation reaction.   We recently proposed a possible approach to solve this problem with a linear black hole perturbation. This paper proposes a non-linear generalization of that method for a general application of this problem. We show that, under a specific gauge condition, the method actually allows us to avoid the gauge problem. 
  Quantum corrections to the Schwarzschild metric generated by loop diagrams have been considered by Bjerrum-Bohr, Donoghue, and Holstein (BHD) [Phys. Rev. D68, 084005 (2003)], and Khriplovich and Kirilin (KK) [J. Exp. Theor. Phys. 98, 1063 (2004)]. Though the same field variables in a covariant gauge are used, the results obtained differ from one another. The reason is that the different sets of diagrams have been used. Here we will argue that the quantum corrections to metric must be independent of the choice of field variables, i.e., must be reparametrization invariant. Using simple reparametrization  transformation, we will show that the contribution considered by BDH, is not invariant under it. Meanwhile the contribution of the complete set of the diagrams, considered by KK, satisfies the requirement of the invariance. 
  In this brief note we reply to the authors of a recent preprint in which an alleged explicit proposal of using the mean anomaly of the LAGEOS satellites to measure the general relativistic Lense-Thirring effect in the gravitational field of the Earth is attributed to the present author. 
  In a vacuum spacetime equipped with the Bondi's radiating metric which is asymptotically flat at spatial infinity including gravitational radiation ({\bf Condition D}), we establish the relation between the ADM total energy-momentum and the Bondi energy-momentum for perturbed radiative spatial infinity. The perturbation is given by defining the "real" time the sum of the retarded time, the Euclidean distance and certain function $f$. 
  We apply the warped product spacetime scheme to the Banados-Teitelboim-Zanelli black holes and the Reissner-Nordstr\"om-anti-de Sitter black hole to investigate their interior solutions in terms of warped products. It is shown that there exist no discontinuities of the Ricci and Einstein curvatures across event horizons of these black holes. 
  The symmetry method is used to derive solutions of Einstein's equations for fluid spheres using an isotropic metric and a velocity four vector that is non-comoving. Initially the Lie, classical approach is used to review and provide a connecting framework for many comoving and so shear free solutions. This provides the basis for the derivation of the classical point symmetries for the more general and mathematicaly less tractable description of Einstein's equations in the non-comoving frame. Although the range of symmetries is restrictive, existing and new symmetry solutions with non-zero shear are derived. The range is then extended using the non-classical direct symmetry approach of Clarkson and Kruskal and so additional new solutions with non-zero shear are also presented. The kinematics and pressure, energy density, mass function of these solutions are determined. 
  By using elementary astrophysical concepts, we show that for any self-luminous astrophysical object, the ratio of radiation energy density inside the body (rho_r) and the baryonic energy density (rho_0) may be crudely approximated, in the Newtonian limit, as rho_r/rho_0 ~ GM/Rc^2, where G is constant of gravitation, c is the speed of light, M is gravitational mass, and R is the radius of the body. The key idea is that radiation quanta must move out in a diffusive manner rather than free stream inside the body of the star. When one would move to the extreme General Realtivistic case i.e., if the surface gravitational redshift, z >> 1, it is found that, rho_r/rho_0 ~ (1+z). Previous works on gravitational collapse, however, generally assumed rho_r/rho_0 << 1. On the other hand, actually, during continued general relativistic gravitational collapse to the Black Hole state (z --> infty), the collapsing matter may essentially become an extremely hot fireball a la the very early universe even though the observed luminosity of the body as seen by a faraway observer, L^\infty ~ (1+z)^{-1} --> 0 as z --> infty, and the collapse might appear as ``adiabatic''. 
  We present results from fully nonlinear simulations of inspiralling, unequal mass binary black holes, concentrating on four cases with mass ratios q = M_2/M_1 = (1,0.96,0.85,0.54), or equivalently with reduced mass parameters eta = M_1M_2/(M_1+M_2)^2 = (0.25, 0.249, 0.248, 0.227). We show waveforms of the dominant l=2,3 modes. The power spectrum of these modes yields insight on how the mass ratio in a binary impacts the degree of complexity of the emitted waveforms. In addition, we provide approximate estimates of energy and angular momentum radiated as well as kick velocities from gravitational radiation recoil. 
  Certain semi-Riemannian metrics may be decomposed into a Riemannian part and an isochronal part. We use this idea and an idea of Kasner to construct a manifold in 6+1 Minkowski space with a well known metric. The full embedding we display is isochronal which simplifies visualizing the properties of the manifold. 
  Many theories are formulated as constrained systems. We provide a mechanism that explains the origin of physical states of a constrained system by a process of selection of noiseless subsystems when the system is coupled to an external environment. Effectively, physical states that solve all the constraints are selected by a passive error correction scheme which has been developed in the context of quantum information processing. We apply this mechanism to several constrained theories including the relativistic particle, electromagnetism, and quantum gravity, and discuss some interesting (and speculative) implications on the problem of time and the status of symmetries in nature. 
  In coalescing neutron star binaries, r-modes in one of the stars can be resonantly excited by the gravitomagnetic tidal field of its companion. This post-Newtonian gravitomagnetic driving of these modes dominates over the Newtonian tidal driving previously computed by Ho and Lai. To leading order in the tidal expansion parameter R/r (where R is the radius of the neutron star and r is the orbital separation), only the l=2, |m|= 1 and |m| = 2 r-modes are excited. The tidal work done on the star through this driving has an effect on the evolution of the inspiral and on the phasing of the emitted gravitational wave signal. For a neutron star of mass M, radius R, spin frequency f_spin, modeled as a Gamma =2 polytrope, with a companion also of mass M, the gravitational wave phase shift for the m=2 mode is (0.1radians)(R/10km)^4(M/1.4M_sun)^{-10/3}(f_spin/100Hz)^{2/3} for optimal spin orientation. For canonical neutron star parameters this phase shift will likely not be detectable by gravitational wave detectors such as LIGO, but if the neutron star radius is larger it may be detectable if the signal-to-noise ratio is moderately large. For neutron star - black hole binaries, the effect is smaller; the phase shift scales as companion mass to the -4/3 power for large companion masses. The net energy transfer from the orbit into the star is negative corresponding to a slowing down of the inspiral. This occurs because the interaction reduces the spin of the star, and occurs only for modes which satisfy the Chandrasekhar-Friedman-Schutz instability criterion. 
  We establish a new algorithm that generates a new solution to the Einstein field equations, with an anisotropic matter distribution, from a seed isotropic solution. The new solution is expressed in terms of integrals of an isotropic gravitational potential; and the integration can be completed exactly for particular isotropic seed metrics. A good feature of our approach is that the anisotropic solutions necessarily have an isotropic limit. We find two examples of anisotropic solutions which generalise the isothermal sphere and the Schwarzschild interior sphere. Both examples are expressed in closed form involving elementary functions only. 
  We have measured the mechanical loss of a dielectric multilayer reflective coating (ion-beam sputtered SiO$_2$ and Ta$_2$O$_5$) in cooled mirrors. The loss was nearly independent of the temperature (4 K $\sim$ 300 K), frequency, optical loss, and stress caused by the coating, and the details of the manufacturing processes. The loss angle was $(4 \sim 6) \times 10^{-4}$. The temperature independence of this loss implies that the amplitude of the coating thermal noise, which is a severe limit in any precise measurement, is proportional to the square root of the temperature. Sapphire mirrors at 20 K satisfy the requirement concerning the thermal noise of even future interferometric gravitational wave detector projects on the ground, for example, LCGT. 
  Even though the energy carried by a gravitational wave is not itself gauge invariant, the interaction with a gravitational antenna of the gravitational wave which carries that energy is. It therefore has to be possible to make some statements which involve the energy which are in fact gauge invariant, and it is the objective of this paper to provide them. In order to develop a gauge invariant treatment of the issues involved, we construct a specific action for gravitational fluctuations which is gauge invariant to second perturbative order. Then, via variation of this action, we obtain an energy-momentum tensor for perturbative gravitational fluctuations around a general curved background whose covariant conservation condition is also fully gauge invariant to second order. Contraction of this energy-momentum tensor with a Killing vector of the background conveniently allows us to convert this covariant conservation condition into an ordinary conservation condition which is also gauge invariant through second order. Then, via spatial integration we are able to obtain a relation involving the time derivative of the total energy of the fluctuation and its asymptotic spatial momentum flux which is also completely gauge invariant through second order. It is only in making the simplification of setting the asymptotic momentum flux to zero that one would actually lose manifest gauge invariance, with only invariance under those particular gauge transformations which leave the asymptotic momentum flux zero then remaining. However, if one works in an arbitrary gauge where the asymptotic momentum flux is non-zero, the gravitational wave will then deliver both energy and momentum to a gravitational antenna in a completely gauge invariant manner, no matter how badly behaved at infinity the gauge function might be. 
  In cosmology it has become usual to introduce new entities as dark matter and dark energy in order to explain otherwise unexplained observational facts. Here, we propose a different approach treating spacetime as a continuum endowed with properties similar to the ones of ordinary material continua, such as internal viscosity and strain distributions originated by defects in the texture. A Lagrangian modeled on the one valid for simple dissipative phenomena in fluids is built and used for empty spacetime. The internal "viscosity" is shown to correspond to a four-vector field. The vector field is shown to be connected with the displacement vector field induced by a point defect in a four-dimensional continuum. Using the known symmetry of the universe, assuming the vector field to be divergenceless and solving the corresponding Euler-Lagrange equation, we directly obtain inflation and a phase of accelerated expansion of spacetime. The only parameter in the theory is the "strength" of the defect. We show that it is possible to fix it in such a way to also quantitatively reproduce the acceleration of the universe. We have finally verified that the addition of ordinary matter does not change the general behaviour of the model. 
  I show that massive-particle dynamics can be simulated by a weak, spherical, external perturbation on a potential flow in an ideal fluid. The effective Lagrangian is of the form mc^2L(U^2/c^2), where U is the velocity of the particle relative to the fluid and c the speed of sound. This can serve as a model for emergent relativistic inertia a la Mach's principle with m playing the role of inertial mass, and also of analog gravity where it is also the passive gravitational mass. m depends on the particle type and intrinsic structure, while L is universal: For D dimensional particles L is proportional to the hypergeometric function F(1,1/2;D/2;U^2/c^2). Particles fall in the same way in the analog gravitational field independent of their internal structure, thus satisfying the weak equivalence principle. For D less or equal 5 they all have a relativistic limit with the acquired energy and momentum diverging as U approaches c. For D less or equal 7 the null geodesics of the standard acoustic metric solve our equation of motion. Interestingly, for D=4 the dynamics is very nearly Lorentzian. The particles can be said to follow the geodesics of a generalized acoustic metric of a Finslerian type that shares the null geodesics with the standard acoustic metric. In vortex geometries, the ergosphere is automatically the static limit. As in the real world, in ``black hole'' geometries circular orbits do not exist below a certain radius that occurs outside the horizon. There is a natural definition of antiparticles; and I describe a mock particle vacuum in whose context one can discuss, e.g., particle Hawking radiation near event horizons. 
  The Laser Astrometric Test Of Relativity (LATOR) is a Michelson-Morley-type experiment designed to achieve a major improvement in the accuracy of the tests of relativistic gravity in the solar system. By using a combination of independent time-series of gravitational deflection of light in the immediate proximity to the Sun, along with measurements of the relativistic time delay on interplanetary scales (to a precision respectively better than 0.1 picoradians and 1 cm), LATOR will measure the key post-Newtonian Eddington parameter \gamma with accuracy of one part in a billion - a factor of 30,000 improvement compared to the present best result, Cassini's 2003 test. LATOR's primary measurement pushes to unprecedented accuracy the search for cosmologically relevant scalar-tensor modifications of gravity by looking for a remnant scalar field in today's solar system. We present a comprehensive discussion of the science objectives, proposed technology, mission and optical designs, as well as the expected performance of this fundamental physics experiment in space. 
  This work presents the first application of the method of Genetic Algorithms (GAs) to data analysis for the Laser Interferometer Space Antenna (LISA). In the low frequency regime of the LISA band there are expected to be tens of thousands galactic binary systems that will be emitting gravitational waves detectable by LISA. The challenge of parameter extraction of such a large number of sources in the LISA data stream requires a search method that can efficiently explore the large parameter spaces involved. As signals of many of these sources will overlap, a global search method is desired. GAs represent such a global search method for parameter extraction of multiple overlapping sources in the LISA data stream. We find that GAs are able to correctly extract source parameters for overlapping sources. Several optimizations of a basic GA are presented with results derived from applications of the GA searches to simulated LISA data. 
  The possibility of reconstructing the whole history of the scale factor of the Universe from the power spectrum of relic gravitational waves (RGWs) makes the study of these waves quite interesting. First, we explore the impact of a hypothetical era -right after reheating- dominated by mini black holes and radiation that may lower the spectrum several orders of magnitude. Next, we calculate the power spectrum of the RGWs taking into account the present stage of accelerated expansion and an hypothetical second dust era. Finally, we study the generalized second law of gravitational thermodynamics applied to the present era of accelerated expansion of the Universe. 
  Quantum inequalities have been established for various quantum fields in both flat and curved spacetimes. In particular, for spin-3/2 fields, Yu and Wu have explicitly derived quantum inequalities for massive case. Employing the similar method developed by Fewster and colleagues, this paper provides an explicit formula of quantum inequalities for massless spin-3/2 field in four-dimensional Minkowski spacetime. 
  Computational techniques which establish the stability of an evolution-boundary algorithm for a model wave equation with shift are incorporated into a well-posed version of the initial-boundary value problem for gravitational theory in harmonic coordinates. The resulting algorithm is implemented as a 3-dimensional numerical code which we demonstrate to provide stable, convergent Cauchy evolution in gauge wave and shifted gauge wave testbeds. Code performance is compared for Dirichlet, Neumann and Sommerfeld boundary conditions and for boundary conditions which explicitly incorporate constraint preservation. The results are used to assess strategies for obtaining physically realistic boundary data by means of Cauchy-characteristic matching. 
  We derive the late-time behaviour of tilted Bianchi VII_0 cosmologies with an irrotational radiation fluid as source, and give the asymptotic form of the general solution as $t \to +\infty$, making comparisons with the dust-filled models. At first sight the radiation-filled models appear to approximate the flat FL model at late times, since the Hubble-normalized shear and the tilt tend to zero and the density parameter tends to one. The Hubble-normalized Weyl curvature diverges, however, indicating that physically significant anisotropy remains. We also discuss the influence of a cosmological constant on this phenomenon. 
  Black hole entropy appears to be ``universal''--many independent calculations, involving models with very different microscopic degrees of freedom, all yield the same density of states. I discuss the proposal that this universality comes from the behavior of the underlying symmetries of the classical theory. To impose the condition that a black hole be present, we must partially break the classical symmetries of general relativity, and the resulting Goldstone boson-like degrees of freedom may account for the Bekenstein-Hawking entropy. In particular, I demonstrate that the imposition of a ``stretched horizon'' constraint modifies the algebra of symmetries at the horizon, allowing the use of standard conformal field theory techniques to determine the asymptotic density of states. The results reproduce the Bekenstein-Hawking entropy without any need for detailed assumptions about the microscopic theory. 
  This paper shows that a hyperbolic equation for heat conduction can be obtained directly using the tenets of linear irreversible thermodynamics in the context of the five dimensional space-time metric originally proposed by T. Kaluza back in 1922. The associated speed of propagation is slightly lower than the speed of light by a factor inversely proportional to the specific charge of the fluid element. Moreover, consistency with the second law of thermodynamics is achieved. Possible implications in the context of physics of clusters of galaxies of this result are briefly discussed. 
  Freeman Dyson has questioned whether any conceivable experiment in the real universe can detect a single graviton. If not, is it meaningful to talk about gravitons as physical entities? We attempt to answer Dyson's question and find it is possible concoct an idealized thought experiment capable of detecting one graviton; however, when anything remotely resembling realistic physics is taken into account, detection becomes impossible, indicating that Dyson's conjecture is very likely true. We also point out several mistakes in the literature dealing with graviton detection and production. 
  Questions about black holes in quantum gravity generally presuppose the presence of a horizon. Recently Carlip has shown that enforcing an initial data surface to be a horizon leads to the correct form for the Bekenstein-Hawking entropy of the black hole. Requiring a horizon also constitutes fixed background geometry, which generically leads to non-conservation of the matter stress tensor at the horizon. In this work, I show that the generated matter energy flux for a Schwarzschild black hole is in agreement with the first law of black hole thermodynamics, $8 \pi G \Delta Q = \kappa \Delta A$. 
  We previously showed that we can use di quark pairs as a model of how nucleation of a new universe occurs. We now can construct a model showing evolution from a dark matter dark energy mix to a pure cosmological constant cosmology due to changes in the slope of the resulting scalar field,using much of Scherrer's k-essence model.This same construction permits a use of the speed of sound,in k-essence models evolving from zero to one. Having the sound speed eventually reach unity permits matching conventional cosmological observations in the aftermath of change of slope of a di quark pair generated scalar field during the nucleation process of a new universe. These results are consistent with applying Bunyi and Hu's semi classical criteria for cosmological potentials to indicate a phase transition alluded to by Dr. Edward Kolbs model of how the initial degrees of freedom declined from over 100 to something approaching what we see today in flat space cosmology. 
  We construct approximate initial data for non-spinning black hole binary systems by asymptotically matching the 4-metrics of two tidally perturbed Schwarzschild solutions in isotropic coordinates to a resummed post-Newtonian 4-metric in ADMTT coordinates. The specific matching procedure used here closely follows the calculation in gr-qc/0503011, and is performed in the so called buffer zone where both the post-Newtonian and the perturbed Schwarzschild approximations hold. The result is that both metrics agree in the buffer zone, up to the errors in the approximations. However, since isotropic coordinates are very similar to ADMTT coordinates, matching yields better results than in the previous calculation, where harmonic coordinates were used for the post-Newtonian 4-metric. In particular, not only does matching improve in the buffer zone, but due to the similarity between ADMTT and isotropic coordinates the two metrics are also close to each other near the black hole horizons. With the help of a transition function we also obtain a global smooth 4-metric which has errors on the order of the error introduced by the more accurate of the two approximations we match. This global smoothed out 4-metric is obtained in ADMTT coordinates which are not horizon penetrating. In addition, we construct a further coordinate transformation that takes the 4-metric from global ADMTT coordinates to new coordinates which are similar to Kerr-Schild coordinates near each black hole, but which remain ADMTT further away from the black holes. These new coordinates are horizon penetrating and lead, for example, to a lapse which is everywhere positive on the t=0 slice. Such coordinates may be more useful in numerical simulations. 
  It is shown that geometric optical description of electromagnetic wave with account of its polarization in curved space-time can be obtained straightforwardly from the classical variational principle for electromagnetic field. For this end the entire functional space of electromagnetic fields must be reduced to its subspace of locally plane monochromatic waves. We have formulated the constraints under which the entire functional space of electromagnetic fields reduces to its subspace of locally plane monochromatic waves. These constraints introduce variables of another kind which specify a field of local frames associated to the wave and contain some congruence of null-curves. The Lagrangian for constrained electromagnetic field contains variables of two kinds, namely, a congruence of null-curves and the field itself. This yields two kinds of Euler-Lagrange equations. Equations of first kind are trivial due to the constraints imposed. Variation of the curves yields the Papapetrou equations for a classical massless particle with helicity 1. 
  The article deals with the results of the research on the realization of the scientific and technical project Dulkyn for detection of low frequency periodic gravitational radiation from the double relativistic astrophysical objects, done by the Scientific Centre of Gravitational-Wave Research Dulkyn of the Academy of Sciences of the Tatarstan Republic (SC GWR Dulkyn AS RT), and the Federal State Unitary Enterprise and R and D Corporation called State Institute of Applied Optics (FSUE NPO GIPO). 
  In this work, gravitational collapse of a spherical cloud, consists of both dark matter and dark energy in the form of modified Chaplygin gas is studied. It is found that dark energy alone in the form of modified Chaplygin gas forms black hole. Also when both components of the fluid are present then the collapse favors the formation of black hole in cases the dark energy dominates over dark matter. The conclusion is totally opposite to the usually known results. 
  In this thesis, we have reviewed how the expansion of the Universe amplifies the quantum vacuum fluctuations, and how the relic GWs spectrum is related with the scale factor. We have evaluated the spectrum in different scenarios: a model of expanding universe with an era dominated by mini black holes and radiation right after the inflation and a model with an era of accelerated expansion right after the radiation era. Next, we have applied the generalized second law of thermodynamics to the latter model. Finally, we have extended the GSL study to a single stage universe model dominated by dark energy (either phantom or not). 
  The examples of ten-dimensional vacuum Einstein spaces composed of four-dimensional Einstein spaces and six-dimensional Ricci-flat base space defined by the solutions of the Korteveg de-Vries equation are constructed. 
  In this work we develop a geometrical unification theory for gravity and the electro-weak model in a Kaluza-Klein approach; in particular, from the curvature dimensional reduction Einstein-Yang-Mills action is obtained. We consider two possible space-time manifolds: 1)$V^{4}\otimes S^{1}\otimes S^{2}$ where isospin doublets are identified with spinors; 2) $V^{4}\otimes S^{1}\otimes S^{3}$ in which both quarks and leptons doublets can be recast into the same spinor, such that the equal number of quark generations and leptonic families is explained. Finally a self-interacting complex scalar field is introduced to reproduce the spontaneous symmetry breaking mechanism; in this respect, at the end we get an Higgs fields whose two components have got opposite hypercharges. 
  We study gravitational lensing by compact objects in gravity theories that can be written in a Post-Post-Newtonian (PPN) framework: i.e., the metric is static and spherically symmetric, and can be written as a Taylor series in m/r, where m is the gravitational radius of the compact object. Working invariantly, we compute corrections to standard weak-deflection lensing observables at first and second order in the ratio of the angular gravitational radius to the angular Einstein ring radius of the lens. We show that the first-order corrections to the total magnification and centroid position vanish universally for gravity theories that can be written in the PPN framework. This arises from some surprising, fundamental relations among the lensing observables in PPN gravity models. We derive these relations for the image positions, magnifications, and time delays. A deep consequence is that any violation of the universal relations would signal the need for a gravity model outside the PPN framework (provided that some basic assumptions hold). In practical terms, the relations will guide observational programs to test general relativity, modified gravity theories, and possibly the Cosmic Censorship conjecture. We use the new relations to identify lensing observables that are accessible to current or near-future technology, and to find combinations of observables that are most useful for probing the spacetime metric. We give explicit applications to the Galactic black hole, microlensing, and the binary pulsar J0737-3039. 
  The consequences of spin-rotation-gravity coupling are worked out for linear gravitational waves. The coupling of helicity of the wave with the rotation of a gravitational-wave antenna is investigated and the resulting modifications in the Doppler effect and aberration are pointed out for incident high-frequency gravitational radiation. Extending these results to the case of a gravitomagnetic field via the gravitational Larmor theorem, the rotation of linear polarization of gravitational radiation propagating in the field of a rotating mass is studied. It is shown that in this case the linear polarization state rotates by twice the Skrotskii angle as a consequence of the spin-2 character of linear gravitational waves. 
  In this paper we investigate the effects that an anomalous acceleration as that experienced by the Pioneer spacecraft after they passed the 20 AU threshold would induce on the orbital motions of the Solar System planets placed at heliocentric distances of 20 AU or larger as Uranus, Neptune and Pluto. It turns out that such an acceleration, with a magnitude of 8.74\times 10^-10 m s^-2, would affect their orbits with secular and short-period signals large enough to be detected according to the latest published results by E.V. Pitjeva, even by considering errors up to 30 times larger than those released. The absence of such anomalous signatures in the latest data rules out the possibility that in the region 20-40 AU of the Solar System an anomalous force field inducing a constant and radial acceleration with those characteristics affects the motion of the major planets. 
  We investigate the conditions under which cosmological variations in physical `constants' and scalar fields are detectable on the surface of local gravitationally-bound systems, such as planets, in non-spherically symmetric background spacetimes. The method of matched asymptotic expansions is used to deal with the large range of length scales that appear in the problem. We derive a sufficient condition for the local time variation of the scalar fields driving variations in 'constants' to track their large-scale cosmological variation and show that this is consistent with our earlier conjecture derived from the spherically symmetric problem. We perform our analysis with spacetime backgrounds that are of Szekeres-Szafron type. They are approximately Schwarzschild in some locality and free of gravitational waves everywhere. At large distances, we assume that the spacetime matches smoothly onto a Friedmann background universe. We conclude that, independent of the details of the scalar-field theory describing the varying `constant', the condition for its cosmological variations to be measured locally is almost always satisfied in physically realistic situations. The very small differences expected to be observed between different scales are quantified. This strengthens the proof given in our previous paper that local experiments see global variations by dropping the requirement of exact spherical symmetry. It provides a rigorous justification for using terrestrial experiments and solar system observations to constraint or detect any cosmological time variations in the traditional `constants' of Nature in the case where non-spherical inhomogeneities exist. 
  Exact solutions are found for the Chandrasekhar Page angular equation which results when the Dirac equation in a Kerr Newman space time is separated into its radial and angular parts. The solutions turn out to be remarkably simple in form while satisfying the asymptotic conditions deduced earlier. The eigenvalues are found to be the square root of the total angular momentum as first found by Dirac for flat space; supplemented by a term which is the product of the mass of the Dirac particle times the specific angular momentum of the black hole. The additional contribution is what is expected from frame dragging. 
  Black holes can be practically located (e.g. in numerical simulations) by trapping horizons, hypersurfaces foliated by marginal surfaces, and one desires physically sound measures of their mass and angular momentum. A generically unique angular momentum can be obtained from the Komar integral by demanding that it satisfy a simple conservation law. With the irreducible (Hawking) mass as the measure of energy, the conservation laws of energy and angular momentum take a similar form, expressing the rate of change of mass and angular momentum of a black hole in terms of fluxes of energy and angular momentum, obtained from the matter energy tensor and an effective energy tensor for gravitational radiation. Adding charge conservation for generality, one can use Kerr-Newman formulas to define combined energy, surface gravity, angular speed and electric potential, and derive a dynamical version of the so-called "first law" for black holes. A generalization of the "zeroth law" to local equilibrium follows. Combined with an existing version of the "second law", all the key quantities and laws of the classical paradigm for black holes (in terms of Killing or event horizons) have now been formulated coherently in a general dynamical paradigm in terms of trapping horizons. 
  Using exact solutions, we show that it is in principle possible to regard waves and particles as representations of the same underlying geometry, thereby resolving the problem of wave-particle duality. 
  This paper has been addressed to a very old but burning problem of energy in General Relativity. We evaluate energy and momentum densities for the static and axisymmetric solutions. This specializes to two metrics, i.e., Erez-Rosen and the gamma metrics, belonging to the Weyl class. We apply four well-known prescriptions of Einstein, Landau-Lifshitz, Papaterou and M$\ddot{o}$ller to compute energy-momentum density components. We obtain that these prescriptions do not provide similar energy density, however momentum becomes constant in each case. The results can be matched under particular boundary conditions. 
  Neutron stars are discussed as laboratories of physics of strong gravitational fields. The mass of a neutron star is split into matter energy and gravitational field energy contributions. The energy of the gravitational field of neutron stars is calculated with three different approaches which give the same result. It is found that up to one half of the gravitational mass of maximum mass neutron stars is comprised by the gravitational field energy. Results are shown for a number of realistic equations of state of neutron star matter. 
  We study the collapse towards the gravitational radius of a macroscopic spherical thick shell surrounding an inner massive core. This overall electrically neutral macroshell is composed by many nested delta-like massive microshells which can bear non-zero electric charge, and a possibly non-zero cosmological constant is also included. The dynamics of the shells is described by means of Israel's (Lanczos) junction conditions for singular hypersurfaces and, adopting a Hartree (mean field) approach, an effective Hamiltonian for the motion of each microshell is derived which allows to check the stability of the matter composing the macroshell. We end by briefly commenting on the quantum effects which may arise from the extension of our classical treatment to the semiclassical level. 
  Within the framework of metric-affine gravity (MAG, metric and an independent linear connection constitute spacetime), we find, for a specific gravitational Lagrangian and by using {\it prolongation} techniques, a stationary axially symmetric exact solution of the vacuum field equations. This black hole solution embodies a Kerr-deSitter metric and the post-Riemannian structures of torsion and nonmetricity. The solution is characterized by mass, angular momentum, and shear charge, the latter of which is a measure for violating Lorentz invariance. 
  This thesis is concerned with formulations of the Einstein equations in axisymmetric spacetimes which are suitable for numerical evolutions. We develop two evolution systems based on the (2+1)+1 formalism. The first is a (partially) constrained scheme with elliptic gauge conditions arising from maximal slicing and conformal flatness. The second is a strongly hyperbolic first-order formulation obtained by combining the (2+1)+1 formalism with the Z4 formalism. A careful study of the behaviour of regular axisymmetric tensor fields enables us to cast the equations in a form that is well-behaved on the axis. Further topics include (non)uniqueness of solutions to the elliptic equations arising in constrained schemes, and comparisons between various boundary conditions used in numerical relativity. The numerical implementation is applied to adaptive evolutions of nonlinear Brill waves, including twist. 
  Higher-dimensional theories of the kind which may unify gravitation with particle physics can lead to significant modifications of general relativity. In five dimensions, the vacuum becomes non-standard, and the Weak Equivalence Principle becomes a geometrical symmetry which can be broken, perhaps at a level detectable by new tests in space. 
  A numerical simulation of fluid flows in a Laval nozzle is performed to observe formations of acoustic black holes and the classical counterpart to Hawking radiation under a realistic setting of the laboratory experiment. We determined the Hawking temperature of the acoustic black hole from obtained numerical data. Some noteworthy points in analyzing the experimental data are clarified through our numerical simulation. 
  Quantum Cosmology and Gravity are formulated here as the primary and secondary quantizations of the energy constraints by analogy with the historical formulation of quantum field theory. New fact is that both the Universe and its matter are created from stable vacuum obtained by the Bogoliubov-type transformation just as it is in the theory of quantum superfluid liquid. Such the Quantum Gravity gives us possibility to explain topical problems of cosmology by the cosmological creation of universes and particles from vacuum. 
  It is shown that four dimensional vacuum Einstein solutions simply embedded in five dimensions obey the Gauss-Bonnet-Einstein field equations: $G_{ab}+\alpha GB_{ab}+\delta^{55}_{ab}\alpha\exp(-2\chi/\sqrt{\alpha})GB_4=0$ and the Pauli-Einstein equations $G_{ab}-3\alpha P_{ab}/5=0$, and the Bach-Einstein equations $B_{ab}=0$. General equations are calculated for which these and similar results follow. It is briefly argued that such field equations could be significant on large distance scales. 
  A criticism sometimes made of the causal set quantum gravity program is that there is no practical scheme for identifying manifoldlike causal sets and finding embeddings of them into manifolds. A computational method for constructing an approximate embedding of a small manifoldlike causal set into Minkowski space (or any spacetime that is approximately flat at short scales) is given, and tested in the 2 dimensional case. This method can also be used to determine how manifoldlike a causal set is, and conversely to define scales of manifoldlikeness. 
  The energy distribution in the Locally Rotationally Symmetric (LRS) Bianchi type II space-time is obtained by considering the Moller energy-momentum definition in both Einstein's theory of general relativity and teleparallel theory of relativity. The energy distribution which includes both the matter and gravitational field is found to be zero in both of these different gravitation theories. This result agrees with previous works of Cooperstock and Israelit, Rosen, Johri et al., Banerjee and Sen, Vargas, and Aydogdu and Salti. Our result that the total energy of the universe is zero supports the view points of Albrow and Tryon. 
  We introduce a linearized bi-metric theory of gravity with two metrics. The metric g_{ab} describes null hypersurfaces of the gravitational field while light moves on null hypersurfaces of the optical metric \bar{g}_{ab}. Bi-metrism naturally arises in vector-tensor theories with matter being non-minimally coupled to gravity via long-range vector field. We derive explicit Lorentz-invariant solution for a light ray propagating in space-time of the bi-metric theory and disentangle relativistic effects associated with the two metrics. This anlysis can be valuable for future spaceborne laser missions ASTROD and LATOR dedicated to map various relativistic gravity parameters in the solar system to unparalleled degree of accuracy. 
  We analyze the problem of parameter estimation for compact binary systems that could be detected by ground-based gravitational wave detectors.   So far this problem has only been dealt with for the inspiral and the ringdown phases separately. In this paper, we combine the information from both signals, and we study the improvement in parameter estimation, at a fixed signal-to-noise ratio, by including the ringdown signal without making any assumption on the merger phase. The study is performed for both initial and advanced LIGO and VIRGO detectors. 
  The purpose of this paper is to discuss the various types of physical universe which could exist according to modern mathematical physics. The paper begins with an introduction that approaches the question from the viewpoint of ontic structural realism. Section 2 takes the case of the 'multiverse' of spatially homogeneous universes, and analyses the famous Collins-Hawking argument, which purports to show that our own universe is a very special member of this collection. Section 3 considers the multiverse of all solutions to the Einstein field equations, and continues the discussion of whether the notions of special and typical can be defined within such a collection. 
  We describe plane-fronted waves in the Yang-Mills type quadratic metric-affine theory of gravity. The torsion and the nonmetricity are both nontrivial, and they do not belong to the triplet ansatz. 
  Recent theoretical work suggests that violation of the Equivalence Principle might be revealed in a measurement of the fractional differential acceleration $\eta$ between two test bodies -of different composition, falling in the gravitational field of a source mass- if the measurement is made to the level of $\eta\simeq 10^{-13}$ or better. This being within the reach of ground based experiments, gives them a new impetus. However, while slowly rotating torsion balances in ground laboratories are close to reaching this level, only an experiment performed in low orbit around the Earth is likely to provide a much better accuracy.   We report on the progress made with the "Galileo Galilei on the Ground" (GGG) experiment, which aims to compete with torsion balances using an instrument design also capable of being converted into a much higher sensitivity space test.   In the present and following paper (Part I and Part II), we demonstrate that the dynamical response of the GGG differential accelerometer set into supercritical rotation -in particular its normal modes (Part I) and rejection of common mode effects (Part II)- can be predicted by means of a simple but effective model that embodies all the relevant physics. Analytical solutions are obtained under special limits, which provide the theoretical understanding. A simulation environment is set up, obtaining quantitative agreement with the available experimental data on the frequencies of the normal modes, and on the whirling behavior. This is a needed and reliable tool for controlling and separating perturbative effects from the expected signal, as well as for planning the optimization of the apparatus. 
  The detection and estimation of gravitational wave burst signals, with {\em a priori} unknown polarization waveforms, requires the use of data from a network of detectors. For determining how the data from such a network should be combined, approaches based on the maximum likelihood principle have proven to be useful. The most straightforward among these uses the global maximum of the likelihood over the space of all waveforms as both the detection statistic and signal estimator. However, in the case of burst signals, a physically counterintuitive situation results: for two aligned detectors the statistic includes the cross-correlation of the detector outputs, as expected, but this term disappears even for an infinitesimal misalignment. This {\em two detector paradox} arises from the inclusion of improbable waveforms in the solution space of maximization. Such waveforms produce widely different responses in detectors that are closely aligned. We show that by penalizing waveforms that exhibit large signal-to-noise ratio (snr) variability, as the corresponding source is moved on the sky, a physically motivated restriction is obtained that (i) resolves the two detector paradox and (ii) leads to a better performing statistic than the global maximum of the likelihood. Waveforms with high snr variability turn out to be precisely the ones that are improbable in the sense mentioned above. The coherent network analysis method thus obtained can be applied to any network, irrespective of the number or the mutual alignment of detectors. 
  We provide some considerations on the excitation of black hole quasinormal modes (QNMs) in different physical scenarios. Considering a simple model in which a stream of particles accretes onto a black hole, we show that resonant QNM excitation by hyperaccretion requires a significant amount of fine-tuning, and is quite unlikely to occur in nature. Then we summarize and discuss present estimates of black hole QNM excitation from gravitational collapse, distorted black holes and head-on black hole collisions. We emphasize the areas that, in our opinion, are in urgent need of further investigation from the point of view of gravitational wave source modeling. 
  Brane-world models, where observers are restricted to a brane in a higher-dimensional spacetime, offer a novel perspective on cosmology. I discuss some approaches to cosmology in extra dimensions and some interesting aspects of gravity and cosmology in brane-world models. 
  Using the action principle we first review how linear density perturbations (sound waves) in an Eulerian fluid obey a relativistic equation: the d'Alembert equation. This analogy between propagation of sound and that of a massless scalar field in a Lorentzian metric also applies to non-homogeneous flows. In these cases, sound waves effectively propagate in a curved four-dimensional ''acoustic'' metric whose properties are determined by the flow. Using this analogy, we consider regular flows which become supersonic, and show that the acoustic metric behaves like that of a black hole. The analogy is so good that, when considering quantum mechanics, acoustic black holes should produce a thermal flux of Hawking phonons.   We then focus on two interesting questions related to Hawking radiation which are not fully understood in the context of gravitational black holes due to the lack of a theory of quantum gravity. The first concerns the calculation of the modifications of Hawking radiation which are induced by dispersive effects at short distances, i.e., approaching the atomic scale when considering sound. We generalize existing treatments and calculate the modifications caused by the propagation near the black hole horizon. The second question concerns backreaction effects. We return to the Eulerian action, compute second order effects, and show that the backreaction of sound waves on the fluid's flow can be expressed in terms of their stress-energy tensor. Using this result in the context of Hawking radiation, we compute the secular effect on the background flow. 
  We present a class of spherically symmetric hypersurfaces in the  Kruskal extension of the Schwarzschild space-time. The hypersurfaces have constant negative scalar curvature, so they are hyperboloidal in the regions of space-time which are asymptotically flat. 
  We describe an improved version of the Hough transform search for continuous gravitational waves from isolated neutron stars assuming the input to be short segments of Fourier transformed data. The method presented here takes into account possible non-stationarities of the detector noise and the amplitude modulation due to the motion of the detector. These two effects are taken into account for the first stage only, i.e. the peak selection, to create the time-frequency map of our data, while the Hough transform itself is performed in the standard way. 
  Using a Hamiltonian formulation of the spherically symmetric gravity-scalar field theory adapted to flat spatial slicing, we give a construction of the reduced Hamiltonian operator. This Hamiltonian, together with the null expansion operators presented in an earlier work, form a framework for studying gravitational collapse in quantum gravity. We describe a setting for its numerical implementation, and discuss some conceptual issues associated with quantum dynamics in a partial gauge fixing. 
  Using methods of Quantum Field Theory in curved spacetime, the first order in hbar quantum corrections to the motion of a fluid in an acoustic black hole configuration are numerically computed. These corrections arise from the non linear backreaction of the emitted phonons. Time dependent (isolated system) and equilibrium configurations (hole in a sonic cavity) are both analyzed. 
  Acoustic black holes are very interesting non-gravitational objects which can be described by the geometrical formalism of General Relativity. These models can be useful to experimentally test effects otherwise undetectable, as for example the Hawking radiation. The back-reaction effects on the background quantities induced by the analogue Hawking radiation could be the key to indirectly observe it. We briefly show how this analogy works and derive the backreaction equations for the linearized quantum fluctuations in the background of an acoustic black hole. A first order in hbar solution is given in the near horizon region. It indicates that acoustic black holes, unlike Schwarzschild ones, get cooler as they radiate phonons. They show remarkable analogies with near-extremal Reissner-Nordstrom black holes. 
  Quantum gravity is expected to be necessary in order to understand situations where classical general relativity breaks down. In particular in cosmology one has to deal with initial singularities, i.e. the fact that the backward evolution of a classical space-time inevitably comes to an end after a finite amount of proper time. This presents a breakdown of the classical picture and requires an extended theory for a meaningful description. Since small length scales and high curvatures are involved, quantum effects must play a role. Not only the singularity itself but also the surrounding space-time is then modified. One particular realization is loop quantum cosmology, an application of loop quantum gravity to homogeneous systems, which removes classical singularities. Its implications can be studied at different levels. Main effects are introduced into effective classical equations which allow to avoid interpretational problems of quantum theory. They give rise to new kinds of early universe phenomenology with applications to inflation and cyclic models. To resolve classical singularities and to understand the structure of geometry around them, the quantum description is necessary. Classical evolution is then replaced by a difference equation for a wave function which allows to extend space-time beyond classical singularities. One main question is how these homogeneous scenarios are related to full loop quantum gravity, which can be dealt with at the level of distributional symmetric states. Finally, the new structure of space-time arising in loop quantum gravity and its application to cosmology sheds new light on more general issues such as time. 
  The introduction of an interaction for dark energy to the standard cosmology offers a potential solution to the cosmic coincidence problem. We examine the conditions on the dark energy density that must be satisfied for this scenario to be realized. Under some general conditions we find a stable attractor for the evolution of the Universe in the future. Holographic conjectures for the dark energy offer some specific examples of models with the desired properties. 
  The theory of canonical linearized gravity is quantized using the Projection Operator formalism, in which no gauge or coordinate choices are made. The ADM Hamiltonian is used and the canonical variables and constraints are expanded around a flat background. As a result of the coordinate independence and linear truncation of the perturbation series, the constraint algebra surprisingly becomes partially second-class in both the classical and quantum pictures after all secondary constraints are considered. While new features emerge in the constraint structure, the end result is the same as previously reported: the (separable) physical Hilbert space still only depends on the transverse-traceless degrees of freedom. 
  This paper delineates the first steps in a systematic quantitative study of the spacetime fluctuations induced by quantum fields in an evaporating black hole. We explain how the stochastic gravity formalism can be a useful tool for that purpose within a low-energy effective field theory approach to quantum gravity. As an explicit example we apply it to the study of the spherically-symmetric sector of metric perturbations around an evaporating black hole background geometry. For macroscopic black holes we find that those fluctuations grow and eventually become important when considering sufficiently long periods of time (of the order of the evaporation time), but well before the Planckian regime is reached. In addition, the assumption of a simple correlation between the fluctuations of the energy flux crossing the horizon and far from it, which was made in earlier work on spherically-symmetric induced fluctuations, is carefully analyzed and found to be invalid. Our analysis suggests the existence of an infinite amplitude for the fluctuations of the horizon as a three-dimensional hypersurface. We emphasize the need for understanding and designing operational ways of probing quantum metric fluctuations near the horizon and extracting physically meaningful information. 
  We consider a Weyssenhoff fluid assuming that the spacetime is homogeneous and isotropic, therefore being relevant for cosmological considerations of gravity theories with torsion. In this paper, it is explicitely shown that the Weyssenhoff fluids obeying the Frenkel condition or the Papapetrou-Corinaldesi condition are incompatible with the cosmological principle, which restricts the torsion tensor to have only a vector and an axial vector component. Moreover it turns out that the Weyssenhoff fluid obeying the Tulczyjew condition is also incompatible with the cosmological principle. Based on this result we propose to reconsider a number of previous works that analysed cosmological solutions of Einstein-Cartan theory, since their spin fluids usually did not obey the cosmological principle. 
  In the gauge theory of gravity based on the Poincare group (the semidirect product of the Lorentz group and the spacetime translations) the mass (energy-momentum) and the spin are treated on an equal footing as the sources of the gravitational field. The corresponding spacetime manifold carries the Riemann-Cartan geometric structure with the nontrivial curvature and torsion. We describe some aspects of the classical Poincare gauge theory of gravity. Namely, the Lagrange-Noether formalism is presented in full generality, and the family of quadratic (in the curvature and the torsion) models is analyzed in detail. We discuss the special case of the spinless matter and demonstrate that Einstein's theory arises as a degenerate model in the class of the quadratic Poincare theories. Another central point is the overview of the so-called double duality method for constructing of the exact solutions of the classical field equations. 
  We have used our new technique for fully numerical evolutions of orbiting black-hole binaries without excision to model the last orbit and merger of an equal-mass black-hole system. We track the trajectories of the individual apparent horizons and find that the binary completed approximately one and a third orbits before forming a common horizon. Upon calculating the complete gravitational radiation waveform, horizon mass, and spin, we find that the binary radiated 3.2% of its mass and 24% of its angular momentum. The early part of the waveform, after a relatively short initial burst of spurious radiation, is oscillatory with increasing amplitude and frequency, as expected from orbital motion. The waveform then transitions to a typical `plunge' waveform; i.e. a rapid rise in amplitude followed by quasinormal ringing. The plunge part of the waveform is remarkably similar to the waveform from the previously studied `ISCO' configuration. We anticipate that the plunge waveform, when starting from quasicircular orbits, has a generic shape that is essentially independent of the initial separation of the binary. 
  In this paper, we study the effects of Generalized Uncertainty Principle(GUP) and Modified Dispersion Relations(MDRs) on the thermodynamics of ultra-relativistic particles in early universe. We show that limitations imposed by GUP and particle horizon on the measurement processes, lead to certain modifications of early universe thermodynamics. 
  We study Aichelburg-Sexl ultrarelativistic limits in higher dimensions. After reviewing the boost of D>=4 Reissner-Nordstrom black holes as a simple illustrative example, we consider the case of D=5 black rings, presenting new results for static charged rings. 
  Explicit expressions for the quasi-Kinnersley tetrad for the quasi-Kerr metric are given. These provide a very clear and simple example of the quasi-Kinnersley tetrad, and may be useful in the future development of a `quasi-Teukolsky' scheme for the analysis of perturbation equations in spacetimes which are Petrov type I but in some sense close to type D. 
  The problem of background independent quantum gravity is the problem of defining a quantum field theory of matter and gravity in the absence of an underlying background geometry. Loop quantum gravity (LQG) is a promising proposal for addressing this difficult task. Despite the steady progress of the field, dynamics remains to a large extend an open issue in LQG. Here we present the main ideas behind a series of proposals for addressing the issue of dynamics. We refer to these constructions as the {\em spin foam representation} of LQG. This set of ideas can be viewed as a systematic attempt at the construction of the path integral representation of LQG.   The {\em spin foam representation} is mathematically precise in 2+1 dimensions, so we will start this chapter by showing how it arises in the canonical quantization of this simple theory. This toy model will be used to precisely describe the true geometric meaning of the histories that are summed over in the path integral of generally covariant theories.   In four dimensions similar structures appear. We call these constructions {\em spin foam models} as their definition is incomplete in the sence that at least one of the following issues remains unclear: 1) the connection to a canonical formulation, and 2) regularization independence (renormalizability). In the second part of this chapter we will describe the definition of these models emphasizing the importance of these open issues. We also discuss the non standard picture of quantum spacetime that follows from background independence. 
  This paper reports on the methods and results of a theoretical analysis to design an insulator which must provide a thermally quiet environment to test on ground delicate temperature sensors and associated electronics. These will fly on board ESA's LISA PathFinder (LPF) mission as part of the thermal diagnostics subsystem of the LISA Test-flight Package (LTP). We evaluate the heat transfer function (in frequency domain) of a central body of good thermal conductivity surrounded by a layer of a very poorly conducting substrate. This is applied to assess the materials and dimensions necessary to meet temperature stability requirements in the metal core, where sensors will be implanted for test. The analysis is extended to evaluate the losses caused by heat leakage through connecting wires, linking the sensors with the electronics in a box outside the insulator. The results indicate that, in spite of the very demanding stability conditions, a sphere of outer diameter of the order one metre is sufficient. 
  The relevance of the Planck scale to a theory of quantum gravity has become a worryingly little examined assumption that goes unchallenged in the majority of research in this area. However, in all scientific honesty, the significance of Planck's natural units in a future physical theory of spacetime is only a plausible, yet by no means certain, assumption. The purpose of this article is to clearly separate fact from belief in this connection. 
  Higher dimensional, direct analogues of the usual d=4 Einstein--Yang-Mills (EYM) systems are studied. These consist of the gravitational and Yang-Mills hierarchies in d=4p dimensional spacetimes, both consisting of 2p-form curvature terms only. Regular and black hole solutions are constructed in $2p+2\le d \le 4p$, in which dimensions the total mass-energy is finite, generalising the familiar Bartnik-McKinnon solutions in EYM theory for p=1. In d=4p, this similarity is complete. In the special case of d=2p+1, just beyond the finite energy range of d, exact solutions in closed form are found. Finally, d=2p+1 purely gravitational systems, whose solutions generalise the static d=3 BTZ solutions, are discussed. 
  We study a space-time finite element approach for the nonhomogeneous wave equation using a continuous time Galerkin method. We present fully implicit examples in 1+1, 2+1, and 3+1 dimensions using linear quadrilateral, hexahedral, and tesseractic elements. Krylov solvers with additive Schwarz preconditioning are used for solving the linear system. We introduce a time decomposition strategy in preconditioning which significantly improves performance when compared with unpreconditioned cases. 
  We consider a modified form of gravity in which the action contains a power alpha of the scalar curvature. It is shown how the presence of a bulk viscosity in a spatially flat universe may drive the cosmic fluid into the phantom region (w<-1) and thus into a Big Rip singularity, even if it lies in the quintessence region (w>-1) in the non-viscous case. The condition for this to occur is that the bulk viscosity contains the power (2 alpha-1) of the scalar expansion. Two specific examples are discussed in detail. The present paper is a generalization of the recent investigation dealing with barrier crossing in Einstein's gravity: I. Brevik and O. Gorbunova, Gen. Relativ. Grav. 37 (2005) 2039. 
  Our purpose is to recast KK model in terms of ADM variables. We examine and solve the problem of the consistency of this approach, with particular care about the role of the cylindricity hypothesis. We show in details how the KK reduction commutes with the ADM slicing procedure and how this leads to a well defined and unique ADM reformulation. This allows us to consider the hamiltonian formulation of the model and can be the first step for the Ashtekar reformulation of the KK scheme. Moreover we show how the time component of the gauge vector arises naturally from the geometrical constraints of the dynamics; this is a positive check for the autoconsistency of the KK theory and for an hamiltonian description of the dynamics which wants to take into account the compactification scenario: this result enforces the physical meaning of KK model. 
  The evolution of the methods used to find solutions of Einstein's field equations during the last 100 years is described. Early papers used assumptions on the coordinate forms of the metrics. Since the 1950s more invariant methods have been deployed in most new papers. The uses to which the solutions found have been put are discussed, and it is shown that they have played an important role in the development of many aspects, both mathematical and physical, of general relativity. 
  We construct globally regular gravitating Skyrmions, which possess only discrete symmetries. In particular, we present tetrahedral and cubic Skyrmions. The SU(2) Skyrme field is parametrized by an improved harmonic map ansatz. Consistency then requires also a restricted ansatz for the metric. The numerical solutions obtained within this approximation are compared to those obtained in dilaton gravity. 
  We obtain a global embedding of the surface of a rapidly rotating Kerr-Newman black hole in an Euclidean 4-dimensional space. 
  In this work it is investigated the evolution of a Universe where a scalar field, non-minimally coupled to space-time curvature, plays the role of quintessence and drives the Universe to a present accelerated expansion. A non-relativistic dark matter constituent that interacts directly with dark energy is also considered, where the dark matter particle mass is assumed to be proportional to the value of the scalar field. Two models for dark matter pressure are considered: the usual one, pressureless, and another that comes from a thermodynamic theory and relates the pressure with the coupling between the scalar field and the curvature scalar. Although the model has a strong dependence on the initial conditions, it is shown that the mixture consisted of dark components plus baryonic matter and radiation can reproduce the expected red-shift behavior of the deceleration parameter, density parameters and luminosity distance. 
  In this work we investigate the evolution of a Universe consisted of a scalar field, a dark matter field and non-interacting baryonic matter and radiation. The scalar field, which plays the role of dark energy, is non-minimally coupled to space-time curvature, and drives the Universe to a present accelerated expansion. The non-relativistic dark matter field interacts directly with the dark energy and has a pressure which follows from a thermodynamic theory. We show that this model can reproduce the expected behavior of the density parameters, deceleration parameter and luminosity distance. 
  We give a correct tensor proof of the positive energy problem for the case including momentum on basis of conditions of existence of the two-to-one correspondence between the Sen-Witten spinor field and the Sen-Witten orthonormal frame. These conditions were obtained in our previous publications, but true significance of our works was not estimated properly by G.Y.Chee, and these were not correct quoted in his publication. On other hand, the main result of our work is key argument in favour of geometrical nature of the Sen-Witten spinor field. 
  Based on a rather general low-energy effective action (interacting quantum fields in classical curved space-times), we calculate potential signatures of new physics (such as quantum gravity) at ultra-high energies (presumably the Planck scale) in the anisotropies of the cosmic microwave background. These Planck-scale interactions create non-Gaussian contributions, where special emphasis is laid on the three-point function as the most promising observable, which also allows the discrimination between models violating and those obeying Lorentz invariance. PACS: 98.80.Cq, 04.62.+v, 98.70.Vc, 98.80.Qc. 
  The second order Killing and conformal tensors are analyzed in terms of their spectral decomposition, and some properties of the eigenvalues and the eigenspaces are shown. When the tensor is of type I with only two different eigenvalues, the condition to be a Killing or a conformal tensor is characterized in terms of its underlying almost-product structure. A canonical expression for the metrics admitting these kinds of symmetries is also presented. The space-time cases 1+3 and 2+2 are analyzed in more detail. Starting from this approach to Killing and conformal tensors a geometric interpretation of some results on quadratic first integrals of the geodesic equation in vacuum Petrov-Bel type D solutions is offered. A generalization of these results to a wider family of type D space-times is also obtained. A generalization of these results to a wider family of type D space-times is also obtained. 
  The theory of relativistic {\em location systems} is sketched. An interesting class of these systems is that of relativistic {\em positioning systems,} which consists in sets of four clocks broadcasting their proper time. Among them, the more important ones are the {\em auto-located positioning systems,} in which every clock broadcasts not only its proper time but the proper times that it receives from the other three. At this level, no reference to any exterior system (the Earth surface, for example) and no synchronization are needed. Some properties are presented. In the SYPOR project, such a structure is proposed, eventually anchored to a classical reference system on the Earth surface, as the best relativistic structure for Global Navigation Satellite Systems. 
  The Fermi transport of the Dirac spinor is considered as a generalization of the parallel transport of this spinor which was introduced by V. Fock and D. Ivanenko (1929). The possible structure of the new variant of the general-relativistic Dirac equation based on the Fermi transport is discussed. 
  The stability features of steady states of the spherically symmetric Einstein-Vlasov system are investigated numerically. We find support for the conjecture by Zeldovich and Novikov that the binding energy maximum along a steady state sequence signals the onset of instability, a conjecture which we extend to and confirm for non-isotropic states. The sign of the binding energy of a solution turns out to be relevant for its time evolution in general. We relate the stability properties to the question of universality in critical collapse and find that for Vlasov matter universality does not seem to hold. 
  A new approximate solution of vacuum and stationary Einstein field equations is obtained. This solution is constructed by means of a power series expansion of the Ernst potential in terms of two independent and dimensionless parameters representing the quadrupole and the angular momentum respectively. The main feature of the solution is a suitable description of small deviations from spherical symmetry through perturbations of the static configuration and the massive multipole structure by using those parameters. This quality of the solution might eventually provide relevant differences with respect to the description provided by the Kerr solution. 
  We discuss possible variations of the effective gravitational constant with length scale, predicted by most of alternative theories of gravity and unified models of physical interactions. After a brief general exposition, we review in more detail the predicted corrections to Newton's law of gravity in diverse brane world models. We consider various configurations in 5 dimensions (flat, de Sitter and AdS branes in Einstein and Einstein-Gauss-Bonnet theories, with and without induced gravity and possible incomplete graviton localization), 5D multi-brane systems and some models in higher dimensions. A common feature of all models considered is the existence of corrections to Newton's law at small radii comparable with the bulk characteristic length: at such radii, gravity on the brane becomes effectively multidimensional. Many models contain superlight perturbation modes, which modify gravity at large scale and may be important for astrophysics and cosmology. 
  Recently, the early superinflation driven by phantom field has been proposed and studied. The detection of primordial gravitational wave is an important means to know the state of very early universe. In this brief report we discuss in detail the gravitational wave background excited during the phantom superinflation. 
  There are several approaches to quantum gravitational corrections of black hole thermodynamics. String theory and loop quantum gravity, by direct analysis on the basis of quantum properties of black holes, show that in the entropy-area relation the leading order correction should be of log-area type. On the other hand, generalized uncertainty principle(GUP) and modified dispersion relations(MDRs) provide perturbational framework for such modifications. Although both GUP and MDRs are common features of all quantum gravity scenarios, their functional forms are quantum gravity model dependent. Since both string theory and loop quantum gravity give more reliable solution of the black hole thermodynamics, one can use their results to test approximate results of GUP and MDRs. In this paper, we find quantum corrected black hole thermodynamics in the framework of GUP and MDR and then we compare our results with string theory solutions. This comparison suggests severe constraints on the functional form of GUP and MDRs. These constraints may reflect characteristic features of ultimate quantum gravity theory. 
  The basic elements of Coll positioning systems (n clocks broadcasting electromagnetic signals in a n-dimensional space-time) are presented in the two-dimensional case. This simplified approach allows us to explain and to analyze the properties and interest of these relativistic positioning systems. The positioning system defined in flat metric by two geodesic clocks is analyzed. The interest of the Coll systems in gravimetry is pointed out. 
  We investigate the effect of deviations from general relativity on approach to the initial singularity by finding exact cosmological solutions to a wide class of fourth-order gravity theories. We present new anisotropic vacuum solutions of modified Kasner type and demonstrate the extent to which they are valid in the presence of non-comoving perfect-fluid matter fields. The infinite series of Mixmaster oscillations seen in general relativity will not occur in these solutions, except in unphysical cases. 
  The Hessian of either the entropy or the energy function can be regarded as a metric on a Gibbs surface. For two parameter families of asymptotically flat black holes in arbitrary dimension one or the other of these metrics are flat, and the state space is a flat wedge. The mathematical reason for this is traced back to the scale invariance of the Einstein-Maxwell equations. The picture of state space that we obtain makes some properties such as the occurence of divergent specific heats transparent. 
  In this work a new set of boundary conditions for Chern Simons gravity that lead to a fully gauge invariant action is analyzed. This particular form of the action reproduces the standard results of black hole thermodynamics and determines that the algebra of charges of diffeomorphisms at the horizon be the Virasoro algebra. 
  The ideas of spacetime discreteness and causality are important in several of the popular approaches to quantum gravity. But if discreteness is accepted as an initial assumption, conflict with Lorentz invariance can be a consequence. The causal set is a discrete structure which avoids this problem and provides a possible history space on which to build a ``path integral'' type quantum gravity theory. Motivation, results and open problems are discussed and some comparisons to other approaches are made. Some recent progress on recovering locality in causal sets is recounted. 
  We use direct products of Einstein Metrics to construct new solutions to Einstein's Equations with cosmological constant. We illustrate the technique with three families of solutions having the geometries Kerr/de Sitter X de Sitter, Kerr/anti-de Sitter X anti-de Sitter and Kerr X Kerr. 
  Many theories of gravity admit formulations in different, conformally related manifolds, known as the Jordan and Einstein conformal frames. Among them are various scalar-tensor theories of gravity and high-order theories with the Lagrangian $f(R)$ where $R$ is the scalar curvature and $f$ an arbitrary function. It may happen that a singularity in the Einstein frame corresponds to a regular surface S_trans in the Jordan frame, and the space-time is then continued beyond this surface. This phenomenon is called a conformal continuation (CC). We discuss the properties of vacuum static, spherically symmetric configurations of arbitrary dimension in scalar-tensor and $f(R)$ theories of gravity and indicate necessary and sufficient conditions for the existence of solutions admitting a CC. Two cases are distinguished, when S_trans is an ordinary regular sphere and when it is a Killing horizon. Two explicit examples of CCs are presented. 
  We discuss the initial-boundary value problem of General Relativity. Previous considerations for a toy model problem in electrodynamics motivate the introduction of a variational principle for the lapse with several attractive properties. In particular, it is argued that the resulting elliptic gauge condition for the lapse together with a suitable condition for the shift and constraint-preserving boundary conditions controlling the Weyl scalar Psi_0 are expected to yield a well posed initial-boundary value problem for metric formulations of Einstein's field equations which are commonly used in numerical relativity.   To present a simple and explicit example we consider the 3+1 decomposition introduced by York of the field equations on a cubic domain with two periodic directions and prove in the weak field limit that our gauge condition for the lapse and our boundary conditions lead to a well posed problem. The method discussed here is quite general and should also yield well posed problems for different ways of writing the evolution equations, including first order symmetric hyperbolic or mixed first-order second-order formulations. Well posed initial-boundary value formulations for the linearization about arbitrary stationary configurations will be presented elsewhere. 
  4 emitters broadcasting an increasing electromagnetic signal generate a system of relativistic coordinates for the space-time, called emission coordinates. Their physical realization requires an apparatus similar to the one of the Global Navigation Satellite Systems (GNSS). Several relativistic corrections are utilized for the current precisions, but the GNSS are conceived as classical (Newtonian) systems, which has deep implications in the way of operating them. The study of emission coordinates is an essential step in order to develop a fully relativistic theory of positioning systems. This talk presents some properties of emission coordinates. In particular, we characterize how any observer sees a configuration of satellites giving a degenerated system and show that the trajectories of the satellites select a unique privileged observer at each point and, for any observer, a set of 3 orthogonal spatial axes. 
  It is proposed a formalism of quantification of the electric charges in the Kaluza Klein theory of five dimensions and a explanation of the cause of the variation of the electromagnetic fine-structure constant in cosmological times.There is a formalism that eliminates the problem that impeached the KK theory to complete the unification of the electromagnetic and the gravitational fields. The complete unification of these two fields is obtained. The radius of compactation is determined as 70,238 lengths of Planck. 
  It is shown that Eppley and Hannah's thought experiment establishing that gravity must be quantized is fatally flawed. The device they propose, even if built, cannot establish their claims, nor is it plausible that it can be built with any materials compatible with the values of c, h, and G. Finally the device, and any reasonable modification of it, would be so massive as to be within its own Schwarzschild radius--a fatal flaw for any thought experiment. 
  In this short article we introduce the mathematical framework of the principle of the fermionic projector and set up a variational principle in discrete space-time. The underlying physical principles are discussed. We outline the connection to the continuum theory and state recent results. In the last two sections, we speculate on how it might be possible to describe quantum gravity within this framework. 
  We investigate novel approach, which improves the sensitivity of gravitational wave (GW) interferometer due to stochastic resonance (SR) phenomenon, performing in additional nonlinear cavity (NC). The NC is installed in the output of interferometer before photodetector, so that optical signal emerging interferometer incidents on the NC and passes through it. Under appropriate circumstances a specific transformation of noisy signal inside the NC takes place, which results in the increase of output signal-to-noise ratio (SNR). As a result optical noisy signal of interferometer becomes less noisy after passing through the NC. The improvement of SNR is especially effective in bistable NC for wideband (several hundreds Hz) detection, when chirp GW signal is detected. Then SNR gain reaches amount ~ 10. When detection bandwidth is narrowed, the influence of SR mechanism gradually disappears, and SNR gain tends to 1. SNR gain also tends to 1 when the NC is gradually transformed to linear cavity. Proposed enhancement of SNR due to the SR is not dependent of noise type, which is prevalent in interferometer. Particularly proposed approach is capable to increase signal-to-displacement noise ratio. 
  A class of exact solutions of the Einstein-Maxwell equations is presented which describes an accelerating and rotating charged black hole in an asymptotically de Sitter or anti-de Sitter universe. The metric is presented in a new and convenient form in which the meaning of the parameters is clearly identified, and from which the physical properties of the solution can readily be interpreted. 
  The Cauchy problem is considered for the scalar wave equation in the Schwarzschild geometry. We derive an integral spectral representation for the solution and prove pointwise decay in time. 
  We consider a braneworld model in which an anisotropic brane is embedded in a dilatonic background. We solve the background solutions and study the behavior of the perturbations when the universe evolves from an inflationary Kasner phase to a Minkowski phase. We calculate the massless mode spectrum, and find that it does not differ from what expected in standard four-dimensional cosmological models. We then evaluate the spectrum of both light (ultrarelativistic) and heavy (nonrelativistic) massive modes, and find that, at high energies, there can be a strong enhancement of the Kaluza-Klein spectral amplitude, which can become dominant in the total spectrum. The presence of the dilaton, on the contrary, decrease the relative importance of the massive modes. 
  To calculate the total energy distribution (due to both matter and fields including gravitation) associated with locally rotationally symmetric (LRS) Bianchi type-II space-times. We use the Bergmann-Thomson energy-momentum complex in both general relativity and teleparallel gravity. We find that the energy density in these different gravitation theories is vanishing at all times. This result is the same as that obtained by one of the present authors who solved the problem of finding the energy-momentum in LRS Bianchi type-II by using the energy-momentum complexes of Einstein and Landau and Lifshitz. The results of this paper also are consistent with those given in the previous works of Cooperstock and Israelit, Rosen, Johri et al., Banerjee-Sen, Vargas, and Salti et al. In this paper, we perform the calculations for a non-diagonal expanding space-time to determine whether the Bergmann-Thomson energy momentum prescription is consistent with the other formulations. (We previously considered diagonal and expanding space-time models.) Our result supports the viewpoints of Albrow and Tryon. 
  We present some results concerning the large volume limit of loop quantum cosmology in the flat homogeneous and isotropic case. We derive the Wheeler-De Witt equation in this limit. Looking for the action from which this equation can also be obtained, we then address the problem of the modifications to be brought to the Friedman's equation of motion and to the equation of motion of the scalar field, in the classical limit. 
  In this work the Poincare-Chern Simons and Anti de Sitter Chern Simons gravities are studied. For both a solution that can be casted as a black hole with manifest torsion is found. Those solutions resemble Schwarzschild and Schwarzschild-AdS solutions respectively. 
  In this paper I examine black hole and cosmological space-times in Born-Infeld-Einstein theory with electric and magnetic charges. The field equations are derived and written in the form $G_{\mu\nu}=-\kappa T_{\mu\nu}$ for spherically symmetric space-times. The energy-momentum tensor is not the Born-Infeld energy-momentum tensor, but can be obtained from Born-Infeld theory by letting $a\to ia$, where $a$ is the Born-Infeld parameter. It is shown that there is a curvature singularity in spherically symmetric space-times at a nonzero radial coordinate and that, as in Reissner-Nordstrom space-times, there are zero, one or two horizons. Charged black holes have either two horizons and a timelike singularity or one horizon with a spacelike, timelike, or null singularity. Anisotropic cosmological solutions with electric and magnetic fields are obtained from the spherically symmetric solutions. 
  We treat two possible phenomenological effects of quantum fluctuations of spacetime geometry: spectral line broadening and angular blurring of the image of a distance source. A geometrical construction will be used to express both effects in terms of the Riemann tensor correlation function. We apply the resulting expressions to study some explicit examples in which the fluctuations arise from a bath of gravitons in either a squeezed state or a thermal state. In the case of a squeezed state, one has two limits of interest: a coherent state which exhibits classical time variation but no fluctuations, and a squeezed vacuum state, in which the fluctuations are maximized. 
  Lorentzian frames may belong to one of the 199 causal classes. Of these numerous causal classes, people are essentially aware only of two of them. Nevertheless, other causal classes are present in some well-known solutions, or present a strong interest in the physical construction of coordinate systems. Here we show the unusual causal classes to which belong so familiar coordinate systems as those of Lema{\^{\i}}tre, those of Eddington-Finkelstein, or those of Bondi-Sachs. Also the causal classes associated to the Coll light coordinates (four congruences of real geodetic null lines) and to the Coll positioning systems (light signals broadcasted by four clocks) are analyzed. The role that these results play in the comprehension and classification of relativistic coordinate systems is emphasized. 
  The iterated Crank-Nicolson is a predictor-corrector algorithm commonly used in numerical relativity for the solution of both hyperbolic and parabolic partial differential equations. We here extend the recent work on the stability of this scheme for hyperbolic equations by investigating the properties when the average between the predicted and corrected values is made with unequal weights and when the scheme is applied to a parabolic equation. We also propose a variant of the scheme in which the coefficients in the averages are swapped between two corrections leading to systematically larger amplification factors and to a smaller numerical dispersion. 
  A proposal for the Hamilton-Jacobi theory in the context of the covariant formulation of Hamiltonian systems is done. The current approach consists in applying Dirac's method to the corresponding action which implies the inclusion of second-class constraints in the formalism which are handled using the procedure of Rothe and Scholtz recently reported. The current method is applied to the nonrelativistic two-dimensional isotropic harmonic oscillator employing the various symplectic structures for this dynamical system recently reported. 
  The Moller energy(due to matter and fields including gravity) distribution of the gamma metric is studied in tele-parallel gravity. The result is the same as those obtained in general relativity by Virbhadra in the Weinberg complex and Yang-Radincshi in the Moller definition. Our result is also independent of the three teleparallel dimensionless coupling constants, which means that it is valid not only in the teleparallel equivalent of general relativity, but also in any teleparallel model. 
  In a paper by Maartens, Lesame and Ellis (Class. Quant. Grav. 15, 1005) it was shown that irrotational dust solutions with vanishing electric part of the Weyl tensor are subject to severe integrability conditions and it was conjectured that the only such solutions are FLRW spacetimes. In their analysis the possibility of a cosmological constant Lambda was omitted. The conjecture is proved, irrespective as to whether Lambda is zero or not, and qualitative differences with the case of vanishing magnetic Weyl curvature are pointed out. 
  It has previously been shown that the Einstein equation can be derived from the requirement that the Clausius relation dS = dQ/T hold for all local acceleration horizons through each spacetime point, where dS is one quarter the horizon area change in Planck units, and dQ and T are the energy flux across the horizon and Unruh temperature seen by an accelerating observer just inside the horizon. Here we show that a curvature correction to the entropy that is polynomial in the Ricci scalar requires a non-equilibrium treatment. The corresponding field equation is derived from the entropy balance relation dS =dQ/T+dS_i, where dS_i is a bulk viscosity entropy production term that we determine by imposing energy-momentum conservation. Entropy production can also be included in pure Einstein theory by allowing for shear viscosity of the horizon. 
  The Schwarzschild--de Sitter space--time describes the gravitational field of a spherically symmetric mass in a universe with cosmological constant $\Lambda$. Based on this space--time we calculate Solar system effects like gravitational redshift, light deflection, gravitational time delay, Perihelion shift, geodetic or de Sitter precession, as well as the influence of $\Lambda$ on a Doppler measurement, used to determine the velocity of the Pioneer 10 and 11 spacecraft. For $\Lambda=\Lambda_0 \sim 10^{-52} {\rm m}^{-2}$ the cosmological constant plays no role for all of these effects, while a value of $\Lambda \sim - 10^{-37} {\rm m}^{-2}$, if hypothetically held responsible for the Pioneer anomaly, is not compatible with the Perihelion shift. 
  In the context of a parametric theory (with the time being a dynamical variable) we consider the coupling between the quantum vacuum and the background gravitation that pervades the universe (unavoidable because of the universality of gravity). In our model the fourth Heisenberg relation introduces a possible source of discrepancy between the marches of atomic and gravitational clocks, which accelerate with respect to one another. This produces in its turn another discrepancy between the observations, performed with atomic time, and the theoretical analysis, which uses parametric astronomical time. Curiously, this approach turns out to be compatible with current physics; lacking a unified theory of quantum physics and gravitation, it cannot be discarded {\it a priori}. It happens that this phenomenon has the same footprint as the Pioneer Anomaly, what suggests a solution to this riddle. This is because the velocity of the Pioneer spaceship with respect to atomic time turns out to be slightly smaller that with astronomical time, so that that the apparent trajectory lags behind the real one. In 1998, after many unsuccessful efforts to account for this phenomenon, the discoverers suggested "the possibility that the origin of the anomalous signal is new physics". 
  This work concerns the loss of energy of a material system due to gravitational radiation in Einstein-Aether theory-an alternative theory of gravity in which the metric couples to a dynamical, timelike, unit-norm vector field. Derived to lowest post-Newtonian order are waveforms for the metric and vector fields far from a nearly Newtonian system and the rate of energy radiated by the system. The expressions depend on the quadrupole moment of the source, as in standard general relativity, but also contain monopolar and dipolar terms. There exists a one-parameter family of Einstein-aether theories for which only the quadrupolar contribution is present, and for which the expression for the damping rate is identical to that of general relativity to the order worked to here. This family cannot yet be declared observationally viable, since effects due to the strong internal fields of bodies in the actual systems used to test the damping rate have not yet been determined. 
  In this paper we very preliminarily investigate the possibility of measuring the post-Newtonian general relativistic gravitoelectric and gravitomagnetic components of the acceleration of gravity on the Earth, in continuous regime, with two absolute measurements at the equator and the south pole with superconducting gravimeters. The magnitudes of such relativistic effects are 10^-10 m s^-2 and 10^-11 m s^-2, respectively. Unfortunately, the present-day uncertainties in the Earth's geodetic parameters which enter the classical Newtonian terms induce systematic errors 1-2 orders of magnitude larger than the relativistic ones. Moreover, a \sim 1 ngal sensitivity can be reached by the currently available superconducting gravimeters, but only for relative measurements. 
  We calculate the exact degeneracy of states corresponding to the area operator in the framework of semiclassical loop quantum gravity, using techniques of combinatorial theory. The degeneracy counting is used to find entropy of apparent horizons derived from generalised coherent states which include a sum over graphs. The correction to the entropy is determined as exponentially decreasing in area. 
  We characterize a general gravitational field near conformal infinity (null, spacelike, or timelike) in spacetimes of any dimension. This is based on an explicit evaluation of the dependence of the radiative component of the Weyl tensor on the null direction from which infinity is approached. The behaviour similar to peeling property is recovered, and it is shown that the directional structure of radiation has a universal character that is determined by the algebraic type of the spacetime. This is a natural generalization of analogous results obtained previously in the four-dimensional case. 
  Today, the motion of spacecrafts is still described according to the classical Newtonian equations plus the so-called "relativistic corrections", computed with the required precision using the Post-(Post-)Newtonian formalism. The current approach, with the increase of tracking precision (Ka-Band Doppler, interplanetary lasers) and clock stabilities (atomic fountains) is reaching its limits in terms of complexity, and is furthermore error prone. In the appropriate framework of General Relativity, we study a method to numerically integrate the native relativistic equations of motion for a weak gravitational field, also taking into account small non-gravitational forces. The latter are treated as perturbations, in the sense that we assume that both the local structure of space-time is not modified by these forces, and that the unperturbed satellite motion follows the geodesics of the local space-time. The use of a symplectic integrator to compute the unperturbed geodesic motion insures the constancy of the norm of the proper velocity quadrivector. We further show how this general relativistic framework relates to the classical one. 
  This paper deals thoroughly with the scalar and electromagnetic fields of uniformly accelerated charges in de Sitter spacetime. It gives details and makes various extensions of our Physical Review Letter from 2002. The basic properties of the classical Born solutions representing two uniformly accelerated charges in flat spacetime are first summarized. The worldlines of uniformly accelerated particles in de Sitter universe are defined and described in a number of coordinate frames, some of them being of cosmological significance, the other are tied naturally to the particles. The scalar and electromagnetic fields due to the accelerated charges are constructed by using conformal relations between Minkowski and de Sitter space. The properties of the generalized `cosmological' Born solutions are analyzed and elucidated in various coordinate systems. In particular, a limiting procedure is demonstrated which brings the cosmological Born fields in de Sitter space back to the classical Born solutions in Minkowski space. In an extensive Appendix, which can be used independently of the main text, nine families of coordinate systems in de Sitter spacetime are described analytically and illustrated graphically in a number of conformal diagrams. 
  We present a new group field theory describing 3d Riemannian quantum gravity coupled to matter fields for any choice of spin and mass. The perturbative expansion of the partition function produces fat graphs colored with SU(2) algebraic data, from which one can reconstruct at once a 3-dimensional simplicial complex representing spacetime and its geometry, like in the Ponzano-Regge formulation of pure 3d quantum gravity, and the Feynman graphs for the matter fields. The model then assigns quantum amplitudes to these fat graphs given by spin foam models for gravity coupled to interacting massive spinning point particles, whose properties we discuss. 
  We study the one-loop radiative corrections for massless fermions in de Sitter space induced by a Yukawa coupling to a light, nearly minimally coupled scalar field. We show that the fermions acquire a mass. Next we construct the corresponding (nonlocal) effective fermionic action, which -- in contrast to the case of a massive Dirac fermion -- preserves chirality. Nevertheless, the resulting fermion dynamics is precisely that of a Dirac fermion with a mass proportional to the expansion rate. Our finding supports the view that an observer or a test particle responds to a scalar field in inflation by shifting its energy rather than seeing a thermal bath. 
  In this paper, considering Einstein, Bergmann-Thomson and Landau-Lifshitz's energy-momentum definitions in both general relativity and teleparallel gravity, we compute the total energy distribution (due to matter and fields including gravitation) of the universe based on generalized inhomogeneous space-times. We obtain that Einstein and Bergmann-Thomson definitions of the energy-momentum complexes give the same results, while Landau-Lifshitz's energy-momentum definition does not provide same results for these type of metric. However, it is shown that the results obtained are reduced to the energy-momentum density of the Robertson-Walker space-times already available in the literature. 
  Familiar textbook quantum mechanics assumes a fixed background spacetime to define states on spacelike surfaces and their unitary evolution between them. Quantum theory has changed as our conceptions of space and time have evolved. But quantum mechanics needs to be generalized further for quantum gravity where spacetime geometry is fluctuating and without definite value. This paper reviews a fully four-dimensional, sum-over-histories, generalized quantum mechanics of cosmological spacetime geometry. This generalization is constructed within the framework of generalized quantum theory. This is a minimal set of principles for quantum theory abstracted from the modern quantum mechanics of closed systems, most generally the universe. In this generalization, states of fields on spacelike surfaces and their unitary evolution are emergent properties appropriate when spacetime geometry behaves approximately classically. The principles of generalized quantum theory allow for the further generalization that would be necessary were spacetime not fundamental. Emergent spacetime phenomena are discussed in general and illustrated with the example of the classical spacetime geometries with large spacelike surfaces that emerge from the `no-boundary' wave function of the universe. These must be Lorentzian with one, and only one, time direction. The essay concludes by raising the question of whether quantum mechanics itself is emergent. 
  We consider a single field governed expansion of the universe from a five dimensional (5D) vacuum state. Under an appropiate change of variables the universe can be viewed in a effective manner as expanding in 4D with an effective equation of state which describes different epochs of its evolution. In the example here worked the universe fistly describes an inflationary phase, followed by a decelerated expansion. Thereafter, the universe is accelerated and describes a quintessential expansion to finally, in the future, be vacuum dominated. 
  A relativistic positioning system is a physical realization of a coordinate system consisting in four clocks in arbitrary motion broadcasting their proper times. The basic elements of the relativistic positioning systems are presented in the two-dimensional case. This simplified approach allows to explain and to analyze the properties and interest of these new systems. The positioning system defined by geodesic emitters in flat metric is developed in detail. The information that the data generated by a relativistic positioning system give on the space-time metric interval is analyzed, and the interest of these results in gravimetry is pointed out. 
  The nature of gravity is fundamental to our understanding of our own solar system, the galaxy and the structure and evolution of the Universe. Einstein's general theory of relativity is the standard model that is used for almost ninety years to describe gravitational phenomena on these various scales. We review the foundations of general relativity, discuss the recent progress in the tests of relativistic gravity, and present motivations for high-accuracy gravitational experiments in space. We also summarize the science objectives and technology needs for the laboratory experiments in space with laboratory being the entire solar system. We discuss the advances in our understanding of fundamental physics anticipated in the near future and evaluate discovery potential for the recently proposed gravitational experiments. 
  The field equations following from a Lagrangian L(R) will be deduced and solved for special cases. If L is a non-linear function of the curvature scalar, then these equations are of fourth order in the metric. In the introduction we present the history of these equations beginning with the paper of H. Weyl from 1918, who first discussed them as alternative to Einstein's theory. In the third part, we give details about the cosmic no hair theorem, i.e., the details how within fourth order gravity with L= R + R^2 the inflationary phase of cosmic evolution turns out to be a transient attractor. Finally, the Bicknell theorem, i.e. the conformal relation from fourth order gravity to scalar-tensor theory, will be shortly presented. 
  A linear vector model of gravitation is introduced in the context of quantum physics as a generalization of electromagnetism. The gravitoelectromagnetic gauge symmetry corresponds to a hyperbolic unitary extension of the usual complex phase symmetry of electromagnetism. The reversed sign for the gravitational coupling is obtained by means of the pseudoscalar of the underlying complex Clifford algebra. 
  The Testbed for LISA Analysis (TLA) Project aims to facilitate the development, validation and comparison of different methods for LISA science data analysis, by the broad LISA Science Community, to meet the special challenges that LISA poses. It includes a well-defined Simulated LISA Data Product (SLDP), which provides a clean interface between the communities that have developed to model and to analyze the LISA science data stream; a web-based clearinghouse (at <http://tla.gravity.psu.edu>) providing SLDP software libraries, relevant software, papers and other documentation, and a repository for SLDP data sets; a set of mailing lists for communication between and among LISA simulators and LISA science analysts; a problem tracking system for SLDP support; and a program of workshops to allow the burgeoning LISA science community to further refine the SLDP definition, define specific LISA science analysis challenges, and report their results. This note describes the TLA Project, the resources it provides immediately, its future plans, and invites the participation of the broader community in the furtherance of its goals. 
  The analytical and numerical solutions of the Mathisson-Papapetrou equations under the Mathisson-Pirani supplementary condition describing highly relativistic (ultrarelativistic) motions of a spinning particle in a Schwarzschild field are investigated. The known condition S/mr<<1, which is necessary for a test particle, holds on all these solutions. The explicit expressions for the non-equatorial circular orbits, in particular for the space boundaries of the region of existence of these orbits, are obtained. The dynamics of the deviation of a spinning particle from the equatorial ultrarelativistic circular orbit with r=3M caused by the non-zero initial value of the radial particle's velocity is studied. It is shown in the concrete cases that spin can considerable influence the shape of an ultrarelativistic trajectory, as compared to the corresponding geodesic trajectory, for the short time, less than the time of one or two revolutions of a spinning particle around a Schwarzschild mass. 
  We present a histories version of the connection formalism of general relativity. Such an approach introduces a spacetime description--a characteristic feature of the histories approach--and we discuss the extent to which the usual loop variables are compatible with a spacetime description. In particular, we discuss the definability of the Barbero connection without any gauge fixing. Although it is not the pullback of a spacetime connection onto the three-surface and it does not have a natural spacetime interpretation, this does not mean that the Barbero connection is not suitable variable for quantisation; it appears naturally in the formalism even in absence of gauge fixing. It may be employed therefore, to define loop variables similar to those employed in loop quantum gravity. However, the loop algebra would have to be augmented by the introduction of additional variables. 
  I derive the Einstein 1915 classical field theory of gravity with what resembles both a massive torsion field and the Calabi Yau degrees of freedom from a conjectured eight Goldstone phases of the cosmic inflation field provided that the full Poincare group is locally gauged and its Lorentz subgroup is spontaneously broken in the vacuum. What looks like both the t Hooft Susskind world hologram conjecture of volume without volume and the quantization of area in Planck units given by Bekenstein and Hawking seem to be natural consequences of the conjecture. Just as the Michelson Morley experiment gave a null result, this model predicts that the LHC will never find any viable dark matter exotic particles as a matter of fundamental principle, neither will any other conceivable dark matter detector. The Cambridge IofA dark matter virial speed of 9km/sec is questioned. A way to detect pocket universes in the cosmic landscape beyond all types of horizons bounded by null geodesics is suggested based on the work of Antony Valentini. 
  It is shown that in the model [3,4] of quantum mechanics besides probability amplitudes, the Planck constant and the Fock space, the cosmological constant also appear in the natural way. The Poisson brackets are generalized for the case of kinetics. 
  The geometrization of the Electro-Weak Model is achieved in a 5-dimensional Riemann-Cartan framework. Matter spinorial fields are extended to 5 dimensions by the choice of a proper dependence on the extra-coordinate and of a normalization factor. U(1) weak hyper-charge gauge fields are obtained from a Kaluza-Klein scheme, while the tetradic projections of the extra-dimensional contortion fields are interpreted as SU(2) weak isospin gauge fields. SU(2) generators are derived by the identification of the weak isospin current to the extra-dimensional current term in the Lagrangian density of the local Lorentz group. The geometrized U(1) and SU(2) groups will provide the proper transformation laws for bosonic and spinorial fields. Spin connections will be found to be purely Riemannian. 
  Using the Hamiltonian constraint derived by Ashtekar and Bojowald, we look for pre-classical wave functions in the Schwarzschild interior. In particular, when solving this difference equation by separation of variables, an inequality is obtained relating the Immirzi parameter $\gamma$ to the quantum ambiguity $\delta$ appearing in the model. This bound is violated when we use a natural value for $\delta$ based on loop quantum gravity together with a recent proposal for $\gamma$. We also present numerical solutions of the constraint. 
  We study dynamics and radiation generation in the last few orbits and merger of a binary black hole system, applying recently developed techniques for simulations of moving black holes. Our analysis of the gravitational radiation waveforms and dynamical black hole trajectories produces a consistent picture for a set of simulations with black holes beginning on circular-orbit trajectories at a variety of initial separations. We find profound agreement at the level of one percent among the simulations for the last orbit, merger and ringdown. We are confident that this part of our waveform result accurately represents the predictions from Einstein's General Relativity for the final burst of gravitational radiation resulting from the merger of an astrophysical system of equal-mass non-spinning black holes. The simulations result in a final black hole with spin parameter a/m=0.69. We also find good agreement at a level of roughly 10 percent for the radiation generated in the preceding few orbits. 
  The periodic standing wave approach to binary inspiral assumes rigid rotation of gravitational fields and hence helically symmetric solutions. To exploit the symmetry, numerical computations must solve for ``helical scalars,'' fields that are functions only of corotating coordinates, the labels on the helical Killing trajectories. Here we present the formalism for describing linearized general relativity in terms of helical scalars and we present solutions to the mixed partial differential equations of the linearized gravity problem (and to a toy nonlinear problem) using the adapted coordinates and numerical techniques previously developed for scalar periodic standing wave computations. We argue that the formalism developed may suffice for periodic standing wave computations for post-Minkowskian computations and for full general relativity. 
  We discuss the Palatini formulation of modified gravity including a Yukawa-like term. It is shown that in this formulation, the Yukawa term offers an explanation for the current exponential accelerated expansion of the universe and reduces to the standard Friedmann cosmology in the appropriate limit. We then discuss the scalar-tensor formulation of the model as a metric theory and show that the Yukawa term predicts a power-law acceleration at late-times. The Newtonian limit of the theory is also discussed in context of the Palatini formalism. 
  It is shown in the article that according to the Relativistic Theory of Gravitation the gravitational field providing slowing down of the time rate nevertheless stops itself this slowing down in strong fields. So a physical tendency of this field to self-restriction of the gravitational potential is demonstrated. This property of the field leads to a stopping of the collapse of massive bodies and to the cyclic evolution of the homogeneous and isotropic Universe. 
  Einstein field equations for anisotropic spheres are solved and exact interior solutions obtained. This paper extends earlier treatments to include anisotropic models which accommodate a wider variety of physically viable energy densities. Two classes of solutions are possible. The first class contains the limiting case $\mu\propto r^{-2}$ for the energy density which arises in many astrophysical applications. In the second class the singularity at the center of the star is not present in the energy density. The models presented in this paper allow for increasing and decreasing profiles in the behavior of the energy density. 
  The condition for pressure isotropy is reduced to a recurrence equation with variable, rational coefficients of order three. We prove that this difference equation can be solved in general. Consequently we can find an exact solution to the field equations corresponding to a static spherically symmetric gravitational potential in terms of elementary functions. The metric functions, the energy density and the pressure are continuous and well behaved which implies that this solution could be used to model the interior of a relativistic sphere. The model satisfies a barotropic equation of state in general which approximates a polytrope close to the stellar centre. 
  Comparing the corrections to Kepler's law with orbital evolution under a self force, we extract the finite, already regularized part of the latter in a specific gauge. We apply this method to a quasi-circular orbit around a Schwarzschild black hole of an extreme mass ratio binary, and determine the first- and second-order conservative gravitational self force in a post Newtonian expansion. We use these results in the construction of the gravitational waveform, and revisit the question of the relative contribution of the self force and spin-orbit coupling. 
  Data from the Laser Interferometer Space Antenna (LISA) is expected to be dominated by frequency noise from its lasers. However the noise from any one laser appears more than once in the data and there are combinations of the data that are insensitive to this noise. These combinations, called time delay interferometry (TDI) variables, have received careful study, and point the way to how LISA data analysis may be performed. Here we approach the problem from the direction of statistical inference, and show that these variables are a direct consequence of a principal component analysis of the problem. We present a formal analysis for a simple LISA model and show that there are eigenvectors of the noise covariance matrix that do not depend on laser frequency noise. Importantly, these orthogonal basis vectors correspond to linear combinations of TDI variables. As a result we show that the likelihood function for source parameters using LISA data can be based on TDI combinations of the data without loss of information. 
  It is stated in many text books that the any metric appearing in general relativity should be locally Lorentzian i.e. of the type $\eta_\mn = {\rm diag} (1,-1,-1,-1)$ this is usually presented as an independent axiom of the theory, which can not be deduced from other assumptions. In this work we show that the above assertion is a consequence of a standard stability analysis of the Einstein \eqs and need not be assumed. 
  We show that Pleba\'nski's equation for self-dual metrics is equivalent to a pair of equations describing canonical transformations in 2-dimensional phase spaces. Examples of linearizations of these equations are given. 
  In this paper we apply second-order gauge-invariant perturbation theory to investigate the possibility that the non-linear coupling between gravitational waves (GW) and a large scale inhomogeneous magnetic field acts as an amplification mechanism in an `almost' Friedmann-Lemaitre-Robertson-Walker (FLRW) Universe. The spatial inhomogeneities in the magnetic field are consistently implemented using the magnetohydrodynamic (MHD) approximation, which yields an additional source term due to the interaction of the magnetic field with velocity perturbations in the plasma. Comparing the solutions with the corresponding results in our previous work indicates that, on super-horizon scales, the interaction with the spatially inhomogeneous field in the dust regime induces the same boost as the case of a homogeneous field, at least in the ideal MHD approximation. This is attributed to the observation that the MHD induced part of the generated field effectively only contributes on scales where the coherence length of the initial field is less than the Hubble scale. At sub-horizon scales, the GW induced magnetic field is completely negligible in relation to the MHD induced field. Moreover, there is no amplification found in the long-wavelength limit. 
  We review the emergence of gravity from gauge theory in the context of AdS/CFT duality. We discuss the evidence for the duality, its lessons for gravitational physics, generalizations, and open questions. 
  In this paper we investigate the structure of the pseudo-Newtonian force and potential about a five dimensional rotating black hole. The conditions for the force character from an attractive to repulsive are considered. It is also found that the force will reach a maximum under certain conditions. 
  The spectrum of small perturbations about oscillating bounce solutions recently discussed in the literature is investigated. Our study supports quite intuitive and expected result: the bounce with N nodes has exactly N homogeneous negative modes. Existence of more than one negative modes makes obscure the relation of these oscillating bounce solutions to the false vacuum decay processes. 
  We present a general study about the relation between the vorticity tensor and the Poynting vector of the electromagnetic field for axially symmetric stationary electrovacuum metrics. The obtained expressions allow to understand the role of the Poynting vector in the dragging of inertial frames. The particular case of the rotating massive charged magnetic dipole is analyzed in detail. In addition, the electric and magnetic parts of the Weyl tensor are calculated and the link between the later and the vorticity is established. Then we show that, in the vacuum case, the necessary and sufficient condition for the vanishing of the magnetic part is that the spacetime be static. 
  After a brief review of arguments in favor of antigravity (as gravitational repulsion between matter and antimatter) we present a simple idea for an experimental test using antiprotons. Different experimental realizations of the same basic idea are considered 
  We analyze the space-time structure of local gauge string with a phenomenological energy momentum tensor, as prescribed by Vilenkin in an arbitrary number of space-time dimensions with a non-zero cosmological constant >. A set of solutions of full non-linear Einstein's equations for the interior region of such a string are presented. 
  We investigate the space-time of a global monopole in a five dimensional space-time in presence of the cosmological term. Also the gravitational properties of the monopole solution are discussed. 
  The classical delta filters used in the current resonant bar experiments for detecting GW bursts are viable when the bandwidth of resonant bars is few Hz. In that case, the incoming GW burst is likely to be viewed as an impulsive signal in a very narrow frequency window. After making improvements in the read-out with new transducers and high sensitivity dc-SQUID, the Explorer-Nautilus have improved the bandwidth ($\sim 20$ Hz) at the sensitivity level of $10^{-20}/\sqrt{Hz}$. Thus, it is necessary to reassess this assumption of delta-like signals while building filters in the resonant bars as the filtered output crucially depends on the shape of the waveform. This is presented with an example of GW signals -- stellar quasi-normal modes, by estimating the loss in SNR and the error in the timing, when the GW signal is filtered with the delta filter as compared to the optimal filter. 
  I show that there are no SU(2)-invariant (time-dependent) tensorial perturbations of Lorentzian Taub-NUT space. It follows that the spacetime is unstable at the linear level against generic perturbations. I speculate that this fact is responsible for so far unsuccessful attempts to define a sensible thermodynamics for NUT-charged spacetimes. 
  We demonstrate that the source of a shear-free null congruence is, generically, a world sheet of a singular string in the complex Minkowski space. In particular cases the string degenerates into the Newman's point singularity. We describe also a time-dependent deformation of the Kerr's congruence which could indicate the unstability of the Kerr (Kerr-Newman) (electro)vacuum solution. Then we consider a 5D extension of Minkowski space obtained by complexification of the time coordinate. Singular locus of a shear-free null congruence on such an extended space-time consists of a (great) number of point particles (``markeons'') and 2D surfaces -- light-like wave fronts. 
  We investigate the late-time behavior of the massive vector field in the background of the Schwarzschild black holes. For Schwarzschild black hole, at intermediately late times, the massive vector field is represented by three functions with different decay law $\Psi \propto t^{-(\ell + 3/2)} \sin{m t}$, $\Psi_{1} \propto t^{-(\ell + 5/2)} \sin{m t}$, $\Psi_{2} \propto t^{-(\ell + 1/2)} \sin{m t}$, while at asymptotically late times the decay law $\Psi \propto t^{-5/6} \sin{(m t)}$ is universal, and does not depend on the multipole number $\ell$. Together with previous study of massive scalar and Dirac fields where the same asymptotically late-time decay law was found, it means, that the asymptotically late-time decay law $\propto t^{-5/6} \sin{(m t)}$ does not depend also on the spin of the field under consideration. 
  The correspondence between conformal covariant fields in Minkowski's space-time and isometric fields in the five dimensional anti-deSitter space-time is extended to a six-dimensional bulk space and its regular sub-manifolds, so as to include the analysis of evaporating Schwarzschild's black holes without loss of quantum unitarity. 
  A new set of tetrads is introduced within the framework of SU(2) X U(1) Yang-Mills field theories in curved spacetimes. Each one of these tetrads diagonalizes separately each term of the Yang-Mills stress-energy tensor. Therefore, three pairs of planes also known as blades, can be defined, and make up the underlying geometrical structure, at each point. These tetrad vectors are gauge dependent on one hand, and also in their definition, there is an additional inherent freedom in the choice of two vector fields. In order to get rid of the gauge dependence, another set of tetrads is defined, such that the only choice we have to make is for the two vector fields. A particular choice is made for these two vector fields such that they are gauge dependent, but the transformation properties of the tetrads are analogous to those already known for curved spacetimes where only electromagnetic fields are present. This analogy allows to establish group isomorphisms between the local gauge group SU(2), and the tensor product of the groups of local Lorentz tetrad transformations, either on blade one or blade two. 
  The DGP brane world model allows us to get the observed late time acceleration via modified gravity, without the need for a ``dark energy'' field. This can then be generalised by the inclusion of high energy terms, in the form of a Gauss-Bonnet bulk. This is the basis of the Gauss-Bonnet-Induced-Gravity (GBIG) model explored here with both early and late time modifications to the cosmological evolution. Recently the simplest GBIG models (Minkowski bulk and no brane tension) have been analysed. Two of the three possible branches in these models start with a finite density ``Big-Bang'' and with late time acceleration. Here we present a comprehensive analysis of more general models where we include a bulk cosmological constant and brane tension. We show that by including these factors it is possible to have late time phantom behaviour. 
  In the harmonic description of general relativity, the principle part of Einstein equations reduces to a constrained system of 10 curved space wave equations for the components of the space-time metric. We use the pseudo-differential theory of systems which are well-posed in the generalized sense to establish the well-posedness of constraint preserving boundary conditions for this system when treated in second order differential form. The boundary conditions are of a generalized Sommerfeld type that is benevolent for numerical calculation. 
  We explore, in the general relativistic context, the properties of the recently introduced GPS coordinates, as well as those of the associated frames and coframes. We show that they are covariant, and completely independent of any observer. We show that standard spectroscopic and astrometric observations allow any observer to measure (i) the values of the GPS coordinates at his position, (ii) the components of his [four-]velocity and (iii) the components of the metric in the GPS frame. This provides to this system an unique value both for conceptual discussion (no frame dependence) and for practical use (involved quantities are directly measurable): localisation, motion monitoring, astrometry, cosmography, tests of gravitation theories. We show explicitly, in the general relativistic context, how an observer may estimate its position and motion, and reconstruct the components of the metric. This arises from two main results: the extension of the velocity fields of the probes to the whole (curved) spacetime; and the identification of the components of the observer's velocity in the GPS frame with the (inversed) observed redshifts of the probes. Specific cases (non relativistic velocities; Minkowski and Friedmann-Lema\^{i}tre spacetimes; geodesic motions) are studied in details. 
  This paper describes a multidimensional hydrodynamic code which can be used for the studies of relativistic astrophysical flows. The code solves the special relativistic hydrodynamic equations as a hyperbolic system of conservation laws based on High Resolution Shock Capturing (HRSC) Scheme. Two standard tests, one of which is the relativistic blast wave tested in our previous paper\cite{DO1}, and the other is the collision of two ultrarelativistic blast waves tested in here, are presented to demonstrate that the code captures correctly and gives solution in the discontinuities, accurately.   The relativistic astrophysical jet is modeled for the ultrarelativistic flow case. The dynamics of jet flowing is then determined by the ambient parameters such as densities, and velocities of the jets and the momentum impulse applied to the computational surface. We obtain solutions for the jet structure, propagation of jet during the time evolution, and variation in the Mach number on the computational domain at a fixed time. 
  A theory of gravitation is proposed, modeled after the notion of a Ricci flow. In addition to the metric an independent volume enters as a fundamental geometric structure. Einstein gravity is included as a limiting case. Despite being a scalar-tensor theory the coupling to matter is different from Jordan-Brans-Dicke gravity. In particular there is no adjustable coupling constant. For the solar system the effects of Ricci flow gravity cannot be distinguished from Einstein gravity and therefore it passes all classical tests. However for cosmology significant deviations from standard Einstein cosmology will appear. 
  The compactification of M theory with time dependent hyperbolic internal space gives an effective scalar field with exponential potential which provides a transient acceleration in Einstein frame in four dimensions. Ordinary matter and radiation are present in addition to the scalar field coming from compactification. We find that we have to fine-tune the initial conditions of the scalar field so that our Universe experiences acceleration now. During the evolution of our Universe, the volume of the internal space increases about 12 times. The time variation of the internal space results in a large time variation of the fine structure constant which violates the observational constraint on the variation of the fine structure constant. The large variation of the fine structure constant is a generic feature of transient acceleration models. 
  The exact solution for the motion of a test particle in a non-spherical polar orbit around a Kerr black hole is derived. Exact novel expressions for frame dragging (Lense-Thirring effect), periapsis advance and the orbital period are produced. The resulting formulae, are expressed in terms of Appell's first hypergeometric function $F_1$, Jacobi's amplitude function, and Appell's $F_1$ and Gau$\ss$ hypergeometric function respectively. The exact expression for frame dragging is applied for the calculation of the Lense-Thirring effect for the orbits of S-stars in the central arcsecond of our Galaxy assuming that the galactic centre is a Kerr black hole, for various values of the Kerr parameter including those supported by recent observations. In addition, we apply our solutions for the calculation of frame dragging and periapsis advance for stellar non-spherical polar orbits in regions of strong gravitational field close to the event horizon of the galactic black hole, e.g. for orbits in the central milliarcsecond of our galaxy. Such orbits are the target of the GRAVITY experiment. We provide examples with orbital periods in the range of 100min - 54 days. Detection of such stellar orbits will allow the possibility of measuring the relativistic effect of periapsis advance with high precision at the strong field realm of general relativity. Further, an exact expression for the orbital period of a test particle in a non-circular equatorial motion around a Kerr black hole is produced. We also derive exact expressions for the periapsis advance and the orbital period for a test particle in a non-circular equatorial motion in the Kerr field in the presence of the cosmological constant in terms of Lauricella's fourth hypergeometric function $F_D$. 
  I give an overview of the motivations for gravitational-wave research, concentrating on the aspects related to ``fundamental'' physics. 
  Parametric transducers, such as superconducting rf cavities, can boost the bandwidth and sensitivity of the next generation resonant antennas, thanks to a readily available technology. We have developed a fully coupled dynamic model of the system "antenna--transducer" and worked out some estimates of signal--to--noise ratio and the stability conditions in various experimental configurations. We also show the design and the prototype of a rf cavity which, together with a suitable read--out electronic, will be used as a test bench for the parametric transducer. 
  It is generally assumed that naked singularities must be physically excluded, as they could otherwise introduce unpredictable influences in their future null cones. Considering geodesics for a naked Reissner-Nordstrom singularity, it is found that the singularity is effectively clothed by its repulsive nature. Regarding electron as naked singularity, the size of the clothed singularity (electron) turns out to be classical electro-magnetic radius of the electron, to an observer falling freely from infinity, initially at rest. The size shrinks for an observer falling freely from infinity, with a positive initial velocity. For geodetic parameters corresponding to negative energy there are trapped geodesics. The similarity of this picture with that arising in the Quantum Theory is discussed. 
  The brane-worlds model was inspired partly by Kaluza-Klein's theory, where the gravitation and the gauge fields are obtained of a geometry of higher dimension (bulk). Such a model has been showing positive in the sense of we find perspectives and probably deep modifications in the physics, such as: Unification in a scale TeV, quantum gravity in this scale and deviation of Newton's law for small distances. One of the principles of this model is to suppose a space-time embedded in a bulk of high dimension. In this note it is shown, basing on the theorem of Collinson-Szekeres, that the space-time of Schwarzschild cannot be embedded locally and isometrically in a bulk of five dimensions with constant curvature,(for example ADS-5). From the point of view of the semi-Riemannian geometry this last result consists constraints to the model brane-world. 
  We consider a Brans-Dicke cosmology in five-dimensional space-time. Neglecting the quadratic and the mixed Brans-Dicke terms in the Einstein equation, we derive a modified wave equation of the Brans-Dicke field. We show that, at high energy limit, the 3-brane Brans-Dicke cosmology could be described as the standard one by changing the equation of state. Finally as an illustration of the purpose, we show that the dark energy component of the universe agrees with the observations data. 
  Gravitational lensing by traversable Lorentzian wormholes is a ew possibility which is analyzed here in the strong field limit. Wormhole solutions are considered in the Einstein minimally coupled theory and in the brane world model. The observables in both the theories show significant differences from those arising in the Schwarzschild black hole lensing. As a corollary, it follows that wormholes with zero Keplerian mass exhibit lensing properties which are qualitatively (though not quantitatively) the same as those of a Schwarzschild black hole. Some special features of the considered solutions are pointed out. 
  We generalize our previous thick shell formalism to incorporate any codimension-1 thick wall with a peculiar velocity and proper thickness bounded by arbitrary spacetimes. Within this new formulation we obtain the equation of motion of a spherically symmetric dust thick shell immersed in vacuum as well as in Friedmann-Robertson-Walker spacetimes. 
  We review the algebraic field theory based completely on a nonlinear generalization of the CR complex analiticity conditions to the noncommutative algebra of biquaternions. Any biquaternionic field possesses natural twistor structure and, in the Minkowski space, gives rise to a shear-free null congruence of rays and to a set of gauge fields associated with it. In the paper we develop the algebrodynamical scheme in the complex extension of Minkowski space-time -- in the full vector space of biquaternion algebra. This primodial space dynamically reduces to the 6D ``observable'' space-time of the complex null cone which in turn decomposes into the 4D physical space-time and the 2D internal ``spin space''. A set of identical point charges (``duplicons'') -- focal points of the congruence -- arises in the procedure. Temporal dynamics of individual duplicons is strongly correlated via fundamental twistor field of the congruence. We briefly discuss some new notions inevitably arising in the considered algebrodynamical scheme, namely those of ``complex time'' and of ``evolutionary curve'', as well as their hypothetical links to the quantum uncertainty phenomenon. 
  The modes of vibration of hanging and partially supported strings provide useful analogies to scalar fields travelling through spacetimes that admit conformally flat spatial sections. This wide class of spacetimes includes static, spherically symmetric spacetimes. The modes of a spacetime where the scale factor depends as a power-law on one of the coordinates provide a useful starting point and yield a new classification of these spacetimes on the basis of the shape of the string analogue. The family of corresponding strings follow a family of curves related to the cycloid, denoted here as hypercycloids (for reasons that will become apparent). Like the spacetimes that they emulate these strings exhibit horizons, typically at their bottommost points where the string tension vanishes; therefore, hanging strings may provide a new avenue for the exploration of the quantum mechanics of horizons. 
  Accurate calculation of the gradual inspiral motion in an extreme mass-ratio binary system, in which a compact-object inspirals towards a supermassive black-hole requires calculation of the interaction between the compact-object and the gravitational perturbations that it induces. These metric perturbations satisfy linear partial differential equations on a curved background spacetime induced by the supermassive black-hole. At the point particle limit the second-order perturbations equations have source terms that diverge as $r^{-4}$, where $r$ is the distance from the particle. This singular behavior renders the standard retarded solutions of these equations ill-defined. Here we resolve this problem and construct well-defined and physically meaningful solutions to these equations. We recently presented an outline of this resolution [E. Rosenthal, Phys. Rev. D 72, 121503 (2005)]. Here we provide the full details of this analysis. These second-order solutions are important for practical calculations: the planned gravitational-wave detector LISA requires preparation of waveform templates for the expected gravitational-waves. Construction of templates with desired accuracy for extreme mass-ratio binaries requires accurate calculation of the inspiral motion including the interaction with the second-order gravitational perturbations. 
  Presented is a description of a Markov chain Monte Carlo (MCMC) parameter estimation routine for use with interferometric gravitational radiational data in searches for binary neutron star inspiral signals. Five parameters associated with the inspiral can be estimated, and summary statistics are produced. Advanced MCMC methods were implemented, including importance resampling and prior distributions based on detection probability, in order to increase the efficiency of the code. An example is presented from an application using realistic, albeit fictitious, data. 
  In the present work, inhomogeneous Tolman-Bondi type dust space-time is studied on the brane. There are two sets of solutions of the above model. The first solution represents either a collapsing model starting from an infinite volume at infinite past to the singularity or a model starting from a singularity and expanding for ever having a transition from decelerating phase to accelerating phase. The first solution shows that the end state of collapse may be black hole or a naked singularity depending signs of various parameters involved. The second solution represents a bouncing model where the bounce occurs at different comoving radii at different epochs. 
  We present arguments for the existence of a new type of solutions of the Euclidean Einstein-Yang-Mills-dilaton theory in $d=4$ dimensions. Possesing nonvanishing nonabelian charges, these nonselfdual configurations have no counterparts on the Lorentzian section. They provide, however, new saddle points in the Euclidean path integral. 
  Using the energy-momentum complexes of Tolman, Papapetrou and Weinberg, the total energy of the universe in Locally Rotationally Symmetric (LRS) Bianchi type II models is calculated . The total energy is found to be zero due to the matter plus field. This result supports the viewpoint of Tryon, Rosen and Albrow. 
  The motion of free nearby test particles relative to a stable equatorial circular geodesic orbit about a Kerr source is investigated. It is shown that the nonlinear generalized Jacobi equation can be transformed in this case to an autonomous form. Tidal dynamics beyond the critical speed c/sqrt(2) is studied. We show, in particular, that a free test particle vertically launched from the circular orbit parallel or antiparallel to the Kerr rotation axis is tidally accelerated if its initial relative speed exceeds c/sqrt(2). Possible applications of our results to high-energy astrophysics are briefly mentioned. 
  After reviewing the covariant description of Hamiltonian dynamics, some applications are done to the non-relativistic isotropic three-dimensional harmonic oscillator, Rovelli's model, and SO(3,1) BF theories. 
  The violation of spacetime symmetries provides a promising candidate signal for underlying physics, possibly arising at the Planck scale. This talk gives an overview over various aspects in the field, including some mechanisms for Lorentz breakdown, the SME test framework, and phenomenological signatures for such effects. 
  For the baseline design of the advanced Laser Interferometer Gravitational-wave Observatory (LIGO), use of optical cavities with non-spherical mirrors supporting flat-top ("mesa") beams, potentially capable of mitigating the thermal noise of the mirrors, has recently drawn a considerable attention. To reduce the severe tilt-instability problems affecting the originally conceived nearly-flat, "Mexican-hat-shaped" mirror configuration, K. S. Thorne proposed a nearly-concentric mirror configuration capable of producing the same mesa beam profile on the mirror surfaces. Subsequently, Bondarescu and Thorne introduced a generalized construction that leads to a one-parameter family of "hyperboloidal" beams which allows continuous spanning from the nearly-flat to the nearly-concentric mesa beam configurations. This paper is concerned with a study of the analytic structure of the above family of hyperboloidal beams. Capitalizing on certain results from the applied optics literature on flat-top beams, a physically-insightful and computationally-effective representation is derived in terms of rapidly-converging Gauss-Laguerre expansions. Moreover, the functional relation between two generic hyperboloidal beams is investigated. This leads to a generalization (involving fractional Fourier transform operators of complex order) of some recently discovered duality relations between the nearly-flat and nearly-concentric mesa configurations. Possible implications and perspectives for the advanced LIGO optical cavity design are discussed. 
  Double Special Relativity theories are the relativistic theories in which the transformations between inertial observers are characterized by two observer-independent scales of the light speed and the Planck length. We study two main examples of these theories and want to show that these theories are not the new theories of relativity, but only are re-descriptions of Einstein's special relativity in the non-conventional coordinates. 
  In this paper we study a new symmetry argument that results in a vacuum state with strictly vanishing vacuum energy. This argument exploits the well-known feature that de Sitter and Anti- de Sitter space are related by analytic continuation. When we drop boundary and hermiticity conditions on quantum fields, we get as many negative as positive energy states, which are related by transformations to complex space. The paper does not directly solve the cosmological constant problem, but explores a new direction that appears worthwhile. 
  We study the dynamics of multiple scalar fields and a barotropic fluid in an FLRW-universe. The scalar potential is a sum of exponentials. All critical points are constructed and these include scaling and de Sitter solutions. A stability analysis of the critical points is performed for generalized assisted inflation, which is an extension of assisted inflation where the fields mutually interact. Effects in generalized assisted inflation which differ from assisted inflation are emphasized. One such a difference is that an (inflationary) attractor can exist if some of the exponential terms in the potential are negative. 
  Numerical results from a study of boson stars under nonspherical perturbations using a fully general relativistic 3D code are presented together with the analysis of emitted gravitational radiation. We have constructed a simulation code suitable for the study of scalar fields in space-times of general symmetry by bringing together components for addressing the initial value problem, the full evolution system and the detection and analysis of gravitational waves. Within a series of numerical simulations, we explicitly extract the Zerilli and Newman-Penrose scalar $\Psi_4$ gravitational waveforms when the stars are subjected to different types of perturbations. Boson star systems have rapidly decaying nonradial quasinormal modes and thus the complete gravitational waveform could be extracted for all configurations studied. The gravitational waves emitted from stable, critical, and unstable boson star configurations are analyzed and the numerically observed quasinormal mode frequencies are compared with known linear perturbation results. The superposition of the high frequency nonspherical modes on the lower frequency spherical modes was observed in the metric oscillations when perturbations with radial and nonradial components were applied. The collapse of unstable boson stars to black holes was simulated. The apparent horizons were observed to be slightly nonspherical when initially detected and became spherical as the system evolved. The application of nonradial perturbations proportional to spherical harmonics is observed not to affect the collapse time. An unstable star subjected to a large perturbation was observed to migrate to a stable configuration. 
  During the inflationary phase of the early universe, quantum fluctuations in the vacuum generate particles as they stretch beyond the Hubble length. These fluctuations are thought to result in the density fluctuations and gravitational radiation that we can try to observe today. It is possible to calculate the quantum-mechanical evolution of these fluctuations during inflation and the subsequent expansion of the universe until the present day. Calculating this evolution in the Schrodinger picture directly exposes the particle creation during accelerated expansion and while a fluctuation is larger than the Hubble length. Because all fluctuations regardless of their scale today began as the vacuum state in the early universe, the current quantum mechanical state of fluctuations is correlated on different scales and in different directions. 
  The SU(1,1)/U(1) symmetry of the equations of motion for particles on radial Schwarzschild geodesics is used to define the momentum space of the particles. Using geometric properties of the SU(1,1)/U(1) coset space it is demonstrated that all particles on radial geodesics share a time-like variable that is related to particle proper time. Hamilton's equations of motion are identified with respect to this time-like variable. The Hamiltonian is quadratic on the canonical position and momentum, suggesting a well defined quantization procedure for the position, momentum, proper time and energy of the particles on radial geodesics. 
  We consider the astrophysical and cosmological implications of the existence of a minimum density and mass due to the presence of the cosmological constant. If there is a minimum length in nature, then there is an absolute minimum mass corresponding to a hypothetical particle with radius of the order of the Planck length. On the other hand, quantum mechanical considerations suggest a different minimum mass. These particles associated with the dark energy can be interpreted as the "quanta" of the cosmological constant. We study the possibility that these particles can form stable stellar-type configurations and the Jeans and Chandrasekhar masses are estimated. From the requirement of the energetic stability of the minimum density configuration on a macroscopic scale one obtains a mass of the order of $10^{55}$ g, of the same order of magnitude as the mass of the universe. Furthermore we present a representation of the cosmological constant in terms of 'classical' fundamental constants. 
  We study the Einstein-Maxwell system of equations in spherically symmetric gravitational fields for static interior spacetimes. The condition for pressure isotropy is reduced to a recurrence equation with variable, rational coefficients. We demonstrate that this difference equation can be solved in general using mathematical induction. Consequently we can find an explicit exact solution to the Einstein-Maxwell field equations. The metric functions, energy density, pressure and the electric field intensity can be found explicitly. Our result contains models found previously including the neutron star model of Durgapal and Bannerji. By placing restrictions on parameters arising in the general series we show that the series terminate and there exist two linearly independent solutions. Consequently it is possible to find exact solutions in terms of elementary functions, namely polynomials and algebraic functions. 
  A scalar self-interacting theory non-linearly coupled with some power of the curvature have a possibility to explain the current smallness of the cosmological constant. Here one concentrate on a massless scalar field in the four-dimensional Fridmann-Robertson-Walker (FRW) spacetime with flat spatial part. One show the phase structure of radiative symmetry breaking and review a dynamical resolution of the cosmological constant problem. 
  To explore the possibility that an inflationary universe can be created out of a stable particle in the laboratory, we consider the classical and quantum dynamics of a magnetic monopole in the thin-shell approximation. Classically there are three types of solutions: stable, collapsing and inflating monopoles. We argue that the transition from a stable monopole to an inflating one could occur either by collision with a domain wall or by quantum tunneling. 
  We study the possibility of occurrence of scalar hair with a non-canonical kinetic term for a static, spherically symmetric asymptotically flat black hole spacetime. We first obtain a general equation for this purpose and then consider various examples for the kinetic term $F(X)$ with $X=-{1\over{2}}\partial^{\mu}\phi\partial_{\mu}\phi$. Our study shows that for a tachyon field with a positive potential, which naturally arises in open string theory, asymptotically flat a static black hole solution does not exist. 
  Some long standing issues concerning the quantum nature of the big bang are resolved in the context of homogeneous isotropic models with a scalar field. Specifically, the known results on the resolution of the big bang singularity in loop quantum cosmology are significantly extended as follows: i) the scalar field is shown to serve as an internal clock, thereby providing a detailed realization of the `emergent time' idea; ii) the physical Hilbert space, Dirac observables and semi-classical states are constructed rigorously; iii) the Hamiltonian constraint is solved numerically to show that the big bang is replaced by a big bounce. Thanks to the non-perturbative, background independent methods, unlike in other approaches the quantum evolution is deterministic across the deep Planck regime. 
  Strong field gravitational lensing in the Brans-Dicke theory has been studied. The deflection angle for photons passing very close to the photon sphere is estimated for the static spherically symmetric space-time of the theory and the position and magnification of the relativistic images are obtained. Modeling the super massive central object of the galaxy by the Brans-Dicke space-time, numerical values of different strong lensing observable are estimated. It is found that against the expectation there is no significant scalar field effect in the strong field observable lensing parameters. This observation raises question on the potentiality of the strong field lensing to discriminate different gravitational theories. 
  We present a scheme of biquaternionic algebrodymamics based on a nonlinear generalization of the Cauchy-Riemann holomorphy conditions considered therein as fundamental field equations. The automorphism group SO(3,C) of the biquaternion algebra acts as a proper Lorentz group on a real space whose coordinates are bilinear in the complex coordinates of biquaternionic vector space. A new invariant of Lorentz transformations then arises - the geometric phase. This invariant can be responsible for the quantum properties of particles associated in this approach with field singularities. Some new notions are introduced, related to ``hidden'' complex dynamics: ``observable'' space-time, the ensemble of identical correlated particles-singularities (``duplicons'') and others. 
  If the Pioneer anomaly has a gravitational origin, it would, according to the equivalence principle, distort the motions of the planets in the Solar System. Since no anomalous motion of the planets have been detected, it is generally believed that the Pioneer anomaly can not originate from a gravitational source in the Solar System. However, this conclusion becomes less obvious when considering models that either imply modifications to gravity at long range or gravitational sources localized to the outer Solar System, given the uncertainty in the orbital parameters of the outer planets. Following the general assumption that the Pioneer spacecrafts move geodesically in a spherically symmetric spacetime metric, we derive the metric disturbance that is needed in order to account for the Pioneer anomaly. We then analyze the residual effects on the astronomical observables of the outer planets that would arise from this metric disturbance, given an arbitrary metric theory of gravity. The computed residuals are much larger than the observed residuals, and we are lead to the conclusion that the Pioneer anomaly can not originate from a metric disturbance and therefore that the motion of the Pioneer spacecrafts must be non-geodesic. Since our results are model independent, they can be applied to rule out any model of the Pioneer anomaly that implies that the Pioneer spacecrafts move geodesically in a perturbed spacetime metric, regardless of the origin of this metric disturbance. 
  In spherically symmetric charged dust, just like in electrically neutral dust, two kinds of singularity may be present: the Big Bang/Crunch (BB/BC) singularity, and shell crossings. Quite unlike in neutral dust, the BB/BC singularity may be avoided. When the absolute value of the charge density r is everywhere small compared to the mass-energy density d (|r| < D = \sqrt{G} d / c^2), the conditions that allow the model to avoid the BB/BC singularity necessarily lead to shell crossings. There exist sets of initial conditions that allow us to avoid both singularities (when either |r| > D everywhere, or |r| = D at the center of symmetry while |r| < D elsewhere). In those cases, both kinds of singularity may be avoided for a sufficiently long period that a body of charged dust may go through the tunnel between the singularities in the maximally extended Reissner -- Nordstrom spacetime, to emerge into another asymptotically flat region. An explicit example of such a configuration is presented and discussed. It does not contradict any astrophysical constraints. 
  We prove that aligned Petrov type D purely magnetic perfect fluids are necessarily locally rotationally symmetric and hence are all explicitly known. 
  We first roughly present a summary of the optico-mechanical analogy, which has always been so profitable in physics. Then we put forward a geometrodynamical formulation of gravity suitable to our intentions, both formally and conceptually. We present difficulties in some approaches to canonically quantize gravity which can be ammended by the idea put forward in this paper, which we introduce in the last section. It consists basically in trying to find an intermediary between the quantization step going from the classical superhamiltonian constraint to the Wheeler-DeWitt equation. This is accomplished by inputing interference beyond the WKB approximation, through a sort of Huygens-Fresnel Principle (HFP) in superspace. It turns out that we can derive wave-like character for both domains from this principle by allowing backward angles of diffraction, and what is more, approximate to a high degree of accuracy Feynman's path integral method in any domain. 
  Considering that gravitational force might deviate from Newton's inverse-square law (ISL) and become much stronger in small scale, we propose a kind of optical spectroscopy experiment to detect this possible deviation and take electronic, muonic and tauonic hydrogen atoms as examples. This experiment might be used to indirectly detect the deviation of ISL down to nanometer scale and to explore the possibility of three extra dimensions in ADD's model, while current direct gravity tests cannot break through micron scale and go beyond two extra dimensions scenario. 
  In this paper we analyze certain aspects of virtual black holes. We see how quantum bubble is the representative of creation and annihilation of a pair of virtual black holes that satisfy the Wheeler-DeWitt equation. We calculate the loss of quantum coherence due to these virtual black holes and also see how these virtual black holes give space-time an intrinsic entropy. 
  In this paper we explicitly work out the effects that a spherically symmetric distribution of dark matter with constant density would induce on the Keplerian orbital elements of the Solar System planets and compare them with the latest results in planetary orbit determination from the EPM2004 ephemerides. It turns out that the longitudes of perihelia and the mean longitudes are affected by secular precessions. The resulting upper bounds on dark matter density, obtained from the EPM2004 formal errors in the determined mean longitude shifts over 90 years, lie in the range 10^-19-10^-20 g cm^-3 with a peak of 10^-22 g cm^-3 for Mars. Suitable combinations of the planetary mean longitudes and perihelia, which cancel out the aliasing impact of some of the unmodelled or mismodelled forces of the dynamical models of EPM2004, yield a global upper bound of 7 10^-20 g cm^-3 and 4 10^-19 g cm^-3, respectively. 
  In an experiment to simulate the conditions in high optical power advanced gravitational wave detectors such as Advanced LIGO, we show that strong thermal lenses form in accordance with predictions and that they can be compensated using an intra-cavity compensation plate heated on its cylindrical surface. We show that high finesse ~1400 can be achieved in cavities with internal compensation plates, and that the cavity mode structure can be maintained by thermal compensation. It is also shown that the measurements allow a direct measurement of substrate optical absorption in the test mass and the compensation plate. 
  A new gauge-invariant criterion for stability against inhomogeneous perturbations of de Sitter space is applied to scenarios of dark energy and inflation in scalar-tensor gravity. The results extend previous studies. 
  In this expository paper we address the question of whether, and to what extent, the cosmological expansion influences the dynamics on small scales (as compared to cosmological ones), particularly in our Solar System. We distinguish between dynamical and kinematical effects and critically review the status of both as presented in the current literature. 
  We consider defining time as a function of a cyclical field, an abstraction of a clock. The definition of time corresponds to a novel interpretation of the relationship between space-time coordinates of observers at different locations in space. As a first test of the utility of this definition, we show that it leads to a Lorentz covariant description of space-time. This derivation of Lorenz covariance provides a starting point for considering more general constructions that relate to physical laws. The definition of time couples time to space, making time not orthogonal to space, and making dynamics a result of geometry, providing a vehicle for curved space-time theories that generalize general relativity. 
  Black Holes have always played a central role in investigations of quantum gravity. This includes both conceptual issues such as the role of classical singularities and information loss, and technical ones to probe the consistency of candidate theories. Lacking a full theory of quantum gravity, such studies had long been restricted to black hole models which include some aspects of quantization. However, it is then not always clear whether the results are consequences of quantum gravity per se or of the particular steps one had undertaken to bring the system into a treatable form. Over a little more than the last decade loop quantum gravity has emerged as a widely studied candidate for quantum gravity, where it is now possible to introduce black hole models within a quantum theory of gravity. This makes it possible to use only quantum effects which are known to arise also in the full theory, but still work in a rather simple and physically interesting context of black holes. Recent developments have now led to the first physical results about non-rotating quantum black holes obtained in this way. Restricting to the interior inside the Schwarzschild horizon, the resulting quantum model is free of the classical singularity, which is a consequence of discrete quantum geometry taking over for the continuous classical space-time picture. This fact results in a change of paradigm concerning the information loss problem. The horizon itself can also be studied in the quantum theory by imposing horizon conditions at the level of states. Thereby one can illustrate the nature of horizon degrees of freedom and horizon fluctuations. All these developments allow us to study the quantum dynamics explicitly and in detail which provides a rich ground to test the consistency of the full theory. 
  In recent twenty years, loop quantum gravity, a background independent approach to unify general relativity and quantum mechanics, has been widely investigated. We consider the quantum dynamics of a real massless scalar field coupled to gravity in this framework. A Hamiltonian operator for the scalar field can be well defined in the coupled diffeomorphism invariant Hilbert space, which is both self-adjoint and positive. On the other hand, the Hamiltonian constraint operator for the scalar field coupled to gravity can be well defined in the coupled kinematical Hilbert space. There are 1-parameter ambiguities due to scalar field in the construction of both operators. The results heighten our confidence that there is no divergence within this background independent and diffeomorphism invariant quantization approach of matter coupled to gravity. Moreover, to avoid possible quantum anomaly, the master constraint programme can be carried out in this coupled system by employing a self-adjoint master constraint operator on the diffeomorphism invariant Hilbert space. 
  The standard cosmological model posits a spatially flat universe of infinite extent. However, no observation, even in principle, could verify that the matter extends to infinity. In this work we model the universe as a finite spherical ball of dust and dark energy, and obtain a lower limit estimate of its mass and present size: the mass is at least 5 x 10^23 solar masses and the present radius is at least 50 Gly. If we are not too far from the dust-ball edge we might expect to see a cold spot in the cosmic microwave background, and there might be suppression of the low multipoles in the angular power spectrum. Thus the model may be testable, at least in principle. We also obtain and discuss the geometry exterior to the dust ball; it is Schwarzschild-de Sitter with a naked singularity, and provides an interesting picture of cosmogenesis. Finally we briefly sketch how radiation and inflation eras may be incorporated into the model. 
  By incorporating the holographic principle in a time-depending Lambda-term cosmology, new physical bounds on the arbitrary parameters of the model can be obtained. Considering then the dark energy as a purely geometric entity, for which no equation of state has to be introduced, it is shown that the resulting range of allowed values for the parameters may explain both the coincidence problem and the universe accelerated expansion, without resorting to any kind of additional structures. 
  We describe a generic infrastructure for time evolution simulations in numerical relativity using multiple grid patches. After a motivation of this approach, we discuss the relative advantages of global and patch-local tensor bases. We describe both our multi-patch infrastructure and our time evolution scheme, and comment on adaptive time integrators and parallelisation. We also describe various patch system topologies that provide spherical outer and/or multiple inner boundaries.   We employ penalty inter-patch boundary conditions, and we demonstrate the stability and accuracy of our three-dimensional implementation. We solve both a scalar wave equation on a stationary rotating black hole background and the full Einstein equations. For the scalar wave equation, we compare the effects of global and patch-local tensor bases, different finite differencing operators, and the effect of artificial dissipation onto stability and accuracy. We show that multi-patch systems can directly compete with the so-called fixed mesh refinement approach; however, one can also combine both. For the Einstein equations, we show that using multiple grid patches with penalty boundary conditions leads to a robustly stable system. We also show long-term stable and accurate evolutions of a one-dimensional non-linear gauge wave. Finally, we evolve weak gravitational waves in three dimensions and extract accurate waveforms, taking advantage of the spherical shape of our grid lines. 
  In this paper, utilizing M{\o}ller's energy-momentum complex, we explicitly evaluate the energy and momentum density associated with a metric describing a four-dimensional, Schwarzschild-like, spacetime derived from an effective gravity coupled with a U(1) gauge field in the context of a D3-brane dynamics in the classical regime, i.e., between the asymptotic and the Planck regime. 
  In this paper we discuss a new method which can be used to obtain arbitrarily accurate analytical expressions for the deflection angle of light propagating in a given metric. Our method works by mapping the integral into a rapidly convergent series and provides extremely accurate approximations already to first order. We have derived a general first order formula for a generic spherically symmetric static metric tensor and we have tested it in four different cases. 
  The energy and momentum distributions in the dyadosphere of a Reissner-Nordstrom black hole are evaluated. The Moller's energy-momentum complex is employed for this computation. The spacetime under study is modified due to the effects of vacuum fluctuations in the dyadosphere. Therefore, the corrected Reissner-Nordstrom black hole metric takes into account the first contribution of the weak field limit of one-loop QED. Furthermore, a comparison and a consequent connection between our results that those already existing in the literature is provided. We hypothesize that when the energy distribution is of specific form there is a relation that connects the coefficients in the Einstein's prescription with those in the Moller's prescription. 
  Expressions for G-dot are considered in a multidimensional model with an Einstein internal space and a multicomponent perfect fluid. In the case of two non-zero curvatures without matter, a mechanism for prediction of small G-dot is suggested. The result is compared with exact  (1+3+6)-dimensional solutions. A two-component example with two matter sources (dust + 5-brane) and two Ricci-flat factor spaces is also considered. 
  We give a model of the higher-dimensional spherically symmetric gravitational collapse of a dust cloud in Einstein-Gauss-Bonnet gravity. A simple formulation of the basic equations is given for the spacetime $M \approx M^2 \times K^{n-2}$ with a perfect fluid and a cosmological constant. This is a generalization of the Misner-Sharp formalism of the four-dimensional spherically symmetric spacetime with a perfect fluid in general relativity. The whole picture and the final fate of the gravitational collapse of a dust cloud differ greatly between the cases with $n=5$ and $n \ge 6$. There are two families of solutions, which we call plus-branch and the minus-branch solutions. Bounce inevitably occurs in the plus-branch solution for $n \ge 6$, and consequently singularities cannot be formed. Since there is no trapped surface in the plus-branch solution, the singularity formed in the case of $n=5$ must be naked. In the minus-branch solution, naked singularities are massless for $n \ge 6$, while massive naked singularities are possible for $n=5$. In the homogeneous collapse represented by the flat Friedmann-Robertson-Walker solution, the singularity formed is spacelike for $n \ge 6$, while it is ingoing-null for $n=5$. In the inhomogeneous collapse with smooth initial data, the strong cosmic censorship hypothesis holds for $n \ge 10$ and for $n=9$ depending on the parameters in the initial data, while a naked singularity is always formed for $5 \le n \le 8$. These naked singularities can be globally naked when the initial surface radius of the dust cloud is fine-tuned, and then the weak cosmic censorship hypothesis is violated. 
  We generalize Starobinskii's stochastic technique to the theory of a massless, minimally coupled scalar interacting with a massless fermion in a locally de Sitter geometry. The scalar is an ``active'' field that can engender infrared logarithms. The fermion is a ``passive'' field that cannot cause infrared logarithms but which can carry them, and which can also induce new interactions between the active fields. The procedure for dealing with passive fields is to integrate them out, then stochastically simplify the resulting effective action following Starobinski\u{\i}. Because Yukawa theory is quadratic in the fermion this can be done explicitly using the classic solution of Candelas and Raine. We check the resulting stochastic formulation against an explicit two loop computation. We also derive a nonperturbative, leading log result for the stress tensor. Because the scalar effective potential induced by fermions is unbounded below, back-reaction from this model might dynamically cancel an arbitrarily large cosmological constant. 
  This paper is a set of notes that we wrote concerning the first version of Emergent Gravity [gr-qc/0602022]. It is our version of an exercise that we proposed to some of our students. The idea was to find mathematical errors and inconsistencies on some recent articles published in scientific journals and in the arXiv, and we did. 
  A dynamical symmetry for spherical collapse has been studied using a linear transformation of the initial data set (mass and kinetic energy function) and the area radius. With proper choice of the initial area radius, the evolution as well as the physical parameters namely energy density and shear remain invariant both initially and at any time instant. Finally, it is found that the final outcome of collapse depends on the initial choice of the area radius. 
  In part I we study quantum modified photon trajectories in a Schwarzschild blackhole spacetime. The photon vacuum polarization effect in curved spacetime leads to birefringence, i.e. the photon velocity becomes c+/-dc depending on its polarization. This velocity shift then results in modified photon trajectories. In this work we give an introduction to this quantum effect in relativity and we study its effects in Schwarzschild spacetime for critical orbits. Some key results are that the critical orbits are shifted depending on polarization and the event horizon remains fixed.   In Part II we use the 2+1d Nambu-Jona-Lasino NJL model to study the superfluid behaviour of two-dimensional quark matter. We begin with an introduction to QCD its symmetries and the NJL model. We then go on to study the 2+1d NJL model. We show that at high density the 2+1d NJL model represents a relativistic gapless thin film BCS superfluid. 
  Motivated by a recent paper of Louko and Molgado, we consider a simple system with a single classical constraint R(q)=0. If q_l denotes a generic solution to R(q)=0, our examples include cases where R'(q_l)\ne 0 (regular constraint) and R'(q_l)=0 (irregular constraint) of varying order as well as the case where R(q)=0 for an interval, such as a \leq q \leq b. Quantization of irregular constraints is normally not considered; however, using the projection operator formalism we provide a satisfactory quantization which reduces to the constrained classical system when \hbar \to 0. It is noteworthy that irregular constraints change the observable aspects of a theory as compared to strictly regular constraints. 
  A numerical solution scheme for the Einstein field equations based on generalized harmonic coordinates is described, focusing on details not provided before in the literature and that are of particular relevance to the binary black hole problem. This includes demonstrations of the effectiveness of constraint damping, and how the time slicing can be controlled through the use of a source function evolution equation. In addition, some results from an ongoing study of binary black hole coalescence, where the black holes are formed via scalar field collapse, are shown. Scalar fields offer a convenient route to exploring certain aspects of black hole interactions, and one interesting, though tentative suggestion from this early study is that behavior reminiscent of "zoom-whirl" orbits in particle trajectories is also present in the merger of equal mass, non-spinning binaries, with appropriately fine-tuned initial conditions. 
  Euclidean quantum measure in Regge calculus with independent area tensors is considered using example of the Regge manifold of a simple structure. We go over to integrations along certain contours in the hyperplane of complex connection variables. Discrete connection and curvature on classical solutions of the equations of motion are not, strictly speaking, genuine connection and curvature, but more general quantities and, therefore, these do not appear as arguments of a function to be averaged, but are the integration (dummy) variables. We argue that upon integrating out the latter the resulting measure can be well-defined on physical hypersurface (for the area tensors corresponding to certain edge vectors, i.e. to certain metric) as positive and having exponential cutoff at large areas on condition that we confine ourselves to configurations which do not pass through degenerate metrics. 
  Using the Effective One Body approach, that includes nonperturbative resummed estimates for the damping and conservative parts of the compact binary dynamics, we compute the recoil during the late inspiral and the subsequent plunge of non-spinning black holes of comparable masses moving in quasi-circular orbits. Further, using a prescription that smoothly connects the plunge phase to a perturbed single black hole, we obtain an estimate for the total recoil associated with the binary black hole coalescence. We show that the crucial physical feature which determines the magnitude of the terminal recoil is the presence of a ``burst'' of linear momentum flux emitted slightly before coalescence. When using the most natural expression for the linear momentum flux during the plunge, together with a Taylor-expanded $(v/c)^4$ correction factor, we find that the maximum value of the terminal recoil is $\sim 74$ km/s and occurs for a mass ratio $m_2/m_1 \simeq 0.38$. We comment, however, on the fact that the above `best bet estimate' is subject to strong uncertainties because the location and amplitude of the crucial peak of linear momentum flux happens at a moment during the plunge where most of the simplifying analytical assumptions underlying the Effective One Body approach are no longer justified. Changing the analytical way of estimating the linear momentum flux, we find maximum recoils that range between 49 and 172 km/s. (Abridged) 
  This article examines how the physical presence of field energy and particulate matter could influence the topological properties of space time. The theory is developed in terms of vector and matrix equations of exterior differential forms. The topological features and the dynamics of such exterior differential systems are studied with respect to processes of continuous topological evolution. The theory starts from the sole postulate that field properties of a Physical Vacuum (a continuum) can be defined in terms of a vector space domain, of maximal rank, infinitesimal neighborhoods, that supports a Basis Frame as a 4 x 4 matrix of C2 functions with non-zero determinant. The basis vectors of such Basis Frames exhibit differential closure. The particle properties of the Physical Vacuum are defined in terms of topological defects (or compliments) of the field vector space defined by those points where the maximal rank, or non-zero determinant, condition fails. The topological universality of a Basis Frame over infinitesimal neighborhoods can be refined by particular choices of a subgroup structure of the Basis Frame, [B]. It is remarkable that from such a universal definition of a Physical Vacuum, specializations permit the deduction of the field structures of all four forces, from gravity fields to Yang Mills fields, and associate the origin of topological charge and topological spin to the Affine torsion coefficients of the induced Cartan Connection matrix [C] of 1-forms. 
  GGR News:   GGR program at the APS April meeting in Dallas   We hear that..., 100 years ago, by Jorge Pullin  Research Briefs:   What's new in LIGO, by David Shoemaker   LISA Pathfinder, by Paul McNamara   Recent progress in binary black hole simulations, by Thomas Baumgarte  Conference reports:   Workshop on Emergence of Spacetime, by Olaf Dreyer   Quantum gravity subprogram at the Isaac Newton Institute, by Jorma Louko   Global problems in Math Relativity at the Newton Institute, by Jim Isenberg   Loops '05, by Thomas Thiemann   Numrel 2005, by Scott Hawley and Richard Matzner   Apples With Apples Workshop in Argentina, by Sascha Husa 
  We explore the possibility of replacing point set topology by higher category theory and topos theory as the foundation for quantum general relativity. We discuss the BC model and problems of its interpretation, and connect with the construction of causal sites. 
  All homothetic self-similar solutions of the Brans-Dicke scalar field in three-dimensional spacetime with circular symmetry are found in closed form. 
  The analysis of singular regions in the NUT solutions carried out in the recent paper (Manko and Ruiz, 2005 Class. Quantum Grav. 22, p.3555) is now extended to the Demianski-Newman vacuum and electrovacuum spacetimes. We show that the effect which produces the NUT parameter in a more general situation remains essentially the same as in the purely NUT solutions: it introduces the semi-infinite singularities of infinite angular momenta and positive or negative masses depending on the interrelations between the parameters; the presence of the electromagnetic field additionally endows the singularities with electric and magnetic charges. The exact formulae describing the mass, charges and angular momentum distributions in the Demianski-Newman solutions are obtained and concise general expressions P_n=(m+i\nu)(ia)^n, Q_n=(q+ib)(ia)^n for the entire set of the respective Beig-Simon multipole moments are derived. These moments correspond to a unique choice of the integration constant in the expression of the metric function \omega which is different from the original choice made by Demianski and Newman. 
  We solve exactly the Regge-Wheeler equation for axial perturbations of the Schwarzschild metric in the black hole interior in terms of Heun functions and give a description of the spectrum and the eigenfunctions of the interior problem. The phenomenon of attraction and repulsion of the discrete eigenvalues of gravitational waves is discovered. 
  We use an idea of Wang and Yau to give a new definition of quasi-local mass for a topological sphere in an initial date set. The new definition modifies Brown-York's definition by using certain spinor norm as lapse function. And it requires mean curvature of the topological sphere satisfies apparent horizon conditions, the topological sphere can be isometrically into Euclidean 3-space and mean curvature of the initial date set does not change sign. The positivity holds if we further assume the image of the topological sphere in Euclidean 3-space has nonnegative mean curvature. 
  This article has been withdrawn by the author. We initially believed we could solve the obvious problem of conservation of energy which affects a model such as the one we proposed by postulating an absence of direct interactions between the two types of matter involved. However, it seems that we cannot escape the conclusion that the introduction of negative action matter obeying our postulates would prevent the formulation of a principle of conservation of energy that would be in line with traditional expectations. Unless we can make sense of a physics in which one type of energy or action can transform into another without any compensation, there is no hope that negative action can ever become an element of a truly consistent model of physical reality. As we do not believe that this goal is attainable, we consider our conclusion as a sufficient motive for withdrawal of this paper. 
  The quantization of the family of linearly polarized Gowdy $T^3$ spacetimes is discussed in detail, starting with a canonical analysis in which the true degrees of freedom are described by a scalar field that satisfies a Klein-Gordon type equation in a fiducial time dependent background. A time dependent canonical transformation, which amounts to a change of the basic (scalar) field of the model, brings the system to a description in terms of a Klein-Gordon equation on a background that is now static, although subject to a time dependent potential. The system is quantized by means of a natural choice of annihilation and creation operators. The quantum time evolution is considered and shown to be unitary, allowing both the Schr\"odinger and Heisenberg pictures to be consistently constructed. This has to be contrasted with previous treatments for which time evolution failed to be implementable as a unitary transformation. Possible implications for both canonical quantum gravity and quantum field theory in curved spacetime are commented. 
  We show that considering time measured by an observer to be a function of a cyclical field (an abstract version of a clock) is consistent with Hamilton's and Lagrange's equations of motion for a one dimensional space manifold. The derivation may provide a simple understanding of the conventions that are used in defining the relationship between independent and dependent variables in the Lagrangian and Hamiltonian formalisms. These derivations of the underlying principles of classical mechanics are steps on the way to discussions of physical laws and interactions in ZM theory. 
  After a brief review of spin networks and their interpretation as wave functions for the (space) geometry, we discuss the renormalisation of the area operator in loop quantum gravity. In such a background independent framework, we propose to probe the structure of a surface through the analysis of the coarse-graining and renormalisation flow(s) of its area. We further introduce a procedure to coarse-grain spin network states and we quantitatively study the decrease in the number of degrees of freedom during this process. Finally, we use these coarse-graining tools to define the correlation and entanglement between parts of a spin network and discuss their potential interpretation as a natural measure of distance in such a state of quantum geometry. 
  This lecture provides some introduction to perfect fluid dynamics within the framework of general relativity. The presentation is based on the Carter-Lichnerowicz approach. It has the advantage over the more traditional approach of leading very straightforwardly to important conservation laws, such as the relativistic generalizations of Bernoulli's theorem or Kelvin's circulation theorem. It also permits to get easily first integrals of motion which are particularly useful for computing equilibrium configurations of relativistic stars in rotation or in binary systems. The presentation is relatively self-contained and does not require any a priori knowledge of general relativity. In particular, the three types of derivatives involved in relativistic hydrodynamics are introduced in detail: this concerns the Lie, exterior and covariant derivatives. 
  We study how inhomogeneities modify large scale parameters in General Relativity. For a particular model, we obtain exact results: we compare an infinite string of extremal black holes to a corresponding smooth line with the same mass and charge in five dimensions. We find that the effective energy density does not differ significantly. 
  We attempt a justification of a generalisation of the consistent histories programme using a notion of probability that is valid for all complete sets of history propositions. This consists of introducing Cox's axioms of probability theory and showing that our candidate notion of probability obeys them. We also give a generalisation of Bayes' theorem and comment upon how Bayesianism should be useful for the quantum gravity/cosmology programmes. 
  We give the equations governing the shear evolution in Bianchi spacetimes for general f(R)-theories of gravity. We consider the case of R^n-gravity and perform a detailed analysis of the dynamics in Bianchi I cosmologies which exhibit local rotational symmetry. We find exact solutions and study their behaviour and stability in terms of the values of the parameter n. In particular, we found a set of cosmic histories in which the universe is initially isotropic, then develops shear anisotropies which approaches a constant value. 
  We study the possibility to construct an observationally viable scenario where both early Inflation and the recently detected accelerated expansion of the universe can be explained by using a single scalar field associated with the Tachyon. The Reheating phase becomes crucial to enable us to have a consistent cosmology and also to get a second accelerated expansion period. A discussion using an exponential potential is presented. 
  Motivated by the conjectured Penrose inequality and by the work of Hawking, Geroch, Huisken and Ilmanen in the null and the Riemannian case, we examine necessary conditions on flows of two-surfaces in spacetime under which the Hawking quasilocal mass is monotone. We focus on a subclass of such flows which we call uniformly expanding, which can be considered for null as well as for spacelike directions. In the null case, local existence of the flow is guaranteed. In the spacelike case, the uniformly expanding condition leaves a 1-parameter freedom, but for the whole family, the embedding functions satisfy a forward-backward parabolic system for which local existence does not hold in general. Nevertheless, we have obtained a generalization of the weak (distributional) formulation of this class of flows, generalizing the corresponding step of Huisken and Ilmanen's proof of the Riemannian Penrose inequality. 
  We compare and contrast the basic principles of two philosophies: Bayesianism and relationalism. These two philosophies are both based upon criteria of rationality. The analogy invoked in such a comparison seems rather apt when discussing tentative proofs of quantum nonlocality. We argue that Bayesianism is almost to quantum theory, what general covariance is to general relativity. This is because the Bayesian interpretation of quantum theory can be given a relational flavour. 
  The Pioneer 10/11 spacecraft yielded a very accurate navigation that was limited only by a small, anomalous frequency drift of their carrier signals received by the NASA Deep Space Network (DSN). This discrepancy, evident in the data for both spacecraft, was interpreted as an approximately constant acceleration and has become known as the Pioneer anomaly. The origin of this anomaly is as of yet unknown. Recent efforts to explain the effect included a search for independent confirmation, analyses of conventional mechanisms, even ideas rooted in new physics and proposals for a dedicated mission. We assert that before any discussion of new physics and/or a dedicated mission can take place, one must analyze the entire set of radiometric Doppler data received from Pioneers 10 and 11. We report on our efforts to recover and utilize the complete set of radio Doppler and telemetry records of both spacecraft. The collection of radio Doppler data for both missions is now complete; we are ready to begin its evaluation. We also made progress utilizing the recently recovered Pioneer telemetry data. We present a strategy for studying the effect of on-board generated small forces with this telemetry data, in conjunction with the analysis of the entire set of the Pioneer Doppler data. We report on the preparations for the upcoming analysis of the newly recovered data with the ultimate goal of determining the origin of the Pioneer anomaly. Finally, we discuss implications of our on-going research of the Pioneer anomaly for other missions, most notably for New Horizons, NASA's recently launched mission to Pluto. 
  In this paper we construct the set of quantum mechanical observables in the Fedosov *-formalism (a coordinate invariant way to do quantum mechanics on any manifold M) of a single free particle that lives on a constant curvature manifold with metric signature (p,q). This was done for most but not all constant curvature manifolds. We show that the algebra of all observables in n=p+q dimensions is SO(p+1,q+1) in a nonperturbative calculation. A subgroup of this group is identified as the analogue of the Poincare group in Minkowski space i.e. it is the space of symmetries on the manifolds considered. We then write down a Klein-Gordon (KG) equation given by the the equation p^2|phi>=m^2|phi> for the set of allowed physical states. This result is consistent with previous results on AdS. Furthermore we lay out the standard scheme for the free KG field from the single particle theory. Furthermore we argue that this scheme will work on a general space-time. 
  We propose a novel formalism for inflation from a 5D vacuum state which could explain both, seeds of matter and magnetic fields in the early universe. 
  We investigate quantum tunnelling methods for calculating black hole temperature, specifically the null geodesic method of Parikh and Wilczek and the Hamilton-Jacobi Ansatz method of Angheben et al. We consider application of these methods to a broad class of spacetimes with event horizons, inlcuding Rindler and non-static spacetimes such as Kerr-Newman and Taub-NUT. We obtain a general form for the temperature of Taub-NUT-Ads black holes that is commensurate with other methods. We examine the limitations of these methods for extremal black holes, taking the extremal Reissner-Nordstrom spacetime as a case in point. 
  It is shown that two gravitating scalar fields may form a thick brane in 5D spacetime. The necessary condition for the existence of such a regular solution is that the scalar fields potential must have local and global minima. 
  We are investigating if the Jeans instability criteria mandading a low entropy,low temperature initial pre inflation state configuration can be recociled with thermal conditions of temperature at or above ten to the 12 power Kelvin, or higher,when cosmic inflation physics takes over. We justify this by pointing to the Akshenkar, Pawlowski, and Singh (2006) article about a prior universe being modeled via their quantum bounce which states that this prior universe geometrically can be modeled via a discretized Wheeler - De Witt equation. This allows a way of getting around the fact that conventional cosmological CMB is limited by a barrier as of a red shift limit of about z = 1000,as to photons, and to come up with a working model of quintessence scalar fields which permits relic generation of dark matter/dark energy as well as relic gravitons. 
  In this short review of Doubly Special Relativity I describe first the relations between DSR and (quantum) gravity. Then I show how, in the case of a field theory with curved momentum space, the Hopf algebra of symmetries naturally emerges. I conclude with some remarks concerning DSR phenomenology and description of open problems. 
  Weak isolated horizon boundary conditions have been relaxed supposedly to their weakest form such that both zeroth and the first law of black hole mechanics still emerge, thus making the formulation more amenable for applications in both analytic and numerical Relativity. As an additional gain it explicitly brings the non-extremal and extremal black holes at the same footing. 
  Bessel X-waves, or Bessel beams, have been extensively studied in last years, especially with regard to the topic of superluminality in the propagation of a signal. However, in spite of many efforts devoted to this subject, no definite answer has been found, mainly for lack of an exact definition of signal velocity. The purpose of the present work is to investigate the field of existence of Bessel beams in order to overcome the specific question related to the definition of signal velocity. Quite surprisingly, this field of existence can be represented in the Minkowski space-time by a Super-Light Cone which wraps itself around the well-known Light Cone. So, the change in the upper limit of the light velocity does not modify the fundamental low of the relativity and the causal principle. 
  Singularities in the dark energy late universe are discussed, under the assumption that the Lagrangian contains the Einstein term R plus a modified gravity term of the form R^\alpha, where \alpha is a constant. It is found, similarly as in the case of pure Einstein gravity [I. Brevik and O. Gorbunova, Gen. Rel. Grav. 37 (2005), 2039], that the fluid can pass from the quintessence region (w>-1) into the phantom region (w<-1) as a consequence of a bulk viscosity varying with time. It becomes necessary now, however, to allow for a two-fluid model, since the viscosities for the two components vary differently with time. No scalar fields are needed for the description of the passage through the phantom barrier. 
  We apply the method of comparison equations to study cosmological perturbations during inflation, obtaining the full power spectra of scalar and tensor perturbations to first and to second order in the slow-roll parameters. We compare our results with those derived by means of other methods, in particular the Green's function method and the improved WKB approximation, and find agreement for the slow-roll structure. The method of comparison equations, just as the improved WKB approximation, can however be applied to more general situations where the slow-roll approximation fails. 
  In this paper, we consider both Einstein's theory of general relativity and the teleparallel gravity (the tetrad theory of gravitation) analogs of the energy-momentum definition of M{\o}ller in order to explicitly evaluate the energy distribution (due to matter and fields including gravity) associated with a general black hole model which includes several well-known black holes. To calculate the special cases of energy distribution, here we consider eight different types of black hole models such as anti-de Sitter C-metric with spherical topology, charged regular black hole, conformal scalar dyon black hole, dyadosphere of a charged black hole, regular black hole, charged topological black hole, charged massless black hole with a scalar field, and the Schwarzschild-de Sitter space-time. Our teleparallel gravitational result is also independent of the teleparallel dimensionless coupling constant, which means that it is valid not only in teleparallel equivalent of general relativity but also in any teleparallel model. This paper also sustains (a) the importance of the energy-momentum definitions in the evaluation of the energy distribution of a given spacetime and (b) the viewpoint of Lessner that the M{\o}ller energy-momentum complex is the powerful concept to calculate energy distribution in a given space-time. 
  An explicit and complete set of constants of the motion are constructed algorithmically for Friedmann-Lema\^{i}tre-Robertson-Walker (FLRW) models consisting of an arbitrary number of non-interacting species. The inheritance of constants of the motion from simpler models as more species are added is stressed. It is then argued that all FLRW models admit what amounts to a unique candidate for a gravitational epoch function (a dimensionless scalar invariant derivable from the Riemann tensor without differentiation which is monotone throughout the evolution of the universe). The same relations that lead to the construction of constants of the motion allow an explicit evaluation of this function. In the simplest of all models, the $\Lambda$CDM model, it is shown that the epoch function exists for all models with $\Lambda > 0$, but for almost no models with $\Lambda \leq 0$. 
  The entropy of a black hole can be obtained by counting states in loop quantum gravity. The dominant term depends on the Immirzi parameter involved in the quantization and is proportional to the area of the horizon, while there is a logarithmic correction with coefficient -1/2. 
  The pure-gravity sector of the minimal Standard-Model Extension is studied in the limit of Riemann spacetime. A method is developed to extract the modified Einstein field equations in the limit of small metric fluctuations about the Minkowski vacuum, while allowing for the dynamics of the 20 independent coefficients for Lorentz violation. The linearized effective equations are solved to obtain the post-newtonian metric. The corresponding post-newtonian behavior of a perfect fluid is studied and applied to the gravitating many-body system. Illustrative examples of the methodology are provided using bumblebee models. The implications of the general theoretical results are studied for a variety of existing and proposed gravitational experiments, including lunar and satellite laser ranging, laboratory experiments with gravimeters and torsion pendula, measurements of the spin precession of orbiting gyroscopes, timing studies of signals from binary pulsars, and the classic tests involving the perihelion precession and the time delay of light. For each type of experiment considered, estimates of the attainable sensitivities are provided. Numerous effects of local Lorentz violation can be studied in existing or near-future experiments at sensitivities ranging from parts in 10^4 down to parts in 10^{15}. 
  In this paper we write down the equation for a scalar conformally coupled field simultaneously for de Sitter (dS), anti-de Sitter (AdS) and Minkowski spacetime in d-dimensions. The curvature dependence appears in a very simple way through a conformal factor. As a consequence the process of curvature free limit, including wave functions limit and two-points functions, turns to be a straightforward issue. We determine a set of modes, that we call de Sitter plane waves, which become ordinary plane waves when the curvature vanishes. 
  We show that non-zero masses for a spin-1 graviton (called graviphoton) leads to considerable gravitomagnetic fields around rotating mass densities, which are not observed. The solution to the problem is found by an equivalent graviphoton mass which depends on the local mass density to ensure the principle of equivalence. This solution, derived from Einstein-Proca equations, has important consequences such as a correction term for the Cooper-pair mass anomaly reported by Tate among many others. Similar results were obtained for the photon mass which is then proportional to the charge density in matter. For the case of coherent matter the predicted effects have been experimentally observed by the authors. 
  It is well known that a rotating superconductor produces a magnetic field proportional to its angular velocity. The authors conjectured earlier, that in addition to this so-called London moment, also a large gravitomagnetic field should appear to explain an apparent mass increase of Niobium Cooper-pairs. This phenomenon was indeed observed and induced acceleration fields outside the superconductor in the order of about 10^-4 g were found. The field appears to be directly proportional to the applied angular acceleration of the superconductor following our theoretical motivations. If confirmed, a gravitomagnetic field of measurable magnitude was produced for the first time in a laboratory environment. These results may open up a new experimental window on testing general relativity and its consequences using coherent matter. 
  We study the numerical propagation of waves through future null infinity in a conformally compactified spacetime. We introduce an artificial cosmological constant, which allows us some control over the causal structure near null infinity. We exploit this freedom to ensure that all light cones are tilted outward in a region near null infinity, which allows us to impose excision-style boundary conditions in our finite difference code. In this preliminary study we consider electromagnetic waves propagating in a static, conformally compactified spacetime. 
  We present a general relativistic framework for studying gravitational effects in quantum mechanical phenomena. We concentrate our attention on the case of ultra-relativistic, spin-1/2 particles propagating in Kerr spacetime. The two-component Weyl equation with general relativistic corrections is obtained in the case of a slowly rotating, weak gravitational field. Our approach is also applied to neutrino oscillations in the presence of a gravitational field. The relative phase of two different mass eigenstates is calculated in radial propagation, and the result is compared with those of previous works. 
  We present a simple approach for obtaining Kerr interior solutions with the help of the Newman-Janis algorithm (NJA) starting with static space-times describing physically sensible interior Schwarzschild solutions. In this context, the Darmois-Israel (DI) junction conditions are analyzed. Starting from the incompressible Schwarzschild solution, a class of Kerr interior solutions is presented, together with a discussion of the slowly rotating limit. The energy conditions are discussed for the solutions so obtained. Finally, the NJA algorithm is applied to the static, anisotropic, conformally flat solutions found by Stewart leading to interior Kerr solutions with oblate spheroidal boundary surfaces. 
  We present a solution to the time discontinuity paradox in rotating reference frames by postulating that time is periodic. A kinematic restriction is enforced that requires the discontinuity to be an integral number of the periodicity of time. Quantized radii emerge for which the associated tangential velocities are less than the speed of light. Using the de Broglie relationship, we show that quantum theory may determine the periodicity of time. A rotating Kerr black hole and a rigidly rotating disk of dust are also considered. 
  Thanks to the new generation of gravitational wave detectors LIGO and VIRGO, the theory of general relativity will face new and important confrontations to observational data with unprecedented precision. Indeed the detection and analysis of the gravitational waves from compact binary star systems requires beforehand a very precise solution of the two-body problem within general relativity. The approximation currently used to solve this problem is the post-Newtonian one, and must be pushed to high order in order to describe with sufficient accuracy (given the sensitivity of the detectors) the inspiral phase of compact bodies, which immediately precedes their final merger. The resulting post-Newtonian ``templates'' are currently known to 3.5PN order, and are used for searching and deciphering the gravitational wave signals in VIRGO and LIGO. 
  We construct a simple accretion model of a rotating pressureless gas sphere onto a Schwarzschild black hole. We show how to build analytic solutions in terms of Jacobi elliptic functions. This construction represents a general relativistic generalisation of the Newtonian accretion model first proposed by Ulrich (1976). In exactly the same form as it occurs for the Newtonian case, the flow naturally predicts the existence of an equatorial rotating accretion disc about the hole. However, the radius of the disc increases monotonically without limit as the flow reaches its maximum allowed angular momentum. 
  The Klein-Gordon equation in the Rindler space-time is studied carefully. It is shown that the stable properties depend on using what time coordinate to define the initial time. If we use the Rindler time, the scalar field is stable. Alternatively, if we use the Minkowski time, the scalar field may be regarded unstable to some extent. Furthermore, the complete extension of the Rindler space time is the Minkowski space time, we could also study the stable problem of the Rindler space time by the Klein-Gordon equation completely in the Minkowski coordinates system. The results support that the Rindler space time is really unstable. This in turn might cast some lights on the stable problem of the Schwarzschild black-hole, which not only in many aspects shares the similar geometrical properties with the Rindler space time but also has the very same situation in stable study as that in Rindler space time. So, it is not unreasonable to infer that the Schwarzschild black hole might really be unstable in comparison with the case in Rindler space time. Of course, one must go further to get the conclusion definitely. 
  We show that analytic solutions $\mcE$ of the Ernst equation with non-empty zero-level-set of $\Re \mcE$ lead to smooth ergosurfaces in space-time. In fact, the space-time metric is smooth near a "Ernst ergosurface" $E_f$ if and only if $\mcE$ is smooth near $E_f$ and does not have zeros of infinite order there. 
  Light propagation is investigated in the context of local anisotropic nonlinear dielectric media at rest with the dielectric coefficients $\epsilon^\mu{}_\nu = \epsilon^\mu{}_\nu (\vec{E},\vec{B})$ and constant $\mu$, in the limit of geometrical optics. Birefringence was examined and the general conditions for its occurrence were presented. A toy model is exhibited, in which uniaxial birefringent media with nonlinear dielectric properties could be driven by external fields in such way that birefringence may be artificially controlled. The effective geometry interpretation is also addressed. 
  The discrete quantum geometric effects play an important role in dynamical evolution in the loop quantum cosmology. These effects which are significant at the high energies lead to the quadratic energy density modifications to the Friedmann equation, as in the Randall-Sundrum braneworld scenarios but with a negative sign. We investigate the scalar field dynamics in this scenario and show the existence of a phase of super-inflation independent of the inverse scale factor modifications as found earlier. In this regime the scalar field mimics the dynamics of a phantom field and vice versa. We also find various symmetries between the expanding phase, the contracting phase and the phantom phase in the loop quantum cosmology. We then construct the scaling solutions in the loop quantum cosmology and show their dual relationship with those of the Randall-Sundrum cosmology. 
  In order to evaluate the energy distribution (due to matter and fields including gravitation) associated with a spacetime model of generalized diagonal metric, we consider the Einstein, Bergmann-Thomson and Landau-Lifshitz energy and/or momentum definitions both in Einstein's theory of general relativity and the teleparallel gravity (the tetrad theory of gravitation). We find the same energy distribution using Einstein and Bergmann-Thomson formulations, but we also find that the energy-momentum prescription of Landau-Lifshitz disagree in general with these definitions. We also give eight different well-known spacetime models as examples, and considering these models and using our results, we calculate the energy distributions associated with them. Furthermore, we show that for the Bianci type-I all the formulations give the same result. This result agrees with the previous works of Cooperstock-Israelit, Rosen, Johri {\it et al.}, Banerjee-Sen, Xulu, Vargas and Salt{\i} {\it et al.} and supports the viewpoints of Albrow and Tryon. 
  We discuss causality properties of asymmetrically warped space-times and argue that such scenarios may allow for timelike curves which can be closed via paths in the extra-dimensional bulk. We find a metric where the null, weak and dominant energy conditions are violated in the bulk, but satisfied on the brane. Such scenarios are interesting, since in principle gravitons or gauge-singlet (``sterile'') fermions propagating in the extra dimension may be manipulated in a way to test the chronology protection conjecture experimentally. 
  The black-hole information paradox has fueled a fascinating effort to reconcile the predictions of general relativity and those of quantum mechanics. Gravitational considerations teach us that black holes must trap everything that falls into them. Quantum mechanically the mass of a black hole leaks away as featureless (Hawking) radiation, but if the black hole vanishes, where is the information about the matter that made it? We treat the states of the in-fallen matter quantum mechanically and show that the black-hole information paradox becomes more severe. Our formulation of the paradox rules out one of the most conservative resolutions: that the state of the in-falling matter might be hidden in correlations between semi-classical Hawking radiation and the internal states of the black hole. As a consequence, either unitarity or Hawking's semi-classical predictions must break down. Any resolution of the black-hole information crisis must elucidate one of these possibilities. 
  Numerical Relativity has been using orbifolds for a long time, although they appear under different names in the literature. We review orbifolds previously used in simulations also discuss some that have not been used yet but are likely to be useful in the future. 
  A new numerical code for computing stationary axisymmetric rapidly rotating stars in general relativity is presented. The formulation is based on a fully constrained-evolution scheme for 3+1 numerical relativity using the Dirac gauge and maximal slicing. We use both the polytropic and MIT bag model equations of state to demonstrate that the code can construct rapidly rotating neutron star and strange star models. We compare numerical models obtained by our code and a well-established code, which uses a different gauge condition, and show that the two codes agree to high accuracy. 
  We investigate the strong deflection limit of gravitational lensing by a Schwarzschild black hole embedded in an external gravitational field. The study of this model, analogous to the Chang & Refsdal lens in the weak deflection limit, is important to evaluate the gravitational perturbations on the relativistic images that appear in proximity of supermassive black holes hosted in galactic centers. By a simple dimensional argument, we prove that the tidal effect on the light ray propagation mainly occurs in the weak field region far away from the black hole and that the external perturbation can be treated as a weak field quadrupole term. We provide a description of relativistic critical curves and caustics and discuss the inversion of the lens mapping. Relativistic caustics are shifted and acquire a finite diamond shape. Sources inside the caustics produce four sequences of relativistic images. On the other hand, retro-lensing caustics are only shifted while remaining point-like to the lowest order. 
  We show that the most general dark energy model that possesses a scaling solution $\rho_\phi\propto a^n$ is the k-essence model, which includes both of the quintessence and tachyon models. The exact scaling solutions are then derived. The potential that gives the tracking solution in which dark energy exactly tracks the background matter field is the inverse squared potential. The quintessence field with exponential potential can be obtained from the k-essence field with the inverse squared potential. We also find the fixed points and study their main properties, whereby the scalar field dominant fixed point is identified. 
  In three dimensions, a phase transition occurs between the non-rotating BTZ black hole and the massless BTZ black hole. Further, introducing the mass of a conical singularity, we show that a transition between the non-rotating BTZ black hole and thermal AdS space is also possible. 
  Modelling the free fall (and radiative phenomenology) of a massive particle, charged or not, in a static and spherically symmetric black hole is a classic, good relativistic dare that produced a remarkable series of papers, mainly in the seventies of the past century. Some formal topics about the mathematical machinery required to perform the task are unfortunately still not very clear; however, with the help of modern computer algebra techniques, some results can at least be tested and corrected. 
  The treatment of the principle of general covariance based on coordinate systems, i.e., on classical tensor analysis suffers from an ambiguity. A more preferable formulation of the principle is based on modern differential geometry: the formulation is coordinate-free. Then the principle may be called ``principle of geometricity.'' In relation to coordinate transformations, there had been confusions around such concepts as symmetry, covariance, invariance, and gauge transformations. Clarity has been achieved on the basis of a group-theoretical approach and the distinction between absolute and dynamical objects. In this paper, we start from arguments based on structures on cosmological manifold rather than from group-theoretical ones, and introduce the notion of setting elements. The latter create a scene on which dynamics is performed. The characteristics of the scene and dynamical structures on it are considered. 
  We propose a class of displacement- and laser-noise free gravitational-wave-interferometer configurations, which does not sense non-geodesic mirror motions and laser noises, but provides non-vanishing gravitational-wave signal. Our interferometer consists of 4 mirrors and 2 beamsplitters, which form 4 Mach-Zehnder interferometers. By contrast to previous works, no composite mirrors are required. Each mirror in our configuration is sensed redundantly, by at least two pairs of incident and reflected beams. Displacement- and laser-noise free detection is achieved when output signals from these 4 interferometers are combined appropriately. Our 3-dimensional interferometer configuration has a low-frequency response proportional to f^2, which is better than the f^3 achievable by previous 2-dimensional configurations. 
  Even when we consider Newtonian stars, i.e., stars with surface gravitational redshift, z<< 1, it is well known that, theoretically, it is possible to have stars, supported against self-gravity, almost entirely by radiation pressure. However, such Newtonian stars must necessarily be supermassive. We point out that this requirement for excessive large M, in Newtonian case, is a consequence of the occurrence of low z<< 1. On the other hand, if we remove such restrictions, and allow for possible occurrence highly general relativistic regime, z >> 1, we show that, it is possible to have radiation pressure supported stars at arbitrary value of M. Since radiation pressure supported stars necessarily radiate at the Eddington limit, in Einstein gravity, they are never in strict hydrodynamical equilibrium. Further, it is believed that sufficiently massive or dense objects undergo continued gravitational collapse to the Black Hole stage characterized by z =infty. Thus, late stages of Black Hole formation, by definition, will have, z >> 1, and hence would be examples of quasi-stable general relativistic RPSSs. This result is also supported by with our previous finding that that trapped surfaces are not formed in gravitational collapse and the value of the integration constant in the vacuum Schwarzschild solution is zero. Hence the supposed observed BHs are actually ECOs. 
  We obtain an efficient description for the dynamics of nonspinning compact binaries moving in inspiralling eccentric orbits to implement the phasing of gravitational waves from such binaries at the 3.5 post-Newtonian (PN) order. Our computation heavily depends on the phasing formalism, presented in [T. Damour, A. Gopakumar, and B. R. Iyer, Phys. Rev. D \textbf{70}, 064028 (2004)], and the 3PN accurate generalized quasi-Keplerian parametric solution to the conservative dynamics of nonspinning compact binaries moving in eccentric orbits, available in [R.-M. Memmesheimer, A. Gopakumar, and G. Sch\"afer, Phys. Rev. D \textbf{70}, 104011 (2004)]. The gravitational-wave (GW) polarizations $h_{+}$ and $h_{\times}$ with 3.5PN accurate phasing should be useful for the earth-based GW interferometers, current and advanced, if they plan to search for gravitational waves from inspiralling eccentric binaries. Our results will be required to do \emph{astrophysics} with the proposed space-based GW interferometers like LISA, BBO, and DECIGO. 
  The vacuum solutions arising from a spontaneous breaking of Lorentz symmetry due to the acquisition of a vacuum expectation value by a vector field are derived. These include the purely radial Lorentz symmetry breaking (LSB), radial/temporal LSB and axial/temporal LSB scenarios. It is found that the purely radial LSB case gives rise to new black hole solutions. Whenever possible, Parametrized Post-Newtonian (PPN) parameters are computed and compared to observational bounds, in order to constrain the Lorentz symmetry breaking scale. 
  The time independent spherically symmetric solutions of General Relativity (GR) coupled to a dynamical unit timelike vector are studied. We find there is a three-parameter family of solutions with this symmetry. Imposing asymptotic flatness restricts to two parameters, and requiring that the aether be aligned with the timelike Killing field further restricts to one parameter, the total mass. These "static aether" solutions are given analytically up to solution of a transcendental equation. The positive mass solutions have spatial geometry with a minimal area 2-sphere, inside of which the area diverges at a curvature singularity occurring at an extremal Killing horizon that lies at a finite affine parameter along a radial null geodesic. Regular perfect fluid star solutions are shown to exist with static aether exteriors, and the range of stability for constant density stars is identified. 
  It is shown numerically that strange matter rings permit a continuous transition to the extreme Kerr black hole. The multipoles as defined by Geroch and Hansen are studied and suggest a universal behaviour for bodies approaching the extreme Kerr solution parametrically. The appearance of a `throat region', a distinctive feature of the extreme Kerr spacetime, is observed. With regard to stability, we verify for a large class of rings, that a particle sitting on the surface of the ring never has enough energy to escape to infinity along a geodesic. 
  The possibility is discussed of superluminal motion of non-tachyonic (i.e. moving with the instantaneous speed v<c) bodies within general relativity. It is shown that in some occasions quantum field theory apparently prohibits such motion, but not - to all appearance - in the general case. 
  Braneworld gravity is a model that endows physical space with an extra dimension. In the type II Randall-Sundrum braneworld gravity model, the extra dimension modifies the spacetime geometry around black holes, and changes predictions for the formation and survival of primordial black holes. We develop a comprehensive analytical formalism for far-field black hole lensing in this model, using invariant quantities to compute all geometric optics lensing observables. We then make the first analysis of wave optics in braneworld lensing, working in the semi-classical limit. We show that wave optics offers the only realistic way to observe braneworld effects in black hole lensing. We point out that if primordial braneworld black holes exist, have mass M, and contribute a fraction f of the dark matter, then roughly 3e5 x f (M/1e-18 Msun)^(-1) of them lie within our Solar System. These objects, which we call "attolenses," would produce interference fringes in the energy spectra of gamma-ray bursts at energies ~100 (M/1e-18 Msun)^(-1) MeV (which will soon be accessible with the GLAST satellite). Primordial braneworld black holes spread throughout the universe could produce similar interference effects; the probability for "attolensing" may be non-negligible. If interference fringes were observed, the fringe spacing would yield a simple upper limit on M. Detection of a primordial black hole with M <~ 1e-19 Msun would challenge general relativity and favor the braneworld model. Further work on lensing tests of braneworld gravity must proceed into the physical optics regime, which awaits a description of the full spacetime geometry around braneworld black holes. 
  A new canonical transformation is found that enables the direct canonical treatment of the conformal factor in general relativity. The resulting formulation significantly simplifies the previously presented conformal geometrodynamics. It provides a further theoretical basis for the conformal approach to loop quantum gravity and offers a generic framework for the conformal analysis of spacetime dynamics. 
  In order to obtain energy and momentum (due to matter and fields including gravitation) distributions of the Gibbons-Maeda dilaton spacetime, we use the M{\o}ller energy and/or momentum prescription both in Einstein's theory of general relativity and teleparallel gravity. We find the same energy distribution for a given metric in both of these different gravitation theories. Under two limits, we also calculate energy associated with two other models such as the Garfinkle-Horowitz-Strominger dilaton spacetime and the Reissner-Nordstrom spacetime. The energy obtained is also independent of the teleparallel dimensionless coupling constant, which means that it is valid in any teleparallel model. Our result also sustains (a) the importance of the energy-momentum definitions in the evaluation of the energy distribution for a given spacetime and (b) the viewpoint of Lessner that the M{\o}ller energy-momentum complex is a powerful concept of energy and momentum (c) the hypothesis of Vagenas that there is a connection between the coefficients of the energy-momentum expression of Einstein and those of Moller. 
  Extremely high precision of near-future radio/optical interferometric observatories like SKA, Gaia, SIM and the unparalleled sensitivity of LIGO/LISA gravitational-wave detectors demands more deep theoretical treatment of relativistic effects in the propagation of electromagnetic signals through variable gravitational fields of the solar system, oscillating and precessing neutron stars, coalescing binary systems, exploding supernova, and colliding galaxies. Especially important for future gravitational-wave observatories is the problem of propagation of light rays in the field of multipolar gravitational waves emitted by a localized source of gravitational radiation. Present paper suggests physically-adequate and consistent mathematical solution of this problem in the first post-Minkowskian approximation of General Relativity which accounts for all time-dependent multipole moments of an isolated astronomical system. 
  In this work a new asymptotically flat solution of the coupled Einstein-Born-Infeld equations for a static spherically symmetric space-time is obtained. When the intrinsic mass is zero the resulting spacetime is regular everywhere, in the sense given by B. Hoffmann and L. Infeld in 1937, and the Einstein-Born-Infeld theory leads to the identification of the gravitational with the electromagnetic mass. This means that the metric, the electromagnetic field and their derivatives have not discontinuities in all the manifold. In particular, there are not conical singularities at the origin, in contrast to well known monopole solution studied by B. Hoffmann in 1935. The lack of uniqueness of the action function in Non-Linear-Electrodynamics is discussed. 
  In this work, the solution of the Einstein equations for a slowly rotating black hole with Born-Infeld charge is obtained. Geometrical properties and horizons of this solution are analyzed. The conditions when the ADM mass (as in the nonlinear static cases) and the ADM angular momentum of the system have been modified by the non linear electromagnetic field of the black hole, are considered. 
  The 5D Cosmological General Relativity theory developed by Carmeli reproduces all of the results that have been successfully tested for Einstein's 4D theory. However the Carmeli theory because of its fifth dimension, the velocity of the expanding universe, predicts something different for the propagation of gravity waves on cosmological distance scales. This analysis indicates that gravitational radiation may not propagate as an unattenuated wave where effects of the Hubble expansion are felt. In such cases the energy does not travel over such large length scales but is evanescent and dissipated into the surrounding space as heat. 
  In the present work the approach - density matrix deformation - earlier developed by the author to study a quantum theory of the Early Universe (Planck's scales) is applied to study a quantum theory of black holes. On this basis the author investigates the information paradox problem, entropy of the black hole remainders after evaporation, and consistency with the holographic principle. The possibility for application of the proposed approach to the calculation of quantum entropy of a black hole is considered. 
  Based on the recent understanding of the role of the densitized lapse function in Einstein's equations and of the proper way to pose the thin sandwich problem, a slight readjustment of the minimal distortion shift gauge in the 3+1 approach to the dynamics of general relativity allows this shift vector to serve as the vector potential for the longitudinal part of the extrinsic curvature tensor in the new approach to the initial value problem, thus extending the initial value decomposition of gravitational variables to play a role in the evolution as well. The new shift vector globally minimizes the changes in the conformal 3-metric with respect to the spacetime measure rather than the spatial measure on the time coordinate hypersurfaces, as the old shift vector did. 
  We study a new field theory effect in the cosmological context in the Two Measures Field Theory (TMT). TMT is an alternative gravity and matter field theory where the gravitational interaction of fermionic matter is reduced to that of General Relativity when the energy density of the fermion matter is much larger than the dark energy density. In this case also the 5-th force problem is solved automatically. In the opposite limit, where the magnitudes of fermionic energy density and scalar field dark energy density become comparable, nonrelativistic fermions can participate in the cosmological expansion in a very unusual manner. Some of the features of such states in a toy model of the late time universe filled with homogeneous scalar field and uniformly distributed nonrelativistic neutrinos: neutrino mass increases as m ~ a^{3/2}; the neutrino gas equation-of-state approaches w=-1, i.e. neutrinos behave as a sort of dark energy; the total (scalar field + neutrino) equation-of-state also approaches w=-1; the total energy density of such universe is less than it would be in the universe filled with the scalar field alone. An analytic solution is presented. A domain structure of the dark energy seems to be possible. We speculate that decays of the CLEP state neutrinos may be both an origin of cosmic rays and responsible for a late super-acceleration of the universe. In this sense the CLEP states exhibit simultaneously new physics at very low densities and for very high particle masses. 
  PPN-limit of higher order theories of gravity represents a still controversial matter of debate and no definitive answer has been provided, up to now, about this issue. By exploiting the analogy between scalar-tensor and fourth-order theories of gravity, one can generalize the PPN-limit formulation.  By using the definition of the PPN-parameters $\gamma$ and $\beta$ in term of the $f(R)$ derivatives, we show that a family of third-order polynomial theories, in the Ricci scalar $R$, turns out to be compatible with the PPN-limit and the deviation from General Relativity theoretically predicted agree with experimental data. 
  As a point of departure it is suggested that Quantum Cosmology is a topological concept independent from metrical constraints. Methods of continuous topological evolution and topological thermodynamics are used to construct a cosmological model of the present universe, using the techniques based upon Cartan's theory of exterior differential systems. Thermodynamic domains, which are either Open, Closed, Isolated, or in Equilibrium, can be put into correspondence with topological systems of Pfaff topological dimension 4, 3, 2 and 1. If the environment of the universe is assumed to be a physical vacuum of Pfaff topological dimension 4, then continuous but irreversible topological evolution can cause the emergence of topologically coherent defect structures of Pfaff topological dimension less than 4. As galaxies and stars exchange radiation but not matter with the environment, they are emergent topological defects of Pfaff topological dimension 3 which are far from equilibrium. DeRham topological theory of period integrals over closed but not exact exterior differential systems leads to the emergence of quantized, deformable, but topologically coherent, singular macrostates at all scales. The method leads to the conjecture that dark matter and energy is represented by those thermodynamic topological defect structures of Pfaff dimension 2 or less. 
  In quantum gravity theories Planckian behavior is triggered by the energy of {\it elementary} particles approaching the Planck energy, $E_P$, but it's also possible that anomalous behavior strikes systems of particles with total energy near $E_P$. This is usually perceived to be pathological and has been labelled ``the soccer ball problem''. We point out that there is no obvious contradiction with experiment if {\it coherent} collections of particles with bulk energy of order $E_P$ do indeed display Planckian behavior, a possibility that would open a new experimental window. Unfortunately field theory realizations of deformed special relativity never exhibit a ``soccer ball problem''; we present several formulations where this is undeniably true. Upon closer scrutiny we discover that the only chance for Planckian behavior to be triggered by large coherent energies involves the details of second quantization. We find a formulation where the quanta have their energy-momentum (mass-shell) relations deformed as a function of the bulk energy of the coherent packet to which they belong, rather than the frequency. Given ongoing developments in Laser technology, such a possibility would be of great experimental interest. 
  Slowly evolving horizons are trapping horizons that are "almost" isolated horizons. This paper reviews their definition and discusses several spacetimes containing such structures. These include certain Vaidya and Tolman-Bondi solutions as well as (perturbatively) tidally distorted black holes. Taking into account the associated mass scales, they also suggest that slowly evolving horizons are the norm rather than the exception in astrophysical processes that involve stellar-scale black holes. 
  We calculated the energy and momentum densities of stiff fluid solutions, using Einstein, Bergmann-Thomson and Landau-Lifshitz energy-momentum complexes, in both general relativity and teleparallel gravity. In our analysis we get different results comparing the aforementioned complexes with each other when calculated in the same gravitational theory, either this is in general relativity and teleparallel gravity. However, interestingly enough, each complex's value is the same either in general relativity or teleparallel gravity. Our results sustain that (i) general relativity or teleparallel gravity are equivalent theories (ii) different energy-momentum complexes do not provide the same energy and momentum densities neither in general relativity nor in teleparallel gravity. In the context of the theory of teleparallel gravity, the vector and axial-vector parts of the torsion are obtained. We show that the axial-vector torsion vanishes for the space-time under study. 
  The most general action, quadratic in the B fields as well as in the curvature F, having SO(3,1) or SO(4) as the internal gauge group for a four-dimensional BF theory is presented and its symplectic geometry is displayed. It is shown that the space of solutions to the equations of motion for the BF theory can be endowed with symplectic structures alternative to the usual one. The analysis also includes topological terms and cosmological constant. The implications of this fact for gravity are briefly discussed. 
  The newly found conformal decomposition in canonical general relativity is applied to drastically simplify the recently formulated parameter-free construction of spin-gauge variables for gravity. The resulting framework preserves many of the main structures of the existing canonical framework for loop quantum gravity related to the spin network and Thiemann's regularization. However, the Barbero-Immirzi parameter is now converted into the conformal factor as a canonical variable. It behaves like a scalar field but is somehow non-dynamical since the effective Hamiltonian constraint does not depend on its momentum. The essential steps of the mathematical derivation of this parameter-free framework for the spin-gauge variables of gravity are spelled out. The implications for the loop quantum gravity programme are briefly discussed. 
  In this paper we review the extent to which one can use classical distribution theory in describing solutions of Einstein's equations. We show that there are a number of physically interesting cases which cannot be treated using distribution theory but require a more general concept. We describe a mathematical theory of nonlinear generalised functions based on Colombeau algebras and show how this may be applied in general relativity. We end by discussing the concept of singularity in general relativity and show that certain solutions with weak singularities may be regarded as distributional solutions of Einstein's equations. 
  The recent formulation of locally covariant quantum field theory may open the way towards a background independent perturbative formulation of Quantum Gravity. 
  A more conventional realization of a symmetry which had been proposed towards the solution of cosmological constant problem is considered. In this study the multiplication of the coordinates by the imaginary number $i$ in the literature is replaced by the multiplication of the metric tensor by minus one. This realization of the symmetry as well forbids a bulk cosmological constant and selects out $2(2n+1)$ dimensional spaces. On contrary to its previous realization the symmetry, without any need for its extension, also forbids a possible cosmological constant term which may arise from the extra dimensional curvature scalar provided that the space is taken as the union of two $2(2n+1)$ dimensional spaces where the usual 4-dimensional space lies at the intersection of these spaces. It is shown that this symmetry may be realized through spacetime reflections that change the sign of the volume element. A possible relation of this symmetry to the E-parity symmetry of Linde is also pointed out. 
  Recently Kordas (1995, Class. Quantum Grav. 12 2037) and Meinel and Neugebauer (1995, Class. Quantum Grav. 12 2045) studied the conditions for reflection symmetry in stationary axisymmetric space--times in vacuum. They found that a solution to the Einstein field equations is reflectionally symmetric if their Ernst's potential ${\cal E}(\rho=0,z)= e(z)$ on a portion of the positive z-axis extending to infinity satisfies the condition $e_{+}(z){e}_{+}^{*}(-z)=1$. In this note, we formulate an analogous conditions for two complex Ernst potentials in electrovacuum. We also present the special case of rational axis potentials. 
  Using conformal coordinates associated with conformal relativity -- associated with de Sitter spacetime homeomorphic projection into Minkowski spacetime -- we obtain a conformal Klein-Gordon partial differential equation, which is intimately related to the production of quasi-normal modes (QNMs) oscillations, in the context of electromagnetic and/or gravitational perturbations around, e.g., black holes. While QNMs arise as the solution of a wave-like equation with a Poschl-Teller potential, here we deduce and analytically solve a conformal radial d'Alembert-like equation, from which we derive QNMs formal solutions, in a proposed alternative to more completely describe QNMs. As a by-product we show that this radial equation can be identified with a Schrodinger-like equation in which the potential is exactly the second Poschl-Teller potential, and it can shed some new light on the investigations concerning QNMs. 
  A transient chaos in a closed FRW cosmological model with a scalar field is studied. We describe two different chaotic regimes and show that the type of chaos in this model depends on the scalar field potential. We have found also that for sufficiently steep potentials or for potentials with large cosmological constant the chaotic behavior disappears. 
  The junction conditions for General Relativity in the presence of domain walls with intrinsic spin are derived in three and higher dimensions. A stress tensor and a spin current can be defined just by requiring the existence of a well defined volume element instead of an induced metric, so as to allow for generic torsion sources. In general, when the torsion is localized on the domain wall, it is necessary to relax the continuity of the tangential components of the vielbein. In fact it is found that the spin current is proportional to the jump in the vielbein and the stress-energy tensor is proportional to the jump in the spin connection. The consistency of the junction conditions implies a constraint between the direction of flow of energy and the orientation of the spin. As an application, we derive the circularly symmetric solutions for both the rotating string with tension and the spinning dust string in three dimensions. The rotating string with tension generates a rotating truncated cone outside and a flat space-time with inevitable frame dragging inside. In the case of a string made of spinning dust, in opposition to the previous case no frame dragging is present inside, so that in this sense, the dragging effect can be "shielded" by considering spinning instead of rotating sources. Both solutions are consistently lifted as cylinders in the four-dimensional case. 
  After a review of exotic statistics for point particles in 3d BF theory, and especially 3d quantum gravity, we show that string-like defects in 4d BF theory obey exotic statistics governed by the 'loop braid group'. This group has a set of generators that switch two strings just as one would normally switch point particles, but also a set of generators that switch two strings by passing one through the other. The first set generates a copy of the symmetric group, while the second generates a copy of the braid group. Thanks to recent work of Xiao-Song Lin, we can give a presentation of the whole loop braid group, which turns out to be isomorphic to the 'braid permutation group' of Fenn, Rimanyi and Rourke. In the context 4d BF theory this group naturally acts on the moduli space of flat G-bundles on the complement of a collection of unlinked unknotted circles in R^3. When G is unimodular, this gives a unitary representation of the loop braid group. We also discuss 'quandle field theory', in which the gauge group G is replaced by a quandle. 
  We use Fermi coordinates to calculate the canonical partition function for an ideal gas in a circular geodesic orbit in Schwarzschild spacetime. To test the validity of the results we prove theorems for limiting cases. We recover the Newtonian gas law subject only to tidal forces in the Newtonian limit. Additionally we recover the special relativistic gas law as the radius of the orbit increases to infinity. We also discuss how the method can be extended to the non ideal gas case. 
  In the first part of this paper I review some of the difficulties that seem to obstruct generally valid definitions of "general covariance" and/or "background independence" The second and more historical part deals with a rather strange argument that Einstein put forward in his 1913 "Entwurf paper" with M. Grossmann to discredit scalar theories of gravity in order to promote general covariance. 
  We argue that, when coupled to Einstein's theory of gravity, the Yukawa theory may solve the cosmological constant problem in the following sense: The radiative corrections of fermions generate an effective potential for the scalar field, such that the effective cosmological term Lambda_eff is dynamically driven to zero. Thence, for any initial positive cosmological constant Lambda_0, Lambda_eff = 0 is an attractor of the semiclassical Einstein theory coupled to fermionic and scalar matter fields. When the initial cosmological term is negative, Lambda_eff=Lambda_0 does not change. Next we argue that the dark energy of the Universe may be explained by a GUT scale fermion with a mass, m = 4.3 * 10^15 (Lambda_0/10^13GeV)^(1/2) GeV.  Finally, we comment on how the inflationary paradigm, BEH mechanism and phase transitions in the early Universe get modified in the light of our findings. 
  We examine the linear stability of various configurations in Bose-Einstein condensates with sonic horizons. These configurations are chosen in analogy with gravitational systems with a black hole horizon, a white hole horizon and a combination of both. We discuss the role of different boundary conditions in this stability analysis, paying special attention to their meaning in gravitational terms. We highlight that the stability of a given configuration, not only depends on its specific geometry, but especially on these boundary conditions. Under boundary conditions directly extrapolated from those in standard General Relativity, black hole configurations, white hole configurations and the combination of both into a black hole--white hole configuration are shown to be stable. However, we show that under other (less stringent) boundary conditions, configurations with a single black hole horizon remain stable, whereas white hole and black hole--white hole configurations develop instabilities associated to the presence of the sonic horizons. 
  We present a discussion of the fundamental loss of unitarity that appears in quantum mechanics due to the use of a physical apparatus to measure time. This induces a decoherence effect that is independent of any interaction with the environment and appears in addition to any usual environmental decoherence. The discussion is framed self consistently and aimed to general physicists. We derive the modified Schroedinger equation that arises in quantum mechanics with real clocks and discuss the theoretical and potential experimental implications of this process of decoherence. 
  A possible candidate for the present accelerated expansion of the Universe is ''phantom energy'', which possesses an equation of state of the form $\omega\equiv p/\rho<-1$, consequently violating the null energy condition. As this is the fundamental ingredient to sustain traversable wormholes, this cosmic fluid presents us with a natural scenario for the existence of these exotic geometries. In this context, we shall construct phantom wormhole geometries by matching an interior wormhole solution, governed by the phantom energy equation of state, to an exterior vacuum at a junction interface. Several physical properties and characteristics of these solutions are further investigated. The dynamical stability of the transition layer of these phantom wormholes to linearized spherically symmetric radial perturbations about static equilibrium solutions is also explored. It is found that the respective stable equilibrium configurations may be increased by strategically varying the wormhole throat radius. 
  In earlier Letters, we adopted a complex approach to quantum processes in the formation and evaporation of black holes. Taking Feynman's $+i\epsilon$ prescription, rather than than one of the more usual approaches, we calculated the quantum amplitude (not just the probability density) for final weak-field configurations following gravitational collapse to a black hole with subsequent evaporation. What we have done is to find quantum amplitudes relating to a pure state at late times following black-hole matter collapse. Such pure states are then shown to be susceptible to a description in terms of coherent and squeezed states - in practice, this description is not very different from that for the well-known highly-squeezed final state of the relic radiation background in inflationary cosmology. The simplest such collapse model involves Einstein gravity with a massless scalar field. The Feynman approach involves making the boundary-value problem for gravity and a massless scalar field well-posed. To define this, let T be the proper-time separation, measured at spatial infinity, between two space-like hypersurfaces on which initial (collapse) and final (evaporation) data are posed. Then, in this approach, one rotates T by a complex phase exp(-i\delta) into the lower half-plane. In an adiabatic approximation, the resulting quantum amplitude may be expressed in terms of generalised coherent states of the quantum oscillator, and a physical interpretation is given. A squeezed-state representation, as above, then follows. 
  Ripplons -- gravity-capillary waves on the free surface of a liquid or at the interfaces between two superfluids -- are the most favourable excitations for simulation of the general-relativistic effects related to horizons and ergoregions. The white-hole horizon for the ``relativistic'' ripplons at the surface of the shallow liquid is easily simulated using the kitchen-bath hydraulic jump. The same white-hole horizon is observed in quantum liquid -- superfluid 4He. The ergoregion for the ``non-relativistic'' ripplons is generated in the experiments with two sliding 3He superfluids. The common property experienced by all these ripplons is the Miles instability inside the ergoregion or horizon. Because of the universality of the Miles instability, one may expect that it could take place inside the horizon of the astrophysical black holes, if there is a preferred reference frame which comes from the trans-Planckian physics. If this is the case, the black hole would evapotate much faster than due to the Hawking radiation. Hawking radiation from the artificial black hole in terms of the quantum tunneling of phonons and ripplons is also discussed. 
  We found type I critical phenomena in the gravitational collapses of a compact object modeled by a polytropic equation of state (EOS) with polytropic index $\Gamma=2$ without the ultra-relativistic assumption. The object is formed in head-on collisions of neutron stars. Further, we showed that the critical collapse can result from a slow change of the EOS, without fine tuning of initial data. This opens the possibility that a neutron star like compact object may undergo a critical collapse under processes which change the EOS, such as cooling. There is evidence that the critical solution, as well as the mode that brings the near critical solution away from the critical one, are spherical symmetric, despite that the initial data is highly non-spherical. This also suggests that the critical collapse analyzed may be relevant to a range of physical systems not limited to a compact object formed in the head-on collision of neutron stars. 
  Non-Abelian gauge fields are traditionally not coupled to torsion due to violation of gauge invariance. However, it is possible to couple torsion to Yang-Mills fields while maintaining gauge invariance provided one accepts that the gauge couplings then become scalar fields. In the past this has been untenable from experimental constraints at the current epoch for the electromagnetic field at least. Recent researches on the "landscape" arising out of string theory provides for many scalar fields which eventually determine the various low energy parameters including gauge couplings in the universe. With this scenario, we argue that the very early universe provides a Riemann-Cartan geometry with non-zero torsion coupling to gauge fields. The torsion is just the derivative of gauge coupling (scalar) fields. As a result, in the evolution of the Universe, when the scalar (moduli) fields determine the geometry of the universe to be Riemannian, torsion goes to zero, implying that the associated modulus (and hence the gauge coupling) has a constant value. An equivalent view is that the modulus fixes the gauge coupling at some constant value causing the torsion to vanish as a consequence. Of course, when torsion vanishes we recover Einstein's theory for further evolution of the universe. 
  As is well known, in order for the Einstein--Hilbert action to have a well defined variation, and therefore to be used for deriving field equation through the stationary action principle, it has to be amended by the addition of a suitable boundary term. It has recently been claimed that, if one constructs an action by adding this term to the matter action, the Einstein field equations can be derived by requiring this action to be invariant under active transformations which are normal to a null boundary. In this paper we re-examine this approach both for the case of pure gravity and in the presence of matter. We show that in the first case this procedure holds for more general actions than the Einstein-Hilbert one and trace the basis of this remarkable attribute. However, it is also pointed out the when matter is rigorously considered the approach breaks down. The reasons for that are thoroughly discussed. 
  A spherically symmetric distribution of classical blackbody radiation is considered, at conditions in which gravitational self-interaction effects become not negligible. Static solutions to Einstein field equations are searched for, for each choice of the assumed central energy density. Spherical cavities at thermodynamic equilibrium, i.e. filled with blackbody radiation, are then studied, in particular for what concerns the relation among the mass M of the ball of radiation contained in them and their temperature at center and at the boundary. For these cavities it is shown, in particular, that: i) there is no absolute limit to M as well to their central and boundary temperatures; ii) when radius R is fixed, however, limits exist both for mass and for boundary energy density rho_B: M <= K M_S(R) and rho_B <= Q/R^2, with K = 0.493 and Q = 0.02718, dimensionless, and M_S(R) the Schwarzschild mass for that radius. Some implications of the existence and the magnitude of these limits are considered. Finally the radial profiles for entropy for these systems are studied, in their dependence on the mass (or central temperature) of the ball of radiation. 
  We use the general formalism of squeezed rotated states to calculate baryon asymmetry in the wake of inflation through parametric amplification. We base our analysis on a B and CP violating Lagrangian in an isotropically expanding universe. The B and CP violating terms originate from the coupling of complex fields with non-zero baryon number to a complex background inflaton field. We show that a differential amplification of particle and anti-particle modes gives rise to baryon asymmetry. 
  We provide a quantum field theoretical derivation of the Abraham-Lorentz-Dirac (ALD) equation, describing the motion of an electric point charge sourcing an electromagnetic field, which back-reacts on the charge as a self-force, and the Mino-Sasaki-Tanaka-Quinn-Wald (MSTQW) equation describing the motion of a point mass with self-force interacting with the linearized metric perturbations caused by the mass off an otherwise vacuous curved background spacetime. We regularize the formally divergent self-force by smearing the direct part of the retarded Green's function and using a quasilocal expansion. We also derive the ALD-Langevin and the MSTQW-Langevin equations with a classical stochastic force accounting for the effect of the quantum fluctuations in the field, which causes small fluctuations on the particle trajectory. These equations will be useful for studying the stochastic motion of charges and small masses under the influence of both quantum and classical noise sources, derived either self-consistently or put in by hand phenomenologically. We also show that history-dependent noise-induced drift motions could arise from such stochastic sources on the trajectory that could be a hidden feature of gravitational wave forms hitherto unknown. 
  Axial vector torsion in the Einstein-Cartan space $U_{4}$ is considered here. By picking a particular term from the SO(4,1) Pontryagin density and then modifying it in a SO(3,1) invariant way, we get a Lagrangian density with Lagrange multipliers. Then considering torsion and torsion-less connection as independent fields, it has been found that $\kappa$ and $\lambda$ of Einstein-Hilbert Lagrangian, appear as integration constants in such a way that $\kappa$ has been found to be linked with the topological Nieh-Yan density of $U_{4}$ space. 
  A large amount of work has been dedicated to studying general relativity coupled to non-Abelian Yang-Mills type theories. It has been shown that the magnetic monopole, a solution of the Yang-Mills-Higgs equations can be coupled to gravitation. For a low Higgs mass there are regular solutions, and for a sufficiently massive monopole the system develops an extremal magnetic Reissner-Nordstrom quasi-horizon. These solutions, called quasi-black holes, although non-singular, are arbitrarily close to having a horizon. However, at the critical value the quasi-black hole turns into a degenerate spacetime. On the other hand, for a high Higgs mass, a sufficiently massive monopole develops also a quasi-black hole, but it turns into an extremal true horizon, with matter fields outside. One can also put a small Schwarzschild black hole inside the magnetic monopole, an example of a non-Abelian black hole. Surprisingly, Majumdar-Papapetrou systems, Abelian systems constructed from extremal dust, also show a resembling behavior. Previously, we have reported that one can find Majumdar-Papapetrou solutions which can be arbitrarily close of being a black hole, displaying quasi-black hole behavior. With the aim of better understanding the similarities between gravitational monopoles and Majumdar-Papapetrou systems, we study a system composed of two extremal electrically charged spherical shells (or stars, generically) in the Einstein--Maxwell--Majumdar-Papapetrou theory. We review the gravitational properties of the monopoles, and compare with the properties of the double extremal electric shell system. These quasi-black holes can help in the understanding of true black holes, and can give insight into the nature of the entropy of black holes in the form of entanglement. 
  Thermal conditions in the LTP, the LISA Technology Package, are required to be very stable, and in such environment precision temperature measurements are also required for various diagnostics objectives. A sensitive temperature gauging system for the LTP is being developed at IEEC, which includes a set of thermistors and associated electronics. In this paper we discuss the derived requirements applying to the temperature sensing system, and address the problem of how to create in the laboratory a thermally quiet environment, suitable to perform meaningful on-ground tests of the system. The concept is a two layer spherical body, with a central aluminium core for sensor implantation surrounded by a layer of polyurethane. We construct the insulator transfer function, which relates the temperature at the core with the laboratory ambient temperature, and evaluate the losses caused by heat leakage through connecting wires. The results of the analysis indicate that, in spite of the very demanding stability conditions, a sphere of outer diameter of the order one metre is sufficient. We provide experimental evidence confirming the model predictions. 
  We study the behaviour of a specific system of relativistic elasticity in its own gravitational field: a static, spherically symmetric shell whose wall is of arbitrary thickness consisting of hyperelastic material. We give the system of field equations and boundary conditions within the framework of the Einsteinian theory of gravity. Furthermore, we analize the situation in the Newtonian theory of gravity and obtain an existence result valid for small gravitational constants and pointwise stability by using the implicit function theorem. If one replaces the elastic material with a fluid, one finds that stable states can not exist. 
  In general relativity, the equation of motion of the spin is given by the equation of parallel transport, which is a result of the space-time geometry. Any result of the space-time geometry can not be directly applied to gauge theory of gravity. In gauge theory of gravity, based on the viewpoint of the coupling between the spin and gravitational field, an equation of motion of the spin is deduced. In the post Newtonian approximation, it is proved that this equation gives out the same result as that of the equation of parallel transport. So, in the post Newtonian approximation, gauge theory of gravity gives out the same prediction on the precession of orbiting gyroscope as that of general relativity. 
  The stability of naked singularities in self-similar collapse is probed using scalar waves. It is shown that the multipoles of a minimally coupled massless scalar field propagating on a spherically symmetric self-similar background spacetime admitting a naked singularity maintain finite $L^2$ norm as they impinge on the Cauchy horizon. It is also shown that each multipole obeys a pointwise bound at the horizon, as does its locally observed energy density. $L^2$ and pointwise bounds are also obtained for the multipoles of a minimally coupled massive scalar wave packet. 
  We use the Conformal Metric as described in Kar-Sinha work on Gravitational Bending of Light in a $5D$ Spacetime to recompute the equations of the $5D$ Force in Basini-Capozziello-Leon Formalism and we arrive at a result that possesses some advantages. The equations of the Extra Force as proposed by Leon are now more elegant in Conformal Formalism and many algebraic terms can be simplified or even suppressed. Also we recompute the Kar-Sinha Gravitational Bending of Light affected by the presence of the Extra Dimension and analyze the Superluminal Chung-Freese Features of this Formalism describing the advantages of the Chung-Freese BraneWorld when compared to other Superluminal spacetime metrics(eg:Warp Drive) and we describe why the Extra Dimension is invisible and how the Extra Dimension could be made visible at least in theory.We also examine the Maartens-Clarkson Black Holes in $5D$(Black Strings) coupled to massive Kaluza-Klein graviton modes predicted by Extra Dimensions theories and we study experimental detection of Extra Dimensions on-board LIGO and LISA Space Telescopes.We also propose the use of International Space Station(ISS) to measure the additional terms(resulting from the presence of Extra Dimensions) in the Kar-Sinha Gravitational Bending of Light in Outer Space to verify if we really lives in a Higher Dimensional Spacetime.Also we demonstrate that Particle $Z$ can only exists if the $5D$ spacetime exists. 
  We introduce the basic elements of SO(n)-local theory of Regge Calculus. A first order formalism, in the sense of Palatini, is defined on the metric-dual Voronoi complex of a simplicial complex. The Quantum Measure exhibits an expansion, in four dimensions, in characters of irreducible representation of SO(4) which has close resemblance and differences as well with the Spin Foam Formalism. The coupling with fermionic matter is easily introduced which could have consequences for the Spin Foam Formalism and Loop Quantum Gravity. 
  The energy (due to matter plus fields including gravity) distribution of the Reissner-Nordstr\"{o}m anti-de Sitter (RN AdS) black holes is studied by using the M{\o}ller energy-momentum definition in general relativity. This result is compered with the energy expression obtained by using the Einstein and Tolman complexes. Total energy depends on the black hole mass $M$ and charge $Q$ and cosmological constant $\Lambda$. Energy distribution of the RN AdS is also calculated by using the M{\o}ller prescription in teleparallel gravity. We get the same result for both of these different gravitation theories. The energy obtained is also independent of the teleparallel dimensionless coupling constant, which means that it is valid not only in the teleparallel equivalent of general relativity, but also in any teleparallel model. Under special cases of our model, we also discuss the energy distributions associated with the Schwarzschild AdS, RN and Schwarzschild black holes, respectively. 
  In the Einstein-Cartan space $U_4$, an axial vector torsion together with a scalar field connected to a local scale factor have been considered. By combining two particular terms from the SO(4,1) Pontryagin density and then modifying it in a SO(3,1) invariant way, we get a Lagrangian density with Lagrange multipliers. Then under FRW-cosmological background, where the scalar field is connected to the source of gravitation, the Euler-Lagrange equations ultimately give the constancy of the gravitational constant together with only three kinds of energy densities representing mass, radiation and cosmological constant. The gravitational constant has been found to be linked with the geometrical Nieh-Yan density. 
  Quantum cosmology in general denotes the application of quantum physics to the whole universe and thus gives rise to many realizations and examples, covering problems at different mathematical and conceptual levels. It is related to quantum gravity and more specifically describes the application to cosmological situations rather than the construction and analysis of quantum field equations. As there are several different approaches to quantum gravity, equations for quantum cosmology are not unique. Most investigations have been performed in the context of canonical quantization, where Wheeler--DeWitt like equations are the prime object. Applications are mostly conceptual, ranging from possible resolutions of classical singularities and explanations of the uniqueness of the universe to the origin of seeds for a classical world and its initial conditions. 
  The eigenvalue problem for the square integrable solutions is studied usually for elliptic equations. In this note we consider such a problem for the hyperbolic Klein-Gordon equation on Lorentzian manifolds. The investigation could help to answer the question why elementary particles have a discrete mass spectrum. An infinite family of square integrable solutions for the Klein-Gordon equation on the Friedman type manifolds is constructed. These solutions have a discrete mass spectrum and a finite action. In particular the solutions on de Sitter space are investigated. 
  One of the biggest unsolved problems in physics is the unification of quantum mechanics and general relativity. The lack of experimental guidance has made the issue extremely evasive, though various attempts have been made to relate the loss of matter wave coherence to quantum spacetime fluctuations. We present a new approach to the gravitational decoherence near the Planck scale, made possible by recently discovered conformal structure of canonical gravity. This leads to a gravitational analogue of the Brownian motion whose correlation length is given by the Planck length up to a scaling factor. With input from recent matter wave experiments, we show that the minimum value of this factor to be well within the expected range for quantum gravity theories. This suggests that the sensitivities of advanced matter wave interferometers may be approaching the fundamental level due to quantum spacetime fluctuations and that investigating Planck scale physics using matter wave interferometry may become a reality in the near future. 
  Carefully analyze influence of the tortoise coordinates r* and t on the stable study of the Schwarzschild black hole. Actually, one should be cautious in using the compact property of the perturbation field: it is true only with respect with the coordinate r and proper time or "good time", not the tortoise coordinates r* and t. Therefore, the mathematical proof used in reference [7] is incorrect because of it relying on the compact property of the perturbation fields. The Schwarzschild black hole might be unstable[1]-[5] 
  The idea of quantum relativity as a generalized, or rather deformed, version of Einstein (special) relativity has been taking shape in recent years. Following the perspective of deformations, while staying within the framework of Lie algebra, we implement explicitly a simple linear realization of the relativity symmetry, and explore systematically the resulting physical interpretations. Some suggestions we make may sound radical, but are arguably natural within the context of our formulation. Our work may provide a new perspective on the subject matter, complementary to the previous approach(es), and may lead to a better understanding of the physics. 
  We discuss data analysis techniques that can be used in the search for gravitational wave bursts from cosmic strings. When data from multiple interferometers are available, we describe consistency checks that can be used to greatly reduce the false alarm rates. We construct an expression for the rate of bursts for arbitrary cosmic string loop distributions and apply it to simple known solutions. The cosmology is solved exactly and includes the effects of a late-time acceleration. We find substantially lower burst rates than previous estimates suggest and explain the disagreement. Initial LIGO is unlikely to detect field theoretic cosmic strings with the usual loop sizes, though it may detect cosmic superstrings as well as cosmic strings and superstrings with non-standard loop sizes (which may be more realistic). In the absence of a detection, we show how to set upper limits based on the loudest event. Using Initial LIGO sensitivity curves, we show that these upper limits may result in interesting constraints on the parameter space of theories that lead to the production of cosmic strings. 
  We consider the possibility of energy being exchanged between the scalar and matter fields in scalar-tensor theories of gravity. Such an exchange provides a new mechanism which can drive variations in the gravitational 'constant' G. We find exact solutions for the evolution of spatially flat Friedman-Roberston-Walker cosmologies in this scenario and discuss their behaviour at both early and late times. We also consider the physical consequences and observational constraints on these models. 
  Conformal connection of scalar field is shown to produce possible non-metricity in affine connection spaces. In case of self-consistent solution the non-metricity is a correction to background Riemannian structure with respect to gravitational constant and its magnitude may be essential in the early Universe. 
  This paper has been withdrawn by the author due to an error in the use of the Cauchy-Kowalevski theorem in section 4., theorem 4.1 which does not allow to prove the final result. When (and if) this error will be cured it will be replaced by a corrected version of the result. 
  A result of existence of homogeneous scalar field solutions between prescribed configurations is given, using a modified version of Euler--Maupertuis least action variational principle. Solutions are obtained as limit of approximating variational problems, solved using techniques introduced by Rabinowitz. 
  Global visibility of naked singularities is analyzed here for a class of spherically symmetric spacetimes, extending previous studies - limited to inhomogeneous dust cloud collapse - to more physical valid situations in which pressures are non-vanishing. Existence of nonradial geodesics escaping from the singularity is shown, and the observability of the singularity from far-away observers is discussed. 
  Without imposing the isolated horizon boundary condition classically for the purpose of determining black hole entropy, loop quantum gravity is a theory that predicts generic degeneracy for black hole states. This degeneracy is within the definition of area operator. This leads to a a more fundamental microscopic description of a quantum black hole from the very geometrical point of view. The entropy of quantum surfaces turns out to be proportional to its area. This entropy if applied on the a black hole spin network states coincides with the Bekenstein-Hawking entropy. 
  Due to its underlying gauge structure, teleparallel gravity achieves a separation between inertial and gravitational effects. It can, in consequence, describe the isolated gravitational interaction without resorting to the equivalence principle, and is able to provide a tensorial definition for the energy-momentum density of the gravitational field. Considering the conceptual conflict between the local equivalence principle and the nonlocal uncertainty principle, the replacement of general relativity by its teleparallel equivalent can be considered an important step towards a prospective reconciliation between gravitation and quantum mechanics. 
  Asking very elementary relativistic quantum mechanics to meet quantums of area and time, it is possible to observe at a general level: a) the seesaw bound for the mass of neutrinos, and b) the need of a gauge group at energies below Planck mass. 
  We present a general study of the dynamical properties of Anisotropic Bianchi Universes in the context of Einstein General Relativity. Integrability results using Kovalevskaya exponents are reported and connected to general knowledge about Bianchi dynamics. Finally, dynamics toward singularity in Bianchi type VIII and IX universes are showed to be equivalent in some precise sence. 
  We raise, and provide an (unsatisfactory) answer to, the title's question: why, unlike all other fields, does the gravitational "metric" variable not have zero vacuum? After formulating, without begging it, we exhibit additions to the conventional action that express existence of the inverse through a field equation. 
  We consider the Regge-Teitelboim model for a relativistic extended object embedded in a fixed background Minkowski spacetime, in which the dynamics is determined by an action proportional to the integral of the scalar curvature of the worldvolume spanned by the object in its evolution. In appearance, this action resembles the Einstein-Hilbert action for vacuum General Relativity: the equations of motion for both are second order; the difference is that here the dynamical variables are not the metric, but the embedding functions of the worldvolume. We provide a novel Hamiltonian formulation for this model. The Lagrangian, like that of General Relativity, is linear in the acceleration of the extended object. As such, the model is not a genuine higher derivative theory, a fact reflected in the order of the equations of motion. Nevertheless, as we will show, it is possible as well as useful to treat it as a `fake' higher derivative system, enlarging the phase space appropriately. The corresponding Hamiltonian on this phase space is constructed: it is a polynomial. The complete set of constraints on the phase space is identified. The fact that the equations of motion are of second order in derivatives of the field variables manifests itself in the Hamiltonian formulation through the appearance of additional constraints, both primary and secondary. These new constraints are second class. In this formulation, the Lagrange multipliers implementing the primary constraints get identified as accelerations. This is a generic feature of any Lagrangian linear in the acceleration possessing reparametrization invariance. 
  We study static, spherically symmetric, Skyrme black holes in the context of the assumption that they can be viewed as bound states between ordinary bare black holes and solitons. This assumption and results stemming from the isolated horizons formalism lead to several conjectures about the static black hole solutions. These conjectures are tested against the Skyrme black hole solutions. It is shown that, while there is in general good agreement with the conjectures, a crucial aspect seems to violate one of the conjectures. 
  The specified constant 4-vector field reproducing the spherically symmetric stationary metric of cold dark matter halo in the region of flat rotation curves results in a constant angle of light deflection at small impact distances. The effective deflecting mass is factor $\pi/2$ greater than the dark matter mass. The perturbation of deflection picture due to the halo edge is evaluated. 
  We discuss the phenomenon of the smooth dynamical gravity induced crossing of the phantom divide line in a framework of simple cosmological models where it appears to occur rather naturally, provided the potential of the unique scalar field has some kind of cusp. The behavior of cosmological trajectories in the vicinity of the cusp is studied in some detail and a simple mechanical analogy is presented. The phenomenon of certain complementarity between the smoothness of the spacetime geometry and matter equations of motion is elucidated. We introduce a network of cosmological histories and qualitatively describe some of its properties. 
  We consider the problem of the nature and possible types of spacetime singularities that can form during the evolution of \emph{FRW} universes in general relativity. We show that by using, in addition to the Hubble expansion rate and the scale factor, the Bel-Robinson energy of these universes we can consistently distinguish between the possible different types of singularities and arrive at a complete classification of the singularities that can occur in the isotropic case. We also use the Bel-Robinson energy to prove that known behaviours of exact flat isotropic universes with given singularities are generic in the sense that they hold true in every type of spatial geometry. 
  In general relativity, for fluids with a linear equation of state (EoS) or scalar fields, the high isotropy of the universe requires special initial conditions, and singularities are anisotropic in general. In the brane world scenario anisotropy at the singularity is suppressed by an effective quadratic equation of state. There is no reason why the effective EoS of matter should be linear at the highest energies, and a non-linear EoS may describe dark energy or unified dark matter (Paper I, astro-ph/0512224). In view of this, here we study the effects of a quadratic EoS in homogenous and inhomogeneous cosmological models in general relativity, in order to understand if in this context the quadratic EoS can isotropize the universe at early times. With respect to Paper I, here we use the simplified EoS P=alpha rho + rho^2/rho_c, which still allows for an effective cosmological constant and phantom behavior, and is general enough to analyze the dynamics at high energies. We first study anisotropic Bianchi I and V models, focusing on singularities. Using dynamical systems methods, we find the fixed points of the system and study their stability. We find that models with standard non-phantom behavior are in general asymptotic in the past to an isotropic fixed point IS, i.e. in these models even an arbitrarily large anisotropy is suppressed in the past: the singularity is matter dominated. Using covariant and gauge invariant variables, we then study linear perturbations about the homogenous and isotropic spatially flat models with a quadratic EoS. We find that, in the large scale limit, all perturbations decay asymptotically in the past, indicating that the isotropic fixed point IS is the general asymptotic past attractor for non phantom inhomogeneous models with a quadratic EoS. (Abridged) 
  In recent years, interest in extra dimensions has experienced a dramatic increase. A common practice has been to look for higher-dimensional generalizations of four-dimensional solutions to the Einstein equations. In this vein, we have found a static, spherically symmetric solution to the 5-d Einstein equations. Certain aspects of this solution are very different from the 4-d Schwarzschild solution. However, in observationally accessible regions, the geodesics of the two solutions are essentially the same. 
  We show that Husain's reduction of the self-dual Einstein equations is equivalent to the Pleba\'nski equation. The B\"acklund transformation between these equations is found. Contact symmetries of the Husain equation are derived. 
  The role of torsion and a scalar field $\phi$ in gravitation, especially, in the presence of a Dirac field in the background of a particular class of the Riemann-Cartan geometry is considered here. Recently, a Lagrangian density with Lagrange multipliers has been proposed by the author which has been obtained by picking some particular terms from the SO(4,1) Pontryagin density, where the scalar field $\phi$ causes the de Sitter connection to have the proper dimension of a gauge field. In this article the scalar field has been linked to the dimension of the Dirac field. Here we get the field equations for the Dirac field and the scalar field in such a way that both of them appear to be mutually non-interacting. In this scenario the scalar field appears to be a natural candidate for the dark matter and the dark radiation. 
  We solve the effective Dirac equation for massless fermions during inflation in the simplest gauge, including all one loop corrections from quantum gravity. At late times the result for a spatial plane wave behaves as if the classical solution were subjected to a time dependent field strength renormalization of Z_2(t) = 1 - \frac{17}{4 \pi} G H^2 \ln(a) + O(G^2). We show that this also follows from making the Hartree approximation, although the numerical coefficients differ. 
  The subject of this study is Quantum and Statistical Mechanics of the Early Universe. In it a new approach to investigation of these two theories - density matrix deformation - is proposed. The distinguishing feature of the proposed approach as compared to the previous ones is the fact that here the density matrix is subjected to deformation, rather than commutators or (that is the same) Heisenberg's Algebra. The deformation is understood as an extension of a particular theory by inclusion of one or several additional parameters in such a way that the initial theory appears in the limiting transition. Some consequences of this approach for unitarity problem in Early Universe,black hole entropy, information paradox problem,for symmetry restoration in the associated deformed field model with scalar field are proposed. 
  We briefly review the recent experimental results on possible variations of the fine structure constant $\alpha$ on the cosmological time scale and its position dependence. We outline the theoretical grounds for the assumption that $\alpha$ might be variable, mention some phenomenological models incorporating a variable $\alpha$ into the context of modern cosmology and discuss the significance of possible $\alpha$ variations for theoretical and practical metrology. 
  After recalling the mathematical structure of Einstein's Hermitian extension of the gravitational theory of 1915, the problem, whether its field equations should admit of phenomenological sources at their right-hand sides, and how this addition should be done, is expounded by relying on a thread of essential insights and achievements by Schr\"odinger, Kursunoglu, Lichnerowicz, H\'ely and Borchsenius. When sources are appended to all the field equations, from the latter and from the contracted Bianchi identities a sort of gravoelectrodynamics appears, that totally departs from the so called Einstein-Maxwell theory, since its constitutive equation, that rules the link between inductions and fields, is a very complicated differential relation that allows for a much wider, still practically unexplored range of possible occurrences. In this sort of theory one can allow for both an electric and a magnetic four-current, which are not a physically wrong replica of each other, like it would occur if both these currents were allowed in Maxwell's vacuum. Particular static exact solutions show that, due to the peculiar constitutive equation, while electric charges with a pole structure behave according to Coulomb's law, magnetic charges with a pole structure interact with forces not depending on their mutual distance. The latter behaviour was already discovered by Treder in 1957 with an approximate calculation, while looking for ordinary electromagnetism in the theory. He also showed that in the Hermitian theory magnetic charges of unlike sign mutually attract, hence they are permanently confined entities. The exact solutions confirm this finding, already interpreted in 1980 by Treder in a chromodynamic sense. 
  Time-series data from multiple gravitational wave (GW) detectors can be linearly combined to form a null-stream, in which all GW information will be cancelled out. This null-stream can be used to distinguish between actual GW triggers and spurious noise transients in a search for GW bursts using a network of detectors. The biggest source of error in the null-stream analysis comes from the fact that the detector data are not perfectly calibrated. In this paper, we present an implementation of the null-stream veto in the simplest network of two co-located detectors. The detectors are assumed to have calibration uncertainties and correlated noise components. We estimate the effect of calibration uncertainties in the null-stream veto analysis and propose a new formulation to overcome this. This new formulation is demonstrated by doing software injections in Gaussian noise. 
  Coherent techniques for searches of gravitational-wave bursts effectively combine data from several detectors, taking into account differences in their responses. The efforts are now focused on the maximum likelihood principle as the most natural way to combine data, which can also be used without prior knowledge of the signal. Recent studies however have shown that straightforward application of the maximum likelihood method to gravitational waves with unknown waveforms can lead to inconsistencies and unphysical results such as discontinuity in the residual functional, or divergence of the variance of the estimated waveforms for some locations in the sky. So far the solutions to these problems have been based on rather different physical arguments. Following these investigations, we now find that all these inconsistencies stem from rank deficiency of the underlying network response matrix. In this paper we show that the detection of gravitational-wave bursts with a network of interferometers belongs to the category of ill-posed problems. We then apply the method of Tikhonov regularization to resolve the rank deficiency and introduce a minimal regulator which yields a well-conditioned solution to the inverse problem for all locations on the sky. 
  General Relativity assumes that spacetime is fully described by the metric alone. An alternative is the so called Palatini formalism where the metric and the connections are taken as independent quantities. The metric-affine theory of gravity has attracted considerable attention recently, since it was shown that within this framework some cosmological models, based on some generalized gravitational actions, can account for the current accelerated expansion of the universe. However we think that metric-affine gravity deserves much more attention than that related to cosmological applications and so we consider here metric-affine gravity theories in which the gravitational action is a general function of the scalar curvature while the matter action is allowed to depend also on the connection which is not {\em a priori} symmetric. This general treatment will allow us to address several open issues such as: the relation between metric-affine $f(R)$ gravity and General Relativity (in vacuum as well as in the presence of matter), the implications of the dependence (or independence) of the matter action on the connections, the origin and role of torsion and the viability of the minimal-coupling principle. 
  One of the unsolved issues in the quantum gravity comes from the Wheeler-DeWitt equation, which is second order functional derivative equation. In this paper, we introduce a new method to solve the Wheeler-DeWitt equation. Usually one treats the state functional of the space of 3-dimensional metrics, which do not contain timelike metrics. However we can expand this state to the state which has support on the space of spacetime metrics with using additional constraint which requires the recovery of 4-dimensional quantum gravity. Enlarging the support of the state functional of the spacetime metrics, we can simply solve the usual Wheeler-DeWitt equation with the additional constraint. Using this method we can solve some unsolved problems, such as the quantization of the black hole. 
  A new kind of uniformly accelerated reference frames with a line-element different from the M{\o}ller and Rindler ones is presented, in which every observer at $x, y, z=$consts. has the same constant acceleration. The laws of mechanics are checked in the new kind of frames. Its thermal property is studied. The comparison with the M{\o}ller and Rindler uniform accelerated reference frames is also made. 
  Re-analyzing the data published by the Berlin and Duesseldorf ether-drift experiments, we have found a clean non-zero daily average for the amplitude of the signal. The two experimental values, A_0\sim (10.5 \pm 1.3) 10^{-16} and A_0\sim (12.1\pm 2.2) 10^{-16}$ respectively, are entirely consistent with the theoretical prediction (9.7\pm 3.5) 10^{-16} that is obtained once the Robertson-Mansouri-Sexl anisotropy parameter is expressed in terms of N_{vacuum}, the effective vacuum refractive index that one would get, for an apparatus placed on the Earth's surface, in a flat-space picture of gravity . 
  In the harmonic description of general relativity, the principle part of Einstein's equations reduces to 10 curved space wave equations for the componenets of the space-time metric. We present theorems regarding the stability of several evolution-boundary algorithms for such equations when treated in second order differential form. The theorems apply to a model black hole space-time consisting of a spacelike inner boundary excising the singularity, a timelike outer boundary and a horizon in between. These algorithms are implemented as stable, convergent numerical codes and their performance is compared in a 2-dimensional excision problem. 
  The main result of the paper is a new representation for the Weyl Lagrangian (massless Dirac Lagrangian). As the dynamical variable we use the coframe, i.e. an orthonormal tetrad of covector fields. We write down a simple Lagrangian - wedge product of axial torsion with a lightlike element of the coframe - and show that variation of the resulting action with respect to the coframe produces the Weyl equation. The advantage of our approach is that it does not require the use of spinors, Pauli matrices or covariant differentiation. The only geometric concepts we use are those of a metric, differential form, wedge product and exterior derivative. Our result assigns a variational meaning to the tetrad representation of the Weyl equation suggested by J.B.Griffiths and R.A.Newing. 
  We present the first fully-nonlinear numerical study of the dynamics of highly spinning black-hole binaries. We evolve binaries from quasicircular orbits (as inferred from Post-Newtonian theory), and find that the last stages of the orbital motion of black-hole binaries are profoundly affected by their individual spins. In order to cleanly display its effects, we consider two equal mass holes with individual spin parameters S/m^2=0.757, both aligned and anti-aligned with the orbital angular momentum (and compare with the spinless case), and with an initial orbital period of 125M. We find that the aligned case completes three orbits and merges significantly after the anti-aligned case, which completes less than one orbit. The total energy radiated for the former case is ~7% while for the latter it is only ~2%. The final Kerr hole remnants have rotation parameters a/M=0.89 and a/M=0.44 respectively, showing the unlikeliness of creating a maximally rotating black hole out of the merger of two spinning holes. 
  Analytical and numerical methods are developed to analyze the quantum nature of the big bang in the setting of loop quantum cosmology. They enable one to explore the effects of quantum geometry both on the gravitational and matter sectors and significantly extend the known results on the resolution of the big bang singularity. Specifically, the following results are established for the homogeneous isotropic model with a massless scalar field: i) the scalar field is shown to serve as an internal clock, thereby providing a detailed realization of the `emergent time' idea; ii) the physical Hilbert space, Dirac observables and semi-classical states are constructed rigorously; iii) the Hamiltonian constraint is solved numerically to show that the big bang is replaced by a big bounce. Thanks to the non-perturbative, background independent methods, unlike in other approaches the quantum evolution is deterministic across the deep Planck regime. Our constructions also provide a conceptual framework and technical tools which can be used in more general models. In this sense, they provide foundations for analyzing physical issues associated with the Planck regime of loop quantum cosmology as a whole. 
  The dynamical systems observability properties of barotropic bosonic and fermionic FRW cosmological oscillators are investigated. Nonlinear techniques for dynamical analysis have been recently developed in many engineering areas but their application has not been extended beyond their standard field. This paper is a small contribution to an extension of this type of dynamical systems analysis to FRW barotropic cosmologies. We find that determining the Hubble parameter of barotropic FRW universes does not allow the observability, i.e., the determination of neither the barotropic FRW zero mode nor of its derivative as dynamical cosmological states. Only knowing the latter ones correspond to a rigorous dynamical observability in barotropic cosmology 
  This paper presents a quasi-local method of studying the physics of dynamical black holes in numerical simulations. This is done within the dynamical horizon framework, which extends the earlier work on isolated horizons to time-dependent situations. In particular: (i) We locate various kinds of marginal surfaces and study their time evolution. An important ingredient is the calculation of the signature of the horizon, which can be either spacelike, timelike, or null. (ii) We generalize the calculation of the black hole mass and angular momentum, which were previously defined for axisymmetric isolated horizons to dynamical situations. (iii) We calculate the source multipole moments of the black hole which can be used to verify that the black hole settles down to a Kerr solution. (iv) We also study the fluxes of energy crossing the horizon, which describes how a black hole grows as it accretes matter and/or radiation.   We describe our numerical implementation of these concepts and apply them to three specific test cases, namely, the axisymmetric head-on collision of two black holes, the axisymmetric collapse of a neutron star, and a non-axisymmetric black hole collision with non-zero initial orbital angular momentum. 
  In this work we show that 3d Feynman amplitudes of standard QFT in flat and homogeneous space can be naturally expressed as expectation values of a specific topological spin foam model. The main interest of the paper is to set up a framework which gives a background independent perspective on usual field theories and can also be applied in higher dimensions. We also show that this Feynman graph spin foam model, which encodes the geometry of flat space-time, can be purely expressed in terms of algebraic data associated with the Poincare group. This spin foam model turns out to be the spin foam quantization of a BF theory based on the Poincare group, and as such is related to a quantization of 3d gravity in the limit where the Newton constant G_N goes to 0. We investigate the 4d case in a companion paper where the strategy proposed here leads to similar results. 
  This paper considers the problem of detecting nonstationary phenomena, and chirps in particular, from very noisy data. Chirps are waveforms of the very general form A(t) exp(i\lambda \phi(t)), where \lambda is a (large) base frequency, the phase \phi(t) is time-varying and the amplitude A(t) is slowly varying. Given a set of noisy measurements, we would like to test whether there is signal or whether the data is just noise. One particular application of note in conjunction with this problem is the detection of gravitational waves predicted by Einstein's Theory of General Relativity.   We introduce detection strategies which are very sensitive and more flexible than existing feature detectors. The idea is to use structured algorithms which exploit information in the so-called chirplet graph to chain chirplets together adaptively as to form chirps with polygonal instantaneous frequency. We then search for the path in the graph which provides the best trade-off between complexity and goodness of fit. Underlying our methodology is the idea that while the signal may be extremely weak so that none of the individual empirical coefficients is statistically significant, one can still reliably detect by combining several coefficients into a coherent chain. This strategy is general and may be applied in many other detection problems. We complement our study with numerical experiments showing that our algorithms are so sensitive that they seem to detect signals whenever their strength makes them detectable. 
  The Laser Interferometric Space Antenna (LISA) will observe supermassive black hole binary mergers with amplitude signal-to-noise ratio of several thousands. We investigate the extent to which such observations afford high-precision tests of Einstein's gravity. We show that LISA provides a unique opportunity to probe the non-linear structure of post-Newtonian theory both in the context of general relativity and its alternatives. 
  This work investigates how a conical singularity can affect the specific heat of systems. A free nonrelativistic particle confined to the lateral surface of a cone -- conical box -- is taken as a toy model. Its specific heat is determined as a function of the deficit angle and the temperature. For a vanishing deficit angle, the specific heat is that of a particle in a flat disk where a characteristic temperature separates quantum and classical behaviors, as usual. By increasing the deficit angle the characteristic temperature increases also, and eventually another characteristic temperature (which does not depend on the deficit angle) arises. When the cone gets sufficiently sharp, at low and intermediate temperatures the azimuthal degree of freedom is suppressed. At low temperatures the specific heat varies discontinuously with the deficit angle. Connections between certain theorems regarding common zeros of the Bessel functions and this discontinuity are reported. 
  Back-reaction of the massless scalar field vacuum to the Universe expansion is considered. Automatic renormalization procedure based on the equations of motion instead of the Friedman equation is used to avoid the cosmological constant problem. It is found, that the vacuum tends to decelerate an expansion of Universe if the conformal acceleration of Universe is equal to zero. In contract, the vacuum acts as the true cosmological constant in Universe expanding with the present day acceleration. Estimation for third derivative of the Universe scale factor with respect to conformal time is presented. 
  We study the negative mass Schwarzschild spacetime, which has a naked singularity, and show that it is perturbatively unstable. This is achieved by first introducing a modification of the well known Regge - Wheeler - Zerilli approach to black hole perturbations to allow for the presence of a ``kinematic'' singularity that arises for negative masses, and then exhibiting exact exponentially growing solutions to the linearized Einstein's equations. The perturbations are smooth everywhere and behave nicely around the singularity and at infinity. In particular, the first order variation of the scalar invariants can be made everywhere arbitrarily small as compared to the zeroth order terms. Our approach is also compared to a recent analysis that leads to a different conclusion regarding the stability of the negative mass Schwarzschild spacetime. We also comment on the relevance of our results to the stability of more general negative mass, nakedly singular spacetimes. 
  Beyond point mass effects various contributions add to the radiative evolution of compact binaries. We present all the terms up to the second post-Newtonian order contributing to the rate of increase of gravitational wave frequency and the number of gravitational wave cycles left until the final coalescence for binary systems with spin, mass quadrupole and magnetic dipole moments, moving on circular orbit. We evaluate these contributions for some famous or typical compact binaries and show that the terms representing the self interaction of individual spins, given for the first time here, are commensurable with the proper spin-spin contributions for the recently discovered double pulsar J0737-3039. 
  We derive source-free Maxwell-like equations in flat spacetime for any helicity "j" by comparing the transformation properties of the 2(2j+1) states that carry the manifestly covariant representations of the inhomogeneous Lorentz group with the transformation properties of the two helicity "j" states that carry the irreducible representations of this group. The set of constraints so derived involves a pair of curl equations and a pair of divergence equations. These reduce to the free-field Maxwell equations for j=1 and the analogous equations coupling the gravito-electric and the gravito-magnetic fields for j=2. 
  Gravitational interactions are treated as non-associative part of a Lagrangian of octonion fields. The Lagrangian is defined in the charge space as squared curvature with respect to the octonion fields. The applications of suggested formalism to homogeneous and isotropic space are studied. 
  Recently the class of purely magnetic non-rotating dust spacetimes has been shown to be empty (Wylleman, Class. Quant. Grav. 23, 2727). It turns out that purely magnetic rotating dust models are subject to severe integrability conditions as well. One of the consequences of the present paper is that also rotating dust cannot be purely magnetic when it is of Petrov type D or when it has a vanishing spatial gradient of the energy density. For purely magnetic and non-rotating perfect fluids on the other hand, which have been fully classified earlier for Petrov type D (Lozanovski, Class. Quant. Grav. 19, 6377), the fluid is shown to be non-accelerating if and only if the spatial density gradient vanishes. Under these conditions, a new and algebraically general solution is found, which is unique up to a constant rescaling, which is spatially homogeneous of Bianchi type $VI_0$, has degenerate shear and is of Petrov type I($M^\infty)$ in the extended Arianrhod-McIntosh classification.   The metric and the equation of state are explicitly constructed and properties of the model are briefly discussed. We finally situate it within the class of normal geodesic flows with degenerate shear tensor. 
  Beginning with a brief sketch of the derivation of Hawking's theorem of horizon area increase, based on the Raychaudhuri equation, we go on to discuss the issue as to whether generic black holes, undergoing Hawking radiation, can ever remain in stable thermal equilibrium with that radiation. We derive a universal criterion for such a stability, which relates the black hole mass and microcanonical entropy, both of which are well-defined within the context of the Isolated Horizon, and in principle calculable within Loop Quantum Gravity. The criterion is argued to hold even when thermal fluctuations of electric charge are considered, within a {\it grand} canonical ensemble. 
  We derive a canonical analysis of a double null 2+2 Hamiltonian description of General Relativity in terms of complex self-dual 2-forms and the associated SO(3) connection variables. The algebra of first class constraints is obtained and forms a Lie algebra that consists of two constraints that generate diffeomorphisms in the two surface, a constraint that generates diffeomorphisms along the null generators and a constraint that generates self-dual spin and boost transformations. 
  In the present paper we will investigate the relation between scalar-tensor theory and $f(R)$ theories of gravity. Such studies have been performed in the past for the metric formalism of $f(R)$ gravity; here we will consider mainly the Palatini formalism, where the metric and the connections are treated as independent quantities. We will try to investigate under which circumstances $f(R)$ theories of gravity are equivalent to scalar-tensor theory and examine the implications of this equivalence, when it exists. 
  We consider the problem of describing the asymptotic behaviour of \textsc{FRW} universes near their spacetime singularities in general relativity. We find that the Bel-Robinson energy of these universes in conjunction with the Hubble expansion rate and the scale factor proves to be an appropriate measure leading to a complete classification of the possible singularities. We show how our scheme covers all known cases of cosmological asymptotics possible in these universes and also predicts new and distinct types of singularities. We further prove that various asymptotic forms met in flat cosmologies continue to hold true in their curved counterparts. These include phantom universes with their recently discovered big rips, sudden singularities as well as others belonging to graduated inflationary models. 
  Static space times with maximal symmetric transverse spaces are classified according to their Ricci collineations. These are investigated for non-degenerate Ricci tensor ($det.(R_{\alpha}) \neq 0$). It turns out that the only collineations admitted by these spaces can be ten, seven, six or four. Some new metrics admitting proper Ricci collineations are also investigated. 
  We study the possibility that a generalised real scalar field minimally coupled to gravity could explain both the galactic and the cosmological dark components of the universe. Within the framework of Einstein's Relativity we model static galactic halos by considering the most general action built from the scalar field and its first derivatives. Although the gravitational configuration is static, the scalar field may be either static, or homogeneous and linear in time. In the case of the static scalar field, the models we look at inevitably posses unphysical negative energies, and we are led to a sort of no-go result. In the case of the homogeneous scalar field, on the contrary, we find that compact objects with flat rotational curves and with the mass and the size of a typical galaxy can be successfully modeled and the Tully-Fisher relation recovered. We further show that the homogeneous scalar field deduced from the galactic halo spacetimes has an action compatible with the kinetic Unified Dark Matter models recently proposed by Scherrer. Therefore, such a homogeneous kinetic Unified Dark Matter not only may correctly mimic the galactic dynamics, but could also be used to model the present-day accelerated expansion in the universe. 
  The signal-to-noise ratios (SNRs) for quasi-circular binary black hole inspirals computed from restricted post-Newtonian waveforms are compared with those attained by more complete post-Newtonian signals, which are superpositions of amplitude-corrected harmonics of the orbital phase. It is shown that if one were to use the best available amplitude-corrected waveforms for detection templates, one should expect SNRs in actual searches to be significantly lower than those suggested by simulations based purely on restricted waveforms. 
  We consider scalar and axial gravitational perturbations of black hole solutions in brane world scenarios. We show that perturbation dynamics is surprisingly similar to the Schwarzschild case with strong indications that the models are stable. Quasinormal modes and late-time tails are discussed. We also study the thermodynamics of these scenarios verifying the universality of Bekenstein's entropy bound as well as the applicability of 't Hooft's brickwall method. 
  Cauchy-Characteristic Matching (CCM), the combination of a central 3+1 Cauchy code with an exterior characteristic code connected across a time-like interface, is a promising technique for the generation and extraction of gravitational waves. While it provides a tool for the exact specification of boundary conditions for the Cauchy evolution, it also allows to follow gravitational radiation all the way to infinity, where it is unambiguously defined.   We present a new fourth order accurate finite difference CCM scheme for a first order reduction of the wave equation around a Schwarzschild black hole in axisymmetry. The matching at the interface between the Cauchy and the characteristic regions is done by transfering appropriate characteristic/null variables. Numerical experiments indicate that the algorithm is fourth order convergent. As an application we reproduce the expected late-time tail decay for the scalar field. 
  We prove that when the equations are restricted to the principal part the standard version of the BSSN formulation of the Einstein equations is equivalent to the NOR formulation for any gauge, and that the KST formulation is equivalent to NOR for a variety of gauges. We review a family of elliptic gauge conditions and the implicit parabolic and hyperbolic drivers that can be derived from them, and show how to make them symmetry-seeking. We investigate the hyperbolicity of ADM, NOR and BSSN with implicit hyperbolic lapse and shift drivers. We show that BSSN with the coordinate drivers used in recent "moving puncture" binary black hole evolutions is ill-posed at large shifts, and suggest how to make it strongly hyperbolic for arbitrary shifts. For ADM, NOR and BSSN with elliptic and parabolic gauge conditions, which cannot be hyperbolic, we investigate a necessary condition for well-posedness of the initial-value problem. 
  An explicit calculation of Casimir effect through an alternative approach of field quantization [1, 2], has been presented in this paper. In this method, the auxiliary negative norm states have been utilized, the modes of which do not interact with the physical states or real physical world. Naturally these modes cannot be affected by the physical boundary conditions. Presence of negative norm states play the rule of an automatic renormalization device for the theory. 
  Gravitational waves from coalescing compact binaries are searched using the matched filtering technique. As the model waveform depends on a number of parameters, it is necessary to filter the data through a template bank covering the astrophysically interesting region of the parameter space. The choice of templates is defined by the maximum allowed drop in signal-to-noise ratio due to the discreteness of the template bank. In this paper we describe the template-bank algorithm that was used in the analysis of data from the Laser Interferometer Gravitational Wave Observatory (LIGO) and GEO 600 detectors to search for signals from binaries consisting of non-spinning compact objects. Using Monte-Carlo simulations, we study the efficiency of the bank and show that its performance is satisfactory for the design sensitivity curves of ground-based interferometric gravitational wave detectors GEO 600, initial LIGO, advanced LIGO and Virgo. The bank is efficient to search for various compact binaries such as binary primordial black holes, binary neutron stars, binary black holes, as well as a mixed binary consisting of a non-spinning black hole and a neutron star. 
  In the gravity without metric formalism of Capovilla, Jacobson and Dell, the topological $\theta$-term appears through a canonical transformation.The origin of this canonical transformation is probed here. It is shown here that when $\theta$-term appears cosmological $\lambda$-term also appears simultaneously. 
  We study the statistical mechanics for quantum scalar fields under the multi-parameter rotating black hole spacetime in arbitrary D dimensions. The method of analysis is general in the sense that the metric does not depend on the explicit black hole solutions. The generalized Stefan-Boltzmann's law for the scalar field is derived by considering the allowed energy region properly. Then the generalized area law for the scalar field entropy is derived by introducing the invariant regularization parameter in the Rindler spacetime. The derived area law is applied to Kerr-AdS black holes in four and five dimensions. Thermodynamic implication is also discussed. 
  This article concerns the fate of local Lorentz invariance in quantum gravity, particularly for approaches in which a discrete structure replaces continuum spacetime. Some features of standard quantum mechanics, presented in a sum-over-histories formulation, are reviewed, and their consequences for such theories are discussed. It is argued that, if the individual histories of a theory give bad approximations to macroscopic continuum properties in some frames, then it is inevitable that the theory violates Lorentz symmetry. 
  Motivation and perspective for an exciting new research direction interconnecting logic, spacetime theory, relativity--including such revolutionary areas as black hole physics, relativistic computers, new cosmology--are presented in this paper. We would like to invite the logician reader to take part in this grand enterprise of the new century. Besides general perspective and motivation, we present initial results in this direction. 
  Riemann-Cartan space time $U_{4}$ is considered here. It has been shown that when we link topological Nieh-Yan density with the gravitational constant then we get Einstein-Hilbert Lagrangian as a consequence. 
  We consider the nonlinear stability of the Kaluza-Klein monopole viewed as the static solution of the five dimensional vacuum Einstein equations. Using both numerical and analytical methods we give evidence that the Kaluza-Klein monopole is asymptotically stable within the cohomogeneity-two biaxial Bianchi IX ansatz recently introduced in \cite{bcs}. We also show that for sufficiently large perturbations the Kaluza-Klein monopole loses stability and collapses to a Kaluza-Klein black hole. The relevance of our results for the stability of BPS states in M/String theory is briefly discussed. 
  We compute some components of the graviton propagator in loop quantum gravity, using the spinfoam formalism, up to some second order terms in the expansion parameter. 
  Introductive chapter of a book on Quantum Gravity, edited by Daniele Oriti, to appear with Cambridge University Press. 
  In Brans-Dicke theory of gravity we explain how the extra constant value in the formula for rotation velocities of stars in a galactic halo can be obtained due to the global monopole field. We argue on a few points of the preceding Comment and discuss improvement of our model. 
  Empirical implications of a teleparallel displacement of momentum between initial and final quantum states, using conformally flat quantum coordinates are investigated. An exact formulation is possible in an FRW cosmology in which cosmological redshift is given by 1+z=a_0^2/a^2(t). This is consistent with current observation for a universe expanding at half the rate and twice as old as indicated by a linear law, and, in consequence, requiring a quarter of the critical density for closure. A no CDM teleconnection model resolves inconsistencies between galactic profiles found from lensing data, rotation curves and analytic models of galaxy evolution. The teleconnection model favours a closed no Lambda cosmology. After rescaling Omega so that Omega=1 is critical density, for 225 supernovae, the best fit teleconnection no Lambda model with Omega=1.89 is marginally preferred to the best fit standard flat space Lambda model with Omega=0.284. It will require many observations of supernovae at z>1.5 to eliminate either the standard or teleconnection magnitude-redshift relation. In quantum coordinates the anomalous Pioneer blueshift and the flattening of galaxies' rotation curves appear as optical effects, not as modifications to classical motions. An exact form of Milgrom's phenomenological law (MOND) is shown. 
  We discuss the relevance, for the search of gravitational-wave bursts, of upper limits on the total mass loss of the Galaxy which come from various astronomical observations. For sub-millisecond bursts we obtain limits on the event rate, as a function of the GW amplitude, which are stronger than the corresponding upper limits set by LIGO in the S2 run. A detection of a burst rate saturating these limits, with the sensitivities of present and near-future runs, would imply that, with some improvement on the accuracy of astronomical observations of the Galaxy, as foreseen with the GAIA mission, it might be possible to detect gravitational waves indirectly from their effect on galactic dynamics. 
  The implications of an Evolutionary Quantum Gravity are addressed in view of formulating a new dark matter candidate. We consider a Schr\"odinger dynamics for the gravitational field associated to a generic cosmological model and then we solve the corresponding eigenvalues problem, inferring its phenomenological issue for the actual Universe. The spectrum of the super-Hamiltonian is determined including a free inflaton field, the ultrarelativistic thermal bath and a perfect gas into the dynamics. We show that, when a Planckian cut-off is imposed in the theory and the classical limit of the ground state is taken, then a dark matter contribution can not arise because its critical parameter $\Omega_{dm}$ is negligible today when the appropriate cosmological implementation of the model is provided. Thus, we show that, from a phenomenological point of view, an Evolutionary Quantum Cosmology overlaps the Wheeler-DeWitt approach and therefore it can be inferred as appropriate to describe early stages of the Universe without significant traces on the later evolution. Finally, we provide indications that the horizon paradox can be solved in the Planck era by the morphology of the Universe wave function. 
  An analytical approach for studying the cosmological scenario with a homogeneous tachyon field within the framework of loop quantum gravity is developed. Our study is based on the semi-classical regime where space time can be approximated as a continuous manifold, but matter Hamiltonian gets non-perturbative quantum corrections. A formal correspondence between classical and loop quantum cosmology is also established. The Hamilton-Jacobi method for getting exact solutions is constructed and some exact power-law as well as bouncing solutions are presented. 
  The low energy effective action of gravity in any even dimension generally acquires non-local terms associated with the trace anomaly, generated by the quantum fluctuations of massless fields. The local auxiliary field description of this effective action in four dimensions requires two additional scalar fields, not contained in classical general relativity, which remain relevant at macroscopic distance scales. The auxiliary scalar fields depend upon boundary conditions for their complete specification, and therefore carry global information about the geometry and macroscopic quantum state of the gravitational field. The scalar potentials also provide coordinate invariant order parameters describing the conformal behavior and divergences of the stress tensor on event horizons. We compute the stress tensor due to the anomaly in terms of its auxiliary scalar potentials in a number of concrete examples, including the Rindler wedge, the Schwarzschild geometry, and de Sitter spacetime. In all of these cases, a small number of classical order parameters completely determine the divergent behaviors allowed on the horizon, and yield qualitatively correct global approximations to the renormalized expectation value of the quantum stress tensor. 
  A collection is made of presently unexplained phenomena within our Solar system and in the universe. These phenomena are (i) the Pioneer anomaly, (ii) the flyby anomaly, (iii) the increase of the Astronomical Unit, (iv) the quadrupole and octupole anomaly, and (v) Dark Energy and (vi) Dark Matter. A new data analysis of the complete set of Pioneer data is announced in order to search for systematic effects or to confirm the unexplained acceleration. We also review the mysterious flyby anomaly where the velocities of spacecraft after Earth swing--bys are larger than expected. We emphasize the scientific aspects of this anomaly and propose systematic and continuous observations and studies at the occasion of future flybys. Further anomalies within the Solar system are the increase of the Astronomical Unit and the quadrupole and octupole anomaly. We briefly mention Dark Matter and Dark Energy since in some cases a relation between them and the Solar system anomalies have been speculated. 
  We analyze the violation of the strong energy condition, including the contribution from gravity sector, in the framework of loop quantum cosmology (LQC). Our discussion is limited to the spatially flat LQC, and we make use of the effective Hamiltonian to study this issue. The central finding is that the effect of gravity sector is to weaken the violation of the strong energy condition when the gravity sector becomes more and more prominent compared to the matter field. 
  We consider universal properties that arise from a local geometric structure of a Killing horizon. We first introduce a non-perturbative definition of such a local geometric structure, which we call an asymptotic Killing horizon. It is shown that infinitely many asymptotic Killing horizons reside on a common null hypersurface, once there exists one asymptotic Killing horizon. The acceleration of the orbits of the vector that generates an asymptotic Killing horizon is then considered. We show that there exists the $\textit{diff}(S^1)$ or $\textit{diff}(R^1)$ sub-algebra on an asymptotic Killing horizon universally, which is picked out naturally based on the behavior of the acceleration. We also argue that the discrepancy between string theory and the Euclidean approach in the entropy of an extreme black hole may be resolved, if the microscopic states responsible for black hole thermodynamics are connected with asymptotic Killing horizons. 
  We study Einstein gravity coupled to a massless scalar field in a static spherically symmetric space-time in four dimensions. Black hole solutions exist when the kinetic energy of the scalar field is negative, that is, for a phantom field. These ``scalar black holes'' have an infinite horizon area and zero temperature. They are related through a conformal transformation with similar objects in the Jordan frames of scalar-tensor theories of gravity. The thermodynamical properties of these solutions are discussed. It is proved that any \ssph \bhs with an infinite horizon area have zero Hawking temperature. 
  We study the stability of general relativistic static thick disks, as an application we consider the thick disk generated by applying the ``displace, cut, fill and reflect" method, usually known as the image method, to the Schwarzschild metric in isotropic coordinates. The isotropic Schwarzschild thick disk obtained from this method is the simplest model to describe, in the context of General Relativity, real thick galaxies. The stability under a general first order perturbation of the disk is investigated. The first order perturbation, when applying to the conservation equations, leads to a set of differential equations that are, in general, not self-consistent. This opens the possibility of performing various kinds of perturbations to transform the resulting system of equations into a self-consistent system. We perform a complete classification of the perturbations as well as the stability analysis for all the relevant physical perturbations. We found that, in general, the isotropic Schwarzschild thick disk is stable under these kinds of perturbations. 
  We consider the possibility that the total dark energy (DE) of the Universe is made out of two dynamical components of different nature: a variable cosmological term, Lambda, and a dynamical ``cosmon'', X, possibly interacting with Lambda but not with matter -- which remains conserved. We call this scenario the LXCDM model. One possibility for X would be a scalar field, but it is not the only one. The overall equation of state (EOS) of the LXCDM model can effectively appear as quintessence or phantom energy depending on the mixture of the two components. Both the dynamics of Lambda and of X could be linked to high energy effects near the Planck scale. In the case of Lambda it may be related to the running of this parameter under quantum effects, whereas X might be identified with some fundamental field (say, a dilaton) left over as a low-energy ``relic'' by e.g. string theory. We find that the dynamics of the LXCDM model can trigger a future stopping of the Universe expansion and can keep the ratio rho_D/rho_m (DE density to matter-radiation density) bounded and of order 1. Therefore, the model could explain the so-called ``cosmological coincidence problem''. This is in part related to the possibility that the present value of the cosmological term can be Lambda<0 in this framework (the current total DE density nevertheless being positive). However, a cosmic halt could occur even if Lambda>0 because of the peculiar behavior of X as ``Phantom Matter''. We describe various cosmological scenarios made possible by the composite and dynamical nature of LXCDM, and discuss in detail their impact on the cosmological coincidence problem. 
  We discuss the issue of quasi-particle production by ``analogue black holes'' with particular attention to the possibility of reproducing Hawking radiation in a laboratory. By constructing simple geometric acoustic models, we obtain a somewhat unexpected result: We show that in order to obtain a stationary and Planckian emission of quasi-particles, it is not necessary to create an ergoregion in the acoustic spacetime (corresponding to a supersonic regime in the flow). It is sufficient to set up a dynamically changing flow either eventually generating an arbitrarily small sonic region v=c, but without any ergoregion, or even just asymptotically, in laboratory time, approaching a sonic regime with sufficient rapidity. 
  We prove existence of static solutions to the cylindrically symmetric Einstein-Vlasov system, and we show that the matter cylinder has finite extension. The same results are also proved for a quite general class of equations of state for perfect fluids coupled to the Einstein equations, extending the class of equations of state considered in \cite{BL}. We also obtain this result for the Vlasov-Poisson system. 
  The most precise test of the post-Newtonian gamma parameter in the solar system has been achieved in measurement of the frequency shift of radio waves to and from the Cassini spacecraft as they passed near the Sun. The JPL model of radio wave propagation includes but does not explicitly evaluate the impact of the orbital motion of the Sun on the Shapiro delay as a radio wave traverses the solar system. The motion of the Sun induces a non-stationary component of the gravitational field that is associated with the Lorentz transformation from the heliocentric to the barycentric frame of the solar system. We analyze the effect of this motion on the propagation of a radio wave passing closely to the solar limb and show that, had the Lorentz-invariance of gravity been broken, it would have affected the measured value of gamma as published by Bertotti, Iess, and Tortora in Nature at the level of about 0.0001. Our analysis suggests that the JPL relativistic model of light propagation must be corrected to include the effect of the orbital motion of the Sun explicitly in order to separate its contribution to the measured value of gamma in such extremely precise experiments as with the Cassini spacecraft. 
  The gauge approach to gravity based on the local Lorentz group with a general independent affine connection A_{\mu cd} is developed. We consider SO(1,3) gauge theory with a Lagrangian quadratic in curvature as a simple model of quantum gravity. The torsion is proposed to represent a dynamic degree of freedom of quantum gravity at scales above the Planckian energy. The Einstein-Hilbert theory is induced as an effective theory due to quantum corrections of torsion via generating a stable gravito-magnetic condensate. We conjecture that torsion possesses an intrinsic quantum nature and can be confined. A minimal Abelian projection for the Lorentz gauge model has been constructed, and an effective theory of the cosmic knot at the Planckian scale is proposed. 
  We discuss the main myths related to the vacuum energy and cosmological constant, such as: ``unbearable lightness of space-time''; the dominating contribution of zero point energy of quantum fields to the vacuum energy; non-zero vacuum energy of the false vacuum; dependence of the vacuum energy on the overall shift of energy; the absolute value of energy only has significance for gravity; the vacuum energy depends on the vacuum content; cosmological constant changes after the phase transition; zero-point energy of the vacuum between the plates in Casimir effect must gravitate, that is why the zero-point energy in the vacuum outside the plates must also gravitate; etc. All these and some other conjectures appear to be wrong when one considers the thermodynamics of the ground state of the quantum many-body system, which mimics macroscopic thermodynamics of quantum vacuum. In particular, in spite of the ultraviolet divergence of the zero-point energy, the natural value of the vacuum energy is comparable with the observed dark energy. That is why the vacuum energy is the plausible candidate for the dark energy. 
  We provide a simple mathematical description of the exchange of energy between two fluids in an expanding Friedmann universe with zero spatial curvature. The evolution can be reduced to a single non-linear differential equation which we solve in physically relevant cases and provide an analysis of all the possible evolutions. Particular power-law solutions exist for the expansion scale factor and are attractors at late times under particular conditions. We show how a number of problems studied in the literature, such as cosmological vacuum energy decay, particle annihilation, and the evolution of a population of evaporating black holes, correspond to simple particular cases of our model. In all cases we can determine the effects of the energy transfer on the expansion scale factor. We also consider the situation in the presence of anti-decaying fluids and so called phantom fluids which violate the dominant energy conditions. 
  Although implicit in the discovery of the Schwarzschild solution 40 years earlier, the issues raised by the theory of what are now known as black holes were so unsettling to physicists of Einstein's generation that the subject remained in a state of semiclandestine gestation until his demise. That turning point -- just half a century after Einstein's original foundation of relativity theory, and just half a century ago today -- can be considered to mark the birth of black hole theory as a subject of systematic development by physicists of a new and less inhibited generation, whose enthusastic investigations have revealed structures of unforeseen mathematical beauty, even though questions about the physical significance of the concomitant singularities remain controversial. 
  By introducing the generalized uncertainty principle (GUP) on the quantum state density, we calculate the statistical entropy of a scalar field on the background of (2+1)-dimensional de Sitter space without artificial cutoff. The desired entropy proportional to the horizon perimeter is obtained. 
  An exact solution of Einsteins equations which represents a pair of accelerating and rotating black holes was presented by J. B. Griffiths and J. Podolsky [2]. In the paper [2] they have shown the explicit form of a spinning C-metric starting from the Plebanski-Demianski metric, and transformed it using NUT and angular velocity parameters in addition to the usual parameters and thus gave a generalized form of such solutions. In the forthcoming discussion, an attempt has been made to realize the Riemann components of the proposed metric. Furthermore, certain optical characteristics of the metric have been analyzed using the Newman-Penrose formalism. 
  Observations of the inspiral of massive binary black holes (BBH) in the Laser Interferometer Space Antenna (LISA) and stellar mass binary black holes in the European Gravitational-Wave Observatory (EGO) offer an unique opportunity to test the non-linear structure of general relativity. For a binary composed of two non-spinning black holes, the non-linear general relativistic effects depend only on the masses of the constituents. In a recent letter, we explored the possibility of a test to determine all the post-Newtonian coefficients in the gravitational wave-phasing.   However, mutual covariances dilute the effectiveness of such a test. In this paper, we propose a more powerful test in which the various post-Newtonian coefficients in the gravitational wave phasing are systematically measured by treating three of them as independent parameters and demanding their mutual consistency. LISA (EGO) will observe BBH inspirals with a signal-to-noise ratio of more than 1000 (100) and thereby test the self-consistency of each of the nine post-Newtonian coefficients that have so-far been computed, by measuring the lower order coefficients to a relative accuracy of $\sim 10^{-5}$ (respectively, $\sim 10^{-4}$) and the higher order coefficients to a relative accuracy in the range $10^{-4}$-0.1 (respectively, $10^{-3}$-1). 
  In this paper an explanation of the Pioneer anomaly is given, using the hypothesis of the time dependent gravitational potential. The implications of this hypothesis on the planetary orbits and orbital periods are given in section 2. In sections 3 and 4 we give a detailed explanation of the Pioneer anomaly, which corresponds to the observed phenomena. 
  This is an updated introductory review of 2 possible gravitational anomalies that has attracted part of the Scientific community: the Allais effect that occur during solar eclipses, and the Pioneer 10 spacecraft anomaly, experimented also by Pioneer 11 and Ulysses spacecrafts. It seems that, to date, no satisfactory conventional explanation exist to these phenomena, and this suggests that possible new physics will be needed to account for them. The main purpose of this review is to announce 3 other new measurements that will be carried on during the 2005 solar eclipses in Panama and Colombia (Apr. 8) and in Portugal (Oct.15). 
  Quantization of a dust-like closed isotropic cosmological model with a cosmological constant is realized by the method of B. DeWitt \cite{1}. It is shown that such quantization leads to interesting results, in particular, to a finite lifetime of the system, and appearance of the Universe in our world as penetration via the barrier. These purely quantum effects appear when $\Lambda>0$. 
  Braneworld models typically predict gravity to grow stronger at short distances. In this paper, we consider braneworlds with two types of additional curvature couplings, a Gauss-Bonnet term in the bulk, and an Einstein-Hilbert (EH) term on the brane. In the regime where these terms are dominant over the bulk EH term, linearized gravity becomes weaker at short distances on the brane. In both models, the weakening of gravity is tied to the presence of ghosts in the graviton mass spectrum. We find that the ordinary coupling of matter to gravity is recovered at low energies/long wavelengths on the brane. We give some implications for cosmology and show its compatibility with observations. We also discuss the stability of compact stars. 
  It has been claimed recently that the black hole information-loss paradox has been resolved: the evolution of quantum states in the presence of a black hole is unitary and information preserving. We point out that, contrary to some claims in literature, information-preserving black holes still violate baryon number and any other quantum number which follows from an effective (and thus approximate) or anomalous symmetry. 
  We analyze in detail the highly damped quasinormal modes of $d$-dimensional Reissner-Nordstr$\ddot{\rm{o}}$m black holes with small charge, paying particular attention to the large but finite damping limit in which the Schwarzschild results should be valid. In the infinite damping limit, we confirm using different methods the results obtained previously in the literature for higher dimensional Reissner-Nordstr$\ddot{\rm{o}}$m black holes. Using a combination of analytic and numerical techniques we also calculate the transition of the real part of the quasinormal mode frequency from the Reissner-Nordstr$\ddot{\rm{o}}$m value for very large damping to the Schwarzschild value of $\ln(3) T_{bh}$ for intermediate damping. The real frequency does not interpolate smoothly between the two values. Instead there is a critical value of the damping at which the topology of the Stokes/anti-Stokes lines change, and the real part of the quasinormal mode frequency dips to zero. 
  In this paper we investigate the quantum nature of a 2+1 dimensional black hole using the method [arXiv: gr-qc/0504030] which earlier revealed the quantum nature of a black hole in 3+1 dimensions. 
  We review different approaches to quantum gravity in which spacetime is emerging. We discuss in some detail the proposals by G. Volovik and S. Lloyd and show how they differ in the way they treat time. We further propose an approach to quantum gravity in which the Einstein equations are derived rather then used. We call this approach Internal Relativity. 
  Exact payload trajectories in the strong gravitational fields of compact masses moving with constant relativistic velocities are calculated. The strong field of a suitable driver mass at relativistic speeds can quickly propel a heavy payload from rest to a speed significantly faster than the driver, a condition called hyperdrive. Hyperdrive thresholds and maxima are calculated as functions of driver mass and velocity. 
  This dissertation aims at studying the braneworld models in the context proposed by Randall and Sundrum. The focus is on the spin-0 perturbations in the Kerr space-time as a 4-dimensional braneworld.   The work deals the main aspects of Einstein General Relativity as well as perturbations of black holes metrics. We also review the Randall-Sundrum models and their motivations and attempts to describe braneworld black holes. In the end the Kerr-Randall-Sundrum black string scalar perturbation and superradiance are obtained. 
  We report on the optical response of a suspended-mass detuned resonant sideband extraction (RSE) interferometer with power recycling. The purpose of the detuned RSE configuration is to manipulate and optimize the optical response of the interferometer to differential displacements (induced by gravitational waves) as a function of frequency, independently of other parameters of the interferometer. The design of our interferometer results in an optical gain with two peaks: an RSE optical resonance at around 4 kHz and a radiation pressure induced optical spring at around 41 Hz. We have developed a reliable procedure for acquiring lock and establishing the desired optical configuration. In this configuration, we have measured the optical response to differential displacement and found good agreement with predictions at both resonances and all other relevant frequencies. These results build confidence in both the theory and practical implementation of the more complex optical configuration being planned for Advanced LIGO. 
  Homogeneous cosmological solutions are obtained in five dimensional space time assuming equations of state $ p = k\rho $ and $ p_{5}= \gamma\rho$ where p is the isotropic 3 - pressure and $p_{5}$, that for the fifth dimension. Using different values for the constants k and $\gamma$ many known solutions are rediscovered. Further the current acceleration of the universe has led us to investigate higher dimensional gravity theory, which is able to explain acceleration from a theoretical view point without the need of introducing dark energy by hand. We argue that the terms containing higher dimensional metric coefficients produce an extra negative pressure that apparently drives an acceleration of the 3D space, tempting us to suggest that the accelerating universe seems to act as a window to the existence of extra spatial dimensions. Interestingly the 5D matter field remains regular while the \emph{effective} negative pressure is responsible for the inflation. Relaxing the assumptions of two equations of state we also present a class of solutions which provide early deceleration followed by a late acceleration in a unified manner. Interesting to point out that in this case our cosmology apparently mimics the well known quintessence scenario fuelled by a generalised Chaplygin-type of fluid where a smooth transition from a dust dominated model to a de Sitter like one takes place. 
  We investigate the possibility of using binary IMBH inspirals to perform the Ryan test of general relativity in a theoretically robust manner using data from early in the detectable part of the inspiral. We find this to be feasible and compute the masses of the most favourable systems. 
  The evolution of gauge invariant second-order scalar perturbations in a general single field inflationary scenario are presented. Different second order gauge invariant expressions for the curvature are considered. We evaluate perturbatively one of these second order curvature fluctuations and a second order gauge invariant scalar field fluctuation during the slow-roll stage of a massive chaotic inflationary scenario, taking into account the deviation from a pure de Sitter evolution and considering only the contribution of super-Hubble perturbations in mode-mode coupling. The spectra resulting from their contribution to the second order quantum correlation function are nearly scale-invariant, with additional logarithmic corrections to the first order spectrum. For all scales of interest the amplitude of these spectra depend on the total number of e-folds. We find, on comparing first and second order perturbation results, an upper limit to the total number of e-folds beyond which the two orders are comparable. 
  In the context of Brans-Dicke scalar tensor theory of gravitation, the cosmological Friedmann equation which relates the expansion rate $H$ of the universe to the various fractions of energy density is analyzed rigorously. It is shown that Brans-Dicke scalar tensor theory of gravitation brings a negligible correction to the matter density component of Friedmann equation. Besides, in addition to $\Omega_{\Lambda}$ and $\Omega_{M}$ in standard Einstein cosmology, another density parameter, $\Omega_{_{\Delta}}$, is expected by the theory. This implies that if $\Omega_{_{\Delta}}$ is found to be nonzero, data will favor this model instead of the standard Einstein cosmological model with cosmological constant and will enable more accurate predictions for the rate of change of Newtonian gravitational constant in the future. 
  Since the Brans-Dicke theory is conformal related to the dilaton gravity theory, by applying a conformal transformation to the dilaton gravity theory, we derived the cosmological constant term in the Brans-Dicke theory and the physical solution of black holes with the cosmological constant. It is found that, in four dimensions, the solution is just the Kerr-Newman-de Sitter solution with a constant scalar field. However, in $n>4$ dimensions, the solution is not yet the $n$ dimensional Kerr-Newman-de Sitter solution and the scalar field is not a constant in general. In Brans-Dicke-Ni theory, the resulting solution is also not yet the Kerr-Newman-de Sitter one even in four dimensions. The higher dimensional origin of the Brans-Dicke scalar field is briefly discussed. 
  In this paper we obtain a 2+2 double null Hamiltonian description of General Relativity using only the (complex) SO(3) connection and the components of the complex densitised self-dual bivectors. We carry out the general canonical analysis of this system and obtain the first class constraint algebra entirely in terms of the self-dual variables. The first class algebra forms a Lie algebra and all the first class constraints have a simple geometrical interpretation. 
  We justify generalisations of weak values from a tentatively relational perspective by deriving them from a generalisation of Bayes' rule. We also argue that these generalisations have implications of quantum nonlocality and may form a novel approach to quantum gravity and cosmology. 
  A new parametrization of the 3-metric allows to find explicitly a York map in canonical ADM tetrad gravity, the two pairs of physical tidal degrees of freedom and 14 gauge variables. These gauge quantities (generalized inertial effects) are all configurational except the trace ${}^3K(\tau ,\vec \sigma)$ of the extrinsic curvature of the instantaneous 3-spaces $\Sigma_{\tau}$ (clock synchronization convention) of a non-inertial frame. The Dirac hamiltonian is the sum of the weak ADM energy $E_{ADM} = \int d^3\sigma {\cal E}_{ADM}(\tau ,\vec \sigma)$ (whose density is coordinate-dependent due to the inertial potentials) and of the first-class constraints. Then: i) The explicit form of the Hamilton equations for the two tidal degrees of freedom in an arbitrary gauge: a deterministic evolution can be defined only in a completely fixed gauge, i.e. in a non-inertial frame with its pattern of inertial forces. ii) A general solution of the super-momentum constraints, which shows the existence of a generalized Gribov ambiguity associated to the 3-diffeomorphism gauge group. It influences: a) the explicit form of the weak ADM energy and of the super-momentum constraint; b) the determination of the shift functions and then of the lapse one. iii) The dependence of the Hamilton equations for the two pairs of dynamical gravitational degrees of freedom (the generalized tidal effects) and for the matter, written in a completely fixed 3-orthogonal Schwinger time gauge, upon the gauge variable ${}^3K(\tau ,\vec \sigma)$, determining the convention of clock synchronization. Therefore it should be possible (for instance in the weak field limit but with relativistic motion) to try to check whether in Einstein's theory the {\it dark matter} is a gauge relativistic inertial effect induced by ${}^3K(\tau ,\vec \sigma)$. 
  "The last remnant of physical objectivity of space-time" is disclosed in the case of a continuous family of spatially non-compact models of general relativity (GR). The {\it physical individuation} of point-events is furnished by the intrinsic degrees of freedom of the gravitational field, (viz, the {\it Dirac observables}) that represent - as it were - the {\it ontic} part of the metric field. The physical role of the {\it epistemic} part (viz. the {\it gauge} variables) is likewise clarified as emboding the unavoidable non-inertial aspects of GR. At the end the philosophical import of the {\it Hole Argument} is substantially weakened and in fact the Argument itself dis-solved, while a specific four-dimensional {\it holistic and structuralist} view of space-time, (called {\it point-structuralism}), emerges, including elements common to the tradition of both {\it substantivalism} and {\it relationism}. The observables of our models undergo real {\it temporal change}: this gives new evidence to the fact that statements like the {\it frozen-time} character of evolution, as other ontological claims about GR, are {\it model dependent}. \medskip Forthcoming in Studies in History and Philosophy of Modern Physics 
  We study black hole solutions in general relativity coupled to a unit timelike vector field dubbed the "aether". To be causally isolated a black hole interior must trap matter fields as well as all aether and metric modes. The theory possesses spin-0, spin-1, and spin-2 modes whose speeds depend on four coupling coefficients. We find that the full three-parameter family of local spherically symmetric static solutions is always regular at a metric horizon, but only a two-parameter subset is regular at a spin-0 horizon. Asymptotic flatness imposes another condition, leaving a one-parameter family of regular black holes. These solutions are compared to the Schwarzschild solution using numerical integration for a special class of coupling coefficients. They are very close to Schwarzschild outside the horizon for a wide range of couplings, and have a spacelike singularity inside, but differ inside quantitatively. Some quantities constructed from the metric and aether oscillate in the interior as the singularity is approached. The aether is at rest at spatial infinity and flows into the black hole, but differs significantly from the the 4-velocity of freely-falling geodesics. 
  The junction conditions between static and non-static space-times are studied for analyzing gravitational collapse in the presence of a cosmological constant. We have discussed about the apparent horizon and their physical significance. We also show the effect of cosmological constant in the collapse and it has been shown that cosmological constant slows down the collapse of matter. 
  We study gravitational collapse in higher dimensional quasi-spherical Szekeres space-time for matter with anisotropic pressure. Both local and global visibility of central curvature singularity has been studied and it is found that with proper choice of initial data it is possible to show the validity of CCC for six and higher dimensions. Also the role of pressure in the collapsing process has been discussed. 
  A simple formula, invariant under the duality rotation \Phi -> exp(i\alpha) \Phi, is obtained for the Poynting vector within the framework of the Ernst formalism, and its application to the known exact solutions for a charged massive magnetic dipole is considered. 
  It has recently been suggested that observed galaxy rotation curves can be accounted for by general relativity without recourse to dark-matter halos. Good fits have been produced to observed galatic rotation curves using this model. We show that the implied total mass is infinite, adding to the evidence opposing the hypothesis. 
  We generalize our previous work on gravitational lensing by a Kerr black hole in the strong deflection limit, removing the restriction to observers on the equatorial plane. Starting from the Schwarzschild solution and adding corrections up to the second order in the black hole spin, we perform a complete analytical study of the lens equation for relativistic images created by photons passing very close to a Kerr black hole. We find out that, to the lowest order, all observables (including shape and shift of the black hole shadow, caustic drift and size, images position and magnification) depend on the projection of the spin on a plane orthogonal to the line of sight. In order to break the degeneracy between the black hole spin and its inclination relative to the observer, it is necessary to push the expansion to higher orders. In terms of future VLBI observations, this implies that very accurate measures are needed to determine these two parameters separately. 
  We consider the linearized nonsymmetric theory of gravitation (NGT) within the background of an expanding universe and near a Schwarzschild metric. We show that the theory always develops instabilities unless the linearized nonsymmetric lagrangian reduces to a particular simple form. This theory contains a gauge invariant kinetic term, a mass term for the antisymmetric metric-field and a coupling with the Ricci curvature scalar. This form cannot be obtained within NGT. Next we discuss NGT beyond linearized level and conjecture that the instabilities are not a relic of the linearization, but are a general feature of the full theory. Finally we show that one cannot add ad-hoc constraints to remove the instabilities as is possible with the instabilities found in NGT by Clayton. 
  In this work we explore the possible existence of static, spherically symmetric and stationary, axisymmetric traversable wormholes coupled to nonlinear electrodynamics. Considering static and spherically symmetric (2+1) and (3+1)-dimensional wormhole spacetimes, we verify the presence of an event horizon and the non-violation of the null energy condition at the throat. For the former spacetime, the principle of finiteness is imposed, in order to obtain regular physical fields at the throat. Next, we analyze the (2+1)-dimensional stationary and axisymmetric wormhole, and also verify the presence of an event horizon, rendering the geometry non-traversable. Relatively to the (3+1)-dimensional stationary and axisymmetric wormhole geometry, we find that the field equations impose specific conditions that are incompatible with the properties of wormholes. Thus, we prove the non-existence of the general class of traversable wormhole solutions, outlined above, within the context of nonlinear electrodynamics. 
  Phase space of General Relativity is extended to a Poisson manifold by inclusion of the determinant of the metric and conjugate momentum as additional independent variables. As a result, the action and the constraints take a polynomial form. New expression for the generating functional for the Green functions is proposed. We show that the Dirac bracket defines degenerate Poisson structure on a manifold, and a second class constraints are the Casimir functions with respect to this structure. As an application of the new variables, we consider the Friedmann universe. 
  We consider static spacetimes whose spatial part admits foliations with the extrinsic curvature tensor K_{ab}=0. There are two complementary cases when the gradient of the lapse function points 1) to the direction of foliation or 2) orthogonally to it. Case 1) gives generalization of metrics like Bertotti-Robinson or Nariai. In case 2) the matter source violates the null energy condition at least on the part of the manifold, having in this sense phantom nature. We also demonstrate that for the manifolds under discussion the horizon can be naked in the sense that certain Weyl components diverge in the free-falling frame although the Kretschmann scalar is finite. The Petrov type is D or O. Explicit solutions for (i) the linear anisotropic equation of state, (ii) Chaplygin gas and (iii) uniform energy density are found. 
  We find a class of exact solutions of differentially rotating dust in the framework of General Relativity. There exist asymptotically flat space-times of the flow with positive mass function that for radii sufficiently large is monotonic and tends to zero at infinity. Some of the space-times may have non-vanishing total angular momentum. The flow is essentially different from another exactly solvable flow described by van Stockum line element. 
  We use recently developed effective field theory techniques to calculate the third order post-Newtonian correction to the spin-spin potential between two spinning objects. This correction represents the first contribution to the spin-spin interaction due to the non-linear nature of general relativity and will play an important role in forthcoming gravity wave experiments. 
  We have implemented a parallel multigrid solver, to solve the initial data problem for 3+1 General Relativity. This involves solution of elliptic equations derived from the Hamiltonian and the momentum constraints. We use the conformal transverse-traceless method of York and collaborators which consists of a conformal decomposition with a scalar $\phi$ that adjusts the metric, and a vector potential $w^i$ that adjusts the longitudinal components of the extrinsic curvature. The constraint equations are then solved for these quantities $\phi$, $w^i$ such that the complete solution fully satisfies the constraints. We apply this technique to compare with theoretical expectations for the spin-orientation- and separation-dependence in the case of spinning interacting (but not orbiting) black holes. We write out a formula for the effect of the spin-spin interaction which includes a result of Wald as well as additional effect due to the rotation of the mass quadrupole moment of a spinning black hole. A subset of these spin-spin effects are confirmed via our numerical calculations, however due to computer time limitations the full parameter space has not yet been surveyed and confirmed. In particular, at the relatively small s eparations ($d \leq 18m$) we are able to consider, we are unable to confirm the expected asymptotic fall-off of $d^{-3}$ for these effects. 
  We show that algebraically special modes lead to the instability of naked singularity spacetimes with negative mass. Four-dimensional negative-mass Schwarzschild and Schwarzschild-de Sitter spacetimes are unstable. Stability of the Schwarzschild-anti-de Sitter spacetime depends on boundary conditions. We briefly discuss the generalization of these results to charged and rotating singularities. 
  In Einstein's general relativity, with its nonlinear field equations, the discoveries and analyzes of various specific explicit solutions made a great impact on understanding many of the unforeseen features of the theory. Some solutions found fundamental applications in astrophysics, cosmology and, more recently, in the developments inspired by string theory. In this short article we survey the invariant characterization and classification of the solutions and describe the properties and role of the most relevant classes: Minkowski, (anti-)de Sitter spacetimes, spherical Schwarzschild and Reissner-Nordstroem metrics, stationary axisymmetric solutions, radiative metrics describing plane and cylindrical waves, radiative fields of uniformly accelerated sources and Robinson-Trautman solutions. Metrics representing regions of spacetimes filled with matter are also discussed and cosmological models are very briefly mentioned. Some parts of the text are based on a detailed survey which appeared in gr-qc/0004016 (see Ref. 2). 
  Using the generalised invariant formalism we derive a class of conformally flat spacetimes whose Ricci tensor has a pure radiation and a Ricci scalar component. The method used is a development of the methods used earlier for pure radiation spacetimes of Petrov types O and N respectively. In this paper we demonstrate how to handle, in the generalised invariant formalism, spacetimes with isotropy freedom and rich Killing vector structure. Once the spacetimes have been constructed, it is straightforward to deduce their Karlhede classification: the Karlhede algorithm terminates at the fourth derivative order, and the spacetimes all have one degree of null isotropy and three, four or five Killing vectors. 
  The fields of rapidly moving sources are studied within nonlinear electrodynamics by boosting the fields of sources at rest. As a consequence of the ultrarelativistic limit the delta-like electromagnetic shock waves are found. The character of the field within the shock depends on the theory of nonlinear electrodynamics considered. In particular, we obtain the field of an ultrarelativistic charge in the Born-Infeld theory. 
  Non-minimally coupling a scalar field to gravity introduces an additional curvature term into the action which can change the general behavior in strong curvature regimes, in particular close to classical singularities. While one can conformally transform any non-minimal model to a minimally coupled one, that transformation can itself become singular. It is thus not guaranteed that all qualitative properties are shared by minimal and non-minimal models. This paper addresses the classical singularity issue in isotropic models and extends singularity removal in quantum gravity to non-minimal models. 
  The quantized Dirac field is known, by a result of Fewster and Verch, to satisfy a Quantum Weak Energy Inequality (QWEI) on its averaged energy density along time-like curves in arbitrary four-dimensional globally hyperbolic spacetimes. However, this result does not provide an explicit form for the bound. By adapting ideas from the earlier work, we give a simplified derivation of a QWEI for the Dirac field leading to an explicit bound. The bound simplifies further in the case of static curves in static spacetimes, and, in particular, coincides with a result of Fewster and Mistry in four-dimensional Minkowski spacetime. We also show that our QWEI is compatible with local covariance and derive a simple consequence. 
  The validity of the Weak Equivalence Principle relative to a local inertial frame is detailed in a scalar-vector gravitation model with Lorentz-Poincar\'e type interpretation. Given the previously established first Post-Newtonian concordance of dynamics with General Relativity, the principle is to this order compatible with GRT. The gravitationally modified Lorentz transformations, on which the observations in physical coordinates depend, are shown to provide a physical interpretation of \emph{parallel transport}. A development of ``geodesic'' deviation in terms of the present model is given as well. 
  We build extended sources for the Reissner-Nordstr\"{o}m metric. Our models describe a neutral perfect fluid core bounded by a charged thin shell, and feature everywhere positive rest mass density and everywhere non-negative active gravitational mass, as well as classical electron radius and electromagnetic total mass. We contrast our results with previously discussed models featuring similar properties at the expense of including anisotropic pressures within the fluid. Our charged thin shells are restricted by the 2D texture equation of state which causes the continuity of the active gravitational mass, in spite of the singularity of the energy-momentum tensor. We mention possible extensions of this study suggested by modified active mass formulae proposed in the literature. 
  There exist corresponding metric perturbations of the relic gravitational waves (GWs) in the region of approximately h~10^(-30)-10^(-32)in the GHz band. A detector for these GWs is described in which we measure the perturbative photon flux (PPF) or signal generated by such high-frequency relic GWs (HFRGWs) via a coupling system of fractal membranes and a Gaussian beam (GB) passing through a static magnetic field. It is found that under the synchro-resonance condition in which the frequency of the GB is set equal to the frequency of the expected HFRGWs (h~2.00*10^(-31), v_g=10^10Hz in the quintessential inflationary models (QIM) and h~6.32*10^(-31), v_g=10^10Hz in the pre-big bang scenario (PBBS) may produce the PPFs of ~4.04*10^2/s and ~1.27*10^3/s in a surface of 100cm^2 area at the waist of the GB, respectively. The relatively weak first-order PPF, directed at right angles to the expected HFRGWs, is reflected by fractal membrane and the resulting reflected PPF (signal) exhibits a very small decay in transit to the detector (tunable microwave receiver) compared with the much stronger background photon flux, which allows for detection of the reflected PPF with signal to background noise ratios greater than one at the distance of the detector. We also discuss the selection capability of system and directional sensitivity for the resonance components from the stochastic relic GW background. The resolution of tiny difference between the PPFs generated by the relic GWs in the QIM and in the PBBS may be established and will be of cosmological significance. PACS numbers: 04.30.Nk, 04.30.Db, and 98.80.Cq. 
  The boson star filled with two interacting scalar fields is investigated. The scalar fields can be considered as a gauge condensate formed by SU(3) gauge field quantized in a non-perturbative manner. The corresponding solution is regular everywhere, has a finite energy and can be considered as a quantum SU(3) version of the Bartnik - McKinnon particle-like solution. 
  In this paper we discuss curvature tensors in the context of Absolute Parallelism geometry. Different curvature tensors are expressed in a compact form in terms of the torsion tensor of the canonical connection. Using the Bianchi identities some other identities are derived from the expressions obtained. These identities, in turn, are used to reveal some of the properties satisfied by an intriguing fourth order tensor which we refer to as Wanas tensor. A further condition on the canonical connection is imposed, assuming it is semi-symmetric. The formulae thus obtained, together with other formulae (Ricci tensors and scalar curvatures of the different connections admitted by the space) are calculated under this additional assumption. Considering a specific form of the semi-symmetric connection causes all nonvanishing curvature tensors to coincide, up to a constant, with the Wanas tensor. Physical aspects of some of the geometric objects considered are mentioned. 
  We are concerned with the issue of quantization of a scalar field in a diffeomorphism invariant manner. We apply the method used in Loop Quantum Gravity. It relies on the specific choice of scalar field variables referred to as the polymer variables. The quantization, in our formulation, amounts to introducing the `quantum' polymer *-star algebra and looking for positive linear functionals, called states. Assumed in our paper homeomorphism invariance allows to derive the complete class of the states. They are determined by the homeomorphism invariant states defined on the CW-complex *-algebra. The corresponding GNS representations of the polymer *-algebra and their self-adjoint extensions are derived, the equivalence classes are found and invariant subspaces characterized. In the preceding letter (the part I) we outlined those results. Here, we present the technical details. 
  We present a new point of view on the problem of the Schwarzschild black hole in the noncommutative spaces, proposed recently by F. Nasseri. We apply our treatment also to the case of the 2+1 dimensional Ba\~ nados-Teitelboim-Zanelli black hole. 
  We propose a toy model for study merger transitions in a curved spaceime with an arbitrary number of dimensions. This model includes a bulk N-dimensional static spherically symmetric black hole and a test D-dimensional brane interacting with the black hole. The brane is asymptotically flat and allows O(D-1) group of symmetry. Such a brane--black-hole (BBH) system has two different phases. The first one is formed by solutions describing a brane crossing the horizon of the bulk black hole. In this case the internal induced geometry of the brane describes D-dimensional black hole. The other phase consists of solutions for branes which do not intersect the horizon and the induced geometry does not have a horizon. We study a critical solution at the threshold of the brane-black-hole formation, and the solutions which are close to it. In particular, we demonstrate, that there exists a striking similarity of the merger transition, during which the phase of the BBH-system is changed, both with the Choptuik critical collapse and with the merger transitions in the higher dimensional caged black-hole--black-string system. 
  I review the status of research, conducted by a variety of independent groups, aimed at the eventual observation of Extreme Mass Ratio Inspirals (EMRIs) with gravitational wave detectors. EMRIs are binary systems in which one of the objects is much more massive than the other, and which are in a state of dynamical evolution that is dominated by the effects of gravitational radiation. Although these systems are highly relativistic, with the smaller object moving relative to the larger at nearly light-speed, they are well described by perturbative calculations which exploit the mass ratio as a natural small parameter. I review the use of such approximations to generate waveforms needed by data analysis algorithms for observation. I also briefly review the status of developing the data analysis algorithms themselves. Although this article is almost entirely a review of previous work, it includes (as an appendix) a new analytical estimate for the time over which the influence of radiation on the binary itself is observationally negligible. 
  We consider a multilevel hydrogen atom in interaction with the quantum electromagnetic field and separately calculate the contributions of the vacuum fluctuation and radiation reaction to the rate of change of the mean atomic energy of the atom for uniform acceleration. It is found that the acceleration disturbs the vacuum fluctuations in such a way that the delicate balance between the contributions of vacuum fluctuation and radiation reaction that exists for inertial atoms is broken, so that the transitions to higher-lying states from ground state are possible even in vacuum. In contrast to the case of an atom interacting with a scalar field, the contributions of both electromagnetic vacuum fluctuations and radiation reaction to the spontaneous emission rate are affected by the acceleration, and furthermore the contribution of the vacuum fluctuations contains a non-thermal acceleration-dependent correction, which is possibly observable. 
  The effects of the gravitational back reaction of cosmological perturbations are investigated in a phantom inflation model. The effective energy-momentum tensor of the gravitational back reaction of cosmological perturbations whose wavelengths are larger than the Hubble radius is calculated. Our results show that the effects of gravitational back reaction will counteract that of the phantom energy. It is demonstrated in a chaotic phantom inflation model that if the phantom field at the end of inflation is larger than a critical value determined by the necessary e-folds, the phantom inflation phase might be terminated by the gravitational back reaction. 
  Alternative theories of relativistic rotation considered viable as of 2004 are compared in the light of experiments reported in 2005. En route, the contentious issue of simultaneity choice in rotation is resolved by showing that only one simultaneity choice, the one possessing continuous time, gives rise, via the general relativistic equation of motion, to the correct Newtonian limit Coriolis acceleration. In addition, the widely dispersed argument purporting Lorentz contraction in rotation and the concomitant curved surface of a rotating disk is analyzed and argued to be lacking for more than one reason. It is posited that not by theoretical arguments, but only via experiment can we know whether such effect exists in rotation or not.   The Coriolis/simultaneity correlation, and the results of the 2005 experiments, support the Selleri theory as being closest to the truth, though it is incomplete in a more general applicability sense, because it does not provide a global metric. Two alternatives, a modified Klauber approach and a Selleri-Klauber hybrid, are presented which are consistent with recent experiment and have a global metric, thereby making them applicable to rotation problems of all types. 
  In this letter we compute the corrections to the Cardy-Verlinde formula of Schwarzschild black holes. These corrections stem from the space noncommutativity. Because the Schwarzschild black holes are non rotating, to the first order of perturbative calculations, there is no any effect on the properties of black hole due to the noncommutativity of space. 
  A modern re-visitation of the consequences of the lack of an intrinsic notion of instantaneous 3-space in relativistic theories leads to a reformulation of their kinematical basis emphasizing the role of non-inertial frames centered on an arbitrary accelerated observer. In special relativity the exigence of predictability implies the adoption of the 3+1 point of view, which leads to a well posed initial value problem for field equations in a framework where the change of the convention of synchronization of distant clocks is realized by means of a gauge transformation. This point of view is also at the heart of the canonical approach to metric and tetrad gravity in globally hyperbolic asymptotically flat space-times, where the use of Shanmugadhasan canonical transformations allows the separation of the physical degrees of freedom of the gravitational field (the tidal effects) from the arbitrary gauge variables. Since a global vision of the equivalence principle implies that only global non-inertial frames can exist in general relativity, the gauge variables are naturally interpreted as generalized relativistic inertial effects, which have to be fixed to get a deterministic evolution in a given non-inertial frame. As a consequence, in each Einstein's space-time in this class the whole chrono-geometrical structure, including also the clock synchronization convention, is dynamically determined and a new approach to the Hole Argument leads to the conclusion that "gravitational field" and "space-time" are two faces of the same entity. This view allows to get a classical scenario for the unification of the four interactions in a scheme suited to the description of the solar system or our galaxy with a deperametrization to special relativity and the subsequent possibility to take the non-relativistic limit. 
  We recently showed (Class. Quantum Grav.\ 23, 3353; gr-qc/0602091) that aligned Petrov type D purely magnetic perfect fluids are necessarily locally rotationally symmetric (LRS) and hence are all explicitly known. We provide here a more transparent proof. 
  Backreactions are considered in a de Sitter spacetime whose cosmological constant is generated by the potential of scalar field. The leading order gravitational effect of nonlinear matter fluctuations is analyzed and it is found that the initial value problem for the perturbed Einstein equations possesses linearization instabilities. We show that these linearization instabilities can be avoided by assuming strict de Sitter invariance of the quantum states of the linearized fluctuations. We furthermore show that quantum anomalies do not block the invariance requirement. This invariance constraint applies to the entire spectrum of states, from the vacuum to the excited states (should they exist), and is in that sense much stronger than the usual Poincare invariance requirement of the Minkowski vacuum alone. Thus to leading order in their effect on the gravitational field, the quantum states of the matter and metric fluctuations must be de Sitter invariant. 
  The gravitational wave signal generated by global, nonaxisymmetric shear flows in a neutron star is calculated numerically by integrating the incompressible Navier--Stokes equation in a spherical, differentially rotating shell. At Reynolds numbers $\Rey \gsim 3 \times 10^{3}$, the laminar Stokes flow is unstable and helical, oscillating Taylor--G\"ortler vortices develop. The gravitational wave strain generated by the resulting kinetic-energy fluctuations is computed in both $+$ and $\times$ polarizations as a function of time. It is found that the signal-to-noise ratio for a coherent, $10^{8}$-{\rm s} integration with LIGO II scales as $ 6.5 (\Omega_*/10^{4}  {\rm rad} {\rm s}^{-1})^{7/2}$ for a star at 1 {\rm kpc} with angular velocity $\Omega_*$. This should be regarded as a lower limit: it excludes pressure fluctuations, herringbone flows, Stuart vortices, and fully developed turbulence (for $\Rey \gsim 10^{6}$). 
  An important question that discrete approaches to quantum gravity must address is how continuum features of spacetime can be recovered from the discrete substructure. Here, we examine this question within the causal set approach to quantum gravity, where the substructure replacing the spacetime continuum is a locally finite partial order. A new topology on causal sets using ``thickened antichains'' is constructed. This topology is then used to recover the homology of a globally hyperbolic spacetime from a causal set which faithfully embeds into it at sufficiently high sprinkling density. This implies a discrete-continuum correspondence which lends support to the fundamental conjecture or ``Hauptvermutung'' of causal set theory. 
  The Hojman-Rosenbaum-Ryan-Shepley dynamical theory of torsion preserves local gauge invariance of electrodynamics and makes minimal coupling compatible with torsion. It also allows propagation of torsion in vacuum. It is known that implications of this model disagree with the Eotvos-Dicke-Braginsky solar tests of the principle of equivalence. We modify this theory and make it consistent with experiment by introducing a massive component of the torsionic potential. 
  Motivated by the generalization of quantum theory for the case of non-Hermitian Hamiltonians with PT symmetry, we show how a classical cosmological model describes a smooth transition from ordinary dark energy to the phantom one. The model is based on a classical complex Lagrangian of a scalar field. Specific symmetry properties analogous to PT in non-Hermitian quantum mechanics lead to purely real equation of motion. 
  It is believed that soon after the Planck era, spacetime should have a semi-classical nature. In this context we consider quantum fields propagating in a classical gravitational field and study the backreaction of these fields, using the expected value of the energy-momentum tensor as source of the gravitational field. According to this theory, the escape from General Relativity theory is unavoidable. Two geometric counter-term are needed to regularize the divergences which come from the expected value. There is a parameter associated to each counter-term and in this work we found numerical solutions of this theory to particular initial conditions, for general Bianchi Type I spaces. We show that even though there are spurious solutions some of them can be considered physical. These physical solutions include de Sitter and Minkowski that are obtained asymptotically. 
  Perturbations in a Chaplygin gas, characterized by an equation of state $p = -A/\rho$, may acquire non-adiabatic contributions if spatial variations of the parameter $A$ are admitted. This feature is shown to be related to a specific internal structure of the Chaplygin gas. We investigate how perturbations of this type modify the adiabatic sound speed. A reduction of the effective sound speed compared with the adiabatic value is expected to suppress oscillations in the matter power spectrum. This text is an abridged version of the following reference: W. Zimdahl and J.C. Fabris, Classical and Quantum Gravity, {\bf 22}, 4311(2005). 
  Existing coherent network analysis techniques for detecting gravitational-wave bursts simultaneously test data from multiple observatories for consistency with the expected properties of the signals. These techniques assume the output of the detector network to be the sum of a stationary Gaussian noise process and a gravitational-wave signal, and they may fail in the presence of transient non-stationarities, which are common in real detectors. In order to address this problem we introduce a consistency test that is robust against noise non-stationarities and allows one to distinguish between gravitational-wave bursts and noise transients. This technique does not require any a priori knowledge of the putative burst waveform. 
  We study the entanglement entropy of a scalar filed in 2+1 spacetime where space is modeled by a fuzzy sphere and a fuzzy disc. In both models we evaluate numerically the resulting entropies and find that they are proportional to the number of boundary degrees of freedom. In the Moyal plan limit of the fuzzy disc the entanglement entropy per unit area (length) diverges if the ignored region is of infinite size. The divergence is (interpreted) of IR-UV mixing origin. In general we expect the entanglement entropy per unit area to be finite on a non-commutative space if the ignored region is of finite size. 
  We apply the standard theory of the elastic body to obtain a set of equations describing the behavior of an acoustic Gravitational Wave detector, fully taking into account the 3-dimensional properties of the mass, the readout and the signal. We show that the advantages given by a Dual detector made by two nested oscillators can also be obtained by monitoring two different acoustic modes of the same oscillator, thus easing the detector realization. We apply these concepts and by means of an optimization process we derive the main figures of such a single-mass Dual detector designed specifically for the frequency interval 2-5kHz. Finally we calculate the SQL sensitivity of this detector. 
  We consider the description of classical oscillatory motion in ZM theory, and explore the relationship of ZM theory to semi-classical Bohr-Sommerfeld quantization. The treatment illustrates some features of ZM theory, especially the inadequacies of classical and semi-classical treatments due to non-analyticity of the mapping of classical trajectories onto the ZM clock field. While the more complete ZM formalism is not developed here, the non-analyticities in the classical treatment resemble issues in the comparison of classical and quantum formalisms. We also show that semi-classical quantization is valid for a periodic manifold in ZM theory, though the quantum number $n=0$ is allowed, as it would be in quantum mechanics for a periodic manifold. Still, this suggests a connection to the first-order success of Bohr theory in describing the phenomenology of atomic quantum states. The approximate nature of the semi-classical treatment of three dimensional atomic orbits is, however, also apparent in relation to ZM theory. These observations are preliminary to a discussion of ZM theory in relation to quantum mechanics and quantum field theory in subsequent papers. 
  This paper concerns sprinklings into Minkowski space (Poisson processes). It proves that there exists no equivariant measurable map from sprinklings to spacetime directions (even locally). Therefore, if a discrete structure is associated to a sprinkling in an intrinsic manner, then the structure will not pick out a preferred frame, locally or globally. This implies that the discreteness of a sprinkled causal set will not give rise to ``Lorentz breaking'' effects like modified dispersion relations. Another consequence is that there is no way to associate a finite-valency graph to a sprinkling consistently with Lorentz invariance. 
  The discovery of scale acceleration evidenced from supernovae luminosities and spatial flatness of feature evolution in the cosmic microwave background presents a challenge to the understanding of the evolution of cosmological vacuum energy. Although some scenarios prefer a fixed cosmological constant with dynamics governed in a Friedman-Robertson-Walker (FRW) geometry, an early inflationary epoch remains a popular model for cosmology. It is therefore advantageous to develop a metric framework that allows a transition from an early inflationary period to a late stage dominated by dark energy. Such a metric is here developed, and some properties of this metric are explored. 
  Some connections between the deviation equations and weak equivalence principle are investigated. 
  Vacuum polarization in QED in a background gravitational field induces interactions which {\it effectively} modify the classical picture of light rays, as the null geodesics of spacetime. These interactions violate the strong equivalence principle and affect the propagation of light leading to superluminal photon velocities. Taking into account the QED vacuum polarization, we study the propagation of a bundle of rays in a background gravitational field. To do so we consider the perturbative deformation of Raychaudhuri equation through the influence of vacuum polarization on photon propagation. We analyze the contribution of the above interactions to the optical scalars namely, shear, vorticity and expansion using the Newman-Penrose formalism. 
  The relativistic fluid is a highly successful model used to describe the dynamics of many-particle, relativistic systems. It takes as input basic physics from microscopic scales and yields as output predictions of bulk, macroscopic motion. By inverting the process, an understanding of bulk features can lead to insight into physics on the microscopic scale. Relativistic fluids have been used to model systems as ``small'' as heavy ions in collisions, and as large as the universe itself, with ``intermediate'' sized objects like neutron stars being considered along the way. The purpose of this review is to discuss the mathematical and theoretical physics underpinnings of the relativistic (multiple) fluid model. We focus on the variational principle approach championed by Brandon Carter and his collaborators, in which a crucial element is to distinguish the momenta that are conjugate to the particle number density currents. This approach differs from the ``standard'' text-book derivation of the equations of motion from the divergence of the stress-energy tensor, in that one explicitly obtains the relativistic Euler equation as an ``integrability'' condition on the relativistic vorticity. We discuss the conservation laws and the equations of motion in detail, and provide a number of (in our opinion) interesting and relevant applications of the general theory. 
  After a brief introduction, basic ideas of the quantum Riemannian geometry underlying loop quantum gravity are summarized. To illustrate physical ramifications of quantum geometry, the framework is then applied to homogeneous isotropic cosmology. Quantum geometry effects are shown to replace the big bang by a big bounce. Thus, quantum physics does not stop at the big-bang singularity. Rather there is a pre-big-bang branch joined to the current post-big-bang branch by a `quantum bridge'. Furthermore, thanks to the background independence of loop quantum gravity, evolution is deterministic across the bridge. 
  In the Kaluza-Klein model with a cosmological constant and a flux, the external spacetime and its dimension of the created universe from a $S^s \times S^{n-s}$ seed instanton can be identified in quantum cosmology. One can also show that in the internal space the effective cosmological constant is most probably zero. 
  We find quasinormal spectrum of the massive scalar field in the background of the Kerr black holes. We show that all found modes are damped under the quasinormal modes boundary conditions when $\mu M$ is not large, thereby implying stability of the massive scalar field. This complements the region of stability determined by the Beyer inequality for large masses of the field. We show that, similar to the case of a non-rotating black holes, the massive term of the scalar field does not contribute in the regime of high damping. Thereby, the high damping asymptotic should be the same as for the massless scalar field. 
  Quantum black holes within the loop quantum gravity (LQG) framework are considered. The number of microscopic states that are consistent with a black hole of a given horizon area $A_0$ are counted and the statistical entropy, as a function of the area, is obtained for $A_0$ up to $550 l^2_{\rm Pl}$. The results are consistent with an asymptotic linear relation and a logarithmic correction with a coefficient equal to -1/2. The Barbero-Immirzi parameter that yields the asymptotic linear relation compatible with the Bekenstein-Hawking entropy is shown to coincide with a value close to $\gamma=0.274$, which has been previously obtained analytically. However, a new and oscillatory functional form for the entropy is found for small, Planck size, black holes that calls for a physical interpretation. 
  We study the classical and quantum cosmology of a 4+1-dimensional space-time with a non-zero cosmological constant coupled to a self interacting massive spinor field. We consider a spatially flat Robertson-Walker universe with the usual scale factor $R(t)$ and an internal scale factor $a(t)$ associated with the extra dimension. For a free spinor field the resulting equations admit exact solutions, whereas for a self interacting spinor field one should resort to a numerical method for exhibiting their behavior. These solutions give rise to a degenerate metric and exhibit signature transition from a Euclidean to a Lorentzian domain. Such transitions suggest a compactification mechanism for the internal and external scale factors such that $a\sim R^{-1}$ in the Lorentzian region. The corresponding quantum cosmology and the ensuing Wheeler-DeWitt equation have exact solutions in the mini-superspace when the spinor field is free, leading to wavepackets undergoing signature change. The question of stabilization of the extra dimension is also discussed. 
  This is a pedagogical introduction to original Kaluza-Klein theory and its salient features. Most of the technical calculations are given in detail and the nature of gravitons is discussed. 
  We study the global structure of Lorentzian manifolds with partial sectional curvature bounds. In particular, we prove completeness theorems for homogeneous and isotropic cosmologies as well as static spherically symmetric spacetimes. The latter result is used to rigorously prove the absence of static spherically symmetric black holes in more than three dimensions. The proofs of these new results are preceded by a detailed exposition of the local aspects of sectional curvature bounds for Lorentzian manifolds, which extends and strengthens previous constructions. 
  We introduce a computational framework which avoids solving explicitly hydrodynamic equations and is suitable to study the pre-merger evolution of black hole-neutron star binary systems. The essence of the method consists of constructing a neutron star model with a black hole companion and freezing the internal degrees of freedom of the neutron star during the course of the evolution of the space-time geometry. We present the main ingredients of the framework, from the formulation of the problem to the appropriate computational techniques to study these binary systems. In addition, we present numerical results of the construction of initial data sets and evolutions that demonstrate the feasibility of this approach. 
  A framework is developed in which one can write down the constraint equations on a three--dimensional hypersurface of arbitrary signature. It is then applied to isolated and dynamical horizons. The derived equations can be used to extract physicaly relevant quantities describing the horizon irrespective to whether it is isolated (null) or dynamical at a given instant of time. Furthermore, small perturbation of isolated horizons are considered, and finally a family of axially--symmetric exact solution of the constraint equations on a dynamical horizon is presented. 
  Perturbations form an important section of black hole analyses. This paper deals with the effect of perturbations as in the delineation of waves that occur. It makes use of the spin coefficients from [3] to represent the general equations of waves in an accelerating black hole proposed in [2]. 
  Laser Interferometer Space Antenna (LISA) will routinely observe coalescences of supermassive black hole (BH) binaries up to very high redshifts. LISA can measure mass parameters of such coalescences to a relative accuracy of $10^{-4}-10^{-6}$, for sources at a distance of 3 Gpc. The problem of parameter estimation of massive nonspinning binary black holes using post-Newtonian (PN) phasing formula is studied in the context of LISA. Specifically, the performance of the 3.5PN templates is contrasted against its 2PN counterpart using a waveform which is averaged over the LISA pattern functions. The improvement due to the higher order corrections to the phasing formula is examined by calculating the errors in the estimation of mass parameters at each order. The estimation of the mass parameters ${\cal M}$ and $\eta$ are significantly enhanced by using the 3.5PN waveform instead of the 2PN one. For an equal mass binary of $2\times10^6M_\odot$ at a luminosity distance of 3 Gpc, the improvement in chirp mass is $\sim 11%$ and that of $\eta$ is $\sim 39%$. Estimation of coalescence time $t_c$ worsens by 43%. The improvement is larger for the unequal mass binary mergers. These results are compared to the ones obtained using a non-pattern averaged waveform. The errors depend very much on the location and orientation of the source and general conclusions cannot be drawn without performing Monte Carlo simulations. Finally the effect of the choice of the lower frequency cut-off for LISA on the parameter estimation is studied. 
  We extensively study the exact solutions of the massless Dirac equation in 3D de Sitter spacetime that we published recently. Using the Newman-Penrose formalism, we find exact solutions of the equations of motion for the massless classical fields of spin s=1/2,1,2 and to the massive Dirac equation in 4D de Sitter metric. Employing these solutions, we analyze the absorption by the cosmological horizon and de Sitter quasinormal modes. We also comment on the results given by other authors. 
  Suggested a non-linear, non-gauge invariant model of Maxwell equations, based on the Kaluza-Klein theory. The spectrum of elementary charges and masses is obtained. 
  At densities below the neutron drip threshold, a purely elastic solid model (including, if necessary, a frozen-in magnetic field) can provide an adequate description of a neutron star crust, but at higher densities it will be necessary to allow for the penetration of the solid lattice by an independently moving current of superfluid neutrons. In order to do this, the previously available category of relativistic elasticity models is combined here with a separately developed category of relativistic superfluidity models in a unified treatment based on the use of an appropriate Lagrangian master function. As well as models of the purely variational kind, in which the vortices flow freely with the fluid, such a master function also provides a corresponding category of non-dissipative models in which the vortices are pinned to the solid structure. 
  There are many different formulations of relativistic elasticity. Most of them are only concerned with formal questions rather than questions regarding the PDE point of view. The aim of this thesis is to obtain various local existence results for dynamical scenarios involving elastic matter. Following Beig and Schmidt we treat relativistic elasticity as a Lagrangian field theory. The basic unknowns are mappings between spacetime and an abstract material manifold. The equations of motion will be the Euler-Lagrange equations arising from the action principle. Under certain (physically reasonable) restrictions on the elastic properties of the material the resulting system is hyperbolic and local existence results can be derived. This way we can show well-posedness for the equations describing elastic matter on a fixed gravitational background, self-gravitating elastic matter and a finite elastic body on a fixed background. The thesis also contains some results on energy conservation and on linearized elasticity. 
  Symmetric non-expanding horizons are studied in arbitrary dimension. The global properties -as the zeros of infinitesimal symmetries- are analyzed particularly carefully. For the class of NEH geometries admitting helical symmetry a quasi-local analog of Hawking's rigidity theorem is formulated and proved: the presence of helical symmetry implies the presence of two symmetries: null, and cyclic.   The results valid for arbitrary-dimensional horizons are next applied in a complete classification of symmetric NEHs in 4-dimensional space-times (the existence of a 2-sphere crossection is assumed). That classification divides possible NEH geometries into classes labeled by two numbers - the dimensions of, respectively, the group of isometries induced in the horizon base space and the group of null symmetries of the horizon. 
  We calculate the exact values of the quasinormal frequencies for an electromagnetic field and a gravitational perturbation moving in $D$-dimensional de Sitter spacetime ($D \geq 4$). We also study the quasinormal modes of a real massive scalar field and we compare our results with those of other references. 
  We carry out two searches for periodic gravitational waves using the most sensitive few hours of data from the second LIGO science run. The first search is targeted at isolated, previously unknown neutron stars and covers the entire sky in the frequency band 160-728.8 Hz. The second search targets the accreting neutron star in the low-mass X-ray binary Scorpius X-1, covers the frequency bands 464-484 Hz and 604-624 Hz, and two binary orbit parameters. Both searches look for coincidences between the Livingston and Hanford 4-km interferometers.   For isolated neutron stars our 95% confidence upper limits on the gravitational wave strain amplitude range from 6.6E-23 to 1E-21 across the frequency band; For Scorpius X-1 they range from 1.7E-22 to 1.3E-21 across the two 20-Hz frequency bands. The upper limits presented in this paper are the first broad-band wide parameter space upper limits on periodic gravitational waves using coherent search techniques. The methods developed here lay the foundations for upcoming hierarchical searches of more sensitive data which may detect astrophysical signals. 
  It is expected that matter composed of a perfect fluid cannot be at rest outside of a black hole if the spacetime is asymptotically flat and static (non-rotating). However, there has not been a rigorous proof for this expectation without assuming spheical symmetry. In this paper, we provide a proof of non-existence of matter composed of a perfect fluid in static black hole spacetimes under certain conditions, which can be interpreted as a relation between the stellar mass and the black hole mass. 
  Recent demonstrations of unexcised black holes traversing across computational grids represent a significant advance in numerical relativity. Stable and accurate simulations of multiple orbits, and their radiated waves, result. This capability is critically undergirded by a careful choice of gauge. Here we present analytic considerations which suggest certain gauge choices, and numerically demonstrate their efficacy in evolving a single moving puncture black hole. 
  We study the scattering of massive spin-half waves by a Schwarzschild black hole using analytical and numerical methods. We begin by extending a recent perturbation theory calculation to next order to obtain Born series for the differential cross section and Mott polarization, valid at small couplings. We continue by deriving an approximation for glory scattering of massive spinor particles by considering classical timelike geodesics and spin precession. Next, we formulate the Dirac equation on a black hole background, and outline a simple numerical method for finding partial wave series solutions. Finally, we present our numerical calculations of absorption and scattering cross sections and polarization, and compare with theoretical expectations. 
  Research on black holes and their physical proprieties has been active on last 90 years. With the appearance of the String Theory and the Braneworld models as alternative descriptions of our Universe, the interest on black holes, in these context, increased. In this work we studied black holes in Braneworld models. A class of spherically symmetric black holes is investigaded as well its stability under general perturbations. Thermodynamic proprieties and quasi-normal modes are discussed. The black holes studied are the SM (zero mass) and CFM solutions, obtained by Casadio {\it et al.} and Bronnikov {\it et al.}. The geometry of bulk is unknown. However the Campbell-Magaard Theorem guarantees the existence of a 5-dimensional solution in the bulk whose projection on the brane is the class of black holes considered. They are stable under scalar perturbations. Quasi-normal modes were observed in both models. The tail behavior of the perturbations is the same. The entropy upper bound of a body absorved by the black holes studied was calculated. This limit turned out to be independent of the black hole parameters. 
  The detection of gravitational waves based on the geodesic deviation equation is discussed. In particular, it is shown that the only non-vanishing components of the wave field in the conventional traceless-transverse gauge in linearized general relativity do not enter the geodesic deviation equation, and therefore, apparently, no effect is predicted by that equation in that specific gauge. The reason is traced back to the fact that the geodesic deviation equation is written in terms of a coordinate distance, which is not a directly measurable quantity. On the other hand, in the proper Lorentz frame of the detector, the conventional result described in standard textbooks holds. 
  Using the monodromy method we calculate the asymptotic quasinormal (QN) frequencies of an electromagnetic field moving in D-dimensional Schwarzschild and Schwarzschild de Sitter (SdS) black holes ($D\geq 4$). For the D-dimensional Schwarzschild anti-de Sitter (SadS) black hole we also compute these frequencies with a similar method. Moreover, we calculate the electromagnetic normal modes of the D-dimensional anti-de Sitter (AdS) spacetime. 
  Results from helically symmetric scalar field models and first results from a convergent helically symmetric binary neutron star code are reported here; these are models stationary in the rotating frame of a source with constant angular velocity omega. In the scalar field models and the neutron star code, helical symmetry leads to a system of mixed elliptic-hyperbolic character. The scalar field models involve nonlinear terms that mimic nonlinear terms of the Einstein equation. Convergence is strikingly different for different signs of each nonlinear term; it is typically insensitive to the iterative method used; and it improves with an outer boundary in the near zone. In the neutron star code, one has no control on the sign of the source, and convergence has been achieved only for an outer boundary less than approximately 1 wavelength from the source or for a code that imposes helical symmetry only inside a near zone of that size. The inaccuracy of helically symmetric solutions with appropriate boundary conditions should be comparable to the inaccuracy of a waveless formalism that neglects gravitational waves; and the (near zone) solutions we obtain for waveless and helically symmetric BNS codes with the same boundary conditions nearly coincide. 
  Recently, it has been shown that Absolute Parallelism (AP) geometry admits paths that are naturally quantized. These paths have been used to describe the motion of spinning particles in a background gravitational field. In case of a weak static gravitational field limits, the paths are applied successfully to interpret the discrepancy in the motion of thermal neutrons in the Earth's gravitational field (COW-experiment). The aim of the present work is to explore the properties of the deviation equations corresponding to these paths. In the present work the deviation equations are derived and compared to the geodesic deviation equation of the Riemannian geometry. 
  We study Doppler effects in curved space-time, i.e. the frequency shifts induced on electromagnetic signals propagating in the gravitational field. In particular, we focus on the frequency shift due to the bending of light rays in weak gravitational fields. We consider, using the PPN formalism, the gravitational field of an axially symmetric distribution of mass. The zeroth order, i.e. the sphere, is studied then passing to the contribution of the quadrupole moment, and finally to the case of a rotating source. We give numerical estimates for situations of physical interest, and by a very preliminary analysis, we argue that analyzing the Doppler effect could lead, in principle, in the foreseeable future, to the measurement of the quadrupole moment of the giant planets of the Solar System. 
  The notion of Lorentz violation in four dimensions is extended to a 5-dimensional brane-world scenario by utilizing a dynamical vector field assumed to point in the bulk direction, with Lorentz invariance holding on the brane. The cosmological consequences of this theory consisting of the time variation in the gravitational coupling $G$ and cosmological term $\Lambda_4$ are explored. The brane evolution is addressed by studying the generalized Friedmann and Raychaudhuri equations. The behavior of the expansion scale factor is then considered for different possible scenarios where the bulk cosmological constant is zero, positive or negative. 
  Static and inflating brane world models are considered in $4+n$-dimensions with a non zero bulk cosmological constant and with a hyper-spherically symmetric topological defect residing in the $n$ extra dimensions. Several vacuum solutions can be constructed explicitely when the bulk and inflating constants are non zero. We study how these solutions are deformed by the presence of a global monopole for generic values of $n$ and a local monopole for $n=3$. New types of solutions, regular and periodic in the radial variable relative to the extra dimensions, are constructed. 
  We study the classical solutions of the Einstein-Yang-Mills model in five dimensions in the presence of a cosmological constant $\Lambda$. Using a spherically symmetric ansatz and assuming that the fields do not depend on the extra dimension, we transform the equations into a set of differential equations that we solve numerically. We construct new types of regular (resp. black holes) solutions which, close to the origin (resp. the event horizon) resemble the 4-dimensional gravitating monopole (resp. non abelian black hole) and study their global properties. 
  Previous work developed a space-time metric with two cosmological scales; one that conveniently describes the classical evolution of the dynamics, and the other describing a scale associated with macroscopic quantum aspects like vacuum energy. The present work expands upon the dynamics of these scales to demonstrate the usefulness of these coordinates for describing early and late time behaviors of our universe. A convenient parameter, the fraction of classical energy density, is introduced as a means to parameterize the various early time models for the microscopic input. 
  This paper studies thin domain walls within the frame work of Lyra Geometry. We have considered two models. First one is the thin domain wall with negligible pressures perpendicular and transverse direction to the wall and secondly, we take a particular type of thin domain wall where the pressure in the perpendicular direction is negligible but transverse pressures are existed. It is shown that the thin domain walls have no particle horizon and the gravitational force due to them is attractive. 
  Gravitoelectromagnetic inflation was recently introduced to describe, in an unified manner, electromagnetic, gravitatory and inflaton fields in the early (accelerated) inflationary universe from a 5D vacuum state. In this paper, we study a stochastic treatment for the gravitoelectromagnetic components $A_B=(A_{\mu},\phi)$, on cosmological scales. We focus our study on the seed magnetic fields on super Hubble scales, which could play an important role in large scale structure formation of the universe. 
  We use perturbation theory and the relativistic Cowling approximation to numerically compute characteristic oscillation modes of rapidly rotating relativistic stars which consist of a perfect fluid obeying a polytropic equation of state. We find the expected infinite pressure mode spectrum extending towards higher frequencies, but also what appears to be an infinite number of inertial modes confined to a finite, well-defined frequency range, depending on the compactness and the rotation frequency of the star. We observe the shift of this range towards negative frequencies for non-axisymmetric modes with respect to the axisymmetric ones, making all m>2 modes unstable. We discuss whether our results indicate the existence of a continuous part of the star's spectrum, or whether they all are discrete solutions. 
  We discuss the properties of the gravitational energy-momentum 3-form within the tetrad formulation of general relativity theory. We derive the covariance properties of the quantities describing the energy-momentum content under Lorentz transformations of the tetrad. As an application, we consider the computation of the total energy (mass) of some exact solutions of Einstein's general relativity theory which describe compact sources with asymptotically flat spacetime geometry. As it is known, depending on the choice of tetrad frame, the formal total integral for such configurations may diverge. We propose a natural regularization method which yields finite values for the total energy-momentum of the system and demonstrate how it works on a number of explicit examples. 
  In this paper, we discuss the internal and external metric of the semi-realistic stars in relativistic MOND theory. We show the Oppenheimer-Volkoff equation in relativistic MOND theory and get the metric and pressure inside the stars to order of post-Newtonian corrections. We study the features of motion around the static, spherically symmetric stars by Hamilton-Jacobi mothod, and find there are only some small corrections in relativistic MOND theory. 
  The structure of the equation of state $\omega$ could be very complicate in nature while a few linear models have been successful in cosmological predictions. Linear models are treated as leading approximation of a complete Taylor series in this paper. If the power series converges quickly, one can freely truncate the series order by order. Detailed convergent analysis on the choices of the expansion parameters is presented in this paper. The related power series for the energy density function, the Hubble parameter and related physical quantities of interest are also computed in this paper. 
  In this paper we investigate the Proca-field in the framework of Loop Quantum Gravity. It turns out that the methods developed there can be applied to the symplectically embedded Proca-field, giving a rigorous, consistent, non-perturbative quantization of the theory. This can be achieved by introducing a scalar field, which has completely different properties than the one used in spontaneous symmetry breaking. The analysis of the kernel of the Hamiltonian suggests that the mass term in the quantum theory has a different role than in the classical theory. 
  The Block Universe idea, representing spacetime as a fixed whole, suggests the flow of time is an illusion: the entire universe just is, with no special meaning attached to the present time. This view is however based on time-reversible microphysical laws and does not represent macro-physical behaviour and the development of emergent complex systems, including life, which do indeed exist in the real universe. When these are taken into account, the unchanging block universe view of spacetime is best replaced by an evolving block universe which extends as time evolves, with the potential of the future continually becoming the certainty of the past. However this time evolution is not related to any preferred surfaces in spacetime; rather it is associated with the evolution of proper time along families of world lines 
  In the context of scalar-tensor models of dark energy and inflation, the dynamics of vacuum scalar-tensor cosmology are analysed without specifying the coupling function or the scalar field potential. A conformal transformation to the Einstein frame is used and the dynamics of general relativity with a minimally coupled scalar field are derived for a generic potential. It is shown that the dynamics are non-chaotic, thus settling an existing debate. 
  We propose intuitive derivations of the Hawking temperature and the Bekenstein-Hawking entropy of a Schwarzschild black hole. 
  Graviton absorption cross sections and emission rates for hydrogen are calculated by both semi-classical and field theoretic methods. We point out several mistakes in the literature concerning spontaneous emission of gravitons and related phenomena, some of which are due to a subtle issue concerning gauge invariance of the linearized interaction Hamiltonian. 
  The construction of initial data for black-hole binaries usually involves the choice of free parameters that define the spins of the black holes and essentially the eccentricity of the orbit. Such parameters must be chosen carefully to yield initial data with the desired physical properties. In this paper, we examine these choices in detail for the quasiequilibrium method coupled to apparent-horizon/quasiequilibrium boundary conditions. First, we compare two independent criteria for choosing the orbital frequency, the "Komar-mass condition" and the "effective-potential method," and find excellent agreement. Second, we implement quasi-local measures of the spin of the individual holes, calibrate these with corotating binaries, and revisit the construction of non-spinning black hole binaries. Higher-order effects, beyond those considered in earlier work, turn out to be important. Without those, supposedly non-spinning black holes have appreciable quasi-local spin; furthermore, the Komar-mass condition and effective potential method agree only when these higher-order effects are taken into account. We compute a new sequence of quasi-circular orbits for non-spinning black-hole binaries, and determine the innermost stable circular orbit of this sequence. 
  We consider a magnetized degenerate gas of fermions as the matter source of a homogeneous but anisotropic Bianchi I spacetime with a Kasner--like metric. We examine the dynamics of this system by means of a qualitative and numerical study of Einstein-Maxwell field equations which reduce to a non--linear autonomous system. For all initial conditions and combinations of free parameters the gas evolves from an initial anisotropic singularity into an asymptotic state that is completely determined by a stable attractor. Depending on the initial conditions the anisotropic singularity is of the ``cigar'' or ``plate'' types. 
  In this paper we provide a classification of plane symmetric kinematic self-similar perfect fluid and dust solutions. In the perfect fluid and dust cases, kinematic self-similar vectors for the tilted, orthogonal and parallel cases have been explored in the first, second, zeroth and infinite kinds with different equations of state. We obtain total of eleven plane symmetric kinematic self-similar solutions out of which six are independent. The perfect fluid case gives two solutions for infinite tilted and infinite orthogonal kinds. In the dust case, we have four independent solutions in the first orthogonal, infinite tilted, infinite orthogonal and infinite parallel kinds. The remaining cases provide contradiction. It is interesting to mention that some of these solutions turn out to be vacuum. 
  We discuss the existence of asymptotically Euclidean initial data sets to the vacuum Einstein field equations which would give rise (modulo an existence result for the evolution equations near spatial infinity) to developments with a past and a future null infinity of different smoothness. For simplicity, the analysis is restricted to the class of conformally flat, axially symmetric initial data sets. It is shown how the free parameters in the second fundamental form of the data can be used to satisfy certain obstructions to the smoothness of null infinity. The resulting initial data sets could be interpreted as those of some sort of (non-linearly) distorted Schwarzschild black hole. Its developments would be so that they admit a peeling future null infinity, but at the same time have a polyhomogeneous (non-peeling) past null infinity. 
  We introduce in the framework of the linear approximation of General relativity a natural distinction between General gauge transformations generated by any vector field and those Special ones for which this vector field is a gradient. This allows to introduce geometrical objects that are not invariant under General gauge transformations but they are under Special ones. We develop then a formalism that strengthens the analogy of the formalisms of the electromagnetic and the gravitational theories in a Special relativity framework. We are thus able to define the energy-momentum tensor of the gravitational field and to fully analyze the gravitational field of isolated point masses or continuous distributions of them obtained by linear superpositions. 
  We demonstrate that the rotating black holes in an arbitrary number of dimensions and without any restrictions on their rotation parameters possess the same `hidden' symmetry as the 4-dimensional Kerr metric. Namely, besides the spacetime symmetries generated by the Killing vectors they also admit the (antisymmetric) Killing-Yano and symmetric Killing tensors. 
  Our review is devoted to three promising research lines in quantum cosmology and the physics of the early universe. The nonperturbative renormalization programme is making encouraging progress that we here assess from the point of view of cosmological applications: Lagrangian and Hamiltonian form of pure gravity with variable G and Lambda; power-law inflation for pure gravity; an accelerating universe without dark energy. In perturbative quantum cosmology, on the other hand, diffeomorphism-invariant boundary conditions lead naturally to a singularity-free one-loop wave function of the universe. Last, but not least, in the braneworld picture one discovers the novel concept of cosmological wave function of the bulk space-time. Its impact on quantum cosmology and singularity avoidance is still, to a large extent, unexplored. 
  We discuss the gravitational wave background produced by bouncing models based on a full quantum evolution of a universe filled with a perfect fluid. Using an ontological interpretation for the background wave function allows us to solve the mode equations for the tensorial perturbations, and we find the spectral index as a function of the fluid equation of state. 
  The Einstein-Cartan-Saa theory of torsion modifies the spacetime volume element so that it is compatible with the connection. The condition of connection compatibility gives constraints on torsion, which are also necessary for the consistence of torsion, minimal coupling, and electromagnetic gauge invariance. To solve the problem of positivity of energy associated with the torsionic scalar, we reformulate this theory in the Einstein conformal frame. In the presence of the electromagnetic field, we obtain the Hojman-Rosenbaum-Ryan-Shepley theory of propagating torsion with a different factor in the torsionic kinetic term. 
  We dispose of some objections raised by Manko et al. (gr-qc/0604091) on a recently published paper on the role of Poynting vector in the ocurrence of vorticity in electrovaccum spacetimes (2006, Class. Quantum Grav. 23, 2395) 
  In Part One, we show that the total energy of the flat Robertson-Walker's Universe, is null. We employ several pseudotensors: Einstein's, Weinberg's and Landau-Lifshitz's. This calculation confirms other conjectures on the same problem. In Part Two and Three, we give two counter-examples which show that, unless we employ Cartesian coordinates, we may get wrong result for the energy; the examples given work with spherical coordinates. In Part Four, we give a counter-counter-example, where the use of polar spherical coordinates does no harm. We remember that the zero total energy of the flat Universe has importance due to the inflationary scenario. 
  Stability of self-similar solutions for gravitational collapse is an important problem to be investigated from the perspectives of their nature as an attractor, critical phenomena and instability of a naked singularity. In this paper we study spherically symmetric non-self-similar perturbations of matter and metrics in spherically symmetric self-similar backgrounds. The collapsing matter is assumed to be a perfect fluid with the equation of state $P=\alpha\rho$. We construct a single wave equation governing the perturbations, which makes their time evolution in arbitrary self-similar backgrounds analytically tractable. Further we propose an analytical application of this master wave equation to the stability problem by means of the normal mode analysis for the perturbations having the time dependence given by $\exp{(i\omega\log|t|)}$, and present some sufficient conditions for the absence of non-oscillatory unstable normal modes with purely imaginary $\omega$. 
  Kucha{\v{r}} showed that the quantum dynamics of (1 polarization) cylindrical wave solutions to vacuum general relativity is determined by that of a free axially-symmetric scalar field along arbitrary axially-symmetric foliations of a fixed flat 2+1 dimensional spacetime. We investigate if such a dynamics can be defined {\em unitarily} within the standard Fock space quantization of the scalar field.   Evolution between two arbitrary slices of an arbitrary foliation of the flat spacetime can be built out of a restricted class of evolutions (and their inverses). The restricted evolution is from an initial flat slice to an arbitrary (in general, curved) slice of the flat spacetime and can be decomposed into (i) `time' evolution in which the spatial Minkowskian coordinates serve as spatial coordinates on the initial and the final slice, followed by (ii) the action of a spatial diffeomorphism of the final slice on the data obtained from (i). We show that although the functional evolution of (i) is unitarily implemented in the quantum theory, generic spatial diffeomorphisms of (ii) are not. Our results imply that a Tomanaga-Schwinger type functional evolution of quantum cylindrical waves is not a viable concept even though, remarkably, the more limited notion of functional evolution in Kucha{\v{r}}'s `half parametrized formalism' is well-defined. 
  By using virial theorem, Helmholtz and Kelvin showed that the contraction of a bound self-gravitating system must be accompanied by release of radiation energy irrespective of the details of the contraction process. This happens because the total Newtonian energy of the system E_N (and not just the Newtonian gravitational potential energy E_g^N) decreases for such contraction. In the era of General Relativity (GR) too, it is justifiably believed that gravitational contraction must release radiation energy. However no GR version of (Newtonian) Helmholtz- Kelvin (HK) process has ever been derived. Here, for the first time, we derive the GR version of the appropriate virial theorem and Helmholtz Kelvin mechanism by simply equating the well known expressions for the gravitational mass and the Inertial Mass of a spherically symmetric static fluid. Simultaneously, we show that the GR counterparts of global ``internal energy'', ``gravitational potential energy'' and ``binding energy'' are actually different from what have been used so far. Existence of this GR HK process asserts that, in Einstein gravity too, gravitational collapse must be accompanied by emission of radiation irrespective of the details of the collapse process. Consequently, all studies of strictly adibatic gravitational collapse are only of academic interest. 
  We explore at phenomenological level a model of the Universe filled with various kinds of matter characterized by different equations of state. We show that introducing of each kind of matter is equivalent to a certain choice of a gauge condition, the gauge condition describing a medium with a given equation of state. The case of a particular interest is when one kind of matter (de Sitter false vacuum) dominates at the early stage of the Universe evolution while another kind (radiation, or ultrarelativistic gas) dominates at its later stage. We can, therefore, consider different asymptotic regimes for the early and later stages of the Universe existence. These regimes are described by solutions to the Wheeler - DeWitt equation for the Universe with matter in that given state, and, at the same time, in the "extended phase space" approach to quantum geometrodynamics the regimes are described by solutions to a Schrodinger equation associated with a choice of some gauge condition. It is supposed that, from the viewpoint of the observer located at the later stage of the Universe evolution, solutions for a Lambda-dominated early Universe would decay. 
  In the present work, gravitational collapse of an inhomogeneous spherical star model, consisting of inhomogeneous dust fluid (dark matter) in the background of dark energy is considered. The collapsing process is examined first separately for both dark matter and dark energy and then under the combined effect of dark matter and dark energy with or without interaction. The dark energy is considered in the form of perfect fluid and both marginally and non-marginally bound cases are considered for the collapsing model. Finally dark energy in the form of anisotropic fluid is investigated and it is found to be similar to ref. [12] 
  The condition for pressure isotropy, for spherically symmetric gravitational fields with charged and uncharged matter, is reduced to a recurrence equation with variable, rational coefficients. This difference equation is solved in general using mathematical induction leading to an exact solution to the Einstein field equations which extends the isotropic model of John and Maharaj. The metric functions, energy density and pressure are well behaved which suggests that this model could be used to describe a relativistic sphere. The model admits a barotropic equation of state which approximates a polytrope close to the stellar centre. 
  We present solutions to the Einstein-Maxwell system of equations in spherically symmetric gravitational fields for static interior spacetimes with a specified form of the electric field intensity. The condition of pressure isotropy yields three category of solutions. The first category is expressible in terms of elementary functions and does not have an uncharged limit. The second category is given in terms of Bessel functions of half-integer order. These charged solutions satisfy a barotropic equation of state and contain Finch-Skea uncharged stars. The third category is obtained in terms of modified Bessel functions of half-integer order and does not have an uncharged limit. The physical features of the charged analogue of the Finch-Skea stars are studied in detail. In particular the condition of causality is satisfied and the speed of sound does not exceed the speed of light. The physical analysis indicates that this analogue is a realistic model for static charged relativistic perfect fluid spheres. 
  The Laser Interferometer Space Antenna (LISA) will produce a data stream containing a vast number of overlapping sources: from strong signals generated by the coalescence of massive black hole binary systems to much weaker radiation form sub-stellar mass compact binaries and extreme-mass ratio inspirals. It has been argued that the observation of weak signals could be hampered by the presence of loud ones and that they first need to be removed to allow such observations. Here we consider a different approach in which sources are studied simultaneously within the framework of Bayesian inference. We investigate the simplified case in which the LISA data stream contains radiation from a massive black hole binary system superimposed over a (weaker) quasi-monochromatic waveform generated by a white dwarf binary. We derive the posterior probability density function of the model parameters using an automatic Reversible Jump Markov Chain Monte Carlo algorithm (RJMCMC). We show that the information about the sources and noise are retrieved at the expected level of accuracy without the need of removing the stronger signal. Our analysis suggests that this approach is worth pursuing further and should be considered for the actual analysis of the LISA data. 
  We present an algorithm for constructing the Wilson operator product expansion (OPE) for perturbative interacting quantum field theory in general Lorentzian curved spacetimes, to arbitrary orders in perturbation theory. The remainder in this expansion is shown to go to zero at short distances in the sense of expectation values in arbitrary Hadamard states. We also establish a number of general properties of the OPE coefficients: (a) they only depend (locally and covariantly) upon the spacetime metric and coupling constants, (b) they satisfy an associativity property, (c) they satisfy a renormalization group equation, (d) they satisfy a certain microlocal wave front set condition, (e) they possess a ``scaling expansion''. The latter means that each OPE coefficient can be written as a sum of terms, each of which is the product of a curvature polynomial at a spacetime point, times a Lorentz invariant Minkowski distribution in the tangent space of that point. The algorithm is illustrated in an example. 
  We can obtain one solution of the Hamiltonian constraint equation in the local sense. The form of the state is suggested from the up-to-down method in our previous work. The up-to-down method works for different way in treating the general metrics. In the mini-superspace approach there appears additional constraint in the 4-dimensional quantum gravity Hilbert space. However, in the general treatment of the metrics this method works as only solving technique. 
  In this essay we wish to seek a unifying thread between the basic forces. We propose that there exists a universal force which is shared by all that physically exists. Universality is characterized by the two properties: (i) universal linkage and (ii) long range. They uniquely identify Einstein gravity as the unversal force. All other forces then arise as these properties are peeled off. For instance, relaxing (i) but retaining (ii) will lead to Maxwell electromagnetic force. This unified outlook makes interesting suggestions and predictions: if there exists a new force, it can only be a short range non-abelian vector or a scalar field, and there should exist in an appropriate space duality relations between weak and electric, and between strong and gravity. 
  Maartens {\it et al.}\@ gave a covariant characterization, in a 1+3 formalism based on a perfect fluid's velocity, of the parts of the first derivatives of the curvature tensor in general relativity which are ``locally free'', i.e. not pointwise determined by the fluid energy momentum and its derivative. The full decomposition of independent curvature derivative components given in earlier work on the spinor approach to the equivalence problem enables analogous general results to be stated for any order: the independent matter terms can also be characterized. Explicit relations between the two sets of results are obtained. The 24 Maartens {\it et al.} locally free data are shown to correspond to the $\nabla \Psi$ quantities in the spinor approach, and the fluid terms are similarly related to the remaining 16 independent quantities in the first derivatives of the curvature. 
  While the generally recognized symmetries of cosmos are preserved, conservation laws for gravitational system are reconsidered and the Lagrangian density of pure gravitational field is revised. From these considerations, some of the theoretical foundations of the current cosmology are extended or revised, and a new theory of cosmology is established. This new theory leads to the following distinct properties of cosmos: the energy of matter field might originate from the gravitational field; the big bang might not have occurred; the fields of the dark energy and some parts of the dark matter would not be matter fields but might be gravitational fields, they would only interact with gravitational force but could not interact with other forces. These distinct properties can be tested by future experiments and observations. 
  We examine the motion in Schwarzschild spacetime of a point particle endowed with a scalar charge. The particle produces a retarded scalar field which interacts with the particle and influences its motion via the action of a self-force. We exploit the spherical symmetry of the Schwarzschild spacetime and decompose the scalar field in spherical-harmonic modes. Although each mode is bounded at the position of the particle, a mode-sum evaluation of the self-force requires regularization because the sum does not converge: the retarded field is infinite at the position of the particle. The regularization procedure involves the computation of regularization parameters, which are obtained from a mode decomposition of the Detweiler-Whiting singular field; these are subtracted from the modes of the retarded field, and the result is a mode-sum that converges to the actual self-force. We present such a computation in this paper. There are two main aspects of our work that are new. First, we define the regularization parameters as scalar quantities by referring them to a tetrad decomposition of the singular field. Second, we calculate four sets of regularization parameters (denoted schematically by A, B, C, and D) instead of the usual three (A, B, and C). As proof of principle that our methods are reliable, we calculate the self-force acting on a scalar charge in circular motion around a Schwarzschild black hole, and compare our answers with those recorded in the literature. 
  We give a rigorous derivation of the general-relativistic formula for the two-way Doppler tracking of a spacecraft in Friedmann-Lemaitre-Robertson-Walker and in McVittie spacetimes. The leading order corrections of the so-determined acceleration to the Newtonian acceleration are due to special-relativistic effects and cosmological expansion. The latter, although linear in the Hubble constant, is negligible in typical applications within the Solar System. 
  The search for signatures of transient, unmodelled gravitational-wave (GW) bursts in the data of ground-based interferometric detectors typically uses `excess-power' search methods. One of the most challenging problems in the burst-data-analysis is to distinguish between actual GW bursts and spurious noise transients that trigger the detection algorithms. In this paper, we present a unique and robust strategy to `veto' the instrumental glitches. This method makes use of the phenomenological understanding of the coupling of different detector sub-systems to the main detector output. The main idea behind this method is that the noise at the detector output (channel H) can be projected into two orthogonal directions in the Fourier space -- along, and orthogonal to, the direction in which the noise in an instrumental channel X would couple into H. If a noise transient in the detector output originates from channel X, it leaves the statistics of the noise-component of H orthogonal to X unchanged, which can be verified by a statistical hypothesis testing. This strategy is demonstrated by doing software injections in simulated Gaussian noise. We also formulate a less-rigorous, but computationally inexpensive alternative to the above method. Here, the parameters of the triggers in channel X are compared to the parameters of the triggers in channel H to see whether a trigger in channel H can be `explained' by a trigger in channel X and the measured transfer function. 
  A model based on simple assumptions about 4-dimensional space-time being closed and isotropic, and embedded in a 5th large-scale dimension $r$ representing the radius of curvature of space-time, has been used in an application of Newton's Second Law to describe a system with angular momentum. It has been found that the equations of MOND used to explain the rotation curves of galaxies appear as a limit within this derivation and that there is a universal acceleration constant, ao, with a well defined value, again consistent with that used by MOND. This approach does not require modification of Newtonian dynamics, only its extension into a fifth large-scale dimension. The transition from the classical Newtonian dynamics to the MOND regime emerges naturally and without the introduction of arbitrary fitting functions, if this five-dimensional model is adopted. 
  A new method is described for recognising internal non-randomness in deep galaxy surveys. 
  We develop a new method for calculation of quasi-normal modes of black holes, when the effective potential, which governs black hole perturbations, is known only numerically in some region near the black hole. This method can be applied to perturbations of a wide class of numerical black hole solutions. We apply it to the black holes in the Einstein-Aether theory, a theory where general relativity is coupled to a unit time-like vector field, in order to observe local Lorentz symmetry violation. We found that in the non-reduced Einstein-Aether theory, real oscillation frequency and damping rate of quasi-normal modes are larger than those of Schwarzschild black holes in the Einstein theory. 
  The dynamics of a universe with an anti-gravitating contribution to the matter content is examined. The modified Friedmann equations are derived, and it is shown that anti-gravitating radiation is the slowest component to dilute when the universe expands. Assuming an interaction between both kinds of matter which becomes important at Planckian densities, it is found that the universe undergoes a periodic cycle of contraction and expansion. Furthermore, the possibility of energy loss in our universe through separation of both types of matter is discussed. 
  By introducing the generalized uncertainty principle, we calculate the entropy of the bulk scalar field on the Randall-Sundrum brane background without any cutoff. We obtain the entropy of the massive scalar field proportional to the horizon area. Here, we observe that the mass contribution to the entropy exists in contrast to all previous results, which is independent of the mass of the scalar field, of the usual black hole cases with the generalized uncertainty principle. 
  The Laser Interferometer Space Antenna will be able to detect the inspiral and merger of Super Massive Black Hole Binaries (SMBHBs) anywhere in the Universe. Standard matched filtering techniques can be used to detect and characterize these systems. Markov Chain Monte Carlo (MCMC) methods are ideally suited to this and other LISA data analysis problems as they are able to efficiently handle models with large dimensions. Here we compare the posterior parameter distributions derived by an MCMC algorithm with the distributions predicted by the Fisher information matrix. We find excellent agreement for the extrinsic parameters, while the Fisher matrix slightly overestimates errors in the intrinsic parameters. 
  A hybrid metric with off-diagonal temporal-radial behavior that was constructed to conveniently parameterized the early and late time behaviors of the universe is shown to have diagonal forms consistent with Robertson-Walker and deSitter geometries. The dynamics of the energy content of the cosmology as parameterized by the classical thermal fraction is briefly discussed as motivation for the comparison of the observables predicted by various micoscopic models of the early evolution of the universe. 
  BF theory is a topological theory that can be seen as a natural generalization of 3-dimensional gravity to arbitrary dimensions. Here we show that the coupling to point particles that is natural in three dimensions generalizes in a direct way to BF theory in d dimensions coupled to (d-3)-branes. In the resulting model, the connection is flat except along the membrane world-sheet, where it has a conical singularity whose strength is proportional to the membrane tension. As a step towards canonically quantizing these models, we show that a basis of kinematical states is given by `membrane spin networks', which are spin networks equipped with extra data where their edges end on a brane. 
  Tolman-Bondi inhomogeneous spacetimes are used as a cosmological model for type Ia supernova data. It is found that with certain parameter choices the model fits the data as well as the standard $\Lambda$CDM cosmology does. 
  In this essay we discuss the difference in views of the Universe as seen by two different observers. While one of the observers follows a geodesic congruence defined by the geometry of the cosmological model, the other observer follows the fluid flow lines of a perfect fluid with a linear equation of state. We point out that the information these observers collect regarding the state of the Universe can be radically different; while one observes a non-inflating ever-expanding ever-lasting universe, the other observer can experience a dynamical behaviour reminiscent to that of quintessence or even that of a phantom cosmology leading to a 'big rip' singularity within finite time (but without the need for exotic forms of matter). 
  Using the Ponce de Leon background metric, which describes a 5D universe in an apparent vacuum: $\bar{G}_{AB}=0$, we study the effective 4D evolution of both, the inflaton and gauge-invariant scalar metric fluctuations, in the recently introduced model of space time matter inflation. 
  We discuss some general characteristics of modifications of the 4D Einstein-Hilbert action that become important for low space-time curvatures. In particular we focus on the chameleon-like behaviour of the massive gravitational degrees of freedom. Generically there is at least one extra scalar that is light on cosmic scales, but for certain models it becomes heavy close to any mass source. 
  Following a paper by Berman and Marinho Jr (2001), where it was established an equation of state (p=-(1/3)rho), for the very early Universe, under which, Einstein's equations with lambda=0, render a scale-factor proportional to the time coordinate, and under which, Berman(2006, 2006a) showed that in the non-null lambda case, the Universe entered automatically into an exponential inflationary phase, we now study the corresponding effect in Brans-Dicke Cosmology. We find that the resulting models are similar to those in General Relativity Cosmology. 
  It was recently proposed to use extra-galactic point sources to constrain space-time quantum fluctuations in the universe. In these proposals, the fundamental "fuzziness" of distance caused by space-time quantum fluctuations have been directly identified with fluctuations in optical paths. Phase-front corrugations deduced from these optical-path fluctuations are then applied to light from extragalactic point sources, and used to constrain various models of quantum gravity. In particular, the so-called random-walk model has been claimed to be ruled out by existing astrophysical observations from the Hubble Space Telescope. However, when a photon propagates in three spatial dimensions, it does not follow a specific ray, but would rather sample a finite, three-dimensional region around that ray -- thereby averaging over space-time quantum fluctuations all through that region. We use a simple, random-walk type model to demonstrate that, once the appropriate wave optics is applied, the averaging of neighboring space-time fluctuations will cause much less distortions on the phase front. In our model, the extra suppression factor due to diffraction is (Planck Length)/(Wavelength), which is at least 19 orders of magnitude for astronomical observations. 
  We present the generalized Friedmann equations describing the cosmological evolution of a finite thick brane immeresed in a five-dimensional Schwarzschild Anti-de Sitter spacetime. A linear term in the density in addition to a quatratic one arises in the Friedmann equation, leading to the standard cosmological evolution at late times without introducing an ad hoc tension term for the brane. The effective four-dimensional cosmological constant is then uplifted similar to the KKLT effect and vanishes for a brane thickness equal to the AdS curvature size, up to the third order of the thickness. The four-dimensional gravitational constant is then equal to the five-dimensional one divided by the AdS curvature radius, similar to that derived by dimensional compactification. An accelerating brane cosmology may emerge at late times provided there is either a negative transverse pressure component in the brane energy-momentum tensor or the effective brane cosmological constant is positive. 
  In recent, Kar.S et.al [ Phys Rev D 67,044005 (2003) ] have obtained static spherically symmetric solutions of the Einstein-Kalb-Ramond field equations. We have shown that their solutions, indeed, represent Wormholes. 
  It is shown that sufficiently smooth initial data for the Einstein-dust or the Einstein-Maxwell-dust equations with non-negative density of compact support develop into solutions representing isolated bodies in the sense that the matter field has spatially compact support and is embedded in an exterior vacuum solution. 
  A classical result by Buchdahl \cite{Bu1} shows that for static solutions of the spherically symmetric Einstein-matter system, the total ADM mass M and the area radius R of the boundary of the body, obey the inequality $2M/R\leq 8/9.$ The proof of this inequality rests on the hypotheses that the energy density is non-increasing outwards and that the pressure is isotropic. In this work neither of Buchdahl's hypotheses are assumed. We consider non-isotropic spherically symmetric shells, supported in $[R_0,R_1], R_0>0,$ of matter models for which the energy density $\rho\geq 0,$ and the radial- and tangential pressures $p\geq 0$ and $q,$ satisfy $p+q\leq\Omega\rho, \Omega\geq 1.$ We show a Buchdahl type inequality for shells which are thin; given an $\epsilon<1/4$ there is a $\kappa>0$ such that $2M/R_1\leq 1-\kappa$ when $R_1/R_0\leq 1+\epsilon.$ It is also shown that for a sequence of solutions such that $R_1/R_0\to 1,$ the limit supremum of $2M/R_1$ of the sequence is bounded by $((2\Omega+1)^2-1)/(2\Omega+1)^2.$ In particular if $\Omega=1,$ which is the case for Vlasov matter, the boumd is $8/9.$ The latter result is motivated by numerical simulations \cite{AR2} which indicate that for non-isotropic shells of Vlasov matter $2M/R_1\leq 8/9,$ and moreover, that the value 8/9 is approached for shells with $R_1/R_0\to 1$. In \cite{An2} a sequence of shells of Vlasov matter is constructed with the properties that $R_1/R_0\to 1,$ and that $2M/R_1$ equals 8/9 in the limit. We emphasize that in the present paper no field equations for the matter are used, whereas in \cite{An2} the Vlasov equation is important. 
  We consider a spacelike two-plane originally at rest with respect to electromagnetic radiation in equilibrium. We find that if the plane is moved with respect to the radiation, the plane shrinks such that the maximum amount of entropy carried by radiation through the plane is, in natural units, exactly one-half of the decrease in the area of the plane. This result suggests that the equivalence between area and entropy may not be limited in black holes, nor even in the spacetime horizons only, but the equivalence between horizon area and entropy may be a special case of some general and simple, still undiscovered fundamental principle of nature. 
  In this paper we derived Hawking radiation as a tuneling of massless particles through a non-singular horizon in the s-wave approximation. The back reaction of emitted modes on the background black hole geometry is self-consistently taken into account. This is arXive copy of the paper published in the Russian journal "Gravitation and Cosmology" in 1999. 
  A strong variable gravitational field of the very early Universe inevitably generates relic gravitational waves by amplifying their zero-point quantum oscillations. We begin our discussion by contrasting the concepts of relic gravitational waves and inflationary `tensor modes'. We explain and summarize the properties of relic gravitational waves that are needed to derive their effects on CMB temperature and polarization anisotropies. The radiation field is characterized by four invariants I, V, E, B. We reduce the radiative transfer equations to a single integral equation of Voltairre type and solve it analytically and numerically. We formulate the correlation functions C^{XX'}_{\ell} for X, X'= T, E, B and derive their amplitudes, shapes and oscillatory features. Although all of our main conclusions are supported by exact numerical calculations, we obtain them, in effect, analytically by developing and using accurate approximations. We show that the TE correlation at lower \ell's must be negative (i.e. an anticorrelation), if it is caused by gravitational waves, and positive if it is caused by density perturbations. This difference in TE correlation may be a signature more valuable observationally than the lack or presence of the BB correlation, since the TE signal is about 100 times stronger than the expected BB signal. We discuss the detection by WMAP of the TE anticorrelation at \ell \approx 30 and show that such an anticorrelation is possible only in the presence of a significant amount of relic gravitational waves (within the framework of all other common assumptions). We propose models containing considerable amounts of relic gravitational waves that are consistent with the measured TT, TE and EE correlations. 
  The evidence for supermassive Kerr black holes in galactic centers is strong and growing, but only the detection of gravitational waves will convincingly rule out other possibilities to explain the observations. The Kerr spacetime is completely specified by the first two multipole moments: mass and angular momentum. This is usually referred to as the ``no-hair theorem'', but it is really a ``two-hair'' theorem. If general relativity is the correct theory of gravity, the most plausible alternative to a supermassive Kerr black hole is a rotating boson star. Numerical calculations indicate that the spacetime of rotating boson stars is determined by the first three multipole moments (``three-hair theorem''). LISA could accurately measure the oscillation frequencies of these supermassive objects. We propose to use these measurements to ``count their hair'', unambiguously determining their nature and properties. 
  This paper presents a new computer code to solve the general relativistic magnetohydrodynamics (GRMHD) equations using distributed parallel adaptive mesh refinement (AMR). The fluid equations are solved using a finite difference Convex ENO method (CENO) in 3+1 dimensions, and the AMR is Berger-Oliger. Hyperbolic divergence cleaning is used to control the $\nabla\cdot {\bf B}=0$ constraint. We present results from three flat space tests, and examine the accretion of a fluid onto a Schwarzschild black hole, reproducing the Michel solution. The AMR simulations substantially improve performance while reproducing the resolution equivalent unigrid simulation results. Finally, we discuss strong scaling results for parallel unigrid and AMR runs. 
  This paper has been withdrawn by the author due to the fact that the results were found to be done previously. 
  We examine embedding diagrams of hypersurfaces in the Reissner-Nordstrom black hole spacetime. These embedding diagrams serve as useful tools to visualize the geometry of the hypersurfaces and of the whole spacetime in general. 
  It is shown how consistent histories quantum cosmology can be realised through Isham's Histories Projection Operator consistent histories scheme. This is done by using an affine algebra instead of a canonical one and also by using cocycle representations. A regularisation scheme allows us to find a history Hamiltonian which exists as a proper self-adjoint operator. The role of a cocycle choice is also discussed. 
  A key result in the proof of black hole uniqueness in 4-dimensions is that a stationary black hole that is ``rotating''--i.e., is such that the stationary Killing field is not everywhere normal to the horizon--must be axisymmetric. The proof of this result in 4-dimensions relies on the fact that the orbits of the stationary Killing field on the horizon have the property that they must return to the same null geodesic generator of the horizon after a certain period, $P$. This latter property follows, in turn, from the fact that the cross-sections of the horizon are two-dimensional spheres. However, in spacetimes of dimension greater than 4, it is no longer true that the orbits of the stationary Killing field on the horizon must return to the same null geodesic generator. In this paper, we prove that, nevertheless, a higher dimensional stationary black hole that is rotating must be axisymmetric. No assumptions are made concerning the topology of the horizon cross-sections other than that they are compact. However, we assume that the horizon is non-degenerate and, as in the 4-dimensional proof, that the spacetime is analytic. 
  Gauge invariant treatments of the second order cosmological perturbation in a four dimensional homogeneous isotropic universe filled with the perfect fluid are completely formulated without any gauge fixing. We derive all components of the Einstein equations in the case where the first order vector and tensor modes are negligible. These equations imply that the tensor and the vector mode of the second order metric perturbations may be generated by the scalar-scalar mode coupling of the linear order perturbations as the result of the non-linear effects of the Einstein equations. 
  Along the general framework of the gauge invariant perturbation theory developed in the papers [K. Nakamura, Prog. Theor. Phys. {\bf 110} (2003), 723; {\it ibid}, {\bf 113} (2005), 481.], we formulate the second order gauge invariant cosmological perturbation theory in a four dimensional homogeneous isotropic universe. We consider the perturbations both in the universe dominated by the single perfect fluid and in that dominated by the single scalar field. We derive the all components of the Einstein equations in the case where the first order vector and tensor modes are negligible. All equations are derived in terms of gauge invariant variables without any gauge fixing. These equations imply that the second order vector and tensor modes may be generated due to the mode-mode coupling of the linear order scalar perturbations. We also briefly discuss the main progress of this work by the comparison with some literatures. 
  It is well known that, in contrast to general relativity, there are two conformally related frames, the Jordan frame and the Einstein frame, in which the Brans-Dicke theory, a prototype of generic scalar-tensor theory, can be formulated. There is a long standing debate on the physical equivalence of the formulations in these two different frames. It is shown here that gravitational deflection of light to second order accuracy may observationally distinguish the two versions of the Brans-Dicke theory. 
  In the present work, an attempt has been made to explain the recent cosmic acceleration with two mutually interacting scalar fields, one being the Brans-Dicke scalar field and the other a quintessence scalar field. Conditions have been derived for which the quintessence scalar field has an early oscillation and it grows during a later time to govern the dynamics of the universe. 
  In this paper we address the following question: do the recent advances in the orbit determination of the major natural satellites of Saturn obtained with the analysis of the first data sets from the Cassini mission allow to detect the general relativistic gravitoelectric orbital precessions of such moons? The answer is still negative. The present-day down-track accuracy would be adequate for Mimas, Enceladus, Thetys, Dione, Rhea and Titan and inadequate for Hyperion, Iapetus and Phoebe. Instead, the size of the systematic errors induced by the mismodelling in the key parameters of the Saturnian gravitational field like the even zonal harmonics Jl are larger than the relativistic down-track shifts by about one order of magnitude, mainly for the inner satellites like Mimas, Enceladus, Thetys, Dione, Rhea, Titan and Hyperion. Iapetus and Phoebe are not sensibly affected by such kind of perturbations. Moreover, the bias due to the uncertainty in Saturn's GM is larger than the relativistic down-track effects for all such moons. Proposed linear combinations of the satellites' orbital elements would allow to cancel out the impact of the mismodelling in the low-degree even zonal harmonics and GM, but the combined down-track errors would be larger than the combined relativistic signature. 
  As it is well known the topology of space is not totally determined by Einstein's equations. It is considered a massless scalar quantum field in a static Euclidean space of dimension 3. The expectation value for the energy density in all compact orientable Euclidean 3-spaces are obtained in this work as a finite summation of Epstein type zeta functions. The Casimir energy density for these particular manifolds is independent of the type of coupling with curvature. A numerical plot of the result inside each Dirichlet region is obtained. 
  We consider the fate of future singularities in the effective dynamics of loop quantum cosmology. Non-perturbative quantum geometric effects which lead to $\rho^2$ modification of the Friedmann equation at high energies result in generic resolution of singularities whenever energy density $\rho$ diverges at future singularities of Friedmann dynamics. Such quantum effects lead to the avoidance of a Big Rip, which is followed by a recollapsing universe stable against perturbations. Resolution of sudden singularity, the case when pressure diverges but energy density approaches a finite value depends on the ratio of the latter to a critical energy density of the order of Planck. If the value of this ratio is greater than unity, the universe escapes the sudden future singularity and becomes oscillatory. 
  It is fair to say that our current mathematical understanding of the dynamics of gravitational collapse to a black hole is limited to the spherically symmetric situation and, in fact, even in this case much remains to be learned. The reason is that Einstein's equations become tractable only if they are reduced to a 1+1 dimensional system of partial differential equations. Due to this technical obstacle, very little is known about the collapse of pure gravitational waves because by Birkhoff's theorem there is no spherical collapse in vacuum. In this essay we describe a new cohomogeneity-two symmetry reduction of the vacuum Einstein equations in five and higher odd dimensions which evades Birkhoff's theorem and admits time dependent asymptotically flat solutions. We argue that this model provides an attractive 1+1 dimensional geometric setting for investigating the dynamics of gravitational collapse in vacuum. 
  We have previously discussed the characteristics of the gravitational waves (GW) and have, theoretically, shown that, like the corresponding electromagnetic (EM) waves, they also demonstrate, under certain conditions, holographic properties. In this work we have expanded this discussion and show that the assumed gravitational holographic images may, theoretically, be related to another property of GW's which is their possible relation to singular (or nonsingular) trapped surfaces. We also show that this possibility may be, theoretically, related even to weak GW's. 
  The spherically symmetric, static spacetime generated by a crossflow of non-interacting radiation streams, treated in the geometrical optics limit (null dust) is equivalent to an anisotropic fluid forming a radiation atmosphere of a star. This reference fluid provides a preferred / internal time, which is employed as a canonical coordinate. Among the advantages we encounter a new Hamiltonian constraint, which becomes linear in the momentum conjugate to the internal time (therefore yielding a functional Schr\"{o}dinger equation after quantization), and a strongly commuting algebra of the new constraints. 
  Due to quantum fluctuations, spacetime is foamy on small scales. For maximum spatial resolution of the geometry of spacetime, the holographic model of spacetime foam stipulates that the uncertainty or fluctuation of distance $l$ is given, on the average, by $(l l_P^2)^{1/3}$ where $l_P$ is the Planck length. Applied to cosmology, it predicts that the cosmic energy is of critical density and the cosmic entropy is the maximum allowed by the holographic principle. In addition, it requires the existence of unconventional (dark) energy/matter and accelerating cosmic expansion in the present era. We will argue that a holographic foam cosmology of this type has the potential to become a full fledged competitor (with distinct testable consequences) for scalar driven inflation. 
  Distorted black holes radiate gravitational waves. In the so-called ringdown phase radiation is emitted in a discrete set of complex quasinormal frequencies, whose values depend only on the black hole's mass and angular momentum. Ringdown radiation could be detectable with large signal-to-noise ratio by the Laser Interferometer Space Antenna LISA. If more than one mode is detected, tests of the black hole nature of the source become possible. The detectability of different modes depends on their relative excitation, which in turn depends on the cause of the perturbation (i.e. on the initial data). A ``universal'', initial data-independent measure of the relative mode excitation is encoded in the poles of the Green's function that propagates small perturbations of the geometry (``excitation factors''). We compute for the first time the excitation factors for general-spin perturbations of Kerr black holes. We find that for corotating modes with $l=m$ the excitation factors tend to zero in the extremal limit, and that the contribution of the overtones should be more significant when the black hole is fast rotating. We also present the first analytical calculation of the large-damping asymptotics of the excitation factors for static black holes, including the Schwarzschild and Reissner-Nordstrom metrics. This is an important step to determine the convergence properties of the quasinormal mode expansion. 
  We consider a single 3-brane situated between two bulk spacetimes that posses the same cosmological constant, but whose metrics do not posses a $Z_{2}$-symmetry. On each side of the brane, the bulk is a solution to Gauss-Bonnet gravity. This asymmetry modifies junction conditions, and so new terms arise in the Friedmann equation. If these terms become dominant, these behave cosmological constant at early times for some case, and might remove the initial singularity for other case. However, we show that these new terms can not become dominant ones under usual conditions when our brane is outside an event horizon. We also show that any brane-world scenarios of this type revert to a $Z_{2}$-symmetric form at late times, and hence rule out certain proposed scenarios. 
  If the expanding and contracting regions coexist in the universe, the speed of cosmic expansion can be accelerated. We construct simple inhomogeneous dust-filled universe models of which the speed of cosmic volume expansion is accelerated for finite periods. These models are made by removing spherical domains from the Einstein-de Sitter universe and filling each domain by the Lema\^{\i}tre-Tolman-Bondi dust sphere with the same gravitational mass that contained in the removed region. This is an exact solution of Einstein equations. We find that the the acceleration of the cosmic volume expansion is realized when the size of the contracting region is smaller than the horizon radius of the Einstein-de Sitter universe. 
  Analogue spacetimes are powerful models for probing the fundamental physical aspects of geometry - while one is most typically interested in ultimately reproducing the pseudo-Riemannian geometries of interest in general relativity and cosmology, analogue models can also provide useful physical probes of more general geometries such as pseudo-Finsler spacetimes. In this chapter we shall see how a 2-component Bose-Einstein condensate can be used to model a specific class of pseudo-Finsler geometries, and after suitable tuning of parameters, both bi-metric pseudo-Riemannian geometries and standard single metric pseudo-Riemannian geometries, while independently allowing the quasi-particle excitations to exhibit a "mass". Furthermore, when extrapolated to extremely high energy the quasi-particles eventually leave the phononic regime and begin to act like free bosons. Thus this analogue spacetime exhibits an analogue of the "Lorentz violation" that is now commonly believed to occur at or near the Planck scale defined by the interplay between quantum physics and gravitational physics. In the 2-component Bose-Einstein analogue spacetime we will show that the mass generating mechanism for the quasi-particles is related to the size of the Lorentz violations. This relates the "mass hierarchy" to the so-called "naturalness problem". In short the analogue spacetime based on 2-component Bose-Einstein condensates exhibits a very rich mathematical and physical structure that can be used to investigate many issues of interest to the high-energy physics, cosmology, and general relativity communities. 
  This work is part of an ongoing research programme to study possible Primordial Black Hole (PBH) formation during the radiation dominated era of the early universe. Working within spherical symmetry, we specify an initial configuration in terms of a curvature profile, which represents initial conditions for the large amplitude metric perturbations, away from the homogeneous Friedmann Robertson Walker model, which are required for PBH formation. Using an asymptotic quasi-homogeneous solution, we relate the curvature profile with the density and velocity fields, which at an early enough time, when the length scale of the configuration is much larger than the cosmological horizon, can be treated as small perturbations of the background values. We present general analytic solutions for the density and velocity profiles. These solutions enable us to consider in a self-consistent way the formation of PBHs in a wide variety of cosmological situations with the cosmological fluid being treated as an arbitrary mixture of different components with different equations of state. We show that the analytical solutions for the density and velocity profiles as functions of the initial time are pure growing modes. We then use two different parametrisation for the curvature profile and follow numerically the evolution of a range of initial configuration. 
  We consider the recent calculation gr-qc/0508124 of the graviton propagator in the spinfoam formalism. Within the 3d toy model introduced in gr-qc/0512102, we test how the spinfoam formalism can be used to construct the perturbative expansion of graviton amplitudes. Although the 3d graviton is a pure gauge, one can choose to work in a gauge where it is not zero and thus reproduce the structure of the 4d perturbative calculations. We compute explicitly the next to leading and next to next to leading orders, corresponding to one-loop and two-loop corrections. We show that while the first arises entirely from the expansion of the Regge action around the flat background, the latter receives contributions from the microscopic, non Regge-like, quantum geometry. Surprisingly, this new contribution reduces the magnitude of the next to next to leading order. It thus appears that the spinfoam formalism is likely to substantially modify the conventional perturbative expansion at higher orders.   This result supports the interest in this approach. We then address a number of open issues in the rest of the paper. First, we discuss the boundary state ansatz, which is a key ingredient in the whole construction. We propose a way to enhance the ansatz in order to make the edge lengths and dihedral angles conjugate variables in a mathematically well-defined way. Second, we show that the leading order is stable against different choices of the face weights of the spinfoam model; the next to leading order, on the other hand, is changed in a simple way, and we show that the topological face weight minimizes it. Finally, we extend the leading order result to the case of a regular, but not equilateral, tetrahedron. 
  The emergence of loop quantum gravity over the past two decades has stimulated a great resurgence of interest in unifying general relativity and quantum mechanics. Amongst a number of appealing features of this approach are the intuitive picture of quantum geometry using spin networks and powerful mathematical tools from gauge field theory. However, the present form of loop quantum gravity suffers from a quantum ambiguity, due to the presence of a free (Barbero-Immirzi) parameter. Following recent progress on the conformal decomposition of gravitational fields, we present a new phase space for general relativity. In addition to spin-gauge symmetry, the new phase space also incorporates conformal symmetry making the description parameter free. The Barbero-Immirzi ambiguity is shown to occur only if the conformal symmetry is gauge-fixed prior to quantization. By withholding its full symmetries, the new phase space offers a promising platform for the future development of loop quantum gravity. This paper aims to provide an exposition, at a reduced technical level, of the above theoretical advances and their background developments. Further details are referred to cited references. 
  We propose an idea in spectroscopy to search for extra spatial dimensions as well as to detect the possible deviation from Newton's inverse-square law at small scale, and we take high-Z hydrogenic systems and muonic atoms as illustrations. The relevant experiments might help to explore more than two extra dimensions scenario in ADD's brane world model and to set constraints for fundamental parameters such as the size of extra dimensions. 
  In the framework of Einstein's General Relativity Theory and of the Relativistic Theory of Gravitation, the equations governing the trajectories of charged particles in the field created by a charged mass point are given. An analysis of the shape of the trajectories in both theories is presented. The first and the second order approximate solutions of the electrogravitational Kepler problem are found in the two theories and the results are compared with each other. I have pointed out the differences between the predictions in the two theories. 
  A Relational Quantum Theory Incorporating Gravity developed the concept of quantum covariance and argued that this is the correct expression of the fundamental physical principle that the behaviour of matter is everywhere the same in the quantum domain, as well as being the required condition for the unification of general relativity with quantum mechanics for non-interacting particles. This paper considers the interactions of elementary particles. Quantum covariance describes families of finite dimensional Hilbert spaces with an inbuilt cut-off in energy-momentum and using flat space metric (quantum coordinates) between initial and final states. It is shown by direct construction that it is possible to construct a quantum field theory of operators on members of these families, obeying locality, suitable for a description of particle interactions, and leading to a general formulation of particle theoretic field theory incorporating qed. The construction is consistent and effectively identical in the continuum limit to classical quantum electrodynamics with all loop divergences removed by the method of Epstein and Glaser. The model avoids the Landau pole. The model does not start from a classical Lagrangian and it is shown that the interaction leads to Maxwell's equations in the classical limit and to Feynman rules as in the standard theory after renormalisation. 
  We study perfect fluid cosmological models with a constant equation of state parameter $\gamma$ in which there are two naturally defined time-like congruences, a geometrically defined geodesic congruence and a non-geodesic fluid congruence. We establish an appropriate set of boost formulae relating the physical variables, and consequently the observed quantities, in the two frames. We study expanding spatially homogeneous tilted perfect fluid models, with an emphasis on future evolution with extreme tilt. We show that for ultra-radiative equations of state (i.e., $\gamma>4/3$), generically the tilt becomes extreme at late times and the fluid observers will reach infinite expansion within a finite proper time and experience a singularity similar to that of the big rip. In addition, we show that for sub-radiative equations of state (i.e., $\gamma < 4/3$), the tilt can become extreme at late times and give rise to an effective quintessential equation of state. To establish the connection with phantom cosmology and quintessence, we calculate the effective equation of state in the models under consideration and we determine the future asymptotic behaviour of the tilting models in the fluid frame variables using the boost formulae. We also discuss spatially inhomogeneous models and tilting spatially homogeneous models with a cosmological constant. 
  A path integral representation of the evolution operator for the four-dimensional Dirac equation is proposed. A quadratic form of the canonical momenta regularizes the original representation of the path integral in the electron phase space. This regularization allows to obtain a representation of the path integral over trajectories in the configuration space, i.e. in the Minkowsky space. This form of the path integral is useful for the formulation of perturbation theory in an external electromagnetic field. 
  The consistency of the constraint with the evolution equations for spatially inhomogeneous and irrotational silent (SIIS) models of Petrov type I, demands that the former are preserved along the timelike congruence represented by the velocity of the dust fluid, leading to \emph{new} non-trivial constraints. This fact has been used to conjecture that the resulting models correspond to the spatially homogeneous (SH) models of Bianchi type I, at least for the case where the cosmological constant vanish. By exploiting the full set of the constraint equations as expressed in the 1+3 covariant formalism and using elements from the theory of the spacelike congruences, we provide a direct and simple proof of this conjecture for vacuum and dust fluid models, which shows that the Szekeres family of solutions represents the most general class of SIIS models. The suggested procedure also shows that, the uniqueness of the SIIS of the Petrov type D is not, in general, affected by the presence of a non-zero pressure fluid. Therefore, in order to allow a broader class of Petrov type I solutions apart from the SH models of Bianchi type I, one should consider more general ``silent'' configurations by relaxing the vanishing of the vorticity and the magnetic part of the Weyl tensor but maintaining their ``silence'' properties i.e. the vanishing of the curls of $E_{ab},H_{ab}$ and the pressure $p$. 
  We study black hole formation in the head-on collision of ultrarelativistic charges. The metric of charged particles is obtained by boosting the Reissner-Nordstr\"om spacetime to the speed of light. Using the slice at the instant of collision, we study formation of the apparent horizon (AH) and derive a condition indicating that a critical value of the electric charge is necessary for formation to take place. Evaluating this condition for characteristic values at the LHC, we find that the presence of charge decreases the black hole production rate in accelerators. We comment on possible limitations of our approach. 
  We show that almost all metric--affine theories of gravity yield Einstein equations with a non--null cosmological constant $\Lambda$. Under certain circumstances and for any dimension, it is also possible to incorporate a Weyl vector field $W_\mu$ and therefore the presence of an anisotropy. The viability of these field equations is discussed in view of recent astrophysical observations. 
  A semi-classical analysis of vacuum energy in the expanding spacetime suggests that the cosmological term decays with time, with a concomitant matter production. For early times we find, in Planck units, $\Lambda \approx H^4$, where H is the Hubble parameter. The corresponding cosmological solution has no initial singularity, existing since an infinite past. During an infinitely long period we have a quasi-de Sitter, inflationary universe, with $H \approx 1$. However, at a given time, the expansion undertakes a phase transition, with H and $\Lambda$ decreasing to nearly zero in a few Planck times, producing a huge amount of radiation. On the other hand, the late-time scenario is similar to the standard model, with the radiation phase followed by a dust era, which tends asymptotically to a de Sitter universe, with vacuum dominating again. 
  A general framework for an emergent universe scenario has been given which makes use of an equation of state. The general features of the model have also been studied and possible primordial composition of the universe have been suggested. 
  The gravitational wave signals from coalescing Supermassive Black Hole Binaries are prime targets for the Laser Interferometer Space Antenna (LISA). With optimal data processing techniques, the LISA observatory should be able to detect black hole mergers anywhere in the Universe. The challenge is to find ways to dig the signals out of a combination of instrument noise and the large foreground from stellar mass binaries in our own galaxy. The standard procedure of matched filtering against a grid of templates can be computationally prohibitive, especially when the black holes are spinning or the mass ratio is large. Here we develop an alternative approach based on Metropolis-Hastings sampling and simulated annealing that is orders of magnitude cheaper than a grid search. We demonstrate our approach on simulated LISA data streams that contain the signals from binary systems of Schwarzschild Black Holes, embedded in instrument noise and a foreground containing 26 million galactic binaries. The search algorithm is able to accurately recover the 9 parameters that describe the black hole binary without first having to remove any of the bright foreground sources, even when the black hole system has low signal-to-noise. 
  As an extension of the Robinson-Trautman solutions of D=4 general relativity, we investigate higher dimensional spacetimes which admit a hypersurface orthogonal, non-shearing and expanding geodesic null congruence. Einstein's field equations with an arbitrary cosmological constant and possibly an aligned pure radiation are fully integrated, so that the complete family is presented in closed explicit form. As a distinctive feature of higher dimensions, the transverse spatial part of the general line element must be a Riemannian Einstein space, but it is otherwise arbitrary. On the other hand, the remaining part of the metric is - perhaps surprisingly - not so rich as in the standard D=4 case, and the corresponding Weyl tensor is necessarily of algebraic type D. While the general family contains (generalized) static Schwarzschild-Kottler-Tangherlini black holes and extensions of the Vaidya metric, there is no analogue of important solutions such as the C-metric. 
  We consider the leading post-Newtonian and quantum corrections to the non-relativistic scattering amplitude of charged spin-1/2 fermions in the combined theory of general relativity and QED. The coupled Dirac-Einstein system is treated as an effective field theory. This allows for a consistent quantization of the gravitational field. The appropriate vertex rules are extracted from the action, and the non-analytic contributions to the 1-loop scattering matrix are calculated in the non-relativistic limit. The non-analytical parts of the scattering amplitude are known to give the long range, low energy, leading quantum corrections, are used to construct the leading post-Newtonian and quantum corrections to the two-particle non-relativistic scattering matrix potential for two massive fermions with electric charge. 
  A simple diffeomorphism invariant theory of connections with the non-compact structure group R of real numbers is quantized. The theory is defined on a four-dimensional 'space-time' by an action resembling closely the self-dual Plebanski action for general relativity. The space of quantum states is constructed by means of projective techniques by Kijowski. Except this point the applied quantization procedure is based on Loop Quantum Gravity methods. 
  We derive the equations of motion of spinning compact binaries including the spin-orbit (SO) coupling terms one post-Newtonian (PN) order beyond the leading-order effect. For black holes maximally spinning this corresponds to 2.5PN order. Our result for the equations of motion essentially confirms the previous result by Tagoshi, Ohashi and Owen. We also compute the spin-orbit effects up to 2.5PN order in the conserved (Noetherian) integrals of motion, namely the energy, the total angular momentum, the linear momentum and the center-of-mass integral. We obtain the spin precession equations at 1PN order beyond the leading term, as well. Those results will be used in a future paper to derive the time evolution of the binary orbital phase, providing more accurate templates for LIGO-Virgo-LISA type interferometric detectors. 
  Motivated by the search for gravitational waves emitted by binary black holes, we investigate the gravitational radiation field of point particles with spins within the framework of the multipolar-post-Newtonian wave generation formalism. We compute: (i) the spin-orbit (SO) coupling effects in the binary's mass and current quadrupole moments one post-Newtonian (1PN) order beyond the dominant effect, (ii) the SO contributions in the gravitational-wave energy flux and (iii) the secular evolution of the binary's orbital phase up to 2.5PN order. Crucial ingredients for obtaining the 2.5PN contribution in the orbital phase are the binary's energy and the spin precession equations, derived in paper I of this series. These results provide more accurate gravitational-wave templates to be used in the data analysis of rapidly rotating Kerr-type black-hole binaries with the ground-based detectors LIGO, Virgo, GEO 600 and TAMA300, and the space-based detector LISA. 
  The time delay effect for planets and spacecraft is obtained from a fully relativistic modified gravity theory including a fifth force skew symmetric field by fitting to the Pioneer 10/11 anomalous acceleration data. A possible detection of the predicted time delay corrections to general relativity for the outer planets and future spacecraft missions is considered. The time delay correction to GR predicted by the modified gravity is consistent with the observational limit of the Doppler tracking measurement reported by the Cassini spacecraft on its way to Saturn, and the correction increases to a value that could be measured for a spacecraft approaching Neptune and Pluto. 
  Nonmetricity derived from a scalar field is shown to exist as a cosmic field, without direct coupling to matter. It leads to a variable cosmological term, a term that dominates the expansion in the early universe but dies away at later time. 
  Assuming the Hubble parameter is a continuous and differentiable function of comoving time, we investigate necessary conditions for quintessence to phantom phase transition in quintom model. For power-law and exponential potential examples, we study the behavior of dynamical dark energy fields and Hubble parameter near the transition time, and show that the phantom-divide-line w=-1 is crossed in these models. 
  The current bounds on the PPN parameters gamma and beta are of the order of 10^-4-10^-5. Various missions aimed at improving such limits by several orders of magnitude have more or less recently been proposed like LATOR, ASTROD, BepiColombo and GAIA. They involve the use of various spacecraft, to be launched along interplanetary trajectories, for measuring the effects of the solar gravity on the propagation of electromagnetic waves. In this paper we explore the possibility of measuring the combination nu=(2+2gamma-beta)/3 of the post-Newtonian gravitoelectric Einstein perigee precession of a test particle to an accuracy of 10^-5-10^-6 with a pair of drag-free spacecraft in the Earth's gravitational field. It turns out that the latest gravity models from the dedicated CHAMP and GRACE missions would allow to reduce the systematic error of gravitational origin just to this demanding level of accuracy. In regard to the non-gravitational errors, the spectral noise density of the drag-free sensors required to reach such level of accuracy would amounts to 10^-8-10^-9 cm s^-2 Hz^-1/2 over very low frequencies. Although not yet obtainable with the present technologies, such level of compensation is much less demanding than those required for, e.g., LISA. As a by-product, an independent measurement of the post-Newtonian gravitomagnetic Lense-Thirring effect with a 0.9% accuracy would be possible as well. The forthcoming Earth gravity models from CHAMP and GRACE will further reduce the systematic gravitational errors in both of such tests. 
  In this paper we calculate the energy distribution of six cases of Vaidya-solutions in the M{\o}ller prescription. Except the energy complex of M{\o}ller for the monopole solution vanishes everywhere, for other solutions have non-zero energy component, only the energy distributions of the de Sitter and anti-de Sitter solution are independent on $v$. For the radiating dyon solution, the difference in energy complex between M{\o}ller's and Einstein's prescription is like the case of Reissner-Nordstr\"{o}m solution. 
  We investigate the existence of time-periodic solutions of the Dirac equation in the Kerr-Newman background metric. To this end, the solutions are expanded in a Fourier series with respect to the time variable $t$ and the Chandrasekhar separation ansatz is applied so that the question of existence of a time-periodic solution is reduced to the solvability of a certain coupled system of ordinary differential equations. First, we prove the already known result that there are no time-periodic solutions in the non-extreme case. Then it is shown that in the extreme case for fixed black hole data there is a sequence of particle masses $(m_N)_{N\in\mathbb N}$ for which a time-periodic solution of the Dirac equation does exist. The period of the solution depends only on the data of the black hole described by the Kerr-Newman metric. 
  The photon vacuum polarization effect in curved spacetime leads to birefringence, i.e. the photon velocity becomes greater than (or less than) the speed of light depending on its polarization. We investigate this phenomenon in a Schwarzschild curved spacetime. 
  Frequently it is argued that the microstates responsible for the Bekenstein-Hawking entropy should arise from some physical degrees of freedom located near or on the black hole horizon. In this Essay we elucidate that instead entropy may emerge from the conversion of physical degrees of freedom, attached to a generic boundary, into unobservable gauge degrees of freedom attached to the horizon. By constructing the reduced phase space it can be demonstrated that such a transmutation indeed takes place for a large class of black holes, including Schwarzschild. 
  Chen and Nester proposed four boundary expressions for the quasilocal quantities using the covariant Hamiltonian formalism. Based on these four expressions, there is a simple generalization that one can consider, so that a two parameter set of boundary expressions can be constructed. Using these modified expressions, a nice result for gravitational energy-momentum can be obtained in holonomic frames. 
  We have studied the famous classical pseudotensors in the small region limit, both inside matter and in vacuum. A recent work [Deser et al.1999 CQG 16, 2815] had found one combination of the Einstein and Landau-Lifshitz expressions which yields the Bel-Robinson tensor in vacuum. Using similar methods we found another independent combination of the Bergmann-Thomson, Papapetrou and Weinberg pseudotensors with the same desired property. Moreover we have constructed an infinite number of additional new holonomic pseudotensors satisfying this important positive energy requirement, all seem quite artificial. On the other hand we found that Moller's 1961 tetrad-teleparallel energy-momentum expression naturally has this Bel-Robinson property. 
  In a previous work \cite{An1} matter models such that the energy density $\rho\geq 0,$ and the radial- and tangential pressures $p\geq 0$ and $q,$ satisfy $p+q\leq\Omega\rho, \Omega\geq 1,$ were considered in the context of Buchdahl's inequality. It was proved that static shell solutions of the spherically symmetric Einstein equations obey a Buchdahl type inequality whenever the support of the shell, $[R_0,R_1], R_0>0,$ satisfies $R_1/R_0<1/4.$ Moreover, given a sequence of solutions such that $R_1/R_0\to 1,$ then the limit supremum of $2M/R_1$ was shown to be bounded by $((2\Omega+1)^2-1)/(2\Omega+1)^2.$ In this paper we show that the hypothesis that $R_1/R_0\to 1,$ can be realized for Vlasov matter, by constructing a sequence of static shells of the spherically symmetric Einstein-Vlasov system with this property. We also prove that for this sequence not only the limit supremum of $2M/R_1$ is bounded, but that the limit is $((2\Omega+1)^2-1)/(2\Omega+1)^2=8/9,$ since $\Omega=1$ for Vlasov matter. Thus, static shells of Vlasov matter can have $2M/R_1$ arbitrary close to $8/9,$ which is interesting in view of \cite{AR2}, where numerical evidence is presented that 8/9 is an upper bound of $2M/R_1$ of any static solution of the spherically symmetric Einstein-Vlasov system. 
  Gravity is one of the most inexplicable forces of nature, controlling everything, from the expansion of the Universe to the ebb and flow of ocean tides. The search for the laws of motion and gravitation began more than two thousand years ago but still we do not have the complete picture of it. In this article, we have outlined how our understanding of gravity is changing drastically with time and how the previous explanations have shaped the most recent developments in the field like superstrings and braneworlds. 
  We show that spin-gravity interaction can distinguish between Dirac and Majorana neutrino wave packets propagating in a Lense-Thirring background. Using time-independent perturbation theory and gravitational phase to generate a perturbation Hamiltonian with spin-gravity coupling, we show that the associated matrix element for the Majorana neutrino differs significantly from its Dirac counterpart. This difference can be demonstrated through significant gravitational corrections to the neutrino oscillation length for a two-flavour system, as shown explicitly for SN1987A. 
  The Laser Interferometer Space Antenna (LISA) is being designed to detect and study in detail gravitational waves from sources throughout the Universe such as massive black hole binaries. The conceptual formulation of the LISA space-borne gravitational wave detector is now well developed. The interferometric measurements between the sciencecraft remain one of the most important technological and scientific design areas for the mission.   Our work has concentrated on developing the interferometric technologies to create a LISA-like optical signal and to measure the phase of that signal using commercially available instruments. One of the most important goals of this research is to demonstrate the LISA phase timing and phase reconstruction for a LISA-like fringe signal, in the case of a high fringe rate and a low signal level. We present current results of a test-bed interferometer designed to produce an optical LISA-like fringe signal previously discussed in the literature. 
  Over the past few years questions have been raised concerning the use of laser communications links between sciencecraft to transmit phase information crucial to the reduction of laser frequency noise in the LISA science measurement. The concern is that applying medium frequency phase modulations to the laser carrier could compromise the phase stability of the LISA fringe signal. We have modified the table-top interferometer presented in a previous article by applying phase modulations to the laser beams in order to evaluate the effects of such modulations on the LISA science fringe signal. We have demonstrated that the phase resolution of the science signal is not degraded by the presence of medium frequency phase modulations. 
  Recently, it was shown that differential rotation is an unavoidable feature of nonlinear r-modes. We investigate the influence of this differential rotation on the detectability of gravitational waves emitted by a newly born, hot, rapidly-rotating neutron star, as it spins down due to the r-mode instability. We conclude that gravitational radiation may be detected by the advanced laser interferometer detector LIGO if the amount of differential rotation at the time the r-mode instability becomes active is not very high. 
  We describe a midi-superspace quantization scheme for generic single horizon black holes in which only the spatial diffeomorphisms are fixed. The remaining Hamiltonian constraint yields an infinite set of decoupled eigenvalue equations: one at each spatial point. The corresponding operator at each point is the product of the outgoing and ingoing null convergences, and describes the scale invariant quantum mechanics of a particle moving in an attractive $1/X^2$ potential. The variable $X$ that is analoguous to particle position is the square root of the conformal mode of the metric. We quantize the theory via Bohr quantization, which by construction turns the Hamiltonian constraint eigenvalue equation into a finite difference equation. The resulting spectrum gives rise to a discrete spatial topology exterior to the horizon. The spectrum approaches the continuum in the asymptotic region. 
  Formulating a dust filled spherically symmetric metric utilizing the 3+1 formalism for general relativity, we show that the metric coefficients are completely determined by the matter distribution, and it's time rate of change on an initial slice. Rather than specifying Schwarzschild coordinates for the exterior of the collapsing region, we let the interior dictate the form of the solution in the exterior, and thus both regions are found to be written in one coordinate patch. This not only alleviates the need for complicated matching schemes at the interface, but also finds a new coordinate system for the Schwarzschild spacetime expressed in generalized Painleve-Gullstrand coordinates. We show the interior metric is equivalent to the class of Tolman-Bondi metrics under a coordinate transformation. In particular, specifying the initial density as a step function the solution reduces to the Oppenheimer-Snyder model. In general, the solution is the class of Tolman-Bondi models, which are known to develop shell crossing singularities. We show these are equivalent to fluid shock waves and suggest methods for extending beyond their initial point of formation. 
  The selfgravity of an infalling gas can alter significantly the accretion of gases. In the case of spherically symmetric steady flows of polytropic perfect fluids the mass accretion rate achieves maximal value when the mass of the fluid is 1/3 of the total mass. There are two weakly accreting regimes, one over-abundant and the other poor in fluid content. The analysis within the newtonian gravity suggests that selfgravitating fluids can be unstable, in contrast to the accretion of test fluids. 
  We examine, in a purely geometrical way, static Ricci-flat 5-manifolds admitting the 2-sphere and an additional hypersurface-orthogonal Killing vector. These are widely studied in the literature, from different physical approaches, and known variously as the Kramer - Gross - Perry - Davidson - Owen solutions. The 2-fold infinity of cases that result are studied by way of new coordinates (which are in most cases global) and the cases likely to be of interest in any physical approach are distinguished on the basis of the nakedness and geometrical mass of their associated singularities. It is argued that the entire class of solutions has to be considered unstable about the exceptional solutions: the black string and soliton cases. Any physical theory which admits the non-exceptional solutions as the external vacuua of a collapsing object has to accept the possibility of collapse to zero volume leaving behind the weakest possible, albeit naked, geometrical singularities at the origin.Finally, it is pointed out that these types of solutions generalize, in a straightforward way, to higher dimensions. 
  In this pedagogical note, we discuss obstacles to the usual Palatini formulations of gauge and gravity theories in presence of odd-derivative order, Chern-Simons, terms. 
  This paper is a further development of the approach to weak cosmic censorship proposed by the authors in Ref. 5. We state and prove a modified version of that work's main result under significantly relaxed assumptions on the asymptotic structure of space--time. The result, which imposes strong constraints on the occurrence of naked singularities of the strong curvature type, is in particular applicable to physically realistic cosmological models. 
  A new technique is presented for modifying the Einstein evolution equations off the constraint hypersurface. With this approach the evolution equations for the constraints can be specified freely. The equations of motion for the gravitational field variables are modified by the addition of terms that are linear and nonlocal in the constraints. These terms are obtained from solutions of the linearized Einstein constraints. 
  The stress energy tensor for the classical non-minimally coupled scalar field is known not to satisfy the point-wise energy conditions of general relativity. In this paper we show, however, that local averages of the classical stress energy tensor satisfy certain inequalities. We give bounds for averages along causal geodesics and show, e.g., that in Ricci-flat background spacetimes, ANEC and AWEC are satisfied. Furthermore we use our result to show that in the classical situation we have an analogue to the phenomenon of quantum interest. These results lay the foundations for analogous energy inequalities for the quantised non-minimally coupled fields, which will be discussed elsewhere. 
  The Friedman universe is re-examined in a context that is non-standard only in that the properties of matter are postulated in the form of an action principle. Applications to equilibrium configurations of ideal stars have already been reported. In this paper we apply the same theory to a fresh examination of the Friedman universe. The results agree with standard theory in the case of low densities. A suggestion is made to replace "vacuum energy" by "external force". 
  In this thesis, we consider two different problems relevant to general relativity. Over the last few years, opinions on physically relevant singularities occurring in FRW cosmologies have considerably changed. We present an extensive catalogue of such cosmological milestones using generalized power series both at the kinematical and dynamical level. We define the notion of "scale factor singularity" and explore its relation to polynomial and differential curvature singularities. We also extract dynamical information using the Friedmann equations and derive necessary and sufficient conditions for the existence of cosmological milestones such as big bangs, big crunches, big rips, sudden singularities and extremality events. Specifically, we provide a complete characterization of cosmological milestones for which the dominant energy condition is satisfied. The second problem looks at one of the very small number of serious alternatives to the usual concept of an astrophysical black hole, that is, the gravastar model developed by Mazur and Mottola. By considering a generalized class of similar models with continuous pressure (no infinitesimally thin shells) and negative central pressure, we demonstrate that gravastars cannot be perfect fluid spheres: anisotropic pressures are unavoidable. We provide bounds on the necessary anisotropic pressure and show that these transverse stresses that support a gravastar permit a higher compactness than is given by the Buchdahl-Bondi bound for perfect fluid stars. We also comment on the qualitative features of the equation of state that such gravastar-like objects without any horizon must have. 
  Using the ansatz of Matos and N\'{u}\~{n}ez, the present article proposes an algorithm for generating several classes, not all independent, of asymptotically flat rotating wormhole solutions in the Brans-Dicke Theory. The algorithm allows us to associate a real number}$n$ \textit{with} \textit{each static and generated rotating solution. We shall also demonstrate how to match a rotating wormhole to a flat spacetime at the matter boundaries. The vacuum string extensions of the solutions are straightforward. The physical interpretations of the solutions are deferred. 
  We study point symmetries of the Robinson--Trautman equation. The cases of one- and two-dimensional algebras of infinitesimal symmetries are discussed in detail. The corresponding symmetry reductions of the equation are given. Higher dimensional symmetries are shortly discussed. It turns out that all known exact solutions of the Robinson--Trautman equation are symmetric. 
  This Letter considers the generalized second law of gravitational thermodynamics in two scenarios featuring a phantom dominated expansion plus a black hole. The law is violated in both scenarios. 
  We analyze the space-times admitting two shear-free geodesic null congruences. The integrability conditions are presented in a plain tensorial way as equations on the volume element $U$ of the time-like 2--plane that these directions define. From these we easily deduce significant consequences. We obtain explicit expressions for the Ricci and Weyl tensors in terms of $U$ and its first and second order covariant derivatives. We study the different compatible Petrov-Bel types and give the necessary and sufficient conditions that characterize every type in terms of $U$. The type D case is analyzed in detail and we show that every type D space-time admitting a conformal Killing tensor also admits a conformal Killing-Yano tensor. 
  We review recent work on the existence and nature of cosmological singularities that can be formed during the evolution of generic as well as specific cosmological spacetimes in general relativity. We first discuss necessary and sufficient conditions for the existence of geodesically incomplete spacetimes based on a tensorial analysis of the geodesic equations. We then classify the possible singularities of isotropic globally hyperbolic universes using the Bel-Robinson slice energy that closely monitors the asymptotic properties of fields near the singularity. This classification includes all known forms of spacetime singularities in isotropic universes and also predicts new types. 
  The dilaton-gravity sector of the Two Measures Field Theory (TMT) is explored in detail in the context of cosmology. The dilaton \phi dependence of the effective Lagrangian appears only as a result of the spontaneous breakdown of the scale invariance. If no fine tuning is made, the effective \phi-Lagrangian p(\phi,X) depends quadratically upon the kinetic energy X. Hence TMT may represent an explicit example of the effective k-essence resulting from first principles without any exotic term in the fundamental action intended for obtaining this result. Depending of the choice of regions in the parameter space, TMT exhibits different possible outputs for cosmological dynamics: a) Possibility of resolution of the old cosmological constant (CC) problem. From the point of view of TMT, it becomes clear why the old CC problem cannot be solved (without fine tuning) in the conventional field theories (i.e theories with only the measure of integration \sqrt{-g} in the action). b) The power law inflation without any fine tuning can end with damped oscillations of \phi around the state with zero CC. d) There is a broad range of the parameters such that: in the late time universe w=p/\rho <-1 and asymptotically (as t\to\infty) approaches -1 from below; \rho approaches a cosmological constant. The smallness of the CC may be achieved without fine tuning of dimensionfull parameters: either by a seesaw type mechanism or due to a correspondence principle between TMT and conventional field theories. 
  It is believed that soon after the Planck era, space time should have a semi-classical nature. According to this, the escape from General Relativity theory is unavoidable. Two geometric counter-terms are needed to regularize the divergences which come from the expected value. These counter-terms are responsible for a higher derivative metric gravitation. Starobinsky idea was that these higher derivatives could mimic a cosmological constant. In this work it is considered numerical solutions for general Bianchi I anisotropic space-times in this higher derivative theory. The approach is ``experimental'' in the sense that there is no attempt to an analytical investigation of the results. It is shown that for zero cosmological constant $\Lambda=0$, there are sets of initial conditions which form basins of attraction that asymptote Minkowski space. The complement of this set of initial conditions form basins which are attracted to some singular solutions. It is also shown, for a cosmological constant $\Lambda> 0$ that there are basins of attraction to a specific de Sitter solution. This result is consistent with Starobinsky's initial idea. The complement of this set also forms basins that are attracted to some type of singular solution. Because the singularity is characterized by curvature scalars, it must be stressed that the basin structure obtained is a topological invariant, i.e., coordinate independent. 
  Graviatom existence conditions have been found. The graviatoms (quantum systems around mini-black-holes) satisfying these conditions contain the following charged particles: the electron, muon, tau lepton, wino, pion and kaon. Electric dipole and quadrupole and gravitational radiations are calculated for the graviatoms and compared with Hawking's mini-hole radiation. 
  Spatially averaged inhomogeneous cosmologies in classical general relativity can be written in the form of effective Friedmann equations with sources that include backreaction terms. In this paper we propose to describe these backreaction terms with the help of a homogeneous scalar field evolving in a potential; we call it the `morphon field'. This new field links classical inhomogeneous cosmologies to scalar field cosmologies, allowing to reinterpret, e.g., quintessence scenarios by routing the physical origin of the scalar field source to inhomogeneities in the Universe. We investigate a one-parameter family of scaling solutions to the backreaction problem. Subcases of these solutions (all without an assumed cosmological constant) include scale-dependent models with Friedmannian kinematics that can mimic the presence of a cosmological constant or a time-dependent cosmological term. We explicitly reconstruct the scalar field potential for the scaling solutions, and discuss those cases that provide a solution to the Dark Energy and coincidence problems. In this approach, Dark Energy emerges from morphon fields, a mechanism that can be understood through the proposed correspondence: the averaged cosmology is characterized by a weak decay (quintessence) or growth (phantom quintessence) of kinematical fluctuations, fed by `curvature energy' that is stored in the averaged 3-Ricci curvature. We find that the late-time trajectories of those models approach attractors that lie in the future of a state that is predicted by observational constraints. 
  Based on the algebraico-categorical (:sheaf-theoretic and sheaf cohomological) conceptual and technical machinery of Abstract Differential Geometry, a new, genuinely background spacetime manifold independent, field quantization scenario for vacuum Einstein gravity and free Yang-Mills theories is introduced. The scheme is coined `third quantization' and, although it formally appears to follow a canonical route, it is fully covariant, because it is an expressly functorial `procedure'. Various current and future Quantum Gravity research issues are discussed under the light of 3rd-quantization. A postscript gives a brief account of this author's personal encounters with Rafael Sorkin and his work. 
  We use the M{\o}ller energy-momentum complex both in general relativity and teleparallel gravity to evaluate energy distribution (due to matter plus fields including gravity) in the dyadosphere region for Reissner-Nordstr{\"o}m black hole. We found the same and acceptable energy distribution in these different approaches of the M{\o}ller energy-momentum complex. Our teleparallel gravitational result is also independent of the teleparallel dimensionless coupling constant, which means that it is valid in any teleparallel model. This paper sustains (a) the importance of the energy-momentum definitions in the evaluation of the energy distribution of a given space-time and (b) the viewpoint of Lessner that the M{\o}ller energy-momentum complex is a powerful concept for energy and momentum. 
  We describe a general approach to the correspondence of ZM theory with quantum electrodynamics. As a first step, we show the correspondence of helical clock-field states with plane wave states of the Dirac equation. Specifically, defining the direction of time as the gradient of the field in the combined field and space dimensions, for constant gradients, results in states that are consistent with conventional plane wave Dirac equation eigenfunctions. Particles and antiparticles, as well as up and down spins are related by axis inversions. The Dirac wavefunction represents the clock-field and the observer defined direction of time and spatial coordinate axes rotation relative to the region of observation. Additional steps showing the correspondence of ZM theory to quantum mechanics and the Dirac equation are deferred to subsequent papers. 
  We show that the cosmological constant at late time places a bound on the entropy of microwave background radiation deposited in the future event horizon of a given observer, $S\leq S_{\Lambda_0}^{3/4}$. This bound is independent of the energy scale of reheating and the FRW evolution after reheating. We also discuss why the entropy of microwave background in our observable universe has its present value. 
  We use covariant and first-order formalism techniques to study the properties of general relativistic cosmology in three dimensions. The covariant approach provides an irreducible decomposition of the relativistic equations, which allows for a mathematically compact and physically transparent description of the 3-dimensional spacetimes. Using this information we review the features of homogeneous and isotropic 3-d cosmologies, provide a number of new solutions and study gauge invariant perturbations around them. The first-order formalism is then used to provide a detailed study of the most general 3-d spacetimes containing perfect-fluid matter. Assuming the material content to be dust with comoving spatial 2-velocities, we find the general solution of the Einstein equations with non-zero (and zero) cosmological constant and generalise known solutions of Kriele and the 3-d counterparts of the Szekeres solutions. In the case of a non-comoving dust fluid we find the general solution in the case of one non-zero fluid velocity component. We consider the asymptotic behaviour of the families of 3-d cosmologies with rotation and shear and analyse their singular structure. We also provide the general solution for cosmologies with one spacelike Killing vector, find solutions for cosmologies containing scalar fields and identify all the PP-wave 2+1 spacetimes. 
  Interactions between outgoing Hawking particles and ingoing matter are determined by gravitational forces and Standard Model interactions. In particular the gravitational interactions are responsible for the unitarity of the scattering against the horizon, as dictated by the holographic principle, but the Standard Model interactions also contribute, and understanding their effects is an important first step towards a complete understanding of the horizon's dynamics. The relation between in- and outgoing states is described in terms of an operator algebra. In this contribution, in which earlier results are rederived and elaborated upon, we first describe the algebra induced on the horizon by U(1) vector fields and scalar fields, including the case of an Englert-Brout-Higgs mechanism, and a more careful consideration of the transverse vector field components. We demonstrate that, unlike classical black holes, the quantized black hole has on its horizon an imprint of its (recent) past history, i.e., quantum hair. The relation between in- and outgoing states depends on this imprint. As a first step towards the inclusion of non-Abelian interactions, we then compute the effects of magnetic monopoles both in the in-states and in the out-states. They completely modify, and indeed simplify, our algebra. 
  We study a system of differential equations that governs the distribution of matter in the theory of General Relativity. The new element in this paper is the use of a dynamical action principle that includes all the degrees of freedom, matter as well as metric. The matter lagrangian defines a relativistic version of non-viscous, isentropic hydrodynamics. The matter fields are a scalar density and a velocity potential; the conventional, four-vector velocity field is replaced by the gradient of the potential and its scale is fixed by one of the eulerian equations of motion, an innovation that significantly affects the imposition of boundary conditions. If the density is integrable at infinity, then the metric approaches the Schwarzschild metric at large distances. There are stars without boundary and with finite total mass; the metric shows rapid variation in the neighbourhood of the Schwarzschild radius and there is a very small core where a singularity indicates that the gas laws break down. For stars with boundary there emerges a new, critical relation between the radius and the gravitational mass, a consequence of the stronger boundary conditions. Tentative applications are suggested, to certain Red Giants, and to neutron stars, but the investigation reported here was limited to polytropic equations of state. Comparison with the results of Oppenheimer and Volkoff on neutron cores shows a close agreement of numerical results. However, in the model the boundary of the star is fixed uniquely by the required matching of the interior metric to the external Schwarzschild metric, which is not the case in the traditional approach. 
  Using the Einstein and Bergmann-Thomson prescriptions, the energy and momentum distributions for the Bianchi type-V bulk viscous space-time are evaluated in both general relativity and the teleparallel gravity (the tetrad theory of gravity). It is shown that for the Bianchi type-V bulk viscous solution, the energy and momentum due to matter and fields including gravity are the same in both the methods used. This paper indicates an important point that these energy-momentum definitions agree with each other not only in general relativity but also in teleparallel gravity and sustains the results obtained by some physicist who show that the energy-momentum definitions of Einstein, Landau-Lifshitz, Papapetrou, Weinberg, Penrose and Bergmann-Thomson complexes give the same energy expression in general relativity. 
  We demonstrate that the recent measurements of the angular diameter distance of 38 cluster of galaxies using Chandra X-ray data and radio observations from the OVRO and BIMA interferometric arrays place new and independent constraints on deviations in the duality relation between angular and luminosity distances. Using only cluster data, we found that the ratio between the two distances defined as $\eta = D_L/D_A(1+z)^2.$ is bound to be $\eta=0.97\pm0.03$ at 68% c.l. with no evidence for distance duality violation. Comparing the cluster angular diameter distance data with luminosity distance data from type Ia Supernovae, we obtain the model independent constraint $\eta=1.01\pm0.07$ at 68% c.l.. Those results provide an useful check for the cosmological concordance model and for the presence of systematics in SN-Ia and clusters data. 
  We investigate the possibility of constraining Chaplygin dark energy models with current Integrated Sachs Wolfe effect data. In the case of a flat universe we found that generalized Chaplygin gas models must have an energy density such that $\Omega_c >0.55$ and an equation of state $w <-0.6$ at 95% c.l.. We also investigate the recently proposed Silent Chaplygin models, constraining $\Omega_c >0.55$ and $w <-0.65$ at 95% c.l.. Better measurements of the CMB-LSS correlation will be possible with the next generation of deep redshift surveys. This will provide independent and complementary constraints on unified dark energy models such as the Chaplygin gas. 
  We discuss inelastic collisions of two rotating disks by using the conservation laws for baryonic mass and angular momentum. In particular, we formulate conditions for the formation of a new disk after the collision and calculate the total energy loss to obtain upper limits for the emitted gravitational energy. 
  Non-perturbative quantum geometric effects in Loop Quantum Cosmology predict a $\rho^2$ modification to the Friedmann equation at high energies. The quadratic term is negative definite and can lead to generic bounces when the matter energy density becomes equal to a critical value of the order of the Planck density. The non-singular bounce is achieved for arbitrary matter without violation of positive energy conditions. By performing a qualitative analysis we explore the nature of the bounce for inflationary and Cyclic model potentials. For the former we show that inflationary trajectories are attractors of the dynamics after the bounce implying that inflation can be harmoniously embedded in LQC. For the latter difficulties associated with singularities in cyclic models can be overcome. We show that non-singular cyclic models can be constructed with a small variation in the original Cyclic model potential by making it slightly positive in the regime where scalar field is negative. 
  Canonical Hamiltonian field theory in curved spacetime is formulated in a manifestly covariant way. Second quantization is achieved invoking a correspondence principle between the Poisson bracket of classical fields and the commutator of the corresponding quantum operators. The Dirac theory is investigated and it is shown that, in contrast to the case of bosonic fields, in curved spacetime, the field momentum does not coincide with the generators of spacetime translations. The reason is traced back to the presence of second class constraints occurring in Dirac theory. Further, it is shown that the modification of the Dirac Lagrangian by a surface term leads to a momentum transfer between the Dirac field and the gravitational background field, resulting in a theory that is free of constraints, but not manifestly hermitian. 
  In this note we study the adiabatic perturbation spectrum of N-flation with power law potential. We show that the scalar spectrum of N-flation is generally redder than that of its corresponding single field. The result obtained for that with unequal massive fields is consistent with the recent numerical investigation of Kim and Liddle. 
  The Dirac electron theory and QED do not take into account gravitational field, while the corresponding Kerr-Newman solution with parameters of electron has very strong stringy, topological and non-local action on the Compton distances, polarizing space-time and deforming the Coulomb field. We discuss the relation of the electron to the Kerr's microgeon model and argue that the Kerr geometry may be hidden beyond the Quantum Theory. In particular, we show that the Foldi-Wouthuysen `mean-position' operator of the Dirac electron is related to a complex representation of the Kerr geometry, and to a complex stringy source. Therefore, the complex Kerr geometry may be hidden beyond the Dirac equation. 
  We study first the Hamiltonian operator H corresponding to the Fock-Weyl extension of the Dirac equation to gravitation. When searching for stationary solutions to this equation, in a static metric, we show that just one invariant Hermitian product appears natural. In the case of a space-isotropic metric, H is Hermitian for that product. Then we investigate the asymptotic post-Newtonian approximation of the stationary Schroedinger equation associated with H, for a slow particle in a weak-field static metric. We rewrite the expanded equations as one equation for a two-component spinor field. This equation contains just the non-relativistic Schroedinger equation in the gravity potential, plus correction terms. Those "correction" terms are of the same order in the small parameter as the "main" terms, but are numerically negligible in the case of ultra-cold neutrons in the Earth's gravity. 
  We construct approximate analytical solutions to the constraint equations of general relativity for binary black holes of arbitrary mass ratio in quasicircular orbit. We adopt the puncture method to solve the constraint equations in the transverse-traceless decomposition and consider perturbations of Schwarzschild black holes caused by boosts and the presence of a binary companion. A superposition of these two perturbations then yields approximate, but fully analytic binary black hole initial data that are accurate to first order in the inverse of the binary separation and the square of the black holes' momenta. 
  The time-redshift relation of Carmeli et al. differs from that of the standard flat LambdaCDM model by more than 500 million years for 1 < z < 4.5. 
  Bekenstein's theory of relativistic gravity is conventionally written as a bi-metric theory. The two metrics are related by a disformal transformation defined by a dynamical vector field and a scalar field. In this comment we show that the theory can be re-written as Vector-Tensor theory akin to Einstein-Aether theories with non-canonical kinetic terms. We discuss some of the implications of this equivalence. 
  In a Kaluza-Klein space-time $V^{4}\otimes S^{3}$, we demonstrate that the dimensional reduction of spinors provides a 4-field, whose associated SU(2) gauge connections are geometrized. However, additional and gauge-violating terms arise, but they are highly suppressed by a factor $\beta$, which fixes the amount of the spinor dependence on extra-coordinates. The application of this framework to the Electro-Weak model is performed, thus giving a lower bound for $\beta$ from the request of the electric charge conservation. Moreover, we emphasize that also the Higgs sector can be reproduced, but neutrino masses are predicted and the fine-tuning on the Higgs parameters can be explained, too. 
  The braneworld model of Dvali-Gabadadze-Porrati (DGP) provides an interesting alternative to a positive cosmological constant by modifying gravity at large distances. We investigate the asymptotic behavior of homogeneous and anisotropic cosmologies on the DGP brane. It is shown that all Bianchi models except type IX isotropize, as in general relativity, if the so called $E_{\mu\nu}$ term satisfies some energy condition. Isotropization can proceed slower in DGP gravity than in general relativity. 
  We write out the geodesic deviations that take place in a $d\geq 4$ dimensional brane world subspace of a higher dimensional spacetime by splitting out the brane and the extra space dynamical quantities from a global metric spacetime of dimension $D\geq 5$. The higher dimensional dynamical quantities are projected onto two orthogonal subspaces, where one of which is identified with a $(d-1)$-brane. This is done by using some technics of the conventional submanifold theory of the Riemannian geometry, applied to pseudo-Riemannian spaces. Using the splitting technic, we obtain the tidal field on $(d-1)$ branes with an arbitrary number of non compact extra dimensions. Later, we analise the geodesic deviations seen by an ordinary observer in a $d=4$ dimensional spacetime and show that deviations from general relativity tidal field due to the existence of the extra dimensions can appear because, (i) - the dependence of the indunced metric on the brane with the extra coordinates and (ii) - deviations of the higher dimensional spacetime metric from spherical symmetry. 
  Some examples of ten-dimensional vacuum Einstein spaces made up on basis of four-dimensional Ricci-flat spaces and six-dimensional Ricci-flat spaces defined by solutions of the Sin-Gordon equation are constructed. The properties of geodesics for such type of the spaces are discussed 
  This paper introduces some general properties of the gravitational metric and the natural basis of vectors and covectors in 4-dimensional emission coordinates. Emission coordinates are a class of space-time coordinates defined and generated by 4 emitters (satellites) broadcasting their proper time by means of electromagnetic signals. They are a constitutive ingredient of the simplest conceivable relativistic positioning systems. Their study is aimed to develop a theory of these positioning systems, based on the framework and concepts of general relativity, as opposed to introducing `relativistic effects' in a classical framework. In particular, we characterize the causal character of the coordinate vectors, covectors and 2-planes, which are of an unusual type. We obtain the inequality conditions for the contravariant metric to be Lorentzian, and the non-trivial and unexpected identities satisfied by the angles formed by each pair of natural vectors. We also prove that the metric can be naturally split in such a way that there appear 2 parameters (scalar functions) dependent exclusively on the trajectory of the emitters, hence independent of the time broadcast, and 4 parameters, one for each emitter, scaling linearly with the time broadcast by the corresponding satellite, hence independent of the others. 
  Two classic field theories of metric gravitation are given as constant-coefficient Exterior Differential Systems (EDS) on the flat orthonormal frame bundle over ten dimensional space. They are derivable by variation of Cartan 4-forms, and shown to be well-posed by calculation of their Cartan characteristic integers. Their solutions are embedded Riemannian 4-spaces. The first theory is generated by torsion 2-forms and Ricci-flat 3-forms and is a constant-coefficient EDS for vacuum tetrad gravity; its Cartan character table is the same as found for an EDS recently given in terms of tetrad frame and connection variables [1] [2]. The second constant-coefficient EDS is generated solely by 2-forms, and has a Cartan form of quadratic Yang-Mills type. Its solutions lie in torsion free 6-spaces and are fibered over 3-spaces. We conjecture that these solutions may be classically related to 10-dimensional quantum field theoretic constructions of cosmological vacua [3]. 
  In this paper we consider a model universe with large extra dimensions to obtain a modified black hole entropy-area relation. We use the generalized uncertainty principle to find a relation between the number of spacetime dimensions and the presence or vanishing of logarithmic prefactor in the black hole entropy-area relation. Our calculations are restricted to the microcanonical ensembles and we show that in the modified entropy-area relation, the microcanonical logarithmic prefactor appears only when spacetime has an even number of dimensions. 
  We show that a previously proposed cosmological model based on general relativity with non vanishing divergence for the energy-momentum tensor is consistent with the observed values for the nucleosynthesis of helium for some values of the arbitrary parameter $\alpha$ presented in this model. Further more values of $\alpha$ can be accommodated if we adopt the Randall-Sundrum single brane model. 
  We address the question how to adapt cosmological constant $\Lambda$ for description of a vacuum dark energy density jumping from the big initial value to the small today value suggested by observations. We find such a possibility in the gauge-noninvariance of quantum cosmology which leads to a connection between a choice of the gauge and quantum spectrum for a certain physical quantity which can be specified in the framework of the minisuperspace model. We introduce a particular gauge in which the cosmological constant $\Lambda$ is quantized and show that making a measurement of $\Lambda$ today one can find its small value with the biggest probability, while at the beginning of the evolution, the biggest probability corresponds to its biggest value. Transitions between quantum levels of $\Lambda$ in the course of the Universe evolution, could be related to several scales for symmetry breaking. 
  The metric of some Lorentzian wormholes in the background of the FRW universe is obtained. It is shown that for a de Sitter space-time the new solution is supported by Phantom Energy. The wave equation for a scalar field in such backgrounds is separable. The form of the potential for the Schr\"{o}dinger type one dimensional wave equation is found. 
  We give an alternative proof of the completeness of the Chandrasekhar ansatz for the Dirac equation in the Kerr-Newman metric. Based on this, we derive an integral representation for smooth compactly supported functions which in turn we use to derive an integral representation for the propagator of solutions of the Cauchy problem with initial data in the above class of functions. As a by-product, we also obtain the propagator for the Dirac equation in the Minkowski space-time in oblate spheroidal coordinates. 
  Starting with the Dirac equation outside the event horizon of a non-extreme Kerr black hole, we develop a time-dependent scattering theory for massive Dirac particles. The explicit computation of the modified wave operators at infinity is done by implementing a time-dependent logarithmic phase shift from the free dynamics to offset the long range term in the full Hamiltonian due to the presence of the gravitational force. Analytical expressions for the wave operators are also given. 
  A study covering some aspects of the Einstein--Rosen metric is presented. The electric and magnetic parts of the Weyl tensor are calculated. It is shown that there are no purely magnetic E--R spacetimes, and also that a purely electric E--R spacetime is necessarily static. The geodesics equations are found and circular ones are analyzed in detail. The super--Poynting and the ``Lagrangian'' Poynting vectors are calculated and their expressions are found for two specific examples. It is shown that for a pulse--type solution, both expressions describe an inward radially directed flow of energy, far behind the wave front. The physical significance of such an effect is discussed. 
  This paper is concerned with the initial-boundary value problem for the Einstein equations in a first-order generalized harmonic formulation. We impose boundary conditions that preserve the constraints and control the incoming gravitational radiation by prescribing data for the incoming fields of the Weyl tensor. High-frequency perturbations about any given spacetime (including a shift vector with subluminal normal component) are analyzed using the Fourier-Laplace technique. We show that the system is boundary-stable. In addition, we develop a criterion that can be used to detect weak instabilities with polynomial time dependence, and we show that our system does not suffer from such instabilities. A numerical robust stability test supports our claim that the initial-boundary value problem is most likely to be well-posed even if nonzero initial and source data are included. 
  We present a solution to the vacuum Einstein Equations which represents a collapse of a gravitational wave in 5 dimensions. Depending on the focal length of the wave, the collapse results, either in a black string covered by a horizon, or in a naked singularity which can be removed. 
  Utilizing the ADM equations, we derive a metric and reduced field equations describing a general, spherically symmetric perfect fluid. The metric describes both the interior perfect fluid region and exterior vacuum Schwarzschild spacetime in a single coordinate patch. The exterior spacetime is in generalized Painleve-Gullstrand coordinates which is an infinite class of coordinate systems. In the static limit the system reduces to a Tolman-Oppenheimer-Volkoff equation on the interior with the exterior in Schwarzschild coordinates. We show the coordinate transformation for the non-static cases to comoving coordinates, where the metric is seen to be a direct generalization of the Lemaitre-Tolman-Bondi spacetime to include pressures. 
  We study cosmological perturbations in the brane models with an induced Einstein-Hilbert term on a brane. We consider an inflaton confined to a de Sitter brane in a five-dimensional Minkowski spacetime. Inflaton fluctuations excite Kaluza-Klein modes of bulk metric perturbations with mass $m^2 = -2(2\ell-1) (\ell +1) H^2$ and $m^2 = -2\ell(2\ell+3) H^2$ where $\ell$ is an integer. There are two branches ($\pm$ branches) of solutions for the background spacetime. In the $+$ branch, which includes the self-accelerating universe, a resonance appears for a mode with $m^2 = 2 H^2$ due to a spin-0 perturbation with $m^2 = 2H^2$. The self-accelerating universe has a distinct feature because there is also a helicity-0 mode of spin-2 perturbations with $m^2 = 2H^2$. In the $-$ branch, which can be thought as the Randall-Sundrum type brane-world with the high energy quantum corrections, there is no resonance. At high energies, we analytically confirm that four-dimensional Einstein gravity is recovered, which is related to the disappearance of van Dam-Veltman-Zakharov discontinuity in de Sitter spacetime.  On sufficiently small scales, we confirm that the lineariaed gravity on the brane is well described by the Brans-Dicke theory with $\omega=3Hr_c$ in $-$ branch and $\omega = -3H r_c$ in $+$ branch, respectively, which confirms the existence of the ghost in $+$ branch. We also study large scale perturbations. In $+$ branch, the resonance induces a non-trivial anisotropic stress on the brane via the projection of Weyl tensor in the bulk, but no instability is shown to exist on the brane. 
  The new scale-covariant formulation of the Dirac's Large Number Hypothesis (LNH) is proposed. The basic equations of LNH are formulated in the scale-covariant and "G-invariant" (invariant on the transformation law for G) form. On the basis of the Scale Covariant Theory of Gravitation and Dirac's LNH the cosmological model is constructed that gives as result the closed static Einstein's Universe in the gravitational system of units (g.s.u.) and the closed expanding Universe in the atomic system of units (a.s.u.). The simple dynamical model of atomic clock is proposed that leads to the same connection between the a.s.u. and the g.s.u. as the LNH and allows to specify the parameters of the model. The question about the equivalence of inertial and gravitational masses in the arbitrary system of units is ivestigated. It is deduced that the equivalence of inertial and gravitational masses in the arbitrary system of units takes place only for the unique transformation law for G even if the equivalence of two kinds of mass in the g.s.u. is assumed. It is shown that the model predicts the conservation of the black-body spectrum of the microvave background radiation independently of choice of the transformation law for G. It is argued that the some paradoxes of the standard cosmology can be overcomed in the given model without the supposition about the existence in the past of a phase of inflationary expansion. 
  Suppose General Relativity, provocatively governed by a dimensional coupling constant, is a spontaneously induced theory of Gravity. Invoking Zee's mechanism, we represent the reciprocal Newton constant by a Brans Dicke scalar field, and let it damped oscillating towards its General Relativistic VEV. The corresponding cosmological evolution, in the Jordan frame, averagely resembles the familiar dark radiation -> dark matter -> dark energy domination sequence. The fingerprints of the theory are fine ripples, hopefully testable, in the FRW scale factor; they die away at the strict General Relativity limit. Also derived is the spherically symmetric static configuration associated with spontaneously induced General Relativity. At the stiff scalar potential limit, the exterior Schwarzschild solution is recovered. However, due to level crossing at the would have been horizon, it now connects with a novel dark core characterized by a locally varying Newton constant. The theory further predicts light Einstein-style gravitational corpuscles (elementary particles?) which become point-like at the GR-limit. 
  We present a new method for the calculation of black hole perturbations induced by extended sources in which the solution of the nonlinear hydrodynamics equations is coupled to a perturbative method based on Regge-Wheeler/Zerilli and Bardeen-Press-Teukolsky equations when these are solved in the frequency domain. In contrast to alternative methods in the time domain which may be unstable for rotating black-hole spacetimes, this approach is expected to be stable as long as an accurate evolution of the matter sources is possible. Hence, it could be used under generic conditions and also with sources coming from three-dimensional numerical relativity codes. As an application of this method we compute the gravitational radiation from an oscillating high-density torus orbiting around a Schwarzschild black hole and show that our method is remarkably accurate, capturing both the basic quadrupolar emission of the torus and the excited emission of the black hole. 
  A new six-parametric, axisymmetric and asymptotically flat exact solution of Einstein-Maxwell field equations having reflection symmetry is presented. It has arbitrary physical parameters of mass, angular momentum, mass--quadrupole moment, current octupole moment, electric charge and magnetic dipole, so it can represent the exterior field of a rotating, deformed, magnetized and charged object; some properties of the closed-form analytic solution such as its multipolar structure, electromagnetic fields and singularities are also presented. In the vacuum case, this analytic solution is matched to some numerical interior solutions representing neutron stars, calculated by Berti & Stergioulas (Mon. Not. Roy. Astron. Soc. 350, 1416 (2004)), imposing that the multipole moments be the same. As an independent test of accuracy of the solution to describe exterior fields of neutron stars, we present an extensive comparison of the radii of innermost stable circular orbits (ISCOs) obtained from Berti & Stergioulas numerical solutions, Kerr solution (Phys. Rev. Lett. 11, 237 (1963)), Hartle & Thorne solution (Ap. J. 153, 807, (1968)), an analytic series expansion derived by Shibata & Sasaki (Phys. Rev. D. 58 104011 (1998)) and, our exact solution. We found that radii of ISCOs from our solution fits better than others with realistic numerical interior solutions. 
  A general paradigm for describing classical (and semiclassical) gravity is presented. This approach brings to the centre-stage a holographic relationship between the bulk and surface terms in a general class of action functionals and provides a deeper insight into several aspects of classical gravity which have no explanation in the conventional approach. After highlighting a series of unresolved issues in the conventional approach to gravity, I show that (i) principle of equivalence, (ii) general covariance and (iii)a reasonable condition on the variation of the action functional, suggest a generic Lagrangian for semiclassical gravity of the form $L=Q_a^{bcd}R^a_{bcd}$ with $\nabla_b Q_a^{bcd}=0$. The expansion of $Q_a^{bcd}$ in terms of the derivatives of the metric tensor determines the structure of the theory uniquely. The zeroth order term gives the Einstein-Hilbert action and the first order correction is given by the Gauss-Bonnet action. Any such Lagrangian can be decomposed into a surface and bulk terms which are related holographically. The equations of motion can be obtained purely from a surface term in the gravity sector. Hence the field equations are invariant under the transformation $T_{ab} \to T_{ab} + \lambda g_{ab}$ and gravity does not respond to the changes in the bulk vacuum energy density. The cosmological constant arises as an integration constant in this approach. The implications are discussed. 
  The Einstein--Cartan Theory (ECT) of gravity is a modification of General Relativity Theory (GRT), allowing space-time to have torsion, in addition to curvature, and relating torsion to the density of intrinsic angular momentum. This modification was put forward in 1922 by Elie Cartan, before the discovery of spin. Cartan was influenced by the work of the Cosserat brothers (1909), who considered besides an (asymmetric) force stress tensor also a moments stress tensor in a suitably generalized continuous medium. 
  A proposal is made to test Newton's inverse-square law using the perihelion shift of test masses (planets) in free fall within a spacecraft located at the Earth-Sun L2 point. Such an Artificial Planetary System In Space (APSIS) will operate in a drag-free environment with controlled experimental conditions and minimal interference from terrestrial sources of contamination. We demonstrate that such a space experiment can probe the presence of a "hidden" fifth dimension on the scale of a micron, if the perihelion shift of a "planet" can be measured to sub-arc-second accuracy. Some suggestions for spacecraft design are made. 
  We prove upper bounds on angular momentum and centre of mass in terms of the Hamiltonian mass and cosmological constant for non-singular asymptotically anti-de Sitter initial data sets satisfying the dominant energy condition. We work in all space-dimensions larger than or equal to three, and allow a large class of asymptotic backgrounds, with spherical and non-spherical conformal infinities; in the latter case, a spin-structure compatibility condition is imposed. We give a large class of non-trivial examples saturating the inequality. We analyse exhaustively the borderline case in space-time dimension four: for spherical cross-sections of Scri, equality together with completeness occurs only in anti-de Sitter space-time. On the other hand, in the toroidal case, regular non-trivial initial data sets saturating the bound exist. 
  The various entropy bounds that exist in the literature suggest that spacetime is fundamentally discrete, and hint at an underlying relationship between geometry and "information". The foundation of this relationship is yet to be uncovered, but should manifest itself in a theory of quantum gravity. We present a measure for the maximal entropy of spherically symmetric spacelike regions within the causal set approach to quantum gravity. In terms of the proposal, a bound for the entropy contained in this region can be derived from a counting of potential "degrees of freedom" associated to the Cauchy horizon of its future domain of dependence. For different spherically symmetric spacelike regions in Minkowski spacetime of arbitrary dimension, we show that this proposal leads, in the continuum approximation, to Susskind's well-known spherical entropy bound. 
  In this paper, we extend Parikh' work to the non-stationary black hole. As an example of the non-stationary black hole, we study the tunnelling effect and Hawking radiation from a Vaidya black hole whose Bondi mass is identical to its mass parameter. We view Hawking radiation as a tunnelling process across the event horizon and calculate the tunnelling probability. We find that the result is different from Parikh's work because $\frac{dr_{H}}{dv}$ is the function of Bondi mass m(v). 
  We analyse within first-order perturbation theory the instantaneous transition rate of an accelerated Unruh-DeWitt particle detector whose coupling to a massless scalar field on four-dimensional Minkowski space is regularised by a spatial profile. For the Lorentzian profile introduced by Schlicht, the zero size limit is computed explicitly and expressed as a manifestly finite integral formula that no longer involves regulators or limits. The same transition rate is obtained for an arbitrary profile of compact support under a modified definition of spatial smearing. Consequences for the asymptotic behaviour of the transition rate are discussed. A number of stationary and nonstationary trajectories are analysed, recovering in particular the Planckian spectrum for uniform acceleration. 
  We give a general procedure to obtain non perturbative evolution operators in closed form for quantized linearly polarized two Killing vector reductions of general relativity with a cosmological interpretation. We study the representation of these operators in Fock spaces and discuss in detail the conditions leading to unitary evolutions. 
  We present a polymer(loop) quantization of a two dimensional theory of dilatonic gravity known as the CGHS model. We recast the theory as a parametrized free field theory on a flat 2-dimensional spacetime and quantize the resulting phase space using techniques of loop quantization. The resulting (kinematical) Hilbert space admits a unitary representation of the spacetime diffeomorphism group. We obtain the complete spectrum of the theory using a technique known as group averaging and perform quantization of Dirac observables on the resulting Hilbert space. We argue that the algebra of Dirac observables gets deformed in the quantum theory. Combining the ideas from parametrized field theory with certain relational observables, evolution is defined in the quantum theory in the Heisenberg picture. Finally the dilaton field is quantized on the physical Hilbert space which carries information about quantum geometry. 
  From the viewpoint of local quantum field theory, this letter investigates the high-order corrections to the holographic entropy bound. As a result, the logarithmic correction term appears naturally with the definite coefficient $-{1/2}$, which thus provides another semi-classical platform to examine such candidates for quantum gravity as Loop Quantum Gravity and String Theory. 
  We obtain rotating anisotropic fluids starting with any vacuum stationary axisymmetric metric. With the help of the Ernst method, the basic equations are derived, together with the expression for the energy-momentum tensor and with the equation of state compatible with the field equations. In principle, we can obtain source matter satisfying all three energy conditions, provided that the parameters of the solutions are chosen appropriately. Further, the method is presented by using different coordinate systems: the cylindrical coordinates $\rho,z$, the quasi-spherical coordinates and the oblate spheroidal ones. Moreover, we study the energy conditions when matching conditions with an exterior solution are considered. Finally, a class of interior solutions in oblate spheroidal coordinates is found matching with any stationary axisymmetric asymptotically flat vacuum solution. 
  We present an investigation of nonlinear interactions between Gravitational Radiation and modified Alfv\'{e}n modes in astrophysical dusty plasmas. Assuming that stationary charged dust grains form neutralizing background in an electron-ion-dust plasma, we obtain the three wave coupling coefficients, and calculate the growth rates for parametrically coupled gravitational radiation and modified Alfv\'{e}n-Rao modes. The threshold value of the gravitational wave amplitude associated with convective stabilization is particularly small if the gravitational frequency is close to twice the modified Alfv\'en wave-frequency. The implication of our results to astrophysical dusty plasmas is discussed. 
  We discuss the status of both cosmological and black hole type singularities in the framework of the brane-world model of gravity. We point out that the Big Bang is not properly understood yet. We also show new features of the gravitational collapse on the brane, the most important being the production of dark energy during the collapse. 
  We construct a macroscopic semiclassical state state for a quantum tetrahedron. The expectation values of the geometrical operators representing the volume, areas and dihedral angles are peaked around assigned classical values, with vanishing relative uncertainties. 
  Tetrads are introduced in order to study the relationship between gravity and particle interactions, specially in weak processes at low energy. Through several examples like inverse Muon decay, elastic Neutrino-Electron scattering, it is explicitly shown how to assign to each vertex of the corresponding low-order Feynman diagram in a weak interaction, a particular set of tetrad vectors. The relationship between the tetrads associated to different vertices is exhibited explicitly to be generated by a SU(2) local gauge transformation. 
  Using some supernovae and CMB data, we constrain the Cardassian, Randall-Sundrum, and Dvali-Gabadadze-Porrati brane-inspired cosmological models. We show that a transient acceleration and an early loitering period are usually excluded by the data. Moreover, the three models are equivalent to some usual quintessence/ghost dark energy models defined by a barotropic index $\gamma_\phi$ depending on the redshift. We calculate this index for each model and show that they mimic a universe close to a $\Lambda CDM$ model today. 
  The general relativistic notion of gravitational and inertial mass is discussed from the general viewpoint of the tidal forces implicit in the curvature and the Einstein field equations within ponderable matter. A simple yet rigorously general derivation is given for the Tolman gravitational mass viewpoint wherein the computation of gravitational mass requires both a rest energy contribution (the inertial mass) and a pressure contribution. The pressure contribution is extremely small under normal conditions which implies the equality of gravitational and inertial mass to a high degree of accuracy. However, the pressure contribution is substantial for conformal symmetric systems such as Maxwell radiation, whose constituent photons are massless. Implications of the Tolman mass for standard cosmology and standard high energy particle physics models are briefly explored. 
  Constructing a classical mechanical system associated with a given quantum mechanical one, entails construction of a classical phase space and a corresponding Hamiltonian function from the available quantum structures and a notion of coarser observations. The Hilbert space of any quantum mechanical system naturally has the structure of an infinite dimensional symplectic manifold (`quantum phase space'). There is also a systematic, quotienting procedure which imparts a bundle structure to the quantum phase space and extracts a classical phase space as the base space. This works straight forwardly when the Hilbert space carries weakly continuous representation of the Heisenberg group and recovers the linear classical phase space $\mathbb{R}^{\mathrm{2N}}$. We report on how the procedure also allows extraction of non-linear classical phase spaces and illustrate it for Hilbert spaces being finite dimensional (spin-j systems), infinite dimensional but separable (particle on a circle) and infinite dimensional but non-separable (Polymer quantization). To construct a corresponding classical dynamics, one needs to choose a suitable section and identify an effective Hamiltonian. The effective dynamics mirrors the quantum dynamics provided the section satisfies conditions of semiclassicality and tangentiality. 
  We present a new numerical code developed for the evolution of binary black-hole spacetimes using different initial data and evolution techniques. The code is demonstrated to produce state-of-the-art simulations of orbiting and inspiralling black-hole binaries with convergent waveforms. We also present the first detailed study of the dependence of gravitational waveforms resulting from three-dimensional evolutions of different types of initial data. For this purpose we compare the waveforms generated by head-on collisions of superposed Kerr-Schild data with those of Brill-Lindquist data over a wide range of initial separations. 
  In this study, in context of general relativity we consider Einstein, Bergmann-Thomson, Moller and Landau-Lifshitz energy momentum definitions and we compute the total energy distribution (due to matter and fields including gravitation) of the universe based on Szekeres class I and class II space-times. We show that Einstein and Bergmann-Thomson definitions of the energy-momentum complexes give the same results, while Moller's and Landau-Lifshitz's energy-momentum definition does not provide same results for Szekeres class II space. The definitions of Einstein, Bergmann-Thomson and Moller definitions of the energy-momentum complexes give similar results in Szekeres class I space time. 
  We give the correct expressions for the spin network evaluations proposed in Class. Quant. Grav. 21 (2004) 3909 as the coefficients of the quantum gravity vacuum wavefunction in the spin network basis. 
  Non-minimal actions with matter represented by a scalar field coupled to gravity are considered in the context of a homogeneous and isotropic universe. The coupling is of the form $-\xi/2 \phi^2 R$. The possibility of successful inflation is investigated taking into account features of loop cosmology. For that end a conformal transformation is performed. That brings the theory into the standard minimally coupled form (Einstein frame) with some effective field and its potential. Both analytical and numerical estimates show that a negative coupling constant is preferable for successful inflation. Moreover, provided fixed initial conditions, larger $|\xi|$ leads to a greater number of {\em e}-folds. The latter is obtained for a reasonable range of initial conditions and the coupling parameter and indicates a possibility for successful inflation. 
  An examples of a Ricci-flat of four-dimensional spaces with a Walker metrics and their generalizations are constructed. The properties of corresponding geodesic equations are discussed. 
  It is known that certain quantum cosmological models present quantum behavior for large scale factors. Since quantization can suppress past singularities, it is natural to inquire whether quantum effects can prevent future singularities. To this end, a Friedmann-Robertson-Walker quantum cosmological model dominated by a phantom energy fluid is investigated. The classical model displays accelerated expansion ending in a Big Rip. The quantization is performed in three different ways, which turn out to lead to the same result, namely that quantum gravitational effects do not, in general, remove the Big Rip. 
  We show how braneworld cosmology with bulk matter can explain structure formation. In this scenario, the nonlocal corrections to the Friedmann equations supply a Weyl fluid that can dominate over matter at late times due to the energy exchange between the brane and the bulk. We demonstrate that the presence of the Weyl fluid radically changes the perturbation equations, which can take care of the fluctuations required to account for the large amount of inhomogeneities observed in the local universe. Further, we show how this Weyl fluid can mimic dark matter. We also investigate the bulk geometry responsible for the scenario. 
  Asymptotic symmetry of the Euclidean 3D gravity with torsion is described by two independent Virasoro algebras with different central charges. Elements of this boundary conformal structure are combined with Cardy's formula to calculate the black hole entropy. 
  The paper presents equations determining the particle spin evolution in the post-Newtonian approximation in the problem of motion of two mass and spin possessing particles. The equations are derived with the Einstein-Infeld-Hoffmann method from the condition of metric tensor symmetry. The consideration uses the condition of coordinate harmonicity and the metric coincidence nearby the particles with expansions of the Kerr solution written in the harmonic coordinates. For gyroscopes on satellites Gravity Probe B, for example, the equations yield a deviation of the axis of revolution, which is, first, two times as small as that obtained by J.L.Anderson in paper gr-qc/0511093 and, second, the deviation is of opposite sense. From the equations it follows that the total angular momentum of the spin particle system, generally speaking, is not conserved, beginning even with the post-Newtonian approximation. The results are briefly discussed. 
  We develop a general formalism for the parameter-space metric of the multi-detector F-statistic, which is a matched-filtering detection statistic for continuous gravitational waves. We find that there exists a whole family of F-statistic metrics, parametrized by the (unknown) amplitude parameters of the gravitational wave. The multi-detector metric is shown to be expressible in terms of noise-weighted averages of single-detector contributions, which implies that the number of templates required to cover the parameter space does not scale with the number of detectors. Contrary to using a longer observation time, combining detectors of similar sensitivity is therefore the computationally cheapest way to improve the sensitivity of coherent wide-parameter searches for continuous gravitational waves.   We explicitly compute the F-statistic metric family for signals from isolated spinning neutron stars, and we numerically evaluate the quality of different metric approximations in a Monte-Carlo study. The metric predictions are tested against the measured mismatches and we identify regimes in which the local metric is no longer a good description of the parameter-space structure. 
  The principal goal of the \emph{LISA Science Analysis Workshop} is to encourage the development and maturation of science analysis technology in preparation for LISA science operations. Exactly because LISA is a pathfinder for a new scientific discipline -- gravitational wave astronomy -- LISA data processing and science analysis methodologies are in their infancy and require considerable maturation if they are to be ready to take advantage of LISA data. Here we offer some thoughts, in anticipation of the LISA Science Analysis Workshop, on analysis research problems that demonstrate the capabilities of different proposed analysis methodologies and, simultaneously, help to push those techniques toward greater maturity. Particular emphasis is placed on formulating questions that can be turned into well-posed problems involving tests run on specific data sets, which can be shared among different groups to enable the comparison of techniques on a well-defined platform. 
  It is shown how states of a quantum mechanical particle in the Schroedinger representation can be approximated by states in the so-called polymer representation. The result may shed some light on the semiclassical limit of loop quantum gravity. 
  We examine the transition of a particle across the singularity of the compactified Milne (CM) space. Quantization of the phase space of a particle and testing the quantum stability of its dynamics are consistent to one another. One type of transition of a quantum particle is described by a quantum state that is continuous at the singularity. It indicates the existence of a deterministic link between the propagation of a particle before and after crossing the singularity. Regularization of the CM space leads to the dynamics similar to the dynamics in the de Sitter space. The CM space is a promising model to describe the cosmological singularity deserving further investigation by making use of strings and membranes. 
  According to general relativity, the gravitomagnetic Lense-Thirring force of Mars would secularly shift the orbital plane of the Mars Global Surveyor (MGS) spacecraft by an amount of 1.5 m, on average, in the cross-track direction over 5 years. The determined cross-track post-fit residuals of MGS, built up by neglecting just the gravitomagnetic force in the dynamical force models and without fitting any empirical cross-track acceleration which could remove the relativistic signal, amount to 1.6 m, on average, over a 5-years time interval spanning from 10 February 2000 to 14 January 2005. The discrepancy with the predictions of general relativity is 6%. 
  Continuing work initiated in an earlier publication [Phys. Rev. D 69, 084007 (2004)], we construct a system of light-cone coordinates based at a geodesic world line of an arbitrary curved spacetime. The construction involves (i) an advanced-time or a retarded-time coordinate that labels past or future light cones centered on the world line, (ii) a radial coordinate that is an affine parameter on the null generators of these light cones, and (iii) angular coordinates that are constant on each generator. The spacetime metric is calculated in the light-cone coordinates, and it is expressed as an expansion in powers of the radial coordinate in terms of the irreducible components of the Riemann tensor evaluated on the world line. The formalism is illustrated in two simple applications, the first involving a comoving world line of a spatially-flat cosmology, the other featuring an observer placed on the axis of symmetry of Melvin's magnetic universe. 
  The geometrical meaning of the Eddington-Finkelstein coordinates of Schwarzschild spacetime is well understood: (i) the advanced-time coordinate v is constant on incoming light cones that converge toward r=0, (ii) the angles theta and phi are constant on the null generators of each light cone, (iii) the radial coordinate r is an affine-parameter distance along each generator, and (iv) r is an areal radius, in the sense that 4 pi r^2 is the area of each two-surface (v,r) = constant. The light-cone gauge of black-hole perturbation theory, which is formulated in this paper, places conditions on a perturbation of the Schwarzschild metric that ensure that properties (i)--(iii) of the coordinates are preserved in the perturbed spacetime. Property (iv) is lost in general, but it is retained in exceptional situations that are identified in this paper. Unlike other popular choices of gauge, the light-cone gauge produces a perturbed metric that is expressed in a meaningful coordinate system; this is a considerable asset that greatly facilitates the task of extracting physical consequences. We illustrate the use of the light-cone gauge by calculating the metric of a black hole immersed in a uniform magnetic field. We construct a three-parameter family of solutions to the perturbative Einstein-Maxwell equations and argue that it is applicable to a broader range of physical situations than the exact, two-parameter Schwarzschild-Melvin family. 
  We calculate the total energy of Taub's 1951 exact solution for a Bianchi type IX geometry using several different energy-localization procedures, including the prescriptions of Einstein, Papapetrou, Landau-Lifshitz and Moller. We compare these results to those for other anisotropic geometries, and comment on their relationship to Rosen's conjecture about the total energy of the universe. 
  The cosmological constant $\Lambda$ modifies certain properties of large astrophysical rotating configurations with ellipsoidal geometries, provided the objects are not too compact. Assuming an equilibrium configuration and so using the tensor virial equation with $\Lambda$ we explore several equilibrium properties of homogeneous rotating ellipsoids. One shows that the bifurcation point, which in the oblate case distinguishes the Maclaurin ellipsoid from the Jacobi ellipsoid, is sensitive to the cosmological constant. Adding to that, the cosmological constant allows triaxial configurations of equilibrium rotating the minor axis as solutions of the virial equations. The significance of the result lies in the fact that minor axis rotation is indeed found in nature. Being impossible for the oblate case, it is permissible for prolate geometries, with $\Lambda$ zero and positive. For the triaxial case, however, an equilibrium solution is found only for non-zero positive $\Lambda$. Finally, we solve the tensor virial equation for the angular velocity and display special effects of the cosmological constant there. 
  On the basis of the Kerr spinning particle, we show that the mass renormalization is perfectly performed by gravity for an arbitrary distribution of source matter. A smooth regularization of the Kerr-Newman solution is considered, leading to a source in the form of a rotating bag filled by a false vacuum. It is shown that gravity controls the phase transition to an AdS or dS false vacuum state inside the bag, providing the mass balance. 
  Following Dirac's brane variation prescription, the brane must not be deformed during the variation process, or else the linearity of the variation may be lost. Alternatively, the variation of the brane is done, in a special Dirac frame, by varying the bulk coordinate system itself. Imposing appropriate Dirac style boundary conditions on the constrained 'sandwiched' gravitational action, we show how Israel junction conditions get relaxed, but remarkably, all solutions of the original Israel equations are still respected. The Israel junction conditions are traded, in the $Z_2$-symmetric case, for a generalized Regge-Teitelboim type equation (plus a local conservation law), and in the generic $Z_2$-asymmetric case, for a pair of coupled Regge-Teitelboim equations. The Randall-Sundrum model and its derivatives, such as the Dvali-Gabadadze-Porrati and the Collins-Holdom models, get generalized accordingly. Furthermore, Randall-Sundrum and Regge-Teitelboim brane theories appear now to be two different faces of the one and the same unified brane theory. Within the framework of unified brane cosmology, we examine the dark matter/energy interpretation of the effective energy/momentum deviations from General Relativity. 
  The puncture method for black holes in numerical relativity has recently been extended to punctures that move across the grid, which has led to significant advances in numerical simulations of black-hole binaries. We examine how and why the method works. The coordinate singularity and hence the geometry at the puncture are found to change during evolution, but sufficient regularity is maintained for the numerics to work. We construct an analytic solution for the stationary state of a black hole in spherical symmetry that matches the numerical result and demonstrates that the numerics are not dominated by artefacts at the puncture but indeed find the analytical result. 
  A local quantum bosonic model on a lattice is constructed whose low energy excitations are gravitons described by linearized Einstein action. Thus the bosonic model is a quantum theory of gravity, at least at the linear level. We find that the compactification and the discretization of metric tenor are crucial in obtaining a quantum theory of gravity. 
  We study the question of local and global uniqueness of completions, based on null geodesics, of Lorentzian manifolds. We show local uniqueness of such boundary extensions. We give a necessary and sufficient condition for existence of unique maximal completions. The condition is verified in several situations of interest. This leads to existence and uniqueness of maximal spacelike conformal boundaries, of maximal strongly causal boundaries, as well as uniqueness of conformal boundary extensions for asymptotically simple space-times. Examples of applications include the definition of mass, or the classification of inequivalent extensions across a Cauchy horizon of the Taub space-time. 
  A time varying space-time metric is shown to be a source of electromagnetic radiation. The post-Newtonian approximation is used as a realistic model of the connection between the space-time metric and a time varying gravitational potential. Large temporal variations in the metric from the coalescence of colliding black holes and neutron stars are shown to be possible progenitors of gamma ray burst and millisecond pulsars. 
  Motivated by recent results on non-vanishing spatial curvature \cite{curve} we employ the holographic model of dark energy to investigate the validity of first and second laws of thermodynamics in non-flat (closed) universe enclosed by apparent horizon $R_A$ and the event horizon measured from the sphere of horizon named $L$. We show that for the apparent horizon the first law is roughly respected for different epochs while the second laws of thermodynamics is respected while for $L$ as the system's IR cut-off first law is broken down and second law is respected for special range of deceleration parameter. It is also shown that at late-time universe $L$ is equal to $R_A$ and the thermodynamic laws are hold, when the universe has non-vanishing curvature. Defining the fluid temperature to be proportional to horizon temperature the range for coefficient of proportionality is obtained provided that the generalized second law of thermodynamics is hold. 
  We present the first simulations in full General Relativity of the head-on collision between a neutron star and a black hole of comparable mass. These simulations are performed through the solution of the Einstein equations combined with an accurate solution of the relativistic hydrodynamics equations via high-resolution shock-capturing techniques. The initial data is obtained by following the York-Lichnerowicz conformal decomposition with the assumption of time symmetry. Unlike other relativistic studies of such systems, no limitation is set for the mass ratio between the black hole and the neutron star, nor on the position of the black hole, whose apparent horizon is entirely contained within the computational domain. The latter extends over 400 M and is covered with six levels of fixed mesh refinement. Concentrating on a prototypical binary system with mass ratio ~6, we find that although a tidal disruption is evident the neutron star is accreted promptly and entirely into the black hole. While the collision is completed before ~300 M, the evolution is carried over up to ~1700 M, thus providing time for the extraction of the gravitational-wave signal produced and allowing for a first estimate of the radiative efficiency of processes of this type. 
  We prove that extreme Kerr initial data set is a unique absolute minimum of the total mass in a (physically relevant) class of vacuum, maximal, asymptotically flat, axisymmetric data for Einstein equations with fixed angular momentum. These data represent non-stationary, axially symmetric, black holes. As a consequence, we obtain that any data in this class satisfy the inequality $\sqrt{J} \leq m$, where $m$ and $J$ are the total mass and angular momentum of the spacetime. 
  A study of proper Weyl collineations in Kantowski-Sachs and Bianchi type III space-times is given by using the rank of the 6X6 Weyl matrix and direct integration techniques. Studying proper Weyl collineations in each of the above space-times, it is shown that there exists no such possibility when the above space-times admit proper Weyl collineations. 
  We show that, in contrast to the flat case, the Maxwell theory is not confining in the background of the three dimensional BTZ black-hole (covering space). We also study the effect of the curvature on screening behavior of Maxwell-Chern-Simons model in this space-time. 
  This article provides a cartoon of the quantization of General Relativity using the ideas of effective field theory. These ideas underpin the use of General Relativity as a theory from which precise predictions are possible, since they show why quantum corrections to standard classical calculations are small. Quantum corrections can be computed controllably provided they are made for the weakly-curved geometries associated with precision tests of General Relativity, such as within the solar system or for binary pulsars. They also bring gravity back into the mainstream of physics, by showing that its quantization (at low energies) exactly parallels the quantization of other, better understood, non-renormalizable field theories which arise elsewhere in physics. Of course effective field theory techniques do not solve the fundamental problems of quantum gravity discussed elsewhere in these pages, but they do helpfully show that these problems are specific to applications on very small distance scales. They also show why we may safely reject any proposals to modify gravity at long distances if these involve low-energy problems (like ghosts or instabilities), since such problems are unlikely to be removed by the details of the ultimate understanding of gravity at microscopic scales. 
  We study the thermodynamic properties of Schwarzschild de Sitter spacetimes with the consideration of quantum effects. It is shown that by considering the cosmological constant as a variable state parameter and adding an extra term which denotes the vacuum energy, both the differential and integral mass formulas of the first law of Schwarzschild de Sitter spacetimes can be directly derived from the general Schwarzschild de Sitter metrics in a simple and natural way. Furthermore, after taking quantum effects into account, we can see that the cosmological constant must decrease and the spontaneous decay of the vacuum energy never makes the entropy of Schwarzschild de Sitter spacetimes decrease. In addition, though the laws of thermodynamics are very powerful, not all the four laws can be applied to the Schwarzschild de Sitter spacetimes. 
  We adopt Leaver's method to determine quasi normal frequencies of the Schwarzschild black hole in higher (D >= 10) dimensions. In D-dimensional Schwarzschild metric, when D increases, more and more singularities, spaced uniformly on the unit circle |r|=1, approach the horizon at r = r_h = 1. Thus, a solution satisfying the outgoing wave boundary condition at the horizon must be continued to some mid point and only then the continued fraction condition can be applied. This prescription is general and applies to all cases for which, due to regular singularities on the way from the point of interest to the irregular singularity, Leaver's method in its original setting breaks down. We illustrate the method calculating gravitational vector and tensor quasinormal frequencies of the Schwarzschild black hole in D=11 and D=10 dimensions. We also give the details for the D=9 case, considered in gr-qc/0511064. 
  We investigate the problem of whether one can anticipate any features of the graviton without a detailed knowledge of a full quantum gravity. Assuming that in linearized gravity the graviton is in a sense similar to the photon, we derive a curious large number coincidence between the number of gravitons emitted by a solar planet during its orbital period and the number of its nucleons. In Einstein's GR the analogy between the graviton and the photon is ill founded. A generic relationship between quanta of a quantum field and plane waves of the corresponding classical field is broken in the case of GR. The graviton cannot be classically approximated by a generic pp wave nor by the exact plane wave. Most important, the ADM energy is a zero frequency characteristic of any asymptotically flat spacetime and this means that any general relationship between energy and frequency is a priori impossible. In particular the formula $E=\hbar \omega$ does not hold. The graviton must have features different from those of the photon and these cannot be predicted from classical general relativity. 
  We have developed a full model to simulate spherical detectors where all main sources of noise are considered. We have built a computer code for determining the source direction and the wave polarization (solution of the inverse problem) in real time acquisition. The digital filter used is a simple bandpass filter. The ``data'' used for testing our code was simulated, which considers both the source signal and detector noise. The detector noise includes the antenna thermal, back action, phase noise, series noise and thermal from transducer coupled masses. The simulated noise takes into account all these noises and the antenna-transducers coupling. The detector transfer function was calculated for a spherical antenna with six two-mode parametric transducers.   From the results we determined that spherical detectors are able to locate an astrophysical source in the sky in real time as long as the signal-to-noise ratio for a burst signal is equal to or higher than 3. 
  Spatially homogeneous models with a scalar field non-minimally coupled to the space-time curvature or to the ordinary matter content are analysed with respect to late-time asymptotic behaviour, in particular to accelerated expansion and isotropization. It is found that a direct coupling to the curvature leads to asymptotic de Sitter expansion in arbitrary exponential potentials, thus yielding a positive cosmological constant although none is apparent in the potential. This holds true regardless of the steepness of the potential or the smallness of the coupling constant. For matter-coupled scalar fields, the asymptotics are obtained for a large class of positive potentials, generalizing the well-known cosmic no-hair theorems for minimal coupling. In this case it is observed that the direct coupling to matter does not impact the late-time dynamics essentially. 
  We give a straightforward and divergence free derivation of the equation of motion for a small but finite object in an arbitrary background using strong field point particle limit. The resulting equation becomes a generalized geodesic for a non-rotating spherical object which is consisitent with previous studies. 
  In this paper the multipole moments of stationary asymptotically flat spacetimes are considered. We show how the tensorial recursion of Geroch and Hansen can be replaced by a scalar recursion on R^2. We also give a bound on the multipole moments. This gives a proof of the "necessary part" of a long standing conjecture due to Geroch. 
  In this paper we revisit Brill's proof of positive mass for three-dimensional, time-symmetric, axisymmetric initial data and generalise his argument in various directions. In 3+1 dimensions, we include an apparent horizon in the initial data and prove the Riemannian Penrose inequality in a wide number of cases by an elementary argument. In the case of 4+1 dimensions we obtain the analogue of Brill's formula for initial data admitting a generalised form of axisymmetry. Including an apparent horizon in the initial data, the Riemannian Penrose inequality is again proved for a large class of cases. The results may have applications in numerical relativity. 
  A brief explanation of the meaning of the anthropic principle - as a prescription for the attribution of a priori probability weighting - is illustrated by various cosmological and local applications, in which the relevant conclusions are contrasted with those that could be obtained from (less plausible) alternative prescriptions such as the vaguer and less restrictive ubiquity principle, or the more sterile and restrictive autocentric principle. 
  Pairs of Planck-mass drops of superfluid helium coated by electrons (i.e., ``Millikan oil drops''), when levitated in a superconducting magnetic trap, can be efficient quantum transducers between electromagnetic (EM) and gravitational (GR) radiation. This leads to the possibility of a Hertz-like experiment, in which EM waves are converted at the source into GR waves, and then back-converted at the receiver from GR waves back into EM waves. Detection of the gravity-wave analog of the cosmic microwave background using these drops can discriminate between various theories of the early Universe. 
  We explore an effective 4D cosmological model for the universe where the variable cosmological constant governs its evolution and the pressure remains negative along all the expansion. This model is introduced from a 5D vacuum state where the (space-like) extra coordinate is considered as noncompact. The expansion is produced by the inflaton field, which is considered as nonminimally coupled to gravity. We conclude from experiental data that the coupling of the inflaton with gravity should be weak, but variable in different epochs of the evolution of the universe. 
  While it is widely believed that gravity should ultimately be treated as a quantum theory, there remains a possibility that general relativity should not be quantized. If this is the case, the coupling of classical gravity to the expectation value of the quantum stress-energy tensor will naturally lead to nonlinearities in the Schrodinger equation. By numerically investigating time evolution in the nonrelativistic "Schrodinger-Newton" approximation, we show that such nonlinearities may be observable in the next generation of molecular interferometry experiments. 
  We present a new method for the quantization of totally constrained systems including general relativity. The method consists in constructing discretized theories that have a well defined and controlled continuum limit. The discrete theories are constraint-free and can be readily quantized. This provides a framework where one can introduce a relational notion of time and that nevertheless approximates in a well defined fashion the theory of interest. The method is equivalent to the group averaging procedure for many systems where the latter makes sense and provides a generalization otherwise. In the continuum limit it can be shown to contain, under certain assumptions, the ``master constraint'' of the ``Phoenix project''. It also provides a correspondence principle with the classical theory that does not require to consider the semiclassical limit. 
  A special relativity based on the de Sitter group is introduced. Like ordinary special relativity, it retains the quotient character of spacetime, and consequently a notion of homogeneity. This means that the underlying spacetime will be a de Sitter spacetime, the quotient space between de Sitter and the Lorentz groups. Since the local symmetry will also be given by the de Sitter group, each tangent space must be replaced by an osculating de Sitter spacetime. As far as the de Sitter group can be considered a particular deformation of the Poincare' group, this theory turns out to be a specific kind of deformed special relativity. The modified notions of energy and momentum are obtained, and the exact relationship between them explicitly exhibited. The causal structure of spacetime, which is modified by the presence of the de Sitter length parameter, is briefly discussed. 
  Using the factorization approach of quantum mechanics, we obtain a family of isospectral scalar potentials for power law inflationary cosmology. The construction is based on a scattering Wheeler-DeWitt solution. These iso-spectrals have new features, they give a mechanism to end inflation, as well as the possibility to have new inflationary epochs. The procedure can be extended to other cosmological models. 
  We consider deformed special relativity (DSR) theories on commutative space-time, perhaps as an first approximation to a noncommutative space-time formulation. The corresponding field theories in general possess derivatives of all orders. From a quantisation procedure and the deformed Dirac equation in momentum space we obtain for a specific DSR model enhanced expressions for momentum and charge and finite vacuum contributions. 
  The second order Coleman - de Luccia instanton and its action in the Randall - Sundrum type II model are investigated and the comparison with the results in Einstein's general relativity is done in the present paper. 
  The problem of deriving a shock-wave geometry with cosmological constant by boosting a Schwarzschild-de Sitter (or anti-de Sitter) black hole is re-examined. Unlike previous work in the literature, we deal with the exact Schwarzschild-de Sitter (or anti-de Sitter) metric. By virtue of peculiar cancellations in this exact calculation, where the metric does not depend linearly on the mass parameter, we find a singularity of distributional nature on a null hypersurface, which corresponds to a shock-wave geometry derived in a fully non-perturbative way. The result agrees with previous calculations, where the metric had been linearized in the mass parameter. 
  The problem of resolving spherical harmonic components from numerical data defined on a rectangular grid has many applications, particularly for the problem of gravitational radiation extraction. A novel method due to Misner improves on traditional techniques by avoiding the need to cover the sphere with a coordinate system appropriate to the grid geometry. This paper will discuss Misner's method and suggest how it can be improved by exploiting local regression techniques. 
  Inspiralling compact binaries are ideally suited for application of a high-order post-Newtonian (PN) gravitational wave generation formalism. To be observed by the LIGO and VIRGO detectors, these very relativistic systems (with orbital velocities of the order of 0.5c in the last rotations) require high-accuracy templates predicted by general relativity theory. Recent calculations of the motion and gravitational radiation of compact binaries at the 3PN approximation using the Hadamard self-field regularization have left undetermined a few dimensionless coefficients called ambiguity parameters. In this article we review the application of dimensional self-field regularization, within Einstein's classical general relativity formulated in D space-time dimensions, which finally succeeded in clearing up the problem, by uniquely fixing the values of all the ambiguity parameters. 
  Following the approach of Julien Lesgourgues [astro-ph/0409426], we analyze the mathematical structure of the time co-ordinate of present day cosmological models, where these models include a cosmological constant term to account for the observed acceleration of the universe: we find that in all cases, except for a set of measure zero in the parameter space, the time is given by an (abelian) integral on a torus; the imaginary period of this integral then gives a natural periodicity in imaginary time for the universe; following Stephen Hawking, this periodicity may be interpreted either as giving a fundamental mass scale for the universe, or (using Planck's constant) a fundamental temperature, or both. The precise structure that emerges suggests that the structure of time can be regarded as an order parameter arising perhaps in a phase transition in the early universe; one might hope that this structure would be predictable in any fundamental theory. 
  The field equation with the cosmological constant term is derived and the energy of the general 4-dimensional stationary axisymmetric spacetime is studied in the context of the hamiltonian formulation of the teleparallel equivalent of general relativity (TEGR). We find that, by means of the integral form of the constraints equations of the formalism naturally without any restriction on the metric parameters, the energy for the asymptotically flat/de Sitter/Anti-de Sitter stationary spacetimes in the Boyer-Lindquist coordinate can be expressed as $E=\frac{1}{8\pi}\int_S d\theta d\phi(sin\theta \sqrt{g_{\theta\theta}}+\sqrt{g_{\phi\phi}}-(1/\sqrt{g_{rr}})(\partial{\sqrt{g_ {\theta\theta} g_{\phi\phi}}}/\partial r))$. It is surprised to learn that the energy expression is relevant to the metric components $g_{rr}$, $g_{\theta\theta}$ and $g_{\phi\phi}$ only. As examples, by using this formula we calculate the energies of the Kerr-Newman (KN), Kerr-Newman Anti-de Sitter (KN-AdS), Kaluza-Klein, and Cveti\v{c}-Youm spacetimes. 
  We compute the geometrical Berry phase for the noncommutative gravitational quantum well. We find that $\Delta\gamma(S)\sim{\eta}^3$, where $\sqrt{\eta}$ is the fundamental momentum scale for the noncommutative gravitational quantum well in a segment S of the path in the configuration space. For the full closed path, we find that $\gamma(C)=0$. 
  In this Master thesis we consider 't Hooft's polygon model for 2+1D gravity. After a detailed review of the polygon model in the classical context, we discuss problems associated with its quantization and calculate the explicitly the full Poisson structure of the constraints.   The calculation of the Poisson structure in chapter 6 introduces corrections and generalizations to earlier published results. We show that the full Poisson structure closes on shell, but not off shell. An attempt is made to interpret the gauge orbits generated by the constraints. 
  We study formal expansions of asymptotically flat solutions to the static vacuum field equations which are determined by minimal sets of freely specifyable data referred to as `null data'. These are given by sequences of symmetric trace free tensors at space-like infinity of increasing order. They are 1:1 related to the sequences of Geroch multipoles. Necessary and sufficient growth estimates on the null data are obtained for the formal expansions to be absolutely convergent. This provides a complete characterization of all asymptotically flat solutions to the static vacuum field equations. 
  We present the possibility that Dirac and Majorana neutrino wave packets can be distinguished when subject to spin-gravity interaction while propagating through vacuum described by the Lense-Thirring metric. By adopting the techniques of gravitational phase and time-independent perturbation theory following the Brillouin-Wigner method, we generate spin-gravity matrix elements from a perturbation Hamiltonian and show that this distinction is easily reflected in well-defined gravitational corrections to the neutrino oscillation length for a two-flavour system. Explicit examples are presented using the Sun and SN1987A as the gravitational sources for the Lense-Thirring metric. This approach offers the possibility to determine the absolute neutrino masses by this method and identify a theoretical upper bound for the absolute neutrino mass difference, where the distinctions between the Dirac and Majorana cases are evident. We discuss the relevance of this analysis to the upcoming attempts to measure the properties of low-energy neutrinos by SNO and other solar neutrino observatories. 
  Initial data for boosted Kerr black hole are constructed in an axially symmetric case. Momentum and hamiltonian constraints are solved numerically using finite element method (FEM) algorithms. Both Bowen-York and puncture boundary conditions are adopted and appropriate results are compared. Past and future apparent horizons are also found numerically and the Penrose inequality is tested in detail. 
  We analyze the decomposition of the enveloping algebra of the conformal algebra in arbitrary dimension with respect to the mass-squared operator. It emerges that the subalgebra that commutes with the mass-squared is generated by its Poincare subalgebra together with a vector operator. The special cases of the conformal algebras of two and three dimensions are described in detail, including the construction of their Casimir operators. 
  The Tolman-Oppenheimer-Volkov [TOV] equation constrains the internal structure of general relativistic static perfect fluid spheres. We develop several "solution generating" theorems for the TOV, whereby any given solution can be "deformed" to a new solution. Because the theorems we develop work directly in terms of the physical observables -- pressure profile and density profile -- it is relatively easy to check the density and pressure profiles for physical reasonableness. This work complements our previous article [Phys. Rev. D71 (2005) 124307; gr-qc/0503007] wherein a similar "algorithmic" analysis of the general relativistic static perfect fluid sphere was presented in terms of the spacetime geometry -- in the present analysis the pressure and density are primary and the spacetime geometry is secondary. In particular, our "deformed" solutions to the TOV equation are conveniently parameterized in terms of delta rho_c and delta p_c, the shift in the central density and central pressure. 
  We present a model unifying general relativity and quantum mechanics. It is based on the (noncommutative) algebra ${\cal A}$ on the groupoid $\Gamma = E \times G$, where $E$ is the total space of the frame bundle over spacetime, and $G$ the Lorentz group. The differential geometry, based on derivations of ${\cal A}$, is constructed. The eigenvalue equation for the Einstein operator plays the role of a generalized Einstein's equation. The algebra ${\cal A}$, when suitably represented in a bundle of Hilbert spaces, is a von Neumann algebra $\mathcal{M}$ of random operators representing the quantum sector of the model. The Tomita-Takesaki theorem allows us to define the dynamics of random operators which depends on the state $\phi $. The same state defines the noncommutative probability measure (in the sense of free probability theory). The state $\phi $ satisfies the Kubo-Martin-Schwinger (KMS) condition, and can be interpreted as describing a generalized equilibrium state. By suitably averaging elements of the algebra ${\cal A}$, one recovers the standard geometry of spacetime. We show that any act of measurement, performed at a given spacetime point, makes the model to collapse to the standard quantum mechanics . We compute the noncommutative version of the closed Friedman world model. Eigenfunctions of the Einstein operator produce the correct components of the energy-momentum tensor. Dynamics of random operators does not ``feel'' singularities. 
  We discuss two effects predicted by the general theory of relativity in the context of Rindler accelerated observers: the gravitational spectral shift and the time delay of light. We show that these effects also appear in a Rindler frame in the absence of gravitational field, in accordance with the Einstein's equivalence principle. 
  A gravitomagnetic analogue of the London moment in superconductors can explain the anomalous Cooper pair mass excess reported by Janet Tate. Ultimately the gravitomagnetic London moment is attributed to the breaking of the principle of general covariance in superconductors. This naturally implies non-conservation of classical energy-momentum. Possible relation with the manifestation of dark energy in superconductors is questioned. 
  We study the phase space of the spherically symmetric solutions of the system obtained from the dimensional reduction of the six-dimensional Einstein-Gauss- Bonnet action. We show the existence of solutions with nonflat asymptotic behavior. 
  We discuss how developments in physics often imply in the need that spacetime acquires an increasingly richer and complex structure. General Relativity was the first theory to show us the way to connect space and time with the physical world. Since then, scrutinizing the ways spacetime might exist is, in a way, the very essence of physics. Physics has thus given substance to the pioneering work of scores of brilliant mathematicians who speculated on the geometry and topology of spaces. 
  One of the most exciting potential sources of gravitational waves for low-frequency, space-based gravitational wave (GW) detectors such as the proposed Laser Interferometer Space Antenna (LISA) is the inspiral of compact objects into massive black holes in the centers of galaxies. The detection of waves from such "extreme mass ratio inspiral" systems (EMRIs) and extraction of information from those waves require template waveforms. The systems' extreme mass ratio means that their waveforms can be determined accurately using black hole perturbation theory. Such calculations are computationally very expensive. There is a pressing need for families of approximate waveforms that may be generated cheaply and quickly but which still capture the main features of true waveforms. In this paper, we introduce a family of such "kludge" waveforms and describe ways to generate them. We assess performance of the introduced approximations by comparing "kludge" waveforms to accurate waveforms obtained by solving the Teukolsky equation in the adiabatic limit (neglecting GW backreaction). We find that the kludge waveforms do extremely well at approximating the true gravitational waveform, having overlaps with the Teukolsky waveforms of 95% or higher over most of the parameter space for which comparisons can currently be made. Indeed, we find these kludges to be of such high quality (despite their ease of calculation) that it is possible they may play some role in the final search of LISA data for EMRIs. 
  We discuss the issue of quasi-particle production by ``analogue black holes'' with particular attention to the possibility of reproducing Hawking radiation in a laboratory. By constructing simple geometric acoustic models, we obtain a somewhat unexpected result: We show that in order to obtain a stationary and Planckian emission of quasi-particles, it is not necessary to create a trapped region in the acoustic spacetime (corresponding to a supersonic regime in the fluid flow). It is sufficient to set up a dynamically changing flow asymptotically approaching a sonic regime with sufficient rapidity in laboratory time. 
  In the present paper we generalize the original work of C.W. Misner \cite{M69q} about the quantum dynamics of the Bianchi type IX geometry near the cosmological singularity. We extend the analysis to the generic inhomogeneous universe by solving the super-momentum constraint and outlining the dynamical decoupling of spatial points. Firstly, we discuss the classical evolution of the model in terms of the Hamilton-Jacobi approach as applied to the super-momentum and super-Hamiltonian constraints; then we quantize it in the approximation of a square potential well after an ADM reduction of the dynamics with respect to the super-momentum constraint only. Such a reduction relies on a suitable form for the generic three-metric tensor which allows the use of its three functions as the new spatial coordinates. We get a functional representation of the quantum dynamics which is equivalent to the Misner-like one when extended point by point, since the Hilbert space factorizes into $\infty^3$ independent components due to the parametric role that the three-coordinates assume in the asymptotic potential term. Finally, we discuss the conditions for having a semiclassical behavior of the dynamics and we recognize that this already corresponds to having mean occupation numbers of order $\mathcal{O}(10^2)$. 
  Hawking temperature is computed for a large class of black holes (with spherical, toroidal and hyperboloidal topologies) using only laws of classical physics plus the "classical" Heisenberg Uncertainty Principle. This principle is shown to be fully sufficient to get the result, and there is no need to this scope of a Generalized Uncertainty Principle. 
  We use the teleparallel geometry analog of the Moller energy-momentum complex to calculate the energy distribution (due to matter plus field including gravity) of a charged black hole solution in heterotic string theory. We find the same energy distribution as obtained by Gad who investigated the same problem by using the Moller energy-momentum complex in general relativity. The total energy depends on the black hole mass M and charge Q. The energy obtained is also independent of the teleparallel dimensionless coupling constant, which means that it is valid not only in the teleparallel equivalent of general relativity, but also in any teleparallel model. Furthermore, our results also sustains (a) the importance of the energy-momentum definitions in the evaluation of the energy distribution of a given spacetime and (b) the viewpoint of Lessner that the Moller energy-momentum complex is a powerful concept of energy and momentum. 
  Sorkin's causet mechanism is generalized to include energy exchange between causet elements and conventional vacuum fluctuations to the inflationary epoch. In this, the dark energy of the adiabatic era is the fluctuating remnant of inflation. The mechanism is also applicable to black hole evaporation. 
  We show how the quantization of two-dimensional gravity leads to an (Euclidean) quantum space-time where the average geometry is that of constant negative curvature and where the Hartle-Hawking boundary condition arises naturally. 
  Since the work of Mac-Dowell-Mansouri it is well known that gravity can be written as a gauge theory for the de Sitter group. In this paper we consider the coupling of this theory to the simplest gauge invariant observables that is, Wilson lines. The dynamics of these Wilson lines is shown to reproduce exactly the dynamics of relativistic particles coupled to gravity, the gauge charges carried by Wilson lines being the mass and spin of the particles. Insertion of Wilson lines breaks in a controlled manner the diffeomorphism symmetry of the theory and the gauge degree of freedom are transmuted to particles degree of freedom. 
  We discuss a robust data analysis method to detect a stochastic background of gravitational waves in the presence of non-Gaussian noise. In contrast to the standard cross-correlation (SCC) statistic frequently used in the stochastic background searches, we consider a {\it generalized cross-correlation} (GCC) statistic, which is nearly optimal even in the presence of non-Gaussian noise. The detection efficiency of the GCC statistic is investigated analytically, particularly focusing on the statistical relation between the false-alarm and the false-dismissal probabilities, and the minimum detectable amplitude of gravitational-wave signals. We derive simple analytic formulae for these statistical quantities. The robustness of the GCC statistic is clarified based on these formulae, and one finds that the detection efficiency of the GCC statistic roughly corresponds to the one of the SCC statistic neglecting the contribution of non-Gaussian tails. This remarkable property is checked by performing the Monte Carlo simulations and successful agreement between analytic and simulation results was found. 
  The dynamical equivalence between modified and scalar-tensor gravity theories is revisited and it is concluded that it breaks down in the limit to general relativity. A gauge-independent analysis of cosmological perturbations in both classes of theories lends independent support to this conclusion. As a consequence, the PPN formalism of scalar-tensor gravity and Solar System experiments do not veto modified gravity, as previously thought. 
  In this article we give a brief outline of the applications of the generalized Heun equation (GHE) in the context of Quantum Field Theory in curved space-times. In particular, we relate the separated radial part of a massive Dirac equation in the Kerr-Newman metric and the static perturbations for the non-extremal Reissner-Nordstr\"{o}m solution to a GHE. 
  In the present work, we prove the validity of two theorems on null-paths in a version of absolute parallelism geometry. A version of these theorems has been originally established and proved by Kermak, McCrea and Whittaker (KMW) in the context of Riemannian geometry. The importance of such theorems is their use, in applications, to derive a general formula for the red-shift of spectral lines coming from distant objects. The formula derived in the present work, can be applied for both cosmological and astrophysical red-shifts. It takes into account the shifts resulting from gravitation, different motions of the source of photons, spin of the moving particle (photons) and the direction of the line of sight. It is shown that this formula cannot be derived in the context of Riemannian geometry, but it can be reduced to a formula given by KMW under certain conditions. 
  Gravitational self-interactions are assumed to be determined by the covariant derivative acting on the Riemann-Christoffel field strength. Once imposed on a metric theory, this Yang-Mills gauge constraint extends the equality of gravitational mass and inertial mass to compact bodies with non-negligible gravitational binding energy. Applied to generalized Brans-Dicke theories, it singles out one tensor theory and one scalar theory for gravity but also suggests a way to implement a minimal violation of the strong equivalence principle. 
  We clarify the conditions for Birkhoff's theorem, that is, time-independence in general relativity. We work primarily at the linearized level where guidance from electrodynamics is particularly useful. As a bonus, we also derive the equivalence principle. The basic time-independent solutions due to Schwarzschild and Kerr provide concrete illustrations of the theorem. Only familiarity with Maxwell's equations and tensor analysis is required. 
  We derive the expression for the jerk parameter in $f(R)$ gravity. We use the Palatini variational principle and the field equations in the Einstein conformal gauge. For the particular case $f(R)=R-\frac{\alpha^2}{3R}$, the predicted value of the jerk parameter agrees with the SNLS SNIa and X-ray galaxy cluster distance data but does not with the SNIa gold sample data. 
  In this thesis three separate problems relevant to general relativity are considered. Methods for algorithmically producing all the solutions of isotropic fluid spheres have been developed over the last five years. A different and somewhat simpler algorithm is discussed here, as well as algorithms for anisotropic fluid spheres. The second and third problems are somewhat more speculative in nature and address the nature of black hole entropy. Specifically, the second problem looks at the genericity of the so-called quasinormal mode conjecture introduced by Hod, while the third problem looks at the near-horizon structure of a black hole in hope of gaining an understanding of why so many different approaches yield the same entropy. A method of finding the asymptotic QNM structure is found based on the Born series, and serious problems for the QNM conjecture are discussed. The work in this thesis does not completely discount the possibility that the QNM conjecture is true. New results released weeks before this thesis was finished showed that the QNM conjecture was flawed. Finally, the near-horizon structure of a black hole is found to be very restricted, adding credence to the ideas put forward by Carlip and Solodukhin that the black hole entropy is related to an inherited symmetry from the classical theory. 
  The relativistic kinetic theory of the phonon gas in superfluids is developed. The technique of the derivation of macroscopic balance equations from microscopic equations of motion for individual particles is applied to an ensemble of quasi-particles. The necessary expressions are constructed in terms of a Hamilton function of a (quasi-)particle. A phonon contribution into superfluid dynamic parameters is obtained from energy-momentum balance equations for the phonon gas together with the conservation law for superfluids as a whole. Relations between dynamic flows being in agreement with results of relativistic hydrodynamic consideration are found. Based on the kinetic approach a problem of relativistic variation of the speed of sound under phonon influence at low temperature is solved. 
  It was recently proposed that deformations of the relativistic symmetry, as those considered in Deformed Special Relativity (DSR), can be seen as the outcome of a measurement theory in the presence of non-negligible (albeit small) quantum gravitational fluctuations [1,2]. In this paper we explicitly consider the case of a spacetime described by a flat metric endowed with stochastic fluctuations and, for a free particle, we show that DSR-like nonlinear relations between the spaces of the measured and classical momenta, can result from the average of the stochastic fluctuations over a scale set be the de Broglie wavelength of the particle. As illustrative examples we consider explicitly the averaging procedure for some simple stochastic processes and discuss the physical implications of our results. 
  The Gerlach and Sengupta (GS) formalism of coordinate-invariant, first-order, spherical and nonspherical perturbations around an arbitrary spherical spacetime is generalized to higher orders, focusing on second-order perturbation theory. The GS harmonics are generalized to an arbitrary number of indices on the unit sphere and a formula is given for their products. The formalism is optimized for its implementation in a computer algebra system, something that becomes essential in practice given the size and complexity of the equations. All evolution equations for the second-order perturbations, as well as the conservation equations for the energy-momentum tensor at this perturbation order, are given in covariant form, in Regge-Wheeler gauge. 
  We study the entropy of the black hole with torsion using the covariant form of the partition function. The regularization of infinities appearing in the semiclassical calculation is shown to be consistent with the grand canonical boundary conditions. The correct value for the black hole entropy is obtained provided the black hole manifold has two boundaries, one at infinity and one at the horizon. However, one can construct special coordinate systems, in which the entropy is effectively associated with only one of these boundaries. 
  It was shown that quantum metric fluctuations smear out the singularities of Green's functions on the light cone [1], but it does not remove other ultraviolet divergences of quantum field theory. We have proved that the quantum field theory in Krein space, {\it i.e.} indefinite metric quantization, removes all divergences of quantum field theory with exception of the light cone singularity [2,3]. In this paper, it is discussed that the combination of quantum field theory in Krein space together with consideration of quantum metric fluctuations, results in quantum field theory without any divergences. 
  The intracavity topologies of laser gravitational-wave detectors are the promising way to obtain sensitivity of these devices significantly better than the Standard Quantum Limit (SQL). The most challenging element of the intracavity topologies is the \emph{local meter} which has to monitor position of a small ($1\div10$ gram) local mirror and which precision defines the sensitivity of the detector.   To overcome the SQL, the quantum variational measurement can be used in the local meter. In this method a frequency-dependent correlation between the meter back-action noise and measurement noise is introduced, which allows to eliminate the back-action noise component from the meter output signal. This correlation is created by means of an additional filter cavity.   In this article the sensitivity limitations of this scheme imposed by the optical losses both in the local meter itself and in the filter cavity are estimated. It is shown that the main sensitivity limitation stems from the filter cavity losses. In order to overcome it, it is necessary to increase the filter cavity length. In a preliminary prototype experiment about 10 meter long filter cavity can be used to obtain sensitivity approximately $2\div3$ times better than the SQL. For future QND gravitational-wave detectors with sensitivity about ten times better than the SQL, the filter cavity length should be within kilometer range. 
  The crucial role played by pressure in general relativity is explored in the mathematically simple context of a static spherically symmetric geometry. By keeping all pressure terms, the standard formalisms of rotation curve and gravitational lensing observations are extended to a first post-Newtonian order. It turns out that both post-Newtonian formalisms encode the gravitational field differently. Therefore, combined observations of rotation curves and gravitational lensing of the same galaxy can in principle be used to infer both the density and pressure profile of the galactic fluid.   The newly introduced post-Newtonian formalism might allow us to make inferences about the equation of state of dark matter. While the Cold Dark Matter paradigm is currently favoured in the astrophysics and cosmology communities, the formalism presented herein offers an unprecedented way of being able to directly observe the equation of state.   In a logically distinct analysis, I investigate the effects of negative pressure in compact objects, motivated by the recently introduced gravastar model. I find that gravastar like objects can in principle mimic the external gravitational field of a black hole. Unlike a black hole, however, gravastars neither exhibit a pathological curvature singularity at the origin nor do they posess an event horizon. Instead they are mathematically well defined everywhere. Finally, another exotic option is considered as a mathematical alternative to black holes: The anti-gravastar, which is characterized by a core that has a negative mass-energy density. 
  We show that the running of gravitational couplings, together with a suitable identification of the renormalization group scale can give rise to modified dispersion relations for massive particles. This result seems to be compatible with both the frameworks of effective field theory with Lorentz Invariance Violation and Deformed Special Relativity. The phenomenological consequences depend on which of the frameworks is assumed. We discuss the nature and strength of the available constraints for both cases and show that in the case of Lorentz Invariance Violation the theory is strongly constrained. 
  The latest test of the general relativistic gravitomagnetic Lense-Thirring effect in the gravitational field of the Earth was performed by analyzing a suitable linear combination of the nodes of the LAGEOS and LAGEOS II satellites. In this note we show that the errors in their inclinations i, assumed to be 0.5 milliarcseconds, couple with the classical even zonal node precessions yielding a further contribution of 9% to the total error budget. This bias must be added to all the previously obtained estimates which mainly accounted for the impact of the even zonal harmonics J_L and their secular variations dot J_L only. In the most optimistic case, the total error now amounts to 19%, while, in more conservative scenarios, it gets close to about 30%. 
  We give a very concise review of the group field theory formalism for non-perturbative quantum gravity, a higher dimensional generalisation of matrix models. We motivate it as a simplicial and local realisation of the idea of 3rd quantization of the gravitational field, equivalently of a quantum field theory of simplicial geometry, in which also the topology of space is fully dynamical. We highlight the basic structure of the formalism, and discuss briefly various models that are being studied, some recent results and the many open issues that future research should face. Finally, we point out the connections with other approaches to quantum gravity, such as loop quantum gravity, quantum Regge calculus and dynamical triangulations, and causal sets. 
  An action functional for the loop quantum cosmology difference equation is presented. It is shown that by guessing the general form of the solution and optimizing the action functional with respect to the parameters in the guessed solution one can obtain approximate solutions which are reasonably good. 
  Here we shall find the green's function of the difference equation of loop quantum cosmology. To illustrate how to use it, we shall obtain an iterative solution for closed model and evaluate its corresponding Bohmian trajectory. 
  Advanced LIGO's sensitivity will be limited by coating noise. Though this noise depends on beam shape, and though nongaussian beams are being seriously considered for advanced LIGO, no published analysis exists to compare the quantitative thermal noise improvement alternate beams offer. In this paper, we derive and discuss a simple integral which completely characterizes the dependence of coating thermal noise on shape. The derivation used applies equally well, with minor modifications, to all other forms of thermal noise in the low-frequency limit. 
  It has been pointed by Hall et al. [1] that matter collinations can be defined by using three different methods. But there arises the question of whether one studies matter collineations by using the ${\cal L}_\xi T_{ab}=0$, or ${\cal L}_\xi T^{ab}=0$ or ${\cal L}_\xi T_a^b=0$. These alternative conditions are, of course, not generally equivalent. This problem has been explored by applying these three definitions to general static spherically symmetric spacetimes. We compare the results with each definition. 
  The basic elements of the relativistic positioning systems in a two-dimensional space-time have been introduced in a previous work [Phys. Rev. D {\bf 73}, 084017 (2006)] where geodesic positioning systems, constituted by two geodesic emitters, have been considered in a flat space-time. Here, we want to show in what precise senses positioning systems allow to make {\em relativistic gravimetry}. For this purpose, we consider stationary positioning systems, constituted by two uniformly accelerated emitters separated by a constant distance, in two different situations: absence of gravitational field (Minkowski plane) and presence of a gravitational mass (Schwarzschild plane). The physical coordinate system constituted by the electromagnetic signals broadcasting the proper time of the emitters are the so called {\em emission coordinates}, and we show that, in such emission coordinates, the trajectories of the emitters in both situations, absence and presence of a gravitational field, are identical. The interesting point is that, in spite of this fact, particular additional information on the system or on the user allows not only to distinguish both space-times, but also to complete the dynamical description of emitters and user and even to measure the mass of the gravitational field. The precise information under which these dynamical and gravimetric results may be obtained is carefully pointed out. 
  We summarize the twelve most important in our view novel concepts that have arisen, based on results that have been obtained, from various applications of Abstract Differential Geometry (ADG) to Quantum Gravity (QG). The present document may be used as a concise, yet informal, discursive and peripatetic conceptual guide-cum-terminological glossary to the voluminous technical research literature on the subject. In a bonus section at the end, we dwell on the significance of introducing new conceptual terminology in future QG research by means of `poetic language' 
  An improved Hamiltonian constraint operator is introduced in loop quantum cosmology. Quantum dynamics of the spatially flat, isotropic model with a massless scalar field is then studied in detail using analytical and numerical methods. The scalar field continues to serve as `emergent time', the big bang is again replaced by a quantum bounce, and quantum evolution remains deterministic across the deep Planck regime. However, while with the Hamiltonian constraint used so far in loop quantum cosmology the quantum bounce can occur even at low matter densities, with the new Hamiltonian constraint it occurs only at a Planck-scale density. Thus, the new quantum dynamics retains the attractive features of current evolutions in loop quantum cosmology but, at the same time, cures their main weakness. 
  We study some aspects of classical & quantum cosmology in the context of two-dimensionsal dilaton gravity theories with matter being described by a perfect fluid. We derive the classical equations obeyed by the metric function & the dilaton field. An impicit solution for the metric function is obtained & used to discuss possible expansion & contraction scenarios for the universe. Quantization is carried out & finite norm exact solutions of the Wheeler-DeWitt equation are obtained. 
  Recent work has shown that the addition of an appropriate covariant boundary term to the gravitational action yields a well-defined variational principle for asymptotically flat spacetimes and thus leads to a natural definition of conserved quantities at spatial infinity. Here we connect such results to other formalisms by showing explicitly i) that for spacetime dimension $d \ge 4$ the canonical form of the above-mentioned covariant action is precisely the ADM action, with the familiar ADM boundary terms and ii) that for $d=4$ the conserved quantities defined by counter-term methods agree precisely with the Ashtekar-Hansen conserved charges at spatial infinity. 
  It was shown recently that, in the case of Schwarschild black hole, one can obtain the correct thermodynamic relations by studying a model quantum system and using a particular duality transformation. We study this approach further for the case a general spherically symmetric horizon. We show that the idea works for a general case only if we define the entropy S as a congruence ("observer") dependent quantity and the energy E as the integral over the source of the gravitational acceleration for the congruence. In fact, in this case, one recovers the relation S=E/2T between entropy, energy and temperature previously proposed by one of us in gr-qc/0308070. This approach also enables us to calculate the quantum corrections of the Bekenstein-Hawking entropy formula for all spherically symmetric horizons. 
  We consider bulk fields coupled to the graviton in a Lorentz violating fashion. We expect that the overly tested Lorentz symmetry might set constraints on the induced Lorentz violation in the brane, and hence on the dynamics of the interaction of bulk fields on the brane. We also use the requirement for Lorentz symmetry to constrain the cosmological constant observed on the brane. 
  In this article we perform von Neumann analysis of the difference equations that arise as a result of loop quantum gravity being applied to models of cosmology and black holes. In particular, we study the numerical stability of Bianchi I LRS (symmetric and non-symmetric constraint) and Schwarzschild interior (symmetric constraint) models, and find that there exist domains over which there are instabilities, generically. We also present explicit evolutions of wave-packets in these models and clearly demonstrate the presence of these instabilities. 
  The interactions of gravitons with matter are calculated in parallel with the familiar photon case. It is shown that graviton scattering amplitudes can be factorized into a product of familiar electromagnetic forms, and cross sections for various reactions are straightforwardly evaluated using helicity methods. 
  We prove that the Schwarzschild black hole is linearly stable under electromagnetic and gravitational perturbations. Our method is to show that for spin $s=1$ or $s=2$, solutions of the Teukolsky equation with smooth, compactly supported initial data outside the event horizon, decay in $L^\infty_{loc}$. 
  Using a post-Newtonian diagnostic tool developed by Mora and Will, we examine numerically generated quasiequilibrium initial data sets that have been used in recently successful numerical evolutions of binary black holes through plunge, merger and ringdown. We show that a small but significant orbital eccentricity is required to match post-Newtonian and quasiequilibrium calculations. If this proves to be a real eccentricity, it could affect the fine details of the subsequent numerical evolutions and the predicted gravitational waveforms. 
  This paper presents a comprehensive study of the fundamental quasinormal modes of all Standard Model fields propagating on a brane embedded in a higher-dimensional rotating black hole spacetime. The equations of motion for fields with spin $s=0, 1/2$ and 1 propagating in the induced-on-the-brane background are solved numerically, and the dependence of their QN spectra on the black hole angular momentum and dimensionality of spacetime is investigated. It is found that the brane-localised field perturbations are longer-lived when the higher-dimensional black hole rotates faster, while an increase in the number of transverse-to-the-brane dimensions reduces their lifetime. Finally, the quality factor $Q$, that determines the best oscillator among the different field perturbations, is investigated and found to depend on properties of both the particular field studied (spin, multipole numbers) and the gravitational background (dimensionality, black hole angular momentum number). 
  The topology of the causal boundary for standard static spacetimes--spacetimes time-invariantly conformal to a metric product of the Lorentz line and a Riemannian manifold--is studied in depth. As this is given in terms of a set of real-valued functions on the Riemannian factor, one could use a function-space topology, but physical reasons recommend a chronological topology instead. The function-space topology has a simple product structure, while the chronological topology might not. This paper examines when the chronological topology coincides with the function-space topology and when it has a simple product structure. A class of standard static spacetimes is examined, all of which yield a simple product structure for the causal boundary; the conformal class of these spacetimes includes classical spacetimes such as external Schwarzschild or Reissner-Nordstrom. 
  We derive from Einstein equation an evolution law for the area of a trapping or dynamical horizon. The solutions to this differential equation show a causal behavior. Moreover, in a viscous fluid analogy, the equation can be interpreted as an energy balance law, yielding to a positive bulk viscosity. These two features contrast with the event horizon case, where the non-causal evolution of the area and the negative bulk viscosity require teleological boundary conditions. This reflects the local character of trapping horizons as opposed to event horizons. Interpreting the area as the entropy, we propose to use an area/entropy evolution principle to select a unique dynamical horizon and time slicing in the Cauchy evolution of an initial marginally trapped surface. 
  A previous evaluation of one-photon loop corrections to the energy-momentum tensor has been extended to particles with unit spin and speculations are presented concerning general properties of such forms. 
  We provide evidence that ``super-extremal'' black hole space-times (either with charge larger than mass or angular momentum larger than mass), which contain naked singularities, are unstable under linearized perturbations. This is given by an infinite family of exact unstable solutions in the charged non rotating case, and by a set of (unstable) numerical solutions in the rotating case. These results may be relevant to the expectation that these space-times cannot be the endpoint of physical gravitational collapse. 
  The purpose of this paper is to discuss in detail the use of scalar matter coupled to linearly polarized Einstein-Rosen waves as a probe to study quantum gravity in the restricted setting provided by this symmetry reduction of general relativity. We will obtain the relevant Hamiltonian and quantize it with the techniques already used for the purely gravitational case. Finally we will discuss the use of particle-like modes of the quantized fields to operationally explore some of the features of quantum gravity within this framework. Specifically we will study two-point functions, the Newton-Wigner propagator, and radial wave functions for one-particle states. 
  a previous evaluation of the one-graviton loop corrections to the energy-momentum tensor has been extended to particles with unit spin and speculations are presented concerning general properties of such forms. 
  Field theories whose full action is Lorentz invariant (or diffeomorphism invariant) can exhibit superluminal behaviors through the breaking of local Lorentz invariance. Quantum induced superluminal velocities are well-known examples of this effect. The issue of the causal behavior of such propagations is somewhat controversial in the literature and we intend to clarify it. We provide a careful analysis of the meaning of causality in classical relativistic field theories, and we stress the role played by the Cauchy problem and the notions of chronology and time arrow. We show that superluminal behavior threaten causality only if a prior chronology on spacetime is chosen. In the case where superluminal propagations occur, however, there is at least two non conformally related metrics on spacetime and thus two available notions of chronology. These two chronologies are on equal footing and it would thus be misleading to choose \textit{ab initio} one of them to define causality. Rather, we provide a formulation of causality in which no prior chronology is assumed. We argue this is the only way to deal with the issue of causality in the case where some degrees of freedom propagate faster than others. We actually show that superluminal propagations do not threaten causality. As an illustration of these conceptual issues, we consider two field theories, namely k-essences scalar fields and bimetric theories of gravity, and we derive the conditions imposed by causality. We discuss various applications such as the dark energy problem, MOND-like theories of gravity and varying speed of light theories. 
  A method is introduced for solving Einstein's equations using two distinct coordinate systems. The coordinate basis vectors associated with one system are used to project out components of the metric and other fields, in analogy with the way fields are projected onto an orthonormal tetrad basis. These field components are then determined as functions of a second independent coordinate system. The transformation to the second coordinate system can be thought of as a mapping from the original ``inertial'' coordinate system to the computational domain. This dual-coordinate method is used to perform stable numerical evolutions of a black-hole spacetime using the generalized harmonic form of Einstein's equations in coordinates that rotate with respect to the inertial frame at infinity; such evolutions are found to be generically unstable using a single rotating coordinate frame. The dual-coordinate method is also used here to evolve binary black-hole spacetimes for several orbits. The great flexibility of this method allows comoving coordinates to be adjusted with a feedback control system that keeps the excision boundaries of the holes within their respective apparent horizons. 
  It is shown that an Einstein cluster of WIMPs, WIMPs on stable circular geodesic orbits generating the static spherically symmetric gravitational field of a galactic halo, can exactly reproduce the rotation curve of any galaxy simply by adjusting the local angular momentum distribution and consequent number distribution of the WIMPs. No new physics is involved (assuming the forthcoming discovery of WIMPs). Further, stability of the orbits can require an inner truncation of the halo and an explicit example of this is given. There is no exact Newtonian counterpart to the model presented since pure Newtonian gravity fails to account for the contribution made by the angular momentum distribution of the particles in creating the gravitational field. In effect, a galactic halo is supported by the hoop stresses created by the orbiting WIMPs. 
  The factorization property of graviton scattering amplitudes is reviewed and show to be valid only if the "natural" value of the gyromagnetic ratio $g_S=2$ is employed -- independent of spin. 
  In the context of loop quantum cosmology, we consider an inflationary era driven by a canonical scalar field and occurring in the semiclassical regime, where spacetime is a continuum but quantum gravitational effects are important. The spectral amplitude and index of scalar perturbations on an unperturbed de Sitter background are computed at lowest order in the slow-roll parameters. The scalar spectrum can be blue-tilted and far from scale invariance, and tuning of the quantization ambiguities is necessary for agreement with observations. The results are extended to a generalized quantization scheme including those proposed in the literature. Quantization of the matter field at sub-horizon scales can provide a consistency check of such schemes. 
  The present work is a review of a series of papers, published in the last ten years, comprising an attempt to find a suitable avenue from geometry to quantum. It shows clearly that, any non-symmetric geometry admits some built-in quantum features. These features disappear completely once the geometry becomes symmetric (torsion-less). It is shown that, torsion of space-time plays an important role in both geometry and physics. It interacts with the spin of the moving particle and with its charge. The first interaction, {\bf{Spin-Torsion Interaction}}, has been used to overcome the discrepancy in the results of the COW-experiment. The second interaction, {\bf{Charge-Torsion Interaction}}, is similar to the Aharonov-Bohm effect.   As a byproduct, a new version of Absolute Parallelism (AP) geometry, the Parameterized Absolute Parallelism (PAP) geometry, has been established and developed. This version can be used to construct field theories that admit some quantum features. Riemannian geometry and conventional AP-geometry are special cases of PAP-geometry. 
  We construct thin shell Lorentzian wormholes in higher dimensional Einstein-Maxwell theory applying the ' Cut and Paste ' technique proposed by Visser. The linearized stability is analyzed under radial perturbations around some assumed higher dimensional spherically symmetric static solution of the Einstein field equations in presence of Electromagnetic field. We determine the total amount of exotic matter, which is concentrated at the wormhole throat. 
  In this thesis the first order formulation of generalized dilaton gravities in two dimensions coupled to a Dirac fermion is considered. After a Hamiltonian analysis of the gauge symmetries and constraints of the theory and fixing Eddington-Finkelstein gauge by use of the Batalin-Vilkovisky-Fradkin method, the system is quantized in the Feynman path integral approach. It turns out that the path integral over the dilaton gravity sector can be evaluated exactly, while in the matter sector perturbative methods are applied. The gravitationally induced four-fermi scattering vertices as well as asymptotic states are calculated, and -- as for dilaton gravities coupled to scalar fields -- a ``virtual black hole'' is found to form as an intermediary geometric state in scattering processes. The results are compared to the well-known scalar case and evidence for bosonization in this context is found. 
  The second order Ordinary Differential Equation which describes the unknown part of the solution space of some vacuum Bianchi Cosmologies is completely integrated for Type III, thus obtaining the general solution to Einstein's Field Equations for this case, with the aid of the sixth Painlev\'{e} transcendent $P_{VI}$. For particular representations of $P_{VI}$ we obtain the known Kinnersley two-parameter space-time and a solution of Euclidean signature. The imposition of the space-time generalization of a "hidden" symmetry of the generic Type III spatial slice, enables us to retrieve the two-parameter subfamily without considering the Painlev\'{e} transcendent. 
  The universe today, with structure such as stars, galaxies and black holes, seems to have evolved from a very homogeneous initial state. From this it appears as if the entropy of the universe is decreasing, in violation of the second law of thermodynamics. It has been suggested by Roger Penrose \cite{grossmann:penrose:wcc} that this inconsistency can be solved if one assigns an entropy to the spacetime geometry. He also pointed out that the Weyl tensor has the properties one would expect to find in a description of a gravitational entropy. In this article we make an attempt to use this so-called Weyl curvature conjecture to describe the Hawking-Bekenstein entropy of black holes and the entropy of horizons due to a cosmological constant. 
  The large scale binary black hole effort in numerical relativity has led to an increasing distinction between numerical and mathematical relativity. This note discusses this situation and gives some examples of succesful interactions between numerical and mathematical methods is general relativity. 
  We present the recent results of a research project aimed at constructing a robust wave extraction technique for numerical relativity. Our procedure makes use of Weyl scalars to achieve wave extraction. It is well known that, with a correct choice of null tetrad, Weyl scalars are directly associated to physical properties of the space-time under analysis in some well understood way. In particular it is possible to associate $\Psi_4$ with the outgoing gravitational radiation degrees of freedom, thus making it a promising tool for numerical wave--extraction. The right choice of the tetrad is, however, the problem to be addressed. We have made progress towards identifying a general procedure for choosing this tetrad, by looking at transverse tetrads where $\Psi_1=\Psi_3=0$.   As a direct application of these concepts, we present a numerical study of the evolution of a non-linearly disturbed black hole described by the Bondi--Sachs metric. This particular scenario allows us to compare the results coming from Weyl scalars with the results coming from the news function which, in this particular case, is directly associated with the radiative degrees of freedom. We show that, if we did not take particular care in choosing the right tetrad, we would end up with incorrect results. 
  We construct the Einstein field equations on a 4-dimensional brane embedded in an $m$-dimensional bulk where the matter fields are confined to the brane by means of a confining potential. As a result, an extra term in the Friedmann equation in a $m$-dimensional bulk appears that may be interpreted as the X-matter, providing a possible phenomenological explanation for the acceleration of the universe. The study of the relevant observational data suggests good agreement with the predictions of this model. 
  Parametrized field theory (PFT) is free field theory on flat spacetime in a diffeomorphism invariant disguise. It describes field evolution on arbitrary foliations of the flat spacetime instead of only the usual flat ones, by treating the `embedding variables' which describe the foliation as dynamical variables to be varied in the action in addition to the scalar field. A formal Dirac quantization turns the constraints of PFT into functional Schrodinger equations which describe evolution of quantum states from an arbitrary Cauchy slice to an infinitesimally nearby one.This formal Schrodinger picture- based quantization is unitarily equivalent to the standard Heisenberg picture based Fock quantization of the free scalar field if scalar field evolution along arbitrary foliations is unitarily implemented on the Fock space. Torre and Varadarajan (TV) showed that for generic foliations emanating from a flat initial slice in spacetimes of dimension greater than 2, evolution is not unitarily implemented, thus implying an obstruction to Dirac quantization.   We construct a Dirac quantization of PFT,unitarily equivalent to the standard Fock quantization, using techniques from Loop Quantum Gravity (LQG) which are powerful enough to super-cede the no- go implications of the TV results. The key features of our quantization include an LQG type representation for the embedding variables, embedding dependent Fock spaces for the scalar field, an anomaly free representation of (a generalization of) the finite transformations generated by the constraints and group averaging techniques. The difference between 2 and higher dimensions is that in the latter, only finite gauge transformations are defined in the quantum theory, not the infinitesimal ones. 
  We consider the anomaly induced effective action in N=4 super Yang-Mills theory in interaction with the Brans-Dicke (BD) field. The generalization of the BD theory so as to permit an energy exchange between the scalar field and ordinary matter fields, was recently worked out by T. Clifton and J. D. Barrow [Phys. Rev. D 73, 104022 (2006)]. We derive the scalar field equations for the dilaton field, and the BD field, and discuss the Friedmann equation in the general case. The present paper is a continuation of an investigation some years ago dealing with the case of conformal anomaly plus ordinary classical gravity [I. Brevik and S. D. Odintsov, Phys. Lett. B 455, 104 (1999)]. 
  This short paper derives the constant of motion of a scalar field in the gravitational field of a Kerr black hole which is associated to a Killing tensor of that space-time. In addition, there is found a related new symmetry operator S for the solutions of the wave equation in that background. That operator is a partial differential operator with a leading order time derivative of the first order that commutes with a normal form of the wave operator. That form is obtained by multiplication of the wave operator from the left with the reciprocal of the coefficient function of its second order time derivative. It is shown that S induces an operator that commutes with the generator of time evolution in a formulation of the initial value problem for the wave equation in the setting of strongly continuous semigroups. 
  We study the gravitational collapse of a dust dark matter star in a $\Lambda$-background. We consider two distinct cases: First we do not have a dark matter and dark energy coupling; second, we consider that $\Lambda $ decay in dark particles. The approach adopted assumes a modified matter expansion rate and we have formation of a black hole, since that, we have the formation of an apparent horizon. A brief comparison of the process of the matter condensation using the gravitational collapse approach and the linear scalar perturbation theory is considered. 
  We investigate the possibility of assigning consistent probabilities to sets of histories characterized by whether they enter a particular subspace of the Hilbert space of a closed system during a given time interval. In particular we investigate the case that this subspace is a region of the configuration space. This corresponds to a particular class of coarse grainings of spacetime regions. We consider with the arrival time problem and the problem of time in reparametrization invariant theories as for example in canonical quantum gravity. Decoherence conditions and probabilities for those application are derived. The resulting decoherence condition does not depend on the explicit form of the restricted propagator that was problematic for generalizations such as application in quantum cosmology. Closely related is the problem of tunnelling time as well as the quantum Zeno effect. Some interpretational comments conclude, and we discuss the applicability of this formalism to deal with the arrival time problem. 
  In this paper causal geodesic completeness of FLRW cosmological models is analysed in terms of generalised power expansions of the scale factor in coordinate time. The strength of the found singularities is discussed following the usual definitions due to Tipler and Krolak. It is shown that while classical cosmological models are both timelike and lightlike geodesically incomplete, certain observationally alllowed models which have been proposed recently are lightlike geodesically complete. 
  We study the Robinson-Trautman-Maxwell Fields in two closely related coordinate systems, the original Robinson-Trautman (RT) coordinates (in a more general context, often referred to as NU coordinates) and Bondi coordinates. In particular, we identify one of the RT variables as a velocity and then from the Bondi energy-momentum 4-vector, we find kinematic expressions for the mass and momentum in terms of this velocity. From these kinematic expressions and the energy-momentum loss equation we obtain surprising equations of motion for `the center of mass' of the source where the motion takes place in the four-dimensional Poincare translation sub-group of the BMS group. 
  In this article, we investigate the possibility of approximating the physical inner product of constrained quantum theories. In particular, we calculate the physical inner product of a simple cosmological model in two ways: Firstly, we compute it analytically via a trick, secondly, we use the complexifier coherent states to approximate the physical inner product defined by the master constraint of the system. We will find that the approximation is able to recover the analytic solution of the problem, which solidifies hopes that coherent states will help to approximate solutions of more complicated theories, like loop quantum gravity. 
  Non-abelian higher gauge theory has recently emerged as a generalization of standard gauge theory to higher dimensional (2-dimensional in the present context) connection forms, and as such, it has been successfully applied to the non-abelian generalizations of the Yang-Mills theory and 2-form electrodynamics. (2+1)-dimensional gravity, on the other hand, has been a fertile testing ground for many concepts related to classical and quantum gravity, and it is therefore only natural to investigate whether we can find an application of higher gauge theory in this latter context. In the present paper we investigate the possibility of applying the formalism of higher gauge theory to gravity in (2+1) dimensions, and we show that a nontrivial model of (2+1)-dimensional gravity coupled to scalar and tensorial matter fields - the $\Sigma\Phi EA$ model - can be formulated both as a standard gauge theory and as a higher gauge theory. Since the model has a very rich structure - it admits as solutions black-hole BTZ-like geometries, particle-like geometries as well as Robertson-Friedman-Walker cosmological-like expanding geometries - this opens a wide perspective for higher gauge theory to be tested and understood in a relevant gravitational context. Additionally, it offers the possibility of studying gravity in (2+1) dimensions coupled to matter in an entirely new framework. 
  In this paper, we find the energy-momentum distribution of stationary axisymmetric spacetimes in the context of teleparallel theory by using M$\ddot{o}$ller prescription. The metric under consideration is the generalization of the Weyl metrics called the Lewis-Papapetrou metric. The class of stationary axisymmetric solutions of the Einstein field equations has been studied by Galtsov to include the gravitational effect of an {\it external} source. Such spacetimes are also astrophysically important as they describe the exterior of a body in equilibrium. The energy density turns out to be non-vanishing and well-defined and the momentum becomes constant except along $\theta$-direction. It is interesting to mention that the results reduce to the already available results for the Weyl metrics when we take $\omega=0$. 
  We discuss magnetic monopole solutions of the Einstein-Yang-Mills-Higgs equations with a positive cosmological constant. These configurations approach asymptotically the de Sitter spacetime background and exist only for a nonzero Higgs potential. We find that the total mass of the solutions within the cosmological horizon is finite. However, their mass evaluated by using the surface counterterm method outside the cosmological horizon at early/late time infinity generically diverges. Magnetic monopole solutions with finite mass and noninteger charge are found however in a truncation of the theory with a vanishing Higgs field. Both solutions with a regular origin and cosmological black holes are studied, special attention being paid to the computation of the global charges. 
  In the macroscopic gravity approach to the averaging problem in cosmology, the Einstein field equations on cosmological scales are modified by appropriate gravitational correlation terms. We study the averaging problem within the class of spherically symmetric cosmological models. That is, we shall take the microscopic equations and effect the averaging procedure to determine the precise form of the correlation tensor in this case. In particular, by working in volume preserving coordinates, we calculate the form of the correlation tensor under some reasonable assumptions on the form for the inhomogeneous gravitational field and matter distribution. We find that the correlation tensor in a FLRW background must be of the form of a spatial curvature. Inhomogeneities and spatial averaging, through this spatial curvature correction term, can have a very significant dynamical effect on the dynamics of the Universe and cosmological observations; in particular, we discuss whether spatial averaging might lead to a more conservative explanation of the observed acceleration of the Universe (without the introduction of exotic dark matter fields). We also find that the correlation tensor for a non-FLRW background can be interpreted as the sum of a spatial curvature and an anisotropic fluid. This may lead to interesting effects of averaging on astrophysical scales. We also discuss the results of averaging an inhomogeneous Lemaitre-Tolman-Bondi solution as well as calculations of linear perturbations (that is, the back-reaction) in an FLRW background, which support the main conclusions of the analysis. 
  We develop a numerical scheme to make a high-frequency skymap of gravitational-wave backgrounds (GWBs) observed via space-based interferometer. Based on the cross-correlation technique, the intensity distribution of anisotropic GWB can be directly reconstructed from the time-ordered data of cross-correlation signals, with full knowledge of detector's antenna pattern functions. We demonstrate how the planned space interferometer, LISA, can make a skymap of GWB for a specific example of anisotropic signals. At the frequency higher than the characteristic frequency $f_*=1/(2\pi L)$, where $L$ is the arm-length of the detector, the reconstructed skymap free from the instrumental noise potentially reaches the angular resolution up to the multipoles $\ell\sim10$. The presence of instrumental noises degrades the angular resolution. The resultant skymap has angular resolution with multipoles $\ell\leq 6\sim7$ for the anisotropic signals with signal-to-noise ratio S/N$>5$. 
  An essentially complete new paradigm for dynamical black holes in terms of trapping horizons is presented, including dynamical versions of the physical quantities and laws which were considered important in the classical paradigm for black holes in terms of Killing or event horizons. Three state functions are identified as surface integrals over marginal surfaces: irreducible mass, angular momentum and charge. There are three corresponding conservation laws, expressing the rate of change of the state function in terms of flux integrals, or equivalently as divergence laws for associated conserved currents. The currents of energy and angular momentum include the matter energy tensor in a physically appropriate way, plus terms attributable to an effective energy tensor for gravitational radiation. Four other state functions are derived: an effective energy, surface gravity, angular speed and electric potential. There follows a dynamical version of the so-called first law of black-hole mechanics. A corresponding zeroth law holds for null trapping horizons. 
  In this work, in order to compute energy and momentum distributions (due to matter plus fields including gravitation) associated with the Brans-Dicke wormhole solutions we consider Moller's energy-momentum complexes both in general relativity and the teleparallel gravity, and the Einstein energy-momentum formulation in general relativity. We find exactly the same energy and momentum in three of the formulations. The results obtained in teleparallel gravity is also independent of the teleparallel dimensionless coupling parameter, which means that it is valid not only in the teleparallel equivalent of general relativity, but also in any teleparallel model. Furthermore, our results also sustains (a) the importance of the energy-momentum definitions in the evaluation of the energy distribution of a given spacetime and (b) the viewpoint of Lessner that the M{\o}ller energy-momentum complex is a powerful concept of energy and momentum. (c) The results calculated supports the hypothesis by Cooperstock that the energy is confined to the region of non-vanishing energy-momentum tensor of matter and all non-gravitational fields. 
  A specific Lemaitre-Tolman model of a spherically symmetric non-rotating white hole model with a few adjustable parameters is investigated to calculate the momentum four-vector distribution (due to matter plus fields including gravity) in the teleparallel gravity. The energy-momentum distributions (due to matter and fields including gravity) of a model are found to be zero. The result that the total energy and momentum components of a white hole model of the big bang are zero supports the viewpoints of Albrow and Tryon. It is also independent of the teleparallel dimensionless coupling constant, which means that it is valid in any teleparallel model. The results we obtained support the viewpoint of Lessner that the Moller energy-momentum formulation is a powerful concept to calculate energy and momentum distributions associated with the universe, and sustains the importance of the energy-momentum definitions in the evaluation of the energy-momentum distribution of a given space-time. Furthermore, the results obtained here are also agree with those calculated in literature by Cooperstock-Israelit, Rosen, Johri et al., Banerjee-Sen, Vargas and Salti et al.. 
  The polarized ${\bf T}^3$ Gowdy model is, in a standard gauge, characterized by a point particle degree of freedom and a scalar field degree of freedom obeying a linear field equation on ${\bf R}\times{\bf S}^1$. The Fock representation of the scalar field has been well-studied. Here we construct the Schrodinger representation for the scalar field at a fixed value of the Gowdy time in terms of square-integrable functions on a space of distributional fields with a Gaussian probability measure. We show that ``typical'' field configurations are slightly more singular than square-integrable functions on the circle. For each time the corresponding Schrodinger representation is unitarily equivalent to the Fock representation, and hence all the Schrodinger representations are equivalent. However, the failure of unitary implementability of time evolution in this model manifests itself in the mutual singularity of the Gaussian measures at different times. 
  It has been suggested that the Dark Energy Coincidence Problem could be interpreted as a possible link between the cosmological constant and a massive graviton. We show that by using that link and models for the graviton mass a dark energy density can be obtained that is indeed very close to measurements by WMAP. As a consequence of the models, the cosmological constant was found to depend on the density of matter. A brief outline of the cosmological consequences such as the effect on the black hole solution is given. 
  Superconductors have often been used to claim gravitational anomalies in the context of breakthrough propulsion. The experiments could not be reproduced by others up to now, and the theories were either shown to be wrong or are often based on difficult to prove assumptions. We will show that superconductors indeed could be used to produce non-classical gravitational fields, based on the established disagreement between theoretical prediction and measured Cooper-pair mass in Niobium. Tate et al failed to measure the Cooper-pair mass in Niobium as predicted by quantum theory. This has been discussed in the literature without any apparent solution. Based on the work from DeWitt to include gravitomagnetism in the canonical momentum of Cooper-pairs, the authors published a number of papers discussing a possibly involved gravitomagnetic field in rotating superconductors to solve Tate's measured anomaly. Although one possibility to match Tate's measurement, a number of reasons were developed by the authors over the last years to show that the gravitomagnetic field in a rotating quantum material must be different from its classical value and that Tate's result is actually the first experimental sign for it. This paper reviews the latest theoretical approaches to solve the Tate Cooper-pair anomaly based on gravitomagnetic fields in rotating superconductors. 
  It is known that, for a static fluid sphere, the GeneralRelativistic (GR) Effective Mass Energy Density (EMD) appears to be (rho + 3 p), where rho is the bare mass density, p is the isotropic pressure, from a purely localized view point. But since there is no truly local definition of ``gravitational field'', such a notion could actually be misleading. On the other hand, by using the Tolman mass formula, we point out that, from a global perspective, the Active Gravitational Mass Energy Density (AGMD) is sqrt{g_{00}} (rho + 3 p) and which is obviously smaller than (rho + 3p) because g_{00} < 1. Then we show that the AGMD eventually is (rho - 3p), i.e., exactly opposite to what is generally believed. We further identify the AGMD to be proportional to the Ricci Scalar. By using this fundamental and intersting property, we obtain the GR virial theorem in terms of appropriate ``proper energies''. 
  The present paper analyses the Einstein-Cartan theory of gravitation with Elko spinors as sources of curvature and torsion. After minimally coupling the Elko spinors to torsion, the spin angular momentum tensor is derived and its structure is discussed. It shows a much richer structure than the Dirac analogue and hence it is demonstrated that spin one half particles do not necessarily yield only an axial vector torsion component. Moreover, it is argued that the presence of Elko spinors partially solves the problem of minimally coupling Maxwell fields to Einstein-Cartan theory. 
  In order to evaluate the energy distribution (due to matter and fields including gravitation) associated with a space-time model of Szekeres class I and II metrics, we consider the Einstein, Bergmann-Thomson and Landau-Lifshitz energy and/or momentum definitions in the tele-parallel gravity (the tetrad theory of gravitation). We find the same energy distribution using Einstein and Bergmann-Thomson formulations, but we also find that the energy-momentum prescription of Landau-Lifshitz disagree in general with these definitions. This results are the same as a previous works of Aygun et al., they investigated the same problem in general relativity by using Einstein, Bergman-Thomson, Moller and Landau-Lifshitz (LL) energy-momentum complexes. 
  The dynamical effect of the cosmological constant $\Lambda$ on a single spherical void evolving in a the universe is investigated within a non linear perturbation of Newton-Friedmann models. The void expands with a huge initial burst which freezes asymptotically with time up to matching Hubble flow. For $\Omega\_{\circ}\sim 0.3$, $\Lambda$-effect on the kinematics intervenes significantly by amplifying the expansion rate at redshift $z\sim 1.7$. As a result, the size increases with the background density and with $\Lambda$, what interprets by the gravitational attraction of borders from outside regions and the gravitational repulsion of borders. The velocity flow within the void region depends solely on $\Lambda$, it reads $\vec{v}=\sqrt{\Lambda/3}\vec{r}$. Hence, the empty regions are swept out for spatially closed Friedmann models what provides us with a stability criterion. 
  We study two types of axially symmetric, stationary and asymptotically flat spacetimes using highly accurate numerical methods. The one type contains a black hole surrounded by a perfect fluid ring and the other a rigidly rotating disc of dust surrounded by such a ring. Both types of spacetime are regular everywhere (outside of the horizon in the case of the black hole) and fulfil the requirements of the positive energy theorem. However, it is shown that both the black hole and the disc can have negative Komar mass. Furthermore, there exists a continuous transition from discs to black holes even when their Komar masses are negative. 
  We study the phenomenological consequences of amplitude-corrected post-Newtonian (PN) gravitational waveforms, as opposed to the more commonly used restricted PN waveforms, for the quasi-circular, adiabatic inspiral of compact binary objects. In the case of initial detectors it has been shown that the use of amplitude-corrected waveforms for detection templates would lead to significantly lower signal-to-noise ratios (SNRs) than those suggested by simulations based exclusively on restricted waveforms. We further elucidate the origin of the effect by an in-depth analytic treatment. The discussion is extended to advanced detectors, where new features emerge. Non-restricted waveforms are linear combinations of harmonics in the orbital phase, and in the frequency domain the $k$th harmonic is cut off at $k f_{LSO}$, with $f_{LSO}$ the orbital frequency at the last stable orbit. As a result, with non-restricted templates it is possible to achieve sizeable signal-to-noise ratios in cases where the dominant harmonic (which is the one at twice the orbital phase) does not enter the detector's bandwidth. This will have important repercussions on the detection of binary inspirals involving intermediate-mass black holes. For sources at a distance of 100 Mpc, taking into account the higher harmonics will double the mass reach of Advanced LIGO, and that of EGO gets tripled. Conservative estimates indicate that the restricted waveforms underestimate detection rates for intermediate mass binary inspirals by at least a factor of twenty. 
  In this paper we study the effects of $f(R)$ Theories of Gravity on Solar System gravitational tests. In particular, starting from an exact solution of the field equation in vacuum, in the Palatini formalism, we work out the effects that the modifications to the Newtonian potential would induce on the Keplerian orbital elements of the Solar System planets, and compare them with the latest results in planetary orbit determination from the EPM2004 ephemerides. It turns out that the longitudes of perihelia and the mean longitudes are affected by secular precessions. We obtain the resulting best estimate of the parameter $k$ which, being simply related to the scalar curvature, measures the non linearity of the gravitational theory. We use our results to constrain the cosmological constant and show how $f(R)$ functions can be constrained, in principle. What we obtain suggests that, in agreement with other recent papers, the Solar System experiments are not effective to set such constraints, if compared to the cosmologically relevant values. 
  We evaluate the one and two loop contributions to the expectation values of two coincident and gauge invariant scalar bilinears in the theory of massless, minimally coupled scalar quantum electrodynamics on a locally de Sitter background. One of these bilinears is the product of two covariantly differentiated scalars, the other is the product of two undifferentiated scalars. The computations are done using dimensional regularization and the Schwinger-Keldysh formalism. Our results are in perfect agreement with the stochastic predictions at this order. 
  In order to evaluate energy and momentum components associated with two different black hole models, e.g. the electric and magnetic black holes, we use the Moller energy-momentum prescriptions both in Einstein's theory of general relativity and the teleparallel gravity. We obtain the same energy and momentum distributions in both of these different gravitation theories. The energy distribution of the electric black hole depends on the mass M and the magnetic black hole energy distribution depends on the mass M and charge Q. In the process, we notice that (a) the energy obtained in teleparallel gravity is also independent of the teleparallel dimensionless coupling parameter, which means that it is valid not only in teleparallel equivalent of general relativity but also in any teleparallel model, (b) our results also sustains the importance of the energy-momentum definitions in the evaluation of the energy distribution of a given spacetime, and (c) the results obtained support the viewpoint of Lessner that the M{{\o}}ller energy-momentum complex is a powerful concept of energy and momentum. 
  Gravitational theories generated from Lagrangians of the form f(R) are considered. The spherically symmetric solutions to these equations are discussed, paying particular attention to features that differ from the standard Schwarzschild solution. The asymptotic form of solutions is described, as is the lack of validity of Birkhoff's theorem. Exact solutions are presented which illustrate these points and their stability and geodesics are investigated. 
  In a metric variable based Hamiltonian quantization, we give a prescription for constructing semiclassical matter-geometry states for homogeneous and isotropic cosmological models. These "collective" states arise as infinite linear combinations of fundamental excitations in an unconventional "polymer" quantization. They satisfy a number of properties characteristic of semiclassicality, such as peaking on classical phase space configurations. We describe how these states can be used to determine quantum corrections to the classical evolution equations, and to compute the initial state of the universe by a backward time evolution. 
  Equations of motion for a real self-gravitating scalar field in the background of a black hole with negative cosmological constant were solved numerically. We obtain a sequence of static axisymmetric solutions representing thick domain wall cosmological black hole systems, depending on the mass of black hole, cosmological parameter and the parameter binding black hole mass with the width of the domain wall. For the case of extremal cosmological black hole the expulsion of scalar field from the black hole strongly depends on it. 
  We introduce a new top down approach to canonical quantum gravity, called Algebraic Quantum Gravity (AQG):The quantum kinematics of AQG is determined by an abstract $*-$algebra generated by a countable set of elementary operators labelled by an algebraic graph. The quantum dynamics of AQG is governed by a single Master Constraint operator. While AQG is inspired by Loop Quantum Gravity (LQG), it differs drastically from it because in AQG there is fundamentally no topology or differential structure. A natural Hilbert space representation acquires the structure of an infinite tensor product (ITP) whose separable strong equivalence class Hilbert subspaces (sectors) are left invariant by the quantum dynamics. The missing information about the topology and differential structure of the spacetime manifold as well as about the background metric to be approximated is supplied by coherent states. Given such data, the corresponding coherent state defines a sector in the ITP which can be identified with a usual QFT on the given manifold and background. Thus, AQG contains QFT on all curved spacetimes at once, possibly has something to say about topology change and provides the contact with the familiar low energy physics. In particular, in two companion papers we develop semiclassical perturbation theory for AQG and LQG and thereby show that the theory admits a semiclassical limit whose infinitesimal gauge symmetry agrees with that of General Relativity. In AQG everything is computable with sufficient precision and no UV divergences arise due to the background independence of the undamental combinatorial structure. Hence, in contrast to lattice gauge theory on a background metric, no continuum limit has to be taken, there simply is no lattice regulator that must be sent to zero. 
  In the previous article a new combinatorial and thus purely algebraical approach to quantum gravity, called Algebraic Quantum Gravity (AQG), was introduced. In the framework of AQG existing semiclassical tools can be applied to operators that encode the dynamics of AQG such as the Master constraint operator. In this article we will analyse the semiclassical limit of the (extended) algebraic Master constraint operator and show that it reproduces the correct infinitesimal generators of General Relativity. Therefore the question whether General Relativity is included in the semiclassical sector of the theory, which is still an open problem in LQG, can be significantly improved in the framework of AQG. For the calculations we will substitute SU(2) by U(1)^3. That this substitution is justified will be demonstrated in the third article of this series 
  In the two previous papers of this series we defined a new combinatorical approach to quantum gravity, Algebraic Quantum Gravity (AQG). We showed that AQG reproduces the correct infinitesimal dynamics in the semiclassical limit, provided one incorrectly substitutes the non -- Abelean group SU(2) by the Abelean group $U(1)^3$ in the calculations. The mere reason why that substitution was performed at all is that in the non -- Abelean case the volume operator, pivotal for the definition of the dynamics, is not diagonisable by analytical methods. This, in contrast to the Abelean case, so far prohibited semiclassical computations. In this paper we show why this unjustified substitution nevertheless reproduces the correct physical result: Namely, we introduce for the first time semiclassical perturbation theory within AQG (and LQG) which allows to compute expectation values of interesting operators such as the master constraint as a power series in $\hbar$ with error control. That is, in particular matrix elements of fractional powers of the volume operator can be computed with extremely high precision for sufficiently large power of $\hbar$ in the $\hbar$ expansion. With this new tool, the non -- Abelean calculation, although technically more involved, is then exactly analogous to the Abelean calculation, thus justifying the Abelean analysis in retrospect. The results of this paper turn AQG into a calculational discipline. 
  Considering the Einstein, Moller, Bergmann-Thomson, Landau-Lifshitz (LL), Papapetrou, Qadir-Sharif and Weinberg's definitions in general relativity, we find the momentum four-vector of the closed universe based on Marder space-time. The momentum four-vector (due to matter plus field) is found to be zero. These results supports the viewpoints of Banerjee-Sen, Xulu and Aydogdu-Salti. Another point is that our study agree with previous works of Cooperstock-Israelit, Rosen, Johri et al. 
  In this study, using the energy momentum definitions of Einstein, Moller, Bergmann-Thomson, Landau-Lifshitz and Papapetrou we compute the total energy-momentum distribution (due to matter and fields including gravitation) of the universe based on general Bianchi type I-III-V-VI(o) space-time and its transforms type I, III, V, VI(o) metrics, respectively. The energy-momentum densities are found exactly same for Einstein and Bergmann-Thomson definitions. The total energy and momentum is found to be zero for Bianchi types I and VI(o) space-times. These results are same as a previous works of Radinschi, Banerjee-Sen, Xulu and Aydogdu-Salti. Another point is that our study agree with previous works of Cooperstock-Israelit, Rosen, Johri et al. 
  In this paper, we have examined strings with monopole and electric field and domain walls with matter and electric field in the spherically symmetric space-time admitting a one-parameter group of conformal motions. For this purpose, we have solved Einstein's field equations for a spherically symmetric space-time via conformal motions. Also, we have discussed the features of the obtained solutions. 
  The non-abelian generalization of the Born-Infeld non-linear lagrangian is extended to the non-commutative geometry of matrices on a manifold. In this case not only the usual SU(n) gauge fields appear, but also a natural generalization of the multiplet of scalar Higgs fields, with the double-well potential as a first approximation. The matrix realization of non-commutative geometry provides a natural framework in which the notion of a determinant can be easily generalized and used as the lowest-order term in a gravitational lagrangian of a new kind. As a result, we obtain a Born-Infeld-like lagrangian as a root of sufficiently high order of a combination of metric, gauge potentials and the scalar field interactions. We then analyze the behavior of cosmological models based on this lagrangian. It leads to primordial inflation with varying speed, with possibility of early deceleration ruled by the relative strength of the Higgs field. 
  It is argued that the Einstein equation on a braneworld at low curvature cannot be derived from its classical brane dynamics (i. e. dynamics of the brane as a classical object in the higher dimensional spacetime) as far as the action of the system has smooth low-curvature limit. We discuss possible solutions to the problem. 
  The studies of influence of spin on a photon's motion in a Schwartzschild spacetime is continued. In the previous paper [13] the first order correction to the geodesic motion is found for the first half of the photon world line. The system of equations for the first order correction to the geodesic motion is reduced to a non-uniform linear ordinary differential equation. The equation obtained is solved by the standard method of integration of the Green function. 
  We give examples where the Heun function exists in general relativity. It turns out that while a wave equation written in the background of certain metric yields Mathieu functions as its solutions in four space-time dimensions, the trivial generalization to five dimensions results in the double confluent Heun function. We reduce this solution to the Mathieu function with some transformations. 
  In order to evaluate the energy distribution (due to matter and fields including gravitation) associated with a space-time model of Marder universe, we consider the Einstein, Bergmann-Thomson and Landau-Lifshitz energy and momentum definitions in the tele-parallel gravity and the energy-momentum distributions are found to be zero. This results are the same as a previous works of Aygun et al., they investigated the same problem in general relativity by using the Einstein, Moller, Bergmann-Thomson, Landau-Lifshitz (LL), Papapetrou, Qadir-Sharif and Weinberg's definitions. These results supports the viewpoints of Banerjee-Sen, Xulu and Aydogdu-Salti. Another point is that our study agree with previous works of Cooperstock-Israelit, Rosen, Johri textit et al. This paper indicates an important point that these energy-momentum definitions agree with each other not only in general relativity but also in tele-parallel gravity. 
  In this paper, considering the tele-parallel gravity versions of the Einstein, Bergmann-Thomson and Landau-Lifshitz energy-momentum prescriptions energy and momentum distribution of the universe based on the general Bianchi type I-III-V-VI0 universe and its transforms type I, III, V, VI0 metrics, respectively which includes both the matter and gravitational fields are found. We obtain that Einstein and Bergmann-Thomson definitions of the energy-momentum complexes give the same results, while Landau-Lifshitz's energy-momentum definition does not provide same results for these type of metrics. This results are the same as a previous works of Aygun et al., the Authors investigate the same problem in general relativity by using the Einstein, Moller, Bergmann-Thomson, Landau-Lifshitz (LL) and Papapetrou's definitions. Furthermore, we show that for the Bianci type-I and type-VIo all the formulations give the same result. These results supports the viewpoints of Banerjee-Sen, Xulu and Aydogdu-Salti. Another point is that our study agree with previous works of Cooperstock-Israelit, Rosen, Johri et al., Radinschi and Aygun et al. This paper indicates an important point that these energy-momentum definitions agree with each other not only in general relativity but also in tele-parallel gravity. 
  The dilaton-gravity sector of the Two Measures Field Theory (TMT)is explored in detail in the context of cosmology. The model possesses scale invariance which is spontaneously broken due to the intrinsic features of the TMT dynamics. The effective model represents an explicit example of the effective k-essence resulting from first principles without any exotic term in the fundamental action. Depending of the choice of regions in the parameter space, TMT exhibits different possible outputs for cosmological dynamics: a)Dynamical protection from the initial singularity without any tuning of parameters and initial conditions. Power law inflation in the subsequent stage of evolution. Depending on the region in the parameter space (but without fine tuning) the inflation ends with a graceful exit either into the state with zero cosmological constant (CC) or into the state driven by both a small CC and the field phi with a quintessence-like potential. b) Possibility of resolution of the old CC problem. From the point of view of TMT, it becomes clear why the old CC problem cannot be solved (without fine tuning) in conventional field theories. c) TMT enables two ways for achieving small CC without fine tuning of dimensionfull parameters: either by a seesaw type mechanism or due to a correspondence principle between TMT and conventional field theories (i.e theories with only the measure of integration sqrt{-g} in the action. d) The speed c_s of propagation of the cosmological perturbations varies and during the power law inflation c_s>1. e) There is a wide range of the parameters such that in the late time universe: the equation-of-state w=p/\rho <-1; w asymptotically approaches -1 from below; rho approaches a constant, the smallness of which does not require fine tuning of dimensionfull parameters. 
  The relativistic theory of gravitation has the considerable difficulties by description of the gravitational field energy. Pseudotensor t00 in the some cases cannot be interpreted as energy density of the gravitational field. In [1] the approach was proposed, which allow to express the energy density of such a field through the components of a metric tensor. This approach based on the consideration of the isothermal compression of the layer consisted of the incoherent matter. It was employ to the cylindrically and spherically symmetrical static gravitational field. In presented paper the approach is developed. 
  Performing a relativistic approximation as the generalization to a curved spacetime of the flat space Klein-Gordon equation, an effective Hamiltonian which includes non-minimial coupling between gravity and scalar field and also quartic self-interaction of scalar field term is obtained. 
  We show that the tetrad field whose metric gives the Reissner-Nordstr\"om anti-de Sitter black holes gives the correct value of energy in M{\o}ller tetrad theory of gravitation. 
  In this study, using the energy momentum definitions of Einstein and Bergmann-Thomson, we compute the energy-momentum distribution (due to matter and fields including gravitation) of the universe based on generalized Bianchi type II-VIII-IX space-time and its transformations type II, VIII, IX metrics, respectively. The energy and momentum distributions are found to be exactly same for Einstein and Bergmann-Thomson definitions in Bianchi types space-times. 
  We obtain the energy and momentum distributions of the universe in the Bianchi type V10 spacetime in Moller's tetrad theory of gravity. The energy-momentum (due to matter and fields including gravity) are found to be zero. The results are exactly the same as obtained by Radinschi for a model of the universe based on the Bianchi type V10 metric using the energy and/or momentum prescriptions of Tolman, Bergmann-Thomson, Moller, Einstein, Landau-Lifshitz in general relativity. The result that the total energy and momentum components of the universe are zero supports the viewpoints of Albrow and Tryon. It is also independent of the teleparallel dimensionless coupling constant, which means that it is valid in any teleparallel model. The results we obtained support the viewpoint of Lessner that the Moller energy-momentum formulation is powerful concept to calculate energy and momentum distributions associated with the universe, and sustains the importance of the energy-momentum definitions in the evaluation of the energy-momentum distribution of a given space-time. Also the results obtained here are also agree with ones calculated in literature by Cooperstock-Israelit, Rosen, Johri et al., Banerjee-Sen, Vargas and Salti et al. 
  We employ Moller's energy-momentum formulation in tetrad theory of gravity in order to compute energy and momentum components (due to matter plus fields including gravity) associated with the six cases of Vaidya black hole solutions (the monopole solution, the de Sitter and anti-de Sitter solutions, the charged Vaidya solution, the monopole-de Sitter-charged Vaidya solution, the radiating dyon solution, the Husain solutions). The results obtained agree with those calculated in general relativity by Yang and Jeng, and are independent of teleparallel dimensionless coupling parameter $\lambda$ which means that these results are valid not only in teleparallel equivalent of general relativity but also in any teleparallel model. The energy-momentum distribution for the monopole solution vanishes everywhere, for the other solutions we have non zero energy component, and only the energy distribution of the de Sitter and anti-de Sitter solutions is independent of $t$. Furthermore, our results also sustain (a) the importance of the energy-momentum definitions in the evaluation of the energy distribution of a given spacetime, and (b) the viewpoint of Lessner that the Moller energy-momentum complex is a powerful concept of energy and momentum. The results support the hypothesis of Cooperstock that the energy is confined to the region of non-vanishing energy-momentum tensor of matter and all non-gravitational fields. 
  In this work we consider the possibility of describing the current evolution of the universe, without the introduction of any cosmological constant or dark energy (DE), by modifying the Einstein-Hilbert (EH) action. In the context of the f(R) gravities within the metric formalism, we show that it is possible to find an action without cosmological constant which exactly reproduces the behavior of the EH action with cosmological constant. In addition the f(R) action is analytical at the origin having Minkowski and Schwarzschild as vacuum solutions. The found f(R) action is highly non-trivial and must be written in terms of hypergeometric functions but, in spite of looking somewhat artificial, it shows that the cosmological constant, or more generally the DE, is not a logical necessity. 
  In this study, using the energy momentum definitions of Landau-Lifshitz, Papapetrou and Moller, we compute the energy-momentum distribution (due to matter and fields including gravitation) of the universe based on generalized Bianchi type II-VIII-IX space-time and its transformations type II, VIII, IX metrics, respectively. We have obtained that there is a constant proportion between Landau-Lifshitz (LL) and Papapetrou energy density in investigated Bianchi universes. 
  Symmetric hyperbolic systems of equations are explicitly constructed for a general class of tensor fields by considering their structure as r-fold forms. The hyperbolizations depend on 2r-1 arbitrary timelike vectors. The importance of the so-called "superenergy" tensors, which provide the necessary symmetric positive matrices, is emphasized and made explicit. Thereby, a unified treatment of many physical systems is achieved, as well as of the sometimes called "higher order" systems. The characteristics of these symmetric hyperbolic systems are always physical, and directly related to the null directions of the superenergy tensor, which are in particular principal null directions of the tensor field solutions. Generic energy estimates and inequalities are presented too. 
  We relate the geometrical and the Chern-Simons description of (2+1)-dimensional gravity for spacetimes of topology $R\times S_g$, where $S_g$ is an oriented two-surface of genus $g>1$, for Lorentzian signature and general cosmological constant and the Euclidean case with negative cosmological constant. We show how the variables parametrising the phase space in the Chern-Simons formalism are obtained from the geometrical description and how the geometrical construction of (2+1)-spacetimes via grafting along closed, simple geodesics gives rise to transformations on the phase space. We demonstrate that these transformations are generated via the Poisson bracket by one of the two canonical Wilson loop observables associated to the geodesic, while the other acts as the Hamiltonian for infinitesimal Dehn twists. For spacetimes with Lorentzian signature, we discuss the role of the cosmological constant as a deformation parameter in the geometrical and the Chern-Simons formulation of the theory. In particular, we show that the Lie algebras of the Chern-Simons gauge groups can be identified with the (2+1)-dimensional Lorentz algebra over a commutative ring, characterised by a formal parameter $\Theta_\Lambda$ whose square is minus the cosmological constant. In this framework, the Wilson loop observables that generate grafting and Dehn twists are obtained as the real and the $\Theta_\Lambda$-component of a Wilson loop observable with values in the ring, and the grafting transformations can be viewed as infinitesimal Dehn twists with the parameter $\Theta_\Lambda$. 
  We derive a partially gauge fixed Hamiltonian for black hole formation via real scalar field collapse. The class of models considered includes many theories of physical interest, including spherically symmetric black holes in $D$ spacetime dimensions. The boundary and gauge fixing conditions are chosen to be consistent with generalized Painleve-Gullstrand coordinates, in which the metric is regular across the black hole future horizon. The resulting Hamiltonian is remarkably simple and we argue that it provides a good starting point for studying the quantum dynamics of black hole formation. 
  The dynamical evolution of the scale factor of FRW cosmological model is presented, when the equation of state of the material content assume the form $\rm p=\gamma \rho$, $\rm \gamma=constant$, including the cosmological term. We use the WKB approximation and the relation with the Einstein-Hamilton-Jacobi equation to obtain the exact solutions. 
  The field equations in $f(R)$ gravity derived from the Palatini variational principle and formulated in the Einstein conformal frame yield a cosmological term which varies with time. Moreover, they break the conservation of the energy--momentum tensor for matter, generating the interaction between matter and dark energy. Unlike phenomenological models of interacting dark energy, $f(R)$ gravity derives such an interaction from a covariant Lagrangian which is a function of a relativistically invariant quantity (the curvature scalar $R$). We derive the expressions for the quantities describing this interaction in terms of an arbitrary function $f(R)$, and examine how the simplest phenomenological models of a variable cosmological constant are related to $f(R)$ gravity. Particularly, we show that $\Lambda c^2=H^2(1-2q)$ for a flat, homogeneous and isotropic, pressureless universe. For the Lagrangian of form $R-1/R$, which is the simplest way of introducing current cosmic acceleration in $f(R)$ gravity, the predicted matter--dark energy interaction rate changes significantly in time, and its current value is relatively weak (on the order of 1% of $H_0$), in agreement with astronomical observations. 
  In this article, we discuss the space-time of a global monopole field as a candidate for galactic dark matter in the context of scalar tensor theory. 
  In this study, using the energy momentum definitions of Einstein, Bergmann-Thomson and Landau-Lifshitz in teleparallel gravity, we compute the energy-momentum distribution (due to matter and fields including gravitation) of the universe based on generalized Bianchi type II-VIII-IX space-time and its transformations type II, VIII, IX metrics, respectively. We obtain that Einstein and Bergmann-Thomson definitions of the energy-momentum complexes give the same results, while Landau-Lifshitz's energy-momentum definition does not provide same results for these type of metrics. This results are the same as a previous works for Einstein, Bergmann-Thomson and Landau-Lifshitz solutions of the Authors, they investigated the same problem in general relativity, also our results agree with previous works of Cooperstock-Israelit, Rosen, Johri et al. 
  Motivated by previous works, we study semi-classical cosmological solutions and the wave function of the Wheeler-DeWitt equation in the Bose-Parker-Peleg model. We obtain the wave function of the universe satisfying the suitable boundary condition of the redefined fields, which has not been considered in previous works. For some limiting cases, the Wheeler-DeWitt equation is reduced to the Liouville equation with a boundary, and its solution can be described by well-known functions. The consistent requirement of the boundary condition is related to the avoidance of the curvature singularity. 
  In computational relativity, critical behaviour near the black hole threshold has been studied numerically for several models in the last decade. In this paper we present a spatial Galerkin method, suitable for finding numerical solutions of the Einstein-Dirac equations in spherically symmetric spacetime (in polar/areal coordinates). The method features exact conservation of the total electric charge and allows for a spatial mesh adaption based on physical arclength. Numerical experiments confirm excellent robustness and convergence properties of our approach. Hence, this new algorithm is well suited for studying critical behaviour. 
  A new class of exact solutions of Einstein's field equations with perfect fluid for an LRS Bianchi type-I spacetime is obtained by using a time dependent deceleration parameter. We have obtained a general solution of the field equations from which three models of the universe are derived: exponential, polynomial and sinusoidal form respectively. The behaviour of these models of the universe are also discussed in the frame of reference of recent supernovae Ia observations. 
  General relativity successfully describes space-times at scales that we can observe and probe today, but it cannot be complete as a consequence of singularity theorems. For a long time there have been indications that quantum gravity will provide a more complete, non-singular extension which, however, was difficult to verify in the absence of a quantum theory of gravity. By now there are several candidates which show essential hints as to what a quantum theory of gravity may look like. In particular, loop quantum gravity is a non-perturbative formulation which is background independent, two properties which are essential close to a classical singularity with strong fields and a degenerate metric. In cosmological and black hole settings one can indeed see explicitly how classical singularities are removed by quantum geometry: there is a well-defined evolution all the way down to, and across, the smallest scales. As for black holes, their horizon dynamics can be studied showing characteristic modifications to the classical behavior. Conceptual and physical issues can also be addressed in this context, providing lessons for quantum gravity in general. Here, we conclude with some comments on the uniqueness issue often linked to quantum gravity in some form or another. 
  Quantum corrections to the classical field equations, induced by a scale dependent gravitational constant, are analyzed in the case of the static isotropic metric. The requirement of general covariance for the resulting non-local effective field equations puts severe restrictions on the nature of the solutions that can be obtained. In general the existence of vacuum solutions to the effective field equations restricts the value of the gravitational scaling exponent $\nu^{-1}$ to be a positive integer greater than one. We give further arguments suggesting that in fact only for $\nu^{-1}=3$ consistent solutions seem to exist in four dimensions. 
  We show by using the method of matched asymptotic expansions that a sufficient condition can be derived which determines when a local experiment will detect the cosmological variation of a scalar field which is driving the spacetime variation of a supposed constant of Nature. We extend our earlier analyses of this problem by including the possibility that the local region is undergoing collapse inside a virialised structure, like a galaxy or galaxy cluster. We show by direct calculation that the sufficient condition is met to high precision in our own local region and we can therefore legitimately use local observations to place constraints upon the variation of "constants" of Nature on cosmological scales. 
  We study quasinormal spectrum of massive scalar field in the $D$-dimensional black hole background. We found the qualitatively different dependence on the field mass of the fundamental modes for $D\geq6$. The behaviour of higher modes is qualitatively the same for all $D$. Thus for some particular values of mass (of the field and of the black hole) the spectrum has two dominating oscillations with a very long lifetime. Also we show that the asymptotically high overtones do not depend on the field mass. In addition, we present the generalisation of the Nollert improvement of the continued fraction technique for the numerical calculation of quasi-normal frequencies of $D$-dimensional black holes. 
  Because no closed timelike curve (CTC) on a Lorentzian manifold can be deformed to a point, any such manifold containing a CTC must have a topological feature, to be called a timelike wormhole, that prevents the CTC from being deformed to a point. If all wormholes have horizons, which typically seems to be the case in space-times without exotic matter, then each CTC must transit some timelike wormhole's horizon. Therefore, a Lorentzian manifold containing a CTC may nevertheless be causally well behaving once its horizon's are deleted. For instance, there may be a Cauchy-like surface through which every timelike curve passes one and only once before crossing a horizon. 
  Vacuum polarization in QED in a background gravitational field induces interactions which {\it effectively} modify the classical picture of light rays as the null geodesics of spacetime. After a short introduction on the main aspects of the quantum gravitational optics, as a nontrivial example, we study this effect in the background of NUT space characterizing the spacetime of a spherical mass endowed with a gravitomagnetic monopole charge, the so called NUT factor. 
  Modulo a homogeneous degree of freedom and a global constraint, the linearly polarised Gowdy $T^3$ cosmologies are equivalent to a free scalar field propagating in a fixed nonstationary background. Recently, a new field parameterisation was proposed for the metric of the Gowdy spacetimes such that the associated scalar field evolves in a flat background in 1+1 dimensions with the spatial topology of $S^1$, although subject to a time dependent potential. Introducing a suitable Fock quantisation for this scalar field, a quantum theory was constructed for the Gowdy model in which the dynamics is implemented as a unitary transformation. A question that was left open is whether one might adopt a different, nonequivalent Fock representation by selecting a distinct complex structure. The present work proves that the chosen Fock quantisation is in fact unique (up to unitary equivalence) if one demands unitary implementation of the dynamics and invariance under the group of constant $S^1$ translations. These translations are precisely those generated by the global constraint that remains on the Gowdy model. It is also shown that the proof of uniqueness in the choice of complex structure can be applied to more general field dynamics than that corresponding to the Gowdy cosmologies. 
  A new argument is presented confirming the point of view that a Schwarzschild black hole formed during a collapse process does not radiate. 
  This paper is devoted to investigate the teleparallel versions of the Friedmann models as well as the Lewis-Papapetrou solution. We obtain the tetrad and the torsion fields for both the spacetimes. It is shown that the axial-vector vanishes for the Friedmann models. We discuss the different possibilities of the axial-vector depending on the arbitrary functions $\omega$ and $\psi$ in the Lewis-Papapetrou metric. The vector related with spin has also been evaluated. 
  In this letter we discuss the connection between so-called homoclinic chaos and the violation of energy conditions in locally rotationally symmetric Bianchi type IX models, where the matter is assumed to be non-tilted dust and a positive cosmological constant. We show that homoclinic chaos in these models is an artifact of unphysical assumptions: it requires that there exist solutions with positive matter energy density $\rho>0$ that evolve through the singularity and beyond as solutions with negative matter energy density $\rho<0$. Homoclinic chaos is absent when it is assumed that the dust particles always retain their positive mass.In addition, we discuss more general models: for solutions that are not locally rotionally symmetric we demonstrate that the construction of extensions through the singularity, which is required for homoclinic chaos, is not possible in general. 
  Some future global properties of cosmological solutions for the Einstein-Vlasov-Maxwell system with surface symmetry are presented. Global existence is proved, the homogeneous spacetimes are future complete for causal trajectories, and the same is true for inhomogeneous plane-symmetric solutions with small initial data. In the latter case some decay properties are also obtained at late times. Similar but slightly weaker results hold for hyperbolic symmetry. 
  LISA may make it possible to test the black-hole uniqueness theorems of general relativity, also called the no-hair theorems, by Ryan's method of detecting the quadrupole moment of a black hole using high-mass-ratio inspirals. This test can be performed more robustly by observing inspirals in earlier stages, where the simplifications used in making inspiral predictions by the perturbative and post-Newtonian methods are more nearly correct. Current concepts for future missions such as DECIGO and BBO would allow even more stringent tests by this same method. Recently discovered evidence supports the existence of intermediate-mass black holes (IMBHs). Inspirals of binary systems with one IMBH and one stellar-mass black hole would fall into the frequency band of proposed maximum sensitivity for DECIGO and BBO. This would enable us to perform the Ryan test more precisely and more robustly. We explain why tests based on observations earlier in the inspiral are more robust and provide preliminary estimates of possible optimal future observations. 
  The general solution of the Einstein equation for higher dimensional (HD) spherically symmetric collapse of inhomogeneous dust in presence of a cosmological term, i.e., exact interior solutions of the Einstein field equations is presented for the HD Tolman-Bondi metrics imbedded in a de Sitter background. The solution is then matched to exterior HD Scwarschild-de Sitter. A brief discussion on the causal structure singularities and horizons is provided. It turns out that the collapse proceed in the same way as in the Minkowski background, i.e., the strong curvature naked singularities form and that the higher dimensions seem to favor black holes rather than naked singularities. 
  This thesis investigates in the time domain a particular class of second order perturbations of a perfect fluid non-rotating compact star: those arising from the coupling between first order radial and non-radial perturbations. This problem has been treated by developing a gauge invariant formalism based on the 2-parameter perturbation theory (Sopuerta, Bruni and Gualtieri, 2004) where the radial and non-radial perturbations have been separately parameterized. The non-linear perturbations obey inhomogeneous partial differential equations, where the structure of the differential operator is given by the previous perturbative orders and the source terms are quadratic in the first order perturbations. In the exterior spacetime the sources vanish, thus the gravitational wave properties are completely described by the second order Zerilli and Regge-Wheeler functions.   As main initial configuration we have considered a first order differentially rotating and radially pulsating star. Although at first perturbative order this configuration does not exhibit any gravitational radiation, we have found a new interesting gravitational signal at non-linear order, in which the radial normal modes are precisely mirrored. In addition, a resonance effect is present when the frequencies of the radial pulsations are close to the first axial w-mode. Finally, we have roughly estimated the damping times of the radial pulsations due to the non-linear gravitational emission. The coupling near the resonance results to be a very effective mechanism for extracting energy from the radial oscillations. 
  Several different but equivalent forms of the spherical black hole (Schwarzschild) are known, as well as several derivations. Here a novel form is derived through the power of the action formalism. The method generalizes to any spherical black hole or brane, and the charged hole case (Reissner-Nordstrom) is worked out in detail. 
  We discuss the effect of quantum stress tensor fluctuation in deSitter spacetime upon the expansion of a congruence of timelike geodesics. We find that this effect tends to grow, in contrast to the situation in flat spacetime. This growth potentially leads to observable consequences in inflationary cosmology in the form of density perturbations which depend upon the duration of the inflationary period. This effect may be used to place upper bounds on this duration. 
  In this work, we explore the possibility of evolving (2+1) and (3+1)-dimensional wormhole spacetimes, conformally related to the respective static geometries, within the context of nonlinear electrodynamics. For the (3+1)-dimensional spacetime, it is found that the Einstein field equation imposes a contracting wormhole solution and the obedience of the weak energy condition. Nevertheless, in the presence of an electric field, the latter presents a singularity at the throat, however, for a pure magnetic field the solution is regular. For the (2+1)-dimensional case, it is also found that the physical fields are singular at the throat. Thus, taking into account the principle of finiteness, which states that a satisfactory theory should avoid physical quantities becoming infinite, one may rule out evolving (3+1)-dimensional wormhole solutions, in the presence of an electric field, and the (2+1)-dimensional case coupled to nonlinear electrodynamics. 
  In the present work the massless vector field in the de Sitter (dS) space has been quantized. "Massless" is used here by reference to conformal invariance and propagation on the dS light-cone whereas "massive" refers to those dS fields which contract at zero curvature unambiguously to massive fields in Minkowski space. Due to the gauge invariance of the massless vector field, its covariant quantization requires an indecomposable representation of the de Sitter group and an indefinite metric quantization. We will work with a specific gauge fixing which leads to the simplest one among all possible related Gupta-Bleuler structures. The field operator will be defined with the help of coordinate independent de Sitter waves (the modes) which are simple to manipulate and most adapted to group theoretical matters. The physical states characterized by the divergencelessness condition will for instance be easy to identify. The whole construction is based on analyticity requirements in the complexified pseudo-Riemanian manifold for the modes and the two-point function. 
  In this note it is proposed a class of non-stationary de Sitter, rotating and non-rotating, solutions of Einstein's field equations with a cosmological term of variable function. 
  We propose thin single-layer grating waveguide structures to be used as high-reflectivity, but low thermal noise, alternative to conventional coatings for gravitational wave detector test mass mirrors. Grating waveguide (GWG) coatings can show a reflectivity of up to 100% with an overall thickness of less than a wavelength. We theoretically investigate GWG coatings for 1064nm based on tantala (Ta2O5) on a Silica substrate focussing on broad spectral response and low thickness. 
  We present the analysis of undesirable effect of parametric oscillatory instability in signal recycled LIGO interferometer. The basis for this effect is the excitation of the additional (Stokes) optical mode, with frequency $\omega_1$, and the mirror elastic mode, with frequency $\omega_m$, when optical energy stored in the main FP cavity mode, with frequency $\omega_0$, exceeds the certain threshold and the frequencies are related as $\omega_0\simeq \omega_1+\omega_m$. We show that possibility of parametric instability in this interferometer is relatively small due to stronger sensitivity to detuning. We propose to ``scan'' the frequency range where parametric instability may take place varying the position of signal recycling mirror. 
  We study the classical and quantum Euclidean wormholes for a flat Friedmann-Robertson-Walker universe with a perfect fluid including an ordinary matter source plus a source playing the role of a decaying cosmological term. It is shown that classical Euclidean wormholes exist for this model provided the strong energy condition is satisfied. Moreover, unlike the model adopted by Kim and Page in which quantum wormholes are incompatible with a cosmological constant, we show in the present model that quantum wormholes are compatible with a perfect fluid source including a decaying cosmological term. 
  Motivated by the recent work of Louko and Molgado, we consider the Ashtekar-Horowitz-Boulware model using the projection operator formalism. This paper uses the techniques developed in a recent paper of Klauder and Little to overcome the potential difficulties of this particular model. We also extend the model by including a larger class of functions than previously considered and evaluate the classical limit of the model. 
  Paper withdrawn. Error lemma 7. 
  We consider colliding wave packets consisting of hybrid mixtures of electromagnetic, gravitational and scalar waves. Irrespective of the scalar field, the electromagnetic wave still reflects from the gravitational wave. Some reflection processes are given for different choice of packets in which the Coulomb-like component $\Psi_2$ vanishes. Exact solution for multiple reflection of an electromagnetic wave from successive impulsive gravitational waves is obtained in a closed form. It is shown that a succesive sign flip in the Maxwell spinor arises as a result of encountering with an impulsive train (i.e. the Dirac's comb curvature) of gravitational waves. Such an observable effect may be helpful in the detection of gravitational wave bursts. 
  We investigate general relativistic effects associated with the gravitomagnetic monopole moment of gravitational source through the analysis of the motion of test particles and electromagnetic fields distribution in the spacetime around nonrotating cylindrical NUT source. We consider the circular motion of test particles in NUT spacetime, their characteristics and the dependence of effective potential on the radial coordinate for the different values of NUT parameter and orbital momentum of test particles. It is shown that the bounds of stability for circular orbits are displaced toward the event horizon with the growth of monopole moment of the NUT object. In addition, we obtain exact analytical solutions of Maxwell equations for magnetized and charged cylindrical NUT stars. 
  The problem of derivation of the equations of motion from the field equations is considered. Einstein's field equations have a specific analytical form: They are linear in the second order derivatives and quadratic in the first order derivatives of the field variables. We utilize this particular form and propose a novel algorithm for thederivation of the equations of motion from the field equations. It is based on the condition of the balance between the singular terms of the field equation. We apply the algorithm to a nonlinear Lorentz invariant scalar field model. We show that it results in the Newton law of attraction between the singularities of the field moved on approximately geodesic curves. The algorithm is applicable to the $N$-body problem of the Lorentz invariant field equations. 
  In order to evaluate the energy distribution (due to matter and fields including gravitation) associated with a spacetime model of Taub solution, we consider the Einstein, Bergmann-Thomson and Landau-Lifshitz energy-momentum definitions in the teleparallel gravity (the tetrad theory of gravitation). In the teleparallel gravity using Einstein and Landau-Lifshitz prescriptions, we find the same energy-momentum distribution which is calculated in general relativity by P. Harpen. We also find that the energy-momentum prescription of Bergmann-Thomson in the tetrad theory of gravitation and general relativity. This result agrees with the previous works of Virbhadra, Xulu, Vargas, Vagenas and Salti et al. and supports the viewpoints of Albrow and Tryon. 
  A definition of the gravitational flow and a short description of the recipe of its calculation are presented. 
  Considering the quintom model with arbitrary potential, it is shown that there always exists a solution which evolves from \omega > -1 region to \omega <-1 region. The problem is restricted to the slowly varying potentials, i.e. the slow-roll approximation. The perturbative solutions of the fields are also obtained. 
  The general relativistic modifications to the resistive state in superconductors of second type in the presence of a stationary gravitational field are studied. Some superconducting devices that can measure the gravitational field by its red-shift effect on the frequency of radiation are suggested. It has been shown that by varying the orientation of a superconductor with respect to the earth gravitational field, a corresponding varying contribution to AC Josephson frequency would be added by gravity. A magnetic flux (being proportional to angular velocity of rotation $\Omega$) through a rotating hollow superconducting cylinder with the radial gradient of temperature $\nabla_r T$ is theoretically predicted. The magnetic flux is assumed to be produced by the azimuthal current arising from Coriolis force effect on radial thermoelectric current. Finally the magnetic flux through the superconducting ring with radial heat flow located at the equatorial plane interior the rotating neutron star is calculated. In particular it has been shown that nonvanishing magnetic flux will be generated due to the general relativistic effect of dragging of inertial frames on the thermoelectric current. 
  Quantum field theory in curved spacetime is a theory wherein matter is treated fully in accord with the principles of quantum field theory, but gravity is treated classically in accord with general relativity. It is not expected to be an exact theory of nature, but it should provide a good approximate description when the quantum effects of gravity itself do not play a dominant role. A major impetus to the theory was provided by Hawking's calculation of particle creation by black holes, showing that black holes radiate as perfect black bodies. During the past 30 years, considerable progress has been made in giving a mathematically rigorous formulation of quantum field theory in curved spacetime. Major issues of principle with regard to the formulation of the theory arise from the lack of Poincare symmetry and the absence of a preferred vacuum state or preferred notion of ``particles''. By the mid-1980's, it was understood how all of these difficulties could be overcome for free (i.e., non-self-interacting) quantum fields by formulating the theory via the algebraic approach and focusing attention on the local field observables rather than a notion of ``particles''. However, these ideas, by themselves, were not adequate for the formulation of interacting quantum field theory, even at a perturbative level, since standard renormalization prescriptions in Minkowski spacetime rely heavily on Poincare invariance and the existence of a Poincare invariant vacuum state. However, during the past decade, great progress has been made, mainly due to the importation into the theory of the methods of ``microlocal analysis''. This article will describe the historical development of the subject and describe some of the recent progress. 
  We study cosmological self-reproduction in models of inflation driven by a scalar field $\phi$ with a noncanonical kinetic term ($k$-inflation). We develop a general criterion for the existence of attractors and establish conditions selecting a class of $k$-inflation models that admit a unique attractor solution. We then consider quantum fluctuations on the attractor background. We show that the correlation length of the fluctuations is of order $c_{s}H^{-1}$, where $c_{s}$ is the speed of sound. By computing the magnitude of field fluctuations, we determine the coefficients of Fokker-Planck equations describing the probability distribution of the spatially averaged field $\phi$. The field fluctuations are generally large in the inflationary attractor regime; hence, eternal self-reproduction is a generic feature of $k$-inflation. This is established more formally by demonstrating the existence of stationary solutions of the relevant FP equations. We also show that there exists a (model-dependent) range $\phi_{R}<\phi<\phi_{\max}$ within which large fluctuations are likely to drive the field towards the upper boundary $\phi=\phi_{\max}$, where the semiclassical consideration breaks down. An exit from inflation into reheating without reaching $\phi_{\max}$ will occur almost surely (with probability 1) only if the initial value of $\phi$ is below $\phi_{R}$. In this way, strong self-reproduction effects constrain models of $k$-inflation. 
  The motion of massless spinning test particles is investigated using the Newman-Penrose formalism within the Mathisson-Papapetrou model extended to massless particles by Mashhoon and supplemented by the Pirani condition. When the "multipole reduction world line" lies along a principal null direction of an algebraically special vacuum spacetime, the equations of motion can be explicitly integrated. Examples are given for some familiar spacetimes of this type in the interest of shedding some light on the consequences of this model. 
  The electromagnetic measurements of general relativistic gravitomagnetic effects which can be performed within a conductor embedded in the space-time of slow rotating gravitational object in the presence of magnetic field are proposed. 
  Class of axially symmetric solutions of vacuum Einstein field equations including the Papapetrou solution as particular case has been found. It has been shown that the derived solution describes the external axial symmetric gravitational field of the source with nonvanishing mass. The general solution is obtained for this class of functions. As an example of physical application, the spacetime metric outside a line gravitomagnetic monopole has been obtained from Papapetrou solution of vacuum equations of gravitational field. 
  Recently, there has been much attention devoted to resolving the quantum corrections to the Bekenstein-Hawking entropy of the black hole. In particular, many researchers have expressed a vested interest in the coefficient of the logarithmic term of the black hole entropy correction term. In this paper, we calculate the correction value of the black hole entropy by utilizing the generalized uncertainty principle and obtain the correction term caused by the generalized uncertainty principle. Because in our calculation we think that the Bekenstein-Hawking area theorem is still valid after considering the generalized uncertainty principle, we derive that the coefficient of the logarithmic term of the black hole entropy correction term is negative. This result is different from the known result at present. Our method is valid not only for single horizon spacetime but also for double horizons spacetime. In the whole process, the physics idea is clear and calculation is simple. It offers a new way for studying the condition that Bekenstein-Hawking area theorem is valid. 
  We use the Moller energy-momentum complex in teleparallel gravity to calculate Marder space time. The energy distribution is found to be zero everywhere. This result agrees with previous works of the authors and Cooperstock-Israelit, Salti et al., Israelit, Rosen, Johri et al. and Banerjee-Sen. 
  The cosmological model with two phantom scalar fields with the special choice of field's potential is considered. The obtained regular solution describes a bouncing off with the subsequent transition at the de Sitter stage of the expansion of the Universe. 
  We study the curvaton dynamics in brane-world cosmologies. Assuming that the inflaton field survives without decay after the end of inflation, we apply the curvaton reheating mechanism to Randall-Sundrum and to its curvature corrections: Gauss-Bonnet, induced gravity and combined Gauss-Bonnet and induced gravity cosmological models. In the case of chaotic inflation and requiring suppression of possible short-wavelength generated gravitational waves, we constraint the parameters of a successful curvaton brane-world cosmological model. If density perturbations are also generated by the curvaton field then, the fundamental five-dimensional mass could be much lower than the Planck mass 
  We investigate the possible decay of protons in geodesic circular motion around neutral compact objects. Weak and strong decay rates and the associated emitted powers are calculated using a semi-classical approach. Our results are discussed with respect to distinct ones in the literature, which consider the decay of accelerated protons in electromagnetic fields. A number of consistency checks are presented along the paper. 
  We analyze the change in position (the position shift) of the wave packet of a charged scalar particle due to radiation reaction in the $\hbar \to 0$ limit of quantum electrodynamics. In particular, we re-express the formula previously obtained for the position shift in terms of Green's functions for the electromagnetic field, thus clarifying the relation between the quantum and classical derivations of the radiation-reaction force. 
  In our paper we compute the energy distribution of a magnetic stringy black hole solution in the Weinberg prescription. The metric under consideration describes the dual solution in the string frame that is known as the magnetic stringy black hole solution. The metric is obtained by multiplying the electric metric in the Einstein frame by a factor . The energy distribution depends on the mass M and charge Q. Also, we make a discussion of the results and we compare our result with those obtained in the Einstein and Landau and Lifshitz prescriptions and investigate the connections between the expressions of the energy obtained in these prescriptions. 
  The virialization of matter overdensities within dark energy subsystems is considered under the restrictive assumptions (i) spherical-symmetric density profiles, (ii) time-independent quintessence equation of state parameter, w, and (iii) nothing but gravitational interaction between dark energy scalar field and matter. In addition, the quintessence subsystem is conceived as made of ``particles'' whose mutual interaction has intensity equal to G(1+3w) and scales as the inverse square of their distance. Then the virial theorem is formulated for subsystems. In the special case of fully clustered quintessence, energy conservation is assumed with regard to either the whole system (global energy conservation), or to the matter subsystem within the tidal potential induced by the quintessence subsystem (partial energy conservation). Further investigation is devoted to a few special values, w=-1/3, -1/2, -2/3, -1. The special case of fully clustered (i.e. collapsing together with the matter) quintessence is studied in detail. The general case of partially clustered quintessence is considered in terms of a degree of quintessence de-clustering, \zeta, ranging from fully clustered (\zeta=0) to completely de-clustered (\zeta=1) quintessence, respectively. The special case of unclustered (i.e. remaining homogeneous) quintessence is also discussed. The trend exhibited by the fractional (virialization to turnaround) radius, \eta, as a function of other parameters, is found to be different from its counterparts reported in earlier attempts. The reasons of the above mentioned discrepancy are discussed. 
  We propose a relativistically covariant model of interacting dark energy based on the principle of least action. The cosmological term $\Lambda$ in the gravitational Lagrangian is a function of the trace of the energy--momentum tensor $T$. We find that the $\Lambda(T)$ gravity is more general than the Palatini $f(R)$ gravity, and reduces to the latter if we neglect the pressure of matter. We show that recent cosmological data favor a variable cosmological constant and are consistent with the $\Lambda(T)$ gravity, without knowing the specific function $\Lambda(T)$. 
  We show that the phase transition from the decelerating universe to the accelerating universe, which is of relevance to the cosmological coincidence problem, is possible in the semiclassically quantized two-dimensional dilaton gravity by taking into account the noncommutative field variables during the finite time. Initially, the quantum-mechanically induced energy from the noncommutativity among the fields makes the early universe decelerate and subsequently the universe is accelerating because the dilaton driven cosmology becomes dominant later. 
  Antoci et al. have argued that the horizons of the boost-rotation, Kerr and Schwarzschild solutions are singular, having shown that a certain invariantly-defined acceleration scalar blows up at the horizons. Their examples do not satisfy the usual definition of a singularity. It is argued that using the same term is seriously misleading and it is shown that such divergent functions are natural concomitants of regular horizons. In particular it is noted that the divergence is given by the special relativistic approximation to the overall metric. Earlier work on characterization of horizons by invariants is revisited, a new invariant criterion for them is proposed, and the relation of the acceleration invariant to the Cartan invariants, which are finite at the horizons and completely determine the spacetimes, is examined for the C-metric, Kerr and Schwarzschild cases. An appendix considers coordinate identifications at axes and horizons. 
  The retarded Green function of the electromagnetic field in spacetime of a straight thin cosmic string is found. It splits into a geodesic part (corresponding to the propagation along null rays) and to the field scattered on the string. With help of the Green function the electric and magnetic fields of simple sources are constructed. It is shown that these sources are influenced by the cosmic string through a self-interaction with their field. The distant field of static sources is studied and it is found that it has a different multipole structure than in Minkowski spacetime. On the other hand, the string suppresses the electric and magnetic field of distant sources--the field is expelled from regions near the string. 
  We obtain Vaidya-like solutions to include both a null fluid and a string fluid in non-spherical (plane symmetric and cylindrical symmetric) anti-de Sitter space-times. Assuming that string fluid diffuse, we find exact solutions of Einstein's field equations. Thus we extend a recent work of Glass and Krisch \cite {gk} to non-spherical anti-de Sitter space-times. 
  Measurement of gravitomagnetic field is of fundamental importance as a test of general relativity. Here we present a new theoretical project for performing such a measurement based on detection of the electric field arising from the interplay between the gravitomagnetic and magnetic fields in the stationary axial-symmetric gravitational field of a slowly rotating massive body. Finally it is shown that precise magnetometers based on superconducting quantum interferometers could not be designed for measurement of the gravitomagnetically induced magnetic field in the cavity of a charged capacitor since they measure the circulation of a vector potential of electromagnetic field, i.e., an invariant quantity including the sum of electric and magnetic fields, and the general-relativistic magnetic part will be totally cancelled by the electric one which is in good agreement with the experimental results. 
  During inflation quantum effects from massless, minimally coupled scalars and gravitons can be strengthened so much that perturbation theory breaks down. To follow the subsequent evolution one must employ a nonperturbative resummation. Starobinski\u{\i} has developed such a technique for simple scalDuring inflation quantum effects from massless, minimally coupled scalars and gravitons can be strengthened so much that perturbation theory breaks down. To follow the subsequent evolution one must employ a nonperturbative resummation. Starobinski\u{\i} has developed such a technique for simple scalar theories. I discuss recent progress in applying this technique to more complicated models. 
  This is the second part of our result on a class of global characteristic problems for the Einstein vacuum equations with small initial data. In the previous work denoted by (I), our attention was focused on prescribing the initial data satisfying the costraints imposed by the characteristic problem. Here we show how the global existence result can be achieved. This part is heavily based on the global results of D.Christodoulou, S.Klainerman and S.Klainerman, F.Nicolo`. 
  We study the dipolar magnetic field configuration in dependence on brane tension and present solutions of Maxwell equations in the internal and external background spacetime of a magnetized spherical star in a Randall-Sundrum II type braneworld. The star is modelled as sphere consisting of perfect highly magnetized fluid with infinite conductivity and frozen-in dipolar magnetic field. With respect to solutions for magnetic fields found in the Schwarzschild spacetime brane tension introduces enhancing corrections both to the interior and the exterior magnetic field. These corrections could be relevant for the magnetic fields of magnetized compact objects as pulsars and magnetars and may provide the observational evidence for the brane tension through the modification of formula for magneto-dipolar emission which gives amplification of electromagnetic energy loss up to few orders depending on the value of the brane tension. 
  We carry out an analytic study of odd-parity perturbations of the self-similar Vaidya space-times that admit a naked singularity. It is found that an initially finite perturbation remains finite at the Cauchy horizon. This holds not only for the gauge invariant metric and matter perturbation, but also for all the gauge invariant perturbed Weyl curvature scalars, including the gravitational radiation scalars. In each case, `finiteness' refers to Sobolev norms of scalar quantities on naturally occurring spacelike hypersurfaces, as well as pointwise values of these quantities. 
  In this article we describe applications of Discrete Differential Forms in computational GR. In particular we consider the initial value problem in vacuum space-times that are spherically symmetric. The motivation to investigate this method is mainly its manifest coordinate independence. Three numerical schemes are introduced, the results of which are compared with the corresponding analytic solutions. The error of two schemes converges quadratically to zero. For one scheme the errors depend strongly on the initial data. 
  Based on the coupling between the spin of a particle and gravitoelectromagnetic field, the equation of motion of a spinning test particle in gravitational field is deduced. From this equation of motion, it is found that the motion of a spinning particle deviates from the geodesic trajectory, and this deviation originates from the coupling between the spin of the particle and gravitoelectromagnetic field, which is also the origin of Lense-Thirring effects. In post-Newtonian approximations, this equation gives out the same results as those of Papapetrou equation. Effect of the deviation of geodesic trajectory is detectable. 
  General relativity is a deterministic theory with non-fixed causal structure. Quantum theory is a probabilistic theory with fixed causal structure. In this paper we build a framework for probabilistic theories with non-fixed causal structure. This combines the radical elements of general relativity and quantum theory. The key idea in the construction is physical compression. A physical theory relates quantities. Thus, if we specify a sufficiently large set of quantities (this is the compressed set), we can calculate all the others. We apply three levels of physical compression. First, we apply it locally to quantities (actually probabilities) that might be measured in a particular region of spacetime. Then we consider composite regions. We find that there is a second level of physical compression for the composite region over and above the first level physical compression for the component regions. Each application of first and second level physical compression is quantified by a matrix. We find that these matrices themselves are related by the physical theory and can therefore be subject to compression. This is the third level of physical compression. This third level of physical compression gives rise to a new mathematical object which we call the causaloid. From the causaloid for a particular physical theory we can calculate verything the physical theory can calculate. This approach allows us to set up a framework for calculating probabilistic correlations in data without imposing a fixed causal structure (such as a background time). We show how to put quantum theory in this framework (thus providing a new formulation of this theory). We indicate how general relativity might be put into this framework and how the framework might be used to construct a theory of quantum gravity. 
  Generally, the Schwarzschild black hole was proved stable through two different methods: the mode-decomposition method and the integral method. In the paper, we show the integral method can only apply to the initial data vanishing at both the horizon and the spatial infinity. It can not treat the initial data only vanishing at the spatial infinity. We give an example to show the misleading information caused by the use of the tortoise coordinates in the perturbation equations. Subsequently, the perturbation equation in the Schwarzschild coordinates is shown not sufficient for the stable study. 
  We carefully study the stable problem of the Rindler space time by the scalar wave perturbation. Using the two different coordinate systems, the scalar wave equation is investigated. The results are different in these two cases. They are analyzed and compared in detail. The conclusions are: (a) the Rindler space time as a whole is not stable; (b) the Rindler space time could exist stably only as a part of the Minkowski space time, and the Minkowski space time could be a real entity independently; (c) there are some defects for the scalar wave equation written by the Rindler coordinates, and it is unsuitable for investigation of the stable properties of the Rindler space time. All these results might shed some lights on the stable properties of the Schwarzschild black hole. It is natural and not unreasonable for one to infer that: (a) perhaps the Regge-Wheeler equation is not sufficient to decide the stable properties; (b) the Schwarzschild black hole as a whole might be really unstable; (c) the Kruskal space time is stable and can exist as a real physical entity ; whereas the Schwarzschild black hole could occur only as part of the Kruskal space time. 
  The tunnel process of the quantum wave from the light cone is carefully discussed. They are applied in the massive quantum particles from the Schwarzschild black hole in the Kruskal metric. The tortoise coordinates prevent one from understanding the tunnel process, and are investigated with care. Furthermore, the massive particles could come out of the black hole either by the Hawking radiation or by the tunnel effect; the tunnel effect might give more information about the black hole. 
  A self-consistent system of interacting nonlinear spinor and scalar fields within the scope of a Bianchi type-I cosmological model filled with perfect fluid is considered. Exact self-consistent solutions to the corresponding field equations are obtained. The role of spinor field in the evolution of the Universe is studied. It is shown that the spinor field gives rise to an accelerated mode of expansion of the Universe. At the early stage of evolution the spinor field nonlinearity generates the acceleration while at the later stage it is done by the nonzero spinor mass. 
  We present analytic solutions of Maxwell equations for infinitely long cylindrical conductors with nonvanishing electric charge and currents in the external background spacetime of a line gravitomagnetic monopole. It has been shown that vertical magnetic field arising around cylindrical conducting shell carrying azimuthal current will be modified by the gravitational field of NUT source. We obtain that the purely general relativistic magnetic field which has no any Newtonian analog will be produced around charged gravitomagnetic monopole. 
  We solve the one loop effective scalar field equations for spatial plane waves in massless, minimally coupled scalar quantum electrodynamics on a locally de Sitter background. The computation is done in two different gauges: a non-de Sitter invariant analogue of Feynman gauge, and in the de Sitter invariant, Lorentz gauge. In each case our result is that the finite part of the conformal counterterm can be chosen so that the mode functions experience no significant one loop corrections at late times. This is in perfect agreement with a recent, all orders stochastic prediction. 
  We consider the energy-momentum definition of the Moller in both general relativity and teleparallel gravity to evaluate the energy distribution (due to both matter and fields including gravitation) associated with the topological black holes with a conformally coupled scalar field. Our results show that the energy depends on the mass M and charge Q of the black holes and cosmological constant $\Lambda$. In the some special limits, the expression of the energy reduces to the energy of the well-known space-times. The results also support the viewpoint of Lessner that the Moller energy-momentum formulation is a powerful concept of the energy-momentum. Furthermore, the energy obtained in teleparallel gravity is also independent of the teleparallel dimensionless coupling constants which means that it is valid not only in the teleparallel equivalent of the general relativity but also in any teleparallel model. 
  We construct exact solutions to the Bianchi equations on a flat spacetime background. When the constraints are satisfied, these solutions represent in- and outgoing linearized gravitational radiation. We then consider the Bianchi equations on a subset of flat spacetime of the form [0,T] x B_R, where B_R is a ball of radius R, and analyze different kinds of boundary conditions on \partial B_R. Our main results are: i) We give an explicit analytic example showing that boundary conditions obtained from freezing the incoming characteristic fields to their initial values are not compatible with the constraints. ii) With the help of the exact solutions constructed, we determine the amount of artificial reflection of gravitational radiation from constraint-preserving boundary conditions which freeze the Weyl scalar Psi_0 to its initial value. For monochromatic radiation with wave number k and arbitrary angular momentum number l >= 2, the amount of reflection decays as 1/(kR)^4 for large kR. iii) For each L >= 2, we construct new local constraint-preserving boundary conditions which perfectly absorb linearized radiation with l <= L. (iv) We generalize our analysis to a weakly curved background of mass M, and compute first order corrections in M/R to the reflection coefficients for quadrupolar odd-parity radiation. For our new boundary condition with L=2, the reflection coefficient is smaller than the one for the freezing Psi_0 boundary condition by a factor of M/R for kR > 1.04. Implications of these results for numerical simulations of binary black holes on finite domains are discussed. 
  We investigate the two-dimensional behavior of gravity coupled to a dynamical unit timelike vector field, i.e. "Einstein-aether theory". The classical solutions of this theory in two dimensions depend on one coupling constant. When this coupling is positive the only solutions are (i) flat spacetime with constant aether, (ii) de Sitter or anti-de Sitter spacetimes with a uniformly accelerated unit vector invariant under a two-dimensional subgroup of SO(2,1) generated by a boost and a null rotation, and (iii) a non-constant curvature spacetime that has no Killing symmetries and contains singularities. In this case the sign of the curvature is determined by whether the coupling is less or greater than one. When instead the coupling is negative only solutions (i) and (iii) are present. This classical study of the behavior of Einstein-aether theory in 1+1 dimensions may provide a starting point for further investigations into semiclassical and fully quantum toy models of quantum gravity with a dynamical preferred frame. 
  We propose the generally covariant action for the theory of a self-coupled complex scalar field and electromagnetism which by virtue of constraints is equivalent, in the regime of long wavelengths, to perfect magnetohydrodynamics (MHD). We recover from it the Euler equation with Lorentz force, and the thermodynamic relations for a prefect fluid. The equation of state of the latter is related to the scalar field's self potential. We introduce 1+3 notation to elucidate the relation between MHD and field variables. In our approach the requirement that the scalar field be single valued leads to the quantization of a certain circulation in steps of $\hbar$; this feature leads, in the classical limit, to the conservation of that circulation. The circulation is identical to that in Oron's generalization of Kelvin's circulation theorem to perfect MHD; we here characterize the new conserved helicity associated with it. We also demonstrate the existence for MHD of two Bernoulli-like theorems for each spacetime symmetry of the flow and geometry; one of these is pertinent to suitably defined potential flow. We exhibit the conserved quantities explicitly in the case that two symmetries are simultaneously present, and give examples. Also in this case we exhibit a new conserved MHD circulation distinct from Oron's, and provide an example. 
  With the help of the boost operator we can model the interaction between a weakly interacting particle(WIMP) of dark matter(DAMA) and an atomic nuclei. Via this "kick" we calculate the total electronic excitation cross section of the helium atom. The bound spectrum of He is calculated through a diagonalization process with a configuration interaction (CI) wavefunction built up from Slater orbitals. All together 19 singly- and doubly-excited atomic sates were taken with total angular momenta of L=0,1 and 2. Our calculation may give a rude estimation about the magnitude of the total excitation cross section which could be measured in later scintillator experiments. The upper limit of the excitation cross section is $9.7\cdot 10^{-8}$ barn. 
  The purpose of this paper is to obtain exact solutions of the Einstein field equations describing traversable wormholes supported by phantom energy. Their relationship to exact solutions in the literature is also discussed, as well as the conditions required to determine such solutions. 
  LRS Bianchi type-I models have been studied in the cosmological theory based on Lyra's geometry. A new class of exact solutions has been obtained by considering a time dependent displacement field for variable deceleration parameter models of the universe. We have compared our models with those of Einstein's field theory with the cosmological term $\Lambda$. Our frame of reference is restricted to the recent Ia observations of supernovae. Some physical behaviour of the models is also examined in the presence of perfect fluids. 
  Consistency of $GL(3,R)$ gauge theory of gravity coupled with an external electromagnetic field, is studied. It is shown that possible restrictions on Maxwell field can be avoided through introduction of auxiliary fields. 
  The primordial perturbations of test scalar fields not affecting the evolution of background may be very interesting since they can be transferred to the curvature perturbations by some mechanisms, and thus under certain condition can be responsible for the structure formation of observable universe. In this brief report we study the primordial perturbations of test scalar fields in various (super)accelerated expanding backgrounds. 
  Our results concern the transition of a quantum string through the singularity of the compactified Milne (CM) space. We restrict our analysis to the string winding around the compact dimension (CD) of spacetime. The CD undergoes contraction to a point followed by re-expansion. We demonstrate that both classical and quantum dynamics of considered string are well defined. Most of presently available calculations strongly suggest that the singularity of a time dependent orbifold is useless as a model of the cosmological singularity. We believe that our results bring, to some extent, this claim into question. 
  The general relativistic kinetic theory including the effect of a stationary gravitational field is applied to the electromagnetic transport processes in conductors. Then it is applied to derive the general relativistic Ohm's law where the gravitomagnetic terms are incorporated. The total electric charge quantity and charge distribution inside conductors carrying conduction current in some relativistic cases are considered. The general relativistic Ohm's law is applied to predict new gravitomagnetic and gyroscopic effects which can, in principle, be used to detect the Lense-Thirring and rotational fields. 
  The linear relation between the entropy and area of a black hole can be derived from the Heisenberg principle, the energy-momentum dispersion relation of special relativity, and general considerations about black holes. There exist results in quantum gravity and related contexts suggesting the modification of the usual dispersion relation and uncertainty principle. One of these contexts is the gravity's rainbow formalism. We analyze the consequences of such a modification for black hole thermodynamics from the perspective of two distinct rainbow realizations built from doubly special relativity. One is the proposal of Magueijo and Smolin and the other is based on a canonical implementation of doubly special relativity put forward recently by the authors. In these scenarios, we obtain modified expressions for the entropy and temperature of black holes. We show that, for a family of doubly special relativity theories satisfying certain properties, the temperature can vanish in the limit of zero black hole mass. For the Magueijo and Smolin proposal, this is only possible for some restricted class of models with bounded energy and unbounded momentum. With the proposal of a canonical implementation, on the other hand, the temperature may vanish for more general theories; in particular, the momentum may also be bounded, with bounded or unbounded energy. This opens new possibilities for the outcome of black hole evaporation in the framework of a gravity's rainbow. 
  The coalescence of massive black holes generates gravitational waves (GWs) that will be measurable by space-based detectors such as LISA to large redshifts. The spins of a binary's black holes have an important impact on its waveform. Specifically, geodetic and gravitomagnetic effects cause the spins to precess; this precession then modulates the waveform, adding periodic structure which encodes useful information about the binary's members. Following pioneering work by Vecchio, we examine the impact upon GW measurements of including these precession-induced modulations in the waveform model. We find that the additional periodicity due to spin precession breaks degeneracies among certain parameters, greatly improving the accuracy with which they may be measured. In particular, mass measurements are improved tremendously, by one to several orders of magnitude. Localization of the source on the sky is also improved, though not as much -- low redshift systems can be localized to an ellipse which is roughly $10- {a few} \times 10$ arcminutes in the long direction and a factor of 2 smaller in the short direction. Though not a drastic improvement relative to analyses which neglect spin precession, even modest gains in source localization will greatly facilitate searches for electromagnetic counterparts to GW events. Determination of distance to the source is likewise improved: We find that relative error in measured luminosity distance is commonly $\sim 0.1-0.4%$ at $z \sim 1$. Finally, with the inclusion of precession, we find that the magnitude of the spins themselves can typically be determined for low redshift systems with an accuracy of about $0.1-10 %$, depending on the spin value, allowing accurate surveys of mass and spin evolution over cosmic time. 
  We present a new development of the causal boundary of spacetimes, originally introduced by Geroch, Kronheimer and Penrose. Given a strongly causal spacetime (or, more generally, a chronological set), we reconsider the GKP ideas to construct a family of completions with a chronology and topology extending the original ones. Many of these completions present undesirable features, like those appeared in previous approaches by other authors. However, we show that all these deficiencies are due to the attachment of an ``unnecessarily big'' boundary. In fact, a notion of ``completion with minimal boundary'' is then introduced in our family such that, when we restrict to these minimal completions, which always exist, all previous objections disappear. The optimal character of our construction is illustrated by a number of satisfactory properties and examples. 
  We discuss conservation laws for gravity theories invariant under general coordinate and local Lorentz transformations. We demonstrate the possibility to formulate these conservation laws in many covariant and noncovariant(ly looking) ways. An interesting mathematical fact underlies such a diversity: there is a certain ambiguity in a definition of the (Lorentz-) covariant generalization of the usual Lie derivative. Using this freedom, we develop a general approach to construction of invariant conserved currents generated by an arbitrary vector field on the spacetime. This is done in any dimension, for any Lagrangian of the gravitational field and of a (minimally or nonminimally) coupled matter field. A development of the "regularization via relocalization" scheme is used to obtain finite conserved quantities for asymptotically nonflat solutions. We illustrate how our formalism works by some explicit examples. 
  The "positive square" of any tensor is presented in a universal and unified manner, valid in Lorentzian manifolds of arbitrary dimension, and independently of any (anti)-symmetry properties of the tensor. For rank-m tensors, the positive square has rank 2m. Positive here means future, that is to say, satisfying the dominant property. The standard energy-momentum and super-energy tensors are recovered as appropriate parts of the general square. A richer structure of principal null directions arises. 
  We use the Ernst-Schwarzschild solution for a black hole immersed in a uniform magnetic field to estimate corrections to the bending angle and time delay due-to presence of weak magnetic fields in galaxies and between galaxies, and also due-to influence of strong magnetic field near supermassive black holes. The magnetic field creates a kind of confinement in space, that leads to increasing of the bending angle and time delay for a ray of light propagating in the equatorial plane. 
  By a simple physical consideration and uncertain principle, we derive that temperature is proportional to the surface gravity and entropy is proportional to the surface area of the black hole. We apply the same consideration to de Sitter space and estimate the temperature and entropy of the space, then we deduce that the entropy is proportional to the boundary surface area. By the same consideration, we estimate the temperature and entropy in the uniformly accelerated system (Rindler coordinate). The cases in higher dimensions are considered. 
  Recently, Jaekel and Reynaud put forth a metric linear extension of general relativity which, in the intentions of its proponents, would be able, among other things, to provide a gravitational mechanism for explaining the Pioneer anomaly without contradicting either the equivalence principle or what we know about the planetary motions. In this paper we perform an independent test of such an hypothesis by showing that the planets' orbits are, in fact, affected by the suggested mechanism as well and comparing the resulting effects with the latest observational determinations. It turns out that the predicted perihelion precessions, expressed in terms of an adjustable free parameter zeta_P set equal to the value used to reproduce the magnitude of the Pioneer anomalous acceleration, are quite different from the observationally determined extra-advances of such Keplerian element for the inner planets. Conversely, the values obtained for zeta_P from the determined perihelion extra-rates of the inner planets turn out to be in disagreement with the value which would be required to accommodate the Pioneer anomaly. As a consequence, the suggested explanation of gravitational origin for the Pioneer anomaly, based on the assumption that zeta_P is constant throughout the Solar System, should be rejected, at least in its present form. 
  We derive the propagator for a massive vector field on a de Sitter background of arbitrary dimension. This propagator is de Sitter invariant and possesses the proper flat spacetime and massless limits. Moreover, the retarded Green's function inferred from it produces the correct classical response to a test source. Our result is expressed in a tensor basis which is convenient for performing quantum field theory computations using dimensional regularization. 
  FRW models of the universe have been studied in the cosmological theory based on Lyra's manifold. A new class of exact solutions has been obtained by considering a time dependent displacement field for variable deceleration parameter from which three models of the universe are derived (i) exponential (ii) polynomial and (iii) sinusoidal form respectively. The behaviour of these models of the universe are also discussed. Finally some possibilities of further problems and their investigations have been pointed out. 
  The stability conditions for the motion of classical test particles in an $% n $-dimensional Induced Matter Kaluza-Klein theory is studied. We show that stabilization requires a variance of the strong energy condition for the induced matter to hold and that it is related to the hierarchy problem. Stabilization of test particles in a FRW universe is also discussed. 
  A new method to constrain gravitational theories depending on the Ricci scalar is presented. It is based on the weak energy condition and yields limits on the parameters of a given theory through the current values of the derivatives of the scale factor of the Friedmann-Robertson-Walker geometry. A further constraint depending on the current value of the snap is also given. Actual constraints (and the corresponding error propagation analysis) are calculated for two examples, which show that the method is useful in limiting the possible $f(R)$ theories. 
  In ``Global existence and scattering for the nonlinear Schrodinger equation on Schwarzschild manifolds'' (math-ph/0002030), ``Semilinear wave equations on the Schwarzschild manifold I: Local Decay Estimates'' (gr-qc/0310091), and ``The wave equation on the Schwarzschild metric II: Local Decay for the spin 2 Regge Wheeler equation'' (gr-qc/0310066), local decay estimates were proven for the (decoupled) Schrodinger, wave, and Regge-Wheeler equations on the Schwarzschild manifold, using commutator methods. Here, we correct a step in the commutator argument. The corrected argument works either for radial semilinear equations or general linear equations. This recovers the results in math-ph/0002030 and gr-qc/0310066, but does not recover the non radial, large data, semilinear result asserted in the gr-qc/0310091. 
  A relativistic modified gravity (MOG) theory leads to a self-consistent, stable gravity theory that can describe the solar system, galaxy and clusters of galaxies data and cosmology. 
  We provide strong numerical evidence for a new no-scalar-hair theorem for black holes in general relativity, which rules out spherical scalar hair of static four dimensional black holes if the scalar field theory, when coupled to gravity, satisfies the Positive Energy Theorem. This sheds light on the no-scalar-hair conjecture for Calabi-Yau compactifications of string theory, where the effective potential typically has negative regions but where supersymmetry ensures the total energy is always positive. In theories where the scalar tends to a negative local maximum of the potential at infinity, we find the no-scalar-hair theorem holds provided the asymptotic conditions are invariant under the full anti-de Sitter symmetry group. 
  We investigate the effects of space noncommutativity and the generalized uncertainty principle on the stability of circular orbits of particles in both a central force potential and Schwarzschild spacetime. We find noncommutative form of the effective potential which up to first order of noncommutativity parameter contains an angular momentum dependent extra term. This angular momentum dependent extra term affects the stability of circular orbits in such a way that the radius of a stable circular orbit in noncommutative space is larger than its commutative counterpart. In the case of large angular momentum, the condition for stability of circular orbits in noncommutative space differs considerably from commutative case. 
  This paper considers the effects of space noncommutativity on the thermodynamics of a Reissner-Nordstr\"{o}m black hole. In the first step, we extend the ordinary formalism of Bekenstein-Hawking to the case of charged black holes in commutative space. In the second step we investigate the effects of space noncommutativity and the generalized uncertainty principle on the thermodynamics of charged black holes. Finally we compare thermodynamics of charged black holes in commutative space with thermodynamics of Schwarzschild black hole in noncommutative space. In this comparison we explore some conceptual relation between charge and space noncommutativity. 
  We show that the cosmological constant favours significantly the growth of voids in the universe. This dynamical effect is investigated within a newtonian approach with an extension to Friedmann-Lema\^{\i}tre model. 
  The analysis of the modifications that the presence of a deformed dispersion relation entails in the roots of the so--called degree of coherence function, for a beam embodying two different frequencies and moving in a Michelson interferometer, is carried out. The conditions to be satisfied, in order to detect this kind of quantum gravity effect, are also obtained. 
  We present a Dirac quantization of generic single-horizon black holes in two-dimensional dilaton gravity. The classical theory is first partially reduced by a spatial gauge choice under which the spatial surfaces extend from a black or white hole singularity to a spacelike infinity. The theory is then quantized in a metric representation, solving the quantum Hamiltonian constraint in terms of (generalized) eigenstates of the ADM mass operator and specifying the physical inner product by self-adjointness of a time operator that is affinely conjugate to the ADM mass. Regularity of the time operator across the horizon requires the operator to contain a quantum correction that distinguishes the future and past horizons and gives rise to a quantum correction in the hole's surface gravity. We expect a similar quantum correction to be present in systems whose dynamics admits black hole formation by gravitational collapse. 
  The performance of optical clocks has strongly progressed in recent years, and accuracies and instabilities of 1 part in 10^18 are expected in the near future. The operation of optical clocks in space provides new scientific and technological opportunities. In particular, an earth-orbiting satellite containing an ensemble of optical clocks would allow a precision measurement of the gravitational redshift, navigation with improved precision, mapping of the earth's gravitational potential by relativistic geodesy, and comparisons between ground clocks. 
  This paper studies Thick domain wall within the framework of Lyra geometry. Their exact solutions are obtained in the background of a five dimensional space-time. The space-time is nonsingular in its both spatial and temporal behavior. The gravitational field of the wall is shown to be attractive in nature. 
  Bermann [ Nuovo Cimento B (1983), 74, 182 ] presented a law of variation of Hubble parameter that yields constant deceleration parameter models of the Universe. In this paper, we study some cosmological models with negative constant deceleration parameter within the framework of Lyra geometry. 
  We analyze two kinds of matched filters for data output of a spherical resonant GW detector. In order to filter the data of a real sphere, a strategy is proposed, firstly using an omnidirectional in-line filter, which is supposed to select periodograms with excitations, secondly by performing a directional filter on such selected periodograms, finding the wave arrival time, direction and polarization. We point out that, as the analytical simplifications occurring in the ideal 6 transducers TIGA sphere do not hold for a real sphere, using a 5 transducers configuration could be a more convenient choice. 
  We give a detailed account of the free field spectrum and the Newtonian limit of the linearized "massive" (Pauli-Fierz), "topologically massive" (Einstein-Hilbert-Chern-Simons) gravity in 2+1 dimensions about a Minkowski spacetime. For a certain ratio of the parameters, the linearized free theory is Jordan-diagonalizable and reduces to a degenerate "Pais-Uhlenbeck" oscillator which, despite being a higher derivative theory, is ghost-free. 
  I construct a finite-dimensional quantum theory from general relativity by a homotopy method. Its quantum history is made up of at least two levels of fermionic elements. Its unitary group has the diffeomorphism group as singular limit. Its gravitational metrical form is the algebraic square. Its spinors are multivectors. 
  A novel technique for solving some head-on collisions of plane homogeneous light-like signals in Einstein-Maxwell theory is described. The technique is a by-product of a re-examination of the fundamental Bell-Szekeres solution in this field of study. Extensions of the Bell-Szekeres collision problem to include light-like shells and gravitational waves are described and a family of solutions having geometrical and topological properties in common with the Bell-Szekeres solution is derived. 
  We present high-angular resolution observations of the circumstellar disk around the massive Herbig Be star R Mon (M~8 Msun) in the continuum at 2.7mm and 1.3mm and the CO 1->0 and 2->1 rotational lines. Based on the new 1.3mm continuum image we estimate a disk mass (gas+dust) of 0.007 Msun and an outer radius of <150 AU. Our CO images are consistent with the existence of a Keplerian rotating gaseous disk around this star. Up to our knowledge, this is the most clear evidence for the existence of Keplerian disks around massive stars reported thus far. The mass and physical characteristics of this disk are similar to thoseof the more evolved T Tauri stars and indicate a shorter timescale for the evolution and dispersal of circumstellar disks around massive stars which lose most of their mass before the star becomes visible. 
  We extend to higher order a recently published method for calculating the deflection angle of light in a general static and spherically symmetric metric. Since the method is convergent we obtain very accurate analytical expressions that we compare with numerical results. 
  Theory of General Relativity is considered from the point of view of its general structure; it has been showed that if one assumes any connection to be a metric connection then Cartan tensor must necessary be completely antisymmetric, and an example of application is considered in the case of Lie group theory. Under a geometrical point of view, Cartan tensor is thought to represent the Torsion of the space; a completely antisymmetric torsion is the superfield which squashes the 7-sphere, inducing the mechanism of spontaneous compactification of the 11-dimensional space, in multidimensional theories, or in alternative, in 4-dimensional theories, torsion is the reaction of the spacetime to the presence of spin, and modified field equations are presented, as well as possible ways to detect torsion from a cosmological and a microscopical point of view. 
  We present numerical results from three-dimensional evolutions of scalar perturbations of Kerr black holes. Our simulations make use of a high-order accurate multi-block code which naturally allows for fixed adaptivity and smooth inner (excision) and outer boundaries. We focus on the quasinormal ringing phase, presenting a systematic method for extraction of the quasinormal mode frequencies and amplitudes and comparing our results against perturbation theory.   The amplitude of each mode depends exponentially on the starting time of the quasinormal regime, which is not defined unambiguously. We show that this time-shift problem can be circumvented by looking at appropriately chosen relative mode amplitudes. From our simulations we extract the quasinormal frequencies and the relative and absolute amplitudes of corotating and counterrotating modes (including overtones in the corotating case). We study the dependence of these amplitudes on the shape of the initial perturbation, the angular dependence of the mode and the black hole spin, comparing against results from perturbation theory in the so-called asymptotic approximation. We also compare the quasinormal frequencies from our numerical simulations with predictions from perturbation theory, finding excellent agreement. Finally we study under what conditions the relative amplitude between given pairs of modes gets maximally excited and present a quantitative analysis of rotational mode-mode coupling. The main conclusions and techniques of our analysis are quite general and, as such, should be of interest in the study of ringdown gravitational waves produced by astrophysical gravitational wave sources. 
  We construct new solutions of the vacuum Einstein field equations in four dimensions via a solution generating method utilizing the SL(2,R) symmetry of the reduced Lagrangian. We apply the method to an accelerating version of the Zipoy-Voorhees solution and generate new solutions which we interpret to be the accelerating versions of the Zipoy-Voorhees generalisation of the Taub-NUT solution (with Lorentzian signature) and the Zipoy-Voorhees generalisation of the Eguchi-Hanson solitons (with Euclidean signature). As an intermediary in the solution-generating process we obtain charged versions of the accelerated Zipoy-Voorhees-like families of solutions. Finally we present the accelerating version of the Taub-NUT solution and discuss its properties. 
  In this work a simple toy model for a free interface between bulk phases in space and time is presented, derived from the balance equations for extensive thermodynamic variables of Meinhold-Heerlein. In this case the free interface represents geodesics in the space-time, allowing the derivation of the Einstein's equations for gravitational fields. The effect of the balance equation is examined and a simple expression for cold dark matter is derived. The thermodynamically meaning of this model is also discussed. 
  We have assumed that in a physical universe a blackhole is created some where. We conjecture that this blackhole will then separate itself from the physical universe and will build up an extra dimensional entity associated with the physical universe. The extra dimensional entity we suppose to be orthogonal to the physical universe. We further conjecture that this blackhole is a Schwartzschild blackhole. We assume that this physical universe and the blackhole span a seven dimensional space with a common time coordinate. We then generate the Einstein equation. Using the time-blackhole and the time-time component of the equation we show that the Hubble parameter is positive and time dependent if we conjecture that both scale factor and the radius of the blackhole reduces exponetially. Under the same assumption we have also calculated the deacceleration parameter and shown that under certain constrain the universe accelerates. 
  We have demonstrated displacement- and frequency-noise free laser interferometry (DFI) by partially implementing a recently proposed optical configuration using bi-directional Mach-Zehnder interferometers (MZI). This partial implementation, the minimum necessary to be called DFI, has confirmed the essential feature of DFI: the combination of two MZI signals can be carried out in a way which cancels displacement noise of the mirrors while maintaining gravitational wave signals. The attained maximum displacement noise suppression allowed a simulated-SNR of 45dB. 
  Noether's theorem is reviewed with a particular focus on an intermediate step between global and local gauge and coordinate transformations, namely linear transformations. We rederive the well known result that global symmetry leads to charge conservation (Noether's first theorem), and show that linear symmetry allows for the current to be expressed as a four divergence. Local symmetry leads to identical conservation of the current and allows for the expression of the charge as two dimensional surface integral (Noether's second theorem). In the context of coordinate transformations, an additional step (Poincare symmetry) is of physical interest and leads to the definition of the symmetric Belinfante stress-energy tensor, which is then shown to be identically zero in generally covariant first order theories. The intermediate step of linear symmetry turns out to be important in general relativity when the customary first order Lagrangian is used, which is covariant only under affine transformations. In addition, we derive explicitely the canonical stress-energy tensor in second order theories in its identically conserved form. Finally, we analyze the relations between the generators of local transformations, the corresponding currents and the Hamiltonian constraints. 
  We report the results of a detailed numerical analysis of a real resonant spherical gravitational wave antenna operating with six resonant two-mode capacitive transducers read out by superconducting quantum interference devices (SQUID) amplifiers. We derive a set of equations to describe the electro-mechanical dynamics of the detector. The model takes into account the effect of all the noise sources present in each transducer chain: the thermal noise associated with the mechanical resonators, the thermal noise from the superconducting impedance matching transformer, the back-action noise and the additive current noise of the SQUID amplifier. Asymmetries in the detector signal-to-noise ratio and bandwidth, coming from considering the transducers not as point-like objects but as sensor with physically defined geometry and dimension, are also investigated. We calculate the sensitivity for an ultracryogenic, 30 ton, 2 meter in diameter, spherical detector with optimal and non-optimal impedance matching of the electrical read-out scheme to the mechanical modes. The results of the analysis is useful not only to optimize existing smaller mass spherical detector like MiniGrail, in Leiden, but also as a technological guideline for future massive detectors. Furthermore we calculate the antenna patterns when the sphere operates with one, three and six resonators. The sky coverage for two detectors based in The Netherlands and Brasil and operating in coincidence is also estimated. Finally, we describe and numerically verify a calibration and filtering procedure useful for diagnostic and detection purposes in analogy with existing resonant bar detectors. 
  We characterize a class of simple FRW models filled by both dark energy and dark matter in notion of a single potential function of the scale factor $a(t)$; $t$ is the cosmological time. It is representing potential of fictitious particle - Universe moving in 1-dimensional well $V(a)$ which the positional variable mimics the evolution of the Universe. Then the class of all dark energy models (called a multiverse) can be regarded as a Banach space naturally equipment in the structure of the Sobolev metric. In this paper we explore notion of $C^{1}$ metric introduced in the multiverse which measure distance between any two dark energy models. If we choose cold dark matter as a reference one then we can find how so far apart are different models offering explanation of present accelerating expansion phase of the Universe. We consider both models with dark energy (models with the generalized Chaplygin gas, models with variable coefficient equation of state $w_{X}=\frac{p_{X}}{\rho_{X}}$ parameterized by redshift $z$, models with phantom matter) as well as models basing on some modification of the Friedmann equation (Cardassian models, Dvali-Gabadadze-Porati brane models). We argue that because observational data still favor the $\Lambda$CDM model all reasonable dark energy models should belong to the nearby neighborhood of this model. 
  In this paper, we transfer the global arrow of time introduced in previous works to the local context, where the arrow turns out to be a local energy flow that points to the same temporal direction in all the universe. This energy flow is what breaks the symmetry between the pairs of time-symmetric processes arising from the time-reversal invariant equations of fundamental physics. We also apply this framework to different kinds of processes, such as scattering processes, quantum measurements and entropy increasing thermodynamic evolutions. 
  An exactly solvable bounce model in loop quantum cosmology is identified which serves as a perturbative basis for realistic bounce scenarios. Its bouncing solutions are derived analytically, demonstrating why recent numerical simulations robustly led to smooth bounces under the assumption of semiclassicality. Several effects, easily included in a perturbative analysis, can however change this smoothness. The effective theory is not only applicable to such situations where numerical techniques become highly involved but also allows one to discuss conceptual issues. For instance, consequences of the notoriously difficult physical inner product can be implemented at the effective level. This indicates that even physical predictions from full quantum gravity can be obtained from perturbative effective equations. 
  Recently, Brownstein and Moffat proposed a gravitational mechanism to explain the Pioneer anomaly based on their scalar-tensor-vector (STVG) metric theory of gravity. In this paper we show that their model, fitted to the presently available data for the anomalous Pioneer 10/11 acceleration, is in contrast with the latest determinations of the perihelion extra-rates of Jupiter, Saturn and Uranus. 
  We consider the critical behavior at the threshold of black hole formation for the five dimensional vacuum Einstein equations satisfying the cohomogeneity-two triaxial Bianchi IX ansatz. Exploiting a discrete symmetry present in this model we predict the existence of a codimension-two attractor. This prediction is confirmed numerically and the codimension-two attractor is identified as a discretely self-similar solution with two unstable modes. 
  We compare two strategies of multi-detector detection of compact binary inspiral signals, namely, the coincidence and the coherent. For simplicity we consider here two identical detectors having the same power spectral density of noise, that of initial LIGO, located in the same place and having the same orientation. We consider the cases of independent noise as well as that of correlated noise. The coincident strategy involves separately making two candidate event lists, one for each detector, and from these choosing those pairs of events from the two lists which lie within a suitable parameter window, which then are called as coincidence detections. The coherent strategy on the other hand involves combining the data phase coherently, so as to obtain a single network statistic which is then compared with a single threshold. Here we attempt to shed light on the question as to which strategy is better. We compare the performances of the two methods by plotting the Receiver Operating Characteristics (ROC) for the two strategies. Several of the results are obtained analytically in order to gain insight. Further we perform numerical simulations in order to determine certain parameters in the analytic formulae and thus obtain the final complete results. We consider here several cases from the relatively simple to the astrophysically more relevant in order to establish our results. The bottom line is that the coherent strategy although more computationally expensive in general than the coincidence strategy, is superior to the coincidence strategy - considerably less false dismissal probability for the same false alarm probability in the viable false alarm regime. 
  A tachyon field having a negative squared-mass $-m_t^2$ can be described in terms of massless fields degenerating infinitely with respect to helicities. This picture for tachyons does not contradict causality. It is seen that the tachyon vector field can be quenched from the interactions with matter fields, and the effects can be represented by a phase factor. The accelerated expansion of the universe and the dark energies are interpreted in terms of the phase factor. 
  We report on a 6% test of the Lense-Thirring effect with the Mars Global Surveyor (MGS) spacecraft and on certain features of motion of Uranus, Neptune and Pluto which contradict the hypothesis that the Pioneer anomaly can be caused by some gravitational mechanism. 
  We present a numerical analysis to simulate the response of a spherical resonant gravitational wave detector and to compute its sensitivity. We compute both the sensitivity of each different transducers and the sensitivity obtain from a coherent analysis of the whole set of transducers. We use our model to work out the transfer function and the strain sensitivity for different designs of spherical detectors. In particular we present the case of 1 meter radius bulk and hollow spheres equipped with transducer in TIGA configuration. 
  A new class of exact solutions of Einstein's field equations with a perfect fluid source, variable gravitational coupling $G$ and cosmological term $\Lambda$ for FRW spacetime is obtained by considering variable deceleration parameter models for the universe. The nature of the variables $G(t)$, $\Lambda(t)$ and the energy density $\rho(t)$ have been examined for three cases : (i) exponential (ii) polynomial and (iii) sinusoidal form. The special types of models for dust, Zel'dovich and radiating universe are also mentioned in all these cases. The behaviour of these models of the universe are also discussed in the light of recent supernovae Ia observations. 
  We consider the Einstein-Maxwell equations in space-dimension $n$. We point out that the Lindblad-Rodnianski stability proof applies to those equations whatever the space-dimension $n\ge 3$. In even space-time dimension $n+1\ge 6$ we use the standard conformal method on a Minkowski background to give a simple proof that the maximal globally hyperbolic development of initial data sets which are sufficiently close to the data for Minkowski space-time and which are Schwarzschildian outside of a compact set lead to geodesically complete space-times, with a complete Scri, with smooth conformal structure, and with the gravitational field approaching the Minkowski metric along null directions at least as fast as $r^{-(n-1)/2}$. 
  We consider a Randall-Sundrum two-brane cosmological model in the low energy gradient expansion approximation by Kanno and Soda. It is a scalar-tensor theory with a specific coupling function and a specific potential. Upon introducing the FLRW metric and perfect fluid matter on both branes in the Jordan frame, the effective dynamical equation for the the A-brane (our Universe) scale factor decouples from the scalar field and B-brane matter leaving only a non-vanishing integration constant (the dark radiation term). We find exact solutions for the A-brane scale factor for four types of matter: cosmological constant, radiation, dust, and cosmological constant plus radiation. We perform a complementary analysis of the dynamics of the scalar field (radion) using phase space methods and examine convergence towards the limit of general relativity. In particular, we find that radion stabilizes at a certain finite value for suitable negative matter densities on the B-brane. Observational constraints from Solar system experiments (PPN) and primordial nucleosynthesis (BBN) are also briefly discussed. 
  We investigate self-gravitating rotating solutions in the Einstein-Skyrme theory. These solutions are globally regular and asymptotically flat. We present a new kind of solutions with zero baryon number, which possess neither a flat limit nor a static limit. 
  Using the energy-momentum complexes of Einstein, Bergmann-Thomson, Landau-Lifshitz (LL), Moller and Papapetrou we have tried to solve energy-momentum and colliding plane wave problems for Bertotti-Robinson (BR) space-time (it has been also a subject of an extensive study in the context of the so-called colliding plane wave problem in General relativity) in General Relativity (GR). Moreover Einstein, Bergmann-Thomson and Landau-Lifshitz (LL) energy-momentum prescriptions have been discussed in Teleparallel Gravity (TP). While Moller and Bergmann-Thomson complexes give exactly same results other energy-momentum complexes do not provide same energy densities. Also we get that both general relativity and teleparallel gravity are equivalent theories. 
  Here we describe a hierarchal and iterative data analysis algorithm used for searching, characterizing, and removing bright, monochromatic binaries from the Laser Interferometer Space Antenna (LISA) data streams. The algorithm uses the F-statistic to provide an initial solution for individual bright sources, followed by an iterative least squares fitting for all the bright sources. Using the above algorithm, referred to as Slice & Dice, we demonstrate the removal of multiple, correlated galactic binaries from simulated LISA data. Initial results indicate that Slice & Dice may be a useful tool for analyzing the forthcoming LISA data. 
  A paradigm deeply rooted in modern numerical relativity calculations prescribes the removal of those regions of the computational domain where a physical singularity may develop. We here challenge this paradigm by performing three-dimensional simulations of the collapse of uniformly rotating stars to black holes without excision. We show that this choice, combined with suitable gauge conditions and the use of minute numerical dissipation, improves dramatically the long-term stability of the evolutions. In turn, this allows for the calculation of the waveforms well beyond what previously possible, providing information on the black-hole ringing and setting a new mark on the present knowledge of the gravitational-wave emission from the stellar collapse to a rotating black hole. 
  A typical approach to developing an analysis algorithm for analyzing gravitational wave data is to assume a particular waveform and use its characteristics to formulate a detection criteria. Once a detection has been made, the algorithm uses those same characteristics to tease out parameter estimates from a given data set. While an obvious starting point, such an approach is initiated by assuming a single, correct model for the waveform regardless of the signal strength, observation length, noise, etc. This paper introduces the method of Bayesian model selection as a way to select the most plausible waveform model from a set of models given the data and prior information. The discussion is done in the scientific context for the proposed Laser Interferometer Space Antenna. 
  The nonuniform black strings branch, which emerges from the critical Gregory-Laflamme string, is numerically constructed in dimensions 6 <= D <= 11 and extended into the strongly non-linear regime. All the solutions are more massive and less entropic than the marginal string. We find the asymptotic values of the mass, the entropy and other physical variables in the limit of large horizon deformations. By explicit metric comparison we verify that the local geometry around the ``waist'' of our most nonuniform solutions is cone-like with less than 10% deviation. We find evidence that in this regime the characteristic length scale has a power-law dependence on a parameter along the branch of the solutions, and estimate the critical exponent. 
  In this paper we tackle the issue of causality in quantum gravity, in the context of 3d spin foam models. We identify the correct procedure for implementing the causality/orientation dependence restriction that reduces the path integral for BF theory to that of quantum gravity in first order form. We construct explicitly the resulting causal spin foam model. We then add matter degrees of freedom to it and construct a causal spin foam model for 3d quantum gravity coupled to matter fields. Finally, we show that the corresponding spin foam amplitudes admit a natural approximation as the Feynman amplitudes of a non-commutative quantum field theory, with the appropriate Feynman propagators weighting the lines of propagation, and that this effective field theory reduces to usual QFT in flat space in the no-gravity limit. 
  Loop quantum cosmology homogeneous models with a massless scalar field show that the big-bang singularity can be replaced by a big quantum bounce. To gain further insight on the nature of this bounce, we study the semi-discrete loop quantum gravity Hamiltonian constraint equation from the point of view of numerical analysis. For illustration purposes, we establish a numerical analogy between the quantum bounces and reflections in finite difference discretizations of wave equations triggered by the use of nonuniform grids or, equivalently, reflections found when solving numerically wave equations with varying coefficients. We show that the bounce is closely related to the method for the temporal update of the system and demonstrate that explicit time-updates in general yield bounces. Finally, we present an example of an implicit time-update devoid of bounces and show back-in-time, deterministic evolutions that reach and partially jump over the big-bang singularity. 
  In a recent paper (gr-qc/0509107) the author and Rick Schoen obtained a generalization to higher dimensions of a classical result of Hawking concerning the topology of black holes. It was proved, for example, that, apart from certain exceptional circumstances, cross sections of the event horizon in stationary black hole spacetimes obeying a standard energy condition are of positive Yamabe type. This implies many well-known restrictions on the topology, and is consistent with recent examples of five dimensional stationary black hole spacetimes with horizon topology $S^2 \times S^1$. In the present paper, we rule out in this setting the possibility of any such exceptional circumstances (which might have permitted, e.g., toroidal cross sections). This follows from the main result, which is a rigidity result for suitably outermost outer marginally trapped surfaces that are not of positive Yamabe type. 
  In this paper we reply to recent claims by Ciufolini and Pavlis about certain aspects of the measurement of the general relativistic Lense-Thirring effect in the gravitational field of the Earth. I) The proposal by such authors of using the existing satellites endowed with some active mechanism of compensation of the non-gravitational perturbations as an alternative strategy to improve the currently ongoing Lense-Thirring tests is unfeasible because of the impact of the uncancelled even zonal harmonics of the geopotential and of some time-dependent tidal perturbations. II) It is shown that their criticisms about the possibility of using the existing altimeter Jason-1 and laser-ranged Ajisai satellites are groundless.III) Ciufolini and Pavlis also claimed that we would have explicitly proposed to use the mean anomaly of the LAGEOS satellites in order to improve the accuracy of the Lense-Thirrring tests. We prove that it is false. In regard to the mean anomaly of the LAGEOS satellites, Ciufolini himself did use such an orbital element in some previously published tests. About the latest test performed with the LAGEOS satellites, IV) we discuss the cross-coupling between the inclination errors and the first even zonal harmonic as another possible source of systematic error affecting it with an additional 9% bias. V) Finally, we stress the weak points of the claims about the origin of the two-nodes LAGEOS-LAGEOS II combination used in that test. 
  In the usual brane-world scenario matter fields are confined to the four-dimensional spacetime, called a 3-brane, embedded in a higher-dimensional space, usually referred to as the bulk spacetime. In this paper we assume that the 3-brane a de Sitter space; there is only one extra spatial dimension, assumed to be time dependent. By using the form of the brane-world energy-momentum tensor suggested by Shiromizu et al. in the five-dimensional Einstein equations, it is shown that whenever the bulk cosmological constant \Lambda is negative, the extra spatial dimension rapidly shrinks during the inflation of the brane. When \Lambda>0, on the other hand, the extra spatial dimension either completely follows the cosmological expansion of the brane or completely ignores it. This behavior resembles the all-or-nothing behavior of ordinary systems in an expanding universe, as recently demonstrated by R.H. Price. 
  It is well-entrenched folklore that torsion gravity theories predict observationally negligible torsion in the solar system, since torsion (if it exists) couples only to the intrinsic spin of elementary particles, not to rotational angular momentum. We argue that this assumption has a logical loophole which can and should be tested experimentally. In the spirit of action=reaction, if a rotating mass like a planet can generate torsion, then a gyroscope should also feel torsion. Using symmetry arguments, we show that to lowest order, the torsion field around a uniformly rotating spherical mass is determined by seven dimensionless parameters. These parameters effectively generalize the PPN formalism and provide a concrete framework for further testing GR. We construct a parametrized Lagrangian that includes both standard torsion-free GR and Hayashi- Shirafuji maximal torsion gravity as special cases. We demonstrate that classic solar system tests rule out the latter and constrain two observable parameters. We show that Gravity Probe B (GPB) is an ideal experiment for further constraining torsion theories, and work out the most general torsion-induced precession of its gyroscope in terms of our torsion parameters 
  Recently, Hawking radiation of the black hole has been studied using the tunnel effect method. It is found that the radiation spectrum of the black hole is not a strictly pure thermal spectrum. How does the departure from pure thermal spectrum affect the entropy? This is a very interesting problem. In this paper, we calculate the partition function by energy spectrum obtained from tunnel effect. Using the partition function, we compute the black hole entropy and derive the expression of the black hole entropy after considering the radiation. And we derive the entropy of charged black hole. In our calculation, we consider not only the correction to the black hole entropy due to fluctuation of energy but also the effect of the change of the black hole charges on entropy. There is no other hypothesis. Our result is more reasonable.According to the fact that the black hole entropy is not divergent, we obtain the lower limit of Banados-Teitelboim-Zanelli black hole energy. That is, the least energy of Banados-Teitelboim-Zanelli black hole, which satisfies the stationary condition in thermodynamics. 
  Recently, Hawking radiation of the black hole has been studied by using the tunnel effect method. It is found that the radiation spectrum of the black hole is not a strictly pure thermal spectrum. How does the departure from pure thermal spectrum affect the entropy? This is a very interesting problem. In this paper, we calculate the partition function through energy spectrum obtained by using the tunnel effect. From the relation between the partition function and canonical entropy, we can derive the entropy of charged black hole. In our calculation, we consider not only the correction to the black hole entropy due to fluctuation of energy, but also the effect of the change in the black hole charges on entropy. There is not any assumption. This makes our result more reliable. 
  Modified dispersion relations(MDRs) as a manifestation of Lorentz invariance violation, have been appeared in alternative approaches to quantum gravity problem. Loop quantum gravity is one of these approaches which evidently requires modification of dispersion relations. These MDRs will affect the usual formulation of the Compton effect. The purpose of this paper is to incorporate the effects of loop quantum gravity MDRs on the formulation of Compton scattering. Using limitations imposed on MDRs parameters from Ultra High Energy Cosmic Rays(UHECR), we estimate the quantum gravity-induced wavelength shift of scattered photons in a typical Compton process. Possible experimental detection of this wavelength shift will provide strong support for underlying quantum gravity proposal. 
  We present the study of exact inhomogeneous cosmological solutions to a four-dimensional low energy limit of string theory containing non-minimal interacting electromagnetic, dilaton and axion fields. We analyze Einstein-Rosen solutions of Einstein-Maxwell-dilaton-axion equations and show, by explicitly taken the asymptotic limits, that they have asymptotically velocity-term dominated (AVTD) singularities. 
  In this paper, we confront the predictions of the power law cardassian model for the baryon power spectrum with the observations of the SDSS galaxy survey. We show that they fit only for very unusual values of the cold dark matter or baryon density parameters, the Hubble parameter or the spectral index of the initial power spectrum. Moreover, the best-fit Cardassian models turn out to be phantom models. If one wants to recover the usual values for these constants, as quoted by the WMAP team, the power law Cardassian model turns out to be indistinguishable from a LCDM model. 
  In this paper we test the hypothesis that the Pioneer anomaly can be of gravitational origin by comparing the predicted model-independent shifts Delta a/a for the semimajor axis of Uranus and Neptune with the Voyager 2 radio-technical distance measurements performed at JPL-NASA. As in the case of other tests based on different methods and data sets (secular perihelion advance, right ascension/declination residuals over about one century), the orbits of the investigated planets are not affected by any anomalous acceleration like that experienced by the Pioneer 10/11 spacecraft. 
  As currently designed, the signal-recycling cavity (SRC) in the Advanced-LIGO interferometer is degenerate. In such a degenerate cavity, the phase fronts of optical fields become badly distorted when the mirror shapes are slightly deformed due to mirror figure error and/or thermal aberration, and this causes significant loss of the signal power and the signal-to-noise ratio (SNR) of a gravitational wave event. Through a numerical modal simulation of the optical fields in a simplified model of an Advanced-LIGO interferometer, We investigate the loss of the SNR and the behavior of both the carrier and signal optical fields, with the SRC at various levels of degeneracy. We show that the loss of the SNR is severe with a degenerate SRC, and a nondegenerate SRC can be used to solve this problem. We identify the optimal level of degeneracy for the SRC, which is achieved when the cavity Gouy phase is between 0.2 and 1.3 radians. We also discuss possible alternative designs of the SRC to achieve this optimal degeneracy. 
  Leopold Halpern, who was a close associate of both Erwin Schroedinger and Paul Dirac before making his own mark as a theoretical physicist of the first rank, died in Tallahassee, Florida on 3 June 2006 after a valiant struggle with cancer. We give an outline of his life and work, including his progress towards a unified gauge theory of gravitation and spin. 
  The classical and relativistic Hamilton-Jacobi approach is applied to the one-dimensional homogeneous potential, $V(q)=\alpha q^n$, where $\alpha$ and $n$ are continuously varying parameters. In the non-relativistic case, the exact analytical solution is determined in terms of $\alpha$, $n$ and the total energy $E$. It is also shown that the non-linear equation of motion can be linearized by constructing a hypergeometric differential equation for the inverse problem $t(q)$. A variable transformation reducing the general problem to that one of a particle subjected to a linear force is also established. For any value of $n$, it leads to a simple harmonic oscillator if $E>0$, an "anti-oscillator" if $E<0$, or a free particle if E=0. However, such a reduction is not possible in the relativistic case. For a bounded relativistic motion, the first order correction to the period is determined for any value of $n$. For $n >> 1$, it is found that the correction is just twice that one deduced for the simple harmonic oscillator ($n=2$), and does not depend on the specific value of $n$. 
  We consider the proposal of gr-qc/0508124 for the extraction of the graviton propagator from the spinfoam formalism. We propose a new ansatz for the boundary state, using which we can write the propagator as an integral over SU(2). The perturbative expansion in the Planck length can be recast into the saddle point expansion of this integral. We compute the leading order and recover the behavior expected from low-energy physics. In particular, we prove that the degenerate spinfoam configurations are suppressed. 
  We obtain the analogue of collapsing Vaidya-like solution to include both a null fluid and a string fluid, with a linear equation of state ($p_{\bot} = k \rho$), in non-spherical (plane symmetric and cylindrically symmetric) anti-de Sitter space-timess. It turns out that the non-spherical collapse of two fluid in anti-de Sitter space-times, in accordance with cosmic censorship, proceed to form black holes, i.e., on naked singularity ever forms, violating hoop conjecture. 
  A physically meaningful local concept of temperature is introduced in quantum field theory on curved spacetime and applied to the example of a massless field on de Sitter space. It turns out in this model that the equilibrium (Gibbs) states which can be prepared by a geodesic observer have in general a varying temperature distribution in the neighborhood of the geodesic and may not even allow for a consistent thermal interpretation close to the horizon. This result, which can be traced back to the Unruh effect, illustrates the failure of a global notion of temperature in curved spacetime and reveals the need for a local concept, as presented here. 
  We study the properties of a future singularity encountered by a perfect fluid observer in tilting spatially homogeneous Bianchi cosmologies. We derive the boost formulae for the Weyl tensor to establish that, for two observers that are asymptotically null with respect to each other, their respective Weyl parameters generally both tend to zero, constant, or infinity together. We examine three classes of typical examples and one exceptional class. Given the behaviour of the Weyl parameter, we can predict that the singularity encountered is a Weyl singularity or a kinematic singularity. The analysis suggests that the kinematic variables are also useful in indicating a singularity in these models. 
  We review the canonical analysis of the Palatini action without going to the time gauge as in the standard derivation of Loop Quantum Gravity. This allows to keep track of the Lorentz gauge symmetry and leads to a theory of Covariant Loop Quantum Gravity. This new formulation does not suffer from the Immirzi ambiguity, it has a continuous area spectrum and uses spin networks for the Lorentz group. Finally, its dynamics can easily be related to Barrett-Crane like spin foam models. 
  We investigate here spherically symmetric gravitational collapse in a spacetime with an arbitrary number of dimensions and with a general {\it type I} matter field, which is a broad class that includes most of the physically reasonable matter forms. We show that given the initial data for matter in terms of the initial density and pressure profiles at an initial surface $t=t_i$ from which the collapse evolves, there exist rest of the initial data functions and classes of solutions of Einstein equations which we construct here, such that the spacetime evolution goes to a final state which is either a black hole or a naked singularity, depending on the nature of initial data and evolutions chosen, and subject to validity of the weak energy condition. The results are discussed and analyzed in the light of the cosmic censorship hypothesis in black hole physics. The formalism here combines the earlier results on gravitational collapse in four dimensions in a unified treatment. Also the earlier work is generalized to higher dimensional spacetimes to allow a study of the effect of number of dimensions on the possible final outcome of the collapse in terms of either a black hole or naked singularity. No restriction is adopted on the number of dimensions, and other limiting assumptions such as self-similarity of spacetime are avoided, in order to keep the treatment general. Our methodology allows to consider to an extent the genericity and stability aspects related to the occurrence of naked singularities in gravitational collapse. 
  We define the notion of a finite-time singularity of a vector field and then discuss a technique suitable for the asymptotic analysis of vector fields and their integral curves in the neighborhood of such a singularity. Having in mind the application of this method to cosmology, we also provide an analysis of the time singularities of an isotropic universe filled with a perfect fluid in general relativity. 
  We revisit the directionally optimal data streams of LISA first introduced in Nayak etal. It was shown that by using appropriate choice of Time delay interferometric (TDI) combinations, a monochromatic fixed source in the barycentric frame can be optimally tracked in the LISA frame. In this work, we study the beaming properties of these optimal streams. We show that all the three streams V+, Vx and Vo with maximum, minimum and zero directional SNR respectively are highly beamed. We study in detail the frequency dependence of the beaming. 
  The multiyear problem of a two-body system consisting of a Reissner-Nordstr\"om black hole and a charged massive particle at rest is here solved by an exact perturbative solution of the full Einstein-Maxwell system of equations. The expressions of the metric and of the electromagnetic field, including the effects of the electromagnetically induced gravitational perturbation and of the gravitationally induced electromagnetic perturbation, are presented in closed analytic formulas. 
  The inspirals of ``small'' ($1 - 100 M_\odot$) compact bodies through highly relativistic orbits of massive (several $\times 10^5 M_\odot -$ several $\times 10^6 M_\odot$) black holes are among the most anticipated sources for the LISA gravitational-wave antenna. The measurement of these waves is expected to map the spacetime of the larger body with high precision, allowing us to test in detail the hypothesis that black hole candidates are described by the Kerr metric of general relativity. In this article, we will briefly describe how these sources can be used to perform such a test. These proposed measurements are often described as ``testing relativity''. This description is at best somewhat glib: Because -- at least to date -- all work related to these measurements assumes general relativity as the theoretical framework in which these tests are performed, the measurements cannot be said to ``test relativity'' in a fundamental way. More accurately, they test the {\it nature of massive compact bodies within general relativity}. A surprising result for such a test could point to deviations from general relativity, and would provide an experimentally motivated direction in which to pursue tests of gravity theories beyond GR. 
  The fractal dimension of large-scale galaxy clustering has been demonstrated to be roughly $D_F \sim 2$ from a wide range of redshift surveys. This statistic is of interest for two main reasons: fractal scaling is an implicit representation of information content, and also the value itself is a geometric signature of area. It is proposed that the fractal distribution of galaxies may thus be interpreted as a signature of holography (``fractal holography''), providing more support for current theories of holographic cosmologies. Implications for entropy bounds are addressed. In particular, because of spatial scale invariance in the matter distribution, it is shown that violations of the spherical entropy bound can be removed. This holographic condition instead becomes a rigid constraint on the nature of the matter density and distribution in the Universe. Inclusion of a dark matter distribution is also discussed, based on theoretical considerations of possible universal CDM density profiles. 
  The accuracy of time-domain solutions of the inhomogeneous Teukolsky equation is improved significantly. Comparing energy fluxes in gravitational waves with highly accurate frequency-domain results for circular equatorial orbits in Schwarzschild and Kerr, we find agreement to within 1% or better, which we believe can be even further improved. We apply our method to orbits for which frequency-domain calculations have a relative disadvantage, specifically high-eccentricity (elliptical and parabolic) "zoom-whirl" orbits, and find the energy fluxes, waveforms, and characteristic strain in gravitational waves. 
  We present a numerical solution of a stationary 5-dimensional spinning cosmic string in the Einstein-Yang-Mills (EYM) model, where the extra bulk coordinate $\psi$ is periodic. It turns out that when $g_{\psi\psi}$ approaches zero, i.e., a closed time-like curve (CTC) would appear, the solution becomes singular. When a negative cosmological constant is incorporated in the model, the singular behaviour is resolved, but the magnetic component of the Yang-Mills field then approaches zero. We also investigated the geometrical structure of the static 5D cosmic string. The matching condition yields no obstruction for an effective angle deficit. Moreover, by considering the angular momentum in bulk space, no helical stucture of time is necessary. Two opposite moving 5D strings could, in contrast with the 4D case, fulfil the Gott condition. So the static geometrical approach don't show the complete picture of CTC forming in this spacetime. 
  We show that the differential-geometric description of matter by differential structures of spacetime leads to a unifying model of the three types of energy in the cosmos: matter, dark matter and dark energy. Using this model we are able to calculate the value of the cosmological constant with Lambda = sqrt(14/27) 8 pi G/c^2 rho_obs = 1.4 10^-52 m^-2. 
  The coordinate system $(\bar{x},\bar{t})$ defined by $r = 2m + K\bar{x}- c K \bar{t}$ and $t=\bar{x}/cK - 1 /cK \int_{r_a}^r (1- 2m/r + K^2)^{1/2} (1 - 2m/r)^{-1}dr$ allow us to write the Schwarzschild metric in the form: \[ds^2=c^2 d\bar{t}^2 + (W^2/K^2 - 2W/K) d\bar{x}^2 + 2c (1 + W/K) d\bar{x}d\bar{t} - r^2 (d\theta^2 + cos^2\theta d\phi^2)\] with $W=(1 - 2m/r + K^2)^{1/2}$, in which the coefficients' pathologies are moved to $r_K = 2m/(1+K^2)$. This new coordinate system is used to study the entrance into a black hole of a rigid line (a line in which the shock waves propagate with velocity c). 
  Motivated by the mounting evidence for dark energy, here we explore the consequences of a fundamental cosmological constant $\Lambda$ for our universe. We show that when the gravitational entropy of a pure DeSitter state ultimately wins over matter, then the thermodynamic arrow of time in our universe must reverse in scales of order a Hubble time. This phenomenon arises from the gravitational instabilities that develop during a DeSitter epoch and turn catastrophic. A reversed arrow of time is clearly in disagreement with observations. Thus we are led to conclude: Nature forbids a fundamental $\Lambda$. Or else general relativity must be modified in the IR regime when $\Lambda$ dominates the expansion of the Universe. 
  The Averaged Null Energy Condition (ANEC) requires that the average along a complete null geodesic of the projection of the stress-energy tensor onto the geodesic tangent vector can never be negative. It is sufficient to rule out many exotic phenomena in general relativity. Subject to certain conditions, we show that the ANEC can never be violated by a quantized minimally coupled free scalar field along a complete null geodesic surrounded by a tubular neighborhood in which the geometry is flat and whose intrinsic causal structure coincides with that induced from the full spacetime. In particular, the ANEC holds in flat space with boundaries, as in the Casimir effect, for geodesics which stay a finite distance away from the boundary 
  Angular momentum can be defined by rearranging the Komar surface integral in terms of a twist form, encoding the twisting around of space-time due to a rotating mass, and an axial vector. If the axial vector is a coordinate vector and has vanishing transverse divergence, it can be uniquely specified under certain generic conditions. Along a trapping horizon, a conservation law expresses the rate of change of angular momentum of a general black hole in terms of angular momentum densities of matter and gravitational radiation. This identifies the transverse-normal block of an effective gravitational-radiation energy tensor, whose normal-normal block was recently identified in a corresponding energy conservation law. Angular momentum and energy are dual respectively to the axial vector and a previously identified vector, the conservation equations taking the same form. Including charge conservation, the three conserved quantities yield definitions of an effective energy, electric potential, angular velocity and surface gravity, satisfying a dynamical version of the so-called first law of black-hole mechanics. A corresponding zeroth law holds for null trapping horizons, resolving an ambiguity in taking the null limit. 
  We present some analytical solutions to the Einstein equations, describing radiating collapsing spheres in the diffusion approximation. Solutions allow for modeling physical reasonable situations. The temperature is calculated for each solution, using a hyperbolic transport equation, which permits to exhibit the influence of relaxational effects on the dynamics of the system. 
  We demonstrate the use of automatic Bayesian inference for the analysis of LISA data sets. In particular we describe a new automatic Reversible Jump Markov Chain Monte Carlo method to evaluate the posterior probability density functions of the a priori unknown number of parameters that describe the gravitational wave signals present in the data. We apply the algorithm to a simulated LISA data set containing overlapping signals from white dwarf binary systems (DWD) and to a separate data set containing a signal from an extreme mass ratio inspiral (EMRI). We demonstrate that the approach works well in both cases and can be regarded as a viable approach to tackle LISA data analysis challenges. 
  We have critically compared different approaches to the cosmological constant problem, which is at the edge of elementary particle physics and cosmology. This problem is deeply connected with the difficulties formulating a theory of quantum gravity. After the 1998 discovery that our universe's expansion is accelerating, the cosmological constant problem has obtained a new dimension. We are mainly interested in the question why the cosmological constant is so small. We have identified four different classes of solutions: a symmetry, a back-reaction mechanism, a violation of (some of) the building blocks of general relativity, and statistical approaches. In this thesis we carefully study all known potential candidates for a solution, but conclude that so far none of the approaches gives a satisfactory solution. A symmetry would be the most elegant solution and we study a new symmetry under transformation to imaginary spacetime. 
  The equations of motion for matter fields are invariant under the shift of the matter lagrangian by a constant. Such a shift changes the energy momentum tensor of matter by T^a_b --> T^a_b +\rho \delta^a_b. In the conventional approach, gravity breaks this symmetry and the gravitational field equations are not invariant under such a shift of the energy momentum tensor. I argue that until this symmetry is restored, one cannot obtain a satisfactory solution to the cosmological constant problem. I describe an alternative perspective to gravity in which the gravitational field equations are [G_{ab} -\kappa T_{ab}] n^an^b =0 for all null vectors n^a. This is obviously invariant under the change T^a_b --> T^a_b +\rho \delta^a_b and restores the symmetry under shifting the matter lagrangian by a constant. These equations are equivalent to G_{ab} = \kappa T_{ab} + Cg_{ab} where C is now an integration constant so that the role of the cosmological constant is very different in this approach. The cosmological constant now arises as an integration constant, somewhat like the mass M in the Schwarzschild metric, the value of which can be chosen depending on the physical context. These equations can be obtained from a variational principle which uses the null surfaces of spacetime as local Rindler horizons and can be given a thermodynamic interpretation. This approach turns out to be quite general and can encompass even the higher order corrections to Einstein's gravity and suggests a principle to determine the form of these corrections in a systematic manner. 
  The generalized Chaplygin gas, characterized by the equation of state $p=-\mathcal{A}/\rho^{\alpha}$, has been considered as a model for dark energy due its dark-energy-like evolution at late times. When dissipative processes are taken account, within the framework of the standard Eckart theory of relativistic irreversible thermodynamics, cosmological analytical solutions are found. Using the truncated causal version of the Israel-Stewart formalism, a suitable model was constructed which cross the $w=-1$ barrier. The future-singularities encounter in both approaches are of a new type, not included in the classification presented by S. Nojiri, S. D. Odintsov and S. Tsujikawa, Phys. Rev. D71, 063004 (2005). 
  We derive conditions for stable tracker solutions for both quintessence and k-essence in a general cosmological background, H^2 \propto f(\rho). We find that tracker solutions are possible only when \eta = d ln f /d ln \rho is constant, aside from a few special cases, which are enumerated. Expressions for the quintessence or k-essence equation of state are derived as a function of \eta and the equation of state of the dominant background component. 
  We introduce a new method for modelling the gravitational wave flux function of a test-mass particle inspiralling into an intermediate mass Schwarzschild black hole which is based on Chebyshev polynomials of the first kind. It is believed that these Intermediate Mass Ratio Inspiral events (IMRI) are expected to be seen in both the ground and space based detectors. Starting with the post-Newtonian expansion from Black Hole Perturbation Theory, we introduce a new Chebyshev approximation to the flux function, which due to a process called Chebyshev economization gives a model with faster convergence than either post-Newtonian or Pad\'e based methods. As well as having excellent convergence properties, these polynomials are also very closely related to the elusive minimax polynomial. We find that at the last stable orbit, the error between the Chebyshev approximation and a numerically calculated flux is reduced, $< 1.8%$, at all orders of approximation. We also find that the templates constructed using the Chebyshev approximation give better fitting factors, in general $> 0.99$, and smaller errors, $< 1/10%$, in the estimation of the Chirp mass when compared to a fiducial exact waveform, constructed using the numerical flux and the exact expression for the orbital energy function, again at all orders of approximation. We also show that in the intermediate test-mass case, the new Chebyshev template is superior to both PN and Pad\'e approximant templates, especially at lower orders of approximation. 
  It is known that Vilenkin's phenomenological equation of state for static straight cosmic strings is inconsistent with Brans-Dicke theory. We will prove that, in the presence of a cosmological constant, this equation of state is consistent with Brans-Dicke theory. The general solution of the full nonlinear field equations, representing the interior of a cosmic string with a cosmological constant is also presented. 
  The dynamics of density and metric perturbations is investigated for the previously developed model where the decay of the vacuum energy into matter (or vice versa) is due to the renormalization group (RG) running of the cosmological constant (CC) term. The evolution of the CC depends on the single parameter \nu, which characterizes the running of the CC produced by the quantum effects of matter fields of the unknown high energy theory below the Planck scale. The sign of \nu indicates whether bosons or fermions dominate in the running. The spectrum of perturbations is computed assuming an adiabatic regime and an isotropic stress tensor. Moreover, the perturbations of the CC term are generated from the simplest covariant form suggested by the RG model under consideration. The corresponding numerical analysis shows that for \nu>0 there is a depletion of the matter power spectrum at low scales (large wave numbers) as compared to the standard LCDM model, whereas for \nu<0 there is an excess of power at low scales. We find that the LSS data rule out the range |\nu|> 10^{-4} while the values |\nu|< 10^{-6} look perfectly acceptable. For \nu<0 the excess of power at low scales grows rapidly and the bound is more severe. From the particle physics viewpoint, the values |\nu|\sim 10^{-6} correspond to the ``desert'' in the mass spectrum above the GUT scale M_X\sim 10^{16} GeV. Our results are consistent with those obtained in other dynamical models admitting an interaction between dark matter and dark energy. We find that the matter power spectrum analysis is a highly efficient method to discover a possible scale dependence of the vacuum energy. 
  We redefine the gravitational angular momentum in the framework of the teleparallel equivalent of general relativity. In similarity to the gravitational energy-momentum, the new definition for the gravitational angular momentum is coordinate independent. By considering the Poisson brackets in the phase space of the theory, we find that the gravitational energy-momentum and angular momentum correspond to a representation of the Poincar\'e group. This result allows us to define Casimir type invariants for the gravitational field. 
  A naive introduction of a dependency of the mass of a black hole on the Schwarzschild time coordinate results in singular behavior of curvature invariants at the horizon, violating expectations from complementarity. If instead a temporal dependence is introduced in terms of a coordinate akin to the river time representation, the Ricci scalar is nowhere singular away from the origin. It is found that for a shrinking mass scale due to evaporation, the null radial geodesics that generate the horizon are slightly displaced from the coordinate singularity. In addition, a changing horizon scale significantly alters the form of the coordinate singularity in diagonal (orthogonal) metric coordinates representing the space-time. A Penrose diagram describing the growth and evaporation of an example black hole is constructed to examine the evolution of the coordinate singularity. 
  This proceeding is intended to be a first introduction to spectral methods. It is written around some simple problems that are solved explicitly and in details and that aim at demonstrating the power of those methods. The mathematical foundation of the spectral approximation is first introduced, based on the Gauss quadratures. The two usual basis of Legendre and Chebyshev polynomials are then presented. The next section is devoted to one dimensional equation solvers using only one domain. Three different methods are described. Techniques using several domains are shown in the last section of this paper and their various merits discussed. 
  The double slit experiment (DSE) is known as an important cornerstone in the foundations of physical theories such as Quantum Mechanics and Special Relativity. A large number of different variants of it were designed and performed over the years. We perform and discuss here a new verion with the somewhat unexpected results of obtaining interference pattern from single-slit screen. This outcome, which shows that the routes of the photons through the array were changed, leads one to discuss it, using the equivalence principle, in terms of geodesics mechanics. We show using either the Brill's version of the canonical formulation of general relativity or the linearized version of it that one may find corresponding and analogous situations in the framework of general relativity. 
  When a black hole is in an empty space on which there is no matter field except that of the Hawking radiation (Hawking field), then the black hole evaporates and the entropy of the black hole decreases. The generalised second law guarantees the increase of the total entropy of the whole system which consists of the black hole and the Hawking field. That is, the increase of the entropy of the Hawking field is faster than the decrease of the black hole entropy. In naive sense, one may expect that the entropy increase of the Hawking field is due to the self-interaction among the composite particles of the Hawking field, and that the "self"-relaxation of the Hawking field results in the entropy increase. Then, when one consider a non-self-interacting matter field as the Hawking field, it is obvious that the self-relaxation does not take place, and one may think that the total entropy does not increase. However, using nonequilibrium thermodynamics which has been developed recently, we find for the non-self-interacting Hawking field that the rate of entropy increase of the Hawking field (the entropy emission rate by the black hole) grows faster than the rate of entropy decrease of the black hole along the black hole evaporation in the empty space. The origin of the entropy increase of the Hawking field is the increase of the black hole temperature. Hence an understanding of the generalised second law in the context of the nonequilibrium thermodynamics is suggested; even if the self-relaxation of the Hawking field does not take place, the temperature increase of the black hole during the evaporation process causes the entropy increase of the Hawking field to result in the increase of the total entropy. 
  In this paper we dynamically determine the mass of the Kuiper Belt Objects by exploiting the latest observational determinations of the orbital motions of the inner planets of the Solar System. Our result, in units of terrestrial masses, is 0.033 +/- 0.115 by modelling the Classical Kuiper Belt Objects as an ecliptic ring of finite thickness. A two-rings model yields for the Resonant Kuiper Belt Objects a value of 0.018 +/- 0.063. Such figures are consistent with recent determinations obtained with ground and space-based optical techniques. Some implications for precise tests of Einsteinian and post-Einsteinian gravity are briefly discussed. 
  We study the (quantum) formation of black holes by spherical domain wall collapse as seen by an asymptotic observer. Using the Wheeler-de Witt equation to describe the collapsing spherical domain wall, we show that the black hole takes an infinite time to form for the asymptotic observer in the quantum theory, just as in the classical treatment. We argue that such observers will therefore see a compact object but never see effects associated with the formation of an event horizon. To explore what signals such observers will see, we study radiation of a scalar quantum field in the collapsing domain wall background. Both the functional Schrodinger approach and an adaptation of Hawking's original calculation indicate that there is radiation from the collapsing domain wall. The radiation may be relevant should gravitational collapse in particle collisions be induced at the LHC if low scale gravity is observed. The radiation is non-thermal, and the total flux radiated diverges when backreaction of the radiation on the collapsing wall is ignored. We discuss the conjecture, based on our analysis, that the domain wall will evaporate completely by non-thermal radiation during the collapse process and never form a black hole or an event horizon. We also attempt to reconcile how it is that an infalling observer, for whom the period of evaporation will be finite and will be completed before any incipient event horizon is crossed, will nevertheless not observe most of the radiation emitted and seen at infinity by an asymptotic observer. Whether or not there is ``evaporation before formation'' no horizon down which information may be lost forms in any finite time, so that gravitational collapse appears to preserve unitarity. (Abridged.) 
  We study the closed universe recollapse conjecture for positively curved Friedmann-Robertson-Walker (FRW) models in the Jordan frame of the second order gravity theory. We analyse the late time evolution of the model with the methods of the dynamical systems. We find that an initially expanding closed FRW universe, starting close to the Minkowski spacetime, may exhibit oscillatory behaviour. 
  Einstein's theory of gravitation that governs the geometry of space-time, coupled with spectacular advance in cosmological observations, promises to deliver a `standard model' of cosmology in the near future. However, local geometry of space constrains, but does not dictate the topology of the cosmos. hence, Cosmic topology has remained an enigmatic aspect of the `standard model' of cosmology. Recent advance in the quantity and quality of observations has brought this issue within the realm of observational query. The breakdown of statistical homogeneity and isotropy of cosmic perturbations is a generic consequence of non trivial cosmic topology arising from to the imposed `crystallographic' periodicity on the eigenstates of the Laplacian. The sky maps of Cosmic Microwave Background (CMB) anisotropy and polarization most promising observations that would carry signatures of a violation of statistical isotropy and homogeneity. Hence, a measurable spectroscopy of cosmic topology is made possible using the Bipolar power spectrum (BiPS) of the temperature and polarization that quantifies violation of statistical isotropy. 
  A precision measurement of the gravitational constant $G$ has been made using a beam balance. Special attention has been given to determining the calibration, the effect of a possible nonlinearity of the balance and the zero-point variation of the balance. The equipment, the measurements and the analysis are described in detail. The value obtained for G is 6.674252(109)(54) 10^{-11} m3 kg-1 s-2. The relative statistical and systematic uncertainties of this result are 16.3 10^{-6} and 8.1 10^{-6}, respectively. 
  LISA is a planned space-based gravitational-wave (GW) detector that would be sensitive to waves from low-frequency sources, in the band of roughly $(0.03 - 0.1) {\rm mHz} \lesssim f \lesssim 0.1 {\rm Hz}$. This is expected to be an extremely rich chunk of the GW spectrum -- observing these waves will provide a unique view of dynamical processes in astrophysics. Here we give a quick survey of some key LISA sources and what GWs can uniquely teach us about these sources. Particularly noteworthy science which is highlighted here is the potential for LISA to track the moderate to high redshift evolution of black hole masses and spins through the measurement of GWs generated from massive black hole binaries (which in turn form by the merger of galaxies and protogalaxies). Measurement of these binary black hole waves has the potential to determine the masses and spins of the constituent black holes with percent-level accuracy or better, providing a unique high-precision probe of an aspect of early structure growth. This article is based on the ``Astrophysics Tutorial'' talk given by the author at the Sixth International LISA Symposium. 
  The comprehensive formulation for loop quantum cosmology in the spatially flat, isotropic model was recently constructed. In this paper, the methods are extended to the anisotropic Bianchi I cosmology. Both the precursor and the improved strategies are applied and the expected results are established: (i) the scalar field again serves as an internal clock and is treated as emergent time; (ii) the total Hamiltonian constraint is derived by imposing the fundamental discreteness and gives the evolution as a difference equation; and (iii) the physical Hilbert space, Dirac observables and semi-classical states are constructed rigorously. It is also shown that the state in the kinematical Hilbert space associated with the classical singularity is decoupled in the difference evolution equation, indicating that the big bounce may take place when any of the area scales undergoes the vanishing behavior. The investigation affirms the robustness of the framework used in the isotropic model by enlarging its domain of validity and provides foundations to conduct the detailed numerical analysis. 
  The above hep-th posting purports -- erroneously -- to be a comment on a Note by me in gr-qc. 
  We apply the generalized uncertainty principle to the thermodynamics of a small black hole. Here we have a black hole system with the UV cutoff. It is shown that the minimal length induced by the GUP interrupts the Gross-Perry-Yaffe phase transition for a small black hole. In order to see whether the black hole remnant takes place a transition to a large black hole, we introduce a black hole in a cavity (IR system). However, we fail to show the phase transition of the remnant to the large black hole. 
  As a toy model for the implementation of the diffeomorphism constraint, the interpretation of the resulting states, and the treatment of ordering ambiguities in loop quantum gravity, we consider the Hilbert space of spatially diffeomorphism invariant states for a scalar field. We give a very explicit formula for the scalar product on this space, and discuss its structure.   Then we turn to the quantization of a certain class of diffeomorphism invariant quantities on that space, and discuss in detail the ordering issues involved. On a technical level these issues bear some similarity to those encountered in full loop quantum gravity. 
  The proposal of this work is to provide an answer to the following question: is it possible to treat the metric of space-time - that in General Relativity (GR) describes the gravitational interaction - as an effective geometry? In other words, to obtain the dynamics of the metric tensor as a consequence of the dynamics of other fields. In this work we will use a slight modfication of the non-linear equation of motion of a spinor field proposed some years ago by Heisenberg, although in a completely distinct context, to obtain a field theory that provides a framework equivalent to the way GR represents the gravitational interaction. In particular we exhibit a solution of the equations of motion that represents the gravitational field of a compact object and compare it with the corresponding Schwarzschild solution of General Relativity. 
  Inhomogeneities are introduced in loop quantum cosmology using regular lattice states, with a kinematical arena similar to that in homogeneous models considered earlier. The framework is intended to encapsulate crucial features of background independent quantizations in a setting accessible to explicit calculations of perturbations on a cosmological background. It is used here only for qualitative insights but can be extended with further more detailed input. One can thus see how several parameters occuring in homogeneous models appear from an inhomogeneous point of view. Their physical roles in several cases then become much clearer, often making previously unnatural choices of values look more natural by providing alternative physical roles. This also illustrates general properties of symmetry reduction at the quantum level and the roles played by inhomogeneities. Moreover, the constructions suggest a picture for gravitons and other metric modes as collective excitations in a discrete theory, and lead to the possibility of quantum gravity corrections in large universes. 
  We analyse the extent of possible computations following Hogarth in Malament-Hogarth (MH) spacetimes, and Etesi and N\'emeti in the special subclass containing rotating Kerr black holes. Hogarth had shown that any arithmetic statement could be resolved in a suitable MH spacetime. Etesi and Nemeti had shown that some \forall \exists relations on natural numbers which are neither universal nor co-universal, can be decided in Kerr spacetimes, and had asked specifically as to the extent of computational limits there. The purpose of this note is to address this question, and further show that MH spacetimes can compute far beyond the arithmetic: effectively Borel statements (so hyperarithmetic in second order number theory, or the structure of analysis) can likewise be resolved: Theorem A. If H is any hyperarithmetic predicate on integers, then there is an MH spacetime in which any query ? n \in H ? can be computed. In one sense this is best possible, as there is an upper bound to computational ability in any spacetime which is thus a universal constant of the space-time M. Theorem C. Assuming the (modest and standard) requirement that space-time manifolds be paracompact and Hausdorff, for any MH spacetime M there will be a countable ordinal upper bound, w(M), on the complexity of questions in the Borel hierarchy resolvable in it. 
  We consider here scalar and electromagnetic perturbations for the Vaidya metric in double-null coordinates. Such an approach allows one to go a step further in the analysis of quasinormal modes for time-dependent spacetimes. Some recent results are refined, and a new non-stationary behavior corresponding to some sort of inertia for quasinormal modes is identified. Our conclusions can enlighten some aspects of the wave scattering by black holes undergoing some mass accretion processes. 
  Semi-classical states in homogeneous loop quantum cosmology (LQC) are constructed by two different ways. In the first approach, we firstly construct an exponentiated annihilation operator. Then a kind of semi-classical (coherent) state is obtained by solving the eigen-equation of that operator. Moreover, we use these coherent states to analyze the semi-classical limit of the quantum dynamics. It turns out that the Hamiltonian constraint operator employed currently in homogeneous LQC has correct classical limit with respect to the coherent states. In the second approach, the other kind of semi-classical state is derived from the mathematical construction of coherent states for compact Lie groups due to Hall. 
  In this work, we study the collapse dynamics of an inhomogeneous spherically symmetric star made of dark matter (DM) and dark energy (DE). The dark matter is taken in the form of a dust cloud while anisotropic fluid is chosen as the candidate for dark energy. It is investigated how dark energy modifies the collapsing process and is examined whether dark energy has any effect on the Cosmic Censorship Conjecture. The collapsing star is assumed to be of finite radius and the space time is divided into three distinct regions $\Sigma$ and $V^{\pm}$, where $\Sigma$ represents the boundary of the star and $V^{-}(V^{+})$ denotes the interior (exterior) of the star. The junction conditions for matching $V^{\pm}$ over $\Sigma$ are specified. Role of Dark energy in the formation of apparent horizon is studied and central singularity is analyzed. 
  In this paper we study the duality in two-field Quintom models of Dark Energy. We find that an expanding universe dominated by Quintom-A field is dual to a contracting universe with Quintom-B field. 
  We study the coupling of massive fermions to the quantum mechanical dynamics of spacetime emerging from the spinfoam approach in three dimensions. We first recall the classical theory before constructing a spinfoam model of quantum gravity coupled to spinors. The technique used is based on a finite expansion in inverse fermion masses leading to the computation of the vacuum to vacuum transition amplitude of the theory. The path integral is derived as a sum over closed fermionic loops wrapping around the spinfoam. The effects of quantum torsion are realised as a modification of the intertwining operators assigned to the edges of the two-complex, in accordance with loop quantum gravity. The creation of non-trivial curvature is modelled by a modification of the pure gravity vertex amplitudes. The appendix contains a review of the geometrical and algebraic structures underlying the classical coupling of fermions to three dimensional gravity. 
  The interaction of a Reissner-Nordstr\"om black hole and a charged massive particle is studied in the framework of perturbation theory. The particle backreaction is taken into account, studying the effect of general static perturbations of the hole following the approach of Zerilli. The solutions of the combined Einstein-Maxwell equations for both perturbed gravitational and electromagnetic fields at first order of the perturbation are exactly reconstructed by summing all multipoles, and are given explicit closed form expressions. The existence of a singularity-free solution of the Einstein-Maxwell system requires that the charge to mass ratios of the black hole and of the particle satisfy an equilibrium condition which is in general dependent on the separation between the two bodies. If the black hole is undercritically charged (i.e. its charge to mass ratio is less than one), the particle must be overcritically charged, in the sense that the particle must have a charge to mass ratio greater than one. If the charge to mass ratios of the black hole and of the particle are both equal to one (so that they are both critically charged, or "extreme"), the equilibrium can exist for any separation distance, and the solution we find coincides with the linearization in the present context of the well known Majumdar-Papapetrou solution for two extreme Reissner-Nordstr\"om black holes. In addition to these singularity-free solutions, we also analyze the corresponding solution for the problem of a massive particle at rest near a Schwarzschild black hole, exhibiting a strut singularity on the axis between the two bodies. The relations between our perturbative solutions and the corresponding exact two-body solutions belonging to the Weyl class are also discussed. 
  The Schwarzschild solution has played a fundamental conceptual role in general relativity, and beyond, for instance, regarding event horizons, spacetime singularities and aspects of quantum field theory in curved spacetimes. However, one still encounters the existence of misconceptions and a certain ambiguity inherent in the Schwarzschild solution in the literature. By taking into account the point of view of an observer in the interior of the event horizon, one verifies that new conceptual difficulties arise. In this work, besides providing a very brief pedagogical review, we further analyze the interior Schwarzschild black hole solution. Firstly, by deducing the interior metric by considering time-dependent metric coefficients, the interior region is analyzed without the prejudices inherited from the exterior geometry. We also pay close attention to several respective cosmological interpretations, and briefly address some of the difficulties associated to spacetime singularities. Secondly, we deduce the conserved quantities of null and timelike geodesics, and discuss several particular cases in some detail. Thirdly, we examine the Eddington-Finkelstein and Kruskal coordinates directly from the interior solution. In concluding, it is important to emphasize that the interior structure of realistic black holes has not been satisfactorily determined, and is still open to considerable debate. 
  To understand the coupling behavior of the spinor with spacetime, the explicit form of the energy-momentum tensor of the spinor in curved spacetime is important. This problem seems to be overlooked for a long time. In this paper we derive the explicit form of energy momentum tensors and display some equivalent but simple forms of the covariant derivatives for both the Weyl spinor and the Dirac bispinor in curved spacetime. 
  This paper is devoted to the computation of compact binaries composed of one black hole and one neutron star. The objects are assumed to be on exact circular orbits. Standard 3+1 decomposition of Einstein equations is performed and the conformal flatness approximation is used. The obtained system of elliptic equations is solved by means of multi-domain spectral methods. Results are compared with previous work both in the high mass ratio limit and for one neutron star with very low compactness parameter. The accuracy of the present code is shown to be greater than with previous codes. Moreover, for the first time, some sequences containing one neutron star of realistic compactness are presented and discussed. 
  Research Briefs:  Singularity Avoidance in Canonical Quantum Gravity, by Viqar Husain  What's New in LIGO, by David Shoemaker   Conference reports:  Scanning New Horizons: GR Beyond 4 dimensions, by Donald Marolf  Quantum Gravity in the Americas III, by Jorge Pullin  New Frontiers in Numerical Relativity, by Luciano Rezzolla  Teaching General Relativity to Undergraduates, by Greg Comer  Ninth Capra Meeting on Radiation Reaction, by Lior Burko 
  Non-uniform black strings coupled to a gauge field are constructed by a perturbative method in a wide range of spacetime dimensions. At the linear order of perturbations, we see that the Gregory-Laflamme instability vanishes at the point where the background solution becomes thermodynamically stable. The emergence/vanishing of the static mode resembles phase transitions, and in fact we find that its critical exponent is nearly 1/2, which means a second-order transition. By employing higher-order perturbations, the physical properties of the non-uniform black strings are investigated in detail. For fixed spacetime dimensions, we find the critical charges at which the stability of non-uniform states changes. For some range of charge, non-uniform black strings are entropically favored over uniform ones. The gauge charge works as a control parameter that controls not only the stability of uniform black strings but also the non-uniform states. In addition, we find that for a fixed background charge the uniform state is not necessarily the state carrying the largest tension. The phase diagram and a comparison with the critical dimension are also discussed. 
  The accuracy and stability of the Caltech-Cornell pseudospectral code is evaluated using the KST representation of the Einstein evolution equations. The basic ``Mexico City Tests'' widely adopted by the numerical relativity community are adapted here for codes based on spectral methods. Exponential convergence of the spectral code is established, apparently limited only by numerical roundoff error. A general expression for the growth of errors due to finite machine precision is derived, and it is shown that this limit is achieved here for the linear plane-wave test. All of these tests are found to be stable, except for simulations of high amplitude gauge waves with nontrivial shift. 
  We study cosmological dynamics of a flat Randall-Sundrum brane with a scalar field and a negative "dark radiation" term. It is shown that in some situations the "dark radiation" can mimic spatial curvature and cause a chaotic behavior which is similar to chaotic dynamics in closed Universe with a scalar field. 
  We study electromagnetic test fields in the background of vacuum black rings using Killing vectors as vector potentials. We consider both spacetimes with a rotating S^1 and with a rotating S^2 and we demonstrate, in particular, that the gyromagnetic ratio of slightly charged black rings takes the value g=3 (this will in fact apply to a wider class of spacetimes). We also observe that a S^2-rotating black ring immersed in an external "aligned" magnetic field completely expels the magnetic flux in the extremal limit. Finally, we discuss the mutual alignment of principal null directions of the Maxwell 2-form and of the Weyl tensor, and the algebraic type of exact charged black rings. In contrast to spherical black holes, charged rings display new distinctive features and provide us with an explicit example of algebraically general (type G) spacetimes in higher dimensions. Appendix A contains some global results on black rings with a rotating 2-sphere. Appendix C shows that g=D-2 in any D>=4 dimensions for test electromagnetic fields generated by a time translation. 
  Cosmology is usually understood as an observational science, where experimentation plays no role. It is interesting, nevertheless, to change this perspective addressing the following question: what should we do to create a universe, in a laboratory? It appears, in fact, that this is, in principle, possible according to at least two different paradigms; both allow to circumvent singularity theorems, i.e. the necessity of singularities in the past of inflating domains which have the required properties to generate a universe similar to ours. The first of them is substantially classical, and is built up considering solitons which collide with surrounding topological defects, generating an inflationary domain of space-time. The second is, instead, partly quantum and considers the possibility of tunnelling of past-non-singular regions of spacetime into an inflating universe, following a well-known instanton proposal.   We are, here, going to review some of these models, as well as highlight possible extensions, generalizations and the open issues (as for instance the detectability of child universes and the properties of quantum tunnelling processes) that still affect the description of their dynamics. In doing so we will remark how the works on this subject can represent virtual laboratories to test the role that fundamental principles of physics (particularly, the interplay of quantum and general relativistic realms) played in the formation of our universe. 
  We investigate the possibility of extending Newton's second law to the general framework of theories in which special relativity is locally valid, and in which gravitation changes the flat Galilean space-time metric into a curved metric. This framework is first recalled, underlining the possibility to uniquely define a space metric and a local time in any given reference frame, hence to define velocity and momentum in terms of the local space and time standards. It is shown that a unique consistent definition can be given for the derivative of a vector (the momentum) along a trajectory. Then the possible form of the gravitation force is investigated. It is shown that, if the motion of free particles has to follow space-time geodesics, then the expression for the gravity acceleration is determined uniquely. It depends on the variation of the metric with space and time, and it involves the velocity of the particle. 
  We propose a new scheme for extracting gravitational radiation from a characteristic numerical simulation of a spacetime. This method is similar in conception to our earlier work but analytical and numerical implementation is different. The scheme is based on direct transformation to the Bondi coordinates and the gravitational waves are extracted by calculating the Bondi news function in Bondi coordinates. The entire calculation is done in a way which will make the implementation easy when we use uniform Bondi angular grid at $\mathcal I^+$. Using uniform Bondi grid for news calculation has added advantage that we have to solve only ordinary differential equations instead of partial differential equation. For the test problems this new scheme allows us to extract gravitational radiation much more accurately than the previous schemes. 
  We construct quasiequilibrium sequences of black hole-neutron star binaries for arbitrary mass ratios by solving the constraint equations of general relativity in the conformal thin-sandwich decomposition. We model the neutron star as a stationary polytrope satisfying the relativistic equations of hydrodynamics, and account for the black hole by imposing equilibrium boundary conditions on the surface of an excised sphere (the apparent horizon). In this paper we focus on irrotational configurations, meaning that both the neutron star and the black hole are approximately nonspinning in an inertial frame. We present results for a binary with polytropic index n=1, mass ratio M_{irr}^{BH}/M_{B}^{NS}=5 and neutron star compaction M_{ADM,0}^{NS}/R_0=0.0879, where M_{irr}^{BH} is the irreducible mass of the black hole, M_{B}^{NS} the neutron star baryon rest-mass, and M_{ADM,0}^{NS} and R_0 the neutron star Arnowitt-Deser-Misner mass and areal radius in isolation, respectively. Our models represent valid solutions to Einstein's constraint equations and may therefore be employed as initial data for dynamical simulations of black hole-neutron star binaries. 
  Causality violations in Bonnor-Steadman solutions of the Einstein equation referring to laboratory situations suggest general relativity does not provide a satisfactory account of certain physical situations. It is argued below that, by contrast, an Einstein vacuum space-time with causality violations has richer behavior and greater explanatory power than one without, by allowing nontrivial and potentially dynamic topology. It is well known that no closed timelike curve on a Lorentz manifold can be continuously deformed among timelike curves (be timelike homotopic) to a point, as that point would not be causally well behaved. It is shown that the topological features which prevent such a deformation may be self propagating; that self propagation and conservation of momentum rules out arbitrary topology change; and that a space-time with many such features may provide a model of collapse that naturally avoids satisfying assumptions of the singularity theorems. 
  Applying the first and generalised second laws of thermodynamics for a realistic process of near critical black hole formation, we derive an entropy bound, which is identical to Bekenstein's one for radiation. Relying upon this bound, we derive an absolute minimum mass $\sim0.04 \sqrt{g_{*}}m_{\rm Pl}$, where $g_{*}$ and $m_{\rm Pl}$ is the effective degrees of freedom for the initial temparature and the Planck mass, respectively. Since this minimum mass coincides with the lower bound on masses of which black holes can be regarded as classical against the Hawking evaporation, the thermodynamical argument will not prohibit the formation of the smallest classical black hole. For more general situations, we derive a minimum mass, which may depend on the initial value for entropy per particle. For primordial black holes, however, we show that this minimum mass can not be much greater than the Planck mass at any formation epoch of the Universe, as long as $g_{*}$ is within a reasonable range. We also derive a size-independent upper bound on the entropy density of a stiff fluid in terms of the energy density. 
  The basic properties of the C-metric are well known. It describes a pair of causally separated black holes which accelerate in opposite directions under the action of forces represented by conical singularities. However, these properties can be demonstrated much more transparently by making use of recently developed coordinate systems for which the metric functions have a simple factor structure. These enable us to obtain explicit Kruskal-Szekeres-type extensions through the horizons and construct two-dimensional conformal Penrose diagrams. We then combine these into a three-dimensional picture which illustrates the global causal structure of the space-time outside the black hole horizons. Using both the weak field limit and some invariant quantities, we give a direct physical interpretation of the parameters which appear in the new form of the metric. For completeness, relations to other familiar coordinate systems are also discussed. 
  Cosmological perturbation equations are derived systematically in a canonical scheme based on Ashtekar variables. A comparison with the covariant derivation and various subtleties in the calculation and choice of gauges are pointed out. Nevertheless, the treatment is more systematic when correction terms of canonical quantum gravity are to be included. This is done throughout the paper for one characteristic modification expected from loop quantum gravity. 
  A relatively simple method of overcoming the Standard Quantum Limit in the next-generation Advanced LIGO gravitational wave detector is considered. It is based on the quantum variational measurement with a single short (a few tens of meters) filter cavity. Estimates show that this method allows to reduce the radiation pressure noise at low frequencies ($<100 \mathrm{Hz}$) to the level comparable with or smaller than the low-frequency noises of non-quantum origin (mirrors suspension noise, mirrors internal thermal noise, and gravity gradients fluctuations). 
  We investigate cosmological solutions of Brans-Dicke theory with both the vacuum energy density and the gravitational constant decaying linearly with the Hubble parameter. A particular class of them, with constant deceleration factor, sheds light on the cosmological constant problems, leading to a presently small vacuum term, and to a constant ratio between the vacuum and matter energy densities. By fixing the only free parameter of these solutions, we obtain cosmological parameters in accordance with observations of both the relative matter density and the universe age. In addition, we have three other solutions, with Brans-Dicke parameter w = -1 and negative cosmological term, two of them with a future singularity of big-rip type. Although interesting from the theoretical point of view, two of them are not in agreement with the observed universe. The third one leads, in the limit of large times, to a constant relative matter density, being also a possible solution to the cosmic coincidence problem. 
  We calculate, in d spacetime dimensions, the relationship between the coefficient 1/K^2 of the Einstein-Hilbert term in the action of general relativity and the coefficient G_N of the force law that results from the Newtonian limit of general relativity. The result is K^2=2[(d-2)/(d-3)]Vol(S^[d-2])G_N, where Vol(S^n) is the volume of the unit n-sphere. While the d=4 case is an elementary calculation in any general relativity text, the arbitrary case presented here is slightly less well known. We discuss the relevance of this result for the definition of the so-called "reduced Planck mass" and comment very briefly on the implications for brane world models. [abstract abridged] 
  We derive upper and lower limits for the basic physical parameters (mass-radius ratio, anisotropy, redshift and total energy) for arbitrary anisotropic general relativistic matter distributions in the presence of a cosmological constant. The values of these quantities are strongly dependent on the value of the anisotropy parameter (the difference between the tangential and radial pressure) at the surface of the star. In the presence of the cosmological constant, a minimum mass configuration with given anisotropy does exist. Anisotropic compact stellar type objects can be much more compact than the isotropic ones, and their radii may be close to their corresponding Schwarzschild radii. Upper bounds for the anisotropy parameter are also obtained from the analysis of the curvature invariants. General restrictions for the redshift and the total energy (including the gravitational contribution) for anisotropic stars are obtained in terms of the anisotropy parameter. Values of the surface redshift parameter greater than two could be the main observational signature for anisotropic stellar type objects. 
  The instantaneous transition rate of an arbitrarily accelerated Unruh-DeWitt particle detector on four-dimensional Minkowski space is ill defined without regularisation. We show that Schlicht's regularisation as the zero-size limit of a Lorentz-function spatial profile yields a manifestly well-defined transition rate with physically reasonable asymptotic properties. In the special case of stationary trajectories, including uniform acceleration, we recover the results that have been previously obtained by a regularisation that relies on the stationarity. Finally, we discuss evidence for the conjecture that the zero-size limit of the transition rate is independent of the detector profile. 
  We discuss a large class of phenomenological models incorporating quantum gravity motivated corrections to electrodynamics. The framework is that of electrodynamics in a birefringent and dispersive medium with non-local constitutive relations, which are considered up to second order in the inverse of the energy characterizing the quantum gravity scale. The energy-momentum tensor, Green functions and frequency dependent refraction indices are obtained, leading to departures from standard physics. The effective character of the theory is also emphasized by introducing a frequency cutoff. The analysis of its effects upon the standard notion of causality is performed, showing that in the radiation regime the expected corrections get further suppressed by highly oscillating terms, thus forbiding causality violations to show up in the corresponding observational effects. 
  Until recently, the physically relevant singularities occurring in FRW cosmologies had traditionally been thought to be limited to the "big bang", and possibly a "big crunch". However, over the last few years, the zoo of cosmological singularities considered in the literature has become considerably more extensive, with "big rips" and "sudden singularities" added to the mix, as well as renewed interest in non-singular cosmological events such as "bounces" and "turnarounds". In this talk, we present an extensive catalogue of such cosmological milestones, both at the kinematical and dynamical level. First, using generalized power series, purely kinematical definitions of these cosmological events are provided in terms of the behaviour of the scale factor a(t). The notion of a "scale-factor singularity" is defined, and its relation to curvature singularities (polynomial and differential) is explored. Second, dynamical information is extracted by using the Friedmann equations (without assuming even the existence of any equation of state) to place constraints on whether or not the classical energy conditions are satisfied at the cosmological milestones. Since the classification is extremely general, and modulo certain technical assumptions complete, the corresponding results are to a high degree model-independent. 
  We study the fate of a collapsing star on the brane in a generalized braneworld gravity with bulk matter. Specifically, we investigate for the possibility of having a static exterior for a collapsing star in the radiative bulk scenario. Here, the nonlocal correction due to bulk matter is manifest in an induced mass that adds up to the physical mass of the star resulting in an effective mass. A Schwarzschild solution for the exterior in terms of this effective mass is obtained, which reveals that even if the star exchanges energy with the bulk, the exterior may appear to be static to a braneworld observer located outside the collapsing region. The possible explanation of the situation from the discussion on the role of bulk matter is provided. The nature of bulk matter and the corresponding bulk geometry have also been obtained and analyzed, which gives a complete picture of both brane and bulk viewpoints. 
  We investigate Isaacson's high-frequency gravitational waves which propagate in some relevant cosmological models, in particular the FRW spacetimes. Their time evolution in Fourier space is explicitly obtained for various metric forms of (anti--)de Sitter universe. Behaviour of high-frequency waves in the anisotropic Kasner spacetime is also described. 
  The paper presents some solutions to the five dimensional Einstein equations due to a perfect fluid on the brane with pure dust filling the entire bulk in one case and a cosmological constant (or vacuum) in the bulk for the second case. In the first case, there is a linear relationship between isotropic pressure, energy density and the brane tension, while in the second case, the perfect fluid is assumed to be in the form of chaplygin gas. Cosmological solutions are found both for brane and bulk scenarios and some interesting features are obtained for the chaplygin gas on the brane which are distinctly different from the standard cosmology in four dimensions. 
  We present the whole set of equations with regularity and matching conditions required for the description of physically meaningful stationary cylindrically symmmetric distributions of matter, smoothly matched to Lewis vacuum spacetime. A specific example is given. The electric and magnetic parts of the Weyl tensor are calculated, and it is shown that purely electric solutions are necessarily static. Then, it is shown that no conformally flat stationary cylindrical fluid exits, satisfying regularity and matching conditions. 
  We derive an expression for the second-order gravitational self-force that acts on a self-gravitating compact-object moving in a curved background spacetime. First we develop a new method of derivation and apply it to the derivation of the first-order gravitational self-force. Here we find that our result conforms with the previously derived expression. Next we generalize our method and derive a new expression for the second-order gravitational self-force. This study also has a practical motivation: The data analysis for the planned gravitational wave detector LISA requires construction of waveforms templates for the expected gravitational waves. Calculation of the two leading orders of the gravitational self-force will enable one to construct highly accurate waveform templates, which are needed for the data analysis of gravitational-waves that are emitted from extreme mass-ratio binaries. 
  Direct integration technique is used to study the proper conformal vector fields in non conformally flat Bianchi type-1 space-times. Using the above mentioned technique we have shown that a very special class of the above space-time admits proper conformal vector fields. 
  The scalar field with the modified dispersion relation under certain conditions may seed the primordial perturbations during a decelerated expansion. In this note we examine whether and how these perturbations can be responsible for the structure formation of observable universe. We find that there seems slightly difficult in matching the requirements of observable cosmology, i.e. obtain simultaneously the scale invariant spectrum of primordial perturbations and enough "efolding number", in a simple case. We discuss possible solutions to this problem. 
  We present new results from our test of Lorentz invariance, which compares two orthogonal cryogenic sapphire microwave oscillators rotating in the lab. We have now acquired over 1 year of data, allowing us to avoid the short data set approximation (less than 1 year) that assumes no cancelation occurs between the $\tilde{\kappa}_{e-}$ and $\tilde{\kappa}_{o+}$ parameters from the photon sector of the standard model extension. Thus, we are able to place independent limits on all eight $\tilde{\kappa}_{e-}$ and $\tilde{\kappa}_{o+}$ parameters. Our results represents up to a factor of 10 improvement over previous non rotating measurements (which independently constrained 7 parameters), and is a slight improvement (except for $\tilde{\kappa}_{e-}^{ZZ}$) over results from previous rotating experiments that assumed the short data set approximation. Also, an analysis in the Robertson-Mansouri-Sexl framework allows us to place a new limit on the isotropy parameter $P_{MM}=\delta-\beta+{1/2}$ of $9.4(8.1)\times10^{-11}$, an improvement of a factor of 2. 
  We use generalized Puisieux series expansions to determine the behaviour of the scale factor in the vicinity of typical cosmological milestones occurring in a FRW universe. We describe some of the consequences of this generalized Puisieux series expansion on other physical observables. 
  We study equilibrium states in relativistic galactic dynamics which are described by solutions of the Einstein-Vlasov system for collisionless matter. We recast the equations into a regular three-dimensional system of autonomous first order ordinary differential equations on a bounded state space. Based on a dynamical systems analysis we derive new theorems that guarantee that the steady state solutions have finite radii and masses. 
  We calculate and compare the response of light wave interferometers and matter wave interferometers to gravitational waves. We find that metric matter wave interferometers will not challenge kilometric light wave interferometers such as Virgo or LIGO, but could be a good candidate for the detection of very low frequency gravitational waves. 
  By the assumption that the thermodynamics second law is valid, we study the possibility of $\omega=-1$ crossing in interacting holographic dark energy model. Depending on the choice of the horizon and the interaction, the transition from quintessence to phantom regime and subsequently from phantom to quintessence phase may be possible. The second transition avoids the big rip singularity. We compute the dark energy density at transition time and show that by choosing appropriate parameters we can alleviate the coincidence problem. 
  We review our recent work on the classification of finite time singularities that arise in isotropic universes. This scheme is based on the exploitation of the Bel Robinson energy in a cosmological setting. We comment on the relation between geodesic completeness and the Bel Robinson energy and present evidence that relates the divergence of the latter to the existence of closed trapped surfaces. 
  We examine the effects of spatial inhomogeneities on irrotational anisotropic cosmologies by looking at the average properties of anisotropic pressure-free models. Adopting the Buchert scheme, we recast the averaged scalar equations in Bianchi-type form and close the standard system by introducing a propagation formula for the average shear magnitude. We then investigate the evolution of anisotropic average vacuum models and those filled with pressureless matter. In the latter case we show that the backreaction effects can modify the familiar Kasner-like singularity and potentially remove Mixmaster-type oscillations. The presence of nonzero average shear in our equations also allows us to examine the constraints that a phase of backreaction-driven accelerated expansion might put on the anisotropy of the averaged domain. We close by assessing the status of these and other attempts to define and calculate `average' spacetime behaviour in general relativity. 
  We show by an almost elementary calculation that the ADM mass of an asymptotically flat space can be computed as a limit involving a rate of change of area of a closed 2-surface. The result is essentially the same as that given by Brown and York. We will prove this result in two ways, first by direct calculation from the original formula as given by Arnowitt, Deser and Misner and second as a corollary of an earlier result by Brewin for the case of simplicial spaces. 
  Recently, there has been a lot of attention devoted to resolving the quantum corrections to the Bekenstein-Hawking entropy of the black hole. In particular, the coefficient of the logarithmic term in the black hole entropy correction has been of great interest. In this paper, the black hole is corresponded to a canonical ensemble in statistics by radiant spectrum, resulted from the black hole tunneling effect studies and the partition function of ensemble is derived. Then the entropy of the black hole is calculated. When the first order approximation is taken into account, the logarithmic term of entropy correction is consistent with the result of the generalized uncertainty principle. In our calculation, there are no uncertainty factors. The prefactor of the logarithmic correction and the one if fluctuation is considered are the same. Our result shows that if the thermal capacity is negative, there is no divergent term. We provide a general method for further discussion on quantum correction to Bekenstein-Hawking entropy. We also offer a theoretical basis for comparing string theory and loop quantum gravitaty and deciding which one is more reliable. 
  This paper is motivated by the recent possibility to find an inexpensive launching vehicle for the LARES satellite, however at an altitude much lower than originally planned for the LAGEOS III/LARES satellite.   We present here a preliminary error analysis corresponding to a lower, quasi-polar, orbit, in particular we analyze the effect on the LARES node of the Earth's static gravitational field, and in particular of the Earth's even zonal harmonics, the effect of the time dependent Earth's gravitational field, and in particular of the K1 tide, and the effect of particle drag. 
  Considered a unified field theory approach describing matter and space (metric tensor) by means of a 3-index tensor $P^{i}_{jk}$. It is shown that if the Lagrangian has partial U(1) gauge (semi-gauge) of the type $P^{i}_{jk}->P^{i}_{jk}+a \delta{^i_j} \phi{_{,k}}+b \delta{^i_k} \phi{_{,j}}+cg_{jk}g^{mi} \phi{_{,m}}$ where a,b,c are constants, then the Euler equations of motion contain the covariant low of conservation in the form $J^{k}_{;k}=0$. 
  In the recent literature on dark energy (DE) model building we have learnt that cosmologies with variable cosmological parameters can mimic more traditional DE pictures exclusively based on scalar fields (e.g. quintessence and phantom). In a previous work we have illustrated this situation within the context of a renormalization group running cosmological term, Lambda. Here we analyze the possibility that both the cosmological term and the gravitational coupling, G, are running parameters within a more general framework (a variant of the so-called ``LXCDM models'') in which the DE fluid can be a mixture of a running Lambda and another dynamical entity X (the ``cosmon'') which may behave quintessence-like or phantom-like. We compute the effective EOS parameter, w, of this composite fluid and show that the LXCDM can mimic to a large extent the standard LCDM model while retaining features hinting at its potential composite nature (such as the smooth crossing of the cosmological constant boundary w=-1). We further argue that the LXCDM models can cure the cosmological coincidence problem. All in all we suggest that future experimental studies on precision cosmology should take seriously the possibility that the DE fluid can be a composite medium whose dynamical features are partially caused and renormalized by the quantum running of the cosmological parameters. 
  We study Einstein gravity minimally coupled to a scalar field in a static, spherically symmetric space-time in four dimensions. Black hole solutions are shown to exist for a phantom scalar field whose kinetic energy is negative. These ``scalar black holes'' have an infinite horizon area and zero Hawking temperature and are termed ``cold black holes'' (CBHs). The relevant explicit solutions are well-known in the massless case (the so-called anti-Fisher solution), and we have found a particular example of a CBH with a nonzero potential $V(\phi)$. All CBHs with $V(\phi) \not \equiv 0$ are shown to behave near the horizon quite similarly to those with a massless field. The above solutions can be converted by a conformal transformation to Jordan frames of a general class of scalar-tensor theories of gravity, but CBH horizons in one frame are in many cases converted to singularities in the other, which gives rise to a new type of conformal continuation. 
  In this work we construct and analyze exact solutions describing Ricci flows and nonholonomic deformations of four dimensional (4D) Taub-NUT spacetimes. It is outlined a new geometric techniques of constructing Ricci flow solutions. Some conceptual issues on spacetimes provided with generic off-diagonal metrics and associated nonlinear connection structures are analyzed. The limit from gravity/Ricci flow models with nontrivial torsion to configurations with the Levi-Civita connection is allowed in some specific physical circumstances by constraining the class of integral varieties for the Einstein and Ricci flow equations. 
  The common assertion that the Ricci flows of Einstein spaces with cosmological constant can be modelled by certain classes of nonholonomic frame, metric and linear connection deformations resulting in nonhomogeneous Einstein spaces is examined in the light of the role played by topological three dimensional (3D) Taub-NUT-AdS/dS spacetimes. 
  We present a new pseudo-spectral code for the simulation of evolution systems that are second order in space. We test this code by evolving a non-linear scalar wave equation. These non-linear waves can be stably evolved using very simple constant or radiative boundary conditions, which we show to be well-posed in the scalar wave case. The main motivation for this work, however, is to evolve black holes for the first time with the BSSN system by means of a spectral method. We use our new code to simulate the evolution of a single black hole using all applicable methods that are usually employed when the BSSN system is used together with finite differencing methods. In particular, we use black hole excision and test standard radiative and also constant outer boundary conditions. Furthermore, we study different gauge choices such as $1+\log$ and constant densitized lapse. We find that these methods in principle do work also with our spectral method. However, our simulations fail after about $100M$ due to unstable exponentially growing modes. The reason for this failure may be that we evolve the black hole on a full grid without imposing any symmetries. Such full grid instabilities have also been observed when finite differencing methods are used to evolve excised black holes with the BSSN system. 
  The first static spherically symmetric perfect fluid solution with constant density was found by Schwarzschild in 1918. Generically, perfect fluid spheres are interesting because they are first approximations to any attempt at building a realistic model for a general relativistic star. Over the past 90 years a confusing tangle of specific perfect fluid spheres has been discovered, with most of these examples seemingly independent from each other. To bring some order to this collection, we develop several new transformation theorems that map perfect fluid spheres into perfect fluid spheres. These transformation theorems sometimes lead to unexpected connections between previously known perfect fluid spheres, sometimes lead to new previously unknown perfect fluid spheres, and in general can be used to develop a systematic way of classifying the set of all perfect fluid spheres. In addition, we develop new ``solution generating'' theorems for the TOV, whereby any given solution can be ``deformed'' to a new solution. Because these TOV-based theorems work directly in terms of the pressure profile and density profile it is relatively easy to impose regularity conditions at the centre of the fluid sphere. 
  We study the deformation of a long cosmic string by a nearby rotating black hole. We examine whether the deformation of a cosmic string, induced by the gravitational field of a Kerr black hole, may lead to the formation of a loop of cosmic string. The segment of the string which enters the ergosphere of a rotating black hole gets deformed and, if it is sufficiently twisted, it can self-intersect chopping off a loop of cosmic string. We find that the formation of a loop, via this mechanism, is a rare event. It will only arise in a small region of the collision phase space, which depends on the string velocity, the impact parameter and the black hole angular momentum. We conclude that generically, the cosmic string is simply scattered or captured by the rotating black hole. 
  We study the possible singularities of isotropic cosmological models that have a varying speed of light as well as a varying gravitational constant. The field equations typically reduce to two dimensional systems which are then analyzed both by dynamical systems techniques in phase space and by applying the method of asymptotic splittings. In the general case we find initially expanding closed models which recollapse to a future singularity and open universes that are eternally expanding towards the future. The precise nature of the singularities is also discussed. 
  The electrovacuum around a rotating massive body with electric charge density is described by its multipole moments (mass moments, mass-current moments, electric moments, and magnetic moments). A small uncharged test particle orbiting around such a body moves on geodesics if gravitational radiation is ignored. The waves emitted by the small body carry information about the geometry of the central object, and hence, in principle, we can infer all its multipole moments. Due to its axisymmetry the source is characterized now by four families of scalar multipole moments: its mass moments $M_l$, its mass-current moments $S_l$, its electrical moments $E_l$ and its magnetic moments $H_l$, where $l=0,1,2,...$. Four measurable quantities, the energy emitted by gravitational waves per logarithmic interval of frequency, the precession of the periastron (assuming almost circular orbits), the precession of the orbital plane (assuming almost equatorial orbits), and the number of cycles emitted per logarithmic interval of frequency, are presented as power series of the newtonian orbital velocity of the test body. The power series coefficients are simple polynomials of the various moments. 
  We investigate the inner structure of an evaporating charged black hole, within the context of semiclassical dilaton gravity in two dimensions. The matter fields are charged, allowing the evaporation of both the mass and charge of the black hole. We find that the semiclassical effects cause the inner horizon to expand (by a finite factor), rather than to shrink to a point singularity. Although this expansion is a quantum phenomenon, the overall expansion factor is found to be independent of the magnitude of the quantum terms in the effective theory. 
  We derive the effective cosmological equations for a non-$\mathbb{Z}_2$ symmetric codimension one brane embedded in an arbitrary D-dimensional bulk spacetime, generalizing the $D=5,6$ cases much studied previously. As a particular case, this may be considered as a regularized codimension (D-4) brane avoiding the problem of curvature divergence on the brane. We apply our results to the case of spherical symmetry around the brane and to partly compactified AdS-Schwarzschild bulks. 
  We study gravitationally collapsing models of pressureless dust, fluids with pressure, and the generalized Chaplygin gas (GCG) shell in (2+1)-dimensional spacetimes. Various collapse scenarios are investigated under a variety of the background configurations such as anti-de Sitter(AdS) black hole, de Sitter (dS) space, flat and AdS space with a conical deficit. As with the case of a disk of dust, we find that the collapse of a dust shell coincides with the Oppenheimer-Snyder type collapse to a black hole provided the initial density is sufficiently large. We also find -- for all types of shell -- that collapse to a naked singularity is possible under a broad variety of initial conditions. For shells with pressure this singularity can occur for a finite radius of the shell. We also find that GCG shells exhibit diverse collapse scenarios, which can be easily demonstrated by an effective potential analysis. 
  We report progress on a program of gravitational physics experiments using cryogenic torsion pendula undergoing large-amplitude torsion oscillation. This program includes tests of the gravitational inverse square law and of the weak equivalence principle. Here we describe our ongoing search for inverse-square-law violation at a strength down to $10^{-5}$ of standard gravity. The low-vibration environment provided by the Battelle Gravitation Physics Laboratory (BGPL) is uniquely suited to this study. 
  An exact charged axially symmetric solution of the coupled gravitational and electromagnetic fields in the teleparallel equivalent of Einstein theory is derived. It is characterized by three parameters ``$ $the gravitational mass $M$, the charge parameter $Q$ and the rotation parameter $a$" and its associated metric gives Kerr-Newman spacetime. The parallel vector field and the electromagnetic vector potential are axially symmetric. We then, calculate the total energy using the gravitational energy-momentum. The energy is found to be shared by its interior as well as exterior. Switching off the charge parameter we find that no energy is shared by the exterior of the Kerr-Newman black hole. 
  In this note we comment on a recent paper by I.Ciufolini about the possibility of placing the proposed LARES satellite in a low-altitude, nearly polar orbit in order to measure the Lense-Thirring effect with its node. While Ciufolini claims that for a departure of 4 deg from the ideal polar configuration the impact of the even zonal harmonics of the geopotential, modelled with EIGEN-GRACE02S, would be nearly zero allowing for a few-percent measurement of the Lense-Thirring effect, we find that, with the same Earth gravity model and for the same values of the inclination, the systematic error due to the even zonals amounts to 64% of the investigated relativistic effect. 
  In this short paper, we drive the explicit relation between the temperature $T$ of classical ideal gas in the universe and the scale factor $a(t)$ of the Friedman-Robertson-Walker metric via kinetic and statistical calculation. This formula is suitable for both the ultra-relativistic and the non-relativistic cases. 
  We report several new transformation theorems that map perfect fluid spheres into perfect fluid spheres. In addition, we report new ``solution generating'' theorems for the TOV, whereby any given solution can be ``deformed'' to a new solution. 
  A characterisation of initial data sets for the Schwarzschild spacetime is provided. This characterisation is obtained by performing a 3+1 decomposition of a certain invariant characterisation of the Schwarzschild spacetime given in terms of concomitants of the Weyl tensor. This procedure renders a set of necessary conditions --which can be written in terms of the electric and magnetic parts of the Weyl tensor and their concomitants-- for an initial data set to be a Schwarzschild initial data set. Our approach also provides a formula for a static Killing initial data set candidate --a KID candidate. Sufficient conditions for an initial data set to be a Schwarzschild initial data set are obtained by supplementing the necessary conditions with the requirement that the initial data set possesses a stationary Killing initial data set of the form given by our KID candidate. Thus, we obtain an algorithmic procedure of checking whether a given initial data set is Schwarzschildean or not. 
  We study energy distribution in the context of teleparallel theory of gravity, due to matter and fields including gravitation, of the universe based on the plane-wave Bianchi VII$_{\delta}$ spacetimes described by the Lukash metric. In order to make this calculation we consider the teleparallel gravity analogs of the energy-momentum formulations of Einstein, Bergmann-Thomson and Landau-Lifshitz. We find that Einstein and Bergmann-Thomson prescriptions agree with each other and give the same results for the energy distribution in a given spacetime, but the Landau-Lifshitz complex does not. Energy density turns out to be non-vanishing in all of these prescriptions. It is interesting to mention that the results can be reduced to the already available results for the Milne universe when we write $\omega=1$ and $\Xi^2=1$ in the metric of the Lukash spacetime, and for this special case, we get the same relation among the energy-momentum formulations of Einstein, Bergmann-Thomson and Landau-Lifshitz as obtained for the Lukash spacetime. Furthermore, our results support the hypothesis by Cooperstock that the energy is confined to the region of non-vanishing energy-momentum tensor of matter and all non-gravitational fields, and also sustain the importance of the energy-momentum definitions in the evaluation of the energy distribution associated with a given spacetime. 
  The two-component formalism in quantum cosmology is revisited with a particular emphasis on the identification of time. Its relation with the appearance of imaginary eigenvalues is demonstrated. Moreover, it is explicitly shown how a good choice of the global time prevents this peculiarity. 
  The explicit violation of the general covariance on the whole and its minimal violation to the unimodular covariance specifically is considered. The proper extension of General Relativity is shown to describe consistently the massive scalar graviton together with the massless tensor one, as the parts of the metric. The bearing of the scalar graviton to the dark matter and dark energy is indicated. 
  A duality transformation that interrelates expanding and contracting cosmological models is shown to single out a duality invariant, interacting two-component description of any irrotational, geodesic and shearfree cosmic medium with vanishing three curvature scalar. We apply this feature to a system of matter and radiation, to a mixture of dark matter and dark energy, to minimal and conformal scalar fields, and to an enlarged Chaplygin gas model of the cosmic substratum. We extend the concept of duality transformations to cosmological perturbations and demonstrate the invariance of adiabatic pressure perturbations under these transformations. 
  The LISA International Science Team Working Group on Data Analysis (LIST-WG1B) is sponsoring several rounds of mock data challenges, with the purposeof fostering the development of LISA data analysis capabilities, and of demonstrating technical readiness for the maximum science exploitation of the LISA data. The first round of challenge data sets were released at the Sixth LISA Symposium. We briefly describe the objectives, structure, and time-line of this programme. 
  The LISA International Science Team Working Group on Data Analysis (LIST-WG1B) is sponsoring several rounds of mock data challenges, with the purpose of fostering development of LISA data-analysis capabilities, and of demonstrating technical readiness for the maximum science exploitation of the LISA data. The first round of challenge data sets were released at this Symposium. We describe the models and conventions (for LISA and for gravitational-wave sources) used to prepare the data sets, the file format used to encode them, and the tools and resources available to support challenge participants. 
  We propose a lattice boson model where a fully covariant Euclidean quantum gravity emerges at low energies. Two phases are obtained, that is, a `gravitational Coulomb phase' with massless graviton and a `gravitational Higgs phase' with massive graviton. 
  I shall discuss some "conditions of possibility" of a quantum theory of gravity, stressing the need for solutions to some of fundamental problems confronting any attempt to apply some method of quantization to the field equations of general relativity. 
  Physically motivated gravitational wave signals are needed in order to study the behaviour and efficacy of different data analysis methods seeking their detection. GravEn, short for Gravitational-wave Engine, is a MATLAB software package that simulates the sampled response of a gravitational wave detector to incident gravitational waves. Incident waves can be specified in a data file or chosen from among a group of pre-programmed types commonly used for establishing the detection efficiency of analysis methods used for LIGO data analysis. Every aspect of a desired signal can be specified, such as start time of the simulation (including inter-sample start times), wave amplitude, source orientation to line of sight, location of the source in the sky, etc. Supported interferometric detectors include LIGO, GEO, Virgo and TAMA. 
  A number of different methods have been proposed to identify unanticipated burst sources of gravitational waves in data arising from LIGO and other gravitational wave detectors. When confronted with such a wide variety of methods one is moved to ask if they are all necessary, i.e. given detector data that is assumed to have no gravitational wave signals present, do they generally identify the same events with the same efficiency, or do they each 'see' different things in the detector? Here we consider three different methods, which have been used within the LIGO Scientific Collaboration as part of its search for unanticipated gravitational wave bursts. We find that each of these three different methods developed for identifying candidate gravitational wave burst sources are, in fact, attuned to significantly different features in detector data, suggesting that they may provide largely independent lists of candidate gravitational wave burst events. 
  We study the behavior under conformal transformations of energy and other charges in generic scalar-tensor models. This enables us to conclude that the ADM/AD masses are invariant under field redefinitions mixing metric and scalar despite the permitted slow asymptotic falloff of massless scalars. 
  The aim of this paper is to dynamically constrain the quadrupole mass moment J_2 of the main-sequence HD 209458 star. The adopted method is the confrontation between the measured orbital period of its transiting planet Osiris and a model of it. Osiris is assumed to move along an equatorial and circular orbit. Our estimate, for given values of the stellar mass M and radius R and by assuming the validity of general relativity, is J_2=(3.5 +/- 385.1) 10^-5. Previous fiducial evaluations based on indirect, spectroscopic measurements yielded J_2\sim 10^-6: such a value is compatible with our result. 
  Future interferometric gravitational wave detectors will make use of the coupling between shot noise and radiation pressure noise that produces a squeezed output for the quantum noise at the dark-port of the interferometer allowing these interferometers to operate with a sensitivity that exceeds the standard quantum limit at certain frequencies. The ability of the detector to take advantage of this squeezed output state depends in part on the readout system used. With the phase modulated laser input planned for these interferometers the quantum-noise limited sensitivity is slightly better when homodyne (rather than heterodyne) detection is used. We show that by modifying the laser input spectrum for these interferometers, readout with heterodyne detection can be made to have the same quantum-noise limited sensitivity as readout with homodyne detection. 
  We generalize the previously studied cosmological solutions in D-dimensional gravity with an antisymmetric (p+2)-form to the case when the spatial part of the metric is Ricci-flat rather than flat. These generalized solutions are characterized by a parallel self-dual or anti-self-dual charge density form Q of rank 2m and satisfy the condition Q^2 >0. As with the previous flat-space case, these electric, S-brane solutions only exist when D=4m+1 = 5, 9, 13, ... and p= 2m -1 = 1, 3, 5, .... We further generalize these solutions by adding Ricci-flat factor-spaces which are not covered by branes. 
  The coincidence problem is studied for the dark energy model of effective Yang-Mills condensate in a flat expanding universe during the matter-dominated stage. The YMC energy $\rho_y(t)$ is taken to represent the dark energy, which is coupled either with the matter, or with both the matter and the radiation components. The effective YM Lagrangian is completely determined by quantum field theory up to 1-loop order. It is found that under very generic initial conditions and for a variety of forms of coupling, the existence of the scaling solution during the early stages and the subsequent exit from the scaling regime are inevitable. The transition to the accelerating stage always occurs around a redshift $z\simeq (0.3\sim 0.5)$. Moreover, when the Yang-Mills condensate transfers energy into matter or into both matter and radiation, the equation of state $w_y$ of the Yang-Mills condensate can cross over -1 around $z\sim 2$, and takes on a current value $\simeq -1.1$. This is consistent with the recent preliminary observations on supernovae Ia. Therefore, the coincidence problem can be naturally solved in the effective YMC dark energy models. 
  Einstein field equations with a cosmological constant are approximated to the second order in the perturbation to a flat background metric. The final result is a set of Einstein-Maxwell-Proca equations for gravity in the weak field regime. This approximation procedure implements the breaking of gauge symmetry in general relativity. A brief discussion of the physical consequences (Pioneer anomalous deceleration) is proposed in the framework of the gauge theory of gravity. 
  The spacetime metric around a rotating SuperConductive Ring (SCR) is deduced from the gravitomagnetic London moment in rotating superconductors. It is shown that theoretically it is possible to generate Closed Timelike Curves (CTC) with rotating SCRs. The possibility to use these CTC's to travel in time as initially idealized by G\"{o}del is investigated. It is shown however, that from a technology and experimental point of view these ideas are impossible to implement in the present context. 
  This paper deals with the reconstruction of the direction of a gravitational wave source using the detection made by a network of interferometric detectors, mainly the LIGO and Virgo detectors. We suppose that an event has been seen in coincidence using a filter applied on the three detector data streams. Using the arrival time (and its associated error) of the gravitational signal in each detector, the direction of the source in the sky is computed using a chi^2 minimization technique. For reasonably large signals (SNR>4.5 in all detectors), the mean angular error between the real location and the reconstructed one is about 1 degree. We also investigate the effect of the network geometry assuming the same angular response for all interferometric detectors. It appears that the reconstruction quality is not uniform over the sky and is degraded when the source approaches the plane defined by the three detectors. Adding at least one other detector to the LIGO-Virgo network reduces the blind regions and in the case of 6 detectors, a precision less than 1 degree on the source direction can be reached for 99% of the sky. 
  The full causal ladder of spacetimes is constructed, and their updated main properties are developed. Old concepts and alternative definitions of each level of the ladder are revisited, with emphasis in minimum hypotheses. The implications of the recently solved ``folk questions on smoothability'', and alternative proposals (as recent isocausality), are also summarized. 
  We analyze asymptotic symmetries on the Killing horizon of the four-dimensional Kerr--Newman black hole. We first derive the asymptotic Killing vectors on the Killing horizon, which describe the asymptotic symmetries, and find that the general form of these asymptotic Killing vectors is the universal one possessed by arbitrary Killing horizons. We then construct the phase space associated with the asymptotic symmetries. It is shown that the phase space of an extreme black hole either has the size comparable with a non-extreme black hole, or is small enough to exclude degeneracy, depending on whether or not the global structure of a Killing horizon particular to an extreme black hole is respected. We also show that the central charge in the Poisson brackets algebra of these asymptotic symmetries vanishes, which implies that there is not an anomaly of diffeomorphism invariance. By taking into account other results in the literature, we argue that the vanishing central charge on a black hole horizon, in a macroscopic effective theory, looks consistent with the thermal feature of a black hole. 
  The dynamics of "dipolar particles", i.e. particles endowed with a four-vector mass dipole moment, is investigated using an action principle in general relativity. The action is a specific functional of the particle's world line, and of the dipole moment vector, considered as a dynamical variable. The first part of the action is inspired by that of a particle with spin moving on an arbitrary gravitational background. The second part is intended to describe, at some effective level, the internal non-gravitational force linking together the "microscopic" constituents of the dipole. We find that some solutions of the equations of motion and evolution of the dipolar particles correspond to an equilibrium state for the dipole moment in a gravitational field. Under some hypothesis we show that a fluid of dipolar particles, supposed to constitute the dark matter, reproduces the modified Newtonian dynamics (MOND) in the non relativistic limit. We then recover the main characteristics of a recently proposed quasi-Newtonian model of "gravitational polarization". 
  Ever since the pioneer works of Bekenstein and Hawking, black hole entropy has been known to have a quantum origin. Furthermore, it has long been argued by Bekenstein that entropy should be quantized in discrete (equidistant) steps given its identification with horizon area in (semi-)classical general relativity and the properties of area as an adiabatic invariant. This lead to the suggestion that black hole area should also be quantized in equidistant steps to account for the discrete black hole entropy. Here we shall show that loop quantum gravity, in which area is not quantized in equidistant steps can nevertheless be consistent with Bekenstein's equidistant entropy proposal in a subtle way. For that we perform a detailed analysis of the number of microstates compatible with a given area and show that an observed oscillatory behavior in the entropy-area relation, when properly interpreted yields an entropy that has discrete, equidistant values that are consistent with the Bekenstein framework. 
  Within the inflationary scenario, Planck scale physics should have affected the comoving modes' initial conditions and early evolution, thereby potentially affecting the inflationary predictions for the cosmic microwave background (CMB). This issue has been studied extensively on the basis of various models for how quantum field theory (QFT) is modified and finally breaks down towards the Planck scale. In one model, in particular, an ultraviolet cutoff was implemented into QFT through generalized uncertainty relations which have been motivated from general quantum gravity arguments and from string theory. Here, we improve upon prior numerical and semi-analytical results by presenting the exact mode solutions for both de Sitter and power-law inflation in this model. This provides an explicit map from the modes' initial conditions, which are presumably set by quantum gravity, to the modes' amplitudes at horizon crossing and thus to the inflationary predictions for the CMB. The solutions' particular behaviour close to the cutoff scale suggests unexpected possibilities for how the degrees of freedom of QFT emerge from the Planck scale. 
  Using the positive energy theorem, we derive some constraints on static steller models in asymptotically flat spacetimes in a general setting without imposing spherical symmetry. We show that there exist no regular solutions under certain conditions on the equation of state. As the contraposition, we obtain some constraints on the pressure and adiabatic index. 
  The exact static and spherically symmetric solutions of the vacuum field equations for a Higgs Scalar-Tensor theory (HSTT) are derived in Schwarzschild coordinates. It is shown that there exists no Schwarzschild horizon and that the massless scalar field acts like a massless field in the conventional theory of gravitation. Only in the center (point-particle) the fields are singular (as naked singularity). However, the Schwarzschild solution is obtained for the limit of vanishing excited Higgs fields. 
  We consider some formulations of the entropy bounds at the semiclassical level. The entropy S(V) localized in a region V is divergent in quantum field theory (QFT). Instead of it we focus on the mutual information I(V,W)=S(V)+S(W)-S(V U W) between two different non-intersecting sets V and W. This is a low energy quantity, independent of the regularization scheme. In addition, the mutual information is bounded above by twice the entropy corresponding to the sets involved. Calculations of I(V,W) in QFT show that the entropy in empty space cannot be renormalized to zero, and must be actually very large. We find that this entropy due to the vacuum fluctuations violates the FMW bound in Minkowski space. The mutual information also gives a precise, cutoff independent meaning to the statement that the number of degrees of freedom increases with the volume in QFT. If the holographic bound holds, this points to the essential non locality of the physical cutoff. Violations of the Bousso bound would require conformal theories and large distances. We speculate the cosmological constant might prevent such a violation. 
  Perturbed stationary axisymmetric isolated bodies, e.g. stars, represented by a matter-filled interior and an asymptotically flat vacuum exterior joined at a surface where the Darmois matching conditions are satisfied, are considered. The initial state is assumed to be static. The perturbations of the matching conditions are derived and used as boundary conditions for the perturbed Ernst equations in the exterior region. The perturbations are calculated to second order. The boundary conditions are overdetermined: necessary and sufficient conditions for their compatibility are derived. The special case of perturbations of spherical bodies is given in detail. 
  A closed vacuum-dominated Friedmann universe is asymptotic to a de Sitter space with a cosmological event horizon for any observer. The holographic principle says the area of the horizon in Planck units determines the number of bits of information about the universe that will ever be available to any observer. The wavefunction describing the probability distribution of mass quanta associated with bits of information on the horizon is the boundary condition for the wavefunction specifying the probability distribution of mass quanta throughout the universe. Local interactions between mass quanta in the universe cause quantum transitions in the wavefunction specifying the distribution of mass throughout the universe, with instantaneous non-local effects throughout the universe. 
  The formalism of rainbow gravity is studied in a cosmological setting. We consider the very early universe which is radiation dominated. A novel treatment in our paper is to look for an ``averaged'' cosmological metric probed by radiation particles themselves. Taking their cosmological evolution into account, we derive the modified Friedmann-Robertson-Walker(FRW) equations which is a generalization of the solution presented by Magueijo and Smolin. Based on this phenomenological cosmological model we argue that the spacetime curvature has an upper bound such that the cosmological singularity is absent. These modified $FRW$ equations can be treated as effective equations in the semi-classical framework of quantum gravity and its analogy with the one recently proposed in loop quantum cosmology is also discussed. 
  The kinematics of particles moving in rainbow spacetime is studied in this paper. In particular the geodesics of a massive particle in rainbow flat spacetime is obtained when the semi-classical effect of its own energy on the background is taken into account. We show that in general the trajectory of a freely falling particle remains unchanged which is still a straight line as in the flat spacetime. The implication to the Unruh effect in rainbow flat spacetime is also discussed. 
  Presented in this paper is a Markov chain Monte Carlo (MCMC) routine for conducting coherent parameter estimation for interferometric gravitational wave observations of an inspiral of binary compact objects using data from multiple detectors. The MCMC technique uses data from several interferometers and infers all nine of the parameters (ignoring spin) associated with the binary system, including the distance to the source, the masses, and the location on the sky. The Metropolis-algorithm utilises advanced MCMC techniques, such as importance resampling and parallel tempering. The data is compared with time-domain inspiral templates that are 2.5 post-Newtonian (PN) in phase and 2.0 PN in amplitude. Our routine could be implemented as part of an inspiral detection pipeline for a world wide network of detectors. Examples are given for simulated signals and data as seen by the LIGO and Virgo detectors operating at their design sensitivity. 
  We consider a planar gravitating thick domain wall of the $\lambda \phi^4$ theory as a spacetime with finite thickness glued to two vacuum spacetimes on each side of it. Darmois junction conditions written on the boundaries of the thick wall with the embedding spacetimes reproduce the Israel junction condition across the wall in the limit of infinitesimal thickness. The thick planar domain wall located at a fixed position is then transformed to a new coordinate system in which its dynamics can be formulated. It is shown that the wall's core expands as if it were a thin wall. The thickness in the new coordinates is not constant anymore and its time dependence is given. 
  We study the phase space of the spherically symmetric solutions of the system obtained from the dimensional reduction of the six-dimensional Einstein-Gauss-Bonnet action with a cosmological constant. We show that all the physical solutions have anti-de Sitter asymptotic behavior. 
  Calculating the spinor connection in curved spacetime is a tiresome and fallible task. This pedagogical paper display an equivalent but simple form of the covariant derivative for both the Weyl spinor and the Dirac bispinor, which is more convenient for calculation, and its geometrical meanings are more distinct. We separate the spinor connection into two parts. One is caused by geometry, but the other results in gravimagnetic effects. For a spinor in skew spacetime, the principle of equivalence seems to be broken. 
  Previous work on exact solutions has been shown that sources need to be appended to the field equation of Einstein's unified field theory in order to achieve physically meaningful results,such sources can be included in a variational formulation by Borchsenius and moffat.The resulting field equations and conservation identities related to the theory that can be used to derive the equations of structure and motion of a pole-dipole particle according to an explicitly covariant approach by Dixon6.In this present paper it is shown that,under certain conditions for the energy tensor of the spinning particle,the equations of structure and motion in an electromagnetic field turn out to be formly identical to those occurring in Einstein-Maxwell theory. 
  We show that a number of problems of modern cosmology may be solved in the framework of multidimensional gravity with high-order curvature invariants, without invoking other fields. We use a method employing a slow-change approximation, able to work with rather a general form of the gravitational action, and consider Kaluza-Klein type space-time with one or several extra factor spaces. A vast choice of effective theories suggested by the present framework may be stressed: even if the initial Lagrangian is entirely fixed, one obtains quite different models for different numbers, dimensions and topologies of the extra factor spaces. Among the problems addressed are (i) explanation of the present accelerated expansion of the Universe, with a reasonably small cosmological constant, (ii) massive primordial black hole production, which is a necessary stage in some scenarios of cosmic structure formation, (iii) sufficient particle production rate at the end of inflation. We also discuss chaotic attractors appearing at possible nodes of the kinetic term of the effective scalar field Lagrangian. 
  Reducing thermal noise from optical coatings is crucial to reaching the required sensitivity in next generation interferometric gravitational-waves detectors. Here we show that adding TiO$_2$ to Ta$_2$O$_5$ in Ta$_2$O$_5$/SiO$_2$ coatings reduces the internal friction and in addition present data confirming it reduces thermal noise. We also show that TiO$_2$-doped Ta$_2$O$_5$/SiO$_2$ coatings are close to satisfying the optical absorption requirements of second generation gravitational-wave detectors. 
  There are two important basic questions in the measurement of time. The first one is how to define the simultaneity of two events occurring at two different places. The second one is how to define the equality of two durations. The first question has been solved by Einstein, Landau and others on the convention that the velocity of light is isotropic and it is a constant in empty space. But no body has answered the second question until today. In this paper, on the same convention about the velocity of light given by Poincar\'{e}, Einstein, Landau and others, we find the solution to the definition of the equality of two durations. Meanwhile, we also find answer to the question about the definition of the synchronization of rate of clocks located at different places. 04.20.Cv, 04.20.-q 
  We study the numerical implementation of a set of boundary conditions derived from the isolated horizon formalism, and which characterize a black hole whose horizon is in quasi-equilibrium. More precisely, we enforce these geometrical prescriptions as inner boundary conditions on an excised sphere, in the numerical resolution of the Conformal Thin Sandwich equations. As main results, we firstly establish the consistency of including in the set of boundary conditions a "constant surface gravity" prescription, interpretable as a lapse boundary condition, and secondly we assess how the prescriptions presented recently by Dain et al. for guaranteeing the well-posedness of the Conformal Transverse Traceless equations with quasi-equilibrium horizon conditions extend to the Conformal Thin Sandwich elliptic system. As a consequence of the latter analysis, we discuss the freedom of prescribing the expansion associated with the ingoing null normal at the horizon. 
  We consider an extended version of the Kerr theorem incorporated in the Kerr-Schild formalism. It allows one to construct the series of exact solutions of the Einstein-Maxwell field equations from a holomorphic generating function $F$ of twistor variables. The exact multiparticle Kerr-Schild solutions are obtained from generating function of the form $F=\prod_i^k F_i, $ where $F_i$ are partial generating functions for the spinning particles $ i=1...k$. Gravitational and electromagnetic interaction of the spinning particles occurs via the light-like singular twistor lines. As a result, each spinning particle turns out to be `dressed' by singular pp-strings connecting it to other particles. Physical interpretation of this solution is discussed. 
  It has been persuasively argued that the number of the effective degrees of freedom of a macroscopic system is proportional to its area rather than to its volume. This entails interesting consequences for cosmology. Here we present a model based on this "holographic principle" that accounts for the present stage of accelerated expansion of the Universe and significantly alleviates the coincidence problem also for non-spatially flat cosmologies. Likewise, we comment on a recently proposed late transition to a fresh decelerated phase. 
  A method is presented which allows the exact construction of conserved (i.e. divergence-free) current vectors from appropriate sets of multipole moments. Physically, such objects may be taken to represent the flux of particles or electric charge inside some classical extended body. Several applications are discussed. In particular, it is shown how to easily write down the class of all smooth and spatially-bounded currents with a given total charge. This implicitly provides restrictions on the moments arising from the smoothness of physically reasonable vector fields. We also show that requiring all of the moments to be constant in an appropriate sense is often impossible; likely limiting the applicability of the Ehlers-Rudolph-Dixon notion of quasirigid motion. A simple condition is also derived that allows currents to exist in two different spacetimes with identical sets of multipole moments (in a natural sense). 
  5-dimensional special relativity can be considered as the 5-dimensional extension of Carmeli's cosmological special relativity, as well as the flat specialization of 5-d brane world theory. To this framework we add a 5-dimensional perfect fluid stress-energy tensor, and unify the equations of perfect hydrodynamics in a single 5-dimensional tensor conservation law. This picture permits to interpret particle production phenomena as cosmological effects, in the spirit of open system cosmology. The source of particle production vanishes if the fluid is isentropic. Moreover the hydrodynamical equations can be interpreted in terms of a scale factor, giving rise to a set of equations which simulate in a sense Friedmann cosmology. 
  Some aspects of lightlike dimensional reduction in flat spacetime are studied with emphasis to classical applications. Among them the Galilean transformation of shadows induced by inertial frame changes is studied in detail by proving that, (i) the shadow of an object has the same shape in every orthogonal-to-light screen, (ii) if two shadows are simultaneous in an orthogonal-to-light screen then they are simultaneous in any such screen. In particular, the Galilean group in 2+1 dimensions is recognized as an exact symmetry of Nature which acts on the shadows of the events instead that on the events themselves. The group theoretical approach to lightlike dimensional reduction is used to solve the reconstruction problem of a trajectory starting from its acceleration history or from its projected (shadow) trajectory. The possibility of obtaining a Galilean projected physics starting from a Poincar\'e invariant physics is stressed through the example of relativistic collisions. In particular, it is shown that the projection of a relativistic collision between massless particles gives a non-relativistic collision in which the kinetic energy is conserved. 
  We find exact cosmological solutions when the Newton parameter and the cosmological term are dynamically evolving in a renormalization-group improved Hamiltonian approach. In our derivation we use the Noether symmetry approach, leading to an interesting variable transformation which yields exact and general integration of the cosmological equations. The functional dependence of Lambda on G is determined by the method itself, therefore generalizing previous results on symmetry principles in cosmology. 
  We use a dynamical systems approach to investigate Bianchi type I and II universes in quadratic theories of gravity. Due to the complicated nature of the equations of motion we focus on the stability of exact solutions and find that there exists an isotropic FRW universe acting as a past attractor. This may indicate that there is an isotropisation mechanism at early times for these kind of theories. We also discuss the Kasner universes, elucidate the associated centre manifold structure, and show that there exists a set of non-zero measure which has the Kasner solutions as a past attractor. Regarding the late-time behaviour, the stability shows a dependence of the parameters of the theory. We give the conditions under which the de Sitter solution is stable and also show that for certain values of the parameters there is a possible late-time behaviour with phantom-like behaviour. New types of anisotropic inflationary behaviour are found which do not have counterparts in general relativity. 
  The Mathisson-Papapetrou method is originally used for derivation of the particle world line equation from the covariant conservation of its stress-energy tensor. We generalize this method to extended objects, such as a string. Without specifying the type of matter the string is made of, we obtain both the equations of motion and boundary conditions of the string. The world sheet equations turn out to be more general than the familiar minimal surface equations. In particular, they depend on the internal structure of the string. The relevant cases are classified by examining canonical forms of the effective 2-dimensional stress-energy tensor. The case of homogeneously distributed matter with the tension that equals its mass density is shown to define the familiar Nambu-Goto dynamics. The other three cases include physically relevant massive and massless strings, and unphysical tahyonic strings. 
  It is well known that a rotating superconductor produces a magnetic field proportional to its angular velocity. The authors conjectured earlier, that in addition to this so-called London moment, also a large gravitomagnetic field should appear to explain an apparent mass increase of Niobium Cooper-pairs. A similar field is predicted from Einstein's general relativity theory and the presently observed amount of dark energy in the universe. An experimental facility was designed and built to measure small acceleration fields as well as gravitomagnetic fields in the vicinity of a fast rotating and accelerating superconductor in order to detect this so-called gravitomagnetic London moment. This paper summarizes the efforts and results that have been obtained so far. Measurements with Niobium superconductors indeed show first signs which appear to be within a factor of 2 of our theoretical prediction. Possible error sources as well as the experimental difficulties are reviewed and discussed. If the gravitomagnetic London moment indeed exists, acceleration fields could be produced in a laboratory environment. 
  The purpose of this paper is to propose an extension to Lee Smolin's hypothesis that our own universe belongs to a population of universes which have evolved by natural selection. Smolin's hypothesis explains why the parameters of physics possess the values we observe them to possess, but depends upon the contingent fact that the universe is a quantum relativistic universe. It is proposed that the prior existence of a quantum relativistic universe can itself be explained by the notion of evolution towards stable (`rigid') mathematical structures. 
  We investigate the action of diffeomorphisms in the context of Hamiltonian Gravity. By considering how the diffeomorphism-invariant Hilbert space of Loop Quantum Gravity should be constructed, we formulate a physical principle by demanding, that the gauge-invariant Hilbert space is a completion of gauge- (i.e. diffeomorphism-)orbits of the classical (configuration) variables, explaining which extensions of the group of diffeomorphisms must be implemented in the quantum theory. It turns out, that these are at least a subgroup of the stratified analytic diffeomorphisms. Factoring these stratified diffeomorphisms out, we obtain that the orbits of graphs under this group are just labelled by their knot classes, which in turn form a countable set. Thus, using a physical argument, we construct a separable Hilbert space for diffeomorphism invariant Loop Quantum Gravity, that has a spin-knot basis, which is labelled by a countable set consisting of the combination of knot-classes and spin quantum numbers. It is important to notice, that this set of diffeomorphism leaves the set of piecewise analytic edges invariant, which ensures, that one can construct flux-operators and the associated Weyl-operators. A note on the implications for the treatment of the Gauss- and the Hamilton-constraint of Loop Quantum Gravity concludes our discussion. 
  The asymptotic safety scenario in quantum gravity is reviewed, according to which a renormalizable quantum theory of the gravitational field is feasible which reconciles asymptotically safe couplings with unitarity.All presently known evidence is surveyed: (a) from the 2+\epsilon expansion,(b) from renormalizable higher derivative gravity theories and a `large N' expansion in the number of matter fields, (c) from the 2-Killing vector reduction, and (d) from truncated flow equations for the effective average action. Special emphasis is given to the role of perturbation theory as a guide to `asymptotic safety'. Further it is argued that as a consequence of the scenario the selfinteractions appear two-dimensional in the extreme ultraviolet. Two appendices discuss the distinct roles of the ultraviolet renormalization in perturbation theory and in the flow equation formalism. 
  The characteristic approach to numerical relativity is a useful tool in evolving gravitational systems. In the past this has been implemented using two patches of stereographic angular coordinates. In other applications, a six-patch angular coordinate system has proved effective. Here we investigate the use of a six-patch system in characteristic numerical relativity, by comparing an existing two-patch implementation (using second-order finite differencing throughout) with a new six-patch implementation (using either second- or fourth-order finite differencing for the angular derivatives). We compare these different codes by monitoring the Einstein constraint equations, numerically evaluated independently from the evolution. We find that, compared to the (second-order) two-patch code at equivalent resolutions, the errors of the second-order six-patch code are smaller by a factor of about 2, and the errors of the fourth-order six-patch code are smaller by a factor of nearly 50. 
  The approximate renormalized one-loop effective action of the quantized massive scalar, spinor and vector field in a large mass limit, i.e., the lowest order of the DeWitt-Schwinger expansion involves the coincidence limit of the Hadamard-DeWitt coefficient a3. Building on this and using Wald's approach we shall construct the general expression describing entropy of the spherically-symmetric static black hole being the solution of the semi-classical field equations. For the concrete case of the quantum-corrected Reissner-Nordstrom black hole this result coincides, as expected, with the entropy obtained by integration of the first law of black hole thermodynamics with a suitable choice of the integration constant. The case of the extremal quantum corrected black hole is briefly considered. 
  We show that when the antisymmetric Kalb-Ramond field is included in the Randall-Sundrum scenario, although the hierarchy problem can be solved, it requires an extreme fine tuning of the Kalb-Ramond field (about 1 part in $10^{62}$). We interpret this as the return of the problem in disguise. Further, we show that the Kalb-Ramond field induces a small negative cosmological constant on the visible brane. 
  We review aspects of black hole thermodynamics, and show how entanglement of a quantum field between the inside and outside of a horizon can account for the area-proportionality of black hole entropy, provided the field is in its ground state. We show that the result continues to hold for Coherent States and Squeezed States, while for Excited States, the entropy scales as a power of area less than unity. We also identify location of the degrees of freedom which give rise to the above entropy. 
  We discuss in detail the uniform discretization approach to the quantization of totally constrained theories. This approach allows to construct the continuum theory of interest as a well defined, controlled, limit of well behaved discrete theories. We work out several finite dimensional examples that exhibit behaviors expected to be of importance in the quantization of gravity. We also work out the case of BF theory. At the time of quantization, one can take two points of view. The technique can be used to define, upon taking the continuum limit, the space of physical states of the continuum constrained theory of interest. In particular we show in models that it agrees with the group averaging procedure when the latter exists. The technique can also be used to compute, at the discrete level, conditional probabilities and the introduction of a relational time. Upon taking the continuum limit one can show that one reproduces results obtained by the use of evolving constants, and therefore recover all physical predictions of the continuum theory. This second point of view can also be used as a paradigm to deal with cases where the continuum limit does not exist. There one would have discrete theories that at least at certain scales reproduce the semiclassical properties of the theory of interest. In this way the approach can be viewed as a generalization of the Dirac quantization procedure that can handle situations where the latter fails. 
  We obtained an exact solution for a uniformly accelerated Unruh-DeWitt detector interacting with a massless scalar field in (3+1) dimensions which enables us to study the entire evolution of the total system, from the initial transient to late-time steady state. We find that the Unruh effect as derived from time-dependent perturbation theory is valid only in the transient stage and is totally invalid for cases with proper acceleration smaller than the damping constant. We also found that, unlike in (1+1)D results, the (3+1)D uniformly accelerated Unruh-DeWitt detector in a steady state does emit a positive radiated power of quantum nature at late-times, but it is not connected to the thermal radiance experienced by the detector in the Unruh effect proper. 
  In this paper we consider photon-photon scattering due to self-induced gravitational perturbations on a Minkowski background. We focus on four-wave interaction between plane waves with weakly space and time dependent amplitudes, since interaction involving a fewer number of waves is excluded by energy-momentum conservation. The Einstein-Maxwell system is solved perturbatively to third order in the field amplitudes and the coupling coefficients are found for arbitrary polarizations in the center of mass system. Comparisons with calculations based on quantum field theoretical methods are made, and the small discrepances are explained. 
  In order to consistently introduce an interaction between gravity and fermions in the Ashtekar-Barbero-Immirzi formalism a non-minimal term is necessary. The non-minimal term together with the Holst modification to the Hilbert-Palatini action reconstruct the Nieh-Yan invariant. The Immirzi parameter does not affect the classical dynamics, which is described by the Einstein-Cartan effective action. 
  We present a method for the Hamiltonian formulation of field theories that are based on Lagrangians containing second derivatives. The new feature of our formalism is that all four partial derivatives of the field variables are initially considered as independent fields, in contrast to the conventional Ostrogradski method, where only the velocity is turned into an independent field variable. The consistency of the formalism is demonstrated by simple unconstrained and constrained second order scalar field theories. Its application to General Relativity is briefly outlined. 
  We present a new test for a possible Mach-Sciama dependence of the Gravitational constant G. According to Ernst Mach (1838-1916), the gravitational interaction depends on the distribution of masses in the universe. A corresponding hypothesis of Sciama (1953) on the gravitational constant, $c^2/G = \sum m_i/r_i$, can be tested since the elliptic earth orbit should then cause minute annual variations in G. The test is performed by analyzing the gravity signals of a network of superconducting gravimeters (SG) which reach a precision of $10^{-10} m/s^2$. After reducing the signal by modelling tidal, meteorologic and geophysical effects, no significant evidence for the above dependence is found. 
  The Hubble law is extended to massive particles based on the de Broglie wavelength. Due to the expansion of the universe the wavelength of an unbound particle would increase according to its cosmological redshift. Based on the navigation anomalies of the Pioneer 10 & 11 spacecraft it is postulated that an unbound massive particle has a cosmological redshift z = (c / v_0) H_0 t, where c is the speed of light in vacuum, v_0 is the initial velocity of the particle, H_0 is Hubble's constant and t is the duration of time that the particle has been unbound. The increase in wavelength of the particle corresponds to a decrease in its speed by delta_v = - c H_0 t. Furthermore, it is hypothesized that the solar system has escaped the gravity of the Galaxy as evidenced by its orbital speed and radial distance and by the visible mass within the solar system radius. This means that spacecraft which become unbound to the solar system would also be galactically unbound and subject to the Hubble law. This hypothesis and the extended Hubble law may explain the anomalous acceleration found to be acting upon the unbound Pioneer 10 & 11 spacecraft. Thus, the Pioneer anomaly may be a counter example to the dark matter hypothesis. Because photons have a speed which make them unbound to the Galaxy, it is predicted that the navigation beam in open space would undergo a cosmological redshift in its frequency which would be detectable with modern clocks. 
  Motivated by the newest progress in geometric flows both in mathematics and physics, we apply the geometric evolution equation to study some black-hole problems. Our results show that, under certain conditions, the geometric evolution equations satisfy the Birkhoff theorem, and surprisingly, in the case of spherically symmetric metric field, the Einstein equation, the Ricci flow, and the hyperbolic geometric flow in vacuum spacetime have the same black-hole solutions, especially in the case of $\Lambda=0$, they all have the Schwarzschild solution. In addition, these results can be generalized to a kind of more general geometric flow. 
  It is shown that not all linear electromagnetic constitutive laws will define almost-complex structure on the bundle of 2-forms on the spacetime manifold when composed with the Poincare duality isomorphism, but only a restricted class of them that includes linear spatially isotropic and some bi-isotropic laws. Although this does not trivialize the formulation of the basic equations equations of pre-metric electromagnetism, it does affect their reduction to metric electromagnetism by its effect on the types of media that are reducible, and possibly its effect on the way that such media support the propagation of electromagnetic waves. 
  We study the geometry and dynamics of both isolated and dynamical trapping horizons by considering the allowed variations of their foliating two-surfaces. This provides a common framework that may be used to consider both their possible evolutions and their deformations as well as derive the well-known flux laws. Using this framework, we unify much of what is already known about these objects as well as derive some new results. In particular we characterize and study the "almost-isolated" trapping horizons known as slowly evolving horizons. It is for these horizons that a dynamical first law holds and this is analogous and closely related to the Hawking-Hartle formula for event horizons. 
  In this paper, the effect of a positive cosmological constant on spherically symmetric collapse with perfect fluid has been investigated. The matching conditions between static exterior and non-static interior spacetimes are given in the presence of a cosmological constant. We also study the apparent horizons and their physical significance. It is concluded that the cosmological constant slows down the collapse of matter and hence limit the size of the black hole. This analysis gives the generalization of the dust case to the perfect fluid. We recover the results of the dust case for $p=0$. 
  There is a longstanding mystery connected with the radiotracking of distant interplanetary spaceprobes like ULYSSES, Galileo and especially the two NASA probes PIONEER 10 and 11. Comparing radiosignals outgoing from the earth to the probe and ingoing again from the probes do show anomalous frequency shifts which up to now have been explained as caused by anomalous non-Newtonian decelerations of these probes recognizable at solar distances beyond 5 AU. In this paper we study cosmological conditions for the transfer of radiosignals between the Earth and these distant probes. Applying general relativity, we derive both the geodetic deceleration as well as the cosmological redshift and compare the resulting frequency shift with the observed effect. We find that anomalous decelerations do act on these probes which are of cosmological nature, but these are, as expected from standard cosmology, much too low to explain the observed effect. In contrast, the cosmological redshift of radiophotons suffered during the itinerary to the probe and back due to the local spacetime expansion reveals a frequency shift which by its magnitude is in surprisingly good agreement with the long registered phenomenon, and thus explains the phenomenon well, except for the sign of the effect. Problems of a local Hubble expansion may give the reason for this. 
  It is argued that constraints on time variation of the gravitational constant (dG/dt)/G, e.g. derived from the lunar laser ranging, cannot be immediately applied to restrict the cosmological expansion at planetary scales, as it was done by Williams, Turyshev, and Boggs [PRL, 93, 261101 (2004), arXiv:gr-qc/0411113]. 
  We outline that, in a Kaluza-Klein framework, not only the electro-magnetic field can be geometrized, but also the dynamics of a charged spinning particle can be inferred from the motion in a 5-dimensional space-time. This result is achieved by the dimensional splitting of Papapetrou equations and by proper identifications of 4-dimensional quantities. 
  We consider Kaluza-Klein theories as candidates for the unification of gravity and the electro-weak model. In particular, we fix how to reproduce geometrically the interaction between fermions and gauge bosons, in the low energy limit. 
  It is shown that cosmological spacetime manifold has the structure of a Lie group and a spinor space. This leads naturally to the Minkowski metric on tangent spaces and the Lorentzian metric on the manifold and makes it possible to dispense with double-valued representations. 
  The gravitational collapse of a cylindrical thick shell of dust matter is investigated. A spacetime singularity forms at the symmetry axis and it is necessarily naked, i.e., observable in principle. We propose a physically reasonable boundary condition at the naked singularity to construct the solution including its causal future. This boundary condition enables us to construct the unique continuation of spacetime beyond the naked singularity and makes the dust shell pass through the naked singularity. When the cylindrical shell has left the symmetry axis away, the naked singularity disappears and regularity is recovered. We succeed in constructing numerical solutions with this feature. This result means that gravity produced by a cylindrical thick shell of dust is too weak to bind itself even if it forms the curvature singularity which is so strong as to satisfy the limiting focusing condition. As a result, this naked singularity is very gentle in the extended spacetime; the metric tensor is $C^{1-}$ even at the naked singularity and the extended spacetime is complete for almost all geodesics. This feature is also seen for singular hypersurfaces. Such an extended spacetime can be regarded as phenomenological in a sense that it is valid providing that the relevant microphysics length scale is sufficiently small compared to the scale of interest. 
  Two general-relativistic hydrodynamical models are considered: a model of self-gravitating static configurations of perfect fluid and a model of steady accretion of fluid onto a black hole. We generalise analytic results obtained for the original polytropic versions of these models onto a wider class of barotropic equations of state. The knowledge about the polytropic solutions is used to establish bounds on certain characteristic quantities appearing in both cases. 
  In second-generation, ground-based interferometric gravitational-wave detectors such as advanced LIGO, the dominant noise at frequencies f ~ 40 Hz to 200 Hz is expected to be due to thermal fluctuations in the mirrors' substrates and coatings which induce random fluctuations in the shape of the mirror face. The laser-light beam averages over these fluctuations; the larger the beam and the flatter its light-power distribution, the better the averaging and the lower the resulting thermal noise. This has led O'Shaughnessy and Thorne to propose flattening and enlarging the beam shape to reduce the thermal noise. In this paper I derive and discuss simple scaling laws that describe the dependence of the thermal noise on the beam's (axisymmetric) light-power distribution. Each of these scaling laws has previously been deduced, from somewhat general arguments rather than detailed calculations, by O'Shaughnessy; independently, the same scaling laws have been found by Vyatchanin [for Brownian coating noise], by by O'Shaughnessy, Strigin and Vyatchanin [for substrate thermoelastic noise], and by Vinet [for substrate Brownian noise]. These scaling laws are valid in the limit that the mirror dimensions are large compared to the beam radius. Recently Agresti has computed the sensitivity improvement when flat-top (or "mesa'') beams are used instead of gaussian beams (with the diffraction loss fixed). When the mirror substrate is fused silica with radius not larger than the baseline radius for advanced LIGO (17 cm), the coating-noise infinite-mirror scaling laws agree with Agresti's finite-mirror calculations within about 10%, and the substrate-noise infinite-mirror scaling laws agree with Agresti's finite-mirror calculations within about 15%. 
  The stability analysis of self-similar solutions is an important approach to confirm whether they act as an attractor in general non-self-similar gravitational collapse. Assuming that the collapsing matter is a perfect fluid with the equation of state $P=\alpha\rho$, we study spherically symmetric non-self-similar perturbations in homogeneous self-similar collapse described by the flat Friedmann solution. In the low pressure approximation $\alpha \ll 1$, we analytically derive an infinite set of the normal modes and their growth (or decay) rate. The existence of one unstable normal mode is found to conclude that the self-similar behavior in homogeneous collapse of a sufficiently low pressure perfect fluid must terminate and a certain inhomogeneous density profile can develop with the lapse of time. 
  It will be shown that a given realization of nonlinear electrodynamics, used as source of Einstein's equations, generates a cosmological model with interesting features, namely a phase of current cosmic acceleration, and the absence of an initial singularity, thus pointing to a way to solve two important problems in cosmology. 
  We consider dark energy cosmology in a de Sitter universe filled with quantum conformal matter. Our model represents a Gauss-Bonnet model of gravity with contributions from quantum effects. To the General Relativity action an arbitrary function of the GB invariant, f(G), is added, and taking into account quantum effects from matter the cosmological constant is studied. For the considered model the conditions for a vanishing cosmological constant are considered. Creation of a de Sitter universe by quantum effects in a GB modified gravity is discussed. 
  We study the constraint equations for the Einstein-scalar field system on compact manifolds. Using the conformal method we reformulate these equations as a determined system of nonlinear partial differential equations. By introducing a new conformal invariant, which is sensitive to the presence of the initial data for the scalar field, we are able to divide the set of free conformal data into subclasses depending on the possible signs for the coefficients of terms in the resulting Einstein-scalar field Lichnerowicz equation. For many of these subclasses we determine whether or not a solution exists. In contrast to other well studied field theories, there are certain cases, depending on the mean curvature and the potential of the scalar field, for which we are unable to resolve the question of existence of a solution. We consider this system in such generality so as to include the vacuum constraint equations with an arbitrary cosmological constant, the Yamabe equation and even (all cases of) the prescribed scalar curvature problem as special cases. 
  One of the most exciting prospects for the Laser Interferometer Space Antenna (LISA) is the detection of gravitational waves from the inspirals of stellar-mass compact objects into supermassive black holes. Detection of these sources is an extremely challenging computational problem due to the large parameter space and low amplitude of the signals. However, recent work has suggested that the nearest extreme mass ratio inspiral (EMRI) events will be sufficiently loud that they might be detected using computationally cheap, template-free techniques, such as a time-frequency analysis. In this paper, we examine a particular time-frequency algorithm, the Hierarchical Algorithm for Clusters and Ridges (HACR). This algorithm searches for clusters in a power map and uses the properties of those clusters to identify signals in the data. We find that HACR applied to the raw spectrogram performs poorly, but when the data is binned during the construction of the spectrogram, the algorithm can detect typical EMRI events at distances of up to $\sim2.6$Gpc. This is a little further than the simple Excess Power method that has been considered previously. We discuss the HACR algorithm, including tuning for single and multiple sources, and illustrate its performance for detection of typical EMRI events, and other likely LISA sources, such as white dwarf binaries and supermassive black hole mergers. We also discuss how HACR cluster properties could be used for parameter extraction. 
  The unprecedented precision of atom interferometry will soon lead to laboratory tests of general relativity to levels that will rival or exceed those reached by astrophysical observations. We propose such an experiment that will initially test the equivalence principle to 1 part in 10^15 (300 times better than the current limit), and 1 part in 10^17 in the future. It will also probe general relativistic effects--such as the non-linear three-graviton coupling, the gravity of an atom's kinetic energy, and the falling of light--to several decimals. Further, in contrast to astrophysical observations, laboratory tests can isolate these effects via their different functional dependence on experimental variables. 
  In d+1 dimensions we solve the equations of motion for the case of gravity minimally or conformally coupled to a scalar field. For the minimally coupled system the equations can either be solved directly or by transforming vacuum solutions, as shown before in 3+1 dimensions by Buchdahl. In d+1 dimensions the solutions have been previously found directly by Xanthopoulos and Zannias. Here we first rederive these earlier results, and then extend Buchdahl's method of transforming vacuum solutions to d+1 dimensions. We also review the conformal coupling case, in which d+1 dimensional solutions can be found by extending Bekenstein's method of conformal transformation of the minimal coupling solution. Combining the extended versions of Buchdahl transformations and Bekenstein transformations we can in arbitrary dimensions always generate solutions of both the minimal and the conformal equations from known vacuum solutions. 
  Contrary to our immediate and vivid sensation of past, present, and future as continually shifting non-relational modalities, time remains as tenseless and relational as space in all of the established theories of fundamental physics. Here an empirically adequate generalized theory of the inertial structure is discussed in which proper time is causally compelled to be tensed within both spacetime and dynamics. This is accomplished by introducing the inverse of the Planck time at the conjunction of special relativity and Hamiltonian mechanics, which necessitates energies and momenta to be invariantly bounded from above, and lengths and durations similarly bounded from below, by their respective Planck scale values. The resulting theory abhors any form of preferred structure, and yet captures the transience of now along timelike worldlines by causally necessitating a genuinely becoming universe. This is quite unlike the scenario in Minkowski spacetime, which is prone to a block universe interpretation. The minute deviations from the special relativistic effects such as dispersion relations and Doppler shifts predicted by the generalized theory remain quadratically suppressed by the Planck energy, but may nevertheless be testable in the near future, for example via observations of oscillating flavor ratios of ultrahigh energy cosmic neutrinos, or of altering pulse rates of extreme energy binary pulsars. 
  In order to satisfy the equivalence principle, any mechanism proposed to gravitationally explain the Pioneer anomaly, in the form in which it is presently known from the so-far analyzed Pioneer 10/11 data, cannot leave out of consideration its impact on the motion of the planets of the Solar System as well, especially those orbiting in the regions in which the anomalous behavior of the Pioneer probes manifested itself. In this paper we, first, use the latest determinations of the secular perihelion advances of some planets in order to put on the test two gravitational mechanisms recently proposed to accommodate the Pioneer anomaly based on two models of modified gravity. Second, we use the radio-technical ranging data to Voyager 2 when it encountered Uranus and Neptune to perform a further, independent test of the hypothesis that a Pioneer-like acceleration can also affect the motion of the outer planets of the Solar System. As in the case of previous tests based on the use of the directly observable planetary right ascension and declination, the obtained answer is negative. 
  Although the observed universe appears to be geometrically flat, it could have one of 18 global topologies. A constant-time slice of the spacetime manifold could be a torus, Mobius strip, Klein bottle, or others. This global topology of the universe imposes boundary conditions on quantum fields and affects the vacuum energy density via Casimir effect. In a spacetime with such a nontrivial topology, the vacuum energy density is shifted from its value in a simply-connected spacetime. In this paper, the vacuum expectation value of the stress-energy tensor for a massless scalar field is calculated in all 17 multiply-connected, flat and homogeneous spacetimes with different global topologies. It is found that the vacuum energy density is lowered relative to the Minkowski vacuum level in all spacetimes and that the stress-energy tensor becomes position-dependent in spacetimes that involve reflections and rotations. 
  Static observers remain on Killing-vector world lines and measure the rest-mass+kinetic energies of particles moving past them, and the flux of that mechanical energy through space and time. The total mechanical energy is the total flux through a spacelike cut at one time. The difference between the total mass-energy and the total mechanical energy is the total gravitational energy, which we prove to be negative for certain classes of systems. For spherical systems, Misner, Thorne and Wheeler define the total gravitational energy this way.   To obtain the gravitational energy density analogous to that of electromagnetism we first use Einstein's equations with integrations by parts to remove second order derivatives. Next we apply a conformal transformation to reexpress the scalar 3-curvature of the 3-space. The resulting density is non-local.   We repeat the argument for mechanical energies as measured by stationary observers moving orthogonally to constant time slices like the "zero angular momentum" observers of Bardeen who exist even within ergospheres. 
  The locality hypothesis is generally considered necessary for the study of the kinematics of non-inertial systems in special relativity. In this paper we discuss this hypothesis, showing the necessity of an improvement, in order to get a more clear understanding of the various concepts involved, like coordinate velocity and standard velocity of light. Concrete examples are shown, where these concepts are discussed. 
  The existence of a non-zero cosmological constant $\Lambda$ gives rise to controversial interpretations. Is $\Lambda$ a universal constant fixing the geometry of an empty universe, as fundamental as the Planck constant or the speed of light in the vacuum? Its natural place is then on the left-hand side of the Einstein equation. Is it instead something emerging from a perturbative calculus performed on the metric $g\_{\mu\nu}$ solution of the Einstein equation and to which it might be given a material status of (dark or bright) "energy"? It should then be part of the content of the right-hand side of the Einstein equations. The purpose of this paper is not to elucidate the fundamental nature of $\Lambda$, but instead we aim to present and discuss some of the arguments in favor of both interpretations of the cosmological constant. We conclude that if the fundamental of the geometry of space-time is minkowskian, then the square of the mass of the graviton is proportional to $\Lambda$; otherwise, if the fundamental state is deSitter/AdS, then the graviton is massless in the deSitterian sense. 
  The static, cylindrically symmetric vacuum solutions with a cosmological constant in the framework of the Brans-Dicke theory are investigated. Some of these solutions admitting Lorentz boost invariance along the symmetry axis correspond to local, straight cosmic strings with a cosmological constant. Some physical properties of such solutions are studied. These strings apply attractive or repulsive forces on the test particles. A smooth matching is also performed with a recently introduced interior thick string solution with a cosmological constant. 
  The generalized uncertainty principle has been described as a general consequence of incorporating a minimal length from a theory of quantum gravity. We consider a simple quantum mechanical model where the operator corresponding to position has discrete eigenvalues and show how the generalized uncertainty principle results for minimum uncertainty wave packets. 
  The aim of this work is to review the concepts of time in quantum mechanics and general relativity to show their incompatibility. We show that the absolute character of Newtonian time is present in quantum mechanics and also partially in quantum field theories which consider the Minkowski metric as the background spacetime. We discuss the problems which this non-dynamical concept of time causes in general relativity that is characterized by a dynamical spacetime. 
  Using a nonlinear electrodynamics coupled to teleparallel theory of gravity, three regular charged spherically symmetric solutions are obtained. The nonlinear theory reduces to the Maxwell one in the weak limit and the solutions correspond to charged spacetimes. The third solution contains an arbitrary function from which we can generate the other two solutions. The metric associated with these spacetimes is the same, i.e., a regular charged static spherically symmetric black hole. In calculating the energy content of the third solution using the gravitational energy-momentum given by M{\o}ller, within the framework of the teleparallel geometry, we find that the resulting form depends on the arbitrary function. Using the regularized expression of the gravitational energy-momentum we get the value of energy. 
  A non-singular cosmology is derived in modified gravity (MOG). The universe begins in the distant past in a contracting de Sitter phase, is non-singular at $t=0$ and then describes approximately the standard radiation dominated solution before the time of decoupling. The spacetime in the neighborhood of $t=0$ is described by Minkowski spacetime in which the Weyl curvature tensor vanishes and the entropy is at a minimum value. The Hubble radius $H^{-1}(t)$ is infinite at $t=0$ and the universe goes into a static or quasi-static period, and the temperature of the hot radiation plasma at $t\sim 0$ does not exceed the Hagedorn temperature. 
  We introduce a general approximation scheme in order to calculate gauge invariant observables in the canonical formulation of general relativity. Using this scheme we will show how the observables and the dynamics of field theories on a fixed background or equivalently the observables of the linearized theory can be understood as an approximation to the observables in full general relativity. Gauge invariant corrections can be calculated up to an arbitrary high order and we will explicitly calculate the first non--trivial correction. Furthermore we will make a first investigation into the Poisson algebra between observables corresponding to fields at different space--time points and consider the locality properties of the observables. 
  This a book is for those who would like to learn something about special and general relativity beyond the usual textbooks, about quantum field theory, the elegant Fock-Schwinger-Stueckelberg proper time formalism, the elegant description of geometry by means of Clifford algebra, about the fascinating possibilities the latter algebra offers in reformulating the existing physical theories, and quantizing them in a natural way. It is shown how Clifford algebra provides much more: it provides room for new physics, with the prospects of resolving certain long standing puzzles. The theory of branes and the idea of how a 3-brane might represent our world is discussed in detail. Much attention is paid to the elegant geometric theory of branes which employs the infinite dimensional space of functions describing branes. Clifford algebra is generalized to the infinite dimensional spaces. In short, this is a book for anybody who would like to explore how the ``theory of everything'' might possibly be formulated. The theory that would describe all the known phenomena, could not be formulated without taking into account ``all'' the theoretical tools which are available. Foundations of those tools and their functional interrelations are described in the book. 
  We present a method that yields three decoupled covariant equations for three complex scalars, which completely govern electromagnetic perturbations of non-vacuum, locally rotationally symmetric class II spacetimes. One of these equations is equivalent to the previously established generalized Regge-Wheeler equation for electromagnetic fields. The remaining two equations are a direct generalization of the Bardeen-Press equations. The approach undertaken makes use of the well established 3+1 (and 2+1+1) formalism, and therefore, it is an ideal setting for specifying interpretable energy-momentum on an initial spacelike three-slice as the perturbation sources to the resultant electromagnetic radiation. 
  We briefly discuss the Hamiltonian formalism of Kantowski-Sachs space-times with vacuum, anisotropic fluid and two cross-streaming radiation field sources. For these models a cosmological time is introduced. New constraints are found in which the fluid momenta are separated from the rest of the variables. In consequence their Poisson brackets give an Abelian algebra. 
  We analyze the data of TAMA300 detector to search for gravitational waves from inspiraling compact star binaries with masses of the component stars in the range 1-3Msolar. In this analysis, 2705 hours of data, taken during the years 2000-2004, are used for the event search. We combine the results of different observation runs, and obtained a single upper limit on the rate of the coalescence of compact binaries in our Galaxy of 20 per year at a 90% confidence level. In this upper limit, the effect of various systematic errors such like the uncertainty of the background estimation and the calibration of the detector's sensitivity are included. 
  In A. Poltorak's concept, the reference frame in General Relativity is a certain manifold equipped with a connection. The question under consideration here is whether it is possible to join two events in the space-time by a time-like geodesic if they are joined by a geodesic of the reference frame connection that has a time-like initial vector. This question is interpreted as whether an event belongs to the proper future of another event in the space-time in case it is so in the reference frame. For reference frames of two special types some geometric conditions are found under which the answer is positive. 
  We give a progress report of our research on spacetime fluctuations induced by quantum fields in an evaporating black hole and a black hole in quasi-equilibrium with its Hawking radiation. We note the main issues involved in these two classes of problems and outline the key steps for a systematic quantitative investigation. This report contains unpublished new ideas for further studies. 
  Born-Infeld strategy to smooth theories having divergent solutions is applied to teleparallel equivalent of General Relativity. Differing from other theories of modified gravity, modified teleparallelism leads to second order equations, since teleparallel Lagrangian only contains first derivatives of the vierbein. We show that Born-Infeld-modified teleparallelism solves the particle horizon problem in a spatially flat FRW universe by providing an initial exponential expansion without resorting to an inflaton field. 
  A new theory of fundamental physics is presented; it predicts that the new dimensions that may be observed by the Large Hadron Collider are timelike. 
  Since antiquity, from Euclid of Alexandria to Galileo Galilei to Immanuel Kant to Hermann Minkowksi to Albert Einstein, the question of the nature of space and time has occupied scientists and philosophers. In the four-dimensional space-time of Einstein's wonderful theory of gravity, the squared interval, in units such that the speed of light is unity, is the difference between a squared time increment and the sum of three squares representing the three dimensions of spatial change. More recently higher dimensional theories have been proposed, which aim to unify gravity with the other forces in nature. Such theories typically have a hyperbolic character in that there is one time variable and many spatial variables (rather than just three) in the formula for the squared interval. Here a new physical theory is advanced, based on spinors, which clearly predicts that the basic extra dimensions are timelike: in its simplest form, there are three timelike and three spatial degrees of freedom. It is expected that devices such as the Large Hadron Collider will be sensitive to these new degrees of freedom and thus one may hope that in the near future, this issue can be settled experimentally 
  Rastall generalized Einstein's field equations relaxing the Einstein's assumption that the covariant divergence of the energy-momentum tensor should vanish. His field equations contain a free parameter alpha and in an empty space, i.e. if T_{\mu\nu}=0, they reduce to the Einstein's equations of standard general relativity. We calculate the elements of the metric tensor given by Rastall' equations for different \alpha assuming that T_{\mu\nu} to be that of a perfect fluid and analyse these model solutions from the point of view of Mach's principle. Since the source terms in Rastall's modified gravity equations include the common energy-momentum tensor T_{\mu\nu} as well as the expressions of the form (1-\alpha)g_{\mu\nu}T/2, these source terms depend on metric determined by the mass and momentum distribution of the external space. Likewise, in the classical limit, the source term of the corresponding Poisson equation for \alpha\neq 1 depends on the gravitational potential in the sense of Mach's principle. 
  This work investigates alternative theories of gravity, the solutions to their field equations and the constraints that can be imposed upon them from observation and experiment. Specifically, we consider the cosmologies and spherically symmetric solutions that can be expected to result from scalar-tensor and fourth-order theories of gravity. We find exact cosmological solutions of various different kinds; isotropic and anisotropic, homogeneous and inhomogeneous. These solutions are used to investigate the behaviour of the Universes at both late and early times, to investigate the effects of corrections to general relativity on approach to an initial singularity and to look for effects which may be observable in the present day Universe. We use physical processes, such as the primordial nucleosynthesis of the light elements, to impose constraints upon any deviations from the standard model. Furthermore, we investigate the vacuum spherically symmetric solutions of these theories. This environment is of particular interest for considerations of the local effects of gravity, where the most accurate experiments and observations of gravitational phenomena can be performed. Exact solutions are obtained for this situation and their stability analysed. It is found that a variety of new behaviour is obtainable in these theories that was not previously possible in the standard model. This new behaviour allows us an extended framework in which to consider gravitational physics, and its cosmological consequences. 
  In this paper, a version of polymer quantum mechanics, which is inspired by loop quantum gravity, is considered and shown to be equivalent, in a precise sense, to the standard, experimentally tested, Schroedinger quantum mechanics. The kinematical cornerstone of our framework is the so called polymer representation of the Heisenberg-Weyl (H-W) algebra, which is the starting point of the construction. The dynamics is constructed as a continuum limit of effective theories characterized by a scale, and requires a renormalization of the inner product. The result is a physical Hilbert space in which the continuum Hamiltonian can be represented and that is unitarily equivalent to the Schroedinger representation of quantum mechanics. As a concrete implementation of our formalism, the simple harmonic oscillator is fully developed. 
  In the present work the thermodynamical properties of bosonic and fermionic gases are analyzed under the condition that a modified dispersion relation is present. This last condition implies a breakdown of Lorentz symmetry. The implications upon the condensation temperature will be studied, as well, as upon other thermodynamical variables such as specific heat, entropy, etc. Moreover, it will be argued that those cases entailing a violation of time reversal symmetry of the motion equations could lead to problems with the concept of entropy. Concerning the fermionic case it will be shown that Fermi temperature suffers a modification due to the breakdown of Lorentz symmetry. The results will be applied to white dwarfs and the consequences upon the Chandrasekhar mass--radius relation will be shown. The possibility of resorting to white dwarfs for the testing of modified dispersion relations is also addressed. It will be shown that the comparison of the current observations against the predictions of our model allows us to discard some values of one of the parameters appearing in the modifications of the dispersion relation. 
  In this paper we study the gravitational collapse in loop quantum gravity. We consider the space-time region inside the Schwarzschild black hole event horizon and we divide this region in two parts, the first one where the matter (dust matter) is localized and the other (outside) where the metric is Kantowski-Sachs type. We calculate the state solving Hamiltonian constraint and we obtain a set of three difference equations that give a regular and natural evolution beyond the classical singularity point in "r=0" localized. 
  We prove strong cosmic censorship for T^2-symmetric cosmological spacetimes (with spatial topology T^3 and vanishing cosmological constant Lambda) with collisionless matter. Gowdy symmetric spacetimes constitute a special case. The formulation of the conjecture is in terms of generic C^2-inextendibility of the metric. Our argument exploits a rigidity property of Cauchy horizons, inherited from Killing fields. 
  Absolute Parallelism (AP) has many interesting features: large symmetry group of equations; field irreducibility with respect to this group; large list of consistent second order equations not restricted with Lagrangian ones.   There is the variant of AP whose solutions seem to be free of arising singularities iff D=5. In the absence of singularities, AP acquires the topological features of nonlinear sigma-model. Starting with topological charge, one can also introduce the topological quasi-charge groups for field configurations having some symmetry. For 4D, considering symmetrical equipped 0-(sub)manifolds in R^3, we find QC-groups for some symmetries, G\subset O_3, and describe their morphisms. The differential 3-form of topological charge (dual to the topological current) is derived, as well as O_3-quasi-charge 1-form. The problem for D=5 is briefly discussed, and results of topological classification of symmetric field configurations are announced. An example of SO_2-symmetric configuration is considered and the quasi-charge forms are obtained. In conclusion, we propose a variant of experiment with single photon interference (or with bi-photon non-local correlations) which should verify a possible non-local (spaghetti-like) 5D ontology of particles. 
  Taking the horizon surface of the black hole as a compact membrane and solving the oscillation equation of this membrane by Klein-Gordon equation, we derive the frequencies of oscillation modes of the horizon surface, which are proportional to the radiation temperature of the black hole. However, the frequencies of oscillation modes are not equidistant. Using the distribution of obtained frequencies of oscillation mode we compute the statistic entropy of the black hole and obtain that the statistic entropy of the black hole is proportional to the area of the horizon. Therefore, it is proven that the quantum statistic entropy of the black hole is consistent with Bekenstein-Hawking entropy. 
  We give an exact solution to the gravitational field in the new general relativity. The solution creates Bonnor spacetime. This spacetime describes the gravitational field of a stationary beam of light. The energy and momentum of this solution is calculated using the energy-momentum complex given by M{\o}ller in (1978) within the framework of the Weitzenb{\rm $\ddot{o}$}ck spacetime. 
  Adopting a non geometrical point of view, we are led to an alternative theory of the order two and symetric gravitational tensor field of GR. The field is no more interpreted as the metric of our space-time. The true metric is globally Minkowskian and describes a flat manifold, a context which justifies a genuine rehabilitation of the global discrete space-time symetries involved in the structure of the Lorentz group along with their 'problematic' representations: the negative energy and tachyonic ones. 
  An interacting dark energy model with interaction term $Q= \lambda_m H\rho_m+\lambda_dH\rho_d$ is considered. By studying the model near the transition time, in which the system crosses the w=-1 phantom-divide-line, the conditions needed to overcome the coincidence problem is investigated. The phantom model, as a candidate for dark energy, is considered and for two specific examples, the quadratic and exponential phantom potentials, it is shown that it is possible the system crosses the w=-1 line, meanwhile the coincidence problem is alleviated, the two facts that have root in observations. 
  We report on a package of routines for the computer algebra system Maple which supports the explicit determination of the geometric quantities, field equations, equations of motion, and conserved quantities of General Relativity in the post-Newtonian approximation. The package structure is modular and allows for an easy modification by the user. The set of routines can be used to verify hand calculations or to generate the input for further numerical investigations. 
  We analyze motion of massless and massive particles around black holes immersed in an asymptotically uniform magnetic field and surrounded by some mechanical structure, which provides the magnetic field. The space-time is described by Preston-Poisson metric, which is the generalization of the well-known Ernst metric with a new parameter, tidal force, characterizing the surrounding structure. The Hamilton-Jacobi equations allow separation of variables in the equatorial plane. The presence of tidal force from surroundings considerably changes parameters of the test particle motion: it increases the radius of circular orbits of particles, increases the binding energy of massive particles going from a given circular orbits to the innermost stable orbit near black hole. In addition, it increases the distance of minimal approach, time delay and bending angle for a ray of light propagating near black hole. 
  We derive the phenomenological Milgrom square-law acceleration, describing the apparent behavior of dark matter, as the reaction to the Big Bang from a model based on the Lorentz-Dirac equation of motion traditionally describing radiation reaction in electromagnetism but proven applicable to expansion reaction in cosmology. The model is applied within the Robertson-Walker hypersphere, and suggests that the Hubble expansion exactly cancels the classical reaction imparted to matter following the Big Bang, leaving behind a residue proportional to the square of the acceleration. The model further suggests that the energy density associated with the reaction acceleration is precisely the critical density for flattening the universe thus providing a potential explanation of dark energy as well. A test of this model is proposed. 
  We consider a Schr\"odinger quantum dynamics for the gravitational field associated to a generic cosmological model and then we solve the corresponding eigenvalue problem. We show that, from a phenomenological point of view, an Evolutionary Quantum Cosmology overlaps the Wheeler-DeWitt approach. 
  A new tetrad introduced within the framework of geometrodynamics for non-null electromagnetic fields allows for the geometrical analysis of the Lorentz equation and its solutions. This tetrad, through the use of the Frenet-Serret formulae and Fermi-Walker transport, exhibits explicitly the set of solutions to the Lorentz equation in a curved spacetime. 
  The gravitational fields of vacuumless global and gauge strings have been investigated in the context of Einstein Cartan theory under the weak field assumption of the field equations. It has been shown that global string and gauge string can have only repulsive gravitational effect on a test particle. 
  We consider particle-like and black holes solutions of the Einstein-Yang-Mills system with positive cosmological constant in d>4 spacetime dimensions. These configurations are spherically symmetric and present a cosmological horizon for a finite value of the radial coordinate, approaching asymptotically the de Sitter background. In the usual Yang--Mills case we find that the mass of these solutions, evaluated outside the cosmological horizon at future/past infinity generically diverges for d>4. Solutions with finite mass are found by adding to the action higher order gauge field terms belonging to the Yang--Mills hierarchy. A discussion of the main properties of these solutions and their differences from those to the usual Yang-Mills model, both in four and higher dimensions is presented. 
  Pointlike objects cause many of the divergences that afflict physical theories. For instance, the gravitational binding energy of a point particle in Newtonian mechanics is infinite. In general relativity, the analog of a point particle is a black hole and the notion of binding energy must be replaced by quasilocal energy. The quasilocal energy (QLE) derived by York, and elaborated by Brown and York, is finite outside the horizon but it was not considered how to evaluate it inside the horizon. We present a prescription for finding the QLE inside a horizon, and show that it is finite at the singularity for a variety of types of black hole. The energy is typically concentrated just inside the horizon, not at the central singularity. 
  Expression for second post-Newtonian level gravitational deflection angle of massive particles is obtained in a model independent framework. Several of its important implications including the possibility of testing gravitational theories at that level are discussed. 
  Symmetric conformal Killing tensors and (skew-symmetric) conformal Yano-Killing tensors for Euclidean Taub-NUT metric are given in explicit form. Relations between Yano and CYK tensors in terms of conformal rescaling are discussed. 
  We develop solution-generating techniques for stationary metrics with one angular momentum and axial symmetry, in the presence of a cosmological constant and in arbitrary spacetime dimension. In parallel we study the related lower dimensional Einstein-Maxwell-dilaton static spacetimes with a Liouville potential. For vanishing cosmological constant, we show that the field equations in more than four dimensions decouple into a four dimensional Papapetrou system and a Weyl system. We also show that given any four dimensional 'seed' solution, one can construct an infinity of higher dimensional solutions parametrised by the Weyl potentials, associated to the extra dimensions. When the cosmological constant is non-zero, we discuss the symmetries of the field equations, and then extend the well known works of Papapetrou and Ernst (concerning the complex Ernst equation) in four-dimensional general relativity, to arbitrary dimensions. In particular, we demonstrate that the Papapetrou hypothesis generically reduces a stationary system to a static one even in the presence of a cosmological constant. We also give a particular class of solutions which are deformations of the (planar) adS soliton and the (planar) adS black hole. We give example solutions of these techniques and determine the four-dimensional seed solutions of the 5 dimensional black ring and the Myers-Perry black hole. 
  Conservation laws in gravitational theories with diffeomorphism and local Lorentz symmetry are studied. Main attention is paid to the construction of conserved currents and charges associated with an arbitrary vector field that generates a diffeomorphism on the spacetime. We further generalize previous results for the case of gravitational models described by quasi-invariant Lagrangians, that is, Lagrangians that change by a total derivative under the action of the local Lorentz group. The general formalism is then applied to the teleparallel models, for which the energy and the angular momentum of a Kerr black hole are calculated. The subsequent analysis of the results obtained demonstrates the importance of the choice of the frame. 
  A method for mapping known cylindrical magnetovac solutions to solutions in torus coordinates is developed. Identification of the cylinder ends changes topology from R1 x S1 to S1 x S1. An analytic Einstein-Maxwell solution for a toroidal magnetic field in tori is presented. The toroidal interior is matched to an asymptotically flat vacuum exterior, connected by an Israel boundary layer. 
  We use $(4+1)$ split to derive the 4D induced energy density $\rho $ and pressure $p$ of the $5D$ Ricci-flat cosmological solutions which are characterized by having a bounce instead of a bang. The solutions contain two arbitrary functions of time $t$ and, therefore, are mathematically rich in giving various cosmological models. By using four known energy conditions (null, weak, strong, and dominant) to pick out and study physically meaningful solutions, we find that the 4D part of the $5D$ solutions asymptotically approach to the standard 4D FRW models and the expansion of the universe is decelerating for normal induced matter for which all the four energy conditions are satisfied. We also find that quintessence might be normal or abnormal, depending on the parameter $w$ of the equation of state. If $-1\leqslant w<-1/3$, the expansion of the universe is accelerating and the quintessence is abnormal because the strong energy condition is violated while other three are satisfied. For phantom, all the four energy conditions are violated. Before the bounce all the four energy conditions are violated, implying that the cosmic matter before the bounce could be explained as a phantom which has a large negative pressure and makes the universe bouncing. In the early times after the bounce, the dominant energy condition is violated while the other three are satisfied, and so the cosmic matter could be explained as a super-luminal acoustic matter. 
  The need to know the force exerted by moving body on ground of intriguing interplay between geometry and dynamics gives a possible introducing of gravitomagnetic (GM) field as an analogous to the magnetic field. The existence of such a field has straightforwardly been presented in two approaches based on special relativity (SR) only and SR plus gravitational time dilation (semi SR) for different cases. We treat these two approaches for when the cases are switched, using appropriate key points. Hence, we demonstrate that the strength of GM field in semi SR approach is twice SR approach. Then, we also discuss that the full linearized general relativity should give the same strength for GM field as semi SR, and hence, through an exact analogy with the electrodynamic equations, we present an argument for the best potential definition amongst those used in this issue. 
  We use results by Kirilin to show that in general relativity the nonleading terms in the energy-momentum tensor of a particle depends on the parameterization of the gravitational field. While the classical metric that is calculated from this source, used to define the leading long-distance corrections to the metric, also has a parameteriztion dependence, it can be removed by a coordinate change. Thus the classical observables are parameterization independent. The quantum effects that emerge within the same calculation of the metric also depend on the parameterization and a full quantum calculation requires the inclusion of further diagrams. However, within a given parameterization the quantum effects calculated by us in a previous paper are well defined. Flaws of Kirilin's proposed alternate metric definition are described and we explain why the diagrams that we calculated are the appropriate ones. 
  Spherically symmetric cosmological equations in the usual FLRW coordinates are explored, with different sources. The first couples a perfect fluid with a quintessence scalar field and the second couples a perfect fluid to a tachyonic scalar field. In both cases, in the inflationary regime, the scale factor a(t) and its first two time derivatives are positive definite. Both sources in the matter phase yield a scale factor and its first derivative as positive definite. In both cases and in each phase, the general solutions of the differential equations together with the algebraic and differential inequalities are obtained. As special cases, exponential, hyperbolic, and power law inflation, as well as power law expansion for the matter phase are all derived from the general solutions. With recent data on baryonic matter, Hubble parameter and deceleration parameter, the relative percentages of baryonic matter, daek matter and dark energy are calculated. The quintessence model yields: 4% baryonic matter, 18% dark matter, and 78% dark energy. The tachyonic model gives: 4% baryonic matter, 36% dark matter, and 60% dark energy. 
  This is a comment on the paper gr-qc/0605153, by Singh, Mobed and Papini. 
  In this paper the internal structure of a neutron star is shown to be inferrable from its gravitational-wave spectrum. Iteratively applying the inverse scheme of the scaled coordinate logarithmic perturbation method for neutron stars proposed by Tsui and Leung [Astrophys. J. {\bf 631}, 495 (2005)], we are able to determine the mass, the radius and the mass distribution of a star from its quasi-normal mode frequencies of stellar pulsation. In addition, accurate equation of state of nuclear matter can be obtained from such inversion scheme. Explicit formulas for the case of axial $w$-mode oscillation are derived here and numerical results for neutron stars characterized by different equations of state are shown. 
  A new 5D thick brane solution is presented. We conjecture that the deduced thick brane is a plane defect in a bulk gauge condensate. 
  We have shown how the quantization of two-dimensional quantum gravity with an action which contains only a positive cosmological constant and boundary cosmological constants leads to the emergence of a spacetime which can be described as a constant negative curvature spacetime with superimposed quantum fluctuations. 
  We investigate the gravitational radiation produced by a linearly accelerated source in general relativity. The investigation is performed by studying the vacuum C metric, which is interpreted as representing the exterior space-time of an uniformly accelerating spherically symmetric gravitational source, and is carried out in the context of the teleparallel equivalent of general relativity. For an observer sufficiently far from both the (modified) Schwarzschild and Rindler horizons, which is a realistic situation, we obtain a simple expression for the total emitted gravitational radiation. We also briefly discuss on the absolute or relative character of the accelerated motion. 
  The capacity to model magnetohydrodynamical (MHD) flows in dynamical, strongly curved spacetimes significantly extends the reach of numerical relativity in addressing many problems at the forefront of theoretical astrophysics. We have developed and tested an evolution code for the coupled Einstein-Maxwell-MHD equations which combines a BSSN solver with a high resolution shock capturing scheme. As one application, we evolve magnetized, differentially rotating neutron stars under the influence of a small seed magnetic field. Of particular significance is the behavior found for hypermassive neutron stars (HMNSs), which have rest masses greater the mass limit allowed by uniform rotation for a given equation of state. The remnant of a binary neutron star merger is likely to be a HMNS. We find that magnetic braking and the magnetorotational instability lead to the collapse of HMNSs and the formation of rotating black holes surrounded by massive, hot accretion tori and collimated magnetic field lines. Such tori radiate strongly in neutrinos, and the resulting neutrino-antineutrino annihilation (possibly in concert with energy extraction by MHD effects) could provide enough energy to power short-hard gamma-ray bursts. To explore the range of outcomes, we also evolve differentially rotating neutron stars with lower masses and angular momenta than the HMNS models. Instead of collapsing, the non-hypermassive models form nearly uniformly rotating central objects which, in cases with significant angular momentum, are surrounded by massive tori. 
  The models that unify dark matter and dark energy based upon the Chaplygin gas fail owing to the suppression of structure formation by the adiabatic speed of sound. Including string theory effects, in particular the Kalb-Ramond field, we show how nonadiabatic perturbations allow a successful structure formation. 
  The 'hole argument'(the English translation of German 'Lochbetrachtung') was formulated by Albert Einstein in 1913 in his search for a relativistic theory of gravitation. The hole argument was deemed to be based on a trivial error of Einstein, until 1980 when John Stachel (Talk on Einsteins Search for General Covariance, 1912-1915 at the GRG meeting in Jena 1980) recognized its highly non-trivial character. Since then the argument has been intensively discussed by many physicists and philosophers of science. (See e.g., Earman & Norton (1987), Gaul & Rovelli (1999), Stachel & Iftime(2005}, and Iftime & Stachel(2006).)   I shall provide here a coordinate-free formulation of the argument using the language of categories and bundles, and generalize the argument for arbitrary covariant and permutable theories (see Iftime & Stachel(2006). In conclusion I shall point out a way of avoiding the hole argument, by looking at the structure of the space of solutions of Einstein's equations on a space-time manifold. This superspace Q(M) is defined as the orbit space of space-time solutions on M under the action of the diffeomorphisms of M, and it plays an important role in the study of the gravitational field and attempts to find a theory of quantum gravity (QG). 
  Studying spacetimes with continuous symmetries by dimensional reduction to a lower dimensional spacetime is a well known technique in field theory and gravity. Recently, its use has been advocated in numerical relativity as an efficient computational technique for the numerical study of axisymmetric asymptotically flat 4-dimensional spacetimes. We prove here that if the dimensionally reduced spacetime is a physically reasonable 3-dimensional asymptotically flat or asymptotically anti-de Sitter spacetime, then, surprisingly, the topology of the higher dimensional spacetime must be one of two product topologies. Reductions of other topologies result in physically pathological spacetimes. In particular, reduction of asymptotically flat 4-dimensional spacetimes must lead to pathologies. These results use only the topological censorship theorem and topological methods and consequently are independent of the field equations and reduction method. 
  We consider a DGP brane scenario where a scalar field is present on the brane through the introduction of a scalar potential, itself motivated by the notion of modified gravity. This theory predicts that the mass appearing in the gravitational potential is modified by the addition of the mass of the scalar field. The cosmological implications that such a scenario entails are examined and shown to be consistent with a universe expanding with power-law acceleration. 
  Originally, the Unruh effect is considered in the Minkowski spacetime. That is, for a uniformly accelerated observer in the Minkowski spacetime, there will be an event horizon, and the observer will detect radiation from it. In this paper, we extend the Unruh effect to the Kruskal spacetime. After defining some special null hypersurface in the Kruskal spacetime, we find that it can be considered as the event horizon for some accelerated observers, too. Moreover, there is radiation from it. In addition, our result shows that these observers by definition are 3-accelerated observers in the Schwarschild space. Thus, the conculsion may be available in the astronomical exploration. 
  The exact solution for dynamic of conform-flat space homogeneous since dynamic equation is given. Conform mode of space metric changing in Global time theory has negative energy density. Swap of energy to this mode from another ones lead to increasing of Universe homogeneity although probability of this swap from local objects is negligibly small. Conform mode is corresponding to "dark energy" in observation astronomy. 
  Scenarios of large extra dimensions have enhanced the importance for the study of black holes in higher dimensions. In this paper, we analyze an axisymmetric system of two black holes. Specifically, the Bowen-York method is generalized for higher dimensions in order to calculate the initial data for head-on collision of two equal-mass black holes. Then, the initial data are evolved adopting the close-slow approximation to study gravitational waves emitted during the collision. We derive an empirical formula for radiation efficiency, which depends weakly on the dimensionality. Possible implications of our results for the black hole formation in particle colliders are discussed. 
  The model of cosmic string formed from two gravitating and interacting scalar fields is considered. It is shown that the regular solutions exist at special choice of the model's parameters only. 
  The known static isotropic metric of Schwarzschild solution of Einstein equation cannot cover with the range of r<2MG, a new isotropic metric of Schwarzschild solution is obtained. The new isotropic metric has the characters: (1) It is dynamic and periodic. (2) It has infinite singularities of the spacetime. (3) It cannot cover with the range of 0<r<r0; On the other hand, r0 can be small discretionarily. (4) It seemed as if the range of negative r could be unavoidable, although this range is meaningless for the Schwarzschild metric. 
  In the current debate referring to the construction of a tenable background independent theory of Quantum Gravity we introduce the notion of topos-theoretic relativization of physical representability and demonstrate its relevance concerning the merging of General Relativity and Quantum Theory. For this purpose we show explicitly that the dynamical mechanism of physical fields can be constructed by purely algebraic means, in terms of connection inducing functors and their associated curvatures, independently of any background substratum. The application of this mechanism in General Relativity is constrained by the absolute representability of the theory in the field of real numbers. The relativization of physical representability inside operationally selected topoi of sheaves forces an appropriate interpretation of the mechanism of connection functors in terms of a generalized differential geometric dynamics of the corresponding fields in the regime of these topoi. In particular, the relativization inside the topos of sheaves over commutative algebraic contexts makes possible the formulation of quantum gravitational dynamics by suitably adapting the functorial mechanism of connections inside that topos. 
  By ignoring the local density fluctuations, we construct an uniform Higgs-field's (inflaton's) quantum theory with varying effective Planck constant ($\hbar_{v}(t) \propto R(t)^{-3}$) for the evolution of the dark energy density during the epoch after inflation. With presumable sufficient inflation in the very early period (time-scale is $t_{inf}$), so that $\hbar_{v}\to 0$, the state of universe decomposes into some decoherent components, which could be the essential meaning of phase transition, and each of them could be well described by classical mechanics for an inharmonic oscillator in the corresponding potential-well with a viscous force. We find that the cosmological constant at present is $\Lambda_{now}\approx 2.05\times 10^{-3}$ eV, which is almost independent of the choice of potential for inflaton, and agrees excellently with the recent observations. In addition, we find that, during the cosmic epoch after inflation, the dark energy is almost conserved as well as the matter's energy, therefore the "why now" problem can be avoided. 
  The generalized Duffin-Kemmer-Petiau equation in curved space-time is proposed for non-minimal coupling to the curvature and external fields. The corresponding scalar and vector fields equation are found. Equations are presented, which are equivalent to those of a scalar field with conformal coupling and electromagnetic field with non-minimal coupling to the curvature. The gauge-invariant Duffin-Kemmer-Petiau equation with non-minimal coupling is given. 
  We consider the problem of (1+1)-dimensional quantum gravity coupled to particles. Working with the canonically reduced Hamiltonian, we obtain its post-Newtonian expansion for two identical particles. We quantize the (1+1)-dimensional Newtonian system first, after which explicit energy corrections to second order in 1/c are obtained. We compute the perturbed wavefunctions and show that the particles are bound less tightly together than in the Newtonian case. 
  Interferometers with kilometer-scale arms have been built for gravitational-wave detections on the ground; ones with much longer arms are being planned for space-based detection. One fundamental motivation for long baseline interferometry is from displacement noise. In general, the longer the arm length L, the larger the motion the gravitational-wave induces on the test masses, until L becomes comparable to the gravitational wavelength. Recently, schemes have been invented, in which displacement noises can be evaded by employing differences between the influence of test-mass motions and that of gravitational waves on light propagation. However, in these schemes, such differences only becomes significant when L approaches the gravitational wavelength, and shot-noise limited sensitivity becomes worse than that of conventional configurations by a factor of at least (f L/c)^(-2), for f<c/L. Such a factor, although can be overcome theoretically by employing high optical powers, makes these schemes quite impractical. In this paper, we explore the use of time delay in displacement-noise-free interferometers, which can improve their shot-noise-limited sensitivity at low frequencies, to a factor of (f L/c)^(-1) of the shot-noise-limited sensitivity of conventional configurations. 
  The van der Waals quintessence equation of state is an interesting scenario for describing the late universe, and seems to provide a solution to the puzzle of dark energy, without the presence of exotic fluids or modifications of the Friedmann equations. In this work, the construction of inhomogeneous compact spheres supported by a van der Waals equation of state is explored. These relativistic stellar configurations shall be denoted as {\it van der Waals quintessence stars}. Despite of the fact that, in a cosmological context, the van der Waals fluid is considered homogeneous, inhomogeneities may arise through gravitational instabilities. Thus, these solutions may possibly originate from density fluctuations in the cosmological background. Two specific classes of solutions, namely, gravastars and traversable wormholes are analyzed. Exact solutions are found, and their respective characteristics and physical properties are further explored. 
  We investigate the geodesic motion in $D-$dimensional Majumdar-Papapetrou multi-black hole spacetimes and find that the qualitative features of the D=4 case are shared by the higher dimensional configurations. The motion of timelike and null particles is chaotic, the phase space being divided into basins of attraction which are separated by a fractal boundary, with a fractal dimension $d_B$. The mapping of the geodesic trajectories on a screen placed in the asymptotic region is also investigated. We find that the fractal properties of the phase space induces a fractal structure on the holographic screen, with a fractal dimension $d_B-1$. 
  We study the appearance of multiple solutions to certain decompositions of Einstein's constraint equations. Pfeiffer and York recently reported the existence of two branches of solutions for identical background data in the extended conformal thin-sandwich decomposition. We show that the Hamiltonian constraint alone, when expressed in a certain way, admits two branches of solutions with properties very similar to those found by Pfeiffer and York. We construct these two branches analytically for a constant-density star in spherical symmetry, but argue that this behavior is more general. In the case of the Hamiltonian constraint this non-uniqueness is well known to be related to the sign of one particular term, and we argue that the extended conformal thin-sandwich equations contain a similar term that causes the breakdown of uniqueness. 
  We discuss the effects of a (possibly) negative $(1+z)^6$ type contribution to the Friedmann equation. No definite answer can be given as to the presence and magnitude of a particular mechanism, because any test using the general relation $H(z)$ is able to estimate only the total of all sources of such a term. That is why we describe four possibilities: 1) geometric effects of loop quantum cosmology, 2) braneworld cosmology, 3) metric-affine gravity, and 4) cosmology with spinning fluid. We find the exact solutions for the models with $\rho^2$ correction in terms of elementary functions, and show all evolutional paths on their phase plane. Instead of the initial singularity, the generic feature is now a bounce. 
  We investigate the dynamics and gravitational-wave (GW) emission in the binary merger of equal-mass black holes as obtained from numerical relativity simulations. Results from the evolution of three sets of initial data are explored in detail, corresponding to different initial separations of the black holes. We find that to a good approximation the inspiral phase of the evolution is quasi-circular, followed by a "blurred, quasi-circular plunge", then merger and ring down. We present first-order comparisons between analytical models of the various stages of the merger and the numerical results. We provide comparisons between the numerical results and analytical predictions based on the adiabatic Newtonain, post-Newtonian (PN), and non-adiabatic resummed-PN models. From the ring-down portion of the GW we extract the fundamental quasi-normal mode and several of the overtones. Finally, we estimate the optimal signal-to-noise ratio for typical binaries detectable by GW experiments. 
  In this article we construct cylindrical thin-shell wormholes in the context of global cosmic strings. We study the stability of static configurations under perturbations preserving the symmetry and we find that the throat tends to collapse or expand, depending only on the direction of the velocity perturbation. 
  In this paper we can solve a Wheeler-DeWitt equation of the some inhomogeneous spacetime models as a local solution. From the previous study of up-to-down method we derived the static restriction relating the problem of the time. Although static restriction does not commute with the general Hamiltonian constraint, the Hamiltonian constraint of some mini-superspace models commute with static restriction. We can quantize such inhomogeneous models. With obtained result we can success to simplify the general local Hamiltonian constraint. 
  In this paper we investigate a class of solutions of Einstein equations for the plane-symmetric perfect fluid case with shear and vanishing acceleration. If these solutions have shear, they must necessarily be non-static. We examine the integrable cases of the field equations systematically. Among the cases with shear we find three classes of solutions. 
  We study parameter estimation with post-Newtonian (PN) gravitational waveforms for the quasi-circular, adiabatic inspiral of spinning binary compact objects. The performance of amplitude-corrected waveforms is compared with that of the more commonly used restricted waveforms, in Advanced LIGO and EGO. With restricted waveforms, the properties of the source can only be extracted from the phasing. For amplitude-corrected waveforms, the spectrum encodes a wealth of additional information, which leads to dramatic improvements in parameter estimation. At distances of $\sim 100$ Mpc, the full PN waveforms allow for high-accuracy parameter extraction for total mass up to several hundred solar masses, while with the restricted ones the errors are steep functions of mass, and accurate parameter estimation is only possible for relatively light stellar mass binaries. At the low-mass end, the inclusion of amplitude corrections reduces the error on the time of coalescence by an order of magnitude in Advanced LIGO and a factor of 5 in EGO compared to the restricted waveforms; at higher masses these differences are much larger. The individual component masses, which are very poorly determined with restricted waveforms, become measurable with high accuracy if amplitude-corrected waveforms are used, with errors as low as a few percent in Advanced LIGO and a few tenths of a percent in EGO. The usual spin-orbit parameter $\beta$ is also poorly determined with restricted waveforms (except for low-mass systems in EGO), but the full waveforms give errors that are small compared to the largest possible value consistent with the Kerr bound. This suggests a way of finding out if one or both of the component objects violate this bound. We also briefly discuss the effect of amplitude corrections on parameter estimation in Initial LIGO. 
  Inspired by Raychaudhuri's work, and using the equation named after him as a basic ingredient, I prove a new singularity theorem: open non-rotating everywhere expanding universes with non-vanishing spatial average of the matter variables are totally geodesically incomplete to the past. Another way of stating the same is that, under the same conditions, any singularity-free model must have a vanishing spatial average of the energy density (and other physical variables). This is very satisfactory and provides a clear decisive difference between singular and non-singular cosmologies. 
  We present single and binary black hole simulations that follow the moving puncture paradigm of simulating black-hole spacetimes without excision, and use moving boxes mesh refinement. Focussing on binary black hole configurations where the simulations cover roughly two orbits, we address five major issues determining the quality of our results: numerical discretization error, finite extraction radius of the radiation signal, physical appropriateness of initial data, gauge choice and computational performance. We also compare results we have obtained with the BAM code described here with the independent LEAN code. 
  Standard methods in non-linear analysis are used to show that there exists a parabolic branching of solutions of the Lichnerowicz-York equation with an unscaled source. We also apply these methods to the extended conformal thin sandwich formulation and show that if the linearised system develops a kernel solution for sufficiently large initial data then we obtain parabolic solution curves for the conformal factor, lapse and shift identical to those found numerically by Pfeiffer and York. The implications of these results for constrained evolutions are discussed. 
  The symmetry algebra of asymptotically flat spacetimes at null infinity in three dimensions is the semi-direct sum of the infinitesimal diffeomorphisms on the circle with an abelian ideal of supertranslations. The associated charge algebra is shown to admit a non trivial classical central extension of Virasoro type closely related to that of the anti-de Sitter case. 
  By combining the numerical solution of the nonlinear hydrodynamics equations with the solution of the linear inhomogeneous Zerilli-Moncrief and Regge-Wheeler equations we investigate the properties of the gravitational radiation emitted during the axisymmetric accretion of matter onto a Schwarzschild black hole. The matter models considered include quadrupolar dust shells and thick accretion disks, permitting us to simulate situations which may be encountered at the end stages of stellar gravitational collapse or binary neutron star merger. We focus on the interference pattern appearing in the energy spectra of the emitted gravitational waves and on the amount of excitation of the quasi-normal modes of the accreting black hole. We show that, quite generically in the presence of accretion, the black hole ringdown is not a simple superposition of quasi-normal modes, although the fundamental mode is usually present and often dominates the gravitational-wave signal. We interpret this as due to backscattering of waves off the non-exponentially decaying part of the black-hole potential and to the finite spatial extension of the accreting matter. Our results suggest that the black-hole QNM contributions to the full gravitational-wave signal should be extremely small and possibly not detectable in generic astrophysical scenarios involving the accretion of extended distributions of matter. 
  We review the vacuum pure-affine gravity with the nonsymmetric connection and metric. We also examine dynamical effects of the second Ricci tensor and covariant second-rank tensors built of the torsion tensor in the gravitational Lagrangian. 
  We derive the expression for the snap parameter in $f(R)$ gravity. We use the Palatini variational principle to obtain the field equations, and regard the Einstein conformal frame as physical. We give the prediction for the present value of the snap parameter for the particular case $f(R)=R-\frac{\alpha^2}{3R}$, which is the simplest $f(R)$ model explaining current acceleration of the universe. 
  We investigate, in Randall-Sundrum braneworld scenario, the relationship between perturbations of gravitational and electromagnetic waves in a black hole neighboorhood, proposing an extra-dimensional braneworld extension of the traditional formalism. Mutual transformations of electromagnetic and gravitational fields due to the strong fields in Reissner-Nordstrom black holes are analyzed from an effective 5-dimensional Randall-Sundrum braneworld perspective. 
  Einstein's general relativity relates the curvature of space time, a second order differential property, to the stress-energy-momentum tensor. In this paper we ask whether it is possible to develop a first order theory relating space-time angles to the energy-momentum vector. We suggest several concepts that would be relevant, including quantum mechanical concepts that are usually treated separately. Phenomenological Lorentz covariance arises from both field and coordinate transformation and the Dirac equation becomes a special case of the space-time field equation. We reinterpret Kaluza-Klein theory in this context by considering the compact fifth dimension as the quantum wavefunction phase. Further directions for development are suggested. 
  This paper has been withdrawn because the part concerning the definition of global hyperbolicity has already been included in an expanded and clearer way in gr-qc/0611138. The remainder will be also extended and posted. 
  The spherically symmetric perturbations in the spatially flat Friedman models are considered. It is assumed that the Friedmannian density and pressure are related through a linear equation of state. The perturbation is joined smoothly with an unperturbed Friedmann's background at the sound horizon of perturbation. Such junction is in accordance with the "birth" of a local perturbation as a result of the redistribution of matter. The solution of the Einstein's equations is obtained in linear approximation on a Friedmann's background near the the sound horizon of perturbation. 
  This paper investigates the particle radiation from Gibbons-Maeda black hole. Taking into account the self-gravitation of the particle, we calculate the tunnelling rate of the massless particle across the horizon, then we promote the work to the radiation of the charged particle. The calculations prove that the rate of tunnelling equals precisely the exponent of the difference of the black hole entropy before and after emission and the radiation spectrum deviates from exact thermal. The conclusion supports the viewpoint of information conservation. 
  Stellar-mass compact binaries in eccentric orbits are almost guaranteed sources of gravitational waves for Laser Interferometer Space Antenna. We present a prescription to compute accurate and efficient gravitational-wave polarizations associated with bound compact binaries of arbitrary eccentricity and mass ratio moving in slowly precessing orbits. We compare our approach with those existing in the literature and present its advantages. 
  We discuss the definition of quantum probability in the context of "timeless" general--relativistic quantum mechanics. In particular, we study the probability of sequences of events, or multi-event probability. In conventional quantum mechanics this can be obtained by means of the ``wave function collapse" algorithm. We first point out certain difficulties of some natural definitions of multi-event probability, including the conditional probability widely considered in the literature. We then observe that multi-event probability can be reduced to single-event probability, by taking into account the quantum nature of the measuring apparatus. In fact, by exploiting the von-Neumann freedom of moving the quantum classical boundary, one can always trade a sequence of non-commuting quantum measurements at different times, with an ensemble of simultaneous commuting measurements on the joint system+apparatus system. This observation permits a formulation of quantum theory based only on single-event probability, where the results of the "wave function collapse" algorithm can nevertheless be recovered. The discussion bears also on the nature of the quantum collapse. 
  The aim of this proceeding is to give a basic introduction to Deformation Quantization (DQ) to physicists. We compare DQ to canonical quantization and path integral methods. It is described how certain issues such as the roles of associativity, coordinate independence, dynamics, and operator orderings are understood in the context of DQ. Convergence issues in DQ are mentioned. Additionally, we formulate the Klein-Gordon (KG) equation in DQ. Original results are discussed which include the exact construction of the Fedosov star-product on the dS and AdS space-times. Also, the KG equation is written down for these space-times. This is a proceedings to the Second International Conference on Quantum Theories and Renormalization Group in Gravity and Cosmology. 
  Non-relativistic quantum mechanics for a free particle is shown to emerge from classical mechanics through the requirement of an invariance principle under transformations that preserve the Heisenberg position-momentum inequality. These transformations acting on the position and momentum uncertainties are induced by isotropic space dilatations. This invariance imposes a change in the laws of classical mechanics that exactly corresponds to the transition to quantum mechanics. Space-time geometry is affected with possible consequences for quantum gravity. 
  This paper continues the analysis of the quantum states determined by the universal asymptotic structure of four-dimensional asymptotically flat vacuum spacetimes at null infinity M. It is now focused on the quantum state lambda_M, of a massles conformally coupled scalar field phi propagating in M. lambda_M is ``holographically'' induced in the bulk by the universal BMS-invariant state lambda at infinity scri of M. It is done by means of the correspondence between observables in the bulk and those on the boundary at null infinity discussed in previous papers. The induction is possible when some requirements are fulfilled, in particular the spacetime M and the associated unphysical one are globally hyperbolic and M admits future time infinity i^+. lambda_M coincides with Minkowski vacuum if M is Minkowski spacetime. It is now proved that, in the general case of a curved spacetime M, the state lambda_M enjoys the following further properties. (1) lambda_M is invariant under the group of isometries of the bulk spacetime M. (2) lambda_M fulfills a natural energy-positivity condition with respect to every notion of Killing time (if any) in the bulk spacetime M: If M admits a complete time-like Killing vector, the associated one-parameter group of isometries is represented by a strongly-continuous unitary group in the GNS representation of lambda_M. The unitary group has positive self-adjoint generator without zero modes in the one-particle space. {3} lambda_M is (globally) Hadamard in M and thus lambda_M can be used as starting point for perturbative renormalization procedure of QFT of phi in M. 
  It is well known that 4-dimensional Kerr-NUT-AdS spacetime possesses the hidden symmetry associated with the Killing-Yano tensor. This tensor is "universal" in the sense that there exist coordinates where it does not depend on any of the free parameters of the metric. Recently the general higher dimensional Kerr-NUT-AdS solutions of the Einstein equations were obtained. We demonstrate that all these metrics with arbitrary rotation and NUT parameters admit a universal Killing-Yano tensor. We give an explicit presentation of the Killing-Yano and Killing tensors and briefly discuss their properties. 
  We address the problem of finding a system in which there would be measurable quantum gravitational effects. Following standard quantum-field methods, we have calculated the first-order radiative correction of graviton exchange on the binding energy of an electron with an ultra cold superfluid droplet of $^4$He with mass about the Planck mass. For two $^4$He droplets with a mass difference of about one microgram, we show that the relative difference in the binding energies is about one percent. 
  Pairs of Planck-mass--scale drops of superfluid helium coated by electrons (i.e., "Millikan oil drops"), when levitated in a superconducting magnetic trap in the presence of strong magnetic fields and at low temperatures, can be efficient quantum transducers between electromagnetic (EM) and gravitational (GR) radiation. A Hertz-like experiment, in which EM waves are converted at the source into GR waves, and then back-converted at the receiver from GR waves back into EM waves, should be practical to perform. As a step in this direction, an experiment to measure the quadrupolar electromagnetic scattering cross-section of a pair of "Millikan oil drops" will be performed first. 
  In this paper, we discuss the system of massive nonlinear dark spinors coupling with the Friedman-Robertson-Walker(FRW) metric in detail, where the thermodynamic movement of spinors is also taken into account. The results reveal that, the nonlinear potential of the spinor field can provide a little negative pressure, which resists the universe to become singular. The model is oscillating in time and closed in space. The corresponding metric approximately takes the following form $g_{\mu\nu}=\bar R^2 (1-\delta\cos t)^2 \diag(1,-1,-\sin^2r,-\sin^2r \sin^2\theta), (\delta<1)$. 
  It is shown that using of the equation of motion of the Universe scale factor allows calculation of the contribution of the vacuum fluctuations to the acceleration of the Universe expansion. Renormalization of the equation is needed only in the case of massive particles. Under a known number of the different kinds of fundamental fields, this provides determination of momentum of the ultraviolet cut-off from the observed value of acceleration. 
  In this thesis four separate problems in general relativity are considered, divided into two separate themes: coordinate conditions and perfect fluid spheres. Regarding coordinate conditions we present a pedagogical discussion of how the appropriate use of coordinate conditions can lead to simplifications in the form of the spacetime curvature -- such tricks are often helpful when seeking specific exact solutions of the Einstein equations. Regarding perfect fluid spheres we present several methods of transforming any given perfect fluid sphere into a possibly new perfect fluid sphere. This is done in three qualitatively distinct manners: The first set of solution generating theorems apply in Schwarzschild curvature coordinates, and are phrased in terms of the metric components: they show how to transform one static spherical perfect fluid spacetime geometry into another. A second set of solution generating theorems extends these ideas to other coordinate systems (such as isotropic, Gaussian polar, Buchdahl, Synge, and exponential coordinates), again working directly in terms of the metric components. Finally, the solution generating theorems are rephrased in terms of the TOV equation and density and pressure profiles. Most of the relevant calculations are carried out analytically, though some numerical explorations are also carried out. 
  We generalize the Raychaudhuri equation for the evolution of a self gravitating fluid to include an Abelian and non-Abelian hybrid magneto fluid at a finite temperature. The aim is to utilize this equation for investigating the dynamics of astrophysical high temperature Abelian and non-Abelian plasmas. 
  We have investigated the thermodynamical properties of the Universe with dark energy. Adopting the usual assumption in deriving the constant co-moving entropy density that the physical volume and the temperature are independent, we observed some strange thermodynamical behaviors. However, these strange behaviors disappeared if we consider the realistic situation that the physical volume and the temperature of the Universe are related. Based on the well known correspondence between the Friedmann equation and the first law of thermodynamics of the apparent horizon, we argued that the apparent horizon is the physical horizon in dealing with thermodynamics problems. We have concentrated on the volume of the Universe within the apparent horizon and considered that the Universe is in thermal equilibrium with the Hawking temperature on the apparent horizon. For dark energy with $w\ge -1$, the holographic principle and the generalized second law are always respected. 
  We discuss theoretical formalisms concerning with experimental verification of General Relativity (GR). Non-metric generalizations of GR are considered and a system of postulates is formulated for metric-affine and Finsler gravitational theories. We consider local observer reference frames to be a proper tool for comparing predictions of alternative theories with each other and with the observational data. Integral formula for geodesic deviation due to the deformation of connection is obtained. This formula can be applied for calculations of such effects as the bending of light and time-delay in presence of non-metrical effects. 
  We develop a non--perturbative method that yields analytical expressions for the deflection angle of light in a general static and spherically symmetric metric. It is an improvement on a method previously devised by the authors, and provides a correct description of the photon sphere already to first order. We also propose an alternative approach that provides general simpler formulas, although with larger errors. We apply our technique to different metrics and verify that the error is at most $0.5 %$ for {\sl all} regimes. We show that our approximation is more accurate than others proposed earlier. 
  When unequal-mass black holes merge, the final black hole receives a ``kick'' due to the asymmetric loss of linear momentum in the gravitational radiation emitted during the merger. The magnitude of this kick has important astrophysical consequences. Recent breakthroughs in numerical relativity allow us to perform the largest parameter study undertaken to date in numerical simulations of binary black hole inspirals. We study non-spinning black-hole binaries with mass ratios from $q=M_1/M_2=1$ to $q =0.25$ ($\eta = q/(1 + q)^2$ from 0.25 to 0.16). We accurately calculate the velocity of the kick to within 6\%, and the final spin of the black holes to within 2\%. A maximum kick of $175.70\pm11$ km\,s$^{-1}$ is achieved for a mass ratio of $\eta = 0.195 \pm 0.005$. 
  The study of post-Einsteinian metric extensions of general relativity (GR), which preserve the metric interpretation of gravity while considering metrics which may differ from that predicted by GR, is pushed one step further. We give a complete description of radar ranging and Doppler tracking in terms of the time delay affecting an electromagnetic signal travelling between the Earth and a remote probe. Results of previous publications concerning the Pioneer anomaly are corrected and an annually modulated anomaly is predicted besides the secular anomaly. Their correlation is shown to play an important role when extracting reliable information from Pioneer observations. The formalism developed here provides a basis for a quantitative analysis of the Pioneer data, in order to assess whether extended metric theories can be the appropriate description of gravity in the solar system. 
  Recently, with an enlighting treatment, Baskaran and Grishchuk have shown the presence and importance of the so-called ``magnetic'' components of gravitational waves (GWs), which have to be taken into account in the context of the total response functions of interferometers for GWs propagating from arbitrary directions. In this paper the analysis of the response functions for the magnetic components is generalized in its full frequency dependence, while in the work of Baskaran and Grishchuk the response functions were computed only in the approximation of wavelength much larger than the linear dimensions of the interferometer. It is also shown that the response functions to the magnetic components grow at high frequencies, differently from the values of the response functions to the well known ordinary components that decrease at high frequencies. Thus the magnetic components could in principle become the dominant part of the signal at high frequencies. This is important for a potential detection of the signal at high frequencies and confirms that the magnetic contributions must be taken into account in the data analysis. More, the fact that the response functions of the magnetic components grow at high frequencies shows that, in principle, the frequency-range of Earth-based interferometers could extend to frequencies over 10000 Hz. 
  In this paper the "Shibata Nakao and Nakamura" (SNN) gauge for scalar gravitational waves (SGWs)is reanalyzed, showing that in [1] there was an error in the geodesic equations of motion. This error conditioned also the analysis of [2] and [3] where wrong equation of motion taken from [1] were used. In the analysis of the response of interferometers the computation is first made in the low frequencies approximation, then the analysis is applied to all SGWs using a generalization to the SNN gauge of the analysis of [4] where the computation was made in the TT gauge for tensorial waves. At the end of this paper the correct detector pattern of interferometers in the SNN gauge is also computed with a further generalization of the analysis of [4] to the angular dependence of the propagating SGW. 
  In this paper we study the orbits of massive bodies moving in the spacetime generated by a spherically symmetric and non-rotating distribution of mass. More specifically, our treatment discusses the more accurate calculation of the precession of pericenter due to general-relativistic effects. Our result is accurate up to terms of second order, while the precession met in the bibliography is accurate only up to first-order terms. 
  The aim of this paper is to give a basic overview of Deformation Quantization (DQ) to physicists. A summary is given here of some of the key developments over the past thirty years in the context of physics, from quantum mechanics to quantum field theory. Also, we discuss some of the conceptual advantages of DQ and how DQ may be related to algebraic quantum field theory. Additionally, our previous results are summarized which includes the construction of the Fedosov star-product on dS/AdS. One of the goals of these results was to verify that DQ gave the same results as previous analyses of these spaces. Another was to verify that the formal series used in the conventional treatment converged by obtaining exact and nonperturbative results for these spaces. 
  Experimental tests of gravity performed in the solar system show a good agreement with general relativity. The latter is however challenged by the Pioneer anomaly which might be pointing at some modification of gravity law at ranges of the order of the size of the solar system. As this question could be related to the puzzles of ``dark matter'' or ``dark energy'', it is important to test it with care. There exist metric extensions of general relativity which preserve the well verified equivalence principle while possibly changing the metric solution in the solar system. Such extensions have the capability to preserve compatibility with existing gravity tests while opening free space for the Pioneer anomaly. They constitute arguments for new mission designs and new space technologies as well as for having a new look at data of already performed experiments. 
  Some solutions of the Heavenly equations and their generalizations are considered 
  Formulating a perfect fluid filled spherically symmetric metric utilizing the 3+1 formalism for general relativity, we show that the metric coefficients are completely determined by the mass-energy distribution, and its time rate of change on an initial spacelike hypersurface. Rather than specifying Schwarzschild coordinates for the exterior of the collapsing region, we let the interior dictate the form of the solution in the exterior, and thus both regions are found to be written in one coordinate patch. This not only alleviates the need for complicated matching schemes at the interface, but also finds a new coordinate system for the Schwarzschild spacetime expressed in generalized Painleve-Gullstrand coordinates. 
  We investigate both the intermediate late-time tail and the asymptotic tail behavior of the charged massive Dirac fields in the background of the Kerr-Newman black hole. We find that the intermediate late-time behavior of charged massive Dirac fields is dominated by an inverse power-law decaying tail without any oscillation, which is different from the oscillatory decaying tails of the scalar field. We note that the dumping exponent depends not only on the angular quantum numbers $m$, the separation constant $\lambda$ and the rotating parameter $a$, but also on the product $seQ$ of the spin weight of the Dirac field and the charges of the black hole and the fields. We also find that the decay rate of the asymptotically late-time tail is $t^{-5/6}$, and the oscillation of the tail has the period of $2\pi / \mu$ which is modulated by two types of long-term phase shifts. 
  From information theory and thermodynamic considerations a universal bound on the relaxation time $\tau$ of a perturbed system is inferred, $\tau \geq \hbar/\pi T$, where $T$ is the system's temperature. We prove that black holes comply with the bound; in fact they actually {\it saturate} it. Thus, when judged by their relaxation properties, black holes are the most extreme objects in nature, having the maximum relaxation rate which is allowed by quantum theory. 
  We consider the linearized nonsymmetric theory of gravitation (NGT) within the background of an expanding universe and near a Schwarzschild mass. We show that the theory always develops instabilities unless the linearized nonsymmetric lagrangian reduces to a particular simple form. This form contains a gauge invariant kinetic term, a mass term for the antisymmetric metric-field and a coupling with the Ricci curvature scalar. This form cannot be obtained within NGT. Based on the linearized lagrangian we know to be stable, we consider the generation and evolution of quantum fluctuations of the antisymmetric gravitational field (B-field) from inflation up to the present day. We find that a B-field with a mass m ~ 0.03(H_I/10^(13)GeV)^4 eV is an excellent dark matter candidate. 
  Whitehead's 1922 theory of gravitation continues to attract the attention of philosophers, despite evidence presented in 1971 that it violates experiment. We demonstrate that the theory strongly fails five quite different experimental tests, and conclude that, notwithstanding its meritorious philosophical underpinnings, Whitehead's theory is truly dead. 
  Strategies intended to resolve the problem of time in quantum gravity by means of emergent or hidden timefunctions are considered in the arena of relational particle toy models. In situations with `heavy' and `light' degrees of freedom, two notions of emergent semiclassical WKB time emerge; these are furthermore equivalent to two notions of emergent classical `Leibniz--Mach--Barbour' time. I futhermore study the semiclassical approach, in a geometric phase formalism, extended to include linear constraints, and with particular care to make explicit those approximations and assumptions used. I propose a new iterative scheme for this in the cosmologically-motivated case with one heavy degree of freedom. I find that the usual semiclassical quantum cosmology emergence of time comes hand in hand with the emergence of other qualitatively significant terms, including back-reactions on the heavy subsystem and second time derivatives. I illustrate my analysis by taking it further for relational particle models with linearly-coupled harmonic oscillator potentials. As these examples are exactly soluble by means outside the semiclassical approach, they are additionally useful for testing the justifiability of some of the approximations and assumptions habitually made in the semiclassical approach to quantum cosmology. Finally, I contrast the emergent semiclassical timefunction with its hidden dilational Euler time counterpart. 
  I apply the preceding paper's semiclassical treatment to geometrodynamics. The analogy between the two papers is quite useful at the level of the quadratic constraints, while I document the differences between the two due to the underlying differences in their linear constraints. I provide a specific minisuperspace example for my emergent semiclassical time scheme and compare it with the hidden York time scheme. Overall, interesting connections are shown between Newtonian, Leibniz--Mach--Barbour, WKB and cosmic times, while the Euler and York hidden dilational times are argued to be somewhat different from these. 
  The Einstein-scalar field theory can be used to model gravitational physics with scalar field matter sources. We discuss the initial value formulation of this field theory, and show that the ideas of Leray can be used to show that the Einstein-scalar field system of partial differential equations is well-posed as an evolutionary system. We also show that one can generate solutions of the Einstein-scalar field constraint equations using conformal methods. 
  Specifying boundary conditions continues to be a challenge in numerical relativity in order to obtain a long time convergent numerical simulation of Einstein's equations in domains with artificial boundaries. In this paper, we address this problem for the Einstein--Christoffel (EC) symmetric hyperbolic formulation of Einstein's equations linearized around flat spacetime. First, we prescribe simple boundary conditions that make the problem well posed and preserve the constraints. Next, we indicate boundary conditions for a system that extends the linearized EC system by including the momentum constraints and whose solution solves Einstein's equations in a bounded domain. 
  A well-posed initial-boundary value problem is formulated for the model problem of vector wave equation subject to the divergence-free constraint. Existence, uniqueness, and stability of the solution is proved by reduction to a system evolving the constraint quantity trivially, namely, the second time derivative of the constraint quantity is zero. A new set of radiation-controlling constraint-preserving boundary conditions is constructed for the free evolution problem. Comparison between the new conditions and the standard constraint-preserving boundary conditions is made using the Fourier-Laplace analysis and the power series in time decomposition. The new boundary conditions satisfy the Kreiss condition and are free from the ill-posed modes growing polynomially in time. 
  For certain models, the energy of the universe which includes the energy of both the matter and the gravitational fields is obtained by using the quasilocal energy-momentum in teleparallel gravity. It is shown that in the case of the Bianchi type I and II universes, not only the total energy but also the quasilocal energy-momentum for any region vanishes independently of the three dimensionless coupling constants of teleparallel gravity. 
  We obtain an approximate global stationary and axisymmetric solution of Einstein's equations which can be considered as a simple star model: a self-gravitating perfect fluid ball with constant mass density rotating in rigid motion. Using the post-Minkowskian formalism (weak-field approximation) and considering rotation as a perturbation (slow-rotation approximation), we find approximate interior and exterior (asymptotically flat) solutions to this problem in harmonic and quo-harmonic coordinates. In both cases, interior and exterior solutions are matched, in the sense of Lichnerowicz, on the surface of zero pressure to obtain a global solution. The resulting metric depends on three arbitrary constants: mass density, rotational velocity and the star radius at the non-rotation limit. The mass, angular momentum, quadrupole moment and other constants of the exterior metric are determined by these three parameters. It is easy to show that this type of fluid cannot be a source of the Kerr metric 
  The Brane-World black hole models are investigated to evaluate their relative energy and momentum components. We consider Einstein and M{\o}ller's energy-momentum prescriptions in general relativity, and also perform the calculation of energy-momentum density in M{\o}ller's tetrad theory of gravity. For the Brane-World black holes we show that although Einstein and M{\o}ller complexes, in general relativity give different energy relations, they yield the same results for the momentum components. In addition, we also make the calculation of the energy-momentum distribution in teleparallel gravity, and calculate exactly the same energy as that obtained by using M{\o}ller's energy-momentum prescription in general relativity. This interesting result supports the viewpoint of Lessner that the M{\o}ller energy-momentum complex is a powerful concept for the energy and momentum. We also give five different examples of Brane-World black holes and find the energy distributions associated with them. The result calculated in teleparallel gravity is also independent of the teleparallel dimensionless coupling constant, which means that it is valid in any teleparallel model. This study also sustains the importance of the energy-momentum definitions in the evaluation of the energy distribution of a given space-time, and supports the hypothesis by Cooperstock that the energy is confined to the region of non-vanishing energy-momentum tensor of matter and all non-gravitational fields. 
  A multidimensional cosmological model describing the dynamics of n+1   Ricci-flat factor-spaces M_i in the presence of a one-component anisotropic fluid is considered. The pressures in all spaces are proportional to the density: p_{i} = w_i \rho, i = 0,...,n. Solutions with accelerated expansion of our 3-space M_0 and small enough variation of the gravitational constant G are found. These solutions exist for two branches of the parameter w_0. The first branch describes superstiff matter with w_0 > 1, the second one may contain phantom matter with w_0 < - 1, e.g., when G grows with time. 
  This is a reply to a comment (gr-qc/0610098) written by Nieves and Pal about our paper (gr-qc/0605153) published in Phys. Rev. Lett. 97, 041101 (2006). 
  The Newtonian approximation for the gravitational field equation should not necessarily involve admission of non-relativistic properties of the source terms in Einstein's equations: it is sufficient to merely consider the weak-field condition for gravitational field. When a source has electromagnetic nature, one simply {\em cannot} ignore its intrinsically relativistic properties, since there cannot be invented any non-relativistic approximation which would describe electromagnetic stress-energy tensor adequately, even at large distances where the fields become naturally weak. But the test particle on which gravitational field is acting, should be treated as non-relativistic (this premise is required for introduction of the Newtonian potential $\Phi_{\rm N}$ from the geodesic equation). 
  Bianchi type III string cosmological models with bulk viscous fluid for massive string are investigated. To get the determinate model of the universe, we have assumed that the coefficient of bulk viscosity ($\xi$) is inversely proportional to the expansion ($\theta$) in the model and expansion ($\theta$) in the model is proportional to the shear ($\sigma$). This leads to $B = \ell C^{n}$, $\ell$ and $n$ are constants. The behaviour of the model in presence and absence of bulk viscosity, is discussed. The physical implications of the models are also discussed in detail. 
  We present the explicit metric forms for higher dimensional vanishing scalar invariant (VSI) Lorentzian spacetimes. We note that all of the VSI spacetimes belong to the higher dimensional Kundt class. We determine all of the VSI spacetimes which admit a covariantly constant null vector, and we note that in general in higher dimensions these spacetimes are of Ricci type III and Weyl type III. The Ricci type N subclass is related to the chiral null models and includes the relativistic gyratons and the higher dimensional pp-wave spacetimes. The spacetimes under investigation are of particular interest since they are solutions of supergravity or superstring theory. 
  We present an approach to cosmological perturbations based on a covariant perturbative expansion between two worldlines in the real inhomogeneous universe. As an application, at an arbitrary order we define an exact scalar quantity which describes the inhomogeneities in the number of e-folds on uniform density hypersurfaces and which is conserved on all scales for a barotropic ideal fluid. We derive a compact form for its conservation equation at all orders and assign it a simple physical interpretation. To make a comparison with the standard perturbation theory, we develop a method to construct gauge-invariant quantities in a coordinate system at arbitrary order, which we apply to derive the form of the n-th order perturbation in the number of e-folds on uniform density hypersurfaces and its exact evolution equation. On large scales, this provides the gauge-invariant expression for the curvature perturbation on uniform density hypersurfaces and its evolution equation at any order. 
  The two body problem in a scalar theory of gravity is investigated. We focus on the closest theory to General Relativity (GR), namely Nordstr\"om's theory of gravity (1913). The gravitational field can be exactly solved for any configuration of point-particles. We then derive the exact equations of motion of two inspiraling bodies including the exact self-forces terms. We prove that there is no innermost circular orbit (ICO) in the exact theory whereas we find (order-dependent) ICOs if post-Newtonian (PN) truncations are used. We construct a solution of the two body problem in an iterative (non-PN) way, which can be viewed as a series in powers of $(v/c)^{5}$. Besides this rapid convergence, each order also provides non-perturbative information. Starting from a circular Newtonian-like orbit, the first iteration already yields the 4.5 PN radiation reaction. These results not only shed light on some non-perturbative effects of relativistic gravity, but may also be useful to test numerical codes. 
  We give a comparative description of different types of regular static, spherically symmetric black holes (BHs) and discuss in more detail their particular type, which we suggest to call black universes. The latter have a Schwarzschild-like causal structure, but inside the horizon there is an expanding Kantowski-Sachs universe and a de Sitter infinity instead of a singularity. Thus a hypothetic BH explorer gets a chance to survive. Solutions of this kind are naturally obtained if one considers static, spherically symmetric distributions of various (but not all) kinds of phantom matter whose existence is favoured by cosmological observations. It also looks possible that our Universe has originated from phantom-dominated collapse in another universe and underwent isotropization after crossing the horizon. An explicit example of a black-universe solution with positive Schwarzschild mass is discussed. 
  In this paper we show that with standard methods it is possible to obtain highly precise results for QNMs. In particular, secondary modes are obtained by numerical integration. We compare several results making a detailed analysis. 
  A common feature of all Quantum Gravity (QG) phenomenology approaches is to consider a modification of the mass shell condition of the relativistic particle to take into account quantum gravitational effects. The framework for such approaches is therefore usually set up in the cotangent bundle (phase space). However it was recently proposed that this phenomenology could be associated with an energy dependent geometry that has been coined ``rainbow metric". We show here that the latter actually corresponds to a Finsler Geometry, the natural generalization of Riemannian Geometry. We provide in this way a new and rigorous framework to study the geometrical structure possibly arising in the semiclassical regime of QG. We further investigate the symmetries in this new context and discuss their role in alternative scenarios like Lorentz violation in emergent spacetimes or Deformed Special Relativity-like models. 
  We address the possibility of taking advantage of high accuracy gravitational space experiments in the Solar System and complementary cosmological tests to distinguish between the usual general relativistic theory from the alternative modified Newtonian dynamics paradigm. 
  In the past, Hawking radiation was viewed as a tunneling process and the barrier was just created by the outgoing particle itself. In this paper, Parikh's recent work is extended to the case of massive particles' tunneling. We investigate the behavior of the tunneling massive particles from a particular black hole solution-G.H Dilaton black hole which is obtained from the string theory, and calculate the emission rate at which massive particles tunnel across the event horizon. We obtain that the result is also consistent with an underlying unitary theory. Furthermore, the result takes the same functional form as that of massless particles. 
  Angular momentum has recently been defined as a surface integral involving an axial vector and a twist 1-form, which measures the twisting around of space-time due to a rotating mass. The axial vector is chosen to be a transverse, divergence-free, coordinate vector, which is compatible with any initial choice of axis and integral curves. Then a conservation equation expresses rate of change of angular momentum along a uniformly expanding flow as a surface integral of angular momentum densities, with the same form as the standard equation for an axial Killing vector, apart from the inclusion of an effective energy tensor for gravitational radiation. 
  Analyzing the tunneling probability of a Schwarzschild black hole with a negative log-area correction to Bekenstein-Hawking entropy, I argue that this correction may be closely related to a black hole remnant. The value for the minimal black hole mass is also discussed. 
  The quantization of the Friedmann-Robertson-Walker spacetime in the presence of a negative cosmological constant was used in a recent paper to conclude that there are solutions that avoid singularities (big bang--big crunch) at the quantum level. We show that a proper study of their model{\sl does not prevent} the occurrence of singularities at the quantum level, in fact the quantum probability of such event is larger than the classical one. Our numerical simulations based on the powerful variational sinc collocation method (VSCM) also show that the precision of the results of that paper is much lower than the 20 significant digits reported by the authors. 
  We report on the measurement of parasitic surface force noise on a hollow replica of a LISA (Laser Interferometer Space Antenna for the observation of gravitational waves) proof mass surrounded by a faithful representation of its in flight surroundings, namely the capacitive sensor used to detect proof-mass motion. Parasitic forces are detected through the corresponding torque exerted on the proof mass and measured with a torsion pendulum in the frequency range 0.1 30 mHz. The sensor electrodes, electrode housing and associated readout electronics have the same nominal design as for the flight hardware, including 4 mm gaps around the proof mass along the sensitive laser interferometry axis. We show that the measured upper limit for surface forces would allow detection of a number of galactic binaries signals with signal to noise ratio up to approximately 40 for 1 year integration. We also discuss how the flight test under development, LISA Pathfinder, will substantially improve this limit, approaching the performance required for LISA. 
  Bianchi type V bulk viscous fluid cosmological models are investigated with dynamic cosmological term $\Lambda(t)$. Using a generation technique (Camci {\it et al.}, 2001), it is shown that the Einstein's field equations are solvable for any arbitrary cosmic scale function. Solutions for particular forms of cosmic scale functions are also obtained. The cosmological constant is found to be decreasing function of time, which is supported by results from recent type Ia supernovae observations. Some physical and geometrical aspects of the models are also discussed. 
  We investigate the integrability of cosmic strings in Bianchi III space-time in presence of a bulk viscous fluid by applying a new technique. The behaviour of the model is reduced to the solution of a single second order nonlinear differential equation. We show that this equation admits an infinite family of solutions. Some physical consequences from these results are also discussed. 
  Mathematical theory of an observer is elaborated upon the basis of A.Poincare's ideas on the nature of geometry and the role of observer's perceptive space. The said theory is generalizing reference frames theory in GR. Physical structure (P-structure) and corresponding physical geometry (P-geometry) notions, representing properties invariance of some physical objects and their relations, are introduced. P-structure of classical physical time and its corresponding chronogeometry is considered as an example. Some quantitative characteristics of observer's visual space geometry are experimentally determined. The affine model of visual geometry is offered to interpret experimentally sampled data. The connection of the obtained results with some problems of theoretical physics is being discussed. 
  Although cosmology is usually considered an observational science, where there is little or no space for experimentation, other approaches can (and have been) also considered. In particular, we can change rather drastically the above, more passive, observational perspective and ask the following question: could it be possible, and how, to create a universe in a laboratory? As a matter of fact, this seems to be possible, according to at least two different paradigms; both of them help to evade the consequences of singularity theorems. In this contribution we will review some of these models and we will also discuss possible extensions and generalizations, by paying a critical attention to the still open issues as, for instance, the detectability of child universes and the properties of quantum tunnelling processes. 
  We consider asymptotically-flat, static and stationary solutions of the Einstein equations representing Einstein-Maxwell space-times in which the Maxwell field is not constant along the Killing vector defining stationarity, so that the symmetry of the space-time is not inherited by the electromagnetic field. We find that static degenerate black hole solutions are not possible and, subject to stronger assumptions, nor are static, non-degenerate or stationary black holes. We describe the possibilities if the stronger assumptions are relaxed. 
  A four mass torsion pendulum facility for testing of the LISA GRS is under development in Trento. With a LISA-like test mass suspended off-axis with respect to the pendulum fiber, the facility allows for a direct measurement of surface force disturbances arising in the GRS. We present here results with a prototype pendulum integrated with very large-gap sensors, which allows an estimate of the intrinsic pendulum noise floor in the absence of sensor related force noise. The apparatus has shown a torque noise near to its mechanical thermal noise limit, and would allow to place upper limits on GRS related disturbances with a best sensitivity of 300 fN/Hz^(1/2) at 1mHz, a factor 50 from the LISA goal. Also, we discuss the characterization of the gravity gradient noise, one environmental noise source that could limit the apparatus performances, and report on the status of development of the facility. 
  The old cosmological-constant (CC) problem indicates an inconsistency of the usual formulation of semiclassical gravity. The usual formulation of semiclassical gravity also seems to be inconsistent with the conventional interpretation of quantum mechanics based on the discontinuous wave-function collapse. By reformulating semiclassical gravity in terms of Bohmian deterministic particle trajectories, the resulting semiclassical theory avoids both the old CC problem and the discontinuous collapse problem of the usual semiclassical theory. The relevance to the new CC problem and to particle creation by classical gravitational fields is also discussed. 
  The SNe type Ia data admit that the Universe today may be dominated by some exotic matter with negative pressure violating all energy conditions. Such exotic matter is called {\it phantom matter} due to the anomalies connected with violation of the energy conditions. If a phantom matter dominates the matter content of the universe, it can develop a singularity in a finite future proper time. Here we show that, under certain conditions, the evolution of perturbations of this matter may lead to avoidance of this future singularity (the Big Rip). At the same time, we show that local concentrations of a phantom field may form, among other regular configurations, black holes with asymptotically flat static regions, separated by an event horizon from an expanding, singularity-free, asymptotically de Sitter universe. 
  A binary system, composed of a compact object orbiting around a massive central body, will emit gravitational waves which will depend on the central body's spacetime geometry. We expect that the gravitational wave observables will somehow ``encode'' the information about the spacetime structure. On the other hand, it has been known for some time that the geometry around an axisymmetric body can be described by its (Geroch-Hansen) multipole moments. Therefore one can speculate that using the multipole moments can prove to be a helpful tool for extracting this information. We will try to demonstrate this in this talk, following the procedure described by [F. D. Ryan, Phys. Rev. D {\bf 52} 5707 (1995)] and [T. P. Sotiriou and T. A. Apostolatos, Phys. Rev. D {\bf 71} 044005 (2005)]. 
  Recently a class of alternative theories of gravity which goes under the name f(R) gravity, has received considerable attention, mainly due to its interesting applications in cosmology. However, the phenomenology of such theories is not only relevant to cosmological scales, especially when it is treated within the framework of the so called Palatini variation, an independent variation with respect to the metric and the connection, which is not considered a priori to be the Levi-Civita connection of the metric. If this connection has its standard geometrical meaning the resulting theory will be a metric-affine theory of gravity, as will be discussed in this talk. The general formalism will be presented and several aspects of the theory will be covered, mainly focusing on the enriched phenomenology that such theories exhibit with respect to General Relativity, relevant not only to large scales (cosmology) but also to small scales (e.g. torsion). 
  In this letter we study the generation of gravitational waves during inflation from a 5D vacuum theory of gravity. Within this formalism, on an effective 4D de Sitter background, we recover the typical results obtained with 4D inflationary theory in general relativity, for the amplitude of gravitational waves generated during inflation. We also obtain a range of values for the amplitude of tensor to scalar ratio which is in agreement with COBE observations. 
  Vacuum 5-D Einstein equations with spherical symmetry and t-dependence are considered. For the case of separating variables several classes of exact solutions are obtained. Effective matter, induced by geometrical scalar field is analyzed. 
  In this paper we calculate modifications to the Schwarzschild solution by using a semiclassical analysis of loop quantum black hole. We obtain a metric inside the event horizon that coincides with the Schwarzschild solution near the horizon but that is substantially different at the Planck scale. In particular we obtain a bounce of the two-sphere for a minimum value of the radius and that it is possible to have another event horizon close to the r=0 point. 
  We use a covariant phase space formalism to give a general prescription for defining Hamiltonian generators of bosonic and fermionic symmetries in diffeomorphism invariant theories, such as supergravities. A simple and general criterion is derived for a choice of boundary condition to lead to conserved generators of the symmetries on the phase space. In particular, this provides a criterion for the preservation of supersymmetries. For bosonic symmetries corresponding to diffeomorphisms, our prescription coincides with the method of Wald et al.   We then illustrate these methods in the case of certain supergravity theories in $d=4$. In minimal AdS supergravity, the boundary conditions such that the supercharges exist as Hamiltonian generators of supersymmetry transformations are unique within the usual framework in which the boundary metric is fixed. In extended ${\mathcal N}=4$ AdS supergravity, or more generally in the presence of chiral matter superfields, we find that there exist many boundary conditions preserving ${\mathcal N}=1$ supersymmetry for which corresponding generators exist. These choices are shown to correspond to a choice of certain arbitrary boundary ``superpotentials,'' for suitably defined ``boundary superfields.'' We also derive corresponding formulae for the conserved bosonic charges, such as energy, in those theories, and we argue that energy is always positive, for any supersymmetry-preserving boundary conditions. We finally comment on the relevance and interpretation of our results within the AdS-CFT correspondence. 
  The Hamiltonian structure of General Relativity (GR), for both metric and tetrad gravity in a definite continuous family of space-times, is fully exploited in order to show that: i) the "Hole Argument" can be bypassed by means of a specific "physical individuation" of point-events of the space-time manifold $M^4$ in terms of the "autonomous degrees of freedom" of the vacuum gravitational field ("Dirac observables"), while the "Leibniz equivalence" is reduced to differences in the "non-inertial appearances" (connected to "gauge" variables) of the same phenomena. ii) the chrono-geometric structure of a solution of Einstein equations for given, gauge-fixed, initial data (a "3-geometry" satisfying the relevant constraints on the Cauchy surface), can be interpreted as an "unfolding" in mathematical global time of a sequence of "achronal 3-spaces" characterized by "dynamically determined conventions" about distant simultaneity. This result stands out as an important "conceptual difference" with respect to the standard chrono-geometrical view of Special Relativity (SR) and allows, in a specific sense, for an "endurantist" interpretations of ordinary "physical objects" in GR. 
  A derivation of the optical axis lenght fluctations due by tilts of the mirrors of the Fabry-Perot cavity of long-baseline interferometers for the detection of gravitational waves in presence of the gravitational field of the earth is discussed. By comparing with the typical tilt-induced noises it is shown that this potential signal, which is considered a weak source of noise, is negligible for the first generation of gravitational waves interferometers, but, in principle, this effect could be used for high precision measures of the gravitational acceleration if advanced projects will achieve an high sensitivity. In that case the precision of the misure could be higher than the gravimeter realized by the Istituto di Metrologia ``Gustavo Colonnetti''. 
  The conical singularity in flat spacetime is mostly known as a model of the cosmic string or the wedge disclination in solids. Its another, equally important, function is to be a representative of quasiregular singularities.   From all these of views it seems interesting to find out whether there exist other similar singularities. To specify what "similar" means I introduce the notion of the string-like singularity, which is, roughly speaking, an absolutely mild singularity concentrated on a curve or on a 2-surface S (depending on whether the space is three- of four-dimensional). A few such singularities are already known: the aforementioned conical singularity, two its Lorentzian versions, the "spinning string", the "screw dislocation", and Tod's spacetime. In all these spacetimes S is a straight line (or a plane) and one may wonder if this is an inherent property of the string-like singularities. The aim of this paper is to construct string-like singularities with less trivial S. These include flat spacetimes in which S is a spiral, or even a loop. If such singularities exist in nature (in particular, as an approximation to gravitational field of strings) their cosmological and astrophysical manifestations must differ drastically from those of the conventional cosmic strings. Likewise, being realized as topological defects in crystals such loops and spirals will probably also have rather unusual properties. 
  In order to include nontrivial spatial topologies in the problem of quantum creation of a universe, it seems to be necessary to generalize the sum over compact, smooth 4-manifolds to a sum over finite-volume, compact 4-orbifolds. We consider in detail the case of a 4-spherical orbifold with a cone-point singularity. This allows for the inclusion of a nontrivial topology in the semiclassical path integral approach to quantum cosmology, in the context of a Robertson-Walker minisuperspace. 
  We construct a covariant phase space for Einstein gravity in dimensions d>=4 with negative cosmological constant, describing black holes in local equilibrium. Thus, space-times under consideration are asymptotically anti-de Sitter and admit an inner boundary representing an isolated horizon. This allows us to derive a first law of black hole mechanics that involves only quantities defined quasi-locally at the horizon, without having to assume that the bulk space-time is stationary. The first law proposed by Gibbons et al. for the Kerr-AdS family follows from a special case of this much more general first law. 
  A theory in which points, lines, areas and volumes are on on the same footing is investigated. All those geometric objects form a 16-dimensional manifold, called C-space, which generalizes spacetime. In such higher dimensional space fundamental interactions can be unified \` a la Kaluza-Klein. The ordinary, 4-dimensional, gravity and gauge fields are incorporated in the metric and spin connection, whilst the conserved gauge charges are related to the isometries of curved C-space. It is shown that a conserved generator of an isometry in C-space contains a part with derivatives, which generalizes orbital angular momentum, and a part with the generators of Clifford algebra, which generalizes spin. 
  The back-reaction of a classical gravitational field interacting with quantum matter fields is described by the semiclassical Einstein equation, which has the expectation value of the quantum matter fields stress tensor as a source. The semiclassical theory may be obtained from the quantum field theory of gravity interacting with N matter fields in the large N limit. This theory breaks down when the fields quantum fluctuations are important. Stochastic gravity goes beyond the semiclassical limit and allows for a systematic and self-consistent description of the metric fluctuations induced by these quantum fluctuations. The correlation functions of the metric fluctuations obtained in stochastic gravity reproduce the correlation functions in the quantum theory to leading order in an 1/N expansion. Two main applications of stochastic gravity are discussed. The first, in cosmology, to obtain the spectrum of primordial metric perturbations induced by the inflaton fluctuations, even beyond the linear approximation. The second, in black hole physics, to study the fluctuations of the horizon of an evaporating black hole. 
  We consider full perturbations to a covariantly defined Schwarzschild spacetime. By constructing complex quantities, we derive two decoupled, covariant and gauge-invariant, wave-like equations for spin-weighted scalars. These arise naturally from the Bianchi identities and comprise a covariant representation of the Bardeen-Press equations for scalars with spin-weight $\pm2$. Furthermore, the covariant and gauge-invariant 1+1+2 formalism is employed, and consequently, the physical interpretation of the energy-momentum perturbations is transparent. They are written explicitly in terms of the energy-momentum specified on spacelike three-slices. Ultimately, a Cauchy problem is constructed whereby, an initial three-slice may be perturbed by an energy-momentum source, which induces resultant gravitational fields. 
  Using both numerical and analytical tools we study various features of static, spherically symmetric solutions of the Einstein-Vlasov system. In particular, we investigate the possible shapes of their mass-energy density and find that they can be multi-peaked, we give numerical evidence and a partial proof for the conjecture that the Buchdahl inequality $\sup_{r > 0} 2 m(r)/r < 8/9$, $m(r)$ the quasi-local mass, holds for all such steady states--both isotropic {\em and} anisotropic--, and we give numerical evidence and a partial proof for the conjecture that for any given microscopic equation of state--both isotropic {\em and} anisotropic--the resulting one-parameter family of static solutions generates a spiral in the radius-mass diagram. 
  Using ideas of STM theory, but starting from a 6D vacuum state, we propose an inflationary model where the universe emerges from the blast of a white hole. Under this approach, the expansion is affected by a geometrical deformation induced by the gravitational attraction of the hole, which should be responsible for the k_{R}-non invariant spectrum of galaxies (and likewise of the matter density) today observed. 
  The Cosmological Constant Problem emerges when Quantum Field Theory is applied to the gravitational theory, due to the enormous magnitude of the induced energy of the vacuum. The unique known solution of this problem involves an extremely precise fine-tuning of the vacuum counterpart. We review a few of the existing approaches to this problem based on the account of the quantum (loop) effects and pay special attention to the ones involving the renormalization group. 
  Recently some authors concluded that the energy and momentum of the Fiedman universes, flat and closed, are equal to zero locally and globally (flat universes) or only globally (closed universes). The similar conclusion was also done for more general only homogeneous universes (Kasner and Bianchi type I). Such conclusions originated from coordinate dependent calculations performed only in comoving Cartesian coordinates by using the so-called {\it energy-momentum complexes}. But it is known that the energy-momentum complexes can be reasonably use only in precisely defined asymptotically flat spacetimes (at null or at spatial infinity) to calculate global energy and momentum. In this paper we show, by using new coordinate independent expressions on energy and momentum that the Friedman and more general universes {\it needn't be energetic nonentity}. 
  It is proven that in Vaidya spacetimes of bounded total mass, the outer boundary, in spacetime, of the region containing outer trapped surfaces, is the event horizon. Further, it is shown that the region containing trapped surfaces in these spacetimes does not always extend to the event horizon. 
  This essay is based on a physics lecture given at the Hong Kong University of Science and Technology on May 24, 2006, Hong Kong. It has 5 sections and one page of references and suggested reading. The section titles are: 1. Origin of the Universe and Quantum Gravity 2. Two major paradigms in Physics and Two directions of Research in Cosmology 3. The Mescoscopic Structures of Spacetime and Stochastic Gravity 4. One Vein of the Hydro View: Spacetime as Condensate? 5. Implications for the Origin of the Universe and other issues. 
  We present an exact black hole solution in a model having besides gravity a dilaton and a monopole field. The solution has three free parameters, one of which can be identified with the monopole charge, and another with the ADM mass. The metric is asymptotically flat and has two horizons and irremovable singularity only at $r=0$. The dilaton field is singular only at $r=0$. The dominant and the strong energy condition are satisfied outside and on the external horizon. According to a formulation of the no hair conjecture the solution is "hairy". Also the well know GHS-GM solution is obtained from our solution for certain values of its parameters. 
  We present an exact black hole solution in a model having besides gravity a dilaton and a monopole field, which is a generalization of a black hole solution we have found. The new solution, as the previous one, has three free parameters, one of which can be identified with the monopole charge, and another with the ADM mass. Its metric is asymptotically flat, has two horizon, irremovable singularity only at $r=0$, and the dilaton field is singular only at $r=0$. The dominant and the strong energy condition are satisfied outside and on the external horizon. According to a formulation of the no hair conjecture the solution is "hairy". Also a reformulation of the model with two monopole fields is given, which results in the appearance of an additional symmetry and therefore in the appearance of a conserved dilaton charge. 
  Exact static spherically symmetric charged black holes in four dimensions are presented. One of them has only electric charge and another electric and magnetic charges. In these solutions the metric is asymptotically flat, has two horizons, irremovable singularity only at $r=0$, and the dilaton field is singular only at $r=0$. The solution with electric charge only is characterized by three free parameters, the ADM mass, the electric charge and an additional free parameter. It can be considered as a modification of the GHS-GM solution obtained by changing the coupling between dilaton and electromagnetic field. The general dyonic solution is again characterized by three free parameters, the ADM mass, the magnetic charge and an additional free parameter, which is not the electric charge. According to a definition of the no-hair conjecture the solutions are "hairy".A very interesting special case of the dyonic solution is characterized by three free parameters, the ADM mass and the electric and the magnetic charges. The solutions satisfy the dominant as well as the strong energy condition outside and on the external horizon. 
  Using non-perturbative results obtained recently for an uniformly accelerated Unruh-DeWitt detector, we discover a very different scenario in the dynamical evolution of the detector's internal degree of freedom after the coupling with a quantum field is turned on. From a calculation of the evolution of the reduced density matrix of the detector, we find that the Unruh effect as originally derived from time-dependent perturbation theory is existent only in transient and under very special limiting conditions. In particular, the detector at late times never sees an exact Boltzmann distribution over the energy eigenstates of the free detector, and in the range of parameters of realistic processes no Unruh temperature can be identified. 
  A new variable in the Riemannian geometry is introduced by the tetrad and the Ricci's coefficients of rotation, the characters of curve of the Riemannian geometry are determined completely by the new variable; for general relativity, all the Einstein-Hilbert action, the Einstein equation in general relativity and the Dirac equation in curved spacetime can be expressed by the new variable, and, further, as well the action of the theory on the interaction of gravitational, electromagnetic and spinor field (TGESF). All the characters of transformations of the new variable, the Einstein-Hilbert action and the action of TGESF under the general coordinate transformations and the local Lorentz transformations are discussed, respectively. After presenting the method of introduction of gravitational field in terms of the principle of gauge invariance based on the Dirac equation, and the ten constraint conditions for the tetrad are given, the vacuum-vacuum transition amplitude with the Faddeev-Popov ghost and the terms of external sources of the pure gravitational field is presented; finally, as well that of TGESF. 
  For evolution of flat universe, we classify late time and future attractors with scaling behavior of scalar field quintessence in the case of potential, which, at definite values of its parameters and initial data, corresponds to exact scaling in the presence of cosmological constant. 
  We present the effective field equations obtained from a generalized gravity action with Euler-Poincare term and a cosmological constant in a $D$ dimensional bulk space-time. A class of plane-symmetric solutions that describe a 3-brane world embedded in a D=5 dimensional bulk space-time are given. 
  We investigate photon emission from a moving particle in an expanding universe. This process is analogous to the radiation from an accelerated charge in the classical electromagnetic theory. Using the framework of quantum field theory in curved spacetime, we demonstrate that the Wentzel-Kramers-Brillouin (WKB) approximation leads to the Larmor formula for the rate of the radiation energy from a moving charge in an expanding universe. Using exactly solvable models in a radiation-dominated universe and in a Milne universe, we examine the validity of the WKB formula. It is shown that the quantum effect suppresses the radiation energy in comparison with the WKB formula. 
  The transition probability in first-order perturbation theory for an Unruh-DeWitt detector coupled to a massless scalar field in Minkowski space is calculated. It has been shown recently that the conventional $i\epsilon$ regularisation prescription for the correlation function leads to non-Lorentz invariant results for the transition rate, and a different regularisation, involving spatial smearing of the field, has been advocated to replace it. We show that the non-Lorentz invariance arises solely from the assumption of sudden switch-on and switch-off of the detector, and that when the model includes a smooth switching function the results from the conventional regularisation are both finite and Lorentz invariant. The sharp switching limit of the model is also discussed, as well as the falloff properties of the spectrum for large frequencies. 
  Quadratic curvature corrections to Einstein-Hilbert action lead in general to higher-order equations of motion, which can induced instability of some unperturbed solutions of General Relativity. We study conditions for stability of de Sitter cosmological solution. We argue that simple form of this condition known for FRW background in 3+1 dimensions changes seriously if at least one of these two assumptions is violated. In the present paper the stability conditions for de Sitter solution have been found for multidimensional FRW background and for Bianchi I metrics in 3+1 dimensions. 
  A gauge invariant metric fluctuations formalism from a non-compact Kaluza-Klein (NKK) theory of gravity is presented in this talk notes. In this analysis we recover the well-known result $\frac{delta \rho}{\rho}\simeq 2\Phi$ obtained typically in the standard 4D semiclassical approach to inflation and also the spectrum of this fluctuations becomes dependent of the fifth (space-like) coordinate. This fact allows to establish an interval of values for the wave number associated with the fifth dimension. 
  The radiation gauges used by Chrzanowski (his IRG/ORG) for metric reconstruction in the Kerr spacetime seem to be over-specified. Their specification consists of five conditions: four (which we treat here as) ``gauge'' conditions plus an additional condition on the trace of the metric perturbation. In this work, we utilize a newly developed form of the perturbed Einstein equations to establish a condition -- on a particular tetrad component of the stress-energy tensor -- under which one can impose the full IRG/ORG. In a Petrov type II background, imposing the IRG/ORG additionally requires (consistently) setting a particular component of the metric perturbation to zero ``by hand''. By contrast, in a generic type D background, gauge freedom can generally be used to achieve this. As a specific example, we work through the process of imposing the IRG in a Schwarzschild background, using a more traditional approach. Implications for metric reconstruction using the Teukolsky curvature perturbations in type D spacetimes are briefly discussed. 
  In this paper we revisit the relation between the Friedmann equations and the first law of thermodynamics. We find that the unified first law firstly proposed by Hayward to treat the "outer" trapping horizon of dynamical black hole can be used to the apparent horizon (a kind of "inner" trapping horizon in the context of the FRW cosmology) of the FRW universe. In Einstein theory, the first law of thermodynamics is always satisfied on the apparent horizon without any approximation. Treating the higher derivative terms in the Lovelock gravity as an effective energy-momentum tensor, we find that this method can give the same entropy formula for the apparent horizon as that of black hole horizon. This implies that the Clausius relation holds for the Lovelock theory. By using the same procedure, we also discuss the scalar-tensor theory and find that the Clausius relation no longer holds in this case. This indicates that the apparent horizon of FRW universe in the scalar-tensor gravity corresponds to a system of non-equilibrium thermodynamics. We show this point by using the method developed recently by Jacobson {\it et al.} for dealing with the $f(R)$ gravity. 
  Gravitational waves from the inspiral of a stellar-size black hole to a supermassive black hole can be accurately approximated by a point particle moving in a Kerr background. This paper presents progress on finding the electromagnetic and gravitational field of a point particle in a black-hole spacetime and on computing the self-force in a ``radiation gauge.'' The gauge is chosen to allow one to compute the perturbed metric from a gauge-invariant component $\psi_0$ (or $\psi_4$) of the Weyl tensor and follows earlier work by Chrzanowski and Cohen and Kegeles (we correct a minor, but propagating, error in the Cohen-Kegeles formalism). The electromagnetic field tensor and vector potential of a static point charge and the perturbed gravitational field of a static point mass in a Schwarzschild geometry are found, surprisingly, to have closed-form expressions. The gravitational field of a static point charge in the Schwarzschild background must have a strut, but $\psi_0$ and $\psi_4$ are smooth except at the particle, and one can find local radiation gauges for which the corresponding spin $\pm 2$ parts of the perturbed metric are smooth. Finally a method for finding the renormalized self-force from the Teukolsky equation is presented. The method is related to the MiSaTaQuWa renormalization and to the Detweiler-Whiting construction of the singular field. It relies on the fact that the renormalized $\psi_0$ (or $\psi_4$) is a {\em sourcefree} solution to the Teukolsky equation; and one can therefore reconstruct a nonsingular renormalized metric in a radiation gauge. 
  The Kodama State is unique in being an exact solution to all the ordinary constraints of canonical quantum gravity that also has a well defined semi-classical interpretation as a quantum version of a classical spacetime, namely (anti)de Sitter space. However, the state is riddled with difficulties which can be tracked down to the complexification of the phase space necessary in its construction. This suggests a generalization of the state to real values of the Immirzi parameter. In this first part of a two paper series we show that one can generalize the state to real variables and the result is surprising in that it appears to open up an infinite class of physical states. We show that these states closely parallel the ordinary momentum eigenstates of non-relativistic quantum mechanics with the Levi-Civita curvature playing the role of the momentum. With this identification, the states inherit many of the familiar properties of the momentum eigenstates including delta-function normalizability. In the companion paper we will discuss the physical interpretation, CPT properties, and an interesting connection between the inner product and the Macdowell-Mansouri formulation of general relativity. 
  In this second part of a two paper series we discuss the properties and physical interpretation of the generalized Kodama states. We first show that the states are the three dimensional boundary degrees of freedom of two familiar 4-dimensional topological invariants: the second Chern class and the Euler class. Using this, we show that the states have the familiar interpretation as WKB states, in this case corresponding not only to de Sitter space, but also to first order perturbations therein. In an appropriate spatial topology, the de Sitter solution has pure Chern-Simons functional form, and is the unique state in the class that is identically diffeomorphism and SU(2) gauge invariant. The q-deformed loop transform of this state yields evidence of a cosmological horizon when the deformation parameter is a root of untiy. We then discuss the behavior of the states under discrete symmetries, showing that the states violate $P$ and $T$ due to the presence of the Immirzi parameter, but they are $CPT$ invariant. We conclude with an interesting connection between the physical inner product and the Macdowell Mansouri formulation of gravity. 
  An outstanding challenge for models of non-perturbative quantum gravity is the consistent formulation and quantitative evaluation of physical phenomena in a regime where geometry and matter are strongly coupled. After developing appropriate technical tools, one is interested in measuring and classifying how the quantum fluctuations of geometry alter the behaviour of matter, compared with that on a fixed background geometry.   In the simplified context of two dimensions, we show how a method invented to analyze the critical behaviour of spin systems on flat lattices can be adapted to the fluctuating ensemble of curved spacetimes underlying the Causal Dynamical Triangulations (CDT) approach to quantum gravity. We develop a systematic counting of embedded graphs to evaluate the thermodynamic functions of the gravity-matter models in a high- and low-temperature expansion. For the case of the Ising model, we compute the series expansions for the magnetic susceptibility on CDT lattices and their duals up to orders 6 and 12, and analyze them by ratio method, Dlog Pad\'e and differential approximants. Apart from providing evidence for a simplification of the model's analytic structure due to the dynamical nature of the geometry, the technique introduced can shed further light on criteria \`a la Harris and Luck for the influence of random geometry on the critical properties of matter systems. 
  It is well known that the Lorentzian length of a timelike curve in Minkowski spacetime is smaller than the Lorentzian length of the geodesic connecting its initial and final endpoints. The difference is known as the 'differential aging' and its calculation in terms of the proper acceleration history of the timelike curve would provide an important tool for the autonomous spacetime navigation of non-inertial observers. I give a solution in 3+1 dimensions which holds whenever the acceleration is decomposed with respect to a lightlike transported frame (lightlike transport will be defined), the analogous and more natural problem for a Fermi-Walker decomposition being still open. 
  We study symmetric teleparallel (STP) gravity model, in which only spacetime non-metricity is nonzero. First we obtain STP equivalent Einstein-Hilbert Lagrangian and give an approach for a generic solution in terms of only metric tensor. Then we obtain a spherically symmetric static solution to the Einstein's equation in STP space-time and discuss the singularities. Finally, we study a model given by a Lagrangian 4-form quadratic in non-metricity. Thus, we seek Schwarzschild-type solutions because of its observational success and obtain some sets of solutions. Finally, we discuss physical relevance of the solutions. 
  The general expression of the angular distance between two point sources as measured by an arbitrary observer is given. The modelling presented here is rigorous, covariant and valid in any space-time. The sources of light may be located at a finite distance from the observer. The aberration and the gravitational deflection of light are treated in a unified way. Assuming the gravitational field to be weak, an explicit expansion of the angular separation within the post-post-Minkowskian approximation is carried out. The angular separation within the post-Newtonian approximation truncated at the order $1/c^3$ is straightforwardly derived. 
  We consider irrotational dust spacetimes in the full non-linear regime which are "purely radiative" in the sense that the gravitational field satisfies the covariant transverse conditions div(H) = div(E) = 0. Within this family we show that the Bianchi class A spatially homogeneous dust models are uniquely characterised by the condition that $H$ is diagonal in the shear-eigenframe. 
  In this series of papers, we propose a new rendition of 3d and 4d state sum models based upon the group field theory (GFT) approach to non-perturbative quantum gravity. We will see that the group field theories investigated in the literature to date are, when judged from the position of quantum field theory, an unusual manifestation of quantum dynamics. They are one in which the Hadamard function for the field theory propagates a-causally the physical degrees of freedom of quantum gravity. This is fine if we wish to define a scalar product on the physical state space, but it is not what we generally think of as originating directly from a field theory. We propose a model in 3d more in line with standard quantum field theory, and therefore the field theory precipitates causal dynamics. Thereafter, we couple the model to point matter, and extract from the GFT the effective non-commutative field theory describing the matter dynamics on a quantum gravity background. We identify the symmetries of our new model and clarify their meaning in the GFT setting. We are aided in this process by identifying the category theory foundations of this GFT which, moreover, propel us towards a categorified version for the 4d case. 
  In this paper we put on the test the new mechanism of gravitational origin recently put forth by Jaekel and Reynaud in order to explain the Pioneer anomaly in the framework of their post-Einsteinian metric extension of general relativity. According to such a proposal, the secular part of the anomalous acceleration experienced by the twin spacecraft of about 1 nm s^-2 could be caused by an extra-potential \delta\Phi_P=c^2\chi r^2, with \chi=4 10^-8 AU^-2, coming from the second sector of the considered model. When applied to the motion of the planets of the Solar System, it would induce anomalous secular perihelion advances which amount to tens-hundreds of arcseconds per century for the outer planets. As for other previously proposed non-conventional gravitational explanations of the Pioneer anomaly, the answer of the latest observational determinations of the residual perihelion rates by RAS IAA is neatly and unambiguously negative. The presence of another possible candidate to explain the Pioneer anomaly, i.e. the extra-potential \delta\Phi_N, linear in distance, from the first sector of the Jaekel and Reynaud model, is ruled out not only by the residuals of the optical data of the outer planets processed with the recent RAS IAA EPM2004 ephemerides but also by those produced with other, older dynamical theories like, e.g., the well known NASA JPL DE200. 
  In (4 + 1) gravity the assumption that the five-dimensional metric is independent of the fifth coordinate authorizes the extra dimension to be either spacelike or timelike. As a consequence of this, the time coordinate and the extra coordinate are interchangeable, which in turn allows the conception of different scenarios in 4D from a single solution in 5D. In this paper, we make a thorough investigation of all possible 4D scenarios, associated with this interchange, for the well-known Kramer-Gross-Perry-Davidson-Owen set of solutions. We show that there are {\it three} families of solutions with very distinct geometrical and physical properties. They correspond to different sets of values of the parameters which characterize the solutions in 5D. The solutions of physical interest are identified on the basis of physical requirements on the induced-matter in 4D. We find that only one family satisfies these requirements; the other two violate the positivity of mass-energy density. The "physical" solutions possess a lightlike singularity which coincides with the horizon. The Schwarzschild black string solution as well as the zero moment dipole solution of Gross and Perry are obtained in different limits. These are analyzed in the context of Lake's geometrical approach. We demonstrate that the parameters of the solutions in 5D are not free, as previously considered. Instead, they are totally determined by measurements in 4D. Namely, by the surface gravitational potential of the astrophysical phenomena, like the Sun or other stars, modeled in Kaluza-Klein theory. This is an important result which may help in observations for an experimental/observational test of the theory. 
  Gravastar models have recently been proposed as an alternative to black holes, mainly to avoid the problematic issues associated with event horizons and singularities. In this work, a wide variety of gravastar models within the context of nonlinear electrodynamics are constructed. Using the $F$ representation, specific forms of Lagrangians are considered describing magnetic gravastars, which may be interpreted as self-gravitating magnetic monopoles with charge $g$. Using the dual $P$ formulation of nonlinear electrodynamics, electric gravastar models are constructed by considering specific structural functions, and the characteristics and physical properties of the solutions are further explored. These interior nonlinear electrodynamic geometries are matched to an exterior Schwarzschild spacetime at a junction interface. 
  In this paper, we study the holographic dark energy model in non-flat universe from the statefinder viewpoint. We plot the evolutionary trajectories of the holographic dark energy model for different values of the parameter $c$ as well as for different contributions of spatial curvature, in the statefinder parameter-planes. The statefinder diagrams characterize the properties of the holographic dark energy and show the discrimination between this scenario and other dark energy models. As we show, the contributions of the spatial curvature in the model can be diagnosed out explicitly by the statefinder diagrams. Furthermore, we also investigate the holographic dark energy model in the $w-w'$ plane, which can provide us with a useful complement as dynamical diagnosis to the statefinder geometrical diagnosis. 
  Originally, Parikh and Wilczek's work is only suitable for the massless particles' tunneling. But their work has been further extended to the cases of massive uncharged and charged particles' tunneling recently. In this paper, as a particular black hole solution, we apply this extended method to reconsider the tunneling effect of the H.S Dilaton black hole. We investigate the behavior of both massive uncharged and charged particles, and respectively calculate the emission rate at the event horizon. Our result shows that their emission rates are also consistent with the unitary theory. Moreover, comparing with the case of massless particles' tunneling, we find that this conclusion is independent of the kind of particles. And it is probably caused by the underlying relationship between this method and the laws of black hole thermodynamics. 
  The bending angle of light is a central quantity in the theory of gravitational lensing. We develop an analytical perturbation framework for calculating the bending angle of light rays lensed by a Schwarzschild black hole. Using a perturbation parameter given in terms of the gravitational radius of the black hole and the light ray's impact parameter, we determine an invariant series for the strong-deflection bending angle that extends beyond the standard logarithmic deflection term used in the literature. In the process, we discover that the expression for the standard logarithmic deflection is not quite correct and provide the appropriate correction. Our perturbation framework is also used to derive as a consistency check, the recently found weak deflection bending angle series. We also reformulate the latter series in terms of a more natural invariant pertubation parameter, one that smoothly transitions between the weak and strong deflection series. We then compare our invariant strong deflection bending-angle series with the numerically integrated exact formal bending angle expression, and find less than 1% discrepancy for light rays as far out as twice the critical impact parameter. The paper concludes by showing that the strong and weak deflection bending angle series together provide an approximation that is within 1% of the exact bending angle value for light rays traversing anywhere between the photon sphere and infinity. 
  There is a statement on the parametrization dependence of the classical metric in the recent paper of N.E.J. Bjerrum-Bohr, J.F. Donoghue, B.R. Holstein, gr-qc/0610096. I completely disagree with this statement. Here I show reparametrization invariance of the classical metric. 
  This paper investigates the late-time behaviour of certain cosmological models where oscillations play an essential role. Rigorous results are proved on the asymptotics of homogeneous and isotropic spacetimes with a linear massive scalar field as source. Various generalizations are obtained for nonlinear massive scalar fields, $k$-essence models and $f(R)$ gravity. The effect of adding ordinary matter is discussed as is the case of nonlinear scalar fields whose potential has a degenerate zero. 
  Considering charged fluid spheres as anisotropic sources and the diffusion limit as the transport mechanism, we suppose that the inner space--time admits self--similarity. Matching the interior solution with the Reissner--Nordstr\"om--Vaidya exterior one, we find an extremely compact and oscillatory final state with a redistribution of the electric charge function and non zero pressure profiles. 
  In this letter we will revise the steps followed by A. Einstein when he first wrote on cosmology from the point of view of the general theory of relativity. We will argue that his insightful line of thought leading to the introduction of the cosmological constant in the equations of motion has only one weakness: The constancy of the cosmological term, or what is the same, its independence of the matter content of the universe. Eliminating this feature, I will propose what I see as a simple and reasonable modification of the cosmological equations of motion. The solutions of the new cosmological equations give place to a cosmological model that tries to approach the Einstein static solution. This model shows very appealing features in terms of fitting current observations. 
  In this review we present the theoretical background for treating General Relativity as an effective field theory and focus on the concrete results of such a treatment. As a result we present the calculations of the low-energy leading gravitational corrections to the Newtonian potential between two sources. 
  It is demonstrated that the Melvin universe representing the spacetime with a strong 'homogeneous' electric field can by obtained from the spacetime of two accelerated charged black holes by a suitable limiting procedure. The behavior of various invariantly defined geometrical quantities in this limit is also studied. 
  We study the G\"odel universe through worldlines associated with motion at constant speed and constant acceleration orthogonal to the instantaneous velocity (WSAs). We show that these worldlines can be used to access every region -- both spatial and temporal -- of the space-time. We capture the insights they accord in a series of sketches, which extend significantly the Hawking and Ellis picture of the G\"odel universe. 
  In this letter, we have considered a model of the universe filled with modified Chaplygin gas and another fluid (with barotropic equation of state) and its role in accelerating phase of the universe. We have assumed that the mixture of these two fluid models is valid from (i) the radiation era to $\Lambda$CDM for $-1\le\gamma\le 1$ and (ii) the radiation era to quiessence model for $\gamma<-1$. For these two fluid models, the statefinder parameters describe different phase of the evolution of the universe. 
  Existing capabilities in laser ranging, optical interferometry and metrology, in combination with precision frequency standards, atom-based quantum sensors, and drag-free technologies, are critical for the space-based tests of fundamental physics; as a result, of the recent progress in these disciplines, the entire area is poised for major advances. Thus, accurate ranging to the Moon and Mars will provide significant improvements in several gravity tests, namely the equivalence principle, geodetic precession, PPN parameters $\beta$ and $\gamma$, and possible variation of the gravitational constant $G$. Other tests will become possible with development of an optical architecture that would allow proceeding from meter to centimeter to millimeter range accuracies on interplanetary distances. Motivated by anticipated accuracy gains, we discuss the recent renaissance in lunar laser ranging and consider future relativistic gravity experiments with precision laser ranging over interplanetary distances. 
  In this paper I discuss the extension of the analogy between gravitation and some systems of condensed matter physics from kinematics to dynamics. I will focus my attention on two applications of the analogy to the dynamics of fluids that have been recently proposed: the study of backreaction effects and the calculation of the depletion in Bose-Einstein condensates, showing how this extension is possible and stressing the main differences with respect to the gravitational context. I will conclude with some remarks about the actual reliability of the proposed scheme, pointing out the basis issues that have still to be addressed. 
  We show how the expectation values of geometrical quantities in 3d quantum gravity can be explicitly computed using grasping rules. We compute the volume of a labelled tetrahedron using the triple grasping. We show that the large spin expansion of this value is dominated by the classical expression, and we study the next to leading order quantum corrections. 
  The theoretical framework established in arXiv:quant-ph/0404103 is extended to deal with possible astrophysical manifestations of phenomena involving reverse, as well as forward, causation in time. The basic idea is that space-time comprises the direct sum of a number of quantized fields, including a distinct physical vacuum for each space in the sum. It is presumed that these fields all contribute to, and are influenced by, gravitation, but generally do not interact electromagnetically, i.e., are mutually invisible. At least one term in the sum is proposed to consist of matter+vacuum evolving backward in time; the expectation values of energies, both of matter and vacuum, of this subspace are by construction negative, with a corresponding change of sign in classical gravitational and inertial masses. 
  Using the Post-Minkowskian formalism and considering rotation as a perturbation, we compute an approximate interior solution for a stationary perfect fluid with constant density and axial symmetry. A suitable change of coordinates allows this metric to be matched to the exterior metric to a particle with a pole-dipole-quadrupole structure, relating the parameters of both. 
  On his way to General Relativity (GR) Einstein gave several arguments as to why a special relativistic theory of gravity based on a massless scalar field could be ruled out merely on grounds of theoretical considerations. We re-investigate his two main arguments, which relate to energy conservation and some form of the principle of the universality of free fall. We find that such a theory-based a priori abandonment not to be justified. Rather, the theory seems formally perfectly viable, though in clear contradiction with (later) experiments. This may be of interest to those who teach GR and/or have an active interest in its history. 
  The familiar Bang/Crunch singularities of classical cosmology have recently been augmented by new varieties: rips, sudden singularities, and so on. These tend to be associated with final states. Here we consider an alternative possibility for the initial state: a singularity which has the novel property of being inaccessible to physically well-defined probes. These singularities arise naturally in cosmologies with toral spatial sections. 
  During the last few years, exact solutions that describe black holes that are bound to a two-brane in a four dimensional anti-de Sitter bulk have been constructed. In situations wherein there is a negative cosmological constant on the brane, for large masses, these solutions are exactly the rotating BTZ black holes on the brane and, in fact, describe rotating BTZ black strings in the bulk. We evaluate the canonical entropy of a free and massless scalar field (at the Hawking temperature) around the rotating BTZ black string using the brick wall model. We explicitly show that the Bekenstein-Hawking `area' law is satisfied both on the brane as well as in the bulk. 
  Through the consideration of spherically symmetric gravitating systems consisting of perfect fluids with linear equation of state constrained to be in a finite volume, an account is given of the properties of entropy at conditions in which it is no longer an extensive quantity. To accomplish this, the methods introduced by Oppenheim [1] to characterize non-extensivity are used, suitably generalized to the case of gravitating systems subject to an external pressure. In particular when, far from the system Schwarzschild limit, both area scaling for conventional entropy and inverse radius law for the temperature set in (i.e. the same properties of the corresponding black hole thermodynamical quantities), the entropy profile is found to behave like 1/r, being r the area radius inside the system. In such circumstances thus entropy heavily resides in internal layers, in opposition to what happens when area scaling is gained while approaching the Schwarzschild mass, in which case conventional entropy lies at the surface of the system. The information content of these systems, even if it globally scales like the area, is then stored in the whole volume, instead to be packed on the boundary. 
  In spite of alleged impossibility proofs, "simple derivations" of the Schwarzschild metric, based solely on Einstein's equivalence principle and Newton's free fall velocity formula, are presented. 
  The construction of ready to use templates for gravitational waves from spinning binaries is an important challenge in the investigation of detectable gravitational wave signals. Here we present a method to evaluate the gravitational wave polarization states for inspiralling compact binaries in the extreme mass ratio limit. We discuss the effects caused by the rotation of the central massive object for eccentric orbits in the Lense-Thirring approximation and give the formal expressions of the polarization states including higher order corrections. Our results are in agreement with existing calculations for the spinless and circular orbit limits. 
  The Friedman Universe with the remnant stochastic scalar field in the hybrid inflation model is examined. It is shown that the small effective cosmological constant appears which increases the cosmological expansion. 
  An instability in the presence of matter in theories of gravity which include a 1/R correction in the gravitational action has been found by Dolgov and Kawasaki. In the present paper this instability is discussed for f(R) gravity in general. We focus on the Palatini formalism of the theory and it is shown that no such instability occurs in this version of f(R) gravity. The reasons for the appearance of the instability in the metric but not in the Palatini formalism are fully investigated. 
  We prove that given a stress-free elastic body there exists, for sufficiently small values of the gravitational constant, a unique static solution of the Einstein equations coupled to the equations of relativistic elasticity. The solution constructed is a small deformation of the relaxed configuration. This result yields the first proof of existence of static solutions of the Einstein equations without symmetries. 
  We carry out a Lie group analysis of the Sachs equations for a time-dependent axisymmetric non-rotating space-time in which the Ricci tensor vanishes. These equations, which are the first two members of the set of Newman-Penrose equations, define the characteristic initial-value problem for the space-time. We find a particular form for the initial data such that these equations admit a Lie symmetry, and so defines a geometrically special class of such spacetimes. These should additionally be of particular physical interest because of this special geometric feature. 
  In the present work we study numerically quasi-equatorial lensing by the charged, stationary, axially-symmetric Kerr-Sen dilaton-axion black hole in the strong deflection limit. In this approximation we compute the magnification and the positions of the relativistic images. The most outstanding effect is that the Kerr-Sen black hole caustics drift away from the optical axis and shift in clockwise direction with respect to the Kerr caustics. The intersections of the critical curves on the equatorial plane as a function of the black hole angular momentum are found, and it is shown that they decrease with the increase of the parameter $Q^{2}/M$. All of the lensing quantities are compared to particular cases as Schwarzschild, Kerr and Gibbons-Maeda black holes. 
  Two different realizations of a symmetry principle that impose a zero cosmological constant in an extra-dimensional set-up are studied. The symmetry is identified by multiplication of the metric by minus one. In the first realization of the symmetry this is provided by a symmetry transformation that multiplies the coordinates by the imaginary number i. In the second realization this is accomplished by a symmetry transformation that multiplies the metric tensor by minus one. In both realizations of the symmetry the requirement of the invariance of the gravitational action under the symmetry selects out the dimensions given by D = 2(2n+1), n=0,1,2,... and forbids a bulk cosmological constant. Another attractive aspect of the symmetry is that it seems to be more promising for quantization when compared to the usual scale symmetry. The second realization of the symmetry is more attractive in that it is posible to make a possible brane cosmological constant zero in a simple way by using the same symmetry, and the symmetry may be identified by reflection symmetry in extra dimensions. 
  Within a perturbative cosmological regime of loop quantum gravity corrections to effective constraints are computed. This takes into account all inhomogeneous degrees of freedom relevant for scalar metric modes around flat space and results in explicit expressions for modified coefficients and of higher order terms. It also illustrates the role of different scales determining the relative magnitude of corrections. Our results demonstrate that loop quantum gravity has the correct classical limit, at least in its sector of cosmological perturbations around flat space, in the sense of perturbative effective theory. 
  The covariant gauge invariant perturbation theory of scalar cosmological perturbations is developed for a general Scalar-Tensor Friedmann-Lemaitre-Robertson-Walker cosmology in a vacuum. The perturbation equations are then solved exactly in the long wavelength limit for a specific coupling, potential and background. Differences with the minimally coupled case are briefly discussed. 
  We study the conditions required for validity of the generalized second law in phantom dominated universe in the presence of Schwarzschild black hole. Our study is independent of the origin of the phantom like behavior of the considered universe. We also discuss the generalized second law in the neighborhood of transition (from quintessence to phantom regime) time. We show that even for a constant equation of state parameter the generalized second law may be satisfied provided the temperature is not taken as de Sitter temperature. It is shown that in models with (only) a transition from quintessence to phantom regime the generalized second law does not hold in the transition epoch. 
  We prove a new global existence result for the asymptotically flat, spherically symmetric Einstein-Vlasov system which describes in the framework of general relativity an ensemble of particles which interact by gravity. The data are such that initially all the particles are moving radially outward and that this property can be bootstrapped. The resulting non-vacuum spacetime is future geodesically complete. 
  This work is an application of the second order gauge theory for the Lorentz group, where a description of the gravitational interaction is obtained which includes derivatives of the curvature. We analyze the form of the second field strenght, $G=\partial F+fAF$, in terms of geometrical variables. All possible independent Lagrangians constructed with quadratic contractions of $F$ and quadratic contractions of $G$ are analyzed. The equations of motion for a particular Lagrangian, which is analogous to Podolsky's term of his Generalized Electrodynamics, are calculated. 
  We analyse the instantaneous transition rate of an accelerated Unruh-DeWitt particle detector whose coupling to a quantum field on Minkowski space is regularised by a finite spatial profile. We show, under mild technical assumptions, that the zero size limit of the detector response is well defined, independent of the choice of the profile function, and given by a manifestly finite integral formula that no longer involves epsilon-regulators or limits. Applications to specific trajectories are discussed, recovering in particular the thermal result for uniform acceleration. Extensions of the model to de Sitter space are also considered. 
  We review the correspondence between symmetric affine theories and the nonlinear Einstein-Proca system that was found by Einstein and Schrodinger. With the use of this correspondence, we investigate static spherically symmetric solutions in symmetric affine theory with no matter fields. Use of the correspondence leads to a significant simplification of the calculation. A special instance of ``no-hair'' theorem for symmetric affine theory is established. 
  The Ruppeiner metric as determined by the Hessian of the Gibbs surface provides a geometric description of thermodynamic systems in equilibrium. An interesting example is a black hole in equilibrium with its own Hawking radiation. In this article, we present results from the Ruppeiner study of various black hole families from different gravity theories e.g. 2D dilaton gravity, BTZ, general relativity and higher-dimensional Einstein-Maxwell gravity. 
  In this paper we attempt to dynamically constrain the quadrupole mass moment Q of the millisecond PSR B1257+12 pulsar from the orbital periods of its three planets, assumed to be in equatorial and circular orbits. Given the present-day accuracy in knowing their ephemerides, no post-Newtonian corrections to their motions are required. By conservatively assuming 1% and 10% uncertainties in the pulsar's mass and planets' semimajor axes, respectively, our determination of the adimensional parameter q=c^4 Q/G^2 M^3 is q=(-0.90 +/- 67.45) 10^12, so that Q=(-1.08 +/- 80.22) 10^49 kg m^2. As an independent check of such a result, we apply the same method to the binary system composed by the millisecond PSR J1909-3744 pulsar and a white dwarf in circular orbit. We find for such a pulsar q=(-0.76 +/- 21.18) 10^12 and Q=(-0.98 +/- 27.26) 10^49 kg m^2, which are consistent with the estimates for PSR B1257+12. 
  In this work we use a recently developed nonintegrability theorem of Morales and Ramis to prove that the Friedmann Robertson Walker cosmological model with a conformally coupled massive scalar field is nonintegrable. 
  The dynamical system approach has recently acquired great importance in the investigation on higher order theories of gravity. In this talk I review the main results and I give brief comments on the perspectives for further developments. 
  We present a brief review on the Raychaudhuri equations. Beginning with a summary of the essential features of the original article by Raychaudhuri and subsequent work of numerous authors, we move on to a discussion of the equations in the context of alternate non--Riemannian spacetimes as well as other theories of gravity, with a special mention on the equations in spacetimes with torsion (Einstein--Cartan--Sciama--Kibble theory). Finally, we give an overview of some recent applications of these equations in General Relativity, Quantum Field Theory, String Theory and the theory of relativisitic membranes. We conclude with a summary and provide our own perspectives on directions of future research. 
  It is stated in many text books that the any metric appearing in general relativity should be locally Lorentzian i.e. of the type $\eta_\mn = {\rm diag} (1,-1,-1,-1)$ this is usually presented as an independent axiom of the theory, which can not be deduced from other assumptions. In this work we show that the above assertion is a consequence of a standard linear stability analysis of the Einstein \eqs and need not be assumed. 
  Some recent results obtained by the author and collaborators about QFT in asymptotically flat spacetimes at null infinity are summarized and reviewed. In particular it is focused on the physical properties of ground states in the bulk induced by the BMS-invariant state defined at null infinity. 
  In this letter we dynamically determine the quadrupole mass moment Q of the magnetic white dwarf WD 0137-349 by analyzing the period of the recently discovered brown dwarf moving around it in a close 2-hr orbit. It turns out that a purely Newtonian model for the orbit of WD 0137-349B, assumed circular and equatorial, is adequate, given the present-day accuracy in knowing the orbital parameters of such a binary system. Our result is Q=(-1.4615 +/- 0.9004) 10^47 kg m^2 for i=35 deg. It is able to accommodate the 3-sigma significant discrepancy of (1.0 +/- 0.3) 10^-8 s^-2 between the inverse square of the phenomenologically measured orbital period and the inverse square of the calculated Keplerian one. The impact of i, for which a range Delta i of possible values close to 35 deg is considered, is investigated as well; it amounts to Delta Q/Q\approx 83% for Delta i/i=11%. 
  We study spherically symmetric solutions in f(R) theories and its compatibility with local tests of gravity. We start by clarifying the range of validity of the weak field expansion and show that for many models proposed to address the Dark Energy problem this expansion breaks down in realistic situations. This invalidates the conclusions of several papers that make inappropriate use of this expansion. For the stable models that modify gravity only at small curvatures we find that when the asymptotic background curvature is large we approximately recover the solutions of Einstein gravity through the so-called Chameleon mechanism, as a result of the non-linear dynamics of the extra scalar degree of freedom contained in the metric. In these models one would observe a transition from Einstein to scalar-tensor gravity as the Universe expands and the background curvature diminishes. Assuming an adiabatic evolution we estimate the redshift at which this transition would take place for a source with given mass and radius. We also show that models of dynamical Dark Energy claimed to be compatible with tests of gravity because the mass of the scalar is large in vacuum (e.g. those that also include R^2 corrections in the action), are not viable. 
  Approximations are commonly employed to find approximate solutions to the Einstein equations. These solutions, however, are usually only valid in some specific spacetime region. A global solution can be constructed by gluing approximate solutions together, but this procedure is difficult because discontinuities can arise, leading to large violations of the Einstein equations. In this paper, we provide an attempt to formalize this gluing scheme by studying transition functions that join approximate solutions together. In particular, we propose certain sufficient conditions on these functions and proof that these conditions guarantee that the joined solution still satisfies the Einstein equations to the same order as the approximate ones. An example is also provided for a binary system of non-spinning black holes, where the approximate solutions are taken to be given by a post-Newtonian expansion and a perturbed Schwarzschild solution. For this specific case, we show that if the transition functions satisfy the proposed conditions, the joined solution does not contain any violations to the Einstein equations larger than those already inherent in the approximations. We further show that if these functions violate the proposed conditions, then the matter content of the spacetime is modified by the introduction of a matter-shell, whose stress-energy tensor depends on derivatives of these functions. 
  These notes provide a self-contained introduction to the derivation of the zero, first and second laws of black hole mechanics. The prerequisite conservation laws in gauge and gravity theories are also briefly discussed. An explicit derivation of the first law in general relativity is performed in appendix. 
  We investigate the thermodynamics of the noncommutative black hole whose static picture is similar to that of the nonsingular black hole known as the de Sitter-Schwarzschild black hole. It turns out that the final remnant of extremal black hole is a thermodynamically stable object. We describe the evaporation process of this black hole by using the noncommutativity-corrected Vaidya metric. It is found that there exists a close relationship between thermodynamic approach and evaporation process. 
  We have developed an action formulation for the Generalized Chaplygin Gas  (GCG). The most general form for the nonrelativistic GCG action is derived consistent with the equation of state. We have also discussed a relativistic formulation for GCG by providing a detailed analysis of the Poincare algebra. 
  We study the interior spacetimes of stars in the Palatini formalism of f(R) gravity and derive a generalized Tolman-Oppenheimer-Volkoff and mass equation for a static, spherically symmetric star. We show that matching the interior solution with the exterior Schwarzschild-De Sitter solution in general gives a relation between the gravitational mass and the density profile of a star, which is different from the one in General Relativity. These modifications become neglible in models for which $\delta f(R) \equiv (R)-R$ is a decreasing function of R however. As a result, both Solar System constraints and stellar dynamics are perfectly consistent with $f(R) = R - \mu^4/R$. 
  It has been suggested that the cosmological constant is a variable dynamical quantity. A class of solution has been presented for the spherically symmetric space time describing wormholes by assuming the erstwhile cosmological constant $\Lambda$ to be a space variable scalar, viz., $\Lambda$ = $\Lambda (r) $ . It is shown that the Averaged Null Energy Condition (ANEC) violating exotic matter can be made arbitrarily small. 
  Using 'Cut and Paste' technique, we develop a thin shell wormhole in heterotic string theory. We determine the surface stresses, which are localized in the shell, by using Darmois-Israel formalism. The linearized stability of this thin wormhole is also analyzed. 
  We use techniques of quantum information theory to analyze the quantum causal histories approach to quantum gravity. We show that while it is consistent to introduce closed timelike curves (CTCs), they cannot generically carry independent degrees of freedom. Moreover, if the effective dynamics of the chronology-respecting part of the system is linear, it should be completely decoupled from the CTCs. In the absence of a CTC not all causal structures admit the introduction of quantum mechanics. It is possible for those and only for those causal structures that can be represented as quantum computational networks. The dynamics of the subsystems should not be unitary or even completely positive. However, we show that other commonly maid assumptions ensure the complete positivity of the reduced dynamics. 
  We study a method to solve stationary axisymmetric vacuum Einstein equations numerically. As an illustration, the five-dimensional doubly spinning black rings that have two independent angular momenta are formulated in a way suitable for fully nonlinear numerical method. Expanding for small second angular velocity, the formulation is solved perturbatively upto second order involving the backreaction from the second spin. The obtained solutions are regular without conical singularity, and the physical properties are discussed with the phase diagram of the reduced entropy vs the reduced angular momenta. Possible extensions of the present approach to constructing the higher dimensional version of black ring and the ring with the cosmological constant are also discussed. 
  By using a massless scalar field we examine the effect of an extra dimension on black hole radiation. Because the equations are coupled, we find that the structure of the fifth dimension (as for membrane and induced-matter theory) affects the nature of the radiation observed in four-dimensional spacetime. In the case of the Schwarzschild-de Sitter solution embedded in a Randall-Sundrum brane model, the extension of the black hole along the fifth dimension looks like a black string. Then it is shown that, on the brane, the potential barrier surrounding the black hole has a quantized as well as a continuous spectrum. In principle, Hawking radiation may thus provide a probe for higher dimensions. 
  The classical definition of {\em global hyperbolicity} for a spacetime $(M,g)$ comprises two conditions: (A) compactness of the diamonds $J^+(p)\cap J^-(q)$, and (B) strong causality. Here we show that condition (B) can be replaced just by causality. In fact, we show first that the classical definition of causal simplicity (which impose to be distinguishing, apart from the closedness of $J^+(p)$, $J^-(q)$) can be weakened in causal instead of distinguishing. So, the full consistency of the causal ladder (recently proved by the authors in a definitive way) yields directly the result. 
  We analyze the quasinormal modes of $D$-dimensional Schwarzschild black holes with the Gauss-Bonnet correction in the large damping limit and show that standard analytic techniques cannot be applied in a straightforward manner to the case of infinite damping. However, by using a combination of analytic and numeric techniques we are able to calculate the quasinormal mode frequencies in a range where the damping is large but finite. We show that for this damping region the famous $\ln(3)$ appears in the real part of the quasinormal mode frequency. In our calculations, the Gauss-Bonnet coupling, $\alpha$, is taken to be much smaller than the parameter $\mu$, which is related to the black hole mass. 
  We investigate matter collineations of plane symmetric spacetimes when the energy-momentum tensor is degenerate. There exists three interesting cases where the group of matter collineations is finite-dimensional. The matter collineations in these cases are either four, six or ten in which four are isometries and the rest are proper. 
  We give an introduction to the canonical formalism of Einstein's theory of general relativity. This then serves as the starting point for one approach to quantum gravity called quantum geometrodynamics. The main features and applications of this approach are briefly summarized. 
  The equations of motion of compact binary systems have been derived in the post-Newtonian (PN) approximation of general relativity. The current level of accuracy is 3.5PN order. The conservative part of the equations of motion (neglecting the radiation reaction damping terms) is deducible from a generalized Lagrangian in harmonic coordinates, or equivalently from an ordinary Hamiltonian in ADM coordinates. As an application we investigate the problem of the dynamical stability of circular binary orbits against gravitational perturbations up to the 3PN order. We find that there is no innermost stable circular orbit or ISCO at the 3PN order for equal masses. 
  We consider static circularly symmetric solution of three-dimensional Einstein's equations with negative cosmological constant (the BTZ black hole). The case of zero cosmological constant corresponding to the interior region of a black hole is analyzed in detail. We prove that the maximally extended BTZ solution with zero cosmological constant coincides with flat three-dimensional Minkowskian space-time without any singularity and horizons. The Euclidean version of this solution is shown to have physical interpretation in the geometric theory of defects in solids describing combined wedge and screw dislocations. 
  We consider the problem of the motion of $N$ bodies in a self-gravitating system. We point out that this system can be mapped onto the quantum-mechanical problem of an N-body generalization of the problem of the $H$_{2}^{+}$\ molecular ion in one dimension. We derive a general algorithm for solving this problem, and show how it reduces to known results for the 2-body and 3-body systems. 
  After describing in short some problems and methods regarding the smoothness of null infinity for isolated systems, I present numerical calculations in which both spatial and null infinity can be studied. The reduced conformal field equations based on the conformal Gauss gauge allow us in spherical symmetry to calculate numerically the entire Schwarzschild-Kruskal spacetime in a smooth way including spacelike, null and timelike infinity and the domain close to the singularity. 
  We present the quasinormal frequencies of the massive scalar field in the background of a Schwarzchild black hole surrounded by quintessence with the third-order WKB method. The mass of the scalar field $u$ plays an important role in studying the quasinormal frequencies, the real part of the frequencies increases linearly as mass $u$ increases, while the imaginary part in absolute value decreases linearly which leads to damping more slowly and the frequencies having a limited value. Moreover, owing to the presence of the quintessence, the massive scalar field damps more slowly. 
  We review the matching conditions for a collapsing anisotropic cylindrical perfect fluid, recently discussed in the literature (2005 {\it Class. Quantum Grav.} {\bf 22} 2407). It is shown that radial pressure vanishes on the surface of the cylinder, contrary to what is asserted in that reference. The origin of this discrepancy is to be found in a mistake made in one step of the calculations. Some comments about the relevance of this result in relation to the momentum of Einstein--Rosen waves are presented. 
  Unitarity is a pillar of quantum theory. Nevertheless, it is also a source of several of its conceptual problems. We note that in a world where measurements are relational, as is the case in gravitation, quantum mechanics exhibits a fundamental level of loss of coherence. This can be the key to solving, among others, the puzzles posed by the black hole information paradox, the formation of inhomogeneities in cosmology and the measurement problem in quantum mechanics. 
  We derive the equilibrium hydrostatic equation of a spherical star for any gravitational Lagrangian density of the form $L=\sqrt{-g}f(R)$. The Palatini variational principle for the Helmholtz Lagrangian in the Einstein gauge is used to obtain the field equations in this gauge. The equilibrium hydrostatic equation is obtained and is used to study the Newtonian limit for $f(R)=R-\frac{a^{2}}{3R}$. The same procedure is carried out for the more generally case $f(R)=R-\frac{1}{n+2}\frac{a^{n+1}}{R^{n}}$ giving a good Newtonian limit. 
  We explore the possibility that traversable wormholes be supported by specific equations of state responsible for the present accelerated expansion of the Universe, namely, phantom energy, the generalized Chaplygin gas, and the van der Waals quintessence equation of state. 
  The Casimir stress on two parallel plates in a de Sitter background corresponding to different metric signatures and cosmological constants is calculated for massless scalar fields satisfying Robin boundary conditions on the plates. Our calculation shows that for the parallel plates with false vacuum between and true vacuum outside, the total Casimir pressure leads to an attraction of the plates at very early universe. 
  Forces defined in the framework of optical reference geometry are introduced in the case of stationary and axially symmetric Kerr black-hole and naked-singularity spacetimes with a repulsive cosmological constant. Properties of the forces acting on test particles moving along circular orbits in the equatorial plane are discussed, whereas it is shown where the gravitational force vanishes and changes its orientation and where the centrifugal force vanishes and changes its orientation independently of the velocity of test particles related to the optical geometry; the Coriolis force does not vanish for the velocity being non-zero. The spacetimes are classified according to the number of circular orbits where the gravitational and centrifugal forces vanish. 
  Equilibrium conditions and spin dynamics of spinning test particles are discussed in the stationary and axially symmetric Kerr-de Sitter black-hole or naked-singularity spacetimes. The general equilibrium conditions are established, but due to their great complexity, the detailed discussion of the equilibrium conditions and spin dynamics is presented only in the simple and most relevant cases of equilibrium positions in the equatorial plane and on the symmetry axis of the spacetimes. It is shown that due to the combined effect of the rotation of the source and the cosmic repulsion the equilibrium is spin dependent in contrast to the spherically symmetric spacetimes. In the equatorial plane, it is possible at the so-called static radius, where the gravitational attraction is balanced by the cosmic repulsion, for the spinless particles as well as for spinning particles with arbitrarily large azimuthal-oriented spin or at any radius outside the ergosphere with a specifically given spin orthogonal to the equatorial plane. On the symmetry axis, the equilibrium is possible at any radius in the stationary region and is given by an appropriately tuned spin directed along the axis. At the static radii on the axis the spin of particles in equilibrium must vanish. 
  The geometric content of the MacDowell-Mansouri formulation of general relativity is best understood in terms of Cartan geometry. In particular, Cartan geometry gives clear geometric meaning to the MacDowell-Mansouri trick of combining the Levi-Civita connection and coframe field, or soldering form, into a single physical field. The Cartan perspective allows us to view physical spacetime as tangentially approximated by an arbitrary homogeneous "model spacetime", including not only the flat Minkowski model, as is implicitly used in standard general relativity, but also de Sitter, anti de Sitter, or other models. A "Cartan connection" gives a prescription for parallel transport from one "tangent model spacetime" to another, along any path, giving a natural interpretation of the MacDowell-Mansouri connection as "rolling" the model spacetime along physical spacetime. I explain Cartan geometry, and "Cartan gauge theory", in which the gauge field is replaced by a Cartan connection. In particular, I discuss MacDowell-Mansouri gravity, as well as its recent reformulation in terms of BF theory, in the context of Cartan geometry. 
  We have investigated the thermodynamical properties of the dark energy. Assuming that the dark energy temperature $T\sim a^{-n}$ and considering that the volume of the universe enveloped by the apparent horizon relates to the temperature, we have derived the dark energy entropy. For the dark energy with constant equation of state $w>-1$ and the generalized Chaplygin gas, the derived entropy can be positive and satisfy the entropy bound. The total entropy, including those of the dark energy, the thermal radiation and the apparent horizon, satisfies the generalized second law of thermodynamics. However, for the phantom with constant equation of state, the positivity of entropy, the entropy bound and the generalized second law cannot be satisfied simultaneously. 
  We explicitly construct and characterize all possible independent loop states in 3+1 dimensional loop quantum gravity by regulating it on a 3-d regular lattice in the Hamiltonian formalism. These loop states, characterized by the (dual) angular momentum quantum numbers, describe SU(2) rigid rotators on the links of the lattice. The loop states are constructed using the Schwinger bosons which are harmonic oscillators in the fundamental (spin half) representation of SU(2). Using generalized Wigner Eckart theorem, we compute the matrix elements of the volume operator in the loop basis. Some simple loop eigenstates of the volume operator are explicitly constructed. 
  We reconsider the consistency constraints on a free massless symmetric, rank 2, tensor field in a background and confirm that they uniquely require it to be the linear deviation about Einstein gravity. Neither adding non-minimal higher derivative terms nor changing the gauge transformations by allowing terms non-analytic in the cosmological constant alters this fact. 
  The way one chooses to couple gravity to matter is an essential characteristic of any gravitational theory. In theories where the gravitational field is allowed to have more degrees of freedom than those of General Relativity (e.g. scalar-tensor theory, f(R) gravity) this issue often becomes even more important. We concentrate here on f(R) gravity treated within the Palatini variational principle and discuss how the coupling between matter and the extra degrees of freedom of gravity (the independent connections in our case) affects not only the resulting phenomenology but even the geometrical meaning of fundamental fields. 
  It is well-known that Bowen-York initial data contain spurious radiation. Although this ``junk'' radiation has been seen to be small for non-spinning black-hole binaries in circular orbit, its magnitude increases when the black holes are given spin. It is possible to reduce the spurious radiation by applying the puncture approach to multiple Kerr black holes, as we demonstrate for examples of head-on collisions of equal-mass black-hole binaries. 
  We study modified theories of gravity of the f(R) type in Palatini formalism. For a generic f(R) lagrangian, we show that the metric can be solved as the product of a scalar function times a rank-two tensor (or auxiliary metric). The scalar function is sensitive to the local energy-momentum density. The auxiliary metric satisfies a set of equations very similar to Einstein's equations and, for weak sources, it can be approximated by the Minkowski metric. According to this, the metric coupled to the matter strongly departs from the Minkowskian one in the neighbourhood of any microscopic physical system. As a consequence, new gravitationally-induced interactions arise and lead to observable effects at microscopic and macroscopic scales. In particular, test body trajectories experience self-accelerations which depend on the internal structure and composition of the body. These facts make very unlikely the viability of Palatini f(R) models designed to change the late-time cosmic evolution. 
  Chaos in the orbits of black hole pairs has by now been confirmed by several independent groups. While the chaotic behavior of binary black hole orbits is no longer argued, it remains difficult to quantify the importance of chaos to the evolutionary dynamics of a pair of comparable mass black holes. None of our existing approximations are robust enough to offer convincing quantitative conclusions in the most highly nonlinear regime. It is intriguing to note that in three different approximations to a black hole pair built of a spinning black hole and a non-spinning companion, two approximations exhibit chaos and one approximation does not. The fully relativistic scenario of a spinning test-mass around a Schwarzschild black hole shows chaos, as does the Post-Newtonian Lagrangian approximation. However, the approximately equivalent Post-Newtonian Hamiltonian approximation does not show chaos when only one body spins. It is well known in dynamical systems theory that one system can be regular while an approximately related system is chaotic, so there is no formal conflict. However,the physical question remains, Is there chaos for comparable mass binaries when only one object spins? We are unable to answer this question given the poor convergence of the Post-Newtonian approximation to the fully relativistic system. A resolution awaits better approximations that can be trusted in the highly nonlinear regime. 
  In analogy with classical electromagnetic theory, where one determines the total charge and both electric and magnetic multipole moments of a source from certain surface integrals of the asymptotic (or far) fields, it has been known for many years - from the work of Hermann Bondi - that energy and momentum of gravitational sources could be determined by similar integrals of the asymptotic Weyl tensor. Recently we observed that there were certain overlooked structures, {defined at future null infinity,} that allowed one to determine (or define) further properties of both electromagnetic and gravitating sources. These structures, families of {complex} `slices' or `cuts' of Penrose's null infinity, are referred to as Universal Cut Functions, (UCF). In particular, one can define from these structures a (complex) center of mass (and center of charge) and its equations of motion - with rather surprising consequences. It appears as if these asymptotic structures contain in their imaginary part, a well defined total spin-angular momentum of the source. We apply these ideas to the type II algebraically special metrics, both twisting and twist-free. 
  There is something missing in our understanding of the origin of the seeds of Cosmic Structuture.   The fact that the fluctuation spectrum can be extracted from the inflationary scenario through an analysis that involves quantum field theory in curved space-time, and that it coincides with the observational data has lead to a certain complacency in the community, which prevents the critical analysis of the obscure spots in the derivation. The point is that the inhomogeneity and anisotropy of our universe seem to emerge from an exactly homogeneous and isotropic initial state through processes that do not break those symmetries. This article gives a brief recount of the problems faced by the arguments based on established physics, which comprise the point of view held by a large majority of researchers in the field.   The conclusion is that we need some new physics to be able to fully address the problem. The article then exposes one avenue that has been used to address the central issue and elaborates on the degree to which, the new approach makes different predictions from the standard analyses.   The approach is inspired on Penrose's proposals that Quantum Gravity might lead to a real, dynamical collapse of the wave function, a process that we argue has the properties needed to extract us from the theoretical impasse described above. 
  The Xi-transform is a new spinor transform arising naturally in Einstein's general relativity. Here the example of conformally flat space-time is discussed in detail. In particular it is shown that for this case, the transform coincides with two other naturally defined transforms: one a two-variable transform on the Lie group SU(2, C), the other a transform on the space of null split octaves. The key properties of the transform are developed. 
  We express Einstein's field equations for a spherically symmetric ball of general fluid such that they are conducive to an initial value problem. We show how the equations reduce to the Vaidya spacetime in a non-null coordinate frame, simply by designating specific equations of state. Furthermore, this reduces to the Schwarzschild spacetime when all matter variables vanish. We then describe the formulation of an initial value problem, whereby a general fluid ball with vacuum exterior is established on an initial spacelike slice. As the system evolves, the fluid ball collapses and emanates null radiation such that a region of Vaidya spacetime develops. Therefore, on any subsequent spacelike slice there exists three regions; general fluid, Vaidya and Schwarzschild, all expressed in a single coordinate patch with two free-boundaries determined by the equations. This implies complicated matching schemes are not required at the interfaces between the regions, instead, one simply requires the matter variables tend to the appropriate equations of state. We also show the reduction of the system of equations to the static cases, and show staticity necessarily implies zero ``heat flux''. Furthermore, the static equations include a generalization of the Tolman-Oppenheimer-Volkoff equations for hydrostatic equilibrium to include anisotropic stresses in general coordinates. 
  An evolution of radiant shock wave front is considered in the framework of a recently presented method to study self-gravitating relativistic spheres, whose rationale becomes intelligible and finds full justification within the context of a suitable definition of the post-quasistatic approximation. The spherical matter configuration is divided into two regions by the shock and each side of the interface having a different equation of state and anisotropic phase. In order to simulate dissipation effects due to the transfer of photons and/or neutrinos within the matter configuration, we introduce the flux factor, the variable Eddington factor and a closure relation between them. As we expected the strength of the shock increases the speed of the fluid to relativistic values and for some critical ones is larger than light speed. In addition, we find that energy conditions are very sensible to the anisotropy, specially the strong one. As a special feature of the model, we find that the contribution of the matter and radiation to the radial pressure are the same order of magnitude as in the mant as in the core, moreover, in the core radiation pressure is larger than matter pressure. 
  In this paper, the quasinormal modes of gravitational perturbation around a Schwarzschild black hole surrounded by quintessence were evaluated by using the third-order WKB approximation. Due to the presence of quintessence, the gravitational wave damps more slowly. 
  We evaluated the quasi-normal modes of electromagnetic perturbation in a Schwarzschild black hole surrounded by quintessence by using the third-order WKB approximation. Due to the presence of quintessence, Maxwellfield damps more slowly. 
  Quasi-stationary (i.e. parametric) transitions from rotating equilibrium configurations of fluid bodies to rotating black holes are discussed. For the idealized model of a rotating disc of dust, analytical results derived by means of the "inverse scattering method" are available. They are generalized by numerical results for rotating fluid rings with various equations of state. It can be shown rigorously that a black hole limit of a fluid body in equilibrium occurs if and only if the gravitational mass becomes equal to twice the product of angular velocity and angular momentum. Therefore, any quasi-stationary route from fluid bodies to black holes passes through the extreme Kerr solution. 
  We show that stationary, asymptotically flat solutions of the electro-vacuum Einstein equations are analytic at $i^0$, for a large family of gauges, in odd space-time dimensions higher than seven. The same is true in space-time dimension five for static vacuum solutions with non-vanishing mass. 
  We discuss the gravitational wave background generated by primordial density perturbations evolving during the radiation era. At second-order in a perturbative expansion, density fluctuations produce gravitational waves. We calculate the power spectra of gravitational waves from this mechanism, and show that, in principle, future gravitational wave detectors could be used to constrain the primordial power spectrum on scales vastly different from those currently being probed by large-scale structure. As examples we compute the gravitational wave background generated by both a power-law spectrum on all scales, and a delta-function power spectrum on a single scale. 
  We give a causal version of Eisenhart's geodesic characterization of classical mechanics. We emphasize the geometric, coordinate independent properties needed to express Eisenhart's theorem in light of modern studies on the Bargmann structures (lightlike dimensional reduction). The construction of the space metric, Coriolis 1-form and scalar potential through which the theorem is formulated is shown in detail, and in particular it is proved a one-to-one correspondence between Newtonian frames and Abelian connections on suitable lightlike principal bundles. The relation of Eisenhart's theorem in the lightlike case with a Fermat type principle is pointed out. The operation of lightlike lift is introduced and the existence of minimizers for the classical action is related to the causal simplicity of Eisenhart's spacetime. 
  There is a deep structural link between acausal spacetimes and quantum theory. As a consequence quantum theory may resolve some "paradoxes" of time travel. Conversely, non-time-orientable spacetimes naturally give rise to electric charges and spin half. If an explanation of quantum theory is possible, then general relativity with time travel could be it. 
  An axially symmetric scalar field is considered in the teleparallel gravity. We calculate, respectively, the tensor, the vector and the axial-vector parts of torsion and energy, momentum and angular momentum in the ASSF. We find the vector parts are in the radial and $\hat{e}_{\theta}$ directions, the axial-vector, momentum and angular momentum vanish identically, but the energy distribution is different from zero. The vanishing axial-vector part of torsion gives us that there occurs no deviation in the spherical symmetry of the spacetime. Consequently, there exists no inertia field with respect to Dirac particle and the spin vector of a Dirac particle becomes constant. The result for the energy is the same as obtained by Radinschi. Next, the work also (a) supports the viewpoint of Lessner that the M{\o}ller energy-momentum complex is a powerful concept for the energy-momentum, (b) sustains the importance of the energy-momentum definitions in the evaluation of the energy distribution of a given space-time, and (c) supports the hypothesis by Cooperstock that the energy is confined to the region of non-vanishing energy-momentum tensor of matter and all non-gravitational fields. 
  First post-Newtonian (PN) approximation of the scalar-tensor theory of gravity is used to discuss the effacing principle in N-body system, that is dependence of equations of motion of spherically-symmetric bodies comprising the system on their internal structure. We demonstrate that the effacing principle is violated by terms which are proportional to the second order rotational moment of inertia of each body coupled with \beta-1, where \beta is the measure of non-linearity of gravitational field. In case of general relativity, where \beta=1, the effacing principle is violated by terms being proportional to the rotational moment of inertia of the forth order. For systems made of neutron stars (NS) and/or black holes (BH) these terms contribute to the orbital equations of motion at the level of the third and fifth PN approximation respectively. 
  The universality of semiclassical gravity is investigated by considering the behavior of the quantities < \phi^2 > and < {T^a}_b >, along with quantum corrections to the effective Newtonian potential in the far field limits of static spherically symmetric objects ranging from stars in the weak field Newtonian limit to black holes. For scalar fields it is shown that when differences occur they all result from the behavior of a single mode with zero frequency and angular momentum and are thus due to a combination of infrared and s-wave effects. An intriguing combination of similarities and differences between the extreme cases of a Schwarzschild black hole and a star in the weak field Newtonian limit is explained. 
  We discuss new exact spherically symmetric static solutions to non-minimally extended Einstein-Yang-Mills equations. The obtained solution to the Yang-Mills subsystem is interpreted as a non-minimal Wu-Yang monopole solution. We focus on the analysis of two classes of the exact solutions to the gravitational field equations. Solutions of the first class belong to the Reissner-Nordstr{\"o}m type, i.e., they are characterized by horizons and by the singularity at the point of origin. The solutions of the second class are regular ones. The horizons and singularities of a new type, the non-minimal ones, are indicated. 
  A self-consistent non-minimal non-Abelian Einstein-Yang-Mills model, containing three phenomenological coupling constants, is formulated. The ansatz of a vanishing Yang-Mills induction is considered as a particular case of the self-duality requirement for the gauge field. Such an ansatz is shown to allow obtaining an exact solution of the self-consistent set of equations when the space-time has a constant curvature. An example describing a pure magnetic gauge field in the de Sitter cosmological model is discussed in detail. 
  Introducing his Chronology Protection Conjecture Stephen Hawking said that it seems that there exists a Chronology Protection Agency making the Universe safe for historians.   Without taking sides about such a conjecture we show that the Chronology Protection Agency is not necessary in order to make the Universe unsafe for historians but safe for logicians. 
  In this paper we dynamically determine the quadrupole mass moment Q of the two-pulsars system PSR J0737-3039A/B by analyzing the orbital period of the relative motion occurring along a close 2.4-hr, elliptic orbit.   By using the timing measurement of sin i, we obtain Q=(-7.7 +/- 3.9) 10^45 kg m^2. The major source of systematic error is the uncertainty in the semimajor axis a mainly due, in turn, to the error in sin i. Our result is capable to accommodate the observed discrepancy \Delta P = -25.894452 +/- 13.153928 s between the phenomenologically measured orbital period P_b and the purely Keplerian period P^(0)=2\pi\sqrt{a^3/G(m_A + m_B)} calculated with the system's parameters which have been determined independently of the third Kepler law. If the value for i obtained from scintillation measurements is used, we get Q=(-6.7 +/- 2.9) 10^45 kg m^2, which is compatible with the timing-based result. The discrepancy Delta P amounts, in this case, to -22.584893 +/- 9.960784 s, i.e. it is incompatible with zero at 2.3 sigma level. 
  We show that the second accelerating expansion of the universe appears smoothly from the decelerating universe remarkably after the initial inflation in the two-dimensional soluble semi-classical dilaton gravity along with the modified Poisson brackets of noncommutativity between the relevant fields. However, the ordinary solution coming from the equations of motion following the conventional Poisson algebra describes permanent accelerating universe without any phase change. In this modified model, it turns out that the phase transition is related to the noncommutative Poisson algebra. 
  General relativity predicts the gravitational radiation signatures of mergers of compact binaries, such as coalescing binary black hole systems. Derivations of waveform predictions for such systems are required for optimal scientific analysis of observational gravitational wave data, and have so far been achieved primarily with the aid of the post-Newtonian (PN) approximation. The quality of this treatment is unclear, however, for the important late inspiral portion. We derive late-inspiral waveforms via a complementary approach, direct numerical simulation of Einstein's equations, which has recently matured sufficiently for such applications. We compare waveform phasing from simulations covering the last $\sim 14$ cycles of gravitational radiation from an equal-mass binary system of non-spinning black holes with the corresponding 3PN and 3.5PN orbital phasing. We find agreement consistent with internal error estimates based on either approach at the level of one radian over $\sim 10$ cycles. The result suggests that PN waveforms for this system are effective roughly until the system reaches its last stable orbit just prior to the final merger. 
  A Newtonian approach to quantum gravity is studied. At least for weak gravitational fields it should be a valid approximation. Such an approach could be used to point out problems and prospects inherent in a more exact theory of quantum gravity, yet to be discovered. Newtonian quantum gravity, e.g., shows promise for prohibiting black holes altogether (which would eliminate singularities and also solve the black hole information paradox), breaks the equivalence principle of general relativity, and supports non-local interactions (quantum entanglement). Its predictions should also be testable at length scales well above the "Planck scale", by high-precision experiments feasible even with existing technology. As an illustration of the theory, it turns out that the solar system, superficially, perfectly well can be described as a quantum gravitational system, provided that the $l$ quantum number has its maximum value, $n-1$. This results exactly in Kepler's third law. If also the $m$ quantum number has its maximum value ($\pm l$) the probability density has a very narrow torus-like form, centered around the classical planetary orbits. However, as the probability density is independent of the azimuthal angle $\phi$ there is, from quantum gravity arguments, no reason for planets to be located in any unique place along the orbit (or even \textit{in} an orbit for $m \neq \pm l$). This is, in essence, a reflection of the ``measurement problem" inherent in all quantum descriptions. 
  We consider a massless, minimally coupled scalar with a quartic self-interaction which is released in Bunch-Davies vacuum in locally de Sitter background of an inflating universe. It was shown, in this system, that quantum effects can induce a temporary phase of super-acceleration causing a violation of the Weak Energy Condition on cosmological scales. In this paper we investigate the system's stability by studying the behavior of linearized perturbations in the quantum-corrected effective field equation at one and two-loop order. We show that the time dependence we infer from the quantum-corrected mode function is in perfect agreement with the system developing a positive mass squared. The maximum induced mass remains perturbatively small and it does not go tachyonic. Thus, the system is stable. 
  We investigate the clustering properties of a dynamical dark energy component. In a cosmic mix of a pressureless fluid and a light scalar field, we follow the linear evolution of spherical matter perturbations. We find that the scalar field tends to form underdensities in response to the gravitationally collapsing matter. We thoroughly investigate these voids for a variety of initial conditions, explain the physics behind their formation and consider possible observational implications. Detection of dark energy voids will clearly rule out the cosmological constant as the main source of the present acceleration. 
  A collection of requirements to the General Relativity that follow from the WMAP observations of the Cosmic Microwave Background radiation anisotropy as an inertial frame are discussed. These obligations include the separation of both the CMB frame from the diffeomorphisms and the diffeo-invariant cosmic evolution from the local scalar metric component in the manner compatible with the canonical Hamiltonian approach to the Einstein--Hilbert theory with the energy constraints. The solution of these constraints in classical and quantum theories and a fit of units of measurements are discussed in the light of the last Supernovae data. 
  Inspirals of stellar-mass compact objects into $\sim 10^6 M_{\odot}$ black holes are especially interesting sources of gravitational waves for LISA. We investigate whether the emitted waveforms can be used to strongly constrain the geometry of the central massive object, and in essence check that it corresponds to a Kerr black hole (BH). For a Kerr BH, all multipole moments of the spacetime have a simple, unique relation to $M$ and $S$, the BH's mass and spin; in particular, the spacetime's mass quadrupole moment is given by $Q=- S^2/M$. Here we treat $Q$ as an additional parameter, independent of $M$ and $S$, and ask how well observation can constrain its difference from the Kerr value. This was already estimated by Ryan, but for simplified (circular, equatorial) orbits, and neglecting signal modulations due to the motion of the LISA satellites. Here we consider generic orbits and include these modulations. We use a family of approximate (post-Newtonian) waveforms, which represent the full parameter space of Inspiral sources, and exhibit the main qualitative features of true, general relativistic waveforms. We extend this parameter space to include (in an approximate manner) an arbitrary value of $Q$, and construct the Fisher information matrix for the extended parameter space. By inverting the Fisher matrix we estimate how accurately $Q$ could be extracted from LISA observations. For 1 year of coherent data from the inspiral of a $10 M_{\odot}$ BH into rotating BHs of masses $10^{5.5} M_{\odot}$, $10^6 M_{\odot}$, or $10^{6.5} M_{\odot}$, we find $\Delta (Q/M^3) \sim 10^{-4}$, $10^{-3}$, or $10^{-2}$, respectively (assuming total signal-to-noise ratio of 100, typical of the brightest detectable EMRIs). These results depend only weakly on the eccentricity of the orbit or the BH's spin. 
  In this work, a generalization of the Mazur-Mottola gravastar model is explored, by considering a matching of an interior solution governed by the dark energy equation of state, $\omega\equiv p/ \rho<-1/3$, to an exterior Schwarzschild vacuum solution at a junction interface, situated near to where the event horizon is expected to form. The motivation for implementing this generalization arises from the fact that recent observations have confirmed an accelerated cosmic expansion, for which dark energy is a possible candidate. 
  In the present work, we quantize a closed Friedmann-Robertson-Walker model in the presence of a positive cosmological constant and radiation. It gives rise to a Wheeler-DeWitt equation for the scale factor which has the form of a Schr\"{o}dinger equation for a potential with a barrier. We solve it numerically and determine the tunneling probability for the birth of a asymptotically DeSitter, inflationary universe, initially, as a function of the mean energy of the initial wave-function. Then, we verify that the tunneling probability increases with the cosmological constant, for a fixed value of the mean energy of the initial wave-function. 
  It is proved that no wormholes can be formed in viable scalar-tensor models of dark energy admitting its phantom-like ($w < -1$) behaviour in cosmology, even in the presence of electric or magnetic fields, if the non-minimal coupling function $f(\Phi)$ is everywhere positive and the scalar field $\Phi$ itself is not a ghost. Some special static, spherically symmetric wormhole solutions may exist if $f(\Phi)$ is allowed to reach zero or to become negative, so that the effective gravitational constant becomes negative in some region making the graviton a ghost. If $f$ remains non-negative, such solutions require severe fine tuning and a very peculiar kind of model. If $f < 0$ is allowed, it is argued (and confirmed by previous investigations) that such solutions are generically unstable under non-static perturbations, the instability appearing right near transition surfaces to negative $f$. 
  We discuss geometrical properties of the horizon surface of five-dimensional rotating black holes and black rings. Geometrical invariants characterizing these 3D geometries are calculated. We obtain a global embedding of the 5D rotating black horizon surface into a flat space. We also describe the Kaluza-Klein reduction of the black ring solution (along the direction of its rotation) which relates this solution to the 4D metric of a static black hole distorted by the presence of external scalar (dilaton) and vector (`electromagnetic') field. The properties of the reduced black hole horizon and its embedding in $\E^3$ are briefly discussed. 
  This paper is devoted to classify the most general plane symmetric spacetimes according to kinematic self-similar perfect fluid and dust solutions. We provide a classification of the kinematic self-similarity of the first, second, zeroth and infinite kinds with different equations of state, where the self-similar vector is not only tilted but also orthogonal and parallel to the fluid flow. This scheme of classification yields twenty four plane symmetric kinematic self-similar solutions. Some of these solutions turn out to be vacuum. These solutions can be matched with the already classified plane symmetric solutions under particular coordinate transformations. As a result, these reduce to sixteen independent plane symmetric kinematic self-similar solutions. 
  We find a new, non-commutative geometry inspired, solution of the coupled Einstein-Maxwell field equations describing a variety of charged, self-gravitating objects, including extremal and non-extremal black holes. The metric smoothly interpolates between deSitter geometry, at short distance, and Reissner-Nordstroem geometry far away from the origin. Contrary to the ordinary Reissner-Nordstroem spacetime there is no curvature singularity in the origin neither "naked" nor shielded by horizons. We investigate both the Hawking process and pair creation in this new scenario. 
  In a foregoing paper, gravity has been interpreted as the pressure force exerted on matter at the scale of elementary particles by a perfect fluid. Under the condition that Newtonian gravity must be recovered in the incompressible case, a scalar field equation has thus been proposed for gravity, giving a new theory in the compressible case. Here the theory is reinterpreted so as to describe the relativistic effects, by extending the Lorentz-Poincar\'e interpretation of special relativity which is first recalled. Gravitational space-contraction and time-dilatation are postulated, as a consequence of the principle of local equivalence between the effects of motion and gravitation. The space-time metric (expressing the proper time along a trajectory) is hence curved also in the proposed theory. As the result of a modified Newton law, it is proved that free test particles follow geodesic lines of this metric. In the spherical static situation, Schwarzschild's exterior metric is exactly recovered and with it the experimental support of general relativity, but the interior solution as well as the problematic of singularities are different in the proposed theory, e.g. the radius of the body cannot be smaller than the Schwarzschild radius. 
  The inspirals of stellar-mass compact objects into supermassive black holes are some of the most exciting sources of gravitational waves for LISA. Detection of these sources using fully coherent matched filtering is computationally intractable, so alternative approaches are required. In Wen & Gair (2005), we proposed a detection method based on searching for significant deviation of power density from noise in a time-frequency spectrogram of the LISA data. The performance of the algorithm was assessed in Gair & Wen (2005) using Monte-Carlo simulations on several trial waveforms and approximations to the noise statistics. We found that typical extreme mass ratio inspirals (EMRIs) could be detected at distances of up to 1-3 Gpc, depending on the source parameters. In this paper, we first give an overview of our previous work in Wen & Gair (2005) and Gair & Wen (2005), and discuss the performance of the method in a broad sense. We then introduce a decomposition method for LISA data that decodes LISA's directional sensitivity. This decomposition method could be used to improve the detection efficiency, to extract the source waveform, and to help solve the source confusion problem. Our approach to constraining EMRI parameters using the output from the time-frequency method will be outlined. 
  For a long time, it is generally believed that spin-spin interactions can only exist in a theory where Lorentz symmetry is gauged, and a theory with spin-spin interactions is not perturbatively renormalizable. But this is not true. By studying the motion of a spinning particle in gravitational field, it is found that there exist spin-spin interactions in gauge theory of gravity. Its mechanism is that a spinning particle will generate gravitomagnetic field in space-time, and this gravitomagnetic field will interact with the spin of another particle, which will cause spin-spin interactions. So, spin-spin interactions are transmitted by gravitational field. The form of spin-spin interactions in post Newtonian approximations is deduced. This result can also be deduced from the Papapetrou equation. This kind of interactions will not affect the renormalizability of the theory. The spin-spin interactions will violate the weak equivalence principle, and the violation effects are detectable. An experiment is proposed to detect the effects of the violation of the weak equivalence principle. 
  We consider a multigravity approach to spacetime foam. As an application we give indications on the computation of the cosmological constant, considered as an eigenvalue of a Sturm-Liouville problem. A variational approach with Gaussian trial wave functionals is used as a method to study such a problem. We approximate the equation to one loop in a Schwarzschild background and a zeta function regularization is involved to handle with divergences. The regularization is closely related to the subtraction procedure appearing in the computation of Casimir energy in a curved background. A renormalization procedure is introduced to remove the infinities together with a renormalization group equation. 
  The second order perturbations in Friedmann-Robertson-Walker universe filled with a perfect fluid are completely formulated in the gauge invariant manner without any gauge fixing. All components of the Einstein equations are derived neglecting the first order vector and tensor modes. These equations imply that the tensor and the vector mode of the second order metric perturbations may be generated by the non-linear effects of the Einstein equations from the first order density perturbations. 
  In this paper, we improve the calculation of the relic gravitational waves (RGW) in two aspects: First, we investigate the transfer function after considering the redshift-suppression effect, the accelerating expansion effect, the damping effect of free-streaming relativistic particles, and the damping effect of cosmic phase transition, and give a simple approximate analytic expression, which clearly illustrates the dependent relations on the cosmological parameters. Second, we develop a numerical method to calculate the primordial power spectrum of RGW at a very wide frequency range, where the observed constraints on $n_s$ (the scalar spectral index) and $P_S(k_0)$ (the amplitude of primordial scalar spectrum) and the Hamilton-Jacobi equation are used. This method is applied to two kinds of inflationary models, which all satisfy the current constraints on $n_s$, $\alpha$ (the running of $n_s$) and $r$ (the tensor-scalar ratio). We plot them in the $r-\Omega_g$ diagram, where $\Omega_g$ is the strength of RGW, and study their detection by the CMB experiments and laser interferometers. 
  Long ago, since 1951, Lyra proposed a modification of Riemannian geometry. Based on the Lyra's modification on Riemannian geometry, Sen and Dunn constructed a field equation which is analogous to Einstein's field equations. Further more, Sen and Dunn gave series type solution to the static vacuum field equations. Retaining only a few terms, we have shown that their solutions correspond to black holes (we call, Lyra black holes). Some interesting properties of the Lyra black holes are studied. 
  In this paper, the problem of finding an axisymmetric stationary spacetime from a specified set of multipole moments, is studied. The condition on the multipole moments, for existence of a solution, is formulated as a convergence condition on a power series formed from the multipole moments. The methods in this paper can also be used to give approximate solutions to any order as well as estimates on each term of the resulting power series. 
  We modified the modal expansion, which is the traditional method used to calculate thermal noise. This advanced modal expansion provides new insight about the discrepancy between the actual thermal noise caused by inhomogeneously distributed loss and the traditional modal expansion. 
  Physics is facing contingency. Not only in facts but also in laws (the frontier becoming extremely narrow). Cosmic natural selection is a tantalizing idea to explain the apparently highly improbable structure of our Universe. In this brief note I will study the creation of Universes by black holes in -string inspired- higher order curvature gravity. 
  The second order perturbative field equations for slowly and rigidly rotating perfect fluid balls of Petrov type D are solved numerically. It is found that all the slowly and rigidly rotating perfect fluid balls up to second order, irrespective of Petrov type, may be matched to a possibly non-asymptotically flat stationary axisymmetric vacuum exterior. The Petrov type D interior solutions are characterized by five integration constants, corresponding to density and pressure of the zeroth order configuration, the magnitude of the vorticity, one more second order constant and an independent spherically symmetric second order small perturbation of the central pressure. A four-dimensional subspace of this five-dimensional parameter space is identified for which the solutions can be matched to an asymptotically flat exterior vacuum region. Hence these solutions are completely determined by the spherical configuration and the magnitude of the vorticity. The physical properties like equations of state, shapes and speeds of sound are determined for a number of solutions. 
  We discuss two aspects of f(R) theories of gravity in metric formalism. We first study the reasons to introduce a scalar-tensor representation for these theories and the behavior of this representation in the limit to General Relativity, f(R)--> R. We find that the scalar-tensor representation is well behaved even in this limit. Then we work out the exact equations for spherically symmetric sources using the original f(R) representation, solve the linearized equations, and compare our results with recent calculations of the literature. We observe that the linearized solutions are strongly affected by the cosmic evolution, which makes very unlikely that the cosmic speedup be due to f(R) models with correcting terms relevant at low curvatures. 
  In this paper, we elaborate the problem of energy-momentum in General Relativity with the help of some well-known solutions. In this connection, we use the prescriptions of Einstein, Landau-Lifshitz, Papapetrou and M\"{o}ller to compute the energy-momentum densities for four exact solutions of the Einstein field equations. We take the gravitational waves, special class of Ferrari-Ibanez degenerate solution, Senovilla-Vera dust solution and Wainwright-Marshman solution. It turns out that these prescriptions do provide consistent results for special class of Ferrari-Ibanez degenerate solution and Wainwright-Marshman solution but inconsistent results for gravitational waves and Senovilla-Vera dust solution. 
  A flexible spectral code for the study of general relativistic magnetohydrodynamics is presented. Aiming at investigating the physics of slowly rotating magnetized compact stars, this new code makes use of various physically motivated approximations. Among them, the relativistic anelastic approximation is a key ingredient of the current version of the code. In this article, we mainly outline the method, putting emphasis on algorithmic techniques that enable to benefit as much as possible of the non-dissipative character of spectral methods, showing also a potential astrophysical application and providing a few illustrative tests. 
  A brane-world universe consists of a 4-dimensional brane embedded into a 5-dimensional space-time (bulk). We apply the Arnowitt-Deser-Misner decomposition to the brane-world, which results in a 3+1+1 break-up of the bulk. We present the canonical theory of brane cosmology based on this decomposition. The Hamiltonian equations allow for the study of any physical phenomena in brane gravity. This method gives new prospects for studying the initial value problem, stability analysis, brane black holes, cosmological perturbation theory and canonical quantization in brane-worlds. 
  The principle part of Einstein equations in the harmonic gauge consists of a constrained system of 10 curved space wave equations for the components of the space-time metric. A new formulation of constraint-preserving boundary conditions of the Sommerfeld type for such systems has recently been proposed. We implement these boundary conditions in a nonlinear 3D evolution code and test their accuracy. 
  The spherically symmetric, static spacetime generated by a cross-flow of non-interacting null dust streams can be conveniently interpreted as the radiation atmosphere of a star, which also receives exterior radiation. Formally, such a superposition of sources is equivalent to an anisotropic fluid. Therefore, there is a preferred time function in the system, defined by this reference fluid. This internal time is employed as a canonical coordinate, in order to linearize the Hamiltonian constraint. This turns to be helpful in the canonical quantization of the geometry of the radiating atmosphere. 
  The minimal metagravity theory, explicitly violating the general covariance but preserving the unimodular one, is applied to study the evolution of the isotropic homogeneous Universe. The massive scalar graviton, contained in the theory in addition to the massless tensor one, is treated as a source of the dark matter and/or dark energy. The modified Friedmann equation for the scale factor of the Universe is derived. The question wether the minimal metagravity can emulate the LCDM concordance model, valid in General Relativity, is discussed. 
  We investigate some quantum properties of a toroidal compactified spacetime modified due this unusual topology. More specifically, we present the vacuum fluctuation and the particle production in a Friedmann cosmological model in R3 x S1 outside a U(1) cosmic string, which throughout this paper, means the local type of these topological defects. The case of a teleparallel Friedmann spacetime is investigated, where we analyze the case with torsion. Also, we present a generalization to toroidal compactification of p extra dimensions, where the topology is given by R3 x T^p. Some implications are presented and discussed. Besides the interest in the dynamics of spacetime we also show the physical consequences of the topological modifications. 
  We examine observational constraints on the generalized Chaplygin gas (GCG) model for dark energy from the 9 Hubble parameter data points, the 115 SNLS Sne Ia data and the size of baryonic acoustic oscillation peak at redshift, $z=0.35$. At a 95.4% confidence level, a combination of three data sets gives $0.67\leq A_s\leq 0.83$ and $-0.21\leq \alpha\leq 0.42$, which is within the allowed parameters ranges of the GCG as a candidate of the unified dark matter and dark energy. It is found that the standard Chaplygin gas model ($\alpha=1$) is ruled out by these data at the 99.7% confidence level. 
  In this paper we directly constrain possible spatial variations of the Newtonian gravitational constant G over ranges 0.01-5 AU in various extrasolar multi-planet systems. By means of the third Kepler's law we determine the quantity \Gamma_ XY=G_X/G_Y for each couple of planets X and Y located at different distances from their parent star: deviations of the measured values of \Gamma from unity would signal variations of G. The obtained results for \eta=1-\Gamma are found to be well compatible with zero within the experimental errors (\eta/\delta\eta= 0.2-0.3). A comparison with an analogous test previously performed in our Solar System is made. 
  The problem of the dynamics of false vacuum bubbles has become more and more widespread in recent years. This is due to its importance to inflationary models in cosmology and the possibility for obtaining stable solitons in particle physics and quantum field theory. In this work, we investigate the behavior of test particles near the domain wall of the bubble. We show that matter is naturally trapped in the vicinity of the domain wall, for the case of a static domain wall. This, we believe, might give a physical explanation for existing models and, also, give rise to new, more realistic, physical models. 
  In this paper, we consider global texture with time dependent displacement vector based on Lyra geometry in normal gauge i.e. displacement vector \phi_i^* = (\beta_0(t),0,0,0). We investigate gravitational field of global texture configuration by solving Einstein equations as well as that for the scalar field due to texture. 
  The Hamilton-Jacobi analysis is applied to the dynamics of the scalar fluctuations about the Friedmann-Robertson-Walker (FRW). The gauge conditions are found from the consistency conditions. The physical degrees of freedom of the model are obtain by symplectic projector method. The role of the linearly dependent Hamiltonians and the gauge variables in Hamilton-Jacobi formalism is discussed. 
  We explore the prospects for Advanced LIGO to detect gravitational waves from neutron stars and stellar mass black holes spiraling into intermediate-mass ($M\sim 50 M_\odot$ to $350 M_\odot$) black holes. We estimate an event rate for such \emph{intermediate-mass-ratio inspirals} (IMRIs) of up to $\sim 10$--$30 \mathrm{yr}^{-1}$. Our numerical simulations show that if the central body is not a black hole but its metric is stationary, axisymmetric, reflection symmetric and asymptotically flat then the waves will likely be tri-periodic, as for a black hole. We report generalizations of a theorem due to Ryan (1995) which suggest that the evolutions of the waves' three fundamental frequencies and of the complex amplitudes of their spectral components encode (in principle) a full map of the central body's metric, full details of the energy and angular momentum exchange between the central body and the orbit, and the time-evolving orbital elements. We estimate that Advanced LIGO can measure or constrain deviations of the central body from a Kerr black hole with modest but interesting accuracy. 
  Moller's tetrad gravitational energy-momentum "tensor" is evaluated for a small vacuum region using an orthonormal frame adapted to Riemann normal coordinates. We find that it does satisfy the highly desired property of being a positive multiple of the Bel-Robinson tensor. 
  Alternatives to the usual general relativity (GR) Riemannian framework include Riemann-Cartan and teleparallel geometry. The ``teleparallel equivalent of GR" (TEGR, aka GR${}_{||}$) has certain virtues, however there have been allegations of serious source coupling limitations. Now it is quite straightforward to show that the coupled dynamical field equations of Einstein's GR with any source can be accurately represented in terms of any other connection, in particular teleparallel geometry. Using an argument similar to one used long ago to show the "effective equivalence" between GR and the Einstein-Cartan theory, we construct the teleparallel action which is equivalent to a given Riemanian one; thereby finding the ``effectively equivalent" coupling principle for all sources, including spinors. No auxiliary field is required. Can one decide which is the real "physical" geometry? Invoking the minimal coupling principle may give a unique answer. 
  A new metric is obtained by applying a complex coordinate trans- formation to the static metric of the self-gravitating Born-Infeld monopole. The behaviour of the new metric is typical of a rotating charged source, but this source is not a spherically symmetric Born-Infeld monopole with rotation. We show that the structure of the energy-momentum tensor obtained with this new metric does not correspond to the typical structure of the energy momentum tensor of Einstein-Born-Infeld theory induced by a rotating spherically symmetric source. This also show, that the complex coordinate transformations have the interpretation given by Newman and Janis only in space-time solutions with linear sources. 
  The stress-energy tensor for the non-minimally coupled scalar field is known not to satisfy the pointwise energy conditions, even on the classical level. We show, however, that local averages of the classical stress-energy tensor satisfy certain inequalities and give bounds for averages along causal geodesics. It is shown that in vacuum background spacetimes, ANEC and AWEC are satisfied. Furthermore we use our result to show that in the classical situation we have an analogue to the so called quantum interest conjecture. These results lay the foundations for averaged energy inequalities for the quantised non-minimally coupled fields. 
  The Parametrized Post-Newtonian expansion of gravitational theories with a scalar field coupled to the Gauss-Bonnet invariant is performed and confrontation of such theories with Solar system experiments is discussed. 
  Within a scalar model theory of gravity, where the interaction between particles is given by the half-retarded + half-advanced solution of the scalar wave equation, we consider an N-body problem: we investigate configurations of N particles which form an equilateral N-angle and are in helical motion about their common center. We prove that there exists a unique equilibrium configuration and compute the equilibrium radius explicitly in a post-Newtonian expansion. 
  We study head-on collisions of boson stars in three dimensions. We consider evolutions of two boson stars which may differ in their phase or have opposite frequencies but are otherwise identical. Our studies show that these phase differences result in different late time behavior and gravitational wave output. 
  We present an approach to the problem of vacuum energy in cosmology, based on dynamical screening of Lambda on the horizon scale. We review first the physical basis of vacuum energy as a phenomenon connected with macroscopic boundary conditions, and the origin of the idea of its screening by particle creation and vacuum polarization effects. We discuss next the relevance of the quantum trace anomaly to this issue. The trace anomaly implies additional terms in the low energy effective theory of gravity, which amounts to a non-trivial modification of the classical Einstein theory, fully consistent with the Equivalence Principle. We show that the new dynamical degrees of freedom the anomaly contains provide a natural mechanism for relaxing Lambda to zero on cosmological scales. We consider possible signatures of the restoration of conformal invariance predicted by the fluctuations of these new scalar degrees of freedom on the spectrum and statistics of the CMB, in light of the latest bounds from WMAP. Finally we assess the prospects for a new cosmological model in which the dark energy adjusts itself dynamically to the cosmological horizon boundary, and therefore remains naturally of order H^2 at all times without fine tuning. 
  We discuss the impact of Lorentz violation on the cosmology. Firstly, we show that the Lorentz violation affects the dynamics of the chaotic inflationary model and gives rise to an interesting feature. Secondly, we propose the Lorentz violating DGP brane models where the Lorentz violating terms on the brane accelerate the current universe. We conjecture that the ghost disappears in the Lorentz violating DGP models. 
  The loop quantization of the negatively curved k=-1 RW model poses several technical challenges. We show that the issues can be overcome and a successful quantization is possible that extends the results of the k=0,+1 models in a natural fashion. We discuss the resulting dynamics and show that for a universe consisting of a massless scalar field, a bounce is predicted in the backward evolution in accordance with the results of the k=0,+1 models. We also show that the model predicts a vacuum repulsion in the high curvature regime that would lead to a bounce even for matter with vanishing energy density. We finally comment on the inverse volume modifications of loop quantum cosmology and show that, as in the k=0 model, the modifications depend sensitively on the introduction of a length scale which a priori is independent of the curvature scale or a matter energy scale. 
  We establish an equivalence between the Hamiltonian formulation of the Plebanski action for general relativity and the covariant canonical formulation of the Hilbert-Palatini action. This is done by comparing the symplectic structures of the two theories through the computation of Dirac brackets. We also construct a shifted connection with simplified Dirac brackets, playing an important role in the covariant loop quantization program, in the Plebanski framework. Implications for spin foam models are also discussed. 
  Using a Hamiltonian treatment, charged thin shells in spherically symmetric spacetimes in d dimensional Lovelock-Maxwell theory are studied. The coefficients of the theory are chosen to obtain a sensible theory, with a negative cosmological constant appearing naturally. After writing the action and the Lagrangian for a spacetime comprised of an interior and an exterior regions, with a thin shell as a boundary in between, one finds the Hamiltonian using an ADM description. For spherically symmetric spacetimes, one reduces the relevant constraints. The dynamic and constraint equations are obtained. The vacuum solutions yield a division of the theory into two branches, d-2k-1>0 (which includes general relativity, Born-Infeld type theories) and d-2k-1=0 (which includes Chern-Simons type theories), where k gives the highest power of the curvature in the Lagrangian. An additional parameter, chi, gives the character of the vacuum solutions. For chi=1 the solutions have a black hole character. For chi=-1 the solutions have a totally naked singularity character. The integration through the thin shell takes care of the smooth junction. The subsequent analysis is divided into two cases: static charged thin shell configurations, and gravitationally collapsing charged dust shells. Physical implications are drawn: if such a large extra dimension scenario is correct, one can extract enough information from the outcome of those collapses as to know, not only the actual dimension of spacetime, but also which particular Lovelock gravity, is the correct one. 
  The probability representation of quantum mechanics including propagators and tomograms of quantum states of the universe and its application to quantum gravity and cosmology are reviewed. The minisuperspaces modeled by oscillator, free pointlike particle and repulsive oscillator are considered. The notion of tomographic entropy and its properties are used to find some inequalities for the tomographic probability determining the quantum state of the universe. The sense of the inequality as a lower bound for the entropy is clarified. 
  The finiteness of black hole entropy suggest that spacetime is fundamentally discrete, and hints at an underlying relationship between geometry and "information". The foundation of this relationship is yet to be uncovered, but should manifest itself in a theory of quantum gravity. We review recent attempts to define a microscopic measure for black hole entropy and for the maximum entropy of spherically symmetric spacelike regions, within the causal set approach to quantum gravity. 
  The issue of the equivalence between Jordan and Einstein conformal frames in scalar-tensor gravity is revisited, with emphasis on implementing running units in the latter. The lack of affine parametrization for timelike worldlines and the cosmological constant problem in the Einstein frame are clarified, and a paradox in the literature about cosmological singularities appearing only in one frame is solved. While, classically, the two conformal frames are physically equivalent, they seem to be inequivalent at the quantum level. 
  We use the `moving puncture' approach to perform fully non-linear evolutions of spinning quasi-circular black-hole binaries with individual spins not aligned with the orbital angular momentum. We evolve configurations with the individual spins (parallel and equal in magnitude) pointing in the orbital plane and 45-degrees above the orbital plane. We introduce a technique to measure the spin direction and track the precession of the spin during the merger, as well as measure the spin flip in the remnant horizon. The former configuration completes 1.75 orbits before merging, with the spin precessing by 98-degrees and the final remnant horizon spin flipped by ~72-degrees with respect to the component spins. The latter configuration completes 2.25 orbits, with the spins precessing by 151-degrees and the final remnant horizon spin flipped ~34-degrees with respect to the component spins. These simulations show for the first time how the spins are reoriented during the final stage of binary black hole mergers verifying the hypothesis of the spin-flip phenomenon. We also compute the track of the holes before merger and observe a precession of the orbital plane with frequency similar to the orbital frequency and amplitude increasing with time. 
  Deser et al. proposed a combination of the Einstein and Landau-Lifshitz pseudotensors such that the second derivatives in vacuum are proportional to the Bel-Robinson tensor. Stimulated by their work, the present paper discuss the gravitational energy-momentum expression which has the same desirable Bel-Robinson tensor property. We find modifications of the Einstein and Landau-Lifshitz pseudotensors that both give the same coefficient of the Bel-Robinson tensor in vacuum in holonomic frames. 
  We perform an analysis where Einstein's field equation is derived from two simple thermodynamical relations. First, we assume that the fundamental thermodynamical relation, $\delta Q = TdS$, is valid at any accelerating spacelike two-plane which moves very close to its local Rindler horizon. The heat flow through the plane, $\delta Q$, is interpreted here as the boost energy of matter which flows across the past Rindler horizon and which is measured by an observer moving along with the plane. The temperature $T$, in turn, is the Unruh temperature experienced by the observer. Secondly, we assume that when matter flows through the accelerating two-plane, the plane shrinks and the entropy content of matter increases in such a way that the maximum increase in the entropy is, in natural units, exactly one-half of the decrease in the area of the plane. Our analysis supports the view that Einstein's field equation is just a thermodynamical equation of state. 
  The Friedmann-Robertson-Walker-Lemaitre metric tensor in matter fields presence in quantum context is studied. New quantization method for this cosmological model is proposed. Four elementary approaches are effectively jointed. The Dirac method for constrained systems is used, then the Fock space is built and the second quantization is carried out. Finally the diagonalization ansatz is proposed - combination of the Bogoliubov transformation and the Heisenberg equation of motion. As a result the Quantum Cosmology as the many-Universe system is formulated. 
  Slowly rotating perfect fluid balls with regular center and asymptotically flat exterior are considered to second order in the rotation parameter. The necessary condition for being Petrov type D is given for general perfect fluid matter. As a special case, fluids with a linear equation of state are considered. Using a power series expansion at the regular center, it is shown that the Petrov D condition is inconsistent with the linear equation of state assumption. 
  We present a multi-domain spectral method to compute initial data of binary systems in General Relativity. By utilizing adapted conformal coordinates, the vacuum region exterior to the gravitational sources is divided up into two subdomains within which the spectral expansion of the field quantities is carried out. If a component of the binary is a neutron star, a further subdomain covering the star's interior is added. As such, the method can be used to construct arbitrary initial data corresponding to binary black holes, binary neutron stars or mixed binaries. In particular, it is possible to describe a black hole component by the puncture ansatz as well as through an excision technique. First examples are given for binary black hole excision data that fulfill the requirements of the quasi-stationary framework, which combines the Conformal Thin Sandwich formulation of the constraint equations with the Isolated Horizon conditions for black holes in quasi-equilibrium. These numerical solutions were obtained to extremely high accuracy with moderate computational effort. Moreover, the method proves to be applicable even when tending toward limiting cases such as large mass ratios of the binary components. 
  We propose a test of the principle of relativity, involving quantum signals between two inertial frames. If the principle is upheld, classical causality will appear to be split in a dramatic and emphatic way. We discuss the existence of quantum horizons, which are barriers to the transmission of any form of quantum information. These must occur in any finite time, inter-frame experiment if quantum causality holds. We conclude with some comments on such experiments involving entangled states. 
  We explore the possibility of dynamic wormhole geometries, within the context of nonlinear electrodynamics. The Einstein field equation imposes a contracting wormhole solution and the obedience of the weak energy condition. Furthermore, in the presence of an electric field, the latter presents a singularity at the throat, however, for a pure magnetic field the solution is regular. Thus, taking into account the principle of finiteness, that a satisfactory theory should avoid physical quantities becoming infinite, one may rule out evolving wormhole solutions, in the presence of an electric field, coupled to nonlinear electrodynamics. 
  In this paper we obtain the black hole metric from a semiclassical analysis of loop quantum black hole. Our solution and the Schwarzschild one tend to match well at large distances from Planck region. In r=0 the semiclassical metric is regular and singularity free in contrast to the classical one. By using the new metric we calculate the Hawking temperature and the entropy. For the entropy we obtain the logarithmic correction to the classical area law. Finally we study the mass evaporation process and we show the mass and temperature tend to zero at infinitive time. 
  We consider EGO as a possible third-generation ground-based gravitational wave detector and evaluate its capabilities for the detection and interpretation of compact binary inspiral signals. We identify areas of astrophysics and cosmology where EGO would have qualitative advantages, using Advanced LIGO as a benchmark for comparison. 
  The radial component of the motion of compact binary systems composed of neutron stars and/or black holes on eccentric orbit is integrated. We consider all type of perturbations that emerge up to second post-Newtonian order. These perturbations are either of relativistic origin or are related to the spin, mass quadrupole and magnetic dipole moments of the binary components. We derive a generalized Kepler equation and investigate its domain of validity, in which it properly describes the radial motion. 
  We investigate the Liouvillian integrability of Hamiltonian systems describing a universe filled with a scalar field (possibly complex). The tool used is the differential Galois group approach, as introduced by Morales-Ruiz and Ramis. The main result is that the generic systems are non-integrable, although there still exist some values of parameters for which integrability remains undecided. We also draw a connection with chaos present in such cosmological models. The first part of the article deals with minimally coupled fields, and the second treats the conformal couping. 
  We speculate about impact of antigravity (i.e. gravitational repulsion between matter and antimatter) on the creation and emission of particles by a black hole. If antigravity is present a black hole made of matter may radiate particles as a black body, but this shouldn't be true for antiparticles. It may lead to radical change of radiation process predicted by Hawking and should be taken into account in preparation of the attempt to create and study mini black holes at CERN. Gravity, including antigravity is more than ever similar to electrodynamics and such similarity with a successfully quantized interaction may help in quantization of gravity. 
  Recently it has shown that Einstein's field equations can be rewritten into a form of the first law of thermodynamics both at event horizon of static spherically symmetric black holes and apparent horizon of Friedmann-Robertson-Walker (FRW) universe, which indicates intrinsic thermodynamic properties of spacetime horizon. In the present paper we deal with the so-called $f(R)$ gravity, whose action is a function of the curvature scalar $R$. In the setup of static spherically symmetric black hole spacetime, we find that at the event horizon, the field equations of $f(R)$ gravity can be written into a form $dE = TdS - PdV + Td\bar{S}$, where $T$ is the Hawking temperature and $S=Af'(R)/4G$ is the horizon entropy of the black hole, $E$ is the horizon energy of the black hole, $P$ is the radial pressure of matter, $V$ is the volume of black hole horizon, and $d\bar S$ can be interpreted as the entropy production term due to nonequilibrium thermodynamics of spacetime. In the setup of FRW universe, the field equations can also be cast to a similar form, $dE=TdS +WdV +Td\bar S$, at the apparent horizon, where $W=(\rho-P)/2$, $E$ is the energy of perfect fluid with energy density $\rho$ and pressure $P$ inside the apparent horizon. Compared to the case of Einstein's general relativity, an additional term $d\bar S$ also appears here. The appearance of the additional term is consistent with the argument recently given by Eling {\it et al.} (gr-qc/0602001) that the horizon thermodynamics is non-equilibrium one for the $f(R)$ gravity. 
  The geodesic equation in induced matter theory is redefined. It is shown that the extra forces do not change the four-impulse of massive particles. We show that the 4D space-time is non-integrable and find the relation between non-integrability and the Mach's principal. 
  In this work we focus on the search and detection of Supermassive black hole binary systems, including systems at high redshift. As well as expanding on previous works where we used a variant of Markov Chain Monte Carlo (MCMC) with simulated annealing, we introduce a new search method based on frequency annealing which leads to a more rapid and robust detection. We compare the two search methods on systems where we do and do not see the merger of the black holes. In the non-merger case, we also examine the posterior distribution exploration using a 7-D MCMC algorithm. We demonstrate that this method is quite effective in dealing with the high correlations between parameters, has a higher acceptance rate than previously proposed methods and produces posterior distribution functions that are extremely close to the prediction from the Fisher matrix. Finally, after carrying out searches where there is only one binary in the data stream, we examine the case where two black hole binaries are present in the same data stream. We demonstrate that we have no problems in pulling out the two binaries, displaying that there is virtually no correlation between overlapping binary black holes, and more importantly showing that we can safely extract the SMBHB sources without contaminating the rest of the data stream. 
  The generalized second law of gravitational thermodynamics is examined in scenarios where the dark energy dominates the cosmic expansion. For quintessence and phantom fields this law is fulfilled but it may fail when the dark energy is in the form of a Chaplygin gas. However, if a black hole is allowed in the picture, the law can be violated if the field is of phantom type. 
  In this paper, by making use of the perturbative expansion around topological field theory we are trying to understand why the standard perturbation theory for General Relativity, which starts with linearized gravity does not see gravitational collapse. We start with investigating classical equations of motion. For zero Immirzi parameter the ambiguity of the standard perturbative expansion is reproduced. This ambiguity is related to the appearance of the linearized diffeomorphism symmetry, which becomes unlinked from the original diffeomorphism symmetry. Introducing Immirzi parameter makes it possible to restore the link between these two symmetries and thus removes the ambiguity, but at the cost of making classical perturbation theory rather intractable. Then we argue that the two main sources of complexity of perturbation theory, infinite number of degrees of freedom and non-trivial curvature of the phase space of General Relativity could be disentangled when studying {\it quantum} amplitudes. As an illustration we consider zero order approximation in quantum perturbation theory. We identify relevant observables, and sketch their quantization. We find some indications that this zero order approximation might be described by Doubly Special Relativity. 
  We study the formation of black strings from a gravitational collapse of cylindrical dust clouds in the three-dimensional low-energy string theory. A new junction condition for the dilaton as well as two junction conditions for metrics and extrinsic curvatures between both regions of the clouds is presented. As a consequence, it is found that the collapsing dust cloud always collapses to a black string within a finite collapse time, and then a curvature singularity formed at origin is cloaked by an event horizon. Moreover, it is also found that the collapse process can form a naked singularity within finite time, regardless of the choice of initial data. 
  Starting from the Hamiltonian formulation for the inhomogeneous Mixmaster dynam- ics, we approach its quantum features through the link of the quasi-classical limit. We fix the proper operator-ordering which ensures that the WKB continuity equation overlaps the Liouville theorem as restricted to the configuration space. We describe the full quantum dynamics of the model in some details, providing a characterization of the (discrete) spectrum with analytic expressions for the limit of high occupation number. One of the main achievements of our analysis relies on the description of the ground state morphology, showing how it is characterized by a non-vanishing zero-point energy associated to the Universe anisotropy degrees of freedom 
  We discuss the transition from quasi-circular inspiral to plunge of a system of two nonrotating black holes of masses $m_1$ and $m_2$ in the extreme mass ratio limit $m_1m_2\ll (m_1+m_2)^2$. In the spirit of the Effective One Body (EOB) approach to the general relativistic dynamics of binary systems, the dynamics of the two black hole system is represented in terms of an effective particle of mass $\mu\equiv m_1m_2/(m_1+m_2)$ moving in a (quasi-)Schwarzschild background of mass $M\equiv m_1+m_2$ and submitted to an ${\cal O}(\mu)$ radiation reaction force defined by Pad\'e resumming high-order Post-Newtonian results. We then complete this approach by numerically computing, \`a la Regge-Wheeler-Zerilli, the gravitational radiation emitted by such a particle. Several tests of the numerical procedure are presented. We focus on gravitational waveforms and the related energy and angular momentum losses. We view this work as a contribution to the matching between analytical and numerical methods within an EOB-type framework. 
  Currently the most popular method to evolve black-hole binaries is the ``moving puncture'' method. It has recently been shown that when puncture initial data for a Schwarzschild black hole are evolved using this method, the numerical slices quickly lose contact with the second asymptotically flat end, and end instead on a cylinder of finite Schwarzschild coordinate radius. These slices are stationary, meaning that their geometry does not evolve further. We will describe these results in the context of maximal slices, and present time-independent puncture-like data for the Schwarzschild spacetime. 
  In this paper we show that a free electromagnetic field living in Minkowski spacetime generates an effective Weitzenbock or an effective Lorentzian spacetime whose properties aredetermined in details. These results are possible because we found using the Clifford bundle formalism the noticeable result that the energy-momentum densities of a free electromagnetic field are sources of the Hodge duals of exact 2-form fields which satisfy Maxwell like equations. 
  The black hole entropy formula applied to local Rindler horizon at each spacetime point has been used in the literature to derive the Einstein field equation as an equation of state of a thermodynamical system of spacetime. In the present paper we argue that due to the key role of causal structure and discrete spacetime in this approach the natural framework is unimodular relativity rather than general relativity. It is shown that the equation of state is trace free unimodular relativity field equation that uniquely determines only the traceless stress tensor. Recent generalization to nonequilibrium thermodynamics is shown to be equivalent to the conformally related spacetime metrics. We suggest that the cosmological constant should possess thermodynamical fluctuations, and at a deeper level the metric may have statistical origin. 
  Activities in data analysis and numerical simulation of gravitational waves have to date largely proceeded independently. In this work we study how waveforms obtained from numerical simulations could be effectively used within the data analysis effort to search for gravitational waves from black hole binaries. We propose measures to quantify the accuracy of numerical waveforms for the purpose of data analysis and study how sensitive the analysis is to errors in the waveforms. We estimate that ~100 templates (and ~10 simulations with different mass ratios) are needed to detect waves from non-spinning binary black holes with total masses in the range 100 Msun < M < 400 Msun using initial LIGO. Of course, many more simulation runs will be needed to confirm that the correct physics is captured in the numerical evolutions. From this perspective, we also discuss sources of systematic errors in numerical waveform extraction and provide order of magnitude estimates for the computational cost of simulations that could be used to estimate the cost of parameter space surveys. Finally, we discuss what information from near-future numerical simulations of compact binary systems would be most useful for enhancing the detectability of such events with contemporary gravitational wave detectors and emphasize the role of numerical simulations for the interpretation of eventual gravitational-wave observations. 
  The basic idea of the LQC applies to every spatially homogeneous cosmological model, however only the spatially flat (so called $k=0$) case has been understood in detail in the literature thus far. In the closed (so called: k=1) case certain technical difficulties have been the obstacle that stopped the development. In this work the difficulties are overcome, and a new LQC model of the spatially closed, homogeneous, isotropic universe is constructed. The topology of the spacelike section of the universe is assumed to be that of SU(2) or SO(3). Surprisingly, according to the results achieved in this work, the two cases can be distinguished from each other just by the local properties of the quantum geometry of the universe. The quantum hamiltonian operator of the gravitational field takes the form of a difference operator, where the elementary step is the quantum of the 3-volume derived in the flat case by Ashtekar, Pawlowski and Singh. The mathematical properties of the operator are studied: it is essentially self-adjoint, bounded from above by 0, the 0 itself is not an eigenvalue, the eigenvectors form a basis. An estimate on the dimension of the spectral projection on any finite interval is provided. 
  The motion of a local source inducing small oscillations in the gravitational field is investigated and shown to exhibit pure rotational kinetic energy. Should the net affect of these slow, revolving oscillations cause large-scale rotations in spacetime it would certainly result in anomalous celestial accelerations. When this angular rotational frequency of spacetime is applied to the anomalous acceleration of the Pioneer 10/11 spacecrafts, the correlation is promising. 
  We study the possibility of experimental testing the manifestations of equivalence principle in spin-gravity interactions. We reconsider the earlier experimental data and get the first experimental bound on anomalous gravitomagnetic moment. The spin coupling to the Earth's rotation may be also explored at the extensions of neutron EDM and g-2 experiments. The spin coupling to the terrestrial gravity produces a considerable effect which may be discovered at the planned deuteron EDM experiment. The Earth's rotation should be also taken into account in optical experiments on a search for axion-like particles. 
  The closed, k=1, FRW model coupled to a massless scalar field is investigated in the framework of loop quantum cosmology using analytical and numerical methods. As in the k=0 case, the scalar field can be again used as emergent time to construct the physical Hilbert space and introduce Dirac observables. The resulting framework is then used to address a major challenge of quantum cosmology: resolving the big-bang singularity while retaining agreement with general relativity at large scales. It is shown that the framework fulfills this task. In particular, for states which are semi-classical at some late time, the big-bang is replaced by a quantum bounce and a recollapse occurs at the value of the scale factor predicted by classical general relativity. Thus, the `difficulties' pointed out by Green and Unruh in the k=1 case do not arise in a more systematic treatment. As in k=0 models, quantum dynamics is deterministic across the deep Planck regime. However, because it also retains the classical recollapse, in contrast to the k=0 case one is now led to a cyclic model. Finally, we clarify some issues raised by Laguna's recent work addressed to computational physicists. 
  The variant of the quintessence theory is proposed in order to get an accelerated expansion of the Friedmannian Universe in the frameworks of relativistic theory of gravitation. The substance of quintessence is built up the scalar field of dark energy. It is shown, that function V(Phi), which factorising scalar field Lagrangian (Phi is a scalar field) has no influence on the evolution of the Universe. Some relations, allowing to find explicit dependence Phi on time, were found, provided given function V(Phi). 
  The vierbien formalism can not be uniquely determined by the metric. This flexibility increases difficulties in studing their relation and properties, and easily leads to ambiguous results. Considering the importance of the interaction between spinor fields and the spacetime, in this paper we derive some explicit expressions for vierbien formalism and the transformation rules between metric and vierbein. 
  A method is given which renders indirect detection of strong gravitational waves possible. This is based on the reflection of a linearly polarized electromagnetic wave from a cross polarized gravitational wave which induces a detectable Faraday rotation in its polarization vector. 
  The generation of gravitational waves during inflation due to the non-linear coupling of scalar and tensor modes is discussed. Two formalisms are used and compared: a covariant and local approach, as well as a metric-based analysis. An application to slow-roll inflation is also described. 
  We consider the general asymptotic expression of stationary space-time. Using Killing equation, we reduce the dynamical freedom of Einstein equation to the in-going gravitational wave $\Psi_0$. The general form of this function can be got. With the help of asymptotically algebraic special condition, we prove that all Newman-Penrose constants vanish. 
  General relativity and quantum mechanics provide a natural explanation for the existence of dark energy with its observed value and predict its dynamics. Dark energy proves to be necessary for the existence of space-time itself and determines the rate of its stability. 
  Deformed Special Relativity is usually presented as a deformation of Special Relativity accommodating a new universal constant, the Planck mass, while respecting the relativity principle. In order to avoid some fundamental problems (e.g. soccer ball problem), we argue that we should switch point of view and consider instead the Newton constant $G$ as the universal constant. 
  For non-flat universe of $k\not=0$, we investigate a model of the interacting holographic dark energy with cold dark matter (CDM). There exists a mixture of two components arisen from decaying of the holographic dark energy into CDM. In this case we use the effective equations of state ($\omega^{\rm eff}_{\rm \Lambda}, \omega^{\rm eff}_{\rm m}$) instead of the native equations of state ($\omega_{\rm \Lambda},\omega_{\rm m})$. Consequently, we show that any interacting holographic energy models in non-flat universe cannot accommodate a transition from the dark energy to the phantom regime. 
  We present two families of exterior differential systems (EDS) for non-isometric embeddings of orthonormal frame bundles over Riemannian spaces of dimension q = 2, 3, 4, 5.... into orthonormal frame bundles over flat spaces of sufficiently higher dimension. We have calculated Cartan characters showing that these EDS satisfy Cartan's test and are well-posed dynamical field theories. The first family includes a constant-coefficient (cc) EDS for classical Einstein vacuum relativity (q = 4). The second family is generated only by cc 2-forms, so these are integrable (but nonlinear) systems of partial differential equations. These calibrated field theories apparently are new, although the simplest case q = 2 turns out to embed a ruled surface of signature (1,1) in flat space of signature (2,1). Cartan forms are found to give explicit variational principles for all these dynamical theories. 
  The influence of the observed relict vacuum energy on the fluctuations of CMBR going through cosmological matter condensations is studied in the framework of the Einstein-Strauss-de Sitter vakuola model. It is shown that refraction of light at the matching surface of the vakuola and the expanding Friedman universe can be very important during accelerated expansion of the universe, when the velocity of the matching surface relative to static Schwarzchildian observers becomes relativistic. Relevance of the refraction effect for the temperature fluctuations of CMBR is given in terms of the redshift and the angular extension of the fluctuating region. 
  Principal ideas of gauge approach applying to gravitational interaction and leading to gravitation theory in Riemann-Cartan space-time are discussed. The principal relations of isotropic cosmology built in the framework of the Poincare gauge theory of gravity and the most important their consequences are presented. 
  The "dark energy" problem is investigated in the framework of the Poincare gauge theory of gravity in 4-dimensional Riemann-Cartan space-time. By using general expression for gravitational Lagrangian homogeneous isotropic cosmological models with pseudoscalar torsion function are built and investigated. It is shown that by certain restrictions on indefinite parameters of gravitational Lagrangian the cosmological equations at asymptotics contain effective cosmological constant and can lead to observable acceleration of cosmological expansion. This effect has geometrical nature and is connected with space-time torsion. 
  Coalescing binary black hole mergers are expected to be the strongest gravitational wave sources for ground-based interferometers, such as the LIGO, VIRGO, and GEO600, as well as the space-based interferometer LISA. Until recently it has been impossible to reliably derive the predictions of General Relativity for the final merger stage, which takes place in the strong-field regime. Recent progress in numerical relativity simulations is, however, revolutionizing our understanding of these systems. We examine here the specific case of merging equal-mass Schwarzschild black holes in detail, presenting new simulations in which the black holes start in the late inspiral stage on orbits with very low eccentricity and evolve for ~1200M through ~7 orbits before merging. We study the accuracy and consistency of our simulations and the resulting gravitational waveforms, which encompass ~14 cycles before merger, and highlight the importance of using frequency (rather than time) to set the physical reference when comparing models. Matching our results to PN calculations for the earlier parts of the inspiral provides a combined waveform with less than half a cycle of accumulated phase error through the entire coalescence. Using this waveform, we calculate signal-to-noise ratios (SNRs) for iLIGO, adLIGO, and LISA, highlighting the contributions from the late-inspiral and merger-ringdown parts of the waveform which can now be simulated numerically. Contour plots of SNR as a function of z and M show that adLIGO can achieve SNR >~ 10 for some IMBBHs out to z ~ 1, and that LISA can see MBBHs in the range 3x10^4 <~ M/M_Sun <~ 10^7 at SNR > 100 out to the earliest epochs of structure formation at z > 15. 
  We examine the ADM reformulation of the 5-D KK model: the dimensional reduction is provided to commute with the ADM splitting and we show how the time component of the gauge vector is given by combination of the Lagrangian multipliers for the 5-D gravitational field. We consider 5D particles motion and after dimensional reduction the definition of charge is recovered within electrodynamic coupling. A time-varying fine structure constant is recognized because an extra scalar field is present in the 4-D theory. 
  We consider the thermodynamics of a charged black hole enclosed in a cavity. The charge in the cavity and the temperature at the walls are fixed so that we have a canonical ensemble. We derive the phase structure and stability of black hole equilibrium states. We compare our results to that of other work which uses asymptotically anti-de Sitter boundary conditions to define the thermodynamics. The thermodynamic properties have extensive similarities which suggest that the idea of AdS holography is more dependent on the existence of the boundary than on the exact details of asymptotically AdS metrics. 
  We show that reheating of the universe occurs spontaneously in a broad class of inflation models with f(phi)R gravity (phi is inflaton). The model does not require explicit couplings between phi and bosonic or fermionic matter fields. The couplings arise spontaneously when phi settles in the vacuum expectation value (vev) and oscillates, with coupling constants given by derivatives of f(phi) at the vev and the mass of resulting bosonic or fermionic fields. This mechanism allows inflaton quanta to decay into any fields which are not conformally invariant in f(phi)R gravity theories. 
  We present a general method for the analysis of the stability of static, spherically symmetric solutions to spherically symmetric perturbations in an arbitrary diffeomorphism covariant Lagrangian field theory. Our method involves fixing the gauge and solving the linearized gravitational field equations to eliminate the metric perturbation variable in terms of the matter variables. In a wide class of cases---which include f(R) gravity, the Einstein-aether theory of Jacobson and Mattingly, and Bekenstein's TeVeS theory---the remaining perturbation equations for the matter fields are second order in time. We show how the symplectic current arising from the original Lagrangian gives rise to a symmetric bilinear form on the variables of the reduced theory. If this bilinear form is positive definite, it provides an inner product that puts the equations of motion of the reduced theory into a self-adjoint form. A variational principle can then be written down immediately, from which stability can be tested readily. We illustrate our method in the case of Einstein's equation with perfect fluid matter, thereby re-deriving, in a systematic manner, Chandrasekhar's variational principle for radial oscillations of spherically symmetric stars. In a subsequent paper, we will apply our analysis to f(R) gravity, the Einstein-aether theory, and Bekenstein's TeVeS theory. 
  There was obtained a numerical external solution for the exact system of the RTG equations with some natural boundary conditions in the static spherically symmetric case. The properties of the solution are discussed. 
  To investigate the imprint on the gravitational-wave emission from extreme mass-ratio inspirals in non-pure Kerr spacetimes, we have studied the ``kludge'' waveforms generated in highly-accurate and numerically-generated spacetimes containing a black hole and a self-gravitating torus with comparable mass and spin. In order to maximize their impact on the produced waveforms, we have considered tori that are compact, massive and close to the central black hole, investigating under what conditions the LISA experiment could detect their presence. Our results show that for a large portion of the space of parameters the waveforms produced by EMRIs in these black hole-torus systems are indistinguishable from pure-Kerr waveforms. Hence, a ``confusion problem'' will be present for observations carried out over a timescale below or comparable to the dephasing time. 
  We developed realistic fully general relativistic computer code for simulation of optical projection in a strong, spherically symmetric gravitational field. Standard theoretical analysis of optical projection for an observer in the vicinity of a Schwarzschild black hole is extended to black hole spacetimes with a repulsive cosmological constant, i.e, Schwarzschild-de Sitter spacetimes. Influence of the cosmological constant is investigated for static observers and observers radially free-falling from static radius. Simulation includes effects of gravitational lensing, multiple images, Doppler and gravitational frequency shift, as well as the amplification of intensity. The code generates images of static observers sky and a movie simulations for radially free-falling observers. Techniques of parallel programming are applied to get high performance and fast run of the simulation code. 
  We apply the Faddeev-Jackiw method to the Hamiltonian analysis of the massless spin-two field. As expected, the reduced Hamiltonian contains only the traceless-transverse tensor, while some, but not all of the non-propagating components are determined by the constraints of the theory. In particular, it is concluded that no gauge choice can be imposed on the fields such that only the propagating modes remain in the theory, meaning that for the spin-2 field there is no direct analogue to the Coulomb gauge of electromagnetism. Implications for General Relativity are discussed. 
  A new class of Bianchi - I cosmological model with magnetic field within the frame work of Lyra geometry has been presented . Exact solutions to the field equations for the model are obtained . The physical and kinematical behaviors of the model have been discussed. 
  Usually, General Relativity (GR) is known to be unrenormalizable perturbatively from the viewpoint of quantum field theory. But in the modern sense of renormalizability, there still remains the possibility to investigate whether GR is "nonpertubatively" renormalizable or not. Here I review the basics and results in this topic based on the "Effective Average Action" approach which was proposed by M.Reuter, and discuss its application to the balck hole geometry. 
  We show that the causal set approach to creating an ever-present cosmological 'constant' in the expanding universe is strongly constrained by the isotropy of the microwave background. Fluctuations generated by stochastic lambda generation which are consistent with COBE and WMAP observations are far too small to dominate the expansion dynamics at z<1000 and so cannot explain the observed late-time acceleration of the universe. We also discuss other observational constraints from the power spectrum of galaxy clustering and show that the theoretical possibility of ever-present lambda arises only in 3+1 dimensional space-times. 
  We propose the mechanism of spontaneous symmetry breaking of a bulk vector field as a way to generate the selection of bulk dimensions invisible to the standard model confined to the brane. By assigning a non-vanishing vacuum value to the vector field, a direction is singled out in the bulk vacuum, thus breaking the bulk Lorentz symmetry. We present the condition for induced Lorentz symmetry on the brane, as phenomenologically required. 
  The mirror relative motion of a suspended Fabry-Perot cavity is studied in the frequency range 3-10 Hz. The experimental measurements presented in this paper, have been performed at the Low Frequency Facility, a high finesse optical cavity 1 cm long suspended to a mechanical seismic isolation system identical to that one used in the VIRGO experiment. The measured relative displacement power spectrum is compatible with a system at thermal equilibrium within its environmental. In the frequency region above 3 Hz, where seismic noise contamination is negligible, the measurement distribution is stationary and Gaussian, as expected for a system at thermal equilibrium. Through a simple mechanical model it is shown that: applying the fluctuation dissipation theorem the measured power spectrum is reproduced below 90 Hz and noise induced by external sources are below the measurement. 
  We present a purely relativistic effect according to which asymmetric oscillations of a quasi-rigid body slow down or accelerate its fall in a gravitational background. 
  The cosmological consequences of a simple scalar field model for the generation of Newton's constant through the spontaneous breaking of scale invariance in a curved space-time are again presented and discussed. Such a model leads to a consistent description wherein the introduction of matter introduces a small perturbation on a de Sitter universe and a time dependence of the gravitational coupling. 
  The evolution equations for tensor perturbations in a generic scalar tensor theory of gravity are presented. Exact solution are given for a specific class of theories and Friedmann-Lema\^{i}tre-Robertson-Walker backgrounds. In these cases it is shown that, although the evolution of tensor models depends on the choice of parameters of the theory, no amplification is possible if the gravitational interaction is attractive. 
  We discuss some lessons from quantum hydrodynamics to quantum gravity. 
  We show that a generalised phantom Chaplygin gas can present a future singularity in a finite future cosmic time. Like the big rip singularity, this singularity would also take place at a finite future cosmic time, but unlike the big rip singularity, it happens for a finite scale factor. In addition, we define a dual of the generalised phantom Chaplygin gas which satisfies the null energy condition. Then, in a Randall-Sundrum 1 brane-world scenario, we show that the same kind of singularity at a finite scale factor arises for a brane filled with a dual of the generalised phantom Chaplygin gas. 
  Einstein's equations are known to lead to the formation of black holes and spacetime singularities. This appears to be a manifestation of the mathematical phenomenon of finite-time blowup: a formation of singularities from regular initial data. We present a simple hyperbolic system of two semi-linear equations inspired by the Einstein equations. We explore a class of solutions to this system which are analogous to static black-hole models. These solutions exhibit a black-hole structure with a finite-time blowup on a characteristic line mimicking the null inner horizon of spinning or charged black holes. We conjecture that this behavior - namely black-hole formation with blow-up on a characteristic line - is a generic feature of our semi-linear system. Our simple system may provide insight into the formation of null singularities inside spinning or charged black holes in the full system of Einstein equations. 
  We study the invariants of spin networks embedded in a three-dimensional manifold which are based on the path integral for SU(2) BF-Theory. These invariants appear naturally in Loop Quantum Gravity, and have been defined as spin-foam state sums. By using the Chain-Mail technique, we give a more general definition of these invariants, and show that the state-sum definition is a special case. This provides a rigorous proof that the state-sum invariants of spin networks are topological invariants. We derive various results about the BF-Theory spin network invariants, and we find a relation with the corresponding invariants defined from Chern-Simons Theory, i.e. the Witten-Reshetikhin-Turaev invariants. We also prove that the BF-Theory spin network invariants coincide with V. Turaev's definition of invariants of coloured graphs embedded in 3-manifolds and thick surfaces, constructed by using shadow-world evaluations. Our framework therefore provides a unified view of these invariants. 
  This paper serves as a preparation of work that focuses on extracting cosmological sectors from Loop Quantum Gravity. We start with studying the extraction of subsystems from classical systems. A classical Hamiltonian system can be reduced to a subsystem of ''relevant observables'' using the pull-back under the Poisson-embedding of the ''relevant part of phase space'' into full phase space. Since a quantum theory can be thought of as a noncommutative phase space, one encounters the problem of embedding noncommutative spaces. We solve this problem for a physically interesting set of quantum systems and embeddings by constructing the noncommutative analogue of the construction of an embedding as the projection to the base space of an embedding of fibre bundles over the involved spaces. This paper focuses on the physical ideas that enter our programme of reduction of quantum theories and tries to explain these on examples rather than abstractly, which will be the focus of a forthcoming paper. 
  It is shown how can be made the classification of all tensors constructed from the Riemann tensor that verify the conservation of gravitational energy momentum. More precisely we explain that there exists a unique tensor of degree n in the Riemann tensor and its contractions that verifies the conservation of energy. We show that this tensor, only because it obeys this degree n structure as well as energy conservation, two facts which are true in all dimensions, verifies in dimension 2n this striking particularity of being Euler gravity. We stick here to the case n=2 but explain briefly why the general case is similar. 
  We propose a new approach to a physical analogy between General Relativity and Electromagnetism, based on comparing tidal tensors of both theories. Using this approach we write a covariant form for the gravitational analogues of the Maxwell equations, from which the regime of validity of the analogy becomes manifest. Two explicit realisations of the analogy are given. The first one matches linearised gravitational tidal tensors to exact electromagnetic tidal tensors in Minkwoski spacetime. The second one matches exact magnetic gravitational tidal tensors for ultra-stationary metrics to exact magnetic tidal tensors of electromagnetism in curved spaces. We then establish a new proof for a class of tensor identities that define invariants of the type $\vec{E}^2-\vec{B}^2$ and $\vec{E}\cdot\vec{B}$, and we exhibit the invariants built from tidal tensors in both gravity and electromagnetism. We contrast our approach with the two gravito-electromagnetic analogies commonly found in the literature, which are reviewed, and argue that our approach makes clear both the limitations and incorrect results within one analogy as well as it trivially solves inconsistencies in the physical interpretation of the other analogy. 
  In this paper I shall extend the formulation of the hole argument to permutable theories. As covariant theories provides a general mathematical framework for classical physics, permutable theories provide the language for quantum physics. This analogy is deeply founded: as covariant theories are defined functorially on the category of manifolds and local diffeomorphisms with values into the category of fibered manifolds and fiber-preserving morphisms, and some rule of selecting sections, permutable theories are functorially defined on the category of sets and permutations with values into the category of fibered sets and fiber-preserving automorphisms, and rules of selecting sections. One of the main features of covariant theories, in particular general relativity, is that the field equation possesses gauge freedom associated with global diffeomorphisms of the underlying manifold. I shall explain here how the hole argument is a reflection of this gauge freedom. Finally I shall point out some implications of the hole argument to current development in physics. 
  We perform a simulation for merger of a black hole (BH)-neutron star (NS) binary in full general relativity preparing a quasicircular state as initial condition. The BH is modeled by a moving puncture with no spin and the NS by the $\Gamma$-law equation of state with $\Gamma=2$. Corotating velocity field is assumed for the NS. The mass of the BH and the rest-mass of the NS are chosen to be $\approx 3.2 M_{\odot}$ and $\approx 1.4 M_{\odot}$ with relatively large radius of the NS $\approx 14$ km. The NS is tidally disrupted near the innermost stable orbit but $\sim 80%$ of the material is swallowed into the BH with small disk mass $\sim 0.3M_{\odot}$ even for such small BH mass $\sim 3M_{\odot}$. The result indicates that the system of a BH and a massive disk of $\sim M_{\odot}$ is not formed from nonspinning BH-NS binaries, although a disk of mass $\sim 0.1M_{\odot}$ is a possible outcome. 
  The arguments were given in a number of our papers that the discrete quantum gravity based on the Regge calculus possesses nonzero vacuum expectation values of the triangulation lengths of the order of Plank scale $10^{-33}cm$. These results are considered paying attention to the possibility of having finite theory within this framework. 
  It is well known that the quantum double structure plays an important role in three dimensional quantum gravity coupled to matter field. In this paper, we show how this algebraic structure emerges in the context of three dimensional Riemannian loop quantum gravity (LQG) coupled to a finite number of massive spinless point particles. In LQG, physical states are usually constructed from the notion of SU(2) cylindrical functions on a Riemann surface $\Sigma$ and the Hilbert structure is defined by the Ashtekar-Lewandowski measure. In the case where $\Sigma$ is the sphere $S^2$, we show that the physical Hilbert space is in fact isomorphic to a tensor product of simple unitary representations of the Drinfeld double DSU(2): the masses of the particles label the simple representations, the physical states are tensor products of vectors of simple representations and the physical scalar product is given by intertwining coefficients between simple representations. This result is generalized to the case of any Riemann surface $\Sigma$. 
  In a companion paper, we have emphasized the role of the Drinfeld double DSU(2) in the context of three dimensional Riemannian Loop Quantum Gravity coupled to massive spinless point particles. We make use of this result to propose a model for a self-gravitating quantum field theory (massive spinless non-causal scalar field) in three dimensional Riemannian space. We start by constructing the Fock space of the free self-gravitating field: the vacuum is the unique DSU(2) invariant state, one-particle states correspond to DSU(2) unitary irreducible simple representations and any multi-particles states is obtained as the symmetrized tensor product between simple representations. The associated quantum field is defined by the usual requirement of covariance under DSU(2). Then, we introduce a DSU(2)-invariant self-interacting potential (the obtained model is a Group Field Theory) and compute explicitely the lowest order terms (in the self-interaction coupling constant $\lambda$) of the propagator and of the three-points function. Finally, we compute the lowest order quantum gravity corrections (in the Newton constant G) to the propagator and to the three-points function. 
  We study the effect of cosmological expansion on orbits--galactic, planetary, or atomic--subject to an inverse-square force law. We obtain the laws of motion for gravitational or electrical interactions from general relativity--in particular, we find the gravitational field of a mass distribution in an expanding universe by applying perturbation theory to the Robertson-Walker metric. Cosmological expansion induces an ($\ddot a/a) \vec r$ force where $a(t)$ is the cosmological scale factor. In a locally Newtonian framework, we show that the $(\ddot a/a) \vec r$ term represents the effect of a continuous distribution of cosmological material in Hubble flow, and that the total force on an object, due to the cosmological material plus the matter perturbation, can be represented as the negative gradient of a gravitational potential whose source is the material actually present. We also consider the effect on local dynamics of the cosmological constant. We calculate the perihelion precession of elliptical orbits due to the cosmological constant induced force, and work out a generalized virial relation applicable to gravitationally bound clusters. 
  We describe preliminary results of a detailed numerical analysis of the volume operator as formulated by Ashtekar and Lewandowski. Due to a simplified explicit expression for its matrix elements, it is possible for the first time to treat generic vertices of valence greater than four. It is found that the vertex geometry characterizes the volume spectrum. 
  We study the renormalized energy-momentum tensor (EMT) of cosmological scalar fluctuations during the slow-rollover regime for power-law inflation and find that it is characterized by a negative energy density at the leading order, with the same time behaviour as the background energy. The average expansion rate appears decreased by the back-reaction of the effective energy of cosmological fluctuations but this value becomes comparable with the energy of background only if inflation starts at a Planckian energy. We also find that, for this particular model, the first and second order inflaton fluctuations are decoupled and satisfy the same equation of motion. To conclude the fourth order adiabatic expansion for the inflaton scalar field is evaluated for a general potential V(\phi). 
  We present a method for extracting gravitational waves from numerical spacetimes which generalizes and refines one of the standard methods based on the Regge--Wheeler--Zerilli perturbation formalism. [abridged] We then present fully nonlinear three-dimensional numerical evolutions of a distorted Schwarzschild black hole in Kerr--Schild coordinates with an odd parity perturbation and analyze the improvement we gain from our generalized wave extraction, comparing our new method to the standard one. [abridged]   We find that, even with observers as far out as $R=80 M$--which is larger than what is commonly used in state-of-the-art simulations--the assumption in the standard method that the background is close to having Schwarzschild-like coordinates increases the error in the extracted waveforms considerably. Even for our coarsest resolutions, our new method decreases the error by between one and two orders of magnitudes. Furthermore, we explicitly see that the errors in the extracted waveforms obtained by the standard method do not converge to zero with increasing resolution. [abridged]   In a general scenario, for example a collision of compact objects, there is no precise definition of gravitational radiation at a finite distance, and gravitational wave extraction methods at such distances are thus inherently approximate. The results of this paper bring up the possibility that different choices in the wave extraction procedure at a fixed and finite distance may result in relative differences in the waveforms which are actually larger than the numerical errors in the solution. 
  We describe an explicit in time, finite-difference code designed to simulate black holes by using the excision method. The code is based upon the harmonic formulation of the Einstein equations and incorporates several features regarding the well-posedness and numerical stability of the initial-boundary problem for the quasilinear wave equation. After a discussion of the equations solved and of the techniques employed, we present a series of testbeds carried out to validate the code. Such tests range from the evolution of isolated black holes to the head-on collision of two black holes and then to a binary black hole inspiral and merger. Besides assessing the accuracy of the code, the inspiral and merger test has revealed that individual apparent horizons can touch and even intersect. This novel feature in the dynamics of the marginally trapped surfaces is unexpected but consistent with theorems on the properties of apparent horizons. 
  We discuss the transition from quasi-circular inspiral to plunge of a system of two nonrotating black holes of masses $m_1$ and $m_2$ in the extreme mass ratio limit $m_1m_2\ll (m_1+m_2)^2$. In this limit, we compare the merger waveforms obtained by two different methods: a {\it numerical} (Regge-Wheeler-Zerilli) one, and an {\it analytical} (Effective One Body) one. This is viewed as a contribution to the matching between analytical and numerical methods. 
  This is a review about LISA and its technology demonstrator, LISA PathFinder. We first describe the conceptual problems which need to be overcome in order to set up a working interferometric detector of low frequency Gravitational Waves (GW), then summarise the solutions to them as currently conceived by the LISA mission team. This will show that some of these solutions require new technological abilities which are still under development, and which need proper test before being fully implemented. LISA PathFinder (LPF) is the the testbed for such technologies. The final part of the paper will address the ideas and concepts behind the PathFinder as well as their impact on LISA. 
  LISA PathFinder (LPF) will be flown with the objective to test in space key technologies for LISA. However its sensitivity goals are, for good reason, one order of magnitude less than those which LISA will have to meet, both in drag-free and optical metrology requirements, and in the observation frequency band. While the expected success of LPF will of course be of itself a major step forward to LISA, one might not forget that a further improvement by an order of magnitude in performance will still be needed. Clues for the last leap are to be derived from proper disentanglement of the various sources of noise which contribute to the total noise, as measured in flight during the PathFinder mission. This paper describes the principles, workings and requirements of one of the key tools to serve the above objective: the diagnostics subsystem. This consists in sets of temperature, magnetic field, and particle counter sensors, together with generators of controlled thermal and magnetic perturbations. At least during the commissioning phase, the latter will be used to identify feed-through coefficients between diagnostics sensor readings and associated actual noise contributions. A brief progress report of the current state of development of the diagnostics subsystem will be given as well. 
  We have found an exact solution of spherically symmetrical Einstein equations describing a black hole with a special type phantom energy source. It is surprising to note that our solution is analogous to Reissner-Nordstr\"{o}m black hole. 
  To illustrate the conceptual problems for the low-energy symmetries in the continuum of spacetime emerging from the discrete quantum geometry, Galileo symmetries are investigated in the polymer particle representation of a non-relativistic particle as a simple toy model. The complete Galileo transformations (translation, rotation and Galileo boost) are naturally defined in the polymer particle Hilbert space and Galileo symmetries are recovered with highly suppressed deviations in the low-energy regime from the underlying polymer particle description. 
  This talk aims at questioning the vanishing of Unruh temperature for an inertial observer in Minkovski spacetime with finite lifetime, arguing that in the non eternal case the existence of a causal horizon is not linked to the non-vanishing of the acceleration. This is illustrated by a previous result, the diamonds temperature, that adapts the algebraic approach of Unruh effect to the finite case. 
  A generalized model of space-time is given, taking into consideration the anisotropic structure of fields which are depended on the position and the direction (velocity).In this framework a generalized FRW-metric the Raychaudhouri and Friedmann equations are studied.A long range vector field of cosmological origin is considered in relation to the physical geometry of space-time in which Cartan connection has a fundamental role.The generalised Friedman equations are produced including anisotropic terms.The variation of anisotropy $z_t$ is expressed in terms of the Cartan torsion tensor of the Finslerian space-time. 
  Quantum radiative characteristics of slowly varying nonstationary Kerr-Newman black holes are investigated by using the method of generalized tortoise coordinate transformation. It is shown that the temperature and the shape of the event horizon of this kind of black holes depend on the time and the angle. Further, we reveal a relationship that is ignored before between thermal radiation and non-thermal radiation, which is that the chemical potential in thermal radiation spectrum is equal to the highest energy of the negative energy state of particles in non-thermal radiation for slowly varying nonstationary Kerr-Newman black holes. Also, we show that the deduced general results can be degenerated to the known conclusion of stationary Kerr-Newman black holes. 
  Astrophysical observations (usually explained by dark matter) suggest that classical mechanics could break down when the acceleration becomes extremely small (the approach known as modified Newtonian dynamics, or MOND). I present the first analysis of MOND manifestations in terrestrial (rather than astrophysical) settings. A new effect is reported: around each equinox date, 2 spots emerge on the Earth where static bodies experience spontaneous acceleration due to the possible violation of the 2nd Newton's law. Preliminary estimates indicate that an experimental search for this effect can be feasible. 
  In this talk we review the perspectives of testing the multidimensional Dvali-Gabadadze-Porrati (DGP) model of modified gravity in the Solar System. The inner planets, contrary to the giant gaseous ones, yield the most promising scenario for the near future. 
  We study in spherical symmetry the conformal compactification for hyperboloidal foliations with nonvanishing constant mean curvature. The conformal factor and the coordinates are chosen such that null infinity is at a fixed radial coordinate location. 
  We consider brane cosmology when the 4D Ricci scalar term is added to the 5D Einstein-Hilbert action. The induced brane dynamics is shown to be the usual Einstein dynamics coupled to a modified energy-momentum tensor which is well defined once the 5D Einstein equations are solved in the bulk. In order to obtain a bulk solution, which will consist of outgoing brane waves, we construct a time-dependent solution for the Randall-Sundrum model with non zero spatial curvature applying boosts along the fifth dimension and patching together isometries broken by the brane. The 5D Einstein equations valid everywhere in the bulk, but not in the brane, are projected on the brane. Then making use for the embedding of the brane in the bulk of the Israel junction conditions, modified by a source term coming from the addition of the intrinsic curvature scalar in the bulk action, it is possible to generate brane cosmological solutions consistent with the bulk geometry for any kind of matter confined to the brane. 
  We investigate whether dark matter can be replaced by various source terms appearing in the effective Einstein equation, valid on a brane embedded into a higher dimensional space-time (the bulk). Such non-conventional source terms include a quadratic (ordinary) matter source term, a geometric source term originating in the Weyl curvature of the bulk, a source term arising from the possible asymmetric embedding, and finally the pull-back to the brane of possible non-standard model bulk fields. 
  In generic models of cosmological inflation, quantum fluctuations strongly influence the spacetime metric and produce infinitely many regions where the end of inflation (reheating) is delayed until arbitrarily late times. The geometry of the resulting spacetime is highly inhomogeneous on scales of many Hubble sizes. The recently developed string-theoretic picture of the "landscape" presents a similar structure, where an infinite number of de Sitter, flat, and anti-de Sitter universes are nucleated via quantum tunneling. Since observers on the Earth have no information about their location within the eternally inflating universe, the main question in this context is to obtain statistical predictions for quantities observed at a random location. I describe the problems arising within this statistical framework, such as the need for a volume cutoff and the dependence of cutoff schemes on time slicing and on the initial conditions. After reviewing different approaches and mathematical techniques developed in the past two decades for studying these issues, I discuss the existing proposals for extracting predictions and give examples of their applications. 
  We investigate here the causal structure of spacetime in the vicinity of a spacetime singularity. The particle and energy emission from such ultra-dense regions forming in gravitational collapse of a massive matter cloud is governed by the nature of non-spacelike paths near the same. These trajectories are examined to show that if a null geodesic comes out from the singularity, then there exist families of future-directed non-spacelike curves which also necessarily escape from the naked singularity. The existence of such families is crucial to the physical visibility of the singularity. We do not assume any underlying symmetries for the spacetime, and earlier considerations on the nature of causal trajectories emerging from a naked singularity are generalized and clarified. 
  The effective evolution of an inhomogeneous universe model in Einstein's theory of gravitation may be described in terms of spatially averaged scalar variables. This evolution can be modeled by solutions of a set of Friedmann equations for an effective scale factor, with matter and backreaction source terms, where the latter can be represented by a minimally coupled scalar field (`morphon field'). We review the basic steps of a description of backreaction effects in relativistic cosmology that lead to refurnishing the standard cosmological equations, but also lay down a number of unresolved issues in connection with their interpretation within observational cosmology. 
  We investigate how deformations of special relativity in momentum space can be extended to position space in a consistent way, such that the dimensionless contraction between wave-vector and coordinate-vector remains invariant. By using a parametrization in terms of an energy dependent speed of light, and an energy dependent Planck's constant, we are able to formulate simple requirements that completely determine the active transformations in position space. These deviate from the standard transformations for large velocities of the observed object. Some examples are discussed, and it is shown how the relativistic mass gain of a massive particle is affected. We finally study the construction of passive Lorentz-transformations. 
  The outline of a recent approach to quantum gravity is presented. Novel ingredients include: (1) Affine kinematical variables; (2) Affine coherent states; (3) Projection operator approach toward quantum constraints; (4) Continuous-time regularized functional integral representation without/with constraints; and (5) Hard core picture of nonrenormalizability. The ``diagonal representation'' for operator representations, introduced by Sudarshan into quantum optics, arises naturally within this program. 
  We discuss a new analytic solution to the Einstein-Maxwell field equations that describes electrically charged black holes with a slow rotation and with a single angular momentum in all higher dimensions. We also compute the gyromagnetic ratio of these black holes. 
  The Ponzano-Regge model of three-dimensional quantum gravity is well-defined when the observables satisfy a certain condition involving the twisted cohomology. In this case, the partition function is defined in terms of the Reidemeister torsion. Some consequences for the special cases of planar graphs and knots are given. 
  A decreasing gravitational constant, G, coupled with angular momentum conservation is expected to increrase a planetary semimajor axis, a, as \dot a/a=-\dot G/G. Analysis of lunar laser ranging data strongly limits such temporal variations and constrains a local (~1 AU) scale expansion of the solar system as \dot a/a=-\dot G/G =-(4\pm9)\times10^{-13} yr^{-1}, including that due to cosmological effects. 
  Dynamics of brane-world models of dark energy is reviewed. We demonstrate that simple dark energy brane models can be represented as 2-dimensional dynamical systems of a Newtonian type. Hence a fictitious particle moving in a potential well characterizes the model. We investigate the dynamics of the brane models using methods of dynamical systems. The simple brane-world models can be successfully unified within a single scheme -- an ensemble of brane dark energy models. We characterize generic models of this ensemble as well as exceptional ones using the notion of structural stability (instability). Then due to the Peixoto theorem we can characterize the class of generic brane models. We show that global dynamics of the generic brane models of dark energy is topologically equivalent to the concordance $\Lambda$CDM model. We also demonstrate that the bouncing models or models in which acceleration of the universe is only transient phenomenon are non-generic (or exceptional cases) in the ensemble. We argue that the adequate brane model of dark energy should be a generic case in the ensemble of FRW dynamical systems on the plane. 
  We study a simple system of two hyperbolic semi-linear equations, inspired by the Einstein equations. The system, which was introduced in gr-qc/0612136, is a model for singularity formation inside black holes. We show for a particular case of the equations that the system demonstrates a finite time blowup. The singularity that is formed is a null singularity. Then we show that in this particular case the singularity has features that are analogous to known features of models of black-hole interiors - which describe the inner-horizon instability. Our simple system may provide insight into the formation of null singularities inside spinning or charged black holes. 
  Quantum radiative characteristics of general nonstationary black holes in the general case are investigated by using the method of generalized tortoise coordinate transformation. It is generally shown that the temperature and the shape of the event horizon of this kind of black holes depend on both the time and different angles. Further, we discover that there is a certain relationship that is ignored before between thermal radiation and non-thermal radiation of black holes, which is that the chemical potential in thermal radiation spectrum is equal to the highest energy of the negative energy state of particles in non-thermal radiation for slowly varying nonstationary black holes. Also, we show that the deduced general results can be applied to different concrete conditions. 
  Quantum radiation of Dirac particles in general nonstationary black holes in the general case is investigated by using the method of generalized tortoise coordinate transformation and considering simultaneously the asymptotic behaviors of the first order and second order forms of Dirac equation near the event horizon. It is generally shown that the temperature and the shape of the event horizon of this kind of black holes depend on both the time and different angles. Further, we give a general expression of the new extra coupling effect in thermal radiation spectrum of Dirac particles which is absent from the thermal radiation spectrum of scalar particles. Also, we reveal a relationship that is ignored before between thermal radiation and non-thermal radiation in the case of scalar particles, which is that the chemical potential in thermal radiation spectrum is equal to the highest energy of the negative energy state of scalar particles in non-thermal radiation for general nonstationary black holes. 
  The accelerating expansion of the present Universe is an exciting challenge to the standard cosmology, which reflects that our current theories on matter have some incompletion. In this paper, we analyze the cosmological model with nonlinear spinor fields source in detail. The results may be able to provide a natural explanation for the puzzles of the acceleration and negative pressure. 
  It is shown that the gravitational field, as a physical field developing in the Minkowsky space, does not lead to unlimited gravitational collapse of massive bodies and, hence, excludes a possibility of the formation of the ``black holes''. 
  We present a method based on the so-called Quantum Energy Inequalities, which allows to compare, and bound, the expectation values of energy-densities of ground states of quantum fields in spacetimes possessing isometric regions. The method supports the conclusion, that the Boulware energy density is universal both: at modest (and far) distances from compact spherical objects, and close to the would-be horizons of the gravastar/QBHO spacetimes. It also provides a natural consistency check for concrete (approximate, numerical) calculations of the expectation values of the energy-momentum tensors. 
  We discuss a new exact solution for self-dual Abelian gauge fields living on the space of the Kerr-Taub-bolt instanton, which is a generalized example of asymptotically flat instantons with non-self-dual curvature. 
  We clarify the conditions under which dark energy models whose Lagrangian densities f are written in terms of the Ricci scalar R are cosmologically viable. The existence of a viable matter dominated epoch prior to a late-time acceleration requires that the variable m=Rf_{,RR}/f_{,R} (where f_{,R}=df/dR) satisfies the conditions m(r) approx +0 and dm/dr>-1 at r approx -1 where r=-Rf_{,R}/f. For the existence of a viable late-time acceleration we require instead either (i) m=-r-1, (sqrt{3}-1)/2 < m le 1 and dm/dr<-1 or (ii) 0< m le 1 at r=-2. These conditions identify two regions in the (r,m) space, one for the matter era and the other for the acceleration. Only models with a m(r) curve that connects these regions and satisfy the requirements above lead to an acceptable cosmology. The models of the type f(R)= alpha R^{-n} and f=R+ alpha R^{-n} do not satisfy these conditions for any n>0 and n<-1 and are thus cosmologically unacceptable. Similar conclusions can be reached for many other examples discussed in the text. In most cases the standard matter era is replaced by a cosmic expansion with scale factor a=t^{1/2}. We show that the cosmological behavior of f(R) models can be understood by a geometrical approach consisting in studying the m(r) curve on the (r,m) plane. This allows us to classify the f(R) models into four general classes, depending on the existence of a standard matter epoch and on the final accelerated stage. Among several other results, we find that f(R) models can have a strongly phantom attractor but in this case there is no acceptable matter era. 
  Full relativistic simulations in three dimensions are known to develop runaway modes that grow exponentially and are accompanied by violations of the Hamiltonian and momentum constraints. We present here a method that controls the violation of these constraints and is tested with simulations of binary neutron stars in circular orbits. We show that this technique improves the overall quality of the simulations. 
  It is shown that the warped product spacetime P=M *_f R and the original spacetime M share necessarily the same causality properties, the only exceptions being the properties of causal continuity and causal simplicity which present some subtleties. In this respect it is shown that the direct product spacetime P=M*R is causally simple if and only if (M,g) is causally simple, has a continuous Lorentzian distance and any two causally related events are connected by a maximizing geodesic. Similar conditions are found for the causally continuous property. Some results concerning the behavior of the Lorentzian distance are obtained. 
  In the current paper we present some new data on the issue of quasi-normal modes (QNMs) of uniform, neutron and quark stars. These questions have already been addressed in the literature before, but we have found some interesting features that have not been discussed so far. We have increased the range of frequency values for the scalar and axial perturbations of such stars and made a comparison between such QNMs and those of the very well-known Schwarzschild black holes. Also addressed in this work was the interesting feature of competing modes, which appear not only for uniform stars, but for quark stars as well. 
  The spherically symmetric steady accretion of polytropic perfect fluids onto a black hole is the simplest flow model that can demonstrate the effects of backreaction. Backreaction keeps intact most of the characteristics of the sonic point. For any such system the mass accretion rate achieves maximal value when the mass of the fluid is 1/3 of the total mass. Fixing the total mass of the system, one observes the existence of two weakly accreting regimes, one overabundant and the other poor in fluid content. 
  Bianchi type V bulk viscous fluid cosmological models are investigated. Using a generation technique (Camci {\it et al.}, 2001), it is shown that the Einstein's field equations are solvable for any arbitrary cosmic scale function. The viscosity coefficient of bulk viscous fluid is assumed to be a power function of mass density. Solutions for particular forms of cosmic scale functions are also obtained. Some physical and geometric aspects of the models are also discussed. 
  Aligned to the Kerr-Schild geometry electromagnetic excitations are investigated, and asymptotically exact solutions are obtained for the low-frequency limit. 
  The Dirac theory of electron and QED neglect gravitational field, while the corresponding to electron Kerr-Newman gravitational field has very strong influence on the Compton distances. It polarizes space-time, deforms the Coulomb field and changes topology. We argue that the Kerr geometry may be hidden beyond the Quantum Theory, representing a complimentary space-time description. 
  Kerr-Schild formalism is generalized by incorporation of the Kerr Theorem with polynomials of higher degrees in $Y\in CP^1.$ It leads to multisheeted twistor spaces and multiparticle solutions. 
  We construct time independent configurations of two gravitating elastic bodies. These configurations either correspond to the two bodies moving in a circular orbit around their center of mass or strictly static configurations. 
  It was shown long ago by T. V. Ruzmaikina and A. A. Ruzmaikin that within the framework of a homogeneous and isotropic cosmological model quadratic corrections of the gravitational field cannot provide solutions that are both regular initially and go over to Friedmann type at later times. We find here, by applying a dynamical systems approach, the general form of the solution to this class of models in the neighborhood of the initial singularity under the above conditions. 
  Using the gradient expansion approach, we formulate a nonlinear cosmological perturbation theory on super-horizon scales valid to $O(\epsilon^2)$, where $\epsilon$ is the expansion parameter associated with a spatial derivative. For simplicity, we focus on the case of a single perfect fluid, but we take into account not only scalar but also vector and tensor modes. We derive the general solution under the uniform-Hubble time-slicing. In doing so, we identify the scalar, vector and tensor degrees of freedom contained in the solution. Then we consider the coordinate transformation to the synchronous gauge to compare our result with the previous result given in the literature. 
  Interactions incorporating the vacuum polarization effects in curved backgrounds modify the null cone structure in such a way that the photon trajectories would not be the spacetime geodesics anymore. The gravitational birefringence introduced as a direct consequence of these effects, will allow shifts in the photon velocities leading to polarization dependent superluminal propagation. Taking these effects into account we study Fermat's principle in the context of the 1+3 (threading) formulation of the spacetime decomposition. Then, as an application of the above ideas, photon velocity shifts in the case of the NUT space as the background gravitational field are calculated. 
  The affine variational principle of Eddington generates the Einstein field equations of general relativity in vacuum with a non-zero cosmological constant. We generalize this principle to include electromagnetism, obtaining the Einstein-Maxwell field equations and the Lorentz equation of motion. We vary the action with respect to the quantities that appear in the definition of the electromagnetic covariant derivative: the affine (nonsymmetric) connection and the electromagnetic potential, while the Lagrangian density is taken to be the square root of the determinant of a linear combination of the symmetrized Ricci tensor and the electromagnetic field tensor. This construction generates a symmetric metric tensor and a connection with torsion that depends only on the torsion vector. The whole formulation is valid only for very weak electromagnetic fields, on the order of the magnetic field in interstellar space. 
  We study light curves and the spectral broadening of the radiation emitted during the finite interval of time by a surface of a collapsing object. We study a simplified model of monochromatic radiations from a spherical surface which is assumed to be falling freely. We discuss the possible way how to infer the physical parameters, such as the mass and radii of emission, from the light curves and spectral broadenings. 
  There is an intriguing analogy between the gravitational dynamics of the horizons and thermodynamics. In case of general relativity, as well as for a wider class of Lanczos-Lovelock theories of gravity, it is possible to interpret the field equations near any spherically symmetric horizon as a thermodynamic identity TdS = dE + PdV. We study this approach further and generalize the results to two more generic cases within the context of general relativity: (i) stationary axis-symmetric horizons and (ii) time dependent evolving horizons. In both the cases, the near horizon structure of Einstein equations can be expressed as a thermodynamic identity under the virtual displacement of the horizon. This result demonstrates the fact that the thermodynamic interpretation of gravitational dynamics is not restricted to spherically symmetric or static horizons but is quite generic in nature and indicates a deeper connection between gravity and thermodynamics. 
  The null surfaces of a spacetime act as one-way membranes and can block information for a corresponding family of observers (time-like curves). Since lack of information can be related to entropy, this suggests the possibility of assigning an entropy to the null surfaces of a spacetime. We motivate and introduce such an entropy functional for any vector field in terms of a fourth-rank divergence free tensor P_{ab}^{cd} with the symmetries of the curvature tensor. Extremising this entropy for all the null surfaces then leads to equations for the background metric of the spacetime. When P_{ab}^{cd} is constructed from the metric alone, these equations are identical to Einstein's equations with an undetermined cosmological constant (which arises as an integration constant). More generally, if P_{ab}^{cd} is allowed to depend on both metric and curvature in a polynomial form, one recovers the Lanczos-Lovelock gravity. In all these cases: (a) We only need to extremise the entropy associated with the null surfaces; the metric is not a dynamical variable in this approach. (b) The extremal value of the entropy agrees with standard results, when evaluated on-shell for a solution admitting a horizon. The role of full quantum theory of gravity will be to provide the specific form of P_{ab}^{cd} which should be used in the entropy functional. With such an interpretation, it seems reasonable to interpret the Lanczos-Lovelock type terms as quantum corrections to classical gravity. 
  Although consensus seems to exist about the validity of equations accounting for radiation reaction in curved space-time, their previous derivations were criticized recently as not fully satisfactory: some ambiguities were noticed in the procedure of integration of the field momentum over the tube surrounding the world-line. To avoid these problems we suggest a purely local derivation dealing with the field quantities defined only {\em on the world-line}. We consider point particle interacting with scalar, vector (electromagnetic) and linearized gravitational fields in the (generally non-vacuum) curved space-time. To properly renormalize the self-action in the gravitational case, we use a manifestly reparameterization-invariant formulation of the theory. Scalar and vector divergences are shown to cancel for a certain ratio of the corresponding charges. We also report on a modest progress in extending the results for the gravitational radiation reaction to the case of non-vacuum background. 
  We investigate higher rank Killing-Yano tensors showing that third rank Killing-Yano tensors are not always trivial objects being possible to construct irreducible Killing tensors from them. We give as an example the Kimura IIC metric were from two rank Killing-Yano tensors we obtain a reducible Killing tensor and from third rank Killing-Yano tensors we obtain three Killing tensors, one reducible and two irreducible. 
  Small violations of spacetime symmetries have recently been identified as promising Planck-scale signals. This talk reviews how such violations can arise in various approaches to quantum gravity, how the emergent low-energy effects can be described within the framework of relativistic effective field theories, how suitable tests can be identified, and what sensitivities can be expected in current and near-future experiments. 
  We consider a d-dimensional spherically symmetric dilatonic R^2 string corrected black hole solution. We study its stability under tensor type gravitational perturbations and compute the absorption cross section for low frequency gravitational waves. 
  The possibility of measuring the second order correlation function of the gravitational waves detectors' currents or photonumbers, and the observation of the gravitational signals by using a spectrum analyzer is discussed. The method is based on complicated data processing and is expected to be efficient for coherent periodic gravitational waves. It is suggested as an alternative method to the conventional one which is used now in the gravitational waves observatories. 
  We study the phase--space of FLRW models derived from Scalar--Tensor Gravity where the non--minimal coupling is $F(\phi)=\xi\phi^2$ and the effective potential is $V(\phi)=\lambda \phi^n$. Our analysis allows to unfold many feature of the cosmology of this class of theories. For example, the evolution mechanism towards states indistinguishable from GR is recovered and proved to depend critically on the form of the potential $V(\phi)$. Also, transient almost--Friedmann phases evolving towards accelerated expansion and unstable inflationary phases evolving towards stable ones are found. Some of our results are shown to hold also for the three level action of string theory. 
  In creating his gravitational field equations Einstein unjustifiedly assumed that inertial mass, and its energy equivalent, is a source of gravity. Denying this assumption allows modifying the field equations to a form in which a positive cosmological constant appears as a uniform density of gravitationally repulsive matter. This repulsive matter is identified as the back sides of the 'drainholes' (called by some 'traversable wormholes') introduced by the author in 1973, which attract on the high, front sides and repel more strongly on the low, back sides. The field equations with a scalar field added produce cosmological models that 'bounce' off a positive minimum of the scale factor and accelerate throughout history. The 'dark drainholes' that radiate nothing visible are hypothesized to constitute the 'dark matter' inferred from observation, their excess of negative active mass over positive active mass driving the accelerating expansion. For a universe with spatial curvature zero, and the ratio of scale factor now to scale factor at bounce equal to the Hubble radius over the Planck length, the model gives an elapsed time since the bounce of two trillion years. The solutions for negative spatial curvature exhibit early stage inflation of great magnitude in short times. Cosmic voids, filaments, and walls are attributed to separation of the back sides of the drainholes from the front, driven by their mutual attractive-repulsive interactions. 
  Geometric objects on manifolds form natural bundles, and one can work with sections of these bundles. In this paper we shall review the structural features of classical fields in the (gauge-)natural bundles framework. We shall discuss various types of interactions between fields, stressing on background dependence and independence. Finally we shall present a gauge natural formulation of general relativity theory, and draw some insights into the structure of the space of 4-geometries. 
  In creating his gravitational field equations Einstein unjustifiedly assumed that inertial mass, and its energy equivalent, is a source of gravity. Denying this assumption allows modifying the field equations to a form in which a positive cosmological constant appears as a uniform density of gravitationally repulsive matter. Field equations with both positive and negative active gravitational mass densities incorporated along with a scalar field coupled to geometry with nostandard polarity yield cosmological solutions that exhibit acceleration, inflation, coasting, and a 'big bounce' instead of a 'big bang'. The repulsive matter is identified as the back sides of the 'drainholes' (called by some 'traversable wormholes') introduced by the author in 1973, solutions of the same field equations, which attract on their high, front sides and repel more strongly on their low, back sides. The front sides serve as the unseen particles of 'dark matter' needed to hold together the large scale structures seen in the universe. Formation of cosmic voids, walls, filaments, and nodes are attributed to separation of the back sides of the drainholes from the front, driven by their mutual attractive-repulsive interactions. One can assert that all of these cosmological entities have been found wrapped in one neat package, namely, the field equations and the variational principle from which they are derived. 
  In this paper, we have considered a model of modified Chaplygin gas in VSL theory with variable gravitational constant $G$. We have shown that the evolution of the universe starts from radiation era to phantom model. The whole evolution of the universe has been shown diagramatically by using statefinder parameters. 
  The issue of radiant spherical black holes being in stable thermal equilibrium with their radiation bath is reconsidered. Using a simple equilibrium statistical mechanical analysis incorporating Gaussian thermal fluctuations in a canonical ensemble of isolated horizons, the heat capacity is shown to diverge at a critical value of the classical mass of the isolated horizon, given (in Planckian units) by the {\it microcanonical} entropy calculated using Loop Quantum Gravity. The analysis reproduces the Hawking-Page phase transition discerned for anti-de Sitter black holes and generalizes it in the sense that nowhere is any classical metric made use of. 
  Paths of test particles, rotating and charged objects in brane-worlds using a modified Bazanski Lagrangian are derived. We also discuss the transition to their corresponding equations in four dimensions. We then make a comparison between the given equations in brane-worlds (BW) and their analog in space-time-matter (STM) theory. 
  This a particularly exciting time for gravitational wave physics. Ground-based gravitational wave detectors are now operating at a sensitivity such that gravitational radiation may soon be directly detected, and recently several groups have independently made significant breakthroughs that have finally enabled numerical relativists to solve the Einstein field equations for coalescing black-hole binaries, a key source of gravitational radiation. The numerical relativity community is now in the position to begin providing simulated merger waveforms for use by the data analysis community, and it is therefore very important that we provide ways to validate the results produced by various numerical approaches. Here, we present a simple comparison of the waveforms produced by two very different, but equally successful approaches--the generalized harmonic gauge and the moving puncture methods. We compare waveforms of equal-mass black hole mergers with minimal or vanishing spins. The results show exceptional agreement for the final burst of radiation, with some differences attributable to small spins on the black holes in one case. 
  Fluctuations of the Cosmic Microwave Background CMB are observed by the WMAP. When expanded into the harmonic eigenmodes of the space part of a cosmological model, they provide insight into the large-scale topology of space. All harmonic polynomials on the multiply connected dodecahedral Poincare space are constructed. Strong and specific selection rules are given by comparing the polynomials to those on the 3-sphere, its simply connected cover. 
  The Cauchy problem is considered for the scalar wave equation in the Kerr geometry. We prove that by choosing a suitable wave packet as initial data, one can extract energy from the black hole, thereby putting supperradiance, the wave analogue of the Penrose process, into a rigorous mathematical framework. We quantify the maximal energy gain. We also estimate the infinitesimal change of mass and angular momentum of the black hole, in agreement with Christodoulou's result for the Penrose process. The main mathematical tool is our previously derived integral representation of the wave propagator. 
  We compute the graviton one loop contribution to a classical energy in a \textit{traversable} wormhole background. The form of the shape function considered is obtained by the equation of state $p=\omega\rho$. We investigate the size of the wormhole as a function of the parameter $\omega$. The investigation is evaluated by means of a variational approach with Gaussian trial wave functionals. A zeta function regularization is involved to handle with divergences. A renormalization procedure is introduced and the finite one loop energy is considered as a \textit{self-consistent} source for the traversable wormhole.The case of the phantom region is briefly discussed. 
  A possible candidate for the late time accelerated expanding Universe is phantom energy, which possesses rather bizarre properties, such as the prediction of a Big Rip singularity and the violation of the null energy condition. The latter is a fundamental ingredient of traversable wormholes, and it has been shown that phantom energy may indeed sustain these exotic geometries. Inspired by the evolving dark energy parameter crossing the phantom divide, we consider in this work a varying equation of state parameter dependent on the radial coordinate, i.e., $\omega(r)=p(r)/\rho(r)$. We shall impose that phantom energy is concentrated in the neighborhood of the throat, to ensure the flaring out condition, and several models are analyzed. We shall also consider the possibility that these phantom wormholes be sustained by their own quantum fluctuations. The energy density of the graviton one loop contribution to a classical energy in a phantom wormhole background and the finite one loop energy density are considered as a self-consistent source for these wormhole geometries. The latter semi-classical approach prohibits solutions with a constant equation of state parameter, which further motivates the imposition of a radial dependent parameter, $\omega(r)$, and only permits solutions with a steep positive slope proportional to the radial derivative of the equation of state parameter, evaluated at the throat. The size of the wormhole throat as a function of the relevant parameters is also explored. 
  Does the Solar System and, more generally, a gravitationally bound system follow the cosmic expansion law ? Is there a cosmological influence on the dynamics or optics in such systems ? The general relativity theory provides an unique and unambiguous answer, as a solution of Einstein equations with local sources (e.g., the Sun), and with the correct (cosmological) limiting conditions. This solution has no analytic expression. A Taylor development of its metric allows a complete treatment of dynamics and optics in gravitationally bound systems, up to the size of galaxy clusters, taking into account both local and cosmological effects. In the solar System, this provides an estimation of the (non zero) cosmological influence on the Pioneer probe: it fails to account for the " Pioneer effect " by about 10 orders of magnitude. We criticize contradictory claims on this topic. 
  The conservative dephasing effects of gravitational self forces for extreme mass-ratio inspirals are studied. Both secular and non-secular conservative effects may have a significant effect on LISA waveforms that is independent of the mass ratio of the system. Such effects need to be included in generated waveforms to allow for accurate gravitational wave astronomy that requires integration times as long as a year. 
  We study the spontaneous excitation of an accelerated multilevel atom in dipole coupling to the derivative of a massless quantum scalar field and separately calculate the contributions of the vacuum fluctuation and radiation reaction to the rate of change of the mean atomic energy of the atom. It is found that, in contrast to the case where a monopole like interaction between the atom and the field is assumed, there appear extra corrections proportional to the acceleration squared, in addition to corrections which can be viewed as a result of an ambient thermal bath at the Unruh temperature, as compared with the inertial case, and the acceleration induced correction terms show anisotropy with the contribution from longitudinal polarization being four times that from the transverse polarization for isotropically polarized accelerated atoms. Our results suggest that the effect of acceleration on the rate of change of the mean atomic energy is dependent not only on the quantum field to which the atom is coupled, but also on the type of the interaction even if the same quantum scalar field is considered. 
  We present a new asymptotically-flat time-machine model made solely of vacuum and dust. The spacetime evolves from a regular spacelike initial hypersurface S and subsequently develops closed timelike curves. The initial hypersurface S is asymptotically flat and topologically trivial. The chronology violation occurs in a compact manner; namely the first closed causal curves form at the boundary of the future domain of dependence of a compact region in S (the core). This central core is empty, and so is the external asymptotically flat region. The intermediate region surrounding the core (the envelope) is made of dust with positive energy density. This model trivially satisfies the weak, dominant, and strong energy conditions. Furthermore it is governed by a well-defined system of field equations which possesses a well-posed initial-value problem. 
  We couple a neutral scalar field and a Majorana fermion field to Einstein gravity represented by the Robertson-Walker metric and find a class of exact cosmological solutions. 
  The search procedure for burst gravitational waves has been studied using 24 hours of simulated data in a network of three interferometers (Hanford 4-km, Livingston 4-km and Virgo 3-km are the example interferometers). Several methods to detect burst events developed in the LIGO Scientific Collaboration (LSC) and Virgo collaboration have been studied and compared. We have performed coincidence analysis of the triggers obtained in the different interferometers with and without simulated signals added to the data. The benefits of having multiple interferometers of similar sensitivity are demonstrated by comparing the detection performance of the joint coincidence analysis with LSC and Virgo only burst searches. Adding Virgo to the LIGO detector network can increase by 50% the detection efficiency for this search. Another advantage of a joint LIGO-Virgo network is the ability to reconstruct the source sky position. The reconstruction accuracy depends on the timing measurement accuracy of the events in each interferometer, and is displayed in this paper with a fixed source position example. 
  Presented in this paper is a detailed and direct comparison of the LIGO and Virgo binary neutron star detection pipelines. In order to test the search programs, numerous inspiral signals were added to 24 hours of simulated detector data. The efficiencies of the different pipelines were tested, and found to be comparable. Parameter estimation routines were also tested. We demonstrate that there are definite benefits to be had if LIGO and Virgo conduct a joint coincident analysis; these advantages include increased detection efficiency and the providing of source sky location information. 
  Optical reference geometry and related concept of inertial forces are investigated in Kerr-de Sitter spacetimes. Properties of the inertial forces are summarized and their typical behaviour is illustrated. The intuitive 'Newtonian' application of the forces in the relativistic dynamics is demonstrated in the case of the test particle circular motion, static equilibrium positions and perfect fluid toroidal configurations. Features of the optical geometry are illustrated by the embedding diagrams of its equatorial plane. The embedding diagrams do not cover whole the stationary regions of the spacetimes, therefore the limits of embeddability are established. A shape of the embedding diagrams is related to the behaviour of the centrifugal force and it is characterized by the number of turning points of the diagrams. Discussion of the number of embeddable photon circular orbits is also included and the typical embedding diagrams are constructed. The Kerr-de Sitter spacetimes are classified according to the properties of the inertial forces and embedding diagrams. 
  We propose that the Universe is filled with a massive vector field, non-minimally coupled to gravitation. The field equations of the model are consistently derived and their application to cosmology is considered. The Friedmann equations acquire an extra dark-energy component, which is proportional to the mass of the vector particle. This leads to a late-time accelerated de Sitter type expansion. The free parameters of the model (gravitational coupling constants and initial value of the cosmological vector field) can be estimated by using the PPN solar system constraints. The mass of the cosmological massive vector particle, which may represent the main component of the Universe, is of the order of 10^-63 g. 
  When a black hole evaporates, there arises a net energy flow from black hole into its outside environment (heat bath). The existence of energy flow means that the thermodynamic state of the whole system, which consists of the black hole and the heat bath, is in a nonequilibrium state. Therefore, in order to study the detail of evaporation process, the nonequilibrium effects of the energy flow should be taken into account. Using the nonequilibrium thermodynamics which has been formulated recently, this paper shows the following: (1) Time scale of black hole evaporation in a heat bath becomes shorter than that of the evaporation in an empty space (a situation without heat bath), because a nonequilibrium effect of temperature difference between the black hole and heat bath appears as a strong energy extraction from the black hole by the heat bath. (2) Consequently a huge energy burst (stronger than that of the evaporation in an empty space) arises at the end of semi-classical stage of evaporation. (3) It is suggested that a remnant of Planck size remains after the quantum stage of evaporation in order to guarantee the increase of total entropy of the whole system. 
  There is nothing to prevent a higher-dimensional anti-de Sitter bulk spacetime from containing various other branes in addition to hosting our universe, presumed to be a positive-tension 3-brane. In particular, it could contain closed, microscopic branes that form the boundary surfaces of void bubbles and thus violate the null energy condition in the bulk. The possible existence of such micro branes can be investigated by considering the properties of the ground state of a pseudo-Wheeler-DeWitt equation describing brane quantum dynamics in minisuperspace. If they exist, a concentration of these micro branes could act as a fluid of exotic matter able to support macroscopic wormholes connecting otherwise distant regions of the bulk. Were the brane constituting our universe to expand into a region of the bulk containing such higher-dimensional macroscopic wormholes, they would likely manifest themselves in our brane as wormholes of normal dimensionality, whose spontaneous appearance and general dynamics would seem inexplicably peculiar. This encounter could also result in the formation of baby universes of a particular type. 
  We propose wormholes solutions by assuming space dependent equation of state parameter. Our models show that the existence of wormholes is supported by arbitrary small quantities of averaged null energy condition (ANEC) violating phantom energy characterized by variable equation state parameter. 
  The purpose of this brief report is to present some results of our on-going project on the asymptotic behaviour of braneworld-type solutions on approach to their possible finite `time' singularities. 
  This paper addresses strong cosmic censorship for spacetimes with self-gravitating collisionless matter, evolving from surface-symmetric compact initial data. The global dynamics exhibit qualitatively different features according to the sign of the curvature $k$ of the symmetric surfaces and the cosmological constant $\Lambda$. With a suitable formulation, the question of strong cosmic censorship is settled in the affirmative if $\Lambda=0$ or $k\le0$, $\Lambda>0$. In the case $\Lambda>0$, $k=1$, we give a detailed geometric characterization of possible "boundary" components of spacetime; the remaining obstruction to showing strong cosmic censorship in this case has to do with the possible formation of extremal Schwarzschild-de Sitter-type black holes. In the special case that the initial symmetric surfaces are all expanding, strong cosmic censorship is shown in the past for all $k,\Lambda$. Finally, our results also lead to a geometric characterization of the future boundary of black hole interiors for the collapse of asymptotically flat data: in particular, in the case of small perturbations of Schwarzschild data, it is shown that these solutions do not exhibit Cauchy horizons emanating from $i^+$ with strictly positive limiting area radius. 
  We discuss the construction of wave packets resulting from the solutions of a class of Wheeler-DeWitt (WD) equations in Robertson-Walker type cosmologies, for arbitrary curvature. We claim that there always exists a unique initial "canonical slope" for a given initial wave function, which optimizes the desirable properties of the resulting wave packet, including good classical-quantum correspondence. This can be properly denoted as a canonical wave packet. We introduce a general method for finding these canonical initial slopes. We present a numerical solutions whenever the problem cannot be solved analytically either in the classical or the quantum domain. 
  We present the complete family of higher dimensional spacetimes that admit a geodesic, shearfree, twistfree and expanding null congruence, thus extending the well-known D=4 class of Robinson-Trautman solutions. Einstein's equations are solved for empty space with an arbitrary cosmological constant and for aligned pure radiation. Main differences with respect to the D=4 case (such as the absence of type III/N solutions, related to ``violations'' of the Goldberg-Sachs theorem in D>4) are pointed out, also in connection with other recent works. A formal analogy with electromagnetic fields is briefly discussed in an appendix, where we demonstrate that multiple principal null directions of null Maxwell fields are necessarily geodesic, and that in D>4 they are also shearing if expanding. 
  The ``moving puncture'' technique has led to dramatic advancements in the numerical simulations of binary black holes. Hannam et.al. have recently demonstrated that, for suitable gauge conditions commonly employed in moving puncture simulations, the evolution of a single black hole leads to a well-known time-independent, maximal slicing of Schwarzschild. They construct the corresponding solution in isotropic coordinates numerically and demonstrate its usefulness, for example for testing and calibrating numerical codes that employ moving puncture techniques. In this Brief Report we point out that this solution can also be constructed analytically, making it even more useful as a test case for numerical codes. 
  When simulating the inspiral and coalescence of a binary black-hole system, special care needs to be taken in handling the singularities. Two main techniques are used in numerical-relativity simulations: A first and more traditional one ``excises'' a spatial neighborhood of the singularity from the numerical grid on each spacelike hypersurface. A second and more recent one, instead, begins with a ``puncture'' solution and then evolves the full 3-metric, including the singular point. While the first approach is mathematically and numerically well-defined, the second one still maintains a non-differentiable point within the black hole. No strong-field evidence has yet been provided to show that the two approaches are indeed dynamically equivalent. To address this question we have used both techniques to evolve a binary system of equal-mass non-spinning black holes and compared the evolution of two curvature 4-scalars with proper time along the invariantly-defined worldline midway between the two black holes. Using Richardson-extrapolation techniques to reduce the influence of the finite-difference truncation error, we find that the moving-punctures and excision evolutions produce the same spacetimes along that worldline. This represents the first strong-field and dynamical evidence that the moving-puncture prescription is robust both mathematically and numerically. 
  We report on the first results of self-consistent second order metric perturbations produced by a point particle moving in the Schwarzschild spacetime. The second order waveforms satisfy a wave equation with an effective source term build up from products of first order metric perturbations and its derivatives. We have explicitly regularized this source term at the particle location as well as at the horizon and at spatial infinity. 
  This paper has been withdrawn by the author, due a crucial formal errors and will replace after corrections. 
  We investigate, from the point of view of a coaccelerated frame, the spontaneous excitation of a uniformly accelerated two-level atom interacting with a scalar field in a thermal state at a finite temperature $T$ and show that the same spontaneous excitation rate for the uniformly accelerated atom in the Minkowski vacuum obtained in the inertial frame can only be recovered in the coaccelerated frame assuming a thermal bath at the Fulling-Davies-Unruh temperature $T_{FDU}=a/2\pi$ for what appears to be the Minkowski vacuum to the inertial observer. Our discussion provides another example of a physical process different from those examined before in the literature to better understand the Fulling-Davies-Unruh effect. 
  In this letter we accurately measure the general relativistic gravitomagnetic Lense-Thirring effect by analyzing the RMS overlap differences of the out-of-plane portion of the orbit of the Martian polar orbiter Mars Global Surveyor (MGS) over a time span of about 5 years (14 November 1999-14 January 2005). Our result is \mu= 1.0018 +/- 0.0053; general relativity predicts \mu=1. A previous test of the Lense-Thirring effect conducted in the Earth's gravitational field with the LAGEOS satellites reached an about 10% level, although it is still controversial. The expected accuracy of the GP-B mission, aimed at the detection of another gravitomagnetic effect in the terrestrial gravitational field, i.e. the Schiff precession of a gyroscope, is about 1%. The precision of our test is 0.5%. 
  We provide details and present additional results on the numerical study of the gravitational-wave emission from the collapse of neutron stars to rotating black holes in three dimensions. More specifically, we concentrate on the advantages and disadvantages of the use of the excision technique and on how alternative approaches to that of excision can be successfully employed. Furthermore, as a first step towards source-characterization, we present a systematic discussion of the influence that rotation and different perturbations have on the waveforms and hence on the energy emitted in gravitational waves. 
  The Mashhoon rotation-spin coupling is studied by means of the parallelism description of general relativity. The relativistic rotational tetrad is exploited, which results in the Minkowski metric, and the torsion axial-vector and Dirac spin coupling will give the Mashhoon rotation-spin term. For the high speed rotating cases, the tangent velocity constructed by the angular velocity $\Ome$ multiplying the distance r may exceed over the speed of light c, i.e., $\Ome r \ge c$, which will make the relativistic factor $\gamma$ infinity or imaginary. In order to avoid this "meaningless" difficulty occurred in $\gamma$ factor, we choose to make the rotation nonuniform and position-dependent in a particular way, and then we find that the new rotation-spin coupling energy expression is consistent with the previous results in the low speed limit. 
  The analogy between general relativity and electromagnetism suggests that there is a galvano-gravitomagnetic effect, which is the gravitational analog of the Hall effect. This new effect takes place when a current carrying conductor is placed in a gravitomagnetic field and the conduction electrons moving inside the conductor are deflected transversally with respect to the current flow. In connection with this galvano-gravitomagnetic effect, we explore the possibility of using current carrying conductors for detecting the gravitomagnetic field of the Earth. 
  It has been shown that a magnetic field proportional to angular velocity of rotation $\omega$ arises around a rotating conductor with the radial gradient of temperature $\nabla_r T$. However, the theoretical value of the proportionality coefficient $10^{-17} {\rm cm}^{5/2}\cdot {\rm g}^{1/2}\cdot {\rm deg}^{-1}$ does not coincide with the experimental one $10^{-8} {\rm cm}^{5/2}\cdot {\rm g}^{1/2}\cdot {\rm deg}^{-1}$ [1]. At least two additional mechanisms act to produce an observable magnetic field in the experiment [1]. First, the fact that the rotation of the conductor is slowed down and second, the nonstationarity of the temperature gradient during the measurements contribute to the vertical magnetic field. The value of the magnetic flux arising from the azimuthal current induced by the ``Coriolis force'' effect on the thermoelectric radial current, is in good agreement with the experimental data [1]. 
  Using post-Newtonian equations of motion for fluid bodies that include radiation-reaction terms at 2.5 and 3.5 post-Newtonian (PN) order O[(v/c)^5] and O[(v/c)^7] beyond Newtonian order), we derive the equations of motion for binary systems with spinning bodies, including spin-spin effects. In particular we determine the effects of radiation-reaction coupled to spin-spin effects on the two-body equations of motion, and on the evolution of the spins. We find that radiation damping causes a 3.5PN order, spin-spin induced precession of the individual spins. This contrasts with the case of spin-orbit coupling, where there is no effect on the spins at 3.5PN order. Employing the equations of motion and of spin precession, we verify that the loss of total energy and total angular momentum induced by spin-spin effects precisely balances the radiative flux of those quantities calculated by Kidder et al. 
  We consider Randall-Sundrum two-brane cosmological model in the low energy gradient expansion approximation by Kanno and Soda. It is a scalar-tensor theory with a specific coupling function. We find a first integral of equations for the A-brane metric and estimate constraints for the dark radiation term. We perform a complementary analysis of the dynamics of the scalar field (radion) using phase space methods and examine convergence towards the limit of general relativity. We find that it is possible to stabilize the radion at a finite value with suitable negative matter densities on the B-brane. 
  We extend Penrose's peeling model for the asymptotic behaviour of solutions to the scalar wave equation at null infinity on asymptotically flat backgrounds, which is well understood for flat space-time, to Schwarzschild and the asymptotically simple space-times of Corvino-Schoen/Chrusciel-Delay. We combine conformal techniques and vector field methods: a naive adaptation of the ``Morawetz vector field'' to a conformal rescaling of the Schwarzschild metric yields a complete scattering theory on Corvino-Schoen/Chrusciel-Delay space-times. A good classification of solutions that peel arises from the use of a null vector field that is transverse to null infinity to raise the regularity in the estimates. We obtain a new characterization of solutions admitting a peeling at a given order that is valid for both Schwarzschild and Minkowski space-times. On flat space-time, this allows large classes of solutions than the characterizations used since Penrose's work. Our results establish the validity of the peeling model at all orders for the scalar wave equation on the Schwarzschild metric and on the corresponding Corvino-Schoen/Chrusciel-Delay space-times. 
  The conventional method to determine the space curvature is to measure the total mass density $\Omega_{tot}$. Unfortunately the observational $\Omega_{tot}$ is closely near the critical value 1. The computation of this paper discloses that $\Omega_{tot}\approx 1$ is an inevitable result of the evolution of the young Universe independent of the space type. So the mass density is not a good criterion to determine if the Universe is open, flat or closed. In this paper we derive a new criterion based on the galaxy counting, which only depends on the cosmological principle and pure geometry. The different type of the curvature will sharply give different results, then the case can be definitely determined. 
  The Sturm-Liouville eigenvalue equation for eigenmodes of the radial oscillations is determined for spherically symmetric perfect fluid configurations in spacetimes with a nonzero cosmological constant and applied in the cases of configurations with uniform distribution of energy density and polytropic spheres. It is shown that a repulsive cosmological constant rises the critical adiabatic index and decreases the critical radius under which the dynamical instability occurs. 
  We study the scattering of light (null geodesics) by two fixed extreme Reissner-Nordstr\"om black holes, in which the gravitational attraction of their masses is exactly balanced with the electrostatic repulsion of their charges, allowing a static spacetime. We identify the set of unstable periodic orbits that constitute the fractal repeller that completely describes the chaotic escape dynamics of photons. In the framework of periodic orbit theory, the analysis of the linear stability of the unstable periodic orbits is used to obtain the main quantities of chaos that characterize the escape dynamics of the photons scattered by the black holes. In particular, the escape rate which is compared with the result obtained from numerical simulations that consider statistical ensembles of photons. We also analyze the dynamics of photons in the proximity of a perturbed black hole and give an analytical estimation for the escape rate in this system. 
  On the basis of a semi-classical analysis of vacuum energy in an expanding spacetime, we describe a non-singular cosmological model in which the vacuum density decays with time, with a concomitant production of matter. During an infinitely long period we have an empty, inflationary universe, with H \approx 1. This primordial era ends in a fast phase transition, during which H and \Lambda decrease to nearly zero in a few Planck times, with release of a huge amount of radiation. The late-time scenario is similar to the standard model, with the radiation phase followed by a long dust era, which tends asymptotically to a de Sitter universe, with vacuum dominating again. An analysis of the redshift-distance relation for supernovas Ia leads to cosmological parameters in agreement with other current estimations. 
  We prove a global foliation result, using areal time, for T^2 symmetric spacetimes with a positive cosmological constant. We then find a class of solutions that exhibit AVTD behavior near the singularity. 
  Propagation of weak plane waves in a universe filled with a general relativistic, homogeneous, isotropic and flat continuum is studied. The continuum is described by analogues of non-relativistic characteristics, namely energy per particle, pressure and Lame coefficients, and considered in the co-moving proper-time gauge. For different modes of plane waves with a given wave covector, differential equations governing the time dependence of the amplitudes are obtained. As an example, we write down equations for plane waves in a universe filled with stiff ultrarigid continuum. 
  We discuss exotic singularities in the evolution of the universe motivated by the progress of observations in cosmology. Among them there are: Big-Rip (BR), Sudden Future Singularities (SFS), Generalized Sudden Future Singularities (GSFS), Finite Density Singularities (FD), type III, and type IV singularities. We relate some of these singularities with higher-order characteristics of expansion such as jerk and snap. We also discuss the behaviour of pointlike objects and classical strings on the approach to these singularities. 
  I discuss the dark energy characterized by the violation of the null energy condition ($\varrho + p \geq 0$), dubbed phantom. Amazingly, it is admitted by the current astronomical data from supernovae. We discuss both classical and quantum cosmological models with phantom as a source of matter and present the phenomenon called phantom duality. 
  Recently some authors concluded that the energy and momentum of the Fiedman universes, flat and closed, are equal to zero locally and globally (flat universes) or only globally (closed universes). The similar conclusion was also done for more general only homogeneous universes (Kasner and Bianchi type I). Such conclusions originated from coordinate dependent calculations performed only in comoving Cartesian coordinates by using the so-called {\it energy-momentum complexes}. By using new coordinate independent expressions on energy and momentum one can show that the Friedman and more general universes {\it needn't be energetic nonentity}. 
  A self-consistent system of interacting spinor and scalar fields is considered within the scope of Bianchi type VI cosmological model filled with a perfect fluid. The contribution of the cosmological constant ($\Lambda$-term) is taken into account as well. Exact self-consistent solutions to the field equations are obtained for a special choice of spatial inhomogeneity and the interaction terms of the spinor and scalar fields. It has been found that some special choice of metric functions can give rise to a singularity-free solutions independent of the value and sign of the $\Lambda$ term. It is also shown that the introduction of a positive $\Lambda$, the most widespread kind of dark energy, leads to the rapid growth of the universe, while the negative one, corresponding to an additional gravitational energy gives rise to an oscillatory or non-periodic mode of expansion. The role of the spatial inhomogeneity in the evolution of the universe is clarified within the scope of the considered models. 
  We give physical explanations of explicit invariant expressions for the energy and angular momentum densities of gravitational fields in stationary space-times. These expressions involve non-locally defined conformal factors. In certain coordinates these become locally defined in terms of the metric. These results are derived via expressions for total gravitational potential energy from the difference between the total energy and the mechanical energy. The latter involves kinetic energy seen in the frame of static observers.   When in the axially symmetric case we consider zero angular momentum observers (who move orthogonally to surfaces of constant time), we find that the angular momentum they attribute to the gravitational field is solely due to their motion. 
  It is explicitly shown that part of the C-metric space-time inside the black hole horizon may be interpreted as the interaction region of two colliding plane waves with aligned linear polarization, provided the rotational coordinate is replaced by a linear one. This is a one-parameter generalization of the degenerate Ferrari-Ibanez solution in which the focussing singularity is a Cauchy horizon rather than a curvature singularity. 
  Certain semi-Riemannian metrics can be decomposed into a Riemannian part and an isochronal part. The properties of such metrics are particularly easy to visualize in a coordinate-free way, using isometric embedding. We present such an isochronal, isometric embedding of the well known Schwarzschild ideal fluid metric in an attempt to see what is happening when the pressure becomes singular. 
  A way for appending sources at the right-hand sides of the field equations of Einstein's unified field theory is recalled. Two exact solutions endowed with point sources in equilibrium are shown, and their physical meaning is discussed. 
  Starting with the whole class of type-D vacuum backgrounds with cosmological constant we show that the separated Teukolsky equation for zero rest-mass fields with spin $s=\pm 2$ (gravitational waves), $s=\pm 1$ (electromagnetic waves) and $s=\pm 1/2$ (neutrinos) is an Heun equation in disguise. 
  The existence of chaotic behavior for the geodesics of the test particles orbiting compact objects is a subject of much current research. Some years ago, Gu\'eron and Letelier [Phys. Rev. E \textbf{66}, 046611 (2002)] reported the existence of chaotic behavior for the geodesics of the test particles orbiting compact objects like black holes induced by specific values of the quadrupolar deformation of the source using as models the Erez--Rosen solution and the Kerr black hole deformed by an internal multipole term. In this work, we are interesting in the study of the dynamic behavior of geodesics around astrophysical objects with intrinsic quadrupolar deformation or nonisotropic stresses, which induces nonvanishing quadrupolar deformation for the nonrotating limit. For our purpose, we use the Tomimatsu-Sato spacetime [Phys. Rev. Lett. \textbf{29} 1344 (1972)] and its arbitrary deformed generalization obtained as the particular vacuum case of the five parametric solution of Manko et al [Phys. Rev. D 62, 044048 (2000)], characterizing the geodesic dynamics throughout the Poincar\'e sections method. In contrast to the results by Gu\'eron and Letelier we find chaotic motion for oblate deformations instead of prolate deformations. It opens the possibility that the particles forming the accretion disk around a large variety of different astrophysical bodies (nonprolate, e.g., neutron stars) could exhibit chaotic dynamics. We also conjecture that the existence of an arbitrary deformation parameter is necessary for the existence of chaotic dynamics. 
  It has been suggested that re-expressing relativity in terms of forces could provide fresh insights. The formalism developed for this purpose only applied to static, or conformally static, space-times. Here we extend it to arbitrary space-times. It is hoped that this formalism may lead to a workable definition of mass and energy in relativity. 
  We write the Mathisson-Papapetrou equations of motion for a spinning particle in a stationary spacetime using the quasi-Maxwell formalism and give an interpretation of the coupling between spin and curvature. 
  We broaden the domain of application of the recently proposed thermal boundary condition of the wave function of the Universe, which has been suggested as the basis of a dynamical selection principle on the landscape of string solutions. 
  We calculate the gravitational self force acting on a pointlike test particle of mass $\mu$, set in a circular geodesic orbit around a Schwarzschild black hole. Our calculation is done in the Lorenz gauge: For given orbital radius, we first solve directly for the Lorenz-gauge metric perturbation using numerical evolution in the time domain; We then compute the (finite) back-reaction force from each of the multipole modes of the perturbation; Finally, we apply the ``mode sum'' method to obtain the total, physical self force. The {\em temporal} component of the self force (which is gauge invariant) describes the dissipation of orbital energy through gravitational radiation. Our results for this component are consistent, to within the computational accuracy, with the total flux of gravitational-wave energy radiated to infinity and through the event horizon. The {\em radial} component of the self force (which is gauge dependent) is calculated here for the first time. It describes a conservative shift in the orbital parameters away from their geodesic values. We thus obtain the $O(\mu)$ correction to the energy and angular momentum parameters (in the Lorenz gauge), as well as the $O(\mu)$ shift in the orbital frequency (which is gauge invariant). 
  We investigate homogeneous and isotropic oscillating cosmologies with multiple fluid components. Transfer of energy between these fluids is included in order to model the effects of non-equilibrium behavior on closed universes. We find exact solutions which display a range of new behaviors for the expansion scale factor. Detailed examples are studied for the exchange of energy from dust or scalar field into radiation. We show that, contrary to expectation, it is unlikely that such models can offer a physically viable solution to the flatness problem. 
  This article reviews a recent work by a couple of colleagues and myself about the shortcomings of the standard explanations of the quantum origin of cosmic structure in the inflationary scenario, and a proposal to address them. The point it that in the usual accounts the inhomogeneity and anisotropy of our universe seem to emerge from an exactly homogeneous and isotropic initial state through processes that do not break those symmetries. We argued that some novel aspect of physics must be called upon to able to address the problem in a fully satisfactory way. The proposed approach is inspired on Penrose's ideas regarding an quantum gravity induced, real and dynamical collapse of the wave function. 
  We discuss the general-relativistic contributions to occur in the electromagnetic properties of a superconductor with a heat flow. The appearance of general-relativistic contribution to the magnetic flux through a superconducting thermoelectric bimetallic circuit is shown. A response of the Josephson junctions to a heat flow is investigated in the general-relativistic framework. Some gravitothermoelectric effects which are observable in the superconducting state in the Earth's gravitational field are considered. 
  The decreasing of the inertial mass density, established in the past for dissipative fluids, is found to be produced by the ``inertial'' term of the transport equation. Once the transport equation is coupled to the dynamical equation one finds that the contribution of the inertial term diminishes the effective inertial mass and the ``gravitational'' force term, by the same factor. An intuitive picture, and prospective applications of this result to astrophysical scenarios are discussed. 
  By choosing the future event horizon as the horizon of the flat FLRW universe, we show that although the interacting holographic dark energy model is able to explain the phantom divide line crossing, but the thermodynamics second law is not respected in this model. We show that if one takes the particle event horizon as the horizon of the universe, besides describing $\omega=-1$ crossing in a consistent way with thermodynamics second law, he is able to determine appropriately the ratio of dark matter to dark energy density at transition time. In this approach, after the first transition from quintessence to phantom, there is another transition from phantom to quintessence phase which avoids the big rip singularity. 
  The effects of noncommutativity in the phase-space of the classical and quantum cosmology of Bianchi models are investigated. Exact solutions in both commutative and noncommutative cases are presented and compared. Further, the Noether symmetries of the Bianchi class A spacetimes are studied in both cases and similarities and differences are discussed. 
  Inspirals of neutron star-neutron star binaries are a promising source of gravitational waves for gravitational wave detectors like LIGO. During the inspiral, the tidal gravitational field of one of the stars can resonantly excite internal modes of the other star, resulting in a phase shift in the gravitational wave signal. We compute using a Fisher-matrix analysis how large the phase shift must be in order to be detectable. For a $1.4 M_\odot, 1.4 M_\odot$ binary the result is $\sim 8.1, 2.9$ and 1.8 radians, for resonant frequencies of $16, 32$ and 64 Hz. The measurement accuracies of the other binary parameters are degraded by inclusion of the mode resonance effect. 
  We show that under gravity the effective masses for neutrino and antineutrino are different which opens a possible window of neutrino-antineutrino oscillation even if the rest masses of the corresponding eigenstates are same. This is due to CPT violation and possible to demonstrate if the neutrino mass eigenstates are expressed as a combination of neutrino and antineutrino eigenstates, as of the neutral kaon system, with the plausible breaking of lepton number conservation. In early universe, in presence of various lepton number violating processes, this oscillation might lead to neutrino-antineutrino asymmetry which resulted baryogenesis from the B-L symmetry by electro-weak sphaleron processes. On the other hand, for Majorana neutrinos, this oscillation is expected to affect the inner edge of neutrino dominated accretion disks around a compact object by influencing the neutrino sphere which controls the accretion dynamics, and then the related type-II supernova evolution and the r-process nucleosynthesis. 
  We aim to study the thermodynamic properties of the spherically symmetric reference frames with uniform acceleration, including the spherically symmetric generalization of Rindler reference frame and the new kind of uniformly accelerated reference frame. We find that, unlike the general studies about the horizon thermodynamics, one cannot obtain the laws of thermodynamics for their horizons in the usual approaches, despite that one can formally define an area entropy (Bekenstein-Hawking entropy). In fact, the common horizon for a set of uniformly accelerated observers is not always exist, even though the Hawking-Unruh temperature is still well-defined. This result indicates that the Hawking-Unruh temperature is only a kinematic effect, to gain the laws of thermodynamics for the horizon, one needs the help of dynamics. Our result is in accordance with those from the various studies about the acoustic black holes. 
  In a Kaluza-Klein background $V^4\otimes S^3$, we provide a way to reproduce, by the dimensional reduction, a 4-spinor with a SU(2) gauge coupling. Since additional gauge violating terms cannot be avoided, we compute their order of magnitude by virtue of the application to the Electro-Weak model. 
  We analyze the behavior of a spinning particle in gravity, both from a quantum and a classical point of view. We infer that, since the interaction between the space-time curvature and a spinning test particle is expected, then the main features of such an interaction can get light on which degrees of freedom have physical meaning in a quantum gravity theory with fermions. Finally, the dimensional reduction of Papapetrou equations is performed in a 5-dimensional Kaluza-Klein background and Dixon-Souriau results for the motion of a charged spinning body are obtained. 
  The scheme developed by Hartle for describing slowly rotating bodies in 1967 was applied to the simple model of constant density by Chandrasekhar and Miller in 1974. The pivotal equation one has to solve turns out to be one of Heun's equations. After a brief discussion of this equation and the chances of finding a closed form solution, a quickly converging series solution of it is presented. A comparison with numerical solutions of the full Einstein equations allows one to truncate the series at an order appropriate to the slow order approximation. The truncated solution is then used to provide explicit expressions for the metric. 
  We review some applications of relativistic shells that are relevant in the context of quantum gravity/quantum cosmology. Using a recently developed approach, the stationary states of this general relativistic system can be determined in the semiclassical approximation. We suggest that this technique might be of phenomenological relevance in the context of the brane-world scenario and we draw a picture of the general set-up and of the possible developments. 
  In this thesis we investigate certain cosmological brane world models of the Randall-Sundrum type. The models are motivated by string theory but we focus on the phenomenology of the cosmology. Two models of specific interest are the Dvali-Gabadadze-Porrati (DGP, induced-gravity) model, where the brane action is modified, and the Gauss-Bonnet model where the bulk action is modified. Both of these modifications maybe motivated by string theory. We provide a brief review of Randall-Sundrum models and then consider the Kaluza-Klein modes on Minkowski and de Sitter branes, in both the two and one brane cases. The spectrum obtained for the de Sitter branes is a new result. We then consider a Friedmann-Robertson-Walker brane in order to investigate the cosmological dynamics on the brane. We present a brief discussion of the DGP and Gauss-Bonnet brane worlds. We then investigate the Gauss-Bonnet-Induced-Gravity (GBIG) model where the Gauss-Bonnet (GB) bulk term is combined with the induced-gravity (IG) brane term of the DGP model. We present a thorough investigation of cosmological dynamics, in particular focusing on GBIG models that behave like self-accelerating DGP models at late times but at early times show the remarkable feature of a finite-temperature Big Bang. We also discuss the constraints from observations, including ages and Big Bang nucleosynthesis. 
  A model of a stasis chamber, slowing the passage of time in its interior down to arbitrarily small rates relative to the outside world, is considered within classical general relativity. Real and apparent (gravitational) forces as perceived by an interior observer are altered, but in opposite ways. Comparison with special-relativistic time dilation shows the use of such a static chamber to be economical only when the most drastic slowing of time relative to the outside world is desired (d(tau)/dt < 10^{-20}) or when one wants to avoid spending the time needed to accelerate to relativistic speeds. 
  A transitional layer matching the asymptotically flat exterior of a charged, massive toroid to an interior with spatially cylindrical symmetry is described. The changes in the geometry, which by themselves would require an energy tensor violating the energy conditions of classical general relativity, are compensated for by an additional strong electric field. Part of its gradient is consumed, depending on how much the exterior toroid departs from local cylindrical symmetry; what is left over can be used to effect further transitions in the cylindrical interior. An example is given of such a transition creating an angular deficit, allowing an otherwise infinite cosmic string to be captured within a bounded source. 
  The ringdown phase following a binary black hole merger is usually assumed to be well described by a linear superposition of complex exponentials (quasinormal modes). In the strong-field conditions typical of a binary black hole merger, non-linear effects may produce mode coupling. Mode coupling can also be induced by the black hole's rotation, or by expanding the radiation field in terms of spin-weighted spherical harmonics (rather than spin-weighted spheroidal harmonics). Observing deviations from the predictions of linear black hole perturbation theory requires optimal fitting techniques to extract ringdown parameters from numerical waveforms, which are inevitably affected by numerical error. So far, non-linear least-squares fitting methods have been used as the standard workhorse to extract frequencies from ringdown waveforms. These methods are known not to be optimal for estimating parameters of complex exponentials. Furthermore, different fitting methods have different performance in the presence of noise. The main purpose of this paper is to introduce the gravitational wave community to modern variations of a linear parameter estimation technique first introduced in 1795 by Prony: the Kumaresan-Tufts and matrix pencil methods. Using ``test'' damped sinusoidal signals in Gaussian white noise we illustrate the advantages of these methods, showing that they have variance and bias at least comparable to standard non-linear least-squares techniques. Then we compare the performance of different methods on unequal-mass binary black hole merger waveforms. The methods we discuss should be useful both theoretically (to monitor errors and search for non-linearities in numerical relativity simulations) and experimentally (for parameter estimation from ringdown signals after a gravitational wave detection). 
  Recently, a spin one half matter field with mass dimension one was discovered, called Elko spinors. The present work shows how to introduce these fields into a curved spacetime by the standard covariantisation scheme. After formulating the coupled Einstein-Elko field equations, the spacetime is assumed to be homogeneous and isotropic in order to simplify the resulting field equations. Analytical ghost Elko solutions are constructed which have vanishing energy-momentum tensor without and with cosmological constant. The cosmological Elko theory is finally related to the standard scalar field theory with self interaction that gives rise to inflation and it is pointed out that the Elko spinors are not only prime dark matter candidates but also prime candidates for inflation. 
  We have studied the quasinormal modes and the late-time tail behaviors of scalar perturbations in the Schwarzschild black hole pierced by a cosmic string. Although the metric is locally identical to that of the Schwarzschild black hole so that the presence of the string will not imprint in the motion of test particles, we found that quasinormal modes and the late time tails can be effective ways to disclose physical signatures of the cosmic string. 
  Using the third-order WKB approximation and monodromy methods, we investigate the influence of Lorentz violating coefficient $b$ (associated with a special axial-vector $b_{\mu}$ field) on Dirac quasinormal modes in the Schwarzschild black hole spacetime. At fundamental overtone, the real part decreases linearly as the parameter $b$ increases. But the variation of the imaginary part with $b$ becomes more complex. For the larger multiple moment $k$, the magnitude of imaginary part increases with the increase of $b$, which means that presence of Lorentz violation makes Dirac field damps more rapidly. At high overtones, it is found that the real part of high-damped quasinormal frequency does not tend to zero, which is quite a different from the symptotic Dirac quasinormal modes without Lorentz violation. 
  Dynamical dark energy (DE) has been proposed to explain various aspects of the cosmological constant (CC) problem(s). For example, it is very difficult to accept that a strictly constant Lambda-term constitutes the ultimate explanation for the DE in our Universe. It is also hard to acquiesce in the idea that we accidentally happen to live in an epoch where the CC contributes an energy density value right in the ballpark of the rapidly diluting matter density. It should perhaps be more plausible to conceive that the vacuum energy, is actually a dynamical quantity as the Universe itself. More generally, we could even entertain the possibility that the total DE is in fact a mixture of vacuum energy and other dynamical components (e.g. fields, higher order terms in the effective action etc) which can be represented collectively by an effective entity X (dubbed the ``cosmon''). The ``cosmon'', therefore, acts as a dynamical DE component different from the vacuum energy. While it can actually behave phantom-like by itself, the overall DE fluid may effectively appear as standard quintessence, or even mimic at present an almost exact CC behavior. Thanks to the versatility of such cosmic fluid we can show that a composite DE system of this sort (``LXCDM'') may have a key to resolving the mysterious coincidence problem. 
  Caianiello's fundamental derivation of Quantum Geometry through an isometric immersion procedure from T^*M in the space-time manifold M is reconsidered. In the new derivation, the non-linear connection and the bundle formalism induce a metric in the 4-dimensional manifold M that is covariant under arbitrary local coordinate transformations. These models have the intrinsic feature that gravity should be supplied with other interactions, if non-trivial models with maximal acceleration arerequired to be covariant and the Equivalence Principle is maintained. 
  We show that in multidimensional gravity vector fields completely determine the structure and properties of singularity. It turns out that in the presence of a vector field the oscillatory regime exists for any number of spatial dimensions and for all homogeneous models. We derive the Poincar\'e return map associated to the Kasner indexes and fix the rules according to which the Kasner vectors rotate. In correspondence to a 4-dimensional space time, the oscillatory regime here constructed overlap the usual Belinski-Khalatnikov-Liftshitz one. 
  We outline the covariant nature of the chaos characterizing the generic cosmological solution near the initial singularity. Our analysis is based on a "gauge" independent ADM-reduction of the dynamics to the physical degrees of freedom, and shows that the dynamics is isomorphic point by point in space to a billiard on a Lobachevsky plane. The Jacobi metric associated to the geodesic flow is constructed and a non-zero Lyapunov exponent is explicitly calculated. The chaos covariance emerges from the independence of the form of the lapse function and the shift vector. 
  We refine Misner's analysis of the classical and quantum Mixmaster in the fully inhomogeneous picture; we both connect the quantum behavior to the ensemble representation, both describe the precise effect of the boundary conditions on the structure of the quantum states. 
  We consider a Schr\"odinger quantum dynamics for the gravitational field associated to a FRW spacetime and then we solve the corresponding eigenvalue problem. We show that, from a phenomenological point of view, an Evolutionary Quantum Cosmology overlaps the Wheeler-DeWitt approach. We also show how a so peculiar solution can be inferred to describe the more interesting case of a generic cosmological model. 
  We study quantum radiation emitted during the collapse of a quantized, gravitating, spherical domain wall. The amount of radiation emitted during collapse now depends on the wavefunction of the collapsing wall and the background spacetime. If the wavefunction is initially in the form of a sharp wavepacket, the expectation value of the particle occupation number is determined as a function of time and frequency. The results are in good agreement with our earlier semiclassical analysis and show that the quantum radiation is non-thermal and evaporation accompanies gravitational collapse. 
  In this work, we develop and apply the WKB approximation to several examples of noncommutative quantum cosmology, obtaining the time evolution of the noncommutative universe, this is done starting from a noncommutative quantum formulation of cosmology where the noncommutativity is introduced by a deformation on the minisuperspace variables. This procedure gives a straightforward algorithm to incorporate noncommutativity to cosmology and inflation. 
  Ehlers and Kundt have provided an approximate procedure to demonstrate that gravitational waves impart momentum to test particles. This was extended to cylindrical gravitational waves by Weber and Wheeler. Here a general, exact, formula for the momentum imparted to test particles in arbitrary spacetimes is presented. 
  We study the stability of black holes that are solutions of the dilaton gravity derived from string-theoretical models in two and five dimensions against to scalar field perturbations, using the Quasinormal Modes (QNMs) approach. In order to find the QNMs corresponding to a black hole geometry, we consider perturbations described by a massive scalar field non-minimally coupled to gravity. We find that the QNM's frequencies turn out to be pure imaginary leading to purely damped modes, that is in agreement with the literature of dilatonic black holes. Our result exhibits the unstable behavior of the considered geometry against the scalar perturbations. We consider both the minimal coupling case, i.e., for which the coupling parameter $\zeta$ vanishes, and the case $\zeta={1/4}$. 
  Classical particle-like solutions of field equations such as general relativity, could account for dark matter. Such particles would not interact quantum mechanically and would have negligible interactions apart from gravitation. As a relic from the big bang, they would be a candidate for cold dark matter consistent with observations. 
  We extract the Weyl scalars $\Psi_0$ and $\Psi_4$ in the quasi-Kinnersley tetrad by finding initially the (gauge--, tetrad--, and background--independent) transverse quasi-Kinnersley frame. This step still leaves two undetermined degrees of freedom: the ratio $|\Psi_0|/|\Psi_4|$, and one of the phases (the product $|\Psi_0|\cdot |\Psi_4|$ and the {\em sum} of the phases are determined by the so-called BB radiation scalar). The residual symmetry ("spin/boost") can be removed by gauge fixing of spin coefficients in two steps: First, we break the boost symmetry by requiring that $\rho$ corresponds to a global constant mass parameter that equals the ADM mass (or, equivalently in perturbation theory, that $\rho$ or $\mu$ equal their values in the no-radiation limits), thus determining the two moduli of the Weyl scalars $|\Psi_0|, |\Psi_4|$, while leaving their phases as yet undetermined. Second, we break the spin symmetry by requiring that the ratio $\pi/\tau$ gives the expected polarization state for the gravitational waves, thus determining the phases. Our method of gauge fixing--specifically its second step--is appropriate for cases for which the Weyl curvature is purely electric. Applying this method to Misner and Brill--Lindquist data, we explicitly find the Weyl scalars $\Psi_0$ and $\Psi_4$ perturbatively in the quasi-Kinnersley tetrad. 
  The Laser Astrometric Test of Relativity (LATOR) experiment is designed to explore general theory of relativity in the close proximity to the Sun -- the most intense gravitational environment in the solar system. Using independent time-series of highly accurate measurements of the Shapiro time-delay (interplanetary laser ranging accurate to 3 mm at 2 AU) and interferometric astrometry (accurate to 0.01 picoradian), LATOR will measure gravitational deflection of light by the solar gravity with accuracy of 1 part in a billion -- a factor ~30,000 better than currently available. LATOR will perform series of highly-accurate tests in its search for cosmological remnants of scalar field in the solar system. We present science, technology and mission design for the LATOR mission. 
  We consider, in Einstein's theory of gravity, smooth (non-degenerate) stationary black hole spacetimes of arbitrary dimension, $n\geq 3$, with matter and allowing non-zero cosmological constant. The matter fields are assumed to satisfy suitable hyperbolic evolution equations and the dominant energy condition, moreover, the energy momentum tensor is required to be `regular'. We assume that the space of null generators of the event horizon is compact. By introducing and making use of the concept of ``spacetime conjugation'' it is shown then that there always exists a Killing vector field in a neighbourhood, as opposed to all the earlier results, on both sides of the event horizon which is normal to the horizon. If, in addition, the domain of outer communication can be smoothly foliated by a 1-parameter family of null hypersurfaces, each generated by congruences of shear free null geodesics, the horizon Killing vector field is shown to exist on the entire of the domain of outer communication, without assuming analyticity. Since the existence of such a foliation can be proved (gr-qc/0701104), e.g., in case of four dimensional asymptotically flat or asymptotically (locally) anti-de-Sitter electrovac spacetimes our result provides an immediate generalisation of the black hole rigidity theorem of Hawking to the corresponding non-analytic setting. 
  In a recent paper (gr-qc/0701103) the present author introduced the concept of spacetime conjugation with the help of which the most important gap of the black hole rigidity argument could be filled up. This paper is to provide another use of this concept in general relativity. Here, attention will be restricted to the case of smooth (non-degenerate) four dimensional electrovac stationary black hole spacetimes in Einstein's theory of gravity. By making use of a combination of the Newman-Penrose formalism and that of the null characteristic initial value formulation of Friedrich, we shall determine first the necessary and sufficient condition guaranteeing the existence of globally well-defined Gaussian null coordinate systems in the domain of outer communication. This justifies the claim of (gr-qc/0701103), that in certain cases the associated domain of outer communication can be smoothly foliated by a 1-parameter family of null hypersurfaces, each generated by congruences of shear free null geodesics. To show the existence of globally ``well-behaving'' Gaussian null coordinates, as an interesting new result, we also prove that the domain of dependence associated with the selected class of spacetimes--in the characteristic initial value problem based on two null hypersurfaces intersecting in a 2-dimensional spacelike surface--is larger then it has been known to be before, it contains at least a full four dimensional elementary spacetime neighbourhood of the initial data surface. 
  The problem of motion in General Relativity has lost its academic status and become an active research area since the next generation of gravity wave detectors will rely upon its solution. Here we will show, within scalar gravity, how ideas borrowed from Quantum Field Theory can be used to solve the problem of motion in a systematic fashion. We will concentrate in Post-Newtonian corrections. We will calculate the Einstein-Infeld-Hoffmann action and show how a systematic perturbative expansion puts strong constraints on the couplings of non-derivative interactions in the theory. 
  NRGR, an Effective Field Theory approach to gravity, has emerged as a powerful tool to systematically compute higher order corrections in the Post-Newtonian expansion. Here we discuss in somehow more detail the recently reported new results for the spin-spin gravitational potential at third Post-Newtonian order. 
  This lecture provides us with Newtonian approaches for the interpretation of two puzzling cosmological observations that are still discussed subject : a bulk flow and a foam like structure in the distribution of galaxies. For the first one, we model the motions describing all planar distortions from Hubble flow, in addition of two classes of planar-axial distortions with or without rotation, when spatial distribution of gravitational sources is homogenous. This provides us with an alternative to models which assume the presence of gravitational structures similar to Great Attractor as origin of a bulk flow. For the second one, the model accounts for an isotropic universe constituted by a spherical void surrounded by a uniform distribution of dust. It does not correspond to the usual embedding of a void solution into a cosmological background solution, but to a global solution of fluid mechanics. The general behavior of the void expansion shows a huge initial burst, which freezes asymptotically up to match Hubble expansion. While the corrective factor to Hubble law on the shell depends weakly on cosmological constant at early stages, it enables us to disentangle significantly cosmological models around redshift z ~ 1.7. The magnification of spherical voids increases with the density parameter and with the cosmological constant. An interesting feature is that for spatially closed Friedmann models, the empty regions are swept out, what provides us with a stability criterion. 
  We show that Einstein equations are compatible with the presence of massive point particle idealization and find the corresponding two parameter family of solutions. They are complete defined by the bare mechanical mass $M>0$ and the Keplerian mass $m>0$ ($m < M$) of the point source of gravity. The global analytical properties of these solutions in the complex plane define a unique preferable radial variable of the one particle problem.   These new solutions are fundamental solutions of the quasi-linear Einstein equations. We introduce and discuss a novel nonlinear superposition principle for solutions of Einstein equations and discover the basic role of the relativistic analog of the Newton gravitational potential. For the relativistic potential we introduce a simple quasi-linear superposition principle as a new physical requirement for the initial conditions for Einstein equations, thus justifying the instant gravistatic case for N particle system.   This superposition principle allows us to sketch a new theory of the gravitational mass defect. In it a specific Mach-like principle for the Keplerian mass $m$ is valid, i.e. it depends on the mass distribution in the universe, in contrast to the bare mass $M$, which remains a true constant. Several basic examples both of discrete and of continuous mass distributions are considered. 
  The accurate modelling of astrophysical scenarios involving compact objects and magnetic fields, such as the collapse of rotating magnetized stars to black holes or the phenomenology of gamma-ray bursts, requires the solution of the Einstein equations together with those of general-relativistic magnetohydrodynamics. We present a new numerical code developed to solve the full set of general-relativistic magnetohydrodynamics equations in a dynamical and arbitrary spacetime with high-resolution shock-capturing techniques on domains with adaptive mesh refinements. After a discussion of the equations solved and of the techniques employed, we present a series of testbeds carried out to validate the code and assess its accuracy. Such tests range from the solution of relativistic Riemann problems in flat spacetime, over to the stationary accretion onto a Schwarzschild black hole and up to the evolution of oscillating magnetized stars in equilibrium and constructed as consistent solutions of the coupled Einstein-Maxwell equations. 
  We construct quasiequilibrium sequences of black hole-neutron star binaries in general relativity. We solve Einstein's constraint equations in the conformal thin-sandwich formalism, subject to black hole boundary conditions imposed on the surface of an excised sphere, together with the relativistic equations of hydrostatic equilibrium. In contrast to our previous calculations we adopt a flat spatial background geometry and do not assume extreme mass ratios. We adopt a Gamma=2 polytropic equation of state and focus on irrotational neutron star configurations as well as approximately nonspinning black holes. We present numerical results for ratios of the black hole's irreducible mass to the neutron star's ADM mass in isolation of M_{irr}^{BH}/M_{ADM,0}^{NS} = 1, 2, 3, 5, and 10. We consider neutron stars of baryon rest mass M_B^{NS}/M_B^{max} = 83% and 56%, where M_B^{max} is the maximum allowed rest mass of a spherical star in isolation for our equation of state. For these sequences, we locate the onset of tidal disruption and, in cases with sufficiently large mass ratios and neutron star compactions, the innermost stable circular orbit. We compare with previous results for black hole-neutron star binaries and find excellent agreement with third-order post-Newtonian results, especially for large binary separations. We also use our results to estimate the energy spectrum of the outgoing gravitational radiation emitted during the inspiral phase for these binaries. 
  We consider the cosmologies that arise in a subclass of f(R) gravity with f(R)=R+\mu ^{2n+2}/(-R)^{n} and -1<n<0 in the metric (as opposed to the Palatini) variational approach to deriving the gravitational field equations. The calculations of the isotropic and homogeneous cosmological models are undertaken in the Jordan frame and at both the background and the perturbation levels. For the former, we also discuss the connection to the Einstein frame in which the extra degree of freedom in the theory is associated with a scalar field sharing some of the properties of a 'chameleon' field. For the latter, we derive the cosmological perturbation equations in general theories of f(R) gravity in covariant form and implement them numerically to calculate the cosmic-microwave-background temperature and matter-power spectra of the cosmological model. The CMB power is shown to reduce at low l's, and the matter power spectrum is almost scale-independent at small scales, thus having a similar shape to that in standard general relativity. These are in stark contrast with what was found in the Palatini f(R) gravity, where the CMB power is largely amplified at low l's and the matter spectrum is strongly scale-dependent at small scales. These features make the present model more adaptable than that arising from the Palatini f(R) field equations, and none of the data on background evolution, CMB power spectrum, or matter power spectrum currently rule it out. 
  Two theorems are proved concerning how stationary axisymmetric electrovac spacetimes that are equatorially symmetric or equatorially antisymmetric can be characterized correctly in terms of the Ernst potentials $\E$ and $\Phi$ or in terms of axis-data. 
  In this second paper devoted to the equatorial symmetry/antisymmetry of stationary axisymmetric electrovac spacetimes we show how two theorems proved in our previous paper (Ernst, Manko and Ruiz 2006 Class. Quantum Grav. 23, 4945) can be utilized to construct exact solutions that are equatorially symmetric or antisymmetric. 
  In this paper we study the behaviour of gravitational wave background (GWB) generated during inflation in the environment of the noncommutative field approach. From this approach we derive out one additive term, and then we find that the dispersion relation of the gravitational wave would be modified and the primordial gravitational wave would obtain an effective mass. Therefore it breaks lorentz symmetry in local. Moreover, this additive term would a little raise up the energy spectrum of GWB in low frequency and then greatly suppress the spectrum at even lower energy scale of which the wave length may be near the current horizon. Therefore, a sharp peak is formed on the energy spectrum in the range of low frequencies. This peak should be a key criterion to detect the possible existence of noncommutativity of space-time in the background of our universe and a critical test for breaking lorentz symmetry in local field theory. Adding all possible effects on the evolution of GWB, we give some new information of the tensor power spectrum and its energy spectrum which may be probed in the future cosmological observations. 
  We discuss how to construct the full Schwarzschild (Kruskal-Szekeres) spacetime in one swoop by using the bundle of orthonormal Lorentz frames and the Einstein equation without the use of coordinates. We never have to write down the Kruskal-Szekeres or an equivalent form of the metric. 
  The Averaging problem in general relativity and cosmology is discussed. The approach of macroscopic gravity to resolve the problem is presented. An exact cosmological solution to the equations of macroscopic gravity is given and its properties are discussed.   Contents:   1. Introduction to General Relativity 2. General Relativity -> Relativistic Cosmology 3. Introduction to Relativistic Cosmology 4. Relativistic Cosmology -> Mathematical Cosmology 5. Averaging Problem in Relativistic Cosmology 6. History of the Averaging Problem 7. Averaging and FLRW Cosmologies 8. Averaging Problem in General Relativity 9. Macroscopic Gravity and General Relativity 10. Macroscopic Gravity and Cosmology 11. The System of Macroscopic Gravity Equations 12. An Exact Cosmological Solution 
  We review a procedure to use semiclassical methods in the quantization of General Relativistic shells and apply these techniques in some simplified models of inflationary cosmology. Some interesting open issues are introduced and the relevance of their solution in the broader context of Quantum Gravity is discussed. 
  Starting with a new theory of symmetries generated by isometries in field theories with spin, one finds the generators of the spinor representation in backgrounds with a given symmetry. In this manner one obtains a collection of conserved operators from which one can chose the complete sets of commuting operators defining quantum modes. In this framework, the quantum modes of the free Dirac field on de Sitter or anti-de Sitter spacetimes can be completely derived in static or moving charts. One presents the discrete quantum modes, in the central static charts of the anti-de Sitter spacetime, whose eigenspinors can be normalized. The consequence is that the second quantization can be done in this case in canonical manner. For the free Dirac on de Sitter manifolds this can not be done in static charts being forced to consider the moving ones. The quantum modes of the free Dirac field in these charts are used for writing down the quantum Dirac field and the its one-particle operators. 
  Approximative analytic solutions of the Dirac equation in the geometry of Schwarzschild black holes are derived obtaining information about the discrete energy levels and the asymptotic behavior of the energy eigenspinors. 
  Recently we have presented a new formulation of the theory of gravity based on an implementation of the Einstein Equivalence Principle distinct from General Relativity. The kinetic part of the theory - that describes how matter is affected by the modified geometry due to the gravitational field - is the same as in General Relativity. However, we do not consider the metric as an independent field. Instead, it is an effective one, constructed in terms of two fundamental spinor fields $\Psi$ and $\Upsilon$ and thus the metric does not have a dynamics of its own, but inherits its evolution through its relation with the fundamental spinors. In the first paper it was shown that the metric that describes the gravitational field generated by a compact static and spherically symmetric configuration is very similar to the Schwarzschild metric. In the present paper we describe the cosmological framework in the realm of the Spinor Theory of Gravity. 
  This note derives the analogue of the Mukhanov-Sasaki variables both for scalar and tensor perturbations in the 1+3 covariant formalism. The possibility of generalizing them to non-flat Friedmann-Lemaitre universes is discussed. 
  The equations describing nonradial adiabatic oscillations of differentially rotating relativistic stars are derived in relativistic slow rotation approximation. The differentially rotating configuration is described by a perturbative version of the relativistic j-constant rotation law. Focusing on the oscillation properties of the stellar fluid, the adiabatic nonradial perturbations are studied in the Cowling approximation with a system of five partial differential equations. In these equations, differential rotation introduces new coupling terms between the perturbative quantites with respect to the uniformly rotating stars. In particular, we investigate the axisymmetric and barotropic oscillations and compare their spectral properties with those obtained in nonlinear hydrodynamical studies. The perturbative description of the differentially rotating background and the oscillation spectrum agree within a few percent with those of the nonlinear studies. 
  Binary black hole simulations have traditionally been computationally very expensive: current simulations are performed in supercomputers involving dozens if not hundreds of processors, thus systematic studies of the parameter space of binary black hole encounters still seem prohibitive with current technology. Here we present results obtained using dual processor workstations with comparable quality to those obtained using much larger computer resources. For this, we use the multi-layered refinement level code BAM, based on the moving punctures method. BAM provides grid structures composed of boxes of increasing resolution near the center of the grid. In the case of binaries, the highest resolution boxes are placed around each black hole and they track them in their orbits until the final merger when a single set of levels surrounds the black hole remnant. This is particular useful when simulating spinning black holes since the gravitational fields gradients are larger. We present simulations of binaries with equal mass black holes with spins parallel to the binary axis and intrinsic magnitude of S/m^2= 0.75. Our results compare favorably to those of previous simulations of this particular system. We show that the moving punctures method produces stable simulations at maximum spatial resolutions up to M/160 and for durations of up to the equivalent of 20 orbital periods. 
  If spacetime undergoes quantum fluctuations, an electromagnetic wavefront will acquire uncertainties in direction as well as phase as it propagates through spacetime. These uncertainties can show up in interferometric observations of distant quasars as a decreased fringe visibility. The Very Large Telescope and Keck interferometers may be on the verge of probing spacetime fluctuations which, we also argue, have repercussions for cosmology, requiring the existence of dark energy/matter, the critical cosmic energy density, and accelerating cosmic expansion in the present era. 
  We address the problem of defining the concept of entropy for anisotropic cosmological models. In particular, we analyze for the Bianchi I and V models the entropy which follows from postulating the validity of the laws of standard thermodynamics in cosmology. Moreover, we analyze the Cardy-Verlinde construction of entropy and show that it cannot be associated with the one following from relativistic thermodynamics. 
  Following the demonstration that gravitational waves impart linear momentum, it is argued that if they are polarized they should impart angular momentum to appropriately placed 'test rods' in their path. A general formula for this angular momentum is obtained and used to provide expressions for the angular momentum imparted by plane and cylindrical gravitational waves. 
  We present cosmic solutions corresponding to universes filled with dark and phantom energy, all having a negative cosmological constant. All such solutions contain infinite singularities, successively and equally distributed along time, which can be either big bang/crunchs or big rips singularities. Classicaly these solutions can be regarded as associated with multiverse scenarios, being those corresponding to phantom energy that may describe the current accelerating universe. 
  We consider a possibility to construct a quantum-mechanical model of spacetime, where Planck size quantum black holes act as the fundamental constituents of space and time. Spacetime is assumed to be a graph, where black holes lie on the vertices. Our model implies that area has a discrete spectrum with equal spacing. At macroscopic length scales our model reproduces Einstein's field equation with a vanishing cosmological constant as a sort of thermodynamical equation of state of spacetime and matter fields. In the low temperature limit, where most black holes are assumed to be in the ground state, our model implies the Unruh and the Hawking effects, whereas in the high temperature limit we find, among other things, that black hole entropy depends logarithmically on the event horizon area, instead of being proportional to the area. 
  It is well-known that Birkhoff's theorem is no longer valid in theories with more than four dimensions. Thus, in these theories the effective 4-dimensional picture allows the existence of different possible, non-Schwarzschild, scenarios for the description of the spacetime outside of a spherical star, contrary to general relativity in 4D. We investigate the exterior spacetime of a spherically symmetric star in the context of Kaluza-Klein gravity. We take a well-known family of static spherically symmetric solutions of the Einstein equations in an empty five-dimensional universe, and analyze possible stellar exteriors that are conformal to the metric induced on four-dimensional hypersurfaces orthogonal to the extra dimension. All these exteriors are continuously matched with the interior of the star. Then, without making any assumptions about the interior solution, we prove the following statement: the condition that in the weak-field limit we recover the usual Newtonian physics singles out an unique exterior. This exterior is "similar" to Scharzschild vacuum in the sense that it has no effect on gravitational interactions. However, it is more realistic because instead of being absolutely empty, it is consistent with the existence of quantum zero-point fields. We also examine the question of how would the deviation from the Schwarzschild vacuum exterior affect the parameters of a neutron star. In the context of a model star of uniform density, we show that the general relativity upper limit M/R < 4/9 is significantly increased as we go away from the Schwarzschild vacuum exterior. We find that, in principle, the compactness limit of a star can be larger than 1/2, without being a black hole. The generality of our approach is also discussed. 
  We consider two-level detectors coupled to a scalar field and moving on arbitrary trajectories in Minkowski space-time. We first derive a generic expression for the response function using a (novel) regularization procedure based on the Feynmann prescription that is explicitly causal, and we compare it to other expressions used in the literature. We then use this expression to study, analytically and numerically, the time dependence of the response function in various non-stationarity situations. We show that, generically, the response function decreases like a power in the detector's level spacing, $E$, for high $E$. It is only for stationary world-lines that the response function decays faster than any power-law, in keeping with the known exponential behavior for some stationary cases. Under some conditions the (time dependent) response function for a non-stationary world-line is well approximated by the value of the response function for a stationary world-line having the same instantaneous acceleration, torsion, and hyper-torsion. While we cannot offer general conditions for this to apply, we discuss special cases; in particular, the low energy limit for linear space trajectories. 
  We discuss head-on collisions of neutron stars and disks of dust ("galaxies") following the ideas of equilibrium thermodynamics, which compares equilibrium states and avoids the description of the dynamical transition processes between them. As an always present damping mechanism, gravitational emission results in final equilibrium states after the collision. In this paper we calculate selected final configurations from initial data of colliding stars and disks by making use of conservation laws and solving the Einstein equations. Comparing initial and final states, we can decide for which initial parameters two colliding neutron stars (non-rotating Fermi gas models) merge into a single neutron star and two rigidly rotating disks form again a final (differentially rotating) disk of dust. For the neutron star collision we find a maximal energy loss due to outgoing gravitational radiation of 2.3% of the initial mass while the corresponding efficiency for colliding disks has the much larger limit of 23.8%. 
  In this paper we address the physical meaning of states in loop quantum cosmology (LQC). A first step in this is the completion of the program begun in [1], applied to LQC. Specifically, we introduce a family of (what are called) b-embeddings of isotropic loop quantum cosmology (LQC) into full loop quantum gravity. As a side note, we exhibit a large class of operators preserving each of these embeddings, and show their consistency with the LQC quantization. Embedding at the gauge and diffeomorphism invariant level is discussed in the conclusion section. 
  The brane cosmology scenario is based on the idea that our Universe is a 3-brane embedded in a five-dimensional bulk. In this work, a general class of braneworld wormholes is explored with $R\neq 0$, where $R$ is the four dimensional Ricci scalar, and specific solutions are further analyzed. A fundamental ingredient of traversable wormholes is the violation of the null energy condition (NEC). However, it is the effective total stress energy tensor that violates the latter, and in this work, the stress energy tensor confined on the brane, threading the wormhole, is imposed to satisfy the NEC. It is also shown that in addition to the local high-energy bulk effects, nonlocal corrections from the Weyl curvature in the bulk may induce a NEC violating signature on the brane. Thus, braneworld gravity seems to provide a natural scenario for the existence of traversable wormholes. 
  We present an approach to experimentally evaluate gravity gradient noise, a potentially limiting noise source in advanced interferometric gravitational wave (GW) detectors. In addition, the method can be used to provide sub-percent calibration in phase and amplitude of modern interferometric GW detectors. Knowledge of calibration to such certainties shall enhance the scientific output of the instruments in case of an eventual detection of GWs. The method relies on a rotating symmetrical two-body mass, a Dynamic gravity Field Generator (DFG). The placement of the DFG in the proximity of one of the interferometer's suspended test masses generates a change in the local gravitational field detectable with current interferometric GW detectors. 
  The renormalized mean value of the quantum Lagrangian and the Energy-Momentum tensor for scalar fields coupled to an arbitrary gravitational field configuration are analytically evaluated in the Schwinger-DeWitt approximation, up to second order in the inverse mass value. The cylindrical symmetry situation is considered. The results furnish the starting point for investigating iterative solutions of the back-reaction problem related with the quantization of cylindrical scalar field configurations. Due to the homogeneity of the equations of motion of the Klein-Gordon field, the general results are also valid for performing the quantization over either vanishing or non-vanishing mean field configurations. As an application, compact analytical expressions are derived here for the quantum mean Lagrangian and Energy-Momentum tensor in the particular background given by the Black-String space-time. 
  The evolution of the effective Yang-Mills (YM) condensate dark energy model with electric and magnetic components is studied. In the case of electric field being dominant, the magnetic field disappears with the expansion of the universe. The total YM condensate tracked the evolution of the radiation in the early universe, but in the later stage, it becomes $\omega_y\sim-1$, so the cosmic coincidence problem is also avoided. But in the case of magnetic field being dominant, $\omega_y>1/3$ holds for all time. So the constraint of $E^2>B^2$ must be satisfied for the YM condensate as a kind of candidate of dark energy. 
  We investigate (2+1)-dimensional black strings in the Kaluza-Klein spacetime. The system is classically stable as long as the horizon size is much larger than the size of the compact space. Semiclassically, however, the horizon size shrinks gradually due to the energy loss through the Hawking radiation. Eventually, the system will enter into the regime of the Gregory-Laflamme instability and get destabilized. Subsequently, the spherically symmetric black hole is formed and evaporated in the usual manner. This standard picture may be altered by the dynamics of the internal space which induced by the Hawking radiation. We argue that the black string is excised from the Kaluza-Klein spacetime before the onset of the Gregory-Laflamme instability and therefore before the evaporation. 
  Motivated by the dark energy issue, a minisuperspace approach to the stability for modified gravitational models in a four dimensional cosmological setting are investigated. Specifically, after revisiting the $f(R)$ case, $R$ being the Ricci curvature, we present a stability condition around a de Sitter solution valid for modified gravitational models of generalized Gauss-Bonnet type $F(R,G)$, $G$ being the Gauss-Bonnet invariant. 
  The Mock LISA Data Challenges (MLDCs) have the dual purpose of fostering the development of LISA data analysis tools and capabilities, and demonstrating the technical readiness already achieved by the gravitational-wave community in distilling a rich science payoff from the LISA data output. The first round of MLDCs has just been completed: nine data sets containing simulated gravitational wave signals produced either by galactic binaries or massive black hole binaries embedded in simulated LISA instrumental noise were released in June 2006 with deadline for submission of results at the beginning of December 2006. Ten groups have participated in this first round of challenges. Here we describe the challenges, summarise the results, and provide a first critical assessment of the entries. 
  We propose a novel but natural definition of conserved quantities for gravity models quadratic and higher in curvature. Based on the spatial asymptotics of curvature rather than of metric, it avoids the GR energy machinery's more egregious problems--such as zero energy "theorems" and failure in flat backgrounds -- in this fourth-derivative realm. In D>4, the present expression indeed correctly discriminates between second derivative Gauss-Bonnet and generic, fourth derivative, actions. 
  We show here that the recent claim of a test of the Lense-Thirring effect with an error of 0.5% using the Mars Global Surveyor is misleading and the quoted error is incorrect by a factor of at least ten thousand. Indeed, the simple error analysis of [1] neglects the role of some important systematic errors affecting the out-of-plane acceleration. The preliminary error analysis presented here shows that even an optimistic uncertainty for this measurement is at the level of, at least, ~ 3026% to ~ 4811%, i.e., even an optimistic uncertainty is about 30 to 48 times the Lense-Thirring effect. In other words by including only some systematic errors we obtained an uncertainty almost ten thousand times larger than the claimed 0.5% error. 
  Quantum gravity places entirely new challenges on the formulation of a consistent theory as well as on an extraction of potentially observable effects. Quantum corrections due to the gravitational field are commonly expected to be tiny because of the smallness of the Planck length. However, a consistent formulation now shows that key features of quantum gravity imply magnification effects on correction terms which are especially important in cosmology with its long stretches of evolution. After a review of the salient features of recent canonical quantizations of gravity and their implications for the quantum structure of space-time a new example for potentially observable effects is given. 
  The inspiral and merger of binary black holes will likely involve black holes with both unequal masses and arbitrary spins. The gravitational radiation emitted by these binaries will carry angular as well as linear momentum. A net flux of emitted linear momentum implies that the black hole produced by the merger will experience a recoil or kick. Previous studies have focused on the recoil velocity from unequal mass, non-spinning binaries. We present results from simulations of equal mass but spinning black hole binaries and show how a significant gravitational recoil can also be obtained in these situations. We consider the case of black holes with opposite spins of magnitude $a$ aligned/anti-aligned with the orbital angular momentum, with $a$ the dimensionless spin parameters of the individual holes. For the initial setups under consideration, we find a recoil velocity of $V = 475 \KMS a$. Supermassive black hole mergers producing kicks of this magnitude could result in the ejection from the cores of dwarf galaxies of the final hole produced by the collision. 
  We obtain solutions of the Klein-Gordon and Dirac equations in the gravitational fields of vacuumless defects. We calculate the energy levels and the current, respectively, in the scalar and spinor cases. In all these situations we emphasize the role played by the defects on the solutions, energy and current. 
  The merger of two neutron stars usually produces a remnant with a mass significantly above the single (nonrotating) neutron star maximum mass. In some cases, the remnant will be stabilized against collapse by rapid, differential rotation. MHD-driven angular momentum transport eventually leads to the collapse of the remnant's core, resulting in a black hole surrounded by a massive accretion torus. Here we present simulations of this process. The plausibility of generating short duration gamma ray bursts through this scenario is discussed. 
  In this note we reply to the criticisms by Krogh concerning some aspects of the recent frame-dragging test performed by Iorio with the Mars Global Surveyor (MGS) spacecraft in the gravitational field of Mars. 
  Some thermodynamic quantities of nonrelativistic ideal boson and fermion gases in the static Taub universe are derived to first order in a small anisotropy parameter $d$ which measuring the deformation from the spherical Einstein universe. They are used to investigate the problem of how the curvature anisotropy affects the thermodynamic behaviors of an ideal gas. It is found that, when the universe is in the oblate configuration (i.e., $d > 0$), the effect of curvature anisotropy is to increase the number of the fraction in the Bose-Einstein condensation and to decrease the fermion distribution function at low temperature. When the universe is in the prolate configuration (i.e., $d < 0$), the effects of curvature anisotropy on the thermodynamic quantities is contrary to that in the oblate configuration. The density matrix of a two particle system is evaluated and it is used to define the "statistical interparticle potential" as an attempt to give a "statistical interpretation" about the found thermodynamic behaviors. It is found that when the universe is in the oblate (prolate) configuration the curvature anisotropy will enhance (reduce) both the "statistic attraction" among the bosons and "statistical repulsion" among the fermions. It is expected that such a behavior will also be shown in the relativistic system. 
  We present the application of a novel method of time-series analysis, the Hilbert-Huang Transform, to the search for gravitational waves. This algorithm is adaptive and does not impose a basis set on the data, and thus the time-frequency decomposition it provides is not limited by time-frequency uncertainty spreading. Because of its high time-frequency resolution it has important applications to both signal detection and instrumental characterization. Applications to the data analysis of the ground and space based gravitational wave detectors, LIGO and LISA, are described. 
  We study a broad class of spacetimes whose metric coefficients reduce to powers of a radius r in the limit of small r. Among these four-parameter "power-law" metrics we identify those parameters for which the spacetimes have classical singularities as r approaches 0. We show that a large set of such classically singular spacetimes is nevertheless nonsingular quantum mechanically, in that the Hamiltonian operator is essentially self-adjoint, so that the evolution of quantum wave packets lacks the ambiguity associated with scattering off singularities. Using these metrics, the broadest class yet studied to compare classical with quantum singularities, we explore the physical reasons why some that are singular classically are "healed" quantum mechanically, while others are not. 
  We explore connections between geometrical properties of null congruences and the algebraic structure of the Weyl tensor in n>4 spacetime dimensions. First, we present the full set of Ricci identities on a suitable ``null'' frame, thus completing the extension of the Newman-Penrose formalism to higher dimensions. Then we specialize to geodetic null congruences and study specific consequences of the Sachs equations. These imply, for example, that Kundt spacetimes are of type II or more special (like for n=4) and that for odd n a twisting geodetic WAND must also be shearing (in contrast to the case n=4). 
  The true- and eccentric-anomaly parametrizations of the Kepler motion are generalized to quasiperiodic orbits by considering perturbations of the radial part of kinetic energy as a series in the negative powers of the orbital radius. A toolbox of methods for averaging observables in terms of the energy $E$ and angular momentum $L$ is developed. A broad range of systems governed by the generic Brumberg force, as well as recent applications of the theory of gravitational radiation involve integrals over a period of motion. These integrals are evaluated by using the residue theorem. It is shown that the pole of the integrand is located in the origin and that under certain circumstances an additional pole emerges. 
  Five tensor equations are obtained for a thin shell in Gauss-Bonnet gravity. There is the well known junction condition for the singular part of the stress tensor intrinsic to the shell. There are also equations relating the geometry of the shell (jump and average of the extrinsic curvature as well as the intrinsic curvature) to the non-singular components of the bulk stress tensor on the sides of the thin shell.   The equations are applied to spherically symmetric thin shells in vacuum. The shells are part of the vacuum, they carry no energy tensor. We classify these solutions of `thin shells of nothingness' in the pure Gauss-Bonnet theory. There are three types of solutions, with one, zero or two asymptotic regions respectively. The third kind of solution are wormholes. Although vacuum solutions, they have the appearance of mass in the asymptotic regions. It is striking that in this theory, exotic matter is not needed in order for wormholes to exist- they can exist even with no matter. 
  A recently proposed approach to the construction of the Vaidya metric in double-null coordinates for generic mass functions is extended to the $n$-dimensional $(n>2)$ case and to allow the inclusion of a cosmological constant. Some new exact solutions are presented and some physical examples are explicitly discussed. The results presented here can simplify considerably the study of spherically symmetric gravitational collapse in arbitrary dimensions and also the quasinormal mode analysis of higher dimensional varying mass black-holes. 
  The gravastar model, which postulates a strongly correlated thin shell of anisotropic matter surrounding a region of anti-de Sitter space, has been proposed as an alternative to black holes. We discuss constraints that present-day observations of well-known black hole candidates place on this model. We focus upon two black hole candidates known to have extraordinarily low luminosities: the supermassive black hole in the Galactic Center, Sagittarius A*, and the stellar-mass black hole, XTE J1118+480. We find that the length scale for modifications of the type discussed in Chapline et al. (2003) must be sub-Planckian. 
  The similarity between the energy spectra of relativistic particles and that of quasi-particles in super-conductivity BCS theory makes us conjecture that the relativistic physical vacuum medium as the ground state of the background field is a super fluid medium, and the rest mass of a relativistic particle is like the energy gap of a quasi-particle. This conjecture is strongly supported by the results of our following investigation: a particle moving through the vacuum medium at a speed less than the speed of light in vacuum, though interacting with the vacuum medium, never feels friction force and thus undergoes a frictionless and inertial motion. The profound and intrinsic relationship between the super fluid property of the relativistic physical vacuum medium and the energy-momentum conservation law as well as the relativistic energy-momentum dispersion relation or the principle of relativity, can be established. 
  We analyze the interaction of plane '+'-polarized gravitational waves with a Fabry-Perot cavity in the local Lorentz frame of the cavity's input mirror outside of range of long-wave approximation with the force of radiation pressure taken into account. The obtained detector's response signal is represented as a sum of two parts: (i) the phase shift due to displacement of movable mirror under the influence of gravitational wave and the force of light pressure, and (ii) the phase shift due to direct interaction of gravitational wave with light wave inside the cavity. We obtain formula for movable mirror's law of motion paying close attention to the phenomena of optical rigidity, radiative friction and direct coupling of gravitational wave to light wave. Some issues concerning the detection of high-frequency gravitational waves and the role of optical rigidity in it are discussed. We also examine in detail special cases of optical resonance and small detuning from it and compare our results with the known ones. 
  The dimensional reduction of Papapetrou equations is performed in a 5-dimensional Kaluza-Klein background and Dixon-Souriau results for the motion of a charged spinning body are obtained. The splitting provides an electric dipole moment, and, for elementary particles, the induced parity and time-reversal violations are explained. 
  In my lectures I will deal with three seemingly unrelated problems: i) to what extent is general relativity exceptional among metric gravity theories? ii) is it possible to define gravitational energy density applying field-theory approach to gravity? and iii) can a consistent theory of a gravitationally interacting spin-two field be developed at all? The connecting link to them is the concept of a fundamental spin-2 field. A linear spin-2 field encounters insurmountable inconsistencies when coupled to gravity. After discussing the inconsistencies of any coupling of the linear spin-2 field to gravity, I exhibit the origin of the fact that a gauge invariant field has the variational metric stress tensor which is gauge dependent. I give a general theorem explaining under what conditions a symmetry of a field Lagrangian becomes also the symmetry of the stress tensor. It is a conclusion of the theorem that any attempt to define gravitational energy density in the framework of a field theory of gravity must fail. Finally I make a very brief introduction to basic concepts of how a certain kind of a necessarily nonlinear spin-2 field arises in a natural way from vacuum higher derivative gravity theories. This specific spin-2 field consistently interacts gravitationally. 
  In this note we reply to the criticisms by Sindoni, Paris and Ialongo concerning some aspects of the recent frame-dragging test performed by Iorio with the Mars Global Surveyor (MGS) spacecraft in the gravitational field of Mars. 
  We explore differences between scalar field, and scalar density solutions by using Robertson-Walker (RW) metric, and also a non-relativistic Hamiltonian is derived for a scalar density field in the post-Newtonian approximation. The results are compared with those of scalar field. The expanding universe in RW metric, and post-Newtonian solution of Klein-Gordon equation are separately discussed. 
  New boundary conditions are imposed on the familiar cylindrical gravitational wave vacuum spacetimes. The new spacetime family represents cylindrical waves in a flat expanding (Kasner) universe. Space sections are flat and nonconical where the waves have not reached and wave amplitudes fall off more rapidly than they do in Einstein-Rosen solutions, permitting a more regular null inifinity. 
  The sensitivity of next-generation gravitational-wave detectors such as Advanced LIGO and LCGT should be limited mostly by quantum noise with an expected technical progress to reduce seismic noise and thermal noise. Those detectors will employ the optical configuration of resonant-sideband-extraction that can be realized with a signal-recycling mirror added to the Fabry-Perot Michelson interferometer. While this configuration can reduce quantum noise of the detector, it can possibly increase laser frequency noise and intensity noise. The analysis of laser noise in the interferometer with the conventional configuration has been done in several papers, and we shall extend the analysis to the resonant-sideband-extraction configuration with the radiation pressure effect included. We shall also refer to laser noise in the case we employ the so-called DC readout scheme. 
  The final evolution of a generic binary black-hole system is expected to give rise to a net recoil velocity as a result of the asymmetry in the beamed gravitational radiation emitted. A quantitative measurement of this effect in the case of binaries with unequal masses has been recently computed by a number of different groups in full numerical-relativity simulations. These have pointed out that kick velocities as large as 175 km/s can be reached for a mass ratio $q\equiv M_1/M_2\simeq 0.36$, where $M_1$ and $M_2$ are the masses of the two black holes. However, a recoil velocity can also be obtained for equal-mass binaries with spinning black holes that have unequal spins. We report here on numerical evolutions of such binary black-hole systems and show, using two independent methods, that even larger kick velocities are possible, with a maximum of $257 \pm 15$ km/s for a system having a spin ratio $a_1/a_2 = -1$ and $a_2\equiv S_2/m^2_2=0.584$. This extrapolates to $\sim 450$ km/s for extremal black holes. Such large velocities are not unexpected and we show that the numerical results reproduce, at least qualitatively, the post-Newtonian predictions. 
  We report the first results from evolutions of a generic black-hole binary, i.e. a binary containing unequal mass black holes with misaligned spins. Our configuration, which has a mass ratio of 2:1, consists of an initially non-spinning hole orbiting a larger, rapidly spinning hole (specific spin a/m = 0.885), with the spin direction oriented -45-degrees with respect to the orbital plane. We track the inspiral and merger for ~2 orbits and find that the remnant receives a substantial kick of 454 km/s, more than twice as large as the maximum kick from non-spinning binaries. Such a large recoil velocity reopens the possibility that a merged binary can be ejected even from the nucleus of a massive host galaxy. The remnant spin direction is flipped by 103-degrees with respect to the initial spin direction of the larger hole. 
  We study the radial propagation of a classical electro-magnetic wave in a Schwarzschild metric. By relaxing the standard assumption which leads to the eikonal equation, that the EM wave has zero spatial extent, we show that EM waves with wavelengths larger that the Schwarzschild radius are not absorbed by the black hole, but are reflected. This is shown to be consistent with the seccond law of thermodynamics under the Bekenstein interpretation of the area of a black hole as a measure of its entropy. The propagation speed is also calculated and seen to differ from the value c, for wavelengths larger than $R_{s}$, in the vicinity of $R_{s}$. As in all optical phenomena, we see that whenever the wavelength of light is larger or comparable to the physical size of elements in the system, in this case changes in the metric, the zero extent 'particle' description of light as photons fails, and its wave nature becomes apparent. This leads to a wide variety of effects, some of which we begin to explore here. 
  Certain aspects of the behaviour of the gravitational field near null and spatial infinity for the developments of asymptotically Euclidean, conformally flat initial data sets are analysed. Ideas and results from two different approaches are combined: on the one hand the null infinity formalism related to the asymptotic characteristic initial value problem and on the other the regular Cauchy initial value problem at spatial infinity which uses Friedrich's representation of spatial infinity as a cylinder. The decay of the Weyl tensor for the developments of the class of initial data under consideration is analysed under some existence and regularity assumptions for the asymptotic expansions obtained using the cylinder at spatial infinity. Conditions on the initial data to obtain developments satisfying the Peeling Behaviour are identified. Further, the decay of the asymptotic shear on null infinity is also examined as one approaches spatial infinity. This decay is related to the possibility of selecting the Poincar\'e group out of the BMS group in a canonical fashion. It is found that for the class of initial data under consideration, if the development peels, then the asymptotic shear goes to zero at spatial infinity. Expansions of the Bondi mass are also examined. Finally, the Newman-Penrose constants of the spacetime are written in terms of initial data quantities and it is shown that the constants defined at future null infinity are equal to those at past null infinity. 
  The Mock LISA Data Challenge is a worldwide effort to solve the LISA data analysis problem. We present here our results for the Massive Black Hole Binary (BBH) section of Round 1. Our results cover Challenge 1.2.1, where the coalescence of the binary is seen, and Challenge 1.2.2, where the coalescence occurs after the simulated observational period. The data stream is composed of Gaussian instrumental noise plus an unknown BBH waveform. Our search algorithm is based on a variant of the Markov Chain Monte Carlo method that uses Metropolis-Hastings sampling and thermostated frequency annealing. We present results from the training data sets and the blind data sets. We demonstrate that our algorithm is able to rapidly locate the sources, accurately recover the source parameters, and provide error estimates for the recovered parameters. 
  It is shown how the gravitational repulsion between matter and antimatter (called antigravity) naturally leads to the emergence of the Planck Length 
  In the paper [F. Nasseri, Phys. Lett. B 632 (2006) 151--154], F. Nasseri supposed that the value of the angular momentum for the Bohr's atom in the presence of the cosmic string is quantized in units of &#8463;. Using this assumption it was obtained an incorrect expression for Bohr radius in this scenario. In this comment I want to point out that this assumption is not correct and present a corrected expression for the Bohr radius in this background. 
  The Mock Data Challenges (MLDCs) have the dual purpose of fostering the development of LISA data analysis tools and capabilities, and demonstrating the technical readiness already achieved by the gravitational-wave community in distilling a rich science payoff from the LISA data. The first round of MLDCs has just been completed and the second round data sets have been released. The round two data sets contain radiation from an entire Galactic population of stellar-mass binary systems, massive black hole binaries and extreme-mass-ratio inspirals. These data sets are designed to capture much of the complexity that is expected in the actual LISA data, and should provide a fairly realistic setting to test advanced data analysis techniques, and in particular the aspect of global data analysis. Here we describe the second round of MLDCs and provide details about its implementation. 
  A family of exact conformal field theories is constructed which describe charged black strings in three dimensions. Unlike previous charged black hole or extended black hole solutions in string theory, the low energy spacetime metric has a regular inner horizon (in addition to the event horizon) and a timelike singularity. As the charge to mass ratio approaches unity, the event horizon remains but the singularity disappears. 
  We show that all W-gravity actions can be easilly constructed and understood from the point of view of the Hamiltonian formalism for the constrained systems. This formalism also gives a method of constructing gauge invariant actions for arbitrary conformally extended algebras. 
  We study the classical version of supersymmetric $W$-algebras. Using the second Gelfand-Dickey Hamiltonian structure we work out in detail $W_2$ and $W_3$-algebras. 
  String theories with two dimensional space-time target spaces are characterized by the existence of a ``ground ring'' of operators of spin $(0,0)$. By understanding this ring, one can understand the symmetries of the theory and illuminate the relation of the critical string theory to matrix models. The symmetry groups that arise are, roughly, the area preserving diffeomorphisms of a two dimensional phase space that preserve the fermi surface (of the matrix model) and the volume preserving diffeomorphisms of a three dimensional cone. The three dimensions in question are the matrix eigenvalue, its canonical momentum, and the time of the matrix model. 
  We discuss when and how the Verlinde dimensions of a rational conformal field theory can be expressed as correlation functions in a topological LG theory. It is seen that a necessary condition is that the RCFT fusion rules must exhibit an extra symmetry. We consider two particular perturbations of the Grassmannian superpotentials. The topological LG residues in one perturbation, introduced by Gepner, are shown to be a twisted version of the $SU(N)_k$ Verlinde dimensions. The residues in the other perturbation are the twisted Verlinde dimensions of another RCFT; these topological LG correlation functions are conjectured to be the correlation functions of the corresponding Grassmannian topological sigma model with a coupling in the action to instanton number.  
  It is shown that, in the three-dimensional lattice gravity defined by Ponzano and Regge, the space of physical states is isomorphic to the space of gauge-invariant functions on the moduli space of flat $SU(2)$ connections over a two-dimensional surface, which gives physical states in the $ISO(3)$ Chern-Simons gauge theory. 
  Starting from a given S-matrix of an integrable quantum field theory in $1+1$ dimensions, and knowledge of its on-shell quantum group symmetries, we describe how to extend the symmetry to the space of fields. This is accomplished by introducing an adjoint action of the symmetry generators on fields, and specifying the form factors of descendents. The braiding relations of quantum field multiplets is shown to be given by the universal $\CR$-matrix. We develop in some detail the case of infinite dimensional Yangian symmetry. We show that the quantum double of the Yangian is a Hopf algebra deformation of a level zero Kac-Moody algebra that preserves its finite dimensional Lie subalgebra. The fields form infinite dimensional Verma-module representations; in particular the energy-momentum tensor and isotopic current are in the same multiplet.  
  We show that various actions of topological conformal theories that were suggested recentely are particular cases of a general action. We prove the invariance of these models under transformations generated by nilpotent fermionic generators of arbitrary conformal dimension, $\Q$ and $\G$. The later are shown to be the $n^{th}$ covariant derivative with respect to ``flat abelian gauge field" of the fermionic fields of those models. We derive the bosonic counterparts $\W$ and $\R$ which together with $\Q$ and $\G$ form a special $N=2$ super $W_\infty$ algebra. The algebraic structure is discussed and it is shown that it generalizes the so called ``topological algebra". 
  A new transverse lattice model of $3+1$ Yang-Mills theory is constructed by introducing Wess-Zumino terms into the 2-D unitary non-linear sigma model action for link fields on a 2-D lattice. The Wess-Zumino terms permit one to solve the basic non-linear sigma model dynamics of each link, for discrete values of the bare QCD coupling constant, by applying the representation theory of non-Abelian current (Kac-Moody) algebras. This construction eliminates the need to approximate the non-linear sigma model dynamics of each link with a linear sigma model theory, as in previous transverse lattice formulations. The non-perturbative behavior of the non-linear sigma model is preserved by this construction. While the new model is in principle solvable by a combination of conformal field theory, discrete light-cone, and lattice gauge theory techniques, it is more realistically suited for study with a Tamm-Dancoff truncation of excited states. In this context, it may serve as a useful framework for the study of non-perturbative phenomena in QCD via analytic techniques. 
  $U(1)$ zero modes in the $SL(2,R)_k/U(1)$ and $SU(2)_k/U(1)$ conformal coset theories, are investigated in conjunction with the string black hole solution. The angular variable in the Euclidean version, is found to have a double set of winding. Region III is shown to be $SU(2)_k/U(1)$ where the doubling accounts for the cut sructure of the parafermionic amplitudes and fits nicely across the horizon and singularity. The implications for string thermodynamics and identical particles correlations are discussed. 
  It has been shown that given a classical background in string theory which is independent of $d$ of the space-time coordinates, we can generate other classical backgrounds by $O(d)\otimes O(d)$ transformation on the solution. We study the effect of this transformation on the known black $p$-brane solutions in string theory, and show how these transformations produce new classical solutions labelled by extra continuous parameters and containing background antisymmetric tensor field. 
  The perturbations of string-theoretic black holes are analyzed by generalizing the method of Chandrasekhar. Attention is focussed on the case of the recently considered charged string-theoretic black hole solutions as a representative example. It is shown that string-intrinsic effects greatly alter the perturbed motions of the string-theoretic black holes as compared to the perturbed motions of black hole solutions of the field equations of general relativity, the consequences of which bear on the questions of the scattering behavior and the stability of string-theoretic black holes. The explicit forms of the axial potential barriers surrounding the string-theoretic black hole are derived. It is demonstrated that one of these, for sufficiently negative values of the asymptotic value of the dilaton field, will inevitably become negative in turn, in marked contrast to the potentials surrounding the static black holes of general relativity. Such potentials may in principle be used in some cases to obtain approximate constraints on the value of the string coupling constant. The application of the perturbation analysis to the case of two-dimensional string-theoretic black holes is discussed.  
  We derive directly from the N=2 LG superpotential the differential equations that determine the flat coordinates of arbitrary topological CFT's.  
  We study the double scaling limit of mKdV type, realized in the two-cut Hermitian matrix model. Building on the work of Periwal and Shevitz and of Nappi, we find an exact solution including all odd scaling operators, in terms of a hierarchy of flows of $2\times 2$ matrices. We derive from it loop equations which can be expressed as Virasoro constraints on the partition function. We discover a ``pure topological" phase of the theory in which all correlation functions are determined by recursion relations. We also examine macroscopic loop amplitudes, which suggest a relation to 2D gravity coupled to dense polymers. 
  A particular $U(N)$ gauge theory defined on the three dimensional dodecahedral lattice is shown to correspond to a model of oriented self-avoiding surfaces. Using large $N$ reduction it is argued that the model is partially soluble in the planar limit. 
  This is a talk given by S.D. at the the workshop on Random Surfaces and 2D Quantum Gravity, Barcelona 10-14 June 1991. It is an updated review of recent work done by the authors on a proposal for non-perturbatively stable 2D quantum gravity coupled to c<1 matter, based on the flows of the (generalised) KdV hierarchy. 
  We find new solutions to the Yang--Baxter equation in terms of the intertwiner matrix for semi-cyclic representations of the quantum group $U_q(s\ell(2))$ with $q= e^{2\pi i/N}$. These intertwiners serve to define the Boltzmann weights of a lattice model, which shares some similarities with the chiral Potts model. An alternative interpretation of these Boltzmann weights is as scattering matrices of solitonic structures whose kinematics is entirely governed by the quantum group. Finally, we consider the limit $N\to\infty$ where we find an infinite--dimensional representation of the braid group, which may give rise to an invariant of knots and links.  
  We describe four types of inner involutions of the Cartan-Weyl basis providing (for $ |q|=1$ and $q$ real) three types of real quantum Lie algebras: $U_{q}(O(3,2))$ (quantum D=4 anti-de-Sitter), $U_{q}(O(4,1))$ (quantum D=4 de-Sitter) and $U_{q}(O(5))$. We give also two types of inner involutions of the Cartan-Chevalley basis of $U_{q}(Sp(4;C))$ which can not be extended to inner involutions of the Cartan-Weyl basis. We outline twelve contraction schemes for quantum D=4 anti-de-Sitter algebra. All these contractions provide four commuting translation generators, but only two (one for $ |q|=1$, second for $q$ real) lead to the quantum \po algebra with an undeformed space rotations O(3) subalgebra. 
  I review some of the recent progress in two-dimensional string theory, which is formulated as a sum over surfaces embedded in one dimension. 
  It is shown that the scattering of spacetime axions with fivebrane solitons of heterotic string theory at zero momentum is proportional to the Donaldson polynomial. 
  We discuss the conformal field theory and string field theory of the NSR superstring using a BRST operator with a nonminimal term, which allows all bosonic ghost modes to be paired into creation and annihilation operators. Vertex operators for the Neveu-Schwarz and Ramond sectors have the same ghost number, as do string fields. The kinetic and interaction terms are the same for Neveu-Schwarz as for Ramond string fields, so spacetime supersymmetry is closer to being manifest. The kinetic terms and supersymmetry don't mix levels, simplifying component analysis and gauge fixing. 
  This review talk focusses on some of the interesting developments in the area of superstring compactification that have occurred in the last couple of years. These include the discovery that ``mirror symmetric" pairs of Calabi--Yau spaces, with completely distinct geometries and topologies, correspond to a single (2,2) conformal field theory. Also, the concept of target-space duality, originally discovered for toroidal compactification, is being extended to Calabi--Yau spaces. It also associates sets of geometrically distinct manifolds to a single conformal field theory.   A couple of other topics are presented very briefly. One concerns conceptual challenges in reconciling gravity and quantum mechanics. It is suggested that certain ``distasteful allegations" associated with quantum gravity such as loss of quantum coherence, unpredictability of fundamental parameters of particle physics, and paradoxical features of black holes are likely to be circumvented by string theory. Finally there is a brief discussion of the importance of supersymmetry at the TeV scale, both from a practical point of view and as a potentially significant prediction of string theory. 
  We extend Felder's construction of Fock space resolutions for the Virasoro minimal models to all irreducible modules with $c\leq 1$. In particular, we provide resolutions for the representations corresponding to the boundary and exterior of the Kac table. 
  We compute the $S$-matrix of the Tricritical Ising Model perturbed by the subleading magnetic operator using Smirnov's RSOS reduction of the Izergin-Korepin model. The massive model contains kink excitations which interpolate between two degenerate asymmetric vacua. As a consequence of the different structure of the two vacua, the crossing symmetry is implemented in a non-trivial way. We use finite-size techniques to compare our results with the numerical data obtained by the Truncated Conformal Space Approach and find good agreement. 
  We calculate three- and four-point functions in super Liouville theory coupled to super Coulomb gas on world sheets with spherical topology. We first integrate over the zero mode and assume that a parameter takes an integer value. After calculating the amplitudes, we formally continue the parameter to an arbitrary real number. Remarkably the result is completely parallel to the bosonic case, the amplitudes being of the same form as those of the bosonic case. 
  We construct superstring theory in two dimensional black hole background based on supersymmetric $SU(1,1)/U(1)$ gauged Wess-Zumino-Witten model. 
  Factorization of the $N$-point amplitudes in two-dimensional $c=1$ quantum gravity is understood in terms of short-distance singularities arising from the operator product expansion of vertex operators after the Liouville zero mode integration. Apart from the tachyon states, there are infinitely many topological states contributing to the intermediate states.  
  These lectures consisted of an elementary introduction to conformal field theory, with some applications to statistical mechanical systems, and fewer to string theory.  Contents:   1. Conformal theories in d dimensions   2. Conformal theories in 2 dimensions   3. The central charge and the Virasoro algebra   4. Kac determinant and unitarity   5. Identication of m = 3 with the critical Ising model   6. Free bosons and fermions   7. Free fermions on a torus   8. Free bosons on a torus   9. Affine Kac-Moody algebras and coset constructions  10. Advanced applications 
  We propose possible new string theories based on local world-sheet symmetries corresponding to extensions of the Virasoro algebra by fractional spin currents. They have critical central charges $c=6(K+8)/(K+2)$ and Minkowski space-time dimensions $D=2+16/K$ for $K\geq2$ an integer. We present evidence for their existence by constructing modular invariant partition functions and the massless particle spectra. The dimension $4$ and $6$ strings have space-time supersymmetry. 
  A review is given of work by Abhay Ashtekar and his colleagues Carlo Rovelli, Lee Smolin, and others, which is directed at constructing a nonperturbative quantum theory of general relativity. 
  We investigate the renormalization of N=2 SUSY L-G models with central charge $c=3p/(2+p)$ perturbed by an almost marginal chiral operator. We calculate the renormalization of the chiral fields up to $gg{^*}$ order and of nonchiral fields up to $g(g^{*})$ order. We propose a formulation of the nonrenormalization theorem and show that it holds in the lowest nontrivial order. It turns out that, in this approximation, the chiral fields can not get renormalized $\Phi^{k}=\Phi^{k}_{0}$. The $\beta$ function then remains unchanged $\beta=\epsilon gr$. 
  We bosonise the complex-boson realisations of the $W_\infty$ and $W_{1+\infty}$ algebras. We obtain nonlinear realisations of $W_\infty$ and $W_{1+\infty}$ in terms of a pair of fermions and a real scalar. By further bosonising the fermions, we then obtain realisations of $W_\infty$ in terms of two scalars. Keeping the most non-linear terms in the scalars only, we arrive at two-scalar realisations of classical $w_\infty$. 
  We show that tree level ``resonant'' $N$ tachyon scattering amplitudes, which define a sensible ``bulk'' S -- matrix in critical (super) string theory in any dimension, have a simple structure in two dimensional space time, due to partial decoupling of a certain infinite set of discrete states. We also argue that the general (non resonant) amplitudes are determined by the resonant ones, and calculate them explicitly, finding an interesting analytic structure. Finally, we discuss the space time interpretation of our results. 
  For (2+1)-dimensional spacetimes with the spatial topology of a torus, the transformation between the Chern-Simons and ADM versions of quantum gravity is constructed explicitly, and the wave functions are compared. It is shown that Chern-Simons wave functions correspond to modular forms of weight 1/2, that is, spinors on the ADM moduli space, and that their evolution (in York's ``extrinsic time'' variable) is described by a Dirac equation. (This version replaces paper 9109006, which was garbled by my mailer.) 
  The deconfining transition in non-Abelian gauge theory is known to occur by a condensation of Wilson lines. By expanding around an appropriate Wilson line background, it is possible at large $N$ to analytically continue the confining phase to arbitrarily high temperatures, reaching a weak coupling confinement regime. This is used to study the high temperature partition function of an $SU(N)$ electric flux tube. It is found that the partition function corresponds to that of a string theory with a number of world-sheet fields that diverges at short distance. 
  We generalize the method of quantizing effective strings proposed by Polchinski and Strominger to superstrings. The Ramond-Neveu-Schwarz string is different from the Green-Schwarz string in non-critical dimensions. Both are anomaly-free and Poincare invariant. Some implications of the results are discussed. The formal analogy with 4D (super)gravity is pointed out. 
  We obtain the bi-Hamiltonian structure of the super KP hierarchy based on the even super KP operator $\Lambda = \theta^{2} + \sum^{\infty}_{i=-2}U_{i} \theta^{-i-1}$, as a supersymmetric extension of the ordinary KP bi-Hamiltonian structure. It is expected to give rise to a universal super $W$-algebra incorporating all known extended superconformal $W_{N}$ algebras by reduction. We also construct the super BKP hierarchy by imposing a set of anti-self-dual constraints on the super KP hierarchy. 
  We use an argument of Romans showing that every Virasoro construction leads to realizations of $W_3$, to construct $W_3$ realizations on arbitrary affine Lie algebras. Solutions are presented for generic values of the level as well as for specific values of the level but with arbitrary parameters. We give a detailed discussion of the $\aff{su}(2)_\ell$-case. Finally, we discuss possible applications of these realizations to the construction of $W$-strings. 
  Gauge systems in the confining phase induce constraints at the boundaries of the effective string, which rule out the ordinary bosonic string even with short distance modifications. Allowing topological excitations, corresponding to winding around the colour flux tube, produces at the quantum level a universal free fermion string with a boundary phase nu=1/4. This coincides with a model proposed some time ago in order to fit Monte Carlo data of 3D and 4D Lattice gauge systems better. A universal value of the thickness of the colour flux tube is predicted. 
  The $c=1$ string in the Liouville field theory approach is shown to possess a nontrivial tree-level $S$-matrix which satisfies factorization property implied by unitary, if all the extra massive physical states are included. 
  A collective field formalism for nonrelativistic fermions in (1+1) dimensions is presented. Applications to the D=1 hermitian matrix model and the system of one-dimensional fermions in the presence of a weak electromagnetic field are discussed. 
  In this paper we examine the bi-Hamiltonian structure of the generalized KdV-hierarchies. We verify that both Hamiltonian structures take the form of Kirillov brackets on the Kac-Moody algebra, and that they define a coordinated system. Classical extended conformal algebras are obtained from the second Poisson bracket. In particular, we construct the $W_n^l$ algebras, first discussed for the case $n=3$ and $l=2$ by A. Polyakov and M. Bershadsky. 
  The set of solutions to the string equation $[P,Q]=1$ where $P$ and $Q$ are differential operators is described.It is shown that there exists one-to-one correspondence between this set and the set of pairs of commuting differential operators.This fact permits us to describe the set of solutions to the string equation in terms of moduli spa- ces of algebraic curves,however the direct description is much simpler. Some results are obtained for the superanalog to the string equation where $P$ and $Q$ are considered as superdifferential operators. It is proved that this equation is invariant with respect to Manin-Radul, Mulase-Rabin and Kac-van de Leur KP-hierarchies. 
  We consider the two dimensional Jackiw-Teitelboim model of gravity. We first couple the model to the Liouville action and $c$ scalar fields and show, treating the combined system as a non linear sigma model, that the resulting theory can be interpreted as a critical string moving in a target space of dimension $D=c+2$. We then analyse perturbatively a generalised model containing a kinetic term and an arbitrary potential for the auxiliary field. We use the background field method and work with covariant gauges. We show that the renormalisability of the theory depends on the form of the potential. For a general potential, the theory can be renormalised as a non linear sigma model. In the particular case of a Liouville-like potential, the theory is renormalisable in the usual sense. 
  Explicit expressions for the singular vectors in the highest weight representations of $A_1^{(1)}$ are obtained using the fusion formalism of conformal field theory. 
  We consider the stochastic quantization scheme for a non-perturbative stabilization of 2D quantum gravity and prove that it does not satisfy the KdV flow equations. It therefore differs from a recently suggested matrix model which allows real solutions to the KdV equations. The behaviour of the Fermi energy, the free energy and macroscopic loops in the stochastic quantization scheme are elucidated. 
  We study the structure of superselection sectors of an arbitrary perturbation of a conformal field theory. We describe how a restriction of the q-deformed $\hat{sl(2)}$ affine Lie algebra symmetry of the sine-Gordon theory can be used to derive the S-matrices of the $\Phi^{(1,3)}$ perturbations of the minimal unitary series. This analysis provides an identification of fields which create the massive kink spectrum. We investigate the ultraviolet limit of the restricted sine-Gordon model, and explain the relation between the restriction and the Fock space cohomology of minimal models. We also comment on the structure of degenerate vacuum states. Deformed Serre relations are proven for arbitrary affine Toda theories, and it is shown in certain cases how relations of the Serre type become fractional spin supersymmetry relations upon restriction.  
  I describe how integrable quantum field theories in 2 spacetime dimensions are characterized by infinite dimensional quantum group symmetries, namely the q-deformations of affine Lie algebras, and their Yangian limit. These symmetries can provide a new non-perturbative formulation of the theories.  
  In the last several years, the Casimir energy for a variety of 1+1-dimensional integrable models has been determined from the exact S-matrix. It is shown here how to modify the boundary conditions to project out the lowest-energy state, which enables one to find excited-state energies. This is done by calculating thermodynamic expectation values of operators which generate discrete symmetries. This is demonstrated with a number of perturbed conformal field theories, including the Ising model, the three-state Potts model, ${\bf Z}_n$ parafermions, Toda minimal S-matrices, and massless Goldstinos. 
  We study general properties of the classical solutions in non-polynomial closed string field theory and their relationship with two dimensional conformal field theories. In particular we discuss how different conformal field theories which are related by marginal or nearly marginal deformations can be regarded as different classical solutions of some underlying string field theory. We also discuss construction of a classical solution labelled by infinite number of parameters in string field theory in two dimensions. For a specific set of values of the parameters the solution can be identified to the black hole solution. 
  The appearance of quantum groups in conformal field theories is traced back to the Poisson-Lie symmetries of the classical chiral theory. A geometric quantization of the classical theory deforms the Poisson-Lie symmetries to the quantum group ones. This elucidates the fundamental role of chiral symmetries that quantum groups play in conformal models. As a byproduct, one obtains a more geometric approach to the representation theory of quantum groups. 
  We evaluate the three point function for arbitrary states in bosonic minimal models on the sphere coupled to quantum gravity in two dimensions. The validity of the formal continuation in the number of Liouville screening charge insertions is shown directly from the Liouville functional integral using semi-classical methods. 
  We compute the three point functions of Neveu--Schwarz primary fields of the minimal models on the sphere when coupled to supergravity in two dimensions. The results show that the three point correlation functions are determined by the scaling dimensions of the fields, as in the bosonic case. 
  Graviton-dilaton background field equations in three space-time dimensions, following from the string effective action are solved when the metric has only time dependence. By taking one of the two space dimensions as compact, our solution reproduces a recently discovered charged black hole solution in two space-time dimensions. Solutions in presence of nonvanishing three dimensional background antisymmetric tensor field are also discussed. 
  On the basis of an area-preserving symmetry in the phase space of a one-dimensional matrix model - believed to describe two-dimensional string theory in a black-hole background which also allows for space-time foam - we give a geometric interpretation of the fact that two-dimensional stringy black holes are consistent with conventional quantum mechanics due to the infinite gauged `W-hair' property that characterises them. 
  We find new special physical operators of $W_3 -$gravity having non trivial ghost sectors. Some of these operators may be viewed as the liouville dressings of the energy operator of the Ising model coupled to {\it 2d~gravity} and this fills in a gap in the connection between pure $W_3 -$gravity and Ising model coupled to 2d gravity found in our previous work. We formulate a selection rule required for the calculation of correlators in $W -$gravity theories. Using this rule, we construct the non ghost part of the new operators of $W_N -$gravity and find that they represent the $(N , N+1)$ minimal model operators from both inside and outside the minimal table. Along the way we obtain the canonical spectrum of $W_N -$gravity for all $N$ .  
  We show that the symmetry algebra of the $SL(2,R)_{k}/U(1)$ coset model is a non-linear deformation of $W_{\infty}$, characterized by $k$. This is a universal $W$-algebra which linearizes in the large $k$ limit and truncates to $W_{N}$ for $k=-N$. Using the theory of non-compact parafermions we construct a free field realization of the non-linear $W_{\infty}$ in terms of two bosons with background charge. The $W$-characters of all unitary $SL(2,R)/U(1)$ representations are computed. Applications to the physics of 2-d black hole backgrounds are also discussed and connections with the KP approach to $c=1$ string theory are outlined. 
  A review is given of recent work on topology changing solutions to the first order form of general relativity. These solutions have metrics which are smooth everywhere, invertible almost everywhere, and have bounded curvature. The importance of considering degenerate metrics is discussed, and the possibility that quantum effects can suppress topology change is briefly examined. 
  We show that the new classical action for two dimensional gravity (the Jackiw-Teitelboim model) possesses a $W_3$ algebra. We quantise the resulting $W_3$ gravity in the presence of matter fields with arbitrary central charges and obtain the critical exponents. The auxiliary field of the model, expressing the constancy of the scalar curvature, can be interpreted as one of the physical degrees of freedom of the $W_3$ gravity. Our expressions are corrections to some previously published results for this model where the $W_3$ symmetry was not accounted for. 
  We study the couplings of discrete states that appear in the string theory embedded in two dimensions, and show that they are given by the structure constants of the group of area preserving diffeomorphisms. We propose an effective action for these states, which is itself invariant under this infinite-dimensional group. 
  Within the 4-dimensional conformal algebra, the presence of two translation operators implies the existence of 3 distinct metrics of definite Weyl weight constructible from the translational gauge fields. If we demand that each of these metrics give rise to a gauge theory of gravity, we are led to extend the symmetry so that each of these three metrics has a corresponding translation operator. Assigning a vierbein to each of these three translations, a different spacetime metric arises for every choice of inner product of the vierbeins. The covering group of the compact part of the minimal transitive group classifying these inner products is $SU(4)$. An additional $SU(2)$ symmetry classifies the antisymmetric parts of the vierbein product. If the metric is chosen as the gauge field of the translations in the standard way, the SU(4) part of this symmetry is broken to the semidirect product of $SU(3)$ with $U(1)$. 
  The self-dual Einstein equations on a compact Riemannian 4-manifold can be expressed as a quadratic condition on the curvature of an $SU(2)$ (spin) connection which is a covariant generalization of the self-dual Yang-Mills equations. Local properties of the moduli space of self-dual Einstein connections are described in the context of an elliptic complex which arises in the linearization of the quadratic equations on the $SU(2)$ curvature. In particular, it is shown that the moduli space is discrete when the cosmological constant is positive; when the cosmological constant is negative the moduli space can be a manifold the dimension of which is controlled by the Atiyah-Singer index theorem. 
  We summarize recent work on the classification of modular invariant partition functions that can be obtained with simple currents in theories with a center (Z_p)^k with p prime. New empirical results for other centers are also presented. Our observation that the total number of invariants is monodromy-independent for (Z_p)^k appears to be true in general as well. (Talk presented in the parallel session on string theory of the Lepton-Photon/EPS Conference, Geneva, 1991.) 
  The Lagrangian Batalin-Vilkovisky (BV) formalism gives the rules for the quantisation of a general class of gauge theories which contain all the theories known up to now. It does, however, not only give a recipe to obtain a gauge fixed action, but also gives a nice understanding of the mechanism behind gauge fixing. It moreover brings together a lot of previous knowledge and recipes in one main concept~: the canonical transformations. We explain the essentials of this formalism and give related results on the superparticle. Also anomalies (in general functions of fields and antifields) can be obtained in this formalism, and it gives the relation between anomalies in different gauges. A Pauli-Villars scheme can be used to obtain a regularised definition of the expressions at the one loop level. The calculations become similar to those of Fujikawa with the extra freedom of using arbitrary variables. A discrepancy between anomalies in light-cone gauge of the Green-Schwarz superstring and in the semi-light-cone gauge is discussed. 
  We compute general three-point functions of minimal superconformal models coupled to supergravity in the Neveu-Schwarz sector for spherical topology thus extending to the superconformal case the results of Goulian and Li and of Dotsenko. 
  We show that, given a classical solution of the heterotic string theory which is independent of $d$ of the space time directions, and for which the gauge field configuration lies in a subgroup that commutes with $p$ of the $U(1)$ generators of the gauge group, there is an $O(d)\otimes O(d+p)$ transformation, which, acting on the solution, generates new classical solutions of the theory. With the help of these transformations we construct black 6-brane solutions in ten dimensional heterotic string theory carrying independent magnetic, electric and antisymmetric tensor gauge field charge, by starting from a black 6-brane solution that carries magnetic charge but no electric or antisymmetric tensor gauge field charge. The electric and the magnetic charges point in different directions in the gauge group. 
  We develop a $\kappa$-symmetry calculus for the d=2 and d=3, N=2 massive superparticles, which enables us to construct higher order $\kappa$-invariant actions. The method relies on a reformulation of these models as supersymmetric sigma models that are invariant under local worldline superconformal transformations. We show that the $\kappa$-symmetry is embedded in the superconformal symmetry so that a calculus for the $\kappa$-symmetry is equivalent to a tensor calculus for the latter. We develop such a calculus without the introduction of a worldline supergravity multiplet. 
  We exhibit a novel solution of the strong CP problem, which does not involve any massless particles. The low energy effective Lagrangian of our model involves a discrete spacetime independent axion field which can be thought of as a parameter labeling a dense set of $\theta$ vacua. In the full theory this parameter is seen to be dynamical, and the model seeks the state of lowest energy, which has $\theta_{eff} = 0$. The processes which mediate transitions between $\theta$ vacua involve heavy degrees of freedom and are very slow. Consequently, we do not know whether our model can solve the strong CP problem in a universe which has been cool for only a finite time. We present several speculations about the cosmological evolution of our model. 
  We elaborate on a previous attempt to prove the irreversibility of the renormalization group flow above two dimensions. This involves the construction of a monotonically decreasing $c$-function using a spectral representation. The missing step of the proof is a good definition of this function at the fixed points. We argue that for all kinds of perturbative flows the $c$-function is well-defined and the $c$-theorem holds in any dimension. We provide examples in multicritical and multicomponent scalar theories for dimension $2<d<4$. We also discuss the non-perturbative flows in the yet unsettled case of the $O(N)$ sigma-model for $2\leq d\leq 4$ and large $N$.  
  We give a direct proof of the relation between vacuum singular vectors and conservation laws for the quantum KdV equation or equivalently for $\Phi_{(1,3)}$-perturbed conformal field theories. For each degree at which a classical conservation law exists, we find a quantum conserved quantity for a specific value of the central charge. Various generalizations ($N=1,2$ supersymmetric, Boussinesq) of this result are presented. 
  Based on the assumption that the target space duality ($T\to 1/T$) is preserved even nonperturbatively, the properties of static string vacua are studied. A discussion of the effective four-dimensional supergravity action based on target-space modular symmetry $SL(2,{\bf Z})$ is presented. The nonperturbative superpotential removes the vacuum degeneracy with respect to the compactification modulus ($T$) generically breaks supersymmetry with negative cosmological constant. Charged matter fields get negative $(mass)^2$ signalling an additional instability of string vacuum and the blowing up of orbifold singularities. In addition for a class of modularly invariant potentials topologically stable stringy domain walls of nontrivial compaction modulus field configuration are found. They are supersymmetric solutions, thus saturating the Bogomolnyi bound. Their physical implications are discussed. 
  We point out that the moduli sector of the $(2,2)$ string compactification with its nonperturbatively preserved non-compact symmetries is a fertile framework to study global topological defects, thus providing a natural source for the large scale structure formation. Based on the target space modular invariance of the nonperturbative superpotential of the four-dimensional N=1 supersymmetric string vacua, topologically stable stringy domain walls are found. They are supersymmetric solutions, thus saturating the Bogomolnyi bound. It is also shown that there are moduli sectors that allow for the global monopole-type and texture-type configurations whose radial stability is ensured by higher derivative terms. 
  We consider the probem of gauging discrete symmetries. All valid constraints on such symmetries can be understood in the low energy theory in terms of instantons. We note that string perturbation theory often exhibits global discrete symmetries, which are broken non-perturbatively. 
  Building on a recent work of \v C. Crnkovi\'c, M. Douglas and G. Moore, a study of multi-critical multi-cut one-matrix models and their associated $sl(2,C)$ integrable hierarchies, is further pursued. The double scaling limits of hermitian matrix models with different scaling ans\"atze, lead, to the KdV hierarchy, to the modified KdV hierarchy and part of the non-linear Schr\"odinger hierarchy. Instead, the anti-hermitian matrix model, in the two-arc sector, results in the Zakharov-Shabat hierarchy, which contains both KdV and mKdV as reductions. For all the hierarchies, it is found that the Virasoro constraints act on the associated tau-functions. Whereas it is known that the ZS and KdV models lead to the Virasoro constraints of an $sl(2,C)$ vacuum, we find that the mKdV model leads to the Virasoro constraints of a highest weight state with arbitrary conformal dimension. 
  Field theoretic and geometric ideas are used to construct a chiral supersymmetric field theory whose ground state is a specified irreducible representation of a centrally extended loop group. The character index of the associated supercharge (an appropriate Dirac operator on $LG/T$) is the Weyl-K\v{a}c character formula which we compute explicitly by the steepest descent approximation.  
  Aspects of string cosmology for critical and non-critical strings are discussed emphasizing the necessity to account for the dilaton dynamics for a proper incorporation of ``large - small" duality. This drastically modifies the intuition one has with Einstein's gravity. For example winding modes, even though contribute to energy density, oppose expansion and if not annihilated will stop the expansion. Moreover we find that the radiation dominated era of the standard cosmology emerges quite naturally in string cosmology. Our analysis of non-critical string cosmology provides a reinterpretation of the (universal cover of the) recently studied two dimensional black hole solution as a conformal realization of cosmological solutions found previously by Mueller. 
  We study the possibility of extended inflation in the effective theory of gravity from strings compactified to four dimensions and find that it strongly depends on the mechanism of supersymmetry breaking. We consider a general class of string--inspired models which are good candidates for successful extended inflation. In particular, the $\omega$--problem of ordinary extended inflation is automatically solved by the production of only very small bubbles until the end of inflation. We find that the inflaton field could belong either to the untwisted or to the twisted massless sectors of the string spectrum, depending on the supersymmetry breaking superpotential. 
  Using the Coulomb Gas formulation of N=1 Superconformal Field Theories, we extend the arguments of Dotsenko and Fateev for the bosonic case to evaluate the structure constants of N=1 minimal Superconformal Algebras in the Neveu-Schwarz sector. 
  When a gluon or a quark is sent through the hot QCD plasma it can be absorbed into the ambient heat bath and so can acquire an effective lifetime. At high temperatures and for weak couplings the inverse lifetime, or damping rate, for energetic quarks and transverse gluons, (those whose momenta satisfy $|\p| \gg gT$) is given by $\gamma(\p) = c\; g^2 \log\left({1\over g}\right)\; T + O(g^2T)$. We show that very simple arguments suffice both to fix the numerical coefficient, $c$, in this expression and to show that the $O(g^2T)$ contribution is incalculable in perturbation theory without further assumptions. For QCD with $N_c$ colours we find (expressed in terms of the casimir invariants $C_a=N_c$ and $C_f=(N_c^2-1)/(2N_c)$): $c_g=+{C_a\over 4\pi}$ for gluons and $c_q=+{C_f\over 4\pi}$ for quarks. These numbers agree with the more detailed calculations of Pisarski \etal\ but disagree with those of Lebedev and Smilga. The simplicity of the calculation also permits a direct verification of the gauge-invariance and physical sign of the result.  
  This is a transcript of lectures given at the Sixth Jorge Andre Swieca Summer School in Theoretical Physics. The subject of these lectures is soliton solutions of string theory. We construct a class of exact conformal field theories possessing a spacetime soliton or instanton interpretation and present a preliminary discussion of their physical properties.  
  We study the evolution of the gauge coupling constants in string unification schemes in which the light spectrum below the compactification scale is exactly that of the minimal supersymmetric standard model. In the absence of string threshold corrections the predicted values $\sin^2\theta _W=0.218$ and $\alpha _s=0.20$ are in gross conflict with experiment, but these corrections are generically important. One can express the string threshold corrections to $\sin^2\theta _W$ and $\alpha_s$ in terms of certain $modular$ $weights$ of quark, lepton and Higgs superfields as well as the $moduli$ of the string model. We find that in order to get agreement with the experimental measurements within the context of this $minimal$ scheme, certain constraints on the $modular$ $weights$ of the quark, lepton and Higgs superfields should be obeyed. Our analysis indicates that this $minimal$ $string$ $unification$ 
  In this paper we analyze one-matrix models by means of the associated discrete linear systems. We see that the consistency conditions of the discrete linear system lead to the Virasoro constraints. The linear system is endowed with gauge invariances. We show that invariance under time-independent gauge transformations entails the integrability of the model, while the double scaling limit is connected with a time-dependent gauge transformation. We derive the continuum version of the discrete linear system, we prove that the partition function is actually the $\tau$-function of the KdV hierarchy and that the linear system completely determines the Virasoro constraints. 
  We discuss three closely related questions; i)~Given a conformal field theory, how may we deform it? ii)~What are the symmetries of string theory? and iii)~Does string theory have free parameters? We show that there is a distinct deformation of the stress tensor for every solution to the linearised covariant equations of motion for the massless modes of the Bosonic string, and use this result to discuss the symmetries of the string. We also find an additional finite dimensional space of deformations which may correspond to free parameters of string theory, or alternatively may be interpreted as topological degrees of freedom, perhaps analogous to the isolated states found in two dimensions. 
  Vertex operators are constructed providing representations of the exchange relations containing either the S-matrix of a real coupling (simply-laced) affine Toda field theory, or its minimal counterpart. One feature of the construction is that the bootstrap relations for the S-matrices follow automatically from those for the conserved quantities, via an algebraic interpretation of the fusing of two particles to form a single bound state. 
  We investigate the possibility to construct extended parafermionic conformal algebras whose generating current has spin $1+\frac{1}{K}$, generalizing the superconformal (spin 3/2) and the Fateev Zamolodchikov (spin 4/3) algebras. Models invariant under such algebras would possess $Z_K$ exotic supersymmetries satisfying (supercharge)$^K$ = (momentum). However, we show that for $K=4$ this new algebra allows only for models at $c=1$, for $K=5$ it is a trivial rephrasing of the ordinary $Z_5$ parafermionic model, for $K=6,7$ (and, requiring unitarity, for all larger $K$) such algebras do not exist. Implications of this result for existence of exotic supersymmetry in two dimensional field theory are discussed. 
  We review various aspects of (infinite) quantum group symmetries in 2D massive quantum field theories. We discuss how these symmetries can be used to exactly solve the integrable models. A possible way for generalizing to three dimensions is shortly described. 
  We use the Virasoro master equation to study the space of Lie h-invariant conformal field theories, which includes the standard rational conformal field theories as a small subspace. In a detailed example, we apply the general theory to characterize and study the Lie h-invariant graphs, which classify the Lie h-invariant conformal field theories of the diagonal ansatz on SO(n). The Lie characterization of these graphs is another aspect of the recently observed Lie group-theoretic structure of graph theory. 
  It is argued that the effective string of whatever 3D gauge system at the deconfining transition is universally described by the minimal $N=2$ extended superconformal theory at $c=1$. A universal value of the critical temperature is predicted. 
  We study the nonunitary diagonal cosets constructed from admissible representations of Kac-Moody algebras at fractional level, with an emphasis on the question of field identification. Generic classes of field identifications are obtained from the analysis of the modular S matrix. These include the usual class related to outer automorphisms, as well as some intrinsically nonunitary field identifications. They allow for a simple choice of coset field representatives where all field components of the coset are associated with integrable finite weights. 
  An $O(d,d)$ symmetry of the manifold of string vacua that do not depend on $d$ (out of $D$) space-time coordinates has been recently identified. Here we write down, for $d=D-1$, the low energy equations of motion and their general solution in a manifestly $O(d,d)$-invariant form, pointing out an amusing similarity with the renormalization group framework. Previously considered cosmological and black hole solutions are recovered as particular examples. 
  It is known that much of the structure of string theory can be derived from three-dimensional topological field theory and gravity. We show here that, at least for simple topologies, the string diffeomorphism ghosts can also be explained in terms of three-dimensional physics. 
  The solution is given for the $c=3$ topological matter model whose underlying conformal theory has Landau-Ginzburg model $W=-\qa (x^4 +y^4)+\af x^2y^2$. While consistency conditions are used to solve it, this model is probably at the limit of such techniques. By using the flatness of the metric of the space of coupling constants I rederive the differential equation that relates the parameter \af\ to the flat coordinate $t$. This simpler method is also applied to the $x^3+y^6$-model. 
  We review some aspects of the free field approach to two-dimensional conformal field theories. Specifically, we discuss the construction of free field resolutions for the integrable highest weight modules of untwisted affine Kac-Moody algebras, and extend the construction to a certain class of admissible highest weight modules. Using these, we construct resolutions of the completely degenerate highest weight modules of W-algebras by means of the quantum Drinfeld-Sokolov reduction. As a corollary we derive character formulae for these degenerate highest weight modules. 
  Some results are presented concerning duality invariant effective string actions and the construction of automorphic functions for general (2,2) string compactifications. These considerations are applied in order to discuss the {\it minimal} unification of gauge coupling constants in orbifold compactifications with special emphasis on string threshold corrections. 
  We exhibit soliton solutions of QCD in two dimensions that have the quantum numbers of quarks. They exist only for quarks heavier than the dimensional gauge coupling, and have infinite energy, corresponding to the presence of a string carrying the non-singlet color flux off to spatial infinity. The quark solitons also disappear at finite temperature, as the temperature-dependent effective quark mass is reduced in the approach to the quark/hadron phase transition.  
  The spectra of $A_r$ affine Toda field theories with imaginary coupling constant, are investigated. Soliton solutions are found, which, despite the non-unitary form of the Lagrangian, have real classical masses and are stable to small perturbations. The quantum corrections to the soliton masses are determined, to lowest order in $\hbar$. The solitons have the same spectrum as the fundamental Toda particles; a feature that is preserved in the quantum theory. 
  This is a non-technical talk given at the Sixth Marcel Grossman Meeting on General Relativity, Kyoto, Japan in June 1991. Some developments in string theory over the last six years are discussed together with their qualitative implications for issues in quantum gravity. 
  Lecture notes on factorizable S-matrices, thermodynamic Bethe Ansatz and integrable perturbations of conformally invariant models; J.A.Swieca Summer School 1991 
  We study the BRST cohomology for two-dimensional supergravity coupled to $\hat c \leq 1$ superconformal matter in the conformal gauge. The super-Liouville and superconformal matters are represented by free scalar fields $\phi^L$ and $\phi^M$ and fermions $\psi^L$ and $\psi^M$, respectively, with suitable background charges, and these are coupled in such a way that the BRST charge is nilpotent. The physical states of the full theory are determined for NS and R sectors. It is shown that there are extra states with ghost number $N_{FP}=0,\pm 1$ for discrete momenta other than the degree of freedom corresponding to the ``center of mass", and that these are closely related to the ``null states" in the minimal models with $\hat c<1$. 
  We first give a complete, albeit brief, review of the discovery of mirror symmetry in $N=2$ string/conformal field theory. In particular, we describe the naturality arguments which led to the initial mirror symmetry conjectures and the subsequent work which established the existence of mirror symmetry through direct construction. We then review a number of striking consequences of mirror symmetry -- both conceptual and calculational. Finally, we describe recent work which introduces a variant on our original proof of the existence of mirror symmetry. This work affirms classical--quantum symmetry duality as well as extends the domain of our initial mirror symmetry construction. 
  We study the spectrum of $W_3$ strings. In particular, we show that for appropriately chosen space-time signature, one of the scalar fields is singled out by the spin-3 constraint and is ``frozen'': no creation operators from it can appear in physical states and the corresponding momentum must assume a specific fixed value. The remaining theory is unitary and resembles an ordinary string theory in $d\ne26$ with anomalies cancelled by appropriate background charges. In the case of the $W_3$ string, however, the spin-two ``graviton'' is massive. 
  We present a family of classical spacetimes in 2+1 dimensions. Such a spacetime is produced by a Nambu-Goto self-gravitating string. Due to the special properties of three-dimensional gravity, the metric is completely described as a Minkowski space with two identified worldsheets. In the flat limit, the standard string is recovered. The formalism is developed for an open string with massive endpoints, but applies to other boundary conditions as well. We consider another limit, where the string tension vanishes in geometrical units but the end-masses produce finite deficit angles. In this limit, our open string reduces to the free-masses solution of Gott, which possesses closed timelike curves when the relative motion of the two masses is sufficiently rapid. We discuss the possible causal structures of our spacetimes in other regimes. It is shown that the induced worldsheet Liouville mode obeys ({\it classically}) a differential equation, similar to the Liouville equation and reducing to it in the flat limit. A quadratic action formulation of this system is presented. The possibility and significance of quantizing the self-gravitating string, is discussed. 
  We give a systematic analysis of forward scattering in 3$+$1-dimensional quantum gravity, at center of mass energies comparable or larger than the Planck energy. We show that quantum gravitational effects in this kinematical regime are described by means of a topological field theory. We find that the scattering amplitudes display a universal behaviour very similar to two dimensional string amplitudes, thereby recovering results obtained previously by 't Hooft. Finally, we discuss the two-particle process in some detail. 
  Studying perturbatively, for large m, the torus partition function of both (A,A) and (A,D) series of minimal models in the Cappelli, Itzykson, Zuber classification, deformed by the least relevant operator $\phi_{(1,3)}$, we disentangle the structure of $\phi_{1,3}$ flows. The results are conjectured on reasonable ground to be valid for all m. They show that (A,A) models always flow to (A,A) and (A,D) ones to (A,D). No hopping between the two series is possible. Also, we give arguments that there exist 3 isolated flows (E,A)-->(A,E) that, together with the two series, should exhaust all the possible $\phi_{1,3}$ flows. Conservation (and symmetry breaking) of non-local currents along the flows is discussed and put in relation to the A,D,E classification. 
  The relation between Einstein gravity and the Chern-Simons gauge theory of the Poincare' group is discussed at the classical level. 
  The classical dynamics of N spinning point sources in 2+1 Einstein-Cartan gravity is considered. It corresponds to the ISO(2,1) Chern-Simons theory, in which the torsion source is restricted to its intrinsic spin part. A class of explicit solutions is found for the dreibein and the spin connection, which are torsionless in the spinless limit. By using the residual local Poincare' invariance of the solutions, we fix the gauge so that the metric is smooth outside the particles and satisfies proper asymptotic conditions at space and time infinity. We recover previous results for test bodies and find new ones for the scattering of two dynamical particles in the massless limit. 
  We present a non-relativistic fermionic field theory in 2-dimensions coupled to external gauge fields. The singlet sector of the $c=1$ matrix model corresponds to a specific external gauge field. The gauge theory is one-dimensional (time) and the space coordinate is treated as a group index. The generators of the gauge algebra are polynomials in the single particle momentum and position operators and they form the group $W^{(+)}_{1+\infty}$. There are corresponding Ward identities and residual gauge transformations that leave the external gauge fields invariant. We discuss the realization of these residual symmetries in the Minkowski time theory and conclude that the symmetries generated by the polynomial basis are not realized. We motivate and present an analytic continuation of the model which realises the group of residual symmetries. We consider the classical limit of this theory and make the correspondence with the discrete states of the $c=1$ (Euclidean time) Liouville theory. We explain the appearance of the $SL(2)$ structure in $W^{(+)}_{1+\infty}$. We also present all the Euclidean classical solutions and the classical action in the classical phase space. A possible relation of this theory to the $N=2$ string theory and also self-dual Einstein gravity in 4-dimensions is pointed out. 
  Metric independent $\sigma$ models are constructed. These are field theories which generalise the membrane idea to situations where the target space has fewer dimensions than the base manifold. Instead of reparametrisation invariance of the independent variables, one has invariance of solutions of the field equations under arbitrary functional redefinitions of the field quantities. Among the many interesting properties of these new models is the existence of a hierarchical structure which is illustrated by the following result. Given an arbitrary Lagrangian, dependent only upon first derivatives of the field, and homogeneous of weight one, an iterative procedure for calculating a sequence of equations of motion is discovered, which ends with a universal, possibly integrable equation, which is independent of the starting Lagrangian. A generalisation to more than one field is given. 
  We examine the inter-relationship of the superpotential containing hidden and observable matter fields and the ensuing condensates in free fermionic string models. These gauge and matter condensates of the strongly interacting hidden gauge groups play a crucial role in the determination of the physical parameters of the observable sector. Supplementing the above information with the requirement of modular invariance, we find that a generic model with only trilinear superpotential allows for a degenerate (and sometimes pathological) set of vacua. This degeneracy may be lifted by higher order terms in the superpotential. We also point out some other subtle points that may arise in calculations of this nature. We exemplify our observations by computing explicitly the modular invariant gaugino and matter condensates in the flipped SU(5) string model with hidden gauge group $SO(10)\times SU(4)$. 
  We report on generalizations of the KdV-type integrable hierarchies of Drinfel'd and Sokolov. These hierarchies lead to the existence of new classical $W$-algebras, which arise as the second Hamiltonian structure of the hierarchies. In particular, we present a construction of the $W_n^{(l)}$ algebras. 
  In this paper we develop two coadjoint orbit constructions for the phase spaces of the generalised $Sl(2)$ and $Sl(3)$ KdV hierachies. This involves the construction of two group actions in terms of Yang Baxter operators, and an Hamiltonian reduction of the coadjoint orbits. The Poisson brackets are reproduced by the Kirillov construction. From this construction we obtain a `natural' gauge fixing proceedure for the generalised hierarchies. 
  We propose using the general structure and properties of conformal field theory amplitudes, in particular those defined on surfaces with boundaries, to explore effective string theory amplitudes for some hadronic processes. Two examples are considered to illustrate the approach. In one a natural mechanism for chiral symmetry breaking within the string picture is proposed. One consequence is that the vertex operator for pion emission (at zero momentum) behaves like a world sheet current evaluated on the string boundary. This fact is used to rederive, in a more general setting, hadronic mass relations found in the early days of string theory by Lovelace, and Ademollo, Veneziano and Weinberg. In the second example, we derive the general structure of the form factor for the emission of a pomeron (interpreted as a closed string) from a meson or baryon. The result reconciles the interpretation of the pomeron as a closed string, emitted from the interior of the meson or baryon world sheet, with the additive quark rules for total hadronic cross sections. We also review the difficulties involved in constructing complete effective string theories for hadrons, and comment on the relation between the intercepts of trajectories and the short distance behavior of the underlying theory. 
  Classical W-symmetry is globally parametrized by the Grassmannian Manifold which is associated with the non-relativistic fermions. We give the bosonization rule which defines the natural higher coordinates system to describe the W-geometry. Generators of the W-algebra can be obtained from a single tau-function by using vertex operators. 
  It is shown that, classically, the W-algebras are directly related to the extrinsic geometry of the embedding of two-dimensional manifolds with chiral parametrisation (W-surfaces) into higher dimensional K\"ahler manifolds. We study the local and the global geometries of such embeddings, and connect them to Toda equations. The additional variables of the related KP hierarchy are shown to yield a specific coordinate system of the target-manifold, and this allows us to prove that W-transformations are simply particular diffeomorphisms of this target space. The W-surfaces are shown to be instantons of the corresponding non-linear $\sigma$-models. 
  We generalize the ground ring structure to all special BRST invariant operators in the right branch in the c=1 Liouville theory. We also discuss correlation functions of special states on the sphere. 
  Recent advances are being discussed on the calculation, within the conformal field theory approach, of the correlation functions for local operators in the theory of 2D gravity coupled to the minimal models of matter. 
  We summarize some recent results on the BRST analysis of physical states of 2D gravity coupled to c<=1 conformal matter and the supersymmetric generalization. 
  It is shown that the two-loop Kac-Moody algebra is equivalent to a two variable loop algebra and a decoupled $\beta$-$\gamma$ system. Similarly WZNW and CSW models having as algebraic structure the Kac-Moody algebra are equivalent to an infinity of versions of the corresponding ordinary models and decoupled abelian fields. 
  We give a review of some recent developments in the quantisation of $W$-gravity theories. In particular, we discuss the construction of anomaly-free $W_\infty$ and $W_3$ gravities. 
  We show that tree level open two dimensional string theory is exactly solvable; the solution exhibits some unusual features, and is qualitatively different from the closed case. The open string ``tachyon'' S -- matrix describes free fermions, which can be interpreted as the quarks at the ends of the string. These ``quarks'' live naturally on a lattice in space-time. We also find an exact vacuum solution of the theory, corresponding to a charged black hole. 
  We show how to write an off-shell action for the $SU(2)\times U(1)$ supersymmetric WZW model in terms of $N=2$ chiral and twisted chiral multiplets. We discuss the $N=4$ supersymmetry of this model and exhibit the $N=4$ superconformal current algebra. Finally, we show that the off-shell formulation makes it possible to perform a duality transformation, which leads to a supersymmetric sigma model on a manifold with a black hole type singularity. 
  We present an account of the early developments that led to the present form of the flipped $SU(5)$ string model. We focus on the method used to decide on this particular string model, as well as the basic steps followed in constructing generic models in the free fermionic formulation of superstrings in general and flipped $SU(5)$ in particular. We then describe the basic calculable features of the model which are used to obtain its low-energy spectrum: doublet and triplet Higgs mass matrices, fermion Yukawa matrices, neutrino masses, and the top-quark mass. We also review the status of proton decay in the model, as well as the hidden sector bound states called cryptons. Finally, we comment on the subject of string threshold corrections and string unification. 
  We develop a stochastic approach to the theory of tunneling with the baby universe formation. This method is applied also to the theory of creation of the universe in a laboratory. 
  An appropriate field configuration in non-polynomial closed string field theory is shown to correspond to a general off-shell field configuration in low energy effective field theory. A set of string field theoretic symmetries that act on the fields in low energy effective field theory as general coordinate transformation and antisymmetric tensor gauge transformation is identified. The analysis is carried out to first order in the fields; thus the symmetry transformations in string field theory reproduce the linear and the first non-linear terms in the gauge transformations in the low energy effective field theory. 
  We consider a string theory based on an SU(1,1) Wess-Zumino-Novikov-Witten model and an arbitrary unitary conformal fild theory. We show that the solutions of the Virasoro conditions, in the unitarity regime of the SU(1,1) theory, are states which lie in the Euclidean coset SU(1,1)/U(1). This shows the validity, at the quantum level, of a time-like type of gauge in these models. 
  We investigate the S-matrix of N=2 supersymmetric sine-Gordon theory based on the N=2 supersymmetry and the quantum group structure. The topological charges play an important role to derive physical contents. 
  We review some recent developments in string theory, emphasizing the importance of vacuum instabilities, their relation to the density of states, and the role of space-time fermions in non-critical string theory. We also discuss the classical dynamics of two dimensional string theory. 
  K\"ahler-Chern-Simons theory describes antiself-dual gauge fields on a four- dimensional K\"ahler manifold. The phase space is the space of gauge potentials, the symplectic reduction of which by the constraints of antiself-duality leads to the moduli space of antiself-dula instantons. We outline the theory highlighting the symmetries, their canonical realization and some properties of the quantum wave functions. The relationship to integrable systems via dimensional reduction is briefly discussed. 
  Redundancies are pointed out in the widely used extension of the crystallographic concept of Bravais class to quasiperiodic materials. Such pitfalls can be avoided by abandoning the obsolete paradigm that bases ordinary crystallography on microscopic periodicity. The broadening of crystallography to include quasiperiodic materials is accomplished by defining the point group in terms of indistinguishable (as opposed to identical) densities.  
  We review the recently proposed string theory in two dimensional black hole background. Especially, the structure of the duality in the target space is discussed. The duality is analogous to \lq\lq $R \rightarrow 1/R$" symmetry of a compactified boson. We consider the duality in more general target space manifolds which have Killing symmetries and we give an explicit formula which connects two different manifolds which are dual to each other. Superstring theory in two dimensional black hole background is also discussed based on supersymmetric $SU(1,1)/U(1)$ gauged Wess-Zumino-Witten model. 
  We present the Wakimoto construction of the super OSp(1,2) and SL(2,1) Kac-Moody algebras and the free field representation of the corresponding WZW models. After imposing suitable constraints, we can lead the Feigin-Fuchs representation of Virasoro algebras and coadjoint actions ofthe N=1 and N=2 conformal symmetries. This formulation corresponds to a supercovariant extension of the Drinfeld and Sokolov Hamiltonian reduction. 
  The dissipative quantum mechanics of a charged particle in a uniform magnetic field and periodic potential has delocalization critical points which correspond to backgrounds for the open string. We study the phase diagram of this system (in the magnetic field/dissipation constant plane) and find a fractal structure which, in the limit of zero dissipation, matches the fractal energy level structure of the pure quantum mechanical version of this problem (Hofstadter model). 
  Using results of the thermodynamic Bethe Ansatz approach and conformal perturbation theory we argue that the $\phi_{1,3}$-perturbation of a unitary minimal $(1+1)$-dimensional conformal field theory (CFT) in the $D$-series of modular invariant partition functions induces a renormalization group (RG) flow to the next-lower model in the $D$-series. An exception is the first model in the series, the 3-state Potts CFT, which under the $\ZZ_2$-even $\phi_{1,3}$-perturbation flows to the tricritical Ising CFT, the second model in the $A$-series. We present arguments that in the $A$-series flow corresponding to this exceptional case, interpolating between the tetracritical and the tricritical Ising CFT, the IR fixed point is approached from ``exactly the opposite direction''. Our results indicate how (most of) the relevant conformal fields evolve from the UV to the IR CFT. 
  We analyze the superstring propagating on a Calabi-Yau threefold. This theory naturally leads to the consideration of Witten's topological non-linear sigma-model and the structure of rational curves on the Calabi-Yau manifold. We study in detail the case of the world-sheet of the string being mapped to a multiple cover of an isolated rational curve and we show that a natural compactification of the moduli space of such a multiple cover leads to a formula in agreement with a conjecture by Candelas, de la Ossa, Green and Parkes. 
  We have constructed and solved various one-dimensional quantum mechanical models which have quantum algebra symmetry. Here we summarize this work, and also present new results on graded models, and on the so-called string solutions of the Bethe Ansatz equations for the $A^{(2)}_2$ model. 
  We have recently constructed a large class of open quantum spin chains which have quantum-algebra symmetry and which are integrable. We show here that these models can be exactly solved using a generalization of the analytical Bethe Ansatz (BA) method. In particular, we determine in this way the spectrum of the transfer matrices of the $U_q [(su(2)]$-invariant spin chains associated with $A^{(1)}_1$ and $A^{(2)}_2$ in the fundamental representation. The quantum-algebra invariance of these models plays an essential role in obtaining these results. The BA equations for these open chains are ``doubled'' with respect to the BA equations for the corresponding closed chains. 
  We analyze the non--perturbative features of 2D quantum gravity defined by stochastic regularization of the unstable matrix model showing, first, that the WKB approximation of the well-defined quantum Fokker-Planck hamiltonian corresponds to the semiclassical eigenvalue density of the former. The double scaled potential exhibits an instanton--like behaviour, which is universal and scales, but whose interpretation in terms of pure gravity is still open. 
  The fractional supersymmetry chiral algebras in two-dimensional conformal field theory are extended Virasoro algebras with fractional spin currents. We show that associativity and closure of these algebras determines their structure constants in the case that the Virasoro algebra is extended by precisely one current. We compute the structure constants of these algebras explicitly and we show that correlators of the currents satisfy non-Abelian braiding relations. 
  We study the generalization of $R\to 1/R$ duality to arbitrary conformally invariant sigma models with an isometry. We show that any pair of dual sigma models can be represented as quotients of a self-dual sigma model obtained by gauging different combinations of chiral currents. This observation is used to clarify the interpretation of the generalized duality as a symmetry of conformal field theory. We extend these results to $N=2$ supersymmetric sigma models. 
  Deformations of gauged WZW actions are constructed for any pair $(G,H)$ by taking different embeddings of the gauge group $H\subset G$ as it acts on the left and right of the group element $g$. This leads to models that are dual to each other, generalizing the axial/vector duality of the two dimensional black hole manifold. The classical equations are completely solved for any pair $(G,H)$ and in particular for the anti de Sitter string based on $SO(d- 1,2)/SO(d-1,1)$ for which the normal modes are determined. Duality is demonstrated for models that have the same set of normal modes. Concentrating on $SO(2,2)/SO(2,1)$, the metric and dilaton fields of the $d=3$ string as well as some of the dual generalizations are obtained. They have curvature singularities and represent new singular solutions of Einstein's general relativity in three dimensions.  
  We show how topological $G_k/G_k$ models can be embedded into the topological matter models that are obtained by perturbing the twisted $N=2$ supersymmetric, hermitian symmetric, coset models. In particular, this leads to an embedding of the fusion ring of $G$ as a sub-ring of the perturbed, chiral primary ring. The perturbation of the twisted $N=2$ model that leads to the fusion ring is also shown to lead to an integrable $N=2$ supersymmetric field theory when the untwisted $N=2$ superconformal field theory is perturbed by the same operator and its hermitian conjugate.  
  We study the perturbation theory for three dimensional Chern--Simons quantum field theory on a general compact three manifold without boundary. We show that after a simple change of variables, the action obtained by BRS gauge fixing in the Lorentz gauge has a superspace formulation. The basic properties of the propagator and the Feynman rules are written in a precise manner in the language of differential forms. Using the explicit description of the propagator singularities, we prove that the theory is finite. Finally the anomalous metric dependence of the $2$-loop partition function on the Riemannian metric (which was introduced to define the gauge fixing) can be cancelled by a local counterterm as in the $1$-loop case. In fact, the counterterm is equal to the Chern--Simons action of the metric connection, normalized precisely as one would expect based on the framing dependence of Witten's exact solution. 
  We study the $q$-deformed su(2) spin network as a 3-dimensional quantum gravity model. We show that in the semiclassical continuum limit the Turaev-Viro invariant obtained recently defines naturally regularized path-integral $\grave{\rm a}$ la Ponzano-Regge, In which a contribution from the cosmological term is effectively included. The regularization dependent cosmological constant is found to be ${4\pi^2\over k^2} +O(k^{-4})$, where $q^{2k}=1$. We also discuss the relation to the Euclidean Chern-Simons-Witten gravity in 3-dimension.  
  Starting from a recently-proposed general formula, various properties of the ADE series of purely elastic S-matrices are rederived in a universal way. In particular, the relationship between the pole structure and the bootstrap equations is shown to follow from properties of root systems. The discussion leads to a formula for the signs of the three-point couplings in the simply-laced affine Toda theories, and a simple proof of a result due to Klassen and Melzer of relevance to Thermodynamic Bethe Ansatz calculations. 
  This paper is devoted to the quantization of the second-ilk superparticle using the Batalin-Vilkovisky method. We show the full structure of the master action. By imposing gauge conditions on the gauge fields rather than on coordinates we find a gauge-fixed quantum action which is free. The structure of the BRST charge is exhibited and the BRST cohomology yields the same physical spectrum as the light- cone quantization of the usual superparticle. 
  We give the complete twisted Yukawa couplings for all the Z_n orbifold constructions in the most general case, i.e. when orbifold deformations are considered. This includes a certain number of tasks. Namely, determination of the allowed couplings, calculation of the explicit dependence of the Yukawa couplings values on the moduli expectation values (i.e. the parameters determining the size and shape of the compactified space), etc. The final expressions are completely explicit, which allows a counting of the DIFFERENT Yukawa couplings for each orbifold (with and without deformations). This knowledge is crucial to determine the phenomenological viability of the different schemes, since it is directly related to the fermion mass hierarchy. Other facts concerning the phenomenological profile of Z_n orbifolds are also discussed, e.g. the existence of non--diagonal entries in the fermion mass matrices, which is related to a non--trivial structure of the Kobayashi--Maskawa matrix. Finally some theoretical results are given, e.g. the no--participation of (1,2) moduli in twisted Yukawa couplings. Likewise, (1,1) moduli associated with fixed tori which are involved in the Yukawa coupling, do not affect the value of the coupling. 
  We discuss the generalization of Abelian Chern-Simons theories when $\theta $-angles and magnetic monopoles are included. We map sectors of two dimensional Conformal Field Theories into these three dimensional theories. 
  We provide an intrinsic description of $N$-super \RS s and $TN$-\SR\ surfaces. Semirigid surfaces occur naturally in the description of topological gravity as well as topological supergravity. We show that such surfaces are obtained by an integrable reduction of the structure group of a complex supermanifold. We also discuss the \s moduli spaces of $TN$-\SR\ surfaces and their relation to the moduli spaces of $N$-\s\ \RS s. 
  Using Chern-Simons gauge theory, we show that the fusion ring of the conformal field theory G_k is isomorphic to P(u)/(\del V), where V is a polynomial in u and (\del V) is the ideal generated by the conditions \del V=0. We also derive a residue-like formula for the correlation functions in the Chern-Simons theory thus providing a RCFT version of the residue formula for the TLG models. An operator that acts like the measure in the residue formula has the ionterpretation of a handle squashing operator and an explicit formula for this operator is given. 
  We show that the one dimensional unitary matrix model with potential of the form $a U + b U^2 + h.c.$ is integrable. By reduction to the dynamics of the eigenvalues, we establish the integrability of a system of particles in one space dimension in an external potential of the form $a \cos (x+\alpha ) + b \cos ( 2x +\beta )$ and interacting through two-body potentials of the inverse sine square type. This system constitutes a generalization of the Sutherland model in the presence of external potentials. The positive-definite matrix model, obtained by analytic continuation, is also integrable, which leads to the integrability of a system of particles in hyperbolic potentials interacting through two-body potentials of the inverse hypebolic sine square type.  
  It is shown that for a translationally invariant solution to string theory, spacetime duality interchanges the momentum in the symmetry direction and the axion charge per unit length. As one application, we show explicitly that charged black strings are equivalent to boosted (uncharged) black strings. The extremal black strings (which correspond to the field outside of a fundamental macroscopic string) are equivalent to plane fronted waves describing strings moving at the speed of light. 
  We establish a previously conjectured connection between $p$-adics and quantum groups. We find in Sklyanin's two parameter elliptic quantum algebra and its generalizations, the conceptual basis for the Macdonald polynomials, which ``interpolate'' between the zonal spherical functions of related real and $p$\--adic symmetric spaces. The elliptic quantum algebras underlie the $Z_n$\--Baxter models. We show that in the $n \air \infty$ limit, the Jost function for the scattering of {\em first} level excitations in the $Z_n$\--Baxter model coincides with the Harish\--Chandra\--like $c$\--function constructed from the Macdonald polynomials associated to the root system $A_1$. The partition function of the $Z_2$\--Baxter model itself is also expressed in terms of this Macdonald\--Harish\--Chandra\ $c$\--function, albeit in a less simple way. We relate the two parameters $q$ and $t$ of the Macdonald polynomials to the anisotropy and modular parameters of the Baxter model. In particular the $p$\--adic ``regimes'' in the Macdonald polynomials correspond to a discrete sequence of XXZ models. We also discuss the possibility of ``$q$\--deforming'' Euler products. 
  We find several classes of exact classical solutions of critical bosonic string theory, constructed as twisted products of one Euclidean and one Minkowskian 2D black hole coset. One class of these solutions leads (after tensoring with free scalars and supersymmetrizing) to a rotating version of the recently discovered exact black fivebrane. Another class represents a one-parameter family of axisymmetric stationary four-dimensional targets with horizons. Global properties and target duality of the 4D solutions are briefly analyzed.  
  We study a two-dimensional conformal field theory coupled to quantum gravity on a disk. Using the continuum Liouville field approach, we compute three-point correlation functions of boundary operators. The structure of momentum singularities is different from that of correlation functions on a sphere and is more complicated. We also compute four-point functions of boundary operators and three-point functions of two boundary operators and one bulk operator. 
  Chern-Simons Theory with gauge group $SU(N)$ is analyzed from a perturbation theory point of view. The vacuum expectation value of the unknot is computed up to order $g^6$ and it is shown that agreement with the exact result by Witten implies no quantum correction at two loops for the two-point function. In addition, it is shown from a perturbation theory point of view that the framing dependence of the vacuum expectation value of an arbitrary knot factorizes in the form predicted by Witten. 
  We extend the classical heterotic instanton solutions to all orders in $\alpha'$ using the equations of anomaly-free supergravity, and discuss the relation between these equations and the string theory $\beta$-functions. 
  It is known that Liouville theory can be represented as an SL(2,R) gauged WZW model. We study a two dimensional field theory which can be obtained by analytically continuing some of the variables in the SL(2,R) gauged WZW model. We can derive Liouville theory from the analytically continued model, ( which is a gauged SL(2,C)/SU(2) model, ) in a similar but more rigorous way than from the original gauged WZW model. We investigate the observables of this gauged SL(2,C)/SU(2) model. We find infinitely many extra observables which can not be identified with operators in Liouville theory. We concentrate on observables which are $(1,1)$ forms and the correlators of their integrals over two dimensional spacetime. At a special value of the coupling constant of our model, the correlators of these integrals on the sphere coincide with the results from matrix models.  
  We develop elementary canonical methods for the quantization of abelian and nonabelian Chern-Simons actions using well known ideas in gauge theories and quantum gravity. Our approach does not involve choice of gauge or clever manipulations of functional integrals. When the spatial slice is a disc, it yields Witten's edge states carrying a representation of the Kac-Moody algebra. The canonical expression for the generators of diffeomorphisms on the boundary of the disc are also found, and it is established that they are the Chern-Simons version of the Sugawara construction. This paper is a prelude to our future publications on edge states, sources, vertex operators, and their spin and statistics in 3D and 4D topological field theories. 
  Starting from SL(3,R) Chern-Simons theory we derive the covariant action for W_3 gravity. 
  We show that the XY quantum chain in a magnetic field is invariant under a two parameter deformation of the SU(1/1) superalgebra. One is led to an extension of the braid group and the Hecke algebra which reduce to the known ones when the two parameter coincide. The physical significance of the two parameters is discussed. 
  Ooguri and Vafa have shown that the open N=2 string corresponds to self-dual Yang-Mills (SDYM) and also that, in perturbation theory, it has has a vanishing four particle scattering amplitude. We discuss how the dynamics of the three particle scattering implies that on shell states can only scatter if their momenta lie in the same self-dual plane and then investigate classical SDYM with the aim of comparing exact solutions with the tree level perturbation theory predictions. In particular for the gauge group SL(2,C) with a plane wave Hirota ansatz SDYM reduces to a complicated set of algebraic relations due to de Vega. Here we solve these conditions and the solutions are shown to correspond to collisions of plane wave kinks. The main result is that for a class of kinks the resulting phase shifts are non-zero, the solution as a whole is not pure gauge and so the scattering seems non-trivial. However the stress energy and Lagrangian density are confined to string like regions in the space time and in particular are zero for the incoming/outgoing kinks so the solution does not correspond to physical four point scattering. 
  We discuss non-compact WZW sigma models, especially the ones with symmetric space $H^{\bf C}/H$ as the target, for $H$ a compact Lie group. They offer examples of non-rational conformal field theories. We remind their relation to the compact WZW models but stress their distinctive features like the continuous spectrum of conformal weights, diverging partition functions and the presence of two types of operators analogous to the local and non-local insertions recently discussed in the Liouville theory. Gauging non-compact abelian subgroups of $H^{\bf C}$ leads to non-rational coset theories. In particular, gauging one-parameter boosts in the $SL(2,\bC)/SU(2)$ model gives an alternative, explicitly stable construction of a conformal sigma model with the euclidean 2D black hole target. We compute the (regularized) toroidal partition function and discuss the spectrum of the theory. A comparison is made with more standard approach based on the $U(1)$ coset of the $SU(1,1)$ WZW theory where stability is not evident but where unitarity becomes more transparent. 
  Based on a study of recently proposed solution of 2 dim. black hole we argue that the space-time singularities of general relativity may be described by topological field theories (TFTs). We also argue that in general TFT is a field theory which decsribes singular configurations with a reduced holonomy in its field space. 
  We construct new multi-field realisations of the $N=2$ super-$W_3$ algebra, which are important for building super-$W_3$ string theories. We derive the structure of the ghost vacuum for such theories, and use the result to calculate the intercepts. These results determine the conditions for physical states in the super-$W_3$ string theory. 
  In low dimensions, conformal anomaly has profound influence on the critical behavior of random surfaces with extrinsic curvature rigidity $1/\a$. We illustrate this by making a small $D$ expansion of rigid random surfaces, where a non-trivial infra-red fixed point is shown to exist. We speculate on the renormalization group flow diagram in the $(\a,D)$ plane. We argue that the qualitative behavior of numerical simulations in $D=3, 4$ could be understood on the basis of the phase diagram. 
  These theories, which are surely some of the simplest possible quantum field theories, were introduced in a paper of Dijkgraaf and Witten. The path integral reduces to a finite sum, so it is quite amenable to direct mathematical study. Although the theory exisits in arbitrary dimensions, it is most interesting in $2+1$~dimensions, where it has a ``modular structure.'' This is related to quantum groups, and the precise details may give clues as to what happens in other contexts.   This paper is written using AMSTeX 2.1, which can be obtained via ftp from the American Mathematical Society (instructions included). 1 encapsulated postscript file was submitted separately in uuencoded tar-compressed format. 
  The $q$--deformation $U_q (h_4)$ of the harmonic oscillator algebra is defined and proved to be a Ribbon Hopf algebra.Associated with this Hopf algebra we define an infinite dimensional braid group representation on the Hilbert space of the harmonic oscillator, and an extended Yang--Baxter system in the sense of Turaev. The corresponding link invariant is computed in some particular cases and coincides with the inverse of the Alexander--Conway polynomial. The $R$ matrix of $U_q (h_4)$ can be interpreted as defining a baxterization of the intertwiners for semicyclic representations of $SU(2)_q$ at $q=e^{2 \pi i/N}$ in the $N \rightarrow \infty$ limit.Finally we define new multicolored braid group representations and study their relation to the multivariable Alexander--Conway polynomial. 
  Theoretical developments during the past several years have shown that large scale properties of the Quantum Hall system can be successfully described by effective field theories which use the Chern-Simons interaction. In this article, we first recall certain salient features of the Quantum Hall Effect and their microscopic explanation. We then review one particular approach to their description based on the Chern-Simons Lagrangian and its variants. 
  It is shown how twisted N=2 (k=1) provides for the first time a complete conformal field theory description of the usual geometrical phase transitions in two dimensions, like polymers, percolation or brownian motion. In particular, four point functions of operators with half integer Kac labels are computed, together with geometrical operator products. In addition to Ramond and Neveu Schwartz, a sector with quarter twists has to be introduced. The role of fermions and their various sectors is geometrically interpreted, modular invariant partition functions are built. The presence of twisted N=2 is traced back to the Parisi Sourlas supersymmetry. It is shown that N=2 leads also to new non trivial predictions; for instance the fractal dimension of the percolation backbone in two dimensions is conjectured to be D=25/16, in good agreement with numerical studies. 
  We introduce in this paper two dimensional lattice models whose continuum limit belongs to the $N=2$ series. The first kind of model is integrable and obtained through a geometrical reformulation, generalizing results known in the $k=1$ case, of the $\Gamma_{k}$ vertex models (based on the quantum algebra $U_{q}sl(2)$ and representation of spin $j=k/2$). We demonstrate in particular that at the $N=2$ point, the free energy of the $\Gamma_{k}$ vertex model can be obtained exactly by counting arguments, without any Bethe ansatz computation, and we exhibit lattice operators that reproduce the chiral ring. The second class of models is more adequately described in the language of twisted $N=2$ supersymmetry, and consists of an infinite series of multicritical polymer points, which should lead to experimental realizations. It turns out that the exponents $\nu=(k+2)/2(k+1)$ for these multicritical polymer points coincide with old phenomenological formulas due to the chemist Flory. We therefore confirm that these formulas are {\bf exact} in two dimensions, and suggest that their unexpected validity is due to non renormalization theorems for the $N=2$ underlying theories. We also discuss the status of the much discussed theta point for polymers in the light of $N=2$ renormalization group flows. 
  We present an alternative derivation and geometrical formulation of Verlinde topological field theory, which may describe scattering at center of mass energies comparable or larger than the Planck energy. A consistent trunckation of 3+1 dimensional Einstein action is performed using the standard geometrical objects, like tetrads and spin connections. The resulting topological invariant is given in terms of differential forms. 
  The Chern-Simons ten-dimensional manifestly supersymmetric non-Abelian gauge theory is presented by performing the second quantization of the superparticle theory. The equation of motion is $F = (d+A)^2 = 0$, where $d$ is the nilpotent fermionic BRST operator of the first quantized theory and $A$ is the anti- commuting connection for the gauge group. This equation can be derived as a condition of the gauge independence of the first quantized theory in a background field $A$, or from the string field theory Lagrangian of the Chern- Simons type. The trivial solutions of the cohomology are the gauge symmetries, the non-trivial solution is given by the D=10 superspace, describing the super Yang-Mills theory on shell 
  We prove that the extrinsic Hausdorff dimension is always greater than or equal to the intrinsic Hausdorff dimension in models of triangulated random surfaces with action which is quadratic in the separation of vertices. We furthermore derive a few naive scaling relations which relate the intrinsic Hausdorff dimension to other critical exponents. These relations suggest that the intrinsic Hausdorff dimension is infinite if the susceptibility does not diverge at the critical point. 
  We investigate unitary one-matrix models coupled to bosonic quarks. We derive a flow equation for the square-root of the specific heat as a function of the renormalized quark mass. We show numerically that the flows have a finite number of solitary waves, and we postulate that their number equals the number of quark flavors. We also study the nonperturbative behavior of this theory and show that as the number of flavors diverges, the flow does not reach two-dimensional gravity. 
  A construction of elements of the BRS cohomology of ghost number +/- 1 in c<1 string theory is described, and their two-point function computed on the sphere. The construction makes precise the relation between these extra states and null vectors. The physical states of ghost number +1 are found to be exact forms with respect to a ``conjugate'' BRS operator. 
  We show that there are solitons with fractional fermion number in integrable $N$=2 supersymmetric models. We obtain the soliton S-matrix for the minimal, $N$=2 supersymmetric theory perturbed in the least relevant chiral primary field, the $\Phi _{(1,3)}$ superfield. The perturbed theory has a nice Landau-Ginzburg description with a Chebyshev polynomial superpotential. We show that the S-matrix is a tensor product of an associated ordinary $ADE$ minimal model S-matrix with a supersymmetric part. We calculate the ground-state energy in these theories and in the analogous $N$=1 case and $SU(2)$ coset models. In all cases, the ultraviolet limit is in agreement with the conformal field theory. 
  We apply the recently developed method of differential renormalization to the Wess-Zumino model. From the explicit calculation of a finite, renormalized effective action, the $\beta$-function is computed to three loops and is found to agree with previous existing results. As a further, nontrivial check of the method, the Callan-Symanzik equations are also verified to that loop order. Finally, we argue that differential renormalization presents advantages over other superspace renormalization methods, in that it avoids both the ambiguities inherent to supersymmetric regularization by dimensional reduction (SRDR), and the complications of virtually all other supersymmetric regulators. 
  We show that the metric and Berry's curvature for the ground states of $N=2$ supersymmetric sigma models can be computed exactly as one varies the Kahler structure. For the case of $CP^n$ these are related to special solutions of affine toda equations. This allows us to extract exact results (including exact instanton corrections). We find that the ground state metric is non-singular as the size of the manifold shrinks to zero thus suggesting that 2d QFT makes sense even beyond zero radius. In other words it seems that manifolds with zero size are non-singular as target spaces for string theory (even when they are not conformal). The cases of $CP^1$ and $CP^2$ are discussed in more detail. 
  Aspects of duality and mirror symmetry in string theory are discussed. We emphasize, through examples, the importance of loop spaces for a deeper understanding of the geometrical origin of dualities in string theory. Moreover we show that mirror symmetry can be reformulated in very simple terms as the statement of equivalence of two classes of topological theories: Topological sigma models and topological Landau-Ginzburg models. Some suggestions are made for generalization of the notion of mirror symmetry. 
  We consider 4-dimensional string models obtained by tensoring N=2 coset theories with non-diagonal modular invariants. We present results from a systematic analysis including moddings by discrete symmetries. 
  Two items are reproduced herein: my `Outlook' talk, an amended version of which was presented at the 1991 joint Lepton--Photon and EPS Conference in Geneva, and an Open Letter addressed to HEPAP. One is addressed primarily to the European high--energy physics community, the other to the American. A common theme of these presentations is a plea for the rational allocation of the limited funds society provides for high--energy physics research. If my `loose cannon' remarks may seem irresponsible to some of my colleagues, my silence would be more so. 
  We find and analyze the Landau-Ginzburg potentials whose critical points determine chiral rings which are exactly the fusion rings of Sp(N)_{K} WZW models. The quasi-homogeneous part of the potential associated with Sp(N)_{K} is the same as the quasi-homogeneous part of that associated with SU(N+1)_{K}, showing that these potentials are different perturbations of the same Grassmannian potential. Twisted N=2 topological Landau-Ginzburg theories are derived from these superpotentials. The correlation functions, which are just the Sp(N)_{K} Verlinde dimensions, are expressed as fusion residues. We note that the Sp(N)_{K} and Sp(K)_{N} topological Landau-Ginzburg theories are identical, and that while the SU(N)_{K} and SU(K)_{N} topological Landau-Ginzburg models are not, they are simply related. 
  We discuss the bosonization of non-relativistic fermions in one space dimension in terms of bilocal operators which are naturally related to the generators of $W$-infinity algebra. The resulting system is analogous to the problem of a spin in a magnetic field for the group $W$-infinity. The new dynamical variables turn out to be $W$-infinity group elements valued in the coset $W$-infinity/$H$ where $H$ is a Cartan subalgebra. A classical action with an $H$ gauge invariance is presented. This action is three-dimensional. It turns out to be similiar to the action that describes the colour degrees of freedom of a Yang-Mills particle in a fixed external field. We also discuss the relation of this action with the one we recently arrived at in the Euclidean continuation of the theory using different coordinates. 
  We construct the restricted sine-Gordon theory by truncating the sine-Gordon multi-soliton Hilbert space for the repulsive coupling constant due to the quantum group symmetry $SL_q(2)$ which we identify from the Korepin's $S$-matrices. We connect this restricted sine-Gordon theory with the minimal ($c<1$) conformal field theory ${\cal M}_{p/p+2}$ ($p$ odd) perturbed by the least relevent primary field $\Phi_{1,3}$. The exact $S$-matrices are derived for the particle spectrum of a kink and neutral particles. As a consistency check, we compute the central charge of the restricted theory in the UV limit using the thermodynamic Bethe ansatz analysis and show that it is equal to that of ${\cal M}_{p/p+2}$. 
  The connection between q-analogs of special functions and representations of quantum algebras has been developed recently. It has led to advances in the theory of q-special functions that we here review. 
  The Ward identities of the Liouville gravity coupled to the minimal conformal matter are investigated. We introduce the pseudo-null fields and the generalized equations of motion, which are classified into series of the Liouville charges. These series have something to do with the W and Virasoro constraints. The pseudo-null fields have non-trivial contributions at the boundaries of the moduli space. We explicitly evaluate the several boundary contributions. Then the structures similar to the W and the Virasoro constraints appearing in the topological and the matrix methods are realized. Although our Ward identities have some different features from the other methods, the solutions of the identities are consistent to the matrix model results. 
  We describe a strategy for computing Yukawa couplings and the mirror map, based on the Picard-Fuchs equation. (Our strategy is a variant of the method used by Candelas, de la Ossa, Green, and Parkes in the case of quintic hypersurfaces.) We then explain a technique of Griffiths which can be used to compute the Picard-Fuchs equations of hypersurfaces. Finally, we carry out the computation for four specific examples (including quintic hypersurfaces, previously done by Candelas et al.). This yields predictions for the number of rational curves of various degrees on certain hypersurfaces in weighted projective spaces. Some of these predictions have been confirmed by classical techniques in algebraic geometry. 
  The methods of conformal field theory are used to compute the crossing probabilities between segments of the boundary of a compact two-dimensional region at the percolation threshold. These probabilities are shown to be invariant not only under changes of scale, but also under mappings of the region which are conformal in the interior and continuous on the boundary. This is a larger invariance than that expected for generic critical systems. Specific predictions are presented for the crossing probability between opposite sides of a rectangle, and are compared with recent numerical work. The agreement is excellent. 
  We review the main topics concerning Fusion Rule Algebras (FRA) of Rational Conformal Field Theories. After an exposition of their general properties, we examine known results on the complete classification for low number of fields ($\leq 4$). We then turn our attention to FRA's generated polynomially by one (real) fundamental field, for which a classification is known. Attempting to generalize this result, we describe some connections between FRA's and Graph Theory. The possibility to get new results on the subject following this ``graph'' approach is briefly discussed. 
  Starting from $W_{\infty}$ as a fundamental symmetry and using the coadjoint orbit method, we derive an action for one dimensional strings. It is shown that on the simplest nontrivial orbit this gives the single scalar collective field theory. On higher orbits one finds generalized KdV type field theories with increasing number of components. Here the tachyon is coupled to higher tensor fields. 
  We investigate the classical phase space of 2-d string theory. We derive the linearised covariant equations for the spacetime fields by considering the most general deformation of the energy-momentum tensor which describes $c=1$ matter system coupled to 2-d gravity and by demanding that it respect conformal invariance. We derive the gauge invariances of the theory, and so investigate the classical phase space, defined as the space of all solutions to the equations of motion modulo gauge transformations. We thus clarify the origins of two classes of isolated states. 
  Some results in random matrices are generalized to supermatrices, in particular supermatrix integration is reduced to an integration over the eigenvalues and the resulting volume element is shown to be equivalent to a one dimensional Coulomb gas of both positive and negative charges.It is shown that,for polynomial potentials, after removing the instability due to the annihilation of opposite charges, supermatrix models are indistinguishable from ordinary matrix models, in agreement with a recent result by Alvarez-Gaume and Manes. It is pointed out however that this may not be true for more general potentials such as for instance the supersymmetric generalization of the Penner model. 
  We show that, in string theory, the quantum evaporation and decay of black holes in two-dimensional target space is related to imaginary parts in higher-genus string amplitudes. These arise from the regularisation of modular infinities due to the sum over world-sheet configurations, that are known to express the instabilities of massive string states in general, and are not thermal in character. The absence of such imaginary parts in the matrix model limit confirms that the latter constitutes the final stage of the evaporation process, at least in perturbation theory. Our arguments appear to be quite generic, related only to the summation over world-sheet surfaces, and hence should also apply to higher-dimensional target spaces. 
  Using the zero-curvature formulation, it is shown that W-algebra transformations are symmetries of corresponding generalised Drinfel'd-Sokolov hierarchies. This result is illustrated with the examples of the KdV and Boussinesque hierarchies, and the hierarchy associated to the Polyakov-Bershadsky W-algebra. 
  Random matrix models based on an integral over supermatrices are proposed as a natural extension of bosonic matrix models. The subtle nature of superspace integration allows these models to have very different properties from the analogous bosonic models. Two choices of integration slice are investigated. One leads to a perturbative structure which is reminiscent of, and perhaps identical to, the usual Hermitian matrix models. Another leads to an eigenvalue reduction which can be described by a two component plasma in one dimension. A stationary point of the model is described. 
  We briefly review some results in the theory of quantum $W_3$ gravity in the chiral gauge. We compare them with similar results in the analogous but simpler cases of $d=2$ induced gauge theories and $d=2$ induced gravity. 
  We formulate simple graphical rules which allow explicit calculation of nonperturbative $c=1$ $S$-matrices. This allows us to investigate the constraint of nonperturbative unitarity, which indeed rules out some theories. Nevertheless, we show that there is an infinite parameter family of nonperturbatively unitary $c=1$ $S$-matrices. We investigate the dependence of the $S$-matrix on one of these nonperturbative parameters. In particular, we study the analytic structure, background dependence, and high-energy behavior of some nonperturbative $c=1$ $S$-matrices. The scattering amplitudes display interesting resonant behavior both at high energies and in the complex energy plane. 
  We study Lie-Poisson actions on symplectic manifolds. We show that they are generated by non-Abelian Hamiltonians. We apply this result to the group of dressing transformations in soliton theories; we find that the non-Abelian Hamiltonian is just the monodromy matrix. This provides a new proof of their Lie-Poisson property. We show that the dressing transformations are the classical precursors of the non-local and quantum group symmetries of these theories. We treat in detail the examples of the Toda field theories and the Heisenberg model. 
  A 1-matrix model is proposed, which nicely interpolates between double-scaling continuum limits of all multimatrix models. The interpolating partition function is always a KP $\tau $-function and always obeys ${\cal L}_{-1}$-constraint and string equation. Therefore this model can be considered as a natural unification of all models of 2d-gravity (string models) with $c\leq 1.$ 
  We discuss two dimensional string theories containing gauge fields introduced either via coupling to open strings, in which case we get a Born-Infeld type action, or via heterotic compactification. The solutions to the modified background field equations are charged black holes which exhibit interesting space-time geometries. We also compute their masses and charges. 
  We summarize some aspects of matrix models from the approaches directly based on their properties at finite N. 
  Neveu-Schwarz-Ramond type heterotic and type-II superstrings in four dimensional curved space-time are constructed as exact $N=1$ superconformal theories. The tachyon is eliminated with a GSO projection. The theory is based on the N=1 superconformal gauged WZW model for the anti-de Sitter coset $SO(3,2)/SO(3,1)$ with integer central extension $k=5$. The model has dynamical duality properties in its space-time metric that are similar to the large-small ($R\rightarrow 1/R$) duality of tori. To first order in a $1/k$ expansion we give expressions for the metric, the dilaton, the Ricci tensor and their dual generalizations. The curvature scalar has several singularities at various locations in the 4-dimensional manifold. This provides a new singular solution to Einstein's equations in the presence of matter in four dimensions. A non-trivial path integral measure which we conjectured in previous work for gauged WZW models is verified.  
  We indicate the tentative source of instability in the two-dimensional black hole background. There are relevant operators among the tachyon and the higher level vertex operators in the conformal field theory. Connection of this instability with Hawking radiation is not obvious. The situation is somewhat analogous to fields in the background of a negative mass Euclidean Schwarzschild solution (in four dimensions). Speculation is made about decay of the Minkowski black hole into finite temperature flat space. 
  It is demonstrated that static, charged, spherically--symmetric black holes in string theory are classically and catastrophically unstable to linearized perturbations in four dimensions, and moreover that unstable modes appear for arbitrarily small positive values of the charge. This catastrophic classical instability dominates and is distinct from much smaller and less significant effects such as possible quantum mechanical evaporation. The classical instability of the string--theoretic black hole contrasts sharply with the situation which obtains for the Reissner--Nordstr\"om black hole of general relativity, which has been shown by Chandrasekhar to be perfectly stable to linearized perturbations at the event horizon. 
  We give a selfcontained introduction to the theory of quantum groups according to Drinfeld highlighting the formal aspects as well as the applications to the Yang-Baxter equation and representation theory. Introductions to Hopf algebras, Poisson structures and deformation quantization are also provided. After having defined Poisson-Lie groups we study their relation to Lie-bi algebras and the classical Yang-Baxter equation. Then we explain in detail the concept of quantization for them. As an example the quantization of $sl_2$ is explicitly carried out. Next we show how quantum groups are related to the Yang-Baxter equation and how they can be used to solve it. Using the quantum double construction we explicitly construct the universal $R$-matrix for the quantum $sl_2$ algebra. In the last section we deduce all finite dimensional irreducible representations for $q$ a root of unity. We also give their tensor product decomposition (fusion rules) which is relevant to conformal field theory. 
  The $SL(2,R)/U(1)$ gauged WZWN model is modified by a topological term and the accompanying change in the geometry of the two dimensional target space is determined. The possibility of this additional term arises from a symmetry in the general formalism of gauging an isometry subgroup of a non-linear sigma model with an antisymmetric tensor. It is shown, in particular, that the space-time exhibits some general singularities for which the recently found black hole is just a special case. From a conformal field theory point of view and for special values of the unitary representations of $SL(2,R)$, this topological term can be interpreted as a small perturbation by a (1,1) conformal operator of the gauged WZWN action. 
  Working in the context of spontaneously broken gauge theories, we show that the magnetically charged Reissner-Nordstrom solution develops a classical instability if the horizon is sufficiently small. This instability has significant implications for the evolution of a magnetically charged black hole. In particular, it leads to the possibility that such a hole could evaporate completely, leaving in its place a nonsingular magnetic monopole. 
  We analyze the W_N^l algebras according to their conjectured realization as the second Hamiltonian structure of the integrable hierarchy resulting from the interchange of x and t in the l^{th} flow of the sl(N) KdV hierarchy. The W_4^3 algebra is derived explicitly along these lines, thus providing further support for the conjecture. This algebra is found to be equivalent to that obtained by the method of Hamiltonian reduction. Furthermore, its twisted version reproduces the algebra associated to a certain non-principal embedding of sl(2) into sl(4), or equivalently, the u(2) quasi-superconformal algebra. The general aspects of the W_N^l algebras are also presented.  
  We prove the no-ghost theorem for the N=2 SUSY strings in (2,2) dimensional flat Minkowski space. We propose a generalization of this theorem for an arbitrary geometry of the N=2 SUSY string theory taking advantage of the N=4 SCA generators present in this model. Physical states are found to be the highest weight states of the N=4 SCA.  
  All solvable two-dimensional quantum gravity models have non-trivial BRST cohomology with vanishing ghost number. These states form a ring and all the other states in the theory fall into modules of this ring. The relations in the ring and in the modules have a physical interpretation. The existence of these rings and modules leads to nontrivial constraints on the correlation functions and goes a long way toward solving these theories in the continuum approach.  
  Factorization of the $N$-tachyon amplitudes in two-dimensional $c=1$ quantum gravity is studied by means of the operator product expansion of vertex operators after the Liouville zero mode integration. Short-distance singularities between two tachyons with opposite chiralities account for all singularities in the $N$-tachyon amplitudes. Although the factorization is valid, other possible short-distance singularities corresponding to other combinations of vertex operators are absent since the residue vanishes. Apart from the tachyon states, there are infinitely many topological states contributing to the intermediate states. This is a more detailed account of our short communication on the factorization. 
  We studied the marginal deformation of the $c=0$ topological conformal field theories (TCFT). We showed that topological $SL(2)$ Wess-Zumino-Witten (WZW) model, topological superconformal ghost system, TCFT constructed from the $N=2$ superconformal system and two dimensional topological gravity belong to the same one parameter family (moduli space) of the $c=0$ TCFT's. We conjectured that the $N=2$ TCFT constructed from the Wolf space realization of $N=4$ superconformal algebra belongs to another family. 
  We use the Kontsevich--Miwa transform to relate the Virasoro constraints on integrable hierarchies with the David-Distler-Kawai formalism of gravity-coupled conformal theories. The derivation relies on evaluating the energy-momentum tensor on the hierarchy at special values of the spectral parameter. We thus obtain in the Kontsevich parametrization the `master equations' which implement the Virasoro constraints and at the same time coincide with null-vector decoupling equations in an `auxiliary' conformal field theory on the complex plane of the spectral parameter. This gives the operators their gravitational scaling dimensions (for one out of four possibilities to choose signs), with the $\alpha_+$ being equal to the background charge $Q$ of an abstract $bc$ system underlying the structure of the Virasoro constraints. The formalism also generalizes to the $N$-KdV hierarchies. 
  In these lecture notes from Strings `91, I briefly sketch the analogy between two dimensional black holes and the s-wave sector of four dimensional black holes, and the physical interest of the latter, particularly in the magnetically charged case. 
  The algebra W_{1+\infty} with central charge c=0 can be identified with the algebra of quantum observables of a particle moving on a circle. Mathematically, it is the universal enveloping algebra of the Euclidean algebra in two dimensions. Similarly, the super W_\infty algebra is found to be the universal enveloping algebra of the super-Euclidean algebra in two dimensions. 
  We show that the Manin-Radul super KP hierarchy is invariant under super W_\infty transformations. These transformations are characterized by time dependent flows which commute with the usual flows generated by the conserved quantities of the super KP hierarchy. 
  It will be argued that among the known systems in three dimensions that have string like excitations periodic U(1) pure gauge theories are the most likely candidates to lead to a string representation of their universal properties. Some recent work with F. David will also be reviewed. 
  A renormalizable theory of quantum gravity coupled to a dilaton and conformal matter in two space-time dimensions is analyzed. The theory is shown to be exactly solvable classically. Included among the exact classical solutions are configurations describing the formation of a black hole by collapsing matter. The problem of Hawking radiation and backreaction of the metric is analyzed to leading order in a $1/N$ expansion, where $N$ is the number of matter fields. The results suggest that the collapsing matter radiates away all of its energy before an event horizon has a chance to form, and black holes thereby disappear from the quantum mechanical spectrum. It is argued that the matter asymptotically approaches a zero-energy ``bound state'' which can carry global quantum numbers and that a unitary $S$-matrix including such states should exist. 
  In this paper we compute the N-point correlation functions of the tachyon operator from the Neveu Schwarz sector of super Liouville theory coupled to matter fields (with $\hat c\le 1$) in the super Coulomb gas formulation, on world sheets with spherical topology. We first integrate over the zero mode assuming that the $s$ parameter takes an integer value, subsequently we continue the parameter to an arbitrary real number. We included an arbitrary number of screening charges (s.c.) and as a result, after renormalizing the s.c., the external legs and the cosmological constant, the form of the final amplitudes do not modify. Remarkably, the result is completely parallel to the bosonic case. We also completed a discussion on the calculation of bosonic correlators including arbitrary screening charges. 
  We study the irreducible unitary highest weight representations, which are obtained from free field realizations, of $W$ infinity algebras ($W_{\infty}$, $W_{1+\infty}$, $W_{\infty}^{1,1}$, $W_{\infty}^M$, $W_{1+\infty}^N$, $W_{\infty}^{M,N}$) with central charges ($2$, $1$, $3$, $2M$, $N$, $2M+N$). The characters of these representations are computed. We construct a new extended superalgebra $W_{\infty}^{M,N}$, whose bosonic sector is $W_{\infty}^M\oplus W_{1+\infty}^N$. Its representations obtained from a free field realization with central charge $2M+N$, are classified into two classes: continuous series and discrete series. For the former there exists a supersymmetry, but for the latter a supersymmetry exists only for $M=N$. 
  We prove that critical and subcritical N=2 string theory gives a realization of an N=2 superfield extension of the topological conformal algebra. The essential observation is the vanishing of the background charge. 
  It is shown that the effective string recently introduced to describe the long distance dynamics of 3D gauge systems in the confining phase has an intriguing description in terms of models of 2D self-avoiding walks in the dense phase. The deconfinement point, where the effective string becomes N=2 supersymmetric, may then be interpreted as the tricritical Theta point where the polymer chain undergoes a collapse transition. As a consequence, a universal value of the deconfinement temperature is predicted. 
  We investigate the explicit construction of the $WB_{2}$ algebra, which is closed and associative for all values of the central charge $c$, using the Jacobi identity and show the agreement with the results studied previously. Then we illustrate a realization of $c=\frac{5}{2}$ free fermion model, which is $m \rightarrow \infty$ limit of unitary minimal series, $c ( WB_{2} )=\frac{5}{2} (1-\frac{12}{ (m+3)(m+4) })$ based on the cosets $( \hat{B_{2}} \oplus \hat{B_{2}}, \hat{B_{2} })$ at level $(1,m).$ We confirm by explicit computations that the bosonic currents in the $ WB_{2}$ algebra are indeed given by the Casimir operators of $\hat{B_{2}}$ . 
  In this paper we consider the structure of general quantum W-algebras. We introduce the notions of deformability, positive-definiteness, and reductivity of a W-algebra. We show that one can associate a reductive finite Lie algebra to each reductive W-algebra. The finite Lie algebra is also endowed with a preferred $sl(2)$ subalgebra, which gives the conformal weights of the W-algebra. We extend this to cover W-algebras containing both bosonic and fermionic fields, and illustrate our ideas with the Poisson bracket algebras of generalised Drinfeld-Sokolov Hamiltonian systems. We then discuss the possibilities of classifying deformable W-algebras which fall outside this class in the context of automorphisms of Lie algebras. In conclusion we list the cases in which the W-algebra has no weight one fields, and further, those in which it has only one weight two field. 
  Three dimensional SU(2) Chern-Simons theory has been studied as a topological field theory to provide a field theoretic description of knots and links in three dimensions. A systematic method has been developed to obtain the link-invariants within this field theoretic framework. The monodromy properties of the correlators of the associated Wess-Zumino SU(2)$_k$ conformal field theory on a two-dimensional sphere prove to be useful tools. The method is simple enough to yield a whole variety of new knot invariants of which the Jones polynomials are the simplest example. 
  The non-perturbative behaviour of macroscopic loop amplitudes in the exactly solvable string theories based on the KdV hierarchies is considered. Loop equations are presented for the real non-perturbative solutions living on the spectral half-line, allowed by the most general string equation $[\tilde{P},Q]=Q$, where $\tilde{P}$ generates scale transformations. In general the end of the half-line (the `wall') is a non-perturbative parameter whose r\^ole is that of boundary cosmological constant. The properties are compared with the perturbative behaviour and solutions of $[P,Q]=1$. Detailed arguments are given for the $(2,2m-1)$ models while generalisation to the other $(p,q)$ minimal models and $c=1$ is briefly addressed. 
  We extend the three dimensional stringy black hole of Horne and Horowitz to four dimensions. After a brief discussion of the global properties of the metric, we discuss the stability of the background with respect to small perturbations, following the methods of Gilbert and of Chandrasekhar. The potential for axial perturbations is found to be positive definite. 
  A survey of ghost techniques in mathematical physics, which can be grouped under the rubric of `cohomological physics', particularly BRST cohomology. 
  We discuss the non-perturbative aspect of zero dimensional superstring. The perturbative expansions of correlation functions diverge as $\sum_l(3l)!\kappa^{2l}$, where $\kappa$ is a string coupling constant. This implies there are non-perturbative contributions of order $\e^{C\kappa^{-{2 \over 3}}}$. (Here $C$ is a constant.) This situation contrasts with those of critical or non-critical bosonic strings, where the perturbative expansions diverge as $\sum_ll!\kappa^{2l}$ and non-perturbative behaviors go as $\e^{C\kappa^{-1}}$. It is explained how such nonperturbative effects of order $\e^{C\kappa^{-{2 \over 3}}}$ appear in zero dimensional superstring theory. Due to these non-perturbative effects, the supersymmetry in target space breaks down spontaneously. 
  We review some formal aspects of cosmological solutions in closed string theory with duality symmetric ``matter'' following recent paper with C. Vafa (HUTP-91/A049). We consider two models : when the matter action is the classical action of the fields corresponding to momentum and winding modes and when the matter action is represented by the quantum vacuum energy of the string compactified on a torus. Assuming that the effective vacuum energy is positive one finds that in both cases the scale factor undergoes oscillations from maximal to minimal values with the amplitude of oscillations decreasing to zero or increasing to infinity depending on whether the effective coupling (dilaton field) decreases or increases with time. The contribution of the winding modes to the classical action prevents infinite expansion. Duality is ``spontaneously broken'' on a solution with generic initial conditions. 
  The path integral of four dimensional quantum gravity is restricted to conformally self-dual metrics. It reduces to integrals over the conformal factor and over the moduli space of conformally self--dual metrics and can be studied with the methods of two dimensional quantum gravity in conformal gauge. The conformal anomaly induces an analog of the Liouville action. The proposal of David, Distler and Kawai is generalized to four dimensions. Critical exponents and the analog of the c=1 barrier of two dimensional gravity are derived. Connections with Weyl gravity and four dimensional topological gravity are suggested. 
  Progress towards the classification of the meromorphic $c=24$ conformal field theories is reported. It is shown if such a theory has any spin-1 currents, it is either the Leech lattice CFT, or it can be written as a tensor product of Kac-Moody algebras with total central charge 24. The total number of combinations of Kac-Moody algebras for which meromorphic $c=24$ theories may exist is 221. The next step towards classification is to obtain all modular invariant combinations of Kac-Moody characters. The presently available results are sufficient to obtain a complete list of all ten-dimensional heterotic strings. Furthermore there are strong indications for the existence of several (probably at least 20) new meromorphic $c=24$ theories. 
  We derive the exact, factorized, purely elastic scattering matrices for the $a_{2n-1}^{(2)}$ family of nonsimply-laced affine Toda theories. The derivation takes into account the distortion of the classical mass spectrum by radiative correction, as well as modifications of the usual bootstrap assumptions since for these theories anomalous threshold singularities lead to a displacement of some single particle poles. 
  We study magnetically charged classical solutions of a spontaneously broken gauge theory interacting with gravity. We show that nonsingular monopole solutions exist only if the Higgs vacuum expectation value $v$ is less than or equal to a critical value $v_{cr}$, which is of the order of the Planck mass. In the limiting case the monopole becomes a black hole, with the region outside the horizon described by the critical Reissner-Nordstrom solution. For $v<v_{cr}$, we find additional solutions which are singular at $r=0$, but which have this singularity hidden within a horizon. These have nontrivial matter fields outside the horizon, and may be interpreted as small black holes lying within a magnetic monopole. The nature of these solutions as a function of $v$ and of the total mass $M$ and their relation to the Reissner-Nordstrom solutions is discussed. 
  Previously we have established that the second Hamiltonian structure of the KP hierarchy is a nonlinear deformation, called $\hat{W}_{\infty}$, of the linear, centerless $W_{\infty}$ algebra. In this letter we present a free-field realization for all generators of $\hat{W}_{\infty}$ in terms of two scalars as well as an elegant generating function for the $\hat{W}_{\infty}$ currents in the classical conformal $SL(2,R)/U(1)$ coset model. After quantization, a quantum deformation of $\hat{W}_{\infty}$ appears as the hidden current algebra in this model. The $\hat{W}_{\infty}$ current algebra results in an infinite set of commuting conserved charges, which might give rise to $W$-hair for the 2d black hole arising in the corresponding string theory at level $k=9/4$. 
  The geometrical structure and the quantum properties of the recently proposed harmonic space action describing self-dual Yang-Mills (SDYM) theory are analyzed. The geometrical structure that is revealed is closely related to the twistor construction of instanton solutions. The theory gets no quantum corrections and, despite having SDYM as its classical equation of motion, its S matrix is trivial. It is therefore NOT the theory of the N=2 string. We also discuss the 5-dimensional actions that have been proposed for SDYM. 
  We examine the modular properties of nonrenormalizable superpotential terms in string theory and show that the requirement of modular invariance necessitates the nonvanishing of certain Nth order nonrenormalizable terms. In a class of models (free fermionic formulation) we explicitly verify that the nontrivial structure imposed by the modular invariance is indeed present. Alternatively, we argue that after proper field redefinition, nonrenormalizable terms can be recast as to display their invariance under the modular group. We also discuss the phenomenological implications of the above observations. 
  We discuss gauge theory with a topological $N=2$ symmetry. This theory        captures the de Rham complex and Riemannian geometry of some underlying moduli space $\cal M$ and the partition function equals the Euler number of $\cal M$. We explicitly deal with moduli spaces of instantons and of  flat connections in two and three dimensions.     To motivate our constructions we explain the relation between the  Mathai-Quillen formalism and supersymmetric quantum mechanics and introduce  a new kind of supersymmetric quantum mechanics based on the Gauss-Codazzi  equations. We interpret the gauge theory actions from the Atiyah-Jeffrey  point of view and relate them to supersymmetric quantum mechanics on spaces  of connections. As a consequence of these considerations we propose the  Euler number of the moduli space of flat connections as a generalization  to arbitrary three-manifolds of the Casson invariant. We also comment on  the possibility of constructing a topological version of the Penner matrix  model.  
  These are introductory lectures for a general audience that give an overview  of the subject of matrix models and their application to random surfaces,  2d gravity, and string theory. They are intentionally 1.5 years out of date.    0. Canned Diatribe, Introduction, and Apologies    1. Discretized surfaces, matrix models, and the continuum limit    2. All genus partition functions    3. KdV equations and other models    4. Quick tour of Liouville theory 
  We show that the $N=2$ superstring in $d=2D\ge6$ real dimensions, with criticality achieved by including background charges in the two real time directions, exhibits a ``coordinate-freezing'' phenomenon, whereby the momentum in one of the two time directions is constrained to take a specific value for each physical state. This effectively removes this time direction as a physical coordinate, leaving the theory with $(1,d-2)$ real spacetime signature. Norm calculations for low-lying physical states suggest that the theory is ghost free. 
  Fractional superstrings are recently-proposed generalizations of the traditional superstrings and heterotic strings. They have critical spacetime dimensions which are less than ten, and in this paper we investigate model-building for the heterotic versions of these new theories. We concentrate on the cases with critical spacetime dimensions four and six, and find that a correspondence can be drawn between the new fractional superstring models and a special subset of the traditional heterotic string models. This allows us to generate the partition functions of the new models, and demonstrate that their number is indeed relatively limited. It also appears that these strings have uniquely natural compactifications to lower dimensions. In particular, the fractional superstring with critical dimension six has a natural interpretation in four-dimensional spacetime. 
  For a large class of hierarchies of integrable equations admitting a classical $r-$matrix, we propose a construction for the Virasoro algebra actionon the Lax operators which commutes with the hierarchy flows. The construction relies on the existence of dressing transformations associated to the $r$-matrix and does not involve the notion of a tau function. The dressing-operator form of the Virasoro action gives the corresponding formulation of the Virasoro constraints on hierarchies of the $r-$matrix type. We apply the general construction to several examples which include KP, Toda and generalized KdV hierarchies, the latter both in scalar and the Drinfeld-Sokolov formalisms. We prove the consistency of Virasoro action on the scalar and matrix (Drinfeld-Sokolov) Lax operators, and make an observation on the difference in the form of string equations in the two formalisms. 
  The universality of the non-perturbative definition of Hermitian one-matrix models following the quantum, stochastic, or $d=1$-like stabilization is discussed in comparison with other procedures. We also present another alternative definition, which illustrates the need of new physical input for $d=0$ matrix models to make contact with 2D quantum gravity at the non-perturbative level. 
  We propose a discrete model whose continuum limit reproduces the string susceptibility and the scaling dimensions of $(2,4m)$-minimal superconformal models coupled to $2D$-supergravity. The basic assumption in our presentation is a set of super-Virasoro constraints imposed on the partition function. We recover the Neveu-Schwarz and Ramond sectors of the theory, and we are also able to evaluate all planar loop correlation functions in the continuum limit. We find evidence to identify the integrable hierarchy of non-linear equations describing the double scaling limit as a supersymmetric generalization of KP studied by Rabin. 
  Continuum and discrete approaches to 2d gravity coupled to $c<1$ matter are reviewed. 
  Electromagnetic fields of a massless charged particle are described by a gauge potential that is almost everywhere pure gauge. Solution of quantum mechanical wave equations in the presence of such fields is therefore immediate and leads to a new derivation of the quantum electrodynamical eikonal approximation. The elctromagnetic action in the eikonal limit is localised on a contour in a two-dimensional Minkowski subspace of four-dimensional space-time. The exact S-matrix of this reduced theory coincides with the eikonal approximation, and represents the generalisatin to electrodynamics of the approach of 't Hooft and the Verlinde's to Planckian scattering. 
  We identify the puncture operator in c=1 Liouville gravity as the discrete state with spin J=1/2. The correlation functions involving this operator satisfy the recursion relation which is characteristic in topological gravity. We derive the recursion relation involving the puncture operator by the operator product expansion. Multiple point correlation functions are determined recursively from fewer point functions by this recursion relation. 
  We derive fusion rules for the composition of $q$-deformed classical representations (arising in tensor products of the fundamental representation) with semi-periodic representations of $SL(N)_q$ at roots of unity. We obtain full reducibility into semi-periodic representations. On the other hand, heterogeneous $\cR$-matrices which intertwine tensor products of periodic or semi-periodic representations with $q$-deformed classical representations are given. These $\cR$-matrices satisfy all the possible Yang Baxter equations with one another and, when they exist, with the $\cR$-matrices intertwining homogeneous tensor products of periodic or semi-periodic representations. This compatibility between these two kinds of representations has never been used in physical models. 
  We ask whether the recently discovered superstring and superfivebrane solutions of D=10 supergravity admit the interpretation of non-singular solitons even though, in the absence of Yang-Mills fields, they exhibit curvature singularities at the origin. We answer the question using a test probe/source approach, and find that the nature of the singularity is probe-dependent. If the test probe and source are both superstrings or both superfivebranes, one falls into the other in a finite proper time and the singularity is real, whereas if one is a superstring and the other a superfivebrane it takes an infinite proper time (the force is repulsive!) and the singularity is harmless. Black strings and fivebranes, on the other hand, always display real singularities. 
  A field theoretical realization of topological gravity is discussed in the semirigid geometry context. In particular, its topological nature is given by the relation between deRham cohomology and equivariant BRST cohomology and the fact that all but one of the physical operators are BRST-exact. The puncture equation and the dilaton equation of pure topological gravity are reproduced, following reference \dil.  
  We review the structure of W_\infty algebras, their super and topological extensions, and their contractions down to (super) w_\infty. Emphasis is put on the field theoretic realisations of these algebras. We also review the structure of w_\infty and W_\infty gravities and comment on various applications of W_\infty symmetry. 
  It is shown, using the Wakimoto representation, that the level zero SU(2) Kac-Moody conformal field theory is topological and can be obtained by twisting an N=2 superconformal theory. Expressions for the associated N=2 superconformal generators are written down and the Kac-Moody generators are shown to be BRST exact. 
  The existing classification of homogeneous quaternionic spaces is not complete. We study these spaces in the context of certain $N=2$ supergravity theories, where dimensional reduction induces a mapping between {\em special} real, K\"ahler and quaternionic spaces. The geometry of the real spaces is encoded in cubic polynomials, those of the K\"ahler and quaternionic manifolds in homogeneous holomorphic functions of second degree. We classify all cubic polynomials that have an invariance group that acts transitively on the real manifold. The corresponding K\"ahler and quaternionic manifolds are then homogeneous. We find that they lead to a well-defined subset of the normal quaternionic spaces classified by \Al\ (and the corresponding special K\"ahler spaces given by Cecotti), but there is a new class of rank-3 spaces of quaternionic dimension larger than 3. We also point out that some of the rank-4 \Al\ spaces were not fully specified and correspond to a finite variety of inequivalent spaces. A simpler version of the equation that underlies the classification of this paper also emerges in the context of $W_3$ algebras. 
  We quantise the classical gauge theory of $N=2\ w_\infty$-supergravity and show how the underlying $N=2$ super-$w_\infty$ algebra gets deformed into an $N=2$ super-$W_\infty$ algebra. Both algebras contain the $N=2$ super-Virasoro algebra as a subalgebra. We discuss how one can extract from these results information about quantum $N=2\ W_N$-supergravity theories containing a finite number of higher-spin symmetries with superspin $s\le N$. As an example we discuss the case of quantum $N=2\ W_3$-supergravity. 
  The tree-level three-point correlation functions of local operators in the general $(p,q)$ minimal models coupled to gravity are calculated in the continuum approach. On one hand, the result agrees with the unitary series ($q=p+1$); and on the other hand, for $p=2, q=2k-1$, we find agreement with the one-matrix model results. 
  These notes are based on lectures given by C. Callan and J. Harvey at the 1991 Trieste Spring School on String Theory and Quantum Gravity. The subject is the construction of supersymmetric soliton solutions to superstring theory. A brief review of solitons and instantons in supersymmetric theories is presented. Yang-Mills instantons are then used to construct soliton solutions to heterotic string theory of various types. The structure of these solutions is discussed using low-energy field theory, sigma-model arguments, and in one case an exact construction of the underlying superconformal field theory. 
  Using nonperturbative techniques, we study the renormalization group trajectory between two conformal field theories. Specifically, we investigate a perturbation of the A3 superconformal minimal model such that in the infrared limit the theory flows to the A2 model. The correlation functions in the topological sector of the theory are computed numerically along the trajectory, and these results are compared to the expected asymptotic behavior. Excellent agreement is found, and the characteristic features of the infrared theory, including the central charge and the normalized operator product expansion coefficients are obtained. We also review and discuss some aspects of the geometrical description of N=2 supersymmetric quantum field theories recently uncovered by S. Cecotti and C. Vafa. 
  We study the renormalization group for nearly marginal perturbations of a minimal conformal field theory M_p with p >> 1. To leading order in perturbation theory, we find a unique one-parameter family of ``hopping trajectories'' that is characterized by a staircase-like renormalization group flow of the C-function and the anomalous dimensions and that is related to a recently solved factorizable scattering theory. We argue that this system is described by interactions of the form t phi_{(1,3)} - t' \phi_{(3,1)} . As a function of the relevant parameter t, it undergoes a phase transition with new critical exponents simultaneously governed by all fixed points M_p, M_{p-1}, ..., M_3. Integrable lattice models represent different phases of the same integrable system that are distinguished by the sign of the irrelevant parameter t'. 
  We study $c<1$ matter coupled to gravity in the Coulomb gas formalism using the double cohomology of the string BRST and Felder BRST charges. We find that states outside the primary conformal grid are related to the states of non-zero ghost number by means of descent equations given by the double cohomology. Some aspects of the Virasoro structure of the Liouville Fock space are studied. As a consequence, states of non-zero ghost number are easily constructed by ``solving'' these descent equations. This enables us to map ghost number conserving correlation functions involving non-zero ghost number states into those involving states outside the primary conformal grid. 
  We introduce a new model describing a bosonic system with chiral properties. It consists of a free boson with two peculiar couplings to the background geometry which generalizes the Feigen-Fuchs-Dotsenko-Fateev construction. By choosing the two background charges of the model, it is possible to achieve any prefixed value of the left and right central charges and, in particular, obtain chiral bosonization. A supersymmetric version of the model is also given. We use the latter to identify the effective action induced by chiral superconformal matter. 
  The computation of anomalies in quantum field theory may be carried out by evaluating path integral Jacobians, as first shown by Fujikawa. The evaluation of these Jacobians can be cast in the form of a quantum mechanical problem, whose solution has a path integral representation. For the case of Weyl anomalies, also called trace anomalies, one is immediately led to study the path integral for a particle moving in curved spaces. We analyze the latter in a manifestly covariant way and by making use of ghost fields. The introduction of the ghost fields allows us to represent the path integral measure in a form suitable for performing the perturbative expansion. We employ our method to compute the Hamiltonian associated with the evolution kernel given by the path integral with fixed boundary conditions, and use this result to evaluate the trace needed in field theoretic computation of Weyl anomalies in two dimensions. 
  We discuss the BSRT quantization of 2D $N=1$ supergravity coupled to superconformal matter with $\hat{c} \leq 1$ in the conformal gauge. The physical states are computed as BRST cohomology. In particular, we consider the ring structure and associated symmetry algebra for the 2D superstring ($\hat{c} = 1$). 
  Rectangular $N\times M$ matrix models can be solved in several qualitatively distinct large $N$ limits, since two independent parameters govern the size of the matrix. Regarded as models of random surfaces, these matrix models interpolate between branched polymer behaviour and two-dimensional quantum gravity. We solve such models in a `triple-scaling' regime in this paper, with $N$ and $M$ becoming large independently. A correspondence between phase transitions and singularities of mappings from ${\bf R}^2$ to ${\bf R}^2$ is indicated. At different critical points, the scaling behavior is determined by: i) two decoupled ordinary differential equations; ii) an ordinary differential equation and a finite difference equation; or iii) two coupled partial differential equations. The Painlev\'e II equation arises (in conjunction with a difference equation) at a point associated with branched polymers. For critical points described by partial differential equations, there are dual weak-coupling/strong-coupling expansions. It is conjectured that the new physics is related to microscopic topology fluctuations. 
  We develop an operator formalism for investigating the properties of nonabelian cosmic strings (and vortices) in quantum field theory. Operators are constructed that introduce classical string sources and that create dynamical string loops. The operator construction in lattice gauge theory is explicitly described, and correlation functions are computed in the strong--coupling and weak--coupling limits. These correlation functions are used to study the long--range interactions of nonabelian strings, taking account of charge--screening effects due to virtual particles. Among the phenomena investigated are the Aharonov--Bohm interactions of strings with charged particles, holonomy interactions between string loops, string entanglement, the transfer of ``Cheshire charge'' to a string loop, and domain wall decay via spontaneous string nucleation. We also analyze the Aharonov--Bohm interactions of magnetic monopoles with electric flux tubes in a confining gauge theory. We propose that the Aharonov--Bohm effect can be invoked to distinguish among various phases of a nonabelian gauge theory coupled to matter. 
  We analyze the unlocalized ``Cheshire charge'' carried by ``Alice strings.'' The magnetic charge on a string loop is carefully defined, and the transfer of magnetic charge from a monopole to a string loop is analyzed using global topological methods. A semiclassical theory of electric charge transfer is also described. 
  We analyze the charges carried by loops of string in models with non-abelian local discrete symmetry. The charge on a loop has no localized source, but can be detected by means of the Aharonov--Bohm interaction of the loop with another string. We describe the process of charge detection, and the transfer of charge between point particles and string loops, in terms of gauge--invariant correlation functions. 
  Present state of the study of nonlinear ``integrable" systems related to the group of area-preserving diffeomorphisms on various surfaces is overviewed. Roles of area-preserving diffeomorphisms in 4-d self-dual gravity are reviewed. Recent progress in new members of this family, the SDiff(2) KP and Toda hierarchies, is reported. The group of area-preserving diffeomorphisms on a cylinder plays a key role just as the infinite matrix group GL($\infty$) does in the ordinary KP and Toda lattice hierarchies. The notion of tau functions is also shown to persist in these hierarchies, and gives rise to a central extension of the corresponding Lie algebra. 
  A continuum limit of the Toda lattice field theory, called the SDiff(2) Toda equation, is shown to have a Lax formalism and an infinite hierarchy of higher flows. The Lax formalism is very similar to the case of the self-dual vacuum Einstein equation and its hyper-K\"ahler version, however now based upon a symplectic structure and the group SDiff(2) of area preserving diffeomorphisms on a cylinder $S^1 \times \R$. An analogue of the Toda lattice tau function is introduced. The existence of hidden SDiff(2) symmetries are derived from a Riemann-Hilbert problem in the SDiff(2) group. Symmetries of the tau function turn out to have commutator anomalies, hence give a representation of a central extension of the SDiff(2) algebra. 
  We study the quantum conserved charges and S-matrices of N=2 supersymmetric sine-Gordon theory in the framework of perturbation theory formulated in N=2 superspace. The quantum affine algebras ${\widehat {sl_{q}(2)}}$ and super topological charges play important roles in determining the N=2 soliton structure and S-matrices of soliton-(anti)soliton as well as soliton-breather scattering. 
  The recently discovered $O(d,d)$ symmetry of the space of slowly varying cosmological string vacua in $d+1$ dimensions is shown to be preserved in the presence of bulk string matter. The existence of $O(d,d)$ conserved currents allows all the equations of string cosmology to be reduced to first-order differential equations. The perfect-fluid approximation is not $O(d,d)$-invariant, implying that stringy fluids possess in general a non-vanishing viscosity. 
  We reformulate the zero-dimensional hermitean one-matrix model as a (nonlocal) collective field theory, for finite~$N$. The Jacobian arising by changing variables from matrix eigenvalues to their density distribution is treated {\it exactly\/}. The semiclassical loop expansion turns out {\it not\/} to coincide with the (topological) ${1\over N}$~expansion, because the classical background has a non-trivial $N$-dependence. We derive a simple integral equation for the classical eigenvalue density, which displays strong non-perturbative behavior around $N\!=\!\infty$. This leads to IR singularities in the large-$N$ expansion, but UV divergencies appear as well, despite remarkable cancellations among the Feynman diagrams. We evaluate the free energy at the two-loop level and discuss its regularization. A simple example serves to illustrate the problems and admits explicit comparison with orthogonal polynomial results. 
  An analogue of the KP hierarchy, the SDiff(2) KP hierarchy, related to the group of area-preserving diffeomorphisms on a cylinder is proposed. An improved Lax formalism of the KP hierarchy is shown to give a prototype of this new hierarchy. Two important potentials, $S$ and $\tau$, are introduced. The latter is a counterpart of the tau function of the ordinary KP hierarchy. A Riemann-Hilbert problem relative to the group of area-diffeomorphisms gives a twistor theoretical description (nonlinear graviton construction) of general solutions. A special family of solutions related to topological minimal models are identified in the framework of the Riemann-Hilbert problem. Further, infinitesimal symmetries of the hierarchy are constructed. At the level of the tau function, these symmetries obey anomalous commutation relations, hence leads to a central extension of the algebra of infinitesimal area-preserving diffeomorphisms (or of the associated Poisson algebra). 
  We construct a class of Heterotic String vacua described by Landau--Ginzburg theories and consider orbifolds of these models with respect to abelian symmetries. For LG--vacua described by potentials in which at most three scaling fields are coupled we explicitly construct the chiral ring and discuss its diagonalization with respect to its most general abelian symmetry. For theories with couplings between at most two fields we present results of an explicit construction of the LG--potentials and their orbifolds. The emerging space of (2,2)--theories shows a remarkable mirror symmetry. It also contains a number of new three--generation models. 
  $O(N)$ invariant vector models have been shown to possess non-trivial scaling large $N$ limits, at least perturbatively within the loop expansion, a property they share with matrix models of 2D quantum gravity. In contrast with matrix models, however, vector models can be solved in arbitrary dimensions. We present here the analysis of field theory vector models in $d$ dimensions and discuss the nature and form of the critical behaviour. The double scaling limit corresponds for $d>1$ to a situation where a bound state of the $N$-component fundamental vector field $\phi$, associated with the $\phi^2$ composite operator, becomes massless, while the field $\phi$ itself remains massive. The limiting model can be described by an effective local interaction for the corresponding $O(N)$ invariant field. It has a physical interpretation as describing the statistical properties of a class of branched polymers.\par It is hoped that the $O(N)$ vector models, which can be investigated in their most general form, can serve as a test ground for new ideas about the behaviour of 2D quantum gravity coupled with $d>1$ matter. 
  We show that in special K\"ahler geometry of $N=2$ space-time supergravity the gauge variant part of the connection is holomorphic and flat (in a Riemannian sense). A set of differential identities (Picard-Fuchs identities) are satisfied on a holomorphic bundle. The relationship with the differential equations obeyed by the periods of the holomorphic three form of Calabi-Yau manifolds is outlined. 
  I discuss several aspects of strings as unified theories. After recalling the difficulties of the simplest supersymmetric grand unification schemes I emphasize the distinct features of string unification. An important role in constraining the effective low energy physics from strings is played by $duality$ symmetries. The discussed topics include the unification of coupling constants (computation of $\sin ^2\theta _W$ and $\alpha _s$ at the weak scale), supersymmetry breaking through gaugino condensation, and properties of the induced SUSY-breaking soft terms. I remark that departures from universality in the soft terms are (in contrast to the minimal SUSY model) generically expected. 
  Topological quantum field theories containing matter fields are constructed by twisting $N=2$ supersymmetric quantum field theories. It is shown that $N=2$ chiral (antichiral) multiplets lead to topological sigma models while $N=2$ twisted chiral (twisted antichiral) multiplets lead to Landau-Ginzburg type topological quantum field theories. In addition, topological gravity in two dimensions is formulated using a gauge principle applied to the topological algebra which results after the twisting of $N=2$ supersymmetry. 
  We explore consequences of $W$-infinity symmetry in the fermionic field theory of the $c=1$ matrix model. We derive exact Ward identities relating correlation functions of the bilocal operator. These identities can be expressed as equations satisfied by the effective action of a {\it three} dimensional theory and contain non-perturbative information about the model. We use these identities to calculate the two point function of the bilocal operator in the double scaling limit. We extract the operator whose two point correlator has a {\it single} pole at an (imaginary) integer value of the energy. We then rewrite the \winf~ charges in terms of operators in the matrix model and use this derive constraints satisfied by the partition function of the matrix model with a general time dependent potential. 
  We generalize Toda--like integrable lattice systems to non--symmetric case. We show that they possess the bi--Hamiltonian structure. 
  It is shown that the breakdown of a {\it global} symmetry group to a discrete subgroup can lead to analogues of the Aharonov-Bohm effect. At sufficiently low momentum, the cross-section for scattering of a particle with nontrivial $\Z_2$ charge off a global vortex is almost equal to (but definitely different from) maximal Aharonov-Bohm scattering; the effect goes away at large momentum. The scattering of a spin-1/2 particle off a magnetic vortex provides an amusing experimentally realizable example. 
  We attempt a direct derivation of a conformal field theory description of 2D quantum gravity~+~matter from the formalism of integrable hierarchies subjected to Virasoro constraints. The construction is based on a generalization of the Kontsevich parametrization of the KP times by introducing Miwa parameters into it. The resulting Kontsevich--Miwa transform can be applied to the Virasoro constraints provided the Miwa parameters are related to the background charge $Q$ of the Virasoro generators on the hierarchy. We then recover the field content of the David-Distler-Kawai formalism, with the matter theory represented by a scalar with the background charge $Q_m=Q-{Q\over 2}$. In particular, the tau function is related to the correlator of a product of the `21' operators of the minimal model with central charge $d=1-3Q_m^2$. 
  These notes are devoted to explaining aspects of the mirror manifold problem that can be naturally understood from the point of view of topological field theory. Basically this involves studying the topological field theories made by twisting $N=2$ sigma models. This is mainly a review of old results, except for the discussion in \S7 of certain facts that may be relevant to constructing the ``mirror map'' between mirror moduli spaces. 
  We derive one-point functions of the loop operators of Hermitian matrix-chain models at finite $N$ in terms of differential operators acting on the partition functions. The differential operators are completely determined by recursion relations from the Schwinger-Dyson equations. Interesting observation is that these generating operators of the one-point functions satisfy $W_{1+\infty}$-like algebra. Also, we obtain constraint equations on the partition functions in terms of the differential operators. These constraint equations on the partition functions define the symmetries of the matrix models at off-critical point before taking the double scaling limit. 
  In the first part of the talk, I review the applications of loop equations to the matrix models and to 2-dimensional quantum gravity which is defined as their continuum limit. The results concerning multi-loop correlators for low genera and the Virasoro invariance are discussed. The second part is devoted to the Kontsevich matrix model which is equivalent to 2-dimensional topological gravity. I review the Schwinger--Dyson equations for the Kontsevich model as well as their explicit solution in genus zero. The relation between the Kontsevich model and the continuum limit of the hermitean one-matrix model is discussed. 
  We investigate the field theory of strings having as a target space an arbitrary discrete one-dimensional manifold. The existence of the continuum limit is guaranteed if the target space is a Dynkin diagram of a simply laced Lie algebra or its affine extension. In this case the theory can be mapped onto the theory of strings embedded in the infinite discrete line $\Z$ which is the target space of the SOS model. On the regular lattice this mapping is known as Coulomb gas picture. ... Once the classical background is known, the amplitudes involving propagation of strings can be evaluated by perturbative expansion around the saddle point of the functional integral. For example, the partition function of the noninteracting closed string (toroidal world sheet) is the contribution of the gaussian fluctuations of the string field. The vertices in the corresponding Feynman diagram technique are constructed as the loop amplitudes in a random matrix model with suitably chosen potential. 
  Starting with three dimensional Chern--Simons theory with gauge group $Sl(N,R)$, we derive an action $S_{cov}$ invariant under both left and right $W_N$ transformations. We give an interpretation of $S_{cov}$ in terms of anomalies, and discuss its relation with Toda theory. 
  We use the W-infinity symmetry of c=1 quantum gravity to compute matrix model special state correlation functions. The results are compared, and found to agree, with expectations from the Liouville model. 
  In previous papers we have shown how strings in a two-dimensional target space reconcile quantum mechanics with general relativity, thanks to an infinite set of conserved quantum numbers, ``W-hair'', associated with topological soliton-like states. In this paper we extend these arguments to four dimensions, by considering explicitly the case of string black holes with radial symmetry. The key infinite-dimensional W-symmetry is associated with the $\frac{SU(1,1)}{U(1)}$ coset structure of the dilaton-graviton sector that is a model-independent feature of spherically symmetric four-dimensional strings. Arguments are also given that the enormous number of string {\it discrete (topological)} states account for the maintenance of quantum coherence during the (non-thermal) stringy evaporation process, as well as quenching the large Hawking-Bekenstein entropy associated with the black hole. Defining the latter as the measure of the loss of information for an observer at infinity, who - ignoring the higher string quantum numbers - keeps track only of the classical mass,angular momentum and charge of the black hole, one recovers the familiar a quadratic dependence on the black-hole mass by simple counting arguments on the asymptotic density of string states in a linear-dilaton background. 
  We present a simple a direct proof of the complete integrability of the quantum KdV equation at $c=-2$, with an explicit description of all the conservation laws. 
  We rederive the recently introduced $N=2$ topological gauge theories, representing the Euler characteristic of moduli spaces ${\cal M}$ of connections, from supersymmetric quantum mechanics on the infinite dimensional spaces ${\cal A}/{\cal G}$ of gauge orbits. To that end we discuss variants of ordinary supersymmetric quantum mechanics which have meaningful extensions to infinite-dimensional target spaces and introduce supersymmetric quantum mechanics actions modelling the Riemannian geometry of submersions and embeddings, relevant to the projections ${\cal A}\rightarrow {\cal A}/{\cal G}$ and inclusions ${\cal M}\subset{\cal A}/{\cal G}$ respectively. We explain the relation between Donaldson theory and the gauge theory of flat connections in $3d$ and illustrate the general construction by other $2d$ and $4d$ examples.  
  We argue the existence of solutions of the Euclidean Einstein equations that correspond to a vortex sitting at the horizon of a black hole. We find the asymptotic behaviours, at the horizon and at infinity, of vortex solutions for the gauge and scalar fields in an abelian Higgs model on a Euclidean Schwarzschild background and interpolate between them by integrating the equations numerically. Calculating the backreaction shows that the effect of the vortex is to cut a slice out of the Euclidean Schwarzschild geometry. Consequences of these solutions for black hole thermodynamics are discussed. 
  The space of all solutions to the string equation of the symmetric unitary one-matrix model is determined. It is shown that the string equation is equivalent to simple conditions on points $V_1$ and $V_2$ in the big cell $\Gr$ of the Sato Grassmannian $Gr$. This is a consequence of a well-defined continuum limit in which the string equation has the simple form $\lb \cp ,\cq_- \rb =\hbox{\rm 1}$, with $\cp$ and $\cq_-$ $2\times 2$ matrices of differential operators. These conditions on $V_1$ and $V_2$ yield a simple system of first order differential equations whose analysis determines the space of all solutions to the string equation. This geometric formulation leads directly to the Virasoro constraints $\L_n\,(n\geq 0)$, where $\L_n$ annihilate the two modified-KdV $\t$-functions whose product gives the partition function of the Unitary Matrix Model. 
  We show that the Yang-Baxter equations for two dimensional models admit as a group of symmetry the infinite discrete group $A_2^{(1)}$. The existence of this symmetry explains the presence of a spectral parameter in the solutions of the equations. We show that similarly, for three-dimensional vertex models and the associated tetrahedron equations, there also exists an infinite discrete group of symmetry. Although generalizing naturally the previous one, it is a much bigger hyperbolic Coxeter group. We indicate how this symmetry can help to resolve the Yang-Baxter equations and their higher-dimensional generalizations and initiate the study of three-dimensional vertex models. These symmetries are naturally represented as birational projective transformations. They may preserve non trivial algebraic varieties, and lead to proper parametrizations of the models, be they integrable or not. We mention the relation existing between spin models and the Bose-Messner algebras of algebraic combinatorics. Our results also yield the generalization of the condition $q^n=1$ so often mentioned in the theory of quantum groups, when no $q$ parameter is available. 
  The structure of Hamiltonian reductions of the Wess-Zumino-Novikov-Witten (WZNW) theory by first class Kac-Moody constraints is analyzed in detail. Lie algebraic conditions are given for ensuring the presence of exact integrability, conformal invariance and $\cal W$-symmetry in the reduced theories. A Lagrangean, gauged WZNW implementation of the reduction is established in the general case and thereby the path integral as well as the BRST formalism are set up for studying the quantum version of the reduction. The general results are applied to a number of examples. In particular, a ${\cal W}$-algebra is associated to each embedding of $sl(2)$ into the simple Lie algebras by using purely first class constraints. The importance of these $sl(2)$ systems is demonstrated by showing that they underlie the $W_n^l$-algebras as well. New generalized Toda theories are found whose chiral algebras are the ${\cal W}$-algebras belonging to the half-integral $sl(2)$ embeddings, and the ${\cal W}$-symmetry of the effective action of those generalized Toda theories associated with the integral gradings is exhibited explicitly. 
  We give a simple derivation of the Virasoro constraints in the Kontsevich model, first derived by Witten. We generalize the method to a model of unitary matrices, for which we find a new set of Virasoro constraints. Finally we discuss the solution for symmetric matrices in an external field. 
  The elements of $O(d,d,\Z)$ are shown to be discrete symmetries of the space of curved string backgrounds that are independent of $d$ coordinates. The explicit action of the symmetries on the backgrounds is described. Particular attention is paid to the dilaton transformation. Such symmetries identify different cosmological solutions and other (possibly) singular backgrounds; for example, it is shown that a compact black string is dual to a charged black hole. The extension to the heterotic string is discussed. 
  We show how the Turaev--Viro invariant can be understood within the framework of Chern--Simons theory with gauge group SU(2). We also describe a new invariant for certain class of graphs by interpreting the triangulation of a manifold as a graph consisiting of crossings and vertices with three lines. We further show, for $S^3$ and $RP^3$, that the Turaev-Viro invariant is the square of the absolute value of their respective partition functions in SU(2) Chern--Simons theory and give a method of evaluating the later in a closed form for lens spaces $L_{p,1}$. 
  We define a physical Hilbert space for the three-dimensional lattice gravity of Ponzano and Regge and establish its isomorphism to the ones in the $ISO(3)$ Chern-Simons theory. It is shown that, for a handlebody of any genus, a Hartle-Hawking-type wave-function of the lattice gravity transforms into the corresponding state in the Chern-Simons theory under this isomorphism. Using the Heegaard splitting of a three-dimensional manifold, a partition function of each of these theories is expressed as an inner product of such wave-functions. Since the isomorphism preserves the inner products, the partition function of the two theories are the same for any closed orientable manifold. We also discuss on a class of topology-changing amplitudes in the lattice gravity and their relation to the ones in the Chern-Simons theory. 
  The $N=2$ minimal superconformal model can be twisted yielding an example of topological conformal field theory. In this article we investigate a Lie theoretic extension of this process.  
  We apply non-linear WKB analysis to the study of the string equation. Even though the solutions obtained with this method are not exact, they approximate extremely well the true solutions, as we explicitly show using numerical simulations. ``Physical'' solutions are seen to be separatrices corresponding to degenerate Riemann surfaces. We obtain an analytic approximation in excellent agreement with the numerical solution found by Parisi et al. for the $k=3$ case. 
  A set of exactly solvable one-dimensional quantum mechanical potentials is described. It is defined by a finite-difference-differential equation generating in the limiting cases the Rosen-Morse, harmonic, and P\"oschl-Teller potentials. General solution includes Shabat's infinite number soliton system and leads to raising and lowering operators satisfying $q$-deformed harmonic oscillator algebra. In the latter case energy spectrum is purely exponential and physical states form a reducible representation of the quantum conformal algebra $su_q(1,1)$. 
  We give a review of the extended conformal algebras, known as $W$ algebras, which contain currents of spins higher than 2 in addition to the energy-momentum tensor. These include the non-linear $W_N$ algebras; the linear $W_\infty$ and $W_{1+\infty}$ algebras; and their super-extensions. We discuss their applications to the construction of $W$-gravity and $W$-string theories. 
  We study algebraic aspects of Kontsevich integrals as generating functions for intersection theory over moduli space and review the derivation of Virasoro and KdV constraints.  1. Intersection numbers  2. The Kontsevich integral   2.1. The main theorem   2.2 Expansion of Z on characters and Schur functions   2.3 Proof of the first part of the Theorem  3. From Grassmannians to KdV  4. Matrix Airy equation and Virasoro highest weight conditions  5. Genus expansion  6. Singular behaviour and Painlev'e equation.  7. Generalization to higher degree potentials 
  The quantum deformed (1+1) Poincare' algebra is shown to be the kinematical symmetry of the harmonic chain, whose spacing is given by the deformation parameter. Phonons with their symmetries as well as multiphonon processes are derived from the quantum group structure. Inhomogeneous quantum groups are thus proposed as kinematical invariance of discrete systems. 
  In these lecture notes we review the various relations between intersection theory on the moduli space of Riemann surfaces, integrable hierarchies of KdV type, matrix models, and topological quantum field theories. We explain in particular why matrix integrals of the type considered by Kontsevich naturally appear as tau-functions associated to minimal models. Our starting point is the extremely simple form of the string equation for the topological (p,1) models, where the so-called Baker-Akhiezer function is given by a (generalized) Airy function. 
  By defining the heterotic Green-Schwarz superstring action on an N=(2,0) super-worldsheet, rather than on an ordinary worldsheet, many problems with the interacting Green-Schwarz superstring formalism can be solved. In the light-cone approach, superconformally transforming the light-cone super-worldsheet onto an N=(2,0) super-Riemann surface allows the elimination of the non-trivial interaction-point operators that complicate the evaluation of scattering amplitudes. In the Polyakov approach, the ten-dimensional heterotic Green-Schwarz covariant action defined on an N=(2,0) super-worldsheet can be gauge-fixed to a free-field action with non-anomalous N=(2,0) superconformal invariance, and integrating the exponential of the covariant action over all punctured N=(2,0) super-Riemann surfaces produces scattering amplitudes that closely resemble amplitudes obtained using the unitary light-cone approach. 
  I study the Ward identities of the $w_\infty$ symmetry of the two-dimensional string theory. It is found that, not just an isolated vertex operator, but also a number of vertex operators colliding at a point can produce local charge non-conservation. The structure of all such contact terms is determined. As an application, I calculate all the non-vanishing bulk tachyon amplitudes directly through the Ward identities for a Virasoro subalgebra of the $w_\infty$. 
  We describe few aspects of the quantum symmetries of some massless two-dimensional field theories. We discuss their relations with recent proposals for the factorized scattering theories of the massless $PCM_1$ and $O(3)_{\theta=\pi}$ sigma models. We use these symmetries to propose massless factorized S-matrices for the $su(2)$ sigma models with topological terms at any level, alias the $PCM_k$ models, and for the $su(2)$-coset massless flows. 
  We study supersymmetric domain walls in N=1 supergravity theories, including those with modular-invariant superpotentials arising in superstring compactifications. Such domain walls are shown to saturate the Bogomol'nyi bound of wall energy per unit area. We find \sl static \rm and \sl reflection asymmetric \rm domain wall solutions of the self-duality equations for the metric and the matter fields. Our result establishes a new class of domain walls beyond those previously classified. As a corollary, we define a precise notion of vacuum degeneracy in the supergravity theories. In addition, we found examples of global supersymmetric domain walls that do not have an analog when gravity is turned on. This result establishes that in the case of extended topological defects gravity plays a crucial, nontrivial role. 
  The Coulomb gas representations are presented for the ${\rm SU(2)}$$_k$-extended $N$=4 superconformal algebras, incorporating the Feigin-Fuchs representation of the\break ${\rm SU(2)}$$_k$ Kac-Moody algebra with {\sl arbitrary} level $k$. Then the long-standing problem of identifying the whole set of charge-screening operators for the $N$=4 superconformal algebras is solved and their explicit expressions are given. The method of achieving a rigorous proof of the $N$=4 Kac determinant formulae following Kato and Matsuda is suggested. The complete proof for them will be given elsewhere. Our results for the screening operators also provide the basis for studying the BRST formalism of the $N$=4 superconformal algebras ${\sl {\grave a}\ la}$ Felder. 
  The simplest toroidally compactified string theories exhibit a duality between large and small radii: compactification on a circle, for example, is invariant under R goes to 1/R. Compactification on more general Lorentzian lattices (i.e. toroidal compactification in the presence of background metric, antisymmetric tensor, and gauge fields) yields theories for which large-small invariance is not so simple. Here an equivalence is demonstrated between large and small geometries for all toroidal compactifications. By repeatedly transforming the momentum mode corresponding to the smallest winding length to another mode on the lattice, it is possible to increase the volume to exceed a finite lower bound. 
  The Ward identities in Kontsevich-like 1-matrix models are used to prove at the level of discrete matrix models the suggestion of Gava and Narain, which relates the degree of potential in asymmetric 2-matrix model to the form of $\cal W$-constraints imposed on its partition function. 
  We demonstrate the equivalence of Virasoro constraints imposed on continuum limit of partition function of Hermitean 1-matrix model and the Ward identities of Kontsevich's model. Since the first model describes ordinary $d = 2$ quantum gravity, while the second one is supposed to coincide with Witten's topological gravity, the result provides a strong implication that the two models are indeed the same. 
  Using the finite-size effects the scaling dimensions and correlation functions of the main operators in continuous and lattice models of 1d spinless Bose-gas with pairwise interaction of rather general form are obtained. The long-wave properties of these systems can be described by the Gaussian model with central charge $c=1$. The disorder operators of the extended Gaussian model are found to correspond to some non-local operators in the {\it XXZ} Heisenberg antiferromagnet. Just the same approach is applicable to fermionic systems. Scaling dimensions of operators and correlation functions in the systems of interacting Fermi-particles are obtained. We present a universal treatment for $1d$ systems of different kinds which is independent of the exact integrability and gives universal expressions for critical exponents through the thermodynamic characteristics of the system. 
  We introduce a new 1-matrix model with arbitrary potential and the matrix-valued background field. Its partition function is a $\tau$-function of KP-hierarchy, subjected to a kind of ${\cal L}_{-1}$-constraint. Moreover, partition function behaves smoothly in the limit of infinitely large matrices. If the potential is equal to $X^{K+1}$, this partition function becomes a $\tau$-function of $K$-reduced KP-hierarchy, obeying a set of ${\cal W} _K$-algebra constraints identical to those conjectured in \cite{FKN91} for double-scaling continuum limit of $(K-1)$-matrix model. In the case of $K=2$ the statement reduces to the early established \cite{MMM91b} relation between Kontsevich model and the ordinary $2d$ quantum gravity . Kontsevich model with generic potential may be considered as interpolation between all the models of $2d$ quantum gravity with $c<1$ preserving the property of integrability and the analogue of string equation. 
  We generalize the known method for explicit construction of mirror pairs of $(2,2)$-superconformal field theories, using the formalism of Landau-Ginzburg orbifolds. Geometrically, these theories are realized as Calabi-Yau hypersurfaces in weighted projective spaces. This generalization makes it possible to construct the mirror partners of many manifolds for which the mirror was not previously known.  
  It is shown explicitly, that a number of solutions for the background field equations of the string effective action in space-time dimension D can be generated from any known lower dimensional solution, when background fields have only time dependence. An application of the result to the two dimensional charged black hole is presented. The case of background with more general coordinate dependence is also discussed. 
  We present a bicovariant differential calculus on the quantum Poincare group in two dimensions. Gravity theories on quantum groups are discussed. 
  In general quantum systems there are two kinds of spacetime modes, those that fluctuate and those that do not. Fluctuating modes have normalizable wavefunctions. In the context of 2D gravity and ``non-critical'' string theory these are called macroscopic states. The theory is independent of the initial Euclidean background values of these modes. Non-fluctuating modes have non-normalizable wavefunctions and correspond to microscopic states. The theory depends on the background value of these non-fluctuating modes, at least to all orders in perturbation theory. They are superselection parameters and should not be minimized over. Such superselection parameters are well known in field theory. Examples in string theory include the couplings $t_k$ (including the cosmological constant) in the matrix models and the mass of the two-dimensional Euclidean black hole. We use our analysis to argue for the finiteness of the string perturbation expansion around these backgrounds. 
  Continuum Virasoro constraints in the two-cut hermitian matrix models are derived from the discrete Ward identities by means of the mapping from the $GL(\infty )$ Toda hierarchy to the nonlinear Schr\"odinger (NLS) hierarchy. The invariance of the string equation under the NLS flows is worked out. Also the quantization of the integration constant $\alpha$ reported by Hollowood et al. is explained by the analyticity of the continuum limit. 
  The coupling of Yang-Mills fields to the heterotic string in bosonic formulation is generalized to extended objects of higher dimension (p-branes). For odd p, the Bianchi identities obeyed by the field strengths of the (p+1)-forms receive Chern-Simons corrections which, in the case of the 5-brane, are consistent with an earlier conjecture based on string/5-brane duality.  
  We present a unified group-theoretical framework for superparticle theories. This explains the origin of the ``twistor-like'' variables that have been used in trading the superparticle's $\kappa$-symmetry for worldline supersymmetry. We show that these twistor-like variables naturally parametrise the coset space ${\cal G}/{\cal H}$, where $\cal G$ is the Lorentz group $SO^\uparrow(1,d-1)$ and $\cal H$ is its maximal subgroup. This space is a compact manifold, the sphere $S^{d-2}$. Our group-theoretical construction gives the proper covariantisation of a fixed light-cone frame and clarifies the relation between target-space and worldline supersymmetries. 
  We discuss some classical and quantum properties of 2d gravity models involving metric and a scalar field. Different models are parametrized in terms of a scalar potential. We show that a general Liouville-type model with exponential potential and linear curvature coupling is renormalisable at the quantum level while a particular model (corresponding to D=2 graviton-dilaton string effective action and having a black hole solution) is finite. We use the condition of a ``split" Weyl symmetry to suggest possible expressions for the ``effective" action which includes the quantum anomaly term. 
  The Polyakov measure for the Abelian gauge field is considered in the Robertson-Walker spacetimes. The measure is concretely represented by adopting two kind of decompositions of the gauge field degrees of freedom which are most familiarly used in the covariant and canonical path integrals respectively. It is shown that the two representations are different by an anomalous Jacobian factor from each other and also that the factor has a direct relationship to an uncancellation factor of the contributions from the Faddeev-Popov ghost and the unphysical part of the gauge field to the covariant one-loop partition function. 
  Two-dimensional gravity in the light-cone gauge was shown by Polyakov to exhibit an underlying $SL(2,R)$ Kac-Moody symmetry, which may be used to express the energy-momentum tensor for the metric component $h_{++}$ in terms of the $SL(2,R)$ currents {\it via}\ the Sugawara construction. We review some recent results which show that in a similar manner, $W_\infty$ and $W_{1+\infty}$ gravities have underlying $SL(\infty,R)$ and $GL(\infty,R)$ Kac-Moody symmetries respectively. 
  We construct a centerless W-infinity type of algebra in terms of a generator of a centerless Virasoro algebra and an abelian spin-1 current. This algebra conventionally emerges in the study of pseudo-differential operators on a circle or alternatively within KP hierarchy with Watanabe's bracket. Construction used here is based on a special deformation of the algebra $w_{\infty}$ of area preserving diffeomorphisms of a 2-manifold. We show that this deformation technique applies to the two-loop WZNW and conformal affine Toda models, establishing henceforth $W_{\infty}$ invariance of these models. 
  The current algebra of classical non-linear sigma models on arbitrary Riemannian manifolds is analyzed. It is found that introducing, in addition to the Noether current $j_\mu$ associated with the global symmetry of the theory, a composite scalar field $j$, the algebra closes under Poisson brackets. 
  This is a detailed development for the $A_n$ case, of our previous article entitled "W-Geometries" to be published in Phys. Lett. It is shown that the $A_n$--W-geometry corresponds to chiral surfaces in $CP^n$. This is comes out by discussing 1) the extrinsic geometries of chiral surfaces (Frenet-Serret and Gauss-Codazzi equations) 2) the KP coordinates (W-parametrizations) of the target-manifold, and their fermionic (tau-function) description, 3) the intrinsic geometries of the associated chiral surfaces in the Grassmannians, and the associated higher instanton- numbers of W-surfaces. For regular points, the Frenet-Serret equations for $CP^n$--W-surfaces are shown to give the geometrical meaning of the $A_n$-Toda Lax pair, and of the conformally-reduced WZNW models, and Drinfeld-Sokolov equations. KP coordinates are used to show that W-transformations may be extended as particular diffeomorphisms of the target-space. This leads to higher-dimensional generalizations of the WZNW and DS equations. These are related with the Zakharov- Shabat equations. For singular points, global Pl\"ucker formulae are derived by combining the $A_n$-Toda equations with the Gauss-Bonnet theorem written for each of the associated surfaces. 
  Coset constructions in the framework of Chern-Simons topological gauge theories are studied. Two examples are considered: models of the types ${U(1)_p\times U(1)_q\over U(1)_{p+q}}\cong U(1)_{pq(p+q)}$ with $p$ and $q$ coprime integers, and ${SU(2)_m\times SU(2)_1\over SU(2)_{m+1}}$. In the latter case it is shown that the Chern-Simons wave functionals can be identified with t he characters of the minimal unitary models, and an explicit representation of the knot (Verlinde) operators acting on the space of $c<1$ characters is obtained. 
  We prove that a class of one-loop partition functions found by Dienes, giving rise to a vanishing cosmological constant to one-loop, cannot be realized by a consistent lattice string. The construction of non-supersymmetric string with a vanishing cosmological constant therefore remains as elusive as ever. We also discuss a new test that any one-loop partition function for a lattice string must satisfy. 
  The effective action of $N=2$, $d=4$ supergravity is shown to acquire no quantum corrections in background metrics admitting super-covariantly constant spinors. In particular, these metrics include the Robinson-Bertotti metric (product of two 2-dimensional spaces of constant curvature) with all 8 supersymmetries unbroken. Another example is a set of arbitrary number of extreme Reissner-Nordstr\"om black holes. These black holes break 4 of 8 supersymmetries, leaving the other 4 unbroken.   We have found manifestly supersymmetric black holes, which are non-trivial solutions of the flatness condition $\cd^{2} = 0$ of the corresponding (shortened) superspace. Their bosonic part describes a set of extreme Reissner-Nordstr\"om black holes. The super black hole solutions are exact even when all quantum supergravity corrections are taken into account. 
  A systematic construction of super W-algebras in terms of the WZNW model based on a super Lie algebra is presented. These are shown to be the symmetry structure of the super Toda models, which can be obtained from the WZNW theory by Hamiltonian reduction. A classification, according to the conformal spin defined by an improved energy-momentum tensor, is dicussed in general terms for all super Lie algebras whose simple roots are fermionic . A detailed discussion employing the Dirac bracket structure and an explicit construction of W-algebras for the cases of $OSP(1,2)$, $OSP(2,2)$ , $OSP(3,2)$ and $D(2,1 \mid \alpha )$ are given. The $N=1$ and $N=2$ super conformal algebras are discussed in the pertinent cases. 
  We present particularly simple new solutions to the Yang--Baxter equation arising from two--dimensional cyclic representations of quantum $SU(2)$. They are readily interpreted as scattering matrices of relativistic objects, and the quantum group becomes a dynamical symmetry. 
  We propose and investigate the thermodynamic Bethe ansatz equations for the minimal $W_p^N$ models~(associated with the $A_{N-1}$ Lie algebra) perturbed by the least~($Z_N$ invariant) primary field $\Phi_N$. Our results reproduce the expected ultraviolet and infrared regimes. In particular for the positive sign of the perturbation our equations describe the behaviour of the ground state flowing from the $W_p^N$ model to the next $W_{p-1}^N$ fixed point. 
  We consider the hermitian matrix model with an external field entering the quadratic term $\tr(\Lambda X\Lambda X)$ and Penner--like interaction term $\alpha N(\log(1+X)-X)$. An explicit solution in the leading order in $N$ is presented. The critical behaviour is given by the second derivative of the free energy in $\alpha$ which appears to be a pure logarithm, that is a feature of $c=1$ theories. Various critical regimes are possible, some of them corresponds to critical points of the usual Penner model, but there exists an infinite set of multi-critical points which differ by values of scaling dimensions of proper conformal operators. Their correlators with the puncture operator are given in genus zero by Legendre polynomials whose argument is determined by an analog of the string equation. 
  We review the BRST analysis of the system of a (super)conformal matter coupled to 2D (super)gravity. The spectrum and its operator realization are reported. In particular, the operators associated with the states of nonzero ghost number are given. We also discuss the ground ring structure of the super-Liouville coupled to ${\hat c}=1$ matter. In appendices, hermiticities, states for $c<1$ conformal matter coupled to gravity and the proof for the spectrum are discussed. 
  The method of separation of variables is shown to apply to both the classical and quantum Neumann model. In the classical case this nicely yields the linearization of the flow on the Jacobian of the spectral curve. In the quantum case the Schr\"odinger equation separates into one--dimensional equations belonging to the class of generalized Lam\'e differential equations. 
  $G/G$ topological field theories based on $G_k$ WZW models are constructed and studied. These coset models are formulated as Complex BRST cohomology in $G^c_k$, the complexified level $k$ current algebra. The finite physical spectrum corresponds to the conformal blocks of $G_k$ .The amplitudes for $G/G$ theories are argued to be given in terms of the $G_k$ fusion rules. The $G_k/G_k$ character is the Kac-Weyl numerator of $G_k$ and is interpreted as an index. The Complex BRST cohomology is found to contain states of arbitrary ghost number. Intriguing similarities of $G/G$ to $c\leq 1$ matter systems coupled to two dimensional gravity are pointed out. 
  We generalize the Marinari-Parisi definition for pure two dimensional quantum gravity $(k = 2)$ to all non unitary minimal multicritical points $(k \geq 3)$. The resulting interacting Fermi gas theory is treated in the collective field framework. Making use of the fact that the matrices evolve in Langevin time, the Jacobian from matrix coordinates to collective modes is similar to the corresponding Jacobian in $d = 1$ matrix models. The collective field theory is analyzed in the planar limit. The saddle point eigenvalue distribution is the one that defines the original multicritical point and therefore exhibits the appropriate scaling behaviour. Some comments on the nonperturbative properties of the collective field theory as well as comments on the Virasoro constraints associated with the loop equations are made at the end of this letter. There we also make some remarks on the fermionic formulation of the model and its integrability, as a nonlocal version of the non linear Schr\"{o}dinger model. 
  The constraints proposed recently by Bershadsky to produce $W^l_n$ algebras are a mixture of first and second class constraints and are degenerate. We show that they admit a first-class subsystem from which they can be recovered by gauge-fixing, and that the non-degenerate constraints can be handled by previous methods. The degenerate constraints present a new situation in which the natural primary field basis for the gauge-invariants is rational rather than polynomial. We give an algorithm for constructing the rational basis and converting the base elements to polynomials. 
  We show that Witten's two-dimensional string black hole metric is exactly conformally invariant in the supersymmetric case. We also demonstrate that this metric, together with a recently proposed exact metric for the bosonic case, are respectively consistent with the supersymmetric and bosonic $\sigma$-model conformal invariance conditions up to four-loop order. 
  Toroidal backgrounds for bosonic strings are used to understand target space duality as a symmetry of string field theory and to study explicitly issues in background independence. Our starting point is the notion that the string field coordinates $X(\sigma)$ and the momenta $P(\sigma)$ are background independent objects whose field algebra is always the same; backgrounds correspond to inequivalent representations of this algebra. We propose classical string field solutions relating any two toroidal backgrounds and discuss the space where these solutions are defined.   String field theories formulated around dual backgrounds are shown to be related by a homogeneous field redefinition, and are therefore equivalent, if and only if their string field coupling constants are identical. Using this discrete equivalence of backgrounds and the classical solutions we find discrete symmetry transformations of the string field leaving the string action invariant. These symmetries, which are spontaneously broken for generic backgrounds, are shown to generate the full group of duality symmetries, and in general are seen to arise from the string field gauge group. 
  Given two conformal field theories related to each other by a marginal perturbation, and string field theories constructed around such backgrounds, we show how to construct explicit redefinition of string fields which relate these two string field theories. The analysis is carried out completely for quadratic and cubic terms in the action. Although a general proof of existence of field redefinitions which relate higher point vertices is not given, specific examples are discussed. Equivalence of string field theories formulated around two conformal field theories which are not close to each other, but are related to each other by a series of marginal deformations, is also discussed. The analysis can also be applied to study the equivalence of different formulation of string field theories around the same background. 
  We use the 5-th time action formalism introduced by Halpern and Greensite to stabilize the unbounded Euclidean 4-D gravity in two simple minisuperspace models. In particular, we show that, at the semiclassical level ($\hbar \rightarrow 0$), we still have as a leading saddle point the $S^4$ solution and the Coleman peak at zero cosmological constant, for a fixed De Witt supermetric. At the quantum (one-loop) level the scalar gravitational modes give a positive semi-definite Hessian contribution to the 5-D partition function, thus removing the Polchinski phase ambiguity. 
  We here calculate the one-loop approximation to the Euclidean Quantum Gravity coupled to a scalar field around the classical Carlini and Miji\'c wormhole solutions. The main result is that the Euclidean partition functional $Z_{EQG}$ in the ``little wormhole'' limit is real. Extension of the CM solutions with the inclusion of a bare cosmological constant to the case of a sphere $S^4$ can lead to the elimination of the destabilizing effects of the scalar modes of gravity against those of the matter. In particular, in the asymptotic region of a large 4-sphere, we recover the Coleman's $\exp \left (\exp \left ({1\over \lambda_{eff}}\right )\right )$ peak at the effective cosmological constant $\lambda_{eff}=0$, with no phase ambiguities in $Z_{EQG}$. 
  We show that, for a class of critical strings in ${\bf R}\times S^{1}$-target space, the description of string theory given by its field content (analog model) breaks down when the radius of $S^{1}$ decreases below $R_{0}=\sqrt{\alpha^{\prime}}$, the self-dual point of the partition function $Z(R)$. We find that $Z(R)$ has a soft singularity at $R_{0}$ (a finite jump in the first derivative of $Z$). 
  The classical orbits of a test string in the transverse space of a singular heterotic fivebrane source are classified. The orbits are found to be either circular or open, but not conic because the inverse square law is not satisfied at long range. This result differs from predictions of General Relativity. The conserved total angular momentum contains an intrinsic component from the fivebrane source, analogous to the electron-monopole case. 
  The classical motion of a test string in the transverse space of two types of heterotic fivebrane sources is fully analyzed, for arbitrary instanton scale size. The singular case is treated as a special case and does not arise in the continuous limit of zero instanton size. We find that the orbits are either circular or open, which is a solitonic analogy with the motion of an electron around a magnetic monopole, although the system we consider is quantitatively different. We emphasize that at long distance this geometry does not satisfy the inverse square law, but satisfies the inverse cubic law. If the fivebrane exists in nature and this structure survives after any proper compactification, this last result can be used to test classical ``stringy'' effects. 
  The duality-invariant gaugino condensation with or without massive matter fields is re-analysed, taking into account the dependence of the string threshold corrections on the moduli fields and recent results concerning one-loop corrected K\"ahler potentials. The scalar potential of the theory for a generic superpotential is also calculated. 
  In a previous work, a straightforward canonical approach to the source-free quantum Chern-Simons dynamics was developed. It makes use of neither gauge conditions nor functional integrals and needs only ideas known from QCD and quantum gravity. It gives Witten's conformal edge states in a simple way when the spatial slice is a disc. Here we extend the formalism by including sources as well. The quantum states of a source with a fixed spatial location are shown to be those of a conformal family, a result also discovered first by Witten. The internal states of a source are not thus associated with just a single ray of a Hilbert space. Vertex operators for both abelian and nonabelian sources are constructed. The regularized abelian Wilson line is proved to be a vertex operator. We also argue in favor of a similar nonabelian result. The spin-statistics theorem is established for Chern-Simons dynamics even though the sources are not described by relativistic quantum fields. The proof employs geometrical methods which we find are strikingly transparent and pleasing. It is based on the research of European physicists about ``fields localized on cones.'' 
  Explicit construction of the light-cone gauge quantum theory of bosonic strings in 1+1 spacetime dimensions reveals unexpected structures. One is the existence of a gauge choice that gives a free action at the price of propagating ghosts and a nontrivial BRST charge. Fixing this gauge leaves a U(1) Kac-Moody algebra of residual symmetry, generated by a conformal tensor of rank two and a conformal scalar. Another is that the BRST charge made from these currents is nilpotent when the action includes a linear dilaton background, independent of the particular value of the dilaton gradient. Spacetime Lorentz invariance in this theory is still elusive, however, because of the linear dilaton background and the nature of the gauge symmetries. 
  We obtain the complete physical spectrum of the $W_N$ string, for arbitrary $N$. The $W_N$ constraints freeze $N-2$ coordinates, while the remaining coordinates appear in the currents only {\it via} their energy-momentum tensor. The spectrum is then effectively described by a set of ordinary Virasoro-like string theories, but with a non-critical value for the central charge and a discrete set of non-standard values for the spin-2 intercepts. In particular, the physical spectrum of the $W_N$ string includes the usual massless states of the Virasoro string. By looking at the norms of low-lying states, we find strong indications that all the $W_N$ strings are unitary.  
  The canonical structure of classical non-linear sigma models on Riemannian symmetric spaces, which constitute the most general class of classical non-linear sigma models known to be integrable, is shown to be governed by a fundamental Poisson bracket relation that fits into the $r$-$s$-matrix formalism for non-ultralocal integrable models first discussed by Maillet. The matrices $r$ and $s$ are computed explicitly and, being field dependent, satisfy fundamental Poisson bracket relations of their own, which can be expressed in terms of a new numerical matrix~$c$. It is proposed that all these Poisson brackets taken together are representation conditions for a new kind of algebra which, for this class of models, replaces the classical Yang-Baxter algebra governing the canonical structure of ultralocal models. The Poisson brackets for the transition matrices are also computed, and the notorious regularization problem associated with the definition of the Poisson brackets for the monodromy matrices is discussed. 
  We study how canonical transfomations in first quantized string theory can be understood as gauge transformations in string field theory. We establish this fact by working out some examples. As a by product, we could identify some of the fields appearing in string field theory with their counterparts in the $\sigma$-model. 
  In the light of recent blackhole solutions inspired by string theory, we review some old statements on field theoretic hair on blackholes. We also discuss some stability issues. In particular we argue that the two dimensional string blackhole solution is semi-classically stable while the naked singularity is unstable to tachyon fluctuations. Finally we comment on the relation between the linear dilaton theory and the $2d$ blackhole solution.  
  The effect of world-sheet boundaries on the temperature-dependence of bosonic string theory is studied to first order in string perturbation theory. The high- temperature behaviour of a theory with Dirichlet boundary conditions has features suggestive of the high-temperature limit of the confining phase of large-$n$ $SU(n)$ Yang--Mills theory, recently discussed by Polchinski. 
  Addendum to the paper Combinatorics of the Modular Group II The Kontsevich integrals, hep-th/9201001, by C. Itzykson and J.-B. Zuber (3 pages) 
  A careful treatment of closed string BRST cohomology shows that there are more discrete states and associated symmetries in $D=2$ string theory than has been recognized hitherto. The full structure, at the $SU(2)$ radius, has a natural description in terms of abelian gauge theory on a certain three dimensional cone $Q$. We describe precisely how symmetry currents are constructed from the discrete states, explaining the role of the ``descent equations.'' In the uncompactified theory, we compute the action of the symmetries on the tachyon field, and isolate the features that lead to nonlinear terms in this action. The resulting symmetry structure is interpreted in terms of a homotopy Lie algebra. 
  Classical W-gravities and the corresponding quantum theories are reviewed. W-gravities are higher-spin gauge theories in two dimensions whose gauge algebras are W-algebras. The geometrical structure of classical W-gravity is investigated, leading to surprising connections with self-dual geometry. The anomalies that arise in quantum W-gravity are discussed, with particular attention to the new types of anomalies that arise for non-linearly realised symmetries and to the relation between path-integral anomalies and non-closure of the quantum current algebra. Models in which all anomalies are cancelled by ghost contributions lead to new generalisations of string theories. 
  We show that the generating functional for hard thermal loops with external gluons in QCD is essentially given by the eikonal for a Chern-Simons gauge theory. This action, determined essentially by gauge invariance arguments, also gives an efficient way of obtaining the hard thermal loop contributions without the more involved calculation of Feynman diagrams. 
  A black hole may carry quantum numbers that are {\it not} associated with massless gauge fields, contrary to the spirit of the ``no-hair'' theorems. We describe in detail two different types of black hole hair that decay exponentially at long range. The first type is associated with discrete gauge charge and the screening is due to the Higgs mechanism. The second type is associated with color magnetic charge, and the screening is due to color confinement. In both cases, we perform semi-classical calculations of the effect of the hair on local observables outside the horizon, and on black hole thermodynamics. These effects are generated by virtual cosmic strings, or virtual electric flux tubes, that sweep around the event horizon. The effects of discrete gauge charge are non-perturbative in $\hbar$, but the effects of color magnetic charge become $\hbar$-independent in a suitable limit. We present an alternative treatment of discrete gauge charge using dual variables, and examine the possibility of black hole hair associated with discrete {\it global} symmetry. We draw the distinction between {\it primary} hair, which endows a black hole with new quantum numbers, and {\it secondary} hair, which does not, and we point out some varieties of secondary hair that occur in the standard model of particle physics. 
  The Lax pair formulation of the two dimensional induced gravity in the light-cone gauge is extended to the more general $w_N$ theories. After presenting the $w_2$ and $w_3$ gravities, we give a general prescription for an arbitrary $w_N$ case. This is further illustrated with the $w_4$ gravity to point out some peculiarities. The constraints and the possible presence of the cosmological constants are systematically exhibited in the zero-curvature condition, which also yields the relevant Ward identities. The restrictions on the gauge parameters in presence of the constraints are also pointed out and are contrasted with those of the ordinary 2d-gravity. 
  We investigate the proposal by Callan, Giddings, Harvey and Strominger (CGHS) that two dimensional quantum fluctuations can eliminate the singularities and horizons formed by matter collapsing on the nonsingular extremal black hole of dilaton gravity. We argue that this scenario could in principle resolve all of the paradoxes connected with Hawking evaporation of black holes. However, we show that the generic solution of the model of CGHS is singular. We propose modifications of their model which may allow the scenario to be realized in a consistent manner. 
  We examine the Wess-Zumino-Novikov-Witten (WZNW) model on a circle and compute the Poisson bracket algebra for left and right moving chiral group elements. Our computations apply for arbitrary groups and boundary conditions, the latter being characterized by the monodromy matrix. Unlike in previous treatments, they do not require specifying a particular parametrization of the group valued fields in terms of angles spanning the group. We do however find it necessary to make a gauge choice, as the chiral group elements are not gauge invariant observables. (On the other hand, the quadratic form of the Poisson brackets may be defined independent of a gauge fixing.) Gauge invariant observables can be formed from the monodromy matrix and these observables are seen to commute in the quantum theory.  
  We make the simple observation that, because of global symmetry violating higher-dimension operators expected to be induced by Planck-scale physics, textures are generically much too short-lived to be of use for large-scale structure formation. 
  We use free field techniques in D=2 string theory to calculate the perturbation of the special state algebras when the cosmologi- cal constant is turned on. In particular, we find that the "ground cone" preserved by the ring structure is promoted to a three dimen- sional hyperboloid as conjectured by Witten. On the other hand, the perturbed (1,1) a three dimensional hyperboloid as conjectured by Witten. On the other hand, the perturbed (1,1) current algebra of moduli deformations is computed completely, and no simple geometrical inter- pretation is found. We also quote some facts concerning the Liouville/matrix model dictio- nary in this class of theories. 
  All the three point couplings involving tachyons and/or discrete states are obtained in $c=1$ two-dimensional (2-D) quantum gravity by means of the operator product expansion (OPE). Cocycle factors are found to be necessary in order to maintain the analytic structure of the OPE, and are constructed explicitly both for discrete states and for tachyons. The effective action involving tachyons and discrete states is worked out to summarize all of these three point couplings. 
  The two-dimensional (2-D) quantum gravity coupled to the conformal matter with $c=1$ is studied. We obtain all the three point couplings involving tachyons and/or discrete states via operator product expansion. We find that cocycle factors are necessary and construct them explicitly. We obtain an effective action for these three point couplings. This is a brief summary of our study of couplings of tachyons and discrete states, reported at the workshop in Tokyo Metropolitan University, December 4-6, 1991. 
  We derive exact, factorized, purely elastic scattering matrices for affine Toda theories based on the nonsimply-laced Lie algebras and superalgebras. 
  We couple the 2D black-hole conformal field theory discovered by Witten to a $D-1$ dimensional Euclidean bosonic string. We demonstrate that the resulting planar (=zero genus) string susceptibility is real for any $0\leq D \leq 4$. 
  The non critical string (2D gravity coupled to the matter with central charge $D$) is quantized taking care of both diffeomorphism and Weyl symmetries. In incorporating the gauge fixing with respect to the Weyl symmetry, through the condition $R_g=const$, one modifies the classical result of Distler and Kawai. In particular one obtains the real string tension for an arbitrary value of central charge $D$. 
  We derive the explicit form of the Wess-Zumino quantum effective action of chiral $\Winf$-symmetric system of matter fields coupled to a general chiral $\Winf$-gravity background. It is expressed as a geometric action on a coadjoint orbit of the deformed group of area-preserving diffeomorphisms on cylinder whose underlying Lie algebra is the centrally-extended algebra of symbols of differential operators on the circle. Also, we present a systematic derivation, in terms of symbols, of the "hidden" $SL(\infty;\IR)$ Kac-Moody currents and the associated $SL(\infty;\IR)$ Sugawara form of energy-momentum tensor component $T_{++}$ as a consequence of the $SL(\infty;\IR)$ stationary subgroup of the relevant $\Winf$ coadjoint orbit. 
  We consider BRST quantized 2D gravity coupled to conformal matter with arbitrary central charge $c^M = c(p,q) < 1$ in the conformal gauge. We apply a Lian-Zuckerman $SO(2,\bbc)$ ($(p,q)$ - dependent) rotation to Witten's $c^M = 1$ chiral ground ring. We show that the ring structure generated by the (relative BRST cohomology) discrete states in the (matter $\otimes$ Liouville $\otimes$ ghosts) Fock module may be obtained by this rotation. We give also explicit formulae for the discrete states. For some of them we use new formulae for $c <1$ Fock modules singular vectors which we present in terms of Schur polynomials generalizing the $c=1$ expressions of Goldstone, while the rest of the discrete states we obtain by finding the proper $SO(2,\bbc)$ rotation. Our formulae give the extra physical states (arising from the relative BRST cohomology) on the boundaries of the $p \times q$ rectangles of the conformal lattice and thus all such states in $(1,q)$ or $(p,1)$ models. 
  I study tachyon condensate perturbations to the action of the two dimensional string theory corresponding to the c=1 matrix model. These are shown to deform the action of the ground ring on the tachyon modules, confirming a conjecture of Witten. The ground ring structure is used to derive recursion relations which relate (N+1) and N tachyon bulk scattering amplitudes. These recursion relations allow one to compute all bulk amplitudes.  
  We discuss the calculation of semi-classical wormhole vertex operators from wave functions which satisfy the Wheeler-deWitt equation and momentum constraints, together with certain `wormhole boundary conditions'. We consider a massless minimally coupled scalar field, initially in the spherically symmetric `mini-superspace' approximation, and then in the `midi-superspace' approximation, where non-spherically symmetric perturbations are linearized about a spherically symmetric mini-superspace background. Our approach suggests that there are higher derivative corrections to the vertex operator from the non-spherically symmetric perturbations. This is compared directly with the approach based on complete wormhole solutions to the equations of motion where it has been claimed that the semi-classical vertex operator is exactly given by the lowest order term, to all orders in the size of the wormhole throat. Our results are also compared with the conformally coupled case. 
  The formation and quantum mechanical evaporation of black holes in two spacetime dimensions can be studied using effective classical field equations, recently introduced by Callan {\it et al.} We find that gravitational collapse always leads to a curvature singularity, according to these equations, and that the region where the quantum corrections introduced by Callan {\it et al.} could be expected to dominate is on the unphysical side of the singularity. The model can be successfully applied to study the back-reaction of Hawking radiation on the geometry of large mass black holes, but the description breaks down before the evaporation is complete. 
  In 2+1 dimensional gravity, a dreibein and the compatible spin connection can represent a space-time containing a closed spacelike surface $\Sigma$ only if the associated SO(2,1) bundle restricted to $\Sigma$ has the same non-triviality (Euler class) as that of the tangent bundle of $\Sigma.$ We impose this bundle condition on each external state of Witten's topology-changing amplitude. The amplitude is non-vanishing only if the combination of the space topologies satisfies a certain selection rule. We construct a family of transition paths which reproduce all the allowed combinations of genus $g \ge 2$ spaces. 
  Dilaton contact terms in the bosonic and heterotic strings are examined following the recent work of Distler and Nelson on the bosonic and semirigid strings. In the bosonic case dilaton two-point functions on the sphere are calculated as a stepping stone to constructing a `good' coordinate family for dilaton calculations on higher genus surfaces. It is found that dilaton-dilaton contact terms are improperly normalized, suggesting that the interpretation of the dilaton as the first variation of string coupling breaks down when other dilatons are present. It seems likely that this can be attributed to the tachyon divergence found in \TCCT. For the heterotic case, it is found that there is no tachyon divergence and that the dilaton contact terms are properly normalized. Thus, a dilaton equation analogous to the one in topological gravity is derived and the interpretation of the dilaton as the string coupling constant goes through. 
  Some elaboration is given to the structure of physical states in 2D gravity coupled to $C \leq 1$ matter, and to the chiral algebra ($w_{\infty}$) of $C_{M} = 1$ theory which has been found recently, in the continuum approach, by Witten and by Klebanov and Polyakov. It is shown then that the chiral algebra is being realized as well in the minimal models of gravity ($C_{M}<1$), so that it stands as a general symmetry of 2D gravity theories. 
  We state and prove various new identities involving the Z_K parafermion characters (or level-K string functions) for the cases K=4, K=8, and K=16. These identities fall into three classes: identities in the first class are generalizations of the famous Jacobi theta-function identity (which is the K=2 special case), identities in another class relate the level K>2 characters to the Dedekind eta-function, and identities in a third class relate the K>2 characters to the Jacobi theta-functions. These identities play a crucial role in the interpretation of fractional superstring spectra by indicating spacetime supersymmetry and aiding in the identification of the spacetime spin and statistics of fractional superstring states. 
  We show how the interplay between the fusion formalism of conformal field theory and the Knizhnik--Zamolodchikov equation leads to explicit formulae for the singular vectors in the highest weight representations of A1{(1)}. 
  In the spirit of the quantum Hamiltonian reduction we establish a relation between the chiral $n$-point functions, as well as the equations governing them, of the $A_1^{(1)}$ WZNW conformal theory and the corresponding Virasoro minimal models. The WZNW correlators are described as solutions of the Knizhnik - Zamolodchikov equations with rational levels and isospins. The technical tool exploited are certain relations in twisted cohomology. The results extend to arbitrary level $k+2 \neq 0$ and isospin values of the type $J=j-j'(k+2)$, $ \ 2j, 2j' \in Z\!\!\!Z_+$. 
  Some remarks are made about free anomaly groups in gauged WZW models. Considering a quite general action, anomaly cancellation is analyzed. The possibility of gauging left and right sectors independently in some cases is remarked. In particular Toda theories can be seen as such a kind of models. 
  It is argued that fractional quantum Hall effect wavefunctions can be interpreted as conformal blocks of two-dimensional conformal field theory. Fractional statistics can be extended to nonabelian statistics and examples can be constructed from conformal field theory. The Pfaffian state is related to the 2D Ising model and possesses fractionally charged excitations which are predicted to obey nonabelian statistics. 
  The Wess-Zumino-Witten (WZW) theory has a global symmetry denoted by $G_L\otimes G_R$. In the standard gauged WZW theory, vector gauge fields (\ie\ with vector gauge couplings) are in the adjoint representation of the subgroup $H \subset G$. In this paper, we show that, in the conformal limit in two dimensions, there is a gauged WZW theory where the gauge fields are chiral and belong to the subgroups $H_L$ and $H_R$ where $H_L$ and $H_R$ can be different groups. In the special case where $H_L=H_R$, the theory is equivalent to vector gauged WZW theory. For general groups $H_L$ and $H_R$, an examination of the correlation functions (or more precisely, conformal blocks) shows that the chiral gauged WZW theory is equivalent to $(G/H)_L\otimes (G/H)_R$ coset models in conformal field theory. The equivalence of the vector gauged WZW theory and the corresponding $G/H$ coset theory then follows. 
  Global-symmetry violating higher-dimension operators, expected to be induced by Planck-scale physics, in general drastically alter the properties of the axion field associated with the Peccei-Quinn solution to the strong-CP problem, and render this solution unnatural. The particle physics and cosmology associated with other global symmetries can also be significantly changed. 
  Particle scattering and radiation by a magnetically charged, dilatonic black hole is investigated near the extremal limit at which the mass is a constant times the charge. Near this limit a neighborhood of the horizon of the black hole is closely approximated by a trivial product of a two-dimensional black hole with a sphere. This is shown to imply that the scattering of long-wavelength particles can be described by a (previously analyzed) two-dimensional effective field theory, and is related to the formation/evaporation of two-dimensional black holes. The scattering proceeds via particle capture followed by Hawking re-emission, and naively appears to violate unitarity. However this conclusion can be altered when the effects of backreaction are included. Particle-hole scattering is discussed in the light of a recent analysis of the two-dimensional backreaction problem. It is argued that the quantum mechanical possibility of scattering off of extremal black holes implies the potential existence of additional quantum numbers - referred to as ``quantum whiskers'' - characterizing the black hole. 
  We construct an exact CFT as an SL(2,R)xSU(2)/U(1)^2 gauged WZW model, which describes a black hole in 4 dimensions. Another exact solution, describing a black membrane in 4D (in the sense that the event horizon is an infinite plane) is found as an SL(2,R)xU(1)^2/U(1) gauged WZW model. Finally, we construct an exact solution of a 4D black hole with electromagnetic field, as an SL(2,R)xSU(2)xU(1)/U(1)^2 gauged WZW model. This black hole carries both electric and axionic charges. 
  We reexamine the external field problem for $N\times N$ hermitian one-matrix models. We prove an equivalence of the models with the potentials $\tr{({1/over2N}X^2 + \log X - \Lambda X)}$ and $\sum_{k=1}^\infty t_k\tr{X^k}$ providing the matrix $\Lambda$ is related to $\{t_k\}$ by $t_k=\fr 1k \tr{\Lambda^{-k}}-\frac N2 \delta_{k2}$. Based on this equivalence we formulate a method for calculating the partition function by solving the Schwinger--Dyson equations order by order of genus expansion. Explicit calculations of the partition function and of correlators of conformal operators with the puncture operator are presented in genus one. These results support the conjecture that our models are associated with the $c=1$ case in the same sense as the Kontsevich model describes $c=0$. 
  We construct the K=8 fractional superconformal algebras. There are two such extended Virasoro algebras, one of which was constructed earlier, involving a fractional spin (equivalently, conformal dimension) 6/5 current. The new algebra involves two additional fractional spin currents with spin 13/5. Both algebras are non-local and satisfy non-abelian braiding relations. The construction of the algebras uses the isomorphism between the Z_8 parafermion theory and the tensor product of two tricritical Ising models. For the special value of the central charge c=52/55, corresponding to the eighth member of the unitary minimal series, the 13/5 currents of the new algebra decouple, while two spin 23/5 currents (level-2 current algebra descendants of the 13/5 currents) emerge. In addition, it is shown that the K=8 algebra involving the spin 13/5 currents at central charge c=12/5 is the appropriate algebra for the construction of the K=8 (four-dimensional) fractional superstring. 
  We analyze topological string theory on a two dimensional torus, focusing on symmetries in the matter sector. Even before coupling to gravity, the topological torus has an infinite number of point-like physical observables, which give rise via the BRST descent equations to an infinite symmetry algebra of the model. The point-like observables of ghost number zero form a topological ground ring, whose generators span a spacetime manifold; the symmetry algebra represents all (ground ring valued) diffeomorphisms of the spacetime. At nonzero ghost numbers, the topological ground ring is extended to a superring, the spacetime manifold becomes a supermanifold, and the symmetry algebra preserves a symplectic form on it. In a decompactified limit of cylindrical target topology, we find a nilpotent charge which behaves like a spacetime topological BRST operator. After coupling to topological gravity, this model might represent a topological phase of $c=1$ string theory. We also point out some analogies to two dimensional superstrings with the chiral GSO projection, and to string theory with $c=-2$. 
  We give various examples of asymmetric orbifold models to possess intertwining currents which convert untwisted string states to twisted ones, and vice versa, and see that such asymmetric orbifold models are severely restricted. The existence of the intertwining currents leads to the enhancement of symmetries in asymmetric orbifold models. 
  An effective Hamiltonian for the study of the quantum Hall effect is proposed. This Hamiltonian, which includes a ``current-current" interaction has the form of a Hamiltonian for a conformal field theory in the large $N$ limit. An order parameter is constructed from which the Hamiltonian may be derived. This order parameter may be viewed as either a collective coordinate for a system of $N$ charged particles in a strong magnetic field; or as a field of spins associated with the cyclotron motion of these particles. 
  We study $N$=2 supersymmetric integrable theories with spontaneously-broken \Zn\ symmetry. They have exact soliton masses given by the affine $SU(n)$ Toda masses and fractional fermion numbers given by multiples of $1/n$. The basic such $N$=2 integrable theory is the $A_n$-type $N$=2 minimal model perturbed by the most relevant operator. The soliton content and exact S-matrices are obtained using the Landau-Ginzburg description. We study the thermodynamics of these theories and calculate the ground-state energies exactly, verifying that they have the correct conformal limits. We conjecture that the soliton content and S-matrices in other integrable \Zn\ $N$=2 theories are given by the tensor product of the above basic $N$=2 \Zn\ scattering theory with various $N$=0 theories. In particular, we consider integrable perturbations of $N$=2 Kazama-Suzuki models described by generalized Chebyshev potentials, $CP^{n-1}$ sigma models, and $N$=2 sine-Gordon and its affine Toda generalizations. 
  Based on the observation that a particle motion in one dimension maps to a two-dimensional motion of a charged particle in a uniform magnetic field, constrained in the lowest Landau level, we formulate a system of one-dimen- sional nonrelativistic fermions by using a Chern-Simons field theory in 2+1 dimensions. Using a hydrodynamical formulation we obtain a two-dimensional droplet picture of one-dimensional fermions. The dynamics involved is that of the boundary between a uniform density of particles and vortices. We use the sharp boundary approximation. In the case of well separated boundaries we derive the one-dimensional collective field Hamiltonian. Symmetries of the theory are also discussed as properties of curves in two dimensions. 
  Within the standard quantum mechanics a q-deformation of the simplest N=2 supersymmetry algebra is suggested. Resulting physical systems do not have conserved charges and degeneracies in the spectra. Instead, superpartner Hamiltonians are q-isospectral, i.e. the spectrum of one can be obtained from another by the q^2-factor scaling. A special class of the self-similar potentials is shown to obey the dynamical conformal symmetry algebra su_q(1,1). These potentials exhibit exponential spectra and corresponding raising and lowering operators satisfy the q-deformed harmonic oscillator algebra of Biedenharn and Macfarlane. 
  It is argued that the qualitative features of black holes, regarded as quantum mechanical objects, depend both on the parameters of the hole and on the microscopic theory in which it is embedded. A thermal description is inadequate for extremal holes. In particular, extreme holes of the charged dilaton family can have zero entropy but non-zero, and even (for $a>1$) formally infinite, temperature. The existence of a tendency to radiate at the extreme, which threatens to overthrow any attempt to identify the entropy as available internal states and also to expose a naked singularity, is at first sight quite disturbing. However by analyzing the perturbations around the extreme holes we show that these holes are protected by mass gaps, or alternatively potential barriers, which remove them from thermal contact with the external world. We suggest that the behavior of these extreme dilaton black holes, which from the point of view of traditional black hole theory seems quite bizarre, can reasonably be interpreted as the holes doing their best to behave like normal elementary particles. The $a<1$ holes behave qualitatively as extended objects. 
  A minimal area problem imposing different length conditions on open and closed curves is shown to define a one parameter family of covariant open-closed quantum string field theories. These interpolate from a recently proposed factorizable open-closed theory up to an extended version of Witten's open string field theory capable of incorporating on shell closed strings. The string diagrams of the latter define a new decomposition of the moduli spaces of Riemann surfaces with punctures and boundaries based on quadratic differentials with both first order and second order poles. 
  We show how to construct path integrals for quantum mechanical systems where the space of configurations is a general non-compact symmetric space. Associated with this path integral is a perturbation theory which respects the global structure of the system. This perturbation expansion is evaluated for a simple example and leads to a new exactly soluble model. This work is a step towards the construction of a strong coupling perturbation theory for quantum gravity. 
  By considering mirror symmetry applied to conformal field theories corresponding to strings propagating in quintic hypersurfaces in projective 4-space, Candelas, de la Ossa, Green and Parkes calculated the ``number of rational curves on the hypersurface'' by comparing three point functions. Actually, the number of curves may be infinite for special examples; what is really being calculated is a path integral. The point of this talk is to give mathematical techniques and examples for computing the finite number that ``should'' correspond to an infinite family of curves (which coincides with that given by the path integral in every known instance), and to suggest that these techniques should provide the answer to the not yet solved problem of how to calculate instanton corrections to the three point function in general. 
  We find a consistent formulation of the constraints of Quantum Gravity with a cosmological constant in terms of the Ashtekar new variables in the connection representation, including the existence of a state that is a solution to all the constraints. This state is related to the Chern-Simons form constructed from the Ashtekar connection and has an associated metric in spacetime that is everywhere nondegenerate. We then transform this state to the loop representation and find solutions to all the constraint equations for intersecting loops. These states are given by suitable generalizations of the Jones knot polynomial for the case of intersecting knots. These are the first physical states of Quantum Gravity for which an explicit form is known both in the connection and loop representations. Implications of this result are also discussed. 
  The D=0 matrix model is reformulated as a 2d nonlocal quantum field theory. The interactions occur on the one-dimensional line of hermitian matrix eigenvalues. The field is conjugate to the density of matrix eigenvalues which appears in the Jevicki-Sakita collective field theory. The classical solution of the field equation is either unique or labeled by a discrete index. Such a solution corresponds to the Dyson sea modified by an entropy term. The modification smoothes the sea edges, and interpolates between different eigenvalue bands for multiple-well potentials. Our classical eigenvalue density contains nonplanar effects, and satisfies a local nonlinear Schr\"odinger equation with similarities to the Marinari-Parisi $D=1$ reformulation. The quantum fluctuations about a classical solution are computable, and the IR and UV divergences are manifestly removed to all orders. The quantum corrections greatly simplify in the double scaling limit, and include both string-perturbative and nonperturbative effects. 
  We propose a Thermodynamic Bethe Ansatz (TBA) for G_k x G_l / G_{k+l} conformal coset models (G any simply-laced Lie algebra) perturbed by their operator \phi_{1,1,Adj}. An interesting adjacency structure appears and can be depicted in a sort of ``product'' of Dynkin diagrams of G and A_{k+l-1}. UV and IR limits are computed and reproduce the expected values for the central charges. For k->\infty, l fixed we obtain the TBA of the G_l WZW model perturbed by J_a\bar{J}_a, and for k,l->\infty, k-l fixed, that of Principal Chiral model with WZ term at level k-l. 
  A general method is presented for deriving on-shell Ward-identities in (2D) string theory. It is shown that all tree-level Ward identities can be summarized in a quadratic differential equation for the generating function of tree-amplitudes. This result is extended to loop amplitudes and leads to a master equation {\it \`{a} la} Batalin-Vilkovisky for the complete partition function. 
  New results from the new variables/loop representation program of nonperturbative quantum gravity are presented, with a focus on results of Ashtekar, Rovelli and the author which greatly clarify the physical interpretation of the quantum states in the loop representation. These include: 1) The construction of a class of states which approximate smooth metrics for length measurements on scales, $L$, to order $l_{Planck}/L$. 2) The discovery that any such state must have discrete structure at the Planck length. 3) The construction of operators for the area of arbitrary surfaces and volumes of arbitrary regions and the discovery that these operators are finite. 4) The diagonalization of these operators and the demonstration that the spectra are discrete, so that in quantum gravity areas and volumes are quantized in Planck units. 5) The construction of finite diffeomorphism invariant operators that measure geometrical quantities such as the volume of the universe and the areas of minimal surfaces. These results are made possible by the use of new techniques for the regularization of operator products that respect diffeomorphism invariance. Several new results in the classical theory are also reviewed including the solution of the hamiltonian and diffeomorphism constraints in closed form of Capovilla, Dell and Jacobson and a new form of the action that induces Chern-Simon theory on the boundaries of spacetime. A new classical discretization of the Einstein equations is also presented. 
  We consider a source of gravitational waves of frequency $\omega$, located near the center of a massive galaxy of mass $M$ and radius $R$, with $\omega\gg R^{-1}$. In the case of a perfect fluid galaxy, and of odd-parity waves, there is no direct matter-wave interaction and the propagation of the waves is affected by the galaxy only through the curvature of the spacetime background through which the waves propagate. We find that, in addition to the expected redshift of the radiation emerging from the galaxy, there is a small amount of backscatter, of order $M/\omega^2R^3$. We show that there is no suppression of radiative power by the factor $1+\omega^2M^2/4$ as has been recently predicted by Kundu. The origin of Kundu's suppression lies in the interpretation of a term in the expansion of the exterior field of the galaxy in inverse powers of radius. It is shown why that term is not related to the source strength or to the strength of the emerging radiation. 
  The path integral approach to representing braid group is generalized for particles with spin. Introducing the notion of {\em charged} winding number in the super-plane, we represent the braid group generators as homotopically constrained Feynman kernels. In this framework, super Knizhnik-Zamolodchikov operators appear naturally in the Hamiltonian, suggesting the possibility of {\em spinning nonabelian} anyons. We then apply our formulation to the study of fractional quantum Hall effect (FQHE). A systematic discussion of the ground states and their quasi-hole excitations is given. We obtain Laughlin, Halperin and Moore-Read states as {\em exact} ground state solutions to the respective Hamiltonians associated to the braid group representations. The energy gap of the quasi-excitation is also obtainable from this approach. 
  A generalization of BRST field theory is presented, based on wave operators for the fields constructed out of, but different from the BRST operator. We discuss their quantization, gauge fixing and the derivation of propagators. We show, that the generalized theories are relevant to relativistic particle theories in the Brink-Di Vecchia-Howe-Polyakov (BDHP) formulation, and argue that the same phenomenon holds in string theories. In particular it is shown, that the naive BRST formulation of the BDHP theory leads to trivial quantum field theories with vanishing correlation functions. 
  The $CP^N$ model in three euclidean dimensions is studied in the presence of a Chern-Simons term using the $1/N$ expansion. The $\beta$-function for the CS coefficient $\theta$ is found to be zero to order $1/N$ in the unbroken phase by an explicit calculation. It is argued to be zero to all orders. Some remarks on the $\theta$ dependence of the critical exponents are also made. 
  The QHE is studied in the context of a CFT. An effective field of $N$ ``spins" associated with the cyclotron motion of particles is taken as an order parameter from which an effective Hamiltonian may be defined. This effective Hamiltonian describes the COM motion of the $N$ particles (with coupling $\kappa_0$) together with a current-current interaction (of strength $\kappa_1$). Such a system gives rise to a CFT in the large $N$ limit when $\kappa_0 = \kappa_1$. The Laughlin wavefunction is derived from this CFT as an $N'$-point correlation function of winding state vertex operators. 
  In this paper we study the renormalization group flow of the $(p,q)$ minimal (non-unitary) CFT perturbed by the $\Phi_{1,3}$ operator with a positive coupling. In the perturbative region $q>>(q-p)$, we find a new IR fixed point which corresponds to the $(2p-q,p)$ minimal CFT. The perturbing field near the new IR fixed point is identified with the irrelevent $\Phi_{3,1}$ operator. We extend this result to show that the non-diagonal ($(A,D)$-type) modular invariant partition function of the $(p,q)$ minimal CFT flows into the $(A,D)$-type partition function of the $(2p-q,p)$ minimal CFT and the diagonal partition function into the diagonal. 
  In this paper we present a theory of Singlet Quantum Hall Effect (SQHE). We show that the Halperin-Haldane SQHE wave function can be written in the form of a product of a wave function for charged semions in a magnetic field and a wave function for the Chiral Spin Liquid of neutral spin-$\12$ semions. We introduce field-theoretic model in which the electron operators are factorized in terms of charged spinless semions (holons) and neutral spin-$\12$ semions (spinons). Broken time reversal symmetry and short ranged spin correlations lead to $SU(2)_{k=1}$ Chern-Simons term in Landau-Ginzburg action for SQHE phase. We construct appropriate coherent states for SQHE phase and show the existence of $SU(2)$ valued gauge potential. This potential appears as a result of ``spin rigidity" of the ground state against any displacements of nodes of wave function from positions of the particles and reflects the nontrivial monodromy in the presence of these displacements. 
  We discuss Stochastic Quantization of $d$=3 dimensional non-Abelian Chern-Simons theory. We demonstrate that the introduction of an appropriate regulator in the Langevin equation yields a well-defined equilibrium limit, thus leading to the correct propagator. We also analyze the connection between $d$=3 Chern-Simons and $d$=4 Topological Yang-Mills theories showing the equivalence between the corresponding regularized partition functions. We study the construction of topological invariants and the introduction of a non-trivial kernel as an alternative regularization. 
  We study static spherically symmetric solutions of Einstein gravity plus an action polynomial in the Ricci scalar, $R$, of arbitrary degree, $n$, in arbitrary dimension, $D$. The global properties of all such solutions are derived by studying the phase space of field equations in the equivalent theory of gravity coupled to a scalar field, which is obtained by a field redefinition and conformal transformation. The following uniqueness theorem is obtained: provided that the coefficient of the $R^2$ term in the Lagrangian polynomial is positive then the only static spherically symmetric asymptotically flat solution with a regular horizon in these models is the Schwarzschild solution. Other branches of solutions with regular horizons, which are asymptotically anti-de Sitter, or de Sitter, are also found. An exact Schwarzschild-de Sitter type solution is found to exist in the $R+aR^2$ if $D>4$. If terms of cubic or higher order in $R$ are included in the action, then such solutions also exist in four dimensions. The general Schwarzschild-de Sitter type solution for arbitrary $D$ and $n$ is given. The fact that the Schwarzschild solution in these models does not coincide with the exterior solution of physical bodies such as stars has important physical implications which we discuss. As a byproduct, we classify all static spherically symmetric solutions of $D$-dimensional gravity coupled to a scalar field with a potential consisting of a finite sum of exponential terms. 
  We derive, based on the Wakimoto realization, the integral formulas for the WZNW correlation functions. The role of the ``screening currents Ward identity'' is demonstrated with explicit examples. We also give a more simple proof of a previous result. 
  The phase space of the Wess-Zumino-Witten model on a circle with target space a compact, connected, semisimple Lie group $G$ is defined and the corresponding symplectic form is given. We present a careful derivation of the Poisson brackets of the Wess-Zumino-Witten model. We also study the canonical structure of the supersymmetric and the gauged Wess-Zumino-Witten models. 
  A (1+1)-dimensional quantum field theory with a degenerate vacuum (in infinite volume) can contain particles, known as kinks, which interpolate between different vacua and have nontrivial restrictions on their multi-particle Hilbert space. Assuming such a theory to be integrable, we show how to calculate the multi-kink energy levels in finite volume given its factorizable $S$-matrix. In massive theories this can be done exactly up to contributions due to off-shell and tunneling effects that fall off exponentially with volume. As a first application we compare our analytical predictions for the kink scattering theories conjectured to describe the subleading thermal and magnetic perturbations of the tricritical Ising model with numerical results from the truncated conformal space approach. In particular, for the subleading magnetic perturbation our results allow us to decide between the two different $S$-matrices proposed by Smirnov and Zamolodchikov. 
  We describe the (chiral) BRST-cohomology of matter with central charge $1<c_M<25$ coupled to a ``Liouville" theory, realized as a free field with a background charge $Q_L$ such that $c_M+c_L=26$. We consider two cases: a) matter is realized by one free field with an imaginary background charge, b) matter is realized by $D$ free fields: $c_M =D$. In case a) the cohomology states can be labelled by integers $r,s$ of a rotated $c_M =1$ theory, but hermiticity imposes $r=s$. Thus there is still a discrete set of momenta $p_M(r,r),\ p_L(r,r)$ such that there are non- trivial (relative) cohomology states at level $r^2$ with ghost-numbers 0 or 1 (for $r>0$) and ghost-numbers 0 or $-1$ (for $r<0$). The (chiral) ground ring is isomorphic to a subring of the $c_M =1$ theory which is $(xy)^n,\ n=0,1,2,\ldots$, and there are {\it no} non-trivial currents acting on the ground ring. In case b) there is no non-trivial relative cohomology for non-zero ghost numbers and, for zero ghost number, the cohomology groups are isomorphic to a $(D-1)$-dimensional on-shell ``transverse" Fock space. The only exceptions are at level 1 for vanishing matter momentum and $p_L=Q_L(1+r)$ with $r=\pm 1$, where one has one more ghost-number zero and a ghost-number $r$ cohomology state. All these results follow quite easily from the existing literature. 
  Irreducible sigma models, i.e. those for which the partition function does not factorise, are defined on Riemannian spaces with irreducible holonomy groups. These special geometries are characterised by the existence of covariantly constant forms which in turn give rise to symmetries of the supersymmetric sigma model actions. The Poisson bracket algebra of the corresponding currents is a W-algebra. Extended supersymmetries arise as special cases. 
  The interior near the horizon of an extremal Reissner-Nordstr\"om black hole is taken as an initial state for quantum mechanical tunneling. An instanton is presented that connects this state with a final state describing the presence of several horizons. This is interpreted as a WKB description of fluctuations due to the throat splitting into several components. 
  Starting from a topological gauge theory in two dimensions with symmetry groups $ISO(2,1)$, $SO(2,1)$ and $SO(1,2)$ we construct a model for gravity with non-trivial coupling to matter. We discuss the equations of motion which are connected to those of previous related models but incorporate matter content. We also discuss the resulting quantum theory and finally present explicit formul\ae $\;$ for topological invariants. 
  We give a criterion for the existence of a non-degenerate quasihomogeneous polynomial in a configuration, i.e. in the space of polynomials with a fixed set of weights, and clarify the relation of this criterion to the necessary condition derived from the formula for the Poincar\'e polynomial. We further prove finiteness of the number of configurations for a given value of the singularity index. For the value 3 of this index, which is of particular interest in string theory, a constructive version of this proof implies an algorithm for the calculation of all non-degenerate configurations. 
  We define and compute explicitly the classical limit of the realizations of $W_n$ appearing as hamiltonian structures of generalized KdV hierarchies. The classical limit is obtained by taking the commutative limit of the ring of pseudodifferential operators. These algebras---denoted $w_n$---have free field realizations in which the generators are given by the elementary symmetric polynomials in the free fields. We compute the algebras explicitly and we show that they are all reductions of a new algebra $w_{\rm KP}$, which is proposed as the universal classical $W$-algebra for the $w_n$ series. As a deformation of this algebra we also obtain $w_{1+\infty}$, the classical limit of $W_{1+\infty}$. 
  The staircase model is a recently discovered one-parameter family of integrable two-dimensional continuum field theories. We analyze the novel critical behavior of this model, seen as a perturbation of a minimal conformal theory M_p: the leading thermodynamic singularities are simultaneously governed by all fixed points M_p, M_{p-1}, ..., M_3. The exponents of the magnetic susceptibility and the specific heat are obtained exactly. Various corrections to scaling are discussed, among them a new type specific to crossover phenomena between critical fixed points. 
  We present a direct derivation of the thermodynamic integral equations of the O(3) nonlinear $\sigma$-model in two dimensions. 
  We study in detail the quantization of a model which apparently describes chiral bosons. The model is based on the idea that the chiral condition could be implemented through a linear constraint. We show that the space of states is of indefinite metric. We cure this disease by introducing ghost fields in such a way that a BRST symmetry is generated. A quartet algebra is seen to emerge. The quartet mechanism, then, forces all physical states, but the vacuum, to have zero norm. 
  We pose a representation-theoretic question motivated by an attempt to resolve the Andrews-Curtis conjecture. Roughly, is there a triangular Hopf algebra with a collection of self-dual irreducible representations $V_i$ so that the product of any two decomposes as a sum of copies of the $V_i$, and $\sum (\rank V_i)^2=0$? This data can be used to construct a `topological quantum field theory' on 2-complexes which stands a good chance of detecting counterexamples to the conjecture. 
  Topologically charged black holes in a theory with a 2-form coupled to a non-abelian gauge field are investigated. It is found that the classification of the ground states is similar to that in the theory of non-abelian discrete quantum hair. 
  We present a systematic study of the constraints coming from target-space duality and the associated duality anomaly cancellations on orbifold-like 4-D strings. A prominent role is played by the modular weights of the massless fields. We present a general classification of all possible modular weights of massless fields in Abelian orbifolds. We show that the cancellation of modular anomalies strongly constrains the massless fermion content of the theory, in close analogy with the standard ABJ anomalies. We emphasize the validity of this approach not only for (2,2) orbifolds but for (0,2) models with and without Wilson lines. As an application one can show that one cannot build a ${\bf Z}_3$ or ${\bf Z}_7$ orbifold whose massless charged sector with respect to the (level one) gauge group $SU(3)\times SU(2) \times U(1)$ is that of the minimal supersymmetric standard model, since any such model would necessarily have duality anomalies. A general study of those constraints for Abelian orbifolds is presented. Duality anomalies are also related to the computation of string threshold corrections to gauge coupling constants. We present an analysis of the possible relevance of those threshold corrections to the computation of $\sin^2\theta_W$ and $\alpha_3$ for all Abelian orbifolds. Some particular {\it minimal} scenarios, namely those based on all ${\bf Z}_N$ orbifolds except ${\bf Z}_6$ 
  We show how to construct, starting from a quasi-Hopf algebra, or quasi-quantum group, invariants of knots and links. In some cases, these invariants give rise to invariants of the three-manifolds obtained by surgery along these links. This happens for a finite-dimensional quasi-quantum group, whose definition involves a finite group $G$, and a 3-cocycle $\om$, which was first studied by Dijkgraaf, Pasquier and Roche. We treat this example in more detail, and argue that in this case the invariants agree with the partition function of the topological field theory of Dijkgraaf and Witten depending on the same data $G, \,\om$. 
  A connection of a variety of tight-binding models of noninteracting electrons on a rectangular lattice in a magnetic field with theta functions is established. A new spectrum generating symmetry is discovered which essentialy reduces the problem of diagonalization of these models. Provided that one knows one eigenvector at one point in the parameter space of the corresponding Harper equation one knows an eigenfunction of the corresponding model in the whole range of momentum singlet out by the Landau gauge. 
  String field theory for the non-critical NSR string is described. In particular it gives string field theory for the 2D super-gravity coupled to a $\hat{c}=1$ matter field. For this purpose double-step pictures changing operators for the non-critical NSR string are constructed. Analogues of the critical supersymmetry transformations are written for $D<10$, they form a closed on-shell algebra, however their action on vertices is defined only for discrete value of the Liouville momentum. For D=2 this means that spinor massless field has its superpartner in the NS sector only if its momentum is fixed.   Starting from string field theory we calculate string amplitudes. These amplitudes for D=2 have poles which are related with discrete set of primary fields, namely 2R$\to$2R amplitude has poles corresponding to the n-level NS excitations with discrete momenta $p_1=n,~~p_2=-1\pm (n+1)$. 
  We derive the supersymmetric collective field theory for the Marinari-Parisi model. For a specific choice of the superpotential, to leading order we find a one parameter family of ground states which can be connected via instantons. At this level of analysis the instanton size implied by the underlying matrix model does not appear. 
  We consider superstring compactifications where both the classical description, in terms of a Calabi-Yau manifold, and also the quantum theory is known in terms of a Landau-Ginzburg orbifold model. In particular, we study (smooth) Calabi-Yau examples in which there are obstructions to parametrizing all of the complex structure cohomology by polynomial deformations thus requiring the analysis based on exact and spectral sequences. General arguments ensure that the Landau-Ginzburg chiral ring copes with such a situation by having a nontrivial contribution from twisted sectors. Beyond the expected final agreement between the mathematical and physical approaches, we find a direct correspondence between the analysis of each, thus giving a more complete mathematical understanding of twisted sectors. Furthermore, this approach shows that physical reasoning based upon spectral flow arguments for determining the spectrum of Landau-Ginzburg orbifold models finds direct mathematical justification in Koszul complex calculations and also that careful point- field analysis continues to recover suprisingly much of the stringy features. 
  Recently, a topological proof of the spin-statistics Theorem has been proposed for a system of point particles which does not require relativity or field theory, but assumes the existence of antiparticles. We extend this proof to a system of string loops in three space dimensions and show that by assuming the existence of antistring loops, one can prove a spin-statistics theorem for these string loops. According to this theorem, all unparametrized strings(such as flux tubes in superconductors and cosmic strings ) should be quantized as bosons. Also, as in the point particle case, we find that the theorem excludes nonabelian statistics. 
  Holonomy algebras arise naturally in the classical description of Yang-Mills fields and gravity, and it has been suggested, at a heuristic level, that they may also play an important role in a non-perturbative treatment of the quantum theory. The aim of this paper is to provide a mathematical basis for this proposal.   The quantum holonomy algebra is constructed, and, in the case of real connections, given the structure of a certain C-star algebra. A proper representation theory is then provided using the Gel'fand spectral theory. A corollory of these general results is a precise formulation of the ``loop transform'' proposed by Rovelli and Smolin. Several explicit representations of the holonomy algebra are constructed. The general theory developed here implies that the domain space of quantum states can always be taken to be the space of maximal ideals of the C-star algebra. The structure of this space is investigated and it is shown how observables labelled by ``strips'' arise naturally. 
  The recently proposed loop representation, used previously to find exact solutions to the quantum constraints of general relativity, is here used to quantize linearized general relativity. The Fock space of graviton states and its associated algebra of observables are represented in terms of functionals of loops. The ``reality conditions'' are realized by an inner product that is chiral asymmetric, resulting in a chiral asymmetric ordering for the Hamiltonian and in an asymmetric description of the left and right handed gravitons. This chirally asymmetric formulation depends on a splitting of the linearized field into self-dual and anti-self dual parts rather than into positive and negative frequency parts; as the former, but not the latter, is meaningful away from flat backgrounds this is expected to be useful in connecting the nonperturbative theory to the linearized theory. The formalism depends on an arbitrary ``averaging'' function that controls certain divergences, but does not appear in the final physical quantities. Inspite of these somewhat unusual features, the loop quntization presented here is completely equivalent to the standard quantization of linearized gravity. 
  We study 2d gravity coupled to $c,1$ matter through canonical quantization of a free scalar field, with background charge, coupled to gravity. Various features of the theory can be more easily understood in the canonical approach, like gauge indipendence of the path-integral results and the absence of the local physical degrees of freedom. By performing a non-canonical transformation of the phase space variables, we show that the theory takes a free-field form, i.e. the constraints become the free-field Virasoro constraints. This implies that the David-Distler-Kawai results can be derived in the gauge indipendent way, and also proves the free-field assumption which was used for obtaining the spectrum of the theory in the conformal gauge. A discussion of the physical spectrum of the theory is presented, with an analysis of the unitarity of the discrete momentum states. 
  New reparametrisation invariant field equations are constructed which describe $d$-brane models in a space of $d+1$ dimensions. These equations, like the recently discovered scalar field equations in $d+1$ dimensions, are universal, in the sense that they can be derived from an infinity of inequivalent Lagrangians, but are nonetheless Lorentz (Euclidean) invariant. Moreover, they admit a hierarchical structure, in which they can be derived by a sequence of iterations from an arbitrary reparametrisation covariant Lagrangian, homogeneous of weight one. None of the equations of motion which appear in the hierarchy of iterations have derivatives of the fields higher than the second. The new sequence of Universal equations is related to the previous one by an inverse function transformation. The particular case of $d=2$, giving a new reparametrisation invariant string equation in 3 dimensions is solved. 
  We formulate one dimensional many-body integrable systems in terms of a new set of phase space variables involving exchange operators. The hamiltonian in these variables assumes a decoupled form. This greatly simplifies the derivation of the conserved charges and the proof of their commutativity at the quantum level. 
  We present a systematic construction of classical extended superconformal algebras from the hamiltonian reduction of a class of affine Lie superalgebras, which include an even subalgebra $sl(2)$. In particular, we obtain the doubly extended $N=4$ superconformal algebra $\tilde{A}_{\gamma}$ from the hamiltonian reduction of the exceptional Lie superalgebra $D(2|1;\gamma/(1-\gamma))$. We also find the Miura transformation for these algebras and give the free field representation. A $W$-algebraic generalization is discussed. 
  We use the linear supermultiplet formalism of supergravity to study axion couplings and chiral anomalies in the context of field-theoretical Lagrangians describing orbifold compactifications beyond the classical approximation. By matching amplitudes computed in the effective low energy theory with the results of string loop calculations we determine the appropriate counterterm in this effective theory that assures modular invariance to all loop order. We use supersymmetry consistency constraints to identify the correct ultra-violet cut-offs for the effective low energy theory. Our results have a simple interpretation in terms of two-loop unification of gauge coupling constants at the string scale. 
  We discuss the physical spectrum for $W$ strings based on the algebras $B_n$, $D_n$, $E_6$, $E_7$ and $E_8$. For a simply-laced $W$ string, we find a connection with the $(h,h+1)$ unitary Virasoro minimal model, where $h$ is the dual Coxeter number of the underlying Lie algebra. For the $W$ string based on $B_n$, we find a connection with the $(2h,2h+2)$ unitary $N=1$ super-Virasoro minimal model. 
  Reductive W-algebras which are generated by bosonic fields of spin-1, a single spin-2 field and fermionic fields of spin-3/2 are classified. Three new cases are found: a `symplectic' family of superconformal algebras which are extended by $su(2)\oplus sp(n)$, an $N=7$ and an $N=8$ superconformal algebra. The exceptional cases can be viewed as arising a Drinfeld-Sokolov type reduction of the exceptional Lie superalgebras $G(3)$ and $F(4)$, and have an octonionic description. The quantum versions of the superconformal algebras are constructed explicitly in all three cases. 
  We study the metric of minimal area on a punctured Riemann surface under the condition that all nontrivial homotopy closed curves be longer than or equal to $2\pi$. By constructing deformations of admissible metrics we establish necessary conditions on minimal area metrics and a partial converse to Beurling's criterion for extremal metrics. We explicitly construct new minimal area metrics that do not arise from quadratic differentials.   Under the physically motivated assumption of existence of the minimal area metrics, we show there exist neighborhoods of the punctures isometric to a flat semiinfinite cylinder of circumference $2\pi$, allowing the definition of canonical complex coordinates around the punctures. The plumbing of surfaces with minimal area metrics is shown to induce a metric of minimal area on the resulting surface. This implies that minimal area string diagrams define a consistent quantum closed string field theory. 
  Quantization of the free Maxwell field in Minkowski space is carried out using a loop representation and shown to be equivalent to the standard Fock quantization. Because it is based on coherent state methods, this framework may be useful in quantum optics. It is also well-suited for the discussion of issues related to flux quantization in condensed matter physics. Our own motivation, however, came from a non-perturbative approach to quantum gravity. The concrete results obtained in this paper for the Maxwell field provide independent support for that approach. In addition, they offer some insight into the physical interpretation of the mathematical structures that play, within this approach, an essential role in the description of the quantum geometry at Planck scale. 
  I discuss how instanton effects can be wiped-out due to the existence of anomalies. I first consider Compact Quantum Electrodynamics in 3 dimensions where confinement of electric charge is destroyed when fermions are added so that a Chern-Simons term is generated as a one-loop effect. I also show that a similar phenomenon occurs in the two-dimensional abelian chiral Higgs model. In both cases anomalies (parity anomaly, gauge anomaly) are responsible of the deconfinement mechanism. 
  Conditions for the construction of polynomial eigen--operators for the Hamiltonian of collective string field theories are explored. Such eigen--operators arise for only one monomial potential $v(x) = \mu x^2$ in the collective field theory. They form a $w_{\infty}$--algebra isomorphic to the algebra of vertex operators in 2d gravity. Polynomial potentials of orders only strictly larger or smaller than 2 have no non--zero--energy polynomial eigen--operators. This analysis leads us to consider a particular potential $v(x)= \mu x^2 + g/x^2$. A Lie algebra of polynomial eigen--operators is then constructed for this potential. It is a symmetric 2--index Lie algebra, also represented as a sub--algebra of $U (s\ell (2)).$  
  Some time ago, Atiyah showed that there exists a natural identification between the k-instantons of a Yang-Mills theory with gauge group $G$ and the holomorphic maps from $CP_1$ to $\Omega G$. Since then, Nair and Mazur, have associated the $\Theta $ vacua structure in QCD with self-intersecting Riemann surfaces immersed in four dimensions. From here they concluded that these 2D surfaces correspond to the non-perturbative phase of QCD and carry the topological information of the $\Theta$ vacua. In this paper we would like to elaborate on this point by making use of Atiyah's identification. We will argue that an effective description of QCD may be more like a $WZW$ model coupled to the induced metric of an immersion of a 2-D Riemann surface in $R^4$. We make some further comments on the relationship between the coadjoint orbits of the Kac-Moody group on $G$ and instantons with axial symmetry and monopole charge. 
  We point out that the moduli sector of the $(2,2)$ string compactification with its nonperturbatively preserved non-compact symmetries is a framework to study global topological defects. Based on the target space modular invariance of the nonperturbative superpotential of the four-dimensional $N=1$ supersymmetric string vacua, topologically stable stringy domain walls are found. Explicit supersymmetric solutions for the modulus field and the metric, which saturate the Bogomol'nyi bound, are presented. They interpolate between {\it non-degenerate} vacua. As a corollary, this defines a new notion of vacuum degeneracy of supersymmetric vacua. Nonsupersymmetric stringy domain walls are discussed as well. The moduli sectors with more than one modulus and the non-compact continous symmetry preserved allow for global monopole-type and texture-type configurations. 
  We investigate properties of two-dimensional asymptotically flat black holes which arise in both string theory and in scale invariant theories of gravity. By introducing matter sources in the field equations we show how such objects can arise as the endpoint of gravitational collapse. We examine the motion of test particles outside the horizons, and show that they fall through in a finite amount of proper time and an infinite amount of coordinate time. We also investigate the thermodynamic and quantum properties, which give rise to a fundamental length scale. The 't Hooft prescription for cutting off eigenmodes of particle wave functions is shown to be source dependent, unlike the four-dimensional case. The relationship between these black holes and those considered previously in $(1+1)$ dimensions is discussed. 
  We study the renormalization and conservation at the quantum level of higher-spin currents in affine Toda theories with particular emphasis on the nonsimply-laced cases. For specific examples, namely the spin-3 current for the $a_3^{(2)}$ and $c_2^{(1)}$ theories, we prove conservation to all-loop order, thus establishing the existence of factorized S-matrices. For these theories, as well as the simply-laced $a_2^{(1)}$ theory, we compute one-loop corrections to the corresponding higher-spin charges and study charge conservation for the three-particle vertex function. For the $a_3^{(2)}$ theory we show that although the current is conserved, anomalous threshold singularities spoil the conservation of the corresponding charge for the on-shell vertex function, implying a breakdown of some of the bootstrap procedures commonly used in determining the exact S-matrix. 
  We consider Quantum Toda theory associated to a general Lie algebra. We prove that the conserved quantities in both conformal and affine Toda theories exhibit duality interchanging the Dynkin diagram and its dual, and inverting the coupling constant. As an example we discuss the conformal Toda theories based on $B_2,B_3$ and $G_2$ and the related affine theories. 
  We examine the constraints and the reality conditions that have to be imposed in the canonical theory of 4--d gravity formulated in terms of Ashtekar variables. We find that the polynomial reality conditions are consistent with the constraints, and make the theory equivalent to Einstein's, as long as the inverse metric is not degenerate; when it is degenerate, reality conditions cannot be consistently imposed in general, and the theory describes complex general relativity. 
  We review some recent developments in the theory of $W_\infty$. We comment on its relevance to lower-dimensional string theory. 
  The scattering of two excitations (both of the simplest kind) in the magnetic model related to the $Z_n$\--Baxter model is naturally described for $n \rightarrow \infty$ in terms of the Macdonald polynomials for root system $A_1$. These polynomials play the role of zonal spherical functions for a two parameter family of quantum symmetric spaces. These spaces ``interpolate'' between various $p$\--adic and real symmetric spaces. 
  A model is proposed which generates all oriented $3d$ simplicial complexes weighted with an invariant associated with a topological lattice gauge theory. When the gauge group is $SU_q(2)$, $q^n=1,$ it is the Turaev-Viro invariant and the model may be regarded as a non-perturbative definition of $3d$ simplicial quantum gravity. If one takes a finite abelian group $G$, the corresponding invariant gives the rank of the first cohomology group of a complex \nolinebreak $C$: $I_G(C) = rank(H^1(C,G))$, which means a topological expansion in the Betti number $b^1$. In general, it is a theory of the Dijkgraaf-Witten type, $i.e.$ determined completely by the fundamental group of a manifold. 
  We argue that, classically, $s$-wave electrons incident on a magnetically charged black hole are swallowed with probability one: the reflection coefficient vanishes. However, quantum effects can lead to both electromagnetic and gravitational backscattering. We show that, for the case of extremal, magnetically charged, dilatonic black holes and a single flavor of low-energy charged particles, this backscattering is described by a perturbatively computable and unitary $S$-matrix, and that the Hawking radiation in these modes is suppressed near extremality. The interesting and much more difficult case of several flavors is also discussed. 
  Using the Ashtekar formulation, it is shown that the G_{Newton} --> 0 limit of Euclidean or complexified general relativity is not a free field theory, but is a theory that describes a linearized self-dual connection propagating on an arbitrary anti-self-dual background. This theory is quantized in the loop representation and, as in the full theory, an infinite dimnensional space of exact solutions to the constraint is found. An inner product is also proposed. The path integral is constructed from the Hamiltonian theory and the measure is explicitly computed nonperturbatively, without relying on a semiclassical expansion. This theory could provide the starting point for a new approach to perturbation theory in $G_{Newton}$ that does not rely on a background field expansion and in which full diffeomorphism invariance is satisfied at each order. 
  A static configuration of point charges held together by the gravitational attraction is known to be given by the Majumdar-Papapetrou solution in the Einstein-Maxwell theory. We consider a generalization of this solution to non-Abelian monopoles of the Yang-Mills Higgs system coupled to gravity. The solution is governed by an analog of the Bogomol'nyi equations that had played a central role in the analysis of non-Abelian monopoles. 
  We propose a new space-time interpretation for c=1 matrix model with potential $V(x)=-x^{2}/2-\m^{2}/2x^{2}$. It is argued that this particular potential corresponds to a black hole background. Some related issues are discussed. 
  Quantum theory of the free Maxwell field in Minkowski space is constructed using a representation in which the self dual connection is diagonal. Quantum states are now holomorphic functionals of self dual connections and a decomposition of fields into positive and negative frequency parts is unnecessary. The construction requires the introduction of new mathematical techniques involving ``holomorphic distributions''. The method extends also to linear gravitons in Minkowski space. The fact that one can recover the entire Fock space --with particles of both helicities-- from self dual connections alone provides independent support for a non-perturbative, canonical quantization program for full general relativity based on self dual variables. 
  We study the Schwinger-Dyson equations of a matrix model for an open-closed string theory. The free energy with source terms for scaling operators satisfies the same Virasoro conditions as those of the pure closed string and is obtained from that of the pure closed string by giving appropriate nonvanishing background values to all of the sources. 
  A matrix model is presented which leads to the discrete ``eigenvalue model'' proposed recently by Alvarez-Gaum\'e {\it et.al.} for 2D supergravity (coupled to superconformal matters). 
  We show that the XY quantum chain in a magnetic field is invariant under a two parameter deformation of the $SU(1/1)$ superalgebra. One is led to an extension of the braid group and the Hecke algebras which reduce to the known ones when the two parameter coincide. The physical significance of the two parameters is discussed. When both are equal to one, one gets a Pokrovski-Talapov phase transition. We also show that the representation theory of the quantum superalgebras indicates how to take the appropriate thermodynamical limits. 
  We investigate the following three consistency conditions for constructing string theories on orbifolds: i) the invariance of the energy-momentum tensors under twist operators, ii) the duality of amplitudes and iii) modular invariance of partition functions. It is shown that this investigation makes it possible to obtain the general class of consistent orbifold models, which includes a new class of orbifold models. 
  We study cocycle properties of vertex operators and present an operator representation of cocycle operators, which are attached to vertex operators to ensure the duality of amplitudes. It is shown that this analysis makes it possible to obtain the general class of consistent string theories on orbifolds. 
  It has recently been shown that the dissipative Hofstadter model (dissipative quantum mechanics of an electron subject to uniform magnetic field and periodic potential in two dimensions) exhibits critical behavior on a network of lines in the dissipation/magnetic field plane. Apart from their obvious condensed matter interest, the corresponding critical theories represent non-trivial solutions of open string field theory, and a detailed account of their properties would be interesting from several points of view. A subject of particular interest is the dependence of physical quantities on the magnetic field since it, much like $\theta_{\rm QCD}$, serves only to give relative phases to different sectors of the partition sum. In this paper we report the results of an initial investigation of the free energy, $N$-point functions and boundary state of this type of critical theory. Although our primary goal is the study of the magnetic field dependence of these quantities, we will present some new results which bear on the zero magnetic field case as well. 
  The w_\infty algebra is a particular generalization of the Virasoro algebra with generators of higher spin 2,3,...,\infty. It can be viewed as the algebra of a class of functions, relative to a Poisson bracket, on a suitably chosen surface. Thus, w_\infty is a special case of area-preserving diffeomorphisms of an arbitrary surface. We review various aspects of area- preserving diffeomorphisms, w_\infty algebras and w_\infty gravity. The topics covered include a) the structure of the algebra of area-preserving diffeomorphisms with central extensions and their relation to w_\infty algebras, b) various generalizations of w_\infty algebras, c) the structure of w_\infty gravity and its geometrical aspects, d) nonlinear realizations of w_\infty symmetry and e) various quantum realizations of w_\infty symmetry. 
  We discuss appropriate arrangement of picture changing operators required to construct gauge invariant interaction vertices involving Neveu-Schwarz states in heterotic and closed superstring field theory. The operators required for this purpose are shown to satisfy a set of descent equations. 
  We define the concepts of topological particles and topological radiation. These are nothing more than connected components of defects of a vector field. To each topological particle we assign an index which is an integer which is conserved under interactions with other particles much as electric charge is conserved. For space-like vector fields of space-times this index is invariant under all coordinate transformations. We propose the following physical principal: For physical vector fields the index changes only when there is radiation. As an implication of this principal we predict that any physical psuedo-vector field has index zero. The definition of the index is quite elementary. It only depends upon the concepts of continuity, compactness, the Euler-Poincare number, and the idea of inward pointing. The proof that this definition is well defined takes up most of the paper. The paper concludes with a list of properties of the index. 
  We consider the correlation functions of the tachyon vertex operator of the super Liouville theory coupled to matter fields in the super Coulomb gas formulation, on world sheets with spherical topology. After integrating over the zero mode and assuming that the $s$ parameter takes an integer value, we subsequently continue it to an arbitrary real number and compute the correlators in a closed form. We also included an arbitrary number of screening charges and, as a result, after renormalizing them, as well as the external legs and the cosmological constant, the form of the final amplitudes do not modify. The result is remarkably parallel to the bosonic case. For completeness, we discussed the calculation of bosonic correlators including arbitrary screening charges. 
  The recent interest in ``time machines'' has been largely fueled by the apparent ease with which such systems may be formed in general relativity, given relatively benign initial conditions such as the existence of traversable wormholes or of infinite cosmic strings. This rather disturbing state of affairs has led Hawking to formulate his Chronology Protection Conjecture, whereby the formation of ``time machines'' is forbidden. This paper will use several simple examples to argue that the universe appears to exhibit a ``defense in depth'' strategy in this regard. For appropriate parameter regimes Casimir effects, wormhole disruption effects, and gravitational back reaction effects all contribute to the fight against time travel. Particular attention is paid to the role of the quantum gravity cutoff. For the class of model problems considered it is shown that the gravitational back reaction becomes large before the Planck scale quantum gravity cutoff is reached, thus supporting Hawking's conjecture. 
  We define a new class of integrable vertex models associated to quantum groups at roots of unit 
  We present a general discussion of strings propagating on noncompact coset spaces $G/H$ in terms of gauged WZW models, emphasizing the role played by isometries in the existence of target space duality. Fixed points of the gauged transformations induce metric singularities and, in the case of abelian subgroups $H$, become horizons in a dual geometry. We also give a classification of models with a single timelike coordinate together with an explicit list for dimensions $D\leq 10$. We study in detail the class of models described by the cosets $SL(2,\IR)\otimes SO(1,1)^{D-2}/SO(1,1)$. For $D\geq 2$ each coset represents two different spacetime geometries: (2D black hole)$\otimes \IR^{D-2}$ and (3D black string)$\otimes \IR^{D-3}$ with nonvanishing torsion. They are shown to be dual in such a way that the singularity of the former geometry (which is not due to a fixed point) is mapped to a regular surface (i.e.\ not even a horizon) in the latter . These cosets also lead to the conformal field theory description of known and new cosmological string models. 
  We analyze the phase structure of topological Calabi--Yau manifolds defined on the moduli space of instantons. We show in this framework that topological vacua describe new phases of the Heterotic String theory in which the flat directions corresponding to complex deformations are lifted. We also briefly discuss the phase structure of non--K\"ahler manifolds. 
  Disk amplitudes of tachyons in two-dimensional open string theories (two-dimensional quantum gravity coupled to $c \leq 1$ conformal field theories) are obtained using the continuum Liouville field approach. The structure of momentum singularities is different from that of sphere amplitudes and is more complicated. It can be understood by factorizations of the amplitudes with the tachyon and the discrete states as intermediate states. 
  We compute the $S$-matrix of the Tricritical Ising Model perturbed by the subleading magnetic operator using Smirnov's RSOS reduction of the Izergin-Korepin model. We discuss some features of the scattering theory we obtain, in particular a non trivial implementation of crossing-symmetry, interesting connections between the asymptotic behaviour of the amplitudes, the possibility of introducing generalized statistics, and the monodromy properties of the OPE of the unperturbed Conformal Field Theory. 
  The depth rule is a level truncation of tensor product coefficients expected to be sufficient for the evaluation of fusion coefficients. We reformulate the depth rule in a precise way, and show how, in principle, it can be used to calculate fusion coefficients. However, we argue that the computation of the depth itself, in terms of which the constraints on tensor product coefficients is formulated, is problematic. Indeed, the elements of the basis of states convenient for calculating tensor product coefficients do not have a well-defined depth! We proceed by showing how one can calculate the depth in an `approximate' way and derive accurate lower bounds for the minimum level at which a coupling appears. It turns out that this method yields exact results for $\widehat{su}(3)$ and constitutes an efficient and simple algorithm for computing $\widehat{su}(3)$ fusion coefficients. 
  We review the Symmetric Unitary One Matrix Models. In particular we discuss the string equation in the operator formalism, the mKdV flows and the Virasoro Constraints. We focus on the $\t$-function formalism for the flows and we describe its connection to the (big cell of the) Sato Grassmannian $\Gr$ via the Plucker embedding of $\Gr$ into a fermionic Fock space. Then the space of solutions to the string equation is an explicitly computable subspace of $\Gr\times\Gr$ which is invariant under the flows. 
  The action of general relativity proposed by Capovilla, Jacobson and Dell is written in terms of $SO(3)$ gauge fields and gives Ashtekar's constraints for Einstein gravity. However, it does not depend on the space-time metric nor its signature explicitly. We discuss how the space-time metric is introduced from algebraic relations of the constraints and the Hamiltonian by focusing our attention on the signature factor. The system describes both Euclidian and Lorentzian metrics depending on reality assignments of the gauge connections. That is, Euclidian metrics arise from the real gauge fields. On the other hand, self-duality of the gauge fields, which is well known in the Ashtekar's formalism, is also derived in this theory from consistency condition of Lorentzian metric. We also show that the metric so determined is equivalent to that given by Urbantke, which is usually accepted as a definition of the metric for this system. 
  Gravitational theta-sectors are investigated in spatially locally homogeneous cosmological models with flat closed spatial surfaces in 2+1 and 3+1 spacetime dimensions. The metric ansatz is kept in its most general form compatible with Hamiltonian minisuperspace dynamics. Nontrivial theta-sectors admitting a semiclassical no-boundary wave function are shown to exist only in 3+1 dimensions, and there only for two spatial topologies. In both cases the spatial surface is nonorientable and the nontrivial no-boundary theta-sector unique. In 2+1 dimensions the nonexistence of nontrivial no-boundary theta-sectors is shown to be of topological origin and thus to transcend both the semiclassical approximation and the minisuperspace ansatz. Relation to the necessary condition given by Hartle and Witt for the existence of no-boundary theta-states is discussed. 
  We study the underlying gauge symmetry algebra of the $N=2$ string, which is broken down to a subalgebra in any spacetime background. For given toroidal backgrounds, the unbroken gauge symmetries (corresponding to holomorphic and antiholomorphic worldsheet currents) generate area-preserving diffeomorphism algebras of null 2-tori. A minimal Lie algebraic closure containing all the gauge symmetries that arise in this way, is the background--independent volume--preserving diffeomorphism algebra of the target Narain torus $T^{4,4}$. The underlying symmetries act on the ground ring of functions on $T^{4,4}$ as derivations, much as in the case of the $d=2$ string. A background--independent spacetime action valid for noncompact metrics is presented, whose symmetries are volume--preserving diffeomorphisms. Possible extensions to $N=2$ and $N=1$ heterotic strings are briefly discussed. 
  We develop an iterative algorithm for the genus expansion of the hermitian $N\times N$ one-matrix model ( = the Penner model in an external field). By introducing moments of the external field, we prove that the genus $g$ contribution to the $m$-loop correlator depends only on $3g-2+m$ lower moments ($3g-2$ for the partition function). We present the explicit results for the partition function and the one-loop correlator in genus one. We compare the correlators for the hermitian one-matrix model with those at zero momenta for $c=1$ CFT and show an agreement of the one-loop correlators for genus zero. 
  We analyze the beta-function equations for string theory in the case when the target space has one spacelike (or timelike) direction and rest is some conformal field theory (CFT) with appropriate central charge and has one nearly marginal operator. We show there always exists a space (time) dependent solution which interpolates between the original background and the background where CFT is replaced by a new conformal field theory, obtained by perturbing CFT by the nearly marginal operator. 
  We recall the classification of the irreducible representations of $SL(2)_q$, and then give fusion rules for these representations. We also consider the problem of $\cR$-matrices, intertwiners of the differently ordered tensor products of these representations, and satisfying altogether Yang--Baxter equations. 
  We have argued previously that the infinitely many gauge symmetries of string theory provide an infinite set of conserved (gauge) quantum numbers ($W$-hair) which characterise black hole states and maintain quantum coherence. Here we study ways of measuring the $W$-hair of spherically-symmetric four-dimensional objects with event horizons, treated as effectively two-dimensional string black holes. Measurements can be done either through the s-wave scattering of light particles off the string black-hole background, or through interference experiments of Aharonov-Bohm type. In the first type of measurement, selection rules 
  We write down a local $CP_1$ model involving two gauge fields, which is exactly equivalent to the O(3) $\sigma$ model with the Hopf term. We impose the $CP_1$ constraint by using the gaussian representation of the delta function. For the coefficient of the Hopf term, $\theta = {\pi \over 2s}$, 2s being an integer, we show that the resulting model is exactly equivalent to an interacting theory of spin-$s$ fields. Thus we conjecture that there should be a fixed point in the spin-$s$ theory near which it is exactly equal to the $\sigma$ model. 
  We discuss the BRST cohomologies of the invariants associated with the description of classical and quantum gravity in four dimensions, using the Ashtekar variables. These invariants are constructed from several BRST cohomology sequences. They provide a systematic and clear characterization of non-local observables in general relativity with unbroken diffeomorphism invariance, and could yield further differential invariants for four-manifolds. The theory includes fluctuations of the vierbein fields, but there exits a non-trivial phase which can be expressed in terms of Witten's topological quantum field theory. In this phase, the descent sequences are degenerate, and the corresponding classical solutions can be identified with the conformally self-dual sector of Einstein manifolds. The full theory includes fluctuations which bring the system out of this sector while preserving diffeomorphism invariance. 
  We consider the three--dimensional BF--model with planar boundary in the axial gauge. We find two--dimensional conserved chiral currents living on the boundary and satisfying Kac--Moody algebras. 
  We present a very simple and explicit procedure for nonlocalizing the action of any theory which can be formulated perturbatively. When the resulting nonlocal field theory is quantized using the functional formalism --- with unit measure factor --- its Green's functions are finite to all orders. The construction also ensures perturbative unitarity to all orders for scalars with nonderivative interactions, however, decoupling is lost at one loop when vector and tensor quanta are present. Decoupling can be restored (again, to all orders) if a suitable measure factor exists. We compute the required measure factor for pure Yang-Mills at order $g^2$ and then use it to evaluate the vacuum polarization at one loop. A peculiar feature of our regularization scheme is that the on-shell tree amplitudes are completely unaffected. This implies that the nonlocal field theory can be viewed as a highly noncanonical quantization of the original, local field equations. 
  A polarization of the Lie algebras $Map(C, G)$ of gauge transformations on the light-cone $C\subset\RM^4$ is introduced, using splitting of the initial data on $C$ for the wave operator to positive and negative frequencies. This generalizes the usual polarization of affine Kac-Moody algebras to positive and negative frequencies and paves the way to a generalization of the highest weight theory to the $3+1$ dimensional setting. 
  In flat space, the extreme Reissner-Nordstr\o m (RN) black hole is distinguished by its coldness (vanishing Hawking temperature) and its supersymmetry. We examine RN solutions to Einstein-Maxwell theory with a cosmological constant $\Lambda$, classifying the cold black holes and, for positive $\Lambda$, the ``lukewarm" black holes at the same temperature as the de Sitter thermal background. For negative $\Lambda$, we classify the supersymmetric solutions within the context of $N=2$ gauged supergravity. One finds supersymmetric analogues of flat-space extreme RN black holes, which for nonzero $\Lambda$ differ from the cold black holes. In addition, there is an exotic class of supersymmetric solutions which cannot be continued to flat space, since the magnetic charge becomes infinite in that limit. 
  The Weingarten lattice gauge model of Nambu-Goto strings is generalised to allow for fluctuations of an intrinsic worldsheet metric through a dynamical quadrilation. The continuum limit is taken for $c\leq1$ matter, reproducing the results of hermitian matrix models to all orders in the genus expansion. For the compact $c=1$ case the vortices are Wilson lines, whose exclusion leads to the theory of non-interacting fermions. As a by-product of the analysis one finds the critical behaviour of SOS and vertex models coupled to 2D quantum gravity. 
  The extension structure of the 2-dimensional current algebra of non-linear sigma models is analysed by introducing Kostant Sternberg $(L,M)$ systems. It is found that the algebra obeys a two step extension by abelian ideals. The second step is a non-split extension of a representation of the quotient of the algebra by the first step of the extension. The cocycle which appears is analysed. 
  Based on a path integral prescription for anomaly calculation, we analyze an effective theory of the two-dimensional $N=2$ supergravity, i.e., $N=2$ super-Liouville theory. We calculate the anomalies associated with the BRST supercurrent and the ghost number supercurrent. From those expressions of anomalies, we construct covariant BRST and ghost number supercurrents in the effective theory. We then show that the (super-)coordinate BRST current algebra forms a superfield extension of the topological conformal algebra for an {\it arbitrary\/} type of conformal matter or, in terms of the string theory, for an arbitrary number of space-time dimensions. This fact is very contrast with $N=0$ and $N=1$ (super-)Liouville theory, where the topological algebra singles out a particular value of dimensions. Our observation suggests a topological nature of the two-dimensional $N=2$ supergravity as a quantum theory. 
  A generalisation of the non--perturbatively stable solutions of string equations which respect the KdV flows, obtained recently for the $(2m-1,2)$ conformal minimal models coupled to two--dimensional quantum gravity, is presented for the $(p,q)$ models. These string equations are the most general string equations compatible with the $q$--th generalised KdV flows. They exhibit a close relationship with the bi-hamiltonian structure in these hierarchies. The Ising model is studied as a particular example, for which a real non-singular numerical solution to the string susceptibility is presented. 
  A variation on the abelian Higgs model, with global SU(2) x local U(1) symmetry broken to global U(1) was recently shown by Vachaspati and Achucarro to admit stable, finite energy cosmic string solutions even though the manifold of minima of the potential energy does not have non-contractible loops. Here the most general solutions, both in the single and multi-vortex cases, are described in the Bogomol'nyi limit. The gravitational field of the vortices considered as cosmic strings is obtained and monopole-like solutions surrounded by an event horizon are found. 
  We discuss new realisations of $W$ algebras in which the currents are expressed in terms of two arbitrary commuting energy-momentum tensors together with a set of free scalar fields. This contrasts with the previously-known realisations, which involve only one energy-momentum tensor. Since realisations of non-linear algebras are not easy to come by, the fact that this new class exists is of intrinsic interest. We use these new realisations to build the corresponding $W$-string theories and show that they are effectively described by two independent ordinary Virasoro-like strings. 
  We present an extensive search for a general class of flipped $SU(5)$ models built within the free fermionic formulation of the heterotic string. We describe a set of algorithms which constitute the basis for a computer program capable of generating systematically the massless spectrum and the superpotential of all possible models within the class we consider. Our search through the huge parameter space to be explored is simplified considerably by the constraint of $N=1$ spacetime supersymmetry and the need for extra $Q,\bar Q$ representations beyond the standard ones in order to possibly achieve string gauge coupling unification at scales of ${\cal O}(10^{18}\GeV)$. Our results are remarkably simple and evidence the large degree of redundancy in this kind of constructions. We find one model with gauge group $SU(5)\times U(1)_\ty\times SO(10)_h\times SU(4)_h\times U(1)^5$ and fairly acceptable phenomenological properties. We study the $D$- and $F$-flatness constraints and the symmetry breaking pattern in this model and conclude that string gauge coupling unification is quite possible. 
  These lecture notes give an introductory account of an approach to cohomological field theory due to Atiyah and Jeffrey which is based on the construction of Gaussian shaped Thom forms by Mathai and Quillen. Topics covered are: an explanation of the Mathai-Quillen formalism for finite dimensional vector bundles; the definition of regularized Euler numbers of infinite dimensional vector bundles; interpretation of supersymmetric quantum mechanics as the regularized Euler number of loop space; the Atiyah-Jeffrey interpretation of Donaldson theory; the construction of topological gauge theories from infinite dimensional vector bundles over spaces of connections. 
  The statistics-altering operators present in the limit $q=-1$ of multiparticle SU_q(2)-invariant subspaces parallel the action of such operators which naturally occur in supersymmetric theories. We illustrate this heuristically by comparison to a toy $N=2$ superymmetry algebra, and ask whether there is a supersymmetry structure underlying SU(2)_q at that limit. We remark on the relevance of such alternating-symmetry multiplets to the construction of invariant hamiltonians. 
  In this note I discuss some features of the topological theory obtained from the Zakharov-Shabat (or general sl(2,C)) hierarchy, and comment on some possible physical and/or mathematical interpretations of it. 
  We discuss the canonical quantization of Quantum Electrodynamics in $2+1$ dimensions, with a Chern-Simons topological mass term and gauge-covariant coupling to a Dirac spinor field. A gauge-fixing term is used which generates a canonical momentum for $A_0$, so that there are no primary constraints on operator-valued fields. Gauss's Law and the gauge condition, $A_0=0$, are implemented by embedding the formulation in an appropriate physical subspace, in which state vectors remain naturally, in the course of time evolution. The photon propagator is derived from the canonical theory. The electric and magnetic fields are separated into parts that reflect the presence of massive photons, and other parts that are rigidly attached to charged fermions and do not consist of any observable, propagating particle excitations. The effect of rotations on charged particle states is analyzed, and the relation between the canonical and the Belinfante ``symmetric'' angular momentum is discussed. It is shown that the rotation operator can be consistently formulated so that charged particles behave like fermions, and do not acquire any arbitrary phases during rotations, even when they are dressed in the electromagnetic fields required for them to obey Gauss's law. 
  We exhibit the multicritical phase structure of the loop gas model on a random surface. The dense phase is reconsidered, with special attention paid to the topological points $g=1/p$. This phase is complementary to the dilute and higher multicritical phases in the sense that dense models contain the same spectrum of bulk operators (found in the continuum by Lian and Zuckerman) but a different set of boundary operators. This difference illuminates the well-known $(p,q)$ asymmetry of the matrix chain models. Higher multicritical phases are constructed, generalizing both Kazakov's multicritical models as well as the known dilute phase models. They are quite likely related to multicritical polymer theories recently considered independently by Saleur and Zamolodchikov. Our results may be of help in defining such models on {\it flat} honeycomb lattices; an unsolved problem in polymer theory. The phase boundaries correspond again to ``topological'' points with $g=p/1$ integer, which we study in some detail. Two qualitatively different types of critical points are discovered for each such $g$. For the special point $g=2$ we demonstrate that the dilute phase $O(-2)$ model does {\it not} correspond to the Parisi-Sourlas model, a result likely to hold as well for the flat case. Instead it is proven that the first {\it multicritical} $O(-2)$ point possesses the Parisi-Sourlas supersymmetry.} 
  The exact black hole solution of 2D closed string theory has, as any other maximally extended Schwarzschild-like geometry, two asymptotically flat spacetime domains. One can get rid of the second domain by gauging the discrete symmetry on the SL(2,R)/U(1) coset that interchanges the two asymptotic domains and preserves the Kruskal time orientation everywhere in the Kruskal plane. Here it is shown that upon performing this orbifold procedure, we obtain a theory of unoriented open and closed strings in a black hole background, with just one asymptotically flat domain and a time-like orbifold singularity at the origin. All of the open string states of the model are confined to the orbifold singularity. We also discuss various physical aspects of the truncated black hole, in particular its target duality -- the model is dual to a conventional open string theory in the black hole geometry. 
  A formal relationship between scattering amplitudes in critical bosonic string theory and correlation functions of operators in topological string theory is found. 
  We discuss time - dependent solutions of the leading order string effective equations for a non-zero central charge deficit and curved maximally symmetric space. Some regular solutions are found for the case of non-trivial antisymmetric tensor and vector backgrounds (in various dimensions) and negative spatial curvature. It remains an open question which conformal theories are exact generalisations of these solutions. 
  A group of volume-preserving diffeomorphisms in 3D turns out to play a key role in an Einstein-Maxwell theory whose Weyl tensor is selfdual and whose Maxwell tensor has algebraically general anti-selfdual part. This model was first introduced by Flaherty and recently studied by Park as an integrable deformation of selfdual gravity. A twisted volume form on the corresponding twistor space is shown to be the origin of volume-preserving diffeomorphisms. An immediate consequence is the existence of an infinite number of symmetries as a generalization of $w_{1+\infty}$ symmetries in selfdual gravity. A possible relation to Witten's 2D string theory is pointed out. 
  We show that the anisotropic Heisenberg-Ising chains with higher spin allow, for special values of the anisotropy, integrable deformations intimately related to the theory of quantum groups at roots of unity. For the spin one case we construct and study the symmetries of the hamiltonian which depends on a spectral variable belonging to an elliptic curve. One of the points of this curve yields the Fateev-Zamolodchikov hamiltonian of spin one and anisotropy $\Delta = \frac{ q^2 + q^{-2}}{2} $ with $q$ a cubic root of unity. In some other special points the spin degrees of freedom as well as the hamiltonian splits into pieces governed by a larger symmetry. 
  A method for quantizing the bidimensional N=2 supersymmetric non-linear sigma model is developed. This method is both covariant under coordinate transformations (concerning the order relevant for calculations) and explicitly N=2 supersymmetric. The OPE of the supercurrent is computed accordingly, including also the dilaton. By imposing the N=2 superconformal algebra the equations for the metric and dilaton are obtained. In particular, they imply that the dilaton is a constant. 
  R-matrices for the semicyclic representations of U_qsl^(2) are found as a limit in the checkerboard chiral Potts model. 
  We present an invariant regularisation scheme to compute two dimensional induced gauge theory actions, that is local in Polyakov's variables, but nonlocal in the original gauge potentials. Our method sheds light on the locality of this induced action, and leads to a straightforward proof that the $\varepsilon$-anomaly in $W_3$-gravity is completely given by the one loop term. 
  NON-ABELIAN TODA THEORIES are shown to provide EXACTLY SOLVABLE conformal systems in the presence of a BLACK HOLE which may be regarded as describing a string propagating in target space with a black-hole metric. These theories are associated with non-canonical $\bf Z$-gradations of simple algebras, where the gradation-zero subgroup is non-abelian. They correspond to gauged WZNW models where the gauge group is nilpotent and are thus basically different from the ones currently considered following Witten. The non-abelian Toda potential gives a cosmological term which may be exactly integrated at the classical level. 
  Solutions to both the diffeomorphism and the hamiltonian constraint of quantum gravity have been found in the loop representation, which is based on Ashtekar's new variables. While the diffeomorphism constraint is easily solved by considering loop functionals which are knot invariants, there remains the puzzle why several of the known knot invariants are also solutions to the hamiltonian constraint. We show how the Jones polynomial gives rise to an infinite set of solutions to all the constraints of quantum gravity thereby illuminating the structure of the space of solutions and suggesting the existance of a deep connection between quantum gravity and knot theory at a dynamical level. 
  Dynamical systems of a new kind are described, which are motivated by the problem of constructing diffeomorphism invariant quantum theories. These are based on the extremization of a non-local and non-additive quantity that we call the variety of a system. In these systems all dynaqmical variables refer to relative coordinates or, more generally, describe relations between particles, so that they are invariant under discrete analogues of diffeomorphisms in which the labels of all particles are permutted arbitrarily. The variety is a measures of how uniquely each of the elements of the system can be distinguished from the others in terms of the values of these relative coordinates. Thus a system with extremal variety is one in which the parts are related to the whole in as distinct a way as possible.    We study numerically several dynamical systems which are defined by setting the action of the system equal to its variety. We find evidence that suggests that such systems may serve as the basis for a new kind of pregeometry theories in which the geometry of low dimensional space emerges in the thermodynamic limit from a system which is defined without the use of any background space.    The mathematical definition of variety may also provide a quantitative tool to study self-organizing systems, because it distinguishes highly structured, but asymmetric, configurations such as one finds in biological systems from both random configurations and highly ordered configurations. 
  Static solutions of large-$N$ quantum dilaton gravity in $1+1$ dimensions are analyzed and found to exhibit some unusual behavior. As expected from previous work, infinite-mass solutions are found describing a black hole in equilibrium with a bath of Hawking radiation. Surprisingly, the finite mass solutions are found to approach zero coupling both at the horizon and spatial infinity, with a ``bounce'' off of strong coupling in between. Several new zero mass solutions -- candidate quantum vacua -- are also described. 
  We represent the partition function of the Generalized Kontsevich Model (GKM) in the form of a Toda lattice $\tau$-function and discuss various implications of non-vanishing "negative"- and "zero"-time variables: the appear to modify the original GKM action by negative-power and logarithmic contributions respectively. It is shown that so deformed $\tau$-function satisfies the same string equation as the original one. In the case of quadratic potential GKM turns out to describe {\it forced} Toda chain hierarchy and, thus, corresponds to a {\it discrete} matrix model, with the role of the matrix size played by the zero-time (at integer positive points). This relation allows one to discuss the double-scaling continuum limit entirely in terms of GKM, $i.e.$ essentially in terms of {\it finite}-fold integrals. 
  We propose to study a generalization of the Klebanov-Polyakov-Witten (KPW) construction for the algebra of observables in the $c = 1$ string model to theories with $c > 1$. We emphasize the algebraic meaning of the KPW construction for $c = 1$ related to occurrence of a {\it model} of {\it SU}(2) as original structure on the algebra of observables. The attempts to preserve this structure in generalizations naturally leads to consideration of $W$-gravities. As a first step in the study of these generalized KPW constructions we design explicitly the subsector of the space of observables in appropriate $W_G$-string theory, which forms the {\it model} of $G$ for any simply laced {\it G}. The {\it model} structure is confirmed by the fact that corresponding one-loop Kac-Rocha-Caridi $W_G$-characters for $c = r_G$ sum into a chiral (open string) $k=1$ $G$-WZW partition function.  
  A list of superconformal chiral operator product expansion algebras with quadratic nonlinearity in two dimensions is completed on the basis of the known classification of little conformal Lie superalgebras. In addition to the previously known cases and the constructed in our previous paper exceptional $N=8$ superalgebra associated with $F(4)$, a novel exceptional $N=7$ superconformal algebra associated with $G(3)$ is found, as well as a whole family of superalgebras containing affine $\widehat{su}_2 \oplus \widehat{usp}_{2N}$. A classification scheme for quasisuperconformal algebras is also outlined. 
  We analyse the fusion, braiding and scattering properties of discrete non-abelian anyons. These occur in (2+1)-dimensional theories where a gauge group G is spontaneously broken down to some discrete subgroup H. We identify the quantumnumbers of the electrically and magnetically charged sectors of the remaining discrete gauge theory, and show that on the quantum level the symmetry group H is extended to the (quasi-triangular) Hopf algebra D(H). Most of our considerations are relevant for discrete gauge theories in (3+1)-dimensional space time as well. 
  We study the effect of a Chern-Simons term in a theory with discrete gauge group H, which in (2+1)-dimensional space time describes (non-abelian) anyons. As in a previous paper, we emphasize the underlying algebraic structure, namely the Hopf algebra D(H). We argue on physical grounds that the addition of a Chern-Simons term in the action leads to a non-trivial 3-cocycle on D(H). Accordingly, the physically inequivalent models are labelled by the elements of the cohomology group H^3(H,U(1)). It depends periodically on the coefficient of the Chern-Simons term which model is realized. This establishes a relation with the discrete topological field theories of Dijkgraaf and Witten. Some representative examples are worked out explicitly. 
  Inhomogeneous quantum groups are shown to be an effective algebraic tool in the study of integrable systems and to provide solutions equivalent to the Bethe ansatz. The method is illustrated on the 1D Heisenberg ferromagnet whose symmetry is shown to be the quantum Galilei group Gamma_q(1) here introduced. Both the single magnon and the s=1/2 bound states of n-magnons are completely described by the algebra. 
  The path-integral measure of linearized gravity around a saddle-point background with the cosmological term is considered in order to study the conformal rotation prescription proposed by Gibbons, Hawking and Perry. It is also argued that the most generally used measure, i.e., the covariant path-integral measure, does not give us a one-loop partition function which the only physical variables contribute and that its path integral fails to keep the cancellation of contributions between the Faddeev-Popov ghosts and the unphysical variables of the linearized gravitational field, although it has a coordinate invariant measure. In de~Sitter spacetime, it is shown that the uncancellation factor can be understood as a nontrivial (anomalous) Jacobian factor under the transformation of the path-integral measure from covariant one to canonical one. 
  We investigate the causal structure of $(1+1)$-dimensional spacetimes. For two sets of field equations we show that at least locally any spacetime is a solution for an appropriate choice of the matter fields. For the theories under consideration we investigate how smoothness of their black hole solutions affects time orientation. We show that if an analog to Hawking's area theorem holds in two spacetime dimensions, it must actually state that the size of a black hole never {\em increases}, contrary to what happens in four dimensions. Finally, we discuss the applicability of the Penrose and Hawking singularity theorems to two spacetime dimensions. 
  We propose a new formulation of the $D=10$ Brink-Schwarz superparticle which is manifestly invariant under both the target-space super-Poincar\'e group and the world-line local $N=8$ superconformal group. This twistor-like construction naturally involves the sphere $S^8$ as a coset space of the $D=10$ Lorentz group. The action contains only a finite set of auxiliary fields, but they appear in unusual trilinear combinations. The origin of the on-shell $D=10$ fermionic $\kappa$ symmetry of the standard Brink-Schwarz formulation is explained. The coupling to a $D=10$ super-Maxwell background requires a new mechanism, in which the electric charge appears only on shell as an integration constant. 
  Callan, Giddings, Harvey and Strominger have proposed an interesting two dimensional model theory that allows one to consider black hole evaporation in the semi-classical approximation. They originally hoped the black hole would evaporate completely without a singularity. However, it has been shown that the semi-classical equations will give a singularity where the dilaton field reaches a certain critical value. Initially, it seems this singularity will be hidden inside a black hole. However, as the evaporation proceeds, the dilaton field on the horizon will approach the critical value but the temperature and rate of emission will remain finite. These results indicate either that there is a naked singularity, or (more likely) that the semi-classical approximation breaks down when the dilaton field approaches the critical value. 
  We provide a non-perturbative geometrical characterization of the partition function of $n$-dimensional quantum gravity based on a coarse classification of riemannian geometries. We show that, under natural geometrical constraints, the theory admits a continuum limit with a non-trivial phase structure parametrized by the homotopy types of the class of manifolds considered. The results obtained qualitatively coincide, when specialized to dimension two, with those of two-dimensional quantum gravity models based on random triangulations of surfaces. 
  The puzzles of black hole evaporation can be studied in the simplified context of 1+1 dimensional gravity. The semi-classical equations of Callan, Giddings, Harvey and Strominger provide a consistent description of the evaporation process which we describe in detail. We consider the possibility that black hole evolution leads to massive stable remnants. We show that such zero temperature remnant solutions exist but we also prove that a decaying black hole cannot evolve into one of them. Finally we consider the issue of loss of quantum information behind the global event horizon which develops in these geometries. An analogy with a well known solvable system shows that there may be less to information than meets the eye. 
  We show that Euclidean 3D-gravity coupled to a Gaussian scalar massive matter field in first-order dreibein formalism gives a quantum theory which has a finite perturbative expansion around a non-vanishing background. We also discuss a possible mechanism to generate a non-trivial background metric starting from Rovelli-Smolin's loop observables. 
  It is shown that the currently studied ``string-inspired'' model for gravity on a line can be formulated as a gauge invariant theory based on the Poincar\'e group with central extension -- a formulation that complements and simplifies H.~Verlinde's construction based on the unextended Poincar\'e group. 
  Considerable interest has recently been expressed in (static spherically symmetric) blackholes in interaction with various classical matter fields (such as electromagnetic fields, dilaton fields, axion fields, Abelian Higgs fields, non--Abelian gauge fields, {\sl etc}). A common feature of these investigations that has not previously been remarked upon is that the Hawking temperature of such systems appears to be suppressed relative to that of a vacuum blackhole of equal horizon area. That is: $k T_H \leq \hbar/(4\pi r_H) \equiv \hbar/\sqrt{4\pi A_H}$. This paper will argue that this suppression is generic. Specifically, it will be shown that  \[ k T_H = {\hbar\over4\pi r_H} \; e^{-\phi(r_H)} \;                \left( 1 - 8\pi G \; \rho_H \; r_H^2 \right). \]  Here $\phi(r_H)$ is an integral quantity, depending on the distribution of matter, that is guaranteed to be positive if the Weak Energy Condition is satisfied. Several examples of this behaviour will be discussed. Generalizations of this behaviour to non--symmetric non--static blackholes are conjectured. 
  Green and Seiberg showed that, in simple treatments of fermionic string theory, it is necessary to introduce contact interactions when vertex operators collide. Otherwise, certain superconformal Ward identities would be violated. In this note, we show how these contact terms arise naturally when proper account is taken of the superconformal geometry involved when punctures collide. More precisely, we show that there is no contact term at all! Rather, corrections arise to the ``na\"\i ve" formula when the boundary of moduli space is described correctly. 
  This paper revisits the conundrum faced when one attempts to understand the dynamics of black hole formation and evaporation without abandoning unitary evolution. Previous efforts to resolve this puzzle assume that information escapes in corrections to the Hawking process, that an arbitrarily large amount of information is transmitted by a planckian energy or contained in a Planck-sized remnant, or that the information is lost to another universe. Each of these possibilities has serious difficulties. This paper considers another alternative: remnants that carry large amounts of information and whose size and mass depend on their information content. The existence of such objects is suggested by attempts to incorporate a Planck scale cutoff into physics. They would greatly alter the late stages of the evaporation process. The main drawback of this scenario is apparent acausal behavior behind the horizon. 
  We find the general solution to Polchinski's classical scattering equations for $1+1$ dimensional string theory. This allows efficient computation of scattering amplitudes in the standard Liouville $\times$ $c=1$ background. Moreover, the solution leads to a mapping from a large class of time-dependent collective field theory backgrounds to corresponding nonlinear sigma models. Finally, we derive recursion relations between tachyon amplitudes. These may be summarized by an infinite set of nonlinear PDE's for the partition function in an arbitrary time-dependent background. 
  We consider the Sine-Gordon model coupled to 2D gravity. We find a nonperturbative expression for the partition function as a function of the cosmological constant, the SG mass and the SG coupling constant. At genus zero, the partition function exhibits singularities which are interpreted as signals of phase transitions. A semiclassical picture of one of these transitions is proposed. According to this picture, a phase in which the Sine-Gordon field and the geometry are frozen melts into another phase in which the fields and geometry become dynamical. 
  Given the two boson representation of the conformal algebra \hat W_\infty, the second Hamiltonian structure of the KP hierarchy, I construct a bi-Hamiltonian hierarchy for the two associated currents. The KP hierarchy appears as a composite of this new and simpler system. The bi-Hamiltonian structure of the new hierarchy gives naturally all the Hamiltonian structures of the KP system. 
  We study string theory in the background of a two-dimensional black hole which is described by an $SL(2, R)/U(1)$ coset conformal field theory. We determine the spectrum of this conformal field theory using supersymmetric quantum mechanics and give an explicit form of the vertex operators in terms of the Jacobi functions. We also discuss the applicability of SUSY quantum mechanics techniques to non-linear $\sigma$-models. 
  We present a new approach to the calculation of thermodynamic functions for crossing-invariant models solvable by Bethe Ansatz. In the case of the XXZ Heisemberg chain we derive, for arbitrary values of the anysotropy, a {\bf single} non--linear integral equation from which the free energy can be exactly calculated. The high--temperature expansion follows in a sistematic and relatively simple way. For low temperatures we obtain the correct central charge and predict the analytic structure of the full expansion around $T=0$. Furthermore, we derive a single non-linear integral equation describing the finite--size ground--state energy of the Sine--Gordon quantum field theory. PACS: 05.30, 03.70. 75.10.5 
  We solve the RSOS($p$) models on the light--cone lattice with fixed boundary conditions by disentangling the type II representations of $SU(2)_q$, at $q=e^{i\pi/p}$, from the full SOS spectrum obtained through Algebraic Bethe Ansatz. The rule which realizes the quantum group reduction to the RSOS states is that there must not be {\it singular} roots in the solutions of the Bethe Ansatz equations describing the states with quantum spin $J<(p-1)/2$. By studying how this rule is active on the particle states, we are able to give a microscopic derivation of the lattice $S-$matrix of the massive kinks. The correspondence between the light--cone Six--Vertex model and the Sine--Gordon field theory implies that the continuum limit of the RSOS($p+1$) model is to be identified with the $p-$restricted Sine--Gordon field theory. 
  We study some aspects of 2d supersymmetric sigma models on orbifolds. It turns out that independently of whether the 2d QFT is conformal the operator products of twist operators are non-singular, suggesting that massive (non-conformal) orbifolds also `resolve singularities' just as in the conformal case. Moreover we recover the OPE of twist operators for conformal theories by considering the UV limit of the massive orbifold correlation functions. Alternatively, we can use the OPE of twist fields at the conformal point to derive conditions for the existence of non-singular solutions to special non-linear differential equations (such as Painleve III). 
  We are able to show that BF theories naturally emerge from the coadjoint orbits of $W_2$ and $w_\infty$ algebras which includes a Kac-Moody sector. Since QCD strings can be identified with a BF theory, we are able to show a relationship between the orbits and monopole-string solutions of QCD. Furthermore, we observe that when 4D gravitation is cast into a BF form through the use of Ashtekar variables, we are able to get order $\hbar$ contributions to gravity which can be associated with the $W_2$ anomaly. We comment on the relationship to gravitational monopoles. 
  This manuscripts corrects some minor error in the paper, Mod. Phys. Lett. A 6 1893 (1991) 
  We carry on the study of the Alexander Conway invariant from the quantum field  theory point of view started in \cite{RS91}.   We first discuss in details $S$ and $T$ matrices for the $U(1,1)$ super WZW model and obtain, for the level $k$ an integer, new finite dimensional representations of the modular group. These have the remarkable property that some of the $S$ matrix elements are infinite. Moreover, typical and  atypical representations as well as indecomposable blocks are mixed: truncation to maximally atypical representations, as advocated  in some recent papers, is not consistent.   The main topological application of this work is the computation of Alexander invariants for 3-manifolds and for links in 3-manifolds. Invariants of 3-manifolds seem to depend trivially on the level $k$, but still contain interesting topological information. For Seifert manifolds for instance, they  coincide with the order  of the first homology group. Examples of invariants of links in 3-manifolds are given. They exhibit interesting arithmetic properties. 
  Power-counting arguments based on extended superfields have been used to argue that two-dimensional supersymmetric sigma models with (4,0) supersymmetry are finite. This result is confirmed up to three loop order in pertubation theory by an explicit calculation using (1,0) superfields. In particular, it is shown that the finite counterterms which must be introduced into the theory in order to maintain (4,0) supersymmetry are precisely the terms that are required to establish ultra-violet finiteness. 
  We study moduli dependent threshold corrections to gravitational couplings in the case of the heterotic string compactified on a symmetric orbifold, for untwisted moduli, extending previous analysis on gauge couplings. Like in the gauge case, the contribution comes entirely from the spacetime $N=2$ sector. As a byproduct, this calculation provides a simple derivation of the trace anomaly coefficients for the different fields coupled to gravity. 
  We consider the midisuperspace of four dimensional spherically symmetric metrics and the Kantowski-Sachs minisuperspace contained in it. We discuss the quantization of the midisuperspace using the fact that the dimensionally reduced Einstein Hilbert action becomes a scalar-tensor theory of gravity in two dimensions. We show that the covariant regularization procedure in the midisuperspace induces modifications into the minisuperspace Wheeler DeWitt equation. 
  A quantum group analysis is applied to the Sine-Gordon model (or may be its version) in a strong-coupling regime. Infinitely many bound states are found together with the corresponding S-matrices. These new solutions of the Yang-Baxter eqations are related to some reducible representations of the quantum $sl(2)$ algebra resembling the Kac-Moody algebra representations in the Wess-Zumino-Witten-Novikov conformal field theory. 
  We make a change of field variables in the J formulation of self-dual Yang--Mills theory. The field equations for the resulting algebra valued field are derivable from a simple cubic action. The cubic interaction vertex is different from that considered previously from the N=2 string, however, perturbation theory with this action shows that the only non-vanishing connected scattering amplitude is for three external particles just as for the string. 
  We derive the current algebra of principal chiral models with a Wess-Zumino term. At the critical coupling where the model becomes conformally invariant (Wess-Zumino-Novikov-Witten theory), this algebra reduces to two commuting Kac-Moody algebras, while in the limit where the coupling constant is taken to zero (ordinary chiral model), we recover the current algebra of that model. In this way, the latter is explicitly realized as a deformation of the former, with the coupling constant as the deformation parameter. 
  The problem of quantizing a class of two-dimensional integrable quantum field theories is considered. The classical equations of the theory are the complex $sl(n)$ affine Toda equations which admit soliton solutions with real masses. The classical scattering theory of the solitons is developed using Hirota's solution techniques. A form for the soliton $S$-matrix is proposed based on the constraints of $S$-matrix theory, integrability and the requirement that the semi-classical limit is consistent with the semi-classical WKB quantization of the classical scattering theory. The proposed $S$-matrix is an intertwiner of the quantum group associated to $sl(n)$, where the deformation parameter is a function of the coupling constant. It is further shown that the $S$-matrix describes a non-unitary theory, which reflects the fact that the classical Hamiltonian is complex. The spectrum of the theory is found to consist of the basic solitons, scalar states (or breathers) and excited (or `breathing') solitons. It is also noted that the construction of the $S$-matrix is valid for any representation of the Hecke algebra, allowing the definition of restricted $S$-matrices, in which case the theory is unitary. 
  Finite versions of W-algebras are introduced by considering (symplectic) reductions of finite dimensional simple Lie algebras. In particular a finite analogue of $W^{(2)}_3$ is introduced and studied in detail. Its unitary and non-unitary, reducible and irreducible highest weight representations are constructed. 
  The $N=2$ supersymmetric {\it self-dual} Yang-Mills theory and the $N=4$ and $N=2$ {\it self-dual} supergravities in $2+2$ space-time dimensions are formulated for the first time. These formulations utilize solutions of the Bianchi identities subject to the super-Yang-Mills or supergravity constraints in the relevant $N$-extended superspace with the space-time signature $(2,2)$. 
  Results that illuminate the physical interpretation of states of nonperturbative quantum gravity are obtained using the recently introduced loop variables. It is shown that: i) While local operators such as the metric at a point may not be well-defined, there do exist {\it non-local} operators, such as the area of a given 2-surface, which can be regulated diffeomorphism invariantly and which are finite {\it without} renormalization; ii)there exist quantum states which approximate a given flat geometry at large scales, but such states exhibit a discrete structure at the Planck scale; iii) these results are tied together by the fact that the spectra of the operators that measure the areas of surfaces are quantized in integral units of the Planck area. 
  We present the Green-Schwarz $\s\-$model coupled to the $N=1$ {\it {supersymmetric}} Yang-Mills and supergravity in a four-dimensional space-time with the indefinite signature $(+,+,-,-)$. We first confirm the $\k\-$invariance of the Green-Schwarz action, and show that all the $\b\-$functions for the backgrounds vanish consistently after the use of their superfield equations. Subsequently, we inspect the supersymmetric {\it self-duality} conditions, that have been developed in our previous paper on the Yang-Mills and supergravity backgrounds. Remarkably, the Majorana-Weyl spinor dictating the {\it supersymmetric self-duality} conditions is consistent with the couplings of Green-Schwarz superstring. Such Green-Schwarz superstring is supposed to be the underlying theory of the {\it supersymmetric self-dual} Yang-Mills theory, which is conjectured to generate {\it all} exactly soluble supersymmetric systems in lower dimensions. 
  The properties of Dirac gamma matrices in a four-dimensional space-time with the $(2,2)$ signature are studied. The basic spinors are classified, and the existence of Majorana-Weyl spinors is noted. Supersymmetry in $2 + 2$ dimensions is discussed, and the existence of the {\it real chiral} scalar supermultiplet is discovered. Supersymmetric {\it self-dual} Yang-Mills theories and {\it self-dual} supergravity model in $2 + 2$ dimensions, that are apparently relevant to integrable systems, are formulated for the first time. 
  Various approaches to high energy forward scattering in quantum gravity are compared using the eikonal approximation. The massless limit of the eikonal is shown to be equivalent to other approximations for the same process, specifically the semiclassical calculation due to G. 't Hooft and the topological field theory due to H. and E. Verlinde. This comparison clarifies these previous results, as it is seen that the amplitude arises purely from a linearised gravitational interaction. The interpretation of poles in the scattering amplitude is also clarified. 
  It is shown that an arbitrarily small amount of angular momentum can qualitatively change the properties of extremal charged black holes coupled to a dilaton. In addition, the gyromagnetic ratio of these black holes is computed and an exact rotating black string solution is presented. 
  We derive the period structure of several one-modulus Calabi-Yau manifolds. With this knowledge we then obtain the generators of the duality group and the mirror map that defines the physical variable $t$ representing the radius of compactification. We also describe the fundamental region of $t$ and discuss its relation with automorphic functions. As a byproduct of our analysis we compute the non-perturbative corrections of Yukawa couplings. 
  A metal ring removed from a soap-water solution encloses a film of soap which can be mathematically described as a minimal surface having the ring as its only boundary. This is known to everybody. In this letter we suggest a relativistic extension of the above fluidodynamic system where the soap film is replaced by a Kalb-Ramond gauge potential $\b(x)$ and the ring by a closed string. The interaction between the $\b$-field and the string current excites a new configuration of the system consisting of a relativistic membrane bounded by the string. We call such a classical solution of the equation of motion an axionic membrane. As a dynamical system, the axionic membrane admits a Hamilton-Jacobi formulation which is an extension of the H-J theory of electromagnetic strings. 
  The two-dimensional theory of gravity describing a graviton-dilaton system is considered. The graviton-dilaton coupling can be fixed such that the quantum theory remains free of the conformal anomaly for any conformal dimension of the coupled matter system, even if the dilaton does not appear as Lagrange multiplier. Interaction terms are introduced and the system is analyzed and solutions are given at the classical level and at the quantum level by using canonical quantization. 
  We find the rules which count the energy levels of the 3 state superintegrable chiral Potts model and demonstrate that these rules are complete. We then derive the complete spectrum of excitations in the thermodynamic limit in the massive phase and demonstrate the existence of excitations which do not have a quasi-particle form. The physics of these excitations is compared with the BCS superconductivity spectrum and the counting rules are compared with the closely related $S=1$ XXZ spin chain. 
  The conformal symmetry of the QCD Lagrangian for massless quarks is broken both by renormalization effects and the gauge fixing procedure. Renormalized primitive divergent amplitudes have the property that their form away from the overall coincident point singularity is fully determined by the bare Lagrangian, and scale dependence is restricted to $\delta$-functions at the singularity. If gauge fixing could be ignored, one would expect these amplitudes to be conformal invariant for non-coincident points. We find that the one-loop three-gluon vertex function $\Gamma_{\mu\nu\rho}(x,y,z)$ is conformal invariant in this sense, if calculated in the background field formalism using the Feynman gauge for internal gluons. It is not yet clear why the expected breaking due to gauge fixing is absent. The conformal property implies that the gluon, ghost and quark loop contributions to $\Gamma_{\mu\nu\rho}$ are each purely numerical combinations of two universal conformal tensors $D_{\mu\nu\rho}(x,y,z)$ and $C_{\mu\nu\rho}(x,y,z)$ whose explicit form is given in the text. Only $D_{\mu\nu\rho}$ has an ultraviolet divergence, although $C_{\mu\nu\rho}$ requires a careful definition to resolve the expected ambiguity of a formally linearly divergent quantity. Regularization is straightforward and leads to a renormalized vertex function which satisfies the required Ward identity, and from which the beta-function is easily obtained. Exact conformal invariance is broken in higher-loop orders, but we outline a speculative scenario in which the perturbative structure of the vertex function is determined from a conformal invariant primitive core by interplay of the renormalization group equation and Ward identities. 
  We redo the quantization of the N=4 string, taking into account the reducibility of the constraints. The result is equivalent to the N=2 string, with critical dimension D=4 and signature (++--). The N=4 formulation has several advantages: the sigma-model field equations are implied classically, rather than by quantum/beta-function calculations; self-duality/chirality is one of the super-Virasoro constraints; SO(2,2) covariance is manifest. This reveals that the theory includes fermions, and is apparently spacetime supersymmetric. 
  A brief overview of strings propagating on noncompact coset spaces G/H is presented in terms of WZW models. The role played by isometries in the existence of target space duality and by fixed points of the gauge transformations in the existence of singularities and horizons, is emphasized. A general classification of the spaces with a single time-like coordinate is presented. The spacetime geometry of a class of models, existing for every dimension and having cosmological and black hole-like interpretations, is discussed. 
  An ``anomalous'' supersymmetry transformation of the gaugino axial current is given in supersymmetric Yang-Mills theory. The contact term is computed to one-loop order by a gauge-invariant point-splitting procedure. We reexamine the supercurrent anomaly in this method. 
  It is shown how to couple non-relativistic matter with a Chern--Simons gauge field that belongs to a non-compact group. We treat in some details the $SL(2,{\bf R})$ and the Poincar\'e $ISO(2,1)$ groups. For suitable self-interactions, we are able to exhibit soliton solutions. 
  We show that 2+1-dimensional Euclidean quantum gravity is equivalent, under some mild topological assumptions, to a Gaussian fermionic system. In particular, for manifolds topologically equivalent to $\Sigma_g\times\RrR$ with $\Sigma_g$ a closed and oriented Riemann surface of genus $g$, the corresponding 2+1-dimensional Euclidean quantum gravity may be related to the 3D-lattice Ising model before its thermodynamic limit.  
  A method for finding Berry's phase is proposed under the Euclidean path integral formalism. It is characterized by picking up the imaginary part from the resultant exponent. Discussion is made on the generalized harmonic oscillator which is shown being so universal in a single degree case. The spin model expressed by creation and annihilation operators is also discussed. A systematic way of expansion in the adiabatic approximation is presented in every example. 
  The background for string propagation is obtained by a chiral gauging of the $SL(2,R)$ Wess-Zumino-Witten model. It is shown explicitly that the resulting background fields satisfy the field equations of the three dimensional string effective action and the target space has curvature singularity. Close connection of our solution with the three dimensional black string is demonstrated.  
  We study in a systematic and modular invariant way gaugino condensation in the hidden sector as a potential source of hierarchical supersymmetry breaking and a non--trivial potential for the dilaton $S$ whose real part corresponds to the tree level gauge coupling constant (${\rm Re}\ S\sim g_{gut}^{-2}$). For the case of pure Yang--Mills condensation, we show that no realistic results (in particular no reasonable values for ${\rm Re}\ S$) can emerge, even if the hidden gauge group is not simple. However, in the presence of hidden matter (i.e. the most frequent case) there arises a very interesting class of scenarios with two or more hidden condensing groups for which the dilaton dynamically acquires a reasonable value (${\rm Re}\ S\sim 2$) and supersymmetry is broken at the correct scale ($m_{3/2}\sim 10^3\ GeV$) with no need of fine--tuning. Actually, good values for ${\rm Re}\ S$ and $m_{3/2}$ are correlated. We make an exhaustive classification of the working possibilities. Remarkably, the results are basically independent from the value of $\delta^{GS}$ (the contributions from the Green--Schwarz mechanism). The radius of the compactified space also acquires an expectation value, breaking duality spontaneously. 
  We propose a random matrix model as a representation for $D=1$ open strings. We show that the model is equivalent to $N$ fermions with spin in a central potential that also includes a long-range ferromagnetic interaction between the fermions that falls off as $1/(r_{ij})^2$. We find two interesting scaling limits and calculate the free energy for both situations. One limit corresponds to Dirichlet boundary conditions for the dual graphs and the other corresponds to Neumann conditions. We compute the boundary cosmological constant and show that it is of order $1/\log(\beta)$. We also briefly discuss a possible analog of the Das-Jevicki field for the open string tachyon. (n.b. This is a revised version of paper previously submitted to xxx@lanl.gov. The original version misidentified the Dirichlet and Neumann cases. This version also includes references to work by Yang that was missing in the original.) 
  Beginning with the work of Dirac and Arnowitt, Deser, Misner in the late fifties and early sixties, and then after subsequent development by Kucha\v r, the canonical dynamical structure of general relativity has often been viewed as that of a parametrized field theory in which the many-fingered spacetime variables are hidden amongst the geometrodynamical field variables. This paradigm of general relativity as an ``already parametrized theory'' forms the basis for one of the most satisfactory resolutions of the problems of time and observables in classical and quantum gravity. However, despite decades of effort, no identification of many-fingered spacetime variables has ever been satisfactorily obtained for vacuum general relativity. We point out that there is an obstruction to identifying the constraint surface of general relativity (for the case of a closed universe) with that of any parametrized theory. Therefore, strictly speaking, general relativity cannot be viewed as a parametrized field theory. We discuss implications for the canonical quantization program. 
  Matter is coupled to three-dimensional gravity such that the topological phase is allowed and the (anti-) de Sitter or Poincar\'e symmetry remains intact. Spontaneous symmetry breaking to the Lorentz group occurs if a scalar field is included. This Higgs field can then be used to couple matter so that the familiar form of the matter coupling is established in the broken phase. We also give the supersymmetrization of this construction. 
  We deduce the $sl_{3}$ Toda realization of classical $W_3$ symmetry on two scalar fields in a geometric way, proceeding from a nonlinear realization of some associate higher-spin symmetry $W_{3}^{\infty}$. The Toda equations are recognized as the constraints singling out a two-dimensional fully geodesic subspace in the initial coset space of $W_{3}^{\infty}$. The proposed geometric approach can be extended to other nonlinear algebras and integrable systems. 
  We show that the surface roughness for $c<1$ matter theories coupled to $2D$ quantum gravity is described by a self-similar structure of baby universes. There exist baby universes whose neck thickness is of the order of the ultraviolet cutoff, the largest of these having a macroscopic area $\sim A^{1 \over {1-\gamma}}$, where $A$ is the total area and $\gamma$ the string susceptibility exponent. 
  In the usual matrix-model approach to random discretized two-dimensional manifolds, one introduces n Ising spins on each cell, i.e. a discrete version of 2D quantum gravity coupled to matter with a central charge n/2. The matrix-model consists then of an integral over $2^{n}$ matrices, which we are unable to solve for $n>1$. However for a fixed genus we can expand in the cosmological constant g for arbitrary values of n, and a simple minded analysis of the series yields for n=0,1 and 2 the expected results for the exponent $\gamma_{string}$ with an amazing precision given the small number of terms that we considered. We then proceed to larger values of n. Simple tests of universality are successfully applied; for instance we obtain the same exponents for n=3 or for one Ising model coupled to a one dimensional target space. The calculations are easily extended to states Potts models, through an integration over $q^{n}$ matrices. We see no sign of the tachyonic instability of the theory, but we have only considered genus zero at this stage. 
  We show that there exists an alternative procedure in order to extract differential hierarchies, such as the KdV hierarchy, from one--matrix models, without taking a continuum limit. To prove this we introduce the Toda lattice and reformulate it in operator form. We then consider the reduction to the systems appropriate for one--matrix model. 
  The fusion of fields in a rational conformal field theory gives rise to a ring structure which has a very particular form. All such rings studied so far were shown to arise from some potentials. In this paper the fusion rings of the WZW models based on the symplectic group are studied. It is shown that they indeed arise from potentials which are described. These potentials give rise to new massive perturbations of superconformal hermitian symmetric models. The metric of the perturbation is computed and is shown to be given by solutions of the sinh--gordon equation. The kink structure of the theories is described, and it is argued that these field theories are integrable. The $S$ matrices for the fusion theories are argued to be non--minimal extensions of the $G_k\times G_1/ G_{k+1}$ $S$ matrices with the adjoint perturbation, in the case of $G=SU(N)$. 
  The low energy limit of an axion field coupled to gauge fields is investigated through the behaviour of the gauge field propagator in a local vaccum angle background. The local (singular) part of the effective action for the axion field is calculated at one loop level. In the case of a timelike, linearly growing axion field, representing a massive axion, we give an asymptotic expansion of the causal propagator and we solve nonlocally for the first coefficient. We show that, for a generic axionic background, short distance propagation of the gauge fields is well defined.  
  Pure (2+1)-dimensional Einstein gravity is analysed in the Ashtekar formulation, when the spatial manifold is a torus. We have found a set of globally defined observables, forming a closed algebra. This allowed us to solve the quantum constraints, and to show that the reduced phase space of the Ashtekar formulation is greater then the corresponding space of the Witten formulation. Furthermore, we have found a globally defined time variable which satisfies all the requiriments of an extrinsic time variable in quantum gravity. 
  We present a manifestly $N=2$ supersymmetric formulation of $N=2$ super-$W_3$ algebra (its classical version) in terms of the spin 1 and spin 2 supercurrents. Two closely related types of the Feigin-Fuchs representation for these supercurrents are found: via two chiral spin $\frac{1}{2}$ superfields generating $N=2$ extended $U(1)$ Kac-Moody algebras and via two free chiral spin 0 superfields. We also construct a one-parameter family of $N=2$ super Boussinesq equations for which $N=2$ super-$W_3$ provides the second hamiltonian structure. 
  Proposals that $O(d,d)$ boosts of trivial backgrounds lead to non-trivial conformally invariant backgrounds are checked to two loop order. We find that conformal invariance can be achieved by adding simple higher order corrections to the metric and dilaton. 
  It is shown that the Dirac operator in the background of a magnetic %Reissner-Nordstr\"om black hole and a Euclidean vortex possesses normalizable zero modes in theories containing superconducting cosmic strings. One consequence of these zero modes is the presence of a fermion condensate around magnetically charged black holes which violates global quantum numbers. 
  We study a general class of two-dimensional theories of the dilaton-gravity type inspired by string theory and show that they admit charged multiple-horizon black holes. These solutions are proved to satisfy scalar no-hair theorems. 
  Picture changed operators are discussed in $N=2$ strings with space-time signature $(2,2)$. A gauge symmetry algebra is derived in a background of torus space-time $T^{2,2}$ and its simple representation on the picture changed operators is given. Simple Ward identities associated with the gauge algebra and their consequences for three and four point amplitudes of arbitrary loops are also discussed. 
  We formulate the $c=1$ matrix model as a quantum fluid and discuss its classical limit in detail, emphasizing the $\hbar$ corrections. We view the fermi fluid profiles as elements of \winf-coadjoint orbit and write down a geometric action for the classical phase space. In the specific representation of fluid profiles as `strings' the action is written in a four-dimensional form in terms of gauge fields built out of the embedding of the `string' in the phase plane. We show that the collective field action can be derived from the above action provided one restricts to quadratic fluid profiles and ignores the dynamics of their `turning points'. 
  A three dimensional generally covariant theory is described that has a 2+1 canonical decomposition in which the Hamiltonian constraint, which generates the dynamics, is absent. Physical observables for the theory are described and the classical and quantum theories are compared with ordinary 2+1 gravity. 
  We study one-loop, moduli-dependent corrections to gauge and gravitational couplings in supersymmetric vacua of the heterotic string. By exploiting their relation to the integrability condition for the associated CP-odd couplings, we derive general expressions for them, both for $(2,2)$ and $(2,0)$ models, in terms of tree level four-point functions in the internal $N=2$ superconformal theory. The $(2,2)$ case, in particular symmetric orbifolds, is discussed in detail. 
  A recent study of supersymmetric domain walls in $N=1$ supergravity theories revealed a new class of domain walls interpolating between supersymmetric vacua with different non-positive cosmological constants. We classify three classes of domain wall configurations and study the geodesic structure of the induced space-time. Motion of massive test particles in such space-times shows that these walls are always repulsive from the anti-deSitter (AdS) side, while on the Minkowski side test particles feel no force. Freely falling particles far away from a wall in an AdS vacuum experience a constant proper acceleration, \ie\ they are Rindler particles. A new coordinate system for discussing AdS space-time is presented which eliminates the use of a periodic time-like coordinate. 
  We study the stability under perturbations of a charged four dimensional stringy black hole arising from gauging a previously studied WZW model. We find that the black hole is stable only in the extremal case $Q=M$. 
  We study the Quantum Field Theory of nonrelativistic bosons coupled to a Chern--Simons gauge field at nonzero particle density. This field theory is relevant to the study of anyon superconductors in which the anyons are described as {\bf bosons} with a statistical interaction. We show that it is possible to find a mean field solution to the equations of motion for this system which has some of the features of bose condensation. The mean field solution consists of a lattice of vortices each carrying a single quantum of statistical magnetic flux. We speculate on the effects of the quantum corrections to this mean field solution. We argue that the mean field solution is only stable under quantum corrections if the Chern--Simons coefficient $N=2\pi\theta/g^2$ is an integer. Consequences for anyon superconductivity are presented. A simple explanation for the Meissner effect in this system is discussed. 
  An investigation is made of the super-Calogero model with particular emphasis on its continuum formulation and possible application in the context of supersymmetrizing the bosonic collective d=1 string field theory. 
  We investigate the system of holomorphic differential identities implied by special K\"ahlerian geometry of four-dimensional N=2 supergravity. For superstring compactifications on \cy threefolds these identities are equivalent to the Picard-Fuchs equations of algebraic geometry that are obeyed by the periods of the holomorphic three-form. For one variable they reduce to linear fourth-order equations which are characterized by classical $W$-generators; we find that the instanton corrections to the Yukawa couplings are directly related to the non-vanishing of $w_4$. We also show that the symplectic structure of special geometry can be related to the fact that the Yukawa couplings can be written as triple derivatives of some holomorphic function $F$. Moreover, we give the precise relationship of the Yukawa couplings of special geometry with three-point functions in topological field theory. 
  We give an integrable extension of the lattice models recently considered by I.Kostov in his study of strings in discrete space. These models are IRF models with spin variables living in any connected graph, the vertex model underlying these models is the Izergin-Korepin model. When the graph is taken to be a simply laced Dynkin diagram, it is conjectured that these models possess critical regimes which are the dilute phase of SOS models of ADE type. 
  The three point correlation functions with twist fields are determined for bosonic $Z_N$ orbifolds. Both the choice of the modular background (compatible with the twist) and of the (higher) twisted sectors involved are fully general. We point out a necessary restriction on the set of instantons contributing to twist field correlation functions not obtained in previous calculations. Our results show that the theory is target space duality invariant. 
  It has been known for some time that $W$ algebras can be realised in terms of an energy-momentum tensor together with additional free scalar fields. Some recent results have shown that more general realisations are also possible. In this paper, we consider a wide class of realisations that may be obtained from the Miura transformation, related to the existence of canonical subalgebras of the Lie algebras on which the $W$ algebras are based. We give explicit formulae for all realisations of this kind, and discuss their applications in $W$-string theory. 
  We suggest a model of induced gravity in which the fundamental object is a relativistic {\it membrane} minimally coupled to a background metric and to an external three index gauge potential. We compute the low energy limit of the two-loop effective action as a power expansion in the surface tension. A generalized bootstrap hypothesis is made in order to identify the physical metric and gauge field with the lowest order terms in the expansion of the vacuum average of the composite operators conjugate to the background fields. We find that the large distance behaviour of these classical fields is described by the Einstein action with a cosmological term plus a Maxwell type action for the gauge potential. The Maxwell term enables us to apply the Hawking-Baum argument to show that the physical cosmological constant is ``~probably~'' zero. 
  We describe the duality group $\Gamma=SU(3,3,Z)$ for the Narain lattice of the $T^6/Z_3$ orbifold and its action on the corresponding moduli space. A symplectic embedding of the momenta and winding numbers allows us to connect the orbifold lattice to the special geometry of the moduli space. As an application, a formal expression for an automorphic function, which is a candidate for a non--perturbative superpotential, is given. 
  We consider the realization of N=2 superconformal models in terms of free first-order $(b,c,\beta,\gamma)$-systems, and show that an arbitrary Landau-Ginzburg interaction with quasi-homogeneous potential can be introduced without spoiling the (2,2)-superconformal invariance. We discuss the topological twisting and the renormalization group properties of these theories, and compare them to the conventional topological Landau-Ginzburg models. We show that in our formulation the parameters multiplying deformation terms in the potential are flat coordinates. After properly bosonizing the first-order systems, we are able to make explicit calculations of topological correlation functions as power series in these flat coordinates by using standard Coulomb gas techniques. We retrieve known results for the minimal models and for the torus. 
  We consider the solutions of the field equations for the large $N$ dilaton gravity model in $1+1$ dimensions recently proposed by Callan, Giddings, Harvey and Strominger (CGHS). We find time dependant solutions with finite mass and vanishing flux in the weak coupling regime, as well as solutions which lie entirely in the Liouville region. 
  We study the twisted version of the supersymmetric $G/T=SU(n)/U(1)^{\otimes(n-1)} gauged Wess-Zumino-Witten model. By studying its fixed points under BRST transformation this model is shown to be reduced to a simple topological field theory, that is, the topological matter system in the K.Li's theory of 2 dimensional gravity for the case of $n=2$, and its generalization for $n \geq 3$. 
  Perturbation theory for a class of topological field theories containing antisymmetric tensor fields is considered. These models are characterized by a supersymmetric structure which allows to establish their perturbative finiteness. 
  We analyze an abelian gauge model in 3 dimensions which includes massless scalar matter fields. By controlling the trace anomalies with a local dilatation Ward identity, we show that, in perturbation theory and within the BPHZL scheme, the Chern-Simons term has no radiative corrections. This implies, in particular, the vanishing of the corresponding $\beta$ function in the renormalization group equation. 
  We construct a solution of the classical equations of motion arising in the low energy effective field theory for heterotic string theory. This solution describes a black hole in four dimensions carrying mass $M$, charge $Q$ and angular momentum $J$. The extremal limit of the solution is discussed. 
  We show that the strong coupling limit of d-dimensional quantum electrodynamics with $2^{d}/2^{[d/2]}$ flavors of fermions can be mapped onto the s=1/2 quantum Heisenberg antiferromagnet in d-1 space dimensions. The staggered N\'eel order parameter is the expectation value of a mass operator in QED and the spin-waves are pions. We speculate that the chiral symmetry breaking phase transition corresponds to a transition between the flux phase and the conventional N\'eel ordered phase of an antiferromagnetic t-J model. 
  We show that it is possible to formulate Abelian Chern-Simons theory on a lattice as a topological field theory. We discuss the relationship between gauge invariance of the Chern-Simons lattice action and the topological interpretation of the canonical structure. We show that these theories are exactly solvable and have the same degrees of freedom as the analogous continuum theories. 
  The ground ring structure of 1+1 dimensional string theory leads to an infinite set of non linear recursion relations among the `bulk' scattering amplitudes of open and closed tachyons on the disk, which fix them uniquely. The relations are generated by the action of the ring on the tachyon modules; associativity of this action determines all structure constants. This algebraic structure may allow one to relate the continuum picture to a matrix model. 
  We study the "topological gauged WZW model associated with $SU(2)/U(1)$",which is defined as the twisted version of the corresponding supersymmetric gauged WZW model. It is shown that this model is equivalent to a topological conformal field theory characterized by two independent topological conformal algebras, one of which is the "twisted Kazama-Suzuki type" and the other is "twisted Coulomb gas type". We further show that our formalism of this gauged WZW model naturally reduces to the well-known formulations of 2D gravity coupled with conformal matter; one of the gauge choices leads to the K.Li's theory, and the alternative choices lead to the KPZ theory or the DDK (Liouville) theory. 
  We calculate gravitational dressed tachyon correlators in non critcal dimensions. The 2D gravity part of our theory is constrained to constant curvature. Then scaling dimensions of gravitational dressed vertex operators are equal to their bare conformal dimensions. Considering the model as d+2 dimensional critical string we calculate poles of generalized Shapiro-Virasoro amplitudes. 
  We rederive the $w_\infty$ Ward identities, starting from the existence of trivial linearized gauge invariances, and using the method of canceled propagators in the operator formalism. Recursion relations for certain classes of correlation functions are derived, and these correlation function are calculated exactly. We clarify the relation of these results with another derivation of the Ward identities, which relies directly on charge conservation. We also emphasize the importance of the kinematics of canceled propagators in ensuring that the Ward identities are non-trivial. Finally, we sketch an extension of Ward identities to open strings. 
  We show that there is (p-1)(p'-1) dimensional semi-relative BRST cohomology at each non-positive ghost number in the (p,p') minimal conformal field theory coupled to two dimensional quantum gravity. These closed string states are related to currents and symmetry charges of `exotic' ghost number. We investigate the symmetry structure generated by the most conventional currents (those of vanishing total ghost number), and make a conjecture about the extended algebra which results from incorporating the currents at negative ghost number. 
  The 1D Heisenberg spin chain with anisotropy of the XXZ type is analyzed in terms of the symmetry given by the quantum Galilei group Gamma_q(1). We show that the magnon excitations and the s=1/2, n-magnon bound states are determined by the algebra. Thus the Gamma_q(1) symmetry provides a description that naturally induces the Bethe Ansatz. The recurrence relations determined by Gamma_q(1) permit to express the energy of the n-magnon bound states in a closed form in terms of Tchebischeff polynomials. 
  Parametrized field theories, which are generally covariant versions of ordinary field theories, are studied from the point of view of the covariant phase space: the space of solutions of the field equations equipped with a canonical (pre)symplectic structure. Motivated by issues arising in general relativity, we focus on: phase space representations of the spacetime diffeomorphism group, construction of observables, and the relationship between the canonical and covariant phase spaces. 
  The two-dimensional self-dual Chern--Simons equations are equivalent to the conditions for static, zero-energy solutions of the $(2+1)$-dimensional gauged nonlinear Schr\"odinger equation with Chern--Simons matter-gauge dynamics. In this paper we classify all finite charge $SU(N)$ solutions by first transforming the self-dual Chern--Simons equations into the two-dimensional chiral model (or harmonic map) equations, and then using the Uhlenbeck--Wood classification of harmonic maps into the unitary groups. This construction also leads to a new relationship between the $SU(N)$ Toda and $SU(N)$ chiral model solutions. 
  An elementary derivation is given for the ``Peierles substitution'' used in projecting dynamics in a strong magnetic field onto the lowest Landau level. The projection of wavefunctions and the ordering prescription for the projected Hamiltonian is explained. 
  We discuss the algebraic structure of the various BRST symmetries associated with topological Yang-Mills theory as a generalization of the BRS analysis developed for the non-Abelian anomaly in the local Yang-Mills theory. We show that our BRST algebra leads to an extended {\it Russian formula\/} and {\it descent equations}, which contains the descent equation of Yang-Mills theory as sub-relations. We propose the non-Abelian anomaly counterpart in Topological Yang-Mills theory using the extended descent equation. We also discuss the geometrical structure of our BRST symmetry and some explicit solutions of the extended descent equation are calculated. 
  Path-integral for theories with degenerate vacua is investigated. The origin of the non Borel-summability of the perturbation theory is studied. A new prescription to deal with small coupling is proposed. It leads to a series, which at low orders and small coupling differs from the ordinary perturbative series by nonperturbative amount, but is Borel-summable. 
  We investigate a class of (2,2) supersymmetric string vacua which may be represented as Landau--Ginzburg theories with a quasihomogeneous potential which has an isolated singularity at the origin. There are at least three thousand distinct models in this class. All vacua of this type lead to Euler numbers which lie in the range $-960 \leq \chi \leq 960$. The Euler characteristics do not pair up completely hence the space of Landau--Ginzburg ground states is not mirror symmetric even though it exhibits a high degree of symmetry. We discuss in some detail the relation between Landau--Ginzburg models and Calabi--Yau manifolds and describe a subtlety regarding Landau--Ginzburg potentials with an arbitrary number of fields. We also show that the use of topological identities makes it possible to relate Landau-Ginzburg theories to types of Calabi-Yau manifolds for which the usual Landau-Ginzburg framework does not apply. 
  A field theoretic formulation of the Marinari-Parisi supersymmetric matrix model is established and shown to be equivalent to a recently proposed supersymmetrization of the bosonic collective string field theory. It also corresponds to a continuum description of super-Calogero models. The perturbation theory of the model is developed and, in this approach, an infinite sequence of vertices is generated. A class of potentials is identified for which the spectrum is that of a massless boson and Majorana fermion. For the harmonic oscillator case, the cubic vertices are obtained in an oscillator basis. For a rather general class of potentials it is argued that one cannot generate from Marinari-Parisi models a continuum limit similar to that of the d=1 bosonic string. 
  It has recently become fashionable to regard black holes as elementary particles. By taking this suggestion seriously it is possible to cobble together an elementary particle physics based estimate for the decay rate $(\hbox{black hole})_i \to (\hbox{black hole})_f + (\hbox{massless quantum})$. This estimate of the spontaneous emission rate contains two free parameters which may be fixed by demanding that the high energy end of the spectrum of emitted quanta match a blackbody spectrum at the Hawking temperature. The calculation, though technically trivial, has important conceptual implications: (1) The existence of Hawking radiation from black holes is ultimately dependent only on the fact that massless quanta (and all other forms of matter) couple to gravity. (2) The thermal nature of the Hawking spectrum depends only on the fact that the number of internal states of a large mass black hole is enormous. (3) Remarkably, the resulting formula for the decay rate gives meaningful answers even when extrapolated to low mass black holes. The analysis strongly supports the scenario of complete evaporation as the endpoint of the Hawking radiation process (no naked singularity, no stable massive remnant). 
  We solve Virasoro constraints on the KP hierarchy in terms of minimal conformal models. The constraints we start with are implemented by the Virasoro generators depending on a background charge $Q$. Then the solutions to the constraints are given by the theory which has the same field content as the David-Distler-Kawai theory: it consists of a minimal matter scalar with background charge $Q$, dressed with an extra `Liouville' scalar. The construction is based on a generalization of the Kontsevich parametrization of the KP times achieved by introducing into it Miwa parameters which depend on the value of $Q$. Under the thus defined Kontsevich-Miwa transformation, the Virasoro constraints are proven to be equivalent to a master equation depending on the parameter $Q$. The master equation is further identified with a null-vector decoupling equation. We conjecture that $W^{(n)}$ constraints on the KP hierarchy are similarly related to a level-$n$ decoupling equation. We also consider the master equation for the $N$-reduced KP hierarchies. Several comments are made on a possible relation of the generalized master equation to {\it scaled} Kontsevich-type matrix integrals and on the form the equation takes in higher genera. 
  We diagonalize the anti-ferroelectric XXZ-Hamiltonian directly in the thermodynamic limit, where the model becomes invariant under the action of affine U_q( sl(2) ).   Our method is based on the representation theory of quantum affine algebras, the related vertex operators and KZ equation, and thereby bypasses the usual process of starting from a finite lattice, taking the thermodynamic limit and filling the Dirac sea. From recent results on the algebraic structure of the corner transfer matrix of the model, we obtain the vacuum vector of the Hamiltonian. The rest of the eigenvectors are obtained by applying the vertex operators, which act as particle creation operators in the space of eigenvectors.   We check the agreement of our results with those obtained using the Bethe Ansatz in a number of cases, and with others obtained in the scaling limit --- the $su(2)$-invariant Thirring model. 
  An extension of the Field-Antifield formalism to treat anomalous gauge theories with a closed, irreducible classical gauge algebra is proposed. Introducing extra degrees of freedom, we construct the gauge transformations for these new fields, the Wess-Zumino term and the corresponding measure. 
  In theories where spacetime is a direct product of Minkowski space ($M^4$) and a d dimensional compact space ($K^d$), there can exist topological solitons that simultaneously wind around $R^3$ (or $R^2$ or $R^1$) in $M^4$ and the compact dimensions. A paradigmatic non-gravitational example of such ``co-winding" solitons is furnished by Yang-Mills theory defined on $M^4 X S^1$. Pointlike, stringlike and sheetlike solitons can be identified by transcribing and generalizing the proceedure used to construct the periodic instanton (caloron) solutions. Asymptotically the classical pointlike objects have non-Abelian magnetic dipole fields together with a non-Abelian scalar potential while the ``color" electric charge is zero. However quantization of collective coordinates associated with zeromodes and coupling to fermions can radically change these quantum numbers due to fermion number fractionalization and its non-Abelian generalization. Interpreting the YM group as color (or the Electroweak SU(2) group) and assuming that an extra circular dimension exists thus implies the existence of topologically stable solitonic objects which carry baryon(lepton) number and a mass O($1/g^2R$), where R is the radius of the compact dimension. 
  We generalize the Lax pair and B\"acklund transformations for Toda and N=1 super Toda equations to the case of arbitrary worldsheet background geometry. We use the fact that the Toda equations express constant curvature conditions, which arise naturally from flatness conditions equivalent to the W--gravity equations of motion. 
  Let H be the corner-transfer-matrix Hamiltonian for the six-vertex model in the anti-ferroelectric regime. It acts on the infinite tensor product W = V . V . V ....., where is the 2-dimensional irreducible representation of the quantum affine sl(2). We observe that H is the derivation of quantum affine sl(2), and conjecture that the eigenvectors of H form the level-1 vacuum representation of quantum affine sl(2). We report on checks in support of our conjecture. 
  A generally covariant gauge theory for an arbitrary gauge group with dimension $\geq 3$, that reduces to Ashtekar's canonical formulation of gravity for SO(3,C), is presented. The canonical form of the theory is shown to contain only first class constraints. 
  We describe the most general treatment of all anomalies both for chiral and massless Dirac fermions, in two-dimensional gravity. It is shown that for this purpose two regularization dependent parameters are present in the effective action. Analogy to the \sc\ model is displayed corresponding to a specific choice of the second parameter, thus showing that the gravitational model contains \a\ relations having no analogy in the \sc\ model. 
  We construct a manifestly $N=(4,0)$ world-sheet supersymmetric twistor-like formulation of the $D=6$ Green-Schwarz superstring, using the principle of double (target-space and world-sheet) Grassmann analyticity. The superstring action contains two Lagrange multiplier terms and a Wess-Zumino term. They are written down in the analytic subspace of the world-sheet harmonic $N=(4,0)$ superspace, the target manifold being too an analytic subspace of the harmonic $D=6\;\; N=1$ superspace. The kappa symmetry of the $D=6$ superstring is identified with a Kac-Moody extension of the world-sheet $N=(4,0)$ superconformal symmetry. It can be enlarged to include the whole world-sheet reparametrization group if one introduces the appropriate gauge Beltrami superfield into the action. To illustrate the basic features of the new $D=6$ superstring construction, we first give some details about the simpler (already known) twistor-like formulations of $D=3, N=(1,0)$ and $D=4, N=(2,0)$ superstrings. 
  We investigate Hawking radiation from two-dimensional dilatonic black holes using standard quantization techniques. In the background of a collapsing black hole solution the Bogoliubov coefficients can be exactly determined. In the regime after the black hole has settled down to an `equilibrium' state but before the backreaction becomes important these give the known result of a thermal distribution of Hawking radiation at temperature ${\lambda\over2\pi}$. The density matrix is computed in this regime and shown to be purely thermal. Similar techniques can be used to derive the stress tensor. The resulting expression agrees with the derivation based on the conformal anomaly and can be used to incorporate the backreaction. Corrections to the thermal density matrix are also examined, and it is argued that to leading order in perturbation theory the effect of the backreaction is to modify the Bogoliubov transformation, but not in a way that restores information lost to the black holes. 
  We study the topological nature of both isotropic and anisotropic SU(N) Thirring model. It is shown that in the isotropic model there exists the special point where the system lives in the topological phase and that in the anisotropic one which is obtained by introducing two coupling constants and has U(1) symmetry, we present a simple mechanism of the dynamical topological phase transition which takes place at the infinite energy scale. 
  Finite Euler hierarchies of field theory Lagrangians leading to universal equations of motion for new types of string and membrane theories and for {\it classical} topological field theories are constructed. The analysis uses two main ingredients. On the one hand, there exists a generic finite Euler hierarchy for one field leading to a universal equation which generalises the Plebanski equation of self-dual four dimensional gravity. On the other hand, specific maps are introduced between field theories which provide a ``triangular duality'' between certain classes of arbitrary field theories, classical topological field theories and generalised string and membrane theories. The universal equations, which derive from an infinity of inequivalent Lagrangians, are generalisations of certain reductions of the Plebanski and KdV equations, and could possibly define new integrable systems, thus in particular integrable membrane theories. Some classes of solutions are constructed in the general case. The general solution to some of the universal equations is given in the simplest cases. 
  We consider the low energy limit of three dimensional Quantum Chromodynamics with an even number of flavors. We show that Parity is not spontaneously broken, but the global (flavor) symmetry is spontaneously broken. The low energy effective lagrangian is a nonlinear sigma model on the Grassmannian. Some Chern--Simons terms are necessary in the lagrangian to realize the discrete symmetries correctly. We consider also another parametrization of the low energy sector which leads to a three dimensional analogue of the Wess--Zumino--Witten--Novikov model. Since three dimensional QCD is believed to be a model for quantum anti--ferromagnetism, our effective lagrangian can describe their long wavelength excitations (spin waves). 
  We show that baryons of three dimensional Quantum Chromodynamics can be understood as solitons of its effective lagrangian. In the parity preserving phase we study, these baryons are fermions for odd $N_c$ and bosons for even $N_c$, never anyons. We quantize the collective variables of the solitons and there by calculate the flavor quantum numbers, magnetic moments and mass splittings of the baryon. The flavor quantum numbers are in agreement with naive quark model for the low lying states. The magnetic moments and mass splittings are smaller in the soliton model by a factor of $\log {F_\pi\over N_c m_\pi}$. We also show that there is a dibaryon solution that is an analogue of the deuteron. These solitons can describe defects in a quantum anti--ferromagnet. 
  The signatures of the inner product matrices on a Lie algebra's highest weight representation are encoded in the representation's signature character. We show that the signature characters of a finite-dimensional Lie algebra's highest weight representations obey simple difference equations that have a unique solution once appropriate boundary conditions are imposed. We use these results to derive the signature characters of all $A_2$ and $B_2$ highest weight representations. Our results extend, and explain, signature patterns analogous to those observed by Friedan, Qiu and Shenker in the Virasoro algebra's representation theory. 
  I consider a D+1 dimensional nonlinear $\sigma$ model based on a possible interpretation of the Liouville field as a physical time. The Weyl invariance of this theory gives us restrictions for the background fields and the parameters of the theory, e.g.\ for trivial background one obtains the known regions for the dimension of the space-time ($\leq$1 or $\geq$25). For a Robertson-Walker space time a special solution of these equations is discussed. 
  The cosmology of the string effective action, including one loop string threshold corrections, is analyzed for static compactifications. The stability of the minima of a general supersymmetry breaking potential is studied in the presence of radiation. In particular, it is shown that the radiation bath makes the minima with negative cosmological constant unstable. 
  We argue that the description of meson-nucleon dynamics based on the boson-exchange approach, is compatible with the description of the nucleon as a soliton in the nonrelativistic limit. Our arguments are based on an analysis of the meson-soliton form factor and the exact meson-soliton and soliton-soliton scattering amplitudes in the Sine-Gordon model.  
  The number of ghost states at each energy level in a non-unitary conformal field theory is encoded in the signature characters of the relevant Virasoro algebra highest weight representations. We give expressions for these signature characters. These results complete Friedan-Qiu-Shenker's analysis of the Virasoro algebra's highest weight representations. 
  In this article we report a preliminary investigation of the large $N$ limit of a generalized one-matrix model which represents an $O(n)$ symmetric model on a random lattice. The model on a regular lattice is known to be critical only for $-2\le n\le 2$. This is the situation we shall discuss also here, using steepest descent. We first determine the critical and multicritical points, recovering in particular results previously obtained by Kostov. We then calculate the scaling behaviour in the critical region when the cosmological constant is close to its critical value. Like for the multi-matrix models, all critical points can be classified in terms of two relatively prime integers $p,q$. In the parametrization $p=(2m+1)q \pm l$, $m,l$ integers such that $0<l<q$, the string susceptibility exponent is found to be $\gamma_{\rm string}=-2l/(p+q-l)$. When $l=1$ we find that all results agree with those of the corresponding $(p,q)$ string models, otherwise they are different.\par We finally explain how to derive the large order behaviour of the corresponding topological expansion in the double scaling limit. 
  Topological gravity is equivalent to physical gravity in two dimensions in a way that is still mysterious, though by now it has been proved by Kontsevich. In this paper it is shown that a similar relation between topological and physical Yang-Mills theory holds in two dimensions; in this case, however, the relation can be explained by a direct mapping between the two path integrals. This (1) explains many strange facts about two dimensional Yang-Mills theory, like the way the partition function can be expressed exactly as a sum over classical solutions, including unstable ones; (2) makes the corresponding topological theory completely computable. 
  We consider the S-matrix of c=1 Liouville theory with vanishing cosmological constant. We examine some of the constraints imposed by unitarity. These completely determine (N,2) amplitudes at tree level in terms of the (N,1) amplitudes when the plus tachyon momenta take generic values. A surprising feature of the matrix model results is the lack of particle creation branch cuts in the higher genus amplitudes. In fact, we show the naive field theory limit of Liouville theory would predict such branch cuts. However, unitarity in the full string theory ensures that such cuts do not appear in genus one (N,1) amplitudes. We conclude with some comments about the genus one (N,2) amplitudes. 
  A direct relation between the conformal formalism for 2d-quantum gravity and the W-constrained KP hierarchy is found, without the need to invoke intermediate matrix model technology. The Kontsevich-Miwa transform of the KP hierarchy is used to establish an identification between W constraints on the KP tau function and decoupling equations corresponding to Virasoro null vectors. The Kontsevich-Miwa transform maps the $W^{(l)}$-constrained KP hierarchy to the $(p^\prime,p)$ minimal model, with the tau function being given by the correlator of a product of (dressed) $(l,1)$ (or $(1,l)$) operators, provided the Miwa parameter $n_i$ and the free parameter (an abstract $bc$ spin) present in the constraints are expressed through the ratio $p^\prime/p$ and the level $l$. 
  Field-theoretic models for fields taking values in quantum groups are investigated. First we consider $SU_q(2)$ $\sigma$ model ($q$ real) expressed in terms of basic notions of noncommutative differential geometry. We discuss the case in which the $\sigma$ models fields are represented as products of conventional $\sigma$ fields and of the coordinate-independent algebra. An explicit example is provided by the $U_q(2)$ $\sigma$ model with $q\sp{N}=1$, in which case quantum matrices $U_q(2)$ are realised as $2N\times 2N$ unitary matrices. Open problems are pointed out. 
  We give an argument that magnetic monopoles should not exist. It is based on the concept of the index of a vector field. The thrust of the argument is that indices of vector fields are invariants of space-time orientation and of coordinate changes, and thus physical vector fields should preserve indices. The index is defined inductively by means of an equation called the Law of Vector Fields. We give extended philosophical arguments that this Law of Vector Fields should play an important role in mathematics, and we back up this contention by using it in a mechanical way to greatly generalize the Gauss--Bonnet theorem and the Brouwer fixed point theorem and get new proofs of many other theorems. We also give some other suggestions for using the Law and index in physics. 
  Starting from a covariant and background independent definition of normal ordered vertex operators we give an alternative derivation of the KPZ relation between conformal dimensions and their gravitational dressed partners. With our method we are able to study for arbitrary genus the dependence of N-point functions on all dimensionful parameters. Implications for the interpretation of gravitational dressed dimensions are discussed. 
  Paragrassmann algebras with one and many paragrassmann variables are considered from the algebraic point of view without using the Green ansatz. Operators of differentiation with respect to paragrassmann variables and a covariant para-super-derivative are introduced giving a natural generalization of the Grassmann calculus to a paragrassmann one. Deep relations between paragrassmann algebras and quantum groups with deformation parameters being roots of unity are established. 
  We examine the two-dimensional spacetimes that emerge from string theory. We find all the solutions with no tachyons, and show that the only non-trivial solution is the black hole spacetime. We examine the role of duality in this picture. We then explore the thermodynamics of these solutions which is complicated by the fact that only in two spacetime dimensions is it impossible to redefine the dilaton field in terms of a canonical scalar field. Finally, we extend our analysis to the heterotic string, and briefly comment on exact, as opposed to perturbative, solutions. 
  A new set of realizations of the Virasoro algebra on a bosonic Fock space are found by explicitly computing the Virasoro representations associated with coadjoint orbits of the form (Diff S1) / S1. Some progress is made in understanding the unitary structure of these representations. The characters of these representations are exactly the bosonic partition functions calculated previously by Witten using perturbative and fixed-point methods. The representations corresponding to the discrete series of unitary Virasoro representations with c <= 1 are found to be reducible in this formulation, confirming a conjecture by Aldaya and Navarro-Salas. 
  It is well-known that solutions to the string equation are generated by elements of Sato's Grassmannian which are invariant under action of some differential operator. Here it is shown that this operator is nothing else than the infinitesimal operator of the group of additional symmetries of the KdV flow. This is done for KdV hierarchies of arbitrary orders. Virasoro constraints are obtained in a slightly more general form than they are usually written. 
  We review some of the recent developments in the construction of $W$-string theories. These are generalisations of ordinary strings in which the two-dimensional ``worldsheet'' theory, instead of being a gauging of the Virasoro algebra, is a gauging of a higher-spin extension of the Virasoro algebra---a $W$ algebra. Despite the complexity of the (non-linear) $W$ algebras, it turns out that the spectrum can be computed completely and explicitly for more or less any $W$ string. The result is equivalent to a set of spectra for Virasoro strings with unusual central charge and intercepts. 
  We discuss in this paper various aspects of the off-critical $O(n)$ model in two dimensions. We find the ground-state energy conjectured by Zamolodchikov for the unitary minimal models, and extend the result to some non-unitary minimal cases. We apply our results to the discussion of scaling functions for polymers on a cylinder. We show, using the underlying N=2 supersymmetry, that the scaling function for one non-contractible polymer loop around the cylinder is simply related to the solution of the Painleve III differential equation. We also find the ground-state energy for a single polymer on the cylinder. We check these results by numerically simulating the polymer system. We also analyze numerically the flow to the dense polymer phase. We find there surprising results, with a $c_{\hbox{eff}}$ function that is not monotonous and seems to have a roaming behavior, getting very close to the values 81/70 and 7/10 between its UV and IR values of 1. 
  An analysis of the BRST cohomology of the G/G topological models is performed for the case of $A_1^{(1)}$. Invoking a special free field parametrization of the various currents, the cohomology on the corresponding Fock space is extracted. We employ the singular vector structure and fusion rules to translate the latter into the cohomology on the space of irreducible representations. Using the physical states we calculate the characters and partition function, and verify the index interpretation. We twist the energy-momentum tensor to establish an intriguing correspondence between the ${SL(2)\over SL(2)}$ model with level $k={p\over q}-2$ and $(p,q)$ models coupled to gravity. 
  We argue that the infinitely many gauge symmetries of string theory provide an infinite set of conserved (gauge) quantum numbers (W-hair) which characterise black hole states and maintain quantum coherence, even during exotic processes like black hole evaporation/decay. We study ways of measuring the W-hair of spherically-symmetric four-dimensional objects with event horizons, treated as effectively two-dimensional string black holes. Measurements can be done either through the s-wave scattering of light particles off the string black-hole background, or through interference experiments of Aharonov-Bohm type. We also speculate on the role of the extended W-symmetries possessed by the topological field theories that describe the region of space-time around a singularity. 
  We give expressions for the singular vectors in the highest weight representations of the Virasoro algebra. We verify that the expressions --- which take the form of a product of operators applied to the highest weight vector --- do indeed define singular vectors. These results explain the patterns of embeddings amongst Virasoro algebra highest weight representations. 
  We use recently derived explicit formulae for the Virasoro algebra's singular vectors to give constructive proofs of three results due to Feigin and Fuchs. The main result, which is needed for a rigorous treatment of fusion, describes the action of the singular vectors on conformal fields. 
  The standard Einstein-Maxwell equations in 2+1 spacetime dimensions, with a negative cosmological constant, admit a black hole solution. The 2+1 black hole -characterized by mass, angular momentum and charge, defined by flux integrals at infinity- is quite similar to its 3+1 counterpart. Anti-de Sitter space appears as a negative energy state separated by a mass gap from the continuous black hole spectrum. Evaluation of the partition function yields that the entropy is equal to twice the perimeter length of the horizon. 
  We obtain lattice models whose continuum limits correspond to $N=2$ superconformal coset models. This is done by taking the well known vertex model whose continuum limit is the $G \times G/G$ conformal field theory, and twisting the transfer matrix and modifying the quantum group truncation. We find that the natural order parameters of the new models are precisely the chiral primary fields. The integrable perturbations of the conformal field theory limit also have natural counterparts in the lattice formulation, and these can be incorporated into an affine quantum group structure. The topological, twisted $N=2$ superconformal models also have lattice analogues, and these emerge as an intermediate part of our analysis. 
  We consider here a generalization of the Abelian Higgs model in curved space, by adding a Chern--Simons term. The static equations are self-dual provided we choose a suitable potential. The solutions give a self-dual Maxwell--Chern--Simons soliton that possesses a mass and a spin. 
  We show that ${\rm Tr}(-1)^F F e^{-\beta H}$ is an index for $N$=2 supersymmetric theories in two dimensions, in the sense that it is independent of almost all deformations of the theory. This index is related to the geometry of the vacua (Berry's curvature) and satisfies an exact differential equation as a function of $\beta$. For integrable theories we can also compute the index thermodynamically, using the exact $S$-matrix. The equivalence of these two results implies a highly non-trivial equivalence of a set of coupled integral equations with these differential equations, among them Painleve III and the affine Toda equations. 
  We summarize results on the reliability of the eikonal approximation in obtaining the high energy behavior of a two particle forward scattering amplitude. Reliability depends on the spin of the exchanged field. For scalar fields the eikonal fails at eighth order in perturbation theory, when it misses the leading behavior of the exchange-type diagrams. In a vector theory the eikonal gets the exchange diagrams correctly, but fails by ignoring certain non-exchange graphs which dominate the asymptotic behavior of the full amplitude. For spin--2 tensor fields the eikonal captures the leading behavior of each order in perturbation theory, but the sum of eikonal terms is subdominant to graphs neglected by the approximation. We also comment on the eikonal for Yang-Mills vector exchange, where the additional complexities of the non-abelian theory may be absorbed into Regge-type modifications of the gauge boson propagators. 
  The expressions for the $\hat{R}$--matrices for the quantum groups SO$_{q^2}$(5) and SO$_q$(6) in terms of the $\hat{R}$--matrices for Sp$_q$(2) and SL$_q$(4) are found, and the local isomorphisms of the corresponding quantum groups are established. 
  We analyze the properties of the q-vertex operators of U_q(sl(2)^) introduced by Frenkel and Reshetikhin. As the condition for the null vector decoupling, we derive the existence condition of the q-vertex operators ( the fusion rules ). 
  The nonlinear reality structure of the derivatives and the differentials for the euclidean q-spaces are found. A real Laplacian is constructed and reality properties of the exterior derivative are given. 
  We use a recent classification of non-degenerate quasihomogeneous polynomials to construct all Landau-Ginzburg (LG) potentials for N=2 superconformal field theories with c=9 and calculate the corresponding Hodge numbers. Surprisingly, the resulting spectra are less symmetric than the existing incomplete results. It turns out that models belonging to the large class for which an explicit construction of a mirror model as an orbifold is known show remarkable mirror symmetry. On the other hand, half of the remaining 15\% of all models have no mirror partners. This lack of mirror symmetry may point beyond the class of LG-orbifolds. 
  It is shown that the local completeness condition introduced in the analysis of the locality of the gauge fixed action in the antifield formalism plays also a key role in the proof of unitarity. 
  Following the reasoning of Claudson and Halpern, it is shown that "fifth-time" stabilized quantum gravity is equivalent to Langevin evolution (i.e. stochastic quantization) between fixed non-singular, but otherwise arbitrary, initial and final states. The simple restriction to a fixed final state at $t_5 \rightarrow \infty$ is sufficient to stabilize the theory. This equivalence fixes the integration measure, and suggests a particular operator-ordering, for the fifth-time action of quantum gravity. Results of a numerical simulation of stabilized, latticized Einstein-Cartan theory on some small lattices are reported. In the range of cosmological constant $\l$ investigated, it is found that: 1) the system is always in the broken phase $<det(e)> \ne 0$; and 2) the negative free energy is large, possibly singular, in the vincinity of $\l = 0$. The second finding may be relevant to the cosmological constant problem. 
  In Rovelli and Smolin's loop representation of nonperturbative quantum gravity in 4 dimensions, there is a space of solutions to the Hamiltonian constraint having as a basis isotopy classes of links in R^3. The physically correct inner product on this space of states is not yet known, or in other words, the *-algebra structure of the algebra of observables has not been determined. In order to approach this problem, we consider a larger space H of solutions of the Hamiltonian constraint, which has as a basis isotopy classes of tangles. A certain algebra T, the ``tangle algebra,'' acts as operators on H. The ``empty state'', corresponding to the class of the empty tangle, is conjectured to be a cyclic vector for T. We construct simpler representations of T as quotients of H by the skein relations for the HOMFLY polynomial, and calculate a *-algebra structure for T using these representations. We use this to determine the inner product of certain states of quantum gravity associated to the Jones polynomial (or more precisely, Kauffman bracket). 
  In bosonic field theories the low-energy scattering of solitons that saturate Bogomol'nyi-type bounds can be approximated as geodesic motion on the moduli space of static solutions. In this paper we consider the analogous issue within the context of supersymmetric field theories. We focus our study on a class of $N=2$ non-linear sigma models in $d=2+1$ based on an arbitrary K\"ahler target manifold and their associated soliton or ``lump" solutions. Using a collective co-ordinate expansion, we construct an effective action which, upon quantisation, describes the low-energy dynamics of the lumps. The effective action is an $N=2$ supersymmetric quantum mechanics action with the target manifold being the moduli space of static charge $N$ lump solutions of the sigma model. The Hilbert space of states of the effective theory consists of anti-holomorphic forms on the moduli space. The normalisable elements of the dolbeault cohomology classes $H^{(0,p)}$ of the moduli space correspond to zero energy bound states and we argue that such states correpond to bound states in the full quantum field theory of the sigma model. 
  We discuss the complete set of one-loop triangle graphs involving the Yang-Mills gauge connection, the \Kahler\ connection and the $\sigma$-model coordinate connection in the effective field theory of $(2,2)$ symmetric $Z_N$ orbifolds. That is, we discuss pure gauge, pure \Kahler\ and pure $\sigma$-model coordinate anomalies as well as the mixed anomalies, such as \Kahler-gauge, some of which have been discussed elsewhere. We propose a mechanism for restoring both \Kahler\ and $\sigma$-model coordinate symmetry based upon the introduction of two types of counterterms. Finally, we enlarge the $\sigma$-model generalization of the Green-Schwarz mechanism to allow the removal of the universal parts of a wider class of anomalies than those previously discussed. 
  Using the reduced formulation of large-N Quantum Field Theories we study strings in space-time dimensions higher than one. Some preliminary results concerning the possible string susceptibilities and general properties of the model are presented. 
  We argue that \CP is a gauge symmetry in string theory. As a consequence, \CP cannot be explicitly broken either perturbatively or non-pertubatively; there can be no non-perturbative \CP-violating parameters. String theory is thus an example of a theory where all $\theta$ angles arise due to spontaneous \CP violation, and are in principle calculable. 
  We establish two-loop (on shell) finiteness of certain supergravity theories in two dimensions. Possible implications of this result are discussed 
  We provide a general description of realisations of W--algebras in terms of smaller W--algebras and free fields. This is based on the definition of the W--algebra as the commutant of a set of screening charges. This is conjectured to be related to partial gauge-fixings in the Hamiltonian reduction model. 
  The critical behaviour of the $D=0$ matrix model with potential perturbed by nonlocal term generating touchings between random surfaces is studied. It is found that the phase diagram of the model has many features of the phase diagram of discretized Polyakov's bosonic string with higher order curvature terms included. It contains the phase of smooth (Liouville) surfaces, the intermediate phase and the phase of branched polymers. The perturbation becomes irrelevant at the first phase and dominates at the third one. 
  The trace anomaly of matter in curved space generates an effective action for the conformal factor of the metric tensor in $D=4$ dimensions, analogous to the Polyakov action for $D=2$. We compute the contributions of the reparameterization ghosts to the central charges for $D=4$, as well as the quantum contribution of the conformal factor itself. The ghost contribution satisfies the necessary Wess-Zumino consistency condition only if combined with the spin-2 modes, whose contributions to the trace anomaly we also discuss. 
  The evolution of a closed bosonic string is envisaged in the time-dependent background of its massless modes. A duality transformation is implemented on the spatial component of string coordinates to obtain a dual string. It is shown that the evolution equations are manifestly $O(d,d)$ invariant. The tree level string effective actions for the original and the dual string theory are shown to be equivalent. 
  We examine the application of $c=1$ conformal field theory to the description of the fractional quantum Hall effect (FQHE). It is found that the Gaussian model together with an appropriate boundary condition for the order parameter furnishes an effective theory for the Laughlin type FQHE. The plateau formation condition corresponds to taking the {\em chiral} portion of the theory. 
  The geometric interpretation of the antibracket formalism given by Witten is extended to cover the anti-BRST symmetry. This enables one to formulate the quantum master equation for the BRST--anti-BRST formalism in terms of integration theory over a supermanifold. A proof of the equivalence of the standard antibracket formalism with the antibracket formalism for the BRST--anti-BRST symmetry is also given. 
  We show that when a Chern-Simons term is added to the action of $SU(N)$ ($N\geq 3$) Yang-Mills theory in 5 dimensions the usual self-dual topological solitons present in the theory necessarily pick up a (topological) electric charge. 
  Scattering amplitudes for discrete states in 2D string theory are considered. Pole divergences of tree-level amplitudes are extracted and residues are interpreted as renormalized amplitudes for discrete states. An effective Lagrangian generating renormalized amplitudes for open string is written and corresponding Ward identities are presented. A relation of this Lagrangian with homotopy Lie algebra is discussed. 
  We analyze the statistical mechanics of a gas of neutral and charged black holes. The microcanonical ensemble is the only possible approach to this system, and the equilibrium configuration is the one for which most of the energy is carried by a single black hole. Schwarzschild black holes are found to obey the statistical bootstrap condition. In all cases, the microcanonical temperature is identical to the Hawking temperature of the most massive black hole in the gas. U(1) charges in general break the bootstrap property. The problems of black hole decay and of quantum coherence are also addressed. 
  In most attempts to compute the Hartle-Hawking ``wave function of the universe'' in Euclidean quantum gravity, two important approximations are made: the path integral is evaluated in a saddle point approximation, and only the leading (least action) extremum is taken into account. In (2+1)-dimensional gravity with a negative cosmological constant, the second assumption is shown to lead to incorrect results: although the leading extremum gives the most important single contribution to the path integral, topologically inequivalent instantons with larger actions occur in great enough numbers to predominate. One can thus say that in 2+1 dimensions --- and possibly in 3+1 dimensions as well --- entropy dominates action in the gravitational path integral. 
  We investigate the stability of charged black holes in two-dimensional heterotic string theories that were recently discussed by McGuidan, Nappi and Yost. In the framework of small time-dependent perturbation, we find that these black holes are linearly stable. 
  We propose and investigate a large class of models possessing resonance factorized S-matrices. The associated Casimir energy describes a rich pattern of renormalization group trajectories related to flows in the coset models based on the simply laced Lie Algebras. From a simplest resonance S-matrix, satisfying the ``$\phi^3$-property'', we predict new flows in non-unitary minimal models. 
  We study the scattering of a massless and neutral test particle in the gravitational field of a body (the string star) made of a large number of scalar states of the superstring. We consider two cases, the one in which these states are neutral string excitations massive already in ten dimensions and the one in which their masses (and charges) originate in the process of compactification on tori. A perturbative calculation based on superstring amplitudes gives us the deflection angle up to the second order in Newton's constant. A comparison with field theory explicitly shows which among the various massless fields of the superstring give a contribution to the scattering process. In both cases, the deflection angle is smaller than the one computed in general relativity. The perturbative series can be resummed by finding the exact solution to the classical equations of motion of the corresponding low-energy action. The space-time metric of our two examples of string stars has no horizon. 
  Starting from a Poincar\'e invariant field theory of a real scalar field with interactions governed by a double-well potential in 2+1 dimensions, the Lorentz representation induced on the collective coordinates describing low-energy excitations about an effective string background is derived. In this representation, Lorentz transformations are given in terms of an infinite series, in powers of derivatives along the worldsheet. Transformations that act on the direction transverse to the string worldsheet involve a universal dimension $-1$ term. As a consequence, Lorentz invariance holds in this theory of long effective strings due to cancellations in the action between irrelevant terms and the dimension two term that describes free massless scalar fields in two dimensions. (in plain tex, no macropackages necessary) 
  In supersymmetric theories the mass of any state is bounded below by the values of some of its charges. The corresponding bounds in case of Schwarzschild and Reissner-Nordstr\"om black holes are known to coincide with the requirement that naked singularities be absent. Here we investigate charged dilaton black holes in this context. We show that the extreme solutions saturate the supersymmetry bound of $N=4\ d=4$ supergravity, or dimensionally reduced superstring theory. Specifically, we have shown that extreme dilaton black holes, with electric and magnetic charges, admit super-covariantly constant spinors. The supersymmetric positivity bound for dilaton black holes, $M \geq \frac{1}{\sqrt 2}(|Q|+|P|)$, takes care of the absence of naked singularities of the dilaton black holes and is, in this sense, equivalent to the cosmic censorship condition. The temperature, entropy and singularity are discussed. The Euclidean action (entropy) of the extreme black hole is given by $2\pi |PQ|$. We argue that this result, as well as the one for Lorentzian signature, is not altered by higher order corrections in the supersymmetric theory. When a black hole reaches its extreme limit, it cannot continue to evaporate by emitting elementary particles, since this would violate the supersymmetric positivity bound. We speculate on the possibility that an extreme black hole may ``evaporate" by emitting smaller extreme black holes. 
  Recently Callan, Giddings, Harvey and the author derived a set of one-loop semiclassical equations describing black hole formation/evaporation in two-dimensional dilaton gravity conformally coupled to $N$ scalar fields. These equations were subsequently used to show that an incoming matter wave develops a black hole type singularity at a critical value $\phi_{cr}$ of the dilaton field. In this paper a modification to these equations arising from the Fadeev-Popov determinant is considered and shown to have dramatic effects for $N<24$, in which case $\phi_{cr}$ becomes complex. The $N<24$ equations are solved along the leading edge of an incoming matter shock wave and found to be non-singular. The shock wave arrives at future null infinity in a zero energy state, gravitationally cloaked by negative energy Hawking radiation. Static black hole solutions supported by a radiation bath are also studied. The interior of the event horizon is found to be non-singular and asymptotic to deSitter space for $N<24$, at least for sufficiently small mass. It is noted that the one-loop approximation is {\it not} justified by a small parameter for small $N$. However an alternate theory (with different matter content) is found for which the same equations arise to leading order in an adjustable small parameter. 
  We investigate the possibility of accommodating neutrino masses compatible with the MSW study of the Solar neutrino deficit within the minimal supersymmetric Standard Model. The ``gravity-induced'' seesaw mechanism based on an interplay of nonrenormalizable and renormalizable terms in the superpotential allows neutrino masses $m_\nu\propto m_u^2/M_I$, with $m_u$ the corresponding quark mass and $M_I\simeq 4\times10^{11}$ GeV, while at the same time ensuring the grand desert with the gauge coupling unification at $M_U\simeq 2\times10^{16}$ GeV. The proposed scenario may be realized in a class of string vacua, {\it i.e.,} large radius ($R^2/\alpha '={\cal O}(20)$) $(0,2)$ Calabi-Yau spaces. In this case $M_U^2=M_C^2/{\cal O} (2R^2/\alpha')$ and $M_I= {\cal O}(e^{-R^2/\alpha'})M_C$. Here $M_C=g\times 5.2\times 10^{17}$GeV is the scale of the tree level (genus zero) gauge coupling ($g$) unification. 
  We study the BRST quantization of the 1+1 dimensional gravity model proposed by Jackiw and Teitelboim and also the topological gauge model which is equivalent to the gravity model at least classically. The gravity model quantized in the light-cone gauge is found to be a free theory with a nilpotent BRST charge. We show also that there exist twisted N=2 superconformal algebras in the Jackiw-Teitelboim's model as well as in the topological gauge model. We discuss the quantum equivalence between the gravity theory and the topological gauge theory. It is shown that these theories are indeed equivalent to each other in the light-cone gauge. 
  The addition of a topological model to the matter content of a conventional closed-string theory leads to the appearance of many perturbatively-decoupled space-time worlds. We illustrate this by classifying topological vertex models on a triangulated surface. We comment on how such worlds could have been coupled in the Planck era. 
  It is shown that the N=2 superconformal transformations are restricted N=1 supergauge transformations of a supergauge theory with Osp(2,2) as a gauge group. Based on this result, a canonical derivation of the Osp(2,2) current algebra in the superchiral gauge formulation of N=2 supergravity is presented. 
  Strong anti-gravity is the vanishing to all orders in Newton's constant of the net force between two massive particles at rest. We study this phenomenon and show that it occurs in any effective theory of gravity which is obtained from a higher-dimensional model by compactification on a manifold with flat directions. We find the exact solution of the Einstein equations in the presence of a point-like source of strong anti-gravity by dimensional reduction of what is a shock-wave solution in the higher-dimensional model. (Latex file, no macros, figures not included) 
  The three-point functions for minimal models coupled to gravity are derived in the operator approach to Liouville theory which is based on its $U_q(sl(2))$ quantum group structure. The result is shown to agree with matrix-model calculations on the sphere. The precise definition of the corresponding cosmological constant is given in the operator solution of the quantum Liouville theory. It is shown that the symmetry between quantum-group spins $J$ and $-J-1$ previously put forward by the author is the explanation of the continuation in the number of screening operators discovered by Goulian and Li. Contrary to the previous discussions of this problem, the present approach clearly separates the emission operators for each leg. This clarifies the structure of the dressing by gravity. It is shown, in particular that the end points are not treated on the same footing as the mid point. Since the outcome is completely symmetric this suggests the existence of a picture-changing mechanism in two dimensional gravity. 
  The Euclidean analogues of the sine-Gordon solitons are used as sources of the heterotic fivebrane solutions in the ten-dimensional heterotic string theory. Some properties of these soliton solutions are discussed. These solitons in principle can appear as string-like objects in 4-dimensional space-time after proper compactifications. 
  Necessary and sufficient conditions are found for any object in $3+1$ dimensions to have integer rather than fractional fermion number. Nontrivial examples include the Jackiw-Rebbi monopole and the already well studied Su-Schrieffer-Heeger soliton, both displaying integer multiples of elementary charges in combinations that normally are forbidden. 
  We present a global analysis of the geometries that arise in non-compact current algebra (or gauged WZW) coset models of strings and particles propagating in curved space-time. The simplest case is the 2d black hole. In higher dimensions these geometries describe new and much more complex singularities. For string and particle theories (defined in the text) we introduce general methods for identifying global coordinates and give the general exact solution for the geodesics for any gauged WZW model for any number of dimensions. We then specialize to the 3d geometries associated with $SO(2,2)/SO(2,1)$ (and also $SO(3,1)/SO(2,1)$) and discuss in detail the global space, geodesics, curvature singularities and duality properties of this space. The large-small (or mirror) type duality property is reformulated as an inversion in group parameter space. The 3d global space has two topologically distinct sectors, with patches of different sectors related by duality. The first sector has a singularity surface with the topology of ``pinched double trousers". It can be pictured as the world sheet of two closed strings that join into a single closed string and then split into two closed strings, but with a pinch in each leg of the trousers. The second sector has a singularity surface with the topology of ``double saddle", pictured as the world sheets of two infinite open strings that come close but do not touch. We discuss the geodesicaly complete spaces on each side of these surfaces and interpret the motion of particles in physical terms. A cosmological interpretation is suggested and comments are mode on possible physical applications. 
  General relativity can be recast as a theory of connections by performing a canonical transformation on its phase space. In this form, its (kinematical) structure is closely related to that of Yang-Mills theory and topological field theories. Over the past few years, a variety of techniques have been developed to quantize all these theories non-perturbatively. These developments are summarized with special emphasis on loop space methods and their applications to quantum gravity. 
  To the Yang-Baxter equation an additional relation can be added. This is the reflection equation which appears in various places, with or without spectral parameter. For example, in factorizable scattering on a half-line, integrable lattice models with non-periodic boundary conditions, non-commutative differential geometry on quantum groups, etc. We study two forms of spectral parameter independent reflection equations, chosen by the requirement that their solutions be comodules with respect to the quantum group coaction leaving invariant the reflection equations. For a variety of known solutions of the Yang-Baxter equation we give the constant solutions of the reflection equations. Various quadratic algebras defined by the reflection equations are also given explicitly. 
  The partition functions of Pasquier models on the cylinder, and the associated intertwiners, are considered. It is shown that earlier results due to Saleur and Bauer can be rephrased in a geometrical way, reminiscent of formulae found in certain purely elastic scattering theories. This establishes the positivity of these intertwiners in a general way and elucidates connections between these objects and the finite subgroups of SU(2). It also offers the hope that analogous geometrical structures might lie behind the so-far mysterious results found by Di Francesco and Zuber in their search for generalisations of these models. 
  We consider Calabi-Yau compactifications with one K\"ahler modulus. Following the method of Candelas et al. we use the mirror hypothesis to solve the quantum theory exactly in dependence of this modulus by performing the calculation for the corresponding complex structure deformation on the mirror manifold. Here the information is accessible by techniques of classical geometry. It is encoded in the Picard-Fuchs differential equation which has to be supplemented by requirements on the global properties of its solutions. 
  A reorganized perturbation expansion with a propagator of soft infrared behavior is used to study the critical behavior of the mass gap. The condition of relativistic covariance fixes the form of the soft propagator. Finite approximants to the correlation critical exponent can be obtained in every order of the modified, soft perturbation expansion. Alternatively, a convergent series of exponents in large orders of the soft perturbation expansion is provided by the renormalization group in all spatial dimensions, $1\leq D\leq3$. The result of the $\epsilon$-expansion is recovered in the $D\rightarrow 3 $ limit. 
  We find that the high temperature limit of the free energy per unit length for the rigid string agrees dimensionally with that of the QCD string (unlike the Nambu-Goto string). The sign, and in fact the phase, do not agree. While this may be a clue to a string theory of QCD, we note that the problem of the fourth derivative action makes it impossible for the rigid string to be a correct description. 
  The quantum Hamiltonian reduction on the OSp(1,2) super Kac-Moody algebra is described in the BRST formalism. Using a free field representation of the KM currents, the super Kac-Moody algebra is shown to be reduced to a superconformal one via the Hamiltonian reduction. This reduction is manifestly supersymmetric because of supersymmetric constraints imposed on the algebra. 
  We study the two-dimensional supersymmetric Toda theory based on the Lie superalgebra $B(1,1) \equiv Osp(3|2)$ and construct its quantum W-currents. We also investigate the fermionic affinization of this model: we show that despite the non-unitary form of the Lagrangian the $B^{(1)}(1,1)$ theory has a real particle mass spectrum which is not renormalized at one-loop. We construct the first higher--spin conserved current, prove its conservation to all-loop order, compute one-loop corrections to the corresponding charge and check consistency between charge and mass renormalization. 
  We discuss the four-dimensional target-space interpretation of bosonic strings based on gauged WZW models, in particular of those based on the non-compact coset space $SL(2,{\bf R})\times SO(1,1)^2 /SO(1,1)$. We show that these theories lead, apart from the recently broadly discussed black-hole type of backgrounds, to cosmological string backgrounds, such as an expanding Universe. Which of the two cases is realized depends on the sign of the level of the corresponding Kac-Moody algebra. We discuss various aspects of these new cosmological string backgrounds. 
  Renormalization group equations for massless GUT's in curved space-time with non-trivial topology are formulated. The asymptotics of the effective action both at high and low energies are obtained. It is shown that the Casimir energy contribution at high curvature (early Universe) becomes non-essential in the effective action. 
  The path integral for higher-derivative quantum gravity with torsion is considered. Applying the methods of two-dimensional quantum gravity, this path integral is analyzed in the limit of conformally self-dual metrics. A scaling law for fixed-volume geometry is obtained. 
  Spontaneous compactification ---on a $R^1\times S^1$ background--- in 2D induced quantum gravity (considered as a toy model for more fundamental quantum gravity) is analyzed in the gauge-independent effective action formalism. It is shown that such compactification is stable, in contradistinction to multidimensional quantum gravity on a $R^D\times S^1 \ (D>2)$ background ---which is known to be one-loop unstable. 
  The calculation of the effective potential for fixed-end and toroidal rigid $p$-branes is performed in the one-loop as well as in the $1/d$ approximations. The analysis of the involved zeta-functions (of inhomogeneous Epstein type) which appear in the process of regularization is done in full detail. Assymptotic formulas (allowing only for exponentially decreasing errors of order $\leq 10^{-3}$) are found which carry all the dependences on the basic parameters of the theory explicitly. The behaviour of the effective potential (specified to the membrane case $p=2$) is investigated, and the extrema of this effective potential are obtained. 
  A multimonopole solution in Yang-Mills field theory is obtained by a modification of the 't Hooft ansatz for a four-dimensional instanton. Although this solution has divergent action near each source, it can be used to construct an exact finite action multimonopole solution of heterotic string theory, in which the divergences from the Yang-Mills sector are precisely cancelled by those from the gravity sector. 
  In the low-velocity limit, multi-soliton solutions trace out geodesics in the static solution manifold with distance defined by a metric on moduli space. For the recently constructed multimonopole solutions of heterotic string theory, we obtain a flat metric to leading order in the impact parameter. This result agrees with the trivial scattering predicted by a test monopole calculation. 
  An action for two dimensional gravity conformally coupled to two dilaton-type fields is analysed. Classically, the theory has some exact solutions. These include configurations representing black holes. A semi-classical theory is obtained by assuming that these singular solutions are caused by the collapse of some matter fields. The semi-classical equations of motion reveal then that any generic solution must have a flat geometry. 
  We construct $N=2$ super-$W_{n+1}$ strings and obtain the complete physical spectrum, for arbitrary $n \ge 2$. We also derive more general realisations of the super-$W_{n+1}$ algebras in terms of $k$ commuting $N=2$ super energy-momentum tensors and $n-k$ pairs of complex superfields, with $0\le k \le [\ft{n+1}{2}]$.  
  A new approach to the correlation functions is presented for the XXZ model in the anti-ferroelectric regime. The method is based on the recent realization of the quantum affine symmetry using vertex operators. With the aid of a boson representation for the latter, an integral formula is found for correlation functions of arbitrary local operators. As a special case it reproduces the spontaneous staggered polarization obtained earlier by Baxter. 
  The string equation for the $[{\tilde P},Q]=Q$ formulation of non--perturbatively stable 2D quantum gravity coupled to the $(2m-1,2)$ models is studied. Global KdV flows between the appropriate solutions are considered as deformations of two compatible linear problems. It is demonstrated that the necessary conditions for such flows to exist are satisfied. A numerical study reveals such flows between the pole--free solutions of pure gravity ($m=2$), the Lee--Yang edge model ($m=3$) and topological gravity ($m=1$). We conjecture that this is the case for all of the $m$--critical models. As the $m=1$ solution is unique these global flows define a {\sl unique} solution for each $m$--critical model. 
  We discuss a new method of integration over matrix variables based on a suitable gauge choice in which the angular variables decouple from the eigenvalues at least for a class of two-matrix models. The calculation of correlation functions involving angular variables is simple in this gauge. Where the method is applicable it also gives an extremely simple proof of the classical integration formula used to reduce multi-matrix models to an integral over the eigenvalues. 
  We consider a two - dimensional Minkowski signature sigma model with a $2+N$ - dimensional target space metric having a null Killing vector. It is shown that the model is finite to all orders of the loop expansion if the dependence of the ``transverse" part of the metric $\ggij (u,x)$ on the light cone coordinate $u$ is subject to the standard renormalization group equation of the $N$ - dimensional sigma model, $ {d\ggij\over du} = \gb_{ij} =R_{ij} + ... $. In particular, we discuss the `one - coupling' case when $\ggij(u,x)$ is a metric of an $N$ - dimensional symmetric space $\gij(x)$ multiplied by a function $f(u)$. The theory is finite if $f(u)$ is equal to the ``running" coupling of the symmetric space sigma model (with $u$ playing the role of the RG ``time"). For example, the geometry of space - time with $\gij$ being the metric of the $N$ - sphere is determined by the form of the $\gb$ - function of the $O(N+1)$ model. The ``asymptotic freedom" limit of large $u$ corresponds to the weak coupling limit of small $2+N$ - dimensional curvature. We prove that there exists a dilaton field which together with the $2+N$ - dimensional metric solves the sigma model Weyl invariance conditions. The resulting backgrounds thus represent new tree level string vacua. We also remark on possible connections with some $2d$ quantum gravity models. 
  The open string with one-dimensional target space is formulated in terms of an SOS, or loop gas, model on a random surface. We solve an integral equation for the loop amplitude with Dirichlet and Neumann boundary conditions imposed on different pieces of its boundary. The result is used to calculate the mean values of order and disorder operators, to construct the string propagator and find its spectrum of excitations. The latter is not sensible neither to the string tension $\L$ nor to the mass $\mu$ of the ``quarks'' at the ends of the string. As in the case of closed strings, the SOS formulation allows to construct a Feynman diagram technique for the string interaction amplitudes. 
  We study cosmological implications of the duality ($PSL(2,{\bf Z})$) invariant potential for the compactification radius $T$, arising in a class of superstring vacua. We show that in spite of having only one minimum in the fundamental domain of the $T$ field there are two types of non-supersymmetric domain walls: one is associated with the discrete Peccei-Quinn symmetry $T\to T+i$, analogous to the axionic domain wall, and another one associated with the noncompact symmetry $T\to 1/T$, analogous to the $Z_2$ domain walls. The first one is bound by stringy cosmic strings. The scale of such domain walls is governed by the scale of gaugino condensation (${\cal O} (10^{16}$ GeV) in the case of hidden $E_8$ gauge group), while the separation between minima is of order $M_{pl}$. We discuss the formation of walls and their cosmological implications: the walls must be gotten rid of, either by chopping by stringy cosmic strings and/or inflation. Since there is no usual Kibble mechanism to create strings, either one must assume they exist $ab\ initio$, or one must conclude that string cosmologies require inflation. The non-perturbative potential dealt with here appears not to give the needed inflationary epoch. 
  We describe the quantum mechanical scattering of slowly moving maximally charged black holes. Our technique is to develop a canonical quantization procedure on the parameter space of possible static classical solutions. With this, we compute the capture cross sections for the scattering of two black holes. Finally, we discuss how quantization on this parameter space relates to quantization of the degrees of freedom of the gravitational field. 
  An exact conformal field theory describing a four dimensional singular string background is obtained by chiral gauging a $U(1)$ subgroup along with translations in $R$ of an $SL(2,R)\times R$ Wess-Zumino-Witten model. It is shown that the target space-time describes a four dimensional black membrane. Furthermore various duality transformed solutions are constructed. These are also shown to correspond to various forms of four dimensional black membranes. 
  This thesis is a study of two dimensional noncritical string theory. The main tool which is used, is the matrix model. Introductions to both the Liouville model and its matrix model formulation are included. In particular the special states are discussed. Some calculations of partition functions on genus one using field theory techniques are given. Nonperturbative issues and string theory at finite radius are discussed. Zero momentum correlation functions are calculated using the matrix model. One important result is a set of recursion relations. The treatment is extended to nonzero momentum. The main result is a clear identification of the special states. Some comments on the Wheeler de Witt equation is given. The matrix model $W_{\infty}$ algebra is introduced. This organizes the previous results. In particular, a simple derivation of the genus zero tachyon correlation functions is given. The results are then extended to higher genus. It is seen how a deformation of the algebra is responsible for much of the higher genus structure. Some very explicit formulae are derived. Then the Liouville and matrix model calculations are compared followed by some general conclusions. 
  Various aspects of recent works on affine quantum group symmetry of integrable 2d quantum field theory are reviewed and further clarified. A geometrical meaning is given to the quantum double, and other properties of quantum groups. Multiplicative presentations of the Yangian double are analyzed. 
  Remarks are given to the structure of physical states in 2D gravity coupled to $C\leq 1$ matter. The operator algebra of the discrete state operators is calculated for the theory with non-vanishing cosmological constant. 
  A kind of topological field theory is proposed as a candidate to describe the global structure of the 2-form Einstein gravity with or without a cosmological constant. Indeed in the former case, we show that a quantum state in the candidate gives an exact solution of the Wheeler-DeWitt equation. The BRST quantization based on the Batalin-Fradkin-Vilkovisky (BFV) formalism is carried out for this topological version of the 2-form Einstein gravity. 
  We discuss questions arising from the recent work of Schellekens, and also from an earlier paper by Schellekens and Yankielowicz. We summarise Schellekens' results, and proceed to discuss the uniqueness of the c=24 self-dual conformal field theory with no weight one states, i.e. the Monster module $V^\natural$. After introducing the concept of complementary representations, we examine $Z_2$-orbifold constructions in general, and then proceed to apply our considerations firstly to the specific case of the FKS constructions $H(\Lambda)$ and then to the reflection twisted theories $\widetilde H(\Lambda)$. Our techniques provide evidence for the existence of several new theories beyond those proven to exist in previous work and conjectured to exist in Schellekens and Yankielowicz. 
  The original paper, as published in Nuclear Physics B in 1988, had a few factor-of-two errors. Some people got confused by those errors. The purpose of these errata is to make things clear. The revised version of the complete article is also posted to hep-th. 
  We discuss the quantization of the 2d gravity theory of Callan, Giddings, Harvey, and Strominger (CGHS), following the procedure of David, and of Distler and Kawai. We find that the physics depends crucially on whether the number of matter fields is greater than or less than 24. In the latter case the singularity pointed out by several authors is absent but the physical interpretation is unclear. In the former case (the one studied by CGHS) the quantum theory which gives CGHS in the linear dilaton semi-classical limit, is different from that which gives CGHS in the extreme Liouville regime. 
  Like grand unification of old, string unification predicts simple tree-level relations between the couplings of all unbroken gauge groups such as $SU(3)_C$ or $SU(2)_W\)$.  I show here how to compute one-loop corrections to these relations for any four-dimensional model based on a classical vacuum of the heterotic string.  The result can be used to calculate both $\sin^2\theta_W$ and $\Lambda_{\rm QCD}$ in terms of $\alpha_{\rm QED}$ and $\mpl\)$.   The original version of this paper was written in 1987 and published in Nuclear Physics in 1988.  That version had a few factor-of-two errors, which lead some people into confusion.  To avoid future confusion, I've written Errata; they are submitted separately to hep-th (article #9205068). This submission is the complete revised version of the paper. 
  The integrability of $R^2$-gravity with torsion in two dimensions is traced to an ultralocal dynamical symmetry of constraints and momenta in Hamiltonian phase space. It may be interpreted as a quadratically deformed $iso(2,1)$-algebra with the deformation consisting of the Casimir operators of the undeformed algebra. The locally conserved quantity encountered in the explicit solution is identified as an element of the centre of this algebra. Specific contractions of the algebra are related to specific limits of the explicit solutions of this model. 
  Modular invariant conformal field theories with just one primary field and central charge $c=24$ are considered. It has been shown previously that if the chiral algebra of such a theory contains spin-1 currents, it is either the Leech lattice CFT, or it contains a Kac-Moody sub-algebra with total central charge 24. In this paper all meromorphic modular invariant combinations of the allowed Kac-Moody combinations are obtained. The result suggests the existence of 71 meromorphic $c=24$ theories, including the 41 that were already known. 
  We show how the stochastic stabilization provides both the weak coupling genus expansion and a strong coupling expansion of 2d quantum gravity. The double scaling limit is described by a hamiltonian which is unbounded from below, but which has a discrete spectrum. 
  In these lectures the relations between symmetries, Lie algebras, Killing vectors and Noether's theorem are reviewed. A generalisation of the basic ideas to include velocity-dependend co-ordinate transformations naturally leads to the concept of Killing tensors. Via their Poisson brackets these tensors generate an {\em a priori} infinite-dimensional Lie algebra. The nature of such infinite algebras is clarified using the example of flat space-time. Next the formalism is extended to spinning space, which in addition to the standard real co-ordinates is parametrized also by Grassmann-valued vector variables. The equations for extremal trajectories (`geodesics') of these spaces describe the pseudo-classical mechanics of a Dirac fermion. We apply the formalism to solve for the motion of a pseudo-classical electron in Schwarzschild space-time. 
  N=2 string amplitudes, when required to have the Lorentz covariance of the equivalent N=4 string, describe a self-dual form of N=4 super Yang-Mills in 2+2 dimensions. Spin-independent couplings and the ghost nature of SO(2,2) spacetime make it a topological-like theory with vanishing loop corrections. 
  The discrete model of the real symmetric one-matrix ensemble is analyzed with a cubic interaction. The partition function is found to satisfy a recursion relation that solves the model. The double-scaling limit of the recursion relation leads to a Miura transformation relating the contributions to the free energy coming from oriented and unoriented random surfaces. This transformation is the same kind as found with a cuartic interaction. 
  We apply elementary canonical methods for the quantization of 2+1 dimensional gravity, where the dynamics is given by E. Witten's $ISO(2,1)$ Chern-Simons action. As in a previous work, our approach does not involve choice of gauge or clever manipulations of functional integrals. Instead, we just require the Gauss law constraint for gravity to be first class and also to be everywhere differentiable. When the spatial slice is a disc, the gravitational fields can either be unconstrained or constrained at the boundary of the disc. The unconstrained fields correspond to edge currents which carry a representation of the $ISO(2,1)$ Kac-Moody algebra. Unitary representations for such an algebra have been found using the method of induced representations. In the case of constrained fields, we can classify all possible boundary conditions. For several different boundary conditions, the field content of the theory reduces precisely to that of 1+1 dimensional gravity theories. We extend the above formalism to include sources. The sources take into account self- interactions. This is done by punching holes in the disc, and erecting an $ISO(2,1)$ Kac-Moody algebra on the boundary of each hole. If the hole is originally sourceless, a source can be created via the action of a vertex operator $V$. We give an explicit expression for $V$. We shall show that when acting 
  An exact conformal field theory describing a four dimensional 2-brane solution is found by considering a chiral gauged Wess-Zumino -Witten theory corresponding to $SL(2, R)\times R$ , where one gauges the one dimensional $U(1)$ subgroup together with a translation in $R$. The backgrounds for string propagation are explicitly obtained and the target space is shown to have a true curvature singularity. 
  We initiate a program to study the relationship between the target space, the spectrum and the scattering amplitudes in string theory. We consider scattering amplitudes following from string theory and quantum field theory on a curved target space, which is taken to be the $SU(2)$ group manifold, with special attention given to the duality between contributions from different channels. We give a simple example of the equivalence between amplitudes coming from string theory and quantum field theory, and compute the general form of a four-scalar field theoretical amplitude. The corresponding string theory calculation is performed for a special case, and we discuss how more general string theory amplitudes could be evaluated. 
  Three elementary canonical transformations are shown both to have quantum implementations as finite transformations and to generate, classically and infinitesimally, the full canonical algebra. A general canonical transformation can, in principle, be realized quantum mechanically as a product of these transformations. It is found that the intertwining of two super-Hamiltonians is equivalent to there being a canonical transformation between them. A consequence is that the procedure for solving a differential equation can be viewed as a sequence of elementary canonical transformations trivializing the super-Hamiltonian associated to the equation. It is proposed that the quantum integrability of a system is equivalent to the existence of such a sequence. 
  An exact multimonopole solution of heterotic string theory is presented. The solution is constructed by a modification of the 't Hooft ansatz for a four-dimensional instanton. An analogous solution in Yang-Mills field theory saturates a Bogomoln'yi bound and possesses the topology and far field limit of a multimonopole configuration, but has divergent action near each source. In the string solution, however, the divergences from the Yang-Mills sector are precisely cancelled by those from the gravity sector. The resultant action is finite and easily computed. The Manton metric on moduli space for the scattering of two string monopoles is found to be flat to leading order in the impact parameter, in agreement with the trivial scattering predicted by a test monopole calculation. 
  A previously proposed two-step algorithm for calculating the expectation values of Chern-Simons graphs fails to determine certain crucial signs. The step which involves calculating tetrahedra by solving certain non- linear equations is repaired by introducing additional linear equations. As a first step towards a new algorithm for general graphs we find useful linear equations for those special graphs which support knots and links. Using the improved set of equations for tetrahedra we examine the symmetries between tetrahedra generated by arbitrary simple currents. Along the way we uncover the classical origin of simple-current charges. The improved skein relations also lead to exact identities between planar tetrahedra in level $K$ $G(N)$ and level $N$ $G(K)$ CS theories, where $G(N)$ denotes a classical group. These results are recast as identities for quantum $6j$-symbols and WZW braid matrices. We obtain the transformation properties of arbitrary graphs and links under simple current symmetries and rank-level duality. For links with knotted components this requires precise control of the braid eigenvalue permutation signs, which we obtain from plethysm and an explicit expression for the (multiplicity free) signs, valid for all compact gauge groups and all fusion products. 
  In view of the expectation that the solitonic sector of the lower dimensional world may be originated from the solitonic sector of string theory, various solitonic solutions are reduced from the heterotic fivebrane solutions in the ten-dimensional heterotic string theory. These solitons in principle can appear after proper compactifications, {\it e.g.} toroidal compactifications. 
  In a previous paper we derived a relation between the operator product coefficients and anomalous dimensions. We explore this relation in the $(\phi^4)_4$ theory and compute the coefficient functions in the products of $\phi^2$ and $\phi^4$ to first order in the parameter $\lambda$. The calculation results in two-loop beta functions. 
  We give a formula for the derivatives of a correlation function of composite operators with respect to the parameters (i.e., the strong fine structure constant and the quark mass) of QCD in four-dimensional euclidean space. The formula is given as spatial integration of the operator conjugate to a parameter. The operator product of a composite operator and a conjugate operator has an unintegrable part, and the formula requires divergent subtractions. By imposing consistency conditions we derive a relation between the anomalous dimensions of the composite operators and the unintegrable part of the operator product coefficients. 
  We present a fractional superspace formulation of the centerless parasuper-Viraso-ro and fractional super-Virasoro algebras. These are two different generalizations of the ordinary super-Virasoro algebra generated by the infinitesimal diffeomorphisms of the superline. We work on the fractional superline parametrized by $t$ and $\theta$, with $t$ a real coordinate and $\theta$ a paragrassmann variable of order $M$ and canonical dimension $1/F$. We further describe a more general structure labelled by $M$ and $F$ with $M\geq F$. The case $F=2$ corresponds to the parasuper-Virasoro algebra of order $M$, while the case $F=M$ leads to the fractional super-Virasoro algebra of order $F$. The ordinary super-Virasoro algebra is recovered at $F=M=2$. The connection with $q$-oscillator algebras is discussed. 
  We investigate non-commutative differential calculus on the supersymmetric version of quantum space where the non-commuting super-coordinates consist of bosonic as well as fermionic (Grassmann) coordinates. Multi-parametric quantum deformation of the general linear supergroup, $GL_q(m|n)$, is studied and the explicit form for the ${\hat R}$-matrix, which is the solution of the Yang-Baxter equation, is presented. We derive the quantum-matrix commutation relation of $GL_q(m|n)$ and the quantum superdeterminant. We apply these results for the $GL_q(m|n)$ to the deformed phase-space of supercoordinates and their momenta, from which we construct the ${\hat R}$-matrix of q-deformed orthosymplectic group $OSp_q(2n|2m)$ and calculate its ${\hat R}$-matrix. Some detailed argument for quantum super-Clifford algebras and the explict expression of the ${\hat R}$-matrix will be presented for the case of $OSp_q(2|2)$. 
  The present paper is devoted to the study of geometry of Batalin-Vilkovisky quantization procedure. The main mathematical objects under consideration are P-manifolds and SP-manifolds (supermanifolds provided with an odd symplectic structure and, in the case of SP-manifolds, with a volume element). The Batalin-Vilkovisky procedure leads to consideration of integrals of the superharmonic functions over Lagrangian submanifolds. The choice of Lagrangian submanifold can be interpreted as a choice of gauge condition; Batalin and Vilkovisky proved that in some sense their procedure is gauge independent. We prove much more general theorem of the same kind. This theorem leads to a conjecture that one can modify the quantization procedure in such a way as to avoid the use of the notion of Lagrangian submanifold. In the next paper we will show that this is really so at least in the semiclassical approximation. Namely the physical quantities can be expressed as integrals over some set of critical points of solution S to the master equation with the integrand expressed in terms of Reidemeister torsion. This leads to a simplification of quantization procedure and to the possibility to get rigorous results also in the infinite-dimensional case. The present paper contains also a compete classification of P-manifolds and SP-manifolds. The classification is interesting by itself, but in this paper it plays also a role of an important tool in the proof of other results. 
  We construct new theories of dilation gravity coupled to conformal matter which are exact $c=26$ conformal field theories and presumably consistent frameworks for discussing black hole physics in two dimensions. They differ from the CGHS equations in the precise dilaton dependence of the cosmological constant. A further modification proposed by Strominger with a view to eliminating unphysical ghost Hawking radiation is also considered. The new classical equations of motion are explicitly soluble, thus permitting an exact analysis of both static and dynamic senarios. While the static solutions are physically reasonable, the dynamical solutions include puzzling examples with wrong-sign Hawking radiation. We indicate how the latter problem may be resolved in the full quantum theory. 
  We define a lattice statistical model on a triangulated manifold in four dimensions associated to a group $G$. When $G=SU(2)$, the statistical weight is constructed from the $15j$-symbol as well as the $6j$-symbol for recombination of angular momenta, and the model may be regarded as the four-dimensional version of the Ponzano-Regge model. We show that the partition function of the model is invariant under the Alexander moves of the simplicial complex, thus it depends only on the piecewise linear topology of the manifold. For an orientable manifold, the model is related to the so-called $BF$ model. The $q$-analogue of the model is also constructed, and it is argued that its partition function is invariant under the Alexander moves. It is discussed how to realize the 't Hooft operator in these models associated to a closed surface in four dimensions as well as the Wilson operator associated to a closed loop. Correlation functions of these operators in the $q$-deformed version of the model would define a new type of invariants of knots and links in four dimensions. 
  In recent work, several classes of solitonic solutions of string theory with higher-membrane structure have been obtained. These solutions can be classified according to the symmetry possessed by the solitons in the subspace of the spacetime transverse to the membrane. Solitons with four-dimensional spherical symmetry represent instanton solutions in string theory, while those with three-dimensional spherical symmetry represent magnetic monopole-type solutions. For both of these classes, we discuss bosonic as well as heterotic solutions. 
  Previous analyses on the gauge invariance of the action for a generally covariant system are generalized. It is shown that if the action principle is properly improved, there is as much gauge freedom at the endpoints for an arbitrary gauge system as there is for a system with ``internal'' gauge symmetries. The key point is to correctly identify the boundary conditions for the allowed histories and to include the appropriate end-point contribution in the action. The path integral is then discussed. It is proved that by employing the improved action, one can use time-independent canonical gauges even in the case of generally covariant theories. From the point of view of the action and the path integral, there is thus no conceptual difference between general covariance and ``ordinary gauge invariance''. The discussion is illustrated in the case of the point particle, for which various canonical gauges are considered. 
  A chiral $(N,0) $ supergravity theory in d=2 dimensions for any $N$ and its induced action can be obtained by constraining the currents of an Osp(N$|$2) WZWN model. The underlying symmetry algebras are the nonlinear SO(N) superconformal algebras of Knizhnik and Bershadsky. The case $N=3$ is worked out in detail. We show that by adding quantum corrections to the classical transformation rules, the gauge algebra on gauge fields and currents closes. Integrability conditions on Ward identities are derived. The effective action is computed at one loop. It is finite, and can be obtained from the induced action by rescaling the central charge and fields by finite Z factors. 
  A distinctive feature of string unification is the possibility of unification by a non-simply-laced group. This occurs most naturally in four dimensional type~II string models where the gauge symmetry is realized by Kac-Moody algebras at different levels. We investigate the running coupling constants and the one-loop thresholds for such general models. As a specific case, we examine a $\rm SU(3)\times U(1)\times U(1)$ model and find that the threshold corrections lead to a small $6\%$ increase in the unification scale. 
  Some exact static solutions for Einstein gravity in 2+1 dimensions coupled to abelian gauge field are discussed. Some of these solutions are three-dimensional analogs of the Schwarzschild black holes. The metrics in the regions inside and outside the horison are connected by the changing of the Planck mass sign. 
  Thermal history of the string universe based on the Brandenberger and Vafa's scenario is examined. The analysis thereby provides a theoretical foundation of the string universe scenario. Especially the picture of the initial oscillating phase is shown to be natural from the thermodynamical point of view. A new tool is employed to evaluate the multi state density of the string gas. This analysis points out that the well-known functional form of the multi state density is not applicable for the important region $T \leq T_H$, and derives a correct form of it. 
  A covariant path integral calculation of the even spin structure contribution to the one-loop N-graviton scattering amplitude in the type-II superstring theory is presented. The apparent divergence of the $N=5$ amplitude is resolved by separating it into twelve independent terms corresponding to different orders of inserting the graviton vertex operators. Each term is well defined in an appropriate kinematic region and can be analytically continued to physical regions where it develops branch cuts required by unitarity. The zero-slope limit of the $N=5$ amplitude is performed, and the Feynman diagram content of the low-energy field theory is examined. Both one-particle irreducible (1PI) and one-particle redicible (1PR) graphs with massless internal states are generated in this limit. One set of 1PI graphs has the same divergent dependence on the cut-off as that found in the four-graviton case, and it is proved that such graphs exist for all~$N$. The 1PR graphs are contributed by the poles in the world-sheet chiral Green functions. 
  The general theory of matching conditions is developed for gravitational theories in two spacetime dimensions. Models inspired from general relativity and from string theory are considered. These conditions are used to study collapsing dust solutions in spacetimes with non-zero cosmological constant, demonstrating how two-dimensional black holes can arise as the endpoint of such collapse processes. 
  In string theory, nilpotence of the BRS operator $\d$ for the string functional relates the Chern-Simons term in the gauge-invariant antisymmetric tensor field strength to the central term in the Kac-Moody algebra. We generalize these ideas to p-branes with odd p and find that the Kac-Moody algebra for the string becomes the Mickelsson-Faddeev algebra for the p-brane. 
  It has long been argued that the continuum limit of the 3D Ising model is equivalent to a string theory. Unfortunately, in the usual starting point for this equivalence -- a certain lattice theory of surfaces -- it is not at all obvious how to take the continuum limit. In this note, I reformulate the lattice theory of surfaces in a fashion such that the continuum limit is straightforward. I go on to discuss how this new formulation may overcome some fundamental objections to the notion that the Ising model is equivalent to a string theory. In an appendix, I also discuss some aspects of fermion doubling, and the lattice fermion formulation of the 2D Ising model. 
  We study the interactions of the discrete states with nonzero ghost number in $c=1$ two-dimensional ($2D$) quantum gravity. By using the vertex operator representations, it is shown that their interactions are given by the structure constants of the group of the area preserving diffeomorphism similar to those of vanishing ghost number. The effective action for these states is also worked out. The result suggests the whole system has a BRST-like symmetry. 
  I show that the generalized Beltrami differentials and projective connections which appear naturally in induced light cone $W_n$ gravity are geometrical fields parametrizing in one-to-one fashion generalized projective structures on a fixed base Riemann surface. I also show that $W_n$ symmetries are nothing but gauge transformations of the flat ${SL}(n,{\bf C})$ vector bundles canonically associated to the generalized projective structures. This provides an original formulation of classical light cone $W_n$ geometry. From the knowledge of the symmetries, the full BRS algebra is derived. Inspired by the results of recent literature, I argue that quantum $W_n$ gravity may be formulated as an induced gauge theory of generalized projective connections. This leads to projective field theory. The possible anomalies arising at the quantum level are analyzed by solving Wess-Zumino consistency conditions. The implications for induced covariant $W_n$ gravity are briefly discussed. The results presented, valid for arbitrary $n$, reproduce those obtained for $n=2,3$ by different methods. 
  We find two different q-generalizations of Yang-Mills theories. The corresponding lagrangians are invariant under the q-analogue of infinitesimal gauge transformations. We explicitly give the lagrangian and the transformation rules for the bicovariant q-deformation of $SU(2) \times U(1)$. The gauge potentials satisfy q-commutations, as one expects from the differential geometry of quantum groups. However, in one of the two schemes we present, the field strengths do commute. 
  We re-examine the geometry and algebraic structure of BRST's of Topological Yang-Mills theory based on the universal bundle formalism of Atiyah and Singer. This enables us to find a natural generalization of the {\it Russian formula and descent equations\/}, which can be used as algebraic method to find the non-Abelian anomalies counterparts in Topological Yang-Mills theory. We suggest that the presence of the non-Abelian anomaly obstructs the proper definition of Donaldson's invariants. 
  An action is constructed that gives an arbitrary equation in the KdV or MKdV hierarchies as equation of motion; the second Hamiltonian structure of the KdV equation and the Hamiltonian structure of the MKdV equation appear as Poisson bracket structures derived from this action. Quantization of this theory can be carried out in two different schemes, to obtain either the quantum KdV theory of Kupershmidt and Mathieu or the quantum MKdV theory of Sasaki and Yamanaka. The latter is, for specific values of the coupling constant, related to a generalized deformation of the minimal models, and clarifies the relationship of integrable systems of KdV type and conformal field theories. As a generalization it is shown how to construct an action for the $SL(3)$-KdV (Boussinesq) hierarchy. 
  We study the algebraic geometrical background of the Penner--Kontsevich matrix model with the potential $N\alpha \tr {\bigl(- \fr 12 \L X\L X +\log (1-X)+X\bigr)}$. We show that this model describes intersection indices of linear bundles on the discretized moduli space right in the same fashion as the Kontsevich model is related to intersection indices (cohomological classes) on the Riemann surfaces of arbitrary genera. The special role of the logarithmic potential originated from the Penner matrix model is demonstrated. The boundary effects which was unessential in the case of the Kontsevich model are now relevant, and intersection indices on the discretized moduli space of genus $g$ are expressed through Kontsevich's indices of the genus $g$ and of the lower genera. 
  Physics in the neighbourhood of a space-time metric singularity is described by a world-sheet topological gauge field theory which can be represented as a twisted $N=2$ superconformal Wess-Zumino model with a $W_{1+\infty} \otimes W_{1+\infty} $ bosonic symmetry. The measurable $W$-hair associated with the singularity is associated with Wilson loop integrals around gauge defects. The breaking of $W_{1+\infty}$ $\otimes $ $W_{1+\infty}$ $\rightarrow $ $W_{1+\infty}$ is associated with expectation values for open Wilson lines that make the metric non-singular away from the singularity. This symmetry breaking is accompanied by massless discrete `tachyon' states that appear as leg poles in $S$-matrix elements. The triviality of the $S$-matrix in the high-energy limit of the $c=1$ string model, after renormalisation by the leg pole factors, is due to the restoration of double $W$-symmetry at the singularity. 
  We review some exact solitonic solutions of string theory with higher-membrane structure. These include an axionic instanton solution of bosonic string theory as well as multi-instanton and multimonopole solutions of heterotic string theory. The heterotic solutions reveal some interesting aspects of string theory as a theory of quantum gravity. 
  A new open spin chain hamiltonian is introduced. It is both integrable (Sklyanin`s type $K$ matrices are used to achieve this) and invariant under ${\cal U}_{\epsilon}(sl(2))$ transformations in nilpotent irreps for $\epsilon^3=1$. Some considerations on the centralizer of nilpotent representations and its representation theory are also presented. 
  The universal Witham hierarchy is considered from the point of view of topological field theories. The $\tau$-function for this hierarchy is defined. It is proved that the algebraic orbits of Whitham hierarchy can be identified with various topological matter models coupled with topological gravity. 
  The Batalin-Vilkovisky antifield action for the BF theories is constructed by means of the extended form method. The BRST invariant BV antifield action is directly written down by making use of the extended forms that involve all the required ghosts and antifields. 
  Time does not obviously appear amongst the coordinates on the constrained phase space of general relativity in the Hamiltonian formulation. Recent work in finite-dimensional models claims that topological obstructions generically make the global definition of time impossible. It is shown here that a time coordinate can be globally defined on a constrained phase space by patching together local time coordinates, just as coordinates are defined on topologically non-trivial manifolds. 
  The $K=4$ fractional superstring Fock space is constructed in terms of $\bZ_4$ parafermions and free bosons. The bosonization of the $\bZ_4$ parafermion theory and the generalized commutation relations satisfied by the modes of various parafermion fields are reviewed. In this preliminary analysis, we describe a Fock space which is simply a tensor product of $\bZ_4$ parafermion and free boson Fock spaces. It is larger than the Lorentz-covariant Fock space indicated by the fractional superstring partition function. We derive the form of the fractional superconformal algebra that may be used as the constraint algebra for the physical states of the FSS. Issues concerning the associativity, modings and braiding properties of the fractional superconformal algebra are also discussed. The use of the constraint algebra to obtain physical state conditions on the spectrum is illustrated by an application to the massless fermions and bosons of the $K=4$ fractional superstring. However, we fail to generalize these considerations to the massive states. This means that the appropriate constraint algebra on the fractional superstring Fock space remains to be found. Some possible ways of doing this are discussed. 
  The approach of 't Hooft to the puzzles of black hole evaporation can be applied to a simpler system with analogous features. The system is $1+1$ dimensional electrodynamics in a linear dilaton background. Analogues of black holes, Hawking radiation and evaporation exist in this system. In perturbation theory there appears to be an information paradox but this gets resolved in the full quantum theory and there exists an exact $S$-matrix, which is fully unitary and information conserving. 't Hooft's method gives the leading terms in a systematic approximation to the exact result. 
  We construct and study an N=3 supersymmetric Chern-Simons Higgs theory. This theory is the maximally supersymmetric one containing the self-dual models with a single gauge field and no gravity. 
  It is known that the 3d Chern-Simons interaction describes the scaling limit of a quantum Hall system and predicts edge currents in a sample with boundary, the currents generating a chiral $U(1)$ Kac-Moody algebra. It is no doubt also recognized that in a somewhat similar way, the 4d $BF$ interaction (with $B$ a two form, $dB$ the dual $^*j$ of the eletromagnetic current, and F the electromagnetic field form) describes the scaling limit of a superconductor. We show in this paper that there are edge excitations in this model as well for manifolds with boundaries. They are the modes of a scalar field with invariance under the group of diffeomorphisms (diffeos) of the bounding spatial two-manifold. Not all of this group seem implementable by operators in quantum theory, the implementable group being a subgroup of volume preserving diffeos. The $BF$ system in this manner can lead to the $w_{1+\infty }$ algebra and its variants. Lagrangians for fields on the bounding manifold which account for the edge observables on quantization are also presented. They are the analogues of the $1+1$ dimentional massless scalar field Lagrangian describing the edge modes of an abelian Chern-Simons theory with a disk as the spatial manifold. We argue that the addition of ``Maxwell'' terms constructed from $F\wedge ^*F$ and $dB\wedge ^*dB$ do not affect the edge states, and that the augmented Lagrangian has an infinite number of conserved charges- the aforementioned scalar field modes- localized at the edges. This Lagrangian is known to describe London equations and a massive vector field. A $(3+1)$ dimensional generalization of the Hall effect involving vortices coupled to $B$ is also proposed. 
  The conformal non-compact $SL(2,R)/U(1)$ coset model in two dimensions has been recently shown to embody a nonlinear $\hat{W}_\infty$ current algebra, consisting of currents of spin $\geq 2$ including the energy-momentum tensor. In this letter we explicitly construct an infinite set of commuting quantum $\hat{W}_\infty$ charges in the model with $k=1$. These commuting quantum charges generate a set of infinitely many compatible flows (quantum KP flows), which maintain the nonlinear $\hat{W}_\infty$ current algebra invariant. 
  We study the thermodynamic properties of a family of integrable 1D spin chain hamiltonians associated with quantum groups at roots of unity. These hamiltonians depend for each primitive root of unit on a parameter $s$ which plays the role of a continuous spin. The model exhibits ferrimagnetism even though the interaction involved is between nearest neighbors. The latter phenomenon is interpreted as a genuine quantum group effect with no ``classical" analog. The discussion of conformal properties is given. 
  We reconsider the construction of solitons by dressing transformations in the sine-Gordon model. We show that the $N$-soliton solutions are in the orbit of the vacuum, and we identify the elements in the dressing group which allow us to built the $N$-soliton solutions from the vacuum solution. The dressed $\tau$-functions can be computed in two different ways~: either using adjoint actions in the affine Lie algebra $\hat {sl_2}$, and this gives the relation with the B\"acklund transformations, or using the level one representations of the affine Lie algebra $\widehat{sl_2}$, and this directly gives the formulae for the $\tau$-functions in terms of vertex operators. 
  Using exact expressions for the Ising form factors, we give a new very simple proof that the spin-spin and disorder-disorder correlation functions are governed by the Painlev\'e III non linear differential equation. We also show that the generating function of the correlation functions of the descendents of the spin and disorder operators is a $N$-soliton, $N\to\infty$, $\tau$-function of the sinh-Gordon hierarchy. We discuss a relation of our approach to isomonodromy deformation problems, as well as further possible generalizations. 
  We investigate the possibility of false vacuum decay in $N=1$ supergravity theories. By establishing a Bogomol'nyi bound for the energy density stored in the domain wall of the $O(4)$ invariant bubble, we show that supersymmetric vacua remain absolutely stable against false vacuum decay into another supersymmetric vacuum. This conforms with and completes the previous perturbative analysis of Weinberg. Implications for dynamical supersymmetry breaking and decompactification instabilities in superstring theory are discussed. In addition, we show that there are no compact static spherical domain walls. 
  An algebraic rule is presented for computing expectation values of products of local nonabelian charge operators for fermions coupled to an external vector potential in $3+1$ space-time dimensions. The vacuum expectation value of a product of four operators is closely related to a cyclic cocycle in noncommutative geometry of Alain Connes. The relevant representation of the current is constructed using Kirillov's method of coadjoint orbits. 
  Using a Hamiltonian approach to gauged WZW models, we present a general method for computing the conformally exact metric and dilaton, to all orders in the $1/k$ expansion, for any bosonic, heterotic, or type-II superstring model based on a coset $G/H$. We prove the following relations: (i) For type-II superstrings the conformally exact metric and dilaton are identical to those of the non-supersymmetric {\it semi-classical} bosonic model except for an overall renormalization of the metric obtained by $k\to k- g$. (ii) The exact expressions for the heterotic superstring are derived from their exact bosonic string counterparts by shifting the central extension $k\to 2k-h$ (but an overall factor $(k-g)$ remains unshifted). (iii) The combination $e^\Phi\sqrt{-G}$ is independent of $k$ and therefore can be computed in lowest order perturbation theory as required by the correct formulation of a conformally invariant path integral measure. The general formalism is applied to the coset models $SO(d-1,2)_{-k}/SO(d-1,1)_{-k}$ that are relevant for string theory on curved spacetime. Explicit expressions for the conformally exact metric and dilaton for the cases $d=2,3,4$ are given. In the semiclassical limit $(k\to \infty)$ our results agree with those obtained with the Lagrangian method up to 1-loop in perturbation theory. 
  We study quantum Chern-Simons theory as the large mass limit of the limit $D\to 3$ of dimensionally regularized topologically massive Yang-Mills theory. This approach can also be interpreted as a BRS-invariant hybrid regularization of Chern-Simons theory, consisting of a higher-covariant derivative Yang-Mills term plus dimensional regularization. Working in the Landau gauge, we compute radiative corrections up to second order in perturbation theory and show that there is no two-loop correction to the one-loop shift $k\rightarrow k+ c_{\scriptscriptstyle V},\,\,k$ being the bare Chern-Simons parameter. In passing we also prove by explicit computation that topologically massive Yang-Mills theory is UV finite. 
  We study the modular invariance of $N=2$ superconformal $SU(1,1)$ models. By decomposing the characters of Kazama-Suzuki model $SU(3)/(SU(2)\times U(1))$ into an infinite sum of the characters of $(SU(1,1)/U(1))\times U(1)$ we construct modular invariant partition functions of $(SU(1,1)/U(1))\times U(1)$. 
  We have studied the quantum Liouville theory on the Riemann sphere with n>3 punctures. While considering the theory on the Riemann surfaces with n=4 punctures, the quantum theory near an arbitrary but fixed puncture can be obtained via canonical quantization and an extra symmetry is explored. While considering more than four distinguished punctures, we have found the exchange relations of the monodromy parameters from which we can get a reasonable quantum theory. 
  We investigate the evaluation of the Dirac index using symplectic geometry in the loop space of the corresponding supersymmetric quantum mechanical model. In particular, we find that if we impose a simple first class constraint, we can evaluate the Callias index of an odd dimensional Dirac operator directly from the quantum mechanical model which yields the Atiyah-Singer index of an even dimensional Dirac operator in one more dimension. The effective action obtained by BRST quantization of this constrained system can be interpreted in terms of loop space symplectic geometry, and the corresponding path integral for the index can be evaluated exactly using the recently developed localization techniques. 
  We construct the Zamolodchikov's c-function for the Chiral Gross-Neveu Model up to two loops. We show that the c-function interpolates between the two known critical points of the theory, it is stationary at them and it decreases with the running coupling constant. In particular one can infer the non-existence of additional critical points in the region under investigation. 
  Minor misprints corrected. 
  We describe the quasitriangular structure (universal $R$-matrix) on the non-standard quantum group $U_q(H_1,H_2,X^\pm)$ associated to the Alexander-Conway matrix solution of the Yang-Baxter equation. We show that this Hopf algebra is connected with the super-Hopf algebra $U_qgl(1|1)$ by a general process of superization. 
  We find the discrete states of the c=1 string in the light-cone gauge of Polyakov. When the state space of the gravitational sector of the theory is taken to be the irreducible representations of the SL(2,R) current algebra, the cohomology of the theory is NOT the same as that in the conformal gauge. In particular, states with ghost numbers up to 4 appear. However, after taking the space of the theory to be the Fock space of the Wakimoto free-field representation of the SL(2,R), the light-cone and conformal gauges are equivalent. This supports the contention that the discrete states of the theory are physical. We point out that the natural states in the theory do not satisfy the KPZ constraints. 
  We propose and study at large N a new lattice gauge model , in which the Yang-Mills interaction is induced by the heavy scalar field in adjoint representation. At any dimension of space and any $ N $ the gauge fields can be integrated out yielding an effective field theory for the gauge invariant scalar field, corresponding to eigenvalues of the initial matrix field. This field develops the vacuum average, the fluctuations of which describe the elementary excitations of our gauge theory. At $N= \infty $ we find two phases of the model, with asymptotic freedom corresponding to the strong coupling phase (if there are no phase transitions at some critical $N$). We could not solve the model in this phase, but in the weak coupling phase we have derived exact nonlinear integral equations for the vacuum average and for the scalar excitation spectrum. Presumably the strong coupling equations can be derived by the same method. 
  Classical solutions of equations of motion in low energy effective field theory, describing fundamental charged heterotic string, are found. These solutions automatically carry an electric current equal to the charge per unit length, and hence are accompanied by both, electric and magnetic fields. Force between two parallel strings vanish due to cancellation between electric and magnetic forces, and also between graviton, dilaton, and antisymmetric tensor field induced forces. Multi-string solutions describing configuration of parallel strings are also found. Finally, the solutions are shown to possess partially broken space-time supersymmetry. 
  Two dimensional charged black hole solution is obtained by implementing an $O(2,2)$ transformation on the three dimensional black string solution. Two different monopole backgrounds in five dimensions are related through an $O(2,2)$ transformation. It has been shown in these examples that the particular $O(2,2)$ transformation corresponds to duality transformation. 
  Campbell et al. demonstrated the existence of axion ``hair'' for Kerr black holes due to the non-trivial Lorentz Chern-Simons term and calculated it explicitly for the case of slow rotation. Here we consider the dilaton coupling to the axion field strength, consistent with low energy string theory and calculate the dilaton ``hair'' arising from this specific axion source. 
  We consider the BRST and superconformal properties of the ghost action of 2-D supergravity. Using the background spin structure on the worldsheet, we show that this action can be transformed by canonical field transformations to reach other conformal models such as the 2-D topological gravity or the chiral models for which the gauge variation of the action reproduces the left or right conformal anomaly. Our method consists in using the gravitino and its ghost as fundamental blocks to build fields with different conformal weights and statistics. This indicates in particular that the twisting of a conformal model into another one can be classically interpreted as a change of "field representation" of the superconformal symmetry. 
  In a previous paper it was shown that the quantum consistency conditions for the dilaton-gravity theory of Callan et al., imply that the cosmological constant term undergoes a dilaton dependent renormalization, in such a manner that the theory can be written as a Liouville-like theory. In this paper we discuss the physical interpretation of the solutions of this theory. In particular we demonstrate explicitly how quantum corrections tame the black hole singularity. Also under the assumption that in asymptotically Minkowski coordinates, there are no incoming or outgoing ghosts, we show that the Hawking radiation rate is independent of the number of matter fields and is determined by the ghost conformal anomaly. 
  There is a large mathematical literature on classical mechanics and field theory, especially on the relationship to symplectic geometry. One might think that the classical Chern-Simons theory, which is topological and so has vanishing hamiltonian, is completely trivial. However, this theory exhibits interesting geometry that is usually absent from ordinary field theories. (The same is true on the quantum level; topological quantum field theories exhibit geometric properties not usually seen in ordinary quantum field theories, and they lack analytic properties which are usually seen.) In this paper we carefully develop this geometry. Of particular interest are the line bundles with connection over the moduli space of flat connections on a 2-manifold. We extend the usual theory to cover 2-manifolds with boundary. We carefully develop ``gluing laws'' in all of our constructions, including the line bundle with connection over moduli space. The corresponding quantum gluing laws are fundamental. Part 1 covers connected and simply connected gauge groups; Part 2 will cover arbitrary compact Lie groups. 
  The properties of a string-inspired two-dimensional theory of gravity are studied. The post-Newtonian and weak-field approximations, `stellar' structure and cosmological solutions of this theory are developed. Some qualitative similarities to general relativity are found, but there are important differences. 
  A truncation of string field theory is compared with the duality invariant effective action of $D=4, N=4$ heterotic strings to cubic order. The three string vertex must satisfy a set of compatibility conditions. Any cyclic three string vertex is compatible with the $D=4, N=4$ effective field theory. The effective actions may be useful in understanding the non--polynomial structure and the underlying symmetry of covariant closed string field theory, and in addressing issues of background independence. We also discuss the effective action and string field theory of the $N=2$ string. 
  The algebra dual to Woronowicz's deformation of the 2-\-di\-men\-sion\-al Euclidean group is constructed. The same algebra is obtained from $SU_{q}(2)$ via contraction on both the group and algebra levels. 
  A finite-dimensional su($N$) Lie algebra equation is discussed that in the infinite $N$ limit (giving the area preserving diffeomorphism group) tends to the two-dimensional, inviscid vorticity equation on the torus. The equation is numerically integrated, for various values of $N$, and the time evolution of an (interpolated) stream function is compared with that obtained from a simple mode truncation of the continuum equation. The time averaged vorticity moments and correlation functions are compared with canonical ensemble averages. 
  We generalize to p-dimensional extended objects and type II superstrings a recently proposed Green-Schwarz type I superstring action in which the tension $T$ emerges as an integration constant of the equations of motion. The action is spacetime scale-invariant but its equations of motion are equivalent to those of the standard super p-brane for $T\ne 0$ and the null super p-brane for $T=0$. We also show that for $p=1$ the action can be written in ``Born-Infeld'' form. 
  Free planar electrons in a uniform magnetic field are shown to possess the symmetry of area-preserving diffeomorphisms ($W$-infinity algebra). Intuitively, this is a consequence of gauge invariance, which forces dynamics to depend only on the flux. The infinity of generators of this symmetry act within each Landau level, which is infinite-dimensional in the thermodynamical limit. The incompressible ground states corresponding to completely filled Landau levels (integer quantum Hall effect) are shown to be infinitely symmetric, since they are annihilated by an infinite subset of generators. This geometrical characterization of incompressibility also holds for fractional fillings of the lowest level (simplest fractional Hall effect) in the presence of Haldane's effective two-body interactions. Although these modify the symmetry algebra, the corresponding incompressible ground states proposed by Laughlin are again symmetric with respect to the modified infinite algebra. 
  Explicit divergences and counterterms do not appear in the differential renormalization method, but they are concealed in the neglected surface terms in the formal partial integration procedure used. A systematic real space cutoff procedure for massless $\phi^4$ theory is therefore studied in order to test the method and its compatibility with unitarity. Through 3-loop order, it is found that cutoff bare amplitudes are equal to the renormalized amplitudes previously obtained using the formal procedure plus singular terms which can be consistently cancelled by adding conventional counterterms to the Lagrangian. Renormalization group functions $\beta (g)$ and $\gamma (g)$ obtained in the cutoff theory also agree with previous results. 
  An exterior derivative, inner derivation, and Lie derivative are introduced on the quantum group $GL_{q}(N)$. $SL_{q}(N)$ is then found by constructing matrices with determinant unity, and the induced calculus is found. 
  Small errors are corrected 
  Many $W$-algebras (e.g. the $W_N$ algebras) are consistent for all values of the central charge except for a discrete set of exceptional values. We show that such algebras can be contracted to new consistent degenerate algebras at these exceptional values of the central charge. 
  A closed and explicit formula for all $\su{(3)}_k$ fusion coefficients is presented which, in the limit $k \rightarrow \infty$, turns into a simple and compact expression for the $su(3)$ tensor product coefficients. The derivation is based on a new diagrammatic method which gives directly both tensor product and fusion coefficients. 
  We argue, that for a general class of nontrivial bosonic theories the path integral can be related to an equivariant generalization of conventional characteristic classes. 
  We propose a system of functional relations having a universal form connected to the $U_q(X^{(1)}_r)$ Bethe ansatz equation. Based on the analysis of it, we conjecture a new sum formula for the Rogers dilogarithm function in terms of the scaling dimensions of the $X^{(1)}_r$ parafermion conformal field theory. 
  Matrix models of 2D quantum gravity are either exactly solvable for matter of central charge $ c\leq 1, $ or not understood. It would be useful to devise an approximate scheme which would be reasonable for the known cases and could be carried to the unsolved cases in order to achieve at least a qualitative understanding of the properties of the models. The double scaling limit is an indication that a change of the length scale induces a flow in the parameters of the theory, the size of the matrix and the coupling constants for matrix models, at constant long distances physics. We construct here these renormalization group equations at lowest orders in various cases to check that we reproduce qualitatively the properties of $ c\leq 1 $ models. 
  Quantum field theory is discussed in M\"obius corner kaleidoscopes using the method of images. The vacuum average of the stress-energy tensor of a free field is derived and is shown to be a simple sum of straight cosmic string expressions, the strings running along the edges of the corners. It does not seem possible to set up a spin-half theory easily. 
  Integrability of equations of topological-antitopological fusion (being proposed by Cecotti and Vafa) describing ground state metric on given 2D topological field theory (TFT) model, is proved. For massive TFT models these equations are reduced to a universal form (being independent on the given TFT model) by gauge transformations. For massive perturbations of topological conformal field theory models the separatrix solutions of the equations bounded at infinity are found by the isomonodromy deformations method. Also it is shown that ground state metric together with some part of the underlined TFT structure can be parametrized by pluriharmonic maps of the coupling space to the symmetric space of real positive definite quadratic forms. 
  We show that there is a very simple relationship between differential and dimensional renormalization of low-order Feynman graphs in renormalizable massless quantum field theories. The beauty of the differential approach is that it achieves the same finite results as dimensional renormalization without the need to modify the space time dimension. 
  We study the nonrelativistic limit of the $N=2$ supersymmetric Chern-Simons matter system. We show that in addition to Galilean invariance the model admits a set of symmetries generated by fermionic charges, which can be interpreted as an {\it extended Galilean supersymmetry }. The system also possesses a hidden conformal invariance and then the full group of symmetries is the {\it extended superconformal Galilean} group. We also show that imposing extended superconformal Galilean symmetry determines the values of the coupling constants in such a way that their values in the bosonic sector agree with the values of Jackiw and Pi for which self-dual equation exist. We finally analyze the second quantized version of the model and the two-particle sector. 
  Classical $W$-algebras in higher dimensions are constructed. This is achieved by generalizing the classical Gel'fand-Dickey brackets to the commutative limit of the ring of classical pseudodifferential operators in arbitrary dimension. These $W$-algebras are the Poisson structures associated with a higher dimensional version of the Khokhlov-Zabolotskaya hierarchy (dispersionless KP-hierarchy). The two dimensional case is worked out explicitly and it is shown that the role of Diff$S(1)$ is taken by the algebra of generators of local diffeomorphisms in two dimensions. 
  S-matrices for integrable perturbations of $N=2$ superconformal field theories are studied. The models we consider correspond to perturbations of the coset theory $G_k \times H_{g-h} /H_{k+g-h} $. The perturbed models are closely related to $\hat G$-affine Toda theories with a background charge tuned to $H$. Using the quantum group restriction of the affine Toda theories we derive the S-matrix. 
  We study symmetries between untwisted and twisted strings on asymmetric orbifolds. We present a list of asymmetric orbifold models to possess intertwining currents which convert untwisted string states to twisted ones, and vice versa. We also present a list of heterotic strings on asymmetric orbifolds with supersymmetry between untwisted and twisted string states. Some of properties inherent in asymmetric orbifolds, which are not shared by symmetric orbifolds, are pointed out. 
  Two-dimensional Maxwell-dilaton quantum gravity, which covers a large family of the actions for two-dimensional gravity (in particular, string-inspired models) is investigated. Charged black holes which appear in the theory are briefly discussed. The one-loop divergences in the linear covariant gauges are calculated. It is shown that for some choices of the dilaton potential and dilaton-Maxwell coupling, the theory is one-loop multiplicatively renormalizable (or even finite). A comparison with the divergences structure of four-dimensional Einstein-Maxwell gravity is given. 
  A very general class of Lagrangians which couple scalar fields to gravitation and matter in two spacetime dimensions is investigated. It is shown that a vector field exists along whose flow lines the stress-energy tensor is conserved, regardless of whether or not the equations of motion are satisfied or if any Killing vectors exist. Conditions necessary for the existence of Killing vectors are derived. A new set of 2D black hole solutions is obtained for one particular member within this class of Lagrangians. One such solution bears an interesting resemblance to the 2D string-theoretic black hole, yet contains markedly different thermodynamic properties. 
  The 1/N expansion for the O(N) vector model in four dimensions is reconsidered. It is emphasized that the effective potential for this model becomes everywhere complex just at the critical point, which signals an unstable vacuum. Thus a critical O(N) vector model cannot be consistently defined in the 1/N expansion for four-dimensions, which makes the existence of a double-scaling limit for this theory doubtful. 
  We show that a class of models for particles with internal degrees of freedom are integrable. These systems are basically generalizations of the models of Calogero and Sutherland. The proofs of integrability are based on a recently developed exchange operator formalism. We calculate the wave-functions for the Calogero-like models and find the ground-state wave-function for a Calogero-like model in a position dependent magnetic field. This last model might have some relevance for matrix models of open strings. 
  We show that the quantum-algebra-invariant open spin chains associated with the affine Lie algebras $A^{(1)}_n$ for $n>1$ are integrable. The argument, which applies to a large class of other quantum-algebra-invariant chains, does not require that the corresponding $R$ matrix have crossing symmetry. 
  Using the conformal invariance of the $SL(2,R)\otimes SO(1,1)^{d-2}/SO(1,1)$ coset models we calculate the conformally exact metric and dilaton, to all orders in the $1/k$ expansion. We consider both vector and axial gauging. We find that these cosets represent two different space--time geometries: ($2d$ black hole)$\otimes \IR^{d-2}$ for the vector gauging and ($3d$ black string)$\otimes \IR^{d-3}$ for the axial one. In particular for $d=3$ and for the axial gauging one obtains the exact metric and dilaton of the charged black string model introduced by Horne and Horowitz. If the value of $k$ is finite we find two curvature singularities which degenerate to one in the semi--classical $k\to \infty$ limit. We also calculate the reflection and transmission coefficients for the scattering of a tachyon wave and using the Bogoliubov transformation we find the Hawking temperature. 
  We solve the N-body Calogero problem, \ie N particles in 1 dimension subject to a two-body interaction of the form $\half \sum_{i,j}[ (x_i - x_j)^2 + g/ {(x_i - x_j)^2}]$, by constructing annihilation and creation operators of the form $ a_i^\mp =\frac 1 {\sqrt 2} (x _i \pm i\hat{p}_i )$, where $\hat{p}_i$ is a modified momentum operator obeying %!!!!!!! Heisenberg-type commutation relations with $x_i$, involving explicitly permutation operators. On the other hand, $ D_j =i\,\hat{p}_j$ can be interpreted as a covariant derivative corresponding to a flat connection. The relation to fractional statistics in 1+1 dimensions and anyons in a strong magnetic field is briefly discussed. 
  We apply the method of differential renormalization to two and three dimensional abelian gauge theories. The method is especially well suited for these theories as the problems of defining the antisymmetric tensor are avoided and the calculus involved is impressively simple. The topological and dynamical photon masses are obtained. 
  An O($\tilde{d}$, $\tilde{d}$) transformation is given which relates ungauged string actions to the gauged ones for a large class of models discussed recently by Giveon and Rocek. Interestingly, the transformation is background independent and has a unique matrix representation in a given space-time dimension. 
  We propose a class of purely elastic scattering theories generalising the staircase model of Al. B. Zamolodchikov, based on the affine Toda field theories for simply-laced Lie algebras g=A,D,E at suitable complex values of their coupling constants. Considering their Thermodynamic Bethe Ansatz equations, we give analytic arguments in support of a conjectured renormalisation group flow visiting the neighbourhood of each W_g minimal model in turn. 
  We construct a Goulian-Li-type continuation in the number of insertions of the cosmological constant operator which is no longer restricted to one dimensional target space. The method is applied to the calculation of the three-point and a special four-point correlation function. Various aspects of the emerging analytical structure are discussed. 
  We compute correlation functions in $N=2$ non critical superstrings on the sphere. Our calculations are restrained to the ($s=0$) bulk amplitudes. We show that the four point function factorizes as a consequence of the non-critical kinematics, but differently from the $N=0,1$ cases no extra discrete state appears in the $\hat c\to 1^-$ limit. 
  We argue that the paradoxes associated with infinitely degenerate states, which plague relic particle scenarios for the endpoint of black hole evaporation, may be absent when the relics are horned particles. Most of our arguments are based on simple observations about the classical geometry of extremal dilaton black holes, but at a crucial point we are forced to speculate about classical solutions to string theory in which the infinite coupling singularity of the extremal dilaton solution is shielded by a condensate of massless modes propagating in its infinite horn. We use the nonsingular $c=1$ solution of (1+1) dimensional string theory as a crude model for the properties of the condensate. We also present a brief discussion of more general relic scenarios based on large relics of low mass. 
  Phenomenological and formal restrictions on the evolution of pure into mixed states are discussed. In particular, it is argued that, if energy is conserved, loss of purity is incompatible with the weakest possible form of Lorentz covariance. 
  We present formulas for the Clebsch-Gordan coefficients and the Racah coefficients for the root of unity representations ($N$-dimensional representations with $q^{2N}=1$) of $U_q(sl(2))$. We discuss colored vertex models and colored IRF (Interaction Round a Face) models from the color representations of $U_q(sl(2))$. We construct invariants of trivalent colored oriented framed graphs from color representations of $U_q(sl(2))$. 
  The self-duality equations for the Riemann tensor are studied using the Ashtekar Hamiltonian formulation for general relativity. These equations may be written as dynamical equations for three divergence free vector fields on a three dimensional surface in the spacetime. A simplified form of these equations, describing metrics with a one Killing field symmetry are written down, and it shown that a particular sector of these equations has a Hamiltonian form where the Hamiltonian is an arbitrary function on a two-surface. In particular, any element of the $w_\infty$ algebra may be chosen as a the Hamiltonian.  For a special choice of this Hamiltonian, an infinite set of solutions of the self-duality equations are given. These solutions are parametrized by elements of the $w_\infty$ algebra, which in turn leads to an explicit form of four dimensional complex self-dual metrics that are in one to one correspondence with elements of this algebra. 
  We examine the relation between Polyakov's formulation of two dimensional supergravity and gauged Wess-Zumino-Novikov-Witten models. 
  The KdV and modified KdV integrable hierarchies are shown to be different descriptions of the same 2D gravitational system -- open-closed string theory. Non-perturbative solutions of the multi-critical unitary matrix models map to non-singular solutions of the `renormalisation group' equation for the string susceptibility, $[\tilde{P},Q]=Q$. We also demonstrate that the large N solutions of unitary matrix integrals in external fields, studied by Gross and Newman, equal the non-singular pure closed-string solutions of $[\tilde{P},Q]=Q$. 
  We present a concise method to construct a BRST invariant action for the topological quantum field theories in the Batalin-Vilkovisky antifield formalism. The BV action that is a solution for the master equation is directly obtained by means of the extended forms that involve all the required ghosts and antifields. The BV actions for the non-abelian $BF$ theories (in 4 and higher dimensions) and the Chern-Simons theory are constructed by means of the extended form method. An extension of the $BF$ theory in 4-dimensions to include a ``cosmological term'' is also examined and the close connection with the topological Yang-Mills theory is indicated. 
  It is shown that a renormalizable nonlinear sigma model gives rise to the effective string theory proposed by Polchinski and Strominger. In the presence of long string background, the model contains massive world-sheet degrees of freedom owing to the spontaneous breaking of conformal invariance. 
  We derive the BRST cohomology of the G/G topological model for the case of A^{(1)}_{N-1} . It is shown that at level k={p/q}-N the latter describes the (p,q) W_N minimal model coupled to $W_N$ gravity (plus some extra ``topological sectors"). 
  We report results of two investigations of the double-scaling equations for the unitary matrix models. First, the fixed area partition functions have all positive coefficients only for the first four critical points. This implies that the critical points at $k\ge5$ describe non-unitary continuum theories. Secondly, we examine a conjectured connection to branched polymers, but find that the scaling solutions of the unitary models do not agree with those of a particular model describing branched polymers. 
  We construct a quantum thermal field theory for scalar particles in the case of infinite statistics. The extension is provided by working out the Fock space realization of a "quantum algebra", and by identifying the hamiltonian as the energy operator. We examine the perturbative behavior of this theory and in particular the possible extension of the KLN theorem, and argue that it appears as a stable structure in a quantum field theory context. 
  We study further the r\^ole of the boundary operator $\O_B$ for macroscopic loop length in the stable definition of 2D quantum gravity provided by the $[{\tilde P},Q]=Q$ formulation. The KdV flows are supplemented by an additional flow with respect to the boundary cosmological constant $\sigma$. We numerically study these flows for the $m=1$, $2$ and $3$ models, solving for the string susceptibility in the presence of $\O_B$ for arbitrary coupling $\sigma$. The spectrum of the Hamiltonian of the loop quantum mechanics is continuous and bounded from below by $\sigma$. For large positive $\sigma$, the theory is dominated by the `universal' $m=0$ topological phase present only in the $[{\tilde P},Q]=Q$ formulation. For large negative $\sigma$, the non--perturbative physics approaches that of the $[P,Q]=1$ definition, although there is no path to the unstable solutions of the $[P,Q]=1$ $m$-even models. 
  We discuss some aspects of string cosmology with an emphasis on the role played by the dilaton.   A cosmological scenario based on the assumption that all spatial dimensions are periodic so that winding modes play an important role is reviewed. A possibility of a transition from a `string phase' to the `standard' cosmology via a radiation dominated era and inflation is analysed. We also summarise some recent results about time dependent solutions of tree level string equations. 
  Under the axisymmetry and under the invarance of simultaneous inversion of time and azimuthal angle, we show that the axionic Kerr black hole is the ${\it unique}$ stationary solution of the minimal coupling theory of gravity and the Kalb-Ramond field, which has a regular event horizon, is asymptotically flat and has a finite axion field strength at event horizon. 
  This lecture surveys a few loosely related topics, ranging from the scarcity of quantum field theories -- and the role that this has played, and still plays, in physics -- to paradoxes involving black holes in soluble two dimensional string theory and the question of whether naked singularities might be of even greater interest to string theorists than black holes. 
  The formation and semi-classical evaporation of two-dimensional black holes is studied in an exactly solvable model. Above a certain threshold energy flux, collapsing matter forms a singularity inside an apparent horizon. As the black hole evaporates the apparent horizon recedes and meets the singularity in a finite proper time. The singularity emerges naked and future evolution of the geometry requires boundary conditions to be imposed there. There is a natural choice of boundary conditions which match the evaporated black hole solution onto the linear dilaton vacuum. Below the threshold energy flux no horizon forms and boundary conditions can be imposed where infalling matter is reflected from a time-like naked singularity. All information is recovered at spatial infinity in this case. 
  We discuss the quantum theory of 1+1 dimensional dilaton gravity, which is an interesting model with analogous features to the spherically symmetric gravitational systems in 3+1 dimensions. The functional measures over the metrics and the dilaton field are explicitly evaluated and the diffeomorphism invariance is completely fixed in conformal gauge by using the technique developed in the two dimensional quantum gravity. We argue the relations to the ADM formalism. The physical state conditions reduce to the usual Wheeler-DeWitt equations when the dilaton $\df^2 ~ (=\e^{-2\phi}) $ is large enough compared with $\kappa =(N-51/2)/12$, where $N $ is the number of matter fields. This corresponds to the large mass limit in the black hole geometry. A singularity appears at $\df^2 =\kappa (>0) $. The final stage of the black hole evaporation corresponds to the region $\df^2 \sim \kappa $, where the Liouville term becomes important, which just comes from the measure of the metrics. If $\kappa < 0 $, the singularity disappears. 
  Revisions: reference added to: G. Gilbert, {\sl Nucl.Phys.} {\bf B328}, 159 (1989) 
  We write a Ginzburg-Landau Hamiltonian for a charged order parameter interacting with a background electromagnetic field in 2+1 dimensions. Using the method of Lund we derive a collective coordinate action for vortex defects in the order parameter and demonstrate that the vortices are charged. We examine the classical dynamics of the vortices and then quantize their motion, demonstrating that their peculiar classical motion is a result of the fact that the quantum motion takes place in the lowest Landau level. The classical and quantum motion in two dimensional regions with boundaries is also investigated. The quantum theory is not invariant under magnetic translations. Magnetic translations add total time derivative terms to the collective action, but no extra constants of the motion result. 
  Thesis includes review on the large order behaviour of perturbation theory in quantum mechanical and field theory models; generalization of the Borel summability and strong asymptotic conditions to various (including horn-shaped) regions; discussion of analytic aspects of perturbation theory; examples which demonstrate differences between the Borel summability and generalized one; application to the Rayleigh-Schr\"{o}dinger perturbation theory and to the definition of the operator valued functions. The new summability methods converges in the whole Mittag-Leffeler star of an analytical function and as such is useful for localization of singularities in the complex plane. Their position can be calculated even analytically provided large order behaviour of the Taylor series is known. Method can be implemented numerically as well. 
  Using the reduced WZNW formulation we analyse the classical $W$ orbit content of the space of classical solutions of the $A_2$ Toda theory. We define the quantized Toda field as a periodic primary field of the $W$ algebra satifying the quantized equations of motion.   We show that this local operator can be constructed consistently only in a Hilbert space consisting of the representations corresponding to the minimal models of the $W$ algebra. 
  We consider two types of generalized self-duality conditions for Yang-Mills fields on paracomplex manifolds of arbitrary dimension. We then specialize to $3+3$ dimensions and show how one can obtain the KP equation from these self-duality conditions on $SL(2,R)$ valued gauge fields. 
  We interpret Minkowski black holes as world-sheet {\it spikes } which are related by world-sheet { \it duality} to {\it vortices } that correspond to Euclidean black holes. These world-sheet defects induce defects in the gauge fields of the corresponding coset Wess-Zumino descriptions of spherically-symmetric black holes. The low-temperature target space-time foam is a Minkowski black hole (spike) plasma with confined Euclidean black holes (vortices). The high-temperature phase is a {\it dense} vortex plasma described by a topological gauge field theory on the world-sheet, which possesses enhanced symmetry as in the target space-time singularity at the core of a black hole. Quantum decay via higher-genus effects induces a back-reaction which causes a Minkowski black hole to lose mass until it is indistinguishable from intrinsic fluctuations in the space-time foam. 
  We present a conformal field theory -- obtained from a gauged WZW model -- that describes a closed, inhomogeneous expanding and recollapsing universe in $3+1$ dimensions. A possible violation of cosmic censorship is avoided because the universe recollapses just when a naked singularity was about to form. The model has been chosen to have $c=4$ (or $\widehat c=4$ in the supersymmetric case), just like four dimensional Minkowski space. 
  The static stationary axially symmetric background ("infinite cosmic string") of the $D=4$ string theory provided with an axion charge is studied in the effective theory approach. The most general exact solution is constructed. It is found that the Kalb-Ramond axion charge, present in the string topology $R^{3} \times S^{1}$, produces nontrivial gravitational field configurations which feature horizons. The corresponding ``no-hair'' theorems are proved which stress uniqueness of black strings. Connection of the solutions with the gauged WZWN sigma model constructions on the world sheet is discussed since they are the only target spaces which hide their singularities behind horizons, and thus obey the cosmic censorship conjecture. 
  We investigate the nature of the ground ring of c=1 string theory at the special A-D-E points in the c=1 moduli space associated to discrete subgroups of SU(2). The chiral ground rings at these points are shown to define the A-D-E series of singular varieties introduced by Klein. The non-chiral ground rings relevant to closed-string theory are 3 real dimensional singular varieties obtained as U(1) quotients of the Kleinian varieties. The unbroken symmetries of the theory at these points are the volume-preserving diffeomorphisms of these varieties. The theory of Kleinian singularities has a close relation to that of complex hyperKahler surfaces, or gravitational instantons. We speculate on the relevance of these instantons and of self-dual gravity in c=1 string theory. 
  We formulate quantum gravity in $2+\epsilon$ dimensions in such a way that the conformal mode is explicitly separated. The dynamics of the conformal mode is understood in terms of the oversubtraction due to the one loop counter term. The renormalization of the gravitational dressed operators is studied and their anomalous dimensions are computed. The exact scaling exponents of the 2 dimensional quantum gravity are reproduced in the strong coupling regime when we take $\epsilon\rightarrow0$ limit. The theory possesses the ultraviolet fixed point as long as the central charge $c<25$, which separates weak and strong coupling phases. The weak coupling phase may represent the same universality class with our Universe in the sense that it contains massless gravitons if we extrapolate $\epsilon$ up to 2. 
  Canonical forms are given for the nilpotent BRS operator $\d$ and the covariant `loop space' derivative ${\cal D}_{\m}$ for the p-brane fields for all odd p. The defining characteristic of ${\cal D}_{\m}$ is that it is a functional derivative operator which generalizes the ordinary functional derivative and also commutes with $\d$. Methods of construction for the canonical forms are discussed. 
  The capability of string theories to reproduce at low energy the observed pattern of quark and lepton masses and mixing angles is examined, focusing the attention on orbifold constructions, where the magnitude of Yukawa couplings depends on the values of the deformation parameters which describe the size and shape of the compactified space. A systematic exploration shows that for $Z_3$, $Z_4$, $Z_6$--I and possibly $Z_7$ orbifolds a correct fit of the physical fermion masses is feasible. In this way the experimental masses, which are low--energy quantities, select a particular size and shape of the compactified space, which turns out to be very reasonable (in particular the modulus $T$ defining the former is $T=O(1)$). The rest of the $Z_N$ orbifolds are rather hopeless and should be discarded on the assumption of a minimal $SU(3)\times SU(2)\times U(1)_Y$ scenario. On the other hand, due to stringy selection rules, there is no possibility of fitting the Kobayashi--Maskawa parameters at the renormalizable level, although it is remarked that this job might well be done by non--renormalizable couplings. 
  The complete quantum theory of covariant closed strings is constructed in detail. The action is defined by elementary vertices satisfying recursion relations that give rise to Jacobi-like identities for an infinite chain of string field products. The genus zero string field algebra is the homotopy Lie algebra $L_\infty$, and the higher genus algebraic structure implies the Batalin-Vilkovisky (BV) master equation. From these structures on the off-shell state space, we show how to derive the $L_\infty$ algebra, and the BV equation on physical states, recently constructed in d=2 string theory. The string diagrams are surfaces with minimal area metrics, foliated by closed geodesics of length $2\pi$. These metrics generalize quadratic differentials in that foliation bands can cross. The string vertices are succinctly characterized; they include the surfaces whose foliation bands are all of height smaller than $2\pi$.   --While this is not a review paper, an effort was made to give a fairly complete and accessible account of the quantum closed string field theory.-- 
  The free energy of the Penner model is shown to be closely related to the integral over the two diagonalizing unitary matrices of a complex rectangular matrix. 
  We propose random matrix models which have $N=\half$ supersymmetry in zero dimension. The supersymmetry breaks down spontaneously. It is shown that the double scaling limit can be defined in these models and the breakdown of the supersymmetry remains in the continuum limit. The exact non-trivial partition functions of the string theories corresponding to these matrix models are also obtained. 
  We consider Callan, Giddings, Harvey and Strominger's (CGHS) two dimensional dilatonic gravity with electromagnetic interactions. This model can be also solved classically. Among the solutions describing static black holes, there exist extremal solutions which have zero temperatures. In the extremal solutions, the space-time metric is not singular. We also obtain the solutions describing charged matter (chiral fermions) collapsing into black holes. Through the collapsing, not only future horizon but past horizon is also shifted. The quantum corrections including chiral anomaly are also discussed. In a way similar to CGHS model, the curvature singularity also appeared, except extremal case, when the matter collapsing. The screening effects due to the chiral anomaly have a tendency to cloak the singularity 
  Macroscopic loop correlators are investigated in the hermitian one matrix model with the potential perturbed by the higher order curvature term. In the phase of smooth surfaces the model is equivalent to the minimal conformal matter coupled to gravity. The properties of the model in the intermediate phase are similar to that of the discretized bosonic string with the central charge $C > 1.$ Loop correlators describe the effect of the splitting of the random surfaces. It is shown, that the properties of the surfaces are changed in the intermediate phase because the perturbation modifies the spectrum of the scaling operators. 
  Starting from the Wess-Zumino action associated to the super Weyl anomaly, we determine the local counterterm which allows to pass from this anomaly to the chirally split superdiffeomorphism anomaly (as defined on a compact super Riemann surface without boundary). The counterterm involves the graded extension of the Verlinde functional and the results can be applied to the study of holomorphic factorization of partition functions in superconformal field theory. 
  We obtain in closed form averages of polynomials, taken over hermitian matrices with the Gaussian measure involved in the Kontsevich integral, and prove a conjecture of Witten enabling one to express analogous averages with the full (cubic potential) measure, as derivatives of the partition function with respect to traces of inverse odd powers of the external argument. The proofs are based on elementary algebraic identities involving a new set of invariant polynomials of the linear group, closely related to the general Schur functions. 
  The hierarchical nonlinear super-differential equations are identified which describe universal behavior of the discretized model of $2d$ supergravity recently proposed. This is done by first taking a double scaling limit of the super Virasoro constraints ( at finite $N$) of the model and by rederiving it from the $\tilde{G}_{-1/2}$ constraint and the two reduction of the super KP hierarchy discussed. The double-scaled constraints are found to be described by a twisted scalar and a Ramond fermion. 
  These days, Franco Iachello is {\it the\/} eminent practitioner applying classical and finite groups to physics. In this he is following a tradition at Yale, established by the late Feza Gursey, and succeeding Gursey in the Gibbs chair; Gursey in turn, had Pauli as a mentor. Iachello's striking achievement has been to find an actual realization of arcane supersymmetry within mundane adjacent even-odd nuclei. Thus far this is the only {\it physical\/} use of supersymmetry, and its fans surely must be surprised at the venue. Here we describe the role of $SO(2,1)$ conformal symmetry in non-relativistic Chern--Simons theory: how it acts, how it controls the nature of solutions, how it expands to an infinite group on the manifold of static solutions thereby rendering the static problem completely integrable. Since Iachello has also used the $SO(2,1)$ group in various contexts, this essay is presented to him on the occasion of his fiftieth birthday. 
  Professor M. C. Polivanov and I met only a few times, during my infrequent visits to the-then Soviet Union in the 1970's and 1980's. His hospitality at the Moscow Steclov Institute made the trips a pleasure, while the scientific environment that he provided made them professionally valuable. But it is the human contact that I remember most vividly and shall now miss after his death. At a time when issues of conscience were both pressing for attention and difficult/dangerous to confront, Professor Polivanov made a deep impression with his quiet but adamant commitment to justice. I can only guess at the satisfaction he must have felt when his goal of gaining freedom for Yuri Orlov was attained, and even more so these days when human rights became defensible in his country; it is regrettable that he cannot now enjoy the future that he strived to attain.   One of our joint interests was the Liouville theory,$^{1,\,2}$ which in turn can be viewed as a model for gravity in two-dimensional space-time. Some recent developments in this field are here summarized and dedicated to Polivanov's memory, with the hope that he would have enjoyed knowing about them. 
  To travel into the past, to observe it, perhaps to influence it and correct mistakes of one's youth, has been an abiding fantasy of mankind for as long as we have been aware of a past. Here are described some recent scientific investigations on this topic. 
  We show that a lattice model for induced lattice QCD which was recently proposed by Kazakov and Migdal has a $Z_N$ gauge symmetry which, in the strong coupling phase, results in a local confinement where only color singlets are allowed to propagate along links and all Wilson loops for non-singlets average to zero. We argue that, if this model is to give QCD in its continuum limit, it must have a phase transition. We give arguments to support presence of such a phase transition. 
  A simple description of the KP hierarchy and its multi-hamiltonian structure is given in terms of two Bose currents. A deformation scheme connecting various W-infinity algebras and relation between two fundamental nonlinear structures are discussed. Properties of Fa\'a di Bruno polynomials are extensively explored in this construction. Applications of our method are given for the Conformal Affine Toda model, WZNW models and discrete KP approach to Toda lattice chain. 
  The Jacobian for infinitesimal BRST transformations of path integrals for pure Yang-Mills theory, viewed as a matrix $\unity +\Delta J$ in the space of Yang-Mills fields and (anti)ghosts, contains off-diagonal terms. Naively, the trace of $\Delta J$ vanishes, being proportional to the trace of the structure constants. However, the consistent regulator $\cR$, constructed from a general method, also contains off-diagonal terms. An explicit computation demonstrates that the regularized Jacobian $Tr\ \Delta J\exp -\cR /M^2$ for $M^2\rightarrow \infty $ is the variation of a local counterterm, which we give. This is a direct proof at the level of path integrals that there is no BRST anomaly. 
  In this paper we study arbitrary $W$ algebras related to embeddings of $sl_2$ in a Lie algebra $g$. We give a simple formula for all $W$ transformations, which will enable us to construct the covariant action for general $W$ gravity. It turns out that this covariant action is nothing but a Fourier transform of the WZW action. The same general formula provides a geometrical interpretation of $W$ transformations: they are just homotopy contractions of ordinary gauge transformations. This is used to argue that the moduli space relevant to $W$ gravity is part of the moduli space of $G$-bundles over a Riemann surface. 
  We compute the critical exponents of $d = 1$ string theory to leading order, using the renormalization group approach recently suggested by Br\'{e}zin and Zinn-Justin. 
  A classical two dimensional theory of gravity which has a number of interesting features (including a Newtonian limit, black holes and gravitational collapse) is quantized using conformal field theoretic techniques. The critical dimension depends upon Newton's constant, permitting models with $d=4$. The constraint algebra and scaling properties of the model are computed. 
  Starting from the new minimal multiplet of supergravity in $2+2$ dimensions, we construct two types of self-dual supergravity theories. One of them involves a self-duality condition on the Riemann curvature and implies the equations of motion following from the Hilbert-Einstein type supergravity action. The other one involves a self-duality condition on a {\it torsionful} Riemann curvature with the torsion given by the field-strength of an antisymmetric tensor field, and implies the equations of motion that follow from an $R^2$-type action. 
  The flow of the action induced by changing $N$ is computed in large $N$ matrix models. It is shown that the change in the action is non-analytic. This non-analyticity appears at the origin of the space of matrices if the action is even. 
  In Hawking's Euclidean path integral approach to quantum gravity, the partition function is computed by summing contributions from all possible topologies. The behavior such a sum can be estimated in three spacetime dimensions in the limit of small cosmological constant. The sum over topologies diverges for either sign of $\Lambda$, but for dramatically different reasons: for $\Lambda>0$, the divergent behavior comes from the contributions of very low volume, topologically complex manifolds, while for $\Lambda<0$ it is a consequence of the existence of infinite sequences of relatively high volume manifolds with converging geometries. Possible implications for four-dimensional quantum gravity are discussed. 
  We generalize our results${}^1$ on charged topological solitons (CTS) in $4+1$ dimensional $SU(3)$ Yang-Mills-Chern-Simons (YMCS) theory to $SU(N)$. The $SU(N)$ multiplet structure of two classes of solitons associated with the maximal embeddings $SU(2)\times U(1)^{N-2}\subset SU(N)$ and $SO(3)\times U(1)^{N-3}\subset SU(N)$ and the vital role of the $SU(N)$ multiplet of topological currents is clarified. In the case of the first embedding one obtains a $^{N}C_{2}-$plet of CTS. In the second, for $N = 3$, one obtains neutral solitons which, though (classically) spinless, have magnetic moments. For $N \geq 4$, after modding out the above mentioned non-particulate feature, one obtains $^{N}C_{3}$ plets of CTS. 
  We investigate the embedding variable approach to geometrodynamics advocated in work by Isham, Kucha\v{r} and Unruh for a general class of coordinate conditions that mirror the Isham-Kucha\v{r} Gaussian condition but allow for arbitrary algebraic complexity. We find that the same essential structure present in the ultralocal Gaussian condition is repeated in the general case. The resultant embedding--extended phase space contains a full representation of the Lie algebra of the spacetime diffeomorphism group as well as a consistent pure gravity sector. 
  We produce the general solution of the Wess-Zumino consistency condition for gauge theories of the Yang-mills type, for any ghost number and form degree. We resolve the problem of the cohomological independence of these solutions. In other words we fully describe the local version of the cohomology of the BRS operator, modulo the differential on space--time. This in particular includes the presence of external fields and non--trivial topologies of space--time. 
  We analyse in detail the $SL(2,R)$ black hole by extending standard techniques of Kac-Moody current algebra to the non-compact case. We construct the elements of the ground ring and exhibit W-infinity type structure in the fusion algebra of the discrete states. As a consequence, we can identify some of the exactly marginal deformations of the black hole. We show that these deformations alter not only the spacetime metric but also turn on non-trivial backgrounds for the tachyon and all of the massive modes of the string. 
  The nonlinear scalar-field realisation of $w_{1+\infty}$ symmetry in $d=2$ dimensions is studied in analogy to the nonlinear realisation of $d=4$ conformal symmetry $SO(4,2)$. The $w_{1+\infty}$ realisation is derived from a coset-space construction in which the divisor group is generated by the non-negative modes of the Virasoro algebra, with subsequent application of an infinite set of covariant constraints. The initial doubly-infinite set of Goldstone fields arising in this construction is reduced by the covariant constraints to a singly-infinite set corresponding to the Cartan-subalgebra generators $v^\ell_{-(\ell+1)}$. We derive the transformation rules of this surviving set of fields, finding a triangular structure in which fields transform into themselves or into lower members of the set only. This triangular structure gives rise to finite-component subrealisations, including the standard one for a single scalar. We derive the Maurer-Cartan form and discuss the construction of invariant actions. 
  We present explicit expressions for the Maurer-Cartan forms of the superdiffeomorphism group associated to a super Riemann surface. As an application to superconformal field theory, we use these forms to evaluate the effective action for the factorized superdiffeomorphism anomaly. 
  In the first part of this paper we investigate the operator aspect of higher-rank supersymmetric model which is introduced as a Lie theoretic extension of the $N=2$ minimal model with the simplest case $su(2)$ corresponding to the $N=2$ minimal model.   In particular we identify the analogs of chirality conditions and chiral ring. In the second part we construct a class of topological conformal field theories starting with this higher-rank supersymmetric model. We show the BRST-exactness of the twisted stress-energy tensor, find out physical observables and discuss how to make their correlation functions. It is emphasized that in the case of $su(2)$ the topological field theory constructed in this paper is distinct from the one obtained by twisting the $N=2$ minimal model through the usual procedure. 
  We analyze the new states that have recently been discovered in 2D string theory by E. Witten and B. Zwiebach. Since the Liouville direction is uncompactified, we show that the deformations by the new ghost number two states generate equivalent classical solutions of the string fields. We argue that the new ghost number one states are responsible for generating transformations which relate such equivalent solutions. We also discuss the possible interpretation of higher ghost number states of those kinds. 
  We present the results of a Monte--Carlo simulation of the $G_2^{(1)}$ Affine Toda field theory action in two dimensions. We measured the ratio of the masses of the two fundamental particles as a function of the coupling constant. Our results strongly support the conjectured duality with the $D_4^{(3)}$ theory, and are consistent with the mass formula of Delius et al. 
  We develop a phase space path-integral approach for deriving the Lagrangian realization of the models defined by Hamiltonian reduction of the WZNW theory. We illustrate the uses of the approach by applying it to the models of non-Abelian chiral bosons, $W$-algebras and the GKO coset construction, and show that the well-known Sonnenschein's action, the generalized Toda action and the gauged WZNW model are precisely the Lagrangian realizations of those models, respectively. 
  By viewing the Sine-Gordon and massive Thirring models as perturbed conformal field theories one sees that they are different (the difference being observable, for instance, in finite-volume energy levels). The UV limit of the former (SGM) is a gaussian model, that of the latter (MTM) a so-called {\it fermionic} gaussian model, the compactification radius of the boson underlying both theories depending on the SG/MT coupling. (These two families of conformal field theories are related by a ``twist''.) Corresponding SG and MT models contain a subset of fields with identical correlation functions, but each model also has fields the other one does not, e.g. the fermion fields of MTM are not contained in SGM, and the {\it bosonic} soliton fields of SGM are not in MTM. Our results imply, in particular, that the SGM at the so-called ``free-Dirac point'' $\beta^2 = 4\pi$ is actually a theory of two interacting bosons with diagonal S-matrix $S=-1$, and that for arbitrary couplings the overall sign of the accepted SG S-matrix in the soliton sector should be reversed. More generally, we draw attention to the existence of new classes of quantum field theories, analogs of the (perturbed) fermionic gaussian models, whose partition functions are invariant only under a subgroup of the modular group. One such class comprises ``fermionic versions'' of the Virasoro minimal models. 
  The covariant form of the field equations for two--dimensional $R^2$--gravity with torsion as well as its Hamiltonian formulation are shown to suggest the choice of the light--cone gauge. Further a one--to--one correspondence between the Hamiltonian gauge symmetries and the diffeomorphisms and local Lorentz transformations is established, thus proving that there are no hidden local symmetries responsible for the complete integrability of the model. Finally the constraint algebra is shown to have no quantum anomalies so that Dirac's quantization should be applicable. 
  We study the implications of duality symmetry on the analyticity properties of the partition function as it depends upon the compactification length. In order to obtain non-trivial compactifications, we give a physical prescription to get the Helmholtz free energy for any heterotic string supersymmetric or not. After proving that the free energy is always invariant under the duality transformation $R\rightarrow \alpha^{'}/(4R)$ and getting the zero temperature theory whose partition function corresponds to the Helmholtz potential, we show that the self-dual point $R_{0}=\sqrt{\alpha^{'}}/2$ is a generic singularity as the Hagedorn one. The main difference between these two critical compactification radii is that the term producing the singularity at the self-dual point is finite for any $R \neq R_{0}$. We see that this behavior at $R_{0}$ actually implies a loss of degrees of freedom below that point. 
  We use path-\-integral methods to derive the ground state wave functions of a number of two-\-dimensional fermion field theories and related systems in one-\-dimensional many body physics. We derive the exact wave function for the Thirring/Luttinger and Coset fermion models and apply our results to derive the universal behavior of the wave functions of the Heisenberg antiferromagnets and of the Sutherland model. We find explicit forms for the wave functions in the density and in the Grassmann representations. We show that these wave functions always have the Jastrow factorized form and calculated the exponent. Our results agree with the exponents derived from the Bethe Ansatz for the Sutherland model and the Haldane-\-Shastri spin chain but apply to all the systems in the same universality class. 
  The BRST invariance condition in a highest-weight representation of the topological ($\equiv$ twisted $N=2$) algebra captures the `invariant' content of two-dimensional gravity coupled to matter. The standard DDK formulation is recovered by splitting the topological generators into $c=-26$ reparametrization ghosts+matter +`Liouville', while a similar splitting involving $c=-2$ ghosts gives rise to the matter dressed in exactly the way required in order that the theory be equivalent to Virasoro constraints on the KP hierarchy. The two dressings of matter with the `Liouville' differ also by their `ghost numbers', which is similar to the existence of representatives of BRST cohomologies with different ghost numbers. The topological central charge $\ctop\neq3$ provides a two-fold covering of the allowed region $d\leq1\cup d\geq25$ of the matter central charge $d$ via $d=(\ctop+1)(\ctop+6)/(\ctop-3)$. The `Liouville' field is identified as the ghost-free part of the topological $U(1)$ current. The construction thus allows one to establish a direct relation (presumably an equivalence) between the Virasoro-constrained KP hierarchies, minimal models, and the BRST invariance condition for highest-weight states of the topological algebra. 
  The tensionless limit of the free bosonic string is space-time conformally symmetric classically. Requiring invariance of the quantum theory in the light cone gauge tests the reparametrization symmetry needed to fix this gauge. The full conformal symmetry gives stronger constraints than the Poincar\'e subalgebra. We find that the symmetry may be preserved in any space-time dimension, but only if the spectrum is drastically reduced (part of this reduction is natural in a zero tension limit of the ordinary string spectrum). The quantum states are required to be symmetric ({\it i.e.} singlets) under space-time diffeomorphisms, except for the centre of mass wave function. 
  We study canonical quantization of a class of 2d dilaton gravity models, which contains the model proposed by Callan, Giddings, Harvey and Strominger. A set of non-canonical phase space variables is found, forming an $SL(2,{\bf R}) \times U(1)$ current algebra, such that the constraints become quadratic in these new variables. In the case when the spatial manifold is compact, the corresponding quantum theory can be solved exactly, since it reduces to a problem of finding the cohomology of a free-field Virasoro algebra. In the non-compact case, which is relevant for 2d black holes, this construction is likely to break down, since the most general field configuration cannot be expanded into Fourier modes. Strategy for circumventing this problem is discussed. 
  We construct the enveloping fundamental spin model of the t-J hamiltonian using the Quantum Inverse Scattering Method (QISM), and present all three possible Algebraic Bethe Ans\"atze. Two of the solutions have been previously obtained in the framework of Coordinate Space Bethe Ansatz by Sutherland and by Schlottmann and Lai, whereas the third solution is new. The formulation of the model in terms of the QISM enables us to derive explicit expressions for higher conservation laws. 
  The two--dimensional topological BF model is considered in the Landau gauge in the framework of perturbation theory. Due to the singular behaviour of the ghost propagator at long distances, a mass term to the ghost fields is introduced as infrared regulator. Relying on the supersymmetric algebraic structure of the resulting massive theory, we study the infrared and ultraviolet renormalizability of the model, with the outcome that it is perturbatively finite. 
  We study the classical and quantum $G$ extended superconformal algebras from the hamiltonian reduction of affine Lie superalgebras, with even subalgebras $G\oplus sl(2)$. At the classical level we obtain generic formulas for the Poisson bracket structure of the algebra. At the quantum level we get free field (Feigin-Fuchs) representations of the algebra by using the BRST formalism and the free field realization of the affine Lie superalgebra. In particular we get the free field representation of the $sl(2)\oplus sp(2N)$ extended superconformal algebra from the Lie superalgebra $osp(4|2N)$. We also discuss the screening operators of the algebra and the structure of singular vectors in the free field representation. 
  We present explicit free field representations for the $N=4$ doubly extended superconformal algebra, $\tilde{\cal{A}}_{\gamma}$. This algebra generalizes and contains all previous $N=4$ superconformal algebras. We have found $\tilde{\cal{A}}_{\gamma}$ to be obtained by hamiltonian reduction of the Lie superalgebra $D(2|1;\alpha)$. In addition, screening operators are explicitly given and the associated singular vectors identified. We use this to present a natural conjecture for the Kac determinant generalizing a previous conjecture by Kent and Riggs for the singly extended case. The results support and illuminate several aspects of the characters of this algebra previously obtained by Taormina and one of us. 
  We apply the method of coadjoint orbits of \winf-algebra to the problem of non-relativistic fermions in one dimension. This leads to a geometric formulation of the quantum theory in terms of the quantum phase space distribution of the fermi fluid. The action has an infinite series expansion in the string coupling, which to leading order reduces to the previously discussed geometric action for the classical fermi fluid based on the group $w_\infty$ of area-preserving diffeomorphisms. We briefly discuss the strong coupling limit of the string theory which, unlike the weak coupling regime, does not seem to admit of a two dimensional space-time picture. Our methods are equally applicable to interacting fermions in one dimension. 
  The Polyakov's "soldering procedure" which shows how two-dimensional diffeomorphisms can be obtained from SL(2,R) gauge transformations is discussed using the free-field representation of SL(2,R) current algebra. Using this formalism, the relation of Polyakov's method to that of the Hamiltonian reduction becomes transparent. This discussion is then generalised to N=1 superdiffeomorphisms which can be obtained from N=1 super Osp(1,2) gauge transformations. It is also demonstrated that the phase space of the Osp(2,2) supercurrent algebra represented by free superfields is connected to the classical phase space of N=2 superconformal algebra via Hamiltonian reduction.} 
  A two-boson realization of the second hamiltonian structure for the KP hierarchy has recently appeared in the literature. Furthermore, it has been claimed that this is also a realization of the hierarchy itself. This is surprising because it would mean that the dynamics of the KP hierarchy---which in its usual formulation requires an infinite number of fields---can be described with only two. The purpose of this short note is to point out the almost obvious fact that the hierarchy described by the two bosons is not the KP hierarchy but rather a reduction thereof---one which is moreover incompatible with the reduction to the KdV-type subhierarchies. 
  We show that, the lattice regularization of chiral gauge theories proposed by Kaplan, when applied to a (2+1)-dimensional domain wall, produces a (1+1)-dimensional theory at low energy even if gauge anomaly produced by chiral fermions does not cancel. But the corresponding statement is not true in higher dimensions. 
  We use the method of the tensor product graph to construct rational (Yangian invariant) solutions of the Yang-Baxter equation in fundamental representations of $c_n$ and thence the full set of $c_n$-invariant factorized $S$-matrices. Brief comments are made on their bootstrap structure and on Belavin's scalar Yangian conserved charges. 
  Noncompact groups, similar to those that appeared in various supergravity theories in the 1970's, have been turning up in recent studies of string theory. First it was discovered that moduli spaces of toroidal compactification are given by noncompact groups modded out by their maximal compact subgroups and discrete duality groups. Then it was found that many other moduli spaces have analogous descriptions. More recently, noncompact group symmetries have turned up in effective actions used to study string cosmology and other classical configurations. This paper explores these noncompact groups in the case of toroidal compactification both from the viewpoint of low-energy effective field theory, using the method of dimensional reduction, and from the viewpoint of the string theory world sheet. The conclusion is that all these symmetries are intimately related. In particular, we find that Chern--Simons terms in the three-form field strength $H_{\mu\nu\rho}$ play a crucial role. 
  This review is devoted to the application of bosonization techniques to two dimensional QCD. We start with a description of the ``abelian bosonization". The methods of the abelian bosonization are applied to several examples like the Thirring model, the Schwinger model and QCD$_2$. The failure of this scheme to handle flavored fermions is explained. Witten's non-abelian bosonization rules are summarized including the generalization to the case of fermions with color and flavor degrees of freedom. We discuss in details the bosonic version of the mass bilinear of colored-flavored fermions in various schemes. The color group is gauged and the full bosonized version of massive multiflavor QCD is written down. The strong coupling limit is taken in the ``product scheme" and then in the $U(N_F\times N_C)$ scheme. Once the multiflavor $QCD_2$ action in the interesting region of the low energies is written down, we extract the semiclassical low lying baryonic spectrum. First classical soliton solutions of the bosonic action are derived. Quantizing the flavor space around those classical solutions produces the masses as well as the flavor properties of the two dimensional baryons. In addition low lying multibaryonic solutions are presented, as well as wave functions and matrix elements of interest, like $q\bar q$ content. 
  We consider $N=1$ supersymmetric Toda theories which admit a fermionic untwisted affine extension, i.e. the systems based on the $A(n,n)$, $D(n+1,n)$ and $B(n,n)$ superalgebras. We construct the superspace Miura trasformations which allow to determine the W-supercurrents of the conformal theories and we compute their renormalized expressions. The analysis of the renormalization and conservation of higher-spin currents is then performed for the corresponding supersymmetric massive theories. We establish the quantum integrability of these models and show that although their Lagrangian is not hermitian, the masses of the fundamental particles are real, a property which is maintained by one-loop corrections. The spectrum is actually much richer, since the theories admit solitons. The existence of quantum conserved higher-spin charges implies that elastic, factorized S-matrices can be constructed. 
  Two series of W-algebras with two generators are constructed from chiral vertex operators of a free field representation. If $c = 1 - 24k$, there exists a W(2,3k) algebra for k in $Z_{+}/2$ and a W(2,8k) algebra for k in $Z_{+}/4$. All possible lowest-weight representations, their characters and fusion rules are calculated proving that these theories are rational. It is shown, that these non-unitary theories complete the classification of all rational theories with effective central charge $c_{eff} = 1$. The results are generalized to the case of extended supersymmetric conformal algebras. 
  We describe an iterative scheme which allows us to calculate any multi-loop correlator for the complex matrix model to any genus using only the first in the chain of loop equations. The method works for a completely general potential and the results contain no explicit reference to the couplings. The genus $g$ contribution to the $m$--loop correlator depends on a finite number of parameters, namely at most $4g-2+m$. We find the generating functional explicitly up to genus three. We show as well that the model is equivalent to an external field problem for the complex matrix model with a logarithmic potential. 
  Within the Quantum Action Principle framework we show the perturbative renormalizability of previously proposed topological lagrangian \`a la Witten-Fujikawa describing polymers, then we perform a 2 loop computation. The theory turns out to have the same predictive power of De Gennes theory, even though its running coupling constants exhibit a very peculiar behaviour. Moreover we argue that the theory presents two phases , a topological and a non topological one. 
  We calculate the low-lying part of the spectrum of the $Z_3$-symmetrical chiral Potts quantum chain in its self-dual and integrable versions, using numerical diagonalisation of the hamiltonian for $N \leq 12$ sites and extrapolation $N \ra \infty$. From the sequences of levels crossing we show that the massive phases have oscillatory correlation functions. We calculate the wave vector scaling exponent. In the high-temperature massive phase the pattern of the low-lying levels can be explained assuming the existence of two particles, with $Z_3$-charge $Q\!=\!1$ and $Q\!=\!2$, and their scattering states. In the superintegrable case the $Q\!=\!2$-particle has twice the mass of the $Q\!=\!1$-particle. Exponential convergence in $N$ is observed for the single particle gaps, while power convergence is seen for the scattering levels. In the high temperature limit of the self-dual model the parity violation in the particle dispersion relation is equivalent to the presence of a macroscopic momentum $P_m = \pm \vph/3$, where $\vph$ is the chiral angle. 
  We study the chiral rings in N=2 and N=4 superconformal algebras. The chiral primary states of N=2 superconformal algebras realized over hermitian triple systems are given. Their coset spaces G/H are hermitian symmetric which can be compact or non-compact. In the non-compact case, under the requirement of unitarity of the representations of G we find an infinite set of chiral primary states associated with the holomorphic discrete series representations of G. Further requirement of the unitarity of the corresponding N=2 module truncates this infinite set to a finite subset. The chiral primary states of the N=2 superconformal algebras realized over Freudenthal triple systems are also studied. These algebras have the special property that they admit an extension to N=4 superconformal algebras with the gauge group SU(2)XSU(2)XU(1). We generalize the concept of the chiral rings to N=4 superconformal algebras. We find four different rings associated with each sector (left or right moving). We also show that our analysis yields all the possible rings of N=4 superconformal algebras. 
  We show that the N=2 open string describes a theory of self-dual Yang Mills (SDYM) in (2,2) dimensions. The coupling to the closed sector is described by SDYM in a Kahler background, with the Yang-Mills fields providing a source term to the self-duality equation in the gravity sector. The four-point S-matrix elements of the theory vanish, so the tree-level unitarity constraints leading to the Chan-Paton construction are relaxed. By considering more general group-theory ansatze the N=2 string can be written for any gauge group, and not just the classical groups allowed for the bosonic and N=1 strings. Such ad hoc group-theory factors can not be appended to the closed N=2 string, explaining why the Z_n closed N=2 strings are trivial extensions of the Z_1 theory. 
  The leading and the subleading Landau singularities in affine Toda field theories are examined in some detail. Formulae describing the subleading simple pole structure of box diagrams are given explicitly. This leads to a new and nontrivial test of the conjectured exact S-matrices for these theories. We show that to the one-loop level the conjectured S-matrices of the $A_n$ Toda family reproduce the correct singularity structure, leading as well as subleading, of the field theoretical amplitudes. The present test has the merit of being independent of the details of the renormalisations. 
  For the special case of the quantum group $SL_q (2,{\bf C})\ (q= \exp \pi i/r,\ r\ge 3)$ we present an alternative approach to quantum gauge theories in two dimensions. We exhibit the similarities to Witten's combinatorial approach which is based on ideas of Migdal. The main ingredient is the Turaev-Viro combinatorial construction of topological invariants of closed, compact 3-manifolds and its extension to arbitrary compact 3-manifolds as given by the authors in collaboration with W. Mueller. 
  On the example of nonabelian Toda type theory associated with the Lie superalgebra $osp(2|4)$ we show that this integrable dynamical system is relevant to a black hole background metric in the corresponding target space. In the even sector the model under consideration reduces to the exactly solvable conformal theory (nonabelian $B_2$ Toda system) in the presence of a black hole recently proposed in the article "Black holes from non-abelian Toda theories" by the last two authors (hep-th 9203039). 
  It is shown that the probability distribution $P(\lambda)$ for the effective cosmological constant is sharply peaked at $\lambda=0$ in stochastic (or "fifth-time") stabilized quantum gravity. The effect is similar to the Baum-Hawking mechanism, except that it comes about due to quantum fluctuations, rather than as a zeroth-order (in $\hbar$) semiclassical effect. 
  $Vect(N)$, the algebra of vector fields in $N$ dimensions, is studied. Some aspects of local differential geometry are formulated as $Vect(N)$ representation theory. There is a new class of modules, {\it conformal fields}, whose restrictions to the subalgebra $sl(N+1) \subset Vect(N)$ are finite-dimensional $sl(N+1)$ representations. In this regard they are simpler than tensor fields. Fock modules are also constructed. Infinities, which are unremovable even by normal ordering, arise unless bosonic and fermionic degrees of freedom match. 
  Conformal fields are a recently discovered class of representations of the algebra of vector fields in $N$ dimensions. Invariant first-order differential operators (exterior derivatives) for conformal fields are constructed. 
  Sets of commuting charges constructed from the current of a U(1) Kac-Moody algebra are found. There exists a set S_n of such charges for each positive integer n > 1; the corresponding value of the central charge in the Feigin-Fuchs realization of the stress tensor is c = 13-6n-6/n. The charges in each series can be written in terms of the generators of an exceptional W-algebra. 
  It is shown that the sl(2,C) KZ equation for (half-) integer isospins recovers, up to a gauge transformation, the matrix system for Virasoro algebra singular vectors of Bauer et al. In the case of Kac-Kazhdan spins the general (infinite matrix) KZ system is truncated due to the decoupling of the A^(1)_1 singular vectors. This suggests an algorithm converting Malikov-Feigin-Fuks singular vectors into Virasoro ones. 
  The particle detector model consisting of a harmonic oscillator coupled to a scalar field in $1+1$ dimensions is investigated in the inertial case. The same approach is then used in the accelerating case. The absence of radiation from a uniformly accelerated detector in a stationnary state is discussed and clarified. 
  Quantization of two-dimensional dilaton gravity coupled to conformal matter is investigated. Working in conformal gauge about a fixed background metric, the theory may be viewed as a sigma model whose target space is parameterized by the dilaton $\phi$ and conformal factor $\rho$. A precise connection is given between the constraint that the theory be independent of the background metric and conformal invariance of the resulting sigma model. Although the action is renormalizable, new coupling constants must be specified at each order in perturbation theory in order to determine the quantum theory. These constants may be viewed as initial data for the beta function equations. It is argued that not all choices of this data correspond to physically sensible theories of gravity, and physically motivated constraints on the data are discussed. In particular a recently constructed subclass of initial data which reduces the full quantum theory to a soluble Liouville-like theory has energies unbounded from below and thus is unphysical. Possibilities for modifying this construction so as to avoid this difficulty are briefly discussed. 
  A unified treatment of both superconformal and quasisuperconformal algebras with quadratic non-linearity is given. General formulas describing their structure are found by solving the Jacobi identities. A complete classification of quasisuperconformal and ${\bf Z}_2\times{\bf Z}_2$-graded algebras are obtained and in addition to the previously known cases five exceotional quasisuperconformal algebras and a series of ${\bf Z}_2\times{\bf Z}_2$-superconformal algebras containing affine $\widehat{sp}_2\otimes\widehat{osp}(N|2M)$ are constructed. 
  Recent work on Euler hierarchies of field theory Lagrangians iteratively constructed {}from their successive equations of motion is briefly reviewed. On the one hand, a certain triality structure is described, relating arbitrary field theories, {\it classical\ts} topological field theories -- whose classical solutions span topological classes of manifolds -- and reparametrisation invariant theories -- generalising ordinary string and membrane theories. On the other hand, {\it finite} Euler hierarchies are constructed for all three classes of theories. These hierarchies terminate with {\it universal\ts} equations of motion, probably defining new integrable systems as they admit an infinity of Lagrangians. Speculations as to the possible relevance of these theories to quantum gravity are also suggested. 
  We prove the title. 
  We discuss the quantization of theories which are formulated using compensating fields. In particular, we discuss the relation between the components formulation and the superspace formulation of supergravity theories. The requirement that the compensating field can be eliminated at the quantum level gives rise to on-shell constraints on the operators of the theory. In some cases, the constraints turn out to be physically unacceptable. Using these considerations we show that new minimal supergravity is in general anomalous. 
  We study a three dimensional analogue of the Wess--Zumino--Witten model, which describes the Goldstone bosons of three dimensional Quantum Chromodynamics. The topologically non--trivial term of the action can also be viewed as a nonlinear realization of Chern--Simons form. We obtain the current algebra of this model by canonical methods. This is a three dimensional generalization of the Kac--Moody algebra. 
  We prove a useful identity valid for all $ADE$ minimal S-matrices, that clarifies the transformation of the relative thermodynamic Bethe Ansatz (TBA) from its standard form into the universal one proposed by Al.B.Zamolodchikov. By considering the graph encoding of the system of functional equations for the exponentials of the pseudoenergies, we show that any such system having the same form as those for the $ADE$ TBA's, can be encoded on $A,D,E,A/Z_2$ only. This includes, besides the known $ADE$ diagonal scattering, the set of all $SU(2)$ related {\em magnonic} TBA's. We explore this class sistematically and find some interesting new massive and massless RG flows. The generalization to classes related to higher rank algebras is briefly presented and an intriguing relation with level-rank duality is signalled. 
  The Vassiliev-Gusarov link invariants of finite type are known to be closely related to perturbation theory for Chern-Simons theory. In order to clarify the perturbative nature of such link invariants, we introduce an algebra V_infinity containing elements g_i satisfying the usual braid group relations and elements a_i satisfying g_i - g_i^{-1} = epsilon a_i, where epsilon is a formal variable that may be regarded as measuring the failure of g_i^2 to equal 1. Topologically, the elements a_i signify crossings. We show that a large class of link invariants of finite type are in one-to-one correspondence with homogeneous Markov traces on V_infinity. We sketch a possible application of link invariants of finite type to a manifestly diffeomorphism-invariant perturbation theory for quantum gravity in the loop representation. 
  We study supersymmetry and self-duality in a four-dimensional space-time with the signature (2,2), that we call the Atiyah-Ward space-time. Dirac matrices and spinors, in particular Majorana-Weyl spinors, are investigated in detail. We formulate $ N\ge 1 $ supersymmetric self-dual Yang-Mills theories and self-dual supergravities. An N=1 ``self-dual'' tensor multiplet is constructed and a possible ten-dimensional theory that gives rise to the four-dimensional self-dual supersymmetric theories is found. Instanton solutions are given as the zero modes in the N=2 self-dual Yang-Mills theory. The N=2 superstrings are conjectured to have no possible counter-terms at quantum level to all orders. These self-dual supersymmetric theories are to generate exactly soluble supersymmetric systems in lower dimensions. 
  As open N=2 or 4 strings describe self-dual N=4 super Yang-Mills in 2+2 dimensions, the corresponding closed (heterotic) strings describe self-dual ungauged (gauged) N=8 supergravity. These theories are conveniently formulated in a chiral superspace with general supercoordinate and local OSp(8|2) gauge invariances. The super-light-cone and covariant-component actions are analyzed. Because only half the Lorentz group is gauged, the gravity field equation is just the vanishing of the torsion. 
  Corrected with some rewording 
  We consider some questions of naturalness which arise when one considers conventional field theories in the presence of gravitation: the problem of global symmetries, the strong CP problem, and the cosmological constant problem. Using string theory as a model, we argue that it is reasonable to postulate weakly broken global discrete symmetries. We review the arguments that gravity is likely to spoil the Peccei-Quinn solution of the strong CP problem, and update earlier analyses showing that discrete symmetries can lead to axions with suitable properties. Even if there are not suitable axions, we note that string theory is a theory in which CP is spontaneously broken and $\theta$ in principle calculable. $\theta$ thus might turn out to be small along lines suggested some time ago by Nelson and by Barr. 
  The quantum properties of two-dimensional matter-dilaton gravity   ---which includes a large family of actions for two-dimensional gravity (in particular, string-inspired models)--- are investigated. The one-loop divergences in linear covariant gauges are calculated and the structure of the one-loop renormalization is studied. The explicit forms of the dilaton potential, dilaton-Maxwell, and dilaton-scalar couplings for which the theory is one-loop multiplicatively renormalizable are found.   A comparison with the one-loop renormalization structure of four-dimensional gravity-matter theory is given. Charged multiple-horizon black holes which appear in the model are also considered. 
  A new quantum mechanical wave equation describing a particle with frictional forces is derived. It depends on a parameter $\alpha$ whose range is determined by the coefficient of friction $\gamma$, that is, $0 \leq \alpha \leq \gamma$. For one extreme value of this parameter, $\alpha = 0$, we recover Kostin's equation. For the other extreme value, $\alpha = \gamma$, we obtain an equation in which friction manifests in "magnetic" type terms. It further exhibits breakdown of translational invariance, manifesting through a symmetry breaking parameter $\beta$, as well as localized stationary states in the absence of external potentials. Other physical properties of this new class of equations are also discussed. 
  A brief review is given of an adaptation of the coadjoint orbit method appropriate for study of models with infinite-dimensional symmetry groups. It is illustrated on several examples, including derivation of the WZNW action of induced $D=2\,$ $(N,0)\,$ supergravity. As a main application, we present the geometric action on a generic coadjoint orbit of the deformed group of area preserving diffeomorphisms. This action is precisely the anomalous effective WZNW action of $D=2 \,$ matter fields coupled to chiral $W_\infty$ gravity background. Similar actions are given which produce the {\em KP} hierarchy as on-shell equations of motion. 
  We derive the basic correlation functions of twist fields coming from arbitrary twisted sectors in symmetric $Z_N$ orbifold conformal field theories, keeping all the admissible marginal perturbations, in particular those corresponding to the antisymmetric tensor background field. This allows a thorough investigation of modular symmetries in this type of string compactification. Such a study is explicitly carried out for the group generated by duality transformations. Thus, apart from being of phenomenological use, our couplings are also interesting from the mathematical point of view as they represent automorphic functions for a large class of discrete groups. 
  We propose a new formulation of the heterotic $D=10$ Green-Schwarz superstring whose worldsheet is a superspace with two even and eight odd coordinates. The action is manifestly invariant under both target-space supersymmetry and a worldsheet reparametrisation supergroup. It contains only a finite set of auxiliary fields. The key ingredient are the commuting spinor (twistor) variables, which naturally arise as worldsheet superpartners of the target space Grassmann coordinates. These spinors parametrise the sphere $S^8$ regarded as a coset space of the $D=10$ Lorentz group. The sphere is associated with the lightlike vector of one of the string Virasoro constraints. The origin of the on-shell $D=10$ fermionic kappa symmetry of the standard Green-Schwarz formulation is explained. An essential and unusual feature is the appearance of the string tension only on shell as an integration constant. 
  We derive an expression in closed form for the action of a finite element of the Virasoro Group on generalized vertex operators. This complements earlier results giving an algorithm to compute the action of a finite string of generators of the Virasoro Algebra on generalized vertex operators. The main new idea is to use a first order formalism to represent the infinitesimal group element as a loop variable. To obtain a finite group element it is necessary to thicken the loop to a band of finite thickness. This technique makes the calculation very simple. 
  A systematic way of formulating the Batalin-Vilkovisky method of quantization was obtained in terms of the ``odd time'' formulation. We show that in a class of gauge theories it is possible to find an ``odd time lagrangian'' yielding, by a Legendre transformation, an ``odd time hamiltonian'' which is the minimal solution of the master equation. This constitutes a very simple method of finding the minimal solution of the master equation which is usually a tedious task. To clarify the general procedure we discussed its application to Yang-Mills theory, massive (abelian) theory in Stueckelberg formalism, relativistic particle and the self-interacting antisymmetric tensor field. 
  The electric-magnetic duality transformation in four dimensional heterotic string theory discussed by Shapere, Trivedi and Wilczek is shown to be an exact symmetry of the equations of motion of low energy effective field theory even after including the scalar and the vector fields, arising due to compactification, in the effective field theory. Using this duality transformation we construct rotating black hole solutions in the effective field theory carrying both, electric and magnetic charges. The spectrum of extremal magnetically charged black holes turns out to be similar to that of electrically charged elementary string excitations. We also discuss the possibility that the duality symmetry is an exact symmetry of the full string theory under which electrically charged elementary string excitations get exchanged with magnetically charged soliton like solutions. This proposal might be made concrete following the suggestion of Dabholkar et. al. that fundamental strings may be regarded as soliton like classical solutions in the effective field theory. 
  Gott spacetime has closed timelike curves, but no locally anomalous stress-energy. A complete orthonormal set of eigenfunctions of the wave operator is found in the special case of a spacetime in which the total deficit angle is $2\pi$. A scalar quantum field theory is constructed using these eigenfunctions. The resultant interacting quantum field theory is not unitary because the field operators can create real, on-shell, particles in the acausal region. These particles propagate for finite proper time accumulating an arbitrary phase before being annihilated at the same spacetime point as that at which they were created. As a result, the effective potential within the acausal region is complex, and probability is not conserved. The stress tensor of the scalar field is evaluated in the neighborhood of the Cauchy horizon; in the case of a sufficiently small Compton wavelength of the field, the stress tensor is regular and cannot prevent the formation of the Cauchy horizon. 
  We consider a soluble model of large $\phi^{4}$-graphs randomly embedded in one compactified dimension; namely the large-order behaviour of finite-temperature perturbation theory for the partition function of the anharmonic oscillator. We solve the model using semi-classical methods and demonstrate the existence of a critical temperature at which the system undergoes a second-order phase transition from $D=1$ to $D=0$ behaviour. Non-trivial windings of the closed loops in a graph around the compactified time direction are interpreted as vortices. The critical point has a natural interpretation as the temperature at which these vortices condense and disorder the system. We show that the vortex density increases rapidly in the critical region indicating the breakdown of the dilute vortex gas approximation at this point. We discuss the relation of this phenomenon to the Berezinskii-Kosterlitz-Thouless transition in the $D=1$ matrix model formulated on a circle. 
  We show that all solutions to the vacuum Einstein field equations may be mapped to instanton configurations of the Ashtekar variables. These solutions are characterized by properties of the moduli space of the instantons. We exhibit explicit forms of these configurations for several well-known solutions, and indicate a systematic way to get new ones. Some interesting examples of these new solutions are described. 
  We discuss In\"on\"u-Wigner contractions of affine Kac-Moody algebras. We show that the Sugawara construction for the contracted affine algebra exists only for a fixed value of the level $k$, which is determined in terms of the dimension of the uncontracted part of the starting Lie algebra, and the quadratic Casimir in the adjoint representation. Further, we discuss contractions of $G/H$ coset spaces, and obtain an affine {\it translation} algebra, which yields a Virasoro algebra (via a GKO construction) with a central charge given by $dim(G/H)$. 
  Additional symmetries of the $p$-reduced KP hierarchy are generated by the Lax operator $L$ and another operator $M$, satisfying $res (M^n L^{m+n/p})$ = 0 for $1 \leq n \leq p-1$ and $m \geq -1$ with the condition that ${\partial L \over {\partial t_{kp}}}$ = 0, $k$ = 1, 2,..... We show explicitly that the generators of these additional symmetries satisfy a closed and consistent W-algebra only when we impose the extra condition that ${\partial M \over {\partial t_{kp}}} = 0$. 
  We consider the effect of vacuum polarization around the horizon of a 4 dimensional axionic stringy black hole. In the extreme degenerate limit ($Q_a=M$), the lower limit on the black hole mass for avoiding the polarization of the surrounding medium is $M\gg (10^{-15}\div 10^{-11})m_p$ ($m_p$ is the proton mass), according to the assumed value of the axion mass ($m_a\simeq (10^{-3}\div 10^{-6})~eV$). In this case, there are no upper bounds on the mass due to the absence of the thermal radiation by the black hole. In the nondegenerate (classically unstable) limit ($Q_a<M$), the black hole always polarizes the surrounding vacuum, unless the effective cosmological constant of the effective stringy action diverges. 
  We re-examine the classification of supersymmetric extended objects in the light of the recently discovered Type II p-branes, previously thought not to exist for p> 1. We find new points on the brane-scan only in D = 10 and then only for p = 3(Type IIB), p = 4 (Type IIA), p = 5 (Type IIA and IIB) and p = 6 (Type IIA). The case D = 10, p = 2 (Type IIA) also exists but is equivalent to the previously classified D = 11 supermembrane. 
  It is shown that the Affine Toda models (AT) constitute a ``gauge fixed'' version of the Conformal Affine Toda model (CAT). This result enables one to map every solution of the AT models into an infinite number of solutions of the corresponding CAT models, each one associated to a point of the orbit of the conformal group. The Hirota's $\tau$-function are introduced and soliton solutions for the AT and CAT models associated to $\hat {SL}(r+1)$ and $\hat {SP}(r)$ are constructed. 
  In this note we investigate the generalised critical $N=2$ superstrings in $(1,2p)$ spacetime signature. We calculate the four-point functions for the tachyon operators of these theories. In contrast to the usual $N=2$ superstring in $(2,2)$ spacetime, the four-point functions do not vanish. The exchanged particles of the four-point function are included in the physical spectrum of the corresponding theory and have vanishing fermion charge. 
  States in the absolute (semi-relative) cohomology but not in the relative cohomology are examined through the component decomposition of the string field theory action for the 2-D string. It is found that they are auxiliary fields without kinetic terms, but are important for instance in the master equation for the Ward-Takahashi identities. The ghost structure is analyzed in the Siegel gauge, but it is noted that the absolute (semi-relative) cohomology states are lost. 
  We argue that the torus partition sum in $2d$ (super) gravity, which counts physical states in the theory, is a decreasing function of the renormalization group scale. As an application we chart the space of $(\hat c\leq1)$ $c\leq1$ models coupled to (super) gravity, confirming and extending ideas due to A. Zamolodchikov, and discuss briefly string theory, where our results imply that the number of degrees of freedom decreases with time. 
  We study the large $N$ limit of an interacting \td\ matrix field theory, whose perturbative expansion generates the sum over planar random graphs embedded in two dimensions. In the \lc\ quantization the theory possesses closed string excitations which become free as $N\to\infty$. If the longitudinal momenta are discretized, then the calculation of the free string spectrum reduces to finite matrix diagonalization, the size of the matrix growing as the cut-off is removed. Our numerical results suggest that, for a critical coupling, the \lc\ string spectrum becomes continuous. This would indicate the massless dynamics of the Liouville mode of \td\ gravity, which would constitute a {\it third} dimension of the string theory. 
  We present a new parametrisation of the space of solutions of the Wess-Zumino-Witten model on a cylinder, with target space a compact, connected Lie group G. Using the covariant canonical approach the phase space of the theory is shown to be the co-tangent bundle of the loop group of the Lie group G, in agreement with the result from the Hamiltonian approach. The Poisson brackets in this phase space are derived. Other formulations in the literature are shown to be obtained by locally-valid gauge-fixings in this phase space. 
  We construct the BRST operator for non-critical $W_3$-strings and discuss the tachyon-like spectrum. For $N$-punctured spheres with $N \leq 5$ we briefly describe a formal definition of the integral over $W_3$-moduli space. 
  Canonical quantisation of rigid particles is considered paying special attention to the restriction on phase space due to causal propagation. A mixed Lorentz-gravitational anomaly is found in the commutator of Lorentz boosts with world-line reparametrisations. The subspace of gauge invariant physical states is therefore not invariant under Lorentz transformations. The analysis applies for an arbitrary extrinsic curvature dependence with the exception of only one case to be studied separately. Consequences for rigid strings are also discussed. 
  We propose the factorizable S-matrices of the massive excitations of the non-unitary minimal model $M_{2,11}$ perturbed by the operator $\Phi_{1,4}$. The massive excitations and the whole set of two particle S-matrices of the theory is simply related to the $E_8$ unitary minimal scattering theory. The counting argument and the Thermodynamic Bethe Ansatz (TBA) are applied to this scattering theory in order to support this interpretation. Generalizing this result, we describe a new family of NON UNITARY and DIAGONAL $ADE$-related scattering theories. A further generalization suggests the magnonic TBA for a large class of non-unitary $\G\otimes\G/\G$ coset models ($\G=A_{odd},D_n,E_{6,7,8}$) perturbed by $\Phi_{id,id,adj}$, described by non-diagonal S-matrices. 
  Hopefully tex-able version. 
  Generalizing the Knizhnik-Zamolodchikov equations, we derive a hierarchy of non-linear Ward identities for affine-Virasoro correlators. The hierarchy follows from null states of the Knizhnik-Zamolodchikov type and the assumption of factorization, whose consistency we verify at an abstract level. Solution of the equations requires concrete factorization ans\"atze, which may vary over affine-Virasoro space. As a first example, we solve the non-linear equations for the coset constructions, using a matrix factorization. The resulting coset correlators satisfy first-order linear partial differential equations whose solutions are the coset blocks defined by Douglas. 
  In this paper we consider extensions of the super Virasoro algebra by one and two super primary fields. Using a non-explicitly covariant approach we compute all SW-algebras with one generator of dimension up to 7 in addition to the super Virasoro field. In complete analogy to W-algebras with two generators most results can be classified using the representation theory of the super Virasoro algebra. Furthermore, we find that the SW(3/2, 11/2)-algebra can be realized as a subalgebra of SW(3/2, 5/2) at c = 10/7. We also construct some new SW-algebras with three generators, namely SW(3/2, 3/2, 5/2), SW(3/2, 2, 2) and SW(3/2, 2, 5/2). 
  Some additional references are included on the last 3 pages. 
  We examine solitons in theories with heavy fermions. These ``quantum'' solitons differ dramatically from semi-classical (perturbative) solitons because fermion loop effects are important when the Yukawa coupling is strong. We focus on kinks in a $(1+1)$--dimensional $\phi^4$ theory coupled to fermions; a large-$N$ expansion is employed to treat the Yukawa coupling $g$ nonperturbatively. A local expression for the fermion vacuum energy is derived using the WKB approximation for the Dirac eigenvalues. We find that fermion loop corrections increase the energy of the kink and (for large $g$) decrease its size. For large $g$, the energy of the quantum kink is proportional to $g$, and its size scales as $1/g$, unlike the classical kink; we argue that these features are generic to quantum solitons in theories with strong Yukawa couplings. We also discuss the possible instability of fermions to solitons. 
  Quantum superalgebras $su_{q}(m\mid n)$ are studied in the framework of $R$-matrix formalism. Explicit parametrization of $L^{(+)}$ and $L^{(-)}$ matrices in terms of $su_{q}(m\mid n)$ generators are presented. We also show that quantum deformation of nonsimple superalgebra $su(n\mid n)$ requires its extension to $u(n\mid n)$. 
  Three simple examples illustrate properties of path integral amplitudes in fixed background spacetimes with closed timelike curves: non-relativistic potential scattering in the Born approximation is non-unitary, but both an example with hard spheres and the exact solution of a totally discrete model are unitary. 
  We consider exactly solvable semi-classical theory of two dimensional dilatonic gravity with electromagnetic interactions. As was done in the paper by Russo, Susskind and Thorlacius, the term which changes the kinetic term is added to the action. The theory contains massless fermions as matter fields and there appear the quantum corrections including chiral anomaly. The screening effect due to the chiral anomaly has a tendency to cloak the singularity. In a region of the parameter space, the essential behavior of the theory is similar to that of Callan, Giddings, Harvey and Strominger's dilatonic black hole theory modified in the paper by Russo, Susskind and Thorlacius and the singularity formed by the collapsing matter emerges naked. We find, however, another region of the parameter space where the singularity disappears in a finite proper time. Furthermore, in the region of the parameter space, there appears a discontinuity in the metric on the trajectory of the collapsing matter, which would be a signal of topology change 
  The ring structure of Lian-Zuckerman states for $(q,p)$ minimal models coupled to gravity is shown to be ${\cal R}={\cal R}_0\otimes {\bf C} [w,w^{-1}]$ where ${\cal R}_0$ is the ring of ghost number zero operators generated by two elements and $w$ is an operator of ghost number $-1$. Some examples are discussed in detail. For these models the currents are also discussed and their algebra is shown to contain the Virasoro algebra. 
  We extend a previous calculation which treated Schwarschild black hole horizons as quantum mechanical objects to the case of a charged, dilaton black hole. We show that for a unique value of the dilaton parameter `a', which is determined by the condition of unitarity of the S matrix, black holes transform at the extremal limit into strings. 
  In this paper, we formulate a generalization of the classical BRST construction which applies to the case of the reduction of a poisson manifold by a submanifold. In the case of symplectic reduction, our procedure generalizes the usual classical BRST construction which only applies to symplectic reduction of a symplectic manifold by a coisotropic submanifold, \ie\ the case of reducible ``first class'' constraints. In particular, our procedure yields a method to deal with ``second-class'' constraints. We construct the BRST complex and compute its cohomology. BRST cohomology vanishes for negative dimension and is isomorphic as a poisson algebra to the algebra of smooth functions on the reduced poisson manifold in zero dimension. We then show that in the general case of reduction of poisson manifolds, BRST cohomology cannot be identified with the cohomology of vertical differential forms. 
  This paper deals with the dispersionless KP hierarchy from the point of view of quasi-classical limit. Its Lax formalism, W-infinity symmetries and general solutions are shown to be reproduced from their counterparts in the KP hierarchy in the limit of $\hbar \to 0$. Free fermions and bosonized vertex operators play a key role in the description of W-infinity symmetries and general solutions, which is technically very similar to a recent free fermion formalism of $c=1$ matrix models. 
  The non-perturbative ultraviolet divergence of the sine-Gordon model is used to study the $k^+ = 0$ region of light-cone perturbation theory. The light-cone vacuum is shown to be unstable at the non-perturbative $\beta^2 = 8\pi$ critical point by a light-cone version of Coleman's variational method. Vacuum bubbles, which are $k^+=0$ diagrams in light-cone field theory and are individually finite and non-vanishing for all $\beta$, conspire to generate ultraviolet divergences of the light-cone energy density. The $k^+ = 0$ region of momentum also contributes to connected Green's functions; the connected two point function will not diverge, as it should, at the critical point unless diagrams which contribute only at $k^+ = 0$ are properly included. This analysis shows in a simple way how the $k^+ =0$ region cannot be ignored even for connected diagrams. This phenomenon is expected to occur in higher dimensional gauge theories starting at two loop order in light-cone perturbation theory. 
  Staggered fermions are constructed for the transverse lattice regularization scheme. The weak perturbation theory of transverse lattice non-compact QED is developed in light-cone gauge, and we argue that for fixed lattice spacing this theory is ultraviolet finite, order by order in perturbation theory. However, by calculating the anomalous scaling dimension of the link fields, we find that the interaction Hamiltonian becomes non-renormalizable for $g^2(a) > 4\pi$, where $g(a)$ is the bare (lattice) QED coupling constant. We conjecture that this is the critical point of the chiral symmetry breaking phase transition in QED. Non-perturbative chiral symmetry breaking is then studied in the strong coupling limit. The discrete remnant of chiral symmetry that remains on the lattice is spontaneously broken, and the ground state to lowest order in the strong coupling expansion corresponds to the classical ground state of the two-dimensional spin one-half Heisenberg antiferromagnet. 
  We give a pedagogical introduction to the differential calculus on quantum groups by stressing at all stages the connection with the classical case ($q \rightarrow 1$ limit). The Lie derivative and the contraction operator on forms and tensor fields are found. A new, explicit form of the Cartan--Maurer equations is presented. The example of a bicovariant differential calculus on the quantum group $GL_q(2)$ is given in detail. The softening of a quantum group is considered, and we introduce $q$-curvatures satisfying q-Bianchi identities, a basic ingredient for the construction of $q$-gravity and $q$-gauge theories. 
  Deviations from scale invariance resulting from small perturbations of a general two dimensional conformal field theory are studied. They are expressed in terms of beta functions for renormalization of general couplings under local change of scale. The beta functions for homogeneous background are given perturbatively in terms of the data of the original conformal theory without any specific assumptions on its nature. The renormalization of couplings to primary operators and to first descendents is considered as well as that of couplings of a dilatonic type which involve explicit dependence on world sheet curvature. 
  We give the exact solution of the Kazakov-Migdal induced gauge model in the case of a D=1 compactified lattice with a generic number $S$ of sites and for any value of N. Due to the peculiar features of the model, the partition function that we obtain also describes the vortex-free sector of the D=1 compactified bosonic string, and it coincides in the continuum limit with the one obtained by Boulatov and Kazakov in this context. 
  We study numerically the gravitational field of a star made of massive and neutral string states for the case in which the dilaton is massive. The solution exhibits very simple scaling properties in the dilaton mass. There is no horizon and the singularity is surrounded by a halo (the physical size of which is inversely proportional to the dilaton mass) where the scalar curvature is very large and proportional to the square of the dilaton mass. 
  The propagation differential for bosonic strings on a complex torus with three symmetric punctures is investigated. We study deformation aspects between two point and three point differentials as well as the behaviour of the corresponding Krichever-Novikov algebras. The structure constants are calculated and from this we derive a central extension of the Krichever-Novikov algebras by means of b-c systems. The defining cocycle for this central extension deforms to the well known Virasoro cocycle for certain kinds of degenerations of the torus.    AMS subject classification (1991): 17B66, 17B90, 14H52, 30F30, 81T40 
  It is shown that self-dual theories generalize to four dimensions both the conformal and analytic aspects of two-dimensional conformal field theories. In the harmonic space language there appear several ways to extend complex analyticity (natural in two dimensions) to quaternionic analyticity (natural in four dimensions). To be analytic, conformal transformations should be realized on $CP^3$, which appears as the coset of the complexified conformal group modulo its maximal parabolic subgroup. In this language one visualizes the twistor correspondence of Penrose and Ward and consistently formulates the analyticity of Fueter. 
  We show that multivariable colored link invariants are derived from the roots of unity representations of $U_q(g)$. We propose a property of the Clebsch-Gordan coefficients of $U_q(g)$, which is important for defining the invariants of colored links. For $U_q(sl_2) we explicitly prove the property, and then construct invariants of colored links and colored ribbon graphs, which generalize the multivariable Alexander polynomial. 
  We show that the isometries of the manifold of scalars in $N=2$ supergravity in $d=5$ space-time dimensions can be broken by the supergravity interactions. The opposite conclusion holds for the dimensionally reduced $d=4$ theories, where the isometries of the scalar manifold are always symmetries of the full theory. These spaces, which form a subclass of the {\em special} K\"ahler manifolds, are relevant for superstring compactifications. 
  The KP hierarchy is hamiltonian relative to a one-parameter family of Poisson structures obtained from a generalized Adler map in the space of formal pseudodifferential symbols with noninteger powers. The resulting $\W$-algebra is a one-parameter deformation of $\W_{\rm KP}$ admitting a central extension for generic values of the parameter, reducing naturally to $\W_n$ for special values of the parameter, and contracting to the centrally extended $\W_{1+\infty}$, $\W_\infty$ and further truncations. In the classical limit, all algebras in the one-parameter family are equivalent and isomorphic to $\w_{\rm KP}$. The reduction induced by setting the spin-one field to zero yields a one-parameter deformation of $\widehat{\W}_\infty$ which contracts to a new nonlinear algebra of the $\W_\infty$-type. 
  In this Letter the method of Lund is applied to formulate a variational principle for the motion of charged vortices in an effective non-linear Schr\"{o}dinger field theory describing finite size two-dimensional quantum Hall samples under the influence of an arbitrary perpendicular magnetic field. Freezing out variations in the modulus of the effective field yields a $U(1)$ sigma-model. A duality transformation on the sigma-model reduces the problem to finding the Green function for a related electrostatics problem. This duality illuminates the plasma analogy to the Laughlin wave function. 
  Certain two dimensional topological field theories can be interpreted as string theory backgrounds in which the usual decoupling of ghosts and matter does not hold. Like ordinary string models, these can sometimes be given space-time interpretations. For instance, three-dimensional Chern-Simons gauge theory can arise as a string theory. The world-sheet model in this case involves a topological sigma model. Instanton contributions to the sigma model give rise to Wilson line insertions in the space-time Chern-Simons theory. A certain holomorphic analog of Chern-Simons theory can also arise as a string theory. 
  We show that a whole class of quantum actions for dilaton-gravity, which reduce to the CGHS theory in the classical limit, can be written as a Liouville-like theory. In a sub-class of this, the field space singularity observed by several authors is absent, regardless of the number of matter fields, and in addition it is such that the dilaton-gravity functional integration range (the real line) transforms into itself for the Liouville theory fields. We also discuss some problems associated with the usual calculation of Hawking radiation, which stem from the neglect of back reaction. We give an alternative argument incorporating back reaction but find that the rate is still asymptotically constant. The latter is due to the fact that the quantum theory does not seem to have a lower bound in energy and Hawking radiation takes positive Bondi (or ADM) mass solutions to arbitrarily large negative mass. 
  This paper supersedes the preprint Superloop Equations in the Double Scaling Limit, CERN-TH-6575. It is an improved version with corrections in the derivation of the continuum limit, a clarification of the doubling of degrees of freedom for even potentials, and with a heuristic argument showing that the part of the free energy independent of fermion couplings in our model satisfies the KdV Hierarchy. 
  This paper discusses the large N limit of the two-Hermitian-matrix model in zero dimensions, using the hidden BRST method. A system of integral equations previously found is solved, showing that it contained the exact solution of the model in leading order of large $N$. 
  We discuss the problem of N anyons in harmonic well, and derive the semi-classical spectrum as an exactly solvable limit of the many-anyon Hamiltonian. The relevance of our result to the solution of the anyon-gas model is discussed. 
  For the many-anyon system in external magnetic field, we derive the energy spectrum as an exact solution of the quantum eigenvalue problem with particular topological constraints. Our results agree with the numerical spectra recently obtained for the 3- and the 4-anyon systems. 
  In the framework of the Caldeira-Leggett model of dissipative quantum mechanics, we investigate the effects of the interaction of the thermal reservoir with an external field. In particular, we discuss how the interaction modifies the conservative dynamics of the central particle, and the mechanism of dissipation. We briefly comment on possible observable consequencies. 
  We quantize $sl_n$ Toda field theories in a periodic lattice. We find the quantum exchange algebra in the diagonal monodromy (Bloch wave) basis in the case of the defining representation. In the $sl_3$ case we extend the analysis also to the second fundamental representation. We clarify, in particular, the relation of Jimbo and Rosso's quantum $R$ matrix with the quantum $R$ matrix in the Bloch wave basis. 
  We present a classification of $W$ algebras and superalgebras arising in Abelian as well as non Abelian Toda theories. Each model, obtained from a constrained WZW action, is related with an $Sl(2)$ subalgebra (resp. $OSp(1|2)$ superalgebra) of a simple Lie algebra (resp. superalgebra) $\cg$. However, the determination of an $U(1)_Y$ factor, commuting with $Sl(2)$ (resp. $OSp(1|2)$), appears, when it exists, particularly useful to characterize the corresponding $W$ algebra. The (super) conformal spin contents of each $W$ (super)algebra is performed. The class of all the superconformal algebras (i.e. with conformal spins $s\leq2$) is easily obtained as a byproduct of our general results. 
  We argue that the light particles in string theory obey an effective quantum mechanics modified by the inclusion of a quantum-gravitational friction term, induced by unavoidable couplings to unobserved massive string states in the space-time foam. This term is related to the $W$-symmetries that couple light particles to massive solitonic string states in black hole backgrounds, and has a formal similarity to simple models of environmental quantum friction. It increases apparent entropy, and may induce the wave functions of macroscopic systems to collapse. 
  Degenerations of Lie algebras of meromorphic vector fields on elliptic curves (i.e. complex tori) which are holomorphic outside a certain set of points (markings) are studied. By an algebraic geometric degeneration process certain subalgebras of Lie algebras of meromorphic vector fields on P^1 the Riemann sphere are obtained. In case of some natural choices of the markings these subalgebras are explicitly determined. It is shown that the number of markings can change. AMS subject classification (1991): 17B66, 17B90, 14F10, 14H52, 30F30, 81T40 
  If an evaporating black hole does not settle down to a non radiating remnant, a description by a semi classical Lorentz metric must contain either a naked singularity or what we call a thunderbolt, a singularity that spreads out to infinity on a spacelike or null path. We investigate this question in the context of various two dimensional models that have been proposed. We find that if the semi classical equations have an extra symmetry that make them solvable in closed form, they seem to predict naked singularities but numerical calculations indicate that more general semi classical equations, such as the original CGHS ones give rise to thunderbolts. We therefore expect that the semi classical approximation in four dimensions will lead to thunderbolts. We interpret the prediction of thunderbolts as indicating that the semi classical approximation breaks down at the end point of black hole evaporation, and we would expect that a full quantum treatment would replace the thunderbolt with a burst of high energy particles. The energy in such a burst would be too small to account for the observed gamma ray bursts. 
  Quantum Electrodynamics can be formulated as the theory of an antisymmetric tensor gauge field. In this formulation the topological current of this field appears as an additional source for the electromagnetic field. The topological charge therefore acts physically as an electric charge. The topologically nontrivial, electrically charged sector contains massless quantum states orthogonal to the vacuum in spite of the absence of classical topological solutions. These states are created by a gauge invariant local operator and can be interpreted as coherent states of photons. The obtainment of a quantity like charge, which is usually associated with matter, as a property of some peculiar states of the gauge field points towards the possibility of describing both the matter and the fields which mediate its interactions within the same unified framework. 
  Chiral densities obeying a $w_{\infty}$ Poisson--bracket algebra are constructed for the $2+1\,\, A_{\infty}$ -- Toda field theory, using its alternative $w_{\infty}$ -- Toda representation. They are obtained from formal traces of powers of the Lax operator. The spin 2 and 3 currents are explicitely derived, and the consistency of their Poisson algebra is checked. 
  We show how to obtain the two-dimensional black hole action by dimensional reduction of the three-dimensional Einstein action with a non-zero cosmological constant. Starting from the Chern-Simons formulation of 2+1 gravity, we obtain the 1+1 dimensional gauge formulation given by Verlinde. Remarkably, the proposed reduction shares the relevant features of the formulation of Cangemi and Jackiw, without the need for a central charge in the algebra. We show how the Lagrange multipliersin these formulations appear naturally as the remnants of the three dimensional connection associated to symmetries that have been lostin the dimensional reduction. The proposed dimensional reduction involves a shift in the three dimensional connection whose effect is to make the length of the extra dimension infinite. 
  We show how the double cohomology of the String and Felder BRST charges naturally leads to the ring structure of $c<1$ strings. The chiral ring is a ring of polynomials in two variables modulo an equivalence relation of the form $x^p \simeq y^{p+1}$ for the (p+1,p) model. We also study the states corresponding to the edges of the conformal grid whose inclusion is crucial for the closure of the ring. We introduce candidate operators that correspond to the observables of the matrix models. Their existence is motivated by the relation of one of the screening operators of the minimal model to the zero momentum dilaton. 
  Causal rigid particles whose action includes an {\it arbitrary} dependence on the world-line extrinsic curvature are considered. General classes of solutions are constructed, including {\it causal tachyonic} ones. The Hamiltonian formulation is developed in detail except for one degenerate situation for which only partial results are given and requiring a separate analysis. However, for otherwise generic rigid particles, the precise specification of Hamiltonian gauge symmetries is obtained with in particular the identification of the Teichm$\ddot{\rm u}$ller and modular spaces for these systems. Finally, canonical quantisation of the generic case is performed paying special attention to the phase space restriction due to causal propagation. A mixed Lorentz-gravitational anomaly is found in the commutator of Lorentz boosts with world-line reparametrisations. The subspace of gauge invariant physical states is therefore not invariant under Lorentz transformations. Consequences for rigid strings and membranes are also discussed. 
  There has been some confusion concerning the number of $(1,1)$-forms in orbifold compactifications of the heterotic string in numerous publications. In this note we point out the relevance of the underlying torus lattice on this number. We answer the question when different lattices mimic the same physics and when this is not the case. As a byproduct we classify all symmetric $Z_N$-orbifolds with $(2,2)$ world sheet supersymmetry obtaining also some new ones. 
  We suggest a method to compute leading contribution at Planckian energies for superstring scattering amplitudes of any genus. In particular we test the method at one-loop level by comparison with previous result for the Regge trajectory renormalization. Modular invariance of these asymptotic terms are also discussed. 
  Corrections to the semiclassical approximation in nearly forward Planckian energy collisions are here reconsidered. Starting from the one-loop superstring amplitude, we are able to disentangle the first subleading high-energy contribution at large impact parameters, and we thus directly compute the one-loop correction to the superstring eikonal. We finally argue, on the basis of analyticity and unitarity, that gravitinos do not contribute at all to the large distance two-loop ACV correction, which thus acquires a universal ``classical'' interpretation. 
  The (1+1)-dimensional bosonization relations for fermionic mass terms are derived by choosing a specific gauge in an enlarged gauge-invariant theory containing both fermionic and bosonic fields. The fermionic part of the generating functional subject to the gauge constraint can be cast into the form of a strongly coupled Schwinger model, which can be solved exactly. The resulting bosonic theory coupled to the scalar sources then exhibits directly the bosonic counterparts of the fermionic scalar and pseudoscalar mass densities. 
  A proposal for the path-integral of pure-spin-connection formulation of gravity is described, based on the two-form formulation of Capovilla et. al. It is shown that the resulting effective-action for the spin-connection, upon functional integration of the two-form field $\Sigma$ and the auxiliary matrix field $\psi$ is {\it non-polynomial}, even for the case of vanishing cosmological constant and absence of any matter couplings. Further, a diagramatic evaluation is proposed for the contribution of the matrix-field to the pure spin connection action. 
  We describe deformations of non-linear (birational) representations of discrete groups generated by involutions, having their origin in the theory of the symmetric five-state Potts model. One of the deformation parameters can be seen as the number $q$ of states of a chiral Potts models. This analogy becomes exact when $q$ is a Fermat number. We analyze the stability of the corresponding dynamics, with a particular attention to orbits of finite order. 
  It is shown that conformal matter with $c_{\ssc L}\not=c_{\ssc R}$ can be consistently coupled to two-dimensional `frame' gravity. The theory is quantized, following David, and Distler and Kawai, using the derivation of their {\it ansatz} due to Mavromatos and Miramontes, and D'Hoker and Kurzepa. New super-selection rules are found by requiring SL(2,{\bf C}) invariance of correlation functions on the plane. There is no analogue of the $c=1$ barrier found in non-chiral non-critical strings. A non-critical heterotic string is constructed---it has 744 states in its spectrum, transforming in the adjoint representation of $(E_8)^3.$ Correlation functions are calculated in this example. 
  We explore the possibility of string theories in only four spacetime dimensions without any additional compactified dimensions. We show that, provided the theory is defined in curved spacetime that has a cosmological interpration, it is possible to construct consistent heterotic string theories based on a few non-compact current algebra cosets. We classify these models. The gauge groups that emerge fall within a remarkably narrow range and include the desirable low energy flavor symmetry of $SU(3)\times SU(2)\times U(1)$. The quark and lepton states, which come in color triplets and $SU(2)$ doublets, are expected to emerge in several families. 
  Pursuing further the recent methods in the algebraic Hamiltonian approach to gauged WZW models, we apply them to the bosonic SL(2,R) X SU(2)/R^2 model recently investigated by Nappi and Witten. We find the global space and compute the conformally exact metric and dilaton fields to all orders in the $1/k$ expansion. The semiclassical limit $k',k\to \infty$ of our exact results agree with the lowest order perturbation computation which was done in the Lagrangian formalism. We also discuss the supersymmetric type-II and heterotic versions of this model and verify the non-renormalization of $e^\Phi\sqrt{-G}$. 
  Migdal and Kazakov have suggested that lattice QCD with an adjoint representation scalar in the infinite coupling limit could induce QCD.   I find an exact saddlepoint of this theory for infinite $N$ in the case of a quadratic scalar potential. I discuss some aspects of this solution and also show how the continuum D=1 matrix model with an arbitrary potential can be reproduced through this approach. 
  The field algebra of the minimal models of W-algebras is amenable to a very simple description as a polynomial algebra generated by few elementary fields, corresponding to order parameters. Using this description, the complete Landau-Ginzburg lagrangians for these models are obtained. Perturbing these lagrangians we can explore their phase diagrams, which correspond to multicritical points with $D_n$ symmetry. In particular, it is shown that there is a perturbation for which the phase structure coincides with that of the IRF models of Jimbo et al. 
  An arbitrary Feynman graph for string field theory interactions is analysed and the homeomorphism type of the corresponding world sheet surface is completely determined even in the non-orientable cases. Algorithms are found to mechanically compute the topological characteristics of the resulting surface from the structure of the signed oriented graph. Whitney's permutation-theoretic coding of graphs is utilized. 
  Classical random walks and Markov processes are easily described by Hopf algebras. It is also known that groups and Hopf algebras (quantum groups) lead to classical and quantum diffusions. We study here the more primitive notion of a quantum random walk associated to a general Hopf algebra and show that it has a simple physical interpretation in quantum mechanics. This is by means of a representation theorem motivated from the theory of Kac algebras: If $H$ is any Hopf algebra, it may be realised in $\Lin(H)$ in such a way that $\Delta h=W(h\tens 1)W^{-1}$ for an operator $W$. This $W$ is interpreted as the time evolution operator for the system at time $t$ coupled quantum-mechanically to the system at time $t+\delta$. Finally, for every Hopf algebra there is a dual one, leading us to a duality operation for quantum random walks and quantum diffusions and a notion of the coentropy of an observable. The dual system has its time reversed with respect to the original system, leading us to a CTP-type theorem. 
  Quantum matrices $A(R)$ are known for every $R$ matrix obeying the Quantum Yang-Baxter Equations. It is also known that these act on `vectors' given by the corresponding Zamalodchikov algebra. We develop this interpretation in detail, distinguishing between two forms of this algebra, $V(R)$ (vectors) and $V^*(R)$ (covectors). $A(R)\to V(R_{21})\tens V^*(R)$ is an algebra homomorphism (i.e. quantum matrices are realized by the tensor product of a quantum vector with a quantum covector), while the inner product of a quantum covector with a quantum vector transforms as a scaler. We show that if $V(R)$ and $V^*(R)$ are endowed with the necessary braid statistics $\Psi$ then their braided tensor-product $V(R)\und\tens V^*(R)$ is a realization of the braided matrices $B(R)$ introduced previously, while their inner product leads to an invariant quantum trace. Introducing braid statistics in this way leads to a fully covariant quantum (braided) linear algebra. The braided groups obtained from $B(R)$ act on themselves by conjugation in a way impossible for the quantum groups obtained from $A(R)$. 
  We construct quantum group-valued canonical connections on quantum homogeneous spaces, including a q-deformed Dirac monopole on the quantum sphere of Podles quantum differential coming from the 3-D calculus of Woronowicz on $SU_q(2)$ . The construction is presented within the setting of a general theory of quantum principal bundles with quantum group (Hopf algebra) fiber, associated quantum vector bundles and connection one-forms. Both the base space (spacetime) and the total space are non-commutative algebras (quantum spaces). 
  Braided groups and braided matrices are novel algebraic structures living in braided or quasitensor categories. As such they are a generalization of super-groups and super-matrices to the case of braid statistics. Here we construct braided group versions of the standard quantum groups $U_q(g)$. They have the same FRT generators $l^\pm$ but a matrix braided-coproduct $\und\Delta L=L\und\tens L$ where $L=l^+Sl^-$, and are self-dual. As an application, the degenerate Sklyanin algebra is shown to be isomorphic to the braided matrices $BM_q(2)$; it is a braided-commutative bialgebra in a braided category. As a second application, we show that the quantum double $D(\usl)$ (also known as the `quantum Lorentz group') is the semidirect product as an algebra of two copies of $\usl$, and also a semidirect product as a coalgebra if we use braid statistics. We find various results of this type for the doubles of general quantum groups and their semi-classical limits as doubles of the Lie algebras of Poisson Lie groups. 
  When a Hamiltonian system is subject to constraints which depend explicitly on time, difficulties can arise in attempting to reduce the system to its physical phase space. Specifically, it is non-trivial to restrict the system in such a way that one can find a Hamiltonian time-evolution equation involving the Dirac bracket. Using a geometrical formulation, we derive an explicit condition which is both necessary and sufficient for this to be possible, and we give a formula defining the resulting Hamiltonian function. Some previous results are recovered as special cases. 
  We introduce a large class of bicovariant differential calculi on any quantum group $A$, associated to $Ad$-invariant elements. For example, the deformed trace element on $SL_q(2)$ recovers Woronowicz' $4D_\pm$ calculus. More generally, we obtain a sequence of differential calculi on each quantum group $A(R)$, based on the theory of the corresponding braided groups $B(R)$. Here $R$ is any regular solution of the QYBE. 
  We introduce the A-D-E resonance factorized models as an appropriate analytical continuation of the Toda S-matrices to the complex values of their coupling constant. An investigation of the associated Casimir energy, via the thermodynamic Bethe ansatz, reveals a rich pattern of renormalization group trajectories interpolating between the central charges of the $G_1 \otimes G_k / G_{k+1}$ GKO coset models. We have also constructed the simplest resonance factorized model satisfying the ``$\phi^3$''-property. From this resonance scattering, we predict new flows in non-unitary minimal models. 
  We investigate a class of operators with non-vanishing averages in a D-dimensional matrix model recently proposed by Kazakov and Migdal. Among the operators considered are ``filled Wilson loops" which are the most reasonable counterparts of Wilson loops in the conventional Wilson formulation of lattice QCD. The averages of interest are represented as partition functions of certain 2-dimensional statistical systems with nearest neighbor interactions. The ``string tension" $\alpha'$, which is the exponent in the area law for the ``filled Wilson loop" is equal to the free energy density of the corresponding statistical system. The continuum limit of the Kazakov--Migdal model corresponds to the critical point of this statistical system. We argue that in the large $N$ limit this critical point occurs at zero temperature. In this case we express $\alpha'$ in terms of the distribution density of eigenvalues of the matrix-valued master field. We show that the properties of the continuum limit and the description of how this limit is approached is very unusual and differs drastically from what occurs in both the Wilson theory ($S\propto({\rm Tr}\prod U +{\rm c.c.})$) and in the ``adjoint'' theory ($S\propto\vert{\rm Tr}\prod U\vert^2$). Instead, the continuum limit of the model appears to be intriguingly similar to a $c>1$ string theory. 
  We establish a relation between the classical non-linear Schr\"odinger equation and the KP hierarchy, and we extend this relation to the quantum case by defining a quantum KP hierarchy. We present evidence that an integrable hierarchy of equations is obtained by quantizing the first Hamiltonian structure of the KdV equation. The connection between infinite-dimensional algebras and integrable models is discussed. 
  A modification of the canonical quantization procedure for systems with time-dependent second-class constraints is discussed and applied to the quantization of the relativistic particle in a plane wave. The time dependence of constraints appears in the problem in two ways. The Lagrangian depends on time explicitly by origin, and a special time-dependent gauge is used. Two possible approaches to the quantization are demonstrated in this case. One is to solve directly a system of operator equations, proposed by Tyutin and one of the authors (Gitman) as a generalization of Dirac canonical quantization in nonstationary case, and another to find first a canonical transformation, which makes it possible to discribe the dynamics in the physical sactor by means of some effective Hamiltonian. Quantum mechanics constructed in both cases proves to be equivalent to Klein-Gordon theory of the relativistic particle in a plane wave. The general conditions of unitarity of the dynamics in the physical sector are discussed. 
  The divergent part of the one-loop off-shell effective action is computed for a single scalar field coupled to the Ricci curvature of 2D gravity ($c \phi R$), and self interacting by an arbitrary potential term $V(\phi)$. The Vilkovisky-DeWitt effective action is used to compute gauge-fixing independent results. In our background field/covariant gauge we find that the Liouville theory is finite on shell. Off-shell, we find a large class of renormalizable potentials which include the Liouville potential. We also find that for backgrounds satisfying $R=0$, the Liouville theory is finite off shell, as well. 
  This paper (completed March 1992) is an extensively revised and expanded version of work which appeared July 1991 on the initial incarnation of the hepth bulletin board, and which was published in the Proceedings of the Workshop on String Theory, Trieste, March 1991. Abstract We present a hamiltonian quantization of the $SL(2,R)$ 3-dimensional Chern-Simons theory with fractional coupling constant $k=s/r$ on a space manifold with torus topology in the ``constrain-first'' framework. By generalizing the ``Weyl-odd'' projection to the fractional charge case, we obtain multi-components holomorphic wave functions whose components are the Kac-Wakimoto characters of the modular invariant admissible representations of ${\hat A}_1$ current algebra with fractional level. The modular representations carried by the quantum Hilbert space satisfy both Verlinde's and Vafa's constraints coming from conformal field theory. They are the ``square-roots'' of the representations associated to the conformal $(r,s)$ minimal models. Our results imply that Chern-Simons theory with $SO(2,2)$ as gauge group, which describes $2+1$-dimensional gravity with negative cosmological constant, has the modular properties of the Virasoro discrete series. On the way, we show that the 2-dimensional counterparts of Chern-Simons $SU(2)$ theories with half-integer charge $k=p/2$ 
  Coherent states $(CS)$ of the $SU(N)$ groups are constructed explicitly and their properties are investigated. They represent a nontrivial generalization of the spining $CS$ of the $SU(2)$ group. The $CS$ are parametrized by the points of the coset space, which is, in that particular case, the projective space $CP^{N-1}$ and plays the role of the phase space of a corresponding classical mechanics. The $CS$ possess of a minimum uncertainty, they minimize an invariant dispersion of the quadratic Casimir operator. The classical limit is ivestigated in terms of symbols of operators. The role of the Planck constant playes $h=P^{-1}$, where $P$ is the signature of the representation. The classical limit of the so called star commutator generates the Poisson bracket in the $CP^{N-1}$ phase space. The logarithm of the modulus of the $CS$ overlapping, being interpreted as a symmetric in the space, gives the Fubini-Study metric in $CP^{N-1}$. The $CS$ constructed are useful for the quasi-classical analysis of the quantum equations of the $SU(N)$ gauge symmetric theories. 
  Spacetimes generated by a lightlike particle source for topologically massive gravity and its limits - Einstein gravity and the pure gravitational Chern-Simons model - are obtained both by solving the field equations and by infinite boosts of static metrics. The resulting geometries are the first known solutions of topologically massive gravity that are asymptotically flat and generated by compact matter sources. Explicit metrics describing various multiphoton solutions are also derived. For Einstein gravity, we also construct such solutions by null boost identifications of Minkowski space and thereby obtain limits on the energies of the sources. 
  We study the sL(3,C) mKDV string theories. We obtain the flows and the string equations. Using the generalized Miura map, we show that we have an unification of these models with the [P,Q]=Q sL(3,C) KDV ones in the framework of open-closed string theories in minimal models backgrounds. 
  We study the lattice gauge model proposed recently by Kazakov and Migdal for inducing QCD. We discuss an extra local Z_N which is a symmetry of the model and propose of how to construct observables. We discuss the role of the large-N phase transition which should occur before the one associated with the continuum limit in order that the model describes continuum QCD. We formulate the mean field approach to study the large-N phase transition for an arbitrary potential and show that no first order phase transition occurs for the quadratic potential. 
  We review how to construct a large class of integrable quantum spin chains with quantum-algebra symmetry, and how to determine their spectra. (To appear in Louis Witten Festschrift) 
  I present a short review of our results with S.Kharchev, A.Mironov, A.Morozov and A.Zabrodin on Generalized Kontsevich model which in a sense can be interpreted as unifying ``string field theory'' for $c < 1$ minimal series coupled to 2d gravity. The problem of interpolation between different models is discussed. It is found that this problem is closely connected with ``deformations'' within the set of solutions to KP hierarchy, described by a sort of reparameterization of a spectral curve and change of asymptotics of the basis in the Grassmannian. The $c \rightarrow 1$ limit is considered along this line. 
  I investigate the role of nonrenormalizable terms, up to order N=8, in a superstring derived standard--like model. I argue that nonrenormalizable terms restrict the gauge symmetry, at the Planck scale, to be $SU(3)\times SU(2)\times U(1)_{B-L}\times U(1)_{T_{3_R}}$ rather than $SU(3)\times SU(2)\times U(1)_Y$. I show that breaking the gauge symmetry directly to the Standard Model leads to breaking of supersymmetry at the Planck scale, or to dimension four, baryon and lepton violating, operators. I show that if the gauge symmetry is broken directly to the Standard Model the cubic level solution to the F and D flatness constraints is violated by higher order terms, while if $U(1)_{Z^\prime}$ remains unbroken at the Planck scale, the cubic level solution is valid to all orders of nonrenormalizable terms. I discuss the Higgs and fermion mass spectrum. I demonstrate that realistic, hierarchical, fermion mass spectrum can be generated in this model. 
  I discuss in detail the construction of realistic superstring standard--like models in the four dimensional free fermionic formulation. The analysis results in a restricted class of models with unique characteristics: (i) Three and only three generations of chiral fermions with their superpartners and the correct Standard Model quantum numbers. (ii) Proton decay from dimension four and dimension five operators is suppressed due to gauged $U(1)$ symmetries. (iii) There exist Higgs doublets from two distinct sectors, which can generate realistic symmetry breaking. (iv) These models explain the top--bottom mass hierarchy. At the trilinear level of the superpotential only the top quark gets a non vanishing mass term. The bottom quark and the lighter quarks and leptons get their mass terms from non renormalizable terms. This result is correlated with the requirement of a supersymmetric vacuum at the Planck scale. (v) The models predict the existence of small hidden gauge groups, like $SU(3)$, with matter spectrum in vector representations. 
  A special Bianchi I universe is constructed in $~4D~$ string theory. Geometrically it represents a $~3D~$ anti-de-Sitter space crossed with a flat direction whereas in terms of an associated conformal field theory it is an extremal case of a gauged WZWN theory with target the coset $~SU(1,1) \times R^2 / R~$. Some of its properties are discussed. 
  Using the reduced formulation on large-N Quantum Field Theories we study strings in space-time dimensions higher that one. We present results on possible string susceptibilities, macroscopic loop operators, 1/N -corrections and other general properties of the model. 
  A framework for background independent open-string field theory is proposed. The approach involves using the BV formalism -- in a way suggested by recent developments in closed-string field theory -- to implicitly define a gauge invariant Lagrangian in a hypothetical ``space of all open-string world-sheet theories.'' It is built into the formalism that classical solutions of the string field theory are BRST invariant open-string world-sheet theories and that, when expanding around a classical solution, the infinitesimal gauge transformations are generated by the world-sheet BRST operator. 
  We discuss the $W_{\infty}$ symmetry in the $2+1$ gauge theory with the Chern-Simons term. It is shown that the generators of this symmetry act on the ground state as the canonical transformations in the phase space. We shall also discuss the analogy between discrete states in $c=1$ string theory and Landau level states in $2+1$ gauge theory with Chern-Simons term. 
  We show that the BRST quantum version of pure D=4 N=2 supergravity can be topologically twisted, to yield a formulation of topological gravity in four dimensions. The topological BRST complex is just a rearrangement of the old BRST complex, that partly modifies the role of physical and ghost fields: indeed, the new ghost number turns out to be the sum of the old ghost number plus the internal U(1) charge. Furthermore, the action of N=2 supergravity is retrieved from topological gravity by choosing a gauge fixing that reduces the space of physical states to the space of gravitational instanton configurations, namely to self-dual spin connections. The descent equations relating the topological observables are explicitly exhibited and discussed. Ours is a first step in a programme that aims at finding the topological sector of matter coupled N=2 supergravity, viewed as the effective Lagrangian of type II superstrings and, as such, already related to 2D topological field-theories. As it stands the theory we discuss may prove useful in describing gravitational instantons moduli-spaces. 
  We solve a class of branched polymer models coupled to spin systems and show that they have no phase transition and are either always magnetized or never magnetized depending on the branching weights. By comparing these results with numerical simulations of two-dimensional quantum gravity coupled to matter fields with central charge $c$ we provide evidence that for $c$ sufficiently large ($c\geq 12$) these models are effectively described by branched polymers. Moreover, the numerical results indicate a remarkable universality in the influence on the geometry of surfaces due to the interaction with matter. For spin systems this influence only depends on the total central charge. 
  We derive a compact and explicit expression for the generating functional of all correlation functions of tachyon operators in 2D string theory. This expression makes manifest relations of the $c=1$ system to KP flow and $W_{1+\infty}$ constraints. Moreover we derive a Kontsevich-Penner integral representation of this generating functional. 
  We consider a discrete model of euclidean quantum gravity in four dimensions based on a summation over random simplicial manifolds. The action used is the Einstein-Hilbert action plus an $R^2$-term. The phase diagram as a function of the bare coupling constants is studied in the search for a sensible continuum limit. For small values of the coupling constant of the $R^2$ term the model seems to belong to the same universality class as the model with pure Einstein-Hilbert action and exhibits the same phase transition. The order of the transition may be second or higher. The average curvature is positive at the phase transition, which makes it difficult to understand the possible scaling relations of the model. 
  Motivated by Harish-Chandra theory, we construct, starting from a simple CDD\--pole $S$\--matrix, a hierarchy of new $S$\--matrices involving ever ``higher'' (in the sense of Barnes) gamma functions.These new $S$\--matrices correspond to scattering of excitations in ever more complex integrable models.From each of these models, new ones are obtained either by ``$q$\--deformation'', or by considering the Selberg-type Euler products of which they represent the ``infinite place''. A hierarchic array of integrable models is thus obtained. A remarkable diagonal link in this array is established.Though many entries in this array correspond to familiar integrable models, the array also leads to new models. In setting up this array we were led to new results on the $q$\--gamma function and on the $q$\--deformed Bloch\--Wigner function. 
  Einstein Gravity in 2+1 dimensions arises as a consequence of the equations of motion of a gauge model in an external metric. Newton's constant appears as an order parameter of a spontaneously broken discrete symmetry. Matter is coupled in a straightforward way. 
  The manifestly SU(4)xU(1) super-Poincare invariant free-field N=2 twistor- string action for the ten-dimensional Green-Schwarz superstring is quantized using standard BRST methods. Unlike the light-cone and semi-light-cone gauge-fixed Green-Schwarz actions, the twistor-string action does not require interaction-point operators at the zeroes of the light-cone momentum, $\dz x^+$, which complicated all previous calculations. After defining the vertex operator for the massless physical supermultiplet, as well as two picture-changing operators and an instanton-number-changing operator, scattering amplitudes for an arbitrary number of loops and external massless states are explicitly calculated by evaluating free-field correlation functions of these operators on N=2 super-Riemann surfaces of the appropriate topology, and integrating over the global moduli. Although there is no sum over spin structures, only discrete values of the global U(1) moduli contribute to the amplitudes. Because the spacetime supersymmetry generators do not contain ghost fields, the amplitudes are manifestly spacetime supersymmetric, there is no multiloop ambiguity, and the non-renormalization theorem is easily proven. By choosing the picture-changing operators to be located at the zeroes of $\dz x^+$, these amplitudes are shown to agree with amplitudes obtained using the manifestly unitary light-cone gauge formalism. 
  We investigate the circumstances under which gravitationally collapsing dust can form a black hole in three-dimensional spacetime. 
  We construct a bicovariant differential calculus on the quantum group $GL_q(3)$, and discuss its restriction to $[SU(3) \otimes U(1)]_q$. The $q$-algebra of Lie derivatives is found, as well as the Cartan-Maurer equations. All the quantities characterizing the non-commutative geometry of $GL_q(3)$ are given explicitly. 
  We demonstrate how to find modular discrete symmetry groups for $Z_N$ orbifolds. The $Z_7$ orbifold is treated in detail as a non-trivial example of a $(2,2)$ orbifold model. We give the generators of the modular group for this case which, surprisingly, does not contain $\sltz^3$ as had been speculated. The treatment models with discrete Wilson lines is also discussed. We consider examples which demonstrate that discrete Wilson lines affect the modular group in a non-trivial manner. In particular, we show that it is possible for a Wilson line to break $SL(2,{\bf Z})$. 
  The BRST cohomology of 1+1 strings in a free light-cone gauge contains only the two-dimensional tachyon, and excludes all excited states of both matter and ghosts, including the special states that arise in the continuum conformal gauge quantization and in the $c = 1$ matrix models. This exclusion takes place at a very basic level, and therefore may signal some serious problems or at least unresolved issues involved in this gauge choice. 
  The twisted G/H models are constructed as twisted supersymmetric gauged WZW models. We analyze the case of $G=SU(N)$, $H=SU(N_1)\times ...\times SU(N_n)\times U(1)^r$ with $rank\ G =\ rank\ H$, and discuss possible generalizations. We introduce a non-abelian bosonization of the $(1,0)$ ghost system in the adjoint of $H$ and in G/H. By computing chiral anomalies in the latter picture we write the quantum action as a decoupled sum of ``matter", gauge and ghost sectors. The action is also derived in the unbosonized version. We invoke a free field parametrization and extract the space of physical states by computing the cohomology of $Q$ , the sum of the BRST gauge-fixing charge and the twisted supersymmetry charge. For a given $G$ we briefly discuss the relation between the various G/H models corresponding to different choices of $H$. The choice $H=G$ corresponds to the topological G/G theory. 
  We develop further a new geometrical model of a discretized string, proposed in [1] and establish its basic physical properties. The model can be considered as the natural extention of the usual Feynman amplitude of the random walks to random surfaces. Both amplitudes coinside in the case, when the surface degenarates into a single particle world line. We extend the model to open surfaces as well. The boundary contribution is proportional to the full length of the boundary and the coefficient of proportionality can be treated as a hopping parameter of the quarks. In the limit, when this parameter tends to infinity, the theory is essentialy simlplified. We prove that the contribution of a given triangulation to the partition function is finite and have found the explicit form for the upper bound. The question of the convergence of the full partition function remains open. In this model the string tension may vanish at the critical point, if the last one exists, and possess a nontrivial scaling limit. The model contains hidden fermionic variables and can be considered as an independent model of hadrons. 
  We find the precise relationship between the loop gas method and the matrix quantum mechanics approach to two-dimensional string theory. The two systems are distinguished by different target spaces ($\Z$ and $\R$, respectively) as far as {\it observables} are concerned. We argue that target space loop correlators should coincide in the two models and demonstrate this for a number of examples. As a consequence some interesting generic observations about the structure of two-dimensional string theory may be made: Restricting to a discrete target space leads to {\it factorization} of amplitudes and thus to very simple sewing rules. It is also demonstrated that the restriction to the discrete target space still allows to calculate the correlation functions of tachyon operators in the unrestricted theory. 
  Conformal fields are a new class of $Vect(N)$ modules which are more general than tensor fields. The corresponding diffeomorphism group action is constructed. Conformal fields are thus invariantly defined. 
  We introduce {\it conformal multi-matrix models} (CMM) as an alternative to conventional multi-matrix model description of two-dimensional gravity interacting with $c < 1$ matter. We define CMM as solutions to (discrete) extended Virasoro constraints. We argue that the so defined alternatives of multi-matrix models represent the same universality classes in continuum limit, while at the discrete level they provide explicit solutions to the multi-component KP hierarchy and by definition satisfy the discrete $W$-constraints. We prove that discrete CMM coincide with the $(p,q)$-series of 2d gravity models in a {\it well}-{\it defined} continuum limit, thus demonstrating that they provide a proper generalization of Hermitian one-matrix model. 
  I study the large-N reduction a la Eguchi--Kawai in the Kazakov--Migdal lattice gauge model. I show that both quenching and twisting prescriptions lead to the coordinate-independent master field. I discuss properties of loop averages in reduced as well as unreduced models and demonstrate those coincide in the large mass expansion. I derive loop equations for the Kazakov--Migdal model at large N and show they are reduced for the quadratic potential to a closed set of two equations. I find an exact strong coupling solution of these equations for any D and extend the result to a more general interacting potential. 
  We consider the deformations of ``monomial solutions'' to Generalized Kontsevich Model \cite{KMMMZ91a,KMMMZ91b} and establish the relation between the flows generated by these deformations with those of $N=2$ Landau-Ginzburg topological theories. We prove that the partition function of a generic Generalized Kontsevich Model can be presented as a product of some ``quasiclassical'' factor and non-deformed partition function which depends only on the sum of Miwa transformed and flat times. This result is important for the restoration of explicit $p-q$ symmetry in the interpolation pattern between all the $(p,q)$-minimal string models with $c<1$ and for revealing its integrable structure in $p$-direction, determined by deformations of the potential. It also implies the way in which supersymmetric Landau-Ginzburg models are embedded into the general context of GKM. From the point of view of integrable theory these deformations present a particular case of what is called equivalent hierarchies. 
  We show that the BF theory in any space-time dimension, when quantized in a certain linear covariant gauge, possesses a vector supersymmetry. The generator of the latter together with those of the BRS transformations and of the translations form the basis of a superalgebra of the Wess-Zumino type. We give a general classification of all possible anomalies and invariant counterterms. Their absence, which amounts to ultraviolet finiteness, follows from purely algebraic arguments in the lower-dimensional cases. 
  We present a superconformally invariant and integrable model based on the twisted affine Kac-Moody superalgebra $\hat{osp(2|2)}^{(2)}$ which is the supersymmetrization of the purely bosonic conformal affine Liouville theory recently proposed by Babelon and Bonora. Our model reduces to the super-Liouville or to the super sinh-Gordon theories under certain limit conditions and can be obtained, via hamiltonian reduction, from a superspace WZNW model with values in the corresponding affine KM supergroup. The reconstruction formulae for classical solutions are given. The classical $r$-matrices in the homogeneous grading and the exchange algebras are worked out. 
  A generalization of the pseudoclassical action of a spinning particle in the presence of an anomalous magnetic momentum is given. The action is written in reparametrization and supergauge invariant form. The Dirac quantization, based on the Hamiltonian analyses of the model, leads to the Dirac-Pauli equation for a particle with an anomalous magnetic momentum in an external electromagnetic field. Due to the structure of first-class constraints in that case, the Dirac quantization demands for consistency to take into account an operators ordering problem. 
  We consider the possibility of using the hierarchical approximation to understand the continuum limit of a reformulation of the 3D Ising model initiated by Polyakov. We introduce several new formulations of the hierarchical model using dual or fermionic variables. We discuss several aspects of the renormalization group transformation in terms of these new variables. We mention a reformulation of the model closely related to string models proposed by Zabrodin. 
  The structure of differential equations as they appear in special \K\ geometry of $N=2$ supergravity and $(2,2)$ vacua of the heterotic string is summarized. Their use for computing couplings in the low energy effective Lagrangians of string compactifications is outlined. (Talk presented at the Workshop on String Theory, April 8--10, 1992, Trieste, Italy) 
  A free non-relativistic particle moving in two dimensions on a half-plane can be described by self-adjoint Hamiltonians characterized by boundary conditions imposed on the systems. The most general boundary condition is parameterized in terms of the elements of an infinite-dimensional matrix. We construct the Brownian functional integral for each of these self-adjoint Hamiltonians. Non-local boundary conditions are implemented by allowing the paths striking the boundary to jump to other locations on the boundary. Analytic continuation in time results in the Green's functions of the Schrodinger equation satisfying the boundary condition characterizing the self-adjoint Hamiltonian. 
  The models of triangulated random surfaces embedded in (extended) Dynkin diagrams are formulated as a gauge-invariant matrix model of Weingarten type. The double scaling limit of this model is described by a collective field theory with nonpolynomial interaction.   The propagator in this field theory is essentially two-loop correlator in the corresponding string theory. 
  We analyze the scalar field sector of the Kazakov--Migdal model of induced QCD. We present a detailed description of the simplest one dimensional {($d$$=$$1$)} model which supports the hypothesis of wide applicability of the mean--field approximation for the scalar fields and the existence of critical behaviour in the model when the scalar action is Gaussian. Despite the ocurrence of various non--trivial types of critical behaviour in the $d=1$ model as $N\rightarrow\infty$, only the conventional large-$N$ limit is relevant for its {\it continuum} limit. We also give a mean--field analysis of the $N=2$ model in {\it any} $d$ and show that a saddle point always exists in the region $m^2>m_{\rm crit}^2(=d)$. In $d=1$ it exhibits critical behaviour as $m^2\rightarrow m_{\rm crit}^2$. However when $d$$>$$1$ there is no critical behaviour unless non--Gaussian terms are added to the scalar field action. We argue that similar behaviour should occur for any finite $N$ thus providing a simple explanation of a recent result of D. Gross. We show that critical behaviour at $d$$>$$1$ and $m^2>m^2_{\rm crit}$ can be obtained by adding a $logarithmic$ term to the scalar potential. This is equivalent to a local modification of the integration measure in the original Kazakov--Migdal model. Experience from previous studies of the Generalized Kontsevich Model implies that, unlike the inclusion of higher powers in the potential, this minor modification should not substantially alter the behaviour of the Gaussian model. 
  According to string/fivebrane duality, the Green-Schwarz factorization of the $D=10$ spacetime anomaly polynomial $I_{12}$ into $X_4\, X_8$ means that just as $X_4$ is the anomaly polynomial of the $d=2$ string worldsheet so $X_8$ should be the anomaly polynomial of the $d=6$ fivebrane worldvolume. To test this idea we perform a fivebrane calculation of $X_8$ and find perfect agreement with the string one--loop result. 
  Quantization of the dilaton gravity in two dimensions is discussed by a semiclassical approximation. We compute the fixed-area partition function to one-loop order and obtain the string susceptibility on Riemann surfaces of arbitrary genus. Our result is consistent with the approach using techniques of conformal field theories. 
  Using Hirota's method, solitons are constructed for affine Toda field theories based on the simply-laced affine algebras. By considering automorphisms of the simply-laced Dynkin diagrams, solutions to the remaining algebras, twisted as well as untwisted, are deduced. 
  The tau-function formalism for a class of generalized ``zero-curvature'' integrable hierarchies of partial differential equations, is constructed. The class includes the Drinfel'd-Sokolov hierarchies. A direct relation between the variables of the zero-curvature formalism and the tau-functions is established. The formalism also clarifies the connection between the zero-curvature hierarchies and the Hirota-type hierarchies of Kac and Wakimoto. 
  The 1-loop anomalies of a d-dimensional quantum field theory can be computed by evaluating the trace of the regulated path integral jacobian matrix, as shown by Fujikawa. In 1983, Alvarez-Gaum\'e and Witten observed that one can simplify this evaluation by replacing the operators which appear in the regulator and in the jacobian by quantum mechanical operators with the same (anti)commutation relations. By rewriting this quantum mechanical trace as a path integral with periodic boundary conditions for a one-dimensional supersymmetric nonlinear sigma model, they obtained the chiral anomalies for spin 1/2 and 3/2 fields and selfdual antisymmetric tensors in d dimensions. In this article, we treat the case of trace anomalies for spin 0, 1/2 and 1 fields in a gravitational and Yang-Mills background. We do not introduce a supersymmetric sigma model, but keep the original Dirac matrices $\g^\m$ and internal symmetry generators $T^a$ in the path integral. As a result, we get a matrix-valued action. Gauge covariance of the path integral then requires a definition of the exponential of the action by time-ordering. We exponentiate the factors $\sqrt g$ in the path integral measure by using vector ghosts in order to exhibit the cancellation of the sigma model divergences more clearly. We compute the trace anomalies in d=2 and d=4. 
  Naked singularities appear naturally in dynamically evolving solutions of Einstein equations involving gravitational collapse of radiation, dust and perfect fluids, provided the rate of accretion is less than a critical value. We propose that the gamma-ray bursters (GRBs) are examples of these naked singularity solutions. For illustration, we show that according to solutions involving spherically symmetric collapse of pure radiation field, the energy $E_\gamma$ and the observed duration $\Delta t_o$ of a GRB should satisfy, $\frac{E_\gamma}{\Delta t_o} \leq 4.5 \times 10^{58} \ f_\gamma $ erg sec$^{-1}$, $f_\gamma$ being the fraction ($10^{-2}$ to $10^{-3}$) of energy released as gamma rays. All the presently observed GRBs satisfy this condition; those satisfying the condition close to equality must necessarily be of cosmological origin with the red-shift factor $z$ not exceeding $\sim 2-10$ depending on exact observed flux. 
  We study representations of Temperley-Lieb algebras associated with the transfer matrix formulation of statistical mechanics on arbitrary  lattices. We first discuss a new hyperfinite algebra, the Diagram algebra $D_{\underline{n}}(Q)$, which is a quotient of the Temperley-Lieb algebra appropriate for Potts models in the mean field case, and in which the algebras appropriate for all transverse lattice shapes $G$ appear as subalgebras. We give the complete structure of this subalgebra in the case ${\hat A}_n$ (Potts model on a cylinder). The study of the Full Temperley Lieb algebra of graph $G$ reveals a vast number of infinite sets of inequivalent irreducible representations characterized by one or more (complex) parameters associated to topological effects such as links. We give a complete classification in the ${\hat A}_n$ case where the only such effects are loops and twists. 
  The order-disorder duality structure is exploited in order to obtain a quantum description of anyons and vortices in: a) the Maxwell theory; b) the Abelian Higgs Model; c) the Maxwell-Chern-Simons theory; d) the Maxwell-Chern-Simons-Higgs theory. A careful construction of a charge bearing order operator($\sigma$) and a magnetic flux bearing disorder operator (vortex operator) ($\mu$) is performed, paying attention to the necessary requirements for locality. An anyon operator is obtained as the product $\varphi=\sigma\mu$. A detailed and comprehensive study of the euclidean correlation functions of $\sigma$, $\mu$ and $\varphi$ is carried on in the four theories above. The exact correlation functions are obtained in cases $\underline{a}$ and $\underline{c}$. The large distance behavior of them is obtained in cases $\underline{b}$ and $\underline{d}$. The study of these correlation functions allows one to draw conclusions about the condensation of charge and magnetic flux, establishing thereby an analogy with the Ising model. The mass of vortex and anyon excitations is explicitly obtained wherever these excitations are present in the spectrum. The independence between the mechanisms of mass generation for the vortices and for the vector field is clearly exposed. 
  The S-matrices for the scattering of two excitations in the XYZ model and in all of its SU(n)-type generalizations are obtained from the asymptotic behavior of Kerov's generalized Hall-Littlewood polynomials. These physical scattering processes are all reduced to geometric s-wave scattering problems on certain quantum-symmetric spaces, whose zonal spherical functions these Hall-Littlewood-Kerov polynomials are. Mathematically, this involves a generalization with an unlimited number of parameters of the Macdonald polynomials. Physically, our results suggest that, of the (1+1)-dimensional models, the integrable ones are those, for which the scattering of excitations becomes geometric in the sense above. 
  We compute N-point correlation functions of pure vertex operator states(DK states) for minimal models coupled to gravity. We obtain agreement with the matrix model results on analytically continuing in the numbers of cosmological constant operators and matter screening operators. We illustrate this for the cases of the $(2k-1,2)$ and $(p+1,p)$ models. 
  We show directly in the Lax operator approach how the Virasoro and W-constraints on the $\tau$-function arise in the $p$-reduced KP hierarchy or generalized KdV hierarchy. In partiacular, we consider the KdV and Boussinesq hierarchy to show that the Virasoro and W-constraints follow from the string equation by expanding the ``additional symmetry" operator in terms of the Lax operator. We also mention how this method could be generalized for higher KdV hierarchies. 
  We study the higher spin anologs of the six vertex model on the basis of its symmetry under the quantum affine algebra $U_q(\slth)$. Using the method developed recently for the XXZ spin chain, we formulate the space of states, transfer matrix, vacuum, creation/annihilation operators of particles, and local operators, purely in the language of representation theory. We find that, regardless of the level of the representation involved, the particles have spin $1/2$, and that the $n$-particle space has an RSOS-type structure rather than a simple tensor product of the $1$-particle space. This agrees with the picture proposed earlier by Reshetikhin. 
  The restricted solid-on-solid models in the anti-ferromagnetic regime is studied in the framework of quantum affine algebras. Following the line developed recently for vertex models, a representation theoretical picture is presented for the structure of the space of states. The local operators and the creation/annihilation operators of quasi-particles are defined using vertex operators, and their commutation relations are calculated. 
  By extending the concept of \mc, I introduce a dual formulation of (classical) nonlinear extensions of the \vir\ algebra. This dual formulation is closely related to three dimensional actions which are analogous to a \cs\ action. I present an explicit construction in terms of superfields of the $N=2$ super \wfour. 
  We investigate extensions of the N=2 super Virasoro algebra by one additional super primary field and its charge conjugate. Using a supersymmetric covariant formalism we construct all N=2 super W-algebras up to spin 5/2 of the additional generator. Led by these first examples we close with some conjectures on the classification of N=2 ${\cal SW}(1,\Dt)$ algebras. 
  We extend the considerations of a previous paper on black hole statistical mechanics to the case of black extended objects such as black strings and black membranes in 10-dimensional space-time. We obtain a general expression for the Euclidean action of quantum black p-branes and derive their corresponding degeneracy of states. The statistical mechanics of a gas of black p-branes is then analyzed in the microcanonical ensemble. As in the case of black holes, the equilibrium state is not thermal and the stable configuration is the one for which a single black object carries most of the energy. Again, neutral black p-branes obey the bootstrap condition and it is then possible to argue that their scattering amplitudes satisfy crossing symmetry. Finally, arguments identifying quantum black p-branes with ordinary quantum branes of different dimensionality are presented. 
  We propose to induce QCD by fermions in the adjoint representation of the gauge group SU(N_c) on the lattice. We consider various types of lattice fermions: chiral, Kogut--Susskind and Wilson ones. Using the mean field method we show that a first order large-N phase transition occurs with decreasing fermion mass. We conclude, therefore, that adjoint fermions induce QCD. We draw the same conclusion for the adjoint scalar or fermion models at large number of flavors N_f when they induce a single-plaquette lattice gauge theory. We find an exact strong coupling solution for the adjoint fermion model and show it is quite similar to that for the Kazakov--Migdal model with the quadratic potential. We discuss the possibility for the adjoint fermion model to be solvable at N_c=\infty in the weak coupling region where the Wilson loops obey normal area law. 
  We study superdifferential operators of order $2n+1$ which are covariant with respect to superconformal changes of coordinates on a compact super Riemann surface. We show that all such operators arise from super M\"obius covariant ones. A canonical matrix representation is presented and applications to classical super W algebras are discussed. 
  Affine transformations (dilatations and translations) are used to define a deformation of one-dimensional $N=2$ supersymmetric quantum mechanics. Resulting physical systems do not have conserved charges and degeneracies in the spectra. Instead, superpartner Hamiltonians are $q$-isospectral, i.e. the spectrum of one can be obtained from another (with possible exception of the lowest level) by $q^2$-factor scaling. This construction allows easily to rederive a special self-similar potential found by Shabat and to show that for the latter a $q$-deformed harmonic oscillator algebra of Biedenharn and Macfarlane serves as the spectrum generating algebra. A general class of potentials related to the quantum conformal algebra $su_q(1,1)$ is described. Further possibilities for $q$-deformation of known solvable potentials are outlined.   Talk presented at the workshop on Harmonic Oscillators, College Park, 25-28 March 1992. 
  We study non-linear sigma models with N local supersymmetries in three space-time dimensions. For N=1 and 2 the target space of these models is Riemannian or Kahler, respectively. All N>2 theories are associated with Einstein spaces. For N=3 the target space is quaternionic, while for N=4 it generally decomposes into two separate quaternionic spaces, associated with inequivalent supermultiplets. For N=5,6,8 there is a unique (symmetric) space for any given number of supermultiplets. Beyond that there are only theories based on a single supermultiplet for N=9,10,12 and 16, associated with coset spaces with the exceptional isometry groups $F_{4(-20)}$, $E_{6(-14)}$, $E_{7(-5)}$ and $E_{16(+8)}$, respectively. For $N=3$ and $N\geq5$ the $D=2$ theories obtained by dimensional reduction are two-loop finite. 
  We describe a self-consistent canonical quantization of Liouville theory in terms of canonical free fields. In order to keep the non-linear Liouville dynamics, we use the solution of the Liouville equation as a canonical transformation. This also defines a Liouville vertex operator. We show, in particular, that a canonical quantized conformal and local quantum Liouville theory has a quantum group structure, and we discuss correlation functions for non-critical strings. 
  A large class of cosmological solutions (of the Einstein equations) in string theory, in the presence of Maxwell fields, is obtained by $O(d,d)$ transformations of simple backgrounds with $d$ toroidal isometries. In all the examples in which we find a (closed) expanding universe, such that the universe admits a smooth, complete initial value hypersurface, a naked singularity may form only at the time when the universe collapses. The discrete symmetry group $O(d,d,Z)$ identifies different cosmological solutions with a background corresponding to a (relatively) simple CFT, and therefore, may be useful in understanding the properties of naked singularities in string theory. 
  We chart out the landscape of $\Winfty$-type algebras using $\Wkpq$---a recently discovered one-parameter deformation of $\W_{\rm KP}$. We relate all hitherto known $\Winfty$-type algebras to $\Wkpq$ and its reductions, contractions, and/or truncations at special values of the parameter. 
  We present a generalization of the $U(1)^{2}$ charged dilaton black holes family whose main feature is that both $U(1)$ fields have electric and magnetic charges, the axion field still being trivial. We show the supersymmetry of these solutions in the extreme case, in which the corresponding generalization of the Bogomolnyi bound is saturated and a naked singularity is on the verge of being visible to external observers. Then we study the action of a subset of the $SL(2,R)$ group of electric-magnetic duality rotations that generates a non-trivial axion field on those solutions. This group of transformations is an exact symmetry of the $N=4$ $d=4$ ungauged supergravity equations of motion. It has been argued recently that it could be an exact symmetry of the full effective string theory. The generalization of the Bogomolnyi bound is invariant under the full $SL(2,R)$ and the solutions explicitly rotated are shown to be supersymmetric if the originals are. We conjecture that any $SL(2,R)$ transformation will preserve supersymmetry. 
  A Fock representation of the quantum affine algebra $U_q(\widehat{\sl}_2)$ is constructed by three bosonic fields for an arbitrary level with the help of the Drinfeld realization. 
  The Liouville approach is applied to the quantum treatment of the dilaton gravity in two dimensions. The physical states are obtained from the BRST cohomology and correlation functions are computed up to three-point functions. For the $N=0$ case (i.e., without matter), the cosmological term operator is found to have the discrete momentum that plays a special role in the $c=1$ Liouville gravity. The correlation functions for arbitrary numbers of operators are found in the $N=0$ case, and are nonvanishing only for specific ``chirality'' configurations. 
  The continuum (Liouville) approach to the two-dimensional (2-D) quantum gravity is reviewed with particular attention to the $c=1$ conformal matter coupling, and new results on a related problem of dilaton gravity are reported. After finding the physical states, we examine the procedure to compute correlation functions. The physical states in the relative cohomology show up as intermediate state poles of the correlation functions. The states in the absolute cohomology but not in the relative cohomology arise as auxiliary fields in string field theory. The Liouville approach is applied also to the quantum treatment of the dilaton gravity. The physical states are obtained from the BRST cohomology and correlation functions are computed in the dilaton gravity. 
  We generalize the Goddard-Kent-Olive (GKO) coset construction to the dimension 5/2 operator for $ \hat{so} (5) $ and compute the fourth order   Casimir invariant in the coset model $\hat{SO} (5)_{1} \times \hat{SO} (5)_{m} / \hat{SO} (5)_{1+m} $ with the generic unitary minimal $ c < 5/2 $ series that can be viewed as perturbations of the $ m \rightarrow \infty $ limit, which has been investigated previously in the realization of $ c= 5/2 $ free fermion model. 
  We give a simple geometrical interpretation of classical $\W$-transformations as deformations of constant energy surfaces by canonical transformations on a two-dimensional phase space. 
  We present field theoretical descriptions of massless (2+1) dimensional nonrelativistic fermions in an external magnetic field, in terms of a fermionic and bosonic second quantized language. An infinite dimensional algebra, $W_{\infty}$, appears as the algebra of unitary transformations which preserve the lowest Landau level condition and the particle number. In the droplet approximation it reduces to the algebra of area-preserving diffeomorphisms, which is responsible for the existence of a universal chiral boson Lagrangian independent of the electrostatic potential. We argue that the bosonic droplet approximation is the strong magnetic field limit of the fermionic theory. The relation to the $c=1$ string model is discussed. 
  We present a notion of symmetry for 1+1-dimensional integrable systems which is consistent with their group theoretic description and reproduces in special cases the known Baecklund transformation for the generalized Korteweg-deVries hierarchies. We also apply it to the relativistic invariance of the Leznov-Saveliev systems. 
  Perturbative renormalization of a non-Abelian Chern-Simons gauge theory is examined. It is demonstrated by explicit calculation that, in the pure Chern-Simons theory, the beta-function for the coefficient of the Chern-Simons term vanishes to three loop order. Both dimensional regularization and regularization by introducing a conventional Yang-Mills component in the action are used. It is shown that dimensional regularization is not gauge invariant at two loops. A variant of this procedure, similar to regularization by dimensional reduction used in supersymmetric field theories is shown to obey the Slavnov-Taylor identity to two loops and gives no renormalization of the Chern-Simons term. Regularization with Yang-Mills term yields a finite integer-valued renormalization of the coefficient of the Chern-Simons term at one loop, and we conjecture no renormalization at higher order. We also examine the renormalization of Chern-Simons theory coupled to matter. We show that in the non-abelian case the Chern-Simons gauge field as well as the matter fields require infinite renormalization at two loops and therefore obtain nontrivial anomalous dimensions. We show that the beta function for the gauge coupling constant is zero to two-loop order, consistent with the topological quantization condition for this constant. 
  The Adler-Kostant-Symes $R$-bracket scheme is applied to the algebra of pseudo-differential operators to relate the three integrable hierarchies: KP and its two modifications, known as nonstandard integrable models. All three hierarchies are shown to be equivalent and connection is established in the form of a symplectic gauge transformation. This construction results in a new representation of the W-infinity algebras in terms of 4 bosonic fields. 
  The general BRST-anti-BRST construction in the framework of the antifield-antibracket formalism is illustrated in the case of the Freedmann-Townsend model. 
  Black hole evaporation may lead to massive or massless remnants, or naked singularities. This paper investigates this process in the context of two quite different two dimensional black hole models. The first is the original CGHS model, the second is another two dimensional dilaton-gravity model, but with properties much closer to physics in the real, four dimensional, world. Numerical simulations are performed of the formation and subsequent evaporation of black holes and the results are found to agree qualitatively with the exactly solved modified CGHS models, namely that the semiclassical approximation breaks down just before a naked singularity appears. 
  We present the extension of the Wakimoto construction to the $su(2)_k$ quantum current algebra and its associated $Z_k$ quantum parafermion algebra. This construction is achieved in terms of various deformations of three classical free boson fields. We also give the vertex operators corresponding to the quantum spin-$j$ representation. 
  We study the discrete state structure of $\hat c=1$ superconformal matter coupled to 2-D supergravity. Factorization properties of scattering amplitudes are used to identify these states and to construct the corresponding vertex operators. For both Neveu-Schwarz and Ramond sectors these states are shown to be organized in   SU(2) multiplets. The algebra generated by the discrete states is computed in the limit of null cosmological constant. 
  We determine explicitly all structure constants of the whole chiral BRST cohomology ring in $D=2$ string theory including both the discrete states and tachyon states. This is made possible by establishing several identities for Schur polynomials with operator argument and exploring associativity. Furthermore we find that the (chiral) symmetry algebra of the charges obtained by using the descent equations can actually be read off from the cohomology ring structure by simple operation involving the ghost field $b$. We also determine the enlarged symmetry algebra which contains the charges having ghost number $-1$ and $1$. Finally the complete symmetry transformation rules are derived for closed string discrete states by carefully combining the left and right sectors. It turns out that the new states introduced recently by Witten and Zwiebach are naturally created when symmetries act on the old states. 
  A weak version of the cosmic censorship hypothesis is implemented as a set of boundary conditions on exact semi-classical solutions of two-dimensional dilaton gravity. These boundary conditions reflect low-energy matter from the strong coupling region and they also serve to stabilize the vacuum of the theory against decay into negative energy states. Information about low-energy incoming matter can be recovered in the final state but at high energy black holes are formed and inevitably lead to information loss at the semi-classical level. 
  Our answer is the latter. Space-time singularities, including the initial one, are described by world-sheet topological Abelian gauge theories with a Chern-Simons term. Their effective $N=2$ supersymmetry provides an initial fixed point where the Bogomolny bound is saturated on the world-sheet, corresponding to an extreme Reissner-Nordstrom solution in space-time. Away from the singularity the gauge theory has world-sheet matter fields, bosons and fermions, associated with the generation of target space-time. Because the fermions are complex (cf the Quantum Hall Effect) rather than real (cf high-$T_c$ superconductors) the energetically-preferred vacuum is not parity or time-reversal invariant, and the associated renormalization group flow explains the cosmological arrow of time, as well as the decay of real or virtual black holes, with a monotonic increase in entropy. 
  A representation of the quantum affine algebra $U_{q}(\hat{sl_{2}})$ of an arbitrary level $k$ is realized in terms of three boson fields, whose $q \rightarrow 1$ limit becomes the Wakimoto representation. An analogue of the screening current is also obtained. It commutes with the action of $U_{q}(\hat{sl_{2}})$ modulo total difference of some fields. 
  A bosonization scheme of the $q$-vertex operators of $\uqa$ for arbitrary level is obtained. They act as intertwiners among the highest weight modules constructed in a bosonic Fock space. An integral formula is proposed for $N$-point functions and explicit calculation for two-point function is presented. 
  We analyze the allowed spectrum of electric and magnetic charges carried by dyons in (toroidally compactified) heterotic string theory in four dimensions at arbitrary values of the string coupling constant and $\theta$ angle. The spectrum is shown to be invariant under electric-magnetic duality transformation, thereby providing support to the conjecture that this is an exact symmetry in string theory. 
  An on-shell formulation of (p,q), 2\leq p \leq 4, 0\leq q\leq 4, supersymmetric coset models with target space the group G and gauge group a subgroup H of G is given. It is shown that there is a correspondence between the number of supersymmetries of a coset model and the geometry of the coset space G/H. The algebras of currents of supersymmetric coset models are superconformal algebras. In particular, the algebras of currents of (2,2) and (4,0) supersymmetric coset models are related to the N=2 Kazama-Suzuki and N=4 Van Proeyen superconformal algebras correspondingly. 
  The construction of mirror symmetry in the heterotic string is reviewed in the context of Calabi-Yau and Landau-Ginzburg compactifications. This framework has the virtue of providing a large subspace of the configuration space of the heterotic string, probing its structure far beyond the present reaches of solvable models. The construction proceeds in two stages: First all singularities/catastrophes which lead to ground states of the heterotic string are found. It is then shown that not all ground states described in this way are independent but that certain classes of these LG/CY string vacua can be related to other, simpler, theories via a process involving fractional transformations of the order parameters as well as orbifolding. This construction has far reaching consequences. Firstly it allows for a systematic identification of mirror pairs that appear abundantly in this class of string vacua, thereby showing that the emerging mirror symmetry is not accidental. This is important because models with mirror flipped spectra are a priori independent theories, described by distinct CY/LG models. It also shows that mirror symmetry is not restricted to the space of string vacua described by theories based on Fermat potentials (corresponding to minimal tensor models). Furthermore it shows the need for a better set of coordinates of the configuration space or else the structure of this space will remain obscure. While the space of LG vacua is {\it not} completely mirror symmetric, results described in the last part suggest that the space of Landau--Ginburg {\it orbifolds} possesses this symmetry. 
  We study the continuum limit of the spin-1 chain in the non-Abelian bosonization approach of Affleck and show that the Hamiltonian of integrable spin-1 chain yields the Lagrangian of supersymmetric sine-Gordon model in the zero lattice spacing limit. We also show that the quantum group generators of the spin-1 chain give non-local charges of the supersymmetric sine-Gordon theory. 
  Four-dimensional string theories predict in general the existence of light exotic particles with fractional electric charges. Such particles could escape present observations if they are confined by a gauge group of the "hidden" sector into integrally charged states. It is conceivable that the same gauge group is also responsible for dynamical supersymmetry breaking, via gaugino and scalar condensation. The communication of the breaking to the observable sector is now mediated by ordinary gauge interactions, implying that the confining scale can be in the TeV-region. We study the main phenomenological implications of this possibility. In particular, we analyze the pattern of supersymmetry breaking and the mass-spectrum of the sparticles. We also show that this scenario can be consistent with the unification of all coupling constants at the string scale. 
  Open superstrings at non-zero temperature are considered. A novel representation for the free energy (Laurent series representation) is constructed. It is shown that the Hagedorn temperature arises in this formalism as the convergence condition (specifically, the radius of convergence) of the Laurent series. 
  Two dimensional quantum gravity coupled to a conformally invariant matter field of central charge c=n/2, is represented, in a discretized version, by n independent Ising spins per cell of the triangulations of a random surface. The matrix integral representation of this model leads to a diagrammatic expansion at large orders, when the Ising coupling constant is tuned to criticality, one extracts the values of the string susceptibility exponent. We extend our previous calculation to order eight for genus zero and investigate now also the genus one case in order to check the possibility of having a well-defined double scaling limit even c>1. 
  We consider a $2d$ sigma model with a $2+N$ - dimensional Minkowski signature target space metric having a covariantly constant null Killing vector. We study solutions of the conformal invariance conditions in $2+N$ dimensions and find that generic solutions can be represented in terms of the RG flow in $N$ - dimensional ``transverse space'' theory. The resulting conformal invariant sigma model is interpreted as a quantum action of the $2d$ scalar (``dilaton") quantum gravity model coupled to a (non-conformal) `transverse' sigma model. The conformal factor of the $2d$ metric is identified with a light cone coordinate of the $2+N$ - dimensional sigma model. We also discuss the case when the transverse theory is conformal (with or without the antisymmetric tensor background) and reproduce in a systematic way the solutions with flat transverse space known before. 
  The first quantum mass corrections for the solitons of complex $sl(n)$ affine Toda field theory are calculated. The corrections are real and preserve the classical mass ratios. The formalism also proves that the solitons are classically stable. 
  We study the dressing of operators and flows of corresponding couplings in models of {\it embedded} random surfaces. We show that these dressings can be obtained by applying the methods of David and Distler and Kawai. We consider two extreme limits. In the first limit the string tension is large and the dynamics is dominated by the Nambu-Goto term. We analyze this theory around a classical solution in the situation where the length scale of the solution is large compared to the length scale set by the string tension. Couplings get dressed by the liouville mode (which is now a composite field) in a non-trivial fashion. However this does {\it not} imply that the excitations around a physical ``long string" have a phase space corresponding to an extra dimension. In the second limit the string tension is small and the dynamics is governed by the extrinsic curvature term. We show, perturbatively, that in this theory the relationship between the induced metric and the worldsheet metric is ``renormalized", while the extrinsic curvature term receives a non-trivial dressing as well. This has the consequence that in a generic situation the dependence of couplings on the physical scale is different from that predicted by their beta functions. 
  We show explicitly that the question of gauge invariance of the effective potential in standard scalar electrodynamics remains unchanged despite the introduction of the Chern-Simons term. The result does not depend on the presence of the Maxwell term in the Chern-Simons territory. 
  We propose a bilocal field theory for mesons in two dimensions obtained as a kind of non local bosonization of two dimensional QCD. Its semi-classical expansion is equivalent to the $1/N_c$ expansion of QCD. Using an ansatz we reduce the classical equation of motion of this theory in the baryon number one sector to a relativistic Hartree equation and solve it numerically. This (non topological) soliton is identified with the baryon. 
  We give a general procedure for extracting the propagators in gauge theories in presence of a sharp gauge fixing and we apply it to derive the propagators in quantum gravity in the radial gauge, both in the first and in the second order formalism in any space-time dimension. In the three dimensional case such propagators vanish except for singular collinear contributions, in agreement with the absence of propagating gravitons. 
  We comment on a program designed for the study of local chiral algebras and their representations in 2D conformal field theory. Based on the algebraic approach described by W. Nahm, this program efficiently calculates arbitrary n-point functions of these algebras. The program is designed such that calculations involving e.g. current algebras, W-algebras and N-Superconformal algebras can be performed. As a non-trivial application we construct an extension of the Virasoro algebra by two fields with spin four and six using the N=1-Super-Virasoro algebra. 
  In this paper we consider the representation theory of N=1 Super-W-algebras with two generators for conformal dimension of the additional superprimary field between two and six. In the superminimal case our results coincide with the expectation from the ADE-classification. For the parabolic algebras we find a finite number of highest weight representations and an effective central charge $\tilde c = 3/2$. Furthermore we show that most of the exceptional algebras lead to new rational models with $\tilde c > 3/2$. The remaining exceptional cases show a new `mixed' structure. Besides a continuous branch of representations discrete values of the highest weight exist, too. 
  In [1,2] we established and discussed the algebra of observables for 2+1 gravity at both the classical and quantum level. Here our treatment broadens and extends previous results to any genus $g$ with a systematic discussion of the centre of the algebra. The reduction of the number of independent observables to $6g-6 (g > 1)$ is treated in detail with a precise classification for $g = 1$ and $g = 2$. 
  We calculate exactly the rate of pair production of open bosonic and supersymmetric strings in a constant electric field. The rate agrees with Schwinger's classic result in the weak-field limit, but diverges when the electric field approaches some critical value of the order of the string tension. (Phyzzx file) 
  New non-perturbative interactions in the effective action of two dimensional string theory are described. These interactions are due to ``stringy" instantons 
  We discuss various techniques for computing the semi-infinite cohomology of highest weight modules which arise in the BRST quantization of two dimensional field theories. In particular, we concentrate on two such theories -- the $G/H$ coset models and $2d$ gravity coupled to $c\leq 1$ conformal matter. (to appear in the proceedings of the XXV Karpacz Winter School) 
  In this paper we discuss the BRST-quantization of anomalous 2d-Yang Mills (YM) theory. Since we use an oscillator basis for the YM-Fock-space the anomaly appears already for a pure YM-system and the constraints form a Kac-Moody algebra with negative central charge. We also discuss the coupling of chiral fermions and find that the BRST-cohomology for systems with chiral fermions in a sufficiently large representation of the gauge group is completely equivalent to the cohomology of the finite dimensional gauge group. For pure YM theory or YM theory coupled to chiral fermions in small representations there exists an infinite number of inequivalent cohomology classes. This is discussed in some detail for the example of $SU(2)$. 
  We propose a generalization of the collective field theory hamiltonian, including interactions between the original bosonic collective field $w_0 (z)$ and supplementary fields ${\bar w}_j (z)$ realizing classically a $w_\infty$ algebra. The latter are then shown to represent a 3--dimensional topological field theory. This generalization follows from a conjectured representation of the $W_{1 + \infty}$ algebra of bilinear fermion operators underlying the original matrix model. It provides an improved bosonization scheme for $1+1$ dimensional fermion theories. 
  We derive the recently proposed BRST charge for non-critical W strings from a Lagragian approach. The basic observation is that, despite appearances, the combination of two classical ``matter'' and ``Toda'' w_3 systems leads to a closed modified gauge algebra, which is of the so-called soft type. Based on these observations, a novel way to construct critical W_3 strings is given. 
  We study a theory of Dirac fermions on a disk in presence of an electromagnetic field. Using the heat-kernel technique we compute the functional determinant which results after decoupling the zero-flux gauge degrees of freedom from the fermions. We also compute the Green functions of the remaining fermionic theory with the appropriate boundary conditions. Finally we analyze the coset model associated to this gauge theory and compute all its correlations functions. 
  We investigate quantum mechanical Hamiltonians with explicit time dependence. We find a class of models in which an analogue of the time independent \S equation exists. Among the models in this class is a new exactly soluble model, the harmonic oscillator with frequency inversely proportional to time. 
  Misprints in the definition of semiclassical tau-function and in the formulae (3.40b) and (4.20) are corrected. 
  In the context of hermitean one--matrix models we show that the emergence of the NLS hierarchy and of its reduction, the KdV hierarchy, is an exact result of the lattice characterizing the matrix model. Said otherwise, we are not obliged to take a continuum limit to find these hierarchies. We interpret this result as an indication of the topological nature of them. We discuss the topological field theories associated with both and discuss the connection with topological field theories coupled to topological gravity already studied in the literature. 
  Thus far in the search for, and classification of, `physical' modular invariant partition functions $\sum N_{LR}\,\c_L\,\C_R$ the attention has been focused on the {\it symmetric} case where the holomorphic and anti-holomorphic sectors, and hence the characters $\c_L$ and $\c_R$, are associated with the same Kac-Moody algebras $\g_L=\g_R$ and levels $k_L=k_R$. In this paper we consider the more general possibility where $(\g_L,k_L)$ may not equal $(\g_R,k_R)$. We discuss which choices of algebras and levels may correspond to well-defined conformal field theories, we find the `smallest' such {\it heterotic} (\ie asymmetric) partition functions, and we give a method, generalizing the Roberts-Terao-Warner lattice method, for explicitly constructing many other modular invariants. We conclude the paper by proving that this new lattice method will succeed in generating all the heterotic partition functions, for all choices of algebras and levels. 
  A natural first step in the classification of all `physical' modular invariant partition functions $\sum N_{LR}\,\c_L\,\C_R$ lies in understanding the commutant of the modular matrices $S$ and $T$. We begin this paper extending the work of Bauer and Itzykson on the commutant from the $SU(N)$ case they consider to the case where the underlying algebra is any semi-simple Lie algebra (and the levels are arbitrary). We then use this analysis to show that the partition functions associated with even self-dual lattices span the commutant. This proves that the lattice method due to Roberts and Terao, and Warner, will succeed in generating all partition functions. We then make some general remarks concerning certain properties of the coefficient matrices $N_{LR}$, and use those to explicitly find all level 1 partition functions corresponding to the algebras $B_n$, $C_n$, $D_n$, and the 5 exceptionals. Previously, only those associated to $A_n$ seemed to be generally known. 
  We study general relativity in the framework of non-commutative differential geometry. In particular, we introduce a gravity action for a space-time which is the product of a four dimensional manifold by a two-point space. In the simplest situation, where the Riemannian metric is taken to be the same on the two copies of the manifold, one obtains a model of a scalar field coupled to Einstein gravity. This field is geometrically interpreted as describing the distance between the two points in the internal space. 
  A canonical Lorentz invariant field theory extension of collective field theory of d=1 matrix models is presented.   We show that the low density, discrete, sector of collective field theory includes single eigenvalue Euclidean instantons which tunnel between different vacua of the extended theory. These "stringy" instantons induce non-perturbative effective operators of strength $e^{-{1/g}}$. The relationship of the world sheet description of string theory and Liouville theory to the effective space-time theory is explained. 
  We describe conformal field theories, correlation functions of which satisfy equations of the two-dimensional fluid mechanics. Prediction for the energy spectrum is given, $E(k) \sim k^{-25/7}$. 
  Exact and explicit string solutions in de Sitter spacetime are found. (Here, the string equations reduce to a sinh-Gordon model). A new feature without flat spacetime analogy appears: starting with a single world-sheet, several (here two) strings emerge. One string is stable and the other (unstable) grows as the universe grows. Their invariant size and energy either grow as the expansion factor or tend to constant. Moreover, strings can expand (contract) for large (small) universe radius with a different rate than the universe. 
  This paper is an expanded version of a talk given at the XIX International Colloquium on Group Theoretical Methods in Physics, Salamanca, July, 1992.   We discuss the geometry of topological terms in classical actions, which by themselves form the actions of topological field theories. We first treat the Chern-Simons action directly. Then we explain how the geometry is best understood via integration of smooth Deligne cocycles. We conclude with some remarks about the corresponding quantum theories in 3 and 4 dimensions.   This paper is written using AMSTeX 2.1, which can be obtained via ftp from the American Mathematical Society (instructions included). 
  We propose gauging matrix models of string theory to eliminate unwanted non-singlet states. To this end we perform a discretised light-cone quantisation of large N gauge theory in 1+1 dimensions, with scalar or fermionic matter fields transforming in the adjoint representation of SU(N). The entire spectrum consists of bosonic and fermionic closed-string excitations, which are free as N tends to infinity. We analyze the general features of such bound states as a function of the cut-off and the gauge coupling, obtaining good convergence for the case of adjoint fermions. We discuss possible extensions of the model and the search for new non-critical string theories. 
  A modified rigid string theory with infrared behaviour governed by a nontrivial fixed point is presented. 
  The BRST cohomology analysis of Lian and Zuckerman leads to physical states at all ghost number for $c<1$ matter coupled to Liouville gravity. We show how these states are related to states at ghost numbers zero(pure vertex operator states -- DK states) and ghost number one(ring elements) by means of descent equations. These descent equations follow from the double cohomology of the String BRST and Felder BRST operators. We briefly discuss how the ring elements allow one to determine all correlation functions on the sphere. 
  Generalizing our previous work, we show how $O(d,d)$ transformations can be used to "boost away" in new dimensions the physical singularities that occur generically in cosmological and/or black-hole string backgrounds. As an example, we show how a recent model by Nappi and Witten can be made singularity-free via $O(3,3)$ boosts involving a fifth dimension. 
  Optimal hypercontractivity bounds for the fermion oscillator semigroup are obtained. These are the fermion analogs of the optimal hypercontractivity bounds for the boson oscillator semigroup obtained by Nelson. In the process, several results of independent interest in the theory of non-commutative integration are established. {}. 
  Quadratic algebras related to the reflection equations are introduced. They are quantum group comodule algebras. The quantum group $F_q(GL(2))$ is taken as the example. The properties of the algebras (center, representations, realizations, real forms, fusion procedure etc) as well as the generalizations are discussed. 
  This review is based on lectures given at the 1992 Trieste Spring School on String Theory and Quantum Gravity and at the 1992 TASI Summer School in Boulder, Colorado. 
  Recent results for tree amplitudes for the $N=2$ noncritical strings are presented and compared with the critical case. Arguments are given which indicate a certain discontinuity in passing from the $\hat c < 1$ model (in a Coulomb gas representation) to the $\hat c = 1$ critical case. 
  We study the correspondence between the linear matrix model and the interacting nonlinear string theory. Starting from the simple matrix harmonic oscillator states, we derive in a direct way scattering amplitudes of 2-dimensional strings, exhibiting the nonlinear equation generating arbitrary N-point tree amplitudes. An even closer connection between the matrix model and the conformal string theory is seen in studies of the symmetry algebra of the system. 
  I review the information loss paradox that was first formulated by Hawking, and discuss possible ways of resolving it. All proposed solutions have serious drawbacks. I conclude that the information loss paradox may well presage a revolution in fundamental physics. (To appear in the proceedings of the International Symposium on Black Holes, Membranes, Wormholes, and Superstrings.) 
  $N=2$ coset models of the type $SU(m+1)/SU(m)\times U(1)$ with nondiagonal modular invariants for both $SU(m+1)$ and $SU(m)$ are considered. Poincar\'e polynomials of the corresponding chiral rings of these algebras are constructed. They are used to compute the number of chiral generations of the associated string compactifications. Moddings by discrete symmetries are also discussed. 
  The physical interpretation of the main notions of the quantum group theory (coproduct, representations and corepresentations, action and coaction) is discussed using the simplest examples of $q$-deformed objects (quantum group $F_q(GL(2))$, quantum algebra $sl_q(2)$, $q$-oscillator and $F_q$-covariant algebra.) Appropriate reductions of the covariant algebra of second rank $q$-tensors give rise to the algebras of the $q$-oscillator and the $q$-sphere. A special covariant algebra related to the reflection equation corresponds to the braid group in a space with nontrivial topology. 
  We study the Turaev-Viro invariant as the Euclidean Chern-Simons-Witten gravity partition function with positive cosmological constant. After explaining why it can be identified as the partition function of 3-dimensional gravity, we show that the initial data of the TV invariant can be constructed from the duality data of a certain class of rational conformal field theories, and that, in particular, the original Turaev-Viro's initial data is associated with the $A_{k+1}$ modular invariant WZW model. As a corollary we then show that the partition function $Z(M)$ is bounded from above by $Z((S^2\times S^1)^{\sharp g}) =(S_{00})^{-2g+2}\sim \Lambda^{-\frac{3g-3}{2}}$, where $g$ is the smallest genus of handlebodies with which $M$ can be presented by Hegaard splitting. $Z(M)$ is generically very large near $\Lambda\sim +0$if $M$ is neither $S^3$ nor a lens space, and many-wormholeconfigurations dominate near $\Lambda\sim +0$ in the sense that $Z(M)$ generically tends to diverge faster as the ``number of wormholes'' $g$ becomes larger. 
  The hamiltonian BRST-anti-BRST theory is developed in the general case of arbitrary reducible first class systems. This is done by extending the methods of homological perturbation theory, originally based on the use of a single resolution, to the case of a biresolution. The BRST and the anti-BRST generators are shown to exist. The respective links with the ordinary BRST formulation and with the $ sp(2) $-covariant formalism are also established. 
  We compute the low-energy classical differential scattering cross-section for BPS $SU(2)$ magnetic monopoles using the geodesic approximation to the actual dynamics and 16K parallel processors on a CM2. Numerical experiments suggest that the quantum BPS magnetic monopole differential cross-section is well-approximated by the classical BPS magnetic monopole differential cross-section. In particular, the expected quantum interference effects for bosons at scattering angle $\theta=\pi/2$ (CoM frame) are contradicted numerically. We argue that this is due to the topology of the classical configuration space for these solitons. We also study the scattering and bounded classical motions of BPS dyons and their global structure in phase space by constructing `escape plots'. The escape plots contain a surprising amount of structure, and suggest that the classical dynamics of two BPS $SU(2)$ magnetic monopoles is chaotic and that there are closed and bounded two dyon motions with isolated energies. 
  Recently, models of two-dimensional dilaton gravity have been shown to admit classical black-hole solutions that exhibit Hawking radiation at the semi-classical level. These classical and semi-classical analyses have been performed in conformal gauge. We show in this paper that a similar analysis in the light--cone gauge leads to the same results. Moreover, quantization of matter fields in light--cone gauge can be naturally extended to include quantizing the metric field {\it \`a la} KPZ. We argue that this may provide a new framework to address many issues associated to black-hole physics. 
  We perform a systematic investigation of free-scalar realisations of the Za\-mo\-lod\-chi\-kov $W_3$ algebra in which the operator product of two spin-three generators contains a non-zero operator of spin four which has vanishing norm. This generalises earlier work where such an operator was required to be absent. By allowing this spin-four null operator we obtain several realisations of the $W_3$ algebra both in terms of two scalars as well as in terms of an arbitrary number $n$ of free scalars. Our analysis is complete for the case of two-scalar realisations. 
  In this set of lectures, we give a pedagogical introduction to the subject of anyons. We discuss 1) basic concepts in anyon physics, 2) quantum mechanics of two anyon systems, 3) statistical mechanics of many anyon systems, 4) mean field approach to many anyon systems and anyon superconductivity, 5) anyons in field theory and 6) anyons in the Fractional Quantum Hall Effect (FQHE). (Based on lectures delivered at the VII SERC school in High Energy Physics at the Physical Research Laboratory, Ahmedabad, January 1992 and at the I SERC school in Statistical Mechanics at Puri, February 1994.) 
  We explicitly demonstrate that the unitary representations of the $w_\infty$ algebra and its truncations are just the unitary representations of the Virasoro algebra. 
  It is shown that the asymmetric chiral gauging of the WZW models give rise to consistent string backgrounds. The target space structure of the ${[{SL(2,\Re)/ {SO(1,1)}}]}_L \bigotimes {[{SL(2,\Re)/U(1)}]}_R$ model is analyzed and the presence of a hidden isometry in this background is demonstrated. A nonlinear coordinate transformation is obtained which transforms the asymmetric model to the symmetric one, analyzed recently by two of the present authors. 
  A topological quantum field theory of non-abelian differential forms is investigated from the point of view of its possible applications to description of polynomial invariants of higher-dimensional two-component links. A path-integral representation of the partition function of the theory, which is a highly on-shell reducible system, is obtained in the framework of the antibracket-antifield formalism of Batalin and Vilkovisky. The quasi-monodromy matrix, giving rise to corresponding skein relations, is formally derived in a manifestly covariant non-perturbative manner. 
  The modifications of dilaton black holes which result when the dilaton acquires a mass are investigated. We derive some general constraints on the number of horizons of the black hole and argue that if the product of the black hole charge $Q$ and the dilaton mass $m$ satisfies $Q m < O(1)$ then the black hole has only one horizon. We also argue that for $Q m > O(1)$ there may exist solutions with three horizons and we discuss the causal structure of such solutions. We also investigate the possible structures of extremal solutions and the related problem of two-dimensional dilaton gravity with a massive dilaton. 
  We study a fermionic coset model $G/H$ when the subgroup $H$ is not simply connected. We show that even when the fermionic zero modes impose selection rules which alters the values of the correlators, the Virasoro central charge of the theory results independent of the topological sector. 
  We incorporate both BRS symmetry and anti-BRS symmetry into the quantisation of topological Yang--Mills theory. This refines previous treatments which consider only the BRS symmetry. Our formalism brings out very clearly the geometrical meaning of topological Yang--Mills theory in terms of connections and curvatures in an enlarged superspace; and its simple relationship to the geometry of ordinary Yang--Mills theory. We also discover a certain SU(3) triality between physical spacetime, and the two ghost directions of superspace. Finally, we demonstrate how to recover the usual gauge-fixed topological Yang--Mills action from our formalism. 
  We show that the $U(1,1)$ (super) Chern Simons theory is one loop exact. This provides a direct proof of the relation between the Alexander polynomial and analytic and Reidemeister torsion. We then proceed to compute explicitely the torsions of Lens spaces and Seifert manifolds using surgery and the $S$ and $T$ matrices of the $U(1,1)$ Wess Zumino Witten model recently determined, with complete agreement with known results. $U(1,1)$ quantum field theories and the Alexander polynomial provide thus "toy" models with a non trivial topological content, where all ideas put forward by Witten for $SU(2)$ and the Jones polynomial can be explicitely checked, at finite $k$. Some simple but presumably generic aspects of non compact groups, like the modified relation between Chern Simons and Wess Zumino Witten theories, are also illustrated. We comment on the closely related case of $GL(1,1)$. 
  An explicit expression is suggested for the average $<U_{ij}U_{kl}^{\dagger}>$ over the unitary group $SU(N)$ with the Itzykson-Zuber measure $[dU] \exp {\rm tr} \Phi U\Psi U^{\dagger}$ 
  By generalizing the Miura transformation for $\Ww_N$ to other classical $\Ww$ algebras obtained by hamiltonian reduction, we find realisations of these algebras in terms of relatively simple non-abelian current algebras, e.g. $\widehat{sl(2)}\times \widehat{u(1)}^N$, generalizing the free field realisation of $\Ww_N$. As an example, we present the $\widehat{sl(2)}\times \widehat{u(1)}$ realisation of $\Ww_3^2$, which we also quantize. By a specific example, we also show how the realisation of $\Ww_N$ with the currents of $\Ww_{N-1}$ and a free boson can be generalized to certain classes of ``extended'' $\Ww_N$ algebras. 
  Quantum groups play a role of symmetries of integrable theories in two dimensions. They may be detected on the classical level as Poisson-Lie symmetries of the corresponding phase spaces. We discuss specifically the Wess-Zumino-Witten conformally invariant quantum field model combining two chiral parts which describe the left- and right-moving degrees of freedom. On one hand side, the quantum group plays the role of the symmetry of the chiral components of the theory. On the other hand, the model admits a lattice regularization (in the Minkowski space) in which the current algebra symmetry of the theory also becomes quantum, providing the simplest example of a quantum group symmetry coupling space-time and internal degrees of freedom. We develop a free field approach to the representation theory of the lattice $sl(2)$-based current algebra and show how to use it to rigorously construct an exact solution of the quantum $SL(2)$ WZW model on lattice. 
  A general method of obtaining linear differential equations having polynomial solutions is proposed. The method is based on an equivalence of the spectral problem for an element of the universal enveloping algebra of some Lie algebra in the "projectivized" representation possessing an invariant subspace and the spectral problem for a certain linear differential operator with variable coefficients. It is shown in general that polynomial solutions of partial differential equations occur; in the case of Lie superalgebras there are polynomial solutions of some matrix differential equations, quantum algebras give rise to polynomial solutions of finite--difference equations. Particularly, known classical orthogonal polynomials will appear when considering $SL(2,{\bf R})$ acting on ${\bf RP_1}$. As examples, some polynomials connected to projectivized representations of $sl_2 ({\bf R})$, $sl_2 ({\bf R})_q$, $osp(2,2)$ and $so_3$ are briefly discussed. 
  For a generic value of the central charge, we prove the holomorphic factorization of partition functions for free superconformal fields which are defined on a compact Riemann surface without boundary. The partition functions are viewed as functionals of the Beltrami coefficients and their fermionic partners which variables parametrize superconformal classes of metrics. 
  A classification of ordinary differential equations and finite-difference equations in one variable having polynomial solutions (the generalized Bochner problem) is given. The method used is based on the spectral problem for a polynomial element of the universal enveloping algebra of $sl_2({\bf R})$ (for differential equations) or $sl_2({\bf R})_q$ (for finite-difference equations) in the "projectivized" representation possessing an invariant subspace. Connection to the recently-discovered quasi-exactly-solvable problems is discussed. 
  A classification theorem for linear differential equations in two variables (one real and one Grassmann) having polynomial solutions(the generalized Bochner problem) is given. The main result is based on the consideration of the eigenvalue problem for a polynomial element of the universal enveloping algebra of the algebra $osp(2,2)$ in the "projectivized" representation (in differential operators of the first order) possessing an invariant subspace. A classification of 2 x 2 matrix differential equations in one real variable possessing polynomial solutions is described. Connection to the recently-discovered quasi-exactly-solvable problems is discussed. 
  We discuss continuous and discrete sectors in the collective field theory of $d=1$ matrix models. A canonical Lorentz invariant field theory extension of collective field theory is presented and its classical solutions in Euclidean and Minkowski space are found. We show that the discrete, low density, sector of collective field theory includes single eigenvalue Euclidean instantons which tunnel between different vacua of the extended theory. We further show that these ``stringy" instantons induce non-perturbative effective operators of strength $e^{-{1\over g}}$ in the extended theory. The relationship of the world sheet description of string theory and Liouville theory to the effective space-time theory is explained. We also comment on the role of the discrete, low density, sector of collective field theory in that framework. 
  Classification theorems for linear differential equations in two real variables, possessing eigenfunctions in the form of the polynomials (the generalized Bochner problem) are given. The main result is based on the consideration of the eigenvalue problem for a polynomial elements of the universal enveloping algebras of the algebras   $sl_3({\bf R})$, $sl_2({\bf R})\oplus sl_2({\bf R})$  and   $gl_2 ({\bf R})\ \triangleright\!\!\!< {\bf R}^{r+1}\ , r>0$  taken in the "projectivized" representations (in differential operators of the first order in two real variables) possessing an invariant subspace. General insight to the problem of a description of linear  differential operators possessing an invariant sub-space with a basis in polynomials is presented. Connection to the recently-discovered quasi-exactly-solvable problems is discussed. 
  The n-point function for the integral over unitary matrices with Itzykson-Zuber measure is reduced to the integral over Gelfand-Tzetlin table; integrand (for generic n) is given by linear exponential times rational function. For $n=2$ and in some cases for $n>2$ later in fact is the polinomial and this allows to give an explicit and simple expression for all 2-point and a set of n-point functions. For the most general n-point function a simple linear differential equation is constructed. 
  We derive the entire KdV hierarchy as well as the recursion relations from the self-duality condition on gauge fields in four dimensions. 
  We establish a direct link between massive Ising model and arbitrary massive $N=2$ supersymmetric QFT's in two dimensions. This explains why the equations which appear in the computation of spin-correlations in the non-critical Ising model are the same as those describing the geometry of vacua in $N=2$ theories. The tau-function appearing in the Ising model (i.e., the spin correlation function) is reinterpreted in the $N=2$ context as a new `index'. In special cases this new index is related to Ray-Singer analytic torsion, and can be viewed as a generalization of that to loop space of K\"ahler manifolds. 
  A generalization of the pseudoclassical action of a spinning particle in the presence of an anomalous magnetic moment is given. The leading considerations, to write the action, are gotten from the path integral representation for the causal Green's function of the generalized (by Pauli) Dirac equation for the particle with anomalous magnetic momentum in an external electromagnetic field. The action can be written in reparametrization and supergauge invariant form. Both operator (Dirac) and path-integral (BFV) quantization are discussed. The first one leads to the Dirac-Pauli equation, whereas the second one gives the corresponding propagator. One of the nontrivial points in this case is that both quantizations schemes demand for consistency to take into account an operators ordering problem. 
  A procedure for constructing topological actions from centrally extended Lie groups is introduced. For a \km\ group, this produces \3al \cs, while for the \vir\ group the result is a new \3al \tft\ whose physical states satisfy the \vir\ \wi. This \tft\ is shown to be a first order formulation of two dimensional induced gravity in the chiral gauge. The extension to $W_3$-gravity is discussed. 
  \def\mon{S^3\stackrel{S^1}{\rightarrow}S^2} \def\inst{S^7\stackrel{S^3}{\rightarrow}S^4} \def\octo{S^{15}\stackrel{S^7}{\rightarrow}S^8} In semilocal theories, the vacuum manifold is fibered in a non-trivial way by the action of the gauge group. Here we generalize the original semilocal theory (which was based on the Hopf bundle $\mon$) to realize the next Hopf bundle $\inst$, and its extensions $S^{2n+1}\stackrel{S^3}\rightarrow \H P^n$. The semilocal defects in this class of theories are classified by $\pi_3(S^3)$, and are interpreted as constrained instantons or generalized sphaleron configurations. We fail to find a field theoretic realization of the final Hopf bundle $\octo$, but are able to construct other semilocal spaces realizing Stiefel bundles over Grassmanian spaces. 
  The renormalization group flow recently found by Br\'ezin and Zinn- Justin by integrating out redundant entries of the $(N+1)\times (N+1)$ hermitian random matrix is studied. By introducing explicitly the RG flow parameter, and adding suitable counter terms to the matrix potential of the one matrix model, we deduce some interesting properties of the RG trajectories. In particular, the string equation for the general massive model interpolating between the UV and IR fixed points turns out to be a consequence of RG flow. An ambiguity in the UV regime of the RG trajectory is remarked to be related to the large order behavior of the one matrix model. 
  We present a new method to find solutions of the Virasoro master equations for any affine Lie algebra $\widehat{g}$. The basic idea is to consider first the simplified case of an In\"on\"u-Wigner contraction $\widehat{g}_c$ of $\widehat{g}$ and to extend the Virasoro constructions of $\widehat{g}_c$ to $\widehat{g}$ by a perturbative expansion in the contraction parameter. The method is then applied to the orthogonal algebras, leading to fixed-level multi-parameter Virasoro constructions, which are the generalisations of the one-parameter Virasoro construction of $\widehat{su}(2)$ at level four. 
  The spectrum of observables in the induced lattice gauge model proposed recently by V.A.Kazakov and A.A.Migdal obeys the local-confinement selection rule. The underlying local continuous symmetry cannot be spontaneously broken within the model. 
  The Feigin-Fuks construction of irreducible lowest-weight Virasoro representations is reviewed using physics terminology. The procedure consists of two steps: constructing invariants and applying them to the Fock vacuum. We attempt to generalize this construction to the diffeomorphism algebra in higher dimensions. The first step is straightforward, but the second is difficult, due to the appearence of infinite Schwinger terms. This might be avoided by imposing constraints on the fields, which should be of the recently discovered conformal type. The resulting representations are reminiscent of quantum gravity. 
  It is argued that chiral algebras of conformal field theory possess a W-algebra structure. A survey of explicitly known W-algebras and their constructions is given. (Talk given at the XIX International Colloquium on ``Group Theoretical Methods in Physics'', Salamanca, Spain, June 29 -- July 4, 1992) 
  We consider a subdivision invariant action for dynamically triangulated random surfaces that was recently proposed (R.V. Ambartzumian et. al., Phys. Lett. B 275 (1992) 99) and show that it is unphysical: The grand canonical partition function is infinite for all values of the coupling constants. We conjecture that adding the area action to the action of Ambartzumian et. al. leads to a well-behaved theory. 
  We prove that the covariant and Hamiltonian phase spaces of the Wess-Zumino-Witten model on the cylinder are diffeomorphic and we derive the Poisson brackets of the theory. 
  We give a geometrical set up for the semiclassical approximation to euclidean field theories having families of minima (instantons) parametrized by suitable moduli spaces ${\cal M}$. The standard examples are of course Yang-Mills theory and non-linear $\sigma$-models. The relevant space here is a family of measure spaces $\tilde {\cal N} \ra {\cal M}$, with standard fibre a distribution space, given by a suitable extension of the normal bundle to ${\cal M}$ in the space of smooth fields. Over $\tilde {\cal N}$ there is a probability measure $d\mu$ given by the twisted product of the (normalized) volume element on ${\cal M}$ and the family of gaussian measures with covariance given by the tree propagator $C_\phi$ in the background of an instanton $\phi \in {\cal M}$. The space of ``observables", i.e. measurable functions on ($\tilde {\cal N}, \, d\mu$), is studied and it is shown to contain a topological sector, corresponding to the intersection theory on ${\cal M}$. The expectation value of these topological ``observables" does not depend on the covariance; it is therefore exact at all orders in perturbation theory and can moreover be computed in the topological regime by setting the covariance to zero. 
  I consider a lattice model of a gauge field interacting with matrix-valued scalars in $D$ dimensions. The model includes an adjustable parameter $\s$, which plays role of the string tension. In the limit $\s=\infty$ the model coincides with Kazakov-Migdal's ``induced QCD", where Wilson loops obey a zero area law. The limit $\s=0$, where Wilson loops $W(C)=1$ independently of the size of the loop, corresponds to the Hermitian matrix model. For $D=2$ and $D=3$ I show that the model obeys the same combinatorics as the standard LGT and therefore one may expect the area law behavior. In the strong coupling expansion such a behavior is demonstrated. 
  We discuss the canonical quantization of Chern-Simons theory in $2+1$ dimensions, minimally coupled to a Dirac spinor field. Gauss's law and the gauge condition, $A_0 = 0$, are implemented by embedding the formulation in an appropriate physical subspace. We find two kinds of charged particle states in this model. One kind has a rotational anomaly in the form of arbitrary phases that develop in $2\pi$ rotations; the other kind rotates ``normally''---i.e., charged states only change sign in $2\pi$ rotations. The rotational anomaly has nothing to do with the implementation of Gauss's law. It is possible to inadvertently produce these anomalous states in the process of implementing Gauss's law, but it is also possible to implement Gauss's law without producing rotational anomalies. Moreover, states with or without rotational anomalies obey ordinary Fermi statistics. 
  Closed string field theory leads to a generalization of Lie algebra which arose naturally within mathematics in the study of deformations of algebraic structures. It also appeared in work on higher spin particles \cite{BBvD}. Representation theoretic analogs arose in the mathematical analysis of the Batalin-Fradkin-Vilkovisky approach to constrained Hamiltonians. A major goal of this paper is to see the relevant formulas, especially in closed string field theory, as a generalization of those for a differential graded Lie algebra, hopefully describing the mathematical essentials in terms accessible to {\it physicists}. 
  The double scaling limit of a new class of the multi-matrix models proposed in \cite{MMM91}, which possess the $W$-symmetry at the discrete level, is investigated in details. These models are demonstrated to fall into the same universality class as the standard multi-matrix models. In particular, the transformation of the W-algebra at the discrete level into the continuum one of the paper \cite{FKN91a} is proposed, the corresponding partition functions being compared. All calculations are demonstrated in full in the first non-trivial case of $W^{(3)}$-constraints. 
  Critical values of Wilson lines and general background fields for toroidal compactifications of heterotic string theories are constructed systematically using Dynkin diagrams. 
  A space consisting of two rapidly moving cosmic strings has recently been constructed by Gott that contains closed timelike curves. The global structure of this space is analysed and is found that, away from the strings, the space is identical to a generalised Misner space. The vacuum expectation value of the energy momentum tensor for a conformally coupled scalar field is calculated on this generalised Misner space. It is found to diverge very weakly on the Chronology horizon, but more strongly on the polarised hypersurfaces. The divergence on the polarised hypersurfaces is strong enough that when the proper geodesic interval around any polarised hypersurface is of order the Planck length squared, the perturbation to the metric caused by the backreaction will be of order one. Thus we expect the structure of the space will be radically altered by the backreaction before quantum gravitational effects become important. This suggests that Hawking's `Chronology Protection Conjecture' holds for spaces with non-compactly generated Chronology horizon. 
  Aspects of a generalized representation theory of current algebras in $3+1$ dimensions are discussed in terms of the Fock bundle method, the sesquilinear form approach (of Langmann and Ruijsenaars), and Hilbert space cocycles. 
  We give explicit expression of recurrency formulae of canonical realization for quantum enveloping algebras $U_{q}(sl(n+1,C))$. In these formulas the generators of the algebra $U_{q}(sl(n+1,C))$ are expressed by means of n-canonical q-boson pairs one auxiliary representation of the algebra $U_{q}(gl(n,C))$. 
  We construct $W_3$ null vectors of a restricted class explicitly in two different forms. The method we use is an extension of that of Bauer et al.~in the Virasoro case. Our results are analogous to the formulae of Benoit and St.~Aubin for the Virasoro null vectors. We derive in the Virasoro case some alternative formulae for the same null vectors involving only the $L_{-1}$ and $L_{-2}$ modes of the Virasoro algebra. } 
  The coupling of a dilaton to the $SU(2)$-Yang-Mills field leads to interesting non-perturbative static spherically symmetric solutions which are studied by mixed analitical and numerical methods. In the abelian sector of the theory there are finite-energy magnetic and electric monopole solutions which saturate the Bogomol'nyi bound. In the nonabelian sector there exist a countable family of globally regular solutions which are purely magnetic but have zero Yang-Mills magnetic charge. Their discrete spectrum of energies is bounded from above by the energy of the abelian magnetic monopole with unit magnetic charge. The stability analysis demonstrates that the solutions are saddle points of the energy functional with increasing number of unstable modes. The existence and instability of these solutions are "explained" by the Morse-theory argument recently proposed by Sudarsky and Wald. 
  We develop a superfield formalism for N=4 superconformal two-dimensional field theory. A list is presented of minimal free superfields, i.e. of multiplets containing four bosons and four fermions. We show that the super-Poincar\'e algebra of the six-dimensional superstring in the light-cone gauge is essentially equivalent to a local N=4 superconformal symmetry, and outline the construction of octonionic superconformal field theory. 
  Non linear sigma models on Riemannian symmetric spaces constitute the most general class of classical non-linear sigma models which are known to be integrable. Using the current algebra structure of these models their canonical structure is analysed and it is shown that their non ultralocal fundamental Poisson bracket relation is governed by a field dependent non antisymmetric r-matrix obeying a dynamical Yang Baxter equation. Contribution presented at the XIX ICGTMP Salamanca June 92 
  It is shown that there is a generalization of the Conley-Zehnder index for periodic trajectories of a classical Hamiltonian system $(Q, \omega, H)$ from the case $Q = T^*R^n$ to arbitrary symplectic manifolds. As it turns out, it is precisely this index which appears as a Maslov phase in the trace formulas by Gutzwiller and Duistermaat-Guillemin. Contribution presented at the XIX ICGTMP Salamanca June 92. 
  Self-dual vortex solutions are studied in detail in the generalized abelian Higgs model with independent Chern-Simons interaction. For special choices of couplings, it reduces to a Maxwell-Higgs model with two scalar fields, a Chern-Simons-Higgs model with two scalar fields, or other new models. We investigate the properties of the static solutions and perform detailed numerical analyses. For the Chern-Simons-Higgs model with two scalar fields in an asymmetric phase, we prove the existence of multisoliton solutions which can be viewed as hybrids of Chern-Simons vortices and $CP^1$ lumps. We also discuss solutions in a symmetric phase with the help of the corresponding exact solutions in its nonrelativistic limit. The model interpolating all three models---Maxwell-Higgs, Chern-Simons-Higgs, and $CP^1$ models--- is discussed briefly. Finally we study the possibility of vortex solutions with half-integer vorticity in the special case of the model. Numerical results are negative. 
  We construct the low-lying discrete states of the two-scalar $W_3$ string. This includes states that correspond to the analogues of the ground ring generators of the ordinary two-dimensional string. These give rise to infinite towers of discrete states at higher levels. 
  We compute N-point correlation functions of non-unitary (2k-1, 2) minimal matter coupled to 2D quantum gravity on a sphere using the continuum Liouville field approach. A gravitational dressing of the matter primary field with the minimum conformal weight is used as the cosmological operator. Our results are in agreement with the correlation functions of the one-matrix model at the k-th critical point. 
  These are lecture notes for the 1992 Erice Workshop on Theoretical Physics. They first present a summary of the paradox of information loss to black holes, of its proposed resolutions, and of the flaws in the proposed resolutions. There follows a review of recent attempts to attack this problem, and other issues in black hole physics, using two-dimensional dilaton gravity theories as toy models. These toy models contain collapsing black holes and have for the first time enabled an explicit semiclassical treatment of the backreaction of the Hawking radiation on the geometry of an evaporating black hole. However, a complete answer to the information conundrum seems to require physics beyond the semiclassical approximation. Preliminary attempts to make progress in this direction, using connections to conformal field theory, are described. 
  Motivated by a formula (due to Zelobenko) for finite Lie algebra tensor products, we propose a reformulation of the Gepner-Witten depth rule. Implementation of this rule remains difficult, however, since the basis states convenient for calculating tensor product coefficients do not have a well-defined depth. To avoid this problem, we present a `crystal depth rule', that gives a lower bound for the minimum level at which a WZNW fusion appears. The bound seems to be quite accurate for $su(N>3),$ and for $su(3)$ the rule is proven to be exact. (Talk presented by M.W. at the XIXth International Colloquium on Group Theoretical Methods in Physics.) 
  The superfield formulation of type II Green-Schwarz superstring with n= (1,0) worldsheet supersymmetry is constructed. It is shown that the inclusion of the second spinor coordinate in the target superspase leads to the possibility of the reparametrization invariant description of the superstring in the absence of any field from the two dimensional supergravity multiplet. The twistor-like action of Chern-Simons type for bosonic string in D=3,4,6,10 is obtained from the component form of the superstring action. 
  Generic $U(1)^2$ 4-d black holes with unbroken $N=1$ supersymmetry are shown to tend to a Robinson-Bertotti type geometry with a linear dilaton and doubling of unbroken supersymmetries near the horizon. Purely magnetic dilatonic black holes, which have unbroken $N=2$ supersymmetry, behave near the horizon as a 2-d linear dilaton vacuum $\otimes \, S^2$. This geometry is invariant under 8 supersymmetries, i.e. half of the original $N=4$ supersymmetries are unbroken. The supersymmetric positivity bound, which requires the mass of the 4-d dilaton black holes to be greater than or equal to the central charge, corresponds to positivity of mass for a class of stringy 2-d black holes. 
  We discuss a study of domain walls in $N=1, d=4$ supergravity. The walls saturate the Bogomol'nyi bound of wall energy per unit area thus proving stability of the classical solution. They interpolate between two vacua whose cosmological constant is non-positive and in general different. The matter configuration and induced geometry are static. We discuss the field theoretic realization of these walls and classify three canonical configurations with examples. The space-time induced by a wall interpolating between the Minkowski (topology $\Re^{4}$) and anti-de~Sitter (topology $S^{1}(time) \times \Re^{3}(space)$) vacua is discussed. (Comments in chapter 6 on AdS-Minkowski wall induced space-time have been slightly changed) 
  We analyze a supergravity theory coupled to a dilaton and superconformal matters in two dimensions. This theory is classically soluble and we find all the solutions appeared in Callan, Giddings, Harvey and Strominger's dilatonic gravity also satisfy the constraints and the equations of motion in this supersymmetric theory. We quantize this theory by following the procedure of Distler, Hlousek and Kawai. In the quantum action, the cosmological term is renormalized to vanish. As a result, any solution corresponding to classical black hole does not appear in the quantum theory, which should be compared with the non-supersymmetric case. 
  We propose a new method for numerical calculation of link plynomials for knots given in 3 dimensions. We calculate derivatives of the Jones polynomial in a computational time proportional to $N^{\alpha}$ with respect to the system size $N$ . This method gives a new tool for determining topology of knotted closed loops in three dimensions using computers. 
  A twistor correspondence for the self-duality equations for supersymmetric Yang-Mills theories is developed. Their solutions are shown to be encoded in analytic harmonic superfields satisfying appropriate generalised Cauchy-Riemann conditions. An action principle yielding these conditions is presented. 
  The effect of de Sitter transformations on Tsamis and Woodard's solutions to the linearized gauge fixed equations of motion of quantum gravity in a de Sitter space background is worked out explicitly. It is shown that these solutions are closed under the transformations of the de Sitter group. To do this it is necessary to use a compensating gauge transformation to return the transformed solution to the original gauge. 
  We study the $SL(2;R)/U(1)$ coset model of two-dimensional black hole and its relation to the Liouville theory coupled to c=1 matter. We uncover a basic isomorphism in the algebraic structures of these theories and show that the black hole model has the same physical spectrum as the c=1 model, i.e. tachyons, $W_\infty$ currents and the ground ring elements. we also identify the operator responsible for the creation of the mass of the black hole. 
  A classification of (super) $W$ algebras arising from non Abelian Toda and super Toda theories is presented. This classification is based on the $Sl(2)$ or $OSp(1|2)$ sub(super)algebras of the simple Lie (super)algebra underlying the model. This allows to compute the conformal spin content of each $W$ (super)algebra. {\em Based on two lectures given by L. Frappat and E. Ragoucy at the\\ XIX International Colloquium on Group Theoretical Methods in Physics\\ Salamanca (Spain), June 29 - July 4, 1992} 
  Certain results related to the cancellation of quadratic divergences, which had been obtained using dimensional reduction, are reconsidered using a nonlocal regulator. The results obtained are shown to depend strongly on the regulator. Specifically, it is shown that a certain result of Al-sarhi, Jack, and Jones no longer holds, even if a nontrivial measure factor is used; also that there are no values of the top and Higgs mass for which the one-loop quadratic divergence in the standard model cancels independently of the renormalization scale, whether or not strong interaction effects are ignored. 
  The heterotic string compactified on a six-torus is described by a low-energy effective action consisting of N=4 supergravity coupled to N=4 super Yang-Mills, a theory that was studied in detail many years ago. By explicitly carrying out the dimensional reduction of the massless fields, we obtain the bosonic sector of this theory. In the Abelian case the action is written with manifest global $O(6,6+n)$ symmetry. A duality transformation that replaces the antisymmetric tensor field by an axion brings it to a form in which the axion and dilaton parametrize an $SL(2,R)/SO(2)$ coset, and the equations of motion have $SL(2,R)$ symmetry. This symmetry, which combines Peccei--Quinn translations with Montonen--Olive duality transformations, has been exploited in several recent papers to construct black hole solutions carrying both electric and magnetic charge. Our purpose is to explore whether, as various authors have conjectured, an $SL(2,Z)$ subgroup could be an exact symmetry of the full quantum string theory. If true, this would be of fundamental importance, since this group transforms the dilaton nonlinearly and can relate weak and strong coupling. 
  Based upon the intrinsic relation between the divergent lower point functions and the convergent higher point ones in the renormalizable quantum field theories, we propose a new method for regularization and renormalization in QFT. As an example, we renormalize the $\phi^{4}$ theory at the one loop order by means of this method. 
  (Talk presented at the 1992 ICTP summer workshop in high energy physics and cosmology) The BRST cohomology ring for $(p,q)$ models coupled to gravity is discussed. In addition to the generators of the ghost number zero ring, the existence of a generator of ghost number $-1$ and its inverse is proven and used to construct the entire ring. Some comments are made regarding the algebra of the vector fields on the ring and the supersymmetric extension. 
  Assuming trivial action of Euclidean translations of the little group, we derive a simple correspondence between massless field representations transforming under the full generalized even dimensional Lorentz group, and highest weight states of the relevant little group. This yields a connection between `helicity' and `chirality' in all dimensions, and highlights the special nature of the restricted representations under `gauge' transformations. 
  The subject of this talk was the review of our study of three ($2+1$) dimensional Quantum Chromodynamics. In our previous works, we showed the existence of a phase where parity is unbroken and the flavor group $U(2n)$ is broken to a subgroup $U(n)\times U(n)$. We derived the low energy effective action for the theory and showed that it has solitonic excitations with Fermi statistic, to be identified with the three dimensional ``baryon''. Finally, we studied the current algebra for this effective action and we found a co-homologically non trivial generalization of Kac-Moody algebras to three dimensions. 
  Free field representation for the classical limit of quantum affine algebra is constructed by simple deformation of the known expressions from WZW theory. 
  It is pointed out that Chern-Simons theories do not allow an anyon interpretation when spin is included. 
  It is shown that AB-like cross sections can be obtained from symmetry breaking which does not require infinite energy, angular dependence in the symmetry breaking term, or a nontrivial $Z_2$ charge. 
  Point particles fall freely along geodesics; strings do not. In string theory all probes of spacetime structure, including photons, are extended objects and therefore always subject to tidal forces. We illustrate how string theory modifies the behavior of light in weak gravitational fields and limits the applicability of the principle of equivalence. This gives in principle a window on the short-distance structure of geometry in quantum gravity where one can see in a model-independent way how some of its predictions differ from those of classical gravity. We compare this with the lessons of high-energy string scattering. 
  The $Z_N$-invariant chiral Potts model is considered as a perturbation of a $Z_N$ conformal field theory. In the self-dual case the renormalization group equations become simple, and yield critical exponents and anisotropic scaling which agree with exact results for the super-integrable lattice models. Although the continuum theory is not Lorentz invariant, it respects a novel type of space-time symmetry which allows for the observed spontaneous breaking of translational symmetry in the ground state. The continuum theory is shown to possess an infinite number of conserved charges on the self-dual line, which remain conserved when the theory is perturbed by the energy operator. 
  The problem of Bose-Einstein condensation for a relativistic ideal gas on a 3+1 dimensional manifold with a hyperbolic spatial part is analyzed in some detail. The critical temperature is evaluated and its dependence of curvature is pointed out. 
  By generalizing the Drinfel'd--Sokolov reduction we construct a large class of W algebras as reductions of Kac--Moody algebras. Furthermore we construct actions, invariant under local left and right W transformations, which are the classical covariant induced actions for W gravity. Talk presented by T. Tjin at the Trieste Summerschool on strings and related topics. 
  Starting with topological field theories we investigate the Ray-Singer analytic torsion in three dimensions. For the lens Spaces L(p;q) an explicit analytic continuation of the appropriate zeta functions is contructed and implemented. Among the results obtained are closed formulae for the individual determinants involved, the large $p$ behaviour of the determinants and the torsion, as well as an infinite set of distinct formulae for zeta(3): the ordinary Riemann zeta function evaluated at s=3.   The torsion turns out to be trivial for the cases L(6,1), L((10,3) and L(12,5) and is, in general, greater than unity for large p and less than unity for a finite number of p and q. 
  We address the possibility of false vacuum decay in $N=1$ supergravity theories, including those corresponding to superstring vacua. By establishing a Bogomol'nyi bound for the energy density stored in the domain wall of the $O(4)$ invariant bubble, we show that supersymmetric vacua remain absolutely stable against false vacuum decay into another supersymmetric vacuum, including those from a Minkowski to an anti-deSitter (AdS) one. As a consequence, there are no compact static spherical domain walls, while on the other hand there exist planar domain walls interpolating between non-degenerate supersymmetric vacua, e.g. between Minkowski (topology $\Re^{4}$) and AdS (topology $S^{1}(time) \times \Re^{3}(space)$) vacua. (Talk presented at the XXVI International Conference on High Energy Physics August 6-12, 1992, Dallas, Texas) 
  For any non-unitary model with central charge c(2,q) the path spaces associated to a certain fusion graph are isomorphic to the irreducible Virasoro highest weight modules. 
  The self-dual superstring has been described previously in a Neveu-Schwarz-Ramond formulation with local N=2 or 4 world-sheet supersymmetry. We present a Green-Schwarz-type formulation, with manifest spacetime supersymmetry. 
  The quantum affine $\CU_q (\hat{sl(2)}) $ symmetry is studied when $q^2$ is an even root of unity. The structure of this algebra allows a natural generalization of N=2 supersymmetry algebra. In particular it is found that the momentum operators $P ,\bar{P}$, and thus the Hamiltonian, can be written as generalized multi-commutators, and can be viewed as new central elements of the algebra $\CU_q (\hat{sl(2)})$. We show that massive particles in (deformations of) integer spin representions of $sl(2)$ are not allowed in such theories. Generalizations of Witten's index and Bogomolnyi bounds are presented and a preliminary attempt in constructing manifestly $\CU_q (\hat{sl(2)})$ invariant actions as generalized supersymmetric Landau-Ginzburg theories is made. 
  We review various aspects of $\cW$-algebra symmetry in two-dimensional conformal field theory and string theory. We pay particular attention to the construction of $\cW$-algebras through the quantum Drinfeld-Sokolov reduction and through the coset construction. 
  (a few typos and an error corrected) 
  We investigate charged black holes coupled to a massive dilaton. It is shown that black holes which are large compared to the Compton wavelength of the dilaton resemble the Reissner-Nordstr\"om solution, while those which are smaller than this scale resemble the massless dilaton solutions. Black holes of order the Compton wavelength of the dilaton can have wormholes outside the event horizon in the string metric. Unlike all previous black hole solutions, nearly extremal and extremal black holes (of any size) repel each other. We argue that extremal black holes are quantum mechanically unstable to decay into several widely separated black holes. We present analytic arguments and extensive numerical results to support these conclusions. 
  The vacuum energy is calculated for a free, conformally-coupled scalar field on the orbifold space-time \R$\times \S^2/\Gamma$ where $\Gamma$ is a finite subgroup of O(3) acting with fixed points. The energy vanishes when $\Gamma$ is composed of pure rotations but not otherwise. It is shown on general grounds that the same conclusion holds for all even-dimensional factored spheres and the vacuum energies are given as generalised Bernoulli functions (i.e. Todd polynomials). The relevant $\zeta$- functions are analysed in some detail and several identities are incidentally derived. The general discussion is presented in terms of finite reflection groups. 
  We study integrable dynamical systems described by a Lax pair involving a spectral parameter. By solving the classical Yang-Baxter equation when the R-matrix has two poles we show that they can be interpreted as natural motions on a twisted loop algebra. 
  We observe that the ratio of determinants of $2d$ Laplacians which appear in the duality transformation relating two sigma models with abelian isometries can be represented as a torsion of an elliptic (DeRham) complex. As a result, this ratio can be computed exactly and is given by the exponential of a local functional of $2d$ metric and target space metric. In this way the well known dilaton shift under duality is reproduced. We also present the exact computation of the determinant which appears in the duality transformation in the path integral. 
  These are lecture notes for the 1992 Erice Workshop on String Quantum Gravity and Physics at the Planck Energy Scale. In this talk a review of earlier work on finite temperature strings was presented. Several topics were covered, including the canonical and microcanonical ensemble of strings, the behavior of strings near the Hagedorn temperature as well as speculations on the possible phases of high temperature strings. The connection of the string ensemble and, more generally, statistical systems with an exponentially growing density of states with number theory was also discussed. 
  An $N = 1$ supersymmetric version of two dimensional dilaton gravity coupled to matter is considered. It is shown that the linear dilaton vacuum spontaneously breaks half the supersymmetries, leaving broken a linear combination of left and right supersymmetries which squares to time translations. Supersymmetry suggests a spinorial expression for the ADM energy $M$, as found by Witten in four-dimensional general relativity. Using this expression it is proven that ${M}$ is non-negative for smooth initial data asymptotic (in both directions) to the linear dilaton vacuum, provided that the (not necessarily supersymmetric) matter stress tensor obeys the dominant energy condition. A {\it quantum} positive energy theorem is also proven for the semiclassical large-$N$ equations, despite the indefiniteness of the quantum stress tensor. For black hole spacetimes, it is shown that $M$ is bounded from below by $e^{- 2 \phi_H}$, where $\phi_H$ is the value of the dilaton at the apparent horizon, provided only that the stress tensor is positive outside the apparent horizon. This is the two-dimensional analogue of an unproven conjecture due to Penrose. Finally, supersymmetry is used to prove positive energy theorems for a large class of generalizations of dilaton gravity which arise in consideration of the quantum theory. 
  I report on the observation of the production of strings (disclination lines and loops) via the Kibble mechanism of domain (bubble) formation in the isotropic to nematic phase transition of a sample of uniaxial nematic liquid crystal. The probablity of string formation per bubble is measured to be $0.33 \pm 0.01$. This is in good agreement with the theoretical value $1/ \pi$ expected in two dimensions for the order parameter space $S^2/{\bf Z}_2$ of a simple uniaxial nematic liquid crystal. ( NB: The only change is a correction of the preprint number for proper referencing and the title page.) 
  We argue that there is no consistent quantisation of the two BPS SU(2) magnetic monopole dynamical system compatible with the correspondence principle. 
  We linearize the Artin representation of the braid group given by (right) automorphisms of a free group providing a linear faithful representation of the braid group. This result is generalized to obtain linear representations for the coloured braid groupoid and pure braid group too. Applications to some areas of two-dimensional physics are discussed. 
  In string theory it is known that abelian isometries in the sigma model lead to target space duality. We generalize this duality to backgrounds with non--abelian isometries. The procedure we follow consists of gauging the isometries of the original action and constraining the field strength $F$ to vanish. This new action generates dual theories by integrating over either the Lagrange multipliers that set F=0 or the gauge fields. We find that this new duality transformation maps spaces with non--abelian isometries to spaces that may have no isometries at all. This suggests that duality symmetries in string theories need to be understood in a more general context without regard to the existence of continuous isometries on the target space (this is also indicated by the existence of duality in string compactifications on Calabi--Yau manifolds which have no continuous isometries). Physically interesting examples to which our formalism apply are the Schwarzschild metric and the 4D charged dilatonic black hole. For these spherically symmetric black holes in four dimensions, the dual backgrounds are presented and explicitly shown to be new solutions of the leading order string equations. Some of these new backgrounds are found to have no continuous isometries (except for time translations) and also have naked singularities. 
  It is shown that a recently proposed relativistic field theory of anyons is mathematically flawed and also does not satisfy reasonable criteria for such a theory. 
  Braided tensor products have been introduced by the author as a systematic way of making two quantum-group-covariant systems interact in a covariant way, and used in the theory of braided groups. Here we study infinite braided tensor products of the quantum plane (or other constant Zamolodchikov algebra). It turns out that such a structure precisely describes the exchange algebra in 2D quantum gravity in the approach of Gervais. We also consider infinite braided tensor products of quantum groups and braided groups. 
  We study the quantum group gauge theory developed elsewhere in the limit when the base space (spacetime) is a classical space rather than a general quantum space. We show that this limit of the theory for gauge quantum group $U_q(g)$ is isomorphic to usual gauge theory with Lie algebra $g$. Thus a new kind of gauge theory is not obtained in this way, although we do find some differences in the coupling to matter. Our analysis also illuminates certain inconsistencies in previous work on this topic where a different conclusion had been reached. In particular, we show that the use of the quantum trace in defining a Yang-Mills action in this setting is not appropriate. 
  We dicuss some curious aspects of the Rogers dilogarithm and the functional relations in integrable systems in two dimensions. This is for the proceedings of the XX1 Differential Geometry Methods in Theoretical Physics, Tianjin, China, 5-9 June 1992. 
  A generalization of the Faddeev-Popov procedure to deal with either anomalous and non-anomalous gauge theories with closed, irreducible gauge algebra is presented. An expression for the Wess-Zumino action is obtained. 
  A selfconsistent definition of quantum free particle on a generic curved manifold emerges naturally by restricting the dynamics to submanifolds of co-dimension one.   PACS 0365 0240 
  We study the simplicial quantum gravity in three dimensions. Motivated by the Boulatov's model which generates a sum over simplicial complexes weighted with the Turaev-Viro invariant, we introduce boundary operators in the simplicial gravity associated to compact orientable surfaces. An amplitude of the boundary operator is given by a sum over triangulations in the interior of the boundary surface. It turns out that the amplitude solves the Schwinger-Dyson equation even if we restrict the topology in the interior of the surface, as far as the surface is non-degenerate. We propose a set of factorization conditions on the amplitudes which singles out a solution associated to triangulations of $S^3$. 
  The relationship between the nonlinear Schrodinger hierarchy and the parafermion and SL(2,R)/U(1) coset models, analogous to the relationship between the KdV hierarchy and the minimal models, is explained. To do this I first present an in depth study of a series of integrable hierarchies related to NLS, and write down an action from which any of these hierarchies, and the associated second Poisson bracket structures, can be obtained. In quantizing the free part of this action we find many features in common with the bosonized parafermion and SL(2,R)/U(1) models, and particularly it is clear that the quantum NLS hamiltonians are conserved quantities in these models. The first few quantum NLS hamiltonians are constructed. 
  The moduli dependent Yukawa couplings between twisted sectors of the $Z_3\times Z_3$ orbifold are studied. 
  Two string-like solutions to the equations of motion of the low-energy effective action for the heterotic string are found, each a source of electric and magnetic fields. The first carries an electric current equal to the electric charge per unit length and is the most general solution which preserves one half of the supersymmetries. The second is the most general charged solution with an event horizon, a `black string'. The relationship of the solutions to fundamental, macroscopic heterotic strings is discussed, and in particular it is shown that any stable state of such a fundamental string also preserves one half of the supersymmetries, in the same manner as the first solution. 
  The notion of $q$-deformed lattice gauge theory is introduced. If the deformation parameter is a root of unity, the weak coupling limit of a 3-$d$ partition function gives a topological invariant for a corresponding 3-manifold. It enables us to define the generalized Turaev-Viro invariant for cell complexes. It is shown that this invariant is determined by an action of a fundamental group on a universal covering of a complex. A connection with invariants of framed links in a manifold is also explored. A model giving a generating function of all simplicial complexes weighted with the invariant is investigated. 
  In the physical interpretation of states in non-perturbative loop quantum gravity the so-called weave states play an important role. Until now only weaves representing flat geometries have been introduced explicitly. In this paper the construction of weaves for non-flat geometries is described; in particular, weaves representing the Schwarzschild solution are constructed. 
  The exact master equation for a harmonic oscillator coupled to a heat bath, derived recently by Hu, Paz and Zhang, is simplified by taking the weak-coupling, late-time limit. The unique time-independent solution to this simplified master equation is the canonical ensemble at the temperature of the bath. The frequency of the oscillator is effectively lowered by the interaction with the bath. 
  The influence functional is derived for a massive scalar field in the ground state, coupled to a uniformly accelerating DeWitt monopole detector in $D+1$ dimensional Minkowski space. This confirms the local nature of the Unruh effect, and provides an exact solution to the problem of the accelerating detector without invoking a non-standard quantization. A directional detector is presented which is efficiently decohered by the scalar field vacuum, and which illustrates an important difference between the quantum mechanics of inertial and non-inertial frames. From the results of these calculations, some comments are made regarding the possibility of establishing a quantum equivalence principle, so that the Hawking effect might be derived from the Unruh effect. 
  In two-dimensional space-time, point particles can experience a geometric, dimension-specific gravity force, which modifies the usual geodesic equation of motion and provides a link between the cosmological constant and the vacuum $\theta$-angle. The description of such forces fits naturally into a gauge theory of gravity based on the extended Poincar\'e group, {\it i.e.\/} ``string-inspired'' dilaton gravity. 
  Generalized Drinfeld-Sokolov (DS) hierarchies are constructed through local reductions of Hamiltonian flows generated by monodromy invariants on the dual of a loop algebra. Following earlier work of De Groot et al, reductions based upon graded regular elements of arbitrary Heisenberg subalgebras are considered. We show that, in the case of the nontwisted loop algebra $\ell(gl_n)$, graded regular elements exist only in those Heisenberg subalgebras which correspond either to the partitions of $n$ into the sum of equal numbers $n=pr$ or to equal numbers plus one $n=pr+1$. We prove that the reduction belonging to the grade $1$ regular elements in the case $n=pr$ yields the $p\times p$ matrix version of the Gelfand-Dickey $r$-KdV hierarchy, generalizing the scalar case $p=1$ considered by DS. The methods of DS are utilized throughout the analysis, but formulating the reduction entirely within the Hamiltonian framework provided by the classical r-matrix approach leads to some simplifications even for $p=1$. 
  Using the generalized hamiltonian method of Batalin, Fradkin and Vilkovisky, we investigate the algebraic structure of anomalies in the Polyakov string theory that appear as the Schwinger terms in super-commutation relations between BRST charge and total hamiltonian. We obtain the most general form of the anomalies in the extended phase space, without any reference to a two dimensional metric. This pregeometri- cal result, refered to as the genelarized Virasoro anomaly, independent of the gauge and the regularization under a minor assumption, is a non-perturbative result, and valid for any space-time dimension. In a configuration space, in which the two dimensional metric can be identified, we can geometrize the result without assuming the weak gravitational field, showing that the most general anomaly exactly exhibits the Weyl anomaly. 
  Using a geometrical approach to the quantum Yang-Baxter equation, the quantum algebra ${\cal U}_{\hbar}(sl_{2})$ and its universal quantum $R$-matrix are explicitely constructed as functionals of the associated classical $r$-matrix. In this framework, the quantum algebra ${\cal U}_{\hbar}(sl_{2})$ is naturally imbedded in the universal envelopping algebra of the $sl_{2}$ current algebra. 
  Using the representation theory of the subgroups SL_2(Z_p) of the modular group we investigate the induced fusion algebras in some simple examples. Only some of these representations lead to 'good' fusion algebras. Furthermore, the conformal dimensions and the central charge of the corresponding rational conformal field theories are calculated. Two series of representations which can be realized by unitary theories are presented. We show that most of the fusion algebras induced by admissible representations are realized in well known rational models. 
  Recent results about topological coset models are summarized. The action of a topological ${G\over H}$ coset model ($rank\ H = rank\ G$) is written down as a sum of ``decoupled" matter, gauge and ghost sectors.   The physical states are in the cohomology of a BRST-like operator that relates these secotrs. The cohomology on a free field Fock space as well as on an irreducible representation of the ``matter" Kac-Moody algebra are extracted. We compare the results with those of $(p,q)$ minimal models coupled to gravity and with $(p,q)$ $W_N$ strings for the case of $A_1^{(1)}$ at level $k={p\over q}-2$ and $A_1^{(N-1)}$ at level $k={p\over q}-N$ respectively. 
  We compute the braided groups and braided matrices $B(R)$ for the solution $R$ of the Yang-Baxter equation associated to the quantum Heisenberg group. We also show that a particular extension of the quantum Heisenberg group is dual to the Heisenberg universal enveloping algebra $U_{q}(h)$, and use this result to derive an action of $U_{q}(h)$ on the braided groups. We then demonstrate the various covariance properties using the braided Heisenberg group as an explicit example. In addition, the braided Heisenberg group is found to be self-dual. Finally, we discuss a physical application to a system of n braided harmonic oscillators. An isomorphism is found between the n-fold braided and unbraided tensor products, and the usual `free' time evolution is shown to be equivalent to an action of a primitive generator of $U_{q}(h)$ on the braided tensor product. 
  By generalizing the Drinfeld-Sokolov reduction a large class of $W$ algebras can be constructed. We introduce 'finite' versions of these algebras by Poisson reducing Kirillov Poisson structures on simple Lie algebras. A closed and coordinate free formula for the reduced Poisson structure is given. These finitely generated algebras play the same role in the theory of $W$ algebras as the simple Lie algebras in the theory of Kac-Moody algebras and will therefore presumably play an important role in the representation theory of $W$ algebras. We give an example leading to a quadratic $sl_2$ algebra. The finite dimensional unitary representations of this algebra are discussed and it is shown that they have Fock realizations. It is also shown that finite dimensional generalized Toda theories are reductions of a system describing a free particle on a group manifold. These finite Toda systems have the non-linear finite $W$ symmetry discussed above. Talk given at the `workshop on low dimensional topology and physics', Cambridge, September 1992. 
  We introduce $*$-structures on braided groups and braided matrices. Using this, we show that the quantum double $D(U_q(su_2))$ can be viewed as the quantum algebra of observables of a quantum particle moving on a hyperboloid in q-Minkowski space (a three-sphere in the Lorentz metric), and with the role of angular momentum played by $U_q(su_2)$. This provides a new example of a quantum system whose algebra of observables is a Hopf algebra. Furthermore, its dual Hopf algebra can also be viewed as a quantum algebra of observables, of another quantum system. This time the position space is a q-deformation of $SL(2,\R)$ and the momentum group is $U_q(su_2^*)$ where $su_2^*$ is the Drinfeld dual Lie algebra of $su_2$. Similar results hold for the quantum double and its dual of a general quantum group. 
  Is large-$N$ QCD equivalent to a string theory? Maybe, maybe not. I review various attempts to answer the question. 
  This is an introduction to the method of effective field theory. As an application, I derive the effective field theory of low energy excitations in a conductor, the Landau theory of Fermi liquids, and explain why the high-$T_c$ superconductors must be described by a different effective field theory. 
  This talk considers possible lessons of string theory for low energy physics. These are of two types. First, assuming that string theory is the correct underlying theory of all interactions, we ask whether there are any generic predictions the theory makes, and we compare the predictions of string theory with those of conventional grand unified theories. Second, string theory offers some possible answers to a number of troubling naturalness questions. These include problems of discrete and continuous symmetries in general, and CP and the strong CP problem in particular. 
  We study the role of rotational symmetry in the systems where nonabelian Berry potentials emerge as a result of integrating out fast degrees of freedom. The conserved angular momentum is constructed in the presence of a non-abelian Berry potential, which is formulated using Grassmann variables. The modifications on conventional angular momentum are discussed in close analogy with monopole systems. The diatomic molecular system discussed by Zygelman is found to have a similar structure to that of a non-abelian $SU(2)$ charge coupled to a 't Hooft - Polyakov monopole. The abelian limit of the Berry potential in diatomic systems is shown to be the same as $U(1)$ monopole and in a large separation limit, we observe the decoupling associated with a vanishing field tensor. 
  Superstrings have been postulated based on parafermionic partition functions which permit spacetime supersymmetry by generalized Jacobi identities. A comprehensive search finds new such identities. Quadrilateral anomaly cancellation gives constraints on allowed chiral fermions. Bosonic left-movers and $Z_4$ parafermionic right-movers combine in a new heterotic superstring, more constrained than the old one, yet equally applicable to physics. 
  In this review, I discuss a general method for constructing classical solutions of the equations of motion arising in the effective low energy string theory, and discuss specific applications of this method. (Based on talks given at the Johns Hopkins Workshop held at Goteborg, June 8-10, 1992, and ICTP Summer Workshop held at Trieste, July 2-3, 1992) 
  Deformed and undeformed KZ equations are considered for $k=0$. It is shown that they allow the same number of solutions, one being the asymptotics of others. Essential difference in analitical properties of the solutions is explained. 
  We study the limit of asymptotically free massive integrable models in which the algebra of nonlocal charges turns into affine algebra. The form factors of fields in that limit are described by KZ equations on level 0. We show the limit to be connected with finite-gap integration of classical integrable equations. 
  We search for a \Lg\ interpretation of non-diagonal modular invariants of tensor products of minimal $n=2$ superconformal models, looking in particular at automorphism invariants and at some exceptional cases. For the former we find a simple description as \lgo s, which reproduce the correct chiral rings as well as the spectra of various Gepner--type models and orbifolds thereof. On the other hand, we are able to prove for one of the exceptional cases that this conformal field theory cannot be described by an orbifold of a \Lg\ model with respect to a manifest linear symmetry of its potential. 
  The components of the position operator, at a fixed time, for a massless and spinning particle with given helicity $\lambda$ described in terms of bosonic degrees of freedom have an anomalous commutator proportional to $\lambda$. The position operator generates translations in momentum space. We show that a ray-representation for these translations emerges due to the non-commuting components of the position operators and relate this to the Berry-phase for photons. The Tomita-Chiao experiment then gives support for this relativistic and quantum mechanical description of photons in terms of non-commuting position operators. 
  We discuss time-dependent perturbations (induced by matter fields) of a black-hole background in tree-level two-dimensional string theory. We analyse the linearized case and show the possibility of having black-hole solutions with time-dependent horizons. The latter exist only in the presence of time-dependent `tachyon' matter fields, which constitute the only propagating degrees of freedom in two-dimensional string theory. For real tachyon field configurations it is not possible to obtain solutions with horizons shrinking to a point. On the other hand, such a possibility seems to be realized in the case of string black-hole models formulated on higher world-sheet genera. We connect this latter result with black hole evaporation/decay at a quantum level.} 
  We report on a new approach to the calculation of thermodynamic functions for crossing-invariant models solvable by Bethe Ansatz. In the case of the XXZ Heisenberg chain we derive, for arbitrary values of the anysotropy, a single non-linear integral equation from which the free energy can be exactly calculated.These equations are shown to be equivalent to an infinite set of algebraic equations of Bethe type which provide alternatively the thermodinamics.   The high-temperature expansion follows in a sistematic and relatively simple way from our non-linear integral equations. For low temperatures we obtain the correct central charge and predict the analytic structure of the full expansion around T=0. Furthermore, we derive a single non-linear integral equation describing the finite-size ground-state energy of the Sine-Gordon quantum field theory. 
  We study string theory on orbifolds in the presence of an antisymmetric constant background field and discuss some of new aspects of the theory. It is shown that the term with the antisymmetric field has a topological nature like a Chern-Simons term or a Wess-Zumino term. Due to this property, the theory exhibits various anomalous behavior: Zero mode variables obey nontrivial quantization conditions. Coordinate transformations which define orbifolds are modified at the quantum level. Wavefunctions of twisted strings acquire phase factors when they move around non-contractible loops on orbifolds. Zero mode eigenvalues are shifted from naively expected values, in favor of modular invariance. 
  We construct new coset realizations of infinite-dimensional linear $W_3^{\infty}$ symmetry associated with Zamolodchikov's $W_3$ algebra which are different from the previously explored $sl_3$ Toda realization of $W_3^{\infty}$. We deduce the Boussinesq and modified Boussinesq equations as constraints on the geometry of the corresponding coset manifolds.The main characteristic features of these realizations are:i. Among the coset parameters there are the space and time coordinates $x$ and $t$ which enter the Boussinesq equations, all other coset parameters are regarded as fields depending on these coordinates;ii. The spin 2 and 3 currents of $W_3$ and two spin 1 $U(1)$ Kac- Moody currents as well as two spin 0 fields related to the $W_3$currents via Miura maps, come out as the only essential parameters-fields of these cosets. The remaining coset fields are covariantly expressed through them;iii.The Miura maps get a new geometric interpretation as $W_3^{\infty}$ covariant constraints which relate the above fields while passing from one coset manifold to another; iv. The Boussinesq equation and two kinds of the modified Boussinesq equations appear geometrically as the dynamical constraints accomplishing $W_3^{\infty}$ covariant reductions of original coset manifolds to their two-dimensional geodesic submanifolds;v. The zero-curvature representations for these equations arise automatically as a consequence of the covariant reduction. The approach proposed could provide a universal geometric description of the relationship between $W$-type algebras and integrable hierarchies. 
  We construct a one-parameter family of N=3 supersymmetric extensions of the KdV equation as a Hamiltonian flow on N=3 superconformal algebra and argue that it is non-integrable for any choice of the parameter. Then we propose a modified N=3 super KdV equation which possesses the higher order conserved quantities and so is a candidate for an integrable system. Upon reduction to N=2, it yields the recently discussed ``would-be integrable'' version of the N=2 super KdV equation. In the bosonic core it contains a coupled system of the KdV type equation and a three-component generalization of the mKdV equation. We give a Hamiltonian formulation of the new N=3 super KdV equation as a flow on some contraction of the direct sum of two N=3 superconformal algebras. 
  In the quantized two-dimensional non-linear supersymmetric $\sigma$-model, the supercurrent supermultiplet, which contains the energy-momentum tensor, is transformed by the nonlocal symmetry of the model into the isospin current supermultiplet. This effect incorporates supersymmetry into the known infinite-dimensional Yangian deformation symmetry of plain $\sigma$-models, leads to precisely the same nontrivial extension of the two-dimensional super-Poincar\'e group as found previously for the Poincar\'e group, and thus determines the theory's mass spectrum. A generalization to all higher-order nonlocal charges is conjectured such that their generating function, the so-called ``master charge'', has a definite Lorentz spin which depends on the spectral parameter. 
  Using canonical methods, we study the invariance properties of a bosonic $p$--brane propagating in a curved background locally diffeomorphic to $M\times G$, where $M$ is spacetime and $G$ a group manifold. The action is that of a gauged sigma model in $p+1$ dimensions coupled to a Yang--Mills field and a $(p+1)$--form in $M$. We construct the generators of Yang-Mills and tensor gauge transformations and exhibit the role of the $(p+1)$--form in cancelling the potential Schwinger terms. We also discuss the Noether currents associated with the global symmetries of the action and the question of the existence of infinite dimensional symmetry algebras, analogous to the Kac-Moody symmetry of the string. 
  A new framework is found for the compactification of supersymmetric string theory. It is shown that the massless spectra of Calabi--Yau manifolds of complex dimension $D_{crit}$ can be derived from noncritical manifolds of complex dimension $2k + D_{crit}$, $k\geq 1$. These higher dimensional manifolds are spaces whose nonzero Ricci curvature is quantized in a particular way. This class is more general than that of Calabi--Yau manifolds because it contains spaces which correspond to critical string vacua with no K\"ahler deformations, i.e. no antigenerations, thus providing mirrors of rigid Calabi--Yau manifolds. The constructions introduced here lead to new insights into the relation between exactly solvable models and their mean field theories on the one hand and Calabi--Yau manifolds on the other. They also raise fundamental questions about the Kaluza--Klein concept of string compactification, in particular regarding the r\^{o}le played by the dimension of the internal theories. 
  We construct a factorized representation of the $\frak g \frak l _n$-Sklyanin algebra from the vertex-face correspondence. Using this representation, we obtain a new solvable model which gives an $\frak s \frak l _n$-generalization of the broken $\bz _N $ model. We further prove the Yang-Baxter equation for this model. 
  Fractional superstrings are non-trivial generalizations of ordinary superstrings and heterotic strings, and have critical spacetime dimensions which are less than ten by virtue of a worldsheet fractional supersymmetry relating worldsheet bosons to worldsheet parafermions. In this paper, I provide a short non-technical survey of the fundamental issues involved in these recently-proposed theories. After introducing the basic ideas which underlie these new string theories, I review their early successes and outline some related outstanding questions.   ( Talk given at the Internal Workshop on String Theory,   Quantum Gravity, and the Unification of the Fundamental   Interactions, held in Rome, Italy, 21-26 September 1992. ) 
  Recently, background independent open-string field theory has been formally defined in the space of all two-dimensional world-sheet theories. In this paper, to make the construction more concrete, I compute the action for an off-shell tachyon field of a certain simple type. From the computation it emerges that, although the string field action does not coincide with the world-sheet (matter) partition function in general, these functions do coincide on shell. This can be demonstrated in general, as long as matter and ghosts are decoupled. 
  We review the construction of generalized integrable hierarchies of partial differential equations, associated to affine Kac-Moody algebras, that include those considered by Drinfel'd and Sokolov. These hierarchies can be used to construct new models of 2D quantum or topological gravity, as well as new $\cal W$-algebras. 
  We describe how the complete solution to the two-dimensional constant quantum Yang-Baxter equation [J. Hietarinta, Phys. Lett. A165,245(1992)] was found. (Talk presented at the XIX International Colloquium on Group Theoretical Methods in Physics.) 
  Using techniques from supergravity and dimensional reduction, we study the full isometry algebra of K\"ahler and quaternionic manifolds with special geometry. These two varieties are related by the so-called c-map, which can be understood from dimensional reduction of supergravity theories or by changing chirality assignments in the underlying superstring theory. An important subclass, studied in detail, consists of the spaces that follow from real special spaces using the so-called r-map. We generally clarify the presence of `extra' symmetries emerging from dimensional reduction and give the conditions for the existence of `hidden' symmetries. These symmetries play a major role in our analysis. We specify the structure of the homogeneous special manifolds as coset spaces $G/H$. These include all homogeneous quaternionic spaces. 
  It is proposed that instead of normal representations one should look at cocycles of group extensions valued in certain groups of unitary operators acting in a Hilbert space (e.g the Fock space of chiral fermions), when dealing with groups associated to current algebras in gauge theories in $3+1$ space-time dimensions. The appropriate cocycle is evaluated in the case of the group of smooth maps from the physical three-space to a compact Lie group. The cocyclic representation of a component $X$ of the current is obtained through two regularizations, 1) a conjugation by a background potential dependent unitary operator $h_A,$ 2) by a subtraction $-h_A^{-1}\Cal L_X h_A,$ where $\Cal L_X$ is a derivative along a gauge orbit. It is only the total operator $h_A^{-1} Xh_A-h_A^{-1}\Cal L_X h_A$ which is quantizable in the Fock space using the usual normal ordering subtraction. 
  Reducible gauge theories with constraints linear in the momenta are quantized. The equivalence of the reduced phase space quantization, Dirac quantization and BRST quantization is established. The ghosts of ghosts are found to play a crucial role in the equivalence proof. 
  It is shown how Darboux coordinates on a reduced symplectic vector space may be used to parametrize the phase space on which the finite gap solutions of matrix nonlinear Schr\"odinger equations are realized as isospectral Hamiltonian flows. The parametrization follows from a moment map embedding of the symplectic vector space, reduced by suitable group actions, into the dual $\tilde\grg^{+*}$ of the algebra $\tilde\grg^+$ of positive frequency loops in a Lie algebra $\grg$. The resulting phase space is identified with a Poisson subspace of $\tilde\grg^{+*}$ consisting of elements that are rational in the loop parameter. Reduced coordinates associated to the various Hermitian symmetric Lie algebras $(\grg,\grk)$ corresponding to the classical Lie algebras are obtained. 
  We discuss the non--perturbative formulation for $c \leq 1$ string theory. The field theory like formulation of topological and non--topological models is presented. The integral representation for arbitrary $(p,q)$ solutions is derived which explicitly obeys $p-q$ duality of these theories. The exact solutions to string equation and various examples are also discussed. 
  These notes provide an introduction to the theory of random matrices. The central quantity studied is $\tau(a)= det(1-K)$ where $K$ is the integral operator with kernel $1/\pi} {\sin\pi(x-y)\over x-y} \chi_I(y)$. Here $I=\bigcup_j(a_{2j-1},a_{2j})$ and $\chi_I(y)$ is the characteristic function of the set $I$. In the Gaussian Unitary Ensemble (GUE) the probability that no eigenvalues lie in $I$ is equal to $\tau(a)$. Also $\tau(a)$ is a tau-function and we present a new simplified derivation of the system of nonlinear completely integrable equations (the $a_j$'s are the independent variables) that were first derived by Jimbo, Miwa, M{\^o}ri, and Sato in 1980. In the case of a single interval these equations are reducible to a Painlev{\'e} V equation. For large $s$ we give an asymptotic formula for $E_2(n;s)$, which is the probability in the GUE that exactly $n$ eigenvalues lie in an interval of length $s$. 
  Scaling level-spacing distribution functions in the ``bulk of the spectrum'' in random matrix models of $N\times N$ hermitian matrices and then going to the limit $N\to\infty$, leads to the Fredholm determinant of the sine kernel $\sin\pi(x-y)/\pi (x-y)$. Similarly a double scaling limit at the ``edge of the spectrum'' leads to the Airy kernel $[{\rm Ai}(x) {\rm Ai}'(y) -{\rm Ai}'(x) {\rm Ai}(y)]/(x-y)$. We announce analogies for this Airy kernel of the following properties of the sine kernel: the completely integrable system of P.D.E.'s found by Jimbo, Miwa, M{\^o}ri and Sato; the expression, in the case of a single interval, of the Fredholm determinant in terms of a Painlev{\'e} transcendent; the existence of a commuting differential operator; and the fact that this operator can be used in the derivation of asymptotics, for general $n$, of the probability that an interval contains precisely $n$ eigenvalues. 
  Explicit general constructions of paragrassmann calculus with one and many variables are given. Relations of the paragrassmann calculus to quantum groups are outlined and possible physics applications are briefly discussed. This paper is the same as the original 9210075 except added Appendix and minor changes in Acknowledgements and References. IMPORTANT NOTE: This paper bears the same title as the Dubna preprint E5-92-392 but is NOT identical to it, containing new results, extended discussions, and references. 
  We study the problem of string propagation in a general instanton background for the case of the complete heterotic superstring. We define the concept of generalized HyperK\" ahler manifolds and we relate it to (4,4) superconformal theories. We propose a generalized h-map construction that predicts a universal SU(6) symmetry for the modes of the string excitations moving in an instanton background. We also discuss the role of abstract $N$=4 moduli and, applying it to the particular limit case of the solvable SU(2) X R instanton found by Callan et al. we show that it admits deformations and corresponds to a point in a 16-dimensional moduli space. The geometrical characterization of the other spaces in the same moduli-space remains an outstanding problem. 
  The sine-Gordon equation is considered in the hamiltonian framework provided by the Adler-Kostant-Symes theorem. The phase space, a finite dimensional coadjoint orbit in the dual space $\grg^*$ of a loop algebra $\grg$, is parametrized by a finite dimensional symplectic vector space $W$ embedded into $\grg^*$ by a moment map. Real quasiperiodic solutions are computed in terms of theta functions using a Liouville generating function which generates a canonical transformation to linear coordinates on the Jacobi variety of a suitable hyperelliptic curve. 
  \noindent The algebraic characterization of classes of locally isomorphic aperiodic tilings, being examples of quantum spaces, is conducted for a certain type of tilings in a manner proposed by A. Connes.   These $2$-dimensional tilings are obtained by application of the strip method to the root lattice of an $ADE$-Coxeter group.   The plane along which the strip is constructed is determined by the canonical Coxeter element leading to the result that a $2$-dimensional tiling decomposes into a cartesian product of two $1$-dimensional tilings.   The properties of the tilings are investigated, including selfsimilarity, and the determination of the relevant algebraic invariant is considered, namely the ordered $K_0$-group of an algebra naturally assigned to the quantum space.   The result also yields an application of the $2$-dimensional abstract gap labelling theorem. 
  In this lecture I summarize recent developments on strings propagating in curved spacetime. Exact conformal field theories that describe gravitational backgrounds such as black holes and more intricate gravitational singularities have been discovered and investigated at the classical and quantum level. These models are described by gauged Wess-Zumino-Witten models, or equivalently current algebra G/H coset models based on non-compact groups, with a single time coordinate. The classification of such models for all dimensions is complete. Furthermore the heterotic superstrings in curved spacetime based on non-compact groups have also been constructed. For many of the $d\le 4$ models the gravitational geometry described by a sigma model has been determined. Some general results outlined here include a global analysis of the geometry and the exact classical geodesics for any G/H model. Moreover, in the quantized theory, the conformally exact metric and dilaton are obtained for all orders in an expansion of $k$ (the central extension). All such models have large-small (or mirror) duality properties which we reformulate as an inversion in group space. To illustrate model building techniques a specific 4-dimensional heterotic string in curved spacetime is presented. Finally the methods for investigating the quantum theory are outlined. The construction and analysis of these models at the classical and quantum level involve some aspects of noncompact groups which are not yet sufficiently well understood. Some of the open problems in the physics and mathematics areas are outlined. 
  Recent work on a free field realization of the Hamiltonian structures of the classical KP hierarchy and of its flows is reviewed. It is shown that it corresponds to a reduction of KP to the NLS system. (Talk given by D.A.D. at the NSERC-CAP Workshop on Quantum Groups, Integrable Models and Statistical Systems, Kingston, Canada July 13-17 1992.) 
  Quantization of two dimensional chiral matter coupled to gravity induces an effective action for the zweibein field which is both Weyl and Lorentz anomalous. Recently, the quantization of this induced action has been analyzed in the light-cone gauge as well as in the conformal gauge. An apparent mismatch between the results obtained in the two gauges is analyzed and resolved by properly treating the Lorentz field as a chiral boson. 
  It is shown that conformal matter with $c_{\ssc L}\not=c_{\ssc R}$ can be consistently coupled to two-dimensional `frame' gravity. The theory is quantized in conformal gauge, following David, and Distler and Kawai. There is no analogue of the $c=1$ barrier found in nonchiral non-critical strings. A non-critical heterotic string is constructed---it has 744 states in its spectrum, transforming in the adjoint representation of $(E_8)^3.$ Correlation functions are calculated in this example, revealing the existence of extra discrete states. 
  The Adler-Kostant-Symes theorem yields isospectral hamiltonian flows on the dual $\tilde\grg^{+*}$ of a Lie subalgebra $\tilde\grg^+$ of a loop algebra $\tilde\grg$. A general approach relating the method of integration of Krichever, Novikov and Dubrovin to such flows is used to obtain finite-gap solutions of matrix Nonlinear Schr\"odinger Equations in terms of quotients of $\tet$-functions. 
  By considering the fermionic realization of $G/H$ coset models, we show that the partition function for the $U(1)/U(1)$ model defines a Topological Quantum Field Theory and coincides with that for a 2-dimensional Abelian BF system. In the non-Abelian case, we prove the topological character of $G/G$ coset models by explicit computation, also finding a natural extension of 2-dimensional BF systems with non-Abelian symmetry. 
  A novel representation ---in terms of a Laurent series--- for the free energy of string theory at non-zero temperature is constructed. The examples of open bosonic, open supersymmetric and closed bosonic strings are studied in detail. In all these cases the Laurent series representation for the free energy is obtained explicitly.   It is shown that the Hagedorn temperature arises in this formalism as the convergence condition (specifically, the radius of convergence) of the corresponding Laurent series. Some prospects for further applications are also discussed. In particular, an attempt to describe string theory above the Hagedorn temperature ---via Borel analytical continuation of the Laurent series representation--- is provided. 
  The free energy of a lattice model, which is a generalization of the Heisenberg $XYZ$ model with the higher spin representation of the Sklyanin algebra, is calculated by the generalized Bethe Ansatz of Takhtajan and Faddeev. (Talk given at the XXI Differential Geometry Methods in Theoretical Physics, Tianjin, China 5-9 June 1992) 
  We compute explicitly the Schr\"odinger picture space of states of SU(2) Chern-Simons theory on $T^2\times R$ in the presence of temporal Wilson lines. Relation with Friedan-Shenker bundle of conformal field theory and the existence of a projective flat connection on this bundle is discussed. Talk given by the first author at the XIX International Colloquium on Group Theoretical Methods in Physics, Salamanca (Spain), June 29-July 4, 1992 
  In this paper we reformulate the dilaton-gravity theory of Callan \etal\ as a new effective conformal field theory which turns out to be a generalization of the so-called $SL_2$-conformal affine Toda (CAT) theory studied some times ago by Babelon and Bonora. We quantize this model, thus keeping in account the dilaton-gravity quantum effects. We then implement a Renormalization Group analysis to study the black hole thermodynamics and the final state of the Hawking evaporation. 
  Darboux coordinates are constructed on rational coadjoint orbits of the positive frequency part $\wt{\frak{g}}^+$ of loop algebras. These are given by the values of the spectral parameters at the divisors corresponding to eigenvector line bundles over the associated spectral curves, defined within a given matrix representation. A Liouville generating function is obtained in completely separated form and shown, through the Liouville-Arnold integration method, to lead to the Abel map linearization of all Hamiltonian flows induced by the spectral invariants. Serre duality is used to define a natural symplectic structure on the space of line bundles of suitable degree over a permissible class of spectral curves, and this is shown to be equivalent to the Kostant-Kirillov symplectic structure on rational coadjoint orbits. The general construction is given for $\frak{g}=\frak{gl}(r)$ or $\frak{sl}(r)$, with reductions to orbits of subalgebras determined as invariant fixed point sets under involutive automorphisms. The case $\frak{g=sl}(2)$ is shown to reproduce the classical integration methods for finite dimensional systems defined on quadrics, as well as the quasi-periodic solutions of the cubically nonlinear Schr\"odinger equation. For $\frak{g=sl}(3)$, the method is applied to the computation of quasi-periodic solutions of the two component coupled nonlinear Schr\"odinger equation. 
  There are two methods to study families of conformal theories in the operator formalism. In the first method we begin with a theory and a family of deformed theories is defined in the state space of the original theory. In the other there is a distinct state space for each theory in the family, with the collection of spaces forming a vector bundle. This paper establishes the equivalence of a deformed theory with that in a nearby state space in the bundle via a connection that defines maps between nearby state spaces. We find that an appropriate connection for establishing equivalence is one that arose in a recent paper by Kugo and Zwiebach. We discuss the affine geometry induced on the space of backgrounds by this connection. This geometry is the same as the one obtained from the Zamolodchikov metric. 
  Supergeneralization of $\DC P(N)$ provided by even and odd K\"ahlerian structures from Hamiltonian reduction are construct.Operator $ \Delta$ which used in Batalin-- Vilkovisky quantization formalism and mechanics which are bi-Hamiltonian under corresponding even and odd Poisson brackets are considered. 
  Supersymmetric theories in four dimensions with chiral superfields have very rich BRS cohomology, which gives rise to potential anomalies in theories that contain composite antichiral spinor superfields. Assuming the coefficients are non-zero, absence of the anomalies would generate new constraints on theories. In addition, the anomalies give rise to a new kind of supersymmetry breaking which is quite different from the known kinds, and also naturally yields a zero cosmological constant after supersymmetry breaking. 
  We give exact solutions for a recently developed ~$N=1$~ locally supersymmetric self-dual gauge theories in $~(2+2)\-$dimensions. We give the exact solutions for an $~SL(2)$~ self-dual Yang-Mills multiplet and what we call ``self-dual tensor multiplet'' on the gravitational instanton background by Eguchi-Hanson. We use a general method to get an $~SL(2)$~ self-dual Yang-Mills solution from any known self-dual gravity solution. Our result is the first example of exact solutions for the coupled system of these $~N=1$~ locally supersymmetric self-dual multiplets in ~$(2+2)\-$dimensions, which is supposed to have strong significance for integrable models in lower-dimensions upon appropriate dimensional reduction or truncation. We also inspect the consistency of our exact solutions as a background for $~N=2$~ superstring coupled to the Wess-Zumino-Witten term in $~\s\-$model formulation. 
  A new string theory is proposed as a candidate for the large N limit of QCD. In this theory, strings are constrained to be non-backtracking; a condition that is essential for the gauge invariance of the underlying field theory.This condition is implemented by first placing the theory on a lattice and then introducing fermionic variables on the world sheet. The naive continuum limit leads to a generalized Thirring model on the world sheet,and it is suggested that the application of the renormalization group should drive the coupling constants to a conformally invariant fixed point. 
  Various Hamiltonian actions of loop groups $\wt G$ and of the algebra $\text{diff}_1$ of first order differential operators in one variable are defined on the cotangent bundle $T^*\wt G$ of a Loop Group. The moment maps generating  the $\text{diff}_1$ actions are shown to factorize through those generating the loop group actions, thereby defining commuting diagrams of Poisson maps  to the duals of the corresponding centrally extended algebras. The maps are  then used to derive a number of infinite commuting families of Hamiltonian flows that are nonabelian generalizations of the dispersive water wave hierarchies. As a further application, sets of pairs of generators of the nonabelian mKdV hierarchies are shown to give a commuting hierarchy on $T^*\wt G$ that contain the WZW system as its first element. 
  We derive a continuum field theory for the Majumdar-Ghosh model in the large-$S$ limit, where the field takes values in the manifold of the $SO(3)$ group. No topological term is induced in the action and the cases for integer spin and half-integer spin appear to be indistinguishable. A one-loop $\beta -$function calculation indicates that the theory flows towards a strong coupling (disordered) phase at long distances. This is verified in the large-$N$ limit, where all excitations are shown to be massive. (Three figures not included) 
  Self-similar potentials and corresponding symmetry algebras are briefly discussed.   Talk presented at the XIXth ICGTMP, Salamanca, 29 June - 4July 1992. 
  We build and investigate a pure gauge theory on arbitrary discrete groups. A systematic approach to the construction of the differential calculus is presented. We discuss the metric properties of the models and introduce the action functionals for unitary gauge theories. A detailed analysis of two simple models based on $\z_2$ and $\z_3$ follows. Finally we study the method of combining the discrete and continuous geometry. 
  We construct the supersymmetric completion of quartic $R+R^4$-actions in the ten-dimensional effective action of the heterotic string. Two invariants, of which the bosonic parts are known from one-loop string amplitude calculations, are obtained. One of these invariants can be generalized to an $R+F^2+F^4$-invariant for supersymmetric Yang-Mills theory coupled to supergravity.   Supersymmetry requires the presence of $B\wedge R\wedge R\wedge   R\wedge R$-terms, ($B\wedge F\wedge F\wedge F\wedge F$ for   Yang-Mills) which correspond to counterterms in the Green-Schwarz anomaly cancellation. Within the context of our calculation the $\zeta(3)R^4$-term from the tree-level string effective action does not allow supersymmetrization. 
  Quantum groups play the role of hidden symmetries of some two-dimensional field theories. We discuss how they appear in this role in the Wess-Zumino-Witten model of conformal field theory. 
  We investigate the quantum theory of 1+1 dimensional dilaton gravity, which is an interesting toy model of the black hole dynamics. The functional measures of gravity part are explicitly evaluated and derive the Wheeler-DeWitt like equations as physical state conditions. In ADM formalism the measures are very ambiguous, but in our formalism they are explicitly defined. Then the new features which are not seen in ADM formalism come out. A singularity appears at $\df^2 =\kappa (>0) $, where $\kappa =(N-51/2)/12 $ and $ N$ is the number of matter fields. At the final stage of the black hole evaporation, the Liouville term becomes important, which just comes from the measures of the fields. Behind the singularity the quantum mechanical region $\kappa > \df^2 >0 $ extends, where the sign of the kinetic term in the Wheeler-DeWitt like equation changes. If $\kappa <0 $, the singularity disappears. We briefly discuss the possibility of gravitational tunneling and the issue of the information loss.   (Talk given at "YITP Workshop on Theories of Quantum Fields -Beyond Perturbation-", Kyoto, Japan, 14-17 July 1992.   Some misleading arguments in the preprint UT-Komaba 92-7 entitled "Quamtum Theory of Dilaton Gravity in 1+1 Dimensions" are corrected. The several remarks on the quantization are included. The difference from the other quantum theory is clarified. 
  The extension of dynamical scaling to local, space-time dependent rescaling factors is investigated. For a dynamical exponent $z=2$, the corresponding invariance group is the Schr\"odinger group. Schr\"odinger invariance is shown to determine completely the two-point correlation function. The result is checked in two exactly solvable models. 
  We consider Chern-Simons gauge theory on a torus with both nonrelativistic and relativistic matter. It is shown that the Hamiltonian and two total momenta commute among themselves only in the physical Hilbert space. We also discuss relations among degenerate physical states, degenerate vacua, and the existence of multicomponent Schrodinger wavefunctions. 
  We construct a four-parameter point-interaction for a non-relativistic particle moving on a line as the limit of a short range interaction with range tending toward zero. For particular choices of the parameters, we can obtain a delta-interaction or the so-called delta'-interaction. The Hamiltonian corresponding to the four-parameter point-interaction is shown to correspond to the four-parameter self-adjoint Hamiltonian of the free particle moving on the line with the origin excluded. 
  We review some of the recent progress in the continuum formulation of two-dimensional string theory, i.e. two-dimensional quantum gravity coupled to $c=1$ matter. Special attention is devoted to the discrete states and to the $w_\infty$ algebra they generate. To demonstrate the power of the infinite symmetry, we use the $w_\infty$ Ward identities to derive recursion relations among certain classes of correlation functions, which allow to calculate them exactly. (Lectures delivered by I.R. Klebanov at the Workshop "String Quantum Gravity and Physics at the Planck Energy Scale", Erice, June 21-28, 1992, and at the 1992 Trieste Summer School of Theoretical Physics.) 
  A topological procedure for computing correlation functions for any (1,q) model is presented. Our procedure can be used to compute any correlation function on the sphere as well as some correlation functions at higher genus. We derive new and simpler recursion relations that extend previously known results based on W constraints. In addition, we compute an effective contact algebra with multiple contacts that extends Verlindes' algebra. Computational techniques based on the KdV approach are developed and used to compute the same correlation functions. A simple and elegant proof of the puncture equation derived directly from the KdV equations is included. We hope that this approach can lead to a deeper understanding of D=1 quantum gravity and non-critical string theory. (Paper uses tex TeX macro package jytex and includes 8 Postscript figures in the text using dvips (and epsf). Instructions for processing are included.) 
  We show that an integral transform of the fluctuations of the collective field of the $d=1$ matrix model satisfy the same linearized equation as that of the massless "tachyon" in the black hole background of the two dimensional critical string. This suggests that the $d=1$ matrix model may provide a non-perturbative description of black holes in two dimensional string theory. 
  Each isometric complex structure on a 2$\ell$-dimensional euclidean space $E$ corresponds to an identification of the Clifford algebra of $E$ with the canonical anticommutation relation algebra for $\ell$ ( fermionic) degrees of freedom. The simple spinors in the terminology of E.~Cartan or the pure spinors in the one of C. Chevalley are the associated vacua. The corresponding states are the Fock states (i.e. pure free states), therefore, none of the above terminologies is very good. 
  We present a new lattice integrable system in one dimension of the Haldane-Shastry type. It consists of spins positioned at the static equilibrium positions of particles in a corresponding classical Calogero system and interacting through an exchange term with strength inversely proportional to the square of their distance. We achieve this by viewing the Haldane-Shastry system as a high-interaction limit of the Sutherland system of particles with internal degrees of freedom and identifying the same limit in a corresponding Calogero system. The commuting integrals of motion of this system are found using the exchange operator formalism. 
  We propose a naive unification of Electromagnetism and General Relativity based on enlarging the gauge group of Ashtekar's new variables. We construct the connection and loop representations and analyze the space of states. In the loop representation, the wavefunctions depend on two loops, each of them carrying information about both gravitation and electromagnetism. We find that the Chern-Simons form and the Jones Polynomial play a role in the model. 
  The following two loosely connected sets of topics are reviewed in these lecture notes: 1) Gauge invariance, its treatment in field theories and its implications for internal symmetries and edge states such as those in the quantum Hall effect. 2) Quantisation on multiply connected spaces and a topological proof the spin-statistics theorem which avoids quantum field theory and relativity. Under 1), after explaining the meaning of gauge invariance and the theory of constraints, we discuss boundary conditions on gauge transformations and the definition of internal symmetries in gauge field theories. We then show how the edge states in the quantum Hall effect can be derived from the Chern-Simons action using the preceding ideas. Under 2), after explaining the significance of fibre bundles for quantum physics, we review quantisation on multiply connected spaces in detail, explaining also mathematical ideas such as those of the universal covering space and the fundamental group. These ideas are then used to prove the aforementioned topological spin-statistics theorem.e of the universal covering space and the fundamental group. 
  We prove the equivalence between anyon quantum mechanics on a torus and Chern-Simons gauge theory. It is also shown that the Hamiltonian and total momenta commute among themselves only in the physical Hilbert space. 
  We analyze some features of the perturbative quantization of Chern-Simons theory (CST) in the Landau gauge. In this gauge the theory is known to be perturbatively finite. We consider the renormalization scheme in which the renormalized parameter $k$ equals the bare or classical one and show that it constitutes a natural parametrization for the quantum theory. The reason is that, although in this renormalization scheme the value of the Green functions depends on the regularization used, comparison among different regularization methods shows that the observables (Wilson loops) are the same function of the shifted monodromy parameter $k+c_v$ for all BRS invariant regulators used so far for CST. We also discuss a particular BRS invariant regularization prescription in which CST is perturbatively defined as the large mass limit of dimensionally regularized topologically massive Yang-Mills theory. With this regularization prescription the radiative corrections induced by two-loop contributions do not entail observable consequences since they can be reabsorbed by a finite rescaling of the fields only. This very mechanism is conjectured to take place at higher perturbative orders. Talk presented by G.G. at the NATO AWR on ``Low dimensional Topology and Quantum Field Theory'', 6-13 September 1992, Cambridge (UK). 
  We show that a single uncharged chiral superfield, canonically coupled to \mbox{$N=1$} supergravity with vanishing superpotential, naturally drives inflation in the early universe for a class of simple Kahler potentials. Inflation occurs due to the one-loop generation of a Kahler anomaly proportional to $\R^2$. The coefficient of this $\R^2$ term is of the correct magnitude to describe all aspects of an inflationary cosmology, including sufficient amplitude perturbations and reheating. Higher order terms proportional to $\R^n$ for $n \geq 3$ are naturally suppressed relative to the $\R^2$ term and, hence, do not destabilize the theory. 
  The geometry of supermanifolds provided with $Q$-structure (i.e. with odd vector field $Q$ satisfying $\{ Q,Q\} =0$), $P$-structure (odd symplectic structure ) and $S$-structure (volume element) or with various combinations of these structures is studied. The results are applied to the analysis of Batalin-Vilkovisky approach to the quantization of gauge theories. In particular the semiclassical approximation in this approach is expressed in terms of Reidemeister torsion. 
  We study bosonic string theory in a gravitational background. We show that either left-movers or right-movers are the only background independent non-perturbative solutions of the field equations for an arbitrary static metric. They are stable and have a conserved topological charge being therefore topological solitons. The action vanishes for these solutions and hence they provide the dominating contribution in a path integral quantization 
  In this paper we study the inter-relationship between the integrable KP hierarchy, nonlinear $\hat{W}_{\infty}$ algebra and conformal noncompact $SL(2,R)/U(1)$ coset model at the classical level. We first derive explicitly the Possion brackets of the second Hamiltonian structure of the KP hierarchy, then use it to define the $\hat{W}_{1+\infty}$ algebra and its reduction $\hat{W}_{\infty}$. Then we show that the latter is realized in the $SL(2,R)/U(1)$ coset model as a hidden current algebra, through a free field realization of $\hat{W}_{\infty}$, in closed form for all higher-spin currents, in terms of two bosons. An immediate consequence is the existence of an infinite number of KP flows in the coset model, which preserve the $\hat{W}_{\infty}$ current algebra. 
  This paper is devoted to constructing a quantum version of the famous KP hierarchy, by deforming its second Hamiltonian structure, namely the nonlinear $\hat{W}_{\infty}$ algebra. This is achieved by quantizing the conformal noncompact $SL(2,R)_{k}/U(1)$ coset model, in which $\hat{W}_{\infty}$ appears as a hidden current algebra. For the quantum $\hat{W}_{\infty}$ algebra at level $k=1$, we have succeeded in constructing an infinite set of commuting quantum charges in explicit and closed form. Using them a completely integrable quantum KP hierarchy is constructed in the Hamiltonian form. A two boson realization of the quantum $\hat{W}_{\infty}$ currents has played a crucial role in this exploration. 
  Solutions to low energy string theory describing black holes and black strings are reviewed. Many of these solutions can be obtained by applying simple solution generating transformations to the Schwarzschild metric. In a few cases, the corresponding exact conformal field theory is known. Various properties of these solutions are discussed including their global structure, singularities, and Hawking temperature. (This review is based on lectures given at the 1992 Trieste Spring School on String Theory and Quantum Gravity.) 
  We discuss the classical 2-dim. black-hole in the framework of the non-perturbative formulation (in terms of non-relativistic fermions) of c=1 string field theory. We identify an off-shell operator whose classical equation of motion is that of tachyon in the classical graviton-dilaton black-hole background. The black-hole `singularity' is identified with the fermi surface in the phase space of a single fermion, and as such is a consequence of the semi-classical approximation. An exact treatment reveals that stringy quantum effects wash away the classical singularity. 
  We show that O(2,2) transformation of SU(2) WZNW model gives rise to marginal deformation of this model by the operator $\int d^2 z J(z)\bar J(\bar z)$ where $J$, $\bar J$ are U(1) currents in the Cartan subalgebra. Generalization of this result to other WZNW theories is discussed. We also consider O(3,3) transformation of the product of an SU(2) WZNW model and a gauged SU(2) WZNW model. The three parameter set of models obtained after the transformation is shown to be the result of first deforming the product of two SU(2) WZNW theories by marginal operators of the form $\sum_{i,j=1}^2 C_{ij} J_i \bar J_j$, and then gauging an appropriate U(1) subgroup of the theory. Our analysis leads to a general conjecture that O(d,d) transformation of any WZNW model corresponds to marginal deformation of the WZNW theory by combination of appropriate left and right moving currents belonging to the Cartan subalgebra; and O(d,d) transformation of a gauged WZNW model can be identified to the gauged version of such marginally deformed WZNW models. 
  We solve the Kac-Moody branching equation to obtain explicit formulae for the characters of coset conformal field theories and then apply these to specific examples to determine the integer shift of the conformal weights of primary fields. We also present an example of coset conformal field theory which cannot be described by the identification current method. 
  We study geodesically complete, singularity free space-times induced by supersymmetric planar domain walls interpolating between Minkowski and anti-de Sitter ($AdS_4$) vacua. A geodesically complete space-time without closed time-like curves includes an infinite number of semi-infinite Minkowski space-times, separated from each other by a region of $AdS_4$ space-time. These space-times are closely related to the extreme Reissner Nordstr\" om (RN) black hole, exhibiting Cauchy horizons with zero Hawking temperature, but in contrast to the RN black hole there is no entropy. Another geodesically complete extension with closed time-like curves involves space-times connecting a finite number of semi-infinite Minkowski space-times. 
  Invariant polynomials for torus links are obtained in the framework of the Chern-Simons topological gauge theory. The polynomials are computed as vacuum expectation values on the three-sphere of Wilson line operators representing the Verlinde algebra of the corresponding rational conformal field theory. In the case of the $SU(2)$ gauge theory our results provide explicit expressions for the Jones polynomial as well as for the polynomials associated to the $N$-state ($N>2$) vertex models (Akutsu-Wadati polynomials). By means of the Chern-Simons coset construction, the minimal unitary models are analyzed, showing that the corresponding link invariants factorize into two $SU(2)$ polynomials. A method to obtain skein rules from the Chern-Simons knot operators is developed. This procedure yields the eigenvalues of the braiding matrix of the corresponding conformal field theory. 
  We calculate the lowest translationally invariant levels of the Z_3- and Z_4-symmetrical chiral Potts quantum chains, using numerical diagonalization of the hamiltonian for N <= 12 and N <= 10 sites, respectively, and extrapolating N to infinity. In the high-temperature massive phase we find that the pattern of the low-lying zero momentum levels can be explained assuming the existence of n-1 particles carrying Z_n-charges Q = 1, ... , n-1 (mass m_Q), and their scattering states. In the superintegrable case the masses of the n-1 particles become proportional to their respective charges: m_Q = Q m_1. Exponential convergence in N is observed for the single particle gaps, while power convergence is seen for the scattering levels. We also verify that qualitatively the same pattern appears for the self-dual and integrable cases. For general Z_n we show that the energy-momentum relations of the particles show a parity non-conservation asymmetry which for very high temperatures is exclusive due to the presence of a macroscopic momentum P_m=(1-2Q/n)/\phi, where \phi is the chiral angle and Q is the Z_n-charge of the respective particle. 
  We present a multi-grid algorithm in order to solve numerically the thermodynamic Bethe ansatz equations. We specifically adapt the program to compute the data of the conformal field theory reached in the ultraviolet limit. Submitted to Computer Physics Communications 
  An interesting feature of some open superstring models in $D < 10$ is the simultaneous presence, in the spectrum, of gauge fields and of a number of antisymmetric tensor fields. In these cases the Green-Schwarz mechanism can (and does) take a generalized form, resulting from the combined action of all the antisymmetric tensors. These novelties are illustrated referring to some simple rational models in six dimensions, and some of their implications for the low-energy effective field theory are pointed out. 
  A classical R-matrix structure is described for the Lax representation of the integrable n-particle chains of Calogero-Olshanetski-Perelo\-mov. This R-matrix is dynamical, non antisymmetric and non-invertible. It immediately triggers the integrability of the Type I, II and III potentials, and the algebraic structures associated with the Type V potential. 
  We use the single particle excitation energies and the completeness rules of the 3-state anti-ferromagnetic Potts chain, which have been obtained from Bethe's equation, to compute the modular invariant partition function. This provides a fermionic construction for the branching functions of the $D_4$ representation of $Z_4$ parafermions which complements the previous bosonic constructions. It is found that there are oscillations in some of the correlations and a new connection with the field theory of the Lee-Yang edge is presented. 
  The recently derived current algebra of classical non-linear sigma models on arbitrary Riemannian manifolds is extended to include the energy-momentum tensor. It is found that in two dimensions the energy-momentum tensor $\theta_{\mu\nu}$, the Noether current $j_\mu$ associated with the global symmetry of the theory and the composite field $j$ appearing as the coefficient of the Schwinger term in the current algebra, together with the derivatives of $j_\mu$ and $j$, generate a closed algebra. The subalgebra generated by the light-cone components of the energy-momentum tensor consists of two commuting copies of the Virasoro algebra, with central charge $\, c\!=\!0 $, reflecting the classical conformal invariance of the theory, but the current algebra part and the semidirect product structure are quite different from the usual Kac-Moody / Sugawara type construction. 
  We study the statistics of vortices which appear in (2+1)--dimensional spontaneously broken gauge theories, where a compact group G breaks to a finite nonabelian subgroup H. Two simple models are presented. In the first, a quantum state which is symmetric under the interchange of a pair of indistinguishable vortices can be transformed into an antisymmetric state after the passage through the system of a third vortex with an appropriate $H$-flux element. Further, there exist states containing two indistinguishable spinless vortices which obey Fermi statistics. These results generalize to loops of nonabelian cosmic string in 3+1 dimensions. In the second model, fractional analogues of the above behaviors occur. Also, composites of vortices in this theory may possess fractional ``Cheshire spin'' which can be changed by passing an additional vortex through the system. 
  This paper reviews the covariant formalism of N=1, D=10 classical superparticle models. It discusses the local invariances of a number of superparticle actions and highlights the problem of finding a covariant quantization scenario. Covariant quantization has proved problematic, but it has motivated in seeking alternative approaches that avoids those found in earlier models. It also shows new covariant superparticle theories formulated in extended spaces that preserve certain canonical form in phase-space, and easy to quantize by using the Batalin-Vilkovisky procedure, as the gauge algebra of their constraints only closes on-shell. The mechanics actions describe particles moving in a superspace consisting of the usual $N=1$ superspace, together with an extra spinor or vector coordinate. A light-cone analysis shows that all these new superparticle models reproduce the physical spectrum of the N=1 super-Yang-Mills theory. 
  In this work we propose an alternative description of the quantum mechanics of a massive and spinning free particle in anti-de~Sitter spacetime, using a phase space rather than a spacetime representation. The regularizing character of the curvature appears clearly in connection with a notion of localization in phase space which is shown to disappear in the zero curvature limit. We show in particular how the anti-de~Sitter optimally localized (coherent) states contract to plane waves as the curvature goes to zero. In the first part we give a detailed description of the classical theory {\it \a la Souriau\/}. This serves as a basis for the quantum theory which is constructed in the second part using methods of geometric quantization. The invariant positive K\"ahler polarization that selects the anti-de~Sitter quantum elementary system is shown to have as zero curvature limit the Poincar\'e polarization which is no longer K\"ahler. This phenomenon is then related to the disappearance of the notion of localization in the zero curvature limit. 
  The group theoretic method is extended to include fields with a background charge. This formalism is used to compute the tree level scattering for $W_3$ strings. The scattering amplitudes involve Ising model correlation functions. A detailed study of the four tachyon amplitude shows that the $W_3$ string must possess additional states in its spectrum associated with intercept $1/2$ and the energy operator of the Ising model. 
  (Replacement because mailer changed `hat' for supercript into something weird. The macro `\sp' has been used in place of the `hat' character in this revised version.) Fermionic Brownian paths are defined as paths in a space para\-metr\-ised by anticommuting variables. Stochastic calculus for these paths, in conjunction with classical Brownian paths, is described; Brownian paths on supermanifolds are developed and applied to establish a Feynman-Kac formula for the twisted Laplace-Beltrami operator on differential forms taking values in a vector bundle. This formula is used to give a proof of the Atiyah-Singer index theorem which is rigorous while being closely modelled on the supersymmetric proofs in the physics literature. 
  We study two dimensional Quantum Chromodynamics with massive quarks on a cylinder in a light--cone formalism. We eliminate the non--dynamical degrees of freedom and express the theory in terms of the quark and Wilson loop variables. It is possible to perform this reduction without gauge fixing. The fermionic Fock space can be defined independent of the gauge field in this light--cone formalism. 
  I report on work on a Lagrangian formulation for the simplest 1+1 dimensional integrable hierarchies. This formulation makes the relationship between conformal field theories and (quantized) 1+1 dimensional integrable hierarchies very clear. 
  When the $q$-deformed creation and annihilation operators are used in a second quantization procedure, the algebra satisfied by basis vectors (orthogonal complete set) should be also deformed such as a field operator remains invariant under the coaction of the quantum group. In the 1+1 dimensional quantum field theories we deform the algebra of the basis vectors and study the $q$-deformation in the second quantization procedure. 
  The equations of motion of anomaly-free supergravity are shown to fulfil (to all orders in $\alpha'$) a differential condition corresponding to the one relating the Weyl anomaly coefficients for a non-linear sigma model representing a (heterotic) string propagating in a non-trivial background. This supports the possibility that anomaly-free supergravity could provide the complete massless effective theory for the heterotic string. 
  We present a two-loop computation of the beta functions and the anomalous dimensions of a $\gamma_5$-Yukawa model using differential renormalization. The calculation is carried out in coordinate space without modifying the space-time dimension and no ad-hoc prescription for $\gamma_5$ is needed.   It is shown that this procedure is specially suited for theories involving $\gamma_5$, and it should be considered in analyzing chiral gauge theories. 
  The $q$-Poincar\'e group of \cite{SWW:inh} is shown to have the structure of a semidirect product and coproduct $B\cocross \widetilde{SO_q(1,3)}$ where $B$ is a braided-quantum group structure on the $q$-Minkowski space of 4-momentum with braided-coproduct $\und\Delta \vecp=\vecp\tens 1+1\tens \vecp$. Here the necessary $B$ is not a usual kind of quantum group, but one with braid statistics. Similar braided-vectors and covectors $V(R')$, $V^*(R')$ exist for a general R-matrix. The abstract structure of the $q$-Lorentz group is also studied. 
  We review recent advances towards the computation of string couplings. Duality symmetry, mirror symmetry, Picard-Fuchs equations, etc. are some of the tools. 
  We discuss the $N=2$ super $W$ algebras from the hamiltonian reduction of affine Lie superalgebras $A(n|n-1)^{(1)}$ and $A(n|n)^{(1)}$. From the quantum hamiltonian reduction of $A(n|n-1)^{(1)}$ we get the free field realization of $N=2$ $CP_{n}$ super coset models. In the case of the affine Lie superalgebras $A(n|n)^{(1)}$, the corresponding conformal field theories do not have $N=2$ superconformal symmetry. However we show that these models are twisted $N=2$ $CP_{n}$ models and may be regarded as topological conformal field theories. (Talk presented at the International Workshop on "String Theory, Quantum Gravity and the Unification of Fundamental Interactions" Rome, September 21--26, 1992) 

  A generalization of the $SU(2)$--spin systems on a lattice and their continuum limit to an arbitrary compact group $G$ is discussed. The continuum limits are, in general, non--relativistic $\sigma$--model type field theories targeted on a homogeneous space $G/H$, where $H$ contains the maximal torus of $G$. In the ferromagnetic case the equations of motion derived from our continuum Lagrangian generalize the Landau--Lifshitz equations with quadratic dispersion relation for small wave vectors. In the antiferromagnetic case the dispersion law is always linear in the long wavelength limit. The models become relativistic only when $G/H$ is a symmetric space. Also discussed are a generalization of the Holstein--Primakoff representation of the $SU(N)$ algebra, the topological term and the existence of the instanton type solutions in the continuum limit of the antiferromagnetic systems. 
  We demonstrate the relation of the infrared anomaly of conformal field theory with entropy considerations of finite temperature thermodynamics for the 3-state Potts chain. We compute the free energy and compute the low temperature specific heat for both the ferromagnetic and anti-ferromagnetic spin chains, and find the central charges for both. 
  During the last few years, interest has arisen in using light-front Tamm-Dancoff field theory to describe relativistic bound states for theories such as QCD. Unfortunately, difficult renormalization problems stand in the way. We introduce a general, non-perturbative approach to renormalization that is well suited for the ultraviolet and, presumably, the infrared divergences found in these systems. We reexpress the renormalization problem in terms of a set of coupled inhomogeneous integral equations, the ``counterterm equation.'' The solution of this equation provides a kernel for the Tamm-Dancoff integral equations which generates states that are independent of any cutoffs. We also introduce a Rayleigh-Ritz approach to numerical solution of the counterterm equation. Using our approach to renormalization, we examine several ultraviolet divergent models. Finally, we use the Rayleigh-Ritz approach to find the counterterms in terms of allowed operators of a theory. 
  In this paper we construct a new quantum double by endowing the l-state bosonalgebra with a non-trivial Hopf algebra structure,which is not a q-deformation of the Lie algebra or superalgebra.The universal R-matrix for the Yang-Baxter equation associated with this new quantum group structure is obtained explicitly.By building the representations of this quantum double,we get some R-matrices ,which can result in new representations of the braid group. 
  By computing anomalous dimensions of gauge invariant composite operators $(\bar\psi\psi)^n$ and $(\phi^*\phi)^n$ in Chern-Simons fermion and boson models, we address that Chern-Simons interactions make these operators more relevant or less irrelevant in the low energy region. We obtain a critical Chern-Simons fermion coupling, ${1\over \kappa_c^2} = {6\over 19}$, for a phase transition at which the leading irrelevant four-fermion operator $(\bar\psi\psi)^2$ becomes marginal, and a critical Chern-Simons boson coupling, ${1\over \kappa_c^2} = {6\over 34}$, for a similar phase transition for the leading irrelevant operator $(\phi^*\phi)^4$. We see this phenomenon also in the $1/N$ expansion. 
  A bicovariant calculus of differential operators on a quantum group is constructed in a natural way, using invariant maps from \fun\ to \uqg\ , given by elements of the pure braid group. These operators --- the `reflection matrix' $Y \equiv L^+ SL^-$ being a special case --- generate algebras that linearly close under adjoint actions, i.e. they form generalized Lie algebras. We establish the connection between the Hopf algebra formulation of the calculus and a formulation in compact matrix form which is quite powerful for actual computations and as applications we find the quantum determinant and an orthogonality relation for $Y$ in $SO_q(N)$. 
  We introduce the covariant forms for the non-Abelian anomaly counterparts in topological Yang-Mills theory, which satisfies the topological descent equation modulo terms that vanish at the space of BRST fixed points. We use the covariant anomalies as a new set of observables, which can absorb both $\dw$ and $\db$ ghost number violations of zero-modes. Then, we study some problems due to the zero-modes originated from the reducible connections. 
  A new type of algebras that represent a generalization of both quantum groups and braided groups is defined. These algebras are given by a pair of solutions of the Yang--Baxter equation that satisfy some additional conditions. Several examples are presented. 
  A reference has been corrected 
  We develop the dual picture for Quantum Electrodynamics in 3+1 dimensions. It is shown that the photon is massless in the Coulomb phase due to spontaneous breaking of the magnetic symmetry group. The generators of this group are the magnetic fluxes through any infinite surface $\Phi_S$. The order parameter for this symmetry breaking is the operator $V(C)$ which creates an infinitely long magnetic vortex. We show that although the order parameter is a stringlike rather than a local operator, the Goldstone theorem is applicable if $<V(C)>\ne 0$. If the system is properly regularized in the infrared, we find $<V(C)>\ne 0$ in the Coulomb phase and $V(C)=0$ in the Higgs phase. The Higgs - Coulomb phase transition is therefore understood as condensation of magnetic vortices. The electric charge in terms of $V(C)$ is topological and is equal to the winding number of the mapping from a circle at spatial infinity into the manifold of possible vacuum expectation values of a magnetic vortex in a given direction. Since the vortex operator takes values in $S^1$ and $\Pi_1(S^1)={\cal Z}$, the electric charge is quantized topologically. 
  As in the first part of this paper (hep-th 9204092), solutions to a string equation are regarded as fixed points of some additional symmetries of a hierarchy of integrable equations. In this part matrix hierarchies are studied: the multi-component KP and KdV hierarchies, and the modified KdV hierarchy as their reduction. In particular, the action of additional symmetries on the Grassmannian is found, as well as Virasoro constraints on the $\tau$-functions. The matrix string equations are known to be involved in some matrix models. 
  This TASI lecture covers the material in hep-th/9205026. It reviewed the theory of effective strings, with particular emphasis on the manner in which Lorentz invariance is represented. The quantum properties of an example of an effective string are derived from the underlying field theory. A comparison is made with what one would expect if one assumed that quantum effective strings were governed by fundamental string actions such as the Nambu-Goto or the Polyakov actions. It is shown that the requirements on dimensions for consistent quantizations of fundamental strings imply no contradictions for effective strings. 
  We investigate quantum deformation of conformal algebras by constructing the quantum space for $sl_q(4,C)$. The differential calculus on the quantum space and the action of the quantum generators are studied. We derive deformed $su(4)$ and $su(2,2)$ algebras from the deformed $sl(4)$ algebra using the quantum 4-spinor and its conjugate spinor. The 6-vector in $so_q(4,2)$ is constructed as a tensor product of two sets of 4-spinors. The reality condition for the 6-vector and that for the generators are found. The q-deformed Poincar{\'e} algebra is extracted as a closed subalgebra. 
  Vacuum polarization effects are non-perturbatively incorporated into the photon propagator to eliminate the severe infrared problems characteristic of QED$_3$. The theory is thus rephrased in terms of a massive vector boson whose mass is $e^2/(8\pi)$. Subsequently, it is shown that electron-electron bound states are possible in QED$_3$. 
  We consider correlation functions in Neveu--Schwarz string theory coupled to two dimensional gravity. The action for the 2D gravity consists of the string induced Liouville action and the Jackiw--Teitelboim action describing pure 2D gravity. Then gravitational dressed dimensions of vertex operators are equal to their bare conformal dimensions. There are two possible interpretations of the model. Considering the 2D dilaton and the Liouville field as additional target space coordinates one gets a $d+2$-dimensional critical string. In the $d$-dimensional non critical string picture gravitational fields retain their original meaning and for $d=4$ one can get a mass spectrum via consistency requirements. In both cases a GSO projection is possible. 
  We show that several WZW coset models can be obtained by applying O(d,d) symmetry transformations (referred to as twisting) on WZW models. This leads to a conjecture that WZW models gauged by U(1)^n subgroup can be obtained by twisting (ungauged) WZW models.  In addition, a class of solutions that describe charged black holes in four dimensions is derived by twisting SL(2,R)\times SU(2) WZW. 
  This paper is dedicated to the study of the existence and the properties of electron-electron bound states in QED$_3$. A detailed analysis of the infrared structure of the perturbative series of the theory is presented. We start by analyzing the two-point Green's function, in the Bloch-Nordsieck approximation. The theory appears to be plagued by severe infrared divergencies, which nevertheless disappear when vacuum-polarization effects are non-perturbatively taken into account. The dynamical induction of a Chern-Simons term is at the root of this mechanism. {}From the inspection of the electron-electron non-relativistic potential it then follows that equally charged fermions may either repel or attract and, moreover, that bound states do in fact exist in the theory. We calculate numerically the binding energies and average radius of the bound states. We find an accidental quasi-degeneracy of the ground state of the system, between the lowest-energy $l=-3$ and $l=-5$ states, which could be related to a radio-frequency resonance in high-$T_c$ superconductors. 
  We give a unified description of our recent results on the the inter-relationship between the integrable infinite KP hierarchy, nonlinear $\hat{W}_{\infty}$ current algebra and conformal noncompact $SL(2,R)/U(1)$ coset model both at the classical and quantum levels. In particular, we present the construction of a quantum version of the KP hierarchy by deforming the second KP Hamiltonian structure through quantizing the $SL(2,R)_k/U(1)$ model and constructing an infinite set of commuting quantum $\hat{W}_{\infty}$ charges (at least at $k$=1). 
  We perform dimensional reductions of recently constructed self-dual $~N=2$~ {\it supersymmetric} Yang-Mills theory in $~2+2\-$dimensions into two-dimensions. We show that the universal equations obtained in these dimensional reductions can embed supersymmetric exactly soluble systems, such as $~N=1$~ and $~N=2$~ supersymmetric Korteweg-de Vries equations, $~N=1$~ supersymmetric Liouville theory or supersymmetric Toda theory. This is the first supporting evidence for the conjecture that the $~2+2\-$dimensional self-dual {\it supersymmetric} Yang-Mills theory generates {\it supersymmetric} soluble systems in lower-dimensions. 
  We consider a string theory with two types of strings with geometric interaction. We show that the theory contains strings with constant Dirichlet boundary condition and those strings are glued together by 2-d topological gravity with macroscopic boundaries. A light-cone string field theory is given and the theory has interactions to all orders. (Postscript files of the figures can be obtained by anonymous ftp uful07.phys.ufl.edu and are in the directory /het/ufift-hep-92-26.) 
  A comprehensive analysis of small fluctuations about two-dimensional string-theoretic and string-inspired black holes is presented. It is shown with specific examples that two-dimensional black holes behave in a radically different way from all known black holes in four dimensions. For both the $SL(2,R)/U(1)$ black hole and the two-dimensional black hole coupled to a massive dilaton with constant field strength, it is shown that there are a {\it continuous infinity} of solutions to the linearized equations of motion, which are such that it is impossible to ascertain the classical linear response. It is further shown that the two-dimensional black hole coupled to a massive, linear dilaton admits {\it no small fluctuations at all}. We discuss possible implications of our results for the Callan-Giddings-Harvey-Strominger black hole. 
  A recently introduced framework for the compactification of supersymmetric string theory involving noncritical manifolds of complex dimension $2k+D_{crit}$, $k\geq 1$, is reviewed. These higher dimensional manifolds are spaces with quantized positive Ricci curvature and therefore do not, a priori, describe consistent string vacua. It is nevertheless possible to derive from these manifolds the massless spectra of critical string groundstates. For a subclass of these noncritical theories it is also possible to explicitly construct Calabi--Yau manifolds from the higher dimensional spaces. Thus the new class of theories makes contact with the standard framework of string compactification. This class of manifolds is more general than that of Calabi--Yau manifolds because it contains spaces which correspond to critical string vacua with no K\"ahler deformations, i.e. no antigenerations, hence providing mirrors of rigid Calabi--Yau manifolds. The constructions reviewed here lead to new insight into the relation between exactly solvable models and their mean field theories on the one hand and Calabi--Yau manifolds on the other, leading, for instance, to a modification of Gepner's conjecture. They also raise fundamental questions about the Kaluza--Klein concept of string compactification, in particular regarding the r\^{o}le played by the dimension of the internal theories. 
  Dynamics of quantized free fields ( of spin 0 and 1/2 ) contained in a subspace $V_*$ of an N+4 dimensional flat space $V$ is studied. The space $V_*$ is considered as a neighborhood of a four dimensional submanifold $M$ arbitrarily embedded into $V$. We study the system as a simple model of unified theory of gravity ($g$), SO(N) gauge fields ($A$) and Higgs fields ($\phi $). In this paper classical treatment of the system is given. We show that, especially when the fields have spin 1/2, the system is described by an infinite number of fields in $M$ interacting with $g$, $A$ and $\phi $. The fields $g$, $A$ and $\phi $ are induced themselves by embedding functions of $M$ and correspond respectively to induced metric, normal connection and extrinsic curvature of $M$. 
  We present a rigorous analysis of the Schr\"{o}dinger picture quantization for the $SU(2)$ Chern-Simons theory on 3-manifold torus$\times$line, with insertions of Wilson lines. The quantum states, defined as gauge covariant holomorphic functionals of smooth $su(2)$-connections on the torus, are expressed by degree $2k$ theta-functions satisfying additional conditions. The conditions are obtained by splitting the space of semistable $su(2)$-connections into nine submanifolds and by analyzing the behavior of states at four codimension $1$ strata. We construct the Knizhnik-Zamolodchikov-Bernard connection allowing to compare the states for different complex structures of the torus and different positions of the Wilson lines. By letting two Wilson lines come together, we prove a recursion relation for the dimensions of the spaces of states which, together with the (unproven) absence of states for spins$\s>{_1\over^2}$level implies the Verlinde dimension formula. 
  We investigate the effects of the Chern-Simons coupling on the high energy behavior in the $(2+1)$-dimensional Chern-Simons QED with a four-Fermi interaction. Using the $1/N$ expansion we discuss the Chern-Simons effects on the critical four-Fermi coupling at $O(1/N)$ and the $\beta$ function around it. High-energy behavior of Green's functions is also discussed. By explicit calculation, we find that the radiative correction to the Chern-Simons coupling vanishes at $O(1/N)$ in the broken phase of the dynamical parity symmetry. We argue that no radiative corrections to the Chern-Simons term arise at higher orders in the $1/N$ expansion. 
  A first step towards a systematic theory of relative line bundles over SUSY-curves is presented. In this paper we deal with the case of relative line bundles over families of ordinary Riemann surfaces. Generalizations of the Gauss-Bonnet theorem and of the flatness theorem for line bundles are discussed. 
  A relative Picard theory in the context of graded manifolds is introduced. A Berezinian calculus and a theory of connections over SUSY-curves are systematically developed, and used to prove a Gauss-Bonnet theorem for line bundles in that setting and to discuss the validity of a flatness theorem 
  We study regular and black hole solutions to the coupled classical Einstein--Yang-Mills--Higgs system. It has long been thought that black hole solutions in the spontaneously broken phase of such a theory could have no nontrivial field structure outside of the horizon. We first show that the standard black hole no-hair theorem underlying this belief, although true in the abelian setting, does not necessarily extend to the non-abelian case. This indicates the possibility of solutions with non-trivial gauge and Higgs configurations decaying exponentially {\it outside} the horizon. We then find such solutions by numerical integration of the classical equations for the case of $SU(2)$ coupled to a Higgs doublet (the standard model less hypercharge).   As a prelude to this work we also study regular and black hole solutions to Einstein--Non-Abelian--Proca theory and as a postscript we briefly discuss the important issue of stability. 
  The conformal anomaly induced sector of four-dimensional quantum gravity (infrared quantum gravity) ---which has been introduced by Antoniadis and Mottola--- is here studied on a curved fiducial background. The one-loop effective potential for the effective conformal factor theory is calculated with accuracy, including terms linear in the curvature. It is proven that a curvature induced phase transition can actually take place. An estimation of the critical curvature for different choices of the parameters of the theory is given. 
  A new 2-parameter quadratic deformation of the quantum oscillator algebra and its 1-parameter deformed Heisenberg subalgebra are considered. An infinite dimensional Fock module representation is presented which at roots of unity contains null vectors and so is reducible to a finite dimensional representation. The cyclic, nilpotent and unitary representations are discussed. Witten's deformation of $sl_2$ and some deformed infinite dimensional algebras are constructed from the $1d$ Heisenberg algebra generators. The deformation of the centreless Virasoro algebra at roots of unity is mentioned. Finally the $SL_q(2)$ symmetry of the deformed Heisenberg algebra is explicitly constructed. 
  We review recent progresses in the study of factorized resonance scattering S-matrices. The resonance amplitudes are introduced through a suitable analytical continuation of the ADE Toda S-matrices. By using the thermodynamic Bethe ansatz approach we are able to compute the ground state energy, which describes a rich pattern of flows interpolating between the central charges of the coset models based on the ADE Lie algebras. We also present the simplest resonance ``$\phi^3$'' scattering model and discuss its relation with new flows in non-unitary minimal models. Further generalizations are discussed in terms of certain asymptotic conditions in a family of ``resonance'' functional hierarchies. 
  Extremal black holes are studied in a two dimensional model motivated by a dimensional reduction from four dimensions. Their quantum corrected geometry is calculated semiclassically and a mild singularity is shown to appear at the horizon.   Extensions of the geometry past the horizon are not unique but there are continuations free from malevolent singularities. A few comments are made about the relevance of these results to four dimensions and to the study of black hole entropy and information loss. 
  In this letter we present constant solutions to the tetrahedron equations proposed by Zamolodchikov. In general, from a given solution of the Yang-Baxter equation there are two ways to construct solutions to the tetrahedron equation. There are also other kinds of solutions. We present some two-dimensional solutions that were obtained by directly solving the equations using either an upper triangular or Zamolodchikov's ansatz. 
  In this work the quantum theory of two dimensional dilaton black holes is studied using the Wheeler De Witt equation. The solutions correspond to wave functions of the black hole. It is found that for an observer inside the horizon, there are uncertainty relations for the black hole mass and a parameter in the metric determining the Hawking flux. Only for a particular value of this parameter, can both be known with arbitrary accuracy. In the generic case there is instead a relation which is very similar to the so called string uncertainty relation. 
  We show how the supersymmetric KdV equation can be obtained from the self-duality condition on Yang-Mills fields in four dimension associated with the graded Lie algebra OSp(2/1). We also obtain the hierarchy of Susy KdV equations from such a condition. We formulate the Susy KdV hierarchy as a vanishing curvature condition associated with the U(1) group and show how an Abelian self-duality condition in four dimension can also lead to these equations. 
  We present a detailed calculation of the entropy and action of $U(1)~2$ dilaton black holes, and show that both quantities coincide with one quarter of the area of the event horizon. Our methods of calculation make it possible to find an explanation of the rule $S = A/4$ for all static, spherically symmetric black holes studied so far. We show that the only contribution to the entropy comes from the extrinsic curvature term at the horizon, which gives $S = A/4$ independently of the charge(s) of the black hole, presence of scalar fields, etc. Previously, this result did not have a general explanation, but was established on a case-by-case basis. The on-shell Lagrangian for maximally supersymmetric extreme dilaton black holes is also calculated and shown to vanish, in agreement with the result obtained by taking the limit of the expression obtained for black holes with regular horizon.The physical meaning of the entropy is discussed in relation to the issue of splitting of extreme black holes. 
  Using recent advances in the understanding of non-critical strings, we construct a unique, conformally invariant continuation to off-shell momenta of Polyakov amplitudes in critical string theory. Three-point amplitudes are explicitly calculated. These off-shell amplitudes possess some unusual, apparently stringy, characteristics, which are unlikely to be reproduced in a string field theory. Thus our results may be an indication that some fundamentally new formulation, other than string field theory, will be required to extend our understanding of critical strings beyond the Polyakov path integral. 
  Using the factorization approach of Gepner and Qiu, I systematically rederive the closed fractional superstring partition functions for K= 4, 8, and 16. For these theories the relationship between the massless graviton and gravi- tino sector and the purely massive sectors is explored. Properties of the massive sectors are investigated. A twist current in these models is found responsible for the occurrence of N=1 space-time supersymmetry. I show this twist current transforms bosonic (fermionic) projection states into fermionic (bosonic) non-projection states and vice-versa. 
  Bauer, Di Francesco, Itzykson and Zuber proposed recently an algorithm to construct all singular vectors of the Virasoro algebra. It is based on the decoupling of (already known) singular fields in the fusion process. We show how to extend their algorithm to the Neveu-Schwarz superalgebra. 
  We consider the extended superconformal algebras of the Knizhnik-Bershadsky type with $W$-algebra like composite operators occurring in the commutation relations, but with generators of conformal dimension 1,$\frac{3}{2}$ and 2, only. These have recently been neatly classified by several groups, and we emphasize the classification based on hamiltonian reduction of affine Lie superalgebras with even subalgebras $G\oplus sl(2)$. We reveiw the situation and improve on previous formulations by presenting generic and very compact expressions valid for all algebras, classical and quantum. Similarly generic and compact free field realizations are presented as are corresponding screening charges. Based on these a discussion of singular vectors is presented. (Based on talk by J.L. Petersen at the Int. Workshop on "String Theory, Quantum Gravity and the Unification of the Fundamental Interactions", Rome Sep. 21-26, 1992) 
  We study $2D$ supergravity in a covariant and gauge independent way. The theory is obtained from $2D$ bosonic gravity following the square root method and the diffeomorphism superalgebra is explicitly computed. We argue that our approach could be a procedure for introducing nontrivial physics in quantum $2D$ (super)gravity. 
  The duality-type symmetries of string cosmology naturally lead to a pre-big-bang phase of accelerated evolution as dual counterpart of the decelerated expansion of standard cosmology. We discuss several properties of this scenario, including the possibility that tracks of the pre-big-bang may be found either in the spectrum of relic gravitons or in the distortion they induce on the cosmic microwave background. 
  Usually in quantum mechanics the Heisenberg algebra is generated by operators of position and momentum. The algebra is then represented on an Hilbert space of square integrable functions. Alternatively one generates the Heisenberg algebra by the raising and lowering operators. It is then natural to represent it on the Bargmann Fock space of holomorphic functions. In the following I show that the Bargmann Fock construction can also be done in the quantum group symmetric case. This leads to a 'q- deformed quantum mechanics' in which the basic concepts, Hilbert space of states and unitarity of time evolution, are preserved. 
  In quantum field theory the creation and annihilation operators that are located at the points in 3-momentum space have commutation relations that are conserved under the action of a $U({\infty})$ group. Here it is shown how to define an appropriate $U_q({\infty})$ which then leads to deformed commutation relations. A formalism that was previously defined for quantum group symmetric quantum mechanics can be applied by imposing simple adjustments. The simplest example of a free scalar $q$-bosonic field is discussed. The deformation occurs as a multi-particle effect, which introduces a kinetic energy (i.e. $({\vec{p}})^2$) - ordering of the involved particles. 
  Requiring that the path integral has the global symmetries of the classical action and obeys the natural composition property of path integrals, and also that the discretized action has the correct naive continuum limit, we find a viable discretization of the (D=3,N=2) superparticle action. 
  In a recent paper, the BRST formalism for the gauge-fixed N=2 twistor-string was used to calculate Green-Schwarz supersring scattering amplitudes with an arbitrary number of loops and external massless states. Although the gauge-fixing procedure preserved the worldsheet N=2 superconformal invariance of the twistor-string, it broke the target-space SO(9,1) super-Poincar\'e invariance down to an SU(4)xU(1) subgroup. In this paper, generators for the SO(9,1) super-Poincar\'e transformations, as well as manifestly covariant vertex operators, are explicitly constructed out of the gauge-fixed matter fields. The earlier calculated amplitudes are then expressed in manifestly Lorentz-covariant notation. 
  We construct for all $N$ a solution of the Frenkel--Moore $N$--simplex equation which generalizes the $R$--matrix for the Jordanian quantum group. 
  We derive the four point correlation function involving four twist fields for arbitrary even dimensional Z_N x Z_M orbifold compactifications. Using techniques from the conformal field theory the three point correlation functions with twist fields are determined. Both the choice of the modular background (compatible with the twists) and of the (higher) twisted sectors involved are fully general. Our results turn out to be target space duality invariant. 
  As compared to the previous version, the example of Gupta-Bleuler quantization of massive electrodynamics was added, and the derivation of path integral for anomalous theories is further elaborated. This is the final version to be published in Annals of Physics. 
  We present an operator formulation of the q-deformed dual string model amplitude using an infinite set of q-harmonic oscillators. The formalism attains the crossing symmetry and factorization and allows to express the general n-point function as a factorized product of vertices and propagators. 
  Magnetically charged dilatonic black holes have a perturbatively infinite ground state degeneracy associated with an infinite volume throat region of the geometry. A simple argument based on causality is given that these states do not have a description as ordinary massive particles in a low-energy effective field theory. Pair production of magnetic black holes in a weak magnetic field is estimated in a weakly-coupled semiclassical expansion about an instanton and found to be finite, despite the infinite degeneracy of states. This suggests that these states may store the information apparently lost in black hole scattering processes. 
  Dynamics of quantized free fields ( of spin 0 and 1/2 ) contained in a subspace $V_*$ of an N+4 dimensional flat space $V$ is studied. The space $V_*$ is considered as a neighborhood of a four dimensional submanifold $M$ arbitrarily embedded into $V$. We show that Einstein SO(N)-Yang-Mills Higgs theory is induced as a low energy effective theory of the system. Gravity, SO(N) gauge fields and Higgs fields are obtained from embedding functions of $M$. 
  We find the R matrix for the inhomogeneous quantum groups whose homogeneous part is $GL_q(n)$, or its restrictions to $SL_q(n)$,$U_q(n)$ and $SU_q(n)$. The quantum Yang-Baxter equation for R holds because of the Hecke relation for the braiding matrix of the homogeneous subgroup. A bicovariant differential calculus on $IGL_q(n)$ is constructed, and its application to the $D=4$ Poincar\'e group $ISL_q(2,\Cb)$ is discussed. 
  : We consider two--dimensional supersymmetric Toda theories based on the Lie superalgebras $A(n,n)$, $D(n+1,n)$ and $B(n,n)$ which admit a fermionic set of simple roots and a fermionic untwisted affine extension. In particular, we concentrate on two simple examples, the $B(1,1)$ and $A(1,1)$ theories. Both in the conformal and massive case we address the issue of quantum integrability by constructing the first non trivial conserved currents and proving their conservation to all--loop orders. While the $D(n+1,n)$ and $B(n,n)$ systems are genuine $N=1$ supersymmetric theories, the $A(n,n)$ models possess a global $N=2$ supersymmetry. In the conformal case, we show that the $A(n,n)$ stress--energy tensor, uniquely determined by the holomorphicity condition, has vanishing central charge and it corresponds to the stress--energy tensor of the associated topological theory. (Invited talk at the International Workshop ``String theory, quantum gravity and the unification of the fundamental interactions'', Roma, September 1992) 
  Dilogarithm identities for the central charges and conformal dimensions exist for at least large classes of rational conformally invariant quantum field theories in two dimensions. In many cases, proofs are not yet known but the numerical and structural evidence is convincing. In particular, close relations exist to fusion rules and partition identities. We describe some examples and ideas, and present some conjectures useful for the classification of conformal theories. The mathematical structures seem to be dual to Thurston's program for the classification of 3-manifolds. 
  We consider vortex dynamics in self-dual Chern-Simons Higgs systems. We show that the naive Aharanov-Bohm phase is the inverse of the statistical phase expected from the vortex spin, and that the self-dual configurations of vortices are degenerate in energy but not in angular momentum. We also use the path integral formalism to derive the dual formulation of Chern-Simons Higgs systems in which vortices appear as charged particles. We argue that besides the electromagnetic interaction, there is an additional interaction between vortices, the so-called Magnus force, and that these forces can be put together into a single `dual electromagnetic' interaction. This dual electromagnetic interaction leads to the right Aharanov-Bohm phase. We also derive and study the effective action for slowly moving vortices, which contains terms both linear and quadratic in the vortex velocity. 
  Radul has recently introduced a map from the Lie algebra of differential operators on the circle to $\W_n$. In this note we extend this map to $\Wkpq$, a recently introduced one-parameter deformation of $\Wkp$---the second hamiltonian structure of the KP hierarchy. We use this to give a short proof that $\W_\infty$ is the symmetry algebra of additional symmetries of the KP equation. (The replaced version clears up some minor points and corrects some notational glitches. Get this one!) 
  A master equation ( $n$ dimensional non--Abelian current conservation law with mutually commuting current components ) is introduced for multi-dimensional non-linear field theories. It is shown that the master equation provides a systematic way to understand 2-d integrable non-linear equations as well as 4-d self-dual equations and, more importantly, their generalizations to higher dimensions. 
  We consider a continuous model of D-dimensional elastic (polymerized) manifold fluctuating in d-dimensional Euclidean space, interacting with a single impurity via an attractive or repulsive delta-potential (but without self-avoidance interactions). Except for D=1 (the polymer case), this model cannot be mapped onto a local field theory. We show that the use of intrinsic distance geometry allows for a rigorous construction of the high-temperature perturbative expansion and for analytic continuation in the manifold dimension D. We study the renormalization properties of the model for 0<D<2, and show that for d<d* where d*=2D/(2-D) is the upper critical dimension, the perturbative expansion is UV finite, while UV divergences occur as poles at d=d*. The standard proof of perturbative renormalizability for local field theories (the BPH theorem) does not apply to this model. We prove perturbative renormalizability to all orders by constructing a subtraction operator based on a generalization of the Zimmermann forests formalism, and which makes the theory finite at d=d*. This subtraction operation corresponds to a renormalization of the coupling constant of the model (strength of the interaction with the impurity). The existence of a Wilson function, of an epsilon-expansion around the critical dimension, of scaling laws for d<d* in the repulsive case, and of non-trivial critical exponents of the delocalization transition for d>d* in the attractive case is thus established. To our knowledge, this provides the first proof of renormalizability for a model of extended objects, and should be applicable to the study of self-avoidance interactions for random manifolds. 
  Integrable hierarchy based on the constrained Osp(2$\mid%2) connection is considered. The connection with 2D supergravity and some analogies with the W$_3^{(2)}$ case are given. It is shown that super Virasoro transformations are symmetries of tha hierarchy. 
  We show that almost all string theories, including the bosonic string, the superstring and $W$-string theories, possess a twisted N=2 superconformal symmetry. This enables us to establish a connection between topological gravity and the field theoretical description of matter coupled to gravity. We also show how the \brs operators of the $W_n$-string can be obtained by hamiltonian reduction of $SL(n|n-1)$. The tachyonic and ground ring states of $W$-strings are described in the light of the N=2 superconformal structure, and the ground ring generators for the non-critical $W_3$-string are explicitly constructed. The relationship to $G/G$ models and quantum integrable systems is also briefly described. 
  The dimensional reduction technique is adopted to derive string effective action. Wormhole solutions corresponding to space-time geometries $R^1\times S^1\times S^2$ and $R^1\times S^3$ are presented. The duality and SL(2,R) symmetries are implemented to generate new wormhole solutions. 
  We present the {\it canonical} set of superspace constraints for self-dual supergravity, a ``self-dual'' tensor multiplet and a self-dual Yang-Mills multiplet with $~N=1~$ supersymmetry in the space-time with signature $(+,+,-,-)$. For this set of constraints, the consistency of the self-duality conditions on these multiplets with supersymmetry is manifest. The energy-momentum tensors of all the self-dual ``matter'' multiplets vanish, to be consistent with the self-duality of the Riemann tensor. In particular, the special significance of the ``self-dual'' tensor multiplet is noted. This result fills the gap left over in our previous series of papers, with respect to the consistent couplings among the self-dual matter multiplets. We also couple these non-trivial backgrounds to a Green-Schwarz superstring $~\s\-$model, under the requirement of invariance under fermionic (kappa) symmetry. The finiteness of the self-dual supergravity is discussed, based on its ``off-shell'' structure. A set of exact solutions for the ``self-dual'' tensor and self-dual Yang-Mills multiplets for the gauge group $~SL(2)$~ on self-dual gravitational instanton background is given, and its consistency with the Green-Schwarz string ~$\s\-$model is demonstrated. 
  In this paper it is shown that a quantum observable algebra, the Heisenberg-Weyl algebra, is just given as the Hopf algebraic dual to the classical observable algebra over classical phase space and the Plank constant is included in this scheme of quantization as a compatible parameter living in the quantum double theory.In this sense,the quantum Yang-Baxter equation naturally appears as a necessary condition to be satisfied by a canonical elements,the universal R-matrix,intertweening the quantum and classical observable algebras. As a byproduct,a new ``quantum group'' is obtained as the quantum double of the classical observable algebra 
  In a scalar field theory, when the tree level potential admits broken symmetry ground states, the quantum corrections to the static effective potential are complex. (The imaginary part is a consequence of an instability towards phase separation and the static effective potential is not a relevant quantity for understanding the dynamics). Instead, we study here the equations of motion obtained from the one loop effective action for slow rollover out of equilibrium.   We considering the case in which a scalar field theory undergoes a rapid phase transition from $T_i>T_c$ to $T_f<T_c$. We find that, for slow rollover initial conditions (the field near the maximum of the tree level potential), the process of phase separation controlled by unstable long-wavelength fluctuations introduces dramatic corrections to the dynamical evolution of the field. We find that these effects slow the rollover even further 
  The free Maxwell field theory is quantized in the Lorentz gauge on a two dimensional manifold $M$ with conformally flat background metric. It is shown that in this gauge the theory is equivalent, at least at the classical level, to a biharmonic version of the bosonic string theory. This equivalence is exploited in order to construct in details the propagator of the Maxwell field theory on $M$. The expectation values of the Wilson loops are computed. A trivial result is obtained confirming in the Lorentz gauge previous calculations. Finally the interacting case is briefly discussed taking the Schwinger model as an example. The two and three point functions of the Schwinger model are explicitly derived at the lowest order on a Riemann surface. 
  We construct the theory of non-abelian gauge antisymmetric tensor fields, which generalize the standard Yang-MIlls fields and abelian gauge p-forms. The corresponding gauge group acts on the space of inhomogeneous differential forms and it is shown to be a supergroup. The wide class of generalized Chern-Simons actions is constructed. 
  We present an algorithm for determining all inequivalent abelian symmetries of non-degenerate quasi-homogeneous polynomials and apply it to the recently constructed complete set of Landau--Ginzburg potentials for $N=2$ superconformal field theories with $c=9$. A complete calculation of the resulting orbifolds without torsion increases the number of known spectra by about one third. The mirror symmetry of these spectra, however, remains at the same low level as for untwisted Landau--Ginzburg models. This happens in spite of the fact that the subclass of potentials for which the Berglund--H\"ubsch construction works features perfect mirror symmetry. We also make first steps into the space of orbifolds with $\ZZ_2$ torsions by including extra trivial fields. 
  We perform an exact renormalization-group analysis of one-dimensional 4-state clock models with complex interactions. Our aim is to provide a simple explicit illustration of the behavior of the renormalization-group flow in a system exhibiting a rich phase diagram. In particular we study the flow in the vicinity of phase transitions with a first-order character, a matter that has been controversial for years. We observe that the flow is continuous and single-valued, even on the phase transition surface, provided that the renormalized Hamiltonian exist. The characteristics of such a flow are in agreement with the Nienhuis-Nauenberg standard scenario, and in disagreement with the ``discontinuity scenario'' proposed by some authors and recently disproved by van Enter, Fern\'andez and Sokal for a large class of models (with real interactions). However, there are some points in the space of interactions for which a renormalized Hamiltonian cannot be defined. This pathological behavior is similar, and in some sense complementary, to the one pointed out by Griffiths, Pearce and Israel for Ising models. We explicitly see that if the transformation is truncated so as to preserve a Hamiltonian description, the resulting flow becomes discontinuous and multivalued at some of these points. This suggests a possible explanation for the numerical results that motivated the ``discontinuity scenario''. 
  We describe time-dependent tunneling of massless particles in 1+1 dimensional string field theory. Polchinski's description of the classical solutions in terms of the Fermi sea is used to identify the leading instanton contribution connecting the two half-lines. The field theory lagrangian is proportional to $1/g^2$, where $g$ is the string coupling constant, but the $S$-matrix for tunneling from one half-line to the other behaves as $\exp(-C/g)$. We note the constant~$C$ involves curious boundary contributions and observe that a prescription connecting the two half-lines unifies treatments of the Fermi level above and below the barrier. We also note the relation to recent work of Brustein and Ovrut. 
  The general problems of three-dimensional quantum gravity are recatitulated here, putting the emphasis on the mathematical problems of defining the measure of the path integral over all three-dimensional metrics.This work should be viewed as an extension of a preceding one on the four dimensional case (\cite{kn:eav5}), where also some general ideas are discussed in detail. We finally put forward some suggestions on the lines one could expect further progress in the field. 
  We present a review of some of the recent developments in the study of the $W_3$ string. One of the interesting features of the theory is that the physical spectrum includes states with non-standard ghost structure, such as excitations of the ghost fields, both for discrete-momentum and continuous-momentum states. (Contribution to the Proceedings of the 16'th Johns Hopkins Workshop on Current Problems in Particle Physics, Gothenburg, Sweden, June 1992) 
  In this paper a class of conformal field theories with nonabelian and discrete group of symmetry is investigated. These theories are realized in terms of free scalar fields starting from the simple $b-c$ systems and scalar fields on algebraic curves. The Knizhnik-Zamolodchikov equations for the conformal blocks can be explicitly solved. Besides of the fact that one obtains in this way an entire class of theories in which the operators obey a nonstandard statistics, these systems are interesting in exploring the connection between statistics and curved space-times, at least in the two dimensional case. 
  Using Watson's and the recursive equations satisfied by matrix elements of local operators in two-dimensional integrable models, we compute the form factors of the elementary field $\phi(x)$ and the stress-energy tensor $T_{\mu\nu}(x)$ of Sinh-Gordon theory. Form factors of operators with higher spin or with different asymptotic behaviour can easily be deduced from them. The value of the correlation functions are saturated by the form factors with lowest number of particle terms. This is illustrated by an application of the form factors of the trace of $T_{\mu\nu}(x)$ to the sum rule of the $c$-theorem. 
  For a 1+1 dimensional theory of gravity with torsion different approaches to the formulation of a quantum theory are presented. They are shown to lead to the same finite dimensional quantum system. Conceptual questions of quantum gravity like e.g.\ the problem of time are discussed in this framework. 
  We argue that 4D gravity is drastically modified at distances larger than the horizon scale, due to the large infrared quantum fluctuations of the conformal part of the metric. The infrared dynamics of the conformal factor is generated by an effective action, induced by the trace anomaly of matter in curved space, analogous to the Polyakov action in two dimensions. The resulting effective scalar theory is renormalizable, and possesses a non-trivial, infrared stable fixed point, characterized by an anomalous scaling dimension of the conformal factor. We argue that this theory describes a large distance scale invariant phase of 4D gravity and provides a framework for a dynamical solution of the cosmological constant problem. 
  Magnetic monopole solutions to heterotic string theory are discussed in toroidal compactifications to four spacetime dimensions. Particular emphasis is placed on the relation to previously studied fivebrane solutions in ten dimensions and on the possibility of constructing exact monopole solutions related to symmetric fivebranes. 
  A suggestion on how black holes may appear in Das-Jevicki Collective field theory is given. We study the behaviour of a `test' particle when energy is sent into the system. A perturbation moving near the potential barrier can create a large-distance black hole geometry where the seeming curvature singularity is at the position of the barrier. In the simplest `static' case the exact $D=2$ black hole metric emerges. 
  We examine a model of non-self-avoiding, fluctuating surfaces as a candidate continuum string theory of surfaces in three dimensions. This model describes Dynamically Triangulated Random Surfaces embedded in three dimensions with an extrinsic curvature dependent action. We analyze, using Monte Carlo simulations, the dramatic crossover behaviour of several observables which characterize the geometry of these surfaces. We then critically discuss whether our observations are indicative of a phase transition. 
  I give an overview of open, closed and heterotic N=2 strings. At the tree level I derive the effective field theories of all the strings, and discuss the group theory of the N=2 open string and the interaction between its open and closed sectors. At one loop N=2 string loop amplitudes and partition functions have incurable infra-red divergences, and show puzzling disagreements on the dimension of spacetime when compared to their effective field theories. I show that the closed-string three-point amplitude can be written directly in terms of a Schwinger parameter, so explicitly exhibiting the inconsistency. I finally discuss the possibility that the puzzles posed by the loop amplitudes could be solved if the N=2 theories were Lorentz invariant and supersymmetric, and I speculate on possible modifications of the string calculations. 
  The $q$-deformed harmonic oscillator within the framework of the recently introduced Schwenk-Wess $q$-Heisenberg algebra is considered. It is shown, that for "physical" values $q\sim1$, the gap between the energy levels decreases with growing energy. Comparing with the other (real) $q$-deformations of the harmonic oscillator, where the gap instead increases, indicates that the formation of the macroscopic energy gap in the Schwenk-Wess $q$-Quantum Mechanics may be avoided. 
  We consider $2d$ sigma models with a $D=2+N$ - dimensional Minkowski signature target space metric having a covariantly constant null Killing vector. These models are UV finite. The $2+N$-dimensional target space metric can be explicitly determined for a class of supersymmetric sigma models with $N$-dimensional `transverse' part of the target space being homogeneous K\"ahler. The corresponding `transverse' sub-theory is an $n=2$ supersymmetric sigma model with the exact $\gb$-function coinciding with its one-loop expression. For example, the finite $D=4$ model has $O(3)$ supersymmetric sigma model as its `transverse' part. Moreover, there exists a non-trivial dilaton field such that the Weyl invariance conditions are also satisfied, i.e. the resulting models correspond to string vacua. Generic solutions are represented in terms of the RG flow in `transverse' theory. We suggest a possible application of the constructed Weyl invariant sigma models to quantisation of $2d$ gravity. They may be interpreted as `effective actions' of the quantum $2d$ dilaton gravity coupled to a (non-conformal) $N$-dimensional `matter' theory. The conformal factor of the $2d$ metric and $2d$ `dilaton' are identified with the light cone coordinates of the $2+N$ - dimensional sigma model. 
  An alternative, pedagogically simpler derivation of the allowed physical wave fronts of a propagating electromagnetic signal is presented using geometric algebra. Maxwell's equations can be expressed in a single multivector equation using 3D Clifford algebra (isomorphic to Pauli algebra spinorial formulation of electromagnetism). Subsequently one can more easily solve for the time evolution of both the electric and magnetic field simultaneously in terms of the fields evaluated only on a 3D hypersurface. The form of the special "characteristic" surfaces for which the time derivative of the fields can be singular are quickly deduced with little effort. 
  In this paper, we present a new formulation of topological conformal gravity in four dimensions. Such a theory was first considered by Witten as a possible gravitational counterpart of topological Yang-Mills theory, but several problems left it incomplete. The key in our approach is to realise a theory which describes deformations of conformally self-dual gravitational instantons. We first identify the appropriate elliptic complex which does precisely this. By applying the Atiyah-Singer index theorem, we calculate the number of independent deformations of a given gravitational instanton which preserve its self-duality. We then quantise topological conformal gravity by BRST gauge-fixing, and discover how the quantum theory is naturally described by the above complex. Indeed, it is a process which closely parallels that of the Yang-Mills theory, and we show how the partition function generates an uncanny gravitational analogue of the first Donaldson invariant. 
  A matrix model describing surfaces embedded in a Bethe lattice is considered. From the mean field point of view, it is equivalent to the Kazakov-Migdal induced gauge theory and therefore, at $N=\infty$ and $d>1$, the latter can be interpreted as a matrix model for infinite-tension strings. We show that, in the naive continuum limit, it is governed by the one-matrix-model saddle point with an upside-down potential. To derive mean field equations, we consider the one-matrix model in external field. As a simple application, its explicit solution in the case of the inverted W potential is given. 
  We consider the relations of generalized commutativity in the algebra of formal series $ M_q (x^i ) $, which conserve a tensor $ I_q $-grading and depend on parameters $ q(i,k) $ . We choose the $ I_q $-preserving version of differential calculus on $ M_q$ . A new construction of the symmetrized tensor product for $ M_q $-type algebras and the corresponding definition of minimally deformed linear group $ QGL(n) $ and Lie algebra $ qgl(n) $ are proposed. We study the connection of $ QGL(n) $ and $ qgl(n) $ with the special matrix algebra $ \mbox{Mat} (n,Q) $ containing matrices with noncommutative elements. A definition of the deformed determinant in the algebra $ \mbox{Mat} (n,Q) $ is given. The exponential parametrization in the algebra $ \mbox{Mat} (n,Q) $ is considered on the basis of Campbell-Hausdorf formula. 
  We propose that the correlation functions of the inhomogeneous eight vertex model in the anti-ferroelectric regime satisfy a system of difference equations with respect to the spectral parameters. Solving the simplest difference equation we obtain the expression for the spontaneous staggered polarization conjectured by Baxter and Kelland. We discuss also a related construction of vertex operators on the lattice. 
  Withdrawn. 
  We review briefly a stream of ideas concerning the role of quantum groups as hidden symmetries in conformal field theories, paying particular attention to the field theoretical representations of quantum groups based on Coulomb gas methods. An extensive bibliography is also included. 
  In this talk we consider the relationship between the conjectured uniqueness of the Moonshine module of Frenkel, Lepowsky and Meurman and Monstrous Moonshine, the genus zero property for Thompson series discovered by Conway and Norton. We discuss some evidence to support the uniqueness of the Moonshine module by considering possible alternative orbifold constructions from a Leech lattice compactified string. Within these constructions we find a new relationship between the centralisers of the Monster group and the Conway group generalising an observation made by Conway and Norton. We also relate the uniqueness of the Moonshine module to Monstrous Moonshine and argue that given this uniqueness, then the genus zero properties hold if and only if orbifolding the Moonshine module with respect to a Monster element reproduces the Moonshine module or the Leech theory. (Talk presented at the Nato Advanced Research Workshop on `Low dimensional topology and quantum field theory`, Cambridge, 6-13 Sept 1992) 
  In recent years it has been shown that many, and possibly all, integrable systems can be obtained by dimensional reduction of self-dual Yang-Mills. I show how the integrable systems obtained this way naturally inherit bihamiltonian structure. I also present a simple, gauge-invariant formulation of the self-dual Yang-Mills hierarchy proposed by several authors, and I discuss the notion of gauge equivalence of integrable systems that arises from the gauge invariance of the self-duality equations (and their hierarchy); this notion of gauge equivalence may well be large enough to unify the many diverse existing notions. 
  The low-lying excitations of a quantum Hall state on a disk geometry are edge excitations. Their dynamics is governed by a conformal field theory on the cylinder defined by the disk boundary and the time variable. We give a simple and detailed derivation of this conformal field theory for integer filling, starting from the microscopic dynamics of $(2+1)$-dimensional non-relativistic electrons in Landau levels. This construction can be generalized to describe Laughlin's fractional Hall states via chiral bosonization, thereby making contact with the effective Chern-Simons theory approach. The conformal field theory dictates the finite-size effects in the energy spectrum. An experimental or numerical verification of these universal effects would provide a further confirmation of Laughlin's theory of incompressible quantum fluids. 
  Motivated by the descent equation in string theory, we give a new interpretation for the action of the symmetry charges on the BRST cohomology in terms of what we call {\em the Gerstenhaber bracket}. This bracket is compatible with the graded commutative product in cohomology, and hence gives rise to a new class of examples of what mathematicians call a {\em Gerstenhaber algebra}. The latter structure was first discussed in the context of Hochschild cohomology theory \cite{Gers1}. Off-shell in the (chiral) BRST complex, all the identities of a Gerstenhaber algebra hold up to homotopy. Applying our theory to the c=1 model, we give a precise conceptual description of the BRST-Gerstenhaber algebra of this model. We are led to a direct connection between the bracket structure here and the anti-bracket formalism in BV theory \cite{W2}. We then discuss the bracket in string backgrounds with both the left and the right movers. We suggest that the homotopy Lie algebra arising from our Gerstenhaber bracket is closely related to the HLA recently constructed by Witten-Zwiebach. Finally, we show that our constructions generalize to any topological conformal field theory. 
  We give two results concerning the construction of modular invariant partition functions for conformal field theories constructed by tensoring together other conformal field theories. First we show how the possible modular invariants for the tensor product theory are constrained if the allowed modular invariants of the individual conformal field theory factors have been classified. We illustrate the use of these constraints for theories of the type $SU(2)_{K_A}\otimes SU(2)_{K_B}$, finding all consistent theories for $K_A,K_B$ odd. Second we show how known diagonal modular invariants can be used to construct some inherently asymmetric ones where the holomorphic and anti-holomorphic theories do not share the same chiral algebra. Some explicit examples are given. 
  We study in this paper a theory of free anyons associated to free conformal field theories defined on Riemann surfaces with a discrete and nonabelian group of authomorphisms. The particles are exchanged according to a nonabelian statistics, in which the $R-$matrix satisfy a multiparametric generalization of the usual Yang$-$Baxter equations. 
  We review and present new results for a string moving on an $SU(1,1)$ group manifold. We discuss two classes of theories which use discrete representations. For these theories the representations forbidden by unitarity decouple and, in addition, one can construct modular invariant partition functions. The partion functions do, however, contain divergencies due to the time-like direction of the $SU(1,1)$ manifold. The two classes of theories have the corresponding central charges $c=9,6,5,9/2,\ldots$ and $c=9,15,21,27,\ldots$. Subtracting two from the latter series of central charges we get the Gervais-Neveu series $c-2=7,13,19,25$. This suggests a relationship between the $SU(1,1)$ string and the Liouville theory, similar to the one found in the $c=1$ string. Modular invariance is also demonstrated for the principal continous representations. Furthermore, we present new results for the Euclidean coset $SU(1,1)/U(1)$. The same two classes of theories will be possible here and will have central charges $c=8,5,4,\dots$ and $c=8,14,20,26,\ldots$, where the latter class includes the critical 2d black hole. The partition functions for the coset theory are convergent.(Talk presented by S.H. at the 16'th Johns Hopkins' Workshop, G\"oteborg, Sweden, June 8-10, 1992) 
  In this paper it is shown how the generating functional for Green's functions in relativistic quantum field theory and in thermal field theory can be evaluated in terms of a standard quantum mechanical path integral. With this calculational approach one avoids the loop-momentum integrals usually encountered in Feynman perturbation theory, although with thermal Green's functions, a discrete sum (over the winding numbers of paths with respect to the circular imaginary time) must be computed. The high-temperature expansion of this sum can be performed for all Green's functions at the same time, and is particularly simple for the static case. The procedure is illustrated by evaluating the two-point function to one-loop order in a $\phi^3_6$ model. 
   Lie groups and quantum algebras are connected through their common universal enveloping algebra. The adjoint action of Lie group on its algebra is naturally extended to related q-algebra and q-coalgebra. In such a way, quantum structure can be dealt more or less as the Lie one and we do not need to introduce the not easy to handle topological groups.    Composed system also is described by the suitably symmetrized q-coalgebra.    A physical application to the phonon, irreducible unitary representation of E_q(1,1), shows both the transformation under the group action of one phonon state and the fusion of two phonons, by means of the coproduct, in only one phonon lying on a branch of the appropriate dispersion relation. 
  Recent advances in non-critical string theory allow a unique continuation, preserving conformal invariance, of critical Polyakov string amplitudes to off-shell momenta. These continuations possess unusual, apparently stringy, characteristics, which are unlikely to be reproduced in a string field theory. Thus our results may be an indication that some fundamentally new formulation, other than string field theory, will be required to extend our understanding of critical strings beyond the Polyakov path integral. Three-point functions are explicitly calculated. The tree-level effective potential is computed for the tachyon. (This preprint includes some computations used to arrive at results mentioned in hep-th/9211016.) 
  (Lecture at the workshop "Basic Problems in String Theory", Yukawa Institute for Theoretical Physics, Kyoto, October 19-21) In this talk, we first review the possibility of matrix models toward a nonperturbative (critical) string theory. We then discuss whether the $c=1$ matrix model can describe the black hole solution of 2D critical string theory. We show that there exists a class of integral transformations which send the Virasoro condition for the tachyon field around the 2D black hole to that around the linear dilaton vacuum. In particular, we construct an explicit integral formula wihich describes a continuous deformation of the linear dilaton vacuum to the black hole background. 
  We study the relation between the $SL(2,R)/U(1)$ black hole and the $c=1$ Liouville theory. A deformation, which interpolates the BRST operators of both models, is explicitly constructed. This interpolation is isomorphic, and the physical spectrum of the black hole is equivalent to that of the $c=1$ model tensored by a topological $U(1)/U(1)$ model. Some implications of the deformation are discussed. 
  The duality symmetries of WZW and coset models are discussed. The exact underlying symmetry responsible for semiclassical duality is identified with the symmetry under affine Weyl transformations. This identification unifies the treatement of duality symmetries and shows that in the compact and unitary case they are exact symmetries of string theory to all orders in $\alpha'$ and in the string coupling constant. Non-compact WZW models and cosets are also discussed. A toy model is analyzed suggesting that duality will not generically be a symmetry. 
  Reduced phase space formulation of CGHS model of 2d dilaton gravity is studied in en extrinsic time gauge. The corresponding Hamiltonian can be promoted into a Hermitian operator acting in the physical Hilbert space, implying a unitary evolution for the system. Consequences for the black hole physics are discussed. In particular, this manifestly unitary theory rules out the Hawking scenario for the endpoint of the black hole evaporation process. 
  Various typos corrected 
  The study of string models including both unoriented closed strings and open strings presents a number of new features when compared to the standard case of models of oriented closed strings only. We review some basic features of the construction of these models, describing in particular how gauge symmetry breaking can be achieved in this case. We also review some peculiar properties of the Green-Schwarz anomaly cancellation mechanism that present themselves in lower-dimensional open-string models.(Figures not included) (Talk presented at the Tenth National General Relativity Conference, Bardonecchia, September 1992) 
  We discuss the basic features of the double scaling limit of the one dimensional matrix model and its interpretation as a two dimensional string theory. Using the collective field theory formulation of the model we show how the fluctuations of the collective field can be interpreted as the massless "tachyon" of the two dimensional string in a linear dilaton background. We outline the basic physical properties of the theory and discuss the nature of the S-matrix. Finally we show that the theory admits of another interpretation in which a certain integral transform of the collective field behaves as the massless "tachyon" in the two dimensional string with a blackhole background. We show that both the classical background and the fluctuations are non-singular at the black hole singularity. 
  The renormalization group equations for a class of non--relativistic quantum $\sigma$--models targeted on flag manifolds are given. These models emerge in a continuum limit of generalized Heisenberg antiferromagnets. The case of the ${SU(3)\over U(1)\times U(1)}$ manifold is studied in greater detail. We show that at zero temperature there is a fixed point of the RG transformations in $(2+\varepsilon )$--dimensions where the theory becomes relativistic. We study the linearized RG transformations in the vicinity of this fixed point and show that half of the couplings are irrelevant. We also show that at this fixed point there is an enlargement of the global isometries of the target manifold. We construct a discrete non--abelian enlargement of this kind. 
  We consider the q-deformed Knizhnik-Zamolodchikov equation for the two point function of q-deformed vertex operators of $U_q(sl_2^)$. We give explicitly the fundamental solutions, the connection matrices and the exchange relations for the q-vertex operators of spin 1/2 and $j \in {1\over 2}{\bf Z}_{\geq 0}$. Consequently, we confirm that the connection matrices are equivalent to the elliptic Boltzman weights of IRF type obtained by the fusion procedure from ABF models. 
  There is a constrained-WZNW--Toda theory for any simple Lie algebra equipped with an integral gradation. It is explained how the different approaches to these dynamical systems are related by gauge transformations. Combining Gauss decompositions in relevent gauges, we unify formulae already derived, and explictly determine the holomorphic expansion of the conformally reduced WZNW solutions - whose restriction gives the solutions of the Toda equations. The same takes place also for semi-integral gradations. Most of our conclusions are also applicable to the affine Toda theories. 
  A new method for constructing flows between distinct Landau-Ginzburg theories at fixed central charge is presented. The essential ingredient of the construction is an enlarged moduli space obtained by adding theories with zero central charge. The flows involve only marginal directions hence they can be applied to transitions between string vacua, in particular to the construction of mirror pairs of string ground states described by RG fixed points of N=2 supersymmetric Landau--Ginzburg theories. In contrast to previous methods this new construction of mirror theories does not depend on particular symmetries of the original theory. 
  It is argued that gravitational descendants in Landau-Ginsburg theory coupled to topological gravity can be constructed from the matter fields only. For example, $\sigma_1 (\Phi) = V'\int \Phi$. This descendants turn out to be connected with K.Saito higher residue pairing. Naive computaitions are corrected with the help of the contact term, the form of this term is found. k-point correlation functions on a sphere are calculated. 
  It is pointed out that the apparent presence of the fusing rule for purely elastic scattering theories in theories with dynamical Yangian invariance implies beautiful and hitherto unseen structure in the finite-dimensional representation theory of Yangians. 
  Fixed angle scattering at high energy in a string theory with boundaries satisfying Dirichlet conditions (Dirichlet strings) in $D=4$ is shown to have logarithmic dependence on energy, in addition to the power-like behavior known before. High temperature free energy also depends logarithmically on temperature. Such a result could provide a matching mechanism between strings at long distance and asymptotic freedom at short distance, which is necessary for the reformulation of large-$N$ QCD as a string theory. 
  We consider GLq(N)-covariant quantum algebras with generators satisfying quadratic polynomial relations. We show that, up to some inessential arbitrariness, there are only two kinds of such quantum algebras, namely, the algebras with q-deformed commutation and q-deformed anticommutation relations. The connection with the bicovariant differential calculus on the linear quantum groups is disscussed. 
  We review the construction of Drinfeld-Sokolov type hierarchies and classical W-algebras in a Hamiltonian symmetry reduction framework. We describe the list of graded regular elements in the Heisenberg subalgebras of the nontwisted loop algebra based on $gl_n$ and deal with the associated hierarchies. We exhibit an $sl_2$ embedding for each reduction of a Kac-Moody Poisson bracket algebra to a W-algebra of gauge invariant differential polynomials. 
  We consider a renormalizable two-dimensional model of dilaton gravity coupled to a set of conformal fields as a toy model for quantum cosmology. We discuss the cosmological solutions of the model and study the effect of including the backreaction due to quantum corrections. As a result, when the matter density is below some threshold new singularities form in a weak coupling region, which suggests that they will not be removed in the full quantum theory. We also solve the Wheeler-DeWitt equation. Depending on the quantum state of the Universe, the singularities may appear in a quantum region where the wave function is not oscillatory, i.e., when there is not a well defined notion of classical spacetime. 
  A generalized version is proposed for the field-antifield formalism. The antibracket operation is defined in arbitrary field-antifield coordinates. The antisymplectic definitions are given for first- and second-class constraints. In the case of second-class constraints the Dirac's antibracket operation is defined. The quantum master equation as well as the hypergauge fixing procedure are formulated in a coordinate-invariant way. The general hypergauge functions are shown to be antisymplectic first-class constraints whose Jacobian matrix determinant is constant on the constraint surface. The BRST-type generalized transformations are defined and the functional integral is shown to be independent of the hypergauge variations admitted. In the case of reduced phase space the Dirac's antibrackets are used instead of the ordinary ones. 
  We find a relation between the spectrum of solitons of massive $N=2$ quantum field theories in $d=2$ and the scaling dimensions of chiral fields at the conformal point. The condition that the scaling dimensions be real imposes restrictions on the soliton numbers and leads to a classification program for symmetric $N=2$ conformal theories and their massive deformations in terms of a suitable generalization of Dynkin diagrams (which coincides with the A--D--E Dynkin diagrams for minimal models). The Landau-Ginzburg theories are a proper subset of this classification. In the particular case of LG theories we relate the soliton numbers with intersection of vanishing cycles of the corresponding singularity; the relation between soliton numbers and the scaling dimensions in this particular case is a well known application of Picard-Lefschetz theory. 
  We propose one possible generalization of the KP hierarchy, which possesses multi bi--hamiltonian structures, and can be viewed as several KP hierarchies coupled together. 
  The mathematical formalism commonly used in treating nonlocal highly singular interactions is revised. The notion of support cone is introduced which replaces that of support for nonlocalizable distributions. Such support cones are proven to exist for distributions defined on the Gelfand-Shilov spaces $S^\beta$, where $0<\beta <1$ . This result leads to a refinement of previous generalizations of the local commutativity condition to nonlocal quantum fields. For string propagators, a new derivation of a representation similar to that of K\"{a}llen-Lehmann is proposed. It is applicable to any initial and final string configurations and manifests exponential growth of spectral densities intrinsic in nonlocalizable theories. 
  The connection between Rational Conformal Field Theory (RCFT), $N=2$ massive supersymmetric field theory, and solvable Interaction Round the Face (IRF) lattice models is explored here. Specifically, one identifies the fusion rings with the chiral rings. The theories so obtained are conjectured, and largely shown, to be integrable. A variety of examples and the structure of the metric in moduli space are given. The kink scattering theory is given by the Boltzmann weights of an IRF model, which is built entirely in terms of the conformal data of the original RCFT. This procedure produces all solvable IRF models in terms of projection operators of the RCFT. The soliton structure and their scattering amplitudes are described. A host of new rational conformal field theories are constructed generalizing most, if not all, of the known ones. 
  We compare results from $\delta$--expansion, in simple theories, with self--consistent calculations as well as calculations involving the principle of minimal sensitivity. We show that the latter methods give relatively more accurate results order by order. 
  We present fermionic quasi-particle sum representations for some of the characters (or branching functions) of ~${(G^{(1)})_1 \times (G^{(1)})_1 \o (G^{(1)})_2}$ ~for all simply-laced Lie algebras $G$. For given $G$ the characters are written as the partition function of a set of rank~$G$ types of massless quasi-particles in certain charge sectors, with nontrivial lower bounds on the one-particle momenta. We discuss the non-uniqueness of the representations for the identity character of the critical Ising model, which arises in both the $A_1$ and $E_8$ cases. 
  The N=1,2 supergravities with non-zero cosmological constants are investigated in the Ashtekar formalism. We solve the constraints of the N=1,2 supergravities semi-classically. The resulting WKB wave functions are expressed by exponentials of supersymmetric-extended SL(2,C) Chern-Simons functional. 
  The non-perturbative canonical quantization of the N=1 supergravity with the non-zero cosmological constant is studied using the Ashtekar formalism. A semi-classical wave function is obtained and it has the form of the exponential of the N=1 supersymmetric extension of the Chern-Simons functional. The N=1 supergravity in the Robertson-Walker universe is also examined and some analytic solutions are obtained. (We omit all graphs in the following LaTex file.) 
  The algebra of monodromy matrices for sl(n) trigonometric R-matrices is studied. It is shown that a generic finite-dimensional polynomial irreducible representation of this algebra is equivalent to a tensor product of L-operators. Cocommutativity of representations is discussed. A special class of representations - factorizable representations is introduced and intertwiners for cocommuting factorizable representations are written through the Boltzmann weights of the sl(n) chiral Potts model.   Let us consider an algebra generated by noncommutative entries of the matrix $T(u)$ satisfying the famous bilinear relation originated from the quantum inverse scattering method: $R(\la-\mu)T(\la)T(\mu)=T(\mu)T(\la) R(\la-\mu)$ where $R(\la)$ is R-matrix. For historical reasons this algebra is called the algebra of monodromy matrices. If $\g$ is a simple finite-dimensional Lie algebra and $R(\la)$ is $\g$-invariant R-matrix the algebra of monodromy matrices after a proper specialization gives the Yangian $Y(\g)$ introduced by Drinfeld. If $R(\la)$ is corresponding trigonometric R-matrix this algebra is closely connected  with $U_q(\g)$ and $U_q(\hat\g)$ at zero level. If $R(\la)$ is $sl(2)$ elliptic R-matrix the algebra of monodromy matrices gives rise to Sklyanin's algebra.   In this paper we shall study algebras of monodromy matrices for $sl(n)$ trigonometric R-matrices at roots of 1. Finite-dimensional cyclic irreducible polynomial representations and their intertwiners are discussed. 
  The non-relativistic quantum field theoretic lagrangian which describes an anyon system in the presence of an electromagnetic field is identified. A non-minimal magnetic coupling to the Chern-Simons statistical field as well as to the electromagnetic field together with a direct coupling between between both fields are the non-trivial ingredients of the lagrangian obtained from the non-relativistic limit of the fermionic relativistic formulation. The results, an electromagnetic gyromagnetic ratio 2 for any spin together with a non-trivial dynamical spin dependent contact interaction between anyons as well as the spin dependence of the electromagnetic effective action, agree with the quantum mechanical formulation. 
  The effective action for chiral $W_3$ gravity is studied. It is shown that the computation of the effective action can be reduced to that of a $SL(3,\re)$ Wess-Zumino-Witten theory. If one assumes that the effective action for the Wess-Zumino-Witten model is identical to the WZW action up to multiplicative renormalizations, then the effective action for $W_3$ gravity is, to all orders, given by a constrained WZW model. The multiplicative renormalization constants of the WZW model are discussed and it is analyzed which particular values of these constants are consistent with previous one-loop calculations, and which reproduce the KPZ formulas for gravity and their generalizations for $W_3$ gravity. 
  Starting from the covariant action for $W_3$ gravity, we discuss the BRST quantization of $W_3$ gravity. Taking the chiral gauge the BRST charge has a natural interpretation in terms of the quantum Drinfeld--Sokolov reduction for $Sl(3,\re)$. Nilpotency of this charge leads to the KPZ formula for $W_3$. In the conformal gauge, where the covariant action reduces to a Toda action, the BRST charge is equivalent to the one recently constructed by Bershadsky et al. 
  In this paper we study the finitely generated algebras underlying $W$ algebras. These so called 'finite $W$ algebras' are constructed as Poisson reductions of Kirillov Poisson structures on simple Lie algebras. The inequivalent reductions are labeled by the inequivalent embeddings of $sl_2$ into the simple Lie algebra in question. For arbitrary embeddings a coordinate free formula for the reduced Poisson structure is derived. We also prove that any finite $W$ algebra can be embedded into the Kirillov Poisson algebra of a (semi)simple Lie algebra (generalized Miura map). Furthermore it is shown that generalized finite Toda systems are reductions of a system describing a free particle moving on a group manifold and that they have finite $W$ symmetry. In the second part we BRST quantize the finite $W$ algebras. The BRST cohomology is calculated using a spectral sequence (which is different from the one used by Feigin and Frenkel). This allows us to quantize all finite $W$ algebras in one stroke. Explicit results for $sl_3$ and $sl_4$ are given. In the last part of the paper we study the representation theory of finite $W$ algebras. It is shown, using a quantum version of the generalized Miura transformation, that the representations of finite $W$ algebras can be constructed from the representations of a certain Lie subalgebra of the original simple Lie algebra. As a byproduct of this we are able to construct the Fock realizations of arbitrary finite $W$ algebras. 
  A fundamental task for the heterotic superstring theory is the determination of the effective action describing the physics of massless string excitations at low energies. This is necessary for the phenomenological applications of string theory, in particular for the unification of gauge interactions and for the gaugino condensation mechanism of supersymmetry breaking. In this talk, I report on the recent progress in computing the effective supergravity action from superstring scattering amplitudes, at the tree level and beyond. I discuss moduli-dependent string loop corrections to gauge and Yukawa couplings. Talk presented at the 7th Meeting of the American Physical Society Division of Particles and Fields, Fermilab, November 10-14, 1992 
  Four lectures given at Nankai Institute of Mathematics, Tianjin, China, 5--13 April 1991 present an elementary introduction into the quantum integrable models aimed for mathematical physicists and mathematicians. The stress is made on the algebraic aspects of the theory and the problem of determining the spectrum of quantum integrals of motion. The XXX magnetic chain is used as the basic example. Two lectures are devoted to a detailed exposition of the Functional Bethe Ansatz --- a new technique alternative to the Algebraic Bethe Ansatz --- and its relation to the separation of variable method. A possibility to extend FBA to the $SL(3)$ is discussed. 
  We study the following problem: can a classical $sl_n$ Toda field theory be represented by means of free bosonic oscillators through a Drinfeld--Sokolov construction? We answer affirmatively in the case of a cylindrical space--time and for real hyperbolic solutions of the Toda field equations. We establish in fact a one--to--one correspondence between such solutions and the space of free left and right bosonic oscillators with coincident zero modes. We discuss the same problem for real singular solutions with non hyperbolic monodromy. 
  The geometric structure of theories with gauge fields of spins two and higher should involve a higher spin generalisation of Riemannian geometry. Such geometries are discussed and the case of $W_\infty$-gravity is analysed in detail. While the gauge group for gravity in $d$ dimensions is the diffeomorphism group of the space-time, the gauge group for a certain $W$-gravity theory (which is $W_\infty$-gravity in the case $d=2$) is the group of symplectic diffeomorphisms of the cotangent bundle of the space-time. Gauge transformations for $W$-gravity gauge fields are given by requiring the invariance of a generalised line element. Densities exist and can be constructed from the line element (generalising $\sqrt { \det g_{\mu \nu}}$) only if $d=1$ or $d=2$, so that only for $d=1,2$ can actions be constructed. These two cases and the corresponding $W$-gravity actions are considered in detail. In $d=2$, the gauge group is effectively only a subgroup of the symplectic diffeomorphism group. Some of the constraints that arise for $d=2$ are similar to equations arising in the study of self-dual four-dimensional geometries and can be analysed using twistor methods, allowing contact to be made with other formulations of $W$-gravity. While the twistor transform for self-dual spaces with one Killing vector reduces to a Legendre transform, that for two Killing vectors gives a generalisation of the Legendre transform. 
  Boundary conditions compatible with integrability are obtained for two dimensional models by solving the factorizability equations for the reflection matrices $K^{\pm}(\theta)$. For the six vertex model the general solution depending on four arbitrary parameters is found. For the $A_{n-1}$ models all diagonal solutions are found. The associated integrable magnetic Hamiltonians are explicitly derived. 
  Systems of integral equations are proposed which generalise those previously encountered in connection with the so-called staircase models. Under the assumption that these equations describe the finite-size effects of relativistic field theories via the Thermodynamic Bethe Ansatz, analytical and numerical evidence is given for the existence of a variety of new roaming renormalisation group trajectories. For each positive integer $k$ and $s=0,\dots, k-1$, there is a one-parameter family of trajectories, passing close by the coset conformal field theories $G^{(k)}\times G^{(nk+s)}/G^{((n+1)k+s)}$ before finally flowing to a massive theory for $s=0$, or to another coset model for $s \neq 0$. 
  This talk contains a summary of our work on dynamical CPT invariance and spontaneous CPT violation in string theories, including the possibility that stringy CPT violation could occur at levels detectable in the next generation of experiments. In particular, we present here an estimate for values of parameters for CPT violation in the kaon system. 
  We treat the horizons of charged, dilaton black extended objects as quantum mechanical objects. We show that the S matrix for such an object can be written in terms of a p-brane-like action. The requirements of unitarity of the S matrix and positivity of the p-brane tension equivalent severely restrict the number of space-time dimensions and the allowed values of the dilaton parameter a. Generally, black objects transform at the extremal limit into p-branes. 
  The dynamics of a class of fivebrane string solitons is considered in the moduli space approximation. The metric on moduli space is found to be flat. This implies that at low energies the solitons do not interact, and their scattering is trivial. The range of validity of the approximation is also briefly discussed. 
  We discuss the toplogical sigma model on an orbifold target space. We describe the moduli space of classical minima for computing correlation functions involving twisted operators, and show, through a detailed computation of an orbifold of ${\bf CP}^1$ by the dihedral group $D_{4},$ how to compute the complete ring of observables. Through this procedure, we compute all the rings from dihedral ${\bf CP}^1$ orbifolds; we note a similarity with rings derived from perturbed $D-$series superpotentials of the $A-D-E$ classification of $N = 2$ minimal models. We then consider ${\bf CP}^2/D_4,$ and show how the techniques of topological-anti-topological fusion might be used to compute twist field correlation functions for nonabelian orbifolds. 
  For the whole set of dilogarithm identities found recently using the thermodynamic Bethe-Ansatz for the $ADET$ series of purely elastic scattering theories we give partition identities which involve characters of those conformal field theories which correspond to the UV-limits of the scattering theories. These partition identities in turn allow to derive the dilogarithm identities using modular invariance and a saddle point approximation. We conjecture on possible generalizations of this correspondance, namely, a lift from dilogarithm to partition identities. 
  In this paper we continue the programme of topologically twisting N=2 theories in D=4, focusing on the coupling of vector multiplets to N=2 supergravity. We show that in the minimal case, namely when the special geometry prepotential F(X) is a quadratic polynomial, the theory has a so far unknown on shell U(1) symmetry, that we name R-duality. R-duality is a generalization of the chiral-dual on shell symmetry of N=2 pure supergravity and of the R-symmetry of N=2 super Yang-Mills theory. Thanks to this, the theory can be topologically twisted and topologically shifted, precisely as pure N=2 supergravity, to yield a natural coupling of topological gravity to topological Yang-Mills theory. The gauge-fixing condition that emerges from the twisting is the self- duality condition on the gauge field-stength and on the spin connection. Hence our theory reduces to intersection theory in the moduli-space of gauge instantons living in gravitational instanton backgrounds. We remark that the topological Yang-Mills theory we obtain by taking the flat space limit of our gravity coupled Lagrangian is different from the Donaldson theory constructed by Witten. Whether this difference is substantial and what its geometrical implications may be is yet to be seen. We also discuss the topological twist of the hypermultiplets leading to topological quaternionic $\sigma$-models. The instantons of these models, named by us hyperinstantons, correspond to a notion of triholomorphic mappings discussed in the paper. 
  Character sumrules associated with the realization of the $N=4$ superconformal algebra $\At$ on manifolds corresponding to the group cosets $SU(3)_{\ktp }/U(1)$ are derived and developed as an important tool in obtaining the modular properties of $\At$ characters as well as information on certain extensions of that algebra. Their structure strongly suggests the existence of rational conformal field theories with central charges in the range $1 \le c\le 4$. The corresponding characters appear in the massive sector of the sumrules and are completely specified in terms of the characters for the parafermionic theory $SU(3)/(SU(2)\times U(1))$ and in terms of the branching functions of massless $\At$ characters into $SU(2)_{\ktp }\times SU(2)_1$ characters. 
  By consireding representation theory for non-associative algebras we construct the fundamental and adjoint representations of the octonion algebra. We then show how these representations by associative matrices allow a consistent octonionic gauge theory to be realised. We find that non-associativity implies the existence of new terms in the transformation laws of fields and the kinetic term of an octonionic Lagrangian. 
  We study chiral symmetry breaking in quenched QED$_4$, using a vertex Ansatz recently proposed by Curtis and Pennington. Bifurcation analysis is employed to establish the existence of a critical coupling and to estimate its value. The main results are in qualitative agreement with the ladder approximation, the numerical changes being minor. 
  The general formula for R-matrices of slq(2,C) for the highest weight repre- sentations both for general q and for q being a root of unity by generalizing G.Gomez's and G.Sierra's one for semiperiodic representations of slq(2,C) at roots of unity is presented. 
  There are two fundamental problems studied by the theory of hamiltonian integrable systems: integration of equations of motion, and construction of action-angle variables. The third problem, however, should be added to the list: separation of variables. Though much simpler than two others, it has important relations to the quantum integrability. Separation of variables is constructed for the $SL(3)$ magnetic chain --- an example of integrable model associated to a nonhyperelliptic algebraic curve. 
  We argue that the field theory that descibes randomly branched polymers is not generally conformally invariant in two dimensions at its critical point. In particular, we show (i) that the most natural formulation of conformal invariance for randomly branched polymers leads to inconsistencies; (ii) that the free field theory obtained by setting the potential equal to zero in the branched polymer field theory is not even classically conformally invariant; and (iii) that numerical enumerations of the exponent $\theta (\alpha )$, defined by $T_N(\alpha )\sim \lambda^NN^{-\theta (\alpha ) +1}$, where $T_N(\alpha )$ is number of distinct configuratations of a branched polymer rooted near the apex of a cone with apex angel $\alpha$, indicate that $\theta (\alpha )$ is not linear in $1/\alpha$ contrary to what conformal invariance leads one to expect. 
  I construct classical superextensions of the Virasoro algebra by employing the Ward identities of a linearly realized subalgebra. For the $N=4$ superconformal algebra, this subalgebra is generated by the $N=2$ $U(1)$ supercurrent and a spin~0 $N=2$ superfield. I show that this structure can be extended to an $N=4$ super $W_3$ algebra, and give the complete form of this algebra. 
  We present the (1+1)-dimensional description of the algebraically special class of space-times of 4-dimensions. It is described by the (1+1)-dimensional Yang-Mills action interacting with matter fields, with diffeomorphisms of 2-surface as the gauge symmetry. Parts of the constraints are identified as the gauge fixing condition. We also show that the representations of $w_{\infty}$-gravity appear naturally as special cases of this description, and discuss the geometry of $w_{\infty}$-gravity in term of the fibre bundle. 
  Many extended conformal algebras with one generator in addition to the Virasoro field as well as Casimir algebras have non-trivial outer automorphisms which enables one to impose `twisted' boundary conditions on the chiral fields. We study their effect on the highest weight representations. We give formulae for the enlarged rational conformal field theories in both series of W-algebras with two generators and conjecture a general formula for the additional models in the minimal series of Casimir algebras. A third series of W-algebras with two generators which includes the spin three algebra at $c=-2$ also has finitely many additional fields in the twisted sector although the model itself is apparently not rational. The additional fields in the twisted sector have applications in statistical mechanics as we demonstrate for $Z_n$-quantum spin chains with a particular type of boundary conditions. 
  The technique of $Q$-polinomials are used to derive the $w$- constraints in the two-matrix and Kontsevich-like model at finite $N$. These constraints are closed and form Lie algebra. They are associated with the matrices, $\lambda ^n{\partial}_\lambda^m$ with $n,m\geq 0$. In the case of two-matrix model they can be reduced to the $W$-constraints of \cite{8}. For the case of Kontsevich-like model and two-matrix model with the finite polinomial potential, the number of constraints are limited by the power of the finite matrix potential i.e. the spin of $w$-s coincide with that power. This statement is the natural consequence of the form of constraints. 
  Recent interest in large N matrix models in the double scaling limit raised new interest also in O(N) vector models. The limit $N \rightarrow \infty$, correlated with the limit $g \rightarrow g_c$, results in an expansion in terms of filamentary surfaces and explicit calculations can be carried out also in dimensions $d\geq 2$. It is shown here that the absence of physical massless bound states in two dimensions sets strong constraints on this limit. 
  We review some aspects of the quantum Yangians as symmetry algebras of two-dimensional quantum field theories. The plan of these notes is the following: 1 - The classical Heisenberg model: Non-Abelian symmetries;   The generators of the symmetries and the semi-classical Yangians;   An alternative presentation of the semi-classical Yangians;   Digression on Poisson-Lie groups. 2 - The quantum Heisenberg chain:   Non-Abelian symmetries and the quantum Yangians;   The transfer matrix and an alternative presentation of the Yangians;   Digression on the double Yangians. Talk given at the "Integrable Quantum Field Theories" conference held at Come, Italy , September 13-19, 1992. 
  We analyze the algebraic structure of $\phi_{1,2}$ perturbed minimal models relating them to graph-state models with an underlying Birman-Wenzl-Murakami algebra. Using this approach one can clarify some physical properties and reformulate the bootstrap equations. These are used to calculate the $S$-matrix elements of higher kinks, and to determine the breather spectrum of the $\phi_{1,2}$ perturbations of the unitary minimal models $\M_{r,r+1}$. 
  We study the quantum theory of 1+1 dimensional dilaton gravity, which is an interesting toy model of the black hole dynamics. The functional measures are explicitly evaluated and the physical state conditions corresponding to the Hamiltonian and the momentum constraints are derived. It is pointed out that the constraints form the Virasoro algebra without central charge. In ADM formalism the measures are very ambiguous, but in our formalism they are explicitly defined. Then the new features which are not seen in ADM formalism come out. A singularity appears at $\df^2 =\kappa (>0) $, where $\kappa =(N-51/2)/12 $ and $ N$ is the number of matter fields. Behind the singularity the quantum mechanical region $\kappa > \df^2 >0 $ extends, where the sign of the kinetic term in the Hamiltonian constraint changes. If $\kappa <0 $, the singularity disappears. We discuss the quantum dynamics of black hole and then give a suggestion for the resolution of the information loss paradox. We also argue the quantization of the spherically symmetric gravitational system in 3+1 dimensions. In appendix the differences between the other quantum dilaton gravities and ours are clarified and our status is stressed. 
  The discovery of black-hole evaporation represented in many respects a revolutionary event in scientific world; as such, in giving answers to open questions, it gave rise to new problems part of which are still not resolved. Here we want to make a brief review of such problems and examine some possible solutions. Invited Talk at the "Workshop on String Theory, Quantum Gravity and the Unification of the Fundamental Interactions" Rome, September 21-26 
  We prove new identities between the values of Rogers dilogarithm function and describe a connection between these identities and spectra in conformal field theory. 
  We reconsider the conjecture by Gepner that the fusion ring of a rational conformal field theory is isomorphic to a ring of polynomials in $n$ variables quotiented by an ideal of constraints that derive from a potential. We show that in a variety of cases, this is indeed true with {\it one-variable} polynomials. 
  We investigate Weyl anomalies on a curved world sheet to second order in a weak field expansion. Using a local version of the exact renormalization group equations, we obtain nonperturbative results for the tachyon/graviton/dilaton system. We discuss the elimination of redundant operators, which play a crucial role for the emergence of target space covariance. Performing the operator product expansion on a curved world sheet allows us to obtain the nonperturbative contributions to the dilaton $\beta$ function. We find the $\beta$ functions, after suitable field redefinitions, to be related to a target space effective action through a $\kappa$ function involving derivatives. Also we can establish a nonperturbative Curci-Paffuti relation including the tachyon $\beta$ function. 
  This is a brief and subjective description of ideas which formed our present field-theoretic understanding of fundamental physics. 
  Scaling level-spacing distribution functions in the ``bulk of the spectrum'' in random matrix models of $N\times N$ hermitian matrices and then going to the limit $N\to\infty$, leads to the Fredholm determinant of the sine kernel $\sin\pi(x-y)/\pi (x-y)$. Similarly a scaling limit at the ``edge of the spectrum'' leads to the Airy kernel $[{\rm Ai}(x) {\rm Ai}'(y) -{\rm Ai}'(x) {\rm Ai}(y)]/(x-y)$. In this paper we derive analogues for this Airy kernel of the following properties of the sine kernel: the completely integrable system of P.D.E.'s found by Jimbo, Miwa, M{\^o}ri and Sato; the expression, in the case of a single interval, of the Fredholm determinant in terms of a Painlev{\'e} transcendent; the existence of a commuting differential operator; and the fact that this operator can be used in the derivation of asymptotics, for general $n$, of the probability that an interval contains precisely $n$ eigenvalues. 
  The main aim of this work is to present the interpretation of the Ising type models as a kind of field theory in the framework of noncommutative geometry. We present the method and construct sample models of field theory on discrete spaces using the introduced tools of discrete geometry. We write the action for few models, then we compare them with various models of statistical physics. We construct also the gauge theory with a discrete gauge group. 
  Quantization of the pure $1+1$ dimensional dilaton gravity is examined in the light-cone gauge. It is found that the total action including ghosts generates a $c=0$ free conformal field theory without modification of the classical action, which is required in the conformal gauge. We also study semiclassical equations of the dilaton gravity coupled to $N$ scalar fields. It is shown that the black hole singularity is not removed even for $N<24$ in the light-cone gauge. This indicates that the semiclassical analysis breaks down for small $N$. 
  It is shown that the BRST charge $Q$ for any gauge model with a Lie algebra symmetry may be decomposed as $$Q=\del+\del^{\dag}, \del^2=\del^{\dag 2}=0, [\del, \del^{\dag}]_+=0$$ provided dynamical Lagrange multipliers are used but without introducing other matter variables in $\del$ than the gauge generators in $Q$. Furthermore, $\del$ is shown to have the form $\del=c^{\dag a}\phi_a$ (or $\phi'_ac^{\dag a}$) where $c^a$ are anticommuting expressions in the ghosts and Lagrange multipliers, and where the non-hermitian operators $\phi_a$ satisfy the same Lie algebra as the original gauge generators. By means of a bigrading the BRST condition reduces to $\del|ph\hb=\del^{\dag}|ph\hb=0$ which is naturally solved by $c^a|ph\hb=\phi_a|ph\hb=0$ (or $c^{\dag a}|ph\hb={\phi'_a}^{\dag}|ph\hb=0$). The general solutions are shown to have a very simple form. 
  We discuss various aspects of the calculation of correlation functions in conformal theories coupled to quantized 2-dimensional gravity. The main emphasis lies on the construction of a continuation in the number of insertions of the cosmological constant operator for $arbitrary$ dimension of the target space. Following closely our paper \cite{DO2} we add a more extended introduction and discussion of the peculiarities in 1-dimensional target space as well as some further remarks about the 4-point function. 
  The Universal Field Equations, recently constructed as examples of higher dimensional dynamical systems which admit an infinity of inequivalent Lagrangians are shown to be linearised by a Legendre transformation. This establishes the conjecture that these equations describe integrable systems. While this construction is implicit in general, there exists a large class of solutions for which an explicit form may be written. 
  These are the lecture notes for the Les Houches Summer School on Quantum Gravity held in July 1992. The notes present some general critical assessment of other (non-string) approaches to quantum gravity, and a selected set of topics concerning what we have learned so far about the subject from string theory. Since these lectures are long (133 A4 pages), we include in this abstract the table of contents, which should help the user of the bulletin board in deciding whether to latex and print the full file.   1-FIELD THEORETICAL APPROACH TO QUANTUM GRAVITY: Linearized gravity; Supergravity; Kaluza-Klein theories; Quantum field theory and classical gravity; Euclidean approach to Quantum Gravity; Canonical quantization of gravity; Gravitational Instantons.   2-CONSISTENCY CONDITIONS: ANOMALIES: Generalities about anomalies; Spinors in 2n dimensions; When can we expect to find anomalies?; The Atiyah-Singer Index Theorem and the computation of anomalies; Examples: Green-Schwarz cancellation mechanism and Witten's SU(2) global anomaly.   3-STRING THEORY I. BOSONIC STRING: Bosonic string; Conformal Field Theory; Quantization of the bosonic string; Interaction in string theory and the characterization of the moduli space; Bosonic strings with background fields. Stringy corrections to Einstein equations; Toroidal compactifications. $R$-duality; Operator formalism   4-STRING THEORY II. FERMIONIC STRINGS: Fermionic String; Heterotic String; Strings at finite temperature; Is string theory finite?   5-OTHER DEVELOPMENTS AND CONCLUSIONS: String ``Phenomenology''; Black Holes and Related Subjects 
  The reflection equations (RE) are a consistent extension of the Yang-Baxter equations (YBE) with an addition of one element, the so-called reflection matrix or $K$-matrix. For example, they describe the conditions for factorizable scattering on a half line just like the YBE give the conditions for factorizable scattering on an entire line. The YBE were generalized to define quadratic algebras, \lq Yang-Baxter algebras\rq\ (YBA), which were used intensively for the discussion of quantum groups. Similarly, the RE define quadratic algebras, \lq the reflection equation algebras\rq\ (REA), which enjoy various remarkable properties both new and inherited from the YBA. Here we focus on the various properties of the REA, in particular, the quantum group comodule properties, generation of a series of new solutions by composing known solutions, the extended REA and the central elements, etc. 
  The spin-projector operators for symmetric rank-2 tensors are reassessed in connection with the issue of topologically massive gravity. The original proposal by Barnes and Rivers is generalised to account for D-dimensional Einstein gravity and 3-dimensional Chern-Simons massive gravitation. 
  We study spontaneous symmetry breaking in phi^4_(1+1) using the light-front formulation of the field theory. Since the physical vacuum is always the same as the perturbative vacuum in light-front field theory the fields must develop a vacuum expectation value through the zero-mode components of the field. We solve the nonlinear operator equation for the zero-mode in the one-mode approximation. We find that spontaneous symmetry breaking occurs at lambda_critical = 4 pi(3+sqrt 3), which is consistent with the value lambda_critical = 54.27 obtained in the equal time theory. We calculate the value of the vacuum expectation value as a function of the coupling constant in the broken phase both numerically and analytically using the delta expansion. We find two equivalent broken phases. Finally we show that the energy levels of the system have the expected behavior within the broken phase. 
  We made a careful study of Polyakov's Diofantian equations for 2D turbulence and found several additional CFTs which meet his criterion. This fact implies that we need further conditions for CFT in order to determine the exponent of the energy spectrum function. 
  We examine some properties of the filled Wilson loop observables in the Kazakov-Migdal model of induced QCD. We show that they have a natural interpretation in a modification of the original model in which the $Z_N$ gauge symmetry is broken explicitly by a Wilson kinetic term for the gauge fields. We argue that there are two large N limits of this theory, one leads to ordinary Wilson lattice gauge theory coupled to a dynamical scalar field and the other leads to a version of the Kazakov-Migdal model in which the large N solution found by Migdal can still be used. We discuss the properties of the string theory which emerges. 
  Recent developments in quantum gravity suggest that wormholes may influence the observed values of the constants of nature. The Euclidean formulation of quantum gravity predicts that wormholes induce a probability distribution in the space of possible fundamental constants. This distribution may computed by evaluating the functional integral about the stationary points of the action. In particular, the effective action on a large spherical space may lead to the vanishing of the cosmological constant and possibly determine the values of other constants of nature. The ability to perform calculations involving interacting quantum fields, particularly non-Abelian models, on a four-sphere is vital if one is to investigate this possibility. In this paper we present a self-consistent formulation of field theory on a four-sphere using the angular momentum space representation of $SO(5)$. We give a review of field theory on a sphere and then show how a matrix element prescription in angular momentum space overcomes previous limitations in calculational techniques.   The standard one-loop graphs of QED are given as examples. 
  The inhomogeneous quantum groups $IGL_q(n)$ are obtained by means of a particular projection of $GL_q(n+1)$. The bicovariant differential calculus on $GL_q(n)$ is likewise projected into a consistent bicovariant calculus on $IGL_q(n)$. Applying the same method to $GL_q(n,\Cb)$ leads to a bicovariant calculus for the complex inhomogeneous quantum groups $IGL_q(n,\Cb)$. The quantum Poincare' group and its bicovariant geometry are recovered by specializing our results to $ISL_q(2,\Cb)$. 
  We present results of a high precision Monte Carlo simulation of dynamically triangulated random surfaces (up to $\approx$ 34,000 triangles) coupled to one scalar field ($c=1$). The mean square extent has been measured for different actions to test the universality of the leading term as a function of the size of the surfaces. Furthermore, the integrated 2-point correlation function for vertex operators is compared with conformal field theory and matrix model predictions. 
  A set of different conformal solutions corresponding to a constant flux of squared vorticity is considered. Requiring constant fluxes of all inviscid vorticity invariants (higher powers of the vorticity), we come to the conclusion that the general turbulence spectrum should be given by Kraichnan's expression $E(k)\propto k\sp{-3}$. This spectrum, in particular, can be obtained as a limit of some subsequences of the conformal solutions. 
  A spectrum generating algebra is constructed and used to find all the physical states of the $W_3$ string with standard ghost number. These states are shown to have positive norm and their partition function is found to involve the Ising model characters corresponding to the weights 0 and 1/16. The theory is found to be modular invariant if , in addition, one includes states that correspond to the Ising character of weight 1/2. It is shown that these additional states are indeed contained in the cohomology of $Q$. 
  We investigate the spectrum of physical states in the $W_3$ string theory, up to level 2 for a multi-scalar string, and up to level 4 for the two-scalar string. The (open) $W_3$ string has a photon as its only massless state. By using screening charges to study the null physical states in the two-scalar $W_3$ string, we are able to learn about the gauge symmetries of the states in the multi-scalar $W_3$ string. 
  We propose a model which represents a four-dimensional version of Ponzano and Regge's three-dimensional euclidean quantum gravity. In particular we show that the exponential of the euclidean Einstein-Regge action for a $4d$-discretized block is given, in the semiclassical limit, by a gaussian integral of a suitable $12j$-symbol. Possible developments of this result are discussed. 
  Polyakov recently showed how to use conformal field theory to describe two-dimensional turbulence. Here we construct an infinite hierarchy of solutions, both for the constant enstrophy flux cascade, and the constant energy flux cascade. We conclude with some speculations concerning the stability and physical meaning of these solutions. 
  Non-extreme walls (bubbles with two insides) and ultra-extreme walls (bubbles of false vacuum decay) are discussed. Their respective energy densities are higher and lower than that of the corresponding extreme (supersymmetric), planar domain wall. These singularity free space-times exhibit non-trivial causal structure analogous to certain non-extreme black holes. We focus on anti-de~Sitter--Minkowski walls and comment on Minkowski--Minkowski walls with trivial extreme limit, as well as walls adjacent to de~Sitter space-times with no extreme limit. 
  Polyakov has suggested that two dimensional turbulence might be described by a minimal model of conformal field theory. However, there are many minimal models satisfying the same physical inputs as Polyakov's solution (p,q)=(2,21). Dynamical magnetic fields and passive scalars pose different physical requirements. For large magnetic Reynolds number other minimal models arise. The simplest one, (p,q)=(2,13) makes reasonable predictions that may be tested in the astrophysical context. In particular, the equipartition theorem between magnetic and kinetic energies does not hold: the magnetic one dominates at larger distances. 
  We obtain explicit expressions for the determinants of the Laplacians on zero and one forms for an infinite class of three dimensional lens spaces $L(p,q)$. These expressions can be combined to obtain the Ray-Singer torsion of these lens spaces. As a consequence we obtain an infinite class of formulae for the Riemann zeta function $\zeta(3)$. The value of these determinants (and the torsion) grows as the size of the fundamental group of the lens space increases and this is also computed. The triviality of the torsion for just the three lens spaces $L(6,1)$, $L(10,3)$ and $L(12,5)$ is also noted. (postscript figures available as a compressed tar file) 
  We use matrix model results to investigate the Sine-Gordon model coupled to two dimensional gravity. For relevant (in the RG sense) potentials, we show that the $c=1$ string, which appears in the ultraviolet limit of this model, flows to a set of decoupled $c=0$ (pure gravity) models in the infrared. The torus partition sum, which was argued previously to count the number of string degrees of freedom and hence satisfy a new $c$ -- theorem, is shown to be a monotonically decreasing function of the scale (given by the quantum area of the world-sheet). 
  We present a picture of confinement based on representation of quarks as pointlike topological defects. The topological charge carried by quarks and confined in hadrons is explicitly constructed in terms of Yang - Mills variables. In 2+1 dimensions we are able to construct a local complex scalar field $V(x)$, in terms of whichthe topological charge is $Q=-\frac{i}{4\pi}\int d^2x \epsilon_{ij}\partial_i(V^*\partial_i V -c.c.)$. The VEV of the field $V$ in the confining phase is nonzero and the charge is the winding number corresponding to homotopy group $\pi_1(S^1)$. Qurks carry the charge $Q$ and therefore are topological solitons. The effective Lagrangian for $V$ is derived in models with adjoint and fundamental quarks. In 3+1 dimensions the explicit expression for $V$ and therefore a detailed picture is not available. However, assuming the validity of the same mechanism wepoint out several interesting qualitative consequences. We argue that in the Georgi - Glashow model or any grand unified model the photon (in the Higgs phase) should have a small nonperturbative mass and $W^\pm$ should be confined although with small string tension. 
  We study the operator product expansion of two non-interacting chiral currents in the presence of external gauge fields in four dimensional euclidean space. We obtain the operator singularity in terms of the beta function of the free energy density and a connection on the space of external gauge fields, by imposing the consistency between the variational formula, introduced previously by the author, and both the renormalization group equations and the equation of motion for the currents. As a byproduct, we derive a euclidean version of the anomalous commutator of two currents in an appendix. 
  We compute the metric on moduli space for the Dabholkar-Harvey string soliton in $D=4$ to lowest nontrivial order in the string tension. The metric is found to be flat, which implies trivial scattering of the solitons. This result is consistent with an earlier test-string calculation of the leading order dynamical force and a computation of the Veneziano amplitude for the scattering of macroscopic strings. 
  We describe a real-time classical solution of $c=1$ string field theory written in terms of the phase space density, $u(p,q,t)$, of the equivalent fermion theory. The solution corresponds to tunnelling of a single fermion above the filled fermi sea and leads to amplitudes that go as $\exp(- C/ \gst)$. We discuss how one can use this technique to describe non-perturbative effects in the Marinari-Parisi model. We also discuss implications of this type of solution for the two-dimensional black hole. 
  We consider the topological theory of Witten type for gauge differential p-forms. It is shown that some topological invariants such as linking numbers appear under quantization of this theory. The non-abelian generalization of the model is discussed. 
  In recent work, Dabholkar {\it et al.} constructed static ``cosmic string" solutions of the low-energy supergravity equations of the heterotic string, and conjectured that these solitons are actually exterior solutions for infinitely long fundamental strings. In this paper we provide compelling dynamical evidence to support this conjecture by computing the dynamical force exerted by a solitonic string on an identical test-string limit, the Veneziano amplitude for the scattering of macroscopic winding states and the metric on moduli space for the scattering of two string solitons. All three methods yield trivial scattering in the low-energy limit. 
  We present plane-wave-type solutions of the lowest order superstring effective action which have unbroken space-time supersymmetries. They describe dilaton, axion and gauge fields in a stringy generalization of the Brinkmann metric. Some conspiracy between the metric and the axion field is required. We show that there exists a special class of these solutions, for which $\alpha^\prime$ stringy corrections to the effective on-shell action, to the equations of motion (and therefore to the solutions themselves), and to the supersymmetry transformations vanish. We call these solutions supersymmetric string waves (SSW). 
  A new method of determining B\"acklund transformations for nonlinear partial differential equations of the evolution type is introduced. Using the Hilbert space approach the problem of finding B\"acklund transformations is brought down to the solution of an abstract equation in Hilbert space. 
  A three dimensional string model is analyzed in the strong coupling regime. The contribution of surfaces with different topology to the partition function is essential. A set of corresponding models is discovered. Their critical indices, which depend on two integers (m,n), are calculated analyticaly. The critical indices of the three dimensioal Ising model should belong to this set. A possible connection with the chain of three dimensional lattice Pott's models is pointed out. 
  The production of o(1/g^2) particles in a weakly-coupled theory is believed to be non-perturbatively suppressed. I comment on the prospects of (a) establishing this rigorously, and (b) estimating the effect to exponential accuracy semiclassically, by discussing two closely-related problems: the large-order behaviour of few-point Green functions, and induced excitation in quantum mechanics. Induced tunneling in the latter case is exponentially enhanced for frequencies of the order of the barrier height. 
  We argue that the collapse of a non-rotating object into a black hole has not been proved to be dynamically stable.  There are unstable modes and one should explore whether they may be excited. This paper is withdrawn for revisions and clarification 
  We present simple, analytic solutions to the Einstein-Maxwell equation, which describe an arbitrary number of charged black holes in a spacetime with positive cosmological constant $\Lambda$. In the limit $\Lambda=0$, these solutions reduce to the well known Majumdar-Papapetrou (MP) solutions. Like the MP solutions, each black hole in a $\Lambda >0$ solution has charge $Q$ equal to its mass $M$, up to a possible overall sign. Unlike the $\Lambda = 0$ limit, however, solutions with $\Lambda >0$ are highly dynamical. The black holes move with respect to one another, following natural trajectories in the background deSitter spacetime. Black holes moving apart eventually go out of causal contact. Black holes on approaching trajectories ultimately merge. To our knowledge, these solutions give the first analytic description of coalescing black holes. Likewise, the thermodynamics of the $\Lambda >0$ solutions is quite interesting. Taken individually, a $|Q|=M$ black hole is in thermal equilibrium with the background deSitter Hawking radiation. With more than one black hole, because the solutions are not static, no global equilibrium temperature can be defined. In appropriate limits, however, when the black holes are either close together or far apart, approximate equilibrium states are established. 
  We discuss basic features and new developments in recently proposed induced gauge theory solvable in any number of dimensions in the limit of infinite number of colours. Its geometrical (string) picture is clarified, using planar graph expansion of the corresponding matrix model. New analytical approach is proposed for this theory which is based on its equivalence to an effective two-matrix model. It is shown on some particular examples how the approach works.   This approach may be applicable to a wide class of matrix models with tree-like quadratic couplings of matrices.   (This talk was presented on the International Symposium on Lattice   Field Theory "LATTICE-92" in Amsterdam, the Netherlands, 15-19   September 1992) 
  We consider, for $p$ odd, a $p$--brane coupled to a $(p+1)$th rank background antisymmetric tensor field and to background Yang-Mills (YM) fields {\it via} a Wess-Zumino term. We obtain the generators of antisymmetric tensor and Yang-Mills gauge transformations acting on $p$--brane wavefunctionals (functions on `$p$-loop space'). The Yang-Mills generators do not form a closed algebra by themselves; instead, the algebra of Yang-Mills and antisymmetric tensor generators is a $U(1)$ extension of the usual algebra of Yang-Mills gauge transformations. We construct the $p$-brane's Hamiltonian and thereby find gauge-covariant functional derivatives acting on $p$--brane wavefunctionals that commute with the YM and $U(1)$ generators. 
  Using the hamiltonian framework, we analyze the Gribov problem for U(N) and SU(N) gauge theories on a cylinder (= (1+1) dimensional spacetime with compact space S^1). The space of gauge orbits is found to be an orbifold. We show by explicit construction that a proper treatment of the Gribov ambiguity leads to a highly non-trivial structure of all physical states in these quantum field theory models. The especially interesting example of massless QCD is discussed in more detail: There, some of the special static gauge transformations which are responsible for the Gribov ambiguity also lead to a spectral flow, and this implies a chiral condensate in all physical states. We also show that the latter is closely related to the Schwinger term and the chiral anomaly. 
  A boson representation of the quantum affine algebra $U_q(\widehat{\sl}_2)$ is realized based on the Wakimoto construction. We discuss relations with the other boson representations. 
  The $q$-vertex operators of Frenkel and Reshetikhin are studied by means of a $q$-deformation of the Wakimoto module for the quantum affine algebra $U_q(\widehat{\sl}_2)$ at an arbitrary level $k\ne 0,-2$. A Fock module version of the $q$-deformed primary field of spin $j$ is introduced, as well as the screening operators which (anti-)commute with the action of $U_q(\widehat{\sl}_2)$ up to a total difference of a field. A proof of the intertwining property is given for the $q$-vertex operators corresponding to the primary fields of spin $j\notin {1 \over2}\Z_{\geq0}$, which is enough to treat a general case. A sample calculation of the correlation function is also given. 
  Time-dependent domain wall solutions with infinitesimal thickness are obtained in the theory of a scalar field coupled to gravity with the dilaton, i.e. the Jordan-Brans-Dicke gravity. The value of the dilaton is determined in terms of the Brans-Dicke parameter $\omega$. In particular, the solutions exist for any $\omega>0$ and as $\omega\to\infty$ we obtain new solutions in general relativity. They have horizons whose sizes depend on $\omega$. 
  We describe a way in which spin of quarks can enter a consistent QCD string theory. We show that the spin factor of the 4d massless, spin 1/2 fermions is related to the self-intersection number of a 2d surfaces immersed in the 4d space. We argue that the latter quantity should appear in a consistent description of the QCD string. We also calculate the chiral anomaly and show that the self-intersection number corresponds to the topological charge $F{\tilde F}$ of QCD. 
  Batalin-Vilkovisky algebras are a new type of algebraic structure on graded vector spaces, which first arose in the work of Batalin and Vilkovisky on gauge fixing in quantum field theory. In this article, we show that there is a natural structure of a Batalin-Vilkovisky algebra on the cohomology of a topological field theory in two dimensions.   Lian and Zuckerman have constructed this Batalin-Vilkovisky structure, in the setting of topological chiral field theories, and shown that the structure is non-trivial in two-dimensional string theory. Our approach is to use algebraic topology, whereas their proofs have a more algebraic character. 
  Recent results on the quantum mechanics of black holes indicate that the spacetime singularity can be avoided if the space of fields is extended in its domain as well as the vanishing of beta functions required as an equation in target space. Such an approach seems to inevitably lead to the quantization in this space that is the so-called third quantization. We discuss some of the implications of third quantization for the physics of blackholes in two dimensions. By discretizing the transverse dimensions describing the horizon it may also be possible to describe regulated four dimensional gravity in this manner and perhaps shed some light on the meaning of black hole entropy. 
  A general method of computing string corrections to the K\"ahler metric and Yukawa couplings is developed at the one-loop level for a general compactification of the heterotic superstring theory. It also provides a direct determination of the so-called Green-Schwarz term. The matter metric has an infrared divergent part which reproduces the field-theoretical anomalous dimensions, and a moduli-dependent part which gives rise to threshold corrections in the physical Yukawa couplings. Explicit expressions are derived for symmetric orbifold compactifications. 
  We present the quantization of the Liouville model defined in light-cone coordinates in (1,1) signature space. We take advantage of the representation of the Liouville field by the free field of the Backl\"{u}nd transformation and adapt the approch by Braaten, Curtright and Thorn.   Quantum operators of the Liouville field $\partial_{+}\phi$, $\partial_{-}\phi$, $e^{g\phi}$, $e^{2g\phi}$ are constructed consistently in terms of the free field. The Liouville model field theory space is found to be restricted to the sector with field momentum $P_{+}=-P_{-}$, $P_{+}> 0$ , which is a closed subspace for the Liouville theory operator algebra. 
  We investigate the $q$-deformation of the BRST algebra, the algebra of the ghost, matter and gauge fields on one spacetime point using the result of the bicovariant differential calculus. There are two nilpotent operations in the algebra, the BRST transformation $\brs$ and the derivative $d$. We show that one can define the covariant commutation relations among the fields and their derivatives consistently with these two operation as well as the $*$-operation, the antimultiplicative inner involution. 
  We point out the existence of an arithmetical symmetry for the commutant of the modular matrices S and T. This symmetry holds for all affine simple Lie algebras at all levels and implies the equality of certain coefficients in any modular invariant. Particularizing to SU(3)_k, we classify the modular invariant partition functions when k+3 is an integer coprime with 6 and when it is a power of either 2 or 3. Our results imply that no detailed knowledge of the commutant is needed to undertake a classification of all modular invariants. 
  We consider whether current notions about superstring theory below the Planck scale are compatible with cosmology. We find that the anticipated form for the dilaton interaction creates a serious roadblock for inflation and makes it unlikely that the universe ever reaches a state with zero cosmological constant and time-independent gravitational constant. 
  The solvable $sl(n)$-chiral Potts model can be interpreted as a three-dimensional lattice model with local interactions. To within a minor modification of the boundary conditions it is an Ising type model on the body centered cubic lattice with two- and three-spin interactions. The corresponding local Boltzmann weights obey a number of simple relations, including a restricted star-triangle relation, which is a modified version of the well-known star-triangle relation appearing in two-dimensional models. We show that these relations lead to remarkable symmetry properties of the Boltzmann weight function of an elementary cube of the lattice, related to spatial symmetry group of the cubic lattice. These symmetry properties allow one to prove the commutativity of the row-to-row transfer matrices, bypassing the tetrahedron relation. The partition function per site for the infinite lattice is calculated exactly. 
  We can recast the Yang-Baxter equation as a triple product equation. Assuming the triple product to satisfy some algebraic relations, we can find new solutions of the Yang-Baxter equation. This program has been completed here for the simplest triple systems which we call octonionic and quaternionic. The solutions are of rational type. 
  We generalize the result of the preceeding paper and solve the Yang-Baxter equation in terms of triple systems called orthogonal and symplectic ternary systems. In this way, we found several other new solutions. 
  Scalar field theories regularized on a $D$ dimensional lattice are found to exhibit double scaling for a class of critical behaviors labeled by an integer $m\geq 2$. The continuum theory reached in the double scaling limit defines a universality class and is of a massless scalar with only a $m+1$ point self-interaction, but in the presence of a constant source. The upper critical dimension for this to occur is the same as that for renormalizability of the $m+1$ point interaction. 
  A recently introduced method for constructing marginal singular flows between distinct Landau--Ginzburg theories at fixed central charge is reviewed. The flows are constructed in an enlarged moduli space obtained by adding theories with zero central charge. This mechanism is used to construct flows between mirror pairs of string vacua described by N$=$2 superconformal Landau--Ginzburg fixed points. In contrast to previous methods this new construction of mirror theories does not depend on particular symmetries of the original theory.   (Based in part on a talk presented at the Trieste Summer   Workshop on Superstrings and Related Topics, July 1992) 
  Manifestly consistent Fock representations of non-central (but ``core-central'') extensions of the $Z^N$-graded algebras of functions and vector fields on the $N$-dimensional torus $T^N$ are constructed by a kind of renormalization procedure. These modules are of lowest-energy type, but the energy is not a linear function of the momentum. Modulo a technical assumption, reducibility conditions are proved for the extension of $vect(T^N)$, analogous to the discrete series of Virasoro representations. 
  The posibility of quantizing the anomalous $SU(N)$ Yang--Mills model preserving the symmetry under the orthogonal subgroup is indicated. The corresponding Wess--Zumino action (1-cocycle) possesses the additional $SO(N)$ symmetry and can be expressed in terms of chiral fields taking values in the homogeneous space $SU(N)/SO(N)$. The modified anomaly and the constraints commutator (2-cocycle) are calculated. 
  We show that $CPT$ is in general violated in a non-quantum-mechanical way in the effective low-energy theory derived from string theory, as a result of apparent world-sheet charge non-conservation induced by stringy monopoles corresponding to target-space black hole configurations. This modification of quantum mechanics does not violate energy conservation. The magnitude of this effective spontaneous violation of $CPT$ may not be be far from the present experimental sensitivity in the neutral kaon system. We demonstrate that our previously proposed stringy modifications to the quantum-mechanical description of the neutral kaon system violate $CPT$, although in a different way from that assumed in phenomenological analyses within conventional quantum mechanics. We constrain the novel $CPT$-violating parameters using available data on $K_L \rightarrow 2\pi$, $K_S \rightarrow 3\pi ^0$ and semileptonic $K_{L,S}$ decay asymmetries. We demonstrate that these data and an approximate treatment of interference effects in $K \rightarrow 2\pi $ decays are consistent with a {\it non-vanishing} amount of $CPT$ violation at a level accessible to a new round of experiments, and further data and/or analysis are required to exclude the extreme possibility that they dominate over $CP$ violation. Could non-quantum-field theoretical and non-quantum-mechanical $CPT$ violation usher in the long-awaited era of string phenomenology?} 
  By demanding that the path integral measure of topological field theories be metric independent, we can derive powerful constraints on the particle content of a topological field theory as well as on the dimensionality of space-time. 
  We investigate the thermal equilibrium properties of kinks in a classical $\phi^4$ field theory in $1+1$ dimensions. The distribution function, kink density, and correlation function are determined from large scale simulations. A dilute gas description of kinks is shown to be valid below a characteristic temperature. A double Gaussian approximation to evaluate the eigenvalues of the transfer operator enables us to extend the theoretical analysis to higher temperatures where the dilute gas approximation fails. This approach accurately predicts the temperature at which the kink description breaks down. 
  A complete classification of the WZNW modular invariant partition functions is known for very few affine algebras and levels, the most significant being all levels of SU(2), and level 1 of all simple algebras. In this paper we solve the classification problem for SU(3) modular invariant partition functions. Our approach will also be applicable to other affine Lie algebras, and we include some preliminary work in that direction, including a sketch of a new proof for SU(2). 
  We discuss the effective action of tachyon in the two dimensional string theory at tree level. We show that already starting from the cubic terms the action is nonlocal and the usually assumed simplest cubic term does not give the correct amplitude. Four point 1PI terms are also discussed. 
  These lectures, given at the 1992 Trieste Spring School, are devoted to some selected topics in N=2 \sm s on Calabi-Yau manifolds and the associated N=2 superconformal field theories. The first lecture is devoted to the ``special geometry" of the moduli space of $c=9$ N=2 superconformal field theories. An important role is played by the extended chiral algebra which appears in theories with integer $U(1)$ charges. The second lecture is devoted to the \sm\ approach. The main focus is an explication of a calculation of Aspinwall and Morrison. 
  A brief review of a self-contained genuinely three-dimensional monodromy-matrix based non-perturbative covariant path-integral approach to {\it polynomial invariants} of knots and links in the framework of (topological) quantum Chern-Simons field theory is given. An idea of ``physical'' observables represented by an auxiliary topological quantum-mechanics model in an external gauge field is introduced substituting rather a limited notion of the Wilson loop. Thus, the possibility of using various generalizations of the Chern-Simons action (also higher-dimensional ones) as well as a purely functional language becomes open. The theory is quantized in the framework of the best suited in this case {\it antibracket-antifield} formalism of Batalin and Vilkovisky. Using the Stokes theorem and formal translational invariance of the path-integral measure a {\it monodromy matrix} corresponding to an arbitrary pair of irreducible representations of an arbitrary semi-simple Lie group is derived. 
  The Clebsch-Gordan and Racah-Wigner coefficients for the positive (or negative) discrete series of irreducible representations for the noncompact form $U_q(SU(1,1))$ of the algebra $U_q(sl(2))$ are computed. 
  In this letter we derive a deformed Dirac equation invariant under the k-Poincare` quantum algebra. A peculiar feature is that the square of the k-Dirac operator is related to the second Casimir (the k-deformed squared Pauli-Lubanski vector). The ``spinorial'' realization of the k-Poincare` is obtained by a contraction of the coproduct of the real form of SO_q(3,2) using the 4-dimensional representation which results to be, up some scalar factors, the same of the undeformed algebra in terms of the usual gamma matrices. 
  The exotic quantum double and its universal R-matrix for quantum Yang-Baxter equation are constructed in terms of Drinfeld's quantum double theory.As a new quasi-triangular Hopf algebra, it is much different from those standard quantum doubles that are the q-deformations for Lie algebras or Lie superalgebras. By studying its representation theory,many-parameter representations of the exotic quantum double are obtained with an explicit example associated with Lie algebra $A_2$ .The multi-parameter R-matrices for the quantum Yang-Baxter equation can result from the universal R-matrix of this exotic quantum double and these representattions. 
  The tensor products of (restricted and unrestricted) finite dimensional irreducible representations of $\uq$ are considered for $q$ a root of unity. They are decomposed into direct sums of irreducible and/or indecomposable representations. 
  The generating functional for hard thermal loops in QCD is important in setting up a resummed perturbation theory, so that all terms of a given order in the coupling constant can be consistently taken into account. It is also the functional which leads to a gauge invariant description of Debye screening and plasma waves in the quark-gluon plasma. We have recently shown that this functional is closely related to the eikonal for a Chern-Simons gauge theory. In this paper, this relationship is explored and explained in more detail, along with some generalizations. 
  A thorough analysis of stochastically stabilised hermitian one matrix models for two dimensional quantum gravity at all its $(2,2k-1)$ multicritical points is made. It is stressed that only the zero fermion sector of the supersymmetric hamiltonian, i.e., the forward Fokker-Planck hamiltonian, is relevant for the analysis of bosonic matter coupled to two dimensional gravity. Therefore, supersymmetry breaking is not the physical mechanism that creates non perturbative effects in the case of points of even multicriticality $k$. Non perturbative effects in the string coupling constant $g_{str}$ result in a loss of any explicit relation to the KdV hierarchy equations in the latter case, while maintaining the perturbative genus expansion. As a by-product of our analysis it is explicitly proved that polynomials orthogonal relative to an arbitrary weight $\exp (-\beta V(x))$ along the whole real line obey an Hartree-Fock equation. 
  We derive the discrete linear systems associated to multi--matrix models, the corresponding discrete hierarchies and the appropriate coupling conditions. We also obtain the $W_{1+\infty}$ constraints on the partition function. We then apply to multi--matrix models the technique, developed in previous papers, of extracting hierarchies of differential equations from lattice ones without passing through a continuum limit. In a q--matrix model we find 2q coupled differential systems. The corresponding differential hierarchies are particular versions of the KP hierarchy. We show that the multi--matrix partition function is a $\tau$--function of these hierarchies. We discuss a few examples in the dispersionless limit. 
  The integration of the exponential of the square of the moment map of the circle action is studied by a direct stationary phase computation and by applying the Duistermaat-Heckman formula. Both methods yield two distinct formulas expressing the integral in terms of contributions from the critical set of the square of the moment map. The cohomological pairings on the symplectic quotient, including its volume (which was known to be a piecewise polynomial), are computed explicitly using the asymptotic behavior of the two formulas. 
  Lian and Zuckerman proved that the homology of a topological chiral algebra can be equipped with the structure of a BV-algebra; \ie one can introduce a multiplication, an odd bracket, and an odd operator $\Delta$ having the same properties as the corresponding operations in Batalin-Vilkovisky quantization procedure. We give a simple proof of their results and discuss a generalization of these results to the non chiral case. To simplify our proofs we use the following theorem giving a characterization of a BV-algebra in terms of multiplication and an operator $\Delta$: {\em If $A$ is a supercommutative, associative algebra and $\Delta$ is an odd second order derivation on $A$ satisfying $\Delta^2=0$, one can provide $A$ with the structure of a BV-algebra.} 
  We consider a random surface representation of the three-dimensional Ising model.The model exhibit scaling behaviour and a new critical index $\k$ which relates $\g_{string}$ for the bosonic string to the exponent $\a$ of the specific heat of the 3d Ising model is introduced. We try to determine $\k$ by numerical simulations. 
  We show that the Kazakov-Migdal (K-M) induced gauge model in $d$ dimensions describes the high temperature limit of ordinary lattice gauge theories in $d+1$ dimensions. The matter fields are related to the Polyakov loops, while the spatial gauge variables become the gauge fields of the K-M model. This interpretation of the K-M model is in agreement with some recent results in high temperature lattice QCD. 
  The Liouville action for two--dimensional quantum gravity coupled to interacting matter contains terms that have not been considered previously. They are crucial for understanding the renormalization group flow and can be observed in recent matrix model results for the phase diagram of the Sine--Gordon model coupled to gravity. These terms insure, order by order in the coupling constant, that the dressed interaction is exactly marginal. They are discussed up to second order. 
  There being no precise definition of the quantum integrability, the separability of variables can serve as its practical substitute. For any quantum integrable model generated by the Yangian Y[sl(3)] the canonical coordinates and the conjugated operators are constructed which satisfy the ``quantum characteristic equation'' (quantum counterpart of the spectral algebraic curve for the L operator). The coordinates constructed provide a local separation of variables. The conditions are enlisted which are necessary for the global separation of variables to take place. 
  In odd-dimensional spaces, gauge invariance permits a Chern-Simons mass term for the gauge fields in addition to the usual Maxwell-Yang-Mills kinetic energy term. We study the Casimir effect in such a (2+1)-dimensional Abelian theory. For the case of parallel conducting lines the result is the same as for a scalar field. For the case of circular boundary conditions the results are completely different, with even the sign of the effect being opposite for Maxwell-Chern-Simons fields and scalar fields. We further examine the effect of finite temperature. The Casimir stress is found to be attractive at both low and high temperature. Possibilities of observing this effect in the laboratory are discussed. 
  We consider gravity in 2+1 dimensions in presence of extended stationary sources with rotational symmetry. We prove by direct use of Einstein's equations that if i) the energy momentum tensor satisfies the weak energy condition, ii) the universe is open (conical at space infinity), iii) there are no CTC at space infinity, then there are no CTC at all. 
  We introduce a gaussian probability density for the space-time distribution of wormholes, thus taking effectively into account wormhole interaction. Using a mean-field approximation for the free energy, we show that giant wormholes are probabilistically suppressed in a homogenous isotropic ``large'' universe. 
  A Lagrangian continuum model for a string theory with central charge $c>1$ is formulated by incorporating Weyl and diffeomorphism gauge fixing. In particular the tachyon scattering amplitudes are deduced generalizing the standard $c\le 1$ computation. 
  We extend the coset space formulation of the one-field realization of $w_{1+\infty}$ to include more fields as the coset parameters. This can be done either by choosing a smaller stability subalgebra in the nonlinear realization of $w_{1+\infty}$ symmetry, or by considering a nonlinear realization of some extended symmetry, or by combining both options. We show that all these possibilities give rise to the multi-field realizations of $w_{1+\infty}$. We deduce the two-field realization of $w_{1+\infty}$ proceeding from a coset space of the symmetry group $\tilde{G}$ which is an extension of $w_{1+\infty}$ by the second self-commuting set of higher spin currents. Next, starting with the unextended $w_{1+\infty}$ but placing all its spin 2 generators into the coset, we obtain a new two-field realization of $w_{1+\infty}$ which essentially involves a $2D$ dilaton. In order to construct the invariant action for this system we add one more field and so get a new three-field realization of $w_{1+\infty}$. We re-derive it within the coset space approach, by applying the latter to an extended symmetry group $\hat{G}$ which is a nonlinear deformation of $\tilde{G}$. Finally we present some multi-field generalizations of our three-field realization and discuss several intriguing parallels with $N=2$ strings and conformal affine Toda theories. 
  We study a quantum Yang-Baxter structure associated with non-ultralocal lattice models. We discuss the canonical structure of a class of integrable quantum mappings, i.e. canonical transformations preserving the basic commutation relations. As a particular class of solutions we present two examples of quantum mappings associated with the lattice analogues of the KdV and MKdV equations, together with their exact quantum invariants. plain LaTeX, equations.sty appended 
  We study the dynamics of phase transitions out of equilibrium in weakly coupled scalar field theories. We consider the case in which there is a rapid supercooling from an initial symmetric phase in thermal equilibrium at temperature $T_i>T_c$ to a final state at low temperature $T_f \approx 0$. In particular we study the formation and growth of correlated domains out of equilibrium. It is shown that the dynamics of the process of domain formation and growth (spinodal decomposition) cannot be studied in perturbation theory, and a non-perturbative self-consistent Hartree approximation is used to study the long time evolution. We find in weakly coupled theories that the size of domains grow at long times as $\xi_D(t) \approx \sqrt{t\xi(0)}$. For very weakly coupled theories, their final size is several times the zero temperature correlation length. For strongly coupled theories the final size of the domains is comparable to the zero temperature correlation length and the transition proceeds faster. 
  We show that the coefficients of decomposition into an irreducible components of the tensor powers of level $r$ symmetric algebra of adjoint representation coincide with the Verlinder numbers. Also we construct (for $sl(2)) the representations of a general linear group those dimensions are given by corresponding Verlinde's numbers. 
  This paper considers interactions between closed strings and open strings satisfying either Neumann or constant (point-like) Dirichlet boundary conditions in a BRST formalism in the critical dimension. With Neumann conditions this reproduces the well-known stringy version of the Higgs mechanism. With Dirichlet conditions the open-string states correspond to either auxiliary or Lagrange multiplier target-space fields and their coupling to the closed-string sector leads to constraints on the closed-string spectrum. 
  We use Hirota's method formulated as a recursive scheme to construct complete set of soliton solutions for the affine Toda field theory based on an arbitrary Lie algebra. Our solutions include a new class of solitons connected with two different type of degeneracies encountered in the Hirota's perturbation approach. We also derive an universal mass formula for all Hirota's solutions to the Affine Toda model valid for all underlying Lie groups. Embedding of the Affine Toda model in the Conformal Affine Toda model plays a crucial role in this analysis. 
  For every partition of a positive integer $n$ in $k$ parts and every point of an infinite Grassmannian we obtain a solution of the $k$ component differential-difference KP hierarchy and a corresponding Baker function. A partition of $n$ also determines a vertex operator construction of the fundamental representations of the infinite matrix algebra $gl_\infty$ and hence a $\tau$ function. We use these fundamental representations to study the Gauss decomposition in the infinite matrix group $Gl_\infty$ and to express the Baker function in terms of $\tau$-functions. The reduction to loop algebras is discussed. 
  We construct a random matrix model that, in the large $N$ limit, reduces to the low energy limit of the QCD partition function put forward by Leutwyler and Smilga. This equivalence holds for an arbitrary number of flavors and any value of the QCD vacuum angle. In this model, moments of the inverse squares of the eigenvalues of the Dirac operator obey sum rules, which we conjecture to be universal. In other words, the validity of the sum rules depends only on the symmetries of the theory but not on its details. To illustrate this point we show that the sum rules hold for an interacting liquid of instantons. The physical interpretation is that the way the thermodynamic limit of the spectral density near zero is approached is universal. However, its value, $i.e.$ the chiral condensate, is not. 
  We discuss the relationship between target space modular invariance and discrete gauge symmetries in four-dimensional orbifold-like strings. First we derive the modular transformation properties of various string vertex operators of the massless string fields. Then we find that for supersymmetric compactifications the action of the duality elements, leaving invariant the multicritical points, corresponds to a combination of finite K\"ahler and gauge transformations. However, those finite gauge transformations are not elements of a remnant discrete gauge symmetry. We suggest that, at least in the case of Gepner models corresponding to tensor products of identical minimal models, the duality element leaving invariant the multicritical point corresponds to the ${\bf Z}_{k+2}$ symmetry of any of the minimal $N=2$ models appearing in the tensor product. 
  The partition function of a two-dimensional quantum gauge theory in the large-$N$ limit is expressed as the functional integral over some scalar field. The large-$N$ saddle point equation is presented and solved. The free energy is calculated as the function of the area and of the Euler characteristic. There is no non-trivial saddle point at genus $g>0$. The existence of a non-trivial saddle point is closely related to the weak coupling behavior of the theory. Possible applications of the method to higher dimensions are briefly discussed. 
  It is shown that a model recently proposed for numerical calculations of bound states in QED$_3$ is in fact an improper truncation of the Aharonov-Bohm potential. 
  Chiral/self-dual restrictions of various super Yang-Mills and supergravity theories in (2,2) dimensions are described. These include the N=1 supergravity with a cosmological term and the N=1 new minimal supergravity theory. In the latter case, a self-duality condition on a torsionful Riemann curvature is possible, and it implies the equations of motion that follow from an $R^2$ type supergravity action. 
  A review of recent developments in the quantum differential calculus. The quantum group $GL_q(n)$ is treated by considering it as a particular quantum space. Functions on $SL_q(n)$ are defined as a subclass of functions on $GL_q(n)$. The case of $SO_q(n)$ is also briefly considered. These notes cover part of a lecture given at the XIX International Conference on Group Theoretic Methods in Physics, Salamanca, Spain 1992. 
  We give a new proof of the dilogarithm identities, associated to the (2,2n+1) minimal models of the Virasoro algebra. 
  We show that the gauged SL(2,R) WZWN model yields arbitrary spacetimes in two dimensions. The c = 1 matter field and the black hole singularity are just two particular cases in these spacetimes. 
  We study the sum $\ds\zeta_H(s)=\sum_j E_j^{-s}$ over the eigenvalues $E_j$ of the Schrdinger equation in a spherical domain with Dirichlet walls, threaded by a line of magnetic flux. Rather than using Green's function techniques, we tackle the mathematically nontrivial problem of finding exact sum rules for the zeros of Bessel functions $J_{\nu}$, which are extremely helpful when seeking numerical approximations to ground state energies. These results are particularly valuable if $\nu$ is neither an integer nor half an odd one. 
  Using a Laurent series representation for the (super)string one-loop free energy, an explicit form for the analytic continuation of the Laurent series beyond the critical (Hagedorn) temperature is obtained. As an additional result, a periodic structure is found in (super)string thermodynamics. A brief physical discussion about the origin and meaning of such structure is carried out. 
  A consistent method for obtaining a well-defined Polyakov action on the supertorus is presented. This method uses the covariantization of derivative operators and enables us to construct a Polyakov action which is globally defined. 
  We investigate the tachyon coupling in a static Robertson--Walker like metric background. For a tachyon and dilaton field which are only time dependent one can rewrite this model as a SU(2) Wess--Zumino--Witten model and a scalar Feigin--Fuchs theory. In this case the restriction to a real exponential tachyon field fixes the level $k$ of the Wess--Zumino--Witten model. For a spatially dependent tachyon the world radius and the dilaton are quantized in terms of $k$ and the tachyon by two integers, i.e. one has a discrete set of fields. The spatial part of the tachyon is given by Chebyshev polynomials of the second kind. An investigation of the tachyon mass shows that the tachyon is massless for $k=1$. 
  The problem of interpreting a set of ${\cal W}$-algebra constraints constructed in terms of an arbitrarily twisted scalar field as the recursion relations of a topological theory is addressed. In this picture, the conventional models of topological gravity coupled to $A$, $D$ or $E$ topological matter, correspond to taking the scalar field twisted by the Coxeter element of the Weyl group. It turns out that not all conjugacy classes of the Weyl group lead to a topological model. For example, it is shown that for the $A$ algebras there are two possible choices for the conjugacy class, giving both the conventional and a new series of topological models. Furthermore, it is shown how the new series of theories contains the conventional series as a subsector. A tentative interpretation of this new series in terms of intersection theory is presented. 
  Replacing the continuous space by a cubic lattice we find a deformation of the Poincar\'e algebra. A deformation of the relativistic mass operator is shown to be a Casimir of the algebra. The real structure is preserved. 
  We consider a model of D-dimensional tethered manifold interacting by excluded volume in R^d with a single point. By use of intrinsic distance geometry, we first provide a rigorous definition of the analytic continuation of its perturbative expansion for arbitrary D, 0 < D < 2. We then construct explicitly a renormalization operation, ensuring renormalizability to all orders. This is the first example of mathematical construction and renormalization for an interacting extended object with continuous internal dimension, encompassing field theory. 
  Recently Strachan introduced a Moyal algebraic deformation of selfdual gravity, replacing a Poisson bracket of the Plebanski equation by a Moyal bracket. The dressing operator method in soliton theory can be extended to this Moyal algebraic deformation of selfdual gravity. Dressing operators are defined as Laurent series with coefficients in the Moyal (or star product) algebra, and turn out to satisfy a factorization relation similar to the case of the KP and Toda hierarchies. It is a loop algebra of the Moyal algebra (i.e., of a $W_\infty$ algebra) and an associated loop group that characterize this factorization relation. The nonlinear problem is linearized on this loop group and turns out to be integrable. 
  We show that that the Jacobi-identities for a W-algebra with primary fields of dimensions 3, 4 and 5 allow two different solutions. The first solution can be identified with WA_4. The second is special in the sense that, even though associative for general value of the central charge, null-fields appear that violate some of the Jacobi-identities, a fact that is usually linked to exceptional W-algebras. In contrast we find for the algebra that has an additional spin 6 field only the solution WA_5. 
  We investigate global properties of the mappings entering the description of symmetries of integrable spin and vertex models, by exploiting their nature of birational transformations of projective spaces. We give an algorithmic analysis of the structure of invariants of such mappings. We discuss some characteristic conditions for their (quasi)--integrability, and in particular its links with their singularities (in the 2--plane). Finally, we describe some of their properties {\it qua\/} dynamical systems, making contact with Arnol'd's notion of complexity, and exemplify remarkable behaviours. 
  The most general large N eigenvalues distribution for the one matrix model is shown to consist of tree-like structures in the complex plane. For the m=2 critical point, such a split solution describes the strong coupling phase of 2d quantum gravity (c=0 non-critical string). It is obtained by taking combinations of complex contours in the matrix integral, and the relative weight of the contours is identified with the non-perturbative theta-parameter that fixes uniquely the solution of the string equation (Painleve I). This allows to recover by instanton methods results on the non-perturbative effects obtained by the Isomonodromic Deformation Method, and to construct for each theta-vacuum the observables (the loop correlation functions) which satisfy the loop equations. The breakdown of analyticity of the large N solution is related to the existence of poles for the loop operators. 
  We present a systematic derivation for a general formula for the n-point coupling constant valid for affine Toda theories related to any simple Lie algebra {\bf g}. All n-point couplings with $n \geq 4$ are completely determined in terms of the masses and the three-point couplings. A general fusing rule, formulated in the root space of the Lie algebra, is derived for all n-point couplings. 
  We study a generating function for the sum over fatgraphs with specified valences of vertices and faces, inversely weighted by the order of their symmetry group. A compact expression is found for general (i.e. non necessarily connected) fatgraphs. This expression admits a matrix integral representation which enables to perform semi--classical computations, leading in particular to a closed formula corresponding to (genus zero, connected) trees. 
  These lecture notes review current progress on the class of conformal theories which may be studied by quantizing the conformal Toda dynamics. After summarizing recent developments in undertanding the quantum group structure of the Liouville theory, one recalls how classically, two-dimensional black holes come out from the non-abelian Toda systems, and reviews the geometrical interpretation of the classical $A_n$-Toda theories, just put forward, that relates W-geometries with the external geometries of holomorphic surfaces in complex projective spaces. 
  Chern-Simons field theory based on a compact non-abelian gauge group is studied as a theory of knots and links in three dimensions. A method to obtain the invariants for links made from braids of upto four strands is developed. This generalizes our earlier work on $SU(2)$ Chern-Simons theory. 
  Massless particle dynamics in $D=10$ Minkowski space is given an $E_7$-covariant formulation, including both space-time and twistor variables. $E_7$ contains the conformal algebra as a subalgebra. Analogous constructions apply to $D=3,4$ and 6. 
  The topological supersymmetry of the pure Chern-Simons model in three dimensions is established in the case where the theory is defined in the axial gauge. 
  We use the Kontsevich-Miwa transform to relate the different pictures describing matter coupled to topological gravity in two dimensions: topological theories, Virasoro constraints on integrable hierarchies, and a DDK-type formalism. With the help of the Kontsevich-Miwa transform, we solve the Virasoro constraints on the KP hierarchy in terms of minimal models dressed with a (free) Liouville-like scalar. The dressing prescription originates in a topological (twisted N=2) theory. The Virasoro constraints are thus related to essentially the N=2 null state decoupling equations. The N=2 generators are constructed out of matter, the `Liouville' scalar, and $c=-2$ ghosts. By a `dual' construction involving the reparametrization $c=-26$ ghosts, the DDK dressing prescription is reproduced from the N=2 symmetry. As a by-product we thus observe that there are two ways to dress arbitrary $d\leq1$ or $d\geq25$ matter theory, that allow its embedding into a topological theory. By th e Kontsevich-Miwa transform, which introduces an infinite set of `time' variables $t_r$, the equations ensuring the vanishing of correlators that involve BRST-exact primary states, factorize through the Virasoro generators expressed in terms of the $t_r$. The background charge of these Virasoro generators is determined by the topological central charge. 
  We compute the exact spectral density of random matrices in the ground state of the quantum hamiltonian corresponding to the matrix model whose double scaling limit describes pure gravity in 2D. We show that the non-perturbative effects are very large and in certain cases dominate the semi-classical WKB contribution studied in the earlier literature. The physical observables in this model are the loop averages with respect to the spectral density. We compute their exact ground-state expectation values and show that they differ significantly from the values obtained in the WKB approximation. Unlike the alternative regularizations of the nonperturbative 2D quantum gravity, based on analytic continuation of the Painlev\'e transcendent, our solution shows no pathologies. 
  We derive (quasi-)quantum groups in 2+1 dimensional topological field theory directly from the classical action and the path integral. Detailed computations are carried out for the Chern-Simons theory with finite gauge group. The principles behind our computations are presumably more general. We extend the classical action in a d+1 dimensional topological theory to manifolds of dimension less than d+1. We then ``construct'' a generalized path integral which in d+1 dimensions reduces to the standard one and in d dimensions reproduces the quantum Hilbert space. In a 2+1 dimensional topological theory the path integral over the circle is the category of representations of a quasi-quantum group. In this paper we only consider finite theories, in which the generalized path integral reduces to a finite sum. New ideas are needed to extend beyond the finite theories treated here. 
  This article reports on a program to obtain and understand coherent states for general systems. Most recently this has included supersymmetric systems. A byproduct of this work has been studies of squeezed and supersqueezed states. To obtain a physical understanding of these systems has always been a primary goal. In particular, in the work on supersymmetry an attempt to understand the role of Grassmann numbers in quantum mechanics has been initiated. 
  We present a procedure for computing gauge-invariant scattering amplitudes in the $W_3$ string, and use it to calculate three-point and four-point functions. We show that non-vanishing scattering amplitudes necessarily involve external physical states with excitations of ghosts as well as matter fields. The crossing properties of the four-point functions are studied, and it is shown that the duality of the Virasoro string amplitudes generalises in the $W_3$ string, with different sets of intermediate states being exchanged in different channels. We also exhibit a relation between the scattering amplitudes of the $W_3$ string and the fusion rules of the Ising model. 
  We consider the Feigin-Fuchs-Felder formalism of the $SU(2)_k\times SU(2)_l/SU(2)_{k+l}$ coset minimal conformal field theory and extend it to higher genus. We investigate a double BRST complex with respect to two compatible BRST charges, one associated with the parafermion sector and the other associated with the minimal sector in the theory. The usual screened vertex operator is extended to the BRST invariant screened three string vertex. We carry out a sewing operation of these string vertices and derive the BRST invariant screened $g$-loop operator. The latter operator characterizes the higher genus structure of the theory. An analogous operator formalism for the topological minimal model is obtained as the limit $ l=0$ of the coset theory. We give some calculations of correlation functions on higher genus. 
  In this talk I discuss the form factor approach used to compute correlation functions of integrable models in two dimensions. The Sinh-Gordon model is our basic example. Using Watson's and the recursive equations satisfied by matrix elements of local operators, I present the computation of the form factors of the elementary field $\phi(x)$ and the stress-energy tensor $T_{\mu\nu}(x)$ of the theory. 
  A general solution to the $D=2$ 1-loop beta functions equations including tachyonic back reaction on the metric is presented. Dynamical black hole (classical) solutions representing gravitational collapse of tachyons are constructed. A discussion on the correspondence with the matrix-model approach is given. 
  The classical limit of quantum q-oscillators suggests an interpretation of the deformation as a way to introduce non linearity. Guided by this idea, we considered q-fields, the partition fumction, and compute a consequence on specific heat and second order correlation function of the q-oscillator which may serve for experimental checks for the non linearity. 
  For time-periodical quantum systems generalized Floquet operator is found to be integral of motion.Spectrum of this invariant is shown to be quasienergy spectrum.Analogs of invariant Floquet operators are found for nonperiodical systems with several characteristic times.Generalized quasienergy states are introduced for these systems. Geometrical phase is shown to be integral of motion. 
  Following a prescription of \cite{4} for a solitonic specialization of the general solutions to the (abelian) periodic Toda field theories, we discuss a construction of the soliton solutions for a wide class of two-dimensional completely integrable systems arising in the framework of the group-algebraic approach, including the \lq\lq non-abelian" version of the affine Toda theory. 
  We review how to obtain the thermodynamic Bethe Ansatz (TBA) equations for the antiferromagnetic Heisenberg ring in an external magnetic field. We review how to solve these equations for low temperature and small field, and calculate the specific heat and magnetic susceptibility. 
  We formulate the thermodynamic Bethe Ansatz (TBA) equations for the closed (periodic boundary conditions) $A^{(2)}_2$ quantum spin chain in an external magnetic field, in the (noncritical) regime where the anisotropy parameter $\eta$ is real. In the limit $\eta \to 0$, we recover the TBA equations of the antiferromagnetic su(3)-invariant chain in the fundamental representation. We solve these equations for low temperature and small field, and calculate the specific heat and magnetic susceptibility. 
  It is a common belief among field theorists that path integrals can be computed exactly only in a limited number of special cases, and that most of these cases are already known. However recent developments, which generalize the WKBJ method using equivariant cohomology, appear to contradict this folk wisdom. At the formal level, equivariant localization would seem to allow exact computation of phase space path integrals for an arbitrary partition function! To see how, and if, these methods really work in practice, we have applied them in explicit quantum mechanics examples. We show that the path integral for the 1-d hydrogen atom, which is not WKBJ exact, is localizable and computable using the more general formalism. We find however considerable ambiguities in this approach, which we can only partially resolve. In addition, we find a large class of quantum mechanics examples where the localization procedure breaks down completely. 
  We study symmetry breaking in $Z_2$ symmetric large $N$ matrix models. In the planar approximation for both the symmetric double-well $\phi^4$ model and the symmetric Penner model, we find there is an infinite family of broken symmetry solutions characterized by different sets of recursion coefficients $R_n$ and $S_n$ that all lead to identical free energies and eigenvalue densities. These solutions can be parameterized by an arbitrary angle $\theta(x)$, for each value of $x = n/N < 1$. In the double scaling limit, this class reduces to a smaller family of solutions with distinct free energies already at the torus level. For the double-well $\phi^4$ theory the double scaling string equations are parameterized by a conserved angular momentum parameter in the range $0 \le l < \infty$ and a single arbitrary $U(1)$ phase angle. 
  We study the treatment of the constraints in stochastic quantization method. We improve the treatment of the stochastic consistency condition proposed by Namiki et al. by suitably taking account of the Ito calculus. Then we obtain an improved Langevin equation and the Fokker-Planck equation which naturally leads to the correct path integral quantization of the constrained system as the stochastic equilibrium state. This treatment is applied to $O(N)$ non-linear $\sigma$ model and it is shown that singular terms appearing in the improved Langevin equation cancel out the $\delta^n(0)$ divergences in one loop order. We also ascertain that the above Langevin equation, rewritten in terms of independent variablesis, actually equivalent to the one in the general-coordinate-transformation-covariant and vielbein-rotation-invariant formalism. 
  Explicit and complete topological solution of SU(2) Chern-Simons theory on S^3 is presented. 
  D-dimensional constrained systems are studied with stochastic Lagrangian and\break Hamiltonian. It is shown that stochastic consistency conditions are second class constraints and Lagrange multiplier fields can be determined in (D+1)-dimensional canonical formulation. The Langevin equations for the constrained system are obtained as Hamilton's equations of motion where conjugate momenta play a part of noise fields. 
  We study the stochastic quantization of two-dimensional nonlinear sigma model in the large $N$ limit. Our main tool is the {\it effective} Langevin equation with which we investigate nonperturbative phenomena and derive the results which are same as the path integral approach gives. 
  In order to investigate dynamical symmetry breaking, we study Nambu$\cdot$Jona-Lasinio model in the large-N limit in the stochastic quantization method. Here in order to solve Langevin equation, we impose specified initial conditions and construct ``effective Langevin equation'' in the large-N limit and give the same non-perturbative results as path-integral approach gives. Moreover we discuss stability of vacuum by means of ``effective potential''. 
  Partition functions of some two-dimensional statistical models can be represented by means of Grassmann integrals over loops living on two-dimensional torus. It is shown that those Grassmann integrals are topological invariants, which depend only on the winding numbers of the loops. The fact makes possible to evaluate the partition functions of the models and the statistical mean values of certain topological characteristics (indices) of the configurations, which behave as the (topological) order parameters. 
  Inspired by the formulation of the Batalin-Vilkovisky method of quantization in terms of ``odd time'', we show that for a class of gauge theories which are first order in the derivatives, the kinetic term is bilinear in the fields, and the interaction part satisfies some properties, it is possible to give the solution of the master equation in a very simple way. To clarify the general procedure we discuss its application to Yang-Mills theory, massive (abelian) theory in the Stueckelberg formalism, relativistic particle and to the self-interacting antisymmetric tensor field. 
  Isotropic XY is considered. It describes interaction of quantum spins on 1-dimesional lattice. Alternatevly one can call the model XXO Hiesenberg antiferromagnet. We solved long standing problem of evaluation of temperature corelations. We proved that correlation function of the model is $\tau $ function of Ablowitz-Ladik PDE. We explicitly evaluated asymptotics. 
  We introduce a large class of modifications of the standard lagrangian for two dimensional dilaton gravity, whose general solutions are nonsingular black holes. A subclass of these lagrangians have extremal solutions which are nonsingular analogues of the extremal Reissner-Nordstrom spacetime. It is possible that quantum deformations of these extremal solutions are the endpoint of Hawking evaporation when the models are coupled to matter, and that the resulting evolution may be studied entirely within the framework of the semiclassical approximation. Numerical work to verify this conjecture is in progress. We point out however that the solutions with non-negative mass always contain Cauchy horizons, and may be sensitive to small perturbations. 
  We generalize the braid algebra to the case of loops with intersections. We introduce the Reidemeister moves for 4 and 6-valent vertices to have a theory of rigid vertex equivalence. By considering representations of the extended braid algebra, we derive skein relations for link polynomials, which allow us to generalize any link Polynomial to the intersecting case. We perturbatively show that the HOMFLY Polynomials for intersecting links correspond to the vacuum expectation value of the Wilson line operator of the Chern Simon's Theory. We make contact with quantum gravity by showing that these polynomials are simply related with some solutions of the complete set of constraints with cosmological constant 
  A $D>2$ topological string is presented by coupling the $2d$ topological gravity with the twisted version of the $N=2$ superconformal matter with $c=3k/(k-2)$. The latter is shown to admit $k+1$ chiral primary fields from the $SL(2,R)_{k}/U(1)$ unitary irreducible representations. The analysis of topological contact interactions along with the consistency requirement lead to recursion relations of correlation functions, that are convertable to the Virasoro constraints on the perturbed partition function. It is further expected to satisfy the nonlinear $\hat{W}^{(k-2)}_{\infty}$ constraints associated with the graded algebra $SL(k,2)$, and thus the model is completely solvable at arbitrary genus of the surface. 
  The renormalization structure of two-dimensional quantum gravity is investigated, in a covariant gauge. One-loop divergences of the effective action are calculated. All the surface divergent terms are taken into account, thus completing previous one-loop calculations of the theory. It is shown that the on-shell effective action contains only surface divergences. The off-shell renormalizability of the theory is discussed and classes of renormalizable dilaton and Maxwell potentials are found. 
  The trace anomaly induced dynamics of the conformal factor is investigated in four-dimensional quantum gravity with torsion. The constraints for the coupling constants of torsion matter interaction are obtained in the infrared stable fixed point of the effective scalar theory. 
  Improving on an earlier proposal, we construct the gauge theories of the quantum groups $U_q(N)$. We find that these theories are consistent also with an ordinary (commuting) spacetime. The bicovariance conditions of the quantum differential calculus are essential in our construction. The gauge potentials and the field strengths are $q$-commuting ``fields", and satisfy $q$-commutation relations with the gauge parameters. The transformation rules of the potentials are given explicitly, and generalize the ordinary infinitesimal gauge variations. The $q$-lagrangian invariant under the $U_q(N)$ variations has the Yang-Mills form $\Fmn^i \Fmn^j g_{ij}$, the ``quantum metric'' $g_{ij}$ being a generalization of the Killing metric. 
  In this paper we discuss the interplay among (~super-~)coordinate, Weyl and \l\ anomaly both in chiral and non-chiral super-gravity represented by $(1,0)$ and $(1,1)$ two-dimensional models. It is shown that for this purpose two regularization dependent parameters are needed in the effective action. We discuss in {\it full generality} the regularization ambiguities of the induced effective action and recover the corresponding general form of the anomalous Ward Identities. Finally, we explain the difference between chiral and non-chiral super-gravity models in terms of the free parameters and establish relation between these two models by projecting $(1,1)$ into $(1,0)$ super-symmetry. 
  The bootstrap equations for the ADE series of purely elastic scattering theories have turned out to be intimately connected with the geometry of root systems and the Coxeter element. An informal review of some of this material is given, mentioning also a couple of other contexts -- the Pasquier models, and the simply-laced affine Toda field theories -- where similar structures are encountered. The relevance of twisted Coxeter elements is indicated, and a construction of these elements inspired by the twisted foldings of the affine Toda models is described. 
  The constraints of $BF$ topological gauge theories are used to construct Hamiltonians which are anti-commutators of the BRST and anti-BRST operators. Such Hamiltonians are a signature of Topological Quantum Field Theories (TQFT's). By construction, both classes of topological field theories share the same phase spaces and constraints. We find that, for 2+1 and 1+1 dimensional space-times foliated as $M=\Sigma\times\IR$, a homomorphism exists between the constraint algebras of our TQFT and those of canonical gravity. The metrics on the two-dimensional hypersurfaces are also obtained. 
  The methods of conformal field theory are used to obtain the series of exact solutions of the fundamental equations of the theory of turbulence. The basic conjecture, proved to be self-consistent ,is the conformal invariance of the inertial range. The resulting physical picture is different from the standard one , since the enstrophy transfer is catalyzed by the large scale motions. The theory gives some unambiguous predictions for the correlations in the inertial range. 
  We investigate the qualitative new features of charged dilatonic black holes which emerge when both the Yang-Mills and Gauss-Bonnet curvature corrections are included in the effective action. We consider perturbative effects by an expansion up to second order in the inverse string tension on the four dimensional Schwarzschild background and determine the backreaction. We calculate the thermodynamical functions and show that for magnetic charge above a critical value, the temperature of the black hole has a maximum and goes to zero for a finite value of the mass. This indicates that the conventional Hawking evaporation law is modified by string theory at a classical level. 
  Quantum mechanical potentials satisfying the property of shape invariance are well known to be algebraically solvable. Using a scaling ansatz for the change of parameters, we obtain a large class of new shape invariant potentials which are reflectionless and possess an infinite number of bound states. They can be viewed as q-deformations of the single soliton solution corresponding to the Rosen-Morse potential. Explicit expressions for energy eigenvalues, eigenfunctions and transmission coefficients are given. Included in our potentials as a special case is the self-similar potential recently discussed by Shabat and Spiridonov. 
  This is a talk delivered at the Meeting on Integrable Quantum Field Theories, Villa Olmo and at STRINGS 1992, Rome, September 1992. I discuss some recent attempts to revive two old ideas regarding an analytic approach to QCD-the development of a string representation of the theory and the large N limit of QCD. 
  I explore the possibility of finding an equivalent string representation of two dimensional QCD. I develop the large N expansion of the ${\rm QCD_2}$ partition function on an arbitrary two dimensional Euclidean manifold. If this is related to a two-dimensional string theory then many of the coefficients of the ${1\over N}$ expansion must vanish. This is shown to be true to all orders, giving strong evidence for the existence of a string representation. 
  We prove new identities betweenthe values of Rogers dilogarithm function and describe a connection between these identities and spectra in conformal field theory. 
  This is a systematic introduction for physicists to the theory of algebras and groups with braid statistics, as developed over the last three years by the author. There are braided lines, braided planes, braided matrices and braided groups all in analogy with superlines, superplanes etc. The main idea is that the bose-fermi $\pm1$ statistics between Grassmannn coordinates is now replaced by a general braid statistics $\Psi$, typically given by a Yang-Baxter matrix $R$. Most of the algebraic proofs are best done by drawing knot and tangle diagrams, yet most constructions in supersymmetry appear to generalise well. Particles of braid statistics exist and can be expected to be described in this way. At the same time, we find many applications to ordinary quantum group theory: how to make quantum-group covariant (braided) tensor products and spin chains, action-angle variables for quantum groups, vector addition on $q$-Minkowski space and a semidirect product q-Poincar\'e group are among the main applications so far. Every quantum group can be viewed as a braided group, so the theory contains quantum group theory as well as supersymmetry. There also appears to be a rich theory of braided geometry, more general than super-geometry and including aspects of quantum geometry. Braided-derivations obey a braided-Leibniz rule and recover the usual Jackson $q$-derivative as the 1-dimensional case. 
  A combinatorial proof of the unimodality of the generalized q-Gaussian coefficients based on the explicit formula for Kostka-Foulkes polynomials is given. 
  Static spherically symmetric asymptotically flat particle-like and black hole solutions are constructed within the SU(2) sector of 4-dimensional heterotic string effective action. They separate topologically distinct Yang-Mills vacua and are qualitatively similar to the Einstein-Yang-Mills spha- lerons and non-abelian black holes discussed recently. New solutions possess quantized values of the dilaton charge. 
  The lattice definition of a two-dimensional topological field theory (TFT) is given generically, and the exact solution is obtained explicitly. In particular, the set of all lattice topological field theories is shown to be in one-to-one correspondence with the set of all associative algebras $R$, and the physical Hilbert space is identified with the center $Z(R)$ of the associative algebra $R$. Perturbations of TFT's are also considered in this approach, showing that the form of topological perturbations is automatically determined, and that all TFT's are obtained from one TFT by such perturbations. Several examples are presented, including twisted $N=2$ minimal topological matter and the case where $R$ is a group ring. 
  We find a principle of harmonic analyticity underlying the quaternionic (quaternion-K\"ahler) geometry and solve the differential constraints which define this geometry. To this end the original $4n$-dimensional quaternionic manifold is extended to a bi-harmonic space. The latter includes additional harmonic coordinates associated with both the tangent local $Sp(1)$ group and an extra rigid $SU(2)$ group rotating the complex structures. Then the constraints can be rewritten as integrability conditions for the existence of an analytic subspace in the bi-harmonic space and solved in terms of two unconstrained potentials on the analytic subspace. Geometrically, the potentials have the meaning of vielbeins associated with the harmonic coordinates. We also establish a one-to-one correspondence between the quaternionic spaces and off-shell $N=2$ supersymmetric sigma-models coupled to $N=2$ supergravity. The general $N=2$ sigma-model Lagrangian when written in the harmonic superspace is composed of the quaternionic potentials. Coordinates of the analytic subspace are identified with superfields describing $N=2$ matter hypermultiplets and a compensating hypermultiplet of $N=2$ supergravity. As an illustration we present the potentials for the symmetric quaternionic spaces. 
  Starting from string field theory for 2d gravity coupled to c=1 matter we analyze the off-shell tree amplitudes of discrete states. The amplitudes exhibit the pole structure and we use the off-shell calculus to extract the residues and prove that just the residues are constrained by the Ward Identities. The residues generate a simple effective action. 
  The paragrassmann calculus proposed earlier is applied to constructing paraconformal transformations and paragrassmann generalizations of the Virasoro-Neveu-Schwarz-Ramond algebras. 
  It is argued that the recently proposed Kazakov-Migdal model of induced gauge theory, at large $N$, involves only the zero area Wilson loops that are effectively trees in the gauge action induced by the scalars. This retains only a constant part of the gauge action excluding plaquettes or anything like them and the gauge variables drop out. 
  Few natural basic principles allow to extend Feynman integral over the paths to an integral over the surfaces so, that they coincide at long time scale, that is when the surface degenerates into a single particle world line. In the first approximation the loop Green functions have perimeter behavior. That corresponds to the case when free quarks interact through one gluon exchange. Quantum fluctuations of the surface generate the area low. Thus in this string theory asymptotic freedom and confinement can coexist. 
  We propose that large quantum fluctuations of the conformal factor drastically modify classical general relativity at cosmological distance scales, resulting in a scale invariant phase of quantum gravity in the far infrared. We derive scaling relations for the partition function and physical observables in this conformal phase, and suggest quantitative tests of these relations in numerical simulations of simplicial four geometries with $S^4$ topology. In particular, we predict the form of the critical curve in the coupling constant plane, and determine the scaling of the Newtonian coupling with volume which permits a sensible continuum limit. The existing numerical results already provide some evidence of this new conformal invariant phase of quantum gravity. 
  Following some recent work by Gross, we consider the partition function for QCD on a two dimensional torus and study its stringiness. We present strong evidence that the free energy corresponds to a sum over branched surfaces with small handles mapped into the target space. The sum is modded out by all diffeomorphisms on the world-sheet. This leaves a sum over disconnected classes of maps. We prove that the free energy gives a consistent result for all smooth maps of the torus into the torus which cover the target space $p$ times, where $p$ is prime, and conjecture that this is true for all coverings. Each class can also contain integrations over the positions of branch points and small handles which act as ``moduli'' on the surface. We show that the free energy is consistent for any number of handles and that the first few leading terms are consistent with contributions from maps with branch points. 
  We discuss the large order behaviour and Borel summability of the topological expansion of models of 2D gravity coupled to general $(p,q)$ conformal matter. In a previous work it was proven that at large order $k$ the string susceptibility had a generic $a^k\Gamma(2k-\ud)$ behaviour. Moreover the constant $a$, relevant for the problem of Borel summability, was determined for all one-matrix models. We here obtain a set of equations for this constant in the general $(p,q)$ model. String equations can be derived from the construction of two differential operators $P,Q$ satisfying canonical commutation relations $[P,Q]=1$. We show that the equation for $a$ is determined by the form of the operators $P,Q$ in the spherical or semiclassical limits. The results for the general one-matrix models are then easily recovered. Moreover, since for the $(p,q)$ string models such $p=(2m+1)q\pm1$ the semiclassical forms of $P,Q$ are explicitly known, the large order behaviour is completely determined. This class contains all unitary $(q+1,q)$ models for which the answer is specially simple. As expected we find that the topological expansion for unitary models is not Borel summable. \preprint{SPhT/92-163}, Plain-TeX, macro harvmac 
  We study duality transformations for two-dimensional sigma models with abelian chiral isometries and prove that generic such transformations are equivalent to integrated marginal perturbations by bilinears in the chiral currents, thus confirming a recent conjecture by Hassan and Sen formulated in the context of Wess-Zumino-Witten models. Specific duality transformations instead give rise to coset models plus free bosons. 
  We review the recently proposed \lc\ quantization of the matrix model which is expected to have a critical point describing 2-d quantum gravity coupled to $c=2$ matter. In the $N\to\infty$ limit, we derive a linear Schroedinger equation for the free string spectrum. Numerical study of this equation suggests that the spectrum is tachyonic, and that the string tension diverges at the critical point. 
  We apply the fusion procedure to a quantum Yang-Baxter algebra associated with time-discrete integrable systems, notably integrable quantum mappings. We present a general construction of higher-order quantum invariants for these systems. As an important class of examples, we present the Yang-Baxter structure of the Gel'fand-Dikii mapping hierarchy, that we have introduced in previous papers, together with the corresponding explicit commuting family of quantum invariants. 
  I provide an extremely simple argument that the kink-type solitons in certain theories are fermionic. The argument is based on the Witten index, but can in fact be used to determine soliton statistics in non-supersymmetric theories as well. 
  Vertex operator algebras are mathematically rigorous objects corresponding to chiral algebras in conformal field theory. Operads are mathematical devices to describe operations, that is, $n$-ary operations for all $n$ greater than or equal to $0$, not just binary products. In this paper, a reformulation of the notion of vertex operator algebra in terms of operads is presented. This reformulation shows that the rich geometric structure revealed in the study of conformal field theory and the rich algebraic structure of the theory of vertex operator algebras share a precise common foundation in basic operations associated with a certain kind of (two-dimensional) ``complex'' geometric object, in the sense in which classical algebraic structures (groups, algebras, Lie algebras and the like) are always implicitly based on (one-dimensional) ``real'' geometric objects. In effect, the standard analogy between point-particle theory and string theory is being shown to manifest itself at a more fundamental mathematical level. 
  It is shown that the previously known $N=3$ and $N=4$ superconformal algebras can be contracted consistently by singular scaling of some of the generators. For the later case, by a contraction which depends on the central term, we obtain a new $N=4$ superconformal algebra which contains an $SU(2)\times {U(1)}^4$ Kac-Moody subalgebra and has nonzero central extension. 
  We present a classification of characteristic Dynkin diagrams for the $A_N$, $B_N$, $C_N$ and $D_N$ algebras. This classification is related to the classification of \cw(\cg,\ck) algebras arising from non-Abelian Toda models, and we argue that it can give new insight on the structure of $W$ algebras. 
  We show that some models with non-local (and non-localizable) interactions have a property, called quasi-locality, which allows for the definition of a transfer matrix. We give the Yang-Baxter equation as a sufficient condition for the existence of a family of commuting transfer matrices and solve them for a loop model with intersections. This solvable model is then analyzed in some detail and its applications to a Lorentz gas are briefly discussed. 
  A representation for the fermionic determinant of the massive Schwinger model, or $QED_2$, is obtained that makes a clean separation between the Schwinger model and its massive counterpart. From this it is shown that the index theorem for $QED_2$ follows from gauge invariance, that the Schwinger model's contribution to the determinant is canceled in the weak field limit, and that the determinant vanishes when the field strength is sufficiently strong to form a zero-energy bound state. 
  We give model-independent arguments, valid in nearly any number of spacetime dimensions, that topological solitons and instantons satisfy Bogomol'nyi-type bounds and, when these bounds are saturated, satisfy self-duality equations. In the supersymmetric case, we also show that, in spacetime dimensions greater than two, theories with topological charges necessarily exhibit extended supersymmetry, in which the topological charge appears as the central charge. The significance of our arguments lies in their generality. In the supersymmetric case, we obtain insight into the contrast observed between topological charges in 1+1 and higher dimensional models. The centerpiece of our method is to require that the supersymmetric extension of a generic (non-supersymmetric) field theory be self-consistent. Our discussion of supersymmetric extensions is quite detailed, and introduces the notion of the "associated superfield" to construct such extensions. 
  We suggest how to derive the exact (all order in $\a'$) expressions for the background fields for string solutions corresponding to gauged WZW models directly at the $2d$ field theory level. One is first to replace the classical gauged WZW action by the quantum effective one and then to integrate out the gauge field. We find the explicit expression for the gauge invariant non-local effective action of the gauged WZW model. The two terms (corresponding to the group and subgroup) which appear with the same coefficients in the classical action get different $k$-dependent coefficients in the effective one. The procedure of integrating out the gauge field is considered in detail for the $SL(2,R)/U(1)$ model and the exact expressions for the $D=2$ metric and the dilaton (originally found in the conformal field theory approach) are reproduced. 
  Using shift vector method we obtain a large class of self-dual lattices of dimension $(l,l)$, which has a one to one correspondence with modular invariants of free bosonic theory compactified on co-root lattice of a rank $l$ Lie group. Then a large number of modular invariants of affine Lie algebras are derived explicitly. As two applications of this method, we give a direct derivation of $D$-series of $SU(N)$ and a new proof for the A-D-E classification of the $SU(2)_k$ partition functions. 
  We present the solution of the discrete super-Virasoro constraints to all orders of the genus expansion. Integrating over the fermionic variables we get a representation of the partition function in terms of the one-matrix model. We also obtain the nonperturbative solution of the super-Virasoro constraints in the double scaling limit but do not find agreement between our flows and the known supersymmetric extensions of KdV. 
  We propose a new $q$-series formula for a character of parafermion conformal field theories associated to arbitrary non-twisted affine Lie algebra $\widehat{g}$. We show its natural origin from a thermodynamic Bethe ansatz analysis including chemical potentials. 
  A relativistic string is usually represented by the Nambu-Goto action in terms of the extremal area of a 2-dimensional timelike submanifold of Minkowski space. Alternatively, a family of classical solutions of the string equation of motion can be globally described in terms of the associated geodesic field. In this paper we propose a new gauge theory for the geodesic field of closed and open strings. Our approach solves the technical and conceptual problems affecting previous attempts to describe strings in terms of local field variables. The connection between the geodesic field, the string current and the Kalb-Ramond gauge potential is discussed and clarified. A non-abelian generalization and the generally covariant form of the model are also discussed. 
  For the two dimensional dilaton-coupled quantum gravity model, we give the local black hole mass, which is an analogue of what was first introduced by Fischler, Morgan and Polchinski in the four dimensional gravitational systems. We analyze the original CGHS model with this local mass and find that the local mass is decreasing in the future direction on the matter shock-wave line, while it stays constant at past null infinity. 
  Extending the work of Park and Strominger, we prove a positive energy theorem for the exactly solvable quantum-corrected 2D dilaton gravity theories. The positive energy functional we construct is shown to be unique (within a reasonably broad class of such functionals). For field configurations asymptotic to the LDV we show that this energy functional (if defined on a space-like surface) yields the usual (classical) definition of the ADM mass {\it plus a certain ``quantum"-correction. If defined on a null surface the energy functional yields the Bondi mass. The latter is evaluated careflly and applied to the RST shock-wave scenario where it is shown to behave as physically expected. Motivated by the existence of a positivity theorem we construct manifestly supersymmetric (semiclassical) extensions of these quantum-corrected dilaton-gravity theories. 
  The field content of the two dimensional string theory consists of the dynamical tachyon field and some nonpropagating fields which consist in the topological sector of this theory. We propose in this paper to study this topological sector as a spacetime gauge theory with a simple centrally extended $w_\infty$ algebra. This $w_\infty$ algebra appears in both the world sheet BRST analysis and the matrix model approach. Since the two dimensional centrally extended Poincar\'e algebra is naturally embedded in the centrally extended $w_\infty$ algebra, the low energy action for the metric and dilaton appears naturally when the model is truncated at this level. We give a plausible explanation of emergence of discrete states in this formulation. This theory is again the effective theory at zero slope limit. To include higher order $\alpha'$ corrections, we speculate that the whole theory is a gauge theory of a deformed $w_\infty$ algebra, and the deformation parameter is just $\alpha'$. 
  The correspondence between the braid group on a solid torus of arbitrary genus and the algebra of Yang-Baxter and reflection equation operators is shown. A representation of this braid group in terms of $R$-matrices is given. The characteristic equation of the reflection equation matrix is considered as an additional skein relation. This could lead to an intrinsic definition of invariant link polynomials on solid tori and, via Heegaard splitting, to invariant link polynomials on arbitrary three-manifolds without boundary. 
  We construct $N=4$ supersymmetric KdV equation as a hamiltonian flow on the $N=4\;SU(2)$ super Virasoro algebra. The $N=4$ KdV superfield, the hamiltonian and the related Poisson structure are concisely formulated in $1D \;N=4$ harmonic superspace. The most general hamiltonian is shown to necessarily involve $SU(2)$ breaking parameters which are combined in a traceless rank 2 $SU(2)$ tensor. First nontrivial conserved charges of $N=4$ super KdV (of dimensions 2 and 4) are found to exist if and only if the $SU(2)$ breaking tensor is a bilinear of some $SU(2)$ vector with a fixed length proportional to the inverse of the central charge of $N=4\;SU(2)$ algebra. After the reduction to $N=2$ this restricted version of $N=4$ super KdV goes over to the $a=4$ integrable case of $N=2$ super KdV and so is expected to be integrable. We show that it is bi-hamiltonian like its $N=2$ prototype. 
  By using lessons learned from modern string studies, we show how interesting non-perturbative features of QCD can be learned from future heavy ion collisions even if the deconfinement density is not reached. 
  An equivalence between generalised restricted solid-on-solid (RSOS) models, associated with sets of graphs, and multi-colour loop models is established. As an application we consider solvable loop models and in this way obtain new solvable families of critical RSOS models. These families can all be classified by the Dynkin diagrams of the simply-laced Lie algebras. For one of the RSOS models, labelled by the Lie algebra pair (A$_L$,A$_L$) and related to the C$_2^{(1)}$ vertex model, we give an off-critical extension, which breaks the Z$_2$ symmetry of the Dynkin diagrams. 
  Physical states of two-dimensional topological gauge theories are studied using the BRST formalism in the light-cone gauge. All physical states are obtained for the abelian theory. There are an infinite number of physical states with different ghost numbers. Simple examples of physical states in a non-abelian theory are also given. 
  These notes summarize the lectures delivered in the V Mexican School of Particle Physics, at the University of Guanajuato. We give a survey of the application of Ashtekar's variables to the quantization of General Relativity in four dimensions with special emphasis on the application of techniques of analytic knot theory to the loop representation. We discuss the role that the Jones Polynomial plays as a generator of nondegenerate quantum states of the gravitational field. 
  We present a topological version of two dimensional dilaton supergravity. It is obtained by gauging an extension of the super-Poincar\'e algebra in two space-time dimensions. This algebra is obtained by an unconventional contraction of the super de Sitter algebra. Besides the generators of the super de Sitter algebra it has one more fermionic generator and two more bosonic generators one of them being a central charge. The gauging of this algebra is performed in the usual way. Unlike some proposals for a dilaton supergravity theory we obtain a model which is non-local in the gravitino field. 
  A critical analysis of the conformal approach to the theory of 2d turbulence is delivered. It is shown, in particular, that conformal minimal models cannot give a general turbulent solution, which should provide for constant fluxes of all vorticity integrals of motion. 
  We consider the quantum-mechanical algebra of observables generated by canonical quantization of $SL(2,R)$ Chern-Simons theory with rational charge on a space manifold with torus topology. We produce modular representations generalizing the representations associated to the $SU(2)$ WZW models and we exhibit the explicit polynomial representations of the corresponding fusion algebras. The relation to Kac-Wakimoto characters of highest weight $\widehat{sl}(2)$ representations with rational level is illustrated. Talk given at the Como Conference on ``Integrable Quantum Field Theories'', September 1992. 
  The spatially inhomogeneous large $N$ solutions to Kazakov--Migdal model are analyzed. The set of nonlinear differential equations is derived in the continuum limit. In one dimensional case these equations has a natural interpretation in terms of the dynamics of a Fermi gas. The multidimensional case seems to be inconsistent because of its instability related to the collapse of eigenvalues of the scalar field. 
  We report on the recent progress in computing the effective supergravity action from superstring scattering amplitudes beyond the tree approximation. We discuss the moduli-dependent string loop corrections to gauge, gravitational and Yukawa couplings. 
  We study the electrodynamics of generic charged particles (bosons, fermions, relativistic or not) constrained to move on an infinite plane. An effective gauge theory in 2+1 dimensional spacetime which describes the real electromagnetic interaction of this particles is obtained. The relationship between this effective theory with the Chern-Simons theory is explored. It is shown that the QED lagrangian {\it per se} produces the Chern-Simons constraint relating the current to the effective gauge field in 2+1 D. It is also shown that the geometry of the system unavoidably induces a contribution from the topological $\theta$-term that generates an explicit Chern-Simons term for the effective 2+1 dimensional gauge field as well as a minimal coupling of the matter to it. The possible relation of the effective three dimensional theory with the bosonization of the Dirac fermion field in 2+1 D is briefly discussed as well as the potential applications in Condensed Matter systems. 
  In recent work a multimonopole solution of heterotic string theory was obtained. The monopoles are noted to be stable, in contrast with analogous solutions of Einstein-Maxwell or Yang-Mills-dilaton theory. The existence of this and other classes of stable solitonic solutions in string theory thus provides a possible test for low-energy string theory as distinct from other gauge + gravity theories. 
  We find an explicit solution of the Schr\"odinger equation for a Chern-Simons theory coupled to charged particles on a Riemann surface, when the coefficient of the Chern-Simons term is a rational number (rather than an integer) and where the total charge is zero. We find that the wave functions carry a projective representation of the group of large gauge transformations. We also examine the behavior of the wave function under braiding operations which interchange particle positions. We find that the representation of both the braiding operations and large gauge transformations involve unitary matrices which mix the components of the wave function. The set of wave functions are expressed in terms of appropriate Jacobi theta functions. The matrices form a finite dimensional representation of a particular (well known to mathematicians) version of the braid group on the Riemann surface. We find a constraint which relates the charges of the particles, $q$, the coefficient of the Chern-Simons term, $k$ and the genus of the manifold, $g$: $q^2(g-1)/k$ must be an integer. We discuss a duality between large gauge transformations and braiding operations. 
  Using the appropriate representation of the Poincare group and a definition of minimal coupling, we discuss some aspects of the electromagnetic interactions of charged anyons. In a nonrelativistic expansion, we derive a Schrodinger-type equation for the anyon wave function which includes spin-magnetic field and spin-orbit couplings. In particular, the gyromagnetic ratio for charged anyons is shown to be 2; this last result is essentially a reflection of the fact that the spin is parallel to the momentum in (2+1) dimensions. 
  We write down and solve a closed set of Schwinger-Dyson equations for the two-matrix model in the large $N$ limit. Our elementary method yields exact solutions for correlation functions involving angular degrees of freedom whose calculation was impossible with previously known techniques. The result sustains the hope that more complicated matrix models important for lattice string theory and QCD may also be solvable despite the problem of the angular integrations. As an application of our method we briefly discuss the calculation of wavefunctions with general matter boundary conditions for the Ising model coupled to $2D$ quantum gravity. Some novel insights into the relationship between lattice and continuum boundary conditions are obtained. 
  We prove that the dimensions of coinvariants of certain nilpotent subalgebras of the Virasoro algebra do not change under deformation in the case of irreducible representations of (2,2r+1) minimal models. We derive a combinatorial description of these representations and the Gordon identities from this result. 
  In the same way the folding of the Dynkin diagram of A_{2n} (resp. A_{2n-1}) produces the B_n (resp. C_n) Dynkin diagram, the symmetry algebra W of a Toda model based on B_n (resp. C_n) can be seen as resulting from the folding of a W-algebra based on A_{2n} (resp. A_{2n-1}). More generally, W algebras related to the B-C-D algebra series can appear from W algebras related to the unitary ones. Such an approach is in particular well adapted to obtain fusion rules of W algebras based on non simply laced algebras from fusion rules corresponding to the A_n case. Anagously, super W algebras associated to orthosymplectic superalgebras are deduced from those relative to the unitary A(m,n) series. 
  In this paper I discuss how the component structure of anyon wave functions arises in theories with non-relativistic matter coupled to a Chern-Simons gauge field on the torus. It is shown that there exists a singular gauge transformation which brings the Hamiltonian to free form. The gauge transformation removes a degree of freedom from the Hamiltonian. This degree of freedom generates only a finite dimensional Hilbert space and is responsible for the component structure of free anyon wave functions. This gives an understanding of the need for multiple component anyon wave functions from the point of view of Chern-Simons theory. 
  This is a study of the Landau-Ginzburg/Calabi-Yau correspondence, and related matters, using linear sigma models. 
  We use mirror symmetry to establish the first concrete arena of spacetime topology change in string theory. In particular, we establish that the {\it quantum theories\/} based on certain nonlinear sigma models with topologically distinct target spaces can be smoothly connected even though classically a physical singularity would be encountered.  We accomplish this by rephrasing the description of these nonlinear sigma models in terms of their mirror manifold partners---a description in which the full quantum theory can be described exactly using lowest order geometrical methods.    We establish that, for the known class of mirror manifolds, the moduli space of the corresponding conformal field theory requires not just two but {\it numerous\/}  topologically distinct Calabi-Yau manifolds for its geometric interpretation.  A {\it single\/} family of continuously connected conformal theories thereby probes a host of topologically distinct geometrical spaces giving rise to {\it multiple mirror manifolds}. 
  We study the real time formalism of non-equilibrium many-body theory, in a first quantised language. We argue that on quantising the relativistic scalar particle in spacetime with Minkowski signature, we should study both propagations $e^{i(p^2-m^2)\tilde \lambda}$ and $e^{-i(p^2-m^2)\tilde \lambda}$ on the particle world line. The path integral needs regulation at the mass shell $p^2=m^2$. If we regulate the two propagations independently we get the Feynman propagator in the vacuum, and its complex conjugate. But if the regulation mixes the two propagations then we get the matrix propagator appropriate to perturbation theory in a particle flux. This formalism unifies the special cases of thermal fluxes in flat space and the fluxes `created' by Cosmological expansion, and also gives covariance under change of particle definition in curved space. We comment briefly on the proposed application to closed strings, where we argue that coherent fields and `exponential of quadratic' particle fluxes must {\it both} be used to define the background for perturbation theory. 
  Braiding matrices in rational conformal field theory are considered. The braiding matrices for any two block four point function are computed, in general, using the holomorphic properties of the blocks and the holomorphic properties of rational conformal field theory. The braidings of $SU(N)_k$ with the fundamental are evaluated and are used as examples. Solvable interaction round the face lattice models are constructed from these braiding matrices, and their Boltzmann weights are given. This allows, in particular, for the derivation of the Boltzmann weights of such solvable height models. 
  We present sum representations for all characters of the unitary Virasoro minimal models. They can be viewed as fermionic companions of the Rocha-Caridi sum representations, the latter related to the (bosonic) Feigin-Fuchs-Felder construction. We also give fermionic representations for certain characters of the general $(G^{(1)})_k \times (G^{(1)})_l \over (G^{(1)})_{k+l}}$ coset conformal field theories, the non-unitary minimal models ${\cal M}(p,p+2)$ and ${\cal M}(p,kp+1)$, the $N$=2 superconformal series, and the $\ZZ_N$-parafermion theories, and relate the $q\to 1$ behaviour of all these fermionic sum representations to the thermodynamic Bethe Ansatz. 
  We present an effective quantum action for the gauged WZW model $G_{-k}/H_{-k}$. It is conjectured that it is valid to all orders of the central extension $(-k)$ on the basis that it reproduces the exact spacetime geometry of the zero modes that was previously derived in the algebraic Hamiltonian formalism. Besides the metric and dilaton, the new results that follow from this approach include the exact axion field and the solution of the geodesics in the exact geometry. It is found that the axion field is generally non-zero at higher orders of $1/k$ even if it vanishes at large $k$. We work out the details in two specific coset models, one non-abelian, i.e. $SO(2,2)/SO(2,1)$ and one abelian, i.e $SL(2,\IR)\otimes SO(1,1)^{d-2}/SO(1,1)$. The simplest case $SL(2,\IR)/\IR$ corresponds to a limit. 
  In this paper we consider symplectic and Hamiltonian structures of systems generated by actions of sigma-model type and show that these systems are naturally connected with specific symplectic geometry on loop spaces of Riemannian and (pseudo)Riemannian manifolds. 
  As shown by Witten the N=1 supersymmetric gauged WZW model based on a group G has an extended N=2 supersymmetry if the gauged subgroup H is so chosen that G/H is Kahler. We extend Witten's result and prove that the N=1 supersymmetric gauged WZW models over G X U(1) are actually invariant under N=4 superconformal transformations if the gauged subgroup H is such that G/HXSU(2) is a quaternionic symmetric space. A previous construction of "maximal" N=4 superconformal algebras with SU(2)XSU(2)XU(1) symmetry is reformulated and further developed so as to relate them to the N=4 gauged WZW models. Based on earlier results we expect the quantization of N=4 gauged WZW models to yield the unitary realizations of maximal N=4 superconformal algebras provided by this construction. 
  We study the conformal groups of Jordan algebras along the lines suggested by Kantor. They provide a natural generalization of the concept of conformal transformations that leave 2-angles invariant to spaces where "p-angles" can be defined. We give an oscillator realization of the generalized conformal groups of Jordan algebras and Jordan triple systems(JTS). These results are extended to Jordan superalgebras and super JTS's. We give the conformal algebras of simple Jordan algebras, hermitian JTS's and the simple Jordan superalgebras as classified by Kac. 
  The level spacing distributions in the Gaussian Unitary Ensemble, both in the ``bulk of the spectrum,'' given by the Fredholm determinant of the operator with the sine kernel ${\sin \pi(x-y) \over \pi(x-y)}$ and on the ``edge of the spectrum,'' given by the Airy kernel ${\rm{Ai}(x) \rm{Ai}'(y) - \rm{Ai}(y) \rm{Ai}'(x) \over (x-y)}$, are determined by compatible systems of nonautonomous Hamiltonian equations. These may be viewed as special cases of isomonodromic deformation equations for first order $ 2\times 2 $ matrix differential operators with regular singularities at finite points and irregular ones of Riemann index 1 or 2 at $\infty$. Their Hamiltonian structure is explained within the classical R-matrix framework as the equations induced by spectral invariants on the loop algebra ${\tilde{sl}(2)}$, restricted to a Poisson subspace of its dual space ${\tilde{sl}^*_R(2)}$, consisting of elements that are rational in the loop parameter. 
  We investigate the evolution of small perturbations around black strings and branes which are low energy solutions of string theory. For simplicity we focus attention on the zero charge case and show that there are unstable modes for a range of time frequency and wavelength in the extra $10-D$ dimensions. These perturbations can be stabililized if the extra dimensions are compactified to a scale smaller than the minimum wavelength for which instability occurs and thus will not affect large astrophysical black holes in four dimensions. We comment on the implications of this result for the Cosmic Censorship Hypothesis. 
  An infinite number of free field realizations of the universal nonlinear $\hat{W}_{\infty}^{(N)}$ ($\hat{W}_{1+\infty}^{(N)}$) algebras, which are identical to the KP Hamiltonian structures, are obtained in terms of $p$ plus $q$ scalars of different signatures with $p-q=N$. They are generalizations of the Miura transformation, and naturally give rise to the modified KP hierarchies via corresponding realizations of the latter. Their characteristic Lie-algebraic origin is shown to be the graded $SL(p,q)$. 
  We study string theory on orbifolds in the presence of an antisymmetric constant background field in detail and discuss some of new aspects of the theory. It is pointed out that the term with the antisymmetric background field can be regarded as a topological term like a Chern-Simons term or a Wess-Zumino term. Detailed analysis in the operator formalism shows that orbifold models with topologically nontrivial twists exhibit various anomalous behavior: Zero mode variables obey nontrivial quantization conditions. Coordinate transformations which define orbifolds are modified at quantum level. Wave functions of twisted strings in general acquire nontrivial phases when they move around non-contractible loops on orbifolds. Zero mode eigenvalues are shifted from naively expected values, in favor of modular invariance. 
  In this talk new formulations of the Green--Schwarz heterotic strings in $D$ dimensions that involve commuting spinors, are reviewed. These models are invariant under $n$--extended, world sheet supersymmetry as well as under $N=1$, target space supersymmetry where $n\leq D-2$ and $D=3,4,6,10$. The world sheet supersymmetry replaces $n$ components (and provides a geometrical meaning) of the $\kappa$--symmetry in the Green--Schwarz approach. The models in $D=10$ for $n=1,2,8$ are discussed explicitly. 
  We show in two simple examples that for one-dimensional quantum chains with quantum group symmetries, the correlation functions of local operators are, in general, infrared divergent. If one considers, however, correlation functions invariant under the quantum group, the divergences cancel out. 
  We propose an approach to statistical systems on lattices with sphere-like topology. Focusing on the Ising model, we consider the thermodynamic limit along a sequence of lattices which preserve the {\em fixed} large scale geometry. The hypothesis of scaling appears to hold at criticality, pointing at a sensible definition of the continuum limit of the model in the curved space. 
  Hawking's radiance, even as computed without account of backreaction, departs from blackbody form due to the mode dependence of the barrier penetration factor. Thus the radiation is not the maximal entropy radiation for given energy. By comparing estimates of the actual entropy emission rate with the maximal entropy rate for the given power, and using standard ideas from communication theory, we set an upper bound on the permitted information outflow rate. This is several times the rates of black hole entropy decrease or radiation entropy production. Thus, if subtle quantum effects not heretofore accounted for code information in the radiance, the information that was thought to be irreparably lost down the black hole may gradually leak back out from the black hole environs over the full duration of the hole's evaporation. 
  Some mistakes have been corrected 
  If the Higgs particle is never found, one will need an alternative theory for vector boson masses. I propose such a theory involving an antisymmetric tensor potential coupled to a gauge field. 
  I propose a new mathematical form for the quantum theory of gravity coupled to matter. The motivation is from the connection between CSW TQFT and the Ashtekar variables. I also connect the algebraic structure of TQFT with some thoughts about the interpretation of quantum mechanics in a general relativistic setting. 
  We construct a four dimensional topological Quantum Field Theory from a modular tensor category. We complete the proof in the case of SU(2)q at a root of unity. Our construction may be important in the physical interpretation of the Chern Simons state in the Ashtekar variables. 
  Given a principal G-bundle over a smooth manifold M, with G a compact Lie group, and given a finite-dimensional unitary representation of G, one may define an algebra of functions on the space of connections modulo gauge transformations, the ``holonomy Banach algebra'' H_b, by completing an algebra generated by regularized Wilson loops. Elements of the dual H_b* may be regarded as a substitute for measures on the space of connections modulo gauge transformations. There is a natural linear map from diffeomorphism- invariant elements of H_b* to the space of complex-valued ambient isotopy invariants of framed oriented links in M. Moreover, this map is one-to-one. Similar results hold for a C*-algebraic analog, the ``holonomy C*-algebra.'' These results clarify the relation between diffeomorphism-invariant gauge theories and link invariants, and the framing dependence of the expectation values of products of Wilson loops. 
  We examine the various linkings in space-time of ``ball-like'' and ``ring-like'' topological solitons in certain nonlinear sigma models in 2+1 and 3+1 dimensions. By going to theories where soliton overlaps are forbidden, these linkings become homotopically nontrivial and can be studied by investigating the topology of the corresponding configuration spaces. Our analysis reveals the existence of topological terms which give the linking number of the world-tubes of distinct species of ball solitons in 2+1 dimensions, or which in 3+1 dimensions count the number of times a ball or ring soliton threads through the center of a ring of a different species. We explicitly construct these terms for our models, and generalize them to cases where soliton overlaps are no longer strictly forbidden so the terms are no longer purely topological. One of the (3+1)-dimensional theories we consider also has topological solitons which consist of two rings (of distinct species) linked in space. 
  A simple lattice model inducing a gauge theory is considered. The model describes an interaction of a gauge field to an $N\times N$ complex matrix scalar field transforming as a field in the fundamental representation. In contrast to the Kazakov-Migdal model the model contains only the linear interaction between scalar and gauge lattice fields. This model does not suffer from extra local U(1) symmetries. In an approximation of a translation invariant master field the large N limit of the model is investigated. At large N the gauge fields can be integrated out yielding an effective theory describing an interaction of eigenvalues of the master field. The reduced model exhibits phase transitions at the points $\beta_{\bar {cr}}$ and $\beta _{\underline{cr}}$ and the region $(\beta_{\bar{cr}}, \beta_{\underline{cr}})$ separates the strong and weak regions of the model. To study the behaviour of the model at large $N$ in more systematic way the quenched momentum prescription with constraints for treating the large N limit of gauge theories is used. With the help of the technique of orthogonal polynomials nonlinear equations describing the large N limit of the reduced model {\it with quenching} are presented. 
  We find an infinite set of new noncommuting conserved charges in a specific class of perturbed CFT's and present a criterion for their existence.They appear to be higher momenta of the already known commuting conserved currents.The algebra they close consists of two noncommuting$W_\infty$ algebras.We find various Virasoro subalgebras of the full symmetry algebra. It is shown on the examples of the perturbed Ising and Potts models that one of them plays an essencial role in the computation of the correlation functions of the fields of the theory. 
  We discuss a Gedanken experiment for the measurement of the area of the apparent horizon of a black hole in quantum gravity. Using rather general and model-independent considerations we find a generalized uncertainty principle which agrees with a similar result obtained in the framework of string theories. The result indicates that a minimum length of the order of the Planck length emerges naturally from any quantum theory of gravity, and that the concept of black hole is not operationally defined if the mass is smaller than the Planck mass. 
  The partition function of two dimensional QCD on a Riemann surface of area $A$ is expanded as a power series in $1/N$ and $A$. It is shown that the coefficients of this expansion are precisely determined by a sum over maps from a two dimensional surface onto the two dimensional target space. Thus two dimensional QCD has a simple interpretation as a closed string theory. 
  We employ the conformal bootstrap to re-examine the problem of finding the critical behavior of four-Fermion theory at its strong coupling fixed point. Existence of a solution of the bootstrap equations indicates self-consistency of the assumption that, in space-time dimensions less than four, the renormalization group flow of the coupling constant of a four-Fermion interaction has a nontrivial fixed point which is generally out of the perturbative regime. We exploit the hypothesis of conformal invariance at this fixed point to reduce the set of the Schwinger-Dyson bootstrap equations for four-Fermion theory to three equations which determine the scale dimension of the Fermion field $\psi$, the scale dimension of the composite field $\bar{\psi}\psi$ and the critical value of the Yukawa coupling constant. We solve the equations assuming this critical value to be small. We show that this solution recovers the fixed point for the four-fermion interaction with $N$-component fermions in the limit of large $N$ at (Euclidean) dimensions $d$ between two and four. We perform a detailed analysis of the $1/N$-expansion in $d=3$ and demonstrate full agreement with the conformal bootstrap. We argue that this is a useful starting point for more sophisticated computations of the critical indices. 
  Previous results on quasi-classical limit of the KP hierarchy and its W-infinity symmetries are extended to the Toda hierarchy. The Planck constant $\hbar$ now emerges as the spacing unit of difference operators in the Lax formalism. Basic notions, such as dressing operators, Baker-Akhiezer functions and tau function, are redefined. $W_{1+\infty}$ symmetries of the Toda hierarchy are realized by suitable rescaling of the Date-Jimbo-Kashiara-Miwa vertex operators. These symmetries are contracted to $w_{1+\infty}$ symmetries of the dispersionless hierarchy through their action on the tau function. (A few errors in the earlier version is corrected.) 
  A new formulation of fermions based on a second order action is proposed. An analysis of the $U(1)$ anomaly allows us to test the validity of the formalism at the quantum level. This formulation gives a new perpective to the introduction of parity non-invariant interactions. 
  A new formulation for fermions on the lattice based on a discretization of a second order formalism is proposed. A comparison with the first order formalism in connection with the $U(1)$ anomaly and the doubling problem is presented. The new formulation allows to eliminate half of the degrees of freedom, which can be important in non-perturbative calculations, and it can new possibities to the formulation of chiral gauge theories. 
  \small The SL$(2,R)/U(1)$ coset model, with $U(1)$ an element of the third conjugacy class of $SL(2,R)$ subgroups, is considered. The resulting theory is seen to collapse to a one dimensional field theory of Liouville. Then the 2 dimensional black hole $SL(2,R)/U(1)$, with $U(1)$ a non-compact subgroup boosted by a Lorentz transformation, is considered. In the limit of high boost, the resulting black hole is found to tend to the Liouville field coupled to $a\ C=1$ matter field. The limit of the vertex operators of the 2 dimensional black hole also tend to those of the $C=1$ two dimensional gravity. 
  The higher-spin geometries of $W_\infty$-gravity and $W_N$-gravity are analysed and used to derive the complete non-linear structure of the coupling to matter and its symmetries. The symmetry group is a subgroup of the symplectic diffeomorphisms of the cotangent bundle of the world-sheet, and the $W_N$ geometry is obtained from a non-linear truncation of the $W_\infty$ geometry. Quantum W-gravity is briefly discussed. (Talk given at {\it Pathways to Fundamental Interactions}, the 16th John Hopkins Workshop on Current Problems in Particle Theory, Gothenborg, 1992.) 
  We present a general scheme for describing su(N)_k fusion rules in terms of elementary couplings, using Berenstein-Zelevinsky triangles. A fusion coupling is characterized by its corresponding tensor product coupling (i.e. its Berenstein-Zelevinsky triangle) and the threshold level at which it first appears. We show that a closed expression for this threshold level is encoded in the Berenstein-Zelevinsky triangle and an explicit method to calculate it is presented. In this way a complete solution of su(4)_k fusion rules is obtained. 
  The Hamiltonian structure of the monodromy preserving deformation equations of Jimbo {\it et al } is explained in terms of parameter dependent pairs of moment maps from a symplectic vector space to the dual spaces of two different loop algebras. The nonautonomous Hamiltonian systems generating the deformations are obtained by pulling back spectral invariants on Poisson subspaces consisting of elements that are rational in the loop parameter and identifying the deformation parameters with those determining the moment maps. This construction is shown to lead to ``dual'' pairs of matrix differential operators whose monodromy is preserved under the same family of deformations. As illustrative examples, involving discrete and continuous reductions, a higher rank generalization of the Hamiltonian equations governing the correlation functions for an impenetrable Bose gas is obtained, as well as dual pairs of isomonodromy representations for the equations of the Painleve transcendents $P_{V}$ and $P_{VI}$. 
  We construct the $N=2$ super $W_4$ algebra as a certain reduction of the second Gel'fand-Dikii bracket on the dual of the Lie superalgebra of $N=1$ super pseudo-differential operators. The algebra is put in manifestly $N=2$ supersymmetric form in terms of three $N=2$ superfields $\Phi_i(X)$, with $\Phi_1$ being the $N=2$ energy momentum tensor and $\Phi_2$ and $\Phi_3$ being conformal spin $2$ and $3$ superfields respectively. A search for integrable hierarchies of the generalized KdV variety with this algebra as Hamiltonian structure gives three solutions, exactly the same number as for the $W_2$ (super KdV) and $W_3$ (super Boussinesq) cases. 
  We define a Chern- Simons Lagrangian for a system of planar particles topologically interacting at a distance. The anyon model appears as a particular case where all the particles are identical. We propose exact N-body eigenstates, set up a perturbative algorithm, discuss the case where some particles are fixed on a lattice, and also consider curved manifolds. PACS numbers: 05.30.-d, 11.10.-z 
  The moduli dependent Yukawa couplings between twisted sectors of ${\bf Z}_M\times {\bf Z}_N$ Coxeter orbifolds are studied. 
  The \nl \cls for the N=1 supersymmetric KdV equation are shown to be related in a simple way to powers of the fourth root of its Lax operator. This provides a direct link between the supersymmetry invariance and the existence of \nl conservation laws. It is also shown that nonlocal conservation laws exist for the two integrable N=2 supersymmetric KdV equations whose recursion operator is known. 
  To study the behavior of the Kazakov-Migdal at large N the quenched momentum prescription with constraints for treating the large N limit of gauge theories is used. It is noted that it leads to a quartic dependence of an action on unitary matrix instead of a quadratic dependence discussed in previous considerations. Therefore the model is not exactly solvable in the weak coupling limit. An approximation procedure for investigation of the model is outlined. In this approximation an indication to a phase transition for $d<4,8$ with $\beta_{cr}=\frac{1}{d-4,8}$ is obtained. 
  In the formulation of Banks, Peskin and Susskind, we show that one can construct evolution equations for the quantum mechanical density matrix $\rho$ with operators which do not commute with hamiltonian which evolve pure states into mixed states, preserve the normalization and positivity of $\rho$ and conserve energy. Furthermore, it seems to be different from a quantum mechanical system with random sources. 
  We show that a special superconformal coset (with $\hat c =3$) is equivalent to $c=1$ matter coupled to two dimensional gravity. This identification allows a direct computation of the correlation functions of the $c=1$ non-critical string to all genus, and at nonzero cosmological constant, directly from the continuum approach. The results agree with those of the matrix model. Moreover we connect our coset with a twisted version of a Euclidean two dimensional black hole, in which the ghost and matter systems are mixed. 
  We consider the $ su(n) $ spin chains with long range interactions and the spin generalization of the Calogero-Sutherland models. We show that their properties derive from a transfer matrix obeying the Yang-Baxter equation. We obtain the expression of the conserved quantities and we diagonalize them. 
  The Symmetry charges associated with the Lian-Zuckerman states for $d<2$ closed string theory are constructed. Unlike in the open string case, it is shown here that the symmetry charges commute among themselves and act trivially on all the physical states. 
  Product forms of characters of Virasoro minimal models are obtained which factorize into $(2,\odd)\times(3,\even)$ characters. These are related by generalized Rogers-Ramanujan identities to sum forms allowing for a quasiparticle interpretation. The corresponding dilogarithm identities are given and the factorization is used to analyse the related path space structure as well as the fusion of the maximally extended chiral algebra. 
  Four apparently different bosonizations of the $U_q(su(2)_k)$ quantum current algebra for arbitrary level $k$ have recently been proposed in the literature. However, the relations among them have so far remained unclear except in one case. Assuming a special standard form for the $U_q(su(2)_k)$ quantum currents, we derive a set of general consistency equations that must be satisfied. As particular solutions of this set of equations, we recover two of the four bosonizations and we derive a new and simpler one. Moreover, we show that the latter three, and the remaining two bosonizations which cannot be derived directly from this set of equations since by construction they do not have the standard form, are all related to each other through some redefinitions of their Heisenberg boson oscillators. 
  These lectures review some of the basic properties of $N=2$ superconformal field theories and the corresponding topological field theories. One of my basic aims is to show how the techniques of topological field theory can be used to compute effective \LG potentials for perturbed $N=2$ superconformal field theories. In particular, I will briefly discuss the application of these ideas to $N=2$ supersymmetric quantum integrable models. (Lectures given at the Summer School on High Energy Physics and Cosmology, Trieste, Italy, June 15th -- July 3rd, 1992. To appear in the proceedings.) 
  Recently, string theory on Calabi--Yau manifolds was constructed and was shown to be a fully consistent, space--time supersymmetric string theory. The physically interesting case is the case of three generations. Intriguingly, it appears at the present that there is a unique manifold which gives rise to three generations. We describe in this paper a full fledged string theory on this manifold in which the complete spectrum and all the Yukawa couplings can be computed exactly. 
  Previous results on quasi-classical limit of the KP and Toda hierarchies are now extended to the BKP hierarchy. Basic tools such as the Lax representation, the Baker-Akhiezer function and the tau function are reformulated so as to fit into the analysis of quasi-classical limit. Two subalgebras $\WB_{1+\infty}$ and $\wB_{1+\infty}$ of the W-infinity algebras $W_{1+\infty}$ and $w_{1+\infty}$ are introduced as fundamental Lie algebras of the BKP hierarchy and its quasi-classical limit, the dispersionless BKP hierarchy. The quantum W-infinity algebra $\WB_{1+\infty}$ emerges in symmetries of the BKP hierarchy. In quasi-classical limit, these $\WB_{1+\infty}$ symmetries are shown to be contracted into $\wB_{1+\infty}$ symmetries of the dispersionless BKP hierarchy. 
  % A new, formal, non-combinatorial approach to invariants of % three-dimensional manifolds of Reshetikhin, Turaev and % Witten in the framework of non-perturbative topological % quantum Chern-Simons theory, corresponding to an arbitrary % compact simple Lie group, is presented. A direct % implementation of surgery instructions in the context of % quantum field theory is proposed. An explicit form of the % specialization of the invariant to the group SU(2) is % derived, and shown to agree with the result of Lickorish. % % AMS subject classifications (1991): 57M25, 57N10, 57R65, 81T13. 
  Star products on the classical double group of a simple Lie group and on corresponding symplectic grupoids are given so that the quantum double and the "quantized tangent bundle" are obtained in the deformation description. "Complex" quantum groups and bicovariant quantum Lie algebras are discused from this point of view. Further we discuss the quantization of the Poisson structure on symmetric algebra $S(g)$ leading to the quantized enveloping algebra $U_{h}(g)$ as an example of biquantization in the sense of Turaev. Description of $U_{h}(g)$ in terms of the generators of the bicovariant differential calculus on $F(G_q)$ is very convenient for this purpose. Finally we interpret in the deformation framework some well known properties of compact quantum groups as simple consequences of corresponding properties of classical compact Lie groups. An analogue of the classical Kirillov's universal character formula is given for the unitary irreducible representation in the compact case. 
  We consider a coset construction of minimal models. We define it rigorously and prove that it gives superconformal minimal models. This construction allows to build all primary fields of superconformal models and to calculate their three-point correlation functions. 
  It is shown that zero ghost conformal blocks of coset theory G/H are determined uniquely by those of G and H theories. G/G theories are considered as an example, their structure constants and correlation functions on sphere are calculated. 
  We study the process of spinodal decomposition in a scalar quantum field theory that is quenched from an equilibrium disordered initial state at $T_i > T_f$ to a final state at $T_f \approx 0$. The process of formation and growth of correlated domains is studied in a Hartree approximation. We find an approximate scaling law for the size of the domains $\xi_D(t) \approx \sqrt{t \xi_0}$ at long times for weakly coupled theories, with $\xi_0$ the zero temperature correlation length. 
  In this paper a new quasi-triangular Hopf algebra as the quantum double of the Heisenberg-Weyl algebra is presented.Its universal R-matrix is built and the corresponding representation theory are studied with the explict construction for the representations of this quantum double. \newpage 
  We investigate the variation of the string field action under changes of the string field vertices giving rise to different decompositions of the moduli spaces of Riemann surfaces. We establish that any such change in the string action arises from a field transformation canonical with respect to the Batalin-Vilkovisky (BV) antibracket, and find the explicit form of the generator of the infinitesimal transformations. Two theories using different decompositions of moduli space are shown to yield the same gauge fixed action upon use of different gauge fixing conditions. We also elaborate on recent work on the covariant BV formalism, and emphasize the necessity of a measure in the space of two dimensional field theories in order to extend a recent analysis of background independence to quantum string field theory. 
  We present a B\"acklund transformation (a discrete symmetry transformation) for the self-duality equations for supersymmetric gauge theories in N-extended super-Minkowski space ${\cal M}^{4|4N}$ for an arbitrary semisimple gauge group. For the case of an $A_1$ gauge algebra we integrate the transformation starting with a given solution and iterating the process we construct a hierarchy of explicit solutions. 
  We present a detailed investigation of scattering processes in $W_3$ string theory. We discover further physical states with continuous momentum, which involve excitations of the ghosts as well as the matter, and use them to gain a better understanding of the interacting theory. The scattering amplitudes display factorisation properties, with states from the different sectors of the theory being exchanged in the various intermediate channels. We find strong evidence for the unitarity of the theory, despite the unusual ghost structure of some of the physical states. Finally, we show that by performing a transformation of the quantum fields that involves mixing the ghost fields with one of the matter fields, the structure of the physical states is dramatically simplified. The new formalism provides a concise framework within which to study the $W_3$ string. 
  Anyonic oscillators with fractional statistics are built on a two-dimensional square lattice by means of a generalized Jordan-Wigner construction, and their deformed commutation relations are thoroughly discussed. Such anyonic oscillators, which are non-local objects that must not be confused with $q$-oscillators, are then combined \`a la Schwinger to construct the generators of the quantum group $SU(2)_q$ with $q=\exp({\rm i}\pi\nu)$, where $\nu$ is the anyonic statistical parameter. 
  We consider the configuration space of the Skyrme model and give a simple proof that loops generated by full spatial rotations are contractible in the even-, and non-contractible in the odd-winding-number sectors. 
  We study four dimensional systems of global, axionic and local strings. By using the path integral formalism, we derive the dual formulation of these systems, where Goldstone bosons, axions and missive vector bosons are described by antisymmetric tensor fields, and strings appear as a source for these tensor fields. We show also how magnetic monopoles attached to local strings are described in the dual formulation. We conclude with some remarks. 
  The antifields of the Batalin-Vilkovisky Lagrangian quantization are standard antighosts of certain collective fields. These collective fields ensure that Schwinger-Dyson equations are satisfied as a consequence of the gauge symmetry algebra. The associated antibracket and its canonical structure appear naturally if one integrates out the corresponding ghost fields. An analogous Master Equation for the action involving these ghosts follows from the requirement that the path integral gives rise to the correct Schwinger-Dyson equations. 
  A class of Poisson embeddings of reduced, finite dimensional symplectic vector spaces into the dual space $\Lg_R^*$ of a loop algebra, with Lie Poisson structure determined by the classical split $R$--matrix $R=P_+ - P_-$ is introduced. These may be viewed as equivariant moment maps inducing natural Hamiltonian actions of the ``dual'' group $\LG_R = \LGp \times \LGm$ of a loop group $\LG$ on the symplectic space. The $R$--matrix version of the Adler-Kostant-Symes theorem is used to induce commuting flows determined by isospectral equations of Lax type. The compatibility conditions determine finite dimensional classes of solutions to integrable systems of PDE's, which can be integrated using the standard Liouville-Arnold approach. This involves an appropriately chosen ``spectral Darboux'' (canonical) coordinate system in which there is a complete separation of variables. As an example, the method is applied to the determination of finite dimensional quasi-periodic solutions of the sine-Gordon equation. 
  The classical black hole background of two-dimensional string theory is examined after including the effect of the tachyon field. Keeping all terms upto $O(T^2)$, and making no other approximations, the only consistent classical solution to the resulting dilaton-graviton theory is found to be flat spacetime with a nontrivial dilaton. 
  The structure of one-loop divergences of two-dimensional dilaton-Maxwell quantum gravity is investigated in two formalisms: one using a convenient effective action and the other a unique effective action. The one-loop divergences (including surface divergences) of the convenient effective action are calculated in three different covariant gauges: (i) De Witt, (ii) $\Omega$-degenerate De Witt, and (iii) simplest covariant. The on-shell effective action is given by surface divergences only (finiteness of the $S$-matrix), which yet depend upon the gauge condition choice.   Off-shell renormalizability is discussed and classes of renormalizable dilaton and Maxwell potentials are found which coincide in the cases of convenient and unique effective actions. A detailed comparison of both situations, i.e. convenient vs. unique effective action, is given. As an extension of the procedure, the one-loop effective action in two-dimensional dilaton-Yang-Mills gravity is calculated. 
  We perform the sewing of two (dual) Ramond reggeon vertices and derive an algorithm by means of which the so obtained four-Ramond reggeon vertex may be explicitly computed at arbitrary oscillator (mass) level. A closed form of the four-vertex is then deduced on the basis of a comparison to all terms obtained by sewing that contain only level zero and one oscillators. Results are presented for both complex fermions and for the previously studied case of real fermions. 
  We derive the current algebra of supersymmetric principal chiral models with a Wess-Zumino term. At the critical point one obtains two commuting super Kac-Moody algebra as expected, but in general there are intertwining fields connecting both right and left sectors, analogously to the bosonic case. Moreover, in the present supersymmetric extension we have a quadratic algebra, rather than an affine Lie algebra, due to the mixing between bosonic and fermionic fields since the purely fermionic sector displays a Lie algebra as well. 
  We consider a string inspired effective Lagrangian for the graviton and dilaton, containing Einstein gravity at the zero slope limit. The numerical solution of the problem shows asymptotically an inflationary universe. The time is measured by the dilaton, as one expects. The result is independent of the introduction of ad-hoc self interactions for the dilaton field. 
  The question as to whether neutron acceleration can occur in uniform electromagnetic fields is examined. Although such an effect has been predicted using the canonical equations of motion some doubt has been raised recently as to whether it is in principle observable for a spin 1/2 particle. To resolve this issue a gedanken experiment is proposed and analyzed using a wave packet construction for the neutron beam. By allowing arbitrary orientation for the neutron spin as well as for the electric and magnetic fields a non vanishing acceleration of the center of the neutron wave packet is found which confirms the predictions of the canonical formalism. 
  We outline the basic principles of canonical formalism for the Nambu mechanics---a generalization of Hamiltonian mechanics proposed by Yoichiro Nambu in 1973. It is based on the notion of Nambu bracket which generalizes the Poisson bracket to the multiple operation of higher order $n \geq 3$ on classical observables and is described by Hambu-Hamilton equations of motion given by $n-1$ Hamiltonians. We introduce the fundamental identity for the Nambu bracket which replaces Jacobi identity as a consistency condition for the dynamics. We show that Nambu structure of given order defines a family of subordinated structures of lower order, including the Poisson structure, satisfying certain matching conditions. We introduce analogs of action from and principle of the least action for the Nambu mechanics and show how dynamics of loops ($n-2$-dimensional objects) naturally appears in this formalism. We discuss several approaches to the quantization problem and present explicit representation of Nambu-Heisenberg commutation relation for $n=3$ case. We emphasize the role higher order algebraic operations and mathematical structures related with them play in passing from Hamilton's to Nambu's dynamical picture. 
  The integrable structure of open--closed string theories in the $(p,q)$ conformal minimal model backgrounds is presented. The relation between the $\tau$--function of the closed string theory and that of the open--closed string theory is uncovered. The resulting description of the open--closed string theory is shown to fit very naturally into the framework of the $sl(q,{\rm C})$ KdV hierarchies. In particular, the twisted bosons which underlie and organise the structure of the closed string theory play a similar role here and may be employed to derive loop equations and correlation function recursion relations for the open--closed strings in a simple way. 
  We apply the self consistency method for determining critical exponents to a model with a four fermi interaction coupled to QED and compute various gauge independent exponents in arbitrary dimensions in the large $N$ expansion at $O(1/N)$. The formalism is developed to include a Chern Simons term in three dimensions and the effect such a term has on the exponents is deduced. 
  A new proof of perturbative renormalizability and infrared finiteness for a scalar massless theory is obtained from a formulation of renormalized field theory based on the Wilson renormalization group. The loop expansion of the renormalized Green functions is deduced from the Polchinski equation of renormalization group. The resulting Feynman graphs are organized in such a way that the loop momenta are ordered. It is then possible to analyse their ultraviolet and infrared behaviours by iterative methods. The necessary subtractions and the corresponding counterterms are automatically generated in the process of fixing the physical conditions for the ``relevant'' vertices at the normalization point. The proof of perturbative renormalizability and infrared finiteness is simply based on dimensional arguments and does not require the usual analysis of topological properties of Feynman graphs. 
  The exactness and universality observed in the quantum Hall effect suggests the existence of a symmetry principle underlying Laughlin's theory. We review the role played by the infinite $W_{\infty }$ and conformal algebras as dynamical symmetries of incompressible quantum fluids and show how they predict universal finite-size effects in the excitation spectrum. 
  The interaction between an ultrarelativistic particle and a linear array made up of $N$ two-level systems (^^ ^^ AgBr" molecules) is studied by making use of a modified version of the Coleman-Hepp Hamiltonian. Energy-exchange processes between the particle and the molecules are properly taken into account, and the evolution of the total system is calculated exactly both when the array is initially in the ground state and in a thermal state. In the macroscopic limit ($N \rightarrow \infty$), the system remains solvable and leads to interesting connections with the Jaynes-Cummings model, that describes the interaction of a particle with a maser. The visibility of the interference pattern produced by the two branch waves of the particle is computed, and the conditions under which the spin array in the $N \rightarrow \infty$ limit behaves as a ^^ ^^ detector" are investigated. The behavior of the visibility yields good insights into the issue of quantum measurements: It is found that, in the thermodynamical limit, a superselection-rule space appears in the description of the (macroscopic) apparatus. In general, an initial thermal state of the ^^ ^^ detector" provokes a more substantial loss of quantum coherence than an initial ground state. It is argued that a system decoheres more as the temperature of the detector increases. The problem of ^^ ^^ imperfect measurements" is also shortly discussed. 
  An arbitrary dimensional expression is given for the anomalous dimension of the fermion field in a model with a four point interaction and a $U(1)$ gauge field, at $O(1/N^2)$ within a large flavour expansion in the Landau gauge. 
  Recently, DiFrancesco and Zuber have characterized the RCFTs which have a description in terms of a fusion potential in one variable, and proposed a generalized potential to describe other theories. In this note we give a simple criterion to determine when such a generalized description is possible. We also determine which RCFTs can be described by a fusion potential in more than one variable, finding that in fact all RCFTs can be described in such a way, as conjectured by Gepner. 
  The formula existing in the literature for the ADM mass of 2D dilaton gravity is incomplete. For example, in the case of an infalling matter shockwave this formula fails to give a time-independent mass, unless a very special coordinate system is chosen. We carefully carry out the canonical formulation of 2D dilaton gravity theories (classical, CGHS and RST). As in 4D general relativity one must add a boundary term to the bulk Hamiltonian to obtain a well-defined variational problem. This boundary term coincides with the numerical value of the Hamiltonian and gives the correct mass which obviously is time-independent. 
  It is known that the Lagrangian for the edge states of a Chern-Simons theory describes a coadjoint orbit of a Kac-Moody (KM) group with its associated Kirillov symplectic form and group representation. It can also be obtained from a chiral sector of a nonchiral field theory. We study the edge states of the abelian $BF$ system in four dimensions (4d) and show the following results in almost exact analogy: 1) The Lagrangian for these states is associated with a certain 2d generalization of the KM group. It describes a coadjoint orbit of this group as a Kirillov symplectic manifold and also the corresponding group representation. 2) It can be obtained from with a ``self-dual" or ``anti-self-dual" sector of a Lagrangian describing a massless scalar and a Maxwell field [ the phrase ``self-dual" here being used essentially in its sense in monopole theory]. There are similar results for the nonabelian $BF$ system as well. These shared features of edge states in 3d and 4d suggest that the edge Lagrangians for $BF$ systems are certain natural generalizations of field theory Lagrangians related to KM groups. 
  We show that the Hopf link invariants for an appropriate set of finite dimensional representations of $ U_q SL(2)$ are identical, up to overall normalisation, to the modular S matrix of Kac and Wakimoto for rational $k$ $\widehat {sl(2)}$ representations. We use this observation to construct new modular Hopf algebras, for any root of unity $q=e^{-i\pi m/r}$, obtained by taking appropriate quotients of $U_q SL(2)$, that give rise to 3-manifold invariants according to the approach of Reshetikin and Turaev. The phase factor correcting for the `framing anomaly' in these invariants is equal to $ e^{- {{i \pi} \over 4} ({ {3k} \over {k+2}})}$, an analytic continuation of the anomaly at integer $k$. As expected, the Verlinde formula gives fusion rule multiplicities in agreement with the modular Hopf algebras. This leads to a proposal, for $(k+2)=r/m$ rational with an odd denominator, for a set of $\widehat {sl(2)}$ representations obtained by dropping some of the highest weight representations in the Kac-Wakimoto set and replacing them with lowest weight representations. For this set of representations the Verlinde formula gives non-negative integer fusion rule multiplicities. We discuss the consistency of the truncation to highest and lowest weight representations in conformal field theory. 
  Anomalies can be anticipated at the classical level without changing the classical cohomology, by introducing extra degrees of freedom. In the process, the anomaly does not quite disappear. We show that, in fact, it is shifted to new symmetries that come with the extra fields. 
  By considering corrections to the asymptotic scaling functions of the photon and electron in quantum electrodynamics with $\Nf$ flavours, we solve the skeleton Dyson equations at $O(1/\Nf)$ in the large $\Nf$ expansion at the $d$-dimensional critical point of the theory and deduce the critical exponent $\beta^\prime(g_c)$, in arbitrary dimensions, and subsequently present a method for computing higher order corrections to $\beta(g)$. 
  Following our earlier work we argue in detail for the equivalence of the nonlinear $\sigma$ model with Hopf term at~$\theta=\pi/2s$ ~and an interacting spin-s theory. We give an ansatz for spin-s operators in the $\sigma$ model and show the equivalence of the correlation functions.We also show the relation between topological and Noether currents. We obtain the Lorentz and discrete transformation properties of the spin-s operator from the fields of the $\sigma$ model. We also explore the connection of this model with Quantum Hall Fluids. 
  A rotating charged black string solution in the low energy effective field theory describing five dimensional heterotic string theory is constructed. The solution is labelled by mass, electric charge, axion charge and angular momentum per unit length. The extremal limit of this solution is also studied. 
  BFV--BRST charge for q-deformed algebras is not unique. Different constructions of it in the classical as well as in the quantum phase space for the $q$-deformed algebra sl_q(2) are discussed. Moreover, deformation of the phase space without deforming the generators of sl(2) is considered. $\hbar$-q-deformation of the phase space is shown to yield the Witten's second deformation. To study the BFV--BRST cohomology problem when both the quantum phase space and the group are deformed, a two parameter deformation of sl(2) is proposed, and its BFV-BRST charge is given. 
  We classify the automorphisms of the (chiral) level-k affine SU(3) fusion rules, for any value of k, by looking for all permutations that commute with the modular matrices S and T. This can be done by using the arithmetic of the cyclotomic extensions where the problem is naturally posed. When k is divisible by 3, the automorphism group (Z_2) is generated by the charge conjugation C. If k is not divisible by 3, the automorphism group (Z_2 x Z_2) is generated by C and the Altsch\"uler--Lacki--Zaugg automorphism. Although the combinatorial analysis can become more involved, the techniques used here for SU(3) can be applied to other algebras. 
  We apply the coadjoint orbit technique to the group of area preserving diffeomorphisms (APD) of a 2D manifold, particularly to the APD of the semi-infinite cylinder which is identified with $w_{\infty}$.  The geometrical action obtained is relevant to both $w$ gravity and 2D turbulence. For the latter we describe the hamiltonian, which appears to be given by the Schwinger mass term, and discuss some possible developments within our approach. Next we show that the set of highest weight orbits of $w_{\infty}$ splits into subsets, each of which consists of highest weight orbits of $\bar{w}_N$ for a given N. We specify the general APD geometric action to an orbit of $\bar{w}_N$ and describe an appropriate set of observables, thus getting an action and observables for $\bar{w}_N$ gravity. We compute also the Ricci form on the $\bar{w}_N$ orbits, what gives us the critical central charge of the $\bar{w}_N$ string, which appears to be the same as the one of the $W_N$ string. 
  We consider the physical properties of four dimensional black hole solutions to the effective action describing the low energy dynamics of the gravitational sector of heterotic superstring theory. We compare the properties of the external field strengths in the perturbative solution to the full $O(\alpha')$ string effective action equations, to those of exact solutions in a truncated action for charged black holes, and to the Kerr-Newman family of solutions of Einstein-Maxwell theory. We contrast the numerical results obtained in these approaches, and discuss limitations of the analyses. Finally we discuss how the new features of classical string gravity affect the standard tests of general relativity. 
  The mechanism underlying any bosonisation or fermionisation is exposed.It is shown that any local theory of fermions on a lattice in any spatial dimension greater than one is equivalent to a local theory of Ising spins coupled to a $Z_{2}$ gauge field.There is a close relation to the description of anyons using a Chern-Simons term. 
  We display a new integrable perturbation for both N=1 and N=2 superconformal minimal models. These perturbations break supersymmetry explicitly. Their existence was expected on the basis of the classification of integrable perturbations of conformal field theories in terms of distinct classical KdV type hierarchies sharing a common second Hamiltonian structure. 
  In this work we extend Onofri and Perelomov's coherent states methods to the recently introduced $OSp(1/2)$ coherent states. These latter are shown to be parametrized by points of a supersymplectic supermanifold, namely the homogeneous superspace $OSp(1/2)/U(1)$, which is clearly identified with a supercoadjoint orbit of $OSp(1/2)$ by exhibiting the corresponding equivariant supermoment map. Moreover, this supermanifold is shown to be a nontrivial example of Rothstein's supersymplectic supermanifolds. More precisely, we show that its supersymplectic structure is completely determined in terms of $SU(1,1)$-invariant (but unrelated) K\"ahler $2$-form and K\"ahler metric on the unit disc. This result allows us to define the notions of a superK\"ahler supermanifold and a superK\"ahler superpotential, the geometric structure of the former being encoded into the latter. 
  In this paper we consider a class of the 2D integrable models. These models are higher spin XXZ chains with an extra condition of the commensurability between spin and anisotropy. The mathematics underlying this commensurability is provided by the quantum groups with deformation parameter being an Nth root of unity. Our discussion covers a range of topics including new integrable deformations, thermodynamics, conformal behaviour, S-matrices and magnetization. The emerging picture strongly depends on the N-parity. For the N even case at the commensurable point, S-matrices factorize into N=2 supersymmetric Sine-Gordon matrix and an RSOS piece. The physics of the N odd case is rather different. Here, the supersymmetry does not manifest itself and the bootstrap hypothesis fails. Away from the commensurable point, we find an unusual behaviour. The magnetization of our chains depends on the sign of the external magnetic field. 
  We discuss the infinite dimensional algebras appearing in integrable perturbations of conformally invariant theories, with special emphasis in the structure of the consequent non-abelian infinite dimensional algebra generalizing $W_\infty$ to the case of a non abelian group. We prove that the pure left-symmetry as well as the pure right-sector of the thus obtained algebra coincides with the conformally invariant case. The mixed sector is more involved, although the general structure seems to be near to be unraveled. We also find some subalgebras that correspond to Kac-Moody algebras. The constraints imposed by the algebras are very strong, and in the case of the massive deformation of a non-abelian fermionic model, the symmetry alone is enough to fix the 2- and 3-point functions of the theory. 
  The Type II Superstring amplitude to 1-loop order is given by an integral of $\vartheta$-functions over the moduli space of tori, which diverges for real momenta. We construct the analytic continuation which renders this amplitude well defined and finite, and we find the expected poles and cuts in the complex momentum plane. 
  A new vector field is introduced into 2-form Einstein gravity in four dimensions to restore a large symmetry of its topological version. Two different expressions for the BRST charge are given in the system: one of them associated with a set of irreducible symmetries and the other with a set of on-shell reducible symmetries. 
  We examine the problem of defining Lagrangian and Hamiltonian mechanics for a particle moving on a quantum plane $Q_{q,p}$. For Lagrangian mechanics, we first define a tangent quantum plane $TQ_{q,p}$ spanned by noncommuting particle coordinates and velocities. Using techniques similar to those of Wess and Zumino, we construct two different differential calculi on $TQ_{q,p}$. These two differential calculi can in principle give rise to two different particle dynamics, starting from a single Lagrangian. For Hamiltonian mechanics, we define a phase space $T^*Q_{q,p}$ spanned by noncommuting particle coordinates and momenta. The commutation relations for the momenta can be determined only after knowing their functional dependence on coordinates and velocities.   Thus these commutation relations, as well as the differential calculus on $T^*Q_{q,p}$, depend on the initial choice of Lagrangian. We obtain the deformed Hamilton's equations of motion and the deformed Poisson brackets, and their definitions also depend on our initial choice of Lagrangian. We illustrate these ideas for two sample Lagrangians. The first system we examine corresponds to that of a nonrelativistic particle in a scalar potential. The other Lagrangian we consider is first order in time derivatives 
  By quantizing the generalized Drinfeld-Sokolov reduction scheme for arbitrary $sl_2$ embeddings we show that a large set $\cal W$ of quantum W algebras can be viewed as (BRST) cohomologies of affine Lie algebras. The set $\cal W$ contains many known $W$ algebras such as $W_N$ and $W_3^{(2)}$. Our formalism yields a completely algorithmic method for calculating the W algebra generators and their operator product expansions, replacing the cumbersome construction of W algebras as commutants of screening operators. By generalizing and quantizing the Miura transformation we show that any $W$ algebra in $\cal W$ can be embedded into the universal enveloping algebra of a semisimple affine Lie algebra which is, up to shifts in level, isomorphic to a subalgebra of the original affine algebra. Therefore {\em any} realization of this semisimple affine Lie algebra leads to a realization of the $W$ algebra. In particular, one obtains in this way a general and explicit method for constructing the free field realizations and Fock resolutions for all algebras in $\cal W$. Some examples are explicitly worked out. 
  We consider vector induced lattice gauge theories, in particular we consider the QED induced and we show that at negative temperature corresponds to the dimer problem while at positive temperature it describes a gas of branched polymers with loops. We argue that these models are critical at $D_{cr}=6$ for $N\ne\infty$ and they are {\sl not} critical for $N=\infty$. 
  We have investigated the standard one-loop quantum corrections for a particularly simple non-commutative geometry model containing fermions interacting with a unique abelian gauge field and a unique scalar through Yukawa couplings. In this model there are certain relations among the different coupling constants quite similar to the ones appearing in the Connes-Lott version of the standard model. We find that it is not possible to implement those relations in a renormalization-group invariant way. 
  We study two-dimensional dilaton gravity coupled to massless scalar fields for static solutions. In addition to the well known black hole, we find another class of solutions that may be understood as that of the black hole in equilibrium with a radiation bath. We claim that there is a solution that is qualitatively unchanged after including Hawking radiation and back-reaction and is furthermore geodesically complete. We compute the thermodynamics of these spacetimes and their mass. We end with a brief discussion of the linear response about these solutions, its significance to stability and noise and a speculation regarding the endpoint of Hawking evaporation in four dimensions. (plain TeX; one figure, available upon request.) 
  We describe a $q$-deformed dynamical system corresponding to the quantum free particle moving along the circle. The algebra of observables is constructed and discussed. We construct and classify irreducible representations of the system. 
  In this paper we investigate in more detail our previous formulation of the dilaton-gravity theory by Bilal--Callan--de~Alwis as a $SL_2$-conformal affine Toda (CAT) theory. Our main results are: i) a field redefinition of the CAT-basis in terms of which it is possible to get the black hole solutions already known in the literature; ii) an investigation the scattering matrix problem for the quantum black hole states. It turns out that there is a range of values of the $N$ free-falling shock matter fields forming the black hole solution, in which the end-point state of the black hole evaporation is a zero temperature regular remnant geometry. It seems that the quantum evolution to this final state is non-unitary, in agreement with Hawking's scenario for the black hole evaporation. 
  Covariant anomalies are studied in terms of the theory of secondary characteristic classes of the universal bundle of Yang-Mills theory. A new set of descent equations is derived which contains the covariant current anomaly and the covariant Schwinger term. The counterterms relating consistent and covariant anomalies are determined. A geometrical realization of the BRS/anti-BRS algebra is presented which is used to understand the relationship between covariant anomalies in different approaches. 
  We study a class of classical dilaton vacua in string theory that depend on the light-cone variable $z=t\pm x$ and, thus, have wavelike behavior. One of the interesting results is the existence of a solution subclass with perfectly regular space-time geometry, where the string coupling constant can be made arbitrarily small. 
  We propose an improved iterative scheme for calculating higher genus contributions to the multi-loop (or multi-point) correlators and the partition function of the hermitian one matrix model. We present explicit results up to genus two. We develop a version which gives directly the result in the double scaling limit and present explicit results up to genus four. Using the latter version we prove that the hermitian and the complex matrix model are equivalent in the double scaling limit and that in this limit they are both equivalent to the Kontsevich model. We discuss how our results away from the double scaling limit are related to the structure of moduli space. 
  We present a quantum geometric framework for stochastic quantisation in the case of a free Klein-Gordon field on Euclidean space. In this approach the noise is part of the background space, spacetime is fuzzy. We extend the notion of sharp point limit and show how fuzzy spacetime and the Klein-Gordon field gives the Euclidean space and the stochastically quantised Klein-Gordon field respectively. 
  Wheeler-DeWitt equation is applied to $k > 0$ Friedmann Robertson Walker metric with various types of matter. It is shown that if the Universe ends in the matter dominated era (e.g., radiation or pressureless gas) with zero cosmological constant, then the resulting Wheeler-DeWitt equation describes a bound state problem. As solutions of a non-degenerate bound state system, the eigen-wave functions are real (Hartle-Hawking) and the usual issue associated with the ambiguity in the boundary conditions for the wave functions is resolved. Furthermore, as a bound state problem, there exists a quantization condition that relates the curvature of the three space with the energy density of the Universe. Incorporating a cosmological constant in the early Universe (inflation) is given as a natural explanation for the large quantum number associated with our Universe, which resulted from the quantization condition.   It is also shown that if there is a cosmological constant $\Lambda > 0$ in our Universe that persists for all time, then the resulting Wheeler-DeWitt equation describes a non-bound state system, regardless of the magnitude of the cosmological constant. As a consequence, the wave functions are in general complex (Vilenkin) and the initial conditions for wave functions are a free parameters not determined by the formalism. 
  The submitted paper regards the example of the Conformal Field Theory on a 2d manifold which metric has a point-like singularity.Since this manifold is not conformally equivalent to that with the flat space-time metric,it's naturally to expect that the theory cannot be trivially reduced to the well-known consideration of the CFT on a plane,and some modifications are needed.Particularly,this paper shows how the vacuum of the theory on a singular surface differs from the vacuum of the BPZ theory.Namely,this vacuum would not be SL(2,C)-invariant and the expressions for the correlation functions should be modified. As a consequence of that,some "effective mass" is brought to the theory. 
  We summarize a study of an Abelian gauge theory in 2+1 dimensions, the gauge field being coupled to nonrelativistic Fermions. The Action for the gauge field is a combination of the Maxwell term and a Chern-Simons (CS) term. We study the limit of vanishing Fermions' charge, keeping fixed the gauge field mass induced by the CS term. By considering a closed surface, in particular a torus to keep translational invariance, we show that the Fermions do not decouple completely from the gauge field, and in fact they behave as Anyons treated in the -translationally invariant formulation of the- Mean Field approximation. We describe the exact solution of this limiting case. 
  We propose a new wiew on the structure of quantum mechanics and postulate a q-deformed algebra of observables. We find equations of motion for this system, which guarantee a unitary time developement. We solve this equations for simple models. We write this formalism in terms of twisted deRham complex. 
  This paper analyzes spacetime symmetries of topological string theory on a two dimensional torus, and points out that the spacetime geometry of the model is that of the Batalin-Vilkovisky formalism. Previously I found an infinite symmetry algebra in the absolute BRST cohomology of the model. Here I find an analog of the BV $\Delta$ operator, and show that it defines a natural semirelative BRST cohomology. In the absolute cohomology, the ghost-number-zero symmetries form the algebra of all infinitesimal spacetime diffeomorphisms, extended at non-zero ghost numbers to the algebra of all odd-symplectic diffeomorphisms on a spacetime supermanifold. In the semirelative cohomology, the symmetries are reduced to $w_\infty$ at ghost number zero, and to a topologically twisted N=2 $w_\infty$ superalgebra when all ghost numbers are included. I discuss deformations of the model that break parts of the spacetime symmetries while preserving the topological BRST symmetry on the worldsheet. In the absolute cohomology of the deformed model, another topological $w_\infty$ superalgebra may emerge, while the semirelative cohomology leads to a bosonic $w_\infty$ symmetry. 
  A direct relationship between the (non-quantum) group SU(2) and the Kauffman bracket in the framework of Chern-Simons theory is explicitly shown. 
  We study the black hole information paradox in the context of a two-dimensional toy model given by dilaton gravity coupled to $N$ massless scalar fields. After making the model well-defined by imposing reflecting boundary conditions at a critical value of the dilaton field, we quantize the theory and derive the quantum quantum $S$-matrix for the case that $N$=$24$. This $S$-matrix is unitary by construction, and we further argue that in the semiclassical regime it describes the formation and subsequent Hawking evaporation of two-dimensional black holes. Finally, we note an interesting correspondence between the dilaton gravity $S$-matrix and that of the $c=1$ matrix model. 
  We discuss several applications and extensions of our previous operator solution of the $N$-body Calogero problem, \ie N particles in 1 dimension subject to a two-body interaction of the form $\half \sum_{i,j}[ (x_i - x_j)^2 + g/ {(x_i - x_j)^2}]$. Using a complex representation of the deformed Heisenberg algebra underlying the Calogero model, we explicitly establish the equivalence between this system and anyons in the lowest Landau level. A construction based on supersymmetry is used to extend our operator method to include fermions, and we obtain an explicit solution of the supersymmetric Calogero model constructed by Freedman and Mende. We also show how the dynamical $OSp(2;2)$ supersymmetry is realized by bilinears of modified creation and annihilation operators, and how to construct a supersymmetic extension of the deformed Heisenberg algebra. 
  We present a new class of quantum two dimensional dilaton gravity model, which is described by $SL(2,R)/U(1)$ gauged Wess-Zumino-Witten model deformed by $(1,1)$-operator. We analyze the model by ${1 \over k}$ expansion ($k$ is the level of $SL(2,R)$ Wess-Zumino-Witten model) and we find that the curvature singularity does not appear when $k$ is large and the Bondi mass is bounded from below. Furthermore, the rate of the Hawking radiation in the quantum black hole created by shock wave goes to zero asymptotically and the radiation stops when the Bondi mass vanishes. 
  We present the crumpling transition in three-dimensional Euclidian space of dynamically triangulated random surfaces with edge extrinsic curvature and fixed topology of a sphere as well as simulations of a dynamically triangulated torus. We used longer runs than previous simulations and give new and more accurate estimates of critical exponents. Our data indicate a cusp singularity in the specific heat. The transition temperature, as well as the exponents are topology dependent. 
  We discuss in detail how string-inspired lineal gravity can be formulated as a gauge theory based on the centrally extended Poincar\'e group in $(1+1)$ dimensions. Matter couplings are constructed in a gauge invariant fashion, both for point particles and Fermi fields. A covariant tensor notation is developed in which gauge invariance of the formalism is manifest. 
  The physical states on the free field Fock space of the ${SL(2,R)\over SL(2,R)$ model at any level are computed. Using a similarity transformation on $Q_{BRST}$, the cohomology of the latter is mapped into a direct sum of simpler cohomologies. We show a one to one correspondence between the states of the $k=-1$ model and those of the $c=1$ string model. A full equivalence between the ${SL(2,R)\over SL(2,R)$ and ${SL(2,R)\over U(1)$ models at the level of their Fock space cohomologies is found. 
  We discuss a simplicial dimension shift which associates to each n-manifold an n-1-manifold. As a corollary we show that an invariant which was recently proposed by Ooguri and by Crane and Yetter for the construction of 4-dimensional quantum field theories out of 3-dimensional theories is trivial. 
  We formulate the conformal bootstrap approach to four--fermion theory at its strong coupling fixed point in dimensions $2<d<4$. We present a solution of the bootstrap equations in the five--vertex approximation. We show that the bootstrap approach gives a particularly simple way to obtain next to leading order corrections to critical exponents in the large $N$ expansion and present the values of the anomalous dimensions of the fermion field $\psi$ and the composite $\bar\psi\psi$ to order $1/N^2$. 
  A physical interpretation is given for some Hermitian Jordan triple systems (HJTS) that were recently discussed by Gunaydin (hep-th/9301050). Quadratic Jordan algebras derived from HJTS provide a formulation of quantum mechanics that is a natural framework within which exceptional structures are identified with physically realistic structures of a quantum field theory that includes both the standard model and MacDowell-Mansouri gravity. The structures allow the calculation of the relative strengths of the four forces, including the fine structure constant ALPHA = 1/137.03608. 
  I study the time evolution of the density matrices of quantum Fermi systems interacting with classic external Fermi fields. This interaction either changes the temperature of the system or it affects the density of particles. For relativistic Dirac fermions, variations of temperature lead to creation (annihilation) of particle - antiparticle pairs. The change of the density (or of the chemical potential) indicates the existence of the incoming (outgoing) flux of fermions from (to) the bath. These changes are independent for the different modes and in order to model the thermalization one should adjust the spectrum of the noise. The linear time dependences of the densities of particles are characteristic for all the processes. 
  We discuss the structure of correlators involving the spinor emission vertex in non critical $N=1$ superstring theory. The technique used in the computation is the zero mode integration to arrive at the integral representation, and later an analysis of the pole structure of the integrals which are thus obtained. Our analysis has been done primarily for the 5-point functions. The result confirms previous expectations and prepares ground for a comparison with computations using matrix models techniques. 
  There is substancial overlap with hepth-9211081. More results are presented for duality in the non-compact case. It is argued that duality persists as a symmetry also in that case. 
  A proof of critical conformal invariance of Green's functions for a quite wide class of models possessing critical scale invariance is given. A simple method for establishing critical conformal invariance of a composite operator, which has a certain critical dimension, is also presented. The method is illustrated with the example of the Gross--Neveu model and the exponents \et\ at order $1/n^3$, \Dl\ and $1/\nu$ at order $1/n^2$ are calculated with the conformal bootstrap method. 
  The F and B matrices associated with Virasoro null vectors are derived in closed form by making use of the operator-approach suggested by the Liouville theory, where the quantum-group symmetry is explicit.  It is found that the entries of the fusing and braiding matrices are not simply equal to quantum-group symbols, but involve additional coupling constants whose derivation is one aim of the present work. Our explicit formulae are new, to our knowledge, in spite of the numerous studies of this problem.  The relationship between the quantum-group-invariant (of IRF type) and quantum-group-covariant (of vertex type) chiral operator-algebras is fully clarified, and connected with the transition to the shadow world for quantum-group symbols. The corresponding 3-j-symbol dressing is shown to reduce to the simpler transformation of Babelon and one of the author (J.-L. G.)  in a suitable infinite limit defined by analytic continuation.  The above two types of operators are found to coincide when applied to states with Liouville momenta going to $\infty$ in a suitable way.  The introduction of quantum-group-covariant operators in the three dimensional picture gives a generalisation of the quantum-group version of discrete three-dimensional gravity that includes tetrahedra associated with 3-j symbols and universal R-matrix elements. Altogether the present work gives the concrete realization of Moore and Seiberg's scheme that describes the chiral operator-algebra of two-dimensional gravity and minimal models. 
  Using independent left and right vierbeins to describe graviton plus axion as suggested by string mechanics, O(d,d) duality can be realized linearly. 
  We study the global structure of the exact two-dimensional space-time which emerges from string theory. Previous work has shown that in the semi-classical limit, this is a black hole similar to the Schwarzschild solution. However, we find that in the exact case, a new Euclidean region appears "between" the singularity and black hole interior. However the boundary between the Lorentzian and Euclidean regions is a coordinate singularity, which turns out to be a surface of time reflection symmetry in an extended space-time. Thus strings having fallen through the black hole horizon would eventually emerge through another one into a new asymptotically flat region. The maximally extended space-time consists of an infinite number of universes connected by wormholes. There are no singularities present in this geometry. We also calculate the mass and temperature associated with the space-time. 
  Postulate of SL(2,\Z) invariance of toroidally compactified heterotic string theory in four dimensions implies the existence of new string (dual string) states carrying both, electric and magnetic charges. In this paper we study the interaction between these dual strings. In particular, we consider scattering of two such strings in the limit where one string passes through the other string without touching it, and show that such a scattering leads to the exchange of a fixed amount of electric and magnetic charges between the two strings. 
  We study the quantum $N=2$ super-$W_{3}$ algebra using the free field realization, which is obtained from the supersymmetric Miura transformation associated with the Lie superalgebra $A(2|1)$. We compute the full operator product expansions of the algebra explicitly. It is found that the results agree with those obtained by the OPE method. 
  The 2D model of gravity with zweibeins $e^{a}$ and the Lorentz connection one-form $\omega^{a}_{\ b}$ as independent gravitational variables is considered and it is shown that the classical equations of motion are exactly integrated in coordinate system determined by components of 2D torsion. For some choice of integrating constant the solution is of the charged black hole type. The conserved charge and ADM mass of the black hole are calculated. 
  Following the approach of Grignani and Nardelli [1], we show how to cast the two-dimensional model $L \sim curv^2 + torsion^2 + cosm.const$ -- and in fact any theory of gravity -- into the form of a Poincare gauge theory. By means of the above example we then clarify the limitations of this approach: The diffeomorphism invariance of the action still leads to a nasty constraint algebra. Moreover, by simple changes of variables (e.g. in a path integral) one can reabsorb all the modifications of the original theory. 
  We develop the approach of Faddeev, Reshetikhin, Takhtajan [1] and of Majid [2] that enables one to associate a quasitriangular Hopf algebra to every regular invertible constant solution of the quantum Yang-Baxter equations. We show that such a Hopf algebra is actually a quantum double. 
  We reformulate the method recently proposed for constructing quasitriangular Hopf algebras of the quantum-double type from the R-matrices obeying the Yang-Baxter equations. Underlying algebraic structures of the method are elucidated and an illustration of its facilities is given. The latter produces an example of a new quasitriangular Hopf algebra. The corresponding universal R-matrix is presented as a formal power series. 
  We give an explicit algebraic description of finite Lorentz transformations of vectors in 10-dimensional Minkowski space by means of a parameterization in terms of the octonions. The possible utility of these results for superstring theory is mentioned. Along the way we describe automorphisms of the two highest dimensional normed division algebras, namely the quaternions and the octonions, in terms of conjugation maps. We use similar techniques to define $SO(3)$ and $SO(7)$ via conjugation, $SO(4)$ via symmetric multiplication, and $SO(8)$ via both symmetric multiplication and one-sided multiplication. The non-commutativity and non-associativity of these division algebras plays a crucial role in our constructions. 
  The Poincare invariance in the temporal gauge canonical quantization of QCD is shown manifestly by verifying the energy-momentum-vector and angular-momentum-tensor satisfy the Poincare algebra in the physical Hilbert space. Two different values of \(\theta\) for the $\theta$-term in QCD lagrangian lead to different representations of the Poincare group, which are, however, connected by an unitary transformation. Thus the parameter \(\theta\) becomes physically irrelevant unless we can further restrict the physical Hilbert space. 
  I present the reduction of phase space of the theory of an antisymmetric tensor potential coupled to an abelian gauge field, using Dirac's procedure. Duality transformations on the reduced phase space are also discussed. 
  We give a complete geometric description of conformal anomalies in arbitrary, (necessarily even) dimension. They fall into two distinct classes: the first, based on Weyl invariants that vanish at integer dimensions, arises from finite -- and hence scale-free -- contributions to the effective gravitational action through a mechanism analogous to that of the (gauge field) chiral anomaly. Like the latter, it is unique and proportional to a topological term, the Euler density of the dimension, thereby preserving scale invariance. The contributions of the second class, requiring introduction of a scale through regularization, are correlated to all local conformal scalar polynomials involving powers of the Weyl tensor and its derivatives; their number increases rapidly with dimension. Explicit illustrations in dimensions 2, 4 and 6 are provided. 
  We discuss physical spectra and correlation functions of topological minimal models coupled to topological gravity. We first study the BRST formalism of these theories and show that their BRST operator $Q=Q_s+Q_v$ can be brought to $Q_s$ by a certain homotopy operator $U$, $UQU^{-1}=Q_s$ ($Q_s$ and $Q_v$ are the $N=2$ and diffeomorphism BRST operators, respectively). The reparametrization (anti)-ghost $b$ mixes with the supercharge operator $G$ under this transformation. Existence of this transformation enables us to use matter fields to represent cohomology classes of the operator $Q$. We explicitly construct gravitational descendants and show that they generate the higher-order KdV flows. We also evaluate genus-zero correlation functions and rederive basic recursion relations of two-dimensional topological gravity. 
  The BRST operator cohomology of $N=2$ $2d$ supergravity coupled to matter is presented. Descent equations for primary superfields of the matter sector are derived. We find one copy of the cohomology at ghost number one, two independent copies at ghost number two, and conjecture that there is a copy at ghost number three. The $N=2$ string has a twisted $N=4$ superconformal symmetry generated by the $N=2$ superstress tensor, the BRST supercurrent, the antighost superfield, and the ghost number supercurrent. 
  Starting from the requirement that a Lagrangian field theory be invariant under both Schwinger-Dyson BRST and Schwinger-Dyson anti-BRST symmetry, we derive the BRST--anti-BRST analogue of the Batalin-Vilkovisky formalism. This is done through standard Lagrangian gauge fixing respecting the extended BRST symmetry. The solutions of the resulting Master Equation and the gauge-fixing procedure for the quantum action can be brought into forms that coincide with those obtained earlier on algebraic grounds by Batalin, Lavrov and Tyutin. 
  The one-loop effective potential for a scalar field defined on an ultrastatic space-time whose spatial part is a compact hyperbolic manifold, is studied using zeta-function regularization for the one-loop effective action. Other possible regularizations are discussed in detail. The renormalization group equations are derived and their connection with the conformal anomaly is pointed out. The symmetry breaking and the topological mass generation are also discussed. 
  Progress on the physics of strings in curved spacetime are comprehensively reviewed.We start by showing through renormalization group arguments that a meaningful quantum theory of gravity must be finite and must include all particle physics.Then, we review classical and quantum string propagation in curved spacetimes.We start by the general expansion method proposed by de Vega and S\'anchez in 1987. The particle transmutation phenomena in asymptotically flat spacetimes are detailed including fermion-boson transitions in supergravity backgrounds.The next chapters review the exactly solvable cases of string propagation: shock waves, singular plane waves, conical spacetimes and de Sitter cosmological spacetime. The calculation of various physical quantities like the string mass and the energy-momentum tensor shows that classical and quantum string propagation in shock-waves and singular plane waves is physically meaningful and full of interesting new phenomena.The important phenomenom of {\bf string stretching} that takes place when strings fall into spacetime singularities and in expanding universes is analyzed.We conclude by reporting on strings in de Sitter spacetime, where the string equations are integrable and reduce to the sinh-Gordon equation and to integrable generalizations of it. Lectures delivered at the ERICE SCHOOL ``STRING QUANTUM   GRAVITY AND PHYSICS AT THE PLANCK ENERGY SCALE'', 21-28 June 1992 , to appear in the Proceedings edited by N. S\'anchez, World Scientific. 
  A systematic classification of Feynman path integrals in quantum mechanics is presented and a table of solvable path integrals is given which reflects the progress made during the last ten years or so, including, of course, the main contributions since the invention of the path integral by Feynman in 1942. An outline of the general theory is given. Explicit formul\ae\ for the so-called basic path integrals are presented on which our general scheme to classify and calculate path integrals in quantum mechanics is based. 
  A specific class of explicitly time-dependent potentials is studied by means of path integrals. For this purpose a general formalism to treat explicitly time-dependent space-time transformations in path integrals is sketched. An explicit time-dependent model under consideration is of the form $V(q,t)=V[q/\zeta(t)]/\zeta^2(t)$, where $V$ is a usual potential, and $\zeta(t)=(at^2+2bt+c)^{1/2}$. A recent result of Dodonov et al.\ for calculating corresponding propagators is incorporated into the path integral formalism by performing a space-time transformation. Some examples illustrate the formalism. 
  A wide class of boundary problems in quantum mechanics is discussed by using path integrals. This includes motion in half-spaces, radial boxes, rings, and moving boundaries. As a preparation the formalism for the incorporation of $\delta$-function perturbations is outlined, which includes the discussion of multiple $\delta$-function perturbations, $\delta$-function perturbations along perpendicular lines and planes, and moving $\delta$-function perturbations. The limiting process, where the strength of the $\delta$-function perturbations gets infinite repulsive, has the effect of producing impenetrable walls at the locations of the $\delta$-function perturbations, i.e.\ a consistent description for boundary problems with Dirichlet boundary-condition emerges. Several examples illustrate the formalism. 
  This paper is the third in a sequel to develop a super-analogue of the classical Selberg trace formula, the Selberg supertrace formula. It deals with bordered super Riemann surfaces. The theory of bordered super Riemann surfaces is outlined, and the corresponding Selberg supertrace formula is developed. The analytic properties of the Selberg super zeta-functions on bordered super Riemann surfaces are discussed, and super-determinants of Dirac-Laplace operators on bordered super Riemann surfaces are calculated in terms of Selberg super zeta-functions. 
  Recently a new supersymmetric extension of the KdV hierarchy has appeared in a matrix-model-inspired approach to $2{-}d$ quantum supergravity. Here we prove that this hierarchy is essentially the KdV hierarchy, where the KdV field is now replaced by an even superfield. This allows us to find the conserved charges and the bihamiltonian structure, and to prove its integrability. We also extend the hierarchy by odd flows in a supersymmetric fashion. 
  We present a way of constructing string solutions around non-trivial gravitational backgrounds. The proposed solutions are constructed using $N = 4$ superconformal building blocks with $\hat c = 4$. We give two different and inequivalent realizations of non-trivial four-dimensional subspaces, and we show the emergence of the $N = 4$ globally defined superconformal symmetry. The existence of $N = 4$ world-sheet symmetry stabilizes our solutions and implies in target space a number of covariantized supersymmetries around space-time dependent gravitational and dilaton backgrounds. 
  We study N=2 SuperVirasoro SCFT for the generic value of the central charge. The main tool is the nonstandard bosonisation suggested in \ref\rRoz{L. Rozansky a letter to M. Bershadsky, 1989}, \ref\rSeBGR{B. Gato-Rivera, A. Semikhatov Phys. Letts. B293 (1992) 72},\ref\rBLNW{M. Bershadsky, W. Lerche, D. Nemeshansky, N. Warner N=2 Extended superconformal structure of Gravity and W Gravity coupled to Matter HUTP-A034/92}. The free field resolutions for the irreducible representations are obtained; the characters of these representations are computed. The quantum hamiltonian reduction from the Kac-Moody $\hat{sl}_k(2|1)$ to N=2 $SVir$ is constructed. 
  We study the spectra of G/G coset models by computing BRST cohomology of affine Lie algebras with coefficients in tensor product of two modules. One-to-one correspondence between the spectra of $A_1^1/A_1^1$ and that of the minimal matter coupled to gravity (including boundary states of the Kac table) is observed. This phenomena is discussed from the point of hamiltonian reduction of BRST complexes of $A_N^1$ Lie algebras. (In the revised version one proof is modified and examplified, some misprints are corrected.) 
  Quantum canonical transformations are defined in analogy to classical canonical transformations as changes of the phase space variables which preserve the Dirac bracket structure. In themselves, they are neither unitary nor non-unitary. A definition of quantum integrability in terms of canonical transformations is proposed which includes systems which have fewer commuting integrals of motion than degrees of freedom. The important role of non-unitary transformations in integrability is discussed. 
  Two quantum theories are physically equivalent if they are related, not by a unitary transformation, but by an isometric transformation. The conditions under which a quantum canonical transformation is an isometric transformation are given. 
  A \rep of \sun, which diverges in the limit of \cl, is investigated. This is an infinite dimensional and a non-unitary \rep, defined for the real value of $ q, \ 0 < q < 1. $ Each \irrep is specified by $ n $ continuous variables and one discrete variable. This \rep gives a new solution of the Yang-Baxter equation, when the R-matrix is evaluated. It is shown that a continuous variables can be regarded as a spectral parameter. 
  A quantum deformation of 4-dimensional superconformal algebra realized on quantum superspace is investigated. We study the differential calculus and the action of the quantum generators corresponding to $sl_q(1|4)$ which act on the quantum superspace. We derive deformed $su(1|2,2)$ algebras from the deformed $sl(1|4)$ algebra. Through a contraction procedure we obtain a deformed super-Poincar{\'e} algebra. 
  An exact solution of the low-energy string theory representing static, spherical symmetric dyonic black hole is found. The solution is labeled by their mass, electric charge, magnetic charge and asymptotic value of the scalar dilaton. Some interesting properties of the dyonic black holes are studied. In particular, the Hawking temperature of dyonic black holes depends on both the electric and magnetic charges, and the extremal ones, which have nonzero electric and magnetic charges, have zero temperature but nonzero entropy. These properties are quite different from those of electrically (or magnetically) charged dilaton black holes found by Gibbons {\it et al.} and Garfinkle {\it et al.}, but are the same as those of the dyonic black holes found by Gibbons and Maeda. After this paper was submitted for publication, D. Wiltshire told us that solutions, eqs.(22)-(28), are related to Gibbons-Maeda dyonic black hole solutions by a coordinate transformation and some parameters reparametization \cite{26}. And, we were also informed that many of our results were previously obtained by Kallosh {\it et al.} \cite{27}. The dyonic black hole solutions, eqs.(22)-(28), are also related to those of reference \cite{27} by another coordinate 
  We described the $q$-deformed phase space. The $q$-deformed Hamilton eqations of motion are derived and discussed. Some simple models are considered. 
  We show that all important features of 2d gravity coupled to $c<1$ matter can be easily understood from the canonical quantization approach a la Dirac. Furthermore, we construct a canonical transformation which maps the theory into a free-field form, i.e. the constraints become free-field Virasoro generators with background charges. This implies the gauge independence of the David-Distler-Kawai results, and also proves the free-field assumption which was used for obtaining the spectrum of the theory in the conformal gauge. A discussion of the unitarity of the physical spectrum is presented and we point out that the scalar products of the discrete states are not well defined in the standard Fock space framework. 
  The requirements of conformal invariance for the two point function of the energy momentum tensor in the neighbourhood of a plane boundary are investigated, restricting the conformal group to those transformations leaving the boundary invariant. It is shown that the general solution may contain an arbitrary function of a single conformally invariant variable $v$, except in dimension 2. The functional dependence on $v$ is determined for free scalar and fermion fields in arbitrary dimension $d$ and also to leading order in the $\vep$ expansion about $d=4$ for the non Gaussian fixed point in $\phi^4$ theory. The two point correlation function of the energy momentum tensor and a scalar field is also shown to have a unique expression in terms of $v$ and the overall coefficient is determined by the operator product expansion. The energy momentum tensor on a general curved manifold is further discussed by considering variations of the metric. In the presence of a boundary this procedure naturally defines extra boundary operators. By considering diffeomorphisms these are related to components of the energy momentum tensor on the boundary. The implications of Weyl invariance in this framework are also derived. 
  Wormhole solutions corresponding to space-time geometries $R^1\times S^1\times S^2$ and $R^1\times S^3$ are obtained from reduced string effective action and the action is written in a manifestly $O(d,d)$ invariant form. A general treatment is given for obtaining wormhole solutions of different topologies from dimensional reduction. For specific ansatz of internal metric and antisymmetric field the reduced action is shown to have a global $SL(2,C)$ symmetry. The $SL(2,C)$ and duality symmetries have been exploited to generate new configurations of internal fields which produce wormhole solutions in four space-time dimensions. The $SL(2,C)$ symmetry discussed in this paper arises due to specific form of the moduli and these transformations belong to a subgroup of $O(d,d)$ global symmetry. 
  We study a string-inspired classical 2-D effective field theory with {\it nonsingular} black holes as well as Witten's black hole among its static solutions. By a dimensional reduction, the static solutions are related to the $(SL(2,R)_{k}\otimes U(1))/U(1)$ coset model, or more precisely its $O\bigl((\alpha')^{0}\bigr)$ approximation known as the 3-D charged black string. The 2-D effective action possesses a propagating degree of freedom, and the dynamics are highly nontrivial. A collapsing shell is shown to bounce into another universe without creating a curvature singularity on its path, and the potential instability of the Cauchy horizon is found to be irrelevent in that some of the infalling observers never approach the Cauchy horizon. Finally a $SL(2,R)_{k}/U(1)$ nonperturbative coset metric, found and advocated by R. Dijkgraaf et.al., is shown to be nonsingular and to coincide with one of the charged spacetimes found above. Implications of all these geometries are discussed in connection with black hole evaporation. 
  We investigate some issues relating to recently proposed fractional superstring theories with $D_{\rm critical}<10$. Using the factorization approach of Gepner and Qiu, we systematically rederive the partition functions of the $K=4,\, 8,$ and $16$ theories and examine their spacetime supersymmetry. Generalized GSO projection operators for the $K=4$ model are found. Uniqueness of the twist field, $\phi^{K/4}_{K/4}$, as source of spacetime fermions is demonstrated. Last, we derive a linear (rather than quadratic) relationship between the required conformal anomaly and the conformal dimension of the supercurrent ghost. 
  We study the massive Schwinger model, quantum electrodynamics of massive, Dirac fermions, in 1+1 dimensions; with space compactified to a circle. In the limit that transitions to fermion--anti-fermion pairs can be neglected, we study the full ground state. We focus on the effect of instantons which mediate tunnelling transitions in the induced potential for the dynamical degree of freedom in the gauge field. 
  This paper is devoted to the exploration of some of the geometrical issues raised by the $N=2$ superstring. We begin by reviewing the reasons that $\beta$-functions for the $N=2$ superstring require it to live in a four-dimensional self-dual spacetime of signature $(--++)$, together with some of the arguments as to why the only degree of freedom in the theory is that described by the gravitational field. We then move on to describe at length the geometry of flat space, and how a real version of twistor theory is relevant to it. We then describe some of the more complicated spacetimes that satisfy the $\beta$-function equations. Finally we speculate on the deeper significance of some of these spacetimes. 
  The renormalization-group improved effective potential for an arbitrary renormalizable massless gauge theory in curved spacetime is found,thus generalizing Coleman-Weinberg's approach corresponding to flat space.Some explicit examples are considered,among of them:scalar self-interacting theory,scalar electrody namics,the asymptotically-free SU(2) gauge model,and the SU(5) GUT theory. The possibility of curvature-induced phase transitions is analyzed.It is shown that such a phase transition may take place in a SU(5) inflationary universe.The inclusion of quantum gravity effects isbriefly discussed. 
  We present a static solution for $d = 2$ critical string theory including the tachyon $T$ but not its potential $V(T)$. This solution thus incorporates tachyon back reaction and, when $T = 0$, reduces to the black hole solution. When $T \neq 0$ one finds that (1) the Schwarzschild horizon of the above black hole splits into two, resembling Reissner-Nordstrom horizons and (2) the curvature scalar develops new singularities at the horizons. These features, as we argue, will persist even with $V(T)$ present. Some possible methods for removing these singularities are discussed. 
  Braided differential operators $\del^i$ are obtained by differentiating the addition law on the braided covector spaces introduced previously (such as the braided addition law on the quantum plane). These are affiliated to a Yang-Baxter matrix $R$. The quantum eigenfunctions $\exp_R(\vecx|\vecv)$ of the $\del^i$ (braided-plane waves) are introduced in the free case where the position components $x_i$ are totally non-commuting. We prove a braided $R$-binomial theorem and a braided-Taylors theorem $\exp_R(\veca|\del)f(\vecx)=f(\veca+\vecx)$. These various results precisely generalise to a generic $R$-matrix (and hence to $n$-dimensions) the well-known properties of the usual 1-dimensional $q$-differential and $q$-exponential. As a related application, we show that the q-Heisenberg algebra $px-qxp=1$ is a braided semidirect product $\C[x]\cocross \C[p]$ of the braided line acting on itself (a braided Weyl algebra). Similarly for its generalization to an arbitrary $R$-matrix. 
  We construct a quantum Hamiltonian operator for the Wess-Zumino-Witten (WZW) model in terms of the Casimir operator. This facilitates the discussion of the reduction of the WZW model to Toda field theory at the quantum level and provides a very straightforward derivation of the quantum central charge for the Toda field theory. 
  The finite-temperature one-loop effective potential for a scalar field in the static de Sitter space-time is obtained. Within this framework, by using zeta-function regularization, one can get, in the conformally invariant case, the explicit expression for the stress tensor anomaly. Its value turns out to depend on the thermal state of the system. This conclusion is different from the one derived by other authors, who considered thermal properties of ultraviolet divergences in static spaces ignoring the effects of horizons. The behaviour of the effective potential in the ground state and in de Sitter-invariant state is also studied, showing the role played by the curvature on the minima. 
  The spin-4/3 fractional superstring is characterized by a chiral algebra involving a spin-4/3 current on the world-sheet in addition to the energy-momentum tensor. These currents generate physical state conditions on the fractional superstring Fock space. Scattering amplitudes of these physical states are described which satisfy both spurious state decoupling and cyclic symmetry (duality). Examples of such amplitudes are calculated using an explicit $c=5$ realization of the spin-4/3 current algebra. This representation has three flat coordinate boson fields and a global SO(2,1) Lorentz symmetry, permitting a particle interpretation of the amplitudes. 
  The low-energy scattering of charged fermions by extremal magnetic Reissner-Nordstrom black holes is analyzed in the large-$N$ and $S$-wave approximations. It is shown that (in these approximations) information is carried into a causally inaccessible region of spacetime, and thereby effectively lost. It is also shown that there is an infinite degeneracy of quantum black hole ground states, or ``remnants", which store --- but will not reveal --- the information. A notable feature of the analysis --- not shared by recent analyses of dilatonic black holes --- is that the key physical questions can be answered within the weak coupling domain. We regard these results as strong evidence that effective information loss occurs in our universe. 
  We extend Dirac's `extensible model of the electron' to include spin and family. $U(1)_{e.m.}$ charge conservation on the bubble is translated into a secondary $U(1)_{g}$ world-manifold gauge principle. Reflecting the secondary magnetic monopole configuration on spatial $S^2$, the harmonic excitations may furnish half integer $SU(2)_{spin}\otimes U(1)_{g}$ representations. Our spin--$1/2$ `electron' is described by four world-manifold scalar fields. Its three varieties are associated with different minima of the (6th--order) surface-tension scalar potential. 
  In this paper we develop the covariant string field theory approach to open 2d strings. Upon constructing the vertices, we apply the formalism to calculate the lowest order contributions to the 4- and 5- point tachyon--tachyon tree amplitudes. Our results are shown to match the `bulk' amplitude calculations of Bershadsky and Kutasov. In the present approach the pole structure of the amplitudes becomes manifest and their origin as coming from the higher string modes transparent. 
  We develop a field-theoretical approach to determination of the background target space fields corresponding to general $G/H$ coset conformal theories described by gauged WZW models. The basic idea is to identify the effective action of a gauged WZW theory with the effective action of a sigma model. The derivation of the quantum effective action in the gauged WZW theory is presented in detail, both in the bosonic and in the supersymmetric cases. We explain why and how one can truncate the effective action by omitting most of the non-local terms (thus providing a justification for some previous suggestions). The resulting metric, dilaton and the antisymmetric tensor are non-trivial functions of $1/k$ (or $\alpha'$) and represent a large class of conformal sigma models. The exact expressions for the fields in the sypersymmetric case are equal to the leading order (`semiclassical') bosonic expressions (with no shift of $k$). An explicit form in which we find the sigma model couplings makes it possible to prove that the metric and the dilaton are equivalent to the fields which are obtained in the operator approach, i.e. by identifying the $L_0$-operator of the conformal theory with a Klein-Gordon operator in a background. The metric can be considered as a `deformation' of an invariant metric on the coset space $G/H$ and the dilaton can be in general represented in terms of the logarithm of the ratio of the determinants of the `deformed' and `round' metrics. 
  The most general R-matrix type state sum model for link invariants is constructed. It contains in itself all R-matrix invariants and is a generating function for "universal" Vassiliev link invariants. This expression is more simple than Kontsevich's expression for the same quantity, because it is defined combinatorially and does not contain any integrals, except for an expression for "the universal Drinfeld's associator". 
  We investigate the possibility of generalizing differential renormalization of D.Z.Freedman, K.Johnson and J.I.Latorre in an invariant fashion to theories with infrared divergencies via an infrared $\tilde{R}$ operation. Two-dimensional $\sigma$ models and the four-dimensional $\phi^4$ theory diagrams with exceptional momenta are used as examples, while dimensional renormalization serves as a test scheme for comparison. We write the basic differential identities of the method simultaneously in co-ordinate and momentum space, introducing two scales which remove ultraviolet and infrared singularities. The consistent set of Fourier-transformation formulae is derived. However, the values for tadpole-type Feynman integrals in higher orders of perturbation theory prove to be ambiguous, depending on the order of evaluation of the subgraphs. In two dimensions, even earlier than this ambiguity manifests itself, renormalization-group calculations based on infrared extension of differential renormalization lead to incorrect results. We conclude that the extended differential renormalization procedure does not perform the infrared $\tilde{R}$ operation in a self-consistent way, as the original recipe does the ultraviolet $R$ operation. 
  We generalize some of the standard homological techniques to $\cW$-algebras, and compute the semi-infinite cohomology of the $\cW_3$ algebra on a variety of modules. These computations provide physical states in $\cW_3$ gravity coupled to $\cW_3$ minimal models and to two free scalar fields. 
  It is shown that the quantum position operator of Newton and Wigner for non-zero mass systems is uniquely determined if one imposes a quantum ''manifest covariance'' condition of the same type as the similar condition of Currie, Jordan and Sudarshan in the the framework of the Hamiltonian formalism. 
  We consider two-state (q^2=-1) and three-state (q^3=1) one-dimensional quantum spin chains with U_q(SL(2)) symmetry. Taking unrestricted representations (periodic, semi-periodic and nilpotent), we show which are the necessary conditions to obtain a Hermitian Hamiltonian. 
  We prove in this paper that the elliptic $R$--matrix of the eight vertex free fermion model is the intertwiner $R$--matrix of a quantum deformed Clifford--Hopf algebra. This algebra is constructed by affinization of a quantum Hopf deformation of the Clifford algebra. 
  The algebraic formulation of the quantum group gauge models in the framework of the $R$-matrix approach to the theory of quantum groups is given. We consider gauge groups taking values in the quantum groups and noncommutative gauge fields transformed as comodules under the coaction of the gauge quantum group $ G_{q}$. Using this approach we construct the quantum deformations of the topological Chern-Simons models, non-abelian gauge theories and the Einstein gravity. The noncommutative fields in these models generate $ G_{q}$-covariant quantum algebras. 
  The $N=2$ fermionic string theory is revisited in light of its recently proposed equivalence to the non-compact $N=4$ fermionic string model. The issues of space-time Lorentz covariance and supersymmetry for the BRST quantized $N=2$ strings living in uncompactified $2 + 2$ dimensions are discussed. The equivalent local quantum supersymmetric field theory appears to be the most transparent way to represent the space-time symmetries of the extended fermionic strings and their interactions. Our considerations support the Siegel's ideas about the presence of $SO(2,2)$ Lorentz symmetry as well as at least one self-dual space-time supersymmetry in the theory of the $N=2(4)$ fermionic strings, though we do not have a compelling reason to argue about the necessity of the {\it maximal} space-time supersymmetry. The world-sheet arguments about the absence of all string massive modes in the physical spectrum, and the vanishing of all string-loop amplitudes in the Polyakov approach, are given on the basis of general consistency of the theory. 
  A new topological invariant of closed connected orientable four-dimensional manifolds is proposed. The invariant, constructed via surgery on a special link, is a four-dimensional counterpart of the celebrated SU(2) three-manifold invariant of Reshetikhin, Turaev and Witten. 
  Levinson's theorem for the Dirac equation is known in the form of a sum of positive and negative energy phase shifts at zero momentum related to the total number of bound states. In this letter we prove a stronger version of Levinson's theorem valid for positive and negative energy phase shifts separately. The surprising result is, that in general the phase shifts for each sign of the energy do not give the number of bound states with the same sign of the energy (in units of $\pi$), but instead, are related to the number of bound states of a certain Schr\"odinger equation, which coincides with the Dirac equation at zero momentum. 
  We determine the structure of two variations on the Temperley-Lieb algebra, both used for dealing with special kinds of boundary conditions in statistical mechanics models.   The first is a new algebra, the `blob' algebra (the reason for the name will become obvious shortly!). We determine both the generic and all the exceptional structures for this two parameter algebra. The second is the periodic Temperley-Lieb algebra. The generic structure and part of the exceptional structure of this algebra have already been studied. Here we complete the analysis, using results from the study of the blob algebra. 
  We determine the structure of the partition algebra $P_n(Q)$ (a generalized Temperley-Lieb algebra) for specific values of $Q \in \C$, focusing on the quotient which gives rise to the partition function of $n$ site $Q$-state Potts models (in the continuous $Q$ formulation) in arbitrarily high lattice dimensions (the mean field case). The algebra is non-semi-simple iff $Q$ is a non-negative integer less than $n$. We determine the dimension of the key irreducible representation in every specialization. 
  Following Hawking, it is usual to mimic the effect of collapse space-time geometry on quantum fields in a semi-classical approximation by imposing suitable boundary conditions at the origin of coordinates, which effectively becomes a moving mirror. Suitable mirror trajectories induces a close analogue to the radiance of black holes, including a flux of outgoing radiation that appears accurately thermal. If the acceleration of the mirror eventually ceases the complete state of the radiation field is a pure quantum state, even though it is indistinguishable from an accurately thermal state for an arbitrarily long period of time and in a precise sense differs little from ``pure thermal'' closely followed by ``vacuum''. Suspicions that the semiclassical calculation of black hole radiance gives evidence for the evolution of pure into mixed states are criticized on this basis. Possible extensions of the model to mimic black holes more accurately (including the effects of back reaction and partial transparency), while remaining within the realm of tractable models, are suggested. 
  In this lecture a short introduction is given into the theory of the Feynman path integral in quantum mechanics. The general formulation in Riemann spaces will be given based on the Weyl- ordering prescription, respectively product ordering prescription, in the quantum Hamiltonian. Also, the theory of space-time transformations and separation of variables will be outlined. As elementary examples I discuss the usual harmonic oscillator, the radial harmonic oscillator, and the Coulomb potential. Lecture given at the graduate college ''Quantenfeldtheorie und deren Anwendung in der Elementarteilchen- und Festk\"orperphysik'', Universit\"at Leipzig, 16-26 November 1992. 
  In this paper the explicit form of the operator of transformation of the vacuum states for the general two-mode Bogolubov transformation is found 
  In models of oriented closed strings, anomaly cancellations are deeply linked to the {\it modular invariance} of the torus amplitude. If open and/or unoriented strings are allowed, there are no non-trivial modular transformations in the additional genus-one amplitudes (Klein bottle, annulus and M\"obius strip). As originally recognized by Green and Schwarz, in the ten-dimensional type-I superstring the anomaly cancellation results from a delicate interplay between the contributions of these additional surfaces. In lower-dimensional models, the possible presence of a number of antisymmetric tensors yields a generalization of the Green-Schwarz mechanism. I illustrate these results by referring to some six-dimensional chiral models, and I conclude by addressing the additional difficulties that one meets when trying to extend the construction to chiral four-dimensional models. (Contribution to ''From Superstrings to Supergravity'', Erice, ITALY, december 5-12, 1992) 
  Using the Polyakov string ansatz for the rectangular Wilson loop we calculate the static potential in the semiclassical approximation. Our results lead to a well defined sum over surfaces in the range $1<d<25$. 
  The quantum theory of the spherically symmetric gravity in 3+1 dimensions is investigated. The functional measures are explicitly evaluated and the physical state conditions are derived by using the technique developed in two dimensional quantum gravity. Then the new features which are not seen in ADM formalism come out. If $\kappa_s > 0 $, where $\kappa_s =(N-27)/12\pi $ and $N$ is the number of matter fields, a singularity appears, while for $\kappa_s <0$ the singularity disappears. The quantum dynamics of black hole seems to be changed by the sign of $\kappa_s $.   (Talk given by K.H. at Workshop on General Relativity and Gravity, Waseda, Tokyo, Japan, 18-20 Jan 1993.) 
  We establish a relationship between the modern theory of Yangians and the classical construction of the Gelfand-Zetlin bases for the complex Lie algebra $\gn$. Our approach allows us to produce the $q$-analogues of the Gelfand-Zetlin formulae in a straightforward way. 
  We study the stringy genus one partition function of $N=2$ SCFT's. It is shown how to compute this using an anomaly in decoupling of BRST trivial states from the partition function. A particular limit of this partition function yields the partition function of topological theory coupled to topological gravity. As an application we compute the number of holomorphic elliptic curves over certain Calabi-Yau manifolds including the quintic threefold. This may be viewed as the first application of mirror symmetry at the string quantum level. 
  Previous studies of high-energy scattering in QCD have shown a remarkable correspondence with two-dimensional field theory. In this paper we formulate a simple effective model in which this two-dimensional nature of the interactions is manifest. Starting from the (3+1)-dimensional Yang-Mills action, we implement the high energy limit $s\! >\! > \! t$ via a scaling argument and we derive from this a simplified effective theory. This effective theory is still (3+1)-dimensional, but we show that its interactions can to leading order be summarized in terms of a two-dimensional sigma-model defined on the transverse plane. Finally, we verify that our formulation is consistent with known perturbative results. This is a revised and extended version of hep-th 9302104. In particular, we have added a section that clarifies the connection with Lipatov's gluon emission vertex. 
  A detailed study is made of super elliptic curves, namely super Riemann surfaces of genus one considered as algebraic varieties, particularly their relation with their Picard groups. This is the simplest setting in which to study the geometric consequences of the fact that certain cohomology groups of super Riemann surfaces are not freely generated modules. The divisor theory of Rosly, Schwarz, and Voronov gives a map from a supertorus to its Picard group, but this map is a projection, not an isomorphism as it is for ordinary tori. The geometric realization of the addition law on Pic via intersections of the supertorus with superlines in projective space is described. The isomorphisms of Pic with the Jacobian and the divisor class group are verified. All possible isogenies, or surjective holomorphic maps between supertori, are determined and shown to induce homomorphisms of the Picard groups. Finally, the solutions to the new super Kadomtsev-Petviashvili (super KP) hierarchy of Mulase-Rabin which arise from super elliptic curves via the Krichever construction are exhibited. 
  We describe some recent progress in understanding and formulating string theory which is based on extensive studies of strings in lower (D=2) dimension. At the center is a large $W_{\infty}$ symmetry that appears most simply in the matrix model picture. In turn the symmetry defines the dynamics giving Ward identities and the complete S-matrix. The integrability aspect where nonlinear string phenomena emerges from linear matrix model dynamics is emphasized. Extensions involving couplings of discrete topological fields to the tachyon are also described. 
  Understanding the nontrivial features of light-front QCD is a central goal in current investigations of nonperturbative light-front field theory. We find that, with the choice of light-front gauge with antisymmetric boundary conditions for the field variables, the residual gauge freedom is fixed and the light-front QCD vacuum is trivial. The nontrivial structure in light-front QCD is determined by non-vanishing asymptotic physical (transverse) gauge fields at longitudinal infinity, which are responsible for nonzero topological winding number. 
  We discuss three classes of solitonic solutions in string theory: instantons, monopoles and string-like solitons. Instantons may provide a nonperturbative understanding of the vacuum structure of string theory, while monopoles may appear in string predictions for grand unification. The particular monopole solution shown has finite action as the result of the cancellation of gauge and gravitational singularities, a feature, which if it survives quantization, may yield insight into the structure of string theory as a finite theory of quantum gravity. In $D=10$, both monopoles and instantons possess fivebrane structure. The string-like solitons represent extended states of fundamental strings.( Talk given at INFN Eloisatron Project: 26th Workshop: ``From Superstrings to Supergravity'', Erice, Italy, Dec. 5-12, 1992.) 
  We present axion-dilaton black-hole and multi-black-hole solutions of the low-energy string effective action. Under $SL(2,R)$ electric-magnetic duality rotations only the ``hair" (charges and asymptotic values of the fields) of our solutions is transformed. The functional form of the solutions is duality-invariant. Axion-dilaton black holes with zero entropy and zero area of the horizon form a family of stable particle-like objects, which we call {\it holons}. We study the quantization of the charges of these objects and its compatibility with duality symmetry. In general the spectrum of black-hole solutions with quantized charges is not invariant under $SL(2,R)$ but only under $SL(2,Z)$ or one of its subgroups $\Gamma_{l}$. Because of their transformation properties, the asymptotic value of the axion-dilaton field of a black hole may be associated with the modular parameter $\tau$ of some complex torus and the integer numbers $(n,m)$ that label its quantized electric and magnetic charges may be associated with winding numbers. 
  Lectures given at the Trieste Summer School, 1992. These notes are an update of my review article "Classical and Quantum W-Gravity", preprint QMW-92-1, published in in "Strings and Symmetries 1991", with some extra material on W-geometry, W-symmetry in conformal field theory and W-strings. 
  By considering a set of $N$ anyonic oscillators ( non-local, intrinsic two-dimensional objects interpolating between fermionic and bosonic oscillators) on a two-dimensional lattice, we realize the $SU_q(N)$ quantum algebra by means of a generalized Schwinger construction. We find that the deformation parameter $q$ of the algebra is related to the anyonic statistical parameter $\nu$ by $q=exp({\rm i}\pi\nu)$. 
  The master equation describing non-equilibrium one-dimensional problems like diffusion limited reactions or critical dynamics of classical spin systems can be written as a Schr\"odinger equation in which the wave function is the probability distribution and the Hamiltonian is that of a quantum chain with nearest neighbor interactions. Since many one-dimensional quantum chains are integrable, this opens a new field of applications. At the same time physical intuition and probabilistic methods bring new insight into the understanding of the properties of quantum chains. A simple example is the asymmetric diffusion of several species of particles which leads naturally to Hecke algebras and $q$-deformed quantum groups. Many other examples are given. Several relevant technical aspects like critical exponents, correlation functions and finite-size scaling are also discussed in detail. 
  Intended for mathematical physicists interested in applications of the division algebras to physics, this article highlights some of their more elegant properties with connections to the theories of Galois fields and quadratic residues. 
  A general formalism for covariant $W_3$ string scattering is given. It is found necessary to use screening charges that are constructed from the $W_3$ fields including ghosts. The scattering amplitudes so constructed contain within them Ising model correlation functions and agree with those found previously by the authors. Using the screening charge and a picture changing operator, an infinite number of states in the cohomology of Q are generated from only three states. We conjecture that, apart from discrete states, these are all the states in the cohomology of Q. 
  RSOS models based on the Lie algebras $B_m$, $C_m$ and $D_m$ are derived from the braiding of conformal field theory. This gives the first systematic derivation of these models earlier described by Jimbo et al. The general two field Boltzmann weights associated to any RCFT are described, giving in particular the off critical thermalized Boltzmann weights. Crossing properties are discussed and are shown to agree with the general theory which connects these with toroidal modular transformations. The soliton systems based on these lattice models are described and are conjectured based on the mass formulae and the spins of the integrals of motions to describe perturbations of the RCFT $G_k\times G_1\over G_{k+1}$, where $G$ is the corresponding Lie algebra. 
  We perform a perturbative analysis of the Aharonov-Bohm problem to one loop in a field-theoretic formulation, and show that contact interactions are necessary for renormalizability. In general, the classical scale invariance of this problem is broken quantum mechanically. There exists however a critical point for which this anomaly disappears. 
  The equations of motion for the position and spin of a classical particle coupled to an external electromagnetic and gravitational potential are derived from an action principle. The constraints insuring a correct number of independent spin components are automatically satisfied. In general the spin is not Fermi-Walker transported nor does the position follow a geodesic, although the deviations are small for most situations. 
  We show that the construction of Ocneanu, which yields 1 for any 4D manifold, is not identical to our construction, which gives different numbers for different manifolds. 
  In the canonical light-front QCD, the elimination of unphysical gauge degrees of freedom leads to a set of boundary integrals which are associated with the light-front infrared singularity. We find that a consistent treatment of the boundary integrals leads to the cancellation of the light-front linear infrared divergences. For physical states, the requirement of finite energy density in the light-front gauge $(A_a^+=0)$ results in equations which determine the asymptotic behavior of the transverse (physical) gauge degrees of freedom at longitudinal infinity. These asymptotic fields are generated by the boundary integrals and they are responsible for the topological winding number. They also involve non-local behavior in the transverse direction that leads to non-local forces. 
  We derive a class of solutions to the string sigma-model equations for the closed bosonic string. The tachyon field is taken to form a constant condensate and the beta-function equations at one-loop level are solved for the evolution of the metric and the dilaton. The solutions represent critical string theories in arbitrary dimensions. The spectrum of the subclass of models with a linearly rising asymptotic dilaton is found using the Feigin-Fuks method. Certain approximate solutions arising in string field theory are used to illustrate the results explicitly. An argument based on conformal invariance leads to the conjecture that that stringy corrections to at least some singular spacetimes in general relativity result in non-singular metrics. We use the singularities of the big-bang/crunch type appearing in our models to examine this conjecture. 
  Intertwined multiple Chern-Simons gauge fields induce matrix statistics among particles. We analyse this theory on a torus, focusing on the vacuum structure and the Hilbert space. The theory can be mimicked, although not completely, by an effective theory with one Chern-Simons gauge field. The correspondence between the Wilson line integrals, vacuum degeneracy and wave functions for these two theories are discussed. Further, it is obtained in both of these cases that the two total momenta and Hamiltonian commute only in the physical Hilbert space. 
  Ladders of field polynomial differential forms obeying systems of descent equations and corresponding to observables and anomalies of gauge theories are renormalized. They obey renormalized descent equations. Moreover they are shown to have vanishing anomalous dimensions. As an application a simple proof of the nonrenormalization theorem for the nonabelian gauge anomaly is given. 
  The proof of the non-renormalization theorem for the gauge anomaly of four-dimensional theories is extended to the case of models with a vanishing one-loop gauge beta function. 
  In 2D conformal quantum field theory, we continue a systematic study of W-algebras with two and three generators and their highest weight representations focussing mainly on rational models. We review the known facts about rational models of W(2,\delta)-algebras. Our new rational models of W-algebras with two generators all belong to one of the known series. The majority of W-algebras with three generators -including the new ones constructed in this letter- can be explained as subalgebras or truncations of Casimir algebras. Nonetheless, for one solution of W(2,4,6) we reveal some features that do not fit into the pattern of Casimir algebras or orbifolds thereof. This shows that there are more W-algebras than those predicted from Casimir algebras (or Toda field theories). However, most of the known rational conformal field theories belong to the minimal series of some Casimir algebra. 
  Bracket preserving gauge equivalence is established between several two-boson generated KP type of hierarchies. These KP hierarchies reduce under symplectic reduction (via Dirac constraints) to KdV, mKdV and Schwarzian KdV hierarchies. Under this reduction the gauge equivalence is taking form of the conventional Miura maps between the above KdV type of hierarchies. 
  A black hole solution to three dimensional general relativity with a negative cosmological constant has recently been found. We show that a slight modification of this solution yields an exact solution to string theory. This black hole is equivalent (under duality) to the previously discussed three dimensional black string solution. Since the black string is asymptotically flat and the black hole is asymptotically anti-de Sitter, this suggests that strings are not affected by a negative cosmological constant in three dimensions. 
  Clarified certain points and related it to other work. 
  We show that the Hamiltonian $ h= H_{QED}+H_2$, where $H_{QED}$ is the spinor QED Hamiltonian and $H_2$ is the positive transversal photon mass term, is unbounded from below if the electromagnetic coupling constant $e^2$ is small enough, $e^2<e^2_0 $, and the transversal photon squared mass parameter $M^2$ is not too large: $0\leq M^2<e^2l^2c$, here, $l$ is the cut-off parameter, and $c$ and $e^2_0$, positive constants which do not depend on any parameters. 
  We discuss the possibility for the spectrum of topologically massive quantum electrodynamics with spinor matter fields to contain unexpected and unusual stable particle excitations for certain values of the topological photon mass. The new field theoretical phenomena arising from this novel spectral structure are briefly discussed. 
  The extended phase space method of Batalin, Fradkin and Vilkovisky is applied to formulate two dimensional gravity in a general class of gauges. A BRST formulation of the light-cone gauge is presented to reveal the relationship between the BRST symmetry and the origin of $SL(2,R)$ current algebra. From the same principle we derive the conformal gauge action suggested by David, Distler and Kawai. 
  The similarity between tree-level string theory scalar amplitudes, the Koba-Nielsen form ($S^{1}$) and the Virasoro-Shapiro form ($S^{2}$) suggests a natural $S^{n}$ generalization for a scalar amplitude. It is shown that the $S^{n}$ amplitude shares many essential properties of the string theory amplitudes, including $SO(n+1,1)$ conformal symmetry and linear Regge trajectories for the mass spectrum. We also discuss factorization and the critical dimension for the amplitude, which are the necessary conditions for the quantum mechanical consistency (unitarity) of the amplitude. 
  The classical 2D cosmological model of Callan, Giddings, Harvey and Strominger possesses a global symmetry that is responsible for decoupling of matter fields. The model is quantized on the basis of the extended phase space method to allow an exhaustive, algebraic analysis to find potential anomalies. Under a certain set of reasonable assumptions we show that neither the BRST symmetry of the theory nor the global symmetry suffers from an anomaly. From this we conclude that there is nothing to recognize the existence of black hole and therefore nothing to radiate in their cosmological model. 
  We develop a transfer matrix formalism for two-dimensional pure gravity. By taking the continuum limit, we obtain a "Hamiltonian formalism'' in which the geodesic distance plays the role of time. Applying this formalism, we obtain a universal function which describes the fractal structures of two dimensional quantum gravity in the continuum limit. 
  We consider a class of $N=2$ supersymmetric non--unitary theories in two--dimensional Minkowski spacetime which admit classical solitonic solutions. We show how these models can be twisted into a topological sector whose energy--momentum tensor is a BRST commutator. There is an infinite number of degrees of freedom associated to the zero modes of the solitons. As explicit realizations of such models we discuss the BRST quantization of a system of free fields, while in the interacting case we study $N=2$ complexified twisted Toda theories. 
  A class of lattice gauge theories is presented which exhibits novel topological properties. The construction is in terms of compact Wilson variables defined on a simplicial complex which models a four dimensional manifold with boundary. The case of Z2 and Z3 gauge groups is considered in detail, and we prove that at certain discrete values of the coupling parameter, the partition function in these models remains invariant under subdivision of the underlying simplicial complex. A variety of extensions is also presented. 
  A new way of solving the descent equations corresponding to the Wess-Zumino consistency conditions is presented. The method relies on the introduction of an operator $\delta$ which allows to decompose the exterior space-time derivative $d$ as a $BRS$ commutator. The case of the Yang-Mills theories is treated in detail. 
  We show that the harmonic Becchi-Rouet-Stora-Tyutin method of quantizing bosonic systems with second-class constraints or first-class holomorphic constraints extends to systems having both bosonic and fermionic second-class or first-class holomorphic constraints. Using a limit argument, we show that the harmonic BRST modified path integral reproduces the correct Senjanovic measure. 
  It is shown that the Lie algebra of the automorphic, meromorphic sl(2, C) -valued functions on a torus is a geometric realization of a certain infinite-dimensional finitely generated Lie algebra. In the trigonometric limit, when the modular parameter of the torus goes to zero, the former Lie algebra goes over into the sl(2,C) -valued loop algebra, while the latter one - into the Lie algebra (sl(2)^)'/(centre) . 
  The quantum bialgebra related to the Baxter's eight-vertex R-matrix is found as a quantum deformation of the Lie algebra of sl(2)-valued automorphic functions on a complex torus. 
  Remarks about highest weight states of the underlying quantum group are corrected. 
  Semi-infinite cohomology is constructed from scratch as the proper generalization of finite dimensional Lie algebra cohomology. The differential d and other operators are realized as universal inner deri- vations of a completed algebra, which acts on any appropriate semi-infinite complex. In particular, d is shown to be the unique derivation satisfying the "Cartan identity" and certain natural degree conditions. The proof that d is square-zero may well be the shortest (arguably, the only) one in print. 
  The wheeler-DeWitt method is applied to the quantization of the 1 + 1 dimensional dilaton gravity coupled with the conformal matter fields. Exact solutions to the WD equations are found, which are interpreted as right(left)-moving black holes. 
  Talk given at the 26th Workshop: ``From Superstrings to Supergravity", Erice - Sicily, 5-12 December 1992. We review the superconformal properties of 2d matter coupled to gravity, and extensions thereof. Focusing on topological strings, we recall how the superconformal structure helps to provide a direct link between Liouville theory coupled to matter, and matrix models. We also construct an infinite class of new theories based on $W$-gravity. 
  Using an argument due to Regge and Teitelboim, an expression for the ADM mass of 2d quantum dilaton gravity is obtained. By evaluating this expression we establish that the quantum theories which can be written as a Liouville-like theory, have a lower bound to energy, provided there is no critical boundary. This fact is then reconciled with the observation made earlier that the Hawking radiation does not appear to stop. The physical picture that emerges is that of a black hole in a bath of quantum radiation. We also evaluate the ADM mass for the models with RST boundary conditions and find that negative values are allowed. The Bondi mass of these models goes to zero for large retarded times, but becomes negative at intermediate times in a manner that is consistent with the thunderpop of RST. 
  A covariant formalism for physical perturbations propagating along a string in an arbitrary curved spacetime is developed. In the case of a stationary string in a static background the propagation of the perturbations is described by a wave-equation with a potential consisting of 2 terms: The first term describing the time-dilation and the second is connected with the curvature of space. As applications of the developed approach the propagation of perturbations along a stationary string in Rindler, de Sitter, Schwarzschild and Reissner-Nordstrom spacetimes are investigated. 
  A quantum version of 2D super dilaton gravity containing a black hole is constructed for $N>8$. A previous disagreement as to whether this is possible or not is resolved. 
  Talk given at the 26th Workshop: ``From Superstrings to Supergravity" Erice - Sicily, 5-12 December 1992: In this talk we discuss string consistency requirements on four dimensional string models, namely the cancellation of target space duality anomalies. The analysis is explicitly performed for (hypothetical) orbifold models assuming the massless spectrum of the supersymmetric standard model. In addition, some phenomenological properties of four-dimensional strings, like the unification of the standard model gauge coupling constants and soft supersymmetry breaking parameters, are investigated. 
  An application of the particular type of nonlinear operator algebras to spectral problems is outlined. These algebras are associated with a set of one-dimensional self-similar potentials, arising due to the q-periodic closure f_{j+N}(x)=qf_j(qx), k_{j+N}=q^2 k_j of a chain of coupled Riccati equations (dressing chain). Such closure describes q-deformation of the finite-gap and related potentials. The N=1 case corresponds to the q-oscillator spectrum generating algebra. At N=2 one gets a q-conformal quantum mechanics, and N=3 set of equations describes a deformation of the Painleve IV transcendent. 
  Higher-derivative generalization of the supersymmetric quantum mechanics is proposed. It is formally based on the standard superalgebra but supercharges involve differential operators of the order $n$. As a result, their anticommutator entails polynomial of a Hamiltonian. The Witten index does not characterize spontaneous supersymmetry breaking in such models. 
  Two-dimensional quantum gravity with an $R^2$ term is investigated in the continuum framework. It is shown that the partition function for small area $A$ is highly suppressed by an exponential factor $exp \{ -2\pi (1-h)^2/(m^2A) \}$, where $1/m^2$ is the coefficient (times $32\pi$) of $R^2$ and $h$ is the genus of the surface. Although positivity is violated, at a short distance scale ( $\ll 1/m$) surfaces are smooth and the problem of the branched polymer is avoided. 
  Some aspects of the rotating three-dimensional Einstein-Anti-de-Sitter black hole solution, constructed recently by Banados, Teitelboim and Zanelli are discussed. It is shown explicitly that this black hole represents the most general black hole type solution of the Einstein-Anti-de-Sitter theory. The interpretation of one of the integrals of motion as the spin is discussed. Its physics relies on the topological structure of the black hole manifold, and the notion of simultaneity of space-like separated intervals. The relationship of the black hole solution to string theory on a $2 + 1$ dimensional target space is examined, and it is shown that the black hole can be understood as a part of the full axion-dilaton-gravity, realized as a WZWN $\sigma $ model. In conclusion, the pertinence of this solution to four-dimensional black strings and topologically massive gravity is pointed out. 
  Nonlinearity of electromagnetic field vibrations described by q-oscillators is shown to produce essential dependence of second correlation functions on intensity and deformation of Planck distribution. Experimental tests of such nonlinearity are suggested. 
  : Algebraic properties of orbifold models on arbitrary Riemann surfaces are investigated. The action of mapping class group transformations and of standard geometric operations is given explicitly. An infinite dimensional extension of the quantum group is presented. 
  By building a general dynamical model for quantum measurement process,it is shown that the factorization of reduced evolution operator sufficiently results in the quantum mechanical realization of the wave packet collapse and the state correlation between the measured system and the measuring instrument-detector.This realizability is largely independent of the details of both the interaction and Hamiltonian of detector. The Coleman-Hepp model and all its generalizations are only the special cases of the more universal model given in this letter.An explicit example of this model is finally given in connection with coherent state. 
  In the framework of communication theory, we analyse the gedanken experiment in which beams of quanta bearing information are flashed towards a black hole. We show that stimulated emission at the horizon provides a correlation between incoming and outgoing radiations consisting of bosons. For fermions, the mechanism responsible for the correlation is the Fermi exclusion principle. Each one of these mechanisms is responsible for the a partial transfer of the information originally coded in the incoming beam to the black--hole radiation. We show that this process is very efficient whenever stimulated emission overpowers spontaneous emission (bosons). Thus, black holes are not `ultimate waste baskets of information'. 
  We investigate the spectral properties of a random matrix model, which in the large $N$ limit, embodies the essentials of the QCD partition function at low energy. The exact spectral density and its pair correlation function are derived for an arbitrary number of flavors and zero topological charge. Their microscopic limit provide the master formulae for sum rules for the inverse powers of the eigenvalues of the QCD Dirac operator as recently discussed by Leutwyler and Smilga. 
  We review recent results concerning the representation of conformal field theory characters in terms of fermionic quasi-particle excitations, and describe in detail their construction in the case of the integrable three-state Potts chain. These fermionic representations are $q$-series which are generalizations of the sums occurring in the Rogers-Ramanujan identities. (To appear in the proceedings of ``Yang-Baxter Equations in Paris'', July 1992, J.-M.~Maillard (ed.).) 
  Exact operator quantization is perfomed of a model of two-dimensional dilaton gravity in Lorentzian spacetime, classically equivalent to the one proposed by Callan, Giddings, Harvey and Strominger, in the special case with 24 massless matter scalars. This is accomplished by developing a non-linear and non-local quantum canonical transformation of basic interacting fields into a set of free fields, rigorously taking into account the spatially closed boundary condition. The quantized model enjoys conformal invariance and the entire set of physical states and operators are obtained in the BRST formalism. In addition, a rather detailed discussion of the nature of the basic issues for exact treatment of models of quantum gravity is provided. 
  The recent classification of Landau--Ginzburg potentials and their abelian symmetries focuses attention on a number of models with large positive Euler number for which no mirror partner is known. All of these models are related to Calabi--Yau manifolds in weighted $\IP_4$, with a characteristic structure of the defining polynomials. A closer look at these potentials suggests a series of non-linear transformations, which relate the models to configurations for which a construction of the mirror is known, though only at certain points in moduli space. A special case of these transformations generalizes the $\ZZ_2$ orbifold representation of the $D$ invariant, implying a hidden symmetry in tensor products of minimal models. 
  Lines generated by marginal deformations of WZW models are considered. The Weyl symmetry at the WZW point implies the existence of a duality symmetry on such lines. The duality is interpreted as a broken gauge symmetry in string theory. It is shown that at the two end points the axial and vector cosets are obtained. This shows that the axial and vector cosets are equivalent CFTs both in the compact and the non-compact cases. Moreover, it is shown that there are $\s$-model deformations that interpolate smoothly between manifolds with different topologies. 
  We derive the moduli dependent threshold corrections to gauge couplings in toroidal orbifold compactifications. The underlying six dimensional torus lattice of the heterotic string theory is not assumed ---as in previous calculations--- to decompose into a direct sum of a four--dimensional and a two--dimensional sublattice, with the latter lying in a plane left fixed by a set of orbifold twists. In this more general case the threshold corrections are no longer automorphic functions of the modular group, but of certain congruence subgroups of the modular group. These groups can also be obtained by studying the massless spectrum; moreover they have larger classes of automorphic functions. As a consequence the threshold corrections cannot be uniquely determined by symmetry considerations and certain boundary conditions at special points in the moduli space, as was claimed in previous publications. 
  There were some errors in paper hep-th/9303018 in formulas 6.1, 6.6, 6.8, 6.11. These errors have been corrected in the present version of this paper. There are also some minor changes in the introduction. 
  We consider the conformal properties of geometries described by higher-rank line elements. A crucial role is played by the conformal Killing equation (CKE). We introduce the concept of null-flat spaces in which the line element can be written as ${ds}^r=r!d\zeta_1\cdots d\zeta_r$. We then show that, for null-flat spaces, the critical dimension, for which the CKE has infinitely many solutions, is equal to the rank of the metric. Therefore, in order to construct an integrable conformal field theory in 4 dimensions we need to rely on fourth-rank geometry. We consider the simple model ${\cal L}={1\over 4} G^{\mu\nu\lambda\rho}\partial_\mu\phi\partial_\nu\phi\partial_\lambda\phi \partial_\rho\phi$ and show that it is an integrable conformal model in 4 dimensions. Furthermore, the associated symmetry group is ${Vir}^4$. 
  The van Vleck determinant is an ubiquitous object, arising in many physically interesting situations such as: (1) WKB approximations to quantum time evolution operators and Green functions. (2) Adiabatic approximations to heat kernels. (3) One loop approximations to functional integrals. (4) The theory of caustics in geometrical optics and ultrasonics. (5) The focussing and defocussing of geodesic flows in Riemannian manifolds. While all of these topics are interrelated, the present paper is particularly concerned with the last case and presents extensive theoretical developments that aid in the computation of the van Vleck determinant associated with geodesic flows in Lorentzian spacetimes. {\sl A fortiori} these developments have important implications for the entire array of topics indicated. PACS: 04.20.-q, 04.20.Cv, 04.60.+n. To appear in Physical Review D47 (1993) 15 March. 
  The two-loop effective action for the SU(3) gauge model in a constant background field ${\bar A}_0(x,t)=B_0^3T_3+B_0^8T_8$ is recalculated for a gauge with an arbitrary $\xi$-parameter. The gauge-invariant thermodynamical potential is found and its extremum points are investigated. Within a two-loop order we find that the stable nontrivial vacuum is completely equivalent to the trivial one but when the high order corrections being taken into account the indifferent equilibrium seems to be broken. Briefly we also discuss the infrared peculiarities and their status for the gauge models with a nonzero condensate. 
  The nonanalytic $g^3$-term is calculated for SU(2)-effective action at finite temperature and the status of a gauge fields condensation is briefly discussed. 
  The recent renaissance of wormhole physics has led to a very disturbing observation: If traversable wormholes exist then it appears to be rather easy to to transform such wormholes into time machines. This extremely disturbing state of affairs has lead Hawking to promulgate his chronology protection conjecture. This paper continues a program begun in an earlier paper [Physical Review {\bf D47}, 554--565 (1993), hepth@xxx/9202090]. An explicit calculation of the vacuum expectation value of the renormalized stress--energy tensor in wormhole spacetimes is presented. Point--splitting techniques are utilized. Particular attention is paid to computation of the Green function [in its Hadamard form], and the structural form of the stress-energy tensor near short closed spacelike geodesics. Detailed comparisons with previous calculations are presented, leading to a pleasingly unified overview of the situation. 
  We study the evolution of cosmic strings taking into account the frictional force due to the surrounding radiation. We consider small perturbations on straight strings, oscillation of circular loops and small perturbations on circular loops. For straight strings, friction exponentially suppresses perturbations whose co-moving scale crosses the horizon before cosmological time $t_*\sim \mu^{-2}$ (in Planck units), where $\mu$ is the string tension. Loops with size much smaller than $t_*$ will be approximately circular at the time when they start the relativistic collapse. We investigate the possibility that such loops will form black holes. We find that the number of black holes which are formed through this process is well bellow present observational limits, so this does not give any lower or upper bounds on $\mu$. We also consider the case of straight strings attached to walls and circular holes that can spontaneously nucleate on metastable domain walls. 
  The sigma model action described in this paper differs in four important features from the usual sigma model action for the four-dimensional Green-Schwarz heterotic superstring in a massless background. Firstly, the action is constructed on an N=(2,0) super-worldsheet using a Kahler potential and an Ogievetsky-Sokatchev constraint; secondly, the target-space background fields are unconstrained; thirdly, the target-space dilaton couples to the two-dimensional curvature; and fourthly, the action reduces in a flat background to a free-field action. A conjecture is made for generalizing this N=(2,0) sigma model action to the ten-dimensional Green-Schwarz heterotic superstring in a manner that preserves these four new features. 
  A new trigonometric degeneration of the Sklyanin algebra is found and the functional realization of its representations in space of polynomials in one variable is studied. A further contraction gives the standard quantum algebra $U_{q}(sl(2))$. It is shown that the degenerate Sklyanin algebra contains a subalgebra isomorphic to algebra of functions on the quantum sphere $(SU(2)/SO(2))_{q^{1\over2}}$. The diagonalization of general quadratic form in its generators leads in the functional realization to the difference equation for Askey-Wilson polynomials. 
  Implementing the requirement that a field theory be invariant under Schwinger-Dyson BRST symmetry in the Hamiltonian formalism, we show the equivalence between Hamiltonian and Lagrangian BRST-formalism at the path integral level. The Lagrangian quantum master equation is derived as a direct consequence of the Fradkin-Vilkovisky theorem in Hamiltonian BRST quantisation. 
  The heat coefficients related to the Laplace-Beltrami operator defined on the hyperbolic compact manifold $H^3/\Ga$ are evaluated in the case in which the discrete group $\Ga$ contains elliptic and hyperbolic elements. It is shown that while hyperbolic elements give only exponentially vanishing corrections to the trace of the heat kernel, elliptic elements modify all coefficients of the asymptotic expansion, but the Weyl term, which remains unchanged. Some physical consequences are briefly discussed in the examples. 
  Considerable interest has recently been expressed in the entropy versus area relationship for ``dirty'' black holes --- black holes in interaction with various classical matter fields, distorted by higher derivative gravity, or infested with various forms of quantum hair. In many cases it is found that the entropy is simply related to the area of the event horizon: S = k A_H/(4\ell_P^2). For example, the ``entropy = (1/4) area'' law *holds* for: Schwarzschild, Reissner--Nordstrom, Kerr--Newman, and dilatonic black holes. On the other hand, the ``entropy = (1/4) area'' law *fails* for: various types of (Riemann)^n gravity, Lovelock gravity, and various versions of quantum hair. The pattern underlying these results is less than clear. This paper systematizes these results by deriving a general formula for the entropy: S = {k A_H/(4\ell_P^2)}   + {1/T_H} \int_\Sigma [rho - {L}_E ] K^\mu d\Sigma_\mu   + \int_\Sigma s V^\mu d\Sigma_\mu. (K^\mu is the timelike Killing vector, V^\mu the four velocity of a co--rotating observer.) If no hair is present the validity of the ``entropy = (1/4) area'' law reduces to the question of whether or not the Lorentzian energy density for the system under consideration is formally equal to the Euclideanized Lagrangian. ****** To appear in Physical Review D 15 July 1993 ****** [Stylistic changes, minor typos fixed, references updated, discussion of the Born-Infeld system excised] 
  The Laughlin states for $N$ interacting electrons at the plateaus of the fractional Hall effect are studied in the thermodynamic limit of large $N$. It is shown that this limit leads to the semiclassical regime for these states, thereby relating their stability to their semiclassical nature. The equivalent problem of two-dimensional plasmas is solved analytically, to leading order for $N\to\infty$, by the saddle point approximation - a two-dimensional extension of the method used in random matrix models of quantum gravity and gauge theories. To leading order, the Laughlin states describe classical droplets of fluids with uniform density and sharp boundaries, as expected from the Laughlin ``plasma analogy''. In this limit, the dynamical $W_\infty$-symmetry of the quantum Hall states expresses the kinematics of the area-preserving deformations of incompressible liquid droplets. 
  The program of induced QCD requires that there exist self-interactions among the heavy matter fields (an adjoint scalar and a few fermions in the fundamental representation) which tend to spoil the asymptotic freedom of SU(N) gauge theory. We consider general interactions between such matter fields and show that, on the contrary, to two loops, they all tend to enhance asymptotic freedom in the perturbative regime. This result casts doubt on whether induced QCD is equivalent to real QCD. 
  We generalize the results of a previous paper by one of the authors to show a relationship among a class of string solutions through $O(\tilde d, \tilde d)$ transformations. The results are applied to a rotating black hole solution of three dimensional general relativity discussed recently.  We extend the black hole solution to string theory and show its connection with the three dimensional black string with nonzero momentum through an $O(\tilde d, \tilde d)$ transformation of the above type. 
  {}~~~We show that the recently constructed $~N=4$~ supersymmetric self-dual Yang-Mills theory as the consistent background of \hbox{$~N=2$} open superstring will generate Witten's topological field theory in two-dimensions as a descendant theory after appropriate twisted dimensional reduction/truncations. We also show that this topological field theory further generates supersymmetric Korteweg de Vries equations, $~SL(n)\-$Toda theory and $~W_\infty\-$gravity in the $~n\rightarrow\infty$~ limit. Considering also that this topological field theory is to generate other integrable and topological systems, our results give supporting evidence for the conjecture that the $~N=2$~ open superstring theory is the underlying ``master theory'' of integrable and topological systems, {\it via} the intermediate $~N=4$~ supersymmetric self-dual Yang-Mills theory as its consistent target space-time background. 
  Classical W-algebras in higher dimensions have been recently constructed. In this letter we show that there is a finitely generated subalgebra which is isomorphic to the algebra of local diffeomorphisms in D dimensions. Moreover, there is a tower of infinitely many fields transforming under this subalgebra as symmetric tensorial one-densities. We also unravel a structure isomorphic to the Schouten symmetric bracket, providing a natural generalization of w_\infty in higher dimensions. 
  We give a construction of Drienfeld's quantum double for a nonstandard deformation of Borel subalgebra of $sl(2)$. We construct explicitly some simple representations of this quantum algebra and from the universal R-matrix we obtain the explicit solutions of the Yang-Baxter equation in those cases. 
  We compute the fundamental group of the "moduli space" of classical solutions of the two dimensional Euclidean $S^n$-model. 
  We obtain some new results on classical solutions of two dimensional Euclidean sigma models. From earlier work of Din-Zakrzewski, Glaser-Stora, and numerous differential geometers, one knows explicit solutions in the case of the $S^n$-model, the $CP^n$-model, and the $U(n)$-model. However, very little is known about the "moduli space" of solutions itself. In this paper we study the connected components of these spaces. In a subsequent paper (with M. Furuta and M. Kotani), we compute the fundamental group, in the case of the $S^n$-model. 
  The Lie-Poisson analogues of the cotangent bundle and coadjoint orbits of a Lie group are considered. For the natural Poisson brackets the symplectic leaves in these manifolds are classified and the corresponding symplectic forms are described. Thus the construction of the Kirillov symplectic form is generalized for Lie-Poisson groups. 
  The tensor product of the division algebras, which is a kernel for the structure of the Standard Model, is also a root for the Clifford algebra of (1,9)-space-time. A conventional Dirac Lagrangian, employing the (1,9)-Dirac operator acting on the Standard Model hyperfield, gives rise to matter into antimatter transitions not mediated by any gauge field. These transitions are eliminated by restricting the dependencies of the components of the hyperfield on the extra six dimensions, which appear in this context as a complex triple. 
  We discuss supersymmetry breakdown in effective supergravities such as emerge in the low-energy limit of superstring theory. Without specifying the precise trigger of the breakdown, we analyse the soft parameters in the Lagrangian of the supersymmetrized Standard Model. 
  We measure the fractal structure of four dimensional simplicial quantum gravity by identifying so-called baby universes. This allows an easy determination of the critical exponent $\g$ connected to the entropy of four-dimensional manifolds. 
  We investigate the phase structure of four-dimensional quantum gravity coupled to Ising spins or Gaussian scalar fields by means of numerical simulations.   The quantum gravity part is modelled by the summation over random simplicial manifolds, and the matter fields are located in the center of the 4-simplices, which constitute the building blocks of the manifolds. We find that the coupling between spin and geometry is weak away from the critical point of the Ising model. At the critical point there is clear coupling, which qualitatively agrees with that of gaussian fields coupled to gravity. In the case of pure gravity a transition between a phase with highly connected geometry and a phase with very ``dilute'' geometry has been observed earlier. The nature of this transition seems unaltered when matter fields are included.   It was the hope that continuum physics could be extracted at the transition between the two types of geometries. The coupling to matter fields, at least in the form discussed in this paper, seems not to improve the scaling of the curvature at the transition point. 
  We consider a two-parameter $(\bar c, \tilde c)$ family of quantum integrable Hamiltonians for a chain of alternating spins of spin $s=1/2$ and $s=1$. We determine the thermodynamics for low-temperature $T$ and small external magnetic field $H$, with $T << H$. In the antiferromagnetic $(\bar c > 0, \tilde c > 0)$ case, the model has two gapless excitations. In particular, for $\bar c = \tilde c$, the model is conformally invariant and has central charge $c_{vir} = 2$. When one of these parameters is zero, the Bethe Ansatz equations admit an infinite number of solutions with lowest energy. 
  We present a systematic implementation of differential renormalization to all orders in perturbation theory. The method is applied to individual Feynamn graphs written in coordinate space. After isolating every singularity. which appears in a bare diagram, we define a subtraction procedure which consists in replacing the core of the singularity by its renormalized form givenby a differential formula. The organizationof subtractions in subgraphs relies in Bogoliubov's formula, fulfilling the requirements of locality, unitarity and Lorentz invariance. Our method bypasses the use of an intermediate regularization andautomatically delivers renormalized amplitudes which obey renormalization group equations. 
  We reexamine the $W_{\infty}$ symmetry of the $sl(N)$ Conformal Affine Toda theories. It is shown that it is possible to reduce (nonuniquely) the zero curvature equation to a Lax equation for a first order pseudodifferential oprator, whose coefficients are the generators of the $W_{\infty}$ algebra. This clarifies the known relation between the Conformal Affine Toda theories and the KP hierarchy. A possible correspondence between the matrix models and the Conformal Affine Toda models is discussed. 
  Many Texo's have been corrected and a reference added. 
  In this paper we generalize some of these results for loop algebras and groups as well as for the Virasoro algebra to the two-dimensional case. We define and study a class of infinite dimensional complex Lie groups which are central extensions of the group of smooth maps from a two dimensional orientable surface without boundary to a simple complex Lie group G. These extensions naturally correspond to complex curves. The kernel of such an extension is the Jacobian of the curve. The study of the coadjoint action shows that its orbits are labelled by moduli of holomorphic principal G-bundles over the curve and can be described in the language of partial differential equations. In genus one it is also possible to describe the orbits as conjugacy classes of the twisted loop group, which leads to consideration of difference equations for holomorphic functions. This gives rise to a hope that the described groups should possess a counterpart of the rich representation theory that has been developed for loop groups. We also define a two-dimensional analogue of the Virasoro algebra associated with a complex curve. In genus one, a study of a complex analogue of Hill's operator yields a description of invariants of the coadjoint action of this Lie algebra. The answer turns out to be the same as in dimension one: the invariants coincide with those for the extended algebra of currents in sl(2). 
  The ground state density matrix for a massless free field is traced over the degrees of freedom residing inside an imaginary sphere; the resulting entropy is shown to be proportional to the area (and not the volume) of the sphere. Possible connections with the physics of black holes are discussed. 
  The two dimensional dilaton gravity with the cosmological term and with an even number of matter fields minimally coupled to the gravity is considered. The exact solutions to the Wheeler-DeWitt equation are obtained in an explicit functional form, which contain an arbitrary holomorphic function of the matter fields. 
  Based upon the formalism of conformal field theory with a boundary, we give a description of the boundary effect on fully developed two dimensional turbulence. Exact one and two point velocity correlation functions and energy power spectrum confined in the upper half plane are obtained using the image method. This result enables us to address the infrared problem of the theory of conformal turbulence. 
  We discuss the conditions under which the BRST operator of a $W$-string can be written as the sum of two operators that are separately nilpotent and anticommute with each other. We illustrate our results with the example of the non-critical $W_3$-string. Furthermore, we apply our results to make a conjecture about a relationship between the spectrum of a non-critical $W_n$-string and a $W_{n-1}$-string. 
  We show how any integrable 2D QFT enjoys the existence of infinitely many non--abelian {\it conserved} charges satisfying a Yang--Baxter symmetry algebra. These charges are generated by quantum monodromy operators and provide a representation of $q-$deformed affine Lie algebras. We review and generalize the work of de Vega, Eichenherr and Maillet on the bootstrap construction of the quantum monodromy operators to the sine--Gordon (or massive Thirring) model, where such operators do not possess a classical analogue. Within the light--cone approach to the mT model, we explicitly compute the eigenvalues of the six--vertex alternating transfer matrix $\tau(\l)$ on a generic physical state, through algebraic Bethe ansatz. In the thermodynamic limit $\tau(\l)$ turns out to be a two--valued periodic function. One determination generates the local abelian charges, including energy and momentum, while the other yields the abelian subalgebra of the (non--local) YB algebra. In particular, the bootstrap results coincide with the ratio between the two determinations of the lattice transfer matrix. 
  The high-energy Regge behavior of gauge theories is studied via the formalism of Analytic Multi-Regge Theory. Perturbative results for spontaneously-broken theories are first organised into reggeon diagrams. Unbroken gauge theories are studied via a reggeon diagram infra-red analysis of symmetry restoration. Massless fermions play a crucial role and the case of QCD involves the Super-Critical Pomeron as an essential intermediate stage. An introductory review of the build up of transverse momentum diagrams and reggeon diagrams from leading log calculations in gauge theories is presented first. It is then shown that the results closely reproduce the general structure for multi-regge amplitudes derived in Part I of the article, allowing the construction of general reggeon diagrams for spontaneously-broken theories. Next it is argued that, with a transverse-momentum cut-off, unbroken gauge theories can be reached through an infra-red limiting process which successively decouples fundamental representation Higgs fields . The first infra-red limit studied is the restoration of SU(2) gauge symmetry. The analysis is dominated by the exponentiation of divergences imposed by Reggeon Unitarity and the contribution of massless quarks ... 
  We discuss two classes of exact (in $\a'$) string solutions described by conformal sigma models. They can be viewed as two possibilities of constructing a conformal model out of the non-conformal one based on the metric of a $D$-dimensional homogeneous $G/H$ space. The first possibility is to introduce two extra dimensions (one space-like and one time-like) and to impose the null Killing symmetry condition on the resulting $2+D$ dimensional metric. In the case when the ``transverse" model is $n=2$ supersymmetric and the $G/H$ space is K\"ahler-Einstein the resulting metric-dilaton background can be found explicitly. The second possibility - which is realised in the sigma models corresponding to $G/H$ conformal theories - is to deform the metric, introducing at the same time a non-trivial dilaton and antisymmetric tensor backgrounds. The expressions for the metric and dilaton in this case are derived using the operator approach in which one identifies the equations for marginal operators of conformal theory with the linearised (near a background) expressions for the `$\b$-functions'. Equivalent results are then reproduced in the direct field-theoretical approach based on computing first the effective action of the $G/H$ gauged WZW model and then solving for the $2d$ gauge field. Both the bosonic and the supersymmetric cases are discussed. ( To be published in the Proceedings of the 26 Workshop ``From Superstrings to Supergravity", Erice, 5-12 December,1992.) 
  We study the random walk representation of the two-point function in statistical mechanics models near the critical point. Using standard scaling arguments we show that the critical exponent $\nu$ describing the vanishing of the physical mass at the critical point is equal to $\nu_\theta/ d_w$. $d_w$ is the Hausdorff dimension of the walk. $\nu_\theta$ is the exponent describing the vanishing of the energy per unit length of the walk at the critical point. For the case of O(N) models, we show that $\nu_\theta=\varphi$, where $\varphi$ is the crossover exponent known in the context of field theory. This implies that the Hausdorff dimension of the walk is $\varphi/\nu$ for O(N) models. 
  We review some of the basic features of the Kazakov-Migdal model of induced QCD. We emphasize the role of $Z_N$ symmetry in determining the observable properties of the model and also argue that it can be broken explicitly without ruining the solvability of induced QCD in the infinite $N$ limit. We outline the sort of critical behavior which the master field must have in order that the model is still solvable. We also review some aspects of the $D=1$ version of the model where the partition function can be obtained analytically. To be published in the Proceedings of "Mathematical Physics, String Theory and Quantum Gravity", Rhakov, Ukraine. October, 1992 
  We show that in heterotic string theory compactified on a six dimensional torus, the lower bound (Bogomol'nyi bound) on the dyon mass is invariant under the SL(2,Z) transformation that interchanges strong and weak coupling limits of the theory. Elementary string excitations are also shown to satisfy this lower bound. Finally, we identify specific monopole solutions that are related via the strong-weak coupling duality transformation to some of the elementary particles saturating the Bogomol'nyi bound, and these monopoles are shown to have the same mass and degeneracy of states as the corresponding elementary particles. 
  We extend the theory of the gauging of classical quadratically nonlinear algebras without a central charge but with a coset structure, to the quantum level. Inserting the minimal anomalies into the classical transformation rules of the currents introduces further quantum corrections to the classical transformation rules of the gauge fields and currents which additively renormalize the structure constants. The corresponding Ward identities are the c -> infinity limit of the full quantum Ward identities, and reveal that the c -> infinity limit of the quantum gauge algebra closes on fields and currents. Two examples are given. 
  A solution of effective string theory in four dimensions is presented which admits interpretation of a rotating black cosmic string. It is constructed by tensoring the three dimensional black hole, extended with the Kalb-Ramond axion, with a flat direction. The physical interpretation of the solution is discussed, with special attention on the axion, which is found to play a role very similar to a Higgs field. Finally, it is pointed out that the solution represents an exact WZWN $\sigma$ model on the string world sheet, to all orders in the inverse string tension $\alpha'$. 
  We propose a new scenario in a class of superstring derived standard--like models that explains the suppression of the left--handed neutrino masses. Due to nonrenormalizable terms and the breaking of the $U(1)_{Z^\prime}$ symmetry a generalized see--saw mechanism takes place. Contrary to the traditional see--saw mechanism in GUTs, the see--saw scale and the right--handed neutrino mass scale are suppressed relative to the $U(1)_{Z^\prime}$ breaking scale. 
  We calculate the one-loop effective potential of a self-interacting scalar field on the spacetime of the form $\reals^2\times H^2/\Gamma$. The Selberg trace formula associated with a co-compact discrete group $\Gamma$ in $PSL(2,\reals )$ (hyperbolic and elliptic elements only) is used. The closed form for the one-loop unrenormalized and renormalized effective potentials is given. The influence of non-trivial topology on curvature induced phase transitions is also discussed. 
  The energy density in Fulling-Rindler vacuum, which is known to be negative "everywhere" is shown to be positive and singular on the horizons in such a fashion as to guarantee the positivity of the total energy. The mechanism of compensation is displayed in detail. 
  I discuss the quantum instability of an electric field in a theory of open strings. 
  We study the Liouville theory on a Riemann surface of genus g by means of their associated Drinfeld--Sokolov linear systems. We discuss the cohomological properties of the monodromies of these systems. We identify the space of solutions of the equations of motion which are single--valued and local and explicitly represent them in terms of Krichever--Novikov oscillators. Then we discuss the operator structure of the quantum theory, in particular we determine the quantum exchange algebras and find the quantum conditions for univalence and locality. We show that we can extend the above discussion to $sl_n$ Toda theories. 
  The partition function corresponding to the "polytopic" action, a new action for the gravitational interaction which we have proposed recently, is computed in the simplest two-dimensional geometries of genus zero and one. The functional integral over the Liouville field is approximated by an ordinary integral over the constant zero mode. We study the dependence on both the coupling constant and the cosmological constant, and compare with recent scaling results in standard 2D quantum gravity. 
  The paper introduces a new geometric interpretation of the quantum Knizhnik-Zamolodchikov equations introduced in 1991 by I.Frenkel and N.Reshetikhin. It turns out that these equations can be linked to certain holomorphic vector bundles on the N-th Cartesian power of an elliptic curve. These bundles are naturally constructed by a gluing procedure from a system of trigonometric quantum affine $R$-matrices. Meromorphic solutions of the quantum KZ equations are interpreted as sections of such a bundle. This interpretation is an analogue of the interpretation of solutions of the classical KZ equations as sections of a flat vector bundle. Matrix elements of intertwiners between representations of the quantum affine algebra correspond to regular (holomorphic) sections. The vector bundle obtained from the quantum KZ system is topologically nontrivial. Its topology can be completely described in terms of crystal bases, using the crystal limit ``q goes to 0''. In the case N=2, this bundle is is essentially a bundle on an elliptic curve which is shown to be semistable (for the case of quantum sl(2)) if the parameters take generic values. The proof makes use of the crystal limit ``q goes to 0''. Finally, we give a vector bundle interpretation of the generalized quantum KZ equations for arbitrary affine root systems defined recently by Cherednik. 
  A recent proposal for a background independent open string field theory is studied in detail for a class of backgrounds that correspond to general quadratic boundary interactions on the world-sheet. A short-distance cut-off is introduced to formulate the theory with a finite number of local and potentially unrenormalizable boundary couplings. It is shown that renormalization of the boundary couplings makes both the world-sheet partition function and the string field action finite and cut-off independent, although the resulting string field action has an unpalatable dependence on the leading unrenormalizable coupling. 
  A general technique is outlined for investigating supersymmetry properties of a charged spin-$\half$ quantum particle in time-varying electromagnetic fields. The case of a time-varying uniform magnetic induction is examined and shown to provide a physical realization of a supersymmetric quantum-mechanical system. Group-theoretic methods are used to factorize the relevant Schr\"odinger equations and obtain eigensolutions. The supercoherent states for this system are constructed. 
  The role in string theory of manifolds of complex dimension $D_{crit} + 2(Q-1)$ and positive first Chern class is described. In order to be useful for string theory, the first Chern class of these spaces has to satisfy a certain relation. Because of this condition the cohomology groups of such manifolds show a specific structure. A group that is particularly important is described by $(D_{crit} + Q-1, Q-1)$--forms because it is this group which contains the higher dimensional counterpart of the holomorphic $(D_{crit}, 0)$--form that figures so prominently in Calabi--Yau manifolds. It is shown that the higher dimensional manifolds do not, in general, have a unique counterpart of this holomorphic form of rank $D_{crit}$. It is also shown that these manifolds lead, in general, to a number of additional modes beyond the standard Calabi--Yau spectrum. This suggests that not only the dilaton but also the other massless string modes, such as the antisymmetric torsion field, might be relevant for a possible stringy interpretation. 
  We study fermionic excitations in a cold ultrarelativistic plasma. We construct explicitly the quantum states associated with the two branches which develop in the excitation spectrum as the chemical potential is raised. The collective nature of the long wavelength excitations is clearly exhibited. Email contact: ollie@amoco.saclay.cea.fr 
  The full non-linear structure of the action and transformation rules for $\W_N$-gravity coupled to matter are obtained from a non-linear truncation of those for $w_ \infty$ gravity. The geometry of the construction is discussed, and it is shown that the defining equations become linear after a twistor-like transform. 
  We consider a biparametric family of BRS invariant regularization methods of SU(N) Chern-Simons theory (the parameters defining the family taking arbitrary values in $\RR^2$) and show that the shift $k\to k + sign(k) N$ of the Chern-Simons parameter $k$ occurs for arbitrary values of the family defining parameters. This supports irrefutably the conjecture that the shift of $k$ is universal for BRS invariant regulators. 
  We present a new supersymmetric integrable model: the $N=2$ superconformal affine Liouville theory. It interpolates between the $N=2$ super Liouville and $N=2$ super sine-Gordon theories and possesses a Lax representation on the complex affine Kac-Moody superalgebra ${\hat {sl(2| 2)^{(1)}}}$. We show that the higher spin $W_{1+\infty}$-type symmetry algebra of ordinary conformal affine Liouville theory extends to a $N=2\; W_{1/2 + \infty}$-type superalgebra. 
  String configurations with nonzero winding number describe soliton string states. We compute the Veneziano amplitude for the scattering of arbitrary winding states and show that in the large radius limit the strings always scatter trivially and with no change in the individual winding numbers of the strings. In this limit, then, these states scatter as true solitons. 
  We present a large $q$ expansion of the 2d $q$-states Potts model free energies up to order 9 in $1/\sqrt{q}$. Its analysis leads us to an ansatz which, in the first-order region, incorporates properties inferred from the known critical regime at $q=4$, and predicts, for $q>4$, the $n^{\rm th}$ energy cumulant scales as the power $(3 n /2-2)$ of the correlation length. The parameter-free energy distributions reproduce accurately, without reference to any interface effect, the numerical data obtained in a simulation for $q=10$ with lattices of linear dimensions up to L=50. The pure phase specific heats are predicted to be much larger, at $q\leq10$, than the values extracted from current finite size scaling analysis of extrema. Implications for safe numerical determinations of interface tensions are discussed. 
  Original Version Corrupted by Mailer 
  Nonlocal regularization of QED is shown to possess an axial anomaly of the same form as other regularization schemes. The Noether current is explicitly constructed and the symmetries are shown to be violated, whereas the identities constructed when one properly considers the contribution from the path integral measure are respected. We also discuss the barrier to quantizing the fully gauged chiral invariant theory, and consequences. 
  We examine an approach to justifying the mean field approximation for the anyon gas, using the scattering of anyons. Parity violation permits a nonzero average scattering angle, from which one can extract a mean radius of curvature for anyons. If this is larger than the interparticle separation, one expects that the graininess of the statistical magnetic field is unimportant, and that the mean field approximation is good. We argue that a non-conventional interaction between anyons is crucial, in which case the criterion for validity of the approximation is identical to the one deduced using a self-consistency argument. 
  I review recent works on the problem of inducing large-N QCD by matrix fields. In the first part of the talk I describe the matrix models which induce large-N QCD and present the results of studies of their phase structure by the standard lattice technology (in particular, by the mean field method). The second part is devoted to the exact solution of these models in the strong coupling region by means of the loop equations. 
  We propose a classical model for the non-Abelian Chern-Simons theory coupled to $N$ point-like sources and quantize the system using the BRST technique. The resulting quantum mechanics provides a unified framework for fractional spin, braid statistics and Knizhnik-Zamolodchikov equation. 
  We modify the $SL(2,{\bf R})/U(1)$ WZW theory, which was shown to describe strings in a 2D black hole, to be invariant under chiral $U(1)$ gauge symmetry by introducing a Steukelberg field. We impose several interesting gauge conditions for the chiral $U(1)$ symmetry. In a paticular gauge the theory is found to be reduced to the Liouville theory coupled to the $c=1$ matter perturbed by the so-called black hole mass operator. Also we discuss the physical states in the models briefly. 
  The values of the Witten invariants, $I_W$, of the lens space $L(p, 1)$ for SU(2) at level $k$ are obtained for arbitrary $p$. A duality relation for $I_W$ when $p$ and $k$ are interchanged, valid for asymptotic $k$, is observed. A method for calculating $I_W$ for any group $G$ is described. It is found that $I_W$ for $Z_m$, even for $m = 2$, distinguishes 3-manifolds quite effectively. 
  We discuss the Dirac quantization of two dimensional gravity with bosonic matter fields. After defining the extended Hamiltonian it is possible to fix the gauge completely. The commutators can all be obtained in closed form; nevertheless, the results are not particularly simple. 
  Using operator sewing techniques we construct the Reggeon vertex involving four external ${\bf Z}_3$-twisted complex fermionic fields. Generalizing a procedure recently applied to the ordinary Ramond four-vertex, we deduce the closed form of the ${\bf Z}_3$ vertex by demanding it to reproduce the results obtained by sewing. 
  We consider a one-dimensional Osp($N|2M$) pseudoparticle mechanical model which may be written as a phase space gauge theory. We show how the pseudoparticle model naturally encodes and explains the two-dimensional zero curvature approach to finding extended conformal symmetries. We describe a procedure of partial gauge fixing of these theories which leads generally to theories with superconformally extended ${\cal W}$-algebras. The pseudoparticle model allows one to derive the finite transformations of the gauge and matter fields occurring in these theories with extended conformal symmetries. In particular, the partial gauge fixing of the Osp($N|2$) pseudoparticle mechanical models results in theories with the SO($N$) invariant $N$-extended superconformal symmetry algebra of Bershadsky and Knizhnik. These algebras are nonlinear for $N \geq 3.$ We discuss in detail the cases of $N=1$ and $N=2,$ giving two new derivations of the superschwarzian derivatives. Some comments are made in the $N=2$ case on how twisted and topological theories represent a significant deformation of the original particle model. The particle model also allows one to interpret superconformal transformations as deformations of flags in super jet bundles over the associated super Riemann surface. 
  A relation between an $Sp(2M)$ gauge particle model and the zero-curvature condition in a two-dimensional gauge theory is presented. For the $Sp(4)$ case we construct finite \W-transformations. 
  The present paper is revised copy of hep-th/9303087 in which higher spin extensions of the nonabelian gauge symmetries for the classical WZNW model are considered. Both linear and nonlinear realizations of the extended affine Kac-Moody algebra are obtained. A characteristic property of the WZNW model is that it admits a higher spin linear realization of the extended affine Kac-Moody algebra which is equivalent to the corresponding higher spin nonlinear realization of the same algebra. However, in both cases the higher spin currents do not form an invariant space with respect to their generating transformations. This makes it imposible for this symmetry to be gauged. 
  The set of degenerate ground states of an arbitrary nonabelian topologically massive gauge theory is shown to be in one-to-one correspondence with the Hilbert space of the associated pure Chern-Simons theory.  (Paper is being withdrawn: original conclusion is incorrect for the nonabelian case.  For a correct treatment, see M. Asorey, S. Carlip, and F. Falceto, hep-th/9304081.) 
  We show that $~N=1$~ {\it supersymmetric} Kadomtsev-Petviashvili (SKP) equations can be embedded into recently formulated $~N=1$~ self-dual {\it supersymmetric} Yang-Mills theories after appropriate dimensional reduction and truncation, which yield three-dimensional supersymmetric Chern-Simons theories. Based on this result, we also give conjectural \hbox{$N=2~$} SKP equations. Subsequently some exact solutions of these systems including fermionic fields are given. 
  The validity of the renormalization group approach for large $N$ is clarified by using the vector model as an example. An exact difference equation is obtained which relates free energies for neighboring values of $N$. The reparametrization freedom in field space provides infinitely many identities which reduce the infinite dimensional coupling constant space to that of finite dimensions. The effective beta functions give exact values for the fixed points and the susceptibility exponents. 
  We study a deformed $su(m|n)$ algebra on a quantum superspace. Some interesting aspects of the deformed algebra are shown. As an application of the deformed algebra we construct a deformed superconformal algebra. {}From the deformed $su(1|4)$ algebra, we derive deformed Lorentz, translation of Minkowski space, $iso(2,2)$ and its supersymmetric algebras as closed subalgebras with consistent automorphisms. 
   Remarkable parallelism between the theory of integrable systems of first-order quasilinear PDE and some old results in projective and affine differential geometry of conjugate nets, Laplace equations, their Bianchi-Baecklund transformations is exposed.    These results were recently applied by I.M.Krichever and B.A.Dubrovin to prove integrability of some models in topological field theories. Within the geometric framework we derive some new integrable (in a sense to be discussed) generalizations describing N-wave resonant interactions. 
  The correspondence claimed by M. Douglas, between the multicritical regimes of the two-matrix model and 2D gravity coupled to (p,q) rational matter field, is worked out explicitly. We found the minimal (p,q) multicritical potentials U(X) and V(Y) which are polynomials of degree p and q, correspondingly. The loop averages W(X) and \tilde W(Y) are shown to satisfy the Heisenberg relations {W,X} =1 and {\tilde W,Y}=1 and essentially coincide with the canonical momenta P and Q. The operators X and Y create the two kinds of boundaries in the (p,q) model related by the duality (p,q) - (q,p). Finally, we present a closed expression for the two two-loop correlators and interpret its scaling limit. 
  We generalize the Lax pair and B\"acklund transformations for Liouville and Toda field theories as well as their supersymmetric generalizations, to the case of arbitrary Riemann surfaces. We make use of the fact that Toda field theory arises naturally and geometrically in a restriction of so called $W$--geometry to ordinary Riemannian geometry. This derivation sheds light on the geometrical structure underlying complete integrability of these systems. (Invited talk presented at the 877th meeting of the American Mathematical Society, USC, November 1992 and at the YITP workshop ``Directions on Quantum Gravity", Kyoto, November 1992.) 
  Explicit formulas of the universal $R$-matrix are given for all quantized nontwisted rank 3 affine Lie algebras $U_q(A_2^{(1)})\,,~U_q(C_2^{(1)})$ and $U_q(G_2^{(1)})$. 
  Let $U_q(\hat{\cal G})$ denote the quantized affine Lie algebra and $U_q({\cal G}^{(1)})$ the quantized {\em nontwisted} affine Lie algebra. Let ${\cal O}_{\rm fin}$ be the category defined in section 3. We show that when the deformation parameter $q$ is not a root of unit all integrable representations of $U_q(\hat{\cal G})$ in the category ${\cal O}_{\rm fin}$ are completely reducible and that every integrable irreducible highest weight module over $U_q({\cal G}^{(1)})$ corresponding to $q>0$ is equivalent to a unitary module. 
  We define a block observable for the $q$-state Potts model which exhibits an intermittent behaviour at the critical point. We express the intermittency indices of the normalised moments in terms of the magnetic critical exponent $\beta /\nu$ of the model. We confirm this relation by a numerical similation of the $q=2$ (Ising) and $q=3$ two-dimensional Potts model. 
  In the quenched approximation, the gauge covariance properties of three vertex Ans\"{a}tze in the Schwinger-Dyson equation for the fermion self energy are analysed in three- and four- dimensional quantum electrodynamics. Based on the Cornwall-Jackiw-Tomboulis effective action, it is inferred that the spectral representation used for the vertex in the gauge technique cannot support dynamical chiral symmetry breaking. A criterion for establishing whether a given Ansatz can confer gauge covariance upon the Schwinger-Dyson equation is presented and the Curtis and Pennington Ansatz is shown to satisfy this constraint. We obtain an analytic solution of the Schwinger-Dyson equation for quenched, massless three-dimensional quantum electrodynamics for arbitrary values of the gauge parameter in the absence of dynamical chiral symmetry breaking. 
  We calculate a large $k$ asymptotic expansion of the exact surgery formula for Witten's $SU(2)$ invariant of Seifert manifolds. The contributions of all flat connections are identified. An agreement with the 1-loop formula is checked. A contribution of the irreducible connections appears to contain only a finite number of terms in the asymptotic series. A 2-loop correction to the contribution of the trivial connection is found to be proportional to Casson's invariant. 
  We study the integral representation for the exact solution to nonperturbative $c le 1$ string theory. A generic solution is determined by two functions $W(x)$ and $Q(x)$ which behaive at infinity like $x^p$ and $x^q$ respectively. The integral model for arbitrary $(p,q)$ models is derived which explicitely demonstrates $p-q$ duality of minimal models coupled to gravity. We discuss also the exact solutions to string equation and reduction condition and present several explicit examples. 
  A review of the appearence of integrable structures in the matrix model description of $2d$-gravity is presented. Most of ideas are demonstrated at the technically simple but ideologically important examples. Matrix models are considered as a sort of "effective" description of continuum $2d$ field theory formulation. The main physical role in such description is played by the Virasoro-$W$ constraints which can be interpreted as a certain unitarity or factorization constraints. Bith discrete and continuum (Generalized Kontsevich) models are formulated as the solutions to those discrete (continuous) Virasoro-$W$ constraints. Their integrability properties are proven using mostly the determinant technique highly related to the representation in terms of free fields. The paper also contains some new observations connected to formulation of more general than GKM solutions and deeper understanding of their relation to $2d$ gravity. 
  The spherical symmetric dyonic black hole solutions of the effective action of heterotic string are studied perturbatively up to second order in the inverse string tension. An expression for the temperature in term of the mass and the electric and magnetic charge of the black hole is derived and it is shown that its behaviour is qualitatively different in the two special cases where either the electric or the magnetic charge vanishes 
  A derivation of the Hawking effect is given which avoids reference to field modes above some cutoff frequency $\omega_c\gg M^{-1}$ in the free-fall frame of the black hole. To avoid reference to arbitrarily high frequencies, it is necessary to impose a boundary condition on the quantum field in a timelike region near the horizon, rather than on a (spacelike) Cauchy surface either outside the horizon or at early times before the horizon forms. Due to the nature of the horizon as an infinite redshift surface, the correct boundary condition at late times outside the horizon cannot be deduced, within the confines of a theory that applies only below the cutoff, from initial conditions prior to the formation of the hole. A boundary condition is formulated which leads to the Hawking effect in a cutoff theory. It is argued that it is possible the boundary condition is {\it not} satisfied, so that the spectrum of black hole radiation may be significantly different from that predicted by Hawking, even without the back-reaction near the horizon becoming of order unity relative to the curvature. 
  It is observed that a large class of $(2,2)$ string vacua with $n>5$ superfields can be rewritten as Landau_Ginzburg orbifolds with discrete torsion and $n=5$. The naive geometric interpretation (if one exists) would be that of a complex 3-fold, not necessarily K\"ahler but still with vanishing first Chern class. 
  n-Ising spins on a random surface represented by a matrix model is studied as a model of the 2D gravity coupled to matter field with the central charge c > 1. The magnetic field is introduced to discuss the scaling exponent $\Delta$, and the value of this magnetic field exponent is estimated by the series expansion. 
  In this talk some essential features of stringy black holes are described. We consider charged four-dimensional axion-dilaton black holes. The Hawking temperature and the entropy of all solutions are shown to be simple functions of the squares of supercharges, defining the positivity bounds. Spherically symmetric and multi black hole solutions are presented. The extreme solutions have some unbroken supersymmetries. Axion-dilaton black holes with zero entropy and zero area of the horizon form a family of stable particle-like objects, which we call holons. We discuss the possibility of splitting of nearly extreme black holes into holons. 
  The two parameters quantum algebra $SU_{p,k}(2)$ can be obtained from a single parameter algebra $SU_q(2)$. This fact gives some relations between $SU_{p,k}(2)$ quantities and the corresponding ones of the $SU_q(2)$ algebra. In this paper are mentioned the relations concerning: Casimir operators, eigenvectors, matrix elements, Clebsch Gordan coefficients and irreducible tensors. 
  We derive the recursive equations for the form factors of the local hermitian operators in the Bullough-Dodd model. At the self-dual point of the theory, the form factors of the fundamental field of the Bullough-Dodd model are equal to those of the fundamental field of the Sinh-Gordon model at a specific value of the coupling constant. 
  We analyze actions for 2D supergravities induced by chiral conformal supermatter. The latter may be thought as described at the classical level by superspace actions invariant under super-reparametrization, super-Weyl and super-Lorentz transformations. Upon quantization various anomalies appear which characterize the non-trivial induced actions for the supergravitational sector. We derive these induced actions using a chiral boson to represent the chiral inducing matter. We show that they can be defined in a super-reparametrization invariant way, but with super-Weyl and super-Lorentz anomalies. We consider the case of $(1,0)$ and $(1,1)$ supergravities by working in their respective superspace formulations and investigate their quantization in the conformal gauge. The actions we consider arise naturally in off-critical heterotic and spinning strings. In the conformal gauge, they correspond to chiral extensions of the super-Liouville theory. 
  We study the subdivision properties of certain lattice gauge theories based on the groups $Z_{2}$ and $Z_{3}$, in four dimensions. The Boltzmann weights are shown to be invariant under all type $(k,l)$ subdivision moves, at certain discrete values of the coupling parameter. The partition function then provides a combinatorial invariant of the underlying simplicial complex, at least when there is no boundary. We also show how an extra phase factor arises when comparing Boltzmann weights under the Alexander moves, where the boundary undergoes subdivision. 
  Recently, Rogers' dilogarithm identities have attracted much attention in the setting of conformal field theory as well as lattice model calculations. One of the connecting threads is an identity of Richmond-Szekeres that appeared in the computation of central charges in conformal field theory. We show that the Richmond-Szekeres identity and its extension by Kirillov-Reshetikhin can be interpreted as a lift of a generator of the third integral homology of a finite cyclic subgroup sitting inside the projective special linear group of all $2 \times 2$ real matrices viewed as a {\it discrete} group. This connection allows us to clarify a few of the assertions and conjectures stated in the work of Nahm-Recknagel-Terhoven concerning the role of algebraic $K$-theory and Thurston's program on hyperbolic 3-manifolds. Specifically, it is not related to hyperbolic 3-manifolds as suggested but is more appropriately related to the group manifold of the universal covering group of the projective special linear group of all $2 \times 2$ real matrices viewed as a topological group. This also resolves the weaker version of the conjecture as formulated by Kirillov. We end with the summary of a number of open conjectures on the mathematical side. 
  Spinning particles in curved space-time can have fermionic symmetries generated by the square root of bosonic constants of motion other than the Hamiltonian. We present a general analysis of the conditions under which such new supersymmetries appear, and discuss the Poisson-Dirac algebra of the resulting set of charges, including the conditions of closure of the new algebra. An example of a new non-trivial supersymmetry is found in black-hole solutions of the Kerr-Newman type and corresponds to the Killing-Yano tensor, which plays an important role in solving the Dirac equation in these black-hole metrics. 
  We show that in 2+1 dimensional Quantum Electrodynamics an external magnetic field applied to a finite density of massless fermions is screened, due to a $2+1$-dimensional realization of the underlying $2$-dimensional axial anomaly of the space components of the electric current. This is shown to imply screening of the magnetic field, i.e., the Meissner effect. We discuss the physical implications of this result. 
  We show that for collinear processes, i.e. processes where the incoming and outgoing momenta are aligned along the same line, the S-matrix of the tree level 2+1 dimensional Thirring model factorizes: any S - matrix element is a product of $2\rightarrow 2$ elements. In particular this means nullification of all collinear $2 \rightarrow n$ amplitudes for $n > 2$. 
  The positive-energy unitary irreducible representations of the $q$-deformed conformal algebra ${\cal C}_q = {\cal U}_q(su(2,2))$ are obtained by appropriate deformation of the classical ones. When the deformation parameter $q$ is $N$-th root of unity, all these unitary representations become finite-dimensional. For this case we discuss in some detail the massless representations, which are also irreducible representations of the $q$-deformed Poincar\'e subalgebra of ${\cal C}_q$. Generically, their dimensions are smaller than the corresponding finite-dimensional non-unitary representation of $su(2,2)$, except when $N=2$, $h=0$ and $N = 2 \vert h\vert +1$, where $h$ is the helicity of the representations. The latter cases include the fundamental representations with $h = \pm 1/2$. 
  We identify a quantity in the $c=1$ matrix model which describes the wavefunction for physical scattering of a tachyon from a black hole of the two dimensional critical string theory. At the semiclassical level this quantity corresponds to the usual picture of a wave coming in from infinity, part of which enters the black hole becoming singular at the singularity, while the rest is scattered back to infinity, with nothing emerging from the whitehole. We find, however, that the exact nonperturbative wavefunction is nonsingular at the singularity and appears to end up in the asymptotic region ``behind'' the singularity. 
  We consider here the Chern-Simons field theory with gauge group SU(N) in the presence of a gravitational background that describes a two-dimensional expanding ``universe". Two special cases are treated here in detail: the spatially flat {\it Robertson-Walker} space-time and the conformally static space-times having a general closed and orientable Riemann surface as spatial section. The propagator and the vertices are explicitely computed at the lowest order in perturbation theory imposing the Coulomb gauge fixing. 
  Generalizing the mapping between the Potts model with nearest neighbor interaction and six vertex model, we build a family of "fused Potts models" related to the spin $k/2$ ${\rm U}_{q}{\rm su}(2)$ invariant vertex model and quantum spin chain. These Potts model have still variables taking values $1,\ldots,Q$ ($\sqrt{Q}=q+q^{-1}$) but they have a set of complicated multi spin interactions. The general technique to compute these interactions, the resulting lattice geometry, symmetries, and the detailed examples of $k=2,3$ are given. For $Q>4$ spontaneous magnetizations are computed on the integrable first order phase transition line, generalizing Baxter's results for $k=1$. For $Q\leq 4$, we discuss the full phase diagram of the spin one ($k=2$) anisotropic and ${\rm U}_{q}{\rm su}(2)$ invariant quantum spin chain (it reduces in the limit $Q=4$ ($q=1$) to the much studied phase diagram of the isotropic spin one quantum chain). Several critical lines and massless phases are exhibited. The appropriate generalization of the Valence Bond State method of Affleck et al. is worked out. 
  We review our work on the relation between integrability and infinite-dimensional algebras. We first consider the question of what sets of commuting charges can be constructed from the current of a \mbox{\sf U}(1) Kac-Moody algebra. It emerges that there exists a set $S_n$ of such charges for each positive integer $n>1$; the corresponding value of the central charge in the Feigin-Fuchs realization of the stress tensor is $c=13-6n-6/n$. The charges in each series can be written in terms of the generators of an exceptional \W-algebra. We show that the \W-algebras that arise in this way are symmetries of Liouville theory for special values of the coupling. We then exhibit a relationship between the \nls equation and the KP hierarchy. From this it follows that there is a relationship between the \nls equation and the algebra \Wi. These examples provide evidence for our conjecture that the phenomenon of integrability is intimately linked with properties of infinite dimensional algebras. 
  The complete set of ground state wave functions for N anyons in an external magnetic field on the torus is found. The cases when the filling factor is less than or equal to one are considered. The single valued description of anyons is employed through out by coupling bosons to a Chern-Simons field. At the end, the Chern-Simons interaction is removed by a singular gauge transformation as a result of which the wave functions become multi-component in agreement with other studies. 
  In this papper, a quantum dynamical model describing the quantum measurement process is presented as an extensive generalization of the Coleman-Hepp model. In both the classical limit with very large quantum number and macroscopic limit with very large particle number in measuring instrument, this model generally realizes the wave packet collapse in quantum measurement as a consequence of the Schrodinger time evolution in either the exactly-solvable case or the non-(exactly-)solvable case.   For the latter, its quasi-adiabatic case is explicitly analysed by making use of the high-order adiabatic approximation method and then manifests the wave packet collapse as well as the exactly-solvable case. By highlighting these analysis, it is finally found that an essence of the dynamical model of wave packet collapse is the factorization of the Schrodinger evolution other than the exact solvability. So many dynamical models including the well-known ones before, which are exactly-solvable or not, can be shown only to be the concrete realizations of this factorizability 
  In two recent papers, a new method was developed for calculating ten-dimensional superstring amplitudes with an arbitrary number of loops and external massless particles, and for expressing them in manifestly Lorentz-invariant form. By explicitly checking for divergences when the Riemann surface degenerates, these amplitudes are proven to be finite. By choosing light-cone moduli for the surface and comparing with the light-cone Green-Schwarz formalism, these amplitudes are proven to be unitary. 
  We formulate a renormalizable quantum gravity in $2+\epsilon$ dimensions by generalizing the nonlinear sigma model approach to string theory. We find that the theory possesses the ultraviolet stable fixed point if the central charge of the matter sector is in the range $0~<~c~<~25$. This may imply the existence of consistent quantum gravity theory in 3 and 4 dimensions. We compute the scaling dimensions of the relevant operators in the theory at the ultraviolet fixed point. We obtain a scaling relation between the cosmological constant and the gravitational constant, which is crucial for searching for the continuum limit in the constructive approach to quantum gravity. 
  The dynamics of relativistic spinning particles in strong external electromagnetic or gravitational fields is discussed. Spin-orbit coupling is shown to affect such relativistic phenomena as time-dilation and perihelion shift. Possible applications include muon decay in a magnetic field and the dynamics of neutron stars in binary systems. 
  A general model of dialton-Maxwell gravity in two dimensions is investigated. The corresponding one-loop effective action and the generalized $\beta$-functions are obtained. A set of models that are fixed points of the renormalization group equations are presented. 
  Making use of inverse Mellin transform techniques for analytical continuation, an elegant proof and an extension of the zeta function regularization theorem is obtained. No series commutations are involved in the procedure; nevertheless the result is naturally split into the same three contributions of very different nature, i.e. the series of Riemann zeta functions and the power and negative exponentially behaved functions, respectively, well known from the original proof. The new theorem deals equally well with elliptic differential operators whose spectrum is not explicitly known. Rigorous results on the asymptoticity of the outcoming series are given, together with some specific examples. Exact analytical formulas, simplifying approximations and numerical estimates for the last of the three contributions (the most difficult to handle) are obtained. As an application of the method, the summation of the series which appear in the analytic computation (for different ranges of temperature) of the partition function of the string ---basic in order to ascertain if QCD is some limit of a string theory--- is PERFORMED. 
  Content:  1. Introduction  2. Regge calculus and dynamical triangulations   Simplicial manifolds and piecewise linear spaces - dual complex and volume elements - curvature and Regge action - topological invariants - quantum Regge calculus - dynamical triangulations  3. Two dimensional quantum gravity, dynamical triangulations and matrix models   continuum formulation - dynamical triangulations and continuum limit - one matrix model - various matrix models - numerical studies - c=1 barrier - intrinsic geometry of 2d gravity - Liouville at c>25  4. Euclidean quantum gravity in three and four dimensions   what are we looking for? - 3d simplicial gravity - 4d simplicial gravity - 3d and 4d Regge calculus  5. Non-perturbative problems in two dimensional quantum gravity   double scaling limit - string equation - non-perturbative properties of the string equation - divergent series and Borel summability - non-perturbative effects in 2d gravity and string theories - stabilization proposals  6. Conclusion 
  The path integral for the propagator is expanded into a perturbation series, which can be exactly summed in the case of $\delta$-function perturbations giving a closed expression for the (energy-dependent) Green function. Making the strength of the $\delta$-function perturbation infinite repulsive, produces a totally reflecting boundary, hence giving a path integral solution in half-spaces in terms of the corresponding Green function. The example of the Wood-Saxon potential serves by an appropriate limiting procedure to obtain the Green function for the step-potential and the finite potential-well in the half-space, respectively. 
  The connection is obtained between Ward identities and W-constraints in Generalized Kontsevich Model with the potential $X^4/4$. We show that Ward identities include W-constraints (and do not include any other constraints) for this potential and make some observations in favour of the same connection for the model with the potential of the form $X^{(K+1)}/(K+1)$ for any $K\geq 2$. 
  The incompressible fluid dynamics is reformulated as dynamics of closed loops $C$ in coordinate space. This formulation allows to derive explicit functional equation for the generating functional $\Psi[C]$ in inertial range of spatial scales, which allows the scaling solutions. The requirement of finite energy dissipation rate leads then to the Kolmogorov index. We find an exact steady solution of the loop equation in inertial range of the loop sizes. The generating functional decreases as $\EXP{-A^{\tt}}$ where $A=\oint_C r \wedge dr$ is the area inside the loop. The pdf for the velocity circulation $\Gamma$ is Lorentzian, with the width $\bar{\Gamma} \propto A^{\tt} $. 
  A general non-relativistic field theory on the plane with couplings to an arbitrary number of abelian Chern-Simons gauge fields is considered. Elementary excitations of the system are shown to exhibit fractional and mutual statistics. We identify the self-dual systems for which certain classical and quantal aspects of the theory can be studied in a much simplified mathematical setting. Then, specializing to the general self-dual system with two Chern-Simons gauge fields (and non-vanishing mutual statistics parameter), we present a systematic analysis for the static vortexlike classical solutions, with or without uniform background magnetic field. Relativistic generalizations are also discussed briefly. 
  A general deformation of the Heisenberg algebra is introduced with two deformed operators instead of just one. This is generalised to many variables, and permits the simultaneous existence of coherent states, and the transposition of creation operators. 
  As a preparation for the study of {\it arbitrary} extensions of $d=2$ gravity we present a detailed investigation of $SO(N)$ supergravity. By gauging a chiral, nilpotent subgroup of the $OSp(N|2)$ Wess-Zumino-Witten model we obtain an all order expression for the effective action. Reality of the coupling constant imposes the usual restrictions on $c$ for $N=0$ and 1. No such restrictions appear for $N\geq 2$. For $N=2$, 3 and 4, no renormalizations of the coupling constant beyond one loop occur. These results are related to non-renormalization theorems for theories with extended supersymmetries. Arbitrary (super)extensions of $d=2$ gravity are then analyzed. The induced theory is represented by a WZW model for which a chiral, solvable group is gauged. From this, we obtain the effective action. All order expressions for both the coupling constant renormalization and the wavefunction renormalization are given. From this we classify all extensions of $d=2$ gravity for which the coupling constant gets at most a one loop renormalization. As an application of the general strategy, $N=4$ theories based on $D(2,1,\a)$ and $SU(1,1|2)$, all $WA$ gravities and the $N=2$ $W_n$ models are treated in some detail. 
  A canonical quantization for two dimensional gravity models, including a dilaton gravity model, is performed in a way suitable for the light-cone gauge. We extend the theory developed by Abdalla {\it et.al.}\cite{AM} and obtain the correlation functions by using the screening charges of the symmetry algebra. It turns out that the role of the cosmological constant term in the Liouville theory is played by the screening charge of the symmetry algebra and the resulting theory looks like a chiral part of the Liouville theory or a non-critical open string theory. Moreover, we show that the dilaton gravity theory has a symmetry realized by the centrally extended Poincar\'{e} algebra instead of the $\slr$ Kac-Moody algebra which is a symmetry of an ordinary two dimensional gravity theory. 
  We clarify the relation between 2-form Einstein gravity and its topological version. The physical space of the topological version is contained in that of the Einstein gravity.   Moreover a new vector field is introduced into 2-form Einstein gravity to restore the large symmetry of its topological version. The wave function of the universe is obtained for each model. (Talk given at the Workshop on General Relativity and Gravitation held at Waseda University, January 18-20, 1993.) 
  An invariant definition of the operator $\Delta $ of the Batalin-Vilkovisky formalism is proposed. It is defined as the divergence of a Hamiltonian vector field with an odd Poisson bracket (antibracket). Its main properties, which follow from this definition, as well as an example of realization on K\"ahlerian supermanifolds, are considered. The geometrical meaning of the Batalin-Vilkovisky formalism is discussed. 
  Based on a simple observation that a classical second order differential equation may be decomposed into a set of two first order equations, we introduce a Hamiltonian framework to quantize the damped systems. In particular, we analyze the system of a linear damped harmonic oscillator and demonstrate that the time evolution of the Schr\"odinger equation is unambiguously determined. 
  I show that the strong coupling solution of the Kazakov--Migdal model with a general interaction potential $V(\Phi)$ in $D$ dimensions coincides at large $N$ with that of the hermitean one-matrix model with the potential $\tilde{V}(\Phi)$: $$ (2D-1)\tilde{V}'= (D-1)V'+ D\sqrt{(V')^2+4(1-2D)\Phi^2}, $$ whose solution is known. The proof is given for an even potential $V(\Phi)=V(-\Phi)$ by solving loop equations. 
  The theory of matrix models is reviewed from the point of view of its relation to integrable hierarchies. Discrete 1-matrix, 2-matrix, ``conformal'' (multicomponent) and Kontsevich models are considered in some detail, together with the Ward identites (``W-constraints''), determinantal formulas and continuum limits, taking one kind of models into another. Subtle points and directions of the future research are also discussed. 
  We describe a static solution for $d = 2$ critical string theory including the tachyon $T$ but with its potential $V (T)$ set to zero. This solution thus incorporates tachyon back reaction and, when $T = 0$, reduces to the black hole solution. When $T \neq 0$ we find that (1) the Schwarzschild horizon of the above black hole splits into two, resembling Reissner-Nordstrom horizons and (2) the curvature scalar develops new singularities at the horizons. We show that these features will persist even when $V (T)$ is nonzero. We present a time dependent extension of our static solution and discuss some possible methods for removing the above singularities. 
  Brownian motion may be embedded in the Fock space of bosonic free field in one dimension.Extending this correspondence to a family of creation and annihilation operators satisfying a q-deformed algebra, the notion of q-deformation is carried from the algebra to the domain of stochastic processes.The properties of q-deformed Brownian motion, in particular its non-Gaussian nature and cumulant structure,are established. 
  We formulate a conjecture, stating that the algebra of $n$ pairs of deformed Bose creation and annihilation operators is a factor-algebra of $U_q[osp(1/2n)]$, considered as a Hopf algebra, and prove it for $n=2$ case. To this end we show that for any value of $q$ $U_q[osp(1/4)]$ can be viewed as a superalgebra, freely generated by two pairs $B_1^\pm$, $B_2^\pm$ of deformed para-Bose operators. We write down all Hopf algebra relations, an analogue of the Cartan-Weyl basis, the "commutation" relations between the generators and a basis in $U_q[osp(1/2n)]$ entirely in terms of $B_1^\pm$, $B_2^\pm$. 
  A direct derivation of the string field theory action in the Witten's background independent open string theory for the case when ghosts and matter are decoupled is given and consequences are discussed. 
  We discuss a possible exact equivalence of the Abelian Higgs model and a scalar theory of a magnetic vortex field in 2+1 dimensions. The vortex model has a current - current interaction and can be viewed as a strong coupling limit of a massive vector theory. The fixed point structure of the theory is discussed and mapped into fixed points of the Higgs model. 
  We perform the complete bosonization of 2+1 dimensional QED with one fermionic flavor in the Hamiltonian formalism. The fermion operators are explicitly constructed in terms of the vector potential and the electric field. We carefully specify the regularization procedure involved in the definition of these operators, and calculate the fermionic bilinears and the energy - momentum tensor. The algebra of bilinears exhibits the Schwinger terms which also appear in perturbation theory. The bosonic Hamiltonian density is a local polynomial function of $A_i$ and $E_i$, and we check explicitly the Lorentz invariance of the resulting bosonic theory. Our construction is conceptually very similar to Mandelstam's construction in 1+1 dimensions, and is dissimilar from the recent bosonization attempts in 2+1 dimensions which hinge crucially on the existence of a Chern - Simons term. 
  We show how to expand the free energy of a matrix model coupled to arbitrary matter in powers of the matter coupling constant. Concentrating on $\nu$ uncoupled Ising models---which have central charge $\nu/2$---we work out the expansion to sixth order for $\nu$ = 1, 2, and 3. Analyzing the series by the ratio method, we exhibit the spin-ordering phase transition. We discuss the limit $\nu \rightarrow \infty$, which is especially clear in the low temperature expansion; we prove that in this limit the dependence of the model on $\nu$ becomes trivial. 
  The method of Hawking to obtain black hole evaporation through Bogoljubov transformation between asymptotic modes (in and out) is generalized. The construction is local in that the in modes (of say positive frequency) are decomposed by Bogoljubov transformation into positive and negative frequency local inertial modes (i.e. those which are solutions of the d'Alembertian in terms of the local normal coordinates). From this follows an interesting reexpression of the local energy momentum tensor, more particularly of the outgoing energy flux. One finds that even in the local description there exists a partial thermal character parametrized by a local temperature. There exists quantum interference effects as well. These become negligible at large distance from the black hole. 
  We introduce the notion of a braided Lie algebra consisting of a finite-dimensional vector space $\CL$ equipped with a bracket $[\ ,\ ]:\CL\tens\CL\to \CL$ and a Yang-Baxter operator $\Psi:\CL\tens\CL\to \CL\tens\CL$ obeying some axioms. We show that such an object has an enveloping braided-bialgebra $U(\CL)$. We show that every generic $R$-matrix leads to such a braided Lie algebra with $[\ ,\ ]$ given by structure constants $c^{IJ}{}_K$ determined from $R$. In this case $U(\CL)=B(R)$ the braided matrices introduced previously. We also introduce the basic theory of these braided Lie algebras, including the natural right-regular action of a braided-Lie algebra $\CL$ by braided vector fields, the braided-Killing form and the quadratic Casimir associated to $\CL$. These constructions recover the relevant notions for usual, colour and super-Lie algebras as special cases. In addition, the standard quantum deformations $U_q(g)$ are understood as the enveloping algebras of such underlying braided Lie algebras with $[\ ,\ ]$ on $\CL\subset U_q(g)$ given by the quantum adjoint action. 
  We investigate the fractal structure of $2d$ quantum gravity, both for pure gravity and for gravity coupled to multiple gaussian fields and for gravity coupled to Ising spins. The roughness of the surfaces is described in terms of baby universes and using numerical simulations we measure their distribution which is related to the string susceptibility exponent $\g_{string}$. 
  Though screened at large distances, a point-like electric charge can still participate in a long-range electromagnetic interaction in the Higgs phase, namely that with the Aharonov-Bohm field produced by a localized magnetic flux. We show that this follows from the fact that the screening charge, induced in the presence of a Higgs condensate, does not interact with the Aharonov-Bohm field. The same phenomenon occurs, if a Chern-Simons term is incorporated in the action. This observation provides a physical basis for the recently proposed classification of the superselection sectors of this model in terms of a quasi-Hopf algebra. 
  Some aspects of finite quantum field theories in 3+1 dimensions are discussed. A model with non--supersymmetric particle content and vanishing one-- and two--loop beta functions for the gauge coupling and one--loop beta functions for Yukawa--couplings is presented. 
  Differential-geometric structures on the space of orbits of a finite Coxeter group, determined by Groth\'endieck residues, are calculated. This gives a construction of a 2D topological field theory for an arbitrary Coxeter group. 
  We consider two dimensional QCD with the spatial dimension compactified to a circle. We show that the states in the theory consist of interacting strings that wind around the circle and derive the Hamiltonian for this theory in the large $N$ limit, complete with interactions. Mapping the winding states into momentum states, we express this Hamiltonian in terms of a continuous field. For a $U(N)$ gauge group with a background source of Wilson loops, we recover the collective field Hamiltonian found by Das and Jevicki for the $c=1$ matrix model, except the spatial coordinate is on a circle. We then proceed to show that two dimensional QCD with a $U(N)$ gauge group can be reduced to a one- dimensional unitary matrix model and is hence equivalent to a theory of $N$ free nonrelativistic fermions on a circle. A similar result is true for the group $SU(N)$, but the fermions must be modded out by the center of mass coordinate. 
  Two-dimensional quantum gravity is identified as a second-class system which we convert into a first-class system via the Batalin-Fradkin (BF) procedure. Using the extended phase space method, we then formulate the theory in most general class of gauges. The conformal gauge action suggested by David, Distler and Kawai is derived from a first principle. We find a local, light-cone gauge action whose Becchi-Rouet-Stora-Tyutin invariance implies Polyakov's curvature equation $\partial_{-}R=\partial_{-}^{3}g_{++}=0$, revealing the origin of the $SL(2,R)$ Kac-Moody symmetry. The BF degree of freedom turns out be dynamically active as the Liouville mode in the conformal gauge, while in the light-cone gauge the conformal degree of freedom plays that r{\^o}le. The inclusion of the cosmological constant term in both gauges and the harmonic gauge-fixing are also considered. 
  The structure of ground states of generic FQH states on a torus is studied by using both effective theory and electron wave function. The relation between the effective theory and the wave function becomes transparent when one considers the ground state structure. We find that the non-abelian Berry's phases of the abelian Hall states generated by twisting the mass matrix are identical to the modular transformation matrix for the characters of a Gaussian conformal field theory. We also show that the Haldane-Rezayi spin singlet state has a ten fold degeneracy on a torus which indicates such a state is a non-abelian Hall state. 
  We propose a new formulation of the D=11 supermembrane theory that involves commuting spinors (twistor--like variables) and exhibits a manifest $n$--extended world volume supersymmetry $(1\leq n\leq 8)$. This supersymmetry replaces $n$ components of the usual $\kappa$--symmetry. We show that this formulation is classically equivalent to the standard one. 
  We study the two-matrix model which represents the sum over closed and open random surfaces coupled to an Ising Model. The boundary conditions are characterized by the fact that the Ising spins sitting at the vertices of the boundaries are all in the same state. We obtain the string equation and discuss the results. (No change in physics, only some misprints are corrected) 
  A general formula is obtained for Yukawa couplings in compactification on \LGO{s} and corresponding \CY\ spaces. Up to the kinetic term normalizations, this equates the classical Koszul ring structure with the \LGO\ chiral ring structure and the true super\CFT\ ring structure. 
  We show how to use quantum mechanics on the group manifold U(N) as a tool for problems in U(N) representation theory. The quantum mechanics reduces to free fermions on the circle, which in the large N limit become relativistic. The theory can be bosonized giving the Das-Jevicki-Sakita collective field theory. The formalism is particularly suited to problems involving tensor product multiplicity (Littlewood-Richardson) coefficients. As examples, we discuss the partition function of two-dimensional Yang-Mills theory on the sphere, and the zero magnetic field limit of D-dimensional Eguchi-Kawai Yang-Mills theory. We give the leading O(N^0) solution of the latter theory, using a method which allows computing corrections. Largely (but not completely) superseded by hep-th/9311130. 
  Conformal field theories with correlation functions which have logarithmic singularities are considered. It is shown that those singularities imply the existence of additional operators in the theory which together with ordinary primary operators form the basis of the Jordan cell for the operator $L_{0}$. An example of the field theory possessing such correlation functions is given. 
  We define multi-colour generalizations of braid-monoid algebras and present explicit matrix representations which are related to two-dimensional exactly solvable lattice models of statistical mechanics. In particular, we show that the two-colour braid-monoid algebra describes the Yang-Baxter algebra of the critical dilute A-D-E models which were recently introduced by Warnaar, Nienhuis, and Seaton as well as by Roche. These and other solvable models related to dense and dilute loop models are discussed in detail and it is shown that the solvability is a direct consequence of the algebraic structure. It is conjectured that the Yang-Baxterization of general multi-colour braid-monoid algebras will lead to the construction of further solvable lattice models. 
  N=2 supersymmetric WZNW models associated with finite-dimensional Manin triples is considered. Physical states in topological phase of these models are computed and them N=2 WZNW representatives are constracted. Ring structure of the physical states is computed. 
  It is shown that "nonintegrable phases of Wilson line integrals" are not true dynamical variables in Chern-Simons field theory. 
  We present some explicit results on the structure of singular vectors in $c=2$ Verma modules of the $\cW_3$ algebra. Using the embedding patterns of those vectors we construct resolutions for the $c=2$ irreducible modules, and thus are able to compute some of the BRST cohomology of $\cW_3$ gravity coupled to $c=2$ matter. In particular, we determine the states in the ground ring of the theory. (To appear in the proceedings of the AMS Special Session on "Geometry and Physics", USC, Los Angeles, November 5-6, 1992) 
  We verify that the fractional KdV equation is a bi-hamiltonian system using the zero curvature equation in $SL(3)$ matrix valued Lax pair representation, and explicitly find the closed form for the hamiltonian operators of the system. The second hamiltonian operator is the classical version of the $W^{(2)}_3$ algebra. We also construct systematically the Miura map of $W^{(2)}_3$ algebra using a gauge transformation of the $SL(3)$ matrix valued Lax operator in a particular gauge, and construct the modified fractional KdV equaiotn as hamiltonian system. 
  We propose a regular way to construct lattice versions of $W$-algebras, both for quantum and classical cases. In the classical case we write the algebra explicitly and derive the lattice analogue of Boussinesq equation from the Hamiltonian equations of motion. Connection between the lattice Faddeev-Takhtadjan-Volkov algebra [1] and q-deformed Virasoro is also discussed. 
  We analyze to all perturbative orders the properties of two possible quantum extensions of classically on-shell equivalent antisymmetric tensor gauge models in four dimensions. The first case, related to the soft breaking of a topological theory wants a gauge field of canonical dimension one. The other possibility, which assigns canonical dimension two to the gauge field, leads to the $\sigma$ model interpretation of the theory. In both instances we find that the models are anomaly free. 
  Recently we investigated a new supersymmetrization procedure for the KdV hierarchy inspired in some recent work on supersymmetric matrix models. We extend this procedure here for the generalized KdV hierarchies. The resulting supersymmetric hierarchies are generically nonlocal, except for the case of Boussinesque which we treat in detail. The resulting supersymmetric hierarchy is integrable and bihamiltonian and contains the Boussinesque hierarchy as a subhierarchy. In a particular realization, we extend it by defining supersymmetric odd flows. We end with some comments on a slight modification of this supersymmetrization which yields local equations for any generalized KdV hierarchy. 
  A very simple minisuperspace describing the Oppenheimer-Snyder collapsing star is found. The semiclasical wave function of that model turn out to describe a bound state. For fixed initial radius of the collapsing star, the corrssponding Bohr-Sommerfeld quantization condition implies mass quantization. An extension of this model, and some consequences, are considered. 
  The picture of S-wave scatering from a 4D extremal dilatonic black hole is examined. Classically, a small matter shock wave will form a non-extremal black hole. In the "throat region" the r-t geometry is exactly that of a collapsing 2D black hole. The 4D Hawking radiation (in this classical background) gives the 2D Hawking radiation exactly in the throat region. Inclusion of the back-reaction changes this picture: the 4D solution can then be matched to the 2D one only if the Hawking radiation is very small and only at the beginning of the radiation. We give that 4D solution. When the total radiating energy approaches the energy carried by the shock wave, the 4D picture breaks down. This happens even before an apparent horizon is formed, which suggests that the 4D semi-classical solution is quite different from the 2D one. 
  Axion strings and domain walls exhibit a number of novel effects in the presence of gauge fields, in particular the electromagnetic field. It is shown how these effects are reproduced in a model of Nambu-Goto-type strings and open or closed membranes coupled to gauge fields. The generalization to `axionic p-branes' is considered and it is shown how worldvolume gauge fields that arise in certain cases can be interpreted as Goldstone fields. 
  The superstring in $D\!=\!3,4$ and 6 is invariant under an $N\!=\!D\!-\!2$ superconformal algebra based on $\widehat{S^{D-3}}$. There is a direct relationship between this (world-sheet) symmetry and the super-Poincar\'e (target space) symmetry. We establish this relationship using the light-cone gauge, show how the statement generalizes to $D\!=\!10$ and examine the properties of the $N\!=\!8$ superconformal algebra and the possible implications of its existence. 
  The string equivalent of a massless particle ($m=0$) is the tensionless string ($T=0$). The study of such strings is of interest when trying to understand the high energy limit of ordinary strings. I discuss the classical $T\to 0$ limit of the bosonic string, the spinning string and the superstring. A common feature is the appearence of a space-time (super-)conformal symmetry replacing the world-sheet Weyl invariance. The question of whether this symmetry may survive quantization is addressed. A lightcone analysis of the quantized bosonic tensionless string leads to severe constraints on the physical states: they are space-time diffeomorphism singlets characterized by their topological properties only. 
  We investigate the hitherto unexplored relation between the superparticle path integral and superfield theory. Requiring that the path integral has the global symmetries of the classical action and obeys the natural composition property of path integrals, and also that the discretized action has the correct naive continuum limit, we find a viable discretization of the (D=3,N=2) free superparticle action. The resulting propagator is not the usual superfield one. We extend the discretization to include the coupling to an external gauge supermultiplet and use this to show the equivalence to superfield theory. This is possible since we are able to reformulate the superfield perturbation theory in terms of our new propagator. 
  Talk given at the ``4th Hellenic School on Elementary Particle Physics", Corfu, 2-20 September 1992: The propagation of strings in cosmological space-time backgrounds is reviewed. We show the relation of a special class of cosmological backgrounds to exact conformal field theory. Particular emphasis is put on the singularity structure of the cosmological space-time and on the discrete duality symmetries of the string background. 
  The Jackiw-Teitelboim gravity with the matter degrees of freedom is considered. The classical model is exactly solvable and its solutions describe non-trivial gravitational scattering of matter wave-packets. For huge amount of the solutions the scattering space-times are free of curvature singularities. However, the quantum corrections to the field equations inevitably cause the formation of (thunderbolt) curvature singularities, vanishing only in the limit $\hbar\to 0$. The singularities cut the space-time and disallow propagation to the future.The model is inspired by the dimensional reduction of 4-d pure Einstein gravity, restricted to the space-times with two commuting space-like Killing vectors. The matter degrees of freedom also stem from the 4-d ansatz. The measures for the continual integrations are judiciously chosen and one loop contributions (including the graviton and the dilaton ones) are evaluated. For the number of the matter fields $N=24$ we obtain even the exact effective action, applying the DDK-procedure. The effective action is nonlocal, but the semiclassical equations can be solved by using some theory of the Hankel transformations. 
  This article, written in honor of Fritz Rohrlich, briefly surveys the role of topology in physics. 
  We analyze two-dimensional large $N_c$ QCD at finite temperature and show explicitly that the free energy has the correct $N_c$ dependence. 
  We discuss the new Coulomb gauge method for testing confinement and measuring the string tension in the context of the Schwinger model and compact QED in 3 dimensions. 
  We analyze the microscopic, topological structure of the interface between domains of opposite magnetization in 3D Ising model near the critical point. This interface exhibits a fractal behaviour with a high density of handles. The mean area is an almost linear function of the genus. The entropy exponent is affected by strong finite-size effects. 
  The string propagation in the two-dimensional stringy black-hole is investigated from a new approach. We completely solve the classical and quantum string dynamics in the lorentzian and euclidean regimes. In the lorentzian case all the physics reduces to a massless scalar particle described by a Klein-Gordon type equation with a singular effective potential. The scattering matrix is found and it reproduces the results obtained by coset CFT techniques. It factorizes into two pieces : an elastic coulombian amplitude and an absorption part. In both parts, an infinite sequence of imaginary poles in the energy appear. The generic features of string propagation in curved D-dimensional backgrounds (string stretching, fall into spacetime singularities) are analyzed in the present case. A new physical phenomenon specific to the present black-hole is found : the quantum renormalization of the speed of light. We find $c_{quantum} = \sqrt{{k\o{k-2}}}~c_{classical}$, where $k$ is the integer in front of the WZW action. This feature is, however, a pathology. Only for $ k \to \infty$ the pathology disappears (although the conformal anomaly is present). We analyze all the classical euclidean string solutions and exactly compute the quantum partition function. No critical Hagedorn temperature appears here. 
  A discrete string theory --a theory of embeddings from ${\bf Z}\times {\bf Z}_C\to {\bf R}^D$, where $C$ is the number of components of the string-- is explored. The closure of the algebra of constraints (`${\bf Z}_C$-Virasoro algebra') is exhibited. The ${\bf Z}_C$-Virasoro `algebra' is shown to be anomaly free in arbitrary number of target space dimensions. We prove the existence of a (manifestly unitary) light-cone gauge with anomaly free Lorentz algebra in any dimensions. The analog of vertex operators are introduced and the physical spectrum is analysed. There are an infinite number of higher-level states repeating a certain mass pattern and leading to an infinite degeneracy. The connection with the continuum string theory (in $D=26$) is investigated. Independently, following a method recently introduced by 't Hooft based on Hilbert-space extension of deterministic systems, a particular one-dimensional cellular automaton submitted to a deterministic evolution is shown to reduce to a massless scalar field theory at long distance scales. This automaton is utilized to define a (`cellular') string theory with world-sheet variables evolving under deterministic rules, which in the framework of `first-quantization' corresponds to the ${\bf Z}_C$ string theory mentioned above. We show that in this theory also the target space motion of free strings is governed by deterministic laws. Finally, we discuss a model for (off-shell) interacting strings where space-time determinism is fully restored. 
  It is argued that the problems of the cosmological constant, stability and renormalizability of quantum gravity can be solved if the space-time manifold arises through spontaneous symmetry breaking. A ``pre-manifold" model is presented in which many points are connected by random bonds. A set of $D$ real numbers assigned to each point are coupled between points connected by bonds. It is then found that the dominant configuration of bonds is a flat $D$-dimensional manifold, on which there is a massless matter field. Long-wavelength fluctuations can describe quantized massless gravity if $D=\;4$, $6$, $8...$. 
  We show that the description of the electroweak interactions based on noncommutative geometry of a continuous and a discrete space gives no special relations between the Higgs mass and other parameters of the model. We prove that there exists a gauge invariant term, linear in the curvature, which is trivial in the standard differential geometry but nontrivial in the case of the discrete geometry. The relations could appear only if one neglects this term, otherwise one gets the Lagrangian of the Standard model with the exact number of free parameters. 
  The dissipative Hofstadter model describes the quantum mechanics of a charged particle in two dimensions subject to a periodic potential, uniform magnetic field, and dissipative force. Its phase diagram exhibits an SL(2,Z) duality symmetry and has an infinite number of critical circles in the dissipation/magnetic field plane. In addition, multi-critical points on a particular critical circle correspond to non-trivial solutions of open string theory. The duality symmetry is expected to provide relations between correlation functions at different multi-critical points. Many of these correlators are contact terms. However we expect them to have physical significance because under duality they transform into functions that are non-zero for large separations of the operators. Motivated by the search for exact, regulator independent solutions for these contact terms, in this paper we derive many properties and symmetries of the coordinate correlation functions at the special multi-critical points. In particular, we prove that the correlation functions are homogeneous, piecewise-linear functions of the momenta, and we prove a weaker version of the anticipated duality transformation. Consequently, the possible forms of the correlation functions are limited to lie in a finite dimensional linear space. We treat the potential perturbatively and these results are valid to all orders in perturbation theory. 
  The Wess-Zumino consistency conditions for Lorentz and diffeomorphism anomalies are discussed by introducing an operator 'delta' which allows to decompose the exterior derivative as a BRS commutator. 
  We show that the normalization integral for the Schr\"odinger and Dirac scattering wave functions contains, besides the usual delta-function, a term proportional to the derivative of the phase shift. This term is of zero measure with respect to the integration over momentum variables and can be discarded in most cases. Yet it carries the full information on phase shifts and can be used for computation and manipulation of quantities which depend on phase shifts. In this paper we prove Levinson's theorem in a most general way which assumes only the completeness of states. In the case of a Dirac particle we obtain a new result valid for positive and negative energies separately. We also make a generalization of known results, for the phase shifts in the asymptotic limit of high energies, to the case of singular potentials. As an application we consider certain equations, which arise in a generalized interaction picture of quantum electrodynamics. Using the above mentioned results for the phase shifts we prove that any solution of these equations, which has a finite number of bound states, has a total charge zero. Furthermore, we show that in these equations the coupling constant is not a free parameter, but rather should be treated as an eigenvalue and hence must have a definite numerical value. 
  Intertwiners between \ade lattice models are presented and the general theory developed. The intertwiners are discussed at three levels: at the level of the adjacency matrices, at the level of the cell calculus intertwining the face algebras and at the level of the row transfer matrices. A convenient graphical representation of the intertwining cells is introduced. The utility of the intertwining relations in studying the spectra of the \ade models is emphasized. In particular, it is shown that the existence of an intertwiner implies that many eigenvalues of the \ade row transfer matrices are exactly in common for a finite system and, consequently, that the corresponding central charges and scaling dimensions can be identified. 
  Macroscopic loop amplitudes are obtained for the dilation gravity in two-dimensions. The dependence on the macroscopic loop length $l$ is completely determined by using the Wheeler-DeWitt equation in the mini-superspace approximation. The dependence on the cosmological constant $\Lambda$ is also determined by using the scaling argument in addition. 
  emphasis is on 2d target space (c=1 coupled to gravity). Contents:   0. Introduction, Overview, and Purpose   1. Loops and States in Conformal Field Theory   2. 2D Euclidean Quantum Gravity I: Path Integral Approach   3. Brief Review of the Liouville Theory   4. 2D Euclidean Quantum Gravity II: Canonical Approach   5. 2D Critical String Theory   6. Discretized surfaces, matrix models, and the continuum limit   7. Matrix Model Technology I: Method of Orthogonal Polynomials   8. Matrix Model Technology II: Loops on the Lattice   9. Matrix Model Technology III: Free Fermions from the Lattice   10. Loops and States in Matrix Model Quantum Gravity   11. Loops and States in the $c=1$ Matrix Model   12. Fermi Sea Dynamics and Collective Field Theory   13. String scattering in two spacetime dimensions   14. Vertex Operator Calculations and Continuum Methods   15. Achievements, Disappointments, Future Prospects   "if you read only one set of lecture notes this year, don't read these." 
  We construct a gauge theory based on general nonlinear Lie algebras. The generic form of `dilaton' gravity is derived from nonlinear Poincar{\' e} algebra, which exhibits a gauge-theoretical origin of the non-geometric scalar field in two-dimensional gravitation theory. 
  It is shown that gauged nonlinear sigma models can be always deformed by terms proportional to the field strength of the gauge fields (nonminimal gauging). These deformations can be interpreted as perturbations, by marginal operators, of conformal coset models. When applied to the SL(2,R)*SU(2)/[U(1)*U(1)] WZWN model, a large class of four-dimensional curved spacetime backgrounds are obtained. In particular, a naked singularity may form at a time when the volume of the universe is different from zero. 
  A Maxwell-Chern-Simons field is minimally coupled to 3D-gravity. Feynman rules are written down and 1-loop corrections to the gauge-field self-energy are calculated. Transversality is verified and gauge-field dynamical mass generation does not take place. 
  We calculate the partition functions of QCD in two dimensions on a cylinder and on a torus in the gauge $\partial_{0} A_{0} = 0$ by integrating explicitly over the non zero modes of the Fourier expansion in the periodic time variable. The result is a one dimensional Kazakov-Migdal matrix model with eigenvalues on a circle rather than on a line. We prove that our result coincides with the standard expansion in representations of the gauge group. This involves a non trivial modular transformation from an expansion in exponentials of $g^2$ to one in exponentials of $1/g^2$. Finally we argue that the states of the $U(N)$ or $SU(N)$ partition function can be interpreted as a gas of N free fermions, and the grand canonical partition function of such ensemble is given explicitly as an infinite product. 
  We use a Hodge decomposition and its generalization to non-abelian flat vector bundles to calculate the partition function for abelian and non- abelian BF theories in $n$ dimensions. This enables us to provide a simple proof that the partition function is related to the Ray-Singer torsion defined on flat vector bundles for all odd-dimensional manifolds, and is equal to unity for even dimensions. 
  We considered a charged quantum mechanical particle with spin ${1\over 2}$ and gyromagnetic ratio $g\ne 2$ in the field af a magnetic string. Whereas the interaction of the charge with the string is the well kown Aharonov-Bohm effect and the contribution of magnetic moment associated with the spin in the case $g=2$ is known to yield an additional scattering and zero modes (one for each flux quantum), an anomaly of the magnetic moment (i.e. $g>2$) leads to bound states. We considered two methods for treating the case $g>2$. \\ The first is the method of self adjoint extension of the corresponding Hamilton operator. It yields one bound state as well as additional scattering. In the second we consider three exactly solvable models for finite flux tubes and take the limit of shrinking its radius to zero. For finite radius, there are $N+1$ bound states ($N$ is the number of flux quanta in the tube).\\ For $R\to 0$ the bound state energies tend to infinity so that this limit is not physical unless $g\to 2$ along with $R\to 0$. Thereby only for fluxes less than unity the results of the method of self adjoint extension are reproduced whereas for larger fluxes $N$ bound states exist and we conclude that this method is not applicable.\\ We discuss the physically interesting case of small but finite radius whereby the natural scale is given by the anomaly of the magnetic moment of the electron $a_e=(g-2)/2\approx 10^{-3}$. 
  Based on the real-time formalism, especially, on Thermo Field Dynamics, we derive the Schwinger-Dyson gap equation for the fermion propagator in QED and Four-Fermion model at finite-temperature and -density. We discuss some advantage of the real-time formalism in solving the self-consistent gap equation, in comparison with the ordinary imaginary-time formalism. Once we specify the vertex function, we can write down the SD equation with only continuous variables without performing the discrete sum over Matsubara frequencies which cannot be performed in advance without further approximation in the imaginary-time formalism. By solving the SD equation obtained in this way, we find the chiral-symmetry restoring transition at finite-temperature and present the associated phase diagram of strong coupling QED.   In solving the SD equation, we consider two approximations: instantaneous-exchange and $p_0$-independent ones. The former has a direct correspondence in the imaginary time formalism, while the latter is a new approximation beyond the former, since the latter is able to incorporate new thermal effects which has been overlooked in the ordinary imaginary-time solution. However both approximations are shown to give qualitatively the same results on the finite-temperature phase transition. 
  We consider the $C_2$ Toda theory in the reduced WZNW framework. Analysing the classical representation space of the symmetry algebra (which is the corresponding $C_2$ $W$ algebra) we determine its classical highest weight representations. We quantise the model promoting only the relevant quantities to operators. Using the quantised equation of motion we determine the selection rules for the $u$ field that corresponds to one of the Toda fields and give restrictions for its amplitude functions and for the structure of the Hilbert space of the model. 
  The effective actions for $d=2$, $N=3,4$ chiral supergravities with a linear and a non-linear gauge algebra are related to each other by a quantum reduction, the latter is obtained from the former by putting quantum currents equal to zero. This implies that the renormalisation factors for the quantum actions are identical. 
  We study a revised version of Witten's 2d black hole, in which the matter and (b,c) ghosts are mixed. The level of the coset model is still 9/4. We show that this model is equivalent to that of Mukhi and Vafa, in which the level of the coset model is taken as 3, and the stress tensor is improved. We argue that the exact metric in such a model is just the semi-classical one, quite different from the exact metric in Witten's black hole, being studied by Dijkgraaf, Verlinde and Verlinde. In addition, there appear ghost-related terms as a part of the background in the world sheet action. 
  A class of generalized Taub-NUT gravitational instantons is reported in five - dimensional Einstein gravity coupled to a non-linear sigma model. The geodesic dynamics of a spinless particle of unit mass on these static gravitational instantons is studied. This is accomplished by finding a generalized Runge-Lenz vector. Unlike the Kepler problem, or, the dynamics of a spinless particle on the familiar Taub-NUT gravitational instantons, the orbits are not conic sections. 
  We construct an $SO(10)$ grand unified theory in the formulation of non-com-\break mutative geometry. The geometry of space-time is that of a product of a continuos four dimensional manifold times a discrete set of points. The properties of the fermionic sector fix almost uniquely the Higgs structure. The simplest model corresponds to the case where the discrete set consists of three points and the Higgs fields are ${\u {16}_s}\times \overline{\u {16}}_s$ and ${\u {16}_s}\times {\u {16}_s} $. The requirement that the scalar potential for all the Higgs fields not vanish imposes strong restrictions on the vacuum expectation values of the Higgs fields and thus the fermion masses. We show that it is possible to remove these constraints by extending the number of discrete points and adding a singlet fermion and a ${\u {16}_s}$ Higgs field. Both models are studied in detail. 
  We consider the gauging of $ SL(2,R) $ WZNW model by its nilpotent subgroup E(1). The resulting space-time of the corresponding sigma model is seen to collapse to a one dimensional field theory of Liouville. Gauging the diagonal subgroup $ E(1) \times U(1) $ of $SL(2,R) \times U(1)$ theory yields an extremal three dimensional black string. We show that these solutions are obtained from the two dimensional black hole of Witten and the three dimensional black string of Horne and Horowitz by boosting the gauge group. 
  In this short note we extend the results of Alfaro and Damgaard on the origin of antifields to theories with a gauge algebra that is open or reducible. 
  The conjecture that $N=2$ minimal models in two dimensions are critical points of a super-renormalizable Landau-Ginzburg model can be tested by computing the path integral of the Landau-Ginzburg model with certain twisted boundary conditions. This leads to simple expressions for certain characters of the $N=2$ models which can be verified at least at low levels. An $N=2$ superconformal algebra can in fact be found directly in the {\it noncritical} Landau-Ginzburg system, giving further support for the conjecture. 
  One possible fate of information lost to black holes is its preservation in black hole remnants. It is argued that a type of effective field theory describes such remnants (generically referred to as informons). The general structure of such a theory is investigated and the infinite pair production problem is revisited. A toy model for remnants clarifies some of the basic issues; in particular, infinite remnant production is not suppressed simply by the large internal volumes as proposed in cornucopion scenarios. Criteria for avoiding infinite production are stated in terms of couplings in the effective theory. Such instabilities remain a problem barring what would be described in that theory as a strong coupling conspiracy. The relation to euclidean calculations of cornucopion production is sketched, and potential flaws in that analysis are outlined. However, it is quite plausible that pair production of ordinary black holes (e.g. Reissner Nordstrom or others) is suppressed due to strong effective couplings. It also remains an open possibility that a microscopic dynamics can be found yielding an appropriate strongly coupled effective theory of neutral informons without infinite pair production. 
  We investigate the statistics of the number $N(R,S)$ of lattice points, $n\in \Z^2$, in a ``random'' annular domain $\Pi(R,w)=\,(R+w)A\,\setminus RA$, where $R,w >0$. Here $A$ is a fixed convex set with smooth boundary and $w$ is chosen so that the area of $\Pi (R,w)$ is $S$. The randomness comes from $R$ being taken as random ( with a smooth denisity ) in some interval $[c_1T,c_2T]$, $c_2>c_1>0$. We find that in the limit $T\to\infty $ the variance and distribution of $\De N=N(R;S)-S$ depends strongly on how $S$ grows with $T$. There is a saturation regime $S/T\to\infty$, as $T\to\infty$ in which the fluctuations in $\Delta N$ coming from the two boundaries of $\Pi $, are independent. Then there is a scaling regime, $S/T\to z$, $0<z<\infty $ in which the distribution depends on $z$ in an almost periodic way going to a Gaussian as $z\to\ 0$. The variance in this limit approaches $z$ for ``generic'' $A$ but can be larger for ``degenerate'' cases. The former behavior is what one would expect from the Poisson limit of a distribution for annuli of finite area. 
  A class of two-dimensional topological conformal field theories (TCFTs) is studied within the framework of gauged WZW models in order to gain some insights on the global geometrical nature of TCFTs. The BRST quantizations of topological G/H gauged WZW models (the twisted versions of SUSY gauged WZW models) are given under fixed back-ground gauge fields. The BRST-cohomology of the system is investigated and the correlation functions among these physical observables are considered under the instanton back-grounds. As a consequence, two-dimensional BF gauge theoretical aspects of TCFTs are revealed. Especially it is shown that two correlation functions under the different instanton back-grounds can change to each other. This process of transmutation is described by the spectral flow. The flow is formulated as a "singular" gauge transformation which creates an appropriate back-ground charge on the physical vacuum of the system. The field identification problem of the system is also discussed from the above point of view. 
  This paper has been heavily revised, the final results now being contained in hep-ph/9311202 and hep-ph/9311203. 
  The functional determinant of an elliptic operator with positive, discrete spectrum may be defined as $e^{-Z'(0)}$, where $Z(s)$, the zeta function, is the sum $\sum_n^{\infty} \lambda_n^{-s}$ analytically continued to $s$ around the origin. In this paper $Z'(0)$ is calculated for the Laplace operator with Dirichlet boundary conditions inside polygons and simplices with the topology of a disc in the Euclidean plane. The domains we consider are hence piece--wise flat with corners on the boundary and in the interior. Our results are complementary to earlier investigations of the determinants on smooth surfaces with smooth boundaries. We have explicit closed integrated expressions for triangles and regular polygons. 
  We examine the dynamics of a free massless scalar field on a figure eight network. Upon requiring the scalar field to have a well defined value at the junction of the network, it is seen that the conserved currents of the theory satisfy Kirchhoff's law, that is that the current flowing into the junction equals the current flowing out. We obtain the corresponding current algebra and show that, unlike on a circle, the left- and right-moving currents on the figure eight do not in general commute in quantum theory. Since a free scalar field theory on a one dimensional spatial manifold exhibits conformal symmetry, it is natural to ask whether an analogous symmetry can be defined for the figure eight. We find that, unlike in the case of a manifold, the action plus boundary conditions for the network are not invariant under separate conformal transformations associated with left- and right-movers. Instead, the system is, at best, invariant under only a single set of transformations. Its conserved current is also found to satisfy Kirchhoff's law at the junction. We obtain the associated conserved charges, and show that they generate a Virasoro algebra. Its conformal anomaly (central charge) is computed for special values of the parameters characterizing the network. 
  Starting from a system of planar electrons in a strong magnetic field normal to the plane, interacting with perturbing electromagnetic fields, an effective Lagrangian for the fermions in the lowest Landau level (L.L.L.) has been derived. By choosing a suitable background electrostatic potential, an incompressible droplet of these electrons is constructed. The gauge invariant effective Lagrangian for the electrons in the L.L.L. is shown to split naturally into a $1+1$ dimensional Lagrangian for the electrons on the surface of the droplet and into a $2+1$ dimensional gauge-field Lagrangian representing the contribution of the interior of the droplet. Upon bosonization, the former represents the surface vibrations of the droplet. Individually neither of these two actions is gauge invariant, but it is shown that the gauge dependence from the two pieces cancels out. This demonstrates that the edge degrees of freedom are essential for maintaining gauge invariance. 
  We consider a class of Calabi-Yau compactifications which are constructed as a complete intersection in weighted projective space. For manifolds with one K\"ahler modulus we construct the mirror manifolds and calculate the instanton sum. 
  Supersymmetry transformations are a kind of square root of spacetime translations. The corresponding Lie superalgebra always contains the supertranslation operator $ \delta = c^{\alpha} \sigma^{\mu}_{\alpha \dot \beta} {\overline c}^{\dot \beta} (\epsilon^{\mu})^{\dag} $. We find that the cohomology of this operator depends on a spin-orbit coupling in an SU(2) group and has a quite complicated structure. This spin-orbit type coupling will turn out to be basic in the cohomology of supersymmetric field theories in general. 
  After reviewing the role of phase in quantum mechanics, I discuss, with the aid of a number of unpublished documents, the development of quantum phase operators in the 1960's. Interwoven in the discussion are the critical physics questions of the field: Are there (unique) quantum phase operators and are there quantum systems which can determine their nature? I conclude with a critique of recent proposals which have shed new light on the problem. 
  The causal boundary of string propagation -- defined as the hypersurface in loop space bordering the timelike(spacelike) domains in which two successive measurements of the string field do(do not) interfere with one another -- is argued to be $0=\int d\sigma\bigl(\delta X(\sigma)\bigr)^2 =   \sum_{\ell=-\infty}^\infty \delta x^\mu_{-\ell}\delta x_{\mu \ell}.$ Some possible consequences are discussed. 
  We rewrite $ N=2$ quantum super $W_{3}$ algebra, a nonlinear extended $N=2$ super Virasoro algebra, containing one additional primary superfield of dimension $2$ which has no $U(1)$ charge, besides the super stress energy tensor of dimension $1$ in $N=2$ superspace. The free superfield realization of this algebra is obtained by two $N=2$ chiral fermionic superfields of dimension $1/2$ satisfying $N=2$ complex $ U(1)$ Kac-Moody algebras. 
  The discrete states in the $c=1$ string are shown to be the physical states of a certain topological sigma model. We define a set of new fields directly from $c=1$ variables, in terms of which the BRST charge and energy-momentum tensor are rewritten as those of the topological sigma model. Remarkably, ground ring generator $x$ turns out to be a coordinate of the sigma model. All of the discrete states realize a graded ring which contains ground ring as a subset. 
  Starting from superdifferential operators in an $N=1$ superfield formulation, we present a systematic prescription for the derivation of classical $N=1$ and $N=2$ super W-algebras by imposing a zero-curvature condition on the connection of the corresponding first order system. We illustrate the procedure on the first non-trivial example (beyond the $N=1$ superconformal algebra) and also comment on the relation with the Gelfand-Dickey construction of $W$-algebras. 
  A new model of bosonic strings is considered. An action of the model is the sum of the standard string action and a term describing an interaction of a metric with a linear (affine) connection. The Lagrangian of this interaction is an arbitrary analytic function $f(R)$ of the scalar curvature. This is a classically integrable model. The space of classical solutions of the theory consists from sectors with constant curvature. In each sector the equations of motion reduce to the standard string equations and to an additional constant curvature equation for the linear connection. A bifurcation in the space of all Lagrangians takes place. Quantization of the model is briefly discussed. In a quasiclassical approximation one gets the standard string model with a fluctuating cosmological constant. The Lagrangian $f(R)$, like Morse function, governs transitions between manifolds with different topologies. 
  The Heisenberg double of a Hopf algebra may be regarded as a quantum analogue of the cotangent bundle of a Lie group. Quantum duality principle describes relations between a Hopf algebra, its dual, and their Heisenberg double in a way which extends both the theory of coadjoint orbits and the classical Fourier transform. We also describe the twisted Heisenberg double which is relevant for the study of nontrivial deformations of the quantized universal enveloping algebras. 
  It is shown that gravity on the line can be described by the two dimensional (2D) Hilbert-Einstein Lagrangian supplemented by a kinetic term for the coframe and a translational {\it boundary} term. The resulting model is equivalent to a Yang-Mills theory of local {\it translations} and frozen Lorentz gauge degrees. We will show that this restricted Poincar\'e gauge model in 2 dimensions is completely integrable. {\it Exact} wave, charged black hole, and `dilaton' solutions are then readily found. In vacuum, the integrability of the {\it general} 2D Poincar\'e gauge theory is formally proved along the same line of reasoning. 
  The global structure of 1 + 1 dimensional compact Universe is studied in two-dimensional model of dilaton gravity.   First we give a classical solution corresponding to the spacetime in which a closed time-like curve appears, and show the instability of this spacetime due to the existence of matters.   We also observe quantum version of such a spacetime having closed timelike curves never reappear unless the parameters are fine-tuned. 
  We describe the mirror of the Z orbifold as a representation of a class of generalized Calabi-Yau manifolds that can be realized as manifolds of dimension five and seven. Despite their dimension these correspond to superconformal theories with $c=9$ and so are perfectly good for compactifying the heterotic string to the four dimensions of space-time. As a check of mirror symmetry we compute the structure of the space of complex structures of the mirror and check that this reproduces the known results for the Yukawa couplings and metric appropriate to the Kahler class parameters on the Z orbifold together with their instanton corrections. 
  We propose a new formulation of the $D=3$ type II superstring which is manifestly invariant under both target-space $N=2$ supersymmetry and worldsheet $N=(1,1)$ super reparametrizations. This gives rise to a set of twistor (commuting spinor) variables, which provide a solution to the two Virasoro constraints. The worldsheet supergravity fields are shown to play the r\^ole of auxiliary fields. 
  We obtain integral representations for the wave functions of Calogero-type systems,corresponding to the finite-dimentional Lie algebras,using exact evaluation of path integral.We generalize these systems to the case of the Kac-Moody algebras and observe the connection of them with the two dimensional Yang-Mills theory.We point out that Calogero-Moser model and the models of Calogero type like Sutherland one can be obtained either classically by some reduction from two dimensional Yang-Mills theory with appropriate sources or even at quantum level by taking some scaling limit.We investigate large k limit and observe a relation with Generalized Kontsevich Model. 
  Cosmological solutions of the beta function equations for the background fields of the closed bosonic string are investigated at the one-loop level. Following recent work of Kostelecky and Perry, it is assumed that the spatial sections of the space-time are conformally flat. Working in the sigma-model frame, the non-trivial tachyon potential is utilized to determine solutions with sufficient inflation to solve the smoothness and flatness problems. The graceful exit and density perturbation constraints can also be successfully implemented. 
  We study the nonunitary representations of N=2 Super Virasoro algebra for the rational central charges c<3. The resolutions for the irreducible representations of N=2 SVir in terms of the "2-d gravity modules" are obtained and their characters are computed. The correspondence between the N=2 nonunitary "minimal" models and the Virasoro minimal models coupled to 2-d gravity is shown at the level of states. We also define the hamiltonian reduction of the BRST complex of sl(N)/sl(N) coset to the BRST complex of the W-gravity coupled to the W matter. The case of sl(2) is considered explicitly. It leads to the presentation of N=2 Super Virasoro algebra by the Lie algebra cohomology. Finally, we reveal the mechanism of the correspondence between sl(2)/sl(2)$ coset and 2-d gravity. 
  The massless flow between successive minimal models of conformal field theory is related to a flow within the sine-Gordon model when the coefficient of the cosine potential is imaginary. This flow is studied, partly numerically, from three different points of view. First we work out the expansion close to the Kosterlitz-Thouless point, and obtain roaming behavior, with the central charge going up and down in between the UV and IR values of $c=1$. Next we analytically continue the Casimir energy of the massive flow (i.e. with real cosine term). Finally we consider the lattice regularization provided by the O(n) model in which massive and massless flows correspond to high- and low-temperature phases. A detailed discussion of the case $n=0$ is then given using the underlying N=2 supersymmetry, which is spontaneously broken in the low-temperature phase. The ``index'' $\tr F(-1)^F$ follows from the Painleve III differential equation, and is shown to have simple poles in this phase. These poles are interpreted as occuring from level crossing (one-dimensional phase transitions for polymers). As an application, new exact results for the connectivity constants of polymer graphs on cylinders are obtained. 
  We study the spectrum, the massless S-matrices and the ground-state energy of the flows between successive minimal models of conformal field theory, and within the sine-Gordon model with imaginary coefficient of the cosine term (related to the minimal models by ``truncation''). For the minimal models, we find exact S-matrices which describe the scattering of massless kinks, and show using the thermodynamic Bethe ansatz that the resulting non-perturbative c-function (defined by the Casimir energy on a cylinder) flows appropriately between the two theories, as conjectured earlier. For the non-unitary sine-Gordon model, we find unusual behavior. For the range of couplings we can study analytically, the natural S-matrix deduced from the minimal one by ``undoing'' the quantum-group truncation does not reproduce the proper c-function with the TBA. It does, however, describe the correct properties of the model in a magnetic field. 
  The connection between a Taylor series and a continued-fraction involves a nonlinear relation between the Taylor coefficients $\{ a_n \}$ and the continued-fraction coefficients $\{ b_n \}$. In many instances it turns out that this nonlinear relation transforms a complicated sequence $\{a_n \}$ into a very simple one $\{ b_n \}$. We illustrate this simplification in the context of graph combinatorics. 
  Motivated by the problem of background independence of closed string field theory we study geometry on the infinite vector bundle of local fields over the space of conformal field theories (CFT's). With any connection we can associate an excluded domain $D$ for the integral of marginal operators, and an operator one-form $\omega_\mu$. The pair $(D, \omega_\mu)$ determines the covariant derivative of any correlator of local fields. We obtain interesting classes of connections in which $\omega_\mu$'s can be written in terms of CFT data. For these connections we compute their curvatures in terms of four-point correlators, $D$, and $\omega_\mu$. Among these connections three are of particular interest. A flat, metric compatible connection $\HG$, and connections $c$ and $\bar c$ having non-vanishing curvature, with $\bar c$ being metric compatible. The flat connection cannot be used to do parallel transport over a finite distance. Parallel transport with either $c$ or $\bar c$, however, allows us to construct a CFT in the state space of another CFT a finite distance away. The construction is given in the form of perturbation theory manifestly free of divergences. 
  We generalize to dimension $p>1$ the notion of string structure and discuss the related obstruction. We apply our results to a model of bosonic $p$-branes propagating on a principal $G$-bundle, coupled to a Yang--Mills field and an antisymmetric tensor field and in the presence of a Wess-Zumino term in the Lagrangian. We construct the quantization line bundle and discuss the action of background gauge transformations on wave functions. 
  We investigate the dependence of the gauge couplings on the dilaton field in string effective theories at the one--loop level. First we resolve the discrepancies between statements based on symmetry considerations and explicit calculations in string effective theories on this subject. A calculation of the relevant one--loop scattering amplitudes in string theory gives us further information and allows us to derive the exact form of the corresponding effective Lagrangian. In particular there is no dilaton dependent one--loop correction to the holomorphic $f$--function arising from massive string modes in the loop. In addition we address the coupling of the antisymmetric tensor field to the gauge bosons at one--loop. While the string S--matrix elements are not reproduced using the usual supersymmetric Lagrangian with the chiral superfield representation for the dilaton field, the analogue Lagrangian with the dilaton in a linear multiplet naturally gives the correct answer. 
  We discuss the relation of the two types of sums in the Rogers-Schur-Ramanujan identities with the Bose-Fermi correspondence of massless quantum field theory in $1+1$ dimensions. One type, which generalizes to sums which appear in the Weyl-Kac character formula for representations of affine Lie algebras and in expressions for their branching functions, is related to bosonic descriptions of the spectrum of the field theory (associated with the Feigin-Fuchs construction in conformal field theory). Fermionic descriptions of the same spectrum are obtained via generalizations of the other type of sums. We here summarize recent results for such fermionic sum representations of characters and branching functions. (To appear in C.N. Yang's 70th birthday Festschrift.) 
  The antibracket in BRST theory is known to define a map $\rm{H^p \times H^q \longrightarrow H^{p+q+1}}$ associating with two equivalence classes of BRST invariant observables of respective ghost number p and q an equivalence class of BRST invariant observables of ghost number p+q+1. It is shown that this map is trivial in the space of all functionals, i.e., that its image contains only the zeroth class. However it is generically non trivial in the space of local functionals. Implications of this result for the problem of consistent interactions among fields with a gauge freedom are then drawn. It is shown that the obstructions to constructing such interactions lie precisely in the image of the antibracket map and are accordingly inexistent if one does not insist on locality. However consistent local interactions are severely constrained. The example of the Chern-Simons theory is considered. It is proved that the only consistent, local, Lorentz covariant interactions for the abelian models are exhausted by the non-abelian Chern-Simons extensions. 
  I consider a Langevin equation with field-dependent kernels and investigate supersymmetry of the stochastic generating functional constructed from the Langevin equation. Moreover I describe the stochastic generating functional in terms of a superfield. In the superfield formalism, it becomes clear that the stochastic quantization method with the field-dependent kernel is equivalent to the path-integral quantization method. 
  The one dimensional Fermi gas of matrix eigenvalues of the Marinari-Parisi model at positive values of the cosmological constant is generalised.The number of matrix eigenvalues (i.e. gas particles) is varied while keeping the effective potential fixed. This model exhibits a transition from a phase whose continuum behaviour is that of $c=1$ conformal matter coupled to gravity to the well known pure gravity phase of the original model.The former phase is character- ised by an extremely large Regge slope $\alpha^\prime$ which scales as $\beta^{2/5}$ causing the scaling regions of the two phases to overlap. In this way a continuous flow from one phase to the other is made poss- ible. This phase transition occurs in the singlet sector of the matrix model. The density of states and the two puncture correlator at non zero momenta are calculated on the sphere and are found to behave very differ- ently in the two phases,a fact which demonstrates the phase transition. We comment on a possible relation between this transition and large \alpha^\prime$ semiclassical expansions in the continuum. 
  We study a topological Yang-Mills theory with $N=2$ fermionic symmetry. Our formalism is a field theoretical interpretation of the Donaldson polynomial invariants on compact K\"{a}hler surfaces. We also study an analogous theory on compact oriented Riemann surfaces and briefly discuss a possible application of the Witten's non-Abelian localization formula to the problems in the case of compact K\"{a}hler surfaces. 
  This is an expanded and updated version of a talk given at the Conference on Topics in Geometry and Physics at the University of Southern California, November 6, 1992. It is a survey talk, aimed at mathematicians AND physicists, which attempts to bring together the topics in the title without assuming much background in any of them. Closed string field theory leads to a (strong homotopy) generalization of Lie algebra, which is strongly related to the way the moduli spaces $\Cal M_{0,N+1}$ fit together as an ``operad''. The latter in turn plays an important role in the understanding of vertex operator algebras. 
  We study dilatonic domain walls specific to superstring theory.   Along with the matter fields and metric the dilaton also changes its value in the wall background. We found supersymmetric (extreme) solutions which in general interpolate between isolated superstring vacua with non-equal value of the matter potential; they correspond to the static, planar domain walls with {\it flat} metric in the string (sigma model) frame.   We point out similarities between the space-time of dilatonic walls and that of charged dilatonic black holes. We also comment on non-extreme solutions corresponding to expanding bubbles. 
  The level spacing distributions which arise when one rescales the Laguerre or Jacobi ensembles of hermitian matrices is studied. These distributions are expressible in terms of a Fredholm determinant of an integral operator whose kernel is expressible in terms of Bessel functions of order $\alpha$. We derive a system of partial differential equations associated with the logarithmic derivative of this Fredholm determinant when the underlying domain is a union of intervals. In the case of a single interval this Fredholm determinant is a Painleve tau function. 
  The spin degrees of freedom for the relativistic particle are described by either Poincar\'{e} group variables (classically) or Grassmann variables (pseudo-classically). The relationship between those two descriptions are given. In doing that, appropriate constraints are constructed to put into the lagrangian. Especially a natural relation of Poincar\'{e} group variables and Grassmann variables is obtained. Hopf fibration relating the spin momentum to the group is just the right transformation of the spin momentum under Poincar\'{e} group. And with the relation just mentioned, pseudoclassical lagrangian is derived naturally from the classical one. 
  The S-matrices for non-simply-laced affine Toda field theories are considered in the context of a generalised bootstrap principle. The S-matrices, and in particular their poles, depend on a parameter whose range lies between the Coxeter numbers of dual pairs of the corresponding non-simply-laced algebras. It is proposed that only odd order poles in the physical strip with positive coefficients throughout this range should participate in the bootstrap. All other singularities have an explanation in principle in terms of a generalised Coleman-Thun mechanism. Besides the S-matrices introduced by Delius, Grisaru and Zanon, the missing case ($f_4^{(1)},e_6^{(2)}$), is also considered and provides many interesting examples of pole generation. 
  A study is made for ${\bf Z}_M\times {\bf Z}_N$ orbifolds of the modification of the form of the twisted sector Yukawa couplings when some of the states involved are excited twisted sectors rather than twisted sector ground states. 
  Let $B$ and $F=\frac 12F_{\mu \nu}dx^\mu \wedge dx^\nu $ be two forms, $F_{\mu \nu}$ being the field strength of an abelian connection $A$. The topological $BF$ system is given by the integral of $B\wedge F$. With "kinetic energy'' terms added for $B$ and $A$, it generates a mass for $A$ thereby suggesting an alternative to the Higgs mechanism, and also gives the London equations. The $BF$ action, being the large length and time scale limit of this augmented action, is thus of physical interest. In earlier work, it has been studied on spatial manifold $\Sigma $ with boundaries $\partial \Sigma $, and the existence of edge states localised at $\partial \Sigma $ has been established. They are analogous to the conformal family of edge states to be found in a Chern-Simons theory in a disc. Here we introduce charges and vortices (thin flux tubes) as sources in the $BF$ system and show that they acquire an infinite number of spin excitations due to renormalization, just as a charge coupled to a Chern-Simons potential acquires a conformal family of spin excitations. For a vortex, these spins are transverse and attached to each of its points, so that it resembles a ribbon. Vertex operators for the creatin of these sources are constructed and interpreted in terms of a Wilson integral involving $A$ and a similar integral involving $B$. The standard spin-statistics theorem is proved for this sources. A new spin-statistics theorem, showing the equality of the ``interchange'' of two identical vortex loops and $2\pi $ rotation of the transverse spins of a constituent vortex, is established. Aharonov-Bohm interactions of charges and vortices are studied. The existence of topologically nontrivial vortex spins is pointed out and their vertex 
  The three dimensional black hole solutions of Ba\~nados, Teitelboim and Zanelli (BTZ) are dimensionally reduced in various different ways. Solutions are obtained to the Jackiw-Teitelboim theory of two dimensional gravity for spinless BTZ black holes, and to a simple extension with a non-zero dilaton potential for black holes of fixed spin. Similar reductions are given for charged black holes. The resulting two dimensional solutions are themselves black holes, and are appropriate for investigating exact ``S-wave'' scattering in the BTZ metrics. Using a different dimensional reduction to the string inspired model of two dimensional gravity, the BTZ solutions are related to the familiar two dimensional black hole and the linear dilaton vacuum. 
  We review the basics of the dynamics of closed strings moving along the discretized line \Z. The string excitations are described by a field \phi_x(\tau) where x is the position of the string in the embedding space and \tau is a semi-infinite ``euclidean time'' parameter related to the longitudinal mode of the string. Interactions due to splitting and joining of closed strings are taken into account by a local potential and occur only along the edge \tau =0 of the semi-plane (x, \tau). 
  A discretized time evolution of the wave function for a Dirac particle on a cubic lattice is represented by a very simple quantum cellular automaton. In each evolution step the updated value of the wave function at a given site depends only on the values at the nearest sites, the evolution is unitary and preserves chiral symmetry. Moreover, it is shown that the relationship between Dirac particles and cellular automata operating on two component objects on a lattice is indeed very close. Every local and unitary automaton on a cubic lattice, under some natural assumptions, leads in the continuum limit to the Weyl equation. The sum over histories is evaluated and its connection with path integrals and theories of fermions on a lattice is outlined. 
  Exact expressions for the partition functions of the rigid string and membrane at any temperature are obtained in terms of hypergeometric functions. By using zeta function regularization methods, the results are analytically continued and written as asymptotic sums of Riemann-Hurwitz zeta functions, which provide very good numerical approximations with just a few first terms. This allows to obtain systematic corrections to the results of Polchinski et al., corresponding to the limits $T\rightarrow 0$ and $T\rightarrow \infty$ of the rigid string, and to analyze the intermediate range of temperatures. In particular, a way to obtain the Hagedorn temperature for the rigid membrane is thus found. 
  We further study the nonperturbative formulation of two-dimensional black holes. We find a nonlinear differential equation satisfied by the tachyon in the black hole background. We show that singularities in the tachyon field configurations are always associated with divergent semiclassical expansions and are absent in the exact theory. We also discuss how the Euclidian black hole emerges from an analytically continued fermion theory that corresponds to the right side up harmonic oscillator potential. 
  A covariant - tensor method for $SU(2)_{q}$ is described. This tensor method is used to calculate q - deformed Clebsch - Gordan coefficients. The connection with covariant oscillators and irreducible tensor operators is established. This approach can be extended to other quantum groups. 
  We show that the Clebsch - Gordan coefficients for the $SU(2)_{p,q}$ - algebra depend on a single parameter Q = $\sqrt{pq}$ ,contrary to the explicit calculation of Smirnov and Wehrhahn [J.Phys.A 25 (1992),5563]. 
  It is shown that the generalized (with nonpolynomial Lagrangian) Chern-Simons membranes and in general $p$-branes moving in $D$-dimensional target space admit an infinite set of secondary constraints. With respect to the Poisson bracket these constraints satisfy closed algebra containing $p$-dimensional classical $W$ algebra as a subalgebra. In the case when the target space dimension $D$ is finite the theory is topological. 
  We consider a simple model of a scalar field with $U(1)$ current algebra gauge symmetry coupled to $2D$-gravity in order to clarify the origin of Stuckelberg symmetry in the $w_{\infty}$-gravity theory. An analogous symmetry takes place in our model too. The possible central extension of the complete symmetry algebra and the corresponding critical dimension have been found. The analysis of the Hamiltonian and the constraints shows that the generators of the current algebra, the reparametrization and the Stuckelberg symmetries are not independent. The connection of the model with $w_{1+\infty}-$ and $W_{1+\infty}$-gravity is discussed. 
  We obtain the formula for intertwining operator(R-matrix) of quantum universal enveloping superalgebra U_qOSP(1,2) for U_qOSP(1,2)-Verma modules. By its restriction we obtain the R-matrix for two semiperiodic(semicyclic), two spin-j and spin-j and semiperiodic representations 
  We study the equations for the quasi-momenta which characterize the wave-functions in the Bethe ansatz for the XXX-Heisenberg model. We show in a simple analytical fashion, that the usual ``string hypothesis" incorrectly predicts the number of real solutions and the number of complex solutions for $N>21$ in the sector with two spins flipped, confirming the work of Essler et al. Two complex pair solutions drop out and form two additional real pair solutions. We also introduce a new set of variables which allows the equations to be written as a single polynomial equation in one variable. We consider in some detail the case of three spins flipped. 
  A criterion to be satisfied by a string theory of QCD is formulated in the ultraviolet regime. It arises from the trace anomaly of the QCD stress tensor computed using instantons. It is sensitive to asymptotic freedom. It appears to be related to the trace anomaly of the QCD string. Our current understanding of noncritical strings in physical dimensions is limited, but remarkably, a formal treatment of the bosonic string yields numerical agreement both in magnitude and sign for the gauge group SU(2). 
  In this talk we report some results about the construction of soliton solutions for the Affine and Conformal Affine Toda models using the Hirota's method. We obtain new classes of solitons connected to the degeneracies of the Cartan matrix eigenvalues as well as to some particular features of the recursive scheme developed here. We obtain an universal mass formula for all those solitons. The examples of $SU(6)$ and $Sp(3)$ are discussed in some detail. ( Talk presented at the VII J.A. Swieca Summer School, Section: Particles and   Fields, Campos do Jord\~ao - Brasil - January/93) 
  In an abelian topologically massive gauge theory, any eigenstate of the Hamiltonian can be decomposed into a factor describing massive propagating gauge bosons and a Chern-Simons wave function describing a set of nonpropagating ``topological'' excitations.  The energy depends only on the propagating modes, and energy eigenstates thus occur with a degeneracy that can be parametrized by the Hilbert space of the pure Chern-Simons theory.  We show that for a {\em nonabelian} topologically massive gauge theory, this degeneracy is lifted: although the Gauss law constraint can be solved with a similar factorization, the Hamiltonian couples the propagating  and nonpropagating (topological) modes. 
  Using a hybrid approach, based on the recursion relations for shape invariant potentials developed by Das and Huang and a time-dependent tranformation of variables, we derive the propagator for a radial oscillator. Although this is not a new result, we explicitly show that time-dependent tranformations are very beneficial even within the context of time-independent Hamiltonians in quantum mechanics. 
  Using perturbative methods we derive new results for the spectrum and correlation functions of the general Z_3-chiral Potts quantum chain in the massive low-temperature phase. Explicit calculations of the ground state energy and the first excitations in the zero momentum sector give excellent approximations and confirm the general statement that the spectrum in the low-temperature phase of general Z_n-spin quantum chains is identical to one in the high-temperature phase where the role of charge and boundary conditions are interchanged. Using a perturbative expansion of the ground state for the Z_3 model we are able to gain some insight in correlation functions. We argue that they might be oscillating and give estimates for the oscillation length as well as the correlation length. 
  By solving the supersymmetry constraints for physical wave-functions, it is shown that the only two allowed bosonic states in $ N = 1 $ supergravity are of the form const.~exp~$(\pm I / \hbar) $, where $ I $ is an action functional of the three-metric. States containing a finite number of fermions are forbidden. In the case that the spatial topology is $ S^3 $, the state const.~exp~$(- I / \hbar)$ is the wormhole ground state, and the state const.~exp~$(I / \hbar)$ is the Hartle--Hawking state. $ N = 1 $ supergravity has no quantum ultraviolet divergences, and no quantum corrections. 
  We present a thorough analysis of the Non Intersecting String (NIS) model and its exact solution. This is an integrable $q$-states vertex model describing configurations of non-intersecting polygons on the lattice. The exact eigenvalues of the transfer matrix are found by analytic Bethe Ansatz. The Bethe Ansatz equations thus found are shown to be equivalent to those for a mixed spin model involving both 1/2 and infinite spin. This indicates that the NIS model provides a representation of the quantum group $SU(2)_{\hat q}$ ($|\hat q|\not= 1$) corresponding to spins $s=1/2$ and $s=\infty$. The partition function and the excitations in the thermodynamic limit are computed. 
  A general family of charge-current carrying cosmic string models is investigated. In the special case of circular configurations in arbitrary axially symmetric gravitational and electromagnetic backgrounds the dynamics is determined by simple point particle Hamiltonians. A certain "duality" transformation relates our results to previous ones, obtained by Carter et. al., for an infinitely long open stationary string in an arbitrary stationary background. 
  In a previous paper [\AS], we used superspace techniques to prove that perturbation theory (around a classical solution with no zero modes) for Chern--Simons quantum field theory on a general $3$-manifold $M$ is finite. We conjectured (and proved for the case of $2$-loops) that, after adding counterterms of the expected form, the terms in the perturbation theory define topological invariants. In this paper we prove this conjecture. Our proof uses a geometric compactification of the region on which the Feynman integrand of Feynman diagrams is smooth as well as an extension of the basic propagator of the theory. 
  We analyze the behavior of the ensemble of surface boundaries of the critical clusters at $T=T_c$ in the $3d$ Ising model. We find that $N_g(A)$, the number of surfaces of given genus $g$ and fixed area $A$, behaves as $A^{-x(g)}$ $e^{-\mu A}$. We show that $\mu$ is a constant independent of $g$ and $x(g)$ is approximately a linear function of $g$. The sum of $N_g(A)$ over genus scales as a power of $A$. We also observe that the volume of the clusters is proportional to its surface area. We argue that this behavior is typical of a branching instability for the surfaces, similar to the ones found for non-critical string theories with $c > 1$. We discuss similar results for the ordinary spin clusters of the $3d$ Ising model at the minority percolation point and for $3d$ bond percolation. Finally we check the universality of these critical properties on the simple cubic lattice and the body centered cubic lattice. 
  The thermal Bogoliubov transformation in thermo field dynamics is generalized in two respects. First, a generalization of the $\alpha$--degree of freedom to tilde non--conserving representations is considered. Secondly, the usual $2\times2$ Bogoliubov matrix is extended to a $4\times4$ matrix including mixing of modes with non--trivial multiparticle correlations. The analysis is carried out for both bosons and fermions. 
  We explore the implications of recent work by Br\'ezin and Zinn-Justin, applying the renormalization group techniques from critical phenomena to the scaling limit of matrix models in two-dimensional quantum gravity. They endeavor to get the lowest order fixed points of the theory giving insight upon the critical points of the theory. We show that at leading order the perturbative result is equal to the saddle-point approximation result. We calculate the next-to-leading order in the perturbative expansion exploring the goodness of the approach. 
  The one-loop effective action corresponding the general model of dilaton gravity given by the Lagrangian $L=-\sqrt{g} \left[ \frac{1}{2}Z(\Phi ) g^{\mu\nu} \partial_\mu \Phi \partial_\nu \Phi + C(\Phi ) R + V (\Phi )\right]$, where $Z(\Phi )$, $ C(\Phi )$ and $V (\Phi )$ are arbitrary functions of the dilaton field, is found. The question of the quantum equivalence of classically equivalent dilaton gravities is studied. By specific calculation of explicit examples it is shown that classically equivalent quantum gravities are also perturbatively equivalent at the quantum level, but only on-shell. The renormalization group equations for the generalized effective couplings $Z(\Phi )$, $ C(\Phi )$ and $V (\Phi )$ are written. An analysis of the equations shows, in particular, that the Callan-Giddings-Harvey-Strominger model is not a fixed point of these equations. 
  We consider a self-interacting scalar field theory in a slowly varying gravitational background field. Using zeta-function regularization and heat-kernel techniques, we derive the one-loop effective Lagrangian up to second order in the variation of the background field and up to quadratic terms in the curvature tensors. Specializing to different spacetimes of physical interest, the influence of the curvature on the phase transition is considered. 
  In the first order form, the model considered by Strobl presents, besides local Lorentz and diffeomorphism invariances, an additional local non-linear symmetry. When the model is realized as a Poincar\'e gauge theory according to the procedure outlined in Refs.[1,2], the generators of the non-linear symmetry are responsible for the ``nasty constraint algebra''. We show that not only the Poincar\'e gauge theoretic formulation of the model is not the cause of the emerging of the undesirable constraint algebra, but actually allows to overcome the problem. In fact one can fix the additional symmetry without breaking the Poincar\'e gauge symmetry and the diffeomorphisms, so that, after a preliminary Dirac procedure, the remaining constraints uniquely satisfy the Poincar\'e algebra. After the additional symmetry is fixed, the equations of motion are unaltered. The objections to our method raised by Strobl in Ref.[3] are then immaterial. Some minor points put forward in Ref.[3] are also discussed. 
  Following the general method discussed in Refs.[1,2], Liouville gravity and the 2 dimensional model of non-Einstenian gravity ${\cal L} \sim curv^2 + torsion^2 + cosm. const.$ can be formulated as ISO(1,1) gauge theories. In the first order formalism the models present, besides the Poincar\'e gauge symmetry, additional local symmetries. We show that in both models one can fix these additional symmetries preserving the ISO(1,1) gauge symmetry and the diffeomorphism invariance, so that, after a preliminary Dirac procedure, the remaining constraints uniquely satisfy the ISO(1,1) algebra. After the additional symmetry is fixed, the equations of motion are unaltered. One thus remarkably simplifies the canonical structure, especially of the second model. Moreover, one shows that the Poincar\'e group can always be used consistently as a gauge group for gravitational theories in two dimensions. 
  We discuss the relationship between geometry, the renormalization group (RG) and gravity. We begin by reviewing our recent work on crossover problems in field theory. By crossover we mean the interpolation between different representations of the conformal group by the action of relevant operators. At the level of the RG this crossover is manifest in the flow between different fixed points induced by these operators. The description of such flows requires a RG which is capable of interpolating between qualitatively different degrees of freedom. Using the conceptual notion of course graining we construct some simple examples of such a group introducing the concept of a ``floating'' fixed point around which one constructs a perturbation theory. Our consideration of crossovers indicates that one should consider classes of field theories, described by a set of parameters, rather than focus on a particular one. The space of parameters has a natural metric structure. We examine the geometry of this space in some simple models and draw some analogies between this space, superspace and minisuperspace. 
  We investigate a solution of the Weyl invariance conditions in open string theory in 4 dimensions. In the closed string sector this solution is a combination of the SU(2) Wess--Zumino--Witten model and a Liouville theory. The investigation is carried out in the $\sigma$ model approach where we have coupled all massless modes (especially an abelian gauge field via the boundary) and tachyon fields. Neglecting all higher derivatives in the field strength we get an exact result which can be interpreted as a monopole configuration living in non--trivial space time. The masses of both tachyon fields are quantized by $c_{wzw}$ and vanish for $c_{wzw} = 1$. 
  We consider a Pauli particle in a Coulomb field. The supersymmetric Hamiltonian is constructed, by explicitly giving the two supercharges $Q_{1}$ and $ Q_{2}$ in the full three-dimensional space and which together with the Hamiltonian, are shown to constitute an $S(2)$ superalgebra. This offers an alternative way of grouping the energy eigenstates into irreducible representations of this symmetry group of the Hamiltonian. 
  Recent work on the classification of conformal field theories with one primary field (the identity operator) is reviewed. The classification of such theories is an essential step in the program of classification of all rational conformal field theories, but appears impossible in general. The last managable case, central charge 24, is considered here. We found a total of 71 such theories (which have not all been constructed yet) including the monster module. The complete list of modular invariant partition functions has already appeared elsewhere \SchM. This paper contains an easily readable account of the method, as well as a few examples and some comments. (This is a compilation of contributions to workshops in Goteborg, Kiev, Corfu and Erice). 
  We prove an inequality for the Kostka-Foulkes polynomials $K_{\lambda ,\mu}(q)$. As a corollary, we obtain a nontrivial lower bound for the Kostka numbers and a new proof of the Berenstein-Zelevinsky weight-multiplicity-one-criterium. 
  We show the factorization of correlation functions of tachyon operators in 2D string theory using the discretized approach of Moore. Our demonstration of the factorization is more general than that of the paper of Sakai and Tanii. We obtain the rules for the factorization of tachyon amplitudes. Our results can be understood in terms of the operator product expansion of tachyon operators. We also give a systematic way of computing correlation functions of tachyon operators and succeed in summarizing the results of the computation in compact form for some simple cases. We confirm that these tachyon amplitudes indeed satisfy our factorization rule. 
  (some errors corrected, slightly extended) 
  We propose two new realizations of the N=4, $\hat{c}=4$ superconformal system based on the compact and non-compact versions of parafermionic algebras. The target space interpretation of these systems is given in terms of four-dimensional target spaces with non-trivial metric and topology different from the previously known four-dimensional semi-wormhole realization. The proposed $\hat{c}=4$ systems can be used as a building block to construct perturbatively stable superstring solutions with covariantized target space supersymmetry around non-trivial gravitational and dilaton backgrounds. 
  We calculate the perturbative correction to every cluster coefficient of a gas of anyons through second order in the anyon coupling constant, as described by Chern-Simons field theory. 
  I show that anomaly cancellation conditions are sufficient to determine the two most important topological numbers relevant for Calabi-Yau compactification to six dimensions. This reflects the fact that K3 is the only non-trivial CY manifold in two complex dimensions. I explicitly construct the Green-Schwarz counterterms and derive sum rules for charges of additional enhanced U(1) factors and compare the results with all possible Abelian orbifold constructions of K3. This includes asymmetric orbifolds as well, showing that it is possible to regain a geometrical interpretation for this class of models. Finally, I discuss some models with a broken $E_7$ gauge group which will be useful for more phenomenological applications. 
  We consider modifications to general relativity due to non-local string effects by using perturbation theory about the 4-dimensional Schwarzschild black hole metric. In keeping with our interpretation in previous works of black holes as quantum p-branes we investigate non-local effects due to a critical bosonic string compactified down to 4 dimensions. We show that non-local effects do not alter the spacetime topology (at least perturbatively), but they do lead to violations of the area law of black hole thermodynamics and to Hawking's first law of black hole thermodynamics. We also consider a simple analytic continuation of our perturbaive result into the non-perturbative region, which yields an ultraviolet-finite theory of quantum gravity. The Hawking temperature goes to zero in the non-perturbative region (zero string tension parameter), which is consistent with the view that Planck-size physics is quantum mechanical. 
  Using the self-dual lattice method, we make a systematic search for modular invariant partition functions of the affine algebras $g\*{(1)}$ of $g=A_2$, $A_1+A_1$, $G_2$, and $C_2$. Unlike previous computer searches, this method is necessarily complete. We succeed in finding all physical invariants for $A_2$ at levels $\le 32$, for $G_2$ at levels $\le 31$, for $C_2$ at levels $\le 26$, and for $A_1+A_1$ at levels $k_1=k_2\le 21$. This work thus completes a recent $A_2$ classification proof, where the levels $k=3,5,6,9,12,15,21$ had been left out. We also compute the dimension of the (Weyl-folded) commutant for these algebras and levels. 
  Properties of the simplest class of self-similar potentials are analyzed. Wave functions of the corresponding Schr\"odinger equation provide bases of representations of the $q$-deformed Heisenberg-Weyl algebra. When the parameter $q$ is a root of unity the functional form of the potentials can be found explicitly. The general $q^3=1$ and the particular $q^4=1$ potentials are given by the equianharmonic and (pseudo)lemniscatic Weierstrass functions respectively. 
  All classical Lie algebras can be realized \`a la Schwinger in terms of fermionic oscillators. We show that the same can be done for their $q$-deformed counterparts by simply replacing the fermionic oscillators with anyonic ones defined on a two dimensional lattice. The deformation parameter $q$ is a phase related to the anyonic statistical parameter. A crucial r\^ole in this construction is played by a sort of bosonization formula which gives the generators of the quantum algebras in terms of the underformed ones. The entire procedure works even on one dimensional chains; in such a case $q$ can also be real. 
  We review some aspects of the non-perturbative formulation of 2-dim. string theory in terms of non-relativistic fermions. We derive the bosonization using $W_\infty$ coherent states in the path-integral formulation. We discuss the classical limit and indicate the precise nature of the truncation of the full theory that leads to collective field theory. As applications we briefly discuss classical solutions reponsible for stringy non-perturbative effects and the 2-dim. black-hole. 
  We described a $q$-deformation of a quantum dynamics in one dimension. We prove that there exists only one essential deforamtion of quantum dynamics. 
  The statistics of $q$-oscillators, quons and to some extent, of anyons are studied and the basic differences among these objects are pointed out. In particular, the statistical distributions for different bosonic and fermionic $q$-oscillators are found for their corresponding Fock space representations in the case when the hamiltonian is identified with the number operator. In this case and for nonrelativistic particles, the single-particle temperature Green function is defined with $q$-deformed periodicity conditions. The equations of state for nonrelativistic and ultrarelativistic bosonic $q$-gases in an arbitrary space dimension are found near Bose statistics, as well as the one for an anyonic gas near Bose and Fermi statistics. The first corrections to the second virial coefficients are also evaluated. The phenomenon of Bose-Einstein condensation in the $q$-deformed gases is also discussed. 
  Elliptic diagonal solutions for the reflection matrices associated to the elliptic $R$ matrix of the eight vertex free fermion model are presented. They lead through the second derivative of the open chain transfer matrix to an XY hamiltonian in a magnetic field which is invariant under a quantum deformed Clifford--Hopf algebra. 
  We examine the quantization of the motion of two charged vortices in a Ginzburg--Landau theory for the fractional quantum Hall effect recently proposed by the first two authors. The system has two second-class constraints which can be implemented either in the reduced phase space or Dirac-Gupta-Bleuler formalism. Using the intrinsic formulation of statistics, we show that these two ways of implementing the constraints are inequivalent unless the vortices are quantized with conventional statistics; either fermionic or bosonic. 
  We discuss the restricted linear group in infinite dimensions modeled by the Schatten class of rank $2p=4$ which contains $(3+1)$-dimensional analog of the loop groups and is closely related to Yang-Mills theory with fermions in $(3+1)$-dimensions. We give an alternative to the construction of the ``highest weight'' representation of this group found by Mickelsson and Rajeev. Our approach is close to quantum field theory, with the elements of this group regarded as Bogoliubov transformations for fermions in an external Yang-Mills field. Though these cannot be unitarily implemented in the physically relevant representation of the fermion field algebra, we argue that they can be implemented by sesquilinear forms, and that there is a (regularized) product of forms providing an appropriate group structure. On the Lie algebra level, this gives an explicit, non-perturbative construction of fermion current algebras in $(3+1)$ space-time dimensions which explicitly shows that the ``wave function renormalization'' required for a consistent definition of the currents and their Lie bracket naturally leads to the Schwinger term identical with the Mickelsson-Rajeev cocycle. Though the explicit form of the Schwinger term is given only for the case $p=2$, our arguments apply also to the restricted linear groups modeled by Schatten classes of rank $2p=6,8,\ldots$ corresponding to current algebras in $(d+1)$- dimensions, $d=5,7,\ldots$. 
  We construct BRST operators for certain higher-spin extensions of the Virasoro algebra, in which there is a spin-$s$ gauge field on the world sheet, as well as the spin-2 gauge field corrresponding to the two-dimensional metric. We use these BRST operators to study the physical states of the associated string theories, and show how they are related to certain minimal models. 
  Usual coset construction $\SU{k}\times\SU{l}/\SU{k+l}$ of Wess--Zumino conformal field theory is presented as a coset construction of minimal models. This new coset construction can be defined rigorously and allows one to calculate easily correlation functions of a number of primary fields. 
  Recent numerical results on the fractal structure of two-dimensional quantum gravity coupled to $c=-2$ matter are reviewed. Analytic derivation of the fractal dimensions based on the Liouville theory and diffusion equation is also discussed. Excellent agreements between the numerical and theoretical results are obtained. Some problems on the non-universal nature of the fractal structure in the continuum limit are pointed out. ============================================================================ This is a review paper on \lq\lq Fractal Structure of Quantum Gravity in Two Dimension. Talk given at Nishinomiya-Yukawa Simposium, Nov. 1992. Figures are available from the author directly as hard copies. 
  We give a new interpretation and proof of the dilogarithm identities, associated to the affine Kac-Moody algebra sl(2)^, using the path description of the corresponding crystal basis. We also discuss connections with algebraic K-theory. 
  Following a suggestion made by Tseytlin, we investigate the case when one replaces the transverse part of the bosonic action by an $n=2$ supersymmetric sigma-model with a symmetric homogeneous K\"ahlerian target space. As conjectured by Tseytlin, the metric is shown to be exactly known since the beta function is known to reduce to its one-loop value. 
  We consider the Statistical Mechanics of systems of particles satisfying the $q$-commutation relations recently proposed by Greenberg and others. We show that although the commutation relations approach Bose (resp.\ Fermi) relations for $q\to1$ (resp.\ $q\to-1$), the partition functions of free gases are independent of $q$ in the range $-1<q<1$. The partition functions exhibit Gibbs' Paradox in the same way as a classical gas without a correction factor $1/N!$ for the statistical weight of the $N$-particle phase space, i.e.\ the Statistical Mechanics does not describe a material for which entropy, free energy, and particle number are extensive thermodynamical quantities. 
  We discuss topological Landau-Ginzburg theories coupled to the 2-dimensional topological gravity. We point out that the basic recursion relations for correlation functions of the 2-dimesional gravity have exactly the same form as the Gauss-Manin differential equations for the period integrals of superpotentials. Thus the one-point functions on the sphere of the Landau-Ginzburg theories are given exactly by the period integrals. We discuss various examples, A-D-E minimal models and the $c=3$ topological theories. 
  Modules over affine Lie superalgebras ${\cal G}$ are studied, in particular, for ${\cal G}=\hat{OSP(1,2)}$. It is shown that on studying Verma modules, much of the results in Kac-Moody algebra can be generalized to the super case. Of most importance are the generalized Kac-Kazhdan formula and the Malikov-Feigin-Fuchs construction, which give the weights and the explicit form of the singular vectors in the Verma module over affine Kac-Moody superalgebras. We have also considered the decomposition of the admissible representation of $\hat{OSP(1,2)}$ into that of $\hat{SL(2)}\otimes$Virasoro algebra, through which we get the modular transformations on the torus and the fusion rules. Different boundary conditions on the torus correspond to the different modings of the current superalgebra and characters or super-characters, which might be relevant to the Hamiltonian reduction resulting in Neveu-Schwarz or Ramond superconformal algebras. Finally, the Felder BRST complex, which consists of Wakimoto modules by the free field realization, is constructed. 
  The $G/G$ gauged supergroup valued WZNW theory is considered. It is shown that for $G=\OSP$, the $G/G$ theory tensoring a ($b$, $c$, $\beta$, $\gamma$) system is equivalent to the non-critical fermionic theory. The relation between integral or half integral moded affine superalgebra and its reduced theory, the NS or R superconformal algebra, is discussed in detail. The physical state space, i.e. the BRST semi-infinite cohomology, is calculated, for the $\OSP/\OSP$ theory. 
  In this note I discuss some aspects of a formulation of quantum mechanics based entirely on the Jordan algebra of observables. After reviewing some facts of the formulation in the \CS -approach I present a Jordan-algebraic Hilbert space construction (inspired by the usual GNS-construction), thereby obtaining a real Hilbert space and a (Jordan-) representation of the algebra of observables on this space. Taking the usual case as a guideline I subsequently derive a Schr\"odinger equation on this Hilbert space. 
  We clarify the notion of the DS --- generalized Drinfeld-Sokolov --- reduction approach to classical ${\cal W}$-algebras. We first strengthen an earlier theorem which showed that an $sl(2)$ embedding ${\cal S}\subset {\cal G}$ can be associated to every DS reduction. We then use the fact that a $\W$-algebra must have a quasi-primary basis to derive severe restrictions on the possible reductions corresponding to a given $sl(2)$ embedding. In the known DS reductions found to date, for which the $\W$-algebras are denoted by ${\cal W}_{\cal S}^{\cal G}$-algebras and are called canonical, the quasi-primary basis corresponds to the highest weights of the $sl(2)$. Here we find some examples of noncanonical DS reductions leading to $\W$-algebras which are direct products of ${\cal W}_{\cal S}^{\cal G}$-algebras and `free field' algebras with conformal weights $\Delta \in \{0, {1\over 2}, 1\}$. We also show that if the conformal weights of the generators of a ${\cal W}$-algebra obtained from DS reduction are nonnegative $\Delta \geq 0$ (which is 
  An experiment that would measure non--commuting quantum mechanical observables without collapsing the wave function has been recently proposed by Y Aharonov and J Anandan. These authors argue that this "protected measurement" may give indication on "the reality of the wave function". We argue that, depending of the precise version of the experiment considered, either the author's prediction is incorrect and the wave function does collapse, or the measurement is not a measurement on a quantum system. In either case, the experiment does not provide a way for measuring non--commuting observables without collapse, and it does not bear on the issue of the "reality of the wave function". 
  We study the renormalization group flow properties of the Wess-Zumino-Witten model in the region of couplings between $g^2=0$ and $g^2=4\pi/k$, by evaluating the two-loop Zamolodchikov's $c$-function. We also discuss the region of negative couplings. 
  We investigate a recently proposed model for a full quantum description of two-dimensional black hole evaporation, in which a reflecting boundary condition is imposed in the strong coupling region. It is shown that in this model each initial state is mapped to a well-defined asymptotic out-state, provided one performs a certain projection in the gravitational zero mode sector. We find that for an incoming localized energy pulse, the corresponding out-going state contains approximately thermal radiation, in accordance with semi-classical predictions. In addition, our model allows for certain acausal strong coupling effects near the singularity, that give rise to corrections to the Hawking spectrum and restore the coherence of the out-state. To an asymptotic observer these corrections appear to originate from behind the receding apparent horizon and start to influence the out-going state long before the black hole has emitted most of its mass. Finally, by putting the system in a finite box, we are able to derive some algebraic properties of the scattering matrix and prove that the final state contains all initial information. 
  A strong-weak coupling duality symmetry of the string equations of motion has been suggested in the literature. This symmetry implies that vacua occur in pairs. Since the coupling constant is a dynamical variable in string theory, tunneling solutions between strong and weak coupling vacua may exist. Such solutions would naturally lead to nonperturbative effects with anomalous coupling dependence. A highly simplified example is given. 
  We investigate $N=2$ supersymmetric sigma model orbifolds of the sphere in the large radius limit. These correspond to $N=2$ superconformal field theories. Using the equations of topological-anti-topological fusion for the topological orbifold, we compute the generalized Dynkin diagrams of these theories - i.e., the soliton spectrum - which was used in the classification of massive superconformal theories. They correspond to the extended Dynkin diagrams associated to finite subgroups of $SO(3).$ 
  The general $\sigma$-model-type string action including both massless and massive higher spins background fields is suggested. Field equations for background fields are followed from the requirement of quantum Weyl invariance. It is shown that renormalization of the theory can be produced level by level. The detailed consideration of background fields structure and corresponding fields equations is given for the first massive level of the closed bosonic string. 
  It is shown that, for any K\"ahler manifold, there exist parametrizations such that the metric takes a block-form identical to the light-cone metric introduced by Polyakov for two-dimensional gravity. Besides its possible relevence for various aspects of K\"ahlerian geometry, this fact allows us to change gauge in W gravities, and explicitly go from the conformal (Toda) gauge to the light-cone gauge using the W-geometry we proposed earlier (this will be discussed in detail in a forthcoming article). 
  The field identification problem, including fixed point resolution, is solved for the non-hermitian symmetric $N=2$ superconformal coset theories. Thereby these models are finally identified as well-defined modular invariant CFTs. As an application, the theories are used as subtheories in tensor products with $c=9$, which in turn are taken as the inner sector of heterotic superstring compactifications. All string theories of this type are classified, and the chiral ring as well as the number of massless generations and anti-generations are computed with the help of the extended Poincare polynomial. Several equivalences between a priori different non-hermitian cosets show up; in particular there is a level-rank duality for an infinite series based on $C$ type Lie algebras. Further, some general results for generic $N=2$ cosets are proven: a simple formula for the number of identification currents is found, and it is shown that the set of Ramond ground states of any $N=2$ coset model is invariant under charge conjugation. 
  We glue together two branched spheres by sewing of two Ramond (dual) two-fermion string vertices and present a rigorous analytic derivation of the closed expression for the four-fermion string vertex. This method treats all oscillator levels collectively and the obtained answer verifies that the closed form of the four vertex previously argued for on the basis of explicit results restricted to the first two oscillator levels is the correct one. 
  We address the problem of constructing the family of (4,4) theories associated with the sigma-model on a parametrized family ${\cal M}_{\zeta}$ of Asymptotically Locally Euclidean (ALE) manifolds. We rely on the ADE classification of these manifolds and on their construction as HyperK\"ahler quotients, due to Kronheimer.   So doing we are able to define the family of (4,4) theories corresponding to a ${\cal M}_{\zeta}$ family of ALE manifolds as the deformation of a solvable orbifold ${\bf C}^2 \, / \, \Gamma$ conformal field-theory, $\Gamma$ being a Kleinian group. We discuss the relation among the algebraic structure underlying the topological and metric properties of self-dual 4-manifolds and the algebraic properties of non-rational (4,4)-theories admitting an infinite spectrum of primary fields. In particular, we identify the Hirzebruch signature $\tau$ with the dimension of the local polynomial ring ${\cal R}=\o {{\bf C}[x,y,z]}{\partial W}$ associated with the ADE singularity, with the number of non-trivial conjugacy classes in the corresponding Kleinian group and with the number of short representations of the (4,4)-theory minus four. 
  A perturbatively renormalized Abelian Higgs-Kibble model with a chirally coupled fermion is considered. The Slavnov identity is fulfilled to all orders of perturbation theory, which is crucial for renormalizability in models with vector bosons. BRS invariance, i.e. the validity of the identity, forces the chiral anomaly to be cancelled by Wess-Zumino counterterms. This procedure preserves the renormalizability in the one-loop approximation but it violates the Froissart bounds for partial wave amplitudes above some energy and destroys renormalizability from the second order in h bar onwards due to the counterterms. (The paper has 3 figs. in postscript which are not included; send request to the author's e-mailbox with subject: figures . The author is willing to mail hard copies of the paper.) 
  By generalizing the Feynman proof of the Lorentz force law, recently reported by Dyson, we derive equations of motion for particles possessing internal degrees of freedom $I^a$ which do not, in general, generate a finite algebra. We obtain consistency criteria for fields which interact with such particles. It is argued that when a particle with internal $SU_q(2)$ degrees of freedom is coupled to $SU(2)$ gauge fields, $SU(2)$ gauge invariance is broken to $U(1)$. We further claim that when such an $SU_q(2)$ particle acts as a source for the field theory, the second rank antisymmetric field tensor, in general, cannot be globally defined. 
  The Askey-Wilson algebra $AW(3)$ with three generators is shown to serve as a hidden symmetry algebra underlying the Racah and (new) generalized Clebsch-Gordan problems for the quantum algebra $sl_q(2)$. On the base of this hidden symmetry a simple method to calculate corresponding coefficients in terms of the Askey-Wilson polynomials is proposed. 
  The connection between supersymmetric quantum mechanics and the Korteweg- de Vries (KdV) equation is discussed, with particular emphasis on the KdV conservation laws. It is shown that supersymmetric quantum mechanics aids in the derivation of the conservation laws, and gives some insight into the Miura transformation that converts the KdV equation into the modified KdV equation. The construction of the $\tau$-function by means of supersymmetric quantum mechanics is discussed. 
  I briefly review the properties of classical affine Toda field theories and indicate how some of this features survive in the quantum theory on-shell. I demonstrate how this knowledge can be extended off-shell, i.e. how to compute correlation functions for completely integrable models via the form factor approach. For the latter I present an axiomatic system and two explicit computation (the Sinh-Gordon theory and the Bullough-Dodd model) which provide a consistent solution of it. (Talk presented at the VII Andr\'e Swieca Summer School, Campos do Jord\~ao, Brasil, 1993) 
  We formulate a general set of consistency requirements, which are expected to be satisfied by the scattering matrices in the presence of reflecting boundaries. In particular we derive an equivalent to the boostrap equation involving the W-matrix, which encodes the reflection of a particle off a wall. This set of equations is sufficient to derive explicit formulas for $W$, which we illustrate in the case of some particular affine Toda field theories. 
  We give a complete analysis of the projective unitary irreducible representations of the Poincar\'e group in 1+2 dimensions applying Mackey theorem and using an explicit formula for the universal covering group of the Lorentz group in 1+2 dimensions. We provide explicit formulae for all representations. 
  Let $U_q(\hat{\cal G})$ be a quantized affine Lie algebra. It is proven that the universal R-matrix $R$ of $U_q(\hat{\cal G})$ satisfies the celebrated conjugation relation $R^\dagger=TR$ with $T$ the usual twist map. As applications, braid generators are shown to be diagonalizable on arbitrary tensor product modules of integrable irreducible highest weight $U_q(\hat{\cal G})$-module and a spectral decomposition formula for the braid generators is obtained which is the generalization of Reshetikhin's and Gould's forms to the present affine case. Casimir invariants are constructed and their eigenvalues computed by means of the spectral decomposition formula. As a by-product, an interesting identity is found. 
  Casimir invariants for quantized affine Lie algebras are constructed and their eigenvalues computed in any irreducible highest weight representation. 
  A bilocal field theory having M\"{o}bius gauge invariance is proposed. In four dimensions there exists a zero momentum state of the first quantized model, which belongs to a non-trivial BRS cohomology class. A field theory lagrangian having a gauge invariance only in four dimensions is constructed. 
  In these lectures I discuss various unsolved problems of string theory and their relations to quantum gravity, 3d Ising model, large N QCD, and quantum cosmology. No solutions are presented but some new and perhaps useful approaches are suggested. 
  We show that it is possible to measure Schrodinger wave of a single quantum system. This provides a strong argument for associating physical reality with the quantum state of a single system, and challenges the usual assumption that the quantum state has physical meaning only for an ensemble of identical systems. 
  Black hole formation/evaporation in two-dimensional dilaton gravity can be described, in the limit where the number $N$ of matter fields becomes large, by a set of second-order partial differential equations. In this paper we solve these equations numerically. It is shown that, contrary to some previous suggestions, black holes evaporate completely a finite time after formation. A boundary condition is required to evolve the system beyond the naked singularity at the evaporation endpoint. It is argued that this may be naturally chosen so as to restore the system to the vacuum. The analysis also applies to the low-energy scattering of $S$-wave fermions by four-dimensional extremal, magnetic, dilatonic black holes. 
  The affine Toda field theories based on the non simply-laced Lie algebras are discussed. By rewriting the S-matrix formulae found by Delius et al, a universal form for the coupling-constant dependence of these models is obtained, and related to various general properties of the classical couplings. This is illustrated via the S-matrix associated with the dual pair of algebras $f_4^{(1)}$ and $e_6^{(2)}$. 
  We obtain new fermionic sum representations for the Virasoro characters of the confromal field theory describing the ferromagnetic three-state Potts spin chain. These arise from the fermionic quasi-particle excitations derived from the Bethe equations for the eigenvalues of the hamiltonian. In the conformal scaling limit, the Bethe equations provide a description of the spectrum in terms of one genuine quasi-particle, and two ``ghost'' excitations with a limited microscopic momentum range. This description is reflected in the structure of the character formulas, and suggests a connection with the integrable perturbation of dimensions (2/3,2/3)$^+$ which breaks the $S_3$ symmetry of the conformal field theory down to $Z_2$. 
  We do a semiclassical analysis for two or three spins which are coupled antiferromagnetically to each other. The semiclassical wave functions transform correctly under permutations of the spins if one takes into account the Wess-Zumino term present in the path integral for spins. The Wess-Zumino term here is a total derivative which has no effect on the energy spectrum. The semiclassical problem is related to that of anyons moving on a sphere with the statistics parameter $\theta$ being $2 \pi S$ for two spins and $3 \pi S$ for three spins. Finally, we present a novel way of deriving the semiclassical wave functions from the spin wave functions. 
  The gauge equivalence between basic KP hierarchies is discussed. The first two Hamiltonian structures for KP hierarchies leading to the linear and non-linear $\Winf$ algebras are derived. The realization of the corresponding generators in terms of two boson currents is presented and it is shown to be related to many integrable models which are bi-Hamiltonian. We can also realize those generators by adding extra currents, coupled in a particular way, allowing for instance a description of multi-layered Benney equations or multi-component non-linear Schroedinger equation. In this case we can have a second Hamiltonian bracket structure which violates Jacobi identity. We consider the reduction to one-boson systems leading to KdV and mKdV hierarchies. A Miura transformation relating these two hierarchies is obtained by restricting gauge transformation between corresponding two-boson hierarchies. Connection to Drinfeld-Sokolov approach is also discussed in the $SL(2,\IR)$ gauge theory. (Lectures presented at the VII J.A. Swieca Summer School, Section: Particles and Fields, Campos do Jord\~ao - Brasil - January/93) 
  We study how the effects of quantum corrections lead to notions of irreversibility and clustering in quantum field theory. In particular, we consider the virtual ``charge" distribution generated by quantum corrections and adopt for it a statistical interpretation. Then, this virtual charge is shown to ($a$) describe a system where the equilibrium state is at its classical limit ($\hbar \rightarrow 0$), ($b$) give rise to spatial diffusion of the virtual cloud that decays as the classical limit is approached and ($c$) lead to a scenario where clustering takes place due to quantum dynamics, and a natural transition from a ``fractal" to a homogeneous regime occurs as distances increase. 
  It is frequently useful to construct dual descriptions of theories containing antisymmetric tensor fields by introducing a new potential whose curl gives the dual field strength, thereby interchanging field equations with Bianchi identities. We describe a general procedure for constructing actions containing both potentials at the same time, such that the dual relationship of the field strengths arises as an equation of motion. The price for doing this is the sacrifice of manifest Lorentz invariance or general coordinate invariance, though both symmetries can be realized nonetheless. There are various examples of global symmetries that have been realized as symmetries of field equations but not actions. These can be elevated to symmetries of the action by our method. The main example that we focus on is the low-energy effective action description of the heterotic string theory compactified on a six-torus to four dimensions. We show that the SL(2,R) symmetry, whose SL(2,Z) subgroup has been conjectured to be an exact symmetry of the full string theory, can be realized on the action in a way that brings out a remarkable similarity to the target space duality symmetry O(6,22). Our analysis indicates that SL(2,Z) symmetry may arise naturally in a dual formulation of the theory. 
  We discuss solutions of the heterotic string theory which are analogous to bosonic and superstring backgrounds related to coset conformal field theories. A class of exact `left-right symmetric' solutions is obtained by supplementing the metric, antisymmetric tensor and dilaton of the superstring solutions by the gauge field background equal to the generalised Lorentz connection with torsion. As in the superstring case, these backgrounds are $\a'$-independent, i.e. have a `semiclassical' form. The corresponding heterotic string sigma model is obtained from the combination of the (1,0) supersymmetric gauged WZNW action with the action of internal fermions coupled to the target space gauge field. The pure (1,0) supersymmetric gauged WZNW theory is anomalous and does not describe a consistent heterotic string solution. We also find (to the order $\alpha'^3$) a two-dimensional perturbative heterotic string solution with the trivial gauge field background. To the leading order in $\alpha'$ it coincides with the known $SL(2,R)/U(1)$ bosonic or superstring solutions. This solution does not correspond to a `heterotic' combination of the left superstring and right bosonic $L_0$-operators at the conformal field theory level. Some duality properties of the heterotic string solutions are studied. 
  We present a definition of the non-abelian generalisations of affine Toda theory related from the outset to vertex operator constructions of the corresponding Kac-Moody algebra $\gh$. Reuslts concerning conjugacy classes of the Weyl group of the finite Lie algebra $\fing$ to embeddings of $A_1$ in $\fing$ are used both to present the theories, and to elucidate their soliton spectrum. We confirm the conjecture of \cite{OSU93} for the soliton specialisation of the Leznov-Saveliev solution. The energy-momentum tensor of such theories is shown to split into a total derivative part and a part dependent only on the free fields which appear in the general solution, and vanish for the soliton solutions. Analogues are provided of the results known for the classical solitons of abelian Toda theories. 
  The gravitating matter is studied within the framework of the non-commutative geometry. The non-commutative Einstein-Hilbert action on the product of a four dimensional manifold with a discrete space gives the models of matter fields coupled to the standard Einstein gravity.The matter multiplet is encoded in the Dirac operator which yields the representation of the algebra of the universal forms. The general form of the Dirac operator depends on a choice of the grading of the corresponding spinor bundle. A choice is given, which leads to the nonlinear vector sigma-model coupled to the Einstein gravity. 
  We consider the one-dimensional random field Ising model, where the spin-spin coupling, $J$, is ferromagnetic and the external field is chosen to be $+h$ with probability $p$ and $-h$ with probability $1-p$. At zero temperature, we calculate an exact expression for the correlation length of the quenched average of the correlation function $\langle s_0 s_n \rangle - \langle s_0 \rangle \langle s_n \rangle$ in the case that $2J/h$ is not an integer. The result is a discontinuous function of $2J/h$. When $p = {1 \over 2}$, we also place a bound on the correlation length of the quenched average of the correlation function $\langle s_0 s_n \rangle$. 
  An ADM mass formula is derived for a wide class of black solutions with certain spherical symmetry. By applying this formula, we calculate the ADM masses for recently discovered black strings and $p$-branes in diverse dimensions. By this, the Bogolmol'nyi equation can be shown to hold explicitly. A useful observation is made for non-extremal black $p$-branes that only for $p = 0$, i.e. for a black hole, can its ADM mass be read directly from the asymptotic behaviour of the metric component $g_{00}$. 
  The first ``Convegno Informale su Quantum Groups'' was held in Florence from February 3 to 6, 1993. This Convegno was conceived as an informal meeting to bring together all the italian people working in the field of quantum groups and related topics. We are very happy indeed that about 30 theoretical physicists decided to take part presenting many aspects of this interesting and live subject of research. We thank all the participants for the stimulating and nice atmosphere that has characterized the meeting. This paper has the intent to give a quick review in english of the contributions and related references. We think useful to include the complete addresses and coordinate data of the participants. It is our intention to diffuse these proceedings by e-mail trough electronic data banks. 
  The Cartan-Maurer equations for any $q$-group of the $A_{n-1}, B_n, C_n, D_n$ series are given in a convenient form, which allows their direct computation and clarifies their connection with the $q=1$ case. These equations, defining the field strengths, are essential in the construction of $q$-deformed gauge theories. An explicit expression $\om ^i\we \om^j= -\Z {ij}{kl}\om ^k\we \om^l$ for the $q$-commutations of left-invariant one-forms is found, with $\Z{ij}{kl} \om^k \we \om^l \qonelim \om^j\we\om^i$. 
  The induced lattice gauge theory with various types of inducing fields in fundamental representation of $SU(N_{c})$ is considered.   In a simple case of one-plaquette lattice the model is solved in the large $N_{c}$ limit by means of loop equations.   Comparison with the solution of usual QCD shows the equivalence of induced and Wilson QCD providing that a mass and a number of flavours of inducing fields are sufficiently large. The possibility to take an asymptotically free continuum limit of induced QCD is discussed. 
  Actions for $D=2$, $N=2$ supergravity coupled to a scalar field are calculated, and it is shown that the most general power-counting renormalizable dilaton gravity action has an $N=2$ locally supersymmetric extension. The presence of chiral terms in the action leads one to hope that non-renormalization theorems similar to those in global SUSY will apply; this would eliminate some of the renormalization ambiguities which plague ordinary bosonic (and $N=1$) dilaton gravity. To investigate this, the model is studied in superconformal gauge, where it is found that one chiral term becomes nonchiral, so that only one term is safe from renormalization. 
  We show that the Hilbert space basis that defines the Ponzano-Regge- Turaev-Viro-Ooguri combinatorial definition of 3-d Quantum Gravity is the same as the one that defines the Loop Representation. We show how to compute lengths in Witten's 3-d gravity and how to reconstruct the 2-d geometry from a state of Witten's theory. We show that the non-degenerate geometries are contained in the Witten's Hilbert space. We sketch an extension of the combinatorial construction to the physical 4-d case, by defining a modification of Regge calculus in which areas, rather than lengths, are taken as independent variables. We provide an expression for the scalar product in the Loop representation in 4-d. We discuss the general form of a nonperturbative quantum theory of gravity, and argue that it should be given by a generalization of Atiyah's topological quantum field theories axioms. 
  In this paper the evolution of a quantum system drived by a non-Hermitian Hamiltonian depending on slowly-changing parameters is studied by building an universal high-order adiabatic approximation(HOAA) method with Berry's phase ,which is valid for either the Hermitian or the non-Hermitian cases. This method can be regarded as a non-trivial generalization of the HOAA method for closed quantum system presented by this author before. In a general situation, the probabilities of adiabatic decay and non-adiabatic transitions are explicitly obtained for the evolution of the non-Hermitian quantum system. It is also shown that the non-Hermitian analog of the Berry's phase factor for the non-Hermitian case just enjoys the holonomy structure of the dual linear bundle over the parameter manifold. The non-Hermitian evolution of the generalized forced harmonic oscillator is discussed as an illustrative examples. 
  It is shown that it is possible to measure the Schr\"odinger wave of a single quantum system. This provides a strong argument for associating physical reality with a quantum state of a single system in sharp contrast with the usual approach in which the physical meaning of a quantum state is related only to an ensemble of identical systems. An apparent paradox between measurability of a quantum state of a single system and the relativistic causality is resolved. 
  A novel manifestation of nonlocality of quantum mechanics is presented. It is shown that it is possible to ascertain the existence of an object in a given region of space without interacting with it. The method might have practical applications for delicate quantum experiments. 
  We solve the equations of motion of the tachyon and the discrete states in the background of Witten's semiclassical black hole and in the exact 2D dilaton-graviton background of Dijkgraaf et al. We find the exact solutions for weak fields, leading to conclusions in disagreement with previous studies of tachyons in the black hole. Demanding that a state in the black hole be well behaved at the horizon implies that it must tend asymptotically to a combination of a Seiberg and an anti-Seiberg c=1 state. For such a state to be well behaved asymptotically, it must satisfy the condition that neither its Seiberg nor its anti-Seiberg Liouville momentum is positive. Thus, although the free-field BRST cohomologies of the underlying SL(2,R)/U(1) theory is the same as that of a c=1 theory, the black hole spectrum is drastically truncated: THERE ARE NO W_INFINITY STATES, AND ONLY TACHYONS WITH X-MOMENTA | P_TACH | <= | M_TACH | ARE ALLOWED. In the Minkowski case only the static tachyon is allowed. The black hole is stable to the back reaction of these remaining tachyons, so they are good perturbations of the black hole, or ``hair''. However, this leaves only 3 tachyonic hairs in the black hole and 7 in the exact solution! Such sparse hair is clearly irrelevant to the maintenance of coherence during black hole evaporation. 
  The QED effective action at finite temperature and density is calculated to all orders in an external homogeneous and time-independent magnetic field in the weak coupling limit. The free energy, obtained explicitly, exhibit the expected de\ Haas -- van\ Alphen oscillations. An effective coupling at finite temperature and density is derived in a closed form and is compared with renormalization group results. 
  We discuss the integrable hierarchies that appear in multi--matrix models. They can be envisaged as multi--field representations of the KP hierarchy. We then study the possible reductions of this systems via the Dirac reduction method by suppressing successively one by one part of the fields. We find in this way new integrable hierarchies, of which we are able to write the Lax pair representations by means of suitable Drinfeld--Sokolov linear systems. At the bottom of each reduction procedure we find an $N$--th KdV hierarchy. We discuss in detail the case which leads to the KdV hierarchy and to the Boussinesque hierarchy, as well as the general case in the dispersionless limit. 
  A 3d generally covariant field theory having some unusual properties is described. The theory has a degenerate 3-metric which effectively makes it a 2d field theory in disguise. For 2-manifolds without boundary, it has an infinite number of conserved charges that are associated with graphs in two dimensions and the Poisson algebra of the charges is closed. For 2-manifolds with boundary there are additional observables that have a Kac-Moody Poisson algebra. It is further shown that the theory is classically integrable and the general solution of the equations of motion is given. The quantum theory is described using Dirac quantization, and it is shown that there are quantum states associated with graphs in two dimensions. 
  Recent results on the vacuum polarization induced by a thin string of magnetic flux lead us to suggest an analogue of the Copenhagen `flux spaghetti' QCD vacuum as a possible mechanism for avoiding the divergence of perturbative QED, thus permitting consistent completion of the full, nonperturbative theory. The mechanism appears to operate for spinor, but not scalar, QED. 
  The renormalization group method is employed to study the effective potential in curved spacetime with torsion. The renormalization-group improved effective potential corresponding to a massless gauge theory in such a spacetime is found and in this way a generalization of Coleman-Weinberg's approach corresponding to flat space is obtained. A method which works with the renormalization group equation for two-loop effective potential calculations in torsionful spacetime is developed. The effective potential for the conformal factor in the conformal dynamics of quantum gravity with torsion is thereby calculated explicitly. Finally, torsion-induced phase transitions are discussed. 
  A geometric global formulation of the higher-order Lagrangian formalism for systems with finite number of degrees of freedom is provided. The formalism is applied to the study of systems with groups of Noetherian symmetries. 
  We give a derivation of the Verlinde formula for the $G_{k}$ WZW model from Chern-Simons theory, without taking recourse to CFT, by calculating explicitly the partition function $Z_{\Sigma\times S^{1}}$ of $\Sigma\times S^{1}$ with an arbitrary number of labelled punctures. By a suitable gauge choice, $Z_{\Sigma\times S^{1}}$ is reduced to the partition function of an Abelian topological field theory on $\Sigma$ (a deformation of non-Abelian BF and Yang-Mills theory) whose evaluation is straightforward. This relates the Verlinde formula to the Ray-Singer torsion of $\Sigma\times S^{1}$. We derive the $G_{k}/G_{k}$ model from Chern-Simons theory, proving their equivalence, and give an alternative derivation of the Verlinde formula by calculating the $G_{k}/G_{k}$ path integral via a functional version of the Weyl integral formula. From this point of view the Verlinde formula arises from the corresponding Jacobian, the Weyl determinant. Also, a novel derivation of the shift $k\ra k+h$ is given, based on the index of the twisted Dolbeault complex. 
  We calculate perturbatively the pressure of a dilute gas of anyons through second order in the anyon coupling constant, as described by Chern-Simons field theory. Near Bose statistics , the divergences in the perturbative expansion are exactly cancelled by a two-body $\delta$-function potential which is not required near Fermi statistics. To the order considered, we find no need for a non-hermitian Hamiltonian. (This paper precedes the article ''Three loop calculation of the full anyonic partition function'', by R. Emparan and M. Valle Basagoiti, hep-th/9304103) 
  We write down the anomalous dimensions of the fields of the Nambu--Jona-Lasinio model or chiral Gross Neveu model with a continuous global chiral symmetry for the two cases $U(1)$ $\times$ $U(1)$ and $SU(M)$ $\times$ $SU(M)$ at $O(1/N^2)$ in a $1/N$ expansion. 
  In this paper, we examine the analogy between topological string theory and equivariant cohomology. We also show that the equivariant cohomology of a topological conformal field theory carries a certain algebraic structure, which we call a gravity algebra. (Error on page 9 corrected: BRS current contains total derivatives.) 
  Unitarily implementable Bogoliubov transformations for charged, relativistic bos\-ons and fermions are discussed, and explicit formulas for the 2-cocycles appearing in the group product of their implementers are derived. In the fermion case this provides a simple field theoretic derivation of the well-known cocycle of the group of unitary Hilbert space operators modeled on the Hilbert Schmidt class and closely related to the loop groups. In the boson case the cocycle is obtained for a similar group of pseudo-unitary (symplectic) operators. I also derive explcite formulas for the phases of one-parameter groups of implementers and, more generally, families of implementers which are unitary propagators with parameter dependent generators. 
  We introduce a ``pre-Bethe-Ansatz'' system of equations for three dimensional vertex models. We bring to the light various algebraic curves of high genus and discuss some situations where these curves simplify. As a result we describe remarkable subvarieties of the space of parameters. 
  A general formula for the entropy of stationary black holes in Lovelock gravity theories is obtained by integrating the first law of black hole mechanics, which is derived by Hamiltonian methods. The entropy is not simply one quarter of the surface area of the horizon, but also includes a sum of intrinsic curvature invariants integrated over a cross section of the horizon. 
  A representation of the quantum affine algebra $U_{q}(\widehat{sl}_3)$ of an arbitrary level $k$ is constructed in the Fock module of eight boson fields. This realization reduces the Wakimoto representation in the $q \rightarrow 1$ limit. The analogues of the screening currents are also obtained. They commute with the action of $U_{q}(\widehat{sl}_3)$ modulo total differences of some fields. 
  A new approach is provided to determine the dilaton--antisymmetric tensor coupling in a supergravity theory by considering the static supersymmetric field configuration around a super extended object, which is consistently formulated in a curved superspace. By this, the corresponding SUSY transformation rules can also be determined for vanishing fermionic fields as well as bosonic fields other than those in the determined coupling. Therefore, we can, in turn, use this determined part of the supergravity theory to study all the related vacuum-like solutions. We have determined the dilaton--antisymmetric tensor couplings, in which each of the antisymmetric tensors is a singlet of the automorphism group of the corresponding superalgebra, for every supergravity multiplet. This actually happens only for $N \leq 2$ supergravity theories, which agrees completely with the spin-content analysis and the classified $N \leq 2$ super $p$-branes, therefore giving more support to the existence of the fundamental Type II $p$-branes. A prediction is made of the $D = 9, N = 2$ supergravity which has not yet been written down so far. 
  We give the description of discretized moduli spaces (d.m.s.) $\Mcdisc$ introduced in \cite{Ch1} in terms of a discrete de Rham cohomologies for each moduli space $\Mgn$ of a genus $g$, $n$ being the number of punctures. We demonstrate that intersection indices (cohomological classes) calculated for d.m.s. coincide with the ones for the continuum moduli space $\Mc$ compactified by Deligne and Mumford procedure. To show it we use a matrix model technique. The Kontsevich matrix model is a generating function for these indices in the continuum case, and the matrix model with the potential $N\alpha \tr {\bigl(- \fr 12 \L X\L X +\log (1-X)+X\bigr)}$ is the one for d.m.s. In the last case the effects of reductions become relevant, but we use the stratification procedure in order to express integrals over open spaces $\Mdisc$ in terms of intersection indices which are to be calculated on compactified spaces. The coincidence of the cohomological classes for both continuum and d.m.s. models enables us to propose the existence of a quantum group structure on d.m.s. Then d.m.s. are nothing but cyclic (exceptional) representations of a quantum group related to a moduli space $\Mc$. Considering the explicit expressions for integrals of Chern classes over $\Mc$ and $\Mcdisc$ we conjecture that each moduli space $\Mc$ in the Kontsevich parametrization can be presented as a coset $\Mc ={\bf T}^d/G$, $d=3g-3+n$, where ${\bf T}^d$ is some $d$--dimensional complex torus and $G$ is a finite order symmetry group of ${\bf T}^d$. 
  In this paper we show that a relativistic membrane admits an equivalent representation in terms of the Kalb-Ramond gauge field $F_{\mu\nu\rho}=\partial_{\,[\,\mu}B_{\nu\rho]}$ encountered in string theory. By `` equivalence '' we mean the following: if $x=X(\xi)$ is a solution of the classical equations of motion derived from the Dirac-Nambu-Goto action, then it is always possible to find a differential form of {\it rank three}, satisfying Maxwell-type equations. The converse proposition is also true. In the first part of the paper, we show that a relativistic membrane, regarded as a mechanical system, admits a Hamilton-Jacobi formulation in which the H-J function describing a family of classical membrane histories is given by $\displaystyle{F=dB=dS^1\wedge dS^2\wedge dS^3}$. In the second part of the paper, we introduce a {\it new} lagrangian of the Kalb-Ramond type which provides a {\it first order} formulation for both open and closed membranes. Finally, for completeness, we show that such a correspondence can be established in the very general case of a p-brane coupled to gravity in a spacetime of arbitrary dimensionality. 
  We explicitly show that the Landau gauge supersymmetry of Chern-Simons theory does not have any physical significance. In fact, the difference between an effective action both BRS invariant and Landau supersymmetric and an effective action only BRS invariant is a finite field redefinition. Having established this, we use a BRS invariant regulator that defines CS theory as the large mass limit of topologically massive Yang-Mills theory to discuss the shift $k \to k+\cv$ of the bare Chern-Simons parameter $k$ in conncection with the Landau supersymmetry. Finally, to convince ourselves that the shift above is not an accident of our regularization method, we comment on the fact that all BRS invariant regulators used as yet yield the same value for the shift. 
  Revision contains rewording of selected text 
  A detailed reexamination is made of the exact operator formalism of two-dimensional Liouville quantum gravity in Minkowski spacetime with the cosmological term fully taken into account.   Making use of the canonical mapping from the interacting Liouville field into a free field, we focus on the problem of how the Liouville exponential operator should be properly defined. In particular, the condition of mutual locality among the exponential operators is carefully analyzed, and a new solution, which is neither smoothly connected nor relatively local to the existing solution, is found. Our analysis indicates that, in Minkowski spacetime, coupling gravity to matter with central charge $d<1$ is problematical. For $d=1$, our new solution appears to be the appropriate one; for this value of $d$, we demonstrate that the operator equation of motion is satisfied to all orders in the cosmological constant with a certain regularization. As an application of the formalism, an attempt is made to study how the basic generators of the ground ring get modified due to the inclusion of the cosmological term. Our investigation, although incomplete, suggests that in terms of the canonically mapped free field the ground ring is not modified. 
  We present an algorithm for the construction of the branching functions in the vacuum sector for affine Lie algebras based on the string hypothesis solution to a system of Bethe equations for generalized RSOS models. We also mention how the ground state structure and features of the excitation spectra like the Brillouin zone schemes of these models (and those in the same universality classes) can be extracted from combinatoric arguments and encoded in Lie algebraic terms. 
  We study the cosmological solutions of the one loop corrected superstring effective action, in a Friedmann-Robertson-Walker background, and in the presence of the dilaton and modulus fields. A particularly interesting class of solutions is found which avoid the initial singularity and are consistent with the perturbative treatment of the effective action. 
  In these introductory notes I explain some basic ideas in string field theory. These include: the concept of a string field, the issue of background independence, the reason why minimal area metrics solve the problem of generating all Riemann surfaces with vertices and propagators, and how Batalin-Vilkovisky structures arise from the state spaces of conformal field theories including ghosts. More advanced topics and recent developments are summarized. (To appear in the proceedings of the 1992 Summer School at Les Houches.)} 
  The isomonodromic deformations underlying the Painlev\'e transcendants are interpreted as nonautonomous Hamiltonian systems in the dual $\gR^*$ of a loop algebra $\tilde\grg$ in the classical $R$-matrix framework. It is shown how canonical coordinates on symplectic vector spaces of dimensions four or six parametrize certain rational coadjoint orbits in $\gR^*$ via a moment map embedding. The Hamiltonians underlying the Painlev\'e transcendants are obtained by pulling back elements of the ring of spectral invariants. These are shown to determine simple Hamiltonian systems within the underlying symplectic vector space. 
  The coupling of topological matter to topological Yang-Mills theory in four dimensions is considered and a model is presented. It is shown that, contrary to the two-dimensional case, this coupling leads to a breaking of the topological symmetry. This means that the vacuum expectation values of the observables of the theory loose their invariance under small deformations of the metric while the action of the model possesses all the symmetries corresponding to the case with no coupling. 
  Multiloop superstring amplitudes are calculated in the explicit form by the solution of Ward identities. A naive generalization of Belavin-Knizhnik theorem to the superstring is found to be incorrect since the period matrix turns out to be depended on the spinor structure over the terms proportional to odd moduli.   These terms appear because fermions mix bosons under the two-dim. supersymmetry transformations. The closed, oriented superstring turns out to be finite, if it possesses the ten-dimensional supersymmetry, as well as the two-dimentional one.   This problem needs a further study. 
  Static black holes in two-dimensional string theory can carry tachyon hair. Configurations which are non-singular at the event horizon have non-vanishing asymptotic energy density. Such solutions can be smoothly extended through the event horizon and have non-vanishing energy flux emerging from the past singularity. Dynamical processes will not change the amount of tachyon hair on a black hole. In particular, there will be no tachyon hair on a black hole formed in gravitational collapse if the initial geometry is the linear dilaton vacuum. There also exist static solutions with finite total energy, which have singular event horizons. Simple dynamical arguments suggest that black holes formed in gravitational collapse will not have tachyon hair of this type. 
  We study by non perturbative techniques a vector, axial--vector theory characterized by a parameter which interpolates between pure vector and chiral Schwinger models. Main results are two windows in the space of parameters which exhibit acceptable solutions. In the first window we find a free massive and a free massless bosonic excitations and interacting left--right fermions endowed with asymptotic \hbox{states}, which feel however a long range interaction. In the second window the massless bosonic excitation is a negative norm state which can be consistently expunged from the ``physical" Hilbert space; fermions are confined. An intriguing feature of our model occurs in the first window where we find that fermionic correlators scale at both short and long distances, but with different critical exponents. The infrared limit in the fermionic sector is nothing but a dynamically generated massless Thirring model. 
  A method to obtain explicit and complete topological solution of SU(2) Chern-Simons theory on $S^3$ is developed. To this effect the necessary aspects of the theory of coloured-oriented braids and duality properties of conformal blocks for the correlators of $SU(2)_k$ Wess-Zumino conformal field theory are presented. A large class of representations of the generators of the groupoid of coloured-oriented braids are obtained. These provide a whole lot of new link invariants of which Jones polynomials are the simplest examples. These new invariants are explicitly calculated as illustrations for knots upto eight crossings and two-component multicoloured links upto seven crossings. 
  (Revised LaTex version).The structure of free field representation and some correlation functions of the SU(3) CFT are considered. 
  Carrying out perturbations around a lattice topological field theory in two dimensions, we show that it is on a first order phase transition fixed point with multiplicity ${n(n-1)/2}$, where $n$ is the number of its independent physical observables. We discuss about the order parameters and the finite size effect for the free energy. The finite size effect is described by the topological field theory. We investigate also the renormalization group flow near the fixed point, and show that the flow agrees with that of the Nienhuis-Nauenberg criterion. 
  We study a model based on $N$ scalar complex fields coupled to a scalar real field, where all fields are treated classically as c-numbers. The model describes a composite particle made up of $N$ constituents with bare mass $m_0$ interacting both with each other and with themselves via the exchange of a particle of mass $\mu_0$. The stationary states of the composite particle are described by relativistic Hartree's equations. Since the self-interaction is included, the case of an elementary particle is a nontrivial special case of this model. Using an integral transform method we derive the exact ground state solution and prove its local stability. The mass of the composite particle is calculated as the total energy in the rest frame. For the case of a massless exchange particle the mass formula is given in closed form. The mass, as a function of the coupling constant, possesses a well pronounced minimum for each value of $\mu_0/m_0$, while the absolute minimum occurs at $\mu_0=0$. 
  We define new quantizations of the Heisenberg group by introducing new quantizations in the universal enveloping algebra of its Lie algebra. Matrix coefficients of the Stone--von Neumann representation are preserved by these new multiplications on the algebra of functions on the Heisenberg group. Some of the new quantizations provide also a new multiplication in the algebra of theta functions; we obtain in this way Sklyanin algebras. 
  We give a direct proof of the new "product" expression for the Ramond sector characters of N=2 minimal models recently suggested by E. Witten. Our construction allows us to generalize these expressions to the D and E series of N=2 minimal models, as well as to other N=2 Kazama--Suzuki coset models such as $SU(N)\times SO(2(N-1))/SU(N-1)\times U(1)$. We verify that these expressions indeed coincide with the corresponding Landau--Ginzburg "elliptic genus", a certain topologically invariant twisted path integral with the effective Landau--Ginzburg action, which we obtain by using Witten's method. We indicate how our approach may be used to construct (or rule out) possible Landau--Ginzburg potentials for describing other N=2 superconformal theories. 
  For a class of first order gauge theories it was shown that the proper solution of the BV-master equation can be obtained straightforwardly. Here we present the general condition which the gauge generators should satisfy to conclude that this construction is relevant. The general procedure is illustrated by its application to the Chern-Simons theory in any odd-dimension. Moreover, it is shown that this formalism is also applicable to BRST field theories, when one replaces the role of the exterior derivative with the BRST charge of first quantization. 
  It is shown that the two dimensional gravity, described either in the conformal gauge (the Liouville theory) or in the light cone gauge, when coupled to matter possesses an infinite number of twisted $N=2$ superconformal symmetries. The central charges of the $N=2$ algebra for the two gauge choices are in general different. Further, it is argued that the physical states in the light cone gauge theory can be obtained from the Liouville theory by a field redefinition. 
  Hawking's 1974 calculation of thermal emission from a classical black hole led to his 1976 proposal that information may be lost from our universe as a pure quantum state collapses gravitationally into a black hole, which then evaporates completely into a mixed state of thermal radiation. Another possibility is that the information is not lost, but is stored in a remnant of the evaporating black hole. A third idea is that the information comes out in nonthermal correlations within the Hawking radiation, which would be expected to occur at too slow a rate, or be too spread out, to be revealed by any nonperturbative calculation. 
  We consider the spherical limit of multi-matrix models on regular target graphs, for instance single or multiple Potts models, or lattices of arbitrary dimension. We show, to all orders in the low temperature expansion, that when the degree of the target graph $\Delta\to\infty$, the free energy becomes independent of the target graph, up to simple transformations of the matter coupling constant. Furthermore, this universal free energy contains contributions only from those surfaces which are made up of ``baby universes'' glued together into trees, all non-universal and non-tree contributions being suppressed by inverse powers of $\Delta$. Each order of the free energy is put into a simple, algebraic form. 
  $S$-matrices associated to the vector representations of the quantum groups for the classical Lie algebras are constructed. For the $a_{m-1}$ and $c_m$ algebras the complete $S$-matrix is found by an application of the bootstrap equations. It is shown that the simplest form for the $S$-matrix which generalizes that of the Gross-Neveu model is not consistent for the non-simply-laced algebras due to the existence of unexplained singularities on the physical strip. However, a form which generalizes the $S$-matrix of the principal chiral model is shown to be consistent via an argument which uses a novel application of the Coleman-Thun mechanism. The analysis also gives a correct description of the analytic structure of the $S$-matrix of the principle chiral model for $c_m$. 
  Chiral vertex-operators are defined for continuous quantum-group spins $J$ from free-field realizations of the Coulomb-gas type. It is shown that these generalized chiral vertex operators satisfy closed braiding relations on the unit circle, which are given by an extension in terms of orthogonal polynomials of the braiding matrix recently derived by Cremmer, Gervais and Roussel. This leads to a natural extension of the Liouville exponentials to continuous powers that remain local. 
  We give a definition for the notion of statistics in the lattice-theoretical (or propositional) formulation of quantum mechanics of Birchoff, von Neumann and Piron. We show that this formalism is compatible only with two types of statistics: Bose-Einstein and Fermi-Dirac. Some comments are made about the connection between this result and the existence of exotic statistics (para-statistics, infinite statistics, braid statistics). 
  The notion of a measure on the space of connections modulo gauge transformations that is invariant under diffeomorphisms of the base manifold is important in a variety of contexts in mathematical physics and topology. At the formal level, an example of such a measure is given by the Chern-Simons path integral. Certain measures of this sort also play the role of states in quantum gravity in Ashtekar's formalism. These measures define link invariants, or more generally multiloop invariants; as noted by Witten, the Chern-Simons path integral gives rise to the Jones polynomial, while in quantum gravity this observation is the basis of the loop representation due to Rovelli and Smolin. Here we review recent work on making these ideas mathematically rigorous, and give a rigorous construction of diffeomorphism-invariant measures on the space of connections modulo gauge transformations generalizing the recent work of Ashtekar and Lewandowski. This construction proceeds by doing lattice gauge theory on graphs analytically embedded in the base manifold. 
  We show that a quantum deformation of quantum mechanics given in a previous work is equivalent to quantum mechanics on a nonlinear lattice with step size $\Delta x=~(1-q)x$. Then, based on this, we develop the basic formalism of quantum group Schr\"{o}dinger field theory in one spatial quantum dimension, and explicitly exhibit the $SU_{q}(2)$ covariant algebras satisfied by the $q$-bosonic and $q$-fermionic Schr\"{o}dinger fields. We generalize this result to an arbitrary number of fields. 
  We compute the exact partition function for pure continuous Yang-Mills theory on the two-sphere in the large $N$ limit, and find that it exhibits a large $N$ third order phase transition with respect to the area $A$ of the sphere. The weak coupling (small A) partition function is trivial, while in the strong coupling phase (large A) it is expressed in terms of elliptic integrals. We expand the strong coupling result in a double power series in $e^{-g^2 A}$ and $g^2 A$ and show that the terms are the weighted sums of branched coverings proposed by Gross and Taylor. The Wilson loop in the weak coupling phase does not show the simple area law. We discuss some consequences for higher dimensions. 
  In recent papers of the author, a method was developed for constructing quasitriangular Hopf algebras (quantum groups) of the quantum-double type. As a by-product, a novel non-standard example of the quantum double has been found. In the present paper, a closed expression (in terms of elementary functions) for the corresponding universal R-matrix is obtained. In reduced form, when the number of generators becomes two instead of four, this quantum group can be interpreted as a deformation of the Lie algebra   [x,h]=2h in the context of Drinfeld's quantization program. 
  The universal field equations introduced by the author and his collaborators, which admit infinitely many inequivalent Lagrangian formulations are shown to arise as consistency conditions for the existence of non-trivial solutions to the quasi-linear equations, called equations of hydrodynamic type by Novikov , Dubrovin and others. The solutions in closed form are only implicit. A method due to Stokes, which is in essence just Fourier Analysis is resurrected for application to those equations. With the benefit of algebraic computation facilities, this method, allows the general structure of power series solutions to be conjectured. 
  It is shown that each integrable mapping is connected with a hierarchical completely integrable sytem of equations of evolution type which are invariant with respect to the transformation described by this mapping. 
  A general formula for physical observables in Chern-Simons theory with an arbitrary compact Lie group $G$, on an arbitrary closed oriented three-dimensional manifold $\cM$ is derived in terms of vacuum expectation values of Wilson loops in ${\cal S}^3$. Surgery presentation of $\cM$ and the Kirby moves are implemented as the main ingredients of the approach. The case of $G={\rm SU}(n)$ is explicitly calculated. 
  We implement the concept of Wilson renormalization in the context of simple quantum mechanical systems. The attractive inverse square potential leads to a $\b$ function with a nontrivial ultraviolet stable fixed point and the Hulthen potential exhibits the crossover phenomenon. We also discuss the implementation of the Wilson scheme in the broader context of one dimensional potential problems. The possibility of an analogue of Zamolodchikov's $C$ function in these systems is also discussed. 
  A universal integrable hierarchy underlying topological Landau-Ginzburg models of D-tye is presented. Like the dispersionless Toda hierarchy, the new hierarchy has two distinct (``positive" and ``negative") set of flows. Special solutions corresponding to topological Landau-Ginzburg models of D-type are characterized by a Riemann-Hilbert problem, which can be converted into a generalized hodograph transformation. This construction gives an embedding of the finite dimensional small phase space of these models into the full space of flows of this hierarchy. One of flat coordinates in the small phase space turns out to be identical to the first ``negative" time variable of the hierarchy, whereas the others belong to the ``positive" flows. 
  Quantum canonical transformations are defined algebraically outside of a Hilbert space context. This generalizes the quantum canonical transformations of Weyl and Dirac to include non-unitary transformations. The importance of non-unitary transformations for constructing solutions of the Schr\"odinger equation is discussed. Three elementary canonical transformations are shown both to have quantum implementations as finite transformations and to generate, classically and infinitesimally, the full canonical algebra. A general canonical transformation can be realized quantum mechanically as a product of these transformations. Each transformation corresponds to a familiar tool used in solving differential equations, and the procedure of solving a differential equation is systematized by the use of the canonical transformations. Several examples are done to illustrate the use of the canonical transformations. [This is an extensively revised version of hep-th-9205080: the first third of the paper is new material; the notation has been simplified, and further discussion has been added to the remainder.] 
  We present a brief discussion of recent work on duality symmetries in non-trivial string backgrounds. Duality is obtained from a gauged non-linear sigma-model with vanishing gauge field strength. Standard results are reproduced for abelian gauge groups, whereas a new type of duality is identified for non--abelian gauge groups. Examples of duals of WZW models and 4-d black holes are given. (Presented at `From Superstrings to Supergravity', Erice 1992) 
  The "body fixed frame" with respect to local gauge transformations is introduced. Rigid gauge "rotations" in QCD and their \Sch equation are studied for static and dynamic quarks. Possible choices of the rigid gauge field configuration corresponding to a nonvanishing static colormagnetic field in the "body fixed" frame are discussed. A gauge invariant variational equation is derived in this frame. For large number N of colors the rigid gauge field configuration is regarded as random with maximally random probability distribution under constraints on macroscopic--like quantities. For the uniform magnetic field the joint probability distribution of the field components is determined by maximizing the appropriate entropy under the area law constraint for the Wilson loop. In the quark sector the gauge invariance requires the rigid gauge field configuration to appear not only as a background but also as inducing an instantaneous quark-quark interaction. Both are random in the large N limit. 
  We consider the relationship between the conjectured uniqueness of the Moonshine Module, ${\cal V}^\natural$, and Monstrous Moonshine, the genus zero property of the modular invariance group for each Monster group Thompson series. We first discuss a family of possible $Z_n$ meromorphic orbifold constructions of ${\cal V}^\natural$ based on automorphisms of the Leech lattice compactified bosonic string. We reproduce the Thompson series for all 51 non-Fricke classes of the Monster group $M$ together with a new relationship between the centralisers of these classes and 51 corresponding Conway group centralisers (generalising a well-known relationship for 5 such classes). Assuming that ${\cal V}^\natural$ is unique, we then consider meromorphic orbifoldings of ${\cal V}^\natural$ and show that Monstrous Moonshine holds if and only if the only meromorphic orbifoldings of ${\cal V}^\natural$ give ${\cal V}^\natural$ itself or the Leech theory. This constraint on the meromorphic orbifoldings of ${\cal V}^\natural$ therefore relates Monstrous Moonshine to the uniqueness of ${\cal V}^\natural$ in a new way. 
  Using the conformal embedding on the torus, we can express some characters of $SU(3)_3$ in terms of $SO(8)_1$ characters. Then with the help of crossing symmetry, modular transformation and factorization properties of Green functions, we will calculate a class of correlators of $SU(3)_3$ on arbitrary Riemann surfaces. This method can apply to all $k>1$ WZW models which can be conformally embedded in some $k=1$ WZW models. 
  The bihamiltonian structure of the N=2 Supersymmetric Boussinesq equation is found. It is not reduced to the corresponding classical structure and hence it describes the pure supersymmetric effect. For the supersymmetric Boussinesq equation which contains the classical partner the Lax pair is given explicitly. Thus we prove the integrability of the equation. 
  The non-perturbative solution to the strong CP problem with magnetic monopoles as originally proposed by the author is described. It is shown that the gauge orbit space with gauge potentials and gauge tranformations restricted on the space boundary and the globally well-defined gauge subgroup in gauge theories with a $\theta$ term has a monopole structure if there is a magnetic monopole in the ordinary space. The Dirac's quantization condition then ensures that the vacuum angle $\theta$ in the gauge theories must be quantized to have a well-defined physical wave functional. The quantization rule for $\theta$ is derived as $\theta=0, 2\pi/n~(n\neq 0)$ with n being the topological charge of the magnetic monopole. Therefore, the strong CP problem is automatically solved with the existence of a magnetic monopole of charge $\pm 1$ with $\theta=\pm 2\pi$. This is also true when the total magnetic charge of monopoles are very large ($|n|\geq 10^92\pi$). The fact that the strong CP violation can be only so small or vanishing may be a signal for the existence of magnetic monopoles and the universe is open. 
  We consider self-avoiding Nambu-Goto open strings on a random surface. We have shown that the partition function of such a string theory can be calculated exactly. The string susceptibility for the disk is evaluated to be $-\frac{1}{2}$. We also consider modifications of the Nambu-Goto action which are exactly soluble on a random surface. 
  Bennett et al. (PRL 70, 1859 (1993)) have shown how to transfer ("teleport") an unknown spin quantum state by using prearranged correlated quantum systems and transmission of classical information. I will show how their results can be obtained in the framework of nonlocal measurements proposed by Aharonov and Albert I will generalize the latter to the teleportation of a quantum state of a system with continuous variables. 
  We use a quite concrete and simple realization of $\slq$ involving finite difference operators. We interpret them as derivations (in the non-commutative sense) on a suitable graded algebra, which gives rise to the double of the projective line as the non commutative version of the standard homogeneous space. 
  Non-perturbative effects of instanton-like solutions are studied within the framework of supergravity theories with field-dependent gauge functions. Fermionic zero modes are constructed and some typical correlation functions are evaluated. The effects of instantons are very similar to those in globally supersymmetric theories: they preserve supersymmetry while breaking a chiral $U(1)$ symmetry. Non-perturbative amplitudes receive corrections which are suppressed at large distances. 
  Non-local (alpha prime) corrections to Schwarzschild black holes are shown to invalidate the thermodynamical interpretation of black holes. In particular, the canonical and Bekenstein-Hawking temperatures are not equal. The particle number density of fields quantized in the (alpha prime modified) black hole background is no longer thermal. In the non-perturbative region (alpha prime going to infinity or mass going to zero), an analytic continuation to the number density is shown to vanish exponentially. 
  We show that macroscopic heterotic strings, formulated as strings which wind around a compact direction of finite but macroscopic extent, exhibit non-trivial scattering at low energies. This occurs at order velocity squared and may thus be described as geodesic motion on a moduli space with a non-trivial metric which we construct. Our result is in agreement with a direct calculation of the string scattering amplitude. 
  Bosonic q-oscillators commute with themselves and so their free distribution is Planckian. In a cavity, their emission and absorption rates may grow or shrink---and even diverge---but they nevertheless balance to yield the Planck distribution via Einstein's equilibrium method, (a careless application of which might produce spurious q-dependent distribution functions). This drives home the point that the black-body energy distribution is not a handle for distinguishing q-excitations from plain oscillators. A maximum cavity size is suggested by the inverse critical frequency of such emission/absorption rates at a given temperature, or a maximum temperature at a given frequency. To remedy fragmentation of opinion on the subject, we provide some discussion, context, and references. 
  It is argued that the low-energy dynamics of $k$ monopoles in N=2 supersymmetric Yang-Mills theory are determined by an N=4 supersymmetric quantum mechanics based on the moduli space of $k$ static monople solutions. This generalises Manton's ``geodesic approximation" for studying the low-energy dynamics of (bosonic) BPS monopoles. We discuss some aspects of the quantisation and in particular argue that dolbeault cohomology classes of the moduli space are related to bound states of the full quantum field theory. 
  A naive dimensional reduction of the $N=1, D=10$ supergravity theory that naturally arises in five-brane models is used to determine the r\^ole of two fields which are basic ingredients of string models: the dilaton and, among the moduli, the breathing mode. It is shown that, under the duality transformation that relates five-branes and strings, these two fields exchange the r\^oles of 10-dimensional dilaton and radius of the compact manifold. A description of this phenomenon in terms of the linear multiplets of the 4-dimensional supergravity is also presented. 
  We study classical $N=2$ super-$W_3$ algebra and its interplay with $N=2$ supersymmetric extensions of the Boussinesq equation in the framework of the nonlinear realization method and the inverse Higgs - covariant reduction approach. These techniques have been previously applied by us in the bosonic $W_3$ case to give a new geometric interpretation of the Boussinesq hierarchy. Here we deduce the most general $N=2$ super Boussinesq equation and two kinds of the modified $N=2$ super Boussinesq equations, as well as the super Miura maps relating these systems to each other, by applying the covariant reduction to certain coset manifolds of linear $N=2$ super-$W_3^{\infty}$ symmetry associated with $N=2$ super-$W_3$. We discuss the integrability properties of the equations obtained and their correspondence with the formulation based on the notion of the second hamiltonian structure. 
  We prove that the deformed oscillator superalgebra $W_q(n)$ (which in the Fock representation is generated essentially by $n$ pairs of $q$-bosons) is a factor algebra of the quantized universal enveloping algebra $U_q[osp(1/2n)]$. We write down a $q$-analog of the Cartan-Weyl basis for the deformed $osp(1/2n)$ and give also an oscillator realization of all Cartan-Weyl generators. 
  The affine-Virasoro Ward identities are a system of non-linear differential equations which describe the correlators of all affine-Virasoro constructions, including rational and irrational conformal field theory. We study the Ward identities in some detail, with several central results. First, we solve for the correlators of the affine-Sugawara nests, which are associated to the nested subgroups $g\supset h_1 \supset \ldots \supset h_n$. We also find an equivalent algebraic formulation which allows us to find global solutions across the set of all affine-Virasoro constructions. A particular global solution is discussed which gives the correct nest correlators, exhibits braiding for all affine-Virasoro correlators, and shows good physical behavior, at least for four-point correlators at high level on simple $g$. In rational and irrational conformal field theory, the high-level fusion rules of the broken affine modules follow the Clebsch-Gordan coefficients of the representations. 
  We extend spacetime duality to superspace, including fermions in the low-energy limits of superstrings. The tangent space is a curved, extended superspace. The geometry is based on an enlarged coordinate space where the vanishing of the d'Alembertian is as fundamental as the vanishing of the curl of a gradient. 
  Following recent work on the effective quantum action of gauged WZW models, we suggest such an action for {\it chiral} gauged WZW models which in many respects differ from the usual gauged WZW models. Using the effective action we compute the conformally exact expressions for the metric, the antisymmetric tensor, and the dilaton fields in the $\s$-model arising from a general {\it chiral } gauged WZW model. We also obtain the general solution of the geodesic equations in the exact geometry. Finally we consider in some detail a three dimensional model which has certain similarities with the three dimensional black string model. Finally we consider in some detail a three dimensional model which has certain similarities with the three dimensional black string model. 
  An analysis of errors in measurement yields new insight into the penetration of quantum particles into classically forbidden regions. In addition to ``physical" values, realistic measurements yield ``unphysical" values which, we show, can form a consistent pattern. An experiment to isolate a particle in a classically forbidden region obtains negative values for its kinetic energy. These values realize the concept of a {\it weak value}, discussed in previous works. 
  Parity violation is a long standing problem in light-cone quantization. \REF\CPT{D. Soper, SLAC-REP-137, 1970, Chap. I . } \refend We propose a new quantization on the light-cone which treats both the $x^{+}$ and the $x^{-}$ coordinates as light-cone 'times.'This quantization respects both parity and time-reversal. We find that now both $P^{-}$ and $P^{+}$ become dynamical. 
  Certain phase space path integrals can be evaluated exactly using equivariant cohomology and localization in the canonical loop space. Here we extend this to a general class of models. We consider hamiltonians which are {\it a priori} arbitrary functions of the Cartan subalgebra generators of a Lie group which is defined on the phase space. We evaluate the corresponding path integral and find that it is closely related to the infinitesimal Lefschetz number of a Dirac operator on the phase space. Our results indicate that equivariant characteristic classes could provide a natural geometric framework for understanding quantum integrability. 
  We study the integrability properties of the one-parameter family of $N=2$ super Boussinesq equations obtained earlier by two of us (E.I. \& S.K., Phys. Lett. B 291 (1992) 63) as a hamiltonian flow on the $N=2$ super-$W_3$ algebra. We show that it admits nontrivial higher order conserved quantities and hence gives rise to integrable hierarchies only for three values of the involved parameter, $\alpha=-2,\;-1/2,\;5/2$. We find that for the case $\alpha = -1/2$ there exists a Lax pair formulation in terms of local $N=2$ pseudo-differential operators, while for $\alpha = -2$ the associated equation turns out to be bi-hamiltonian. 
  We study flows on the space of topological Landau-Ginzburg theories coupled to topological gravity. We argue that flows corresponding to gravitational descendants change the target space from a complex plane to a punctured complex plane and lead to the motion of punctures.It is shown that the evolution of the topological theory due to these flows is given by dispersionless limit of KP hierarchy. We argue that the generating function of correlators in such theories are equal to the logarithm of the tau-function of Generalized Kontsevich Model. 
  We construct and classify topological lattice field theories in three dimensions. After defining a general class of local lattice field theories, we impose invariance under arbitrary topology-preserving deformations of the underlying lattice, which are generated by two new local lattice moves. Invariant solutions are in one--to--one correspondence with Hopf algebras satisfying a certain constraint. As an example, we study in detail the topological lattice field theory corresponding to the Hopf algebra based on the group ring $\C[G]$, and show that it is equivalent to lattice gauge theory at zero coupling, and to the Ponzano--Regge theory for $G=$SU(2). 
  Hawking has shown that the emission of gravitational radiation cannot prevent circular loops of gauged cosmic strings from collapsing into black holes. Here we consider the corresponding question for global strings: can Goldstone boson emission prevent circular loops of global cosmic strings from forming black holes? Our results show that for every value of the string tension there is a certain critical size below which the circular loop does not collapse to form a black hole. For GUT scale strings, this critical size is much larger than the current horizon. 
  We analyze transition potentials $(V(r) \stackrel{r\sim 0}{\rightarrow} {\alpha r^{-2}})$ in non-relativistic quantum mechanics using the techniques of supersymmetry. For the range $-1/4 < \alpha < 3/4$, the eigenvalue problem becomes ill-defined (since it is not possible to choose a unique eigenfunction based on square integrability and boundary conditions). It is shown that supersymmetric quantum mechanics (SUSYQM) provides a natural prescription for a unique determination of the spectrum. Interestingly, our SUSYQM based approach picks out the same "less singular" wave functions as the conventional approach, and thus provides a simple justification for the usual practice in the literature. Two examples (the P\"oschl-Teller II potential and a two anyon system on the plane) have been worked out for illustrative purposes. 
  Exact solutions of heterotic string theory corresponding to four-dimensional charge Q magnetic black holes are constructed as tensor products of an SU(2)/Z(2Q+2) WZW orbifold with a (0,1) supersymmetric SU(1,1)/U(1) WZW coset model. The spectrum is analyzed in some detail. ``Bad'' marginal operators are found which are argued to deform these theories to asymptotically flat black holes. Surprising behaviour is found for small values of Q, where low-energy field theory is inapplicable. At the minimal value Q=1, the theory degenerates. Renormalization group arguments are given that suggest the potential gravitational singularity of the low-energy field theory is resolved by a massive two-dimensional field theory. At Q=0, a stable, neutral ``remnant,'' of potential relevance to the black hole information paradox, is found. 
  It is shown that the finite dimensional irreducible representaions of the quantum matrix algebra $ M_{ q,p}(2) $ ( the coordinate ring of $ GL_{q,p}(2) $) exist only when both q and p are roots of unity. In this case th e space of states has either the topology of a torus or a cylinder which may be thought of as generalizations of cyclic representations. 
  It is shown that the finite dimensional ireducible representations of the quantum matrix algebra $ M_q(3) $ ( the coordinate ring of $ GL_q(3) $ ) exist only when q is a root of unity ( $ q^p = 1 $ ). The dimensions of these representations can only be one of the following values: $ p^3 , { p^3 \over 2 } , { p^3 \over 4 } $ or $ { p^3 \over 8 } $ . The topology of the space of states ranges between two extremes , from a 3-dimensional torus $ S^1 \times S^1 \times S^1 $ ( which may be thought of as a generalization of the cyclic representation ) to a 3-dimensional cube $ [ 0 , 1 ]\times [ 0 , 1 ]\times [ 0 , 1 ] $ . 
  {Although q-oscillators have been used extensively for realization of quantum universal enveloping algebras,such realization do not exist for quantum matrix algebras ( deformation of the algebra of functions on the group ). In this paper we first construct an infinite dimensional representation of the quantum matrix algebra $ M_q ( 3 ) $(the coordinate ring of $ GL_q (3)) $ and then use this representation to realize $ GL_q ( 3 ) $ by q-bosons.} 
  We show that the Kaehler structure can be naturally incorporated in the Batalin-Vilkovisky formalism. The phase space of the BV formalism becomes a fermionic Kaehler manifold. By introducing an isometry we explicitly construct the fermionic irreducible hermitian symmetric space. We then give some solutions of the master equation in the BV formalism. 
  A new approach to the quantization of constrained or otherwise reduced classical mechanical systems is proposed. On the classical side, the generalized symplectic reduction procedure of Mikami and Weinstein, as further extended by Xu in connection with symplectic equivalence bimodules and Morita equivalence of Poisson manifolds, is rewritten so as to avoid the use of symplectic groupoids, whose quantum analogue is unknown. A theorem on symplectic reduction in stages is given. This allows one to discern that the `quantization' of the generalized moment map consists of an operator-valued inner product on a (pre-) Hilbert space (that is, a structure similar to a Hilbert $C^*$-module). Hence Rieffel's far-reaching operator-algebraic generalization of the notion of an induced representation is seen to be the exact quantum counterpart of the classical idea of symplectic reduction, with imprimitivity bimodules and strong Morita equivalence of $C^*$-algebras falling in the right place. Various examples involving groups as well as groupoids are given, and known difficulties with both Dirac and BRST quantization are seen to be absent in our approach. 
  We discuss some recent work in the field of classical solitonic solutions in string theory. In particular, we construct instanton and monopole solutions and discuss the dynamics of string-like solitons. Some of the motivation behind this work is that instantons may provide a nonperturbative understanding of the vacuum structure of string theory, while monopoles may appear in string predictions for grand unification. The string-like solitons represent extended states of fundamental strings. The essential role of supersymmetry in both the saturation of the Bogomol'nyi bound and in the cancellation of higher order corrections is emphasized. (Talk given at the International Workshop: ``Recent Advances in the Superworld'', Houston Advanced Research Center, The Woodlands, TX, April 14-16, 1993.) 
  Gauged Wess-Zumino-Witten theory for compact groups is considered. It is shown that this theory has fermionic BRST-like symmetry and may be exactly solved using localization approach. As an example we calculate functional integral for the case of SU(2) group on the arbitrary Riemann surface. The answer is the particular case of Verlinde formula for the number of conformal blocks. 
  We analyse the spectral problem for the q-Knizhnik-Zamolodchikov equations for $U_q(\widehat{sl_2}) (0 < q \leq 1)$ at level zero. The case of 2-point functions in the fundamental representation is studied in detail. The scattering states are found explicitly in terms of continuous q-Jacobi polynomials. The corresponding S-matrix is shown to coincide, up to a trivial factor, with the kink-antikink S-matrix in the spin-1/2 XXZ antiferromagnet. 
  We discuss the existence of Gribov ambiguities in $SU(m)\times U(1)$ gauge theories over the $n-$spheres. We achieve our goal by showing that there is exactly one conjugacy class of groups of gauge transformations for the theories given above. This implies that these transformation groups are conjugate to the ones of the trivial $SU(m)\times U(1)$ fiber bundles over the $n-$spheres. By using properties of the space of maps $Map_{\ast}(S^n,G)$ where $G$ is one of $U(1)$, $SU(m)$ we are able to determine the homotopy type of the groups of gauge transformations in terms of the homotopy groups of $G$. The non-triviality of these homotopy groups gives the desired result. 
  Fractional superstrings in the tensor-product formulation experience ``internal projections'' which reduce their effective central charges.  Simple expressions for the characters of the resulting effective worldsheet theory are found. All states in the effective theory can be consistently assigned definite spacetime statistics. The projection to the effective theory is shown to be described by the action of a dimension-three current in the original tensor-product theory. 
  Two of the important unresolved issues concerning fractional superstrings have been the appearance of new massive sectors whose spacetime statistics properties are unclear, and the appearance of new types of ``internal projections'' which alter or deform the worldsheet conformal field theory in a highly non-trivial manner. In this paper we provide a systematic analysis of these two connected issues, and explicitly map out the effective post-projection worldsheet theories for each of the fractional-superstring sectors. In this way we determine their central charges, highest weights, fusion rules, and characters, and find that these theories are isomorphic to those of free worldsheet bosons and fermions.   We also analyze the recently-discovered parafermionic ``twist current'' which has been shown to play an important role in reorganizing the fractional-superstring Fock space, and find that this current can be expressed directly in terms of the primary fields of the post-projection theory. This thereby enables us to deduce some of the spacetime statistics properties of the surviving states. 
  We propose $N=2$ holomorphic Yang-Mills theory on compact K\"{a}hler manifolds and show that there exists a simple mapping from the $N=2$ topological Yang-Mills theory. It follows that intersection parings on the moduli space of Einstein-Hermitian connections can be determined by examining the small coupling behavior of the $N=2$ holomorphic Yang-Mills theory. This paper is a higher dimensional generalization of the Witten's work on physical Yang-Mills theory in two dimensions. 
  In development of the started activity on lattice analogues of $W$-algebras, we define the notion of lattice $W_{\infty}$-algebra, accociated with lattice integrable system with infinite set of fields. Various kinds of reduction to lattice $W_N$-algebras, related to discrete $N$-KdV hierarchies are described. We also discuss the connection of our results with those obtained in the papers of Xiong [13] and Bonora [14]. 
  We show that two-dimensional SO(N) and Sp(N) Yang-Mills theories without fermions can be interpreted as closed string theories. The terms in the 1/N expansion of the partition function on an orientable or nonorientable manifold M can be associated with maps from a string worldsheet onto M. These maps are unbranched and branched covers of M with an arbitrary number of infinitesimal worldsheet cross-caps mapped to points in M. These string theories differ from SU(N) Yang-Mills string theory in that they involve odd powers of 1/N and require both orientable and nonorientable worldsheets. 
  The quantization of a free boson whose momentum satisfies a cubic constraint leads to a $c=\ha$ conformal field theory with a BRST symmetry. The theory also has a $W_\infty $ symmetry in which all the generators except the stress-tensor are BRST-exact and so topological. The BRST cohomology includes states of conformal dimensions $0,\si,\ha$, together with \lq copies' of these states obtained by acting with picture-changing and screening operators. The 3-point and 4-point correlation functions agree with those of the Ising model, suggesting that the theory is equivalent to the critical Ising model. At tree level, the $W_3$ string can be viewed as an ordinary $c=26$ string whose conformal matter sector includes this realisation of the Ising model. The two-boson $W_3$ string is equivalent to the Ising model coupled to two-dimensional quantum gravity. Similar results apply for other W-strings and minimal models. 
  The canonical theory of $N=1$ supergravity is applied to Bianchi class A spatially homogeneous cosmologies. The full set of quantum constraints are then solved with the possible ordering ambiguity taken into account by introducing a free parameter. The wave functions are explicitly given for all the Bianchi class A models in a unified way. Some comments are made on the Bianchi type IX cases. 
  We formulate the basic properties of q-vertex operators in the context of the Andrews-Baxter-Forrester (ABF) series, as an example of face-interaction models, derive the q-difference equations satisfied by their correlation functions, and establish their connection with representation theory. We also discuss the q-difference equations of the Kashiwara-Miwa (KM) series, as an example of edge-interaction models. Next, the Ising model--the simplest special case of both ABF and KM series--is studied in more detail using the Jordan-Wigner fermions. In particular, all matrix elements of vertex operators are calculated. 
  An explicit derivation of the elements of the representation ring of SU(2) needed to implement the four-dimensional Kirby calculus is sketched. 
  We re-examine the justification for the imposition of regular boundary conditions on the wavefunction at the Coulomb singularity in the treatment of the hydrogen atom in non-relativistic quantum mechanics. We show that the issue of the correct boundary conditions is not independent of the physical structure of the proton. Under the physically reasonable assumption that the finite size and structure of the proton can be represented as a positive correction to the Coulomb potential, we give a justification for the regular boundary condition, which, in contrast to the usual treatments, is physically motivated and mathematically rigorous. We also describe how irregular boundary conditions can be used to model non-positive corrections to the Coulomb potential. 
  Replaced with revised (uncorrupted) version We present a general scheme for generating (2,2) symmetric fermionic string models and classify the models in D=8 and D=6 space time dimensions. We discuss the relationship to other compactification schemes. 
  Talk given at the 6th Philosophy-and-Physics-Workshop ``Epistemological Aspects of the Role of Mathematics in Physical Science'', FEST, Heidelberg, Feb. 1993 
  We study the Batalin-Vilkovisky master equation for both open and closed string field theory with special attention to anomalies. Open string field theory is anomaly free once the minimal coupling to closed strings induced by loop amplitudes is considered. In closed string field theory the full-fledged master equation has to be solved order by order in perturbation theory. The existence of a solution implies the absence of anomaly. We briefly discuss the relation of the iterative process of solution to methods used in the first quantized formalism and comment on some possible non-perturbative corrections. 
  The interconnection between self-duality, conformal invariance and Lie-Poisson structure of the two dimensional non-abelian Thirring model is investigated in the framework of the hamiltonian method. 
  We discuss the low-energy effective string theory when moduli of the compactified manifold are present. Assuming a nontrivial coupling of the moduli to the Maxwell tensor, we find a class of regular black hole solutions. Both the thermodynamical and the geometrical structure of these solutions are discussed 
  The totality of all Zakharov-Shabat equations (ZS), i.e., zero-curvature equations with rational dependence on a spectral parameter, if properly defined, can be considered as a hierarchy. The latter means a collection of commuting vector fields in the same phase space. Further properties of the hierarchy are discussed, such as additional symmetries, an analogue to the string equation, a Grassmannian related to the ZS hierarchy, and a Grassmannian definition of soliton solutions. 
  The refrerences are corrected. 
  We present a new application of affine Lie algebras to massive quantum field theory in 2 dimensions, by investigating the $q\to 1$ limit of the q-deformed affine $\hat{sl(2)}$ symmetry of the sine-Gordon theory, this limit occuring at the free fermion point. We describe how radial quantization leads to a quasi-chiral factorization of the space of fields. The conserved charges which generate the affine Lie algebra split into two independent affine algebras on this factorized space, each with level 1 in the anti-periodic sector. The space of fields in the anti-periodic sector can be organized using level-$1$ highest weight representations, if one supplements the $\slh$ algebra with the usual local integrals of motion. Using the integrals of motion, a momentum space bosonization involving vertex operators is formulated. This leads to a new way of computing form-factors, as vacuum expectation values in momentum space. 
  We construct a model of gravity in 1+1 spacetime dimensions in which the solutions approach the Schwarzschild metric at large $r$ and de Sitter space far inside the horizon. Our model may be viewed as a two dimensional application of the `Limiting Curvature Construction' recently proposed by two of the authors. 
  Static spherically symmetric asymptotically flat charged black hole solutions are constructed within the magnetic SU(3) sector of the 4-dimensional heterotic string effective action. They possess non-abelian hair in addition to the Coulomb magnetic field and are qualitatively similar to the Einstein-Yang-Mills colored SU(3) black holes except for the extremal case. In the extremality limit the horizon shrinks and the resulting geometry around the origin coincides with that of an extremal abelian dilatonic black hole with magnetic charge. Non-abelian hair exibits then typical sphaleron structure. 
  The properties of the canonical symmetry of the nonlinear Schr\"odinger equation are investigated. The densities of the local conservation laws for this system are shown to change under the action of the canonical symmetry by total space derivatives. 
  We present a ``natural finitization'' of the fermionic q-series (certain generalizations of the Rogers-Ramanujan sums) which were recently conjectured to be equal to Virasoro characters of the unitary minimal conformal field theory (CFT) M(p,p+1). Within the quasi-particle interpretation of the fermionic q-series this finitization amounts to introducing an ultraviolet cutoff, which -- contrary to a lattice spacing -- does not modify the linear dispersion relation. The resulting polynomials are conjectured (proven, for p=3,4) to be equal to corner transfer matrix (CTM) sums which arise in the computation of order parameters in regime III of the r=p+1 RSOS model of Andrews, Baxter, and Forrester. Following Schur's proof of the Rogers-Ramanujan identities, these authors have shown that the infinite-lattice limit of the CTM sums gives what later became known as the Rocha-Caridi formula for the Virasoro characters. Thus we provide a proof of the fermionic q-series representation for the Virasoro characters for p=4 (the case p=3 is ``trivial''), in addition to extending the remarkable connection between CFT and off-critical RSOS models. We also discuss finitizations of the CFT modular-invariant partition functions. 
  The exact free energy of SU($N$) Chern-Simons theory at level $k$ is expanded in powers of $(N+k)^{-2}.$ This expansion keeps rank-level duality manifest, and simplifies as $k$ becomes large, keeping $N$ fixed (or vice versa)---this is the weak-coupling (strong-coupling) limit. With the standard normalization, the free energy on the three-sphere in this limit is shown to be the generating function of the Euler characteristics of the moduli spaces of surfaces of genus $g,$ providing a string interpretation for the perturbative expansion. A similar expansion is found for the three-torus, with differences that shed light on contributions from different spacetime topologies in string theory. 
  We exhibit string contributions to the $\nd{S}$ matrix relating in- and out- state density matrices that do not factorize as a product of $S$ and $S^\dagger$ matrices. They are associated with valley trajectories between topological defects on the string world sheet, that appear as quantum fluctuations in the space-time foam. Through their ultraviolet cut-off dependences these valleys cause non-Hamiltonian time evolution and suppress off-diagonal entries in the density matrix at large times. 
  We show that, in string theory, as a result of the $W_{\infty}$-symmetries that preserve quantum coherence in the {\it full} string theory by coupling different mass levels, transitions between initial- and final-state density matrices for the effective light-particle theory involve non-Hamiltonian terms $\nd{\delta H}$ in their time evolution, and are described by a $\nd{S}$ matrix that is not factorizable as a product of field-theoretical $S$ and $S^\dagger$ matrices. We exhibit non-trivial string contributions to $\nd{\delta H}$ and the $\nd{S}$ matrix associated with topological fluctuations related to the coset model that describes an s-wave black hole. These include monopole-antimonopole configurations on the world-sheet that correspond to black hole creation and annihilation, and instantons that represent back-reaction via quantum jumps between black holes of different mass, both of which make the string supercritical. The resulting Liouville mode is interpreted as the time variable, and the arrow of time is associated with black hole decay. Since conformal invariance is broken in the non-critical string theory, monopole and antimonopole, or instanton and anti-instanton, are not separable, and the 
  Quantum de Rham complexes on the quantum plane and the quantum group itself are constructed for the Zakrewski deformation of $ Fun ( SL(2)) $. As a by-product a new deformation of the two dimensional Heisenberg algeb ra is constructed which can be used to construct models of h-deformed quantum mechanics. 
  The most general form of a marginal extended perturbation in a two-dimensional system is deduced from scaling considerations. It includes as particular cases extended perturbations decaying either from a surface, a line or a point for which exact results have been previously obtained. The first-order corrections to the local exponents, which are functions of the amplitude of the defect, are deduced from a perturbation expansion of the two-point correlation functions. Assuming covariance under conformal transformation, the perturbed system is mapped onto a cylinder. Working in the Hamiltonian limit, the first-order corrections to the lowest gaps are calculated for the Ising model. The results confirm the validity of the gap-exponent relations for the perturbed system. 
  In these lecture notes we explain a connection between Yang-Mills theory on arbitrary Riemann surfaces and two types of topological field theory, the so called $BF$ and cohomological theories. The quantum Yang-Mills theory is solved exactly using path integral techniques. Explicit expressions, in terms of group representation theory, are obtained for the partition function and various correlation functions. In a particular limit the Yang-Mills theory devolves to the topological models and the previously determined correlation functions give topological information about the moduli spaces of flat connections. In particular, the partition function yields the volume of the moduli space for which an explicit expression is derived. These notes are self contained, with a basic introduction to the various ideas underlying the topological field theories. This includes some relatively new work on handling problems that arise in the presence of reducible connections which in turn forms the bridge between the various models under consideration. These notes are identical to those made available to participants of the 1992 summer school in Trieste, except for one or two additions added circa January 1993. 
  A new approach for investigating the classical dynamics of the relativistic string model with rigidity is proposed. It is based on the embedding of the string world surface into the space of a constant curvature. It is shown that the rigid string in flat space-time is described by the Euler-Lagrange equation for the Willmore functional in a space-time of the constant curvature K=-g/(2 a), where g and a are constants in front of the Nambu-Goto term and the curvature term in the rigid string action respectively. For simplicity the Euclidean version of the rigid string in the three-dimensional space-time is considered. The Willmore functional (the action for the "Willmore string") is obtained by dropping the Nambu-Goto term in the Polyakov-Kleinert action for the rigid string. Such "reduction" of the rigid string model would be useful, for example, by applying some results about the Numbu-Goto string dynamics in the de Sitter universe to the rigid string model in the Minkowski space-time. It allows also to use numerous mathematical results about the Willmore surfaces in the context of the physical problem. 
  Coherent states path integral formalism for the simplest quantum algebras, q-oscillator, SU_q(2) and SU_q(1,1) is introduced. In the classical limit canonical structure is derived with modified symplectic and Riemannian metric. Non-constant deformation induced curvature for the phase spaces is obtained. 
  One of the paradoxes associated with the theory of the formation and subsequent Hawking evaporation of a black hole is the disappearance of conserved global charges. It has long been known that metric fluctuations at short distances (wormholes) violate global-charge conservation; if global charges are apparently conserved at ordinary energies, it is only because wormhole-induced global-charge-violating terms in the low-energy effective Lagrangian are suppressed by large mass denominators. However, such suppressed interactions can become important at the high energy densities inside a collapsing star. We analyze this effect for a simple model of the black-hole singularity. (Our analysis is totally independent of any detailed theory of wormhole dynamics; in particular it does not depend on the wormhole theory of the vanishing of the cosmological constant.) We find that in general all charge is extinguished before the infalling matter crosses the singularity. No global charge appears in the outgoing Hawking radiation because it has all gone down the wormholes. 
  This is intended to be a simple discussion of work done in collaboration with S. Cecotti, K. Intriligator and C. Vafa; and with H. Saleur. I discuss how $ Tr F (-1)^F e^{-\beta H}$ can be computed exactly in any N=2 supersymmetric theory in two dimensions. It gives exact information on the soliton spectrum of the theory, and corresponds to the partition function of a single self-avoiding polymer looped once around a cylinder of radius $\beta$. It is independent of almost all deformations of the theory, and satisfies an exact differential equation as a function of $\beta$. For integrable theories it can also be computed from the exact S-matrix. This implies a highly non-trivial equivalence of a set of coupled integral equations with the classical sinh-Gordon and the affine Toda equations. 
  We examine the evaporation of two--dimensional black holes, the classical space--times of which are extended geometries, like for example the two--dimensional section of the extremal Reissner--Nordstrom black hole. We find that the evaporation in two particular models proceeds to a stable end--point. This should represent the generic behavior of a certain class of two--dimensional dilaton--gravity models. There are two distinct regimes depending on whether the back--reaction is weak or strong in a certain sense. When the back--reaction is weak, evaporation proceeds via an adiabatic evolution, whereas for strong back--reaction, the decay proceeds in a somewhat surprising manner. Although information loss is inevitable in these models at the semi--classical level, it is rather benign, in that the information is stored in another asymptotic region. 
  The one-plaquette Hamiltonian of large N lattice gauge theory offers a constructive model of a $1+1$-dimensional string theory with a stable ground state.   The free energy is found to be equivalent to the partition function of a string where the world sheet is discretized by even polygons with signature and the link factor is given by a non-Gaussian propagator.   At large, but finite, N we derive the nonperturbative density of states from the WKB wave function and the dispersion relations. This is expressible as an infinite, but convergent, series with the inverse of the hypergeometric function replacing the harmonic oscillator spectrum of the $1+1$-dimensional string. In the scaling limit, the series is shown to be finite, containing both the perturbative (asymptotic) expansion of the inverted harmonic oscillator model, and a nonperturbative piece that survives the scaling limit. 
  We construct bosonized vertex operators (VOs) and conjugate vertex operators (CVOs) of $U_q(su(2)_k)$ for arbitrary level $k$ and representation $j\leq k/2$. Both are obtained directly as two solutions of the defining condition of vertex operators - namely that they intertwine $U_q(su(2)_k)$ modules. We construct the screening charge and present a formula for the n-point function. Specializing to $j=1/2$ we construct all VOs and CVOs explicitly. The existence of the CVO allows us to place the calculation of the two-point function on the same footing as $k=1$; that is, it is obtained without screening currents and involves only a single integral from the CVO. This integral is evaluated and the resulting function is shown to obey the q-KZ equation and to reduce simply to both the expected $k=1$ and $q=1$ limits. 
  We present a set of quantum-mechanical Hamiltonians which can be written as the $F^{\,\rm th}$ power of a conserved charge: $H=Q^F$ with $[H,Q]=0$ and $F=2,3,...\, .$ This new construction, which we call {\it fractional}\/ supersymmetric quantum mechanics, is realized in terms of \pg\ variables satisfying $\t^F=0$. Furthermore, in a pseudo-classical context, we describe {\it fractional}\/ supersymmetry transformations as the $F^{\,\rm th}$ roots of time translations, and provide an action invariant under such transformations. 
  Recently, we presented a new class of quantum-mechanical Hamiltonians which can be written as the $F^{th}$ power of a conserved charge: $H=Q^F$ with $F=2,3,...\,.$ This construction, called fractional supersymmetric quantum mechanics, was realized in terms of a paragrassmann variable $\theta$ of order $F$, which satisfies $\theta^F=0$. Here, we present an alternative realization of such an algebra in which the internal space of the Hamiltonians is described by a tensor product of two paragrassmann variables of orders $F$ and $F-1$ respectively. In particular, we find $q$-deformed relations (where $q$ are roots of unity) between different conserved charges. (To appear in "Mod.Phys.Lett.A") 
  Supersymmetric (pseudo-classical) mechanics has recently been generalized to {\it fractional}\/ supersymmetric mechanics. In such a construction, the action is invariant under fractional supersymmetry transformations, which are the $F^{\,\rm th}$ roots of time translations (with $F=1,2,...$). Associated with these symmetries, there are conserved charges with fractional canonical dimension $1+1/F$. Using \pg\ variables satisfying $\t^F=0$, we present a fractional-superspace formulation of this construction. 
  The $N=\infty$ vector $O(N)$ model is a solvable, interacting field theory in three dimensions ($D$). In a recent paper with A. Chubukov and J. Ye~\cite{self}, we have computed a universal number, $\tilde{c}$, characterizing the size dependence of the free energy at the conformally-invariant critical point of this theory. The result~\cite{self} for $\tilde{c}$ can be expressed in terms of polylogarithms. Here, we use non-trivial polylogarithm identities to show that $\tilde{c}/N = 4/5$, a rational number; this result is curiously parallel to recent work on dilogarithm identities in $D=2$ conformal theories. The amplitude of the stress-stress correlator of this theory, $c$ (which is the analog of the central charge), is determined to be $c/N=3/4$, also rational. Unitary conformal theories in $D=2$ always have $c = \tilde{c}$; thus such a result is clearly not valid in $D=3$. 
  The Green functions of the Chern-Simons theory quantized in the axial gauge are shown to be calculable as the unique, exact solution of the Ward identities which express the invariance of the theory under the topological supersymmetry of Delduc, Gieres and Sorella. 
  The ordinary Poisson brackets in field theory do not fulfil the Jacobi identity if boundary values are not reasonably fixed by special boundary conditions. We show that these brackets can be modified by adding some surface terms to lift this restriction. The new brackets generalize a canonical bracket considered by Lewis, Marsden, Montgomery and Ratiu for the free boundary problem in hydrodynamics. Our definition of Poisson brackets permits to treat boundary values of a field on equal footing with its internal values and directly estimate the brackets between both surface and volume integrals. This construction is applied to any local form of Poisson brackets. A prescription for delta-function on closed domains and a definition of the {\it full} variational derivative are proposed. 
  The free energy and local height probabilities of the dilute A models with broken $\Integer_2$ symmetry are calculated analytically using inversion and corner transfer matrix methods. These models possess four critical branches. The first two branches provide new realisations of the unitary minimal series and the other two branches give a direct product of this series with an Ising model. We identify the integrable perturbations which move the dilute A models away from the critical limit. Generalised order parameters are defined and their critical exponents extracted. The associated conformal weights are found to occur on the diagonal of the relevant Kac table. In an appropriate regime the dilute A$_3$ model lies in the universality class of the Ising model in a magnetic field. In this case we obtain the magnetic exponent $\delta=15$ directly, without the use of scaling relations. 
  We construct level-0 modules of the quantum affine algebra $\Uq$, as the $q$-deformed version of the Lie algebra loop module construction. We give necessary and sufficient conditions for the modules to be irreducible. We construct the crystal base for some of these modules and find significant differences from the case of highest weight modules. We also consider the role of loop modules in the recent scheme for diagonalising certain quantum spin chains using their $\Uq$ symmetry. 
  For generic $q$ we give expressions for the transformations of all essentially typical finite-dimensional modules of the Hopf superalgebra $U_q[gl(3/2)]$. The latter is a deformation of the universal enveloping algebra of the Lie superalgebra $gl(3/2)$. The basis within each module is similar to the Gel'fand-Zetlin basis for $gl(5)$. We write down expressions for the transformations of the basis under the action of the Chevalley generators. 
  We investigate the cosmological consequences of having quantum fields living in a space with compactified dimensions. We will show that the equation of state is not modified by topological effects and so the dynamics of the universe remains as it is in the infinite volume limit. On the contrary the thermal history of the universe depends on terms that are associated with having non-trivial topology. In the conclusions we discuss some issues about the relationship between the $c=1$ non-critical string-inspired cosmology and the result obtained with matter given by a hot massless field in $S^{1}\times \mbox{\bf   R}$. 
  We study the cosmological solutions of the two-dimensional Brans-Dicke equations considering a gas of $c=1$ strings in $S^{1}\times \mbox{\bf R}$ as the source of the gravitational field. We also study the implications of the $R$-duality invariance on the solutions. To this purpose we conjecture that, as it happens for massless fields in finite boxes, the free energy of a gas of massless string excitations is not given by the corresponding toroidal compactification. 
  We study toroidal compactifications of string theories which include compactification of a timelike coordinate. Some new features in the theory of toroidal compactifications arise. Most notably, Narain moduli space does not exist as a manifold since the action of duality on background data is ergodic. For special compactifications certain infinite dimensional symmetries, analogous to the infinite dimensional symmetries of the $2D$ string are unbroken. We investigate the consequences of these symmetries and search for a universal symmetry which contains all unbroken gauge groups. We define a flat connection on the moduli space of toroidally compactified theories. Parallel transport by this connection leads to a formulation of broken symmetry Ward identities. In an appendix this parallel transport is related to a definition of conformal perturbation theory. 
  We use the formal Lie algebraic structure in the ``space'' of hamiltonians provided by equal time commutators to define a Kirillov-Konstant symplectic structure in the coadjoint orbits of the associated formal group. The dual is defined via the natural pairing between operators and states in a Hilbert space. 
  In the framework of the $2(2S+1)$-- theory of Joos-Weinberg for massless particles, the dynamical invariants have been derived from the Lagrangian density which is considered to be a 4-- vector. A l\'a Majorana interpretation of the 6-- component spinors, the field operators of $S=1$ particles, as the left-- and right--circularly polarized radiation, leads us to the conserved quantities which are analogous to ones obtained by Lipkin and Sudbery. The scalar Lagrangian of Joos-Weinberg theory is shown to be equivalent to the Lagrangian of a free massless field, introduced by Hayashi. As a consequence of a new "gauge" invariance this skew-symmetric field describes physical particles with the longitudinal components only. 
  We present supersymmetric soliton solutions of the four-dimensional heterotic string corresponding to monopoles, strings and domain walls. These solutions admit the $D=10$ interpretation of a fivebrane wrapped around $5$, $4$ or $3$ of the $6$ toroidally compactified dimensions and are arguably exact to all orders in $\alpha'$. The solitonic string solution exhibits an $SL(2,Z)$ {\it strong/weak coupling} duality which however corresponds to an $SL(2,Z)$ {\it target space} duality of the fundamental string. 
  We derive an exact string-like soliton solution of D=10 heterotic string theory. The solution possesses $SU(2)\times SU(2)$ instanton structure in the eight-dimensional space transverse to the worldsheet of the soliton. 
  A generalization of the Yang-Baxter equation is proposed. It enables to construct integrable two-dimensional lattice models with commuting two-layer transfer matrices, while single-layer ones are not necessarily commutative. Explicit solutions to the generalized equations are found. They are related with Botzmann weights of the $sl(3)$ chiral Potts model. 
  As is known, tetrahedron equations lead to the commuting family of transfer-matrices and provide the integrability of corresponding three-dimensional lattice models. We present the modified version of these equations which give the commuting family of more complicated two-layer transfer-matrices. In the static limit we have succeeded in constructing the solution of these equations in terms of elliptic functions. 
  We construct a realization of the quantum affine algebra $U_q(\widehat{sl_N})$ of an arbitrary level $k$ in terms of free boson fields. In the $q\!\rightarrow\! 1$ limit this realization becomes the Wakimoto realization of $\widehat{sl_N}$. The screening currents and the vertex operators(primary fields) are also constructed; the former commutes with $U_q(\widehat{sl_N})$ modulo total difference, and the latter creates the $U_q(\widehat{sl_N})$ highest weight state from the vacuum state of the boson Fock space. 
  We show generally that in thermal gravity, the one-particle irreducible 2-point function depends on the choice of the basic graviton fields. We derive the relevant properties of a physical graviton self-energy, which is independent of the parametrization of the graviton field. An explicit expression for the graviton self-energy at high-temperature is given to one-loop order. 
  We show that the bosonic string theory quantized in the Beltrami parametrization possesses a supersymmetric structure like the vector-supersymmetry already observed in topological field theories. 
  We show that a large class of physical theories which has been under intensive investigation recently, share the same geometric features in their Hamiltonian formulation. These dynamical systems range from harmonic oscillations to WZW-like models and to the KdV dynamics on $Diff_oS^1$. To the same class belong also the Hamiltonian systems on groups of maps.   The common feature of these models are the 'chiral' equations of motion allowing for so-called chiral decomposition of the phase space. 
  Quantum fluctuations, through quantum corrections, have the potential to lead to irreversibility in quantum field theory. We consider the virtual ``charge" distribution generated by quantum corrections in the leading log, short range approximation, and adopt for it a statistical interpretation. This virtual charge density has fractal structure, and it is seen that, independently of whether the theory is or is not asymptotically free, it describes a system where the equilibrium state is at its classical limit ($\hbar \rightarrow 0$). We also present a simple analysis of how diffusion of the charge density proceeds as a function of the distance at which the system is probed. 
  The dynamics of the 3 dimensional perfect fluid is equivalent to the motion of vortex filaments or "strings". We study the action principle and find that it is described by the Hopf term of the nonlinear sigma model. The Poisson bracket structure is described by the loop algebra, for example, the Virasoro algebra or the analogue of O(3) current algebra. As a string theory, it is quite different from the standard Nambu-Goto string in its coupling to the extrinsic geometry. We also analyze briefly the two dimsensional case and give some emphasis on the $w_{1+\infty}$ structure. 
  We discuss an infinite--dimensional k\"ahlerian manifold associated with the area--preserving diffeomorphisms on two--dimensional torus, and, correspondingly, with a continuous limit of the $A_r$--Toda system. In particular, a continuous limit of the $A_r$--Grassmannians and a related Pl\"ucker type formula are introduced as relevant notions for $W_{\infty}$--geometry of the self--dual Einstein space with the rotational Killing vector. 
  Operator angle-action variables are studied in the frame of the SU(2) algebra, and their eigenstates and coherent states are discussed. The quantum mechanical addition of action-angle variables is shown to lead to a novel non commutative Hopf algebra. The group contraction is used to make the connection with the harmonic oscillator. 
  Affine Toda theory is a relativistic integrable theory in two dimensions possessing solutions describing a number of different species of solitons when the coupling is chosen to be imaginary. These nevertheless carry real energy and momentum. To each species of soliton there has to correspond an antisoliton species. There are two different ways of realising the antisoliton whose equivalence is shown to follow from a surprising identity satisfied within the underlying affine Kac-Moody group. This is the classical analogue of the crossing property of analytic S-matrix theory. Since a complex parameter related to the coordinate of the soliton is inverted, this identity implies a sort of modular transformation property of the soliton solution. The results simplify calculations of explicit soliton solutions. 
  We investigate the canonical quantization of gravity coupled to pointlike matter in 2+1 dimensions. Starting from the usual point particle action in the first order formalism, we introduce auxiliary variables which make the action locally Poincar\'e invariant. A Hamiltonian analysis shows that the gauge group is actually larger than the Poincar\'e group -- certain additional gauge constraints are present which act on the matter degrees of freedom. These additional constraints are necessary to mimic the diffeomorphism invariance present if the theory is formulated with a spacetime metric. The additional gauge constraints are realized projectively in the quantum theory, with a phase in the composition law for finite gauge transformations. That phase is responsible for the braid invariance of physical observables (holonomies). 
  Starting from the characteristic polynomial for ordinary matrices we give a combinatorial deduction of the Mandelstam identities and viceversa, thus showing that the two sets of relations are equivalent. We are able to extend this construction to supermatrices in such a way that we obtain the Mandelstam identities in this case, once the corresponding characteristic equation is known. 
  A new local world volume supersymmetric Lagrangian for the bosonic membrane is presented. The starting Lagrangian is the one constructed by Dolan and Tchrakian with vanishing cosmological constant, with quadratic and quartic derivative terms. Our Lagrangian differs from the one constructed by Lindstrom and Rocek in the fact that it is polynomial in the fields facilitating the quantization process. It is argued, rigorously, that if one wishes to construct polynomial actions without a curvature scalar term and, where supersymmetry is linearly realized in the space of physical fields, after the elimination of auxiliary fields, one must relinquish $S$ supersymmetry, altogether, and concentrate solely on the $Q$ supersymmetry associated with the superconformal algebra in three dimensions. A full $''Q+S''$ supersymmetry cannot be implemented in a linearly realized way satisfying all of the above-mentioned requirements, unless a non-polynomial action is chosen.   PACS:04.65.+e, 04.20.Fy. 
  We develop a stochastic approach to a non de Sitter Universe in a gauge-invariant way and obtain a system of Langevin-type equations which may be considered to be renormalization group equations for the long wave parts of the scalar fields and metric. We investigate in detail the case of generalized power-law inflation that appears in the model of the deflationary exit from the inflationary stage. For this case the above system is simplified greatly, we derive a Fokker-Planck equation for the scalar field driving inflation, find its stationary solutions and solve the FP equation for a delta-function initial distribution of the field. 
  We generalize the stochastic approach to quasi-power-law inflationary Universes,obtain the corresponding Langevin and Fokker-Planck equations for the scalar field driving inflation and find stationary solutions to the above FP equation. 
  Affine Toda theories with imaginary couplings associate with any simple Lie algebra ${\bf g}$ generalisations of Sine Gordon theory which are likewise integrable and possess soliton solutions. The solitons are \lq\lq created" by exponentials of quantities $\hat F^i(z)$ which lie in the untwisted affine Kac-Moody algebra ${\bf\hat g}$ and ad-diagonalise the principal Heisenberg subalgebra. When ${\bf g}$ is simply-laced and highest weight irreducible representations at level one are considered, $\hat F^i(z)$ can be expressed as a vertex operator whose square vanishes. This nilpotency property is extended to all highest weight representations of all affine untwisted Kac-Moody algebras in the sense that the highest non vanishing power becomes proportional to the level. As a consequence, the exponential series mentioned terminates and the soliton solutions have a relatively simple algebraic expression whose properties can be studied in a general way. This means that various physical properties of the soliton solutions can be directly related to the algebraic structure. For example, a classical version of Dorey's fusing rule follows from the operator product expansion of two $\hat F$'s, at least when ${\bf g}$ is simply laced. This adds to the list of resemblances of the solitons with respect to the particles which are the quantum excitations of the fields. 
  We study the anyon statistics of a $2 + 1$ dimensional Maxwell-Chern-Simons (MCS) gauge theory by using a systemmetic metheod, the Breit Hamiltonian formalism. 
  It is shown that the joint measurements of some physical variables corresponding to commuting operators performed on pre- and post-selected quantum systems invariably disturb each other. The significance of this result for recent proofs of the impossibility of realistic Lorentz invariant interpretation of quantum theory (without assumption of locality) is discussed. 
  We investigate the relationship between the generalized uncertainty principle in quantum gravity and the quantum deformation of the Poincar\'e algebra. We find that a deformed Newton-Wigner position operator and the generators of spatial translations and rotations of the deformed Poincar\'e algebra obey a deformed Heisenberg algebra from which the generalized uncertainty principle follows. The result indicates that in the $\kappa$-deformed Poincar\'e algebra a minimal observable length emerges naturally. 
  We give the general solution of the stationary problem of 2+1 dimensional gravity in presence of extended sources, also endowed with angular momentum. We solve explicitly the compact support property of the energy momentum tensor and we apply the results to the study of closed time-like curves. In the case of rotational symmetry we prove that the weak energy condition combined with the absence of closed time-like curves at space infinity prevents the existence of closed time-like curves everywhere in an open universe (conical space at infinity). 
  The formation and evaporation of two dimensional black holes are discussed. It is shown that if the radiation in minimal scalars has positive energy, there must be a global event horizon or a naked singularity. The former would imply loss of quantum coherence while the latter would lead to an even worse breakdown of predictability. CPT invariance would suggest that there ought to be past horizons as well. A way in which this could happen with wormholes is described. 
  In this letter we present an operator formalism for Closed String Field Theory based on closed half-strings. Our results indicate that the restricted polyhedra of the classical non-polynomial string field theory, can be represented as traces of infinite matrices, with operator insertions that reparametrise the half-strings. 
  The unified constrained dynamics is formulated without making use of the Dirac splitting of constraint classes. The strengthened, completely--closed, version of the unified constraint algebra generating equations is given. The fundamental phase variable supercommutators are included into the unified algebra as well. The truncated generating operator is defined to be nilpotent in terms of which the Unitarizing Hamiltonian is constructed. 
  Recently \REF\dk{Simon Dalley and Igor Klebanov,'Light Cone Quantization of the $c=2$ Matrix Model', PUPT-1333, hepth@xxx/920705} \refend Dalley and Klebanov proposed a light-cone quantized study of the $c=2$ matrix model, but which ignores $k^{+}=0$ contributions. Since the non-critical string limit of the matrix model involves taking the parameters $\lambda$ and $\mu$ of the matrix model to a critical point, zero modes of the field might be important in this study. The constrained light-cone quantization (CLCQ) approach of Heinzl, Krusche and Werner is applied . It is found that there is coupling between the zero mode sector and the rest of the theory, hence CLCQ should be implemented. 
  A higher dimensional analogue of the KP hierarchy is presented. Fundamental constituents of the theory are pseudo-differential operators with Moyal algebraic coefficients. The new hierarchy can be interpreted as large-$N$ limit of multi-component ($\gl(N)$ symmetric) KP hierarchies. Actually, two different hierarchies are constructed. The first hierarchy consists of commuting flows and may be thought of as a straightforward extension of the ordinary and multi-component KP hierarchies. The second one is a hierarchy of noncommuting flows, and related to Moyal algebraic deformations of selfdual gravity. Both hierarchies turn out to possess quasi-classical limit, replacing Moyal algebraic structures by Poisson algebraic structures. The language of W-infinity algebras provides a unified point of view to these results. 
  The discussions on the modular invariance in section 5 are refined. 
  We consider the N to infinity limits of the N-state chiral Potts model. We find new weights that satisfy the star-triangle relations with spin variables either taking all the integer values or having values from a continous interval. The models provide chiral generalizations of Zamolodchikov's Fishnet Model. (For the more complete version, see math.QA/9906029, where the misprints in eq. (12) are also corrected.) 
  In a four dimensional theory of gravity with lagrangian quadratic in curvature and torsion, we compute the effective action for metrics of the form $g_{\mu\nu}=\rho^2\delta_{\mu\nu}$, with $\rho$ constant. Using standard field-theoretic methods we find that one loop quantum effects produce a nontrivial effective potential for $\rho$. We explain this unexpected result by showing how our regularization procedure differs from the one that is usually adopted in Quantum Gravity. Using the method of the average effective potential, we compute the scale dependence of the v.e.v. of the conformal factor. 
  We use Chen iterated line integrals to construct a topological algebra ${\cal A}_p$ of separating functions on the {\it Group of Loops} ${\bf L}{\cal M}_p$. ${\cal A}_p$ has an Hopf algebra structure which allows the construction of a group structure on its spectrum. We call this topological group, the group of generalized loops $\widetilde {{\bf L}{\cal M}_p}$.   Then we develope a {\it Loop Calculus}, based on the {\it Endpoint} and {\it Area Derivative Operators}, providing a rigorous mathematical treatment of early heuristic ideas of Gambini, Trias and also Mandelstam, Makeenko and Migdal. Finally we define a natural action of the "pointed" diffeomorphism group $Diff_p({\cal M})$ on $ \widetilde {{\bf L}{\cal M}_p}$, and consider a {\it Variational Derivative} which allows the construction of homotopy invariants.   This formalism is useful to construct a mathematical theory of {\it Loop Representation} of Gauge Theories and Quantum Gravity. Figures available by request. 
  We consider a BRST approach to G/H coset WZNW models, {\it i.e.} a formulation in which the coset is defined by a BRST condition. We will give the precise ingrediences needed for this formulation. Then we will prove the equivalence of this approach to the conventional coset formulation by solving the the BRST cohomology. This will reveal a remarkable connection between integrable representations and a class of non-integrable representations for negative levels. The latter representations are also connected to string theories based on non-compact WZNW models. The partition functions of G/H cosets are also considered. The BRST approach enables a covariant construction of these, which does not rely on the decomposition of G as $G/H\times H$. We show that for the well-studied examples of $SU(2)_k \times SU(2)_1/SU(2)_{k+1}$ and $SU(2)_k/U(1)$, we exactly reproduce the previously known results. 
  To overcome the difficulties with the energy indefiniteness in field theories with higher derivatives, it is supposed to use the mechanical analogy, the Timoshenko theory of the transverse flexural vibrations of beams or rods well known in mechanical engineering. It enables one to introduce the notion of a "mechanical" energy in such field models that is wittingly positive definite. This approach can be applied at least to the higher derivative models which effectively describe the extended localized solutions in usual first order field theories (vortex solutions in Higgs models and so on). Any problems with a negative norm ghost states and unitarity violation do not arise here. 
  To construct a quantum group gauge theory one needs an algebra which is invariant under gauge transformations. The existence of this invariant algebra is closely related with the existence of a differential algebra $\delta _{{\cal H}} G_{q}$ compatible with the Hopf algebra structure. It is shown that $\delta _{{\cal H}} G_{q}$ exists only for the quantum group $U_{q}(N)$ and that the quantum group $SU_q(N)$ as a quantum gauge group is not allowed. The representations of the algebra $\delta _{{\cal H}} G_{q}$ are con- structed. The operators corresponding to the differentials are realized via derivations on the space of all irreducible *-representations of $U_q(2)$. With the help of this construction infinitesimal gauge transformations in two-dimensional classical space-time are described. 
  Two integrals along the world trajectory of its curvature and torsion are added to the standard action for the point-like spinless relativistic particle. Since here the three-dimensional space-time is considered at the beginning, the torsion of the world curve is defined with a sign in contrast to the previous consideration: V. V. Nesterenko, J. Math. Phys. 32, 3315 (1991). Upon obtaining a complete set of constraints in the phase space a generalized Hamiltonian description of a new version of the model is constructed. This enables one to quantize the model canonically and to derive exactly the relation between the spin and mass of the states. 
  For the quark-gluon plasma, an energy-momentum tensor is found corresponding to the high-temperature Braaten-Pisarski effective action. The tensor is found by considering the interaction of the plasma with a weak gravitational field and the positivity of the energy is studied. In addition, the complete effective action in curved spacetime is written down. 
  We study the BRST cohomology for $SL(2,R)/U(1)$ coset model, which describes an exact string black hole solution. It is shown that the physical spectrum could contain not only the extra discrete states corresponding to those in $c=1$ two-dimensional gravity but also many additional new states with ghost number $N_{FP}= -1 \sim 2$. We also discuss characters for nonunitary representations and the relation of our results to other approaches. 
  Representations of Quantum Groups U_q (g_n), g_n any semi simple Lie algebra of rank n, are constructed from arbitrary representations of rank n-1 quantum groups for q a root of unity. Representations which have the maximal dimension and number of free parameters for irreducible representations arise as special cases. 
  The transparent way for the invariant (Hamiltonian) description of equivariant localization of the integrals over phase space is proposed. It uses the odd symplectic structure, constructed over tangent bundle of the phase space and permits straightforward generalization for the path integrals. Simultaneously the method of supersymmetrization for a wide class of the Hamiltonian systems is derived. 
  Recently, a class of solvable interaction round the face lattice models (IRF) were constructed for an arbitrary rational conformal field theory (RCFT) and an arbitrary field in it. The Boltzmann weights of the lattice models are related in the extreme ultra violet limit to the braiding matrices of the rational conformal field theory. In this note we use these new lattice models to construct a link invariant for any such pair of an RCFT and a field in it. Using the properties of RCFT and the IRF lattice models, we prove that the invariants so constructed always obey the Markov properties, and thus are true link invariants. Further, all the known link invariants, such as the Jones, HOMFLY and Kauffman polynomials arise in this way, along with giving a host of new invariants, and thus also a unified approach to link polynomials. It is speculated that all link invariants arise from some RCFT, and thus the problem of classifying link and knot invariants is equivalent to that of classifying two dimensional conformal field theory. 
  In the present paper we construct all typical finite-dimensional representations of the quantum Lie superalgebra $U_{q}[gl(2/2)]$ at generic deformation parameter $q$. As in the non-deformed case the finite-dimensional $U_{q}[gl(2/2)]$-module $W^{q}$ obtained is irreducible and can be decomposed into finite-dimensional irreducible $U_{q}[gl(2)\oplus gl(2)]$-submodules $V^{q}_{k}$ 
  This paper is devoted to a proof of a generalized Ray-Singer conjecture for a manifold with boundary (the Dirichlet and the Neumann boundary conditions are independently given on each connected component of the boundary and the transmission boundary condition is given on the interior boundary). The Ray-Singer conjecture \cite{RS} claims that for a closed manifold the combinatorial and the analytic torsion norms on the determinant of the cohomology are equal. For a manifold with boundary the ratio between the analytic torsion and the combinatorial torsion is computed. Some new general properties of the Ray-Singer analytic torsion are found. The proof does~not use any computation of eigenvalues and its asymptotic expansions or explicit expressions for the analytic torsions of any special classes of manifolds. 
  Target space duality (T duality), which interchanges Kaluza--Klein and winding-mode excitations of the compactified heterotic string, is realized as a symmetry of a world-sheet action. Axion-dilaton duality (S duality), a conjectured nonperturbative SL(2,Z) symmetry of the same theory, plays an analogous role for five-branes. We describe a soliton spectrum possessing both duality symmetries and argue that the theory has an infinite number of dual string descriptions. 
  We investigate the critical behavior of the gauged NJL model (QED plus 4-fermion interaction) and the gauged Yukawa model by use of the inversion method.   By calculating the gauge-invariant chiral condensate in the inversion method to the lowest order, we derive the critical line which separates the spontaneous chiral-symmetry breaking phase from the chiral symmetric one. The critical exponent for the chiral order parameter associated with the second order chiral phase transition is shown to take the mean-field value together with possible logarithmic correction to the mean-field prediction.   All the above results are gauge-parameter independent and are compared with the previous results obtained from the Schwinger-Dyson equation for the fermion propagator. 
  Integral and differential mass formulae of 4-dimensional stationary and axisymmetric Einstein-Maxwell-dilaton systems are derived. The total mass (energy) of these systems are expressed in terms of other physical quantities such as electric charge of the black hole suitably modified due to the existence of the dilaton field. It is shown that when we vary slightly the fields (metric of the spacetime $g_{\mu\nu}$, $U(1)-$gauge potential $A_{\mu}$, and dilaton $\phi$) in such a way as they obey classical equations of motion, the variation of the dilaton does not contribute explicitly to the variation of the total mass, but contributes only through the variation of the electric charge of the black hole. 
  We study integrals of motion and factorizable S-matrices in two-dimensional integrable field theory with boundary. We propose the ``boundary cross-unitarity equation'' which is the boundary analog of the cross-symmetry condition of the ``bulk'' S-matrix. We derive the boundary S-matrices for the Ising field theory with boundary magnetic field and for the boundary sine-Gordon model. 
  We consider a pair of coupled non-linear partial differential equations describing a biochemical model system. The Weiss-algorithm for the Painle\'{e} test, that has been succesfully used in mathematical physics for the KdV-equation, Burgers equation, the sine-Gordon equation etc., is applied, and we find that the system possesses only the "conditional" Painlev\'{e} property. We use the outcome of the analysis to construct an auto-B\"{a}cklund transformation, and we find a variety of one and two-parameter families of special solutions. 
  We study bosons interacting with an abelian Chern-Simons field on Riemann surfaces of genus $g>0$. It is shown that a singular gauge transformation brings the hamiltonian to free form. The transformed wave functions furnish a multi-component representation of the braid group studied by Imbo and March-Russell. The construction constitutes a proof of the equivalence of bosons coupled to a Chern-Simons field and anyons and generalizes the well known equivalence of the two pictures on the plane. 
  New development of the theory of Grothendieck polynomials, based on an exponential solution of the Yang-Baxter equation in the algebra of projectors are given. 
  We study the interactions of non-abelian vortices in two spatial dimensions. These interactions have novel features, because the Aharonov-Bohm effect enables a pair of vortices to exchange quantum numbers. The cross section for vortex-vortex scattering is typically a multi-valued function of the scattering angle. There can be an exchange contribution to the vortex-vortex scattering amplitude that adds coherently with the direct amplitude, even if the two vortices have distinct quantum numbers. Thus two vortices can be ``indistinguishable'' even though they are not the same. 
  We calculate in this article the transport coefficients which characterize the dynamics of solitons in quantum field theory using the methods of dissipative quantum systems. We show how the damping and diffusion coefficients of soliton-like excitations can be calculated using the integral functional formalism. The model obtained in this article has new features which cannot be obtained in the standard models of dissipation in quantum mechanics. 
  We discuss gauge transformations in QED coupled to a charged spinor field, and examine whether we can gauge-transform the entire formulation of the theory from one gauge to another, so that not only the gauge and spinor fields, but also the forms of the operator-valued Hamiltonians are transformed. The discussion includes the covariant gauge, in which the gauge condition and Gauss's law are not primary constraints on operator-valued quantities; it also includes the Coulomb gauge, and the spatial axial gauge, in which the constraints are imposed on operator-valued fields by applying the Dirac-Bergmann procedure. We show how to transform the covariant, Coulomb and spatial axial gauges to what we call   ``common form,'' in which all particle excitation modes have identical properties. We also show that, once that common form has been reached, QED in different gauges has a common time-evolution operator that defines time-translation for states that represent systems of electrons and photons.   By combining gauge transformations with changes of representation from standard to common form, the entire apparatus of a gauge theory can be transformed from one gauge to another. 
  We propose a generalization of Heisenberg picture quantum mechanics in which a Lagrangian and Hamiltonian dynamics is formulated directly for dynamical systems on a manifold with non--commuting coordinates, which act as operators on an underlying Hilbert space. This is accomplished by defining the Lagrangian and Hamiltonian as the real part of a graded total trace over the underlying Hilbert space, permitting a consistent definition of the first variational derivative with respect to a general operator--valued coordinate. The Hamiltonian form of the equations is expressed in terms of a generalized bracket operation, which is conjectured to obey a Jacobi identity. The formalism permits the natural implementation of gauge invariance under operator--valued gauge transformations. When an operator Hamiltonian exists as well as a total trace Hamiltonian, as is generally the case in complex quantum mechanics, one can make an operator gauge transformation from the Heisenberg to the Schr\"odinger picture. When applied to complex quantum mechanical systems with one bosonic or fermionic degree of freedom, the formalism gives the usual operator equations of motion, with the canonical commutation relations emerging as constraints associated with the operator gauge invariance. More generally, our methods permit the formulation of quaternionic quantum field theories with operator--valued gauge invariance, in which we conjecture that the operator constraints act as a generalization of the usual canonical commutators. 
  We give the Heisenberg realization for the quantum algebra $U_q(sl_n)$, which is written by the $q$-difference operator on the flag manifold. We construct it from the action of $U_q(sl_n)$ on the $q$-symmetric algebra $A_q(Mat_n)$ by the Borel-Weil like approach. Our realization is applicable to the construction of the free field realization for the $U_q(\widehat{sl_n})$ [AOS]. 
  The relation between Geisteswissenschaft and Naturwissenschaft has been discussed by Munster in hep-th/9305104. The plan of this paper is to begin with the empty set; use it to form sets and quivers (sets of points plus sets of arrows between pairs of points); and then use them to make complex vector spaces and to get the A-D-E Coxeter-Dynkin diagrams. The Dn Spin(2n) Lie algebras have spinor representations to describe fermions. D4 Spin(8) triality gives automorphisms among its vector and two half-spinor representations. D5 Spin(10) contains both Spin(8) and the complexification of the vector representation of Spin(8). E6 contains both Spin(10) and the two half-spinor representations of Spin(10), and therefore contains the adjoint representation of Spin(8) and the complexifications of the vector and the two half-spinor representations of Spin(8). E6 is the basis for construction of a fundamental model of physics that is consistent with experiment (see hep-th/9302030, hep-ph/9301210). 
  The 2D lattice gauge theory with a quantum gauge group $SL_q(2)$ is considered. When $q=e^{i\frac{2\pi}{k+2}}$, its weak coupling partition function coincides with the one of the G/G coset model ({\em i.e.} equals the Verlinde numbers). However, despite such a remarkable coincidence, these models are not equivalent but, in some certain sense, dual to each other. 
  We study $2d$ QCD coupled to fermions in the adjoint representation of the gauge group $SU(N)$ at large $N$, and its relation to string theory. It is shown that the model undergoes a deconfinement transition at a finite temperature (analogous to the Hagedorn transition in string theory), with certain winding modes in the Euclidean time direction turning tachyonic at high temperature. The theory is supersymmetric for a certain ratio of quark mass and gauge coupling. For other values of that ratio, supersymmetry is softly broken. The spectrum of bound states contains an infinite number of approximately linear Regge trajectories, approaching at large mass $M$, $\alpha^\prime M^2=\sum_i i l_i$ $(l_i\in{\bf Z_+})$. Thus, the theory exhibits an exponentially growing density of bosonic and fermionic states at high energy. We discuss these results in light of string expectations. 
  Different aspects of the Verlinde and Verlinde relation between high-energy effective scattering in QCD and a two-dimensional sigma-model are discussed. Starting from a lattice version of the truncated 4-dimensional Yang-Mills action we derive an effective theory with non-trivial longitudinal dynamics which has a form of the lattice two-dimensional chiral field model with non-trivial boundary conditions. To get quantum corrections coming from non-trivial longitudinal dynamics to transversal high-energy effective action one has to solve the two-dimensional chiral field model with non-trivial boundary conditions. We do this within an approximation scheme which takes into account one-dimensional excitations. Contributions of the one-dimensional excitations to quantum corrections for the high-energy effective action are calculated in the large N limit using the character expansion method. 
  Recent developments in unifying treatment of domain wall configurations and their global space-time structure is presented. Domain walls between vacua of non-equal cosmological constant fall in three classes depending on the value of their energy density $\sigma$: (i) extreme walls with $\sigma=\sigma_{ext}$ are planar, static walls corresponding to the supersymmetric configurations, (ii) non-extreme walls with $\sigma>\sigma_{ext}$ are expanding bubbles with two insides, (iii) ultra-extreme walls with $\sigma<\sigma_{ext}$ are bubbles of false vacuum decay. As a prototype exhibiting all three types of configurations vacuum walls between Minkowski and anti-deSitter vacua are discussed. Space-times associated with these walls exhibit non-trivial causal structure closely related to the one of the corresponding extreme and non-extreme charged black holes, however, without singularities. Recently discovered extreme dilatonic walls, pertinent to string theory, are also addressed. They are static, planar domain walls with metric in the string frame being {\it flat} everywhere. Intriguing similarities between the global space-time of dilatonic walls and that of charged dilatonic black holes are pointed out. 
  The observation that $n$ pairs of para-Bose (pB) operators generate the universal enveloping algebra of the orthosymplectic Lie superalgebra $osp(1/2n)$ is used in order to define deformed pB operators. It is shown that these operators are an alternative to the Chevalley generators. On this background $U_q[osp(1/2n)]$, its "Cartan-Weyl" generators and their "supercommutation" relations are written down entirely in terms of deformed pB operators. An analog of the Poincare- Birkhoff-Witt theorem is formulated. 
  The constrained Hamiltonian systems admitting no gauge conditions are considered. The methods to deal with such systems are discussed and developed. As a concrete application, the relationship between the Dirac and reduced phase space quantizations is investigated for spin models belonging to the class of systems under consideration. It is traced out that the two quantization methods may give similar, or essentially different physical results, and, moreover, a class of constrained systems, which can be quantized only by the Dirac method, is discussed. A possible interpretation of the gauge degrees of freedom is given. 
  We show that the connection between certain integrable perturbations of $N=2$ superconformal theories and graphs found by Lerche and Warner extends to a broader class. These perturbations are such that the generators of the perturbed chiral ring may be diagonalized in an orthonormal basis. This allows to define a dual ring, whose generators are labelled by the ground states of the theory and are encoded in a graph or set of graphs, that reproduce the pattern of the ground states and interpolating solitons. All known perturbations of the $ADE$ potentials and some others are shown to satisfy this criterion. This suggests a test of integrability. 
  We give the explicite form of the BRST charge Q for the algebra W_4=WA_3 in the basis where the spin-3 and the spin-4 field are primary as well as for a basis where the algebra closes quadratically. 
  Local and global properties of the moduli space of Calabi--Yau type compactifications determine the low energy parameters of the string effective action. We show that the moduli space geometry is entirely encoded in the Picard--Fuchs equations for the periods of the Calabi--Yau $H^{(3)}$--cohomology. 
  We give the operadic formulation of (weak, strong) topological vertex algebras, which are variants of topological vertex operator algebras studied recently by Lian and Zuckerman. As an application, we obtain a conceptual and geometric construction of the Batalin-Vilkovisky algebraic structure (or the Gerstenhaber algebra structure) on the cohomology of a topological vertex algebra (or of a weak topological vertex algebra) by combining this operadic formulation with a theorem of Getzler (or of Cohen) which formulates Batalin-Vilkovisky algebras (or Gerstenhaber algebras) in terms of the homology of the framed little disk operad (or of the little disk operad). 
  A generalization of the differential geometry of forms and vector fields to the case of quantum Lie algebras is given. In an abstract formulation that incorporates many existing examples of differential geometry on quantum groups, we combine an exterior derivative, inner derivations, Lie derivatives, forms and functions all into one big algebra. In particular we find a generalized Cartan identity that holds on the whole quantum universal enveloping algebra of the left-invariant vector fields and implicit commutation relations for a left-invariant basis of 1-forms. 
  The phase structure of Nambu-Jona-Lasinio model with N-component fermions in curved space-time is studied in the leading order of the 1/N expansion.   The effective potential for composite operator $\bar{\psi}\psi$ is calculated by using the normal coordinate expansion in the Schwinger proper-time method.   The existence of the first-order phase transition caused by the change of the space-time curvature is confirmed and the dynamical mass of the fermion is calculated as a simultaneous function of the curvature and the four-fermion coupling constant.   The phase diagram in the curvature and the coupling constant is obtained. 
  The partition function of the discretized superstring in a target superspace of three (Euclidean) bosonic dimensions, is shown, for a fixed triangulation of the random world sheet, to be derived from the partition function of a discretized bosonic string with an external field present in the action in the form of a specific constant matrix, using first order forms of the actions. This latter partition function appears more amenable to an exact analytical treatment. 
  Starting from the well-known quantum Miura transformation for the Lie algebra $A_n$, we compute explicitly the OPEs for $n=3$ and 4. The primary fields with spin 3, 4 and 5 are found (for general $n$). By using these primary fields and the OPEs from quantum Miura transformation, we derive the complete structure of the nonlinear $W_4$ and $W_5$ algebras. 
  We construct the BRST operator for the nonlinear $WB_2$ and $W_4$ algebras. Contrary to the general belief, the nilpotent condition of the BRST operator doesn't determine all the coefficients. We find a three and seven parameter family of nilpotent BRST operator for $WB_2$ and $W_4$ respectively. These free parameters are related to the canonical transformation of the ghost antighost fields. 
  We calculate the one-loop perturbative correction to the coefficient of the \cs term in non-abelian gauge theory in the presence of Higgs fields, with a variety of symmetry-breaking structures. In the case of a residual $U(1)$ symmetry, radiative corrections do not change the coefficient of the \cs term. In the case of an unbroken non-abelian subgroup, the coefficient of the relevant \cs term (suitably normalized) attains an integral correction, as required for consistency of the quantum theory. Interestingly, this coefficient arises purely from the unbroken non-abelian sector in question; the orthogonal sector makes no contribution. This implies that the coefficient of the \cs term is a discontinuous function over the phase diagram of the theory. 
  We use the recently conjectured exact $S$-matrix of the massive ${\rm O}(n)$ model to derive its form factors and ground state energy. This information is then used in the limit $n\to0$ to obtain quantitative results for various universal properties of self-avoiding chains and loops. In particular, we give the first theoretical prediction of the amplitude ratio $C/D$ which relates the mean square end-to-end distance of chains to the mean square radius of gyration of closed loops. This agrees with the results from lattice enumeration studies to within their errors, and gives strong support for the various assumptions which enter into the field theoretic derivation. In addition, we obtain results for the scaling function of the structure factor of long loops, and for various amplitude ratios measuring the shape of self-avoiding chains. These quantities are all related to moments of correlation functions which are evaluated as a sum over $m$-particle intermediate states in the corresponding field theory. We show that in almost all cases, the restriction to $m\leq2$ gives results which are accurate to at least one part in $10^3$. This remarkable fact is traced to a softening of the $m>2$ branch cuts relative to their behaviour based on phase space arguments alone, a result which follows from the threshold behaviour of the two-body $S$-matrix, $S(0)=-1$. Since this is a general property of interacting 2d field theories, it suggests that similar approximations may well hold for other models. However, we also study the moments of the area of self-avoiding loops, 
  We give the general form of the vertex corresponding to the interaction of an arbitrary number of strings. The technique employed relies on the ``comma" representation of String Field Theory where string fields and interactions are represented as matrices and operations between them such as multiplication and trace. The general formulation presented here shows that the interaction vertex of N strings, for any arbitrary N, is given as a function of particular combinations of matrices corresponding to the change of representation between the full string and the half string degrees of freedom. 
  We show that the isotropic harmonic oscillator in the ordinary euclidean space ${\bf R}^N$ ($N\ge 3$) admits a natural q-deformation into a new quantum mechanical model having a q-deformed symmetry (in the sense of quantum groups), $SO_q(N,{\bf R})$. The q-deformation is the consequence of replacing $ R^N$ by ${\bf R}^N_q$ (the corresponding quantum space). This provides an example of quantum mechanics on a noncommutative geometrical space. To reach the goal, we also have to deal with a sensible definition of integration over ${\bf R}^N_q$, which we use for the definition of the scalar product of states. 
  We provide a proof of a formula conjectured in \cite{OU93} for some coefficients relevant in the principal vertex operator construction of a simply-laced affine algebra $\gh$. These coefficients are important for the study of the topological charges of the solitons of affine Toda theories, and the construction of representations of non-simply-laced $\gh$ and their associated Toda solitons. 
  D'Eath's proof (hep-th/9304084) that there can be at most two allowed quantum states of N=1 supergravity with zero or a finite number of fermions can be extended to show that there are no such states. 
  To any non-trivial embedding of sl(2) in a (super) Lie algebra, one can associate an extension of the Virasoro algebra. We realize the extended Virasoro algebra in terms of a WZW model in which a chiral, solvable group is gauged, the gauge group being determined by the sl(2) embedding. The resulting BRST cohomology is computed and the field content of the extended Virasoro algebra is determined. The (quantum) closure of the extended Virasoro algebra is shown. Applications such as the quantum Miura transformation and the effective action of the associated extended gravity theory are discussed in detail. 
  We consider the Ramond sector of the $N=1$ superconformal algebra and find expressions for the singular vectors in reducible highest weight Verma module representations by the fusion principle of Bauer et al. 
  The $2M$-boson representations of KP hierarchy are constructed in terms of $M$ mutually independent two-boson KP representations for arbitrary number $M$. Our construction establishes the multi-boson representations of KP hierarchy as consistent Poisson reductions of standard KP hierarchy within the $R$-matrix scheme. As a byproduct we obtain a complete description of any finitely-many-field formulation of KP hierarchy in terms of Darboux coordinates with respect to the first Hamiltonian structure. This results in a series of representations of $\Win1\,$ algebra made out of arbitrary even number of boson fields. 
  A method is developed to construct irreducible representations(irreps) of the quantum supergroup $U_q(C(n+1))$ in a systematic fashion. It is shown that every finite dimensional irrep of this quantum supergroup at generic $q$ is a deformation of a finite dimensional irrep of its underlying Lie superalgebra $C(n+1)$, and is essentially uniquely characterized by a highest weight. The character of the irrep is given. When $q$ is a root of unity, all irreps of $U_q(C(n+1))$ are finite dimensional; multiply atypical highest weight irreps and (semi)cyclic irreps also exist. As examples, all the highest weight and (semi)cyclic irreps of $U_q(C(2))$ are thoroughly studied. 
  We derive loop equations for the one-link correlators of gauge and scalar fields in the Kazakov-Migdal model. These equations determine the solution of the model in the large N limit and are similar to analogous equations for the Hermitean two-matrix model. We give an explicit solution of the equations for the case of a Gaussian, quadratic potential. We also show how similar calculations in a non-Gaussian case reduce to purely algebraic equations. 
  We perform a BRST analysis of the N=2 superconformal minimal unitary models. A bosonic as well as fermionic BRST operators are used to construct irreducible representations of the N=2 superconformal algebra on the Fock space as BRST cohomology classes of the BRST operators. Also a character formula is rederived by using the BRST analysis. 
  The statistical mechanics of an anyon gas in a magnetic field is addressed. An harmonic regulator is used to define a proper thermodynamic limit. When the magnetic field is sufficiently strong, only exact $N$-anyon groundstates, where anyons occupy the lowest Landau level, contribute to the equation of state. Particular attention is paid to the interval of definition of the statistical parameter $\alpha\in[-1,0]$ where a gap exists. Interestingly enough, one finds that at the critical filling $\nu=-{1/\alpha}$ where the pressure diverges, the external magnetic field is entirely screened by the flux tubes carried by the anyons. 
  A formalism describing the dynamics of classical and quantum systems from a group theoretical point of view is presented. We apply it to the simple example of the classical free particle. The Galileo group $G$ is the symmetry group of the free equations of motion. Consideration of the free particle Lagrangian semi-invariance under $G$ leads to a larger symmetry group, which is a central extension of the Galileo group by the real numbers. We study the dynamics associated with this group, and characterize quantities like Noether invariants and evolution equations in terms of group geometric objects. An extension of the Galileo group by $U(1)$ leads to quantum mechanics. 
  A brief review of the confrontation between black hole physics and quantum-mechanical unitarity is presented. Possibile reconciliations are modifying the laws of physics to allow fundamental loss of information, escape of information during the Hawking process, or black hole remnants. Each of these faces serious objections. A better understanding of the problem and its possible solutions can be had by studying two-dimensional models of dilaton gravity. Recent developments in these investigations are summarized. (Linear superposition of talks presented at the 7th Nishinomiya Yukawa Memorial Symposium and at the 1992 YITP Workshop on Quantum Gravity, November 1992.) 
  Orthogonal polynomial random matrix models of NxN hermitian matrices lead to Fredholm determinants of integral operators with kernel of the form (phi(x) psi(y) - psi(x) phi(y))/x-y. This paper is concerned with the Fredholm determinants of integral operators having kernel of this form and where the underlying set is a union of open intervals. The emphasis is on the determinants thought of as functions of the end-points of these intervals. We show that these Fredholm determinants with kernels of the general form described above are expressible in terms of solutions of systems of PDE's as long as phi and psi satisfy a certain type of differentiation formula. There is also an exponential variant of this analysis which includes the circular ensembles of NxN unitary matrices. 
  I consider the Hermitean two-matrix model with a logarithmic potential which is associated in the one-matrix case with the Penner model. Using loop equations I find an explicit solution of the model at large N (or in the spherical approximation) and demonstrate that it solves the corresponding Riemann-Hilbert problem. I construct the potential of the Kazakov-Migdal model on a D-dimensional lattice, which turns out to be a sum of two logarithms as well, whose large-N solution is given by the same formulas. In the "naive" continuum limit this potential recovers in D<4 dimensions the standard scalar theory with quartic self-interaction. I exploit the solution to calculate explicitly the pair correlator of gauge fields in the Kazakov-Migdal model with the logarithmic potential. 
  We classify the operator content of local hermitian scalar operators in the Sinh-Gordon model by means of independent solutions of the form-factor bootstrap equations. The corresponding linear space is organized into a tower-like structure of dimension $n$ for the form factors $F_{2n}$ and $F_{2n-1}$. Analyzing the cluster property of the form factors, a particular class of these solutions can be identified with the matrix elements of the operators $e^{k g\phi}$. We also present the complete expression of the form factors of the elementary field $\phi(x)$ and the trace of the energy-momentum tensor $\Theta(x)$. 
  In this note, based on a conference talk, we show how a 3 dimensional topological field theory leads to an algebraic gadget roughly equivalent to a quantum group. This is an expository version of some material in hep-th/9212115 (where we also carry out computations for a specific finite example). We also explain how to incorporate the central extensions usually explained via ``framings'', and we show how to recover invariants of framed tangles. This paper is written using AMSTeX 2.1, which can be obtained via ftp from the American Mathematical Society (instructions included). 2 encapsulated postscript files were submitted separately in uuencoded tar-compressed format. 
  We comment on some calculations concerning the finite-thickness corrections to the (generalized) Nambu action for a curved domain wall in Minkowski space. Contrary to a recent claim in the literature, we find no first order corrections in the wall-thickness, and only one second order correction proportional to the Ricci curvature of the wall. These results are obtained by consistently expanding the action and the equations of motion for the scalar field. 
  We prove, using arguments relying only on the "special K\"ahler" structure of the moduli space of the Calabi-Yau three-fold, that in the case of one single modulus the quantum modular group of the string effective action corresponding to Calabi-Yau vacua can not be SL(2,${Z\kern -4.6pt Z}$). 
  We consider open strings in an external constant magnetic field $H$. For an (infinite) sequence of critical values of $H$ an increasing number of (highest spin component) states lying on the first Regge trajectory becomes tachyonic. In the limit of infinite $H$ all these states are tachyons (with a common tachyonic mass) both in the case of the bosonic string and for the Neveu-Schwarz sector of the fermionic string. This result generalizes to extended object the same instability which occurs in ordinary non-Abelian gauge theories. The Ramond states have always positive square masses as is the case for ordinary QED. The weak field limit of the mass spectrum is the same as for a field theory with gyromagnetic ratio $g_S=2$ for all charged spin states. This behavior suggests a phase transition of the string as it has been argued for the ordinary electroweak theory. 
  We discuss a class of transfer matrix built by a particular combination of isomorphic and non-isomorphic GL(N) invariant vertex operators. We construct a conformally invariant magnet co nstituted of an alternating mixture of GL(N) ``spins'' operators at different order of represent ation. The corresponding central charge is calculated by analysing the low temperature beha viour of the associated free energy. We also comment on possible extensions of our results for more general classes of mixed systems. 
  We examine the problem of determining which representations of the braid group on a Riemann surface are carried by the wave function of a quantized Abelian Chern-Simons theory interacting with non-dynamical matter. We generalize the quantization of Chern-Simons theory to the case where the coefficient of the Chern-Simons term, $k$, is rational, the Riemann surface has arbitrary genus and the total matter charge is non-vanishing. We find an explicit solution of the Schr\"odinger equation. We find that the wave functions carry a representation of the braid group as well as a projective representation of the discrete group of large gauge transformations. We find a fundamental constraint which relates the charges of the particles, $q_i$, the coefficient $k$ and the genus of the manifold, $g$. 
  We study the Abelian Thirring Model when the fermionic fields have non-conserved chiral charge: $\Delta {\cal Q}_5 =N$. One of the main features we find for this model is the dependence of the Virasoro central charge on both the Thirring coupling constant and $N$. We show how to evaluate correlation functions and in particular we compute the conformal dimensions for fermions and fermionic bilinears, which depend on the fermionic chiral charge. Finally we build primary fields with arbitrary conformal weight. 
  We present a generic Lagrangian, in arbitrary spacetime dimension $D$, describing the interaction of a dilaton, a graviton and an antisymmetric tensor of arbitrary rank $d$. For each $D$~and~$d$, we find ``solitonic'' black $\tilde{p}$-brane solutions where $\tilde{p} = \tilde{d} - 1$~and~ $\tilde d = D - d - 2$. These solutions display a spacetime singularity surrounded by an event horizon, and are characterized by a mass per unit $\tilde p$-volume, ${\cal M}_{\tilde{d}}$, and topological ``magnetic'' charge $g_{\tilde{d}}$, obeying $\kappa {\cal M}_{\tilde{d}} \geq g_{\tilde{d}}/ \sqrt{2}$. In the extreme limit $\kappa {\cal M}_{\tilde{d}}=g_{\tilde{d}}/ \sqrt{2}$, the singularity and event horizon coalesce. For specific values of $D$~and~$d$, these extreme solutions also exhibit supersymmetry and may be identified with previously classified heterotic, Type IIA and Type IIB super $\tilde p$-branes. The theory also admits elementary $p$-brane solutions with ``electric'' Noether charge $e_d$, obeying the Dirac quantization rule $e_d g_{\tilde{d}} = 2\pi n$, $n =$~integer. We also present the Lagrangian describing the theory dual to the original theory, whose antisymmetric tensor has rank $\tilde{d}$ and for which the roles of topological and elementary solutions are interchanged. The super $p$-branes and their duals are mutually non-singular. As special cases of our general solution we recover the black $p$-branes of Horowitz and Strominger $(D = 10)$, Guven $(D = 11)$ and Gibbons et al $(D = 4)$, the $N = 1$, $N = 2a$~and~$N = 2b$ super-$p$-branes of Dabholkar et al $(4 \leq D \leq 10)$, Duff and Stelle $(D = 11)$, Duff and Lu $(D = 10)$ and Callan, Harvey and Strominger $(D = 10)$, and the axionic instanton of Rey $(D = 4)$. In particular, the electric/magnetic duality of Gibbons and Perry in $D = 4$ is seen to be a consequence of particle/sixbrane duality in $D = 10$. Among the new solutions is a self-dual superstring in $D = 6$. 
  Combinatorial $B_n$-analogues of Schubert polynomials and corresponding symmetric functions are constructed from an exponential solution of the $B_n$-Yang-Baxter equation that involves the nilCoxeter algebra of the hyperoctahedral group. 
  All anomaly candidates and the form of the most general invariant local action are given for old and new minimal supergravity, including the cases where additional Yang--Mills and chiral matter multiplets are present. Furthermore nonminimal supergravity is discussed. In this case local supersymmetry itself may be anomalous and some of the corresponding anomaly candidates are given explicitly. The results are obtained by solving the descent equations which contain the consistency equation satisfied by integrands of anomalies and invariant actions. 
  A Chern-Simons gauged Nonlinear Schr\"odinger Equation is derived from the continuous Heisenberg model in 2+1 dimensions. The corresponding planar magnets can be analyzed whithin the anyon theory. Thus, we show that static magnetic vortices correspond to the self-dual Chern - Simons solitons and are described by the Liouville equation. The related magnetic topological charge is associated with the electric charge of anyons. Furthermore, vortex - antivortex configurations are described by the sinh-Gordon equation and its conformally invariant extension. Physical consequences of these results are discussed. 
  The method of quantization of magnetic monopoles based on the order-disorder duality existing between the monopole operator and the lagrangian fields is applied to the description of the quantum magnetic monopoles of `t Hooft and Polyakov in the SO(3) Georgi-Glashow model.   The commutator of the monopole operator with the magnetic charge is computed explicitly, indicating that indeed the quantum monopole carries $4\pi/g$ units of magnetic charge. An explicit expression for the asymptotic behavior of the monopole correlation function is derived. From this, the mass of the quantum monopole is obtained. The tree-level result for the quantum monopole mass is shown to satisfy the Bogomolnyi bound ($M_{\rm mon} \geq 4 \pi \frac{M}{g^2}$) and to be within the range of values found for the energy of the classical monopole solution. 
  In a general superstring vacuum configuration, the `internal' space (sector) varies in spacetime. When this variation is non-trivial only in two space-like dimensions, the vacuum contains static cosmic strings with finite energy per unit length and which is, up to interactions with matter, an easily computed topological invariant. The total spacetime is smooth although the `internal' space is singular at the center of each cosmic string. In a similar analysis of the Wick-rotated Euclidean model, these cosmic strings acquire expected self-interactions. Also, a possibility emerges to define a global time in order to rotate back to the Lorentzian case. 
  It is shown that the finite dimensional irreducible representations of the quantum matrix algebra $ M_q(n) $ ( the coordinate ring of $ GL_q(n) $ ) exist only when q is a root of unity ( $ q^p = 1 $ ). The dimensions of these representations can only be one of the following values: $ {p^N \over 2^k } $ where $ N = {n(n-1)\over 2 } $ and $ k \in \{ 0, 1, 2, . . . N \} $ For each $ k $ the topology of the space of states is $ (S^1)^{\times(N-k)} \times [ 0 , 1 ] ^{(\times (k)} $ (i.e. an $ N $ dimensional torus for $ k=0 $ and an $ N $ dimensional cube for $ k = N $ ). 
  After reviewing some aspects of gravity in two dimensions, it is shown that non-trivial embeddings of sl(2) in a semi-simple (super) Lie algebra give rise to a very large class of extensions of 2D gravity. The induced action is constructed as a gauged WZW model and an exact expression for the effective action is given. (Talk presented at the Journees Relativistes '93, Brussels, April, 1993). 
  We use separation of variables as a tool to identify and to analyze exactly soluble time-dependent quantum mechanical potentials. By considering the most general possible time-dependent re-definition of the spatial coordinate, as well as general transformations on the wavefunctions, we show that separation of variables applies and exact solubility occurs only in a very restricted class of time-dependent models. We consider the formal structure underlying our findings, and the relationship between our results and other work on time-dependent potentials. As an application of our methods, we apply our results to the calculations of propagators. 
  Quantum multiparameter deformation of real Clifford algebras is proposed. The corresponding irreducible representations are found. 
  We calculate the degree 2 and 6 Casimirs operators in explicit form, with the generators of G2 written in terms of the subalgebra A2 
  In this paper the $c=1$ matrix model deformed by a $1/x^{2}$ piece is discussed. Tachyon correlation functions are calculated up to genus two using methods similar to those for the undeformed case. The possible connection with the two dimensional black hole is also considered. In particular, restrictions on naked singularities imposed by the matrix model are found. 
  The chiral phase dependence of fermion partition function in spherically symmetric U(1) gauge field background is analyzed in two dimensional space-time. A well-defined method to calculated the path integral which apply to the continuous fermion spectrum is described. The one-to-one correspondence between the nonzero energy continuous spectra of two pertinent hamiltonians, which are defined by the Dirac operator to make the path integral well-defined, is shown to be exact. The asymptotic expansion for the chiral phase dependence in \(1/|m|^2\) (\(m\) is mass of the fermion.) is proposed and the coefficients in the expansion are evaluated up to the next-to-next leading term. Up to this order, the chiral phase dependence is given only by the winding number of background field and the corrections vanish. 
  We perform the complete bosonization of 2+1 dimensional QED with one fermionic flavor in the Hamiltonian formalism. The Fermi operators are explicitly constructed in terms of the vector potential and the electric field. We carefully specify the regularization procedure involved in the definition of these operators, and calculate the fermionic bilinears and the energy - momentum tensor. The algebra of bilinears exhibits the Schwinger terms which also appear in perturbation theory. The bosonic Hamiltonian is a local, polynomial functional of $A_i$ and $E_i$, and we check explicitly the Lorentz invariance of the resulting bosonic theory. Our construction is conceptually very similar to Mandelstam's construction in 1+1 dimensions, and is dissimilar from the recent bosonization attempts in 2+1 dimensions, which hinge crucially on the presence of a Chern - Simons term. 
  R.P. Feynman showed F.J. Dyson a proof of the Lorentz force law and the homogeneous Maxwell equations, which he obtained starting from Newton's law of motion and the commutation relations between position and velocity for a single nonrelativistic particle. We formulate both a special relativistic and a general relativistic versions of Feynman's derivation. Especially in the general relativistic version we prove that the only possible fields that can consistently act on a quantum mechanical particle are scalar, gauge and gravitational fields. We also extend Feynman's scheme to the case of non-Abelian gauge theory in the special relativistic context. 
  Abelian anomaly is examined by means of the recently proposed gauge invariant regularization for SO(10) chiral gauge theory and its generalization for a theory of arbitrary gauge group with anomaly-free chiral fermion contents. For both cases it is shown that the anomaly with correct normalization can be obtained in a gauge invariant form without any counterterms. 
  We present a formulation of Quantum Electrodynamics in terms of an antisymmetric tensor gauge field. In this formulation the topological current of this field appears as a source for the electromagnetic field and the topological charge therefore acts physically as an electric charge. The charged states of QED lie in the sector where the topological charge is identical to the matter charge.  The antisymmetric field theory, however, admits new sectors where the topological charge is more general.  These nontrivial, electrically charged, sectors contain massless states orthogonal to the vacuum which are created by a gauge invariant operator and can be interpreted as coherent states of photons. We evaluate the correlation functions of these states in the absence of matter. The new states have a positive definite norm and do interact with the charged states of QED in the usual way. It is argued that if these new sectors are in fact realized in nature then a very intense background electromagnetic field is necessary for the experimental observation of them. The order of magnitude of the intensity threshold is estimated. 
  Three postulates asserting the validity of conventional quantum theory, semi-classical general relativity and the statistical basis for thermodynamics are introduced as a foundation for the study of black hole evolution. We explain how these postulates may be implemented in a ``stretched horizon'' or membrane description of the black hole, appropriate to a distant observer. The technical analysis is illustrated in the simplified context of 1+1 dimensional dilaton gravity. Our postulates imply that the dissipative properties of the stretched horizon arise from a course graining of microphysical degrees of freedom that the horizon must possess. A principle of black hole complementarity is advocated. The overall viewpoint is similar to that pioneered by 't~Hooft but the detailed implementation is different. 
  We apply CGHS-type dilaton gravity model to (1+1)-dimensional cosmological situations.   First the behavior of a compact 1-dimensional universe (i.e. like a closed string) is classified on the assumption of homogeneity of universe.   Several interesting solutions are found, which include a Misner-type universe having closed time-like curves, and an asymptotically de Sitter universe first pointed out by Yoshimura.   In the second half of this talk, we discuss the modification of the classical homogeneous solutions, considering inhomogeneity of classical conformal matters and also quantum back-reaction respectively.   (An expanded version of the talk presented by T. Mishima at Yukawa Institute of Theoretical Physics workshop `Quantum Gravity' 24-27, November 1992.) 
  We present an alternative 2-parametric deformation $ GL(2)_{h,h'} $ , and construct the differential calculus on the quantum plane on which this quantum group acts. Also we give a new deformation of the two dimensional Heisenberg algebra 
  We introuduce a unified method which can be applied to any WZW model at arbitrary level to search systematically for modular invariant physical partition functions. Our method is based essentially on modding out a known theory on group manifold $G$ by a discrete group $\Gamma$.   We apply our method to $\widehat {su(n)}$ with $n=2,3,4,5,6$, and to $\widehat {g_2}$ models, and obtain all the known partition functions and some new ones, and give explicit expressions for all of them. 
  We construct a lattice model of compact (2+1)-dimensional Maxwell-Chern- Simons theory, starting from its formulation in terms of gauge invariant quantities proposed by Deser and Jackiw. We thereby identify the topological excitations and their interactions. These consist of monopolo- antimonopole pairs bounded by strings carrying both magnetic flux and electric charge. The electric charge renders the Dirac strings observable and endows them with a finite energy per unit length, which results in a linearly confining string tension. Additionally, the strings interact via an imaginary, topological term measuring the (self-) linking number of closed strings. 
  After pointing out the role of the compactification lattice for spectrum calculations in orbifold models, I discuss modular discrete symmetry groups for $Z_N$ or\-bi\-folds. I consider the $Z_7$ orbifold as a nontrivial example of a (2,2) model and give the generators of the modular group for this case, which does not contain $[SL(2,{\bf Z})]^3$ as had been speculated. I also discuss how to treat cases where quantized Wilson lines are present. I consider in detail an example, demonstrating that quantized Wilson lines affect the modular group in a nontrivial manner. In particular, I show that it is possible for a Wilson line to break $SL(2,{\bf Z})$.} 
  An alternative to Dirac's constrained quantization procedure is explained. 
  Exact wavefunctions for N non-Abelian Chern-Simons (NACS) particles are obtained by the ladder operator approach. The same method has previously been applied to construct exact wavefunctions for multi-anyon systems. The two distinct base states of the NACS particles that we use are multi-valued and are defined in terms of path ordered line integrals. Only strings of operators that preserve the monodromy properties of these base states are allowed to act on them to generate new states. 
  We study the large N limit of the Itzykson -- Zuber integral and show that the leading term is given by the exponent of an action functional for the complex inviscid Burgers (Hopf) equation evaluated on its particular classical solution; the eigenvalue densities that enter in the IZ integral being the imaginary parts of the boundary values of this solution. We show how this result can be applied to ``induced QCD" with an arbitrary potential $U(x)$. We find that for a nonsingular $U(x)$ in one dimension the eigenvalue density $\rho(x)$ at the saddle point is the solution of the functional equation $G_{+}(G_{-}(x))=G_{-}(G_{+}(x))=x$, where $G_{\pm}(x) \equiv {1\over{2}}U^{\prime}(x)\pm i\pi \rho(x)$. As an illustration we present a number of new particular solutions of the $c=1$ matrix model on a discrete real line. 
  We give a general expression for the static potential energy of the gravitational interaction of two massive particles, in terms of an invariant vacuum expectation value of the quantized gravitational field. This formula holds for functional integral formulations of euclidean quantum gravity, regularized to avoid conformal instability. It could be regarded as the analogue of the Wilson loop for gauge theories and allows in principle, through numerical simulations or other approximation techniques, non perturbative evaluations of the potential or of the effective coupling constant. The geometrical meaning of this expression is quite simple, as it represents the ``average proper-time delay'', respect to two neighboring lines, of a very long geodesic with unit timelike tangent vector. 
  I show that the classical Toda models built on superalgebras, and obtained from a reduction with respect to an $Sl(2)$ algebra, are "linearly supersymmetrizable" (by adding spin 1/2 fields) if and only if the $Sl(2)$ is the bosonic part of an $OSp(1|2)$ algebra. In that case, the model is equivalent to the one constructed from a reduction with respect to the $OSp(1|2)$ algebra, up to spin 1/2 fields. The corresponding $W$ algebras are related through a factorization of spin 1/2 fields (bosons and fermions). I illustrate this factorization on an example: the superconformal algebra built on $Sl(n+1|2)$. 
  It is well-known that topological sigma-models in 2 dimensions constitute a path-integral approach to the study of holomorphic maps from a Riemann surface S to an almost complex manifold K, the most interesting case being that where K is a Kahler manifold. We show that, in the same way, topological sigma-models in 4 dimensions introduce a path integral approach to the study of triholomorphic maps q:M-->N from a 4dimensional Riemannian manifold M to an almost quaternionic manifold N. The most interesting cases are those where M, N are hyperKahler or quaternionic Kahler. BRST-cohomology translates into intersection theory in the moduli-space of this new class of instantonic maps, named by us hyperinstantons. The definition of triholomorphicity that we propose is expressed by the equation q_*-J_u q_* j_u = 0, u=1,2,3, where {j_u} is an almost quaternionic structure on M and {J_u} is an almost quaternionic structure on N. This is a generalization of the Cauchy-Fueter equations. For M, N hyperKahler, this generalization naturally arises by obtaining the topological sigma-model as a twisted version of the N=2 globally supersymmetric sigma-model. We discuss various examples of hyperinstantons, in particular on the torus and the K3 surface. We also analyse the coupling of the topological sigma-model to topological gravity. The study of 
  No change at all but this time is possible to typset it!. 
  We study the mass-charge relation for the semiclassical extremal black hole of the $S$-wave sector Einstein-Maxwell theory coupled to $N$ conformal scalars. The classical ratio $M/{|Q|}=1$ is shown to be modified to $M/{|Q|} \simeq 1-k/6$ for small $ k \equiv N\hbar/(12\pi Q^{2})$. Furthermore, numerical study for $k<2$ shows that $M/|Q|$ is a monotonically decreasing function of $k$. We speculate on the consequence of such a modification in the 4-dimensional context. 
  If black hole formation and evaporation can be described by an $S$ matrix, information would be expected to come out in black hole radiation. An estimate shows that it may come out initially so slowly, or else be so spread out, that it would never show up in an analysis perturbative in $M_{Planck}/M$, or in 1/N for two-dimensional dilatonic black holes with a large number $N$ of minimally coupled scalar fields. 
  We solve explicitly a closed, linear loop equation for the SU(2) Wilson loop average on a two-dimensional plane and generalize the solution to the case of the SU(N) Wilson loop average with an arbitrary closed contour. Furthermore, the flat space solution is generalized to any two-dimensional manifold for the SU(2) Wilson loop average and to any two-dimensional manifold of genus 0 for the SU(N) Wilson loop average. 
  We have found generic Killing spinor identities which bosonic equations of motion have to satisfy in supersymmetric theories if the solutions admit Killing spinors. Those identities constrain possible quantum corrections to bosonic solutions with unbroken supersymmetries. As an application we show that purely electric static extreme dilaton black holes may acquire specific quantum corrections, but the purely magnetic ones cannot. 
  We report results of an investigation of relativistic causality constraints on the measurability of nonlocal variables. We show that measurability of certain nondegenerate variables with entangled eigenstates contradicts the principle of causality, but that there are other, certainly nonlocal, variables which can be measured without breaking causality. We show that any causal measurement of nonlocal variables must erase certain local information. For example, for a system of two spin-1/2 particles, even if we take the weakest possible definition of verification measurement, verification of an entangled state must erase all local information. 
  Consequences of relativistic causality for measurements of nonlocal characteristics of composite quantum systems are investigated. It is proved that verification measurements of entangled states necessarily erase local information. A complete analysis of measurability of nondegenerate spin operators of a system of two spin-1/2 particles is presented. It is shown that measurability of certain projection operators which play an important role in axiomatic quantum theory contradicts the causality principle. 
  A new parameterisation of the solutions of Toda field theory is introduced. In this parameterisation, the solutions of the field equations are real, well-defined functions on space-time, which is taken to be two-dimensional Minkowski space or a cylinder. The global structure of the covariant phase space of Toda theory is examined and it is shown that it is isomorphic to the Hamiltonian phase space. The Poisson brackets of Toda theory are then calculated. Finally, using the methods developed to study the Toda theory, we extend these results to the non-Abelian Toda field theories. 
  The general solutions for the factorization equations of the reflection matrices $K^{\pm}(\theta)$ for the eight vertex and six vertex models (XYZ, XXZ and XXX chains) are found. The associated integrable magnetic Hamiltonians are explicitly derived, finding families dependig on several continuous as well as discrete parameters. 
  In the first quantised description of strings, we integrate over target space co-ordinates $X^\mu$ and world sheet metrics $g_{\alpha\beta}$. Such path integrals give scattering amplitudes between the `in' and `out' vacuua for a time-dependent target space geometry. For a complete description of   `particle creation' and the corresponding backreaction, we need instead the causal amplitudes obtained from an `initial value formulation'. We argue, using the analogy of a scalar particle in curved space, that in the first quantised path integral one should integrate over $X^\mu$ and world sheet {\it zweibiens}. This extended formalism can be made to yield causal amplitudes; it also naturally allows incorporation of density matrices in a covariant manner. (This paper is an expanded version of hep-th 9301044) 
  We rewrite the action for $QCD_2$ in the light cone gauge only in terms of a bilocal mesonic field. In this formalism the $1/N$ expansion can be done in a straightforward way by a saddle point technique that determines the master field to be identified with the vacuum expectation value of the bilocal field. Finally we show that the equation of motion for the fluctuations around the master field is identical with the 't Hooft meson equation. 
  A gas of electrons confined to a plane is examined in both the relativistic and nonrelativistic case. Using a (0+1)-dimensional effective theory, a remarkably simple method is proposed to calculate the spin density induced by an uniform magnetic background field. The physical properties of possible fluxon excitations are determined. It is found that while in the relativistic case they can be considered as half-fermions (semions) in that they carry half a fermion charge and half the spin of a fermion, in the nonrelativistic case they should be thought of as fermions, having the charge and spin of a fermion. 
  Exponential solutions of the Yang-Baxter equation give rise to generalized Schubert polynomials and corresponding symmetric functions. We provide several descriptions of the local stationary algebra defined by this equation. This allows to construct various exponential solutions of the YBE. The $B_n$ and $G_2$ cases are also treated. 
  Using the symmetry properties of the three-anyon spectrum, we obtain exactly the multiplicities of states with given energy and angular momentum. The results are shown to be in agreement with the proper quantum mechanical and semiclassical considerations, and the unexplained points are indicated. 
  We consider static black holes, which are bosonic solutions of supersymmetric theories. We will show that supersymmetry provides a natural framework for a discussion of various properties of such static black holes. The most fundamental property of simple global supersymmetry, non-negativeness of the energy, is generalized in the black hole theory to cosmic censorship. The SUSY classification of static black holes will be presented in terms of central charges of supersymmetry algebra. The mass, the temperature and the entropy of these black holes are simple functions of supersymmetry charges. The extreme black holes have zero temperature and some unbroken supersymmetries. 
  Recently Witten proposed to consider elliptic genus in $N=2$ superconformal field theory to understand the relation between $N=2$ minimal models and Landau-Ginzburg theories. In this paper we first discuss the basic properties satisfied by elliptic genera in $N=2$ theories. These properties are confirmed by some fundamental class of examples. Then we introduce a generic procedure to compute the elliptic genera of a particular class of orbifold theories, {\it i.e.\/} the ones orbifoldized by $e^{2\pi iJ_0}$ in the Neveu-Schwarz sector. This enables us to calculate the elliptic genera for Landau-Ginzburg orbifolds. When the Landau-Ginzburg orbifolds allow an interpretation as target manifolds with $SU(N)$ holonomy we can compare the expressions with the ones obtained by orbifoldizing tensor products of $N=2$ minimal models. We also give sigma model expressions of the elliptic genera for manifolds of $SU(N)$ holonomy. 
  We analyze quantum two dimensional dilaton gravity model, which is described by $SL(2,R)/U(1)$ gauged Wess-Zumino-Witten model deformed by $(1,1)$ operator. We show that the curvature singularity does not appear when the central charge $c_{\rm matter}$ of the matter fields is given by $22<c_{\rm matter}<24$. When $22<c_{\rm matter}<24$, the matter shock waves, whose energy momentum tensors are given by $T_{\rm matter} \propto \delta(x^+ - x^+_0)$, create a kind of wormholes, {\it i.e.,} causally disconnected regions. Most of the quantum informations in past null infinity are lost in future null infinity but the lost informations would be carried by the wormholes.   We also discuss about the problem of defining the mass of quantum black holes. On the basis of the argument by Regge and Teitelboim, we show that the ADM mass measured by the observer who lives in one of asymptotically flat regions is finite and does not vanish in general. On the other hand, the Bondi mass is ill-defined in this model. Instead of the Bondi mass, we consider the mass measured by observers who live in an asymptotically flat region at first. A class of the observers finds the mass of the black hole created by a shock wave changes as the observers' proper time goes by, i.e. they observe the Hawking radiation. The measured mass vanishes after the infinite proper time and the black hole evaporates completely. Therefore the total Hawking radiation is positive even when $N<24$. 
  We consider the uncertainty relation between position and momentum of a particle on $ S^1 $ (a circle). Since $ S^1 $ is compact, the uncertainty of position must be bounded. Consideration on the uncertainty of position demands delicate treatment. Recently Ohnuki and Kitakado have formulated quantum mechanics on $ S^D $ (a $D$-dimensional sphere). Armed with their formulation, we examine this subject. We also consider parity and find a phenomenon similar to the spontaneous symmetry breaking. We discuss problems which we encounter when we attempt to formulate quantum mechanics on a general manifold. 
  In this paper we study the quantum Clifford-Hopf algebras $\widehat{CH_q(D)}$ for even dimensions $D$ and obtain their intertwiner $R-$matrices, which are elliptic solutions to the Yang- Baxter equation. In the trigonometric limit of these new algebras we find the possibility to connect with extended supersymmetry. We also analyze the corresponding spin chain hamiltonian, which leads to Suzuki's generalized $XY$ model. 
  When one uses the Dirac bracket, second class constraints become first class. Hence, they are amenable to the BRST treatment characteristic of ordinary first class constraints. This observation is the starting point of a recent investigation by Batalin and Tyutin, in which all the constraints are put on the same footing. However, because second class constraints identically vanish as operators in the quantum theory, they are quantum-mechanically reducible and require therefore ghosts of ghosts. Otherwise, the BRST cohomology would not yield the correct physical spectrum. We discuss how to incorporate this feature in the formalism and show that it leads to an infinite tower of ghosts of ghosts. An alternative treatment, in which the brackets of the ghosts are modified, is also mentioned. 
  A simplified proof of a theorem by Joglekar and Lee on the renormalization of local gauge invariant operators in Yang-Mills theory is given. It is based on (i) general properties of the antifield-antibracket formalism; and (ii) well-established results on the cohomology of semi-simple Lie algebras. 
  We use the definition of the Calogero-Moser models as Hamiltonian reductions of geodesic motions on a group manifold to construct their $R$-matrices. In the Toda case, the analogous construction yields constant $R$-matrices. By contrast, for Calogero-Moser models they are dynamical objects. 
  The $S$-matrix of the Ising Model can be obtained as particular limit of the roaming trajectories associated to of the $S$-matrix of the Sinh-Gordon model. Using the form factors of the Sinh-Gordon, we analyse the correspondence between the operators of the two theories. 
  Simple mean field methods can be used to describe transitions between different space-time models in string theory. These include transitions between different Calabi-Yau manifolds, and more exotic things such as the Calabi-Yau/Landau-Ginzberg correspondence. 
  We solve the gauged Nambu--Jona-Lasinio model at leading order in the large $N$ expansion by computing the anomalous dimensions of all the fields of the model and other gauge independent critical exponents by examining the scaling behaviour of the Schwinger Dyson equation. We then restrict to the three dimensional model and include a Chern Simons term to discover the $\theta$-dependence of the same exponents where $\theta$ is the Chern Simons coupling. 
  By using the corrections to the asymptotic scaling forms of the fields of the $O(N)$ Gross Neveu model to solve the dressed skeleton Schwinger Dyson equations, we deduce the critical exponent corresponding to the $\beta$-function of the model at $O(1/N^2)$. 
  We solve the conformal bootstrap equations of the four fermi model or $O(N)$ Gross Neveu model to deduce the fermion anomalous dimension of the theory at $O(1/N^3)$ in arbitrary dimensions. 
  The interaction of the spinor field with the Weinberg's $2(2S+1)$- component massless field is considered. New interpretation of the Weinberg's spinor is proposed. The equation analogous to the Dirac oscillator is obtained. 
  By replacing two of the bosonic scalar superfields of the N=2 string with fermionic scalar superfields (which shifts $d_{critical}$ from (2,2) to (9,1)), a quadratic action for the ten-dimensional Green-Schwarz superstring is obtained. Using the usual N=2 super-Virasoro ghosts, one can construct a BRST operator, picture-changing operators, and covariant vertex operators for the Green-Schwarz superstring. Superstring scattering amplitudes with an arbitrary number of loops and external massless states are then calculated by evaluating correlation functions of these vertex operators on N=2 super-Riemann surfaces, and integrating over the N=2 super-moduli. These multiloop superstring amplitudes have been proven to be SO(9,1) super-Poincar\'e invariant (by constructing the super-Poincar\'e generators and writing the amplitudes in manifest SO(9,1) notation), unitary (by showing agreement with amplitudes obtained using the light-cone gauge Green-Schwarz formalism), and finite (by explicitly checking for divergences in the amplitudes when the Riemann surface degenerates). There is no multiloop ambiguity in these Green-Schwarz scattering amplitudes since spacetime-supersymmetry is manifest (there is no sum over spin structures), and therefore the moduli space can be compactified. 
  The U(N) gauge theory on a D-dimensional lattice is reformulated as a theory of lattice strings (a statistical model of random surfaces). The Boltzmann weights of the surfaces can have both signs and are tuned so that the longitudinal modes of the string are elliminated. The U(\infty) gauge theory is described by noninteracting planar surfaces and the 1/N corrections are produced by surfaces with higher topology as well as by contact interactions due to microscopic tubes, trousers, handles, etc. We pay special attention to the case D=2 where the sum over surfaces can be performed explicitly, and demonstrate that it reproduces the known exact results for the free energy and Wilson loops in the continuum limit. In D=4 dimensions, our lattice string model reproduces the strong coupling phase of the gauge theory. The weak coupling phase is described by a more complicated string whose world surface may have windows. A possible integration measure in the space of continuous surfaces is suggested. 
  Using odd symplectic structure constructed over tangent bundle of the symplectic manifold, we construct the simple supergeneralization of an arbitrary Hamiltonian mechanics on it. In the case, if the initial mechanics defines Killing vector of some Riemannian metric, corresponding supersymmetric mechanics can be reformulated in the terms of even symplectic structure on the supermanifold. 
  We describe the $R$-matrix structure associated with integrable extensions, containing both one-body and two-body potentials, of the $A_N$ Calogero-Moser $N$-body systems. We construct non-linear, finite dimensional Poisson algebras of observables. Their $N   \rightarrow \infty$ limit realize the infinite Lie algebras Sdiff$({\Bbb R} \times S_1 )$ in the trigonometric case and   Sdiff$({\Bbb R }^2)$ in the rational case. It is then isomorphic to the algebra of observables constructed in the two-dimensional collective string field theory. 
  It is shown that the St\"uckelberg formalism can be regarded as a field-enlarging transformation that introduces an additional gauge symmetry to the considered model. The appropriate BRST charge can be defined. The physical state condition, demanding that that a physical state is to be anihilated by the BRST charge, is shown to be equivalent to the St\"uckelberg condition. Several applications of the new approach to the formalism are presented. The comparison with the BFV procedure is given. (The author field has been corrected.) 
  A field-enlarging transformation in the chiral electrodynamics is performed. This introduces an additional gauge symmetry to the model that is unitary and anomaly-free and allows for comparison of different models discussed in the literature. The problem of superfluous degrees of freedom and their influence on quantization is discussed. Several "mysteries" are explained from this point of view. 
  It is argued that the noncommutative geometry construction of the standard model predicts a nonlinear symmetry breaking mechanism rather than the orthodox Higgs mechanism. Such models have experimentally verifiable consequences. 
  We show that the master equation governing the dynamics of simple diffusion and certain chemical reaction processes in one dimension give time evolution operators (Hamiltonians) which are realizations of Hecke algebras. In the case of simple diffusion one obtains, after similarity transformations, reducible hermitian representations while in the other cases they are non-hermitian and correspond to supersymmetric quotients of Hecke algebras. 
  Differential regularization is applied to a field theory of a non-relativistic charged boson field $\phi$ with $\lambda (\phi {}^{*} \phi)^2$ self-interaction and coupling to a statistics-changing $U(1)$ Chern-Simons gauge field. Renormalized configuration-space amplitudes for all diagrams contributing to the $\phi {}^{*} \phi {}^{*} \phi \phi$ 4-point function, which is the only primitively divergent Green's function, are obtained up to 3-loop order. The renormalization group equations are explicitly checked, and the scheme dependence of the $\beta$-function is investigated. If the renormalization scheme is fixed to agree with a previous 1-loop calculation, the 2- and 3-loop contributions to $\beta(\lambda,e)$ vanish, and $\beta(\lambda,e)$ itself vanishes when the ``self-dual'' condition relating $\lambda$ to the gauge coupling $e$ is imposed. 
  We introduce the notion of rational Hopf algebras that we think are able to describe the superselection symmetries of two dimensional rational quantum field theories. As an example we show that a six dimensional rational Hopf algebra $H$ can reproduce the fusion rules, the conformal weights, the quantum dimensions and the representation of the modular group of the chiral Ising model. $H$ plays the role of the global symmetry algebra of the chiral Ising model in the following sense: 1) a simple field algebra $\FA$ and a representation $\pi$ on $\HS_\pi$ of it is given, which contains the $c=1/2$ unitary representations of the Virasoro algebra as subrepresentations; 2) the embedding $U\colon H\to\BOH$ is such that the observable algebra $\pi(\OA)^-$ is the invariant subalgebra of $\BOH$ with respect to the left adjoint action of $H$ and $U(H)$ is the commutant of $\pi(\OA)$; 3) there exist $H$-covariant primary fields in $\BOH$, which obey generalized Cuntz algebra properties and intertwine between the inequivalent sectors of the observables. 
  By studying the geometric properties of correlation functions on the theory space, we are naturally led to a connection for the infinite dimensional vector bundle of composite fields over the theory space. We show how the short distance singularities of the theory are determined by the geometry of the theory space, i.e., the connection, beta functions, and anomalous dimensions. (This is a summary of the talk given at Strings '93 in Berkeley. The unnecessary blank lines in the original version have been removed in this revised version.) 
  We give a prescription for embedding classical solutions and, in particular, topological defects in field theories which are invariant under symmetry groups that are not necessarily simple. After providing examples of embedded defects in field theories based on simple groups, we consider the electroweak model and show that it contains the $Z$ string and a one parameter family of strings called the $W(\alpha )$ string. It is argued that, although the members of this family are gauge equivalent when considered in isolation, each member should be considered distinct when multi-string solutions are considered. We then turn to the issue of stability of embedded defects and demonstrate the instability of a large class of such solutions in the absence of bound states or condensates. The $Z$ string is shown to be unstable when the Weinberg angle ($\theta_w$) is $\pi /4$ for all values of the Higgs mass. The $W$ strings are also shown to be unstable for a large range of parameters. Embedded monopoles suffer from the Brandt-Neri-Coleman instability. A simple physical understanding of this instability is provided in terms of the phenomenon of W-condensation. Finally, we connect the electroweak string solutions to the sphaleron: ``twisted'' loops of W string and finite segments of W and Z strings collapse into the sphaleron configuration, at least, for small values of $\theta_w$. 
  We consider one dimensional lattice gauge theories constructed by the minimal coupling prescription. It is shown that these theories are exactly solvable in the thermodynamic limit. After considering the most general case, we discuss some special cases on finite lattices, and also work out some examples. There is no phase transition in these minimally coupled theories. 
  Not only in physical string theories, but also in some highly simplified situations, background independence has been difficult to understand. It is argued that the ``holomorphic anomaly'' of Bershadsky, Cecotti, Ooguri, and Vafa gives a fundamental explanation of some of the problems. Moreover, their anomaly equation can be interpreted in terms of a rather peculiar quantum version of background independence: in systems afflicted by the anomaly, background independence does not hold order by order in perturbation theory, but the exact partition function as a function of the coupling constants has a background independent interpretation as a state in an auxiliary quantum Hilbert space. The significance of this auxiliary space is otherwise unknown. 
  The Mellin-Barnes representation for the free energy of the genus-$g$ string is constructed. It is shown that the interactions of the open bosonic string do not modify the critical (Hagedorn) temperature. However,for the sectors having a spinor structure, the critical temperature exists also for all $g$ and depends on the windings. The appearance of a periodic structure is briefly discussed. 
  The quantum dynamics of a simplest dissipative system, a particle moving in a constant external field , is exactly studied by taking into account its interaction with a bath of Ohmic spectral density. We apply the main idea and methods developed in our recent work [1] to quantum dissipative system with constant external field. Quantizing the dissipative system we obtain the simple and exact solutions for the coordinate operator of the system in Heisenberg picture and the wave function of the composite system of system and bath in Schroedinger picture. An effective Hamiltonian for the dissipative system is explicitly derived from these solutions with Heisenberg picture method and thereby the meaning of the wavefunction governed by it is clarified by analyzing the effect of the Brownian motion. Especially, the general effective Hamiltonian for the case with arbitrary potential is directly derived with this method for the case when the Brownian motion can be ignored. Using this effective Hamiltonian, we show an interesting fact that the dissipation suppresses the wave packet spreading. 
  We consider the compactification of the dual form of $N=1$ $D=10$ supergravity on a six-dimensional Calabi-Yau manifold. An $N=1$ off-shell supergravity effective Lagrangian in four dimensions can be constructed in a dual version of the gravitational sector (new-minimal supergravity form). Superspace duality has a simple interpretation in terms of Poincar\'{e} duality of two-form cohomology. The resulting $4D$ Lagrangian may describe the low-energy point-field limit of a five-brane theory, dual to string theory, provided Calabi-Yau spaces are consistent vacua of such dual theory. 
  It is shown that $SL(n,R)$ KdV hierarchy can be expressed as definite nonpolynomials in Kac Moody currents and their derivatives by the action of Borel subgroup of $SL(n,R)$ on the phase space of centrally extended $sl(n,R)$ Kac Moody currents. Construction of Lax pair is shown, confirming Drinfeld Sokolov type Hamiltonian reduction. This suggests an example of a moduli space with symplectic structure corresponding to extended conformal symmetries. 
  A number of examples of Hamiltonian systems that are integrable by classical means are cast within the framework of isospectral flows in loop algebras. These include: the Neumann oscillator, the cubically nonlinear Schr\"odinger systems and the sine-Gordon equation. Each system has an associated invariant spectral curve and may be integrated via the Liouville-Arnold technique. The linearizing map is the Abel map to the associated Jacobi variety, which is deduced through separation of variables in hyperellipsoidal coordinates. More generally, a family of moment maps is derived, identifying certain finite dimensional symplectic manifolds with rational coadjoint orbits of loop algebras. Integrable Hamiltonians are obtained by restriction of elements of the ring of spectral invariants to the image of these moment maps. The isospectral property follows from the Adler-Kostant-Symes theorem, and gives rise to invariant spectral curves. {\it Spectral Darboux coordinates} are introduced on rational coadjoint orbits, generalizing the hyperellipsoidal coordinates to higher rank cases. Applying the Liouville-Arnold integration technique, the Liouville generating function is expressed in completely separated form as an abelian integral, implying the Abel map linearization in the general case. 
  The method of geometrical quantization of symplectic manifolds is applied to constructing infinite dimensional irreducible unitary representations of the algebra of functions on the compact quantum group $SU_q(2)$. A formulation of the method for the general case $SU_q(n)$ is suggested. (This work is the English version of the article submitted for publication in Algebra Analiz.) 
  For a generic $\Ww$ algebra, we give an algorithmic procedure for factoring out all fields of dimension $1/2$, both bosonic and fermionic, and some fields of dimension $1$. This generalizes and makes more explicit the Goddard-Schwimmer theorem for free fermions. We also show how the induced gravity theory for the original $\Ww$ algebra containing the free fields relates to the theory where the fields are factored out. 
  We propose a topological version of four-dimensional (Euclidean) Einstein gravity, in which anti-self-dual 2-forms and an SU(2) connection are used as fundamental fields. The theory describes the moduli space of conformally self-dual Einstein manifolds. In the presence of a cosmological constant, we evaluate the index of the elliptic complex associated with the moduli space. 
  The true variables in QED are the transverse photon components and Dirac's physical electron, constructed out of the fermionic field and the longitudinal components of the photon. We calculate the propagators in terms of these variables to one loop and demonstrate their gauge invariance. The physical electron propagator is shown not to suffer from infrared divergences in any gauge. In general, all physical Green's functions are gauge invariant and infrared-finite. 
  We demonstrate that QED exhibits a previously unobserved symmetry. Some consequences are discussed. 
  In this talk I want to explain the operator substractions needed to regularize gauge currents in a second quantized theory. The case of space-time dimension $3+1$ is considered in detail. In presence of chiral fermions the regularization effects a modification of the local commutation relations of the currents by local Schwinger terms. In $1+1$ dimensions one gets the usual central extension (Schwinger term does not depend on background gauge field) whereas in $3+1$ dimensions one gets an anomaly linear in the background potential. 
  We show that any covariant scattering amplitude of the $W_3$ string will contain, as part of its integrand, a factor that obeys the differential equations satisfied by an Ising model correlation function. This factor can thus be identified with such a correlation function, in agreement with a previous result of the authors. The $W_3$ string is also shown to contain an $N=2$ parafermion theory, and hence to contain in addition the non-linear infinite-dimensional $W$-algebra corresponding to this parafermion theory. The physical states form a representation of this algebra. 
  Taking the induced action for gauge fields coupled to affine currents as an example, we show how loop calculations in non-local two-dimensional field theories can be regulated. Our regularisation method for one loop is based on the method of Pauli and Villars. We use it to calculate the renormalisation factors for the corresponding effective actions, clearing up some discrepancies in the literature. In particular, it will be shown explicitly that vector gauge transformation invariance and Haar invariance of functional integral measures impose different requirements, but they are related by a counterterm (which is local in terms of group variables). For higher loops, we use the method of covariant derivatives combined with Pauli-Villars to argue that the one loop result remains unaltered. 
  We analyze the subdivision properties of certain lattice gauge theories for the discrete abelian groups $Z_{p}$, in four dimensions. In these particular models we show that the Boltzmann weights are invariant under all $(k,l)$ subdivision moves, when the coupling scale is a $p$th root of unity. For the case of manifolds with boundary, we demonstrate analytically that Alexander type $2$ and $3$ subdivision of a bounding simplex is equivalent to the insertion of an operator which equals a delta function on trivial bounding holonomies. The four dimensional model then gives rise to an effective gauge invariant three dimensional model on its boundary, and we compute the combinatorially invariant value of the partition function for the case of $S^{3}$ and $S^{2}\times S^{1}$. 
  We study a family of modules over Kac-Moody algebras realized in multi-valued functions on a flag manifold and find integral representations for intertwining operators acting on these modules. These intertwiners are related to some expressions involving complex powers of Lie algebra generators.   When applied to affine Lie algebras, these expressions give integral formulas for correlation functions with values in not necessarily highest weight modules.   We write related formulas out in an explicit form in the case of $\hat{\gtsl_{2}}$. The latter formulas admit q-deformation producing an integral representation of q-correlation functions. We also discuss a relation of complex powers of Lie algebra (quantum group) generators and Casimir operators to ($q-$)special functions. 
  We introduce and study action of quantum groups on skew polynomial rings and related rings of quotients. This leads to a ``q-deformation'' of the Gel'fand-Kirillov conjecture which we partially prove. We propose a construction of automorphisms of certain non-commutaive rings of quotients coming from complex powers of quantum group generators; this is applied to explicit calculation of singular vectors in Verma modules over $U_{q}(\gtsl_{n+1})$.   We finally give a definition of a $q-$connection with coefficients in a ring of skew polynomials and study the structure of quantum group modules twisted by a $q-$connection. 
  We explicitly write dowm integral formulas for solutions to Knizhnik-Zamolodchikov equations with coefficients in non-bounded -- neither highest nor lowest weight -- $\gtsl_{n+1}$-modules. The formulas are closely related to WZNW model at a rational level. 
  The finite form of the $N=2$ super-Weyl transformations in the chiral and twisted-chiral irreducible formulations of the two-dimensional $N=2$ superfield supergravity are found in $N=2$ superspace. The super-Weyl anomaly of the $N=2$ extended fermionic string theory is computed in terms of the $N=2$ superfields, by using a short time expansion of the $N=2$ chiral heat kernel. The super-Weyl invariant $N=2$ superconformal structure is introduced, and a new definition of the $N=2$ super-Riemannian surfaces is proposed. 
  We derive an equation which gives the tree-level scattering amplitudes for tachyons in the black hole background using the exact states of the collective field hamiltonian corresponding to a deformed matrix model recently proposed by Jevicki and Yoneya. Using directly the symmetry algebra we obtain explicit expression for a class of amplitudes in the tree approximation. We also study the quantum effects in the corresponding collective field theory. In particular, we compute the ground state energy and the free energy at finite temperature up to two loops, and the first quantum correction to the two-point function. 
  It is shown how the canonical symmetry is used to look for the hierarchy of the Hamiltonian operators relevant to the system under consideration. It appears that only the invariance condition can be used to solve the problem. 
  Recently, a class of interaction round the face (IRF) solvable lattice models were introduced, based on any rational conformal field theory (RCFT). We investigate here the connection between the general solvable IRF models and the fusion ones. To this end, we introduce an associative algebra associated to any graph, as the algebra of products of the eigenvalues of the incidence matrix. If a model is based on an RCFT, its associated graph algebra is the fusion ring of the RCFT. A number of examples are studied. The Gordon--generalized IRF models are studied, and are shown to come from RCFT, by the graph algebra construction. The IRF models based on the Dynkin diagrams of A-D-E are studied. While the $A$ case stems from an RCFT, it is shown that the $D-E$ cases do not. The graph algebras are constructed, and it is speculated that a natural isomorphism relating these to RCFT exists. The question whether all solvable IRF models stems from an RCFT remains open, though the $D-E$ cases shows that a mixing of the primary fields is needed. 
  A definition of quantum mechanics on a manifold $ M $ is proposed and a method to realize the definition is presented. This scheme is applicable to a homogeneous space $ M = G / H $. The realization is a unitary representation of the transformation group $ G $ on the space of vector bundle-valued functions. When $ H \ne \{ e \} $, there exist a number of inequivalent realizations. As examples, quantum mechanics on a sphere $ S^n $, a torus $ T^n $ and a projective space $ \RP $ are studied. In any case, it is shown that there are an infinite number of inequivalent realizations. 
  We give a complete classification of all simple current modular invariants, extending previous results for $(\Zbf_p)^k$ to arbitrary centers. We obtain a simple explicit formula for the most general case. Using orbifold techniques to this end, we find a one-to-one correspondence between simple current invariants and subgroups of the center with discrete torsions.  As a by-product, we prove the conjectured monodromy independence of the total number of such invariants. The orbifold approach works in a straightforward way for symmetries of odd order, but some modifications are required to deal with symmetries of even order. With these modifications the orbifold construction with discrete torsion is complete within the class of simple current invariants. Surprisingly, there are cases where discrete torsion is a necessity rather than a possibility. 
  We study on-shell and off-shell properties of the supersymmetric sinh-Gordon and perturbed SUSY Yang-Lee models using the thermodynamic Bethe ansatz and form factors. Identifying the supersymmetric models with the Eight Vertex Free Fermion Model, we derive inversion relation for inhomogeneous transfer matrix and TBA equations and get correct UV results. We obtain two-point form factors of the trace of energy-momentum tensor using the Watson equations and their SUSY transformations. As an application, we compute the UV central charge using these form factors and spectral representation of the $C$-theorem. 
  The Batalin--Vilkovisky (BV) formalism is a useful framework to study gauge theories. We summarize a simple procedure to find a a gauge--fixed action in this language and a way to obtain one--loop anomalies. Calculations involving the antifields can be greatly simplified by using a theorem on the antibracket cohomology. The latter is based on properties of a `Koszul--Tate differential', namely its acyclicity and nilpotency. We present a new proof for this acyclicity, respecting locality and covariance of the theory. This theorem then implies that consistent higher ghost terms in various expressions exist, and it avoids tedious calculations.   This is illustrated in chiral $W_3$ gravity.  We compute the one--loop anomaly without terms of negative ghost number. Then the mentioned theorem and the consistency condition imply that the full anomaly is determined up to local counterterms. Finally we show how to implement background charges into the BV language in order to cancel the anomaly with the appropriate counterterms. Again we use the theorem to simplify the calculations, which agree with previous results. 
  In a previous paper we have shown how, for bosonic fields, the generating functional in both relativistic quantum field theory and thermal field theory can be evaluated by use of a standard quantum mechanical path integral. In this paper we extend this method to include fermionic fields. A particular problem is posed by Green's functions with external fermionic lines, where the different boundary conditions of bosons and fermions in imaginary time have to be accommodated within one path integral expression. The general procedure is worked out in the example of scalar and spinor self-energies in a simple model with a Yukawa coupling of a scalar to a Majorana spinor. 
  Explicit expressions for the generators of the quantum superalgebra $U_q[gl(n/m)]$ acting on a class of irreducible representations are given. The class under consideration consists of all essentially typical representations: for these a Gel'fand-Zetlin basis is known. The verification of the quantum superalgebra relations to be satisfied is shown to reduce to a set of $q$-number identities. 
  This is the first part of an investigation concerning the formulation of 2D gravity in the framework of the uniformization theory of Riemann surfaces. As a first step in this direction we show that the classical Liouville action appears in the expression of the correlators of topological gravity. Next we derive an inequality involving the cutoff of 2D gravity and the background geometry. Another result, always related to uniformization theory, concerns a relation between the higher genus normal ordering and the Liouville action. Furthermore, we show that the chirally split anomaly of CFT is equivalent to the Krichever-Novikov cocycle. By means of the inverse map of uniformization we give a realization of the Virasoro algebra on arbitrary Riemann surfaces and find the eigenfunctions for {\it holomorphic} covariant operators defining higher order cocycles and anomalies which are related to $W$-algebras. Finally we attack the problem of considering the positivity of $e^\sigma$, with $\sigma$ the Liouville field, by proposing an explicit construction for the Fourier modes on compact Riemann surfaces. 
  T-Duality invariant worldsheet string actions, recently written down by Schwarz and Sen, are coupled to the worldsheet gauge fields. It is shown that the integration of the dual coordinates gives the conventional, vector, axial and chiral, gauged string actions for the appropriate choice of the gauged isometries. Alternatively, the gauge field integration is shown to give a T-duality invariant action which matches with the corresponding results known earlier. 
  We develop the formulation of turbulence in terms of the functional integral over the phase space configurations of the vortex cells. The phase space consists of Clebsch coordinates at the surface of the vortex cells plus the Lagrange coordinates of this surface plus the conformal metric. Using the Hamiltonian dynamics we find an invariant probability distribution which satisfies the Liouville equation. The violations of the time reversal invariance come from certain topological terms in effective energy of our Gibbs-like distribution. We study the topological aspects of the statistics and use the string theory methods to estimate intermittency. 
  We review recent progress in 2D gravity coupled to $d<1$ conformal matter, based on a representation of discrete gravity in terms of random matrices. We discuss the saddle point approximation for these models, including a class of related $O(n)$ matrix models. For $d<1$ matter, the matrix problem can be completely solved in many cases by the introduction of suitable orthogonal polynomials. Alternatively, in the continuum limit the orthogonal polynomial method can be shown to be equivalent to the construction of representations of the canonical commutation relations in terms of differential operators. In the case of pure gravity or discrete Ising--like matter, the sum over topologies is reduced to the solution of non-linear differential equations (the Painlev\'e equation in the pure gravity case) which can be shown to follow from an action principle. In the case of pure gravity and more generally all unitary models, the perturbation theory is not Borel summable and therefore alone does not define a unique solution. In the non-Borel summable case, the matrix model does not define the sum over topologies beyond perturbation theory. We also review the computation of correlation functions directly in the continuum formulation of matter coupled to 2D gravity, and compare with the matrix model results. Finally, we review the relation between matrix models and topological gravity, and as well the relation to intersection theory of the moduli space of punctured Riemann surfaces. 
  The effective action on an orbifolded sphere is computed for minimally coupled scalar fields. The results are presented in terms of derivatives of Barnes zeta-functions and it is shown how these may be evaluated. Numerical values are shown. An analytical, heat-kernel derivation of the Ces\`aro-Fedorov formula for the number of symmetry planes of a regular solid is also presented. 
  We consider a hierarchy of the natural type Hamiltonian systems of $n$ degrees of freedom with polynomial potentials separable in general ellipsoidal and general paraboloidal coordinates. We give a Lax representation in terms of $2\times 2$ matrices for the whole hierarchy and construct the associated linear $r$-matrix algebra with the $r$-matrix dependent on the dynamical variables. A Yang-Baxter equation of dynamical type is proposed. Using the method of variable separation we provide the integration of the systems in classical mechanics conctructing the separation equations and, hence, the explicit form of action variables. The quantisation problem is discussed with the help of the separation variables. 
  Fractional superstrings experience new types of ``internal projections'' which alter or deform their underlying worldsheet conformal field theories. In this talk I summarize some recent results concerning both the worldsheet theory which remains after the internal projections have acted, and the spacetime statistics properties of its various sectors. 
  We compute the elliptic genera of orbifolds associated with $N=2$ super--conformal theories which admit a Landau-Ginzburg description. The identification of the elliptic genera of the macroscopic Landau-Ginzburg orbifolds with those of the corresponding microscopic $N=2$ orbifolds further supports the conjectured identification of these theories. For $SU(N)$ Kazama-Suzuki models the orbifolds are associated with certain $\IZ_p$ subgroups of the various coset factors. Based on our approach we also conjecture the existence of "$E$-type" variants of these theories, their elliptic genera and the corresponding Landau-Ginzburg potentials. 
  Drawing an analogy with the Dirac theory of fermions interacting with electromagnetic and gravitational field we write down supersymmetric equations of motion and construct a superfield action for particles with spin 1/4 and 3/4 (quartions), where the role of quartion momentum in effective (2+1)--dimensional space-time is played by an abelian gauge superfield propagating in a basic two-dimensional Grassmann-odd space with a cosmological constant showing itself as the quartion mass. So, the (0|2) (0 even and 2 odd) dimensional model of quartions interacting with the gauge and gravitational field manifests itself as an effective (2+1)-dimensional supersymmetric theory of free quartions. 
  For the electromagnetic fields, hydrodynamic media and turbulent flows we consider the relationship between a topological characteristic of vector fields known as helicity and the angular momentum of the medium, and discuss, in this respect, the problem of helicity and angular momentum transfer from the electromagnetic field to a medium. 
  We introduce the notion of ortho-symplectic super triple system, and apply it to find solutions of super Yang-Baxter equation. Also, the para-statistics are formulated as a Lie-super triple system. 
  We discuss the functional Schroedinger picture for fermions in external fields for both stationary and time-dependent problems. We give formal results for the ground state and the solution of the time-dependent Schroedinger equation for QED in arbitrary dimensions, while more explicit results are obtained in two dimensions. For both the massless and massive Schwinger model we give an explicit expression for the ground state functional as well as for the expectation values of energy, electric and axial charge. We also give the corresponding results for non-abelian fields. We solve the functional Schroedinger equation for a constant external field in four dimensions and obtain the amount of particle creation. We solve the Schroedinger equation for arbitrary external fields for massless QED in two dimensions and make a careful discussionof the anomalous particle creation rate. Finally, we discuss some subtleties connected with the interpretation of the quantized Gauss constraint. 
  Several aspects of fusion rings and fusion rule algebras, and of their manifestations in twodimensional (conformal) field theory, are described: diagonalization and the connection with modular invariance; the presentation in terms of quotients of polynomial rings; fusion graphs; various strategies that allow for a partial classification; and the role of the fusion rules in the conformal bootstrap programme. 
  Extended super self-dual systems have a structure reminiscent of a ``matreoshka''. For instance, solutions for N=0 are embedded in solutions for N=1, which are in turn embedded in solutions for N=2, and so on. Consequences of this phenomenon are explored. In particular, we present an explicit construction of local solutions of the higher-N super self-duality equations starting from any N=0 self-dual solution. Our construction uses N=0 solution data to produce N=1 solution data, which in turn yields N=2 solution data, and so on; each stage introducing a dependence of the solution on sufficiently many additional arbitrary functions to yield the most general supersymmetric solution having the initial N=0 solution as the helicity +1 component. The problem of finding the general local solution of the $N>0$ super self-duality equations therefore reduces to finding the general solution of the usual (N=0) self-duality equations. Another consequence of the matreoshka phenomenon is the vanishing of many conserved currents, including the supercurrents, for super self-dual systems. 
  We find a canonical $N{=}2$ superconformal algebra (SCA) in the BRST complex associated to any affine Lie algebra $\ghhat$ with $\gh$ semisimple. In contrast with the similar known results for the Virasoro, $N{=}1$ supervirasoro, and $\W_3$ algebras, this SCA does not depend on the particular ``matter'' representation chosen. Therefore it follows that every gauged WZNW model with data $(\gg\supset\gh, k)$ has an $N{=}2$ SCA with central charge $c=3\dim\gh$ independent of the level $k$. In particular, this associates to every embedding $sl(2) \subset \gg$ a one-parameter family of $c{=}9$ $N{=}2$ supervirasoro algebras. As a by-product of the construction, one can deduce a new set of ``master equations'' for generalized $N{=}2$ supervirasoro constructions which is simpler than the one considered thus far. 
  String theory implies a relatively modest growth in computational complexity for perturbative gravity calculations as compared to gauge theory calculations, contrary to field theory expectations. An explicit string-based calculation, which would be extremely difficult using conventional techniques, is presented to illustrate this. 
  I review the main properties of four-dimensional strings constructed with free-fermions on the world-sheet. In particular I discuss possible model independent low energy predictions related to the existence of states with fractional electric charges, the computation of the string unification scale, the string model building, and the perturbative approach to supersymmetry breaking which makes the spectacular prediction of a new large dimension at the TeV scale. 
  We present a second order gravity action which consists of ordinary Einstein action augmented by a first-order, vector like, Chern-Simons quasi topological term.This theory is ghost-free and propagates a pure spin-2 mode. It is diffeomorphism invariant, although its local Lorentz invariance has been spontaneously broken. 
  We present self-dual spin-3 and 4 actions using relevant dreibein fields. since these actions start with a Chern-Simons like kinetic term (and therefore cannot be obtained through dimensional reduction) one might wonder whether they need the presence of auxiliary ghost-killings fields. It turns out that they must contain, also in this three dimensional case, auxiliary fields. They do not break self-duality: their free actions do not contain kinetic second order terms. 
  Motivated by the possibility of an effective string description for the infrared limit of pure Yang-Mills theory, we present a toy model for an effective theory of random surfaces propagating in a target space of $D>2$. We show that the scaling exponents for the fixed area partition function of the theory are apparently well behaved. We make some observations regarding the usefulness of this toy model. 
  We extend the bosonization of $2+1$ - dimensional QED with one fermionic flavor performed previously to the case of QED with an induced Chern - Simons term. The coefficient of this term is quantized: $e^2n/8\pi$, $n\in {\bf Z}$. The fermion operators are constructed in terms of the bosonic fields $A_i$ and $E_i$. The construction is similar to that in the $n=0$ case. The resulting bosonic theory is Lorentz invariant in the continuum limit and has Maxwell's equations as its equations of motion. The algebra of bilinears exhibits nontrivial operatorial mixing with lower dimensional operators, which is absent for $n=0$. 
  Using the previous obtained universal $R$-matrix for the quantized nontwisted affine Lie algebras $U_q(A_1^{(1)})$ and $U_q(A_2^{(1)})$, we determine the explicitly spectral-dependent universal $R$-matrix for the corresponding quantum Lie algebras $U_q(A_1)$ and $U_q(A_2)$. As their applications, we reproduce the well-known results in the fundamental representations and we also derive an extreamly explicit formula of the spectral-dependent $R$-matrix for the adjoint representation of $U_q(A_2)$, the simplest non-trival case when the tensor product decomposition of the representation with itself has finite multiplicity. 
  This paper is an extended version of our previous short letter \cite{ZG2} and is attempted to give a detailed account for the results presented in that paper. Let $U_q({\cal G}^{(1)})$ be the quantized nontwisted affine Lie algebra and $U_q({\cal G})$ be the corresponding quantum simple Lie algebra. Using the previous obtained universal $R$-matrix for $U_q(A_1^{(1)})$ and $U_q(A_2^{(1)})$, we determine the explicitly spectral-dependent universal $R$-matrix for $U_q(A_1)$ and $U_q(A_2)$. We apply these spectral-dependent universal $R$-matrix to some concrete representations. We then reproduce the well-known results for the fundamental representations and we are also able to derive for the first time the extreamly explicit and compact formula of the spectral-dependent $R$-matrix for the adjoint representation of $U_q(A_2)$, the simplest nontrival case when the tensor product of the representations is {\em not} multiplicity-free. 
  In order to construct the inverse mapping of the period mapping for the primitive form for the semi-universal deformation of a simple elliptic singularity, K.Saito introduced in [5] the ``flat structure'' for the extended affine root system. In section 3, we construct explicitly the flat theta invariants in the case of type $E_6$ using the Jacobi form introduced by Wirthm\"uller [7]. Combining the results of Kato [3], Noumi [4] (explicit description of the flat coordinates), this gives an answer to Jacobi's inversion problem (up to linear isomorphism) of this period mapping for a simple elliptic singularity of type $\tilde E_6$ (see also [6]). The details will be published elsewhere. 
  The requirements of conformal invariance for two and three point functions for general dimension $d$ on flat space are investigated. A compact group theoretic construction of the three point function for arbitrary spin fields is presented and it is applied to various cases involving conserved vector operators and the energy momentum tensor. The restrictions arising from the associated conservation equations are investigated. It is shown that there are, for general $d$, three linearly independent conformal invariant forms for the three point function of the energy momentum tensor, although for $d=3$ there are two and for $d=2$ only one. The form of the three point function is also demonstrated to simplify considerably when all three points lie on a straight line. Using this the coefficients of the conformal invariant point functions are calculated for free scalar and fermion theories in general dimensions and for abelian vector fields when $d=4$. Ward identities relating three and two point functions are also discussed. This requires careful analysis of the singularities in the short distance expansion and the method of differential regularisation is found convenient. For $d=4$ the coefficients appearing in the energy momentum tensor three point function are related to the coefficients of the two possible terms in the trace anomaly for a conformal theory on a curved space background. 
  The quantum mechanics of a system of charged particles interacting with a magnetic field on Riemann surfaces is studied. We explicitly construct the wave functions of ground states in the case of a metric proportional to the Chern form of the $\theta$-bundle, and the wave functions of the Landau levels in the case of the the Poincar{\' e} metric. The degeneracy of the the Landau levels is obtained by using the Riemann-Roch theorem. Then we construct the Laughlin wave function on Riemann surfaces and discuss the mathematical structure hidden in the Laughlin wave function. Moreover the degeneracy of the Laughlin states is also discussed. 
  Contribution to the Proceedings in honour of C.N. Yang, editor S-T Yau. To 
  Using (1,0) superfield methods, we determine the general scalar potential consistent with off-shell (p,0) supersymmetry and (1,1) supersymmetry in two-dimensional non-linear sigma models with torsion. We also present an extended superfield formulation of the (p,0) models and show how the (1,1) models can be obtained from the (1,1)-superspace formulation of the gauged, but massless, (1,1) sigma model. 
  We study a regularization of the Pauli-Villars kind of the one loop gravitational divergences in any dimension. The Pauli-Villars fields are massive particles coupled to gravity in a covariant and nonminimal way, namely one real tensor and one complex vector. The gauge is fixed by means of the unusual gauge-fixing that gives the same effective action as in the context of the background field method. Indeed, with the background field method it is simple to see that the regularization effectively works. On the other hand, we show that in the usual formalism (non background) the regularization cannot work with each gauge-fixing.In particular, it does not work with the usual one. Moreover, we show that, under a suitable choice of the Pauli-Villars coefficients, the terms divergent in the Pauli-Villars masses can be corrected by the Pauli-Villars fields themselves. In dimension four, there is no need to add counterterms quadratic in the curvature tensor to the Einstein action (which would be equivalent to the introduction of new coupling constants). The technique also works when matter is coupled to gravity. We discuss the possible consequences of this approach, in particular the renormalization of Newton's coupling constant and the appearance of two parameters in the effective action, that seem to have physical implications. 
  Developing on the ideas of R. Stora and coworkers, a formulation of two dimensional field theory endowed with extended conformal symmetry is given, which is based on deformation theory of holomorphic and Hermitian spaces. The geometric background consists of a vector bundle $E$ over a closed surface $\Sigma$ endowed with a holomorphic structure and a Hermitian structure subordinated to it. The symmetry group is the semidirect product of the automorphism group ${\rm Aut}(E)$ of $E$ and the extended Weyl group ${\rm Weyl}(E)$ of $E$ and acts on the holomorphic and Hermitian structures. The extended Weyl anomaly can be shifted into an automorphism chirally split anomaly by adding to the action a local counterterm, as in ordinary conformal field theory. The dependence on the scale of the metric on the fiber of $E$ is encoded in the Donaldson action, a vector bundle generalization of the Liouville action. The Weyl and automorphism anomaly split into two contributions corresponding respectively to the determinant and projectivization of $E$. The determinant part induces an effective ordinary Weyl or diffeomorphism anomaly and the induced central charge can be computed. 
  The periodic flag manifold (in the Sato Grassmannian context) description of the modified Korteweg--de Vries hierarchy is used to analyse the translational and scaling self--similar solutions of this hierarchy. These solutions are characterized by the string equations appearing in the double scaling limit of the symmetric unitary matrix model with boundary terms. The moduli space is a double covering of the moduli space in the Sato Grassmannian for the corresponding self--similar solutions of the Korteweg--de Vries hierarchy, i.e. of stable 2D quantum gravity. The potential modified Korteweg--de Vries hierarchy, which can be described in terms of a line bundle over the periodic flag manifold, and its self--similar solutions corresponds to the symmetric unitary matrix model. Now, the moduli space is in one--to--one correspondence with a subset of codimension one of the moduli space in the Sato Grassmannian corresponding to self--similar solutions of the Korteweg--de Vries hierarchy. 
  In this paper the Galilean, scaling and translational self--similarity conditions for the AKNS hierarchy are analysed geometrically in terms of the infinite dimensional Grassmannian. The string equations found recently by non--scaling limit analysis of the one--matrix model are shown to correspond to the Galilean self--similarity condition for this hierarchy. We describe, in terms of the initial data for the zero--curvature 1--form of the AKNS hierarchy, the moduli space of these self--similar solutions in the Sato Grassmannian. As a byproduct we characterize the points in the Segal--Wilson Grassmannian corresponding to the Sachs rational solutions of the AKNS equation and to the Nakamura--Hirota rational solutions of the NLS equation. An explicit 1--parameter family of Galilean self--similar solutions of the AKNS equation and the associated solution to the NLS equation is determined. 
  We present a generalized reduction procedure which encompasses the one based on the momentum map and the projection method. By using the duality between manifolds and ring of functions defined on them, we have cast our procedure in an algebraic context. In this framework we give a simple example of reduction in the non-commutative setting. 
  Realistic quantum mechanics based on complex probability theory is shown to have a frequency interpretation, to coexist with Bell's theorem, to be linear, to include wavefunctions which are expansions in eigenfunctions of Hermitian operators and to describe both pure and mixed systems. Illustrative examples are given. The quantum version of Bayesian inference is discussed. Postscript version of hep-th/9307019. 
  The string equation appearing in the double scaling limit of the Hermitian one--matrix model, which corresponds to a Galilean self--similar condition for the KdV hierarchy, is reformulated as a scaling self--similar condition for the Ur--KdV hierarchy. A non--scaling limit analysis of the one--matrix model has led to the complexified NLS hierarchy and a string equation. We show that this corresponds to the Galilean self--similarity condition for the AKNS hierarchy and also its equivalence to a scaling self--similar condition for the Heisenberg ferromagnet hierarchy. 
  The relation between two--dimensional integrable systems and four--dimen\-sional self--dual Yang--Mills equations is considered. Within the twistor description and the zero--curvature representation a method is given to associate self--dual Yang-Mills connections with integrable systems of the   Korteweg--de Vries and non--linear Schr\"odinger type or principal chiral models.   Examples of self--dual connections are constructed that as points in the moduli do not have two independent conformal symmetries. 
  The geometric interpretation of the Batalin-Vilkovisky antibracket as the Schouten bracket of functional multivectors is examined in detail. The identification is achieved by the process of repeated contraction of even functional multivectors with fermionic functional 1-forms. The classical master equation may then be considered as a generalisation of the Jacobi identity for Poisson brackets, and the cohomology of a nilpotent even functional multivector is identified with the BRST cohomology. As an example, the BRST-BV formulation of gauge fixing in theories with gauge symmetries is reformulated in the jet bundle formalism. (Hopefully this version will be TeXable) 
  The renormalisation group (RG) flow on the space of couplings of a simple model with two couplings is examined. The model considered is that of a single component scalar field with $\phi^4$ self interaction coupled, via Yukawa coupling, to a fermion in flat four dimensional space. The RG flow on the two dimensional space of couplings, ${\cal G}$, is shown to be derivable from a potential to sixth order in the couplings, which requires two loop calculations of the $\beta$-functions. The identification of a potential requires the introduction of a metric on ${\cal G}$ and it is shown that the metric defined by Zamalodchikov, in terms of two point correlation functions of composite operators, gives potential flow to this order. 
  It is shown that the renormalisation group (RG) equation can be viewed as an equation for Lie transport of physical amplitudes along the integral curves generated by the $\beta$-functions of a quantum field theory. The anomalous dimensions arise from Lie transport of basis vectors on the space of couplings. The RG equation can be interpreted as relating a particular diffeomorphism of flat space-(time), that of dilations, to a diffeomorphism on the space of couplings generated by the vector field associated with the $\beta$-functions. 
  Closing a gap in the literature on the subject, the local solutions of 2D-gravity with torsion are given for Euclidian signature. For the topology of a cylinder the system is quantized. 
  We give a physicist oriented survey of Poisson-Lie symmetries of classical systems. We consider finite dimensional geometric actions and the chiral WZNW model as examples for the general construction. An essential point is that quadratic Poisson bracets appear for group--like variables. It is believed that after quantization they lead to quadratic exchange algebras. 
  We study perturbatively the (conformal) WZNW model. At one loop we compute one-particle irreducible two- and three-point current correlation functions, both in the conventional version and in the classically equivalent, chiral, nonlocal, induced version of the model. At two loops we compute the two-point function and find that it vanishes (modulo infrared-induced logarithms). We use dimensional regularization and the $R^*$ operation for removing infrared divergences. The outcome of the calculations is insensitive to the treatment of the $\varepsilon^{\m\n}$ tensor as a two-dimensional or $d$-dimensional object. Our results indicate that the one-particle irreducible current correlation functions constitute an effective action equal to the original WZNW action with the familiar level shift, $k\to k+\tilde h$. 
  Abtract: 2-dimensional fermions are coupled to extrinsic geometry of a conformally immersed surface in ${\bf R}^3$ through gauge coupling. By integrating out the fermions, we obtain a WZNW action involving extrinsic curvature of the surface. Restricting the resulting effective action to surfaces of $h\sqrt g=1$, an explicit form of the action invariant under Virasaro symmetry is obtained. This action is a sum of the geometric action for the Virasaro group and the light-cone action of 2-d gravity plus an interaction term. The central charges of the theory in both the left and right sectors are calculated. 
  We investigate $N=2$ extended superconformal symmetry, using the half-twisted Landau-Ginzburg models. The first example is the $D_{2n+2}$ -type minimal model. It has been conjectured that this model has a spin $n$ super $W$ current. We checked this by the direct computations of the BRS cohomology class up to $n=4$. We observe for $n\le 3$ the super W currents generate the ring isomorphic to the chiral ring of the model with respect to the classical product. We thus conjecture that this isomorphism holds for any $n$. The next example is $ CP_{n}$ coset model. In this case we find a sort of Miura transformation which gives the simple formula for the super W currents of spin \{1,2,...,n\} in terms of the chiral superfields. Explicit form of the super W currents and their Poisson brackets are obtained for $CP_{2},CP_{3}$ case. We also conjecture that as long as the classical product is concerned, these super W currents generate the ring isomorphic to the chiral ring of the model and this is checked for $CP_2$ model. 
  (Minor corrections and reference added) 
  We prove a generalization of the Verlinde formula to fermionic rational conformal field theories. The fusion coefficients of the fermionic theory are equal to sums of fusion coefficients of its bosonic projection. In particular, fusion coefficients of the fermionic theory connecting two conjugate Ramond fields with the identity are either one or two. Therefore, one is forced to weaken the axioms of fusion algebras for fermionic theories. We show that in the special case of fermionic W(2,d)-algebras these coefficients are given by the dimensions of the irreducible representations of the horizontal subalgebra on the highest weight. As concrete examples we discuss fusion algebras of rational models of fermionic W(2,d)-algebras including minimal models of the $N=1$ super Virasoro algebra as well as $N=1$ super W-algebras SW(3/2,d). 
  We argue that the algebra $W_q(n)$, generated by $n$ pairs of deformed $q$-bosons, does not allow a Hopfalgebra structure. To this end we show that it is impossible to define a comultiplication even for the usual, nondeformed case. We indicate how the comultiplication on $U_q[osp(1/2n)]$ can be used in order to construct representations of deformed (not necessarily Hopf) algebras in tensor products of Fock spaces. 
  We analyze the symplectic structure of two-dimensional dilaton gravity by evaluating the symplectic form on the space of classical solutions. The case when the spatial manifold is compact is studied in detail. When the matter is absent we find that the reduced phase space is a two-dimensional cotangent bundle and determine the Hilbert space of the quantum theory. In the non-compact case the symplectic form is not well defined due to an unresolved ambiguity in the choice of the boundary terms. 
  {}From the one-loop effective potential for a gas of non-relativistic bosons in two spatial dimensions interacting via a delta-function potential at zero-temperature and finite chemical potential, the anomaly of the energy-momentum tensor follows directly. It is also similarly derived when the bosons have an additional Chern-Simons interaction. In the special case of anyons, the scale anomaly vanishes to one-loop order in the effective potential and also to second order in the statistical angle. 
  The topological charges of the \an affine Toda solitons are considered. A general formula is presented for the number of charges associated with each soliton, as well as an expression for the charges themselves. For each soliton the charges are found to lie in the corresponding fundamental representation, though in general these representations are not filled. Each soliton's topological charges are invariant under cyclic permutations of the simple roots plus the extended root or equivalently, under the action of the Coxeter element (with a particular ordering). Multisolitons are considered and are found to have topological charges filling the remainder of the fundamental representations as well as the entire weight lattice. The article concludes with a discussion of some of the other affine Toda theories. 
  We greatly simplify the light-cone gauge description of a relativistic membrane moving in Minkowski space by performing a field-dependent change of variables which allows the explicit solution of all constraints and a Hamiltonian reduction to a $SO(1,3)$ invariant $2+1$-dimensional theory of isentropic gas dynamics, where the pressure is inversely proportional to (minus) the mass-density. Simple expressions for the generators of the Poincar\'e group are given. We also find a generalized Lax pair which involves as a novel feature complex conjugation. The extension to the supersymmetric case, as well as to higher-dimensional minimal surfaces of codimension one is briefly mentioned. 
  The conformal field theory for the $gl(N,N)$ affine Lie superalgebra in two space-time dimensions is studied. The energy-momentum tensor of the model, with vanishing Virasoro anomaly, is constructed. This theory has a topological symmetry generated by operators of dimensions 1, 2 and 3, which are represented as normal-ordered products of $gl(N,N)$ currents. The topological algebra they satisfy is linear and differs from the one obtained by twisting the $N=2$ superconformal models. It closes with a set of $gl(N)$ bosonic and fermionic currents. The Wess-Zumino-Witten model for the supergroup $GL(N,N)$ provides an explicit realization of this symmetry and can be used to obtain a free-field representation of the different generators. In this free-field representation, the theory decomposes into two uncoupled components with $sl(N)$ and $U(1)$ symmetries. The non-abelian component is responsible for the extended character of the topological algebra, and it is shown to be equivalent to an $SL(N)/ SL(N)$ coset model. In the light of these results, the $G/ G$ coset models are interpreted as topological sigma models for the group manifold of $G$ 
  We develop techniques to compute the complete the massless spectrum in heterotic string compactification on N=2 supersymmetric Landau-Ginzburg orbifolds. This includes not just the familiar charged fields, but also the gauge singlets. The number of gauge singlets can vary in the moduli space of a given compactification and can differ from what it would be in the large radius limit of the corresponding Calabi-Yau. Comparison with exactly soluble Gepner models provides a confirmation of our results at Gepner points. Our methods carry over straightforwardly to $(0,2)$ Landau-Ginzburg models. 
  We propose a twistor--like formulation of N=1, D=3,4,6 and 10 null superstrings. The model possesses N=1 target space supersymmetry and n=D-2 local worldsheet supersymmetry, the latter replaces the kappa-symmetry of the conventional approach to the strings. Adding a Wess--Zumino term to a null superstring action we observe a string tension generation mechanism: the induced worldsheet metric becomes non-degenerate and the resulting model turns out to be classically equivalent to the heterotic string. 
  Witten recently gave further evidence for the conjectured relationship between the $A$ series of the $N=2$ minimal models and certain Landau-Ginzburg models by computing the elliptic genus for the latter. The results agree with those of the $N=2$ minimal models, as can be calculated from the known characters of the discrete series representations of the $N=2$ superconformal algebra. The $N=2$ minimal models also have a Lagrangian representation as supersymmetric gauged WZW models. We calculate the elliptic genera, interpreted as a genus one path integral with twisted boundary conditions, for such models and recover the previously known result. 
  This work was inspired by the article of Parkhomenko, who drew attention to the central role played in the work of Spindel, Sevrin, Troust and van Proyen, by Manin triples. These authors have shown how to associate to a Manin triple an $N=2$ superconformal field theory (the work of Kazama-Suzuki is a special case of their results). In this paper, we construct a deformation of their theory, with continuously varying central charge, analogous to the Fock representations of the Virasoro algebra with stress-energy tensor $-(\phi')^2/2+\alpha\phi''$. 
  We consider modifications to general relativity by the non-local (classical and quantum) string effects for the case of a D-dimensional Scwarzschild black hole. The classical non-local effects do not alter the spacetime topology (the horizon remains unshifted, at least perturbatively). We suggest a simple analytic continuation of the perturbative result into the non-perturbative domain, which eliminates the black hole singularity at the origin and yields an ultraviolet-finite theory of quantum gravity. We investigate the quantum non- local effects (including massive modes) and argue that the inclusion of these back reactions resolves the problem of the thermal spectrum in the semi- classical approach of field quantization in a black hole background, through the bootstrap condition. The density of states for both the quantum and thermal interpretation of the WKB formula are finally shown to differ quant- itatively when including the non-local effects. 
  We construct the quantum BRST operators for a large class of superconformal and quasi--superconformal algebras with quadratic nonlinearity. The only free parameter in these algebras is the level of the (super) Kac-Moody sector. The nilpotency of the quantum BRST operator imposes a condition on the level. We find this condition for (quasi) superconformal algebras with a Kac-Moody sector based on a simple Lie algebra and for the $Z_2\times Z_2$--graded superconformal algebras with a Kac-Mody sector based on the superalgebra $osp(N\vert 2M)$ or $s\ell(N+2\vert N)$. 
  For a dissipative system with Ohmic friction, we obtain a simple and exact solution for the wave function of the system plus the bath. It is described by the direct product in two independent Hilbert space. One of them is described by an effective Hamiltonian, the other represents the effect of the bath, i.e., the Brownian motion, thus clarifying the structure of the wave function of the system whose energy is dissipated by its interaction with the bath. No path integral technology is needed in this treatment. The derivation of the Weisskopf-Wigner line width theory follows easily. 
  We construct the Hamiltonian operator of the string field theory for $c=0$ string theory. It describes how strings evolve in the coordinate frame, which is defined by using the geodesic distance on the worldsheet. The Hamiltonian consists of three-string interaction terms and a tadpole term. We show that one can derive the loop amplitudes of $c=0$ string theory from this Hamiltonian. 
  We perform a classical BRST analysis of the symmetries corresponding to a generic $w_N$-algebra. An essential feature of our method is that we write the $w_N$-algebra in a special basis such that the algebra manifestly has a ``nested'' set of subalgebras $v_N^N \subset v_N^{N-1} \subset \dots \subset v_N^2 \equiv w_N$ where the subalgebra $v_N^i\ (i=2, \dots ,N)$ consists of generators of spin $s=\{i,i+1,\dots ,N\}$, respectively. In the new basis the BRST charge can be written as a ``nested'' sum of $N-1$ nilpotent BRST charges. In view of potential applications to (critical and/or non-critical) $W$-string theories we discuss the quantum extension of our results. In particular, we present the quantum BRST-operator for the $W_4$-algebra in the new basis. For both critical and non-critical $W$-strings we apply our results to discuss the relation with minimal models. 
  These lectures discuss the formulation of quantum mechanics with fractional spin and statistics in 2+1 dimensions in a relativistic setting, emphasizing the path-integral approach. The non-relativistic theory is reviewed from a path-integral viewpoint. The group-theoretical underpinnings of relativistic fractional spin are discussed, then the path-integral quantization of spin and of massive fermions without using spinors is reviewed. The path integral for a system of n relativistic particles with fractional spin is constructed, the spin-statistics relation and the Lorentz and Poincare' representation content of physical states are discussed. Some problems in formulating a field theory with fractional statistics are presented, and their resolution in the operator cocycle approach is reviewed. 
  We describe how a natural lattice analogue of the abelian current algebra combined with free discrete time dynamics gives rise to the lattice Virasoro algebra and corresponding hierarchy of conservation laws. 
  We show that many of the recently proposed supersymmetric p-brane solutions of d=10 and d=11 supergravity have the property that they interpolate between Minkowski spacetime and a compactified spacetime, both being supersymmetric supergravity vacua. Our results imply that the effective worldvolume action for small fluctuations of the super p-brane is a supersingleton field theory for $(adS)_{p+2}$, as has been often conjectured in the past. 
  We present a simple derivation of the Callan-Harvey-Naculich effect, {\it i.e.} the compensation of charge violation on axion strings due to gauge anomalies by accretion of charge onto the string from the surrounding space. We then show, in the case of axion fields without a potential, that an alternative explanation is possible in which no reference to the surrounding space is necessary because the anomalies are cancelled by a version of the Green-Schwarz mechanism. We prove that such an alternative explanation is always possible in the more general context of p-brane defects in d-dimensional field theories, and hence that there always exists an anomaly-free effective worldvolume action whenever the spacetime theory is anomaly free. Our results have implications, which we discuss, for heterotic and type II fivebranes. 
  The Hilbert bundle for the massless fermions of the Schwinger model on a circle, over the space of gauge field configurations, is topologically non-trivial (twisted). The corresponding bundle for massive fermions is topologically trivial (periodic). Since the structure of the fermionic Hilbert bundle changes discontinuously the possibility of perturbing in the mass is thrown into doubt. In this article, we show that a direct application of the anti-adiabatic theorem of Low, allows the structure of the massless theory to be dynamically preserved in the strong coupling limit, ${e\over m}>>1$. This justifies the use of perturbation theory in the bosonized version of the model, in this limit. 
  After reviewing the basic aspects of the exactly solvable quantum-corrected dilaton-gravity theories in two dimensions, we discuss a (subjective) selection of other aspects: a) supersymmetric extensions, b) canonical formalism, ADM-mass, and the functional integral measure, and c) a positive energy theorem and its application to the ADM- and Bondi-masses. 
  The technique of (discretised) light-cone quantisation, as applied to matrix models of relativistic strings, is reviewed. The case of the c=2 non-critical bosonic string is discussed in some detail to clarify the nature of the continuum limit. Futher applications for the technique are then outlined. (To appear in proceedings of the NATO Advanced Workshop on Recent Developments in Strings, Conformal Models, and Topological Field Theory, Cargese 12-21 May 1993.) 
  We study higher order approximations in the renormalization group approach to matrix models. We use constraint equations on the free energy resulting from a freedom of field redefinitionsand obtain the effective beta function for a single coupling constant to the fifth order. The fixed point and the string susceptibility exponent are shown to approach the values obtained in the exact solution as the order of approximations becomes higher. 
  To entirely determine the resulting functions of one-loop integrals it is necessary to find the correct analytic continuation to all relevant kinematical regions. We argue that this continuation procedure may be performed in a general and mathematical accurate way by using the ${\cal R}$ function notation of these integrals. The two- and three-point cases are discussed explicitly in this manner. 
  Recently dilogarithm identities have made their appearance in the physics literature. These identities seem to allow to calculate structure constants like, in particular, the effective central charge of certain conformal field theories from their fusion rules. In Nahm, Recknagel, Terhoeven (1992) a proof of identities of this type was given by considering the asymptotics of character functions in the so-called Rogers-Ramanujan sum form and comparing with the asymptotics predicted by modular covariance. Refining the argument, we obtain {\it the general connection of quantum dimensions of certain conformal field theories to the arguments of the dilogarithm function} in the identities in question and {\it an infinite set of consistency conditions on the parameters of Rogers-Ramanujan type partitions for them to be modular covariant}. 
  By analysing the infinite dimensional midisuperspace of spherically symmetric dust universes, and aply it to collapsing dust stars, one finds that the general quantum state is a bound state. This leads to discrete spectrum. In the case of a Schwarzschild black hole, the discrete spectrum implies Bekenstein area quantization: the area of the horizon is an integer multiple of the Planck area. Knowing the microscopic (quantum) states, we suggest a microscopic interpretation of the thermodynamics of black holes: the degeneracy of the quantum states forming a black hole, gives the Bekenstein- Hawking entropy. All other thermodynamical quantities can be derived by using the standard definitions. 
  The focus of this thesis is on (1) the role of Ka\v c-Moody (KM) algebras in string theory and the development of techniques for systematically building string theory models based on higher level ($K\geq 2$) KM algebras and (2) fractional superstrings. In chapter two we review KM algebras and their role in string theory. In the next chapter, we present two results concerning the construction of modular invariant partition functions for conformal field theories built by tensoring together other conformal field theories. First we show how the possible modular invariants for the tensor product theory are constrained if the allowed modular invariants of the individual conformal field theory factors have been classified. We illustrate the use of these constraints for theories of the type $SU(2)_{K_A}xSU(2)_{K_B}$, finding all consistent theories for $K_A$ and $K_B$ odd. Second we show how known diagonal modular invariants can be used to construct inherently asymmetric invariants where the holomorphic and anti- holomorphic theories do not share the same chiral algebra. Explicit examples are given. Next, in chapter four we investigate some issues relating to proposed fractional superstring theories with $D_{\rm critical}<10$. Using the factorization approach of Gepner and Qiu, we systematically rederive the partition functions of the $K=4,\, 8,$ and $16$ theories and examine their spacetime supersymmetry. Generalized GSO projection operators for the $K=4$ model are found. Uniqueness of the twist field, $\phi^{K/4}_{K/4}$, as source of spacetime fermions, is demonstrated. 
  The quantum statistics of charged, extremal black holes is investigated beginning with the hypothesis that the quantum state is a functional on the space of closed three-geometries, with each black hole connected to an oppositely charged black hole through a spatial wormhole. From this starting point a simple argument is given that a collection of extremal black holes obeys neither Bose nor Fermi statistics. Rather they obey an exotic variety of particle statistics known as ``infinite statistics'' which resembles that of distinguishable particles and is realized by a $q$-deformation of the quantum commutation relations. 
  The proposed by Bastianelli and van Nieuwenhuizen new method of calculations of trace anomalies is applied in the conformal gauge field case. The result is then reproduced by the heat equation method. An error in previous calculation is corrected. It is pointed out that the introducing gauge symmetries into a given system by a field-enlarging transformation can result in unexpected quantum effects even for trivial configurations. 
  In the present article we present exact solutions of the Dirac equation for electric neutral particles with anomalous electric and magnetic moments. Using the algebraic method of separation of variables, the Dirac equation is separated in cartesian, cylindrical and spherical coordinates, and exact solutions are obtained in terms of special functions. 
  We prove that the wave vectors of the off-shell Bethe Ansatz equation for the inhomogeneous SU(2) lattice vertex model render in the quasiclassical limit the solution of the Knizhnik-Zamolodchikov equation. 
  Loop equations of matrix models express the invariance of the models under field redefinitions. We use loop equations to prove that it is possible to define continuum times for the generic hermitian {1-matrix} model such that all correlation functions in the double scaling limit agree with the corresponding correlation functions of the Kontsevich model expressed in terms of kdV times. In addition the double scaling limit of the partition function of the hermitian matrix model agree with the $\tau$-function of the kdV hierarchy corresponding to the Kontsevich model (and not the square of the $\tau$-function) except for some complications at genus zero. 
  Invariance under non-linear ${\sf {\hat W}}_{\infty}$ algebra is shown for the two-boson Liouville type of model and its algebraic generalizations, the extended conformal Toda models. The realization of the corresponding generators in terms of two boson currents within KP hierarchy is presented. 
  We summarize our renormalization group approach for the vector model as well as the matrix model which are the discretized quantum gravity in one- and two-dimensional spacetime. A difference equation is obtained which relates free energies for neighboring values of $N$. The reparametrization freedom in field space is formulated by means of the loop equation. The reparametrization identities reduce the flow in the infinite dimensional coupling constant space to that in finite dimensions. The matrix model gives a nonlinear differential equation as an effective renormalization group equation. The fixed point and the susceptibility exponents can be determined even for the matrix models in spite of the nonlinearity. They agree with the exact result. 
  We determine the general scalar potential consistent with (p,q) supersymmetry in two-dimensional non-linear sigma models with torsion, generalizing previous results for special cases. We thereby find many new supersymmetric sigma models with potentials, including new (2,2) and (4,4) models. 
  We consider a superconformal quantum mechanical system which has been chosen on the basis of a local BRST topological invariance. We suggest that it truly leads to topological observables which we compute. The absences of a ground state and of a mass gap are special features of this system. 
  In the present work we argue that the usual assumption that magnetic currents possess the vector structure characteristic of electric currents may be the source of several difficulties in the theory of magnetic monopoles. We propose an {\it axial} magnetic current instead and show that such difficulties are solved. Charge quantization is shown to be intimately connected with results of theories of discrete space time. 
  In the present contribution we propose a gedankenexperiment in which the restriction of rational values on the velocities emerges as a necessary condition from Classical Electromagnetism and Quantum Mechanics. This restriction is shown to be intimately connected to Dirac's electric charge quantization condition. 
  We present a possible solution for the long standing problem of the incompatibility of Dirac's charge quantization condition with integer values for the angular momentum of the electromagnetic field. 
  We obtain the exact Dirac algebra obeyed by the conserved non-local charges in bosonic non-linear sigma models. Part of the computation is specialized for a symmetry group $O(N)$. As it turns out the algebra corresponds to a cubic deformation of the Kac-Moody algebra. The non-linear terms are computed in closed form. In each Dirac bracket we only find highest order terms (as explained in the paper), defining a saturated algebra. We generalize the results for the presence of a Wess-Zumino term. The algebra is very similar to the previous one, containing now a calculable correction of order one unit lower. 
  The classical field equations of a Liouville field coupled to gravity in two spacetime dimensions are shown to have black hole solutions. Exact solutions are also obtained when quantum corrections due to back reaction effects are included, modifying both the ADM mass and the black hole entropy. The thermodynamic limit breaks down before evaporation of the black hole is complete, indicating that higher-loop effects must be included for a full description of the process. A scenario for the final state of the black hole spacetime is suggested. 
  We prove that all the correlation functions in the $(1,q)$ models are calculable using only the Virasoro and the $W^{(3)}$ constraints. This result is based on the invariance of correlators with respect to an interchange of the order of the operators they contain.   In terms of the topological recursion relations, it means that only two and three contacts and the corresponding degenerations of the underlying surfaces are relevant. An algorithm to compute correlators for any $q$ and at any genus is presented and demonstrated through some examples. On route to these results, some interesting polynomial identities, which are generalizations of Abel's identity, were discovered. } 
  We derive curvature counterterms in two-dimensional gravity coupled to conformal matter up to infinite order. By construction the higher-order action is equivalent to a finite first-order theory with auxiliary scalar. Due to this equivalence it shares the following remarkable properties: There is no need for gravitational dressing of the cosmological constant, quantization is consistent for any conformal anomaly $c$ of the coupled matter system, and if the coupled matter system is a $c=d~$-dimensional string theory in a Euclidean background then the effective string theory is $D=d+2~$-dimensional with Minkowski signature $(1,D-1)$. The resulting quantum theory favours flat geometries and suppresses both parabolic and hyperbolic singularities. 
  A $q$-discretization of \vi\ algebra is studied which reduces to the ordinary \vi\ algebra in the limit of $q \ra 1$. This is derived starting from the Moyal bracket algebra, hence is a kind of quantum deformation different from the quantum groups. Representation of this new algebra by using $q$-parametrized free fields is also given. 
  We describe a pair of constructions of Eisenstein lattices from ternary codes, and a corresponding pair of constructions of conformal field theories from lattices which turn out to have a string theoretic interpretation. These are found to interconnect in a similar way to results for binary codes, which led to a generalisation of the triality structure relevant in the construction of the Monster module. We therefore make some comments regarding a series of constructions of $V^\natural$. In addition, we present a complete construction of the Niemeier lattices from ternary codes, which in view of the above analogies should prove to be of great importance in the problem of the classification of self-dual $c=24$ conformal field theories. Other progress towards this problem is summarised, and some comments arise from this discussion regarding the uniqueness of the Monster conformal field theory. (Talk presented at the "Monster Bash", Ohio State University, May 1993.) 
  Low-harmonic formulas for closed relativistic strings are given. General parametrizations are presented for the addition of second- and third-harmonic waves to the fundamental wave. The method of determination of the parametrizations is based upon a product representation found for the finite Fourier series of string motion in which the constraints are automatically satisfied. The construction of strings with kinks is discussed, including examples. A procedure is laid out for the representation of kinks that arise from self-intersection, and subsequent intercommutation, for harmonically parametrized cosmic strings. 
  We consider a cell-complex in an arbitrary Hausdorff space as a dynamical object that can be coupled to a field defined on the complex. The Langevin equation is then derived for this field. In other words, a noise-field is created resulting from the field/geometry interactions. 
  It is argued that a unitarity-violating but weakly CPT invariant superscattering matrix exists for leading-order large-$N$ dilaton gravity, if and only if one includes in the Hilbert space planckian ``thunderpop" excitations which create white holes. CPT apparently cannot be realized in a low-energy effective theory in which such states have been integrated out. Rules for computing the leading-large-$N$ superscattering are described in terms of quantum field theory on a single multiply-connected spacetime obtained by sewing the future (past) horizons of the original spacetime with the past (future) horizons of its CPT conjugate. Some difficulties which may arise in going beyond leading order in $1/N$ are briefly discussed. 
  We derive the boson--fermion bound state equation in a two dimensional gauge theory in the large--$\nc$ limit. We analyze the properties of this equation and in particular, find that the mass trajectory is linear with respect to the bound state level for the higher mass states. 
  The structure of topological quantum field theories on the compact   Kahler manifold is interpreted. The BRST transformations of fields are derived from universal bundle and the observables are found from the second Chern class of universal bundle. 
  By dimensional reduction, Einstein-Hermitian equations of n + 1 dimensional closed Kahler manifolds lead to vortex equations of n dimensional closed Kahler manifolds. A Yang-Mills-Higgs functional to unitary bundles over closed Kahler manifolds has topological invariance by adding the additional terms which have ghost fields.   Henceforth we achieve the matter (Higgs field) coupled topological field theories in higher dimension. 
  A method for obtaining complex analytic realizations for a class of deformed algebras based on their respective deformation mappings and their ordinary coherent states is introduced. Explicit results of such realizations are provided for the cases of the $q$-oscillators ($q$-Weyl-Heisenberg algebra) and for the $su_{q}(2)$ and $su_{q}(1,1)$ algebras and their co-products. They are given in terms of a series in powers of ordinary derivative operators which act on the Bargmann-Hilbert space of functions endowed with the usual integration measure. In the $q\rightarrow 1$ limit these realizations reduce to the usual analytic Bargmann realizations for the three algebras. 
  The canonical quantization of $D=2n$ dimensional Dirac spinning particle in the external electromagnetic field is carried out in the gauge which allows to describe simultaneously particles and antiparticles (massive and massless) already at the classical level. Pseudoclassical Foldy-Wouthuysen transformation is used to obtain canonical (Newton-Wigner) coordinates and in terms of this variables the theory is quantized. The connection of this quantization with the Blount picture of Dirac particle in the external electromagnetic field is discussed. 
  We write down all orders large $N$ expansions for the dimensions of irreducible representations of $O(N)$ and $Sp(N)$. We interpret all the terms in these expansions as symmetry factors for singular worldsheet configurations, involving collapsed crosscaps and tubes.   We use it to complete the interpretation of two dimensional Yang Mills Theories with these gauge groups, on arbitrary two dimensional manifolds, in terms of a String Theory of maps of the type considered by Gross and Taylor. We point out some intriguing similarities to the case of $U(N)$ and discuss their implications. 
  Chiral lagrangians describing the interactions of Goldstone bosons in a theory possessing spontaneous symmetry breaking are effective, non-renormalizable field theories in four dimensions. Yet, in a momentum expansion one is able to extract definite, testable predictions from perturbation theory. These techniques have yielded in recent years a wealth of information on many problems where the physics of Goldstone bosons plays a crucial role, but theoretical issues concerning chiral perturbation theory remain, to this date, poorly treated in the literature. We present here a rather comprehensive analysis of the regularization and renormalization ambiguities appearing in chiral perturbation theory at the one loop level. We discuss first on the relevance of dealing with tadpoles properly. We demonstrate that Ward identities severely constrain the choice of regulators to the point of enforcing unique, unambiguous results in chiral perturbation theory at the one-loop level for any observable which is renormalization-group invariant. We comment on the physical implications of these results and on several possible regulating methods that may be of use for some applications. 
  We construct a $W_{\infty}$ gauge field theory of electrons in the lowest Landau level. For this purpose we introduce an external gauge potential $\cal A $ such that its $W_{\infty}$ gauge transformations cancel against the gauge transformation of the electron field. We then show that the electromagnetic interactions of electrons in the lowest Landau level are obtained through a non-linear realization of $\cal A$ in terms of the $U(1)$ gauge potential $A^{\m}$. As applications we derive the effective Lagrangians for circular droplets and for the $\n =1$ quantum Hall system. 
  We give a complete proof of local background independence of the classical master action for closed strings by constructing explicitly, for any two nearby conformal theories in a CFT theory space, a symplectic diffeomorphism between their state spaces mapping the corresponding non-polynomial string actions into each other. We uncover a new family of string vertices, the lowest of which is a three string vertex satisfying exact Jacobi identities with respect to the original closed string vertices. The homotopies between the two sets of string vertices determine the diffeomorphism establishing background independence. The linear part of the diffeomorphism is implemented by a CFT theory-space connection determined by the off-shell three closed string vertex, showing how string field theory induces a natural interplay between Riemann surface geometry and CFT theory space geometry. (Three figures are contained in a separate tar compressed uuencoded figures file. See the TeX file for instructions for printing the figures.) 
  The existence of the invariant measure in nonlocal regularized actions is discussed. It is shown that the measure for nonlocally regularized QED, as presented in\cite{Moff-Wood}, exists to all orders, and is precisely what is required to maintain gauge invariance at one loop and guarantees perturbative unitarity. We also demonstrate how the given procedure breaks down in anomalous theories, and discuss its generalization to other actions. 
  The annihilation poles for the form factors in XXZ model are studied using vertex operators introduced in \cite{DFJMN}. An annihilation pole is the property of form factors according to which the residue of the $2n$-particle form factor in such a pole can be expressed through linear combination of the $2n-2$-particle form factors. To prove this property we use the bosonization of the vertex operators in XXZ model which was invented in \cite{JMMN}. 
  Exact solutions of conformal turbulence restricted on a upper half plane are obtained. We show that the inertial range of homogeneous and isotropic turbulence with constant enstrophy flux develops in a distant region from the boundary. Thus in the presence of an anisotropic boundary, these exact solutions of turbulence generalize Kolmogorov's solution consistently and differ from the Polyakov's bulk case which requires a fine tunning of coefficients. The simplest solution in our case is given by the minimal model of $p=2, q=33$ and moreover we find a fixed point of solutions when $p,q$ become large. 
  We obtain the high energy, small angle, 2-particle gravitational scattering amplitudes in topologically massive gravity (TMG) and its two non-dynamical constituents, Einstein and Chern--Simons gravity. We use 't Hooft's approach, formally equivalent to a leading order eikonal approximation: one of the particles is taken to scatter through the classical spacetime generated by the other, which is idealized to be lightlike. The required geometries are derived in all three models; in particular, we thereby provide the first explicit asymptotically flat solution generated by a localized source in TMG. In contrast to $D$=4, the metrics are not uniquely specified, at least by naive asymptotic requirements -- an indeterminacy mirrored in the scattering amplitudes. The eikonal approach does provide a unique choice, however. We also discuss the discontinuities that arise upon taking the limits, at the level of the solutions, from TMG to its constituents, and compare with the analogous topologically massive vector gauge field models. 
  Some issues in the loop variable renormalization group approach to gauge invariant equations for the free fields of the open string are discussed. It had been shown in an earlier paper that this leads to a simple form of the gauge transformation law. We discuss in some detail some of the curious features encountered there. The theory looks a little like a massless theory in one higher dimension that can be dimensionally reduced to give a massive theory. We discuss the origin of some constraints that are needed for gauge invariance and also for reducing the set of fields to that of standard string theory. The mechanism of gauge invariance and the connection with the Virasoro algebra is a little different from the usual story and is discussed. It is also shown that these results can be extended in a straightforward manner to closed strings. 
  The pseudoclassical hamiltonian and action of the $D=2n$ dimensional Dirac particle with anomalous magnetic moment interacting with the external electromagnetic field is found. The Bargmann-Michel-Telegdi equation of motion for the Pauli-Lubanski vector is deduced. The canonical quantization of $D=2n$ dimensional Dirac spinning particle with anomalous magnetic moment in the external electromagnetic field is carried out in the gauge which allows to describe simultaneously particles and antiparticles (massive and massless) already at the classical level. Pseudoclassical Foldy-Wouthuysen transformation is used to obtain canonical (Newton-Wigner) coordinates and in terms of this variables the theory is quantized. The connection of this quantization with the deGroot and Suttorp's description of Dirac particle with anomalous magnetic moment in the external electromagnetic field is discussed. 
  We study the gravitational interaction involving the dilaton and the anti-symmetrical $B_{\mu\nu}$ fields that arises in the low-energy limit of string theory. It is shown that such interaction can be derived from a geometrical action principle, with the scalar of curvature of a non-Riemannian, but {\em metric-compatible}, connection as the lagrangian, and with a non-parallel volume-element. This action is contrasted with the recently proposed geometrical action for the 4-dimensional axi-dilaton gravity. 
  We show that recently formulated four-dimensional self-dual supersymmetric Yang-Mills theory, which is consistent background for open $~N=2$~ superstring, generates two-dimensional $~N=(1,1),~\, N=(1,0) $~ and $~N=(2,0)$~ supersymmetric gauged Wess-Zumino-Novikov-Witten $~\s\-$models on coset manifolds $~G/H$, after appropriate dimensional reductions. This is supporting evidence for the conjecture that the self-dual supersymmetric Yang-Mills theory will generate lower-dimensional supersymmetric integrable models after dimensional reductions. 
  We continue study of the connection of classical limit of integrable asymptotically free field theory to the finite-gap solutions of classical integrable equations. In the limit the momenta of particles turn into the moduli of Riemann surfaces while their isotopic structure is related to the period lattices. In this paper we explain that the classical limit of the local operators can be understood as a measure induced on the phase space by embedding into the projective space of "classical fields". 
  The introduction and quantization of a center-of-mass coordinate is demonstrated for the one-soliton sector of nonlinear field theories in (1+1) dimensions. The present approach strongly emphazises the gauge and BRST-symmetry aspects of collective coordinate quantization. A gauge is presented which is independent of any approximation scheme and which allows to interpret the new degree of freedom as the {\em quantized} center of mass coordinate of a soliton. Lorentz invariance is used from the beginning to introduce fluctuations of the collective coordinate in the {\em rest frame} of the {\em moving} soliton. It turns out that due to the extended nature of the soliton retardation effects lead to differences in the quantum mechanics of the soliton as compared to a point-like particle. Finally, the results of the semiclassical expansion are used to analyse effective soliton-meson vertices and the coupling to an external source. Such a coupling in general causes acceleration as well as internal excitation of the soliton. 
  We investigate the conditions under which systems of two differential eigenvalue equations are quasi exactly solvable. These systems reveal a rich set of algebraic structures. Some of them are explicitely described. An exemple of quasi exactly system is studied which provides a direct counterpart of the Lam\'e equation. 
  A scalar field theory is investigated within the context of orthodox quantum gravity. 
  We show how to obtain the ultraviolet central charge from the exact S-matrix for a wide variety of models with a $U(1)$ symmetry. This is done by coupling the $U(1)$ current $J$ to a background field. In an $N$=2 superconformal theory with $J$ the fermion number current, the OPE of $J$ with itself and hence the free energy are proportional to $c$. By deforming the supersymmetry into affine $\widehat SU(2)_q$ quantum-group symmetry, this result can be generalized to many $U(1)$-invariant theories, including the N=0 and N=1 sine-Gordon models and the $SU(2)_k$ WZW models. This provides a consistency check on a conjectured S-matrix completely independent of the finite-size effects expressed in terms of dilogarithms resulting from the thermodynamic Bethe ansatz. 
  We present three groups of noncanonical quantum oscillators. The position and the momentum operators of each of the groups generate basic Lie superalgebras, namely $sl(1/3)$, $osp(1/6)$ and $osp(3/2)$. The $sl(1/3)$-oscillators have finite energy spectrum and finite-dimensions. The $osp(1/6)$-oscillators are related to the para-Bose statistictics. The internal angular momentum $s$ of the $osp(3/2)$-oscillators takes no more than three (half)integer values. In a particular representation $s=1/2$. 
  We consider an extension of the (t-U) Hubbard model taking into account new interactions between the numbers of up and down electrons. We confine ourselves to a one-dimensional open chain with L sites (4^L states) and derive the effective Hamiltonian in the strong repulsion (large U) regime. This Hamiltonian acts on 3^L states. We show that the spectrum of the latter Hamiltonian (not the degeneracies) coincides with the spectrum of the anisotropic Heisenberg chain (XXZ model) in the presence of a Z field (2^L states). The wave functions of the 3^L-state system are obtained explicitly from those of the 2^L-state system, and the degeneracies can be understood in terms of irreducible representations of U_q(\hat{sl(2)}). 
  Two types of semiclassical calculations have been used to study quantum effects in black hole backgrounds, the WKB and the mean field approaches. In this work we systematically reconstruct the logical implications of both methods on quantum black hole physics and provide the link between these two approaches. Our conclusions completely support our previous findings based solely on the WKB method: quantum black holes are effectively p-brane excitations and, consequently, no information loss paradox exists in this problem. 
  Two dimensional induced quantum gravity with matter central charge $c>1$ is studied taking a careful consideration of both diffeomorphism and Weyl symmetries . It is shown that, for the gauge fixing condition $R(g)$ (scalar curvature)=$const$, one obtains a modification of the David-Distler-Kawai version of KPZ scaling. We obtain a class of models with the real string tension for all values $c>1$. They contain an indeterminate parameter which is, however, strongly constrained by the requirement of non triviality of such a model. The possible physical significance of the new model is discussed. In particular we note that it describes smooth surfaces imbedded in $d$-dimensional flat space time for arbitrary $d$, consistently with recent numerical result for $d=3$. 
  Factoring out the spin $1$ subalgebra of a $ W $ algebra leads to a new $ W $ structure which can be seen either as a rational finitely generated $ W $ algebra or as a polynomial non-linear $ W_\infty$ realization. 
  Mappings between certain infinite series of N=2 superconformal coset models are constructed. They make use of level-rank dualities for B, C and D type theories. While the WZW level-rank dualities do not constitute isomorphisms of the theories, they lead to level-rank dualities of N=2 coset models that preserve the cft properties in such a manner that the coset models related by duality are expected to be isomorphic as conformal field theories. The construction also gives some further insight in the nature of the resolution of field identification fixed points of coset theories. 
  {}From the ordinary tensile string we derive a geometric action for the tensionless ($T=0$) string and discuss its symmetries and field equations. The Weyl symmetry of the usual string is shown to be replaced by a global space-time conformal symmetry in the $T\to 0$ limit. We present the explicit expressions for the generators of this group in the light-cone gauge. Using these, we quantize the theory in an operator form and require the conformal symmetry to remain a symmetry of the quantum theory. Modulo details concerning zero-modes that are discussed in the paper, this leads to the stringent restriction that the physical states should be singlets under space-time diffeomorphisms, hinting at a topological theory. We present the details of the calculation that leads to this conclusion. 
  We discuss the properties of the supersymmetric $t$-$J$ model in the formalism of the slave operators. In particular we introduce a generalized abelian bosonization for the model in two dimensions, and show that holons and spinons can be anyons of arbitrary complementary statistics (slave anyon representation). The braiding properties of these anyonic operators are thoroughly analyzed, and are used to provide an explicit linear realization of the superalgebra $SU(1|2)$. Finally, we prove that the Hamiltonian of the $t$-$J$ model in the slave anyon representation is invariant under $SU(1|2)$ for $J=2\,t$. 
  We start from the quantum Miura transformation [7] for the $W$-algebra associated with $GL(n)$ group and find an evident formula for quantum L-operator as well as for the action of $W_l$ currents (l=1,..,n) on elements of the completely degenerated n-dimensional representation. Quantum formulae are obtained through the deformation of the pseudodifferential symbols. This deformation is independent of $n$ and preserves Adler's trace. Our main instrument of the proof is the notation of pseudodifferential symbol with right action which has no counterpart in classical theory. 
  We consider 1+1-dimensional QCD coupled to Majorana fermions in the adjoint representation of the gauge group $SU(N)$. Pair creation of partons (fermion quanta) is not suppressed in the large-$N$ limit, where the glueball-like bound states become free. In this limit the spectrum is given by a linear \lc\ Schr\" odinger equation, which we study numerically using the discretized \lcq. We find a discrete spectrum of bound states, with the logarithm of the level density growing approximately linearly with the mass. The wave function of a typical excited state is a complicated mixture of components with different parton numbers. A few low-lying states, however, are surprisingly close to being eigenstates of the parton number, and their masses can be accurately calculated by truncated diagonalizations. 
  It is shown that the difficulties in formulating the quantum field theory on discrete spacetime appear already in classical dynamics of one degree of freedom on discrete time. The difference equation of motion which maintains a conserved quantity like energy has a very restricted form that is not probably derived by the least action principle. On the other hand, the classical dynamics is possible to be canonically formulated and quantized, if the equation is derived from an action. The difficulties come mainly from this incompatibility of the conserved quantity and the action principle. We formulate a quantum field theory canonically on discrete spacetime in the case where the field equation is derived from an action, though there may be no exactly conserved quantity. It may, however, be expected that a conserved quantity exists for a low "energy" region. 
  Mean Field Theory has been extensively used in the study of systems of anyons in two spatial dimensions. In this paper we study the physical grounds for the validity of this approximation by considering the Quantum Mechanical scattering of a charged particle from a two dimensional array of magnetic flux tubes. The flux tubes are arranged on a regular lattice which is infinitely long in the ``$y$'' direction but which has a (small) finite number of columns in the ``$x$'' direction. Their physical size is assumed to be infinitesimally small. We develop a method for computing the scattering angle as well as the reflection and transmission coefficients to lowest order in the Aharonov--Bohm interaction. The results of our calculation are compared to the scattering of the same particle from a region of constant magnetic field whose magnitude is equal to the mean field of all the flux tubes. For an incident plane wave, the Mean Field approximation is shown to be valid provided the flux in each tube is much less than a single flux quantum. This is precisely the regime in which Mean Field Theory for anyons is expected to be valid. When the flux per tube becomes of order 1, Mean Field Theory is no longer valid. 
  Recent algebraic structures of string theory, including homotopy Lie algebras, gravity algebras and Batalin-Vilkovisky algebras, are deduced from the topology of the moduli spaces of punctured Riemann spheres. The principal reason for these structures to appear is as simple as the following. A conformal field theory is an algebra over the operad of punctured Riemann surfaces, this operad gives rise to certain standard operads governing the three kinds of algebras, and that yields the structures of such algebras on the (physical) state space naturally. 
  We study the relation between topological string theory and singularity theory using the partition function of $A_{N-1}$ topological string defined by matrix integral of Kontsevich type. Genus expansion of the free energy is considered, and the genus $g=0$ contribution is shown to be described by a special solution of $N$-reduced dispersionless KP system. We show a universal correspondences between the time variables of dispersionless KP hierarchy and the flat coordinates associated with versal deformations of simple singularities of type $A$. We also study the behavior of topological matter theory on the sphere in a topological gravity background, to clarify the role of the topological string in the singularity theory. Finally we make some comment on gravitational phase transition. 
  An exact renormalization group equation is derived for the free energy of matrix models. The renormalization group equation turns out to be nonlinear for matrix models, as opposed to linear for vector models. An algorithm for determining the critical coupling constant and the critical exponent is obtained. As concrete examples, one-matrix models with one and two coupling constants are analyzed and the exact values of the critical coupling constant and the associated critical exponent are found. 
  We give a unified derivation of the propagator in the gauges $n.A=0$ for $n^2$ timelike, spacelike or lightlike. We discuss the physical states and other physical questions. 
  We present a formalism that extends the Majorana-construction to arbitrary spin (j,0)+(0,j) representation spaces. For the example case of spin-1, a wave equation satisfied by the Majorana-like (1,0)+(0,1) spinors is constructed and its physical content explored. The (j,0)+(0,j) Majorana-construct is found to possess an unusual classical and quantum field theoretic structure. Relevance of our formalism to parity violation, hadronic phenomenologies, and grand unified field theories is briefly pointed out. 
  We consider the even parity superLax operator for the supersymmetric KP hierarchy of the form $L~=~D^2 + \sum_{i=0}^\infty u_{i-2} D^{-i+1}$ and obtain the two Hamiltonian structures following the standard method of Gelfand and Dikii. We observe that the first Hamiltonian structure is local and linear whereas the second Hamiltonian structure is non-local and nonlinear among the superfields appearing in the Lax operator. We discuss briefly on their connections with the super $w_{\infty}$ algebra. 
  Recently, the free energy of the target space mean field (TSMF) matrix model has been calculated in the low temperature phase, order-by-order in a low temperature expansion. The TSMF model is a matrix model whose discrete target space has an infinite coordination number, and whose free energy assumes a universal form, corresponding to baby universes joined into a tree. Here the free energy is summed to all orders, and expressed through a transcendental algebraic equation, using which we analyze the critical phenomena that occur in the TSMF model. There are two critical curves, at which the matter and the geometry become critical, and for which the critical exponents are alpha = 1/2 and gamma_str = -1/2, respectively. There is a bicritical point where the curves meet, at which gamma_str = +1/3. 
  The heterotic string theory, compactified to four dimensions, has been conjectured to have a duality symmetry (S duality) that transforms the dilaton nonlinearly. If valid, this symmetry could provide an important means of obtaining information about nonperturbative features of the theory. Even though it is inherently nonperturbative, S duality exhibits many similarities with the well-established target-space duality symmetry (T duality), which does act perturbatively. These similarities are manifest in a new version of the low-energy effective field theory and in the soliton spectrum obtained by saturating the Bogomol'nyi bound. Curiously, there is evidence that the roles of the S and T dualities are interchanged in passing to a five-brane formulation. 
  Strings propagating along surfaces with Dirichlet boundaries are studied in this paper. Such strings were originally proposed as a possible candidate for the QCD string. Our approach is different from previous ones and is simple and general enough, with which basic problems can be easily addressed. The Green function on a surface with Dirichlet boundaries is obtained through the Neumann Green function on the same surface, by employing a simple approach to Dirichlet conditions. An easy consequence of the simple calculation of the Green function is that in the simplest model, namely the bosonic Dirichlet string, the critical dimension is still 26, and the tachyon is still present in the spectrum, while the scattering amplitudes differ dramatically from those in the usual string theory. We discuss the high energy, fixed angle behavior of the four point scattering amplitudes on the disk and the annulus. We argue for general power-like behavior of arbitrary high energy, fixed angle scattering amplitudes. We also discuss the high temperature property of the finite temperature partition function on an arbitrary surface, and give an explicit formula of the one on the annulus. 
  We find charged, abelian, spherically symmetric solutions (in flat space-time) corresponding to the effective action of $D=4$ heterotic string theory with scale-dependent dilaton $\p$ and modulus $\vp$ fields. We take into account perturbative (genus-one), moduli-dependent `threshold' corrections to the coupling function $f(\p,\vp)$ in the gauge field kinetic term $f(\p,\vp) F^2_{\m\n}$, as well as non-perturbative scalar potential $V(\p, \vp)$, e.g. induced by gaugino condensation in the hidden gauge sector. Stable, finite energy, electric solutions (corresponding to on abelian subgroup of a non-abelian gauge group) have the small scale region as the weak coupling region ($\phi\ra -\infty$) with the modulus $\vp$ slowly varying towards smaller values. Stable, finite energy, abelian magnetic solutions exist only for a specific range of threshold correction parameters. At small scales they correspond to the strong coupling region ($\p\ra \infty$) and the compactification region ($\vp\ra 0$). The non-perturbative potential $V$ plays a crucial role at large scales, where it fixes the asymptotic values of $\phi$ and $\vp$ to be at the minimum of $V$. 
  We extend the recent approach of M. Jimbo, K. Miki, T. Miwa, and A. Nakayashiki to derive an integral formula for the N-point correlation functions of arbitrary local operators of the antiferromagnetic spin-1 XXZ model. For this, we realize the quantum affine symmetry algebra $U_q(su(2)_2)$ of level 2 and its corresponding type I vertex operators in terms of a deformed bosonic field free of a background charge, and a deformed fermionic field. Up to GSO type projections, the Fock space is already irreducible and therefore no BRST projections are involved. This means that no screening charges with their Jackson integrals are required. Consequently, our N-point correlation functions are given in terms of usual classical integrals only, just as those derived by Jimbo et al in the case of the spin-1/2 XXZ model through the Frenkel-Jing bosonization of $U_q(su(2)_1)$. 
  We write down matrix models for Ising spins with zero external field on the vertices of dynamical triangulated random surfaces (DTRS) and dynamically quadrangulated random surfaces (DQRS) and compare these with the standard matrix model approach which places the spins on the dual $\phi^3$ and $\phi^4$ graphs. We show that the critical temperatures calculated in the DTRS and DQRS models agree with those deduced from duality arguments in the standard approach. Using the DQRS model we observe that the Ising antiferromagnet still undergoes a phase transition to a Neel (checkerboard) ordered ground state which is absent because of frustration in the other cases. 
  We summarize the application of the field--antifield formalism to the quantization of gauge field theories. After the gauge fixing, the main issues are the regularization and anomalies. We illustrate this for chiral $W_3$ gravity. We discuss also gauge theories with a non-local action, induced by a matter system with an anomaly. As an illustration we use induced WZW models, including a discussion of non--local regularization, the importance of the measure and regularization of multiple loops. In these theories antifields become propagating --- one of many roles assigned to them in the BV formalism. 
  We represent Feigin's construction [22] of lattice W algebras and give some simple results: lattice Virasoro and $W_3$ algebras. For simplest case $g=sl(2)$ we introduce whole $U_q(sl(2))$ quantum group on this lattice. We find simplest two-dimensional module as well as exchange relations and define lattice Virasoro algebra as algebra of invariants of $U_q(sl(2))$. Another generalization is connected with lattice integrals of motion as the invariants of quantum affine group $U_q(\hat{n}_{+})$. We show that Volkov's scheme leads to the system of difference equations for the function from non-commutative variables. 
  We propose a perturbative improvement of the hierarchical approximation for gaussian models. The procedure is based on a relabeling of the momenta which allows one to express the symmetries of the hierarchical model using a simple multiplication group. The representations of this group are used to expand the action. The perturbative expansion is treated as a problem of symmetry breaking using Ward identity techniques. 
  Exact integral representations of spin one-point functions (ground state expectation values) are reported for the spin-1 analog of the XXZ model in the region $-1<q<0$. The method enables one to calculate arbitrary $n$-point functions in principle. We also report a construction of level 2 irreducible highest weight representations of $U_q(\hat{sl}_2)$ in terms of boson and fermion operators, and explicit forms of related vertex operators. 
  We study the algebra $B_q(\ge)$ presented by Kashiwara and introduce intertwiners similar to $q$-vertex operators. We show that a matrix determined by 2-point functions of the intertwiners coincides with a quantum R-matrix (up to a diagonal matrix) and give the commutation relations of the intertwiners. We also introduce an analogue of the universal R-matrix for the Kashiwara algebra. 
  Preliminary investigations are made for the stability of the $1/N$ expansion in three-dimensional gravity coupled to various matter fields, which are power-counting renormalizable. For unitary matters, a tachyonic pole appears in the spin-2 part of the leading graviton propagator, which implies the unstable flat space-time, unless the higher-derivative terms are introduced. As another possibility to avoid this spin-2 tachyon, we propose Einstein gravity coupled to non-unitary matters. It turns out that a tachyon appears in the spin-0 or -1 part for any linear gauges in this case, but it can be removed if non-minimally coupled scalars are included. We suggest an interesting model which may be stable and possess an ultraviolet fixed point. 
  We present a large-$N$ collective field formalism for anyons in external magnetic fields interacting with arbitrary two-body potential. We discuss how the Landau level is reproduced in our framework. We apply it to the soluble model for anyons proposed by Girvin et al., and obtain the dispersion relation of collective modes for arbitrary statistical parameters. 
  We approximate analytically the semi-classical differential cross-section for low-energy solitonic BPS SU(2) magnetic monopoles using the geodesic approximation. The semi-classical scattering amplitude, f(\theta), can be expressed as a conditionally convergent infinite series which is made absolutely convergent by analytic continuation of the generalised zeta function. Our results suggest that the classical solitonic cross-section (computed numerically in hep-th:9209063) and the semi-classical cross-section are in good agreement over a wide range of scattering angles, \pi/3<\theta<\pi/2 and \pi/2<\theta<2\pi/3. 
  We show that there is only one operator having some minimal properties enabling it to be a one photon position operator. These proerties are stated, and the solution is shown to be the photon position operator proposed by Pryce. This operator has non-commuting components. Nevertheless, it is shown that one can find states localized with an arbitrary precision. 
  The relativistic semi-classical approximation for a free massive particle is studied using the Wigner-Weyl formalism. A non-covariant Wigner function is proposed using the Newton-Wigner position operator. The perturbative solution for the time evolution is found. Causality is found to be perturbatively respected. 
  Using the hamiltonian formalism, we investigate the smooth bosonization method in which bosonization and fermionization are carried out through a specific gauge-fixing of an enlarged gauge invariant theory. The generator of the local gauge symmetry, which cannot be derived from the lagrangian of the enlarged theory, is obtained by making a canonical transformation. We also show that the massless Thirring model possesses a similar local gauge symmetry for a speific value of the coupling constant. 
  We consider the propagation of perturbations along an infinitely long stationary open string in the background of a Schwarzschild black hole. The equations of motion for the perturbations in the 2 transverse physical directions are solved to second order in a weak field expansion. We then set up a scattering formalism where an ingoing wave is partly transmitted and partly reflected due to the interaction with the gravitational field of the black hole. We finally calculate the reflection coefficient to third order in our weak field expansion. 
  By making use of Schwinger's oscillator model of angular momentum, we put forward an interesting connection among three solvable Hamiltonians, widely used for discussions on the quantum measurement problem. This connection implies that a particular macroscopic limit has to be taken for these models to be physically sensible. 
  We consider the topological interactions of vortices on general surfaces. If the genus of the surface is greater than zero, the handles can carry magnetic flux. The classical state of the vortices and the handles can be described by a mapping from the fundamental group to the unbroken gauge group. The allowed configurations must satisfy a relation induced by the fundamental group. Upon quantization, the handles can carry ``Cheshire charge.'' The motion of the vortices can be described by the braid group of the surface. How the motion of the vortices affects the state is analyzed in detail. 
  In this paper an attempt is made to understand the passage from the exact quantum treatment of the CGHS theory to the semi-classical physics discussed by many authors. We find first that to the order of accuracy to which Hawking effects are calculated in the theory, it is inconsistent to ignore correlations in the dilaton gravity sector. Next the standard Dirac or BRST procedure for implementing the constraints is followed. This leads to a set of physical states, in which however the semi-classical physics of the theory seems to be completely obscured. As an alternative, we construct a coherent state formalism, which is the natural framework for understanding the semi-classical calculations, and argue that it satisfies all necessary requirements of the theory, provided that there exist classical ghost configurations which solve an infinite set of equations. If this is the case it may be interpreted as a spontaneous breakdown of general covariance. 
  We obtain off-critical (elliptic) Boltzmann weights for lattice models whose continuum limits correspond to massive, $N=2$ supersymmetric, quantum integrable field theories. We also compute the free energies of these models and show that they are analytic in the region of parameter space where we believe that the supersymmetry is unbroken. While the supersymmetry is not directly realized on the lattice, there is still a very close connection between the models described here and topological lattice models. A simple example is discussed in detail and some corner transfer matrix computations are also presented. 
  For the case of a first-class constrained system with an equivariant momentum map, we study the conditions under which the double process of reducing to the constraint surface and dividing out by the group of gauge transformations $G$ is equivalent to the single process of dividing out the initial phase space by the complexification $G_C$ of $G$. For the particular case of a phase space action that is the lift of a configuration space action, conditions are found under which, in finite dimensions, the physical phase space of a gauge system with first-class constraints is diffeomorphic to a manifold imbedded in the physical configuration space of the complexified gauge system. Similar conditions are shown to hold in the infinite-dimensional example of Yang-Mills theories. As a physical application we discuss the adequateness of using holomorphic Wilson loop variables as (generalized) global coordinates on the physical phase space of Yang-Mills theory. 
  A common belief is that further quantum corrections near the singularity of a large black hole should not substantially modify the semiclassical picture of black hole evaporation; in particular, the outgoing spectrum of radiation should be very close to the thermal spectrum predicted by Hawking. In this paper we explore a possible counterexample: in the context of dilaton gravity, we find that non-perturbative quantum corrections which are important in strong coupling regions may completely alter the semiclassical picture, to the extent that the presumptive space-like boundary becomes time-like, changing in this way the causal structure of the semiclassical geometry. As a result, only a small fraction of the total energy is radiated outside the fake event horizon; most of the energy comes in fact at later retarded times and there is no information loss problem. Thus we propose that this may constitute a general characteristic of quantum black holes, that is, quantum gravity might be such as to prevent the formation of global event horizons. We argue that this is not unnatural from the viewpoint of quantum mechanics. 
  The probability distribution functions of the circulation of velocity in three-dimensional decaying isotropic turbulence are examined by the database of the numerical simulation based on the pseudospectral method. It is shown that the standard deviation increases nearly as $A^{2/3}$ where $A$ is the area of the loop. The PDFs change from exponential to gaussian as the size becomes large. The former agrees well with the prediction by Migdal (1993), although the latter does not match. A modification is proposed to explain the gaussian distribution with the $A^{2/3}$ dependence. 
  We complete the classification of (2,2) string vacua that can be constructed by diagonal twists of tensor products of minimal models with ADE invariants. Using the \LG\ framework, we compute all spectra from inequivalent models of this type. The completeness of our results is only possible by systematically avoiding the huge redundancies coming from permutation symmetries of tensor products. We recover the results for (2,2) vacua of an extensive computation of simple current invariants by Schellekens and Yankielowitz, and find 4 additional mirror pairs of spectra that were missed by their stochastic method. For the model $(1)^9$ we observe a relation between redundant spectra and groups that are related in a particular way. 
  We investigate the light-cone quantization of $\phi^3$ theory in 1+1 dimensions with a regularization of discretized light-cone momentum $k^+$. Solving a second-class constraint associated with the $k^+=0$ mode, we show that the $k^+=0$ mode propagates along the internal lines of Feynman diagrams in any order of perturbation, hence our theory recovers the Lorentz invariance. 
  The discrete models of the Toda and Volterra chains are being constructed out of the continuum two-boson KP hierarchies. The main tool is the discrete symmetry preserving the Hamiltonian structure of the continuum models. The two-boson currents of KP hierarchy are being associated with sites of the corresponding chain by successive actions of discrete symmetry. 
  A Wilson loop is defined, in 4-D pure Einstein gravity, as the trace of the holonomy of the Christoffel connection or of the spin connection, and its invariance under the symmetry transformations of the action is showed (diffeomorphisms and local Lorentz transformations). We then compute the loop perturbatively, both on a flat background and in the presence of an external source; we also allow some modifications in the form of the action, and test the action of ``stabilized'' gravity. A geometrical analysis of the results in terms of the gauge group of the euclidean theory, $SO(4)$, leads us to the conclusion that the correspondent statistical system does not develope any configuration with localized curvature at low temperature. This ``non-local'' behavior of the quantized gravitational field strongly contrasts with that of usual gauge fields. Our results also provide an explanation for the absence of any invariant correlation of the curvature in the same approximation. 
  We consider two different types of deformations for the linear group $ GL(n)$ which correspond to using of a general diagonal R-matrix. Relations between braided and quantum deformed algebras and their coactions on a quantum plane are discussed. We show that tensor-grading-preserving differential calculi can be constructed on braided groups , quantum groups and quantum planes for the case of the diagonal R-matrix. 
  We introduce the analogue of the metric tensor in case of $q$-deformed differential calculus. We analyse the consequences of the existence of such metric, showing that this enforces severe restrictions on the parameters of the theory. We discuss in detail the examples of the Manin plane and the $q$-deformation of $SU(2)$. Finally we touch the topic of relations with the Connes' approach. 
  We present an efficient method for computing the duality group $\Gamma$ of the moduli space \cM for strings compactified on a Calabi-Yau manifold described by a two-moduli deformation of the quintic polynomial immersed in $\CP(4)$, $\cW={1\over5}(\iy_1^5+\cdots+\iy_5^5)-a\,\iy_4^3 \iy_5^2 -b\, \iy_4^2 \iy_5^3$. We show that $\Gamma$ is given by a $3$--dimensional representation of a central extension of $B_5$, the braid group on five strands. 
  The original version of this paper contains an error; when this is corrected the basic conclusion changes. A revised manuscript will be submitted shortly. 
  It is typical for a semi-infinite cohomology complex associated with a graded Lie algebra to occur as a vertex operator (or chiral) superalgebra where all the standard operators of cohomology theory, in particular the differential, are modes of vertex operators (fields). Although vertex operator superalgebras -with the inherent Virasoro action- are regarded as part of Conformal Field Theory (CFT), a VOSA may exhibit a square-zero operator (often, but not always, the semi-infinite cohomology differential) for which the Virasoro algebra acts trivially in the cohomology. Capable of shedding its CFT features, such a VOSA is called a ``topological chiral algebra'' (TCA). We investigate the semi-infinite cohomology of the vertex operator Weil algebra and indicate a number of differentials which give rise to TCA structures. 
  The renormalization group approach is studied for large $N$ models. The approach of Br\'ezin and Zinn-Justin is explained and examined for matrix models. The validity of the approach is clarified by using the vector model as a similar and simpler example. An exact difference equation is obtained which relates free energies for neighboring values of $N$. The reparametrization freedom in field space provides infinitely many identities which reduce the infinite dimensional coupling constant space to that of finite dimensions. The effective beta functions give exact values for the fixed points and the susceptibility exponents. A detailed study of the effective renormalization group flow is presented for cases with up to two coupling constants. We draw the two-dimensional flow diagram. 
  Recent results on the cohomological reformulation of the problem of consistent interactions between gauge fields are illustrated in the case of the Yang-Mills models. By evaluating the local BRST cohomology through descent equation techniques, it is shown (i) that there is a unique local, Poincar\'e invariant cubic vertex for free gauge vector fields which preserves the number of gauge symmetries to first order in the coupling constant; and (ii) that consistency to second order in the coupling constant requires the structure constants appearing in the cubic vertex to fulfill the Jacobi identity. The known uniqueness of the Yang-Mills coupling is therefore rederived through cohomological arguments. 
  We present the formalism for computing the critical exponent corresponding to the $\beta$-function of the Nambu--Jona-Lasinio model with $SU(M)$ $\times$ $SU(M)$ continuous chiral symmetry at $O(1/N^2)$ in a large $N$ expansion, where $N$ is the number of fermions. We find that the equations can only be solved for the case $M$ $=$ $2$ and subsequently an analytic expression is then derived. This contrasting behaviour between the $M$ $=$ $2$ and $M$ $>$ $2$ cases, which appears first at $O(1/N^2)$, is related to the fact that the anomalous dimensions of the bosonic fields are only equivalent for $M$ $=$ $2$. 
  We present a review of the canonical quantization approach to the problem of non-perturbative 2d dilaton gravity. In the case of chiral matter we describe a method for solving the constraints by constructing a Kac-Moody current algebra. For the models of interest, the relevant Kac-Moody algebras are based on SL(2,R) X U(1) group and on an extended 2d Poincare group. As a consequence, the constraints become free-field Virasoro generators with background charges. We argue that the same happens in the non-chiral case. The problem of the corresponding BRST cohomology is discussed as well as the unitarity of the theory. One can show that the theory is unitary by chosing a physical gauge, and hence the problem of transitions from pure into mixed sates is absent. Implications for the physics of black holes are discussed. (Based on the talks presented at Trieste conference on Gauge Theories, Applied Supersymmetry and Quantum Gravity, May 1993 and at Danube '93 Workshop, Belgrade, Yugoslavia, June 1993) 
  We show that certain type II string amplitudes at genus $g$ are given by the topological partition function $F_g$ discussed recently by Bershadsky, Cecotti, Ooguri and Vafa. These amplitudes give rise to a term in the four-dimensional effective action of the form $\sum_g F_g W^{2g}$, where $W$ is the chiral superfield of $N=2$ supergravitational multiplet. The holomorphic anomaly of $F_g$ is related to non-localities of the effective action due to the propagation of massless states. This result generalizes the holomorphic anomaly of the one loop case which is known to lead to non-harmonic gravitational couplings. 
  We discuss properties and interpretation of recently found globally regular and black hole solutions of a Einstein-Yang-Mills-dilaton theory. Talk given at the 15th annual MRST meeting on High Energy Physics, Syracuse University, NY, May 14-15, 1993. 
  We investigate two dimensional supergravity theories, which can be built from a topological and gauge invariant action defined on an ordinary surface. We concentrate on four models. The first model is the $N=1$ supersymmetric extension of Jackiw-Teiltelboim model presented by Chamseddine in a superspace formalism. We complement the proof of Montano, Aoaki, and Sonnenschein that this extension is topological and gauge invariant, based on the graded de Sitter algebra. Not only do the equation of motions correspond to the supergravity ones and gauge transformations encompass local supersymmetries, but also we identify the $\int \langle \eta, F\rangle$-theory with the superfield formalism action written by Chamseddine. Next, we show that the $N=1$ supersymmetric extension of string inspired two dimensional dilaton gravity put forward by Park and Strominger is a theory that satisfies a non-vanishing curvature condition and cannot be written as a $\int\langle \eta,F\rangle$-theory. As an alternative, we propose two examples of topological and gauge invariant theories that are based on graded extension of the extended Poincar\'e algebra and satisfy a vanishing curvature condition. Both models are interpreted as supersymmetric extensions of the string inspired dilaton gravity. 
  The properties of discrete nonlinear symmetries of integrable equations are investigated. These symmetries are shown to be canonical transformations. On the basis of the considered examples, it is concluded, that the densities of the conservation laws are changed under these transformations by spatial divergencies. 
  An elementary method of determination of the character of the hot phase transition in 4d four-fermion NJL-type models is applied to non-supersymmetric and supersymmetric versions of simple NJL model. We find that in the non-susy case the transition is usually of the second order. It is weakly first order only in the region of parameters which correspond to fermion masses comparable to the cut-off. In the supersymmetric case both kinds of phase transitions are possible. For sufficiently strong coupling and sufficiently large susy-breaking scale the transition is always of the first order. 
  The Lie-Poisson structure of non-Abelian Thirring models is discussed and the Hamiltonian quantization of these theories is carried out. The consistency of the Hamiltonian quantization with the path integral method is established. It is shown that the space of non-Abelian Thirring models contains the nonperturbative conformal points which are in one-to-one correspondence with general solutions of the Virasoro master equation. A BRST nature of the mastert equation is clarified. 
  The representations of the observable algebra of a low dimensional quantum field theory form the objects of a braided tensor category. The search for gauge symmetry in the theory amounts to finding an algebra which has the same representation category. In this paper we try to establish that every quantum field theory satisfying some basic axioms posseses a weak quasi Hopf algebra as gauge symmetry. The first step is to construct a functor from the representation category to the category of finite dimensional vector spaces. Given such a functor we can use a generalized reconstruction theorem to find the symmetry algebra. It is shown how this symmetry algebra is used to build a gauge covariant field algebra and we investigate the question why this generality is necessary. 
  The thermodynamics of vortices in the critically coupled abelian Higgs model, defined on the plane, are investigated by placing $N$ vortices in a region of the plane with periodic boundary conditions: a torus. It is noted that the moduli space for $N$ vortices, which is the same as that of $N$ indistinguishable points on a torus, fibrates into a $CP_{N-1}$ bundle over the Jacobi manifold of the torus. The volume of the moduli space is a product of the area of the base of this bundle and the volume of the fibre. These two values are determined by considering two 2-surfaces in the bundle corresponding to a rigid motion of a vortex configuration, and a motion around a fixed centre of mass. The partition function for the vortices is proportional to the volume of the moduli space, and the equation of state for the vortices is $P(A-4\pi N)=NT$ in the thermodynamic limit, where $P$ is the pressure, $A$ the area of the region of the plane occupied by the vortices, and $T$ the temperature. There is no phase transition. 
  We find exactly solvable N=2-supersymmetric flows whose infrared fixed points are the N=2 minimal models. The exact S-matrices and the Casimir energy (a c-function) are determined along the entire renormalization group trajectory. The c-function runs from c=3 (asymptotically) in the UV to the N=2 minimal model values of the central charge in the IR, leading us to interpret these theories as the Landau-Ginzburg models with superpotential $X^{k+2}$. Consideration of the elliptic genus gives further support for this interpretation. We also find an integrable model in this hierarchy which has spontaneously-broken supersymmetry and superpotential $X$, and a series of integrable models with (0,2) supersymmetry. The flows exhibit interesting behavior in the UV, including a relation to the N=2 super sine-Gordon model. We speculate about the relation between the kinetic term and the cigar target-space metric. 
  A study is presented of classical field configurations describing nonabelian vortices in two spatial dimensions, when a global \( SO(3) \) symmetry is spontaneously broken to a discrete group \( \IK \) isomorphic to the group of integers mod 4. The vortices in this model are characterized by the nonabelian fundamental group \(\pi_1 (SO(3)/{\IK}) \), which is isomorphic to the group of quaternions. We present an ansatz describing isolated vortices and prove that it is stable to perturbations. Kinematic constraints are derived which imply that at a finite temperature, only two species of vortices are stable to decay, due to `dissociation'. The latter process is the nonabelian analogue of the instability of charge \(|q| >1 \) abelian vortices to dissociation into those with charge \(|q| = 1\). The energy of configurations containing at maximum two vortex-antivortex pairs, is then computed. When the pairs are all of the same type, we find the usual Coulombic interaction energy as in the abelian case. When they are different, one finds novel interactions which are a departure from Coulomb like behavior. Therefore one can compute the grand canonical partition function (GCPF) for thermal pair creation of nonabelian vortices, in the approximation where the fugacities for vortices of each type are small. It is found that the vortex fugacities depend on a real continuous parameter \( a\) which characterize the degeneracy of the vacuum. Depending on the relative sizes of these fugacities, the vortex gas will be dominated by one of either of the two types mentioned above. In these regimes, we expect the standard Kosterlitz-Thouless phase transitions to occur, as in systems of abelian vortices in 2-dimensions. Between these two regimes, the gas contains pairs of both types, so nonabelian effects will be important. 
  String theory provides an example of the kind of apparent inconsistency that the {\it Principle of Black Hole Complementarity\/} deals with. To a freely infalling observer a string falling through a black hole horizon appears to be a Planck size object. To an outside observer the string and all the information it carries begin to spread as the string approaches the horizon. In a time of order the ``information retention time'' it fills the entire area of the horizon. 
  The new method for solving the descent equations for gauge theories proposed in \cite{s} is shown to be equivalent with that based on the {\em "Russian formula"}. Moreover it allows to obtain in a closed form the expressions of the consistent anomalies in any space-time dimension. 
  We show that the structure constants of W-algebras can be grouped according to the lowest (bosonic) spin(s) of the algebra. The structure constants in each group are described by a unique formula, depending on a functional parameter h(c) that is characteristic for each algebra. As examples we give the structure constants C_{33}^4 and C_{44}^4 for the algebras of type W(2,3,4,...) (that include the WA_{n-1}-algebras) and the structure constant C_{44}^4 for the algebras of type W(2,4,...), especially for all the algebras WD_n, WB(0,n), WB_n and WC_n. It also includes the bosonic projection of the super-Virasoro algebra and a yet unexplained algebra of type W(2,4,6) found previously. 
  The interpretation of the apparent unification of gauge couplings within supersymmetric theories depends on uncertainties induced through heavy particle thresholds. While in standard grand unified theories these effects can be estimated easily, the corresponding calculations are quite complicated in string unified theories and do exist only in models with unbroken $E_6$. We present results for heavy particle thresholds in more realistic models with gauge group $SU(3)\times SU(2)\times U(1)$. Effects of Wilson line background fields as well as the universal part of the (rather mild) threshold corrections indicate a strong model dependence. We discuss the consequences of our results for the idea of string unification without a grand unified gauge group. 
  We study the Yukawa couplings among excited twist fields which might arise in the low-energy effective field theory obtained by compactifying the heterotic string on ${\bf Z}_N$ and ${\bf Z}_M\times {\bf Z}_N$ orbifolds. 
  We find the exact matrix model description of two dimensional Yang-Mills theories on a cylinder or on a torus and with an arbitrary compact gauge group. This matrix model is the singlet sector of a $c =1$ matrix model where the matrix field is in the fundamental representation of the gauge group. We also prove that the basic constituents of the theory are Sutherland fermions in the zero coupling limit, and this leads to an interesting connection between two dimensional gauge theories and one dimensional integrable systems. In particular we derive for all the classical groups the exact grand canonical partition function of the free fermion system corresponding to a two dimensional gauge theory on a torus. 
  We analyze a formulation of QED based on the Wilson renormalization group. Although the ``effective Lagrangian'' used at any given scale does not have simple gauge symmetry, we show that the resulting renormalized Green's functions correctly satisfies Ward identities to all orders in perturbation theory. The loop expansion is obtained by solving iteratively the Polchinski's renormalization group equation. We also give a new simple proof of perturbative renormalizability. The subtractions in the Feynman graphs and the corresponding counterterms are generated in the process of fixing the physical conditions. 
  Hirota's method is used to construct multi--soliton and plane--wave solutions for affine Toda field theories with imaginary coupling. 
  Two-dimensional topological field theories possessing a non-abelian current symmetry are constructed. The topological conformal algebra of these models is analysed. It differs from the one obtained by twisting the $N=2$ superconformal models and contains generators of dimensions $1$, $2$ and $3$ that close a linear algebra. Our construction can be carried out with one and two bosonic currents and the resulting theories can be interpreted as topological sigma models for group manifolds 
  We investigate extremal electrically charged black holes in Einstein-Maxwell-dilaton theory with a cosmological constant inspired by string theory. These solutions are not static, and a timelike singularity eventually appears which is not surrounded by an event horizon. This suggests that cosmic censorship may be violated in this theory. 
  We present a new method of formulating the classical theory of $SU(N+1)$ non-Abelian Chern-Simons (NACS) particles for arbitrary $N\geq 1$ using the symplectic reduction of $CP(N)$ manifold from $S^{2N+1}$. Quantizing the theory using BRST formulation and coherent state path integral method , we obtain a quantum mechanical model for $SU(N+1)$ NACS particles. 
  Since the first attempts to quantize Gauge Theories and Gravity in the loop representation, the problem of the determination of the corresponding classical actions has been raised. Here we propose a general procedure to determine these actions and we explicitly apply it in the case of electromagnetism. Going to the lattice we show that the electromagnetic action in terms of loops is equivalent to the Wilson action, allowing to do Montecarlo calculations in a gauge invariant way. In the continuum these actions need to be regularized and they are the natural candidates to describe the theory in a ``confining phase''. 
  We investigate various boundary conditions in two dimensional turbulence systematically in the context of conformal field theory. Keeping the conformal invariance, we can either change the shape of boundaries through finite conformal transformations, or insert boundary operators so as to handle more general cases. Effects of such operations will be reflected in physically measurable quantities such as the energy power spectrum $E(k)$ or the average velocity profiles. We propose that these effects can be used as a possible test of conformal turbulence in an experimental setting. We also study the periodic boundary conditions, i.e. turbulence on a torus geometry. The dependence of moduli parameter $q$ appears explictly in the one point functions in the theory, which can also be tested. 
  It is shown in a variant of two dimensional dilaton gravity theories that an arbitrary, localized massive source put in an initially regular spacetime gives rise to formation of the wormhole classically, without accompanying the curvature singularity. The semiclassical quantum correction under this wormhole spacetime yields Hawking radiation. It is expected, with the quantum back reaction added to the classical equation, that the information loss paradox may be resolved in this model. 
  The amplitude for the scattering of a point magnetic monopole and a point charge, at centre-of-mass energies much larger than the masses of the particles, and in the limit of low momentum transfer, is shown to be proportional to the (integer-valued) monopole strength, assuming the Dirac quantization condition for the monopole-charge system. It is demonstrated that, for small momentum transfer, charge-monopole electromagnetic effects remain comparable to those due to the gravitational interaction between the particles even at Planckian centre-of-mass energies. 
  Following a recent proposal of Richard Borcherds to regard fusion as the ring-like tensor product of modules of a {\em quantum ring}, a generalization of rings and vertex operators, we define fusion as a certain quotient of the (vector space) tensor product of representations of the symmetry algebra ${\cal A}$. We prove that this tensor product is associative and symmetric up to equivalence. We also determine explicitly the action of ${\cal A}$ on it, under which the central extension is preserved. \\ Having given a precise meaning to fusion, determining the fusion rules is now a well-posed algebraic problem, namely to decompose the tensor product into irreducible representations. We demonstrate how to solve it for the case of the WZW- and the minimal models and recover thereby the well-known fusion rules. \\ The action of the symmetry algebra on the tensor product is given in terms of a comultiplication. We calculate the $R$-matrix of this comultiplication and find that it is triangular. This seems to shed some new light on the possible r\^{o}le of the quantum group in conformal field theory. 
  The general notion of distance dependent statistics in anyon-like systems is discussed. The two-body problem for such statistics is considered, the general formula for the second virial coefficient is derived and it is shown that in the limiting cases it reproduces the known results for ideal anyons. 
  We discuss the properties of the Liouville theory coupled to the c=1 matter when perturbed by an operator, the screening operator of the $SL(2;R)$ current algebra, which is supposed to generate the mass of the two-dimensional black hole. Mimicking the standard KPZ scaling theory of the Liouville system perturbed by the cosmological constant operator, we develop a scaling theory of correlation functions as functions of the mass of the black hole. Contrary to the case of KPZ, the present theory does not have the $c=1$ barrior and seems somewhat insensitive to the delatils of the matter content of the theory; the string succeptibility equals 1 independent of the matter central charge. It turns out that our scaling exponents agree with those of the deformed matrix model proposed recently by Jevicki and Yoneya. 
  We analyze $(2+1)$-dimensional vector-vector type four-Fermi interaction (Thirring) model in the framework of the $1/N$ expansion. By solving the Dyson-Schwinger equation in the large-$N$ limit, we show that in the two-component formalism the fermions acquire parity-violating mass dynamically in the range of the dimensionless coupling $\alpha$, $0 \leq \alpha \leq \alpha_c \equiv {1\over16} {\rm exp} (- {N \pi^2 \over 16})$. The symmetry breaking pattern is, however, in a way to conserve the overall parity of the theory such that the Chern-Simons term is not induced at any orders in $1/N$. $\alpha_c$ turns out to be a non-perturbative UV-fixed point in $1/N$. The $\beta$ function is calculated to be $\beta (\alpha) = -2 (\alpha - \alpha_c)$ near the fixed point, and the UV-fixed point and the $\beta$ function are shown exact in the $1/N$ expansion. 
  We study how the Chern-Simons term effects the dynamically generated fermion mass in $(2+1)D$ Quantum Electrodynamics in the framework of large $N$ expansion. We find that when the Chern-Simons term is present half of the fermions get mass $M+m$ and half get $M-m$. The parity-preserving mass $m$ is generated only when $N<\tilde{N}_c$. Both the critical number, $\tilde{N}_c$, of fermion flavor and the magnitude, $m$, reduce when the effect of the Chern-Simons term dominates. 
  We compute the mapping class group action on cycles on the configuration space of the torus with one puncture, with coefficients in a local system arising in conformal field theory. This action commutes with the topological action of the quantum group $U_q(sl_2)$, and is given in vertex form. 
  We develop the method of the hamiltonian reduction of affine Lie superalgebras to obtain explicit and general expressions both for the classical and the quantum extended superconformal algebras. By performing the gauge transformation which connects the diagonal gauge with the Drinfeld-Sokolov gauge and considering the quantum corrections, we get generic expressions for the classical and quantum free field realizations of the algebras. 
  We simplify and generalize an argument due to Bowcock and Watts showing that one can associate a finite Lie algebra (the `classical vacuum preserving algebra') containing the M\"obius $sl(2)$ subalgebra to any classical $\W$-algebra. Our construction is based on a kinematical analysis of the Poisson brackets of quasi-primary fields. In the case of the $\W_\S^\G$-algebra constructed through the Drinfeld-Sokolov reduction based on an arbitrary $sl(2)$ subalgebra $\S$ of a simple Lie algebra $\G$, we exhibit a natural isomorphism between this finite Lie algebra and $\G$ whereby the M\"obius $sl(2)$ is identified with $\S$. 
  We study the U(1) Higgs model in spacetime-dependent background fields (a background metric and a background scalar field).   Particle creation can occur because of the time-dependence of these background fields. In gauge theories, there is a unphysical sector and consequently unphysical particles may be produced. However, it is shown that produced unphysical particles have no contribution to backreaction to background fields. 
  By considering `coloured' braid group representation we have obtained a quantum group, which reduces to the standard $GL_q(2)$ and $GL_{p,q}(2)$ cases at some particular limits of the `colour' parameters. In spite of quite complicated nature, all of these new quantum group relations can be expressed neatly in the Heisenberg-Weyl form, for a nontrivial choice of the basis elements. Furthermore, it is possible to associate invariant Manin planes, parametrised by the `colour' variables, with such quantum group structure. 
  A wide class of models involve the fine--tuning of significant hierarchies between a strong--coupling ``compositeness'' scale, and a low energy dynamical symmetry breaking scale. We examine the issue of whether such hierarchies are generally endangered by Coleman--Weinberg instabilities. A careful study using perturbative two--loop renormalization group methods finds that consistent large hierarchies are not generally disallowed. 
  It is by now clear that the naive rule for the entropy of a black hole, {entropy} = 1/4 {area of event horizon}, is violated in many interesting cases. Indeed, several authors have recently conjectured that in general the entropy of a dirty black hole might be given purely in terms of some surface integral over the event horizon of that black hole. A formal proof of this conjecture, using Lorentzian signature techniques, has recently been provided by Wald. This note performs two functions. Firstly, a rather different proof of this result is presented --- a proof based on Euclidean signature techniques. The total entropy is S = 1/4 {k A_H / l_P^2} + \int_H {S} \sqrt{g} d^2x. The integration runs over a spacelike cross-section of the event horizon H. The surface entropy density, {S}, is related to the behaviour of the matter Lagrangian under time dilations.  Secondly, I shall consider the specific case of Einstein-Hilbert gravity coupled to an effective Lagrangian that is an arbitrary function of the Riemann tensor (though not of its derivatives). In this case a more explicit result is obtained S = 1/4 {k A_H / l_P^2} + 4 pi {k/hbar} \int_H {partial L / partial R_{\mu\nu\lambda\rho}} g^\perp_{\mu\lambda} g^\perp_{\nu\rho} \sqrt{g} d^2x . The symbol $g^\perp_{\mu\nu}$ denotes the projection onto the two-dimensional subspace orthogonal to the event horizon. 
  An n=8 worldsheet superfield action is proposed for describing chiral fermions in the twistor-like formulation of an N=1, D=10 heterotic superstring. 
  A new approach to massive integrable models is considered. It allows one to find symmetry algebras which define spaces of local operators and to get general integral representations for form-factors in the\ $ SU(2)$\ Thirring and Sine-Gordon models. 
  In this paper we give an explicit formula for level 1 vertex operators related to $U_q(\widehat{sl}(n))$ as operators on the Fock spaces. We derive also their commutation relations. As an applications we culculate the one point functions of the one-dimensional spin chain associated with the vector representation of $U_q(\widehat{sl}(n))$, thereby extending the recent work on the staggered polarization of the $XXZ$-model. 
  We propose a new formulation of gauge theories as a quantum theory which has the gauge theory action $S$ as its dynamical variable. This system is described by a simple actional $I(S)$ (that is, an action for the action $S$) whose equation of motion gives the Batalin-Vilkovisky (BV) master equation for $S$. Upon quantization we find that our new formulation is reduced to something like a topological field theory having a BRST exact gauge-fixed actional. Therefore the present formulation can reproduce ordinary gauge theories since the path-integral over $S$ is dominated by the classical configuration which satisfies the BV master equation. This ``theory of theories'' formulation is intended to be applied to closed string field theory. 
  The physical phase space of gauge field theories on a cylindrical spacetime with an arbitrary compact simple gauge group is shown to be the quotient $ {\bf R}^{2r}/W_A, $ $ r $ a rank of the gauge group, $ W_A $ the affine Weyl group. The PI formula resulting from Dirac's operator method contains a symmetrization with respect to $ W_A $ rather than the integration domain reduction. It gives a natural solution to Gribov's problem. Some features of fermion quantum dynamics caused by the nontrivial phase space geometry are briefly discussed. 
  In 1948, Feynman showed Dyson how the Lorentz force and Maxwell equations could be derived from commutation relations coordinates and velocities. Several authors noted that the derived equations are not Lorentz covariant and so are not the standard Maxwell theory. In particular, Hojman and Shepley proved that the existence of commutation relations is a strong assumption, sufficient to determine the corresponding action, which for Feynman's derivation is of Newtonian form. Tanimura generalized Feynman's derivation to a Lorentz covariant form, however, this derivation does not lead to the standard Maxwell theory either. Tanimura's force equation depends on a fifth ({\it scalar}) electromagnetic potential, and the invariant evolution parameter cannot be consistently identified with the proper time of the particle motion. Moreover, the derivation cannot be made reparameterization invariant; the scalar potential causes violations of the mass-shell constraint which this invariance should guarantee. In this paper, we examine Tanimura's derivation in the framework of the proper time method in relativistic mechanics, and use the technique of Hojman and Shepley to study the unconstrained commutation relations. We show that Tanimura's result then corresponds to the five-dimensional electromagnetic theory previously derived from a Stueckelberg-type quantum theory in which one gauges the invariant parameter in the proper time method. This theory provides the final step in Feynman's program of deriving the Maxwell theory from commutation relations; the Maxwell theory emerges as the ``correlation limit'' of a more general gauge theory, in which it is properly contained. 
  For one-mode light described by the Wigner function of generic Gaussian form the photon distribution function is obtained explicitly and expressed in terms of Hermite polynomials of two variables.The mean values and dispersions of photon numbers are obtained for this generic %mixed state.Generating function for photon distribution is discussed.Known partial cases of thermal state,correlated state,squeezed state and coherent state are considered.The connection of Schrodinger uncertainty relation for quadratures with photon distribution is demonstrated explicitly. 
  The complete structure of the moduli space of \cys\ and the associated Landau-Ginzburg theories, and hence also of the corresponding low-energy effective theory that results from (2,2) superstring compactification, may be determined in terms of certain holomorphic functions called periods. These periods are shown to be readily calculable for a great many such models. We illustrate this by computing the periods explicitly for a number of classes of \cys. We also point out that it is possible to read off from the periods certain important information relating to the mirror manifolds. 
  We generalize to composite operators concepts and techniques which have been successful in proving renormalization of the effective Action in light-cone gauge. Gauge invariant operators can be grouped into classes, closed under renormalization, which is matrix-wise. In spite of the presence of non-local counterterms, an ``effective" dimensional hierarchy still guarantees that any class is endowed with a finite number of elements. The main result we find is that gauge invariant operators under renormalization mix only among themselves, thanks to the very simple structure of Lee-Ward identities in this gauge, contrary to their behaviour in covariant gauges. 
  In a previous paper \cite{Simple} it was shown that the BRST charge $Q$ for any gauge model with a Lie algebra symmetry may be decomposed as $Q=\del+\del^{\dag},\;\;\;\del^2=\del^{\dag 2}=0,\;\;\;[\del, \del^{\dag}]_+=0$ provided dynamical Lagrange multipliers are used but without introducing other matter variables in $\del$ than the gauge generators in $Q$. In this paper further decompositions are derived but now by means of gauge fixing operators. As in \cite{Simple} it is shown that $\del=c^{\dag a}\phi_a$ where $c^a$ are new ghosts and $\phi_a$ are nonhermitian variables satisfying the gauge algebra. However, in distinction to \cite{Simple} also solutions of the form $\del=c^{\dag a}A_a$ where $A_a$ satisfy an abelian algebra is derived (abelianization). By means of a bigrading the BRST condition reduces to $\del|ph\hb=\del^{\dag}|ph\hb=0$ on inner product spaces whose general solutions are expressed in terms of the solutions to a proper Dirac quantization. Thus, the procedure provides for inner products for the solutions of a Dirac quantization. 
  The formulation and resolution of integrable lattice statistical models in a quantum group covariant way is the subject of this review. The Bethe Ansatz turns to be remarkably useful to implement quantum group symmetries and to provide quantum group representations even when $q$ is a root of unity. We start by solving the six-vertex model with fixed boundary conditions (FBC) that guarantee exact $SU(2)_q$ invariance on the lattice. The algebra of the Yang-Baxter (YB) and $SU(2)_q$ generators turns to close. The infinite spectral parameter limit of the YB generators yields {\bf cleanly} the $SU(2)_q$ generators. The Bethe Ansatz states constructed for FBC are shown to be {\bf highest weights} of $SU(2)_q$. The higher level Bethe Ansatz equations (BAE, describing the physical excitations) are explicitly derived for FBC. We then solve the RSOS($p$) models on the light--cone lattice with fixed boundary conditions by disentangling the type II representations of $SU(2)_q$, at $q=e^{i\pi/p}$, from the full SOS spectrum obtained through Algebraic Bethe Ansatz. The RSOS states are those with quantum spin $J<(p-1)/2$ and no {\bf singular} roots in the solutions of the BAE. We thus give a microscopic derivation of the lattice $S-$matrix of the massive kinks and show that the continuum limit of the RSOS($p+1$) model is the $p-$restricted Sine--Gordon field theory. 
  Some recent work has attempted to show that the singular solutions which are known to occur in the Dirac description of spin-1/2 Aharonov-Bohm scattering can be eliminated by the inclusion of strongly repulsive potentials inside the flux tube. It is shown here that these calculations are generally unreliable since they necessarily require potentials which lead to the occurrence of Klein's paradox. To avoid that difficulty the problem is solved within the framework of the Galilean spin-1/2 wave equation which is free of that particular complication.   It is then found that the singular solutions can be eliminated provided that the nongauge potential is made energy dependent. The effect of the inclusion of a Coulomb potential is also considered with the result being that the range of flux parameter for which singular solutions are allowed is only half as great as in the pure Aharonov-Bohm limit. Expressions are also obtained for the binding energies which can occur in the combined Aharonov-Bohm-Coulomb system. 
  We discuss quantum deformation of the affine transformation algebra. It is shown that the quantum algebra has a non-cocommutative Hopf algebra structure, simple realizations and quantum tensor operators. 
  We discuss the interpertation of the $c=1$ matrix model as two-dimensional string theory in a dilaton-black hole background. The nonperturbative formulation of $c=1$ matrix model in terms of an integrable model of nonrelativistic fermions enables us to study the quantum fate of the classical black hole singularity. We find that the classical singularity is wiped out by quantum corrections when summed to all orders. 
  We provide a BRST symmetric version of Yokoyama's Type I gaugeon formalism for quantum electrodynamics; the similar theory by Izawa can be considered as a BRST symmetrized Type II theory. With the help of the BRST symmetry, Yokoyama's physical subsidiary conditions are replaced by the Kugo-Ojima type condition. As a result, the formalism becomes applicable even in the background gravitational field. We show how the Hilbert spaces of standard formalism in various gauges are embedded in the single Hilbert space of the present formalism. We also give a path integral derivation of the Lagrangian. 
  We examine the BRS cohomology of chiral matter in $N=1$, $D=4$ supersymmetry to determine a general form of composite superfield operators which can suffer from supersymmetry anomalies. Composite superfield operators $\Y_{(a,b)}$ are products of the elementary chiral superfields $S$ and $\ov S$ and the derivative operators $D_\a$, $\ov D_{\dot \b}$ and $\pa_{\a \dot \b}$. Such superfields $\Y_{(a,b)}$ can be chosen to have `$a$' symmetrized undotted indices $\a_i$ and `$b$' symmetrized dotted indices $\dot \b_j$. The result derived here is that each composite superfield $\Y_{(a,b)}$ is subject to potential supersymmetry anomalies if $a-b$ is an odd number, which means that $\Y_{(a,b)}$ is a fermionic superfield. 
  We compare the vortex-like solutions of two different theories in (2+1) dimensions. In the first a nonrelativistic field self-interacts through a Chern-Simons gauge connection. It is $P$ and $T$ violating. The second is the standard Maxwell scalar electrodynamics. We show that for specific values of some parameters the same vortex-configurations provide solutions for both theories. 
  We consider $SU(N)$ Yang-Mills on a circle (cylindrical space-time) and quantize the eigenvalues of the holonomy. In this way the Mandelstam identities associated with the holonomy are trivially solved. Furthermore we indicate that there are exactly two physically inequivalent representations of the algebra of gauge-invariant operators, resulting in different spectra. 
  The higher spin symmetry for both Dirac and Majorana massless free fermionic field models are considered. An infinite Lie algebra which is a linear realization of the higher spin extension of the cross products of the Virasoro and affine Kac-Moody algebras is obtained. The corresponding current algebra is closed which is not the case of annalogous current algebra in the WZNW model. The gauging procedure for the higher spin symmetry is given also. 
  The Chern-Simons membranes and in general the Chern-Simons p-branes moving in $D$-dimensional target space admit an infinite set of secondary constraints. With respect to the Poisson bracket these constraints form a closed algebra which contains classical \ $W_{1+\infty }$ \ algebra in $p$-dimensions as a subalgebra. Corresponding gauged theory in the phase-space is constructed in a Hamilton gauge as an analog of the ordinary $W$-gravity. 
  We construct new heterotic string backgrounds which are analogous to superstring solutions corresponding to coset models but are not simply the `embeddings'of the latter. They are described by the (1,0) supersymmetric extension of the $G/H$ chiral gauged WZNW models. The `chiral gauged' WZNW action differs from the standard gauged WZNW action by the absence of the $A\bar A$-term (and thus is not gauge invariant in the usual sense) but can still be expressed as a combination of WZNW actions and is conformal invariant. We explain a close relation between gauged and chiral gauged WZNW models and prove that in the case of the abelian $H$ the $G/H$ chiral gauged theory is equivalent to a particular $(G\times H)/H$ gauged WZNW theory. In contrast to the gauged WZNW model, the chiral gauged one admits a (1,0) supersymmetric extension which is consistent at the quantum level. Integrating out the $2d$ gauge field we determine the exact (in $\alpha'$) form of the couplings of the corresponding heterotic sigma model. While in the bosonic (superstring) cases all the fields depend (do not depend) non-trivially on $\alpha'$ here the metric receives only one $O(\alpha')$ correction while the antisymmetric tensor and the dilaton remain semiclassical. As a simplest example, we discuss the basic $D=3$ solution which is the heterotic string counterpart of the `black string' $SL(2,R) \times R/ R $ background. 
  The effective potential which describes the conformal dynamics of quantum gravity with torsion is discussed. The phase transitions induced by the combination of torsion and curvature are investigated. The mechanism for fixing the vacuum expectation values of the metric and torsion is presented. 
  Two dimensional dilaton gravity interacting with a four-fermion model and scalars is investigated, all the coefficients of the Lagrangian being arbitrary functions of the dilaton field. The one-loop covariant effective action for 2D dilaton gravity with Majorana spinors (including the four-fermion interaction) is obtained, and the technical problems which appear in an attempt at generalizing such calculations to the case of the most general four-fermion model described by Dirac fermions are discussed. A solution to these problems is found, based on its reduction to the Majorana spinor case.   The general covariant effective action for 2D dilaton gravity with the four-fermion model described by Dirac spinors is given. The one-loop renormalization of dilaton gravity with Majorana spinors is carried out and the specific conditions for multiplicative renormalizability are found. A comparison with the same theory but with a classical gravitational field is done. 
  We present a simple unifying gauge theoretical formulation of gravitational theories in two dimensional spacetime. This formulation includes the effects of a novel matter-gravity coupling which leads to an extended de Sitter symmetry algebra on which the gauge theory is based. Contractions of this theory encompass previously studied cases. 
  This paper is concerned with theories describing spinning particles that are formulated in terms of actions possessing either local world-line supersymmetry or local fermionic {\it kappa} invariance. These classical actions are obtained by adding a finite number of spinor or vector coordinates to the usual space-time coordinates. Generalizing to superspace leads to corresponding types of \lq spinning superparticle' theories in which the wave-functions are superfields in some (generally reducible) representation of the Lorentz group. A class of these spinning superparticle actions possesses the same spectrum as ten-dimensional supersymmetric Yang--Mills theory, which it is shown can be formulated in terms of either vector or spinor superfields satisfying supercovariant constraints. The models under consideration include some that were known previously and some new ones. 
  If string theory controls physics at the string scale, the dynamics of the early universe before the GUT era will be governed by the low-energy string equations of motion. Studying these equations for FRW spacetimes, we find that depending on the initial conditions when the stringy era starts, and on the time when it ends, there are a wide variety of qualitatively distinct types of evolution. We classify these, and present the general solution to the equations of motion. 
  We further develop the reduced action formalism of the SU(2)-Higgs model originally given by Aoyama et.al.. Our new ansatz for the sphaleron solution makes it possible to apply this formalism to all range of the Higgs self coupling constant. Based on the formalism, we construct a bounce solution oscillating around the sphaleron. 
  While it is possible to introduce quantum group symmetry into the framework of quantum mechanics, the general problem of how to implement quantum group symmetry into $(3+1)$ dimensional quantum field theory has not yet been solved. Here we try to estimate some features of the behaviour of bosonic modes. 
  Quantum Groups can be constructed by applying the quantization by deformation procedure to Lie groups endowed with a suitable Poisson bracket. Here we try to develop an understanding of these structures by investigating dynamical systems which are associated with this bracket. We look at $SU(2)$ and $SU(1,1)$, as submanifolds of a 4--dimensional phase space with constraints, and deal with two classes of problems. In the first set of examples we consider some hamiltonian systems associated with Lie-Poisson structures and we investigate the equations of the motion. In the second set of examples we consider systems which preserve the chosen bracket, but are dissipative. However in this approach, they survive the quantization procedure. 
  It has been known for some time that there are many inequivalent quantizations possible when the configuration space of a system is a coset space G/H. Viewing this classical system as a constrained system on the group G, we show that these inequivalent quantizations can be recovered from a generalization of Dirac's approach to the quantization of such a constrained system within which the classical first class constraints (generating the H-action on G) are allowed to become anomalous (second class) when quantizing. The resulting quantum theories are characterized by the emergence of a Yang-Mills connection, with quantized couplings, and new 'spin' degrees of {}freedom. Various applications of this procedure are presented in detail: including a new account of how spin can be described within a path-integral formalism, and how on S^4 chiral spin degrees of {}freedom emerge, coupled to a BPST instanton. 
  Hawking's zeta function regularization procedure is shown to be rigorously and uniquely defined, thus putting and end to the spreading lore about different difficulties associated with it. Basic misconceptions, misunderstandings and errors which keep appearing in important scientific journals when dealing with this beautiful regularization method ---and other analytical procedures--- are clarified and corrected. 
  We propose a ladder-operator method for obtaining the squeezed states of general symmetry systems. It is a generalization of the annihilation-operator technique for obtaining the coherent states of symmetry systems. We connect this method with the minimum-uncertainty method for obtaining the squeezed and coherent states of general potential systems, and comment on the distinctions between these two methods and the displacement-operator method. 
  Canonical quantization of local field theories is classical black hole spacetimes with a single horizon leads to a particle number density with a thermal distribution in equilibrium at the Hawking temperature. A complete treatment including non-local quantum gravity effects has shown however that the full "thermal vacuum" of the theory is the false vacuum. In this work we find the true vacuum consistent with the complete semiclassical analysis of quantum black holes. The theory is described by a "microcanonical" quantum field theory with fixed energy E = M, the mass of the black hole. Considerations making use of the microcanonical density matrix as well as the idempotency condition show that particles in black hole backgrounds are described by pure states, unlike the canonical formulation. 
  There is a general mechanism by which certain matter fields coupled to gravity can generate a nontrivial effective potential for the conformal factor of the metric. It is based on a nonstandard regularization method, with the cutoff being defined independently of the conformal factor. This mechanism produces a coupling of the matter fields to a dilaton, and a complicated interaction between matter, dilaton and metric. When it is applied to the standard model, it gives an effective potential which can be used to predict the top and Higgs masses. If the purely gravitational contribution to the potential is added, the mass of the dilaton is of the order of Planck's mass and the large hierarchy between the Planck and Fermi scales appears to be due to the smallness of the Higgs-dilaton coupling. 
  The even and odd coherent states are generalized for multimode case. The explicit forms for the photon distribution, Q-function and Wigner function are derived. In particular, it is shown that for two-mode case there exist strong correlations between these modes, under certain conditions, which are responsible for two-mode squeezing in case of even coherent states 
  For N-mode light described by the Wigner function of generic Gaussian form the photon distribution function is obtained explicitly and expressed in terms of Hermite polynomial of $2N$ variables with equal pairs of indices.The mean values and dispersions of photon numbers are obtained for this generic mixed state.Generating function for photon distribution is obtained explicitly. The expression for $N$-mode photon distribution function for squeezed photon number states in terms of Hermite polynomials of $2N$ variables and for squeezed coherent states in terms of Hermite polynomials of $N$ variables is discussed. 
  The dynamics of Liouville fields coupled to gravity are investigated by applying the principle of general covariance to the Liouville action in the context of a particular form of two-dimensional dilaton gravity. The resultant field equations form a closed system for the Liouville/gravity interaction. A large class of asymptotically flat solutions to the field equations is obtained, many of which can be interpreted as black hole solutions. The temperature of such black holes is proportional to their mass-parameters. An exact solution to the back reaction problem is obtained to one-loop order, both for conformally coupled matter fields and for the quantized metric/Liouville system. Quantum effects are shown to map the space of classical solutions into one another. A scenario for the end-point of black-hole radiation is discussed. 
  At the simple example of a massless scalar field propagating in the static background we study the resummed expressions for the effective action at zero and finite temperature that are free from a usual sickness of the effective action induced by massless particles. 
  We consider the $c=1$ matrix model deformed by the operator ${1\over 2} M\Tr\Phi^{-2}$, which was conjectured by Jevicki and Yoneya to describe a two-dimensional black hole of mass $M$. We calculate the exact non-perturbative $S$-matrix and show that all the amplitudes involving an odd number of particles vanish at least to all orders of perturbation theory. We conjecture that these amplitudes vanish non-perturbatively and prove this for the $2n \to 1$ scattering. For the 2-- and 4--particle amplitudes we give some leading terms of the perturbative expansion. 
  The BRST quantisation of the Drinfeld - Sokolov reduction is exploited to recover all singular vectors of the Virasoro algebra Verma modules from the corresponding $A^{(1)}_1\,$ ones. The two types of singular vectors are shown to be identical modulo terms trivial in the $Q_{BRST}$ cohomology. The main tool is a quantum version of the DS gauge transformation. 
  The problem of describing the singular vectors of $\cW_3$ and $\cW_3^{(2)}$ Verma modules is addressed, viewing these algebras as BRST quantized Drinfeld-Sokolov (DS) reductions of $A^{(1)}_2\,$. Singular vectors of an $A^{(1)}_2\,$ Verma module are mapped into $\W$ algebra singular vectors and are shown to differ from the latter by terms trivial in the BRST cohomology. These maps are realized by quantum versions of the highest weight DS gauge transformations. 
  In this article we formulate a `topological' field theory by employing a generalization of the Duistermaat-Heckman Theorem to localize the path-integral of the `topological action' C^2 , where C is a slight modification of the Zamolodchikov C-Function, over the space of all two-dimensional field theories to the fixed points of the renormalization group's identity component. Also, we propose an interpretation of the background independent classical closed string field theory action S in terms of the Zamolodchikov C-Function's modification. 
  The eigenvalues of the Corner Transfer Matrix Hamiltonian associated to the elliptic $R$ matrix of the eight vertex free fermion model are computed in the anisotropic case for magnetic field smaller than the critical value. An argument based on generating functions is given, and the results are checked numerically. The spectrum consists of equally spaced levels. 
  I describe a new method for computing trace anomalies in quantum field theories which makes use of path-integrals for particles moving in curved spaces. After presenting the main ideas of the method, I discuss how it is connected to the first quantized approach of particle theory and to heat kernel techniques. (Talk given at Journees Relativistes '93). 
  String backgrounds associated with gauged $G/H$ WZNW models in general depend non-trivially on $\alpha'$. We note, however, that there exists a local covariant $\a'$-dependent field redefinition that relates the exact metric-dilaton background corresponding to the $SL(2,R)/U(1)$ model to its leading-order form ($D=2$ black hole). As a consequence, there exists a `scheme' in which the string effective equations have the latter as an exact solution. However, the corresponding equation for the tachyon (which, like other Weyl anomaly coefficients, has scheme-dependent form) still contains corrections of all orders in $\alpha'$. As a result, the `probes' (the tachyons) still feel the $\alpha'$-corrected background. The field redefinitions we discuss contain the dilaton terms in the metric transformation law. We comment on exact forms of the duality transformation in different `schemes'. 
  We prove that unitary two-dimensional topological field theories are uniquely characterized by $n$ positive real numbers $\lambda _1,\ldots \lambda _n$ which can be regarded as the eigenvalues of a hermitean handle creation operator. The number $n$ is the dimension of the Hilbert space associated with the circle and the partition functions for closed surfaces have the form $$ Z_g=\sum_{i=1}^{n}\lambda _i^{g-1} $$ where $g$ is the genus. The eigenvalues can be arbitary positive numbers. We show how such a theory can be constructed on triangulated surfaces. 
  The electric charge of a wormhole mouth and the magnetic flux ``linked'' by the wormhole are non-commuting observables, and so cannot be simultaneously diagonalized. We use this observation to resolve some puzzles in wormhole electrodynamics and chromodynamics. Specifically, we analyze the color electric field that results when a colored object traverses a wormhole, and we discuss the measurement of the wormhole charge and flux using Aharonov-Bohm interference effects. We suggest that wormhole mouths may obey conventional quantum statistics, contrary to a recent proposal by Strominger. 
  In a wide class of three-dimensional Abelian gauge theories with a bare Chern-Simons term, the Lorentz invariance is spontaneously broken by dynamical generation of a non-vanishing magnetic field. A detailed computation of an energy density of the true vacuum is given. The originally massive photon becomes massless, fulfilling the role of a Nambu-Goldstone boson associated with the spontaneous breaking of the Lorentz invariance. 
  Self-consistent Hamiltonian formulation of scalar theory on the null plane is constructed following Dirac method. The theory contains also {\it constraint equations}. They would give, if solved, to a nonlinear and nonlocal Hamiltonian. The constraints lead us in the continuum to a different description of spontaneous symmetry breaking since, the symmetry generators now annihilate the vacuum. In two examples where the procedure lacks self-consistency, the corresponding theories are known ill-defined from equal-time quantization. This lends support to the method adopted where both the background field and the fluctuation above it are treated as dynamical variables on the null plane. We let the self-consistency of the Dirac procedure determine their properties in the quantized theory. The results following from the continuum and the discretized formulations in the infinite volume limit do agree. 
  We study differential calculus on h-deformed bosonic and fermionic quantum space. It is shown that the fermionic quantum space involves a parafermionic variable as well as a classical fermionic one. Further we construct the classical $su(2)$ algebra on the fermionic quantum space and discuss a mapping between the classical $su(2)$ and the h-deformed $su(2)$ algebras. 
  The vacuum energy density (Casimir energy) corresponding to a massless scalar quantum field living in different universes (mainly no-boundary ones), in several dimensions, is calculated. Hawking's zeta function regularization procedure supplemented with a very simple binomial expansion is shown to be a rigorous and well suited method for performing the analysis. It is compared with other, much more involved techniques. The principal-part prescription is used to deal with the poles that eventually appear. Results of the analysis are the absence of poles at four dimensions (for a 4d Riemann sphere and for a 4d cylinder of 3d Riemann spherical section), the total coincidence of the results corresponding to a 3d and a 4d cylinder (the first after pole subtraction), and the fact that the vacuum energy density for cylinders is (in absolute value) over an order of magnitude smaller than for spheres of the same dimension. 
  We discuss quantum deformation of the affine transformation group and its Lie algebra. It is shown that the quantum algebra has a non-cocommutative Hopf algebra structure, simple realizations and quantum tensor operators. The deformation of the group is achieved by using the adjoint representation. The elements of quantum matrix form a Hopf algebra. Furthermore, we construct a differential calculus which is covariant with respect to the action of the quantum matrix. 
  This is the second in a series of papers devoted to open string field theory in two dimensions. In this paper we aim to clarify the origin and the role of discrete physical states in the theory. To this end, we study interactions of discrete states and generic tachyons. In particular, we discuss at length four point amplitudes. We show that behavior of the correlation functions is governed by the number of generic tachyons involved and values of the kinematic invariants $s$, $t$ and $u$. Divergence of certain classes of correlators is shown to be the consequence of the fact certain kinematic invariants are non--positive integers in that case. Explicit examples are included. We check our results by standard conformal technique. 
  We derive the loop equations for the one Hermitian matrix model in any dimension. These are a consequence of the Schwinger-Dyson equations of the model. Moreover we show that in leading order of large $N$ the loop equations form a closed set. 
  This is a summary of a talk based on hep-th/9305139 and presented at the SUSY-93 International Workshop. We study infinite dimensional unbroken gauge symmetries which arise when time is toroidally compactified in string theory. 
  We show that correlation functions of the $\bz _n $-Baxter model in the principal regime satisfy a system of difference equations. We obtain the spontaneous polarization of the $\bz _n $-Baxter model as a solution of the simplest difference equation. 
  Eigenfunctions of total angular momentum for a charged vector field interacting with a magnetic monopole are constructed and their properties studied. In general, these eigenfunctions can be obtained by applying vector operators to the monopole spherical harmonics in a manner similar to that often used for the construction of the ordinary vector spherical harmonics. This construction fails for the harmonics with the minimum allowed angular momentum. These latter form a set of vector fields with vanishing covariant curl and covariant divergence, whose number can be determined by an index theorem. 
  On an oriented, compact, connected, real four-dimensional manifold, $M$, we introduce a topological Lagrangian gauge field theory with a Bogomol'nyi structure that leads to non-singular, finite-Action, stable solutions to the variational field equations. These soliton-like solutions are analogous to the instanton in Yang-Mills theory. Unlike Yang-Mills instantons, however, `topological' instantons are independent of any underlying metric structure, and, in particular, they are independent of the metric signature. We show that when the topology of the underlying manifold, $M$, is equipped with a complex K\"ahler structure, and $M$ is interpreted as space-time, then the moduli space of topological instantons---the space of motions---is a finite-dimensional, smooth, Hausdorff manifold with a natural symplectic structure. We identify space-time topologies which lead to the physical stability of topological instanton field configurations compatible with the additional geometric structures. The spaces of motion for $U(1)$ topological instantons over either minimal elliptic or algebraic complex space-times with irregularity $q=2$ are examined. 
  The recently obtained results in \cite{ZG2} are used to compute the explicitly spectral-dependent $R$-matrix (or the intertwiners) on $V_{(6)}(x)\otimes V_{(6)}(y)$ and $V_{(3)}(x)\otimes V_{(6)}(y)$, where $V_{(6)}$ and $V_{(3)}$ are the 6-dimensional and fundamental representations of $U_q(A_2)$, respectively. It appears that the $R$-matrix on $V_{(3)}(x)\otimes V_{(6)}(y)$ depends on $q$ in the different way from what one might usually think: $q$ occurs in the $R$-matrix in fractional powers. It seems to be the first example in literatures of $R$-matrix with the new feature. 
  The short distance behaviour of massive integrable quantum field theories is analyzed in terms of the form factor approach. We show that the on-shell dynamics is compatible with different definitions of the stress-energy tensor $T_{\mu\nu}(x)$ of the theory. In terms of form factors, this is equivalent to having a possible non-zero matrix element $F_1$ of the trace of $T_{\mu\nu}$ on one-particle state. Each choice of $F_1$ induces a different scaling behaviour of the massive theory in the ultraviolet limit. 
  Quantum Poincar\'e-Weyl group in two dimensional quantum Minkowski space-time is considered and an appriopriate relativistic kinematics is investigated. It is claimed that a consistent approach to the above questions demands a kind of a ``quantum geometry'' in the $q$-deformed space-time. 
  An example of a toy model of $D=2$ Minkowski space and Poincar\'e group with real deformation parameter $q$ is considered. A notion of free motion is defined. The kinematics and phase-space are constructed and the ``uncertainity'' ralations are found. 
  For the integrable $N$-particle Calogero-Moser system with elliptic potential it is shown that the Lax operator found by Krichever possesses a classical $r$-matrix structure. The $r$-matrix is a natural generalisation of the matrix found recently by Avan and Talon (hep-th/9210128) for the trigonometric potential. The $r$-matrix depends on the spectral parameter and only half of the dynamical variables (particles' coordinates). It satisfies a generalized Yang-Baxter equation involving another dynamical matrix. 
  Using a simple version of the model for the quantum measurement of a two level system, the contention of Aharonov, Anandan, and Vaidman that one must in certain circumstances give the wavefunction an ontological as well as an epistemological significance is examined. I decide that their argument that the wave function of a system can be measured on a single system fails to establish the key point and that what they demonstrate is the ontological significance of certain operators in the theory, with the wave function playing its usual epistemological role. 
  We discuss the effect of perturbations on the ground rings of $c=1$ string theory at the various compactification radii defining the $A_N$ points of the moduli space. We argue that perturbations by plus-type moduli define ground varieties which are equivalent to the unperturbed ones under redefinitions of the coordinates and hence cannot smoothen the singularity. Perturbations by the minus-type moduli, on the other hand, lead to semi-universal deformations of the singular varieties that can smoothen the singularity under certain conditions. To first order, the cosmological perturbation by itself can remove the singularity only at the self-dual ($A_1$) point.} 
  By introducing two kinds of gaugeon fields, we extend Yokoyama's Type I gaugeon formalism for quantum electrodynamics. The theory admits a q-number gauge transformation by which we can shift the gauge parameter into arbitrary numerical value; whereas in the original theory we cannot change the sign of the parameter. The relation to the Type II theory is also discussed. 
  We study the long-distance relevance of vortices (instantons) in an $N$-component axially U(1)-gauged four-Fermi theory in $1+1$ dimensions, in which a naive use of $1/N$ expansion predicts the dynamical Higgs phenomenon. Its general effective lagrangian is found to be a frozen U(1) Higgs model with the gauge-field mass term proportional to an anomaly parameter ($b$). The dual-transformed versions of the effective theory are represented by sine-Gordon systems and recursion-relation analyses are performed. The results suggest that in the gauge-invariant scheme ($b=0$) vortices are always relevant at long distances, while in non-invariant schemes ($b>0$) there exists a critical $N$ above which the long-distance behavior is dominated by a free massless scalar field. 
  In this paper we investigate one Wakimoto-type construction of affine Kac-Moody algebras. We obtain a version of the regular representation, on which the affine algebra acts from the left and from the right with the sum of levels equal to minus two dual Coxeter numbers. 
  We present a manifestly supersymmetric procedure for calculating the contributions from matter loops to the mixed K\"{a}hler-gauge and to the mixed K\"{a}hler- Lorentz anomalies in $N=1, D=4$ supergravity-matter systems. We show how this procedure leads to the well-known result for the mixed K\"{a}hler-gauge anomaly. For general supergravity-matter systems the mixed K\"{a}hler-Lorentz anomaly is found to contain a term proportional to ${\cal R}^2$ with a background field dependent coefficient as well as terms proportional to $(C_{mnpq})^2$ and to the Gauss-Bonnet topological density. We briefly comment on the relationship between the mixed K\"{a}hler-Lorentz anomaly and the moduli dependent threshold corrections to gravitational couplings in $Z_N$ orbifolds. 
  We analyse the interacting theory of charged fermions, scalars, pseuso-scalars and photons propagating in 2-dimensional curved spacetime in detail. For certain values of the coupling constants the theory reduces to the gauged Thirring model and for others the Schwinger model incurved spacetime. It is shown that the interaction of the fermions with the pseudo-scalars shields the electromagnetic interaction, and that the non-minimal coupling of the scalars to the gravitational field amplifies the Hawking radiation. We solve the finite temperature and density model by using functional techniques and in particular derive the exact equation of state. The explicit temperature and curvature dependence of the chiral condensate is found. When the electromagnetic field is switched off the model reduces to a conformal field theory. We determine the physically relevant expectation values and conformal weights of the fundamental fields in the theory. 
  An infinite family of cornucopions is found within the $SU(2)\times U(1)$ sector of the 4--d heterotic string low-energy theory, the extremal $U(1)$ magnetic dilatonic black hole being the lowest energy state. Non-abelian cornucopions are interpreted as sphalerons associated with potential barriers separating topologically distinct Yang-Mills vacua on the $U(1)$ cornucopion background. A mass formula for non-abelian dilatonic black holes is derived, and the free energy is calculated through the Euclidean action. 
  We study the model of massless $1+1$ electrodynamics with nonconstant coupling, introduced by Peet, Susskind and Thorlacius as the `charge hole'. But we take the boundary of the strong coupling region to be first timelike, then spacelike for a distance $X$, and then timelike again (to mimic the structure of a black hole). For an incident charge pulse entering this `charge trap' the charge and information get separated. The charge comes out near the endpoint of the singularity. The `information' travels a well localised path through the strong coupling region and comes out later. 
  We study the entropy generation and particle production in scalar quantum field theory in expanding spacetimes with many-particle mixed initial states. The recently proposed coarse-grained entropy approach by Brandenberger et. al. is applied to systems which may have a non-zero initial entropy. We find that although the the particle production is amplified as a result of boson statistics, the (coarse-grained) entropy generation is {\em attenuated} when initial particles are present. 
  We give a unified view of the relation between the $SL(2)$ KdV, the mKdV, and the Ur-KdV equations through the Fr\'{e}chet derivatives and their inverses. For this we introduce a new procedure of obtaining the Ur-KdV equation, where we require that it has no non-local operators. We extend this method to the $SL(3)$ KdV equation, i.e., Boussinesq(Bsq) equation and obtain the hamiltonian structure of Ur-Bsq equationin a simple form. In particular, we explicitly construct the hamiltonian operator of the Ur-Bsq system which defines the poisson structure of the system, through the Fr\'{e}chet derivative and its inverse. 
  We introduce an associative glueing operation $\oplus_q$ on the space of solutions of the Quantum Yang-Baxter Equations of Hecke type. The corresponding glueing operations for the associated quantum groups and quantum vector spaces are also found. The former involves $2\times 2$ quantum matrices whose entries are themselves square or rectangular quantum matrices. The corresponding glueing operation for link-invariants is introduced and involves a state-sum model with Boltzmann weights determined by the link invariants to be glued. The standard $su(n)$ solution, its associated quantum matrix group, quantum space and link-invariant arise at once by repeated glueing of the one-dimensional case. 
  Using knowledge of the explicit $n$ dependence of the RG functions and expressions of critical exponents in the framework of large $N$ expansion in the Gross Neveu model we derive RG functions in 4- and 5-loop approximation. 
  The critical dimension of the bosonic string in the harmonic and the deDonder gauge may be calculated from the time ordered product of two energy momentum tensors. We show that recently found ambiguities within that method in nonconformal gauges can be resolved by a treatment respecting background covariance. 
  In 1973 two Salam prot\'{e}g\'{e}s (Derek Capper and the author) discovered that the conformal invariance under Weyl rescalings of the metric tensor $g_{\mu\nu}(x)\rightarrow\Omega^2(x)g_{\mu\nu}(x)$ displayed by classical massless field systems in interaction with gravity no longer survives in the quantum theory. Since then these Weyl anomalies have found a variety of applications in black hole physics, cosmology, string theory and statistical mechanics. We give a nostalgic review. (Talk given at the {\it Salamfest}, ICTP, Trieste, March 1993.) 
  The equivalence between the covariant and the non-covariant version of a constrained system is shown to hold after quantization in the framework of the field-antifield formalism. Our study covers the cases of Electromagnetism and Yang-Mills fields and sheds light on some aspects of the Faddeev-Popov method, for both the coratiant and non-covariant approaches, which had not been fully clarified in the literature. 
  We consider solutions to low energy string theory which have a horizon and a spacelike symmetry. Each of these solutions has a geometrically different dual description. We show that the dual solution has a horizon with exactly the same Hawking temperature (surface gravity) and entropy (area) as the original solution. 
  A Lagrangian treatment of the quantization of first class Hamiltonian systems with constraints and Hamiltonian linear and quadratic in the momenta respectively is performed. The ``first reduce and then quantize'' and the ``first quantize and then reduce'' (Dirac's) methods are compared. A new source of ambiguities in this latter approach is revealed and its relevance on issues concerning self-consistency and equivalence with the ``first reduce'' method is emphasized. One of our main results is the relation between the propagator obtained {\it \`a la Dirac} and the propagator in the full space, eq. (5.25).As an application of the formalism developed, quantization on coset spaces of compact Lie groups is presented. In this case it is shown that a natural selection of a Dirac quantization allows for full self-consistency and equivalence. Finally, the specific case of the propagator on a two-dimensional sphere $S^2$ viewed as the coset space $SU(2)/U(1)$ is worked out. 
  We give a new interpretation and proof of the ``quasi-particle'' type character formulas for integrable representations of the simply-laced affine Kac-Moody algebras through a new ``semi-infinite'' construction of such representations. We compare formulas of this kind to other formulas obtained using the geometry of the corresponding flag manifold and in particular give a new proof to the Gordon type identities. 
  We derive an explicit, exactly conformally invariant form for the action for the non-abelian Toda field theory. We demonstrate that the conformal invariance conditions, expressed in terms of the $\beta$-functions of the theory, are satisfied to all orders, and we use our results to obtain a value for the central charge agreeing with previous calculations. 
  Invited talk given at the ``International Workshop on `Symmetry Methods in Physics' in memory of Ya.\ A.\ Smorodinsky, 5--10 July 1993, Dubna, Russia; to appear in the proceedings. In this contribution I present further results on steps towards a Table of Feynman Path Integrals. Whereas the usual path integral solutions of the harmonic oscillator (Gaussian path integrals), of the radial harmonic oscillator (Besselian path integrals), and the (modified) P\"oschl-Teller potential(s) (Legendrian path integrals) are well known and can be performed explicitly by exploiting the convolution properties of the various types, a perturbative method opens other possibilities for calculating path integrals. Here I want to demonstrate the perturbation expansion method for point interactions and boundary problems in path integrals. 
  The incorporation of two- and three-dimensional $\delta$-function perturbations into the path-integral formalism is discussed. In contrast to the one-dimensional case, a regularization procedure is needed due to the divergence of the Green-function $G^{(V)}(\vec x,\vec y;E)$, ($\vec x,\vec y\in\bbbr^2,\bbbr^3$) for $\vec x=\vec y$, corresponding to a potential problem $V(\vec x)$. The known procedure to define proper self-adjoint extensions for Hamiltonians with deficiency indices can be used to regularize the path integral, giving a perturbative approach for $\delta$-function perturbations in two and three dimensions in the context of path integrals. Several examples illustrate the formalism. 
  We study, by means of mirror symmetry, the quantum geometry of the K\"ahler-class parameters of a number of Calabi-Yau manifolds that have $b_{11}=2$. Our main interest lies in the structure of the moduli space and in the loci corresponding to singular models. This structure is considerably richer when there are two parameters than in the various one-parameter models that have been studied hitherto. We describe the intrinsic structure of the point in the (compactification of the) moduli space that corresponds to the large complex structure or classical limit. The instanton expansions are of interest owing to the fact that some of the instantons belong to families with continuous parameters. We compute the Yukawa couplings and their expansions in terms of instantons of genus zero. By making use of recent results of Bershadsky et al. we compute also the instanton numbers for instantons of genus one. For particular values of the parameters the models become birational to certain models with one parameter. The compactification divisor of the moduli space thus contains copies of the moduli spaces of one parameter models. Our discussion proceeds via the particular models $\P_4^{(1,1,2,2,2)}[8]$ and $\P_4^{(1,1,2,2,6)}[12]$. Another example, $\P_4^{(1,1,1,6,9)}[18]$, that is somewhat different is the subject of a companion paper. 
  Galilean transformation relates a physical system under mutually perpendicular uniform magnetic and electric fields to that under uniform magnetic field only. This allows a complete specification of quantum states in the former case in terms of those for the latter. Based on this observation, we consider the Hall effect and the behavior of a neutral composite system in the presence of uniform electromagnetic fields. 
  Local modes and local particles are defined at any point in curved space time as those that most resemble Minkowsky modes at that point. It is shown that the renormalised stress tensor is the difference of energy between the physical vacuum and that defined by these local modes. 
  We construct infinite sets of local conserved charges for the conformal affine Toda model. The technique involves the abelianization of the two-dimensional gauge potentials satisfying the zero-curvature form of the equations of motion. We find two infinite sets of chiral charges and apart from two lowest spin charges all the remaining ones do not possess chiral densities. Charges of different chiralities Poisson commute among themselves. We discuss the algebraic properties of these charges and use the fundamental Poisson bracket relation to show that the charges conserved in time are in involution. Connections to other Toda models are established by taking particular limits. 
  Through the analyses of volume-forms in differentiable manifolds, it is shown that the usual way of defining minimal action principles for field theory on curved space-times is not appropriate on non-riemannian manifolds. An alternative approach, based in a new volume-form, is proposed and confronted with the standard one.   The new volume element is explicitly used in the study of Einstein-Cartan theory of gravity and its relation to string theory, in connection with some recent results on the subject. 
  The established results concerning the BRS cohomology of supersymmetric theories in four space-time dimensions are briefly reviewed. The current status of knowledge concerning supersymmetry anomalies and the possibility that supersymmetry breaks itself through anomalies in local composite operators is then discussed.   It turns out that the simplest allowable supersymmetry anomalies occur only in conjunction with the spontaneous breaking of gauge symmetry. A simple example of such a possible supersymmetry anomaly is presented. 
  Leading corrections to Planck's formula and photon thermodynamics arising from the pair-mediated photon-photon interaction are calculated. This interaction is attractive and causes an increase in occupation number for all modes. Possible consequences, including the role of the cosmic photon gas in structure formation, are considered. 
  We extend earlier calculations of the one-loop contributions to the effective bose Lagrangian in supergravity coupled to chiral matter. We evaluate all logarithmically divergent contributions for arbitrary background scalar fields and space-time metric. We show that, with a judicious choice of gauge fixing and of the definition of the action expansion, much of the result can be absorbed into a redefinition of the metric and a renormalization of the K\"ahler potential. Most of the remaining terms depend on the curvature of the K\"ahler metric. Further simplification occurs in models obtained from superstrings in which the K\"ahler Riemann tensor is covariantly constant. 
  We present the results of an extension of our previous work on large-scale simulations of dynamically triangulated toroidal random surfaces embedded in $R^3$ with extrinsic curvature. We find that the extrinsic-curvature specific heat peak ceases to grow on lattices with more than 576 nodes and that the location of the peak $\lam_c$ also stabilizes. The evidence for a true crumpling transition is still weak. If we assume it exists we can say that the finite-size scaling exponent $\frac {\alpha} {\nu d}$ is very close to zero or negative. On the other hand our new data does rule out the observed peak as being a finite-size artifact of the persistence length becoming comparable to the extent of the lattice. 
  A new solution of the Yang-Baxter equation, that is related to the adjoint representation of the quantum enveloping algebra $U_{q}B_{2}$, is obtained by fusion formulas from a non-standard solution. 
  A new deformation of the of the Poincar\'e group and of the Minkowski space-time is given. From the mathematical point of view this deformation is rather quantum-braided group. Global and local structure of this quantum-braided Poincar\'e group is investigated. A kind of ``quantum metrics'' is introduced in the $q$-Minkowski space. 
  We formulate a new geometrical string on the euclidean lattice. It is possible to find such spin systems with local interaction which reproduce the same surface dynamics.In the three-dimensional case this spin system is a usual Ising ferromagnet with additional diagonal antiferromagnetic interaction and with specially adjusted coupling constants. In the four-dimensional case the spin system coincides with the gauge Ising system with an additional double-plaquette interaction and also with specially tuned coupling constants. We extend this construction to random walks and random hypersurfaces (membrane and p-branes) of high dimensionality. We compare these spin systems with the eight-vertex model and BNNNI models. 
  In two-dimensional dilaton gravity theories, there may exist a global Weyl invariance which makes black hole spurious. If the global invariance and the local Weyl invariance of the matter coupling are intact at the quantum level, there is no Hawking radiation. We explicitly verify the absence of anomalies in these symmetries for the model proposed by Callan, Giddings, Harvey and Strominger. The crucial observation is that the conformal anomaly can be cohomologically trivial and so not truly anomalous in such dilaton gravity models. 
  Considering a multi-dimensional $q$-oscillator invariant under the (non quantum) group $U(n)$, we construct a $q$-deformed Levi-Civita epsilon tensor from the inner product states. The invariance of this $q$-epsilon tensor is shown to yield the quantum group $SL_{q}(n)$ and establishes the relationship of the $U(n)$ invariant $q$-oscillator to quantum groups and quantum group related oscillators. Furthermore the $q$-epsilon tensor provides the connection between $SL_{q}(n)$ and the volume element of the quantum hyper plane. 
  We use the recently discovered universality of Einstein equations in the first order formalism to suggest a positive definite Euclidean action. Possible implications for quantum gravity are considered. We discuss the Hawking and Coleman approach to the vanishing of the cosmological constant by using the new action and find that the cosmological constant is probably zero. A possible scenario for obtaining such an action from superstring field theory is discussed. 
  Second-order equations of motion on a group manifold that appear in a large class of so-called chiral theories are presented. These equations are presented and explicitely solved for cases of semi-simple, finite-dimensional Lie groups. With three figures avaliable from the authors upon request. 
  A model for quantized gravity coupled to matter in the form of a single scalar field is investigated in four dimensions. For the metric degrees of freedom we employ Regge's simplicial discretization, with the scalar fields defined at the vertices of the four-simplices. We examine how the continuous phase transition found earlier, separating the smooth from the rough phase of quantized gravity, is influenced by the presence of scalar matter. A determination of the critical exponents seems to indicate that the effects of matter are rather small, unless the number of scalar flavors is large. Close to the critical point where the average curvature approaches zero, the coupling of matter to gravity is found to be weak. The nature of the phase diagram and the values for the critical exponents suggest that gravitational interactions increase with distance. \vspace{24pt} \vfill 
  Analysis of several gedanken experiments indicates that black hole complementarity cannot be ruled out on the basis of known physical principles. Experiments designed by outside observers to disprove the existence of a quantum-mechanical stretched horizon require knowledge of Planck-scale effects for their analysis. Observers who fall through the event horizon after sampling the Hawking radiation cannot discover duplicate information inside the black hole before hitting the singularity. Experiments by outside observers to detect baryon number violation will yield significant effects well outside the stretched horizon. 
  Starting from a self-dual $SU(\infty)$ Yang-Mills theory in $(2+2)$ dimensions, the Plebanski second heavenly equation is obtained after a suitable dimensional reduction. The self-dual gravitational background is the cotangent space of the internal two-dimensional Riemannian surface required in the formulation of $SU(\infty)$ Yang-Mills theory. A subsequent dimensional reduction leads to the KP equation in $(1+2)$ dimensions after the relationship from the Plebanski second heavenly function, $\Omega$, to the KP function, $u$, is obtained. Also a complexified KP equation is found when a different dimensional reduction scheme is performed . Such relationship between $\Omega$ and $u$ is based on the correspondence between the $SL(2,R)$ self-duality conditions in $(3+3)$ dimensions of Das, Khviengia, Sezgin (DKS) and the ones of $SU(\infty)$ in $(2+2)$ dimensions . The generalization to the Supersymmetric KP equation should be straightforward by extending the construction of the bosonic case to the previous Super-Plebanski equation, found by us in [1], yielding self-dual supergravity backgrounds in terms of the light-cone chiral superfield, $\Theta$, which is the supersymmetric analog of $\Omega$. The most important consequence of this Plebanski-KP correspondence is that $W$ gravity can be seen as the gauge theory of $\phi$-diffeomorphisms in the space of dimensionally-reduced $D=2+2,~SU^*(\infty)$ Yang-Mills instantons. These $\phi$ diffeomorphisms preserve a volume-three-form and are, precisely, the ones which provide the Plebanski-KP correspondence. 
  We conjecture that $W$ gravity can be interpreted as the gauge theory of $\phi$ diffeomorphisms in the space of dimensionally-reduced $D=2+2$ $SU^*(\infty)$ Yang-Mills instantons. These $\phi$ diffeomorphisms preserve a volume-three form and are those which furnish the correspondence between the dimensionally-reduced Plebanski equation and the KP equation in $(1+2)$ dimensions. A supersymmetric extension furnishes super-$W$ gravity. The Super-Plebanski equation generates self-dual complexified super gravitational backgrounds (SDSG) in terms of the super-Plebanski second heavenly form. Since the latter equation yields $N=1~D=4~SDSG$ complexified backgrounds associated with the complexified-cotangent space of the Riemannian surface, $(T^*\Sigma)^c$, required in the formulation of $SU^*(\infty)$ complexified Self-Dual Yang-Mills theory, (SDYM ); it naturally follows that the recently constructed $D=2+2~N=4$ SDSYM theory- as the consistent background of the open $N=2$ superstring- can be embedded into the $N=1~SU^*(\infty)$ complexified Self-Dual-Super-Yang-Mills (SDSYM) in $D=3+3$ dimensions. This is achieved after using a generalization of self-duality for $D>4$. We finally comment on the the plausible relationship between the geometry of $N=2$ strings and the moduli of $SU^*(\infty)$ complexified SDSYM in $3+3$ dimensions. 
  We investigate the thermal equilibrium properties of kinks in a classical $\F^4$ field theory in $1+1$ dimensions. From large scale Langevin simulations we identify the temperature below which a dilute gas description of kinks is valid. The standard dilute gas/WKB description is shown to be remarkably accurate below this temperature. At higher, ``intermediate'' temperatures, where kinks still exist, this description breaks down. By introducing a double Gaussian variational ansatz for the eigenfunctions of the statistical transfer operator for the system, we are able to study this region analytically. In particular, our predictions for the number of kinks and the correlation length are in agreement with the simulations. The double Gaussian prediction for the characteristic temperature at which the kink description ultimately breaks down is also in accord with the simulations. We also analytically calculate the internal energy and demonstrate that the peak in the specific heat near the kink characteristic temperature is indeed due to kinks. In the neighborhood of this temperature there appears to be an intricate energy sharing mechanism operating between nonlinear phonons and kinks. 
  A calculational scheme of quantum-gravitational effects on the physical quantities is proposed. The calculations are performed in 4-$\epsilon$ dimension with $1/N$-expansion scheme, where the Einstein gravity is renormalizable and it has an ultraviolet fixed-point within the 1/N-expansion. In order to perform a consistent perturbation in $4-\epsilon$ dimension, spin-3/2 fields should be adopted as the N matter-fields whose loop-corrections are included in the effective action. After calculating the physical quantities at $4-\epsilon$ dimension, the four-dimensional aspects of them can be seen by taking the limit of $\epsilon=0$. In taking this limit, any higher derivative terms are not introduced as the counter terms since no divergence appears at $\epsilon=0$ in our scheme. According to this approach, we have examined the effective potential of a scalar field to see the possibility of the spontaneous symmetry breaking due to the gravitational loop corrections. 
  We study the problem of constructing N=2 superconformal algebras out of an N=1 affine Lie algebra. Following a recent independent observation of Getzler and the author, we derive a simplified set of N=2 master equations, which we then proceed to solve for the case of sl(2). There is a unique construction for all noncritical values of the level, which can be identified as the Kazama-Suzuki coset associated to the hermitian symmetric space SO(3)/SO(2). We also identify the construction with a generalized parafermionic construction or, after bosonization, with a bosonic construction of the type analyzed by Kazama and Suzuki. A mild generalization of this construction can be associated to any embedding of sl(2) in G. 
  Free and self-interacting scalar fields in the presence of conical singularities are analized in some detail. The role of such a kind of singularities on free and vacuum energy and also on the one-loop effective action is pointed out using $\zeta$-function regularization and heat-kernel techniques. 
  We analyze the quantum hair model proposed recently by Coleman,   Preskill and Wilczek. We give arguments suggesting that the potential hair is expected to be destroyed by the instability of the black-hole. We also discuss the general implications of such arguments on the prospects for formulating a quantum extension of the classical ``no hair'' theorems. 
  We investigate $2d$ sigma-models with a $2+N$ dimensional Minkowski signature target space metric and Killing symmetry, specifically supersymmetrized, and see under which conditions they might lead to corresponding exact string vacua. It appears that the issue relies heavily on the properties of the vector $M_{\mu}$, a reparametrization term, which needs to possess a definite form for the Weyl invariance to be satisfied. We give, in the $n = 1$ supersymmetric case, two non-renormalization theorems from which we can relate the $u$ component of $M_{\mu}$ to the $\beta^G_{uu}$ function. We work out this $(u,u)$ component of the $\beta^G$ function and find a non-vanishing contribution at four loops. Therefore, it turns out that at order $\alpha^{\prime 4}$, there are in general non-vanishing contributions to $M_u$ that prevent us from deducing superstring vacua in closed form. 
  We consider integrable systems that are connected with orthogonal separation of variables in complex Riemannian spaces of constant curvature. An isomorphism with the hyperbolic Gaudin magnet, previously pointed out by one of us, extends to coordinates of this type. The complete classification of these separable coordinate systems is provided by means of the corresponding $L$-matrices for the Gaudin magnet. The limiting procedures (or $\epsilon $ calculus) which relate various degenerate orthogonal coordinate systems play a crucial result in the classification of all such systems. 
  A geometric generalization of first-order Lagrangian formalism is used to analyse a conformal field theory for an arbitrary primary field. We require that global conformal transformations are Noetherian symmetries and we prove that the action functional can be taken strictly invariant with respect to these transformations. In other words, there does not exists a "Chern-Simons" type Lagrangian for a conformally invariant Lagrangian theory. 
  Talk given at the Workshop on "Constraint Theory and Quantization Methods"; Montepulciano, Italy, June 1993 --- Instead of attempting to give a summary or to identify highlights of the workshop, the history of the development of analytical mechanics is outlined, with an emphasis on the themes of \lq natural motion' and the variational principle. 
  We investigate duality transformations in a class of backgrounds with non-Abelian isometries, i.e. Bianchi-type (homogeneous) cosmologies in arbitrary dimensions. Simple duality transformations for the metric and the antisymmetric tensor field, generalizing those known from the Abelian isometry (Bianchi I) case, are obtained using either a Lagrangian or a Hamiltonian approach. Applying these prescriptions to a specific conformally invariant $\s$-model, we show that no dilaton transformation leads to a new conformal background. Some possible ways out of the problem are suggested. 
  Several new results on the multicritical behavior of rectangular matrix models are presented. We calculate the free energy in the saddle point approximation, and show that at the triple-scaling point, the result is the same as that derived from the recursion formulae. In the triple-scaling limit, we obtain the string equation and a flow equation for arbitrary multicritical points. Parametric solutions are also examined for the limit of almost-square matrix models. This limit is shown to provide an explicit matrix model realization of the scaling equations proposed to describe open-closed string theory. 
  We study the spectrum of physical states for higher-spin generalisations of string theory, based on two-dimensional theories with local spin-2 and spin-$s$ symmetries. We explore the relation of the resulting effective Virasoro string theories to certain $W$ minimal models. In particular, we show how the highest-weight states of the $W$ minimal models decompose into Virasoro primaries. 
  We analyze the Gribov problem for $\SU(N)$ and $\U(N)$ Yang-Mills fields on $d$-dimensional tori, $d=2,3,\ldots$. We give an improved version of the axial gauge condition and find an infinite, discrete group $\cG'=\Z^{dr}\rtimes({\Z_2}^{N-1}\rtimes\Z_2)$, where $r=N-1$ for $\GG=\SU(N)$ and $r=N$ for $\GG=\U(N)$, containing all gauge transformations compatible with that condition. This residual gauge group $\cG'$ provides (generically) all Gribov copies and allows to explicitly determine the space of gauge orbits which is an orbifold. Our results apply to Yang-Mills gauge theories either in the Lagrangian approach on $d$-dimensional space-time $T^d$, or in the Hamiltonian approach on $(d+1)$-dimensional space-time $T^d\times \R$. Using the latter, we argue that our results imply a non-trivial structure of all physical states in any Yang-Mills theory, especially if also matter fields are present. 
  We solve Schr\"odinger's equation for the ground-state of {\it four}-dimensional Yang-Mills theory as an expansion in inverse powers of the coupling. Expectation values computed with the leading order approximation are reduced to a calculation in {\it two}-dimensional Yang-Mills theory which is known to confine. Consequently the Wilson loop in the four-dimensional theory obeys an area law to leading order and the coupling becomes infinite as the mass-scale goes to zero. 
  Many field theories of physical interest have configuration spaces consisting of disconnected components. Quantum mechanical amplitudes are then expressed as sums over these components. We use the Faddeev-Popov approach to write the terms in this topological expansion as moduli space integrals. A cut-off is needed when these integrals diverge. This introduces a dependence on the choice of parametrisation of configuration space which must be removed if the theory is to make physical sense. For theories that have a local symmetry this also leads to a breakdown in BRST invariance. We discuss in detail the cases of Bosonic Strings and Yang-Mills theory, showing how this arbitrariness may be removed by the use of a counter-term in the former case, and by compactification on $S\sp 4$ in the latter. 
  We discuss a model for interferometric GW antennas without dissipative or active elements.   It is predicted that the even and odd coherent states may play an alternative role to squeezed vacuum states in reducing the optimal power of the input laser. 
  Coherent states on the quantum group $SU_q(2)$ are defined by using harmonic analysis and representation theory of the algebra of functions on the quantum group. Semiclassical limit $q\rightarrow 1$ is discussed and the crucial role of special states on the quantum algebra in an investigation of the semiclassical limit is emphasized. An approach to $q$-deformation as a $q$-Weyl quantization and a relavence of contact geometry in this context is pointed out. Dynamics on the quantum group parametrized by a real time variable and corresponding to classical rotations is considered. 
  The renormalization group flow in two--dimensional field theories that are coupled to gravity is discussed at the example of the sine-Gordon model. In order to derive the phase diagram in agreement with the matrix model results, it is necessary to generalize the theory of David, Distler and Kawai. 
  The theory of embedded random surfaces, equivalent to two--dimensional quantum gravity coupled to matter, is reviewed, further developed and partly generalized to four dimensions. It is shown that the action of the Liouville field theory that describes random surfaces contains terms that have not been noticed previously. These terms are used to explain the phase diagram of the Sine--Gordon model coupled to gravity, in agreement with recent results from lattice computations. It is also demonstrated how the methods of two--dimensional quantum gravity can be applied to four--dimensional Euclidean gravity in the limit of infinite Weyl coupling. Critical exponents are predicted and an analog of the ``$c=1$ barrier'' of two--dimensional gravity is derived. 
  Mirror Symmetry, Picard-Fuchs equations and instanton corrected Yukawa couplings are discussed within the framework of toric geometry. It allows to establish mirror symmetry of Calabi-Yau spaces for which the mirror manifold had been unavailable in previous constructions. Mirror maps and Yukawa couplings are explicitly given for several examples with two and three moduli. 
  We show that the usual fixed point for 3-d rigid string with topological term appears to be a trivial one, consisting of two decoupled conformal field theories. We further argue that by involving an additional term allowed by symmetries and thus generated by RG, the theory appears to exhibit a new fixed point with expected symmetriy. The new fixed point is studied in the weak- and string coupling limit. 
  A large class of new 4-D superstring vacua with non-trivial/singular geometries, spacetime supersymmetry and other background fields (axion, dilaton) are found. Killing symmetries are generic and are associated with non-trivial dilaton and antisymmetric tensor fields. Duality symmetries preserving N=2 superconformal invariance are employed to generate a large class of explicit metrics for non-compact 4-D Calabi-Yau manifolds with Killing symmetries. 
  Inspired by Polyakov's original formulation of quantum Liouville theory through functional integral, we analyze perturbation expansion around a classical solution. We show the validity of conformal Ward identities for puncture operators and prove that their conformal dimension is given by the classical expression. We also prove that total quantum correction to the central charge of Liouville theory is given by one-loop contribution, which is equal to 1. Applied to the bosonic string, this result ensures the vanishing of total conformal anomaly along the lines different from those presented by KPZ and Distler-Kawai. 
  Motivated by the similarity between CSW theory and the Chern Simons state for General Relativity in the Ashtekar variables, we explore what the universe would look like if it were in a state corresponding to a 3D TQFT. We end up with a construction of propagating ststes for parts of the universe and a Hilbert space corresponding to a certain approximation. The construction avoids path integrals, using instead recombination diagrams in a certain tensor category. 
  Starting from a $D=3$, $N=4$ supersymmetric theory for matter fields, a twist with a Grassmann parity change is defined which maps the theory into a gauge fixed, abelian $BF$ theory on curved 3-manifolds. After adding surface terms to this theory, the twist is seen to map the resulting supersymmetric action to two uncoupled copies of the gauge fixed Chern-Simons action. In addition, we give a map which takes the $BF$ and Chern-Simons theories into Donaldson-Witten TQFT's. A similar construction, but with $N=2$ supersymmetry, is given in two dimensions. 
  After adding seven auxiliary scalar fields, the action for ten-dimensional super-Yang-Mills contains an equal number of bosonic and fermionic non-gauge fields. Besides being manifestly Lorentz and gauge-invariant, this action contains nine spacetime supersymmetries whose algebra closes off-shell. Octonions provide a convenient notation for displaying these symmetries. 
  An action for the ten-dimensional Green-Schwarz superstring with N=2 worldsheet superconformal invariance has recently been used to calculate superstring scattering amplitudes and prove their finiteness. In this paper, it is shown that the N=2 stress-energy tensor for this Green-Schwarz action can be constructed out of the stress-energy tensor and ghosts of the Neveu- Schwarz-Ramond action by the standard twisting procedure. In other words, a field redefinition is found from the GS matter fields into the NSR matter and ghost fields which transforms the matter part of the two fermionic GS super- conformal generators into the $b$ ghost and shifted BRST current of the NSR string. In light-cone gauge, this field redefinition reduces to the usual one relating the light-cone GS and NSR fields.   Although this proves the equivalence of physical vertex operators in the two superstring formalisms, multiloop amplitudes are easier to calculate using the Green-Schwarz formalism since manifest spacetime supersymmetry eliminates the need for spin cuts, GSO projections, and cutoffs in moduli space. 
  We investigate Thirring-like models containing fermionic and scalar fields propagating in 2-dimensional space time. The corresponding conformal algebra is studied and we disprove a conjecture relating the finite size effects to the central charge. Some new results concerning the fermionic determinant on the torus with chirally twisted boundary conditions and a chemical potential are presented. In particular we show how the thermodynamics of the Thirring model depends on the current-current interaction. 
  We obtain cosmological four dimensional solutions of the low energy effective string theory by reducing a five dimensional black hole, and black hole--de Sitter solution of Einstein gravity down to four dimensions. The appearance of a cosmological constant in the five dimensional Einstein--Hilbert action produces a special dilaton potential in the four dimensional effective string action. Cosmological scenarios implemented by our solutions are discussed. 
  Two local macros are included (gothic.sty and fleqn.sty) 
  String theories inspire a new formalism for their low-energy limits. In this approach to these field theories, spacetime duality and stringy left/right handedness are manifest. Enlarged tangent-space symmetries allow the different fields (graviton, axion, Yang-Mills) to be treated as a single multiplet, even in the bosonic case, except for the dilaton (multiplet), which appears as the measure. (Based on a talk given at Strings '93, May 24-29, Berkeley, CA. A section added after the talk discusses modifications for nonabelian Yang-Mills.) 
  First, it is proven that the three main operator-approaches to the quantum Liouville exponentials --- that is the one of Gervais-Neveu (more recently developed further by Gervais), Braaten-Curtright-Ghandour-Thorn, and Otto-Weigt --- are equivalent since they are related by simple basis transformations in the Fock space of the free field depending upon the zero-mode only. Second, the GN-G expressions for quantum Liouville exponentials, where the $U_q(sl(2))$ quantum group structure is manifest, are shown to be given by q-binomial sums over powers of the chiral fields in the $J=1/2$ representation. Third, the Liouville exponentials are expressed as operator tau-functions, whose chiral expansion exhibits a q Gauss decomposition, which is the direct quantum analogue of the classical solution of Leznov and Saveliev. It involves q exponentials of quantum group generators with group "parameters" equal to chiral components of the quantum metric. Fourth, we point out that the OPE of the $J=1/2$ Liouville exponential provides the quantum version of the Hirota bilinear equation. 
  Some old and new evidence for the existence of the string (planar random surfaces) representation of multicolour QCD are reviewed. They concern the random surface representation of the strong coupling expansion in lattice multicolour gauge theory in any dimension. Our old idea of modified strong coupling expansion in terms of planar random surfaces, valid for the physical weak coupling phase of the four-dimensional QCD, is explained in detail. Some checks of the validity of this expansion are proposed. (The lectures given in the Trieste Spring School and Workshop-1993 on String Theory). 
  We prove that the Jacobi identity for the generalized Poisson bracket is satisfied in the generalization of Heisenberg picture quantum mechanics recently proposed by one of us (SLA). The identity holds for any combination of fermionic and bosonic fields, and requires no assumptions about their mutual commutativity. 
  Starting from free charged fermions we give equivalent definitions of the $n\/$-component KP hierarchy, in terms of $\tau\/$-functions $\tau_\alpha\/$ (where $\alpha \in M =\/$ root lattice of $sl_n\/$), in terms of $n \times n\/$ matrix valued wave functions $W_\alpha(\alpha\in M)\/$, and in terms of pseudodifferential wave operators $P_\alpha(\alpha\in M)\/$. These imply the deformation and the zero curvature equations. We show that the 2-component KP hierarchy contains the Davey-Stewartson system and the $n\geq3\/$ component KP hierarchy continues the $n\/$-wave interaction equations. This allows us to construct theis solutions. 
  In this paper we analyze the structure of some subalgebras of quantized enveloping algebras corresponding to unipotent and solvable subgroups of a simple Lie group G. These algebras have the non--commutative structure of iterated algebras of twisted polynomials with a derivation, an object which has often appeared in the general theory of non-commutative rings. In particular, we find maximal dimensions of their irreducible representations. Our results confirm the validity of the general philosophy that the representation theory is intimately connected to the Poisson geometry. 
  Consistency of quantum mechanics in black hole physics requires unusual Lorentz transformation properties of the size and shape of physical systems with momentum beyond the Planck scale. A simple parton model illustrates the kind of behavior which is needed. It is then shown that conventional fundamental string theory shares these features. 
  The structure of physical operators and states of the unified constraint dynamics is studied. The genuine second--class constraints encoded are shown to be the superselection operators. The unified constrained dynamics is established to be physically--equivalent to the standard BFV--formalism with constraints split. 
  We extend the Poisson bracket from a Lie bracket of phase space functions to a Lie bracket of functions on the space of canonical histories and investigate the resulting algebras. Typically, such extensions define corresponding Lie algebras on the space of Lagrangian histories via pull back to a space of partial solutions. These are the same spaces of histories studied with regard to path integration and decoherence. Such spaces of histories are familiar from path integration and some studies of decoherence. For gauge systems, we extend both the canonical and reduced Poisson brackets to the full space of histories. We then comment on the use of such algebras in time reparameterization invariant systems and systems with a Gribov ambiguity, though our main goal is to introduce concepts and techniques for use in a companion paper. 
  The representation theory of the group U(1,q) is discussed in detail because of its possible application in a quaternion version of the Salam-Weinberg theory.   As a consequence, from purely group theoretical arguments we demonstrate that the eigenvalues must be right-eigenvalues and that the only consistent scalar products are the complex ones. We also define an explicit quaternion tensor product which leads to a set of additional group representations for integer ``spin''. 
  We consider a hierarchy of many particle systems on the line with polynomial potentials separable in parabolic coordinates. Using the Lax representation, written in terms of $2\times 2$ matrices for the whole hierarchy, we construct the associated linear $r$-matrix algebra with the $r$-matrix dependent on the dynamical variables. A dynamical Yang-Baxter equation is discussed. 
  It is shown that the local quantum field theory of the chiral energy- momentum tensor with central charge $c=1$ coincides with the gauge invariant subtheory of the chiral $SU(2)$ current algebra at level 1, where the gauge group is the global $SU(2)$ symmetry. At higher level, the same scheme gives rise to $W$-algebra extensions of the Virasoro algebra. 
  Some facts about von Neumann algebras and finite index inclusions of factors are viewed in the context of local quantum field theory. The possibility of local fields intertwining superselection sectors with braid group statistics is explored. Conformal embeddings and coset models serve as examples. The associated symmetry concept is pointed out. 
  In this paper we present a systematic study of $W$ algebras from the Hamiltonian reduction point of view. The Drinfeld-Sokolov (DS) reduction scheme is generalized to arbitrary $sl_2$ embeddings thus showing that a large class of W algebras can be viewed as reductions of affine Lie algebras. The hierarchies of integrable evolution equations associated to these classical W algebras are constructed as well as the generalized Toda field theories which have them as Noether symmetry algebras. The problem of quantising the DS reductions is solved for arbitrary $sl_2$ embeddings and it is shown that any W algebra can be embedded into an affine Lie algebra. This also provides us with an algorithmic method to write down free field realizations for arbitrary W algebras. Just like affine Lie algebras W algebras have finite underlying structures called `finite W algebras'. We study the classical and quantum theory of these algebras, which play an important role in the theory of ordinary W algebras, in detail as well as some aspects of their representation theory. The symplectic leaves (or W-coadjoint orbits) associated to arbitrary finite W algebras are determined as well as their realization in terms of bosoic oscillators. Apart from these technical aspects we also review the potential applications of W symmetry to string theory, 2-dimensional critical phenomena, the quantum Hall effect and solitary wave phenomena. This work is based on the Ph.D. thesis of the author. 
  The thermodynamic Bethe Ansatz equations that have been proposed to describe massive integrable deformations of the coset conformal field theories $g_k\times g_l/g_{k+l}$ are shown to result directly by applying the usual thermodynamic Bethe Ansatz arguments to the trigonometric $S$-matrices for the algebras $g=a_{m-1}$. 
  We introduce an addition law for the usual quantum matrices $A(R)$ by means of a coaddition $\underline{\Delta} t=t\otimes 1+1\otimes t$. It supplements the usual comultiplication $\Delta t=t\otimes t$ and together they obey a codistributivity condition. The coaddition does not form a usual Hopf algebra but a braided one. The same remarks apply for rectangular $m\times n$ quantum matrices. As an application, we construct left-invariant vector fields on $A(R)$ and other quantum spaces. They close in the form of a braided Lie algebra. As another application, the wave-functions in the lattice approximation of Kac-Moody algebras and other lattice fields can be added and functionally differentiated. 
  In string theory, there are no continuous global symmetries. Discrete symmetries frequently appear, and these can often be understood as unbroken subgroups of larger, spontaneously broken gauge symmetries (discrete gauge symmetries). In cases which have been studied previously, anomalies in these symmetries could always be cancelled by a Green-Schwarz mechanism. In the present work, we describe results of an investigation of a large number of $\ZZ_3$ orbifold models with two and three Wilson lines. We find that the discrete gauge anomalies can always be cancelled.} 
  We first extend the Peierls algebra of gauge invariant functions from the space ${\cal S}$ of classical solutions to the space ${\cal H}$ of histories used in path integration and some studies of decoherence. We then show that it may be generalized in a number of ways to act on gauge dependent functions on ${\cal H}$. These generalizations (referred to as class I) depend on the choice of an ``invariance breaking term," which must be chosen carefully so that the gauge dependent algebra is a Lie algebra. Another class of invariance breaking terms is also found that leads to an algebra of gauge dependent functions, but only on the space ${\cal S}$ of solutions. By the proper choice of invariance breaking term, we can construct a generalized Peierls algebra that agrees with any gauge dependent algebra constructed through canonical or gauge fixing methods, as well as Feynman and Landau ``gauge." Thus, generalized Peierls algebras present a unified description of these techniques. We study the properties of generalized Peierls algebras and their pull backs to spaces of partial solutions and find that they may posses constraints similar to the canonical case. Such constraints are always first class, and quantization may proceed accordingly. 
  The theory of vertex algebras constitutes a mathematically rigorous axiomatic formulation of the algebraic origins of conformal field theory. In this context Borcherds algebras arise as certain ``physical'' subspaces of vertex algebras. The aim of this review is to give a pedagogical introduction into this rapidly-developing area of mathemat% ics. Based on the machinery of formal calculus we present the axiomatic definition of vertex algebras. We discuss the connection with conformal field theory by deriving important implications of these axioms. In particular, many explicit calculations are presented to stress the eminent role of the Jacobi identity axiom for vertex algebras. As a class of concrete examples the vertex algebras associated with even lattices are constructed and it is shown in detail how affine Lie algebras and the fake Monster Lie algebra naturally appear. This leads us to the abstract definition of Borcherds algebras as generalized Kac-Moody algebras and their basic properties. Finally, the results about the simplest generic Borcherds algebras are analysed from the point of view of symmetry in quantum theory and the construction of the Monster Lie algebra is sketched. 
  We consider the evolution of circular string loops in power law expanding universes represented by a spatially flat Friedman-Robertson-Walker metric with scale factor $a(t)\propto t^p$ where $t$ is the cosmic time and $p\geq 0$. Our main result is the existence of a "magic" power $p_m=3+2\sqrt{2}$. In spacetimes with $p<p_m$ a circular string expands either forever or to a maximal radius and then contracts until it collapses into a point (black hole). For $p>p_m$, however, we find additional types of solutions. They include configurations which contract from a positive initial radius to a minimal one and then expand forever. Their existence we interpret as an indication for the presence of a finite potential barrier. Equivalently the new solutions signal string nucleation and tunneling, phenomena recently shown to occur in de Sitter space. 
  We classify positive energy representations with finite degeneracies of the Lie algebra $W_{1+\infty}\/$ and construct them in terms of representation theory of the Lie algebra $\hatgl ( \infty R_m )\/$ of infinite matrices with finite number of non-zero diagonals over the algebra $R_m = \C [ t ] / ( t^{m + 1} )\/$.  The unitary ones are classified as well. Similar results are obtained for the sin-algebras. 
  We show that nonabelian duality is not a symmetry of a conformal field theory, but rather a symmetry between different theories. We expose a nonlocal symmetry of nonabelian dual theories. We show how, in the case with vanishing isotropy, it can be used to find the inverse dual transformation. Finally, we consider a number of new examples. 
  It is shown that the models of 2D Liouville Gravity, 2D Black Hole- and $R^2$-Gravity are {\em embedded} in the Katanaev-Volovich model of   2D NonEinsteinian Gravity. Different approaches to the formulation of a quantum theory for the above systems are then presented: The Dirac constraints can be solved exactly in the momentum representation, the path integral can be integrated out, and the constraint algebra can be {\em explicitely} canonically abelianized, thus allowing also for a (superficial) reduced phase space quantization. Non--trivial dynamics are obtained by means of time dependent gauges. All of these approaches lead to the {\em same} finite dimensional quantum mechanical system. 
  Preliminary investigations of the topological phase of string theory along the lines of a (restricted) $\dot{w}_{\infty}$ non-linear sigma model are provided. Gauge fixing the w gravity gauge fields by preserving a geometric identity Lorenz covariantizes the w-particle and gauge covariantizes the YM. The notion of foliation ghosts is introduced. Connection between $\dot{w}_{\infty}$ and homotopy associative algebras is indicated. The second quantized twistor space of string theory is constructed and compared with the first quantised one. Speculations about the relevance of the massive string fields to the compactification of the bosonic string down to four dimensions times the standard model group as well as to the solution of the background independence problem and the integration over moduli space problem are also provided. 
  This work can be considered as a continuation of our previous one (J.Phys., 26 (1993) 313), in which an explicit form of coherent states (CS) for all SU(N) groups was constructed by means of representations on polynomials. Here we extend that approach to any SU(l,1) group and construct explicitly corresponding CS. The CS are parametrized by dots of a coset space, which is, in that particular case, the open complex ball $CD^{l}$. This space together with the projective space $CP^{l}$, which parametrizes CS of the SU(l+1) group, exhausts all complex spaces of constant curvature. Thus, both sets of CS provide a possibility for an explicit analysis of the quantization problem on all the spaces of constant curvature. 
  We explain, in a slightly modified form, an old construction allowing to reformulate the U(N) gauge theory defined on a D-dimensional lattice as a theory of lattice strings (a statistical model of random surfaces). The world surface of the lattice string is allowed to have pointlike singularities (branch points) located not only at the sites of the lattice, but also on its links and plaquettes. The strings become noninteracting when $N\to\infty$. In this limit the statistical weight a world surface is given by exp[ $-$ area] times a product of local factors associated with the branch points. In $D=4$ dimensions the gauge theory has a nondeconfining first order phase transition dividing the weak and strong coupling phase. From the point of view of the string theory the weak coupling phase is expected to be characterized by spontaneous creation of ``windows'' on the world sheet of the string. 
  The critical Boltzmann weights for lattice analogues of the $N=2$ superconformal coset models $\frac{G_1 \times SO(dim(G/H))}{H}$ were given in \cite{nick}. In this paper Bethe Ansatz methods are employed to calculate the spectrum of the transfer matrix obtained from these Boltzmann weights. {}From this the central charge and conformal weights are obtained by calculating finite-size corrections to the free energy per site. The results agree with those obtained from the superconformal model. 
  Recently derived general formal solutions of a BRST quantization on inner product spaces of irreducible Lie group gauge theories are applied to trivial models and relativistic particle models for particles with spin 0, 1/2 and 1. In the process general quantization rules are discovered which make the formal solutions exact. The treatment also give evidence that the formal solutions are directly generalizable to theories with graded gauge symmetries. For relativistic particles reasonable results are obtained although there exists no completely Lorentz covariant quantization of the coordinate and momenta on inner product spaces. There are two inequivalent procedures depending on whether or not the time coordinate is quantized with positive or indefinite metric states. The latter is connected to propagators. 
  Recent formal solutions of BRST quantization on inner product spaces within the operator method are shown to lead to an unexpected interpretation of the conventional path integral formulation. The relation between the Hamiltonians in the two formulations is nontrivial. For the operator method the correspondence requires certain quantum rules which make the formal solutions exact, and for the path integral the correspondence yields a precise connection between boundary conditions and the choice of gauge fixing. 
  Recent results of BRST quantization on inner product spaces are reviewed. It is shown how relativistic particle models may be quantized with finite norms and that the relation between the operator method and the conventional path integral treatments is nontrivial. 
  The saddle point equation described by the eigenvalues of N by N Hermitian matrices is analized for a finite N case and the scaling relation for the large N is considered. The critical point and the critical exponents of matrix model are obtained by the finite N scaling. One matrix model and two matrix model are studied in detail. Small N behavior for n-Ising model on a random surface is investigated. 
  The Schwartz kernel of the multiplication operation on a quantum torus is shown to be the distributional boundary value of a classical multivariate theta function. The kernel satisfies a Schr\"odinger equation in which the role of time is played by the deformation parameter $\hbar$ and the role of the hamiltonian by a Poisson structure. At least in some special cases, the kernel can be written as a sum of products of single-variable theta functions. 
  The study of the integrability properties of the N=2 Landau- Ginzburg models leads naturally to a graph generalization of the Yang-Baxter equation which synthetizes the well known vertex and RSOS Yang-Baxter equations. A non trivial solution of this equation is found for the $t_2$ perturbation of the A-models, which turns out to be intimately related to the Boltzmann weights of a Chiral- Potts model. 
  The $d = 2$ string admits a black hole solution and also a singular solution when tachyon back reaction is included. It is of importance to know if the former solution can evolve into a later one. An explicit solution describing this process is difficult to obtain. We present here a scenario in which such an evolution is very likely to occur. In essence, it takes place when a derivative discontinuity is seeded in the dilaton field by an incident tachyon pulse. An application of this scenario to $1 + 1$ dimensional toy models suggests that a black hole can evolve into a massive remnant, strengthening its candidacy for the end state of a black hole. 
  A new construction is presented for point interactions (PI) and generalised point interactions (GPI). The construction is an inverse scattering procedure, using integral transforms suggested by the required scattering theory. The usual class of PI in 3 dimensions (i.e. the self adjoint extensions of the Laplacian on the domain of smooth functions compactly supported away from the origin) is reconstructed. In addition a 1-parameter family of GPI models termed resonance point interactions (RPI) is constructed, labelled by $M$. The case $M<0$ coincides with a special case of a known GPI model; the case $M>0$ appears to be new. In both cases, the Hilbert space of states must be extended, for $M<0$, a larger Hilbert space is required, whilst for $M>0$, the Hilbert space is extended to a Pontryagin space. In the latter case, the space of physical states is identified as a positive definite invariant subspace. Complete M{\o}ller wave operators are constructed for the models considered, using a two space formalism where necessary, which confirm that the PI and RPI models exhibit the required scattering theory. The physical interpretation of RPI as models for quantum mechanical systems exhibiting zero energy resonances is described. 
  Boundary conditions may change the phase diagram of non-equilibrium statistical systems like the one-dimensional asymmetric simple exclusion process with and without particle number conservation. Using the quantum Hamiltonian approach, the model is mapped onto an XXZ quantum chain and solved using the Bethe ansatz. This system is related to a two-dimensional vertex model in thermal equilibrium. The phase transition caused by a point-like boundary defect in the dynamics of the one-dimensional exclusion model is in the same universality class as a continous (bulk) phase transition of the two-dimensional vertex model caused by a line defect at its boundary. (hep-th/yymmnnn) 
  The multilevel geometrically--covariant generalization of the field--antifield BV--formalism is suggested. The structure of quantum generating equations and hypergauge conditions is studied in details. The multilevel formalism is established to be physically--equivalent to the standard BV--version. 
  We study fermionic one-matrix, two-matrix and $D$-dimensional gauge invariant matrix models. In all cases we derive loop equations which unambiguously determine the large-$N$ solution. For the one-matrix case the solution is obtained for an arbitrary interaction potential and turns out to be equivalent to the one for the Hermitean one-matrix model with a logarithmic potential and, therefore, belongs to the same universality class. The explicit solutions for the fermionic two-matrix and $D$-dimensional matrix models are obtained at large $N$ (or in the spherical approximation) for the quadratic potential. 
  Some quantum mechanical potentials, singular at short distances, lead to ultraviolet divergences when used in perturbation theory. Exactly as in quantum field theories, but much simpler, regularization and renormalization lead to finite physical results, which compare correctly to the exact ones. The Dirac delta potential, because of its relevance to triviality, and the Aharonov-Bohm potential, because ot its relevance to anyons, are used as examples here. 
  Using the fusion principle of Bauer et al. we give explicit expressions for some null vectors in the highest weight representations of the \bc algebra in two different forms. These null vectors are the generalization of the Virasoro ones described by Benoit and Saint-Aubin and analogues of the $W_3$ ones constructed by Bowcock and Watts. We find connection between quantum Toda models and the fusion method. 
  The issue of gauge invariances in the sigma model formalism is discussed at the free and interacting level. The problem of deriving gauge invariant interacting equations can be addressed using the proper time formalism. This formalism is discussed, both for point particles and strings. The covariant Klein Gordon equation arises in a geometric way from the boundary terms. This formalism is similar to the background independent open string formalism introduced by Witten. 
  The string equations of motion and constraints are solved for a ring shaped Ansatz in cosmological and black hole spacetimes. In FRW universes with arbitrary power behavior [$R(X^0) = a\;|X^0|^{\a}\, $], the asymptotic form of the solution is found for both $X^0 \to 0$ and $X^0 \to \infty$ and we plot the numerical solution for all times. Right after the big bang ($X^0 = 0$), the string energy decreasess as $ R(X^0)^{-1} $ and the string size grows as $ R(X^0) $ for $ 0 < \a < 1 $ and as $ X^0 $ for $ \a < 0 $ and $ \a > 1 $. Very soon [ $ X^0 \sim 1 $] , the ring reaches its oscillatory regime with frequency equal to the winding and constant size and energy. This picture holds for all values of $ \a $ including string vacua (for which, asymptotically, $ \a = 1$). In addition, an exact non-oscillatory ring solution is found. For black hole spacetimes (Schwarzschild, Reissner-Nordstr\oo m and stringy), we solve for ring strings moving towards the center. Depending on their initial conditions (essentially the oscillation phase), they are are absorbed or not by Schwarzschild black holes. The phenomenon of particle transmutation is explicitly observed (for rings not swallowed by the hole). An effective horizon is noticed for the rings. Exact and explicit ring solutions inside the horizon(s) are found. They may be interpreted as strings propagating between the different universes described by the full black hole manifold. 
  Correlation functions for holonomic fields on the Poincare' disk are analyzed. The two point functions are shown to be expressible in terms of Painleve' functions of type VI. 
  We apply the method of collective coordinate quantization to a model of solitons in two spacetime dimensions with a global $U(1)$ symmetry. In particular we consider the dynamics of the charged states associated with rotational excitations of the soliton in the internal space and their interactions with the quanta of the background field (mesons). By solving a system of coupled saddle-point equations we effectively sum all tree-graphs contributing to the one-point Green's function of the meson field in the background of a rotating soliton. We find that the resulting one-point function evaluated between soliton states of definite $U(1)$ charge exhibits a pole on the meson mass shell and we extract the corresponding S-matrix element for the decay of an excited state via the emission of a single meson using the standard LSZ reduction formula. This S-matrix element has a natural interpretation in terms of an effective Lagrangian for the charged soliton states with an explicit Yukawa coupling to the meson field. We calculate the leading-order semi-classical decay width of the excited soliton states discuss the consequences of these results for the hadronic decay of the $\Delta$ resonance in the Skyrme model. 
  Two solutions of stringy gravity in three and four dimensions which admit interpretation as a black hole and a black string, respectively, are discussed. It is demonstrated that they are exact WZWN nonlinear sigma models to all orders in the inverse string tension, and hence represent exact conformal field theories on the world-sheet. Furthermore, since the dilaton for these two solutions is constant, they also solve the equations of motion of standard GR with a minimally coupled three form field strength. (Based on a talk presented at the Conference on Quantum Aspects of Black Holes, U. of California, Santa Barbara CA June 21-27 '93, and a poster presented at 5th Canadian Conference on General Relativity and Relativistic Astrophysics, Waterloo, Ont., May 13-15, 1993.) 
  Consistent and covariant Lorentz and diffeomorphism anomalies are investigated in terms of the geometry of the universal bundle for gravity. This bundle is explicitly constructed and its geometrical structure will be studied. By means of the local index theorem for families of Bismut and Freed the consistent gravitational anomalies are calculated. Covariant gravitational anomalies are shown to be related with secondary characteristic classes of the universal bundle and a new set of descent equations which also contains the covariant Schwinger terms is derived. The relation between consistent and covariant anomalies is studied. Finally a geometrical realization of the gravitational BRS, anti-BRS transformations is presented which enables the formulation of a kind of covariance condition for covariant gravitational anomalies. 
  Consideration of the model of the relativistic particle with curvature and torsion in the three-dimensional space-time shows that the squaring of the primary constraints entails a wrong result. The complete set of the Hamiltonian constraints arising here correspond to another model with an action similar but not identical with the initial action. 
  The Nested Bethe Ansatz is generalized to open boundary conditions. This is used to find the exact eigenvectors and eigenvalues of the $A_{n-1}$ vertex model with fixed open boundary conditions and the corresponding $SU_{q}(n)$ invariant hamiltonian.   The Bethe Ansatz equations obtained are solved in the thermodynamic limit giving the vertex model free energy and the hamiltonian ground state energy including the corresponding boundary contributions. 
  The inflationary scenarios suggested by the duality properties of string cosmology in the Brans-Dicke (or String) frame are shown to correspond to accelerated contraction (deflation) when   Weyl-transformed to the Einstein frame. We point out that the basic virtues of inflation (solving the flatness and horizon problems, amplifying vacuum fluctuations, etc.) have physically equivalent counterparts in the deflationary (Einstein-frame) picture. This could be the answer to some objections recently raised to superstring cosmology. 
  On basis of generalized 6j-symbols we give a formulation of topological quantum field theories for 3-manifolds including observables in the form of coloured graphs. It is shown that the 6j-symbols associated with deformations of the classical groups at simple even roots of unity provide examples of this construction. Calculational methods are developed which, in particular, yield the dimensions of the state spaces as well as a proof of the relation, previously announced for the case of $SU_q(2)$ by V.Turaev, between these models and corresponding ones based on the ribbon graph construction of Reshetikhin and Turaev. 
  By explicitly eliminating all gauge degrees of freedom in the $3+1$-gauge description of a classical relativistic (open) membrane moving in $\Real^3$ we derive a $2+1$-dimensional nonlinear wave equation of Born-Infeld type for the graph $z(t,x,y)$ which is invariant under the Poincar\'e group in four dimensions. Alternatively, we determine the world-volume of a membrane in a covariant way by the zeroes of a scalar field $u(t,x,y,z)$ obeying a homogeneous Poincar\'e-invariant nonlinear wave-equation. This approach also gives a simple derivation of the nonlinear gas dynamic equation obtained in the light-cone gauge. 
  We describe a representation of the $q$--hypergeometric functions of one variable in terms of correlators of vertex operators made out of free scalar fields on the Riemann sphere. 
  We give a generally covariant description, in the sense of symplectic geometry, of gauge transformations in Batalin-Vilkovisky quantization. Gauge transformations exist not only at the classical level, but also at the quantum level, where they leave the action-weighted measure $d\mu_S = d\mu e^{2S/\hbar}$ invariant. The quantum gauge transformations and their Lie algebra are $\hbar$-deformations of the classical gauge transformation and their Lie algebra. The corresponding Lie brackets $[ , ]^q$, and $[ , ]^c$, are constructed in terms of the symplectic structure and the measure $d\mu_S$. We discuss closed string field theory as an application. 
  We complete the proof of bosonization of noninteracting nonrelativistic fermions in one space dimension by deriving the bosonized action using $W_\infty$ coherent states in the fermion path-integral. This action was earlier derived by us using the method of coadjoint orbits. We also discuss the classical limit of the bosonized theory and indicate the precise nature of the truncation of the full theory that leads to the collective field theory. 
  We discuss multivariable invariants of colored links associated with the $N$-dimensional root of unity representation of the quantum group. The invariants for $N>2$ are generalizations of the multi-variable Alexander polynomial. The invariants vanish for disconnected links. We review the definition of the invariants through (1,1)-tangles. When $(N,3)=1$ and $N$ is odd, the invariant does not vanish for the parallel link (cable) of the knot $3_1$, while the Alexander polynomial vanishes for the cable link. 
  We investigate the seven-sphere as a group-like manifold and its extension to a Kac-Moody-like algebra. Covariance properties and tensorial composition of spinors under $S^7$ are defined. The relation to Malcev algebras is established. The consequences for octonionic projective spaces are examined. Current algebras are formulated and their anomalies are derived, and shown to be unique (even regarding numerical coefficients) up to redefinitions of the currents. Nilpotency of the BRST operator is consistent with one particular expression in the class of (field-dependent) anomalies. A Sugawara construction is given. 
  A star-product formalism describing deformations of the standard quantum mechanical harmonic oscillator is introduced. A number of existing generalized oscillators occur as particular choises of star-products between the elements of the ordinary oscillator algebra. Star dynamics and coherent states are introduced and studied. 
  In the analytic-Bargmann representation associated with the harmonic oscillator and spin coherent states, the wavefunction as entire complex functions can be factorized in terms of their zeros in a unique way. The Schr\"odinger equation of motion for the wavefunction is turned to a system of equations for its zeros. The motion of these zeros as a non-linear flow of points is studied and interpreted for linear and non-linear bosonic and spin Hamiltonians. Attention is given to the study of the zeros of the Jaynes-Cummings model and to its finite analoque. Numerical solutions are derived and discussed. 
  The classical $R$-matrix structure for the $n$-particle Calogero-Moser models with (type IV) elliptic potentials is investigated. We show there is no momentum independent $R$-matrix (without spectral parameter) when $n\ge4$. The assumption of momentum independence is sufficient to reproduce the dynamical $R$-matrices of Avan and Talon for the type I,II,III degenerations of the elliptic potential. The inclusion of a spectral parameter enables us to find $R$-matrices for the general elliptic potential. 
  We show that a deformation of the Heisenberg algebra which depends on a dimensionful parameter $\kappa$ is the algebraic structure which underlies the generalized uncertainty principle in quantum gravity. The deformed algebra and therefore the form of the generalized uncertainty principle are fixed uniquely by rather simple assumptions. The string theory result is reproduced expanding our result at first order in $\Delta p/M_{\rm PL}$. We also briefly comment on possible implications for Lorentz invariance at the Planck scale. 
  We present a grapical way to describe invariants and covariants in the (4 dim) general relativity. This makes us free from the complexity of suffixes . Two new off-shell relations between (mass)$\ast\ast 6$\ invariants are obtained. These are important for 2-loop off-shell calculation in the perturbative quantum gravity. We list up all independent invarians with dimensions of (mass)$\ast\ast 4$\ and (mass)$\ast\ast 6$. Furthermore the explicit form of 6 dim Gauss-Bonnet identity is obtained. 
  We express the defining relations of the $q$-deformed Minkowski space algebra as well as that of the corresponding derivatives and differentials in the form of reflection equations. This formulation encompasses the covariance properties with respect the quantum Lorentz group action in a straightforward way. 
  This is a revised version of our previous paper by the same name and preprint number. It contains various changes, two figures and new results in sect.5. We propose a new approach to four-dimensional Planckian-energy scattering in which the phase of the ${\cal S}$-matrix is written---to leading order in $\hbar$ and to all orders in $R/b =Gs/J$---in terms of the surface term of the gravity action and of a boundary term for the colliding quanta. The proposal is checked at the leading order in $R/b$ and also against some known examples of scattering in strong gravitational fields. 
  We apply the procedure of Magri and Weinstein to produce an infinity of compatible Poisson structures on a bihamiltonian manifold, to the case of the KdV phase space. The higher Gel'fand-Dikii structures thus obtained contain non local terms, which we express with the help of the r.h.s. of the KdV hierarchy. We also give a generating function for all these Poisson structues, in terms of the Baker-Akhiezer functions. Finally we describe the symplectic leaves of these Poisson structures. 
  We explore some of the global aspects of duality transformations in String Theory and Field Theory. We analyze in some detail the equivalence of dual models corresponding to different topologies at the level of the partition function and in terms of the operator correspondence for abelian duality. We analyze the behavior of the cosmological constant under these transformations. We also explore several examples of non-abelian duality where the classical background interpretation can be maintained for the original and the dual theories. In particular we construct a non-abelian dual of $SL(2,R)$ which turns out to be a three-dimensional black hole 
  We compute all string tree level correlation functions of vertex operators in $c<1$ string theory. This is done by using the ring structure of the theory. In order to study the multicritical behaviour, we calculate the correlation functions after perturbation by physical vertex operators. We show that the $(2k-1,2)$ models can be obtained from the $(1,2)$ model and the minimal models can be obtained from the $(1,p)$ model by perturbing the action by appropriate physical operators. Our results are consistent with known results from matrix models. 
  We present a simple procedure for constructing the complete cohomology of the BRST operator of the two-scalar and multi-scalar $W_3$ strings. The method consists of obtaining two level--15 physical operators in the two-scalar $W_3$ string that are invertible, and that can normal order with all other physical operators. They can be used to map all physical operators into non-trivial physical operators whose momenta lie in a fundamental unit cell. By carrying out an exhaustive analysis of physical operators in this cell, the entire cohomology problem is solved. 
  Based on lectures delivered at (1) the AMS meeting at USC, Nov. 1992   (2) Conference on Quantum Aspects of Black Holes, ITP, UC- Santa %Barbara, %%June 1993.   (3) 25th Summer Institute, Ecole Normale Superieure, % Paris, Aug. 1992.   To appear in "Interface Between Mathematics and Physics", Ed. S.-T. Yau.   1--Itroduction to String Theory in Curved Spacetime   2--G/H Coset Conformal Field Theory and String Theory   3--Time, Space and Classification of Non-Compact Cosets   4--Heterotic Strings in Curved Spacetime as gauged WZW Models   5--The Spacetime Manifold and the Geometry   6--Examples in 2D, 3D and 4D 
  A derivation of the Ces\`aro-Fedorov relation from the Selberg trace formula on an orbifolded 2-sphere is elaborated and extended to higher dimensions using the known heat-kernel coefficients for manifolds with piecewise-linear boundaries. Several results are obtained that relate the coefficients, $b_i$, in the Shephard-Todd polynomial to the geometry of the fundamental domain. For the 3-sphere we show that $b_4$ is given by the ratio of the volume of the fundamental tetrahedron to its Schl\"afli reciprocal. 
  We consider two dimensional U(N) QCD on the cylinder with a timelike Wilson line in an arbitrary representation. We show that the theory is equivalent to N fermions with internal degrees of freedom which interact among themselves with a generalized Sutherland-type interaction. By evaluating the expectation value of the Wilson line in the original theory we explicitly find the spectrum and degeneracies of these particle systems. 
  The Gauss law constraint in the Hamiltonian form of the $SU(2)$ gauge theory of gluons is satisfied by any functional of the gauge invariant tensor variable $\phi^{ij} = B^{ia} B^{ja}$. Arguments are given that the tensor $G_{ij} = (\phi^{-1})_{ij}\,\det B$ is a more appropriate variable. When the Hamiltonian is expressed in terms of $\phi$ or $G$, the quantity $\Gamma^i_{jk}$ appears. The gauge field Bianchi and Ricci identities yield a set of partial differential equations for $\Gamma$ in terms of $G$. One can show that $\Gamma$ is a metric-compatible connection for $G$ with torsion, and that the curvature tensor of $\Gamma$ is that of an Einstein space. A curious 3-dimensional spatial geometry thus underlies the gauge-invariant configuration space of the theory, although the Hamiltonian is not invariant under spatial coordinate transformations. Spatial derivative terms in the energy density are singular when $\det G=\det B=0$. These singularities are the analogue of the centrifugal barrier of quantum mechanics, and physical wave-functionals are forced to vanish in a certain manner near $\det B=0$. It is argued that such barriers are an inevitable result of the projection on the gauge-invariant subspace of the Hilbert space, and that the barriers are a conspicuous way in which non-abelian gauge theories differ from scalar field theories. 
  We present the static, spherically symmetric solutions of the low energy $d = 4$ string. They form a simple generalisation of the Schwarzschild solution and describe the gravitational field of a spherical star in perturbative string theory. If the Robertson parameter $\gamma$ is different from one, these solutions describe singular objects with naked singularities at their horizons, and whose Hawking temperature is infinite. These objects might represent ``massive remnants'', the end states of black holes. Our analysis also implies that Brans-Dicke theory has a naked singularity. Invoking cosmic censorship conjecture will rule out Brans-Dicke theory. 
  We analyse a $2+1$ dimensional model with charged, relativistic fermions interacting through a four-Fermi term. Taking advantage of its large-$N$ renormalizability, the various phases of this model are studied at finite temperature and beyond the leading order in $1/N$. Although the vacuum expectation value (VEV) of a charged order parameter is zero at any non-zero temperature, the model nevertheless exhibits a rich phase structure in the strong coupling r\'egime, because of the non-vanishing VEV of a neutral order parameter and due to the non-trivial dynamics of the vortex excitations on the plane. These are: a confined-vortex phase which is superconducting at low temperatures, an intermediate-temperature phase with deconfined vortices, and a high-temperature phase, where the neutral order parameter vanishes. The manifestation of superconductivity at low-temperatures and its disappearance above a critical temperature is explicitly shown to be due to the vortex confinement/deconfinement mechanism of Kosterlitz and Thouless. The ground state does not break parity or time-reversal symmetries and the ratio of the energy gap to $T_c$ is bigger than the conventional BCS value, for $N\ltwid 22$. 
  We introduce a particle mechanics model with Sp($2M$) gauge invariance. Different partial gauge-fixings by means of sl(2) embeddings on the gauge algebra lead to reduced models which are invariant under diffeomorphisms and classical non-linear \W-transformations as the residual gauge symmetries thus providing a set of models of gauge and matter fields coupled in a \W-invariant way. The equations of motion for the matter variables give Lax operators in a matrix form. We examine several examples in detail and discuss the issue of integration of infinitesimal \W-transformations. 
  We show that the renormalized vacuum expectation value of the Wilson loop for topologically massive abelian gauge theory in $\RR^3$ can be defined so that its large-mass limit be the renormalized vacuum expectation value of the Wilson loop for abelian Chern-Simons theory also in $\RR^3$. 
  An asymptotic expansion of the trace of the heat kernel on a cone where the heat coefficients have a delta function behavior at the apex is obtained. It is used to derive the renormalized effective action and total energy of a self-interacting quantum scalar field on the cosmic string space-time. Analogy is pointed out with quantum theory with boundaries. The surface infinities in the effective action are shown to appear and are removed by renormalization of the string tension. Besides, the total renormalized energy turns out to be finite due to cancelation of the known non-integrable divergence in the energy density of the field with a counterterm in the bare string tension. 
  I study the quantum mechanics of a spin interacting with an ``apparatus''. Although the evolution of the whole system is unitary, the spin evolution is not. The system is chosen so that the spin exhibits loss of quantum coherence, or ``wavefunction collapse'', of the sort usually associated with a quantum measurement. The system is analyzed from the point of view of the spin density matrix (or ``Schmidt paths''), and also using the consistent histories approach. These two points of view are contrasted with each other. Connections between the results and the form of the Hamiltonian are discussed in detail. 
  A new linear system is constructed for Poincar\'e supergravities in two dimensions. In contrast to previous results, which were based on the conformal gauge, this linear system involves the topological world sheet degrees of freedom (the Beltrami and super-Beltrami differentials). The associated spectral parameter likewise depends on these and is itself subject to a pair of differential equations, whose integrability condition yields one of the equations of motion. These results suggest the existence of an extension of the Geroch group mixing propagating and topological degrees of freedom on the world sheet. We also develop a chiral tensor formalism for arbitrary Beltrami differentials, in which the factorization of $2d$ diffeomorphisms is always manifest. 
  Generalizing the notion of continuous Hilbert space representations of compact topological groups we define unitary continuous correpresentations of $C^*$-completions of compact quantum group Hopf algebras on arbitrary Hilbert spaces. It is proved that the unitary continuous correpresentations decompose in finite dimensional irreducible correpresentations. 
  An alternative perturbative expansion in quantum mechanics which allows a full expression of the scaling arbitrariness is introduced. This expansion is examined in the case of the anharmonic oscillator and is conveniently resummed using a method which consists in introducing an energy cut-off that is carefully removed as the order of the expansion is increased. We illustrate this technique numerically by computing the asymptotic behavior of the ground state energy of the anharmonic oscillator for large couplings, and show how the exploitation of the scaling arbitrariness substantially improves the convergence of this perturbative expansion. 
  Strassler's formulation of the string-derived Bern-Kosower formalism is reconsidered with particular emphasis on effective actions and form factors. Two- and three point form factors in the nonabelian effective action are calculated and compared with those obtained in the heat kernel approach of Barvinsky, Vilkovisky et al. We discuss the Fock-Schwinger gauge and propose a manifestly covariant calculational scheme for one-loop effective actions in gauge theory. 
  We review some attempts of reformulating both gauge theory and general relativity in terms of holonomy-dependent loop variables. The emphasis lies on exhibiting the underlying mathematical structures, which often are not given due attention in physical applications. An extensive list of references is included. 
  Kazama has described an extension of the N=2 superconformal algebra in which the operator product of G^- with itself is singular. In this paper, we relate actions of this chiral algebra to Drinfeld's theory of Manin pairs, or equivalently, quasi-Lie bialgebras. We also show how to couple topological conformal field theories with this symmetry to topological gravity. As an application, we demonstrate the equivalence of the SL(2)/SL(2) model to a deformed SL(2)/SO(2) model tensored with a free field theory. (Revisions correct some minor errors.) 
  We investigate the additional symmetries of several supersymmetric KP hierarchies: the SKP hierarchy of Manin and Radul, the $\hbox{SKP}_2$ hierarchy, and the Jacobian SKP hierarchy. In all three cases we find that the algebra of symmetries is isomorphic to the algebra of superdifferential operators, or equivalently $\SW_{1+\infty}$. These results seem to suggest that despite their realization depending on the dynamics, the additional symmetries are kinematical in nature. 
  The Peierls bracket quantization scheme is applied to the supersymmetric system corresponding to the twisted spin index theorem. A detailed study of the quantum system is presented, and the Feynman propagator is exactly computed. The Green's function methods provide a direct derivation of the index formula. Note: This is essentially a new SUSY proof of the index theorem. 
  The topology of the non-adiabatic parameter space bundle is discussed for evolution of exact cyclic state vectors in Berry's original example of split angular momentum eigenstates. It turns out that the change in topology occurs at a critical frequency. The first Chern number that classifies these bundles is proportional to angular momentum. The non-adiabatic principal bundle over the parameter space is not well-defined at the critical frequency. 
  The quantization of the superclassical system used in the proof of the index theorem results in a factor of $\hbar^{2}R/8 $ in the Hamiltonian. The path integral expression of the kernel is analyzed up to and including 2-loop order. The existence of the scalar curvature term is confirmed by comparing the linear term in the heat kernel expansion with the 2-loop order terms in the loop expansion. 
  Abtract: The 2D model of gravity with zweibeins $e^{a}$ and the Lorentz connection one-form $\omega^{a}_{\ b}$ as independent gravitational variables is considered. The solutions of classical equations of motion which can be interpreted as cosmological ones are studied. 
  We provide an explicit formula for the invariant of 4-manifolds introduced by Crane and Yetter (in hep-th 9301062). A consequence of our result is the existence of a combinatorial formula for the signature of a 4-manifold in terms of local data from a triangulation. Potential physical applications of our result exist in light of the fact that the Crane-Yetter invariant is a rigorous version of ideas of Ooguri on B wedge F theory. 
  Duality symmetries for strings moving in non-trivial spacetime backgrounds are analysed. It is shown that for backgrounds generated from WZW and coset CFT models such duality symmetries are exact to all orders in string perturbation theory. Their implications for string dynamics in non-trivial/singular spacetimes are discussed. (Talk given at the EPS 93 Conference, held at Marseille, July 22-27. To appear in the Proceedings.) 
  Yang-Mills theories on a 1+1 dimensional cylinder are considered. It is shown that canonical quantization can proceed following different routes, leading to inequivalent quantizations. The problem of the non-free action of the gauge group on the configuration space is also discussed. In particular we re-examine the relationship between ``$\theta$-states" and the fundamental group of the configuration space. It is shown that this relationship does or does not hold depending on whether or not the gauge transformations not connected to the identity act freely on the space of connections modulo connected gauge transformations. 
  An infinite number of topological conformal algebras with varying central charges are explicitly shown to be present in $2d$ gravity (treated both in the conformal gauge and in the light-cone gauge) coupled to minimal matter. The central charges of the underlying $N=2$ theory in two different gauge choices are generically found to be different. The physical states in these theories are briefly discussed in the light of the $N=2$ superconformal symmetry. 
  The loop representation of quantum gravity has many formal resemblances to a background-free string theory. In fact, its origins lie in attempts to treat the string theory of hadrons as an approximation to QCD, in which the strings represent flux tubes of the gauge field. A heuristic path-integral approach indicates a duality between background-free string theories and generally covariant gauge theories, with the loop transform relating the two. We review progress towards making this duality rigorous in three examples: 2d Yang-Mills theory (which, while not generally covariant, has symmetry under all area-preserving transformations), 3d quantum gravity, and 4d quantum gravity. $SU(N)$ Yang-Mills theory in 2 dimensions has been given a string-theoretic interpretation in the large-$N$ limit by Gross, Taylor, Minahan and Polychronakos, but here we provide an exact string-theoretic interpretation of the theory on $\R\times S^1$ for finite $N$. The string-theoretic interpretation of quantum gravity in 3 dimensions gives rise to conjectures about integrals on the moduli space of flat connections, while in 4 dimensions there may be connections to the theory of 2-tangles. 
  The problem of a periodic scalar field on a two-dimensional dynamical random lattice is studied with the inclusion of vortices in the action. Using a random matrix formulation, in the continuum limit for genus zero surfaces the partition function is found exactly, as a function of the chemical potential for vortices of unit winding number, at a specific radius in the plasma phase. This solution is used to describe the Kosterlitz- Thouless phenomenon in the presence of 2D quantum gravity as one passes from the ultra-violet to the infra-red. 
  We study the construction of the minimal supersymmetric standard model from the $Z_8$ orbifold models. We use a target-space duality anomaly cancellation and a unification of gauge couplings as constraints. It is shown that some models obtained through a systematical search realize the unification of SU(3) and SU(2) coupling constants. 
  A study of diff($S^1$) covariant properties of pseudodifferential operator of integer degree is presented. First, it is shown that the action of diff($S^1$) defines a hamiltonian flow defined by the second Gelfand-Dickey bracket if and only if the pseudodifferential operator transforms covariantly. Secondly, the covariant form of a pseudodifferential operator of degree n not equal to 0, 1, -1 is constructed by exploiting the inverse of covariant derivative. This, in particular, implies the existence of primary basis for W_{KP}^{(n)} (n not equal to 0, 1, -1). 
  The zoo of two-dimensional conformal models has been supplemented by a series of nonunitary conformal models obtained by cosetting minimal models. Some of them coincide with minimal models, some do not have even Kac spectrum of conformal dimensions. 
  We review some aspects of the relation between ordinary coherent states and q-deformed generalized coherent states with some of the simplest cases of quantum Lie algebras. In particular, new properties of (q-)coherent states are utilized to provide a path integral formalism allowing to study a modified form of q-classical mechanics, to probe some geometrical consequences of the q-deformation and finally to construct Bargmann complex analytic realizations for some quantum algebras. 
  A non-principal value prescription is used to define the spurious singularities of Yang-Mills theory in the temporal gauge. Typical one-loop dimensionally-regularized temporal-gauge integrals in the prescription are explicitly calculated, and a regularization for the spurious gauge divergences is introduced. The divergent part of the one-loop self-energy is shown to be local and has the same form as that in the spatial axial gauge with the principal-value prescription. The renormalization of the theory is also briefly mentioned. 
  Two sets of identities between unitary minimal Virasoro characters at levels $m=3,4,5$ are presented and proven. The first identity suggests a connection between the Ising and tricritical Ising models since the $m=3$ Virasoro characters are obtained as bilinears of $m=4$ Virasoro characters. The second identity gives the tricritical Ising model characters as bilinears in the Ising model characters and the six combinations of $m=5$ Virasoro characters which do not appear in the spectrum of the three state Potts model. The implication of these identities on the study of the branching rules of $N=4$ superconformal characters into $\widehat{SU(2)} \times \widehat{SU(2)}$ characters is discussed. 
  We consider dilaton gravity theories in four spacetime dimensions parametrised by a constant $a$, which controls the dilaton coupling, and construct new exact solutions. We first generalise the C-metric of Einstein-Maxwell theory ($a=0$) to solutions corresponding to oppositely charged dilaton black holes undergoing uniform acceleration for general $a$. We next develop a solution generating technique which allows us to ``embed" the dilaton C-metrics in magnetic dilaton Melvin backgrounds, thus generalising the Ernst metric of Einstein-Maxwell theory. By adjusting the parameters appropriately, it is possible to eliminate the nodal singularities of the dilaton C-metrics. For $a<1$ (but not for $a\ge 1$), it is possible to further restrict the parameters so that the dilaton Ernst solutions have a smooth euclidean section with topology $S^2\times S^2-{\rm\{pt\}}$, corresponding to instantons describing the pair production of dilaton black holes in a magnetic field. A different restriction on the parameters leads to smooth instantons for all values of $a$ with topology $S^2\times \R^2$. 
  This is the first part in a series of papers developing a tensor product theory for modules for a vertex operator algebra. The goal of this theory is to construct a ``vertex tensor category'' structure on the category of modules for a suitable vertex operator algebra. The notion of vertex tensor category is essentially a ``complex analogue'' of the notion of symmetric tensor category, and in fact a vertex tensor category produces a braided tensor category in a natural way. The theory applies in particular to many familiar ``rational'' vertex operator algebras, including those associated with WZNW models, minimal models and the moonshine module. In this paper (Part I), we introduce the notions of $P(z)$- and $Q(z)$-tensor product, where $P(z)$ and $Q(z)$ are two special elements of the moduli space of spheres with punctures and local coordinates, and we present the fundamental properties and constructions of $Q(z)$-tensor products. 
  Feynman's path integrals provide a hidden variable description of quantum mechanics (and quantum field theories). The expectation values defined through path integrals obey Bell's inequalities in Euclidean time, but not in Minkowski time. This observation allows us to pinpoint the origin of violation of Bell's inequalities in quantum mechanics. % This article is not precise enough for mathematicians and not vague % enough for philosophers, but it should be interesting for physicists. % Contributed to the XVI International Symposium on Lepton-Photon % Interactions, Cornell University, August 10-15, 1993. 
  The partition function of 2+1 Chern-Simons Witten topological gravity has an attractive interpretation in terms of the unbroken and broken phases of gravity. We make this physical interpretation manifest using the background field method. 
  Starting from the well-known expression for the trace anomaly we derive the $T\cdot T$ operator product expansion of the energy-momentum tensor in 2D conformal theories defined in the upper halfplane $without$ making use of the additional condition of no energy-momentum flux across the boundary. The OPE turns out to be the same as in the absence of the boundary. For this result it is crucial that the trace anomaly is proportional to the Gau\ss-Bonnet density. Some relations to the $\sigma$ - model approach for open strings are discussed. 
  We introduce a noncommutative calculus on the odd-symplectic superspace $\scri$ of fields and antifields. To this end we have to extend $\scri$ to $\exscri$ by including an extra anticommuting field $\eta$. As a consequence we show that the commutator induced on $\exnice\times T^\star(\exnice)$ is proportional to the antibracket. The $\Delta$-operator is an element of the quotient space of derivations twisted by the antibracket $A\der$ and $\der$. The natural measure on $\exscri$ is shown to be invariant under canonical transformations provided a certain 'wave equation' is satisfied. 
  The symmetries of the topological Yang-Mills theory are studied in the Hamiltonian formalism and the generators of the twisted N=2 superPoincar\'e algebra are explicitly constructed. Noting that the twisted Lorentz generators do not generate the Lorentz symmetry of the theory, we relate the two by extracting from the latter the twisted version of the internal SU(2) generator. The hermiticity properties of the various generators are also considered throughout, and the boost generators are found to be non-hermitian. We then recover the BRST cohomology condition on physical states from representation theory arguments. 
  Here is summarized the gauge theoretical formulation and quantization of two popular gravity theories in (1+1)-dimensional time. 
  We have generalized recent results of Cappelli, Trugenberger and Zemba on the integer quantum Hall effect constructing explicitly a ${\cal W}_{1+\infty}$ for the fractional quantum Hall effect such that the negative modes annihilate the Laughlin wave functions. This generalization has a nice interpretation in Jain's composite fermion theory. Furthermore, for these models we have calculated the wave functions of the edge excitations viewing them as area preserving deformations of an incompressible quantum droplet, and have shown that the ${\cal W}_{1+\infty}$ is the underlying symmetry of the edge excitations in the fractional quantum Hall effect. Finally, we have applied this method to more general wave functions. 
  We present a new collective field formalism with two rather than one collective field to derive the antifield formalism for extended BRST invariant quantisation. This gives a direct and physical proof of the scheme of Batalin, Lavrov and Tyutin, derived on algebraic grounds. The importance of the collective field in the quantisation of open algebras in both the BRST and extended BRST invariant way is stressed. 
  We consider the problem of removing the divergences in an arbitrary gauge-field theory (possibly nonrenormalizable). We show that this can be achieved by performing, order by order in the loop expansion, a redefinition of some parameters (possibly infinitely many) and a canonical transformation (in the sense of Batalin and Vilkovisky) of fields and BRS sources. Gauge-invariance is turned into a suitable quantum generalization of BRS-invariance. We define quantum observables and study their properties. We apply the result to renormalizable gauge-field theories that are gauge-fixed with a nonrenormalizable gauge-fixing and prove that their predictivity is retained. A corollary is that topological field theories are predictive. Analogies and differences with the formalisms of classical and quantum mechanics are pointed out. 
  We consider a macroscopic charge-current carrying (cosmic) string in the background of a Schwarzschild black hole. The string is taken to be circular and is allowed to oscillate and to propagate in the direction perpendicular to its plane (that is parallel to the equatorial plane of the black hole). Nurmerical investigations indicate that the system is non-integrable, but the interaction with the gravitational field of the black hole anyway gives rise to various qualitatively simple processes like "adiabatic capture" and "string transmutation". 
  In this talk, I will review the foundations of irrational conformal field theory (ICFT), which includes rational conformal field theory as a small subspace. Highlights of the review include the Virasoro master equation, the Ward identities for the correlators of ICFT and solutions of the Ward identities. In particular, I will discuss the solutions for the correlators of the $g/h$ coset constructions and the correlators of the affine-Sugawara nests on $g\supset h_1 \supset \ldots \supset h_n$. Finally, I will discuss the recent global solution for the correlators of all the ICFT's in the master equation. 
  Quantum superintegrable systems in two dimensions are obtained from their classical counterparts, the quantum integrals of motion being obtained from the corresponding classical integrals by a symmetrization procedure. For each quantum superintegrable systema deformed oscillator algebra, characterized by a structure function specific for each system, is constructed, the generators of the algebra being functions of the quantum integrals of motion. The energy eigenvalues corresponding to a state with finite dimensional degeneracy can then be obtained in an economical way from solving a system of two equations satisfied by the structure function, the results being in agreement to the ones obtained from the solution of the relevant Schrodinger equation. The method shows how quantum algebraic techniques can simplify the study of quantum superintegrable systems, especially in two dimensions. 
  Dynamics of cylindrical and spherical relativistic domain walls is investigated with the help of a new method based on Taylor expansion of the scalar field in a vicinity of the core of the wall. Internal oscillatory modes for the domain walls are found. These modes are non-analytic in the "width" of the domain wall. Rather non-trivial transformation to a special coordinate system, widely used in investigations of relativistic domain walls, is studied in detail. 
  Closed bosonic string theory on toroidal orbifolds is studied in a Lagrangian path integral formulation. It is shown that a level one twisted WZW action whose field value is restricted to Cartan subgroups of simply-laced Lie groups on a Riemann surface is a natural and nontrivial extension of a first quantized action of string theory on orbifolds with an antisymmetric background field. 
  Canonical quantisation gives a new and convenient finite-temperature perturbation theory in covariant gauges, and solves the problem of the zero-frequency mode in the temporal gauge. [Talk at Workshop on Thermal Field Theories and their Applications, Banff, August 1993] 
  The critical exponent corresponding to the renormalization of the composite operator $\bar{\psi}\psi$ is computed in quantum electrodynamics at $O(1/\Nf^2)$ in arbitrary dimensions and covariant gauge at the non-trivial zero of the $\beta$-function in the large $\Nf$ expansion and the exponent corresponding to the anomalous dimension of the electron mass which is a gauge independent object is deduced. Expanding in powers of $\epsilon$ $=$ $2$ $-$ $d/2$ we check it is consistent with the known three loop perturbative structure and determine the subsequent coefficients in the coupling constant 
  We present coset conformal field theories whose spectrum is not determined by the identification current method. In these ``maverick'' cosets there is a larger symmetry identifying primary fields than under the identification current. We find an A-D-E classification of these mavericks. } 
  The nonlinear structures in 2D quantum gravity coupled to the $(q+1,q)$ minimal model are studied in the Liouville theory to clarify the factorization and the physical states. It is confirmed that the dressed primary states outside the minimal table are identified with the gravitational descendants. Using the discrete states of ghost number zero and one we construct the currents and investigate the Ward identities which are identified with the W and the Virasoro constraints. As nontrivial examples we derive the $L_0$, $L_1$ and $W_{-1}^{(3)}$ equations exactly. $L_n$ and $W^{(k)}_n$ equations are also discussed. We then explicitly show the decoupling of the edge states $O_j ~(j=0~ {\rm mod}~ q) $. We consider the interaction theory perturbed by the cosmological constant $O_1$ and the screening charge $S^+ =O_{2q+1}$. The formalism can be easily generalized to potential models other than the screening charge. 
  Recent progress on the physical states and scattering amplitudes of the $W_3$ string is reviewed with particular emphasis on the relation between this string theory and the Ising model. 
  Based on a recent paper by Takhtajan, we propose a formulation of 2D quantum gravity whose basic object is the Liouville action on the Riemann sphere $\Sigma_{0,m+n}$ with both parabolic and elliptic points. The identification of the classical limit of the conformal Ward identity with the Fuchsian projective connection on $\Sigma_{0,m+n}$ implies a relation between conformal weights and ramification indices. This formulation works for arbitrary $d$ and admits a standard representation only for $d\le 1$. Furthermore, it turns out that the integerness of the ramification number constrains $d=1-24/(n^2-1)$ that for $n=2m+1$ coincides with the unitary minimal series of CFT. 
  We analyze the moduli spaces of Calabi-Yau threefolds and their associated conformally invariant nonlinear sigma-models and show that they are described by an unexpectedly rich geometrical structure. Specifically, the Kahler sector of the moduli space of such Calabi-Yau conformal theories admits a decomposition into adjacent domains some of which correspond to the (complexified) Kahler cones of topologically distinct manifolds. These domains are separated by walls corresponding to singular Calabi-Yau spaces in which the spacetime metric has degenerated in certain regions. We show that the union of these domains is isomorphic to the complex structure moduli space of a single topological Calabi-Yau space---the mirror. In this way we resolve a puzzle for mirror symmetry raised by the apparent asymmetry between the Kahler and complex structure moduli spaces of a Calabi-Yau manifold. Furthermore, using mirror symmetry, we show that we can interpolate in a physically smooth manner between any two theories represented by distinct points in the Kahler moduli space, even if such points correspond to topologically distinct spaces. Spacetime topology change in string theory, therefore, is realized by the most basic operation of deformation by a truly marginal operator. Finally, this work also yields some important insights on the nature of orbifolds in string theory. 
  We study the one-loop contributions of matter and radiation to the gravitational polarization tensor at finite temperatures. Using the analytically continued imaginary-time formalism, the contribution of matter is explicitly given to next-to-leading ($T^2$) order. We obtain an exact form for the contribution of radiation fields, expressed in terms of generalized Riemann zeta functions. A general expression is derived for the physical polarization tensor, which is independent of the parametrization of graviton fields. We investigate the effective thermal masses associated with the normal modes of the corresponding graviton self-energy. 
  The phase space of a particle on a group manifold can be split in left and right sectors, in close analogy with the chiral sectors in Wess Zumino Witten models. We perform a classical analysis of the sectors, and the geometric quantization in the case of $SU(2)$. The quadratic relation, classically identifying $SU(2)$ as the sphere $S^3$, is replaced quantum mechanically by a similar condition on non-commutative operators ('quantum sphere'). The resulting quantum exchange algebra of the chiral group variables is quartic, not quadratic. The fusion of the sectors leads to a Hilbert space that is subtly different from the one obtained by a more direct (un--split) quantization. 
  The matrix model with a Bethe-tree embedding space (coinciding at large $N$ with the Kazakov-Migdal ``induced QCD'' model \cite{KM}) is investigated. We further elaborate the Riemann-Hilbert approach of \rf{Mig1} assuming certain holomorphic properties of the solution. The critical scaling (an edge singularity of the density) is found to be $\gamma_{str} = -\frac{1}{\pi} \arcos D$, for $|D|<1$, and $\gamma_{str} = -\frac{1}{\pi} \arcos \frac{D}{2D-1}$, for $D>1$. Explicit solutions are constructed at $D=\frac{1}{2}$ and $D=\infty$. 
  Duality symmetries for strings moving in non-trivial spacetime backgrounds are analysed. It is shown that, for backgrounds generated from compact WZW and coset models, such duality symmetries are exact to all orders in string perturbation theory. A global treatment of duality symmetries is given, by associating them to the known symmetries of affine current algebras (affine-Weyl group and external automorphisms). It is argued that self-duality symmetries of WZW and coset models generate the duality symmetries of their moduli space. Some remarks are presented, concerning the survival of such symmetries in the non-compact case. The implications of duality symmetries for string dynamics in non-trivial/singular spacetimes are discussed. (Talk given at the "Strings 93" Conference at Berkeley, May 1993) 
  Recent work of Roberts has shown that the first surgical 4-manifold invariant of Broda and (up to an unspecified normalization factor) the state-sum invariant arising from the TQFT of Crane-Yetter are equivalent to the signature of the 4-manifold. Subsequently Broda defined another surgical invariant in which the 1- and 2-handles are treated differently. We use a refinement of Roberts' techniques developped by the authors in hep-th/9309063 to show that the "improved" surgical invariant of Broda also depends only on the signature and Euler character 
  We quantize the topological $\sigma$-model. The quantum master equation of the Batalin-Vilkovisky formalism $\Delta_\rho \Psi = 0$ appears as a condition which eliminates the exact states from the BRST invariant states $\Psi$ defined by $Q \Psi = 0$. The phase space of the BV formalism is a supermanifold with a specific symplectic structure, called the fermionic K\"ahler manifold. 
  We consider a generalization of the abelian Higgs model with a Chern-Simons term by modifying two terms of the usual Lagrangian. We multiply a dielectric function with the Maxwell kinetic energy term and incorporate nonminimal interaction by considering generalized covariant derivative. We show that for a particular choice of the dielectric function this model admits topological vortices satisfying Bogomol'nyi bound for which the magnetic flux is not quantized even though the energy is quantized. Furthermore, the vortex solution in each topological sector is infinitely degenerate. 
  Recently (hep-th/9307183) we showed that for the case of the WZW- and the minimal models fusion can be understood as a certain ring-like tensor product of the symmetry algebra. In this paper we generalize this analysis to arbitrary chiral algebras. We define the tensor product of conformal field theory in the general case and prove that it is associative and symmetric up to equivalence. We also determine explicitly the action of the chiral algebra on this tensor product. In the second part of the paper we demonstrate that this framework provides a powerful tool for calculating restrictions for the fusion rules of chiral algebras. We exhibit this for the case of the $W_{3}$-algebra and the $N=1$ and $N=2$ NS superconformal algebras. 
  We study the gravitational dressing of renormalizable two-dimensional field theories. Our main result is that the one-loop $\beta$-function is finitely renormalized by the factor ${k+2\over k+1}$, where $k$ is the central charge of the gravitational $SL(2, R)$ current algebra. 
  The partition functions of QCD2 on simple surfaces admit representations in terms of exponentials of the inverse coupling, that are modular transforms of the usual character expansions. We review the construction of such a representation in the case of the cylinder, and show how it leads to a formulation of QCD2 as a $c=1$ matrix model of the Kazakov-Migdal type. The eigenvalues describe the positions of $N$ Sutherland fermions on a circle, while their discretized momenta label the representations in the corresponding character expansion. Using this language, we derive some new results: we give an alternative description of the Douglas-Kazakov phase transition on the sphere, and we argue that an analogous phase transition exists on the cylinder. We calculate the large $N$ limit of the partition function on the cylinder with boundary conditions given by semicircular distributions of eigenvalues, and we find an explicit expression for the large $N$ limit of the Itzykson-Zuber integral with the same boundary conditions. (Talk given at the ``Workshop on high energy physics and cosmology'' at Trieste, July 1993.) 
  We show that string theory with Dirichlet boundaries is equivalent to string theory containing surfaces with certain singular points. Surface curvature is singular at these points. A singular point is resolved in conformal coordinates to a circle with Dirichlet boundary conditions. We also show that moduli parameters of singular surfaces coincide with those of smooth surfaces with boundaries. Singular surfaces with saddle points indeed arise in the strong coupling expansion in lattice QCD. The kind of saddle point, which may be the origin of a singular point we need, is of infinite order. 
  We give an action that can be used to describe the production of global vortices in 2+1 dimensions by scattering Nambu-Goldstone bosons. At strong self-coupling the action reduces to scalar QED with particular values of the coupling constants and the production cross-section is explicitly found. We also consider the production of gauged vortices by scattering particles that have an Aharanov-Bohm interaction with the vortex. 
  A large class of (0,2) Calabi-Yau $\sigma$-models and Landau-Ginzburg orbifolds are shown to arise as different ``phases'' of supersymmetric gauge theories. We find a phenomenon in the Landau-Ginzburg phase which may enable one to understand which Calabi-Yau $\sigma$-models evade destabilization by worldsheet instantons. Examples of (0,2) Landau-Ginzburg vacua are analyzed in detail, and several novel features of (0,2) models are discussed. In particular, we find that (0,2) models can have different quantum symmetries from the (2,2) models built on the same Calabi-Yau manifold, and that a new kind of topology change can occur in (0,2) models of string theory. 
  Chern-Simons theory is analyzed with a gauge-fixing which allows to discuss the Landau gauge and the light-cone gauge at the same time. 
  A model interaction between a two-state quantum system and a classical switching device is analysed and shown to lead to the quantum Zeno effect for large values of the coupling constant k . A minimal piecewise deterministic random process compatible with the Liouville equation is described, and it is shown that 1/k can be interpreted as the jump frequency of the classical device 
  Model interactions between classical and quantum systems are briefly discussed. These include: general measurement-like couplings, Stern-Gerlach experiment, model of a counter, quantum Zeno effect, SQUID-tank model. 
  Various properties of open strings in external constant E.M. fields are reviewed. In particular, the charged-particle pair production rate in an external electric field is evaluated, and shown to reduce to Schwinger's formula in the limit of low-intensity fields. Open strings in external magnetic fields are shown to undergo an infinite number of phase transitions as the strenght of the field increases. (Talk given at the conference ``Strings '93'', Berkeley, CA, (May 24-29, 1993) to appear in the proceedings.) 
  Lectures presented at the Spring School, Trieste, Italy 1993. 
  In this paper we consider 1-D non-local field theories with a particular $1/r^2$ interaction, a constant gauge field and an arbitrary scalar potential. We show that any such theory that is at a renormalization group fixed point also satisfies an infinite set of reparametrization invariance Ward identities. We also prove that, for special values of the gauge field, the value of the potential that satisfies the Ward identities to first order in the potential strength remains a solution to all orders in the potential strength, summed over all loops. These theories are of interest because they describe dissipative quantum mechanics with an arbitrary potential and a constant magnetic field. They also give solutions to open string theory in the presence of a uniform gauge field and an arbitrary tachyon field. 
  The Sine-Gordon theory at $\frac{\beta^{2}}{8\pi} = \frac{2}{(2n+3)},\; n= 1,2,3 \cdots $ has a higher spin generalization of the $N=2$ supersymmetry with the central terms which arises from the affine quantum group $U_{q}( \hat{s \ell} (2))$. Observing that the algebraic determination of $S$ matrices $( \approx {\rm quantum~ integrability })$ requires the saturation of the generalized Bogomolny bound, we construct a variant of the Sine-Gordon theory at this value of the coupling in the framework of $S$ matrix theory. The spectrum consists of a doublet of fractionally charged solitons as well as that of anti-solitons in addition to the ordinary breathers. The construction demonstrates the existence of the theory other than the one by the truncation to the breathers considered by Smirnov. The allowed values for the fractional part of the fermion number is also determined.  The central charge in the massless limit is found to be $c= 1$ from the TBA calculation for nondiagonal S matrices.  The attendant $c=1$ conformal field theory is the gaussian model with ${\bf Z_{2}}$ graded chiral algebra at the radius parameter $r= \sqrt{2n+3}$. In the course of the calculation, we find $4n+2$ zero modes from the (anti-)soliton distributions. 
  We study the quantum Knizhnik-Zamolodchikov equation of level $0$ associated with the spin $1/2$ representation of $U_q \bigl(\widehat{\frak s \frak l _{2}}\bigr)$. We find an integral formula for solutions in the case of an arbitrary total spin and $|q|<1$. In the formula, different solutions can be obtained by taking different integral kernels with the cycle of integration being fixed. 
  We consider $U(N)$ and $SU(N)$ gauge theory on the sphere. We express the problem in terms of a matrix element of $N$ free fermions on a circle. This allows us to find an alternative way to show Witten's result that the partition function is a sum over classical saddle points. We then show how the phase transition of Douglas and Kazakov occurs from this point of view. By generalizing the work of Douglas and Kazakov, we find other `stringy' solutions for the $U(N)$ case in the large $N$ limit. Each solution is described by a net $U(1)$ charge. We derive a relation for the maximum charge for a given area and we also describe the critical behavior for these new solutions. Finally, we describe solutions for lattice $SU(N)$ which are in a sense dual to the continuum $U(N)$ solutions. (Parts of this paper were presented at the Strings '93 Workshop, Berkeley, May 1993.) 
  We find a class of nonlocal operators constructed by attaching a disorder operator to fermionic degrees of freedom, which can be used to generate q-deformed algebras following the Schwinger approach. This class includes the recently proposed anyonic operators defined on a lattice. 
  The self-similar potentials are formulated in terms of the shape-invariance. Based on it, a coherent state associated with the shape-invariant potentials is calculated in case of the self-similar potentials. It is shown that it reduces to the q-deformed coherent state. 
  Renormalization in quantum statistics in the presence of a charge associated to a spontaneously broken symmetry is discussed for the scalar field model. In contrast to the case of non-broken symmetry, the renormalization mass counterterm depends on the chemical potential. We argue that this is connected to the ill-defined character of the charge operator. 
  Not only does Chern-Simons (CS) coupling characterize statistics, but also spin and scaling dimension of matter fields. We demonstrate spin transmutation in relativistic CS matter theory, and moreover show equivalence of several models. We study CS vector model in some details, which provide consistent check to the assertion of the equivalence. 
  We identify and examine a generalization of topological sigma models suitable for coupling to topological open strings. The targets are Kahler manifolds with a real structure, i.e. with an involution acting as a complex conjugation, compatible with the Kahler metric. These models satisfy axioms of what might be called ``equivariant topological quantum field theory,'' generalizing the axioms of topological field theory as given by Atiyah. Observables of the equivariant topological sigma models correspond to cohomological classes in an equivariant cohomology theory of the targets. Their correlation functions can be computed, leading to intersection theory on instanton moduli spaces with a natural real structure. An equivariant $CP^1\times CP^1$ model is discussed in detail, and solved explicitly. Finally, we discuss the equivariant formulation of topological gravity on surfaces of unoriented open and closed string theory, and find a $Z_2$ anomaly explaining some problems with the formulation of topological open string theory. 
  We present a review of the status of $W$ string theories, their physical spectra, and their interactions. (Based on review talks given at the Trieste Spring Workshop, and the Strings 93 meeting at Berkeley, May 1993.) 
  We demonstrate that the Hagedorn-like growth of the number of observed meson states can be used to constrain the degrees of freedom of the underlying effective QCD string. We find that the temperature relevant for such string theories is not given by the usual Hagedorn value $T_H\approx 160$ MeV, but is considerably higher. This resolves an apparent conflict with the results from a static quark-potential analysis, and suggests that conformal invariance and modular invariance are indeed reflected in the hadronic spectrum. We also find that the $D_\perp=2$ scalar string is in excellent agreement with data. 
  A technical point regarding the invariance of Polyakov's nonlocal form of the effective action under uniform rescalings is briefly addressed. 
  The massive Schwinger model with two flavors is studied in the strong coupling region by using light-front Tamm-Dancoff approximation. The mass spectrum of the lightest particles is obtained numerically. We find that the mass of the lightest isotriplet (``pion'') behaves as $m^{0.50}$ for the strong couplings, where $m$ is the fermion mass. We also find that the lightest isosinglet is not in the valence state (``eta'') which is much heavier in the strong coupling region, but can be interpreted as a bound state of two pions. It is 1.762 times heavier than pion at $m=1.0\times10^{-3}(e/\sqrt\pi)$, while Coleman predicted that the ratio is $\sqrt3$ in the strong coupling limit. The ``pion decay constant'' is calculated to be 0.3945. 
  We study the residual symmetry $SL(2,R)\otimes U(1)$ of the chiral gravity in the light-cone gauge. Quantum gravitational effects renormalize the Kac-Moody central charge and introduce, through the Lorentz anomaly, an arbitrary parameter. Due to the presence of this free parameter the Kac-Moody central charge has no forbidden range of values, and the strong gravity regime is open to investigations. 
  The autonomous renormalization of the O(N)-symmetric scalar theory is based on an infinite re-scaling of constant fields, whereas finite-momentum modes remain finite. The natural framework for a detailed analysis of this method is a system of finite size, where all non-constant modes can be integrated out perturbatively and the constant mode is treated by a saddle-point approximation in the thermodynamic limit. The calculation provides a better understanding of the properties of of the effective action and corroborates earlier findings concerning a heavy Higgs particle at about 2 TeV. 
  The anyonic behaviour of massive spinning point particles coupled to linearized massive vector Chern-Simons gravity is studied. This model constitutes the uniform spin-2 generalization of the vector model formed by coupling charged point particles to the topological massive Maxwell-CS action. It turns out that, for this model, the linearized first order triadic Chern-Simons term is the source of the anyonic behaviour we found. This is in constrast with the third order topologically massive gravity, where the anyonic behaviour does not stem in its third-order Lorentz-Chern-Simons term, the second order Einstein's action . 
  We present a second order gravity action which consists of ordinary Einstein action augmented by a first-order, vector like, Chern-Simons quasi topological term. This theory is ghost-free and propagates a pure spin-2 mode. It is diffeomorphism invariant, although its local Lorentz invariance has been spontaneuosly broken. We perform the light-front (LF) analysis for both the linearized system and the exact curved model. In constrast to the 2+1 canonical analysis, in the quasi LF coordinates the differential constraints can be solved. Its solution is presented here. 
  We derive the duality symmetries relevant to moduli dependent gauge coupling constant threshold corrections, in Coxeter $ {\bf Z_N} $ orbifolds. We consider those orbifolds for which the point group leaves fixed a 2-dimensional sublattice $\Lambda_2$, of the six dimensional torus lattice $\Lambda_6$, where $\Lambda_6 $ cannot be decomposed as $\Lambda_2 \bigoplus{\Lambda_4}.$ 
  For general compact K\"ahler manifolds it is shown that both Toeplitz quantization and geometric quantization lead to a well-defined (by operator norm estimates) classical limit. This generalizes earlier results of the authors and Klimek and Lesniewski obtained for the torus and higher genus Riemann surfaces, respectively. We thereby arrive at an approximation of the Poisson algebra by a sequence of finite-dimensional matrix algebras $gl(N)$, $N\to\infty$. 
  We show that integrable vertex and RSOS models with trigonometric Boltzmann weights and appropriate inhomogeneities provide a convenient lattice regularization for massive field theories and for the recently studied massless field theories that interpolate between two non trivial conformal field theories. Massive and massless S matrices are computed from the lattice Bethe ansatz. 
  An extension of the Artin Braid Group with new operators that generate double and triple intersections is considered. The extended Alexander theorem, relating intersecting closed braids and intersecting knots is proved for double and triple intersections, and a counter example is given for the case of quadruple intersections. Intersecting knot invariants are constructed via Markov traces defined on intersecting braid algebra representations, and the extended Turaev representation is discussed as an example. Possible applications of the formalism to quantum gravity are discussed. 
  We study a system of functional relations among a commuting family of row-to-row transfer matrices in solvable lattice models. The role of exact sequences of the finite dimensional quantum group modules is clarified. We find a curious phenomenon that the solutions of those functional relations also solve the so-called thermodynamic Bethe ansatz equations in the high temperature limit for $sl(r+1)$ models.  Based on this observation, we propose possible functional relations for models associated with all the simple Lie algebras.  We show that these functional relations certainly fulfill strong constraints coming from the fusion procedure analysis.  The application to the calculations of physical quantities will be presented in the subsequent publication. 
  Quantum back reaction due to $N$ massless fields may be worked out to a considerable detail in a variant of integrable dilaton gravity model in two dimensions. It is shown that there exists a critical mass of collapsing object of order $\hbar N \times$ (cosmological constant)$^{1/2}$, above which the end point of Hawking evaporation is two disconnected remnants of infinite extent, each separated by a mouth from the outside region. Deep inside the mouth there is a universal flux of radiation in all directions, in a form different from Hawking radiation. Below the critical mass no remnant is left behind, implying complete Hawking evaporation or even showing no sign of Hawking radiation. Existence of infinitely many static states of quantum nature is also demonstrated in this model. 
  We remark that the weak coupling regime of the stochastic stabilization of 2D quantum gravity has a unique perturbative vacuum, which does not support instanton configurations. By means of Monte Carlo simulations we show that the nonperturbative vacuum is also confined in one potential well. Nonperturbative effects can be assessed in the loop equation. This can be derived from the Ward identities of the stabilized model and is shown to be modified by nonperturbative terms. 
  We develop techniques to compute higher loop string amplitudes for twisted $N=2$ theories with $\hat c=3$ (i.e. the critical case). An important ingredient is the discovery of an anomaly at every genus in decoupling of BRST trivial states, captured to all orders by a master anomaly equation. In a particular realization of the $N=2$ theories, the resulting string field theory is equivalent to a topological theory in six dimensions, the Kodaira--Spencer theory, which may be viewed as the closed string analog of the Chern--Simon theory. Using the mirror map this leads to computation of the `number' of holomorphic curves of higher genus curves in Calabi--Yau manifolds. It is shown that topological amplitudes can also be reinterpreted as computing corrections to superpotential terms appearing in the effective 4d theory resulting from compactification of standard 10d superstrings on the corresponding $N=2$ theory. Relations with $c=1$ strings are also pointed out. 
  A self-consistent string field theory with interaction is formulated. The symmetry algebra of this theory includes, in the low-energy limit, local space-time symmetries, and the Brans-Dicke equation describes a class of low-energy solutions. (Talk given at the conference Journees Relativistes'93 held on April 5-7 1993, To be published in the International Journal of Modern Physics D.) 
  We show that the ``boundary crossing-unitarity equation" recently proposed by Ghoshal and Zamolodchikov is a consequence of the boundary bootstrap equation for the S-matrix and the wall-bootstrap equation. We solve this set of equations for all affine Toda theories related to simply laced Lie algebras, obtaining explicit formulas for the W-matrix which encodes the scattering of a particle with the boundary in the ground state. For each theory there are two solutions to these equations, related by CDD-ambiguities, each giving rise to different kind of physics. 
  We present a unified treatment in superspace of the two dual formulations of $D=10$, $N=1$ {\it pure} supergravity based on a strictly super-geometrical framework: the only fundamental objects are the super Riemann curvature and torsion, and the related Bianchi identities are sufficient to set the theory on shell; there is no need to introduce, from the beginning, closed three- or seven-superforms. This formulation extends also to {\it non minimal} models.   Moreover, in this framework the algebraic analogy between pure super Yang--Mills theories and pure supergravity in $D=10$ is manifest. As an additional outcome in the present formulation the supersymmetric partner of the ABBJ-Lorentz anomaly in pure $D=10$ supergravity can be computed in complete analogy to the ABBJ-gauge anomaly in supersymmetric Yang--Mills theories in ten dimensions.   In the same framework we attack the issue of duality in $N=1$, $D=11$ supergravity showing in detail that duality holds at the kinematical level in superspace while it is broken by the dynamics. We discuss also possible extensions of this theory which could be related to quantum corrections of the eleven dimensional membrane. 
  The spectral problem for the q-Knizhnik-Zamolodchikov equations for $U_{q}(\widehat{sl_2}) (0<q<1)$ at arbitrary level $k$ is considered. The case of two-point functions in the fundamental representation is studied in detail.The scattering states are given explicitly in terms of continuous q-Jacobi polynomials, and the $S$-matrix is derived from their asymptotic behavior. The level zero $S$-matrix is shown to coincide, up to a trivial factor, with the kink-antikink $S$-matrix for the spin-$\frac{1}{2}$ XXZ antiferromagnet. In the limit of infinite level we observe connections with harmonic analysis on $p$-adic groups with the prime $p$ given by $p=q^{-2}$. 
  The classical Bekenstein entropy of a black hole is argued to arise from configurations of strings with ends which are frozen on the horizon. Quantum corrections to this entropy are probably finite unlike the case in quantum field theory. Finally it is speculated that all black holes are single string states. The level density of strings is of the right order of magnitude to reproduce the Bekenstein entropy. 
  We consider 3-point and 4-point correlation functions in a conformal field theory with a W-algebra symmetry. Whereas in a theory with only Virasoro symmetry the three point functions of descendants fields are uniquely determined by the three point function of the corresponding primary fields this is not the case for a theory with $W_3$ algebra symmetry. The generic 3-point functions of W-descendant fields have a countable degree of arbitrariness. We find, however, that if one of the fields belongs to a representation with null states that this has implications for the 3-point functions. In particular if one of the representations is doubly-degenerate then the 3-point function is determined up to an overall constant. We extend our analysis to 4-point functions and find that if two of the W-primary fields are doubly degenerate then the intermediate channels are limited to a finite set and that the corresponding chiral blocks are determined up to an overall constant. This corresponds to the existence of a linear differential equation for the chiral blocks with two completely degenerate fields as has been found in the work of Bajnok~et~al. 
  The identity of classical motion is established for two physically different models, one of which is the relativistic particle with torsion, whose action contains higher derivatives and which is the effective system for the statistically charged particle interacting with the Chern-Simons U(1) gauge field, and another is the (2+1)-dimensional relativistic charged particle in external constant electromagnetic field. 
  With the help of the deformed Heisenberg algebra involving Klein operator, we construct the minimal set of linear differential equations for the (2+1)-dimensional relativistic field with arbitrary fractional spin, whose value is defined by the deformation parameter. 
  It is shown that topological massive gravity augmented by the triadic gravitational Chern-Simons first order term is a curved a pure spin-2 action. This model contains two massive spin-2 excitations. However, since its light-front energy is not semidefinite positive, this double CS-action does not have any physical relevance.In other words, topological massive gravity cannot be spontaneously broken down by the presence of the triadic CS term. 
  We consider a new perturbation scheme in nonabelian gauge theory. Pure Yang-Mills theory in three dimensions is taken as a concrete example. The zeroth-order in the perturbative expansion is given by BF theory coupled to a St{\" u}ckelberg-like field. The effective coupling for the expansion can be small in the infrared regime, which implies that nonperturbative treatment of Yang-Mills theory may be partially reduced to that of BF theory. 
  We propose a simple model for a free relativistic particle of fractional spin in 2+1 dimensions which satisfies all the necessary conditions. The canonical quantization of the system leads to the description of one- particle states of the Poincare group with arbitrary spin. Using the Hamil- tonian formulation with the set of constraints, we introduce the electro- magnetic interaction of a charged anyon and obtain the Lagrangian. The Casimir operator of the extended algebra, which is the first-class constraint, is obtained and gives the equation of motion of the anyon. In particular, from the latter it follows that the gyromagnetic ratio for a charged anyon is two due to the parallelness of spin and momentum of the particle in 2+1 dimensions. The canonical quantization is also considered in this case. 
  Solutions of bosonic string theory are constructed which correspond to four-dimensional black holes with axionic quantum hair. The basic building blocks are the renormalization group flows of the CP1 model with a theta term and the SU(1,1)/U(1) WZW coset conformal field theory. However the solutions are also found to have negative energy excitations, and are accordingly expected to decay to the vacuum. 
  An algebraic treatment of shape-invariant potentials in supersymmetric quantum mechanics is discussed. By introducing an operator which reparametrizes wave functions, the shape-invariance condition can be related to a oscillator-like algebra. It makes possible to define a coherent state associated with the shape-invariant potentials. For a large class of such potentials, it is shown that the introduced coherent state has the property of resolution of unity. 
  We consider the N-soliton solutions in the sine-Gordon model as a N-body problem. This leads to a relativistic generalization of the Calogero model first introduced by Ruijsenaars. We show that the fundamental Poisson bracket of the Lax matrix is quadratic, and the $r$-matrix is a dynamical one. This is in contrast to the Calogero model where the fundamental Poisson bracket of the Lax matrix is linear. 
  In the present contribution we show that the introduction of axial currents in electrodynamics can explain the quantization of electric charge and introduces a dynamical discreteness of space-time, justifying thus the regularization of Feymman's integrals. 
  (some corrections in the semiclassical study and one reference added). 
  The $1/x^{2}$ deformed $c=1$ matrix model is studied at finite radius and non-zero cosmological constant. Calculational techniques are presented and illustrated in some examples. Furthermore, a new kind of $R \rightarrow 1/R$ duality is discovered which mixes different genus. 
  Quantum mechanical transition amplitudes are calculated within the stochastic quantization scheme for the free nonrelativistic particle, the harmonic oscillator and the nonrelativistic particle in a constant magnetic field; we close with free Grassmann quantum mechanics. 
  This is the second part in a series of papers presenting a theory of tensor products for module categories for a vertex operator algebra. In Part I (hep-th/9309076), the notions of $P(z)$- and $Q(z)$-tensor product of two modules for a vertex operator algebra were introduced and under a certain hypothesis, two constructions of a $Q(z)$-tensor product were given, using certain results stated without proof. In Part II, the proofs of those results are supplied. 
  We consider the $q$-deformed Schr\"odinger equation of the harmonic oscillator on the $N$-dimensional quantum Euclidian space. The creation and annihilation operator are found, which systematically produce all energy levels and eigenfunctions of the Schr\"odinger equation. In order to get the $q$-series representation of the eigenfunction, we also give an alternative way to solve the Schr\"odinger equation which is based on the $q$-analysis. We represent the Schr\"odinger equation by the $q$-difference equation and solve it by using $q$-polynomials and $q$-exponential functions. 
  Using differential and integral calculi on the quantum plane which are invariant with respect to quantum inhomogeneous Euclidean group E(2)q , we construct path integral representation for the quantum mechanical evolution operator kernel of q-oscillator. 
  It is shown that the bosonic angular degrees of freedom in the one dimensional Marinari-Parisi superstring can be integrated out exactly in the Hamiltonian formulation without having to perform the Dabholkar truncation. The resulting Hamiltonian is that of a supersymmetric Calogero system plus a four fermions interaction. This extra interaction vanishes for all physical states with fermion number zero or one where supersymmetry is manifest. We confirm that supersymmetry is nonperturbativly broken by instanton effects. 
  General covariance in quantum gravity is seen once one integrates over all possible metrics. In recent years topological field theories have given us a different route to general covariance without integrating over all possible metrics. Here we argue that Einstein quantum gravity may be viewed as a topological field theory provided a certain constrant from the path integral measure is satisfied. 
  We define and calculate the fusion algebra of WZW model at a rational level by cohomological methods. As a byproduct we obtain a cohomological characterization of admissible representations of $\widehat{\gtsl}_{2}$. 
  The Born-Infeld equation in two dimensions is generalised to higher dimensions whilst retaining Lorentz Invariance and complete integrability. This generalisation retains homogeneity in second derivatives of the field. 
  Necessary conditions for a field theoretic equation of motion to be the consequence of variation of an infinite number of inequivalent Lagrangians are examined. 
  Observables in the quantum field theories of $(D-1)$-form fields, $\ca$, on $D$-dimensional, compact and orientable manifolds, $M$, are computed. Computations of the vacuum value of $T_{ab}$ find it to be the metric times a function of the volume of spacetime, $\Omega(M)$. Part of this function of $\Omega$ is a finite zero-mode contribution. The correlation functions of another set of operators give intersection numbers on $M$. Furthermore, a similar computation for products of Wilson area operators results in a function of the volumes of the intersections of the submanifolds the operators are defined on. In addition, scalar field couplings are introduced and potentials are induced after integrating out the $\ca$ field. Lastly, the thermodynamics of the pure theories is found to be analogous to the zero-point motion of a scalar particle. The coupling of a Gaussian scalar field to the $\ca$ field is found to manifest itself on the free energy at high temperatures and/or small volumes. 
  The moduli dependence of string loop threshold corrections to gauge coupling constants is investigated for those ${\bf Z}_N$ Coxeter orbifolds with the property that some twisted sectors have fixed planes for which the six-torus ${\bf T}_6$ can not be decomposed into a direct sum ${\bf T}_4 \bigoplus{\bf T}_2$ with the fixed plane lying in ${\bf T}_2.$ 
  Withdrawn due to error. See D. Lowe, L. Susskind and J. Uglum, hep-th/9402136, for correct treatment. Apologies to all recipients. 
  In the present article we have found the complete energy spectrum and the corresponding eigenfunctions of the Dirac oscillator in two spatial dimensions. We show that the energy spectrum depends on the spin of the Dirac particle. 
  The effective action of string theory in three dimensions is investigated, incorporating the Lorentz and gauge Chern-Simons terms in the definition of the Kalb-Ramond axion field strength. Since in three dimensions any three-form is trivial, the action can be reformulated by properly integrating the axion out. The circumstances under which it can be recast in form of topologically massive gravity coupled to a topologically massive gauge theory are pointed out. Finally, the strong coupling limit of the resulting action is inspected, with the focus on the roles played by the axion and dilaton fields. 
  We introduce the Virasoro symmetry in the BV formalism and give an explicit construction of the anti-bracket, which is Virasoro invariant. It is shown that the master equation with this anti-bracket has an infinite number of solutions. The base space of the BV formalism is a fermionic version of the Virasoro manifold $Diff(S^1)/S^1$. We discuss also the Ricci tensor of this fermionic manifold. 
  It is shown, that the geometrical objects of Batalin-Vilkovisky formalism-- odd symplectic structure and nilpotent operator $\Delta$ can be naturally uncorporated in Duistermaat--Heckman localization procedure. The presence of the supersymmetric bi-Hamiltonian dynamics with even and odd symplectic structure in this procedure is established. These constructions can be straightly generalized for the path-integral case. 
  The WZNW model at the Witten conformal point is perturbed by the sigma model term. It is shown that in the large level k limit the perturbed WZNW system with negative k arrives at the WZNW model with positive k. 
  There exists a physically well motivated method for approximating manifolds by certain topological spaces with a finite or a countable set of points. These spaces, which are partially ordered sets (posets) have the power to effectively reproduce important topological features of continuum physics like winding numbers and fractional statistics, and that too often with just a few points. In this work, we develop the essential tools for doing quantum physics on posets. The poset approach to covering space quantization, soliton physics, gauge theories and the Dirac equation are discussed with emphasis on physically important topological aspects. These ideas are illustrated by simple examples like the covering space quantization of a particle on a circle, and the sine-Gordon solitons. 
  Duality groups as (spontaneously broken) gauge symmetries for toroidal backgrounds, and their role in ($\infty$-dimensional) underlying string gauge algebras are reviewed. For curved backgrounds, it is shown that there is a duality in the moduli space of WZNW sigma-models, that can be interpreted as a broken gauge symmetry. In particular, this duality relates the backgrounds corresponding to axially gauged abelian cosets, $G/U(1)_a$, to vectorially gauged abelian cosets, $G/U(1)_v$. Finally, topology change in the moduli space of WZNW sigma-models is discussed. 
  We discuss $d=1, {\cal N}=2$ supersymmetric matrix models and exhibit the associated $d=2$ collective field theory in the limit of dense eigenvalues. From this field theory we construct, by the addition of several new fields, a $d=2$ supersymmetric effective field theory, which reduces to the collective field theory when the new fields are replaced with their vacuum expectation values. This effective theory is Poincare invariant and contains perturbative and non-perturbative information about the associated superstrings. 
  We discuss lowest-weight representations of the $S_N$-Extended Heisenberg Algebras underlying the $N$-body quantum-mechanical Calogero model. Our construction leads to flat derivatives interpolating between Knizhnik-Zamolodchikov and Dunkl derivatives. It is argued that based on these results one can establish new links between solutions of the Knizhnik-Zamolodchikov equations and wave functions of the Calogero model. 
  We consider N=2 and N=4 supersymmetric gauge theories in two-dimensions, coupled to matter multiplets. In analogy with the N=2 case also in the N=4 case one can introduce Fayet-Iliopoulos terms.The associated three-parameters have the meaning of momentum-map levels in a HyperK\"ahler quotient construction. Differently from the N=2 case, however, the N=4 has a single phase corresponding to an effective $\sigma$-model. There is no Landau-Ginzburg phase. The main possible application of our N=4 model is to an effective Lagrangian construction of a $\sigma$-model on ALE-manifolds. We discuss the A and B topological twists of these models clarifying some issues not yet discussed in the literature, in particular the identification of the topological systems emerging from the twist. Applying our results to the case of ALE-manifolds we indicate how one can use the topologically twisted theories to study the K\"ahler class and complex structure deformations of these gravitational instantons. 
  We describe the idea of studying quantum gravity by means of dynamical triangulations and give examples of its implementation in 2, 3 and 4 space time dimensions. For $d=2$ we consider the generic hermitian 1-matrix model. We introduce the so-called moment description which allows us to find the complete perturbative solution of the generic model both away from and in the continuum. Furthermore we show how one can easily by means of the moment variables define continuum times for the model so that its continuum partition function agrees with the partition function of the Kontsevich model except for some complications at genus zero. Finally we study the non perturbative definition of 2D quantum gravity provided by stochastic stabilization, showing how well known matrix model characteristica can be given a simple quantum mechanical interpretation and how stochastic quantization seems to hint to us the possibility of a strong coupling expansion of 2D quantum gravity. For $d=3$ and $d=4$ we consider the numerical approach to dynamically triangulated gravity. We present the results of simulating pure gravity as well as gravity interacting with matter fields. For $d=4$ we describe in addition the effect of adding to the Einstein Hilbert action a higher derivative term. 
  The leading, planar diagrams of the $1/N_c$ expansion and the usual string description suggest that quarks propagate on the boundary of a two-dimensional world surface. We restrict the quarks to the boundary of the world surface by giving them infinitely large mass on the interior of the surface and zero mass on its boundary and show that in this picture the QCD $\theta$--vacua can be represented by the self-intersection number (or equivalently by the first Chern number of the normal bundle) of the surface. 
  A homological construction of integrals of motion of the classical and quantum Toda field theories is given. Using this construction, we identify the integrals of motion with cohomology classes of certain complexes, which are modeled on the BGG resolutions of the associated Lie algebras and their quantum deformations. This way we prove that all classical integrals of motion can be quantized. For the Toda field theories associated to finite-dimensional Lie algebras, the algebra of integrals of motions is the corresponding W-algebra. For affine Toda field theories this algebra is a commutative subalgebra of a W-algebra; it consists of quantum KdV hamiltonians. 
  We study the minimal supersymmetric standard model derived from the $Z_8$ orbifold models and its hidden sectors. We use a target-space duality anomaly cancellation so as to investigate hidden sectors consistent with the MSSM unification. For the allowed hidden sectors, we estimate the running gauge coupling constants making use of threshold corrections due to the higher massive modes. The calculation is important from the viewpoint of gaugino condensations, which is one of the most promissing mechanism to break the supersymmetry. 
  In this paper general abelian gauge field theories interacting with matter fields are quantized on a closed and orientable Riemann surface $\Sigma$. The approach used is that of small perturbations around topologically nontrivial classical solutions $A^I_\mu$ of the Maxwell equations. The propagator of the gauge fields and the form of the fields $A^I_\mu$ are explicitly computed in the Feynman gauge. Despite of the fact that the quantization procedure presented here is mainly perturbative, in the case of the Schwinger model, or two dimensional quantum electrodynamics, it is also possible to derive nonperturbative results. As an example, we show that, at very short distances, the electromagnetic forces on a Riemann surface vanish up to zero modes. 
  We discuss general properties of the conservation law associated with a local symmetry. Using Noether's theorem and a generalized Belinfante symmetrization procedure in 3+1 dimensions, a symmetric energy-momentum (pseudo) tensor for the gravitational Einstein-Hilbert action is derived and discussed in detail. In 2+1 dimensions, expressions are obtained for energy and angular momentum arising in the $ISO(2,1)$ gauge theoretical formulation of Einstein gravity. In addition, an expression for energy in a gauge theoretical formulation of the string-inspired 1+1 dimensional gravity is derived and compared with the ADM definition of energy. 
  The bracket operation on mutually local BRST classes may be combined with Lorentz invariance and analyticity to write an infinite set of finite difference relations on string scattering amplitudes. When combined with some simple physical criteria these relations uniquely determine the genus zero string $S$-matrix for $N\leq 26$-particle scattering in $\IR^{25,1}$ in terms of a single parameter, $\kappa$, the string coupling. We propose that the high-energy limit of the relations are the Ward identities for the high-energy symmetries of string theory. 
  The induced \6s of the $\kappa$-\1 for the massive case are described. It is shown that it extends many of the features of the classical case. 
  The two ways of resumming the efffective action for the massless test particles in inhomogeneous external field at zero and finite temperature providing the infrared finite answer are discussed. The case of the massive test particles having a mass which is parametrically small with respect to a scale set by the inhomogeneous external field is briefly considered. 
  We derive an exact expression for the tachyon $\beta$-function for the Wess-Zumino-Witten model. We check our result up to three loops by calculating the three-loop tachyon $\beta$-function for a general non-linear $\sigma$-model with torsion, and then specialising to the case of the WZW model. 
  When the parameter of deformation q is a m-th root of unity, the centre of U_q(sl(N))$ contains, besides the usual q-deformed Casimirs, a set of new generators, which are basically the m-th powers of all the Cartan generators of U_q(sl(N)). All these central elements are however not independent. In this letter, generalising the well-known case of U_q(sl(2)), we explicitly write polynomial relations satisfied by the generators of the centre. Application to the parametrization of irreducible representations and to fusion rules are sketched. 
  Recent proofs of the convergence of the linear delta expansion in zero and in one dimensions have been limited to the analogue of the vacuum generating functional in field theory. In zero dimensions it was shown that with an appropriate, $N$-dependent, choice of an optimizing parameter $\l$, which is an important feature of the method, the sequence of approximants $Z_N$ tends to $Z$ with an error proportional to ${\rm e}^{-cN}$. In the present paper we establish the convergence of the linear delta expansion for the connected vacuum function $W=\ln Z$. We show that with the same choice of $\l$ the corresponding sequence $W_N$ tends to $W$ with an error proportional to ${\rm e}^{-c\sqrt N}$. The rate of convergence of the latter sequence is governed by the positions of the zeros of $Z_N$. 
  Through appropriate projections of an exact renormalization group equation, we study fixed points, critical exponents and nontrivial renormalization group flows in scalar field theories in $2<d<4$. The standard upper critical dimensions $d_k={2k\over k-1}$, $k=2,3,4,\ldots$ appear naturally encoded in our formalism, and for dimensions smaller but very close to $d_k$ our results match the $\ee$-expansion. Within the coupling constant subspace of mass and quartic couplings and for any $d$, we find a gradient flow with two fixed points determined by a positive-definite metric and a $c$-function which is monotonically decreasing along the flow. 
  We discuss $d=1, {\cal N}=2$ supersymmetric matrix models and exhibit the associated $d=2$ collective field theory in the limit of dense eigenvalues. From this theory we construct, by the addition of several new fields, a $d=2$ supersymmetric effective field theory, which reduces to the collective field theory when the new fields are replaced with their vacuum expectation values. This effective theory is Poincare invariant and contains perturbative and non-perturbative information about the associated superstrings. We exhibit instanton solutions corresponding to the motion of single eigenvalues and discuss their possible role in supersymmetry breaking. 
  We study some generic aspects of the winding angle distribution around a point in two dimensions for Brownian and self avoiding walks (SAW) using corner transfer matrix and conformal field theory. 
  Topological defects constructed out of scalar fields and possessing chiral fermion zero modes are known to exhibit an anomaly inflow mechanism which cancels the anomaly in the effective theory of the zero modes through an inflow of current from the space in which the defect is embedded. We investigate the analog of this mechanism for defects constructed out of gauge fields in higher dimensions. In particular we analyze this mechanism for string (one-brane) defects in six dimensions and for fivebranes in ten dimensions.} 
  The effective action is derived for a self-interacting theory with a finite fixed $O(2)$ charge at finite temperature in curved spacetime. We obtain the high temperature expansion of the effective action in the weak coupling limit. In the relativistic temperature, we discuss about the phase transition in a homogeneous spacetime. 
  In this paper a two dimensional non-linear sigma model with a general symplectic manifold with isometry as target space is used to study symplectic blowing up of a point singularity on the zero level set of the moment map associated with a quasi-free Hamiltonian action. We discuss in general the relation between symplectic reduction and gauging of the symplectic isometries of the sigma model action. In the case of singular reduction, gauging has the same effect as blowing up the singular point by a small amount. Using the exponential mapping of the underlying metric, we are able to construct symplectic diffeomorphisms needed to glue the blow-up to the global reduced space which is regular, thus providing a transition from one symplectic sigma model to another one free of singularities. 
  We give a quantum field theoretic derivation of the formula obeyed by the Ray-Singer torsion on product manifolds. Such a derivation has proved elusive up to now. We use a BRST formalism which introduces the idea of an infinite dimensional Universal Gauge Fermion, and is of independent interest being applicable to situations other than the ones considered here. We are led to a new class of Fermionic topological field theories. Our methods are also applicable to combinatorially defined manifolds and methods of discrete approximation such as the use of a simplicial lattice or finite elements. The topological field theories discussed provide a natural link between the combinatorial and analytic torsion. 
  We have numerically calculated topological andnon-topological solitons in two spatial dimensions with Chern-Simons term. Their quantum stability, as well as that of the Maxwell vortex, is analyzed by means of bounce instantons which involve three-dimensional strings and non-topological solitons. 
  This thesis describes a new approach to conformal field theory. This approach combines the method of coadjoint orbits with resolutions and chiral vertex operators to give a construction of the correlation functions of conformal field theories in terms of geometrically defined objects. Explicit formulae are given for representations of Virasoro and affine algebras in terms of a local gauge choice on the line bundle associated with geometric quantization of a given coadjoint orbit; these formulae define a new set of explicit bosonic realizations of these algebras. The coadjoint orbit realizations take the form of dual Verma modules, making it possible to avoid the technical difficulties associated with the two-sided resolutions which arise from Feigin-Fuchs and Wakimoto realizations. Formulae are given for screening and intertwining operators on the coadjoint orbit representations. Chiral vertex operators between Virasoro modules are constructed, and related directly to Virasoro algebra generators in certain cases. From the point of view taken in this thesis, vertex operators have a geometric interpretation as differential operators taking sections of one line bundle to sections of another. A suggestion is made that by connecting this description with recent work deriving field theory actions from coadjoint orbits, a deeper understanding of the geometry of conformal field theory might be achieved. 
  We give the formula for a simple Wilson loop on a sphere which is valid for an arbitrary QCD$_2$ saddle-point $\rho(x)$: \mbox{$W(A_1,A_2)=\oint \frac{dx}{2\pi i} \exp(\int dy \frac{\rho(y)}{y-x}+A_2x)$}. The strong-coupling-phase solution is investigated. 
  We present a comprehensive discussion of the consistency of the effective quantum field theory of a single $Z_2$ symmetric scalar field. The theory is constructed from a bare Euclidean action which at a scale much greater than the particle's mass is constrained only by the most basic requirements; stability, finiteness, analyticity, naturalness, and global symmetry. We prove to all orders in perturbation theory the boundedness, convergence, and universality of the theory at low energy scales, and thus that the theory is perturbatively renormalizable in the sense that to a certain precision over a range of such scales it depends only on a finite number of parameters. We then demonstrate that the effective theory has a well defined unitary and causal analytic S--matrix at all energy scales. We also show that redundant terms in the Lagrangian may be systematically eliminated by field redefinitions without changing the S--matrix, and discuss the extent to which effective field theory and analytic S--matrix theory are actually equivalent. All this is achieved by a systematic exploitation of Wilson's exact renormalization group flow equation, as used by Polchinski in his original proof of the renormalizability of conventional $\varphi^4$-theory. 
  The $CP^{N-1}$ model is quantised in the generalised canonical formalism of Batalin and Tyutin by converting the original second class system into first class. Operator ordering ambiguities present in the conventional quantisation scheme of Dirac are thereby avoided. The first class constraints, the involutive Hamiltonian and the BRST charge are explicitly computed. The partition function is defined and evaluated in the unitary gauge. 
  We quantise the $O(N)$ nonlinear sigma model using the Batalin Tyutin (BT) approach of converting a second class system into first class. It is a {\it nontrivial} application of the BT method since the quantisation of this model by the conventional Dirac procedure suffers from operator ordering ambiguities. The first class constraints, the BRST Hamiltonian and the BRST charge are explicitly computed. The partition function is constructed and evaluated in the unitary gauge and a multiplier (ghost) dependent gauge. 
  Lipman Bers' universal Teichm\"uller space, classically denoted by $T(1)$, plays a significant role in Teichm\"uller theory, because all the Teichm\"uller spaces $T(G)$ of Fuchsian groups $G$ can be embedded into it as complex submanifolds. Recently, $T(1)$ has also become an object of intensive study in physics, because it is a promising geometric environment for a non-perturbative version of bosonic string theory. We provide a non-technical survey of what is currently known about the geometry of $T(1)$ and what is conjectured about its physical meaning. 
  We calculate two- and three-point tachyon amplitudes of the SL(2,R)/U(1) two-dimensional Euclidean black hole for spherical topologies in the continuum approach proposed by Bershadsky and Kutasov. We find an interesting relation to the tachyon scattering amplitudes of standard non-critical string theory. 
  A class of 2-dimensional models including 2-d dilaton gravity, spherically symmetric reduction of d-dimensional Einstein gravity and other related theories are classically analyzed. The general analytic solutions in the absence of matter fields other than a U(1) gauge field are obtained under a new gauge choice and recast in the conventional conformal gauge. These solutions imply that Birkhoff's theorem, originally applied to spherically symmetric 4-d Einstein gravity, can be applied to all models we consider. Some issues related to the coupling of massless scalar fields and the quantization are briefly discussed. 
  We introduce the massive gauge invariant, second order pure spin-3 theory in three dimensions. It consists of the addition of the second order gauge invariant massless pure spin-3 action with the first order topological(generalized) Chern-Simons spin-3 term corrected with lower spin auxiliary actions which avoid lower spin ghosts propagation. This second order intermediate action completes the catalogue of massive spin-3 actions having topological structure. We also consider its spontaneous break down through the addition of the inertial spin-3 nontopological Fierz-Pauli mass term. It is shown that this non gauge invariant pure spin-3 system is the uniform generalization of linearized massive vector Chern-Simons gravity and propagates just two spin $3^{\pm}$ excitations having different masses. 
  We investigate the spectral properties of a random matrix model, which in the large $N$ limit, embodies the essentials of the QCD partition function at low energy. The exact spectral density and its pair correlation function are derived for an arbitrary number of flavors and zero topological charge. Their microscopic limit provide the master formulae for sum rules for the inverse powers of the eigenvalues of the QCD Dirac operator as recently discussed by Leutwyler and Smilga. 
  We study the scattering of Skyrmions at low energy and large separation using the method proposed by Manton of truncation to a finite number of degree freedom. We calculate the induced metric on the manifold of the union of gradient flow curves, which for large separation, to first non-trivial order is parametrised by the variables of the product ansatz. 
  There is a completely natural and intimate relationship between the diffeomorphism group of the circle and the Teichm\"uller spaces of Riemann surfaces discovered by us in 1988. Such a relationship had been sought-after by physicists from conjectures connecting the loop-space approach to string theory with the path-integral approach. Precisely, the remarkable homogeneous space Diff$(S^1)$/$SL(2,R)$ (which is one of the two possible quantizable coadjoint orbits of Diff$(S^1)$), embeds as a complex analytic and K\"ahler submanifold of the universal Teichm\"uller space. Furthermore, this very homogeneous space, Diff$(S^1)$/$SL(2,R)$, considered by the previous work as a K\"ahler submanifold of the universal Teichm\"uller space, allows on it a natural holomorphic period mapping, $\Pi$, that generalises the classical map associating to a genus $g$ Riemann surface its period matrix. Utilising the fact that the group of quasiconformal homeomorphisms of $S^1$ acts symplectically on the Sobolev space of order $1/2$ on the circle, we (with Dennis Sullivan) have recently extended $\Pi$ to the entire universal Teichm\"uller space. All this is related to non-perturbative string theory. 
  We study the scaling limit of a model of a tethered crumpled D-dimensional random surface interacting through an exclusion condition with a fixed impurity in d-dimensional Euclidean space by the methods of Wilson's renormalization group. In this paper we consider a hierarchical version of the model and we prove rigorously the existence of the scaling limit and convergence to a non-Gaussian fixed point for $1 \leq D< 2$ and $\epsilon >0$ sufficiently small, where $\epsilon = D - (2-D) {d\over 2}$. 
  For all three--dimensional Lie algebras the construction of generators in terms of functions on 4-dimensional real phase space is given with a realization of the Lie product in terms of Poisson brackets. This is the classical Jordan--Schwinger map which is also given for the deformed algebras ${\cal {SL}}_{q}(2,\R)$, ${\cal E}_{q} (2)$ and ${\cal H}_q(1)$. The ${\cal U}_{q}(n)$ algebra is discussed in the same context. 
  Using the theory of fibre bundles, we provide several equivalent intrinsic descriptions for the Hilbert spaces of $n$ ``free'' nonrelativistic and relativistic plektons in two space dimensions. These spaces carry a ray representation of the Galilei group and a unitary representation of the Poincar\'{e} group respectively. In the relativistic case we also discuss the situation where the braid group is replaced by the ribbon braid group. 
  In this paper, the massless Schwinger model or two dimensional quantum electrodynamics is exactly solved on a Riemann surface. The partition function and the generating functional of the correlation functions involving the fermionic currents are explicitly derived using a method of quantization valid for any abelian gauge field theory and explained in the recent references [F. Ferrari, {\it Class. Quantum Grav.} {\bf 10} (1993), 1065], [F. Ferrari, hep-th 9310024]. In this sense, the Schwinger model is one of the few examples of interacting and nontopological field theories that are possible to quantize on a Riemann surface. It is also shown here that the Schwinger model is equivalent to a nonlocal integrable model which represents a generalization of the Thirring model. Apart from the possible applications in string theory and integrable models, we hope that this result can be also useful in the study of quantum field theories in curved space-times. 
  A generalized theory of two-dimensional isotropic turbulence is developed based on conformal symmetry. A number of minimal models of conformal turbulence are solved under an extended constraint including both the enstrophy cascade by Kraichnan and the discontinuity of vorticity by Saffman. There are an infinite number of solutions which fall into two different categories. An explicit relation is derived in one of the categories between the central charge of Virasoro algebra, the lowest anomalous dimension and the power of the energy spectrum. Some statistical properties such as energy spectrum, skewness, flatness and Casimir invariants are predicted and compared with numerical simulation by the pseudospectral method. 
  The application of the Tomita-Takesaki modular theory to the Haag-Kastler net approach in QFT yields external (space-time) symmetries as well as internal ones (internal ``gauge para-groups") and their dual counterparts (the ``super selection para-group"). An attempt is made to develop a (speculative) picture on ``quantum symmetry" which links space-time symmetries in an inexorable way with internal symmetries. In the course of this attempt, we present several theorems and in particular derive the Kac-Wakimoto formula which links Jones inclusion indices with the asymptotics of expectation values in physical temperature states. This formula is a special case of a new asymptotic Gibbs-state representation of mapping class group matrices (in a Haag-Kastler net indexed by intervals on the circle!) as well as braid group matrices. 
  These lecture notes provide an elementary introduction to the study of massless integrable quantum field theory in 1+1 dimensions using ``massless scattering''. Some previously unpublished results are also presented, including a non-perturbative study of Virasoro conserved quantities. 
  We consider a two-dimensional model of gravity with the cosmological constant as a dynamical variable. The effective cosmological constant is derived when the universe has no initial boundary. It turns out to be extremely small if the universe is sufficiently large. 
  Reported are two applications of the functional relations ($T$-system) among a commuting family of row-to-row transfer matrices proposed in the previous paper Part I. For a general simple Lie algebra $X_r$, we determine the correlation lengths of the associated massive vertex models in the anti-ferroelectric regime and central charges of the RSOS models in two critical regimes. The results reproduce known values or even generalize them, demonstrating the efficiency of the $T$-system. 
  It is shown how the theory of classical $W$--algebras can be formulated on a higher genus Riemann surface in the spirit of Krichever and Novikov. An intriguing relation between the theory of $A_1$ embeddings into simple Lie algebras and the holomorphic geometry of Riemann surfaces is exihibited. 
  We show how to formulate $2$-dimensional supersymmetric $N=1,2$ theories, both massive and conformal, within a manifestly supersymmetric hamiltonian framework, via the introduction of a (super)-Poisson brackets structure defined on superfields. In this approach, as distinct from the previously known superfield hamiltonian formulations, the dynamics is not separated into two unrelated $2D$ light-cone superspaces, but is recovered by specifying boundary conditions at a given ``super-time" coordinate. So the approach proposed provides a natural generalization of canonical hamiltonian formalism. One of its interesting features is that the physical and auxiliary fields equations appear on equal footing as the Hamilton ones. 
  By considering the scaling behaviour of various Feynman graphs at leading order in large $\Nf$ at the non-trivial fixed point of the $d$-dimensional $\beta$-function of QCD we deduce the critical exponents corresponding to the quark, gluon and ghost anomalous dimensions as well as the anomalous dimensions of the quark-quark-gluon and ghost-ghost-gluon vertices in the Landau gauge. As the exponents encode all orders information on the perturbation series of the corresponding renormalization group functions we find agreement with the known three loop structure and, moreover, we provide new information at all subsequent orders. 
  The spontaneous symmetry breaking theory of gravity is examined, assuming that the vacuum expectation value of the standard model Higgs is also responsible for the generation of the Planck mass. In this model the physical Higgs couples only with gravitational strength to matter. At presently accessible energies the theory is indistinguishable from the standard model without Higgs boson and is in agreement with all existing data.At energies above the Fermi scale new dynamics should occur. 
  Time evolution of the expectation values of various dynamical operators of the harmonic oscillator with dissipation is analitically obtained within the framework of the Lindblad's theory for open quantum systems. We deduce the density matrix of the damped harmonic oscillator from the solution of the Fokker-Planck equation for the coherent state representation, obtained from the master equation for the density operator. The Fokker-Planck equation for the Wigner distribution function is also solved by using the Wang-Uhlenbeck method of transforming it into a linearized partial differential equation for the Wigner function, subject to either the Gaussian type or the $\delta$-function type of initial conditions. The Wigner functions which we obtain are two-dimensional Gaussians with different widths. 
  An Ansatz for the Poincar\'e metric on compact Riemann surfaces is proposed. This implies that the Liouville equation reduces to an equation resembling a non chiral analogous of the higher genus relationships (KP equation) arising in the framework of Schottky's problem solution. This approach connects uniformization (Fuchsian groups) and moduli space theories with KP hierarchy. Besides its mathematical interest, the Ansatz has some applications in the framework of quantum Riemann surfaces arising in 2D gravity. 
  The moduli space of the Calabi-Yau three-folds, which play a role as superstring ground states, exhibits the same {\em special geometry} that is known from nonlinear sigma models in $N=2$ supergravity theories. We discuss the symmetry structure of special real, complex and quaternionic spaces. Maps between these spaces are implemented via dimensional reduction. We analyze the emergence of {\em extra} and {\em hidden} symmetries. This analysis is then applied to homogeneous special spaces and the implications for the classification of homogeneous quaternionic spaces are discussed. 
  We give a constructive account of the fundamental ingredients of Poisson Lie theory as the basis for a description of the classical double group $D$. The double of a group $G$ has a pointwise decomposition $D\sim G\times G^*$, where $G$ and $G^*$ are Lie subgroups generated by dual Lie algebras which form a Lie bialgebra. The double is an example of a factorisable Poisson Lie group, in the sense of Reshetikhin and Semenov-Tian-Shansky [1], and usually the study of its Poisson structures is developed only in the case when the subgroup $G$ is itself factorisable. We give an explicit description of the Poisson Lie structure of the double without invoking this assumption. This is achieved by a direct calculation, in infinitesimal form, of the dressing actions of the subgroups on each other, and provides a new and general derivation of the Poisson Lie structure on the group $G^*$. For the example of the double of SU(2), the symplectic leaves of the Poisson Lie structures on SU(2) and SU(2$)^*$ are displayed. 
  We review aspects of spacetime singularities from the view point of string theory. Examples considered include cosmological, cosmic string and black-hole singularities. We also discuss the consistency of viewing black-holes as excited states of fundamental strings (based on talk presented at Salamfest, March 1993, Trieste). 
  A quantum generalization of Rogers' five term, or ``pentagon'' dilogarithm identity is suggested. It is shown that the classical limit gives usual Rogers' identity. The case where the quantum identity is realized in finite dimensional space is also considered and the quantum dilogarithm is constructed as a function on Fermat curve, while the identity itself is equivalent to the restricted star-triangle relation introduced by Bazhanov and Baxter. 
  We describe the harmonic superspace formulation of the Witten-Manin supertwistor correspondence for N=3 extended super Yang-Mills theories. The essence is that on being sufficiently supersymmetrised (up to the N=3 extension), the Yang-Mills equations of motion can be recast in the form of Cauchy-Riemann-like holomorphicity conditions for a pair of prepotentials in the appropriate harmonic superspace. This formulation makes the explicit construction of solutions a rather more tractable proposition than previous attempts. 
  In this talk we review the harmonic space formulation of the twistor transform for the supersymmetric self-dual Yang-Mills equations. The recently established harmonic-twistor correspondence for the N-extended supersymmetric gauge theories is described. It affords an explicit construction of solutions to these equations which displays a remarkable matreoshka-like structure determined by the N=0 core. 
  {\bf Exact} solutions of the string equations of motion and constraints are {\bf systematically} constructed in de Sitter spacetime using the dressing method of soliton theory. The string dynamics in de Sitter spacetime is integrable due to the associated linear system. We start from an exact string solution $q_{(0)}$ and the associated solution of the linear system $\Psi^{(0)} (\lambda)$, and we construct a new solution $\Psi(\lambda)$ differing from $\Psi^{(0)}(\lambda)$ by a rational matrix in $\lambda$ with at least four poles $\lambda_{0},1/\lambda_{0},\lambda_{0}^*,1/\lambda_{0}^*$. The periodi- city condition for closed strings restrict $\lambda _{0}$ to discrete values expressed in terms of Pythagorean numbers. Here we explicitly construct solu- tions depending on $(2+1)$-spacetime coordinates, two arbitrary complex numbers (the 'polarization vector') and two integers $(n,m)$ which determine the string windings in the space. The solutions are depicted in the hyperboloid coor- dinates $q$ and in comoving coordinates with the cosmic time $T$. Despite of the fact that we have a single world sheet, our solutions describe {\bf multi- ple}(here five) different and independent strings; the world sheet time $\tau$ turns to be a multivalued function of $T$.(This has no analogue in flat space- time).One string is stable (its proper size tends to a constant for $T\to\infty $, and its comoving size contracts); the other strings are unstable (their proper sizes blow up for $T\to\infty$, while their comoving sizes tend to cons- tants). These solutions (even the stable strings) do not oscillate in time. The interpretation of these solutions and their dynamics in terms of the sinh- Gordon model is particularly enlighting. 
  Representations of propagators by means of path integrals over velocities are discussed both in nonrelativistic and relativistic quantum mechanics. It is shown that all the propagators can only be expressed through bosonic path integrals over velocities of space-time coordinates. In the representations the integration over velocities is not restricted by any boundary conditions; matrices, which have to be inverted in course of doing Gaussian integrals, do not contain any derivatives in time, and spinor and isospinor structures of the propagators are given explicitly. One can define universal Gaussian and quasi-Gaussian integrals over velocities and rules of handling them. Such a technique allows one effectively calculate propagators in external fields. Thus, Klein-Gordon propagator is found in a constant homogeneous electromagnetic field and its combination with a plane wave field. 
  We review and extend the computation of scattering amplitudes of tachyons in the $c=1$ matrix model using a manifestly finite prescription for the collective field hamiltonian. We give further arguments for the exactness of the cubic hamiltonian by demonstrating the equality of the loop corrections in the collective field theory with those calculated in the fermionic picture. 
  We study the hamiltonian reduction of affine Lie superalgebra $sl(2|1)^{(1)}$. Based on a scalar Lax operator formalism, we derive the free field realization of the classical topological topological algebra which appears in the $c\leq1$ non-critical strings. In the quantum case, we analyze the BRST cohomology to get the quantum free field expression of the algebra. 
  The topological coset model appraoch to non-critical string models is summarized. The action of a topological twisted ${G\over H}$ coset model ($rank\ H = rank\ G$) is written down. A ``topological coset algebra" is derived and compared with the algebraic structure of the $N=2$ twisted models.   The cohomology on a free field Fock space as well as on the space of irreducible representation of the ``matter" \ala are extracted. We compare the results of the $A_1^{(N-1)}$ at level $k={p\over q}-N$ with those of $(p,q)$ $W_N$ strings. A ${SL(2,R)\over SL(2,R)}$ model which corresponds to the $c=1$ is written down. A similarity transformation on the BRST charge enables us to extract the full BRST cohomoloy. One to one correspondence between the physical states of the $c=1$ and the corresponding coset model is found. Talk presented at ``Strings' 93", May 93 Berekely.} 
  The purpose of this paper is to give a reciprocity between $U_q(h)$ and $\Cal H_{n,r}$, the Hecke algebra of $(\Bbb Z / r\Bbb Z)\wr \frak S_n$ introduced by Ariki and Koike. Let $K=\Bbb Q(q,u_1,\dots ,u_r)$ be the field of rational funcitons in variables $q,u_1,\dots ,u_r$. We adopt $K$ as the base field for both the quantized universal enveloping algebra $U_q(gl_r)$ and the Hecke algebra $\Cal H_n$. We denote by $U_q(h)$ the $K$-subalgebra of $U_q(gl_r)$ generated by $q^{E_{ii}}\;$'s $(1\le i \le r)$. In this paper, we show that the commutant of $U_q(h)$ in $End((K^r)^{\otimes n})$ is isomorphic to a quotient of $\Cal H_{n,r}$. We also determine the irreducible decomposition of $(K^r)^{\otimes n}$ under the action of $\Cal H_{n,r}$. As a consequence, we obtain the reciprocity for $U_q(h)$ and $\Cal H_{n,r}$. 
  I construct the ground state, up to first nonperturbative order, of the stochastic stabilization of the zero dimensional matrix model which defines 2D Quantum Gravity. It is given by the linear combination of a perturbative wave function and a nonperturbative one. The nonperturbative behaviour which arise from the stabilized model and from the string equation are similar. I show the modification of the loop equation by nonperturbative contribution. 
  It is shown that generally the consistency equation for anomalies of quantum field theories has solutions which depend nontrivially on the sources of the (generalized) BRS-transformations of the fields. Explicit previously unknown examples of such solutions are given for Yang-Mills and super Yang-Mills theories. 
  The extension of strongly anisotropic or dynamical scaling to local scale invariance is investigated. For the special case of an anisotropy or dynamical exponent $\theta=z=2$, the group of local scale transformation considered is the Schr\"odinger group, which can be obtained as the non-relativistic limit of the conformal group. The requirement of Schr\"odinger invariance determines the two-point function in the bulk and reduces the three-point function to a scaling form of a single variable. Scaling forms are also derived for the two-point function close to a free surface which can be either space-like or time-like. These results are reproduced in several exactly solvable statistical systems, namely the kinetic Ising model with Glauber dynamics, lattice diffusion, Lifshitz points in the spherical model and critical dynamics of the spherical model with a non-conserved order parameter. For generic values of $\theta$, evidence from higher order Lifshitz points in the spherical model and from directed percolation suggests a simple scaling form of the two-point function. 
  The discovery of integrable $N=2$ supersymmetric Landau-Ginzburg theories whose chiral rings are fusion rings suggests a close connection between fusion rings, the related Landau-Ginzburg superpotentials, and $N=2$ quantum integrability. We examine this connection by finding the natural $SO(N)_K$ analogue of the construction that produced the superpotentials with $Sp(N)_K$ and $SU(N)_K$ fusion rings as chiral rings. The chiral rings of the new superpotentials are not directly the fusion rings of any conformal field theory, although they are natural quotients of the tensor subring of the $SO(N)_K$ fusion ring.   The new superpotentials yield solvable (twisted $N=2$) topological field theories. We obtain the integer-valued correlation functions as sums of $SO(N)_K$ Verlinde dimensions by expressing the correlators as fusion residues. The $SO(2n+1)_{2k+1}$ and $SO(2k+1)_{2n+1}$ related topological Landau-Ginzburg theories are isomorphic, despite being defined via quite different superpotentials. 
  We consider correlation functions for the Wess-Zumino-Witten model on the torus with the insertion of a Cartan element; mathematically this means that we consider the function of the form $F=\Tr (\Phi_1 (z_1)\ldots \Phi_n (z_n)q^{-\d}e^{h})$ where $\Phi_i$ are intertwiners between Verma modules and evaluation modules over an affine Lie algebra $\ghat$, $\d$ is the grading operator in a Verma module and $h$ is in the Cartan subalgebra of $\g$. We derive a system of differential equations satisfied by such a function. In particular, the calculation of $q\frac{\d} {\d q} F$ yields a parabolic second order PDE closely related to the heat equation on the compact Lie group corresponding to $\g$. We consider in detail the case $n=1$, $\g = \sltwo$. In this case we get the following differential equation ($q=e^{\pi \i \tau}$): $ \left( -2\pi\i (K+2)\frac{\d}{\d\tau} +\frac{\d^2}{\d x^2}\right) F = (m(m+1)\wp(x+\frac{\tau}{2}) +c)F$, which for $K=-2$ (critical level) becomes Lam\'e equation. For the case $m\in\Z$ we derive integral formulas for $F$ and find their asymptotics as $K\to -2$, thus recovering classical Lam\'e functions. 
  We consider the quantum Lobachevsky space ${\bf L}_q^3$, which is defined as subalgebra of the Hopf algebra ${\cal A}_q(SL_2({\bf C}))$. The Iwasawa decomposition of ${\cal A}_q(SL_2({\bf C}))$ introduced by Podles and Woronowicz allows to consider the quantum analog of the horospheric coordinates on ${\bf L}_q^3$. The action of the Casimir element, which belongs to the dual to ${\cal A}_q$ quantum group $U_q(SL_2({\bf C}))$, on some subspace in ${\bf L}_q^3$ in these coordinates leads to a second order difference operator on the infinite one-dimensional lattice. In the continuos limit $q\rightarrow 1$ it is transformed into the Schr\"{o}dinger Hamiltonian, which describes zero modes into the Liouville field theory (the Liouville quantum mechanics). We calculate the spectrum (Brillouin zones) and the eigenfunctions of this operator. They are $q$-continuos Hermit polynomials, which are particular case of the Macdonald or Rogers-Askey-Ismail polynomials. The scattering in this problem corresponds to the scattering of first two level dressed excitations in the $Z_N$ Baxter model in the very peculiar limit when the anisotropy parameter $\ga$ and $N~\rightarrow\infty$, or, equivalently, $(\ga, N)\rightarrow 0$. 
  The partition function and the order parameter for the chiral symmetry breaking are computed for a family of 2-dimensional interacting theories containing the gauged Thirring model. In particular we derive non-perturbative expressions for the dependence of the chiral condensate on the temperature and the curvature. Both, high temperature and high curvature supress the condensate exponentially and we can associate an effective temperature to the curvature. 
  Series for the Wilson functions of an ``environmentally friendly'' renormalization group are computed to two loops, for an $O(N)$ vector model, in terms of the ``floating coupling'', and resummed by the Pad\'e method to yield crossover exponents for finite size and quantum systems. The resulting effective exponents obey all scaling laws, including hyperscaling in terms of an effective dimensionality, ${d\ef}=4-\gl$, which represents the crossover in the leading irrelevant operator, and are in excellent agreement with known results. 
  We present a topological Lagrangian field theory that is geometrically similar to the Yang-Mills(-Higgs) Lagrangian, and study the Bogomol'nyi solitons contained within this theory. The topological field theory may provide an example of a dual field theory to Yang-Mills(-Higgs). The existence of a dual field theory to Yang-Mills(-Higgs) was conjectured by Montonen and Olive. 
  This is the extended version of the preprint \ct{Loop}, based on the lectures given in Cargese Summer School and Chernogolovka Summer School in 93. The incompressible fluid dynamics is reformulated as dynamics of closed loops $C$ in coordinate space. We derive explicit functional equation for the pdf of the circulation $P_C(\Gamma)$ which allows the scaling solutions in inertial range of spatial scales. The pdf decays as exponential of some power of $ \Gamma^3/A^2 $ where $A$ is the minimal area inside the loop. 
  Vertex operators associated with level two $U_q(\widehat{sl}_2)$ modules are constructed explicitly using bosons and fermions. An integral formula is derived for the trace of products of vertex operators. These results are applied to give $n$-point spin correlation functions of an integrable $S=1$ quantum spin chain, extending an earlier work of Jimbo {\em et al} for the case $S=1/2$. 
  We study the stochastic quantization of the system with first class constraints in phase space. Though the Langevin equations of the canonical variables are defined without ordinary gauge fixing procedure, gauge fixing conditions are automatically selected and introduced by imposing stochastic consistency conditions upon the first class constraints. Then the equilibrium solution of the Fokker-Planck equation is identical with corresponding path integral distribution. 
  I present a $q$-analog of the discrete Painlev\'e I equation, and a special realization of it in terms of $q$-orthogonal polynomials. 
  The usual extensions of supersymmetry require the existence of a complex structure and the formulation of the theory on K\"{a}hler manifolds. It is shown, that by relaxing the constraints on the algebra of supercharges we can get new supersymmetries whenever a manifold possesses a structure admitting the existence of a Killing-Yano tensor field. Examples of such manifolds are the Kerr-Newman space-times describing spinning black holes in four dimensions. 
  We analyze a possibility of experimental detection of the contribution of the Kaluza-Klein tower of heavy particles to scattering cross-section in a six-dimensional scalar model with two dimensions being compactified to the torus with the radii $R$. It is shown that there is a noticeable effect even for the energies of colliding particles below $R^{-1}$ which may be observed in future collider experiments if $R^{-1}$ is of the order of $1 TeV$. 
  We present a generalization of the non-Abelian version of the $CP^{N-1}$ models (also known as Grassmannian models) that involve composite gauge fields to accommodate partial breaking of the non-Abelian gauge symmetry. For this to be possible, in most cases, the constituent fields need to belong to an anomaly free complex representation. Symmetry is broken dynamically for large $N$ primarily by a naturally generated composite scalar which simulates Higgs mechanism. In the example studied in some detail, the gauge group SO(10) gets broken down to subgroups like SU(5) or SU(5)$\times$U(1). 
  The spectrum and partition function of a model consisting of SU(n) spins positioned at the equilibrium positions of a classical Calogero model and interacting through inverse-square exchange are derived. The energy levels are equidistant and have a high degree of degeneracy, with several SU(n) multiplets belonging to the same energy eigenspace. The partition function takes the form of a q-deformed polynomial. This leads to a description of the system by means of an effective parafermionic hamiltonian, and to a classification of the states in terms of "modules" consisting of base-n strings of integers. 
  We consider a generalization of the abelian Higgs model with a Chern-Simons term by modifying two terms of the usual Lagrangian. We multiply a dielectric function with the Maxwell kinetic energy term and incorporate nonminimal interaction by considering generalized covariant derivative. We show that for a particular choice of the dielectric function this model admits both topological as well as nontopological charged vortices satisfying Bogomol'nyi bound for which the magnetic flux, charge and angular momentum are not quantized. However the energy for the topolgical vortices is quantized and in each sector these topological vortex solutions are infinitely degenerate. In the nonrelativistic limit, this model admits static self-dual soliton solutions with nonzero finite energy configuration. For the whole class of dielectric function for which the nontopological vortices exists in the relativistic theory, the charge density satisfies the same Liouville equation in the nonrelativistic limit. 
  Based on the representation theory of the $q$-deformed Lorentz and Poincar\'e symmeties $q$-deformed relativistic wave equation are constructed. The most important cases of the Dirac-, Proca-, Rarita-Schwinger- and Maxwell- equations are treated explicitly. The $q$-deformed wave operators look structurally like the undeformed ones but they consist of the generators of a non-commu\-ta\-tive Minkowski space. The existence of the $q$-deformed wave equations together with previous existence of the $q$-deformed wave equations together with previous results on the representation theory of the $q$-deformed Poincar\'e symmetry solve the $q$-deformed relativistic one particle problem. 
  We apply the recently proposed transfer matrix formalism to 2-dimensional quantum gravity coupled to $(2, 2k-1)$ minimal models. We find that the propagation of a parent universe in geodesic (Euclidean) time is accompanied by continual emission of baby universes and derive a distribution function describing their sizes. The $k\to \infty~ (c\to -\infty)$ limit is generally thought to correspond to classical geometry, and we indeed find a classical peak in the universe distribution function. However, we also observe dramatic quantum effects associated with baby universes at finite length scales. 
  We study dynamics of non-relativistic Chern-Simons solitons, both in the absence and in the presence of external fields. We find that a phase, related to the $1$-cocyle of the Galileo group, must be included to give the correct dynamical behavior. We show that interactions among Chern-Simons solitons are mediated by an effective Chern-Simons gauge field induced by the solitons. In the two soliton case, we evaluate analytically the effective interaction Lagrangian, which previously was found numerically. 
  We review the construction of the cross product algebra $\A\rtimes\U$ from two dually paired Hopf algebras $\U$ and $\A$. The canonical element in $\U \otimes \A$ is then introduced, and its properties examined. We find that it is useful for giving coactions on $\A\rtimes\U$, and it allows the construction of objects with specific invariance properties under these coactions. A ``vacuum operator'' is found which projects elements of $\A\rtimes\U$ onto said objects. We then discuss bicovariant vector fields in the context of the canonical element. 
  The information loss and remnant proposals for resolving the black hole information paradox are reconsidered. It is argued that in typical cases information loss implies energy loss, and thus can be thought of in terms of coupling to a spectrum of ``fictitious'' remnants. This suggests proposals for information loss that do not imply planckian energy fluctuations in the low energy world. However, if consistency of gravity prevents energy non-conservation, these remnants must then be considered to be real. In either case, the catastrophe corresponding to infinite pair production remains a potential problem. Using Reissner-Nordstrom black holes as a paradigm for a theory of remnants, it is argued that couplings in such a theory may give finite production despite an infinite spectrum. Evidence for this is found in analyzing the instanton for Schwinger production; fluctuations from the infinite number of states lead to a divergent stress tensor, spoiling the instanton calculation. Therefore naive arguements for infinite production fail. 
  In this work, the following conjectures are proven in the case of a Riemann surface with abelian group of symmetry: a) The $b-c$ systems on a Riemann surface $M$ are equivalent to a multivalued field theory on the complex plane if $M$ is represented as an algebraic curve; b) the amplitudes of the $b-c$ systems on a Riemann surface $M$ with discrete group of symmetry can be derived from the operator product expansions on the complex plane of an holonomic quantum field theory a la Sato, Jimbo and Miwa. To this purpose, the solutions of the Riemann-Hilbert problem on an algebraic curve with abelian monodromy group obtained by Zamolodchikov, Knizhnik and Bershadskii-Radul are used in order to expand the $b-c$ fields in a Fourier-like basis. The amplitudes of the $b-c$ systems on the Riemann surface are then recovered exploiting simple normal ordering rules on the complex plane. 
  The three-body problem in one-dimension with a repulsive inverse square potential between every pair was solved by Calogero. Here, the known results of supersymmetric quantum mechanics are used to propose a number of new three-body potentials which can be solved algebraically. Analytic expressions for the eigenspectrum and the eigenfunctions are given with and without confinement. 
  Using the ideas of supersymmetry and shape invariance we show that the eigenvalues and eigenfunctions of a wide class of noncentral potentials can be obtained in a closed form by the operator method. This generalization considerably extends the list of exactly solvable potentials for which the solution can be obtained algebraically in a simple and elegant manner. As an illustration, we discuss in detail the example of the potential $$V(r,\theta,\phi)={\omega^2\over 4}r^2 + {\delta\over r^2}+{C\over r^2 sin^2\theta}+{D\over r^2 cos^2\theta} + {F\over r^2 sin^2\theta sin^2 \alpha\phi} +{G\over r^2 sin^2\theta cos^2\alpha\phi}$$ with 7 parameters.Other algebraically solvable examples are also given. 
  We show that the string representation of the QCD$_2$ partition function satisfies, by virtue of a Young-tableau-transposition symmetry, the topological constraint that any branched covering of an orientable or nonorientable surface without boundary must have an {\em even} branch point multiplicity. This statement holds for each chiral sector and requires multiple branch point behavior for the twist points, since cross-terms appear that couple twist points with odd powers of simple branch points. We obtain the same result for the complete partition function of $\son$ and $\spn$ Yang-Mills$_2$ theory. 
  For three anyons confined in a harmonic oscillator, only the class of states that interpolate nonlinearly with the statistical parameter contributes to the third virial coefficient of a free anyon gas. Rather than evaluating the full three-body partition function as was done in an earlier publication (Phys. Rev. {\bf A46}, 4693 (1992)), here only the nonlinear contribution is calculated, thus avoiding delicate cancellations between the irrelevant linear part and the two-body partition function. Our numerical results are consistent with the simple analytical form suggested recently by Myrheim and Olaussen. 
  The immersion of the string world sheet, regarded as a Riemann surface, in $R^3$ and $R^4$ is described by the generalized Gauss map. When the Gauss map is harmonic or equivalently for surfaces of constant mean curvature, we obtain Hitchin's self-dual equations, by using $SO(3)$ and $SO(4)$ gauge fields constructed in our earlier studies. This complements our earlier result that $h\surd g\ =\ 1$ surfaces exhibit Virasaro symmetry. The self-dual system so obtained is compared with self-dual Chern-Simons system and a generalized Liouville equation involving extrinsic geometry is obtained.   The immersion in $R^n, \ n>4$ is described by the generalized Gauss map. It is shown that when the Gauss map is harmonic, the mean curvature of the immersed surface is constant. $SO(n)$ gauge fields are constructed from the geometry of the surface and expressed in terms of the Gauss map. It is found Hitchin's self- duality relations for the gauge group $SO(2)\times SO(n-2)$. 
  Using free field representation of quantum affine algebra $U_q(\widehat{sl_2})$, we investigate the structure of the Fock modules over $U_q(\widehat{sl_2})$. The analisys is based on a $q$-analog of the BRST formalism given by Bernard and Felder in the affine Kac-Moody algebra $\widehat {sl_2}$. We give an explicit construction of the singular vectors using the BRST charge. By the same cohomology analysis as the classical case ($q=1$), we obtain the irreducible highest weight representation space as a nontrivial cohomology group. This enables us to calculate a trace of the $q$-vertex operators over this space. 
  We show that uncertainty relations, as well as minimum uncertainty coherent and squeezed states, are structural properties for diffusion processes. Through Nelson stochastic quantization we derive the stochastic image of the quantum mechanical coherent and squeezed states. 
  We generalize the previously established connection between the off-shell Bethe ansatz equation for inhomogeneous SU(2) lattice vertex models in the quasiclassical limit and the solutions of the SU(2) Knizhnik-Zamolodchikov equations to the case of arbitrary simple Lie algebras. 
  We consider a model which effectively restricts the functional integral of Yang--Mills theories to the fundamental modular region. Using algebraic arguments, we prove that this theory has the same divergences as ordinary Yang Mills theory in the Landau gauge and that it is unitary. The restriction of the functional integral is interpreted as a kind of spontaneous breakdown of the $BRS$ symmetry. 
  We present a conformal field theory which desribes a homogeneous four dimensional Lorentz-signature space-time. The model is an ungauged WZW model based on a central extension of the Poincar\'e algebra. The central charge of this theory is exactly four, just like four dimensional Minkowski space. The model can be interpreted as a four dimensional monochromatic plane wave. As there are three commuting isometries, other interesting geometries are expected to emerge via $O(3,3)$ duality. 
  This is mainly a brief review of some key achievements in a `hot'' area of theoretical and mathematical physics. The principal aim is to outline the basic structures underlying {\em integrable} quantum field theory models with {\em infinite-dimensional} symmetry groups which display a radically new type of {\em quantum group} symmetries. Certain particular aspects are elaborated upon with some detail: integrable systems of Kadomtsev-Petviashvili type and their reductions appearing in matrix models of strings; Hamiltonian approach to Lie-Poisson symmetries; quantum field theory approach to two-dimensional relativistic integrable models with dynamically broken conformal invariance. All field-theoretic models in question are of primary relevance to diverse branches of physics ranging from nonlinear hydrodynamics to string theory of fundamental particle interactions at ultra-high energies. 
  Photons propagating in curved spacetime may, depending on their direction and polarisation, have velocities exceeding the ``speed of light'' c. This phenomenon arises through vacuum polarisation in QED and is a tidal gravitational effect depending on the local curvature. It implies that the Principle of Equivalence does not hold for interacting quantum field theories in curved spacetime and reflects a quantum violation of local Lorentz invariance. These results are illustrated for the propagation of photons in the Reissner-Nordstr\"om spacetime characterising a charged black hole. A general analysis of electromagnetic as well as gravitational birefringence is presented. 
  A very basic introduction is given to the r\^oles of division algebras and the related sphere algebras concerning the structure of space-time in the dimensionalities $D\is 3,4,6$ and $10$, with special emphasis on twistors transformations for light-likeness conditions and Hopf maps, together with some outlook for particle and string theory. 
  Ideas recently put forward by Y. Matsuo and the author are summarized on the example of the simplest ($W_3$) generalization of two-dimensional gravity. These notes are based on lectures given at the workshop `` Strings, Conformal Models and Topological Field Theories'', Cargese 12-21 May 1993; and at the meeting ``String 93'', Berkeley, 24-29 May 1993. 
  The $\kappa$-deformation of the D-dimensional Poincar\'e algebra $(D\geq 2)$ with any signature is given. Further the quadratic Poisson brackets, determined by the classical $r$-matrix are calculated, and the quantum Poincar\'e group "with noncommuting parameters" is obtained. 
  We show that the `instantonic' soliton of five-dimensional Yang-Mills theory and the closely related BPS monopole of four-dimensional Yang-Mills/Higgs theory continue to be exact static, and stable, solutions of these field theories even after the inclusion of gravitational, electromagnetic and, in the four-dimensional case, dilatonic interactions, provided that certain non-minimal interactions are included. With the inclusion of these interactions, which would be required by supersymmetry, these exact self-gravitating solitons saturate a gravitational version of the Bogomol'nyi bound on the energy of an arbitrary field configuration. 
  Using $U_q[SU(2)]$ tensor calculus we compute the two-point scalar operators (TPSO), their averages on the ground-state give the two-point correlation functions. The TPSOs are identified as elements of the Temperley-Lieb algebra and a recurrence relation is given for them. We have not tempted to derive the analytic expressions for the correlation functions in the general case but got some partial results. For $q=e^{i \pi/3}$, all correlation functions are (trivially) zero, for $q=e^{i \pi/4}$, they are related in the continuum to the correlation functions of left-handed and right-handed Majorana fields in the half plane coupled by the boundary condition. In the case $q=e^{i \pi/6}$, one gets the correlation functions of Mittag's and Stephen's parafermions for the three-state Potts model. A diagrammatic approach to compute correlation functions is also presented. 
  Decoherence and dissipation in quantum systems has been studied extensively in the context of Quantum Brownian Motion. Effective decoherence in coarse grained quantum systems has been a central issue in recent efforts by Zurek and by Hartle and Gell-Mann to address the Quantum Measurement Problem. Although these models can yield very general classical phenomenology, they are incapable of reproducing relevant characteristics expected of a local environment on a quantum system, such as the characteristic dependence of decoherence on environment spatial correlations. I discuss the characteristics of Quantum   Brownian Motion in a local environment by examining aspects of first principle calculations and by the construction of phenomenological models.   Effective quantum Langevin equations and master equations are presented in a variety of representations. Comparisons are made with standard results such as the Caldeira-Leggett master equation. 
  We consider a bosonic string propagating in 4--dim Minkowski space. We show that in the orthonormal gauge the classical system exhibits a hidden $W_{\infty}$ chiral symmetry, arising from the equivalence of its transverse modes with the $SU(2)/U(1)$ coset model defined on the string world--sheet. Generalizations to other string backgrounds are proposed. We also define a Liouville--like transformation that maps solutions of the $SU(2)/U(1)$ coset model into the solution space of two decoupled Liouville theories. Inverting this transformation, however, remains an open problem. 
  We present a Lagrangian description of the $SU(2)/U(1)$ coset model perturbed by its first thermal operator. This is the simplest perturbation that changes sign under Krammers--Wannier duality. The resulting theory, which is a 2--component generalization of the sine--Gordon model, is then taken in Minkowski space. For negative values of the coupling constant $g$, it is classically equivalent to the $O(4)$ non--linear $\s$--model reduced in a certain frame. For $g > 0$, it describes the relativistic motion of vortices in a constant external field. Viewing the classical equations of motion as a zero curvature condition, we obtain recursive relations for the infinitely many conservation laws by the abelianization method of gauge connections. The higher spin currents are constructed entirely using an off--critical generalization of the $W_{\infty}$ generators. We give a geometric interpretation to the corresponding charges in terms of embeddings. Applications to the chirally invariant $U(2)$ Gross--Neveu model are also discussed. 
  For a large class of gauge theories a nilpotent BRS-operator $s$ is constructed and its cohomology in the space of local functionals of the off-shell fields is shown to be isomorphic to the cohomology of $\4s=s+d$ on functions $f(\4C,\PH)$ of tensor fields $\PH$ and of variables $\4C$ which are constructed of the ghosts and the connection forms. The result allows general statements about the structure of invariant classical actions and anomaly candidates whose BRS-variation vanishes off-shell. The assumptions under which the result holds are thoroughly discussed. 
  This short note is closely related to Sen-Zwiebach paper on gauge transformations in Batalin-Vilkovisky theory (hep-th 9309027). We formulate some conditions of physical equivalence of solutions to the quantum master equation and use these conditions to give a very transparent analysis of symmetry transformations in BV-approach. We prove that in some sense every quantum observable (i.e. every even function $H$ obeying $\Delta_{\rho}(He^S)=0$) determines a symmetry of the theory with the action functional $S$ satisfying quantum master equation $\Delta_{\rho}e^S=0$ \end 
  A certain generalization of the algebra $gl(N,{\bf R})$ of first-order differential operators acting on a space of inhomogeneous polynomials in ${\bf R}^{N-1}$ is constructed. The generators of this (non)Lie algebra depend on permutation operators. It is shown that the Hamiltonian of the $N$-body Calogero model can be represented as a second-order polynomial in the generators of this algebra. Given representation implies that the Calogero Hamiltonian possesses infinitely-many, finite-dimensional invariant subspaces with explicit bases, which are closely related to the finite-dimensional representations of above algebra. This representation is an alternative to the standard representation of the Bargmann-Fock type in terms of creation and annihilation operators. 
  We review various aspects of a fermionic gauge symmetry, known as the $\kappa$--symmetry, which plays an important role in formulations of superstrings, supermembranes and higher dimensional extended objects. We also review some aspects of the connection between $\kappa$--symmetric theories and their twistor-like formulations. 
  We study a dynamically broken U(1) "left" gauge theory endowed with a composite scalar doublet (one scalar and one pseudoscalar); its Lagrangian only differs from that of an abelian `Standard Model' by the addition of a derivative coupling between a Wess-Zumino field, linked to the previous scalars, and the fermionic current. Yet, in the Feynman path integral, the non independence of the fermionic and scalar variables of integration requires the introduction of constraints. When the gauge symmetry is broken by the vacuum expectation value of the scalar field, they freeze all degrees of freedom but those of a massive gauge field, including a (abelian) pion. The anomaly disappears and the gauge current is conserved. This is shown, and renormalizability studied, in the `Nambu-Jona-Lasinio approximation'. Unitarity is demonstrated on general grounds. 
  The standard formula for the change in the effective action under a conformal transformation is extended to the case when the boundary is piecewise smooth. We then find the functional determinants of the scalar Laplacian on regions of the plane obtained by stereographic projection of the fundamental domains on an orbifolded two-sphere. Examples treated are the sector of a disk and a circular crescent. The effective action on a spherical cap is also determined. 
  A new Langevin equation with a field-dependent kernel is proposed to deal with bottomless systems within the framework of the stochastic quantization of Parisi and Wu. The corresponding Fokker-Planck equation is shown to be a diffusion-type equation and is solved analytically. An interesting connection between the solution with the ordinary Feynman measure, which in this case is not normalizable, is clarified. 
  The two-point correlation function of the stress-energy tensor for the $\Phi_{1,3}$ massive deformation of the non-unitary model ${\cal M}_{3,5}$ is computed. We compare the ultraviolet CFT perturbative expansion of this correlation function with its spectral representation given by a summation over matrix elements of the intermediate asymptotic massive particles. The fast rate of convergence of both approaches provides an explicit example of an accurate interpolation between the infrared and ultraviolet behaviours of a Quantum Field Theory. 
  Scattering amplitudes of the spin-4/3 fractional superstring are shown to satisfy spurious state decoupling and cyclic symmetry (duality) at tree-level in the string perturbation expansion. This fractional superstring is characterized by the spin-4/3 fractional superconformal algebra---a parafermionic algebra studied by Zamolodchikov and Fateev involving chiral spin-4/3 currents on the world-sheet in addition to the stress-energy tensor. Examples of tree scattering amplitudes are calculated in an explicit c=5 representation of this fractional superconformal algebra realized in terms of free bosons on the string world-sheet. The target space of this model is three-dimensional flat Minkowski space-time with a level-2 Kac-Moody so(2,1) internal symmetry, and has bosons and fermions in its spectrum. Its closed string version contains a graviton in its spectrum. Tree-level unitarity (i.e., the no-ghost theorem for space-time bosonic physical states) can be shown for this model. Since the critical central charge of the spin-4/3 fractional superstring theory is 10, this c=5 representation cannot be consistent at the string loop level. The existence of a critical fractional superstring containing a four-dimensional space-time remains an open question. 
  By resorting to the Fock--Bargmann representation, we incorporate the quantum Weyl--Heisenberg ($q$-WH) algebra into the theory of entire analytic functions. The main tool is the realization of the $q$--WH algebra in terms of finite difference operators. The physical relevance of our study relies on the fact that coherent states (CS) are indeed formulated in the space of entire analytic functions where they can be rigorously expressed in terms of theta functions on the von Neumann lattice. The r\^ole played by the finite difference operators and the relevance of the lattice structure in the completeness of the CS system suggest that the $q$--deformation of the WH algebra is an essential tool in the physics of discretized (periodic) systems. In this latter context we define a quantum mechanics formalism for lattice systems. 
  The sector of zero $Z_{N}$-charge is studied for the ferromagnetic (FM) and antiferromagnetic (AFM) version of the $Z_{N}\times Z_{2}$ invariant Fateev-Zamolodchikov quantum spin chain. We conjecture that the relevant Bethe ansatz equations should admit, beside the usual string-like solutions, exceptional multiplets, and a number of non-physical solutions. Once the physical ones are identified, we show how to get completeness and the gapless excitation spectrum. The central charge is computed from the specific heat and found to be $c=2\frac{N-1}{N+2}$ (FM) and $c=1$ (AFM). 
  Wilson loop averages of pure gauge QCD at large N on a sphere are calculated by means of Makeenko-Migdal loop equation. 
  The gauging of isometries in general sigma-models which include fermionic terms which represent the interaction of strings with background Yang-Mills fields is considered. Gauging is possible only if certain obstructions are absent. The quantum gauge anomaly is discussed, and the (1,0) supersymmetric generalisation of the gauged action given. 
  To investigate the properties of $c=1$ matter coupled to $2$d{--}gravity we have performed large-scale simulations of two copies of the Ising Model on a dynamical lattice. We measure spin susceptibility and percolation critical exponents using finite-size scaling. We show explicitly how logarithmic corrections are needed for a proper comparison with theoretical exponents. We also exhibit correlations, mediated by gravity, between the energy and magnetic properties of the two Ising species. The prospects for extending this work beyond $c=1$ are addressed. 
  We study percolation on the worldsheets of string theory for $c=0,1/2,1$ and $2$. For $c<1$ we find that critical exponents measured from simulations agree quite well with the theoretical values. For $c=1$ we show how log corrections determined from the exact solution reconcile numerical results with the KPZ predictions. We extend this analysis to the large $c$ regime and estimate how finite-size effects will effectively raise the ground state energy, masking the presence of the tachyon for moderate values of $c > 1$. It thus appears likely that simulations for $c=2,3 \ldots$ on numerically accessible lattices will fail to even capture the qualitative behavior of the continuum limit. 
  We describe the symplectic structure and Hamiltonian dynamics for a class of Grassmannian manifolds. Using the two dimensional sphere ($S^2$) and disc ($D^2$) as illustrative cases, we write their path integral representations using coherent state techniques. These path integrals can be evaluated exactly by semiclassical methods, thus providing examples of localisation formula. Along the way, we also give a local coordinate description for a class of Grassmannians. 
  Borcherds algebras represent a new class of Lie algebras which have almost all the properties that ordinary Kac-Moody algebras have, and the only major difference is that these generalized Kac-Moody algebras are allowed to have imaginary simple roots. The simplest nontrivial examples one can think of are those where one adds ``by hand'' one imaginary simple root to an ordinary Kac-Moody algebra. We study the fundamental representation of this class of examples and prove that an irreducible module is given by the full tensor algebra over some integrable highest weight module of the underlying Kac-Moody algebra. We also comment on possible realizations of these Lie algebras in physics as symmetry algebras in quantum field theory. 
  A global model of $q$-deformation for the quasi--orthogonal Lie algebras generating the groups of motions of the four--dimensional affine Cayley--Klein geometries is obtained starting from the three dimensional deformations. It is shown how the main algebraic classical properties of the CK systems can be implemented in the quantum case. Quantum deformed versions of either the space--time or space symmetry algebras (Poincar\'e (3+1), Galilei (3+1), 4D Euclidean as well as others) appear in this context as particular cases and several $q$-deformations for them are directly obtained. 
  The finite temperature one-loop effective potential for a scalar field defined on an ultrastatic spacetime, whose spatial part is a compact hyperbolic manifold, is studied. Different analytic expressions, especially valuable at low and high temperature are derived. Based on these results, the symmetry breaking and the topological mass generation are discussed. 
  Representations by means of path integrals are used to find spinor and isospinor structure of relativistic particle propagators in external fields. For Dirac propagator in an external electromagnetic field all grassmannian integrations are performed and a general result is presented via a bosonic path integral. The spinor structure of the integrand is given explicitly by its decomposition in the independent $\gamma$-matrix structures. Similar technique is used to get the isospinor structure of the scalar particle propagator in an external non-Abelian field. 
  Non-perturbative interactions in the effective action of two-dimensional bosonic string theory are described. These interactions are due to ``stringy" instantons that are associated with a space-varying coupling parameter. We present progress towards identifying similar non-perturbative interactions in the effective action of two-dimensional superstring theory. We discuss the possible relation to higher dimensional string theory. 
  These are lecture notes of lectures presented at the 1993 Trieste Summer School, dealing with two classes of two-dimensional field theories, (topological) Yang-Mills theory and the G/G gauged WZW model. The aim of these lectures is to exhibit and extract the topological information contained in these theories, and to present a technique (a Weyl integral formula for path integrals) which allows one to calculate directly their partition function and topological correlation functions on arbitrary closed surfaces. Topics dealt with are (among others): solution of Yang-Mills theory on arbitrary surfaces; calculation of intersection numbers of moduli spaces of flat connections; coupling of Yang-Mills theory to coadjoint orbits and intersection numbers of moduli spaces of parabolic bundles; derivation of the Verlinde formula from the G/G model; derivation of the shift k to k+h in the G/G model via the index of the twisted Dolbeault complex. 
  We find a new Hamiltonian formulation of the classical isotropic rotator where left and right $SU(2)$ transformations are not canonical symmetries but rather Poisson Lie group symmetries. The system corresponds to the classical analog of a quantum mechanical rotator which possesses quantum group symmetries. We also examine systems of two classical interacting rotators having Poisson Lie group symmetries. 
  The Schwinger model is studied with a new one - parameter class of gauge invariant regularizations that generalizes the usual point - splitting or Fujikawa schemes. The spectrum is found to be qualitatively unchanged, except for a limiting value of the regularizing parameter, where free fermions appear in the spectrum. 
  We define currents on a Grassmann algebra $Gr(N)$ with $N$ generators as distributions on its exterior algebra (using the symmetric wedge product). We interpret the currents in terms of ${\Z}_2$-graded Hochschild cohomology and closed currents in terms of cyclic cocycles (they are particular multilinear forms on $Gr(N)$). An explicit construction of the vector space of closed currents of degree $p$ on $Gr(N)$ is given by using Berezin integration. 
  For any manifold $M$, we introduce a $\ZZ $-graded differential algebra $\Xi$, which, in particular, is a bi-module over the associative algebra $C(M\cup M)$. We then introduce the corresponding covariant differentials and show how this construction can be interpreted in terms of Yang-Mills and Higgs fields. This is a particular example of noncommutative geometry. It differs from the prescription of Connes in the following way: The definition of $\Xi$ does not rely on a given Dirac-Yukawa operator acting on a space of spinors. 
  We connect Liouville theory, anyons and Higgs model in a purely geometrical way. 
  We consider the deformation of the Schwarzschild solution in general relativity due to spherically symmetric quantum fluctuations of the metric and the matter fields. In this case, the 4D theory of gravity with Einstein action reduces to the effective two-dimensional dilaton gravity. We have found that the Schwarzschild singularity at $r=0$ is shifted to the finite radius $r_{min} \sim r_{Pl}$, where the scalar curvature is finite, so that the space-time looks regular and consists of two asymptotically flat sheets glued at the hypersurface of constant radius. 
  Perturbative breaking of supersymmetry in four-dimensional string theories predict in general the existence of new large dimensions at the TeV scale. Such dimensions can be consistent with perturbative unification up to the Planck scale in a class of string models and open the exciting possibility of lowering a part of the massive string spectrum at energies accessible to future accelerators. The main signature is the production of Kaluza-Klein excitations which have a very particular structure, strongly correlated with the supersymmetry breaking mechanism. We present a model independent analysis of the physics of these states in the context of orbifold compactifications of the heterotic superstring. In particular, we compute the limits on the size of large dimensions used to break supersymmetry. 
  I discuss some results we have obtained recently in a lattice model for quantized gravity coupled to scalar matter in four dimensions. We have looked at how the continuous phase transition separating the smooth from the rough phase of gravity is influenced by the presence of the scalar field. We find that close to the critical point, where the average curvature approaches zero, the effects of the scalar field are small and the coupling of matter to gravity seems to be weak. The nature of the phase diagram and the values for the critical exponents would suggest that gravitational interactions increase with distance. 
  (In the revised version the relevant aspect of noncompactness of the moduli of instantons is discussed. It is shown nonperturbatively that any BRST trivial deformation of A-model which does not change the ranks of BRST cohomology does not change the topological correlation functions either) We show that the Floer cohomology and quantum cohomology rings of the almost Kahler manifold M, both defined over the Novikov ring of the loop space LM of M, are isomorphic. We do it using a BRST trivial deformation of the topological A-model. As an example we compute the Floer = quantum cohomology of the 3-dimensional flag space Fl_3. 
  We represent Feigin's construction [11] of lattice W algebras and give some simple results: lattice Virasoro and $W_3$ algebras. For simplest case $g=sl(2)$ we introduce whole $U_q(sl(2))$ quantum group on this lattice. We find simplest two-dimensional module as well as exchange relations and define lattice Virasoro algebra as algebra of invariants of $U_q(sl(2))$. Another generalization is connected with lattice integrals of motion as the invariants of quantum affine group $U_q(\hat{n}_{+})$. We show that Volkov's scheme leads to the system of difference equations for the function from non-commutative variables.Continium limit of this lattice algebras are considered. 
  Based on our previous work, in which a model of two dimensional dilaton gravity of the type proposed by Callan, Giddings, Harvey and Strominger was rigorously quantized, we explicitly demonstrate how one can extract space-time geometry in exactly solvable theory of quantum gravity. In particular, we have been able to produce a prototypical configuration in which a ( smeared ) matter shock wave generates a black hole without naked sigularity. 
  We describe in detail how one can extract space-time geometry in an exactly solvable model of quantum dilaton gravity of the type proposed by Callan, Giddings, Harvey and Strominger ( CGHS ). Based on our previous work, in which a model with 24 massless matter scalars was quantized rigorously in BRST operator formalism, we compute, without approximation, mean values of the matter stress-energy tensor, the inverse metric and some related quantities in a class of coherent physical states constructed in a specific gauge within the conformal gauge. Our states are so designed as to describe a variety of space-time in which in-falling matter energy distribution produces a black hole with or without naked sigularity. In particular, we have been able to produce the prototypical configuration first discovered by CGHS, in which a ( smeared ) matter shock wave produces a black hole without naked sigularity. 
  We discuss the idea of black hole complementarity, recently suggested by Susskind et al., and the notion of stretched horizon, in the light of the generalized uncertainty principle of quantum gravity. We discuss implications for the no-hair theorem and we show that within this approach quantum hair arises naturally. 
  We discovered new hidden symmetry of the one-dimensional Hubbard model. We showthat the one-dimensional Hubbard model on the infinite chain has the infinite-dimensional algebra of symmetries. This algebra is a direct sum of two $ sl(2) $-Yangians. This $ Y(sl(2)) \oplus Y(sl(2)) $ symmetry is an extension of the well-known $ sl(2) \oplus sl(2) $ . The deformation parameters of the Yangians are equal up to the signs to the coupling constant of the Hubbard model hamiltonian. 
  String backgrounds associated with gauged $G/H$ WZNW models generically depend on $\alpha'$ or $1/k$. The exact expressions for the corresponding metric $G_{\m\n}$, antisymmetric tensor $B_{\m\n}$, and dilaton $\phi$ can be obtained by eliminating the $2d$ gauge field from the local part of the effective action of the gauged WZNW model. We show that there exists a manifestly gauge-invariant prescription for the derivation of the antisymmetric tensor coupling. When the subgroup $H$ is one-dimensional and $G$ is simple the antisymmetric tensor is given by the semiclassical ($\alpha'$-independent) expression. We consider in detail the simplest non-trivial example with non-trivial $B_{\m\n}$ -- the D=3 sigma model corresponding to the $[SL(2,R) x R]/R$ gauged WZNW theory (`charged black string') and show that the exact expressions for $G_{\m\n}$, $B_{\m\n}$ and $\phi$ solve the Weyl invariance conditions in the two-loop approximation. Similar conclusion is reached for the closely related $SL(2,R)/R$ chiral gauged WZNW model. We find that there exists a scheme in which the semiclassical background is also a solution of the two-loop conformal invariance equations (but the tachyon equation takes a non-canonical form). We discuss in detail the role of field redefinitions (scheme dependence) in establishing a correspondence between the sigma model and conformal field theory results. 
  We derive the Kac and new determinant formulae for an arbitrary (integer) level $K$ fractional superconformal algebra using the BRST cohomology techniques developed in conformal field theory. In particular, we reproduce the Kac determinants for the Virasoro ($K=1$) and superconformal ($K=2$) algebras. For $K\geq3$ there always exist modules where the Kac determinant factorizes into a product of more fundamental new determinants. Using our results for general $K$, we sketch the non-unitarity proof for the $SU(2)$ minimal series; as expected, the only unitary models are those already known from the coset construction. We apply the Kac determinant formulae for the spin-4/3 parafermion current algebra ({\em i.e.}, the $K=4$ fractional superconformal algebra) to the recently constructed three-dimensional flat Minkowski space-time representation of the spin-4/3 fractional superstring. We prove the no-ghost theorem for the space-time bosonic sector of this theory; that is, its physical spectrum is free of negative-norm states. 
  This paper is removed from the network permanently. The content in this paper has now been put together with hep-th/9310183. See the recently replaced and widely revised version of the latter. 
  The path integral on the single-sheeted hyperboloid, i.e.\ in $D$-dimensional imaginary Lobachevsky space, is evaluated. A potential problem which we call ``Kepler-problem'', and the case of a constant magnetic field are also discussed. 
  An effective action is obtained for the $N=1$, $2D-$induced supergravity on a compact super Riemann surface (without boundary) $\hat\Sigma$ of genus $g>1$, as the general solution of the corresponding superconformal Ward identity. This is accomplished by defining a new super integration theory on $\hat\Sigma$ which includes a new formulation of the super Stokes theorem and residue calculus in the superfield formalism. Another crucial ingredient is the notion of polydromic fields. The resulting action is shown to be well-defined and free of singularities on $\sig$. As a by-product, we point out a morphism between the diffeomorphism symmetry and holomorphic properties. 
  This paper is a study of monoidal categories with duals where the tensor product need not be commutative. The motivating examples are categories of representations of Hopf algebras and the motivating application is the definition of 6j-symbols as used in topological field theories.     We introduce the new notion of a spherical category. In the first section we prove a coherence theorem for a monoidal category with duals following MacLane (1963). In the second section we give the definition of a spherical category, and construct a natural quotient which is also spherical.     In the third section we define spherical Hopf algebras so that the category of representations is spherical. Examples of spherical Hopf algebras are involutory Hopf algebras and ribbon Hopf algebras. Finally we study the natural quotient in these cases and show it is semisimple. 
  We calculate various Wilson loop averages in a pure $SU(N)$-gauge theory on a two-dimensional sphere, in the large $N$ limit. The results can be expressed through the density of rows in the most probable Young tableau. They are valid in both phases (small and large areas of the sphere). All averages for self-intersecting loops can be reproduced from the average for a simple (non self-intersecting) loop by means of loop equations. 
  We discuss in detail the construction of topological field theories using the Batalin--Vilkovisky (BV) quantisation scheme. By carefully examining the dependence of the antibracket on an external metric, we show that differentiating with respect to the metric and the BRST charge do not commute in general. We introduce the energy momentum tensor in this scheme and show that it is BRST invariant, both for the classical and quantum BRST operators. It is antifield dependent, guaranteeing gauge independence. For topological field theories, this energy momentum has to be quantum BRST exact. This leads to conditions at each order in $\hbar$. As an example of this procedure, we consider topological Yang--Mills theory. We show how the reducible set of symmetries used in topological Yang--Mills can be recovered by means of trivial systems and canonical transformations. Self duality of the antighosts is properly treated by introducing an infinite tower of auxiliary fields. Finally, it is shown that the full energy momentum tensor is classically BRST exact in the antibracket sense. 
  The quantum action principle of renormalisation theory is applied to the antibracket-antifield formalism for Hamiltonian systems. General results on the local BRST cohomology allow one to prove that the anomalies appear in the time development of the BRST charge and violate the nilpotency of this charge. Furthermore they are equivalent to those of the Lagrangian formalism. The analysis provides a completely gauge and regularisation independent proof of Faddeev's conjecture on the relationship between gauge anomalies and Schwinger terms in the context of descent equations. 
  Quantum canonical transformations are used to derive the integral representations and Kummer solutions of the confluent hypergeometric and hypergeometric equations. Integral representations of the solutions of the non-periodic three body Toda equation are also found. The derivation of these representations motivate the form of a two-dimensional generalized hypergeometric equation which contains the non-periodic Toda equation as a special case and whose solutions may be obtained by quantum canonical transformation. 
  The implications of gauging the Wess-Zumino-Novikov-Witten (WZNW) model using the Gauss decomposition of the group elements are explored. We show that, contrary to standard gauging of WZNW models, this gauging is carried out by minimally coupling the gauge fields. We find that this gauging, in the case of gauging an abelian vector subgroup, differs from the standard one by terms proportional to the field strength of the gauge fields. We prove that gauging an abelian vector subgroup does not have a nonlinear sigma model interpretation. This is because the target-space metric resulting from the integration over the gauge fields is degenerate. We demonstrate, however, that this kind of gauging has a natural interpretation in terms of Wakimoto variables. 
  We show that bosonic strings may be viewed as a particular class of vacua for N=1 superstrings, and N=1 superstrings may be viewed as a particular class of vacua for N=2 strings. Continuing this line of string hierarchies, we are led to search for a universal string theory which includes all the rest as a special vacuum selection. 
  The non-relativistic Chern-Simons theory with the single-valued anyonic field is proposed as an example of q-deformed field theory. The corresponding q-deformed algebra interpolating between bosons and fermions,both in position and momentum spaces, is analyzed.A possible generalization to a space with an arbitrary dimension is suggested. 
  Solvable Natanzon potentials in nonrelativistic quantum mechanics are known to group into two disjoint classes depending on whether the Schr\"odinger equation can be reduced to a hypergeometric or a confluent hypergeometric equation. All the potentials within each class are connected via point canonical transformations. We establish a connection between the two classes with appropriate limiting procedures and redefinition of parameters, thereby inter-relating all known solvable potentials. 
  The recently introduced $\kappa$-Poincare-Dirac equation is gauged to treat the $\kappa$-Dirac-Coulomb problem. For the resulting equation, we prove that the perturbation to first order in the quantum group parameter vanishes identically. The second order perturbation is singular, but assuming a heuristic cut-off allows a qualitative estimate of the quantum group parameter. 
  Static spherically symmetric solutions of the Einstein-Maxwell gravity with the dilaton field are described. The solutions correspond to black holes and are generalizations of the previously known dilaton black hole solution. In addition to mass and electric charge these solutions are labeled by a new parameter, the dilaton charge of the black hole. Different effects of the dilaton charge on the geometry of space-time of such black holes are studied. It is shown that in most cases the scalar curvature is divergent at the horizons. Another feature of the dilaton black hole is that there is a finite interval of values of electric charge for which no black hole can exist. 
  We derive the topological obstruction to spin-Klein cobordism. This result has implications for signature change in general relativity, and for the $N=2$ superstring. 
  The concept of ``elements of reality" is analyzed within the framework of quantum theory. It is shown that elements of reality fail to fulfill the product rule. This is the core of recent proofs of the impossibility of a Lorentz-invariant interpretation of quantum mechanics. A generalization and extension of the concept of elements of reality is presented. Lorentz-invariance is restored by giving up the product rule. The consequences of giving up the ``and" rule, which must be abandoned together with the product rule, are discussed. 
  A formulation of $D\is 10$ superparticle dynamics is given that contain space-time and twistor variables. The set of constraints is entirely first class, and gauge conditions may be imposed that reduces the system to a Casalbuoni-Brink-Schwarz superparticle, a spinning particle or a twistor particle. 
  We study $E_8 \times E_8$-heterotic string on asymmetric orbifolds associated with semi-simple simply-laced Lie algebras. Using the fact that $E_6$-model allows different twists, we present a new N=1 space-time supersymmetric model whose supercurrent appears from twisted sectors but not untwisted sector. 
  The BRS transformations for gravity with torsion are discussed by using the Maurer-Cartan horizontality conditions. With the help of an operator $\d$ which allows to decompose the exterior space-time derivative as a BRS commutator we solve the Wess-Zumino consistency condition corresponding to invariant Lagrangians and anomalies. 
  We analyze the relation between a topological coset model based on super $SL(2,R)/U(1)$ coset and non-critical string theory by using free field realization. We show that the twisted $N=2$ algebra of the coset model can be naturally transformed into that of non-critical string. The screening operators of the coset models can be identified either with those of the minimal matters or with the cosmological constant operator. We also find that another screening operator, which is intrinsic in our approach, becomes the BRST nontrivial state of ghost number $0$ (generator of the ground ring for $c=1$ gravity). 
  We show how two-dimensional incompressible quantum fluids and their excitations can be viewed as $\ W_{1+\infty}\ $ edge conformal field theories, thereby providing an algebraic characterization of incompressibility. The Kac-Radul representation theory of the $\ W_{1+\infty}\ $ algebra leads then to a purely algebraic complete classification of hierarchical quantum Hall states, which encompasses all measured fractions. Spin-polarized electrons in single-layer devices can only have Abelian anyon excitations. 
  The two-dimensional self-dual Chern-Simons equations are equivalent to the conditions for static, zero-energy vortex-like solutions of the (2+1) dimensional gauged nonlinear Schr\"odinger equation with Chern-Simons matter-gauge coupling. The finite charge vacuum states in the Chern-Simons theory are shown to be gauge equivalent to the finite action solutions to the two-dimensional chiral model (or harmonic map) equations. The Uhlenbeck-Wood classification of such harmonic maps into the unitary groups thereby leads to a complete classification of the vacuum states of the Chern-Simons model. This construction also leads to an interesting new relationship between $SU(N)$ Toda theories and the $SU(N)$ chiral model. 
  Associated with the fundamental representation of a quantum algebra such as $U_q(A_1)$ or $U_q(A_2)$, there exist infinitely many gauge-equivalent $R$-matrices with different spectral-parameter dependences. It is shown how these can be obtained by examining the infinitely many possible gradations of the corresponding quantum affine algebras, such as $U_q(A_1^{(1)})$ and $U_q(A_2^{(1)})$, and explicit formulae are obtained for those two cases. Spectral-dependent similarity (gauge) transformations relate the $R$-matrices in different gradations. Nevertheless, the choice of gradation can be physically significant, as is illustrated in the case of quantum affine Toda field theories. 
  We identify the untwisted moduli of heterotic orbifold compactifications for the case, when the gauge twist is realized by a rotation. The Wilson lines are found to have both continuous and discrete parts. For the case of the standard Z(3) orbifold we classify all possibilities of breaking the gauge group E(6) times SU(3) by nine of the eighteen Wilson moduli and by additional discrete Wilson lines. 
  We present a simple alternative to Mackey's account of the (infinite) inequivalent quantizations possible on a coset space G/H. Our reformulation is based on the reduction ${\rm G \rightarrow G/H}$ and employs a generalized form of Dirac's approach to the quantization of constrained systems. When applied to the four-sphere $S^4 \simeq {\rm Spin(5)/Spin(4)}$, the inequivalent quantizations induce relativistic spin and a background BPST instanton; thus they might provide a natural account of both of these physical entities. 
  It is shown that the application of Lax-Phillips scattering theory to quantum mechanics provides a natural framework for the realization of the ideas of the Many-Hilbert-Space theory of Machida and Namiki to describe the development of decoherence in the process of measurement. We show that if the quantum mechanical evolution is pointwise in time, then decoherence occurs only if the Hamiltonian is time-dependent. If the evolution is not pointwise in time (as in Liouville space), then the decoherence may occur even for closed systems. These conclusions apply as well to the general problem of mixing of states. 
  What is quantum geometry? This question is becoming a popular leitmotiv in theoretical physics and in mathematics. Conformal field theory may catch a glimpse of the right answer. We review global aspects of the geometry of conformal fields, such as duality and mirror symmetry, and interpret them within Connes' non-commutative geometry. Extended version of lectures given by the 2nd author at the Mathematical Quantum Theory Conference, Vancouver, Canada, August 4 to 8, 1993 
  We study the boundary S-matrix for the reflection of bound states of the two-dimensional sine-Gordon integrable field theory in the presence of a boundary. 
  The momentum operator representation of nonrelativistic anyons is developed in the Chern - Simons formulation of fractional statistics. The connection between anyons and the q-deformed bosonic algebra is established. 
  We study the time-dependent tachyon backgrounds of the string collective field theory using the formalism of the $S$-matrix generating functional. In the process we clarify the connection between two ways of calculating the $S$-matrix, the one using the Feynman rule and the other using the classical solution to the nonlinear equation of motion. We develop the formalism for general backgrounds and apply it to the gravitational sine-Gordon model in detail. We reproduce the conformal field theory calculation which was based on expanding around the static $c=1$ theory. Furthermore, we prove that the tree- level partition function of this model shows the scaling behavior corresponding to $c=0$ model in the limit $p \to 0$ of sine-Gordon `momentum' $p$. 
  I review some recent works on the Hermitean one-matrix and d-dimensional gauge-invariant matrix models. Special attention is paid to solving the models at large-N by the loop equations. For the one-matrix model the main result concerns calculations of higher genera, while for the d-dimensional model the large-N solution for a logarithmic potential is described. Some results on fermionic matrix models are briefly reviewed. Talk at the Workshop on Quantum Field Theoretical Aspects of High Energy Physics, Kyffhaeuser, Germany, September 20-24, 1993 
  A study of the superconformal covariantization of superdifferential operators defined on $(1|1)$ superspace is presented. It is shown that a superdifferential operator with a particular type of constraint can be covariantized only when it is of odd order. In such a case, the action of superconformal transformation on the superdifferential operator is nothing but a hamiltonian flow defined by the corresponding supersymmetric second Gelfand-Dickey bracket. The covariant form of a superdifferential operator of odd order is given. 
  We investigate $O(N)$-symmetric vector field theories in the double scaling limit. Our model describes branched polymeric systems in $D$ dimensions, whose multicritical series interpolates between the Cayley tree and the ordinary random walk. We give explicit forms of residual divergences in the free energy, analogous to those observed in the strings in one dimension. 
  The supersymmetry properties of the asymptotically anti-de Sitter black holes of Einstein theory in 2+1 dimensions are investigated. It is shown that (i) the zero mass black hole has two exact super- symmetries; (ii) extreme $lM=|J|$ black holes with $M \not= 0$ have only one; and (iii) generic black holes do not have any. It is also argued that the zero mass hole is the ground state of (1,1)-adS supergravity with periodic (``Ramond") boundary conditions on the spinor fields. 
  Constrained effective potentials in hot gauge theory give the probability that a configuration p of the order parameter (Polyakov loop) occurs. They are important in the analysis of surface effects and bubble formation in the plasma. The vector potential appears non-linearly in the loop; in weak coupling the linear term gives rise to the traditional free energy graphs. But the non-linear terms generate insertions of the constrained modes into the free energy graphs, through renormalisations of the Polyakov loop. These insertions are gauge dependent and are necessary to cancel the gauge dependence of the free energy graphs. The latter is shown, through the BRST identities, to have again the form of constrained mode insertions. It also follows, that absolute minima of the potential are at the centergroup values of the loop. We evaluate the two-loop contributions for SU(N) gauge theories, with and without quarks, for the full domain of the N-1 variables. 
  For any Hecke symmetry $R$ there is a natural quantization $A_n(R)$ of the Weyl algebra $A_n$. The aim of this paper is to study some general ring-theoretic aspects of $A_n(R)$ and its relations to formal deformations of $A_n$. We also obtain further information on those quantizations obtained from some well-known Hecke symmetries. 
  We show that the WZNW model with arbitrary $\sigma$-model coupling constant may be viewed as a $\sigma$-model perturbation of the WZNW theory around the Witten conformal point. In order for the $\sigma$-model perturbation to be relevant, the level $k$ of the underlying affine algebra has to be negative. We prove that in the large $|k|$ limit the perturbed WZNW system with negative $k$ flows to the conformal WZNW model with positive level. The flow appears to be integrable due to the existence of conserved currents satisfying the Lax equation. This fact is in a favorable agreement with the integrability of the WZNW model discovered by Polyakov and Wiegmann within the Bethe ansatz technique. 
  We analyze the renormalization of systems whose effective degrees of freedom are described in terms of fluctuations which are ``environment'' dependent. Relevant environmental parameters considered are: temperature, system size, boundary conditions, and external fields. The points in the space of \lq\lq coupling constants'' at which such systems exhibit scale invariance coincide only with the fixed points of a global renormalization group which is necessarily environment dependent. Using such a renormalization group we give formal expressions to two loops for effective critical exponents for a generic crossover induced by a relevant mass scale $g$. These effective exponents are seen to obey scaling laws across the entire crossover, including hyperscaling, but in terms of an effective dimensionality, $d\ef=4-\gl$, which represents the effects of the leading irrelevant operator. We analyze the crossover of an $O(N)$ model on a $d$ dimensional layered geometry with periodic, antiperiodic and Dirichlet boundary conditions. Explicit results to two loops for effective exponents are obtained using a [2,1] Pad\'e resummed coupling, for: the ``Gaussian model'' ($N=-2$), spherical model ($N=\infty$), Ising Model ($N=1$), polymers ($N=0$), XY-model ($N=2$) and Heisenberg ($N=3$) models in four dimensions. We also give two loop Pad\'e resummed results for a three dimensional Ising ferromagnet in a transverse magnetic field and corresponding one loop results for the two dimensional model. One loop results are also presented for a three dimensional layered Ising model with Dirichlet and antiperiodic boundary conditions. Asymptotically the effective exponents are in excellent agreement with known results. 
  We discuss how to implement an ``environmentally friendly'' renormalization in the context of finite temperature field theory. Environmentally friendly renormalization provides a method for interpolating between the different effective field theories which characterize different asymptotic regimes. We give explicit two loop Pad\'e resummed results for $\l\ff$ theory for $T>T_c$. We examine the implications for non-Abelian gauge theories. 
  Very recently Berkovits and Vafa have argued that the $N{=}0$ string is a particular choice of background of the $N{=}1$ string. Under the assumption that the physical states of the $N{=}0$ string theory came essentially from the matter degrees of freedom, they proved that the amplitudes of both string theories agree. They also conjectured that this should persist whatever the form of the physical states. The aim of this note is to prove that both theories have the same spectrum of physical states without making any assumption on the form of the physical states. We also notice in passing that this result is reminiscent of a well-known fact in the theory of induced representations and we explore what repercussions this may have in the search for the universal string theory. 
  Let P -> M be a principal G-bundle. Using techniques from the loop representation of gauge theory, we construct well-defined substitutes for ``Lebesgue measure'' on the space A of connections on P and for ``Haar measure'' on the group Ga of gauge transformations. More precisely, we define algebras of ``cylinder functions'' on the spaces A, Ga, and A/Ga, and define generalized measures on these spaces as continuous linear functionals on the corresponding algebras. Borrowing some ideas from lattice gauge theory, we characterize generalized measures on A, Ga, and A/Ga in terms of graphs embedded in M. We use this characterization to construct generalized measures on A and Ga, respectively. The ``uniform'' generalized measure on A is invariant under the group of automorphisms of P. It projects down to the generalized measure on A/Ga considered by Ashtekar and Lewandowski in the case G = SU(n). The ``generalized Haar measure'' on Ga is right- and left-invariant as well as Aut(P)-invariant. We show that averaging any generalized measure on A against generalized Haar measure gives a gauge-invariant generalized measure on A. 
  The Conformal Field Theory of the current algebra of the centrally extended 2-d Euclidean group is analyzed. Its representations can be written in terms of four free fields (without background charge) with signature ($-$+++). We construct all irreducible representations of the current algebra with unitary base out of the free fields and their orbifolds. This is used to investigate the spectrum and scattering of strings moving in the background of a gravitational wave. We find that all the dynamics happens in the transverse space or the longitunal one but not both. 
  In this paper path integration in two- and three-dimensional spaces of constant curvature is discussed: i.e.\ the flat spaces $\bbbr^2$ and $\bbbr^3$, the two- and three-dimensional sphere and the two- and three dimensional pseudosphere. The Laplace operator in these spaces admits separation of variables in various coordinate systems. In all these coordinate systems the path integral formulation will be stated, however in most of them an explicit solution in terms of the spectral expansion can be given only on a formal level. What can be stated in all cases, are the propagator and the corresponding Green function, respectively, depending on the invariant distance which is a coordinate independent quantity. This property gives rise to numerous identities connecting the corresponding path integral representations and propagators in various coordinate systems with each other. 
  We consider a Kaluza-Klein theory whose ground state is ${\bf R}^4 \times {\bf M } \times {\bf K}$ where ${\bf M}$ and ${\bf K}$ are compact, irreducible, homogenous internal mani folds. This is the simplest ground state compatible with the existence of the graviton, gauge fields, massless scalar fields and the absence of the cosmological constan t. The requirement for these conditions to be satisfied are the odd dimensionality of ${\bf M}$ and ${\bf K}$, and the choice of a dimensionally continued Euler form action whose dimension is the same as the dimension of ${\bf M} \times {\bf K}$. We show that in such a theory, which is not simple due to presence of two internal manifolds, the gauge couplings $g^2_M$ and $g^2_K$ are actually unified provided that the internal space sizes are constant. For ${\bf M} \times {\bf K} = S^{2m+ 1} \times S^{2k+1}$ this gauge coupling unification relation reads $g^2_M / g^2_K = m / k$. 
  We apply the theory of superselection sectors in the same way as done by G.Mack and V.Schomerus for the Ising model to generalizations of this model described by J.Fr\"{o}hlich and T.Kerler. 
  We present a new application of affine Lie algebras to massive quantum field theory in 2 dimensions, by investigating the $q\to 1$ limit of the q-deformed affine $\hat{sl(2)}$ symmetry of the sine-Gordon theory, this limit occurring at the free fermion point. Working in radial quantization leads to a quasi-chiral factorization of the space of fields. The conserved charges which generate the affine Lie algebra split into two independent affine algebras on this factorized space, each with level 1 in the anti-periodic sector, and level $0$ in the periodic sector. The space of fields in the anti-periodic sector can be organized using level-$1$ highest weight representations, if one supplements the $\slh$ algebra with the usual local integrals of motion. Introducing a particle-field duality leads to a new way of computing form-factors in radial quantization. Using the integrals of motion, a momentum space bosonization involving vertex operators is formulated. Form-factors are computed as vacuum expectation values in momentum space. (Based on talks given at the Berkeley Strings 93 conference, May 1993, and the III International Conference on Mathematical Physics, String Theory, and Quantum Gravity, Alushta, Ukraine, June 1993.) 
  Using a combination of analytical and numerical methods, we obtain a two-dimensional spacetime describing a black hole with tachyon hair. The physical ADM mass of the black hole is finite. The presence of tachyon hair increases the Hawking temperature. 
  We show that the super-Lax operator for $~N=1$~ supersymmetric Kadomtsev-Petviashvili equation of Manin and Radul in three-dimensions can be embedded into recently developed self-dual supersymmetric Yang-Mills theory in $~2+2\-$dimensions, based on general features of its underlying super-Lax equation. The differential geometrical relationship in superspace between the embedding principle of the super-Lax operator and its associated super-Sato equation is clarified. This result provides a good guiding principle for the embedding of other integrable sub-systems in the super-Lax equation into the four-dimensional self-dual supersymmetric Yang-Mills theory, which is the consistent background for $~N=2$~ superstring theory, and potentially generates other unknown supersymmetric integrable models in lower-dimensions. 
  A simple quantum mechanical model of $N$ free scalar fields interacting with a dynamical moving mirror is formulated and shown to be equivalent to two-dimensional dilaton gravity. We derive the semi-classical dynamics of this system, by including the back reaction due to the quantum radiation. We develop a hamiltonian formalism that describes the time evolution as seen by an asymptotic observer, and write a scattering equation that relates the in-falling and out-going modes at low energies. At higher incoming energy flux, however, the classical matter-mirror dynamics becomes unstable and the mirror runs off to infinity. This instability provides a useful paradigm for black hole formation and introduces an analogous information paradox. Finally, we propose a new possible mechanism for restoring the stability in the super-critical situation, while preserving quantum coherence. This mechanism is based on the notion of an effective time evolution, that takes into account the quantum mechanical effect of the measurement of the Hawking radiation on the state of the infalling matter. 
  We argue that a consistent quantization of the Floreanini-Jackiw model, as a constrained system, should start by recognizing the improper nature of the constraints. Then each boundary conditon defines a problem which must be treated sparately. The model is settled on a compact domain which allows for a discrete formulation of the dynamics; thus, avoiding the mixing of local with collective coordinates. For periodic boundary conditions the model turns out to be a gauge theory whose gauge invariant sector contains only chiral excitations. For antiperiodoc boundary conditions, the mode is a second-class theory where the excitations are also chiral. In both cases, the equal-time algebra of the quantum energy-momentum densities is a Virasoro algebra. The Poincar\'e symmetry holds for the finite as well as for the infinite domain. 
  We prove local background independence of the complete quantum closed string field theory using the recursion relations for string vertices and the existence of connections in CFT theory space. Indeed, with this data we construct an antibracket preserving map between the state spaces of two nearby conformal theories taking the corresponding string field measures $d\mu e^{2S/\hbar}$ into each other. A geometrical construction of the map is achieved by introducing a Batalin-Vilkovisky (BV) algebra on spaces of Riemann surfaces, together with a map to the BV algebra of string functionals. The conditions of background independence show that the field independent terms of the master action arise from vacuum vertices $\V_{g,0}$, and that the overall $\hbar$-independent normalization of the string field measure involves the theory space connection. Our result puts on firm ground the widely believed statement that string theories built from nearby conformal theories are different states of the same theory. 
  We consider gauged WZW models based on a four dimensional non-semi-simple group. We obtain conformal $\s$-models in $D=3$ spacetime dimensions (with exact central charge $c=3$) by axially and vectorially gauging a one-dimensional subgroup. The model obtained in the axial gauging is related to the $3D$ black string after a correlated limit is taken in the latter model. By identifying the CFT corresponding to these $\s$-models we compute the exact expressions for the metric and dilaton fields. All of our models can be mapped to flat spacetimes with zero antisymmetric tensor and dilaton fields via duality transformations. 
  It has previously been proved that the lowest order supersymmetric WKB approximation reproduces the exact bound state spectrum of shape invariant potentials. We show that this is not true for a new, recently discovered class of shape invariant potentials and analyse the reasons underlying this breakdown of the usual proof. 
  We consider a class of $2+D$ - dimensional string backgrounds with a target space metric having a covariantly constant null Killing vector and flat `transverse' part. The corresponding sigma models are invariant under $D$ abelian isometries and are transformed by $O(D,D)$ duality into models belonging to the same class. The leading-order solutions of the conformal invariance equations (metric, antisymmetric tensor and dilaton), as well as the action of $O(D,D)$ duality transformations on them, are exact, i.e. are not modified by $\a'$-corrections. This makes a discussion of different space-time representations of the same string solution (related by $O(D,D|Z)$ duality subgroup) rather explicit. We show that the $O(D,D)$ duality may connect curved $2+D$-dimensional backgrounds with solutions having flat metric but, in general, non-trivial antisymmetric tensor and dilaton. We discuss several particular examples including the $2+D=4$ - dimensional background that was recently interpreted in terms of a WZW model. 
  We describe a few properties of the XXX spin chain with long range interaction. The plan of these notes is:  1. The Hamiltonian  2. Symmetry of the model  3. The irreducible multiplets  4. The spectrum  5. Wave functions and statistics  6. The spinon description  7. The thermodynamics 
  By means of toric geometry we study hypersurfaces in weighted projective space of dimension four. In particular we compute for a given manifold its intrinsic topological coupling. We find that the result agrees with the calculation of the corresponding coupling on the mirror model in the large complex structure limit. 
  We consider (1+1)-dimensional QCD coupled to scalars in the adjoint representation of the gauge group SU($N$). This model results from dimensional reduction of the (2+1)-dimensional pure glue theory. In the large-$N$ limit we study the spectrum of glueballs numerically, using the discretized \lcq. We find a discrete spectrum of bound states, with the density of levels growing approximately exponentially with the mass. A few low-lying states are very close to being eigenstates of the parton number, and their masses can be accurately calculated by truncated diagonalizations. 
  Recently, we proposed a new front-form quantization which treated both the $x^{+}$ and the $x^{-}$ coordinates as front-form 'times.' This quantization was found to preserve parity explicitly. In this paper we extend this construction to fermion fields in the context of the Yukawa theory. We quantize this theory using a method proposed originally by Faddeev and Jackiw . We find that $P^-$ {\it and} $P^+$ become dynamical and that the theory is manifestly invariant under parity. 
  Recently, we have proposed a new front-form quantization which treated both the $x^{+}$ and the $x^{-}$ coordinates as front-form 'times.' This quantization was found to preserve parity explicitly. In this paper we extend this construction to local Abelian gauge fields . We quantize this theory using a method proposed originally by Faddeev and Jackiw . We emphasize here the feature that quantizing along both $x^+$ and $x^-$ , gauge theories does not require extra constraints (also known as 'gauge conditions') to determine the solution uniquely. 
  A set of Green functions ${\cal G}_{\alpha}(x-y), \alpha \in [0, 2 \pi [$, for free scalar field theory is introduced, varying between the Hadamard Green function $\Delta_1(x-y) \equiv \linebreak[2] \lsta{0} \hspace{-0.1cm} \{ \varphi(x), \varphi(y) \} \hspace{-0.1cm} \rsta{0}$ and the causal Green function ${\cal G}_{\pi}(x-y) = i \Delta(x-y) \equiv [\varphi(x), \varphi(y)]$. For every $\alpha \in [0, 2 \pi [$ a path-integral representation for ${\cal G}_{\alpha}$ is obtained both in the configuration space and in the phase space of the classical relativistic particle. Especially setting $\alpha = \pi$ a sum-over-histories representation for the causal Green function is obtained. Furthermore using BRST theory an alternative path-integral representation for ${\cal G}_{\alpha}$ is presented. From these path integral representations the composition laws for the ${\cal G}_{\alpha}$'s are derived using a modified path decomposition expansion. 
  We show how to apply post selection in the context of weak measurement of Aharonov and collaborators to construct the quantum back reaction on a classical field. The particular case which we study in this paper is pair creation in an external electric field and the back reaction is the counter field produced by the pair \underline {as} it is made. The construction leads to a complex electric field obtained from non diagonal matrix elements of the current operator, the interpretation of which is clear in terms of weak measurement. The analogous construction applied to black hole physics (thereby leading to a complex metric) is relegated to a future paper. 
  In this paper we present a new series of 3-dimensional integrable lattice models with $N$ colors. The case $N=2$ generalizes the elliptic model of our previous paper. The weight functions of the models satisfy modified tetrahedron equations with $N$ states and give a commuting family of two-layer transfer-matrices. The dependence on the spectral parameters corresponds to the static limit of the modified tetrahedron equations and weights are parameterized in terms of elliptic functions. The models contain two free parameters: elliptic modulus and additional parameter $\eta$. Also we briefly discuss symmetry properties of weight functions of the models. 
  A brief review of the motivation and the present status of Fractional Superstring is presented. Talk at ``Strings 93'', Berkeley, May 1993. 
  We study the classical theory of a non-Abelian gauge field (gauge group $SU(2)$) coupled to a massive dilaton, massive axion and Einstein gravity. The theory is inspired by the bosonic part of the low-energy heterotic string action for a general Yang-Mills field, which we consider to leading order after compactification to $(3+1)$ dimensions. We impose the condition that spacetime be static and spherically symmetric, and we introduce masses via a dilaton-axion potential associated with supersymmetry (SUSY)-breaking by gaugino condensation in the hidden sector. In the course of describing the possible non-Abelian solutions of the simplified theory, we consider in detail two candidates: a massive dilaton coupled to a purely magnetic Yang-Mills field, and a massive axion field coupled to a non-Abelian dyonic configuration, in which the electric and magnetic fields decay too rapidly to correspond to any global gauge charge. We discuss the feasibility of solutions with and without a nontrivial dilaton for the latter case, and present numerical regular and black hole solutions for the former. 
  I discuss a comprehensive approach to the spacelike physics in high temperature QCD in three dimensions. The approach makes use of dimensional reduction. I suggest that this approach is useful for high temperature QCD in four dimensions. 
  Some first results are presented regarding the behavior of invariant correlations in simplicial gravity, with an action containing both a bare cosmological term and a lattice higher derivative term. The determination of invariant correlations as a function of geodesic distance by numerical methods is a difficult task, since the geodesic distance between any two points is a function of the fluctuating background geometry, and correlation effects become rather small for large distances. Still, a strikingly different behavior is found for the volume and curvature correlation functions. While the first one is found to be negative definite at large geodesic distances, the second one is always positive for large distances. For both correlations the results are consistent in the smooth phase with an exponential decay, turning into a power law close to the critical point at $G_c$. Such a behavior is not completely unexpected, if the model is to reproduce the classical Einstein theory at distances much larger than the ultraviolet cutoff scale. 
  We argue that a basic modification must be made to the first quantised formalism of string theory if the physics of `particle creation' is to be correctly described. The analogous quantisation of the relativistic particle is performed, and it is shown that the proper time along the world line must go both forwards and backwards (in the usual quantisation it only goes forwards). The matrix propagator of the real time formalism is obtained from the two directions of proper time. (Talk given at the Thermal Fields Workshop held at Banff, Canada (August 1993).) 
  We discuss new results in QCD obtained with string-based methods. These methods were originally derived from superstring theory and are significantly more efficient than conventional Feynman rules. This technology was a key ingredient in the first calculation of the one-loop five-gluon amplitude. We also present a conjecture for a particular one-loop helicity amplitude with an arbitrary number of external gluons. 
  An algebraic approach to integrable quantum field theory with a boundary (a half line) is presented and interesting algebraic equations, Reflection equations (RE) and Reflection Bootstrap equations (RBE) are discussed. The Reflection equations are a consistent generalisation of Yang-Baxter equations for factorisable scatterings on a half line (or with a reflecting boundary). They determine the so-called reflection matrices. However, for Toda field theory and/or other theories with diagonal S-matrices, the Reflection-Bootstrap equations proposed by Fring and K\"oberle determine the reflection matrices, since the reflection equations and the Yang-Baxter equations become trivial in these cases. The explicit forms of the reflection matrices together with their symmetry properties are given for various Toda field theories, simply laced and non-simply laced. 
  Matrix generalizations of the N-particle quantum-mechanical Calogero model classifying according to representations of the symmetric group $S_N$ are considered. Symmetry properties of the eigen-wave functions in the matrix Calogero models are analyzed. Latex. 
  A noncommutative-geometric generalization of the theory of principal bundles is sketched. A differential calculus over corresponding quantum principal bundles is analysed. The formalism of connections is presented. In particular, operators of covariant derivative and horizontal projection are described and analysed. Quantum counterparts for the Bianchi identity and the Weil's homomorphism are found. Illustrative examples are considered. (Lecture presented at the XXII-th Conference on Differential Geometric Methods in Theoretical Physics, Ixtapa-Zihuatanejo, Mexico, September 1993). 
  There are several ways to establish and study thermal properties of black holes. I review here method of Fulling and Ruijsenaars, based on the analytic structure of Green functions on the complex plane. This method provides a clear distinction between zero and finite temperature field theories, and allows for quick evaluation of black hole temperature. (Lectures presented at the Danube Workshop '93, June 1993, Belgrade, Yugoslavia.) 
  The Liouville action emerges as the effective action of 2-d gravity in the process of path integral quantization of the bosonic string. It yields a measure of the violation of classical symmetries of the theory at the quantum level. Certain aspects of the residual SL(2,R) invariance of the gauge-fixed Liouville theory are disscussed. ( Lectures presented at the Danube Workshop '93,   June 1993, Belgrade, Yugoslavia.) 
  We consider a Hamiltonian analysis of the Liouville theory and construction of symmetry generators using Castellani's method. We then discuss the light-cone gauge fixed theory. In particular, we clarify the difference between Hamiltonian approaches based on different choices of time, $\xi^0$ and $\xi^+$. Our main result is the construction of SL(2,R) symmetry generators in both cases. ( Lectures presented at the Danube Workshop '93,   June 1993, Belgrade, Yugoslavia.) 
  We extend previous work on the IR regime approximation of QCD in which the dominant contribution comes from a dressed two-gluon effective metric-like field $G_{\mu\nu} = g_{ab} B^{a}_{\mu} B^{b}_{\nu}$ ($g_{ab}$ a color $SU(3)$ metric). The ensuring effective theory is represented by a pseudo-diffeomorphisms gauge theory. The second -quantized $G_{\mu\nu}$ field, together with the Lorentz generators close on the $\overline{SL}(4,R)$ algebra. This algebra represents a spectrum generating algebra for the set of hadron states of a given flavor - hadronic "manifields" transforming w.r.t. $\overline{SL}(4,R)$ (infinite-dimensional) unitary irreducible representations. The equations of motion for the effective pseudo-gravity are derived from a quadratic action describing Riemannian pseudo-gravity in the presence of shear ($\overline{SL}(4,R)$ covariant) hadronic matter currents. These equations yield $p^{-4}$ propagators, i.e. a linearly rising confining potential $H(r) \sim r$, as well as linear $J \sim m^{2}$ Regge trajectories. The $\overline{SL}(4,R)$ symmetry based dynamical theory for the QCD IR region is applied to hadron resonances. All presently known meson and baryon resonances are successfully accommodated and various missing states predicted. (Lectures presented at the Danube Workshop '93, June 1993, Belgrade, Yugoslavia.) 
  The dissipative Hofstadter model, which describes a particle in 2-D subject to a periodic potential, uniform magnetic field, and dissipation, is also related to open string boundary states. This model exhibits an SL(2,Z) duality symmetry and hidden reparametrization invariance symmetries. These symmetries are useful for finding exact solutions for correlation functions. 
  On an equation associated with the contact Lie algebras/ Mikhail V. Saveliev/ In the framework of a Lie algebraic approach we study a nonlinear equation associated with the contact Lie algebra ${\bf K}K_m$, that seems to be relavant for some solvable models of field theory and gravity in higher dimensions. LPTENS--93/43. 
  Invited talk at the International Symposium on Generalized Symmetries in Physics at the Arnold-Sommerfeld-Institute, Clausthal, Germany, July 26 -- July 29, 1993. This talk reviews results on the structure of algebras consisting of meromorphic differential operators which are holomorphic outside a finite set of points on compact Riemann surfaces. For each partition into two disjoint subsets of the set of points where poles are allowed, a grading of the algebra and of the modules of lambda - forms is introduced. With respect to this grading the Lie structure of the algebra and of the modules are almost graded ones. Central extensions and semi-infinite wedge representations are studied. If one considers only differential operators of degree 1 then these algebras are generalizations of the Virasoro algebra in genus zero, resp. of Krichever Novikov algebras in higher genus. 
  The quantum inverse scattering method is applied to the solution of equations for wave functions of compound states of $n$ reggeized gluons in the multicolour QCD in a generalized leading logarithmic approximation. 
  We construct the Bardeen anomaly and its related Wess-Zumino term in the supersymmetric standard model. In particular we show that it can be written in terms of a composite linear superfield related to supersymmetrized Chern-Simons forms, in very much the same way as the Green-Schwarz term in four-dimensional string theory. Some physical applications, such as the contribution to the g-2 of gauginos when a heavy top is integrated out, are briefly discussed. 
  Both the coherent states and also the squeezed states of the harmonic oscillator have long been understood from the three classical points of view: the 1) displacement operator, 2) annihilation- (or ladder-) operator, and minimum-uncertainty methods. For general systems, there is the same understanding except for ladder-operator and displacement-operator squeezed states. After reviewing the known concepts, I propose a method for obtaining generalized minimum-uncertainty squeezed states, give examples, and relate it to known concepts. I comment on the remaining concept, that of general displacement-operator squeezed states. 
  The quantized Knizhnik-Zamolodchikov equations associated with the trigonometric R-matrix or the rational R-matrix of the A-type are considered. Jackson integral representations for solutions of these equations are described. Asymptotic solutions for a holonomic system of difference equations are constructed. Relations between the integral representations and the Bethe ansatz are indicated. 
  Let $U_q(\hat{\cal G})$ be an infinite-dimensional quantum affine Lie algebra. A family of central elements or Casimir invariants are constructed and their eigenvalues computed in any integrable irreducible highest weight representation. These eigenvalue formulae are shown to absolutely convergent when the deformation parameter $q$ is such that $|q|>1$. It is proven that the universal R-matrix $R$ of $U_q(\hat{\cal G})$ satisfies the celebrated conjugation relation $R^\dagger=TR$ with $T$ the usual twist map. As applications, the braid generator is shown to be diagonalizable on arbitrary tensor product modules of integrable irreducible highest weight $U_q(\hat{\cal G})$-modules and a spectral decomposition formula for the braid generator is obtained which is the generalization of Reshetikhin's and Gould's forms to the present affine case. Casimir invariants acting on a specified module are also constructed and their eigenvalues, again absolutely convergent for $|q|>1$, computed by means of the spectral decomposition formula. 
  We analyze global aspects of the moduli space of K\"ahler forms for $N$=(2,2) conformal $\sigma$-models. Using algebraic methods and mirror symmetry we study extensions of the mathematical notion of length (as specified by a K\"ahler structure) to conformal field theory and calculate the way in which lengths change as the moduli fields are varied along distinguished paths in the moduli space. We find strong evidence supporting the notion that, in the robust setting of quantum Calabi-Yau moduli space, string theory restricts the set of possible K\"ahler forms by enforcing ``minimal length'' scales, provided that topology change is properly taken into account. Some lengths, however, may shrink to zero. We also compare stringy geometry to classical general relativity in this context. 
  The review of the following results of the Refs. \cite{Sem} - \cite{Ans} is presented: For mixed state light of $N$-mode electromagnetic field described by Wigner function which has generic Gaussian form the photon distribution function is obtained and expressed expliciltly in terms of Hermite polynomials of $2N$-variables.The momenta of this distribution are calculated and expressed as functions of matrix invariants of the dispersion matrix.The role of new uncertainty relation depending on photon state mixing parameter is elucidated.New sum rules for Hermite polynomials of several variables are found.The photon statistics of polymode even and odd coherent light and squeezed polymode Schr\"odinger cat light is qiven explicitly.Photon distribution for polymode squeezed number states expressed in terms of multivariable Hermite polynomials is discussed. 
  The global counterpart of $k$-Poincare algebra is considered. The induced representations of this group are described. The explicit form of the covariant wave functions in the `minimal' (in Weinberg's sense) case is given. 
  Supergravity theory in $2+\epsilon$ dimensions is studied. It is invariant under supertransformations in 2 and 3 dimensions. One-loop divergence is explicitly computed in the background field method and a nontrivial fixed point is found. In quantizing the supergravity, a gauge fixing condition is devised which explicitly isolates conformal and superconformal modes. The renormalization of the gravitationally dressed operators is studied and their anomalous dimensions are computed. Problems to use the dimensional reduction are also examined. 
  The classification of quasi - primary fields is outlined. It is proved that the only conserved quasi - primary currents are the energy - momentum tensor and the O(N) - Noether currents. Derivation of all quasi - primary fields and the resolution of degeneracy is sketched. Finally the limits d=2 and d=4 of the space dimension are discussed. Whereas the latter is trivial the former is only almost so. 
  We investigate the Ward identities of the $W_{\infty}$ symmetry in the Liouville theory coupled to the $(p,q)$ conformal matter. The correlation functions are defined by applying the analytic continuation procedure for the matter sector as well as the Liouville one. We then find that the Ward identities are equivalent to the $W_q$ algebra constraints deduced from the matrix model. 
  It is shown that new leading ($\al'$) as well as all-order solutions of String theory can be obtained by taking appropriate singular limits of the known solutions. We give several leading order solutions for the bosonic as well as the heterotic string. We then present all-order forms of the previously known two dimensional cosmological solutions. An all-order form for the cosmological solution in three dimensions is also predicted. The physical implications of our results are discussed. 
  By properly accounting for the invariance of a Calabi-Yau sigma-model under shifts of the $B$-field by integral amounts (analagous to the $\theta$-angle in QCD), we show that the moduli spaces of such sigma-models can often be enlarged to include ``large radius limit'' points. In the simplest cases, there are holomorphic coordinates on the enlarged moduli space which vanish at the limit point, and which appear as multipliers in front of instanton contributions to Yukawa couplings. (Those instanton contributions are therefore suppressed at the limit point.) In more complicated cases, the instanton contributions are still suppressed but the enlarged space is singular at the limit point. This singularity may have interesting effects on the effective four-dimensional theory, when the Calabi-Yau is used to compactify the heterotic string. 
  A Fermion in 2+1 dimensions, with a mass function which depends on one spatial coordinate and passes through a zero ( a domain wall mass), is considered. In this model, originally proposed by Callan and Harvey, the gauge variation of the effective gauge action mainly consists of two terms. One comes from the induced Chern-Simons term and the other from the chiral fermions, bound to the 1+1 dimensional wall, and they are expected to cancel each other. Though there exist arguments in favour of this, based on the possible forms of the effective action valid far from the wall and some facts about theories of chiral fermions in 1+1 dimensions, a complete calculation is lacking. In this paper we present an explicit calculation of this cancellation at one loop valid even close to the wall. We show that, integrating out the ``massive'' modes of the theory does produce the Chern-Simons term, as appreciated previously. In addition we show that it generates a term that softens the high energy behaviour of the 1+1 dimensional effective chiral theory thereby resolving an ambiguity present in a general 1+1 dimensional theory. 
  Results obtained by us are overviewed from a general set up. The universal $R$-matrix is exploited to obtain various important relations and structures involved in quantum group algebra, which are used subsequently for generating different classes of quantum integrable systems through a systematic scheme. This recovers known models as well as discovers a series of new ones. 
  We describe the Hamiltonian reduction of the coajoint Kac-Moody orbits to the Virasoro coajoint orbits explicitly in terms of the Lagrangian approach for the Wess-Zumino-Novikov-Witten theory. While a relation of the coajoint Virasoro orbit $Diff \; S^1 /SL(2,R)$ to the Liouville theory has been already studied we analyse the role of special coajoint Virasoro orbits $Diff \; S^1/\tilde{T}_{\pm ,n}$ corresponding to stabilizers generated by the vector fields with double zeros. The orbits with stabilizers with single zeros do not appear in the model. We find an interpretation of zeros $x_i$ of the vector field of stabilizer $\tilde{T}_{\pm ,n}$ and additional parameters $q_i$, $i = 1,...,n$, in terms of quantum mechanics for $n$ point particles on the circle. We argue that the special orbits are generated by insertions of "wrong sign" Liouville exponential into the path integral. The additional parmeters $q_i$ are naturally interpreted as accessory parameters for the uniformization map. Summing up the contributions of the special Virasoro orbits we get the integrable sinh-Gordon type theory. 
  Considering anyonic oscillators in a two-dimensional lattice, we realize the quantum semi-group $sl_{(q,s)}(2)$ by means of a generalized Schwinger construction. We find that the parameter $q$ of the algebra is connected to the statistical parameter, whereas the $s$ parameter is related to a $s$-deformed oscillator introduced at each point of the lattice. 
  A topological gravity is obtained by twisting the effective $(2,0)$ super\-gravity. We show that this topological gravity has an infinite number of BRST invariant quantities with conformal weight $0$. They are a tower of OSp$(2,2)$ multiplets and satisfy the classical exchange algebra of OSp$(2,2)$. We argue that these BRST invariant quantities become physical operators in the quantum theory and their correlation functions are braided according to the quantum OSp$(2,2)$ group. These properties of the topological effective gravity are not shared by the standard topological gravity. 
  The N=2 supersymmetric extension of the 2+1 dimensional Abelian Higgs model is discussed. By analysing the resulting supercharge algebra, the connection between supersymmetry and Bogomol'nyi equations is clarified. Analogous results are presented when the model is considered in 2-dimensional (Euclidean) space. 
  Monopole-mediated baryon number violation, the Callan-Rubakov effect, is reexamined using boundary conformal field theory techniques. It is shown that the low-energy behaviour is described simply by free fermions with a conformally invariant boundary condition at the dyon location. When the number of fermion flavours is greater than two, this boundary condition is of a non-trivial type which has not been elucidated previously. 
  We consider quon statistics in a dynamically evolving curved spacetime in which prior to some initial time and subsequent to some later time is flat. By considering the Bogoliubov transformations associated with gravitationally induced particle creation, we find that the consistent evolution of the generalized commutation relations from the first flat region to the second flat region can only occur if $q = \pm 1$. 
  The finite-volume spectrum of an integrable massive perturbation of a rational conformal field theory interpolates between massive multi-particle states in infinite volume (IR limit) and conformal states, which are approached at zero volume (UV limit). Each state is labeled in the IR by a set of `Bethe Ansatz quantum numbers', while in the UV limit it is characterized primarily by the conformal dimensions of the conformal field creating it. We present explicit conjectures for the UV conformal dimensions corresponding to any IR state in the $\phi_{1,3}$-perturbed minimal models $M(2,5)$ and $M(3,5)$. The conjectures, which are based on a combinatorial interpretation of the Rogers-Ramanujan-Schur identities, are consistent with numerical results obtained previously for low-lying energy levels. 
  A non-parametric gauge for supermembranes is introduced which substantially simplifies previous treatments and directly leads to the supersymmetric extension of a Karman-Tsien gas. 
  A recently-proposed technique, called the dimensional expansion, uses the space-time dimension $D$ as an expansion parameter to extract nonperturbative results in quantum field theory. Here we apply dimensional-expansion methods to examine the Ising limit of a self-interacting scalar field theory. We compute the first few coefficients in the dimensional expansion for $\gamma_{2n}$, the renormalized $2n$-point Green's function at zero momentum, for $n\!=\!2$, 3, 4, and 5. Because the exact results for $\gamma_{2n}$ are known at $D\!=\!1$ we can compare the predictions of the dimensional expansion at this value of $D$. We find typical errors of less than $5\%$. The radius of convergence of the dimensional expansion for $\gamma_{2n}$ appears to be ${{2n}\over {n-1}}$. As a function of the space-time dimension $D$, $\gamma_{2n}$ appears to rise monotonically with increasing $D$ and we conjecture that it becomes infinite at $D\!=\!{{2n}\over {n-1}}$. We presume that for values of $D$ greater than this critical value, $\gamma_{2n}$ vanishes identically because the corresponding $\phi^{2n}$ scalar quantum field theory is free for $D\!>\!{{2n}\over{n-1}}$. 
  We construct quantum deformation of Poincar\'e group using as a starting point $SU(2,2)$ conformal group and twistor-like definition of the Minkowski space. We obtain quantum deformation of $SU(2,2)$ as a real form of multiparametric $GL(4,C)_{q_{ij},r}$. It is shown that Poincar\'e subgroup exists for special nonstandard one-parametric deformation only, the deformation parameter $r$ being equal to unity. This leads to commuting affine structure of the corresponding Minkowski space and simple structure of the corresponding Lie algebra, the deformation of the group being non-trivial. 
  We consider a class of sigma models that appears from a generalisation of the gauged WZW model parametrised by a constant matrix $Q$. Particular values of $Q$ correspond to the standard gauged WZW models, chiral gauged WZW models and a bosonised version of the non-abelian Thirring model. The condition of conformal invariance of the models (to one loop or $1/k$-order but exactly in $Q$) is derived and is represented as an algebraic equation on $Q$. Solving this equation we demonstrate explicitly the conformal invariance of the sigma models associated with arbitrary $G/H$ gauged and chiral gauged WZW theories as well as of the models that can be represented as WZW model perturbed by integrably marginal operators (constructed from currents of the Cartan subalgebra $H_c$ of $G$). The latter models can be also interpreted as $G x H/H$ gauged WZW models and have the corresponding target space couplings (metric, antisymmetric tensor and dilaton) depending on an arbitrary constant matrix which parametrises an embedding of the abelian subgroup $H$ (isomorphic to $H_c$) into $G x H$. We discuss the relation of our conformal invariance equation to the large $k$ form of the master equation of the affine-Virasoro construction. Our equation describes `reducible' versions of some `irreducible' solutions (cosets) of the master equation. We suggest a classically non-Lorentz-invariant sigma model that may correspond to other solutions of the master equation. 
  We introduce a topological field theory with a Bogomol'nyi structure permitting BPS electric, magnetic and dyonic monopoles. From the general arguments given by Montonen and Olive the particle spectrum and mass compare favourably with that of the intermediate vector bosons. In most, if not in all, of its essential features the topological field theory introduced here provides an example of a dual field theory, the existence of which was conjectured by Montonen and Olive. 
  We study a model of quantum Yang-Mills theory with a finite number of gauge invariant degrees of freedom. The gauge field has only a finite number of degrees of freedom since we assume that space-time is a two dimensional cylinder. We couple the gauge field to matter, modeled by either one or two nonrelativistic point particles. These problems can be solved {\it without any gauge fixing}, by generalizing the canonical quantization methods of Ref.\[rajeev] to the case including matter. For this, we make use of the geometry of the space of connections, which has the structure of a Principal Fiber Bundle with an infinite dimensional fiber. We are able to reduce both problems to finite dimensional, exactly solvable, quantum mechanics problems. In the case of one particle, we find that the ground state energy will diverge in the limit of infinite radius of space, consistent with confinement. In the case of two particles, this does not happen if they can form a color singlet bound state (`meson'). 
  Light-cone quantization of gauge field theory is considered. With a careful treatment of the relevant degrees of freedom and where they must be initialized, the results obtained in equal-time quantization are recovered, in particular the Mandelstam-Leibbrandt form of the gauge field propagator. Some aspects of the ``discretized'' light-cone quantization of gauge fields are discussed. 
  Two-dimensional quantum cromodynamics in the light-front frame is studied following hamiltonian methods. The theory is quantized using the path integral formalism and an effective theory similar to the Nambu-Jona Lasinio model is obtained. Confinement in two dimensions is derived analyzing directly the constraints in the path integral. 
  The non-isospectral symmetries of a general class of integrable hierarchies are found, generalizing the Galilean and scaling symmetries of the Korteweg--de Vries equation and its hierarchy. The symmetries arise in a very natural way from the semi-direct product structure of the Virasoro algebra and the affine Kac--Moody algebra underlying the construction of the hierarchy. In particular, the generators of the symmetries are shown to satisfy a subalgebra of the Virasoro algebra. When a tau-function formalism is available, the infinitesimal symmetries act directly on the tau-functions as moments of Virasoro currents. Some comments are made regarding the r\^ole of the non-isospectral symmetries and the form of the string equations in matrix-model formulations of quantum gravity in two-dimensions and related systems. 
  We review the construction of particle physics models in the framework of non-commutative geometry. We first give simple examples, and then progress to outline the Connes-Lott construction of the standard Weinberg-Salam model and our construction of the SO(10) model. We then discuss the analogue of the Einstein-Hilbert action and gravitational matter couplings. Finally we speculate on some experimental signatures of predictions specific to the non-commutative approach. 
  These notes correspond rather accurately to the translation of the lectures given at the Fifth Mexican School of Particles and Fields, held in Guanajuato, Gto., in December~1992. They constitute a brief and elementary introduction to quantum symmetries from a physical point of view, along the lines of the forthcoming book by C. G\'omez, G. Sierra and myself. 
  We discuss a differential integrable hierarchy, which we call the (N, M)$--th KdV hierarchy, whose Lax operator is obtained by properly adding $M$ pseudo--differential terms to the Lax operator of the N--th KdV hierarchy. This new hierarchy contains both the higher KdV hierarchy and multi--field representation of KP hierarchy as sub--systems and naturally appears in multi--matrix models. The N+2M-1 coordinates or fields of this hierarchy satisfy two algebras of compatible Poisson brackets which are {\it local} and {\it polynomial}. Each Poisson structure generate an extended W_{1+\infty} and W_\infty algebra, respectively. We call W(N, M) the generating algebra of the extended W_\infty algebra. This algebra, which corresponds with the second Poisson structure, shares many features of the usual $W_N$ algebra. We show that there exist M distinct reductions of the (N, M)--th KdV hierarchy, which are obtained by imposing suitable second class constraints. The most drastic reduction corresponds to the (N+M)--th KdV hierarchy. Correspondingly the W(N, M) algebra is reduced to the W_{N+M} algebra. We study in detail the dispersionless limit of this hierarchy and the relevant reductions. 
  We describe a canonical covariant approach to the quantization of the Green-Schwarz superstring. The approach is first applied to the canonical covariant quantization of the Brink and Schwarz superparticle. The Kallosh action is obtained in this case, with the correct BRST cohomology. 
  A review is given of recent research on two-dimensional gauge theories, with particular emphasis on the equivalence between these theories and certain string theories with a two-dimensional target space. Some related open problems are discussed. 
  We study the two-dimensional Ising model with a defect line and evaluate multipoint energy correlation functions using non-perturbative field-theoretical methods. We also discuss the evaluation of the two spin correlator on the defect line. 
  We consider the moduli space of flat connections on the Riemann surface with marked points. The new efficient parametrization is suggested and used to construct an integrable model on the moduli space. A family of commuting Hamiltonians is extracted from the trace of the transfer matrix built from the Wilson line observables of the Chern-Simons theory. Our model appears to be gauge equivalent to XXZ magnetic chain with finite number of sites. 
  Pairing between the universal enveloping algebra $U_q(sl(2))$ and the algebra of functions over $SL_q(2)$ is obtained in explicit terms. The regular representation of the quantum double is constructed and investigated. The structure of the root subspaces of the Casimir operator is revealed and described in terms of $SL_q(2)$ elements. 
  Analysing a 3+1 dimensional model with four-Fermi interactions, we show that topological $\BF$ terms (both abelian and non-abelian) can be induced radiatively by massive fermions at the one-loop level. It is further pointed out that a mechanism of photon (or non-abelian gauge field) mass generation distinct from the usual Higgs mechanism, through the $\BF$ term, is also implemented in the long-distance effective action of this model, provided a gap equation is satisfied. 
  We review briefly the thermodynamical interpretation of black hole physics and discuss the problems and inconsistencies in this approach. We provide an alternative interpretation of black holes as quantum objects and investigate the statistical mechanics of a gas of such objects in the microcanonical ensemble. We argue that the theory of black holes has the conformal properties of duality and satisfaction of the statistical bootstrap condition. We show in the context of mean field theory that the thermal vacuum is the false vacuum for a black hole and define a microcanonical vacuum which leads to a number density characteristic of pure states for the Hawking radiation. 
  In the weak field expansion of euclidean quantum gravity, an analysis of the Wilson loops in terms of the gauge group, $SO(4)$, shows that the correspondent statistical system does not develope any configuration with localized curvature at low temperature. Such a ``non-local'' behavior contrasts strongly with that of usual gauge fields. We point out a possible relation between this result and those of the numerical simulations of quantum Regge Calculus. We also give a general quantum formula for the static potential energy of the gravitational interaction of two masses, which is compatible with the mentioned vacuum structure. 
  A theory of higher-derivative 2D dilaton gravity which has its roots in the massive higher-spin mode dynamics of string theory is suggested. The divergences of the effective action to one-loop are calculated, both in the covariant and in the conformal gauge. Some technical problems which appear in the calculations are discussed. An interpretation of the theory as a particular D=2 higher-derivative $\sigma$-model is given. For a specific case of higher-derivative 2D dilaton gravity, which is one loop multiplicatively renormalizable, static configurations corresponding to black holes are shown to exist. 
  We study the cosmological meaning of duality symmetry by considering a two dimensional model of string cosmology. We find that as seen by an internal observer in this universe, the scale factor rebounds at the self-dual length. This rebound is a consequence of the adiabatic expansion. Furthermore, in this situation there are four mathematically different scenarios which describe physically equivalent universes which are in fact indistinguishable. We also stress that $R$-duality suffices to prove that all possible evolutions present a maximum temperature. 
  A class of string backgrounds associated with non semi-simple groups is obtained as a special large level limit of ordinary WZW models. The models have an integer Virasoro central charge and they include the background recently studied by Nappi and Witten. 
  Motivated by suggestions of Paolo Cotta-Ramusino's work at the physical level of rigor relating BF theory to the Donaldson polynomials, we provide a construction applicable to the Turaev/Viro and Crane/Yetter invariants of *a priori* finer invariants dependent on a choice of (co)homology class on the manifold 
  Recent advances in non-critical string theory allow a unique continuation of critical Polyakov string amplitudes to off-shell momenta, while preserving conformal invariance. These continuations possess unusual, apparently stringy, characteristics, as we illustrate with our results for three-point functions. (Talk by R.C.M. at Strings '93) 
  Nilpotent BRST operators for higher-spin $W_{2,s}$ strings, with currents of spins 2 and $s$, have recently been constructed for $s=4$, 5 and 6. In the case of $W_{2,4}$, this operator can be understood as being the BRST operator for the critical $W\!B_2$ string. In this paper, we construct a generalised BRST operator that can be associated with a non-critical $W_{2,4}$ string, in which $W\!B_2$ matter is coupled to the $W\!B_2$ gravity of the critical case. We also obtain the complete cohomology of the critical $W_{2,4}$ BRST operator, and investigate the physical spectra of the $s=5$ and $s=6$ string theories. 
  We investigate the connection between curved spacetime and the emergence of string-instabilities, following the approach developed by Loust\'{o} and S\'{a}nchez for de Sitter and black hole spacetimes. We analyse the linearised equations determining the comoving physical (transverse) perturbations on circular strings embedded in Schwarzschild, Reissner-Nordstr\"{o}m and de Sitter backgrounds. In all 3 cases we find that the "radial" perturbations grow infinitely for $r\rightarrow 0$ (ring-collapse), while the "angular" perturbations are bounded in this limit. For $r\rightarrow\infty$ we find that the perturbations in both physical directions (perpendicular to the string world-sheet in 4 dimensions) blow up in the case of de Sitter space. This confirms results recently obtained by Loust\'{o} and S\'{a}nchez who considered perturbations around the string center of mass. 
  We derive the loop equations for the d-dimensional n-Hermitian matrix model. These are a consequence of the Schwinger-Dyson equations of the model. Moreover we show that in leading order of large $N$ the loop equations form a closed set. In particular we derive the loop equations for the $n=2^k$ matrix model recently proposed to describe the coupling of Two-dimensional quantum gravity to conformal matter with $c> 1$. 
  A method for finding the renormalization group (RG) improved effective Lagrangian for a massive interacting field theory in curved spacetime is presented. As a particular example, the $\lambda \varphi^4$-theory is considered and the RG improved effective Lagrangian is explicitly found up to second order in the curvature tensors. As a further application, the curvature-induced phase transitions are discussed for both the massive and the massless versions of the theory. The problems which appear when calculating the RG improved effective Lagrangian for gauge theories are discussed, taking as example the asymptotically free SU(2) gauge model. 
  I argue that the complete partition function of 3D quantum gravity is given by a path integral over gauge-inequivalent manifolds times the Chern-Simons partition function. In a discrete version, it gives a sum over simplicial complexes weighted with the Turaev-Viro invariant. Then, I discuss how this invariant can be included in the general framework of lattice gauge theory (qQCD$_3$). To make sense of it, one needs a quantum analog of the Peter-Weyl theorem and an invariant measure, which are introduced explicitly. The consideration here is limited to the simplest and most interesting case of $SL_q(2)$, $q=e^{i\frac{2\pi}{k+2}}$. At the end, I dwell on 3D generalizations of matrix models. 
  We show how to calculate correlation functions of two matrix models. Our method consists in making full use of the integrable hierarchies and their reductions, which were shown in previous papers to naturally appear in multi--matrix models. The second ingredient we use are the $W$--constraints. In fact an explicit solution of the relevant hierarchy, satisfying the $W$--constraints (string equation), underlies the explicit calculation of the correlation functions. In the course of our derivation we do not use any continuum limit tecnique. This allows us to find many solutions which are invisible to the latter technique. 
  Model interactions between classical and quantum systems are briefly reviewed. These include: general measurement - like couplings, Stern-Gerlach experiment, model of a counter, quantum Zeno effect, piecewise deterministic Markov processes and meaning of the wave function. 
  In this letter, we discuss the extension of Feynman's derivation of the equation of motion to the case of spinning particles. We show that a spinning particle interacts only with the electromagnetic and gravitational fields. In the absence of the electromagnetic interactions, we rederive Papapetrou's equations for spinning particles in the background of the conformal gravity. We also find that the effect of spin coupled to non-constant electromagnetic fields leads to further corrections to the Lorentz force equations. Some discussions of these results are given at the end. 
  We present a solution of the problem of a free massless scalar field on the half line interacting through a periodic potential on the boundary. For a critical value of the period, this system is a conformal field theory with a non-trivial and explicitly calculable S-matrix for scattering from the boundary. Unlike all other exactly solvable conformal field theories, it is non-rational ({\it i.e.} has infinitely many primary fields). It describes the critical behavior of a number of condensed matter systems, including dissipative quantum mechanics and of barriers in ``quantum wires''. 
  We formulate WZW models based on a centrally extended version of the Euclidean group in $d$-dimensions. We obtain string backgrounds corresponding to conformal $\s$-models in $D=d^2$ space-time dimensions with exact central charge $c=d^2$ and $d(d-1)/2$ null Killing vectors. By identifying the corresponding conformal field theory we show that the one loop results coincide with the exact ones up to a shifting of a parameter. 
  B. Fedosov has given a simple and very natural construction of a deformation quantization for any symplectic manifold, using a flat connection on the bundle of formal Weyl algebras associated to the tangent bundle of a symplectic manifold. The connection is obtained by affinizing, nonlinearizing, and iteratively flattening a given torsion free symplectic connection. In this paper, a classical analog of Fedosov's operations on connections is analyzed and shown to produce the usual exponential mapping of a linear connection on an ordinary manifold. A symplectic version is also analyzed. Finally, some remarks are made on the implications for deformation quantization of Fedosov's index theorem on general symplectic manifolds. 
  We introduce and study the Koszul complex for a Hecke $R$-matrix. Its cohomologies, called the Berezinian, are used to define quantum superdeterminant for a Hecke $R$-matrix. Their behaviour with respect to Hecke sum of $R$-matrices is studied. Given a Hecke $R$-matrix in $n$-dimensional vector space, we construct a Hecke $R$-matrix in $2n$-dimensional vector space commuting with a differential. The notion of a quantum differential supergroup is derived. Its algebra of functions is a differential coquasitriangular Hopf algebra, having the usual algebra of differential forms as a quotient. Examples of superdeterminants related to these algebras are calculated. Several remarks about Woronowicz's theory are made. 
  We show that the external algebra $\cal M$ on $GL(N)$ can be equipped with the graded Poisson brackets compatible with the group action. We prove that there are only two graded Poisson-Lie structures (brackets) on $\cal M$ and we obtain their explicit description. We realize that just these two structures appear as the quasiclassical limit of the bicovariant differential calculi on the quantum linear group $GL_q (N)$. 
  We provide a simple lagrangian interpretation of the meaning of the $b_0^-$ semi-relative condition in closed string theory. Namely, we show how the semi-relative condition is equivalent to the requirement that physical operators be cohomology classes of the BRS operators acting on the space of local fields {\it covariant} under world-sheet reparametrizations. States trivial in the absolute BRS cohomology but not in the semi-relative one are explicitly seen to correspond to BRS variations of operators which are not globally defined world-sheet tensors. We derive the covariant expressions for the observables of topological gravity. We use them to prove a formula that equates the expectation value of the gravitational descendant of ghost number 4 to the integral over the moduli space of the Weil-Peterson K\"ahler form. 
  Cosmological solutions with a homogeneous Yang-Mills field which oscillates and passes between topologically distinct vacua are discussed. These solutions are used to model the collapsing Bartnik-McKinnon gravitational sphaleron and the associated anomalous production of fermions. The Dirac equation is analyzed in these backgrounds. It is shown explicitly that a fermion energy level crosses from the negative to positive energy spectrum as the gauge field evolves between the topologically distinct vacua. The cosmological solutions are also generalized to include an axion field. 
  The Letter reconsiders a result obtained by Chr\'etien and Peierls in 1954 within nonlocal QED in 4D [Proc. Roy. Soc. London A 223, 468]. Starting from secondly quantized fermions subject to a nonlocal action with the kernel $[ i\not\partial_x a(x) - m b(x)]$ and gauge covariantly coupled to an external U(1) gauge field they found that for $a = b$ the induced gauge field action cannot be made finite irrespectively of the choice of the nonlocality $a$ $(= b)$. But, the general case $a \neq b$ naturally to be studied admits a finitely induced gauge field action, as the present Letter demonstrates. 
  We introduce selfdual Maxwell-Higgs systems with uniform background electric charge density and show that the selfdual equations satisfied by topological vortices can be reduced to the original Bogomol'nyi equations without any background. These vortices are shown to carry no spin but to feel the Magnus force due to the shielding charge carried by the Higgs field. We also study the dynamics of slowly moving vortices and show that the spin-statistics theorem holds to our vortices. 
  Light front field theories are known to have the usual infra-red divergences of the equal time theories, as wellas new `spurious' infra-red divergences. The formar kind of IR divergences are usually treated by giving a small mass to the gauge particle. An alternative method to deal with these divergences is to calculate the transition matrix elements in a coherent state basis. In this paper we present, as a model calculation the lowest order correction to the three point vertex in QED using a coherent state basis in the light cone formalism. The relevant transition matrix element is shown to be free of the true IR divergences up to $O(e^2)$. 
  We study the infrared limit of two dimensional QCD, with massless dynamical Dirac fermions that are in the fundamental representation of the gauge group. We find that the theory reduces to a spin generalization of the Calogero model with an additional magnetic coupling which is of the Pauli type. 
  Two canonical formulations of the Einstein gravity in 2+1 dimensions, namely, the ADM formalism and the Chern-Simons gravity, are investigated in the case of nonvanishing cosmological constant. General arguments for reducing phase spaces of the two formalisms are given when spatial hypersurface is compact. In particular when the space has the topology of a sphere $S^{2}$ or a torus $T^{2}$, the spacetimes constructed from these two formulations can be identified and the classical equivalence between the ADM and the CSG is shown. Moreover in the $g=1$ case the relations between their phase spaces, and therefore between their quantizations, are given in almost the same form as that in the case when the cosmological constant vanishes. There are, however, some modifications, the most remarkable one of which is that the phase space of the CSG is in 1 to 2 correspondence with the one of the ADM when the cosmological constant is negative. 
  After a short review of the algebraic setting of N=2 superconformal field theories, their chiral ring and their perturbations, I present some recent results on curious relations between the integrability of their perturbations and algebraic properties of their deformed chiral ring. (Lecture given at Hang-zhou, China, Sept 1993) 
  A system of electrons in the two-dimensional honeycomb lattice with Coulomb interactions is described by a renormalizable quantum field theory similar but not equal to QED_3. Renormalization group techniques are used to investigate the infrared behaviour of the system that flows to a fixed point with non-Fermi liquid characteristics. There are anomalous dimensions in the fermionic observables, no quasiparticle pole, and anomalous screening of the Coulomb interaction. These results are robust as the Fermi level is not changed by the interaction. The system resembles in the infrared the one-dimensional Luttinger liquid. 
  In the present contribution we show that the introduction of a conserved axial current in electrodynamics can explain the quantization of electric charge, preserving parity conservation, and introduces a dynamical discreteness into space-time. 
  We study chiral vertex operators in the sine-Gordon [SG] theory, viewed as an off-conformal system. We find that these operators, which would have been primary fields in the conformal limit, have interesting and, in some ways, unexpected properties in the SG model. Some of them continue to have scale- invariant dynamics even in the presence of the non-conformal cosine interaction. For instance, it is shown that the Mandelstam operator for the bosonic representation of the Fermi field does {\it not} develop a mass term in the SG theory, contrary to what the real Fermi field in the massive Thirring model is expected to do. It is also shown that in the presence of the non-conformal interactions, some vertex operators have unique Lorentz spins, while others do not. 
  In the light front quantisation scheme initial conditions are usually provided on a single lightlike hyperplane. This, however, is insufficient to yield a unique solution of the field equations. We investigate under which additional conditions the problem of solving the field equations becomes well posed. The consequences for quantisation are studied within a Hamiltonian formulation by using the method of Faddeev and Jackiw for dealing with first-order Lagrangians. For the prototype field theory of massive scalar fields in 1+1 dimensions, we find that initial conditions for fixed light cone time {\sl and} boundary conditions in the spatial variable are sufficient to yield a consistent commutator algebra. Data on a second lightlike hyperplane are not necessary. Hamiltonian and Euler-Lagrange equations of motion become equivalent; the description of the dynamics remains canonical and simple. In this way we justify the approach of discretised light cone quantisation. 
  We show that every Lie algebra or superLie algebra has a canonical braiding on it, and that in terms of this its enveloping algebra appears as a flat space with braided-commuting coordinate functions. This also gives a new point of view about $q$-Minkowski space which arises in a similar way as the enveloping algebra of the braided Lie algebra $gl_{2,q}$. Our point of view fixes the signature of the metric on $q$-Minkowski space and hence also of ordinary Minkowski space at $q=1$. We also describe an abstract construction for left-invariant integration on any braided group. 
  Recently it has been argued, that Poincar\'{e} supersymmetric field theories admit an underlying loop space hamiltonian (symplectic) structure. Here shall establish this at the level of a general $N=1$ supermultiplet. In particular, we advocate the use of a superloop space and explain the necessity of using nonconventional auxiliary fields. As an example we consider the nonlinear $\sigma$-model. Due to the quartic fermionic term, we conclude that the use of superloop space variables is necessary for the action to have a hamiltonian loop space interpretation. 
  We find a quantum group structure in two-dimensional motions of a nonrelativistic electron in a uniform magnetic field and in a periodic potential. The representation basis of the quantum algebra is composed of wavefunctions of the system. The quantum group symmetry commutes with the Hamiltonian and is relevant to the Landau level degeneracy. The deformation parameter $q$ of the quantum algebra turns out to be given by the fractional filling factor $\nu=1/m$ ($m$ odd integer). 
  We consider the possible covariant external algebra structures for Cartan's 1-forms on GL_q(N) and SL_q(N). We base upon the following natural postulates:  1. the invariant 1-forms realize an adjoint representation of quantum group;  2. all monomials of these forms possess the unique ordering.   For the obtained external algebras we define the exterior derivative possessing the usual nilpotence condition, and the generally deformed version of Leibniz rules. The status of the known examples of GL_q(N)-differential calculi in the proposed classification scheme, and the problems of SL_q(N)-reduction are discussed. 
  We study the realisation of global symmetries in a polynomial formulation of the non-linear sigma-model. We show that there are global symmetries whose corresponding Noether currents are the topological currents in the usual formulation. The usual Noether currents associated with the internal symmetry group are reproduced, but part of them become non-local in terms of the dynamical variables of the polynomial formulation. 
  The concept of universal T matrix, recently introduced by Fronsdal and Galindo in the framework of quantum groups, is here discussed as a generalization of the exponential mapping. New examples related to inhomogeneous quantum groups of physical interest are developed, the duality calculations are explicitly presented and it is found that in some cases the universal T matrix, like for Lie groups, is expressed in terms of usual exponential series. 
  We study the Regge regime of QCD as a special regime of lattice gauge theory on an asymmetric lattice. This lattice has a spacing $a_0 $ in the longitudinal direction and a spacing $a_t $ in the transversal direction. The limit $\frac{a_{0}}{a_{t}} \to 0$ corresponds to correlation functions with small longitudinal and large transversal coordinates, i.e. large $s$ and small $t$. On this lattice the longitudinal dynamics is described by the usual two-dimensional chiral field in finite volume and the transversal dynamics is emerged through an effective interaction of boundary terms of the longitudinal dynamics. The effective interaction depends crucially on the spectrum of the two-dimensional chiral field. Massless exitations produce an effective 2-dimensional action which is different from the action recently proposed by H.Verlinde and E.Verlinde. Massive exitations give raise to an effective action located on the contour in the longitudinal plane. 
  A large class of Thermodynamic Bethe Ansatz equations governing the Renormalization Group evolution of the Casimir energy of the vacuum on the cylinder for an integrable two-dimensional field theory, can often be encoded on a tensor product of two graphs. We demonstrate here that in this case the two graphs can only be of $ADE$ type. We also give strong numerical evidence for a new large set of Dilogarithm sum Rules connected to $ADE\times ADE$ and a simple formula for the ultraviolet perturbing operator conformal dimensions only in terms of rank and Coxeter numbers of $ADE\times ADE$. We conclude with some remarks on the curious case $ADE\times D$. [Talk given by F.R. at the Cargese Workshop "New Developments in String Theory, Conformal Models and Topological Field Theory" (May 1993)] 
  The two-dimensional supersymmetric $\s$-model on a K\"ahler manifold has a non-vanishing $\b$-function at four loops, but the $\b$-function at five loops can be made to vanish by a specific choice of renormalisation scheme. We investigate whether this phenomenon persists at six loops, and conclude that it does not; there is a non-vanishing six-loop $\b$-function irrespective of renormalisation scheme ambiguities. 
  Abstr.: The classical r-matrix implied by the quantum k-Poincare algebra of Lukierski,Nowicki and Ruegg is used to generate a Poisson structure on the ISL(2,C) group. A quantum deformation of the ISL(2,C) group ( on the Hopf algebra level ) is obtained by a trivial quantization. 
  Exploiting the formulation of the Self Dual Yang-Mills equations as a Riemann-Hilbert factorization problem, we present a theory of pulling back soliton hierarchies to the Self Dual Yang-Mills equations. We show that for each map $ \C^4 \to \C^{\infty } $ satisfying a simple system of linear equations formulated below one can pull back the (generalized) Drinfeld-Sokolov hierarchies to the Self Dual Yang-Mills equations. This indicates that there is a class of solutions to the Self Dual Yang-Mills equations which can be constructed using the soliton techniques like the $\tau$ function method. In particular this class contains the solutions obtained via the symmetry reductions of the Self Dual Yang-Mills equations. It also contains genuine 4 dimensional solutions . The method can be used to study the symmetry reductions and as an example of that we get an equation exibiting breaking solitons, formulated by O. Bogoyavlenskii, as one of the $2 + 1 $ dimensional reductions of the Self Dual Yang-Mills equations. 
  We present solutions of the low-energy four-dimensional heterotic string corresponding to $p$-branes with $p=0,1,2$, which are characterized by a mass per unit $p$-volume, ${\cal M}_{p+1}$, and topological ``magnetic'' charge, $g_{p+1}$. In the extremal limit, $\sqrt{2} \kappa {\cal M}_{p+1} = g_{p+1}$, they reduce to the recently discovered non-singular supersymmetric monopole, string and domain wall solutions. A novel feature is that the solutions involve both the dilaton and the modulus fields. In particular, the effective scalar coupling to the Maxwell field, $e^{-\alpha \phi} F_{\mu\nu} F^{\mu\nu}$, gives rise to a new string black hole with $\alpha = \sqrt{3}$, in contrast to the pure dilaton black hole solution which has $\alpha=1$. This means that electric/magnetic duality in $D=4$ may be seen as a consequence of string/fivebrane duality in $D=10$. 
  We discuss the differential algebras used in Connes' approach to Yang-Mills theories with spontaneous symmetry breaking. These differential algebras generated by algebras of the form functions $\otimes$ matrix are shown to be skew tensorproducts of differential forms with a specific matrix algebra. For that we derive a general formula for differential algebras based on tensor products of algebras. The result is used to characterize differential algebras which appear in models with one symmetry breaking scale. 
  We derive a charged black hole solution in four dimensions described by $SL(2,R)\times SU(2)\times U(1)/U(1)^2$ WZW coset model. Using the algebraic Hamiltonian method we calculate the corresponding solution that is exact to all orders in ${1\over k}$. It is shown that unlike the 2D black hole, the singularity remains also in the exact solution, and moreover, in some range of the gauge parameter the space-time does not fulfil the cosmic censor conjecture, $i.e.$ we find a naked singularity outside the black hole. Exact dual models are derived as well, one of them describes a 4D space-time with a naked singularity. Using the algebraic Hamiltonian approach we also find the exact to all orders $O(d,d)$ transformation of the metric and the dilaton field for general WZW coset models and show the correction with respect to the transformations in one loop order. 
  Families of operator identities appeared as a consequence of an existence of finite-dimensional representation of (super) Lie algebras of first-order differential operators and $q$-deformed (quantum) algebras of first-order finite-difference operators are presented. It is shown that those identities can be rewritten in terms of creation/annihilation operators and it leads to a simplification of the problem of the normal ordering in the second quantization method. 
  It is shown that in the P,T-invariant model with the mixed Chern-Simons term the interaction of charge carriers leads to effective changing of their statistics, which depends on distance between them. In particular, in the limit of large distances fermions effectively turn into bosons. 
  The Lie algebra of pseudodifferential symbols on the circle has a nontrivial central extension (by the ``logarithmic'' 2-cocycle) generalizing the Virasoro algebra. The corresponding extended subalgebra of integral operators generates the Lie group of classical symbols of all real (or complex) degrees. It turns out that this group has a natural Poisson-Lie structure whose restriction to differential operators of an arbitrary integer order coincides with the second Adler-Gelfand-Dickey structure. Moreover, for any real (or complex) \alpha there exists a hierarchy of completely integrable equations on the degree \alpha pseudodifferential symbols, and this hierarchy for \alpha=1 coincides with the KP one, and for an integer \alpha=n>1$ and purely differential symbol gives the n-KdV-hierarchy. 
  It is shown how the Mandelstam constraints for an $SU(2)$ pure lattice gauge theory with $3{\cal N}$ physical degrees of freedom may be solved completely in terms of $3{\cal N}$ Wilson and Polyakov loop variables and ${\cal N}-1$ gauge invariant discrete +/-1 variables, thus enabling a manifestly gauge invariant formulation of the theory. 
  We construct the Poincare polynomials for Landau-Ginzburg orbifolds with projection operators.Using them we show that special types of dualities including Poincare duality are realized under certain conditions. When Calabi-Yau interpretation exists, two simple formulae for Hodge numbers $h^{2,1}$ and $h^{1,1}$ are obtained. 
  Some errors in section 4 are corrected. No change in the results. 
  Notes of the author's talk at Cargese, May 19,1993. The calculation of the string field commutator is simplified by relating it to the short proper time, semiclassical string path integral. 
  Following some motivating comments on large N two-dimensional Yang-Mills theory, we discuss techniques for large N group representation theory, using quantum mechanics on the group manifold U(N), its equivalence to a quasirelativistic two-dimensional free fermion theory, and bosonization. As applications, we compute the free energy for two-dimensional Yang-Mills theory on the torus to O(1/N^2), and an interesting approximation to the leading answer for the sphere. We discuss the question of whether the free energy for the torus has R -> 1/R invariance. A substantially revised version of hep-th/9303159 with many new results. 
  The spin-4/3 fractional superstring is characterized by a world-sheet chiral algebra involving spin-4/3 currents. The discussion of the tree-level scattering amplitudes of this theory presented in hepth/9310131 is expanded to include amplitudes containing two twisted-sector states. These amplitudes are shown to satisfy spurious state decoupling. The restriction to only two external twisted-sector states is due to the absence of an appropriate dimension-one vertex describing the emission of a single twisted-sector state. This is analogous to the ``old covariant'' formalism of ordinary superstring amplitudes in which an appropriate dimension-one vertex for the emission of a Ramond-sector state is lacking. Examples of tree scattering amplitudes are calculated in a c=5 model of the spin-4/3 chiral algebra realized in terms of free bosons on the string world-sheet. The target space of this model is three-dimensional flat Minkowski space-time and the twisted-sector physical states are fermions in space-time. Since the critical central charge of the spin-4/3 fractional superstring theory is 10, this c=5 model is not consistent at the string loop level. 
  In this paper the quantum integrals of the Hamiltonian of the quantum many-body problem with the interaction potential K/sinh^2(x) (Sutherland operator) are constructed as images of higher Casimirs of the Lie algebra gl(N) under a certain homomorphism from the center of U(gl(N)) to the algebra of differential operators in N variables. A similar construction applied to the affine gl(N) at the critical level k=-N defines a correspondence between higher Sugawara operators and quantum integrals of the Hamiltonian of the quantum many-body problem with the potential equal to constant times the Weierstrass function. This allows one to give a new proof of the Olshanetsky-Perelomov theorem stating that this Hamiltonian defines a completely integrable quantum system. We also give a new expression for eigenfunctions of the quantum integrals of the Sutherland operator as traces of intertwining operators between certain representations of gl(N). 
  We describe a classical configuration of conformal matter forming a naked singularity and discuss its subsequent Hawking evaporation within the context of two dimensional dilaton gravity. The one loop analysis is credible for a large mass naked singularity and suggests the existence of a weak cosmological censorship that would cause it to explode into radiation upon forming. (Hardcopies of figures available on request) 
  The generalized form of the Kac formula for Verma modules associated with linear brackets of hydrodynamics type is proposed. Second cohomology groups of the generalized Virasoro algebras are calculated. Connection of the central extensions with the problem of quntization of hydrodynamics brackets is demonstrated. 
  We present the method for finding of the nonlinear Poisson-Lie groups structures on the vector spaces and for their quantization. For arbitrary central extension of Lie algebra explicit formulas of quantization are proposed. 
  We consider a simplified model of particles with effectively distance dependent statistics, that is particles coupled to a gauge field the Lagrangian of which contains the Chern-Simons term. We analyze the low-lying states of the two-particle system and show that under certain conditions they can exhibit negative compressibility, hinting on a possible \`a la van der Vaals picture. 
  We present results for the BRST cohomology of $\cW[\bfg]$ minimal models coupled to $\cW[\bfg]$ gravity, as well as scalar fields coupled to $\cW[\bfg]$ gravity. In the latter case we explore an intricate relation to the (twisted) $\bfg$ cohomology of a product of two twisted Fock modules. 
  We review the construction of exactly solvable lattice models whose continuum limits are $N=2$ supersymmetric models. Both critical and off-critical models are discussed. The approach we take is to first find lattice models with natural topological sectors, and then identify the continuum limits of these sectors with topologically twisted $N=2$ supersymmetric field theories. From this, we then describe how to recover the complete lattice versions of the $N=2$ supersymmetric field theories. We discuss a number of simple physical examples and we describe how to construct a broad class of models. We also give a brief review of the scattering matrices for the excitations of these models. (Contribution to the procedings of the Cargese meeting on ``String Theory, Conformal Models and Topological Field Theories'', May 12-21, 1993.) 
  A similarity transformation, which brings a particular class of the $N=1$ string to the $N=0$ one, is explicitly constructed. It enables us to give a simple proof for the argument recently proposed by Berkovits and Vafa. The $N=1$ BRST operator is turned into the direct sum of the corresponding $N=0$ BRST operator and that for an additional topological sector. As a result, the physical spectrum of these $N=1$ vacua is shown to be isomorphic to the tensor product of the $N=0$ spectrum and the topological sector which consists of only the vacuum. This transformation manifestly keeps the operator algebra. 
  A general procedure for deriving the path integral representation of a transition amplitude on the gauge orbit space having a non-trivial topology is proposed. The path integral formula appears to be modified by including trajectories reflected from the physical configuration space boundary into the sum over paths. A solution of the Gribov problem of gauge fixing ambiguities is given in the framework of the path integral modified. Email contact: shabanov@amoco.saclay.cea.fr 
  The $N\to\infty$ limit of the edges of finite planar electron densities is discussed for higher Landau levels. For full filling, the particle number is correlated with the magnetic flux, and hence with the boundary location, making the $N\to\infty$ limit more subtle at the edges than in the bulk. In the $n^{\rm th}$ Landau level, the density exhibits $n$ distinct steps at the edge, in both circular and rectangular samples. The boundary characteristics for individual Landau levels, and for successively filled Landau levels, are computed in an asymptotic expansion. 
  Non-abelian coordinate ring of $U_q(SL(N))$ (quantum deformation of the algebra of functions) for $N=2,3$ is represented in terms of conventional creation and annihilation operators. This allows to construct explicitly representations of this algebra, which were earlier described in somewhat more abstract algebraic fashion. Generalizations to $N>3$ and Kac-Moody algebras are not discussed but look straightforward. 
  Motivated by recent developments in the computation of periods for string compactifications with $c=9$, we develop a complementary method which also produces a convenient basis for related calculations. The models are realized as Calabi--Yau hypersurfaces in weighted projective spaces of dimension four or as Landau-Ginzburg vacua. The calculation reproduces known results and also allows a treatment of Landau--Ginzburg orbifolds with more than five fields. 
  By considering the most general metric which can occur on a contractable two dimensional symplectic manifold, we find the most general Hamiltonians on a two dimensional phase space to which equivariant localization formulas for the associated path integrals can be applied. We show that in the case of a maximally symmetric phase space the only applicable Hamiltonians are essentially harmonic oscillators, while for non-homogeneous phase spaces the possibilities are more numerous but ambiguities in the path integrals occur. In the latter case we give general formulas for the Darboux Hamiltonians, as well as the Hamiltonians which result naturally from a generalized coherent state formulation of the quantum theory which shows that again the Hamiltonians so obtained are just generalized versions of harmonic oscillators. Our analysis and results describe the quantum geometry of some two dimensional systems. 
  Two possibile applications of the optimized expansion for the free energy of the quantum-mechanical anharmonic oscillator are discussed. The first method is for the finite temperature effective potential; the second one, for the classical effective potential. The results of both methods show a quick convergence and agree well with the exact free energy in the whole range of temperatures. Postscript figures are available under request to AO email ROZYNEK@FUW.EDU.PL 
  The conformal symmetry in the Liouville theory is analysed by using the Hamiltonian light--front formalism. The boundary conditions of dynamical variables are seen to involve an arbitrary function of time, so that the standard methods for studying gauge symmetries do not work. We develop a general method for constructing the gauge generators, which enables a consistent treatment of the boundary conditions present in the case of the conformal symmetry. 
  We study the commutation relations, uncertainty relations and spectra of position and momentum operators within the framework of quantum group % symmetric Heisenberg algebras and their (Bargmann-) Fock representations. As an effect of the underlying noncommutative geometry, a length and a momentum scale appear, leading to the existence of minimal nonzero uncertainties in the positions and momenta. The usual quantum mechanical behaviour is recovered as a limiting case for not too small and not too large distances and momenta. 
  In the non-critical string framework that we have proposed recently, the time $t$ is identified with a dynamical local renormalization group scale, the Liouville mode, and behaves as a statistical evolution parameter, flowing irreversibly from an infrared fixed point - which we conjecture to be a topological string phase - to an ultraviolet one - which corresponds to a static critical string vacuum. When applied to a toy two-dimensional model of space-time singularities, this formalism yields an apparent renormalization of the velocity of light, and a $t$-dependent form of the uncertainty relation for position and momentum of a test string. We speculate within this framework on a stringy alternative to conventional field-theoretical inflation, and the decay towards zero of the cosmological constant in a maximally-symmetric space. 
  Citations are updated; referred papers are increased. An error right after the eq.~(27) is corrected, and several chages (not serious) are made. 
  We present a detailed study of the representations of the algebra of functions on the quantum group $ GL_q(n) $. A q-analouge of the root system is constructed for this algebra which is then used to determine explicit matrix representations of the generators of this algebra. At the end a q-boson realization of the generators of $ GL_q(n) $ is given. 
  We construct the space of vector fields on quantum groups . Its elements are products of the known left invariant vector fields with the elements of the quantum group itself. We also study the duality between vector fields and 1-forms. The construction is easily generalized to tensor fields. A Lie derivative along any (also non left invariant) vector field is proposed. These results hold for a generic Hopf algebra. 
  We consider representations of quadratic $R$-matrix algebras by means of certain first order ordinary differential operators. These operators turn out to act as parameter shifting operators on the Gauss hypergeometric function and its limit cases and on classical orthogonal polynomials. The relationship with W. Miller's treatment of Lie algebras of first order differential operators will be discussed. 
  It is shown that the matrix KP hierarchy can yield new integrable equations in $(2+1)$-dimensions along with the corresponding Lax pair. For particular gauge choice this may result derivative and also a higher order nonlinear extension of the Devay-Stewartson equation (DSE),the higher order DSE being a higher dimensional generalisation of the Kundu- Eckhaus equation. Such gauge transformation is shown also to produce significant extensions to the constrained matrix KP system. 
  Exploiting the residual gauge freedom in the formulation of constrained KP hierarchy a number of new integrable systems are derived including hierarchies of Kundu-Eckhaus equation and higher order nonlinear extensions of Yajima-Oikawa and Melnikov hierarchy. In the multicomponent case such gauge freedom generates novel multicomponent as well as vector generalisations of the above systems, while the constrained modified KP hierarchy is found to yield another set of equations like derivative NLS, Gerdjikov-Ivanov equation and chen-Lee-Liu equation depending on the gauge choice. 
  In this paper we develop a theory for constructing an invariant of closed oriented 3-manifolds, given a certain type of Hopf algebra. Examples are given by a quantised enveloping algebra of a semisimple Lie algebra, or by a semisimple involutory Hopf algebra. The invariant is defined by a state sum model on a triangulation. In some cases, the invariant is the partition function of a topological quantum field theory. 
  I present a new class of topological string theories, and discuss them in two dimensions as candidates for the string description of large-$N$ QCD. The starting point is a new class of topological sigma models, whose path integral is localized to the moduli space of harmonic maps from the worldsheet to the target. The Lagrangian is of fourth order in worldsheet derivatives. After gauging worldsheet diffeomorphisms in this ``harmonic topological sigma model,'' we obtain a topological string theory dominated by minimal-area maps. The bosonic part of this ``topological rigid string'' Lagrangian coincides with the Lagrangian proposed by Polyakov for the QCD string in higher dimensions. (talk given at the Cargese conference on ``Recent Developments in String Theory, CFT and Topological Field Theory'' (May 1993)) 
  Recently Berkovits and Vafa have shown that the bosonic string can be viewed as the fermionic string propagating in a particular background. Such a background is described by a somewhat unusual $N=1$ superconformal system. By coupling it to $N=1$ supergravity I construct a local supersymmetric action for the bosonic string. 
  Invariance under finite renormalization group (RG) transformations is used to structure the invariant charge in models with one coupling in the 4 lowest orders of perturbation theory. In every order there starts a RG-invariant, which is uniquely continued to higher orders. Whereas in massless models the RG-invariants are power series in logarithms, there is no such requirement in a massive model. Only, when one applies the Callan-Symanzik (CS) equation of the respective theories, the high-energy behavior of the RG-invariants is restricted. In models, where the CS-equation has the same form as the RG-equation, the massless limit is reached smoothly, i.e. the beta-functions are constants in the asymptotic limit and the RG-functions starting the new invariant tend to logarithms. On the other hand in the spontaneously broken models with fermions the CS-equation contains a beta-function of a physical mass. As a consequence the beta-functions depend on the normalization point also in the asymptotic region and a mass independent limit does not exist anymore. 
  The effective action for the multi-Regge asymptotics is considered as a first step in calculating the unitarity correction to the perturbative pomeron. It can be derived from the original QCD action by intgrating out certain modes of the fields in the functional integral. The derivation is described for the case without fermions. 
  The oscillations of photon distribution function for squeezed and correlated light are shown to decrease when the temperature increases.The influence of the squeezing parameter and photon quadrature correlation coefficient on the photon distribution oscillations at nonzero temperatures is studied. The connection of deformation of Planck distribution formula with oscillations of distribution for squeezed and correlated light is discussed. 
  We note that a version ``with spectral parameter'' of the Drinfeld-Sokolov reduction gives a natural mapping from the KdV phase space to the group of loops with values in $\widehat N_{+}/A, \widehat N_{+}$~: affine nilpotent and $A$ principal commutative (or anisotropic Cartan) subgroup~; this mapping is connected to the conserved densities of the hierarchy. We compute the Feigin-Frenkel action of $\widehat n_{+}$ (defined in terms of screening operators) on the conserved densities, in the $sl_2$ case. 
  In the Higgs phase we may be left with a residual finite symmetry group H of the condensate. The topological interactions between the magnetic- and electric excitations in these so-called discrete H gauge theories are completely described by the Hopf algebra or quantumgroup D(H). In 2+1 dimensional space time we may add a Chern-Simons term to such a model. This deforms the underlying Hopf algebra D(H) into a quasi-Hopf algebra by means of a 3-cocycle H. Consequently, the finite number of physically inequivalent discrete H gauge theories obtained in this way are labelled by the elements of the cohomology group H^3(H,U(1)). We briefly review the above results in these notes. Special attention is given to the Coulomb screening mechanism operational in the Higgs phase. This mechanism screens the Coulomb interactions, but not the Aharonov-Bohm interactions. (Invited talk given by Mark de Wild Propitius at `The III International Conference on Mathematical Physics, String Theory and Quantum Gravity', Alushta, Ukraine, June 13-24, 1993. To be published in Theor. Math. Phys.) 
  The observation that n pairs of para-Fermi (pF) operators generate the universal enveloping algebra of the orthogonal Lie algebra so(2n+1) is used in order to define deformed pF operators. It is shown that these operators are an alternative to the Chevalley generators. On this background Uq[so(2n+1)] and its "Cartan-Weyl" generators are written down entirely in terms of deformed pB operators. 
  In this paper the several aspects of the $Z_{N}$ symmetry in gauge theories at high temperatures are discussed.  The metastable $Z_{N}$ bubbles in the $SU(N)$ gauge theories with fermions may have, generically, unacceptable thermodynamic behavior. Their free energy $F \propto T^4$ with a positive proportionality constant. This leads not only to negative pressure but also to negative specific heat and, more seriously, to negative entropy. We argue that although such domains are important in the Euclidean theory, they cannot be interpreted as physical domains in Minkowski space.  The related problem is connected with the analysis of the high-temperature limit of the confining phase. Using the two-dimensional QCD with adjoint fermions as a toy model we shall demonstrate that in the light fermion limit in this theory there is no breaking of the $Z_{N}$ symmetry in the high-temperature limit and thus there are no $Z_{N}$ bubbles. 
  We discribe a simple way to derive spin correlation functions in 2D Ising model at critical temperature but with nonzero magnetic field at the boundary. Local magnetization (i.e. one-point function) is computed explicitly for half-plane and disk geometries. 
  We study a minimal string model possessing the same massless spectra as the   MSSM on $Z_N\times Z_M$ orbifolds. Threshold corrections of the gauge coupling constants of SU(3), SU(2) and   U(1)$_Y$ are investigated in a case of an overall modulus. Using computer analyses, we search ranges of levels of U(1)$_Y$ allowed by the LEP experiments. It is found that $Z_3\times Z_3$ can not derive the minimal string model for a $M_Z$ SUSY breaking scale. The minimum values of the overall moduli are estimated within the ranges of the levels. 
  In the present paper we give a differential geometry formulation of the basic dynamical principle of the group--algebraic approach \cite{LeS92} --- the grading condition --- in terms of some holomorphic distributions on flag manifolds associated with the parabolic subgroups of a complex Lie group; and a derivation of the corresponding nonlinear integrable systems, and their general solutions. Moreover, the reality condition for these solutions is introduced. For the case of the simple Lie groups endowed with the canonical gradation, when the systems in question are reduced to the abelian Toda equations, we obtain the generalised Pl\"ucker representation for the pseudo--metrics specified by the K\"ahler metrics on the flag manifolds related to the maximal nonsemisimple parabolic subgroups; and the generalised infinitesimal Pl\"ucker formulas for the Ricci curvature tensors of these pseudo--metrics. In accordance with these formulas, the fundamental forms of the pseudo--metrics and the Ricci curvature tensors are expressed directly in terms of the abelian Toda fields, which have here the sense of K\"ahler potentials. 
  We review the coset construction of conformal field theories; the emphasis is on the construction of the Hilbert spaces for these models, especially if fixed points occur. This is applied to the $N=2$ superconformal cosets constructed by Kazama and Suzuki. To calculate heterotic string spectra we reformulate the Gepner con- struction in terms of simple currents and introduce the so-called extended Poincar\'e polynomial. We finally comment on the various equivalences arising between models of this class, which can be expressed as level rank dualities. (Invited talk given at the III. International Conference on Mathematical Physics, String Theory and Quantum Gravity, Alushta, Ukraine, June 1993. To appear in Theor. Math. Phys.) 
  A set of simple exactly solvable potentials are shown to have convergent WKB series. The resulting all-orders quantisation conditions provide a unified description of all known cases where an exact WKB quantisation condition has been obtained by modifying the potential with Langer-style terms, together with several new examples. 
  In this talk I want to explain the operator substractions needed to renormalize gauge currents in a second quantized theory. The case of space-time dimensions $3+1$ is considered in detail. In presence of chiral fermions the renormalization effects a modification of the local commutation relations of the currents by local Schwinger terms. In $1+1$ dimensions on gets the usual central extension (Schwinger term does not depend on background gauge field) whereas in $3+1$ dimensions one gets an anomaly linear in the background potential. We extend our method to the spatial components of currents. Since the bose-fermi interaction hamiltonian is of the form $j^k A_k$ (in the temporal gauge) we get a new renormalization scheme for the interaction. The idea is to define a field dependent conjugation for the fermi hamiltonian in the one-particle space such that after the conjugation the hamiltonian can be quantized just by normal ordering prescription. 
  A new approach to integrability of affine Toda field theories and closely related to them KdV hierarchies is proposed. The flows of a hierarchy are explicitly identified with infinitesimal action of the principal abelian subalgebra of the nilpotent part of the corresponding affine algebra on a homogeneous space.   This is an extended version of the paper "Generalized KdV flows and nilpotent subgroups of affine Kac-Moody groups"; it has been accepted for publication in Inventiones Mathematicae. 
  (This talk was presented at the Third International Wigner Symposium on Group Theory, Oxford, September, 1993.) Matrix models provides us with the most powerful framework in which to analyze D=2 string theory, yet some of its miraculous features, such as discrete states and $w(\infty)$, remain rather obscure, because the string degrees of freedom have been removed. Liouville theory, on the other hand, has all its string degrees of freedom intact, yet is notoriously difficult to solve. In this paper, we present the second quantized formulation of Liouville theory in D=2, where discrete states and $w(\infty)$ have a natural, field theoretic interpretation. We generalize the non-polynomial closed string field theory, first developed by the author and the Kyoto and MIT groups, to the D=2 case. We find that, in second quantized field theory language, the rather mysterious features of matrix models have an intuitively transparent interpretation, similar to standard gauge theory. Latex file. 
  We construct the second quantized action for sub-critical closed string field theory with zero cosmological constant in dimensions $ 2 \leq D < 26$, generalizing the non-polynomial closed string field theory action proposed by the author and the Kyoto and MIT groups for $D = 26$. The proof of gauge invariance is considerably complicated by the presence of the Liouville field $\phi$ and the non-polynomial nature of the action. However, we explicitly show that the polyhedral vertex functions obey BRST invariance to all orders. By point splitting methods, we calculate the anomaly contribution due to the Liouville field, and show in detail that it cancels only if $D - 26 + 1 + 3 Q ^ 2 = 0 $, in both the bosonized and unbosonized polyhedral vertex functions. We also show explicitly that the four point function generated by this action reproduces the shifted Shapiro-Virasoro amplitude found from $c=1$ matrix models and Liouville theory in two dimensions. LATEX file. 
  In the numerical simulation of certain field theoretical models, the complex Langevin simulation has been believed to fail due to the violation of ergodicity. We give a detailed analysis of this problem based on a toy model with one degree of freedom ($S=-\beta\cos\theta$). We find that the failure is not due to the defect of complex Langevin simulation itself, but rather to the way how one treats the singularity appearing in the drift force. An effective algorithm is proposed by which one can simulate the ${1/\beta}$ behaviour of the expectation value $<\cos\theta>$ in the small $\beta$ limit. 
  A generalized gauge invariant Ising model on random surfaces with non-trivial topology is proposed and investigated with the dual transformation. It is proved that the model is self-dual in case of a self-dual lattice. In special cases the model reduces to the known solvable Ising-type models. 
  We study vacuum domain walls in a class of four-dimensional $N=1$ supergravity theories where along with the matter field, forming the wall, there is more than one ``dilaton'', each respecting $SU(1,1)$ symmetry in their sub-sector. We find {\it supersymmetric} (planar, static) walls, interpolaing between Minkowski vacuum and a new class of supersymmetric vacua which have a naked (planar) singularity. Such walls provide the first example of supersymmetric classical ``solitons'' with naked singularities, and thus violate the (strong) cosmic censorship conjecture within a supersymmetric theory. 
  The problems with background independence are discussed in the example of open string theory. Based on the recent proposal by Witten I calculate the String Field Theory action in conformal perturbation theory to second order and demonstrate that the proper treatment of contact terms leads to nontrivial equations of motion. I conjecture the form of the field theory action to all orders. 
  In this revised version we correct some mistakes, realizing the supersymmetry algebra on the exact S matrix, taking into account several phase factros. We study the constraint imposed by supersymmetry on the exact $S$-matrix of $\Complex P^{n-1}$ model, and compute a non-trivial phase factor in the relation between the $S$-matrix and one of the supersymmetry generators. We discuss several features connected with the physical interpretation of the result. The supersymmetry current is studied in such context as well, and we find some operators appearing in the conservation equation of the supersymmetry current that might be connected with such phase factors. The relation with the literature on the subject is also discussed. hep-th/yymmnnn 
  Recent development in noncommutative geometry generalization of gauge theory is reviewed. The mathematical apparatus is reduced to minimum in order to allow the non-mathematically oriented physicists to follow the development in the interesting field of research. (Lectures presented at the Silesian School of Theoretical Physics: Standard Model and Beyond'93, Szczyrk (Poland), September 1993.) 
  It is argued that singular vectors of the topological conformal (twisted $N=2$) algebra are identical with singular vectors of the $sl(2)$ Kac--Moody algebra. An arbitrary matter theory can be dressed by additional fields to make up a representation of either the $sl(2)$ current algebra or the topological conformal algebra. The relation between the two constructions is equivalent to the Kazama--Suzuki realisation of a topological conformal theory as $sl(2)\oplus u(1)/u(1)$. The Malikov--Feigin--Fuchs (MFF) formula for the $sl(2)$ singular vectors translates into a general expression for topological singular vectors. The MFF/topological singular states are observed to vanish in Witten's free-field construction of the (twisted) $N=2$ algebra, derived from the Landau--Ginzburg formalism. 
  Quantum gravitational effects on the renormalization group equation are studied in the $(2+\epsilon)$-dimensional approach. Divergences in a matter one-loop effective action do not receive gravitational radiative corrections. The renormalization factor for beta functions recently found by Klebanov, Kogan and Polyakov is obtained by using the renormalized cosmological constant to define the physical scale transformation. 
  This paper constructs exact classical solutions of the equations of QED. These are constructed in 4+2 dimensional space, which fibers over the usual 3+1 dimensional space-time. The solution is stationary and localised about a topological singularity in space time. The electromagnetic field is that of a point electric charge, positioned at the singularity. Away from the singularity, all the conserved currents vanish. The solution comes in 8 varieties, corresponding to any choice of positive or negative charge, spin or mass (though it is presumed that the negative mass solutions are not physical). The charge is quantised, and determined up to sign by the coupling constant in the theory. The main element in the construction is simply the requirement that the solution be rotationally symmetric. 
  We extend to non-orientable surfaces previous work on sewing constraints in Conformal Field Theory. A new constraint, related to the real projective plane, is described and is used to illustrate the correspondence with a previous construction of open-string spectra. 
  This is an introduction to work on the generalisation to quantum groups of Mackey's approach to quantisation on homogeneous spaces. We recall the bicrossproduct models of the author, which generalise the quantum double. We describe the general extension theory of Hopf algebras and the nonAbelian cohomology spaces $\CH^2(H,A)$ which classify them. They form a new kind of topological quantum number in physics which is visible only in the quantum world. These same cross product quantisations can also be viewed as trivial quantum principal bundles in quantum group gauge theory. We also relate this nonAbelian cohomology $\CH^2(H,\C )$ to Drinfeld's theory of twisting. 
  We identify exact gauge-instanton-like solutions to (super)-string theory using the method of dimensional reduction. We find in particular the Polyakov instanton of 3d QED, and a class of generalized Yang-Mills merons. We discuss their marginal deformations, and show that for the $3d$ instanton they correspond to a dissociation of vector- and axial-magnetic charges. 
  We review recent work which has significantly sharpened our geometric understanding and interpretation of the moduli space of certain $N$=2 superconformal field theories. This has resolved some important issues in mirror symmetry and has also established that string theory admits physically smooth processes which can result in a change in topology of the spatial universe. 
  String theory in an exact plane wave background is explored. A new example of singularity in the sense of string theory for nonsingular spacetime metric is presented. The 4-tachyon scattering amplitude is constructed. The spectrum of states found from the poles in the factorization turns out to be equivalent to that of the theory in flat space-time. The massless vertex operator is obtained from the residue of the first order pole. 
  We derive a generalized Stokes' theorem, valid in any dimension and for arbitrary loops, even if self intersecting or knotted. The generalized theorem does not involve an auxiliary surface, but inherits a higher rank gauge symmetry from the invariance under deformations of the surface used in the conventional formulation. 
  In this paper the quantum direct scattering problem is solved for the Sine-Gordon model. Correlators of the Jost functions are derived by the angular quantization method. 
  Explicit relation between Laughlin state of the quantum Hall effect and one-dimensional(1D) model with long-ranged interaction ($1/r^2$) is discussed. By rewriting lowest Landau level wave functions in terms of 1D representation, Laughlin state can be written as a deformation of the ground state of Calogero-Sutherland model. Corresponding to Laughlin state on different geometries, different types of 1D $1/r^2$ interaction models are derived. 
  We consider two-dimensional quantum gravity coupled to matter fields which are renormalizable, but not conformal invariant. Questions concerning the $\b$ function and the effective action are addressed, and the effective action and the dressed renormalization group equations are determined for various matter potentials. 
  We study 2-d $\phi F$ gauge theories with the objective to understand, also at the quantum level, the emergence of induced gravity. The wave functionals - representing the eigenstates of a vanishing flat potential - are obtained in the $\phi$ representation. The composition of the space they describe is then analyzed: the state corresponding to the singlet representation of the gauge group describes a topological universe. For other representations a metric which is invariant under the residual gauge group is induced, apart from possible topological obstructions. Being inherited from the group metric it is rather rigid. 
  We consider canonical symplectic structure on the moduli space of flat ${\g}$-connections on a Riemann surface of genus $g$ with $n$ marked points. For ${\g}$ being a semisimple Lie algebra we obtain an explicit efficient formula for this symplectic form and prove that it may be represented as a sum of $n$ copies of Kirillov symplectic form on the orbit of dressing transformations in the Poisson-Lie group $G^{*}$ and $g$ copies of the symplectic structure on the Heisenberg double of the Poisson-Lie group $G$ (the pair ($G,G^{*}$) corresponds to the Lie algebra ${\g}$). 
  It is argued that the sphaleron solutions appearing in the Einstein-Yang-Mills theory are important in the transition processes at extremely high energies. Namely, when the energy exceeds the sphaleron mass, the existence of these solutions ensures constructive interference on the set of the overbarrier transition histories interpolating between distinct topological sectors of the theory. This has to lead to an enhancement of the transition rate and the related non-conservation of the fermion number. 
  A new method to perform numerical simulations of light-front Hamiltonians formulated on transverse lattices is introduced. The method is based on a DLCQ formulation for the (continuous) longitudinal directions. The hopping term in the transverse direction introduces couplings between fields defined on neighboring $1+1$-dimensional sheets. Within each sheet, the light-cone imaginary time evolution operator is calculated numerically with high precision using DLCQ. The coupling between neighboring sheets is taken into account using an initial value random walk algorithm based on the ensemble projector Monte Carlo technique and a checkerboard decomposition for the time evolution operator. The structure functions of $\lambda \phi^4$ theory in $2+1$ dimensions are studied as a trial application. The calculations are performed with up to 64 transverse lattice sites. No Tamm-Dancoff truncations are necessary. 
  We argue that theories with fundamental fermions which undergo chiral symmetry breaking have several universal features which are qualitatively different than those of theories with fundamental scalars. Several bounds on the critical indices $\delta$ and $\eta$ follow. We observe that in four dimensions the logarithmic scaling violations enter into the Equation of State of scalar theories, such as $\lambda\phi^4$, and fermionic models, such as Nambu-Jona-Lasinio, in qualitatively different ways. These observations lead to useful approaches for analyzing lattice simulations of a wide class of model field theories. Our results imply that $\lambda\phi^4$ {\it cannot} be a good guide to understanding the possible triviality of spinor $QED$. 
  An $U_q(sl(n))$ invariant transfer matrix with periodic boundary conditions is analysed by means of the algebraic nested Bethe ansatz for the case of $q$ being a root of unity. The transfer matrix corresponds to a 2-dimensional vertex model on a torus with topological interaction w.r.t. the 3-dimensional interior of the torus. By means of finite size analysis we find the central charge of the corresponding Virasoro algebra as $c=(n-1) \left[1-n(n+1)/(r(r-1))\right] $. 
  A bivertical classical field theory include the Newtonian mechanics and Maxwell's electromagnetic field theory as the special cases. This unification allows to recognize the formal analogies among the notions of Newtonian mechanics and Maxwell's electrodynamics. 
  We show how to relate the parafermions that occur in the $W_3$ string to the standard construction of parafermions. This result is then used to show that one of the screening charges that occurs in parafermionic theories is precisely the non-trivial part of the $W_3$ string BRST charge. A way of generalizing this pattern for a $W_N$ string is explained. This enables us to construct the full BRST charge for a $W_{2,N}$ string and to prove the relation of such a string to the algebra $W_{N-1}$ for arbitrary $N$, We also show how to calculate part of the BRST charge for a $W_N$ string, and we explain how our method might be extended to obtain the full BRST charge for such a string. 
  In the limit where $N\to\infty$ and the coupling constant $g \to g_{c}$ in a correlated manner, O(N) symmetric vector models represent filamentary surfaces. The purpose of these studies is to gain intuition for the long lasting search for a possible description of quantum field theory in terms of extended objects. It is shown here that a certain limiting procedure has to be followed in order to avoid several difficulties in establishing the theory at a critical negative coupling constant. It is also argued that at finite temperature a certain metastable-false vacuum disappears as the temperature is increased. 
  The effective formulas reducing the two-dimensional Hermite polynomials to the classical (one-dimensional) orthogonal polynomials are given. New one-parameter generating functions for these polynomials are derived. Asymptotical formulas for large values of indices are found. The applications to the squeezed one-mode states and to the time-dependent quantum harmonic oscillator are considered. 
  It has recently been shown that the ordinary bosonic string can be represented by a special background of N=1 or N=2 strings. In this paper, it will be shown that the bosonic string can also be represented by a special background of $W$-strings. 
  We study dilaton-electrodynamics in flat spacetime and exhibit a set of global cosmic string like solutions in which the magnetic flux is confined. These solutions continue to exist for a small enough dilaton mass but cease to do so above a critcal value depending on the magnetic flux. There also exist domain wall and Dirac monopole solutions. We discuss a mechanism whereby magnetic monopolesmight have been confined by dilaton cosmic strings during an epoch in the early universe during which the dilaton was massless. 
  A short proof is given to the fact that the additional symmetries of the KP hierarchy defined by their action on pseudodifferential operators, according to Fuchssteiner-Chen-Lee-Lin-Orlov-Shulman, coincide with those defined by their action on $\tau$-functions as Sato's B\"acklund transformations. Also a new simple formula for the generator of additional symmetries is presented. 
  We study the representation theory of finite W-algebras. After introducing parabolic subalgebras to describe the structure of W-algebras, we define the Verma modules and give a conjecture for the Kac determinant. This allows us to find the completely degenerate representations of the finite W-algebras. To extract the irreducible representations we analyse the structure of singular and subsingular vectors, and find that for W-algebras, in general the maximal submodule of a Verma module is not generated by singular vectors only. Surprisingly, the role of the (sub)singular vectors can be encapsulated in terms of a `dual' analogue of the Kazhdan-Lusztig theorem for simple Lie algebras. These involve dual relative Kazhdan-Lusztig polynomials. We support our conjectures with some examples, and briefly discuss applications and the generalisation to infinite W-algebras. 
  In certain two-dimensional models, collapsing matter forms a black hole if and only if the incoming energy flux exceeds the Hawking radiation rate. Near the critical threshold, the black hole mass is given by a universal formula in terms of the distance from criticality, and there exists a scaling solution describing the formation and evaporation of an arbitrarily small black hole. 
  In two dimensions large N QCD with quarks, defined on the plane, is equivalent to a modified string theory with quarks at the ends and taken in the zero fold sector. The equivalence that was established in 1975 was expressed in the form of an interacting string action that reproduces the spectrum and the 1/N interactions of 2D QCD. This action may be a starting point for an analytic continuation to a four dimensional string version of QCD. After reviewing the old work I discuss relations to recent developments in the pure QCD-string equivalence on more complicated background %geometries. 
  We find a quantum group structure in two-dimensional motion of nonrelativistic electrons in a uniform magnetic field on a torus. The representation basis of the quantum algebra is composed of the quantum Hall wavefunctions proposed by Haldane-Rezayi at the Landau-level filling factor $\nu=1/m$ ($m$ odd). It is also shown that the quantum group symmetry is relevant to the degenerate Landau states and the deformation parameter of the quantum algebra is given by the filling factor. 
  Using a formalism of minitwistors, we derive infinitely many conserved charges for the $sl(\infty )$-Toda equation which accounts for gravitational instantons with a rotational Killing symmetry. These charges are shown to form an infinite dimensional algebra through the Poisson bracket which is isomorphic to two dimensional area preserving diffeomorphism with central extentions. 
  We study a vacuum polarization effect in the background of certain dilatonic extremal black hole, known as the cornucopion. Whenever the charged fermions are of any finite mass, the gravitational backreaction to a generic value of a $CP$ nonconserving vacuum angle $\theta$ is shown to be important owing to a vacuum energy density which does not vanish deep inside the cornucopion. When this energy density is positive, this effect creates an extremal horizon at finite physical distance, closing off the infinite neck. We study the geometry near this horizon in some detail and find different physical interpretations for small and large fermion mass. Also, we argue that the conclusion is qualitatively correct despite the strong coupling. 
  The (co)homology theory of n-ary (co)compositions is a functor associating to $n$-ary (co)composition a complex. We present unified approach to the cohomology theory of coassociative and Lie coalgebras and for $2n$-ary cocompositions. This approach points to a possible generalization. 
  A free differential for an arbitrary associative algebra is defined as a differential with a uniqueness property. The existence problem for such a differential is posed. The notion of optimal calculi for given commutation rules is introduced and an explicit construction of it for a homogenous case is provided. Some examples are presented. 
  The $SU(2,2)$-harmonic oscillator on the phase space ${\cal A}(2,2)= {SU(2,2)}/{S(U(2)\times U(2))}$ is quantized using the coherent states. The quantum Hamiltonian is the Toeplitz operator corresponding to the square of the distance with respect to the $SU(2,2)$-invariant K\"ahler metric on the phase space. Its spectrum, depending on the choice of representation of $SU(2,2)$, is computed. 
  We construct new four-dimensional superstring vacua with extended superconformal symmetries. A non-trivial dilaton background implies the existence of Abelian killing symmetries. These are used to construct dual equivalent backgrounds in a way preserving the N=2 superconformal invariance. 
  The quantum symmetry of a rational quantum field theory is a finite- dimensional multi-matrix algebra. Its representation category, which determines the fusion rules and braid group representations of superselection sectors, is a braided monoidal C^*-category. Various properties of such algebraic structures are described, and some ideas concerning the classification programme are outlined. (Invited talk given at the III. International Conference on Mathematical Physics, String Theory and Quantum Gravity, Alushta, Ukraine, June 1993. To appear in Teor.Mat.Fiz.) 
  We review the derivation of the Atiyah-Singer and Callias index theorems using the recently developed localization method to calculate exactly the relevant supersymmetric path integrals. (Talk given at the III International Conference on Mathematical Physics, String Theory and Quantum Gravity, Alushta, Ukraine, June 13-24, 1993) 
  The Lee-Friedrichs model has been very useful in the study of decay-scattering systems in the framework of complex quantum mechanics. Since it is exactly soluble, the analytic structure of the amplitudes can be explicitly studied. It is shown in this paper that a similar model, which is also exactly soluble, can be constructed in quaternionic quantum mechanics. The problem of the decay of an unstable system is treated here. The use of the Laplace transform, involving quaternion-valued analytic functions of a variable with values in a complex subalgebra of the quaternion algebra, makes the analytic properties of the solution apparent; some analysis is given of the dominating structure in the analytic continuation to the lower half plane. A study of the corresponding scattering system will be given in a succeeding paper. 
  In a previous paper, it was shown that a soluble model can be constructed for the description of a decaying system in analogy to the Lee-Friedrichs model of complex quantum theory. It is shown here that this model also provides a soluble scattering theory, and therefore constitutes a model for a decay scattering system. Generalized second resolvent equations are obtained for quaternionic scattering theory. It is shown explicitly for this model, in accordance with a general theorem of Adler, that the scattering matrix is complex subalgebra valued. It is also shown that the method of Adler, using an effective optical potential in the complex sector to describe the effect of the quaternionic interactions, is equivalent to the general method of Green's functions described here. 
  We study the canonical quantization of the induced 2d-gravity and the pure gravity CGHS-model on a closed spatial section. The Wheeler-DeWitt equations are solved in (spatially homogeneous) choices of the internal time variable and the space of solutions is properly truncated to provide the physical Hilbert space. We establish the quantum equivalence of both models and relate the results with the covariant phase-space quantization. We also discuss the relation between the quantum wavefunctions and the classical space-time solutions and propose the wave function representing the ground state. 
  The classical (dynamical) $R$-matrices for the 2- and 3-body Calogero-Moser models with elliptic potentials are given. The 3-body case has an interesting nontrivial structure that goes beyond the known ansatz for momentum independent $R$-matrices. The $R$-matrices presented include the dynamical $R$-matrices of Avan and Talon as degenerate cases of the elliptic potential. 
  In this paper we propose a criteria to establish the integrability of N=2 supersymmetric massive theories.The basic data required are the vacua and the spectrum of Bogomolnyi solitons, which can be neatly encoded in a graph (nodes=vacua and links= Bogomolnyi solitons). Integrability is then equivalent to the existence of solutions of a generalized Yang-Baxter equation which is built up from the graph (graph-Yang-Baxter equation). We solve this equation for two general types of graphs: circular and daisy, proving, in particular, the inte- grability of the following Landau-Ginzburg superpotentials: A_n(t_1), A_n(t_2), D_n(\tau),E_6(t_7), E_8(t_16). For circular graphs the solutions are intertwiners of the affine Hopf algebra $\tilde(U)_q(A^{(1)}_1)$, while for daisy graphs the solution corresponds to a susy generalization of the Boltzmann weights of the chiral Potts model in the trigonometric regime. A chiral Potts like solution is conjectured for the more tricky case $ D_n(t_2)$. The scattering theory of circular models, for instance $A_n(t_1)$ or $D_n(\tau)$, is Toda like. The physical spectrum of daisy models, as $A_n(t_2), E_6(t_7)$ or $E_8(t_16)$, is given by confined states of radial solitons. The scattering theory of the confined states is again Toda like. Bootstrap factors for the confined solitons are given by fusing the susy chiral Potts S-matrices of the elementary constituents, i.e. the radial solitons of the daisy graph. 
  It has been realised recently that there is no unique way to describe the physical states of a given string theory. In particular, it has been shown that any bosonic string theory can be embedded in a particular $N{=}1$ string background in such a way that the spectrum and the amplitudes of both theories agree. Similarly, it is also known that the amplitudes of any $N{=}1$ string theory can be obtained from a particular $N{=}2$ string background. When rephrased in the language of BRST cohomology, these results suggest a close connection to the theory of induced representations. The purpose of this note is to investigate this connection further and at the same time to reveal the mechanism behind these embeddings between string theories. We will first analyze the embedding of an affine algebra $\gghat$ in the $N{=}1$ affine algebra associated to $\gg$. Given any BRST cohomology theory for $\gghat$ we will be able to construct one for the $N{=}1$ affine algebra associated to $\gg$ such that the cohomologies agree as operator product algebras. This is proven in two different ways. This example is the simplest in its kind and, in a sense that is made precise in the paper, all other similar embeddings are deformations of this one. We conclude the paper with a brief treatment of the general case, where we prove that for a particular class of ``good'' embeddings, the cohomologies are again isomorphic. 
  We investigate recent claims concerning a new class of cosmic string solutions in the Weinberg-Salam model. They have the general form of previously discussed semi-local and electroweak strings, but are modified by the presence of a non-zero W-condensate in the core of the string. We explicitly construct such solutions for arbitrary values of the winding number $N$. We then prove that they are gauge equivalent to bare electroweak strings with winding number $N-1$. We also develop new asymptotic expressions for large-$N$ strings. 
  Classical integrable Hamiltonian systems generated by elements of the Poisson commuting ring of spectral invariants on rational coadjoint orbits of the loop algebra $\wt{\gr{gl}}^{+*}(2,{\bf R})$ are integrated by separation of variables in the Hamilton-Jacobi equation in hyperellipsoidal coordinates. The canonically quantized systems are then shown to also be completely integrable and separable within the same coordinates. Pairs of second class constraints defining reduced phase spaces are implemented in the quantized systems by choosing one constraint as an invariant, and interpreting the other as determining a quotient (i.e., by treating one as a first class constraint and the other as a gauge condition). Completely integrable, separable systems on spheres and ellipsoids result, but those on ellipsoids require a further modification of order $\OO(\hbar^2)$ in the commuting invariants in order to assure self-adjointness and to recover the Laplacian for the case of free motion. For each case - in the ambient space ${\bf R}^{n}$, the sphere and the ellipsoid - the Schr\"odinger equations are completely separated in hyperellipsoidal coordinates, giving equations of generalized Lam\'e type. 
  It is shown that the general solution of classical equations of motion in two dimensional dilaton gravity proposed by Callan, Giddings, Harvey and Strominger (CGHS) includes a Lorentzian wormhole solution in addition to a black hole solution. We also show that matter perturbation of the wormhole by a shock wave leads to the formation of a black hole where the curvature singularity is cloaked by the global event horizon. It is also argued that the classical wormhole would be stable against quantum corrections. 
  We show that solutions of Pentagon equations lead to solutions of the Tetrahedron equation. The result is obtained in the spectral parameter dependent case. 
  We study classical radiation and quantum bremsstrahlung effect of a moving point scalar source. Our classical analysis provides another example of resolving a well-known apparent paradox, that of whether a constantly accelerating source radiates or not. Quantum mechanically, we show that for a scalar source with arbitrary motion, the tree level emission rate of scalar particles in the inertial frame equals the sum of emission and absorption rates of zero-energy Rindler particles in the Rindler frame. We then explicitly verify this result for a source undergoing constant proper acceleration. 
  Some mistaken reasonings at the end of the paper omitted. 
  The present paper describes the $W$--geometry of the Abelian finite non-periodic (conformal) Toda systems associated with the $B,C$ and $D$ series of the simple Lie algebras endowed with the canonical gradation. The principal tool here is a generalization of the classical Pl\"ucker embedding of the $A$-case to the flag manifolds associated with the fundamental representations of $B_n$, $C_n$ and $D_n$, and a direct proof that the corresponding K\"ahler potentials satisfy the system of two--dimensional finite non-periodic (conformal) Toda equations. It is shown that the $W$--geometry of the type mentioned above coincide with the differential geometry of special holomorphic (W) surfaces in target spaces which are submanifolds (quadrics) of $CP^N$ with appropriate choices of $N$. In addition, these W-surfaces are defined to satisfy quadratic holomorphic differential conditions that ensure consistency of the generalized Pl\"ucker embedding. These conditions are automatically fulfiled when Toda equations hold. 
  After some definitions, we review in the first part of this talk the construction and classification of classical $W$ (super)algebras symmetries of Toda theories. The second part deals with more recently obtained properties. At first, we show that chains of $W$ algebras can be obtained by imposing constraints on some $W$ generators: we call secondary reduction such a gauge procedure on $W$ algebras. Then we emphasize the role of the Kac-Moody part, when it exists, in a $W$ (super) algebra. Factorizing out this spin 1 subalgebra gives rise to a new $W$ structure which we interpret either as a rational finitely generated $W$ algebra, or as a polynomial non linear $W_\infty$ realization. (Plenary talk presented by P. SORBA at the $XXII^{th}$ International Conference on Differential Geometric Methods in Theoretical Physics. Ixtapa Mexico, September 1993.) 
  We construct the $r$-matrix for the generalization of the Calogero-Moser system introduced by Gibbons and Hermsen. By reduction procedures we obtain the $r$-matrix for the $O(N)$ Euler-Calogero-Moser model and for the standard $A_N$ Calogero-Moser model. 
  We prove an identity between three infinite families of polynomials which are defined in terms of `bosonic', `fermionic', and `one-dimensional configuration' sums. In the limit where the polynomials become infinite series, they give different-looking expressions for the characters of the two integrable representations of the affine $su(2)$ algebra at level one. We conjecture yet another fermionic sum representation for the polynomials which is constructed directly from the Bethe-Ansatz solution of the Heisenberg spin chain. 
  We analize the algebraic structure of consistent and covariant anomalies in gauge and gravitational theories: using a complex extension of the Lie algebra it is possible to describe them in a unified way. Then we study their representations by means of functional determinants, showing how the algebraic solution determines the relevant operators for the definition of the effective action. Particular attention is devoted to the Lorentz anomaly: we obtain by functional methods the covariant anomaly for the spin-current and for the energy-momentum tensor in presence of a curved background. With regard to the consistent sector we are able to give a general functional solution only for $d=2$: using the characterization derived from the extended algebra, we find a continuous family of operators whose determinant describes the effective action of chiral spinors in curved space. We compute this action and we generalize the result in presence of a $U(1)$ gauge connection. Accepted for publication in Fortschritte der Physik. 
  Finite rational $\cw$ algebras are very natural structures appearing in coset constructions when a Kac-Moody subalgebra is factored out. In this letter we address the problem of relating these algebras to integrable hierarchies of equations, by showing how to associate to a rational $\cw$ algebra its corresponding hierarchy. We work out two examples: the $sl(2)/U(1)$ coset, leading to the Non-Linear Schr\"{o}dinger hierarchy, and the $U(1)$ coset of the Polyakov-Bershadsky $\cw$ algebra, leading to a $3$-field representation of the KP hierarchy already encountered in the literature. In such examples a rational algebra appears as algebra of constraints when reducing a KP hierarchy to a finite field representation. This fact arises the natural question whether rational algebras are always associated to such reductions and whether a classification of rational algebras can lead to a classification of the integrable hierarchies. 
  The Feynman-Vernon formalism is used to obtain a microscopic, quantum mechanical derivation of black body radiation, for a massless scalar field in 1+1 dimensions, weakly coupled to an environment of finite size. The model exhibits the absorption, thermal equilibrium, and emission properties of a canonical black body, but shows that the thermal radiation propagates outwards from the body, with the Planckian spectrum applying inside a wavefront region of finite thickness. The black body environment used in the derivation can be considered to represent a very fine, granular medium, such as lampblack. In the course of developing the model for black body radiation, thermalization of a single harmonic oscillator by a heat bath with slowly varying spectral density is demonstrated. Bargmann-Fock coherent state variables, being convenient for problems involving harmonic oscillators and free fields, are reviewed and then used throughout the paper. An appendix reviews the justification for using baths of independent harmonic oscillators to model generic quantum environments. 
  We construct a string field Hamiltonian for a noncritical string theory with the continuum limit of the Ising model or its generalization as the matter theory on the worldsheet. It consists of only three string vertices as in the case for $c=0$. We also discuss a general consistency condition that should be satisfied by this kind of string field Hamiltonian. 
  We give an elementary analysis of the multiplicator group of the Galilei group in 1+2 dimensions $G^{\uparrow}_{+}$. For a non-trivial multiplicator we give a list of all the corresponding projective unitary irreducible representations of $G^{\uparrow}_{+}$. 
  There is a relatively well understood class of deformable W-algebras, resulting from Drinfeld-Sokolov (DS) type reductions of Kac-Moody algebras, which are Poisson bracket algebras based on finitely, freely generated rings of differential polynomials in the classical limit. The purpose of this paper is to point out the existence of a second class of deformable W-algebras, which in the classical limit are Poisson bracket algebras carried by infinitely, nonfreely generated rings of differential polynomials. We present illustrative examples of coset constructions, orbifold projections, as well as first class Hamiltonian reductions of DS type W-algebras leading to reduced algebras with such infinitely generated classical limit. We also show in examples that the reduced quantum algebras are finitely generated due to quantum corrections arising upon normal ordering the relations obeyed by the classical generators. We apply invariant theory to describe the relations and to argue that classical cosets are infinitely, nonfreely generated in general. As a by-product, we also explain the origin of the previously constructed and so far unexplained deformable quantum W(2,4,6) and W(2,3,4,5) algebras. 
  Irrational conformal field theory (ICFT) includes rational conformal field theory as a small subspace, and the affine-Virasoro Ward identities describe the biconformal correlators of ICFT. We reformulate the Ward identities as an equivalent linear partial differential system with flat connections and new non-local conserved quantities. As examples of the formulation, we solve the system of flat connections for the coset correlators, the correlators of the affine-Sugawara nests and the high-level $n$-point correlators of ICFT. 
  We compute the gauge field functional integral giving the scalar product of the SU(2) Chern-Simons theory states on a Riemann surface of genus > 1. The result allows to express the higher genera partition functions of the SU(2) WZNW conformal field theory by explicit finite dimensional integrals. Our calculation may also shed new light on the functional integral of the Liouville theory. 
  We define an entropy for a quantum field theory by combining quantum fluctuations, scaling and the maximum entropy concept. This entropy has different behavior in asymptotically free and non--asymptotically free theories. We find that the transition between the two regimes (from the asymptotically free to the non--asymptotically free) takes place via a continuous phase transition. For asymptotically free theories there exist regimes where the ``temperatures" are negative. In asymptotically free theories there exist maser--like states mostly in the infrared; furthermore, as one goes into the ultraviolet and more matter states contribute to quantum processes, the quantum field system can shed entropy and cause the formation of thermodynamically stable {\it entropy--ordered} states. It is shown how the known heavier quarks can be thus described. 
  In the framework of the compactified closed bosonic string theory with the extra spatial coordinates being circular with radius $R$, we perform both the zero-slope limit and the $R \rightarrow 0$ limit of the tree scattering amplitude of four massless scalar particles. We explicitly show that this double limit leads to amplitudes involving scalars which interact through the exchange of a scalar, spin 1 and spin 2 particle. In particular, this latter case reproduces the same result obtained in linearized quantum gravity. 
  The most disappointing aspect of $\W$-strings is probably the fact that at least for the known models one does not recover a physical spectrum that differs much from that of ordinary string theory. It is hoped that this is not an intrinsic shortcoming of $\W$-string theory, but rather that it is a consequence of the $\W$-algebra realizations that have been chosen. In this note we point out a whole new class of possible $\W$-strings built from representations of affine Lie algebras via (quantum) Drinfel'd-Sokolov reduction. Explicitly, we construct a BRST operator in terms of $\widehat{sl}_3$ currents which computes the physical spectrum of a $\W_3$-string. As a special case of this construction, if we take a free-field realization of $\widehat{sl}_3$, we recover the 2-scalar $\W_3$-string. These results generalize to any $\W$-algebra which can be obtained via quantum Drinfel'd-Sokolov reduction from an affine algebra. 
  We present a detailed evaluation of $\eta$, the critical exponent corresponding to the electron anomalous dimension, at $O(1/N^2_{\!f})$ in a large flavour expansion of QED in arbitrary dimensions in the Landau gauge. The method involves solving the skeleton Dyson equations with dressed propagators in the critical region of the theory. Various techniques to compute massless two loop Feynman diagrams, which are of independent interest, are also given. 
  I compare heatkernel regularization with sharp gauge invariant cutoffs in the Hamiltonian formulation of the Coulomb gauged Schwinger model on a circle. The effective potential for the zero mode of the gauge field in a given fermionic configuration is different in these two regularizations, the difference being independent of the chosen fermionic configuration. In the continuum limit the gauge field can be localized or delocalized depending on the regulator. 
   The purpose of this paper is to introduce and study a q-analogue of the holonomic system of differential equations associated to the Belavin's classical r-matrix (elliptic r-matrix equations), or, equivalently, to define an elliptic deformation of the quantum Knizhnik-Zamolodchikov equations of Frenkel and Reshetikhin. In hep-th 9303018, it was shown that solutions of the elliptic r-matrix equations admit a representation as traces of products of intertwining operators between certain modules over affine sl(N). In this paper, this construction is generalized to quantum affine sl(N).    The main object of study in the paper is a family of meromorphic matrix functions of n complex variables z_1,...,z_n and three additional parameters p,q,s -- (modified) traces of products of intertwiners between modules over quantum affine sl(N). They are a new class of transcendental functions which can be degenerated into many interesting special functions -- hypergeometric and q-hypergeometric functions, elliptic and modular functions, transcendental functions of an elliptic curve, vector-valued modular forms, solutions of the Bethe ansatz equations etc.    The main result of the paper states that these functions satisfy two holonomic systems of difference equations -- the first one has shift parameter p and elliptic modulus s, and the second one has shift parameter s and elliptic modulus p.   The paper also contains a short proof of the quantum KZ equations. 
  Exact solutions of heterotic string theory corresponding to four-dimensional magnetic black holes in $N=4$ supergravity are described. The solutions describe the black holes in the throat limit, and consist of a tensor product of an $SU(2)$ WZW orbifold with the linear dilaton vacuum, supersymmetrized to $(1,0)$ world sheet SUSY. One dimension of the $SU(2)$ model is interpreted as a compactified fifth dimension, leading to a four dimensional solution with a Kaluza-Klein gauge field having a magnetic monopole background; this corresponds to a solution in $N=4$ supergravity, since that theory is obtained by dimensional reduction of string theory. 
  We review nonlinear gauge theory and its application to two-dimensional gravity. We construct a gauge theory based on nonlinear Lie algebras, which is an extension of the usual gauge theory based on Lie algebras. It is a new approach to generalization of the gauge theory. The two-dimensional gravity is derived from nonlinear Poincar{\' e} algebra, which is the new Yang-Mills like formulation of the gravitational theory. As typical examples, we investigate $R^2$ gravity with dynamical torsion and generic form of 'dilaton' gravity. The supersymmetric extension of this theory is also discussed. 
  Eigenvalues of the commuting family of transfer matrices are expected to obey the $T$-system, a set of functional relation, proposed recently. Here we obtain the solution to the $T$-system for $C^{(1)}_2$ vertex models. They are compatible with the analytic Bethe ansatz and Yang-Baxterize the classical characters. 
  The generalised quasienergy states are introduced as eigenstates of the new integral of motion for periodically and nonperiodically kicked quantum systems.The photon distribution function of polymode generalised correlated light expressed in terms of multivariable Hermite polynomials is discussed and the relation of its properties to Schrodinger uncertainty relation is given. 
  We present a simple renormalizable abelian gauge model which includes antisymmetric second-rank tensor fields as matter fields rather than gauge fields known for a long time. The free action is conformally rather than gauge invariant. The quantization of the free fields is analyzed and the one-loop renormalization-group functions are evaluated. Transverse free waves are found to convey no energy. The coupling constant of the axial-vector abelian gauge interaction exhibits asymptotically free ultraviolet behavior, while the self-couplings of the tensor fields do not asymptotically diminish. 
  We study the thermal behavior of the negative dimensional harmonic oscillator of Dunne and Halliday that at zero temperature, due to a hidden BRST symmetry of the classical harmonic oscillator, is shown to be equivalent to the Grassmann oscillator of Finkelstein and Villasante. At finite temperature we verify that although being described by Grassmann numbers the thermal behavior of the negative dimensional oscillator is quite different from a Fermi system. 
  The field theory quantized on the {\it light-front} is compared with the conventional equal-time quantized theory. The arguments based on the {\it microcausality} principle imply that the light-front field theory may become nonlocal with respect to the longitudinal coordinate even though the corresponding equal-time formulation is local. This is found to be the case for the scalar theory which is quantized by following the Dirac procedure. In spite of the different mechanisms of the spontaneous symmetry breaking in the two forms of dynamics they result in the same physical content. The phase transition in {$(\phi^{4})_{2}$} theory is also discussed. The symmetric vacuum state for vanishingly small couplings is found to turn into an unstable symmetric one when the coupling is increased and may result in a phase transition of the {\it second order} in contrast to the first order transition concluded from the usual variational methods. 
  After giving some definitions for vertex operator SUPERalgebras and their modules, we construct an associative algebra corresponding to any vertex operator superalgebra, such that the representations of the vertex operator algebra are in one-to-one correspondence with those of the corresponding associative algebra. A way is presented to decribe the fusion rules for the vertex operator superalgebras via modules of the associative algebra. The above are generalizations of Zhu's constructions for vertex operator algebras. Then we deal in detail with vertex operator superalgebras corresponding to Neveu-Schwarz algebras, super affine Kac-Moody algebras, and free fermions. We use the machinery established above to find the rationality conditions, classify the representations and compute the fusion rules. In the appendix, we present explicit formulas for singular vectors and defining relations for the integrable representations of super affine algebras. These formulas are not only crucial for the theory of the corresponding vertex operator superalgebras and their representations, but also of independent interest. 
  We study whether orbifold models are equivalently rewritten into torus models in the case of fermionic string theories. It is pointed out that symmetric orbifold models cannot be rewritten into torus models in the case of fermionic string theories because of the absence of twist-untwist intertwining currents on the orbifold models. We present a list of current algebras on asymmetric $Z_N$-orbifold models of type II superstring theories with inner automorphisms of Lie algebra lattices of the $A_n$ series. It turns out that whether an asymmetric orbifold model is rewritten into a torus model depends on the specific choice of a momentum lattice and an inner automorphism of the lattice. 
  We point out that in a recent paper by Chaichian et al. (Phys. Rev.Lett71(1993)3405) Dirac quantization procedure is not properly followed as a result of which the resulting quantum theory is different from what one will get by a straightforward application of Dirac's formulation. For example at quantum level the model in I has noncommutative geometry, i.e., $[x_\mu, x_\nu] \neq 0$ while a correct treatment would give $[x_\mu, x_\nu] = 0$. Moreover, their assertion about $\alpha$ being an arbitrary constant in $S_\mu = -\alpha p_\mu /{\sqrt{p^2}} $ is not correct and consequently one can not say whether the model describes anyons. 
  We consider the twisting of Hopf structure for classical enveloping algebra $U(\hat{g})$, where $\hat{g}$ is the inhomogenous rotations algebra, with explicite formulae given for $D=4$ Poincar\'{e} algebra $(\hat{g}={\cal P}_4).$ The comultiplications of twisted $U^F({\cal P}_4)$ are obtained by conjugating primitive classical coproducts by $F\in U(\hat{c})\otimes U(\hat{c}),$ where $\hat{c}$ denotes any Abelian subalgebra of ${\cal P}_4$, and the universal $R-$matrices for $U^F({\cal P}_4)$ are triangular. As an example we show that the quantum deformation of Poincar\'{e} algebra recently proposed by Chaichian and Demiczev is a twisted classical Poincar\'{e} algebra. The interpretation of twisted Poincar\'{e} algebra as describing relativistic symmetries with clustered 2-particle states is proposed. 
  After briefly reviewing how the (proper-time) Schwinger's formula works for computing the Casimir energy in the case of "scalar electrodynamics" where the boundary conditions are dictated by two perfectly conducting parallel plates with separation "a" in the Z-axis, we propose a slightly modification in the previous approach based on an analytical continuation method. As we will see, for the case at hand our formula does not need the use of Poisson summation to get a (renormalized) finite result. 
  We derive the two equations of Davey-Stewartson type from a zero curvature condition associated with SL(2,{\bf R}) in $2+1$ dimensions. We show in general how a $2+1$ dimensional zero curvature condition can be obtained from the self-duality condition in $3+3$ dimensions and show in particular how the Davey-Stewartson equations can be obtained from the self-duality condition associated with SL(2,{\bf R}) in $3+3$ dimensions. 
  Most nonabelian gauge theories admit the existence of conserved and quantized topological charges as generalizations of the Dirac monopole. Their interactions are dictated by topology. In this paper, previous work in deriving classical equations of motion for these charges is extended to quantized particles described by Dirac wave functions. The resulting equations show intriguing similarities to the Yang-Mills theory. Further, although the system is not dual symmetric, its gauge symmetry is nevertheless doubled as in the abelian case from $G$ to $G \times G$, where the second $G$ has opposite parity to the first but is mediated instead by an antisymmetric second-rank tensor potential. (hep-th/yymmnnn) 
  Abstrac: It is shown that an antisymmetric rank-two tensor gauge potential of the type first found in string and supersymmetry theories occurs also in ordinary Yang-Mills theory when formulated in loop space, where it appears as a Lagrange multiplier for a zero curvature constraint necessary and sufficient for removing the inherent redundancy of loop variables. It is then further shown that the tensor potential acts there as the parallel `phase' transport for monopoles. (hep-th/yymmnnn) 
  A generalization of the differential geometry of forms and vector fields to the case of quantum Lie algebras is given. In an abstract formulation that incorporates many existing examples of differential geometry on quantum spaces we combine an exterior derivative, inner derivations, Lie derivatives, forms and functions all into one big algebra, the ``Cartan Calculus''. (This is an extended version of a talk presented by P. Schupp at the XXII$^{th}$ International Conference on Differential Geometric Methods in Theoretical Physics, Ixtapa, Mexico, September 1993) 
  We study the scattering of fermions in a ``global monopole'' background metric. This is the four-dimensional analogue of the scattering on a cone in three dimensions. The scattering amplitude is exactly obtained. We then study massless fermion-dyon systems in such a background metric. The density of the $S$-wave fermion condensate is found to be given by a constant times the flat space value of Callan and Rubakov. 
  The topic of this thesis is the development of a versatile and geometrically motivated differential calculus on non-commutative or quantum spaces, providing powerful but easy-to-use mathematical tools for applications in physics and related sciences. A generalization of unitary time evolution is proposed and studied for a simple 2-level system, leading to non-conservation of microscopic entropy, a phenomenon new to quantum mechanics. A Cartan calculus that combines functions, forms, Lie derivatives and inner derivations along general vector fields into one big algebra is constructed for quantum groups and then extended to quantum planes. The construction of a tangent bundle on a quantum group manifold and an BRST type approach to quantum group gauge theory are given as further examples of applications. The material is organized in two parts: Part I studies vector fields on quantum groups, emphasizing Hopf algebraic structures, but also introducing a `quantum geometric' construction. Using a generalized semi-direct product construction we combine the dual Hopf algebras \A\ of functions and \U\ of left-invariant vector fields into one fully bicovariant algebra of differential operators. The pure braid group is introduced as the commutant of $\Delta (\U)$. It provides invariant maps $\A \to \U$ and thereby bicovariant vector fields, casimirs and metrics. This construction allows the translation of undeformed matrix expressions into their less obvious quantum algebraic counter parts. We study this in detail for quasitriangular Hopf algebras, giving the determinant and 
  A Cartan Calculus of Lie derivatives, differential forms, and inner derivations, based on an undeformed Cartan identity, is constructed. We attempt a classification of various types of quantum Lie algebras and present a fairly general example for their construction, utilizing pure braid methods, proving orthogonality of the adjoint representation and giving a (Killing) metric and the quadratic casimir. A reformulation of the Cartan calculus as a braided algebra and its extension to quantum planes, directly and induced from the group calculus, are provided. 
  The derived modular invariant one-loop string effective coupling constant has been used to discuss the weak scale measurement constraints on minimal superstring models, minimal SUSY left-right string models and minimal left-right SUSY string models. The string intermediate gauge symmetry breaking models are proposed. The possible schemes to predict $M_{GUT} = 10^{16}GeV$ and $M_{SUSY}= 1 TeV$ in string models have been discussed. It is also demonstrated that string unification is more restricted by the experiments than any other unification models are. 
  The main goal of these lectures is to introduce and review the Hamiltonian formalism for classical constrained systems and in particular gauge theories. Emphasis is put on the relation between local symmetries and constraints and on the relation between Lagrangean and Hamiltonian symmetries. 
  Highest weight representations of $U_q(su(1,1))$ with $q=\exp \pi i/N$ are investigated. The structures of the irreducible hieghesat weight modules are discussed in detail. The Clebsch-Gordan decomposition for the tensor product of two irreducible representations is discussed. By using the results, a representation of $SL(2,R)\otimes U_q(su(2))$ is also presented in terms of holomorphic sections which also have $U_q(su(2))$ index. Furthermore we realise $Z_N$-graded supersymmetry in terms of the representation. An explicit realization of $Osp(1 \vert 2)$ via the heighest weight representation of $U_q(su(1,1))$ with $q^2=-1$ is given. 
  The scalar functional determinants on sectors of the two-dimensional disc and spherical cap are determined for arbitrary angles (rational factors of $\pi$). The wholesphere and hemisphere expressions are also given, in low dimensions, for both the ordinary and the conformal Laplacian. Comparisons are made with other work. 
  This is a brief review of our recent work attempted at a generalization of the Grassmann algebra to the paragrassmann ones. The main aim is constructing an algebraic basis for representing `fractional' symmetries appearing in $2D$ integrable models and also introduced earlier as a natural generalization of supersymmetries. We have shown that these algebras are naturally related to quantum groups with $q = {\rm root \;of \; unity}$. By now we have a general construction of the paragrassmann calculus with one variable and preliminary results on deriving a natural generalization of the Neveu--Schwarz--Ramond algebra. The main emphasis of this report is on a new general construction of paragrassmann algebras with any number of variables, N. It is shown that for the nilpotency indices $(p + 1) = 3, 4, 6$ the algebras are almost as simple as the Grassmann algebra (for which $(p + 1) = 2$). A general algorithm for deriving algebras with arbitrary p and N is also given. However, it is shown that this algorithm does not exhaust all possible algebras, and the simplest example of an `exceptional' algebra is presented for $p = 4, N = 4$. 
  Using the heat kernel regularization we show that the Abelian chiral anomaly in the limit of infinite temperature it is not a well defined quantity, contrary to what happens at any finite temperature. We show that there is an ambiguity in the ordering of the limits of infinite temperature and removal of the cut-off so that changing this ordering we find different results for the chiral anomaly. We discuss these cases and their possible interpretation. 
  The dilaton free energy density in external static gravitational field is found. We use the real time formulation of the finite temperature field theory and the free energy density is computed to the first order of the string parameter $\alpha '$. We obtain the thermal corrections to the $\alpha'$ modified Einstein gravity action. 
  This is a review on infinite non-abelian symmetries in two-dimensional field theories. We show how any integrable QFT enjoys the existence of infinitely many {\bf conserved} charges. These charges {\bf do not commute} between them and satisfy a Yang--Baxter algebra. They are generated by quantum monodromy operators and provide a representation of $q-$deformed affine Lie algebras $U_q({\hat\G})$. We review the work by de Vega, Eichenherr and Maillet on the bootstrap construction of the quantum monodromy operators in classically scale invariant theories where the classical monodromy matrix is conserved. Then, the recent generalization to the sine--Gordon (or massive Thirring) model, where such operators do not possess a classical analogue is given (This provides a representation of $S{\hat U}(2)_q$). It is then reported on the recent work by Destri and de Vega, where both commuting and non-commuting integrals of motion are systematically obtained by Bethe Ansatz in the light-cone approach. The eigenvalues of the six--vertex alternating transfer matrix $\tau(\l)$ are explicitly computed on a generic physical state through algebraic Bethe ansatz. In the thermodynamic limit $\tau(\l)$ turns out to be a two--valued periodic function. One determination generates the local abelian charges, including energy and momentum, while the other yields the abelian subalgebra of the (non--local) YB algebra. In particular, the bootstrap results coincide with the ratio between the two determinations of the lattice transfer matrix.[Based on a Lecture at the Clausthal Symposium]. 
  The exact general solution for a sigma model having the $2-d$ stringy black hole (SBH) as internal manifold is found in closed form. We also give the exact solution for the massless complex Sine-Gordon (MCSG) model. Both, models and their solutions are related by analytic continuation. The solution is expressed in terms of four arbitrary functions of one variable. (hep-th 9312085). 
  The representation of scalar products of Bethe wave functions in terms of the Dual Fields, proven by A.G.Izergin and V.E.Korepin in 1987, plays an important role in the theory of completely integrable models. The proof in \cite{Izergin87} and \cite{Korepin87} is based on the explicit expression for the "senior" coefficient which was guessed in \cite{Izergin87} and then proven to satisfy some recurrent relations, which determine it unambiguously. In this paper we present an alternative proof based on the direct computation. 
  The vector supersymmetry recently found for the bosonic string is generalized to superstring theories quantized in the super-Beltrami parametrization. 
   We introduce a Lie bialgebra structure on the central extension of the Lie algebra of differential operators on the line and the circle (with scalar or matrix coefficients). This defines a Poisson---Lie structure on the dual group of pseudodifferential symbols of an arbitrary real (or complex) order. We show that the usual (second) Benney, KdV (or GL_n--Adler---Gelfand---Dickey) and KP Poisson structures are naturally realized as restrictions of this Poisson structure to submanifolds of this ``universal''  Poisson---Lie group.    Moreover, the reduced (=SL_n) versions of these manifolds (W_n-algebras  in physical terminology) can be viewed as subspaces of the quotient (or Poisson reduction) of this  Poisson---Lie group by the dressing action of the group of functions.    Finally, we define an infinite set of functions in involution on the Poisson---Lie group  that give the standard families of Hamiltonians when restricted to the submanifolds  mentioned above. The Poisson structure  and Hamiltonians on the whole  group interpolate between the Poisson structures and Hamiltonians of  Benney, KP and KdV flows. We also discuss the geometrical meaning of  W_\infty as a limit of Poisson algebras  W_\epsilon as \epsilon goes to 0. 
  We begin with a few remarks on an explicit construction of a Bargmann-Wightman-Wigner-type quantum field theory [Phys. Lett. B {\bf 316}, 102 (1993)] in which bosons and associated antibosons have opposite relative intrinsic parities. We then construct $(1,0)\oplus(0,1)$ Majorana ($CP$ self conjugate) and Majorana-like ($C\Gamma^5$ self conjugate, $\Gamma^5=$ chirality operator) fields. We point out that this new structure in the space time symmetries may be relevant to $P$ and $CP$ violation. 
  This is first of the two invited lectures presented (by D. V. Ahluwalia) at the ``XVII International School of Theoretical Physics: Standard Model and Beyond' 93.'' The text is essentially based on a recent publication by the present authors [Phys. Lett. B 316 (1993) 102]. Here we show that the Dirac-like construct in the $(j,0)\oplus(0,j)$ representation space supports a Bargmann-Wightman-Wigner-type quantum field theory. 
  This is second of the two invited lectures presented (by D. V. Ahluwalia) at the ``XVII International School of Theoretical Physics: Standard Model and Beyond' 93.'' The text is essentially based on a recent publication by the present authors [Mod. Phys. Lett. A (in press)]. Here, after briefly reviewing the $(j,0)\oplus(0,j)$ Dirac-like construct in the front form, we present a detailed construction of the $(j,0)\oplus(0,j)$ Majorana-like fields. 
  By considering the parity-transformation properties of the $(1/2,\,0)$ and $(0,\,1/2)$ fields in the {\it front form} we find ourselves forced to study the front-form evolution both along $x^+$ and $x^-$ directions. As a by product, we find that half of the dynamical degrees of freedom of a full theory live on the $x^+=0$ surface and the other half on the $x^-=0$ surface. Elsewhere, Jacob shows how these results are required to build a satisfactory, and internally consistent, front-form quantum field theory. 
  We discuss geometrical aspects of Higgs systems and Toda field theory in the framework of the theory of vector bundles on Riemann surfaces of genus greater than one. We point out how Toda fields can be considered as equivalent to Higgs systems -- a connection on a vector bundle $E$ together with an End($E$)--valued one form both in the standard and in the Conformal Affine case. We discuss how variations of Hodge structures can arise in such a framework and determine holomorphic embeddings of Riemann surfaces into locally homogeneous spaces, thus giving hints to possible realizations of $W_n$--geometries. } 
  Halpern and Yamron have given a Lorentz, conformal, and Diff S$_2$-invariant world-sheet action for the generic irrational conformal field theory, but the action is highly non-linear. In this paper, we introduce auxiliary fields to find an equivalent linearized form of the action, which shows in a very clear way that the generic affine-Virasoro action is a Diff S$_2$-gauged WZW model. In particular, the auxiliary fields transform under Diff S$_2$ as local Lie $g$ $\times$ Lie $g$ connections, so that the linearized affine-Virasoro action bears an intriguing resemblance to the usual (Lie algebra) gauged WZW model. 
  We discuss the large $N$ limit of the supersymmetric $CP^N$ models as an illustration of Cecotti and Vafa's $tt^*$ formalism. In this limit the `$tt^*$ equation' becomes the long wavelength limit of the $2D$ Toda lattice, an equation first studied in the context of self-dual gravity. We show how simple finite temperature and large $N$ techniques determine the relevant solution, and verify analytically that it solves the $tt^*$ equation, using Legendre transform techniques from self-dual gravity. 
  We discuss relations of Vafa's quantum cohomology with Floer's homology theory, introduce equivariant quantum cohomology, formulate some conjectures about its general properties and, on the basis of these conjectures, compute quantum cohomology algebras of the flag manifolds. The answer turns out to coincide with the algebra of regular functions on an invariant lagrangian variety of a Toda lattice. 
  The moduli space of all rational conformal quantum field theories with effective central charge c_eff = 1 is considered. Whereas the space of unitary theories essentially forms a manifold, the non unitary ones form a fractal which lies dense in the parameter plane. Moreover, the points of this set are shown to be in one-to-one correspondence with the elements of the modular group for which an action on this set is defined. 
  An algebraic treatment of shape-invariant potentials is discussed. By introducing an operator which reparametrizes wavefunctions, the shape-invariance condition can be related to a generalized Heisenberg- Weyl algebra. It is shown that this makes it possible to define a coherent state associated with the shape-invariant potentials. 
  We define the $osp(1,2)$ Gaudin algebra and consider integrable models described by it. The models include the $osp(1,2)$ Gaudin magnet and the Dicke model related to it. Detailed discussion of the simplest cases of these models is presented. The effect of the presence of fermions on the separation of variables is indicated. 
  A topological field theory is used to study the cohomology of mapping space. The cohomology is identified with the BRST cohomology realizing the physical Hilbert space and the coboundary operator given by the calculations of tunneling between the perturbative vacua. Our method is illustrated by a simple example. 
   A representation-theoretic approach to special functions was developed in the 40-s and 50-s in the works of I.M.Gelfand, M.A.Naimark, N.Ya.Vilenkin, and their collaborators. The essence of this approach is the fact that most classical special functions can be obtained as suitable specializations of matrix elements or characters of representations of groups. Another rich source of special functions is the theory of Clebsch-Gordan coefficients which describes the geometric juxtaposition of irreducible components inside the tensor product of two representations. Finally, in recent works on representations of (quantum) affine Lie algebras it was shown that matrix elements of intertwining operators between certain representations of these algebras are interesting special functions -- (q-)hypergeometric functions and their generalizations.    In this paper we suggest a general method of getting special functions from representation theory which unifies the three methods mentioned above and allows one to define and study many new special functions. We illustrate this method by a number of examples -- Macdonald's polynomials, eigenfunctions of the Sutherland operator, Lame functions. 
  We study massless QED_2 with N flavors using path integrals. We identify the sector that is generated by the N^2 classically conserved vector currents. One of them (the U(1) current) creates a massive particle, while the others create massless ones. We show that the mass spectrum obeys a Witten-Veneziano type formula. Two theorems on n-point functions clarify the structure of the Hilbert space. Evaluation of the Fredenhagen-Marcu order parameter indicates that a confining force exists only between charges that are integer multiples of +/- Ne, whereas charges that are nonzero mod(N) screen their confining forces and lead to non-vacuum sectors. Finally we identify operators that violate clustering, and decompose the theory into clustering theta vacua. 
  In this paper we present a formula for Macdonald's polynomials for the root system A(n-1) which arises from the representation theory of quantum sl(n). This formula expresses Macdonald's polynomials via (weighted) traces of intertwining operators between certain modules over quantum sl(n). We also describe the commutative system of Macdonald's difference operators using the generators of the center of the quantum universal enveloping algebra, and use this description to prove a trace formula for generic eigenfunctions of these operators. These functions are generalized q-hypergeometric functions which are related to solutions of the quantum Knizhnik-Zamolodchikov equations. 
  The article is devoted to a quantum field theory explanation of the relationship (noticed some years ago by Gepner) between the Verlinde algebra of the group $U(k)$ at level $N-k$ and the cohomology of the Grassmannian. The argument proceeds by starting with the two dimensional sigma model whose target space is the Grassmannian and integrating out some fields in a standard way. It has long been known that the resulting low energy effective action describes a theory with a mass gap; the novelty here is that this theory in fact is equivalent at long distances to a gauged WZW model of $U(k)/U(k)$, and hence is related to the Verlinde algebra. 
  The Landau potentials of $W_3$-algebra models are analyzed with algebraic-geometric methods. The number of ground states and the number of independent perturbations of every potential coincide and can be computed. This number agrees with the structure of ground states obtained in a previous paper, namely, as the phase structure of the IRF models of Jimbo et al. The singularities associated to these potentials are identified. 
  The multimode even and odd coherent states (multimode Schroedinger cat states) are constructed for polymode parametric oscillators of the electromagnetic field. The evolution of the photon distribution function is evaluated explicitly. The distribution function is expressed in terms of the multivariable Hermite polynomials, its means and dispersions are calculated. The conditions for the existence of squeezing are formulated. The correlations among the different modes of Schroedinger cat states are studied. 
  This paper examines the causal structure of the commutator of two string fields, in free light-cone string field theory. By treating the commutator as a distribution on infinite dimensional loop space, it is shown that the commutator vanishes when $\int d\ss (\delta X(\ss))^2 <0$. Of more direct physical interest is the commutator of finite mass fields, obtained by smearing the string fields with appropriate wave functions. This is shown to vanish at spacelike separations, reproducing the usual point particle field theory result. The implications of this for the information spreading mechanism proposed by Susskind to solve the black hole information problem are discussed. Finally, it is verified that the above conclusions also hold for the superstring. 
  The present paper studies a completely integrable conformally invariant model in 1+1 dimensions that corresponds to string propagation on the two-dimensional black hole background (semi-ininite cigar). Besides the two space-time string fields there is a third (internal) field with a very specific Liouville-type interaction leading to the complete integrability. This system is known as non-abelian Toda theory. I give the general explicit classical solution. It realizes a rather involved transformation expressing the interacting string fields in terms of (three) functions $\varphi_j(u)$ and $\bar\varphi_j(v)$ of one light-cone variable only. The latter are shown to lead to standard harmonic oscillator (free field) Poisson brackets thus paving the way towards quantization. There are three left-moving and three right-moving conserved quantities. The right (left)-moving conserved quantities form a new closed non-linear, non-local Poisson bracket algebra. This algebra is a Virasoro algebra extended by two conformal dimension-two primaries. 
  We present the scalar field effective potential for nonrelativistic self-interacting scalar and fermion fields coupled to an Abelian Chern-Simons gauge field. Fermions are non-minimally coupled to the gauge field via a Pauli interaction. Gauss's law linearly relates the magnetic field to the matter field densities; hence, we also include radiative effects from the background gauge field. However, the scalar field effective potential is transparent to the presence of the background gauge field to leading order in the perturbative expansion. We compute the scalar field effective potential in two gauge families. We perform the calculation in a gauge reminiscent of the $R_\xi$-gauge in the limit $\xi\rightarrow 0$ and in the Coulomb family gauges. The scalar field effective potential is the same in both gauge-fixings and is independent of the gauge-fixing parameter in the Coulomb family gauge. The conformal symmetry is spontaneously broken except for two values of the coupling constant, one of which is the self-dual value. To leading order in the perturbative expansion, the structure of the classical potential is deeply distorted by radiative corrections and shows a stable minimum around the origin, which could be of interest when searching for vortex solutions. We regularize the theory with operator regularization and a cutoff to demonstrate that the results are independent of the regularization scheme. 
  The known calculations of the fermion condensate $<\bar{\psi}\psi>$ and the correlator $<\bar{\psi}\psi(x) ~\bar{\psi}\psi(0)>$ have been interpreted in terms of {\em localized} instanton solutions minimizing the {\em effective} action. Their size is of order of massive photon Compton wavelength $\mu^{-1}$. At high temperature, these instantons become quasistatic and present the 2-dimensional analog of the `walls' found recently in 4-dimensional gauge theories. In spite of the static nature of these solutions, they should not be interpreted as `thermal solitons' living in Minkowski space: the mass of these would-be solitons does not display itself in the physical correlators. At small but nonzero fermion mass, the high-T partition function of $QED_2$ is saturated by the rarefied gas of instantons and antiinstantons with density $\propto m~\exp\{-S^{inst.}\}~=~m~\exp\{-\pi T/\mu\}$ to be confronted with the dense strongly correlated instanton-antiinstanton liquid saturating the partition function at $T=0$. 
  Recent work on the relation between a special class of K\"ahler manifolds with positive first Chern class and critical N$=$2 string vacua with c$=$9 is reviewed and extended. (Based on a talk presented at the International Workshop on   Supersymmetry and Unification of Fundamental Interactions   (SUSY 93), Boston, MA, March 1993) 
  Symplectic and Poisson structures of certain moduli spaces/Huebschmann,J./ Abstract: Let $\pi$ be the fundamental group of a closed surface and $G$ a Lie group with a biinvariant metric, not necessarily positive definite. It is shown that a certain construction due to A. Weinstein relying on techniques from equivariant cohomology may be refined so as to yield (i) a symplectic structure on a certain smooth manifold $\Cal M(\Cal P,G)$ containing the space $\roman{Hom}(\pi,G)$ of homomorphisms and, furthermore, (ii) a hamiltonian $G$-action on $\Cal M(\Cal P,G)$ preserving the symplectic structure, with momentum mapping $\mu \colon \Cal M(\Cal P,G) \to g^*$, in such a way that the reduced space equals the space $\roman{Rep}(\pi,G)$ of representations. Our approach is somewhat more general in that it also applies to twisted moduli spaces; in particular, it yields the {\smc Narasimhan-Seshadri} moduli spaces of semistable holomorphic vector bundles by {\it symplectic reduction in finite dimensions}.This implies that, when the group $G$ is compact, such a twisted moduli space inherits a structure of {\it stratified symplectic space}, and that the strata of these twisted moduli spaces have finite symplectic volume. 
  In earlier work we have shown that the moduli space $N$ of flat connections for the (trivial) $\roman{SU(2)}$-bundle on a closed surface of genus $\ell \geq 2$ inherits a structure of stratified symplectic space with two connected strata $N_Z$ and $N_{(T)}$ and $2^{2\ell}$ isolated points. In this paper we show that, close to each point of $N_{(T)}$, the space $N$ and its Poisson algebra look like a product of $\bold C^{\ell}$ endowed with the standard symplectic Poisson structure with the reduced space and Poisson algebra of the system of $(\ell-1)$ particles in the plane with total angular momentum zero, while close to one of the isolated points, the Poisson algebra on $N$ looks like that of the reduced system of $\ell$ particles in $\bold R^3$ with total angular momentum zero. Moreover, in the genus two case where the space $N$ is known to be smooth we locally describe the Poisson algebra and the various underlying symplectic structures on the strata and their mutual positions explicitly in terms of the Poisson structure. 
  We study the formulation of the Wilson renormalization group (RG) method for a non-Abelian gauge theory. We analyze the simple case of $SU(2)$ and show that the local gauge symmetry can be implemented by suitable boundary conditions for the RG flow. Namely we require that the relevant couplings present in the physical effective action, \ie the coefficients of the field monomials with dimension not larger than four, are fixed to satisfy the Slavnov-Taylor identities. The full action obtained from the RG equation should then satisfy the same identities. This procedure is similar to the one we used in QED. In this way we avoid the cospicuous fine tuning problem which arises if one gives instead the couplings of the bare Lagrangian. To show the practical character of this formulation we deduce the perturbative expansion for the vertex functions in terms of the physical coupling $g$ at the subtraction point $\mu$ and perform one loop calculations. In particular we analyze to this order some ST identities and compute the nine bare couplings. We give a schematic proof of perturbative renormalizability. 
  The exact general evolution of circular strings in $2+1$ dimensional de Sitter spacetime is described closely and completely in terms of elliptic functions. The evolution depends on a constant parameter $b$, related to the string energy, and falls into three classes depending on whether $b<1/4$ (oscillatory motion), $b=1/4$ (degenerated, hyperbolic motion) or $b>1/4$ (unbounded motion). The novel feature here is that one single world-sheet generically describes {\it infinitely many} (different and independent) strings. The world-sheet time $\tau$ is an infinite-valued function of the string physical time, each branch yields a different string. This has no analogue in flat spacetime. We compute the string energy $E$ as a function of the string proper size $S$, and analyze it for the expanding and oscillating strings. For expanding strings $(\dot{S}>0)$: $E\neq 0$ even at $S=0$, $E$ decreases for small $S$ and increases $\propto\hspace*{-1mm}S$ for large $S$. For an oscillating string $(0\leq S\leq S_{max})$, the average energy $<E>$ over one oscillation period is expressed as a function of $S_{max}$ as a complete elliptic integral of the third kind. 
  A new, formal, non-combinatorial approach to invariants of three-dimensional manifolds of Reshetikhin, Turaev and Witten in the framework of non-perturbative topological quantum Chern-Simons theory, corresponding to an arbitrary compact simple Lie group, is presented. A direct implementation of surgery instructions in the context of quantum field theory is proposed. An explicit form of the specialization of the invariant to the group SU(2) is derived. 
  We clarify the nature of the graviton as a bound state in open-string field theory: The flat metric in the action appears as the vacuum value of an OPEN string field. The bound state appears as a composite field in the FREE field theory. 
  We consider the loop space representation of multi-matrix models. Explaining the origin of a time variable through stochastic quantization we make contact with recent proposals of Ishibashi and Kawai. We demonstrate how collective field theory with its loop space interactions generates a field theory of non-critical strings. 
  The quasiclassical asymptotics of the Knizhnik-Zamolodchikov equation with values in the tensor product of sl(2)- representations are considered. The first term of asymptotics is an eigenvector of a system of commuting operators. We show that the norm of this vector with respect to the Shapovalov form is equal to the determinant of the matrix of second derivatives of a suitable function. This formula is an analog of the Gaudin and Korepin formulae for the norm of the Bethe vectors. We show that the eigenvectors form a basis under certain conditions. 
  $D=2$ free string in linear dilaton background is considered in so called target space/world-sheet light cone gauge $ X^{+}=0,~g_{++}=0,~g_{+-}=1$. After gauge fixing the theory has the residual Virasoro and $U(1)$ current symmetries. The physical spectrum related to $SL_2$ invariant vacuum is found to be trivial. We find that the theory has a nontrivial spectrum if the states in different non-equivalent representations ("pictures") of CFT algebra of matter fields are considered. 
  Using a duality between the space of particles and the space of fields, we show how one can compute form factors directly in the space of fields. This introduces the notion of vertex operators, and form factors are vacuum expectation values of such vertex operators in the space of fields. The vertex operators can be constructed explicitly in radial quantization. Furthermore, these vertex operators can be exactly bosonized in momentum space. We develop these ideas by studying the free-fermion point of the sine-Gordon theory, and use this scheme to compute some form-factors of some non-free fields in the sine-Gordon theory. This work further clarifies earlier work of one of the authors, and extends it to include the periodic sector. 
  Target space modular symmetries relevant to string loop threshold corrections are studied for $Z_N$ orbifold compactified string theories containing Wilson line background fields. 
  Conjecture 9B from the previous version of the paper stating that any holomorphic vector bundle on an elliptic curve can be realized by a scalar differential equation has now been proved by the authors. The proof is included in the new version. 
  I investigate non-perturbative aspects of zero-dimensional matrix models. Subtleties in the large-$N$ limit of the semiclassical picture are pointed out. The tunneling of eigenvalues is seen to correspond to a chaotic sequence of recursion coefficients determining the orthogonal polynomials. 
  We perform a deformation quantization of the classical isotropic rigid rotator. The resulting quantum system is not invariant under the usual $SU(2)\times SU(2)$ chiral symmetry, but instead $SU_{q^{-1}}(2) \times SU_q(2)$. 
  We study $\Bbb Z_2^{\otimes N}$ graded contractions of the real compact simple Lie algebra $so(N+1)$, and we identify within them the Cayley-Klein algebras as a naturally distinguished subset. 
  We develop the quantization of a macroscopic string which extends radially from a Schwarzschild black hole. The Hawking process excites a thermal bath of string modes that causes the black hole to lose mass. The resulting typical string configuration is a random walk in the angular coordinates. We show that the energy flux in string excitations is approximately that of spacetime field modes. 
  Symmetries of multi-anyon wavefunctions are analysed with the help of a second quantized formulation. Analogues of Slater determinants are constructed. It is shown that the Pauli principle is not enforced. 
  A *-product compatible with the comultiplication of the Hopf algebra of the functions on the Heisenberg group is determined by deforming a coboundary Lie-Poisson structure defined by a classical r-matrix satisfying the modified Yang-Baxter equation. The corresponding quantum group is studied and its R-matrix is explicitly calculated. 
  Algebraic framework for construction of a commuting set of operators that can be interpreted as integrals of motion of the open spin chain with boundary conditions and nearest neighbour interaction is investigated. 
  We investigate the phase accumulated by a charged particle in an extended quantum state as it encircles one or more magnetic fluxons, each carrying half a flux unit. A simple, essentially topological analysis reveals an interplay between the Aharonov-Bohm phase and Berry's phase. 
  The formalism of Greensite for treating the spacetime signature as a dynamical degree of freedom induced by quantum fields is considered for spacetimes with nontrivial topology of the kind ${\bf R}^{D-1} \times {\bf T}^1$, for varying $D$. It is shown that a dynamical origin for the Lorentzian signature is possible in the five-dimensional space ${\bf R}^4 \times {\bf T}^1$ with small torus radius (periodic boundary conditions), as well as in four-dimensional space with trivial topology. Hence, the possibility exists that the early universe might have been of the Kaluza-Klein type, \ie multidimensional and of Lorentzian signature. 
  The problem of diagonalization of the quantum mechanical Hamiltonian, governing dynamics of an electron on a two-dimensional triangular or square lattice in external uniform magnetic field, applied perpendicularly to the lattice plane, the flux through lattice cell, divided by the elementary quantum flux, being rational number, is reduced to generalized Bethe ansatz like equations on high genus algebraic curve. Our formulae for the trigonometric case, where genus of the curve vanishes, contain as a particular case recent result of Wiegmann and Zabrodin. 
  Several interesting features of coset models "without fixed points" are easily understood via Chern-Simons theory. In this paper we derive explicit formulae for the handle-squashing operator in these cosets. These operators are fixed, linear combinations of the irreducible representations of the coset. As a simple application of these curious formulae, we compute the traces of all genus-one operators for several common cosets. (1 Figure, at end in encapsulated postscript- to locate use 'find figs') 
  For the series associated to a group or coset R.C.F.T. there is a simple universal form for the inverse of the handle operator in the ring of fusions. These formulae may be easily understood from the quantization of the associated Chern-Simons theory. (Transcript of talk given at STRINGS '93 (Berkeley) Conference) 
  Black hole spacetimes are semiclassically not static. For black holes whose lifetime is larger than the age of the universe we compute, in leading order, the power spectrum of deviations of the electromagnetic charge from it's average value, zero. Semiclassically the metric itself has a statistical interpretation and we compute a lowerbound on its variance. (1 figure, at end in encapsulated postscript - to locate use 'find figs') 
  Based on a methodological analysis of the effective action approach certain conceptual foundations of quantum field theory are reconsidered to establish a quest for an equation for the effective action. Relying on the functional integral formulation of Lagrangian quantum field theory a functional integral equation for the complete effective action is proposed which can be understood as a certain fixed point condition. This is motivated by a critical attitude towards the distinction artificial from an experimental point of view between classical and effective action. While for free field theories nothing new is accomplished, for interacting theories the concept differs from the established paradigm. The analysis of this new concept is concentrated on gauge field theories treating QED as the prototype model. An approximative approach to the functional integral equation for the complete effective action of QED is exploited to obtain certain nonperturbative information about the quadratic kernels of the action. As particular application the approximative calculation of the QED coupling constant $\alpha$ is explicitly studied. It is understood as one of the characteristics of a fixed point given as a solution of the functional integral equation proposed. Finally, within the present approach the vacuum energy problem is considered and possible implications on the induced gravity concept are contemplated. 
  The first three coefficients in an expansion of the heat kernel of a nonminimal nonabelian kinetic operator taken in an arbitrary background gauge in arbitrary space-time dimension are calculated 
  The properties of the quantum Minkowski space algebra are discussed. Its irreducible representations with highest weight vectors are constructed and relations to other quantum algebras: $su_{q}(2)$, $q$-oscillator, $q$-sphere are pointed out. 
  We obtain an R-matrix or matrix representation of the Artin braid group acting in a canonical way on the vector space of every (super)-Lie algebra or braided-Lie algebra. The same result applies for every (super)-Hopf algebra or braided-Hopf algebra. We recover some known representations such as those associated to racks. We also obtain new representations such as a non-trivial one on the ring $k[x]$ of polynomials in one variable, regarded as a braided-line. Representations of the extended Artin braid group for braids in the complement of $S^1$ are also obtained by the same method. 
  We review some of the recent developments in two dimensional statistical mechanics in which corner transfer matrices provide the vital link between the physical system and the representation theory of quantum affine algebras. This opens many new possibilities, because the eigenstates may be described using the properties of q-vertex operators. 
  The Laughlin function of quantum Hall effect is shown to satisfy Hirota's bilinear difference equation with certain coefficients a little different from the KP hierarchy. Vertex operators which constitute blocks of solutions generate a B\"acklund transformation. Besides the Laughlin function, the equation admits soliton solutions. 
  A large class of new 4-D superstring vacua with non-trivial/singular geometries, spacetime supersymmetry and other background fields (axion, dilaton) are found. Killing symmetries are generic and are associated with non-trivial dilaton and antisymmetric tensor fields. Duality symmetries preserving N=2 superconformal invariance are employed to generate a large class of explicit metrics for non-compact 4-D Calabi-Yau manifolds with Killing symmetries. 
  We show that the multi-boson KP hierarchies possess a class of discrete symmetries linking them to the discrete Toda systems. These discrete symmetries are generated by the similarity transformation of the corresponding Lax operator. This establishes a canonical nature of the discrete transformations. The spectral equation, which defines both the lattice system and the corresponding Lax operator, plays a key role in determining pertinent symmetry structure. We also introduce a concept of the square-root lattice leading to a family of new pseudo-differential operators with covariance under additional B\"{a}cklund transformations. 
  We examine the properties of observables in the Kazakov-Migdal model. We present explicit formulae for the leading asymptotics of adjoint Wilson loops as well as some other observables for the model with a Gaussian potential. We discuss the phase transiton in the large $N$ limit of the $d=1$ model. One of appendices is devoted to discussion of the $N =\infty$ Itzykson-Zuber integrals for arbitrary eigenvalue densities. 
  In this paper alternative formulations of the conventional uncertainty relation are studied in the context of decoherent histories. The results are given in terms of Shannon information. A variety of methods are developed to evaluate the upper bound for the probability of two or more projection histories. The methods employed give improved limits for the maximal achievable probability and an improved lower bound for the Shannon information.   The results are then applied to a number of physically relevant situations. 
  Instead of imposing the Schr\"{o}dinger equation to obtain the configuration space propagator $\csprop$ for a quantum mechanical nonlinear sigma model, we directly evaluate the phase space propagator $\psprop$ by expanding the exponent and pulling all operators $\hat p$ to the right and $\hat x$ to the left. Contrary to the widespread belief that it is sufficient to keep only terms linear in $\Delta t$ in the expansion if one is only interested in the final result through order $\Delta t$, we find that all terms in the expansion must be retained. We solve the combinatorical problem of summing the infinite series in closed form through order $\Delta t$. Our results straightforwardly generalize to higher orders in $\Delta t$. We then include fermions for which we use coherent states in phase space. For supersymmetric $N{=}1$ and $N{=}2$ quantum mechanics, we find that if the super Van Vleck determinant replaces the original Van Vleck determinant the propagator factorizes into a classical part, this super determinant and the extra scalar curvature term which was first found by DeWitt for the purely bosonic case by imposing the Schr\"{o}dinger equation. Applying our results to anomalies in $n$-dimensional quantum field theories, we note that the operator ordering in the corresponding quantum mechanical Hamiltonians is fixed in these cases. We present a formula for the path integral action, which corresponds one to one to any given covariant or noncovariant $\hat H$. We then evaluate these path integrals through two loop order, and reobtain the same propagators in all cases. 
  Expressions are given for the Casimir operators of the exceptional group $F_4$ in a concise form similar to that used for the classical groups. The chain $B_4\subset F_4\subset D_{13}$ is used to label the generators of $F_4$ in terms of the adjoint and spinor representations of $B_4$ and to express the 26-dimensional representation of $F_4$ in terms of the defining representation of $D_{13}$. Casimir operators of any degree are obtained and it is shown that a basis consists of the operators of degree 2, 6, 8 and 12. 
  I investigate solutions to the Euclidean Einstein-matter field equations with topology $S^1 \times S^2 \times R$ in a theory with a massless periodic scalar field and electromagnetism. These solutions carry winding number of the periodic scalar as well as magnetic flux. They induce violations of a quasi-topological conservation law which conserves the product of magnetic flux and winding number on the background spacetime. I extend these solutions to a model with stable loops of superconducting cosmic string, and interpret them as contributing to the decay of such loops. 
  The most general homogeneous monodromy conditions in $N{=}2$ string theory are classified in terms of the conjugacy classes of the global symmetry group $U(1,1)\otimes{\bf Z}_2$. For classes which generate a discrete subgroup $\G$, the corresponding target space backgrounds ${\bf C}^{1,1}/\G$ include half spaces, complex orbifolds and tori. We propose a generalization of the intercept formula to matrix-valued twists, but find massless physical states only for $\Gamma{=}{\bf 1}$ (untwisted) and $\Gamma{=}{\bf Z}_2$ (\`a la Mathur and Mukhi), as well as for $\Gamma$ being a parabolic element of $U(1,1)$. In particular, the sixteen ${\bf Z}_2$-twisted sectors of the $N{=}2$ string are investigated, and the corresponding ground states are identified via bosonization and BRST cohomology. We find enough room for an extended multiplet of `spacetime' supersymmetry, with the number of supersymmetries being dependent on global `spacetime' topology. However, world-sheet locality for the chiral vertex operators does not permit interactions among all massless `spacetime' fermions. 
  Chern-Simons formulation of the 2+1 dimensional Einstein gravity with negative cosmological constant is investigated when the spacetime has the topology ${\bf R}\times T^{2}$. The physical phase space is shown to be a direct product of two sub-phase spaces having symplectic structures with opposite signs. Each sub-phase space is found to be a non-Hausdorff manifold plus a set of measure zero. Geometrical interpretation of each point in the phase space is also given. A prescription to quantize this phase space is proposed. 
  We investigate a two parameter quantum deformation of the universal enveloping orthosymplectic superalgebra U(osp(2/2)) by extending the Faddeev-Reshetikhin-Takhtajan formalism to the supersymetric case. It is shown that $U_{p,q}(osp(2/2))$ possesses a non-commutative, non-cocommutative Hopf algebra structure. All the results are expressed in the standard form using quantum Chevalley basis. 
  We consider the Hamiltonian and Lagrangian formalism describing free \k-relativistic particles with their four-momenta constrained to the \k-deformed mass shell. We study the modifications of the formalism which follow from the introduction of space coordinates with nonvanishing Poisson brackets and from the redefinitions of the energy operator. The quantum mechanics of free \k-relativistic particles and of the free \k-relativistic oscillator is also presented. It is shown that the \k-relativistic oscillator describes a quantum statistical ensemble with finite Hagedorn temperature. The relation to a \k-deformed Schr\"odinger quantum mechanics in which the time derivative is replaced by a finite difference derivative is also discussed. 
  Using the algebraic Hamiltonian approach, we derive the exact to all orders O(d,d) transformations of the metric and the dilaton field in WZW and WZW coset models for both compact and non-compact groups. It is shown that under the exact $O(d)\times O(d)$ transformation only the leading order of the inverse metric $G^{-1}$ is transformed. The quantity $\sqrt{G}\exp(\Phi)$ is the same in all the dual models and in particular is independent of k. We also show that the exact metric and dilaton field that correspond to G/U(1)^d WZW can be obtained by applying the exact O(d,d) transformations on the (ungauged) WZW, a result that was known to one loop order only. As an example we give the O(2,2) transformations in the $SL(2,R)$ WZW that transform to its dual exact models. These include also the exact 3D black string and the exact 2D black hole with an extra $U(1)$ free field. 
  We examine the correspondence between the conformal field theory of boundary operators and two-dimensional hyperbolic geometry. By consideration of domain boundaries in two-dimensional critical systems, and the invariance of the hyperbolic length, we motivate a reformulation of the basic equation of conformal covariance. The scale factors gain a new, physical interpretation. We exhibit a fully factored form for the three-point function. A doubly-infinite discrete series of central charges with limit c=-2 is discovered. A correspondence between the anomalous dimension and the angle of certain hyperbolic figures emerges. Note: email after 12/19: kleban@maine.maine.edu 
  We investigate in details how the Virasoro algebra appears in the scaling limit of the simplest lattice models of XXZ or RSOS type. Our approach is straightforward but to our knowledge had never been tried so far. We simply formulate a conjecture for the lattice stress-energy tensor motivated by the exact derivation of lattice global Ward identities. We then check that the proper algebraic relations are obeyed in the scaling limit. The latter is under reasonable control thanks to the Bethe-ansatz solution. The results, which are mostly numerical for technical reasons, are remarkably precise. They are also corroborated by exact pieces of information from various sources, in particular Temperley-Lieb algebra representation theory. Most features of the Virasoro algebra (like central term, null vectors, metric properties...) can thus be observed using the lattice models. This seems of general interest for lattice field theory, and also more specifically for finding relations between conformal invariance and lattice integrability, since basis for the irreducible representations of the Virasoro algebra should now follow (at least in principle) from Bethe-ansatz computations. 
  We investigate the fractal structure of $2d$ quantum gravity coupled to matter by measuring the distributions of so-called baby universes. We demonstrate that the method works well as long as $c \leq 1$. For $c >1$ it is not clear what distribution to expect. However, we observe strikingly similar distributions for various kinds of matter fields with the same $c$. This indicate that there might be some range of $c >1$ where the central charge of the matter fields alone determines the fractal structure of gravity coupled to matter. The hypothesis that the string susceptibility $\g = 1/3$ is found to be compatible with the data for $1 < c \leq 4$. 
  A two-loop (cylinder) amplitude of the 2d pure gravity theory is obtained in the proper-time gauge ($g_{00}=1$, $g_{01}=g_{10}=0$) in the continuum formulation. The constraint $T_{01}=0$ is solved and used to reduce the problem of field theory to that of quantum mechanics. This reduction can also be proved by using a conformal Ward identity. The amplitude depends on the lengths $l_1, l_2$ of the boundaries, the proper time $T$ and a non-negative integer $m$ associated with winding modes around the boundaries. 
  We discuss a K3 and torus from view point of "mirror symmetry". We calculate the periods of the K3 surface and obtain the mirror map, the two-point correlation function, and the prepotential. Then we find there is no instanton correction on K3 (also torus), which is expected from view point of Algebraic geometry. 
  The role of a physical phase space structure in a classical and quantum dynamics of gauge theories is emphasized. In particular, the gauge orbit space of Yang-Mills theories on a cylindrical spacetime (space is compactified to a circle) is shown to be the Weyl cell for a semisimple compact gauge group, while the physical phase space coincides with the quotient $\R^{2r}/W_A$, $r$ a rank of a gauge group, $W_A$ the affine Weyl group. The transition amplitude between two points of the gauge orbit space (between two Wilson loops) is represented via a Hamiltonian path integral over the physical phase space and explicitly calculated. The path integral formula appears to be modified by including trajectories reflected from the boundary of the physical configuration space (of the Weyl cell) into the sum over pathes. The Gribov problem of gauge fixing ambiguities is considered and its solution is proposed in the framework of the path integral modified. Artifacts of gauge fixing are qualitatively analyzedwith a simple mechanical example. A relation between a gauge-invariant description and a gauge fixing procedure is established. 
  We construct a classical action for a system of $N$ point-like sources which carry SU(2) non-Abelian charges coupled to non-Abelian Chern-Simons gauge fields, and develop a quantum mechanics for them. Adopting the coherent state quantization and solving the Gauss' constraint in an appropriately chosen gauge, we obtain a quantum mechanical Hamiltonian given in terms of the Knizhnik-Zamolodchikov connection. Then we study the non-Abelian Aharonov-Bohm effect, employing the obtained Hamiltonian for two-particle sector. An explicit evaluation of the differential cross section for the non-Abelian Aharonov-Bohm scattering is given. 
  The analogue of the Poisson bracket for the De Donder-Weyl (DW) Hamiltonian formulation of field theory is proposed. We start from the Hamilton- Poincar\'{e}-Cartan (HPC) form of the multidimensional variational calculus and define the bracket on the differential forms over the space-time (=horizontal forms). This bracket is related to the Schouten-Nijenhuis bracket of the multivector fields which are associated with the horizontal forms by means of the "polysymplectic form". The latter is given by the HPC form and generalizes the symplectic form to field theory. We point out that the algebra of forms with respect to our Poisson bracket and the exterior product has the structure of the Gerstenhaber graded algebra. It is shown that the Poisson bracket with the DW Hamiltonian function generates the exterior differential thus leading to the bracket representation of the DW Hamiltonian field equations. Few illustrative examples are also presented. 
  It is shown that from the expectation values of obervables, which can be measured for a single system using protective measurements, the linear structure, inner product, and observables in the Hilbert space can be reconstructed. A universal method of measuring the wave function of a single particle using its gravitational field is given. Protective measurement is generalized to the measurement of a degenerate state and to many particle systems. The question of whether the wave function is real is examined, and an argument of Einstein in favor of the ensemble interpretation of quantum theory is refuted. 
  We discuss the phase structure (in the $1/N$-expansion) of the Nambu-Jona-Lasinio model in curved spacetime with non-trivial topology ${\cal M}^3 \times {\rm S}^1$. The evaluation of the effective potential of the composite field $\bar{\psi} \psi$ is presented in the linear curvature approximation (topology is treated exactly) and in the leading order of the $1/N$-expansion. The combined influence of topology and curvature to the phase transitions is investigated. It is shown, in particular, that at zero curvature and for small radius of the torus there is a second order phase transition from the chiral symmetric to the chiral non-symmetric phase. When the curvature grows and (or) the radius of ${\rm S}^1$ decreases, then the phase transition is in general of first order. The dynamical fermionic mass is also calculated in a number of different situations. 
  Four-dimensional quantum electrodynamics has been formulated on a hypercubic Minkowski finite-element lattice. The equations of motion have been derived so as to preserve lattice gauge invariance and have been shown to be unitary. In addition, species doubling is avoided due to the nonlocality of the interactions. The model is used to investigate the lattice current algebra. Regularization of the current is shown to arise in a natural and nonarbitrary way. The commutators of the lattice current are calculated and shown to have the expected qualitative behavior. These lattice results are compared to various continuum calculations. (Five figures available from author.) 
  We study the symplectic quantization of Abelian gauge theories in $2+1$ space-time dimensions with the introduction of a topological Chern-Simons term. 
  We obtain cosmological four dimensional solutions of the low energy effective string theory by reducing a five dimensional black hole, and black hole--de Sitter solution of the Einstein gravity down to four dimensions. The appearance of a cosmological constant in the five dimensional Einstein--Hilbert action produces a special dilaton potential in the four dimensional effective string action. Cosmological scenarios implemented by our solutions are discussed. (Talk presented at: ``27th Symposium on the Theory of Elementary Particles'' Wendisch--Rietz, September 7--11, 1993) 
  Closed super (p+2)-forms in target superspace are relevant for the construction of the usual super p-brane actions. Here we construct closed super (p+1)-forms on a {\it worldvolume superspace}. They are built out of the pull-backs of the Kalb-Ramond super (p+1)-form and its curvature. We propose a twistor-like formulation of a class of super p-branes which crucially depends on the existence of these closed super (p+1)-forms. 
  Zamolodchikov found an integrable field theory related to the Lie algebra E$_8$, which describes the scaling limit of the Ising model in a magnetic field. He conjectured that there also exist solvable lattice models based on E$_8$ in the universality class of the Ising model in a field. The dilute A$_3$ model is a solvable lattice model with a critical point in the Ising universality class. The parameter by which the model can be taken away from the critical point acts like a magnetic field by breaking the $\Integer_2$ symmetry between the states. The expected direct relation of the model with E$_8$ has not been found hitherto. In this letter we study the thermodynamics of the dilute A$_3$ model and show that in the scaling limit it exhibits an appropriate E$_8$ structure, which naturally leads to the E$_8$ scattering theory for massive excitations over the ground state. 
  We clarify the relation between the approach to $q$-Minkowski space of Carow-Watamura et al. with an approach based on the idea of $2\times 2$ braided Hermitean matrices. The latter are objects like super-matrices but with Bose-Fermi statistics replaced by braid statistics. We also obtain new R-matrix formulae for the $q$-Poincare group in this framework. 
  We discuss both classical and semiclassical properties of extremal black holes in theories where the dilaton and a modulus field are present. We find that the corresponding 2-dim geometry is asymptotically anti-de Sitter rather then asymptotically flat as in the purely dilatonic case. This fact has many important consequences, which we analyze at length, both for the classical behaviour and for the thermodynamical properties of the black hole. We also study the Hawking evaporation process in the semiclassical approximation. The calculations strongly indicates the emergence of a stable ground state as the end point of the process. Some comments are made about the relevance of our results for the problem of information loss in black hole physics. 
  Classical solutions describing charged dilaton black holes accelerating in a background magnetic field have recently been found. They include the Ernst metric of the Einstein-Maxwell theory as a special case. We study the extremal limit of these solutions in detail, both at the classical and quantum levels. It is shown that near the event horizon, the extremal solutions reduce precisely to the static extremal black hole solutions. For a particular value of the dilaton coupling, these extremal black holes are five dimensional Kaluza-Klein monopoles. The euclidean sections of these solutions can be interpreted as instantons describing the pair creation of extremal black holes/Kaluza-Klein monopoles in a magnetic field. The action of these instantons is calculated and found to agree with the Schwinger result in the weak field limit. For the euclidean Ernst solution, the action for the extremal solution differs from that of the previously discussed wormhole instanton by the Bekenstein-Hawking entropy. However, in many cases quantum corrections become large in the vicinity of the black hole, and the precise description of the creation process is unknown. 
  The line bundles which arise in the holonomy interpretations of the geometric phase display curious similarities to those encountered in the statement of the Borel-Weil-Bott theorem of the representation theory. The remarkable relation of the geometric phase to the classification of complex line bundles provides the necessary tools for establishing the relevance of the Borel-Weil-Bott theorem to Berry's adiabatic phase. This enables one to define a set of topological charges for arbitrary compact connected semisimple dynamical Lie groups. In this paper, the problem of the determination of the parameter space of the Hamiltonian is also addressed. A simple topological argument is presented to indicate the relation between the Riemannian structure on the parameter space and Berry's connection. The results about the fibre bundles and group theory are used to introduce a procedure to reduce the problem of the non-adiabatic (geometric) phase to Berry's adiabatic phase for cranked Hamiltonians. Finally, the possible relevance of the topological charges of the geometric phase to those of the non-abelian monopoles is pointed out. 
  We discuss a generalization of the quantum group $\su$ to the $q$-Virasoro algebra in two-dimensional electrons system under uniform magnetic field. It is shown that the integral representations of both algebras are reduced to those in a (1+1)-dimensional fermion. As an application of the quantum group symmetry, we discuss a model of quantum group current on the analogy of the Hall current. 
  We propose a new type of gauge in two-dimensional quantum gravity. We investigate pure gravity in this gauge, and find that the system reduces to quantum mechanics of loop length $l$. Furthermore, we rederive the $c\!=\!0$ string field theory which was discovered recently. In particular, the pregeometric form of the Hamiltonian is naturally reproduced. 
  Recently obtained results for two and three point functions for quasi-primary operators in conformally invariant theories in arbitrary dimensions {\absit d} are described. As a consequence the three point function for the energy momentum tensor has three linearly independent forms for general {\absit d} compatible with conformal invariance. The corresponding coefficients may be regarded as possible generalisations of the Virasoro central charge to {\absit d} larger than 2. Ward identities which link two linear combinations of the coefficients to terms appearing in the energy momentum tensor trace anomaly on curved space are discussed. The requirement of positivity for expectation values of the energy density is also shown to lead to positivity conditions which are simple for a particular choice of the three coefficients. Renormalisation group like equations which express the constraints of broken conformal invariance for quantum field theories away from critical points are postulated and applied to two point functions.\hfill\break Talk presented at the XXVII Ahrenshoop International Symposium. 
  We define a new mathematical structure ( graph quantum group) which combines the tower of algebras associated with a graph ${\cal G}$ and the structure of a Hopf algebra {\cal A}. In this structure Ocneanu's string operators become Hopf algebra intertwiners. We present some examples of graph quantum groups. 
  We study a canonical quantization of the Wess--Zumino--Witten (WZW) model which depends on two integer parameters rather than one. The usual theory can be obtained as a contraction, in which our two parameters go to infinity keeping the difference fixed. The quantum theory is equivalent to a generalized   Thirring model, with left and right handed fermions transforming under different representations of the symmetry group. We also point out that the classical WZW model with a compact target space has a canonical formalism in which the current algebra is an affine Lie algebra of non--compact type.   Also, there are some non--unitary quantizations of the WZW model in which there is invariance only under half the conformal algebra (one copy of the   Virasoro algebra). 
  We present a general method to deform the inhomogeneous algebras of the $B_n,C_n,D_n$ type, and find the corresponding bicovariant differential calculus. The method is based on a projection from $B_{n+1}, C_{n+1}, D_{n+1}$. For example we obtain the (bicovariant) inhomogeneous $q$-algebra $ISO_q(N)$ as a consistent projection of the (bicovariant) $q$-algebra $SO_q(N+2)$. This projection works for particular multiparametric deformations of $SO(N+2)$, the so-called ``minimal" deformations. The case of $ISO_q(4)$ is studied in detail: a real form corresponding to a Lorentz signature exists only for one of the minimal deformations, depending on one parameter $q$. The quantum Poincar\'e Lie algebra is given explicitly: it has 10 generators (no dilatations) and contains the {\sl classical} Lorentz algebra. Only the commutation relations involving the momenta depend on $q$. Finally, we discuss a $q$-deformation of gravity based on the ``gauging" of this $q$-Poincar\'e algebra: the lagrangian generalizes the usual Einstein-Cartan lagrangian. 
  We first reformulate para-statistics in terms of Lie-super triple systems. In this way, we reproduce various new kinds of para-statistics discovered recently by Palev in addition to the standard one. Also, bosonic and fermionic operators may not necessarily commute with each other. 
  Jordan as well as related triple systems have been used to find several solutions of the Yang-Baxter equation, which are of rational as well as trigonometric type. 
  We present a systematic approach to constructing current algebras based on non-semi-simple groups. The Virasoro central charges corresponding to these current algebras are not, in general, given by integer numbers. The key point in this construction is that the bilinear form appearing in the current algebra can be different from the bilinear form used to raise and lower group indices. The action which realises this current algebra as its symmetry is also found. 
  The BRST structure of polynomial Poisson algebras is investigated. It is shown that Poisson algebras provide non trivial models where the full BRST recursive procedure is needed. Quadratic Poisson algebras may already be of arbitrarily high rank. Explicit examples are provided, for which the first terms of the BRST generator are given. The calculations are cumbersome but purely algorithmic, and have been treated by means of the computer algebra system REDUCE. Our analysis is classical ($=$ non quantum) throughout. 
  Within four dimensional (4d) N=1 supergravity theories we present extreme dilatonic domain wall solutions with a general overall coupling $\alpha$ in the dilaton K\" ahler potential. We concentrate on extreme Type I walls, which are static, planar configurations, interpolating between Minkowski space-time and a new type of space-time with a varying dilaton field. $\alpha=0$ case yields extreme ``ordinary'' supergravity walls between Minkowski and anti-de Sitter space-times. For $\alpha=1$ the walls are extreme ``stringy dilatonic'' walls of the three level superstring vacua interpolating between the constant and linear dilaton vacuum. Extreme stringy dilatonic walls ($\alpha =1$) serve as a dividing line between the walls ($0<\alpha<1$) with the (planar) singularity covered by the horizon and those ($\alpha>1$) with the naked singularity. Striking similarities between the global space-time structure of such domain walls and and the one of the corresponding extreme charged black holes with a general dilaton coupling are pointed out. 
  We investigate the cohomology structure of a general noncritical $W_N$-string. We do this by introducing a new basis in the Hilbert space in which the BRST operator splits into a ``nested'' sum of nilpotent BRST operators. We give explicit details for the case $N=3$. In that case the BRST operator $Q$ can be written as the sum of two, mutually anticommuting, nilpotent BRST operators: $Q=Q_0+Q_1$. We argue that if one chooses for the Liouville sector a $(p,q)$ $W_3$ minimal model then the cohomology of the $Q_1$ operator is closely related to a $(p,q)$ Virasoro minimal model. In particular, the special case of a (4,3) unitary $W_3$ minimal model with central charge $c=0$ leads to a $c=1/2$ Ising model in the $Q_1$ cohomology. Despite all this, noncritical $W_3$ strings are not identical to noncritical Virasoro strings. 
  Alain Connes' applications of non-commutative geometry to interaction physics are described for the purpose of model building. 
  We give a simple proof that a particular class of $N=2$ superstrings are equivalent to the $N=1$ superstrings. This is achieved by constructing a similarity transformation which transforms the $N=2$ BRST operators into a direct sum of the BRST operators for the $N=1$ string and topological sectors. 
  We investigate the algebraic structure of integrable hierarchies that, we propose, underlie models of $W$-gravity coupled to matter. More precisely, we concentrate on the dispersionless limit of the topological subclass of such theories, by making use of a correspondence between Drinfeld-Sokolov systems, principal $s\ell(2)$ embeddings and certain chiral rings. We find that the integrable hierarchies can be viewed as generalizations of the usual matrix Drinfeld-Sokolov systems to higher fundamental representations of $s\ell(n)$. The underlying Heisenberg algebras have an intimate connection with the quantum cohomology of grassmannians. The Lax operators are directly given in terms of multi-field superpotentials of the associated topological LG theories. We view our construction as a prototype for a multi-variable system and suspect that it might be useful also for a class of related problems. 
  We study a topological Landau-Ginzburg model with superpotential W(X)=X^{-1}. This is argued to be equivalent to c=1 string theory compactified at the self-dual radius. We compute the tree-level correlation function of N tachyons in this theory and show their agreement with matrix-model results. We also discuss the nature of contact terms, the perturbed superpotential and the flow of operators in the small phase space. The role of gravitational descendants in this theory is examined, and the tachyon two-point function in genus 1 is obtained using a conjectured modification of the gravitational recursion relations. 
  A certain topological field theory is shown to be equivalent to the compactified c=1 string. This theory is described in both Kazama-Suzuki coset and Landau-Ginzburg formulations. The genus-g partition function and genus-0 multi-tachyon correlators of the c=1 string are shown to be calculable in this approach. The KPZ formulation of non-critical string theory has a natural relation to this topological model. (Talk given at the Nato Advanced Research Workshop on `New Developments in String Theory, Conformal Models and Topological Field Theory', Cargese, May 12-21 1993.) 
  In the context of two-dimensional quantum cosmology, we consider the path-integral of a string on annulus which contains the Liouville field and conformal matter fields. We show that, in the transition amplitude of the string universe, the non-zero modes of the fields are all cancelled out only when we take the $c=1$ conformal matter field and impose the Neumann boundary condition on the system. The transition amplitude obtained obeys the minisuperspace Wheeler-DeWitt equation. In our treatment, the modular parameter on annulus plays the role of time variable to integrate out. 
  We study quasi-finite representation of the $\Winf$ algebra recently proposed by Kac and Radul. When the central charge is integer, we show that they are represented by free fermions and bosonic ghosts. There are some nontrivial representations with vanishing central charge. We discuss that they may be described by large $N$ limit of topological models. We calculate their operator algebras explicitly. 
  We give an elementary explicit construction of cell decomposition of the moduli space of projective structures on a two dimensional surface analogous to the decomposition of Penner/Strebel for moduli space of complex structures. The relations between projective structures and $PGL(2,{\bf C})$ flat connections are also described. (in the revised version uuencoded pictures are made printable) 
  We study the Hawking radiation for the geometry of an evaporating 1+1 dimensional black hole. We compute Bogoliubov coefficients and the stress tensor. We use a recent result of Srednicki to estimate the entropy of entanglement produced in the evaporation process, for the 1+1 dimensional hole and for the 3+1 dimensional hole. It is found that the one space dimensional result of Srednicki is the pertinent one to use, in both cases. 
  A coloured braid group representation (CBGR) is constructed with the help of some modified universal ${\cal R}$-matrix, associated to $U_q(gl(2))$ quantised algebra. Explicit realisation of Faddeev-Reshetikhin-Takhtajan (FRT) algebra is built up for this CBGR and new solutions of quantum Yang-Baxter equation are subsequently found through Yang-Baxterisation of FRT algebra. These solutions are interestingly related to nonadditive type quantum $R$-matrix and have a nontrivial $q\rightarrow 1$ limit. Lax operators of several concrete integrable models, which may be considered as some `coloured' extensions of lattice nonlinear Schr${\ddot {\rm o}}$dinger model and Toda chain, are finally obtained by taking different reductions of such solutions. 
  We construct a cohomology theory controlling the deformations of a general Drinfel'd algebra. The picture presented here has two sides -- the combinatorial one related with the fact of the existence of a graded Lie algebra structure on the simplicial cochain complex of the associahedra, and the algebraic one related with the algebra of derivations on the bar construction. 
  A dynamical system with discrete time is studied by means of algebraic geometry. The system admits a reduction that is interpreted as a classical field theory in 2+1-dimensional wholly discrete space-time. The integrals of motion of a particular case of the reduced system are shown to coincide, in essence, with the statistical sum of the well-known (inhomogeneous) 2-dimensional dimer model (the statistical sum is here a function of two parameters). Possible generalizations of the system are examined. 
  We show that $N=8$ {\it self-dual} supergravity theory, which is the consistent background for $N=2$ closed superstring theory in $2+2\-$dimensions, can accommodate the recently discovered two-dimensional dilaton gravity black hole solution, {\it via} appropriate dimensional reductions and truncations. Interestingly, the usual dilaton field in this set of solutions emerges from the scalar field in the ${\bf 70}\-$dimensional representation of an intrinsic global $SO(8)$ group. We also give a set of exact solutions, which can be interpreted as the dilaton field on Eguchi-Hanson gravitational instanton background, realized in an $N=1$ self-dual supergravity theory. This suggests that the $N=2$ superstring has a close (even closer) relationship with the two-dimensional black hole solution, which was originally developed in the context of bosonic string and $N=1$ superstring. Our result also provides supporting evidence for the conjecture that the $N=2$ superstring theory is the ``master theory'' of supersymmetric integrable systems in lower-dimensions. 
  We study a relation between two integrability conditions, namely the Yang-Baxter and the pair propagation equations, in 2D lattice models. While the two are equivalent in the 8-vertex models, discrepancies appear in the 16-vertex models. As explicit examples, we find the exactly solvable 16-vertex models which do not satisfy the Yang-Baxter equations. 
  Swimming of microorganisms is studied from a viewpoint of extended objects (strings and membranes) swimming in the incompressible f luid of low Reynolds number. The flagellated motion is analyzed in two dimensional fluid, by using the method developed in the ciliated motion with the Joukowski transformation. Discussion is given on the conserved charges and the algebra which are associated with the area (volume)- preserving diffeomorphisms giving the swimming motion of microorganisms. It is also suggested that the $N$-point string- and membrane-like amplitudes are useful for studying the collective swimming motion of microorganisms when fluctuation of the vorticity distribution exists in the sticky or rubber-like fluid. 
  We discuss the structure, realizations and quantum BRST operators of a class of nonlinear superconformal algebras with N > 4. 
  Using the variational formula for operator product coefficients a method for perturbative calculation of the short-distance expansion of the Spin-Spin correlation function in the two dimensional Ising model is presented. Results of explicit calculation up to third order agree with known results from the scaling limit of the lattice calculation. 
  We generalize Einstein's Lagrangian in a non-polynomial (in R) way. The usual Lagrangian (linear in R) is the zero $\alpha'$ limit of our theory, where $\alpha'$ is a parameter that is interpreted as the inverse cosmological costant before the Planck time. The theory space of this lagrangian admits a ${\bf Z_{2}}$ modular group, namely $R \leftrightarrow 1/R$. Independence of the modular invariant expectation values from the number of `Big Bangs' enforces a quantization condition for the cosmological constant. At the semiclassical approximation we obtain $\Lambda =0$, and a vacuum equation which is equivalent to inflation cosmology. D=4 and D=1 universes are obtained as unique (and topologically separated by the D=2 semiclassical barrier) integer dimension solutions. They correspond to the first excited level and the ground state respectively of our projective gravity. 
  We quantize the abelian Chern-Simons theory coupled to non-relativistic matter field on a torus without invoking the flux quantization. Through a series of canonical transformations which is equivalent to solving the Gauss constraint, we obtain an effective hamiltonian density with periodic matter field. We also obtain the many-anyon Schr\"odinger equation with periodic Aharonov-Bohm potentials and analyze the periodic property of the wavefunction. Some comments are given on the different features of our approach from the previous ones. 
  We present the q-deformed ^M para-bose oscillators associated with (two-body) Calogero model.^M q-deformed coherent state is also constructed and its resolution of unity ^M is demonstrated. 
  A long-standing conjecture on the structure of renormalized, gauge invariant, integrated operators of arbitrary dimension in Yang-Mills theory is established. The general solution of the consistency condition for anomalies with sources included is also derived. This is achieved by computing explicitely the cohomology of the full unrestricted BRST operator in the space of local polynomial functionals with ghost number equal to zero or one. The argument does not use power counting and is purely cohomological. It relies crucially on standard properties of the antifield formalism. 
  In the spirit of classic works of Wilson on the renormalization group and operator product expansion, a new framework for the study of the theory space of euclidean quantum field theories has been introduced. This formalism is particularly useful for elucidating the structure of the short-distance expansions of the $n$-point functions of a renormalizable quantum field theory near a non-trivial fixed point. We review and apply this formalism in the study of the scaling limit of the two dimensional massive Ising model. Renormalization group analysis and operator product expansions determine all the non-analytic mass dependence of the short-distance expansion of the correlation functions. An extension of the first order variational formula to higher orders provides a manifestly finite scheme for the perturbative calculation of the operator product coefficients to any order in parameters. A perturbative expansion of the correlation functions follows. We implement this scheme for a systematic study of correlation functions involving two spin operators. We show how the necessary non-trivial integrals can be calculated. As two concrete examples we explicitly calculate the short-distance expansion of the spin-spin correlation function to third order and the spin-spin-energy density correlation function to first order in the mass. We also discuss the applicability of our results to perturbations near other non-trivial fixed points corresponding to other unitary minimal models. 
  By using the free field realizations, we analyze the representation theory of the W_{1+infinity} algebra with c=1. The eigenvectors for the Cartan subalgebra of W_{1+infinity} are parametrized by the Young diagrams, and explicitly written down by W_{1+infinity} generators. Moreover, their eigenvalues and full character formula are also obtained. 
  I show that the new topological field theories recently associated by Dubrovin with each Coxeter group may be all obtained in a simple way by a ``restriction'' of the standard ADE solutions. I then study the Chebichev specializations of these topological algebras, examine how the Coxeter graphs and matrices reappear in the dual algebra and mention the intriguing connection with the operator product algebra of conformal field theories. A direct understanding of the occurrence of Coxeter groups in that context is highly desirable. 
  The Kazakov-Migdal model, if considered as a functional of external fields, can be always represented as an expansion over characters of $GL$ group. The integration over "matter fields" can be interpreted as going over the {\it model} (the space of all highest weight representations) of $GL$. In the case of compact unitary groups the integrals should be substituted by {\it discrete} sums over weight lattice. The $D=0$ version of the model is the Generalized Kontsevich integral, which in the above-mentioned unitary (discrete) situation coincides with partition function of the $2d$ Yang-Mills theory with the target space of genus $g=0$ and $m=0,1,2$ holes. This particular quantity is always a bilinear combination of characters and appears to be a Toda-lattice $\tau$-function. (This is generalization of the classical statement that individual $GL$ characters are always singular KP $\tau$-functions.) The corresponding element of the Universal Grassmannian is very simple and somewhat similar to the one, arising in investigations of the $c=1$ string models. However, under certain circumstances the formal sum over representations should be evaluated by steepest descent method and this procedure leads to some more complicated elements of Grassmannian. This "Kontsevich phase" as opposed to the simple "character phase" deserves further investigation. 
  We discuss an interpretation of the projection postulate that implies collapse of the wavefunction along the lightcone. 
  Some approaches to $2d$ gravity developed for the last years are reviewed. They are physical (Liouville) gravity, topological theories and matrix models. A special attention is paid to matrix models and their interrelations with different approaches. Almost all technical details are omitted, but examples are presented. 
  We first review the properties of the conventional $\tau$-functions of the KP and Toda-lattice hierarchies. A straightforward generalization is then discussed. It corresponds to passing from differential to finite-difference equations; it does not involve however the concept of operator-valued $\tau$-function nor the one associated with non-Cartanian (level $k\ne1$) algebras. The present study could be useful to understand better $q$-free fields and their relation to ordinary free fields. 
  Representations of braid group obtained from rational conformal field theories can be used to obtain explicit representations of Temperley-Lieb-Jones algebras. The method is described in detail for SU(2)$_k$ Wess - Zumino conformal field theories and its generalization to an arbitrary rational conformal field theory outlined. Explicit definition of an associated linear trace operation in terms of a certain matrix element in the space of conformal blocks of such a conformal theory is presented. Further for every primary field of a rational conformal field theory, there is a subfactor of hyperfinite II$_1$ factor with trivial relative commutant. The index of the subfactor is given in terms of identity - identity element of certain duality matrix for conformal blocks of four-point correlators. Jones formula for index ( $<$ 4 ) for subfactors corresponds to spin ${\frac{1}{2}}$ representation of SU(2)$_k$ Wess-Zumino conformal field theory. Definition of the trace operation also provides a method of obtaining link invariants explicitly. 
  A framework for studying knot and link invariants from any rational conformal field theory is developed. In particular, minimal models, superconformal models and $W_N$ models are studied. The invariants are related to the invariants obtained from the Wess-Zumino models associated with the coset representations of these models. Possible Chern-Simons representation of these models is also indicated. This generalises the earlier work on knot and link invariants from Chern-Simons theories. 
  We point out the question of ordering momentum operator in the canonical \break quantization of the SU(2) Skyrme Model. Thus, we suggest a new definition for the momentum operator that may solve the infrared problem that appears when we try to minimize the Quantum Hamiltonian. 
  First, we investigate the static non-Abelian Kubo equation. We prove that it does not possess finite energy solutions; thereby we establish that gauge theories do not support hard thermal solitons. A similar argument shows that "static" instantons are absent. In addition, we note that the static equations reproduce the expected screening of the non-Abelian electric field by a gauge invariant Debye mass m=gT sqrt((N+N_F/2)/3). Second, we derive the non-Abelian Kubo equation from the composite effective action. This is achieved by showing that the requirement of stationarity of the composite effective action is equivalent, within a kinematical approximation scheme, to the condition of gauge invariance for the generating functional of hard thermal loops. 
  In the same spirit as done for N=2 and N=4 supersymmetric non-linear $\si$ models in 2 space-time dimensions by Zumino and Alvarez- Gaum\'e and Freedman, we analyse the (2,0) and (4,0) heterotic geometry in holomorphic coordinates. We study the properties of the torsion tensor and give the conditions under which (2,0) geometry is conformally equivalent to a (2,2) one. Using additional isometries, we show that it is difficult to equip a manifold with a closed torsion tensor, but for the real 4 dimensional case where we exhibit new examples. We show that, contrarily to Callan, Harvey and Strominger 's claim for real 4 dimensional manifolds, (4,0) heterotic geometry is not necessarily conformally equivalent to a (4,4) K\"ahler Ricci flat geometry. We rather prove that, whatever the real dimension be, they are special quasi Ricci flat spaces, and we exemplify our results on Eguchi-Hanson and Taub-NUT metrics with torsion. 
  We prove the uncloseability of the free vertex operators used in conformal field theory for the BRST--construction of primary fields. Our proof includes minimal models as well as WZNW--models. 
  The actions for all classical (and consequently quantum) $BF$ theories on $n$-manifolds is proven to be given by anti-commutators of hermitian, nilpotent, scalar fermionic charges with Grassmann-odd functionals. In order to show this, the space of fields in the theory must be enlarged to include ``mass terms'' for new, non-dynamical, Grassmann-odd fields. The implications of this result on observables are examined. 
  We discuss the pseudodual chiral model to illustrate a class of two-dimensional theories which have an infinite number of conservation laws but allow particle production, at variance with naive expectations. We describe the symmetries of the pseudodual model, both local and nonlocal, as transmutations of the symmetries of the usual chiral model. We refine the conventional algorithm to more efficiently produce the nonlocal symmetries of the model, and we discuss the complete local current algebra for the pseudodual theory. We also exhibit the canonical transformation which connects the usual chiral model to its fully equivalent dual, further distinguishing the pseudodual theory. 
  Liouville and Toda gravity theories with non-vanishing interaction potentials have spectra obtained by dividing the free-field spectra for these cases by the Weyl group of the corresponding $A_1$ or $A_2$ Lie algebra. We study the canonical transformations between interacting and free fields using the technique of intertwining operators, giving explicit constructions for the wavefunctions and showing that they are invariant under the corresponding Weyl groups. These explicit constructions also permit a detailed analysis of the operator-state maps and of the nature of the Seiberg bounds. 
  We conjecture that the $O(N)$-symmetric non-linear sigma model in the semi-infinite $(1+1)$-dimensional space is ``integrable'' with respect to the ``free'' and the ``fixed'' boundary conditions. We then derive, for both cases, the boundary S-matrix for the reflection of massive particles of this model off the boundary at $x=0$. 
  While in general there is no one-to-one correspondence between complex and quaternion quantum mechanics (QQM), there exists at least one version of QQM in which a {\em partial} set of {\em translations} may be made. We define these translations and use the rules to obtain rapid quaternion counterparts (some of which are new) of standard quantum mechanical results. 
  We consider the canonical quantization scheme for $c \leq 1$ ($(p,q)$ -) string theories and compare it with what is known from matrix model approach. We derive explicitly a trivial ($\equiv $ topological) solution. We discuss a ``dressing" operator which in principle allows one to obtain a non-trivial solution, but an explicit computation runs into a problem of analytic continuation of the formal expressions for $\tau $-functions. We discuss also the application of proposed scheme to the case of discrete matrix model and consider some parallels with mirror symmetry and background independence in string theory. 
  We study the problem of covariant separation between first and second class constraints for the $D=10$ Brink-Schwarz superparticle. Opposite to the supersymmetric light-cone frame separation, we show here that there is a Lorentz covariant way to identify the second class constraints such that, however, supersymmetry is broken. Consequences for the $D=10$ superstring are briefly discussed. 
  The fermion flavor $N_f$ dependence of non-perturbative solutions in the strong coupling phase of the gauge theory is reexamined based on the interrelation between the inversion method and the Schwinger-Dyson equation approach. Especially we point out that the apparent discrepancy on the value of the critical coupling in QED will be resolved by taking into account the higher order corrections which inevitably lead to the flavor-dependence. In the quenched QED, we conclude that the gauge-independent critical point $\alpha_c=2\pi/3$ obtained by the inversion method to the lowest order will be reduced to the result $\alpha_c=\pi/3$ of the Schwinger-Dyson equation in the infinite order limit, but its convergence is quite slow. This is shown by adding the chiral-invariant four-fermion interaction. 
  For $2D$ string theory, the perturbative $S$-matrices are not well-defined due to a zero mode divergence. Although there exist formal procedures to make the integral convergent, their physical meanings are unclear. We describe a method to obtain finite $S$-matrices physically to justify the formal schemes. The scheme uses asymptotic states by wave packets which fall faster than exponentials. It is shown that the scheme gives well-defined $S$-matrices and justifies the formal shifted Virasoro-Shapiro amplitude for simple processes. The tree-level unitarity for these processes is also shown. We point out a problem in this scheme. 
  Classical bosonic open string models in fourdimensional Minkowski spacetime are discussed. A special attention is paid to the choice of edge conditions, which can follow consistently from the action principle. We consider lagrangians that can depend on second order derivatives of worldsheet coordinates. A revised interpretation of the variational problem for such theories is given. We derive a general form of a boundary term that can be added to the open string action to control edge conditions and modify conservation laws. An extended boundary problem for minimal surfaces is examined. Following the treatment of this model in the geometric approach, we obtain that classical open string states correspond to solutions of a complex Liouville equation. In contrast to the Nambu-Goto case, the Liouville potential is finite and constant at worldsheet boundaries. The phase part of the potential defines topological sectors of solutions. 
  We show that the Classical Constraint Algebra of a Parametrized Relativistic Gauge System induces a natural structure of Conformal Foliation on a Transversal Gauge. Using the theory of Conformal Foliations, we provide a natural Factor Ordering for the Quantum Operators associated to the Canonical Quantization of such Gauge System. 
  We give a generalization of the Reshetikhin-Turaev functor for tangles to get a combinatorial formula for the universal Vassiliev-Kontsevich invariant of framed oriented links which is coincident with the Kontsevich integral. The universal Vassiliev-Kontsevich invariant is constructed using the Drinfeld associator. We prove the uniqueness of the Drinfeld associator. As a corollary one gets the rationality of the Kontsevich integral. Many properties of the universal Vassiliev-Kontsevich invariant are established. Connections to quantum group invariants and to multiple zeta values are discussed. 
  We study quantum intergrable systems of interacting particles from the point of view, proposed in our previous paper. We obtain Calogero-Moser and Sutherland systems as well their Ruijsenaars relativistic generalization by a Hamiltonian reduction of integrable systems on the cotangent bundles over semi-simple Lie algebras, their affine algebras and central extensions of loop groups respectively. The corresponding 2d field theories form a tower of deformations. The top of this tower is gauged G/G WZW model on a cylinder with inserted Wilson line in appropriate representation. 
  These notes are based on a lecture given by S. L. Woronowicz at the Institute of Mathematics, Polish Academy of Sciences. 
  We introduce the notion of locally trivial quantum principal bundles. The base space and total space are compact quantum spaces (unital $C^{\star}$-algebras), the structure group is a compact matrix quantum group. We prove that a quantum bundle admits sections if and only if it is trivial. Using a quantum version of \v{C}ech cocycles, we obtain a reconstruction theorem for quantum principal bundles. The classification of bundles over a given quantum space as a base space is reduced to the corresponding problem, but with an ordinary classical group playing the role of structure group. Some explicit examples are considered. 
  In $N+1$ dimensions, false vacuum decay at zero temperature is dominated by the $O(N+1)$ symmetric instanton, a sphere of radius $R_0$, whereas at temperatures $T>>R_0^{-1}$, the decay is dominated by a `cylindrical' (static) $O(N)$ symmetric instanton. We study the transition between these two regimes in the thin wall approximation. Taking an $O(N)$ symmetric ansatz for the instantons, we show that for $N=2$ and $N=3$ new periodic solutions exist in a finite temperature range in the neighborhood of $T\sim R_0^{-1}$. However, these solutions have higher action than the spherical or the cylindrical one. This suggests that there is a sudden change (a first order transition) in the derivative of the nucleation rate at a certain temperature $T_*$, when the static instanton starts dominating. For $N=1$, on the other hand, the new solutions are dominant and they smoothly interpolate between the zero temperature instanton and the high temperature one, so the transition is of second order. The determinantal prefactors corresponding to the `cylindrical' instantons are discussed, and it is pointed out that the entropic contributions from massless excitations corresponding to deformations of the domain wall give rise to an exponential enhancement of the nucleation rate for $T>>R_0^{-1}$. 
  We show that elliptic Calogero-Moser system and its Lax operator found by Krichever can be obtained by Hamiltonian reduction from the integrable Hamiltonian system on the cotangent bundle to the central extension of the algebra of SL(N,C) currents.Elliptic deformation of Yang-Mills theory is presented. 
  In this paper we solve two matrix models, using standard and new techniques. The two models are represented by special form of antisymmetric matrices and are classified in the DIII generator ensemble. It is shown that, in the double scaling limit, their free energy has the same behavior as previous models describing oriented and unoriented surfaces. We also found an additional solution for the first model. 
  String backgrounds are described as purely geometric objects related to moduli spaces of Riemann surfaces, in the spirit of Segal's definition of a conformal field theory. Relations with conformal field theory, topological field theory and topological gravity are studied. For each field theory, an algebraic counterpart, the (homotopy) algebra satisfied by the tree level correlators, is constructed. 
  We {\em derive} the exact configuration space path integral, together with the way how to evaluate it, from the Hamiltonian approach for any quantum mechanical system in flat spacetime whose Hamiltonian has at most two momentum operators. Starting from a given, covariant or non-covariant, Hamiltonian, we go from the time-discretized path integral to the continuum path integral by introducing Fourier modes. We prove that the limit $N \rightarrow \infty$ for the terms in the perturbation expansion (``Feynman graphs'') exists, by demonstrating that the series involved are uniformly convergent. {\em All} terms in the expansion of the exponent in $<x| \exp (- \Delta t \hat{H} / \hbar) |y>$ contribute to the propagator (even at order $\Delta t$!). However, in the time-discretized path integral the only effect of the terms with $\hat{H}^2$ and higher is to cancel terms which naively seem to vanish for $N \rightarrow \infty$, but, in fact, are nonvanishing. The final result is that the naive correspondence between the Hamiltonian and the Lagrangian approach is correct, after all. We explicitly work through the example of a point particle coupled to electromagnetism. We compute the propagator to order $(\Delta t)^2$ both with the Hamiltonian and the path integral approach and find agreement. 
  We formulate sigma-model duality transformations in terms of spin connection. This allows to investigate the symmetry of the string action including higher order $\alpha'$ corrections. An important feature of the new duality transformations is a simple homogeneous transformation rule of the spin connection (with torsion) and specifically adjusted transformation of the Yang-Mills field. We have found that under certain conditions this duality is a symmetry of the full effective string action in the target space, free of $\alpha'$ corrections. We demonstrate how the exact duality generates new fundamental string solutions from supersymmetric string waves. 
  The present paper is devoted to various objects of the infinite dimensional W-geometry of a second quantized free string. Our purpose is to include the W-symmetries into the general infinite dimensional geometrical picture related to the quantum field theory of strings, which was described in the first part of the paper (Algebras Groups Geom.11(1994)[to appear]). It is done by the change of the Lie algebra of all infinitesimal reparametrizations of a string world-sheet on the Lie quasi(pseudo)algebra of classical W-transformations (Gervais-Matsuo quasi(pseudo)algebra) as well as of the Virasoro algebra on the central extended enlarged Gervais-Matsuo quasi(pseudo)algebra. A way to obtain W-algebras from classical W-transformations (i.e. Gervais-Matsuo quasi (pseudo)algebra) is proposed. The relation of Gervais-Matsuo differential W-geometry to the Batalin-Weinstein-Karasev-Maslov approach to nonlinear Poisson brackets as well as to L.V.Sabinin program of "nonlinear geometric algebra" are mentioned. 
  We postulate a new type of operator algebra with a non-abelian extension. This algebra generalizes the Kac--Moody algebra in string theory and the Mickelsson--Faddeev algebra in three dimensions to higher-dimensional extended objects ($p$-branes). We then construct new BRST operators, covariant derivatives and curvature tensors in the higher-dimensional generalization of loop space. 
  Higher spin extensions of nonabelian gauge symmetries for both free fermionic model and WZNW model are considered on a classical level. It is characteristic property of the WZNW model that the higher spin currents which correspond to linear realized higher spin extended affine algebra do not form an invariant space. An extended invariant current space is obtained which allows us to gauge the symmetry. 
  We compute the elliptic genus for arbitrary two dimensional $N=2$ Landau-Ginzburg orbifolds. This is used to search for possible mirror pairs of such models. We show that if two Landau-Ginzburg models are conjugate to each other in a certain sense, then to every orbifold of the first theory corresponds an orbifold of the second theory with the same elliptic genus (up to a sign) and with the roles of the chiral and anti-chiral rings interchanged. These orbifolds thus constitute a possible mirror pair. Furthermore, new pairs of conjugate models may be obtained by taking the product of old ones. We also give a sufficient (and possibly necessary) condition for two models to be conjugate, and show that it is satisfied by the mirror pairs proposed by one of the authors and~H\"ubsch. 
  We construct a topological Landau-Ginzburg formulation of the two-dimensional string at the self-dual radius. The model is an analytic continuation of the $A_{k+1}$ minimal model to $k=-3$. We compute the superpotential and calculate tachyon correlators in the Landau-Ginzburg framework. The results are in complete agreement with matrix model calculations. We identify the momentum one tachyon as the puncture operator, non-negative momentum tachyons as primary fields, and negative momentum ones as descendants. The model thus has an infinite number of primary fields, and the topological metric vanishes on the small phase space when restricted to these. We find a parity invariant multi-contact algebra with irreducible contact terms of arbitrarily large number of fields. The formulation of this Landau-Ginzburg description in terms of period integrals coincides with the genus zero $W_{1+\infty}$ identities of two-dimensional string theory. We study the underlying Toda lattice integrable hierarchy in the Lax formulation and find that the Landau-Ginzburg superpotential coincides with a derivative of the Baker-Akhiezer wave function in the dispersionless limit. This establishes a connection between the topological and integrable structures. Guided by this connection we derive relations formally analogous to the string equation. 
  The effective action for 2d-gravity with manifest area-preserving invariance is obtained in the flat and in the general gravitational background. The cocyclic properties of the last action are proved, and generalizations on higher dimensions are discussed. 
  The geometric action on a certain orbit of the group of the area-preserving diffeomorphisms is considered, and it is shown, that it coincides with a special reduction of the three-dimensional Chern-Simons theory, under which group and space coordinates are identified. 
  Using a simple model, a new sphaleron solution which incorporates finite fermionic density effects is obtained. The main result is that the height of the potential barrier (sphaleron energy) decreases as the fermion density increases. This suggests that the rate of sphaleron-induced transitions increases when the fermionic density increases. However the rate increase is not expected to change significantly the predictions from the standard sphaleron-induced baryogenesis scenarios. 
  Some aspects and applications of $ \sigma$-models in particle and condensed matter physics are briefly reviewed. 
  Zero modes of first class secondary constraints in the two-dimensional electrodynamics and the four-dimensional SU(2) Yang-Mills theory are considered by the method of reduced phase space quantization in the context of the problem of a stable vacuum. We compare the description of these modes in the Dirac extended method and reveal their connection with the topological structure of the gauge symmetry group. Within the framework of the "reduced" quantization we construct a new global realization of the homotopy group representation in the Yang-Mills theory, where the role of the stable vacuum with a finite action plays the Prasad-Sommerfield solution. 
  The geodesic motion of pseudo-classical spinning particles in the Euclidean Taub-NUT space is analysed. The generalized Killing equations for spinning space are investigated and the constants of motion are derived in terms of the solutions of these equations. A simple exact solution, corresponding to trajectories lying on a cone, is given. 
  A complete list of all transitive symplectic manifolds of the Poincar\'e and Galilei group in 1+2 dimensions is given. 
  By a mapping to the bosonic string theory, we present an exact solution to the O(26) sigma model coupled to 2-D quantum gravity. In particular, we obtain the exact gravitational dressing to the various matter operators classified by the irreducible representations of O(26). We also derive the exact form of the gravitationally modified beta function for the original coupling constant $e^2$. The relation between our exact solution and the asymptotic solution given in ref[3] is discussed in various aspects. 
  Fermions on a cylinder coupled to gravity and gauge fields are examined by studying the geometric action associated with the symmetries of such a system. The gauge coupling constant is shown to be constrained and the effect of gravity on the masses is discussed. Furthermore, we introduce a new gravitational theory which couples to the fermions by promoting the coadjoint vector of the diffeomorphism sector to a dynamical variable. This system, together with the gauge sector is shown to be equivalent to a one dimensional system. 
  We study the scattering of two Skyrmions at low energy and large separation. We use the method proposed by Manton for truncating the degrees of freedom of the system from infinite to a manageable finite number. This corresponds to identifying the manifold consisting of the union of the low energy critical points of the potential along with the gradient flow curves joining these together and by positing that the dynamics is restricted here. The kinetic energy provides an induced metric on this manifold while restricting the full potential energy to the manifold defines a potential. The low energy dynamics is now constrained to these finite number of degrees of freedom. For large separation of the two Skyrmions the manifold is parametrised by the variables of the product ansatz. We find the interaction between two Skyrmions coming from the induced metric, which was independently found by Schroers. We find that the static potential is actually negligible in comparison to this interaction. Thus to lowest order, at large separation, the dynamics reduces to geodesic motion on the manifold. We consider the scattering to first order in the interaction using the perturbative method of Lagrange and find that the dynamics in the no spin or charge exchange sector reduces to the Kepler problem. 
  Minor corrections made and several references changed. 
  It has been discussed earlier that ( weak quasi-) quantum groups allow for conventional interpretation as internal symmetries in local quantum theory. From general arguments and explicit examples their consistency with (braid-) statistics and locality was established. This work addresses to the reconstruction of quantum symmetries and algebras of field operators. For every algebra $\A$ of observables satisfying certain standard assumptions, an appropriate quantum symmetry is found. Field operators are obtained which act on a positive definite Hilbert space of states and transform covariantly under the quantum symmetry. As a substitute for Bose/Fermi (anti-) commutation relations, these fields are demonstrated to obey local braid relation. 
  This talk presents the review of forgotten but attractive formalism proposed by Joos and Weinberg in the sixties for description of high-spin particles. Problems raised in the recent works [Ahluwalia {\it et al.}] are discussed. New results obtained by the author in his preceding papers ["Hadronic J.", 1993, v. 16, No. 5, pp. 423-428; No. 6, pp. 459-467; Preprints IFUNAM FT-93-19, 24, 35] are reported. In {\it Appendix}, bibliography of publications related with mentioned $2(2S+1)$- component formalism is presented. 
  Two dimensional QCD coupled to fermions in the adjoint representation of the gauge group $SU(N)$, a useful toy model of QCD strings, is supersymmetric for a certain ratio of quark mass and gauge coupling constant. Here we study the theory in the vicinity of the supersymmetric point; in particular we exhibit the algebraic structure of the model and show that the mass splittings as one moves away from the supersymmetric point obey a universal relation of the form ${M_i}^2(B)-{M_i}^2(F)=M_i\delta m+O(\delta m^3)$. We discuss the connection of this relation to string and quark model expectations and verify it numerically for large $N$. At least for low lying states the $O(\delta m^3)$ corrections are extremely small. We also discuss a natural generalization of QCD$_2$ with an infinite number of couplings, which preserves SUSY. This leads to a Landau -- Ginzburg description of the theory, and may be useful for defining a scaling limit in which smooth worldsheets appear. 
  We prove inequality (1) for the modified Steiner functional A(M), which extends the notion of the integral of mean curvature for convex surfaces.We also establish an exression for A(M) in terms of an integral over all hyperplanes intersecting the polyhedralral surface M. 
  It is shown that in a formulation of Yang-Mills theory in two dimensions in terms of $A=if^{-1}\pa f$, $\bar A=i\bar f\bpa\bar f^{-1}$ with $f(z,\bar z)$, $\bar f(z,\bar z)\in[SU(N_C)]^c$ the complexification of $SU(N_C)$ , reveals certain subtleties. ``Physical" massive color singlet states seem to exist. When coupled to $N_F$ quarks the coupling constant is renormalized in such a way that it vanishes for the pure Yang- Mills case. This renders the above states massless and unphysical. In the abelian case, on the other hand, the known results of the Schwinger model are reproduced with no need of such a renormalization.   The massless $QCD_2$ theory is analyzed in similar terms and peculiar massive states appear, with a mass of $e_c\sqrt {N_F \over 2\pi}$. 
  This paper is devoted to an interaction of 2 objects: the 1st of them is octonions, the classical structure of pure mathematics, the 2nd one is Mobilevision, the recently developped technique of computer graphics. Namely, it is shown that the binocular Mobilevision maybe elaborated by use of the octonionic colour space - the 7-dimensional extension of the classical one, which includes a strange overcolour besides two triples of ordinary ones (blue,green, red for left and right eyes).  Contents.   I. Interpretational geometry, anomalous virtual realities, quantum projective field theory and Mobilevision:(1.1. Interpretational geometry; 1.2. Anomalous virtual realities; 1.3. Colours in anomalous virtual realities; 1.4. Quantum projective field theory; 1.5. Mobilevision).   II. Quantum conformal and q_R-conformal field theories, an infinite dimensional quantum group and quantum field analogs of Euler-Arnold top:(2.1. Quantum conformal field theory; 2.2. Lobachevskii algebra, the   quantization of the Lobachevskii plane; 2.3. Quantum q_R-conformal field theory; 2.4. An infinite dimensional quantum group; 2.5. Quantum-field Euler-Arnold top and Virasoro master equation).   III. Octonionic colour space and binocular Mobilevision:(3.1. Quaternionic description of ordinary colour space; 3.2. Octonionic colour space and binocular Mobilevision). 
  We present a left-right symmetric model with gauge group $U(@)_L\times U(@)_R$ in the Connes-Lott non-commutative frame work. Its gauge symmetry is broken spontaneously, parity remains unbroken. 
  A generalisation of the classical Calogero-Moser model obtained by coupling it to the Gaudin model is considered. The recently found classical dynamical r-matrix [E. Billey, J. Avan and O. Babelon, PAR LPTHE 93-55] for the Euler-Calogero-Moser model is used to separate variables for this generalised Calogero-Moser model in the case in which there are two Calogero-Moser particles. The model is then canonically quantised and the same classical r-matrix is employed to separate variables in the Schr\"odinger equations. 
  The aim of the paper is to build a universal R-matrix for the multiparameter deformation of any reductive Lie algebra. Such deformations, formulated in the recent past by Truini and Varadarajan, have the property of universality in a certain class and are shown by the present paper to be quasitriangular Hopf algebras. In order to build the R-matrix we exploit the twisting method for introducing new parameters as well as for making the transition to the reductive case. The physical motivation behind this construction is in the theory of integrable models: we intend to use such R-matrix for building the associated quantum R-matrix and Lax operators with spectral as well as color parameters - the color parameters being provided by the eigenvalues of the central generators of the reductive Lie algebra in a given representation. In this letter we present only the explicit form of the universal R-matrix of reductive Lie algebras and postpone its application to a forthcoming paper. 
  In this paper we give a new derivation of the quark-antiquark potential in the Wilson loop context. This makes more explicit the approximations involved and enables an immediate extension to the three-quark case. In the $q\overline{q}$ case we find the same semirelativistic potential obtained in preceding papers but for a question of ordering. In the $3q$ case we find a spin dependent potential identical to that already derived in the literature from the ad hoc and non correct assumption of scalar confinement. Furthermore we obtain the correct form of the spin independent potential up to the $1/m^2$ order. 
  The colour dipole cross section is the principal quantity in the lightcone $s$-channel description of the diffractive scattering. Recently we have shown that the dipole cross section satisfies the generalized BFKL equation. In this paper we discuss properties and solutions of our generalized BFKL equation with allowance for the finite gluon correlation radius $R_{c}$. The latter is introduced in a gauge invariant manner. We present estimates of the intercept of the pomeron and find the asymptotic form of the dipole cross section. 
  We prove that $\be$-function of the gauge coupling in $2+1D$ gauge theory coupled to any renormalizable system of spinor and scalar fields is zero. This result holds both when the gauge field action is the Chern-Simons action and when it is the topologically massive action. 
  When a hydrogen-like atom is treated as a two dimensional system whose configuration space is multiply connected, then in order to obtain the same energy spectrum as in the Bohr model the angular momentum must be half-integral. 
  We present a formal but simple calculational scheme to relate the expectation value of Wilson loops in Chern-Simons theory to the Jones polynomial. We consider the exponential of the generator of homotopy transformations which produces the finite loop deformations that define the crossing change formulas of knot polynomials. Applying this operator to the expectation value of Wilson loops for an unspecified measure we find a set of conditions on the measure and the regularization such that the Jones polynomial is obtained. 
  The phase structure of the $d=3$ Nambu-Jona-Lasinio model in curved spacetime is considered to leading order in the $1/N$--expansion and in the linear curvature approximation. The possibility of a curvature-induced first-order phase transition is investigated numerically. The dynamically generated fermionic mass is calculated for some values of the curvature. 
  The renormalization group (RG) is used in order to obtain the RG improved effective potential in curved spacetime. This potential is explicitly calculated for the Yukawa model and for scalar electrodynamics, i.e. theories with several (namely, more than one) mass scales, in a space of constant curvature. Using the $\lambda \varphi^4$-theory on a general curved spacetime as an example, we show how it is possible to find the RG improved effective Lagrangian in curved spacetime. As specific applications, we discuss the possibility of curvature induced phase transitions in the Yukawa model and the effective equations (back-reaction problem) for the $\lambda \varphi^4$-theory on a De Sitter background. 
  We present a new form of the multi-boson reduction of KP hierarchy with Lax operator written in terms of boson fields abelianizing the second Hamiltonian structure. This extends the classical Miura transformation and the Kupershmidt-Wilson theorem from the (m)KdV to the KP case. A remarkable relationship is uncovered between the higher Hamiltonian structures and the corresponding Miura transformations of KP hierarchy, on one hand, and the discrete integrable models living on {\em refinements} of the original lattice connected with the underlying multi-matrix models, on the other hand. For the second KP Hamiltonian structure, worked out in details, this amounts to finding a series of representations of the nonlinear $\hWinf$ algebra in terms of arbitrary finite number of canonical pairs of free fields. 
  We argue that the spectrum of the QCD Dirac operator near zero virtuality can be described by random matrix theory. As in the case of classical random matrix ensembles of Dyson we have three distinct classes: the chiral orthogonal ensemble (chGOE), the chiral unitary ensemble (chGUE) and the chiral symplectic ensemble (chGSE). They correspond to gauge groups $SU(2)$ in the fundamental representation, $SU(N_c), N_c \ge 3$ in the fundamental representation, and gauge groups for all $N_c$ in the adjoint representation, respectively. The joint probability density reproduces Leutwyler-Smilga sum rules. 
  We review two different methods of calculating Witten's invariant: a stationary phase approximation and a surgery calculus. We give a detailed description of the 1-loop approximation formula for Witten's invariant and of the technics involved in deriving its exact value through a surgery construction of a manifold. Finally we compare the formulas produced by both methods for a 3-dimensional sphere S^3 and a lens space L(p,1). 
    We use the Chern-Simons quantum field theory in order to prove a recently conjectured limitation on the 1/K expansion of the Jones polynomial of a knot and its relation to the Alexander polynomial. This limitation allows us to derive a surgery formula for the loop corrections to the contribution of the trivial connection to Witten's invariant. The 2-loop part of this formula coincides with Walker's surgery formula for Casson-Walker invariant. This proves a conjecture that Casson-Walker invariant is a 2-loop correction to the trivial connection contribution to Witten's invariant of a rational homology sphere. A contribution of the trivial connection to Witten's invariant of a manifold with nontrivial rational homology is calculated for the case of Seifert manifolds. 
  We study two-dimensional gauge theories with fundamental fermions and a general first order gauge-field Lagrangian. For the case of U(1) we show how standard bosonization of the Schwinger model generalizes to give mesons interacting through a general Landau-Ginzburg potential. We then show how for a subclass of SU(N) theories, 't Hooft's solution of large N two-dimensional QCD can be generalized in a consistent and natural manner. We finally point out the possible relevance of studying these theories to the string formulation of two-dimensional QCD as well as to understanding QCD in higher dimensions. 
  We derive general form of finite-dimensional approximations of path integrals for both bosonic and fermionic canonical systems in terms of symbols of operators determined by operator ordering. We argue that for a system with a given quantum Hamiltonian such approximations are independent of the type of symbols up to terms of $O(\epsilon)$, where $\epsilon$ is infinitesimal time interval determining the accuracy of the approximations. A new class of such approximations is found for both c-number and Grassmannian dynamical variables. The actions determined by the approximations are non-local and have no classical continuum limit except the cases of $pq$- and $qp$-ordeeing. As an explicit example the fermionic oscillator is considered in detail. 
  We examine whether a minimal string model possessing the same massless spectra as the MSSM can be obtained from $Z_4$, $Z_6$ and $Z_8$ orbifold constructions. Using an anomaly cancellation condition of the target space duality symmetry, we derive allowable values of a level $k_1$ of U(1)$_Y$ for the minimal string model on the orbifolds through computer analyses. We investigate threshold corrections of the gauge coupling constants of SU(3),   SU(2) and U(1)$_Y$ and examine consistencies of the model with the LEP experiments. It is found that $Z_4$ and $Z_8$-II can not derive the minimal string model but $Z_6$-I, $Z_6$-II and $Z_8$-I are possible to derive it with $13/12   \leq k_1\leq 41/30$, $16/15\leq k_1\leq 17/12$ and $1\leq k_1\leq 41/21$ respectively. The minimum values of the moduli on unrotated planes are estimated within the ranges of the levels. 
  The non-relativistic conformal symmetry found by Jackiw and Pi for the coupled Chern-Simons and gauged nonlinear Schr\"odinger equations in the plane is derived in a non-relativistic Kaluza-Klein framework. 
  A lattice version of quantum nonlinear Schrodinger (NLS) equation is considered, which has significantly simple form and fullfils most of the criteria desirable for such lattice variants of field models. Unlike most of the known lattice NLS, the present model belongs to a class which does not exhibit the usual symmetry properties. However this lack of symmetry itself seems to be responsible for the remarkable simplification of the relevant objects in the theory, such as the Lax operator, the   Hamiltonian and other commuting conserved quantities as well as their spectrum.   The model allows exact quantum solution through algebraic Bethe ansatz and also a straightforward and natural generalisation to the vector case, giving thus a new exact lattice version of the vector NLS model. A deformation representing a new quantum integrable system involving Tamm-Dancoff like $q$-boson operators is constructed. 
  This article is devoted to some interesting geometric and informatic interpretations of peculiarities of 2D quantum field theory, which become re- vealed after its visualization. Contents.   I. Geometry of Mobilevision:   1.1. Interpretational geometry and anomalous virtual realities;   1.2. Quantum projective field theory and Mobilevision;   1.3. Quantum conformal and q_R conformal field theories; quantum-field analogs of Euler-Arnold top;   1.4. Organizing MV cyberspace;   1.5. Non-Alexandrian geometry of Mobilevision.   II. Informatics of Mobilevision:   2.1. Information transmission via anomalous virtual realities: AVR-photodosy;   2.2. Information transmission via intentional anomalous virtual realities: IAVR-teleaesthesy. 
  We calculate the Riemann curvature tensor and sectional curvature for the Lie group of volume-preserving diffeomorphisms of the Klein bottle and projective plane. In particular, we investigate the sign of the sectional curvature, and find a possible disagreement with a theorem of Lukatskii. We suggest an amendment to this theorem. 
  It is pointed out that string-loop modifications of the low-energy matter couplings of the dilaton may provide a mechanism for fixing the vacuum expectation value of a massless dilaton in a way which is naturally compatible with existing experimental data. Under a certain assumption of universality of the dilaton coupling functions , the cosmological evolution of the graviton-dilaton-matter system is shown to drive the dilaton towards values where it decouples from matter (``Least Coupling Principle"). Quantitative estimates are given of the residual strength, at the present cosmological epoch, of the coupling to matter of the dilaton. The existence of a weakly coupled massless dilaton entails a large spectrum of small, but non-zero, observable deviations from general relativity. In particular, our results provide a new motivation for trying to improve by several orders of magnitude the various experimental tests of Einstein's Equivalence Principle (universality of free fall, constancy of the constants,\dots). 
  In this paper the entropy of an eternal Schwarzschild black hole is studied in the limit of infinite black hole mass. The problem is addressed from the point of view of both canonical quantum gravity and superstring theory. The entropy per unit area of a free scalar field propagating in a fixed black hole background is shown to be quadratically divergent near the horizon. It is shown that such quantum corrections to the entropy per unit area are equivalent to the quantum corrections to the gravitational coupling. Unlike field theory, superstring theory provides a set of identifiable configurations which give rise to the classical contribution to the entropy per unit area. These configurations can be understood as open superstrings with both ends attached to the horizon. The entropy per unit area is shown to be finite to all orders in superstring perturbation theory. The importance of these conclusions to the resolution of the problem of black hole information loss is reiterated. 
  It is shown that in the Einstein-Yang-Mills (EYM) theory, as well as in the pure flat space Yang-Mills (YM) theory, there always exists an opportunity to pass over the potential barrier separating homotopically distinct vacuum sectors, because the barrier height may be arbitrarily small. However, at low energies all the overbarrier histories are suppressed by the destructive interference. In the pure YM theory the situation remains the same for any energies. In the EYM theory on the other hand, when the energy is large and exceeds the ground state EYM sphaleron mass, the constructive interference occurs instead. This means that in the extreme high energy limit the exponential suppression of the fermion number violation in pure YM theory is removed due to gravitational effects. 
  We show that a geometrical notion of entropy, definable in flat space, governs the first quantum correction to the Bekenstein-Hawking black hole entropy. We describe two methods for calculating this entropy -- a straightforward Hamiltonian approach, and a less direct but more powerful Euclidean (heat kernel) method. The entropy diverges in quantum field theory in the absence of an ultraviolet cutoff. Various related finite quantities can be extracted with further work. We briefly discuss the corresponding question in string theory. 
  We try to draw lessons for higher dimensions from the string representations recently derived for large $N$ Yang-Mills theory by Gross and Taylor, Kostov, and others, and call attention to three characteristics that should be expected of a string theory precisely equivalent to a higher dimensional gauge theory: continuous world-sheets; strong coupling at short distances; and negative weights. To appear in the proceedings of the Strings '93 Berkeley conference. 
  We develop general techniques for computing the fundamental group of the configuration space of $n$ identical particles, possessing a generic internal structure, moving on a manifold $M$. This group generalizes the $n$-string braid group of $M$ which is the relevant object for structureless particles. In particular, we compute these generalized braid groups for particles with an internal spin degree of freedom on an arbitrary $M$. A study of their unitary representations allows us to determine the available spectrum of spin and statistics on $M$ in a certain class of quantum theories. One interesting result is that half-integral spin quantizations are obtained on certain manifolds having an obstruction to an ordinary spin structure. We also compare our results to corresponding ones for topological solitons in $O(d+1)$-invariant nonlinear sigma models in $(d+1)$-dimensions, generalizing recent studies in two spatial dimensions. Finally, we prove that there exists a general scalar quantum theory yielding half-integral spin for particles (or $O(d+1)$ solitons) on a closed, orientable manifold $M$ if and only if $M$ possesses a ${\rm spin}_c$ structure. 
  Symmetry transformations of the space-time fields of string theory are generated by certain similarity transformations of the stress-tensor of the associated conformal field theories. This observation is complicated by the fact that, as we explain, many of the operators we habitually use in string theory (such as vertices and currents) have ill-defined commutators. However, we identify an infinite-dimensional subalgebra whose commutators are not singular, and explicitly calculate its structure constants. This constitutes a subalgebra of the gauge symmetry of string theory, although it may act on auxiliary as well as propagating fields. We term this object a {\it weighted tensor algebra}, and, while it appears to be a distant cousin of the $W$-algebras, it has not, to our knowledge, appeared in the literature before. 
  The tetrahedron equation arises as a generalization of the famous Yang--Baxter equation to the 2+1-dimensional quantum field theory and the 3-dimensional statistical mechanics. Very little is still known about its solutions. Here a systematic method is described that does produce non-trivial solutions to the tetrahedron equation with spin-like variables on the links. The essence of the method is the use of the so-called tetrahedral Zamolodchikov algebras. 
  Basic representations of A_{2l}^(2) and D_{l+1}^(2) are studied. The weight vectors are represented in terms of Schur's $Q$-functions. The method to get the polynomial solutions to the reduced BKP hierarchies is shown to be equivalent to a certain rule in Maya game. 
  The exact eigenvalues of the infinite set of conserved charges on the multi-particle states in affine Toda theories are determined. This is done by constructing a free field realization of the Zamolodchikov-Faddeev algebra in which the conserved charges are realized as derivative operators. The resulting eigenvalues are renormalization group (RG) invariant, have the correct classical limit and pass checks in first order perturbation theory. For $n=1$ one recovers the (RG invariant form of the) quantum masses of Destri and DeVega. 
  It is shown that the inhomogeneous saddle points of scale invariant theories make the semiclassical expansion sensitive on the choice of non-renormalizable operators. In particular, the instanton fugacity and the beta function of the two dimensional non-linear sigma model depends on apparently non-renormalizable operators. This represents a non-perturbative breakdown of that concept of universality which is based on low dimensional operators. 
  We derive the expressions for $\psi$-functions and generic solutions of lattice principal chiral equations, lattice KP hierarchy and hierarchy including lattice N-wave type equations. $\tau$-function of $n$ free fermions plays fundamental role in this context. Miwa's coordinates in our case appear as the lattice parameters. 
  In this note we consider some consequences of quantum gravity on the process of black hole evaporation. In particular, we will explain the suggestion by 't Hooft that quantum gravitational interactions effectively exclude simultaneous measurements of the Hawking radiation and of the matter falling into the black hole. The complementarity of these measurements is supported by the fact that the commutators between the corresponding observables can be shown to grow uncontrollably large. The only assumption that is needed to obtain this result is that the creation and annihilation modes of the in-falling and out-going matter act in the same Hilbert space. We further illustrate this phenomenon in the context of two-dimensional dilaton gravity. 
  It is nearly twenty years that there exist computer programs to reduce products of Lie algebra irreps. This is a contribution in the field that uses a modern computer language (``C'') in a highly structured and object-oriented way. This gives the benefits of high portability, efficiency, and makes it easier to include the functions in user programs. Characteristic of this set of routines is the {\it all-dynamic} approach for the use of memory, so that the package only uses the memory resources as needed. 
  In this paper we show how the well-know local symmetries of Lagrangeans systems, and in particular the diffeomorphism invariance, emerge in the Hamiltonian formulation. We show that only the constraints which are linear in the momenta generate transformations which correspond to symmetries of the corresponding Lagrangean system. The nonlinear constraints (which we have, for instance, in gravity, supergravity and string theory) rather generate the dynamics of the corresponding Lagrangean system. Only in a very special combination with "trivial" transformations proportional to the equations of motion do they lead to symmetry transformations. We reveal the importance of these special "trivial" transformations for the interconnection theorems which relate the symmetries of a system with its dynamics. We prove these theorems for general Hamiltonian systems. We apply the developed formalism to concrete physically relevant systems and in particular those which are diffeomorphism invariant. The connection between the parameters of the symmetry transformations in the Hamiltonian- and Lagrangean formalisms is found. The possible applications of our results are discussed. 
  We examine the relation between two known classes of solutions of the sine--Gordon equation, which are expressed by theta functions on hyperelliptic Riemann surfaces. The first one is a consequence of the Fay's trisecant identity. The second class exists only for odd genus hyperelliptic Riemann surfaces which admit a fixed--point--free automorphism of order two. We show that these two classes of solutions coincide. The hyperelliptic surfaces corresponding to the second class appear to be double unramified coverings of the Riemann surfaces corresponding to the first class of solutions. We also discuss the soliton limits of these solutions. 
  It is shown that the principle of locality and noncommutative geometry can be connnected by a sheaf theoretical method. In this framework quantum spaces are introduced and examples in mathematical physics are given. With the language of quantum spaces noncommutative principal and vector bundles are defined and their properties are studied. Important constructions in the classical theory of principal fibre bundles like associated bundles and differential calculi are carried over to the quantum case. At the end $q$-deformed instanton models are introduced for every integral index. 
  We consider a wide class of two-dimensional models as gauge theories, Gross-Neveu model, $O(N)$ and $CP^{N-1}$-like models using a formalism based on the introduction of bilocal fields that permits to perform easily the large-N expansion of this set of models in a unified and general way. We mainly discuss the $SU(N)$ gauge field theory minimally coupled to fermionic plus bosonic matter in the fundamental representation, and we obtain within the path integral approach exact equations for the particle spectrum, also in presence of renormalizable polynomial potentials. Finally, we discuss the correspondence between this new approach and the one previously used in the context of the $O(N)$ vector models. 
  The dimension of the third homogeneous component of a matrix quantum bialgebra, determined by pair of quantum spaces, is calculated. The Poincar\'{e} series of some deformations of $GL(n)$ is calculated. A new deformation of $GL(3)$ with the correct dimension is given. 
  The classical soliton solution, quantized by means of suitable translational and rotational collective coordinates, is embedded into the one-particle irreductible representation of the Poincare group corresponding to a definite spin. It is shown, that within the conventional quasiclassical expansion such embedding leads to a set of nontrivial consistency conditions imposed on the classical solution. The validity of these relations is considered for a number of soliton models in 2+1- and 3+1-dimensions. 
  A set of integral relations for rotational and translational zero modes in the vicinity of the soliton solution are derived from the particle-like properties of the latter and verified for a number of models (solitons in 1+1-dimensions, skyrmeons in 2+1- and 3+1-dimensions, non-abelian monopoles). It is shown, that by consistent quantization within the framework of collective coordinates these relations ensure the correct diagonal expressions for the kinetic and centrifugal terms in the Hamiltonian in the lowest orders of the perturbation expansion. The connection between these properties and virial relations is also determined. 
  We obtain expressions for the current operator in the lowest Landau level (L.L.L.) field theory, where higher Landau level mixing due to various external and interparticle interactions is sytematically taken into account. We consider the current operators in the presence of electromagnetic interactions, both Coulomb and time-dependent, as well as local four-fermi interactions. The importance of Landau level mixing for long-range interactions is especially emphasized. We also calculate the edge-current for a finite sample. 
  In this note, we construct new representations of D=2, N=4 supersymmetry which do not involve chiral or twisted chiral multiplets. These multiplets may make it possible to circumvent no-go theorems about N=4 superspace formulations of WZWN-models. 
  We study the spectrum of the QCD Dirac operator near zero virtuality for $N_c =2$. According to a universality argument, it can be described by a random matrix theory with the chiral structure of QCD, but with $real$ matrix elements. Using results derived by Mehta and Mahoux and Nagao and Wadati, we are able to obtain an analytical result for the microscopic spectral density that in turn is the generating function for Leutwyler-Smilga type spectral sum rules. 
  We discuss a quantum \qa symmetry in Landau problem, which naturally arises due to the relation between the \qa and the group of magnetic translations. The last one is connected with the \w and area-preserving (symplectic) diffeomorphisms which are the canonical transformations in the two-dimenssional phase space. We shall discuss the hidden quantum symmetry in a $2+1$ gauge theory with the Chern-Simons term and in aQuantum Hall system which are both connected with the Landau problem. 
  We investigate the neighborhood of Topological Lattice Field Theories (TLFTs) in the parameter space of general lattice field theories in dimension $D\geq 2$, and discuss the phase structures associated to them. We first define a volume-dependent TLFT, and discuss its decomposition to a direct sum of irreducible TLFTs, which cannot be decomposed anymore. Using this decomposed form, we discuss phase structures and renormalization group flows of volume-dependent TLFTs. We find that TLFTs are on multiple first order phase transition points as well as on fixed points of the flow. The phase structures are controlled by the physical states on $(D-1)$-sphere of TLFTs. The flow agrees with the Nienhuis-Nauenberg criterion. We also discuss the neighborhood of a TLFT in general directions by a perturbative method, so-called cluster expansion. We investigate especially the $Z_p$ analogue of the Turaev-Viro model, and find that the TLFT is in general on a higher order discrete phase transition point. The phase structures depend on the topology of the base manifold and are controlled by the physical states on topologically non-trivial surfaces. We also discuss the correlation lengths of local fluctuations, and find long-range modes propagating along topological defects. Thus various discrete phase transitions are associated to TLFTs. 
  Upto ten crossing number, there are two knots, $9_{42}$ and $10_{71}$ whose chirality is not detected by any of the known polynomials, namely, Jones invariants and their two variable generalisations, HOMFLY and Kauffman invariants. We show that the generalised knot invariants, obtained through $SU(2)$ Chern-Simons topological field theory, which give the known polynomials as special cases, are indeed sensitive to the chirality of these knots. 
  We propose a new method which analyzes the dynamical triangulation from the viewpoint of the non-critical string field theory. By using the transfer matrix formalism, we construct the non-critical string field theory (including $c>1$ cases) at the discrete level. For pure quantum gravity, we succeed in taking the continuum limit and obtain the $c=0$ non-critical string field theory at the continuous level. We also study about the universality of the non-critical string field theory. 
  Contrary to the conventional view point of quantization that breaks the gauge symmetry, a gauge invariant formulation of quantum electrodynamics is proposed. Instead of fixing the gauge, some frame is chosen to yield the locally invariant fields. We show that all the formulations, such as the Coulomb, the axial, and the Lorentz gauges, can be constructed and that the explicit LSZ mapping connecting Heisenberg operators to those of the asymptotic fields is possible. We also make some comments on gauge transformations in quantized field theory. 
  We obtain new duality transformations relating some exact string backgrounds, by defining the nilpotent duality. We show that the ungauged $SL(2, R)$ WZW model transforms by its action into the three dimensional plane wave geometry. We also give the inverse transformation from the plane wave to the $SL(2, R)$ model and discuss the implications of the results. 
  We study the evolution under the renormalization group of the restrictions on the parameters of the standard model coming from Non-Commutative Geometry, namely $m_{top} = 2\,m_W$ and $m_{Higgs} = 3.14 \, m_W$. We adopt the point of view that these relations are to be interpreted as {\it tree level} constraints and, as such, can be implemented in a mass independent renormalization scheme only at a given energy scale $\mu_0$. We show that the physical predictions on the top and Higgs masses depend weakly on $\mu_0$. 
  The first quantization of the relativistic Brink-DiVecchia-Howe-Polyakov (BDHP) string in the range $1<d<25$ is considered. It is shown that using the Polyakov sum over bordered surfaces in the Feynman path integral quantization scheme one gets a consistent quantum mechanics of relativistic 1-dim extended objects in the range $1<d<25$. In particular the BDHP string propagator is exactly calculated for arbitrary initial and final string configurations and the Hilbert space of physical states of noncritical BDHP string is explicitly constructed. The resulting theory is equivalent to the Fairlie-Chodos-Thorn massive string model. In contrast to the conventional conformal field theory approach to noncritical string and random surfaces in the Euclidean target space the path integral formulation of the Fairlie-Chodos-Thorn string obtained in this paper does not rely on the principle of conformal invariance. Some consequences of this feature for constructing a consistent relativistic string theory based on the "splitting-joining" interaction are discussed.} 
  Explicit isomorphism is established between quasitriangular Hopf algebras studied recently by O.Ogievetsky and by the present author. 
  We study black hole formation in a model of two dimensional dilaton gravity and 24 massless scalar fields with a boundary. We find the most general boundary condition consistent with perfect reflection of matter and the constraints. We show that in the semiclassical approximation and for the generic value of the parameter which characterizes the boundary conditions, the boundary starts receeding to infinity at the speed of light whenever the total energy of the incoming matter flux exceeds a certain critical value. This is also the critical energy which marks the onset of black hole formation. We then compute the quantum fluctuations of the boundary and of the rescaled scalar curvature and show that as soon as the incoming energy exceeds this critical value, an asymptotic observer using normal time resolutions will always measure large fluctuations of space-time near the horizon, even though the freely falling observer does not. This is an aspect of black hole complementarity relating directly the quantum gravity effects. 
  We compute the quantum cohomology rings of the partial flag manifolds F_{n_1\cdots n_k}=U(n)/(U(n_1)\times \cdots \times U(n_k)). The inductive computation uses the idea of Givental and Kim. Also we define a notion of the vertical quantum cohomology ring of the algebraic bundle. For the flag bundle F_{n_1\cdots n_k}(E) associated with the vector bundle E this ring is found. 
  New proves of decoupling of massive fields in several quantum field theories are derived in the effective Lagrangian approach based on Wilson renormalization group. In the most interesting case of gauge theories with spontaneous symmetry breaking, the approach, combined with the quantum action principle, leads to a rather simple proof to all orders. 
  We show that bosonization in two dimensions can be derived as a special case of the duality transformations that have recently been used to good effect in string theory. This allows the construction of the bosonic counterpart of any fermionic theory simply by `following your nose' using the standard duality transformation rules. We work through the bosonization of the Dirac fermion, the massive and massless Thirring models, and a fermion on a cylindrical spacetime as illustrative examples. 
  In renormalizable theories, we define equal-time commutators (ETC'S) in terms of the equal-time limit and investigate its convergence in perturbation theory. We find that the equal-time limit vanishes for amplitudes with the effective dimension $d_{\em eff} \leq -2$ and is finite for those with $d_{\em eff} =-1$ but without nontrivial discontinuity. Otherwise we expect divergent equal-time limits. We also find that, if the ETC's involved in verifying an Jacobi identity exist, the identity is satisfied.   Under these circumstances, we show in the Yang-Mills theory that the ETC of the $0$ component of the BRST current with each other vanishes to all orders in perturbation theory if the theory is free from the chiral anomaly, from which we conclude that $[\, Q\,,\,Q\,]=0$, where $Q$ is the BRST charge. For the case that the chiral anomaly is not canceled, we use various broken Ward identities to show that $[\, Q\,,\,Q\,]$ is finite and $[\,Q\,,\,[\, Q\,,\,Q]\,]$ vanishes at the one-loop level and that they start to diverge at the two-loop level unless there is some unexpected cancellation mechanism that improves the degree of convergence. 
  The loop equations in the $U(N)$ lattice gauge theory are represented in the form of constraints imposed on a generating functional for the Wilson loop correlators. These constraints form a closed algebra with respect to commutation. This algebra generalizes the Virasoro one, which is known to appear in one-matrix models in the same way. The realization of this algebra in terms of the infinitesimal changes of generators of the loop space is given. The representations on the tensor fields on the loop space, generalizing the integer spin conformal fields, are constructed. The structure constants of the algebra under consideration being independent of the coupling constants, almost all the results are valid in the continuum. 
  In the process of investigating classical realizations of W_3 in terms of free bosons, Romans unveiled a relation to finite-dimensional Jordan algebras with a cubic norm. These algebras have been classified and consist of an infinite series (yielding the ``generic'' realizations) and four sporadic algebras associated to the real division algebras (which yield the ``magical'' realizations). The generic realizations were shown by Romans to quantize, who left the problem of the quantization of the magical realizations open. In later work, Mohammedi showed that the first two magical realizations did not survive quantization. In this note we close the problem by showing that neither do the other two magical realizations. 
  A consistent Euclidean semi classical calculation is given for the superscattering operator $\$ $ in the RST model for states with a constant flux of energy. The $\$ $ operator is CPT invariant. There is no loss of quantum coherence when the energy flux is less than a critical rate and complete loss when the energy flux is critical. 
  We investigate the phase space of a typical model of 1+1 dimensional gravity (Jackiw-Teitelboim model with cylindrical topology) using its reformulation as a non abelian gauge theory based on the sl(2,R) algebra. Modifying the conventional approach we argue that one should take the universal covering of SL(2,R) rather than PSL(2,R) as the gauge group of the theory. We discuss the consequences for the quantization of the model and find that the spectrum of the Dirac observables is sensible to this modification. Our analysis further provides an example for a gravity theory where the standard Hamiltonian formulation identifies gravitationally inequivalent solutions. 
  High energy string scattering at fixed momentum transfer, known to be dominated by Regge trajectory exchange, is interpreted by identifying families of string states which induce each type of trajectory exchange. These include the usual leading trajectory $\alpha(t)=\alpha^\prime t+1$ and its daughters as well as the ``sister'' trajectories $\alpha_m(t)=\alpha(t)/m-(m-1)/2$ and their daughters. The contribution of the sister $\alpha_m$ to high energy scattering is dominated by string excitations in the $m^{th}$ mode. Thus, at large $-t$, string scattering is dominated by wee partons, consistently with a picture of string as an infinitely composite system of ``constituents'' which carry zero energy and momentum. 
  We study the quantum matrix algebra $R_{21}x_1x_2=x_2x_1 R$ and for the standard $2\times 2$ case propose it for the co-ordinates of $q$-deformed Euclidean space. The algebra in this simplest case is isomorphic to the usual quantum matrices $M_q(2)$ but in a form which is naturally covariant under the Euclidean rotations $SU_q(2)\otimes SU_q(2)$. We also introduce a quantum Wick rotation that twists this system precisely into the approach to $q$-Minkowski space based on braided-matrices and their associated spinorial $q$-Lorentz group. 
  The Abelian Higgs model with or without external particles is considered in curved space. Using the dual transformation, we rewrite the model in terms of dual gauge fields and derive the Bogomol'nyi-type bound. We examine cylindrically symmetric solutions to Einstein equations and the first-order Bogomol'nyi equations, and find vortex solutions and vortex-particle composites which lie on the spatial manifold with global geometry described by a cylinder asymptotically or a two sphere in addition to the well-known cone. 
  A tutorial introduction is given to general Hopf algebras and to general compact quantum groups. In the definition and further treatment of compact quantum groups C*-algebras are avoided. Contact with Woronowicz's compact matrix quantum groups is made at a later stage. 
  A nonlocal and nonlinear theory of hadrons, equivalent to the color singlet sector two dimensional QCD, is constructed. The phase space space of this theory is an infinite dimensional Grassmannian. The baryon number of QCD corresponds to a topological invariant (`virtual rank') of the Grassmannian. It is shown that the hadron theory has topological solitons corresponding to the baryons of QCD. ${1\over N_c}$ plays the role of $\hbar$ in this theory; $N_c$ must be an integer for topological reasons. We also describe the quantization of a toy model with a finite dimensional Grassmannian as the phase space. In an appendix, we show that the usual Hartree--Fock theory of atomic and condensed matter physics has a natural formulation in terms of infinite dimensional Grassmannians. 
  The distribution of quasiprimary fields of fixed classes characterized by their O$(N)$ representations $Y$ and the number $p$ of vector fields from which they are composed at $N=\infty$ in dependence on their normal dimension $[\delta]$ is shown to obey a Hardy-Ramanujan law at leading order in a $\frac{1}{N}$-expansion. We develop a method of collective fusion of the fundamental fields which yields arbitrary \qps and resolves any degeneracy. 
  We compute the $r$-matrix for the elliptic Euler-Calogero-Moser model. In the trigonometric limit we show that the model possesses an exact Yangian symmetry. 
  We consider integrable open--boundary conditions for the supersymmetric t--J model commuting with the number operator $n$ and $S^{z}$. Four families, each one depending on two arbitrary parameters, are found. We find the relation between Sklyanin's method of constructing open boundary conditions and the one for the quantum group invariant case based on Markov traces. The eigenvalue problem is solved for the new cases by generalizing the Nested Algebraic Bethe ansatz of the quantum group invariant case (which is obtained as a special limit). For the quantum group invariant case the Bethe ansatz states are shown to be highest weights of $spl_{q}(2,1)$. 
  We introduce the main concepts and announce the main results in a theory of tensor products for module categories for a vertex operator algebra. This theory is being developed in a series of papers including hep-th 9309076 and hep-th 9309159. The theory applies in particular to any ``rational'' vertex operator algebra for which products of intertwining operators are known to be convergent in the appropriate regions, including the vertex operator algebras associated with the WZNW models, the minimal models and the moonshine module for the Monster. In this paper, we provide background and motivation; we present the main constructions and properties of the tensor product operation associated with a particular element of a suitable moduli space of spheres with punctures and local coordinates; we introduce the notion of ``vertex tensor category,'' analogous to the notion of tensor category but based on this moduli space; and we announce the results that the category of modules for a vertex operator algebra of the type mentioned above admits a natural vertex tensor category structure, and also that any vertex tensor category naturally produces a braided tensor category structure. 
  It is shown that a model with a spontaneously broken global symmetry can support defects analogous to Alice strings, and a process analogous to Cheshire charge exchange can take place. A possible realization in superfluid He-3 is pointed out. 
  (Talk given at Strings '93, Berkeley, and at XXVII. Internationales Symposium \"uber Elementarteilchentheorie, Wendisch-Rietz, 1993) We review the superconformal properties of matter coupled to $2d$ gravity, and $W$-extensions thereof. We show in particular how the \nex2 structure provides a direct link between certain matter-gravity systems and matrix models. We also show that much, probably all, of this can be generalized to $W$-gravity, and this leads to an infinite class of new exactly solvable systems. These systems are governed by certain integrable hierarchies, which are generalizations of the usual KdV hierarchy and whose algebraic structure is given in terms of quantum cohomology rings of grassmannians. 
  We present the anyon equation on a cylinder and in an infinite potential wall from the abelian Chern-Simons theory coupled to non-relativistic matter field by obtaining the effective hamiltonian through the canonical transformation method used for the theory on a plane and on a torus. We also give the periodic property of the theory on the cylinder. 
  We apply to the massive scalar field a method recently proposed by Schwinger to calculate the Casimir effect. The method is applied with two different regularization schemes: the Schwinger original one by means of Poisson formula and another one by means of analytical continuation. 
  The topological sigma model with the semi-infinite cigar-like target space (black hole geometry) is considered. The model is shown to possess unsuppressed instantons. The noncompactness of the moduli space of these instantons is responsible for an unusual physics. There is a stable vacuum state in which the vacuum energy is zero, correlation functions are numbers thus the model is in the topological phase. However, there are other vacuum states in which correlation functions show the coordinate dependence. The estimation of the vacuum energy indicates that it is nonzero. These states are interpreted as the ones with broken BRST-symmetry. 
  For an arbitrary quantum field in flat space with a planar boundary, an entropy of entanglement, associated with correlations across the boundary, is present when the field is in its vacuum state. The vacuum state of the same quantum field appears thermal in Rindler space, with an associated thermal entropy. We show that the density matrices describing the two situations are identical, and therefore that the two entropies are equal. We comment on the generality and significance of this result, and make use of it in analyzing the area and cutoff dependence of the entropy. The equivalence of the density matrices leads us to speculate that a planar boundary in Minkowski space has a classical entropy given by the Bekenstein--Hawking formula. 
  When high energy strings scatter at fixed angle, their amplitudes characteristically fall off exponentially with energy, ${\cal A} \sim \exp(-s \times const.)$. We show that in a compact space this suppression disappears for certain kinematic configurations. Amplitudes are power-law behaved and therefore greatly enhanced. In spacetime this corresponds to fixed-angle scattering, with fixed transfer in the compact dimensions. On the worldsheet this process is described by a stationary configuration of effective charges and vortices with vanishing total energy. It is worldsheet duality---and not spacetime duality---that plays a role. 
  All possible graded Poisson-Lie structures on the external algebra of $SL(2)$ are described. We prove that differential Poisson-Lie structures prolonging the Sklyanin brackets do not exist on $SL(2)$. There are two and only two graded Poisson-Lie structures on $SL (2)$ and neither of them can be obtained by a reduction of graded Poisson-Lie structures on the external algebra of $GL(2)$. Both of them can be quantized and as a result we get a new graded algebra of quantum right-invariant forms on $SL_q(2)$ with three generators. 
  We consider the generalized momentum-depending quon algebra in a dynamically evolving curved spacetime and perform a type of analysis similar to that of J.W.Goodison and D.J.Toms. We find that, at least in principle, all kinds of statistics may occur in some regions, i.e. phases in momentum space, depending on Bogoliubov coefficients determined by a specific dynamical model. 
  We examine the geometrical and topological properties of surfaces surrounding clusters in the 3--$d$ Ising model. For geometrical clusters at the percolation temperature and Fortuin--Kasteleyn clusters at $T_c$, the number of surfaces of genus $g$ and area $A$ behaves as $A^{x(g)}e^{-\mu(g)A}$, with $x$ approximately linear in $g$ and $\mu$ constant. We observe that cross--sections of spin domain boundaries at $T_c$ decompose into a distribution $N(l)$ of loops of length $l$ that scales as $l^{-\tau}$ with $\tau \sim 2.2$. We address the prospects for a string--theoretic description of cluster boundaries. (To appear in proceedings for the Cargese Workshop on "String Theory, Conformal Models and Topological Field Theories", May 1993) 
  The action of the total cohomology $H^*(M)$ of the almost Kahler manifold $M$ on its Floer cohomology, int roduced originally by Floer, gives a new ring structure on $H^*(M)$. We prove that the total cohomology space $H^* (M)$, provided with this new ring structure, is isomorphic to the quantum cohomology ring. As a special case, we prove the the formula for the Floer cohomology ring of the complex grassmanians conjectured by Vafa and Witten. 
  Charged dilaton black hole solutions have recently been found for an action with two $U(1)$ gauge fields and a dilaton field. I investigate new exact solutions of this theory analogous to the C-metric and Ernst solutions of classical general relativity. The parameters in the latter solution may be restricted so that it has a smooth Euclidean section with topology $S^2 \times S^2 - \{pt\}$, which gives an instanton describing pair production of the charged dilaton black holes. These instantons generalise those found recently by Dowker {\it et al}. 
  A consistent procedure of canonical quantization of pseudoclassical model for spin one relativistic particle is considered. Two approaches to treat the quantization for the massless case are discussed, the limit of the massive case and independent quantization of a modified action. Quantum mechanics constructed for the massive case proves to be equivalent to the Proca theory and for massless case to the Maxwell theory. Results obtained are compared with ones for the case of spinning (spin one half) particle. 
  A supersymmetric collective coordinate expansion of the monopole solution of $N=4$ Yang-Mills theory is performed resulting in an $N=4$ supersymmetric quantum mechanics on the moduli space of monopole solutions. 
  High energy scattering in 2+1 QCD is studied using the recent approach of Verlinde and Verlinde. We calculate the color singlet part of the quark-quark scattering exactly within this approach, and discuss some physical implication of this result. We also demonstrate, by two independent methods, that reggeization fails for the color singlet channel. We briefly comment on the problem in 3+1 QCD. 
  In this paper the $c=1$ string theory is studied from the point of view of topological field theories. Calculations are done for arbitrary genus. A change in the prescription is proposed, which reproduces the results of the $1/x^2$ deformed matrix model. It is proposed that the deformed matrix model is related to a D-series Landau-Ginzburg superpotential. 
  Alain Connes' construction of the standard model is based on a generalized Dirac-Yukawa operator and the K-cycle $(\HD ,D)$, with $\HD$ a fermionic Hilbert space. If this construction is reformulated at the level of the differential algebra then a direct comparison with the alternative approach by the Marseille-Mainz group becomes possible. We do this for the case of the toy model based on the structure group $U(1)\times U(1)$ and for the $SU(2)\times U(1)$ of electroweak interactions. Connes' results are recovered without the somewhat disturbing $\gamma_{5}$-factors in the fermion mass terms and Yukawa couplings. We discuss both constructions in the same framework and, in particular, pinpoint the origin of the difference in the Higgs potential obtained by them. 
  We consider a model of discretized 2d gravity interacting with Ising spins where phase boundaries are restricted to have minimal length and show analytically that the critical exponent $\gamma= 1/3$ at the spin transition point. The model captures the numerically observed behavior of standard multiple Ising spins coupled to 2d gravity. 
  We construct a class of quantum mechanical theories which are invariant under fermionic transformations similar to supersymmetry transformations. The generators of the transformations in this case, however, satisfy a BRST-like algebra. 
  A review article submitted to Physics Report: Target space duality and discrete symmetries in string theory are reviewed in different settings. 
  The field equations of the Chern-Simons theory quantized in the axial gauge are shown to be completely determined by supersymmetry Ward identities which express the invariance of the theory under the topological supersymmetry of Delduc, Gieres and Sorella together with the usual Slavnov identity without requiring any action principle. 
  The non-commutative differential calculus on quantum groups can be extended by introducing, in analogy with the classical case, inner product operators and Lie derivatives. For the case of $\GL$ we show how this extended calculus induces by coaction a similar extended calculus, covariant under $\GL$, on the quantum plane. In this way, inner product operators and Lie derivatives can be introduced on the plane as well. The situation with other quantum groups and quantum spaces is briefly discussed. Explicit formulas are given for the two dimensional quantum plane. 
  In the paper the nilpotent conditions of BRST operator for new superconformal string model were found. This string includes anticommutation $2-d$ fields additional to the standard Neveu-Schwarz superconformal set which carry quark quantum numbers. In this case the superconformal symmetry is realized by a non-linear way. In the superconformal composite string new constraints for 1 and 1/2 conformal dimension should be added to the standard system of Virasoro superalgebra constraints for 2 and 3/2 conformal dimensions. The number $N$ of the constraints and numbers $D$ and $D'$ of bosonic and fermionic $2-d$ fields are connected by a simple relationship $D'/2+D-3N-15=0.$ Also perspectives of the critical composite superconformal string are discussed. 
  The grand-canonical ensemble of dynamically triangulated surfaces coupled to four species of Ising spins (c=2) is simulated on a computer. The effective string susceptibility exponent for lattices with up to 1000 vertices is found to be $\gamma = - 0.195(58)$. A specific scenario for $c > 1$ models is conjectured. 
  An effective sigma model describing behavior of the 3d rigid string with a $\theta$-term at $\theta=\pi$ is proposed. It contains non-perturbative corrections resulting from summation over different genera of the 2d surfaces. The effective theory is the $SU(2)$ WZW model coupled to the Nambu-Goto action. RG analysis shows the existence of a IR fixed point at which the normal to the surface has long range correlations. A similar model can describe critical behaviour of the 3d Y-M fields or the Ising model. 
  We introduce the linear connection in the noncommutative geometry model of the product of continuous manifold and the discrete space of two points. We discuss its metric properties, define the metric connection and calculate the curvature. We define also the Ricci tensor and the scalar curvature. We find that the latter differs from the standard scalar curvature of the manifold by a term, which might be interpreted as the cosmological constant and apart from that we find no other dynamical fields in the model. Finally we discuss an example solution of flat linear connection, with the nontrivial scaling dependence of the metric tensor on the discrete variable. We interpret the obtained solution as confirmed by the Standard Model, with the scaling factor corresponding to the Weinberg angle. 
  We build a toy model of differential geometry on the real line, which includes derivatives of the second order. Such construction is possible only within the framework of noncommutative geometry. We introduce the metric and briefly discuss two simple physical models of scalar field theory and gauge theory in this geometry. 
  The evolution of a closed NSR string is considered in the background of constant graviton and antisymmetric fields. The $\sigma$-model action is written in a manifestly supersymmetric form in terms of superfields. The first order formalism adopted for the closed bosonic string is generalised to implement duality transformations and the constant dual backgrounds are obtained for the dual theory. We recover the $G \rightarrow G^{-1}$ duality for the case when antisymmetric tensor field is set to zero. Next, the case when the backgrounds depend on one superfield, is also analysed. This scenario is similar to the cosmological case envisaged for the bosonic string. The explicit form of the duality transformation is given for this case. 
  The recently proposed generalized field method for solving the master equation of Batalin and Vilkovisky is applied to a gauge theory of quadratic Lie algebras in 2-dimensions. The charge corresponding to BRST symmetry derived from this solution in terms of the phase space variables by using the Noether procedure, and the one found due to the BFV-method are compared and found to coincide. $W_3$ algebra, formulated in terms of a continuous variable is emploied in the mentioned gauge theory to construct a $W_3$ topological gravity. Moreover, its gauge fixing is briefly discussed. 
  We develop differential calculus and gauge theory on a finite set G. An elegant formulation is obtained when G is supplied with a group structure and in particular for a cyclic group. Connes' two-point model (which is an essential ingredient of his reformulation of the standard model of elementary particle physics) is recovered in our approach. Reductions of the universal differential calculus to `lower-dimensional' differential calculi are considered. The `complete reduction' leads to a differential calculus on a periodic lattice which is related to q-calculus. 
  There is a deformation of the ordinary differential calculus which leads from the continuum to a lattice (and induces a corresponding deformation of physical theories). We recall some of its features and relate it to a general framework of differential calculus on discrete sets. This framework generalizes the usual (lattice) discretization. 
  We explore a differential calculus on the algebra of smooth functions on a manifold. The former is `noncommutative' in the sense that functions and differentials do not commute, in general. Relations with bicovariant differential calculus on certain quantum groups and stochastic calculus are discussed. A similar differential calculus on a superspace is shown to be related to the Batalin-Vilkovisky antifield formalism. 
  For the two-parameter matrix quantum group GLp,q(2) all bicovariant differential calculi (with a four-dimensional space of 1-forms) are known. They form a one-parameter family. Here, we give an improved presentation of previous results by using a different parametrization. We also discuss different ways to obtain bicovariant calculi on the quantum subgroup SLq(2). For those calculi, we do not obtain the ordinary differential calculus on SL(2) in the classical limit. The structure which emerges here can be generalized to a nonstandard differential calculus on an arbitrary differentiable manifold and exhibits relations with stochastic calculus and `proper time' relativistic quantum theories. 
  In this work the determination of low-energy bound states in Quantum Chromodynamics is recast so that it is linked to a weak-coupling problem. This allows one to approach the solution with the same techniques which solve Quantum Electrodynamics: namely, a combination of weak-coupling diagrams and many-body quantum mechanics. The key to eliminating necessarily nonperturbative effects is the use of a bare Hamiltonian in which quarks and gluons have nonzero constituent masses rather than the zero masses of the current picture. The use of constituent masses cuts off the growth of the running coupling constant and makes it possible that the running coupling never leaves the perturbative domain. For stabilization purposes an artificial potential is added to the Hamiltonian, but with a coefficient that vanishes at the physical value of the coupling constant. The weak-coupling approach potentially reconciles the simplicity of the Constituent Quark Model with the complexities of Quantum Chromodynamics. The penalty for achieving this perturbative picture is the necessity of formulating the dynamics of QCD in light-front coordinates and of dealing with the complexities of renormalization which such a formulation entails. We describe the renormalization process first using a qualitative phase space cell analysis, and we then set up a precise similarity renormalization scheme with cutoffs on constituent momenta and exhibit calculations to second order. We outline further computations that remain to be carried out. There is an initial nonperturbative but nonrelativistic calculation of the hadronic masses that determines the artificial potential, with binding energies required to be fourth order in the coupling as in QED. Next there is a calculation of the leading radiative corrections to these masses, which requires our renormalization program. Then the real struggle of finding the right extensions to perturbation theory to study the strong-coupling behavior of bound states can begin. 
  We show that the $SU(N)$, level-1 Wess-Zumino-Witten conformal field theory provides a natural realization of the Yangian $Y(sl_N)$ for $N\geq 3$. We also construct a hamiltonian $H_2$ which commutes with the Yangian generators and study its spectrum. Our results, which generalize work by Haldane et al.\ \cite{hhtbp}, provide the field theory extension of the algebraic structure of the $SU(N)$ Haldane-Shastry spin chains with $1/r^2$ exchange. 
  We discuss the canonical quantization of $N=1$ supergravity in the functional Schrodinger representation. Although the form of the supersymmetry constraints suggests that there are solutions of definite order $n$ in the fermion fields, we show that there are no such states for any finite $n$. For $n=0$, a simple scaling argument definitively excludes the purely bosonic states discussed by D'Eath. For $n>0$, the argument is based on a mode expansion of the gravitino field on the quantization 3-surface. It is thus suggested that physical states in supergravity have infinite Grassmann number. This is confirmed for the free spin-3/2 field, for which we find that states satisfying the gauge constraints contain an infinite product of fermion mode operators. 
  We consider the anisotropic quantum Heisenberg antiferromagnet (with anisotropy $\lambda$) on a square lattice using a Chern-Simons (or Wigner-Jordan) approach. We show that the Average Field Approximation (AFA) yields a phase diagram with two phases: a Ne{\`e}l state for $\lambda>\lambda_c$ and a flux phase for $\lambda<\lambda_c$ separated by a second order transition at $\lambda_c<1$. We show that this phase diagram does not describe the $XY$ regime of the antiferromagnet. Fluctuations around the AFA induce relevant operators which yield the correct phase diagram. We find an equivalence between the antiferromagnet and a relativistic field theory of two self-interacting Dirac fermions coupled to a Chern-Simons gauge field. The field theory has a phase diagram with the correct number of Goldstone modes in each regime and a phase transition at a critical coupling $\lambda^* > \lambda_c$. We identify this transition with the isotropic Heisenberg point. It has a non-vanishing Ne{\` e}l order parameter, which drops to zero discontinuously for $\lambda<\lambda^*$. 
  We have presented canonical and path integral formulations of a theory of loops and closed strings with the matter field quanta transforming in the adjoint representation of the SU(N) gauge group. The physical processes arising out of the interactions of loops and closed strings are discussed. 
  By boosting the vertex operators of Witten's $SL(2,R)/U(1)$ black hole, we show that in the region V they lead to the primary fields of $c=1$ matter coupled to gravity at nonzero cosmological constant, while there is no such correspondence in the region I, showing that Witten's black hole corresponds to $2d$ gravity only in a certain region and in a specific limit. By using the free field representation , we will show that the stress tensor of $SL(2,R)$ gauged by its nilpotent subgroup is equivalent to that of the Liouville theory with zero cosmological constant. 
  The recently discussed notion of geometric entropy is shown to be related to earlier calculations of thermal effects in Rindler space. The evaluation is extended to de Sitter space and to a two-dimensional black hole. 
  There is a large class of classical null-fronted metrics in which a free scalar field has an infinite number of conservation laws. In particular, if the scalar field is quantized, the number of particles is conserved. However, with more general null-fronted metrics, field quantization cannot be interpreted in terms of particle creation and annihilation operators, and the physical meaning of the theory becomes obscure. 
  Quantization of anomalous gauge theories with closed, irreducible gauge algebra within the extended Field-Antifield formalism is further pursued. Using a Pauli-Villars (PV) regularization of the generating functional at one loop level, an alternative form for the anomaly is found which involves only the regulator. The analysis of this expression allows to conclude that recently found ghost number one cocycles with nontrivial antifield dependence can not appear in PV regularization. Afterwards, the extended Field-Antifield formalism is further completed by incorporating quantum effects of the extra variables, i.e., by explicitly taking into account the regularization of the extra sector. In this context, invariant PV regulators are constructed from non-invariant ones, leading to an alternative interpretation of the Wess-Zumino action as the local counterterm relating invariant and non-invariant regularizations. Finally, application of the above ideas to the bosonic string reproduces the well-known Liouville action and the shift $(26-D)\rightarrow(25-D)$ at one loop. 
  We consider the forced harmonic oscillator quantized according to infinite statistics ( a special case of the `quon' algebra proposed by Greenberg ). We show that in order for the statistics to be consistently evolved the forcing term must be identically zero for all time. Hence only the free harmonic oscillator may be quantized according to infinite statistics. 
  In the context of the fractional quantum Hall effect, we investigate Laughlin's celebrated ansatz for the groud state wave function at fractional filling of the lowest Landau level. Interpreting its normalization in terms of a one component plasma, we find the effect of an additional quadrupolar field on the free energy, and derive estimates for the thermodynamically equivalent spherical plasma. In a second part, we present various methods for expanding the wave function in terms of Slater determinants, and obtain sum rules for the coefficients. We also address the apparently simpler question of counting the number of such Slater states using the theory of integral polytopes. 
  We study the cohomology of the critical $W_4$ string using the $W_4$ BRST charge in a special basis in which it contains three separately nilpotent BRST charges. This allows us to obtain the physical operators in three steps. In the first step we obtain the cohomology associated to a spin-four constraint only, and it contains operators of the $c={4\over5}$ $W_3$ minimal model. In the next step, where the spin-three constraint is added, these operators get dressed to operators of the $c={7\over10}$ Virasoro minimal model. Finally, the Virasoro constraint is added to obtain the cohomology of the critical $W_4$ string. We describe the structure of the complete cohomology and compare with other results. 
  A recursive method is derived to calculate all eigenvalue correlation functions of a random hermitian matrix in the large size limit, and after smoothing of the short scale oscillations. The property that the two-point function is universal, is recovered and the three and four-point functions are given explicitly. One observes that higher order correlation functions are linear combinations of universal functions with coefficients depending on an increasing number of parameters of the matrix distribution. 
  An explicit transformation formula for chiral conformal fields under arbitrary holomorphic coordinate transformations is established. As an application I calculate the transformation law of the general quasiprimary field at level 4. 
  I prove the recently conjectured relation between the $2\times 2$-matrix differential operator $L=\partial^2-U$, and a certain non-linear and non-local Poisson bracket algebra ($V$-algebra), containing a Virasoro subalgebra, which appeared in the study of a non-abelian Toda field theory. Here, I show that this $V$-algebra is precisely given by the second Gelfand-Dikii bracket associated with $L$. The Miura transformation is given which relates the second to the first Gelfand-Dikii bracket. The two Gelfand-Dikii brackets are also obtained from the associated (integro-) differential equation satisfied by fermion bilinears. The asymptotic expansion of the resolvent of $(L-\xi)\Psi=0$ is studied and its coefficients $R_l$ yield an infinite sequence of hamiltonians with mutually vanishing Poisson brackets. I recall how this leads to a matrix KdV hierarchy which are flow equations for the three component fields $T, V^+, V^-$ of $U$. For $V^\pm=0$ they reduce to the ordinary KdV hierarchy. The corresponding matrix mKdV equations are also given, as well as the relation to the pseudo- differential operator approach. Most of the results continue to hold if $U$ is a hermitian $n\times n$-matrix. Conjectures are made about $n\times n$-matrix $m^{\rm th}$-order differential operators $L$ and associated $V_{(n,m)}$-algebras. 
  We construct the number operator for particles obeying infinite statistics, defined by a generalized q-deformation of the Heisenberg algebra, and prove the positivity of the norm of linearly independent state vectors. 
  We calculate the Kac determinant for the quasi-finite representation of \Winf algebra up to level 8. It vanishes only when the central charge is integer. We give an algebraic construction of null states and propose the character formulae. The character of the Verma module is related to free fields in three dimensions which has rather exotic modular properties. 
  We present several pieces of evidence for strong-weak coupling duality symmetry in the heterotic string theory, compactified on a six dimensional torus. These include symmetry of the 1) low energy effective action, 2) allowed spectrum of electric and magnetic charges in the theory, 3) allowed mass spectrum of particles saturating the Bogomol'nyi bound, and 4) Yukawa couplings between massless neutral particles and massive charged particles saturating the Bogomol'nyi bound. This duality transformation exchanges the electrically charged elementary string excitations with the magnetically charged soliton states in the theory. It is shown that the existence of a strong-weak coupling duality symmetry in four dimensional string theory makes definite prediction about the existence of new stable monopole and dyon states in the theory with specific degeneracies, including certain supersymmetric bound states of monopoles and dyons. The relationship between strong-weak coupling duality transformation in string theory and target space duality transformation in the five-brane theory is also discussed. (Based on a talk given at the workshop on Strings and Gravity, Madras, India.) 
  The Lagrangian action for the D4-D5-E6 model of hep-th/9306011 has 8-dim spacetime V8 of the vector representation of Spin(0,8); 8-dim fermion fields S8+ = S8- of the half-spinor reps of Spin(0,8); and 28 gauge boson fields of the bivector adjoint rep of Spin(0,8). In this paper, the structure of the positive definite Clifford algebra Cl(0,8) of Spin(0,8), and the triality automorphism V8 = S8+ = S8-, are used to reduce the spacetime to 4 dimensions and thereby change the gauge group from Spin(0,8) to the realistic SU(3)xSU(2)xU(1), Higgs, and Gravity. The effect of dimensional reduction on fermions, to introduce 3 generations, has been described in hep-ph/9301210. The global geometry of manifolds V8 = S8+ = S8- = RP1xS7, the effects of dimensional reduction on them, and the calculation of force strength constants, has been described in hep-th/9302030. 
  General arguments based on curved space-time thermodynamics show that any extensive quantity, like the free energy or the entropy of thermal matter, always has a divergent boundary contribution in the presence of event horizons, and this boundary term comes with the Hawking-Bekenstein form. Although the coefficients depend on the particular geometry we show that intensive quantities, like the free energy density are universal in the vicinity of the horizon. {} From the point of view of the matter degrees of freedom this divergence is of infrared type rather than ultraviolet, and we use this remark to speculate about the fate of these pathologies in String Theory. Finally we interpret them as instabilities of the Canonical Ensemble with respect to gravitational collapse via the Jeans mechanism. 
  We investigate the field dependence of the gauge couplings of locally supersymmetric effective quantum field theories. We find that the Weyl rescaling of supergravity gives rise to Wess-Zumino terms that affect the gauge couplings at the one-loop level. These Wess-Zumino terms are crucial in assuring supersymmetric consistency of both perturbative and non-perturbative gauge interactions. At the perturbative level, we distinguish between the holomorphic Wilsonian gauge couplings and the physically-measurable momentum-dependent effective gauge couplings; the latter are affected by the Konishi and the super-Weyl anomalies and their field-dependence is non-holomorphic. At the non-perturbative level, we show how consistency of the scalar potential generated by infrared-strong gauge interactions with the local supersymmetry requires a very specific form of the effective superpotential. We use this superpotential to determine the dependence of the supersymmetric condensates of a strongly interacting gauge theory on its (field-dependent) Wilsonian gauge coupling and the Yukawa couplings of the matter fields. The article concludes with the discussion of the field-dependent non-perturbative phenomena in the context of string unification. 
  We investigate the generic distribution of bosonic and fermionic states at all mass levels in non-supersymmetric string theories, and find that a hidden ``misaligned supersymmetry'' must always appear in the string spectrum. We show that this misaligned supersymmetry is ultimately responsible for the finiteness of string amplitudes in the absence of full spacetime supersymmetry, and therefore the existence of misaligned supersymmetry provides a natural constraint on the degree to which spacetime supersymmetry can be broken in string theory without destroying the finiteness of string amplitudes. Misaligned supersymmetry also explains how the requirements of modular invariance and absence of physical tachyons generically affect the distribution of states throughout the string spectrum, and implicitly furnishes a two-variable generalization of some well-known results in the theory of modular functions. 
  In 4-D heterotic superstrings, the dilaton and antisymmetric tensor fields belong to a linear N=1 supersymmetric multiplet L. We study the lagrangian describing the coupling of one linear multiplet to chiral and gauge multiplets in global and local supersymmetry, with particular emphasis on string tree-level and loop-corrected effective actions. This theory is dual to an equivalent one with chiral multiplets only. But the formulation with a linear multiplet appears to have decisive advantages beyond string tree-level since, in particular, <L> is the string loop-counting parameter and the duality transformation is in general not exactly solvable beyond tree-level. This formulation allows us to easily deduce some powerful non-renormalization theorems in the effective theory and to obtain explicitly some loop corrections to the string effective supergravity for simple compactifications. Finally, we discuss the issue of supersymmetry breaking by gaugino condensation using this formalism. 
  Using Selberg's integral formula we derive all Leutwyler-Smilga type sum rules for one and two flavors, and for each of the three chiral random matrix ensembles. In agreement with arguments from effective field theory, all sum rules for $N_f = 1$ coincide for the three ensembles. The connection between spectral correlations and the low-energy effective partition function is discussed. 
  In the sigma model approach, the $\beta$-function equations for non critical strings contain a term which acts like a tree level cosmological constant, $\Lambda$. We analyse the static, spherically symmetric solutions to these equations in $d = 4$ space time and show that the curvature scalar seen by the strings is singular if $\Lambda \ne 0$. This singularity is naked. Requiring its absence in our universe imposes the constraint $| \Lambda | < 10^{- 120}$ in natural units. {}From another point of view, our analysis implies that low energy $d = 4$ non critical strings lead to naked singularities. 
  We construct the effective Lagrangian describing QCD in the multi-Regge kinematics. It is obtained from the original QCD   Lagrangian by eliminating modes of gluon and quark fields not appearing in this underlying kinematics. 
  We show that Belavin's solutions of the quantum Yang--Baxter equation can be obtained by restricting an infinite $R$-matrix to suitable finite dimensional subspaces. This infinite $R$-matrix is a modified version of the Shibukawa--Ueno $R$-matrix acting on functions of two variables. 
  Covariantly we reformulate the description of a spinning particle in terms of the Poincar\'{e} group. We also construct a Lagrangian which entails all possible constraints explicitly; all constraints can be obtained just from the Lagrangian. Furthermore, in this covariant reformulation, the Lorentz element is to be considered to evolve the momentum or spin component from an arbitrary fixed frame and not just from the particle rest frame. In distinction with the usual formulation, our system is directly comparable with the pseudo-classical formulation. We get a peculiar symmetry which resembles the supersymmetry of the pseudo-classical formulation. 
  We define a class of deformed multimode oscillator algebras which possess number operators and can be mapped to multimode Bose algebra.We construct and discuss the states in the Fock space and the corresponding number operators. 
  The ``position'' and ``momentum'' operators for the q-deformed oscillator with q being a root of unity are proved to have discrete eigenvalues which are roots of deformed Hermite polynomials. The Fourier transform connecting the ``position'' and ``momentum'' representations is also found   The phase space of this oscillator has a lattice structure, which is a non-uniformly distributed grid. Non-equidistant lattice structures also occur in the cases of the truncated harmonic oscillator and of the q-deformed parafermionic oscillator, while the parafermionic oscillator corresponds to a uniformly distributed grid. 
  A perturbative renormalization group is formulated for the study of Hamiltonian light-front field theory near a critical Gaussian fixed point. The only light-front renormalization group transformations found that can be approximated by dropping irrelevant operators and using perturbation theory near Gaussian fixed volumes, employ invariant-mass cutoffs. These cutoffs violate covariance and cluster decomposition, and allow functions of longitudinal momenta to appear in all relevant, marginal, and irrelevant operators. These functions can be determined by insisting that the Hamiltonian display a coupling constant coherence, with the number of couplings that explicitly run with the cutoff scale being limited and all other couplings depending on this scale only through perturbative dependence on the running couplings. Examples are given that show how coupling coherence restores Lorentz covariance and cluster decomposition, as recently speculated by Wilson and the author. The ultimate goal of this work is a practical Lorentz metric version of the renormalization group, and practical renormalization techniques for light-front quantum chromodynamics. 
  The 1--loop string background equations with axion and dilaton fields are shown to be integrable in four dimensions in the presence of two commuting Killing symmetries and $\delta c = 0$. Then, in analogy with reduced gravity, there is an infinite group that acts on the space of solutions and generates non--trivial string backgrounds from flat space. The usual $O(2,2)$ and $S$--duality transformations are just special cases of the string Geroch group, which is infinitesimally identified with the $O(2,2)$ current algebra. We also find an additional $Z_{2}$ symmetry interchanging the field content of the dimensionally reduced string equations. The method for constructing multi--soliton solutions on a given string background is briefly discussed. 
  We study representations of the mapping class group of the punctured torus on the double of a finite dimensional possibly non-semisimple Hopf algebra that arise in the construction of universal, extended topological field theories. We discuss how for doubles the degeneracy problem of TQFT's is circumvented. We find compact formulae for the ${\cal S}^{\pm 1}$-matrices using the canonical, non degenerate forms of Hopf algebras and the bicrossed structure of doubles rather than monodromy matrices. A rigorous proof of the modular relations and the computation of the projective phases is supplied using Radford's relations between the canonical forms and the moduli of integrals. We analyze the projective $SL(2, Z)$-action on the center of $U_q(sl_2)$ for $q$ an $l=2m+1$-st root of unity. It appears that the $3m+1$-dimensional representation decomposes into an $m+1$-dimensional finite representation and a $2m$-dimensional, irreducible representation. The latter is the tensor product of the two dimensional, standard representation of $SL(2, Z)$ and the finite, $m$-dimensional representation, obtained from the truncated TQFT of the semisimplified representation category of $U_q(sl_2)\,$. 
  We investigate invertible elements and gradings in braided tensor categories. This leads us to the definition of theta-, product-, subgrading and orbitcategories in order to construct new families of BTC's from given ones. We use the representation theory of Hecke algebras in order to relate the fusionring of a BTC generated by an object $X$ with a two component decomposition of its tensorsquare to the fusionring of quantum groups of type $A$ at roots of unity. We find the condition of `local isomorphie' on a special fusionring morphism implying that a BTC is obtained from the above constructions applied to the semisimplified representation category of a quantum group. This family of BTC's contains new series of twisted categories that do not stem from known Hopf algebras. Using the language of incidence graphs and the balancing structure on a BTC we also find strong constraints on the fusionring morphism. For Temperley Lieb type categories these are sufficient to show local isomorphie. Thus we obtain a classification for the subclass of Temperley Lieb type categories. 
  This is a thesis for Rigaku-Hakushi($\simeq$ Ph. D.). It clarifies the geometric meaning and field theoretical consequences of the spectral flows acting on the space of states of the `$G/H$ coset model'. As suggested by Moore and Seiberg, the spectral flow is realized as the response of states to certain change of background gauge field together with the gauge transformation on a circle. Applied to the boundary circle of a disc with field insertion, such a realization leads to a certain relation among correlators of the gauged WZW model for various principal $H$-bundles. In the course of derivation, we find an expression of a (dressed) gauge invariant field as an integral over the flag manifold of $H$ and an expression of a correlator as an integral over a certain moduli space of holomorphic $H_{\bf   C}$-bundles with quasi-flag structure at the insertion point. We also find that the gauge transformation on the circle corresponding to the spectral flow determines a bijection of the set of isomorphism classes of holomorphic $H_{\bf C}$-bundles with quasi-flag structure of one topological type to that of another. As an application, it is pointed out that problems arising from the field identification fixed points may be resolved by taking into account of all principal $H$-bundles. 
  Bosonic end perturbative calculations for quantum mechanical anyon systems require a regularization. I regularize by adding a specific $\delta$-function potential to the Hamiltonian. The reliability of this regularization procedure is verified by comparing its results for the 2-anyon in harmonic potential system with the known exact solutions. I then use the $\delta$-function regularized bosonic end perturbation theory to test some recent conjectures concerning the unknown portion of the many-anyon spectra. 
  The space of solutions of the rational Calogero-Moser hierarchy, and the space of solutions of the KP hierarchy whose tau functions are monic polynomials in $t_1$ with coefficients depending on $t_n$, $n > 1$, are identified, generalizing earlier results of Airault-McKean-Moser and Krichever. 
  We propose a new method of diagonalization of hamiltonians of the Gaudin model associated to an arbitrary simple Lie algebra, which is based on Wakimoto modules over affine algebras at the critical level. We construct eigenvectors of these hamiltonians by restricting certain invariant functionals on tensor products of Wakimoto modules. In conformal field theory language, the eigenvectors are given by certain bosonic correlation functions. Analogues of Bethe ansatz equations naturally appear as Kac-Kazhdan type equations on the existence of certain singular vectors in Wakimoto modules. We use this construction to expalain a connection between Gaudin's model and correlation functions of WZNW models. 
  A canonical basis of global Dirac's observables for Yang-Mills theory with fermions are obtained in a functional space in which Gribov ambiguity is absent and Gauss' laws can be solved exactly. In terms of these observables, one can express the Lagrangian, the Hamiltonian, non-Abelian and topological charges. The problem of the covariantization of these non-local and non-polynomial quantities is solved. From a relativistic presymplectic approach to the localization of the relativistic center-of-mass of the field configuration, a ultraviolet cutoff is deduced. 
  We study a possibility to define the (braided) comultiplication for the GLq(N)-covariant differential complexes on some quantum spaces. We discover such `differential bialgebras' (and Hopf algebras) on the bosonic and fermionic quantum hyperplanes (with additive coproduct) and on the braided matrix algebra BMq(N) with both multiplicative and additive coproducts. The latter case is related (for N=2) to the q-Minkowski space and q-Poincare algebra. 
  Present quantum theory, which is statistical in nature, does not predict joint probability distribution of position and momentum because they are noncommuting. We propose a deterministic quantum theory which predicts a joint probability distribution such that the separate probability distributions for position and momentum agree with usual quantum theory. Unlike the Wigner distribution the suggested distribution is positive definite. The theory predicts a correlation between position and momentum in individual events. 
  Nonlinear integrable equations, such as the KdV equation, the Boussinesq equation and the KP equation, have the close relation with many-body problem. The solutions of such equations are the same as the restricted flows of the classical Calogero model, which is one-dimensional particle system with inverse square interactions. The KP hierarchy and the Calogero model share the same structure called ``additional symmetry''. This symmetry plays a crucial role in this relation. 
  In this paper, we develop several general techniques to investigate modular invariants of conformal field theories whose algebras of the holomorphic and anti-holomorphic sectors are different. As an application, we find all such ``heterotic'' WZNW physical invariants of (horizontal) rank four: there are exactly seven of these, two of which seem to be new. Previously, only those of rank $\le 3$ have been completely classified. We also find all physical modular invariants for $su(2)_{k_1}\times su(2)_{k_2}$, for $22>k_1>k_2$, and $k_1=28$, $k_2<22$, completing the classification of ref.{} \SUSU. 
  The spontaneous breaking of Z(N) symmetry in hot QCD and the appearance of domain walls is reviewed. 
  A light-like Wilson loop is computed in perturbation theory up to ${\cal O} (g^4)$ for pure Yang--Mills theory in 1+1 dimensions, using Feynman and light--cone gauges to check its gauge invariance. After dimensional regularization in intermediate steps, a finite gauge invariant result is obtained, which however does not exhibit abelian exponentiation. Our result is at variance with the common belief that pure Yang--Mills theory is free in 1+1 dimensions, apart perhaps from topological effects. 
  We show that introducing a periodic time coordinate in the models of Penner-Kontsevich type generalizes the corresponding constructions to the case of the moduli space ${\cal S}_{gn}^k$ of curves $C$ with homology chains $\gamma\in H_1(C,\zet_k)$. We make a minimal extension of the resulting models by adding a kinetic term, and we get a new matrix model which realizes a simple dynamics of $\zet_k$-chains on surfaces. This gives a representation of $c=1$ matter coupled to two-dimensional quantum gravity with the target space being a circle of finite radius, as studied by Gross and Klebanov. 
  We consider WZW models based on the non-semi-simple algebras that they were recently constructed as contractions of corresponding algebras for semi-simple groups. We give the explicit expression for the action of these models, as well as for a generalization of them, and discuss their general properties. Furthermore we consider gauged WZW models based on these non-semi-simple algebras and we show that there are equivalent to non-abelian duality transformations on WZW actions. We also show that a general non-abelian duality transformation can be thought of as a limiting case of the non-abelian quotient theory of the direct product of the original action and the WZW action for the symmetry gauge group $H$. In this action there is no Lagrange multiplier term that constrains the gauge field strength to vanish. A particular result is that the gauged WZW action for the coset $(G_k \otimes H_l)/H_{k+l}$ is equivalent, in the limit $l\to \infty$, to the dualized WZW action for $G_k$ with respect to the subgroup $H$. 
  Existence of SL(2,Z) duality in toroidally compactified heterotic string theory (or in the N=4 supersymmetric gauge theories), that includes the strong weak coupling duality transformation, implies the existence of certain supersymmetric bound states of monopoles and dyons. We show that the existence of these bound states, in turn, requires the existence of certain normalizable, (anti-)self-dual, harmonic forms on the moduli space of BPS multi-monopole configurations, with specific symmetry properties. We give an explicit construction of this harmonic form on the two monopole moduli space, thereby proving the existence of all the required bound states in the two monopole sector. 
  The gauging of the q-Poincar\'e algebra of ref. hep-th 9312179 yields a non-commutative generalization of the Einstein-Cartan lagrangian. We prove its invariance under local q-Lorentz rotations and, up to a total derivative, under q-diffeomorphisms. The variations of the fields are given by their q-Lie derivative, in analogy with the q=1 case. The algebra of q-Lie derivatives is shown to close with field dependent structure functions. The equations of motion are found, generalizing the Einstein equations and the zero-torsion condition. 
  A class of generalized Ising models is examined with a view to extracting a low energy sector comprising   Dirac fermions coupled to Yang-Mills vectors. The main feature of this approach is a set of gap equations, covariant with respect to one of the $4$-dimensional crystallographic space groups. 
  By a Sugawara construction we mean a generalized Virasoro construction in which the currents are primary fields of conformal weight one. For simple Lie algebras, this singles out the standard Sugawara construction out of all the solutions to the Virasoro master equation. Examples of nonsemisimple Sugawara constructions have appeared recently. They share the properties that the Virasoro central charge is an integer equal to the dimension of the Lie algebra and that they can be obtained by high-level contraction of reductive Sugawara constructions: they thus correspond to free bosons. Exploiting a recent structure theorem for Lie algebras with an invariant metric, we are able to unify all the known constructions under the same formalism and, at the same time, to prove several results about the Sugawara constructions. In particular, we prove that all such constructions factorize into a standard (semisimple) Sugawara construction and a nonsemisimple one (with integral central charge) of a form which generalizes the nonsemisimple examples known so far. 
  We use the canonical quantization of spherically symetric dust universes, and calculate the expectation value of the quantized metric, $<\Psi|\hat{g}|\Psi>$. Though the classical solutions are singular, and the wave functions have no zero support on singular geometries, the expectation values are everywhere regular. For a quasi-classical (coherent) state, the metric expectation value describes a universe (or star) that collapses to a minimum radius, the Planck radius, and re-expands again. 
  We introduce an algebraic theory of integration on quantum planes and other braided spaces. In the one dimensional case we obtain a novel picture of the Jackson $q$-integral as indefinite integration on the braided group of functions in one variable $x$. Here $x$ is treated with braid statistics $q$ rather than the usual bosonic or Grassmann ones. We show that the definite integral $\int x$ can also be evaluated algebraically as multiples of the integral of a $q$-Gaussian, with $x$ remaining as a bosonic scaling variable associated with the $q$-deformation. Further composing our algebraic integration with a representation then leads to ordinary numbers for the integral. We also use our integration to develop a full theory of $q$-Fourier transformation $F$. We use the braided addition $\Delta x=x\otimes 1+1\otimes x$ and braided-antipode $S$ to define a convolution product, and prove a convolution theorem. We prove also that $F^2=S$. We prove the analogous results on any braided group, including integration and Fourier transformation on quantum planes associated to general R-matrices, including $q$-Euclidean and $q$-Minkowski spaces. 
  A general Lagrangian formulation of integrably deformed G/H-coset models is given. We consider the G/H-coset model in terms of the gauged Wess-Zumino-Witten action and obtain an integrable deformation by adding a potential energy term $Tr(gTg^{-1}\Tb )$, where algebra elements $T, \Tb $ belong to the center of the algebra {\bf h} associated with the subgroup H. We show that the classical equation of motion of the deformed coset model can be identified with the integrability condition of certain linear equations which makes the use of the inverse scattering method possible. Using the linear equation, we give a systematic way to construct infinitely many conserved currents as well as soliton solutions. In the case of the parafermionic SU(2)/U(1)-coset model, we derive $n$-solitons and conserved currents explicitly. 
  Fusion is defined for arbitrary lowest weight representations of $W$-algebras, without assuming rationality. Explicit algorithms are given. A category of quasirational representations is defined and shown to be stable under fusion. Conjecturally, it may coincide with the category of representations of finite quantum dimensions. 
  It is shown that the well known Racah sum rule and Biedenharn-Elliott identity satisfied by the recoupling coefficients or by the $6-j$ symbols of the usual rotation $SO(3)$ algebra can be extended to the corresponding features of the super-rotation $osp(1|2)$ superalgebra. The structure of the sum rules is completely similar in both cases, the only difference concerns the signs which are more involved in the super-rotation case. 
  In this article we review the Duistermaat-Heckman integration formula and the ensuing equivariant cohomology structure, in the finite dimensional case. In particular, we discuss the connection between equivariant cohomology and classical integrability. We also explain how the integration formula is derived, and explore some possible new directions that could eventually yield novel integration formulas for nontrivial integrable models. 
  We show that the $N=2$ superstrings may be viewed as a special class of the $N=4$ superstrings and demonstrate their equivalence. This allows us to realize all known string theories based on linear algebras and with $N<4$ supersymmetries as special choices of the vacua in the $N=4$ superstring. 
  A completely integrable dynamical system in discrete time is studied by means of algebraic geometry. The system is associated with factorization of a linear operator acting in a direct sum of three linear spaces into a product of three operators, each acting nontrivially only in a direct sum of two spaces, and the following reversing of the order of factors. There exists a reduction of the system interpreted as a classical field theory in 2+1-dimensional space-time, the integrals of motion coinciding, in essence, with the statistical sum of an inhomogeneous 6-vertex free-fermion model on the 2-dimensional kagome lattice (here the statistical sum is a function of two parameters). Thus, a connection with the ``local'', or ``generalized'', quantum Yang--Baxter equation is revealed. 
  We consider four dimensional quantum field theories which have a continuous manifold of inequivalent exact ground states -- a moduli space of vacua. Classically, the singular points on the moduli space are associated with extra massless particles. Quantum mechanically these singularities can be smoothed out. Alternatively, new massless states appear there. These may be the elementary massless particles or new massless bound states. 
  We show how to derive exact boundary $S$ matrices for integrable quantum field theories in 1+1 dimensions using lattice regularization. We do this calculation explicitly for the sine-Gordon model with fixed boundary conditions using the Bethe ansatz for an XXZ-type spin chain in a boundary magnetic field. Our results agree with recent conjectures of Ghoshal and Zamolodchikov, and indicate that the only solutions to the Bethe equations which contribute to the scaling limit are the standard strings. 
  We present a unified description of gravity and electromagnetism in the framework of a $Z_2$ noncommutative differential calculus. It can be considered as a ``discrete version" of Kaluza-Klein theory, where the fifth continuous dimension is replaced by two discrete points. We derive an action which coincides with the dimensionally reduced one of the ordinary Kaluza-Klein theory. 
  Weinberg-Salam theory and $SU(5)$ grand unified theory are reconstructed using the generalized differential calculus extended on the discrete space $M_4\times Z_{\mathop{}_{N}}$. Our starting point is the generalized gauge field expressed by $A(x,n)=\!\sum_{i}a^\dagger_{i}(x,n){\bf d}a_i(x,n), (n=1,2,\cdots N)$, where $a_i(x,n)$ is the square matrix valued function defined on $M_4\times Z_{\mathop{}_{N}}$ and ${\bf d}=d+\sum_{m=1}^{\mathop{}_{N}}d_{\chi_m}$ is generalized exterior derivative. We can construct the consistent algebra of $d_{\chi_m}$ which is exterior derivative with respect to $Z_{\mathop{}_{N}}$ and the spontaneous breakdown of gauge symmetry is coded in ${d_{\chi_m}}$. The unified picture of the gauge field and Higgs field as the generalized connection in non-commutative geometry is realized. Not only Yang-Mills-Higgs lagrangian but also Dirac lagrangian, invariant against the gauge transformation, are reproduced through the inner product between the differential forms. Three sheets ($Z_3)$ are necessary for Weinberg-Salam theory including strong interaction and $SU(5)$ Gut. Our formalism is applicable to more realistic model like $SO(10)$ unification model. 
  Standard model is reconstructed using the generalized differential calculus extended on the discrete space $M_4\times Z_3$. $Z_3$ is necessary for the inclusion of strong interaction. Our starting point is the generalized gauge field expressed as $A(x,y)=\!\sum_{i}a^\dagger_{i}(x,y){\bf d}a_i(x,y), (y=0,\pm)$, where $a_i(x,y)$ is the square matrix valued function defined on $M_4\times Z_3$ and ${\bf d}=d+{\chi}$ is generalized exterior derivative. We can construct the consistent algebra of $d_{\chi}$ with the introduction of the symmetry breaking function $M(y)$ and the spontaneous breakdown of gauge symmetry is coded in ${d_{\chi}}$.   The gauge field $A_\mu(x,y)$ and Higgs field $\Phi(x,y)$ are written in terms of $a_i(x,y)$ and $M(y)$, which might suggest $a_i(x,y)$ to be more fundamental object. The unified picture of the gauge field and Higgs field as the generalized connection in non-commutative geometry is realized. Not only Yang-Mills-Higgs lagrangian but also Dirac lagrangian, invariant against the gauge transformation, are reproduced through the inner product between the differential forms. Two model constructions are presented, which are distinguished in the particle assignment of Higgs field $\Phi(x,y)$. 
  Dynamics of a BPS dyon in a weak, constant, electromagnetic field is studied through a perturbative analysis of appropriate non-linear field equations. The full Lorentz force law for a BPS dyon is established. Also derived are the radiation fields accompanying the motion. 
  Using (2+$\epsilon$)-dimensional quantum gravity recently formulated by Kawai, Kitazawa and Ninomiya, we calculate the scaling dimensions of manifestly generally covariant operators in two-dimensional quantum gravity coupled to $(p,q)$ minimal conformal matter. Although the spectrum includes all the scaling dimensions of the scaling operators in the matrix model except the boundary operators, there are also many others which do not appear in the matrix model. We argue that the partial agreement of the scaling dimensions should be considered as accidental and that the operators considered give a new series of operators in two-dimensional quantum gravity. 
  The correspondence of the braid group on a handlebody of arbitrary genus to the algebra of Yang-Baxter and extended reflection equation operators is shown. Representations of the infinite dimensional extended reflection equation algebra in terms of direct products of quantum algebra generators are derived, they lead to a representation of this braid group in terms of $R$-matrices. Restriction to the reflection equation operators only gives the coloured braid group. The reflection equation operators, describing the effect of handles attached to a 3-ball, satisfy characteristic equations which give rise to additional skein relations and thereby invariants of links on handlebodies. The origin of the skein relations is explained and they are derived from an adequately adapted handlebody version of the Jones polynomial. Relevance of these results to the construction of link polynomials on closed 3-manifolds via Heegard splitting and surgery is indicated. 
  We consider a class of spin systems on randomly triangulated surfaces as discrete approximations to conformal matter fields coupled to 2d gravity. On the basis of certain universality assumptions we argue that at critical points with diverging string susceptibility the model either exhibits mean field behaviour or it can effectively be described by a conformal matter system with central charge less than or equal to 1 coupled to 2d gravity. As a particular consequence we conclude in the unitary case that the string susceptibility exponent is limited to possible values of the form 1/n, n=2,3,4,..., where n=2 corresponds to mean field behaviour. 
  We determine the excitations and $S$ matrix of an integrable isotropic antiferromagnetic quantum spin chain of alternating spin 1/2 and spin 1. There are two types of gapless one-particle excitations: the usual spin 1/2 (``spinor'') kink, and a new spin 0 (``scalar'') kink. Remarkably, the scalar-spinor scattering is nontrivial, yet the spinor-spinor scattering is the same as if the scalar kinks were absent. Moreover, there is no scalar-scalar scattering. 
  We compute the Poisson bracket relations for the monodromy matrix of the auxiliary linear problem. If the basic Poisson bracket relations of the model contain derivatives, this computation leads to a peculiar type of symmetry breaking which accounts for a 'spontaneous quantization of the underlying global gauge group. A classification of possible patterns of symmetry breaking is outlined. 
  We give the first operator solution of the Schwinger model that obeys the canonical commutation relations in a covariant guage. 
  We prove that the supersymmetric BKP-hierarchy of Yu (SBKP_2) is hamiltonian with respect to a nonlinear extension of the N=1 Super-Virasoro algebra (W_SBKP) by fields of spin k, where k>3/2 and 2k = 0,3 mod 4. Moreover, we show how to associate in a similar manner an N=1 W-superalgebra with every integrable hierarchy of the SKdV-type. We also show using dressing transformations how to extend, in a way which is compatible with the hamiltonian structure, the SBKP_2-hierarchy by odd flows, as well as the equivalence of this extended hierarchy to the SBKP-hierarchy of Manin-Radul. 
  A many--body Schr\"odinger equation for non--Abelian Chern--Simons particles is obtained from both point--particle and field--theoretic pictures. We present a particle Lagrangian and a field theoretic Lagrange density, and discuss their properties. Both are quantized by the symplectic method of Hamiltonian reduction. An $N$--body Schr\"odinger equation for the particles is obtained from both starting points. It is shown that the resulting interaction between particles can be replaced by non--trivial boundary conditions. Also, the equation is compared with the one given in the literature. 
  An elementary proof is given for the existence of infinite dimensional abelian subalgebras in quantum W-algebras. In suitable realizations these subalgebras define the conserved charges of various quantum integrable systems. We consider all principle W-algebras associated with the simple Lie algebras. The proof is based on the more general result that for a class of vertex operators the quantum operators are related to their classical counterparts by an equivalence transformation. 
  We comment on a recent paper by Chaichian et al. (Phys.Rev.Lett. 71(1993)3405). 
  The algebraic formulation of the quantum group covariant noncommutative geometry in the framework of the $R$-matrix approach to the theory of quantum groups is given. We consider structure groups taking values in the quantum groups and introduce the notion of the noncommutative connections and curvatures transformed as comodules under the "local" coaction of the structure group which is exterior extension of $GL_{q}(N)$. These noncommutative connections and curvatures generate $ GL_{q}(N)$-covariant quantum algebras. For such algebras we find combinations of the generators which are invariants under the coaction of the "local" quantum group and one can formally consider these invariants as the noncommutative images of the Lagrangians for the topological Chern-Simons models, non-abelian gauge theories and the Einstein gravity. We present also an explicit realization of such covariant quantum algebras via the investigation of the coset construction $GL_{q}(N+1)/(GL_{q}(N)\otimes GL(1))$. 
  We present a new class of 2d integrable models obtained as perturbations of minimal CFT with W-symmetry by fundamental weight primaries. These models are generalisations of well known $(1,2)$-perturbed Virasoro minimal models. In the large $p$ (number of minimal model) limit they coincide with scalar perturbations of WZW theories. The algebra of conserved charges is discussed in this limit. We prove that it is noncommutative and coincides with twisted affine algebra $G$ represented in a space of asymptotic states. We conjecture that scattering in these models for generic $p$ is described by $S$-matrix of the $q$-deformed $G$ - algebra with $q$ being root of unity. 
  There are many striking phenomena which are attributed to   ``quantum coherence''. It is natural to wonder if there are new quantum coherence effects waiting to be discovered which could lead to interesting results and perhaps even practical applications. A useful starting point for such discussions is a definition of ``quantum coherence''. In this article I give a definition of quantum coherence and use a number of illustrations to explore the implications of this definition. I point to topics of current interest in the fields of cosmology and quantum computation where questions of quantum coherence arise, and I emphasize the impact that interactions with the environment can have on quantum coherence. 
  We describe a bigraded generalization of the Weil algebra, of its basis and of the characteristic homomorphism which besides ordinary characteristic classes also maps on Donaldson invariants. 
  We construct the nonlinear $N=2$ super-$W_3^{(2)}$ algebra with an arbitrary central charge at the classical level in the framework of Polyakov "soldering" procedure. It contains two non-intersecting subalgebras: $N=2$ superconformal algebra and $W_3^{(2)}$ and their closure gives the $N=2$ super-$W_3^{(2)}$ algebra. Besides the currents of $N=2$ superconformal and $W_3^{(2)}$ algebras, it comprises two pairs of fermionic currents with spins 1 and 2. The hybrid field realization and contractions to the zero central charge are constructed. 
  We discuss solutions of the spherically symmetric wave equation and Klein Gordon equation in an arbitrary number of spatial and temporal dimensions. Starting from a given solution, we present various procedures to generate futher solutions in the same or in different dimensions. The transition from odd to even or non integer dimensions can be performed by fractional derivation or integration. The dimensional shift, however, can also be interpreted simply as a modification of the dynamics. We also discuss the analytic continuation to arbitrary real powers of the D'Alembert operator. There, particular peculiarities in the pole structure show up when $p$ and $q$ are both even. Finally we give operators which transform a time into a space coordinate and v.v. and comment on their possible relation to black holes. In this context, we describe a few aspects of the extension of our discussion to a curved metrics. 
  We show that $QCD_2$ with adjoint fermions involves instantons due to nontrivial $\pi_1[SU(N)/Z_N]~=~Z_N$. At high temperatures, quasiclassical approximation works and the action and the form of effective (with account of quantum corrections) instanton solution can be evaluated. Instanton presents a localized configuration with the size $\propto g^{-1}$. At $N=2$, it involves exactly 2 zero fermion modes and gives rise to fermion condensate $<\bar{\lambda}^a \lambda^a>_T$ which falls off $\propto \exp\{-\pi^{3/2} T/g\}$ at high $T$ but remains finite.   At low temperatures, both instanton and bosonization arguments also exhibit the appearance of fermion condensate $<\bar{\lambda}^a \lambda^a>_{T=0} ~\sim ~g$. For $N>2$, the situation is paradoxical. There are $2(N-1)$ fermion zero modes in the instanton background which implies the absence of the condensate in the massless limit. From the other hand, bosonization arguments suggest the appearance of the condensate for any $N$. Possible ways to resolve this paradox (which occurs also in some 4-dim gauge theories) are discussed. 
  A regular way to define an additive coproduct (or ``coaddition'') on the q-deformed differential complexes is proposed for quantum groups and quantum spaces related to the Hecke-type R-matrices. Several examples of braided coadditive differential bialgebras (Hopf algebras) are presented. 
  Some general techniques and theorems on the spacetime locality of the antifield formalism are illustrated in the familiar cases of the free scalar field, electromagnetism and Yang-Mills theory. The analysis explicitly shows that recent criticisms of the usual approach to dealing with locality are ill-founded. 
  We prove that each action of a compact matrix quantum group on a compact quantum space can be decomposed into irreducible representations of the group. We give the formula for the corresponding multiplicities in the case of the quotient quantum spaces. We describe the subgroups and the quotient spaces of quantum SU(2) and SO(3) groups. 
  The non-commutative differential calculus on the quantum groups $SL_q(N)$ is constructed. The quantum external algebra proposed contains the same number of generators as in the classical case. The exterior derivative defined in the constructive way obeys the modified version of the Leibnitz rules. 
  The generally accepted ``triviality'' of $\lambda\Phi^4$ theories does not forbid Spontaneous Symmetry Breaking but implies a trivially free shifted field which becomes effectively governed by a quadratic hamiltonian. As a consequence, one expects the one-loop potential to be exact . We present a lattice computation of the effective potential for massless $\lambda\Phi^4$ theory which nicely confirms the expectations based on ``triviality''. Our results imply that the magnitude of the Higgs boson mass, beyond perturbation theory, does not represent a measure of the observable interactions in the scalar sector of the standard model. 
  A system of Goldstone bosons - stemming from a symmetry breaking $O(N) \to O(N-1)$ - in a finite volume at finite temperature is considered. In the framework of dimensional regularization, the partition function is calculated to 3 loops for 3 and 4 dimensions, where Polyakov's measure for the functional integration is applied. Although the underlying theory is the non-linear $\sigma $ model, the 3 loop result turns out to be renormalizable in the sense that all the singularities can be absorbed by the couplings occuring so far. In finite volume, this property is highly non trivial and confirms the method for the measure. We also show that the result coincides with the one obtained using the Faddeev- Popov measure. This is also true for the maximal generalization of Polyakov's measure: none of the additional invariant terms that can be added contributes to the dimensionally regularized system. Our phenomenological Lagrangian describes e.g. 2 flavor chiral QCD as well as the classical Heisenberg model, but there are also points of contact with the Higgs model, superconductors etc. Moreover the finite size corrections to the susceptibility might improve the inerpretation of Monte Carlo results on the lattice. 
  We construct a new class of exact and stable superstring solutions based on $N=4$ superconformal world-sheet symmetry. In a subclass of these, the full spectrum of string excitations is derived in a modular-invariant way. In the weak curvature limit, our solutions describe a target space with non-trivial metric and topology, and generalize the previously known (semi) wormhole. The effective field theory limit is identified in certain cases, with solutions of the $N=4$ and $N=8$ extended gauged supergravities, in which the number of space-time supersymmetries is reduced by a factor of 2 because of the presence of non-trivial dilaton, gravitational and/or gauge backgrounds. In the context of string theory, our solutions correspond to stable non-critical superstrings in the strong coupling region; the super-Liouville field couples to a unitary matter system with central charge $5\le{\hat c}_M\le 9$. 
  The complete classification of WZNW modular invariant partition functions is known for very few affine algebras and levels, the most significant being all levels of $A_1$ and $A_2$ and level 1 of all simple algebras. Here, we address the classification problem for the nicest high rank semi-simple affine algebras: $(A_1^{(1)})^{\oplus_r}$. Among other things, we explicitly find all automorphism invariants, for all levels $k=(k_1,\ldots,k_r)$, and complete the classification for $A_1^{(1)}\oplus A_1^{(1)}$, for all levels $k_1,k_2$. We also solve the classification problem for $(A_1^{(1)})^{\oplus_r}$, for any levels $k_i$ with the property that for $i\ne j$ each $gcd(k_i+2,k_j+2)\leq 3$. In addition, we find some physical invariants which seem to be new. Together with some recent work by Stanev, the classification for all $(A^{(1)}_1)^{\oplus_r}_k$ could now be within sight. 
  The scattering of vortices at a critical value of the coupling constant in the Lagrangian can be approximated by a geodesic motion in the moduli space of classical static configurations of vortices. In this paper we give a scheme for generalising this idea to couplings that are near to the critical value. By perturbing a critically coupled field, we show that scattering of vortices at near-critical coupling can be approximated by motion in the original moduli space with a perturbed metric, and a potential. We apply this method to the scattering of two vortices, and compare our results to recent numerical simulations, and find good agreement where the scattering is not highly sensitive to radiation into other field modes. We also investigate the possibility of bound stable orbits of two vortices in the quantum field theory. 
  A ``dilute'' generalisation of the Birman--Wenzl--Murakami algebra is considered. It can be ``Baxterised'' to a solution of the Yang--Baxter algebra. The $D^{(2)}_{n+1}$ vertex models are examples of corresponding solvable lattice models and can be regarded as the dilute version of the $B^{(1)}_{n}$ vertex models. 
  Flow equations describe the evolution of the effective action $\Gamma_k$ in the process of varying an infrared cutoff $k$. The presence of the infrared cutoff explicitly breaks gauge and hence BRS invariance. We derive modified Slavnov-Taylor identities, which are valid for nonvanishing $k$. They guarantee the BRS invariance of $\Gamma_k$ for $k\to0$, and hence allow the study of non-abelian gauge theories by integrating the flow equations. Within a perturbative expansion of $\Gamma_k$, we derive an equation for a $k$ dependent mass term for the gauge fields implied by the modified Slavnov-Taylor identities. 
  We correct an error in our treatment of the tadpole contribution to the fluctuation determinant of the sphaleron, and also a minor mistake in a previous estimate. Thereby the overall agreement between the two existing exact computations and their consistency with the estimate is improved considerably. 
  We give new formulations of the solutions of the field equations of the affine Toda and conformal affine Toda theories on a cylinder and two-dimensional Minkowski space-time. These solutions are parameterised in terms of initial data and the resulting covariant phase spaces are diffeomorphic to the Hamiltonian ones. We derive the fundamental Poisson brackets of the parameters of the solutions and give the general static solutions for the affine theory. 
  We construct a class of superstring solutions in non trivial space-time. The existence of an $N=4$ world-sheet superconformal symmetry stabilizes our solutions under perturbative string loop corrections and implies in target space some unbroken space-time supersymmetries. 
  We obtain nonperturbative results in the framework of continuous Liouville theory. In particular, we express the specific heat ${\cal Z}$ of pure gravity in terms of an expansion of integrals on moduli spaces of punctured Riemann spheres. The integrands are written in terms of the Liouville action. We show that ${\cal Z}$ satisfies the Painlev\'e I. 
  We discuss charged string solutions of the effective equations of D=4 heterotic string theory with non-constant dilaton and modulus fields. The effective action contains a generic moduli-dependent coupling function in the gauge field kinetic term and a non-perturbative scalar potential. This is a review of hep-th/9307123 by M. Cvetic and the author. To be published in the Proceedings of 27-th International Symposium on Theory of Elementary Particles, Wendisch - Rietz, September 1993. 
  We review the relation between the matrix model and Liouville approaches to two-dimensional gravity as elaborated by Moore, Seiberg and Staudacher. Then, based on the supersymmetric Liouville formulation and the discrete eigenvalue model proposed by Alvarez-Gaum\'e, Itoyama, Ma\~nes and Zadra, we extend the previous relation to the supersymmetric case. The minisuperspace approximation for the supersymmetric case is formulated, and the corresponding wave equation is found. 
  An exact expression for the mass-gap, the ratio of the physical particle mass to the $\Lambda$-parameter, is found for the principal chiral sigma models associated to all the classical Lie algebras. The calculation is based on a comparison of the free-energy in the presence of a source coupling to a conserved charge of the theory computed in two ways: via the thermodynamic Bethe Ansatz from the exact scattering matrix and directly in perturbation theory. The calculation provides a non-trivial test of the form of the exact scattering matrix. 
  Inflation can occur in the cores of topological defects, where the scalar field is forced to stay near the maximum of its potential. This topological inflation does not require fine-tuning of the initial conditions. 
  Possible experimental manifestations of the contribution of heavy Kaluza-Klein particles, within a simple scalar model in six dimensions with spherical compactification, are studied. The approach is based on the assumption that the inverse radius $L^{-1}$ of the space of extra dimensions is of the order of the scale of the supersymmetry breaking $M_{SUSY} \sim 1 \div 10$ TeV. The total cross section of the scattering of two light particles is calculated to one loop order and the effect of the Kaluza-Klein tower is shown to be noticeable for energies $\sqrt{s} \geq 1.4 L^{-1}$. 
  The Schwinger-Dyson equations in the ladder approximation for $2D$ induced gravity coupled to fermions on a flat background are obtained in conformal gauge. A numerical study of these equations shows the possiblity of chiral symmetry breaking in this theory. 
  The implications of string theory for understanding the dimension of uncompactified spacetime are investigated. Using recent ideas in string cosmology, a new model is proposed to explain why three spatial dimensions grew large. Unlike the original work of Brandenberger and Vafa, this paradigm uses the theory of random walks. A computer model is developed to test the implications of this new approach. It is found that a four-dimensional spacetime can be explained by the proper choice of initial conditions. 
  We study supersymmetric (extreme) domain walls in four-dimensional (4d) N=1 supergravity theories with a general dilaton coupling $\alpha > 0$. Type I walls, which are static, planar (say, in ($x,y$) plane) configurations, interpolate between Minkowski space-time and a vacuum with a varying dilaton field. We classify their global space-time with respect to the value of the coupling $\alpha$. $N=1$ supergravity with $\alpha =1$, an effective theory from superstrings, provides a dividing line between the theories with $\alpha>1$, where there is a naked (planar) singularity on one side of the wall, and the theories with $\alpha<1$, where the singularity of the of the wall is covered by the horizon. The global space-time (in $(t,z$) direction) of the extreme walls with the coupling $\alpha$ is the same as the global space-time (in ($t,r)$ direction) of the extreme magnetically charged black holes with the coupling $1/\alpha$. 
  The simple algebras of a dressed operator, which is composed of a dressing and a residual operators, are averaged following a proper statistics of the dressing one. In the Bose-Einstein statistics, a (fermionic) Calogero-Vasiliev oscillator, $q$-boson (fermion), and (fermionic) $su_q(1,1)$ are obtained for each bosonic (fermionic) residual operator. In the Fermi-Dirac statistics, new similar algebras are derived for each residual operator. Constructions of dual $q$-algebras, such as a dual Calogero-Vasiliev oscillator, a dual $q$-boson and a $su_q(2)$, and prospects are discussed. 
  We compute, by free field techniques, the scalar product of the SU(2) Chern-Simons states on genus > 1 surfaces. The result is a finite-dimensional integral over positions of ``screening charges'' and one complex modular parameter. It uses an effective description of the CS states closely related to the one worked out by Bertram. The scalar product formula allows to express the higher genus partition functions of the WZW conformal field theory by finite-dimensional integrals. It should provide the hermitian metric preserved by the Knizhnik-Zamolodchikov-Bernard connection describing the variations of the CS states under the change of the complex structure of the surface. 
  Two-dimensional SU($N$) gauge theory is accurately analyzed with the light-front Tamm-Dancoff approximation, both numerically and analytically. The light-front Einstein-Schr\"odinger equation for mesonic mass reduces to the 't Hooft equation in the large $N$ limit, $g^2N$ fixed, where $g$ is the coupling constant. Hadronic masses are numerically obtained in the region of $m^2 \ll g^2N$, where $m$ is the bare quark (q) mass. The lightest mesonic and baryonic states are almost in valence. The second lightest mesonic state is highly relativistic in the sense that it has a large 4-body ($ {\rm qq} \bar{\rm q} \bar{\rm q} $) component in addition to the valence (${\rm q} \bar{\rm q}$) one. In the strong coupling limit our results are consistent with the prediction of the bosonization for ratios of the lightest and second lightest mesonic masses to the lightest baryonic one. Analytic solutions to the lightest hadronic masses are obtained, with a reasonable approximation, as $\sqrt{2Cm}(1-1/N^2)^{1/4}$ in the mesonic case and $\sqrt{CmN(N-1)}(1-1/N^2)^{1/4}$ in the baryonic case, where $C=(g^2N\pi/6)^{1/2}$. The solutions well reproduce the numerical ones. The $N$- and $m$-dependences of the hadronic masses are explicitly shown by the analytical solutions. 
  Polynomial invariants corresponding to the fundamental representation of the gauge group $SU(N)$ are computed for arbitrary torus knots and links in the framework of Chern-Simons gauge theory making use of knot operators. As a result, a formula for the HOMFLY polynomial for arbitrary torus links is presented. 
  Explicit expressions for three series of $R$ matrices which are related to a ``dilute'' generalisation of the Birman--Wenzl--Murakami are presented. Of those, one series is equivalent to the quantum $R$ matrices of the $D^{(2)}_{n+1}$ generalised Toda systems whereas the remaining two series appear to be new. 
  We investigate nonperturbative canonical quantization of two dimensional dilaton gravity theories with an emphasis on the CGHS model. We use an approach where a canonical transformation is constructed such that the constraints take a quadratic form. The required canonical transformation is obtained by using a method based on the B\"acklund transformation from the Liouville theory. We quantize dilaton gravity in terms of the new variables, where it takes a form of a bosonic string theory with background charges. Unitarity is then established by going into a light-cone gauge. As a direct consequence, black holes in this theory do not violate unitarity, and there is no information loss. We argue that the information escapes during the evaporation process. We also discuss the implications of this quantization scheme for the quantum fate of real black holes. The main conclusion is that black holes do not have to violate quantum mechanics. 
  We re-examine three-dimensional gauge theory with a Chern-Simons term in which the Lorentz invariance is spontaneously broken by dynamical generation of a magnetic field. A non-vanishing magnetic field leads, through the Nambu-Goldstone theorem, to the decrease of zero-point energies of photons, which accounts for a major part of the mechanism. The asymmetric spectral flow plays an important role. The instability in pure Chern-Simons theory is also noted. 
  Motivated by some questions in the path integral approach to (topological) gauge theories, we are led to address the following question: given a smooth map from a manifold $M$ to a compact group $G$, is it possible to smoothly `diagonalize' it, i.e.~conjugate it into a map to a maximal torus $T$ of $G$? We analyze the local and global obstructions and give a complete solution to the problem for regular maps. We establish that these can always be smoothly diagonalized locally and that the obstructions to doing this globally are non-trivial Weyl group and torus bundles on $M$. We show how the patching of local diagonalizing maps gives rise to non-trivial $T$-bundles, explain the relation to winding numbers of maps into $G/T$ and restrictions of the structure group and examine the behaviour of gauge fields under this diagonalization. We also discuss the complications that arise for non-regular maps and in the presence of non-trivial $G$-bundles. In particular, we establish a relation between the existence of regular sections of a non-trivial adjoint bundle and restrictions of the structure group of a principal $G$-bundle to $T$. We use these results to justify a Weyl integral formula for functional integrals which, as a novel feature not seen in the finite-dimensional case, contains a summation over all those topological $T$-sectors which arise as restrictions of a trivial principal $G$ bundle and which was used previously to solve completely Yang-Mills theory and the $G/G$ model in two dimensions. 
  Systematic use of the infinite-dimensional spin representation simplifies and rigorizes several questions in Quantum Field Theory. This representation permutes ``Gaussian'' elements in the fermion Fock space, and is necessarily projective: we compute its cocycle at the group level, and obtain Schwinger terms and anomalies from infinitesimal versions of this cocycle. Quantization, in this framework, depends on the choice of the ``right'' complex structure on the space of solutions of the Dirac equation. We show how the spin representation allows one to compute exactly the S-matrix for fermions in an external field; the cocycle yields a causality condition needed to determine the phase. 
  The symmetry algebra of the two-dimensional quantum harmonic oscillator with rational ratio of frequencies is identified as a non-linear extension of the u(2) algebra. The finite dimensional representation modules of this algebra are studied and the energy eigenvalues are determined using algebraic methods of general applicability to quantum superintegrable systems. 
  The quantization of the induced 2d-gravity on a compact spatial section is carried out in three different ways. In the three approaches the supermomentum constraint is solved at the classical level but they differ in the way the hamiltonian constraint is imposed. We compare these approaches establishing an isomorphism between the resulting Hilbert spaces. 
  We translate effectively our earlier quantum constructions to the classical language and using Yang-Baxterisation of the Faddeev-Reshetikhin-Takhtajan algebra are able to construct Lax operators and associated $r$-matrices of classical integrable models. Thus new as well as known lattice systems of different classes are generated including new types of collective integrable models and canonical models with nonstandard $r$ matrices. 
  We introduce in a natural and straigthforward way the $loop$ (Lagrangian) $representation$ for the partition function of pure compact lattice QED. The corresponding classical lattice loop action is proportional to the quadratic area of the loop world sheets. We discuss the parallelism between the $loop$ formulation of this model in terms of world sheets of loops and the $topological$ representation of the Higgs (broken) phase for the non-compact lattice QED in terms of world sheets of Nielsen-Olesen strings. 
  We present the extension of the Lagrangian $loop$ representation in such a way to introduce matter fields. The partition function of lattice compact U(1) Gauge-Higgs model is expressed as a sum over closed as much as open surfaces. These surfaces correspond to world sheets of loop-like pure electric flux excitations and open electric flux tubes carrying matter fields at their ends. There is a duality transformation between this description in terms of loop world sheets and the $topological$ representation in terms of world sheets of Nielsen-Olesen strings both closed and open joining pairs of monopoles. 
  We consider the implications of some simple assumptions about the nature of the quantum theory of gravity which are plausible for a class of possible theories I have been attempting to construct. The simple assumptions turn out to have surprizingly wide implications of a type one might call philosophical. The paper is short and nontechnical. ( This builds on recent unpublished work of Lee Smolin, but comes to opposite conclusions). 
  In this note we present an explicit procedure for the regularization of tree level amplitudes involving discrete states, using open string field theory. We show that there is a natural correspondence between the discrete states and off--shell states, later acting as a regularized version of the former. A general off-shell state corresponds to several physical states. In order to obtain the well--defined $S$ matrix elements one has to choose representatives close but not equal to the desired values of external momenta. The procedure renders finite all $4$-point amplitudes with an even number of (naively) divergent channels even after the regularization is removed. The rest of the amplitudes can be defined by means of such regularization. 
  This work describes the formulation of the manifestly ghost-free (spacetime) light-cone gauge for bosonic string theory with non-trivial spacetime metric, antisymmetric tensor, dilaton and tachyon fields. The action is a general two-dimensional sigma model, corresponding to a closed string theory with a second order action in the Polyakov picture. The spacetime fields must have a symmetry generated by a null, covariantly constant spacetime vector in order for the light-cone gauge to be accessible. Also, the theory must be Weyl invariant. The conditions for Weyl invariance are computed within the light-cone gauge, reproducing the usual beta functions. The calculation of the dilaton beta function and the critical dimension is somewhat novel in this ghost-free theory. Some exactly solvable light-cone theories are discussed. 
  We describe a topological string theory which reproduces many aspects of the 1/N expansion of SU(N) Yang-Mills theory in two spacetime dimensions in the zero coupling (A=0) limit. The string theory is a modified version of topological gravity coupled to a topological sigma model with spacetime as target. The derivation of the string theory relies on a new interpretation of Gross and Taylor's ``\Omega^{-1} points.'' We describe how inclusion of the area, coupling of chiral sectors, and Wilson loop expectation values can be incorporated in the topological string approach. 
  Based on the extended BRST formalism of Batalin, Fradkin and Vilkovisky, we perform a general algebraic analysis of the BRST anomalies in superstring theory of Neveu-Schwarz-Ramond. Consistency conditions on the BRST anomalies are completely solved. The genuine super-Virasoro anomaly is identified with the essentially unique solution to the consistency condition without any reference to a particular gauge for the 2D supergravity fields. In a configuration space where metric and gravitino fields are properly constructed, general form of the super-Weyl anomaly is obtained from the super-Virasoro anomaly as its descendant.   We give a novel local action of super-Liouville type, which plays a role of Wess-Zumino-Witten term shifting the super-Virasoro anomaly into the super-Weyl anomaly. These results reveal a hierarchial relationship in the BRST anoamlies. 
  It is proposed that gravity may arise in the low energy limit of a model of matter fields defined on a special kind of a dynamical random lattice. Time is discretized into regular intervals, whereas the discretization of space is random and dynamical. A triangulation is associated to each distribution of the spacetime points using the flat metric of the embedding space. We introduce a diffeomorphism invariant, bilinear scalar action, but no ``pure gravity'' action.   Evidence for the existence of a non-trivial continuum limit is provided by showing that the zero momentum scalar excitation has a finite energy in the limit of vanishing lattice spacing. Assuming the existence of localized low energy states which are described by a natural set of observables, we show that an effective curved metric will be induced dynamically. The components of the metric tensor are identified with quasi-local averages of certain microscopic properties of the quantum spacetime. The Planck scale is identified with the highest mass scale of the matter sector. 
  $\delta'$-function perturbations and Neumann boundary conditions are incorporated into the path integral formalism. The starting point is the consideration of the path integral representation for the one dimensional Dirac particle together with a relativistic point interaction. The non-relativistic limit yields either a usual $\delta$-function or a $\delta'$-function perturbation; making their strengths infinitely repulsive one obtains Dirichlet, respectively Neumann boundary conditions in the path integral. 
  We consider quantum analogs of the relativistic Toda lattices and give new $2\times 2$ $L$-operators for these models. Making use of the variable separation the spectral problem for the quantum integrals of motion is reduced to solving one-dimensional separation equations. 
  Various reductions, and soime solutions of the classical equations of motion of a relativistic membrane are given 
  We study the conformal field theory of a free massless scalar field living on the half line with interactions introduced via a periodic potential at the boundary. An SU(2) current algebra underlies this system and the interacting boundary state is given by a global SU(2) rotation of the left-moving fields in the zero-potential (Neumann) boundary state. As the potential strength varies from zero to infinity, the boundary state interpolates between the Neumann and the Dirichlet values. The full S-matrix for scattering from the boundary, with arbitrary particle production, is explicitly computed. To maintain unitarity, it is necessary to attribute a hidden discrete ``soliton'' degree of freedom to the boundary. The same unitarity puzzle occurs in the Kondo problem, and we anticipate a similar solution. 
  Calculating the (a,c) ring of the maximal phase orbifold for `invertible' Landau--Ginzburg models, we show that the Berglund--H"ubsch construction works for all potentials of the relevant type. The map that sends a monomial in the original model to a twisted state in the orbifold representation of the mirror is constructed explicitly. Via this map, the OP selection rules of the chiral ring exactly correspond to the twist selection rules for the orbifold. This shows that we indeed arrive at the correct point in moduli space, and that the mirror map can be extended to arbitrary orbifolds, including non-abelian twists and discrete torsion, by modding out the appropriate quantum symmetries. 
  We investigate the global structure of inflationary universe both by analytical methods and by computer simulations of stochastic processes in the early Universe. We show that the global structure of the universe depends crucially on the mechanism of inflation. In the simplest models of chaotic inflation the Universe looks like a sea of thermalized phase surrounding permanently self-reproducing inflationary domains. In the theories where inflation occurs near a local extremum of the effective potential corresponding to a metastable state, the Universe looks like de Sitter space surrounding islands of thermalized phase. A similar picture appears even if the state $\phi = 0$ is unstable but the effective potential has a discrete symmetry $\phi \to =-\phi$. In this case the Universe becomes divided into domains containing different phases. These domains will be separated from each other by domain walls. However, unlike ordinary domain walls, these domain walls will inflate, and their thickness will exponentially grow. In the theories with continuous symmetries inflation generates exponentially expanding strings and monopoles surrounded by thermalized phase. Inflating topological defects will be stable, and they will unceasingly produce new inflating topological defects. This means that topological defects may play a role of indestructible seeds for eternal inflation. 
  The N-dimensional Cayley-Klein scheme allows the simultaneous description of $3^N$ geometries (symmetric orthogonal homogeneous spaces) by means of a set of Lie algebras depending on $N$ real parameters. We present here a quantum deformation of the Lie algebras generating the groups of motion of the two and three dimensional Cayley-Klein geometries. This deformation (Hopf algebra structure) is presented in a compact form by using a formalism developed for the case of (quasi) free Lie algebras. Their quasitriangularity (i.e., the most usual way to study the associativity of their dual objects, the quantum groups) is also discussed. 
  Using Wilson-Polchinski renormalization group equations, we give a simple new proof of decoupling in a $\phi^4$-type scalar field theory involving two real scalar fields (one is heavy with mass $M$ and the other light). Then, to all orders in perturbation theory, it is shown that effects of virtual heavy particles up to the order $1/M^{2N_0}$ can be systematically incorporated into light-particle theory via effective local vertices of canonical dimension at most $4+2N_0$. The couplings for vertices of dimension $4+2N$ are of order $1/M^{2N}$ and are systematically calculable. All this is achieved through intuitive dimensional arguments without resorting to complicated graphical arguments or convergence theorems. 
  We review the detailed structure of the large $N=4$ superconformal algebra, and construct its BRST operator which constitutes the main object for analyzing $N=4$ strings. We then derive the general condition for the nilpotency of the BRST operator and show that there exists a line of critical $N=4$ string theories. 
  We describe mirror manifolds in dimensions different from the familiar case of complex threefolds. We emphasize the simplifying features of dimension three and supply more robust methods that do not rely on such special characteristics and hence naturally generalize to other dimensions. The moduli spaces for Calabi--Yau $d$-folds are somewhat different from the ``special K\"ahler manifolds'' which had occurred for $d=3$, and we indicate the new geometrical structures which arise. We formulate and apply procedures which allow for the construction of mirror maps and the calculation of order-by-order instanton corrections to Yukawa couplings. Mathematically, these corrections are expected to correspond to calculating Chern classes of various parameter spaces (Hilbert schemes) for rational curves on Calabi--Yau manifolds. Our results agree with those obtained by more traditional mathematical methods in the limited number of cases for which the latter analysis can be carried out. Finally, we make explicit some striking relations between instanton corrections for various Yukawa couplings, derived from the associativity of the operator product algebra. 
  We construct a new class of exact string solutions with a four dimensional target space metric of signature ($-,+,+,+$) by gauging the independent left and right nilpotent subgroups with `null' generators of WZNW models for rank 2 non-compact groups $G$. The `null' property of the generators (${\rm Tr }(N_n N_m)=0$) implies the consistency of the gauging and the absence of $\a'$-corrections to the semiclassical backgrounds obtained from the gauged WZNW models. In the case of the maximally non-compact groups ($G= SL(3), SO(2,2), SO(2,3), G_2$) the construction corresponds to gauging some of the subgroups generated by the nilpotent `step' operators in the Gauss decomposition. The rank 2 case is a particular example of a general construction leading to conformal backgrounds with one time-like direction. The conformal theories obtained by integrating out the gauge field can be considered as sigma model analogs of Toda models (their classical equations of motion are equivalent to Toda model equations). The procedure of `null gauging' applies also to other non-compact groups. 
  Path integral formulations for the Smorodinsky-Winternitz potentials in two- and three-dimen\-sional Euclidean space are presented. We mention all coordinate systems which separate the Smorodinsky-Winternitz potentials and state the corresponding path integral formulations. Whereas in many coordinate systems an explicit path integral formulation is not possible, we list in all soluble cases the path integral evaluations explicitly in terms of the propagators and the spectral expansions into the wave-functions. 
  We study the $N = 2$ coset models in their formulation as supersymmetric gauged Wess-Zumino-Witten models. A model based on the coset $G/H$ is invariant under a symmetry group isomorphic to $\Z_{k+Q}$, where $k$ is the level of the model and $Q$ is the dual Coxeter number of $G$. Using a duality-like relationship, we show that the $\Z_m$ orbifold of the vectorially gauged model and the $\Z_{\tilde{m}}$ orbifold of the axially gauged model are each others mirror partners when $m \tilde{m} = k + Q$. 
  The vacuum energy density is obtained for the $O(N)$ nonlinear sigma model. It is shown that non-perturbative contributions are connected with the square of the symmetry current of the group $O(N)$. This result is valid for $\sigma$- fields which are subject to the constraint. 
  We develop the basic formalism of complex $q$-analysis to study the solutions of second order $q$-difference equations which reduce, in the $q\rightarrow 1$ limit, to the ordinary Laplace equation in Euclidean and Minkowski space. After defining an inner product on the function space we construct and study the properties of the solutions, and then apply this formalism to the Schr\"{o}dinger equation and two-dimensional scalar field theory. 
  Susskind has recently shown that a relativistic string approaching the event horizon of a black hole spreads in both the transverse and longitudinal directions in the reference frame of an outside observer. The transverse spreading can be described as a branching diffusion of wee string bits. This stochastic process provides a mechanism for thermalizing the quantum state of the string as it spreads across the stretched horizon. 
  The quasiclassical asymptotics of the Knizhnik-Zamolodchikov system is studied. Solutions to this system in this limit are related naturally to Bethe vectors in the Gaudin model of spin chains. 
  A universal R--matrix for the quantum Heisenberg algebra h(1)q is presented. Despite of the non--quasitriangularity of this Hopf algebra, the quantum group induced from it coincides with the quasitriangular deformation already known. 
  An approach to understand Fractional Quantum Hall Effect (FQHE) using anomalies is studied in this paper. More specifically, this is done by looking at the anomaly in the current conservation equation of a WZNW theory describing fields living at the edge of the two dimensional Hall sample. This WZNW theory itself comes from the non-Abelian bosonisation of fermions living at the edge. It is shown that this model can describe both integer and fractional quantization of conductivities in a unified manner. 
  Physical content of the nonrelativistic quantum field theory with non-Abelian Chern-Simons interactions is clarified with the help of the equivalent first- quantized description which we derive in any physical gauge. 
  The short note is devoted to the setting free of hidden symmetries in Verma modules over sl(2,C) by the noncommutative Veronese mappings. 
  A model describing Ising spins with short range interactions moving randomly in a plane is considered. In the presence of a hard core repulsion, which prevents the Ising spins from overlapping, the model is analogous to a dynamically triangulated Ising model with spins constrained to move on a flat surface. It is found that as a function of coupling strength and hard core repulsion the model exhibits multicritical behavior, with first and second order transition lines terminating at a tricritical point. The thermal and magnetic exponents computed at the tricritical point are consistent with the exact two-matrix model solution of the random Ising model, introduced previously to describe the effects of fluctuating geometries. 
  We introduce a new construction of bilinear invariant forms on Lie algebras, based on the method of graded contractions. The general method is described and the $\Bbb Z_2$-, $\Bbb Z_3$-, and $\Bbb Z_2\otimes\Bbb Z_2$-contractions are found. The results can be applied to all Lie algebras and superalgebras (finite or infinite dimensional) which admit the chosen gradings. We consider some examples: contractions of the Killing form, toroidal contractions of $su(3)$, and we briefly discuss the limit to new WZW actions. 
  BRST operators for two-dimensional theories with spin-2 and spin-$s$ currents, generalising the $W_3$ BRST operator of Thierry-Mieg, have previously been obtained. The construction was based on demanding nilpotence of the BRST operators, making no reference to whether or not an underlying $W$ algebra exists. In this paper, we analyse the known cases ($s=3$, 4, 5 and 6), showing that the two $s=4$ BRST operators are associated with the $W\!B_2$ algebra, and that two of the four $s=6$ BRST operators are associated with the $W\!G_2$ algebra. We discuss the cohomology of all the known higher-spin BRST operators, the Weyl symmetry of their physical states, and their relation with certain minimal models. We also obtain the BRST operator for the case $s=7$. 
  We extend the universal differential calculus on an arbitrary Hopf algebra to a ``universal Cartan calculus''. This is accomplished by introducing inner derivations and Lie derivatives which act on the elements of the universal differential envelope. A new algebra is formulated by incorporating these new objects into the universal differential calculus together with consistent commutation relations. We also explain how to include nontrivial commutation relations into this formulation to obtain the ``generalized Cartan calculus''. 
  We relate the semiclassical limit of the quantum Yang-Mills partition function on a compact oriented surface to the symplectic volume of the moduli space of flat connections, by using an explicit expression for the symplectic form. This gives an independent proof of some recent results of Witten and Forman. 
  The commutator of string fields is considered in the context of light cone string field theory. It is shown that the commutator is in general non--vanishing outside the string light cone. This could have profound implications for our understanding of the localization of information in quantum gravity. 
  It is shown that time-independent solutions to the (2+1)-dimensional non- linear O(3) sigma model may be placed in correspondence with surfaces of constant mean curvature in three-dimensional Euclidean space. The tools required to establish this correspondence are provided by the classical differential geometry of surfaces. A constant-mean-curvature surface induces a solution to the O(3) model through the identification of the Gauss map, or normal vector, of the surface with the field vector of the sigma model. Some explicit solutions, including the solitons and antisolitons discovered by Belavin and Polyakov, and a more general solution due to Purkait and Ray, are considered and the surfaces giving rise to them are found explicitly. It is seen, for example, that the Belavin-Polyakov solutions are induced by the Gauss maps of surfaces which are conformal to their spherical images, i.e. spheres and minimal surfaces, and that the Purkait-Ray solution corresponds to the family of constant-mean-curvature helicoids first studied by do Carmo and Dajczer in 1982. A generalisation of this method to include time-dependence may shed new light on the role of the Hopf invariant in this model. 
  We explicitly construct a series of lattice models based upon the gauge group $Z_{p}$ which have the property of subdivision invariance, when the coupling parameter is quantized and the field configurations are restricted to satisfy a type of mod-$p$ flatness condition. The simplest model of this type yields the Dijkgraaf-Witten invariant of a $3$-manifold and is based upon a single link, or $1$-simplex, field. Depending upon the manifold's dimension, other models may have more than one species of field variable, and these may be based on higher dimensional simplices. 
  Zamolodchikov's tetrahedron equations, which were derived by considering the scattering of straight strings, can be written in three different labeling schemes: one can use as labels the states of the vacua between the strings, the states of the string segments, or the states of the particles at the intersections of the strings. We give a detailed derivation of the three corresponding tetrahedron equations and show also how the Frenkel-Moore equations fits in as a {\em nonlocal} string labeling. We discuss then how an analog of the Wu-Kadanoff duality can be defined between each pair of the above three labeling schemes. It turns out that there are two cases, for which one can simultaneously construct a duality between {\em all} three pairs of labelings. 
  We prove that for $SU(2)$ and $SO(3)$ quantum gauge theory on a torus, holonomy expectation values with respect to the Yang-Mills measure $d\mu_T(\o) =N_T^{-1}e^{-S_{YM}(\o)/T}[{\cal D}\o]$ converge, as $T\downarrow 0$, to integrals with respect to a symplectic volume measure $\mu_0$ on the moduli space of flat connections on the bundle. These moduli spaces and the symplectic structures are described explicitly. 
  Being inspired by Kaplan's proposal for simulating chiral fermions on a lattice, we examine the continuum analog of his domain-wall construction for two-dimensional chiral Schwinger models. Adopting slightly unusual dimensional regularization, we explicitly evaluate the one-loop effective action in the limit that the domain-wall mass goes to infinity. For anomaly-free cases, the effective action turns out to be gauge invariant in two-dimensional sense. 
  A novel Hopf algebra $ ( {\tilde G}_{r,s} )$, depending on two deformation parameters and five generators, has been constructed. This $ {\tilde G}_{r,s}$ Hopf algebra might be considered as some quantisation of classical $GL(2) \otimes GL(1) $ group, which contains the standard $GL_q(2)$ quantum group (with $ q=r^{-1} $) as a Hopf subalgebra. However, we interestingly observe that the two parameter deformed $GL_{p,q}(2)$ quantum group can also be realised through the generators of this $ {\tilde G}_{r,s}$ algebra, provided the sets of deformation parameters $p,~q$ and $r,~s$ are related to each other in a particular fashion. Subsequently we construct the invariant noncommutative planes associated with $ {\tilde G}_{r,s}$ algebra and show how the two well known Manin planes corresponding to $GL_{p,q}(2)$ quantum group can easily be reproduced through such construction. Finally we consider the `coloured' extension of $GL_{p,q}(2)$ quantum group as well as corresponding Manin planes and explore their intimate connection with the `coloured' extension of ${\tilde G}_{r,s}$ Hopf structure. 
  Using finite-size-scaling methods, we study the quantum chain version of the spin-$1$-Blume-Capel model coupled to an imaginary field. The aim is to realize higher order non-unitary conformal field theories in a simple Ising-type spin model. We find that the first ground-state level crossing in the high-temperature phase leads to a second-order phase transition of the Yang-Lee universality class (central charge $c=-22/5$). The Yang-Lee transition region ends at a line of a new type of tricriticality, where the {\em three} lowest energy levels become degenerate. The analysis of the spectrum at two points on this line gives good evidence that this line belongs to the universality class of the ${\cal M}_{2,7}$-conformal theory with $c=-68/7$. 
  We demonstrate that for the sine-Gordon theory at the free fermion point, the 2-point correlation functions of the fields $\exp (i\al \Phi )$ for $0< \al < 1$ can be parameterized in terms of a solution to a sinh-Gordon-like equation. This result is derived by summing over intermediate multiparticle states and using the form factors to express this as a Fredholm determinant. The proof of the differential equations relies on a $\Zmath_2$ graded multiplication law satisfied by the integral operators of the Fredholm determinant. Using this methodology, we give a new proof of the differential equations which govern the spin and disorder field correlators in the Ising model. 
  We propose a quantum lattice version of Feigin and E. Frenkel's constructions, identifying the KdV differential polynomials with functions on a homogeneous space under the nilpotent part of $\widehat{s\ell}_2$. We construct an action of the nilpotent part $U_q\widehat n_+$ of $U_q\widehat{s\ell}_2$ on their lattice counterparts, and embed the lattice variables in a $U_q\widehat n_+$-module, coinduced from a quantum version of the principal commutative subalgebra, which is defined using the identification of $U_q\widehat n_+$ with its coordinate algebra. 
  We construct the chiral Wess-Zumino term as a solution for the Batalin-Vilkovisky master equation for anomalous two-dimensional gauge theories, working in an extended field-antifield space, where the gauge group elements are introduced as additional degrees of freedom.   We analyze the Abelian and the non-Abelian cases, calculating in both cases the BRST generator in order to show the physical equivalence between this chiral solution for the master equation and the usual (non-chiral) one. 
  The paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry. In particular, it contains an axiomatic treatment of Gromov-Witten classes, and a discussion of their properties for Fano varieties. Cohomological Field Theories are defined, and it is proved that tree level theories are determined by their correlation functions. Applications to counting rational curves on del Pezzo surfaces and projective spaces are given. 
  Several aspects of (0,2) Landau-Ginzburg orbifolds are investigated. Especially the elliptic genera are computed in general and, for a class of models recently invented by Distler and Kachru, they are compared with the ones from (0,2) sigma models. Our formalism gives an easy way to calculate the generation numbers for lots of Distler-Kachru models even if they are based on singular Calabi-Yau spaces. We also make some general remarks on the Born-Oppenheimer calculation of the ground states elucidating its mathematical meaning in the untwisted sector. For Distler-Kachru models based on non-singular Calabi-Yau spaces we show that there exist `residue' type formulas of the elliptic genera as well. 
  Rotating stringy black hole solutions with non-vanishing dilaton $\phi$, antisymmetric tensor $B_{\mu\nu}$, and $U(1)$ gauge field $A_{\mu}$ are investigated. Both Boyer-Lindquist-like and Kerr-Schild-like coordinate are constructed. The latter is utilised to construct the analytically extended spacetime. The global structure of the resulting extended spacetime is almost identical to that of the Kerr. In carrying out the analytic extension, the radial coordinate should be suitably chosen so that we can avoid singularity caused by the twisting. The thermodynamic property of the stringy black hole is examined through the injection of test bodies into the black hole. It is shown that one cannot change a black hole configuration into a naked singularity by way of throwing test bodies into the black hole. The global $O(2,3)$ symmetry and the preservation of the asymptotic flatness are discussed. When we impose stationarity, axisymmetry, and asymptotic flatness, there is no other twisting than the one pointed out by A.Sen\cite{sen}. All the other elements of $O(2,3)$ either break the asymptotic flatness, or cause only coordinate transformations and gives no physical change. 
  In the framework of the recently proposed asymptotically finite gauge models the cosmological constant is essentially weakened by quantum effects. The next (and more general) claim is that the coupling between quantum fields may suppress their contributions to the induced cosmological constant. 
  In this letter we consider the nonlinear realizations of the classical Polyakov's algebra $W_3^{(2)}$. The coset space method and the covariant reduction procedure allow us to deduce the Boussinesq equation with interchanged space and evolution coordinates. By adding one more space coordinate and introducing two copies of the $W_3^{(2)}$ algebra, the same method yields the $sl(3,R)$ Toda lattice equations. 
  We study the Chern-Simons theory coupled to matter field by means of an effective Lagrangian obtained from the Batalin-Fradkin-Vilkovisky formalism. We show that there is no rotational anomaly for any proper gauge we choose. 
  Rational Hopf algebras (certain quasitriangular weak quasi-Hopf $^*$-algebras) are expected to describe the quantum symmetry of rational field theories. In this paper methods are developped which allow for a classification of all rational Hopf algebras that are compatible with some prescribed set of fusion rules. The algebras are parametrized by the solutions of the square, pentagon and hexagon identities. As examples, we classify all solutions for fusion rules with two or three sectors, and for the level three affine $A_1$ fusion rules. We also establish several general properties of rational Hopf algebras, and we present a graphical description of the coassociator in terms of labelled tetrahedra which allows to make contact with conformal field theory fusing matrices and with invariants of three-manifolds and topological lattice field theory. 
  A higher-derivative, interacting, scalar field theory in curved spacetime with the most general action of sigma-model type is studied. The one-loop counterterms of the general theory are found. The renormalization group equations corresponding to two different, multiplicatively renormalizable variants of the same are derived. The analysis of their asymptotic solutions shows that, depending on the sign of one of the coupling constants, we can construct an asymptotically free theory which is also asymptotically conformal invariant at strong (or small) curvature. The connection that can be established between one of the multiplicatively renormalizable variants of the theory and the effective theory of the conformal factor, aiming at the description of quantum gravity at large distances, is investigated. 
  The two-dimensional inhomogeneous zeta-function series (with homogeneous part of the most general Epstein type): \[ \sum_{m,n \in \mbox{\bf Z}} (am^2+bmn+cn^2+q)^{-s}, \] is analytically continued in the variable $s$ by using zeta-function techniques. A simple formula is obtained, which extends the Chowla-Selberg formula to inhomogeneous Epstein zeta-functions. The new expression is then applied to solve the problem of computing the determinant of the basic differential operator that appears in an attempt at quantizing gravity by using the Wheeler-De Witt equation in 2+1 dimensional spacetime with the torus topology. 
  The $c=1$ matrix model is equivalent to $1+1$ dimensional string theory. However, the tachyon self-interaction in the former is local, while in the latter it is nonlocal due to the gravitational, dilaton and higher string fields. By studying scattering of classical pulses we show that the appropriate nonlocal field redefinition converts the local matrix model interaction into the expected string form. In particular, we see how the asymptotic behavior of the gravitational field appears in the scattering. 
  A new integrable class of quantum models representing a family of different discrete-time or relativistic generalisations of the periodic Toda chain (TC), including that of a recently proposed classical close to TC model [7] is presented. All such models are shown to be obtainable from a single ancestor model at different realisations of the underlying quantised algebra. As a consequence the $(2\times 2)$ Lax operators and the associated quantum $R$-matrices for these models are easily derived ensuring their quantum integrability.   It is shown that the functional Bethe ansatz developed for the quntum TC is trivially generalised to achieve seperation of variables also for the present models. 
  We use Schwinger's formula, introduced by himself in the early fifties to compute effective actions for QED, and recently applied to the Casimir effect, to obtain the partition functions for both the bosonic and fermionic harmonic oscillator. 
  The representation theory of the quantum group su$_q(2)$ is used to introduce $q$-analogues of the Wigner rotation matrices, spherical functions, and Legendre polynomials. The method amounts to an extension of variable separation from Laplace equations to certain differential-dilation equations. 
  The relation between the $\widehat\gl(\infty)$ symmetry of the Kadomtsev-Petviashvili hierarchy and the Kac-Moody-Virasoro Lie point symmetries of the individual equations is established. The Lie point symmetries are shown to be the only local ones. 
  The relation between the $\widehat{\Sl}(\infty)$ algebra of flows commuting with the KP hierarchy and the Kac-Moody-Virasoro Lie point symmetries of individual equations is established. This is used to calculate the point symmetries for all equations in the hierarchy. 
  We review the application of the loop representation to gauge theories and general relativity. The emphasis lies on exhibiting the loop calculus techniques, and their application to the canonical quantization. We discuss the role that knot theory and loop coordinates play in the determination of nondegenerate quantum states of the gravitational field. 
  This paper discusses the Higgs and spinor fermion terms of the D4-D5-E6 model of a series of papers (hep-ph/9301210, hep-th/9302030, hep-th/9306011, and hep-th/9402003) an 8-dimensional spacetime is reduced to 4-dimensions. The gauge boson terms give SU(3)xSU(2)xU(1) for the color, weak, and electromagnetic forces and gravity of the MacDowell-Mansouri type, which has recently been shown by Nieto, Obregon, and Socorro (gr-qc/9402029) to be equivalent, up to a Pontrjagin topological term, to the Ashtekar formulation. 
  The 2+1 black hole anti de Sitter solution, obtained as a special limit of the conformally exact $SL(2,R)\otimes SO(1,1)/SO(1,1)$ coset WZW model to all orders in 1/k, is investigated with respect to tachyon scattering. We calculate the off-shell reflection and transmission coefficients and we find an expression for the Hawking temperature, which in a certain limit reduces to the statistical result. 
  Using Dirac's approach to constrained dynamics, the Hamiltonian formulation of regular higher order Lagrangians is developed. The conventional description of such systems due to Ostrogradsky is recovered. However, unlike the latter, the present analysis yields in a transparent manner the local structure of the associated phase space and its local sympletic geometry, and is of direct application to {\em constrained\/} higher order Lagrangian systems which are beyond the scope of Ostrogradsky's approach. 
  In their recent letter, Chaichian et.~al. present a Lagrangian for a massive ($m$) point particle on the plane, with which they claim to realize anyon statistics. However, we find that there are some inaccuracies in their formulation and, when these are taken care of, well-known results are reproduced, which already exist in the literature. 
  Solutions to the twisted Yang-Baxter equation, arising from intertwiners for cyclic representations of $U_q(\widehat{sl}_n)$ are described via two coupled the lattice current algebras. 
  Recently the quantum hamiltonian reduction was done in the case of general $s\ell(2)$ embeddings into Lie algebras and superalgebras. In this paper we extend the results to the quantum hamiltonian reduction of $N=1$ affine Lie superalgebras in the superspace formalism. We show that if we choose a gauge for the supersymmetry, and consider only certain equivalence classes of fields, then our quantum hamiltonian reduction reduces to quantum hamiltonian reduction of non-supersymmetric Lie superalgebras. We construct explicitly the super energy-momentum tensor, as well as all generators of spin 1 (and $\hf$); thus we construct explicitly all generators in the superconformal, quasi-superconformal and $\Z_2 \times \Z_2$ superconformal algebras. 
  We continue the study of quantum Liouville theory through Polyakov's functional integral \cite{Pol1,Pol2}, started in \cite{T1}. We derive the perturbation expansion for Schwinger's generating functional for connected multi-point correlation functions involving stress-energy tensor, give the ``dynamical'' proof of the Virasoro symmetry of the theory and compute the value of the central charge, confirming previous calculation in \cite{T1}. We show that conformal Ward identities for these correlation functions contain such basic facts from K\"{a}hler geometry of moduli spaces of Riemann surfaces, as relation between accessory parameters for the Fuchsian uniformization, Liouville action and Eichler integrals, K\"{a}hler potential for the Weil-Petersson metric, and local index theorem. These results affirm the fundamental role, that universal Ward identities for the generating functional play in Friedan-Shenker modular geometry \cite{FS}. 
  A mechanism of supersymmetry breaking in two or four-dimensions is given, in which the breaking is related to the Fermat's last theorem. It is shown that supersymmetry is exact at some irrational number points in parameter space, while it is broken at all rational number points except for the origin. Accordingly, supersymmetry is exact {\it almost everywhere}, as well as broken {\it almost everywhere} on the real axis in the parameter space at the same time. This is the first explicit mechanism of supersymmetry breaking with an arbitrarily small change of parameters around any exact supersymmetric model, which is possibly useful for realistically small non-perturbative supersymmetry breakings in superstring model building. As a byproduct, we also give a convenient superpotential for supersymmetry breaking only for irrational number parameters. Our superpotential can be added as a ``hidden'' sector to other useful supersymmetric models. 
  We reformulate the problem of the "interpretation of quantum mechanics" as the problem of DERIVING the quantum mechanical formalism from a set of simple physical postulates. We suggest that the common unease with taking quantum mechanics as a fundamental description of nature could derive from the use of an incorrect notion, as the unease with the Lorentz transformations before Einstein derived from the notion of observer independent time. Following an an analysis of the measurement process as seen by different observers, we propose a reformulation of quantum mechanics in terms of INFORMATION THEORY. We propose three different postulates out of which the formalism of the theory can be reconstructed; these are based on the notion of information about each other that systems contain. All systems are assumed to be equivalent: no observer-observed distinction, and information is interpreted as correlation. We then suggest that the incorrect notion that generates the unease with quantum mechanichs is the notion of OBSERVER INDEPENDENT state of a system. 
  Usually quantum chains with quantum group symmetry are associated with representations of quantized universal algebras $U_q(g) $ . Here we propose a method for constructing quantum chains with $C_q(G)$ global symmetry , where $C_q(G)$ is the algebra of functions on the quantum group. In particular we will construct a quantum chain with $GL_q(2)$ symmetry which interpolates between two classical Ising chains.It is shown that the Hamiltonian of this chain satisfies in the generalised braid group algebra. 
  It is shown that $D=4$ $N=1$ SUSY Yang-Mills theory with an appropriate supermultiplet of matter can be twisted on compact K\"ahler manifold. The conditions of cancellation of anomalies of BRST charge are found. The twisted theory has an appropriate BRST charge. We find a non-trivial set of physical operators defined as classes of the cohomology of this BRST \op . We prove that the physical correlators are independent on external K\"ahler metric up to a power of a ratio of two Ray-Singer torsions for the Dolbeault cohomology complex on a K\"ahler manifold. The correlators of local physical \op s turn out to be independent of anti-holomorphic coordinates defined with a complex structure on the K\"ahler manifold. However a dependence of the correlators on holomorphic coordinates can still remain. For a hyperk\"ahler metric the physical correlators turn out to be independent of all coordinates of insertions of local physical \op s. 
  We determine the $s$-waves contribution of a scalar field to the four dimensional effective action for arbitrary spherically symmetric external gravitational fields. The result is applied to $4d$-black holes and it is shown that the energy momentum tensor derived from the (nonlocal) effective action contains the Hawking radiation. The luminosity is close to the expected one in the $s$-channel. The energy momentum tensor may be used as starting point to study the backreaction problem. 
  We apply various conventional tests of integrability to the supersymmetric nonlinear Schr\"odinger equation. We find that a matrix Lax pair exists and that the system has the Painlev\'e property only for a particular choice of the free parameters of the theory. We also show that the second Hamiltonian structure generalizes to superspace only for these values of the parameters. We are unable to construct a zero curvature formulation of the equations based on OSp(2$|$1). However, this attempt yields a nonsupersymmetric fermionic generalization of the nonlinear Schr\"odinger equation which appears to possess the Painlev\'e property. 
  We use Feynman diagrams to prove a formula for the Jones polynomial of a link derived recently by N.~Reshetikhin. This formula presents the colored Jones polynomial as an integral over the coadjoint orbits corresponding to the representations assigned to the link components. The large $k$ limit of the integral can be calculated with the help of the stationary phase approximation. The Feynman rules allow us to express the phase in terms of integrals over the manifold and the link components. Its stationary points correspond to flat connections in the link complement. We conjecture a relation between the dominant part of the phase and Milnor's linking numbers. We check it explicitly for the triple and quartic numbers by comparing their expression through the Massey product with Feynman diagram integrals. 
    We extend the results of our previous paper from knots to links by using a formula for the Jones polynomial of a link derived recently by N. Reshetikhin. We illustrate this formula by an example  of  a  torus link. A relation between the parameters of Reshetikhin's  formula  and the multivariable Alexander polynomial is established. We  check  that our expression for the Alexander polynomial satisfies some of its basic properties. Finally we derive a link  surgery  formula  for  the  loop corrections to the trivial connection contribution to Witten's invariant of rational homology spheres. 
  The Thirring model with imaginary mass (or the sine-Gordon model with imaginary coupling) is deeply related to all the flows between minimal conformal theories. We solve this model explicitely using the Bethe ansatz. We find that there are Left and Right moving massless excitations with non trivial LR scattering. We compute the S matrix and recover the result conjectured by Fendley et al. 
  The Tracy-Widom equations associated with level spacing distributions are realized as a special case of monodromy preserving deformations. 
  We give a complete description of the genus expansion of the one-cut solution to the generalized Penner model. The solution is presented in a form which allows us in a very straightforward manner to localize critical points and to investigate the scaling behaviour of the model in the vicinity of these points. We carry out an analysis of the critical behaviour to all genera addressing all types of multi-critical points. In certain regions of the coupling constant space the model must be defined via analytical continuation. We show in detail how this works for the Penner model. Using analytical continuation it is possible to reach the fermionic 1-matrix model. We show that the critical points of the fermionic 1-matrix model can be indexed by an integer, $m$, as it was the case for the ordinary hermitian 1-matrix model. Furthermore the $m$'th multi-critical fermionic model has to all genera the same value of $\gamma_{str}$ as the $m$'th multi-critical hermitian model. However, the coefficients of the topological expansion need not be the same in the two cases. We show explicitly how it is possible with a fermionic matrix model to reach a $m=2$ multi-critical point for which the topological expansion has alternating signs, but otherwise coincides with the usual Painlev\'{e} expansion. 
  We investigate the $W_{\infty}$ algebra in the integer quantum Hall effects. Defining the simplest vacuum, the Dirac sea, we evaluate the central extension for this algebra. A new algebra which contains the central extension is called the $W_{1+\infty}$ algebra. We show that this $W_{1+\infty}$ algebra is an origin of the Kac-Moody algebra which determines the behavior of edge states of the system. We discuss the relation between the $W_{1+\infty}$ algebra and the incompressibility of the integer quantum Hall system. 
  Progress along the line of a previous article are reported. One main point is to include chiral operators with fractional quantum group spins (fourth or sixth of integers) which are needed to achieve modular invariance. We extend the study of the chiral bootstrap (recently completed by E. Cremmer, and the present authors) to the case of semi-infinite quantum-group representations which correspond to positive integral screening numbers. In particular, we prove the Bidenharn-Elliot and Racah identities for q-deformed 6-j symbols generalized to continuous spins. The decoupling of the family of physical chiral operators (with real conformal weights) at the special values C_{Liouville}= =7, 13, and 19, is shown to provide a full solution of Moore and Seiberg's equations, only involving operators with real conformal weights. Moreover, our study confirms the existence of the strongly coupled topological models. The three-point functions are shown to be given by a product of leg factors similar to the ones of the weakly coupled models. However, contrary to this latter case, the equality between the quantum group spins of the holomorphic and antiholomorphic components is not preserved by the local vertex operator. Thus the ``c=1'' barrier appears as connected with a deconfinement of chirality. 
  A repeatedly discussed gedanken experiment, proposed by Fermi to check Einstein causality, is reconsidered. It is shown that, contrary to a recent statement made by Hegerfeldt, there appears no causality paradoxon in a proper theoretical description of the experiment. 
  We present $S^7$-algebras as generalized Kac-Moody algebras. A number of free-field representations is found. We construct the octonionic projective spaces ${\O}P^N$. 
  We give an example of topological theory whose Hilbert space contains physical objects: the N=2 supersymmetric Lagrangian of spin-one particles moving in D-dimensional space-time equals the Lagrangian of a topological sigma model in a (D+2)-dimensional target. The equivalence is valid for a curved space-time. As an application, we calculate the deviation of spin-one particles in a Schwarzschild background (gravitational Stern-Gerlach effect). 
  The Functional Bethe Ansatz (FBA) proposed by Sklyanin is a method which gives separation variables for systems for which an $R$-matrix is known. Previously the FBA was only known for $SL(2)$ and $SL(3)$ (and associated) $R$-matrices. In this paper I advance Sklyanin's program by giving the FBA for certain systems with $SL(N)$ $R$-matrices. This is achieved by constructing rational functions $\A(u)$ and $\B(u)$ of the matrix elements of $T(u)$, so that, in the generic case, the zeros $x_i$ of $\B(u)$ are the separation coordinates and the $P_i=\A(x_i)$ provide their conjugate momenta. The method is illustrated with the magnetic chain and the Gaudin model, and its wider applicability is discussed. 
  Several examples are given of continuous families of SU(2) vector potentials $A_i^a(x)$ in 3 space dimensions which generate the same magnetic field $B^{ai}(x)$ (with det $B\neq 0$). These Wu-Yang families are obtained from the Einstein equation $R_{ij}=-2G_{ij}$ derived recently via a local map of the gauge field system into a spatial geometry with $2$-tensor $G_{ij}=B^a{}_i B^a{}_j\det B$ and connection $\Gamma_{jk}^i$ with torsion defined from gauge covariant derivatives of $B$. 
  The null vectors of an arbitrary highest weight representation of the $WA_2$ algebra are constructed. Using an extension of the enveloping algebra by allowing complex powers of one of the generators, analysed by Kent for the Virasoro theory, we generate all the singular vectors indicated by the Kac determinant. We prove that the singular vectors with given weights are unique up to normalisation and consider the case when $W_0$ is not diagonalisable among the singular vectors. 
  We realize the Hopf algebra $U_{q^{-1}}(so(N))$ as an algebra of differential operators on the quantum Euclidean space ${\bf R}_q^N$. The generators are suitable q-deformed analogs of the angular momentum components on ordinary ${\bf R}^N$. The algebra $Fun({\bf R}_q^N)$ of functions on ${\bf R}_q^N$ splits into a direct sum of irreducible vector representations of $U_{q^{-1}}(so(N))$; the latter are explicitly constructed as highest weight representations. 
  We analyze quasi-topological solitons winding around a mexican-hat potential in two space-time dimensions.   They are prototypes for a large number of physical excitations, including Skyrmions of the Higgs sector of the standard electroweak model, magnetic bubbles in thin ferromagnetic films, and strings in certain non-trivial backgrounds.   We present explicit solutions, derive the conditions for classical stability, and show that contrary to the naive expectation these can be satisfied in the weak-coupling limit. In this limit we can calculate the soliton properties reliably, and estimate their lifetime semiclassically. We explain why gauge interactions destabilize these solitons, unless the scalar sector is extended. 
  The representation theories of the SU(2)$_k$-extended $N$=4 superconformal algebras (SCAs) with $arbitrary$ level $k$ are developed being based on their Feigin-Fuchs representations found recently by the present author. A basic unit of the representation blocks consisting of eight \lq\lq boson-like\rq\rq\ and eight \lq\lq fermion-like\rq\rq\ conformal fields is found to describe arbitrary representations of the $N$=4 SU(2)$_k$ SCAs, including {\it unitary} and {\it nonunitary} representations. The transformation properties of the fundamental sets of the conformal fields under the $N$=4 SU(2)$_k$ superconformal symmetries are given. Then, the whole sets of the charge-screening operators of the $N$=4 SU(2)$_k$ SCAs are identified out of the sixteen conformal fields in the basic unit of the representation blocks. The conditions for the {\it eligible} charge-screening operators are analyzed in terms of the continuous parameters which enter in our vertex-operator forms for the fundamental conformal fields of the representation blocks. 
  Developing upon the ideas of ref. \ref{6}, it is shown how the theory of classical $W$ algebras can be formulated on a higher genus Riemann surface in the spirit of Krichever and Novikov. The basic geometric object is the Drinfeld--Sokolov principal bundle $L$ associated to a simple complex Lie group $G$ equipped with an $SL(2,\Bbb C)$ subgroup $S$, whose properties are studied in detail. On a multipunctured Riemann surface, the Drinfeld--Sokolov--Krichever--Novikov spaces are defined, as a generalization of the customary Krichever--Novikov spaces, their properties are analyzed and standard bases are written down. Finally, a WZWN chiral phase space based on the principal bundle $L$ with a KM type Poisson structure is introduced and, by the usual procedure of imposing first class constraints and gauge fixing, a classical $W$ algebra is produced. The compatibility of the construction with the global geometric data is highlighted. 
  We study the condition for the appearance of space-time supercharges in twisted sectors of asymmetric orbifold models. We present a list of the asymmetric $Z_N$-orbifold models which satisfy a simple condition necessary for the appearance of space-time supercharges in twisted sectors. We investigate whether or not such asymmetric orbifold models possess $N=1$ space-time supersymmetry and obtain a new class of $N=1$ asymmetric orbifold models. It is pointed out that the result of space-time supersymmetry does not depend on any choice of a shift if the order of the left-moving degrees of freedom is preserved. 
  We consider light-cone quantized ${\rm{QCD}}_{1+1}$ on a `cylinder' with periodic boundary conditions on the gluon fields. This is the framework of discretized light-cone quantization. We review the argument that the light-cone gauge $A^+=0$ is not attainable. The zero mode is a dynamical and gauge invariant field. The attainable gauge has a Gribov ambiguity. We exactly solve the problem of pure glue theory coupled to some zero mode external sources. We verify the identity of the front and the more familiar instant form approaches. We obtain a discrete spectrum of vacuum states and their wavefunctions. 
  We develop a rigorous framework for constructing Fock representations of quantum fields obeying generalized statistics associated with certain solutions of the spectral quantum Yang-Baxter equation. The main features of these representations are investigated. Various aspects of the underlying mathematical structure are illustrated by means of explicit examples. 
  This paper considers closed-string states of type 2b superstring theory in which the whole string is localized at a single point in superspace. Correlation functions of these (scalar and pseudoscalar) states possess an infinite number of position-space singularities inside and on the light-cone as well as a space-like singularity outside the light-cone.\foot{This paper was previously produced as preprint QMW-91-02.} 
  A WZW model for the non-semi-simple D-dimensional Heisenberg group, which directly generalizes the E_{2}^{c} group, is constructed. It is found to correspond to an D-dimensional string background of plane-wave type with physical Lorentz signature D-2. Perturbative and non-perturbative considerations lead both to an integer central charge c=D. 
  It is shown that q-deformed quantum mechanics (q-deformed Heisenberg algebra) can be interpreted as quantum mechanics on Kaehler manifolds, or as a quantum theory with second (or first-) class constraints. (Saclay, T93/027). 
  In a recent paper Dargis and Mathieu introduced integrodifferential odd flows for the supersymmetric KdV equation. These flows are obtained from the nonlocal conservation laws associated with the fourth root of its Lax operator. In this note I show that only half of these flows are of the standard Lax form, while the remaining half provide us with hamiltonians for an SKdV-type reduction of a new supersymmetric hierarchy. This new hierarchy is shown to be closely related to the Jacobian supersymmetric KP-hierarchy of Mulase and Rabin. A detailed study of the algebra of additional symmetries of this new hierarchy reveals that it is isomorphic to the super-W_{1+\infty} algebra, thus making it a candidate for a possible interrelationship between superintegrability and two-dimensional supergravity. 
  We describe birational representations of discrete groups generated by involutions, having their origin in the theory of exactly solvable vertex-models in lattice statistical mechanics. These involutions correspond respectively to two kinds of transformations on $q \times q$ matrices: the inversion of the $q \times q$ matrix and an (involutive) permutation of the entries of the matrix. We concentrate on the case where these permutations are elementary transpositions of two entries. In this case the birational transformations fall into six different classes. For each class we analyze the factorization properties of the iteration of these transformations. These factorization properties enable to define some canonical homogeneous polynomials associated with these factorization properties. Some mappings yield a polynomial growth of the complexity of the iterations. For three classes the successive iterates, for $q=4$, actually lie on elliptic curves. This analysis also provides examples of integrable mappings in arbitrary dimension, even infinite. Moreover, for two classes, the homogeneous polynomials are shown to satisfy non trivial non-linear recurrences. The relations between factorizations of the iterations, the existence of recurrences on one or several variables, as well as the integrability of the mappings are analyzed. 
  We construct quantum states for a (1+1) dimensional gravity-matter model that is also a gauge theory based on the centrally extended Poincar\'e group. Explicit formulas are found, which exhibit interesting structures. For example wave functionals are gauge invariant except for a gauge non-invariant phase factor that is the Kirillov-Kostant 1-form on the (co-) adjoint orbit of the group. However no evidence for gravity-matter forces is found. 
  Topological Structures in the Standard Model at high $T$ are discussed. 
  We show how special forms of an $N=2$ Landau-Ginzburg potential directly imply the presence of an $N=2$ super-$W$ algebra. If the Landau-Ginzburg model has a super-$W$ algebra, we show how the elliptic genus can be refined so as to give much more complete information about the structure of the model. We study the super-$W_3$ model in some detail, and present some results and conjectures about more general models. 
  The purpose of this work is to show that there exists an additional invariance of the $\beta$-function equations of string theory on $d+1$-dimensional targets with $d$ toroidal isometries. It corresponds to a shift of the dilaton field and a scaling of the lapse function, and is reminiscent of string field redefinitions. While it preserves the form of the $\beta$-function equations, it changes the effective action and the solutions. Thus it can be used as a solution generating technique. It is particularly interesting to note that there are field redefinitions which map solutions with non-zero string cosmological constant to those with zero cosmological constant. Several simple examples involving two- and three-dimensional black holes and black strings are provided to illustrate the role of such field redefinitions. 
  The factorization condition for the scattering amplitudes of an integrable model with a line of defect gives rise to a set of Reflection-Transmission equations. The solutions of these equations in the case of diagonal $S$-matrix in the bulk are only those with $S =\pm 1$. The choice $S=-1$ corresponds to the Ising model. We compute the transmission and reflection amplitudes relative to the interaction of the Majorana fermion with the defect and we discuss their relevant features. 
  We consider the formulation of two dimensional QCD in terms of gauge invariant bilocal operators (string field) which satisfy a $W_\infty$ algebra. In analogy with our work on the $c=1$ string field theory we derive an action and associated constraints for the bilocal field using the method of coadjoint orbits. The $1/N$ perturbation theory around a classical solution that corresponds to the filled Dirac sea leads to the 'tHooft equation for meson fluctuations. It is shown that the spectrum of mesons, which are the higher string modes, transform as a representation of the wedge subalgebra $W_{\infty+} \otimes W_{\infty-}$. We briefly discuss the baryon as a stringy solitonic configuration and its characterization in terms of $W_\infty$ algebra. 
  The interaction of various algebraic structures describing fusion, braiding and group symmetries in quantum projective field theory is an object of an investigation in the paper. Structures of projective Zamolodchikov al- gebras, their represntations, spherical correlation functions, correlation characters and envelopping QPFT-operator algebras, projective \"W-algebras, shift algebras, braiding admissible QPFT-operator algebras and projective G-hypermultiplets are explored. It is proved (in the formalism of shift algebras) that sl(2,C)-primary fields are characterized by their projective weights and by the hidden additive weight, a hidden quantum number discovered in the paper (some discussions on this fact and its possible relation to a hidden 4-dimensional QFT maybe found in the note by S.Bychkov, S.Plotnikov and D.Juriev, Uspekhi Matem. Nauk 47(3) (1992)[in Russian]). The special attention is paid to various constructions of projective G-hyper- multiplets (QPFT-operator algebras with G-symmetries). 
  We introduce an integrable time-discretized version of the classical Calogero-Moser model, which goes to the original model in a continuum limit. This discrete model is obtained from pole solutions of a discretized version of the Kadomtsev-Petviashvili equation, leading to a finite-dimensional symplectic mapping. Lax pair, symplectic structure and sufficient set of invariants of the discrete Calogero-Moser model are constructed. The classical $r$-matrix is the same as for the continuum model. 
  Using properties of ordered exponentials and the definition of the Drinfeld associator as a monodromy operator for the Knizhnik-Zamolodchikov equations, we prove that the analytic and the combinatorial definitions of the universal Vassiliev invariants of links are equivalent. 
  We derive a condition under which (0,2) linear sigma models possess a ``left-moving'' conformal stress tensor in $\bq$ cohomology (i.e. which leaves invariant the ``right-moving'' ground states) even away from their critical points. At the classical level this enforces quasihomogeneity of the superpotential terms. The persistence of this structure at the quantum level on the worldsheet is obstructed by an anomaly unless the charges and superpotential degrees satisfy a condition which is equivalent to the condition for the cancellation of the anomaly in a particular ``right-moving'' U(1) R-symmetry. 
  This paper provides some background to the theory of operads, used in the first author's papers on 2d topological field theory (hep-th/921204, CMP 159 (1994), 265-285; hep-th/9305013). It is intended for specialists. 
  We propose a new nonrelativistic Chern-Simons theory based on a simple modification of the standard Lagrangian. This admits asymptotically nonvanishing field configurations and is applicable to the description of systems of repulsive bosons. The new model supports topological vortices and has a self-dual limit, both in the pure Chern-Simons and in the mixed Chern-Simons-Maxwell cases. The analysis is based on a new formulation of the Chern-Simons theories as constrained Hamiltonian systems. 
  We develop a perturbation theory of four-dimensional topological 2-form gravity without cosmological constant.   A 2-form and an $SU(2)$ connection 1-form are used as fundamental variables instead of metric.   There is no quantum correction from two-loop and higher orders in covariant gauge, in Landau gauge and in background covariant gauge.   We improve naive dimensional regularization and calculate an exact quantum correction at one-loop order.   We observe that the renormalizability depends on gauge choice in topological 2-form gravity.   It is unrenormalizable in the covariant gauge and in the background covariant gauge.   On the other hand in the Landau gauge, we obtain finite but $SU(2)$ non-covariant quantum corrections.   There may exist anomaly for the $SU(2)$ gauge symmetry.   (Talk given by A. Nakamichi at the Workshop on General Relativity and Gravitation held at Tokyo University, January 17-20,1994.) 
  We examine some novel physical consequences of the general structure of moduli spaces of string vacua. These include (1) finiteness of the volume of the moduli space and (2) chaotic motion of the moduli in the early universe. To fix ideas we examine in detail the example of the (conjectural) dilaton-axion ``$S$-duality'' of four-dimensional string compactifications. The facts (1) and (2) together might help to solve some problems with the standard scenarios for supersymmetry breaking and vacuum selection in string theory. 
  An operation of a coproduct of representations of a bialgebra is defined. The coproduct operation for representations of the Hopf algebra of functions on the quantum group $SU_{q}(2)$ is investigated. A notion of a stable representation $\Pi$ is introdused. This means that the representation $\Pi$ is invariant under coproduct by arbitrary representation. Algebra of functions on the quantum group $SU_{q}(2)$ in the representation $\Pi$ is a factor of a type II$_{\infty}$. Formula for the trace in the representation $\Pi$ is given . The invariant integral of Woronovich on $SU_{q}(2)$ will take the form $\int f d\mu = tr(f cc^{*})$. 
  Isotropic string cosmology solutions can be classified into two duality related branches: one can be connected smoothly to an expanding Friedman-Robertson-Walker Universe, the other describes accelerated inflation or contraction. We show that, if the dilaton potential has certain generic properties, the two branches can evolve smoothly into each other. We find, however, that just the effects of a potential do not allow for a graceful exit from accelerated inflation in the weak curvature regime. We explain how conformal field theory techniques could be used to give a decisive answer on whether graceful exit from accelerated inflation is possible at all. 
  We present new covariant phase space formulations of superparticles moving on a group manifold, deriving the fundamental Poisson brackets and current algebras. We show how these formulations naturally generalise to the supersymmetric Wess-Zumino-Witten models. 
  The one-loop effective potential in $2D$ dilaton gravity in conformal gauge on the topologically non-trivial plane $\reals \times S^1$ and on the hyperbolic plane $H^2/\Gamma$ is calculated. For arbitrary choice of the tree scalar potential it is shown, that the one-loop effective potential explicitly depends on the reference metric (through the dependence on the radius of the torus or the radius of $H^2/\Gamma$). This phenomenon is absent only for some special choice of the tree scalar potential corresponding to the Liouville potential and leading to one-loop ultraviolet finite theory. The effective equations are discussed and some interpretation of the reference metric dependence of the effective potential is made. 
  Higher Derivative (HD) Field Theories can be transformed into second order equivalent theories with a direct particle interpretation. In a simple model involving abelian gauge symmetries we examine the fate of the possible gauge fixings throughout this process. This example is a useful test bed for HD theories of gravity and provides a nice intuitive interpretation of the "third ghost" occurring there and in HD gauge theories when a HD gauge fixing is adopted. 
  We construct central elements in a completion of the quantum affine algebra at the critical level c=-g from the universal R-matrix (g being the dual Coxeter number of the corresponding simple Lie algebra), using the method of Reshetikhin and Semenov-Tian-Shansky. This construction defines an action of the Grothendieck algebra of the category of finite-dimensional representations of the quantum affine algebra on any module over this algebra from category O with c=-g. We explain the connection between the central elements and transfer matrices in statistical mechanics. In the quasiclassical approximation this connection was explained by Feigin, E.Frenkel, and Reshetikhin in hep-th 9402022, and it was mentioned that one could generalize it to the quantum case to get Bethe vectors for transfer matrices. Using this connection, we prove that the central elements   (for all finite dimensional representations) applied to the highest weight vector of a generic Verma module at the critical level generate the whole space of singular vectors in this module. We also compute the first term of the quasiclassical expansion of the central elements near q=1, and show that it always gives the Sugawara current with a certain coefficient. 
  The question as to whether helicity conservation in spin one-half Aharonov Bohm scattering is sufficient in itself to determine uniquely the form of the spinor wave function near the origin is examined. Although it is found that a one parameter family of solutions is compatible with this conservation law, there must nonetheless be singular solutions which break the symmetry $\alpha \rightarrow \alpha +1$ required for an anyon interpretation. Thus the free parameter which occurs does not allow one to eliminate the singular solutions even though it does in principle mean that they can be transferred at will between the spin up and spin down configurations. 
  Motivated by a recent paper of Fock and Rosly \cite{FoRo} we describe a mathematically precise quantization of the Hamiltonian Chern-Simons theory. We introduce the Chern-Simons theory on the lattice which is expected to reproduce the results of the continuous theory exactly. The lattice model enjoys the symmetry with respect to a quantum gauge group. Using this fact we construct the algebra of observables of the Hamiltonian Chern-Simons theory equipped with a *-operation and a positive inner product. 
  Conventional discrete approximations of a manifold do not preserve its nontrivial topological features. In this article we describe an approximation scheme due to Sorkin which reproduces physically important aspects of manifold topology with striking fidelity. The approximating topological spaces in this scheme are partially ordered sets (posets). Now, in ordinary quantum physics on a manifold $M$, continuous probability densities generate the commutative C*-algebra $\cc(M)$ of continuous functions on $M$. It has a fundamental physical significance, containing the information to reconstruct the topology of $M$, and serving to specify the domains of observables like the Hamiltonian. For a poset, the role of this algebra is assumed by a noncommutative C*-algebra $\ca $. As noncommutative geometries are based on noncommutative C*-algebras, we therefore have a remarkable connection between finite approximations to quantum physics and noncommutative geometries. Various methods for doing quantum physics using $\ca $ are explored. Particular attention is paid to developing numerically viable approximation schemes which at the same time preserve important topological features of continuum physics. 
  There are investigated several objects of an INFINITE DIMENSIONAL GEOMETRY appearing from the second quantization of a free string. The paper contains 2 chapters: 1st is devoted to the infinite dimensional geometry of flag, fundamental and $\Pi$-spaces for Virasoro-Bott group and its nonassociative deformation defined by Gelfand-Fuchs 3-cocycle (Gelfand-Fuchs loop) as well as of infinite-dimensional non-Euclidean symplectic grassmannian, to the constructions of Verma modules, their models and skladens over Virasoro algebra; an infinite dimensional geometry of the configuration space for the second quantized free string in flat and curved backgrounds as well as author version of Bowick- Rajeev formalism of the separation of internal and external degrees of freedom of a closed string are described in 2nd chapter. In the 1st chapter the main objects are infinite dimensional Lie algebras, groups and loops, homogeneous, K\"ahler, Finsler, contact and symmetric spaces, complex, real and CR-manifolds, determinant sheaves, manifolds with subsymmetries, polarizations and Fock spaces, bibundles and objects of integral geometry, nonholonomic spaces, deformations of geometric structures and moduli spaces. In the 2nd chapter they are gauge fields, Faddeev-Popov ghosts, Gauss-Manin connections, Kostant-Blattner-Sternberg pairings, BRST-operators. 
  We review the Batalin-Tyutin approach of quantising second class systems which consists in enlarging the phase space to convert such systems into first class. The quantisation of first class systems, it may be mentioned, is already well founded. We show how the usual analysis of Batalin-Tyutin may be generalised, particularly if one is dealing with nonabelian theories. In order to gain a deeper insight into the formalism we have considered two specific examples of second class theories-- the massive Maxwell theory (Proca model) and its nonabelian extension. The first class constraints and the involutive Hamiltonian are explicitly constructed. The connection of our Hamiltonian approach with the usual Lagrangian formalism is elucidated. For the Proca model we reveal the importance of a boundary term which plays a significant role in establishing an exact identification of the extra fields in the Batalin-Tyutin approach with the St\"uckelberg scalar. Some comments are also made concerning the corresponding identification in the nonabelian example. 
  The renormalisation group equation for $N$-point correlation functions can be interpreted in a geometrical manner as an equation for Lie transport of amplitudes in the space of couplings. The vector field generating the diffeomorphism has components given by the $\beta$-functions of the theory. It is argued that this simple picture requires modification whenever any one of the points at which the amplitude is evaluated becomes close to any other. This modification requires the introduction of a connection on the space of couplings and new terms appear in the renormalisation group equation involving co-variant derivatives of the $\beta$-function and the curvature associated with the connection. It is shown how the connection is related to the operator expansion co-efficients, but there remains an arbitrariness in its definition. 
  An $N=2$ supersymmetric model using K\"{a}hler fields is proposed. It is a modified version of two-dimensional Benn-Tucker model. It indicates a geometrical origin of $N=2$ supersymmetry. 
  Paper is withdrawn and superseded by EFI-94-36 which will appear shortly with the new hep-th number hep-th/9407111 . 
  The Hilbert space of an RSOS-model, introduced by Andrews, Baxter, and Forrester, can be viewed as a space of sequences (paths) {a_0,a_1,...,a_L}, with a_j-integers restricted by 1<=a_j<=\nu, |a_j-a_{j+1}|=1, a_0=s, a_L=r. In this paper we introduce different basis which, as shown here, has the same dimension as that of an RSOS-model. Following McCoy et al, we call this basis -- fermionic (FB). Our first theorem Dim(FB)=Dim(RSOS-basis) can be succinctly expressed in terms of some identities for binomial coefficients. Remarkably, these binomial identities can be q-deformed. Here, we give a simple proof of these q-binomial identities in the spirit of Schur's proof of the Rogers-Ramanujan identities. Notably, the proof involves only the elementary recurrences for the q-binomial coefficients and a few creative observations. Finally, taking the limit L --> \infty in these q-identities, we derive an expression for the character formulas of the unitary minimal series M(\nu,\nu+1) "Bosonic Sum = Fermionic Sum". Here, Bosonic Sum denotes Rocha-Caridi representation (\chi_{r,s=1}^{\nu,\nu+1}(q)) and Fermionic Sum stands for the companion representation recently conjectured by the Stony Brook group. 
  The global chiral symmetry of a $SU(2)$ gauge theory is studied in the framework of renormalization group (RG). The theory is defined by the RG flow equations in the infrared cutoff $\L$ and the boundary conditions for the relevant couplings. The physical theory is obtained at $\L=0$. In our approach the symmetry is implemented by choosing the boundary conditions for the relevant couplings not at the ultraviolet point $\L=\L_0\to\infty$ but at the physical value $\L=0$. As an illustration, we compute the triangle axial anomalies. 
  The BRST quantisation of the Drinfeld - Sokolov reduction applied to the case of $A^{(1)}_2\,$ is explored to construct in an unified and systematic way the general singular vectors in ${\cal W}_3$ and ${\cal W}_3^{(2)}$ Verma modules. The construction relies on the use of proper quantum analogues of the classical DS gauge fixing transformations. Furthermore the stability groups $\overline W^{(\eta)}\,$ of the highest weights of the ${\cal W}\,$ - Verma modules play an important role in the proof of the BRST equivalence of the Malikov-Feigin-Fuks singular vectors and the ${\cal W}$ algebra ones. The resulting singular vectors are essentially classified by the affine Weyl group $W\, $ modulo $\overline W^{(\eta)}\,$. This is a detailed presentation of the results announced in a recent paper of the authors (Phys. Lett. B318 (1993) 85). 
  We begin with a prior observation by one of us that Thomas precession in the nonrelativistic limit of the Dirac equation may be attributed to a nonabelian Berry vector potential. We ask what object produces the nonabelian potential in parameter space, in the same sense that the abelian vector potential arising in the adiabatic transport of a nondegenerate level is produced by a monopole, (centered at the point where the level becomes degenerate with another), as shown by Berry. We find that it is a {\em meron}, an object in four euclidean dimensions with instanton number ${1 \over 2}$, centered at the point where the doubly degenerate positive and negative energy levels of the Dirac equation become fourfold degenerate. 
  Quantum theory of conformal factor coupled with matter fields is investigated. The more simple case of the purely classical scalar matter is considered. It is calculated the conformal factor contribution to the effective potential of scalar field. Then the possibility of the first order phase transition is explored and the induced values of Newtonian and cosmological constants are calculated. 
  We perform the numerical simulation of the two dimensional chiral Gross Neveu model using the Kogut-Susskind(KS) fermion. In the case of SU(4), the Kosterlitz-Thouless phase transition happens at some critical value of the coupling constant. In the case of one flavour, there exists the strong coupling phase in which the correlation functions vanish and the general covariance is realized in the quantum field thoery through the dynamical process. 
  Geodesic completness and self-adjointness imply that the Hamiltonian for anyons is the Laplacian with respect to the Weil-Petersson metric. This metric is complete on the Deligne-Mumford compactification of moduli (configuration) space. The structure of this compactification fixes the possible anyon configurations. This allows us to identify anyons with singularities (elliptic points with ramification $q^{-1}$) in the Poincar\'e metric implying that anyon spectrum is chaotic for $n\ge 3$. Furthermore, the bound on the holomorphic sectional curvature of moduli spaces implies a gap in the energy spectrum. For $q=0$ (punctures) anyons are infinitely separated in the Poincar\'e metric (hard-core). This indicates that the exclusion principle has a geometrical intepretation. Finally we give the differential equation satisfied by the generating function for volumes of the configuration space of anyons. 
  To every partition $n=n_1+n_2+\cdots+n_s$ one can associate a vertex operator realization of the Lie algebras $a_{\infty}$ and $\hat{gl}_n$. Using this construction we make reductions of the $s$--component KP hierarchy, reductions which are related to these partitions. In this way we obtain matrix KdV type equations. Now assuming that (1) $\tau$ is a $\tau$--function of the $[n_1,n_2,\ldots,n_s]$--th reduced KP hierarchy and (2) $\tau$ satisfies a `natural' string equation, we prove that $\tau$ also satisfies the vacuum constraints of the $W_{1+\infty}$ algebra. 
  We show how bosonic (free field) representations for so-called degenerate conformal theories are built by singular vectors in Verma modules. Based on this construction, general expressions of conformal blocks are proposed. As an example we describe new modules for the $SL(2)$ Wess-Zumino -Witten model. They are, in fact, the simplest non-trivial modules in a full set of bosonized highest weight representations of $\hat {sl}_2$ algebra. The Verma and Wakimoto modules appear as boundary modules of this set. Our construction also yields a new kind of bosonization in 2d conformal field theories. 
  We study the effective action describing high-energy scattering processes in the multi-Regge limit of QCD, which should provide the starting point for a new attempt to overcome the limitations of the leading logarithmic and the eikonal approximations. The action can be obtained via simple graphical rules or by integrating in the QCD functional integral over momentum modes of gluon and quark fields that do not appear explicitely as scattering or exchanged particles in the considered processes. The supersymmetry is used to obtain the terms in the action involving quarks fields from the pure gluonic ones. We observe a Weizs\"acker - Williams type relations between terms describing scattering and production of particles. 
  We prove that the distributions defined on the Gelfand-Shilov spaces, and hence more singular than hyperfunctions, retain the angular localizability property. Specifically, they have uniquely determined support cones. This result enables one to develop a distribution-theoretic techniques suitable for the consistent treatment of quantum fields with arbitrarily singular ultraviolet and infrared behavior. The proofs covering the most general case are based on the use of the theory of plurisubharmonic functions and Hormander's estimates. 
  A four part series of lectures on the connection of statistical mechanics and quantum field theory. The general principles relating statistical mechanics and the path integral formulation of quantum field theory are presented in the first lecture. These principles are then illustrated in lecture 2 by a presentation of the theory of the Ising model for $H=0$, where both the homogeneous and randomly inhomogeneous models are treated and the scaling theory and the relation with Fredholm determinants and Painlev{\'e} equations is presented. In lecture 3 we consider the Ising model with $H\neq 0$, where the relation with gauge theory is used to discuss the phenomenon of confinement. We conclude in the last lecture with a discussion of quantum spin diffusion in one dimensional chains and a presentation of the chiral Potts model which illustrates the physical effects that can occur when the Euclidean and Minkowski regions are not connected by an analytic continuation. (To be published as part of the Proceedings of the Sixth Annual Theoretical Physics Summer School of the Australian National University which was held in Canberra during Jan. 1994.) 
  A class of explicit exact solutions of Einstein Yang Mills Chern Simons (EYMCS) theory corresponding to topological solitons carrying non-Abelian topological electric charge is obtained. This verifies a conjecture made in Ref.[1,2] regarding the stabilization of the corresponding charged configurations in the theory without gravity . 
  It has been shown that there is a sequential embedding structure in a $w_N$\ string theory based on a linearized $W_N$\ algebra. The $w_N$\ string theory is obtained as a special realization of the $w_{N+1}$\ string. The $w_{\infty}$\ string theory is a universal string theory in this sense. We have also shown that there is a similar sequence for $N=1$\ string theory. The $N=1\ w_N$\ string can be given as a special case of the $N=1\ w_{N+1}$\ string. In addition, we show that the $w_3$\ string theory is obtained as a special realization of the $N=1\ w_3$\ string. We conjecture that the $w_N$\ string can be obtained as a special $N=1\ w_N$\ string for general $w_N$. If this is the case, $N=1\ w_{\infty}$\ string theory is more universal since it includes both $N=0$\ and $N=1$\ $w_N$\ string theories. 
  Inflationary Cosmology makes the universe ``eternal" and provides for recurrent universe creation, ad infinitum -- making it also plausible to assume that ``our" Big Bang was also preceeded by others, etc.. However, GR tells us that in the ``parent" universe's reference frame, the newborn universe's expansion will never start. Our picture of ``reality" in spacetime has to be enlarged. 
  We discuss questions arising from the work of Schellekens. After introducing the concept of complementary representations, we examine $Z_2$-orbifold constructions in general, and propose a technique for identifying the orbifold theory without knowledge of its explicit construction. This technique is then generalised to twists of order 3, 5 and 7, and we proceed to apply our considerations to the FKS constructions $H(\Lambda)$ ($\Lambda$ an even self-dual lattice) and the reflection-twisted orbifold theories $\widetilde H(\Lambda)$, which together remain the only $c=24$ theories which have so far been proven to exist. We also make, in the course of our arguments, some comments on the automorphism groups of the theories $H(\Lambda)$ and $\widetilde H(\Lambda)$, and of meromorphic theories in general, introducing the concept of deterministic theories. 
  Motivated by understanding the phase structure of $d >1$ strings we investigate the $c=1$ matrix model with $g' (\tr M(t)^{2})^{2}$ interaction which is the simplest approximation of the model expected to describe the critical phenomena of the large-$N$ reduced model of odd-dimensional matrix field theory. We find three distinct phases: (i) an ordinary $c=1$ gravity phase, (ii) a branched polymer phase and (iii) an intermediate phase. Further we can also analyse the one with slightly generalized $ g^{(2)} (\frac{1}{N}\tr M(t)^{2})^{2} +g^{(3)} (\frac{1}{N}\tr M(t)^{2})^{3} + \cdots + g^{(n)} (\frac{1}{N}\tr M(t)^{2})^{n} $ interaction. As a result the multi-critical versions of the phase (ii) are found. 
  We obtain an expression for the curvature of the Lie group SDiff$\cal M$ and use it to derive Lukatskii's formula for the case where $\cal M$ is locally Euclidean. We discuss qualitatively some previous findings for SDiff$S^{2}$ in conjunction with our result. 
  We derive the first order canonical formulation of cosmological perturbation theory in a Universe filled by a few scalar fields. This theory is quantized via well-defined Hamiltonian path integral. The propagator which describes the evolution of the initial (for instance, vacuum) state, is calculated. 
  We study the realization of conformal symmetry in the QHE as part of the $W_\infty$ algebra. Conformal symmetry can be realized already at the classical level and implies the complexification of coordinate space. Its quantum version is not unitary. Nevertheless, it can be rendered unitary by a suitable modification of its definition which amounts to taking proper care of the quantum measure. The consequences of unitarity for the Chern-Simons theory of the QHE are also studied, showing the connection of non-unitarity with anomalies. Finally, we discuss the geometrical paradox of realizing conformal transformations as area preserving diffeomorphisms. 
  We show that the algebra of the bicovariant differential calculus on a quantum group can be understood as a projection of the cross product between a braided Hopf algebra and the quantum double of the quantum group. The resulting super-Hopf algebra can be reproduced by extending the exterior derivative to tensor products. 
  An elliptic deformation of $\widehat{sl}_2$ is proposed. Our presentation of the algebra is based on the relation $RLL=LLR^*$, where $R$ and $R^*$ are eight-vertex $R$-matrices with the elliptic moduli chosen differently. In the trigonometric limit, this algebra reduces to a quotient of that proposed by Reshetikhin and Semenov-Tian-Shansky. Conjectures concerning highest weight modules and vertex operators are formulated, and the physical interpretation of $R^*$ is discussed. 
  The expression for the first recoil correction to the Dirac-Coulomb spectrum is obtained employing the gauge invariance. 
  We give an introduction to mirror symmetry of strings on Calabi-Yau manifolds with an emphasis on its applications e.g. for the computation of Yukawa couplings. We introduce all necessary concepts and tools such as the basics of toric geometry, resolution of singularities, construction of mirror pairs, Picard-Fuchs equations, etc. and illustrate all of this on a non-trivial example. Extended version of a lecture given at the Third Baltic Student Seminar, Helsinki September 1993 
  We discuss special perturbations of the gauged level $k$ WZNW model inspired by the $\sigma$-model perturbation of the nonunitary WZNW model. In the large $k$ limit there is a second conformal point in the vicinity of the ultarviolet fixed point. At the second critical point the conformal model has a rational Virasoro central charge which no longer corresponds to a coset construction. In spite of this fact the perturbative conformal model appears to be a unitary system as long as the underlying coset construction at the ultraviolet critical point is unitary. 
  We apply Stochastic Quantization Method to dissipative systems at finite temperature. Especially, the relation of SQM to the Caldeira-Leggett model is clarified ensuring that the naive Wick rotation is improved in this context. We show that the Langevin system obtained by the \lq\lq Improved Wick Rotation " prescription is equivalent to an ideal friction case ( low temperature limit) in the C-L model. We derive, based on our approach, a general formula on the fluctuation-dissipation theorem for higher derivative frictions. 
  We present the exact and explicit solution of the principal chiral field in two dimensions for an infinitely large rank group manifold. The energy of the ground state is explicitly found for the external Noether's fields of an arbitrary magnitude. The exact Gell-Mann - Low function exhibits the asymptotic freedom behaviour at large value of the field in agreement with perturbative calculations. Coefficients of the perturbative expansion in the renormalized charge are calculated. They grow factorially with the order showing the presence of renormalons. At small field we found an inverse logarithmic singularity in the ground state energy at the mass gap which indicates that at $N=\infty$ the spectrum of the theory contains extended objects rather then pointlike particles. 
  The general structure of the representation theory of a $Z_2$-graded coalgebra is discussed. The result contains the structure of Fourier analysis on compact supergroups and quantisations thereof as a special case. The general linear supergroups serve as an explicit illustration and the simplest example is carried out in detail. 
  We consider O(3) sigma-model as a reduction of the principal chiral field. This approach allows to introduce the currents with ultralocal   Poisson brackets and to obtain the zero-curvature equation which admits the fundamental Poisson bracket. 
  The Knizhnik-Zamolodchikov equation associated with $s\ell_2$ is considered. The transition functions between asymptotic solutions to the Knizhnik-Zamolodchikov equation are described. A connection between asymptotic solutions and the crystal base in the tensor product of modules over the quantum group $U_qs\ell_2$ is established, in particular, a correspondence between the Bethe vectors of the Gaudin model of an inhomogenious magnetic chain and the $\Bbb Q-$basis of the crystal base. 
  For the `classical' formulation of a massive spinning particle, the propagator is obtained along with the spin factor. We treat the system with two kinds of constraints that were recently shown to be concerned with the reparametrization invariance and `quasi-supersymmetry'. In the path integral, the BRST invariant Lagrangian is used and the same spin factor is obtained as in the pseudo-classical formulation. 
  We perform a perturbative analysis of the nonabelian Aharonov-Bohm problem to one loop in a field theoretic framework, and show the necessity of contact interactions for renormalizability of perturbation theory. Moreover at critical values of the contact interaction strength the theory is finite and preserves classical conformal invariance. 
  An important property of a Hopf algebra is its quasitriangularity and it is useful various applications. This property is investigated for quantum groups $sl_2$ at roots of 1. It is shown that different forms of the quantum group $sl_2$ at roots of 1 are either quasitriangular or have similar structure which will be called autoquasitriangularity. In the most interesting cases this property means that "braiding automorphism" is a combination of some Poisson transformation and an adjoint transformation with certain element of the tensor square of the algebra. 
  Behavior of the Euclidean path integral at large orders of the perturbation series is studied. When the model allows tunneling, the path-integral functional in the zero instanton sector is known to be dominated by bounce-like configurations at large order of the perturbative series, which causes non-convergence of the series. We find that in addition to this bounce the perturbative functional has a subleading peak at the instanton and anti-instanton pair, and its sum reproduces the non-perturbative valley. 
  We study the Bethe ansatz equations for a generalized $XXZ$ model on a one-dimensional lattice. Assuming the string conjecture we propose an integer version for vacancy numbers and prove a combinatorial completeness of Bethe's states for a generalized $XXZ$ model. We find an exact form for inverse matrix related with vacancy numbers and compute its determinant. This inverse matrix has a tridiagonal form, generalizing the Cartan matrix of type $A$. 
  In statistical physics, useful notions of entropy are defined with respect to some coarse graining procedure over a microscopic model. Here we consider some special problems that arise when the microscopic model is taken to be relativistic quantum field theory. These problems are associated with the existence of an infinite number of degrees of freedom per unit volume. Because of these the microscopic entropy can, and typically does, diverge for sharply localized states. However the difference in the entropy between two such states is better behaved, and for most purposes it is the useful quantity to consider. In particular, a renormalized entropy can be defined as the entropy relative to the ground state. We make these remarks quantitative and precise in a simple model situation: the states of a conformal quantum field theory excited by a moving mirror. From this work, we attempt to draw some lessons concerning the ``information problem'' in black hole physics 
  Generalized coherent states provide a means of connecting square integrable representations of a semi-simple Lie group with the symplectic geometry of some of its homogeneous spaces. In the first part of the present work this point of view is extended to the supersymmetric context, through the study of the OSp(2/2) coherent states. These are explicitly constructed starting from the known abstract typical and atypical representations of osp(2/2). Their underlying geometries turn out to be those of supersymplectic OSp(2/2) homogeneous spaces. Moment maps identifying the latter with coadjoint orbits of OSp(2/2) are exhibited via Berezin's symbols. When considered within Rothstein's general paradigm, these results lead to a natural general definition of a super K\"ahler supermanifold, the supergeometry of which is determined in terms of the usual geometry of holomorphic Hermitian vector bundles over K\"ahler manifolds. In particular, the supergeometry of the above orbits is interpreted in terms of the geometry of Einstein-Hermitian vector bundles. In the second part, an extension of the full geometric quantization procedure is applied to the same coadjoint orbits. Thanks to the super K\"ahler character of the latter, this procedure leads to explicit super unitary irreducible representations of OSp(2/2) in super Hilbert spaces of $L^2$ superholomorphic sections of prequantum bundles of the Kostant type. This work lays the foundations of a program aimed at classifying Lie supergroups' coadjoint orbits and their associated irreducible representations, ultimately leading to harmonic superanalysis. For this purpose a set of consistent conventions is exhibited. 
  Invertible universal R-matrices of quantum Lie algebras do not exist at roots of unity. There exist however quotients for which intertwiners of tensor products of representations always exist, i.e. R-matrices exist in the representations. One of these quotients, which is finite dimensional, has a universal R-matrix. In this paper, we answer the following question: on which condition are the different quotients of U_q(sl(2)) (Hopf)-equivalent? In the case when they are equivalent, the universal R-matrix of one can be transformed into a universal R-matrix of the other. We prove that this happens only when q^4=1, and we explicitly give the expressions for the automorphisms and for the transformed universal R-matrices in this case. 
  It is argued that universality is severely limited for models with multiple fixed points. As a demonstration the renormalization group equations are presented for the potential and the wave function renormalization constants in the $O(N)$ scalar field theory. Our equations are superior compared with the usual approach which retains only the contributions that are non-vanishing in the ultraviolet regime. We find an indication for the existence of relevant operators at the infrared fixed point, contrary to common expectations. This result makes the sufficiency of using only renormalizable coupling constants in parametrizing the long distance phenomena questionable. 
  We show that the Ashtekar-Isham extension of the classical configuration space of Yang-Mills theories (i.e. the moduli space of connections) is (topologically and measure-theoretically) the projective limit of a family of finite dimensional spaces associated with arbitrary finite lattices. These results are then used to prove that the classical configuration space is contained in a zero measure subset of this extension with respect to the diffeomorphism invariant Ashtekar-Lewandowski measure.   Much as in scalar field theory, this implies that states in the quantum theory associated with this measure can be realized as functions on the ``extended" configuration space. 
  We discuss the properties of lorentzian and euclidean black hole solutions of a generalized 2-dim dilaton-gravity action containing a modulus field, which arises from the compactification of heterotic string models. The duality symmetries of these solutions are also investigated. 
  Coherent states are introduced and their properties are discussed for all simple quantum compact groups. The multiplicative form of the canonical element for the quantum double is used to introduce the holomorphic coordinates on a general quantum dressing orbit and interpret the coherent state as a holomorphic function on this orbit with values in the carrier Hilbert space of an irreducible representation of the corresponding quantized enveloping algebra. Using Gauss decomposition, the commutation relations for the holomorphic coordinates on the dressing orbit are derived explicitly and given in a compact R--matrix formulation (generalizing this way the $q$--deformed Grassmann and flag manifolds). The antiholomorphic realization of the irreducible representations of a compact quantum group (the analogue of the Borel--Weil construction) are described using the concept of coherent state. The relation between representation theory and non--commutative differential geometry is suggested.} 
  The theory of the string in interaction with a dilaton background field is analyzed. In the action considered, the metric in the world sheet of the string is the induced metric, and the theory presents second order time derivatives. The canonical formalism is developed and it is showed that first and second class constraints appear. The degrees of freedoom are the same than for the free bosonic string. The light cone gauge is used to reduce to the physical modes and to compute the physical hamiltonian. 
  We introduce a new family of gauge invariant regularizations of Chern-Simons theories which generate one-loop renormalizations of the coupling constant of the form $k\to k+2 s c_v$ where $s$ can take any arbitrary integer value. In the particular case $s=0$ we get an explicit example of a gauge invariant regularization which does not generate radiative corrections to the bare coupling constant. This ambiguity in the radiative corrections to $k$ is reminiscent of the Coste-L\"uscher results for the parity anomaly in (2+1) fermionic effective actions. 
  The universality of radiative corrections to the gauge coupling constant $k$ of Chern-Simons theory is studied in a very general regularization scheme. We show that the effective coupling constant $k$ induced by radiative corrections depends crucially on the balance between the ultraviolet behavior of scalar and pseudoscalar terms in the regularized action. There are three different regimes. When the ultraviolet leading term is scalar the coupling $k$ is shifted to $k+h^{\vee}$.However, if the leading term is pseudoscalar the shift is $k+s h^{\vee}$ with $s=0$ or $s=2$ depending on the sign of such a term. In the borderline case when the scalar and pseudoscalar terms have the same ultraviolet behavior the shift of $k$ becomes arbitrary (even non-integer) and depends on the parameters of the regularization. We also show that the coefficient of the induced gravitational Chern-Simons term is different for the three regimes and has the same universality properties than the effective coupling constant $k$. The results open the possibility of a connection with non-rational two-dimensional conformal theories in the borderline regime. 
  A general method for the construction of solutions of the d'Alambertian and double d'Alambertian (harmonic and bi-harmonic) equations with local dependence of arbitrary functions upon two independent arguments is proposed. The connection between solutions of this kind and self-dual configurations of gauge fields having no singularities is established. 
  We reinterpret U(N) Chern-Simons-Witten theory quantized on a torus as a free fermion system. Its Hilbert space and some observables are simply related to those of group quantum mechanics, even at finite N and k. Its large N limit can be described using techniques developed for matrix quantum mechanics and two-dimensional Yang-Mills theory. We discuss the bosonization of this theory, which for YM_2 gave a precise interpretation of Wilson loop operators in terms of string creation and annihilation operators, and examine its consequences for a string interpretation here. The formalism seems entirely adequate for the leading large N results and in a sense can be thought of as a `classical string field theory'. In considering subleading orders in 1/N, we identify some major differences between CSW and YM_2, which must be dealt with to find a CSW gauge string interpretation. Although these particular differences are probably not relevant for `QCD string,' they do illustrate some of the issues there, and we comment on this. We also propose an approach to dealing with large N transitions. 
  We investigate zero modes of a multivortex background which are due to the energetic degeneracy of the planar static n-vortex solution in Bogomol'nyi limit of the Abelian Higgs model parametrised by a set of 2n real parameters. The zero modes take the form of waves travelling with a speed of light independently along each of the vortices irrespectively to their mutual separations. String description of these zero modes is constructed. We also comment on the remnants of the modes a little outside of the Bogomol'nyi limit. 
  In two dimensions a large class of gravitational systems including, e.g., $R^2$-gravity can be quantized exactly also when coupled dynamically to a Yang-Mills theory. Some previous considerations on the quantization of pure gravity theories are improved and generalized. 
  In this review article, we review the recent developments in constructing string field theories that have been proposed, all of which correctly reproduce the correlation functions of two-dimensional string theory. These include: (a) free fermion field theory (b) collective string field theory (c) temporal gauge string field theory (d) non-polynomial string field theory. We analyze discrete states, the $w(\infty)$ symmetry, and correlation functions in terms of these different string field theories. We will also comment on the relationship between these various field theories. (To appear in Int. J. of Mod. Phys. Written in LATEX.) 
  In this paper the relationship between the classical description of the resolution of quotient singularities and the string picture is reviewed in the context of N=(2,2) superconformal field theories. A method for the analysis of quotients locally of the form C^d/G where G is abelian is presented. Methods derived from mirror symmetry are used to study the moduli space of the blowing-up process. The case C^2/Z_n is analyzed explicitly. This is largely a review paper to appear in "Essays on Mirror Manifolds, II". 
  We derive a set of recursion formulae to construct singular vectors for the $N=2$ (untwisted) algebra, by using the approach of Bauer, di Francesco, Itzykson and Zuber. Applying these formulae, we obtain explicit expressions for the charged singular vectors and for a class of uncharged singular vectors. 
  We study a class of models in which $N$ flavors of massless fermions on the half line are coupled by an arbitrary orthogonal matrix to $N$ rotors living on the boundary. Integrating out the rotors, we find the exact partition function and Green's functions. We demonstrate that the coupling matrix must satisfy a certain rationality constraint, so there is an infinite, discrete set of possible coupling matrices. For one particular choice of the coupling matrix, this model reproduces the low-energy dynamics of fermions scattering from a magnetic monopole. A quick survey of the Green's functions shows that the S-matrix is nonunitary. This nonunitarity is present in previous results for the monopole-fermion system, although it appears not to have been noted. We indicate how unitarity may be restored by expanding the Fock space to include new states that are unavoidably introduced by the boundary interaction. 
  The structure of equivariant cohomology in non-abelian localization formulas and topological field theories is discussed. Equivariance is formulated in terms of a nilpotent BRST symmetry, and another nilpotent operator which restricts the BRST cohomology onto the equivariant, or basic sector. A superfield formulation is presented and connections to reducible (BFV) quantization of topological Yang-Mills theory are discussed. 
  We apply the adiabatic approximation to investigate the low energy dynamics of vortices in the parity invariant double self-dual Higgs model with only mutual Chern-Simons interaction. When distances between solitons are large they are particles subject to the mutual interaction. The dual formulation of the model is derived to explain the sign of the statistical interaction. When vortices of different types pass one through another they behave like charged particles in magnetic field. They can form a bound state due to the mutual magnetic trapping. Vortices of the same type exhibit no statistical interaction. Their short range interactions are analysed. Possible quantum effects due to the finite width of vortices are discussed. 
  We derive gauge covariant self-dual equations for the $SU(2) \times U(1)_Y$ theory of electro-weak interactions and show that they admit solutions describing a periodic lattice of Z-strings.} \newpage 
  The coupling of $N=1$ SCFT of type $(4m,2)$ to two-dimensional supergravity can be formulated non-perturbatively in terms of a discrete super-eigenvalue model proposed by Alvarez-Gaum\'e, et al. We derive the superloop equations that describe, in the double scaling limit, the non-perturbative solution of this model. These equations are equivalent to the double scaled super-Virasoro constraints satisfied by the partition function. They are formulated in terms of a $\widehat c=1$ theory, with a $\IZ_2$-twisted scalar field and a Weyl-Majorana fermion in the Ramond sector. We have solved the superloop equations to all orders in the genus expansion and obtained the explicit expressions for the correlation functions of gravitationally dressed scaling operators in the NS- and R-sector. In the double scaling limit, we obtain a formulation of the model in terms of a new supersymmetric extension of the KdV hierarchy. 
  We resolve the `baxterization' problem with the help of the automorphism group of the Yang-Baxter (resp. star-triangle, tetrahedron, \dots) equations. This infinite group of symmetries is realized as a non-linear (birational) Coxeter group acting on matrices, and exists as such, {\em beyond the narrow context of strict integrability}. It yields among other things an unexpected elliptic parametrization of the non-integrable sixteen-vertex model. It provides us with a class of discrete dynamical systems, and we address some related problems, such as characterizing the complexity of iterations. 
  We examine an extension of the ideas of quantum cosmology and, in particular, the proposal of Hartle and Hawking for the boundary conditions of the Universe, to models which incorporate Yang-Mills fields. Inhomogeneous perturbations about a homogeneous, isotropic minisuperspace background model are considered, by expanding the Yang-Mills fields in harmonics of the spatial directions which are taken to be three-spheres. The expansions are made explicit for $SO(N)$ gauge fields thereby obtaining formulae compatible with the formalism conventionally used in quantum cosmology. We apply these results to the gauge group $SO(3)$ and derive the Lagrangian and the semi-classical wave function for this special case. 
  The renormalization-group improved effective potential ---to leading-log and in the linear curvature approximation--- is constructed for ``finite'' theories in curved spacetime. It is not trivial and displays a quite interesting, exponential-like structure ---in contrast with the case of flat spacetime where it coincides with the classical potential. Several possible cosmological applications, as curvature-induced phase transitions and modifications of the values of the gravitational and cosmological constants, are briefly discussed. 
  We review our approach to time and quantum dynamics based on non-critical string theory, developing its relationship to previous work on non-equilibrium quantum statistical mechanics and the microscopic arrow of time. We exhibit specific non-factorizing contributions to the ${\nd S}$ matrix associated with topological defects on the world sheet, explaining the r\^ole that the leakage of ${W_{\infty}}$ charges plays in the loss of quantum coherence. We stress the analogy with the quantum Hall effect, discuss the violation of $CPT$, and also apply our approach to cosmology. 
  It is shown that the general solution of a homogeneous Monge-Amp\`{e}re equation in $n$-dimensional space is closely connected with the exactly (but only implicitly) integrable system   \frac {\partial \xi_{j}}{\partial x_0}+\sum_{k=1}^{n-1} \xi_{k} \frac {\partial \xi_{j}}{\partial x_{k}}=0 \label{1}   Using the explicit form of solution of this system it is possible to construct the general solution of the Monge-Amp\`{e}re equation. 
  In this paper we study integrability and algebraic integrability properties of certain matrix Schr\"odinger operators. More specifically, we associate such an operator (with rational, trigonometric, or elliptic coefficients) to every simple Lie algebra g and every representation U of this algebra with a nonzero but finite dimensional zero weight subspace. The Calogero-Sutherland operator is a special case of this construction. Such an operator is always integrable. Our main result is that it is also algebraically integrable in the rational and trigonometric case if the representation U is highest weight. This generalizes the corresponding result for Calogero-Sutherland operators proved by Chalyh and Vaselov. We also conjecture that this is true for the elliptic case as well, which is a generalization of the corresponding conjecture by Chalyh and Vaselov for Calogero-Sutherland operators. 
  The elliptic-matrix quantum Olshanetsky-Perelomov problem is introduced for arbitrary root systems by means of an elliptic generalization of the Dunkl operators. Its equivalence with the double affine generalization of the Knizhnik-Zamolodchikov equation (in the induced representations) is established. 
  Black hole evaporation is investigated in a (1+1)-dimensional model of quantum gravity. Quantum corrections to the black hole entropy are computed, and the fine-grained entropy of the Hawking radiation is studied. A generalized second law of thermodynamics is formulated, and shown to be valid under suitable conditions. It is also shown that, in this model, a black hole can consume an arbitrarily large amount of information. 
  The transition amplitudes for the free spinless and spinning relativistic particles are obtained by applying an operator method developed long ago by Dirac and Schwinger to the BFV form of the BRST theory for constrained systems. 
  The moving mirror model is designed to extract essential features of the black hole formation and the subsequent Hawking radiation by neglecting complication due to a finite curvature. We extend this approach to dynamically treat back reaction against the mirror motion due to Hawking radiation. It is found that a unique model in two spacetime dimensions exists in which Hawking radiation completely stops and the end point of evaporation contains a disconnected remnant. When viewed from asymptotic observers at one side of the spacetime, quantum mechanical correlation is recovered in the end. Although the thermal stage accompanying short range correlation may last for an arbitrarily long period, at a much longer time scale a long tail of non-thermal correlation is clearly detected. 
  It is shown that there exists an isomorphism between q-oscillator systems covariant under $ SU_q(n) $ and $ SU_{q^{-1}}(n) $. By the isomorphism, the defining relations of $ SU_{q^{-1}}(n) $ covariant q-oscillator system are transmuted into those of $ SU_q(n) $. It is also shown that the similar isomorphism exists for the system of q-oscillators covariant under the quantum supergroup $ SU_q(n/m) $. Furthermore the cases of q-deformed Lie (super)algebras constructed from covariant q-oscillator systems are considered. The isomorphisms between q-deformed Lie (super)algebras can not obtained by the direct generalization of the one for covariant q-oscillator systems. 
  Based on our generalization of the Goulian-Li continuation in the power of the 2D cosmological term we construct the two and three-point correlation functions for Liouville exponentials with generic real coefficients. As a strong argument in favour of the procedure we prove the Liouville equation of motion on the level of three-point functions. The analytical structure of the correlation functions as well as some of its consequences for string theory are discussed. This includes a conjecture on the mass shell condition for excitations of noncritical strings. We also make a comment concerning the correlation functions of the Liouville field itself. 
  The Brattelli diagram associated with a given bicolored Dynkin-Coxeter graph of type $A_n$ determines planar fractal sets obtained by infinite dissections of a given triangle. All triangles appearing in the dissection process have angles that are multiples of $\pi/ (n+1).$ There are usually several possible infinite dissections compatible with a given $n$ but a given one makes use of $n/2$ triangle types if $n$ is even. Jones algebra with index $[ 4 \ \cos^2{\pi \over n+1}]^{-1}$ (values of the discrete range) act naturally on vector spaces associated with those fractal sets. Triangles of a given type are always congruent at each step of the dissection process. In the particular case $n=4$, there are isometric and the whole structure lead, after proper inflation, to aperiodic Penrose tilings. The ``tilings'' associated with other values of the index are discussed and shown to be encoded by equivalence classes of infinite sequences (with appropriate constraints) using $n/2$ digits (if $n$ is even) and generalizing the Fibonacci numbers. 
  Only the first 15 percent of the file survived the censor that is hidden in the network resources in a previous attempt to submit this thesis to the bulletin board. Hopefully this makes an uncorrupted file available. 
  We find a simpler formulation of the Green-Schwarz action, for which the Wess-Zumino term is the square of supersymmetric currents, like the rest of the action. On a random lattice it gives Feynman diagrams of a particle superfield theory. 
  The generating functional for hard thermal loops in QCD is important in setting up a resummed perturbation theory. I review how this functional is related to the eikonal for a Chern-Simons theory, and using an auxiliary field, to the gauged WZNW-action. The induced current due to hard thermal loops, properly incorporating damping effects, is also briefly discussed. (Invited talk at the Third Worshop on Thermal Field Theories, Banff, Canada, August, 1993.) 
  The effective action for hard thermal loops in QCD is related to a gauged WZNW theory. Some of the technical issues of this approach are clarified and the Hamiltonian formulation is presented. The two-point correlation function for the induced current in QCD is obtained; some simplifications of the dynamics of the longitudinal modes are also pointed out. 
  BRST invariance supplies a sufficient condition for the observability of fields. We show that there is a global obstruction to the observability of quarks and gluons and argue that they will not become observables at finite temperature. We give expressions for quarks and gluons that are, however, {\it perturbatively\/} BRST invariant, and hence locally observable, up to order~$g^2$ and~$g$ respectively. 
  A geometric interpretation of quantum self-interacting string field theory is given. Relations between various approaches to the second quantization of an interacting string are described in terms of the geometric quantization. An algorithm to construct a quantum nonperturbative interacting string field theory in the quantum group formalism is proposed. problems of a metric background (in)dependence are discussed. 
  An explicit expression for all the quantum integrals of motion for the isotropic Heisenberg $s=1/2$ spin chain is presented.   The conserved quantities are expressed in terms of a sum over simple polynomials in spin variables. This construction is direct and independent of the transfer matrix formalism. Continuum limits of these integrals in both ferrromagnetic and antiferromagnetic sectors are briefly discussed. 
  We construct a hierarchy of supersymmetric string theories by showing that the general N-extended superstrings may be viewed as a special class of the (N+1)-extended superstrings. As a side result, we find a twisted (N+2) superconformal algebra realized in the N-extended string. 
  Some aspects of Mirror symmetry are reviewed, with an emphasis on more recent results extending mirror transform to higher genus Riemann surfaces and its relation to the Kodaira-Spencer theory of gravity (talk given in the Geometry and Topology Conference, April 93, Harvard, in honor of Raoul Bott) 
  We construct a Grassmannian-like formulation for the potential KP-hierarchy including additional ``negative'' flows. Our approach will generalize the notion of a tau-function to include negative flows. We compare the resulting hierarchy with results by Hirota, Satsuma and Bogoyavlenskii. 
  In this paper, from the $q$-gauge covariant condition we define the $q$-deformed Killing form and the second $q$-deformed Chern class for the quantum group $SU_{q}(2)$. Developing Zumino's method we introduce a $q$-deformed homotopy operator to compute the $q$-deformed Chern-Simons and the $q$-deformed cocycle hierarchy. Some recursive relations related to the generalized $q$-deformed Killing forms are derived to prove the cocycle hierarchy formulas directly. At last, we construct the $q$-gauge covariant Lagrangian and derive the $q$-deformed Yang-Mills equation. We find that the components of the singlet and the adjoint representation are separated in the $q$-deformed Chern class, $q$-deformed cocycle hierarchy and the $q$-deformed Lagrangian, although they are mixed in the commutative relations of BRST algebra. 
  We show that the differential complex $\Omega_{B}$ over the braided matrix algebra $BM_{q}(N)$ represents a covariant comodule with respect to the coaction of the Hopf algebra $\Omega_{A}$ which is a differential extension of $GL_{q}(N)$. On the other hand, the algebra $\Omega_{A}$ is a covariant braided comodule with respect to the coaction of the braided Hopf algebra $\Omega_{B}$. Geometrical aspects of these results are discussed. 
  A general study of non-abelian duality is presented. We first identify a possible obstruction to the conformal invariance of the dual theory for non-semisimple groups. We construct the exact non-abelian dual for any Wess-Zumino-Witten (WZW) model for any anomaly free subgroup, and the corresponding extension for coset conformal field theories. We characterize the exact non-abelian dual for $\sigma$-models with chiral isometries and extend the standard notion of duality to anomalous subgroups of   WZW models, thus giving a way of constructing dual transformations for different groups on the left and on the right. We also present some new examples of non-abelian duality for four-dimensional gravitational instantons. 
  We show that the argument in Phys Rev Lett 70 (1993) 1360 is correct and consistent, and that Hagen & Sudarshan's solution has inconsistency leading to non-vanishing commutators of $[P^1, P^2]$ and $[P^j, H]$ even in physical states. This proves that many of HS's statements in their Comment are based merely on incorrect guess, but not on careful algebra. 
  We investigate the W-algebra resulting from Drinfel'd-Sokolov reduction of a $B_2$ WZW model with respect to the grading induced by a short root. The quantum algebra, which is generated by three fields of spin-2 and a field of spin-1, is explicitly constructed. A `free field' realisation of the algebra in terms of the zero-grade currents is given, and it is shown that these commute with a screening charge. We investigate the representation theory of the algebra using a combination of the explicit fusion method of Bauer et al. and free field methods. We discuss the fusion rules of degenerate primary fields, and give various character formulae and a Kac determinant formula for the algebra 
  We propose a multiloop generalization of the Bern-Kosower formalism, based on Strassler's approach of evaluating worldline path integrals by worldline Green functions. Those Green functions are explicitly constructed for the basic two-loop graph, and for a loop with an arbitrary number of propagator insertions. For scalar and abelian gauge theories, the resulting integral representations allow to combine whole classes of Feynman diagrams into compact expressions. 
  We study here the spectrum of soliton scattering theories based on interaction round the face lattice models. We take for the admissibility condition the fusion rules of each of the simple Lie algebras. It is found that the mass spectrum is given by that of the corresponding Toda theory, or, that the mass ratios of the different kinks in the model are described by the Perron--Frobenius vector of the Cartan matrix. The scalar part of the soliton amplitude is shown to be identical with the minimal part of the corresponding Toda amplitude. 
  The previous discussion \cite{ezawa} on reducing the phase space of the first order Einstein gravity in 2+1 dimensions is reconsidered. We construct a \lq\lq correct" physical phase space in the case of positive cosmological constant, taking into account the geometrical feature of SO(3,1) connections. A parametrization which unifies the two sectors of the physical phase space is also given. 
  We investigate the nucleation of circular cosmic strings in models of generalized inflationary universes with an accelerating scale factor. We consider toy cosmological models of a smooth inflationary exit and transition into a flat Minkowski spacetime. Our results establish that an inflationary expanding phase is necessary but not sufficient for quantum nucleation of circular cosmic strings to occur. 
  We give a general construction for finite dimensional representations of $U_q(\hat{\G})$ where $\hat{\G}$ is a non-twisted affine Kac-Moody algebra with no derivation and zero central charge. At $q=1$ this is trivial because $U(\hat{\G})=U({\G})\otimes \C(x,x^{-1})$ with $\G$ a finite dimensional Lie algebra. But this fact no longer holds after quantum deformation. In most cases it is necessary to take the direct sum of several irreducible $U_q({\G})$-modules to form an irreducible $U_q(\hat{\G})$-module which becomes reducible at $q = 1$. We illustrate our technique by working out explicit examples for $\hat{\G}=\hat{C}_2$ and $\hat{\G}=\hat{G}_2$. These finite dimensional modules determine the multiplet structure of solitons in affine Toda theory. 
  We develop the notions of fusion for representations of the W_3 algebra along the lines of Feigin and Fuchs. We present some explicit calculations for a W_3 minimal model. 
  A nonperturbative method is proposed for the approximative solution of the exact evolution equation which describes the scale dependence of the effective average action. It consists of a combination of exact evolution equations for independent couplings with renormalization group improved one loop expressions of secondary couplings. Our method is illustrated by an example: We compute the beta-function of the quartic coupling lambda of an O(N) symmetric scalar field theory to order lambda^3 as well as the anomalous dimension to order lambda^2 using only one loop expressions and find agreement with the two loop perturbation theory. We also treat the case of very strong coupling and confirm the existence of a "triviality bound". 
  The equations for various spin particles with oscillator-like interactions are discussed in this talk. Contents: 1. Comment on "The Klein-Gordon Oscillator"; 2. The Dirac oscillator in quaternion form; 3. The Dirac-Dowker oscillator; 4. The Weinberg oscillator; 5. Note on the two-body Dirac oscillator. 
  A certain class of one-dimensional classical lattice models is considered. Using the method of abstract harmonic analysis explicit thermostatic properties of such models are derived. In particular, we discuss the low-temperature behavior of some of these models. 
  We quantize a generalized electromagnetism in 2 + 1 dimensions which contains a higher-order derivative term by using Dirac's method. By introducing auxiliary fields we transform the original theory in a lower-order derivative one which can be treated in a usual way. 
  We define the analogue of Jack's (Jacobi) polynomials, which were defined for finite-dimensional root system by Heckman and Opdam as eigenfunctions of trigonometric Sutherland operator for the affine root system $\hat A_{n-1}$. In the affine case, we define the polynomials as eigenfunctions of "affine Sutherland operator", which is Calogero-Sutherland operator with elliptic potential plus the term involving derivative with respect to the modular parameter. We show that such polynomials can be constructed explicitly as traces of certain intertwiners for affine Lie algebra. Also, we define the q-analogue of this construction, which gives affine analogues of Macdonald's polynomials, and show the (conjectured) relation between the Macdonald's inner product identities for affine case and scalar product of conformal blocks in the WZW model. 
  We present Dirac's method for using dual potentials to solve classical electrodynamics for an oppositely charged pair of particles, with a view to extending these techniques to non-Abelian gauge theories. 
  Quantum field planes furnish a noncommutative differential algebra $\Omega$ which substitutes for the commutative algebra of functions and forms on a contractible manifold. The data required in their construction come from a quantum field theory. The basic idea is to replace the ground field ${\bf C}$ of quantum planes by the noncommutative algebra ${\cal A}$ of observables of the quantum field theory. 
  Blocking transformation is performed in quantum field theory at finite temperature. It is found that the manner temperature deforms the renormalized trajectories can be used to understand better the role played by the quantum fluctuations. In particular, it is conjectured that domain formation and mass parameter generation can be observed in theories without spontaneous symmetry breaking. 
  The standard Hamiltonian machinery, being applied to field theory, leads to infinite-dimensional phase spaces. It is not covariant. In this article, we present covariant finite-dimensional multimomentum Hamiltonian formalism for field theory. This is the multisymplectic generalization of the Hamiltonian formalism in mechanics. In field theory, multimomentum canonical variables are field functions and momenta corresponding to derivatives of fields with respect all world coordinates, not only the time. In case of regular Lagrangian densities, the multimomentum Hamiltonian formalism is equivalent to the Lagrangian formalism, otherwise for degenerate Lagrangian densities. In this case, the Euler-Lagrange equations become undetermined and require additional conditions which remain elusive. In the framework of the multimomentum Hamiltonian machinery, one obtaines them automatically as a part of Hamilton equations. The key point consists in the fact that, given a degenerate Lagrangian density, one must consider a family of associated multimomentum Hamiltonian forms in order to exaust solutions of the Euler-Lagrange equations. We spell out degenerate quadratic and affine Lagrangian densities. The most of field models are of these types. As a result, we get the general procedure of describing constraint field systems. 
  Applying the techniques of nonabelian duality to a system of Majorana fermions in 1+1 dimensions we obtain the level-one Wess-Zumino-Witten model as the dual theory. This makes nonabelian bosonization a particular case of a nonabelian duality transformation, generalizing our previous result (hep-th/9401105) for the abelian case. 
  A four-dimensional topological field theory is introduced which generalises $B\wedge F$ theory to give a Bogomol'nyi structure. A class of non-singular, finite-Action, stable solutions to the variational field equations is identified. The solitonic solutions are analogous to the instanton in Yang-Mills theory. The solutions to the Bogomol'nyi equations in the topologically least complicated $U(1)$ theory have a well-behaved (covariant) phase space of dimension four---the same as that for photons. The dimensional reduction of the four-dimensional Lagrangian is also examined. Bogomol'nyi $U(2)$ solitons resembling the intermediate vector bosons $Z_o$, $W^\pm$ are identified. 
  We propose a string theory model which explains several features of two dimensional YM theory. Folds are suppressed. This in turn leads to the empty theory in flat target spaces. The Nambu-Goto action appears in the usual way. The model naturally splits into two (chiral) sectors: orientation preserving maps and orientation reversing maps. Moreover it has a straightforward extension to 3 and 4 dimensional space-times, which could be the rigid string with the self-intersection number at $\theta=\pi$. 
  We analyze the $t$-$J$ model using the ${\rm CP}^1$ representation for the slave operators (holons and spinons) which is particularly suited to study the phenomenon of the spin-charge separation in strongly correlated electron systems. In particular, we show that for the one-dimensional $t$-$J$ model below half-filling the low energy effective dynamics of the spin and charge degrees of freedom is represented in the continuum limit by a ${\rm CP}^1$ model with a topological term, minimally coupled to a massless Dirac field with a four-fermion interaction. The bosonic term of this action describes the spin waves produced by the spinons, while the fermionic term represents the low energy charge excitations. This theory exhibits explicitly a local abelian gauge invariance. 
  An off-shell manifestly (8,0) worldsheet supersymmetric formulation of a multiplet describing physical chiral fermions is given. The multiplet can be used to complete the doubly supersymmetric (twistor-like) action for the heterotic string. Correction of technical errors. 
  We consider generalizations of Dunkl's differential-difference operators associated with groups generated by reflections. The commutativity condition is equivalent to certain functional equations. These equations are solved in many cases. In particular, solutions associated with elliptic curves are constructed. In the $A_{n-1}$ case, we discuss the relation with elliptic Calogero-Moser integrable $n$-body problems, and discuss the quantization ($q$-analogue) of our construction. 
  We generalize a five dimensional black hole solution of low energy effective string theory to arbitrary constant spatial curvature. After interchanging the signature of time and radius we reduce the 5d solution to four dimensions and obtain that way a four dimensional isotropic cosmological space time. The solution contains a dilaton, modulus field and torsion. Several features of the solution are discussed. 
  The multilevel field-antifield formalism is constructed in a geometrically covariant way without imposing the unimodularity conditions on the hypergauge functions. Thus the previously given version [1,2] is extended to cover the most general case of Lagrangian surface bases. It is shown that the extra measure factors, required to enter the gauge-independent functional integrals, can be included naturally into the multilevel scheme by modifying the boundary conditions to the quantum master equation. 
  In $D=3,4,6$ and 10 space--time dimensions considered is a string model invariant under transformations of $N=1$ space--time supersymmetry and $n=D-2$ local worldsheet supersymmetry with the both Virasoro constraints solved in the twistor form. The twistor solution survives in a modified form even in the presence of the heterotic fermions. 
  A Poisson--Hopf algebra of smooth functions on the (1+1) Cayley--Klein groups is constructed by using a classical $r$--matrix which is invariant under contraction. The quantization of this algebra for the Euclidean, Galilei and Poincar\'e cases is developed, and their duals are also computed. Contractions on these quantum groups are studied. 
  We study the equivariant cohomology of a class of multi-field topological LG models, and find that such systems carry intrinsic information about $W$-gravity. As a result, we can construct the gravitational chiral ring in terms of LG polynomials. We find, in particular, that the spectrum of such theories seems to be richer than so far expected. We also briefly discuss the BRST operator for non-linear topological $W$-gravity. 
  Making use of some results concerning the theory of partitions, relevant in number theory, the complete asymptotic behavior, for large $n$, of the level density of states for a parabosonic string is derived. It is also pointed out the similarity between parabosonic strings and membranes. 
  By enforcing locality we relate the cohomology found in parafermionic theories to that occurring in $W$ strings. This link provides a systematic method of finding states in the cohomology of $W_{2,s}$ strings. 
  Conformal field theories corresponding to two-dimensional electrically charged black holes and to two-dimensional anti-de Sitter space with a covariantly constant electric field are simply constructed as $SL(2,R)/Z$ WZW coset models. The two-dimensional electrically charged black holes are related by Kaluza-Klein reduction to the 2+1-dimensional rotating black hole of Banados, Teitelboim and Zanelli, and our construction is correspondingly related to its realization as a WZW model. Four-dimensional spacetime solutions are obtained by tensoring these two-dimensional theories with $SU(2)/Z(m)$ coset models. These describe a family of dyonic black holes and the Bertotti--Robinson universe. 
  We describe in detail the space of the two K\"ahler parameters of the Calabi--Yau manifold $\P_4^{(1,1,1,6,9)}[18]$ by exploiting mirror symmetry. The large complex structure limit of the mirror, which corresponds to the classical large radius limit, is found by studying the monodromy of the periods about the discriminant locus, the boundary of the moduli space corresponding to singular Calabi--Yau manifolds. A symplectic basis of periods is found and the action of the $Sp(6,\Z)$ generators of the modular group is determined. From the mirror map we compute the instanton expansion of the Yukawa couplings and the generalized $N=2$ index, arriving at the numbers of instantons of genus zero and genus one of each degree. We also investigate an $SL(2,\Z)$ symmetry that acts on a boundary of the moduli space. 
  The (2+1) dimensional nonabelian Chern-Simons theory coupled to complex scalar fields is quantized by using the Batalin-Tyutin canonical Hamiltonian method which systematically embeds second-class constraint system into first-class one. We obtain the gauge-invariant nonabelian Wess-Zumino type action in the extended phase space. 
  We construct cocycles on the Lie algebra of pseudo- and q-pseudodifferential symbols of one variable and on their close relatives: the sine-algebra and the Poisson algebra on two-torus. A ``quantum'' Godbillon-Vey cocycle on (pseudo)-differential operators appears in this construction as a natural generalization of the Gelfand-Fuchs 3-cocycle on periodic vector fields. We describe a nontrivial embedding of the Virasoro algebra into (a completion of) q-pseudodifferential symbols, and propose q-analogs of the KP and KdV-hierarchies admitting an infinite number of conserved charges. 
  The dispersionless Toda hierarchy turns out to lie in the heart of a recently proposed Landau-Ginzburg formulation of two-dimensional string theory at self-dual compactification radius. The dynamics of massless tachyons with discrete momenta is shown to be encoded into the structure of a special solution of this integrable hierarchy. This solution is obtained by solving a Riemann-Hilbert problem. Equivalence to the tachyon dynamics is proven by deriving recursion relations of tachyon correlation functions in the machinery of the dispersionless Toda hierarchy. Fundamental ingredients of the Landau-Ginzburg formulation, such as Landau-Ginzburg potentials and tachyon Landau-Ginzburg fields, are translated into the language of the Lax formalism. Furthermore, a wedge algebra is pointed out to exist behind the Riemann-Hilbert problem, and speculations on its possible role as generators of ``extra'' states and fields are presented. 
  We analyze the coset model $(E_2^c \ti E_2^c)/E_2^c$ and construct a class of exact string vacua which describe plane gravitational waves and their duals, generalizing the plane wave background found by Nappi and Witten. In particular, the vector gauging describes a two-parameter family of singular geometries with two isometries, which is dual to plane gravitational waves. In addition, there is a mixed vector-axial gauging which describes a one-parameter family of plane waves with five isometries. These two backgrounds are related by a duality transformation which generalizes the known axial-vector duality for abelian subgroups. 
  Families of exact $(0,2)$ supersymmetric conformal field theories of magnetically and electrically charged extremal 4D black hole solutions of heterotic string theory are presented. They are constructed using a $(0,1)$ supersymmetric $SL(2,R)\times SU(2)$ WZW model where anomalously embedded $U(1)\times U(1)$ subgroups are gauged. Crucial cancelations of the $U(1)$ anomalies coming from the supersymmetric fermions, the current algebra fermions and the gauging ensure that there is a consistency of these models at the quantum level. Various 2D models, which may be considered as building blocks for extremal 4D constructions, are presented. They generalise the class of 2D models which might be obtained from gauging $SL(2,R)$ and coincide with known heterotic string backgrounds. The exact conformal field theory presented by Giddings, Polchinski and Strominger describing the angular sector of the extremal magnetically charged black hole is a special case of this construction. An example where the radial and angular theories are mixed non--trivially is studied in detail, resulting in an extremal dilatonic Taub--NUT--like dyon. 
  It was shown by Garriga and Vilenkin that the circular shape of nucleated cosmic strings, of zero loop-energy in de Sitter space, is stable in the sense that the ratio of the mean fluctuation amplitude to the loop radius is constant. This result can be generalized to all expanding strings (of non-zero loop-energy) in de Sitter space. In other curved spacetimes the situation, however, may be different.   In this paper we develop a general formalism treating fluctuations around circular strings embedded in arbitrary spatially flat FRW spacetimes. As examples we consider Minkowski space, de Sitter space and power law expanding universes. In the special case of power law inflation we find that in certain cases the fluctuations grow much slower that the radius of the underlying unperturbed circular string. The inflation of the universe thus tends to wash out the fluctuations and to stabilize these strings. 
  We examine the conditions for superconformal invariance and the specific form of the K\"ahler potential for a two-dimensional lagrangian model with $N=2$ supersymmetry and superpotential $gX^k$. Away from the superconformal point we study the renormalization group flow induced by a particular class of K\"ahler potentials. We find trajectories which, in the infrared, reach the fixed point with a central charge whose value is that of the $N=2$, $A_{k-1}$ minimal model. 
  By exploiting standard facts about $N=1$ and $N=2$ supersymmetric Yang-Mills theory, the Donaldson invariants of four-manifolds that admit a Kahler metric can be computed. The results are in agreement with available mathematical computations, and provide a powerful check on the standard claims about supersymmetric Yang-Mills theory. 
  A new model of relativistic massive particle with arbitrary spin (($m,s$)-particle) is suggested. Configuration space of the model is a product of Minkowski space and two-dimensional sphere, ${\cal M}^6 = {\Bbb R}^{3,1} \times S^2$. The system describes Zitterbevegung at the classical level. Together with explicitly realized Poincar\'e symmetry, the action functional turns out to be invariant under two types of gauge transformations having their origin in the presence of two Abelian first-class constraints in the Hamilton formalism. These constraints correspond to strong conservation for the phase-space counterparts of the Casimir operators of the Poincar\'e group. Canonical quantization of the model leads to equations on the wave functions which prove to be equivalent to the relativistic wave equations for the massive spin-$s$ field. 
  There is a standard way to define two symplectic (hamiltonian) structures, the first and second Gelfand-Dikii brackets, on the space of ordinary linear differential operators of order $m$, $L = -d^m + U_1 d^{m-1} + U_2 d^{m-2} + \ldots + U_m$. In this paper, I consider in detail the case where the $U_k$ are $n\times n$-matrix-valued functions, with particular emphasis on the (more interesting) second Gelfand-Dikii bracket. Of particular interest is the reduction to the symplectic submanifold $U_1=0$. This reduction gives rise to matrix generalizations of (the classical version of) the {\it non-linear} $W_m$-algebras, called $V_{m,n}$-algebras. The non-commutativity of the matrices leads to {\it non-local} terms in these $V_{m,n}$-algebras. I show that these algebras contain a conformal Virasoro subalgebra and that combinations $W_k$ of the $U_k$ can be formed that are $n\times n$-matrices of conformally primary fields of spin $k$, in analogy with the scalar case $n=1$. In general however, the $V_{m,n}$-algebras have a much richer structure than the $W_m$-algebras as can be seen on the examples of the {\it non-linear} and {\it non-local} Poisson brackets of any two matrix elements of $U_2$ or $W_3$ which I work out explicitly for all $m$ and $n$. A matrix Miura transformation is derived, mapping these complicated second Gelfand-Dikii brackets of the $U_k$ to a set of much simpler Poisson brackets, providing the analogue of the free-field realization of the $W_m$-algebras. 
  Supersymmetric gauge theories in four dimensions can display interesting non-perturbative phenomena. Although the superpotential dynamically generated by these phenomena can be highly nontrivial, it can often be exactly determined. We discuss some general techniques for analyzing the Wilsonian superpotential and demonstrate them with simple but non-trivial examples. 
  The familiar generating functionals in quantum field theory fail to be true measures and, so they make the sense only in the framework of the perturbation theory. In our approach, generating functionals are defined strictly as the Fourier transforms of Gaussian measures in nuclear spaces of multimomentum canonical variables when field momenta correspond to derivatives of fields with respect to all world coordinates, not only to time. 
  We consider the Krein realization of the Hilbert space for a massless scalar field in 1+1 dimensions. We find convergence criteria and the completion of the space of test functions ${\cal S}$ with the topology induced by the Krein scalar product. Finally, we show that the interpretation for the Fourier components as probability amplitudes for the momentum operator is lost in this case. 
  We analyse with the algebraic, regularisation independent, cohomological B.R.S. methods, the renormalisability of torsionless N=2 supersymmetric non-linear sigma models built on Kahler spaces. Surprisingly enough with respect to the common wisdom, we obtain an anomaly candidate, at least in the compact Ricci-flat case. In the compact homogeneous Kahler case, as expected, the anomaly candidate disappears. 
  The $1/N$ expansion of the weak coupling phase of two-dimensional QCD on a sphere is constructed. It is demonstrated that the phase transition from the weak to the strong coupling phase is induced by instantons. A double scaling limit of the theory at the point of the phase transition is constructed, the value of string susceptibility is determined to be $\gamma_{str}=-1$. The possibility of instanton induced large $N$ phase transitions in four dimensional QCD is explored. 
  We advocate a new approach to the study of unitary matrix models in external fields which emphasizes their relationship to Generalized Kontsevich Models (GKM) with non-polynomial potentials. For example, we show that the partition function of the Brezin-Gross-Witten Model (BGWM), which is defined as an integral over unitary $N\times N$ matrices, $\int [dU] e^{\rm{Tr}(J^\dagger U + JU^\dagger)}$, can also be considered as a GKM with potential ${\cal V}(X) = \frac{1}{X}$. Moreover, it can be interpreted as the generating functional for correlators in the Penner model. The strong and weak coupling phases of the BGWM are identified with the "character" (weak coupling) and "Kontsevich" (strong coupling) phases of the GKM, respectively. This sort of GKM deserves classification as $p=-2$ one (i.e. $c=28$ or $c=-2$) when in the Kontsevich phase. This approach allows us to further identify the Harish-Chandra-Itzykson-Zuber (IZ) integral with a peculiar GKM, which arises in the study of $c=1$ theory and, further, with a conventional 2-matrix model which is rewritten in Miwa coordinates. Inspired by the considered unitary matrix models, some further extensions of the GKM treatment which are inspired by the unitary matrix models which we have considered are also developed. In particular, as a by-product, a new simple method of fixing the Ward identities for matrix models in an external field is presented. 
  We discuss stationary supersymmetric bosonic configurations of the Einstein-Maxwell theory embedded in $N=2$ supergravity. Some of these configurations, including the Kerr-Newman solutions with $m = |q|$ and arbitrary angular momentum per unit mass $a$, exhibit naked singularities. However, $N=2$ supergravity has trace anomaly. The nonvanishing anomalous energy-momentum tensor of these Kerr-Newman solutions violates a consistency condition for a configuration to admit unbroken supersymmetry. Thus, the trace anomaly of this theory prevents the supersymmetric solutions from exhibiting naked singularities. 
  A new derivation of the quantum deformation of the 2 dimensional Euclidean Poincare group (cf S. Zakrzewski) is proposed. It is based on a contraction of the Hopf algebra Fun(SO_q(3)). The deformation parameter q is sent to one, as in the construction of the $\kappa$-Poincare deformed algebra. The quantum group obtained is dual to that algebra. 
  The theory of a massless two-dimensional scalar field with a periodic boundary interaction is considered. At a critical value of the period this system defines a conformal field theory and can be re-expressed in terms of free fermions, which provide a simple realization of a hidden $SU(2)$ symmetry of the original theory. The partition function and the boundary $S$-matrix can be computed exactly as a function of the strength of the boundary interaction. We first consider open strings with one interacting and one Dirichlet boundary, and then with two interacting boundaries. The latter corresponds to motion in a periodic tachyon background, and the spectrum exhibits an interesting band structure which interpolates between free propagation and tight binding as the interaction strength is varied. 
  The $N=2$ topological Yang-Mills and holomorphic Yang-Mills theories on simply connected compact K\"{a}hler surfaces with $p_g\geq 1$ are reexamined. The $N=2$ symmetry is clarified in terms of a Dolbeault model of the equivariant cohomology. We realize the non-algebraic part of Donaldson's polynomial invariants as well as the algebraic part. We calculate Donaldson's polynomials on $H^{2,0}(S,\BZ)\oplus H^{0,2}(S,\BZ)$. 
  We discuss the notion of a Batalin-Vilkovisky (BV) algebra and give several classical examples from differential geometry and Lie theory. We introduce the notion of a quantum operator algebra (QOA) as a generalization of a classical operator algebra. In some examples, we view a QOA as a deformation of a commutative algebra. We then review the notion of a vertex operator algebra (VOA) and show that a vertex operator algebra is a QOA with some additional structures. Finally, we establish a connection between BV algebras and VOAs. 
  We present a new class of topological conformal field theories (TCFT) characterized by a rational $W$ potential, which includes the minimal models of A and D types as its subclasses. An explicit form of the $W$ potential is found by solving the underlying dispersionless KP hierarchy in a particular small phase space. We discuss also the dispersionless KP hierarchy in large phase spaces by reformulating the hierarchy, and show that the $W$ potential takes a universal form, which does not depend on a specific form of the solution in a large space. 
  We express the correlation functions of the SU(2) WZW conformal field theory on Riemann surfaces of genus >1 by finite dimensional integrals. 
  In these lectures the introduction to algebraic aspects of Bethe Ansatz is given. The applications to the seminal spin 1/2 XXX model is discussed in detail and the generalization to higher spin as well as XXZ and lattice Sine-Gordon model are indicated. The origin of quantum groups and their appearance in CFT models is explained. The text can be considered as a guide to the research papers in this field. 
  The behavior of finite temperature planar electrodynamics is investigated. We calculate the static as well as dynamic characteristic functions using real time formalism. The temperature and density dependence of dielectric and permeability functions, plasmon frequencies and their relation to the screening length is determined. The radiative correction to the fermion mass is also calculated. We also calculate the temperature dependence of the electron (anyon) magnetic moment. Our results for the gyromagnetic ratio go smoothly to the known result at zero temperature, $g=2$, in accordance with the general expectation. 
  We obtain both topological as well as nontopological self-dual charged vortex solutions of finite energy per unit length in a generalized abelian Higgs model in $3+1$ dimensions. In this model the Bogomol'nyi bound on the energy per unit length is obtained as a linear combination of the magnetic flux and the electric charge per unit length. 
  The time dependent-integrals of motion, linear in position and momentum operators, of a quantum system are extracted from Noether's theorem prescription by means of special time-dependent variations of coordinates. For the stationary case of the generalized two-dimensional harmonic oscillator, the time-independent integrals of motion are shown to correspond to special Bragg-type symmetry properties. A detailed study for the non-stationary case of this quantum system is presented. The linear integrals of motion are constructed explicitly for the case of varying mass and coupling strength. They are obtained also from Noether's theorem. The general treatment for a multi-dimensional quadratic system is indicated, and it is shown that the time-dependent variations that give rise to the linear invariants, as conserved quantities, satisfy the corresponding classical homogeneous equations of motion for the coordinates. 
  A two-dimensional generalized oscillator with time-dependent parameters is considered to study the two-mode squeezing phenomena. Specific choices of the parameters are used to determine the dispersion matrix and analytic expressions, in terms of standard hermite polynomials, of the wavefunctions and photon distributions. (to be publish in the Third Workshop on Squeezed States and Uncertainty Relations, Baltimore, USA, (August 1993)) 
  The vanishing of the anomaly in the recent example of Nappi and Witten, constructed from the Wess-Zumino-Witten model based on a certain non-semisimple group, follows from a more general result valid for gravitational waves. The construction of the metric is explained.} 
  This article is based on an invited talk given at the Workshop on Mathematical Physics Towards XXIst Century, held at Beer-Sheva, Israel in 1993. It contains an introduction to quantum gravity for mathematical physicists with an emphasis on the difference between the structure of this theory from more familiar, Minkowskian quantum field theories which arise due to the absence of a background space-time geometry. 
  We discuss the ``gravitationally dressed'' beta functions in the Gross--Neveu model interacting with 2d Liouville theory and in $SU(N)$ gauge theory interacting with the conformal sector of 4d quantum gravity. Among the effects that we suggest may feel the gravitational dressing are the minimum of the effective potential and the running of the gauge coupling. 
  We construct five independent screening currents associated with the $U_q(\widehat{sl(3)})$ quantum current algebra. The screening currents are expressed as exponentials of the eight basic deformed bosonic fields that are required in the quantum analogue of the Wakimoto realization of the current algebra. Four of the screening currents are `simple', in that each one is given as a single exponential field. The fifth is expressed as an infinite sum of exponential fields. For reasons we discuss, we expect that the structure of the screening currents for a general quantum affine algebra will be similar to the $U_q(\widehat{sl(3)})$ case. 
  (2+1) dimensional gravity is equivalent to an exactly soluble non-Abelian Chern-Simons gauge field theory (E Witten 1988). Regarding this as the topological phase of quantum gravity in (2+1)d, we suggest a topological symmetry breaking by introducing a mass term for the gauge fields, which carries a space-time metric and local dynamical degrees of freedom. We consider the finite temperature behavior of the symmetry broken phase, and claim a restoration of the topological invariance at some critical temperature. The phase transition is shown of the zeroth order. 
  We construct the deformed Dirac monopole on the quantum sphere for arbitrary charge using two different methods and show that it is a quantum principal bundle in the sense of Brzezinski and Majid. We also give a connection and calculate the analog of its Chern number by integrating the curvature over $S^2_q$. 
  We regularize the 3-D quantum electrodynamics by a parity invariant Pauli-Villars regularization method. We find that in the perturbation theory the Chern-Simons term is not induced in the massless fermion case and induced in the massive fermion case. 
  In a previous paper we showed that bracket relations uniquely fix the tree-level bosonic string $S$-matrix for $N\leq 26$ particle scattering. In this note we extend the proof to $N$-particle scattering for all $N$. 
  We define a discretized Langevin equation in Stratonovich-{\it type} calculus. We show that a generating functional with a field-dependent kernel can be written in mid-point prescription only when we calculate in the calculus. Moreover we investigate whether supersymmetry of the stochastic action with field-dependent kernel exists or not. 
  A new general Lie-algebraic approach is proposed to solving evolution tasks in some nonlinear problems of quantum physics with polynomially deformed Lie algebras $su_{pd}(2)$ as their dynamic symmetry algebras.   The method makes use of an expansion of the evolution operators by power series in the $su_{pd}(2)$ shift operators and a (recursive) reduction of finding coefficient functions to solving auxiliary exactly solvable $su(2)$ problems with quadratic Hamiltonians.   PACS numbers: 03.70; 02.20; 42.50 
  We construct free fields of arbitrary spin in 1+2 dimensions i.e. free fields for which the one-particle Hilbert space carries a projective isometric irreducible representation of the Poincar\'e group in 1+2 dimensions. We analyse in detail these representations in the fiber bundle formalism and afterwards we apply Weinberg procedure to construct the free fields. Some comments concerning axiomatic field theory in 1+2 dimensions are also made. 
  The observable algebra O of SO_q(3)-symmetric quantum mechanics is generated by the coordinates of momentum and position spaces (which are both isomorphic to the SO_q(3)-covariant real quantum space R_q^3). Their interrelations are determined with the quantum group covariant differential calculus. For a quantum mechanical representation of O on a Hilbert space essential self- adjointness of specified observables and compatibility of the covariance of the observable algebra with the action of the unitary continuous corepresent- ation operator of the compact quantum matrix group SO_q(3) are required. It is shown that each such quantum mechanical representation extends uniquely to a self-adjoint representation of O. All these self-adjoint representations are constructed. As an example an SO_q(3)-invariant Coulomb potential is intro- duced, the corresponding Hamiltonian proved to be essentially self-adjoint and its negative eigenvalues calculated with the help of a q-deformed Lenz-vector. 
  Affine Toda field theories in two dimensions constitute families of integrable, relativistically invariant field theories in correspondence with the affine Kac-Moody algebras. The particles which are the quantum excitations of the fields display interesting patterns in their masses and coupling and which have recently been shown to extend to the classical soliton solutions arising when the couplings are imaginary. Here these results are extended from the untwisted to the twisted algebras. The new soliton solutions and their masses are found by a folding procedure which can be applied to the affine Kac-Moody algebras themselves to provide new insights into their structures. The relevant foldings are related to inner automorphisms of the associated finite dimensional Lie group which are calculated explicitly and related to what is known as the twisted Coxeter element. The fact that the twisted affine Kac-Moody algebras possess vertex operator constructions emerges naturally and is relevant to the soliton solutions. 
  Based on the chiral symmetry breaking pattern and the corresponding low-energy effective lagrangian, we determine the fermion mass dependence of the partition function and derive sum rules for eigenvalues of the QCD Dirac operator in finite Euclidean volume. Results are given for $N_c = 2$ and for Yang-Mills theory coupled to several light adjoint Majorana fermions. They coincide with those derived earlier in the framework of random matrix theory. 
  We establish an explicit form of the Backlund transformation for the most known integrable systems. 
  When the supersymmetric theory contains the ``anomalous" $U(1)$ gauge symmetry with Green-Schwarz anomaly cancellation mechanism in 4 dimensions, its Fayet-Iliopoulos $D$-term generates non-universal scalar masses and the positive cosmological constant after the supersymmetry breaking. Both give the new contributions to the known results from $F$-term. Our mechanism is naturally realized in many string models and in some cases, leads to remarkable cancellations between $F$- and $D$-term contributions, providing the universal scalar mass and vanishing cosmological constant. We illustrate how such a possibility can arise by taking a simple orbifold example. 
  We consider the 2D Poincar\'e gravity and show its exact integrability. The choice of the gauge is discussed. The Euclidean solutions on compact closed differential manifolds are studied. 
  Under $SL(2,R)$ electric-magnetic duality transformations the Bogomolnyi bound of dilaton-axion black holes is known to be invariant. In this paper we show that this invariance corresponds to the covariance of the $N=4$ supersymmetry transformation rules and their parameters. In particular this implies that Killing spinors transform covariantly into Killing spinors. As an example, we work out completely the case of the largest known family of axion-dilaton black holes which is $SL(2,R)$-invariant, finding the Killing spinors with the announced properties. 
  We show that some factors of the universal R-matrix generate a family of twistings for the standard Hopf structure of any quantized contragredient Lie (super)algebra of finite growth. As an application we prove that any two isomorphic superalgebras with different Cartan matrices have isomorphic q-deformations (as associative superalgebras) and their standard comultiplications are connected by such twisting. We present also an explicit relation between the generators of the second Drinfeld's realization and Cartan-Weyl generators of quantized affine nontwisted Kac-Moody algebras. Further development of the theory of quantum Cartan-Weyl basis, closely related with this isomorphism, is discussed. We show that Drinfeld's formulas of a comultiplication for the second realization are a twisting of the standard comultiplication by factors of the universal R-matrix. Finally, properties of the Drinfeld's comultiplication are considered. 
  We perform an Hamiltonian reduction on a classical \cw(\cg, \ch) algebra, and prove that we get another \cw(\cg, \ch$'$) algebra, with $\ch\subset\ch'$. In the case $\cg=S\ell(n)$, the existence of a suitable gauge, called Generalized Horizontal Gauge, allows to relate in this way two \cw-algebras as soon as their corresponding \ch-algebras are related by inclusion. 
  The general formula for the universal R-matrix for quantized nontwisted affine algebras by Khoroshkin and Tolstoy is applied for zero central charge highest weight modules of the quantized affine algebras. It is shown how the universal R-matrix produces the Gauss decomposition of trigonomitric R-matrix in tensor product of these modules. Explicit calculations for the simplest case of $A_1^{(1)}$ are presented. As a consequence new formulas for the trigonometric R-matrix with a parameter in tensor product of $U_q(sl_2)$-Verma modules are obtained. 
  A general framework for the deformation of the single-mode oscillators is presented and all deformed single-mode oscillators are unified. The extensions of the Aric-Coon, genon, the para-Bose and the para-Fermi oscillators are proposed. The generalized harmonic oscillator considered by Brzezinski et al. is rederived in a simple way.Some remarks on deformation of $SU(1,1)$ and supersymmetry are made. 
  We construct polynomial Poisson algebras of observables for the classical Euler-Calogero-Moser (ECM) models. The conserved Hamiltonians and symmetry algebras derived in a previous work are subsets of these algebras. We define their linear, $N \rightarrow \infty$ limits, realizing $\w_{\infty}$ type algebras coupled to current algebras. 
  We study quasifinite highest weight modules over the supersymmetric extension of the $W_{1+\infty}$ algebra on the basis of the analysis by Kac and Radul. We find that the quasifiniteness of the modules is again characterized by polynomials, and obtain the differential equations for highest weights. The spectral flow, free field realization over the $(B,C)$--system, and the embedding into $\Glinf$ are also presented. 
  We give an analytical description of the locus of the two-gap elliptic potentials associated with the corresponding flow of the Calogero--Moser system. We start with the description of Treibich--Verdier two--gap elliptic potentials. The explicit formulae for the covers, wave functions and Lam\'e polynomials are derived, together with a new Lax representation for the particle dynamics on the locus. Then we consider more general potentials within the Weierstrass reduction theory of theta functions to lower genera. The reduction conditions in the moduli space of the genus two algebraic curves are given. This is some subvariety of the Humbert surface, which can be singled out by the condition of the vanishing of some theta constants. 
  Tetramodule is a vector space supplied with the bimodule and bicomodule structures over a Hopf algebra. The exact definition is given. Some properties and applications to quantum groups are discussed. 
  We obtain new stronger bounds by orders of magnitude, on the ultimate temperature of the universe by exploiting the copious production of gravitinos at finite temperature. 
  The 2D model of gravity with zweibeins $e^{a}$ and the Lorentz connection one-form $\omega^{a}_{\ b}$ as independent gravitational variables coupled to 2d massless Dirac matter is considered. It is shown that the classical equations of motion are exactly integrated in the case of chiral fermions. 
  Determinants of invertible pseudo-differential operators (PDOs) close to positive self-adjoint ones are defined throughthe zeta-function regularization. We define a multiplicative anomaly as the ratio $\det(AB)/(\det(A)\det(B))$ considered as a functionon pairs of elliptic PDOs. We obtained an explicit formula for the multiplicative anomaly in terms of symbols of operators. For a certain natural classof PDOs on odd-dimensional manifolds generalizing the class of ellipticdifferential operators, the multiplicative anomaly is identically $1$. For elliptic PDOs from this class a holomorphic determinant and a determinant for zero orders PDOs are introduced. Using various algebraic, analytic, and topological tools we study local and global properties of the multiplicative anomaly and of the determinant Lie group closely related with it. The Lie algebra for the determinant Lie group has a description in terms of symbols only. Our main discovery is that there is a {\em quadratic non-linearity} hidden in the definition of determinants of PDOs through zeta-functions. The natural explanation of this non-linearity follows from complex-analytic properties of a new trace functional TR on PDOs of non-integer orders. Using TR we easily reproduce known facts about noncommutative residues of PDOs and obtain several new results. In particular, we describe a structure of derivatives of zeta-functions at zero as of functions on logarithms of elliptic PDOs. We propose several definitions extending zeta-regularized determinants to general elliptic PDOs. For elliptic PDOs of nonzero complex orders we introduce a canonical determinant in its natural domain of definition. 
  The construction and the symmetries of Chern-Simons vortices in harmonic and uniform magnetic force backgrounds found by Ezawa, Hotta and Iwazaki, and by Jackiw and Pi are generalized using the non-relativistic Kaluza-Klein-type framework presented in our previous paper. All Schr\"odinger-symmetric backgrounds are determined. (10 pages. PlainTex) 
  We study scattering processes on $p$-adic multiloop surfaces represented as multiloop infinite graphs with total valence in each vertex equal $p+1$. They all are spaces of the constant negative curvature since they are quotients of the $p$-adic hyperbolic plane over free acting discrete subgroup of $PGL(2, {\bf Q}_p)$. Releasing the closed part of this graph containing all loops which is called reduced graph $T_{red}$ we can obtain $L$-function corresponding to this closed graph. For the total graph we introduce the notion of the spherical functions being eigenfunctions of the Laplace operator acting on the graph and consider $s$--wave scattering processes therefore defining scattering matrices $c_i$. The number of possibilities coincides with $|\T_{red}|$ --- the number of vertices of the reduced graph. Taking the product over all $c_i$ we define the total scattering matrix which appears to be essentially presented as a ratio of $L$--functions: $C\sim L(\alpha_+)/L(\alpha_-)$, where the function $L$ itself depends only on the shape of $\Tr$ and not on the initial infinite graph, and the only dependence of initial $p$ is contained in arguments $\alpha_\pm$ defined by $p$ and eigenvalue $t$ of the Laplacian. We also present a proof by H.Bass of the theorem expressing $L$--functions on arbitrary finite graphs via determinants of some local operators on these graphs. 
  The non-perturbative autonomous renormalization of the scalar $\Phi^4$-model is applied in the framework of stochastic quantization. I show that this requires a selective, momentum-dependent renormalization of the Onsager coefficient $\lambda$, a direct consequence of the characteristic wavefunction renormalization applied. As a result, I obtain a Langevin equation for the renormalized constant mode of the field, which is solved numerically. It is demonstrated for temperature zero that, starting from specified initial conditions, the system relaxes to its equilibrium state, the symmetry-breaking vacuum of the ``static'' $\Phi^4$-theory. 
  We study the $su(2)$ conformal field theory in its spinon description, adapted to the Yangian invariance. By evaluating the action of the Yangian generators on the primary fields, we find a new connection between this conformal field theory and the Calogero-Sutherland model with $su(2)$ spin. We use this connection to describe how the spinons are the quasi-particles spanning the irreducible Yangian multiplet, and also to exhibit operators creating the $N$-spinon highest weight vectors. 
  All the hermitian representations of the ``symmetric" $q$-oscillator are obtained by means of expansions. The same technique is applied to characterize in a systematic way the $k$-order boson realizations of the $q$-oscillator and $su(1,1)_q$. The special role played by the quadratic realizations of $su(1,1)_q$ in terms of boson and $q$-boson operators is analysed and clarified. 
  An embedding method to get $q$-deformations for the non--semisimple algebras generating the motion groups of $N$--dimensional flat spaces is presented. This method gives a global and simultaneous scheme of $q$-deformation for all $iso(p,q)$ algebras and for those ones obtained from them by some In\"on\"u--Wigner contractions, such as the $N$--dimensional Euclidean, Poincar\'e and Galilei algebras. 
  We show that a general miraculous cancellation formula, the divisibility of certain characteristic numbers and some other topologiclal results are con- sequences of the modular invariance of elliptic operators on loop spaces. Previously we have shown that modular invariance also implies the rigidity of many elliptic operators on loop spaces. 
  We give a systematic account of the exterior algebra of forms on q-Minkowski space, introducing the q-exterior derivative, q-Hodge star operator, q-coderivative, q-Laplace-Beltrami operator and the q-Lie-derivative. With these tools at hand, we then give a detailed exposition of the q-d'Alembert and q-Maxwell equation. For both equations we present a q-momentum-indexed family of plane wave solutions. We also discuss the gauge freedom of the q-Maxwell field and give a q-spinor analysis of the q-field strength tensor. 
  The large-$N$ nonlinear $O(N)$, $CP^{N-1}$ $\sigma$ models are studied on $R^2 \times S^1$. The $N$-components scalar fields of the models are supposed to acquire a phase $e^{i2\pi\delta}$ $(0\leq \delta <1)$, along the circulation of the circle, $S^1$. We evaluate the effective potentials to the leading order of the $1/N$ expansion. It is shown that, on $R^2\times S^1$ the $O(N)$ model has rich phase structure while the phase of $CP^{N-1}$ model is just that of the model at finite temperature. 
  The Toda lattice hierarchy is discussed in connection with the topological description of the $c=1$ string theory compactified at the self-dual radius. It is shown that when special constraints are imposed on the Toda hierarchy, it reproduces known results of the $c=1$ string theory, in particular the $W_{1+\infty}$ relations among tachyon correlation functions. These constraints are the analogues of string equations in the topological minimal theories. We also point out that at $c=1$ the Landau-Ginzburg superpotential becomes simply a $U(1)$ current operator. 
  Classification of finite dimensional representations of the q-deformed Heisenberg algebra $H_q(3)$ is made by the help of Clifford algebra of polynomials and generalized Grassmann algebra. Special attention is paid when $q$ is a primitive $n$ root of unity. As a further application we obtain finite dimensional representations of $ sl(2)_q$ using its embedding into $H_q(3)$. 
  The space-time light-cone Hamiltonian P^- of large-N matrix models for dynamical triangulations may be viewed as that of a quantum spin chain and analysed in a mean field approximation. As N -> infinity, the properties of the groundstate as a function of the bare worldsheet cosmological constant exhibit a parton phase and a critical string phase, separated by a transition with non-trivial scaling at which P- -> -infinity. 
  We investigate the structure of the configuration space of gauge theories and its description in terms of the set of absolute minima of certain Morse functions on the gauge orbits. The set of absolute minima that is obtained when the background connection is a pure gauge is shown to be isomorphic to the orbit space of the pointed gauge group. We also show that the stratum of irreducible orbits is geodesically convex, i.e. there are no geometrical obstructions to the classical motion within the main stratum. An explicit description of the singularities of the configuration space of SU(2) theories on a topologically simple space-time and on the lattice is obtained; in the continuum case we find that the singularities are conical and that the singular stratum is isomorphic to a Z_2 orbifold of the configuration space of electrodynamics. 
  The idea of minimum distance, familiar from R <-> 1/R duality when the string target space is a circle, is analyzed for less trivial geometries. The particular geometry studied is that of a blown-up quotient singularity within a Calabi-Yau space and mirror symmetry is used to perform the analysis. It is found that zero distances can appear but that in many cases this requires other distances within the same target space to be infinite. In other cases zero distances can occur without compensating infinite distances. 
  Extensions of standard one-dimensional supersymmetric quantum mechanics are discussed. Supercharges involving higher order derivatives are introduced leading to an algebra which incorporates a higher order polynomial in the Hamiltonian. We study scattering amplitudes for that problem. We also study the role of a dilatation of the spatial coordinate leading to a q-deformed supersymmetric algebra. An explicit model for the scattering amplitude is constructed in terms of a hypergeometric function which corresponds to a reflectionless potential with infinitely many bound states. 
  We present an analysis of the cocycle appearing in the vertex operator representation of simply-laced, affine, Kac-Moody algebras. We prove that it can be described in the context of $R$-commutative geometry, where $R$ is a Yang-Baxter operator, as a strong $R$-commutative algebra. We comment on the Hochschild, cyclic and dihedral homology theories that appear in non-commutative geometry and their potential relation to string theory. 
  We present a number of D=4 bosonic and heterotic string solutions with a covariantly constant null Killing vector which, like the solution of Nappi and Witten (NW), correspond to (gauged) WZW models and thus have a direct conformal field theory interpretation. A class of exact plane wave solutions (which includes the NW solution) is constructed by `boosting' the twisted products of two D=2 `cosmological' or `black-hole' cosets of SL(2,R) and SU(2). We describe a general limiting procedure by which one can construct new solutions with a covariantly constant null Killing vector starting with known string backgrounds. By applying a non-abelian duality transformation to the NW model we find a D=4 solution which has a covariantly constant null Killing vector but is not a plane wave. Higher dimensional bosonic backgrounds with isometries can be interpreted as lower dimensional backgrounds with extra gauge fields. Some of them are at the same time solutions of the heterotic string theory. In particular, the NW model represents also a D=3 `gravi-electromagnetic' heterotic string plane wave. In addition to the (1,1) supersymmetric embeddings of bosonic solutions we construct a number of non-trivial (1,0) supersymmetric exact D=4 heterotic string plane wave solutions some of which are related (by a boost and analytic continuation) to limiting cases of D=4 heterotic black hole solutions. 
  We consider a multiplicatively renormalizable higher-derivative scalar theory which is used as an effective theory for quantum gravity at large distances (infrared phase of quantum gravity). The asymptotic regimes (in particular, the asymptotically free infrared regime) for the coupling constants ---specifically the Newtonian and the cosmological constant--- are obtained. The running of the Newton and cosmological constants in the infrared asymptotically free regime may be relevant for solving the cosmological constant problem and for estimating the leading-log corrections to the static gravitational potential. 
  We calculate the quantum mass corrections to the solitons in the C_2^(1) Affine Toda field theory. We find that the ratio of the masses of the two solitons is not constant. 
  When discussing the black hole information problem the term ``information flow'' is frequently used in a rather loose fashion. In this article I attempt to make this notion more concrete. I consider a Hilbert space which is constructed as a tensor product of two subspaces (representing for example inside and outside the black hole). I discuss how the system has the capacity to contain information which is in NEITHER of the subspaces. I attempt to quantify the amount of information located in each of the two subspaces, and elsewhere, and analyze the extent to which unitary evolution can correspond to ``information flow''. I define the notion of ``overlap information'' which appears to be well suited to the problem. 
  We give a coordinate-free description of real manifolds occurring in certain four dimensional supergravity theories with antisymmetric tensor fields. The relevance of the linear multiplets in the compactification of string and five-brane theories is also discussed. 
  The general expression for the bicovariant bracket for odd generators of the external algebra on a Poisson-Lie group is given. It is shown that the graded Poisson-Lie structures derived before for $GL(N)$ and $SL(N)$ are the special cases of this bracket. The formula is the universal one and can be applied to the case of any matrix Lie group. 
  We have performed some explicit calculations of the conservation laws for classical (affine) Toda field theories, and some generalizations of these models. We show that there is a huge class of generalized models which have an infinite set of conservation laws, with their integrated charges being in involution. Amongst these models we find that only the $A_m$ and $A_m^{(1)}$ ($m\ge 2$) Toda field theories admit such conservation laws for spin-3. We report on our explicit calculations of spin-4 and spin-5 conservation laws in the (affine) Toda models. Our perhaps most interesting finding is that there exist conservation laws in the $A_m$ models ($m\ge4)$ which have a different origin than the exponents of the corresponding affine theory or the energy-momentum tensor of a conformal theory. 
  It is shown, that a spectrum generating algebras and wave functions for the integral and fractional quantum Hall effect are related by the non-unitary similarity transformation. This transformation corresponds to the introduction of the complex Chern-Simons gauge fields, in terms of which the second quantized form of FQHE can be developed 
  We investigate the evolution of small perturbations around charged black strings and branes which are solutions of low energy string theory. We give the details of the analysis for the uncharged case which was summarized in a previous paper. We extend the analysis to the small charge case and give also an analysis for the generic case, following the behavior of unstable modes as the charge is modified. We study specifically a magnetically charged black 6-brane, but show how the instability is generic, and that charge does not in general stabilise black strings and p-branes. 
  We investigate the possible form of local translation invariant conservation laws associated with the relativistic field equations $\partial\bar\partial\phi_i=-v_i(\bphi)$ for a multicomponent field $\bphi$. Under the assumptions that (i)~the $v_i$'s can be expressed as linear combinations of partial derivatives $\partial w_j/\partial\phi_k$ of a set of functions $w_j(\bphi)$, (ii)~the space of functions spanned by the $w_j$'s is closed under partial derivations, and (iii)~the fields $\bphi$ take values in a simply connected space, the local conservation laws can either be transformed to the form $\partial{\bar{\cal P}}=\bar\partial\sum_j w_j {\cal Q}_j$ (where $\bar{\cal P}$ and ${\cal Q}_j$ are homogeneous polynomials in the variables $\bar\partial\phi_i$, $\bar\partial^2\phi_i$,\ldots), or to the parity transformed version of this expression $\partial\equiv(\partial_t+\partial_x)/ \sqrt{2}\rightleftharpoons\bar\partial \equiv (\partial_t-\partial_x)/\sqrt{2}$. 
  The string theory defined by gauging $\W_{\infty/2}$, the sub-algebra of $\W_{\infty}$ generated by the currents of even spin, is discussed. The critical value of the central charge is calculated using zeta-function techniques and shown to be $c=1$. A critical string theory is constructed by coupling a free boson to $\W_{\infty/2}$-gravity and the physical states are analysed. 
  The divergences that arise in the regularized partition function for closed bosonic string theory in flat space lead to three types of perturbation series expansions, distinguished by their genus dependence. This classification of infinities can be traced to geometrical characteristics of the string worldsheet. Some categories of divergences may be eliminated in string theories formulated on compact manifolds. 
  I render the substance of the discussions I had with Robert E. Marshak shortly before his death, wherein the kinship between the ``neutrino paradigm'' ---espoused by Marshak--- and the central notion of K-cycle in noncommutative geometry (NCG) was found. In that context, we give a brief account of the Connes--Lott reconstruction of the Standard Model (SM). 
  This article is based on a talk given at the IInd International Colloquium on Modern Quantum Field Theory, Bombay 1994. The Ernst solution of dilaton gravity describes charged black holes undergoing uniform accleration in a background magnetic field. By analytically continuing the Ernst solution one obtains instantons that describe the pair production of black holes in the background field. We review various aspects of these solutions paying special attention to the Einstein-Maxwell, low-energy string and $d=5$ Kaluza-Klein theories. It is based on work done in collaboration with {}F. Dowker, S. Giddings, G. Horowitz, D. Kastor and J. Traschen \refs{\DGKT,\DGGH}. 
  Polyakov has argued that the QCD string should have long range order. We show that a phase transition does exist in a generalization of string theory characterized by the addition of the curvature of the world sheet (rigidity) and the long range Kalb-Ramond interactions to the Nambu-Goto action. Although rigid strings coupled to long range interactions exhibit the typical pathologies of higher derivative theories at the classical level, we comment based on previous results in rigid paths coupled to long range Coulomb interactions that both phases of the quantum theory are free of ghosts and tachyons. 
  Ordinary QED formulated in the Feynman's space-time picture is equivalent to a one dimensional field theory. In the large N limit there is no phase transition in such a theory. In this letter, we show a phase transition does exist in a generalization of QED characterized by the addition of the curvature of the world line (rigidity) to the Feynman's space-time action. The large distance scale of the disordered phase essentially coincides with ordinary QED, while the ordered phase is strongly coupled. Although rigid QED exhibits the typical pathologies of higher derivative theories at the classical level, we show that both phases of the quantum theory are free of ghosts and tachyons. Quantum fluctuations prevent taking the naive classical limit and inherting the problems of the classical theory. 
  We describe a new geometrical solution to the Wheeler-DeWitt equation in two dimensional quantum gravity. The solution is the amplitude of a surface whose boundary consists of two tangent loops. We further discuss a new method for estimating singular geometries amplitudes, which uses explicit recursive counting of discrete surfaces. 
  Based only on simple principles of renormalization in coordinate space, we derive closed renormalized amplitudes and renormalization group constants at 1- and 2-loop orders for scalar field theories in general backgrounds. This is achieved through a generic renormalization procedure we develop exploiting the central idea behind differential renormalization, which needs as only inputs the propagator and the appropriate laplacian for the backgrounds in question. We work out this generic coordinate space renormalization in some detail, and subsequently back it up with specific calculations for scalar theories both on curved backgrounds, manifestly preserving diffeomorphism invariance, and at finite temperature. 
  The simplest $N=2$ supersymmetric quantum mechanical system is realized in terms of the bosonic creation and annihilation operators obeying either ordinary or deformed Heisenberg algebra involving Klein operator. The construction comprises both the exact and spontaneously broken supersymmetry cases with the scale of supersymmetry breaking being governed by the deformation parameter. Proceeding from the broken supersymmetry case, we realize the Bose-Fermi transformation and obtain spin-$1/2$ representation of $SU(2)$ group in terms of one bosonic oscillator. We demonstrate that the constructions can be generalized to the more complicated $N=2$ supersymmetric systems, in particular, corresponding to the Witten supersymmetric quantum mechanics. 
  The renormalization group equation describing the evolution of the metric of the non linear sigma models poses some nice mathematical problems involving functional analysis, differential geometry and numerical analysis. We describe the techniques which allow a numerical study of the solutions in the case of a two-dimensional target space (deformation of the $O(3)\; \sigma$--model. Our analysis shows that the so-called sausages define an attracting manifold in the U(1) symmetric case, at one-loop level. The paper describes i) the known analytical solutions, ii) the spectral method which realizes the numerical integrator and allows to estimate the spectrum of zero--modes, iii) the solution of variational equations around the solutions, and finally iv) the algorithms which reconstruct the surface as embedded in $R^3$. 
  Using the BFV approach we quantize a pseudoclassical model of the spin one half relativistic particle that contains a single bosonic constraint, contrary to the usual locally supersymmetric models that display first and second class constraints. 
  A derivation of the one-loop effective Lagrangian in the self-interacting $O(N)$ scalar theory, in slowly varying gravitational fields, is presented (using $\zeta$-regularization and heat-kernel techniques). The result is given in terms of the expansion in powers of the curvature tensors (up to quadratic terms) and their derivatives, as well as in derivatives of the background scalar field (up to second derivatives). The renormalization group improved effective Lagrangian is studied, what gives the leading-log approach of the whole perturbation theory. An analysis of the effective equations (back-reaction problem) on the static hyperbolic spacetime $\reals^2 \times H^2/\Gamma$ is carried out for the simplest version of the theory: $m^2=0$ and $N=1$. The existence of the solution $\reals^2 \times H^2/\Gamma$, induced by purely quantum effects, is shown. 
  Using the renormalisation group and a conjecture concerning the perturbation series for the effective potential, the leading logarithms in the effective potential are exactly summed for $O(N)$ scalar and Yukawa theories. 
  Effective potential for scalar $\lambda\phi^4$ theory is obtained using the exact renormalization group method which includes both the usual one-loop contribution as well as the dominant higher loop effects. Our numerical calculation indicates a breakdown of naive one-loop result for sufficiently large renormalized coupling constant. 
  We show explicitly that Schwinger's formula for one-loop effective actions corresponds to the summation of energies associated with the zero-point oscillations of the fields. We begin with a formal proof, and after that we confirm it using a regularization prescription. 
  We present an alternative to the Higgs mechanism to generate masses for non-abelian gauge fields in (3+1)-dimensions. The initial Lagrangian is composed of a fermion with current-current and dipole-dipole type self-interactions minimally coupled to non-abelian gauge fields. The mass generation occurs upon the fermionic functional integration. We show that by fine-tuning the coupling constants the effective theory contains massive non-abelian gauge fields without any residual scalars or other degrees of freedom. 
  We propose the action for a massive $N$-extended superparticle with a pure (half)integer superspin $Y,~Y = 0, 1/2, 1, 3/2, \ldots$. Regardless of the superspin value, the configuration space is ${\Bbb R}^{4|4N} \times S^2$, where $S^2$ corresponds to spinning degrees of freedom. Being explicitly super-Poincar\'e invariant, the model possesses two gauge symmetries implying strong conservation of the squared momentum and superspin. Hamilton constrained dynamics is developed and canonical quantization is studied. For $N$ = 1 we show that the physical super wave-functions are to be on-shell massive chiral superfields. Central-charges generalizations of the model are given. 
  We consider non-unitary similarity transformation, interconnecting the $W_{1+\infty}$ algebra representations for the fractional $\nu=\frac{1}{2p+1}$ and integer $\nu=1$ filling fractions. This transformation corresponds to the introduction of the complex abelian Chern-Simons gauge potentials, in terms of which the field-theoretic description of FQHE can be developed. The Jain's composite fermion approach and Lopez-Fradkin equivalence assertion are considered from the point of view of unitary and similarity transformations. As an application the second-quantized form of Laughlin function is derived. 
  We review the construction of field operators of the N=1 supersymmetric Liouville theory in terms of the components of a quantized free superfield. 
  Exact string solutions are presented, providing backgrounds where a dynamical change of topology is occuring. This is induced by the time variation of a modulus field. Some lessons are drawn concerning the region of validity of effective theories and how they can be glued together, using stringy information in the region where the topology changes. 
  The commutator anomalies (Schwinger terms) of current algebras in $3+1$ dimensions are computed in terms of the Wodzicki residue of pseudodifferential operators; the result can be written as a (twisted) Radul 2-cocycle for the Lie algebra of PSDO's. The construction of the (second quantized) current algebra is closely related to a geometric renormalization of the interaction Hamiltonian $H_I=j_{\mu} A^{\mu}$ in gauge theory. 
  One discusses the validity and equivalence of various perturbative approaches for the Aharonov -Bohm and Anyon models. 
  The $N=1$, $D=10$ Supergravity--Super--Yang--Mills (SUGRA-SYM) theory is plagued by ABBJ gauge and Lorentz anomalies which are cancelled via the Green-Schwarz anomaly cancellation mechanism. Due to the fact that the ABBJ anomalies are not invariant under supersymmetry (SUSY) transformations one concludes that the theory is plagued also by a SUSY anomaly. For the gauge groups $SO(32)$ and $E_8\times E_8$ we compute this SUSY anomaly, by solving a coupled cohomology problem, and we show that it can be cancelled by subtracting from the action the known Green--Schwarz counterterm, the same which cancels also the ABBJ anomaly, the expected result. Finally, we argue that the corresponding mechanism does not apply in the dual SUGRA-SYM, related to the heterotic five-brane. 
  The behaviour of the master field in ``induced QCD" near the edge of its support is studied. An extended scaling domain, where the shape of the master field is a universal function, is found. This function is determined explicitly for the case of dimensions, close to one, and the $D-1$-expansion is constructed. The problem of the meson spectrum corresponding to this solution is analyzed. As a byproduct of these calculations, a new, explicit equation for the meson spectrum in ``induced QCD" with a general potential is derived. 
  There are infinitely many topological solitons in any given complex affine Toda theories and most of them have complex energy density. When we require the energy density of the solitons to be real, we find that the reality condition is related to a simple ``pairing condition.'' Unfortunately, rather few soliton solutions in these theories survive the reality constraint, especially if one also demands positivity. The resulting implications for the physical applicability of these theories are briefly discussed. 
  We define basics of $(4,4)\;\; 2D$ harmonic superspace with two independent sets of $SU(2)$ harmonic variables and apply it to construct new superfield actions of $(4,4)$ supersymmetric two-dimensional sigma models with torsion and mutually commuting left and right complex structures, as well as of their massive deformations. We show that the generic off-shell sigma model action is the general action of constrained analytic superfields $q^{(1,1)}$ representing twisted $N=4$ multiplets in $(4,4)$ harmonic superspace. The massive term of $q^{(1,1)}$ is shown to be unique; it generates a scalar potential the form of which is determined by the metric on the target bosonic manifold. We discuss in detail $(4,4)$ supersymmetric group manifold $SU(2)\times U(1)$ WZNW sigma model and its Liouville deformation. A deep analogy of the relevant superconformally invariant analytic superfield action to that of the improved tensor $N=2\;\;4D$ multiplet is found. We define $(4,4)$ duality transformation and find new off-shell dual representations of the previously constructed actions via {\it unconstrained} analytic $(4,4)$ superfields. The dual representation suggests some hints of how to describe $(4,4)$ models with non-commuting complex structures in the harmonic superspace. 
  Black hole complementarity is incompatible with the existence of traversable wormholes. In fact, traversable wormholes cause problems for any theory where information comes out in the Hawking radiation. 
  In this essay, an {\it ab initio} study of the self/anti-self charge conjugate $(1/2,\,0)\oplus(0,\,1/2)$ representation space is presented. Incompatibility of self/anti-self charge conjugation with helicity eigenstates and gauge interactions is demonstrated. Parity violation is seen as an intrinsic part of the self/anti-self charge conjugate construct. From a phenomenological point of view, an essential part of the theory is the Bargmann-Wightman-Wigner-type boson, where a boson and its antiboson carry opposite relative intrinsic parity. 
  Chern-Simons gauge theory is formulated on three dimensional $Z_2$ orbifolds. The locus of singular points on a given orbifold is equivalent to a link of Wilson lines. This allows one to reduce any correlation function on orbifolds to a sum of more complicated correlation functions in the simpler theory on manifolds. Chern-Simons theory on manifolds is known to be related to 2D CFT on closed string surfaces; here I show that the theory on orbifolds is related to 2D CFT of unoriented closed and open string models, i.e. to worldsheet orbifold models. In particular, the boundary components of the worldsheet correspond to the components of the singular locus in the 3D orbifold. This correspondence leads to a simple identification of the open string spectra, including their Chan-Paton degeneration, in terms of fusing Wilson lines in the corresponding Chern-Simons theory. The correspondence is studied in detail, and some exactly solvable examples are presented. Some of these examples indicate that it is natural to think of the orbifold group $Z_2$ as a part of the gauge group of the Chern-Simons theory, thus generalizing the standard definition of gauge theories. 
  A compact graph rule for the effective action $\Gamma[\phi]$ of a local composite operator is given in this paper. This long-standing problem of obtaining $\Gamma[\phi]$ in this case is solved directly without using the auxiliary field. The rule is first deduced with help of the inversion method, which is a technique for making the Legendre transformation perturbatively. It is then proved by using a topological relation and also by the sum-up rule. Explicitly derived are the rules for the effective action of $\langle \varphi(x)^2 \rangle$ in the $\varphi^4$ theory, of the number density $\langle n_{{\bf r}\sigma} \rangle$ in the itinerant electron model, and of the gauge invariant operator $\langle \bar{\psi}\gamma^\mu\psi \rangle$ in QED. 
  Developing ideas based on combinatorial formulas for characteristic classes we introduce the algebra modeling secondary characteristic classes associated to $N$ connections. Certain elements of the algebra correspond to the ordinary and secondary characteristic classes.That construction allows us to give easily the explicit formulas for some known secondary classes and to construct the new ones. We write how $i$-th differential and $i$-th homotopy operator in the algebra are connected with the Poisson bracket defined in this algebra. There is an analogy between this algebra and the noncommutative symplectic geometry of Kontsevich. We consider then an algebraic model of the action of the gauge group. We describe how elements in the algebra corresponding to the secondary characteristic classes change under this action. 
  In the nonrelativistic case we find that whenever the relation $mc^2/e^2 <X(\al,g_m)$ is satisfied, where $\al$ is a flux in the units of the flux quantum, $g_m$ is magnetic moment, and $X(\al,g_m)$ is some function that is nonzero only for $g_m>2$ (note that $g_m=2.00232$ for the electron), then the matter is unstable against formation of the flux $\al$. The result persists down to $g_m=2$ provided the Aharonov-Bohm potential is supplemented with a short range attractive potential. We also show that whenever a bound state is present in the spectrum it is always accompanied by a resonance with the energy proportional to the absolute value of the binding energy. is considered.   For the Klein-Gordon equation with the Pauli coupling which exists in (2+1) dimensions without any reference to a spin the matter is again unstable for $g_m>2$. The results are obtained by calculating the change of the density of states induced by the Aharonov-Bohm potential. The Krein-Friedel formula for this long-ranged potential is shown to be valid when supplemented with zeta function regularization. PACS : 03.65.Bz, 03-70.+k, 03-80.+r, 05.30.Fk 
  There are only two quantum group structures on the space of two by two unimodular matrices, these are the $SL_q(2)$ and the $SL_h(2)$ [9-13] quantum groups. One can not construct a differential geometry on $ SL_q(2)$, which at the same time is bicovariant, has three generators, and satisfies the Liebnitz rule. We show that such a differential geometry exists for the quantum group $SL_h(2)$ and derive all of its properties. 
  The one--loop effective action for a slowly varying electromagnetic field is computed at finite temperature and density using a real-time formalism. We discuss the gauge invariance of the result. Corrections to the Debye mass from an electric field are computed at high temperature and high density. The effective coupling constant, defined from a purely electric weak--field expansion, behaves at high temperature very differently from the case of a magnetic field, and does not satisfy the renormalization group equation. The issue of pair production in the real--time formalism is discussed and also its relevance for heavy--ion collisions. 
  We study a relativistic anyon model with a spin-$j$ matter field minimally coupled to a statistical gauge potential governed by the Chern-Simons dynamics with a statistical parameter $\alpha$. A spin and statistics transmutation is shown in terms of a continuous random walk method. An integer or odd-half-integer part of $\alpha$ can be reabsorbed by change of $j$. We discuss the equivalence of a large class of (infinite number) Chern-Simons matter models for given $j$ and $\alpha$. 
  The question of the integrability of real-coupling affine toda field theory on a half-line is addressed. It is found, by examining low-spin conserved charges, that the boundary conditions preserving integrability are strongly constrained. In particular, for the $a_n\ (n>1)$ series of models there can be no free parameters introduced by the boundary condition; indeed the only remaining freedom (apart from choosing the simple condition $\partial_1\phi =0$), resides in a choice of signs. For a special case of the boundary condition, it is argued that the classical boundary bound state spectrum is closely related to a consistent set of reflection factors in the quantum field theory. 
  In this paper we show that there exists a new class of topological field theories, whose correlators are intersection numbers of cohomology classes in a constrained moduli space. Our specific example is a formulation of 2D topological gravity. The constrained moduli-space is the Poincare' dual of the top Chern-class of the bundle $E_\rightarrow {\cal M}_g$, whose sections are the holomorphic differentials. Its complex dimension is $2g-3$, rather then $3g-3$. We derive our model by performing the A-topological twist of N=2 supergravity, that we identify with N=2 Liouville theory, whose rheonomic construction is also presented. The peculiar field theoretical mechanism, rooted in BRST cohomology, that is responsible for the constraint on moduli space is discussed, the key point being the fact that the graviphoton becomes a Lagrange multiplier after twist. The relation with conformal field theories is also explored. Our formulation of N=2 Liouville theory leads to a representation of the N=2 superconformal algebra with $c=6$, instead of the value $c=9$ that is obtained by untwisting the Verlinde and Verlinde formulation of topological gravity. The reduced central charge is the shadow, in conformal field theory, of the constraint on moduli space. 
  The problem of introducing a dependence of elements of quantum group on classical parameters is considered. It is suggested to interpret a homomorphism from the algebra of functions on quantum group to the algebra of sections of a sheaf of algebras on a classical manifold as describing such a dependence. It is argued that the functorial point of view of group schemes is more appropriate in quantum group field theory. A sheaf of the Hopf algebras over the manifold (quantum sheaf) is constructed by using bosonization formulas for the algebra of functions on the quantum group $SU_{q}(2)$ and the theory of repre- sentations of canonical commutation relations. A family of automorphisms of the Hopf algebra depending on classical variables is described. Quantum manifolds, i.e. manifolds with commutative and non-commutative coordinates are discussed as a generalization of supermanifolds. Quantum group chiral fields and relations with algebraic differential calculus are discussed. 
  In this note, a new class of representations of the braid groups $B_{N}$ is constructed. It is proved that those representations contain three kinds of irreducible representations: the trivial (identity) one, the Burau one, and an $N$-dimensional one. The explicit form of the $N$-dimensional irreducible representation of the braid group $B_{N}$ is given here. 
  Differential calculus on discrete sets is developed in the spirit of noncommutative geometry. Any differential algebra on a discrete set can be regarded as a `reduction' of the `universal differential algebra' and this allows a systematic exploration of differential algebras on a given set. Associated with a differential algebra is a (di)graph where two vertices are connected by at most two (antiparallel) arrows. The interpretation of such a graph as a `Hasse diagram' determining a (locally finite) topology then establishes contact with recent work by other authors in which discretizations of topological spaces and corresponding field theories were considered which retain their global topological structure. It is shown that field theories, and in particular gauge theories, can be formulated on a discrete set in close analogy with the continuum case. The framework presented generalizes ordinary lattice theory which is recovered from an oriented (hypercubic) lattice graph. It also includes, e.g., the two-point space used by Connes and Lott (and others) in models of elementary particle physics. The formalism suggests that the latter be regarded as an approximation of a manifold and thus opens a way to relate models with an `internal' discrete space ({\`a} la Connes et al.) to models of dimensionally reduced gauge fields. Furthermore, also a `symmetric lattice' is studied which (in a certain continuum limit) turns out to be related to a `noncommutative differential calculus' on manifolds. 
  We show that quantum Casimir W-algebras truncate at degenerate values of the central charge c to a smaller algebra if the rank is high enough: Choosing a suitable parametrization of the central charge in terms of the rank of the underlying simple Lie algebra, the field content does not change with the rank of the Casimir algebra any more. This leads to identifications between the Casimir algebras themselves but also gives rise to new, `unifying' W-algebras. For example, the kth unitary minimal model of WA_n has a unifying W-algebra of type W(2,3,...,k^2 + 3 k + 1). These unifying W-algebras are non-freely generated on the quantum level and belong to a recently discovered class of W-algebras with infinitely, non-freely generated classical counterparts. Some of the identifications are indicated by level-rank-duality leading to a coset realization of these unifying W-algebras. Other unifying W-algebras are new, including e.g. algebras of type WD_{-n}. We point out that all unifying quantum W-algebras are finitely, but non-freely generated. 
  We analyse the stringy gravitational wave background based on the current algebra $E^{c}_{2}$. We determine its exact spectrum and construct the modular invariant vacuum energy. The corresponding N=1 extension is also constructed. The algebra is again mapped to free bosons and fermions and we show that this background has N=4 (N=2) unbroken spacetime supersymmetry in the type II (heterotic case). 
  Topological models involving matter couplings to Donaldson-Witten theory are presented. The construction is carried using both, the topological algebra and its central extension, which arise from the twisting of $N=2$ supersymmetry in four dimensions. The framework in which the construction is based is constituted by the superspace associated to these algebras. The models show new features of topological quantum field theories which could provide either a mechanism for topological symmetry breaking, or the analog of two-dimensional mirror symmetry in four dimensions. 
  We outline the relationship between the thermodynamic densities and quasi-particle descriptions of spectra of RSOS models with an underlying Bethe equation. We use this to prove completeness of states in some cases and then give an algorithm for the construction of branching functions of their emergent conformal field theories. Starting from the Bethe equations of $D_n$ type, we discuss some aspects of the $Z_n$ lattice models. 
  It is shown that linear time-dependent invariants for arbitrary multi\-dimensional quadratic systems can be obtained from the Lagrangian and Hamiltonian formulation procedures by considering a variation of coordinates and momenta that follows the classical trajectory and defines a noetherian symmetry transformation. 
  The 2D off-critical q-state Potts model with boundaries was studied as a factorizable relativistic scattering theory. The scattering S-matrices for particles reflecting off the boundaries were obtained for the cases of ``fixed'' and ``free'' boundary conditions. In the Ising limit, the computed results agreed with recent work[5]. 
  An analysis of the state space in the BRST--quantization in the Schroedinger representation is performed on the basis of the results obtained earlier in the framework of the Fock space representation. It is shown that to get satisfactory results it is necessary to have from the very beginning a meaningful definition of the total state space. 
  We give the classification of second-order polynomial SUSY Quantum Mechanics in one and two dimensions. The particular attention is paid to the irreducible supercharges which cannot be built by repetition of ordinary Darboux transformations. In two dimensions it is found that the binomial superalgebra leads to the dynamic symmetry generated by a central charge operator. 
  In this paper we regard the dynamics obtained from Fermat principle as begin the classical theory of light. We (first-)quantize the action and show how close we can get to the Maxwell theory. We show that Quantum Geometric Optics is not a theory of fields in curved space. Considering Classical Mechanics to be on the same footing, we show the parallelism between Quantum Mechanics and Quantum Geometric Optics. We show that, due to the reparametrization invariance of the classical theories, the dynamics of the quantum theories is given by a Hamiltonian constraint. Some implications of the above analogy in the quantization of true reparameterization invariant systems are discussed. 
  A Heisenberg-Clifford realization of a deformed $U(sl_{2})$ by two parameters $p$ and $q$ is discussed. The commutation relations for this deformed algebra have interesting connection with the theta functions. 
  We discuss time measurement in quantum gravity. Using general relativity for large distances and the uncertainty principle we find a minimum time interval of the order of the Planck time, therefore the uncertainty in time measurment is bounded from below. 
  A first order form of Regge calculus is defined in the spirit of Palatini's action for general relativity. The extra independent variables are the interior dihedral angles of a simplex, with conjugate variables the areas of the triangles. There is a discussion of the extent to which these areas can be used to parameterise the space of edge lengths of a simplex. 
  Abstrct: We show that the action of self-dual supersymmetric Yang-Mills theory in four-dimensions, which describes the consistent massless background fields for $~N=2$~ superstring, generates the actions for $~N=1$~ and $~N=2$~ supersymmetric non-Abelian Chern-Simons theories in three-dimensions after appropriate dimensional reductions. Since the latters play important roles for supersymmetric integrable models, this result indicates the fundamental significance of the $~N=2$~ superstring theory controlling (possibly all) supersymmetric integrable models in lower-dimensions. 
  We discuss different formulations and approaches to string theory and $ 2d$ quantum gravity. The generic idea to get a unique description of {\it many} different string vacua altogether is demonstrated on the examples in $ 2d$ conformal, topological and matrix formulations. The last one naturally brings us to the appearance of classical integrable systems in string theory. Physical meaning of the appearing structures is discussed and some attempts to find directions of possible generalizations to ``higher-dimensional" models are made. We also speculate on the possible appearence of quantum integrable structures in string theory. 
  Ten dimensional supersymmetric Yang-Mills theory may be described, in the light-cone gauge, in terms of either a vector or spinor superfield satisfying certain projection conditions (type I or II). These have been presented in a $ SO(9,1) $ form, and used to construct spinning superparticle theories in extended spaces. This letter presents the covariant quantisation of a "spinor-twin" type II superparticle theory by using the standard techniques of Batalin and Vilkovisky. The quantum action defines a quadratic field theory, whose ghost-independent BRST cohomology class gives the spectrum of N=1 super Yang-Mills. 
  We analyze the analytic continuation of the formally divergent one-loop amplitude for scattering of the graviton multiplet in the Type II Superstring. In particular we obtain explicit double and single dispersion relations, formulas for all the successive branch cuts extending out to plus infinity, as well as for the decay rate of a massive string state of arbitrary mass 2N into two string states of lower mass. We compare our results with the box diagram in a superposition of $\phi^3$-like field theories. The stringy effects are traced to a convergence problem in this superposition. 
  The harmonic oscillator with dissipation is studied within the framework of the Lindblad theory for open quantum systems. By using the Wang-Uhlenbeck method, the Fokker-Planck equation, obtained from the master equation for the density operator, is solved for the Wigner distribution function, subject to either the Gaussian type or the $\delta$-function type of initial conditions. The obtained Wigner functions are two-dimensional Gaussians with different widths. Then a closed expression for the density operator is extracted. The entropy of the system is subsequently calculated and its temporal behaviour shows that this quantity relaxes to its equilibrium value. 
  It is shown that a recent claim of equivalence between three Chern-Simons theories is not tenable. 
  In a recent work it has been shown that the bosonic strings could be embedded into a special class of $N=1$ fermionic strings. We argue that the superpartners of any physical state in the spectrum of this fermionic string is non-physical. So, there is no supersymmetry in the space of physical states and the embedding is, in this sense, ``trivial''. We here propose two different constructions as possible candidates of non-trivial embeddings of the non-critical bosonic strings into some special class of $N=1$ fermionic strings of which one is the non-critical NSR string. The BRST charge of the $N=1$ fermionic strings in both cases decompose as $Q_{N=1} = Q_B + {\tilde Q}$, where $Q_B$ is the BRST charge of the bosonic string. 
  The affine current algebra for Lie superalgebras is examined. The bilinear invariant forms of the Lie superalgebra can be either degenerate or non-degenerate. We give the conditions for a Virasoro construction, in which the currents are primary fields of weight one, to exist. In certain cases, the Virasoro central charge is an integer equal to the super dimension of the group supermanifold. A Wess-Zumino-Novikov-Witten action based on these Lie superalgebras is also found. 
  We show that self-dual Nielsen Olesen (NO) vortices in $3$ dimensions give rise to a class of exact solutions when coupled to Einstein Maxwell Dilaton gravity obeying the Majumdar-Papapetrou(MP) relation between gravitational and Maxwell couplings , provided certain Chern-Simons type interactions are present. The metric may be solved for explicitly in terms of the NO vortex function and is asymptotic to Euclidean space with signature (-1,-1,-1). The MP electric field is long range but, strictly speaking, the charge of the vortices is zero since the field dies off as $O(1/r (ln r)^2)$. The total ADM energy integral of such vortices is {\it{zero}}.   These peculiarities are due to the nature of the two dimensional Greens function. 
  \noindent We propose a set of rules for constructing composite leptons and quarks as triply occupied quasiparticles, in the quaternionic quantum mechanics of a pair of Harari-Shupe preons $T$ and $V$. The composites fall into two classes, those with totally antisymmetric internal wave functions, and those with internal wave functions of mixed symmetry. The mixed symmetry states consist of precisely the three spin 1/2 quark lepton families used in the standard model (48 particle states, {\it not} counting the doubling arising from antiparticles), plus one doublet of spin 3/2 quarks (24 particle states). The antisymmetric states consist of a set of spin 3/2 leptonic states with charges as in a standard model family (16 particle states), and a spin 1/2 leptonic fractionally charged doublet (4 particle states). We sketch ideas for deriving our rules from a fundamental quaternionic preonic field theory. 
  We compute the entropy of the Hawking radiation for an evaporating black hole, in 1+1 dimensions and in 3+1 dimensions. We investigate the validity of the semiclassical approximation for the evaporation process. It appears that there might be a large entropy of entanglement between the classical degrees of freedom describing the black hole and the radiation fields when the theory of quantum gravity plus matter is considered. 
  A model is proposed which can be regarded as a mean field approximation for pure lattice QCD and chiral field. It always possesses a phase transition between a strong coupling phase (where it reduces to a one-plaquette integral) and a non-trivial weak coupling one. For the U(N) gauge group, it is equivalent to some multi-matrix model. This analogy allows for determining possible large N critical regimes thus generalizing the Gross-Witten phase transition in the one-plaquetee model. 
  The purpose of this very short note, which maybe considered as a comment on "hep-th/9401067", is to prove the following proposition. PROPOSITION. There exist certain models of Mobilevision, in which interpretational figures are observable only in a multi--user mode in contrast to single--user ones. This result maybe interesting as for applications as for the understanding of theoretical foundations of quantum interactive processes (hep-th/9403015). 
  We show that $N=2$ and $N=4$ extended supersymmetric Yang-Mills theories in four space-time dimensions could be derived as action functionals for non-commutative spaces. The coupling of $N=1$ and $N=2$ super Yang-Mills to $N=1$ and $N=2$ matter could be derived as action functionals of non-commutative spaces only for a restricted class of models where a general superpotential is not allowed. 
  In the event symmetric approach to quantum gravity it is assumed that the fundamental laws of physics must be invariant under exchange of any two space-time events. The fact that this symmetry if obviously not observed is attributed to the possibility of a spontaneous symmetry breaking mechanism in which the residual symmetry is diffeomorphism invariance on a continuum. The approach has a discrete nature while dimensionality, continuity of space-time and causality are abandoned as fundamental principles. Consequently space-time is a dynamical object which may undergo changes of topology or dimension. A number of different types of Event Symmetric model are described. The most interesting models are event symmetric superstring models based on discrete string supergroups. 
  It is shown how to treat the degrees of freedom of Nielsen-Olesen vortices in the $3+1$-dimensional $U(1)$ higgs model by a collective coordinate method. In the london limit, where the higgs mass becomes infinite, the gauge and goldstone degrees of freedom are integrated out, resulting in the vortex world-sheet action. Introducing an ultraviolet cut-off mimics the effect of finite higgs mass. This action is non-polynomial in derivatives and depends on the extrinsic curvature of the surface. Flat surfaces are stable if the coherence length is less than the penetration depth. It is argued that in the quantum abelian higgs model, vortex world-sheets are dominated by branched polymers. 
  The Nested Bethe Ansatz is generalized to open and independent boundary conditions depending on two continuous and two discrete free parameters. This is used to find the exact eigenvectors and eigenvalues of the $A_{n-1}$ vertex models and $SU(n)$ spin chains with such boundary conditions. The solution is found for all diagonal families of solutions to the reflection equations in all possible combinations. The Bethe ansatz equations are used to find de first order finite size correction. 
  This paper defines a new sequence of finite dimensional algebras as quotients of the group algebras of the braid groups. This sequence depends on three homogeneous parameters and has a one-parameter family of Markov traces, and so gives a three parameter invariant of oriented links. 
  We discuss O(N) invariant scalar field theories in 0+1 and 1+1 space-time dimensions. Combining ordinary ``Large N" saddle point techniques and simple properties of the diagonal resolvent of one dimensional Schr\"odinger operators we find {\it exact} non-trivial (space dependent) solutions to the saddle point equations of these models in addition to the saddle point describing the ground state of the theory. We interpret these novel saddle points as collective O(N) singlet excitations of the field theory, each embracing a host of finer quantum states arranged in O(N) multiplets, in an analogous manner to the band structure of molecular spectra. We comment on the relation of our results to the classical work of Dashen, Hasslacher and Neveu and to a previous analysis of bound states in the O(N) model by Abbott.} 
  The different ways of description of the $S=0$ particle with oscillator-like interaction are considered. The results are in conformity with the previous paper of S. Bruce and P. Minning. 
  The oscillator-like interaction is introduced in the equation for the particle of arbitrary spin, given by Dirac and re-written to a matrix form by Dowker. 
  We investigate the classical phase space structure of $N$ $SU(n+1)$ non-Abelian Chern-Simons (NACS) particles by first constructing the product space of associated $SU(n+1)$ bundle with ${\bf CP}^n$ as the fiber. We calculate the Poisson bracket using the symplectic structure on the associated bundle and find that the minimal substitution in the presence of external gauge fields is equivalent to the modification of symplectic structure by the addition of field strength two form. Then, we take a direct product of the associated bundle by the space of all connections and choose a specific connection by the condition of vanishing momentum map corresponding to the gauge transformation, thus recovering the quantum mechanical model of NACS particles in Ref.\cite{lo1}. 
  An exponential behavior at all times is derived for a solvable dynamical model in the weak-coupling, macroscopic limit. Some implications for the quantum measurement problem are discussed, in particular in connection with dissipation. 
  We argue that the Virasoro algebra for the closed bosonic string can be cast in a form which is suitable for the limit of vanishing string tension. In this form the limit of the Virasoro algebra gives the null string algebra. The anomalous central extension is seen to vanish as well when $T\to 0$. 
  We study maps from a 2D world-sheet to a 2D target space which include folds. The geometry of folds is discussed and a metric on the space of folded maps is written down. We show that the latter is not invariant under area preserving diffeomorphisms of the target space. The contribution to the partition function of maps associated with a given fold configuration is computed. We derive a description of folds in terms of Feynman diagrams. A scheme to sum up the contributions of folds to the partition function in a special case is suggested and is shown to be related to the Baxter-Wu lattice model. An interpretation of folds as trajectories of particles in the adjoint representation of $SU(N)$ gauge group in the large $N$ limit which interact in an unusual way with the gauge fields is discussed. 
  Irreversibility of RG flows in two dimensions is shown using conserved vector currents. Out of a conserved vector current, a quantity decreasing along the RG flow is built up such that it is stationary at fixed points where it coincides with the constant coefficient of the two current correlation function. For Wess-Zumino-Novikov-Witten models this constant coefficient is the level of their associated affine Lie algebra. Extensions to higher dimensions using the spectral decomposition of the two current correlation function are studied. 
  The moduli space of N=(4,4) string theories with a K3 target space is determined, establishing in particular that the discrete symmetry group is the full integral orthogonal group of an even unimodular lattice of signature (4,20). The method combines an analysis of the classical theory of K3 moduli spaces with mirror symmetry. A description of the moduli space is also presented from the viewpoint of quantum geometry, and consequences are drawn concerning mirror symmetry for algebraic K3 surfaces. 
  We use thermodynamic Bethe ansatz to study nonrelativistic scattering theory of low energy excitations of 1D Hubbard model, using the $S$-matrices proposed by E\ss ler and Korepin. This model can be described by two types of excitation states, holons and spinons, as asymptotic states. In the attractive regime, the spinon is massive while the holon is massless. The situation is reversed with a repulsive coupling. We derive that the central charge of the Hubbard model in the IR limit is $c=1$ while it vanishes in the UV limit. The contribution is due to the massless degree of freedom, i.e. the holon for the attractive regime, and the massive mode decouples completely. This result is consistent with various known results based on lattice Bethe ansatz computations. Our results make it possible to use the $S$-matrices of the excitations to compute more interesting quantities like correlation functions. 
  This paper is part of the lecture given at the TH Division of CERN and devoted to the CXXV anniversary of the birthday of Elie Cartan (1869-1951). It is shown how the methods of differential geometry, due to E. Cartan, were applied to the construction of the supersymmetry transformation law and to the actions for Goldstone fermions and supergravity. 
  A soluble model for the relativistically description of an unstable system is given in terms of relativistic quantum field theory, with a structure similar to Van Hove's generalization of the Lee model in the non-relativistic theory. 
  We consider the infrared and ultraviolet behaviour of the effective quantum field theory of a single $Z_2$ symmetric scalar field. In a previous paper we proved to all orders in perturbation theory the renormalizability of massive effective scalar field theory using Wilson's exact renormalization group equation. Here we show that away from exceptional momenta the massless theory is similarly renormalizable, and we prove detailed bounds on Green's functions as arbitrary combinations of exceptional Euclidean momenta are approached. As a corollary we also prove Weinberg's Theorem for the massive effective theory, in the form of bounds on Green's functions at Euclidean momenta much greater than the particle mass but below the naturalness scale of the theory. 
  We consider decoupling in the context of an effective quantum field theory of two scalar fields with well separated mass scales and a $Z_2\times Z_2$ symmetry. We first prove, using Wilson's exact renormalization group equation, that the theory is renormalizable, in the same way that we showed in a previous paper that theories with a single mass scale were renormalizable. We then state and prove a decoupling theorem: at scales below the mass of the heavy particle the full theory may be approximated arbitrarily closely by an effective theory of the light particle alone, with naturalness scale the heavy particle mass. We also compare our formulation of effective field theory with the more conventional local formulation. 
  The SL(2,R)/U(1) gauged Wess-Zumino-Witten model is an exact conformal field theory describing a black hole in two-dimensional space-time. The free field approach of Bershadsky and Kutasov is a suitable formulation of this CFT in order to compute physically interesting quantities of this black hole. We find the space-time interpretation of this model for k=9/4 and show that it reproduces the metric and the dilaton found by Dijkgraaf, E. Verlinde and H. Verlinde in the mini-superspace approximation. We compute the two- and  three-point functions of tachyons interacting in the black hole background and analyse in detail the form of the four-point tachyon scattering amplitude. We discuss the connection to the c=1 matrix model and the deformed matrix model of Jevicki and Yoneya. 
  The contribution to the perturbative Regge asymptotics of the exchange of two reggeized fermions with opposite helicity is investigated. The methods of conformal symmetry known for the case of gluon exchange are extended to this case where double-logarithmic contributions dominate the asymptotics. The Regge trajectories at large momentum transfer are calculated. 
  We use the Hamiltonian framework to study massless QCD$_{1+1}$, i.e.\ Yang-Mills gauge theories with massless Dirac fermions on a cylinder (= (1+1) dimensional spacetime $S^1\times \R$) and make explicite the full, non-perturbative structure of these quantum field theory models. We consider $N_F$ fermion flavors and gauge group either $\U(N_C)$, $\SU(N_C)$ or another Lie subgroup of $\U(N_C)$. In this approach, anomalies are traced back to kinematical requirements such as positivity of the Hamiltonian, gauge invariance, and the condition that all observables are represented by well-defined operators on a Hilbert space. We also give equal time commutators of the energy momentum tensor and find a gauge-covariant form of the (affine-) Sugawara construction. This allows us to represent massless QCD$_{1+1}$ as a gauge theory of Kac-Moody currents and prove its equivalence to a gauged Wess-Zumino-Witten model with a dynamical Yang-Mills field. 
  The model which generalizes Ponzano and Regge $3D$ and Carfora-Martellini-Marzuoli $4D$ euclidean quantum gravity is considered. The euclidean Einstein-Regge action for a $D$-simplex is given in the semiclassical limit by a gaussian integral of a suitable $3nj$-symbol. 
  A formal uniform asymptotic solution of the system of differential equations $ h^{2}\frac{d^{2}U_{1}}{dz^{2}}+\Phi_{1} U_{1}=\alpha U_{2} $ , $ h^{2}\frac{d^{2}U_{2}}{dz^{2}}+\Phi_{2} U_{2}=\alpha U_{1}$ , for $ z\in D$ and for h real, large is obtained, when D contains curve-crossing point. Asymptotic approximations for the solutions are constructed in terms of parabolic cylinder functions. Analytical properties of the expansion's coefficients are investigated.The case of potantial barier is also considered. 
  After adding a scalar chiral boson to the usual superspace variables, the four-dimensional Green-Schwarz superstring is quantized in a manifestly SO(3,1) super-Poincar\'e covariant manner. The constraints are all first-class and form an N=2 superconformal algebra with $c=-3$. Since the Calabi-Yau degrees of freedom are described by an N=2 superconformal field theory with $c=9$, the combined Green-Schwarz and Calabi-Yau systems form the $c=6$ matter sector of a critical N=2 string.   Using the standard N=2 super-Virasoro ghosts, a nilpotent BRST charge is defined and vertex operators for the massless supermultiplets are constructed. Four-dimensional superstring amplitudes can be calculated with manifest SO(3,1) super-Poincar\'e invariance by evaluating correlation functions of these BRST-invariant vertex operators on N=2 super-Riemann surfaces. 
  We introduce a model detector which registers the passage of a particle through the detector location, without substantially perturbing the particle wave function. (The exact time of passage is not determined in such measurements.) We then show that our detector can operate in a classically forbidden region and register particles passing through a certain point under a potential barrier. We show that it should be possible to observe the particle's track under the barrier. 
  One of the interesting features about field theories in odd dimensions is the induction of parity violating terms and well-defined {\em finite} topological actions via quantum loops if a fermion mass term is originally present and conversely. Aspects of this issue are illustrated for electrodynamics in 2+1 and 4+1 dimensions. (3 uuencoded Postscript Files are appended at the end of the TexFile.) 
  We lift the lattice of translations in the extended affine Weyl group to a braid group action on the quantum affine algebra. This action fixes the Heisenberg subalgebra pointwise. Loop like generators are found for the algebra which satisfy the relations of Drinfel$'$d's new realization. Coproduct formulas are given and a PBW type basis is constructed. 
  Auxiliary string fields are introduced in light-cone gauge string field theory in order to express contact interactions as contractions of cubic vertices. The auxiliary field in the purely closed-string bosonic theory may be given a non-zero expectation value, leading to a phase in which world-sheets have boundaries. 
  The stochastic quantization of dissipative systems is discussed. It is shown that in order to stochastically quantize a system with dissipation, one has to restrict the Fourier transform of the space-time variable to the positive half domain in the complex plane. This breaks the time-reversal invariance, which manifests in the formulation through the resulting noninvariant forms for the propagators. The relation of the stochastic approach with the Caldeira and Leggett path-integral method is also analyzed. 
  Topological interpretation of the link invariants associated with the Weinstein--Xu classical solutions of the quantum Yang-Baxter equation are provided. 
  We analyze the quantization of dynamical systems that do not involve any background notion of space and time. We give a set of conditions for the introduction of an intrinsic time in quantum mechanics. We show that these conditions are a generalization of the usual procedure of deparametrization of relational theories with hamiltonian constraint that allow to include systems with an evolving Hilbert Space. We apply our quantization procedure to the parametrized free particle and to some explicit examples of dynamical system with an evolving Hilbert space. Finally, we conclude with some considerations concerning the quantum gravity case. 
  We reformulate the BRST quantisation of chiral Virasoro and $W_3$ worldsheet gravities. Our approach follows directly the classic BRST formulation of Yang-Mills theory in employing a derivative gauge condition instead of the conventional conformal gauge condition, supplemented by an introduction of momenta in order to put the ghost action back into first-order form. The consequence of these simple changes is a considerable simplification of the BRST formulation, the evaluation of anomalies and the expression of Wess-Zumino consistency conditions. In particular, the transformation rules of all fields now constitute a canonical transformation generated by the BRST operator $Q$, and we obtain in this reformulation a new result that the anomaly in the BRST Ward identity is obtained by application of the anomalous operator $Q^2$, calculated using operator products, to the gauge fermion. 
  We study quantum gravity in $2+\epsilon$ dimensions in such a way to preserve the volume preserving diffeomorphism invariance. In such a formulation, we prove the following trinity: the general covariance, the conformal invariance and the renormalization group flow to Einstein theory at long distance. We emphasize that the consistent and macroscopic universes like our own can only exist for matter central charge $0<c<25$. We show that the spacetime singularity at the big bang is resolved by the renormalization effect and universes are found to bounce back from the big crunch. Our formulation may be viewed as a Ginzburg-Landau theory which can describe both the broken and the unbroken phase of quantum gravity and the phase transition between them. 
  We describe the horizon of a quantum black hole in terms of a dynamical surface which defines the boundary of space-time as seen by external static observers, and we define a path integral in the presence of this dynamical boundary. Using renormalization group arguments, we find that the dynamics of the horizon is governed by the action of the relativistic bosonic membrane. {}From the thermodynamical properties of this bosonic membrane we derive the entropy and the temperature of black holes, and we find agreement with the standard results. With this formalism we can also discuss the corrections to the Hawking temperature when the mass $M$ of the black hole approaches the Planck mass $M_{\rm Pl}$. When $M$ becomes as low as $(10-100) M_{\rm Pl}$ a phase transition takes place and the specific heat of the black hole becomes positive. 
  We show that the one-dimensional lattice model proposed by Lipatov to describe the high energy scattering of hadrons in multicolor QCD is completely integrable. We identify this model as the XXX Heisenberg chain of noncompact spin $s=0$ and find the conservation laws of the model. A generalized Bethe ansatz is developed for the diagonalization of the hamiltonian and for the calculation of hadron-hadron scattering amplitude. 
  We study a model in which p independent Ising spins are coupled to 2d quantum gravity (in the form of dynamical planar phi-cubed graphs). Consideration is given to the p tends to infinity limit in which the partition function becomes dominated by certain graphs; we identify most of these graphs. A truncated model is solved exactly providing information about the behaviour of the full model in the limit of small beta. Finally, we derive a bound for the critical value of the coupling constant, beta_c and examine the magnetization transition in the limit p tends to zero. 
  Several string theories related to QCD in two dimensions are studied. For each of these theories the large $N$ free energy on a (target) sphere of area $A$ is calculated. By considering theories with different subsets of the geometrical structures involved in the full QCD${}_2$ string theory, the different contributions of these structures to the string free energy are calculated using both analytic and numerical methods. The equivalence between the leading terms in the $SU(N)$ and $U(N)$ free energies is simply demonstrated from the string formulation. It is shown that when $\Omega$-points are removed from the theory, the free energy is convergent for small and large values of $A$ but divergent in an intermediate range. Numerical results indicate that the free energy for the full QCD${}_2$ string fails to converge at the Douglas-Kazakov phase transition point. Similar results for a single chiral sector of the theory, such as has recently been studied by Cordes, Moore, and Ramgoolam, indicate that there are three distinct phases in that theory. These results indicate that from the point of view of the strong coupling phase, the phase transition in the full QCD${}_2$ string arises from the entropy of branch-point singularities. 
  We present a classification of (2,2) free field compactifations with one twist in which only 95 distinct models (generations and antigenerations) are found. Models with three generations and no antigenerations are given. 
  We show that general cutoff scalar field theories in four dimensions are perturbatively renormalizable through the use of diagrammatic techniques and an adapted BPH renormalization method. Weinberg's convergence theorem is used to show that operators in the Lagrangian with dimension greater than four, which are divided by powers of the cutoff, produce perturbatively only local divergences in the two-, three-, and four-point correlation functions. We also show that the renormalized Green's functions are the same as in ordinary $\Phi^4$ theory up to corrections suppressed by inverse powers of the cutoff. These conclusions are consistent with those of existing proofs based on the renormalization group. 
  An ambiguity is pointed out in J.S. Bell's argument that the distinction between quantum mechanics and hidden variable theories cannot be found in the behavior of single-particle beams. Within the context of theories for which states are unambiguously defined it is shown that the question of whether quantum mechanics or a locally realistic theory is valid may indeed be answered by single-particle beam measurements. It is argued that two-particle correlation experiments are required to answer the more fundamental question of whether or not the notion of a state can be unambiguously defined. As a byproduct of the discussion the general form of completely entangled states is deduced. 
  We study the supersymmetric Gelfand-Dickey algebras associated with the superpseudodifferential operators of positive as well as negative leading order. We show that, upon the usual constraint, these algebras contain the N=2 super Virasoro algebra as a subalgebra as long as the leading order is odd. The decompositions of the coefficient functions into N=1 primary fields are then obtained by covariantizing the superpseudodifferential operators. We discuss the problem of identifying N=2 supermultiplets and work out a couple of supermultiplets by explicit computations. 
  Time-dependent solutions of bosonic string theory resemble renormalisation group trajectories in the space of 2d field theories: they often interpolate between repulsive and attractive static solutions. It is shown that the attractive static solutions are those whose spatial sections are minima of |\bar c-25|, where \bar c is the `c-function'. The size of the domain of attraction of such a solution may be a measure of the probability of the corresponding string vacuum. Our discussion has also an implication for the RG flow in theories coupled to dynamical 2d gravity: the flow from models with c>25 to models with c<25 is forbidden. 
  A new aspect of the vacuum structure of 2+1-dimensional Thirring model is presented. Using the Fierz identity, we split the current-current four-Fermi interaction in terms of a matrix valued auxiliary scalar field and compute its effective potential. Energy consideration shows that contrary to earlier expectations, parity in general is spontaneously broken at any finite order of N, where N is the number of the two component spinors. In the large N limit, there does not exist a stable vacuum of the theory thereby making the application of the large N limit to Thirring model dangerous. A detailed analysis for parity breaking solutions in N=2,3 cases is given. 
  We present two examples of parity-invariant $[U(1)]^{2}$ Chern-Simons-Higgs models with spontaneously broken symmetry. The models possess topological vortex excitations. It is argued that the smallest possible flux quanta are composites of one quantum of each type $(1,1)$. These hybrid anyons will dominate the statistical properties near the ground state. We analyse their statistical interactions and find out that unlike in the case of Jackiw-Pi solitons there is short range magnetic interaction which can lead to formation of bound states of hybrid anyons. In addition to mutual interactions they possess internal structure which can lead upon quantisation to discrete spectrum of energy levels. 
  The problem of maintaining scale and conformal invariance in Maxwell and general N-form gauge theories away from their critical dimension d=2(N+1) is analyzed.We first exhibit the underlying group-theoretical clash between locality,gauge,Lorentz and conformal invariance require- ments. "Improved" traceless stress tensors are then constructed;each violates one of the above criteria.However,when d=N+2,there is a duality equivalence between N-form models and massless scalars.Here we show that conformal invariance is not lost,by constructing a quasilocal gauge invariant improved stress tensor.The correlators of the scalar theory are then reproduced,including the latter's trace anomaly. 
  It is shown that in string theory mirror duality is a gauge symmetry (a Weyl transformation) in the moduli space of $N=2$ backgrounds on group manifolds, and we conjecture on the possible generalization to other backgrounds, such as Calabi-Yau manifolds. 
  The SU(3) modular invariant partition functions were first completely classified in Ref.\ \SU. The purpose of these notes is four-fold: \item{(i)} Here we accomplish the SU(3) classification using only the most basic facts: modular invariance; $M_{\la\mu}\in{\bf Z}_{\ge}$; and $M_{00}=1$. In \SU{} we made use of less elementary results from Moore-Seiberg, in addition to these 3 basic facts. \item{(ii)} Ref.\ \SU{} was completed well over a year ago. Since then I have found a number of significant simplifications to the general argument. They are all included here. \item{(iii)} A number of people have complained that some of the arguments in \SU{} were hard to follow. I have tried here to be as explicit and as clear as possible. \item{(iv)} Hidden in \SU{} were a number of smaller results which should be of independent value. These are explicitly mentioned here. 
  By providing a general correspondence between Landau-Ginzburg orbifolds and non-linear sigma models, we find that the elusive mirror of a rigid manifold is actually a supermanifold. We also discuss when sigma models with super-target spaces are conformally invariant and describe their chiral rings. Both supermanifolds with and without Kahler moduli are considered. This work leads us to conclude that mirror symmetry should be viewed as a relation among super-varieties rather than bosonic varieties. 
  The one-loop effective action for a generic set of quantum fields is calculared as a nonlocal expansion in powers of the curvatures (field strengths). This expansion is obtained to third order in the curvature. It is stressed that the covariant vertices are finite. The trace anomaly in four dimensions is obtained directly by varying the effective action. The nonlocal terms in the action, producing the anomaly, contain non-trivial functions of three operator arguments. The trace anomaly is derived also by making the conformal transformation in the heat kernel. 
  We demonstrate that the generalization of the Coleman-Thun mechanism may be applied to the situation, when considering scattering processes in 1+1-dimensions in the presence of reflecting boundaries. For affine Toda field theories we find that the binding energies of the bound states are always half the sum over a set of masses having the same colour with respect to the bicolouration of the Dynkin diagram. For the case of $E_6$-affine Toda field theory we compute explicitly the spectrum of all higher boundary bound states. The complete set of states constitutes a closed bootstrap. 
  We discuss multicritical behavior of $c=1$ matrix model, extending the recent work of ref. \cite{CIO} on a nonperturbative completion of the density of states function. For the odd orders of multicriticality, we are able to determine the higher genus contributions and a nonperturbative completion from the WKB wave function of the multicritical periodic potential.  The expression for the contributions as a function of the scaled chemical potential is found to be the same as the one at the lowest critical point. We point out a strange scaling behavior. 
  We consider ways in which conventional supersymmetry can be embedded in the set of more general fermionic transformations proposed recently [\Ref{B}] as a framework in which to study $d=10$ super Yang-Mills. Solutions are exhibited which involve closed algebras of various numbers of supersymmetries together with their invariance groups: nine supersymmetries with $\GT {\times}\SO (1,1)$ invariance; eight supersymmetries with $\SO (7){\times}\SO (1,1)$ invariance; four supersymmetries with $\SO (3,1){\times}\U (3)$ invariance. We recover in this manner all previously known ways of adding finite numbers of bosonic auxiliary fields so as to partially close the $d=10$ superalgebra. A crucial feature of these solutions is that the auxiliary fields transform non-trivially under the residual Lorentz symmetry, even though they are originally introduced as Lorentz scalars. 
  Given a static string solution with some free constant parameters (`moduli') it may be possible to construct a time-dependent solution by just replacing the moduli by some functions of time. We present several examples when such `rolling moduli' ansatz is consistent. In particular, the anisotropic D=4 cosmological solution of Nappi and Witten can be reinterpreted as a time-dependent generalisation of the (analytic continuation of) D=3 `charged black string' background with the `charge' changing with time. We find some new D=4 cosmological solutions which are `two rolling moduli' generalisations of the previously known ones. We also comment on interplay between duality transformations and the replacement of moduli by functions of time. 
  Using arguments from two dimensional Yang-Mills theory and the collective coordinate formulation of the Calogero-Sutherland model, we conjecture the dynamical density correlation function for coupling $l$ and $1/l$, where $l$ is an integer. We present overwhelming evidence that the conjecture is indeed correct. 
  't Hooft has recently developed a discretisation of (2+1) gravity which has a multiple-valued Hamiltonian and which therefore admits quantum time evolution only in discrete steps. In this paper, we describe several models in the continuum with single-valued equations of motion in classical physics, but with multiple-valued Hamiltonians. Their time displacements in quantum theory are therefore obliged to be discrete. Classical models on smooth spatial manifolds are also constructed with the property that spatial displacements can be implemented only in discrete steps in quantum theory. All these models show that quantization can profoundly affect classical topology. 
  A certain class of surface motions, including those of a relativistic membrane minimizing the 3-dimensional volume swept out in Minkowski-space, is shown to be equivalent to 3-dimensional steady-state irrotational inviscid isentropic gas-dynamics. The SU($\infty $) Nahm equations turn out to correspond to motions where the time $t$ at which the surface moves through the point $\ssatop\rhup{r}$ is a harmonic function of the three space-coordinates. This solution also implies the linearisation of a non-trivial-looking scalar field theory. 
  We present the coset structure of the untwisted moduli space of heterotic $(0,2) \; Z_N$ orbifold compactifications with continuous Wilson lines. For the cases where the internal 6-torus $T_6$ is given by the direct sum $T_4 \oplus T_2$, we explicitly construct the K\"{a}hler potentials associated with the underlying 2-torus $T_2$. We then discuss the transformation properties of these K\"{a}hler potentials under target space modular symmetries. For the case where the $Z_N$ twist possesses eigenvalues of $-1$, we find that holomorphic terms occur in the K\"{a}hler potential describing the mixing of complex Wilson moduli. As a consequence, the associated $T$ and $U$ moduli are also shown to mix under target space modular transformations. 
  We compute the combined two and three loop order correction to the spin-spin correlation functions for the 2D Ising and q-states Potts model with random bonds at the critical point. The procedure employed is the renormalisation group approach for the perturbation series around the conformal field theories representing the pure models. We obtain corrections for the correlations functions which produce crossover in the amplitude but don't change the critical exponent in the case of the Ising model and which produce a shift in the critical exponent, due to randomness, in the case of the Potts model. Comparison with numerical data is discussed briefly. 
  We show that the most general two--matrix model with bilinear coupling underlies $c=1$ string theory. More precisely we prove that $W_{1+\infty}$ constraints, a subset of the correlation functions and the integrable hierarchy characterizing such two--matrix model, correspond exactly to the $W_{1+\infty}$ constraints, to the discrete tachyon correlation functions and to the integrable hierarchy of the $c=1$ string. 
  We suggest that the spectral properties near zero virtuality of three dimensional QCD, follow from a Hermitean random matrix model. The exact spectral density is derived for this family of random matrix models both for even and odd number of fermions. New sum rules for the inverse powers of the eigenvalues of the Dirac operator are obtained. The issue of anomalies in random matrix theories is discussed. 
  As was shown by Leutwyler and Smilga, the fact that chiral symmetry is broken and the existence of a effective finite volume partition function leads to an infinite number of sum rules for the eigenvalues of the Dirac operator in QCD. In this paper we argue these constraints, together with universality arguments from quantum chaos and universal conductance fluctuations, completely determine its spectrum near zero virtuality. As in the classical random matrix ensembles, we find three universality classes, depedending on whether the color representation of the gauge group is pseudo-real, complex or real. They correspond to $SU(2)$ with fundamental fermions, $SU(N_c)$, $N_c \ge 3$, with fundamental fermions, and $SU(N_c)$, $N_c \ge 3$, with adjoint fermions, respectively.} 
  We present a semi-classical model for the formation and evaporation of a four dimensional black hole. We solve the equations numerically and obtain solutions describing the entire the space-time geometry from the collapse to the end of the evaporation. The solutions satisfy the evaporation law: $\dot M \propto -M^{-2}$ which confirms dynamically that black holes do evaporate thermally. We find that the evaporation process is in fact the shrinking of a throat that connects a macroscopic interior ``universe" to the asymptotically flat exterior. It ends either by pinching off the throat leaving a closed universe and a Minkowskian exterior or by freezing up when the throat's radius approaches a Planck size. In either case the macroscopic inner universe is the region where the information lost during the evaporation process is hidden. 
  We compute the dynamical Green function and density-density correlation in the Calogero-Sutherland model for all integer values of the coupling constant. An interpretation of the intermediate states in terms of quasi-particles is found. 
  The spin-1/2 Aharonov-Bohm problem is examined in the Galilean limit for the case in which a Coulomb potential is included. It is found that the application of the self-adjoint extension method to this system yields singular solutions only for one-half the full range of flux parameter which is allowed in the limit of vanishing Coulomb potential. Thus one has a remarkable example of a case in which the condition of normalizability is necessary but not sufficient for the occurrence of singular solutions. Expressions for the bound state energies are derived. Also the conditions for the occurrence of singular solutions are obtained when the non-gauge potential is $\xi/r^p (0\leq p<2)$. 
  In the bulk scaling limit of the Gaussian Unitary Ensemble of Hermitian matrices the probability that an interval of length $s$ contains no eigenvalues is the Fredholm determinant of the sine kernel $\sin(x-y)\over\pi(x-y)$ over this interval. A formal asymptotic expansion for the determinant as $s$ tends to infinity was obtained by Dyson. In this paper we replace a single interval of length $s$ by $sJ$ where $J$ is a union of $m$ intervals and present a proof of the asymptotics up to second order. The logarithmic derivative with respect to $s$ of the determinant equals a constant (expressible in terms of hyperelliptic integrals) times $s$, plus a bounded oscillatory function of $s$ (zero of $m=1$, periodic if $m=2$, and in general expressible in terms of the solution of a Jacobi inversion problem), plus $o(1)$. Also determined are the asymptotics of the trace of the resolvent operator, which is the ratio in the same model of the probability that the set contains exactly one eigenvalue to the probability that it contains none. The proofs use ideas from orthogonal polynomial theory. 
  This paper begins investigation of the concept of ``generalized $\tau$-function'', defined as a generating function of all the matrix elements of a group element $g \in G$ in a given highest-weight representation of a universal enveloping algebra ${\cal G}$. In the generic situation, the time-variables correspond to the elements of maximal nilpotent subalgebras rather than Cartanian elements. Moreover, in the case of quantum groups such $\tau$-``functions'' are not $c$-numbers but take their values in non-commutative algebras (of functions on the quantum group $G$). Despite all these differences from the particular case of conventional $\tau$-functions of integrable (KP and Toda lattice) hierarchies (which arise when $G$ is a Kac-Moody (1-loop) algebra of level $k=1$), these generic $\tau$-functions also satisfy bilinear Hirota-like equations, which can be deduced from manipulations with intertwining operators. The most important applications of the formalism should be to $k>1$ Kac-Moody and multi-loop algebras, but this paper contains only illustrative calculations for the simplest case of ordinary (0-loop) algebra $SL(2)$ and its quantum counterpart $SL_q(2)$, as well as for the system of fundamental representations of $SL(n)$. 
  We study eta-invariants on odd dimensional manifolds with boundary.  The dependence on boundary conditions is best summarized by viewing the (exponentiated) eta-invariant as an element of the (inverse) determinant line of the boundary.  We prove a gluing law and a variation formula for this invariant.  This yields a new, simpler proof of the holonomy formula for the determinant line bundle of a family of Dirac operators, also known as the ``global anomaly'' formula.   This paper is written using AMSTeX 2.1, which can be obtained via ftp from the American Mathematical Society (instructions included).  A postscript file with figures was submitted separately in uuencoded tar-compressed format. 
  The scattering process of two particles at Planck energies or beyond is calculated using the gravitational shock wave metric for a massive black hole. Then, the scattering between a heavy mass particle and a small mass one is deal with. The cross section contains an extra new term of $\sigma \propto G^3$ as the correction term of the leading term derived by 't Hooft. The ultrarelativistic limit of the Lorentz boosted Reissner--Norstrom black hole is also calculated. 
  The form-factor bootstrap approach is applied to the perturbed minimal models $M_{2,2n+3}$ in the direction of the primary field $\phi_{1,3}$. These theories are integrable and contain $n$ massive scalar particles, whose $S$--matrix is purely elastic. The form-factor equations do not refer to a specific operator. We use this fact to classify the operator content of these models. We show that the perturbed models contain the same number of primary fields as the conformal ones. Explicit solutions are constructed and conjectured to correspond to the off-critical primary fields $\phi_{1,k}$. 
  We determine the canonical structure of two-dimensional black-hole solutions arising in $2D$ dilaton gravity. By choosing the Cauchy surface appropriately we find that the canonically conjugate variable to the black hole mass is given by the difference of local (Schwarzschild) time translations at right and left spatial infinities. This can be regarded as a generalization of Birkhoff's theorem. 
  We discuss a new type of unitary perturbations around conformal theories inspired by the $\sigma$-model perturbation of the nonunitary WZNW model. We show that the nonunitary level $k$ WZNW model perturbed by its sigma model term goes to the unitary level $-k$ WZNW model. When plugged into the gauged WZNW model the given perturbation results in the perturbed gauged WZNW model which no longer describes a coset construction. We consider the BRST invariant generalization of the sigma model perturbation around the gauged WZNW model. In this way we obtain perturbed coset constructions. In the case of the $SU_{m-2}(2)\times SU_1(2)/SU_{m-1}(2)$ coset, the BRST invariant sigma model perturbation is identical to Zamolodchikov's $\Phi_{(3,1)}$ perturbation of the minimal conformal series. The existence of general geometry flows is clarified. 
  We present the theory describing supersymmetrical vortices in the curved superspace of the (1,0) supergravity. The action is defined as a (1,0) locally supersymmetric $SU(2)/U(1)$ coset perturbed by the cosmological constant-like term. The perturbation is such that it preserves the integrability of the coset model. Because of supersymmetry the perturbed theory is an exact quantum system provided a proper dilaton is taken into account. The exact value of the dilaton is determined in the supersymmetric case by the quasi-classical background of the bosonic coset. 
  A simple property of the integrals over the hyperelliptic surfaces of arbitrary genus is observed. Namely, the derivatives of these integrals with respect to the branching points are given by the linear combination of the same integrals. We check that this property is responsible for the solution to the level zero Knizhnik-Zamolodchikov equation given in terms of hyperelliptic integrals. 
  Fusion hierarchies of \ade face models are constructed. The fused critical $D$, $E$ and elliptic $D$ models yield new solutions of the Yang-Baxter equations with bond variables on the edges of faces in addition to the spin variables on the corners. It is shown directly that the row transfer matrices of the fused models satisfy special functional equations. Intertwiners between the fused \ade models are constructed by fusing the cells that intertwine the elementary face weights. As an example, we calculate explicitly the fused $2\times 2$ face weights of the 3-state Potts model associated with the $D_4$ diagram as well as the fused intertwiner cells for the $A_5$--$D_4$ intertwiner. Remarkably, this $2\times 2$ fusion yields the face weights of both the Ising model and 3-state CSOS models. 
  Delta-function potentials in two- and three-dimensional quantum mechanics are analyzed by the incorporation of the self-adjoint extension method to the path integral formalism. The energy-dependent Green functions for free particle plus delta-function potential systems are explicitly calculated. Also the energy-dependent Green function for the spin-1/2 Aharonov-Bohm problem is evaluated. It is found that the only one special value of the self-adjoint extension parameter gives a well-defined and non-trivial time-dependent propagator. This special value corresponds to the viewpoint of the spin-1/2 Aharonov-Bohm problem when the delta-function is treated as a limit of the infinitesimal radius. 
  Newtonian dynamical systems which accept the normal shift on an arbitrary Riemannian manifold are considered. For them the determinating equations making the weak normality condition are derived. The expansion for the algebra of tensor fields is constructed. 
  Using the differential calculus on discrete group, we study the general relativity in the space-time which is the product of a four dimensional manifold by a two-point space. We generalize the usual concept of frame and connection in our space-time, and from the generalized torsion free condition we obtain an action of a scalar field coupled to Einstein gravity, which may be related to the Jordan-Brans-Dicke theory. 
  The $SU_{q}(n)$ generators are obtained as large spectral parameter limit of the Yang-Baxter operators in the integrable $SU_{q}(n)$ invariant vertex model. The commutation relations, including Serre relations, are obtained as limits of the Yang-Baxter equations. The recently found eigenvectors of the $SU_{q}(n)$ invariant spin chains are shown to be Highest Weight vectors of the corresponding quantum group. 
  In four-dimensional compactifications of the heterotic superstring theory the K\"ahler potential has a form which generically induces a superpotential mass term for Higgs particles once supersymmetry is broken at low energies. This ``$\mu$-term'' is analyzed in a model-independent way at the tree level and in the one-loop approximation, and explicit expressions are obtained for orbifold compactifications. Additional contributions which arise in the case of supersymmetry breaking induced by gaugino condensation are also discussed. 
  We construct free field representations of the $SW$-algebras SW(3/2,2) and SW(3/2,3/2,2) by using the corresponding Toda-Field-Theories. In constructing the series of minimal models using covariant vertex operators, we find a necessary restriction on the Cartan matrix of the Super-Lie-Algebra, also for the general case. Within this framework, this restriction claims that there be a minimum of one non-vanishing element on the diagonal of the Cartan matrix, which is without parallel in bosonic conformal field theory. As a consequence only two series of SSLA's yield minimal models, namely Osp(2n|2n-1) and Osp(2n|2n+1). Subsequently some general aspects of degenerate representations of SW-algebras, notably the fusion rules, are investigated. As an application we discuss minimal models of SW(3/2,2), which were constructed with independent methods, in this framework. Covariant formulation is used throughout this paper. 
  We study the string propagation in the 2+1 black hole anti de Sitter background (2+1 BH-ADS). We find the first and second order fluctuations around the string center of mass and obtain the expression for the string mass. The string motion is stable, all fluctuations oscillate with real frequencies and are bounded, even at $r=0.$ We compare with the string motion in the ordinary black hole anti de Sitter spacetime, and in the black string background, where string instabilities develop and the fluctuations blow up at $r=0.$ We find the exact general solution for the circular string motion in all these backgrounds, it is given closely and completely in terms of elliptic functions. For the non-rotating black hole backgrounds the circular strings have a maximal bounded size $r_m,$ they contract and collapse into $r=0.$ No indefinitely growing strings, neither multi-string solutions are present in these backgrounds. In rotating spacetimes, both the 2+1 BH-ADS and the ordinary Kerr-ADS, the presence of angular momentum prevents the string from collapsing into $r=0.$ The circular string motion is also completely solved in the black hole de Sitter spacetime and in the black string background (dual of the 2+1 BH-ADS spacetime), in which expanding unbounded strings and multi-string solutions appear. 
  Some general features of the scattering of boson-based anyons with an added non-statistical interaction are discussed. Periodicity requirements of the phase shifts are derived, and used to illustrate the danger inherent in separating these phase shifts into the well-known pure Aharanov-Bohm phase shifts, and an additional set which arise due to the interaction. It is proven that the added phase shifts, although due to the non-statistical interaction, necessarily change as the statistical parameter is varied, keeping the interaction fixed. A hard-disk interaction provides a concrete illustration of these general ideas. In the latter part of the paper, scattering with an additional hard-disk interaction is studied in detail, with an eye towards providing a criterion for the validity of the mean-field approximation for anyons, which is the first step in virtually any treatment of this system. We find, consistent with previous work, that the approximation is justified if the statistical interaction is weak, and that it must be more weak for boson-based than for fermion-based anyons. 
  A unitary transformation $\Ps [E]=\exp (i\O [E]/g) F[E]$ is used to simplify the Gauss law constraint of non-abelian gauge theories in the electric field representation. This leads to an unexpected geometrization because $\o^a_i\equiv -\d\O [E]/\d E^{ai}$ transforms as a (composite) connection. The geometric information in $\o^a_i$ is transferred to a gauge invariant spatial connection $\G^i_{jk}$ and torsion by a suitable choice of basis vectors for the adjoint representation which are constructed from the electric field $E^{ai}$. A metric is also constructed from $E^{ai}$. For gauge group $SU(2)$, the spatial geometry is the standard Riemannian geometry of a 3-manifold, and for $SU(3)$ it is a metric preserving geometry with both conventional and unconventional torsion. The transformed Hamiltonian is local. For a broad class of physical states, it can be expressed entirely in terms of spatial geometric, gauge invariant variables. 
  In the past ten years, the ideas of supersymmetry have been profitably applied to many nonrelativistic quantum mechanical problems. In particular, there is now a much deeper understanding of why certain potentials are analytically solvable and an array of powerful new approximation methods for handling potentials which are not exactly solvable. In this report, we review the theoretical formulation of supersymmetric quantum mechanics and discuss many applications. Exactly solvable potentials can be understood in terms of a few basic ideas which include supersymmetric partner potentials, shape invariance and operator transformations. Familiar solvable potentials all have the property of shape invariance. We describe new exactly solvable shape invariant potentials which include the recently discovered self-similar potentials as a special case. The connection between inverse scattering, isospectral potentials and supersymmetric quantum mechanics is discussed and multi-soliton solutions of the KdV equation are constructed. Approximation methods are also discussed within the framework of supersymmetric quantum mechanics and in particular it is shown that a supersymmetry inspired WKB approximation is exact for a class of shape invariant potentials. Supersymmetry ideas give particularly nice results for the tunneling rate in a double well potential and for improving large $N$ expansions. We also discuss the problem of a charged Dirac particle in an external magnetic field and other potentials in terms of supersymmetric quantum mechanics. Finally, we discuss structures more general than supersymmetric quantum mechanics such as parasupersymmetric quantum mechanics in which there is a symmetry between a boson and a para-fermion of order $p$. 
  We describe the construction of trigonometric R-matrices corresponding to the (multiplicity-free) tensor product of any two irreducible representations of a quantum algebra $U_q(\G)$. Our method is a generalization of the tensor product graph method to the case of two different representations. It yields the decomposition of the R-matrix into projection operators. Many new examples of trigonometric R-matrices (solutions to the spectral parameter dependent Yang-Baxter equation) are constructed using this approach. 
  The values of the $T$ and $U$ moduli are studied for those ${\bf Z}_N $ Coxeter orbifolds with the property that some of the twisted sectors have fixed planes for which the six-torus ${\bf T}^6 $ can not be decomposed into a direct sum ${\bf T}^2\bigoplus {\bf T}^4 $ with the fixed plane lying in ${\bf T}^2 $. Such moduli in general transform under a subgroup of the modular group $SL(2,Z).$ The moduli are determined by minimizing the effective potential derived from a duality invariant gaugino condensate. 
  The scattering of massless fermions off magnetically charged dilatonic black holes is reconsidered and a violation of unitarity is found. Even for a single species of fermion it is possible for a particle to disappear into the black hole with its information content. 
  The Dirac equation in spherically symmetric fields is separated in two different tetrad frames. One is the standard cartesian (fixed) frame and the second one is the diagonal (rotating) frame. After separating variables in the Dirac equation in spherical coordinates, and solving the corresponding eingenvalues equations associated with the angular operators, we obtain that the spinor solution in the rotating frame can be expressed in terms of Jacobi polynomials, and it is related to the standard spherical harmonics, which are the basis solution of the angular momentum in the Cartesian tetrad, by a similarity transformation. 
  In a space-time of two dimensions the overall effect of the collision of two solitons is a time delay (or advance) of their final trajectories relative to their initial trajectories. For the solitons of affine Toda field theories, the space-time displacement of the trajectories is proportional to the logarithm of a number $X$ depending only on the species of the colliding solitons and their rapidity difference. $X$ is the factor arising in the normal ordering of the product of the two vertex operators associated with the solitons. $X$ is shown to take real values between $0$ and $1$. This means that, whenever the solitons are distinguishable, so that transmission rather than reflection is the only possible interpretation of the classical scattering process, the time delay is negative and so an indication of attractive forces between the solitons. 
  This paper contains an attempt to formulate rigorously and to check predictions in enumerative geometry of curves following from Mirror Symmetry. The main tool is a new notion of stable map. We give an outline of a contsruction of Gromov-Witten invariants for all algebraic projective or closed symplectic manifolds. Mirror Symmetry in the basic example of rational curves on a quintic 3-folds is reduced to certain complicated but explicit identity. The strategy of computations can be described as follows: 1) we reduce counting problems to questions concerning Chern classes on spaces of curves on the ambient projective space, 2) using Bott's residue formula we pass to the space of (degenerate) curves invariant under the action of the group of diagonal matrices, 3) we get a sum over trees and evaluate it using the technique of Feynman diagrams. Our computation scheme gives ``closed'' formulas for generating functions in topological sigma-model for a wide class of manifolds, covering many Calabi-Yau and Fano varieties. 
  We introduce the Wigner functional representing a quantum field in terms of the field amplitudes and their conjugate momenta. The equation of motion for the functional of a scalar field point out the relevance of solutions of the classical field equations to the time evolution of the quantum field. We discuss the field in thermodynamical equilibrium and find the explicit solution of the equations of motion for the so-called ``roll-over'' phase transition. Finally, we briefly discuss the approximate methods for the evaluation of the Wigner functional that may be used to numerically simulate the initial value problem.. 
  The infinite number of time-dependent linear in field and conjugated momenta invariants is derived for the scalar field using the Noether's theorem procedure. 
  We derive explicit integral formulas for eigenfunctions of quantum integrals of the Calogero-Sutherland-Moser operator with trigonometric interaction potential. In particular, we derive explicit formulas for Jack's symmetric functions. To obtain such formulas, we use the representation of these eigenfunctions by means of traces of intertwining operators between certain modules over the Lie algebra $\frak gl_n$, and the realization of these modules on functions of many variables. 
  By "untwisting" the construction of Berkovits and Vafa, one can see that the N=1 superstring contains a topological twisted N=2 algebra, with central charge c^ = 2. We discuss to what extent the superstring is actually a topological theory. 
  The well-known geometric approach to field theory is based on description of classical fields as sections of fibred manifolds, e.g. bundles with a structure group in gauge theory. In this approach, Lagrangian and Hamiltonian formalisms including the multiomentum Hamiltonian formalism are phrased in terms of jet manifolds. Then, configuration and phase spaces of fields are finite-dimensional. Though the jet manifolds have been widely used for theory of differential operators, the calculus of variations and differential geometry, this powerful mathematical methods remains almost unknown for physicists. This Supplementary to our previous article (hep-th/9403172) aims to summarize necessary requisites on jet manifolds and general connections. 
  Starting from a detailed analysis of the structure of pathspaces of the ${\cal A}$-fusion graphs and the corresponding irreducible Virasoro algebra quotients $V(c,h)$ for the ($2,q$ odd) models, we introduce the notion of an admissible pathspace representation. The pathspaces ${\cal P_A}$ over the ${\cal A}$-Graphs are isomorphic to the pathspaces over Coxeter $A$-graphs that appear in FB models. We give explicit construction algorithms for admissible representations. From the finitedimensional results of these algorithms we derive a decomposition of $V(c,h)$ into its positive and negative definite subspaces w.r.t. the Shapovalov form and the corresponding signature characters. Finally, we treat the Virasoro operation on the lattice induced by admissible representations adopting a particle point of view. We use this analysis to decompose the Virasoro algebra generators themselves. This decomposition also takes the nonunitarity of the ($2,q$) models into account. 
  It is shown that elementary indistinguishability properties of partially polarized mixtures are consistent only with the conventional Hilbert space model of quantum mechanics and a few exotic alternatives. This applies even in low dimensions where quantum logic and Gleason's theorem give either weak or no constraints. Experimental methods for eliminating the exotic cases (which include quaternionic and octonionic variants of quantum mechanics) are described. 
  This paper extends an earlier high-temperature lattice calculation of the renormalized Green's functions of a $D$-dimensional Euclidean scalar quantum field theory in the Ising limit. The previous calculation included all graphs through sixth order. Here, we present the results of an eleventh-order calculation. The extrapolation to the continuum limit in the previous calculation was rather clumsy and did not appear to converge when $D>2$. Here, we present an improved extrapolation which gives uniformly good results for all real values of the dimension between $D=0$ and $D=4$. We find that the four-point Green's function has the value $0.620 \pm 0.007$ when $D=2$ and $0.98 \pm 0.01$ when $D=3$ and that the six-point Green's function has the value $0.96 \pm 0.03$ when $D=2$ and $1.2 \pm 0.2$ when $D=3$. 
  The elements of the wide class of quantum universal enveloping algebras are prooved to be Hopf algebras $H$ with spectrum $Q(H)$ in the category of groups. Such quantum algebras are quantum groups for simply connected solvable Lie groups $P(H)$. This provides utilities for a new algorithm of constructing quantum algebras especially useful for nonsemisimple ones. The quantization procedure can be carried out over an arbitrary field. The properties of the algorithm are demonstrated on examples. 
  For a certain class of Lie bialgebras $(A,A^*)$ the corresponding quantum universal enveloping algebras $U_q(A)$ are prooved to be equivalent to quantum groups Fun$_q(F^*)$, $F^*$ being the factor group for the dual group $G^*$. This property can be used to simplify the process of quantization. The described class appears to be wide enough to contain all the standard quantizations of infinite series. The properties of the groups $F^*$ are explicitly demonstrated for the standard deformations $U_q(SL(n))$. It is shown that for different $A^*$ (remaining in the described class of Lie bialgebras) the same algorithm leads to the nonstandard quantizations. 
  We study the interaction between a $p$-brane and $BF$ system constituted by a $(p+1)$-form and a $n-(p+2)$-form B with a metric independent action on a manifold $M^n$. We identify the allowed $(p+1)$-world manifolds sweeped by the $p$-brane as those restricted by the topological condition of being the boundaries of chains in $M^n$. We find the general solutions of the equation of motion for the field configurations. The solutions for the $B$ field are closely related to the external field configurations that describe the defects needed for the computation of soliton correlation functions.The particular cases of the particle and string evolution are discussed with some detail. 
  We describe a canonical covariant formulation of the Green-Schwarz Superstring which allows the construction of a new covariant action canonically equivalent to the Green-Schwarz action but subjected only to first class constraints. From this action the correct BRST operator for the quantization of the Green-Schwarz Superstring may be constructed. Also the gauge fixed action in the Light-Cone gauge may be reobtained. The action presented in this letter generalizes in a non-trivial form the action introduced by Kallosh for the Brink-Schwarz-Casalbuoni Superparticle. 
  A new supersymmetric proof of the Atiyah-Singer index theorem is presented. The Peierls bracket quantization scheme is used to quantize the supersymmetric classical system corresponding to the index problem for the twisted Dirac operator. The problem of factor ordering is addressed and the unique quantum system that is relevant to the index theorem is analyzed in detail. The Hamiltonian operator is shown to include a scalar curvature factor, $\hbar^2R/8$. The path integral formulation of quantum mechanics is then used to obtain a formula for the index. For the first time, the path integral "measure" and the Feynman propagator of the system are exactly computed. The derivation of the index formula relies solely on the definition of a Gaussian superdeterminant. The two-loop analysis of the path integral is also carried out. The results of the loop and heat kernel expansions of the path interal are in complete agreement. This confirms the existence of the scalar curvature factor in the Schr\"odinger equation and validates the supersymmetric proof of the index theorem. Many other related issues are addressed. Finally, reviews of the index theorem and the supersymmetric quantum mechanics are presented. 
  We obtain a new free field realization of $N=2$ super $W_{3}$ algebra using the technique of quantum hamiltonian reduction. The construction is based on a particular choice of the simple root system of the affine Lie superalgebra $sl(3|2)^{(1)}$ associated with a non-standard $sl(2)$ embedding. After twisting and a similarity transformation, this $W$ algebra can be identified as the extended topological conformal algebra of non-critical $W_{3}$ string theory. 
  These three topics are an attempt to explicate some curiosities of the inverse problem of representation theory (i.e. having a set of operators to describe the "correct" algebraic object, which is represented by them) on simple examples related to the Lie algebra sl(2,C). Such consideration maybe regarded as toy one for analogous infinite dimensional problems in the modern quantum field theory (conformal field theory, integrable models, field theory in non-trivial backgrounds). Some constructions inspired by these topics and related to infinite dimensional hidden symmetries, which are produced by vertex operator fields, will be discussed in the forthcoming article. 
  Within the framework of second order derivative (one dimensional) SUSYQM we discuss particular realizations which incorporate large energy shifts between the lowest states of the spectrum of the superhamiltonian (of Schr\"odinger type). The technique used in this construction is based on the "gluing" procedure. We study the limit of infinite energy shift for the charges of the Higher Derivative SUSY Algebra, and compare the results with those of the standard SUSY Algebra. We conjecture that our results can suggest a construction of a toy model where large energy splittings between fermionic and bosonic partners do not affect the SUSY at low energies. 
  In this paper, correlation functions of tachyons in the two-dimensional black-hole are studied. The results are shown to be consistent with the prediction of the deformed matrix model. 
  A generalized chiral Schwinger model is studied by means of perturbative techniques. Explicit expressions are obtained, both for bosonic and fermionic propagators, and compared to the ones derived by means of functional techniques. In particular a consistent recipe is proposed to describe the ambiguity occurring in the regularization of the fermionic determinant. The role of the gauge fixing term, which is needed to develop perturbation theory and the behaviour of the spectrum as a function of the parameters are clarified together with ultraviolet and infrared properties of the model. 
  \noindent We briefly discuss some algebraic and geometric aspects of the generalized Poisson bracket and non--commutative phase space for generalized quantum dynamics, which are analogous to properties of the classical Poisson bracket and ordinary symplectic structure. 
  We introduce Chern-Simons interaction into the three dimensional four-fermi theory, ad suggest a possible line of non-Gaussian infrared stable fixed points of the four-fermi operator, this line is characterized by the Chern-Simons coupling. 
  As a generalization of our previous paper [GK], we formulate a residue formula and some simple behaviors of equivariant quantum cohomology applying to compute the quantum cohomology of partial flag manifolds $F_{k_1,\cdots , k_l} $with a try to give a rigorous definition of equivariant quantum cohomology. 
  We show that a large class of incompressible quantum Hall states correspond to different representations of the \Winf algebra by explicit construction of the second quantized generators of the algebra in terms of fermion and vortex operators. These are parametrized by a set of integers which are related to the filling fraction. The class of states we consider includes multilayer Hall states and the states proposed by Jain to explain the hierarchical filling fractions. The corresponding second quantized order parameters are also given. 
  We discuss a construction of highest weight modules for the recently defined elliptic algebra ${\cal A}_{q,p}(\widehat{sl}_2)$, and make several conjectures concerning them. The modules are generated by the action of the components of the operator $L$ on the highest weight vectors. We introduce the vertex operators $\Phi$ and $\Psi^*$ through their commutation relations with the $L$-operator. We present ordering rules for the $L$- and $\Phi$-operators and find an upper bound for the number of linearly independent vectors generated by them, which agrees with the known characters of $\widehat{sl}_2$-modules. 
  We study $R^2$ gravity in $(2+\epsilon)$--dimensional quantum gravity. Taking care of the oversubtraction problem in the conformal mode dynamics, we perform a full order calculation of string susceptibility in the $\epsilon \rightarrow 0$ limit. The result is consistent with that obtained through Liouville approach. 
  We compute the one-loop potential (the Casimir energy) for scalar, spinor and vectors fields on the spaces $\,R^{m+1}\, \times\,Y$ with $\,Y=\,S^N\,,CP^2$. As a physical model we consider spinor electrodynamics on four-dimensional product manifolds. We examine the cancelation of a divergent part of the Casimir energy on even-dimensional spaces by means of including the parameter $\,M$ in original action. For some models we compare our results with those found in the literature. 
  The possibility of control of phenomena at microscopic level compatible with quantum mechanics and quantum field theory is outlined. The theory could be used in nanotechnology. 
  We examine one-dimensional Bose gas interacting with delta-function potential using the large-$N$ collective field theory. We show that in the case of attractive potential the uniform ground state is unstable to small perturbations and the instability is cured by formation of a collective ground state, \lq\lq bright soliton'' configuration in corresponding nonlinear Schr\"odinger field theory. 
  It is well known that magnetic monopoles are related to the first Chern class. In this note electric charge is used to construct an analogous characteristic class: the charge class. 
  It is shown that B\"acklund transformations (BTs) and zero-curvature representations (ZCRs) of systems of partial differential equations (PDEs) are closely related. The connection is established by nonlinear representations of the symmetry group underlying the ZCR which induce gauge transformations relating different BTs. This connection is used to construct BTs from ZCRs (and vice versa). Furthermore a procedure is outlined which allows a systematic search for ZCRs of a given system of PDEs. 
  A recollection of some theoretical developments that preceded and followed the first formulation of supergravity theory is presented. Special emphasis is placed on the impact of supergravity on the search for a unified theory of fundamental interactions. 
  Superstring theory predicts the existence of a scalar field, the dilaton. I review some basic features of the dilaton interactions and explain their possible consequences in cosmology and particle physics. 
  A noncommutative geometric generalisation of the quantum field theoretical framework is developed by generalising the Heisenberg commutation relations. There appear nonzero minimal uncertainties in positions and in momenta. As the main result it is shown with the example of a quadratically ultraviolet divergent graph in $\phi^4$ theory that nonzero minimal uncertainties in positions do have the power to regularise. These studies are motivated with the ansatz that nonzero minimal uncertainties in positions and in momenta arise from gravity. Algebraic techniques are used that have been developed in the field of quantum groups. 
  Spin models on quenched random graphs are related to many important optimization problems. We give a new derivation of their mean-field equations that elucidates the role of the natural order parameter in these models. 
  We argue that string theory should have a formulation for which stability and causality are evident. Rather than regard strings as fundamental objects, we suggest they should be regarded as composite systems of more fundamental point-like objects. A tentative scheme for such a reinterpretation is described along the lines of 't Hooft's $1/N$ expansion and the light-cone parametrization of the string. 
  Several stationary solutions of the low energy string equations are dualized with respect to their timelike symmetry. Many of the duals have simple physical interpretations. Those of the nonextremal three dimensional black hole and black string are negative mass black strings. The extremal cases of these, and extremal higher dimensional black strings also, give negative energy plane fronted waves. In fact, all of the duals of positive mass solutions that will be considered here have nonpositive energies, but an argument is given which suggests that this is not true in general. 
  The special geometry ($(t,{\bar t})$-equations) for twisted $N=2$ strings are derived as consistency conditions of a new contact term algebra. The dilaton appears in the contact terms of topological and antitopological operators. The holomorphic anomaly, which can be interpreted as measuring the background dependence, is obtained from the contact algebra relations. 
  Quantum theory of dilaton gravity is studied in $2+\epsilon$ dimensions. Divergences are computed and renormalized at one-loop order. The mixing between the Liouville field and the dilaton field eliminates $1/\epsilon$ singularity in the Liouville-dilaton propagator. This smooth behavior of the dilaton gravity theory in the $\epsilon \rightarrow 0$ limit solves the oversubtraction problem which afflicted the higher orders of the Einstein gravity in $2+\epsilon$ dimensions. As a nontrivial fixed point, we find a dilaton gravity action which can be transformed to a CGHS type action. 
  Using a $q$-deformed Moyal algebra associated with the group of area preserving diffeomorphisms of th two-dimensional torus $T^2$, sdiff$_q (T^2)$, a $q$-deformed version for the Heavenly equations is given. Finally, the two-dimensional chiral version of Self-dual gravity in this $q$-deformed context is briefly discussed. 
  The twistor--like formulation of the type IIA superstring $\sigma$--model in D=10 is obtained by performing a dimensional reduction of the recently proposed twistor--like action of the supermembrane in D=11. The superstring action is invariant under local, worldsheet $(n,n)$ supersymmetry where $3\leq n \leq 8$ and is classical equivalent to the standard Green--Schwarz action (at least for $n=8)$ 
  A problem of constructing quantum groups from classical r-matrices is discussed. 
  The classical $r$-matrix for $N=1$ superPoincar{\'e} algebra, given by Lukierski, Nowicki and Sobczyk is used to describe the graded Poisson structure on the $N=1$ Poincar{\'e} supergroup. The standard correspondence principle between the even (odd) Poisson brackets and (anti)commutators leads to the consistent quantum deformation of the superPoincar{\'e} group with the deformation parameter $q$ described by fundamental mass parameter $\kappa \quad (\kappa^{-1}=\ln{q})$. The $\kappa$-deformation of $N=1$ superspace as dual to the $\kappa$-deformed supersymmetry algebra is discussed. 
  We apply the methods of Field Theory to study the turbulent regimes of statistical systems. First we show how one can find their probability densities. For the case of the theory of wave turbulence with four-wave interaction we calculate them explicitly and study their properties. Using those densities we show how one can in principle calculate any correlation function in this theory by means of direct perturbative expansion in powers of the interaction. Then we give the general form of the corrections to the kinetic equation and develop an appropriate diagrammatic technique. This technique, while resembling that of $\varphi^4$ theory, has many new distinctive features. The role of the $\epsilon=d-4$ parameter is played here by the parameter $\kappa=\beta + d - \alpha - \gamma$ where $\beta$ is the dimension of the interaction, $d$ is the space dimension, $\alpha$ is the dimension of the energy spectrum and $\gamma$ is the ``classical'' wave density dimension. If $\kappa > 0$ then the Kolmogorov index is exact, and if $\kappa < 0$ then we expect it to be modified by the interaction. For $\kappa$ a small negative number, $\alpha<1$ and a special form of the interaction we compute this modification explicitly with the additional assumption of the irrelevance of the IR divergencies which still needs to be verified. 
  \noindent It is find a non-linear partial differential equation which we show contains the first Heavenly equation of Self-dual gravity and generalize the second one. This differential equation we call "Weak Heavenly Equation" (${\cal WH}$-equation). For the two-dimensional case the ${\cal WH}$-equation is brought into the evolution form (Cauchy-Kovalevski form) using the Legendre transformation. Finally, we find that this transformed equation ("Evolution Weak Heavenly Equation") does admit very simple solutions. 
  Irreducible representations of quantum groups $SL_q(2)$ (in Woronowicz' approach) were classified in J.Wang, B.Parshall, Memoirs AMS 439 in the~case of $q$ being an~odd root of unity. Here we find the~irreducible representations for all roots of unity (also of an~even degree), as well as describe "the~diagonal part" of tensor product of any two irreducible representations. An~example of not completely reducible representation is given. Non--existence of Haar functional is proved. The~corresponding representations of universal enveloping algebras of Jimbo and Lusztig are provided. We also recall the~case of general~$q$. Our computations are done in explicit way. 
  The lattice model of principal chiral field interacting with the gauge fields, which have no kinetic term, is considered. This model can be regarded as a strong coupling limit of lattice gauge theory at finite temperature. The complete set of equations for collective field variables is derived in the large N limit and the phase structure of the model is studied. 
  We generalize the geometric structures generated by Witten's ground ring. It is shown that these generalized structures involve in a natural way some geometric constructions from Self-dual gravity [1,12]. The formal twistor construction on full quantum ground ring manifold is also given. 
  We give a methodology for solving the chiral equations $(\alpha g_{,z} g^{-1})_{,\overline z} + (\alpha g_{,\overline z} g^{-1})_{,z} \ = \ 0 $ where $g$ belongs to some Lie group $G$. The solutions are writing in terms of Harmonic maps. The method could be used even for some infinite Lie groups. (Preprint CIEA-gr-94/06) 
  Using a gauge covariant operator technique we deduce the path integral for a charged particle in a stationary magnetic field, verifying the "midpoint rule" for the discrete form of the interaction term with the vector potential. 
  Quantum mechanical boundary conditions along a timelike line, corresponding to the origin in radial coordinates, in two-dimensional dilaton gravity coupled to $N$ matter fields, are considered. Conformal invariance and vacuum stability severely constrain the possibilities. The simplest choice found corresponds to a nonlinear Liouville-type boundary interaction. The scattering of low-energy matter off the boundary can be computed perturbatively. It is found that weak incident pulses induce damped oscillations at the boundary while large incident pulses produce black holes. The response of the boundary to such pulses is semi-classically characterized by a second order, nonlinear ordinary differential equation which is analyzed numerically. 
  The relation between solutions to Helmholtz's equation on the sphere $S^{n-1}$ and the $[{\gr sl}(2)]^n$ Gaudin spin chain is clarified. The joint eigenfuctions of the Laplacian and a complete set of commuting second order operators suggested by the $R$--matrix approach to integrable systems, based on the loop algebra $\wt{sl}(2)_R$, are found in terms of homogeneous polynomials in the ambient space. The relation of this method of determining a basis of harmonic functions on $S^{n-1}$ to the Bethe ansatz approach to integrable systems is explained. 
  Because of the existence of rigid Calabi--Yau manifolds, mirror symmetry cannot be understood as an operation on the space of manifolds with vanishing first Chern class. In this article I continue to investigate a particular type of K\"ahler manifolds with positive first Chern class which generalize Calabi--Yau manifolds in a natural way and which provide a framework for mirrors of rigid string vacua. This class is comprised of a special type of Fano manifolds which encode crucial information about ground states of the superstring. It is shown in particular that the massless spectra of $(2,2)$--supersymmetric vacua of central charge $\hat{c}=D_{crit}$ can be derived from special Fano varieties of complex dimension $(D_{crit}+2(Q-1))$, $Q>1$, and that in certain circumstances it is even possible to embed Calabi--Yau manifolds into such higher dimensional spaces. The constructions described here lead to new insight into the relation between exactly solvable models and their mean field theories on the one hand and their corresponding Calabi--Yau manifolds on the other. It is furthermore shown that Witten's formulation of the Landau--Ginzburg/Calabi--Yau relation can be applied to the present framework as well. 
  The hamiltonian formalism is developed for the sine-Gordon model on the space-time light-like lattice, first introduced by Hirota. The evolution operator is explicitely constructed in the quantum variant of the model, the integrability of the corresponding classical finite-dimensional system is established. 
  It is shown how the BRST quantization can be applied to a gauge invariant sector of theories with anomalously broken symmetries. This result is used to show that shifting the anomalies to a classically trivial sector of fields (Wess-Zumino mechanism) makes it possible to quantize the physical sector using a standard BRST procedure, as for a non anomalous theory. The trivial sector plays the role of a topological sector if the system is quantized without shifting the anomalies. 
  We study the quasi-particle spectrum of the integrable three-state chiral Potts chain in the massive phase by combining a numerical study of the zeroes of associated transfer matrix eigenvalues with the exact results of the ferromagnetic three-state Potts chain and the three-state superintegrable chiral Potts model. We find that the spectrum is described in terms of quasi-particles with momenta restricted only to segments of the Brillouin zone $0\leq P \leq 2\pi$ where the boundaries of the segments depend on the chiral angles of the model. 
  An attempt is made to compare the asymptotic state density of twisted $p$-branes and the related state density of mass level $M$ of a $D$-dimensional neutral black hole. To this aim, the explicit form of the twisted $p$-brane total level degeneracy is calculated. The prefactor of the degeneracy, in contrast to the leading behaviour, is found to depend on the winding number of the $p$-brane. 
  In the first part of this lecture I will give an introduction to light-cone field theory, focussing on the ``zero mode problem''. In the second part I discuss $\phi^4$-theory in 1+1 dimensions. I will show how the dynamics of the zero modes can give rise to spontaneous symmetry breaking in spite of the trivial vacuum structure on the light-cone. 
  We present a classical integrable model of $SU(N)$ isospin defined on complex projective phase space in the external magnetic field and solve it exactly by constructing the action-angle variables for the system. We quantize the system using the coherent state path integral method and obtain an exact expression for quantum mechanical propagator by solving the time-dependent Schr\"odinger equations. 
  We diagonalize the Hilbert space of some subclass of the quasifinite module of the \Winf algebra. States are classified according to their eigenvalues for infinitely many commuting charges and the Young diagrams. The parameter dependence of their norms is explicitly derived. The full character formulae of the degenerate representations are given as summation of the bilinear combinations of the Schur polynomials. 
  This paper has been withdrawn pending revision. 
  We define consistent finite-superfields reductions of the $N=1,2$ super-KP hierarchies via the coset approach we already developped for reducing the bosonic KP-hierarchy (generating e.g. the NLS hierarchy from the $sl(2)/U(1)-{\cal KM}$ coset). We work in a manifestly supersymmetric framework and illustrate our method by treating explicitly the $N=1,2$ super-NLS hierarchies. W.r.t. the bosonic case the ordinary covariant derivative is now replaced by a spinorial one containing a spin ${\textstyle {1\over 2}}$ superfield. Each coset reduction is associated to a rational super-$\cw$ algebra encoding a non-linear super-$\cw_\infty$ algebra structure. In the $N=2$ case two conjugate sets of superLax operators, equations of motion and infinite hamiltonians in involution are derived. Modified hierarchies are obtained from the original ones via free-fields mappings (just as a m-NLS equation arises by representing the $sl(2)-{\cal KM}$ algebra through the classical Wakimoto free-fields). 
  Analogues of the KP and the Toda lattice hierarchy called dispersionless KP and Toda hierarchy are studied. Dressing operations in the dispersionless hierarchies are introduced as a canonical transformation, quantization of which is dressing operators of the ordinary KP and Toda hierarchy. An alternative construction of general solutions of the ordinary KP and Toda hierarchy is given as twistor construction which is quatization of the similar construction of solutions of dispersionless hierarchies. These results as well as those obtained in previous papers are presented with proofs and necessary technical details. 
  By investigating the symplectic geometry and geometric quantization on a class of supermanifolds, we exhibit BRST structures for a certain kind of algebras. We discuss the undeformed and q-deformed cases in the classical as well as in the quantum cases. 
  The BH algebra is defined by two sets of generators one of which satisfy the relations of the braid group and the other the relations of the Hecke algebra of projectors.These algebras are then combined by additional relations in a way which generalizes the Birman-Wenzl algebra.In this paper we Yang-Baxterize the algebra BH and compute solutions of the Yang-Baxter equation.The solutions found are expressed algebraically in terms of the generators of the algebra.The expression generalizes the known one for the Birman-Wenzl algebra. 
  We investigate the 3d lattice topological field theories defined by Chung, Fukuma and Shapere. We concentrate on the model defined by taking a deformation $\D{G}$ of the quantum double of a finite commutative group $G$ as the underlying Hopf algebra. It is suggested that Chung-Fukuma-Shapere partition function is related to that of Dijkgraaf-Witten by $\zcfs = |\zdw|^2$ when $G=\Z_{2N+1}$. For $G=\Z_{2N}$, such a relation does not hold. 
  We explicitly compute the anomalous magnetic moment at one loop level for an ``electron'' in a 4d heterotic string theory. The anomalous magnetic moment vanishes if the model is spacetime supersymmetric, as required by the supersymmetric sum rules. 
  Nous presentons une introduction aux concepts de la supersymetrie par l'intermediaire de trois exemples: (i) Mecanique quantique supersymetrique, (ii) Superalgebres de Lie, (iii) Superconnexions de Quillen. Les points communs a toutes ces notions sont soulignes et des applications sont indiquees. En particulier nous esquissons la demonstration du theoreme de Gauss et Bonnet d'apres Patodi et la demonstration des inegalites de Morse d'apres Witten. 
  Novel infinite-dimensional algebras of the Virasoro/Kac-Moody/ Floratos-Iliopoulos type are introduced, which involve special functions in their structure constants 
  Strominger has proposed an interesting concrete realization of Hawking's idea that information is lost in black hole evaporation. In this note we demonstrate that a straightforward interpretation of Strominger's model leads to a complete breakdown of the conditions for using statistics for analyzing the results of experiments. The probabilities produced by the theory are operationally meaningless. 
  In these lectures, I review some recent results on the Calogero-Sutherland model and the Haldane Shastry-chain. The list of topics I cover are the following: 1) The Calogero-Sutherland Hamiltonian and fractional statistics. The form factor of the density operator. 2) The Dunkl operators and their relations with monodromy matrices, Yangians and affine-Hecke algebras. 3) The Haldane-Shastry chain in connection with the Calogero-Sutherland Hamiltonian at a specific coupling constant. 
  We derive the Kac Wakimoto formula in the algebraic framework of chiral conformal QFT s a consequence of the universal behaviour of high-temperature Gibbs-states on intertwiner subalgebras. By studying selfintertwiners in the presence of other charges, we obtain charge-polarization coefficients which generalize the entries of Verlinde matrix S and belong to higher genus mapping class group matrices. 
  A systematic formulation of the higher genus expansion in topological string theory is considered. We also develop a simple way of evaluating genus zero correlation functions. At higher genera we derive some interesting formulas for the free energy in the $A_1$ and $A_2$ models. We present some evidence that topological minimal models associated with Lie algebras other than the A-D-E type do not have a consistent higher genus expansion beyond genus one. We also present some new results on the $CP^1$ model at higher genera. 
  We show that the $\kappa$-deformed Poincar\'e quantum algebra proposed for elementary particle physics has the structure of a Hopf agebra bicrossproduct $U(so(1,3))\cobicross T$. The algebra is a semidirect product of the classical Lorentz group $so(1,3)$ acting in a deformed way on the momentum sector $T$. The novel feature is that the coalgebra is also semidirect, with a backreaction of the momentum sector on the Lorentz rotations. Using this, we show that the $\kappa$-Poincar\'e acts covariantly on a $\kappa$-Minkowski space, which we introduce. It turns out necessarily to be deformed and non-commutative. We also connect this algebra with a previous approach to Planck scale physics. 
  We study a class of subdivision invariant lattice models based on the gauge group $Z_{p}$, with particular emphasis on the four dimensional example. This model is based upon the assignment of field variables to both the $1$- and $2$-dimensional simplices of the simplicial complex. The property of subdivision invariance is achieved when the coupling parameter is quantized and the field configurations are restricted to satisfy a type of mod-$p$ flatness condition. By explicit computation of the partition function for the manifold $RP^{3} \times S^{1}$, we establish that the theory has a quantum Hilbert space which differs from the classical one. 
  We establish general theorems on the cohomology $H^*(s|d)$ of the BRST differential modulo the spacetime exterior derivative, acting in the algebra of local $p$-forms depending on the fields and the antifields (=sources for the BRST variations). It is shown that $H^{-k}(s|d)$ is isomorphic to $H_k(\delta |d)$ in negative ghost degree $-k\ (k>0)$, where $\delta$ is the Koszul-Tate differential associated with the stationary surface. The cohomological group $H_1(\delta |d)$ in form degree $n$ is proved to be isomorphic to the space of constants of the motion, thereby providing a cohomological reformulation of Noether theorem. More generally, the group $H_k(\delta|d)$ in form degree $n$ is isomorphic to the space of $n-k$ forms that are closed when the equations of motion hold. The groups $H_k(\delta|d)$ $(k>2)$ are shown to vanish for standard irreducible gauge theories. The group $H_2(\delta|d)$ is then calculated explicitly for electromagnetism, Yang-Mills models and Einstein gravity. The invariance of the groups $H^{k}(s|d)$ under the introduction of non minimal variables and of auxiliary 
  A class of two dimensional field theories, based on (generically degenerate) Poisson structures and generalizing gravity-Yang-Mills systems, is presented. Locally, the solutions of the classical equations of motion are given. A general scheme for the quantization of the models in a Hamiltonian formulation is found. 
  We generalize the construction of Gelfand-Dikii brackets to the case of nonstandard Lax equations. We also discuss the possible origin of Kac-Moody algebras present in such systems. 
  We discuss the geometry of the Lagrangian quantization scheme based on (generalized) Schwinger-Dyson BRST symmetries. When a certain set of ghost fields are integrated out of the path integral, we recover the Batalin-Vilkovisky formalism, now extended to arbitrary functional measures for the classical fields. Keeping the ghosts reveals the crucial role played by a natural connection on the space of fields. 
  Unique twistor--like Lorentz harmonic formulation for all N=1 supersymmetric extended objects (super-p-branes) moving in the space--time of arbitrary dimension D (admissible for given $p$) are suggested. The equations of motion are derived, explicit form of the \kappa-symmetry transformations is presented and the classical equivalence to the standard formulation is proved.   The cases with minimal world--sheet dimensions $p=1,2$, namely of D=10 heterotic string and D=11 supermembrane, are considered in details. In particular, the explicit form of irreducible \kappa-symmetry transformations for D=11 supermembrane is derived. 
  A non-relativistic scalar field coupled minimally to electromagnetism supports in the presence of a homogeneous background electric charge density the existence of smooth, finite-energy topologically stable flux vortices. The static properties of such vortices are studied numerically in the context of a two parameter model describing this system as a special case. It is shown that the electrostatic and the mexican hat potential terms of the energy are each enough to ensure the existence of vortex solutions. The interaction potential of two minimal vortices is obtained for various values of the parameters. It is proven analytically that a free isolated vortex with topological charge $N\ne 0$ is spontaneously pinned, while in the presence of an external force it moves at a calculable speed and in a direction $(N/|N|) 90^0$ relative to it. In a homogeneous external current $\tilde {\bf J}$ the vortex velocity is $\bf V=-\tilde J$. Other theories with the same vortex behaviour are briefly discussed. 
  By reducing a split $G_2$ Kac-Moody algebra by a non-maximal set of first-class constraints we produce W-algebras which (i) contain fields of negative conformal spin and (ii) are not trivial extensions of canonical W-algebras. 
  We construct affinization of the algebra $gl_{\lambda}$ of ``complex size'' matrices, that contains the algebras $\hat{gl_n}$ for integral values of the parameter. The Drinfeld--Sokolov Hamiltonian reduction of the algebra $\hat{gl_{\lambda}}$ results in the quadratic Gelfand--Dickey structure on the Poisson--Lie group of all pseudodifferential operators of fractional order. This construction is extended to the simultaneous deformation of orthogonal and simplectic algebras that produces self-adjoint operators, and it has a counterpart for the Toda lattices with fractional number of particles. 
  We calculate the density of states with given mass and spin in string theory and obtain asymptotic formulas. We also compute the tree-level gyromagnetic couplings for arbitrary physical states in the heterotic string theory. These results are then applied to study whether fundamental strings can consistently describe the microphysics of the black hole horizon in the case of a general classical solution characterized by mass, charge and angular momentum. 
  We study the thermal ensemble of a gas of free strings in presence of a magnetic field. We find that the thermodynamical partition function diverges when the magnetic field exceeds some critical value $B_{\rm cr}$, which depends on the temperature. We argue that there is a first-order phase transition with a large latent heat. At the critical value an infinite number of states -all states in a Regge trajectory- seem to become massless, which may be an indication of recuperation of spontaneously broken symmetries. 
  We show that string-inspired lineal gravity interacting with matter fields cannot be Dirac-quantized owing to the well known anomaly in energy-momentum tensor commutators. 
  We study the problem of identifying the moduli fields in fermionic four-dimensional string models. We deform a free-fermionic model by introducing exactly marginal operators in the form of Abelian Thirring interactions on the world-sheet, and show that their couplings correspond to the untwisted moduli fields. We study the consequences of this method for simple free-fermionic models which correspond to $Z_2\times Z_2$ orbifolds and obtain their moduli space and K\"ahler potential by symmetry arguments and by direct calculation of string scattering amplitudes. We then generalize our analysis to more complicated fermionic structures which arise in constructions of realistic models corresponding to asymmetric orbifolds, and obtain the moduli space and K\"ahler potential for this case. Finally we extend our analysis to the untwisted matter sector and derive expressions for the full K\"ahler potential to be used in phenomenological applications, and the target space duality transformations of the corresponding untwisted matter fields. 
  We study representations of the central extension of the Lie algebra of differential operators on the circle, the W-infinity algebra. We obtain complete and specialized character formulas for a large class of representations, which we call primitive; these include all quasi-finite irreducible unitary representations. We show that any primitive representation with central charge N has a canonical structure of an irreducible representation of the W-algebra W(gl_N) with the same central charge and that all irreducible representations of W(gl_N) with central charge N arise in this way. We also establish a duality between "integral" modules of W(gl_N) and finite-dimensional irreducible modules of gl_N, and conjecture their fusion rules. 
  Value of generalized hypergeometric function at a special point is calculated. More precisely, value of certain multiple integral over vanishing cycle (all arguments collapse to unity) is calculated. The answer is expressed in terms of $\Gamma$-functions. The constant is relevant to the part of $\rho$ in the Gindikin-Karpelevich formula for c-function of Harish-Chandra. Calculation is an adaptation of classical calculations of Gelfand and Naimark (1950) to the Heckman-Opdam hypergeometric functions in the case of root system of type $A_{n-1}$. 
  In a recent publication we have investigated the spectrum of anomalous dimensions for arbitrary composite operators in the critical N-vector model in 4-epsilon dimensions. We could establish properties like upper and lower bounds for the anomalous dimensions in one-loop order. In this paper we extend these results and explicitly derive parts of the one-loop spectrum of anomalous dimensions. This analysis becomes possible by an explicit representation of the conformal symmetry group on the operator algebra. Still the structure of the spectrum of anomalous dimensions is quite complicated and does generally not resemble the algebraic structures familiar from two dimensional conformal field theories. 
  It is known that, in string sigma-model metric, the `extreme' fivebrane solution of D=10 supergravity interpolates between D=10 Minkowski spacetime and a supersymmetric $S^3$ compactification to a linear dilaton vacuum. We show here that, in {\it fivebrane} sigma-model metric, the extreme string solution of D=10 supergravity interpolates between Minkowski spacetime and a hitherto unknown supersymmetric $S^7$ compactification of d=10 supergravity to a three-dimensional anti-de Sitter generalization of the linear dilaton vacuum, which may be invariantly characterized in terms of conformal   Killing vectors. The dilaton field diverges near the string core but this divergence may be eliminated by re-interpreting the string solution as the extreme membrane solution of 11-dimensional supergravity. We show that the latter has an analytic extension through a regular degenerate event horizon to an interior region containing a curvature singularity. We obtain analogous results for other extended object solutions of supergravity theories. 
  The properties of the three-dimensional noncanonical osp(3/2) oscillators, introduced in J.Phys. A: Math. Gen. {\bf 27} (1994) 977, are further studied. The angular momentum M of the oscillators can take at most three values M=p-1,p,p+1, which are either all integers or all half-integers. The coordinates anticommute with each other. Depending on the state space the energy spectrum can coincide or can be essentially different from those of the canonical oscillator. The ground state is in general degenerated. 
  In this article we analyze a two dimensional lattice gauge theory based on a quantum group.The algebra generated by gauge fields is the lattice algebra introduced recently by A.Yu.Alekseev,H.Grosse and V.Schomerus we define and study wilson loops and compute explicitely the partition function on any Riemann surface. This theory appears to be related to Chern-Simons Theory. 
  The quark loop contribution to the reggeon-reggeon-gluon vertex is calculated in QCD, where the reggeon is the reggeized gluon. Compared with the vertex in the Born approximation, this contribution exhibits a new spin structure as well as the gluon loop one. A remarkable but not complete cancellation between gluon and quark contributions to this new spin structure takes place for the case of three massless quark flavours. 
  The non-perturbative mapping between different Quantum Field Theories and other features of two-dimensional massive integrable models are discussed by using the Form Factor approach. The computation of ultraviolet data associated to the massive regime is illustrated by taking as an example the scattering theory of the Ising Model with boundary. 
  We construct all instantons for the \nlsig\ on a cylinder, known not to exist on a finite time interval. We show that the widest instantons go through sphalerons. A re-interpretation of moduli-space transforms the scale parameter $\rho$ to a boundary condition in time. This may give a handle on the $\rho\rightarrow0$ divergent instanton gas. 
  We investigate the Ward identities of the $\W_{\infty}$ symmetry in the super-Liouville theory coupled to the super-conformal matter of central charge ${\hat c}_M = 1-2(p-q)^2 /pq$. The theory is classified into two chiralities. For the positive chirality, all gravitationally dressed scaling operators are generated from the $q-1$ gravitational primaries by acting one of the ring generators in the R-sector on them repeatedly. After fixing the normalizations of the dressed scaling operators, we find that the Ward identities are expressed in the form of the {\it usual} $\W_q$ algebra constraints as in the bosonic case: $\W^{(k+1)}_n \tau =0$, $(k=1,\cdots,q-1 ;~ n \in {\bf Z}_{\geq 1-k})$, where the equations for even and odd $n$ come from the currents in the NS- and the R-sector respectively. The non-linear terms come from the anomalous contributions at the boundaries of moduli space. The negative chirality is defined by interchanging the roles of $p$ and $q$. Then we get the $\W_p $ algebra constraints. 
  Reduction of the left regular representation of quantum algebra $sl_q(3)$ is studied and ~$q$-difference intertwining operators are constructed. The irreducible representations correspond to the spaces of local sections of certain line bundles over the q-flag manifold. 
  The 2D gravity described by the action which is an arbitrary function of the scalar curvature $f(R)$ is considered. The classical vacuum solutions are analyzed. The one-loop renormalizability is studied. For the function $f=R \ln R$ the model coupled with scalar (conformal) matter is exactly integrated and is shown to describe a black hole of the 2D dilaton gravity type. The influence of one-loop quantum corrections on a classical black hole configuration is studied by including the Liouville-Polyakov term. The resulting model turns out to be exactly solvable. The general solution is analyzed and shown to be free from the space-time singularities for a certain number of scalar fields. 
  The BRST transformations for gravity with torsion including Weyl symmetry are discussed by using the so-called Maurer-Cartan horizontality conditions. Also the coupling of scalar matter fields to gravity is incorporated in this analysis. With the help of an operator $\d$ which allows to decompose the exterior space-time derivative as a BRST commutator we solve the Wess-Zumino consistency condition corresponding to invariant Lagrangians and anomalies for the cases with and without Weyl symmetry. 
  The matrix model of random surfaces with c = inf. has recently been solved and found to be identical to a random surface coupled to a q-states Potts model with q = inf. The mean field-like solution exhibits a novel type of tree structure. The natural question is, down to which--if any--finite values of c and q does this behavior persist? In this work we develop, for the Potts model, an expansion in the fluctuations about the q = inf. mean field solution. In the lowest--cubic--non-trivial order in this expansion the corrections to mean field theory can be given a nice interpretation in terms of structures (trees and ``galaxies'') of spin clusters. When q drops below a finite q_c, the galaxies overwhelm the trees at all temperatures, thus suppressing mean field behavior. Thereafter the phase diagram resembles that of the Ising model, q=2. 
  It is shown that the perturbative expansions of the correlation functions of a relativistic quantum field theory at finite temperature are uniquely determined by the equations of motion and standard axiomatic requirements, including the KMS condition. An explicit expression as a sum over generalized Feynman graphs is derived. The canonical formalism is not used, and the derivation proceeds from the beginning in the thermodynamic limit. No doubling of fields is invoked. An unsolved problem concerning existence of these perturbative expressions is pointed out. 
  We construct the exponentials of the Liouville field with continuous powers within the operator approach. Their chiral decomposition is realized using the explicit Coulomb-gas operators we introduced earlier. {}From the quantum-group viewpoint, they are related to semi-infinite highest or lowest weight representations with continuous spins. The Liouville field itself is defined, and the canonical commutation relations verified, as well as the validity of the quantum Liouville field equations.   In a second part, both screening charges are considered. The braiding of the chiral components is derived and shown to agree with the ansatz of a parallel paper of J.-L. G. and Roussel: for continuous spins the quantum group structure $U_q(sl(2)) \odot U_{\qhat}(sl(2))$ is a non trivial extension of $U_q(sl(2))$ and $U_{\qhat}(sl(2))$. We construct the corresponding generalized exponentials and the generalized Liouville field. 
  The form factors of the flavor changing vector current between a $\bar q Q$-meson and the lightest $\bar q q$ (pseudoscalar-)meson are computed exactly and explicitly in the 't~Hooft model (planar QCD in $1+1$ dimensions) in the limit that the mass of $q$-quark vanishes. 
  We present a systematic technique to construct solutions to the Yang-Baxter equation which depend not only on a spectral parameter but in addition on further continuous parameters. These extra parameters enter the Yang-Baxter equation in a similar way to the spectral parameter but in a non-additive form. We exploit the fact that quantum non-compact algebras such as $U_q(su(1,1))$ and type-I quantum superalgebras such as $U_q(gl(1|1))$ and $U_q(gl(2|1))$ are known to admit non-trivial one-parameter families of infinite-dimensional and finite dimensional irreps, respectively, even for generic $q$. We develop a technique for constructing the corresponding spectral-dependent R-matrices. As examples we work out the the $R$-matrices for the three quantum algebras mentioned above in certain representations. 
  Second quantization is revisited and creation and annihilation operators areshown to be related, on the same footing both to the algebra h(1), and to the superalgebra osp(1|2) that are shown to be both compatible with Bose and Fermi statistics.   The two algebras are completely equivalent in the one-mode sector but, because of grading of osp(1|2), differ in the many-particle case.   The same scheme is straightforwardly extended to the quantum case h_q(1) and osp_q(1|2). 
  We investigate the functional determinant of the laplacian on piece-wise flat two-dimensional surfaces, with conical singularities in the interior and/or corners on the boundary. Our results extend earlier investigations of the determinants on smooth surfaces with smooth boundaries. The differences to the smooth case are: a) different ``interaction energies'' between pairs of conical singularities than one would expect from a naive extrapolation of the results for a smooth surface; and b) ``self-energies'' of the singularities. 
  The edge states of a sample displaying the quantum Hall effect (QHE) can be described by a 1+1 dimensional (conformal) field theory of $d$ massless scalar fields taking values on a $d$-dimensional torus. It is known from the work of Naculich, Frohlich et al.\@ and others that the requirement of chirality of currents in this \underline{scalar} field theory implies the Schwinger anomaly in the presence of an electric field, the anomaly coefficient being related in a specific way to Hall conducvivity. The latter can take only certain restricted values with odd denominators if the theory admits fermionic states. We show that the duality symmetry under the $O(d,d;{\bf Z})$ group of the free theory transforms the Hall conductivity in a well-defined way and relates integer and fractional QHE's. This means, in particular, that the edge spectra for dually related Hall conductivities are identical, a prediction which may be experimentally testable. We also show that Haldane's hierarchy as well as certain of Jain's fractions can be reproduced from the Laughlin fractions using the duality transformations. We thus find a framework for a unified description of the QHE's occurring at different fractions. We also give a derivation of the wave functions for fractions in Haldane's hierarchy. 
  We review the classical boson-fermion correspondence in the context of the $\widehat{sl(2)}$ current algebra at level 2. This particular algebra is ideal to exhibit this correspondence because it can be realized either in terms of three real bosonic fields or in terms of one real and one complex fermionic fields. We also derive a fermionic realization of the quantum current algebra $U_q(\widehat{sl(2)})$ at level 2 and by comparing this realization with the existing bosonic one we extend the classical correspondence to the quantum case. 
  Geometrical form of the one-loop divergences induced by conical singularities of background manifolds is studied. To this aim the heat kernel asymptotic expansion on spaces having the structure $C_{\alpha}\times \Sigma$ near singular surface $\Sigma$ is analysed. Surface corrections to standard second and third heat coefficients are obtained explicitly in terms of angle $\alpha$ of a cone $C_{\alpha}$ and components of the Riemann tensor. These results are compared to ones to be already known for some particular cases. Physical aspects of the surface divergences are shortly discussed. 
  It was previously shown that at critical central charge, $N$-extended superstrings can be embedded in $(N+1)$-extended superstrings. In other words, $(N=0,c=26)\to (N=1,c=15)\to (N=2,c=6)\to (N=3,c=0) \to (N=4,c=0) $. In this paper, we show that similar embeddings are also possible for $N$-extended superstrings at non-critical central charge. For any $x$, the embedding is $(N=0,c=26+x) \to (N=1,c=15+x) \to (N=2,c=6+x) \to (N=3,c=x) \to (N=4,c=x)$. As was conjectured by Vafa, the $(N=2,c=9) \to (N=3,c=3)$ embedding can be used to prove that $N=0$ topological strings are special vaccua of N=1 topological strings. 
  We present a toy model study of the high temperature deconfining transition in Yang-Mills theory as a breakdown of the confinement condition proposed by Kugo and Ojima. Our toy model is a kind of topological field theory obtained from the Yang-Mills theory by taking the limit of vanishing gauge coupling constant $g_{\rm YM}\to 0$, and therefore the gauge field $A_\mu$ is constrained to the pure-gauge configuration $A_\mu=g^{\dagger}\partial_\mu g$. At zero temperature this model has been known to satisfy the confinement condition of Kugo and Ojima which requires the absence of the massless Nambu-Goldstone-like mode coupled to the BRST-exact color current. In the finite temperature case based on the real-time formalism, our model in 3+1 dimensions is reduced, by the Parisi-Sourlas mechanism, to the ``sum'' of chiral models in 1+1 dimensions with various boundary conditions of the group element $g(t,x)$ at the ends of the time contour. We analyze the effective potential of the $SU(2)$ model and find that the deconfining transition in fact occurs due to the contribution of the sectors with non-periodic boundary conditions. 
  We relate the Teichmuller spaces obtained by Hitchin to the Teichmuller spaces of $WA_{n}$-gravity. The relationship of this space to $W$-gravity is obtained by identifying the flat $PSL(n+1,{\BR})$ connections of Hitchin to generalised vielbeins and connections. This is explicitly demonstrated for $WA_2=W_3$ gravity. We show how $W$-diffeomorphisms are obtained in this formulation. We find that particular combinations of the generalised connection play the role of projective connections. We thus obtain $W$-diffeomorphisms in a geometric fashion without invoking the presence of matter fields. This description in terms of vielbeins naturally provides the measure for the gravity sector in the Polyakov path integral for $W$-strings. 
  We consider the abelian vector-field models in the presence of the Wess-Zumino interaction with the pseudoscalar matter. The occurence of the dynamic breaking of Lorentz symmetry at classical and one-loop level is described for massless and massive vector fields. This phenomenon appears to be the non-perturbative counterpart of the perturbative renormalizability and/or unitarity breaking in the chiral gauge theories. 
  We study systems without quenched disorder with a complex landscape, and we use replica symmetry theory to describe them. We discuss the Golay-Bernasconi-Derrida approximation of the low autocorrelation model, and we reconstruct it by using replica calculations. Then we consider the full model, its low $T$ properties (with the help of number theory) and a Hartree-Fock resummation of the high-temperature series. We show that replica theory allows to solve the model in the high $T$ phase. Our solution is based on one-link integral techniques, and is based on substituting a Fourier transform with a generic unitary transformation. We discuss this approach as a powerful tool to describe systems with a complex landscape in the absence of quenched disorder. 
  We consider possible definitions of physical variables in QED. We demonstrate that the condition $\partial_i A_i$$=0$ is the most convenient one because it leads to path integral over physical components with local action. However, other choices, as $A_3=0$, are also possible. The standard expression for configuration space path integral in $A_3=0$ gauge is obtained starting with reduced phase space formulation. Contrary to the claims of the paper [M.Lavelle and D.McMullan,Phys. Lett. B316 (1993)172] the $A_3=0$ gauge is not overconstrained. 
  We construct representations $\hat\pi_{\br}$ of the quantum algebra $U_q(sl(n))$ labelled by $n-1$ complex numbers $r_i$ and acting in the space of formal power series of $n(n-1)/2$ non-commuting variables. These variables generate a flag manifold of the matrix quantum group $SL_q(n)$ which is dual to $U_q(sl(n))$. The conditions for reducibility of $\hat\pi_{\br}$ and the procedure for the construction of the $q$ - difference intertwining operators are given. The representations and $q$ - difference intertwining operators are given in the most explicit form for $n=3$. In the Note Added some general results for arbitrary $n$ are given. 
  Perturbing usual type B topological matter with vector $(0,1)$-forms we find a topological theory which contains explicitly Kodaira-Spencer deformation theory. It is shown that, in genus zero, three-point correlation functions give the Yukawa couplings for a generic point in the moduli space of complex structures. This generalization of type B topological matter seems to be the correct framework to understand mirror symmetry in terms of two-dimensional topological field theories. 
  The relativistic distribution for indistinguishable events is considered in the mass-shell limit $m^2\cong M^2,$ where $M$ is a given intrinsic property of the events. The characteristic thermodynamic quantities are calculated and subject to the zero-mass and the high-temperature limits. The results are shown to be in agreement with the corresponding expressions of an on-mass-shell relativistic kinetic theory. The Galilean limit $c\rightarrow \infty ,$ which coincides in form with the low-temperature limit, is considered. The theory is shown to pass over to a nonrelativistic statistical mechanics of indistinguishable particles. 
  We show that Galois theory of cyclotomic number fields provides a powerful tool to construct systematically integer-valued matrices commuting with the modular matrix S, as well as automorphisms of the fusion rules. Both of these prescriptions allow the construction of modular invariants and offer new insight in the structure of known exceptional invariants. 
  After reviewing the general properties of zero-energy quantum states, we give the explicit solutions of the \seq with $E=0$ for the class of potentials $V=-|\gamma|/r^{\nu}$, where $-\infty < \nu < \infty$. For $\nu > 2$, these solutions are normalizable and correspond to bound states, if the angular momentum quantum number $l>0$. [These states are normalizable, even for $l=0$, if we increase the space dimension, $D$, beyond 4; i.e. for $D>4$.] For $\nu <-2$ the above solutions, although unbound, are normalizable. This is true even though the corresponding potentials are repulsive for all $r$. We discuss the physics of these unusual effects. 
  In supersymmetric theories, one can obtain striking results and insights by exploiting the fact that the superpotential and the gauge coupling function are holomorphic functions of the model parameters. The precise meaning of this holomorphy is subtle, and has been explained most clearly by Shifman and Vainshtein, who have stressed the role of the Wilsonian effective action. In this note, we elaborate on the Shifman-Vainshtein program, applying it to examples in grand unification, supersymmetric QCD and string theory. We stress that among the ``model parameters" are the cutoffs used to define the Wilsonian action itself, and that generically these must be defined in a field-dependent manner to obtain holomorphic results. 
  We find all the classical solutions (minimal surfaces) of open or closed strings in {\it any} two dimensional curved spacetime. As examples we consider the SL(2,R)/R two dimensional black hole, and any 4D black hole in the Schwarzschild family, provided the motion is restricted to the time-radial components. The solutions, which describe longitudinaly oscillating folded strings (radial oscillations in 4D), must be given in lattice-like patches of the worldsheet, and a transfer operation analogous to a transfer matrix determines the future evolution. Then the swallowing of a string by a black hole is analyzed. We find several new features that are not shared by particle motions. The most surprizing effect is the tunneling of the string into the bare singularity region that lies beyond the black hole that is classically forbidden to particles. 
  The `classical' model for a massive spinning particle, which was recently proposed, is derived from the isotropic rotator model. Through this derivation, we note that the spin can be understood as the relativistic extension of the isotropic rotator. Furthermore, the variables $t_\m$ corresponding to the $\p^*$ of the `pseudo-classical' model, are necessary for the covariant formulation. The dynamical term for these extra variables is naturally obtained and the meaning of the constraint term $p^\s\L_{\s\n}+mt_\n =0$, which was recently shown to give `quasi-supersymmetry', is clarified. 
  Using a simple identity between various partial derivatives of the energy of the vector model in 0+0 dimensions, we derive explicit results for the coefficients of the large N expansion of the model. These coefficients are functions in a variable $\rho^2$, which is the expectation value of the two point function in the limit $N=\infty$. These functions are analytic and have only one (multiple) pole in $\rho^2$. We show to all orders that these expressions obey a given general formula. Using this formula it is possible to derive the double scaling limit in an alternative way. All the results obtained for the double scaling limit agree with earlier calculations. (to be published in Physics Letters B) 
  Recently an algorithm was found by means of which one can calculate terms at arbitrary oscillator level in the four-Ramond vertex obtained by sewing. Here we show that this algorithm is applicable also to the case of ${\bf Z}_2$-twisted scalars and derive the full propagator for scalars on the Riemann sphere with two branch cuts. The relation to similar results previously derived in the literature by other means is discussed briefly. 
  It is shown that the BRST path integral for reducible gauge theories, with appropriate boundary conditions on the ghosts, is a solution of the constraint equations. This is done by relating the BRST path integral to the kernel of the evolution operator projected on the physical subspace. 
  The structure and properties of possible $q$-Minkowski spaces is discussed, and the corresponding non-commutative differential calculi are developed in detail and compared with already existing proposals. This is done by stressing its covariance properties as described by appropriate reflection equations. Some isomorphisms among the space-time and derivative algebras are demonstrated, and their representations are described briefly. Finally, some physical consequences and open problems are discussed. 
  Black hole entropy is studied for an exactly solvable model of two-dimensional gravity\cite{rst1}, using recently developed Noether charge techniques\cite{wald1}. This latter approach is extended to accomodate the non-local form of the semiclassical effective action. In the two-dimensional model, the final black hole entropy can be expressed as a local quantity evaluated on the horizon. This entropy is shown to satisfy an increase theorem on either the global or apparent horizon of a two-dimensional black hole. 
  We study the thermodynamics of the relativistic Quantum Field Theory of massive fermions in three space-time dimensions coupled to an Abelian Maxwell-Chern-Simons gauge field. We evaluate the specific heat at finite temperature and density and find that the variation with the statistical angle is consistent with the non-relativistic ideas on generalized statistics. 
  Gauge invariant complex covariant actions for superparticles are derived from the field equations for the chiral superfields in a precise manner. The massive and massless cases in four dimensions are treated both free and in interaction with an external super Maxwell field. By means of a generalized BRST quantization these complex actions are related to real actions with second class constraints which are new in some cases. 
  Representations of $SO(5)_{q}$ are constructed explicitly on the Chevalley basis for all $q$, generic and root of unity. Matrix elements of the generators are obtained for all representations depending on three variable indices, the maximal number being 4. A prescription for contraction is given such that a complete Hopf algebra is immediately obtained for the non-semisimple contracted case. For $q$ a root of unity the periodic representations for $SO(5)_{q}$ and the contracted algebra are obtained directly in the "fractional part" formalism which unifies the treatments for the generic and root of unity cases. The $q$-deformed quadratic Casimir operator is explicitly evaluated for the representations presented. 
  In the generalized Pauli-Villars regularization of chiral gauge theory proposed by Frolov and Slavnov , it is important to specify how to sum the contributions from an infinite number of regulator fields. It is shown that an explicit sum of contributions from an infinite number of fields in anomaly-free gauge theory essentially results in a specific choice of regulator in the past formulation of covariant anomalies. We show this correspondence by reformulating the generalized Pauli- Villars regularization as a regularization of composite current operators. We thus naturally understand why the covariant fermion number anomaly in the Weinberg-Salam theory is reproduced in the generalized Pauli-Villars regularization. A salient feature of the covariant regularization,which is not implemented in the lagrangian level in general but works for any chiral theory and gives rise to covariant anomalies , is that it spoils the Bose symmetry in anomalous theory. The covariant regularization however preserves the Bose symmetry as well as gauge invariance in anomaly-free gauge theory. 
  An example of a finite dimensional factorizable ribbon Hopf C-algebra is given by a quotient H=u_q(g) of the quantized universal enveloping algebra U_q(g) at a root of unity q of odd degree. The mapping class group M_{g,1} of a surface of genus g with one hole projectively acts by automorphisms in the H-module H^{*\otimes g}, if H^* is endowed with the coadjoint H-module structure. There exists a projective representation of the mapping class group M_{g,n} of a surface of genus g with n holes labelled by finite dimensional H-modules X_1,...,X_n in the vector space Hom_H(X_1\otimes...\otimes X_n,H^{*\otimes g}). An invariant of closed oriented 3-manifolds is constructed. Modifications of these constructions for a class of ribbon Hopf algebras satisfying weaker conditions than factorizability (including most of u_q(g) at roots of unity q of even degree) are described. The results are motivated by CFT. 
  A category N of labeled (oriented) trivalent graphs (nets) or ribbon graphs is extended by new generators called fusing, braiding, twist and switch with relations which can be called Moore--Seiberg relations. A functor to N is constructed from the category Surf of oriented surfaces with labeled boundary and their homeomorphisms. Given an (eventually non-semisimple) k-linear abelian ribbon braided category C with some finiteness conditions we construct a functor from a central extension of N with the set of labels ObC to k-vector spaces. Composing the functors we get a modular functor from a central extension of Surf to k-vector spaces. This is a mathematical paper which explains how to get proofs for its hep-th companion paper, which should be read first. Complete proofs are not given here. (Talk at Second Gauss Simposium, Munich, August 1993.) 
  The relation between the spin and the mass of an infinite number of particles in a $q$-deformed dual string theory is studied. For the deformation parameter $q$ a root of unity, in addition to the relation of such values of $q$ with the rational conformal field theory, the Fock space of each oscillator mode in the Fubini-Veneziano operator formulation becomes truncated. Thus, based on general physical grounds, the resulting spin-(mass)$^2$ relation is expected to be below the usual linear trajectory. For such specific values of $q$, we find that the linear Regge trajectory turns into a square-root trajectory as the mass increases. 
  Some cosmological consequences of including the adequate conserved quantities in the density matrix of the electroweak theory are investigated. Several arguments against including the charges associated to the spontaneously broken symmetry are presented. Special attention is focused on the phenomenon of $W$-boson condensation and its interplay with the phase transition for the symmetry restoration is considered. The emerging cosmological implications, such as on the baryon and lepton number densities, are of interest. 
  We use the Regge-Teitelboim method to treat surface integrals in gauge theories to find global charges in Chern-Simons theory. We derive the affine and Virasoro generators as global charges associated with symmetries of the boundary. The role of boundary conditions is clarified. We prove that for diffeomorphisms that do not preserve the boundary there is a classical contribution to the central charge in the Virasoro algebra. The example of anti-de Sitter 2+1 gravity is considered in detail. 
  Field Theory on Event Symmetric space-time is constructed using the gauge group of discrete open strings. Models with invariant actions can be viewed as natural extensions of Matrix Models. The objective is to find a fundamental non-perturbative pre-theory for superstrings. 
  We study a generalization of the Gaussian effective potential for self-interacting scalar fields in one and two spatial dimensions. We compute the two-loop corrections and discuss the renormalization of the generalized Gaussian effective potential. 
  We derive a model of constrained topological gravity, a theory recently introduced by us through the twist of N=2 Liouville theory, starting from the general BRST algebra and imposing the moduli space constraint as a gauge fixing. To do this, it is necessary to introduce a formalism that allows a careful treatment of the global and the local degrees of freedom of the fields. Surprisingly, the moduli space constraint arises from the simplest and most natural gauge-fermion ({\sl antighost} $\times$ {\sl Lagrange multiplier}), confirming the previous results. The simplified technical set-up provides a deeper understanding for constrained topological gravity and a convenient framework for future investigations, like the matter coupling and the analysis of the effects of the constraint on the holomorphic anomaly. 
  In this talk we review some results concerning a mechanism for reducing the moduli space of a topological field theory to a proper submanifold of the ordinary moduli space. Such mechanism is explicitly realized in the example of constrained topological gravity, obtained by topologically twisting the N=2 Liouville theory. 
  Discretized light-cone quantization of (3+1)-dimensional electrodynamics is discussed, with careful attention paid to the interplay between gauge choice and boundary conditions. In the zero longitudinal momentum sector of the theory a general gauge fixing is performed, and the corresponding relations that determine the zero modes of the gauge field are obtained. One particularly natural gauge choice in the zero mode sector is identified, for which the constraint relations are simplest and the fields may be taken to satisfy the usual canonical commutation relations. The constraints are solved in perturbation theory, and the Poincar\'e generators $P^\mu$ are constructed. The effect of the zero mode contributions on the one-loop fermion self-energy is studied. 
  A new $(1,1)$-dimensional super vector bundle which exists on any super Riemann surface is described. Cross-sections of this bundle provide a new class of fields on a super Riemann surface which closely resemble holomorphic functions on a super Riemann surface, but which (in contrast to the case with holomorphic functions) form spaces which have a well defined dimension which does not change as odd moduli become non-zero. 
  We consider $SU(N)$ Yang-Mills theories in $(2n+1)$-dimensional Euclidean spacetime, where $N\geq n+1$, coupled to an even flavour number of Dirac fermions. After integration over the fermionic degrees of freedom the wave functional for the gauge field inherits a non-trivial $U(1)$-connection which we compute in the limit of infinite fermion mass. Its Chern-class turns out to be just half the flavour number so that the wave functional now becomes a section in a non-trivial complex line bundle. The topological origin of this phenomenon is explained in both the Lagrangean and the Hamiltonian picture. 
  A general functional definition of the infinite dimensional quantum $R$-matrix satisfying the Yang-Baxter equation is given. A procedure for the extracting a finite dimensional $R$-matrix from the general definition is demonstrated in a particular case when the group $SU(2)$ takes place. 
  We investigate the spin $1/2$ fermions on quantum two spheres. It is shown that the wave functions of fermions and a Dirac Operator on quantum two spheres can be constructed in a manifestly covariant way under the quantum group $SU(2)_q$. The concept of total angular momentum and chirality can be expressed by using $q$-analog of Pauli-matrices and appropriate commutation relations. 
  The minimal (reduced) and extended canonical formulations for (2+1)-dimensional fractional spin particles are considered. We investigate the relationship between them, clearing up the meaning of the coordinates for such particles, and analyse the related question of correlation between spin and momentum. The classical lagrangian corresponding to the extended canonical formulation is constructed, and its gauge symmetries are identified. 
  We reply a Comment (hep-th/xxx) on our recent paper published in Phys Rev Lett 72 (1994) 2527-2530. We point out that the author of the Comment overlooks the highly non-trivial Chern-Simons interactions and his hand-waving arguments are totally groundless. 
  We propose a new mwthod of constructing 4D-TQFTs. The method uses a new type of algebraic structure called a Hopf Category. We also outline the construction of a family of Hopf categories related to the quantum groups, using the canonical bases. 
  The paper describes a dynamical analogy for the renormalization group which leads to insights into its structure. 
  In this note we consider a stringy description of black hole horizon. We start with a nonlinear sigma model defined on a two dimensional Euclidean surface with background Rindler metric. By solving the field equations, we show that to the leading order the Bekenstein-Hawking formula of black hole entropy can be produced. We also point out a relation between the present formalism and the 'tHooft formalism. 
  We report work done with T. Jayaraman (hep-th/9405146) in this talk. In a recent paper, Hitchin introduced generalisations of the Teichmuller space of Riemann surfaces. We relate these spaces to the Teichmuller spaces of W-gravity. We show how this provides a covariant description of W-gravity and naturally leads to a Polyakov path integral prescription for W-strings. (Talk presented at the International Colloquium on Modern Quantum Field Theory II at the Tata Institute, Bombay during Jan. 5-11, 1994, to appear in the proceedings) 
  The theory of reheating of the Universe after inflation is developed. The transition from inflation to the hot Universe turns out to be strongly model-dependent and typically consists of several stages. Immediately after inflation the field $\phi$ begins rapidly rolling towards the minimum of its effective potential. Contrary to some earlier expectations, particle production during this stage does not lead to the appearance of an extra friction term $\Gamma\dot\phi$ in the equation of motion of the field $\phi$. Reheating becomes efficient only at the next stage, when the field $\phi$ rapidly oscillates near the minimum of its effective potential. We have found that typically in the beginning of this stage the classical inflaton field $\phi$ very rapidly (explosively) decays into $\phi$-particles or into other bosons due to broad parametric resonance. This stage cannot be described by the standard elementary approach to reheating based on perturbation theory. The bosons produced at this stage, as well as some part of the classical field $\phi$ which survives the stage of explosive reheating, should further decay into other particles, which eventually become thermalized. The last stages of decay can be described in terms of perturbation theory. Complete reheating is possible only in those theories where a single massive $\phi$-particle can decay into other particles. This imposes strong constraints on the structure of inflationary models. On the other hand, this means that a scalar field can be a cold dark matter candidate even if it is strongly coupled to other fields. 
  We study some conditions for the hierarchy $m_{3/2} << M_P$ to occur naturally in a generic effective supergravity theory. Absence of fine-tuning and perturbative calculability require that the effective potential has a sliding gravitino mass and vanishing cosmological constant, up to ${\cal O}(m_{3/2}^4)$ corrections. In particular, cancellation of quadratically divergent contributions to the one-loop effective potential should take place, including the `hidden sector' of the theory. We show that these conditions can be met in the effective supergravities derived from four-dimensional superstrings, with supersymmetry broken either at the string tree level via compactification, or by non-perturbative effects such as gaugino condensation. A crucial role is played by some approximate scaling symmetries, which are remnants of discrete target-space dualities in the large moduli limit. We derive explicit formulae for the soft breaking terms arising from this class of `large hierarchy compatible' (LHC) supergravities. 
  We study the $O(N)$ nonlinear $\sigma$ model on a three-dimensional compact space $S^1 \times S^2$ (of radii $L$ and $R$ respectively) by means of large $N$ expansion, focusing on the finite size effects and conformal symmetries of this model at the critical point. We evaluate the correlation length and the Casimir energy of this model and study their dependence on $L$ and $R$. We examine the modular transformation properties of the partition function, and study the dependence of the specific heat on the mass gap in view of possible extension of the $C-$theorem to three dimensions. 
  We investigate the Exact Renormalization Group (ERG) description of ($Z_2$ invariant) one-component scalar field theory, in the approximation in which all momentum dependence is discarded in the effective vertices. In this context we show how one can perform a systematic search for non-perturbative continuum limits without making any assumption about the form of the lagrangian. Concentrating on the non-perturbative three dimensional Wilson fixed point, we then show that the sequence of truncations $n=2,3,\dots$, obtained by expanding about the field $\varphi=0$ and discarding all powers $\varphi^{2n+2}$ and higher, yields solutions that at first converge to the answer obtained without truncation, but then cease to further converge beyond a certain point. No completely reliable method exists to reject the many spurious solutions that are also found. These properties are explained in terms of the analytic behaviour of the untruncated solutions -- which we describe in some detail. 
  Electromagnetism would be a ``more unified'' theory if there were elementary magnetic monopoles and/or particles with both electric and magnetic charges (dyons). I discuss the simplest possibilities for the addition of these entities onto the Standard Model, and their empirical consequences. Lower limits on the masses of monopoles and dyons stemming from their quantum effects on current observables turn out to be much stronger than the existing limits from direct searches. Anomalies in the three-photon decay of the $Z$ constitute good specific signatures for monopoles or dyons. $T$-odd observables in the $e^+e^-\!\rightarrow\! W^+W^-$ process are signatures for dyons, but they are severely constrained by existing data. The subjects of monopolium, monopole cosmology and non-elementary monopoles are also discussed. 
  The main properties of indefinite Kac-Moody and Borcherds algebras, considered in a unified way as Lorentzian algebras, are reviewed. The connection with the conformal field theory of the vertex operator construction is discussed. By the folding procedure a class of subalgebras is obtained. 
  The vector system of linear differential equations for a field with arbitrary fractional spin is proposed using infinite-dimensional half-bounded unitary representations of the $\overline{SL(2,R)}$ group. In the case of $(2j+1)$-dimensional nonunitary representations of that group, $0<2j\in Z$, they are transformed into equations for spin-$j$ fields. A local gauge symmetry associated to the vector system of equations is identified and the simplest gauge invariant field action, leading to these equations, is constructed. 
  Yang-Mills models with compact gauge group coupled to matter fields are considered. The general tools developed in a companion paper are applied to compute the local cohomology of the BRST differential $s$ modulo the exterior spacetime derivative $d$ for all values of the ghost number, in the space of polynomials in the fields, the ghosts, the antifields (=sources for the BRST variations) and their derivatives. New solutions to the consistency conditions $sa+db=0$ depending non trivially on the antifields are exhibited. For a semi-simple gauge group, however, these new solutions arise only at ghost number two or higher. Thus at ghost number zero or one, the inclusion of the antifields does not bring in new solutions to the consistency condition $sa+db=0$ besides the already known ones. The analysis does not use power counting and is purely cohomological. It can be easily extended to more general actions containing higher derivatives of the curvature, or Chern-Simons terms. 
  The tensionless limit of classical string theory may be formulated as a topological theory on the world-sheet. A vector density carries geometrical information in place of an internal metric. It is found that path-integral quantization allows for the definition of several, possibly inequivalent quantum theories. String amplitudes are constructed from vector densities with zeroes for each in- or out-going string. It is shown that independence of a metric in quantum mechanical amplitudes implies that the dependence on such vector density zeroes is purely topological. For example, there is no need for integration over their world-sheet positions. 
  We discuss some physical aspects of our Liouville approach to non-critical strings, including the emergence of a microscopic arrow of time, effective field theories as classical ``pointer'' states in theory space, $CPT$ violation and the possible apparent non-conservation of angular momentum. We also review the application of a phenomenological parametrization of this formalism to the neutral kaon system. 
  The ground state of the $\SO(2n)_{1} \times \SO(2n)_{1} \over \SO(2n)_{2}$ coset theories, perturbed by the $\phi^{id,id}_{adj}$ operator and those of the sine-Gordon theory, for special values of the coupling constant in the attracting regime, is the same. In the first part of this paper we extend these results to the $\SO(2n-1)$ cases. In the second part, we analyze the Algebraic Bethe Ansatz procedure for special points in the repulsive region. We find a one-to-one ``duality'' correspondence between these theories and those studied in the first part of the paper. We use the gluing procedure at the massive node proposed by Fendley and Intriligator in order to obtain the TBA systems for the generalized parafermionic supersymmetric sine-Gordon model. In the third part we propose the TBA equations for the whole class of perturbed coset models $G_k \times G_l \over G_{k+l}$ with the operator $\phi^{id,id}_{adj}$ and $G$ a non-simply-laced group generated by one of the $\G_2,\F_4,\B_n,\C_n$ algebras. 
  Target space symmetries are studied for orbifold compactified string theories containing Wilson line background fields. The symmetries determined are for those moduli which contribute to the string loop threshold corrections of the gauge coupling constants. The groups found are subgroups of the modular group $PSL(2, Z)$ and depend on the choice of discrete Wilson lines and the shape of the underlying six-dimensional lattice. 
  We review the insights into black hole entropy that arise from the formulation of gravitation theory in terms of dimensional continuation. The role of the horizon area and the deficit angle of a conical singularity at the horizon as canonically conjugate dynamical variables is analyzed. The path integral and the extension of the Wheeler-De Witt equation for black holes are discussed. 
  Based on the formulation of Drinfel'd, Chari, and Pressley, a technique to analyze the structure of tensor products of the Yangian algebra representations is presented. We then apply the results to the $S$-matrix theory of the $G\tensor G$-invariant nonlinear $\sigma$-model ($G$-principal chiral model) by Ogievetsky, Reshetikhin, and Wiegmann. We show how the physical data such as mass formula, fusion angle, and the spins of integrals of motion can be extracted from the Yangian highest weight representations. 
  We introduce fusion $U_q(G^{(1)}_2)$ vertex models related to fundamental representations. The eigenvalues of their row to row transfer matrices are derived through analytic Bethe ans{\"a}tze. By combining these results with our previous studies on functional relations among transfer matrices(the $T$-system), we conjecture explicit eigenvalues for a wide class of fusion models. These results can be neatly expressed in terms of a Yangian analogue of the Young tableaux. 
  It is shown that a static $(1+3)$ anti-de Sitter metric defines, in a natural way, a relativistic harmonic oscillator in Minkowski space. The quantum theory can be solved exactly and leads to wave functions having a significantly different behaviour with respect to the non-relativistic ones. The energy spectrum coincides, up to the ground state energy, with that of the non-relativistic oscillator. 
  A functorial derivation is presented of a heat-kernel expansion coefficient on a manifold with a singular fixed point set of codimension two. The existence of an extrinsic curvature term is pointed out. 
  We study abelian gauge theories with anisotropic couplings in $4+D$ dimensions. A layered phase is present, in the absence as well as in the presence of fermions. A line of second order transitions separates the layered from the Coulomb phase, if $D\leq 3$. 
  A realistic analysis shows that constraining a quantomechanical system produces the effective dynamics to be coupled with {\sl abelian/non-abelian gauge fields} and {\sl quantum potentials} induced by the {\sl intrinsic} and {\sl extrinsic geometrical properties} of the constraint's surface.   This phenomenon is observable in the effective rotational motion of some simple polyatomic molecules. By considering specific examples it is shown that the effective Hamiltonians for the nuclear rotation of linear and symmetric top molecules are equivalent to that of a charged system moving in a background magnetic-monopole field. For spherical top molecules an explicit analytical expression of a non-abelian monopole-like field is found. Quantum potentials are also relevant for the description of rotovibrational interactions. 
  We show that the Zamolodchikov's and Polyakov-Bershadsky nonlinear algebras $W_3$ and $W_3^{(2)}$ can be embedded as subalgebras into some {\em linear} algebras with finite set of currents. Using these linear algebras we find new field realizations of $W_3^{(2)}$ and $W_3$ which could be a starting point for constructing new versions of $W$-string theories. We also reveal a number of hidden relationships between $W_3$ and $W_3^{(2)}$. We conjecture that similar linear algebras can exist for other $W$-algebras as well. 
  We find the explicit operatorial form of renormalon-type singularities in abelian gauge theory. Local operators of dimension six take care of the first U.V. renormalon, non local operators are needed for I.R. singularities. In the effective lagrangian constructed with these operators non local imaginary parts appearing in the usual perturbative expansion at large orders are cancelled. 
  The short-distance singularity of the product of a composite scalar field that deforms a field theory and an arbitrary composite field can be expressed geometrically by the beta functions, anomalous dimensions, and a connection on the theory space. Using this relation, we compute the connection perturbatively for the O(N) non-linear sigma model in two dimensions. We show that the connection becomes free of singularities at zero temperature only if we normalize the composite fields so that their correlation functions have well-defined limits at zero temperature. 
  We review various constructions of mirror symmetry in terms of Landau-Ginzburg orbifolds for arbitrary central charge $c$ and \CY\ hypersurfaces and complete intersections in toric varieties. In particular it is shown how the different techniques are related 
  Extreme 4-dimensional dilaton black holes embedded into 10-dimensional geometry are shown to be dual to the gravitational waves in string theory. The corresponding gravitational waves are the generalization of pp-fronted waves, called supersymmetric string waves. They are given by Brinkmann metric and the two-form field, without a dilaton. The non-diagonal part of the metric of the dual partner of the wave together with the two-form field correspond to the vector field in 4-dimensional geometry of the charged extreme black holes. 
  We study a three dimensional conformal field theory in terms of its partition function on arbitrary curved spaces. The large $N$ limit of the nonlinear sigma model at the non-trivial fixed point is shown to be an example of a conformal field theory, using zeta--function regularization. We compute the critical properties of this model in various spaces of constant curvature ($R^2 \times S^1$, $S^1\times S^1 \times R$, $S^2\times R$, $H^2\times R$, $S^1 \times S^1 \times S^1$ and $S^2 \times S^1$) and we argue that what distinguishes the different cases is not the Riemann curvature but the conformal class of the metric. In the case $H^2\times R$ (constant negative curvature), the $O(N)$ symmetry is spontaneously broken at the critical point. In the case $S^2\times R$ (constant positive curvature) we find that the free energy vanishes, consistent with conformal equivalence of this manifold to $R^3$, although the correlation length is finite. In the zero curvature cases, the correlation length is finite due to finite size effects. These results describe two dimensional quantum phase transitions or three dimensional classical ones. 
  In this note, we extend the string theoretic calculation of the black hole entropy, first performed by Susskind and Uglum, away from the infinite mass limit. It is shown that the result agrees with that obtained from the classical action of string theory, using the Noether charge method developed by Wald. Also shown in the process is the equivalence of two general techniques for finding black hole entropies-the Noether charge method, and the method of conical singularities. 
  The moduli associated with boundaries in a Riemann surface are parametrized by the positions and strengths of electric charges. This suggests a method for summing over orientable Riemann surfaces with Dirichlet boundary conditions on the embedding coordinates. A light-cone parameterization of such boundaries is also discussed. 
  A general definition of Chern-Simons actions in non-commutative geometry is proposed and illustrated in several examples. These are based on ``space-times'' which are products of even-dimensional, Riemannian spin manifolds by a discrete (two-point) set. If the *algebras of operators describing the non-commutative spaces are generated by functions over such ``space-times'' with values in certain Clifford algebras the Chern-Simons actions turn out to be the actions of topological gravity on the even-dimensional spin manifolds. By contrasting the space of field configurations in these examples in an appropriate manner one is able to extract dynamical actions from Chern-Simons actions. 
  An effective action for $N=1$, 2d-induced supergravity in the chiral gauge is obtained on a compact super Riemann surface without boundary and of arbitrary genus. 
  We recently obtained the conditions on the couplings of the general two-dimensional massive sigma-model required by (p,q)-supersymmetry. Here we compute the Poisson bracket algebra of the supersymmetry and central Noether charges, and show that the action is invariant under the automorphism group of this algebra. Surprisingly, for the (4,4) case the automorphism group is always a subgroup of SO(3), rather than SO(4). We also reanalyse the conditions for the (2,2) and (4,4) supersymmetry of the zero torsion models without assumptions about the central charge matrix. 
  A rather simple and non-technical exposition of our new approach to {\em Time, Quantum Physics, Black-Hole dynamics}, and {\em Cosmology}, based on non-critical string theory, is provided. A new fundamental principle, the {\em Procrustean Principle}, that catches the essence of our approach is postulated: the low-energy world is {\em unavoidably} an ``open" system due to the spontaneous truncation of the {\em delocalized, topological} string modes in continuous interaction with the low-lying-{\em localized} string modes. The origin of space-time, the expansion of the Universe, the entropy increase and accompanied irreversibility of time, as well as the collapse of the wavefunction are all very neatly tied together. Possible observable consequences include: quantum relaxation with time of the Universal, fundamental constants, like the velocity of light $c$ and the Planck constant $\hbar$ decreasing towards their asymptotic values, and the cosmological constant $\Lambda_C$ diminishing towards zero; possible violation of {\em CPT} invariance in the $K^0-\bar K^0$ system, possible apparent non-conservation of angular momentum, and possible loss of quantum coherence in SQUID-type experiments. 
  The fermion energy spectrum along paths which connect topologically distinct vacua in the Einstein-Yang-Mills theory passing through the gravitational sphaleron equilibrium solutions is investigated. 
  It is shown that the Coulomb correlation problem for a system of two electrons (two charged particles) in an external oscillator potential possesses a hidden $sl_2$-algebraic structure being one of recently-discovered quasi-exactly-solvable problems. The origin of existing exact solutions to this problem, recently discovered by several authors, is explained. A degeneracy of energies in electron-electron and electron-positron correlation problems is found. It manifests the first appearence of hidden $sl_2$-algebraic structure in atomic physics. 
  The invariants of 3-manifolds defined by Kuperberg for involutory Hopf algebras and those defined by the authors for spherical Hopf algebras are the same for Hopf algebras on which they are both defined. 
  We study the description of the $SU(2)$, level $k=1$, Wess-Zumino-Witten conformal field theory in terms of the modes of the spin-1/2 affine primary field $\phi^\alpha$. These are shown to satisfy generalized `canonical commutation relations', which we use to construct a basis of Hilbert space in terms of representations of the Yangian $Y(sl_2)$. Using this description, we explicitly derive so-called `fermionic representations' of the Virasoro characters, which were first conjectured by Kedem et al.~\cite{kedem}. We point out that similar results are expected for a wide class of rational conformal field theories. 
  The existence of nonsingular classical magnetic monopole solutions is usually understood in terms of topologically nontrivial Higgs field configurations. We show that finite energy magnetic monopole solutions also exist within a class of purely Abelian gauge theories containing charged vector mesons, even though the possibility of nontrivial topology does not even arise. provided that certain relationships among the parameters of the theory are satisfied. These solutions are singular if these relationships do not hold, but even then become meaningful once the theory is coupled to gravity, for they then give rise to an interesting new class of magnetically charged black holes with hair. 
  We demonstrate in detail how the space of two-dimensional quantum field theories can be parametrized by off-shell states of a free closed string moving in a flat background. The dynamic equation corresponding to the condition of conformal invariance includes an infinite number of higher order terms, and we give an explicit procedure for their calculation. The symmetries corresponding to equivalence relations of theories are described. In this framework we show how to perform nonperturbative analysis in the low-energy limit and prove that it corresponds to the Brans-Dicke theory of gravity interacting with a skew symmetric tensor field. 
  This set of lecture notes presents a pedantic derivation of the connection between the $ {\hat A} $-genus of spacetime's loop space and the genus one partition function of the $ N=1/2 $ sigma model. It concludes with some remarks on possible generalizations of the $ {\hat A} $-genus which follow naturally from the `stringy' point-of-view but have yet to be explored mathematically. This set of lecture notes is geared towards a mathematical audience unfamiliar with the $ N=1/2 $ sigma model. 
  We generalize the oscillator model of a particle interacting with a thermal reservoir by introducing arbitrary nonlinear couplings in the particle coordinates.The equilibrium positions of the heat bath oscillators are promoted to space-time functions, which are shown to represent a modulation of the internal noise by the external forces. The model thus provides a description of classical and quantum dissipation in non homogeneous environments. In the classical case we derive a generalized Langevin equation with nonlinear multiplicative noise and a position-dependent fluctuation- dissipation theorem associated to non homogeneous dissipative forces. When time-modulation of the noise is present, a new force term is predicted besides the dissipative and random ones. The model is quantized to obtain the non homogenous influence functional and master equation for the reduced density matrix of the Brownian particle. The quantum evolution equations reproduce the correct Langevin dynamics in the semiclassical limit. The consequences for the issues of decoherence and localization are discussed. 
  We consider SU(N) Yang-Mills theories in 2n+1 Euclidean dimensions coupled to an even flavour-number of Dirac fermions. After integrating out the fermions the wave functional for the effective Yang-Mills theory inherits a non-trivial U(1)-connection which is computed in the limit of infinite fermion mass (adiabatic connection). Its Chern-class turns out to be just half the flavour number. Its topological origin is explained in the Lagrangean and Hamiltonian picture. 
  It is argued on general ground and demonstrated in the particular example of the Chiral Schwinger Model that there is nothing wrong with apparently anomalous chiral gauge theory. If quantised correctly, there should be no gauge anomaly and chiral gauge theory should be renormalisable and unitary, even in higher dimensions and with non-abelian gauge groups. Furthermore, mass terms for gauge bosons and chiral fermions can be generated without spoiling the gauge invariance. 
  At high energy scale the only quantum effect of any asymptotic free and asymptotically conformal invariant GUT is the trace anomaly of the energy-momentum tensor. Anomaly generates the new degree of freedom, that is propagating conformal factor. At lower energies conformal factor starts to interact with scalar field because of the violation of conformal invariance. We estimate the effect of such an interaction and find the running of the nonminimal coupling from conformal value $\frac{1}{6}$ to $0$. Then we discuss the possibility of the first order phase transition induced by curvature in a region close to the stable fixed point and calculate the induced values of Newtonian and cosmological constants. 
  We study solutions of the tree level string effective action in the presence of the tachyon mode.We find that the 2-dim. static black hole is stable against tachyonic perturbations.For a particular ansatz for the tachyon field we find an exact solution of the equations of motion which exhibits a naked singularity.In the case of static fields we find numerically that the full system has a black hole solution,with the tachyon regular at the horizon. 
  We compute the string energy-momentum tensor and {\bf derive} the string equation of state from exact string dynamics in cosmological spacetimes. $1+1,~2+1$ and $D$-dimensional universes are treated for any expansion factor $R$. Strings obey the perfect fluid relation $ p = (\gamma -1) \rho $ with three different behaviours: (i) {\it Unstable} for $ R \to \infty $ with growing energy density $ \rho \sim R^{2-D} $, {\bf negative} pressure, and $ \gamma =(D-2)/(D-1) $; (ii){\it Dual} for $ R \to 0 $, with $ \rho \sim R^{-D} $, {\bf positive} pressure and $\gamma = D/(D-1) $ (as radiation); (iii) {\it Stable} for $ R \to \infty $ with $ \rho \sim R^{1-D} $, {\bf vanishing} pressure and $\gamma = 1 $ (as cold matter). We find the back reaction effect of these strings on the spacetime and we take into account the quantum string decay through string splitting. This is achieved by considering {\bf self-consistently} the strings as matter sources for the Einstein equations, as well as for the complete effective string equations. String splitting exponentially suppress the density of unstable strings for large $R$. The self-consistent solution to the Einstein equations for string dominated universes exhibits the realistic matter dominated behaviour $ R \sim (X^0)^{2/(D-1)}\; $ for large times and the radiation dominated behaviour $ R \sim (X^0)^{2/D}\; $ for early times. De Sitter universe does not emerge as solution of the effective string equations. The effective string action (whatever be the dilaton, its potential and the central charge term) is not the appropriate framework in which to address the question of string driven inflation. 
  In order to study in a regularisation free manner the renormalisability of d=2 supersymmetric non-linear $\si$ models, one has to use the algebraic BRS methods ; moreover, in the absence of an off-shell formulation, one has often to deal with open algebras. We then recall in a pedagogical and non technical manner the standard methods used to handle these questions and illustrate them on N=1 supersymmetric non-linear $\si$ model in component fields, giving the first rigorous proof of their renormalisability. In the special case of compact homogeneous manifolds (non-linear $\si$ model on a coset space G/H), we obtain the supersymmetric extension of the analysis done some years ago in the bosonic case. A further publication will be devoted to extended supersymmetry. 
  We analyse with the algebraic, regularisation independant, cohomological B.R.S. methods, the renormalisability of torsionless N=2 and N= 4 supersymmetric non-linear $\si$ models built on K\"ahler spaces. Surprisingly enough with respect to the common wisdom, in the case of N=2 supersymmetry, we obtain an anomaly candidate, at least in the compact K\"ahler Ricci-flat case. If its coefficient does differ from zero, such anomaly would imply the breaking of global N=2 supersymmetry and get into trouble some schemes of superstring compactification as such non-linear $\si$ models offer candidates for the superstring vacuum state. In the compact homogeneous K\"ahler case, as expected, the anomaly candidate disappears. The same phenomena occurs when one enforces N=4 supersymmetry : in that case, we obtain the first rigorous proof of the expected all-orders renormalisability -`` in the space of metrics"- of the corresponding non-linear $\si$ models. PAR/LPTHE/94-11 
  The Kahler moduli space of a particular non-simply-connected Calabi-Yau manifold is mapped out using mirror symmetry. It is found that, for the model considered, the chiral ring may be identical for different associated conformal field theories. This ambiguity is explained in terms of both A-model and B-model language. It also provides an apparent counterexample to the global Torelli problem for Calabi-Yau threefolds. 
  We discuss parity violation in the 3-dimensional (N flavour) Thirring model. We find that the ground state fermion current in a background gauge field does not posses a well defined parity transformation. We also investigate the connection between parity violation and fermion mass generation, proving that radiative corrections force the fermions to be massive. 
  We consider a family of local BRS-invariant higher covariant derivative regularizations of $SU(N)$ Chern-Simons theory that do not shift the value of the Chern-Simons parameter $k$ to $k+\,{\rm sign}(k)\,\cv$ at one loop. 
  We consider the linear space of composite fields as an infinite dimensional vector bundle over the theory space whose coordinates are simply the parameters of a renormalized field theory. We discuss a geometrical expression for the short distance singularities of the composite fields in terms of beta functions, anomalous dimensions, and a connection. (Based upon a talk given at the International Colloquim on Modern Quantum Field Theory, 5--11 January 1994, Bombay, India.) 
  Let $X$ be a compact Kahler manifold and $L\to X$ a quantizing holomorphic Hermitian line bundle. To immersed Lagrangian submanifolds $\Lambda$ of $X$ satisfying a Bohr-Sommerfeld condition we associate sequences $\{ |\Lambda, k\rangle \}_{k=1}^\infty$, where $\forall k$ $|\Lambda, k\rangle$ is a holomorphic section of $L^{\otimes k}$. The terms in each sequence concentrate on $\Lambda$, and a sequence itself has a symbol which is a half-form, $\sigma$, on $\Lambda$. We prove estimates, as $k\to\infty$, of the norm squares $\langle \Lambda, k|\Lambda, k\rangle$ in terms of $\int_\Lambda \sigma\overline{\sigma}$. More generally, we show that if $\Lambda_1$ and $\Lambda_2$ are two Bohr-Sommerfeld Lagrangian submanifolds intersecting cleanly, the inner products $\langle\Lambda_1, k|\Lambda_2, k\rangle$ have an asymptotic expansion as $k\to\infty$, the leading coefficient being an integral over the intersection $\Lambda_1\cap\Lambda_2$. Our construction is a quantization scheme of Bohr-Sommerfeld Lagrangian submanifolds of $X$. We prove that the Poincar\'e series on hyperbolic surfaces are a particular case, and therefore obtain estimates of their norms and inner products. 
  Chiral, conformal and ghost number anomalies are discussed from the viewpoint of the quantum vacuum in Hamiltonian formalism. After introducing the energy cut-off, we derive known anomalies in a new way. We show that the physical origin of the anomalies is the zero point fluctuation of bosonic or fermionic field. We first point out that the chiral U(1) anomaly is understood as the creation of the chirality at the bottom of the regularized Dirac sea in classical electromagnetic field. In the study of the (1+1) dimensional quantum vacuum of matter field coupled to the gravity, we give a physically intuitive picture of the conformal anomaly. The central charges are evaluated from the vacuum energy. We clarify that the non-Hermitian regularization factor of the vacuum energy is responsible for the ghost number anomaly. 
  We examine the thermal behavior of a theory with charged massive vector matter coupled to Chern-Simons gauge field. We obtain a critical temperature Tc, at which the effective mass of vector field vanishes, and the system transfers from a symmetry broken phase to topological phase. The phase transition is suggested to be of the zeroth order, as the free energy of the system is discontinuous at Tc. Application to the (2+1) dimensional quantum gravity is briefly discussed. 
  We investigate on the plane the axial anomaly for euclidean Dirac fermions in the presence of a background Aharonov--Bohm gauge potential. The non perturbative analysis depends on the self--adjoint extensions of the Dirac operator and the result is shown to be influenced by the actual way of understanding the local axial current. The role of the quantum mechanical parameters involved in the expression for the axial anomaly is discussed. A derivation of the effective action by means of the stereographic projection is also considered. 
  We investigate the outcomes of measurements on correlated, few-body quantum systems described by a quaternionic quantum mechanics that allows for regions of quaternionic curvature. We find that a multi-particle interferometry experiment using a correlated system of four nonrelativistic, spin-half particles has the potential to detect the presence of quaternionic curvature. Two-body systems, however, are shown to give predictions identical to those of standard quantum mechanics when relative angles are used in the construction of the operators corresponding to measurements of particle spin components. 
  The method of geometric quantization is applied to a particle moving on an arbitrary Riemannian manifold $Q$ in an external gauge field, that is a connection on a principal $H$-bundle $N$ over $Q$. The phase space of the particle is a Marsden-Weinstein reduction of $T^*N$, hence this space can also be considered to be the reduced phase space of a particular type of constrained mechanical system. An explicit map is found from a subalgebra of the classical observables to the corresponding quantum operators. These operators are found to be the generators of a representation of the semi-direct product group, Aut~$N\lx C^\infty_c(Q)$. A generalised Aharanov-Bohm effect is shown to be a natural consequence of the quantization procedure. In particular the r\^ole of the connection in the quantum mechanical system is made clear. The quantization of the Hamiltonian is also considered. Additionally, our approach allows the related quantization procedures proposed by Mackey and by Isham to be fully understood.\\ 
  The purely algebraic notion of CQG algebra (algebra of functions on a compact quantum group) is defined. In a straightforward algebraic manner, the Peter-Weyl theorem for CQG algebras and the existence of a unique positive definite Haar functional on any CQG algebra are established. It is shown that a CQG algebra can be naturally completed to a $C^\ast$-algebra. The relations between our approach and several other approaches to compact quantum groups are discussed. 
  The dual resonance model, which was a precursor of string theory was based upon the idea that two-particle scattering amplitudes should be expressible equivalently as a sum of contributions of an infinite number of $s$ channel poles each corresponding to a finite number of particles with definite spin, or as a similar sum of $t$ channel poles. The famous example of Veneziano \cite{ven} satisfies all these requirements, and is additionally ghost free.We recall other trajectories which provide solutions to the duality constraints, e.g. the general Mobi\"us trajectories and the logarithmic trajectories, which were thought to be lacking this last feature. We however demonstrate, partly empirically, the existence of a regime within a particular deformation of the Veneziano amplitude for logarithmic trajectories for which the amplitude remains ghost free. 
  In this paper all deformations of the general linear group, subject to certain restrictions which in particular ensure a smooth passage to the Lie group limit, are obtained. Representations are given in terms of certains sets of creation and annihilation operators. These creation and annihilation operators may belong to a generalisation of the $q$-quark type or $q$-hadronic type, of $q$-boson or $q$-fermion type. We are also led to a natural definition of $q$-direct sums of q-algebras. 
  The elliptic genus for arbitrary two dimensional $N=2$ Landau-Ginzburg orbifolds is computed. This is used to search for possible mirror pairs of such models. An important aspect of this work is that there is no restriction to theories for which the conformal anomaly is $\hat c\in\ZZ$, nor are the results only valid at the conformal fixed point. 
  We discuss the general structure of effective hamiltonians for systematic $1/N$ expansion in QCD using the light-cone quantization. These are second-quantized hamiltonians acting on the Fock space of mesons and glueballs defined by the solution of the $N=\infty$ problem. In the two-dimensional case we find only cubic and quartic interaction terms, and give explicit expressions for the vertex functions as integrals of solutions of 't Hooft equation. As examples of possible applications of our formalism, we study $1/N$ corrections to meson mass and form factors for decays of $Q\bar{q}$ states, recently discussed by Grinstein and Mende in the large-$N$ limit. We find that $1/N$ is a good small expansion parameter. 
  It is shown that the heat kernel operator for the Laplace operator on any covariantly constant curved background, i.e. in symmetric spaces, may be presented in form of an averaging over the Lie group of isometries with some nontrivial measure. Using this representation the heat kernel diagonal, i.e. the heat kernel in coinciding points is obtained. Related topics concerning the structure of symmetric spaces and the calculation of the effective action are discussed. 
  The Casimir force on a $D$-dimensional sphere due to the confinement of a massless scalar field is computed as a function of $D$, where $D$ is a continuous variable that ranges from $-\infty$ to $\infty$. The dependence of the force on the dimension is obtained using a simple and straightforward Green's function technique. We find that the Casimir force vanishes as $D\to +\infty$ ($D$ non-even integer) and also vanishes when $D$ is a negative even integer. The force has simple poles at positive even integer values of $D$. 
  We explore self-dual Chern-Simons Higgs systems with the local $SU(3)$ and global $U(1)$ symmetries where the matter field lies in the adjoint representation. We show that there are three degenerate vacua of different symmetries and study the unbroken symmetry and particle spectrum in each vacuum. We classify the self-dual configurations into three types and study their properties. 
  We present a general theory of non-perturbative quantization of a class of hermitian symmetric supermanifolds. The quantization scheme is based on the notion of a super Toeplitz operator on a suitable Z_2 -graded Hilbert space of superholomorphic functions. The quantized supermanifold arises as the C^* -algebra generated by all such operators. We prove that our quantization framework reproduces the invariant super Poisson structure on the classical supermanifold as Planck's constant tends to zero. 
  The purpose of this paper is to apply the framework of non- commutative differential geometry to quantum deformations of a class of Kahler manifolds. For the examples of the Cartan domains of type I and flat space, we construct Fredholm modules over the quantized manifolds using the supercharges which arise in the quantization of supersymmetric generalizations of the manifolds. We compute the explicit formula for the Chern character on generators of the Toeplitz C^* -algebra. 
  We apply the concepts of superanalysis to present an intrinsically supersymmetric formulation of the Chern character in entire cyclic cohomology. We show that the cocycle condition is closely related to the invariance under supertranslations. Using the formalism of superfields, we find a path integral representation of the index of the generalized Dirac operator. 
  We derive and study a class of cosmological and wormhole solutions of low-energy effective string field theory. We consider a general four-dimensional string effective action where moduli of the compactified manifold and the electromagnetic field are present. The cosmological solutions of the two-dimensional effective theory obtained by dimensional reduction of the former are discussed. In particular we demonstrate that the two-dimensional theory possesses a scale-factor duality invariance. Euclidean four-dimensional instantons describing the nucleation of the baby universes are found and the probability amplitude for the nucleation process is given. 
  We present the new extended supersymmetrization of the   Nonlinear Schrodinger Equation by introducing two superbosons fields with the different gradation. Our model is different from this considered by Toppan and in the reduced case (nonextended supersymmetry) is also different from this considered by Kulish and Roelofs and Kersten. We prove that our model is integrable by presenting its Lax formulation. 
  We extend the discussion of mirror symmetry, Picard-Fuchs equations, instanton-corrected Yukawa couplings, and the topological one-loop partition function to the case of complete intersections with higher-dimensional moduli spaces. We will develop a new method of obtaining the instanton-corrected Yukawa couplings through a close study of the solutions of the Picard-Fuchs equations. This leads to closed formulas for the prepotential for the K\"ahler moduli fields induced from the ambient space for all complete intersections in non singular weighted projective spaces. As examples we treat part of the moduli space of the phenomenologically interesting three-generation models that are found in this class. We also apply our method to solve the simplest model in which a topology change was observed and discuss examples of complete intersections in singular ambient spaces. 
  We study the $(1,q=-1)$ model coupled to topological gravity as a candidate to describing $2d$ string theory at the self-dual radius. We define the model by analytical continuation of $q>1$ topological recursion relations to $q=-1$. We show that at genus zero the $q=-1$ recursion relations yield the $W_{1+\infty}$ Ward identities for tachyon correlators on the sphere. A scheme for computing correlation functions of $q=-1$ gravitational descendants is proposed and applied for the computation of several correlators. It is suggested that the latter correspond to correlators of discrete states of the $c=1$ string. In a similar manner to the $q>1$ models, we show that there exist topological recursion relations for the correlators in the $q=-1$ theory that consist of only one and two splittings of the Riemann surface. Using a postulated regularized contact, we prove that the genus one $q=-1$ recursion relations for tachyon correlators coincide with the $W_{1+\infty}$ Ward identities on the torus. We argue that the structure of these recursion relations coincides with that of the $W_{1+\infty}$ Ward identities for any genus. 
  Consideration of the geometric quantization of the phase space of a particle in an external Yang-Mills field allows the results of the Mackey-Isham quantization procedure for homogeneous configuration spaces to be reinterpreted. In particular, a clear physical interpretation of the `inequivalent' quantizations occurring in that procedure is given. 
  The effect of the inevitable coupling to external degrees of freedom of a quantum computer are examined. It is found that for quantum calculations (in which the maintenance of coherence over a large number of states is important), not only must the coupling be small but the time taken in the quantum calculation must be less than the thermal time scale, $\hbar/k_B T$. For longer times the condition on the strength of the coupling to the external world becomes much more stringent. 
  New stationary solutions of $4$-dimensional dilaton-axion gravity are presented, which correspond to the charged Taub-NUT and Israel-Wilson-Perjes (IWP) solutions of Einstein-Maxwell theory. The charged axion-dilaton Taub-NUT solutions are shown to have a number of interesting properties: i) manifest $SL(2,R)$ symmetry, ii) an infinite throat in an extremal limit, iii) the throat limit coincides with an exact CFT construction.   The IWP solutions are shown to admit supersymmetric Killing spinors, when embedded in $d=4,N=4$ supergravity. This poses a problem for the interpretation of supersymmetric rotating solutions as physical ground states. In the context of $10$-dimensional geometry, we show that dimensionally lifted versions of the IWP solutions are dual to certain gravitational waves in string theory. 
  Asymptotic solutions to the quantized Knizhnik-Zamolodchikov equation associated with $\frak{gl}_{N+1}$ are constructed. The leading term of an asymptotic solution is the Bethe vector -- an eigenvector of the transfer-matrix of a quantum spin chain model. We show that the norm of the Bethe vector is equal to the product of the Hessian of a suitable function and an explicitly written rational function. This formula is an analogue of the Gaudin-Korepin formula for the norm of the Bethe vector. It is shown that, generically, the Bethe vectors form a base for the $\frak{gl}_2$ case. 
  It is argued that renormalisation group flow can be interpreted as being a Hamiltonian vector flow on a phase space which consists of the couplings of the theory and their conjugate \lq\lq momenta", which are the vacuum expectation values of the corresponding composite operators. The Hamiltonian is linear in the conjugate variables and can be identified with the vacuum expectation value of the trace of the energy-momentum operator. For theories with massive couplings the identity operator plays a central role and its associated coupling gives rise to a potential in the flow equations. The evolution of any quantity , such as $N$-point Green functions, under renormalisation group flow can be obtained from its Poisson bracket with the Hamiltonian. Ward identities can be represented as constants of the motion which act as symmetry generators on the phase space via the Poisson bracket structure. 
  Using a reformulation of the nonlinear multiplet as a gauge multiplet, we discuss its dynamics. We show that the nonlinear ``duality'' that appears to relate the model to a conventional $\sigma$-model introduces a new sector into the theory. 
  We find the complex structure on the dual of a complex target space. For $N=(2,2)$ systems, we prove that the space orthogonal to the kernel of the commutator of the left and right complex structures is {\em always} integrable, and hence the kernel is parametrized by chiral and twisted chiral superfield coordinates. We then analyze the particular case of $SU(2)\times SU(2)$, and are led to a new $N=2$ superspace formulation of the $SU(2)\times U(1)$ WZW-model. 
  We apply Kac's theory of elements of finite order (EFO) in   Lie groups to the description of discrete gauge symmetries in various supersymmetric grand unified models. Taking into account the discrete anomaly cancellation conditions, we identify the EFO which generate certain matter parities in the context of the supersymmetric $SO(10)$ and $E_6$ models. 
  This paper studies invariants of 3-manifolds derived from certain fin ite dimensional Hopf algebras. The invariants are based on right integrals for these algebras. It is shown that the resulting class of invariants is distinct from the class of Witten-Reshetikhin-Turaev invariants. 
  The algebraic structure of the Green's ansatz is analyzed in such a way that its generalization to the case of q-deformed para-Bose and para-Fermi operators is becoming evident. To this end the underlying Lie (super)algebraic properties of the parastatistics are essentially used. 
  We construct two new classes of exact solutions to string theory which are not of the standard plane wave or gauged WZW type. Many of these solutions have curvature singularities. The first class includes the fundamental string solution, for which the string coupling vanishes near the singularity. This suggests that the singularity may not be removed by quantum corrections. The second class consists of hybrids of plane wave and gauged WZW solutions. We discuss a four dimensional example in detail. 
  We investigate an equivariant generalization of Morse theory for a general class of integrable models. In particular, we derive equivariant versions of the classical Poincar\'e-Hopf and Gauss-Bonnet-Chern theorems and present the corresponding path integral generalizations. Our approach is based on equivariant cohomology and localization techniques, and is closely related to the formalism developed by Matthai and Quillen in their approach to Gaussian shaped Thom forms. 
  Starting with the Taub--NUT solution to Einstein's equations, together with a constant dilaton, a dyonic Taub--NUT solution of low energy heterotic string theory with non--trivial dilaton, axion and $U(1)$ gauge fields is constructed by employing $O(1,1)$ transformations. The electromagnetic dual of this solution is constructed, using $SL(2,\rline)$ transformations. By an appropriate change to scaled variables, the extremal limit of the dual solution is shown to correspond to the low energy limit of an exact conformal field theory presented previously. 
  The (2+1)-dimensional Thirring model is studied by using the Gaussian approximation method in the functional Schr\"odinger picture. Although the dynamical symmetry breaking does not occur in the large N limit, it does occur in the Gaussian approximation which includes the higher order contributions in 1/N. 
  We propose an action describing chiral fermions with an arbitrary gauge group and with manifest $(8,0)$ worldsheet supersymmetry. The form of the action is inspired by and adapted for completing the twistor-like formulation of the $D=10$ heterotic superstring. 
  We show that W_3 is the algebra of symmetries of the ``rigid-particle'', whose action is given by the integrated extrinsic curvature of its world line. This is easily achived by showing that its equation of motion can be written in terms of the Boussinesq operator. We also show how to obtain the equations of motion of the standard relativistic particle provided it is consistent to impose the ``zero-curvature gauge'', and comment about its connection with the KdV operator. 
  We consider the hard thermal loops of QCD for a moving quark-gluon plasma. Generalizing from this we suggest a candidate for the magnetic mass term. This term may also be useful in understanding the mass gap of three-dimensional non-Abelian gauge theories. 
  We analyze the renormalization of wave functionals and energy eigenvalues in field theory. A discussion of the structure of the renormalization group equation for a general Hamiltonian system is also given. 
  We discuss a possible approach to the problem of a gauge theory with a strong coupling constant. It is seen that, instead of plane waves, we have to consider the adiabatic eigenstates of the perturbation in order to get a meaningful perturbation approach. 
  Following the paradigm on the sphere, we begin the study of irrational conformal field theory (ICFT) on the torus. In particular, we find that the affine-Virasoro characters of ICFT satisfy heat-like differential equations with flat connections. As a first example, we solve the system for the general $g/h$ coset construction, obtaining an integral representation for the general coset characters. In a second application, we solve for the high-level characters of the general ICFT on simple $g$, noting a simplification for the subspace of theories which possess a non-trivial symmetry group. Finally, we give a geometric formulation of the system in which the flat connections are generalized Laplacians on the centrally-extended loop group. 
  The Painlev\'e transcendents $P_{\rom{I}}$--$P_{\rom{V}}$ and their representations as isomonodromic deformation equations are derived as nonautonomous Hamiltonian systems from the classical $R$--matrix Poisson bracket structure on the dual space $\wt{\frak{sl}}_R^*(2)$ of the loop algebra $\wt{\frak{sl}}_R(2)$. The Hamiltonians are obtained by composing elements of the Poisson commuting ring of spectral invariant functions on $\wt{\frak{sl}}_R^*(2)$ with a time--dependent family of Poisson maps whose images are $4$--dimensional rational coadjoint orbits in $\wt{\frak{sl}}_R^*(2)$. Each system may be interpreted as describing a particle moving on a surface of zero curvature in the presence of a time--varying electromagnetic field. The Painlev\'e equations follow from reduction of these systems by the Hamiltonian flow generated by a second commuting element in the ring of spectral invariants. 
  Viewing the Knizhnik--Zamolodchikov equations as multi--time, nonautonomous Shr\"odinger equations, the transformation to the Heisenberg representation is shown to yield the quantum Schlesinger equations. These are the quantum form of the isomonodromic deformations equations for first order operators of the form $\DD_\l = {\di \over \di \l} - \wh{\NN}(\l)$, where $\wh{\NN}(\l)$ is a rational $r\times r$ matrix valued function of $\l$ having simple poles only, and the matrix entries are interpreted as operators on a module of the rational $R$--matrix loop algebra $\wt{\frak{gl}}(r)_R$. This provides a simpler formulation of a construction due to Reshetikhin, relating the KZ equations to quantum isomonodromic deformations. 
  We study the order $\alpha'$ correction to the string black hole found by Garfinkle, Horowitz, and Strominger. We include all operators of dimension up to four in the Lagrangian, and use the field redefinition technique which facilitates the analysis. A mass correction, which is implied by the work of Giddings, Polchinski, and Strominger, is found for the extremal GHS black hole. 
  In this paper we construct manifestly covariant relativistic coherent states on the entire complex plane which reproduce others previously introduced on a given $SL(2,R)$ representation, once a change of variables $z\in C\rightarrow z_D \in $ unit disk is performed. We also introduce higher-order, relativistic creation and annihilation operators, $\C,\Cc$, with canonical commutation relation $[\C,\Cc]=1$ rather than the covariant one $[\Z,\Zc]\approx$ Energy and naturally associated with the $SL(2,R)$ group. The canonical (relativistic) coherent states are then defined as eigenstates of $\C$. Finally, we construct a canonical, minimal representation in configuration space by mean of eigenstates of a canonical position operator. 
  For gauge field propagators, the asymptotic behavior is obtained in all directions of the complex $k^2$-plane, and for general, linear, covariant gauges. Asymptotically free theories are considered. Except for coefficients, the functional form of the leading asymptotic terms is gauge-independent. Exponents are determined exactly by one-loop expressions. Sum rules are derived, which generalize the superconvergence relations obtained in the Landau gauge. (To appear in Physics Letters B) 
  We consider the conformal gauging of non-abelian groups. In such cases there are inequivalent ways of gauging (generalizing the axial and vector cases for abelian groups) corresponding to external automorphisms of the group. Different $\s$-models obtained this way correspond to the same conformal field theory. We use the method of quotients to formulate this equivalence as a new duality symmetry. 
  A particle which lives in a d-dimensional ordinary and a d-dimensional Grassmann space manifests itself in an ordinary four-dimensional subspace as a spinor, a scalar or a vector with charges. Operators of the Lorentz transformations and translations in both spaces form the super- Poincar\' e algebra. It is the super-Pauli-Ljubanski vector which generates spinors. Vielbeins and spin connections with the Lorentz index larger than or equal to five may manifest in a four-dimensional subspace as an electromagnetic, a weak and a colour field. 
  We consider a problem which may be viewed as an inverse one to the Schwinger realization of Lie algebra, and suggest a procedure of deforming the so-obtained algebra. We illustrate the method through a few simple examples extending Schwinger's $su(1,1)$ construction. As results, various q-deformed algebras are (re-)produced as well as their undeformed counterparts. Some extensions of the method are pointed out briefly. 
  Entropy arises in strong interactions by a dynamical separation of ``partons'' from unobservable ``environment'' modes due to confinement. For interacting scalar fields we calculate the statistical entropy of the observable subsystem. Diagonalizing its functional density matrix yields field pointer states and their probabilities in terms of Wightman functions. It also indicates how to calculate a finite geometric entropy proportional to a surface area. 
  This paper discusses the phenomenon of spontaneous symmetry breaking in the Schr\"odinger representation formulation of quantum field theory. The analysis is presented for three-dimensional space-time abelian gauge theories with either Maxwell, Maxwell-Chern-Simons, or pure Chern-Simons terms as the gauge field contribution to the action, each of which leads to a different form of mass generation for the gauge fields. 
  We have found some new solutions of both rational and trigonometric types by rewriting Yang-Baxter equation as a triple product equation in a vector space of matrices. 
  For zero energy, $E=0$, we derive exact, classical and quantum solutions for {\em all} power-law oscillators with potentials $V(r)=-\gamma/r^\nu$, $\gamma>0$ and $-\infty <\nu<\infty$. When the angular momentum is non-zero, these solutions lead to the classical orbits $\r(t)= [\cos \mu (\th(t)-\th_0(t))]^{1/\mu}$, with $\mu=\nu/2-1 \ne 0$. For $\nu>2$, the orbits are bound and go through the origin. We calculate the periods and precessions of these bound orbits, and graph a number of specific examples. The unbound orbits are also discussed in detail. Quantum mechanically, this system is also exactly solvable. We find that when $\nu>2$ the solutions are normalizable (bound), as in the classical case. Further, there are normalizable discrete, yet {\it unbound}, states. They correspond to unbound classical particles which reach infinity in a finite time. Finally, the number of space dimensions of the system can determine whether or not an $E=0$ state is bound. These and other interesting comparisons to the classical system will be discussed. 
  Lagrangian and Hamiltonian formulations of a free spinning particle in 2+1-dimensions or {\it anyon} are established, following closely the analysis of Hanson and Regge. Two viable (and inequivalent) Lagrangians are derived. It is also argued that one of them is more favourable. In the Hamiltonian analysis non-triviaal Dirac Brackets of the fundamental variables are computed for both the models. Important qualitative differences with a recently proposed model for anyons are pointed out. 
  We use the quantum symmetries present in string compactification on Landau-Ginzburg orbifolds to prove the existence of a large class of exactly marginal (0,2) deformations of (2,2) superconformal theories. Analogous methods apply to the more general (0,2) models introduced in \DK, lending further credence to the fact that the corresponding \LG\ models represent bona-fide (0,2) SCFTs. We also use the large symmetry groups which arise when the worldsheet superpotential is turned off to constrain the dependence of certain correlation functions on the untwisted moduli. This allows us to approach the problem of what happens when one tries to deform away from the \LG\ point. In particular, we find that the masses and three-point couplings of the massless $E_{6}$ singlets related to ${\rm H^{1}}(\ET)$ vanish at all points in the quintic \Ka\ moduli space. Putting these results together, and invoking some plausible dynamical assumptions about the corresponding linear \sm s, we show that one can deform these \LG\ theories to arbitrary values of the \Ka\ moduli. 
  The quantum symmetry of many \LG\ orbifolds appears to be broken by Yang-Mills instantons. However, isolated Yang-Mills instantons are not solutions of string theory: They must be accompanied by gauge anti-instantons, gravitational instantons, or topologically non-trivial configurations of the $H$ field. We check that the configurations permitted in string theory do in fact preserve the quantum symmetry, as a result of non-trivial cancellations between symmetry breaking effects due to the various types of instantons. These cancellations indicate new constraints on \LG\ orbifold spectra and require that the dilaton modulus mix with the twisted moduli in some \LG\ compactifications. We point out that one can find similar constraints at all fixed points of the modular group of the moduli space of vacua. 
  A recently introduced approach for the dynamical analysis and quantization of field theoretical models with second class constraints is ilustrated applied to linearized gravity in 3-D. The canonical structure of two different models of linerized gravity in 3-D, the intermediate and the self dual models, is discussed in detail. It is shown that the first order self dual model whose constraints are all second class may be regarded as a gauge fixed version of the second order gauge invariant intermediate model. In particular it is shown how to construct the gauge invariant hamiltonian of the intermediate model starting from the one of the self dual model. The relation with the t opologically massive linearized gravity is also discussed. 
  We study the gravitational waves in the 10-dimensional target space of the superstring theory. Some of these waves have unbroken supersymmetries. They consist of Brinkmann metric and of a 2-form field. Sigma-model duality is applied to such waves. The corresponding solutions we call dual partners of gravitational waves, or dual waves. Some of these dual waves upon Kaluza-Klein dimensional reduction to 4 dimensions become equivalent to the conformo-stationary solutions of axion-dilaton gravity. Such solutions include dilaton extreme black holes, axion-dilaton Israel-Werner-Perjes-type spacetimes and extreme charged axion-dilaton Taub-Nut solutions. The unbroken supersymmetry of the gravitational waves transfers to the unbroken supersymmetry of axion-dilaton IWP solutions. More general supersymmetric 4-dimensional configurations derivable from 10-dimensional waves are described. 
  I present a solution to the Schwinger model in the light-cone representation which corrects an error in a previous work. I emphasize the details of the mechanism by which the physical vacuum is different than the perturbative vacuum. I suggest that the method of analyzing vacuum structure presented here may be of use in more complicated theories such as QCD. 
  The present paper can be thought of as a continuation of the paper "Introduction to sh Lie algebras for physicists" by T. Lada and J. Stasheff (International Journal of Theoretical Physics Vol. 32, No. 7 (1993), 1087--1103, appeared also as preprint hep-th/9209099) which provided an exposition of the basic ingredients of the theory of strongly homotopy Lie algebras sufficient for the underpinnings of the physically relevant examples. We demonstrate the `strong homotopy' analog of the usual relation between Lie and associative algebras and investigate the universal enveloping algebra functor emerging as the left adjoint of the symmetrization functor. We show that the category of homotopy associative algebras carries a natural monoidal structure such that the universal enveloping algebra is a unital coassociative cocommutative coalgebra with respect to this monoidal structure. The last section is concerned with the relation between homotopy modules and weak homotopy maps. The present paper is complementary to what currently exists in the literature, both physical and mathematical. 
  We reconsider two-dimensional topological gravity in a functional and lagrangian framework. We derive its Slavnov-Taylor identities and discuss its (in)dependence on the background gauge. Correlators of reparamerization invariant observables are shown to be globally defined forms on moduli space. The potential obstruction to their gauge-independence is the non-triviality of the line bundle on moduli space ${\cal L}_x$, whose first Chern-class is associated to the topological invariants of Mumford, Morita and Miller. Based on talks given at the Fubini Fest, Torino, 24-26 February 1994, and at the Workshop on String Theory, Trieste, 20-22 April 1994. 
  In this paper we outline an approach to calculus over quasitriangular Hopf algebras. We study differential operators in the framework of monoidal categories equipped with a braiding or symmetry. To be more concrete, we choose as an example the category of modules over quasitriangular Hopf algebra. We introduce braided differential operators in a pure algebraic manner.This gives us a possibility to develop calculus in an intrinsic way without enforcing any type of Leibniz rule. A general notion of quantization in monoidal categories, proposed in this paper, is a natural isomorphism of the tensor product bifunctor equipped with some natural coherence conditions. The quantization "deforms" all algebraic and differential objects in the monoidal category. We suggest two ways for calculation of quantizations. One of them reduces the calculation to non-linear cohomologies. THe other describes quantizations in terms of multiplicative Hochschild cohomologies of the Grothendieck ring of the given monoidal category. These constructions are illustrated by some examples. 
  A new "non-standard" quantization of the universal enveloping algebra of the split (natural) real form $so(2,2)$ of $D_2$ is presented. Some (classical) graded contractions of $so(2,2)$ associated to a $Z_2 \times Z_2$ grading are studied, and the automorphisms defining this grading are generalized to the quantum case, thus providing quantum contractions of this algebra. This produces a new family of "non-standard" quantum algebras. Some of these algebras can be realized as (2+1) kinematical algebras; we explicitly introduce a new deformation of Poincar\'e algebra, which is naturally linked to the null plane basis. Another realization of these quantum algebras as deformations of the conformal algebras for the two-dimensional Euclidean, Galilei and Minkowski spaces is given, and its new properties are emphasized. 
  This paper is meant to be an informal introduction to Quantum Groups, starting from its origins and motivations until the recent developments. We call in particular the attention on the newly descovered relationship among quantum groups, integrable models and Jordan structures. 
  We demonstrate that the large-N expansion of Wilson loop expectation values in SO(N) and Sp(N) Yang-Mills theory on orientable and nonorientable surfaces has a natural description as a weighted sum over covers of the given surface. The sum takes the form of the perturbative expansion of an open string theory. The derivation makes contact with the classification of branched covers by Gabai and Kazez. Comparison with the analogous results for the chiral sectors of QCD_2 is instructive for both cases. 
  We investigate in detail the critical $N{=}2$ fermionic string with and without a global ${\bf Z}_2$ twist. An analysis of BRST cohomology shows that twisted sectors contain massless `spacetime' fermions which are {\it non-local\/} with respect to the standard massless boson. However, two distinct GSO projections exist, one (untwisted) retaining merely the usual boson and its spectral-flow partner, the other (twisted) yielding two fermions and one boson, on the massless level. The corresponding chiral BRST-invariant vertex operators are constructed in certain pictures, and their fusion and picture-changing are investigated, including the construction of inverse picture-changing operators. The $N{=}2$ `spacetime supersymmetry' generators are {\it null\/} operators, since the twisted massless states fail to interact. The untwisted three- and four-point functions are recalculated at tree-level. 
  The implications of string theory for understanding the dimension of decompactified spacetime are discussed. Results from a computer model designed to simulate expansion of the early universe during the string dominated phase are presented. This model focuses on the effects of string winding modes on inflation and is based on the theory of random walks. We demonstrate that our decompactified four-dimensional spacetime can be explained by the proper choice of initial conditions. 
  We present a Mathai-Quillen interpretation of topological sigma models. The key to the construction is a natural connection in a suitable infinite dimensional vector bundle over the space of maps from a Riemann surface (the world sheet) to an almost complex manifold (the target). We show that the covariant derivative of the section defined by the differential operator that appears in the equation for pseudo-holomorphic curves is precisely the linearization of the operator itself. We also discuss the Mathai-Quillen formalism of gauged topological sigma models. 
  The Tomita-Takesaki modular theory is used to establish a cluster estimate extending and modifying that of Thomas and Wichmann, so as to extend it to regions within which the relevant observables are not necessarily spacelike separated. This sort of estimate is then applied to the case of a massive free field, to show that wavefunctions localized in a certain sense are analytic functions of momentum. 
  We consider the Schwarz-Sen spectrum of elementary electrically charged massive $N_R=1/2$ states of the four-dimensional heterotic string and show the maximum spin $1$ supermultiplets to correspond to extreme black hole solutions. The $N_L=1$ states and $N_L>1$ states (with vanishing left-moving internal momentum) admit a single scalar-Maxwell description with parameters $a=\sqrt 3$ or $a=1$, respectively. The corresponding solitonic magnetically charged spectrum conjectured by Schwarz and Sen on the basis of $S$-duality is also described by extreme black holes. 
  The quantum mechanical version of the four kinds of classical canonical transformations is investigated by using non-hermitian operator techniques. To help understand the usefulness of this appoach the eigenvalue problem of a harmonic oscillator is solved in two different types of canonical transformations. The quantum form of the classical Hamiton-Jacobi theory is also employed to solve time dependent Schr\"odinger wave equations, showing that when one uses the classical action as a generating function of the quantum canonical transformation of time evolutions of state vectors, the corresponding propagator can easily be obtained. 
  Starting from classical algebraic geometry over the complex numbers (as it can be found for example in Griffiths and Harris it was the goal of these lectures to introduce some concepts of the modern point of view in algebraic geometry. Of course, it was quite impossible even to give an introduction to the whole subject in such a limited time. For this reason the lectures and now the write-up concentrate on the substitution of the concept of classical points by the notion of ideals and homomorphisms of algebras. 
  A fundamental length is introduced into physics in a way which respects the principles of relativity and quantum field theory. This improves the properties of quantum field theory: divergences are removed. How to quantize gravity is also indicated. When the fundamental length tends to zero the present version of quantum field theory is recovered. 
  Swimming of microorganisms is further developed from a viewpoint of strings and membranes swimming in the incompressible fluid of low Reynolds number. In our previous paper the flagellated motion was analyzed in two dimensional fluid, by using the method developed in the ciliated motion with the Joukowski transformation. This method is further refined by incorporating the inertia term of fluid as the perturbation. Understanding of the algebra controlling the deformation of microorganisms in the fluid is further developed, obtaining the central extension of the algebra with the help of the recent progress on the $W_{1+\infty}$ algebra. Our previous suggestion on the usefulness of the $N$-point string- and membrane-like amplitudes for studying the collective swimming motion of $N-1$ microorganisms is also examined. 
  Inspired by the work of Wheeler among others, we have studied the problem of quantum measurements of space-time distances by applying the general principles of quantum mechanics as well as those of general relativity. Contrary to the folklore, the minimum error in the measurement of a length is shown to be proportional to the one-third power of the length itself. This uncertainty in space-time measurements implies an uncertainty of the space-time metric and yields quantum decoherence for particles heavier than the Planck mass. There is also a corresponding minimum error in energy-momentum measurements. 
  We propose a series of new subalgebras of the $W_{1+\infty}$ algebra parametrized by polynomials $p(w)$, and study their quasifinite representations. We also investigate the relation between such subalgebras and the $\hat{\mbox{gl}}(\infty)$ algebra. As an example, we investigate the $\Win$ algebra which corresponds to the case $p(w)=w$, presenting its free field realizations and Kac determinants at lower levels. 
  The applications of parabosons by Schmutz (1980) and of supersymmetry by Jarvis {\em et al.} (1984) in Jahn-Teller systems are compared and contrasted. Although a parasupersymmetric Jahn-Teller system has not yet been identified, the method of Schmutz is used here to show that the $E \times \epsilon $ supersymmetric Jahn-Teller Hamiltonian can be written in terms of paraboson operators. 
  New algebraic structure on the orbits of dressing transformations of the quasitriangular Poisson Lie groups is provided. This give the topological interpretation of the link invariants associated with the Weinstein--Xu classical solutions of the quantum Yang-Baxter equation. Some applications to the three-dimensional topological quantum field theories are discussed. 
  We show that the quantum Heisenberg group $H_{q}(1)$ can be obtained by means of contraction from quantum $SU_q(2)$ group. Its dual Hopf algebra is the quantum Heisenberg algebra $U_{q}(h(1))$. We derive left and right regular representations for $U_{q}(h(1))$ as acting on its dual $H_{q}(1)$. Imposing conditions on the right representation the left representation is reduced to an irreducible holomorphic representation with an associated quantum coherent state. By duality, left and right regular representations for quantum Heisenberg group with the quantum Heisenberg algebra as representation module are also constructed. As before reduction of left representations leads to finite dimensional irreducible ones for which the intertwinning operator is also investigated. 
  Dependence on the gauge parameters is an important issue in gauge theories: physical quantities have to be independent. Extending BRS transformations by variation of the gauge parameter into a Grassmann variable one can control gauge parameter dependence algebraically. As application we discuss the anomaly coefficient in the Slavnov-Taylor identity, $S$-matrix elements, the vector two-point-function and the coefficients of renormalization group and Callan-Symanzik equation. 
  The weights are computed for the Bethe vectors of an RSOS type model with periodic boundary conditions obeying $U_q[sl(n)]$ ($q=\exp(i\pi/r)$) invariance. They are shown to be highest weight vectors. The q-dimensions of the corresponding irreducible representations are obtained. 
  We study the thermodynamics of massive Gross-Neveu models with explicitly broken discrete or continuous chiral symmetries for finite temperature and fermion densities. The large $N$ limit is discussed bearing attention to the no-go theorems for symmetry breaking in two dimensions which apply to the massless cases. The main purpose of the study is to serve as analytical orientation for the more complex problem of chiral transition in $4-$dimensional QCD with quarks. For any non-vanishing fermion mass we find, at finite densities, lines of first order phase transitions. For small mass values traces of would-be second order transitions and a tricritical point are recognizable. We study the thermodynamics of these models, and in the model with broken continuous chiral symmetry we examine the properties of the pion like state. 
  It is known that 2D field theories admit several sectors of mutually local fields so as two fields from different sectors are mutually nonlocal. We show that any one-partical integrable model with ${\bf Z}_2$ symmetry contains three sectors: bosonic, fermionic and `disorder' one. We generalize the form factor axioms to fermionic and `disorder' sectors. For the particular case of the sinh-Gordon model we obtain several form factors in these sectors. 
  We consider a massive fermion interacting with a U(1) gauge field in the limit of a large coupling constant. It is found that the current has a generalized London term that can originate massive excitations for two of the three components of the gauge field, which disappear for a free particle at rest. The origin of the superconductive term is due to a partial breaking of the gauge symmetry in the limit of a large coupling constant. Beside, the scalar potential generated by the particle, the only component of the gauge field that keeps gauge invariance, increases with the square of the distance. These results should give a path towards the derivation of quark confinement from QCD. 
  The paper contains a short review of the theory of symplectic and contact manifolds and of the generalization of this theory to the case of supermanifolds. It is shown that this generalization can be used to obtain some important results in quantum field theory. In particular, regarding $N$-superconformal geometry as particular case of contact complex geometry, one can better understand $N=2$ superconformal field theory and its connection to topological conformal field theory. The odd symplectic geometry constitutes a mathematical basis of Batalin-Vilkovisky procedure of quantization of gauge theories.   The exposition is based mostly on published papers. However, the paper contains also a review of some unpublished results (in the section devoted to the axiomatics of $N=2$ superconformal theory and topological quantum field theory). The paper will be published in Berezin memorial volume. 
  We study equivariant localization formulas for phase space path integrals when the phase space is a multiply connected compact Riemann surface. We consider the Hamiltonian systems to which the localization formulas are applicable and show that the localized partition function for such systems is a topological invariant which represents the non-trivial homology classes of the phase space. We explicitly construct the coherent states in the canonical quantum theory and show that the Hilbert space is finite dimensional with the wavefunctions carrying a projective representation of the discrete homology group of the phase space. The corresponding coherent state path integral then describes the quantum dynamics of a novel spin system given by the quantization of a non-symmetric coadjoint Lie group orbit. We also briefly discuss the geometric structure of these quantum systems. 
  We construct the theoretical base for the search of the possible experimental manifestations of the torsion field at low energies. The weakrelativistic approximation to the Dirac equation in an external torsion field is considered. For the sake of generality we introduce the external electromagnetic field in parallel. The generalized (due to torsion dependent terms) Pauly equation contains new terms which have the different structure if compared with standard electromagnetic ones. Just the same takes place for the weakrelativistic equations for spin $\frac{1}{2}$ particle in an external torsion and electromagnetic field. It is given the brief description of the possible experiments. 
  A particular class of random walks with a spin factor on a three dimensional cubic lattice is studied. This three dimensional random walk model is a simple generalization of random walk for the two dimensional Ising model. All critical diffusion constants and associated critical exponents are calculated. Continuum field theories such as Klein-Gordon, Dirac and massive Chern-Simons theories are constructed near several critical points. 
  A nonlinear realization of super $W_{\infty}$ algebra is shown to exist through a consistent superLax formulation of super KP hierarchy. The reduction of the superLax operator gives rise to the Lax operators for $N=2$ generalized super KdV hierarchies, proposed by Inami and Kanno. The Lax equations are shown to be Hamiltonian and the associated Poisson bracket algebra among the superfields, consequently, gives rise to a realization of nonlinear super $W_{\infty}$ algebra. 
  We solve exactly the "boundary sine-Gordon" system of a massless scalar field \phi with a \cos[\beta\phi/2] potential at a boundary. This model has appeared in several contexts, including tunneling between quantum-Hall edge states and in dissipative quantum mechanics. For \beta^2 < 8\pi, this system exhibits a boundary renormalization-group flow from Neumann to Dirichlet boundary conditions. By taking the massless limit of the sine-Gordon model with boundary potential, we find the exact S matrix for particles scattering off the boundary. Using the thermodynamic Bethe ansatz, we calculate the boundary entropy along the entire flow. We show how these particles correspond to wave packets in the classical Klein-Gordon equation, thus giving a more precise explanation of scattering in a massless theory. 
  Static and spherically symmetric black hole solutions with non-zero cosmological constant are investigated. A formal power series solution is found. It is proved that the number of regular horizons is less than or equal to 2 for positive cosmological constant and is less than or equal to 1 for negative cosmological constant. This shows a striking contrast to the fact that the Reissner-Nordstr{\o}m-de Sitter black hole with positive cosmological horizon has 3 regular horizons. 
  The unified theory of string and two-dimensional quantum gravity is considered. The action for two-dimensional gravity is choosen in a well-known induced form and thus gravity posesses it's oun nontrivial dynamics even on the classical level. Three classically equivalent forms of the action of the unified theory are described, that is $D$, $D+1$ and $D+2$ formulations where the last one corresponds to the special kind of nonlinear sigma model on the background of classical two-dimensional metric. In all cases we find the action of the theory to be invariant under generalized symmetry transformations, and point out the arbitrariness of renormalization which reflects this symmetry. The one - loop counterterms are calculated in a general covariant gauge and also in conformal gauge. The difference between the counterterms for all classically equivalent variants of the theory is trivial on mass-shell. Since the different versions of the theory lead to the one-loop contributions which coincide on mass shell, the corresponding arbitrariness is related with the only generalized reparametrizations of the theory. Hence one can formulate the conditions of Weyl symmetry for any equivalent form of the action. Such a conditions are written for the $D+2$ formulation and then are reduced to other ones. The equations for the background interaction fields differ from the ones in standard approach and look much more complicated. 
  We study strong-weak coupling duality (S-duality) in N=4 supersymmetric non-Abelian Yang-Mills theories. These theories arise naturally as the low-energy limit of four-dimensional toroidal compactifications of the heterotic string. Firstly, we define the free energy in the presence of electric and magnetic fluxes using 't Hooft's prescription, i.e. through functional integrals at finite volume in the presence of twisted boundary conditions. Then, we compute those free energies in two limiting cases: small and large coupling constant. Finally, we extend the free-energies to all values of the coupling constant (and the theta angle) by presenting a fully S-duality invariant ansatz. This ansatz obeys all relevant consistency conditions; in particular, it obeys 't Hooft duality equations and Witten's magnetic-electric transmutation. The existence of an S-duality invariant, consistent definition of free energies supports the claim made in the literature that S-duality is a duality symmetry of N=4 SUSY Yang-Mills. Our ansatz also suggests that N=4, irrespective of whether partially broken or not, is in a self-dual phase: no phase transitions occur between the strong and weak coupling regimes. 
  A gauge invariant notion of a strong connection is presented and characterized. It is then used to justify the way in which a global curvature form is defined. Strong connections are interpreted as those that are induced from the base space of a quantum bundle. Examples of both strong and non-strong connections are provided. In particular, such connections are constructed on a quantum deformation of the fibration $S^2 -> RP^2$. A certain class of strong $U_q(2)$-connections on a trivial quantum principal bundle is shown to be equivalent to the class of connections on a free module that are compatible with the q-dependent hermitian metric. A particular form of the Yang-Mills action on a trivial $U\sb q(2)$-bundle is investigated. It is proved to coincide with the Yang-Mills action constructed by A.Connes and M.Rieffel. Furthermore, it is shown that the moduli space of critical points of this action functional is independent of q. 
  It is shown, that in the general case the Kustaanheimo-Stiefel transformation (reduction of $4d$ oscillator on the twistor space by the Hamiltonian action of $U(1)$ with further time-reparametrization) leads to the integrable system, describing the interaction of a spinless electrically charged particle with a dyon . 
  By using a bosonization we uncover the topological gravity structure of Labastida, Pernici and Witten in ordinary $2d$ gravity coupled to $(p,q)$ minimal models. We study the cohomology class associated with the fermionic charge of the topological gravity which is shown to be isomorphic to that of the total $BRST$ charge. One of the ground ring generators of $c_M <1$ string theory is found to be in the equivariant cohomology of this fermionic charge. 
  String theory, if it describes nature, is probably strongly coupled. As a result, one might despair of making any statements about the theory. In the framework of a set of clearly spelled out assumptions, we show that this is not necessarily the case. Certain discrete gauge symmetries, combined with supersymmetry, tightly constrain the form of the effective action. Among our assumptions are that the true ground state can be obtained from some perturbative ground state by varying the coupling, and that the actual numerical value of the low energy field theoretic coupling ${g^2 \over 4\pi}$ is small. It follows that the low energy theory is approximately supersymmetric; corrections to the superpotential and gauge coupling function are small, while corrections to the Kahler potential are large; the spectrum of light particles is the same at strong as at weak coupling. We survey the phenomenological consequences of this viewpoint. We also note that the string axion can serve as QCD axion in this framework (modulo cosmological problems). 
  We consider correlation functions of the spin-$\half$ XXX and XXZ Heisenberg chains in a magnetic field. Starting from the algebraic Bethe Ansatz we derive representations for various correlation functions in terms of determinants of Fredholm integral operators. 
  Mass independent renormalization group functions in massive theories are related to normalization properties of the Green functions in the asymptotic region, where mass effects are neglected. In this special form the renormalization group invariance is restricted to the asymptotic region and consequently mass effects cannot be recoverd by a integration of the renormalziation group equation. It is shown, that mass independence results in the limit of a large normalization point, whenever a Callan-Symanzik equation exists and contains the same differential operators as the renormalization group equation. 
  The light-cone approach is reviewed. This method allows to find the underlying quantum field theory for any integrable lattice model in its gapless regime. The relativistic spectrum and S-matrix follows straightforwardly in this way through the Bethe Ansatz. We show here how to derive the infinite number of local commuting and non-local and non-commuting conserved charges in integrable QFT, taking the massive Thirring model (sine-Gordon) as an example. They are generated by quantum monodromy operators and provide a representation of $q-$deformed affine Lie algebras $U_q({\hat\G})$. Based on lectures delivered at the $XXX_q$ Karpacz Winter School, Poland, February 14-26, 1994. 
  We present here all the real algebras $\cal{A}$ with dim$\cal{A}\leq $5 and all 6-dimensional nilpotent ones with symmetric, invariant and non-degenerate metrics for which a WZW model can be constructed.   In three and four dimensions there are no other algebras than the well known $SU(2)$, $SU(1,1)$, $E_2^c$ and $H_4$. There exist only one five-dimensional and one six-dimensional nilpotent algebra with invariant non-degenerate metric and central charge $c=5, \,6$, respectively. We examine in details the five-dimensional case and, by gauging an appropriate subgroup, four-dimensional plane-wave string backgrounds are obtained. The corresponding background for the six-dimensional case is flat. 
  We propose a definition of a Poincar\'e algebra for a two dimensional space--time with one discretized dimension. This algebra has the structure of a Hopf algebra. We use the link between Onsager's uniformization of the Ising model and the dispersion relation of a free particle in this space--time, together with the rapidity representation of the quantum deformation of the Poincar\'e enveloping algebra. 
  We investigate what happens to the third order ferromagnetic phase transition displayed by the Ising model on various dynamical planar lattices (ie coupled to 2D quantum gravity) when we introduce annealed bond disorder in the form of either antiferromagnetic couplings or null couplings. We also look at the effect of such disordering for the Ising model on general $\phi^3$ and $\phi^4$ Feynman diagrams. 
  It is possible to extract values for critical couplings and gamma_string in matrix models by deriving a renormalization group equation for the variation of the of the free energy as the size N of the matrices in the theory is varied. In this paper we derive a ``renormalization group equation'' for the Penner model by direct differentiation of the partition function and show that it reproduces the correct values of the critical coupling and gamma_string and is consistent with the logarithmic corrections present for g=0,1. 
  This paper is essentially a short version of hep-th/9404046. We compute multiplicative anomaly det(AB)/(detA detB) =F(A,B) for elliptic pseudo-differential operators (PDOs) A, B on a closed manifold M in terms of their symbols. We prove that F(A,B)=1 for elliptic differential operators close to positive-definite ones on an odd-dimensional M. For such M we introduce a holomorphic determinant. Its monodromy lies in a finite group of roots of unity.   In general case, we relate the multiplicative anomaly with a central extension of the group of elliptic symbols and with an invariant quadratic form on this extension. We compute the Lie algebra of the central extension in terms of logarithmic symbols. The main tool is a new trace-type functional defined on classical PDOs of non-integer orders. A canonical det of elliptic PDOs generalizing the zeta-regularized det is introduced. 
  We review some ideas from a recent construction which introduced the notion of vertex operators and form factors as vacuum expectation values of related vertex operators in the space of fields. The vertex operators are constructed explicitly in radial quantization. These ideas are explained at the free-fermion point of the sine-Gordon theory. 
  A closed expression for the density operator of the damped harmonic oscillator is extracted from the master equation based on the Lindblad theory for open quantum systems. The entropy and effective temperature of the system are subsequently calculated and their temporal behaviour is surveyed by showing how these quantities relax to their equilibrium values. The entropy for a state characterized by a Wigner distribution function which is Gaussian in form is found to depend only on the variance of the distribution function. 
  The cumulants and factorial moments of photon distribution for squeezed and correlated light are calculated in terms of Chebyshev, Legendre and Laguerre polynomials. The phenomenon of strong oscillations of the ratio of the cumulant to factorial moment is found. 
  The specific form of the constant term in the asymptotic expansion of the heat-kernel on an axially-symmetric space with a codimension two fixed-point set of conical singularities is used to determine the associated conformal change of the effective action in four dimensions. Another derivation of the relevant coefficient is presented. 
  The effective potential for a dynamical Wick field (dynamical signature) induced by the quantum effects of massive fields on a topologically non-trivial $D$ dimensional background is considered. It is shown that when the radius of the compactified dimension is very small compared with $\Lambda^{1/2}$ (where $\Lambda$ is a proper-time cutoff), a flat metric with Lorentzian signature is preferred on ${\bf R}^4 \times {\bf S}^1$. When the compactification radius becomes larger a careful analysis of the 1-loop effective potential indicates that a Lorentzian signature is preferred in both $D=6$ and $D=4$ and that these results are relatively stable under metrical perturbations. 
  A representation theory of the quantized Poincar\'e ($\kappa$-Poincar\'e) algebra (QPA) is developed. We show that the representations of this algebra are closely connected with the representations of the non-deformed Poincar\'e algebra. A theory of tensor operators for QPA is considered in detail. Necessary and sufficient conditions are found in order for scalars to be invariants. Covariant components of the four-momenta and the Pauli-Lubanski vector are explicitly constructed.These results are used for the construction of some q-relativistic equations. The Wigner-Eckart theorem for QPA is proven. 
  The number state method is used to study soliton bands for three anharmonic quantum lattices: i) The discrete nonlinear Schr\"{o}dinger equation, ii) The Ablowitz-Ladik system, and iii) A fermionic polaron model. Each of these systems is assumed to have $f$-fold translational symmetry in one spatial dimension, where $f$ is the number of freedoms (lattice points). At the second quantum level $(n=2)$ we calculate exact eigenfunctions and energies of pure quantum states, from which we determine binding energy $(E_{\rm b})$, effective mass $(m^{*})$ and maximum group velocity $(V_{\rm m})$ of the soliton bands as functions of the anharmonicity in the limit $f \rightarrow \infty$. For arbitrary values of $n$ we have asymptotic expressions for $E_{\rm b}$, $m^{*}$, and $V_{\rm m}$ as functions of the anharmonicity in the limits of large and small anharmonicity. Using these expressions we discuss and describe wave packets of pure eigenstates that correspond to classical solitons. 
  We give a non-static exact solution of the Einstein-Maxwell equations (with null fluid), which is a non-static magnetic charge generalization to the Bonnor-Vaidya solution and describes the gravitational and electromagnetic fields of a nonrotating massive radiating dyon. In addition, using the energy-momentum pseudotensors of Einstein and Landau and Lifshitz we obtain the energy, momentum, and power output of the radiating dyon and find that both prescriptions give the same result. 
  The representation theory of affine Kac-Moody Lie algebras has grown tremendously since their independent introduction by Robert V. Moody and Victor G. Kac in 1968. Inspired by mathematical structures found by theoretical physicists, and by the desire to understand the ``monstrous moonshine'' of the Monster group, the theory of vertex operator algebras (VOA's) was introduced by Borcherds, Frenkel, Lepowsky and Meurman. An important subject in this young field is the study of modules for VOA's and intertwining operators between modules. Feingold, Frenkel and Ries defined a structure, called a vertex operator para-algebra(VOPA), where a VOA, its modules and their intertwining operators are unified. In this work, for each $l\geq 1$, we begin with the bosonic construction (from a Weyl algebra) of four level $-\shf$ irreducible representations of the symplectic affine Kac-Moody Lie algebra $C_l^{(1)}$. The direct sum of two of these is given the structure of a VOA, and the direct sum of the other two is given the structure of a twisted VOA-module. In order to define intertwining operators so that the whole structure forms a VOPA, it is necessary to separate the four irreducible modules, taking one As the VOA and the others as modules for it. This work includes the bosonic analog of the fermionic construction of a vertex operator superalgebra from the four level 1 irreducible modules of type $D_l^{(1)}$ given by Feingold, Frenkel and Ries. While they only get a VOPA when $l = 4$ using classical triality, the techniques in this work apply to any $l\geq 1$. 
  On a supersymmetric sigma model the covariantly constant forms are related to the conserved currents that are generators of a super W-algebra extending the superconformal algebra. The existence of covariantly constant forms restricts the holonomy group of the manifold. Via duality transformation we get new covariantly constant forms, thus restricting the holonomy group of the new manifold. 
  We verify the Goldstone theorem in the Gaussian functional approximation to the $\phi^{4}$ theory with internal O(2) symmetry. We do so by reformulating the Gaussian approximation in terms of Schwinger-Dyson equations from which an explicit demonstration of the Goldstone theorem follows directly. 
  The dual version of the D=10 N=1 supergravity (SUGRA) is considered in the superspace approach. The superstring (anomaly compensating) corrections are described by the 3-form superfield $A_{abc}$ . The complete set of dynamical equations for the $A$-field and for physical fields of the theory are presented. The solution of the $A$-field equations as a finite order polynomial in terms of curvature and graviphoton superfields is given. It makes possible to incorporate some of the superstring corrections in the dual supergravity in the explicit, supersymmetric and closed form. 
  Matrix models of 2d quantum gravity coupled to matter field are investigated by the renormalized perturbational method, in which the matrix model Hamiltonian is represented by the equivalent vector model. By the saddle point method, the renormalization group beta-function is obtained in the successive approximation. 
  We present a general method to bosonize systems of Fermions with infinitely many degrees of freedom, in particular systems of non-relativistic electrons at positive density, by expressing the quantized conserved electric charge- and current density in terms of a bosonic antisymmetric tensorfield of a rank d--1, where d is the dimension of space. This enables us to make concepts and tools from gauge theory available for the purpose of analyzing electronic structure of non-relativistic matter. We apply our bosonization identities and concepts from gauge theory, such as Wegner -'t Hooft duality, to a variety of systems of condensed matter physics: Landau-Fermi liquids, Hall fluids, London superconductors, etc.. Among our results are an exact formula for the plasmon gap in a metal, a simple derivation of the Anderson-Higgs mechanism in superconductors, and an analysis of the orthogonality catastrophe for static sources. 
  We study the general structure of Smirnov's axioms on form factors of local operators in integrable models. We find various consistency conditions that the form factor functions have to satisfy. For the special case of the $O(3)$ $\sigma$-model we construct simple polynomial solutions for the operators of the spin-field, current, energy-momentum tensor and topological charge density. 
  We demonstrate that a supersymmetric theory twisted on a K\"ahler four manifold $M=\Sigma_1 \times \Sigma_2 ,$ where $\Sigma_{1,2}$ are 2D Riemann surfaces, possesses a "left-moving" conformal stress tensor on $\Sigma_1$ ($\Sigma_2$) in the BRST cohomology. The central charge of the Virasoro algebra has a purely geometric origin and is proportional to the Euler characteristic of the $\Sigma_2$ ($\Sigma_1$) surface. This structure is shown to be invariant under renormalization group. We also give a representation of the algebra $W_{1+\infty}$ in terms of a free chiral supermultiplet. 
  The machinery of braided geometry introduced previously is used now to construct the $\epsilon$ `totally antisymmetric tensor' on a general braided vector space determined by R-matrices. This includes natural $q$-Euclidean and $q$-Minkowski spaces. The formalism is completely covariant under the corresponding quantum group such as $\widetilde{SO_q(4)}$ or $\widetilde{SO_q(1,3)}$. The Hodge $*$ operator and differentials are also constructed in this approach. 
  A method to construct Hamiltonian theories for systems of both ordinary and partial differential equations is presented. The knowledge of a Lagrangian is not at all necessary to achieve the result. The only ingredients required for the construction are one solution of the symmetry (perturbation) equation and one constant of the motion of the original system. It turns out that the Poisson bracket structure for the dynamical variables is far from being uniquely determined by the differential equations of motion. Examples in classical mechanics as well as in field theory are presented. 
  Let $\Xi$ stand for a finite abelian spin structure group of four-dimensional superstring theory in free fermionic formulation whose elements are 64-dimensional vectors (spin structure vectors) with rational entries belonging to $\rbrack -1,\, 1\rbrack $ and the group operation is the $mod\, \, 2 $ entry by entry summation $\oplus $ of these vectors. Let $B=\{b_i,\, i= 1,\cdots ,k+1\}$ be a set of spin structure vectors such that $b_i$ have only entries 0 and 1 for any $\, i= 1,\cdots ,k$, while $b_{k+1}$ is allowed to have any rational entries belonging to $\rbrack -1,\, 1\rbrack $ with even $N_{k+1}$, where $N_{k+1}$ stands for the least positive integer such that $N_{k+1}b_{k+1}= 0\,mod\,2$. Let $B$ be a basis of $\Xi$, i.e., let $B$ generate $\Xi$, and let $\Lambda_{m, n}$ stand for the transformation of $B$ which replaces $b_n$ by $b_m\oplus b_n$ for any $m \ne k+1$, $n \ne 1$, $m \ne n$. We prove that if $B$ satisfies the axioms for a basis of spin structure group $\Xi$, then $B'=\Lambda_{m, n}B$ also satisfies the axioms. Since the transformations $\Lambda_{m,n}$ for different $m$ and $n$ generate all nondegenerate transformations of the basis $B$ that preserve the vector $b_1$ and a single vector $ b_{k+1} $ with general rational entries, we conclude that the axioms are conditions for the whole group $\Xi$ and not just conditions for a particular choice of its basis. Hence, these transformations generate the discrete symmetry group of four-dimensional superstring models in free fermionic formulation. 
  The Skyrme model can be generalised to a situation where static fields are maps from one Riemannian manifold to another. Here we study a Skyrme model where physical space is two-dimensional euclidean space and the target space is the two-sphere with its standard metric. The model has topological soliton solutions which are exponentially localised. We describe a superposition procedure for solitons in our model and derive an expression for the interaction potential of two solitons which only involves the solitons' asymptotic fields. If the solitons have topological degree 1 or 2 there are simple formulae for their interaction potentials which we use to prove the existence of solitons of higher degree. We explicitly compute the fields and energy distributions for solitons of degrees between one and six and discuss their geometrical shapes and binding energies. 
  It is shown that the phase space path integral for a system with arbitrary second class constraints (primary, secondary ...) can be rewritten as a configuration space path integral of the exponent of the Lagrangian action with some local measure. 
  We compute the $O(1/N^2)$ correction to the critical exponent $2\lambda$ $=$ $-$ $\beta^\prime(g_c)$ for the chiral Gross Neveu model in arbitrary dimensions by substituting the corrections to the asymptotic scaling forms of the propagators into the Schwinger Dyson equations and solving the resulting consistency equations. 
  We show how the Newtonian potential between two heavy masses can be computed in simplicial quantum gravity. On the lattice we compute correlations between Wilson lines associated with the heavy particles and which are closed by the lattice periodicity. We check that the continuum analog of this quantity reproduces the Newtonian potential in the weak field expansion. In the smooth anti-de Sitter-like phase, which is the only phase where a sensible lattice continuum limit can be constructed in this model, we attempt to determine the shape and mass dependence of the attractive potential close to the critical point in $G$. It is found that non-linear graviton interactions give rise to a potential which is Yukawa-like, with a mass parameter that decreases towards the critical point where the average curvature vanishes. In the vicinity of the critical point we give an estimate for the effective Newton constant. 
  Considered are superparticle and superstring models invariant under supersymmetry in a target superspace and local extended worldsheet supersymmetry the latter replacing the fermionic $\kappa$--symmetry of the conventional Green--Schwarz formulations. (Talk given at the Conference on Geometry of Constrained Dynamical Systems,   Cambridge, June 15-18, 1994). 
  In terms of the gauged nonlinear $\sigma$-models, we describe some results and implications of solving the following problem: Given a smooth symplectic manifold as target space with a quasi-free Hamiltonian group action, perform the symplectic blowing up of the point singularity and identify the blow-up modes in the corresponding (gauged) $\sigma$-model. Both classical and quantum aspects of the construction are explained, along with illustrating examples from the toric projective space and the K\"ahler manifold. We also discuss related problems such as the origin of Mirror symmetry and the quantum cohomologies.(Talk to be given at ICHEP94, Glasgow, July 20-27.) 
  We characterize the derivation $d:A\to \Omega^1_{\der}(A)$ by a universal property introducing a new class of bimodules. 
  This paper is a generalization of previous work on the use of classical canonical transformations to evaluate Hamiltonian path integrals for quantum mechanical systems. Relevant aspects of the Hamiltonian path integral and its measure are discussed and used to show that the quantum mechanical version of the classical transformation does not leave the measure of the path integral invariant, instead inducing an anomaly. The relation to operator techniques and ordering problems is discussed, and special attention is paid to incorporation of the initial and final states of the transition element into the boundary conditions of the problem. Classical canonical transformations are developed to render an arbitrary power potential cyclic. The resulting Hamiltonian is analyzed as a quantum system to show its relation to known quantum mechanical results. A perturbative argument is used to suppress ordering related terms in the transformed Hamiltonian in the event that the classical canonical transformation leads to a nonquadratic cyclic Hamiltonian. The associated anomalies are analyzed to yield general methods to evaluate the path integral's prefactor for such systems. The methods are applied to several systems, including linear and quadratic potentials, the velocity-dependent potential, and the time-dependent harmonic oscillator. 
  We solve the problem of Fourier transformation for the one-dimensional $q$-deformed Heisenberg algebra. Starting from a matrix representation of this algebra we observe that momentum and position are unbounded operators in the Hilbert space. Therefore, in order to diagonalise the position operator in a momentum eigenbasis we have to study self-adjoint extensions of these operators. It turns out that there exist a whole family of such extensions for the position operator. This leads, correspondingly, to a one-parametric family of Fourier transformations. These transformations, which are related to continued fractions, are constructed in terms of $q$-deformed trigonometric functions. The entire family of the Fourier transformations turns out to be characterised by an elliptic function. 
  A two-dimensional nonrelativistic fermion system coupled to both electromagnetic gauge fields and Chern-Simons gauge fields is analysed. Polarization tensors relevant in the quantum Hall effect and anyon superconductivity are obtained as simple closed integrals and are evaluated numerically for all momenta and frequencies. The correction to the energy density is evaluated in the random phase approximation (RPA), by summing an infinite series of ring diagrams. It is found that the correction has significant dependence on the particle number density.   In the context of anyon superconductivity, the energy density relative to the mean field value is minimized at a hole concentration per lattice plaquette (0.05 \sim 0.06) (p_c a/\hbar)^2 where p_c and a are the momentum cutoff and lattice constant, respectively. At the minimum the correction is about -5 % \sim -25 %, depending on the ratio (2m \omega_c)/(p_c^2) where \omega_c is the frequency cutoff.   In the Jain-Fradkin-Lopez picture of the fractional quantum Hall effect the RPA correction to the energy density is very large. It diverges logarithmically as the cutoff is removed, implying that corrections beyond RPA become important at large momentum and frequency. 
  We show how the Moyal product of phase-space functions, and the Weyl correspondence between symbols and operator kernels, may be obtained directly using the procedures of geometric quantization, applied to the symplectic groupoid constructed by ``doubling'' the phase space. 
  $\chat=1$ fermionic string theory, which is considered as a fermionic string theory in two dimension, is shown to decompose into two mutually independent parts, one of which can be viewed as a topological model and the other is irrelevant for the theory. The physical contents of the theory is largely governed by this topological structure, and the discrete physical spectrum of $\chat=1$ string theory is naturally explained as the physical spectrum of the topological model. This topological structure turns out to be related with a novel hidden $N=2$ superconformal algebra (SCA) in the enveloping algebra of the $N=3$ SCA in fermionic string theories. 
  We show how to construct central and grouplike quantum determinants for FRT algebras A(R). As an application of the general construction we give a quantum determinant for the q-Lorentz group. 
  It is shown that the standard mod-$p$ valued intersection form can be used to define Boltzmann weights of subdivision invariant lattice models with gauge group $Z_{p}$. In particular, we discuss a four dimensional model which is based upon the assignment of field variables to the $2$-simplices of the simplicial complex. The action is taken to be the intersection form defined on the second cohomology group of the complex, with coefficients in $Z_{p}$. Subdivision invariance of the theory follows when the coupling constant is quantized and the field configurations are restricted to those satisfying a mod-$p$ flatness condition. We present an explicit computation of the partition function for the manifold $\pm CP^{2}$, demonstrating non-triviality. 
  A new version of NLQM is formulated in terms of the generalized Nambu dynamics. The generalization is free from the difficulties of earlier approaches. The paper is a second part of "Elements of NLQM (I): NL Schrodinger equation and two-level atoms". 
  An attempt is made to understand the root spaces of Kac Moody algebras of hyperbolic type, and in particular $E_{10}$, in terms of a DDF construction appropriate to a subcritical compactified bosonic string. While the level-one root spaces can be completely characterized in terms of transversal DDF states (the level-zero elements just span the affine subalgebra), longitudinal DDF states are shown to appear beyond level one. In contrast to previous treatments of such algebras, we find it necessary to make use of a rational extension of the self-dual root lattice as an auxiliary device, and to admit non-summable operators (in the sense of the vertex algebra formalism). We demonstrate the utility of the method by completely analyzing a non-trivial level-two root space, obtaining an explicit and comparatively simple representation for it. We also emphasize the occurrence of several Virasoro algebras, whose interrelation is expected to be crucial for a better understanding of the complete structure of the Kac Moody algebra. 
  It is shown that non-trivial topological sectors can prevent the quantum mechanical implementation of the symmetries of the classical field equations of sigma models with torsion. The associated anomaly is computed, and it is shown that it depends on the homotopy class of the topological sector of the theory and the group action on the sigma model manifold that generates the symmetries of the classical field equations. 
  We present some recently discovered infinite dimensional Lie algebras that can be understood as extensions of the algebra Map(M,g) of maps from a compact p-dimensional manifold to some finite dimensional Lie algebra g. In the first part of the paper, we describe the physical motivations for the study of these algebras. In the second part, we discuss their realization in terms of pseudo-differential operators and comment on their possible representation theory. 
  We describe the duality between different geometries which arises by considering the classical and quantum harmonic map problem. To appear in ``Essays on Mirror Manifolds II''. 
  We perform an explicit calculation of the lowest order effects of single eigenvalue instantons on the continuous sector of the collective field theory derived from a $d=1$ bosonic matrix model. These effects consist of certain induced operators whose exact form we exhibit. 
  We study the analytic Bethe ansatz in solvable vertex models associated with the Yangian $Y(X_r)$ or its quantum affine analogue $U_q(X^{(1)}_r)$ for $X_r = B_r, C_r$ and $D_r$. Eigenvalue formulas are proposed for the transfer matrices related to all the fundamental representations of $Y(X_r)$. Under the Bethe ansatz equation, we explicitly prove that they are pole-free, a crucial property in the ansatz. Conjectures are also given on higher representation cases by applying the $T$-system, the transfer matrix functional relations proposed recently. The eigenvalues are neatly described in terms of Yangian analogues of the semi-standard Young tableaux. 
  Supersymmetry breaking in string theory is expected to occur when moduli fields acquire non-trivial expectation values. In the early universe these fields start out displaced from their final destinations. I present some recent ideas about the cosmological evolution of the dilaton modulus field on the way to its vacuum expectation value. 
  A mathematically consistent procedure for coupling quasiclassical and quantum variables through coupled Hamilton-Heisenberg equations of motion is derived from a variational principle. During evolution, the quasiclassical variables become entangled with the quantum variables with the result that the value of the quasiclassical variables depends on the quantum state. This provides a formalism to compute the backreaction of any quantum system on a quasiclassical one. In particular, it leads to a natural candidate for a theory of gravity coupled to quantized matter in which the gravitational field is not quantized. 
  The $sl_q(2)$-quantum group invariant spin 1/2 XXZ-Heisenberg model with open boundary conditions is investigated by means of the Bethe ansatz. As is well known, quantum groups for $q$ equal to a root of unity possess a finite number of ``good'' representations with non-zero q-dimension and ``bad'' ones with vanishing q-dimension.  Correspondingly, the state space of an invariant Heisenberg chain decomposes into ``good'' and ``bad'' states. A ``good'' state may be described by a path of only ``good'' representations. It is shown that the ``good'' states are given by all ``good'' Bethe ansatz solutions with roots restricted to the first periodicity strip, i.e.  only positive parity strings (in the language of Takahashi) are allowed. Applying Bethe's string counting technique completeness of the ``good'' Bethe states is proven, i.e. the same number of states is found as the number of all restricted path's on the $sl_q(2)$-Bratteli diagram. It is the first time that a ``completeness" proof for an anisotropic quantum invariant reduced Heisenberg model is performed. 
  Bayesian complex probability theory is shown to be consistent with Bell's theorem and with other recent limitations on local realistic theories which agree with the predictions of quantum mechanics. 
  We give an analogue for vertex operator algebras and superalgebras of the notion of endomorphism ring of a vector space by means of a notion of ``local system of vertex operators'' for a (super) vector space. We first prove that any local system of vertex operators on a (super) vector space $M$ has a natural vertex (super)algebra structure with $M$ as a module. Then we prove that for a vertex (operator) superalgebra $V$, giving a $V$-module $M$ is equivalent to giving a vertex (operator) superalgebra homomorphism from $V$ to some local system of vertex operators on $M$. As applications, we prove that certain lowest weight modules for some well-known infinite-dimensional Lie algebras or Lie superalgebras have natural vertex operator superalgebra structures. We prove the rationality of vertex operator superalgebras associated to standard modules for an affine algebra. We also give an analogue of the notion of the space of linear homomorphisms from one module to another for a Lie algebra by introducing a notion of ``generalized intertwining operators.'' We prove that $G(M^{1},M^{2})$, the space of generalized intertwining operators from one module $M^{1}$ to another module $M^{2}$ for a vertex operator superalgebra $V$, is a generalized $V$-module. Furthermore, we prove that for a fixed vertex operator superalgebra $V$ and 
  We wish to report here on a recent approach to the non-commutative calculus on $q$-Minkowski space which is based on the reflection equations with no spectral parameter. These are considered as the expression of the invariance (under the coaction of the $q$-Lorentz group) of the commutation properties which define the different $q$-Minkowski algebras. This approach also allows us to discuss the possible ambiguities in the definition of $q$-Minkowski space ${\cal M}_q$ and its differential calculus. The commutation relations among the generators of ${\cal M}_q$ (coordinates), ${\cal D}_q$ (derivatives), $\Lambda_q$ (one-forms) and a few invariant (scalar) operators are established and compared with earlier results. 
  We present the effective potential for nonrelativistic matter coupled to non-Abelian Chern-Simons gauge fields. We perform the calculation using a functional method in constant background fields to satisfy Gauss's law and to simplify the computation. Both the quantum gauge and matter fields are integrated over. The gauge fixing is achieved with an $R_\xi$-gauge in the $\xi\to 0$ limit. Divergences appearing in the matter sector are regulated via operator regularization. We find no corrections to the Chern-Simons coupling constant and the system experiences conformal symmetry breaking to one-loop order except at the known value of self-duality. These results agree with previous analysis of the non-Abelian Aharonov-Bohm scattering. 
  In this paper, we examine the coupling of matter fields to gravity within the framework of the Standard Model of particle physics. The coupling is described in terms of Weyl fermions of a definite chirality, and employs only (anti)self-dual or left-handed spin connection fields. It is known from the work of Ashtekar and others that such fields can furnish a complete description of gravity without matter. We show that conditions ensuring the cancellation of perturbative chiral gauge anomalies are not disturbed. We also explore a global anomaly associated with the theory, and argue that its removal requires that the number of fundamental fermions in the theory must be multiples of 16. In addition, we investigate the behavior of the theory under discrete transformations P, C and T; and discuss possible violations of these discrete symmetries, including CPT, in the presence of instantons and the Adler-Bell-Jackiw anomaly. 
  Worldsheet supersymmetric string action is written in duality invariant form for flat as well as curved backgrounds. First the action in flat backgrounds is written by introducing auxiliary fields. We also give the superfield form of this action and obtain the offshell supersymmetry algebra. The action has a modified Lorentz invariance and supersymmetry and reduces to the usual form when the auxiliary fields are eliminated using their equations of motion. Supersymmetric nonlinear sigma model in curved backgrounds is also written in manifestly duality invariant form when the background metric and tortion fields are independent of some of the coordinates. 
  We describe a natural structure of an abelian intertwining algebra (in the sense of Dong and Lepowsky) on the direct sum of the untwisted vertex operator algebra constructed {}from the Leech lattice and its (unique) irreducible twisted module. When restricting ourselves to the moonshine module, we obtain a new and conceptual proof that the moonshine module has a natural structure of a vertex operator algebra. This abelian intertwining algebra also contains an irreducible twisted module for the moonshine module with respect to the obvious involution. In addition, it contains a vertex operator superalgebra and a twisted module for this vertex operator superalgebra with respect to the involution which is the identity on the even subspace and is $-1$ on the odd subspace. It also gives the superconformal structures observed by Dixon, Ginsparg and Harvey. 
  The complex unit appearing in the equations of quantum mechanics is generalised to a quaternionic structure on spacetime, leading to the consideration of complex quantum mechanical particles whose dynamical behaviour is governed by inhomogeneous Dirac and Schr\"{o}dinger equations. Mixing of hyper-complex components of wavefunctions occurs through their interaction with potentials dissipative into the extra quaternionic degrees of freedom. An interferometric experiment is analysed to illustrate the effect. 
  \noindent Using hydrodynamic collective field theory approach we show that one-particle density matrix of the $\nu=1/m$ fractional quantum Hall edge state interpolates between chiral Luttinger liquid behavior $\langle \psi^{\dagger}(r) \psi(0) \rangle \sim r^{-m} $ and Calogero-Sutherland model behavior $\langle \psi^{\dagger}(r) \psi(0) \rangle \sim r^{-(m+1/m)/2} $ as the droplet width is varied continuously. Low-energy excitations are described by $c=1$ conformal field theory of a compact boson of radius $\sqrt m$. The result suggests complementary relation between the two-dimensional quantum Hall droplet and the one-dimensional Calogero-Sutherland model. 
  A new theory for the conformal factor in R$^2$-gravity is developed. The infrared phase of this theory, which follows from the one-loop renormalization group equations for the whole quantum R$^2$-gravity theory is described. The one-loop effective potential for the conformal factor is found explicitly and a mechanism for inducing Einstein gravity at the minimum of the effective potential for the conformal factor is suggested. A comparison with the effective theory of the conformal factor induced by the conformal anomaly, and also aiming to describe quantum gravity at large distances, is done. 
  Studying the algebraic structure of the double ${\cal D}Y(g)$ of the yangian $Y(g)$ we present the triangular decomposition of ${\cal D}Y(g)$ and a factorization for the canonical pairing of the yangian with its dual inside ${\cal D}Y(g)$. As a consequence we obtain an explicit formula for the universal R-matrix $R$ of ${\cal D}Y(g)$ and demonstrate how it works in evaluation representations of $Y(sl_2)$. We interprete one-dimensional factor arising in concrete representations of $R$ as bilinear form on highest weight polynomials of irreducible representations of $Y(g)$ and express this form in terms of {\it gamma-functions}. 
  We reexamine a unitary-transformation method of extracting a physical Hamiltonian from a gauge field theory after quantizing all degrees of freedom including redundant variables. We show that this {\it quantum Hamiltonian reduction} method suffers from crucial modifications arising from regularization of composite operators. We assess the effects of regularization in the simplest gauge field theory, the Schwinger model. Without regularization, the quantum reduction yields the identical Hamiltonian with the classically reduced one. On the other hand, with regularization incorporated, the resulting Hamiltonian of the quantum reduction disagrees with that of the classical reduction. However, we find that the discrepancy is resolved by redefinitions of fermion currents and that the results are again consistent with those of the classical reduction. 
  It is known that the involution corresponding to the compact form is incompatible with comultiplication for quantum groups at $|q|=1$. In this paper we consider the quantum algebra of functions on the deformed space $T^{*}G_{q}$ which includes both the quantum group and the quantum universal enveloping algebra as subalgebras. For this extended object we construct an anti-involution which reduces to the compact form $*$-operation in the limit $q\rightarrow 1$. The algebra of functions on $T^{*}G_{q}$ endowed with the $*$-operation may be viewed as an algebra of observables of a quantum mechanical system. The most natural interpretation for such a system is a deformation of the quantum symmetric top. We suggest a discrete dynamics for this system which imitates the free motion of the top. 
  Perturbations of $WD_n$ and $W_3$ conformal theories which generalize the $(1,2)$ perturbations of conformal minimal models are shown to be integrable by counting argument. $A_{2n-1,q}^{(2)}$ and $D_{4,q}^ {(3)}$ symmetries of corresponding S-matrices are conjectured and proved by explicit construction of conserved nonlocal charges in the $WD_3$ case with the proper quantum group of symmetry. 
  After simultaneous compactification of spacetime and worldvolume on $K3$, the $D=10$ heterotic fivebrane with gauge group $SO(32)$ behaves like a $D=6$ heterotic string with gauge group $SO(28) \times SU(2)$, but with Kac--Moody levels different from those of the fundamental string. Thus the string/fivebrane duality conjecture in $D=10$ gets replaced by a string/string duality conjecture in $D=6$. Since $D=6$ strings are better understood than $D=10$ fivebranes, this provides a more reliable laboratory in which to test the conjecture. According to string/string duality, the Green--Schwarz factorization of the $D=6$ spacetime anomaly polynomial $I_{8}$ into $X_4\, \tilde{X}_4$ means that just as $X_4$ is the $\sigma$-model anomaly polynomial of the fundamental string worldsheet so $\tilde{X}_4$ should be the corresponding polynomial of the dual string worldsheet. To test this idea we perform a classical dual string calculation of $\tilde{X}_4$ and find agreement with the quantum fundamental string result. This also provides an {\it a posteriori} justification for assumptions made in a previous paper on string/fivebrane duality. Finally we speculate on the relevance of string/string duality to the vacuum degeneracy problem. 
  We search for alternatives to the trivial $\phi^4$ field theory by including arbitrary powers of the self-coupling. Such theories are renormalizable when the natural cutoff dependencies of the coupling constants are taken into account. We find a continuum of fixed points, which includes the well-known Gaussian fixed point. The fixed point density has a maximum at a location corresponding to a theory with a Higgs mass of approximately 2700 GeV. The Gaussian fixed point is UV stable in some directions in the extended parameter space. Along such directions we obtain non-trivial asymptotically free theories. 
  The central structure in various versions of noncommutative geometry is a differential calculus on an associative algebra. This is an analogue of the calculus of differential forms on a manifold. In this short review we collect examples of differential calculi on commutative algebras (which can be regarded as algebras of functions on some topological space). We explain how these are related to relevant structures in physics. 
  An accelerating observer sees a thermal bath of radiation at the Hawking temperature which is proportional to the acceleration. Also, in string theory there is a Hagedorn temperature beyond which one cannot go without an infinite amount of energy. Several authors have shown that in the context of Hawking radiation a limiting temperature for string theory leads to a limiting acceleration, which for a black hole implies a minimum distance from the horizon for an observer to remain stationary. We argue that this effectively introduces a cutoff in Rindler space or the Schwarzschild geometry inside of which accelerations would exceed this maximum value. Furthermore, this natural cutoff in turn allows one to define a finite entropy for Rindler space or a black hole as all divergences were occurring on the horizon. In all cases if a particular relationship exists between Newton's constant and the string tension then the entropy of the string modes agrees with the Bekenstein-Hawking formula. 
  It is pointed out that the action recently proposed by Ba\~nados et al. for gravitation in odd dimensions higher (and lower) than four, provides a natural quantization for the gravitational constant. These theories possess no dimensionful parameters and hence they may be power counting renormalizable. 
  We construct several quantum coset W-algebras, e.g. sl(2,R)/U(1) and sl(2,R)+sl(2,R) / sl(2,R), and argue that they are finitely nonfreely generated. Furthermore, we discuss in detail their role as unifying W-algebras of Casimir W-algebras. We show that it is possible to give coset realizations of various types of unifying W-algebras, e.g. the diagonal cosets based on the symplectic Lie algebras sp(2n) realize the unifying W-algebras which have previously been introduced as `WD_{-n}'. In addition, minimal models of WD_{-n} are studied. The coset realizations provide a generalization of level-rank-duality of dual coset pairs. As further examples of finitely nonfreely generated quantum W-algebras we discuss orbifolding of W-algebras which on the quantum level has different properties than in the classical case. We demonstrate in some examples that the classical limit according to Bowcock and Watts of these nonfreely finitely generated quantum W-algebras probably yields infinitely nonfreely generated classical W-algebras. 
  The paper consists of two parts. In the first part Schroedinger's equation for a charged quantum particle in a Galilei-Newton curved space-time is derived in a fully geometrical way. Gravitational and electromagnetic fields are coded into space metric and space-time connection. The fundamental geometrical object is a quantum connection in a Hermitian line bundle over the 7-dimensional jet space of 3-velocities. The secondary object is the bundle of Hilbert spaces over absolute time. Time appears as a superselection quantity while Shroedinger equation is interpreted as parallel transport in this bundle. In the second part the problem of measurement in quantum theory is discussed as a part of a more general problem of coupling between quantum and classical systems. The standard framework of quantum theory is extended so as to allow for dynamical central observables within dissipative dynamics. It is shown that within this approach one obtains not only Liouville equation that describes statistical ensembles, but also a piecewise-deterministic random process describing sequences of "events" that can be monitored by a continuous observation of the single, coupled classical system. It also describes "quantum jumps" or "wave packet reductions" that accompany these events. Two example are worked out in some details. The last one deals with the problem oof "how to determine the wave function ?". 
  The aim of this lecture is to give a pedagogical explanation of the notion of a Poisson Lie structure on the external algebra of a Poisson Lie group which was introduced in our previous papers. Using this notion as a guide we construct quantum external algebras on $SL_q(N)$ with proper (classical) dimension. 
  We show that Buscher's abelian duality transformation rules can be recovered in a very simple way by performing a canonical transformation first suggested by Giveon, Rabinovici and Veneziano. We explore the properties of this transformation, and also discuss some aspects of non-abelian duality. 
  We construct the string field Hamiltonian for $c=1-\frac{6}{m(m+1)}$ string theory in the temporal gauge. In order to do so, we first examine the Schwinger-Dyson equations of the matrix chain models and propose the continuum version of them. Results of boundary conformal field theory are useful in making a connection between the discrete and continuum pictures. The $W$ constraints are derived from the continuum Schwinger-Dyson equations. We also check that these equations are consistent with other known results about noncritical string theory. The string field Hamiltonian is easily obtained from the continuum Schwinger-Dyson equations. It looks similar to Kaku-Kikkawa's Hamiltonian and may readily be generalized to $c>1$ cases. 
  We present a ``topological'' formulation of arbitrarily shaped vortex strings in four dimensional field theory. By using a large Higgs mass expansion, we then evaluate the effective action of the closed Abrikosov-Nielsen-Olesen vortex string. It is shown that the effective action contains the Nambu-Goto term and an extrinsic curvature squared term with negative sign. We next evaluate the topological $\FtF$ term and find that it becomes the sum of an ordinary self-intersection number and Polyakov's self-intersection number of the world sheet swept by the vortex string. These self-intersection numbers are related to the self-linking number and the total twist number, respectively. Furthermore, the $\FtF$ term turns out to be the difference between the sum of the writhing numbers and the linking numbers of the vortex strings at the initial time and the one at the final time. When the vortex string is coupled to fermions, the chiral fermion number of the vortex string becomes the writhing number (modulo $\bZ$) through the chiral anomaly. Our formulation is also applied to ``global'' vortex strings in a model with a broken global $U(1)$ symmetry. 
  We discuss the semiclassical approximation to the level density of (super) strings propagating in non-compact coset manifolds $G/H$. We show that the WKB ansatz agrees with heuristic red-shift arguments with respect to the ``exact" sigma model metric, up to some deviations from minimal coupling, parametrized by the dilaton. This approximation is used to study thermal ensembles of free strings in black holes, with the ``brick wall" regularization of `t Hooft. In two dimensions the entropy diverges logarithmically with the horizon thickness, and a local Hagedorn transition occurs in higher dimensional models. We also observe that supersymmetry improves the regularity of strings at the horizon. 
  The area formula for entropy is extended to the case of a dilatonic black hole. The entropy of a scalar field in the background of such a black hole is calculated semiclassically. The area and cutoff dependences are normal {\it except in the extremal case}, where the area is zero but the entropy nonzero. 
  We first formulate a definition of tensor product for two modules for a vertex operator algebra in terms of a certain universal property and then we give a construction of tensor products. We prove the unital property of the adjoint module and the commutativity of tensor products, up to module isomorphism. We relate this tensor product construction with Frenkel and Zhu's $A(M)$-theory. We give a proof of a formula of Frenkel and Zhu for fusion rules. We also give the analogue of the ``Hom''-functor of classical Lie algebra theory for vertex operator algebra theory by introducing a notion of ``generalized intertwining operator.'' We prove that the space of generalized intertwining operators from one module to another for a vertex operator algebra is a generalized module. From this result we derive a general form of Tsuchiya and Kanie's ``nuclear democracy theorem'' for any rational vertex operator algebra. This proves that the fusion rules obtained from our construction of tensor products are the same as the fusion rules obtained by using Tsuchiya and Kanie's method, for both WZW models and minimal models. We prove that if $V$ satisfies certain ``finiteness'' and ``semisimplicity'' conditions, then there exists a unique maximal submodule inside the generalized module. Furthermore, we prove that this maximal submodule is isomorphic to the contragredient module of a certain tensor product module. This gives another construction of tensor product modules and this result turns out to be closely related to Huang and Lepowsky's construction. 
  In bosonic end perturbative calculations for quantum mechanical anyon systems a regularization and renormalization procedure, analogous to those used in field theory, is necessary. I examine the reliability and the physical interpretation of the most commonly used bosonic end regularization procedures. I then use the regularization procedure with the most transparent physical interpretation to derive some bosonic end perturbation theory results on anyon spectra, including a 3-anyon ground state energy. 
  We argue that the results obtained using the quantum mechanical method of self-adjoint extension of the Schr\"odinger Hamiltonian can also be derived using Feynman perturbation theory in the investigation of the corresponding non-relativistic field theories. We show that this is indeed what happens in the study of an anyon system, and, in doing so, we establish a field theoretical description for ``colliding anyons", {\it i.e.} anyons whose quantum mechanical wave functions satisfy the non-conventional boundary conditions obtained with the method of self-adjoint extension. We also show that analogous results hold for a system of non-abelian Chern-Simons particles. 
  We construct the most general supersymmetric two boson system that is integrable. We obtain the Lax operator and the nonstandard Lax representation for this system. We show that, under appropriate redefinition of variables, this reduces to the supersymmetric nonlinear Schr\"odinger equation without any arbitrary parameter which is known to be integrable. We show that this supersymmetric system has three local Hamiltonian structures just like the bosonic counterpart and we show how the supersymmetric KdV equation can be embedded into this system. 
  A scheme suitable for describing quantum nonultralocal models including supersymmetric ones is proposed. Braided algebras are generalised to be used through Baxterisation for constructing braided quantum Yang--Baxter equations.   Supersymmetric and some known nonultralocal models are derived in the framework of the present approach. 
  We give the general analytic solutions derived from the low energy string effective action for four dimensional Friedmann-Robertson-Walker models with dilaton and antisymmetric tensor field, considering both long and short wavelength modes of the $H$-field. The presence of a homogeneous $H$-field significantly modifies the evolution of the scale factor and dilaton. In particular it places a lower bound on the allowed value of the dilaton. The scale factor also has a lower bound but our solutions remain singular as they all contain regions where the spacetime curvature diverges signalling a breakdown in the validity of the effective action. We extend our results to the simplest Bianchi I metric in higher dimensions with only two scale factors. We again give the general analytic solutions for long and short wavelength modes for the $H$ field restricted to the three dimensional space, which produces an anisotropic expansion. In the case of $H$ field radiation (wavelengths within the Hubble length) we obtain the usual four dimensional radiation dominated FRW model as the unique late time attractor. 
  We use nonlinear realizations to describe the spontaneous breaking of $N=2$ supersymmetry to $N=1$ in four dimensions. We identify the Goldstone multiplet with an $N=1$ chiral superfield, and show that chiral $N=1$ matter is consistent with the partially broken $N=2$ supersymmetry. We find that the chiral matter can be in any representation of the gauge group; no mirror particles are required. We present the Goldstone action and the general couplings to $N=1$ matter to the first nontrivial order in the scale of symmetry breaking. 
  Contribution to the proceedings of the NATO workshop on the Electroweak Phase Transition and the Early Universe, Sintra, March 1994. To be published by Plenum Press. 
  For general finite temperature different from the Hawking one there appears a well known conical singularity in the Euclidean classical solution of gravitational equations. The method of regularizing the cone by regular surface is used to determine the curvature tensors for such a metrics. This allows one to calculate the one-loop matter effective action and the corresponding one-loop quantum corrections to the entropy in the framework of the path integral approach of Gibbons and Hawking. The two-dimensional and four-dimensional cases are considered. The entropy of the Rindler space is shown to be divergent logarithmically in two dimensions and quadratically in four dimensions that coincides with results obtained earlier. For the eternal 2D black hole we observe finite, dependent on the mass, correction to the entropy. The entropy of the 4D Schwarzschild black hole is shown to possess an additional (in comparison with the 4D Rindler space) logarithmically divergent correction which does not vanish in the limit of infinite mass of the black hole. We argue that infinities of the entropy in four dimensions are renormalized by the renormalization of the gravitational coupling. 
  A complete RST quantization of a CGHS model plus Strominger term is carried out. In so doing a conformal invariant theory with $\kappa=\frac{N}{12}$ is found, that is, without ghosts contribution. The physical consequences of the model are analysed and positive definite Hawking radiation is found. 
  This note has two purposes. First we establish that the map defined in [L, $\S 40.2.5$ (a)] is an isomorphism for certain admissible sequences. Second we show the map gives rise to a convex basis of Poincar\'e--Birkhoff--Witt (PBW) type for $\bup$, an affine untwisted quantized enveloping algebra of Drinfel$'$d and Jimbo. The computations in this paper are made possible by extending the usual braid group action by certain outer automorphisms of the algebra. 
  The four-dimensional superstring solutions define at low energy effective supergravity theories. A class of them extends successfully the validity of the standard model up to the string scale (${\cal O}(10^{17})~TeV$). We stress the importance of string corrections which are relevant for low energy (${\cal O}(1)~TeV$) predictions of gauge and Yukawa couplings as well as the spectrum of the supersymmetric particles. A class of exact string solutions are also presented, providing non trivial space-time backgrounds, from which we can draw some lessons concerning the regions of space-time where the notion of the effective field theory prescription make sense. We show that the string gravitational phenomena may induce during the cosmological evolution, transitions from one effective field theory prescription to a different one where the geometrical and topological data, as well as the relevant observable states are drastically different. Talk presented by C. Kounnas in the XXIX Moriond Meeting, M\'eribel, France. 
  Exact string solutions are presented, where moduli fields are varying with time. They provide examples where a dynamical change of the topology of space is occurring. Some other solutions give cosmological examples where some dimensions are compactified dynamically or simulate pre-big bang type scenarios. Some lessons are drawn concerning the region of validity of effective theories and how they can be glued together, using stringy information in the region where the geometry and topology are not well defined from the low energy point of view. Other time dependent solutions are presented where a hierarchy of scales is absent. Such solutions have dynamics which is qualitatively different and resemble plane gravitational waves. Talk presented by E. Kiritsis in the 2\`eme Journ\'ee Cosmologie, Observatoire de Paris, 2-4 June 1994. 
  We consider the anomalies of $W_\infty$ gravity in the context of path-integralquantization. We derive the ghost-loop anomalies to all orders in $\hbar$ directly from the path-integral measure by the Fujikawa method. We also show that in the matter sector the higher-loop anomalies can be obtained by implementation of the Wess-Zumino consistency condition using the one-loop anomaly. Cancellation of the anomalies between these two sectors then leaves the theory anomaly-free. 
  We consider two-dimensional QED with several fermion flavors on a finite spatial circle. A modified version of the model with {\em flavor-dependent} boundary conditions $\psi_p(L) = e^{2\pi ip/ N} \psi_p(0)$, $p = 1, \ldots , N$ is discussed ($N $ is the number of flavors). In this case a non-contactable contour in the space of the gauge fields is {\em not} determined by large gauge transformations. The Euclidean path integral acquires the contribution from the gauge field configurations with fractional topological charge. The configuration with $\nu = 1/N$ is responsible for the formation of the fermion condensate $\langle\bar{\psi}_p \psi_p\rangle_0$. The condensate dies out as a power of $L^{-1}$ when the length $L$ of the spatial box is sent to infinity. Implications of this result for non-abelian gauge field theories are discussed in brief. 
  The problem of information loss is considered under the assumption that the process of black hole evaporation terminates in the decay of the black hole interior into a baby universe. We show that such theories can be decomposed into superselection sectors labeled by eigenvalues of the third-quantized baby universe field operator, and that scattering is unitary within each superselection sector. This result relies crucially on the quantum-mechanical variability of the decay time. It is further argued that the decay rate in the black hole rest frame is necessarily proportional to $e^{-S_{tot}}$, where $S_{tot}$ is the total entropy produced during the evaporation process, entailing a very long-lived remnant. 
  Expressions are given for the elliptic genera of the Kazama-Suzuki models associated with hermitian symmetric spaces when the problems of field identifications are absent. We use the models' known Coulomb gas descriptions. 
  We give an explicit description of "bundles of conformal blocks" in Wess-Zumino-Witten models of Conformal field theory and prove that integral representations of Knizhnik-Zamolodchikov equations constructed earlier by the second and third authors are in fact sections of these bundles. 
  Noticing that really the fermions of the Standard Model are best thought of as Weyl - rather than Dirac - particles (relative to fundamental scales located at some presumably very high energies) it becomes interesting that the experimental space-time dimension is singled out by the Weyl equation: It is observed that precisely in the experimentally true space-time dimensionality 4=3+1 the number of linearly independent matrices $n_{Weyl}^2$ dimensionized as the matrices in the Weyl equation equals the dimension $d$. So just in this dimension (in fact, also in a trivial case $d=1$) do the sigma-matrices of the Weyl-equation form a basis. It is also characteristic for this dimension that there is no degeneracy of helicity states of the Weyl spinor for all nonzero momenta. We would like to interpret these features to signal a special ``form stability'' of the Weyl equation in the phenomenologically true dimension of space-time. In an attempt of making this stability to occur in an as large as possible basin of allowed modifications we discuss whether it is possible to define what we could possibly mean by ``stability of Natural laws''. 
  We look at various features of the Standard Model with the purpose of exploring some possibilities of how to seek physical laws beyond it, i.e. at even smaller distances. Only parameters and structure which are not calculable from the Standard Model is considered useful information. Ca. $90$ bits of information contained in the system of representations in the Standard Model are explained by four reasonable postulates. A crude estimate is that there is of the order of $\sim 2 \times 10^2$ useful bits of unexplained information left today. There are several signs of the fact that the Standard Model is a low energy tail of a more fundamental theory (not yet known). However, some worries are expressed as concerns how far the exploration of the physics beyond the Standard Model can proceed - if we are to be inspired from these $\sim 2 \times 10^2$ bits alone. 
  We describe the role of Rational Hopf Algebras as the symmetries of rational field theories and discuss their relation with algebraic field theory, braided monoidal categories and modular fusion rule algebras. 
  By introducing a $\int dt \, g\left(\Tr \Phi^2(t)\right)^2$ term into the action of the $c=1$ matrix model of two-dimensional quantum gravity, we find a new critical behavior for random surfaces. The planar limit of the path integral generates multiple spherical ``bubbles'' which touch one another at single points. At a special value of $g$, the sum over connected surfaces behaves as $\Delta^2 \log\Delta$, where $\Delta$ is the cosmological constant (the sum over surfaces of area $A$ goes as $A^{-3}$). For comparison, in the conventional $c=1$ model the sum over planar surfaces behaves as $\Delta^2/ \log\Delta$. 
  We study the dynamics of the boundary dilaton gravity coupled to N massles scalars. We rederive the boundary conditions of [1] and [3] in a way which makes the requirement of reparametrization invariance and role of conformal anomaly explicit. We then study the semiclassical behaviour of the boundary in the N = 24 theory in the presence of an incoming matter wave with a constant energy flux spreaded over a finite interval. There is a critical value of the matter energy density below which the boundary is stable and all the matter is reflected back. For energy densities greater than this critical value there is a similar behaviour for small values of total energy thrown in. However, when the total energy exceeds another critical value, the boundary exibits a runaway behaviour and the spacetime devolopes singularities and horizons. 
  A straightforward relationship between the two approaches to 3-dimensional topological invariants, one of them put forward by Witten in the framework of topological quantum field theory, and the second one proposed by Kohno in terms of rational conformal field theory, is established. 
  We show that the 2-matrix string model corresponds to a coupled system of $2+1$-dimensional KP and modified KP ($\KPm$) integrable equations subject to a specific ``symmetry'' constraint. The latter together with the Miura-Konopelchenko map for $\KPm$ are the continuum incarnation of the matrix string equation. The $\KPm$ Miura and B\"{a}cklund transformations are natural consequences of the underlying lattice structure. The constrained $\KPm$ system is equivalent to a $1+1$-dimensional generalized KP-KdV hierarchy related to graded ${\bf SL(3,1)}$. We provide an explicit representation of this hierarchy, including the associated ${\bf W(2,1)}$-algebra of the second Hamiltonian structure, in terms of free currents. 
  These lecture notes are devoted to the theory of equations of associativity describing geometry of moduli spaces of 2D topological field theories. Introduction. Lecture 1. WDVV equations and Frobenius manifolds. {Appendix A.} Polynomial solutions of WDVV. {Appendix B.} Symmetriies of WDVV. Twisted Frobenius manifolds. {Appendix C.} WDVV and Chazy equation. Affine connections on curves with projective structure. Lecture 2. Topological conformal field theories and their moduli. Lecture 3. Spaces of isomonodromy deformations as Frobenius manifolds. {Appendix D.} Geometry of flat pencils of metrics. {Appendix E.} WDVV and Painlev\'e-VI. {Appendix F.} Branching of solutions of the equations of isomonodromic deformations and braid group. {Appendix G.} Monodromy group of a Frobenius manifold. {Appendix H.} Generalized hypergeometric equation associated to a Frobenius manifold and its monodromy. {Appendix I.} Determination of a superpotential of a Frobenius manifold. Lecture 4. Frobenius structure on the space of orbits of a Coxeter group. {Appendix J.} Extended complex crystallographic groups and twisted Frobenius manifolds. Lecture 5. Differential geometry of Hurwitz spaces. Lecture 6. Frobenius manifolds and integrable hierarchies. Coupling to topological gravity. 
  The heat kernel for the spin-3/2 Rarita-Schwinger gauge field on an arbitrary Ricci flat space-time ($d>2$) is investigated in a family of covariant gauges with one gauge parameter $\alpha$. The $\alpha$-dependent term of the kernel is expressed by the spin-1/2 heat kernel. It is shown that the axial anomaly and the one-loop divegence of the action are $\alpha$-independent, and that the conformal anomaly has an $\alpha$-dependent total derivative term in $d=2m\geq6$ dimensions. 
  We consider BF-type topological field theory coupled to non-dynamical particle and string sources on spacetime manifolds of the form $\IR^1\times\MT$, where $\MT$ is a 3-manifold without boundary. Canonical quantization of the theory is carried out in the Hamiltonian formalism and explicit solutions of the Schr\"odinger equation are obtained. We show that the Hilbert space is finite dimensional and the physical states carry a one-dimensional projective representation of the local gauge symmetries. When $\MT$ is homologically non-trivial the wavefunctions in addition carry a multi-dimensional projective representation, in terms of the linking matrix of the homology cycles of $\MT$, of the discrete group of large gauge transformations. The wavefunctions also carry a one-dimensional representation of the non-trivial linking of the particle trajectories and string surfaces in $\MT$. This topological field theory therefore provides a phenomenological generalization of anyons to (3 + 1) dimensions where the holonomies representing fractional statistics arise from the adiabatic transport of particles around strings. We also discuss a duality between large gauge transformations and these linking operations around the homology cycles of $\MT$, and show that this canonical quantum field theory provides novel quantum representations of the cohomology of $\MT$ and its associated motion group. 
  We study the target-space duality transformations in $p$--branes as transformations which mix the worldvolume field equations with Bianchi identities. We consider an $(m+p+1)$-dimensional spacetime with $p+1$ dimensions compactified, and a particular form of the background fields. We find that while a $GL(2)=SL(2)\times R$ group is realized when $m=0$, only a two parameter group is realized when $m>0$. 
  We present an extension of ``smooth bosonization'' to the non-Abelian case. We construct an enlarged theory containing both bosonic and fermionic fields which exhibits a local chiral gauge symmetry. A gauge fixing function depending on one real parameter allows us to interpolate smoothly between a purely fermionic and a purely bosonic representation. The procedure is, in the special case of bosonization, complementary to the approach based on duality. 
  We go on in the program of investigating the removal of divergences of a generical quantum gauge field theory, in the context of the Batalin-Vilkovisky formalism. We extend to open gauge-algebrae a recently formulated algorithm, based on redefinitions $\delta\lambda$ of the parameters $\lambda$ of the classical Lagrangian and canonical transformations, by generalizing a well- known conjecture on the form of the divergent terms. We also show that it is possible to reach a complete control on the effects of the subtraction algorithm on the space ${\cal M}_{gf}$ of the gauge-fixing parameters. A principal fiber bundle ${\cal E}\rightarrow {\cal M}_{gf}$ with a connection $\omega_1$ is defined, such that the canonical transformations are gauge transformations for $\omega_1$. This provides an intuitive geometrical description of the fact the on shell physical amplitudes cannot depend on ${\cal M}_{gf}$. A geometrical description of the effect of the subtraction algorithm on the space ${\cal M}_{ph}$ of the physical parameters $\lambda$ is also proposed. At the end, the full subtraction algorithm can be described as a series of diffeomorphisms on ${\cal M}_{ph}$, orthogonal to ${\cal M}_{gf}$ (under which the action transforms as a scalar), and gauge transformations on ${\cal E}$. In this geometrical context, a suitable concept of predictivity is formulated. We give some examples of (unphysical) toy models that satisfy this requirement, though being neither power counting renormalizable, nor finite. 
  In this letter some properties of the Gauss decomposition of quantum group $GL_q(n)$ with application to q-bosonization are considered. 
  The condition of having an $N=1$ spacetime supersymmetry for heterotic string leads to 4 distinct possibilities for compactifications namely compactifications down to 6,4,3 and 2 dimensions. Compactifications to 6 and 4 dimensions have been studied extensively before (corresponding to $K3$ and a Calabi-Yau threefold respectively). Here we complete the study of the other two cases corresponding to compactification down to 3 on a 7 dimensional manifold of $G_2$ holonomy and compactification down to 2 on an 8 dimensional manifold of $Spin(7)$ holonomy. We study the extended chiral algebra and find the space of exactly marginal deformations. It turns out that the role the $U(1)$ current plays in the $N=2$ superconformal theories, is played by tri-critical Ising model in the case of $G_2$ and Ising model in the case of $Spin(7)$ manifolds.   Certain generalizations of mirror symmetry are found for these two cases. We also discuss a topological twisting in each case. 
  We consider 2-dimensional QCD on a cylinder, where space is a circle of length $L$. We formulate the theory in terms of gauge-invariant gluon operators and multiple-winding meson (open string) operators. The meson bilocal operators satisfy a $W_\infty$ current algebra. The gluon sector (closed strings) contains purely quantum mechanical degrees of freedom. The description of this sector in terms of non-relativistic fermions leads to a $W_\infty$ algebra. The spectrum of excitations of the full theory is therefore organized according to two different algebras: a wedge subalgebra of $W_\infty$ current algebra in the meson sector and a wedge subalgebra of $W_\infty$ algebra in the glueball sector. In the large $N$ limit the theory becomes semiclassical and an effective description for the gluon degrees of freedom can be obtained. We have solved the effective theory of the gluons in the small $L$ limit. We get a glueball spectrum which coincides with the `discrete states' of the (Euclidean) $c=1$ string theory. We remark on the implications of these results for (a) QCD at finite temperature and (b) string theory. 
  We prove that the linear delta expansion for energy eigenvalues of the quantum mechanical anharmonic oscillator converges to the exact answer if the order dependent trial frequency $\Omega$ is chosen to scale with the order as $\Omega=CN^\gamma$; $1/3<\gamma<1/2$, $C>0$ as $N\rightarrow\infty$. It converges also for $\gamma=1/3$, if $C\geq\alpha_c g^{1/3}$, $\alpha_c\simeq 0.570875$, where $g$ is the coupling constant in front of the operator $q^4/4$. The extreme case with $\gamma=1/3$, $C=\alpha_cg^{1/3}$ corresponds to the choice discussed earlier by Seznec and Zinn-Justin and, more recently, by Duncan and Jones. 
  We consider the Nielsen identities for the two-point functions of full QCD and QED in the class of Lorentz gauges. For pedagogical reasons the identities are first derived in QED to demonstrate the gauge independence of the photon self-energy, and of the electron mass shell. In QCD we derive the general identity and hence the identities for the quark, gluon and ghost propagators. The explicit contributions to the gluon and ghost identities are calculated to one-loop order, and then we show that the quark identity requires that in on-shell schemes the quark mass renormalisation must be gauge independent. Furthermore, we obtain formal solutions for the gluon self-energy and ghost propagator in terms of the gauge dependence of other, independent Green functions. 
  We present an alternative to the Higgs mechanism to generate masses for non-abelian gauge fields in (3+1)-dimensions. The initial Lagrangian is composed of a fermion with current-current and dipole-dipole type self-interactions minimally coupled to non-abelian gauge fields. The mass generation occurs upon the fermionic functional integration. We show that by fine-tuning the coupling constants the effective theory contains massive non-abelian gauge fields without any residual scalars or other degrees of freedom. 
  We present the one-loop scalar field effective potential for the $N=2$ supersymmetric nonrelativistic self-interacting matter fields coupled to an Abelian Chern-Simons gauge field and for its generalization when bosonic matter fields are coupled to non-Abelian Chern-Simons field. In both models, Gauss's law linearly relates the magnetic field to the matter field densities; hence, we also include radiative effects from the background gauge field. We compute the scalar field effective potentials in two gauge families, a gauge reminiscent of the $R_\xi$-gauge in the limit $\xi\rightarrow 0$ and in the Coulomb family gauges. We regularize the theory with operator regularization and a cutoff to demonstrate that the results are independent of the regularization scheme. 
  We investigate the possibility that stringy nonperturbative effects appear as holes in the world-sheet. We focus on the case of Dirichlet string theory, which we argue should be formulated differently than in previous work, and we find that the effects of boundaries are naturally weighted by $e^{-O(1/g_{\rm st})}$. 
  In this paper I construct lattice models with an underlying $U_q osp(2,2)$ superalgebra symmetry. I find new solutions to the graded Yang-Baxter equation. These {\it trigonometric} $R$-matrices depend on {\it three} continuous parameters, the spectral parameter, the deformation parameter $q$ and the $U(1)$ parameter, $b$, of the superalgebra. It must be emphasized that the parameter $q$ is generic and the parameter $b$ does not correspond to the `nilpotency' parameter of \cite{gs}. The rational limits are given; they also depend on the $U(1)$ parameter and this dependence cannot be rescaled away. I give the Bethe ansatz solution of the lattice models built from some of these $R$-matrices, while for other matrices, due to the particular nature of the representation theory of $osp(2,2)$, I conjecture the result. The parameter $b$ appears as a continuous generalized spin. Finally I briefly discuss the problem of finding the ground state of these models. 
  We construct the action-angle variables of a classical integrable model defined on complex projective phase space and calculate the quantum mechanical propagator in the coherent state path integral representation using the stationary phase approximation. We show that the resulting expression for the propagator coincides with the exact propagator which was obtained by solving the time-dependent Schr\"odinger equation. 
  We consider a two matrix model with gaussian interaction involving the term $tr ABAB$, which is quartic in angular variables. It describes a vertex model (in particular case - of F-model type) on the lattice of fluctuating geometry and is the simplest representative of the class of matrix models describing coupling to two-dimensional gravity of general vertex models. This class includes most of physically interesting matrix models, such as lattice gauge theories and matrix models describing extrinsic curvature strings. We show that the system of loop (Schwinger-Dyson) equations of the model decouples in the planar limit and allows one to find closed equations for arbitrary correlators, including the ones involving angular variables. This provides a solution of the model in the planar limit. We write down the equations for the two-point function and the eigenvalue density and sketch the calculation of perturbative corrections to the free case. 
  The kinematics of SL(2,R) Yang-Mills theory on a circle is considered, for reasons that are spelled out. The gauge transformations exhibit hyperbolic fixed points, and this results in a physical configuration space with a non-Hausdorff "network" topology. The ambiguity encountered in canonical quantization is then much more pronounced than in the compact case, and can not be resolved through the kind of appeal made to group theory in that case. 
  The electron-positron `box' diagram produces an effective action which is fourth order in the electromagnetic field. We examine the behaviour of this effective action at high-temperature (in analytically continued imaginary-time thermal perturbation theory). We argue that there is a finite, nonzero limit as $T\rightarrow \infty$ (where $T$ is the temperature). We calculate this limit in the nonrelativistic static case, and in the long-wavelength limit. We also briefly discuss the self-energy in 2-dimensional QED, which is similar in some respects. 
  This paper provides a description of an algebraic setting for the Lagrangian formalism over graded algebras and is intended as the necessary first step towards the noncommutative C-spectral sequence (variational bicomplex). A noncommutative version of integration procedure, the notion of adjoint operator, Green's formula, the relation between integral and differential forms, conservation laws, Euler operator, Noether's theorem is considered. 
  An explanation for the so-called constrained hierarhies is presented by linking them with the symmetries of the KP hierarchy. While the existence of ordinary symmetries (belonging to the hierarchy) allows one to reduce the KP hierarchy to the KdV hierarchies, the existence of additional symmetries allows to reduce KP to the constrained KP. 
  In this paper we investigate the vanishing of cosmological singularities by quantization. Starting from a 5d Kaluza--Klein approach we quantize, as a first step, the non--spherical metric part and the dilaton field. These fields which are classically singular become smooth after quantization. In addition, we argue that the incorporation of non perturbative quantum corrections form a dilaton potential. Technically, the procedure corresponds to the quantization of 2d dilaton gravity and we discuss several models. From the 4d point of view this procedure is a semiclassical approach where only the dilaton and moduli matter fields are quantized. 
  We discuss the possibility to suppress the collapse in the nonlinear 2+1 D Schr\"odinger equation by using the gauge theory of strong phase correlations. It is shown that invariance relative to $q$-deformed Hopf algebra with deformation parameter $q$ being the fourth root of unity makes the values of the Chern-Simons term coefficient, $k=2$, and of the coupling constant, $g=1/2$, fixed; no collapsing solutions are present at those values. 
  Quadratic Poisson brackets on a vector space equipped with a bilinear multiplication are studied. A notion of a bracket compatible with the multiplication is introduced and an effective criterion of such compatibility is given. Among compatible brackets, a subclass of coboundary brackets is described, and such brackets are enumerated in a number of examples. 
  We analyze in detail the equivariant supersymmetry of the $G/G$ model. In spite of the fact that this supersymmetry does not model the infinitesimal action of the group of gauge transformations, localization can be established by standard arguments. The theory localizes onto reducible connections and a careful evaluation of the fixed point contributions leads to an alternative derivation of the Verlinde formula for the $G_{k}$ WZW model. We show that the supersymmetry of the $G/G$ model can be regarded as an infinite dimensional realization of Bismut's theory of equivariant Bott-Chern currents on K\"ahler manifolds, thus providing a convenient cohomological setting for understanding the Verlinde formula. We also show that the supersymmetry is related to a non-linear generalization (q-deformation) of the ordinary moment map of symplectic geometry in which a representation of the Lie algebra of a group $G$ is replaced by a representation of its group algebra with commutator $[g,h] = gh-hg$. In the large $k$ limit it reduces to the ordinary moment map of two-dimensional gauge theories. 
  The integrability of two symplectic maps, that can be considered as discrete-time analogs of the Garnier and Neumann systems is established in the framework of the $r$-matrix approach, starting from their Lax representation. In contrast with the continuous case, the $r$-matrix for such discrete systems turns out to be of dynamical type; remarkably, the induced Poisson structure appears as a linear combination of compatible ``more elementary" Poisson structures. It is also shown that the Lax matrix naturally leads to define separation variables, whose discrete and continuous dynamics is investigated. 
  This is a synopsis and extension of Phys.~Rev.~{\em D49} 5408 (1994). The Pseudodual Chiral Model illustrates 2-dimensional field theories which possess an infinite number of conservation laws but also allow particle production, at variance with naive expectations---a folk theorem of integrable models. We monitor the symmetries of the pseudodual model, both local and nonlocal, as transmutations of the symmetries of the (very different) usual Chiral Model. We refine the conventional algorithm to more efficiently produce the nonlocal symmetries of the model. We further find the Canonical transformation which connects the usual chiral model to its fully equivalent dual model, thus contradistinguishing the pseudodual theory. 
  The theory of vectors and spinors in 9+1 dimensional spacetime is introduced in a completely octonionic formalism based on an octonionic representation of the Clifford algebra $\Cl(9,1)$. The general solution of the classical equations of motion of the CBS superparticle is given to all orders of the Grassmann hierarchy. A spinor and a vector are combined into a $3 \times 3$ Grassmann, octonionic, Jordan matrix in order to construct a superspace variable to describe the superparticle. The combined Lorentz and supersymmetry transformations of the fermionic and bosonic variables are expressed in terms of Jordan products. 
  In this paper we present representations of the recently introduced dilute Birman-Wenzl-Murakami algebra. These representations, labelled by the level-$l$ B$^{(1)}_n$, C$^{(1)}_n$ and D$^{(1)}_n$ affine Lie algebras, are Baxterized to yield solutions to the Yang-Baxter equation.   The thus obtained critical solvable models are RSOS counterparts of the, respectively, D$^{(2)}_{n+1}$, $A^{(2)}_{2n}$ and B$^{(1)}_n$ $R$-matrices of Bazhanov and Jimbo. For the D$^{(2)}_{n+1}$ and B$^{(1)}_n$ algebras the RSOS models are new. An elliptic extension which solves the Yang-Baxter equation is given for all three series of dilute RSOS models. 
  We describe vector valued conjugacy equivariant functions on a group K in two cases -- K is a compact simple Lie group, and K is an affine Lie group.   We construct such functions as weighted traces of certain intertwining operators between representations of K. For a compact group $K$, Peter-Weyl theorem implies that all equivariant functions can be written as linear combinations of such traces. Next, we compute the radial parts of the Laplace operators of $K$ acting on conjugacy equivariant functions and obtain a comple- tely integrable quantum system with matrix coefficients, which in a special case coincides with the trigonometric Calogero-Sutherland-Moser multi-particle system. In the affine Lie group case, we prove that the space of equivariant functions having a fixed homogeneity degree with respect to the action of the center of the group is finite-dimensional and spanned by weighted traces of intertwining operators. This space coincides with the space of Wess-Zumino-Witten conformal blocks on an elliptic curve. We compute the radial part of the second order Laplace operator on the affine Lie group acting on equivariant functions, and find that it is a certain parabolic partial differential operator, which degenerates to the elliptic Calogero-Sutherland-Moser hamiltonian as the central charge tends to minus the dual Coxeter number (the critical level). Quantum integrals of this hamiltonian are obtained as radial part of the higher Sugawara operators which are central at the critical level. 
  We analyze the Chern-Simons field theory coupled to non-relativistic matter field on a sphere using canonical transformation on the fields with special attention to the role of the rotation symmetry: SO(3) invariance restricts the Hilbert space to the one with a definite number of charges and dictates Dirac quantization condition to the Chern-Simons coefficient, whereas SO(2) invariance does not. The corresponding Schr\"odinger equation for many anyons (and for multispecies) on the sphere are presented with appropriate boundary condition. In the presence of an external magnetic monopole source, the ground state solutions of anyons are compared with monopole harmonics. The role of the translation and modular symmetry on a torus is also expounded. 
  ${\bf Z}_2\times {\bf Z}_2$ Coxeter orbifolds are constructed with the property that some twisted sectors have fixed planes for which the six-torus can not be decomposed into a direct sum ${\bf T}^2\bigoplus{\bf T}^4 $ with the fixed plane lying in ${\bf T}^2$. The string loop threshold corrections to the gauge coupling constants are derived, and display symmetry groups for the $T$ and $U$ moduli that are subgroups of the full modular group $PSL(2,Z)$. The effective potential for duality invariant gaugino condensate in the presence of hidden sector matter is constructed and minimized for the values of the moduli. The effect of Wilson lines on the modular symmetries is also studied. 
  Some consequences of a $qp$-quantization of a point group invariant developed in the enveloping algebra of $SU(2)$ are examined in the present note. A set of open problems concerning such invariants in the $U_{qp}(u(2))$ quantum algebra picture is briefly discussed. 
  We examine the behavior of the non-linear interactions between electromagnetic fields at high temperature. It is shown that, in general, the log(T) dependence on the temperature of the Green functions is simply related to their UV behavior at zero-temperature. We argue that the effective action describing the nonlinear thermal electromagnetic interactions has a finite limit as T tends to infinity. This thermal action approaches, in the long wavelength limit, the negative of the corresponding zero-temperature action. 
  The nonlinear realization of the superconformal symmetry in two dimensions is considered. The superconformal symmetry is realized by means of dimension $-1/2$\ Nambu-Goldstone fermion $\xi$\ and its dimension 3/2 conjugate $\eta$. A matter coupling of these Nambu- Goldstone fermions reproduce the realization found by Berkovits and Vafa to show the equivalence between the bosonic string and a $N=1$\ superstring. This indicates that this equivalence is a result of the super Higgs mechanism as a two dimensional field theory, in which the NG fermions become unphysical. 
  In this paper, we introduce an $N\times N$ matrix $\epsilon^{a\bar{b}}$ in the quantum groups $SU_{q}(N)$ to transform the conjugate representation into the standard form so that we are able to compute the explicit forms of the important quantities in the bicovariant differential calculus on $SU_{q}(N)$, such as the $q$-deformed structure constant ${\bf C}_{IJ}^{~K}$ and the $q$-deformed transposition operator $\Lambda$. From the $q$-gauge covariant condition we define the generalized $q$-deformed Killing form and the $m$-th $q$-deformed Chern class $P_{m}$ for the quantum groups $SU_{q}(N)$. Some useful relations of the generalized $q$-deformed Killing form are presented. In terms of the $q$-deformed homotopy operator we are able to compute the $q$-deformed Chern-Simons $Q_{2m-1}$ by the condition $dQ_{2m-1}=P_{m}$, Furthermore, the $q$-deformed cocycle hierarchy, the $q$-deformed gauge covariant Lagrangian, and the $q$-deformed Yang-Mills equation are derived. 
  A rotational model is developed from a new version of the two-parameter quantum algebra $U_{qp}({\rm u}_2)$. This model is applied to the description of some recent experimental data for the rotating superdeformed nuclei $^{192-194-196-198}{\rm Pb}$ and $^{192-194 }{\rm Hg}$. A comparison between the $U_{qp}({\rm u}_2)$ model presented here and the Raychev-Roussev-Smirnov model with $U_{q }({\rm su}_2)$ symmetry shows the relevance of the introduction of a second parameter of a ``quantum algebra'' type. 
  We relate in a novel way the modular matrices of GKO diagonal cosets without fixed points to those of WZNW tensor products. Using this we classify all modular invariant partition functions of $su(3)_k\oplus su(3)_1/su(3)_{k+1}$ for all positive integer level $k$, and $su(2)_k\oplus su(2)_\ell/su(2)_{k+\ell}$ for all $k$ and infinitely many $\ell$ (in fact, for each $k$ a positive density of $\ell$). Of all these classifications, only that for $su(2)_k\oplus su(2)_1/su(2)_{k+1}$ had been known. Our lists include many new invariants. 
  Light-front coordinates offer a scenario in which a constituent picture of hadron structure can emerge from QCD, after several difficulties are addressed. Field theoretic difficulties force us to introduce cutoffs that violate Lorentz covariance and gauge invariance, and a new renormalization group formalism based on a similarity transformation is used with coupling coherence to fix cuonterterms that restore these symmetries. The counterterms contain functions of longitudinal momentum fractions that severely complicate renormalization, but they also offer possible resolutions of apparent contradictions between the constituent picture and QCD. The similarity transformation and coupling coherence are applied to QED; and it is shown that the resultant Hamiltonian leads to standard lowest order bound state results, with the Coulomb interaction emerging naturally. The same techniques are applied to QCD and with physically motivated assumptions it is shown that a simple confinement mechanism appears. 
  In the first part of the paper we give the denominator identity for all simple finite-dimensional Lie super algebras $\frak g\/$ with a non-degenerate invariant bilinear form. We give also a character and (super) dimension formulas for all finite-dimensional irreducible $\frak g\/$-modules of atypicality $\leq 1\/$ . In the second part of the paper we give the denominator identity for the affine superalgebras $\hat{\frak g}\/$ associated to $\frak g\/$. Specializations of this identity give almost all old and many new formulas for the number of representations of an integer as sums of squares and sums of triangular numbers. At the end, we introduce the notion of an integrable $\hat{\frak g}\/$-module and give a classification of irreducible integrable highest weight $\hat{\frak g}\/$-modules. 
  We study the non-critical string field theory with non-orientable string interactions by using the transfer matrix formalism in the dynamical triangulation. For any value of $c$ (total central charge of matter), we have constructed the non-orientable string field theory at the discrete level. Two coupling constants $G$ and $\Gx$ which satisfy the relation $(\Gx)^3 = G \Gx$ are introduced, where $G$ counts the number of untwisted handles of the world-sheet while $\Gx$ counts the number of cross-caps. In the case of the non-critical string theory which correspond to multicritical one matrix models (including $c=0$ case), we have succeeded in taking the continuum limit, and then have obtained the continuous string field theory. 
  A quantum theory is constructed for the system of a relativistic particle with mass m moving freely on the SL(2,R) group manifold. Applied to the cotangent bundle of SL(2,R), the method of Hamiltonian reduction allows us to split the reduced system into two coadjoint orbits of the group. We find that the Hilbert space consists of states given by the discrete series of the unitary irreducible representations of SL(2,R), and with a positive-definite, discrete spectrum. 
  We investigate a modification of the 2+1 dimensional abelian Chern-Simons theory, obtained by adding a Proca mass term to the gauge field. We are particularly interested in the infrared limit, which can be described by two {\it a priori} different "topological" quantum mechanical models. We apply methods of equivariant cohomology and the ensuing supersymmetry to analyze the partition functions of these quantum mechanical models. In particular, we find that a previously discussed phase-space reductive limiting procedure which relates these two models can be seen as a direct consequence of our supersymmetry. 
  We present results of a computation of the BRST-antibracket cohomology in the space of local functionals of the fields and antifields for a class of 2d gravitational theories which are conformally invariant at the classical level. In particular all classical local action functionals, all candidate anomalies and all BRST--invariant functionals depending nontrivially on antifields are given and discussed for these models. 
  We perform a spectral decomposition of the dynamical correlation function of the spin $1/2$ XXZ model into an infinite sum of products of form factors. Beneath the four-particle threshold in momentum space the only non-zero contributions to this sum are the two-particle term and the trivial vacuum term. We calculate the two-particle term by making use of the integral expressions for form factors provided recently by the Kyoto school. We evaluate the necessary integrals by expanding to twelfth order in $q$. We show plots of $S(w,k)$, for $k=0$ and $\pi$ at various values of the anisotropy parameter, and for fixed anisotropy at various $k$ around $0$ and $\pi$. 
  We present a summary of the results of an explicit calculation of the strength of non-perturbative interactions in matrix models and string effective Lagrangians. These interactions are induced by single eigenvalue instantons in the $d=1$ bosonic matrix model. A well defined approximation scheme is used to obtain induced operators whose exact form we exhibit. We briefly discuss the possibility that similar instantons in a supersymmetric version of the theory may break supersymmetry dynamically. 
  We point out that using the heat kernel on a cone to compute the first quantum correction to the entropy of Rindler space does not yield the correct temperature dependence. In order to obtain the physics at arbitrary temperature one must compute the heat kernel in a geometry with different topology (without a conical singularity). This is done in two ways, which are shown to agree with computations performed by other methods. Also, we discuss the ambiguities in the regularization procedure. 
  We report in this article three- and four-term recursion relations for Clebsch-Gordan coefficients of the quantum algebras $U_q(su_2)$ and $U_q(su_{1,1})$. These relations were obtained by exploiting the complementarity of three quantum algebras in a $q$-deformation of $sp(8, \gr)$. 
  The algebra of volume-preserving vector fields is considered. The potentials for that fields are introduced, and induced algebra of potentials is considered. It is shown, that this algebra fails to satisfy the Jacoby identity. Analogy with hamiltonian mechanics is developed, as well as 3-cocycle interpretation of corresponding expressions. 
  The classical dynamics of antisymmetric second-rank tensor matter fields is analyzed. The conformally invariant action for the tensor field leads to a positive-definite hamiltonian on the class of the solutions that are bounded at the time infinity (plane waves). Only the longitudinal waves contribute to the energy and momentum. The helicity proves to be equal to zero. 
  We investigate the representation theory of some recently constructed N=2 super W-algebras with two generators. Except for the central charges in the unitary minimal series of the N=2 super Virasoro algebra we find no new rational models. However, from our results it is possible to arrange all known N=2 super W-algebras with two generators and vanishing self-coupling constant into four classes. For the algebras existing for c >= 3 which can be understood by the spectral flow of the N=2 super Virasoro algebra we find that the representations have quantized U(1) charge. 
  We consider a class of non-unitary Toda theories based on the Lie superalgebras $A^{(1)}(n,n)$ in two-dimensional Minkowski spacetime, which can be twisted into a topological sector. In particular we study the simplest example with $n=1$ and real fields, and show how this theory can be treated as an integrable perturbation of the $A(1,0)$ superconformal model. We construct the conserved currents and the vertex operators which are chiral primary fields in the conformal theory. We then define the chiral ring of the $A^{(1)}(1,1)$ Toda theory and compute topological correlation functions in the twisted sector. The calculation is performed using a $N=2$ off--shell superspace approach. 
  We study 3-dimensional BF theories and define observables related to knots and links. The quantum expectation values of these observables give the coefficients of the Alexander-Conway polynomial. 
  From a 2-parametric deformation of the harmonic oscillator algebra we construct a 4-point dual amplitude with nonlinear trajectories. The earlier versions of the q-deformed dual models are reproduced as limiting cases of the present model. 
  The nonlinear $\sigma$-model is considered to be useful in describing hadrons (Skyrmions) in low energy hadron physics and the approximate behavior of the global texture. Here we investigate the properties of the static solution of the nonlinear $\sigma$-model equation coupled with gravity. As in the case where gravity is ignored, there is still no scale parameter that determines the size of the static solution and the winding number of the solution is $1/2$. The geometry of the spatial hyperspace in the asymptotic region of large $r$ is explicitly shown to be that of a flat space with some missing solid angle. 
  The Feigenbaum constants $\alpha$ and $\delta$ for the three-site antiferromagnetic Ising spin model on Husimi tree are calculated. It is shown that the numerical values of these constants for this real physical system coincide with the famous universal Feigenbaum constants with high accuracy. The quantitative description from ordering to chaos is also obtained. 
  We show that the conformal characters of various rational models of W-algebras can be already uniquely determined if one merely knows the central charge and the conformal dimensions. As a side result we develop several tools for studying representations of SL(2,Z) on spaces of modular functions. These methods, applied here only to certain rational conformal field theories, may be useful for the analysis of many others. 
  In the context of a relativistic quantum mechanics with invariant evolution parameter, solutions for the relativistic bound state problem have been found, which yield a spectrum for the total mass coinciding with the nonrelativistic Schr\"odinger energy spectrum. These spectra were obtained by choosing an arbitrary spacelike unit vector $n_\mu$ and restricting the support of the eigenfunctions in spacetime to the subspace of the Minkowski measure space, for which $(x_\perp )^2 = [x-(x \cdot n) n ]^2 \geq 0$. In this paper, we examine the Zeeman effect for these bound states, which requires $n_\mu$ to be a dynamical quantity. We recover the usual Zeeman splitting in a manifestly covariant form. 
  Chern-Simons gauge theory for compact semisimple groups is analyzed from a perturbation theory point of view. The general form of the perturbative series expansion of a Wilson line is presented in terms of the Casimir operators of the gauge group. From this expansion new numerical knot invariants are obtained. These knot invariants turn out to be of finite type (Vassiliev invariants), and to possess an integral representation. Using known results about Jones, HOMFLY, Kauffman and Akutsu-Wadati polynomial invariants these new knot invariants are computed up to type six for all prime knots up to six crossings. Our results suggest that these knot invariants can be normalized in such a way that they are integer-valued. 
  The content of this work is concerned with Jordan-Schwinger calculus using $q$-deformed bosons. 
  Using the recently discovered connection between bosonization and duality transformations (hep-th/9401105 and hep-th/9403173), we give an explicit path-integral representation for the bosonization of a massive fermion coupled to a U(1) gauge potential (such as electromagnetism) in d space (D=d+1 spacetime) dimensions. The bosonic theory is described by a rank d-1 antisymmetric Kalb-Ramond-type gauge potential. We construct the bosonized lagrangian explicitly in the limit of large fermion mass. We find that the resulting action is local for d=2 (and given by a Chern-Simons action), but nonlocal for d larger than 3. By coupling to a statistical Chern-Simons field for d=2, we obtain a bosonized formulation of anyons. The bosonic theory may be further dualized to a theory involving purely scalars, for any d, and we show this to be governed by a higher-derivative lagrangian for which the scalar decouples from the U(1) gauge potential. 
  Based on the treatment of the chiral Ising model by Mack and Schomerus, we present examples of localized endomorphisms $\varrho_1^{\rm loc}$ and $\varrho_{1/2}^{\rm loc}$. It is shown that they lead to the same superselection sectors as the global ones in the sense that unitary equivalence $\pi_0\circ\varrho_1^{\rm loc}\cong\pi_1$ and $\pi_0\circ\varrho_{1/2}^{\rm loc}\cong\pi_{1/2}$ holds. Araki's formalism of the selfdual CAR algebra is used for the proof. We prove local normality and extend representations and localized endomorphisms to a global algebra of observables which is generated by local von Neumann algebras on the punctured circle. In this framework, we manifestly prove fusion rules and derive statistics operators. 
  Eleven different types of "maximally superintegrable" Hamiltonian systems on the real hyperboloid $(s^0)^2-(s^1)^2+(s^2)^2-(s^3)^2=1$ are obtained. All of them correspond to a free Hamiltonian system on the homogeneous space $SU(2,2)/U(2,1)$, but to reductions by different maximal abelian subgroups of $SU(2,2)$. Each of the obtained systems allows 5 functionally independent integrals of motion, from which it is possible to form two or more triplets in involution (each of them includes the hamiltonian). The corresponding classical and quantum equations of motion can be solved by separation of variables on the $O(2,2)$ space. 
  We examine some of the standard features of primary fields in the framework of a $q$-deformed conformal field theory. By introducing a $q$-OPE between the energy momentum tensor and a primary field, we derive the $q$-analog of the conformal Ward identities for correlation functions of primary fields. We also obtain solutions to these identities for the two-point function. 
  We compute branching functions of $G/H$ coset models using a BRST invariant branching function formulae, i.e. a branching function that respects a BRST invariance of the model. This ensures that only the coset degrees of freedom will propagate. We consider $G/H$ for rank$(G/H)=0$ models which includes the Kazama-Suzuki construction, and $G_{k_1}\times G_{k_2}/G_{k_1+k_2}$ models. Our calculations here confirm in part previous results for those models which have been obtained under an assumption in a free field approach. We also consider $G_{k_1}\times H_{k_2}/H_{k_1+k_2}$, where $H$ is a subgroup of $G$, and $\prod_{a=1}^mG_{k_a}/G_{\sum_{a=1}^nk_a}$, whose branching functions, to our knowledge, has not been calculated before. 
  The relation between discrete topological field theories on triangulations of two-dimensional manifolds and associative algebras was worked out recently. The starting point for this development was the graphical interpretation of the associativity as flip of triangles. We show that there is a more general relation between flip-moves with two $n$-gons and $Z_{n-2}$-graded associative algebras. A detailed examination shows that flip-invariant models on a lattice of $n$-gons can be constructed {}from $Z_{2}$- or $Z_{1}$-graded algebras, reducing in the second case to triangulations of the two-dimensional manifolds. Related problems occure naturally in three-dimensional topological lattice theories. 
  Some ideas about phenomenological applications of quantum algebras to physics are reviewed. We examine in particular some applications of the algebras $U_ q (su_2)$ and $U_{qp}({\rm u}_2)$ to various dynamical systems and to atomic and nuclear spectroscopy. The lack of a true (unique) $q$- or $qp$-quantization process is emphasized. 
  We study the pairwise interaction of reggeized gluons and quarks in the Regge limit of perturbative QCD. The interactions are represented as integral kernels in the transverse momentum space and as operators in the impact parameter space. We observe conformal symmetry and holomorphic factorization in all cases. 
  The renormalization group flow in a general renormalizable gauge theory with a simple gauge group in 3+1 dimensions is analyzed. The flow of the ratios of the Yukawa couplings and the gauge coupling is described in terms of a bounded potential, which makes it possible to draw a number of non-trivial conclusions concerning the asymptotic structure of the theory. A classification of possible flow patterns is given. 
  We study the vacuum structure and dyon spectrum of N=2 supersymmetric gauge theory in four dimensions, with gauge group SU(2). The theory turns out to have remarkably rich and physical properties which can nonetheless be described precisely; exact formulas can be obtained, for instance, for electron and dyon masses and the metric on the moduli space of vacua. The description involves a version of Olive-Montonen electric-magnetic duality. The ``strongly coupled'' vacuum turns out to be a weakly coupled theory of monopoles, and with a suitable perturbation confinement is described by monopole condensation. 
  Within the context of the recently formulated classical gauge theory of relativistic p-branes minimally coupled to general relativity in D-dimensional spacetimes, we obtain solutions of the field equations which describe black objects. Explicit solutions are found for two cases: D > p+1 (true p-branes) and D = p+1 (p-bags). 
  We calculate the boundary $S$ matrix for the open antiferromagnetic spin $1/2$ isotropic Heisenberg chain with boundary magnetic fields. Our approach, which starts from the model's Bethe Ansatz solution, is an extension of the Korepin-Andrei-Destri method. Our result agrees with the boundary $S$ matrix for the boundary sine-Gordon model with $\beta^2 \rightarrow 8\pi$ and with ``fixed'' boundary conditions. 
  We discuss the canonical quantization of Chern-Simons theory in $2+1$ dimensions, minimally coupled to a Dirac spinor field, first in the temporal gauge and then in the Coulomb gauge. In our temporal gauge formulation, Gauss's law and the gauge condition, $A_0 = 0$, are implemented by embedding the formulation in an appropriate physical subspace. We construct a Fock space of charged particle states that satisfy Gauss's law, and show that they obey fermion, not fractional statistics. The gauge-invariant spinor field that creates these charged states from the vacuum obeys the anticommutation rules that generally apply to spinor fields. The Hamiltonian, when described in the representation in which the charged fermions are the propagating particle excitations that obey Gauss's law, contains an interaction between charge and transverse current densities. We observe that the implementation of Gauss's law and the gauge condition does not require us to use fields with graded commutator algebras or particle excitations with fractional statistics. In our Coulomb gauge formulation, we implement Gauss's law and the gauge condition, $\partial_lA_l=0$, by the Dirac-Bergmann procedure. In this formulation, the constrained gauge fields become functionals of the spinor fields, and are not independent degrees of freedom. The formulation in the Coulomb gauge confirms the results we obtained in the temporal gauge: The ``Dirac-Bergmann'' anticommutation rule for the charged spinor fields $\psi$ and $\psi^\dagger$ that have both been constrained to obey Gauss's law, is precisely identical to the canonical spinor anticommutation rule that generates standard fermion statistics. And we also show that the Hamiltonians for charged particle states in our temporal and Coulomb gauge formulations are identical, once Gauss's law 
  We consider a model of 2D gravity with the coefficient of the Einstein-Hilbert action having an imaginary part $\pi/2$. This is equivalent to introduce a $\Theta$-vacuum structure in the genus expansion whose effect is to convert the expansion into a series of alternating signs, presumably Borel summable. We show that the specific heat of the model has a physical behaviour. It can be represented nonperturbatively as a series in terms of integrals over moduli spaces of punctured spheres and the sum of the series can be rewritten as a unique integral over a suitable moduli space of infinitely punctured spheres. This is an explicit realization \`a la Friedan-Shenker of 2D quantum gravity. We conjecture that the expansion in terms of punctures and the genus expansion can be derived using the Duistermaat-Heckman theorem. We briefly analyze expansions in terms of punctured spheres also for multicritical models. 
  The Berezin quantization on a simply connected homogeneous K\"{a}hler manifold, which is considered as a phase space for a dynamical system, enables a description of the quantal system in a (finite-dimensional) Hilbert space of holomorphic functions corresponding to generalized coherent states. The Lie algebra associated with the manifold symmetry group is given in terms of first-order differential operators. In the classical theory, the Lie algebra is represented by the momentum maps which are functions on the manifold, and the Lie product is the Poisson bracket given by the K\"{a}hler structure. The K\"{a}hler potentials are constructed for the manifolds related to all compact semi-simple Lie groups. The complex coordinates are introduced by means of the Borel method. The K\"{a}hler structure is obtained explicitly for any unitary group representation. The cocycle functions for the Lie algebra and the Killing vector fields on the manifold are also obtained. 
  In 1974, Berezin proposed a quantum theory for dynamical systems having a K\"{a}hler manifold as their phase space. The system states were represented by holomorphic functions on the manifold. For any homogeneous K\"{a}hler manifold, the Lie algebra of its group of motions may be represented either by holomorphic differential operators (``quantum theory"), or by functions on the manifold with Poisson brackets, generated by the K\"{a}hler structure (``classical theory"). The K\"{a}hler potentials and the corresponding Lie algebras are constructed now explicitly for all unitary representations of any compact simple Lie group. The quantum dynamics can be represented in terms of a phase-space path integral, and the action principle appears in the semi-classical approximation. 
  In this paper we define a new family of groups which generalize the {\it classical braid groups on} $\C $. We denote this family by $\{B_n^m\}_{n \ge m+1}$ where $n,m \in \N$. The family $\{ B_n^1 \}_{n \in \N}$ is the set of classical braid groups on $n$ strings. The group $B_n^m$ is the set of motions of $n$ unordered points in $\C^m$, so that at any time during the motion, each $m+1$ of the points span the whole of $\C^m$ as an affine space. There is a map from $B_n^m$ to the symmetric group on $n$ letters. We let $P_n^m$ denote the kernel of this map. In this paper we are mainly interested in understanding $P_n^2$. We give a presentation of a group $PL_n$ which maps surjectively onto $P_n^2$. We also show the surjection $PL_n \to P_n^2$ induces an isomorphism on first and second integral homology and conjecture that it is an isomorphism. We then find an infinitesimal presentation of the group $P_n^2$. Finally, we also consider the analagous groups where points lie in $\P^m$ instead of $\C^m$. These groups generalize of the classical braid groups on the sphere. 
  We show that under the operation of parity the {\it front-form} $(1/2,\,0)$ and $(0,\,1/2)$ Weyl spinors (massive or massless) do not get interchanged. This has the important consequence that if a front-form theory containing $(1/2,\,0)\oplus(0,\,1/2)$ representation space has to be parity covariant then one must study the evolution of a physical system not only along $x^+$ but also along the $x^-$ direction. As a result of our analysis, we find an indication that there may be no halving of the degrees of freedom in the front form of field theories. 
  The analogue of the string equation which specifies the partition function of $c=1$ string with a compactification radius $\beta \in \mbox{$\bf{Z}$}_{\geq 1} $ is described in the framework of the Toda lattice hierarchy. 
  We discuss the relations between (topological) quantum field theories in 4 dimensions and the theory of 2-knots (embedded 2-spheres in a 4-manifold). The so-called BF theories allow the construction of quantum operators whose trace can be considered as the higher-dimensional generalization of Wilson lines for knots in 3-dimensions. First-order perturbative calculations lead to higher dimensional linking numbers, and it is possible to establish a heuristic relation between BF theories and Alexander invariants. Functional integration-by-parts techniques allow the recovery of an infinitesimal version of the Zamolodchikov tetrahedron equation, in the form considered by Carter and Saito. 
  A higher dimensional analogue of the dispersionless KP hierarchy is introduced. In addition to the two-dimensional ``phase space'' variables $(k,x)$ of the dispersionless KP hierarchy, this hierarchy has extra spatial dimensions compactified to a two (or any even) dimensional torus. Integrability of this hierarchy and the existence of an infinite dimensional of ``additional symmetries'' are ensured by an underlying twistor theoretical structure (or a nonlinear Riemann-Hilbert problem). An analogue of the tau function, whose logarithm gives the $F$ function (``free energy'' or ``prepotential'' in the contest of matrix models and topological conformal field theories), is constructed. The infinite dimensional symmetries can be extended to this tau (or $F$) function. The extended symmetries, just like those of the dispersionless KP hierarchy, obey an anomalous commutation relations. 
  We review known exact classical solutions in (bosonic) string theory. The main classes of solutions are `cosets' (gauged WZW models), `plane wave'-type backgrounds (admitting a covariantly constant null Killing vector) and `$F$-models' (backgrounds with two null Killing vectors generalising the `fundamental string' solution). The recently constructed $D=4$ solutions with Minkowski signature are given explicitly. We consider various relations between these solutions and, in particular, discuss some aspects of the duality symmetry. [To appear in the Proceedings of the 2nd Journe'e Cosmologie, Observatoire de Paris, June 2-4, 1994.] 
  Phase transition in spherically symmetric collapse of a massless scalar field is studied in 4-d Einstein gravity. A class of exact solutions that show the evolution of a constant incoming energy flux turned on at a point in the past null infinity are constructed to serve as an explicit example. The recently discovered phase transition in this system via numerical methods \cite{choptuik} is manifest; above a threshold value of the incooming energy flux, a black hole is dynamically formed and below that, the incoming flux is reflected forward into the future null infinity. The critical exponent is evaluated and discussed using the solutions. 
  An inflationary epoch driven by the kinetic energy density in a dynamical Planck mass is studied. In the conformally related Einstein frame it is easiest to see the demands of successful inflation cannot be satisfied by kinetic inflation alone. Viewed in the original Jordan-Brans-Dicke frame, the obstacle is manifest as a kind of graceful exit problem and/or a kind of flatness problem. These arguments indicate the weakness of only the simplest formulation. {}From them can be gleaned directions toward successful kinetic inflation. 
  We discuss the spectral curves and rational maps associated with $SU(2)$ Bogomolny monopoles of arbitrary charge $k$. We describe the effect on the rational maps of inverting monopoles in the plane with respect to which the rational maps are defined, and discuss the monopoles invariant under such inversion. We define the strongly centred monopoles, and show they form a geodesic submanifold of the $k$-monopole moduli space. The space of strongly centred $k$-monopoles invariant under the cyclic group of rotations about a fixed axis, $C_k$, is shown to consist of several surfaces of revolution, generalizing the two surfaces obtained by Atiyah and Hitchin in the 2-monopole case. Geodesics on these surfaces give a novel type of $k$-monopole scattering. We present a number of curves in $TP_1$ which we conjecture are the spectral curves of monopoles with the symmetries of a regular solid. These conjectures are based on analogies with Skyrmions. 
  $M$-dimensional extended objects $\Sigma$ can be described by projecting a Diff $\Sigma$ invariant Hamiltonian of time-independent Hamiltonian density {\cal H} onto the Diff $\Sigma$- singlet sector, which after Hamiltonian reduction, using {\cal H} itself for one of the gauge-fixing conditions, results in a non-local description that may enable one to extend the non-local symmetries for strings to higher dimensions and make contact with gravity at an early stage. 
  We investigate the semiclassical limit and quantum corrections to the metric in a unitary quantum gravity formulation of the CGHS 2d dilaton gravity model. A new method for calculating the back-reaction effects has been introduced, as an expansion of the effective metric in powers of the matter energy-momentum tensor. In the semiclassical limit the quantum corrections can be neglected, and we show that physical states exists which contain the Hawking radiation. The first order back-reaction effect is entirely due to the Hawking radiation. It causes the black-hole mass to monotonically decrease, and it makes it unbounded from bellow as the horizon is approched. The second order quantum corrections have been estimated. Since the matter is propagating freely in this unitary theory, we expect that the higher order corrections will stabilize the mass, and the black hole will completely evaporate leaving a nearly flat space. 
  We reconsider the algebraic BRS renormalization of Witten's topological Yang-Mills field theory by making use of a vector supersymmetry Ward identity which improves the finiteness properties of the model. The vector supersymmetric structure is a common feature of several topological theories. The most general local counterterm is determined and is shown to be a trivial BRS-coboundary. 
  We discuss integrating out matter fields and integrating in matter fields in four dimensional supersymmetric gauge theories. Highly nontrivial exact superpotentials can be easily obtained by starting from a known theory and integrating in matter. 
  Motivated by topological bidimensional quantum models for distinguishable particles, and by Haldane's definition of mutual statistics for different species of particles, we propose a new class of one-dimensional $1/r_{ij}^2$ Calogero model with coupling constants $g_{ij}$ depending on the labels of the particles. We solve the groundstate problem, and show how to build some classes of excited states. 
  The quantisation of the Wess-Zumino-Witten model on a circle is discussed in the case of $SU(N)$ at level $k$. The quantum commutation of the chiral vertex operators is described by an exchange relation with a braiding matrix, $Q$. Using quantum consistency conditions, the braiding matrix is found explicitly in the fundamental representation. This matrix is shown to be related to the Racah matrix for $U_t(SL(N))$. From calculating the four-point functions with the Knizhnik-Zamolodchikov equations, the deformation parameter $t$ is shown to be $t=\exp({i\pi /(k+N)})$ when the level $k\ge 2$. For $k=1$, there are two possible types of braiding, $t=\exp({i\pi /(1+N)})$ or $t=\exp(i\pi)$. In the latter case, the chiral vertex operators are constructed explicitly by extending the free field realisation a la Frenkel-Kac and Segal. This construction gives an explicit description of how to chirally factorise the $SU(N)_{k=1}$ WZW model. 
  We study a supersymmetric theory twisted on a K\"ahler four manifold $M=\Sigma_1 \times \Sigma_2 ,$ where $\Sigma_{1,2}$ are 2D Riemann surfaces. We demonstrate that it possesses a "left-moving" conformal stress tensor on $\Sigma_1$ ($\Sigma_2$) in a BRST cohomology, which generates the Virasoro algebra with the conventional commutation relations. The central charge of the Virasoro algebra has a purely geometric origin and is proportional to the Euler characteristic $\c$ of the $\Sigma_2$ ($\Sigma_1$) surface. It is shown that this construction can be extended to include a realization of a Kac-Moody algebra in BRST cohomology with a level proportional to the Euler characteristic $\c .$ This structure is shown to be invariant under renormalization group. A representation of the algebra $W_{1+\infty}$ in terms of a free chiral supermultiplet is also given. We discuss the role of instantons and a possible relation between the dynamics of 4D Yang-Mills theories and those of 2D sigma models. 
  This is (hopefully) a Latexable version of a talk given at the XXX Winter School in Theoretical Physics at Karpacz in February 1994. It discusses the use of non-commutative differential calculus to construct a Lie algebra of a quantum group. Usually the result has a different dimension from the classical Lie algebra. This is illustrated by menas of the orthogonal quantum group, and various other possible ways of constructing an orthogonal Lie algebra are described. 
  We discuss the predictions of S-duality for the monopole spectrum of four-dimensional heterotic string theory resulting from toroidal compactification. We discuss in detail the spectrum of "H-monopoles", states that are magnetically charged with respect to the U(1) groups arising from the dimensional reduction of the ten dimensional antisymmetric tensor field. Using an assumption concerning the correct treatment of collective coordinates in string theory we find results which are consistent with S-duality. 
  Toda lattice hierarchy and the associated matrix formulation of the $2M$-boson KP hierarchies provide a framework for the Drinfeld-Sokolov reduction scheme realized through Hamiltonian action within the second KP Poisson bracket. By working with free currents, which abelianize the second KP Hamiltonian structure, we are able to obtain an unified formalism for the reduced $SL(M+1,M-k)$-KdV hierarchies interpolating between the ordinary KP and KdV hierarchies. The corresponding Lax operators are given as superdeterminants of graded $SL (M+1,M-k)$ matrices in the diagonal gauge and we describe their bracket structure and field content. In particular, we provide explicit free-field representations of the associated $W(M,M-k)$ Poisson bracket algebras generalizing the familiar nonlinear $W_{M+1}$-algebra. Discrete B\"{a}cklund transformations for $SL(M+1,M-k)$-KdV are generated naturally from lattice translations in the underlying Toda-like hierarchy. As an application we demonstrate the equivalence of the two-matrix string model to the $SL (M+1,1)$-KdV hierarchy. 
  A series of sigma models with torsion are analysed which generate their mass dynamically but whose ultra-violet fixed points are non-trivial conformal field theories -- in fact SU(2) WZW models at level $k$. In contrast to the more familiar situation of asymptotically free theories in which the fixed points are trivial, the sigma models considered here may be termed ``asymptotically CFT''. These theories have previously been conjectured to be quantum integrable; this is confirmed by postulating a factorizable S-matrix to describe their infra-red behaviour and then carrying out a stringent test of this proposal. The test involves coupling the theory to a conserved charge and evaluating the response of the free-energy both in perturbation theory to one loop and directly from the S-matrix via the Thermodynamic Bethe Ansatz with a chemical potential at zero temperature. Comparison of these results provides convincing evidence in favour of the proposed S-matrix; it also yields the universal coefficients of the beta-function and allows for an evaluation of the mass gap (the ratio of the physical mass to the $\Lambda$-parameter) to leading order in $1/k$. 
  We calculate the partition function of the $SU(N)$ ( and $U(N)$) generalized $YM_2$ theory defined on an arbitrary Riemann surface. The result which is expressed as a sum over irreducible representations generalizes the Rusakov formula for ordinary YM_2 theory. A diagrammatic expansion of the formula enables us to derive a Gross-Taylor like stringy description of the model. A sum of 2D string maps is shown to reproduce the gauge theory results. Maps with branch points of degree higher than one, as well as ``microscopic surfaces'' play an important role in the sum. We discuss the underlying string theory. 
  In this paper we obtain an expression for the residue as it occurs in the non-Abelian localization formula due to Jeffrey and Kirwan. This expression can be used in cohomology computations for symplectic quotients. 
  The Hilbert space of a free massless particle moving on a group manifold is studied in details using canonical quantisation. While the simplest model is invariant under a global symmetry, $G \times G$, there is a very natural way to ``factorise" the theory so that only one copy of the global symmetry is preserved. In the case of $G=SU(2)$, a simple deformation of the quantised theory is proposed to give a realisation of the quantum group, $U_t(SL(2))$. The symplectic structures of the corresponding classical theory is derived. This can be used, in principle, to obtain a Lagrangian formulation for the $U_t(SL(2))$ symmetry. 
  We present a unified approach to the Thermodynamic Bethe Ansatz (TBA) for magnetic chains and field theories that includes the finite size (and zero temperature) calculations for lattice BA models. In all cases, the free energy follows by quadratures from the solution of a {\bf single} non-linear integral equation (NLIE). [A system of NLIE appears for nested BA]. We derive the NLIE for: a) the six-vertex model with twisted boundary conditions; b) the XXZ chain in an external magnetic field $h_z$ and c) the sine-Gordon-massive Thirring model (sG-mT) in a periodic box of size $\b \equiv 1/T $ using the light-cone approach. This NLIE is solved by iteration in one regime (high $T$ in the XXZ chain and low $T$ in the sG-mT model). In the opposite (conformal) regime, the leading behaviors are obtained in closed form. Higher corrections can be derived from the Riemann-Hilbert form of the NLIE that we present. 
  We have found that for extreme dilaton black holes an inner boundary must be introduced in addition to the outer boundary to give an integer value to the Euler number. The resulting manifolds have (if one identifies imaginary time) topology $S^1 \times R \times S^2 $ and Euler number $\chi = 0$ in contrast to the non-extreme case with $\chi=2$. The entropy of extreme $U(1)$ dilaton black holes is already known to be zero. We include a review of some recent ideas due to Hawking on the Reissner-Nordstr\"om case. By regarding all extreme black holes as having an inner boundary, we conclude that the entropy of {\sl all} extreme black holes, including $[U(1)]^2$ black holes, vanishes. We discuss the relevance of this to the vanishing of quantum corrections and the idea that the functional integral for extreme holes gives a Witten Index. We have studied also the topology of ``moduli space'' of multi black holes. The quantum mechanics on black hole moduli spaces is expected to be supersymmetric despite the fact that they are not HyperK\"ahler since the corresponding geometry has torsion unlike the BPS monopole case. Finally, we describe the possibility of extreme black hole fission for states with an energy gap. The energy released, as a proportion of the initial rest mass, during the decay of an electro-magnetic black hole is 300 times greater than that released by the fission of an ${}^{235} U$ nucleus. 
  Coadjoint orbits of the Virasoro and Kac-Moody algebras provide geometric actions for matter coupled to gravity and gauge fields in two dimensions. However, the Gauss' law constraints that arise from these actions are not necessarily endemic to two-dimensional topologies. Indeed the constraints associated with Yang-Mills naturally arise from the coadjoint orbit construction of the WZW model. One may in fact use a Yang-Mills theory to provide dynamics to the otherwise fixed coadjoint vectors that define the orbits. In this letter we would like to exhibit an analogue of the Yang-Mills classical action for the diffeomorphism sector. With this analogue one may postulate a 4D theory of gravitation that is related to an underlying two dimensional theory. Instead of quadratic differentials, a (1,3) pseudo tensor becomes the dynamical variable. We briefly discuss how this tensor may be classically coupled to matter. 
  We formulate the Schwinger-Dyson equations in the ladder approximation for 2D induced quantum gravity with fermions using covariant gauges of harmonic type. It is shown that these equations can be formulated consistently in a gauge of Landau type (for negative cosmological constant). A numerical analysis of the equations hints towards the possibility of chiral symmetry breaking, depending on the value of the coupling constant. 
  An exact solution of Einstein - Maxwell - conformal scalar field equations is given, which is a black hole solution and has three parameters: scalar charge, electric charge, and magnetic charge. Switching off the magnetic charge parameter yields the solution given by Bekenstein. In addition the energy of the conformal scalar dyon black hole is obtained. 
  We consider the spin $k/2$ XXZ model in the antiferomagnetic regime using the free field realization of the quantum affine algebra $\uqa$ of level $k$. We give a free field realization of the type II $q$-vertex operator, which describes creation and annihilation of physical particles in the model. By taking a trace of the type I and the type II $q$-vertex operators over the irreducible highest weight representation of $\uqa$, we also derive an integral formula for form factors in this model. Investigating the structure of poles, we obtain a residue formula for form factors, which is a lattice analog of the higher spin extension of the Smirnov's formula in the massive integrable quantum field theory. This result as well as the quantum deformation of the Knizhnik-Zamolodchikov equation for form factors shows a deep connection in the mathematical structure of the integrable lattice models and the massive integrable quantum field theory. 
  We define pure gauge $QCD$ on an infinite strip of width $L$. Techniques similar to those used in finite $TQCD$ allow us to relate $3D$-observables to pure $QCD_2$ behaviors. The non triviality of the $L \arrow 0$ limit is proven and the generalization to four dimensions described. The glueball spectrum of the theory in the small width limit is calculated and compared to that of the two dimensional theory. 
  This paper presents a noncommutative theory of symmetric functions, based on the notion of quasi-determinant. We begin with a formal theory, corresponding to the case of symmetric functions in an infinite number of independent variables. This allows us to endow the resulting algebra with a Hopf structure, which leads to a new method for computing in descent algebras. It also gives unified reinterpretation of a number of classical constructions. Next, we study the noncommutative analogs of symmetric polynomials. One arrives at different constructions, according to the particular kind of application under consideration. For example, when a polynomial with noncommutative coefficients in one central variable is decomposed as a product of linear factors, the roots of these factors differ from those of the expanded polynomial. Thus, according to whether one is interested in the construction of a polynomial with given roots or in the expansion of a product of linear factors, one has to consider two distinct specializations of the formal symmetric functions. A third type appears when one looks for a noncommutative generalization of applications related to the notion of characteristic polynomial of a matrix. This construction can be applied, for instance, to the noncommutative matrices formed by the generators of the universal enveloping algebra $U(gl_n)$ or of 
  We construct the gauge invariant fermion correlator in the Schwinger model on the torus. At zero temperature, this correlator falls off with a rate given by the Coulomb energy of an infinitely heavy charge. At high temperature, the screening mass approaches $\pi T/2$, and this in the presence of a mass gap. The fractional Matsubara frequency arises from the action of a pair of induced merons at high temperature that are localized over a range on the order of the meson Compton wavelength $1/m=\sqrt{\pi}/g$. We discuss the quenched approximation in this model, and comment on the possible relevance of some of these results to higher dimensions. 
  FQHE is presented in the form of non-unitary singular similarity transformation, which relates the Laughlin wave function (and its particle-hole conjugate) to the composite quasi- particle incompressible ground state. (ENSLAPP-A-478/94) 
  We have new solutions to the Yang-Baxter equation, from which we have constructed new link invariants containing more than two arbitrary parameters. This may be regarded as a generalization of the Jones' polynomial. We have also found another simpler invariant which discriminates only the linking structure of knots with each other, but not details of individual knot. 
  We consider bosonized $QCD_2$, and prove that after rewritting the theory in terms of gauge invariant fields, there exists an integrability condition valid for the quantum theory as well. Furthermore, performing a duality type transformation we obtain an appropriate action for the description of the strong coupling limit, which is still integrable. We also prove that the model displays a complicated set of constraints, restricting the dynamics of part of the theory, but which are necessary to maintain the positive metric Hilbert space. 
  We define a natural generalized symmetry of the Yang-Mills equations as an infinitesimal transformation of the Yang-Mills field, built in a local, gauge invariant, and Poincar\'e invariant fashion from the Yang-Mills field strength and its derivatives to any order, which maps solutions of the field equations to other solutions. On the jet bundle of Yang-Mills connections we introduce a spinorial coordinate system that is adapted to the solution subspace defined by the Yang-Mills equations. In terms of this coordinate system the complete classification of natural symmetries is carried out in a straightforward manner. We find that all natural symmetries of the Yang-Mills equations stem from the gauge transformations admitted by the equations. 
  The generalized Killing equations and the symmetries of Taub-NUT spinning space are investigated. For spinless particles the Runge-Lenz vector defines a constant of motion directly, whereas for spinning particles it now requires a non-trivial contribution from spin. The generalized Runge-Lenz vector for spinning Taub-NUT space is completely evaluated. 
  The conformal fixed points of the generalized Thirring model are investigated with the help of bosonization, the large N limit and the operator product expansion. Necessary conditions on the coupling constants for conformal invariance are derived. 
  We realize the $U_q(\widehat{sl(2)})$ current algebra at arbitrary level in terms of one deformed free bosonic field and a pair of deformed parafermionic fields. It is shown that the operator product expansions of these parafermionic fields involve an infinite number of simple poles and simple zeros, which then condensate to form a branch cut in the classical limit $q\rightarrow 1$. Our realization coincides with those of Frenkel-Jing and Bernard when the level $k$ takes the values 1 and 2 respectively. 
  We consider wormhole solutions in $2+1$ Euclidean dimensions. A duality transformation is introduced to derive a new action from magnetic wormhole action of Gupta, Hughes, Preskill and Wise. The classical solution is presented. The vertex operators corresponding to the wormhole are derived. Conformally coupled scalars and spinors are considered in the wormhole background and the vertex operators are computed. ( To be published in Phys. Rev. D15) 
  We discuss the topological $CP^1$ model which consists of the holomorphic maps from Riemann surfaces onto $CP^1$. We construct a large-$N$ matrix model which reproduces precisely the partition function of the $CP^1$ model at all genera of Riemann surfaces. The action of our matrix model has the form ${\rm Tr}\, V(M)=-2{\rm Tr}\, M(\log M -1) +2\sum t_{n,P}{\rm Tr}\, M^n(\log M-c_n) +\sum 1/n\cdot t_{n-1,Q}{\rm Tr}\, M^n~(c_n=\sum_1^n 1/j )$ where $M$ is an $N\times N$ hermitian matrix and $t_{n,P}\, (t_{n,Q}),~(n=0,1,2,\cdots)$ are the coupling constants of the $n$-th descendant of the puncture (K\"ahler) operator. 
  The quantum super Yangian $Y_q(gl(M|N))$ associated with the Perk - Schultz solution of the Yang - Baxter equation is introduced. Its structural properties are investigated, in particular, an extensive study of its central algebra is carried out. A $Z_2$ graded associative algebra epimorphism $Y_q(gl(M|N))--> U_q(gl(M|N))$ is established and constructed explicitly. Images of the central elements of the quantum super Yangian under this epimorphism yield the Casimir operators of the quantum supergroup $U_q(gl(M|N))$ constructed in an earlier publication. 
  Open and Closed super-string field theories are constructed in an event-symmetric target space. The partition functions of Statistical and Quantum models are constructed in terms of invariants defined on Lie-algebra representations. An attractive feature of the closed string models is the elegant unification of the space-time symmetries with the gauge symmetries. 
  A Higgsless model for strong, electro-weak and gravitational interactions is proposed. This model is based on the local symmetry group SU(3)xSU(2)xU(1)xC where C is the local conformal symmetry group. The natural minimal conformally invariant form of total lagrangian is postulated. It contains all Standard Model fields and gravitational interaction. Using the unitary gauge and the conformal scale fixing conditions we can eliminate all four real components of the Higgs doublet in this model. However the masses of vector mesons, leptons and quarks are automatically generated and are given by the same formulas as in the conventional Standard Model. The gravitational sector is analyzed and it is shown that the model admits in the classical limit the Einsteinian form of gravitational interactions. No figures. 
  The aim of this work is to construct a cohomology theory controlling the deformations of a general Drinfel'd algebra. The task is accomplished in three steps. The first step is the construction of a modified cobar complex adapted to a non-coassociative comultiplication. The following two steps each involve a new, highly non-trivial, construction. The first construction, essentially combinatorial, defines a differential graded Lie algebra structure on the simplicial chain complex of the associahedra. The second construction, of a more algebraic nature, is the definition of map of differential graded Lie algebras from the complex defined above to the algebra of derivations on the bar resolution. Using the existence of this map and the acyclicity of the associahedra we can define a so-called homotopy comodule structure on the bar resolution of a general Drinfeld algebra. This in turn allows us to define the desired cohomology theory in terms of a complex which consists, roughly speaking, of the bimodule and bicomodule maps from the bar resolution to the modified cobar resolution. The complex is bigraded but not a bicomplex as in the Gerstenhaber-Schack theory for bialgebra deformations. The new components of the coboundary operator are defined via the constructions mentioned above. As an application we show that the Drinfel'd deformation of the universal enveloping algebra of a simple Lie algebra is not a jump deformation. The results of the paper were announced in the paper "Drinfel'd algebra deformations and the associahedra" (IMRN, Duke Math. Journal, 4(1994), 169-176, appeared also as preprint hep-th/9312196). 
  We use arguments taken from the electrodynamics of media to deduce the QCD trace anomaly from the expression for the vacuum energy in the presence of an external color magnetic field. 
  It is shown that smooth maps $f: S^3 \rightarrow S^3$ contain two countable families of harmonic representatives in the homotopy classes of degree zero and one. 
  We show how the two--matrix model and Toda lattice hierarchy presented in a previous paper can be solved exactly: we obtain compact formulas for correlators of pure tachyonic states at every genus. We then extend the model to incorporate a set of discrete states organized in finite dimensional $sl_2$ representations. We solve also this extended model and find the correlators of the discrete states by means of the $W$ constraints and the flow equations. Our results coincide with the ones existing in the literature in those cases in which particular correlators have been explicitly calculated. We conclude that the extented two--matrix model is a realization of the discrete states of $c=1$ string theory. 
  We study the relationships among the various forms of the $q$ oscillator algebra and consider the conditions under which it supports a Hopf structure. We also present a generalization of this algebra together with its corresponding Hopf structure. Its multimode extensions are also considered. 
  The structure of the quark propagator of $QCD$ in a confining background is not known. We make an Ansatz for it, as hinted by a particular mechanism for confinement, and analyze its implications in the meson and baryon correlators. We connect the various terms in the K\"allen-Lehmann representation of the quark propagator with appropriate combinations of hadron correlators, which may ultimately be calculated in lattice $QCD$. Furthermore, using the positivity of the path integral measure for vector like theories, we reanalyze some mass inequalities in our formalism. A curiosity of the analysis is that, the exotic components of the propagator (axial and tensor), produce terms in the hadron correlators which, if not vanishing in the gauge field integration, lead to violations of fundamental symmetries. The non observation of these violations implies restrictions in the space-time structure of the contributing gauge field configurations. In this way, lattice $QCD$ can help us analyze the microscopic structure of the mechanisms for confinement. 
  The high-temperature expansion for closed superstring one-loop free energy is studied. The Laurent series representation is obtained and its sum is analytically continued in order to investigate the nature of the critical (Hagedorn) temperature. It is found that beyond this critical temperature the statistical sum contribution of the free energy is finite but has an imaginary part, signalling a possible metastability of the system. 
  The notion of a translation map in a quantum principal bundle is introduced. A translation map is then used to prove that the cross sections of a quantum fibre bundle $E(B,V,A)$ associated to a quantum principal bundle $P(B,A)$ are in bijective correspondence with equivariant maps $V\to P$, and that a quantum principal bundle is trivial if it admits a cross section which is an algebra map. The vertical automorphisms and gauge transformations of a quantum principal bundle are discussed. In particular it is shown that vertical automorphisms are in bijective correspondence with $\ad$-covariant maps $A\to P$. 
  Using a modified version of the tetrahedron equations we construct a new family of $N$-state three-dimensional integrable models with commuting two-layer transfer-matrices. We investigate a particular class of solutions to these equations and parameterize them in terms of elliptic functions. The corresponding models contain one free parameter $k$ -- an elliptic modulus. 
  The absence of fermion kinetic terms in supersymmetric-BF gauge theories is established. We do this by means of explicit off-shell (superspace) constructions. As part of our study we give the superspace constraints for D=3, N=4 super Yang-Mills along with the D=3, N=4 superconformal algebra. The puzzle we are interested in solving is the fact that the topological cousins, known as super-BF gauge theories, of certain supersymmetric-BF theories have kinetic terms for the twisted fermions. We show that the map which takes the latter to the former includes a Hodge decomposition of the twisted fermions. In conjunction with this result, we argue that it is natural to modify the naive path integral measure of supersymmetric-BF theories to include the Ray-Singer analytic torsion. 
  The question of the integrability of real-coupling affine toda field theory on a half line is discussed. It is shown, by examining low-spin conserved charges, that the boundary conditions preserving integrability are strongly constrained. In particular, among the cases treated so far, $e_6^{(1)}$, $d_n^{(1)}$ and $a_n^{(1)}, \ n\ge 2$, there can be no free parameters introduced by such boundary conditions; indeed the only remaining freedom (apart from choosing the simple condition $\partial_1\phi =0$), resides in a choice of signs. For a special case of the boundary condition, accessible only for $a_n^{(1)}$, it is pointed out that the classical boundary bound state spectrum may be related to a set of reflection factors in the quantum field theory. Some preliminary calculations are reported for other boundary conditions, demonstrating that the classical scattering data satisfies the weak coupling limit of the reflection bootstrap equation. (Invited talk at \lq Quantum field theory, integrable models and beyond', Yukawa Institute for Theoretical Physics, Kyoto University, 14-18 February 1994.) 
  We re-examine the problem of gauging the Wess-Zumino term of a d-dimensional bosonic sigma-model. We phrase this problem in terms of the equivariant cohomology of the target space and this allows for the homological analysis of the obstruction. As a check, we recover the obstructions of Hull and Spence and also a generalization of the topological terms found by Hull, Rocek and de Wit. When the symmetry group is compact, we use topological tools to derive vanishing theorems which guarantee the absence of obstructions for low dimension (d<=4) but for a variety of target manifolds. For example, any compact semisimple Lie group can be gauged in a three-dimensional sigma-model with simply connected target space. When the symmetry group is semisimple but not necessarily compact, we argue in favor of the persistence of these vanishing theorems by making use of (conjectural) equivariant minimal models (in the sense of Sullivan). By way of persuasion, we construct by hand a few such equivariant minimal models, which may be of independent interest. We illustrate our results with two examples: d=1 with a symplectic target space, and d=2 with target space a Lie group admitting a bi-invariant metric. An alternative homological interpretation of the obstruction is obtained by a closer study of the Noether method. This method displays the obstruction as a class in BRST cohomology at ghost number 1. We comment on the relationship with consistent anomalies. 
  We show that the topological $B \wedge F$ term in $3+1$ dimensions can be generated via spontaneous symmetry breaking in a generalized Abelian Higgs model. Further, we also show that even in $D$ dimensions $ ( D \geq 3 ) $, a $B \wedge F$ term gives rise to the topological massive excitations of the Abelian gauge field and that such a $B \wedge F$ term can also be generated via Higgs mechanism. 
  We study correlation functions in two-dimensional conformal field theory coupled to induced gravity in the light-cone gauge. Focussing on the fermion four-point function, we display an unexpected non-perturbative singularity structure: coupling to gravity {\it qualitatively} changes the perturbative $(x_1-x_2)^{-1}(x_3-x_4)^{-1}$ singularity into a logarithmic one plus a non-singular piece. We argue that this is related to the appearence of new logarithmic operators in the gravitationally dressed operator product expansions. We also show some evidence that non-conformal but integrable models may remain integrable when coupled to gravity. 
  Topological conformal field theories based on superconformal current algebras are constructed. The models thus obtained are the supersymmetric version of the $G/G$ coset theories. Their topological conformal algebra is generated by operators of dimensions $1$, $2$ and $3$ and can be regarded as an extension of the twisted $N=2$ superconformal algebra. These models possess an extended supersymmetry whose generators are exact in the topological BRST cohomology. 
  This is an expository talk about the relation between gauging the WZ term of a one-dimensional sigma-model with a symplectic target and the existence of an equivariant moment mapping for symplectic group actions. The punch line is that the obstructions for gauging coincide with the obstructions for the existence of the moment mapping. This paper can be thought of a "prequel" of hep-th/9407149. 
  An elliptic version of quantum groups is proposed. It comes form the quantization of the Knizhnik-Zamolodchikov- Bernard equation on the torus. The relation with elliptic IRF models is explained. 
  The ten--parametric internal symmetry group is found in the $D=4$ Einstein--Maxwell--Dilaton--Axion theory restricted to space--times admitting a Killing vector field. The group includes dilaton--axion $SL(2,R)$ duality and Harrison--type transformations which are similar to some target--space duality boosts, but act on a different set of variables. New symmetry is used to derive a seven--parametric family of rotating dilaton--axion Taub--NUT dyons. 
  The Hamiltonian description for a wide class of mechanical systems, having local symmetry transformations depending on time derivatives of the gauge parameters of arbitrary order, is constructed. The Poisson brackets of the Hamiltonian and constraints with each other and with arbitrary function are explicitly obtained. The constraint algebra is proved to be the first class. 
  The law of track formation in cloud chambers is derived from the Liouville equation with a simple Lindblad's type generator that describes coupling between a quantum particle and a classical, continuous, medium of two--state detectors. Piecewise deterministic random process (PDP) corresponding to the Liouville equation is derived. The process consists of pairs (classical event,quantum jump), interspersed with random periods of continuous (in general, non--linear) Schroedinger's--type evolution. The classical events are flips of the detectors -- they account for tracks. Quantum jumps are shown, in the simplest, homogeneous case, to be identical to those in the early spontaneous localization model of Ghirardi, Rimini and Weber (GRW). The methods and results of the present paper allow for an elementary derivation and numerical simulation of particle track formation and provide an additional perspective on GRW's proposal 
  We examine some of the implications of implementing the usual boundary conditions on the closed bosonic string in the hamiltonian framework. Using the KN formalism, it is shown that at the quantum level, the resulting constraints lead to relations among the periods of the basis 1-forms. These are compared with those of Riemanns' which arise from a different consideration. 
  A suggestion is made for quantizing gravity perturbatively, and is illustrated for the example of a massive scalar field with gravity. 
  Some part of the local gauge symmetries in the low energy region, say, lower than GUT or the Planck energy can be an induced symmetry describable with the holonomy fields associated with a topologically non-trivial structure of partially compactified space. In the case where a six dimensional space is compactified by the Kaluza-Klein mechanism into a product of the four dimensional Minkowski space $M_{4}$ and a two dimensional Riemann surface with the genus $g$, $\Sigma_{g}$, we show that, in a limit where the compactification mass scale is sent to infinity, a model lagrangian with a U(1) gauge symmetry produces the dynamical gauge fields in $M_{4}$ with a product of $g$ U(1)'s symmetry, i.e., U(1)$\times \cdots\times$U(1). These fields are induced by a Berry phase mechanism, not by the Kaluza-Klein. The dynamical degrees of freedom of the induced fields are shown to come from the holonomies, or the solenoid potentials, associated with the cycles of $\Sigma_{g}$. The production mechanism of kinetic energy terms for the induced fields are discussed in detail. 
  We discuss the connection between anyons (particles with fractional statistics) and deformed Lie algebras (quantum groups). After a brief review of the main properties of anyons, we present the details of the anyonic realization of all deformed classical Lie algebras in terms of anyonic oscillators. The deformation parameter of the quantum groups is directly related to the statistics parameter of the anyons. Such a realization is a direct generalization of the Schwinger construction in terms of fermions and is based on a sort of bosonization formula which yields the generators of the deformed algebra in terms of the undeformed ones. The entire procedure is well defined on two-dimensional lattices, but it can be consistently reduced also to one-dimensional chains. 
  The minimal Standard Model exhibits a nontrivial chiral U(2) symmetry if the vev and the hypercharge splitting (Delta) of right-handed leptons (quarks) in a family vanish and Q=T_0 + Y independently in each helicity sector. As a generalization, we start with SU(2)_L x SU(2)_R x U(1)_{(B-L)} and introduce Delta as a continuous parameter which is a measure of explicit symmetry breakdown. Values of Delta between zero and 1/2 take the neutral generator of the isospin-1/2 representation to the singlet representation, i.e. `deformes' the LR representation into the minimal Standard one. The corresponding classical O(3)-breaking term is a magnetic field perpendicular to the x_3-axis. A simple mapping on the fundamental Drinfeld-Jimbo q-deformed SU(2) representation is given. 
  We discuss some implications of the gravitational dressing of the renormalization group for conformal field theories perturbed by relevant operators. The renormalization group flows are defined with respect to the dilatation operator associated with the $J_0^{(0)}$ mode of the $SL(2,R)$ affine algebra. We discuss the possibility of passing under the $c=25$ barrier along renormalization group flows in some models. 
  We show that the continuum limit of one-dimensional N=2 supersymmetric matrix models can be described by a two-dimensional interacting field theory of a massless boson and two chiral fermions. We interpret this field theory as a two-dimensional N=1 supersymmetric theory of two chiral superfields, in which one of the chiral superfields has a non-trivial vacuum expectation value. 
  In the framework of Nelson stochastic quantization we derive exact non-stationary states for a class of time-dependent potentials. The wave-packets follow a classical motion with constant dispersion. The new states thus define a possible extension of the harmonic-oscillator coherent states to more general potentials. As an explicit example we give a detailed treatement of a sestic oscillator potential. 
  A model describing Ising spins with short range interactions moving randomly in a plane is considered. In the presence of a hard core repulsion, which prevents the Ising spins from overlapping, the model is analogous to a dynamically triangulated Ising model with spins constrained to move on a flat surface. As a function of coupling strength and hard core repulsion the model exhibits multicritical behavior, with first and second order transition lines terminating at a tricritical point. The thermal and magnetic exponents computed at the tricritical point are consistent with the KPZ values associated with Ising spins, and with the exact two-matrix model solution of the random Ising model, introduced previously to describe the effects of fluctuating geometries. 
  Large $N$ matrix models modified by terms of the form $ g(\Tr\Phi^n)^2$ generate random surfaces which touch at isolated points. Matrix model results indicate that, as $g$ is increased to a special value $g_t$, the string susceptibility exponent suddenly jumps from its conventional value $\gamma$ to ${\gamma\over\gamma-1}$. We study this effect in \L\ gravity and attribute it to a change of the interaction term from $O e^{\alpha_+ \phi}$ for $g<g_t$ to $O e^{\alpha_- \phi}$ for $g=g_t$ ($\alpha_+$ and $\alpha_-$ are the two roots of the conformal invariance condition for the \L\ dressing of a matter operator $O$). Thus, the new critical behavior is explained by the unconventional branch of \L\ dressing in the action. 
  It is shown that in $2+1$ dimensions, a constant magnetic field is a strong catalyst of dynamical flavor symmetry breaking, leading to generating a fermion dynamical mass even at the weakest attractive interaction between fermions. The essence of this effect is that in a magnetic field, in $2+1$ dimensions, the dynamics of fermion pairing is essentially one-dimensional. The effect is illustrated in the Nambu-Jona-Lasinio model in a magnetic field. The low-energy effective action in this model is derived and the thermodynamic properties of the model are considered. The relevance of this effect for planar condensed matter systems and for $3+1$ dimensional theories at high temperature is pointed out. 
  We explore the application of approximation schemes from many body physics, including the Hartree-Fock method and random phase approximation (RPA), to the problem of analyzing the low energy excitations of a polymer chain made up of bosonic string bits. We accordingly obtain an expression for the rest tension $T_0$ of the bosonic relativistic string in terms of the parameters characterizing the microscopic string bit dynamics. We first derive an exact connection between the string tension and a certain correlation function of the many-body string bit system. This connection is made for an arbitrary interaction potential between string bits and relies on an exact dipole sum rule. We then review an earlier calculation by Goldstone of the low energy excitations of a polymer chain using RPA. We assess the accuracy of the RPA by calculating the first order corrections. For this purpose we specialize to the unique scale invariant potential, namely an attractive delta function potential in two (transverse) dimensions. We find that the corrections are large, and discuss a method for summing the large terms. The corrections to this improved RPA are roughly 15\%. 
  It is showed how the Hamiltonian lattice $loop$ $representation$ can be cast straightforwardly in the Lagrangian formalism. The procedure is general and here we present the simplest case: pure compact QED. This connection has been shaded by the non canonical character of the algebra of the fundamental loop operators. The loops represent tubes of electric flux and can be considered the dual objects to the Nielsen-Olesen strings supported by the Higgs broken phase. The lattice loop classical action corresponding to the Villain form is proportional to the quadratic area of the loop world sheets and thus it is similar to the Nambu string action. This loop action is used in a Monte Carlo simulation and its appealing features are discussed. 
  It is shown that the even and odd coherent light and other nonclassical states of light like superposition of coherent states with different phases may replace the squeezed light in interferometric gravitational wave detector to increase its sensitivity. (Contribution to the Second Workshop on Harmonic Oscillator, Cocoyoc, Mexico, March 1994) 
  Photon distribution function, means and dispersions are found explicitly for the nonclassical state of light which is created from the photon--added coherent state $\vert \alpha,m \rangle$ due to a time--dependence of the frequency of the electromagnetic field oscillator. Generating function for factorial momenta is obtained. The Wigner function and Q--function are constructed explicitly for the excited photon--added coherent state of light. Influence of added photons on known oscillations of photon distribution function for squeezed light is demonstrated. oscillator are considered. 
  Pair-production of magnetic Reissner-Nordstr\"{o}m black holes (of charges $\pm q$) was recently studied in the leading WKB approximation. Here, we consider generic quantum fluctuations in the corresponding instanton geometry given by the Euclidean Ernst metric, in order to simulate the behaviour of the one-loop tunneling rate. A detailed study of the Ernst metric suggests that for sufficiently weak field $B$, the problem can be reduced to that of quantum fluctuations around a single near-extremal Euclidean black hole in thermal equilibrium with a heat bath of finite size. After appropriate renormalization procedures, typical one-loop contributions to the WKB exponent are shown to be inversely proportional to $B$, as $B\rightarrow 0$, indicating that the leading Schwinger term is corrected by a small fraction $\sim \hbar /q^2$. We demonstrate that this correction to the Schwinger term is actually due to a semiclassical shift of the black hole mass-to-charge ratio that persists even in the extremal limit. Finally we discuss a few loose ends. 
  It is shown that every Feynman integral can be interpreted as Green function of some linear differential operator with constant coefficients. This definition is equivalent to usual one but needs no regularization and application of $R$-operation. It is argued that presented formalism is convenient for practical calculations of Feynman integrals. 
  It is presented the general method that allows to formulate 4D $SU(N)$ Yang - Mills theory in terms of only local gauge invariant variables. For the case N=2, that is discussed in details, this gauge invariant formulation appears to be very similar to $ R^2 $-gravity. 
  We study the free energy of the pure glue QCD string with a torus target space and the gauge groups $SU(N)$ and (chiral) $U(N)$. It is highly constrained by a strong/weak gauge coupling duality which results in modular covariance. The string free energy is computed exactly in terms of modular forms for worldsheet genera 1 - 8. It has a surprisingly mild singularity in the weak gauge coupling/small area limit. 
  A procedure for obtaining correlation function densities and wavefunctionals for quantum gravity from the Donaldson polynomial invariants of topological quantum field theories, is given. We illustrate how our procedure may be applied to three and four dimensional quantum gravity. Detailed expressions, derived from \sbft{}, are given in the three dimensional case. A procedure for normalizing these wavefunctionals is proposed. 
  We bosonize fermions by identifying their occupation numbers as the binary digits of a Bose occupation number. Unlike other schemes, our method allows infinitely many fermionic oscillators to be constructed from just one bosonic oscillator. 
  The theory of representations of Clifford algebras is extended to employ the division algebra of the octonions or Cayley numbers. In particular, questions that arise from the non-associativity and non-commutativity of this division algebra are answered. Octonionic representations for Clifford algebras lead to a notion of octonionic spinors and are used to give octonionic representations of the respective orthogonal groups. Finally, the triality automorphisms are shown to exhibit a manifest $\perm_3 \times SO(8)$ structure in this framework. 
  The $\beta\gamma$ system is generalized by complex(rational) powers of the fields, which leads to a corresponding extension on the Fock space. Two different approaches to compute the Green functions of the physical operators are proposed. First the complex(rational) powers are defined via an integral representation,that allows to compute the conformal blocks, Green functions and structure constants of OPA. Next an approach based on a system of recursion equations for the Green functions is developed. A number of solutions of the system is found. A lot of possible applications is briefly discussed. 
  All the causally regular geometries obtained from (2+1)-anti-de Sitter space by identifications by isometries of the form $P \rightarrow (\exp \pi\xi) P$, where $\xi$ is a self-dual Killing vector of $so(2,2)$, are explicitely constructed. Their remarkable properties (Killing vectors, Killing spinors) are listed. These solutions of Einstein gravity with negative cosmological constant are also invariant under the string duality transformation applied to the angular translational symmetry $\phi \rightarrow \phi+a$ The analysis is made particularly convenient through the construction of {\em global} coordinates adapted to the identifications.} 
  We discuss bosonization in three dimensions by establishing a connection between the massive Thirring model and the Maxwell-Chern-Simons theory. We show, to lowest order in inverse fermion mass, the identity between the corresponding partition functions; from this, a bosonization identity for the fermion current, valid for length scales long compared with the Compton wavelength of the fermion, is inferred. We present a non-local operator in the Thirring model which exhibits fractional statistics. 
  A class of algebras is constructed using free fermions and the invariant antisymmetric tensors associated with irreducible holonomy groups. (This version contains minor typographical corrections and some additional references. ) 
  The central extension of a new infinite dimensional algebra which has both $W_\infty$ and affine $sl(2,R)$ as subalgebras is found. The critical dimension of the corresponding string model is $D=5$. 
  We derive the supersymmetric low-energy effective theory of the D-flat directions of a supersymmetric gauge theory. The Kahler potential of Affleck, Dine and Seiberg is derived by applying holomorphic constraints which manifestly maintain supersymmetry. We also present a simple procedure for calculating all derivatives of the Kahler potential at points on the flat direction manifold. Together with knowledge of the superpotential, these are sufficient for a complete determination of the spectrum and the interactions of the light degrees of freedom. We illustrate the method on the example of a chiral abelian model, and comment on its application to more complicated calculable models with dynamical supersymmetry breaking. 
  It is shown how a chiral Wess-Zumino-Witten theory with globally defined vertex operators and a one-to-one correspondence between fields and states can be constructed. The Hilbert space of this theory is the direct sum of tensor products of representations of the chiral algebra and finite dimensional internal parameter spaces. On this enlarged space there exists a natural action of Drinfeld's quasi quantum group $A_{g,t}$, which commutes with the action of the chiral algebra and plays the r\^{o}le of an internal symmetry algebra. The $R$ matrix describes the braiding of the chiral vertex operators and the coassociator $\Phi$ gives rise to a modification of the duality property.   For generic $q$ the quasi quantum group is isomorphic to the coassociative quantum group $U_{q}(g)$ and thus the duality property of the chiral theory can be restored. This construction has to be modified for the physically relevant case of integer level. The quantum group has to be replaced by the corresponding truncated quasi quantum group, which is not coassociative because of the truncation. This exhibits the truncated quantum group as the internal symmetry algebra of the chiral WZW model, which therefore has only a modified duality property. The case of $g=su(2)$ is worked out in detail. 
  It is known that certain properties of charged dilaton black holes depend on a free parameter $\beta$ which controls the strength of the coupling of the dilaton to the Maxwell field. We obtain the energy associated with static spherically symmetric charged dilaton black holes for arbitrary value of the coupling parameter and find that the energy distribution depends on the value of $\beta$. With increasing radial distance, the energy in a sphere increases for $\beta = 0$ as well as for $\beta < 1$, decreases for $\beta > 1$, and remains constant for $\beta = 1$. However, the total energy turns out to be the same for all values of $\beta$. 
  In a class of three-dimensional abelian gauge theory the Lorentz invariance is spontaneously broken by dynamical generation of a magnetic field. An originally topologically massive photon becomes gapless, i.e. p_0=0 at {\vec p}=0. Indeed, the photon is the Nambu-Goldstone boson associated with the spontaneous breaking of the Lorentz invariance. Although symmetry generated by two Lorentz boost generators is broken, there seems to appear only one physical Nambu-Goldstone boson, namely a photon. We argue that the Ward identities in the Nambu-Goldstone theorem are saturated by the photon pole. 
  This thesis is roughly organized into two parts. The first one (the first three chapters), expository in nature, attempts to place the current work in context: at first historically, but then focusing on the Lax formalism and the Adler--Gel'fand--Dickey scheme for hierarchies of the KdV type. The second part (the last four chapters) comprises the main body of this work. It begins by developing the supersymmetric Lax formalism, introducing the ring of formal superpseudodifferential operators and the associated Poisson structures. We discuss three supersymmetric extensions of the KP hierarchy (MRSKP, \SKP2, and JSKP). We define and compute their additional symmetries and we find that the algebra of additional symmetries are in all three cases isomorphic to the Lie algebra of superdifferential operators. We discuss a new reduction of \SKP2 and the relation between MRSKP and \SKP2 is clarified. Finally we consider the (so far) only integrable hierarchy to have played a role in noncritical superstring theory (sKdV-B). We identify it, prove its bihamiltonian integrability, and extend it by odd flows. We close with a discussion of new integrable supersymmetrizations of the KdV-like hierarchies suggested by the study of sKdV-B. (This is the author's PhD Thesis from the Physics Deparment of the University of Bonn, July 1994.) 
  We show how to make a topological string theory starting from an $N=4$ superconformal theory. The critical dimension for this theory is $\hat c= 2$ ($c=6$). It is shown that superstrings (in both the RNS and GS formulations) and critical $N=2$ strings are special cases of this topological theory. Applications for this new topological theory include: 1) Proving the vanishing to all orders of all scattering amplitudes for the self-dual $N=2$ string with flat background, with the exception of the three-point function and the closed-string partition function; 2) Showing that the topological partition function of the $N=2$ string on the $K3$ background may be interpreted as computing the superpotential in harmonic superspace generated upon compactification of type II superstrings from 10 to 6 dimensions; and 3) Providing a new prescription for calculating superstring amplitudes which appears to be free of total-derivative ambiguities. 
  Presented are polynomial identities which imply generalizations of Euler and Rogers--Ramanujan identities. Both sides of the identities can be interpreted as generating functions of certain restricted partitions. We prove the identities by establishing a graphical one-to-one correspondence between those two kinds of restricted partitions. 
  We give close formulas for the counting functions of rational curves on complete intersection Calabi-Yau manifolds in terms of special solutions of generalized hypergeometric differential systems. For the one modulus cases we derive a differential equation for the Mirror map, which can be viewed as a generalization of the Schwarzian equation. We also derive a nonlinear seventh order differential equation which directly governs the instanton corrected Yukawa coupling. 
  We discuss 2-cocycles of the Lie algebra $\Map(M^3;\g)$ of smooth, compactly supported maps on 3-dimensional manifolds $M^3$ with values in a compact, semi-simple Lie algebra $\g$. We show by explicit calculation that the Mickelsson-Faddeev-Shatashvili cocycle $\f{\ii}{24\pi^2}\int\trac{A\ccr{\dd X}{\dd Y}}$ is cohomologous to the one obtained from the cocycle given by Mickelsson and Rajeev for an abstract Lie algebra $\gz$ of Hilbert space operators modeled on a Schatten class in which $\Map(M^3;\g)$ can be naturally embedded. This completes a rigorous field theory derivation of the former cocycle as Schwinger term in the anomalous Gauss' law commutators in chiral QCD(3+1) in an operator framework. The calculation also makes explicit a direct relation of Connes' non-commutative geometry to (3+1)-dimensional gauge theory and motivates a novel calculus generalizing integration of $\g$-valued forms on 3-dimensional manifolds to the non-commutative case. 
  We construct composite operators in two-dimensional bosonized QCD, which obey a $W_\infty$ algebra, and discuss their relation to analogous objects recently obtained in the fermionic language. A complex algebraic structure is unravelled, supporting the idea that the model is integrable. For singlets we find a mass spectrum obeying the Regge behavior. 
  We construct the Euclidean Hopf algebra $U_q(e^N)$ dual of $Fun(\rn_q^N\lcross SO_{q^{-1}}(N))$ by realizing it as a subalgebra of the differential algebra $\DFR$ on the quantum Euclidean space $\rn_q^N$; in fact, we extend our previous realization \cite{fio4} of $U_{q^{-1}}(so(N))$ within $\DFR$ through the introduction of q-derivatives as generators of q-translations. The fundamental Hilbert space representations of $U_q(e^N)$ turn out to be of highest weight type and rather simple `` lattice-regularized '' versions of the classical ones. The vectors of a basis of the singlet (i.e. zero-spin) irrep can be realized as normalizable functions on $\rn_q^N$, going to distributions in the limit $q\rightarrow 1$. 
  We summarize some results obtained on the problem of gauging the Wess--Zumino term of a d-dimensional bosonic sigma-model. We show that gauged WZ-like terms are in one-to-one correspondence with equivariant cocycles of the target space. By the same token, the obstructions to gauging a WZ term can be understood in terms of the equivariant cohomology of the target space and this allows us to use topological tools to derive some a priori vanishing theorems guaranteeing the absence of obstructions for a large class of target spaces and symmetry groups in the physically interesting dimensions d<=4. (This is an expository summary of the results of hep-th/9407149.) 
  We analyse with the algebraic, regularisation independent, cohomological B.R.S. methods, the renormalisability of torsionless N=2 supersymmetric non-linear $\si$ models built on compact K\"ahler spaces. Surprisingly enough with respect to the common wisdom, we obtain an anomaly candidate, when the Hodge number $h^{3,0}$ of the target space manifold is different from zero : this occurs in particular in the Calabi-Yau case. On the contrary, in the compact homogeneous K\"ahler case, the anomaly candidate disappears. 
  We perform a marginal deformation of the SL(2,R) WZW model in a null direction. If we send the deformation parameter to infinity we obtain a linear dilaton background plus two free bosons. We show in addition that such a background can be obtained by a duality transformation of the undeformed WZW model. In the end we indicate how to generalize the given procedure. 
  We give field theoretic realizations of both the $q$-Heisenberg and the $q$-Virasoro algebra. In particular, we obtain the operator product expansions among the current and the energy momentum tensor obtained using the Sugawara construction. 
  In a space of d $( d > 5) $ ordinary and d Grassmann coordinates, fields manifest in an ordinary four-dimensional subspace as spinor (1/2, 3/2), scalar, vector or tensor fields with the corresponding charges , according to two kinds of generators of the Lorentz transformations in the Grassmann space. Vielbeins and spin connections define gauge fields-gravitational and Yang-Mills. 
  We review the elementary theory of gauge fields and the Becchi-Rouet-Stora- Tyutin symmetry in the context of differential geometry. We emphasize the topological nature of this symmetry and discuss a double Chevalley-Eilenberg complex for it. 
  We consider the sine-Gordon model on a half-line, with an additional potential term of the form $-M\cos{\beta\over 2}(\varphi-\varphi_0)$ at the boundary. We compute the classical time delay for general values of $M$, $\beta$ and $\varphi_0$ using $\tau$-function methods and show that in the classical limit, the method of images still works, despite the non-linearity of the problem. We also perform a semi-classical analysis, and find agreement with the exact quantum S-matrix conjectured by Ghoshal and Zamolodchikov. 
  We consider the continuum string theory corresponding to the Marinari-Parisi supersymmetric matrix model. We argue that the world-sheet physics is exotic, and different from any known supersymmetric string theory. The embedding superspace coordinates become disordered on the world-sheet, but because of the noncompactness of the embedding time the disorder becomes complete only at asymptotic world-sheet scales. 
  The type-I simple Lie-superalgebras are $sl(m|n)$ and $osp(2|2n)$. We study the quantum deformations of their untwisted affine extensions $U_q(sl(m|n)^{(1)})$ and $U_q(osp(2|2n)^{(1)})$. We identify additional relations between the simple generators (``extra $q$-Serre relations") which need to be imposed to properly define $\uqgh$ and $U_q(osp(2|2n)^{(1)})$. We present a general technique for deriving the spectral parameter dependent R-matrices from quantum affine superalgebras. We determine the R-matrices for the type-I affine superalgebra $U_q(sl(m|n)^{(1)})$ in various representations, thereby deriving new solutions of the spectral-dependent Yang-Baxter equation. In particular, because this algebra possesses one-parameter families of finite-dimensional irreps, we are able to construct R-matrices depending on two additional spectral-like parameters, providing generalizations of the free-fermion model. 
  It is shown that the N-extended super $w_n$\ string can be obtained as a special class of vacua of both the N-extended super $w_{n+1}$\ string and the (N+1)-extended super $w_n$\ string. This hierarchy of super $w$\ string theories includes as subhierarchies both the superstring hierarchy given by Bastianelli, Ohta and Petersen and the $w$\ string hierarchy given by the authors. 
  A geometric characterization of transition amplitudes between coherent states, or equivalently, of the hermitian scalar product of holomorphic cross sections in the associated D - M tilda - module, in terms of the embedding of the cohe- rent state manifold M-tilda into a projective Hilbert space is proposed. Cohe- rent state manifolds endowed with a homogeneous kaehler structure are conside- red. Using the coherent state approach, an effective method to find the cut loci on symmetric manifolds and generalized symmetric manifolds M-tilda is proposed. The CW - complex structure of coherent state manifolds of flag type is discussed. Recent results of Anandan and Aharonov are commented vis-a-vis of last century constructions in projective geometry. Calculations with signi- ficance in coherent state approch furnish explicit proofs of the results announ- ced by Y. C. Wong on conjugate locus in complex Grassmann manifold. 
  The new inversion formula for the Laplace transformation of the tempered distributions with supports in the closed positive semiaxis is obtained. The inverse Laplace transform of the tempered distribution is defined by means of the limit of the special distribution constructed from this distribution. The weak spectral condition on the Euclidean Green's functions requires that some of the limits needed for the inversion formula exist for any Euclidean Green's function with even number of variables. We prove that the initial Osterwalder--Schrader axioms (1973) and the weak spectral condition are equivalent with the Wightman axioms. 
  We present an evaluation of the fluctuation determinant which appears as a prefactor in the instanton transition rate for the two-dimensional Abelian Higgs model. The corrections are found to change the rate at most by a factor of 2 for 0.4 < M_W/M_H < 2.0. 
  Thermodynamic properties of non-relativistic bosons and fermions in two spatial dimensions and without interactions are derived. All the virial coefficients are the same except for the second, for which the signs are opposite. This results in the same specific heat for the two gases. Existing equations of state for the free anyon gas are also discussed and shown to break down at low temperatures or high densities. 
  We prove that the number of 3-dimensional simplicial complexes having the spherical topology grows exponentially as a function of a volume. It is suggested that the 3d simplicial quantum gravity has qualitatively the same phase structure as the O(n) matrix model in the dense phase. 
  Holomorphy of the superpotential and of the coefficient of the gauge kinetic terms in supersymmetric theories lead to powerful results. They are the underlying conceptual reason for the important non-renormalization theorems. They also enable us to study the exact non-perturbative dynamics of these theories. We find explicit realizations of known phenomena as well as new ones in four dimensional strongly coupled field theories. These shed new light on confinement and chiral symmetry breaking. This note is based on a talk delivered at the PASCOS (94) meeting at Syracuse University. 
  Techniques based upon the string organisation of amplitudes may be used to simplify field theory calculations. We apply these techniques to perturbative gravity and calculate all one-loop amplitudes for four-graviton scattering with arbitrary internal particle content. Decomposing the amplitudes into contributions arising from supersymmetric multiplets greatly simplifies these calculations. We also discuss how unitarity may be used to constrain the amplitudes. 
  Fisher's phenomenological renormalization method is used to calculate the mass gap and the correlation length of the $O(N)$ nonlinear $\sigma$ model on a semi-compact space $S^{1}\times {\bf R}^{2}$. This shows that the ultraviolet momentum cut-off does not conflict with the infrared cut-off along the $S^{1}$ direction. The mass gap on $S^{2}\times {\bf R}$ is also discussed. 
  We unify all existing results on the change of the speed of low--energy photons due to modifications of the vacuum, finding that it is given by a universal constant times the quotient of the difference of energy densities between the usual and modified vacua over the mass of the electron to the fourth power.  Whether photons move faster or slower than $c$ depends only on the lower or higher energy density of the modified vacuum, respectively. Physically, a higher energy density is characterized by the presence of additional particles (real or virtual) in the vacuum whereas a lower one stems from the absence of some virtual modes.  We then carry out a systematic study of the speed of propagation of massless particles for several field theories up to two loops on a thermal vacuum. Only low--energy massless particles corresponding to a massive theory show genuine modifications of their speed while remaining massless. All other modifications are mass-related, or running mass-related. We also develop a formalism for the Casimir vacuum which parallels the thermal one and check that photons travel faster than $c$ between plates. 
  We construct representations of the enveloping algebra $U_q osp(2,2)$ in terms of finite difference operators and we discuss this result in the framework of quasi-exactly-solvable equations. 
  We use the bilocal method to derive the large $N$ solution of the most general $QCD_2$. 
  We prove that $\W_3$ is the gauge symmetry of the scale-invariant rigid particle, whose action is given by the integrated extrinsic curvature of its world line. This is achieved by showing that its equations of motion can be written in terms of the Boussinesq operator. The $\W_3$ generators $T$ and $W$ then appear respectively as functions of the induced world line metric and the extrinsic curvature. We also show how the equations of motion for the standard relativistic particle arise from those of the rigid particle whenever it is consistent to impose the ``zero-curvature gauge'', and how to rewrite them in terms of the $\KdV$ operator. The relation between particle models and integrable systems is further pursued in the case of the spinning particle, whose equations of motion are closely related to the $\SKdV$ operator. We also partially extend our analysis in the supersymmetric domain to the scale invariant rigid particle by explicitly constructing a supercovariant version of its action. Comment: This is an expanded version of hep-th/9406072 (to be published in the Proceedings of the Workshop on the Geometry of Constrained Dynamical Systems, held at the Isaac Newton Institute for Mathematical Sciences, Cambridge, June 14-18, 1994.). 
  We propose a definite meaning to the concepts of "experiment", "measurement" and "event" in the event-enhanced formalism of quantum theory. A minimal piecewise deterministic process is given that can be used for a computer simulation of real time series of experiments on single quantum objects. As an example a generalized cloud chamber is described, including multiparticle case. Relation to the GRW spontaneous localization model is discussed. 
  We propose a definite meaning to the concepts of "experiment", "measurement" and "event" in the event-enhanced formalism of quantum theory. A minimal piecewise deterministic process is given that can be used for a computer simulation of real time series of experiments on single quantum objects. As an example a generalized cloud chamber is described, including multiparticle case. Relation to the GRW spontaneous localization model is discussed. The second revised version of the paper contains references to papers by other authors that are are aiming in the same direction: to enhance quantum theory in such a way that it will provide stochastic description of events triggered by individual quantum systems. 
  We describe the bound state and scattering properties of a quantum mechanical particle in a scalar $N$-prong potential. Such a study is of special interest since these situations are intermediate between one and two dimensions. The energy levels for the special case of $N$ identical prongs exhibit an alternating pattern of non-degeneracy and $(N-1)$ fold degeneracy. It is shown that the techniques of supersymmetric quantum mechanics can be used to generate new solutions. Solutions for prongs of arbitrary lengths are developed. Discussions of tunneling in $N$-well potentials and of scattering for piecewise constant potentials are given. Since our treatment is for general values of $N$, the results can be studied in the large $N$ limit. A somewhat surprising result is that a free particle incident on an $N$-prong vertex undergoes continuously increased backscattering as the number of prongs is increased. 
  We describe several different regimes which are possible in inflationary cosmology. The simplest one is inflation without self-reproduction of the universe. In this scenario the universe is not stationary. The second regime, which exists in a broad class of inflationary models, is eternal inflation with the self-reproduction of inflationary domains. In this regime local properties of domains with a given density and given values of fields do not depend on the time when these domains were produced. The probability distribution to find a domain with given properties in a self-reproducing universe may or may not be stationary, depending on the choice of an inflationary model. We give examples of models where each of these possibilities can be realized, and discuss some implications of our results for quantum cosmology. In particular, we propose a new mechanism which may help solving the cosmological constant problem. 
  It is shown that Weyl spinors in 4D Minkowski space are composed of primary fields of half-integer conformal weights. This yields representations of fermionic 2-point functions in terms of correlators of primary fields with a factorized transformation behavior under the Lorentz group. I employ this observation to determine the general structure of the corresponding Lorentz covariant correlators by methods similar to the methods employed in conformal field theory to determine 2- and 3-point functions of primary fields. In particular, the chiral symmetry breaking terms resemble fermionic 2-point functions of 2D CFT up to a function of the product of momenta. The construction also permits for the formulation of covariant meromorphy constraints on spinors in 3+1 dimensions. 
  We investigate the coupled system of gravity and a scalar with exponential potential. The energy momentum tensor of the scalar field induces a time-dependent cosmological ``constant''. This adjusts itself dynamically to become in the ``late'' universe (including today) proportional to the energy density of matter and radiation. Possible consequences for the present cosmology are shortly discussed. We also address the question of naturalness of the cosmon model. 
  String propagation is investigated in de Sitter and black hole backgrounds using both exact and approximative methods. The circular string evolution in de Sitter space is discussed in detail with respect to energy and pressure, mathematical solution and physical interpretation, multi-string solutions etc. We compare with the circular string evolution in the 2+1 dimensional black hole anti de Sitter spacetime and in the equatorial plane of ordinary 3+1 dimensional stationary axially symmetric spacetime solutions of Einstein general relativity. Using an approximative string perturbation approach we consider also generic string evolution and propagation in all these curved spacetimes. (To appear in the proceedings of the 2eme Journee Cosmologie held in Paris, June 1994) 
  We compare several parametrized analytic expressions for an arbitrary off-shell one-loop $n$-point function in scalar field theory in $D$-dimensional space-time, and show their equivalence both directly and through path-integral methods. 
  We show that the BV (Batalin Vilkovisky) action, formulated with an extended BRST symmetry (including the shift symmetry), is also invariant under an extended anti-BRST transformation (where the antifields are the parameters of the transformation), when the gauge fixing Lagrangian is both BRST and anti-BRST invariant. We show that for a general gauge fixing Lagrangian, the BV action can be written in a manifestly extended BRST invariant manner in a superspace with one Grassmann coordinate whereas it can be expressed in a manifestly extended BRST and anti-BRST invariant manner in a superspace with two Grassmann coordinates when the gauge fixing Lagrangian is invariant under both BRST and anti-BRST transformations. 
  I investigate the Kazakov-Migdal (KM) model -- the Hermitean gauge-invariant matrix model on a D-dimensional lattice. I utilize an exact large-N solution of the KM model with a logarithmic potential to examine its critical behavior. I find critical lines associated with gamma_{string}=-1/2 and gamma_{string}=0 as well as a tri-critical point associated with a Kosterlitz-Thouless phase transition. The continuum theories are constructed expanding around the critical points. The one associated with gamma_{string}=0 coincides with the standard d=1 string while the Kosterlitz-Thouless phase transition separates it from that with gamma_{string}=-1/2 which is indistinguishable from pure 2D gravity for local observables but has a continuum limit for correlators of extended Wilson loops at large distances due to a singular behavior of the Itzykson-Zuber correlator of the gauge fields. I reexamine the KM model with an arbitrary potential in the large-D limit and show that it reduces at large N to a one-matrix model whose potential is determined self-consistently. A relation with discretized random surfaces is established via the gauged Potts model which is equivalent to the KM model at large N providing the coordination numbers coincide. 
  The BRST transformations for the Yang-Mills gauge fields in the presence of gravity with torsion are discussed by using the so-called Maurer-Cartan horizontality conditions. With the help of an operator $\d$ which allows to decompose the exterior spacetime derivative as a BRST commutator we solve the Wess-Zumino consistency condition corresponding to invariant Chern-Simons terms and gauge anomalies. 
       A  review of the multiparametric linear quantum group GL_qr(N), its real forms, its dual algebra U(gl_qr(N)) and its bicovariant differential calculus is given in the first part of the paper.       We then construct the (multiparametric) linear inhomogeneous quantum group IGL_qr(N) as a projection from GL_qr(N+1), or equivalently, as a quotient of GL_qr(N+1) with respect to a suitable Hopf algebra ideal.       A bicovariant differential calculus on IGL_qr(N) is explicitly obtained as a projection from the one on GL_qr(N+1). Our procedure unifies in a single structure the quantum plane coordinates and the q-group matrix elements T^a_b, and allows to deduce without effort the differential calculus on the q-plane IGL_qr(N) / GL_qr(N).       The general theory is illustrated on the example of IGL_qr(2). 
  We investigate the influence of the projective invariance on the renormalization properties of the theory. One-loop counterterms are calculated in the most general case of interaction of gravity with scalar field. 
  In any string theory there is a hidden, twisted superconformal symmetry algebra, part of which is made up by the BRST current and the anti-ghost. We investigate how this algebra can be systematically constructed for strings with $N\!-\!2$ supersymmetries, via quantum Hamiltonian reduction of the Lie superalgebras $osp(N|2)$. The motivation is to understand how one could systematically construct generalized string theories from superalgebras. We also briefly discuss the BRST algebra of the topological string, which is a doubly twisted $N\!=\!4$ superconformal algebra. 
  It is shown that the master equation of the affine-Virasoro construction on the unitary affine algebra naturally emerges in the fusion algebra of the nonunitary level $k$ WZNW model. Operators corresponding to solutions of the master equation are suitable for performing one-parametrical renormalizable perturbation around the given conformal nonunitary WZNW model. In the large $|k|$ limit, the infrared fixed point of the renormalization group beta function is found. There are as many infrared conformal points as $\frac{1}{2}N(N+1)$, where $N$ is the total number of solutions of the master equation. 
  For any two arbitrary positive integers `$n$' and `$m$', using the $m$--th KdV hierarchy and the $(n+m)$--th KdV hierarchy as building blocks, we are able to construct another integrable hierarchy (referred to as the $(n,m)$--th KdV hierarchy). The $W$--algebra associated to the \shs\, of the $(n,m)$--th KdV hierarchy (called $W(n,m)$ algebra) is isomorphic via a Miura map to the direct sum of $W_m$--algebra, $W_{n+m}$--algebra and an additional $U(1)$ current algebra. In turn, from the latter, we can always construct a representation of $W_\infty$--algebra. 
  We consider generic properties of the moduli space of vacua in $N=2$ supersymmetric Yang--Mills theory recently studied by Seiberg and Witten. We find, on general grounds, Picard--Fuchs type of differential equations expressing the existence of a flat holomorphic connection, which for one parameter (i.e. for gauge group $G=SU(2)$), are second order equations. In the case of coupling to gravity (as in string theory), where also ``gravitational'' electric and magnetic monopoles are present, the electric--magnetic S duality, due to quantum corrections, does not seem any longer to be related to $Sl(2,\IZ)$ as for $N=4$ supersymmetric theory. 
  We study aspects of the theory of generalized Kac-Moody Lie algebras (or Borcherds algebras) and their standard modules. It is shown how such an algebra with no mutually orthogonal imaginary simple roots, including Borcherds' Monster Lie algebra $\frak m$, can be naturally constructed from a certain Kac-Moody subalgebra and a module for it. We observe that certain generalized Verma (induced) modules for generalized Kac-Moody algebras are standard modules and hence irreducible. In particular, starting from the moonshine module for the Monster group $M$, we construct a certain $\{frak gl}_2$- and $M$-module, the tensor algebra over which carries a natural structure of irreducible module for $\frak m$, which is realized as an explicitly prescribed $M$-covariant Lie algebra of operators on this tensor algebra. The existence of large free subalgebras of $\frak m$ is further exploited to provide a simplification of Borcherds' proof of the Conway-Norton conjectures for the McKay-Thompson series of the moonshine module. The coefficients of these series are shown to satisfy natural recursion relations (replication formulas) equivalent to, but different from, those obtained by Borcherds. 
  We use the method due to Batalin, Fradkin and Tyutin (BFT) for the quantization of chiral boson theories. We consider the Floreanini-Jackiw (FJ) formulation as well as others with linear constraints. 
  The definition of matter states on spacelike hypersurfaces of a 1+1 dimensional black hole spacetime is considered. The effect of small quantum fluctuations of the mass of the black hole due to the quantum nature of the infalling matter is taken into account. It is then shown that the usual approximation of treating the gravitational field as a classical background on which matter is quantized, breaks down near the black hole horizon. Specifically, on any hypersurface that captures both infalling matter near the horizon and Hawking radiation, quantum fluctuations in the background geometry become important, and a semiclassical calculation is inconsistent. An estimate of the size of correlations between the matter and gravity states shows that they are so strong that a fluctuation in the black hole mass of order exp[-M/M_{Planck}] produces a macroscopic change in the matter state. 
  We show that the leading order solution describing an extremal electrically charged black hole in string theory is, in fact, an exact solution to all orders in $\a'$ when interpreted in a Kaluza-Klein fashion. This follows from the observation that it can be obtained via dimensional reduction from a five dimensional background which is proved to be an exact string solution. 
  Lectures delivered at the International School of Physics "Enrico Fermi", held in Villa Monastero, Varenna, Italy, 94. 
  In the present note we give an explicit integration of some two--dimensionalised Lotka--Volterra type equations associated with simple Lie algebras, other than the familiar $A_n$ case, possessing a representation without branching. This allows us, in particular, to treat the first fundamental representations of $A_r$, $B_r$, $C_r$, and $G_2$ on the same footing. 
  We consider an Electroweak string in the background of a uniform distribution of cold fermionic matter. As a consequence of the fermion number non-conservation in the Weinberg-Salam model, the string produces a long-range magnetic field. 
  We study an extension of the symplectic formalism in order to quantize reducible systems. We show that a procedure like {\it ghost-of-ghost} of the BFV method can be applied in terms of Lagrange multipliers. We use the developed formalism to quantize the antisymmetric Abelian gauge fields. 
  We quantize the Einstein gravity in the formalism of weak gravitational fields by using the constrained Hamiltonian method. Special emphasis is given to the 2+1 spacetime dimensional case where a (topological) Chern-Simons term is added to the Lagrangian. 
  Using matrix model techniques we investigate the large N limit of generalized 2D Yang-Mills theory. The model has a very rich phase structure. It exhibits multi-critical behavior and reveals a third order phase transitions at all genera besides {\it torus}. This is to be contrasted with ordinary 2D Yang-Mills which, at large N, exhibits phase transition only for spherical topology. 
  The exact quantization of two models, the massive vector meson model and the Higgs model in the London limit, both describing massive photons, is presented. Even though naive arguments (based on gauge-fixing) may indicate the equivalence of these models, it is shown here that this is not true in general when we consider these theories on manifolds with boundaries. We show, in particular, that they are equivalent only for a special choice of the boundary conditions that we are allowed to impose on the fields. 
  We examine the problem of constructing N=2 superconformal algebras out of N=1 non-semi-simple affine Lie algebras. These N=2 superconformal theories share the property that the super Virasoro central charge depends only on the dimension of the Lie algebra. We find, in particular, a construction having a central charge c=9. This provides a possible internal space for string compactification and where mirror symmetry might be explored. 
  We show that the supersymmetric nonlinear Schr\"odinger equation can be written as a constrained super KP flow in a nonstandard representation of the Lax equation. We construct the conserved charges and show that this system reduces to the super mKdV equation with appropriate identifications. We construct various flows generated by the general nonstandard super Lax equation and show that they contain both the KP and mKP flows in the bosonic limits. This nonstandard supersymmetric KP hierarchy allows us to construct a new super KP equation which is nonlocal. 
  We review the Exact Renormalization Group equations of Wegner and Houghton in an approximation which permits both numerical and analytical studies of nonperturbative renormalization flows. We obtain critical exponents numerically and with the local polynomial approximation (LPA), and discuss the advantages and shortcomings of these methods, and compare our results with the literature. In particular, convergence of the LPA is discussed in some detail. We finally integrate the flows numerically and find a $c$-function which determines these flows to be gradient in this approximation. 
  The partition function of $D_{N+1}$ topological string, the coupled system of topological gravity and $D_{N+1}$ topological minimal matter , is proposed in the framework of KP hierarchy. It is specified by the elements of $GL(\infty)$ which constitute the deformed family from the $A_{2N-1}$ topological string. Its dispersionless limit is investigated from the view of both dispersionless KP hierarchy and singularity theory. In particular the free energy restricted on the small phase space coincides with that for the topological Landau-Ginzburg model of type $D_{N+1}$. 
  The Hamiltonians of $SU(2)$ and $SU(3)$ gauge theories in 3+1 dimensions can be expressed in terms of gauge invariant spatial geometric variables, i.e., metrics, connections and curvature tensors which are simple local functions of the non-Abelian electric field. The transformed Hamiltonians are local. New results from the same procedure applied to the $SU(2)$ gauge theory in 2+1 dimensions are also given. 
  We construct a Batalin-Vilkovisky (BV) algebra on moduli spaces of Riemann surfaces. This algebra is background independent in that it makes no reference to a state space of a conformal field theory. Conformal theories define a homomorphism of this algebra to the BV algebra of string functionals. The construction begins with a graded-commutative free associative algebra $\C$ built from the vector space whose elements are orientable subspaces of moduli spaces of punctured Riemann surfaces. The typical element here is a surface with several connected components. The operation $\Delta$ of sewing two punctures with a full twist is shown to be an odd, second order derivation that squares to zero. It follows that $(\C, \Delta)$ is a Batalin-Vilkovisky algebra. We introduce the odd operator $\delta = \partial + \hbar\Delta$, where $\partial$ is the boundary operator. It is seen that $\delta^2=0$, and that consistent closed string vertices define a cohomology class of $\delta$. This cohomology class is used to construct a Lie algebra on a quotient space of $\C$. This Lie algebra gives a manifestly background independent description of a subalgebra of the closed string gauge algebra. 
  For realistic values of the Higgs boson mass the high temperature electroweak phase transition cannot be described perturbatively. The symmetric phase is governed by a strongly interacting $SU(2)$ gauge theory. Typical masses of excitations and scales of condensates are set by the ``high temperature confinement scale'' $\approx 0.2\ T$. For a Higgs boson mass around 100 GeV or above all aspects of the phase transition are highly nonperturbative. Near the critical temperature strong electroweak interactions are a dominant feature also in the phase with spontaneous symmetry breaking. Depending on the value of the Higgs boson mass the transition may be a first order phase transition or an analytical crossover. 
  We describe how geometrical methods can be applied to a system with explicitly time-dependent second-class constraints so as to cast it in Hamiltonian form on its physical phase space. Examples of particular interest are systems which require time-dependent gauge fixing conditions in order to reduce them to their physical degrees of freedom. To illustrate our results we discuss the gauge-fixing of relativistic particles and strings moving in arbitrary background electromagnetic and antisymmetric tensor fields. 
  The Faddeev-Reshetikhin-Takhtajan method to construct matrix bialgebras from non-singular solutions of the quantum Yang-Baxter equation is extended to the anyonic or $\Z_n$-graded case. The resulting anyonic quantum matrices are braided groups in which the braiding is given by a phase factor. 
  For zero energy, $E=0$, we derive exact, classical solutions for {\em all} power-law potentials, $V(r)=-\gamma/r^\nu$, with $\gamma>0$ and $-\infty <\nu<\infty$. When the angular momentum is non-zero, these solutions lead to the orbits $\r(t)= [\cos \mu (\th(t)-\th_0(t))]^{1/\mu}$, for all $\mu \equiv \nu/2-1 \ne 0$. When $\nu>2$, the orbits are bound and go through the origin. This leads to discrete discontinuities in the functional dependence of $\th(t)$ and $\th_0(t)$, as functions of $t$, as the orbits pass through the origin. We describe a procedure to connect different analytic solutions for successive orbits at the origin. We calculate the periods and precessions of these bound orbits, and graph a number of specific examples. Also, we explain why they all must violate the virial theorem. The unbound orbits are also discussed in detail. This includes the unusual orbits which have finite travel times to infinity and also the special $\nu = 2$ case. 
  For zero energy, $E=0$, we derive exact, quantum solutions for {\it all} power-law potentials, $V(r) = -\gamma/r^{\nu}$, with $\gamma > 0$ and $-\infty < \nu < \infty$. The solutions are, in general, Bessel functions of powers of $r$. For $\nu > 2$ and $l \ge 1$ the solutions are normalizable; they correspond to states which are bound by the angular-momentum barrier. Surprisingly, the solutions for $\nu < -2$ are also normalizable, They are discrete states but do not correspond to bound states. For $2> \nu \geq -2$ the states are unnormalizable continuum states. The $\nu=2$ solutions are also unnormalizable, but are exceptional solutions. Finally, we find that by increasing the dimension of the \seq beyond 4 an effective centrifugal barrier is created, due solely to the extra dimensions, which is enough to cause binding. Thus, if $D>4$, there are $E=0$ bound states for $\nu > 2$ even for $l=0$. We discuss the physics of the above solutions are compare them to the classical solutions of the preceding paper. 
  Using the multi-parametric deformation of the algebra of functions on $ \GL{n+1} $ and the universal enveloping algebra $ \U{\igl{n+1}} $, we construct the multi-parametric quantum groups $ \IGLq{n} $ and $ \Uq{\igl{n}} $. 
  It is shown that the O(d,d;R) deformations of the superstring vacua and the O(d,d+16;R) deformations of the heterotic string vacua preserve extended worldsheet supersymmetry and, hence, generate superconformal deformations. The transformations of the complex structures are given explicitly and the action of the discrete duality subgroup is discussed. The results are valid when the complex structures are independent of the d coordinates which appear in the transformations. It is shown that generic deformations do not preserve the known superfield formulations of (2,2) extended supersymmetry. The analysis is performed by decomposing the transformations in terms of the metric vielbein and by introducing space-time connections induced due to the non-linear action of the O(d,d;R) and O(d,d+16;R) deformations on the background fields. 
  We study the $SU(N)$ self-dual Chern-Simons-Higgs systems with adjoint matter coupling, and show that the sixth order self-dual potential has $p(N)$ gauge inequivalent degenerate minima, where $p(N)$ is the number of partitions of $N$. We compute the masses of the gauge and scalar excitations in these different vacua, revealing an intricate mass structure which reflects the self-dual nature of the model. 
  Harmonic superspaces for spacetimes of dimension $d\leq 3$ are constructed. Some applications are given. 
  We investigate the cosmological solutions in closed bosonic string theory in the presence of non zero tachyon condensate. We specifically obtain time dependent solutions which describe an anisotropic universe. We also discuss the nature of such time dependent solutions when small tachyon fluctuations around the condensate are taken into account. 
  We introduce the notion of cyclic cohomology of an A-infinity algebra and show that the deformations of an A-infinity algebra which preserve an invariant inner product are classified by this cohomology. We use this result to construct some cycles on the moduli space of algebraic curves. The paper also contains a review of some well known notions and results about Hochschild and cyclic cohomology of associative algebras, A-infinity algebras, and deformation theory of algebras, and includes a discussion of the homology of the graph complex of metric ribbon graphs which is associated to the moduli space of Riemann surfaces with marked points. 
  It is well-known that the Toda Theories can be obtained by reduction from the Wess-Zumino-Novikov-Witten (WZNW) model, but it is less known that this WZNW $\rightarrow$ Toda reduction is \lq incomplete'. The reason for this incompleteness being that the Gauss decomposition used to define the Toda fields from the WZNW field is valid locally but not globally over the WZNW group manifold, which implies that actually the reduced system is not just the Toda theory but has much richer structures. In this note we furnish a framework which allows us to study the reduced system globally, and thereby present some preliminary results on the global aspects. For simplicity, we analyze primarily 0 $+$ 1 dimensional toy models for $G = SL(n, {\bf R})$, but we also discuss the 1 $+$ 1 dimensional model for $G = SL(2, {\bf R})$ which corresponds to the WZNW $\rightarrow$ Liouville reduction. 
  Ultraviolet regime in quantum theory with horizons, contrary to ordinary theory, depends on the temperature of the system due to additional surface divergences in the effective action. We evaluate their general one-loop structure paying attention to effects of the curvature of the space-time near the horizon. In particular, apart from the area term, the entropy of a black hole is shown to acquire a topological correction in the form of the integral curvature of the horizon. To get the entropy, heat capacity and other thermodynamical quantities finite, such a kind of singularities should be removed by renormalization of a number of constants in a surface functional introduced in the effective action at arbitrary temperature. We also discuss a discrepancy in the different regularization techniques. 
  The algebraic renormalization of a recently proposed abelian axial gauge model with antisymmetric tensor matter fields is presented. 
  The quantum corrections to the entropy of charged black holes are calculated. The Reissner-Nordstrem and dilaton black holes are considered. The appearance of logarithmically divergent terms not proportional to the horizon area is demonstrated. It is shown that the complete entropy which is sum of classical Bekenstein-Hawking entropy and the quantum correction is proportional to the area of quantum-corrected horizon. 
  The quantum group structure of the Liouville theory is reviewd and shown to be an important tool for solving the theory. 
  We introduce some new classes of algebras and estabilish in these algebras Campbell--Hausdorff like formula. We describe the application of these constructions to the problem of the connectivity of the Feynman graphs corresponding to the Green functions in Quantum Fields Theory. 
  Models of relativistic particle with Lagrangian ${\cal L}(k_1)$, depending on the curvature of the worldline $k_1$, are considered. By making use of the Frenet basis, the equations of motion are reformulated in terms of the principal curvatures of the worldline. It is shown that for arbitrary Lagrangian function ${\cal L}(k_1)$ these equations are completely integrable, i.e., the principal curvatures are defined by integrals. The constants of integration are the particle mass and its spin. The developed method is applied to the study of a model of relativistic particle with maximal proper acceleration, whose Lagrangian is uniquely determined by a modified form of the invariant relativistic interval. This model gives us an example of a consistent relativistic dynamics obeying the principle of a superiorly limited value of the acceleration, advanced recently. 
  We investigate classical strings defined by the Nambu-Goto action with the boundary term added. We demonstrate that the latter term has a significant bearing on the string dynamics. It is confirmed that new action terms that depend on higher order derivatives of string coordinates cannot be considered as continuous perturbations from the starting string functional. In the case the boundary term reduces to the Gauss-Bonnet term, a stability analysis is performed on the rotating rigid string solution. We determined the most generic solution that the fluctuations grow to. Longitudinal string excitations are found. The Regge trajectories are nonlinear. 
  We analyse the high-energy behavior of tree-level graviton Compton amplitudes for particles of mass m and arbitrary spin, concentrating on a combination of forward amplitudes that will be unaffected by eventual cross- couplings to other, higher spins. We first show that for any spin larger than 2, tree-level unitarity is already violated at energies well below the Planck scale M, if m << M. We then restore unitarity to this amplitude up to M by adding non-minimal couplings that depend on the curvature and its derivatives, and modify the minimal description - including particle gravitational quadrupole moments - at scales O(1/m). 
  By studying the partition function of $N=4$ topologically twisted supersymmetric Yang-Mills on four-manifolds, we make an exact strong coupling test of the Montonen-Olive strong-weak duality conjecture. Unexpected and exciting links are found with two-dimensional rational conformal field theory. 
  We show that for a system containing a set of general second class constraints which are linear in the phase space variables, the Abelian conversion can be obtained in a closed form and that the first class constraints generate a generalized shift symmetry. We study in detail the example of a general first order Lagrangian and show how the shift symmetry noted in the context of BV quantization arises. 
  The strong coupling physics of two dimensional gravity at $C=7$, $13$, $19$ is deciphered, by building up on previous works along the same line (for a recent review, of the background material, see hep-th/9408069). It is shown that chirality becomes deconfined. The string suceptibility is derived, and found to be real contrary to the continuation of the KPZ formula. Topological Liouville string theories (without transverse degree of freedom) are explicitely solved. Altough they involve strongly coupled gravity, they share many features with the standard matrix models. 
  In the framework of the classical field theory a mapping between antisymmetric tensor matter fields and Weinberg's $2(2j+1)$ component "bispinor" fields is considered. It is shown that such a mapping exists and equations which describe the $j=1$ antisymmetric tensor field coincide with the Hammer-Tucker equations completely and with the Weinberg ones within a subsidiary condition, the Klein-Gordon equation. A new Lagrangian for the Weinberg theory is proposed. It is scalar, Hermitian and contains only the first-order time derivatives of the fields. The remarkable feature of this Lagrangian is the presence of dual field functions, considered as parts of a parity doublet. I study then origins of appearance of the dual solutions in the Weinberg equations on the basis of spinorial analysis and point out the topics which have to be explained in the framework of a secondary quantization scheme. 
  In this note we illustrate by a few examples the general principle: interesting algebras and representations defined over Z_+ come from category theory, and are best understood when their categorical origination has been discovered. We show that indecomposable Z_+-representations of the character ring of SU(2) satisfying certain conditions correspond to affine and infinite Dynkin diagrams with loops. We also show that irreducible Z_+-representations of the Verlinde algebra (the character ring of the quantum group SU(2)_q, where q is a root of unity), satisfying similar conditions correspond to usual (non-affine) Dynkin diagrams with loops. Conjecturedly, the last result is related to the ADE classification of conformal field theories with the chiral algebra \hat{sl(2)}. 
  We study one-loop corrections to the Chern-Simons coefficient $\kappa$ in abelian self-dual Chern-Simons Higgs systems and their $N=2$ and $N=3$ supersymmetric generalizations in both symmetric and asymmetric phases. One-loop corrections to the Chern-Simons coefficient of these systems turn out to be integer multiples of $1/4\pi$ in both phases. Especially in the maximally supersymmetric $N=3$ case, the correction in symmetric phase vanishes and that in asymmetric phase is $\kappa/(2\pi |\kappa|)$. Our results suggest that nonabelian self-dual systems might enjoy similar features. We also discuss various issues arising from our results. 
  We show that the dynamical symmetry of the hydrogen atom leads in a natural way to an infinite-dimensional algebra, which we identify as the positive subalgebras of twisted Kac-Moody algebras of $ so(4)$. We also generalize our results to the $N$-dimensional hydrogen atom. For odd $N$, we identify the dynamical algebra with the positive part of the twisted algebras $\hat {so}(N+1)^\tau$. However, for even $N$ this algebra corresponds to a parabolic subalgebra of the untwisted loop algebra $\hat{so}(N+1)$. 
  We perform a variational calculation in the SU(N) Yang Mills theory in 3+1 dimensions. Our trial variational states are explicitly gauge invariant, and reduce to simple Gaussian states in the zero coupling limit. Our main result is that the energy is minimized for the value of the variational parameter away form the perturbative value. The best variational state is therefore characterized by a dynamically generated mass scale $M$. This scale is related to the perturbative scale $\Lambda_{QCD}$ by the following relation: $\alpha_{QCD}(M)={\pi\over 4}{1\over N}$. Taking the one loop QCD $\beta$- function and $\Lambda_{QCD}=150 Mev$ we find (for N=3) the vacuum condensate ${\alpha\over \pi}<F^2>= 0.008 Gev^4$. 
  We explicitly construct the extension of the N=2 super Virasoro algebra by two super primary fields of dimension two and three with vanishing u(1)-charge. Using a super covariant formalism we obtain two different solutions both consistent for generic values of the central charge c. The first one can be identified with the super W_4-algebra - the symmetry algebra of the CP(3) Kazama-Suzuki model. With the help of unitarity arguments we predict the self-coupling constant of the field of dimension two for all super W_n-algebras. The second solution is special in the sense that it does not have a finite classical limit c->infinity and generic null fields appear. In the spirit of recent results in the N=0 case it can be understood as a unifying N=2 super W-algebra for all CP(n) coset models. It does not admit any unitary representation. 
  Following an old result of Marcus and Schwarz we argue that in the heterotic string theory compactified on a seven dimensional torus, the target space duality group O(7,23;Z) and the strong-weak coupling duality transformations combine into the group O(8,24;Z). We discuss symmetry of the combined spectrum of elementary particles and solitons in the theory, and also show that the existence of this symmetry predicts the number of harmonic forms on the moduli space of periodic arrays of BPS monopoles in (3+1) dimensions. Finally, we show that the O(8,24;Z) transformations relate the soliton solutions of Dabholkar et. al. representing the fundamental string to the soliton solutions of Greene et. al. representing `stringy cosmic strings'. 
  We consider a superradiance effect around rotating dilatonic black holes. We analyze two cases: one is an exact solution with the coupling constant $\alpha=\sqrt{3}$, which effective action is reduced from the 5-dimensional Kaluza-Klein theory, and the other is a slowly rotating dilatonic black holes with arbitrary coupling constant. We find that there exists a critical value ($\alpha \sim 1$), which is predicted from a superstring model, and the superradiant emission rate with coupling larger than the critical value becomes much higher than the Kerr-Newman case ($\alpha=0$) in the maximally charged limit. Consequently, 4-dimensional primordial black holes in higher dimensional unified theories are either rotating but almost neutral or charged but effectively non-rotating. 
  The anomalous magnetic moment of anyons is calculated to leading order in a 1/N expansion. It is shown that the gyromagnetic ratio g remains 2 to the leading order in 1/N. This result strongly supports that obtained in \cite{poly}, namely that g=2 is in fact exact. 
  We prove $q$-series identities between bosonic and fermionic representations of certain Virasoro characters.  These identities include some of the conjectures made by the Stony Brook group as special cases. Our method is a direct application of Andrews' extensions of Bailey's lemma to recently obtained polynomial identities. 
  Explicit expressions are presented for general branching functions for cosets of affine Lie algebras $\hat{g}$ with respect to subalgebras $\hat{g}^\prime$ for the cases where the corresponding finite dimensional algebras $g$ and $g^\prime$ are such that $g$ is simple and $g^\prime$ is either simple or sums of $u(1)$ terms. A special case of the latter yields the string functions. Our derivation is purely algebraical and has its origin in the results on the BRST cohomology presented by us earlier. We will here give an independent and simple proof of the validity our results. The method presented here generalizes in a straightforward way to more complicated $g$ and $g^\prime$ such as {\it e g } sums of simple and $u(1)$ terms. 
  In this paper we study the dynamics of a statistical ensemble of strings, building on a recently proposed gauge theory of the string geodesic field. We show that this stochastic approach is equivalent to the Carath\'eodory formulation of the Nambu-Goto action, supplemented by an averaging procedure over the family of classical string world-sheets which are solutions of the equation of motion. In this new framework, the string geodesic field is reinterpreted as the Gibbs current density associated with the string statistical ensemble. Next, we show that the classical field equations derived from the string gauge action, can be obtained as the semi-classical limit of the string functional wave equation. For closed strings, the wave equation itself is completely analogous to the Wheeler-DeWitt equation used in quantum cosmology. Thus, in the string case, the wave function has support on the space of all possible spatial loop configurations. Finally, we show that the string distribution induces a multi-phase, or {\it cellular} structure on the spacetime manifold characterized by domains with a purely Riemannian geometry separated by domain walls over which there exists a predominantly Weyl geometry. 
  We develop techniques for calculating the ground state wave functional and the geometric entropy for some simple field theories. Special attention is devoted to fermions, which present special technical difficulties in this regard. Explicit calculations are carried through for free mass bosons and fermions in two dimensions, using an adaptation of Unruh's technique to treat black hole radiance. 
  The split involution quantization scheme, proposed previously for pure second--class constraints only, is extended to cover the case of the presence of irreducible first--class constraints. The explicit Sp(2)--symmetry property of the formalism is retained to hold. The constraint algebra generating equations are formulated and the Unitarizing Hamiltonian is constructed. Physical operators and states are defined in the sense of the new equivalence criterion that is a natural counterpart to the Dirac's weak equality concept as applied to the first--class quantities. 
  The Coleman-Hill theorem prohibits the appearance of radiative corrections to the topological mass (more precisely, to the parity-odd part of the vacuum polarization tensor at zero momentum) in a wide class of abelian gauge theories in 2+1 dimensions. We re-express the theorem in terms of the effective action rather than in terms of the vacuum polarization tensor. The theorem so restated becomes somewhat stronger: a known exception to the theorem, spontaneously broken scalar Chern-Simons electrodynamics, obeys the new non-renormalization theorem. Whereas the vacuum polarization {\sl does} receive a one-loop, parity-odd correction, this does not translate to a radiative contribution to the Chern-Simons term in the effective action. We also point out a new situation, involving scalar fields and parity-odd couplings, which was overlooked in the original analysis, where the conditions of the theorem are satisfied and where the topological mass does, in fact, get a radiative correction. 
  We study the conserved charges of affine Toda field theories by making use of the conformally invariant extension of these theories. We compute the values of all charges for the single soliton solutions, and show that these are related to eigenvectors of the Cartan matrix of the finite-dimensional Lie algebra underlying the theory. 
  On the basis of the Andrews--Bailey construction, we derive fermionic sum representations of Virasoro characters of non unitary minimal models ${\cal M}(k,kp+p-1)$ and ${\cal M}(k,kp+1)$. These expressions include certain expressions conjectured by the Stony Brook group as special cases. 
  We show that it is possible to extend Moore's analysis of the classical scattering amplitudes of the bosonic string to those of the N=1 superstring. Using the bracket relations we are able to show that all possible amplitudes involving both bosonic and fermionic string states at arbitrary mass levels can be expressed in terms of amplitudes involving only massless states. A slight generalization of Moore's original definition of the bracket also allows us to determine the 4-point massless amplitudes themselves using only the bracket relations and the usual assumptions of analyticity. We suggest that this should be possible for the higher point massless amplitudes as well. 
  The one-loop effective action for 4-dimensional gauged supergravity with negative cosmological constant, is investigated in space-times with compact hyperbolic spatial section. The explicit expansion of the effective action as a power series of the curvature on hyperbolic background is derived, making use of heat-kernel and zeta-regularization techniques. The induced cosmological and Newton constants are computed. 
  It is known that there exist an infinite number of inequivalent quantizations on a topologically nontrivial manifold even if it is a finite-dimensional manifold. In this paper we consider the abelian sigma model in (1+1) dimensions to explore a system having infinite degrees of freedom. The model has a field variable $ \phi : S^1 \to S^1 $. An algebra of the quantum field is defined respecting the topological aspect of this model. A central extension of the algebra is also introduced. It is shown that there exist an infinite number of unitary inequivalent representations, which are characterized by a central extension and a continuous parameter $ \alpha $ $ ( 0 \le \alpha < 1 ) $. When the central extension exists, the winding operator and the zero-mode momentum obey a nontrivial commutator. 
  This paper further develops the combinatorial approach to quantization of the Hamiltonian Chern Simons theory advertised in \cite{AGS}. Using the theory of quantum Wilson lines, we show how the Verlinde algebra appears within the context of quantum group gauge theory. This allows to discuss flatness of quantum connections so that we can give a mathe- matically rigorous definition of the algebra of observables $\A_{CS}$ of the Chern Simons model. It is a *-algebra of ``functions on the quantum moduli space of flat connections'' and comes equipped with a positive functional $\omega$ (``integration''). We prove that this data does not depend on the particular choices which have been made in the construction. Following ideas of Fock and Rosly \cite{FoRo}, the algebra $\A_{CS}$ provides a deformation quantization of the algebra of functions on the moduli space along the natural Poisson bracket induced by the Chern Simons action. We evaluate a volume of the quantized moduli space and prove that it coincides with the Verlinde number. This answer is also interpreted as a partition partition function of the lattice Yang-Mills theory corresponding to a quantum gauge group. 
  String propagation on a cone with deficit angle $2\pi (1- \frac{1}{N} ) $ is described by constructing a non-compact orbifold of a plane by a $Z_{N}$ subgroup of rotations. It is modular invariant and has tachyons in the twisted sectors that are localized to the tip of the cone. A possible connection with the quantum corrections to the black hole entropy is outlined. The entropy computed by analytically continuing in N would receive contribution only from the twisted sectors and be naturally proportional to the area of the event horizon. Evidence is presented for a new duality for these orbifolds similar to the ${\scriptstyle R} \rightarrow {1\over R} $ duality. 
  We study four dimensional $N=2$ supersymmetric gauge theories with matter multiplets. For all such models for which the gauge group is $SU(2)$, we derive the exact metric on the moduli space of quantum vacua and the exact spectrum of the stable massive states. A number of new physical phenomena occur, such as chiral symmetry breaking that is driven by the condensation of magnetic monopoles that carry global quantum numbers. For those cases in which conformal invariance is broken only by mass terms, the formalism automatically gives results that are invariant under electric-magnetic duality. In one instance, this duality is mixed in an interesting way with $SO(8)$ triality. 
  We investigate the issue of coordinate redefinition invariance by carefully performing nonlinear transformations in the discretized quantum mechanical path integral. By resorting to hamiltonian path integral methods, we provide the first complete derivation of the extra terms (beyond the usual jacobian term) which arises in the action when a nonlinear transformation is made. We comment on possible connections with the renormalization group, by showing that these extra terms may emerge from a ``blocking" procedure. Finally, by performing field redefinitions before and after dimensional reduction of a two dimensional field theory, we derive an explicit form fo an extra term appearing in a field theory. 
  We suggest to consider conformal factor dynamics as applying to composite boundstates, in frames of the $1/N$ expansion. In this way, a new model of effective theory for quantum gravity is obtained. The renormalization group (RG) analysis of this model provides a framework to solve the cosmological constant problem, since the value of this constant turns out to be suppressed, as a result of the RG contributions. The effective potential for the conformal factor is found too. 
  Let $H$ be a Hopf algebra in braided category $\cal C$. Crossed modules over $H$ are objects with both module and comodule structures satisfying some comatibility condition. Category ${\cal C}^H_H$ of crossed modules is braided and is concrete realization of general categorical construction. For quantum braided group $(H,{\cal R})$ corresponding braided category ${\cal C}^{\cal R}_H$ of modules is identifyed with full subcategory in ${\cal C}_H^H$. Connection with crossproducts is discussed. Correct cross product in the class of quantum braided groups is built. Radford's--Majid's theorem gives equivalent condition for usual Hopf algebra to be crossproduct. Braided variant and analog of this theorem for quantum braided qroups are obtained. 
  We give a general expression for the static potential energy of the gravitational interaction of two massive particles, in terms of an invariant vacuum expectation value of the quantized gravitational field. This formula holds for functional integral formulations of euclidean quantum gravity, regularized to avoid conformal instability. It could be regarded as the analogue of the Wilson loop of gauge theories and allows in principle, through numerical lattice simulations or other approximation techniques, non perturbative evaluations of the potential or of the effective coupling constant. 
  We present an affine $sl (n+1)$ algebraic construction of the basic constrained KP hierarchy. This hierarchy is analyzed using two approaches, namely linear matrix eigenvalue problem on hermitian symmetric space and constrained KP Lax formulation and we show that these approaches are equivalent. The model is recognized to be the generalized non-linear Schr\"{o}dinger ($\GNLS$) hierarchy and it is used as a building block for a new class of constrained KP hierarchies. These constrained KP hierarchies are connected via similarity-B\"{a}cklund transformations and interpolate between $\GNLS$ and multi-boson KP-Toda hierarchies. Our construction uncovers origin of the Toda lattice structure behind the latter hierarchy. 
  We present a general method to obtain a closed, finite formula for the exponential map from the Lie algebra to the Lie group, for the defining representation of the orthogonal groups. Our method is based on the Hamilton-Cayley theorem and some special properties of the generators of the orthogonal group, and is also independent of the metric. We present an explicit formula for the exponential of generators of the $SO_+(p,q)$ groups, with $p+q = 6$, in particular we are dealing with the conformal group $SO_+(2,4)$, which is homomorphic to the $SU(2,2)$ group. This result is needed in the generalization of U(1) gauge transformations to spin gauge transformations, where the exponential plays an essential role. We also present some new expressions for the coefficients of the secular equation of a matrix. 
  This is the second part of the paper. Results of the first part about crossed modules are applied here to study of quantum groups in braided categories. Correct cross product in the class of quantum braided groups is built. Criterion when quantum braided group is cross product is otained. 
  A computer program has been developed which generates Feynman graphs automatically for scattering and decay processes in non-Abelian gauge theory of high-energy physics. A new acceleration method is presented for both generating and eliminating graphs. This method has been shown to work quite efficiently for any order of coupling constants in any kind of theoretical model. A utility program is also available for drawing generated graphs. These programs consist of the most basic parts of the GRACE system, which is now used to automatically calculate tree and one-loop processes. 
  In gauge theories, physical histories are represented by space-time connections modulo gauge transformations. The space of histories is thus intrinsically non-linear. The standard framework of constructive quantum field theory has to be extended to face these {\it kinematical} non-linearities squarely. We first present a pedagogical account of this problem and then suggest an avenue for its resolution. 
  Invited lecture at the International Congress of Mathematicians, Zuerich, August 3-11, 1994 (extended version), reviews free field realizations of affine Kac-Moody and W-algebras and their applications. 
  Greene, Morrison and Plesser \cite{GMP} have recently suggested a general method for constructing a mirror map between a $d$-dimensional Calabi-Yau hypersurface and its mirror partner for $d > 3$. We apply their method to smooth hypersurfaces in weighted projective spaces and compute the Chern numbers of holomorphic curves on these hypersurfaces. As anticipated, the results satisfy nontrivial integrality constraints. These examples differ from those studied previously in that standard methods of algebraic geometry which work in the ordinary projective space case for low degree curves are not generally applicable. In the limited special cases in which they do work we can get independent predictions, and we find agreement with our results. 
  Using a representation of the q-deformed Lorentz algebra as differential operators on quantum Minkowski space, we define an algebra of observables for a q-deformed relativistic quantum mechanics with spin zero. We construct a Hilbert space representation of this algebra in which the square of the mass $ p^2 $ is diagonal. 
  We consider two level $k$ WZNW models coupled to each other through a generalized Thirring-like current-current interaction. It is shown that in the large $k$ limit, this interacting system can be presented as a two-parameter perturbation around a nonunitary WZNW model. The perturbation operators are the sigma model kinetic terms with metric related to the Thirring coupling constants. The renormalizability of the perturbed model leads to an algebraic equation for couplings. This equation coincides with the master Virasoro equation. We find that the beta functions of the two-parameter perturbation have nontrivial zeros depending on the Thirring coupling constants. Thus we exhibit that solutions to the master equation provide nontrivial conformal points to the system of two interacting WZNW models. 
  We study the dilogarithm identities from algebraic, analytic, asymptotic, $K$-theoretic, combinatorial and representation-theoretic points of view. We prove that a lot of dilogarithm identities (hypothetically all !) can be obtained by using the five-term relation only. Among those the Coxeter, Lewin, Loxton and Browkin ones are contained. Accessibility of Lewin's one variable and Ray's multivariable (here for $n\le 2$ only) functional equations is given. For odd levels the $\hat{sl_2}$ case of Kuniba-Nakanishi's dilogarithm conjecture is proven and additional results about remainder term are obtained. The connections between dilogarithm identities and Rogers-Ramanujan-Andrews-Gordon type partition identities via their asymptotic behavior are discussed. Some new results about the string functions for level $k$ vacuum representation of the affine Lie algebra $\hat{sl_n}$ are obtained. Connection between dilogarithm identities and algebraic $K$-theory (torsion in $K_3({\bf R})$) is discussed. Relations between crystal basis, branching functions $b_{\lambda}^{k\Lambda_0}(q)$ and Kostka-Foulkes polynomials (Lusztig's $q$-analog of weight multiplicity) are considered. The Melzer and Milne conjectures are proven. In some special cases we are proving that the branching functions $b_{\lambda}^{k\Lambda_0}(q)$ are equal to an appropriate limit of Kostka polynomials (the so-called Thermodynamic Bethe Ansatz limit). Connection between "finite-dimensional part of crystal base" and Robinson-Schensted-Knuth correspondence is considered. 
  A `discrete differential manifold' we call a countable set together with an algebraic differential calculus on it. This structure has already been explored in previous work and provides us with a convenient framework for the formulation of dynamical models on networks and physical theories with discrete space and time. We present several examples and introduce a notion of differentiability of maps between discrete differential manifolds. Particular attention is given to differentiable curves in such spaces. Every discrete differentiable manifold carries a topology and we show that differentiability of a map implies continuity. 
  A matrix model is constructed which describes a chiral version of the large $N$ $U(N)$ gauge theory on a two-dimensional sphere of area $A$. This theory has three separate phases. The large area phase describes the associated chiral string theory. An exact expression for the free energy in the large area phase is used to derive a remarkably simple formula for the number of topologically inequivalent covering maps of a sphere with fixed branch points and degree $n$. 
  The B-twisted topological sigma model coupled to topological gravity is supposed to be described by an ordinary field theory: a type of holomorphic Chern-Simons theory for the open string, and the Kodaira-Spencer theory for the closed string. We show that the B model can be represented as a PARTICLE theory, obtained by reducing the sigma model to one dimension, and replacing the coupling to topological gravity by a coupling to a twisted one-dimensional supergravity. The particle can be defined on ANY Kahler manifold--it does not require the Calabi-Yau condition--so it may provide a more generalized setting for the B model than the topological sigma model. The one-loop partition function of the particle can be written in terms of the Ray-Singer torsion of the manifold, and agrees with that of the original B model. After showing how to deform the Kahler and complex structures in the particle, we prove the independence of this partition function on the Kahler structure, and investigate the origin of the holomorphic anomaly. To define other amplitudes, one needs to introduce interactions into the particle. The particle will then define a field theory, which may or may not be the Chern-Simons or Kodaira-Spencer theories. 
  It has been reported that, within the hard-thermal-loop resummation scheme, the production rate of soft real photons from a hot quark-gluon plasma exhibits unscreened mass singularities. We show that still higher-order resummations screen the mass singularities and obtain the finite soft-photon production rate to leading order at logarithmic accuracy ${\cal O} (\alpha \alpha_s \ln^2 \alpha_s)$. 
  We measure by Monte Carlo simulations $\g_{string}$ for a model of random surfaces embedded in three dimensional Euclidean space-time. The action of the string is the usual Polyakov action plus an extrinsic curvature term. The system undergoes a phase transition at a finite value $\l_c$ of the extrinsic curvature coupling and at the transition point the numerically measured value of $\g_{string}(\l_c) \approx 0.27\pm 0.06$. This is consistent with $\g_{string}(\l_c)=1/4$, i.e. equal to the first of the non-trivial values of $\g_{string}$ between 0 and $1/2$. 
  In this paper we find all permutations of the level k weights of the affine algebra A_r^{(1)} which commute with both its S and T modular matrices. We find that all of these are simple current automorphisms and their conjugations. Previously, the A_{r,k}^{(1)} automorphism invariants were known only for r=1,2 \forall k, and k=1 \forall r. This is a major step toward the full classification of all A_{r,k}^{(1)} modular invariants; the simplicity of this proof strongly suggests that the full classification should be accomplishable. In an appendix we collect some new results concerning the A_{r,k}^{(1)} fusion ring. 
  We investigate static space dependent $\sigx=\lag\bar\psi\psi\rag$ saddle point configurations in the two dimensional Gross-Neveu model in the large N limit. We solve the saddle point condition for $\sigx$ explicitly by employing supersymmetric quantum mechanics and using simple properties of the diagonal resolvent of one dimensional Schr\"odinger operators rather than inverse scattering techniques. The resulting solutions in the sector of unbroken supersymmetry are the Callan-Coleman-Gross-Zee kink configurations. We thus provide a direct and clean construction of these kinks. In the sector of broken supersymmetry we derive the DHN saddle point configurations. Our method of finding such non-trivial static configurations may be applied also in other two dimensional field theories. 
  We discuss duality and mirror symmetry phenomena of Landau-Ginzburg orbifolds considering their elliptic genera. Under the duality (or mirror) transform performed by orbifoldizing the Landau-Ginzburg model via some discrete group of the superpotential we observe that the roles of the untwisted and twisted sectors are exchanged. As explicit evidence detailed orbifold data are presented for $N=2$ minimal models, Arnold's exceptional singularities, $K3$ surfaces constructed from Arnold's singularities and Fermat hypersurfaces. (To appear in the proceedings of the workshop, ``Quantum Field Theory, Integrable Models and Beyond'', Yukawa Institute for Theoretical Physics, Kyoto University, 14-18 February 1994.) 
  The Kalb-Ramond action for an interacting string is generalized to the case of a high-dimensional object (p-brane). The interaction is found to be mediated by a gauge boson of a completely antisymmetric tensor of rank $p+1$. 
  We show how Wigner's little group approach to the representation theory of Poincar\'e group may be generalized to the case of $\kappa$-deformed   Poincar\'e group. We also derive the deformed Lorentz transformations of energy and momentum. We find that if the $\kappa$-deformed Poincar\'e group is adopted as the fundamental symmetry of nature, it results in deviations from predictions of the Poincar\'e symmetry at large energies, which may be experimentally observable. 
  We study numerically the fractal structure of the intrinsic geometry of random surfaces coupled to matter fields with $c=1$. Using baby universe surgery it was possible to simulate randomly triangulated surfaces made of 260.000 triangles. Our results are consistent with the theoretical prediction $d_H = 2+\sqrt{2}$ for the intrinsic Hausdorff dimension. 
  We reformulate the proof of the renormalization of a spontaneously broken gauge theory by multiplicatively renormalizing the vacuum expectation value of the Higgs field in the $SU(2)$~Higgs model. 
  I give an interpretation of the fundamental theorem of algebra based on supersymmetry and the Witten index. The argument gives a physical explanation of why a real polynomial of degree $n$ need not have $n$ real zeroes, while a complex polynomial of degree $n$ must have $n$ complex zeroes. This paper also addresses in a general and model-independent way the statistics of the perturbative ground states (the states which correspond to classical vacua) in supersymmetric theories with complex and with real superfields. 
  The anomalies which occur in chiral WW_{3} gravity are characterized by solving the BRS consistency condition. 
  We present a construction of fermionic operators in 3+1 dimensions in terms of bosonic fields in the framework of $QED_4$. The basic bosonic variables are the electric fields $E_i$ and their conjugate momenta $A_i$. Our construction generalizes the analogous constuction of fermionic operators in 2+1 dimensions. Loosely speaking, a fermionic operator is represented as a product of an operator that creates a pointlike charge and an operator that creates an infinitesimal t'Hooft loop of half integer strength. We also show how the axial $U(1)$ transformations are realized in this construction. 
  I discuss recent progress in our understanding of two barriers in quantum gravity: $c > 1$ in the case of 2d quantum gravity and $D > 2$ in the case of Euclidean Einstein-Hilbert gravity formulated in space-time dimensions $D >2$. 
  We study the generalization of second Gelfand-Dickey bracket to the superdifferential operators with matrix-valued coefficients. The associated Miura transformation is derived. Using this bracket we work out a nonlocal and nonlinear N=2 superalgebra which contains the N=2 super Virasoro algebra as a subalgebra. The bosonic limit of this algebra is considered. We show that when the spin-1 fields in this bosonic algebra are set to zero the resulting Dirac bracket gives precisely the recently derived $V_{2,2}$ algebra. 
  The Kyoto group (Jimbo, Miwa, Nakayashikiet al.) showed that the partition function and correlation funtions of the eight-vertex model in antiferromagnetic phases can be calculated using simple analytical properties of the $R$-matrix. We extend these methods to ferromagnetic and disordered phases. We use Baxter's symmetries to obtain appropriate parametrizations of the $R$-matrix and to substantiate the validity of the analytical approach for these phases. These symmetries allow one to relate correlation functions in different phases. 
  We explore the Coleman-Weinberg phase transition in regions outside the validity of perturbation theory. For this purpose we study a Euclidean field theory with two scalars and discrete symmetry in four dimensions. The phase diagram is established by a numerical solution of a suitable truncation of exact non-perturbative flow equations. We find regions in parameter space where the phase transition (in dependence on the mass term) is of the second or the first order, separated by a triple point. Our quantitative results for the first order phase transition compare well to the standard perturbative Coleman-Weinberg calculation of the effective potential. 
  We present evidence for an undiscovered link between N=2 supersymmetric quantum field theories and the mathematical theory of helices of coherent sheaves. We give a thorough review for nonspecialists of both the mathematics and physics involved, and invite the reader to take up the search for this elusive connection. 
  Above the Hagedorn energy density closed fundamental strings form a long string phase. The dynamics of weakly interacting long strings is described by a simple Boltzmann equation which can be solved explicitly for equilibrium distributions. The average total number of long strings grows logarithmically with total energy in the microcanonical ensemble. This is consistent with calculations of the free single string density of states provided the thermodynamic limit is carefully defined. If the theory contains open strings the long string phase is suppressed. 
  Functional relations are proposed for transfer matrices of solvable vertex models associated with the twisted quantum affine algebras $U_q(X^{(\kappa)}_n)$ where $X^{(\kappa)}_n = A^{(2)}_n, D^{(2)}_n, E^{(2)}_6$ and $D^{(3)}_4$. Their solutions are obtained for $A^{(2)}_n$ and conjectured for $D^{(3)}_4$ in the dressed vacuum form in the analytic Bethe ansatz. 
  The dilute A$_3$ lattice model in regime 2 is in the universality class of the Ising model in a magnetic field. Here we establish directly the existence of an E$_8$ structure in the dilute A$_3$ model in this regime by expressing the 1-dimensional configuration sums in terms of fermionic sums which explicitly involve the E$_8$ root system.  In the thermodynamic limit, these polynomial identities yield a proof of the E$_8$ Rogers--Ramanujan identity recently conjectured by Kedem {\em et al}.   The polynomial identities also apply to regime 3, which is obtained by transforming the modular parameter by $q\to 1/q$. In this case we find an A$_1\times\mbox{E}_7$ structure and prove a Rogers--Ramanujan identity of A$_1\times\mbox{E}_7$ type. Finally, in the critical $q\to 1$ limit, we give some intriguing expressions for the number of $L$-step paths on the A$_3$ Dynkin diagram with tadpoles in terms of the E$_8$ Cartan matrix. All our findings confirm the E$_8$ and E$_7$ structure of the dilute A$_3$ model found recently by means of the thermodynamic Bethe Ansatz. 
  We investigate quantum corrections for a cosmological solution of the string effective action. Starting point is a classical solution containing an antisymmetric tensor field, a dilaton and a modulus field which has singularities in the scalar fields. As a first step we quantize the scalar fields near the singularity with the result that the singularities disappear and that in general non-perturbative quantum corrections form a potential in the scalar fields. 
  In this paper the three-dimensional vertex model is given, which is the duality of the three-dimensional Baxter-Bazhanov (BB) model. The braid group corresponding to Frenkel-Moore equation is constructed and the transformations $R, I$ are found. These maps act on the group and denote the rotations of the braids through the angles $\pi$ about some special axes. The weight function of another three-dimensional vertex model related the 3D lattice integrable model proposed by Boos, Mangazeev, Sergeev and Stroganov is presented also, which can be interpreted as the deformation of the vertex model corresponding to the BB model. 
  In this letter we show that the restricted star-triangle relation introduced by Bazhanov and Baxter can be obtained either from the star-triangle relation of chiral Potts model or from the star-square relation which is proposed by Kashaev $et ~al$ and give a response of the guess which is suggested by Bazhanov and Baxter in Ref. \cite{b2}. 
  This text, written as dissertation within the M.Sc. course in particle theory at the Centre for Particle Theory, University of Durham, during the academic year 1993/94, reviews two articles by D.Gross and by D.Gross and W.Taylor which interpret the 1/N- expansion of the partition function of $QCD_2$ as string perturbation series. For this required mathematical and physical background is presented. 
  Anomalies can be viewed as arising from the cohomology of the Lie algebra of the group of gauge transformations and also from the topological cohomology of the group of connections modulo gauge transformations. We show how these two approaches are unified by the transgression map. We discuss the geometry behind the current commutator anomaly and the Faddeev- Mickelsson anomaly using the recent notion of a gerbe. Some anomalies (notably 3-cocycles) do not have such a geometric origin. We discuss one example and a conjecture on how these may be related to geometric anomalies. 
  We present an explicit solution of superstring effective equations, represented by gravitational waves and dilaton backgrounds. Particular solutions will be examined in a forthcoming note. 
  A $q$-analogue of the Hurwitz zeta-function is introduced through considerations on the spectral zeta-function of quantum group $SU_{q}(2)$, and its analytic aspects are studied via the Euler-MacLaurin summation formula. Asymptotic formulas of some relevant $q$-functions are discussed. 
  We have examined effective theory induced by gauged WZW models, in which the tachyon field is added as a marginal operator. Due to this operator added, we must further add the higher order corrections, which modifies the original configuration, to make the theory full-conformally invariant. It has been found that 2d is a critical dimension in the sense that the metric obtained from gauged WZW is modified by the tachyon condensation for $d>2$, but not for $d\le 2$. 
  In this article we extend our previous results for the orthogonal group, $SO(2,4)$, to its homomorphic group $SU(2,2)$. Here we present a closed, finite formula for the exponential of a $4\times 4$ traceless matrix, which can be viewed as the generator (Lie algebra elements) of the $SL(4,C)$ group. We apply this result to the $SU(2,2)$ group, which Lie algebra can be represented by the Dirac matrices, and discuss how the exponential map for $SU(2,2)$ can be written by means of the Dirac matrices. 
  The main goal of the paper is to study the origins of a contradiction between the Weinberg theorem B-A=\lambda and the `longitudinality' of an antisymmetric tensor field (and of a Weinberg field which is equivalent to it), transformed on the (1,0)\oplus (0,1) Lorentz group representation. On the basis of analysis of dynamical invariants in the Fock space situation has been partly clarified. 
  The physical variables for pure Yang - Mills theory in four - dimensional Minkowskian space time are constructed without using a gauge fixing condition} by the explicit resolving of the non - Abelian Gauss constraint and by the Bogoliubov transformation that diagonalizes the kinetic term in reduced action (action on constraint shell). As a result, the reduced action is expressed in terms of gauge invariant field variables including an additional global (only time - dependent) one, describing zero mode dynamics of the Gauss constraint. This additional variable reflects the symmetry group of topologically nontrivial transformations remaining after the reduction. ( It gives also the characteristic of the Gribov ambiguity from the point of view of the gauge fixing method.) The perturbation theory in terms of quasiparticles with the new stable vacuum, which is defined through the zero mode configuration, is proposed. It is shown, that the averaging of Green's functions for quasiparticles over the global variable leads to the mechanism of color confinement. 
  In a class of renormalizable three-dimensional abelian gauge theory the Lorentz invariance is spontaneously broken by dynamical generation of a magnetic field $B$. The true ground state resembles that of the quantum Hall effect. An originally topologically massive photon becomes gapless, fulfilling the role of the Nambu-Goldstone boson associated with the spontaneous breaking of the Lorentz invariance. We give a simple explanation and a sufficient condition for the spontaneous breaking of the Lorentz invariance with the aid of the Nambu-Goldstone theorem. The decrease of the energy density by $B \not= 0$ is understood mostly due to the shift in zero-point energy of photons. For PASCOS'94. 
  It is shown that the one-loop quadratic divergences of standard supergravity can be regulated by the introduction of heavy Pauli-Villars fields belonging to chiral and abelian gauge multiplets. The resulting one-loop correction can be interpreted as a renormalization of the K\"ahler potential. Regularization of the dilaton couplings to the Yang-Mills sector requires special care, and may shed some light on chiral/linear multiplet duality of the dilaton supermultiplet. 
  In terms of non-commutative geometry, we show that the $\sigma$--model can be built up by the gauge theory on discrete group $Z_2$. We introduce a constraint in the gauge theory, which lead to the constraint imposed on linear $\sigma$ model to get nonlinear $\sigma$ model . 
  By means of the non-commutative differential geometry, we construct an $SU(2)$ generalized gauge field model. It is of $SU(2) \times \pi_4(SU(2))$ gauge invariance. We show that this model not only includes the Higgs field automatically on the equal footing with ordinary Yang-Mills gauge potentials but also is stable against quantum correlation. 
  Based upon a first principle, the generalized gauge principle, we construct a general model with $G_L\times G'_R \times Z_2$ gauge symmetry, where $Z_2=\pi_4(G_L)$ is the fourth homotopy group of the gauge group $G_L$, by means of the non-commutative differential geometry and reformulate the Weinberg-Salam model and the standard model with the Higgs field being a gauge field on the fourth homotopy group of their gauge groups. We show that in this approach not only the Higgs field is automatically introduced on the equal footing with ordinary Yang-Mills gauge potentials and there are no extra constraints among the parameters at the tree level but also it most importantly is stable against quantum correlation. 
  Protective measurements, which we have introduced recently, allow to measure properties of the state of a single quantum system and even the Schr\"odinger wave itself. These measurements require a protection, sometimes due to an additional procedure and sometimes due to the potential of the system itself. The analysis of the protective measurements is presented and it is argued, contrary to recent claims, that they measure the quantum state and not the protective potential. Some other misunderstandings concerning our proposal are clarified. 
  Weak measurement is a standard measuring procedure with two changes: it is performed on pre- and post-selected quantum systems and the coupling to the measuring device is weakened. The outcomes of weak measurements, ``weak values'' are very different for the eigenvalues of the measured operators. The weak values yield novel rich structure of the quantum world. Weak values help explaining peculiar quantum phenomena and finding new effects which might have practical applications. 
  We exhibit $N=1$ supersymmetric field theories in confining, Coulomb and Higgs phases. The superpotential and the gauge kinetic terms are holomorphic and can be determined exactly in the various phases. The Coulomb phase generically has points with massless monopoles. When they condense, the theory undergoes a phase transition to a confining phase. When there are points in the Coulomb phase with massless electric charges, their condensation leads to a transition to a Higgs phase. When the Higgs and confinement phases are distinct, we expect to find massless interacting gluons at the transition point between them. 
  The valley structure associated with quantum meta-stability is examined. It is defined by the new valley equation, which enables consistent evaluation of the imaginary-time path-integral. We study the structure of this new valley equation and solve these equations numerically. The valley is shown to contain the bounce solution, as well as other bubble structures. We find that even when the bubble solution has thick wall, the outer region of the valley is made of large-radius, thin-wall bubble, which interior is occupied by the true-vacuum. Smaller size bubbles, which contribute to decay at higher energies, are also identified. 
  We construct a $2\times 2$ matrix algebra as representation of functions on discrete group $Z_2$ and develop the gauge theory on discrete group proposed by Starz in the matrix algebra. Accordingly, we show that the non-commutative geometry model built by R.Conquereax, G.Esposito-Farese and G.Vaillant results from this approach directly.   For the purpose of Physical model building, we introduce a free fermion   Lagrangian on $M_4\times Z_2$ and study Yang-Mills like gauge theory. 
  We review the recent development in the representation theory of the $W_{1+\infty}$ algebra. The topics that we concern are, Quasifinite representation, Free field realizations, (Super) Matrix Generalization, Structure of subalgebras such as $W_\infty$ algebra, Determinant formula, Character formula. (Invited talk at ``Quantum Field Theory, Integrable Models and Beyond", YITP, 14-17 February 1994. To appear in Progress of Theoretical Physics Proceedings Supplement.) 
  In the spirit of Non-commutative differential calculus on discrete group, we construct a toy model of spontaneous $CP$ violation (SCPV). Our model is different from the well-known Weinberg-Branco model although it involves three Higgs doublets and preserve neutral flavor current conservation (NFC) after using the $Z_2 \ti Z_2 \ti Z_2$ discrete symmetry and imposing some constraints on Yukawa couplings. 
  We introduce the notion of (nondegenerate) strong-modular fusion algebras. Here strongly-modular means that the fusion algebra is induced via Verlinde's formula by a representation of the modular group SL(2,Z) whose kernel contains a congruence subgroup. Furthermore, nondegenerate means that the conformal dimensions of possibly underlying rational conformal field theories do not differ by integers. Our main result is the classification of all strongly-modular fusion algebras of dimension two, three and four and the classification of all nondegenerate strongly-modular fusion algebras of dimension less than 24. We use the classification of the irreducible representations of the finite groups SL(2,Z_{p^l}) where p is a prime and l a positive integer. Finally, we give polynomial realizations and fusion graphs for all simple nondegenerate strongly-modular fusion algebras of dimension less than 24. 
  Small momentum expansion of the "sunset" diagram with three different masses is obtained. Coefficients at powers of $p^2$ are evaluated explicitly in terms of dilogarithms and elementary functions. Also some power expansions of "sunset" diagram in terms of different sets of variables are given. 
  We dicuss functorial consequences of way the eta invariant of Dirac operators behaves under gluing and change of boundary conditions. 
  Certain nontopological magnetic monopoles, recently found by Lee and Weinberg, are reinterpreted as topological solitons of a non-Abelian gauged Higgs model. Our study makes the nature of the Lee-Weinberg monopoles more transparent, especially with regard to their singularity structure. 
  It is possible to define new, gauge invariant variables in the Hilbert space of Yang-Mills theories which manifestly implement Gauss' law on physical states. These variables have furthermore a geometrical meaning, and allow one to uncover further constraints physical states must satisfy. For gauge group $SU(2)$, the underlying geometry is Riemannian and based on the group $GL(3)$. The formalism allows also for the inclusion of static color sources and the extension to gauge groups $SU(N>2)$, both of which are discussed here. 
  A real representation theory of real Clifford algebra has been studied in further detail, especially in connection with Fierz identities. As its application, we have constructed real octonion algebras as well as related octonionic triple system in terms of 8-component spinors associated with the Clifford algebras $C(0,7)$ and $C(4,3)$. 
  A peculiar representation of the Lorentz group is suggested as a starting point for a consistent approach to relativistic quantum theory. 
  We introduce a generalization of $A_{r}$-type Toda theory based on a non-abelian group G, which we call the $(A_{r},G)$-Toda theory, and its affine extensions in terms of gauged Wess-Zumino-Witten actions with deformation terms. In particular, the affine $(A_{1},SU(2))$-Toda theory describes the integrable deformation of the minimal conformal theory for the critical Ising model by the operator $\Phi_{(2,1)}$. We derive infinite conserved charges and soliton solutions from the Lax pair of the affine $(A_{1}, SU(2))$-Toda theory. Another type of integrable deformation which accounts for the $\Phi_{(3,1)}$-deformation of the minimal model is also found in the gauged Wess-Zumino-Witten context and its infinite conserved charges are given. 
  It has been proposed by Atiyah and Manton that the dynamics of Skyrmions may be approximated by motion on a finite dimensional manifold obtained from the moduli space of SU(2) Yang-Mills instantons. Motivated by this work we describe how similar results exist for other soliton and instanton systems. We describe in detail two examples for the approximation of the infinite dimensional dynamics of sine-Gordon solitons by finite dimensional dynamics on a manifold obtained from instanton moduli. In the first example we use the moduli space of CP1 instantons and in the second example we use the moduli space of SU(2) Yang-Mills instantons. The metric and potential functions on these manifolds are constructed and the resulting dynamics is compared with the explicit exact soliton solutions of the sine-Gordon theory. 
  Some years ago Dray and 't Hooft found the necessary and sufficient conditions to introduce a gravitational shock wave in a particular class of vacuum solutions to Einstein's equations. We extend this work to cover cases where non-vanishing matter fields and cosmological constant are present. The sources of gravitational waves are massless particles moving along a null surface such as a horizon in the case of black holes. After we discuss the general case we give many explicit examples. Among them are the $d$-dimensional charged black hole (that includes the 4-dimensional Reissner-Nordstr\"om and the $d$-dimensional Schwarzschild solution as subcases), the 4-dimensional De-Sitter and Anti-De-Sitter spaces (and the Schwarzschild-De-Sitter black hole), the 3-dimensional Anti-De-Sitter black hole, as well as backgrounds with a covariantly constant null Killing vector. We also address the analogous problem for string inspired gravitational solutions and give a few examples. 
  A rigid framework for the Cartan calculus of Lie derivatives, inner derivations, functions, and forms is proposed. The construction employs a semi-direct product of two graded Hopf algebras, the respective super-extensions of the deformed universal enveloping algebra and the algebra of functions on a quantum group. Relations in the Cartan calculus follow as consistency conditions. The approach is not a priori based on the Leibniz rule for the exterior derivative and might hence also be of interest in the recent work on its deformations. The Cartan identity for the Lie derivatives is proven. (This article is based on a lecture given at the Enrico Fermi Summer School on Quantum Groups, Varenna, June 1994) 
  Duality Symmetry is studied for heterotic string orbifold compactifications in the presence of a general background which in addition to the metric and antisymmetric tensor contains both discrete and continuous Wilson lines background. 
  The interference pattern of coherent electrons is effected by coupling to the quantized electromagnetic field. The amplitudes of the interference maxima are changed by a factor which depends upon a double line integral of the photon two-point function around the closed path of the electrons. The interference pattern is sensitive to shifts in the vacuum fluctuations in regions from which the electrons are excluded. Thus this effect combines aspects of both the Casimir and the Aharonov-Bohm effects. The coupling to the quantized electromagnetic field tends to decrease the amplitude of the interference oscillations, and hence is a form of decoherence. The contributions due to photon emission and to vacuum fluctuations may be separately identified. It is to be expected that photon emission leads to decoherence, as it can reveal which path an electron takes. It is less obvious that vacuum fluctuations also can cause decoherence. What is directly observable is a shift in the fluctuations due, for example, to the presence of a conducting plate. In the case of electrons moving parallel to conducting boundaries, the dominant decohering influence is that of the vacuum fluctuations. The shift in the interference amplitudes can be of the order of a few percent, so experimental verification of this effect may be possible. The possibility of using this effect to probe the interior of matter, e.g., to determine the electrical conductivity of a rod by means of electrons encircling it is discussed. (Presented at the Conference on Fundamental Problems in Quantum Theory, University of Maryland, Baltimore County, June 18-22, 1994.) 
  A unified and systematic scheme for constraction of differential opreator realization of any irreducible representation of $sl(n)$ is developed. The $q$-analogue of this unified scheme is used to constract $q$-difference operator realization of any irreducible representation of $U_q(sl(n))$. Explicit results for $U_q(sl(2))$, $U_q(sl(3))$ and $U_q(sl(n))$ are given. 
  The recent solution of the Mandelstam constraints for SU(2) is reviewed. This enables the subspace of physical configurations of an SU(2) pure gauge theory on the lattice (introduced solely to regulate the number of fields) with 3N physical degrees of freedom to be fully described in terms of 3N gauge-invariant continuous loop variables and N-1 gauge-invariant discrete +/-1 variables. The conceptual simplicity of the solution and the essential role of the discrete variables are emphasized. (Talk presented at QCD '94, Montpellier, France, 7-13 July 1994.) 
  The basic ideas and the important role of gauge principles in modern elementary particle physics are outlined. There are three theoretically consistent gauge principles in quantum field theory: the spin-1 gauge principle of electromagnetism and the standard model, the spin-2 gauge principle of general relativity, and the spin-3/2 gauge principle of supergravity. (Dirac Prize lecture, November 1993. Two figures are unavailable, but should not be essential.) 
  An analog of the $j=1/2$ Feynman-Dyson propagator is presented in the framework of the $j=1$ Weinberg's theory. The basis for this construction is the concept of the Weinberg field as a system of four field functions differing by parity and by dual transformations. 
  The Reshetikhin - Turaeve approach to topological invariants of three - manifolds is generalized to quantum supergroups. A general method for constructing three - manifold invariants is developed, which requires only the study of the eigenvalues of certain central elements of the quantum supergroup in irreducible representations. To illustrate how the method works, $U_q(gl(2|1))$ at odd roots of unity is studied in detail, and the corresponding topological invariants are obtained. 
  A topological version of four-dimensional (Euclidean) Einstein gravity which we propose regards anti-self-dual 2-forms and an anti-self-dual part of the frame connections as fundamental fields. The theory describes the moduli spaces of conformally self-dual Einstein manifolds for the non-zero cosmological constant case and Einstein-Kahlerian manifold with the vanishing real first Chern class for the zero cosmological constant. In the non-zero cosmological constant case, we evaluate the index of the elliptic complex associated with the moduli space and calculate the partition function. We also clarify the moduli space and its dimension for the zero cosmological constant case which are related to the Plebansky's heavenly equations. 
  A brief review of the construction and classifiaction of the bicovariant differential calculi on quantum groups is given. 
  In the superspace $z^M = (x^\mu,\theta_R,\theta_L)$ the global symmetries for $d$ = 10 superparticle model with kinetic terms both for Bose and Fermi variables are shown to form a superalgebra, which includes the Poincar\'e superalgebra as a subalgebra. The subalgebra is realized in the space of variables of the theory by a nonstandard way. The local version of this model with off-shell closed Lagrangian algebra of gauge symmetries and off-shell global supersymmetry is presented. It is shown that the resulting model is dynamically equivalent to the Siegel superparticle. 
  We present a method for extracting effective Lagrangians from QCD. The resulting effective Lagrangians are based on exact rewrites of cut-off QCD in terms of these new collective field degrees of freedom. These cut-off Lagrangians are thus ``effective'' in the sense that they explicitly contain some of the physical long-distance degrees of freedom from the outset. As an example we discuss the introduction of a new collective field carrying the quantum numbers of the $\eta'$-meson. (Contribution presented by R. Sollacher at the workshop ``QCD'94'', Montpellier, France, July 7-13, 1994. To appear in those proceedings.) 
  We study asymptotically non-free gauge theories and search for renormalization group invariant   (i.e. technically natural) relations among the couplings which lead to successful gauge-Yukawa unification. To be definite, we consider a supersymmetric model based on $SU(4)\times SU(2)_{R}\times SU(2)_{L}$. It is found that among the couplings of the model, which can be expressed in this way by a single one in the lowest order approximation, are the tree gauge couplings and the Yukawa coupling of the third generation. The corrections to the lowest order results are computed, and we find that the predictions on the low energy parameters resulting from those relations are in agreement with the measurements at LEP and Tevatron for a certain range of supersymmetry breaking scale. 
  The running coupling constants are introduced in Quantum Mechanics and their evolution is described by the help of the renormalization group equation. The harmonic oscillator and the propagation on curved spaces are presented as examples. The hamiltonian and the lagrangian scaling relations are obtained. These evolution equations are used to construct low energy effective models. 
  We investigate the quantum nucleation of pairs of charged circular cosmic strings in de Sitter space. By including self-gravity we obtain the classical potential energy barrier and compute the quantum mechanical tunneling probability in the semiclassical approximation. We also discuss the classical evolution of charged circular strings after their nucleation. 
  We study the various quantum aspects of the $N=2$ supersymmetric Maxwell Chern-Simons vortex systems. The fermion zero modes around the vortices will give rise the degenerate states of vortices. We analyze the angular momentum of these zero modes and apply the result to get the supermultiplet structures of the vortex. The leading quantum correction to the mass of the vortex coming from the mode fluctuations is also calculated using various methods depending on the value of the coefficient of the Chern-Simons term $\kappa$ to be zero, infinite and finite, separately. The mass correction is shown to vanish for all cases. Fermion numbers of vortices are also discussed. 
  The quantum BRST charges for the Bershadsky-Knizhnik orthogonal quasi-superconformal algebras are constructed. These two-dimensional superalgebras have the $N$-extended non-linearly realised supersymmetry and the $SO(N)$ internal symmetry. The BRST charge nilpotency conditions are shown to have a unique solution at $N>2$, namely, $N=4$ and $k=-2$, where $k$ is central extension parameter of the Ka\v{c}-Moody subalgebra. We argue about the existence of a new string theory with the non-linearly realised $N=4$ world-sheet supersymmetry and negative `critical dimension'. 
  \hspace{.2in}We consider the Darboux type transformations for the spectral problems of supersymmetric KdV systems. The supersymmetric analogies of Darboux and Darboux-Levi transformations are established for the spectral problems of Manin-Radul-Mathieu sKdV and Manin-Radul sKdV. Several B\"acklund transformations are derived for the MRM sKdV and MR sKdV systems. 
  Large-$N$ renormalization group equations for one- and two-matrix models are derived. The exact renormalization group equation involving infinitely many induced interactions can be rewritten in a form that has a finite number of coupling constants by taking account of reparametrization identities. Despite the nonlinearity of the equation, the location of fixed points and the scaling exponents can be extracted from the equation. They agree with the spectrum of relevant operators in the exact solution. A linearized $\beta$-function approximates well the global phase structure which includes several nontrivial fixed points. The global renormalization group flow suggests a kind of $c$-theorem in two-dimensional quantum gravity. 
  We consider a theory of a scalar one-component field $\phi$ coupled to a scalar $N$-component field $\chi$. Using large $N$ techiques we calculate the effective potential in the leading order in $1/N$. We show that this is equivalent to a resummation of an infinite subclass of graphs in perturbation theory, which involve fluctuations of the $\chi$ field only. We study the temperature dependence of the expectation value of the $\phi$ field and the resulting first and second order phase transitions. 
  A class of two-dimensional globally scale-invariant, but not conformally invariant, theories is obtained. These systems are identified in the process of discussing global and local scaling properties of models related by duality transformations, based on non-semisimple isometry groups. The construction of the dual partner of a given model is followed through; non-local as well as local versions of the former are discussed. 
  The concept of a quantum algebra is made easy through the investigation of the prototype algebras $u_{qp}(2)$, $su_q(2)$ and $u_{qp}(1,1)$. The latter quantum algebras are introduced as deformations of the corresponding Lie algebras~; this is achieved in a simple way by means of $qp$-bosons. The Hopf algebraic structure of $u_{qp}(2)$ is also discussed. The basic ingredients for the representation theory of $u_{qp}(2)$ are given. Finally, in connection with the quantum algebra $u_{qp}(2)$, we discuss the $qp$-analogues of the harmonic oscillator and of the (spherical and hyperbolical) angular momenta. 
  The stability of the magnetically charged Reissner-Nordstrom black hole solution is investigated in the context of a theory with massive charged vector mesons. By exploiting the spherical symmetry of the problem, the linear perturbations about the Reissner-Nordstrom solution can be decomposed into modes of definite angular momentum $J$. For each value of $J$, unstable modes appear if the horizon radius is less than a critical value that depends on the vector meson gyromagnetic ratio $g$ and the monopole magnetic charge $q/e$. It is shown that such a critical radius exists (except in the anomalous case $q={1\over 2}$ with $0 \le g \le 2$), provided only that the vector meson mass is not too close to the Planck mass. The value of the critical radius is determined numerically for a number of values of $J$. The instabilities found here imply the existence of stable solutions with nonzero vector fields (``hair'') outside the horizon; unless $q=1$ and $g>0$, these will not be spherically symmetric. 
  Following the construction of the $\kappa$-Minkowski space from the bicrossproduct structure of the $\kappa$-Poincare group, we investigate possible differential calculi on this noncommutative space. We discuss then the action of the Lorentz quantum algebra and prove that there are no 4D bicovariant differential calculi, which are Lorentz covariant. We show, however, that there exist a five-dimensional differential calculus, which satisfies both requirements. We study also a toy example of 2D $\kappa$-Minkowski space and and we briefly discuss the main properties of its differential calculi. 
  We derive an explicit formula for the evaluation of the classical closed string action for any off-shell string field, and for the calculation of arbitrary off-shell amplitudes. The formulae require a parametrization, in terms of some moduli space coordinates, of the family of local coordinates needed to insert the off-shell states on Riemann surfaces. We discuss in detail the evaluation of the tachyon potential as a power series in the tachyon field. The expansion coefficients in this series are shown to be geometrical invariants of Strebel quadratic differentials whose variational properties imply that closed string polyhedra, among all possible choices of string vertices, yield a tachyon potential which is as small as possible order by order in the string coupling constant. Our discussion emphasizes the geometrical meaning of off-shell amplitudes. 
  We derive sufficient conditions under which the ``second'' Hamiltonian structure of a class of generalized KdV-hierarchies defines one of the classical $\cal W$-algebras obtained through Drinfel'd-Sokolov Hamiltonian reduction. These integrable hierarchies are associated to the Heisenberg subalgebras of an untwisted affine Kac-Moody algebra. When the principal Heisenberg subalgebra is chosen, the well known connection between the Hamiltonian structure of the generalized Drinfel'd-Sokolov hierarchies - the Gel'fand-Dickey algebras - and the $\cal W$-algebras associated to the Casimir invariants of a Lie algebra is recovered. After carefully discussing the relations between the embeddings of $A_1=sl(2,{\Bbb C})$ into a simple Lie algebra $g$ and the elements of the Heisenberg subalgebras of $g^{(1)}$, we identify the class of $\cal W$-algebras that can be defined in this way. For $A_n$, this class only includes those associated to the embeddings labelled by partitions of the form $n+1= k(m) + q(1)$ and $n+1= k(m+1) + k(m) + q(1)$. 
  The quantum BRST charge for the most general, two-dimensional, non-linear, $N=4$ quasi-superconformal algebra $\hat{D}(1,2;\a)$, whose linearisation is the so-called `large' $N=4$ superconformal algebra, is constructed. The $\hat{D}(1,2;\a)$ algebra has $\Hat{su(2)}_{k^+}\oplus \Hat{su(2)}_{k^-}\oplus\Hat{u(1)}$ Ka\v{c}-Moody component, and $\a=k^-/k^+$. As a pre-requisite to our construction, we check the $\hat{D}(1,2;\a)$ Jacobi identities and construct a classical BRST charge. Then, we analyse the quantum BRST charge nilpotency conditions and find the only solution, $k^+=k^-=-2$. The $\hat{D}(1,2;1)$ algebra is actually isomorphic to the $SO(4)$-based Bershadsky-Knizhnik non-linear quasi-superconformal algebra. We argue about the existence of a new string theory with (i) the non-linearly realised $N=4$ world-sheet supersymmetry, (ii) non-unitary matter in a $\hat{D}(1,2;\a)$ representation of $k=-2$ and $c=-6$, and (iii) negative `critical dimension'. 
  A complete geometric unification of gravity and electromagnetism is proposed by considering two aspects of torsion: its relation to spin established in Einstein--Cartan theory and the possible interpretation of the torsion trace as the electromagnetic potential. Starting with a Lagrangian built of Dirac spinors, orthonormal tetrads, and a complex rather than a real linear connection we define an extended spinor derivative by which we obtain not only a very natural unification, but can also fully clarify the nontrivial underlying fibre bundle structure. Thereby a new type of contact interaction between spinors emerges, which differs from the usual one in Einstein--Cartan theory. The splitting of the linear connection into a metric and an electromagnetic part together with a characteristic length scale in the theory strongly suggest that gravity and electromagnetism have the same geometrical origin. 
  We investigate (1+1)-dimensional $\phi^4$ field theory in the symmetric and broken phases using discrete light-front quantization. We calculate the perturbative solution of the zero-mode constraint equation for both the symmetric and broken phases and show that standard renormalization of the theory yields finite results. We study the perturbative zero-mode contribution to two diagrams and show that the light-front formulation gives the same result as the equal-time formulation. In the broken phase of the theory, we obtain the nonperturbative solutions of the constraint equation and confirm our previous speculation that the critical coupling is logarithmically divergent. We discuss the renormalization of this divergence but are not able to find a satisfactory nonperturbative technique. Finally we investigate properties that are insensitive to this divergence, calculate the critical exponent of the theory, and find agreement with mean field theory as expected. 
  Possible ways of constructing extended fermionic strings with $N=4$ world-sheet supersymmetry are reviewed. String theory constraints form, in general, a non-linear quasi(super)conformal algebra, and can have conformal dimensions $\geq 1$. When $N=4$, the most general $N=4$ quasi-superconformal algebra to consider for string theory building is $\hat{D}(1,2;\a)$, whose linearisation is the so-called `large' $N=4$ superconformal algebra. The $\hat{D}(1,2;\a)$ algebra has $\Hat{su(2)}_{k^+}\oplus \Hat{su(2)}_{k^-}\oplus\Hat{u(1)}$ Ka\v{c}-Moody component, and $\a=k^-/k^+$. We check the Jacobi identities and construct a BRST charge for the $\hat{D}(1,2;\a)$ algebra. The quantum BRST operator can be made nilpotent only when $k^+=k^-=-2$. The $\hat{D}(1,2;1)$ algebra is actually isomorphic to the $SO(4)$-based Bershadsky-Knizhnik non-linear quasi-superconformal algebra. We argue about the existence of a string theory associated with the latter, and propose the (non-covariant) hamiltonian action for this new $N=4$ string theory. Our results imply the existence of two different $N=4$ fermionic string theories: the old one based on the `small' linear $N=4$ superconformal algebra and having the total ghost central charge $c_{\rm gh}=+12$, and the new one with non-linearly realised $N=4$ supersymmetry, based on the $SO(4)$ quasi-superconformal algebra and having $c_{\rm gh}=+6$. Both critical string theories have negative `critical dimensions' and do not admit unitary matter representations. 
  We prove that a large class of leading order string solutions which generalize both the plane-wave and fundamental string backgrounds are, in fact, exact solutions to all orders in \alpha'. These include, in particular, the traveling waves along the fundamental string. The key features of these solutions are a null symmetry and a chiral coupling of the string to the background. Using dimensional reduction, one finds that the extremal electric dilatonic black holes and their recently discovered generalizations with NUT charge and rotation are also exact solutions. We show that our bosonic solutions are also exact solutions of the heterotic string theory with no extra gauge field background. 
  Given a list of $N$ states with probabilities $0<p_1\leq\cdots\leq p_N$, the average conditional algorithmic information $\bar I$ to specify one of these states obeys the inequality $H\leq\bar I<H+O(1)$, where $H=-\sum p_j\log_2p_j$ and $O(1)$ is a computer-dependent constant. We show how any universal computer can be slightly modified in such a way that the inequality becomes $H\leq\bar I<H+1$, thereby eliminating the computer-dependent constant from statistical physics. 
  We discuss the issue of going off-shell in the proper time formalism. This is done by keeping a finite world sheet cutoff. We construct one example of an off-shell covariant Klein Gordon type interaction. For a suitable choice of the gauge transformation of the scalar field, gauge invariance is maintained off mass shell. However at second order in the gauge field interaction, one finds that (U(1)) gauge invariance is violated due to the finite cutoff. Interestingly, we find, to lowest order, that by adding a massive mode with appropriate gauge transformation laws to the sigma model background, one can restore gauge invariance. The gauge transformation law is found to be consistent, to the order calculated, with what one expects from the interacting equation of motion of the massive field. We also extend some previous discussion on applying the proper time formalism for propagating gauge particles, to the interacting (i.e. Yang Mills) case. 
  Classical {\W}$_3$ transformations are discussed as restricted diffeomorphism transformations (\W-Diff) in two-dimensional space. We formulate them by using Riemannian geometry as a basic ingredient. The extended {\W}$_3$ generators are given as particular combinations of Christoffel symbols. The defining equations of \W-Diff are shown to depend on these generators explicitly. We also consider the issues of finite transformations, global $SL(3)$ transformations and \W-Schwarzians. 
  We study in detail the structure of the Yangian Y(gl(N)) and of some new Yangian-type algebras called twisted Yangians. The algebra Y(gl(N)) is a `quantum' deformation of the universal enveloping algebra U(gl(N)[x]), where gl(N)[x] is the Lie algebra of gl(N)-valued polynomial functions. The twisted Yangians are quantized enveloping algebras of certain twisted Lie algebras of polynomial functions which are naturally associated to the B, C, and D series of the classical Lie algebras. 
  The Lie algebra specified by space of local functionals with commutator determined by the Gardner bracket was under survey. Problem of classification of deformations of this bracket over local infinitesimal transformations of functionals was interpreted as a problem of computing the appropriate cohomology group of order 2. The most interesting class of deformations was investigated. 
  Coherent state operators (CSO) are defined as operator valued functions on G=SL(n,C), homogeneous with respect to right multiplication by lower triangular matrices. They act on a model space containing all holomorphic finite dimensional representations of G with multiplicity 1. CSO provide an analytic tool for studying G invariant 2- and 3-point functions, which are written down in the case of $SU_3$. The quantum group deformation of the construction gives rise to a non-commutative coset space. We introduce a "standard" polynomial basis in this space (related to but not identical with the Lusztig canonical basis) which is appropriate for writing down $U_q(sl_3)$ invariant 2-point functions for representaions of the type $(\lambda,0)$ and $(0,\lambda)$. General invariant 2-point functions are written down in a mixed Poincar\'e-Birkhoff-Witt type basis. 
  Removed because of inappropriateness for e-print archives. 
  We consider an (N-2)-dimensional Calabi-Yau manifold which is defined as the zero locus of the polynomial of degree N (of Fermat type) in CP^{N-1} and its mirror manifold. We introduce the (N-2)-point correlation function (generalized Yukawa coupling) and evaluate it both by solving the Picard-Fuchs equation for period integrals in the mirror manifold and also by explicitly calculating the contribution of holomorphic maps of degree 1 to the Yukawa coupling in the Calabi-Yau manifold using the method of Algebraic geometry... 
  We analyse a new class of statistical systems, which simulate different systems of random surfaces on a lattice. Geometrical hierarchy of the energy functionals on which these theories are based produces corresponding hierarchy of the surface dynamics and of the phase transitions. We specially consider 3D gonihedric system and have found that it is equivalent to the propagation of almost free 2D Ising fermions. We construct dual statistical system with new matchbox spin variable $G_{\xi}$, high temperature expansion of which equally well describe these surfaces. 
  We analyse statistical system with interface energy proportional to the length of the edges of interface. We have found the dual system high temperature expansion of which equally well generates surfaces with linear amplitude. These dual systems are in the same relation as 3D Ising ferromagnet to the 3D Gauge spin system. 
  In unified gauge theories there exist renormalization group invariant relations among gauge and Yukawa couplings that are compatible with perturbative renormalizability, which could be considered as a Gauge-Yukawa Unification. Such relations are even necessary to ensure all-loop finiteness in Finite Unified Theories, which have vanishing $\beta$-functions beyond the unification point. We elucidate this alternative way of unification, and then present its phenomenological consequences in $SU(5)$-based models. 
  This paper deals with a dynamical system that generalizes the Kepler-Coulomb system and the Hartmann system. It is shown that the Schr\"odinger equation for this generalized Kepler-Coulomb system can be separated in prolate spheroidal coordinates. The coefficients of the interbasis expansions between three bases (spherical, parabolic and spheroidal) are studied in detail. It is found that the coefficients for the expansion of the parabolic basis in terms of the spherical basis, and vice-versa, can be expressed through the Clebsch-Gordan coefficients for the group SU(2) analytically continued to real values of their arguments. The coefficients for the expansions of the spheroidal basis in terms of the spherical and parabolic bases are proved to satisfy three-term recursion relations. 
  We define an algebra of contravariant symbols on $S^2$ and give an algebraic proof of the Correspondence Principle for that algebra. 
  Explicit constructions of $a_n^{(1)}$ affine Toda field theory breather solutions are presented. Breathers arise either from two solitons of the same species or from solitons which are anti-species of each other. In the first case, breathers carry topological charges. These topological charges lie in the tensor product representation of the fundamental representations associated with the topological charges of the constituent solitons. In the second case, breathers have zero topological charge. The breather masses are, as expected, less than the sum of the masses of the constituent solitons. 
  We study the structure of quantized enveloping algebras called twisted Yangians, which are naturally associated with the B, C, and D series of the classical Lie algebras. We obtain an explicit formula for the formal series (the Sklyanin determinant) whose coefficients are free generators of the center of the twisted Yangian. As a corollary we obtain a new system of algebraically independent generators of the center of the universal enveloping algebra for the orthogonal and symplectic Lie algebras and find the characteristic polynomial for the matrix formed by the generators of these Lie algebras. 
  We show that the classical equations of motion of the low-energy effective field theory describing the massless modes of the heterotic (or type I) string admit two classes of supersymmetric self--dual backgrounds. The first class, which was already considered in the literature, consists of solutions with a (conformally) flat metric coupled to axionic instantons. The second includes Asymptotically Locally Euclidean (ALE) gravitational instantonic backgrounds coupled to gauge instantons through the so--called ``standard embedding''. We show that some elements of these two classes of solutions are dual to each other in the sense of Buscher's duality. We give a world--sheet interpretation of the heterotic ALE istanton solutions in terms of superconformal $N=(4,4)$ $\s$--models and argue for their validity to all orders in $\alpha^\prime$. Specializing the gravitational background to the Eguchi--Hanson instanton, we compute the indices of the fermionic operators and give the explicit form of all the relevant fermionic and bosonic zero--modes.} 
  The general theory of a massless fermion coupled to a massive vector meson in two dimensions is formulated and solved to obtain the complete set of Green's functions. Both vector and axial vector couplings are included. In addition to the boson mass and the two coupling constants, a coefficient which denotes a particular current definition is required for a unique specification of the model.   The resulting four parameter theory and its solution are shown to reduce in appropriate limits to all the known soluble models, including in particular the Schwinger model and its axial vector variant. 
  We investigate the CGHS model through numerical calculation. The behavior of the mass function, which we introduced in our previous work as a ``local mass'', is examined. We found that the mass function takes negative values, which means that the amount of Hawking radiation becomes greater than the initial mass of the black hole as in the case of the RST model. 
  With the aim to construct a dynamical model with quantum group symmetry, the $q$-deformed Schr\"odinger equation of the harmonic oscillator on the $N$-dimensional quantum Euclidian space is investigated. After reviewing the differential calculus on the $q$-Euclidian space, the $q$-analog of the creation-annihilation operator is constructed. It is shown that it produces systematically all eigenfunctions of the Schr\"odinger equation and eigenvalues. We also present an alternative way to solve the Schr\"odinger equation which is based on the $q$-analysis. We represent the Schr\"odinger equation by the $q$-difference equation and solve it by using $q$-polynomials and $q$-exponential functions. The problem of the involution corresponding to the reality condition is discussed. 
  An analytical proof of the existence of negative modes in the odd--parity perturbation sector is given for all known non-abelian Einstein--Yang--Mills black holes. The significance of the normalizability condition in the functional stability analysis is emphasized. The role of the odd--parity negative modes in the sphaleron interpretation of the Bartnik--McKinnon solutions is discussed. 
  We describe flux tubes and their interactions in a low energy sigma model induced by $SU(\NF) \goto SO(\NF)$ flavor symmetry breaking in $SO(N_c)$ QCD. Gauge confinement manifests itself in the low energy theory through flux tube interactions with unscreened sources. The flux tubes which mediate confinement also illustrate an interesting ambiguity in defining global Alice strings. 
  We present a complete solution of the constraints for two-dimensional, N=2 supergravity in N=2 superspace. We obtain explicit expressions for the covariant derivatives in terms of the vector superfield $H^m$ and, for the two versions of minimal (2,2) supergravity, a chiral or twisted chiral scalar superfield $\phi$. 
  Contribution to the Proceedings of the International Congress of Mathematicians 1994. We review recent developments in the physics and mathematics of Yang-Mills theory in two dimensional spacetimes. This is a condensed version of a forthcoming review by S. Cordes, G. Moore, and S. Ramgoolam. 
  We construct wave functions and Dirac operator of spin $1/2$ fermions on quantum four-spheres. The construction can be achieved by the q-deformed differential calculus which is manifestly $SO(5)_q$ covariant. We evaluate the engenvalue of the Dirac operator on wave functions of the spinors and show that we can define the chiral fermions in such a way that the massless Dirac operator anti-commutes with $\gamma_5$. 
  The BRST transformations for gravity in Ashtekar variables are obtained by using the Maurer-Cartan horizontality conditions. The BRST cohomology in Ashtekar variables is calculated with the help of an operator $\delta$ introduced by S.P. Sorella, which allows to decompose the exterior derivative as a BRST commutator. This BRST cohomology leads to the differential invariants for four-dimensional manifolds. 
  By calculating the response function, we study the Hawking radiation of massless Dirac fields in the (2+1)-dimensional black hole geometry. We find that the response function has Planck distributions, with the temperature that agrees with the previous results obtained for the scalar field cases. We also find the Green's functions in (2+1)-dimensional Einstein static universe and anti de-Sitter space. 
  After a preliminary review of the definition and the general properties of the homogeneous spaces of quantum groups, the quantum hyperboloid qH and the quantum plane qP are determined as homogeneous spaces of Fq(E(2)). The canonical action of Eq(2) is used to define a natural q-analog of the free Schro"dinger equation, that is studied in the momentum and angular momentum bases. In the first case the eigenfunctions are factorized in terms of products of two q-exponentials. In the second case we determine the eigenstates of the unitary representation, which, in the qP case, are given in terms of Hahn-Exton functions. Introducing the universal T-matrix for Eq(2) we prove that the Hahn-Exton as well as Jackson q-Bessel functions are also obtained as matrix elements of T, thus giving the correct extension to quantum groups of well known methods in harmonic analysis. 
  It is proposed that an event that constitutes a quantum measurement corresponds to the spontaneous breaking of a symmetry in the measuring device over time. 
  We propose a dynamical mechanism to induce gauge fields in four dimensional space-time from a single scalar field or a spinor field in higher dimensions. The Born-Oppenheimer treatment of the extra dimensions is an essential ingredient in our approach. A possible applications of the idea to low dimensional condensed matter systems and high temperature field theory are also pointed out. This paper is an extended version of our previous unpublished work   (SUNY-NTG-89-48, Jan. 1990). 
  A bibliography on the Hurwitz transformations is given. We deal here, with some details, with two particular Hurwitz transformations, viz, the $\grq \to \grt$ Kustaanheimo-Stiefel transformation and its $\grh \to \grc$ compact extension. These transformations are derived in the context of Fock-Bargmann-Schwinger calculus with special emphasis on angular momentum theory. 
  This short summary of recent developments in quantum compact groups and star products is divided into 2 parts. In the first one we recast star products in a more abstract form as deformations and review its recent developments. The second part starts with a rapid presentation of standard quantum group theory and its problems, then moves to their completion by introduction of suitable Montel topologies well adapted to duality. Preferred deformations (by star products and unchanged coproducts) of Hopf algebras of functions on compact groups and their duals, are of special interest. Connection with the usual models of quantum groups and the quantum double is then presented. 
  We first review the introduction of star products in connection with deformations of Poisson brackets and the various cohomologies that are related to them. Then we concentrate on what we have called ``closed star products" and their relations with cyclic cohomology and index theorems. Finally we shall explain how quantum groups, especially in their recent topological form, are in essence examples of star products. 
  Using duality and topological theory of well behaved Hopf algebras (as defined in [2]) we construct star-product models of non compact quantum groups from Drinfeld and Reshetikhin standard deformations of enveloping Hopf algebras of simple Lie algebras. Our star-products act not only on coefficient functions of finite-dimensional representations, but actually on all $C^\infty$ functions, and they exist even for non linear (semi-simple) Lie groups. 
  We discuss the possibility of the existence of finite algorithms that may give distinct knot classes. In particular we present two attempts for such algorithms which seem promising, one based on knot projections on a plane, the other on placing knots on a cubic lattice. 
  We recall the abstract theory of Hopf algebra bicrossproducts and double cross products due to the author. We use it to develop some less-well known results about the quantum double as a twisting, as an extension and as $q$-Lorentz group. 
  We give an overview of a new kind symmetry in physics which exists between observables and states and which is made possible by the language of Hopf algebras and quantum geometry. It has been proposed by the author as a feature of Planck scale physics. More recent work includes corresponding results at the semiclassical level of Poisson-Lie groups and at the level of braided groups and braided geometry. 
  $*$-structures on quantum and braided spaces of the type defined via an R-matrix are studied. These include $q$-Minkowski and $q$-Euclidean spaces as additive braided groups. The duality between the $*$-braided groups of vectors and covectors is proved and some first applications to braided geometry are made. 
  The effective action for Q.E.D in external magnetic field is constructed using the method of inhomogeneity expansion. We first treat the non-relativistic case where a Chern-Simons like term is generated. We then consider the full relativistic theory and derive the effective action for the $A_{\mu}$ fields.   In the non-relativistic case we also add a 4-fermi type interaction and show that under certain circumstances, it corresponds to a Zeeman type term in the effective action. 
  Extreme magnetic dilaton black holes are promoted to exact solutions of heterotic string theory with unbroken supersymmetry. With account taken of alpha' corrections this is accomplished by supplementing the known solutions with SU(2) Yang-Mills vectors and scalars in addition to the already existing Abelian U(1) vector field. The solution has a simple analytic form and includes multi-black-holes. The issue of exactness of other black-hole-type solutions, including extreme dilaton electrically charged black holes and Taub-NUT solutions is discussed. 
  A description is given of how to construct $(0,2)$ supersymmetric conformal field theories as coset models. These models may be used as non--trivial backgrounds for Heterotic String Theory. They are realised as a combination of an anomalously gauged Wess--Zumino--Witten model, right--moving supersymmetric fermions, and left--moving current algebra fermions. Requiring the sum of the gauge anomalies from the bosonic and fermionic sectors to cancel yields the final model. Applications discussed include exact models of extremal four--dimensional charged black holes and Taub--NUT solutions of string theory. These coset models may also be used to construct important families of $(0,2)$ supersymmetric Heterotic String compactifications. The Kazama--Suzuki models are the left--right symmetric special case of these models. 
  A description is given of how to construct (0,2) supersymmetric conformal field theories as coset models. These models may be used as non-trivial backgrounds for Heterotic String Theory. They are realised as a combination of an anomalously gauged Wess-Zumino-Witten model, right-moving supersymmetric fermions, and left-moving current algebra fermions. Requiring the sum of the gauge anomalies from the bosonic and fermionic sectors to cancel yields the final model. Applications discussed include exact models of extremal four-dimensional charged black holes and Taub-NUT solutions of string theory. These coset models may also be used to construct important families of (0,2) supersymmetric Heterotic String compactifications. The Kazama-Suzuki models are the left-right symmetric special case of these models. 
  This paper reemphasizes the ubiquitous role of moduli spaces of algebraic curves in associative algebra and algebraic topology. The main results are: (1) the space of an operad with multiplication is a homotopy G- (i.e., homotopy graded Poisson) algebra; (2) the singular cochain complex is naturally an operad; (3) the operad of decorated moduli spaces acts naturally on the de Rham complex $\Omega^\bullet X$ of a K\"{a}hler manifold $X$, thereby yielding the most general type of homotopy G-algebra structure on $\Omega^\bullet X$. One of the reasons to put this operadic nonsense on the physics bulletin board is that we use a typical construction of supersymmetric sigma-model, the construction of Gromov-Witten invariants in Kontsevich's version. 
  We present a non-perturbative solution of large $N$ matrix models modified by terms of the form $ g(\Tr\Phi^4)^2$, which add microscopic wormholes to the random surface geometry. For $g<g_t$ the sum over surfaces is in the same universality class as the $g=0$ theory, and the string susceptibility exponent is reproduced by the conventional Liouville interaction $\sim e^{\alpha_+ \phi}$. For $g=g_t$ we find a different universality class, and the string susceptibility exponent agrees for any genus with Liouville theory where the interaction term is dressed by the other branch, $e^{\alpha_- \phi}$. This allows us to define a double-scaling limit of the $g=g_t$ theory. We also consider matrix models modified by terms of the form $g O^2$, where $O$ is a scaling operator. A fine-tuning of $g$ produces a change in this operator's gravitational dimension which is, again, in accord with the change in the branch of the Liouville dressing. 
  The space of linear differential operators on a smooth manifold $M$ has a natural one-parameter family of $Diff(M)$ (and $Vect(M)$)-module structures, defined by their action on the space of tensor-densities. It is shown that, in the case of second order differential operators, the $Vect(M)$-module structures are equivalent for any degree of tensor-densities except for three critical values: $\{0,{1\over 2},1\}$. Second order analogue of the Lie derivative appears as an intertwining operator between the spaces of second order differential operators on tensor-densities. 
  We introduce a parametrization of the coupling constant space of the generalized Kontsevich models in terms of a set of moments equivalent to those introduced recently in the context of topological gravity. For the simplest generalization of the Kontsevich model we express the moments as elementary functions of the susceptibilities and the eigenvalues of the external field. We furthermore use the moment technique to derive a closed expression for the genus zero multi-loop correlators for $(3,3m-1)$ and $(3,3m-2)$ rational matter fields coupled to gravity. We comment on the relation between the two-matrix model and the generalized Kontsevich models 
  We classify non-trivial (non-central) extensions of the group $Diff^+(S^1)$ of all diffeomorphisms of the circle preserving its orientation and of the Lie algebra $Vect (S^1)$ of vector fields on $S^1$, by the modules of tensor-densities on $S^1$. The result is: 4 non-trivial extensions of $Diff^+(S^1)$ and 7 non-trivial extensions of $Vect (S^1)$. Analogous results hold for the Virasoro group and the Virasoro algebra. We also classify central extensions of constructed Lie algebras. CPT-94/P.3024 
  A general classification of linear differential and finite-difference operators possessing a finite-dimensional invariant subspace with a polynomial basis is given. The main result is that any operator with the above property must have a representation as a polynomial element of the universal enveloping algebra of the algebra of differential (difference) operators in finite-dimensional representation. In one-dimensional case a classification is given by algebras $sl_2({\bold R})$ (for differential operators in ${\bold R}$) and $sl_2({\bold R})_q$ (for finite-difference operators in ${\bold R}$), $osp(2,2)$ (operators in one real and one Grassmann variable, or equivalently, $2 \times 2$ matrix operators in ${\bold R}$) and $gl_2 ({\bold R})_K$ ( for the operators containing the differential operators and the parity operator). A classification of linear operators possessing infinitely many finite-dimensional invariant subspaces with a basis in polynomials is presented. 
  We investigate how specific free-field realizations of twisted N=2 supersymmetric coset models give rise to natural extensions of the ``matter'' Hilbert spaces in such a manner as to incorporate the various gravitational excitations. In particular, we show that adopting a particular screening prescription is equivalent to imposing the requisite equivariance condition on cohomology. We find a simple algebraic characterization of the $W_n$-gravitational ground ring spectra of these theories in terms of affine-$SU(n)$ highest weights.. 
  In this paper which is the completion of [1], we construct the $A_0(q)$-algebra of $Q$-meromorphic functions on the quantum plane. This is the largest non-commutative, associative, $A_0(q)$-algebra of functions constructed on the quantum plane. We also define the notion of quantum subsets of R$^2$ which is a generalization of the notion of quantum disc and charactrize some of their properties. In the end we study the $Q$-homomorphisms of the quantum plane. 
  We introduce an integrable time-discretized version of the classical Calogero-Moser model, which goes to the original model in a continuum limit. This discrete model is obtained from pole solutions of a semi-discretized version of the Kadomtsev-Petviashvili equation, leading to a finite-dimensional symplectic mapping. Lax pair, symplectic structure and sufficient set of invariants of the discrete Calogero-Moser model are constructed for both the rational and elliptic cases. The classical $r$-matrix is the same as for the continuum model. An exact solution of the initial value problem is given for the rational discrete-time Calogero-Moser model. The pole-expansion and elliptic solutions of the fully discretized Kadomtsev-Petviashvili equation are also discussed. 
  We study $N=2$ supersymmetric $SU(2)/U(1)$ and $SL(2,R)/U(1)$ gauged Wess-Zumino-Witten models. It is shown that the vector gauged model is transformed to the axial gauged model by a mirror transformation. Therefore the vector gauged model and the axial gauged model are equivalent as $N=(2,2)$ superconformal field theories. In the $SL(2,R)/U(1)$ model, it is known that axial-vector duality relates a background with a singularity to that without a singularity. Implications of the equivalence of these two models to space-time singularities are discussed. 
  We use an auxiliary field construction to discuss the hard thermal loop effective action associated with massless thermal SU(N) QCD interacting with a weak gravitational field. It is demonstrated that the previous attempt to derive this effective action has only been partially successful and that it is presently only known to first order in the graviton coupling constant. This is still sufficient to enable a calculation of a symmetric traceless quark gluon plasma energy momentum tensor. Finally, we comment on the conserved currents and charges of the derived energy momentum tensor. 
  Chern-Simons formulation of 2+1 dimensional Einstein gravity with a negative cosmological constant is investigated when the spacetime has the topology $ R\times T^{2}$. The physical phase space is shown to be a direct product of two sub-phase spaces each of which is a non-Hausdorff manifold plus a set with nonzero codimensions. Spacetime geometrical interpretation of each point in the phase space is also given and we explain the 1 to 2 correspondence with the ADM formalism from the geometrical viewpoint. In quantizing this theory, we construct a "modified phase space" which is a cotangnt bundle on a torus. We also provide a modular invariant inner product and investigate the relation to the quantum theory which is directly related to the spinor representation of the ADM formalism. (This paper is the revised version of a previous paper(hep-th/9312151). The wrong discussion on the topology of the phase space is corrected.) 
  We compute the local integrals of motions of the classical limit of the lattice sine-Gordon system, using a geometrical interpretation of the local sine-Gordon variables. Using an analogous description of the screened local variables, we show that these integrals are in involution. We present some remarks on relations with the situation at roots of 1 and results on another latticisation (linked to the principal subalgebra of $\widehat{s\ell}_{2}$ rather than the homogeneous one). Finally, we analyse a module of ``screened semilocal variables'', on which the whole $\widehat{s\ell}_{2}$ acts. 
  The scattering theory of the integrable statistical models can be generalized to the case of systems with extended lines of defect. This is done by adding the reflection and transmission amplitudes for the interactions with the line of inhomegeneity to the scattering amplitudes in the bulk. The factorization condition for the new amplitudes gives rise to a set of Reflection-Transmission equations. The solutions of these equations in the case of diagonal $S$-matrix in the bulk are only those with $S =\pm 1$. The choice $S=-1$ corresponds to the Ising model. We compute the exact expressions of the transmission and reflection amplitudes relative to the interaction of the Majorana fermion of the Ising model with the defect. These amplitudes present a weak-strong duality in the coupling constant, the self-dual points being the special values where the defect line acts as a reflecting surface. We also discuss the bosonic case $S=1$ which presents instability properties and resonance states. Multi-defect systems which may give rise to a band structure are also considered. The exact expressions of correlation functions is obtained in terms of Form Factors of the bulk theory and matrix elements of the defect operator. 
  The paper commented upon gives the impression that whether the gauged version of a sigma model gives rise to the original or the dual model depends on the choice of gauge fixing. It is demonstrated here that this is not so. 
  The classical (1,0) superparticle in a curved superspace is considered. The minimal set of constraints to be imposed on the background for correct inclusion of interaction is found. The most general form of Siegel-type local fermionic symmetry is presented. Local algebra of the theory is shown to be closed off-shell and nontrivially deformed as compared to the flat one. The requirements leading to the full set of (1,0) supergravity constraints are presented. 
  Here is a list of chapters:    1 Introduction    2 Notation and preliminaries  Part I: Finite quantum groups    3 2x2 Matrix quantum groups and the quantum plane    4 Quantum enveloping algebras at a root of unity  Part II: q-Oscillators    5 Representations of q-oscillators at a root of unity    6 qr-Oscillator at a root of unity  Part III: Infinite quantum groups    7 Quantum affine algebras    8 Quantum affine algebras at a root of unity 
  A complete perturbative expansion for the Hamiltonian describing the motion of a quantomechanical system constrained to move on an arbitrary submanifold of its configuration space $R^n$ is obtained. 
  The BRST transformations for the Yang-Mills gauge fields in the presence of gravity described by Ashtekar variables are obtained by using the so-called Maurer-Cartan horizontality conditions. The BRST cohomology group expressed by the Wess-Zumino consistency condition is solved with the help of an operator $\delta$ introduced by S.P. Sorella which in our case has a very simple form and generates, together with the differential $d$ and the BRST operator $s$, a simpler algebra than in the pure Yang-Mills theory. In this way we shall find the Yang-Mills Lagrangians, the Chern-Simons terms and the gauge anomalies. 
  Feynman diagram expressions in ordinary field theories can be written in a string-like manner. The methods and the advantages for doing so are briefly discussed. 
  String theories should reduce to ordinary four-dimensional field theories at low energies. Yet the formulation of the two are so different that such a connection, if it exists, is not immediately obvious. With the Schwinger proper-time representation, and the spinor helicity technique, it has been shown that field theories can indeed be written in a string-like manner, thus resulting in simplifications in practical calculations, and providing novel insights into gauge and gravitational theories. This paper continues the study of string organization of field theories by focusing on the question of local duality. It is shown that a single expression for the sum of many diagrams can indeed be written for QED, thereby simulating the duality property in strings. The relation between a single diagram and the dual sum is somewhat analogous to the relation between a old- fashioned perturbation diagram and a Feynman diagram. Dual expressions are particularly significant for gauge theories because they are gauge invariant while expressions for single diagrams are not. 
  We discuss the light-front formulation of SU(2) Yang-Mills theory on a torus. The gauge choice we use allows for an exact and unambiguous solution of Gauss's law. 
  We extend the path-integral approach to bosonization to the case in which the fermionic interaction is non-local. In particular we obtain a completely bosonized version of a Thirring-like model with currents coupled by general (symmetric) bilocal potentials. The model contains the Tomonaga-Luttinger model as a special case; exploiting this fact we study the basic properties of the 1-d spinless fermionic gas: fermionic correlators, the spectrum of collective modes, etc. Finally we discuss the generalization of our procedure to the non-Abelian case, thus providing a new tool to be used in the study of 1-d many-body systems with spin-flipping interactions. 
  A generalized approach of the Born-Oppenheimer approximation is developed to analytically deal with the influence exercised by the spatial motion of atom's mass-center on a two-level atom in an optical ring cavity with a quantized single-mode electromagnetic field. The explicit expressions of tunneling rate are obtained for various cases, such as that with initial coherent state and thermal equilibrium state at finite temperature. Therefore, the studies for Doppler and recoil effects of the spatial motion on the scheme controlling atomic tunneling should be reconsidered in terms of the initial momentum of atom's mass center. 
  The systematical studies on the dynamical approach of wavefunction collapse in quantum measurement are reported in this paper based on the Hepp-Coleman's model and its generalizations. Under certain physically reasonable conditions, which are easily satisfied by the practical problems, it is shown that the off-diagonal elements of the reduced density matrix vanish in quantum mechanical evolution process in the macroscopic limit with a very large particle number N. Various examples with detector made up of oscillators of different spectrum distribution are used to illustrate this observations . With the two-level system as an explicit illustration, the quantum information entropy is exactly obtained to quantitatively describe the degree of decoherence for the so-called partial coherence caused by detector. The entropy for the case with many levels is computed based on perturbation method in the limits with very large and very small N. As an application of this general approach for quantum measurement, a dynamical realization of the quantum Zeno effect are present to analyse its recent testing experiment in connection with a description of transition in quantum information entropy. Finally, the Cini's model for the correlation between the states of the measured system and the detector is generalized for the case with many energy-level. 
  Lectures given at International School of Physics ``Enrico Fermi'', Varenna, Villa Monastero, June 28-July 7 1994 
  According to 't Hooft the combination of quantum mechanics and gravity requires the three dimensional world to be an image of data that can be stored on a two dimensional projection much like a holographic image. The two dimensional description only requires one discrete degree of freedom per Planck area and yet it is rich enough to describe all three dimensional phenomena. After outlining 't Hooft's proposal I give a preliminary informal description of how it may be implemented. One finds a basic requirement that particles must grow in size as their momenta are increased far above the Planck scale. The consequences for high energy particle collisions are described. The phenomena of particle growth with momentum was previously discussed in the context of string theory and was related to information spreading near black hole horizons. The considerations of this paper indicate that the effect is much more rapid at all but the earliest times. In fact the rate of spreading is found to saturate the bound from causality. Finally we consider string theory as a possible realization of 't Hooft's idea. The light front lattice string model of Klebanov and Susskind is reviewed and its similarities with the holographic theory are demonstrated. The agreement between the two requires unproven but plausible assumptions about the nonperturbative behavior of string theory. Very similar ideas to those in this paper have been long held by Charles Thorn. 
  New systems of Laplace (Casimir) operators for the orthogonal and symplectic Lie algebras are constructed. The operators are expressed in terms of paths in graphs related to matrices formed by the generators of these Lie algebras with the use of some properties of the noncommutative symmetric functions associated with a matrix. The decomposition of the Sklyanin determinant into a product of quasi-determinants play the main role in the construction. Analogous decomposition for the quantum determinant provides an alternative proof of the known construction for the Lie algebra gl(N). 
  We use the metric on the space of gravity fields given by DeWitt to construct a unique kinematic measure on the space of FRW simple fluids and show that when the mass parameter $\Omega$ is used as a coordinate this measure is singular at $\Omega = 1$. This singularity, combined with the time evolution of $\Omega$, distorts distributions of $\Omega$ values to be concentrated in the neighborhood of 1 at early times. It is a distorted distribution of $\Omega$ values that sometimes misleads the casual observer to conclude that $\Omega$ must be exactly equal to 1. 
  We consider the effect of curved two-dimensional space-time on Witten's $N=2$ supersymmetric sigma models interpolating Calabi-Yau hypersurfaces to Landau-Ginzburg models. In order for the former models to have significant connection to superstring theory, only the $N=(1,1)$ or $N=(1,0)$ part of the total $N=(2,2)$ world-sheet supersymmetry is made local. Even though there arises an additional minimizing condition due to a scalar auxiliary field in the supergravity multiplet on curved two-dimensions, the essential feature of the sigma-model relating Calabi-Yau and Landau-Ginzburg models will be maintained. This indicates the validity of these sigma models formulated on curved two-dimensions or curved world-sheets. As a by-product, the coupling of $N=(2,2)$ vector multiplets to other multiplets with $N=(1,1)$ local supersymmetry is developed. 
  A representation of the group element (also known as ``universal ${\cal T}$-matrix'') which satisfies $\Delta(g) = g\otimes g$, is given in the form $$ g = \left(\prod_{s=1}^{d_B}\phantom.^>\ {\cal E}_{1/q_{i(s)}}(\chi^{(s)}T_{-i(s)})\right) q^{2\vec\phi\vec H} \left(\prod_{s=1}^{d_B}\phantom.^<\ {\cal E}_{q_{i(s)}}(\psi^{(s)} T_{+i(s)})\right)$$ where $d_B = \frac{1}{2}(d_G - r_G)$, $q_i = q^{|| \vec\alpha_i||^2/2}$ and $H_i = 2\vec H\vec\alpha_i/||\vec\alpha_i||^2$ and $T_{\pm i}$ are the generators of quantum group associated respectively with Cartan algebra and the {\it simple} roots. The ``free fields'' $\chi,\ \vec\phi,\ \psi$ form a Heisenberg-like algebra: $\psi^{(s)}\psi^{(s')} = q^{-\vec\alpha_{i(s)} \vec\alpha_{i(s')}} \psi^{(s')}\psi^{(s)}, & \chi^{(s)}\chi^{(s')} = q^{-\vec\alpha_{i(s)}\vec\alpha_{i(s')}} \chi^{(s')}\chi^{(s)}& {\rm for} \ s<s', \\ q^{\vec h\vec\phi}\psi^{(s)} = q^{\vec h\vec\alpha_{i(s)}} \psi^{(s)}q^{\vec h\vec\phi}, & q^{\vec h\vec\phi}\chi^{(s)} = q^{\vec h \vec\alpha_{i(s)}}\chi^{(s)}q^{\vec h\vec\phi}, & \\ &\psi^{(s)} \chi^{(s')} = \chi^{(s')}\psi^{(s)} & {\rm for\ any}\ s,s'.$ We argue that the $d_G$-parametric ``manifold'' which $g$ spans in the operator-valued universal envelopping algebra, can also be invariant under the group multiplication $g \rightarrow g'\cdot g''$. The universal ${\cal R}$-matrix with the property that ${\cal R} (g\otimes I)(I\otimes g) = (I\otimes g)(g\otimes I){\cal R}$ is given by the usual formula $${\cal R} = q^{-\sum_{ij}^{r_G}||\vec\alpha_i||^2|| \vec\alpha_j||^2 (\vec\alpha\vec\alpha)^{-1}_{ij}H_i \otimes H_j}\prod_{ \vec\alpha > 0}^{d_B}{\cal E}_{q_{\vec\alpha}}\left(-(q_{\vec\alpha}- q_{\vec\alpha}^{-1})T_{\vec\alpha}\otimes T_{-\vec\alpha}\right).$$ 
  Combinatoric formulas for cluster expansions have been improved many times over the years. Here we develop some new combinatoric proofs and extensions of the tree formulas of Brydges and Kennedy, and test them on a series of pedagogical examples. 
  We discuss the spontaneous supersymetry breaking within the low-energy effective supergravity action of four-dimensional superstrings. In particular, we emphasize the non-universality of the soft supersymmetry breaking parameters, the $\mu$-problem and the duality symmetries. 
  I discuss how effective grand unified theories requiring adjoint Higgs fields for breaking to the standard model can be contained within string theory. Initial findings are presented in a search for and classification of effective three generation SO(10) SUSY-GUT models built using the free-fermionic string approach. Based on talk presented at PASCOS '94, Syracuse, NY. 
  We show that the the generalized Calogero-Moser model with boundary potential of the P\"oschl-Teller type describes the non-relativistic limit of the quantum sine-Gordon model on a half-line with Dirichlet boundary condition. 
  We use solvable two-dimensional gauge theories to illustrate the issues in relating large N gauge theory to string theory. We also give an introduction to recent mathematical work which allows constructing master fields for higher dimensional large N theories. We illustrate this with a new derivation of the Hopf equation governing the evolution of the spectral density in matrix quantum mechanics. Based on lectures given at the 1994 Trieste Spring School on String Theory, Gauge Theory and Quantum Gravity. 
  We show how the spontaneous breaking of local N=1 supersymmetry and of the SU(2)xU(1) gauge symmetry can be simultaneously realized, with naturally vanishing tree-level vacuum energy, in superstring effective supergravities. Both the gravitino mass m_{3/2} and the electroweak scale m_Z are classically undetermined, and slide along moduli directions that include the Higgs flat direction |H_1^0| = |H_2^0|. There are important differences with conventional supergravity models: the goldstino has components along the higgsino direction; SU(2) x U(1) breaking occurs already at the classical level; the scales m_{3/2} and m_Z, the gauge couplings, the lightest Higgs mass and the cosmological constant are entirely determined by quantum corrections. 
  We propose a contraction of the de Sitter quantum group leading to the quantum Poincare group in any dimensions. The method relies on the coaction of the de Sitter quantum group on a non--commutative space, and the deformation parameter $q$ is sent to one. The bicrossproduct structure of the quantum Poincar\'e group is exhibited and shown to be dual to the one of the $\kappa$--Poincar\'e Hopf algebra, at least in two dimensions. 
  In this note, we review the recent developments in the string field theory in the temporal gauge. (Based on a talk presented by N.I. in the workshop {\it Quantum Field Theory, Integrable Models and Beyond}, Yukawa Institute for Theoretical Physics, Kyoto University, 14-18 February 1994.) 
  In the present article we analyze the bound states of an electron in a Coulomb field when an Aharonov-Bohm field as well as a magnetic Dirac monopole are present. We solve, via separation of variables, the Schr\"odinger equation in spherical coordinates and we show how the Hydrogen energy spectrum depends on the Aharonov-Bohm and the magnetic monopole strengths. In passing, the Klein-Gordon equation is solved. 
  The axial anomaly is computed for Euclidean Dirac fermions on the plane. The dependence upon the self-adjoint extensions of the Dirac operator is investigated and its relevance concerning the second virial coefficient of the anyon gas is discussed. 
  The Wess-Zumino consistency condition for four-dimensional Einstein gravity is investigated in the space of local forms involving the fields, the ghosts, the antifields and their derivatives. Its general solution is constructed for all values of the form degree and of the ghost number. It is shown in particular that the antifields (= sources for the BRST variations) can occur only through cohomologically trivial terms. 
  A symplectic structure on the space of nondegenerate and nonparametrized curves in a locally affine manifold is defined. We also consider several interesting spaces of nondegenerate projective curves endowed with Poisson structures. This construction connects the Virasoro algebra and the Gel'fand-Dikii bracket with the projective differential geometry. 
  We examine deformed Poincar\'e algebras containing the exact Lorentz algebra. We impose constraints which are necessary for defining field theories on these algebras and we present simple field theoretical examples. Of particular interest is a case that exhibits improved renormalization properties. 
  The computation of $\kappa$-anomalies in the Green-Schwarz heterotic superstring sigma-model and the corresponding Wess-Zumino consistency condition constitute a powerful alternative approach for the derivation of manifestly supersymmetric string effective actions. With respect to the beta-function approach this technique presents the advantage that a result which is obtained with the computation of beta-functions at $n$ loops can be obtained through the calculation of $\kappa$-anomalies at \hbox{$n-1$} loops. In this paper we derive by a direct one-loop perturbative computation the $\kappa$-anomaly associated to the Yang-Mills Chern-Simons threeform and, for the first time, the one associated to the Lorentz Chern-Simons threeform. Contrary to what is often stated in the literature we show that the Lorentz $\kappa$-anomaly gets contributions from the integration over both the fermionic {\it and\/} bosonic degrees of freedom of the string. A careful analysis of the absolute coefficients of all these anomalies reveals that they can be absorbed by setting $dH={\alpha'\over4}(\trace F^2-\trace R^2)$, where $\alpha'$ is the string tension, the expected result. We show that this relation ensures also the absence of gauge and Lorentz anomalies in the sigma-model effective action. We evidenciate the presence of infrared divergences. 
  Recently the scaling function of the dilute non-contractible self-avoiding 2D polymer loop on a cylinder was related to the Painleve III transcendent. Using the perturbation theory, the thermidynamic Bethe ansatz and numerical calculations we argue a similar relation for the contractible self-avoiding loop. 
  We describe flux tubes and their interactions in a low energy sigma model induced by $SU({N_f}) \rightarrow SO({N_f})$ flavor symmetry breaking in $SO(N_c)$ QCD. Unlike standard QCD, this model allows gauge confinement to manifest itself in the low energy theory, which has unscreened spinor color sources and global $Z_2$ flux tubes. We construct the flux tubes and show how they mediate the confinement of spinor sources. We further examine the flux tubes' quantum stability, spectrum and interactions. We find that flux tubes are Alice strings, despite ambiguities in defining parallel transport. Furthermore, twisted loops of flux tube support skyrmion number, just as gauged Alice strings form loops that support monopole charge. This model, while phenomenologically nonviable, thus affords a perspective on both the dynamics of confinement and on subtleties which arise for global Alice strings. 
  We clarify certain important issues relevant for the geometric interpretation of a large class of N = 2 superconformal theories. By fully exploiting the phase structure of these theories (discovered in earlier works) we are able to clearly identify their geometric content. One application is to present a simple and natural resolution to the question of what constitutes the mirror of a rigid Calabi-Yau manifold. We also discuss some other models with unusual phase diagrams that highlight some subtle features regarding the geometric content of conformal theories. 
  In 2 + 1 dimensions, in the presence of gravity, supersymmetry can ensure the vanishing of the cosmological constant without requiring the equality of bose and fermi masses. 
  We show that heavy fermions decouple from the low energy physics also in non-perturbative instanton effects. Provided the heavy fermions are lighter than the symmetry breaking scale, all the instanton effects should be expressed as local operators in the effective Lagrangian. The effective theory itself doesn't admit instantons. We present the mechanism which suppresses instantons in the effective theory. 
  We investigate the effective action of 2+1 dimensional charged spin 1/2 fermions and spin 0 bosons in the presence of a $U(1)$ gauge field. We evaluate terms in an expansion up to second order in derivatives of the field strength, but exactly in the mass parameter and in the magnitude of the nonvanishing constant field strength. We find that in a strong uniform magnetic field background, space-derivative terms lower the energy, and there arises an instability toward inhomogeneous magnetic fields. 
  We provide a non-technical introduction to "misaligned supersymmetry", a generic phenomenon in string theory which describes how the arrangement of bosonic and fermionic states at all string energy levels conspires to preserve finite string amplitudes even in the absence of spacetime supersymmetry. Misaligned supersymmetry thus naturally constrains the degree to which spacetime supersymmetry can be broken in string theory while preserving the finiteness of string amplitudes, and explains how the requirements of modular invariance and absence of physical tachyons affect the distribution of states throughout the string spectrum. 
  Starting from linear equations for the complex scalar field, the two- and three-point Green's functions are obtained in the infrared approximation. We show that the infrared singularity factorizes in the vertex function as in spinorial QED, reproducing in a straightforward way the result of lenghty perturbative calculations. 
  Using the generalized coherent states we argue that the path integral formulae for $SU(2)$ and $SU(1,1)$ (in the discrete series) are WKB exact,if the starting point is expressed as the trace of $e^{-iT\hat H}$ with $\hat H$ being given by a linear combination of generators. In our case,WKB approximation is achieved by taking a large ``spin'' limit: $J,K\rightarrow \infty$. The result is obtained directly by knowing that the each coefficient vanishes under the $J^{-1}$($K^{-1}$) expansion and is examined by another method to be legitimated. We also point out that the discretized form of path integral is indispensable, in other words, the continuum path integral expression leads us to a wrong result. Therefore a great care must be taken when some geometrical action would be adopted, even if it is so beautiful, as the starting ingredient of path integral. 
  The first and second covariant quantization of a free massless $d=4$ superparticle with pure gauge auxiliary spinor variables have been carried out. Arised chiral harmonic superfield corresponds to superparticle with finite superspin and it is interpreted as Radon transformation of free massless finite superspin superfield. A prescription for the second quantization procedure of harmonic fields has been formulated. Microcausality is achievable only under description of the particles by cohomology class of harmonic superfields with integer negative homogeneity index and ordinary connection between spin and statistics. 
  In this talk we will describe a string solution which contains a self-dual (instantonic) metric and study its properties. Talk given by M.Bianchi at the Seventh Marcel Grossmann Meeting, Stanford July 24-30, 1994. 
  We study supersymmetric, four-dimensional (4-d), Abelian charged black holes (BH's) arising in (4+n)-d (1 \le n \le 7) Kaluza-Klein (KK) theories. Such solutions, which satisfy supersymmetric Killing spinor equations (formally satisfied for any n) and saturate the corresponding Bogomol'nyi bounds, can be obtained if and only if the isometry group of the internal space is broken down to the U(1)_E \times U(1)_M gauge group; they correspond to dyonic BH's with electric Q and magnetic P charges associated with {\it different} U(1) factors. The internal metric of such configurations is diagonal with (n-2) internal radii constant, while the remaining two radii (associated with the respective electric and magnetic U(1) gauge fields) and the 4-d part of the metric turn out to be independent of n, i.e., solutions are effectively those of supersymmetric 4-d BH's of 6-d KK theory. For Q \ne 0 and P \ne 0, 4-d space-time has a null singularity, finite temperature (T_H \propto 1 / \sqrt{|QP|}) and zero entropy. Special cases with either Q=0 or P=0 correspond to the supersymmetric 4-d BH's of 5-d KK theory, first derived by Gibbons and Perry, which have a naked singularity and infinite temperature. 
  The Moyal--Weyl description of quantum mechanics provides a comprehensive phase space representation of dynamics. The Weyl symbol image of the Heisenberg picture evolution operator is regular in $\hbar$. Its semiclassical expansion `coefficients,' acting on symbols that represent observables, are simple, globally defined differential operators constructed in terms of the classical flow. Two methods of constructing this expansion are discussed. The first introduces a cluster-graph expansion for the symbol of an exponentiated operator, which extends Groenewold's formula for the Weyl product of symbols. This Poisson bracket based cluster expansion determines the Jacobi equations for the semiclassical expansion of `quantum trajectories.' Their Green function solutions construct the regular $\hbar\downarrow0$ asymptotic series for the Heisenberg--Weyl evolution map. The second method directly substitutes such a series into the Moyal equation of motion and determines the $\hbar$ coefficients recursively. The Heisenberg--Weyl description of evolution involves no essential singularity in $\hbar$, no Hamilton--Jacobi equation to solve for the action, and no multiple trajectories, caustics or Maslov indices. 
  We investigate quantum corrections of the 2-d dilaton gravity near the singularity. Our motivation comes from a s-wave reduced cosmological solution which is classically singular in the scalar fields (dilaton and moduli). As result we find, that the singularity disappears and a dilaton/moduli potential is created. 
  The massive high-temperature phase of the chiral Potts quantum chain is studied using perturbative methods. For the Z_3-chain we present high-temperature expansions for the groundstate energy and the dispersion relations of the two single-particle states as well as two-particle states at general values of the parameters. We also present a perturbative argument showing that a large class of massive Z_n-spin quantum chains have quasiparticle spectra with n-1 fundamental particles. It is known from earlier investigations that -at special values of the parameters- some of the fundamental particles exist only for limited ranges of the momentum. In these regimes our argument is not rigorous as one can conclude from a discussion of the radius of convergence of the perturbation series. We also derive correlation functions from a perturbative evaluation of the groundstate for the Z_3-chain. In addition to an exponential decay we observe an oscillating contribution. The oscillation length seems to be related to the asymmetry of the dispersion relations. We show that this relation is exact at special values of the parameters for general Z_n using a form factor expansion. 
  We study toroidal orbifold models with topologically invariant terms in the path integral formalism and give physical interpretations of the terms from an operator formalism point of view. We briefly discuss a possibility of a new class of modular invariant orbifold models. 
  We present the new explicit geometrical knowledge of the Landau-Ginzburg orbifolds, when a typical type of superpotential is considered. Relying on toric geometry, we show the one-to-one correspondence between some of the $(a,c)$ states with $U(1)$ charges $(-1,1)$ and the $(1,1)$ forms coming from blowing-up processes. Consequently, we find the monomial-divisor mirror map for Landau-Ginzburg orbifolds. The possibility of the application of the models of other types is briefly discussed. 
  The structure of classical non-linear $\cw$ algebras closing on rational functions is analyzed both for the ordinary and the supersymmetric case. Such algebras appear as a result of a coset construction. Their relevance to physical applications is pointed out. 
  In this talk the class of multi-fields reductions of the KP and super-KP hierarchies (leading to non-purely differential Lax operators) is revisited from the point of view of coset construction. This means in particular that all the hamiltonian densities of the infinite tower belong to a coset algebra of a given Poisson brackets structure. 
  The usual Laurent expansion of the analytic tensors on the complex plane is generalized to any closed and orientable Riemann surface represented as an affine algebraic curve. As an application, the operator formalism for the $b-c$ systems is developed. The physical states are expressed by means of creation and annihilation operators as in the complex plane and the correlation functions are evaluated starting from simple normal ordering rules. The Hilbert space of the theory exhibits an interesting internal structure, being splitted into $n$ ($n$ is the number of branches of the curve) independent Hilbert spaces. Exploiting the operator formalism a large collection of explicit formulas of string theory is derived. 
  We discuss the effective action for Polyakov-Wilson loops winding around compact Euclidean time, which serve as order parameters for the finite temperature deconfinement transition in $SU(N)$ Yang-Mills gauge theory. We then apply our results to the study of the high temperature continuation of the confining phase, and to the analysis of certain $Z_N$ domain walls that have been argued to play a role in cosmology. We argue that the free energy of these walls is much larger than previously thought. 
  The conventional quantization of w_3 strings gives theories which are equivalent to special cases of bosonic strings. We explore whether a more general quantization can lead to new generalized W_3 string theories by seeking to construct quantum BRST charges directly without requiring the existence of a quantum W_3 algebra. We study W_3-like strings with a direct spacetime interpretation---that is, with matter given by explicit free field realizations. Special emphasis is placed on the attempt to construct a quantum W-string associated with the magic realizations of the classical w_3 algebra. We give the general conditions for the existence of W_3-like strings, and comment how the known results fit into our general construction. Our results are negative: we find no new consistent string theories, and in particular rule out the possibility of critical strings based on the magic realizations. 
  We propose an expression for the eigenvalues of the transfer matrix for the $U_q(B_n)$-invariant open quantum spin chain associated with the fundamental representation of $A^{(2)}_{2n}$. By assumption, the Bethe Ansatz equations are ``doubled'' with respect to those of the corresponding closed chain with periodic boundary conditions. We verify that the transfer matrix eigenvalues have the correct analyticity properties and asymptotic behavior. We also briefly discuss the structure of the eigenstates of the transfer matrix. 
  The main subject of the paper is the so-called Discrete Painlev\'e-1 Equation (DP1). Solutions of the DP1 are classified under criterion of their behavior while argument tends to infinity. The appropriate theorems of existence are proved. 
  An essential prerequisite for the study of q-deformed physics are particle states in position and momentum representation. In order to relate x- and p-space by Fourier transformations the appropriate q-exponential series related to orthogonal quantum symmetries is constructed. It turns out to be a new q-special function giving rise to q-plane wave solutions that transform with a noncommuting phase under translations. 
  We quantize Quantum Electrodynamics in $2+1$ dimensions coupled to a Chern-Simons (CS) term and a charged spinor field, in covariant gauges and in the Coulomb gauge. The resulting Maxwell-Chern-Simons (MCS) theory describes charged fermions interacting with each other and with topologically massive propagating photons. We impose Gauss's law and the gauge conditions and investigate their effect on the dynamics and on the statistics of $n$-particle states. We construct charged spinor states that obey Gauss's law and the gauge conditions, and transform the theory to representations in which these states constitute a Fock space. We demonstrate that, in these representations, the nonlocal interactions between charges and between charges and transverse currents, as well as the interactions between currents and massive propagating photons, are identical in the different gauges we analyze in this and in earlier work. We construct the generators of the Poincar\'e group, show that they implement the Poincar\'e algebra, and explicitly demonstrate the effect of rotations and Lorentz boosts on the particle states. We show that the imposition of Gauss's law does not produce any ``exotic'' fractional statistics. In the case of the covariant gauges, this demonstration makes use of unitary transformations that provide charged particles with the gauge fields required by Gauss's law, but that leave the anticommutator algebra of the spinor fields untransformed. In the Coulomb gauge, we show that the anticommutators of the spinor fields apply to the Dirac-Bergmann constraint surfaces, on which Gauss's law and the gauge conditions obtain. We examine MCS theory in the large CS coupling constant limit, and compare that limiting form with CS theory, in which the Maxwell kinetic energy term is not included in the 
  Continuing our recent argument where we constructed a FNBWW-type spin-$1$ boson having opposite relative intrinsic parity to that of the associated antiparticle, we now study eigenstates of the Charge Conjugation operator. Based on the observation that if $\phi_{_{L}}(p^\mu)$ transforms as a $(0,\,j)$ spinor under Lorentz boosts, then $\Theta_{[j]}\,\phi_{_{L}}^\ast(p^\mu)$ transforms as a $(j,\,0)$ spinor (with a similar relationship existing between $\phi_{_{R}}(p^\mu)$ and $\Theta_{[j]}\,\phi_{_{R}}^\ast(p^\mu)$; where $ \Theta_{[j]}\,{\bf J}\,\Theta_{[j]}^{-1}\,=\,-\,{\bf J}^\ast $ with $\Theta_{[j]}$ the well known Wigner matrix involved in the operation of time reversal) we introduce McLennan-Case type $(j,\,0)\oplus(0,\,j)$ spinors. Relative phases between $\phi_{_{R}}(p^\mu)$ and $\Theta_{[j]}\,\phi_{_{R}}^\ast(p^\mu)$, and $\Theta_{[j]}\,\phi_{_{L}}^\ast(p^\mu)$ and $\phi_{_{L}}(p^\mu)$, turn out to have physical significance and are fixed by appropriate requirements. Explicit construction, and a series of physically relevant properties, for these spinors are obtained for spin-$1/2$ and spin-$1$ culminating in the construction of a new wave equation and introduction of Dirac-like and Majorana-like quantum fields. 
  The chiral model for self-dual gravity given by Husain in the context of the chiral equations approach is discussed. A Lie algebra corresponding to a finite dimensional subgroup of the group of symplectic diffeomorphisms is found, and then use for expanding the Lie algebra valued connections associated with the chiral model. The self-dual metric can be explicitly given in terms of harmonic maps and in terms of a basis of this subalgebra. 
  Within the framework of braided or quasisymmetric monoidal categories braided Q-supersymmetry is investigated, where Q is a certain functorial isomorphism in a braided symmetric monoidal category. For an ordinary (co-)quasitriangular Hopf algebra (H,R) a braided monoidal category of H-(co-)modules with braiding induced by the R-matrix is considered. It can be shown for a specific class of Q-supersymmetries in this category that every braided Q-super-Hopf algebra B admits an ordinary Q-super-Hopf algebra structure on the cross product BxH such that H is a sub-Hopf algebra and B is a subalgebra in BxH. Applying this Q-bosonization to the quantum Koszul complex (K(q,g),d) of the quantum enveloping algebra Uq(g) for Lie algebras g associated with the root systems A, B, C and D one obtains a classical super-Hopf algebra structure on (K(q,g),d) where the structure maps are morphisms of modules with differentiation. 
  We construct the odd symplectic structure and the equivariant even (pre)symplectic one from it on the space of differential forms on the Riemann manifold. The Poincare -- Cartan like invariants of the second structure define the equivariant generalizations of the Euler classes on the surfaces. 
  We consider a general class of boundary terms of the open XYZ spin-1/2 chain compatible with integrability. We have obtained the general elliptic solution of $K$-matrix obeying the boundary Yang-Baxter equation using the $R$-matrix of the eight vertex model and derived the associated integrable spin-chain Hamiltonian. 
  The supersymmetric extension of Taub-NUT space admits 4 standard supersymmetries plus several additional non-standard ones. The geometrical origin of these symmetries is traced. The result has applications to fermion modes in gravitational instantons as well as in long-range monopole dynamics. 
  New kinds of supersymmetry arise in supersymmetric $\sg$-models describing the motion of spinning point-particles in classical backgrounds, for example black-holes, or the dynamics of monopoles. Their geometric origin is the existence of Killing-Yano tensors. The relation between these concepts is explained and examples are given. --- Contribution to Proceedings Quarks-94; Vladimir, Russia (1994). 
  A formula for the mass-gap of the supersymmetric O($N$) sigma model ($N>4$) in two dimensions is derived: $m/\Lambda_{\overline{\rm MS}}=2^{2\Delta}\sin(\pi\Delta)/(\pi\Delta)$, where $\Delta=1/(N-2)$ and $m$ is the mass of the fundamental vector particle in the theory. This result is obtained by comparing two expressions for the free-energy density in the presence of a coupling to a conserved charge; one expression is computed from the exact S-matrix of Shankar and Witten via the the thermodynamic Bethe ansatz and the other is computed using conventional perturbation theory. These calculations provide a stringent test of the S-matrix, showing that it correctly reproduces the universal part of the beta-function and resolving the problem of CDD ambiguities. 
  A formula for the mass-gap of the supersymmetric $\CP^{n-1}$ sigma model ($n > 1$) in two dimensions is derived: $m/\Lambda_{\overline{\rm MS}}=\sin(\pi\Delta)/(\pi\Delta)$ where $\Delta=1/n$ and $m$ is the mass of the fundamental particle multiplet. This result is obtained by comparing two expressions for the free-energy density in the presence of a coupling to a conserved charge; one expression is computed from the exact S-matrix of K\"oberle and Kurak via the thermodynamic Bethe ansatz and the other is computed using conventional perturbation theory. These calculations provide a stringent test of the S-matrix, showing that it correctly reproduces the universal part of the beta-function and resolving the problem of CDD ambiguities. 
  I give a very brief introduction to the use of effective field theory techniques in quantum calculations of general relativity. The gravitational interaction is naturally organized as a quantum effective field theory and a certain class of quantum corrections can be calculated. 
  We extend the action for evolution equations of KdV and MKdV type which was derived in [Capel/Nijhoff] to the case of not periodic, but only equivariant phase space variables, introduced in [Faddeev/Volkov]. The difference of these variables may be interpreted as reduced phase space variables via a Marsden-Weinstein reduction where the monodromies play the role of the momentum map. As an example we obtain the doubly discrete sine-Gordon equation and the Hirota equation and the corresponding symplectic structures. 
  It has been shown recently that the motion of solitons at couplings around a critical coupling can be reduced to the dynamics of particles (the zeros of the Higgs field) on a curved manifold with potential. The curvature gives a velocity dependent force, and the magnitude of the potential is proportional to the distance from a critical coupling. In this paper we apply this approximation to determining the equation of state of a gas of vortices in the Abelian Higgs model. We derive a virial expansion using certain known integrals of the metric, and the second virial coefficient is calculated, determining the behaviour of the gas at low densities. A formula for determining higher order coefficients is given. At low densities and temperatures $T \gg \l$ the equation of state is of the Van der Waals form $(P+b\frac{N^{2}}{A^{2}})(A-aN) = NT$ with $a=4\pi$ and $b=-4.89\pi\l$ where $\l$ is a measure of the distance from critical coupling. It is found that there is no phase transition in a low density type-II gas, but there is a transition in the type-I case between a condensed and gaseous state. We conclude with a discussion of the relation of our results to vortex behaviour in superconductors. 
  Wavefunctionals of three dimensional quantum gravity are extracted from the 3D field theoretic analogs of the four dimensional Donaldson polynomials. Our procedure is generalizable to four and other dimensions. This is a summary of a talk presented at the Seventh Marcel Grossmann Meeting on General Relativity, Stanford University, Stanford, CA, USA, July 24 - 30, 1994 
  (Talk presented at the 7th Marcel Grossmann Meeting on General Relativity, Stanford, CA, July 24-30, 1994) We study the semi-classical limit of the solution of the Dirac equation in a background electromagnetic/gravitational plane wave. We show that the exact solution corresponding to an asymptotically fixed incoming momentum satisfies constraints consistent with the classical notion of a spinning particle. In order to further analyze the motion of a spinning particle in this external inhomogeneous field one has to consider wave-packet superpositions of these exact solutions. We are currently investigating the existence of a classical theory of a phenomenological spin tensor which reproduces our quantum-mechanical results. 
  (Presented at conference on Fundamental Problems in Physics - UMBC - June 1994) It is shown that among the orthogonal sets of EPR (completely entangled) states there is a unique basis (up to equivalence) that is a also a perfectly resolved set of coherent states with respect to a pair of complementary observables. This basis defines a lattice phase space in which quadratic Hamiltonians constructed from the observables induce site-to-site hopping at discrete time intervals. When recently suggested communication schemes\cite{BENa} are adapted to the lattice they are greatly enhanced, because the finite Heisenberg group structure allows dynamic generation of signal sequences using the quadratic Hamiltonians. We anticipate the possibility of interferometry by determining the relative phases between successive signals produced by the simplest Hamiltonians of this type, and we show that they exhibit a remarkable pattern determined by the number-theoretic Legendre symbol. 
  It is shown that the algebraic structure of finite Heisenberg groups associated with the tensor product of two Hilbert spaces leads to a simple demonstration valid in all Hilbert space dimensions of the impossibility of non-contextual hidden variables. 
  Completely entangled quantum states are shown to factorize into tensor products of entangled states whose dimensions are powers of prime numbers. The entangled states of each prime-power dimension transform among themselves under a finite Heisenberg group. We are thus led to examine processes in which factors are exchanged between entangled states and so consider canonical ensembles in which these processes occur. It is shown that the Riemann zeta function is the appropriate partition function and that the Riemann hypothesis makes a prediction about the high temperature contribution of modes of large dimension. 
  Phenomenologically viable string vacua may require incorporating Kac-Moody algebras at level $\geq 2$. We exploit the free fermionic formulation to construct N=(0,2) world-sheet supersymmetric string models with specific phenomenological input: N=1 spacetime supersymmetry, three generations of chiral fermions in gauge groups $SO(10)$ or $SU(5)$, adjoint Higgses, and a single Yukawa coupling of a fundamental Higgs to the third generation. In this talk, we will show models of gauge group $SO(10)$ and of $SU(5)$ without any gauge singlet moduli, and show some novel features appearing in the connection of these two models. The accompanying, and rather non-trivial, discrete chiral sub-algebras can determine hierarchies in the fermion mass matrix. Our approach to string phenomenology opens up the possibility of {\it concrete} explorations of a wide range of proposals both for dynamical supersymmetry breaking and for the dynamics of the dilaton and other stringy moduli. (Talk presented at DPF 94, Albuquerque, New Mexico) 
  We solve the Wheeler-DeWitt equation for {\it four}-dimensional Einstein gravity as an expansion in powers of the Planck mass by means of a heat kernel regularization. Our results suggest that in the universe with a very small radius or with a very large curvature beyond a Planck scale expectation values of operators are reduced to calculations in a path integral representation of {\it three}-dimensional Einstein gravity. 
  We diagonalize the transfer matrix of the inhomogeneous vertex models of the 6-vertex type in the anti-ferroelectric regime intoducing new types of q-vertex operators. The special cases of those models were used to diagonalize the s-d exchange model\cite{W,A,FW1}. New vertex operators are constructed from the level one vertex operators by the fusion procedure and have the description by bosons. In order to clarify the particle structure we estabish new isomorphisms of crystals. The results are very simple and figure out representation theoretically the ground state degenerations. 
  This paper explores a recent idea of Nambu to generate hierarchies among Yukawa couplings in the context of effective supergravity and superstrings models. The Yukawa couplings are homogeneous functions of the moduli and a geometrical constraint between them with a crucial role in the Nambu mechanism is found in a class of models of no-scale type. The Yukawas are dynamical variables at low energy to be determined by a minimization process. (Based on the talk given at ICHEP94, Glasgow, july 1994) 
  The representations of the degenerate affine Hecke algebra in which the analogues of the Dunkl operators are given by finite-difference operators are introduced. The non-selfadjoint lattice analogues of the spin Calogero-Sutherland hamiltonians are analysed by Bethe-Ansatz. The $ sl(m)$-Yangian representations arising from the finite-difference representations of the degenerate affine Hecke algebra are shown to be related to the Yangian representation of the 1-d Hubbard Model. 
  Interacting quantum fields on spacetimes containing regions of closed timelike curves (CTCs) are subject to a non-unitary evolution $X$. Recently, a prescription has been proposed, which restores unitarity of the evolution by modifying the inner product on the final Hilbert space. We give a rigorous description of this proposal and note an operational problem which arises when one considers the composition of two or more non-unitary evolutions. We propose an alternative method by which unitarity of the evolution may be regained, by extending $X$ to a unitary evolution on a larger (possibly indefinite) inner product space. The proposal removes the ambiguity noted by Jacobson in assigning expectation values to observables localised in regions spacelike separated from the CTC region. We comment on the physical significance of the possible indefiniteness of the inner product introduced in our proposal. 
  We summarize our recent results on the large N renormalization group (RG) approach to matrix models for discretized two-dimensional quantum gravity. We derive exact RG equations by solving the reparametrization identities, which reduce infinitely many induced interactions to a finite number of them. We find a nonlinear RG equation and an algorithm to obtain the fixed points and the scaling exponents. They reproduce the spectrum of relevant operators in the exact solution. The RG flow is visualized by the linear approximation. 
  The one-loop contribution to the entropy of a black hole from field modes near the horizon is computed in string theory. It is modular invariant and ultraviolet finite. There is an infrared divergence that signifies an instability near the event horizon of the black hole. It is due to the exponential growth of the density of states and the associated Hagedorn transition characteristic of string theory. It is argued that this divergence is indicative of a tree level contribution, and the Bekenstein-Hawking-Gibbons formula for the entropy should be understood in terms of string states stuck near the horizon. 
  The multimomentum Hamiltonian formalism is applied to field systems represented by sections of composite manifolds $Y\to\Si\to X$ where sections of $\Si\to X$ are parameter fields, e.g., Higgs fields and gravitational fields. Their values play the role of coordinate parameters, besides the world coordinates. 
  Reconsidering the harmonic space description of the self-dual Einstein equations, we streamline the proof that all self-dual pure gravitational fields allow a local description in terms of an unconstrained analytic prepotential in harmonic space. Our formulation yields a simple recipe for constructing self-dual metrics starting from any explicit choice of such prepotential; and we illustrate the procedure by producing a metric related to the Taub-NUT solution from the simplest monomial choice of prepotential. 
  Using the technique developed by Fronsdal and Galindo (Lett. Math. Phys. 27 (1993) 57) for studying the Hopf duality between the quantum algebras $Fun_{p,q}(GL(2))$ and $U_{p,q}(gl(2))$, the Hopf structure of $U_{p,q}(gl(1|1))$, dual to $Fun_{p,q}(GL(1|1))$, is derived and the corresponding universal ${\cal T}$-matrix of $Fun_{p,q}(GL(1|1))$, embodying the suitably modified exponential relationship $U_{p,q}(gl(1|1))$ $\rightarrow$ $Fun_{p,q}(GL(1|1))$, is obtained. 
  A theory of integration for anticommuting paths is described. This is combined with standard It\^o calculus to give a geometric theory of Brownian paths on curved supermanifolds. (Invited lecture given at meeting on `Espaces de Lacets', Institut de Recherche Math\'ematique Advanc\'ee, Universit\'e Louis Pasteur, Strasbourg, June 1994.) 
  The deformed algebra $\cal{A(R)}$, depending upon a Yang-Baxter R- matrix, is considered. The conditions under which the algebra is associative are discussed for a general number of oscillators. Four types of solutions satisfying these conditions are constructed and two of them can be represented by generalized Jordan-Wigner transformations.Our analysis is in some sense an extension of the boson realization of fermions from single-mode to multimode oscillators. 
  In two dimensional fluid, there are only two classes of swimming ways of micro-organisms, {\it i.e.}, ciliated and flagellated motions. Towards understanding of this fact, we analyze the swimming problem by using $w_{1+\infty}$ and/or $W_{1+\infty}$ algebras. In the study of the relationship between these two algebras, there appear the wave functions expressing the shape of micro-organisms. In order to construct the well-defined quantum mechanics based on $W_{1+\infty}$ algebra and the wave functions, essentially only two different kinds of the definitions are allowed on the hermitian conjugate and the inner products of the wave functions. These two definitions are related with the shapes of ciliates and flagellates. The formulation proposed in this paper using $W_{1+\infty}$ algebra and the wave functions is the quantum mechanics of the fluid dynamics where the stream function plays the role of the Hamiltonian. We also consider the area-preserving algebras which arise in the swimming problem of micro-organisms in the two dimensional fluid. These algebras are larger than the usual $w_{1+\infty}$ and $W_{1+\infty}$ algebras. We give a free field representation of this extended $W_{1+\infty}$ algebra. 
  Pairs $\aa \subset \bb$ of local quantum field theories are studied, where $\aa$ is a chiral conformal \qft and $\bb$ is a local extension, either chiral or two-dimensional. The local correlation functions of fields from $\bb$ have an expansion with respect to $\aa$ into \cfb s, which are non-local in general. Two methods of computing characteristic invariant ratios of structure constants in these expansions are compared: $(a)$ by constructing the monodromy \rep of the braid group in the space of solutions of the Knizhnik-Zamolodchikov differential equation, and $(b)$ by an analysis of the local subfactors associated with the extension with methods from operator algebra (Jones theory) and algebraic quantum field theory. Both approaches apply also to the reverse problem: the characterization and (in principle) classification of local extensions of a given theory. 
  We study the topological-antitopological fusion equations for supersymmetric sigma models on Grassmannian manifolds G(k,N). We find a basis in which the metric becomes diagonal and the $tt^*$ equations become tractable. The solution for the metric of G(k,N) can then be described in terms of the metric for the $CP^{N-1}$ models. The IR expansion helps clarify the picture of the vacua and gives the soliton numbers and masses. We also show that the $tt^*$ equation for G(k,N) in the large N limit is solvable, for any k. 
  We provide, with proofs, a complete description of the authors' construction of state-sum invariants announced in [CY], and its generalization to an arbitrary (artinian) semisimple tortile category. We also discuss the relationship of these invariants to generalizations of Broda's surgery invariants [Br1,Br2] using techniques developed in the case of the semi-simple sub-quotient of $Rep(U_q(sl_2))$ ($q$ a principal $4r^{th}$ root of unity) by Roberts [Ro1]. We briefly discuss the generalizations to invariants of 4-manifolds equipped with 2-dimensional (co)homology classes introduced by Yetter [Y6] and Roberts [Ro2], which are the subject of the sequel. (citations refer to bibliography in the paper) 
  An infinite number of distinct $d=1$ matrix models reproduce the perturbation theory of $d=2$ string theory. Due to constraints of causality, however, we argue that none of the existing constructions gives a consistent nonperturbative definition of the $d=2$ string. 
  The Lagrangian model for anyon, presented in [6], is extended to include interactions with external, homogeneous electromagnetic field. Explicit electric and magnetic moment terms for the anyon are introduced in the Lagrangian. The 2+1-dimensional BMT equation as well as the correct value (2) of the gyromagnetic ratio is rederived, in the Hamiltonian framework. 
  We study the excitation spectrum and the correlation functions of the Z_3- chiral Potts model in the massive high-temperature phase using perturbation expansions and numerical diagonalization. We are mainly interested in results for general chiral angles but we consider also the superintegrable case. For the parameter values considered, we find that the band structure of the low- lying part of the excitation spectrum has the form expected from a quasiparticle picture with two fundamental particles. Studying the N-dependence of the spectrum, we confirm the stability of the second fundamental particle in a limited range of the momentum, even when its energy becomes so high that it lies very high up among the multiparticle scattering states. This is not a phenomenon restricted to the superintegrable line. Calculating a non-translationally invariant correlation function, we give evidence that it is oscillating. Within our numerical accuracy we find a relation between the oscillation length and the dip position of the momentum dispersion of the lightest particle which seems to be quite independent of the chiral angles. 
  It is argued that the nonintegrably singular energy density of the electron's electromagnetic field (in both the classical point-charge model and quantum electrodynamics) must entail very strong self-gravitational effects, which, via black hole phenomena at finite radii, could well cut off the otherwise infinite electromagnetic contribution to the electron's mass. The general- relativistic equations for static, spherically symmetric stellar structure are specialized to treat the self-gravitational effects of static, spheri- cally symmetric, nonnegative, localized energy densities which may exhibit nonintegrable singularities at zero radius. It is demonstrated that in many situations, including the electromagnetic ones of interest here, such a system has a black hole whose Schwarzschild radius is that where the original energy per radial distance (the spherical shell area times the original energy density) reaches the inverse of (2G). The total mass of the system is that of this black hole (which follows in the usual way from its Schwarz- schild radius) plus the integrated original energy density outside this black hole. These results produce, for the classical point-charge model of the electron, an electrostatic contribution to its mass which is many orders of magnitude larger than its measured mass. For quantum electrodynamics, how- ever, the result is an electromagnetic mass contribution which is approxi- mately equal to its bare mass -- thus about half of its measured mass. 
  Reflection and braid equations for rank two $q$-tensors are derived from the covariance properties of quantum vectors by using the $R$-matrix formalism. 
  A test on the numerical accuracy of the semiclassical approximation as a function of the principal quantum number has been performed for the Pullen--Edmonds model, a two--dimensional, non--integrable, scaling invariant perturbation of the resonant harmonic oscillator. A perturbative interpretation is obtained of the recently observed phenomenon of the accuracy decrease on the approximation of individual energy levels at the increase of the principal quantum number. Moreover, the accuracy provided by the semiclassical approximation formula is on the average the same as that provided by quantum perturbation theory. 
  Pure Yang-Mills theory at high temperature is considered. We show that no distinct $Z_N$- phases separated by domain walls do exist in the physical Minkowski space. That means the absense of the spontaneous breaking of $Z_N$- symmetry in the physical meaning of this word. 
  We construct the U(N) spinning particle theories, which describe particles moving on Kahler spaces. These particles have the same relation to the N=2 string as usual spinning particles have to the NSR string. We find the restrictions on the target space of the theories coming from supersymmetry and from global anomalies. Finally, we show that the partition functions of the theories agree with what is expected from their spectra, unlike that of the N=2 string in which there is an anomalous dependence on the proper time. 
  The general one-dimensional ``log-sine'' gas is defined by restricting the positive and negative charges of a two-dimensional Coulomb gas to live on a circle. Depending on charge constraints, this problem is equivalent to different boundary field theories. We study the electrically neutral case, which is equivalent to a two-dimensional free boson with an impurity cosine potential. We use two different methods: a perturbative one based on Jack symmetric functions, and a non-perturbative one based on the thermodynamic Bethe ansatz and functional relations. The first method allows us to compute explicitly all coefficients in the virial expansion of the free energy and the experimentally-measurable conductance. Some results for correlation functions are also presented. The second method provides in particular a surprising fluctuation-dissipation relation between the free energy and the conductance. 
  In the low-energy limit, string theory has two remarkable symmetries, O(d,d+p) and SL(2,R). We illustrate the use of these transformations as techniques to generate new solutions by applying them to the Taub--NUT metric. 
  Yang-Baxterising a braid group representation associated with multideformed version of $GL_{q}(N)$ quantum group and taking the corresponding $q\rightarrow 1$ limit, we obtain a rational $R$-matrix which depends on $\left ( 1+ {N(N-1) \over 2} \right ) $ number of deformation parameters. By using such rational $R$-matrix subsequently we construct a multiparameter dependent extension of $Y(gl_N)$ Yangian algebra and find that this extended algebra leads to a modification of usual asymptotic condition on monodromy matrix $T(\lambda )$, at $ \lambda \rightarrow \infty $ limit. Moreover, it turns out that, there exists a nonlinear realisation of this extended algebra through the generators of original $Y(gl_N)$ algebra.  Such realisation interestingly provides a novel $\left ( 1 + { N(N-1) \over 2 } \right ) $ number of deformation parameter dependent coproduct for standard $Y(gl_N)$ algebra. 
  A large class of solvable models of dilaton gravity in two space-time dimensions, capable of describing black hole geometry, are analyzed in a unified way as non-linear sigma models possessing a special symmetry. This symmetry, which can be neatly formulated in the target-space-covariant manner, allows one to decompose the non-linearly interacting dilaton-gravity system into a free field and a field satisfying the Liouville equation with in general non-vanishing cosmological term. In this formulation, all the existent models are shown to fall into the category with vanishing cosmological constant. General analysis of the space-time structureinduced by a matter shock wave is performed and new models, with and without the cosmological term, are discussed. 
  Generalized quons interpolating between Bose, Fermi, para-Bose, para-Fermi, and anyonic statistics are proposed. They follow from the R-matrix approach to deformed associative algebras. It is proved that generalized quons have the same main properties as quons. A new result for the number operator is presented and some physical features of generalized quons are discussed in the limit $|q_{ij}^{2}| \rightarrow 1$. 
  We describe how the procedure of calculating approximate solitons from instanton holonomies may be extended to the case of soliton crystals. It is shown how sine-Gordon kink chains may be obtained from CP1 instantons on a torus. These kink chains turn out to be remarkably accurate approximations to the true solutions. Some remarks on the relevance of this work to Skyrme crystals are also made. 
  We describe how an approximation to the minimal energy Skyrme crystal can be obtained from the holonomy of a Yang-Mills instanton. The appropriate instanton is twisted on a four-torus and has instanton number equal to one half. It generates a Skyrme field with the correct topological and symmetry properties of the crystal. An explicit solution for the instanton is not known, but an analytical fit to numerical data is available and using this we obtain a Skyrme crystal whose energy is only 2 percent above that of the (numerically) known solution. 
  We quantize the self-dual massive theory by using the Batalin-Tyutin Hamiltonian method, which systematically embeds second class constraint system into first class one in the extended phase space by introducing the new fields. Through this analysis we obtain simultaneously the St\"uckelberg scalar term related to the explicit gauge-breaking effect and the new type of Wess-Zumino action related to the Chern-Simons term. 
  Moddings by cyclic permutation symmetries are performed in 4-dimensional strings, built up from N=2 coset models of the type $CP_m=SU(m+1)/SU(m)\times U(1)$. For some exemplifying cases, the massless chiral and antichiral states of $E_6$ are computed. The extent of the equivalence between different conformal invariant theories which possess equal chiral rings is analyzed. 
  The aim of the present letter is to critically review the stability of the Bartnik-McKinnon solutions of the Einstein-Yang-Mills theory. The stability question was already studied by several authors, but there seems to be some confusion about the nature and the number of unstable modes. We suggest to distinguish two different kind of instabilities, which we call `gravitational' respectively `sphaleron' instabilities. We claim that the $n^{\rm th}$ Bartnik-McKinnon solution has exactly $2n$ unstable modes, $n$ of either type. 
  We discuss a realization of the bosonic string as a noncritical $W_3$-string. The relevant noncritical $W_3$-string is characterized by a Liouville sector which is restricted to a (non-unitary) $(3,2)$ $W_3$ minimal model with central charge contribution $c_l = - 2$. Furthermore, the matter sector of this $W_3$-string contains $26$ free scalars which realize a critical bosonic string. The BRST operator for this $W_3$-string can be written as the sum of two, mutually anticommuting, nilpotent BRST operators: $Q = Q_0 + Q_1$ in such a way that the scalars which realize the bosonic string appear only in $Q_0$ while the central charge contribution of the fields present in $Q_1$ equals zero. We argue that, in the simplest case that the Liouville sector is given by the identity operator only, the $Q_1$-cohomology is given by a particular (non-unitary) $(3,2)$ Virasoro minimal model at $c=0$. 
  A new pseudoclassical model to describe Weyl particles is proposed. Different ways of its quantization are presented. They all lead to the theory of Weyl particle; namely, the massless Dirac equation and the Weyl condition are reproduced. In contrast with models discussed previously, this one admits both the Dirac quantization and quasicanonical quantization, with the fixation of almost all gauge freedom on the classical level. 
  We consider the interpretation in classical geometry of conformal field theories constructed from orbifolds with discrete torsion. In examples we can analyze, these spacetimes contain ``stringy regions'' that from a classical point of view are singularities that are to be neither resolved nor blown up. Some of these models also give particularly simple and clear examples of mirror symmetry. 
  The standard formalism of quantum theory is enhanced and definite meaning is given to the concepts of experiment, measurement and event. Within this approach one obtains a uniquely defined piecewise deterministic algorithm generating quantum jumps, classical events and histories of single quantum objects. The wave-function Monte Carlo method of Quantum Optics is generalized and promoted to the level of a fundamental process generating all the real events in Nature. The already worked out applications include SQUID-tank model and generalized cloud chamber model with GRW spontaneous localization as a particular case. Differences between the present approach and quantum measurement theories based on environment induced master equations are stressed. Questions: what is classical, what is time, and what are observers are addressed. Possible applications of the new approach are suggested, among them connection between the stochastic commutative geometry and Connes'noncommutative formulation of the Standard Model, as well as potential applications to the theory and practice of quantum computers. 
  This paper is a brief review of recent results on the concept of ``generalized $\tau$-function'', defined as a generating function of all the matrix elements in a given highest-weight representation of a universal enveloping algebra ${\cal G}$. Despite the differences from the particular case of conventional $\tau$-functions of integrable (KP and Toda lattice) hierarchies, these generic $\tau$-functions also satisfy bilinear Hirota-like equations, which can be deduced from manipulations with intertwining operators. The main example considered in details is the case of quantum groups, when such $\tau$-``functions'' are not $c$-numbers but take their values in non-commutative algebras (of functions on the quantum group $G$). The paper contains only illustrative calculations for the simplest case of the algebra SL(2) and its quantum counterpart $SL_q(2)$, as well as for the system of fundamental representations of SL(n). 
  We consider the large-N Calogero model in the \h\ collective-field approach based on the $1/N$ expansion. The Bogomol'nyi limit appears and the corresponding equation for the semiclassical configuration gives the correct ground-state energy. Using the method of the orthogonal polynomial we find the excitation spectrum of density fluctuations around the semiclassical solution for any value of the statistical parametar $\l$. The wave functions of the excited states are explicitly constructed as a product of Hermite polynomials in terms of the collective modes.The two-point correlation function is calculated as a series expansion in $1/\rho$ for any intermediate statistics. 
  We study algebras with a distributive law and the Koszulness of associated operads. As a corollary of our theory we prove that the homology operad of the little cubes operad is Koszul, the result which was originally proven by E. Getzler and J.D.S. Jones in their preprint "Operads, homotopy algebra, and iterated integrals for double loop spaces" using the combinatorics of the stratification of the Fulton-MacPherson compactification of the configuration space. Since the above mentioned result whose purely algebraic proof we give plays and important role in closed string field theory and especially the methods used in the original proof of Getzler and Jones are very relevant for this part of mathematical physics, we decided to put our paper here, though some 80% of its material belongs rather to universal and homological algebra. 
  We study the Hamilton formalism for Connes-Lott models, i.e., for Yang-Mills theory in non-commutative geometry. The starting point is an associative $*$-algebra $\cA$ which is of the form $\cA=C(I,\cAs)$ where $\cAs$ is itself a associative $*$-algebra. With an appropriate choice of a k-cycle over $\cA$ it is possible to identify the time-like part of the generalized differential algebra constructed out of $\cA$. We define the non-commutative analogue of integration on space-like surfaces via the Dixmier trace restricted to the representation of the space-like part $\cAs$ of the algebra. Due to this restriction it possible to define the Lagrange function resp. Hamilton function also for Minkowskian space-time. We identify the phase-space and give a definition of the Poisson bracket for Yang-Mills theory in non-commutative geometry. This general formalism is applied to a model on a two-point space and to a model on Minkowski space-time $\times$ two-point space. 
  We derive an $su(1,1)$ coherent state path integral formula for a system of two one-dimensional anyons in a harmonic potential. By a change of variables we transform this integral into a coherent states path integral for a harmonic oscillator with a shifted energy. The shift is the same as the one obtained for anyons by other methods. We justify the procedure by showing that the change of variables corresponds to a $su(1,1)$ version of the Holstein-Primakoff transformation. 
  Three illustrated lectures given by Stephen Hawking as part of a series of six lectures with Roger Penrose on the nature of space and time sponsored by Princeton University Press. 
  Preliminary results concerning non-quadratic (and non-bijective) transformations that exibit a degree of parentage with the well known Levi-Civita, Kustaanheimo-Stiefel, and Fock transformations are reported in this article. Some of the new transformations are applied to non-relativistic quantum dynamical systems in two dimensions. 
  A proposal for constructing a universal nonlinear ${\hat W}_{\infty}$ algebra is made as the symmetry algebra of a rotational Killing-symmetry reduction of the nonlinear perturbations of Moyal-Integrable deformations of $D=4$ Self Dual Gravity (IDSDG). This is attained upon the construction of a nonlinear bracket based on nonlinear gauge theories associated with infinite dimensional Lie algebras. A Quantization and supersymmetrization program can also be carried out. The relevance to the Kadomtsev-Petviashvili hierarchy, $2D$ dilaton gravity, quantum gravity and black hole physics is discussed in the concluding remarks. 
  A formulation of abelian and non-abelian chiral gauge theories is presented together with arguments for the unitarity and renormalisability in four dimensions. IASSNS-HEP-94/70, UM-P-94/96, and RCHEP-94/26. 
  In this paper, using a model of N=2 supergravity - vector multiplets interaction with the scalar field geometry $SU(1,m)/SU(m)\otimes U(1)$ as an example, we show that even when the geometry is fixed one can have a whole family of the Lagrangians that differ by the vector field duality transformations. As a byproduct, for this geometry we have constructed a model of (m-1) vector multiplets interacting with the hidden sector admitting spontaneous supersymmetry breaking with two arbitrary scales and without a cosmological term. 
  The Weinberg-Tucker-Hammer equations are shown to substitute the common-used $j=1$ massless equations. Meantime, the old equations preserve their significance as a particular case.   Possible consequences are discussed. 
  We study the possible stationary persistent supercurrents flowing on a cylindrical sample supporting a two-dimensional charged fluid. The internal dynamics of the fluid is obtained by means of an effective theory in which the fluid self-interacts through a $U(1)$ gauge field. We find that the presence of persistent supercurrents depends on what kind of gauge field it is. In particular the current is zero if it is a Maxwell gauge field, and it is maximal if it is a Chern-Simons gauge field. There is an intermediate behaviour in presence of both Maxwell and Chern-Simons term. Therefore it appears that persistent supercurrents are possible only if the fluid is chiral. 
  Some results about non-bijective quadratic transformations generalizing the Kustaanheimo-Stiefel and the Levi-Civita transformations are reviewed in \S 1. The three remaining sections are devoted to new results: \S 2 deals with the Lie algebras under constraints associated to some Hurwitz transformations; \S 3 and \S 4 are concerned with several applications of some Hurwitz transformations to wave equations for various potentials in $R^3$ and $R^5$. 
  We present a central extension of the $(m,n)$ super-Poincar\'e algebra in two dimensions. Besides the usual Poincar\'e generators and the $(m,n)$ supersymmetry generators we have $(m,n)$ Grassmann generators, a bosonic internal symmetry generator and a central charge. We then build up the topological gauge theory associated to this algebra. We can solve the classical field equations for the fields which do not belong to the supergravity multiplet and to a Lagrange multiplier multiplet. The resulting topological supergravity theory turns out to be non-local in the fermionic sector. 
  Pure gauge lattice QCD at arbitrary D is considered. Exact integration over link variables in an arbitrary D-volume leads naturally to an appearance of a set of surfaces filling the volume and gives an exact expression for functional of their boundaries. The interaction between each two surfaces is proportional to their common area and is realized by a non-local matrix differential operator acting on their boundaries. The surface self-interaction is given by the QCD$_2$ functional of boundary. Partition functions and observables (Wilson loop averages) are written as an averages over all configurations of an integer-valued field living on a surfaces. 
  In this paper, we examine the conditions under which a higher-spin string theory can be quantised. The quantisability is crucially dependent on the way in which the matter currents are realised at the classical level. In particular, we construct classical realisations for the $W_{2,s}$ algebra, which is generated by a primary spin-$s$ current in addition to the energy-momentum tensor, and discuss the quantisation for $s\le8$. From these examples we see that quantum BRST operators can exist even when there is no quantum generalisation of the classical $W_{2,s}$ algebra. Moreover, we find that there can be several inequivalent ways of quantising a given classical theory, leading to different BRST operators with inequivalent cohomologies. We discuss their relation to certain minimal models. We also consider the hierarchical embeddings of string theories proposed recently by Berkovits and Vafa, and show how the already-known $W$ strings provide examples of this phenomenon. Attempts to find higher-spin fermionic generalisations lead us to examine the whether classical BRST operators for $W_{2,{n\over 2}}$ ($n$ odd) algebras can exist. We find that even though such fermionic algebras close up to null fields, one cannot build nilpotent BRST operators, at least of the standard form. 
  In this paper we construct a $(2,2)$ dimensional string theory with manifest $N=1$ spacetime supersymmetry. We use Berkovits' approach of augmenting the spacetime supercoordinates by the conjugate momenta for the fermionic variables. The worldsheet symmetry algebra is a twisted and truncated ``small'' $N=4$ superconformal algebra. The physical spectrum of the open string contains an infinite number of massless states, including a supermultiplet of a self-dual Yang-Mills field and a right-handed spinor and a supermultiplet of an anti-self-dual Yang-Mills field and a left-handed spinor. The higher-spin multiplets are natural generalisations of these self-dual and anti-self-dual multiplets. 
  A definition is given, in the framework of stochastic quantization, for the dynamics of a system composed of classical and quantum degrees of freedom mutually interacting. It is found that the theory breaks reflection positivity, and hence it is unphysical. The Feynman rules for the Euclidean vacuum expectation values are derived and the perturbative renormalizability of the theory is analyzed. Contrary to the naive expectation, the semiquantized theory turns out to be less renormalizable, in general, than the corresponding completely quantized theory. 
  We review some recent results about exact classical solutions in string theory. In particular, we consider four dimensional extremal electric black holes which are related via dimensional reduction to the exact five dimensional fundamental string solutions. We also comment on the issue of \a' corrections to non-extremal black holes. (To appear in Proceedings of the Conference on Current Topics in Astrofundamental Physics, September 1994, Erice) 
  The conjugacy classes of the involutive automorphisms of the affine Kac-Moody algebras \Cli\ for $\ell\geq 2$ are determined using the matrix formulation of automorphisms of an affine Kac-Moody algebra. 
  The set of modular invariants that can be obtained from Galois transformations is investigated systematically for WZW models. It is shown that a large subset of Galois modular invariants coincides with simple current invariants. For algebras of type B and D infinite series of previously unknown exceptional automorphism invariants are found. 
  In this paper we discuss the connection on a space of $N=2$ TCFT's that appears in the context of background (in)dependence. We formulate a family of {\it target space field theories} with a similar connection on it. Each theory is a gauge theory (with the gauge group being ${\cal SD}iff $ in the case of $3$-fold). It describes deformations of K\"ahler structures much like Kodaira Spencer theory describes deformations of the complex structures. It is manifestly background independent. It appears to be a target space field theory for supersymmetric quantum mechanics. 
  The supersymmetric collective field theory with the potential $v'(x)=\omega x-{\eta\over x}$ is studied, motivated by the matrix model proposed by Jevicki and Yoneya to describe two dimensional string theory in a black hole background. Consistency with supersymmetry enforces a two band solution. A supersymmetric classical configuration is found, and interpreted in terms of the density of zeros of certain Laguerre polynomials. The spectrum of the model is then studied and is seen to correspond to a massless scalar and a majorana fermion. The $x$ space eigenfunctions are constructed and expressed in terms of Chebyshev polynomials. Higher order interactions are also discussed. 
  Recently it has been discovered that the W-algebras (orbifold of) WD_n can be defined even for negative integers n by an analytic continuation of their coupling constants. In this letter we shall argue that also the algebras WA_{-n-1} can be defined and are finitely generated. In addition, we show that a surprising connection exists between already known W-algebras, for example between the CP(k)-models and the U(1)-cosets of the generalized Polyakov-Bershadsky-algebras. 
  This talk deals with the old problem of formulatingn a covariant quantum theory of superstrings, ``covariant'' here meaning having manifest Lorentz symmetry and supersymmetry. The advantages and disadvantages of several quantization methods are reviewed. Special emphasis is put on the approaches using twistorial variables, and the algebraic structures of these. Some unsolved problems are identified. 
  An explicit expression for continuum annulus amplitudes having boundary lengths $\ell_{1}$ and $\ell_{2}$ is obtained from the two-matrix model for the case of the unitary series; $(p,q) = (m + 1, m)$. In the limit of vanishing cosmological constant, we find an integral representation of these amplitudes which is reproduced, for the cases of the $m = 2~(c=0)$ and the $m \rightarrow \infty~(c=1)$, by a continuum approach consisting of quantum mechanics of loops and a matter system integrated over the modular parameter of the annulus. We comment on a possible relation to the unconventional branch of the Liouville gravity. 
  We study the cohomology of the Schwinger term arising in second quantization of the class of observables belonging to the restricted general linear algebra. We prove that, for all pseudodifferential operators in 3+1 dimensions of this type, the Schwinger term is equivalent to the ``twisted'' Radul cocycle, a modified version of the Radul cocycle arising in non-commutative differential geometry. In the process we also show how the ordinary Radul cocycle for any pair of pseudodifferential operators in any dimension can be written as the phase space integral of the star commutator of their symbols projected to the appropriate asymptotic component. 
  In a Fock representation, a non-surjective Bogoliubov transformation of CAR leads to a reducible representation. For the case that the corresponding Bogoliubov operator has finite corank, the decomposition into irreducible subrepresentations is clarified. In particular, it turns out that the number of appearing subrepresentations is completely determined by the corank. 
  We investigate the dependence of the number and type of untwisted moduli on the boundary condition vectors of relistic free fermionic strings. The number of moduli is given by six minus the number of complex internal world--sheet fermions and the type of moduli is determined by the details of the world--sheet left--right asymmetry of the boundary conditions for the internal fermions. We give a geometrical description of our results in terms of the transformations of the compactified dimensions of $Z_2 \times Z_2$ orbifolds. We investigate all possible boundary conditions for the internal fermions and prove our results in general by showing that world--sheet supersymmetry eliminates those boundary conditions which violate our results. 
  It is proven that the "horizon condition", which was found to characterize the fundamental modular region in continuum theory and the thermodynamic limit of gauge theory on a periodic lattice, holds for every (transverse) configuration on a finite lattice with free boundary conditions. 
  We revise the sequences of SUSY for a cyclic adiabatic evolution governed by the supersymmetric quantum mechanical Hamiltonian. The condition (supersymmetric adiabatic evolution) under which the supersymmetric reductions of Berry (nondegenerated case) or Wilczek-Zee (degenerated case) phases of superpartners are taking place is pointed out. The analogue of Witten index (supersymmetric Berry index) is determined. As the examples of suggested concept of supersymmetric adiabatic evolution the Holomorphic quantum mechanics on complex plane and Meromorphic quantum mechanics on Riemann surface are considered. The supersymmetric Berry indexes for the models are calculated. 
  Some relations between different objects associated with quantum affine algebras are reviewed. It is shown that the Frenkel-Jing bosonization of a new realization of quantum affine algebra $\Uqa$ as well as bosonization of $L$-operators for this algebra can be obtained from Zamolodchikov-Faddeev algebras defined by the quantum $R$-matrix satisfying unitarity and crossing-symmetry conditions. (Talk given at the International Coference "Modern Problems of Quantum Field Theory, Quantum Gravity and Strings", Alushta, June 10--20, 1994.) 
  We define the notion of a minimal affinization of an irreducible representation of $U_q(g)$. We prove that minimal affinizations exist and establish their uniqueness in the rank 2 case. 
  We study CP in orbifold models. It is found that the orbifolds always have some automorphisms as CP symmetry. The symmetries are restricted non-trivially due to geometrical structure of the orbifolds. Explicit analysis on Yukawa couplings also shows that CP is not violated in orbifold models. 
  A story on how an attempt to realize W. Heizenberg idea that the neutrino might be a Goldstone particle had led in its development to the discovery of supergravity action. 
  The dissipative models in string theory can have more broad range of application: 1) Noncritical strings are dissipative systems in the "coupling constant" phase space. 2) Bosonic string in the affine-metric curved space is dissipative system.     But the quantum descriptions of the dissipative systems have well known ambiguities. In order to solve the problems of the quantum description of dissipative systems we suggest to introduce an operator W in addition to usual (associative) operators. The suggested operator algebra does not violate Heisenberg algebra because we extend the canonical commutation relations by introducing an operator W of the nonholonomic quantities in addition to the usual (associative) operators of usual (holonomic) coordinate -momentum functions. To satisfy the generalized commutation relations the operator W must be nonassociative nonLieble (does not satisfied the Jacobi identity) operator. As the result of these properties the total time derivative of the multiplication and commutator of the operators does not satisfies the Leibnitz rule. This lead to compatibility of quantum equations of motion for dissipative systems and canonical commutation relations. The suggested generalization of the von Neumann equation is connected with classical Liouville equation for dissipative systems. 
  The possible boundary conditions consistent with the integrability of the classical sine-Gordon equation are studied. A boundary value problem on the half-line $x\leq 0$ with local boundary condition at the origin is considered. The most general form of this boundary condition is found such that the problem be integrable. For the resulting system an infinite number of involutive integrals of motion exist. These integrals are calculated and one is identified as the Hamiltonian. The results found agree with some recent work of Ghoshal and Zamolodchikov. 
  The product of two free scalar fields on a manifold is shown to be a well defined operator valued distribution on the GNS Hilbert space of a globally Hadamard product state. Viewed as a new field all n-point distributions exist, giving a new example for a Wightman field on a manifold. 
  Quantum gravitational corrections to the effective potential, at one-loop level and in the leading-log approximation, for scalar quantum electrodynamics with higher-derivative gravity ---which is taken as an effective theory for quantum gravity (QG)--- are calculated. We point out the appearence of relevant phenomena caused by quantum gravity, like dimensional transmutation, QG-driven instabilities of the potential, QG corrrections to scalar-to-vector mass ratios, and curvature-induced phase transitions, whose existence is shown by means of analytical and numerical study. 
  An account is given of the structure and representations of chiral bosonic meromorphic conformal field theories (CFT's), and, in particular, the conditions under which such a CFT may be extended by a representation to form a new theory. This general approach is illustrated by considering the untwisted and $Z_2$-twisted theories, $H(\Lambda)$ and $\tilde H(\Lambda)$ respectively, which may be constructed from a suitable even Euclidean lattice $\Lambda$. Similarly, one may construct lattices $\Lambda_C$ and $\tilde\Lambda_C$ by analogous constructions from a doubly-even binary code $C$. In the case when $C$ is self-dual, the corresponding lattices are also. Similarly, $H(\Lambda)$ and $\tilde H(\Lambda)$ are self-dual if and only if $\Lambda$ is. We show that $H(\Lambda_C)$ has a natural ``triality'' structure, which induces an isomorphism $H(\tilde\Lambda_C)\equiv\tilde H(\Lambda_C)$ and also a triality structure on $\tilde H(\tilde\Lambda_C)$. For $C$ the Golay code, $\tilde\Lambda_C$ is the Leech lattice, and the triality on $\tilde H(\tilde\Lambda_C)$ is the symmetry which extends the natural action of (an extension of) Conway's group on this theory to the Monster, so setting triality and Frenkel, Lepowsky and Meurman's construction of the natural Monster module in a more general context. The results also serve to shed some light on the classification of self-dual CFT's. We find that of the 48 theories $H(\Lambda)$ and $\tilde H(\Lambda)$ with central charge 24 that there are 39 distinct ones, and further that all 9 coincidences are accounted for by the isomorphism detailed above, induced by the existence of a doubly-even self-dual binary code. 
  The definition of matter states on spacelike hypersurfaces of a 1+1 dimensional black hole spacetime is considered. Because of small quantum fluctuations in the mass of the black hole, the usual approximation of treating the gravitational field as a classical background on which matter is quantized, breaks down near the black hole horizon. On any hypersurface that captures both infalling matter near the horizon and Hawking radiation, a semiclassical calculation is inconsistent. An estimate of the size of correlations between the matter and gravity states shows that they are so strong that a fluctuation in the black hole mass of order exp[-M/M_{Planck}] produces a macroscopic change in the matter state. (Talk given at the 7th Marcel Grossmann Meeting on work in collaboration with E. Keski-Vakkuri, G. Lifschytz and S. Mathur.) 
  The solubility of a general two dimensional model, which reduces to various models in different limits, is studied within the path integral formalism. Various subtleties and interesting features are pointed out. 
  We consider the non-local energy-momentum tensor of quantum scalar and spinor fields in $2 w$-dimensional curved spaces. Working to lowest order in the curvature we show that, while the non-local terms proportional to $\Box {\cal R}$, $\Box \Box{\cal R}$, $\ldots, \Box^{w-2} {\cal R}$ are fully determined by the early-time behaviour of the heat kernel, the terms proportional to ${\cal R}$ depend on the asymptotic late-time behaviour. This fact explains a discrepancy between the running of the Newton constant dictated by the RG equations and the quantum corrections to the Newtonian potential. 
  We revisit the solvable lattice models described by Andrews Baxter and Forrester and their generalizations. The expressions for the local state probabilities were shown to be related to characters of the minimal models. We recompute these local state probabilities by a different method. This yields generalized Rogers Ramanujan identities, some of which recently conjectured by Kedem et al. Our method provides a proof for some cases, as well as generating new such identities. 
  We investigate Landau-Ginzburg string theory with the singular superpotential X^{-1} on arbitrary Riemann surfaces. This theory, which is a topological version of the c=1 string at the self-dual radius, is solved using results from intersection theory and from the analysis of matter Landau-Ginzburg systems, and consistency requirements. Higher-genus amplitudes decompose as a sum of contributions from the bulk and the boundary of moduli space. These amplitudes generate the W-infinity algebra. 
  We present a derivation of abelian and non-abelian bosonization in a path integral setting by expressing the generating functional for current-current correlation functions as a product of a $G/G$-coset model, which is dynamically trivial, and a bosonic part which contains the dynamics. A BRST symmetry can be identified which leads to smooth bosonization in both the abelian and non-abelian cases. 
  We continue our study of minimal affinizations for algebras of type D, E. 
  It is known that classical electromagnetic radiation at a frequency in resonance with energy splittings of atoms in a dielectric medium can be described using the classical sine-Gordon equation. In this paper we quantize the electromagnetic field and compute quantum corrections to the classical results by using known results from the sine-Gordon quantum field theory. 
  Early developments leading to renormalizable non-Abelian gauge theories for the weak, electromagnetic and strong interactions, are discussed from a personal viewpoint. They drastically improved our view of the role of field theory, symmetry and topology, as well as other branches of mathematics, in the world of elementary particles. 
  Associated to the standard $SU_{q}(n)$ R-matrices, we introduce quantum spheres $S_{q}^{2n-1}$, projective quantum spaces $CP_{q}^{n-1}$, and quantum Grassmann manifolds $G_{k}(C_{q}^{n})$. These algebras are shown to be homogeneous quantum spaces of standard quantum groups and are also quantum principle bundles in the sense of T Brzezinski and S. Majid (Comm. Math. Phys. 157,591 (1993)). 
  In two space-time dimensions a class of classical multicomponent scalar field theories with discrete, in general non-Abelian global symmetry is considered. The corresponding soliton solutions are given for the cases of 2, 3, and 4 components. 
  A four-dimensional dilaton-gravity action whose spherical reduction to two dimensions leads to the Jackiw-Teitelboim theory is presented. A nonsingular black hole solution of the theory is obtained and its physical interpretation is discussed. The classical and semiclassical properties of the solution and of its 2d counterpart are analysed. The 2d theory is also used to model the evaporation process of the near-extremal 4d black hole. We describe in detail the peculiarities of the black hole solutions, in particular the purely topological nature of the Hawking radiation, in the context of the Jackiw-Teitelboim theory. 
  The procedure for obtaining integrable vertex models via reflection matrices on the square lattice with open boundaries is reviewed and explicitly carried out for a number of two- and three-state vertex models. These models include the six-vertex model, the 15-vertex $A_2^{(1)}$ model and the 19-vertex models of Izergin-Korepin and Zamolodchikov-Fateev. In each case the eigenspectra is determined by application of either the algebraic or the analytic Bethe ansatz with inhomeogeneities. With suitable choices of reflection matrices, these vertex models can be associated with integrable loop models on the same lattice. In general, the required choices {\em do not} coincide with those which lead to quantum group-invariant spin chains. The exact solution of the integrable loop models -- including an $O(n)$ model on the square lattice with open boundaries -- is of relevance to the surface critical behaviour of two-dimensional polymers. 
  It is shown that the quantization of the unphysical degrees of freedom, which leads to the Mandelstam--Leibbrandt prescription for the infrared spurious singularities in the continuum light cone gauge, does indeed suggest some quite natural recipe to treat the zero modes in the Discretized Light Front Quantization of gauge theories. 
  A general discussion of the construction of free fields based on Weinberg anszatz is provided and various applications appearing in the litterature are considered. 
  The Abelian current algebra on the lattice is given from a series of the independent Weyl pairs and the shift operator is constructed by this algebra. So the realization of the operators of the braid group is obtained. For $|q|\neq 1$ the shift operator is the product of the theta functions of the generators $w_n$ of the current algebra. For $|q|=1$ it can be expressed by the quantum dilogarithm of $w_n$. 
  The Kaluza-Klein idea of extra spacetime dimensions continues to pervade current attempts to unify the fundamental forces, but in ways somewhat different from that originally envisaged. We present a modern perspective on the role of internal dimensions in physics, focussing in particular on superstring theory. A novel result is the interpretation of Kaluza-Klein string states as extreme black holes.(Talk delivered at the Oskar Klein Centenary Nobel Symposium, Stockholm, September 19-21, 1994.) 
  An important example of a multi-dimensional integrable system is the anti-self-dual Einstein equations. By studying the symmetries of these equations, a recursion operator is found and the associated hierarchy constructed. Owing to the properties of the recursion operator one may construct a hierarchy of symmetries and find the algebra generated by them. In addition, the Lax pair for this hierarchy is constructed. 
  A new Lax equation is introduced for the KP hierarchy which avoids the use of pseudo-differential operators, as used in the Sato approach. This Lax equation is closer to that used in the study of the dispersionless KP hierarchy, and is obtained by replacing the Poisson bracket with the Moyal bracket. The dispersionless limit, underwhich the Moyal bracket collapses to the Poisson bracket, is particularly simple. 
  In this paper we formulate a new N-state spin integrable model on a three-dimensional lattice with spins interacting round each elementary cube of the lattice. This model can be also reformulated as a vertex type model. Weight functions of the model satisfy tetrahedron equations. 
  We investigate the stability of the extremal black p-brane which contains a n-form and a dilaton. We show that the instability due to the s-mode, which was present in the uncharged and non-extremal p-brane, disappears in the extreme case. This is shown to be consistent with an entropy argument which shows that the zero entropy of the extremal black hole is approached more rapidly than the zero entropy of the black p-brane, which would mean an instability would violate the second law of thermodynamics. 
  The requirement of an $SL(2)$ duality symmetry, mixing the worldvolume field equations with Bianchi identities, leads to a highly nonlinear equation involving the transformation parameters and certain worldvolume currents. In general, this equation seems to admit a solution only for a two parameter subgroup of the seeked $SL(2)$. These transformations also leave invariant the first class constraints generating the worldvolume reparametrizations. In the special case of $p$--branes in $p+1$ dimensions, the full $SL(2)$ is realized. 
  This paper is devoted to the construction of a family of linear sigma models with $(0,4)$ supersymmetry which should flow in the infrared to the stringy version of Yang-Mills instantons on ${\bf R}^4$. The family depends on the full set of expected parameters and is obtained by using the data that appear in the ADHM construction of instantons. 
  We formulate the canonical structure of Yang--Mills theory in terms of Poisson brackets of gauge invariant observables analogous to Wilson loops. This algebra is non--trivial and tractable in a light--cone formulation. For U(N) gauge theories the result is a Lie algebra while for SU(N) gauge theories it is a quadratic algebra. We also study the identities satsfied by the gauge invariant observables. We suggest that the phase space of a Yang--Mills theory is a co--adjoint orbit of our Poisson algebra; some partial results in this direction are obtained. 
  Large $N$ two-dimensional QCD on a cylinder and on a vertex manifold (a sphere with three holes) is investigated. The relation between the saddle-point description and the collective field theory of QCD$_2$ is established. Using this relation, it is shown that the Douglas--Kazakov phase transition on a cylinder is associated with the presence of a gap in the eigenvalue distributions for Wilson loops. An exact formula for the phase transition on disc with an arbitrary boundary holonomy is found. The role of instantons in inducing such transitions is discussed. The zero-area limit of the partition function on a vertex manifold is studied. It is found that this partition function vanishes unless the boundary conditions satisfy a certain selection rule which is an analogue of momentum conservation in field theory. 
  In this paper differential operators on various moduli spaces (e.g. of holomorphic vector bundles) are described in a canonical way in terms of the geometry of a certain distinguished completion of an appropriate configuration space. 
  It is pointed out that massive states in D=4, N=1 supergravity-matter theories can, in general, at the 1-loop level contribute non-holomorphic terms to quadratic gravitational couplings. It is then shown in the context of $(2,2)$-symmetric $Z_N$-orbifold theories that, for constant moduli backgrounds, the inclusion of such contributions can result in the cancellation of naked $C^2$-terms. ${\cal R}^2$-terms can also arise but, being ghost free, need not cancel. 
  Without assuming rotational invariance, we derive Bogomol'nyi equations for the solitons in the abelian Chern-Simons gauge theories with the anomalous magnetic moment interaction. We also evaluate the number of zero modes around a static soliton configuration. 
  This paper is the first one in the series devoted to the calculation of particle mass spectrum in Topological GeometroDynamics. TGD Universe is critical at quantum level and an attractive idea to realize criticality is via conformal invariance. Ordinary real numbers do not allow this but if one assumes that in long length scales p-adic topology replaces real topology as effective topology situation changes. The existence of square root in the vicinity of p-adic real axis implies 4-dimensional algebraic extension of p-adic numbers ($p>2$), which can be regarded as padic counterpart of light cone and consists of convergence-cubes of p-adic square root function. Later work has demonstrated that convergence cubes of square root function serve as natural quantization volumes in p-adic field theory limit of TGD. 
  This paper belongs to the series devoted to the calculation of particle masses in p-adic conformal field theory limit of TGD. The concept of topological condensate generalizes the concept of 3-space. Various hierarchically ordered levels of the condensate obey effective p-adic topology and fractal considerations motivate the hypothesis that physically interesting values of p correspond to primes near prime powers of two, in particular Mersenne primes. This hypothesis relates succesfully the fundamental elementary par- ticle mass scales to Planck mass scale. The fundamental description of Higgs mechanism is in terms of p-adic thermodynamics for the Virasoro generator $L^0$ (mass squared). Massivation follows from the small thermal mixing of massless ground state with Planck mass excitations. The quantization of temperature at low temperature limit gives earlier length scale hypothesis as a prediction. In this paper the general theory of Higgs mechanism is described and calculation of elementary fermion and boson masses is left to the third paper of the series. 
  This paper belongs to the series devoted to the calculation of particle masses in the framework of p-adic conformal field theory limit of Topological GeometroDynamics. In paper II the general formulation of p-adic Higgs mechanism was given. In this paper the calculation of the fermionic and bosonic masses is carried out. The calculation of the masses necessitates the evaluation of dege- neracies for states as a function of conformal weight in certain tensor product of Super Virasoro algebras. The masses are very sen- sitive to the degeneracy ratios: Planck mass results unless the ratio for the degeneracies for first excited states and massless states is an integer multiple of 2/3. For leptons, quarks and gauge bosons this miracle occurs. The main deviation from standard model is the prediction of light color excited leptons and quarks as well as colored boson exotics. Higgs particle is absent from spectrum as is also graviton: the latter is due to the basic approximation of p-adic TGD. Reason for replacement: the recently identified light colored boson exotics making theory asymptotically free in standard sense. 
  This paper belongs to a series devoted to the calculation of particle masses in p-adic conformal field theory limit of TGD. In paper III elementary particle masses were calculated. In this paper the results are analyzed. One can reproduce lepton and gauge boson masses exactly by taking into account Coulomb self energy associated with the interior of 3-surface and a small mixing of boundary topologies of charged leptons. Hadronic mass formula in- volves boundary contributions of quarks calculated previously plus interior term consisting isospin-isopin, color magnetic spin-spin and color Coulombic terms. One must take also into account mixing of boundary topologies, the mixing for primary condensate levels (quark spends part of time at lower condensate level) and mixing of I=0,J=0 mesons. Topological mixing implies the nontriviality of CKM matrix. An open question is how uniquely rationality require- ment fixes CKM matrix. It is possible to reproduce CKM matrix sa- tisfying the empirical constraints. CP breaking is a number theo- retical necessity. The large mixing of u and c quarks solves the spin crisis of proton. The parameters associated with other O(p) contributions can be fixed by no Planck mass condition plus some empirical inputs. O(p^2) contributions are also discu- ssed and observed isospin splittings can be reproduced. 
  TGD suggests the existence of two new branches of physics, namely M_{89} hadron physics and M_{127} leptohadron physics. Leptons and U type quarks are predicted to have light colored excitations. The anomalous production of e^+e^- pairs in heavy ion collisions sup- ports the existence of light leptomeson but there are objections against light exotics. a) Asymptotic freedom in the standard sense might be lost: the recently identified exotic color bosons however save the situation. b) Z^0 decay width seems to exclude light exotics. The solution of the problem relies on p-adic probability concept. The real counterpart (\sum_jP(ito j))_R for the sum of p-adic probabi- lities differs from the sum \sum_j P(i toj)_R for the real counter- parts of p-adic probabilities. Interpretation:(\sum_jP(i to j))_R is used, when only a common signature for final states is used. \sum_j(P(i to j)_R is used, when each final state is monitored separately. The total decay rate of $Z^0$ to unmonitored exotic leptons is sensitive to the value of \theta_W(eff) and vanishes for sin^2(\theta_W(eff))=0.2324 and sensible value of \alpha_s(L)! In TGD the observed top quark candidate most probably corresponds to u and d quarks of M_{89} hadron physics. The details of the identification and signatures of the new Physics are discussed. 
  Dispersionless Hirota type equations are extracted from the dispersionless limit of the Fay differential identity due to Takasaki- Takebe. A few other results are sketched between inverse scattering, dKdV, and gravity. 
  A general expression for the conductivity in the QED$_{2+1}$ with nonzero fermion density in the uniform magnetic field is derived. It is shown that the conductivity is entirely determined by the Chern-Simons coefficient: $\sigma_{ij}=\varepsilon_{ij}~{\cal C}$ and is a step-function of the chemical potential and the magnetic field. 
  In these lectures I review classical aspects of the self-dual Chern-Simons systems which describe charged scalar fields in $2+1$ dimensions coupled to a gauge field whose dynamics is provided by a pure Chern-Simons Lagrangian. These self-dual models have one realization with nonrelativistic dynamics for the scalar fields, and another with relativistic dynamics for the scalars. In each model, the energy density may be minimized by a Bogomol'nyi bound which is saturated by solutions to a set of first-order self-duality equations. In the nonrelativistic case the self-dual potential is quartic, the system possesses a dynamical conformal symmetry, and the self-dual solutions are equivalent to the static zero energy solutions of the equations of motion. The nonrelativistic self-duality equations are integrable and all finite charge solutions may be found. In the relativistic case the self-dual potential is sixth order and the self-dual Lagrangian may be embedded in a model with an extended supersymmetry. The self-dual potential has a rich structure of degenerate classical minima, and the vacuum masses generated by the Chern-Simons Higgs mechanism reflect the self-dual nature of the potential. 
  The transfer matrix of the 6-vertex model of two-dimensional statistical physics commutes with many (more complicated) transfer matrices, but these latter, generally, do not commute between each other. The studying of their action in the eigenspaces of the 6-vertex model transfer matrix becomes possible due to a ``multiplicative property'' of the {\em vacuum curves} of $\cal L$-operators from which transfer matrices are built. This approach allowed, in particular, to discover for the first time the fact that the dimensions of abovementioned eigenspaces must be multiples of (big enough) degrees of the number 2. 
  We apply a simple mean field like variational calculation to compact QED in 2+1 dimensions. Our variational ansatz explicitly preserves compact gauge invariance of the theory. We reproduce in this framework all the known results, including dynamical mass generation, Polyakov scaling and the nonzero string tension. It is hoped that this simple example can be a useful reference point for applying similar approximation techniques to nonabelian gauge theories. 
  Thermodynamic relations for a class of 2D black holes are obtained corresponding to observations made from finite spatial distances. We also study the thermodynamics of the charged version of the Jackiw-Teitelboim black holes found recently by Lowe and Strominger. Our results corroborate, in appropriate limits, to those obtained previously by other methods. We also analyze the stability of these black holes thermodynamically. 
  We study the effect of S-duality and target-space duality tranformations of $N=4,d=4$ and $N=1,d=10$ supersymmetric configurations on their Killing spinors. We find that, under reasonable assumptions, the dual configurations are also supersymmetric and that the Killing spinors transform in a simple way. 
  We present the exact solution of the $Osp(1|2)$ invariant magnet by the Bethe ansatz approach. The associated Bethe ansatz equation exhibit a new feature by presenting an explicit and distinct phase behaviour in even and odd sectors of the theory. The ground state, the low-lying excitations and the critical properties are discussed by exploiting the Bethe ansatz solution. 
  We construct two $Osp(n|2m)$ solutions of the graded Yang-Baxter equation by using the algebraic braid-monoid approach. The factorizable S-matrix interpretation of these solutions is also discussed. 
  Self-dual gravity may be reformulated as the two dimensional principal chiral model with the group of area preserving diffeomorphisms as its gauge group. Using this formulation, it is shown that self-dual gravity contains an infinite dimensional hidden symmetry whose generators form the Affine (Kac-Moody) algebra associated with the Lie algebra of area preserving diffeomorphisms. This result provides an observable algebra and a solution generating technique for self-dual gravity. 
  We show that the four-dimensional extreme dilaton black hole with dilaton coupling constant $a= \sqrt{p/(p+2)}$ can be interpreted as a {\it completely non-singular}, non-dilatonic, black $p$-brane in $(4+p)$ dimensions provided that $p$ is {\it odd}. Similar results are obtained for multi-black holes and dilatonic extended objects in higher spacetime dimensions. The non-singular black $p$-brane solutions include the self-dual three brane of ten-dimensional N=2B supergravity and a multi-fivebrane solution of eleven-dimensional supergravity. In the case of a supersymmetric non-dilatonic $p$-brane solution of a supergravity theory, we show that it saturates a bound on the energy per unit $p$-volume. 
  We give some examples in which neglecting the interactions between particles or truncating the description of a black hole to the spherically symmetric mode leads to unphysical results. The restoration of the interactions and higher angular momentum modes resolves these problems. It is argued that mathematical consistency of the description of black holes in the Schwarzschild coordinate system requires that we neither truncate the theory nor ignore the interactions. We present two hypotheses on how matter must behave under large Lorentz boosts in order for black holes to be consistent with quantum mechanics. Finally, we argue that string theory exhibits these properties. Talk presented at the PASCOS meeting in Syracuse, New York, May 1994. 
  Dynamical evolution of the quantum ground state (vacuum) is analyzed for time variant harmonic oscillators characterized by asymptotically constant frequency. The oscillatory density matrix in the asymptotic future is uniquely determined by a constant number of produced particles, independent of other details of transient behavior at intermediate times. Time average over one oscillation period yields a classical, in some cases even an almost thermal behavior. In an analytically soluble model the created particle number obeys the Planck distribution in a parameter limit. This suggests a new way of understanding the Gibbons-Hawking temperature in the de Sitter spacetime. 
  The first-order phase transition of $O(3)$ symmetric model is considered in the limit of high temperature. It is shown that this model supports a new bubble solution where the global monopole is formed at the center of the buble in addition to the ordinary $O(3)$ bubble. Though the free energy of it is larger than that of normal bubble, the production rate can considerably large at high temperatures. 
  We describe the construction of vector valued modular forms transforming under a given congruence representation of the modular group SL(2,Z) in terms of theta series. We apply this general setup to obtain closed and easily computable formulas for conformal characters of rational models of W-algebras. 
  A model of two-dimensional quantum gravity that is the analog of the tensionless string is proposed. The gravitational constant ($k$) is the analog of the Regge slope ($\alpha^{'}$) and it shows that when $k \rightarrow \infty$, $2D$ quantum gravity can be understood as a tensionless string theory embeded in a two-dimensional target space. The temporal coordinate of the target space play the role of time and the wave function can be interpreted as in standard quantum mechanics. 
  A consistent quantization scheme for imaginary-mass field is proposed. It is related to an appriopriate choice of the synchronization procedure (definition of time), which guarantee an absolute causality. In that formulation a possible existence of field exctitations (tachyons) distinguish an inertial frame (tachyon privileged frame of reference) via spontaneous breaking of the so called synchronization group. 
  The problem of causality is analyzed in the context of Local Quantum Field Theory. Contrary to recent claims, it is shown that apparent noncausal behaviour is due to a lack of the notion of sharp localizability for a relativistic quantum system. (Replaced corrupted file) 
  We propose a decomposition of the S-matrix into individually gauge invariant sub-amplitudes, which are kinematically akin to propagators, vertices, boxes, etc. This decompsition is obtained by considering limits of the S-matrix when some or all of the external particles have masses larger than any other physical scale. We show at the one-loop level that the effective gluon self-energy so defined is physically equivalent to the corresponding gauge independent self-energy obtained in the framework of the pinch technique. The generalization of this procedure to arbitrary gluonic $n$-point functions is briefly discussed. 
  A general overview of the history and present status of inflationary cosmology is given. Recent developments discussed in this paper include the hybrid inflation scenario, the theory of reheating, and the theory of inflating tolopogical defects. We also discuss the theory of a stationary self-reproducing inflationary universe, and its possible implications for evaluating the most probable values of the gravitational and cosmological constants. Finally, we discuss some preliminary results concerning possible nonperturbative large-scale deviations of density from the critical density in inflationary universe. 
  A functional integral representation is given for a large class of quantum mechanical models with a non--L2 ground state. As a prototype the particle in a periodic potential is discussed: a unique ground state is shown to exist as a state on the Weyl algebra, and a functional measure (spectral stochastic process) is constructed on trajectories taking values in the spectrum of the maximal abelian subalgebra of the Weyl algebra isomorphic to the algebra of almost periodic functions. The thermodynamical limit of the finite volume functional integrals for such models is discussed, and the superselection sectors associated to an observable subalgebra of the Weyl algebra are described in terms of boundary conditions and/or topological terms in the finite volume measures. 
  Based on considerations in conformal gauge I derive up to nextleading order a relation between the coefficients of beta-functions in 2D renormalizable field theories before and after coupling to gravity. The result implies a coupling constant dependence of the ratio of both beta-functions beyond leading order. 
  Laymen and sometimes even physicists think of natural sciences, in particular of theoretical and mathematical physics often as subjects, which unfold according to an intrinsic logical pattern, with the limitations being set only by the conceptual and (in case of mathematical physics) mathematical developments of the times. This view certainly cannot be maintained in view of the present stagnation and crisis which in particular affects QFT, an area which in the past has been most innovating and fruitful, also in relation to other important areas of theoretical physics. 
  The vacuum correlations of the gravitational field are highly non-trivial to be defined and computed, as soon as their arguments and indices do not belong to a background but become dynamical quantities. Their knowledge is essential however in order to understand some physical properties of quantum gravity, like virtual excitations and the possibility of a continuum limit for lattice theory. In this review the most recent perturbative and non-perturbative advances in this field are presented. (To appear on Riv. Nuovo Cim.) 
  The large N phase transition point is investigated in the heat kernel on the $U(N)$ group with respect to arbitrary boundary conditions. A simple functional relation is found relating the density of eigenvalues of the boundary field to the saddle point shape of the typical Young tableaux in the large $N$ limit of the character expansion of the heat kernel. Both strong coupling and weak coupling phases are investigated for some particular cases of the boundary holonomy. 
  A possibility of strong coupling quantum Liouville gravity is investigated via infinite dimensional representations of $\qslc$ with $q$ at a root of unity. It is explicitly shown that vertex operator in this model can be written by a tensor product of a vertex operator of the classical Liouville theory and that of weak coupling quantum Liouville theory. Some discussions about the strong coupling Liouville gravity within this formulation are given. 
  The fusion rules and modular matrix of a rational conformal field theory obey a list of properties. We use these properties to classify rational conformal field theories with not more than six primary fields and small values of the fusion coefficients. We give a catalogue of fusion rings which can arise for these field theories. It is shown that all such fusion rules can be realized by current algebras. Our results support the conjecture that all rational conformal field theories are related to current algebras. 
  A rigorous definition of a path integral for a spinning particle in three dimensions is given on a regular cubic lattice. The critical diffusion constant and the associated critical exponents in each spin are calculated. Continuum field theories such as Klein-Gordon, Dirac and massive Chern-Simons theories are constructed near these critical points. The universality of obtained results is argued on some other lattices. 
  Knizhnik-Zamolodchikov-Bernard equations for twisted conformal blocks on compact Riemann surfaces with marked points are written explicitly in a general projective structure in terms of correlation functions in the theory of twisted b-c systems. It is checked that on the moduli space the equations provide a flat connection with the spectral parameter. 
  We discuss the Bogoliubov transformation of the scalar wave functions caused by the change of coordinates in 4 dimensional de Sitter space. It is shown that the exact Bogoliubov coefficients can be obtained from the global coordinates to the static coordinates where there exist manifest horizon. We consider two type of global coordinates. In one global coordinates, it is shown that the Bogoliubov transformation to the static coordinates can be expressed by the discontinuous integral of Weber and Schafheitlin. The positive and negative energy states in the global coordinates degenerate in the static coordinates. In the other global coordinates, we obtain the Bogoliubov coefficients by using the analytic continuation of the hypergeometric functions in two variables. 
  We discuss conserved currents and operator product expansions (OPE's) in the context of a $O(N)$ invariant conformal field theory. Using OPE's we find explicit expressions for the first few terms in suitable short-distance limits for various four-point functions involving the fundamental $N$-component scalar field $\phi^{\alpha}(x)$, $\alpha=1,2,..,N$. We propose an alternative evaluation of these four-point functions based on graphical expansions. Requiring consistency of the algebraic and graphical treatments of the four-point functions we obtain the values of the dynamical parameters in either a free theory of $N$ massless fields or a non-trivial conformally invariant $O(N)$ vector model in $2<d<4$, up to next-to-leading order in a $1/N$ expansion. Our approach suggests an interesting duality property of the critical $O(N)$ invariant theory. Also, solving our consistency relations we obtain the next-to-leading order in $1/N$ correction for $C_{T}$ which corresponds to the normalisation of the energy momentum tensor two-point function. 
  We investigate numerically the configurational statistics of strings. The algorithm models an ensemble of global $U(1)$ cosmic strings, or equivalently vortices in superfluid $^4$He. We use a new method which avoids the specification of boundary conditions on the lattice. We therefore do not have the artificial distinction between short and long string loops or a `second phase' in the string network statistics associated with strings winding around a toroidal lattice. Our lattice is also tetrahedral, which avoids ambiguities associated with the cubic lattices of previous work. We find that the percentage of infinite string is somewhat lower than on cubic lattices, 63\% instead of 80\%. We also investigate the Hagedorn transition, at which infinite strings percolate, controlling the string density by rendering one of the equilibrium states more probable. We measure the percolation threshold, the critical exponent associated with the divergence of a suitably defined susceptibility of the string loops, and that associated with the divergence of the correlation length. 
  The most usual procedure for deriving curvature corrections to effective actions for topological defects is subjected to a critical reappraisal. A logically unjustified step (leading to overdetermination) is identified and rectified, taking the standard domain wall case as an illustrative example. Using the appropriately corrected procedure, we obtain a new exact (analytic) expression for the corresponding effective action contribution of quadratic order in the wall width, in terms of the intrinsic Ricci scalar $R$ and the extrinsic curvature scalar $K$. The result is proportional to $cK^2-R$ with the coefficient given by $c\simeq 2$. The resulting form of the ensuing dynamical equations is obtained in terms of the second fundamental form and the Dalembertian of its trace, K. It is argued that this does not invalidate the physical conclusions obtained from the "zero rigidity" ansatz $c=0$ used in previous work. 
  We calculate the canonical partition function $Z_N$ for a system of $N$ free particles obeying so-called `quon' statistics where $q$ is real and satisfies $|q|<1$ by using simple counting arguments. We observe that this system is afflicted by the Gibbs paradox and that $Z_N$ is independent of $q$. We demonstrate that such a system of particles obeys the ideal gas law and that the internal energy $U$ ( and hence the specific heat capacity $C_V$ ) is identical to that of a system of $N$ free particles obeying Maxwell-Boltzmann statistics. 
  It is well known that the centerless W_{1+\infty} algebra provides a hamiltonian structure for the KP hierarchy. In this letter we address the question whether the centerful version plays a similar r\^ole in any related integrable system. We find that, surprisingly enough, the centrally extended W_{1+\infty} algebra yields yet another Poisson structure for the same standard KP hierarchy. This is proven by explicit construction of the infinitely many new hamiltonians in closed form. 
  We compute the partition function of an anyon-like harmonic oscillator. The well known results for both the bosonic and fermionic oscillators are then reobtained as particular cases as ours. The technique we employ is a non-relativistic version of the Green function method used in the computation of one-loop effective actions of quantum field theory. 
  From a geometric point of view, massless spinors in $3+1$ dimensions are composed of primary fields of weights $(\frac{1}{2},0)$ and $(0,\frac{1}{2})$, where the weights are defined with respect to diffeomorphisms of a sphere in momentum space. The Weyl equation thus appears as a consequence of the transformation behavior of local sections of half--canonical bundles under a change of charts. As a consequence, it is possible to impose covariant constraints on spinors of negative (positive) helicity in terms of (anti--)holomorphy conditions. Furthermore, the identification with half--differentials is employed to determine possible extensions of fermion propagators compatible with Lorentz covariance. This paper includes in particular the full derivation of the primary correlators needed in order to determine the fermion correlators. 
  We use the worldline path-integral approach to the Bern-Kosower formalism for developing a new algorithm for calculation of the sum of all diagrams with one spinor loop and fixed numbers of external and internal photons. The method is based on worldline supersymmetry, and on the construction of generalized worldline Green functions. The two-loop QED $\beta$ -- function is calculated as an example. 
  The 1-loop effective potential in a scalar theory with quartic interaction on the space $M^{4} \times T^{n}$ for $n=2$ is calculated and is shown to be unbounded from below. This is an indication of a possible instability of the vacuum of the $\lambda \phi^{4}$ theory on $M^{4}$, when it is regarded as a result of the dimensional reduction of the original six-dimensional model. The issue of stability for other values of the number $n$ of extra dimensions is also discussed. 
  We consider travelling periodical and quasiperiodical waves in single mode fibres, with weak birefringence and under the action of cross-phase modulation. The problem is reduced to the ``1:2:1" integrable case of the two-particle quartic potential. A general approach for finding elliptic solutions is given. New solutions which are associated with two-gap Treibich-Verdier potentials are found. General quasiperiodic solutions are given in terms of two dimensional theta functions with explicit expressions for frequencies in terms of theta constants. The reduction of quasiperiodic solutions to elliptic functions is discussed. 
  The Hamiltonian actions for extreme and non-extreme black holes are compared and contrasted and a simple derivation of the lack of entropy of extreme black holes is given. In the non-extreme case the wave function of the black hole depends on horizon degrees of freedom which give rise to the entropy. Those additional degrees of freedom are absent in the extreme case. 
  The S--duality transformations of the lowest order string effective theory admit a space time interpretation for 4-dim backgrounds with one Killing symmetry. Starting from pure gravity and performing a sequence of intertwined T-S-T duality transformations we obtain new solutions which are always pure gravitational. In this fashion, S-duality induces an $SL(2,R)$ transformation in the space of target space metrics which coincides with the action of the Ehlers-Geroch group and interchanges the electric with the magnetic aspects of gravity. Specializing to gravitational instanton backgrounds we show that ALE instantons are mapped to (multi) Taub-NUT backgrounds and vice-versa. We find, however, that the self-duality of the metric is not generically preserved, unless the corresponding Killing vector field has self-dual covariant derivatives. Thus, the T-S-T transformations are not always compatible with the world-sheet supersymmetry of $N=4$ superconformal string vacua. We also provide an algebraic characterization of the corresponding obstruction and associate it with a breakdown of space time supersymmetry under rotational T-duality transformations. 
  We consider 2-dimensional QCD on a cylinder, where space is a circle. We find the ground state of the system in case of massless quarks in a $1/N$ expansion. We find that coupling to fermions nontrivially modifies the large $N$ saddle point of the gauge theory due to the phenomenon of `decompactification' of eigenvalues of the gauge field. We calculate the vacuum energy and the vacuum expectation value of the Wilson loop operator both of which show a nontrivial dependence on the number of quarks flavours at the leading order in $1/N$. 
  We study the theory of representations of a multiparameter deformation of the function algebra of a simple algebraic group (as defined by Reshetikhin) when the quantum parameter is a root of unity. We extend the technics of De Concini-Lyubashenko in the standard quantum case. 
  The familiar generating functionals in QFT fail to be true measures since the Lebesgue measure in infinite-dimensional spaces is not defined in general. The problem lies in constructing representations of topological $^*$-algebras of quantum fields which are not normed. We restrict our consideration only to chronological forms on quantum field algebras. In this case, since chronological forms of boson fields are symmetric, the algebra of quantum fields can be replaced with the commutative tenzor algebra of the corresponding infinite-dimensional nuclear space $\Phi$. This is the enveloping algebra of the abelian Lie group of translations in $\Phi$. The generating functions of unitary representations of this group play the role of Euclidean generating functionals in algebraic QFT. They are the Fourier transforms of measures in the dual $\Phi'$ to the space $\Phi$. By analogy with the case of boson fields, the corresponding anticommutative algebra of fermion fields is defined to be the algebra of functions taking their values into an infinite Grassman algebra. 
  The purpose of this paper is to introduce the cohomology of various algebras over an operad of moduli spaces including the cohomology of conformal field theories (CFT's) and vertex operator algebras (VOA's). This cohomology theory produces a number of invariants of CFT's and VOA's, one of which is the space of their infinitesimal deformations. 
  It is well known that many interesting realisations of string theories can be obtained via hamiltonian reduction from WZW models. I want to point out that string theories do in certain cases also provide the recipe to reconstruct the ambient space of the hamiltonian reduction, including Kac--Moody currents and the associated ghosts. The procedure of reconstructing the Kac--Moody currents is closely related to properties of matter+gravity multiplets in noncritical string theories. In application to KPZ gravity and its N=1 supersymmetric extension, the `inverted hamiltonian reduction' constructions serve to establish relation with the DDK-type formalism for matter + gravity. 
  We give a simple combinatoric proof of an exponential upper bound on the number of distinct 3-manifolds that can be constructed by successively identifying nearest neighbour pairs of triangles in the boundary of a simplicial 3-ball and show that all closed simplicial manifolds that can be constructed in this manner are homeomorphic to $S^3$. We discuss the problem of proving that all 3-dimensional simplicial spheres can be obtained by this construction and give an example of a simplicial 3-ball whose boundary triangles can be identified pairwise such that no triangle is identified with any of its neighbours and the resulting 3-dimensional simplicial complex is a simply connected 3-manifold. 
  The cosmological term prevents perturbation based on derivative expansion in Einstein gravity. We consider quantum theory of gravitation invariant under volume-preserving diffeomorphism and Weyl transformation, which is suitable for derivative expansion. 
  The one-loop effective action for D-dimensional quantum gravity with negative cosmological constant, is investigated in space-times with compact hyperbolic spatial section. The explicit expansion of the effective action as a power series of the curvature on hyperbolic background is derived, making use of heat-kernel and zeta-regularization techniques. It is discussed, at one-loop level, the Coleman-Weinberg type suppression of the cosmological constant, proposed by Taylor and Veneziano. 
  \noindent{\large\bf Abstract.} We develop a general formalism to study the renormalization group (RG) improved effective potential for renormalizable gauge theories ---including matter-$R^2$-gravity--- in curved spacetime. The result is given up to quadratic terms in curvature, and one-loop effective potentials may be easiliy obtained from it. As an example, we consider scalar QED, where dimensional transmutation in curved space and the phase structure of the potential (in particular, curvature-induced phase trnasitions), are discussed. For scalar QED with higher-derivative quantum gravity (QG), we examine the influence of QG on dimensional transmutation and calculate QG corrections to the scalar-to-vector mass ratio. The phase structure of the RG-improved effective potential is also studied in this case, and the values of the induced Newton and cosmological coupling constants at the critical point are estimated. Stability of the running scalar coupling in the Yukawa theory with conformally invariant higher-derivative QG, and in the Standard Model with the same addition, is numerically analyzed. We show that, in these models, QG tends to make the scalar sector less unstable. 
  New realizations of finite W algebras are constructed by relaxing the usual constraint conditions. Then, finite W algebras are recognized in the Heisenberg quantization recently proposed by Leinaas and Myrheim, for a system of two identical particles in d dimensions. As the anyonic parameter is directly associated to the W-algebra involved in the d=1 case, it is natural to consider that the W-algebra framework is well-adapted for a possible generalization of the anyon statistics. 
  A Haag-Ruelle scattering theory for particles with braid group statistics is developed, and the arising structure of the Hilbert space of multiparticle states is analyzed. 
  We show that any BRST invariant quantum action with open or closed gauge algebra has a corresponding local background gauge invariance. If the BRST symmetry is anomalous, but the anomaly can be removed in the antifield formalism, then the effective action possesses a local background gauge invariance. The presence of antifields (BRST sources) is necessary. As an example we analyze chiral $W_3$ gravity. 
  A non-perturbative method based on the Form Factor bootstrap approach is proposed for the analysis of correlation functions of 2-D massless integrable theories and applied to the massless flow between the Tricritical and the Critical Ising Models. 
  The derivation of the $ q \bar q $ and the $ 3q $ potential for two dynamical quarks in a Wilson--loop context is reviewed. Some improvements are introduced. Only the usual assumptions in the evaluation of the Wilson loop integrals and expansions in the quark velocities are required for the result. It is shown that under the same assumptions it is possible to obtain the relativistic flux--tube lagrangian and a $ q \bar{q}$ Bethe--Salpeter equation with a confining kernel for spinless quarks. IFUM 482/FT 
  There has been considerable interest in the possible cosmic string configurations and Y. Yang has recently derived intriguing obstructions to the existence of cylindrically symmetric solutions in the Bogomol'nyi critical phase of the Einstein-Maxwell-Higgs system. In this comment, we would like to mention some earlier results and recent progress which were not reflected there but are important in connection with the obstructions he claimed. Some of his obstructions are not consistent with the ealier results. Also, we give a new condition stronger than those found before. 
  A model of random triangulations of a domain in $R^{(4)}$ is presented. The global symmetries of the model include SL(4) transformations and translations. If a stable microscopic scale exists for some range of parameters, the model should be in a translation invariant phase where SL(4) is spontaneously broken to O(4). In that phase, SL(4) Ward identities imply that the correlation length in the spin two channel of a symmetric tensor field is infinite. Consequently, it may be possible to identify the continuum limit of four dimensional Quantum Gravity with points inside that phase. 
  Using the full conformal bootstrap method an analytic expression is given in d-dimensions for the anomalous dimension of the fermion at O(1/N^3) in a large N expansion of the Nambu--Jona-Lasinio model with SU(2) x SU(2) continuous chiral symmetry. 
  We study the four-dimensional {\it N}=8 maximal supergravity in the context of Lie superalgebra SU(8/1). All possible successive superalgebraic truncations from four-dimensional {\it N}=8 theory to {\it N}=7, 6, $\cdots$, 1 supergravity theories are systematically realized as sub-superalgebra chains of SU(8/1) by using the Kac-Dynkin weight techniques. 
  In this letter a new gauge invariant, metric independent action is introduced from which Witten's Topological Quantum Field Theory may be obtained after gauge fixing using standard BRST techniques. In our model the BRST algebra of transformations, under which the effective action is invariant, close off-shell in distintion with what occurs in the one proposed by Labastida and Pernici. Our approach provides the geometrical principle for the quantum theory. We also compare our results with an alternative formulation presented by Baulieu and Singer. 
  This is a brief review where some basic elements of non-commutative geometry are given. The rules and ingredients that enter in the construction of the standard model and grand unification models in non-commutative geometry are summarized. A connection between some space-time supersymmetric models and non-commutative geometry is made. The advantages and problems of this direction are discussed. 
  In this talk methods for a rigorous control of the renormalization group (RG) flow of field theories are discussed. The RG equations involve the flow of an infinite number of local partition functions. By the method of exact beta-function the RG equations are reduced to flow equations of a finite number of coupling constants. Generating functions of Greens functions are expressed by polymer activities. Polymer activities are useful for solving the large volume and large field problem in field theory. The RG flow of the polymer activities is studied by the introduction of polymer algebras. The definition of products and recursive functions replaces cluster expansion techniques. Norms of these products and recursive functions are basic tools and simplify a RG analysis for field theories. The methods will be discussed at examples of the $\Phi^4$-model, the $O(N)$ $\sigma$-model and hierarchical scalar field theory (infrared fixed points). 
  The renormalized fermionic determinant of QED in 3 + 1 dimensions, $\mbox{det}_{{ren}}$, in a static, unidirectional, inhomogeneous magnetic field with finite flux can be calculated from the massive Euclidean Schwinger model's determinant, $\mbox{det}_{{Sch}}$, in the same field by integrating $\mbox{det}_{{Sch}}$, over the fermion's mass. Since $\mbox{det}_{{ren}}$ for general fields is central to QED, it is desirable to have nonperturbative information on this determinant, even for the restricted magnetic fields considered here. To this end we continue our study of the physically relevant determinant $\mbox{det}_{{Sch}}$. It is shown that the contribution of the massless Schwinger model to $\mbox{det}_{{Sch}}$ is cancelled by a contribution from the massive sector of QED in 1 + 1 dimensions and that zero modes are suppressed in $\mbox{det}_{{Sch}}$. We then calculate $\mbox{det}_{{Sch}}$ analytically in the presence of a finite flux, cylindrical magnetic field. Its behaviour for large flux and small fermion mass suggests that the zero-energy bound states of the two-dimensional Pauli Hamiltonian are the controlling factor in the growth of $\ln \mbox{det}_{{Sch}}$. Evidence is presented that $\mbox{det}_{{Sch}}$ does not converge to the determinant of the massless Schwinger model in the small mass limit for finite, nonzero flux magnetic fields. 
  This talk summarizes our recent work establishing an algebraic, model-independent basis for the existence of \B bounds and \B equations for topologically non-trivial solitons and instantons. Our arguments use supersymmetry in an essential way to understand both supersymmetric and non-supersymmetric theories. Our arguments are constructive and work in nearly any number of dimensions. Presented at and to appear in the proceedings of the XX^{th} International Colloquium on Group Theoretical Methods in Physics, Osaka, July 1994. 
  Polynomial relations between the generators of $q$--deformed Heisenberg algebra  invariant under the quantization and $q$-deformation are discovered. One of the examples of such relations is the following:  if two elements $a$ and $b$, obeying the relation \[ ab - q ba  = p, \] where $p, q$ are any complex numbers, then for any $p,q$ and natural $n$ \[ (aba)^n = a^n b^n a^n \] 
  In this paper we consider non-relativistic quantum mechanics on a space with an additional internal compact dimension, i.e. $R^3\otimes S^1$ instead of $R^3$. More specifically we study potential scattering for this case and the analyticity properties of the forward scattering amplitude, $T_{nn}(K)$, where $K^2$ is the total energy and the integer n denotes the internal excitation of the incoming particle. The surprising result is that the analyticity properties which are true in $R^3$ do not hold in $R^3\otimes S^1$.   For example, $T_{nn}(K)$, is \underline{not} analytic in K for $ImK>0$, for n such that $(|n|/R)>\mu$, where R is the radius of $S^1$, and $\mu^{-1}$ is the exponential range of the potential, $V(r,\phi)=O(e^{-\mu r})$ for large r. We show by explicit counterexample that $T_{nn}(K)$ for $n\neq0$, can have singularities on the physical energy sheet. We also discuss the motivation for our work, and briefly the lesson it teaches us. 
  A new integration technique for multi-loop Feynman integrals, called the matrix method, is developed and then applied to the divergent part of the overlapping two-loop quark self-energy function $\,i\Sigma\,$ in the light- cone gauge. It is shown that the coefficient of the double-pole term is strictly local, even off mass-shell, while the coefficient of the single-pole term contains local as well as nonlocal parts. On mass-shell, the single-pole part is local, of course. It is worth noting that the original overlapping self-energy integral reduces eventually to 10 covariant and 38 noncovariant- gauge integrals. We were able to verify explicitly that the divergent parts of the 10 double covariant-gauge integrals agreed precisely with those currently used to calculate radiative corrections in the Standard Model.   Our new technique is amazingly powerful, being applicable to massive and massless integrals alike, and capable of handling both covariant-gauge integrals and the more difficult noncovariant-gauge integrals. Perhaps the most important feature of the matrix method is the ability to execute the $4\omega$-dimensional momentum integrations in a single operation, exactly and in analytic form. The method works equally well for other axial-type gauges, notably the temporal gauge ($n^2>0$) and the pure axial gauge ($n^2<0$). 
  The assumption that the Weinberg rotation between the gauge fields associated with the third component of the ``weak isospin" ($T_3$) and the hypercharge ($Y$) proceeds in a natural way from a global homomorphism of the $SU(2)\otimes U(1)$ gauge group in some locally isomorphic group (which proves to be $U(2)$), imposes strong restrictions so as to fix the single value $\sin^2\theta_W=1/2$. This result can be thought of only as being an asymptotic limit corresponding to an earlier stage of the Universe. It also lends support to the idea that $e^2/g^2$ and $1-M_W^2/M_Z^2$ are in principle unrelated quantities. 
  We perform string quantization in anti de Sitter (AdS) spacetime. The string motion is stable, oscillatory in time with real frequencies $\omega_n= \sqrt{n^2+m^2\alpha'^2H^2}$ and the string size and energy are bounded. The string fluctuations around the center of mass are well behaved. We find the mass formula which is also well behaved in all regimes. There is an {\it infinite} number of states with arbitrarily high mass in AdS (in de Sitter (dS) there is a {\it finite} number of states only). The critical dimension at which the graviton appears is $D=25,$ as in de Sitter space. A cosmological constant $\Lambda\neq 0$ (whatever its sign) introduces a {\it fine structure} effect (splitting of levels) in the mass spectrum at all states beyond the graviton. The high mass spectrum changes drastically with respect to flat Minkowski spacetime. For $\Lambda<0,$ we find $<m^2>\sim \mid\Lambda\mid N^2,$ {\it independent} of $\alpha',$ and the level spacing {\it grows} with the eigenvalue of the number operator, $N.$ The density of states $\rho(m)$ grows like $\mbox{Exp}[(m/\sqrt{\mid\Lambda\mid}\;)^{1/2}]$ (instead of $\rho(m)\sim\mbox{Exp}[m\sqrt{\alpha'}]$ as in Minkowski space), thus {\it discarding} the existence of a critical string temperature.   For the sake of completeness, we also study the quantum strings in the black string background, where strings behave, in many respects, as in the ordinary black hole backgrounds. The mass spectrum is equal to the mass spectrum in flat Minkowski space. 
  Using the superspace formalism, we compute for the two-dimensional N=1 supersymmetric non-linear $\sigma$-model, the order $(\alpha^{\prime})^{2}$ $(R_{mnpq})^2$ (three-loop) correction to the central charge via the operator product expansion of the supercurrent with itself. The contribution vanishes, in agreement with previous results obtained from the usual $\sigma$-model $\beta$-function approach. 
  It is shown that the Topological Massive and ``Self-dual'' theories, which are known to provide locally equivalent descriptions of spin 1 theories in 2+1 dimensions, have different global properties when formulated over topologically non-trivial regions of space-time. The partition function of these theories, when constructed on an arbitrary Riemannian manifold, differ by a topological factor, which is equal to the partition function of the pure Chern-Simons theory. This factor is related to the space of solutions of the field equations of the Topological Massive Theory for which the connection is asymptotically flat but not gauge equivalent to zero. A new covariant, first order, gauge action,which generalize the ``Self-dual'' action, is then proposed. It is obtained by sewing local self-dual theories. Its global equivalence to the Topological Massive gauge theory is shown. 
  We show that $h$-deformation can be obtained, by a singular limit of a similarity transformation, from $q$-deformation; to be specefic, we obtain $\GL_h(2)$, its differential structure, its inhomogenous extension, and $\Uh{\sl(2)}$ from their $q$-deformed counterparts. 
  We present a class of exact solutions for homogeneous, anisotropic cosmologies in four dimensions derived from the low-energy string effective action including a homogeneous dilaton $\phi$ and antisymmetric tensor potential $B_{\mu\nu}$. Making this potential time-dependent produces an anisotropic energy-momentum tensor, and leads us to consider a Bianchi I cosmology. The solution for the axion field must then only be a linear function of one spatial coordinate. This in turn places an upper bound on the product of the two scale factors evolving perpendicular to the gradient of the axion field. The only late-time isotropic solution is then a {\em contracting} universe. 
  Lattice gauge theory with gauge group $Z_{P}$ is reconsidered in four dimensions on a simplicial complex $K$. One finds that the dual theory, formulated on the dual block complex $\hat{K}$, contains topological modes which are in correspondence with the cohomology group $H^{2}(\hat{K},Z_{P})$, in addition to the usual dynamical link variables. This is a general phenomenon in all models with single plaquette based actions; the action of the dual theory becomes twisted with a field representing the above cohomology class. A similar observation is made about the dual version of the three dimensional Ising model. The importance of distinct topological sectors is confirmed numerically in the two dimensional Ising model where they are parameterized by $H^{1}(\hat{K},Z_{2})$. 
  In the exact renormalization group (RG) flow in the infrared cutoff $\Lambda$ one needs boundary conditions. In a previous paper on $SU(2)$ Yang-Mills theory we proposed to use the nine physical relevant couplings of the effective action as boundary conditions at the physical point $\Lambda=0$ (these couplings are defined at some non-vanishing subtraction point $\mu \ne 0$). In this paper we show perturbatively that it is possible to appropriately fix these couplings in such a way that the full set of Slavnov-Taylor (ST) identities are satisfied. Three couplings are given by the vector and ghost wave function normalization and the three vector coupling at the subtraction point; three of the remaining six are vanishing (\eg the vector mass) and the others are expressed by irrelevant vertices evaluated at the subtraction point. We follow the method used by Becchi to prove ST identities in the RG framework. There the boundary conditions are given at a non-physical point $\Lambda=\Lambda' \ne 0$, so that one avoids the need of a non-vanishing subtraction point. 
  { This letter discusses the BRST cohomology of superparticles type I and II. It was used an extended super-space to construct $S0(9,1)$ superparticle actions that lead to super-wave functions whose spinor components satisfy $S0(9,1)$ covariant constraints. Their BRST charges were found by using BV methods, since the models present a large number of symmetries and only close on-shell. It is shown that the zero ghost-number cohomology class of both models reproduce the same spectrum as that of N=1 ten dimensional super-Yang-Mills theory. } 
  The scattering of charged solitons in the complex sine-Gordon field theory is investigated. An exact factorizable S-matrix for the theory is proposed when the renormalized coupling constant takes the values $\lambda^{2}_{R}=4\pi/k$ for any integer $k>1$: the minimal S-matrix associated with the Lie algebra $a_{k-1}$. It is shown that the proposed S-matrix reproduces the leading semiclassical behaviour of all amplitudes in the theory and is the minimal S-matrix which is consistent with the semiclassical spectrum of the model. The results are completely consistent with the description of the complex sine-Gordon theory as the SU$(2)/{\rm U}(1)$ coset model at level $k$ perturbed by its first thermal operator. 
  Direct verification of the existence of an infinite set of multicritical non-perturbative FPs (Fixed Points) for a single scalar field in two dimensions, is in practice well outside the capabilities of the present standard approximate non-perturbative methods. We apply a derivative expansion of the exact RG (Renormalization Group) equations in a form which allows the corresponding FP equations to appear as non-linear eigenvalue equations for the anomalous scaling dimension $\eta$. At zeroth order, only continuum limits based on critical sine-Gordon models, are accessible. At second order in derivatives, we perform a general search over all $\eta\ge.02$, finding the expected first ten FPs, and {\sl only} these. For each of these we verify the correct relevant qualitative behaviour, and compute critical exponents, and the dimensions of up to the first ten lowest dimension operators. Depending on the quantity, our lowest order approximate description agrees with CFT (Conformal Field Theory) with an accuracy between 0.2\% and 33\%; this requires however that certain irrelevant operators that are total derivatives in the CFT are associated with ones that are not total derivatives in the scalar field theory. 
  We reconsider here the problem of finding the general 4D spherically symmetric, asymptotically flat and time-independent solutions to the lowest-order string equations in the $\ap$ expansion. Our construction includes earlier work, but differs from it in three ways. (1) We work with general background metric, dilaton, axion and $U(1)$ gauge fields. (2) Much of the original solutions were required to be nonsingular at the apparent horizon, motivated by an interest in finding string corrections to black hole spacetimes. We relax this condition throughout, motivated by the realization that string theory has a less restrictive notion of what constitutes a singular field configuration than do point particle theories. (3) We can construct the general solution from a particularly simple one, by generating it from successive applications of the {\it noncommuting} \sltwor\ and \ooneone\ symmetries of the low-energy string equations containing $S$ and target--space dualities respectively. This allows its construction using relatively simple, purely algebraic, techniques. The general solution is determined by the asymptotic behaviour of the various fields: \ie\ by the mass, dilaton charge, axion charge, electric charge, magnetic charge, and Taub-NUT parameter. 
  We present a new formulation of the tensionless string ($T= 0$) where the space-time conformal symmetry is manifest. Using a Hamiltonian BRST scheme we quantize this {\em Conformal String} and find that it has critical dimension $D=2$. This is in keeping with our classical result that the model describes massless particles in this dimension. It is also consistent with our previous results which indicate that quantized conformally symmetric tensionless strings describe a topological phase away {}from $D=2$. We reach our result by demanding nilpotency of the BRST charge and consistency with the Jacobi identities. The derivation is presented in two different ways: in operator language and using mode expansions. Careful attention is payed to regularization, a crucial ingredient in our calculations. 
  We consider the Quantum Inverse Scattering Method with a new R-matrix depending on two parameters $q$ and $t$. We find that the underlying algebraic structure is the two-parameter deformed algebra $SU_{q,t}(2)$ enlarged by introducing an element belonging to the centre. The corresponding Hamiltonian describes the spin-1/2 XXZ model with twisted periodic boundary conditions. 
  We consider the algebra $R$ generated by three elements $A,B,H$ subject to three relations $[H,A]=A$, $[H,B]=-B$ and $\{A,B\}=f(H)$. When $f(H)=H$ this coincides with the Lie superalgebra $osp(1/2)$; when $f$ is a polynomial we speak of polynomial deformations of $osp(1/2)$. Irreducible representations of $R$ are described, and in the case $\deg(f)\leq 2$ we obtain a complete classification, showing some similarities but also some interesting differences with the usual $osp(1/2)$ representations. The relation with deformed oscillator algebras is discussed, leading to the interpretation of $R$ as a generalized paraboson algebra. 
  We construct a class of representations of the quadratic R-matrix algebra, given by the reflection equation with the spectral parameter, in terms of certain ordinary difference operators. These operators turn out to act as parameter shifting operators on the 3_F_2(1) hypergeometric function and its limit cases and on classical orthogonal polynomials. The relationship with the factorization method will be discussed. 
  There exist many four dimensional integrable theories. They include self-dual gauge and gravity theories, all their extended supersymmetric generalisations, as well the full (non-self-dual) N=3 super Yang-Mills equations. We review the harmonic space formulation of the twistor transform for these theories which yields a method of producing explicit connections and metrics. This formulation uses the concept of harmonic space analyticity which is closely related to that of quaternionic analyticity. (Talk by V. Ogievetsky at the G\"ursey Memorial Conference I, Istanbul, June 1994) 
  (Talk given at the Oskar Klein centenary symposium 19-21 September 1994 in Stockholm, Sweden, to appear in the proceedings.) The first half of this talk is a non-technical discussion of some general aspects of string theory, in particular the problem of compactification. We also give an introduction to mirror symmetry. The second half is a brief account of two recent papers on this subject; one by the author on mirror symmetry for Kazama-Suzuki models and one by P. Berglund and the author on a search for possible mirror pairs of Landau-Ginzburg orbifolds. 
  We demonstrate the existence of three off-shell distinct N = 4 superstrings at the level of manifest locally supersymmetric Lagrangian field theories on the world sheet. 
  By utilizing a new procedure (the RADIO method) for deriving on-shell 2D, 2N-extended multiplets from off-shell 2D, N-extended multiplets, we derive a new on-shell 2D, N = 8 representation; the ultra-multiplet. By twisting with respect to parity, we show that many variant versions of this supermultiplet also exist. 
  25th anniversary and new building dedication Centre de Recherches Math\'{e}matiques Montr\'{e}al, Canada, October 1994 
  In theories of closed oriented superstrings, the one loop amplitude is given by a single diagram, with the topology of a torus. Its interpretation had remained obscure, because it was formally real, converged only for purely imaginary values of the Mandelstam variables, and had to account for the singularities of both the box graph and the one particle reducible graphs in field theories. We present in detail an analytic continuation method which resolves all these difficulties. It is based on a reduction to certain minimal amplitudes which can themselves be expressed in terms of double and single dispersion relations, with explicit spectral densities. The minimal amplitudes correspond formally to an infinite superposition of box graphs on $\phi ^3$ like field theories, whose divergence is responsible for the poles in the string amplitudes. This paper is a considerable simplification and generalization of our earlier proposal published in Phys. Rev. Lett. 70 (1993) p 3692. 
  Franson showed that Aspect's experiment to test Bell's inequality did not rule out local realistic theories with delayed determinism. A class of local, deterministic discrete mathematical models with delayed determinism is described that may be consistent with existing experiments. These are not hidden variables theories in the sense that they are not theories of particles plus hidden variables. They are theories of `hidden' distributed information stored holographic like throughout a space time region. This information cannot be uniquely associated with individual particles although it determines the results observed in particle interactions. The classical parameters of an interaction are determined as {\it focal points} of continuous nonlinear changes in the wave function and not as discrete events. In addition to not violating Bell's inequality this class of theories can in principle be distinguished from standard quantum mechanics by other experiments. These differences and the experimental constraints on a test of Bell's inequality to discriminate between the existing theory and this class of models are discussed. 
  The two-dimensional non-linear sigma model approach to Self-dual Yang-Mills theory and to Self-dual gravity given by Q-Han Park is an example of the deep interplay between two and four dimensional physics. In particular, Husain's two-dimensional chiral model approach to Self-dual gravity is studied. We show that the infinite hierarchy of conservation laws associated to the Husain model carries implicitly a hidden infinite Hopf algebra structure. 
  By analytically continuing the time variable in a black hole background, and requiring unitary evolution, it is found that quantum mechanical states at the horizon develop a thermal factor under suitable identification of the physical time. The thermal factor is found to be determined exactly by the Hawking's temperature. This can be interpreted as the Hawking's radiation and offers an alternative understanding to the process and the information loss puzzle. 
  We examine the conformal property of the second Hamiltonian structure of constrained KP hierarchy derived by Oevel and Strampp. We find that it naturallygives a family of nonlocal extended conformal algebras. We give two examples of such algebras and find that they are similar to Bilal's V algebra. By taking a gauge transformation one can map the constrained KP hierarchy to Kuperschmidt's nonstandard Lax hierarchy. We consider the second Hamiltonian structure in this representation. We show that after mapping the Lax operator to a pure differential operator the second structure becomes the sum of the second and the third Gelfand-Dickey brackets defined by this differential operator. We show that this Hamiltonian structure defines the W-U(1)-Kac-Moody algebra by working out its conformally covariant form. 
  The discrete states of $c=1$ string theory at the self-dual radius are associated with modes of $W_{1+\infty}$ currents and their genus zero correlators are computed. An analogy to a recent suggestion based on the integrable structure of the theory is found. An iterative method for deriving the dependence of the currents on the full space of couplings is presented and applied. The dilaton equation of the theory is derived. 
  We consider the standard vector and chiral gauged WZNW models by their gauged maximal null subgroups and show that they can be mapped to each other by a special transformation. We give an explicit expression for the map in the case of the classical Lie groups $ A_N $, $ B_N $, $ C_N $, $ D_N $, and note its connection with the duality map for the Riemmanian globally symmetric spaces. 
  Our understanding of the mechanism by which topological defects are formed in symmetry breaking phase transitions has recently changed. We examine the non-equilibrium dynamics of defect formation for weakly-coupled global O(N) theories possessing vortices (strings) and monopoles. It is seen that, as domains form and grow, defects are swept along on their boundaries at a density of about one defect per coherence area (strings) or per coherence volume (monopoles). 
  It is shown, that extended particle-like objects should infinitely long collapse into some discontinuous configurations of the same topology, but vanishing mass. Analytic results concerning the general properties and asymptotic rates of such a process are given for 1+1-dimensional soliton models. 
  A recent suggestion has been made that the hydrogen bound state spectrum should not depend on the number of spatial dimensions. It is pointed out here that the uncertainty principle implies that such differences must exist and that a perturbation expansion in the dimensionality parameter yields a precise quantitative confirmation of the effect. 
  We review the Lagrangian Batalin--Vilkovisky method for gauge theories. This includes gauge fixing, quantisation and regularisation. We emphasize the role of cohomology of the antibracket operation. Our main example is $d=2$ gravity, for which we also discuss the solutions for the cohomology in the space of local integrals. This leads to the most general form for the action, for anomalies and for background charges. 
  A quantum algebra $U_{p,q}(\zeta ,H,X_\pm )$ associated with a nonstandard $R$-matrix with two deformation parameters$(p,q)$ is studied and, in particular, its universal ${\cal R}$-matrix is derived using Reshetikhin's method. Explicit construction of the $(p,q)$-dependent nonstandard $R$-matrix is obtained through a coloured generalized boson realization of the universal ${\cal R}$-matrix of the standard $U_{p,q}(gl(2))$ corresponding to a nongeneric case. General finite dimensional coloured representation of the universal ${\cal R}$-matrix of $U_{p,q}(gl(2))$ is also derived. This representation, in nongeneric cases, becomes a source for various $(p,q)$-dependent nonstandard $R$-matrices. Superization of $U_{p,q}(\zeta , H,X_\pm )$ leads to the super-Hopf algebra $U_{p,q}(gl(1|1))$. A contraction procedure then yields a $(p,q)$-deformed super-Heisenberg algebra $U_{p,q}(sh(1))$ and its universal ${\cal R}$-matrix. 
  We study a $U(N)$-invariant vector+matrix chain with the color structure of a lattice gauge theory with quarks and interpret it as a theory of open andclosed strings with target space $\Z$. The string field theory is constructed as a quasiclassical expansion for the Wilson loops and lines in this model. In a particular parametrization this is a theory of two scalar massless fields defined in the half-space $\{x\in \Z , \tau >0\} $. The extra dimension $\tau$ is related to the longitudinal mode of the strings. The topology-changing string interactions are described by a local potential. The closed string interaction is nonzero only at boundary $\tau =0$ while the open string interaction falls exponentially with $\tau$. 
  We show that the KdV and the NLS equations are tri-Hamiltonian systems. We obtain the third Hamiltonian structure for these systems and prove Jacobi identity through the method of prolongation. The compatibility of the Hamiltonian structures is verified directly through prolongation as well as through the shifting of the variables. We comment on the properties of the recursion operator as well as the connection with the two boson hierarchy. 
  We describe the general framework for constructing collective--theory Hamiltonians whose hermicity requirements imply a Kac--Moody algebra of constraints on the associated Jacobian. We give explicit examples for the algebras $sl(2)_k$ and $sl(3)_k$. The reduction to $W_n$--constraints, relevant to $n$-matrix models, is described for the Jacobians. 
  The effective action for type II string theory compactified on a six torus is $N=8$ supergravity, which is known to have an $E_{7}$ duality symmetry. We show that this is broken by quantum effects to a discrete subgroup, $E_7(\Z)$, which contains both the T-duality group $SO(6,6;\Z)$ and the S-duality group $SL(2;\Z)$. We present evidence for the conjecture that $E_7(\Z)$ is an exact \lq U-duality' symmetry of type II string theory. This conjecture requires certain extreme black hole states to be identified with massive modes of the fundamental string. The gauge bosons from the Ramond-Ramond sector couple not to string excitations but to solitons. We discuss similar issues in the context of toroidal string compactifications to other dimensions, compactifications of the type II string on $K_3\times T^2$ and compactifications of eleven-dimensional supermembrane theory. 
  Quantum measurement theory has fallen under the resticting influence of the attempt to explain the fundamental axioms of quantum theory in terms of the theory itself. This has not only led to confusion but has also restricted our attention to a limited class of measurements. This paper outlines some of the novel types of measurements which fall outside the usual textbook description. 
  This paper is a continuation of our papers \cite{EK1, EK2}. In \cite{EK2} we showed that for the root system $A_{n-1}$ one can obtain Macdonald's polynomials as weighted traces of intertwining operators between certain finite-dimensional representations of $U_q(sl_n)$. The main goal of the present paper is to use this construction to give a representation-theoretic proof of Macdonald's inner product and symmetry identities for the root system $A_{n-1}$. The proofs are based on the techniques of ribbon graphs developed by Reshetikhin and Turaev. We also use the symmetry identities to derive recursive relations for Macdonald's polynomials. 
  We discuss the nonlinear extension of $N=2$ superconformal algebra by generalizing Sugawara construction and coset construction built from $N=2$ currents based on Kazama-Suzuki $N=2$ coset model $\frac{SU(3)}{SU(2) \times U(1)}$ in $N=2$ superspace. For the generic unitary minimal series $c = 6(1-\frac{3}{k+3})$ where $k$ is the level of $SU(3)$ supersymmetric Wess-Zumino-Witten model, this algebra reproduces exactly $N=2$ $W_3$ algebra which has been worked out by Romans in component formalism. 
  We have examined solutions of tetrahedron equations from the elliptic free fermion model by using Korepanov mechanism based on tetrahedral Zamolodchikov algebras. As a byproduct, we have found a new integrable 2-dim. lattice model. We have also studied the relation between tetrahedral Zamolodchikov algebras and tetrahedron equations. 
  We reconstruct the Lagrangian of a left-right symmetric model with the gauge group $SU(2)_L\times SU(2)_R\times U(1)_Y \times \pi_4(SU(2)_L\times SU(2)_R\times U(1)_Y) $. The Higgs fields appear as gauge fields on discrete gauge group $\pi_4(SU(2)_L\times SU(2)_R\times U(1)_Y) $ and are assigned in a way complying with the principle that both the original gauge group $G_{YM}$ and the discrete group $\pi_4(G_{YM})$ should be taken as gauge groups in sense of non-commutative geometry. 
  A $q$-deformed free spinning relativistic particle is discussed in the framework of the Lagrangian formalism. Three equivalent Lagrangians are obtained for this system which are endowed with $q$-deformed local (super)gauge symmetries and reparametrization invariance. It is demonstrated that these symmetries are on-shell equivalent only for $ q = \pm1 $ under particular identification of the transformation parameters. The same condition ($ q=\pm1 $) emerges due to the requirement that the $q$-commutator of two supersymmetric gauge transformations should generate a reparametrization plus a supersymmetric gauge transformation. For a specific gauge choice, the solutions for equations of motion respect $GL_{\surd q}(1|1)$ and $GL_{q}(2)$ invariances for any arbitrary value of the evolution parameter characterizing the quantum super world-line. 
  It is shown that the massless $j=1$ Weinberg-Tucker-Hammer equations reduce to the Maxwell's equations for electromagnetic field under the definite choice of field functions and initial and boundary conditions. Thus, the former appear to be of use in a description of some physical processes for which that could be necessitated or be convenient. The possible consequences are discussed. 
  A difference analogue of the logistic equation, which preserves integrability, is derived from Hirota's bilinear difference equation. The integrability of the map is shown to result from the large symmetry associated with the B\"acklund transformation of the KP hierarchy. We introduce a scheme which interpolates between this map and the standard logistic map and enables us to study integrable and nonintegrable systems on an equal basis. In particular we study the behavior of Julia set at the point where the nonintegrable map passes to the integrable map. 
  I critically examine the notion of ``irreversibility,'' and discuss in what sense it applies to the spontaneous creation of particles in external fields. The investigation reveals that particle creation in very strong fields can only be described by a non-Markovian transport theory. 
  We calculate the B-model on the mirror pair of $X_{2N-2}(2,2,\cdots,2,1,1)$ , which is an $(N-2)$-dimensional Calabi-Yau manifold and has two marginal operators i.e. $h^{1,1}(X_{2N-2}(2,2,\cdots,2,1,1))=2$. In \cite{nagandjin} we have discussed about mirror symmetry on $X_N(1,1,\cdots,1)$ and its mirror pair. However, $X_N(1,1,\cdots,1)$ had only one moduli. In this paper we extend its methods to the case with a few moduli using toric geometry. 
  We discuss the properties and interpretation of a discrete sequence of a static spherically symmetric solutions of the Yang-Mills-dilaton theory. This sequence is parametrized by the number $n$ of zeros of a component of the gauge field potential. It is demonstrated that solutions with odd $n$ posses all the properties of the sphaleron. It is shown that there are normalizable fermion zero modes in the background of these solutions. The question of instability is critically analysed. 
  The symmetries associated with the closed bosonic string partition function are examined so that the integration region in Teichmuller space can be determined. The conditions on the period matrix defining the fundamental region can be translated to relations on the parameters of the uniformizing Schottky group. The growth of the lower bound for the regularized partition function is derived through integration over a subset of the fundamental region. 
  A thorough analysis of the general features of $(p=2)$ parasupersymmetric quantum mechanics is presented. It is shown that for both Rubakov--Spiridonov and Beckers--Debergh formulations of $(p=2)$-parasupersymmetric quantum mechanics, the degeneracy structure of the energy spectrum can be derived using the defining parasuperalgebras. Thus the results of the present article is independent of the details of the Hamiltonian. In fact, they are valid for arbitrary systems based on arbitrary dimensional coordinate manifolds. In particular, the Rubakov--Spiridonov (R-S) and Beckers--Debergh (B-D) systems possess identical degeneracy structures. For a subclass of R-S (alternatively B-D) systems, a new topological invariant is introduced. This is a counterpart of the Witten index of the supersymmetric quantum mechanics. 
  We consider a simple quantum system subjected to a classical random force. Under certain conditions it is shown that the noise-averaged Wigner function of the system follows an integro-differential stochastic Liouville equation. In the simple case of polynomial noise-couplings this equation reduces to a generalized Fokker-Planck form. With nonlinear noise injection new ``quantum diffusion'' terms arise that have no counterpart in the classical case. Two special examples that are not of a Fokker-Planck form are discussed: the first with a localized noise source and the other with a spatially modulated noise source. 
  In topologically massive QED$_{2+1}$ with $N$ flavours, there is the possibility that two equal-charged fermions can form a bound state pair in either s-wave or p-wave. We are concerned about the s-wave pairs and obtain the low energy effective action describing them. It is shown that the fermion pairs behave like doubly charged spin-1 bosons and, when they condense, the gauge field aquires the longitudinal mass. The approximate $SU(2)$ symmetry due to the similarity between the fermion pairs and the gauge field is discussed. 
  Talk given at the 7th Marcel Grossmann Meeting on General Relativity, Stanford University, July 24-30, 1994. 
  We study a two parameter family of Calabi-Yau d-fold by means of mirror symmetry. We construct mirror maps and calculate correlation functions associated with {\kae} moduli in the original manifold. We find there are more complicated instanton corrections of these couplings than threefolds, which is expected to reflect families of instantons with continous parameters. 
  Within the setting of algebraic quantum field theory a relation between phase-space properties of observables and charged fields is established. These properties are expressed in terms of compactness and nuclearity conditions which are the basis for the characterization of theories with physically reasonable causal and thermal features. Relevant concepts and results of phase space analysis in algebraic quantum field theory are reviewed and the underlying ideas are outlined. 
  The realizations of the Lie algebra corresponding to the dynamical symmetry group SO(2,1) of the Kepler and oscillator potentials are q-deformed. The q-canonical transformation connecting two realizations is given and a general definition for q-canonical transformation is deduced. q-Schr\"{o}dinger equation for a Kepler like potential is obtained from the q-oscillator Schr\"{o}dinger equation. Energy spectrum and the ground state wave function are calculated. 
  Hawking has proposed non-unitary rules for computing the probabilistic outcome of black hole formation. It is shown that the usual interpretation of these rules violates the superposition principle and energy conservation. Refinements of Hawking's rules are found which restore both the superposition principle and energy conservation, but leave completely unaltered Hawking's prediction of a thermal emission spectrum prior to the endpoint of black hole evaporation. These new rules violate clustering. They further imply the existence of superselection sectors, within each of which clustering is restored and a unitary $S$-matrix is shown to exist. -- This is an expanded version of a talk given at the Seventh Marcel Grossman Meeting on General Relativity, Stanford CA. 
  Recently a new technique in the harmonic analysis on symmetric spaces was suggested based on certain remarkable representations of affine and double affine Hecke algebras in terms of Dunkl and Demazure operators instead of Lie groups and Lie algebras. In the classic case it resulted (among other applications) in a new theory of radial part of Laplace operators and their deformations including a related concept of the Fourier transform.   In the present paper we demonstrate that the new technique works well even in the most general difference-elliptic case conjecturally corresponding to the $q$-Kac-Moody algebras. We discuss here only the construction of the generalized radial (zonal) Laplace operators and connect them with the difference-elliptic Ruijsenaars operators generalizing in its turn the Olshanetsky-Perelomov differential elliptic operators. 
  We describe explicitly the canonical map $\chi:$ Spec $\ue(\a{g})\ \rightarrow \ $Spec $\ze$, where $\ue(\a{g})$ is a quantum loop algebra at an odd root of unity $\ve$. Here $\ze$ is the center of $\ue(\a{g})$ and Spec $R$ stands for the set of all finite--dimensional irreducible representations of an algebra $R$. We show that Spec $\ze$ is a Poisson proalgebraic group which is essentially the group of points of $G$ over the regular adeles concentrated at $0$ and $\infty$. Our main result is that the image under $\chi$ of Spec $\ue(\a{g})$ is the subgroup of principal adeles. 
  We study the 1-form diffeomorphism cohomologies within a local conformal Lagrangian Field Theory model built on a two dimensional Riemann surface with no boundary. We consider the case of scalar matter fields and the complex structure is parametrized by Beltrami differential. The analysis is first performed at the Classical level, and then we improve the quantum extension, introducing the current in the Lagrangian dynamics, coupled to external source fields. We show that the anomalies which spoil the current conservations take origin from the holomorphy region of the external fields, and only the differential spin 1 and 2 currents (as well their c.c) could be anomalous. 
  We discuss the conformal factor dynamics in $D=6$. Accepting the proposal that higher-derivative dimensionless terms in the anomaly-induced effective action may be dropped, we obtain a superrenormalizable (like in $D=4$) effective theory for the conformal factor. The one-loop analysis of this theory gives the anomalous scaling dimension for the conformal factor and provides a natural mechanism to solve the cosmological constant problem. 
  We study an extension of the axial model where local gauge symmetries are taken into account. The anomaly of the axial current is calculated by the Fujikawa formalism and the model is also solved. Besides the well known features of the particular models (axial and Schwinger) it was obtained an effective interaction of scalar and gauge fields via a topological current. 
  The Hoft structure of the central extension of the $U_q \left( \widehat{sl\left( n \right) }\right)$ algebra is considered. The intertwine matrix induces new integrable spin chain models. We show the relation of these models and the biparametric spin chain $\widehat{sl_{p,q} \left( n \right)}$ models. The cases $n=2$ are $n=3$ are discussed and for $n=2$ we obtain the model of Dasgupta and Chowdhury . The case $n=3$ is solved with nested Bethe ansatz method and it is showed the dependence of the Bethe equations in the second parameter introduced 
  A new topology is proposed on the space of holonomy equivalence classes of loops, induced by the topology of the space $\Sigma$ in which the loops are embedded. The possible role for the new topology in the context of the work by Ashtekar et al. is discussed. 
  A general discussion of equations with universal invariance for a scalar field is provided in the framework of Lagrangian theory of first-order systems. 
  In this letter I study the universality of the nonperturbative effects and the vacua structure of the stochastic stabilization of the matrix models which defines Pure 2D Quantum Gravity. I show also that there is not tunneling, in the continuum limit, between the one-arc and three-arc solutions of the simplest matrix model which defines the flow between Pure Gravity and the Lee-Yang model. 
  The mechanism by which the physical vacuum can be different from the perturbative vacuum in the light-cone representation is described and illustrated. 
  We show that the physical requirement of flux conservation can substitute for the usual matching conditions in point interactions. The study covers an arbitrary superposition of $\delta$ and $\delta'$ potentials on the real line and can be easily applied to higher dimensions. Our procedure can be seen as a physical interpretation of the deficiency index of some symmetric, but not self-adjoint operators. 
  A general definition has been proposed recently of a linear connection and a metric in noncommutative geometry. It is shown that to within normalization there is a unique linear connection on the quantum plane and there is no metric. 
  The four-point function arising in the scattering of closed bosonic strings in their tachyonic ground state is evaluated on a surface of infinite genus. The amplitude has poles corresponding to physical intermediate states and divergences at the boundary of moduli space, but no new types of divergences result from the infinite number of handles. The implications for the universal moduli space approach are briefly discussed. 
  A construction is proposed for linear connections on non-commutative algebras. The construction relies on a generalisation of the Leibnitz rules of commutative geometry and uses the bimodule structure of $\Omega^1$. A special role is played by the extension to the framework of non-commutative geometry of the permutation of two copies of $\Omega^1$. The construction of the linear connection as well as the definition of torsion and curvature is first proposed in the setting of the derivations based differential calculus of Dubois- Violette and then a generalisation to the framework proposed by Connes as well as other non-commutative differential calculi is suggested. The covariant derivative obtained admits an extension to the tensor product of several copies of $\Omega^1$. These constructions are illustrated with the example of the algebra of $ n \times n$ matrices. 
  The octonionic X-product gives the octonions a flexibility not found in the other real division algebras. The pattern of that flexibility is investigated here. 
  Generalized KdV hierarchies associated by Drinfeld-Sokolov reduction to grade one regular semisimple elements from non-equivalent Heisenberg subalgebras of a loop algebra $\G\otimes{\bf C}[\lambda,\lambda^{-1}]$ are studied. The graded Heisenberg subalgebras containing such elements are labelled by the regular conjugacy classes in the Weyl group ${\bf W}(\G)$ of the simple Lie algebra $\G$. A representative $w\in {\bf W}(\G)$ of a regular conjugacy class can be lifted to an inner automorphism of $\G$ given by $\hat w=\exp\left(2i\pi {\rm ad I_0}/m\right)$, where $I_0$ is the defining vector of an $sl_2$ subalgebra of $\G$.The grading is then defined by the operator $d_{m,I_0}=m\lambda {d\over d\lambda} + {\rm ad} I_0$ and any grade one regular element $\Lambda$ from the Heisenberg subalgebra associated to $[w]$ takes the form $\Lambda = (C_+ +\lambda C_-)$, where $[I_0, C_-]=-(m-1) C_-$ and $C_+$ is included in an $sl_2$ subalgebra containing $I_0$. The largest eigenvalue of ${\rm ad}I_0$ is $(m-1)$ except for some cases in $F_4$, $E_{6,7,8}$. We explain how these Lie algebraic results follow from known results and apply them to construct integrable systems.If the largest ${\rm ad} I_0$ eigenvalue is $(m-1)$, then using any grade one regular element from the Heisenberg subalgebra associated to $[w]$ we can construct a KdV system possessing the standard $\W$-algebra defined by $I_0$ as its second Poisson bracket algebra. For $\G$ a classical Lie algebra, we derive pseudo-differential Lax operators for those non-principal KdV systems that can be obtained as discrete reductions of KdV systems related to $gl_n$. Non-abelian Toda systems are also considered. 
  Some calculational errors in expressions derived previously by the first author for the effective action, or equivalently for the functional determinant, on sectors of a spherical cap are corrected. The formula for the change in the effective action under Weyl rescalings in the three dimensional case is also amended. 
  The principle of S-duality is used to incorporate gaugino condensates into effective supergravity (superstring) Lagrangians. We discuss two implementations of S-duality which differ in the way the coupling constant is transformed. Both solve the problem of the runaway dilaton and lead to satisfactory supersymmetry breaking in models with a {\em single} gaugino condensate. The breakdown of supersymmetry is intimately related to a nontrivial transformation of the condensate under T-duality. 
  Standard SUSY-GUTs such as those based on $SU(5)$ or $SO(10)$ lead to predictions for the values of $\alpha _s$ and $sin^2\theta _W$ in amazing agreement with experiment. In this article we investigate how these models may be obtained from string theory, thus bringing them into the only known consistent framework for quantum gravity. String models with matter in standard GUT representations require the realization of affine Lie algebras at higher levels. We start by describing some methods to build level $k=2$ orbifold string models with gauge groups $SU(5)$ or $SO(10)$. We present several examples and identify generic features of the type of models constructed. Chiral fields appropriate to break the symmetry down to the standard model generically appear in the massless spectrum. However, unlike in standard SUSY-GUTs, they often behave as string moduli, i.e., they do not have self-couplings. We also discuss briefly the doublet-triplet Higgs splitting. We find that, in some models, built-in sliding-singlet type of couplings exist. 
  The concept of $Diff^4$ invariant Poincare transformations is a cornerstone of T(opological) G(eometro)D(ynamics). This concept makes it possible to understand the concept of subjective time and irreversibelity as well as nontriviality of S-matrix at quantum level. In this paper the possibility of identifying $Diff^4$ invariant Poincare transformations as the recently discovered Lorentz invariant deformation of Poincare algebra with the basic property that 'new' energy is some function of 'old' energy, is considered. 
  In this note we prove the Davies-Foda-Jimbo-Miwa-Nakayashiki conjecture on the asymptotics of the composition of n quantum vertex operators for the quantum affine algebra U_q(\hat sl_2), as n goes to infinity. For this purpose we define and study the leading eigenvalue and eigenvector of the product of two components of the quantum vertex operator. This eigenvector and the corresponding eigenvalue were recently computed by M.Jimbo. The results of his computation are given in Section 4. 
  Structure constants of minimal conformal theories are reconsidered. It is shown that {\it ratios} of structure constants of spin zero fields of a non-diagonal theory over the same evaluated in the diagonal theory are given by a simple expression in terms of the components of the eigenvectors of the adjacency matrix of the corresponding Dynkin diagram. This is proved by inspection, which leads us to carefully determine the {\it signs} of the structure constants that had not all appeared in the former works on the subject. We also present a proof relying on the consideration of lattice correlation functions and speculate on the extension of these identities to more complicated theories. 
  String theory requires two kinds of loop expansion: classical $(\alpha')$ worldsheet loops with expansion parameter $<T>$ where $T$ is a modulus field, and quantum $(\hbar)$ spacetime loops with expansion parameter $<S>$ where $S$ is the dilaton field. Four-dimensional string/string duality (a corollary of ten-dimensional string/fivebrane duality) interchanges the roles of $S$ and $T$ and hence interchanges classical and quantum. 
  In comparison with the $WT$ chiral identity which is indispensable for renormalization theory, relations deduced from the non-linear chiral transformation have a totally different physical significance. We wish to show that non-linear chiral transformations are powerful tools to deduce useful integral equations for propagators. In contrast to the case of linear chiral transformations, identities derived from non-linear ones contain more involved radiative effects and are rich in physical content. To demonstrate this fact we apply the simplest non-linear chiral transformation to the Nambu-Jona-Lasinio model, and show how our identity is related to the Dyson-Schwinger equation and Bethe-Salpeter amplitudes of the Higgs and $\pi$. Unlike equations obtained from the effective potential, our resultant equation is exact and can be used for events beyond the LEP energy. 
  We construct a new class of exact and stable superstring solutions in which our four-dimensional spacetime is taken to be curved . We derive in this space the full one-loop partition function in the presence of non-zero $\langle F^a_{\mu\nu}F_a^{\mu\nu}\rangle=F^2$ gauge background as well as in an $\langle R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\rangle=\R^2$ gravitational background and we show that the non-zero curvature, $Q^2=2/(k+2)$, of the spacetime provides an infrared regulator for all $\langle[F^a_{\mu\nu}]^n[R_{\mu\nu\rho\sigma}]^m\rangle$ correlation functions. The string one-loop partition function $Z(F,\R, Q)$ can be exactly computed, and it is IR and UV finite. For $Q$ small we have thus obtained an IR regularization, consistent with spacetime supersymmetry (when $F=0,\R=0$) and modular invariance. Thus, it can be used to determine, without any infrared ambiguities, the one-loop string radiative corrections on gravitational, gauge or Yukawa couplings necessary for the string superunification predictions at low energies. (To appear in the Proceedings of the Trieste Spring 94 Workshop) 
  The cyclic evolutions and associated geometric phases induced by time-independent Hamiltonians are studied for the case when the evolution operator becomes the identity (those processes are called {\it evolution loops}). We make a detailed treatment of systems having equally-spaced energy levels. Special emphasis is made on the potentials which have the same spectrum as the harmonic oscillator potential (the generalized oscillator potentials) and on their recently found coherent states. 
  We study the large-$N$ limit of adjoint fermion one-matrix models. We find one-cut solutions of the loop equations for the correlators of these models and show that they exhibit third order phase transitions associated with $m$-th order multi-critical points with string susceptibility exponents $\gamma_{\rm str}=-1/m$. We also find critical points which can be interpreted as points of first order phase transitions, and we discuss the implications of this critical behaviour for the topological expansion of these matrix models. 
  Orbifold techniques are used to study bosonic, type II and heterotic strings in Rindler space at integer multiples N of the Rindler temperature, and near a black hole horizon at integer multiples of the Hawking temperature, extending earlier results of Dabholkar. It is argued that a Hagedorn transition occurs nears the horizon for all N>1. 
  A class of two dimensional conformal field theories is known to correspond to three dimensional Chern-Simons theory. Here we claim that there is an analogous class of four dimensional field theories corresponding to five dimensional Chern-Simons theory. The four dimensional theories give a coupling between a scalar field and an external divergenceless vector field and they may have some application in magnetohydrodynamics. Like in conformal theories they possess a diffeomorphism symmetry, which for us is along the direction of the vector field, and their generators are analogous to Virasoro generators. Our analysis of the abelian Chern-Simons system uses elementary canonical methods for the quantization of field theories defined on manifolds with boundaries. Edge states appear for these systems and they yield a four dimensional current algebra. We examine the quantization of these algebras in several special cases and claim that a renormalization of the $5D$ Chern-Simons coupling is necessary for removing divergences. 
  Static axisymmetric Einstein-Maxwell-Dilaton and stationary axisymmetric Einstein-Maxwell-Dilaton-Axion (EMDA) theories in four space-time dimensions are shown to be integrable by means of the inverse scattering transform method. The proof is based on the coset-space representation of the 4-dim theory in a space-time admitting a Killing vector field. Hidden symmetry group of the four-dimensional EMDA theory, unifying T and S string dualities, is shown to be Sp(2, R) acting transitively on the coset Sp(2, R)/U(2). In the case of two-parameter Abelian space-time isometry group, the hidden symmetry is the corresponding infinite-dimensional group of the Geroch-Kinnersley-Chitre type. 
  Following on from a general observation in an earlier paper, we consider the continuous symmetries of a certain class of conformal field theories constructed from lattices and their reflection-twisted orbifolds. It is shown that the naive expectation that the only such (inner) symmetries are generated by the modes of the vertex operators corresponding to the states of unit conformal weight obtains, and a criterion for this expectation to hold in general is proposed. 
  We compute the {\it exact} equation of state of circular strings in the (2+1) dimensional de Sitter (dS) and anti de Sitter (AdS) spacetimes, and analyze its properties for the different (oscillating, contracting and expanding) strings. The string equation of state has the perfect fluid form $P=(\gamma-1)E,$ with the pressure and energy expressed closely and completely in terms of elliptic functions, the instantaneous coefficient $\gamma$ depending on the elliptic modulus. We semi-classically quantize the oscillating circular strings. The string mass is $m=\sqrt{C}/(\pi H\alpha'),\;C$ being the Casimir operator, $C=-L_{\mu\nu}L^{\mu\nu},$ of the $O(3,1)$-dS [$O(2,2)$-AdS] group, and $H$ is the Hubble constant. We find $\alpha'm^2_{\mbox{dS}}\approx 5.9n,\;(n\in N_0),$ and a {\it finite} number of states $N_{\mbox{dS}}\approx 0.17/(H^2\alpha')$ in de Sitter spacetime; $m^2_{\mbox{AdS}}\approx 4H^2n^2$ (large $n\in N_0$) and $N_{\mbox{AdS}}=\infty$ in anti de Sitter spacetime. The level spacing grows with $n$ in AdS spacetime, while is approximately constant (although larger than in Minkowski spacetime) in dS spacetime. The massive states in dS spacetime decay through tunnel effect and the semi-classical decay probability is computed. The semi-classical quantization of {\it exact} (circular) strings and the canonical quantization of generic string perturbations around the string center of mass strongly agree. 
  Relativistic Quantum Mechanics suffers from structural problems which are traced back to the lack of a position operator $\hat{x}$, satisfying $[\hat{x},\hat{p}]=i\hbar\hat{1}$ with the ordinary momentum operator $\hat{p}$, in the basic symmetry group -- the Poincar\'e group. In this paper we provide a finite-dimensional extension of the Poincar\'e group containing only one more (in 1+1D) generator $\hat{\pi}$, satisfying the commutation relation $[\hat{k},\hat{\pi}]=i\hbar\hat{1}$ with the ordinary boost generator $\hat{k}$. The unitary irreducible representations are calculated and the carrier space proves to be the set of Shapiro's wave functions. The generalized equations of motion constitute a simple example of exactly solvable finite-difference set of equations associated with infinite-order polarization equations. 
  We obtain an explicit realization of all the primary fields of the Ising model in terms of a conformal field theory of constrained fermions. The four-point correlators of the energy, order and disorder operators are explicitly calculated. 
  We show that the classical equations of motion of the low-energy effective field theory describing the massless modes of the heterotic (or type I) string admit two classes of supersymmetric self--dual backgrounds. The first class, which was already considered in the literature, consists of solutions with a (conformally) flat metric coupled to axionic instantons. The second includes Asymptotically Locally Euclidean (ALE) gravitational instantonic backgrounds coupled to gauge instantons through the so--called ``standard embedding''. Talk given by F.Fucito at the XI SIGRAV meeting, Trieste, September 27-30, 1994 
  We compute the beta function at one loop for Yang-Mills theory using as regulator the combination of higher covariant derivatives and Pauli-Villars determinants proposed by Faddeev and Slavnov. This regularization prescription has the appealing feature that it is manifestly gauge invariant and essentially four-dimensional. It happens however that the one-loop coefficient in the beta function that it yields is not $-11/3,$ as it should be, but $-23/6.$ The difference is due to unphysical logarithmic radiative corrections generated by the Pauli-Villars determinants on which the regularization method is based. This no-go result discards the prescription as a viable gauge invariant regularization, thus solving a long-standing open question in the literature. We also observe that the prescription can be modified so as to not generate unphysical logarithmic corrections, but at the expense of losing manifest gauge invariance. 
  We present a general argument for the construction of BRST charges of the `non-critical' $\W_{2,4}$, $\W_{2,5}$, $\W_{2,6}$, and $\W_{2,8}$ strings. This evidences the existence of BRST charges for a kind of soft-type algebras which can be constructed from two copies of quantum $\W_{2,s}$ algebras, (s=3,4,5,6,8). 
  The Aharonov-Bohm effect is analyzed for a spin-1/2 particle in the case that a $1/r$ potential is present. Scalar and vector couplings are each considered. It is found that the approach in which the flux tube is given a finite radius that is taken to zero only after a matching of boundary conditions does not give physically meaningful results. Specifically, the operations of taking the limit of zero flux tube radius and the Galilean limit do not commute. Thus there appears to be no satisfactory solution of the relativistic Aharonov-Bohm-Coulomb problem using the finite radius flux tube method. 
  Symplectic modular invariance of the bosonic string partition function has been verified at genus 2 and 3 using the period matrix coordinatization of moduli space. A calculation of the transformation of the holomorphic part of the differential volume element shows that an extra phase arises together with the factor associated with a specific modular weight; the phase is cancelled in the transformation of the entire volume element including the complex conjugate. An argument is given for modular invariance of the reggeon measure at genus twelve. 
  The time-energy uncertainty relation of Anandan-Aharonov is generalized to a relation involving a set of quantum state vectors. This is achieved by obtaining an explicit formula for the distance between two finitely separated points in the Grassmann manifold. 
  We review recent results which clarify the role of the integrable many-body problems in the quantum field theory framework.They describe the dynamics of the topological degrees of freedom in the theories which are obtained by perturbing the topological ones by the proper Hamiltonians and sources. Interpretation of the many-body dynamics as a motion on the different moduli spaces as well as the property of duality is discussed.Tower of many-body systems can be derived from a tower of the field theories with appropriate phase spaces which have a transparent interpretation in terms of the group theory.The appearance of Calogero-type systems in different physical phenomena is mentioned. 
  Electrically as well as magnetically charged states are constructed in the 2+1-dimensional Euclidean Z_N-Higgs lattice gauge model, the former following ideas of Fredenhagen and Marcu and the latter using duality transformations on the algebra of observables. The existence of electrically and of magnetically charged particles is also established. With this work we prepare the ground for the constructive study of anyonic statistics of multiparticle scattering states of electrically and magnetically charged particles in this model (work in progress). 
  We study the interplay between T-duality, compactification and supersymmetry. We prove that when the original configuration has unbroken space-time supersymmetries, the dual configuration also does if a special condition is met: the Killing spinors of the original configuration have to be independent on the coordinate which corresponds to the isometry direction of the bosonic fields used for duality. Examples of ``losers" (T-duals are not supersymmetric) and ``winners" (T-duals are supersymmetric) are given. 
  We consider the non-commutative generalization of the chiral perturbation theory. The resultant coupling constants are severely restricted by the model and in good agreement with the data. When applied to the Skyrme model, our scheme reproduces the non-Skyrme term with the right coefficient. We comment on a similar treatment of the linear $\sigma $-model. 
  We consider a version of generalised $q$-oscillators and some of their applications. The generalisation includes also "quons" of infinite statistics and deformed oscillators of parastatistics. The statistical distributions for different $q$-oscillators are derived for their corresponding Fock space representations. The deformed Virasoro algebra and SU(2) algebra are also treated. 
  It has been argued by Ishikawa and Kato that by making use of a specific bosonization, $c_M=1$ string theory can be regarded as a constrained topological sigma model. We generalize their construction for any $(p,q)$ minimal model coupled to two dimensional (2d) gravity and show that the energy--momentum tensor and the topological charge of a constrained topological sigma model can be mapped to the energy--momentum tensor and the BRST charge of $c_M<1$ string theory at zero cosmological constant. We systematically study the physical state spectrum of this topological sigma model and recover the spectrum in the absolute cohomology of $c_M<1$ string theory. This procedure provides us a manifestly topological representation of the continuum Liouville formulation of $c_M<1$ string theory. 
  Based on the scaling relation for the dynamics at the early time, a new method is proposed to measure both the static and dynamic critical exponents. The method is applied to the two dimensional Ising model. The results are in good agreement with the existing results. Since the measurement is carried out in the initial stage of the relaxation process starting from independent initial configurations, our method is efficient. 
  Using a loop formulation approach of QCD$_2$, we study the potential between two heavy quarks in the presence of adjoint scalar fields, and demonstrate how 't Hooft's planar rule is manifested in this formulation. Based on some physical assumptions, we argue that large adjoint loops ``confined'' inside an external fundamental one give a Casimir type contribution to the potential energy, while the small loops only renormalize the string tension. We also extend the results to the case of massive adjoint fields. 
  A description of elementary particles should be based on irreducible representations of the Poincar\'e group. In the theory of massive representations of the full Poincar\'e group there are essentially four different cases. One of them corresponds to the ordinary Dirac theory. The extension of Dirac theory to the remaining three cases makes it possible to describe an anomalous electric dipole moment of elementary particles without breaking the reflections. 
  In these lectures a general introduction to T-duality is given. In the abelian case the approaches of Buscher, and Ro\u{c}ek and Verlinde are reviewed. Buscher's prescription for the dilaton transformation is recovered from a careful definition of the gauge integration measure. It is also shown how duality can be understood as a quite simple canonical transformation. Some aspects of non-abelian duality are also discussed, in particular what is known on relation to canonical transformations. Some implications of the existence of duality on the cosmological constant and the definition of distance in String Theory are also suggested. 
  We discuss a field theoretical extension of the basic structures of classical analytical mechanics within the framework of the De Donder--Weyl (DW) covariant Hamiltonian formulation. The analogue of the symplectic form is argued to be the {\em polysymplectic} form of degree $(n+1)$, where $n$ is the dimension of space-time, which defines a map between multivector fields or, more generally, graded derivation operators on exterior algebra, and forms of various degrees which play a role of dynamical variables. The Schouten-Nijenhuis bracket on multivector fields induces the graded analogue of the Poisson bracket on forms, which turns the exterior algebra of (horizontal) forms to a Gerstenhaber algebra. The equations of motion are written in terms of the Poisson bracket on forms and it is argued that the bracket with $H\vol$, where $H$ is the DW Hamiltonian function and $\vol$ is the horizontal (i.e. space-time) volume form, is related to the operation of exterior differentiation of forms. 
  Division algebras are used to explain the existence and symmetries of various sets of auxiliary fields for super Yang-Mills in dimensions $d=3,4,6,10$. (Contribution to G\"ursey Memorial Conference I: Strings and Symmetries) 
  Light-cone quantization of (3+1)-dimensional electrodynamics is discussed, using discretization as an infrared regulator and paying careful attention to the interplay between gauge choice and boundary conditions. In the zero longitudinal momentum sector of the theory a general gauge fixing is performed and the corresponding relations that determine the constrained modes of the gauge field are obtained. The constraints are solved perturbatively and the structure of the theory is studied to lowest nontrivial order. (Talk presented at ``Theory of Hadrons and Light-Front QCD,'' Polana Zgorzelisko, Poland, August 1994.) 
  We present a systematic introduction to the geometry of linear braided spaces. These are versions of $\R^n$ in which the coordinates $x_i$ have braid-statistics described by an R-matrix. From this starting point we survey the author's braided-approach to $q$-deformation: braided differentiation, exponentials, Gaussians, integration and forms, i.e. the basic ingredients for $q$-deformed physics are covered. The braided approach includes natural $q$-Euclidean and $q$-Minkowski spaces in R-matrix form. 
  Given a renormalization scheme of QCD, one can define a mass scale $\Lambda_{\rm QCD}$ in terms of the beta function. Under a change of the renormalization scheme, however, $\Lambda_{\rm QCD}$ changes by a multiplicative constant. We introduce a scheme independent $\Lambda_{\rm QCD}$ using a connection on the space of the coupling constant. 
  A calculational framework for determining masses of low lying hadrons using light front quantization is discussed. The method is based upon four theoretical tools: discrete light cone quantization, which has been very successful in 1+1 dimensional models, a projector Monte Carlo method to extract low lying state data with 2 additional transverse dimensions, the transverse spatial lattice Hamiltonian of QCD, and exact form factors of the SU(3) x SU(3) 1+1 dimensional non-linear sigma model (NLSM). How these tools are to be put together to provide a description of hadrons is the main topic of this lecture. I also focus on the NLSM form factors, which are given new physical relevance via this picture of QCD. (Based on a lecture given at the ``Theory of Hadrons and Light-front QCD'' workshop in Zakopane, Poland, August 1994.) 
  We investigate a class of localized, stationary, particular numerical solutions to the Maxwell-Dirac system of classical nonlinear field equations. The solutions are discrete energy eigenstates bound predominantly by the self-produced electric field. 
  Four-dimensional twisted group lattices are used as models for space-time structure. Compared to other attempts at space-time deformation, they have two main advantages: They have a physical interpretation and there is no difficulty in putting field theories on these structures. We present and discuss ordinary and gauge theories on twisted group lattices. We solve the free field theory case by finding all the irreducible representations. The non-abelian gauge theory on the two-dimensional twisted group lattice is also solved. On twisted group lattices, continuous space-time translational and rotational symmetries are replaced by discrete counterparts. We discuss these symmetries in detail. Four-dimensional twisted group lattices can also be used as models for non-trivial discrete compactifactions of certain ten-dimensional spaces. 
  It is shown that the $SU(2)$ Yang-Mills theory in $3$-dimensional Riemann-Cartan space-time can be completely reformulated as a gravity-like theory in terms of gauge invariant variables. The resulting Yang-Mills induced equations are found, and it is demonstrated that they can be derived from a torsion-square type of action. 
  A simple geometric description of T-duality is given by identifying the cotangent bundles of the original and the dual manifold. Strings propagate naturally in the cotangent bundle and the original and the dual string phase spaces are obtained by different projections. Buscher's transformation follows readily and it is literally projective. As an application of the formalism, we prove that the duality is a symplectomorphism of the string phase spaces. 
  We give an explicit description of the vector bundle of WZW conformal blocks on elliptic curves with marked points as subbundle of a vector bundle of Weyl group invariant vector valued theta functions on a Cartan subalgebra. We give a partly conjectural characterization of this subbundle in terms of certain vanishing conditions on affine hyperplanes. In some cases, explicit calculation are possible and confirm the conjecture. The Friedan--Shenker flat connection is calculated, and it is shown that horizontal sections are solutions of Bernard's generalization of the Knizhnik--Zamolodchikov equation. 
  A constrained KP hierarchy is discussed that was recently suggested by Aratyn et al. and by Bonora et al. This hierarchy is a restriction of the KP to a submanifold of operators which can be represented as a ratio of two purely differential operators of prescribed orders. Explicit formulas for action of vector fields on these two differential operators are written which gives a new description of the hierarchy and provides a new, more constructive, proof of compatibility of the constraint with the hierarchy. Also the Poisson structure of the constrained hierarchy is discussed. 
  We propose a nonstandard approach to solving the apparent incompatibility between the coalgebra structure of some inhomogeneous quantum groups and their natural complex conjugation. In this work we sketch the general idea and develop the method in detail on a toy-model; the latter is a q-deformation of the Hopf algebra of 1-dim translations + dilatations. We show how to get all Hilbert space representations of the latter from tensor products of the fundamental ones; physically, this corresponds to constructing composite systems of many free distinct q-particles in terms of the basic one-particle ones. The spectrum of the total momentum turns out to be the same as that of a one-particle momentum, i.e. of the form $\{\mu q^n\}_{n\in\zn}$. 
  A general strategy is described for deriving a constituent approximation to QCD, inspired by the constituent quark model and based on light-front quantization. Some technical aspects of the approach are discussed, including a mechanism for obtaining a confining potential and ways in which spontaneous chiral symmetry breaking can be manifested. (Based on a talk presented by K.G. Wilson at ``Theory of Hadrons and Light-Front QCD,'' Polana Zgorzelisko, Poland, August 1994.) 
  This paper consists of an overview of the discussions on chiral symmetry breaking followed by a transcript of the discussions themselves. (Based on discussions at ``Theory of Hadrons and Light-Front QCD,'' Polana Zgorzelisko, Poland, August 1994.) 
  We conjecture polynomial identities which imply Rogers--Ramanujan type identities for branching functions associated with the cosets $({\cal G}^{(1)})_{\ell-1}\otimes ({\cal G}^{(1)})_{1} / ({\cal G}^{(1)})_{\ell}$, with ${\cal G}$=A$_{n-1}$ \mbox{$(\ell\geq 2)$}, D$_{n-1}$ $(\ell\geq 2)$, E$_{6,7,8}$ $(\ell=2)$. In support of our conjectures we establish the correct behaviour under level-rank duality for $\cal G$=A$_{n-1}$ and show that the A-D-E Rogers--Ramanujan identities have the expected $q\to 1^{-}$ asymptotics in terms of dilogarithm identities. Possible generalizations to arbitrary cosets are also discussed briefly. 
  An argument is given which exhibits color confinement in nonabelian gauge theory. 
  We use the functional representation of Heisenberg-Weyl group and obtain equation for the spectrum of the model, which is more complicated than Bethes ones, but can be written explicitly through theta functions. 
  We study the effective theory of the conformal factor near its infrared stable fixed point.The renormalization group equations for the effective coupling constants are found and their solutions near the critical point are obtained, providing the logarithmic corrections to scaling relations.Some cosmological applications of the running of coupling constants are briefly discussed. 
  We construct a random surface model with a string susceptibility exponent one quarter by taking an Ising model on a random surface and introducing an additional degree of freedom which amounts to allowing certain outgrowths on the surfaces. Fine tuning the Ising temperature and the weight factor for outgrowths we find a triple point where the susceptibility exponent is one quarter. At this point magnetized and nonmagnetized gravity phases meet a branched polymer phase. 
  The recursion relations of 2D quantum gravity coupled to the Ising model discussed by the author previously are reexamined. We study the case in which the matter sector satisfies the fusion rules and only the primary operators inside the Kac table contribute. The theory involves unregularized divergences in some of correlators. We obtain the recursion relations which form a closed set among well-defined correlators on sphere, but they do not have a beautiful structure that the bosonized theory has and also give an inconsistent result when they include an ill-defined correlator with the divergence. We solve them and compute the several normalization independent ratios of the well-defined correlators, which agree with the matrix model results. 
  We present the technical tools needed to compute any one-loop amplitude involving external spacetime fermions in a four-dimensional heterotic string model a` la Kawai-Lewellen-Tye. As an example, we compute the one-loop three-point amplitude with one "photon" and two external massive fermions ("electrons"). As a check of our computation, we verify that the one-loop contribution to the Anomalous Magnetic Moment vanishes if the model has spacetime supersymmetry, as required by the supersymmetric sum rules. 
  The Thirring model, that is, a relativistic field theory of fermions with a contact interaction between vector currents, is studied for dimensionalities 2<d<4 using the 1/N_f expansion, where N_f is the number of fermion species. The model is found to have no ultraviolet divergences at leading order provided a regularization respecting current conservation is used. Explicit O(1/N_f) corrections are computed, and the model shown to be renormalizable at this order in the massless limit; renormalizability appears to hold to all orders due to a special case of Weinberg's theorem. This implies there is a universal amplitude for four particle scattering in the asymptotic regime. Comparisons are made with both the Gross-Neveu model and QED. 
  In the first part, we introduce the notion of fractional statistics in the sense of Haldane. We illustrate it on simple models related to anyon physics and to integrable models solvable by the Bethe ansatz. In the second part, we describe the properties of the long-range interacting spin chains. We describe its infinite dimensional symmetry, and we explain how the fractional statistics of its elementary excitations is an echo of this symmetry. In the third part, we review recent results on the Yangian representation theory which emerged from the study of the integrable long-range interacting models. 
  Limits for the applicability of the equivalence principle are considered in the context of low-energy effective field theories. In particular, we find a class of higher-derivative interactions for the gravitational and electromagnetic fields which produce dispersive photon propagation. The latter is illustrated by calculating the energy-dependent contribution to the deflection of light rays. 
  The free energy is shown to decrease along Wilson renormalization group trajectories, in a dimension-independent fashion, for $d>2.$ The argument assumes the monotonicity of the cutoff function, and positivity of a spectral representation of the two point function. The argument is valid to all orders in perturbation theory. 
  Black hole formation and evaporation is studied in the semiclassical approximation in simple 1+1-dimensional models, with emphasis on issues related to Hawking's information paradox. Exact semiclassical solutions are described and questions of boundary conditions and vacuum stability are discussed. The validity of the semiclassical approximation has been called into question in the context of the information puzzle. A different approach, where black hole evolution is assumed to be unitary, is described. It requires unusual causal properties and kinematic behavior of matter that may be realized in string theory. Based on lectures given at the 1994 Trieste Spring School 
  The basic concepts of non-commutative probability theory are reviewed and applied to the large $N$ limit of matrix models. We argue that this is the appropriate framework for constructing the master field in terms of which large $N$ theories can be written. We explicitly construct the master field in a number of cases including QCD$_2$. There we both give an explicit construction of the master gauge field and construct master loop operators as well. Most important we extend these techniques to deal with the general matrix model, in which the matrices do not have independent distributions and are coupled. We can thus construct the master field for any matrix model, in a well defined Hilbert space, generated by a collection of creation and annihilation operators---one for each matrix variable---satisfying the Cuntz algebra. We also discuss the equations of motion obeyed by the master field. 
  We examine for standard-like orbifold compactification models the constraints due to quarks and leptons generation non-universality of soft supersymmetry breaking interactions. We follow the approach initiated by Ibanez and Lust and developed by Brignole, Ibanez and Munoz. The breaking of supersymmetry is represented in terms of dilaton and moduli auxiliary field components and, consistently with a vanishing cosmological constant, is parametrized in terms of the dilaton-moduli mixing angle $\theta $ and the gravitino mass scale $m_g$. The soft breaking interactions (gaugino masses, squarks and sleptons mass matrices, scalars interactions A and B coupling constants) are calculable as a function of these parameters and of the discrete set of modular weight parameters specifying the modular transformation properties of the low-energy fields. We solve the renormalization group one-loop equations for the full set of gauge, Yukawa and supersymmetry breaking coupling constants. 
  The pseudoclassical hamiltonian and action of the $D=2n$ dimensional Dirac particle with anomalous magnetic moment interacting with the external Yang-Mills field are found. The Bargmann-Michel-Telegdi equation of motion for the Pauli-Lubanski vector is deduced. The canonical quantization of $D=2n$ dimensional Dirac spinning particle with anomalous magnetic moment in the external Yang-Mills field is carried out in the gauge which allows to describe simultaneously particles and antiparticles (massive and massless) already at the classical level. Pseudoclassical Foldy-Wouthuysen transformation is used to obtain canonical (Newton-Wigner) coordinates and in terms of these variables the theory is quantized. 
  We calculate the quantum corrections to the entropy of a very large black hole, coming from the theory of a $D$-dimensional, non-critical bosonic string. We show that, for $D >2$, as a result of modular invariance the entropy is ultraviolet finite, although it diverges in the infrared (while for $D=2$ the entropy contains both ultraviolet and infrared divergences). The issue of modular invariance in field theory, in connection with black-hole entropy, is also investigated. 
  We treat the stochastic equation for large N master fields proposed by Greensite and Halpern using a construction of master fields modelled after work of Voiculescu, and show that it contains the same information as the usual factorized Schwinger-Dyson equations. 
  We apply the supersymmetric procedure to one-step random walks in one dimension at the level of the usual master equation, extending a study initiated by H.R. Jauslin [Phys. Rev. A {\bf 41}, 3407 (1990)]. A discussion of the supersymmetric technique for this discrete case is presented by introducing a formal second-order discrete master derivative and its ``square root", and we solve completely, and in matrix form, the cases of homogeneous random walks (constant jumping rates). A simple generalization of Jauslin's results to two uncorrelated axes is also provided. There may be many applications, especially to bistable and multistable one-step processes. 
  It is shown that the $W_{1+\infty}$ algebra is nothing but the simplest subalgebra of a $q$-discretized \vi\ algebra, in the language of the KP hierarchy explicitly. 
  The first part is an introduction to conformal field theory and string perturbation theory. The second part deals with the search for a deeper answer to the question posed in the title. Contents:   1. Conformal Field Theory   2. String Theory   3. Vacua and Dualities   4. String Field Theory or Not String Field Theory   5. Matrix Models 
  It is shown that if the momenta belong to an integral lattice, then every physical state of string theory leads to a symmetry of the scattering amplitudes. We discuss the role of this symmetry when the momenta are those provided by the usual D.D.F construction and show that the string compactified on the torus associated with the self-dual Lorentzian lattice, $\Pi^{25,1}$ possess the Fake Monster Lie algebra as a symmetry. 
  The renormalization algorithm based on regularization methods with two regulators is analyzed by means of explicit computations. We show in particular that regularization by higher covariant derivative terms can be complemented with dimensional regularization to obtain a consistent renormalized 4-dimensional Yang-Mills theory at the one-loop level. This shows that hybrid regularization methods can be applied not only to finite theories, like \eg\ Chern-Simons, but also to divergent theories. 
  Starting from Laughlin type wave functions with generalized periodic boundary conditions describing the degenerate groundstate of a quantum Hall system we explictly construct $r$ dimensional vector bundles. It turns out that the filling factor $\nu$ is given by the topological quantity $c_1 \over r$ where $c_1$ is the first chern number of these vector bundles. In addition, we managed to proof that under physical natural assumptions the stable vector bundles correspond to the experimentally dominating series of measured fractional filling factors $\nu = {n \over 2pn\pm 1}$. Most remarkably, due to the very special form of the Laughlin wave functions the fluctuations of the curvature of these vector bundles converge to zero in the limit of infinitely many particles which shows a new mathematical property. Physically, this means that in this limit the Hall conductivity is independent of the boundary conditions which is very important for the observabilty of the effect. Finally we discuss the relation of this result to a theorem of Donaldson. 
  It is shown that the vacuum value of the $2d$ gravity stress tensor in the free field theory is singular in the fundamental region on the complex plane where the genus-$n>1$ Riemann surface are mapped. Because of the above singularity, one can not construct modular invariant multi-loop amplitudes. The discussed singularity is due to the singularity in the vacuum value of the $2d$ gravity field that turns out to be on the genus-$n>1$ Riemann surfaces. 
  The full duality between the $\kappa$-Poincar\'e algebra and $\kappa$-Poincar\'e group is proved. 
  Given a simple Lie algebra $\gggg$, we consider the orbits in $\gggg^*$ which are of R-matrix type, i.e., which possess a Poisson pencil generated by the Kirillov-Kostant-Souriau bracket and the so-called R-matrix bracket. We call an algebra quantizing the latter bracket a quantum orbit of R-matrix type. We describe some orbits of this type explicitly and we construct a quantization of the whole Poisson pencil on these orbits in a similar way. The notions of q-deformed Lie brackets, braided coadjoint vector fields and tangent vector fields are discussed as well. 
  The systems of differential equations whose solutions coincide with Bethe ansatz solutions of generalized Gaudin models are constructed. These equations we call the {\it generalized spectral Riccati equations}, because the simplest equation of this class has a standard Riccatian form. The general form of these equations is $R_{n_i}[z_1(\lambda),\ldots, z_r(\lambda)] = c_{n_i}(\lambda), \ i=1,\ldots, r$, where $R_{n_i}$ denote some homogeneous polynomials of degrees $n_i$ constructed from functional variables $z_i(\lambda)$ and their derivatives. It is assumed that $\deg \partial^k z_i(\lambda) = k+1$. The problem is to find all functions $z_i(\lambda)$ and $c_{n_i}(\lambda)$ satisfying the above equations under $2r$ additional constraints $P \ z_i(\lambda)=F_i(\lambda)$ and $(1-P) \ c_{n_i}(\lambda)=0$, where $P$ is a projector from the space of all rational functions onto the space of rational functions having their singularities at {\it a priori} given points. It turns out that this problem has solutions only for very special polynomials $R_{n_i}$ called {\it Riccatians}. There exist a one-to-one correspondence between systems of Riccatians and simple Lie algebras. Functions $c_{n_i}(\lambda)$ satisfying the system of equations constructed from Riccatians of the type ${\cal L}_r$ exactly coincide with eigenvalues of the Gaudin spectral problem associated with algebra ${\cal L}_r$. This result suggests that the generalized Gaudin models admit a total separation of variables. 
  In the present note we suggest an affinization of a theorem by Kostrikin et.al. about the decomposition of some complex simple Lie algebras ${\cal G}$ into the algebraic sum of pairwise orthogonal Cartan subalgebras. We point out that the untwisted affine Kac-Moody algebras of types $A_{p^m-1}$ ($p$ prime, $m\geq 1$), $B_r, \, C_{2^m}, D_r,\, G_2,\, E_7,\, E_8$ can be decomposed into the algebraic sum of pairwise or\-tho\-go\-nal Heisenberg subalgebras. The $A_{p^m-1}$ and $G_2$ cases are discussed in great detail. Some possible applications of such decompositions are also discussed. 
  Light-front field theory offers a scenario in which a constituent picture of hadrons may arise, but only if cutoffs that violate explicit covariance and gauge invariance are used. The perturbative renormalization group can be used to approximate the cutoff QCD hamiltonian, and even at lowest orders the resultant hamiltonian displays interesting phenomenological features. A general scheme for computing and using these hamiltonians is discussed and it is explicitly shown that a confining interaction appears when the hamiltonian is computed to second order. 
  In this paper we construct explicitly natural (from the geometrical point of view) Fock space representations (contragradient Verma modules) of the quantized enveloping algebras. In order to do so, we start from the Gauss decomposition of the quantum group and introduce the differential operators on the corresponding $q$-deformed flag manifold (asuumed as a left comodule for the quantum group) by a projection to it of the right action of the quantized enveloping algebra on the quantum group. Finally, we express the representatives of the elements of the quantized enveloping algebra corresponding to the left-invariant vector fields on the quantum group as first-order differential operators on the $q$-deformed flag manifold. 
  The quantum commutations $RTT=TTR$ and the orthogonal (symplectic) conditions for the inhomogeneous multiparametric $q$-groups of the $B_n,C_n,D_n$ type are found in terms of the $R$-matrix of $B_{n+1},C_{n+1},D_{n+1}$. A consistent Hopf structure on these inhomogeneous $q$-groups is constructed by means of a projection from $B_{n+1},C_{n+1},D_{n+1}$. Real forms are discussed: in particular we obtain the $q$-groups $ISO_{q,r}(n+1,n-1)$, including the quantum Poincar\'e group. 
  The one-dimensional harmonic oscillator wave functions are solutions to a Sturm-Liouville problem posed on the whole real line. This problem generates the Hermite polynomials. However, no other set of orthogonal polynomials can be obtained from a Sturm-Liouville problem on the whole real line. In this paper we show how to characterize an arbitrary set of polynomials orthogonal on $(-\infty,\infty)$ in terms of a system of integro-differential equations of Hartree-Fock type. This system replaces and generalizes the linear differential equation associated with a Sturm-Liouville problem. We demonstrate our results for the special case of Hahn-Meixner polynomials. 
  We investigate the consistency of the background-field formalism when applying various regularizations and renormalization schemes. By an example of a two-dimensional $\sigma$ model it is demonstrated that the background-field method gives incorrect results when the regularization (and/or renormalization) is noninvariant. In particular, it is found that the cut-off regularization and the differential renormalization belong to this class and are incompatible with the background-field method in theories with nonlinear symmetries. 
  In the description of the extrinsic geometry of the string world sheet regarded as a conformal immersion of a 2-d surface in $R^3$, it was previously shown that, restricting to surfaces with $h\surd{g}\ =\ 1$, where $h$ is the mean scalar curvature and $g$ is the determinant of the induced metric on the surface, leads to Virasaro symmetry. An explicit form of the effective action on such surfaces is constructed in this article which is the extrinsic curvature analog of the WZNW action. This action turns out to be the gauge invariant combination of the actions encountered in 2-d intrinsic gravity theory in light-cone gauge and the geometric action appearing in the quantization of the Virasaro group. This action, besides exhibiting Virasaro symmetry in $z$-sector, has $SL(2,C)$ conserved currents in the $\bar{z}$-sector. This allows us to quantize this theory in the $\bar{z}$-sector along the lines of the WZNW model. The quantum theory on $h\surd{g}\ =\ 1$ surfaces in $ R^3$ is shown to be in the same universality class as the intrinsic 2-d gravity theory. 
  The string world sheet, regarded as Riemann surface, in background $R^3$ and $R^4$ is described by the generalised Gauss map. When the Gauss map is harmonic or equivalently for surfaces of constant mean scalar curvature, we obtain an Abelian self-dual system, using $SO(3)$ and $SO(4)$ gauge fields constructed in our earlier studies. This compliments our earlier result that $h\surd g\ =\ 1$ surfaces exhibit Virasaro symmetry. The self-dual system so obtained is compared with self-dual Chern-Simons system and a generalized Liouville equation involving extrinsic geometry is obtained. \vspace{0.2cm} The world sheet in background $R^n, \ n>4$ is described by the generalized Gauss map. It is first shown that when the Gauss map is harmonic, the scalar mean curvature is constant. $SO(n)$ gauge fields are constructed from the geometry of the surface and expressed in terms of the Gauss map. It is shown that the harmonic map satisfies a non-Abelian self-dual system of equations for the gauge group $SO(2)\times SO(n-2)$. 
  Without using Gabber's theorem, the finite-dimensionality of the space of conformal blocks in the WZNW-models is proved. 
  We present a detailed analysis of the structure of the conservation laws in quantum integrable chains of the XYZ-type and in the Hubbard model. With the use of the boost operator, we establish the general form of the XYZ conserved charges in terms of simple polynomials in spin variables and derive recursion relations for the relative coefficients of these polynomials. For two submodels of the XYZ chain - namely the XXX and XY cases, all the charges can be calculated in closed form. For the XXX case, a simple description of conserved charges is found in terms of a Catalan tree. This construction is generalized for the su(M) invariant integrable chain. We also indicate that a quantum recursive (ladder) operator can be traced back to the presence of a hamiltonian mastersymmetry of degree one in the classical continuous version of the model. We show that in the quantum continuous limits of the XYZ model, the ladder property of the boost operator disappears. For the Hubbard model we demonstrate the non-existence of a ladder operator. Nevertheless, the general structure of the conserved charges is indicated, and the expression for the terms linear in the model's free parameter for all charges is derived in closed form. 
  A covariant pseudodifferential calculus on Riemann surfaces, based on the Krichever-Novikov global picture, is presented. It allows defining scalar and matrix KP operators, together with their reductions, in higher genus. Globally defined Miura maps are considered and the arising of polynomial or rational ${\cal W}$ algebras on R.S. associated to each reduction are pointed out. The higher genus NLS hierarchy is analyzed in detail. 
  The zero-momentum ghost-dilaton is a non-primary BRST physical state present in every bosonic closed string background. It is given by the action of the BRST operator on another state $\x$, but remains nontrivial in the semirelative BRST cohomology. When local coordinates arise from metrics we show that dilaton and $\x$ insertions compute Riemannian curvature and geodesic curvature respectively. A proper definition of a CFT deformation induced by the dilaton requires surface integrals of the dilaton and line integrals of $\x$. Surprisingly, the ghost number anomaly makes this a trivial deformation. While dilatons cannot deform conformal theories, they actually deform conformal string backgrounds, showing in a simple context that a string background is not necessarily the same as a CFT. We generalize the earlier proof of quantum background independence of string theory to show that a dilaton shift amounts to a shift of the string coupling in the field-dependent part of the quantum string action. Thus the ``dilaton theorem'', familiar for on-shell string amplitudes, holds off-shell as a consequence of an exact symmetry of the string action. 
  We present a first step towards generalizing the work of Seiberg and Witten on N=2 supersymmetric Yang-Mills theory to arbitrary gauge groups. Specifically, we propose a particular sequence of hyperelliptic genus $n-1$ Riemann surfaces to underly the quantum moduli space of $SU(n)$ N=2 supersymmetric gauge theory. These curves have an obvious generalization to arbitrary simply laced gauge groups, which involves the A-D-E type simple singularities. To support our proposal, we argue that the monodromy in the semiclassical regime is correctly reproduced. We also give some remarks on a possible relation to string theory. 
  I develop a formalism for solving topological field theories explicitly, in the case when the explicit expression of the instantons is known. I solve topological Yang-Mills theory with the $k=1$ Belavin {\sl et al.} instanton and topological gravity with the Eguchi-Hanson instanton. It turns out that naively empty theories are indeed nontrivial. Many unexpected interesting hidden quantities (punctures, contact terms, nonperturbative anomalies with or without gravity) are revealed. Topological Yang-Mills theory with $G=SU(2)$ is not just Donaldson theory, but contains a certain {\sl link} theory. Indeed, local and non-local observables have the property of {\sl marking} cycles. From topological gravity one learns that an object can be considered BRST exact only if it is so all over the moduli space ${\cal M}$, boundary included. Being BRST exact in any interior point of ${\cal M}$ is not sufficient to make an amplitude vanish. Presumably, recursion relations and hierarchies can be found to solve topological field theories in four dimensions, in particular topological Yang-Mills theory with $G=SU(2)$ on ${\bf R}^4$ and topological gravity on ALE manifolds. 
  This paper is the first in a series. The main goal of the series is to present a geometric construction of certain remarkable tensor categories arising from quantum groups coresponding to the value of deformation parameter $q$ equal to a root of unity.    In the present paper we study perverse sheaves over a complex affine space which are smooth along the stratification determined by a finite arrangement of complex affine hyperplanes defined by real equations. In particular, we construct explicitely (in terms of combinatorial data) complexes computing cohomology of Goresky-MacPherson extensions of one-dimensional local systems over the complement of hyperplanes. 
  We study the $c=1$ conformal field theory of a free compactified boson with radius $r=\sqrt{\beta}$ ($\beta$ is an integer). The Fock space of this boson is constructed in terms of anyon vertex operators and each state is labeled by an infinite set of pseudo-momenta of filled particles in pseudo-Dirac sea. Wave function of multi anyon state is described by an eigenfunction of the Calogero-Sutherland (CS) model. The $c=1$ conformal field theory at $r=\sqrt{\beta}$ gives a field theory of CS model. This is a natural generalization of the boson-fermion correspondence in one dimension to boson-anyon correspondence. There is also an interesting duality between anyon with statistics $\theta=\pi/\beta$ and particle with statistics $\theta=\beta \pi$. 
  The operator and the functional formulations of the dynamics of constrained systems are explored for determining unambiguously the quantum Hamiltonian of a nonrelativistic particle in a curved space. 
  On the basis of the collective field method, we analyze the Calogero--Sutherland model (CSM) and the Selberg--Aomoto integral, which defines, in particular case, the partition function of the matrix models. Vertex operator realizations for some of the eigenstates (the Jack polynomials) of the CSM Hamiltonian are obtained. We derive Virasoro constraint for the generalized matrix models and indicate relations with the CSM operators. Similar results are presented for the $q$--deformed case (the Macdonald operator and polynomials), which gives the generating functional of infinitely many conserved charges in the CSM. 
  We examine a collection of particles interacting with inverse-square two-body potentials in the thermodynamic limit. We find explicit large-amplitude density waves and soliton solutions for the motion of the system. Waves can be constructed as coherent states of either solitons or phonons. Therefore, either solitons or phonons can be considered as the fundamental excitations. The generic wave is shown to correspond to a two-band state in the quantum description of the system, while the limiting cases of solitons and phonons correspond to particle and hole excitations. 
  We review the salient features of spacetime and worldvolume supersymmetric super p-brane actions. These are sigma models for maps from a worldvolume superspace to the target superspace. For p-branes, the symmetries of the model depend crucially on the existence of closed super (p+1)-forms on a worldvolume superspace, built out of the pull-backs of the Kalb-Ramond super (p+1)-form in target superspace and its curvature. This formulation of super p-branes is usually referred to as the twistor-like formulation. 
  A review on topological strings and the geometry of the space of two dimensional theories. (Lectures given by C. Gomez at the Enrico Fermi Summer School, Varenna, July 1994) 
  We present an exact description of the metric on the moduli space of vacua and the spectrum of massive states for four dimensional N=2 supersymmetric SU(n) gauge theories. The moduli space of quantum vacua is identified with the moduli space of a special set of genus n-1 hyperelliptic Riemann surfaces. 
  It is shown that for a large class of non-holonomic quantum mechanical systems one can make the computation of BRST charge fully algorithmic. Two computer algebra programs written in the language of {\tt REDUCE} are described. They are able to realize the complex calculations needed to determine the charge for general nonlinear algebras. Some interesting specific solutions are discussed. 
  We show that a chiral sector of a symplectic group manifold possesses a symmetry similar to, but somewhat weaker than the Lie--Poisson one. 
  A crucial requirement for the standard interpretation of Monte Carlo simulations of simplicial quantum gravity is the existence of an exponential bound that makes the partition function well-defined. We present numerical data favoring the existence of an exponential bound, and we argue that the more limited data sets on which recently opposing claims were based are also consistent with the existence of an exponential bound. 
  The scattering of pointlike particles at very large center of mass energies and fixed low momentum transfers, occurring due to both their electromagnetic and gravitational interactions is re-examined in the particular case when one of the particles carries magnetic charge. At Planckian center-of-mass energies, when gravitational dominance is normally expected, the presence of magnetic charge is shown to produce dramatic modifications to the scattering cross section as well as to the holomorphic structure of the scattering amplitude. 
  We consider a spontaneously broken nonabelian topologically massive gauge theory in a broken phase possessing a residual nonabelian symmetry. Recently there has been some question concerning the renormalization of the Chern-Simons coefficient in such a broken phase. We show that, in this broken vacuum, the renormalized ratio of the Chern-Simons coupling to the gauge coupling is shifted by $1/4\pi$ times an integer, preserving the usual integer quantization condition on the bare parameters. 
  In this episode, it is shown how the octonion X-product is related to E8 lattices, integral domains, sphere fibrations, and other neat stuff. 
  We discuss a global anomaly associated with the coupling of chiral Weyl fermions to gravity. The Standard Model based upon $SU(3){\times}SU(2){\times}{U(1)}$ which has 15 fermions per generation is shown to be inconsistent if all background spin manifolds with signature invariant $\tau=8k$ are allowed. Similarly, GUTs based on odd number of Weyl fermions are inconsistent. Consistency can be achieved by adding an extra Weyl fermion which needs to couple only to gravity. For arbitrary $\tau$'s, generalized spin structures are needed, and the global anomaly cancellation requires that the net index of the total Dirac operator with spin and internal gauge connections be even. As a result GUTs with fundamental multiplets which contain multiples of 16 Weyls per generation are selected. The simplest consistent GUT is the SO(10) model with a multiplet of 16 Weyls per generation. The combined gravity and internal symmetry gauge group of the theory is then $[Spin(3,1){\times}Spin(10)]/Z_2$. Physical implications of these results are commented on. 
  The order $g^5$ contribution to the pressure of massless $g^2 \phi^4$ theory at nonzero temperature is obtained explicitly. Lower order contributions are reconsidered and two issues leading to the optimal choice of rearranged Lagrangian for such calculations are clarified. 
  An index relation $dim\ ker\ a^{\dagger}a - dim\ ker\ aa^{\dagger} = 1$ is satisfied by the creation and annihilation operators $a^{\dagger}$ and $a$ of a harmonic oscillator. A hermitian phase operator, which inevitably leads to $dim\ ker\ a^{\dagger}a - dim\ ker\ aa^{\dagger} = 0$, cannot be consistently defined. If one considers an $s+1$ dimensional truncated theory, a hermitian phase operator of Pegg and Barnett which carries a vanishing index can be defined. However, for arbitrarily large $s$, we show that the vanishing index of the hermitian phase operator of Pegg and Barnett causes a substantial deviation from minimum uncertainty in a characteristically quantum domain with small average photon numbers. We also mention an interesting analogy between the present problem and the chiral anomaly in gauge theory which is related to the Atiyah-Singer index theorem. It is suggested that the phase operator problem related to the above analytic index may be regarded as a new class of quantum anomaly. From an anomaly view point ,it is not surprising that the phase operator of Susskind and Glogower, which carries a unit index, leads to an anomalous identity and an anomalous commutator. 
  We study the BKP hierarchy and its $n$--reduction, for the case that $n$ is odd. This is related to the principal realization of the basic module of the twisted affine Lie algebra $\hat{sl}_n^{(2)}$. We show that the following two statements for a BKP $\tau$ function are equivalent: (1) $\tau$ is is $n$--reduced and satisfies the string equation, i.e. $L_{-1}\tau=0$, where $L_{-1}$ is an element of some `natural' Virasoro algebra. (2) $\tau$ satisfies the vacuum constraints of the $BW_{1+\infty}$ algebra. Here $BW_{1+\infty}$ is the natural analog of the $W_{1+\infty}$ algebra, which plays a role in the KP case. 
  The general structure of trace anomaly, suggested recently by Deser and Shwimmer, is argued to be the consequence of the Wess-Zumino consistency condition. The response of partition function on a finite Weyl transformation, which is connected with the cocycles of the Weyl group in $d=2k$ dimensions is considered, and explicit answers for $d=4,6$ are obtained. Particularly, it is shown, that addition of the special combination of the local counterterms leads to the simple form of that cocycle, quadratic over Weyl field $\sigma$, i.e. the form, similar to the two-dimensional Lioville action. This form also establishes the connection of the cocycles with conformal-invariant operators of order $d$ and zero weight. Beside that, the general rule for transformation of that cocycles into the cocycles of diffeomorphisms group is presented. 
  Adler, Shiota and van Moerbeke obtained for the KP and Toda lattice hierarchies a formula which translates the action of the vertex operator on tau--functions to an action of a vertex operator of pseudo-differential operators on wave functions. This relates the additional symmetries of the KP and Toda lattice hierarchyto the $W_{1+\infty}$--, respectively $W_{1+\infty}\times W_{1+\infty}$--algebra symmeties. In this paper we generalize the results to the $s$--component KP hierarchy. The vertex operators generate the algebra $W_{1+\infty}(gl_s)$, the matrix version of $W_{1+\infty}$. Since the Toda lattice hierarchy is equivalent to the $2$--component KP hierarchy, the results of this paper uncover in that particular case a much richer structure than the one obtained by Adler, Shiota and van Moerbeke. 
  We introduce a notion  of elliptic differential graded Lie algebra. The class of elliptic algebras contains such examples as the algebra of differential forms with values in endomorphisms of a flat vector bundle over a compact manifold, etc.   For elliptic differential graded algebra we construct a complete set of deformations.   We show that for several deformation problems the existence of a formal power series solution guarantees the existence of an analytic solution. 
  To every partition $n=n_1+n_2+\cdots+n_s$ one can associate a vertex operator realization of the Lie algebras $a_{\infty}$ and $\hat{gl}_n$. Using this construction we obtain reductions of the $s$--component KP hierarchy, reductions which are related to these partitions. In this way we obtain matrix KdV type equations. We show that the following two constraints on a KP $\tau$--function are equivalent (1) $\tau$ is a $\tau$--function of the $[n_1,n_2,\ldots,n_s]$--th reduced KP hierarchy which satisfies string equation, $L_{-1}\tau=0$, (2) $\tau$ satisfies the vacuum constraints of the $W_{1+\infty}$ algebra. Talk given at the V International Conference on Mathematical Physics, String Theory and Quantum Gravity at Alushta, June 10-20 1994 
  We consider the conversion of gravitons into photons in the $ TE_{mo} $ mode. Cross sections in different directions are given. 
  Closed expressions for the Green function and amplitude of the scalar particle scattering in the external gravitational field $g_{\mu\nu}(x)$ are found in the form of functional integrals. It is shown that, as compared with the scattering on the vector potential, the tensor character of the gravitational field leads to a more rapid increase of the cross section with increasing energy. Discrete energy levels of particles are obtained in the Newton potential. 
  The asymptotic conformal Yano--Killing tensor proposed in J. Jezierski, On the relation between metric and spin-2 formulation of linearized Einstein theory [GRG, in print (1994)] is analyzed for Schwarzschild metric and tensor equations defining this object are given. The result shows that the Schwarzschild metric (and other metrics which are asymptotically ``Schwarzschildean'' up to O(1/r^2) at spatial infinity) is among the metrics fullfilling stronger asymptotic conditions and supertranslations ambiguities disappear. It is also clear from the result that 14 asymptotic gravitational charges are well defined on the ``Schwarzschildean'' background. 
  We study the noncommutative geometry of a two-leaf Parisi--Sourlas supermanifold in Connes' formalism using different $K$-cycles over the Grassmann algebra valued functions on the supermanifold. We find that the curvature of the trivial noncommutative vector bundle defines in the simplest case the super Yang--Mills action coupled to a scalar field. By considering a modified Dirac operator and a suitable limit of its parameters we then obtain an action that turns out to be the continuum limit of the induced QCD in Kazakov--Migdal model. 
  Alternatives to Einstein's theory of general relativity can be distinguished by measuring the parametrised post Newtonian parameters. Two such parameters $\beta$ and $\gamma$, equal to one in Einstein theory, can be obtained from static spherically symmetric solutions. For the graviton-dilaton system, as in Brans-Dicke or low energy string theory, we find that if $\gamma \ne 1$ for a charge neutral point star, then there exist naked singularities. Thus, if $\gamma$ is measured to be different from one, then it cannot be explained by these theories, without implying naked singularities. We also couple a cosmological constant $\Lambda$ to the graviton-dilaton system, a la string theory. We find that static spherically symmetric solutions in low energy string theory, which describe the gravitational field of a point star in the real universe atleast upto a distance $r_* \simeq {\cal O} ({\rm pc})$, always lead to curvature singularities. These singularities are stable and much worse than the naked ones. Requiring their absence upto a distance $r_*$ implies a bound $| \Lambda | < 10^{- 102} (\frac{r_*}{{\rm pc}})^{- 2}$ in natural units. If $r_* \simeq 1 {\rm Mpc}$ then $| \Lambda | < 10^{- 114}$, and if $r_*$ extends all the way upto the edge of the universe ($10^{28} {\rm cm}$) then $| \Lambda | < 10^{- 122}$ in natural units. 
  A subtheory of a quantum field theory specifies von~Neumann subalgebras $\aa(\oo)$ (the `observables' in the space-time region $\oo$) of the von~Neumann algebras $\bb(\oo)$ (the `fields' localized in $\oo$). Every local algebra being a (type $\III_1$) factor, the inclusion $\aa(\oo) \subset \bb(\oo)$ is a subfactor. The assignment of these local subfactors to the space-time regions is called a `net of subfactors'. The theory of subfactors is applied to such nets. In order to characterize the `relative position' of the subtheory, and in particular to control the restriction and induction of superselection sectors, the canonical endomorphism is studied. The crucial observation is this: the canonical endomorphism of a local subfactor extends to an endomorphism of the field net, which in turn restricts to a localized endomorphism of the observable net. The method allows to characterize, and reconstruct, local extensions $\bb$ of a given theory $\aa$ in terms of the observables. Various non-trivial examples are given. 
  Two dimensional classical string theory is solved in any curved spacetime. The complete spacetime required to describe the classical string motions turns out to be larger than the global space required by complete particle geodesics. The solutions are fully classified by their behavior in the asymptotically flat region of spacetime. When the curvature is smooth, the string solutions are deformed folded string solutions as compared to flat spacetime folded strings that were known for 19 years. However, surprizing new stringy behavior becomes evident at curvature singularities such as black holes. The global properties of the classical string theory require that the ``bare singularity region'' of the black hole be included along with the usual black hole spacetime. The mathematical structure needed to describe the solutions include a recursion relation that is analogous to the transfer matrix of lattice theories. This encodes lattice-like properties on the worldsheet on the one hand and the geometry of spacetime on the other hand. A case is made for the presence of folded strings in the quantum theory of non-critical strings for $d\geq 2$. 
  For a manifold M we define a structure on the group action of Diff(M) on the smooth functions on M which reduces to the usual differential geometry upon differentiation at zero along the one-parameter groups of Diff(M). This ``integrated differential geometry'' generalises to all group actions on associative algebras, including noncommutative ones, and defines an ``integrated de Rham cohomology,'' which provides a new set of invariants for group actions. We calculate the first few integrated de Rham cohomologies for two examples;- a discrete group action on a commutative algebra, and a continuous Lie group action on a noncommutative matrix algebra. 
  We present the causal construction of perturbative Yang-Mills theories in four(3+1)-dimensional space-time. We work with free quantum fields throughout. The inductive causal method by Epstein and Glaser leads directly to a finite perturbation series and does not rely on an intermediary regularization of the theory. The causal method naturally separates the physical infrared problem of massless theories from ultraviolet-sensitive features like normalizability by regarding the distributional character of the S-matrix. We prove the normalizability of the Yang-Mills theory with fermionic matter fields and study the discrete symmetry transformations in the causal formalism. We introduce a definition of nonabelian gauge invariance which only involves the free asymptotic field operators and give mathematically rigorous and conceptually simple proofs of nonabelian gauge invariance and of the physical unitarity of the S-matrix in all orders of perturbation theory. 
  We build a toy model of the Wilson-Kogut renormalization group in one dimensional Quantum Mechanics. With it, we show how the RG flow in the space of 1-D S matrices of finite range defines, as renormalized interactions, the known four parametric set of contact interactions, thus providing a suitable framework for its study. 
  We discuss how a large class of incompressible quantum Hall states can be characterized as highest weight states of different representations of the \Winf algebra. Second quantized expressions of the \Winf generators are explicitly derived in the cases of multilayer Hall states, the states proposed by Jain to explain the hierarchical filling fractions and the ones related by particle-hole conjugation. 
  In this note we strenghten a theorem by Esnault-Schechtman-Viehweg which states that one can compute the cohomology of a complement of hyperplanes in a complex affine space with coefficients in a local system using only logarithmic global differential forms, provided certain "Aomoto non-resonance conditions" for monodromies are fulfilled at some "edges" (intersections of hyperplanes). We prove that it is enough to check these conditions on a smaller subset of edges.    We show that for certain known one dimensional local systems over configuration spaces of points in a projective line defined by a root system and a finite set of affine weights (these local systems arise in the geometric study of Knizhnik-Zamolodchikov differential equations), the Aomoto resonance conditions at non-diagonal edges coincide with Kac-Kazhdan conditions of reducibility of Verma modules over affine Lie algebras. 
  Given an irreducible inclusion of infinite von-Neumann-algebras $\cn \subset \cm$ together with a conditional expectation $ E : \cm \rightarrow \cm $ such that the inclusion has depth 2, we show quite explicitely how $\cn $ can be viewed as the fixed point algebra of $\cm$ w.r.t. an outer action of a compact Kac-algebra acting on $\cm$. This gives an alternative proof, under this special setting of a more general result of M. Enock and R. Nest, [E-N], see also S. Yamagami, [Ya2]. 
  We give evidence, by use of the Thermodynamic Bethe Ansatz approach, of the existence of both massive and massless behaviours for the $\phi_{2,1}$ perturbation of the $M_{3,5}$ non-unitary minimal model, thus resolving apparent contradictions in the previous literature. The two behaviours correspond to changing the perturbing bare coupling constant from real values to imaginary ones. Generalizations of this picture to the whole class of non-unitary minimal models $M_{p,2p\pm 1}$, perturbed by their least relevant operator lead to a cascade of flows similar to that of unitary minimal models perturbed by $\phi_{1,3}$. Various aspects and generalizations of this phenomenon and the links with the Izergin-Korepin model are discussed. 
  The complex-time method for quantum tunneling is studied. In one-dimensional quantum mechanics, we construct a reduction formula for a Green function in the number of turning points based on the WKB approximation. This formula yields a series, which can be interpreted as a sum over the complex-time paths. The weights of the paths are determined. 
  All bicovariant first order differential calculi on the quantum group GLq(3,C) are determined. There are two distinct one-parameter families of calculi. In terms of a suitable basis of 1-forms the commutation relations can be expressed with the help of the R-matrix of GLq(3,C). Some calculi induce bicovariant differential calculi on SLq(3,C) and on real forms of GLq(3,C). For generic deformation parameter q there are six calculi on SLq(3,C), on SUq(3) there are only two. The classical limit q-->1 of bicovariant calculi on SLq(3,C) is not the ordinary calculus on SL(3,C). One obtains a deformation of it which involves the Cartan-Killing metric. 
  In Yang-Mills theory massless point sources lead naturally to shock-wave configurations. Their magnetic counterparts endow the vacuum of the four-dimensional compact abelian model with a Coulomb-gas behaviour whose physical implications are briefly discussed. (Contribution to ``Quark Confinement and the Hadron Spectrum'', Como 20-24 June 1994. Revised version.) 
  The fibre bundle formulation of gauge theory is generally accepted. The jet manifold machinery completes this formulation and provides the adequate mathematical description of dynamics of fields represented by sections of fibre bundles. Theory of differential operators, Lagrangian and Hamiltonian formalisms on bundles have utilized widely the language of jet manifolds. Moreover, when not restricted to principal connections, differential geometry also is phrased in jet terms. However, this powerful tool remains almost unknown to physicists. These Lectures give introduction to jet manifolds, Lagrangian and Hamiltonian formalisms in jet manifolds and their application to a number of fundamental field models. 
  Gravity can arise in a conventional non-Abelian gauge theory in which a specific phenomenon takes place. Suppose there is a condensation of polarized instantons and antiinstantons in the vacuum state. Then the excitations of the gauge field in the classical approximation are described through the variables of Riemann geometry satisfying the Einstein equations of general relativity. There are no dimensional coupling constants in the theory. 
  Utilizing (4,0) superfields, we discuss aspects of supersymmetric sigma models and the ADHM construction of instantons a' la Witten. 
  We investigate all the four-body graviton interaction processes: $gX\rightarrow \gamma X$, $gX\rightarrow gX$, and $gg\rightarrow gg$ with $X$ as an elementary particle of spin less than two in the context of linearized gravity except the spin-3/2 case. We show explicitly that gravitational gauge invariance and Lorentz invariance cause every four-body graviton scattering amplitude to be factorized. We explore the implications of this factorization property by investigating polarization effects through the covariant density matrix formalism in each four-body graviton scattering process. 
  We construct new realizations of the Virasoro algebra inspired by the Calogero model. The Virasoro algebra we find acts as a kind of spectrum-generating algebra of the Calogero model. We furthermore present the superextension of these results and introduce a class of higher-spin extensions of the Virasoro algebra which are of the $W_\infty$ - type. 
  A physical system is in local equilibrium if it cannot be distinguished from a global equilibrium by ``infinitesimally localized measurements''. This should be a natural characterization of local equilibrium, but the problem is to give a precise meaning to the qualitative phrase ``infinitesimally localized measurements''. A solution is suggested in form of a Local Equilibrium Condition, which can be applied to linear relativistic quantum field theories but not directly to selfinteracting quantum fields. The concept of local temperature resulting from LEC is compared to an old approach to local temperature based on the principle of maximal entropy. It is shown that the principle of maximal entropy does not always lead to physical states if it is applied to relativistic quantum field theories. 
  In this letter we discuss the relation of four-dimensional, $N=2$ supersymmetric string backgrounds to integrable models. In particular we show that non-K\"ahlerian gravitational backgrounds with one $U(1)$ isometry plus non-trivial antisymmetric tensor and dilaton fields arise as the solutions of the Liouville equation or, for the case of vanishing central charge deficit, as the solutions of the continual Toda equation. When performing an Abelian duality transformation, a particular class of solutions of the continual Toda equation leads to the well-known gravitational Eguchi-Hanson instanton background with self-dual curvature tensor. 
  In this paper we investigate the problem of spontaneous supersymmetry breaking without a cosmological term in $N=3$ supergravity with matter vector multiplets, scalar fields geometry being $SU(3,m)/SU(3)\otimes SU(m)\otimes U(1)$. At first, we consider the case of minimal coupling with different possible gaugings (compact as well as non-compact). Then we show that there exist dual version of such a theory (with the same scalar field geometry), which turns out to be the generalization of the $N=3$ hidden sector, constructed some time ago by one of us, to the case of arbitrary number of vector multiplets. We demonstrate that spontaneous supersymmetry breaking is still possible in the presence of matter multiplets. 
  I discuss some current thoughts on how low-energy measurements and consistency constrain theories at high energies with emphasis on string theory. I also discuss some recent work on the dynamics of supersymmetric gauge theories. 
  The construction approach proposed in the previous paper Ref. 1 allows us there and in the present paper to construct at generic deformation parameter $q$ all finite--dimensional representations of the quantum Lie superalgebra $U_{q}[gl(2/2)]$. The finite--dimensional $U_{q}[gl(2/2)]$-modules $W^{q}$ constructed in Ref. 1 are either irreducible or indecomposible. If a module $W^{q}$ is indecomposible, i.e. when the condition (4.41) in Ref. 1 does not hold, there exists an invariant maximal submodule of $W^{q}$, to say $I_{k}^{q}$, such that the factor-representation in the factor-module $W^{q}/I_{k}^{q}$ is irreducible and called nontypical. Here, in this paper, indecomposible representations and nontypical finite--dimensional representations of the quantum Lie superalgebra $U_{q}[gl(2/2)]$ are considered and classified as their module structures are analized and the matrix elements of all nontypical representations are written down explicitly. 
  We consider a simple model describing a closed bosonic string in a constant magnetic field. Exact conformal invariance demands also the presence of a non-trivial metric and antisymmetric tensor (induced by the magnetic field). The model is invariant under target space duality in a compact Kaluza-Klein direction introduced to couple the magnetic field. Like open string theory in a constant gauge field, or closed string theory on a torus, this model can be straightforwardly quantized and solved with its spectrum of states and partition function explicitly computed. Above some critical value of the magnetic field an infinite number of states become tachyonic, suggesting a presence of phase transition. We also construct heterotic string generalisations of this bosonic model in which the constant magnetic field is embedded either in the Kaluza-Klein or internal gauge group sector. 
  Treating the QCD Wilson loop as amplitude for the propagation of the first quantized particle we develop the second quantization of the same propagation. The operator of the particle position $\hat{\cal X}_{\mu}$ (the endpoint of the "open string") is introduced as a limit of the large $N$ Hermitean matrix. We then derive the set of equations for the expectation values of the vertex operators $\VEV{ V(k_1)\dots V(k_n)} $. The remarkable property of these equations is that they can be expanded at small momenta (less than the QCD mass scale), and solved for expansion coefficients. This provides the relations for multiple commutators of position operator, which can be used to construct this operator. We employ the noncommutative probability theory and find the expansion of the operator $\hat{\cal X}_\mu $ in terms of products of creation operators $ a_\mu^{\dagger}$. In general, there are some free parameters left in this expansion. In two dimensions we fix parameters uniquely from the symplectic invariance. The Fock space of our theory is much smaller than that of perturbative QCD, where the creation and annihilation operators were labelled by continuous momenta. In our case this is a space generated by $d = 4$ creation operators. The corresponding states are given by all sentences made of the four letter words. We discuss the implication of this construction for the mass spectra of mesons and glueballs. 
  We study the $N=2$ string with a real structure on the $(2,2)$ spacetime, using BRST methods. Several new features emerge. In the diagonal basis, the operator $\exp(\lambda \int^z J^{\rm tot})$, which is associated with the moduli for the $U(1)$ gauge field on the world-sheet, is local and it relates the physical operators in the NS and R sectors. However, the picture-changing operators are non-invertible in this case, and physical operators in different pictures cannot be identified. The three-point interactions of all physical operators leads to three different types of amplitudes, which can be effectively described by the interactions of a scalar NS operator and a bosonic spinorial R operator. In the off-diagonal bases for the fermionic currents, the picture-changing operators are invertible, and hence physical operators in different pictures can be identified. However, now there is no local operator $\exp(\lambda \int^z J^{\rm tot})$ that relates the physical operators in different sectors. The physical spectrum is thus described by one scalar NS operator and one spinorial R operator. The NS and R operators give rise to different types of three-point amplitudes, and thus cannot be identified. 
  Recent developments in the understanding of $N=2$ supersymmetric Yang-Mills theory in four dimensions suggest a new point of view about Donaldson theory of four manifolds: instead of defining four-manifold invariants by counting $SU(2)$ instantons, one can define equivalent four-manifold invariants by counting solutions of a non-linear equation with an abelian gauge group. This is a ``dual'' equation in which the gauge group is the dual of the maximal torus of $SU(2)$. The new viewpoint suggests many new results about the Donaldson invariants. 
  Kink-kink S-matrix for integrable vector perturbed $WD_{n}^{(k)}$ minimal models is constructed from the Boltzmann weights of $A_{2n-1}^{(2)}$  RSOS model and checked in two limit cases  $k=1$ and $k\rightarrow \infty$ 
  In this paper we consider the problem of spontaneous supersymmetry breaking in $N=4$ supergravity interacting with vector multiplets. We start with the ordinary version of such model with the scalar field geometry $SU(1,1)/U(1)\otimes SO(6,m)/SO(6)\otimes SO(m)$. Then we construct a dual version of this theory with the same scalar field geometry, which corresponds to the interaction of arbitrary number of vector multiplets with the hidden sector, admitting spontaneous supersymmetry breaking without a cosmological term. We show that supersymmetry breaking is still possible in the presence of matter fields. 
  We carry out a parallel study of the covariant phase space and the conservation laws of local symmetries in two-dimensional dilaton gravity. Our analysis is based on the fact that the Lagrangian can be brought to a form that vanishes on-shell giving rise to a well-defined covariant potential for the symplectic current. We explicitly compute the symplectic structure and its potential and show that the requirement to be finite and independent of the Cauchy surface restricts the asymptotic symmetries. 
  Representations of $SO(5)_{q}$ can be constructed on bases such that either the Chevalley triplet $(e_{1},\;f_{1},\;h_{1})$ or $(e_{2},\;f_{2},\;h_{2})$ has the standard $SU(2)_{q}$ matrix elements. The other triplet in each cases has a more complicated action. The $q$-deformation of such representations present striking differences. In one case a {\bf non-minimal} deformation is found to be essential. This is explained and illustrated below. Broader interests of a parallel use of the two bases are pointed out. 
  This letter studies the Sp(2) covariant quantisation of gauge theories. The geometrical interpretation of gauge theories in terms of quasi principal fibre bundles $Q(M_S, G_S)$ is reviewed. It is then described the Sp(2) algebra of ordinary Yang-Mills theory. A consistent formulation of covariant lagrangian quantisation for general gauge theories based on Sp(2) BRST symmetry is established. The original N=1, ten dimensional superparticle is considered as an example of infinitely reducible gauge algebras, and given explicitly its Sp(2) BRST invariant action. 
  Path integration over Euclidean chiral fermions is replaced by the quantum mechanics of an auxiliary system of non--interacting fermions. Our construction avoids the no--go theorem and faithfully maintains all the known important features of chiral fermions, including the violation of some perturbative conservation laws by gauge field configurations of non--trivial topology. 
  Using the effective potential, we study the one-loop renormalization of a massive self-interacting scalar field at finite temperature in flat manifolds with one or more compactified spatial dimensions. We prove that, owing to the compactification and finite temperature, the renormalized physical parameters of the theory (mass and coupling constant) acquire thermal and topological contributions. In the case of one compactified spatial dimension at finite temperature, we find that the corrections to the mass are positive, but those to the coupling constant are negative. We discuss the possibility of triviality, i.e. that the renormalized coupling constant goes to zero at some temperature or at some radius of the compactified spatial dimension. 
  The contributions of Emmy Noether to particle physics fall into two categories. One is given under the rubric of Noether's theorem, and the other may be described as her important contributions to modern mathematics. These are discussed along with an historical account of her work and what its impact has been. In addition a brief biography is given. (To be published in the Proceedings of the Int'l Conf. on The History of Original Ideas and Basic Discoveries in Particle Physics, Erice, Italy, 29 July - 4 Aug., 1994.) 
  We investigate the distribution of energy density in a stationary self-reproducing inflationary universe. We show that the main fraction of volume of the universe in a state with a given density $\rho$ at any given moment of time $t$ in synchronous coordinates is concentrated near the centers of deep exponentially wide spherically symmetric holes in the density distribution. A possible interpretation of this result is that a typical observer should see himself living in the center of the world. Validity of this interpretation depends on the choice of measure in quantum cosmology. Our investigation suggests that unexpected (from the point of view of inflation) observational data, such as possible local deviations from $\Omega = 1$, or possible dependence of the Hubble constant on the length scale, may tell us something important about quantum cosmology and particle physics at nearly Planckian densities. 
  The $\XXZ$ spin chain with a boundary magnetic field $h$ is considered, using the vertex operator approach to diagonalize the Hamiltonian. We find explicit bosonic formulas for the two vacuum vectors with zero particle content. There are three distinct regions when $h\geq0$, in which the structure of the vacuum states is different. Excited states are given by the action of vertex operators on the vacuum states. We derive the boundary $S$-matrix and present an integral formula for the correlation functions. The boundary magnetization exhibits boundary hysteresis. We also discuss the rational limit, the $\XXX$ model. 
  In this paper we discuss a generalization of the classica PBW-theorem to the case of Koszul algebras. Our result is a slight generalization of that obtained by A.Polischuk and L.Positselsky, but the proof is different and uses deformation theory. 
  I discuss the construction of the effective action for QCD suitable for the description of high-energy and small momentum transfer diffractive processes. 
  The $\kappa$-deformed $D=4$ Poincar{\'e} superalgebra written in Hopf superalgebra form is transformed to the basis with classical Lorentz subalgebra generators. We show that in such a basis the $\kappa$-deformed $D=4$ Poincare superalgebra can be written as graded bicrossproduct. We show that the $\kappa$-deformed $D=4$ superalgebra acts covariantly on $\kappa$-deformed chiral superspace. 
  We discuss a relativistic free particle with fractional spin in 2+1 dimensions, where the dual spin components satisfy the canonical angular momentum algebra $\left\{ S_\mu , S_\nu \right\}\,=\,\epsilon_{\mu \nu \gamma}S^\gamma $. It is shown that it is a general consequence of these features that the Poincar\`e invariance is broken down to the Lorentz one, so indicating that it is not possible to keep simultaneously the free nature of the anyon and the translational invariance. 
  This is about new unexpected features of the Mandestam-Leibbrandt prescription found as applied to spacelike Wilson lines. The regularization parameter $ \omega $ in the M-L denominator for the spurious poles has to be kept throughout the calculation till the very end or else the integrals do not make sense. We get various `ambiguous' terms of the form $ \omega ^{-{\epsilon\over2}} \epsilon^{-2} $ which are not controlled by any sort of Ward identity. These terms cancel out in the sum and the final result is independent of $ \omega $. However, for the self energy on the spacelike Wilson line we obtain an unexpected double pole at $ \epsilon=0 $, using dimensional regularization in $ 4-\epsilon $ dimensions. 
  We examine the properties and symmetries of the lowest order effective theory of 4-dim string backgrounds with axion and dilaton fields and zero cosmological constant. The dimensional reduction yields an O(2,2) current group of transformations in the presence of two commuting Killing symmetries. Special emphasis is placed on the identification of the T and S string duality symmetries, and their intertwining relations. (Contributed to the Proceedings of the Satellite Colloquium "Topology, Strings and Integrable Models" to the XIth International Congress of Mathematical Physics, Paris, 25--28 July 1994; Diderot Editeur.) 
    The "minimal matter + scalar" system can be embedded into the twisted N=2 topological algebra in two ways: a la DDK or a la KM. Here we present some results concerning the topological descendants and their DDK and KM realizations. In particular, we prove four "no-ghost" theorems (two for null states) regarding the reduction of the topological descendants into secon- daries of the "minimal matter + scalar" conformal field theory. We write down the relevant expressions for the case of level 2 descendants. 
  We represent the two - dimensional planar classical continuous Heisenberg spin model as a constrained Chern-Simons gauged nonlinear Schr\"odinger system. The hamiltonian structure of the model is studied, allowing the quantization of the theory by the gauge invariant approach. A preliminary study of the quantum states is displayed and several physical consequences in terms of anyons are discussed. 
  Some physical aspects of $q$-deformed spacetimes are discussed. It is pointed out that, under certain standard assumptions relating deformation and quantization, the classical limit (Poisson bracket description) of the dynamics is bound to contain unusual features. At the same time, it is argued that the formulation of an associated $q$-deformed field theory is fraught with serious difficulties. 
  We compute critical exponents in a $Z_2$ symmetric scalar field theory in three dimensions, using Wilson's exact renormalization group equations expanded in powers of derivatives. A nontrivial relation between these exponents is confirmed explicitly at the first two orders in the derivative expansion. At leading order all our results are cutoff independent, while at next-to-leading order they are not, and the determination of critical exponents becomes ambiguous. We discuss the possible ways in which this scheme ambiguity might be resolved. 
  From the global chiral SU(2)*U(1) that appears in the electroweak Left-Right Model in the fundamental representation, a continuous transition to the representation of the minimal Standard Model is considered in the Cartan subalgebra of the right-handed sector. The connecting parameter is the splitting of U(1)_R quantum numbers. It is a deformation parameter, breaks SU(2)_R and parity and is proportional to an isomagnetic field, leading to Copenhagen type vacuum. A simple mapping on the fundamental representation of SU(2)_q gives the splitting in terms of q. 
  An algebraic restriction of the nonabelian self-dual Chern-Simons-Higgs systems leads to coupled abelian models with interesting mass spectra. The vacua are characterized by embeddings of $SU(2)$ into the gauge algebra, and in the broken phases the gauge and real scalar masses coincide, reflecting the relation of these self-dual models to $N=2$ SUSY. The masses themselves are related to the exponents of the gauge algebra, and the self-duality equation is a deformation of the classical Toda equations. 
  Starting from flat two-dimensional gauge potentials we propose the notion of ${\cal W}$-gauge structure in terms of a nilpotent BRS differential algebra. The decomposition of the underlying Lie algebra with respect to an $SL(2)$ subalgebra is crucial for the discussion conformal covariance, in particular the appearance of a projective connection. Different $SL(2)$ embeddings lead to various ${\cal W}$-gauge structures. We present a general soldering procedure which allows to express zero curvature conditions for the ${\cal W}$-currents in terms of conformally covariant differential operators acting on the ${\cal W}$ gauge fields and to obtain, at the same time, the complete nilpotent BRS differential algebra generated by ${\cal W}$-currents, gauge fields and the ghost fields corresponding to ${\cal W}$-diffeomorphisms. As illustrations we treat the cases of $SL(2)$ itself and to the two different $SL(2)$ embeddings in $SL(3)$, {\it viz.} the ${\cal W}_3^{(1)}$- and ${\cal W}_3^{(2)}$-gauge structures, in some detail. In these cases we determine algebraically ${\cal W}$-anomalies as solutions of the consistency conditions and discuss their Chern-Simons origin. 
  Let g be a semisimple Lie algebra over an algebraically closed field k of characteristic 0. Let V be a simple finite-dimensional g-module and let y\in V be a highest weight vector. It is a classical result of B. Kostant that the algebra of functions on the closure of the orbit of y under the simply connected group which corresponds to g is quadratic (i.e. the closuree of the orbit is a quadratic cone). In the present paper we extend this result of Kostant to the case of the quantized universal enveloping algebra U_q(g). The result uses certain information about spectrum of braiding operators for U_q(g) due to Reshetikhin and Drinfeld. 
  A general definition of a linear connection in noncommutative geometry has been recently proposed. Two examples are given of linear connections in noncommutative geometries which are based on matrix algebras. They both possess a unique metric connection. 
  We reinterpret recent work on extremal Reissner- Nordstr\o m black holes to argue that the mass is a measure of the gravitational entropy. We also find that the entropy of scalar fields in this background has a stronger divergence than usual. 
  We consider possible mixing of electromagnetic and gravitational shock waves, in the Planckian energy scattering of point particles in Minkowski space. By boosting a Reissner-Nordstr\"om black hole solution to the velocity of light, it is shown that no mixing of shock waves takes place for arbitrary finite charge carried by the black hole. However, a similar boosting procedure for a charged black hole solution in dilaton gravity yields some mixing : the wave function of even a neutral test particle, acquires a small additional phase factor depending on the dilatonic black hole charge. Possible implications for poles in the amplitudes for the dilaton gravity case are discussed. 
  It has been shown that the high-temperature limit of perturbative thermal QCD is easily obtained from the Boltzmann transport equation for `classical' coloured particles. We generalize this treatment to curved space-time. We are thus able to construct the effective stress-energy tensor. We give a construction for an effective action. As an example of the convenience of the Boltzmann method, we derive the high-temperature 3-graviton function. We discuss the static case. 
  A residue-theoretic representation is given for massless matter fields in (quotients) of (weighted) \CY\ complete intersection models and the corresponding chiral operators in \LGO{s}.   The well known polynomial deformations are thus generalized and the universal but somewhat abstract Koszul computations acquire a concrete realization and a general but more heuristic reinterpretation.   A direct correspondence with a BRST-type analysis of constrained systems also emerges naturally. 
  Using a cohomological characterization of the consistent and the covariant Lorentz and gauge anomalies, derived from the complexification of the relevant algebras, we study in $d=2$ the definition of the Weyl determinant for a non-abelian theory with Riemannian background. We obtain two second order operators that produce, by means of $\zeta$-function regularization, respectively the consistent and the covariant Lorentz and gauge anomalies, preserving diffeomorphism invariance. We compute exactly their functional determinants and the W-Z-W terms: the ``consistent'' determinant agrees with the non-abelian generalization of the classical Leutwyler's result, while the ``covariant'' one gives rise to a covariant version of the usual Wess-Zumino-Witten action. 
  Owing to subtle issues concerning quantum fluctuations and gauge fixing, a formulation of a general procedure to specify the realization of non-Abelian gauge symmetry has evaded all earlier attempts. In this Letter, we discuss these subtleties and present two sets of order parameters for non-Abelian gauge theories. Such operators directly probe the manifest low energy symmetry group and are crucial for the study of the phase diagram of a non-Abelian gauge theory. 
  A study of the gauged Wess-Zumino-Witten models is given focusing on the effect of topologically non-trivial configurations of gauge fields. A correlation function is expressed as an integral over a moduli space of holomorphic bundles with quasi-parabolic structure. Two actions of the fundamental group of the gauge group is defined: One on the space of gauge invariant local fields and the other on the moduli spaces. Applying these in the integral expression, we obtain a certain identity which relates correlation functions for configurations of different topologies. It gives an important information on the topological sum for the partition and correlation functions. 
  New relations of correlation functions are found in topological string theory; one for each second cohomology class of the target space. They are close cousins of the Deligne-Dijkgraaf-Witten's puncture and dilaton equations. When combined with the dilaton equation and the ghost number conservation, the equation for the first chern class of the target space gives a constraint on the topological sum (over genera and (multi-)degrees) of partition functions. For the $\CP^1$ model, it coincides with the dilatation constraint which is derivable in the matrix model recently introduced by Eguchi and Yang. 
  In the Bogomol'nyi limit of the Calogero-Sutherland collective-field model we find static-soliton solutions. The solutions of the equations of motion are moving solitons, having no static limit for $\l>1$. They describe holes and lumps, depending on the value of the statistical parametar $\l$. 
  We test perturbatively a recent scheme for implementing chiral fermions on the lattice, proposed by Kaplan and modified by Narayanan and Neuberger, using as our testing ground the chiral Schwinger model. The scheme is found to reproduce the desired form of the effective action, whose real part is gauge invariant and whose imaginary part gives the correct anomaly in the continuum limit, once technical problems relating to the necessary infinite extent of the extra dimension are properly addressed. The indications from this study are that the Kaplan--Narayanan--Neuberger (KNN) scheme has a good chance at being a correct lattice regularization of chiral gauge theories. 
  It is shown that the algebraic--geometrical (or quasiperiodic) solutions of the Conformal Affine $sl(2)$ Toda model are generated from the vacuum via dressing transformations. This result generalizes the result of Babelon and Bernard which states that the $N$--soliton solutions are generated from the vacuum by dressing transformations. 
    I start from the Bargmann-Wigner equations and introduce an interaction in the form which is similar to a $j=1/2$ case [M. Moshinsky \& A. Szczepaniak, {\it J. Phys. \,A}{\bf 22}  (1989) L817]. By means of the expansion of the wave function in the complete set of $\gamma$- matrices one can obtain the equations for a system which could be named as the $j=0$ Kemmer-Dirac oscillator.     The equations for the components $\phi_1$ and $\phi_2$ are different from the ones obtained by Y. Nedjadi \& R. Barrett for the $j=0$ Duffin-Kemmer-Petiau oscillator [{\it J. Phys.\,A} {\bf 27} (1994) 4301]. This fact leads to the dissimilar energy spectrum of the $j=0$ relativistic oscillator. 
  We introduce a general theory of twisting algebraic structures based on actions of a bialgebra. These twists are closely related to algebraic deformations and also to the theory of quasi-triangular bialgebras. In particular, a deformation produced from a universal deformation formula (UDF) is a special case of a twist. The most familiar example of a deformation produced from a UDF is perhaps the "Moyal product" which (locally) is the canonical quantization of the algebra of functions on a symplectic manifold in the direction of the Poisson bracket. In this case, the derivations comprising the Poisson bracket mutually commute and so this quantization is essentially obtained by exponentiating this bracket. For more general Poisson manifolds, this formula is not applicable since the associated derivations may no longer commute. We provide here generalizations of the Moyal formula which (locally) give canonical quantizations of various Poisson manifolds. Specifically, whenever a certain central extension of a Heisenberg Lie group acts on a manifold, we obtain a quantization of its algebra of functions in the direction of a suitable Poisson bracket obtained from noncommuting derivations. 
  Symmetries in the Lagrangian formalism of arbitrary order are analysed with the help of the so-called Anderson-Duchamp-Krupka equations. For the case of second order equations and a scalar field we establish a polynomial structure in the second order derivatives. This structure can be used to make more precise the form of a general symmetry. As an illustration we analyse the case of Lagrangian equations with Poincar\'e invariance or with universal invariance. 
  We discuss the equivalence between the path integral representations of spin dynamics for anti-commuting (Grassmann) and commuting variables and establish a bosonization dictionary for both generators of spin and single fermion operators. The content of this construction in terms of the representations of the spin algebra is discussed in the path integral setting. Finally it is shown how a `free field realization' (Dyson mapping) can be constructed in the path integral. 
  Compact string expressions are found for non-intersecting Wilson loops in SU(N) Yang-Mills theory on any surface (orientable or nonorientable) as a weighted sum over covers of the surface. All terms from the coupled chiral sectors of the 1/N expansion of the Wilson loop expectation values are included. 
  We describe the underlying U_q(g)--module structure of representations of quantum affine algebras. 
  We prove a highest weight theorem classifying irerducible finite--dimensional representations of quantum affine algebras and survey what is currently known about the structure of these representations. 
  We derive explicit forms of the two--point correlation functions of the $O(N)$ non-linear sigma model at the critical point, in the large $N$ limit, on various three dimensional manifolds of constant curvature. The two--point correlation function, $G(x, y)$, is the only $n$-point correlation function which survives in this limit. We analyze the short distance and long distance behaviour of $G(x, y)$. It is shown that $G(x, y)$ decays exponentially with the Riemannian distance on the spaces $R^2 \times S^1,~S^1 \times S^1 \times R, ~S^2 \times R,~H^2 \times R$. The decay on $R^3$ is of course a power law. We show that the scale for the correlation length is given by the geometry of the space and therefore the long distance behaviour of the critical correlation function is not necessarily a power law even though the manifold is of infinite extent in all directions; this is the case of the hyperbolic space where the radius of curvature plays the role of a scale parameter. We also verify that the scalar field in this theory is a primary field with weight $\delta=-{1 \over 2}$; we illustrate this using the example of the manifold $S^2 \times R$ whose metric is conformally equivalent to that of $R^3-\{0\}$ up to a reparametrization. 
  The cyclic quantum dilogarithm is interpreted as a cyclic 6j-symbol of the Weyl algebra, considered as a Borel subalgebra $BU_q(sl(2))$. Using modified 6j-symbols, an invariant of triangulated links in triangulated 3-manifolds is constructed. Apparently, it is an ambient isotopy invariant of links. 
  Some general properties of the relativistic $p$-dimensional surface imbedded into $D$-dimensional spacetime and its reduction to the sim\-plest case of the quadratic Lagrangian (the linearized model) are considered. The solutions of the equations of motion of the linearized model for the $p$-brane with arbitrary topology and massless eigenstates, as well as with critical dimension after quantization are presented. Some generalizations for the supermembrane are discussed. 
  We demonstrate electric-magnetic duality in N=1 supersymmetric non-Abelian gauge theories in four dimensions by presenting two different gauge theories (different gauge groups and quark representations) leading to the same non-trivial long distance physics. The quarks and gluons of one theory can be interpreted as solitons (non-Abelian magnetic monopoles) of the elementary fields of the other theory. The weak coupling region of one theory is mapped to a strong coupling region of the other. When one of the theories is Higgsed by an expectation value of a squark, the other theory is confined. Massless glueballs, baryons and Abelian magnetic monopoles in the confining description are the weakly coupled elementary quarks (i.e.\ solitons of the confined quarks) in the dual Higgs description. 
  Berkovits and Vafa recently showed that critical string theories can be considered as critical superstring theories with a special choice of background. The embedding of the Virasoro algebra into the super-Virasoro algebra involved in this construction has been extended to the noncritical case by Berkovits and Ohta. It is shown that the resulting nonlinear realization of the super-Virasoro algebra can be interpreted using standard techniques from the theory of nonlinear realizations. This extends earlier work of Kunitomo. 
  We investigate how exotic differential structures may reveal themselves in particle physics. The analysis is based on the A. Connes' construction of the standard model. It is shown that, if one of the copies of the spacetime manifold is equipped with an exotic differential structure, compact object of geometric origin may exist even if the spacetime is topologically trivial. Possible implications are discussed. An $SU(3)\otimes SU(2)\otimes U(1)$ gauge model is constructed. This model may not be realistic but it shows what kind of physical phenomena might be expected due to the existence of exotic differential structures on the spacetime manifold. 
  We consider gauged Wess-Zumino models based on the non compact group $SU(2,1)$. It is shown that by vector gauging the maximal compact subgroup $U(2)$ the resulting backgrounds obey the gravity-dilaton one loop string vacuum equations of motion in four dimensional euclidean space. The torsionless solution is then interpreted as a pseudo-instanton of the $d=4$ Liouville theory coupled to gravity. The presence of a traslational isometry in the model allows to get another string vacuum backgrounds by using target duality that we identify with those corresponding to the axial gauging. We also compute the exact backgrounds. Depending on the value of $k$, they may be interpreted as instantons connecting a highly singular big bang like universe with a static singular or regular black plane geometry. 
  We consider two-dimensional dilaton-gravity theories with a generic exponential potential for the dilaton, and obtain the most general black hole solutions in the Schwarzshild form. We discuss their geometrical and thermodynamical properties. We also study these models from the point of view of gauge theories of the extended Poincar\'e group and show that they can be considered as gauge theories with broken symmetry. Finally, we examine the theory in a hamiltonian formalism and discuss its quantization and its symmetries. 
  In this letter, we present the action for the massive super-{\QED}. A pair of chiral and a pair of anti-chiral superfields with opposite U(1)-charges are required. We also carry out a dimensional reduction {\it{\`a la}} Scherk from (2+2) to (1+2) dimensions, and we show that, after suitable truncations are performed, the supersymmetric extension of the ${\tau}_{3}$QED$_{1+2}$ naturally comes out. 
  Spinors for an arbitrary Minkowski space with signature ($t$, $s$) are reassessed in connection with $D$-dimensional free Dirac action. The possibility of writing down kinetic and mass terms for charge-conjugated spinors is discussed in terms of the number of time-like directions of the space-time. 
  In order to investigate to what extent string theories are different vacua of a general string theory (the ``universal string"), we discuss realizations of the bosonic string as particular background of certain types of $W$-strings. Our discussions include linearized $W_3^{lin}$, non-critical $W_3$, linearized $W_3^{(2)lin}$ and critical $W_3^{(2)}$ realizations of the bosonic string. 
  This is an additional remark to the paper (hep-th 9411005) concerning a Hamiltonian structure of suggested there system of equations. The remark is inspired by a letter from L. Feher and I. Marshall. 
  We show that the $N=2$ and $N=4$ SUSY Yang-Mills action can be reformulated in the sense of non-commutative geometry on $M^4\times (Z_2\oplus Z_2)$ in a rather simple way. In this way the scalars or pseudoscalars are viewed as gauge fields along two directions in the space of one-forms on $Z_2\oplus Z_2$. 
  We study the BKP hierarchy and prove the existence of an Adler--Shiota--van Moerbeke formula. This formula relates the action of the $BW_{1+\infty}$--algebra on tau--functions to the action of the ``additional symmetries'' on wave functions. 
  The complete solutions of the spin generalization of the elliptic Calogero Moser systems are constructed. They are expressed in terms of Riemann theta-functions. The analoguous constructions for the trigonometric and rational cases are also presented. 
  Beginning with a review of the arguments leading to the so-called c=1 barrier in the continuum formulation of noncritical string theory, the pathology is then exhibited in a discretized version of the theory, formulated through dynamical triangulation of two dimensional random surfaces. The effect of embedding the string in a superspace with fermionic coordinates is next studied in some detail. Using techniques borrowed from the theory of random matrices, indirect arguments are presented to establish that such an embedding may stabilize the two dimensional world sheet against degeneration into a branched polymer-like structure, thereby leading to a well-defined continuum string theory in a spacetime of dimension larger than 2. 
  We propose the action of d=4 Anti-de Sitter (AdS) spinning particle with arbitrary fixed quantum numbers. Regardless of the spin value, the configuration space of the model is a direct product of d=4 AdS space and two-dimensional sphere corresponds to spinning degrees of freedom. The model is an AdS counterpart of the massive spinning particle in the Minkowski space proposed earlier. Being AdS-invariant, the model possesses two gauge symmetries implying identical conservation law for AdS-counterparts of mass and spin. Two equivalent forms of the action functional, minimal and manifestly covariant, are given. 
  A new class of two dimensional integrable field theories, based on the mathematical notion of Poisson manifolds, and containing gravity-Yang-Mills systems as well as the G/G gauged Wess-Zumino Witten-model, are presented. The local solutions of the classical equations of motions as well as a scheme for the quantization in a Hamiltonian formulation is presented for the general model. Partial results of a calculation of the partition function on arbitrary Riemann surfaces via path integral techniques are given. (Contribution to the proceedings of the Conference on Integrable Systems at the JINR, Dubna, July 1994). 
  Gauge theories on manifolds with spatial boundaries are studied. It is shown that observables localized at the boundaries (edge observables) can occur in such models irrespective of the dimensionality of spacetime. The intimate connection of these observables to charge fractionation, vertex operators and topological field theories is described. The edge observables, however, may or may not exist as well-defined operators in a fully quantized theory depending on the boundary conditions imposed on the fields and their momenta. The latter are obtained by requiring the Hamiltonian of the theory to be self-adjoint and positive definite. We show that these boundary conditions can also have nice physical interpretations in terms of certain experimental parameters such as the penetration depth of the electromagnetic field in a surrounding superconducting medium. The dependence of the spectrum on one such parameter is explicitly exhibited for the Higgs model on a spatial disc in its London limit. It should be possible to test such dependences experimentally, the above Higgs model for example being a model for a superconductor. Boundary conditions for the 3+1 dimensional $BF$ system confined to a spatial ball are studied. Their physical meaning is clarified and their influence on the edge states of this system (known to exist under certain conditions) is discussed. It is pointed out that edge states occur for topological solitons of gauge theories such as the 't Hooft-Polyakov monopoles. 
  The gauge symmetries that underlie string theory arise from inner automorphisms of the algebra of observables of the associated conformal field theory. In this way it is possible to study broken and unbroken symmetries on the same footing, and exhibit an infinite-dimensional supersymmetry algebra that includes space-time diffeomorphisms and an infinite number of spontaneously broken level-mixing symmetries. We review progress in this area, culminating in the identification of a weighted tensor algebra as a subalgebra of the full symmetry. We also briefly describe outstanding problems. Talk presented at the Gursey memorial conference, Istanbul, Turkey, June, 1994. 
  Setting an ansats that the metric is expressible by a power series of the inverse radius and taking a particular gauge choice, we construct a "general solution" of (2+1)-dimensional Einstein's equations with a negative cosmological constant in the case where the spacetime is asymptotically anti-de Sitter. Our general solution turns out to be parametrized by two centrally extended quadratic differentials on $S^{1}$. In order to include 3-dimensional Black Holes naturally into our general solution, it is necessary to exclude the region inside the horizon. We also discuss the relation of our general solution to the moduli space of flat $\tilde{SL}(2,R)\times\tilde{SL}(2,R)$ connections. 
  It is shown that the Sugawara-type construction for W3-algebra associated with the four magical Jordan algebras leads to the anomalous theory of W3-gravity. 
  It has recently been shown that the $W_3$ and $W_3^{(2)}$ algebras can be considered as subalgebras in some linear conformal algebras. In this paper we show that the nonlinear algebras $W_{2,4}$ and $WB_2$ as well as Zamolodchikov's spin $5/2$ superalgebra also can be embedded as subalgebras into some linear conformal algebras with a finite set of currents. These linear algebras give rise to new realizations of the nonlinear algebras which could be suitable in the construction of $W$-string theories. 
  We calculate the first quantum corrections to the masses of solitons in imaginary-coupling affine Toda theories using the semi-classical method of Dashen, Hasslacher and Neveu. The theories divide naturally into those based on the simply-laced, the twisted and the untwisted non-simply-laced algebras. We find that the classical relationships between soliton and particle masses found by Olive {\em et al.\ }persist for the first two classes, but do not appear to do so naively for the third. 
  We investigate gravitational effects of extreme, non-extreme and ultra- extreme domain walls in the presence of a dilaton field. The dilaton is a scalar field without self-interaction that couples to the matter po- tential that is responsible for the formation of the wall. Motivated by superstring and supergravity theories, we consider both an exponential dilaton coupling (parametrized with the coupling constant alpha and the case where the coupling is self-dual, i.e. it has an extremum for a fi- nite value of the dilaton. For an exponential dilaton coupling, extreme walls (which are static planar configurations with surface energy density sigma_ext saturating the corresponding Bogomol'nyi bound) have a naked (planar) singularity outside the wall for alpha>1, while for alpha smaller or equal to 1 the singularity is null. On the other hand, non-extreme walls (bubbles with two insides and sigma_non > sigma_ext and ultra-extreme walls bubbles of false vacuum decay with sigma_ultra < sigma_ext always have naked singularities. There are solutions with self-dual couplings, which reduce to singularity-free vacuum domain wall space--times. However, only non- and ultra-extreme walls of such a type are dynamically stable. 
  The irreducible unitary representations of the Banach Lie group $U_0(\H)$ (which is the norm-closure of the inductive limit $\cup_k U(k)$) of unitary operators on a separable Hilbert space $\H$, which were found by Kirillov and Ol'shanskii, are reconstructed from quantization theory. Firstly, the coadjoint orbits of this group are realized as Marsden-Weinstein symplectic quotients in the setting of dual pairs. Secondly, these quotients are quantized on the basis of the author's earlier proposal to quantize a more general symplectic reduction procedure by means of Rieffel induction (a technique in the theory of operator algebras). As a warmup, the simplest such orbit, the projective Hilbert space, is first quantized using geometric quantization, and then again with Rieffel induction. Reduction and induction have to be performed with either $U(M)$ or $U(M,N)$. The former case is straightforward, unless the half-form correction to the (geometric) quantization of the unconstrained system is applied. The latter case, in which one induces from holomorphic discrete series representations, is problematic. For finite-dimensional $\H=\C^k$, the desired result is only obtained if one ignores half-forms, and induces from a representation, `half' of whose highest weight is shifted by $k$ (relative to the naive orbit correspondence). This presumably poses a problem for any theory of quantizing constrained systems. 
  These notes present an introduction to an analytic version of deformation quantization. The central point is to study algebras of physical observables and their irreducible representations. In classical mechanics one deals with real Poisson algebras, whereas in quantum mechanics the observables have the structure of a real non-associative Jordan-Lie algebra. The non-associativity is proportional to $\pl^2$, hence for $\pl\raw 0$ one recovers a real Poisson algebra. This observation lies at the basis of `strict' deformation quantization, where one deforms a given Poisson algebra into a $C^*$-algebra, in such a way that the basic algebraic structures are preserved. Our main interest lies in degenerate Poisson algebras and their quantization by non-simple Jordan-Lie algebras. The traditional symplectic manifolds of classical mechanics, and their quantum counterparts (Hilbert spaces and operator algebras which act irreducibly) emerge from a generalized representation theory. This two-step procedure sheds considerable light on the subject. We discuss a large class of examples, in which the Poisson algebra canonically associated to an (integrable) Lie algebroid is deformed into the Jordan-Lie algebra of the corresponding Lie groupoid. A special case of this construction, which involves the gauge groupoid of a principal fibre bundle, describes the classical and quantum mechanics of a particle moving in an external gravitational and Yang-Mills field. 
  We attempt to clarify the main conceptual issues in approaches to `objectification' or `measurement' in quantum mechanics which are based on superselection rules. Such approaches venture to derive the emergence of classical `reality' relative to a class of observers; those believing that the classical world exists intrinsically and absolutely are advised against reading this paper. The prototype approach (Hepp) where superselection sectors are assumed in the state space of the apparatus is shown to be untenable. Instead, one should couple system and apparatus to an environment, and postulate superselection rules for the latter. These are motivated by the locality of any observer or other (actual or virtual) monitoring system. In this way `environmental' solutions to the measurement problem (Zeh, Zurek) become consistent and acceptable, too. Points of contact with the modal interpretation are briefly discussed. We propose a minimal value attribution to observables in theories with superselection rules, in which only central observables have properties. In particular, the eigenvector-eigenvalue link is dropped. This is mainly motivated by Ockham's razor. 
  The connection between space-time covariant representations (obtained by inducing from the Lorentz group) and irreducible unitary representations (induced from Wigner's little group) of the Poincar\'{e} group is re-examined in the massless case. In the situation relevant to physics, it is found that these are related by Marsden-Weinstein reduction with respect to a gauge group. An analogous phenomenon is observed for classical massless relativistic particles. This symplectic reduction procedure can be (`second') quantized using a generalization of the Rieffel induction technique in operator algebra theory, which is carried through in detail for electro- magnetism. Starting from the so-called Fermi representation of the field algebra generated by the free abelian gauge field, we construct a new (`rigged') sesquilinear form on the representation space, which is positive semi-definite, and given in terms of a Gaussian weak distribution (promeasure) on the gauge group (taken to be a Hilbert Lie group). This eventually constructs the algebra of observables of quantum electro- magnetism (directly in its vacuum representation) as a representation of the so-called algebra of weak observables induced by the trivial representation of the gauge group. 
  The Aharonov-Bohm scattering of a localized wave packet is considered. A careful analysis of the forward direction points out new results: according to the time-dependent solution obtained by means of the asymptotic representation for the propagator (kernel), a phenomenon of auto-interference occurs along the forward direction, where, also, the probability density current is evaluated and found finite. 
  We compute quantum corrections to soliton masses in affine Toda theories with imaginary exponentials based on the nonsimply-laced Lie algebras $c_n^{(1)}$. We find that the soliton mass ratios renormalize nontrivially, in the same manner as those of the fundamental particles of the theories with real exponentials based on the nonsimply-laced algebras $b_n^{(1)}$. This gives evidence that the conjectured relation between solitons in one Toda theory and fundamental particles in a dual Toda theory holds also at the quantum level. This duality can be seen as a toy model for S-duality. 
  The one--loop effective action for the case of a massive scalar loop in the background of both a scalar potential and an abelian or non--abelian gauge field is written in a one--dimensional path integral representation. From this the inverse mass expansion is obtained by Wick contractions using a suitable Green function, which allows the computation of higher order coefficients. For the scalar case, explicit results are presented up to order O(T**8) in the proper time expansion. The relation to previous work is clarified. 
  String theory appears to admit a group of discrete field transformations -- called $S$ dualities -- as exact non-perturbative quantum symmetries. Mathematically, they are rather analogous to the better-known $T$ duality symmetries, which hold perturbatively. In this talk the evidence for $S$ duality is reviewed and some speculations are presented. 
  Contents:   1. Introduction   2. Bosonic propagators and random paths   3. Random surfaces and strings   4. Matrix models and two-dimensional quantum gravity   5. The mystery of $c > 1$   6. Euclidean quantum gravity in $d > 2$   7. Discussion 
  A new local and gauge invariant quantum vortex operator is constructed in three-dimensional gauge field theories. The correlation functions of this operator are evaluated exactly in pure Maxwell theory and by means of a loop expansion in the Abelian Higgs model. In the broken symmetry phase of the latter an explicit expression for the mass of the quantum vortices is obtained from the long distance exponential decay of the two-point function. 
  We give an integral representation for solutions to the quantized Knizhnik- Zamolodchikov equation (qKZ) associated with the Lie algebra $gl_{N+1}$. Asymptotic solutions to qKZ are constructed. The leading term of an asymptotic solution is the Bethe vector -- an eigenvector of the transfer-matrix of a quantum spin chain model. We show that the norm of the Bethe vector is equal to the product of the Hessian of a suitable function and an explicitly written rational function. This formula is a generalization of the Gaudin-Korepin formula for a norm of the Bethe vector. We show that, generically, the Bethe vectors form a base for the $gl_2$ case. 
  Canonical quantization of the Wess-Zumino (WZ) model including chiral interaction with gauge field is considered for the case of a degenerate action. The two-dimensional SU(2) Yang-Mills model and the four-dimensional SU(3) chiral gauge model proposed in the paper \cite{fss} are studied in details. Gauge invariance of the quantum theory is established at the formal level. 
  We consider certain N=4 supersymmetric gauge theories in D=2 coupled to quaternionic matter multiplets in a minimal way. These theories admit as effective theories sigma-models on non-trivial HyperK\"ahler manifolds obtained as HyperK\"ahler quotients. The example of ALE manifolds is discussed. (Based on a talk given by P. Fr\'e at the F. Gursey Memorial Conference, Istanbul, June 1994). 
  We discuss geometrical aspects of Toda Fields generalizing the links between Liouville gravity and uniformization of Riemann surfaces of genus greater than one. The framework is the interplay between the hermitian and the holomorphic geometry of vector bundles on such Riemann surfaces. Pointing out how Toda fields can be considered as equivalent to Higgs systems, we show how the theory of Variations of Hodge Structures enters the game inducing local holomorphic embeddings of Riemann surfaces into homogeneous spaces. The relations of such constructions with previous realizations of $W_n$--geometries are briefly discussed. 
  A conformal field theory can be recovered, via the Kontsevich-Miwa transform, as a solution to the Virasoro constraints on the KP tau function. That theory, which we call KM CFT, consists of d \leq 1 matter plus a scalar and a dressing prescription: \Delta = 0 for every primary field. By adding a spin-1 bc system the KM CFT provides a realization of the N=2 twisted topological algebra. The other twist of the corresponding untwisted N=2 superconformal theory is a DDK realization of the N=2 twisted topological algebra. Talk given by Beatriz Gato-Rivera at the "28th International Symposium on the Theory of Elementary Particles", Wendisch-Rietz (Germany), August 30 - September 3, 1994. 
  We discuss the gauge theory for quantum group $SU_q (2)\times U(1) $ on the quantum Euclidean space. This theory contains three physical gauge fields and one$\; U(1)-$gauge field with a zero field strength. We construct the quantum-group self-duality equation (QGSDE) in terms of differential forms and with the help of the field-strength decomposition. A deformed analog of the BPST-instanton solution is obtained. We consider a harmonic (twistor) interpretation of QGSDE in terms of $SU_q (2)/U(1) $ quantum harmonics. The quantum harmonic gauge equations are formulated in the framework of a left-covariant 3D differential calculus on the quantum group $SU_q (2)$. 
  We construct the general electrically charged, rotating black hole solution in the heterotic string theory compactified on a six dimensional torus and study its classical properties. This black hole is characterized by its mass, angular momentum, and a 28 dimensional electric charge vector. We recover the axion-dilaton black holes and Kaluza-Klein black holes for special values of the charge vector. For a generic black hole of this kind, the 28 dimensional magnetic dipole moment vector is not proportional to the electric charge vector, and we need two different gyromagnetic ratios for specifying the relation between these two vectors. We also give an algorithm for constructing a 58 parameter rotating dyonic black hole solution in this theory, characterized by its mass, angular momentum, a 28 dimensional electric charge vector and a 28 dimensional magnetic charge vector. This is the most general asymptotically flat black hole solution in this theory consistent with the no-hair theorem. 
  We present a nontechnical introduction to the hyperbolic Kac Moody algebra E_{10} and summarize our recent attempt to understand the root spaces of Kac Moody algebras of hyperbolic type in terms of a DDF construction appropriate to a subcritical compactified bosonic string. 
  This work proposes a natural extension of the Bargmann-Fock representation to a SUSY system. The main objective is to show that all essential structures of the n-dimensional SUSY oscillator are supplied by basic differential geometrical notions on an analytical R^n, except for the scalar product which is the only additional ingredient. The restriction to real numbers implies only a minor loss of structure but makes the essential features clearer. In particular, euclidean evolution is enforced naturally by identification with the 1-parametric group of dilations. 
  It is shown that the space of cohomology classes of the $SU(1,1)/U(1)$ coset at negative level $k$ contains states of relevant conformal dimensions. These states correspond to the energy density operator of the associated nonlinear sigma model. We exhibit that there exists a subclass of relevant operators forming a closed fusion algebra. We make use of these operators to perform renormalizable perturbations of the $SU(1,1)/U(1)$ coset. In the infra-red limit, the perturbed theory flows to another conformal model. We identify one of the perturbative conformal points with the $SU(2)/U(1)$ coset at positive level. {}From the point of view of the string target space geometry, the given renormalization group flow maps the euclidean black hole geometry described by the $SU(1,1)/U(1)$ coset into the sphere described by the $SU(2)/U(1)$ coset. 
  We propose a method using perturbation theory in the running coupling constant and the idea of scaling to determine improved actions for lattice field theories combining Wilson's renormalization group with Symanzik's improvement program . The method is based on the analysis of a single renormalization group transformation. We test it on the hierarchical $O(N)$ invariant $\sigma$ model in two dimensions. 
  For certain situations we give a geometrical background for quasiclassical KP calculations based on an explicit connection to quantum mechanics and the collapse of coherent states to coadjoint orbits for classical operators. 
  In 1976 Stephen Hawking proposed that information may be lost from our universe as a pure quantum state collapses gravitationally into a black hole, which then evaporates completely into a mixed state of thermal radiation. Although this proposal is controversial, it is tempting to consider analogous processes that might occur in certain theories of consciousness. For example, one might postulate that independent degrees of freedom be ascribed to the mental world to help explain the feeling of a correlation between one's desires and one's choice of actions. If so, one might ask whether information in the physical world can be lost to such postulated degrees of freedom in the mental world. Or, one might hypothesize that the mental world can affect the physical world by modifying the quantum action for the physical world in a coordinate-invariant way (analogous to the alpha parameters in wormhole theory). 
  We present several identities involving quasi-minors of noncommutative generic matrices. These identities are specialized to quantum matrices, yielding q-analogues of various classical determinantal formulas. 
  We consider the effects of abelian duality transformations on static, spherically-symmetric, asymptotically flat string spacetimes in four dimensions, where the dilaton, axion, metric, and gauge fields are allowed to be nonzero. Independent of the alpha' expansion, there is a six-parameter family of such configurations, labelled by the charges characterizing the asymptotic behaviour of the various fields: ie their mass, dilaton charge, axion charge, electric charge, magnetic charge, and Taub-NUT parameter. We show that duality, based on time-translation invariance, maps these solutions amongst themselves, with the effect of interchanging two pairs of these six labels, namely: (1) the mass and dilaton charge, and (2) the axion charge and the Taub-NUT parameter. We consider in detail the special case of the purely Schwarzschild black hole, for which the mass of the dual configuration vanishes to leading order in alpha'. Working to next-to-leading order in alpha' for the bosonic and heterotic strings, we find that duality takes a black hole of mass M to a (singular) solution having mass - 1/(alpha' M). Finally, we argue that two solutions which are related by duality based on a noncompact symmetry are {\it not} always physically equivalent. 
  Protective measurements yield properties of the quantum state of a single quantum system without affecting the quantum state. A protective measurement involves adiabatic coupling to the measuring device together with a procedure to protect the state from changing. For nondegenerate energy eigenstates the protection is provided by the system itself. In this case it is actually possible to measure the Schr\"odinger wave via measurements on a single system. This fact provides an argument in favor of associating physical reality with a quantum state of a single system, challenging the usual ensemble interpretation. We also believe that the complete description of a quantum system requires a two-state vector formalism involving (in addition to the usual one)a future quantum state evolving backwards in time. Protective measurements testing the two-state vector reality are constructed. 
  Recently R. G. Newton published a comment criticizing the methods and the results of a paper published by the author. His criticism touches on a few key points of the subject and hence deserves a detailed reply. Here is the reply, point by point, to his criticism. 
  We identify a string theory counterpart of the dilatonic Melvin D=4 background describing a "magnetic flux tube" in low-energy field theory limit. The corresponding D=5 bosonic string model containing extra compact Kaluza-Klein dimension is a direct product of the D=2 Minkowski space and a D=3 conformal sigma model. The latter is a singular limit of the [SL(2,R) x R]/R gauged WZW theory. This implies, in particular, that the dilatonic Melvin background is an exact string solution to all orders in \a'. Moreover, the D=3 model is formally related by an abelian duality to a flat space with a non-trivial topology. The conformal field theory for the Melvin solution is exactly solvable (and for special values of magnetic field parameter is equivalent to CFT for a $Z_N$ orbifold of 2-plane times a circle) and should exhibit tachyonic instabilities. 
  A preferred form for the path integral discretization is suggested that allows the implementation of canonical transformations in quantum theory. 
  Using the Fronsdal-Galindo formula for the exponential mapping from the quantum algebra $U_{p,q}(gl(2))$ to the quantum group $GL_{p,q}(2)$, we show how the $(2j+1)$-dimensional representations of $GL_{p,q}(2)$ can be obtained by `exponentiating' the well-known $(2j+1)$-dimensional representations of $U_{p,q}(gl(2))$ for $j$ $=$ $1,\frac{3}{2},\ldots $; $j$ $=$ $\frac{1}{2}$ corresponds to the defining 2-dimensional $T$-matrix. The earlier results on the finite-dimensional representations of $GL_q(2)$ and $SL_q(2)$ (or $SU_q(2)$) are obtained when $p$ $=$ $q$. Representations of $U_{\bar{q},q}(2)$ $(q$ $\in$ $\C \backslash \R$ and $U_q(2)$ $(q$ $\in$ $\R \backslash \{0\})$ are also considered. The structure of the Clebsch-Gordan matrix for $U_{p,q}(gl(2))$ is studied. The same Clebsch-Gordan coefficients are applicable in the reduction of the direct product representations of the quantum group $GL_{p,q}(2)$. 
  We reformulate the Thirring model in $D$ $(2 \le D < 4)$ dimensions as a gauge theory by introducing $U(1)$ hidden local symmetry (HLS) and study the dynamical mass generation of the fermion through the Schwinger-Dyson (SD) equation. By virtue of such a gauge symmetry we can greatly simplify the analysis of the SD equation by taking the most appropriate gauge (``nonlocal gauge'') for the HLS.   In the case of even-number of (2-component) fermions, we find the dynamical fermion mass generation as the second order phase transition at certain fermion number, which breaks the chiral symmetry but preserves the parity in (2+1) dimensions ($D=3$). In the infinite four-fermion coupling (massless gauge boson) limit in (2+1) dimensions, the result coincides with that of the (2+1)-dimensional QED, with the critical number of the 4-component fermion being $N_{\rm cr} = \frac{128}{3\pi^{2}}$. As to the case of odd-number (2-component) fermion in (2+1) dimensions, the regularization ambiguity on the induced Chern-Simons term may be resolved by specifying the regularization so as to preserve the HLS.   Our method also applies to the (1+1) dimensions, the result being consistent with the exact solution. The bosonization mechanism in (1+1) dimensional Thirring model is also reproduced in the context of dual-transformed theory for the HLS. 
  It is shown that in the absence of free abelian gauge fields, the conserved currents of (classical) Yang-Mills gauge models coupled to matter fields can be always redefined so as to be gauge invariant. This is a direct consequence of the general analysis of the Wess-Zumino consistency condition for Yang-Mills theory that we have provided recently. 
  We describe a new infinite family of multi-parameter functional equations for the Rogers dilogarithm, generalizing Abel's and Euler's formulas. They are suggested by the Thermodynamic Bethe Ansatz approach to the Renormalization Group flow of 2D integrable, ADE-related quantum field theories. The known sum rules for the central charge of critical fixed points can be obtained as special cases of these. We conjecture that similar functional identities can be constructed for any rational integrable quantum field theory with factorized S-matrix and support it with extensive numerical checks. 
  Motivated by a possible relativistic string description of hadrons we use a discretised light-cone quantisation and Lanczos algorithm to investigate the phase structure of phi^3 matrix field theory in the large N limit. In 1+1 dimensions we confirm the existence of Polyakov's non-critical string theory at the boundary between parton-like and string-like phases, finding critical exponents for longitudinal oscillations equal to or consistent with those given by a mean field argument. The excitation spectrum is finite, possibly discrete. We calculate light-cone structure functions and find evidence that the probability Q(x) of a parton in the string carrying longitudinal momentum fraction between x and x+dx has support on all 0<x<1, despite the average number of partons being infinite. 
  The monopole equations in the dual abelian theory of the N=2 gauge-theory, recently proposed by Witten as a new tool to study topological invariants, are shown to be the simplest elements in a class of instanton equations that follow from the improved topological twist mechanism introduced by the authors in previous papers. When applied to the N=2 sigma-model, this twisting procedure suggested the introduction of the so-called hyperinstantons, or triholomorphic maps. When gauging the sigma-model by coupling it to the vector multiplet of a gauge group G, one gets gauged hyperinstantons that reduce to the Seiberg-Witten equations in the flat case and G=U(1). The deformation of the self-duality condition on the gauge-field strength due to the monopole-hyperinstanton is very similar to the deformation of the self-duality condition on the Riemann curvature previously observed by the authors when the hyperinstantons are coupled to topological gravity. In this paper the general form of the hyperinstantonic equations coupled to both gravity and gauge multiplets is presented. 
  Riemann surfaces with nodes can be described by introducing simple composite operators in matrix models. In the case of the Kontsevich model, it is sufficient to add the quadratic, but ``non-propagating'', term (tr[X])^2 to the Lagrangian. The corresponding Jenkins-Strebel differentials have pairwise identified simple poles. The result is in agreement with a conjecture formulated by Kontsevich and recently investigated by Arbarello and Cornalba that the set ${\cal M}_{m*,s}$ of ribbon graphs with s faces and $m*=(m_0,m_1,\ldots,m_j,\ldots)$ vertices of valencies $(1,3,\ldots,2j+1,\ldots)$ ``can be expressed in terms of Mumford-Morita classes'': one gets an interpretation for univalent vertices. I also address the possible relationship with a recently formulated theory of constrained topological gravity. 
  We extend the concept of conjugate vertex operators, first introduced by Dotsenko in the case of the bosonization of the $su(2)$ conformal field theory, to the bosonization of the dynamical vertex operators (type II in the classification of the Kyoto school) of the higher spin XXZ model. We show that the introduction of the conjugate vertex operators leads to simpler expressions for the N-point matrix elements of the dynamical vertex operators, that is, without redundant Jackson integrals that arise from the insertion of screening charges. In particular, the two-point matrix element can be represented without any integral. 
  We study properties of differential graded (dg) operads modulo weak equivalences, that is, modulo the relation given by the existence of a chain of dg operad maps inducing a homology isomorphism. This approach, naturally arising in string theory, leads us to consider various versions of models. Besides of some applications in topology and homological algebra we show that our theory enables one to prove the existence of homotopy structures on physically relevant spaces. For example, we prove that a closed string-field theory induces a homotopy Lie algebra structure on the space of relative states, which is one of main results of T. Kimura, A. Voronov and J. Stasheff (see [16]). Our theory gives a systematic way to prove statements of this type. The paper is a corrected version of a preprint which began to circulate in March 1994, with some new examples added. 
  I review the foundations of irrational conformal field theory (ICFT), which includes rational conformal field theory as a small subspace. Highlights of the review include the Virasoro master equation and the generalized Knizhnik-Zamolodchikov equations for the correlators of ICFT on the sphere and the torus. 
  These are expository lectures reviewing   (1) recent developments in two-dimensional Yang-Mills theory, and   (2) the construction of topological field theory Lagrangians. Topological field theory is discussed from the point of view of infinite-dimensional differential geometry. We emphasize the unifying role of equivariant cohomology both as the underlying principle in the formulation of BRST transformation laws and as a central concept in the geometrical interpretation of topological field theory path integrals. 
  The main subject of the paper is the so-called Discrete Painlev\'e-1 Equation (DP1). Solutions of DP1 are classified under criterion of their behavior while argument tends to infinity. The Isomonodromic Deformations Method yields asymptotic formulae for regular solutions of DP1. DP1 is an integrable system, what allows to develop appropriate Whitham theory. Asymptotics of singular solutions of DP1 are calculated by using the Whitham method. 
  A new approach to relativistic elasticity theory is proposed. In this approach the theory becomes a gauge--type theory, with the diffeomorphisms of the material space playing the role of gauge transformations. The dynamics of the elastic material is expressed in terms of three independent, hyperbolic, second order partial differential equations imposed on three (independent) gauge potentials. The relationship with the Carter-Quintana approach is discussed. 
  We review the gauge invariant formulation of 2-dim. QCD. We show that the non-linear gauge invariant phase space is the coset $W_\infty/W_{+\infty}\times W_{-\infty}$ ,which is specified by the $N=\infty$ master-field of this model. The meson fields correspond to the local coordinates of the coset. We comment on the stringy collective coordinates of the solitons (baryons) in this model. 
  We derive the asymptotic expansion of the heat kernel for a Laplace operator acting on deformed spheres. We calculate the coefficients of the heat kernel expansion on two- and three-dimensional deformed spheres as functions of deformation parameters. We find that under some deformation the conformal anomaly for free scalar fields on $R^4\times \tilde S^2$ and $R^6\times \tilde S^2$ is canceled. 
  It is pointed out that the universality might seriously be violated by models with several fixed points. 
  We discuss the phase structure of a higher derivative four-fermion model in four dimensions in curved spacetime in frames of the $\frac{1}{N_c}$-expansion. First, we evaluate in our model the effective potential of two composite scalars in the linear curvature approximation using a local momentum representation in curved spacetime for the higher-derivative propagator which naturally appears. The symmetry breaking phenomenon and phase transition induced by curvature are numerically investigated. A numerical study of the dynamically generated fermionic mass, which depends on the coupling constants and on the curvature, is also presented. 
  The complete set of solutions of two dimensional classical string theory are constructed for any curved spacetime. They describe folded strings moving in curved spacetime. Surprizing stringy behavior becomes evident at singularities such as black holes.The solutions are given in the form ofa map from the world sheet to target spacetime, where the world sheet has to be divided into lattice -like patches corresponding to different maps. A recursion relation analogous to a transfer matrix connects these maps into a single continuous map. This ``transfer matrix'' encodes the properties of the world sheet lattice as well as the geometry of spacetime. The solutions are completely classified by their behavior in the asymptotically flat region of spacetime where they reduce, as boundary conditions, to the folded string solutions that have been known for 19 years. 
  The symmetry algebra of the N-dimensional anisotropic quantum harmonic oscillator with rational ratios of frequencies is constructed by a method of general applicability to quantum superintegrable systems. The special case of the 3-dim oscillator is studied in more detail, because of its relevance in the description of superdeformed nuclei and nuclear and atomic clusters. In this case the symmetry algebra turns out to be a nonlinear extension of the u(3) algebra. A generalized angular momentum operator useful for labeling the degenerate states is constructed, clarifying the connection of the present formalism to the Nilsson model in nuclear physics. 
  We extend our previous analysis of the modification of the spectrum of black hole radiance due to the simplest and probably most quantitatively important back-reaction effect, that is self-gravitational interaction, to the case of charged holes. As anticipated, the corrections are small for low-energy radiation when the hole is well away from extremality, butbecome qualitatively important near extremality. A notable result is that radiation which could leave the hole with mass and charge characteristic of a naked singularity, predicted in the usual approximation of fixed space-time geometry, is here suppressed. We discuss the nature of our approximations, and show how they work in a simpler electromagnetic analogue problem. 
  Summation over hard thermal loops, by themselves and as insertions in higher order Feynman diagrams, is important in thermal perturbation theory for Quantum Chromodynamics, so that all contributions of a given order in the coupling constant can be consistently taken into account. I review some of the basic properties of hard thermal loops and how the generating functional for them is related to the eikonal for a Chern-Simons gauge theory, and using an auxiliary field, to the gauged Wess-Zumino-Novikov-Witten action. The Hamiltonian analysis of the effective action and a discussion of plasma waves are given. It is also pointed out that a possible expression for the magnetic mass term can be written in a closely related way. 
  A large class of non-critical string theories with extended worldsheet gauge symmetry are described by two coupled, gauged Wess-Zumino-Witten Models. We give a detailed analysis of the gauge invariant action and in particular the gauge fixing procedure and the resulting BRST symmetries. The results are applied to the example of ${\cal W}_3$ strings. 
  We use the off-shell string effective action method developed by E.S. Fradkin and A.A. Tseytlin to obtain the formula for all-genus string effective action with and without compactification at the low-energy approximation in the massless background fields. We find that for the bosonic string, one can determine the dilaton vacuum expectation value from the all-genus effective action because of the nontrivial dependence of potential energy on dilaton. For compactified four-dimensional string models, if one requires that the target-space dilaton field lie on a K\"ahler manifold, we obtain a constraint which will specify the worldsheet dilaton in terms of the constant background fields. We also show that under this constraint, the tree-level k\"ahlar potential and superpotential are not changed by the higher-genus effect. This proves again the non-renormalization theorem for a string moving in massless background fields in the low-energy approximation. 
  In this paper the free gauge field theories on a Riemann surface of any genus are quantized in the covariant gauge. The propagators of the gauge fields are explicitly derived and their properties are analysed in details. As an application, the correlation functions of a Yang-Mills field theory with gauge group $SU(N)$ are computed at the lowest order. 
  The Maxwell-Chern-Simons theory is canonically quantized in the Coulomb gauge by using the Dirac bracket quantization procedure. The determination of the Coulomb gauge polarization vector turns out to be intrincate. A set of quantum Poincar\'e densities obeying the Dirac-Schwinger algebra, and, therefore, free of anomalies, is constructed. The peculiar analytical structure of the polarization vector is shown to be at the root for the existence of spin of the massive gauge quanta.The Coulomb gauge Feynman rules are used to compute the M\"oller scattering amplitude in the lowest order of perturbation theory. The result coincides with that obtained by using covariant Feynman rules. This proof of equivalence is, afterwards, extended to all orders of perturbation theory. The so called infrared safe photon propagator emerges as an effective propagator which allows for replacing all the terms in the interaction Hamiltonian of the Coulomb gauge by the standard field-current minimal interaction Hamiltonian. 
  We show explicitly that there is particle creation in a static spacetime. This is done by studying the field in a coordinate system based on a physical principle which has recently been proposed. There the field is quantized by decomposing it into positive and negative frequency modes on a particular spacelike surface. This decomposition depends explicitly on the surface where the decomposition is performed, so that an observer who travels from one surface to another will observe particle production due to the different vacuum state. 
  We study $\l\f^4$ theory using an environmentally friendly finite-temperature renormalization group. We derive flow equations, using a fiducial temperature as flow parameter, develop them perturbatively in an expansion free from ultraviolet and infrared divergences, then integrate them numerically from zero to temperatures above the critical temperature. The critical temperature, at which the mass vanishes, is obtained by integrating the flow equations and is determined as a function of the zero-temperature mass and coupling. We calculate the field expectation value and minimum of the effective potential as functions of temperature and derive some universal amplitude ratios which connect the broken and symmetric phases of the theory. The latter are found to be in good agreement with those of the three-dimensional Ising model obtained from high- and low-temperature series expansions. 
  We employ nonperturbative flow equations for the description of the effective action in Yang-Mills theories. We find that the perturbative vacuum with vanishing gauge field strength does not correspond to the minimum of the Euclidean effective action. The true ground state is characterized by a nonvanishing gluon condensate. 
  A lattice regularization procedure for gauge theories is proposed in which fermions are given a special treatment such that all chiral flavor symmetries that are free of Adler-Bell-Jackiw anomalies are kept intact. There is no doubling of fermionic degrees of freedom. A price paid for this feature is that the number of fermionic degrees of freedom per unit cell is still infinite, although finiteness of the complete functional integrals can be proven (details are outlined in an Appendix). Therefore, although perhaps of limited usefulness for numerical simulations, our scheme can be applied for studying aspects such as analytic convergence questions, spontaneous symmetry breakdown and baryon number violation in non-Abelian gauge theories. 
  A family of theories which interpolate between vector and chiral Schwinger models is studied on the two--sphere $S^{2}$. The conflict between the loss of gauge invariance and global geometrical properties is solved by introducing a fixed background connection. In this way the generalized Dirac--Weyl operator can be globally defined on $S^{2}$. The generating functional of the Green functions is obtained by taking carefully into account the contribution of gauge fields with non--trivial topological charge and of the related zero--modes of the Dirac determinant. In the decompactification limit, the Green functions of the flat case are recovered; in particular the fermionic condensate in the vacuum vanishes, at variance with its behaviour in the vector Schwinger model. 
  It is shown that tachyons are associated with unitary representations of Poincare mappings induced from SO(2) little group instead of SO(2,1) one. This allows us to treat more seriously possibility that neutrinos are fermionic tachyons according to the present experimental data. 
  The hitherto controversial proposition that a ``wiggly" Goto-Nambu cosmic string can be effectively represented by an elastic string model of exactly transonic type (with energy density $U$ inversely proportional to its tension $T$) is shown to have a firm mathematical basis. 
  We apply stochastic quantization method to matrix models for the second quantization of loops in both discretized and continuum levels. The fictitious time evolution described by the Langevin equation is interpreted as the time evolution in a field theory of loops. The corresponding Fokker-Planck hamiltonian defines a non-critical string field theory. We study both orientable and non-orientable interactions of loops in terms of matrix models and take the continuum limit for one-matrix case. As a consequence, we show the equivalence of stochastic quantization of matrix models in loop space to the transfer-matrix formalism in dynamical triangulation of random surfaces. We also clarifies the origin of Virasoro algebra in this context. 
  In this talk, delivered at the Oscar Klein Centenary Symposium in Stockholm, I review the 1938 conference held in Warsaw devoted to \lq \lq New Theories in Physics". I review all of the talks presented at this meeting and discuss in detail Klein's paper where he proposed a unified model of electromagnetism and the nuclear force that foreshadowed the later developments of non-Abelian gauge theories. 
  We study some arithmetic properties of the mirror maps and the quantum Yukawa coupling for some 1-parameter deformations of Calabi-Yau manifolds. First we use the Schwarzian differential equation, which we derived previously, to characterize the mirror map in each case. For algebraic K3 surfaces, we solve the equation in terms of the $J$-function. By deriving explicit modular relations we prove that some K3 mirror maps are algebraic over the genus zero function field ${\bf Q}(J)$. This leads to a uniform proof that those mirror maps have integral Fourier coefficients. Regarding the maps as Riemann mappings, we prove that they are genus zero functions. By virtue of the Conway-Norton conjecture (proved by Borcherds using Frenkel-Lepowsky-Meurman's Moonshine module), we find that these maps are actually the reciprocals of the Thompson series for certain conjugacy classes in the Griess-Fischer group. This also gives, as an immediate consequence, a second proof that those mirror maps are integral. We thus conjecture a surprising connection between K3 mirror maps and the Thompson series. For threefolds, we construct a formal nonlinear ODE for the quantum coupling reduced $mod\ p$. Under the mirror hypothesis and an integrality assumption, we derive $mod~p$ congruences for the Fourier coefficients. For the quintics, we deduce (at least for $5\not{|}d$) that the degree $d$ instanton numbers $n_d$ are divisible by $5^3$ -- a fact first conjectured by Clemens. 
  A manifestly Lorentz and diffeomorphism invariant form for the abelian gauge field action with local duality symmetry of Schwarz and Sen is given. Some of the underlying symmetries of the covariant action are further considered. The Noether conserved charge under continuous local duality rotations is found. The covariant couplings with gravity and the axidilaton field are discussed. 
  The thermodynamic distribution function for exclusion statistics is derived. Creation and annihilation operators for particles obeying such statistics are discussed. A connection with anyons is pointed out. 
  In this paper we calculate the particle creation as seen by a stationary observer in an anisotropic universe. By using an observer and geometry dependent time to quantise a massive scalar field we show that a discrete energy spectrum shift occurs. The length scale associated with the geometry provides the energy scale by which the spectrum is shifted. The $\beta(p,q)$ coefficient for the Bogolubov transformation calculated is proportional to a series of delta functions whose argument contains $p$ and $q$ and half multiples of the root of the curvature. 
  We study parity violation in $2+1$-dimensional gauge theories coupled to massive fermions. Using the $\zeta$-function regularization approach we evaluate the ground state fermion current in an arbitrary gauge field background, showing that it gets two different contributions which violate parity invariance and induce a Chern-Simons term in the gauge-field effective action. One is related to the well-known {\em classical} parity breaking produced by a fermion mass term in 3 dimensions; the other one, already present for massless fermions, is related to peculiarities of gauge invariant regularization in odd-dimensional spaces. 
  A proof of Goldstone's theorem is given for the case in which global chiral symmetry is dynamically broken. The proof highlights a needed consistency between the exact Schwinger--Dyson equation for the fermion propagator and the exact Bethe--Salpeter equation for fermion--antifermion bound states. A criterion, based on the Cornwall, Jackiw and Tomboulis effective action for composite operators, is provided for maintaining the consistency when the equations are modified by approximations. For gauge theories in which partial conservation of the axial current (PCAC) should hold, a constraint on the approximations to the fermion--gauge boson vertex function is discussed, and a vertex model is given which satisfies both the PCAC constraint and the vector Ward--Takahashi identity. 
  A new ansatz is presented for a Lax pair describing systems of particles on the line interacting via (possibly nonsymmetric) pairwise forces. Particular cases of this yield the known Lax pairs for the Calogero-Moser and Toda systems, as well as their relativistic generalisations. The ansatz leads to a system of functional equations. Several new functional equations are described and the general analytic solution to some of these is given. New integrable systems are described. 
  The type-I quantum superalgebras are known to admit non-trivial one-parameter families of inequivalent finite dimensional irreps, even for generic $q$. We apply the recently developed technique to construct new solutions to the quantum Yang-Baxter equation associated with the one-parameter family of irreps of $U_q(gl(m|n))$, thus obtaining R-matrices which depend not only on a spectral parameter but in addition on further continuous parameters. These extra parameters enter the Yang-Baxter equation in a similar way to the spectral parameter but in a non-additive form. 
  We have calculated the first-order beta-functions for a sigma-model ( with dilaton) dualized with respect to an arbitrary Lie group that acts without isotropy. We find that non-abelian duality preserves conformal invariance for semi-simple groups, but in general there is an extra contribution to the beta-function proportional to the trace of the structure constants, which cannot be absorbed into an additional dilaton shift. Two particular examples, a Bianchi V cosmological background and the G \otimes G WZW model, are discussed. 
  We present in detail the classification of the finite dimensional irreducible representations of the super Yangian associated with the Lie superalgebra $gl(1|1)$. 
  The upper bound for asymptotic behavior of the coefficients of expansion of the evolution operator kernel in powers of the time interval $\Dt$ was obtained. It is found that for the nonpolynomial potentials the coefficients may increase as $n!$. But increasing may be more slow if the contributions with opposite signs cancel each other. Particularly, it is not excluded that for number of the potentials the expansion is convergent. For the polynomial potentials $\Dt$-expansion is certainly asymptotic one. The coefficients increase in this case as $\Gamma(n \frac{L-2}{L+2})$, where $L$ is the order of the polynom. It means that the point $\Dt=0$ is singular point of the kernel. 
  The solutions of Einstein's equations admitting one non-null Killing vector field are best studied with the projection formalism of Geroch. When the Killing vector is lightlike, the projection onto the orbit space still exists and one expects a covariant theory with degenerate contravariant metric to appear, its geometry is presented here. Despite the complications of indecomposable representations of the local Euclidean subgroup, one obtains an absolute time and a canonical, Galilean and so-called Newtonian, torsionless connection. The quasi-Maxwell field (Kaluza Klein one-form) that appears in the dimensional reduction is a non-separable part of this affine connection, in contrast to the reduction with a non-null Killing vector. One may define the Kaluza Klein scalar (dilaton) together with the absolute time coordinate after having imposed one of the equations of motion in order to prevent the emergence of torsion. We present a detailed analysis of the dimensional reduction using moving frames, we derive the complete equations of motion and propose an action whose variation gives rise to all but one of them. Hidden symmetries are shown to act on the space of solutions. 
  We present arguments pointing to a behavior of the string free energy in the presence of a black hole horizon similar to the Atick-Witten dependence on temperature beyond the Hagedorn transition. We give some evidence based on orbifold techniques applied to Rindler space and further support is found within a Hamiltonian approach. However, we argue that the interpretation in terms of a reduction of degrees of freedom is confronted by serious problems. Finally, we point out the problems concerning heuristic red-shift arguments and the local interpretation of thermodynamical quantities. 
  In this paper we construct topological sigma models which include a potential and are related to twisted massive supersymmetric sigma models. Contrary to a previous construction these models have no central charge and do not require the manifold to admit a Killing vector. We use the topological massive sigma model constructed here to simplify the calculation of the observables. Lastly it is noted that this model can be viewed as interpolating between topological massless sigma models and topological Landau-Ginzburg models. 
  A generalized AKNS systems introduced and discussed recently in \cite{dGHM} are considered. It was shown that the dressing technique both in matrix pseudo-differential operators and formal series with respect to the spectral parameter can be developed for these hierarchies. 
  We compute exact solutions of two--matrix models, i.e. detailed genus by genus expressions for the correlation functions of these theories, calculated without any approximation. We distinguish between two types of models, the unconstrained and the constrained ones. Unconstrained two--matrix models represent perturbations of $c=1$ string theory, while the constrained ones correspond to topological field theories coupled to topological gravity. Among the latter we treat in particular detail the ones based on the KdV and on the Boussinesq hierarchies. 
  Complex vector fields with Maxwell, Chern-Simons and Proca terms are minimally coupled to an Abelian gauge field. The consistency of the spectrum is analysed and 1-loop quantum corrections to the self-energy are explicitly computed and discussed. The incorporation of 2-loop contributions and the behaviour of tree-level scattering amplitudes in the limit of high center-of-mass energies are also commented. 
  Let G be a finite-dimensional Lie algebra (not necessarily semisimple). It is known that if G is self-dual (that is, if it possesses an invariant metric) then there is a canonical N=1 superconformal algebra associated to its N=1 affinization---that is, it admits an N=1 (affine) Sugawara construction. Under certain additional hypotheses, this N=1 structure admits an N=2 extension. If this is the case, G is said to possess an N=2 structure. It is also known that an N=2 structure on a self-dual Lie algebra G is equivalent to a vector space decomposition G = G_+ \oplus G_- where G_\pm are isotropic Lie subalgebras. In other words, N=2 structures on G are in one-to-one correspondence with Manin triples (G,G_+,G_-). In this paper we exploit this correspondence to obtain a classification of the c=9 N=2 structures on self-dual solvable Lie algebras. In the process we also give some simple proofs for a variety of Lie algebraic results concerning self-dual Lie algebras admitting symplectic or K\"ahler structures. 
  The recently introduced Galois symmetries of RCFT are generalized, for the WZW case, to `quasi-Galois symmetries'. These symmetries can be used to derive a large number of equalities and sum rules for entries of the modular matrix S, including some that previously had been observed empirically. In addition, quasi-Galois symmetries allow to construct modular invariants and to relate S-matrices as well as modular invariants at different levels. They also lead us to an extremely plausible conjecture for the branching rules of the conformal embeddings of g into so(dim g). 
  In this paper we calculate the massive particle creation as seen by a stationary observer in a $1+1$ dimensional spacetime compact in space. The Bogolubov transformation relating the annihilation and creation operators between two spacelike surfaces is calculated. The particle creation, as observed by a stationary observer who moves from the first spacelike surface to the second is then calculated, and shown to be finite, as is expected for a spacetime with finite spatial volume. 
  We reformulate self-dual supersymmetric theories directly in conformal chiral superspace, where superconformal invariance is manifest. The superspace can be interpreted as the generalization of the usual Atiyah-Drinfel'd-Hitchin-Manin twistors (the quaternionic projective line), the real projective light-cone in six dimensions, or harmonic superspace, but can be reduced immediately to four-dimensional chiral superspace. As an example, we give the 't Hooft and ADHM multi-instanton constructions for self-dual super Yang-Mills theory. In both cases, all the parameters are represented as a single, irreducible, constant tensor. 
  We compute the Itzykson-Zuber (IZ) integral for the superunitary group U(m|n). As a consequence, we are able to find the non-zero correlations of superunitary matrices 
  {}From the cyclic quantum dilogarithm the shift operator is constructed with $q$ is a root of unit and the representation is given for the current algebra introduced by Faddeev $et ~al$. It is shown that the theta-function is factorizable also in this case by using the star-square equation of the Baxter-Bazhanov model. 
  Entropy for two dimensional extremal black holes is computed explicitly in a finite-space formulation of the black hole thermodynamics and is shown to be zero {\it locally}. Our results are in conformity with the recent one by Hawking et al in four dimensions. 
  We give an indication that gravity coupled to an infinite number of fields might be a renormalizable theory. A toy model with an infinite number of interacting fermions in four-dimentional space-time is analyzed. The model is finite at any order in perturbation theory. However, perturbation theory is valid only for external momenta smaller than $\lambda ^{-\frac{1}{2}}$ , where $\lambda$ is the coupling constant. 
  We discuss a Gedanken experiment in which we construct a physical coordinate system which covers the universe. Using general properties of quantum gravity we find that the minimum uncertainty of the coordinate system is $\sqrt[5]{R}$ where $R$ is the radius of the universe 
  We discuss quantum algebraic structures of the systems of electrons or quasiparticles on a sphere of which center a magnetic monople is located on. We verify that the deformation parameter is related to the filling ratio of the particles in each case. 
  We investigate multi-flavour gauge theories confined in $d\es 2n$-dimensional Euclidean bags. The boundary conditions for the 'quarks' break the axial flavour symmetry and depend on a parameter $\theta$. We determine the $\theta$-dependence of the fermionic correlators and determinants and find that a $CP$-breaking $\theta$-term is generated dynamically. As an application we calculate the chiral condensate in multi-flavour $QED_2$ and the abelian projection of $QCD_2$. In the second model a condensate is generated in the limit where the number of colours, $N_c$, tends to infinity. We prove that the condensate in $QCD_2$ decreases with increasing bag radius $R$ at least as $\sim R^{-1/N_cN_f}$. Finally we determine the correlators of mesonic currents in $QCD_2$. 
  The Kalb-Ramond action, derived for interacting strings through an action-at-a-distance force, is generalized to the case of interacting p-dimensional objects (p-branes) in D-dimensional space-time. The open p-brane version of the theory is especially taken up. On account of the existence of their boundary surface, the fields mediating interactions between open p-branes are obtained as massive gauge fields, quite in contrast to massless gauge ones for closed p-branes. 
  One-loop divergences appearing in the entropy of a quantum black hole are proven to be completely eliminated by the standard renormalization of both the gravitational constant and other coefficients by the $R^2$-terms in the effective gravitational action. The essential point of the proof is that due to the higher order curvature terms the entropy differs from the Bekenstein-Hawking one in the Einstein gravity by the contributions depending on the internal and external geometry of the horizon surface. 
  The dynamics of massive particles in the collinear high momentum regime is investigated. Methods hitherto exploiting large time asymptotic Hamiltonians in the Dirac picture for the treatment of infrared divergences are adapted to the collinear asymptotic dynamics. The essential role of time ordering of the dressing operators is brought out. 
  We propose a new model for interacting (electrically charged) anyons, where the 2+1-dimensional Darwin term is responsible for interactions. The Hamiltonian is comparable with the one used previously (in the RPA calculation). 
  We construct a two dimensional nonlinear $\sigma$-model that describes the Hamiltonian flow in the loop space of a classical dynamical system. This model is obtained by equivariantizing the standard N=1 supersymmetric nonlinear $\sigma$-model by the Hamiltonian flow. We use localization methods to evaluate the corresponding partition function for a general class of integrable systems, and find relations that can be viewed as generalizations of standard relations in classical Morse theory. 
  We discuss the properties of the large phase space of the genus-0 topological minimal $A_{k + 1}$ model coupled to 2d topological gravity. The minimal action is perturbed by adding all possible gravitational descendants with non-trivial couplings which form the infinite-dimensional large phase space. Some general identities (valid on the large phase space) are derived and the puncture and the dilaton equations are obtained as special cases of these results. Finally we show how the couplings to the gravitational descendants can actually be expressed in terms of the flat coordinates defining the small phase space of the theory. Thus eventually the large phase space becomes determined solely by the LG superpotential characterising the matter sector of the coupled model. 
  We show that reasonably well behaved 3d and 4D TQFts must contain certain algebraic structures. In 4D, we find both Hopf categories and trialgebras. 
  We present a systematic study of high energy scattering of non-abelian gauge particles in (3+1) dimensional Einstein gravity using semi-classical techniques of Verlinde and Verlinde. It is shown that the BRST gauge invariance of the Yang-Mills action in presence of quantum gravity at Planckian energy regime is maintained and the vertex operator is invariant under the BRST transformations. The presence of gravitational shock wave describing the gauge particles is discussed in the resulting (3+1) dimensional effective theory of Yang-Mills gravity. 
  We show that the partition function for a scalar field in a static spacetime background can be expressed as a functional integral in the corresponding optical space, and point out that the difference between this and the functional integral in the original metric is a Liouville type action. A general formula for the free energy is derived in the high temperature approximation and applied to various cases. In particular we find that thermodynamics in the extremal Reissner-Nordstr\"om space has extra singularities that make it ill-defined. 
  It is shown, that the correct account of quark interaction generated by instanton medium gives the possibility of description on the same ground of both pseudoscalar channel and low lying bound states in the vector channel. It must be noted, that all calculations are made in the approximation, taking into account only zero modes. 
  Contracting the $h$-deformation of $\SL(2,\Real)$, we construct a new deformation of two dimensional Poincar\'e algebra, the algebra of functions on its group and its differential structure. It is also shown that the Hopf algebra is triangular, and its universal R matrix is also constructed explicitly. Then, we find a deformation map for the universal enveloping algebra, and at the end, give the deformed mass shells and Lorentz transformation. 
  We present fermionic sum representations of the characters $\chi^{(p,p')}_{r,s}$ of the minimal $M(p,p')$ models for all relatively prime integers $p'>p$ for some allowed values of $r$ and $s$. Our starting point is binomial (q-binomial) identities derived from a truncation of the state counting equations of the XXZ spin ${1\over 2}$ chain of anisotropy $-\Delta=-\cos(\pi{p\over p'})$. We use the Takahashi-Suzuki method to express the allowed values of $r$ (and $s$) in terms of the continued fraction decomposition of $\{{p'\over p}\}$ (and ${p\over p'}$) where $\{x\}$ stands for the fractional part of $x.$ These values are, in fact, the dimensions of the hermitian irreducible representations of $SU_{q_{-}}(2)$ (and $SU_{q_{+}}(2)$) with $q_{-}=\exp (i \pi \{{p'\over p}\})$ (and $q_{+}=\exp ( i \pi {p\over p'})).$ We also establish the duality relation $M(p,p')\leftrightarrow M(p'-p,p')$ and discuss the action of the Andrews-Bailey transformation in the space of minimal models. Many new identities of the Rogers-Ramanujan type are presented. 
  Let $G$ be a semisimple Lie group, ${\frak g}$ its Lie algebra. For any symmetric space $M$ over $G$ we construct a new (deformed) multiplication in the space $A$ of smooth functions on $M$. This multiplication is invariant under the action of the Drinfeld--Jimbo quantum group $U_h{\frak g}$ and is commutative with respect to an involutive operator $\tilde{S}: A\otimes A \to A\otimes A$. Such a multiplication is unique. Let $M$ be a k\"{a}hlerian symmetric space with the canonical Poisson structure. Then we construct a $U_h{\frak g}$-invariant multiplication in $A$ which depends on two parameters and is a quantization of that structure. 
  We present the (0,4) superspace version of Witten's sigma model construction for ADHM instantons. We use the harmonic superspace formalism, which exploits the three complex structures common to both (0,4) supersymmetry and self-dual Yang-Mills theory. A novel feature of the superspace formulation is the manifest interplay between the ADHM construction and its twistor counterpart. 
  We complete the classification of (2,2) vacua that can be constructed from Landau--Ginzburg models by abelian twists with arbitrary discrete torsions. Compared to the case without torsion the number of new spectra is surprisingly small. In contrast to a popular expectation mirror symmetry does not seem to be related to discrete torsion (at least not in the present compactification framework): The Berglund-H"ubsch construction naturally extends to orbifolds with torsion; for more general potentials, on the other hand, the new spectra neither have nor provide mirror partners in our class of models. 
  There exist certain intrinsic relations between the ultraviolet divergent graphs and the convergent ones at the same loop order in renormalizable quantum field theories. Whereupon we may establish a new method, the intrinsic regularization method, to regulate those divergent graphs. In this note, we present a proposal, the inserter proposal, to the method. The $\phi^4$ theory and QED at the one loop order are dealt with in some detail. Inserters in the standard model are given. Some applications to SUSY-models are also made at the one loop order. 
  This paper is devoted to the study of closed string field theory in two dimensions. We compare two different approaches: BRST closed string field theory and the string effective Lagrangian. We show that the quadratic action and the pole structures of tree-level scattering amplitudes agree. We study merits and drawbacks of various gauge fixing procedures. In particular, we discuss conformal gauge in the context of the effective Lagrangian, and Siegel and Lorentz-like gauge in the general BRST approach. We discuss the ways in which discrete states survive a particular gauge fixing both by directly solving the equations of motion, and by analyzing pole structure of the scattering amplitudes. 
  We consider the application of Abelian orbifold constructions in Meromorphic Conformal Field Theory (MCFT) towards an understanding of various aspects of Monstrous Moonshine and Generalised Moonshine. We review some of the basic concepts in MCFT and Abelian orbifold constructions of MCFTs and summarise some of the relevant physics lore surrounding such constructions including aspects of the modular group, the fusion algebra and the notion of a self-dual MCFT. The FLM Moonshine Module, $V^\natural$, is historically the first example of such a construction being a $Z_2$ orbifolding of the Leech lattice MCFT, $V^\Lambda$. We review the usefulness of these ideas in understanding Monstrous Moonshine, the genus zero property for Thompson series which we have shown is equivalent to the property that the only meromorphic $Z_n$ orbifoldings of $V^\natural$ are $V^\Lambda$ and $V^\natural$ itself (assuming that $V^\natural$ is uniquely determined by its characteristic function $J(\tau)$. We show that these constraints on the possible $Z_n$ orbifoldings of $V^\natural$ are also sufficient to demonstrate the genus zero property for Generalised Moonshine functions in the simplest non-trivial prime cases by considering $Z_p\times Z_p$ orbifoldings of $V^\natural$. Thus Monstrous Moonshine implies Generalised Moonshine in these cases. 
  For a simple vertex operator algebra $V$ and a finite automorphism group $G$ of $V$ then $V$ is a direct sum of $V^{\chi}$ where $\chi$ are irreducible character of $G$ and $V^{\chi}$ is the subspace of $V$ which $G$ acts according to the character $\chi.$ We prove the following: 1. Each $V^{\chi}$ is nonzero. 2. $V^{\chi}$ is a tensor product $M_{\chi}\otimes V_{\chi}$ where $M_{\chi}$ is an irreducible $G$-module affording $\chi$ and $V_{\chi}$ is a $V^G$-module. If $G$ is solvable, $V_{\chi}$ is a simple $V^G$-module and $M_{\chi}\mapsto $V_{\chi}$ is a bijection from the set of irreducible $G$-modules to the set of (inequivalent) simple $V^G$-modules which are contained in $V.$ 
  For a vertex operator algebra $V$ and a vertex operator subalgebra $V'$ which is invarinant under an automorphism $g$ of $V$ of finite order, we introduce a $g$-twisted induction functor from the category of $g$-twisted $V'$-modules to the category of $g$-twisted $V$-modules. This functor satisfies the Frobenius reciprocity and transitivity. The results are illustrated with $V$ being simple or with $V'$ being $g$-rational. 
  We examine a simple heuristic test of integrability for quantum chains. This test is applied to a variety of systems, including a generic isotropic spin-1 model with nearest-neighbor interactions and a multiparameter family of spin-1/2 models generalizing the XYZ chain, with next-to-nearest neighbor interactions and bond alternation. Within the latter family we determine all the integrable models with an o(2) symmetry. 
  For gauge theories with confinement, the analytic structure of amplitudes is explored. It is shown that the analytic properties of physical amplitudes are the same as those obtained on the basis of an effective theory involving only the composite, physical fields. The corresponding proofs of dispersion relations remain valid. Anomalous thresholds are considered. They are related to the composite structure of particles. It is shown, that there are no such thresholds in physical amplitudes which are associated with confined constituents, like quarks and gluons in QCD. Unphysical amplitudes are considered briefly, using propagator functions as an example. For general, covariant, linear gauges, it is shown that these functions must have singularities at finite, real points, which may be associated with confined states. 
  In this paper we calculate the particle creation as seen by a stationary observer in 3+1 de Sitter space. This particle creation is calculated using an observer dependent geometrically based definition of time which is used to quantize a field on two different spacelike surfaces. The Bogolubov transformation relating these two quantizations is then calculated and the resulting particle creation is shown to be finite. 
  The running coupling constants are introduced in Quantum Mechanics and their evolution is described by the help of the renormalization group equation. 
  By applying the Lehmann-Symanzik-Zimmermann (LSZ) reduction formalism, we study the S matrix of collective field theory in which fermi energy is larger than the height of potential. We consider the spatially symmetric and antisymmetric boundary conditions. The difference is that S matrices are proportional to momenta of external particles in antisymmetric boundary condition, while they are proportional to energies in symmetric boundary condition. To the order of $g_{st}^2$, we find simple formulas for the S matrix of general potential. As an application, we calculate the S matrix of a case which has been conjectured to describe a "naked singularity". 
  Recent progress on the complete set of solutions of two dimensional classical string theory in any curved spacetime is reviewed. When the curvature is smooth the string solutions are deformed folded string solutions as compared to flat spacetime folded strings that were known for 19 years. However, surprizing new stringy behavior becomes evident at singularities such as black holes. The global properties of these solutions require that the ``bare singularity region"of the black hole be included along with the usual black hole spacetime. The mathematical structure needed to describe the solutions include a recursion relation that is analogous to the transfer matrix of lattice theories. This encodes lattice properties on the worldsheet on the one hand and the geometry of spacetime on the other hand. A case is made for the presence of folded strings in the quantum theory of non-critical strings for $d\geq 2$. 
  We investigate to what extent renormalization can be understood as an algebraic manipulation on concatenated one-loop integrals. We find that the resulting algebra indicates a useful connection to knot theory. 
  Quantum coherence allows the computation of an arbitrary number of distinct computational paths in parallel. Based on quantum parallelism it has been conjectured that exponential or even larger speedups of computations are possible. Here it is shown that, although in principle correct, any speedup is accompanied by an associated attenuation of detection rates. Thus, on the average, no effective speedup is obtained relative to classical (nondeterministic) devices. 
  The Hilbert space formalism of quantum mechanics is reviewed with emphasis on applications to quantum computing. Standard interferomeric techniques are used to construct a physical device capable of universal quantum computation. Some consequences for recursion theory and complexity theory are discussed. 
  Due to the continuity of quantum states, classical diagonalization has to be revised for quantum recursion theory. 
  I consider the dilute monopole gas expansion of the three dimensional Yang-Mills-Higgs system in the symmetry broken phase. The functional determinants which occur in such an expansion are computed in the heat kernel approximation for an arbitrary $SU(N)$ gauge group. Explicit expressions for the gauge boson mass in the unbroken gauge sector and the string tension are obtained for the $SU(2)$ gauge model and are evaluated numerically. The results show a strong dependence on the ratio $m_{\rm Higgs}/m_W$. 
  We present an inverse scattering construction of generalised point interactions (GPI) -- point-like objects with non-trivial scattering behaviour. The construction is developed for single centre $S$-wave GPI models with rational $S$-matrices, and starts from an integral transform suggested by the scattering data. The theory of unitary dilations is then applied to construct a unitary mapping between Pontryagin spaces which extend the usual position and momentum Hilbert spaces. The GPI Hamiltonian is defined as a multiplication operator on the momentum Pontryagin space and its free parameters are fixed by a physical locality requirement. We determine the spectral properties and domain of the Hamiltonian in general, and construct the resolvent and M{\o}ller wave operators thus verifying that the Hamiltonian exhibits the required scattering behaviour. The physical Hilbert space is identified. The construction is illustrated by GPI models representing the effective range approximation. For negative effective range we recover a known class of GPI models, whilst the positive effective range models appear to be new. We discuss the interpretation of these models, along with possible extensions to our construction. 
  The renormalization group flow in two--dimensional field theories is modified if they are coupled to gravity. Beta function coefficients are changed, the $c$--theorem is no longer strictly valid, and flows from fixed points with central charge $c>25$ to fixed points with $c<25$ are forbidden. This is discussed in general and at two examples, the Kosterlitz--Thouless phase transition and the Wess--Zumino--Witten model. A possible application to string cosmology is pointed out. 
  In this brief review we introduce the methods of quantum field theory out of equilibrium and study the non-equilibrium aspects of phase transitions. Specifically we critically study the picture of the ``slow-roll'' phase transition in the new inflationary models, we show that the instabilities that are the hallmark of the phase transition, that is the formation of correlated domains, dramatically change this picture. We analyze in detail the dynamics of phase separation in strongly supercooled phase transitions in Minkowski space. We argue that this is typically the situation in weakly coupled scalar theories. The effective evolution equations for the expectation value and the fluctuations of an inflaton field in a FRW cosmology are derived both in the loop expansion and in a self-consistent non-perturbative scheme. Finally we use these non-equilibrium techniques and concepts to study the influence of quantum and thermal fluctuations on the dynamics of a proposed mechanism for the formation of disoriented chiral condensates during a rapid phase transition out of equilibrium. This last topic may prove to be experimentally relevant at present accelerator energies. To appear in the Proceedings of the `2nd. Journ\'ee Cosmologie', Observatoire de Paris, 2-4, June 1994. H J de Vega and N. S\'anchez, Editors, World Scientific. 
  We study the possible phase transitions between (2+1)-dimensional abelian Chern-Simons theories. We show that they may be described by non-unitary rational conformal field theories with c_eff = 1. As an example we choose the fractional quantum Hall effect and derive all its main features from the discrete fractal structure of the moduli space of these non-unitary transition conforma lfield theories and some large scale principles. Rationality of these theories is intimately related to the modular group yielding sever conditions on the possible phase transitions. This gives a natural explanation for both, the values and the widths, of the observed fractional Hall plateaux. 
  This is the first of a series of papers studying combinatorial (with no ``subtractions'') bases and characters of standard modules for affine Lie algebras, as well as various subspaces and ``coset spaces'' of these modules. In part I we consider certain standard modules for the affine Lie algebra $\ga,\;\g := sl(n+1,\C),\;n\geq 1,$ at any positive integral level $k$ and construct bases for their principal subspaces (introduced and studied recently by Feigin and Stoyanovsky [FS]). The bases are given in terms of partitions: a color $i,\;1\leq i \leq n,$ and a charge $s,\; 1\leq s \leq k,$ are assigned to each part of a partition, so that the parts of the same color and charge comply with certain difference conditions. The parts represent ``Fourier coefficients'' of vertex operators and can be interpreted as ``quasi-particles'' enjoying (two-particle) statistical interaction related to the Cartan matrix of $\g.$ In the particular case of vacuum modules, the character formula associated with our basis is the one announced in [FS]. New combinatorial characters are proposed for the whole standard vacuum $\ga$-modules at level one. 
  We develop a general formalism to calculate the force between beads attached to a flat $d$-dimensional membrane due to the quantum fluctuations of the membrane. The interaction potential is derived as a function of $d$ and the membrane energy density, tension, stiffness and temperature. We find that the induced interactions turn off when $d$ exceeds a certain critical dimension. The potential is attractive in all cases where it is non-zero and at finite temperature falls off exponentially for large distances. 
   We start with the observation that the quantum group SL_q(2), described in terms of its algebra of functions has a quantum subgroup, which is just a usual Cartan group.    Based on this observation we develop a general method of constructing quantum groups with similar property. We also describe this method in the language of quantized universal enveloping algebras, which is another common method of studying quantum groups.    We carry our method in detail for root systems of type SL(2); as a byproduct we find a new series of quantum groups - metaplectic groups of SL(2)-type. Representations of these groups can provide interesting examples of bimodule categories over monoidal category of representations of SL_q(2). 
  I investigate the effects of the Chern-Simons coupling on high-energy behavior in $2+1$ dimensional U(1) gauged $\eta(\phi^\dagger\phi)^3$ theory with a Chern-Simons term. The effective potential and the $\beta$ function for $\eta$ are calculated to the next-to-leading order of the $1/N$ expansion as functions of $\theta$ (the Chern-Simons coupling). For all $\theta$, the theory is found to be driven to instability region at high momenta. It is briefly discussed on radiative corrections to $\theta$. 
  We investigate inhomogeneous quantum groups G built from a quantum group H and translations. The corresponding commutation relations contain inhomogeneous terms. Under certain conditions (which are satisfied in our study of quantum Poincare groups [12]) we prove that our construction has correct `size', find the R-matrices and the analogues of Minkowski space for G. 
  Using the general theory of [10] ( hep-th 9412058 ), quantum Poincar\'e groups (without dilatations) are described and investigated. The description contains a set of numerical parameters which satisfy certain polynomial equations. For most cases we solve them and give the classification of quantum Poincar\'e groups. Each of them corresponds to exactly one quantum Minkowski space. The Poincar\'e series of these objects are the same as in the classical case. We also classify possible $R$-matrices for the fundamental representation of the group. 
  The Lie algebra $so(2n+1)$ and the Lie superalgebra $osp(1/2n)$ are quantized in terms of $3n$ generators, called preoscillator generators. Apart from $n$ "Cartan" elements the preoscillator generators are deformed para-Fermi operators in the case of $so(2n+1)$ and deformed para-Bose operators in the case of $osp(1/2n)$. The corresponding deformed universal enveloping algebras $U_q[so(2n+1)]$ and $U_q[osp(1/2n)]$ are the same as those defined in terms of Chevalley operators. The name "preoscillator" is to indicate that in a certain representation these operators reduce to the known deformed Fermi and Bose operators. 
  The most general version of a renormalizable $d=4$ theory corresponding to a dimensionless higher-derivative scalar field model in curved spacetime is explored. The classical action of the theory contains $12$ independent functions, which are the generalized coupling constants of the theory. We calculate the one-loop beta functions and then consider the conditions for finiteness. The set of exact solutions of power type is proven to consist of precisely three conformal and three nonconformal solutions, given by remarkably simple (albeit nontrivial) functions that we obtain explicitly. The finiteness of the conformal theory indicates the absence of a conformal anomaly in the finite sector. The stability of the finite solutions is investigated and the possibility of renormalization group flows is discussed as well as several physical applications. 
  Massive integrable field theories in $1+1$ dimensions are defined at the Lagrangian level, whose classical equations of motion are related to the ``non-abelian'' Toda field equations. They can be thought of as generalizations of the sine-Gordon and complex sine-Gordon theories. The fields of the theories take values in a non-abelian Lie group and it is argued that the coupling constant is quantized, unlike the situation in the sine-Gordon theory, which is a special case since its field takes values in an abelian group. It is further shown that these theories correspond to perturbations of certain coset conformal field theories. The solitons in the theories will, in general, carry non-abelian charges. 
  General results on asymptotic expansions of Feynman diagrams in momenta and/or masses are reviewed. It is shown how they are applied for calculation of massive diagrams. 
  General prescriptions of differential renormalization are presented. It is shown that renormalization group functions are straightforwardly expressed through some constants that naturally arise within this approach. The status of the action principle in the framework of differential renormalization is discussed. 
  Starting with exact solutions to string theory on curved spacetimes we obtain deformations that represent gravitational shock waves. These may exist in the presence or absence of sources. Sources are effectively induced by a tachyon field that randomly fluctuates around a zero condensate value. It is shown that at the level of the underlying conformal field theory (CFT) these deformations are marginal and moreover all \a'-corrections are taken into account. Explicit results are given when the original undeformed 4-dimensional backgrounds correspond to tensor products of combinations of 2-dimensional CFT's, for instance SL(2,R)/R \times SU(2)/U(1). 
  On the basis of a new approach proposed in our previous work we develope a formalism for calculating of the effective action for some models containing fermion fields. This method allows us to calculate the fermionic part of the effective action (up to two-loop level) properly. The two-loop contribution to the effective potential for the Nambu-Jona-Lasinio model is calculated and is shown to vanish. 
  We study the Photon statistics in the superpositions of coherent states $|\alpha\rangle$ and $|\alpha^*\rangle$ named ``Schr\"odinger real and imaginary cat states''. The oscillatory character of photon distribution function (PDF) emerging due to the quantum interference between the two components is shown, and the phenomenon of the quadrature squeezing is observed for the moderate values of $|\alpha|\sim 1$. Despite the quantity ${\langle\triangle n^2\rangle}/{\langle n\rangle}$ tends to the unit value (like in the Poissonian PDF) at $|\alpha|\gg 1$, the photon statistics is essentially non-Poissonian for all values of $|\alpha|$. The factorial moments and cumulants of the PDF are calculated, and the oscillations of their ratio are demonstrated. 
  We present a study of the decay of metastable states of a scalar field via thermal activation, in the presence of a finite density of fermions. The process we consider is the nucleation of ``{\it droplets}'' of true vacuum inside the false one. We analyze a one-dimensional system of interacting bosons and fermions, considering the latter at finite temperature and with a given chemical potential. As a consequence of a non-equilibrium formalism previously developed, we obtain time-dependent decay rates. 
  A differential calculus on an associative algebra A is an algebraic analogue of the calculus of differential forms on a smooth manifold. It supplies A with a structure on which dynamics and field theory can be formulated to some extent in very much the same way we are used to from the geometrical arena underlying classical physical theories and models. In previous work, certain differential calculi on a commutative algebra exhibited relations with lattice structures, stochastics, and parametrized quantum theories. This motivated the present systematic investigation of differential calculi on commutative and associative algebras. Various results about their structure are obtained. In particular, it is shown that there is a correspondence between first order differential calculi on such an algebra and commutative and associative products in the space of 1-forms. An example of such a product is provided by the Ito calculus of stochastic differentials.   For the case where the algebra A is freely generated by `coordinates' x^i, i=1,...,n, we study calculi for which the differentials dx^i constitute a basis of the space of 1-forms (as a left A-module). These may be regarded as `deformations' of the ordinary differential calculus on R^n. For n < 4 a classification of all (orbits under the general linear group of) such calculi with `constant structure functions' is presented. We analyse whether these calculi are reducible (i.e., a skew tensor product of lower-dimensional calculi) or whether they are the extension (as defined in this article) of a one dimension lower calculus. Furthermore, generalizations to arbitrary n are obtained for all these calculi. 
  In previous articles it was demonstrated that the total cross section of the scattering of two light particles (zero modes of the Kaluza-Klein tower) in the six-dimensional $\lambda \phi^{4}$ model differs significantly from the cross section of the same process in the conventional $\lambda \phi^{4}$ theory in four space-time dimensions even for the energies below the threshold of the first heavy particle. Here the analytical structure of the cross section in the same model with torus compactification for arbitrary radii of the two-dimensional torus is studied. Further amplification of the total cross section due to interaction of the scalar field with constant background Abelian gauge potential in the space of extra dimensions is shown. 
  We discuss recent results on orbifold compactifications with (0,2) world sheet supersymmetry and continuous Wilson lines, emphasizing the role of modular symmetries. (This work is a contribution to the proceedings of the joint US Polish Workshop on Physics from the Planck scale to the Electro-Weak scale, Warsaw, 21.9. - 24.9.1994) 
  It is shown that the vacuum Einstein equations for an arbitrary stationary axisymmetric space-time can be completely separated by re-formulating the Ernst equation and its associated linear system in terms of a non-autonomous Schlesinger-type dynamical system. The conformal factor of the metric coincides (up to some explicitly computable factor) with the $\tau$-function of the Ernst equation in the presence of finitely many regular singularities. We also present a canonical formulation of these results, which is based on a ``two-time" Hamiltonian approach, and which opens new avenues for the quantization of such systems. 
  In a quantum mechanical treatment of gauge theories (including general relativity), one is led to consider a certain completion, $\agb$, of the space $\ag$ of gauge equivalent connections. This space serves as the quantum configuration space, or, as the space of all Euclidean histories over which one must integrate in the quantum theory. $\agb$ is a very large space and serves as a ``universal home'' for measures in theories in which the Wilson loop observables are well-defined. In this paper, $\agb$ is considered as the projective limit of a projective family of compact Hausdorff manifolds, labelled by graphs (which can be regarded as ``floating lattices'' in the physics terminology). Using this characterization, differential geometry is developed through algebraic methods. In particular, we are able to introduce the following notions on $\agb$: differential forms, exterior derivatives, volume forms, vector fields and Lie brackets between them, divergence of a vector field with respect to a volume form, Laplacians and associated heat kernels and heat kernel measures. Thus, although $\agb$ is very large, it is small enough to be mathematically interesting and physically useful. A key feature of this approach is that it does not require a background metric. The geometrical framework is therefore well-suited for diffeomorphism invariant theories such as quantum general relativity. 
  An interpretation of spacelike singularities in string theory uses target space duality to relate the collapsing Schwarzschild geometry near the singularity to an inflationary cosmology in dual variables. An appealing picture thus results whereby gravitational collapse seeds the formation of a new universe. 
  We derive the large k asymptotics of the surgery formula for SU(2) Witten's invariants of general Seifert manifolds. The contributions of connected components of the moduli space of flat connections are identified. The contributions of irreducible connections are presented in a residue form. This form is similar to the one used by A. Szenes, L. Jeffrey and F. Kirwan. This similarity allows us to express the contributions of irreducible connections in terms of intersection numbers on their moduli spaces. 
  The properties of static spherically symmetric black holes, which are either electrically or magnetically charged, and which are coupled to the dilaton in the presence of a cosmological constant, are considered. It is shown that such solutions do not exist if the cosmological constant is positive (in arbitrary spacetime dimension >= 4). However, asymptotically anti-de Sitter black hole solutions with a single horizon do exist if the cosmological constant is negative. These solutions are studied numerically in four dimensions and the thermodynamic properties of the solutions are derived. The extreme solutions are found to have zero entropy and infinite temperature for all non-zero values of the dilaton coupling constant. 
  In Calabi--Yau compactifications of the heterotic string there exist quantities which are universal in the sense that they are present in every Calabi--Yau string vacuum. It is shown that such universal characteristics provide numerical information, in the form of scaling exponents, about the space of ground states in string theory. The focus is on two physical quantities. The first is the Yukawa coupling of a particular antigeneration, induced in four dimensions by virtue of supersymmetry. The second is the partition function of the topological sector of the theory, evaluated on the genus one worldsheet, a quantity relevant for quantum mirror symmetry and threshold corrections. It is shown that both these quantities exhibit scaling behavior with respect to a new scaling variable and that a scaling relation exists between them as well. 
  We formulate the uniformisation problem underlying the geometry of W_n-gravity using the differential equation approach to W-algebras. We construct W_n-space (analogous to superspace in supersymmetry) as an (n-1) dimensional complex manifold using isomonodromic deformations of linear differential equations. The W_n-manifold is obtained by the quotient of a Fuchsian subgroup of PSL(n,R) which acts properly discontinuously on a simply connected domain in CP^{n-1}. The requirement that a deformation be isomonodromic furnishes relations which enable one to convert non-linear W-diffeomorphisms to (linear) diffeomorphisms on the W_n-manifold. We discuss how the Teichmuller spaces introduced by Hitchin can then be interpreted as the space of complex structures or the space of projective structures with real holonomy on the W_n-manifold. The projective structures are characterised by Halphen invariants which are appropriate generalisations of the Schwarzian. This construction will work for all ``generic'' W-algebras. 
  String effective theories with N=1 supersymmetry in four dimensions are subject of the discussion. Gaugino condensation in the chiral representation of the dilaton is reviewed in the truncated formalism in the $U_{K}(1)$-superspace. Using the supersymmetric duality of the dilaton the same investigation is made in the linear representation of the dilaton. We show that for the simple case of one gaugino condensate the results concerning supersymmtry breaking are independent of the representation of the dilaton. 
  A matrix model to describe dynamical loops on random planar graphs is analyzed. It has similarities with a model studied by Kazakov, few years ago, and the O(n) model by Kostov and collaborators. The main difference is that all loops are coherently oriented and empty. The free energy is analytically evaluated and the two critical phases are analyzed, where the free energy exhibits the same critical behaviour of Kazakov's model, thus confirming the universality of the description in the continuum limit (surface with small holes, and the tearing phase). A third phase occurs on the boundary separating the above phase regions, and is characterized by a different singular behaviour, presumably due to the orientation of loops. 
  We study quantum strings in strong gravitational fields. The relevant small parameter is $g=R_c{\sqrt T_0}$, where $R_c$ is the curvature of the spacetime and $T_0$ is the string tension. Within our systematic expansion we obtain to zeroth order the null string (string with zero tension), while the first order correction incorporates the string dynamics. We apply our formalism to quantum null strings in de Sitter spacetime. After a reparametrization of the world-sheet coordinates, the equations of motion are simplified. The quantum algebra generated by the constraints is considered, ordering the momentum operators to the right of the coordinate operators. No critical dimension appears. It is anticipated however that the conformal anomaly will appear when the first order corrections proportional to $T_0$, are introduced. 
  A set of compatible formulas for the Clebsch-Gordan coefficients of the quantum algebra $U_{q}({\rm su}_2)$ is given in this paper. These formulas are $q$-deformations of known formulas, as for instance: Wigner, van der Waerden, and Racah formulas. They serve as starting points for deriving various realizations of the unit tensor of $U_{q}({\rm su}_2)$ in terms of $q$-boson operators. The passage from the one-parameter quantum algebra $U_{q }({\rm su}_2)$ to the two-parameter quantum algebra $U_{qp}({\rm u}_2)$ is discussed at the level of Clebsch-Gordan coefficients. 
  Lie bialgebra contractions are introduced and classified. A non-degenerate coboundary bialgebra structure is implemented into all pseudo-orthogonal algebras $so(p,q)$ starting from the one corresponding to $so(N+1)$. It allows to introduce a set of Lie bialgebra contractions which leads to Lie bialgebras of quasi-orthogonal algebras. This construction is explicitly given for the cases $N=2,3,4$. All Lie bialgebra contractions studied in this paper define Hopf algebra contractions for the Drinfel'd-Jimbo deformations $U_z so(p,q)$. They are explicitly used to generate new non-semisimple quantum algebras as it is the case for the Euclidean, Poincar\'e and Galilean algebras. 
  We show that any of the new knot invariants obtained from Chern-Simons theory based on an arbitrary non-abelian gauge group do not distinguish isotopically inequivalent mutant knots and links. In an attempt to distinguish these knots and links, we study Murakami (symmetrized version) $r$-strand composite braids. Salient features of the theory of such composite braids are presented. Representations of generators for these braids are obtained by exploiting properties of Hilbert spaces associated with the correlators of Wess-Zumino conformal field theories. The $r$-composite invariants for the knots are given by the sum of elementary Chern-Simons invariants associated with the irreducible representations in the product of $r$ representations (allowed by the fusion rules of the corresponding Wess-Zumino conformal field theory) placed on the $r$ individual strands of the composite braid. On the other hand, composite invariants for links are given by a weighted sum of elementary multicoloured Chern-Simons invariants. Some mutant links can be distinguished through the composite invariants, but mutant knots do not share this property. The results, though developed in detail within the framework of $SU(2)$ Chern-Simons theory are valid for any other non-abelian gauge group. 
     Representations of a group $G$ in vector spaces over a field $K$ form a category. One can reconstruct the given group $G$ from its representations to vector spaces as the full group of monoidal automorphisms of the underlying functor. This is a special example of Tannaka-Krein theory. This theory was used in recent years to reconstruct quantum groups (quasitriangular Hopf algebras) in the study of algebraic quantum field theory and other applications.      We show that a similar study of representations in spaces with additional structure (super vector spaces, graded vector spaces, comodules, braided monoidal categories) produces additional symmetries, called ``hidden symmetries''. More generally, reconstructed quantum groups tend to decompose into a smash product of the given quantum group and a quantum group of ``hidden'' symmetries of the base category. 
  Quantum symmetries of four-dimensional superstrings are frequently realized in an anomaly-cancellation mode in the effective low-energy supergravity. The massless antisymmetric tensor plays an important r\^ole in this mechanism and the choice of its supersymmetric description, using either a chiral or a linear multiplet, appears to introduce significant conceptual and practical differences at the string loop level. This paper reviews the construction of loop-corrected string effective supergravities with the dilaton and antisymmetric tensor embedded in a linear multiplet. Using anomaly cancellation and the linear multiplet allows to obtain an all-order renormalization-group invariant effective lagrangian for a pure gauge sector with field-dependent gauge coupling constant.   Presented at the Workshop "Physics from Planck Scale to Electroweak Scale",   Warsaw, Poland, September 1994. 
  The multi-Regge effective action is derived directly from the linearized gravity action. After excluding the redundant field components we separate the fields into momentum modes and integrate over modes which correspond neither to the kinematics of scattering nor to the one of exchanged particles. The effective vertices of scattering and of particle production are obtained as sums of the contributions from the triple and quartic interaction terms and the fields in the effective action are defined in terms of the two physical components of the metric fluctuation. 
  We introduce the chemical potential in a system of two-dimensional massless fermions, confined to a finite region, by imposing twisted boundary conditions in the Euclidean time direction. We explore in this simple model the application of functional techniques which could be used in more complicated situations. 
  A model for quantum gravity in one (time) dimension is discussed, based on Regge's discrete formulation of gravity. The nature of exact continuous lattice diffeomorphisms and the implications for a regularized gravitational measure are examined. After introducing a massless scalar field coupled to the edge lengths, the scalar functional integral is performed exactly on a finite lattice, and the ensuing change in the measure is determined. It is found that the renormalization of the cosmological constant due to the scalar field fluctuations vanishes identically in one dimension. A simple decimation renormalization group transformation is performed on the partition function and the results are compared with the exact solution. Finally the properties of the spectrum of the scalar Laplacian are compared with results obtained for a Poissonian distribution of edge lengths. 
  Considering one-dimensional nonminimally-coupled lattice gauge theories, a class of nonlocal one-dimensional systems is presented, which exhibits a phase transition. It is shown that the transition has a latent heat, and, therefore, is a first order phase transition. 
  It is shown that a kind of solutions of n-simplex equation can be obtained from representations of braid group. The symmetries in its solution space are also discussed. 
  N=2 supersymmetric Yang--Mills theories coupled to matter are considered in the Wess--Zumino gauge. The supersymmetries are realized nonlinearly and the anticommutator between two susy charges gives, in addition to translations, gauge transformations and equations of motion. The difficulties hidden in such an algebraic structure are well known: almost always auxiliary fields can be introduced in order to put the formalism off--shell, but still the field dependent gauge transformations give rise to an infinite dimensional algebra quite hard to deal with. However, it is possible to avoid all these problems by collecting into an unique nilpotent operator all the symmetries defining the theory, namely ordinary BRS, supersymmetries and translations. According to this method the role of the auxiliary fields is covered by the external sources coupled, as usual, to the nonlinear variations of the quantum fields. The analysis is then formally reduced to that of ordinary Yang--Mills theory. 
  The complex-time formalism is developed in the framework of the path-integral formalism, to be used for analysis of the quantum tunneling phenomena. We show that subleading complex-time saddle-points do not account for the right WKB result. Instead, we develop a reduction formula, which enables us to construct Green functions from simple components of the potential, for which saddle-point method is applicable. This method leads us to the valid WKB result, which incorporates imaginary-time instantons and bounces, as well as the real-time boundary conditions. 
  Certain criteria are demonstrated for a spatial derivation of a von Neumann algebra to generate a one-parameter semigroup of endomorphisms of that algebra. These are then used to establish a converse to recent results of Borchers and of Wiesbrock on certain one-parameter semigroups of endomorphisms of von Neumann algebras (specifically, Type III_1 factors) that appear as lightlike translations in the theory of algebras of local observables. 
  We give a general formula for gauge states at the discrete momenta in Liouville theory. These discrete gauge states carry the $w_\infty$ charges. As in the case of the 26D (or 10D) string theory, they are decoupled from the correlation functions and can be considered as the symmetry parameters in the old covariant quantization of the theory. 
  We evaluate the partition functions in the neighbourhood of catastrophes by saddle point integration and express them in terms of generalized Airy functions. 
  We discuss the counting of minimal geodesic ball coverings of $n$-dimensional riemannian manifolds of bounded geometry, fixed Euler characteristic and Reidemeister torsion in a given representation of the fundamental group. This counting bears relevance to the analysis of the continuum limit of discrete models of quantum gravity. We establish the conditions under which the number of coverings grows exponentially with the volume, thus allowing for the search of a continuum limit of the corresponding discretized models. The resulting entropy estimates depend on representations of the fundamental group of the manifold through the corresponding Reidemeister torsion. We discuss the sum over inequivalent representations both in the two-dimensional and in the four-dimensional case. Explicit entropy functions as well as significant bounds on the associated critical exponents are obtained in both cases. 
  The dependence of velocity on momentum for the free massive particle obeying the $\kappa $-Poincar\'{e} Poisson symmetry is calculated in terms of intrinsic non-commuting space-time coordinates and shown to have a monotonic character, with upper limit of velocity equal to 1. 
  We present almost complete list of normal forms of classical $r$-matrices on the Poincar\'{e} group. 
  Extended phase space of an elementary (relativistic) system is introduced in the spirit of the Souriau's definition of the `space of motions' for such system. Our formulation is generally applicable to any homogeneous space-time (e.g. de Sitter) and also to Poisson actions. Calculations concerning the Minkowski case for non-zero spin particles show an intriguing alternative: we should either accept two-dimensional trajectories or (Poisson) noncommuting space-time coordinates. 
  General framework for Poisson homogeneous spaces of Poisson groups is introduced. Poisson Minkowski spaces are discussed as a particular example. 
  In this paper a fundamental length is introduced into physics. This is done in a way which respects special relativity and quantum field theory. The theory has formal similarity to quantum field theory though its properties are far better: divergences are got rid of. The problem of quantizing gravity is straightforward in the approach. 
  The p-adic description of Higgs mechanism in TGD framework provides excellent predictions for elementary particle and hadrons masses (hep-th@xxx.lanl.gov 9410058-62). The gauge group of TGD is just the gauge group of the standard model so that it makes sense to study the p-adic counterpart of the standard model as a candidate for low energy effective theory. Momentum eigen states can be constructed purely number theoretically and the infrared cutoff implied by the finite size of the convergence cube of p-adic square root function leads to momentum discretization. Discretization solves ultraviolet problems: the number of momentum states associated with a fixed value of the propagator expression in the loop is integer and has p-adic norm not larger than one so that the contribution of momentum squared with p-adic norm $p^{k}$ converges as $p^{-2k-2}$ for boson loop. The existence of the action exponential forces number theoretically the decomposition into free and interacting parts. The free part is of order $O(p^0)$ and must vanish (and does so by equations of motion) and interaction part is at most of order $O(\sqrt{p})$ p-adically. p-Adic coupling constants are of form $g\sqrt{p}$: their real counterparts are obtained by canonical identification between p-adic and real numbers. The discretized version of Feynmann rules of real theory should give S-matrix elements but Feynmann rules guarantee unitarity in formal sense only. The unexpected result is the upper bound $L_p=L_0/\sqrt{p}$ ($L_0\sim 10^4\sqrt{G}$) for the size of p-adic convergence cube from the cancellation of infrared divergences so that p-adic field theory doesn't make sense above length scale $L_p$. 
  The solution of the O$(N) \phi^4$ scalar field theory in the broken phase is given in the framework of light cone quantization and a 1/N expansion. It involves the successive building of operator solutions to the equation of motion and constraints including operator zero modes of the fields which are the LC counterpart to the equal time non trivial vacuum effects. The renormalization of the procedure is accomplished up to $2^{nd}$ order in the 1/N expansion for the equation of motion and constraints. In addition the renormalization of the divergent contributions of the 2-point and 4-point functions is performed in a covariant way. The presence of a zero modes leads to genuine non perturbative renormalization features. 
  The concept of the one -- dimensional quantum mechanics in the relativistic configurational space (RQM) is reviewed briefly. The Relativistic Schroedinger equation (RSE) arising here is the finite-difference equation with the step equal to the Compton wave length of the particle. The different generalizations of the Dirac -- Infeld -- Hall factorizarion method for this case are constructed. This method enables us to find out all possible finite-difference generalizations of the most important nonrelativistic integrable case -- the harmonic oscillator. As it was shown (\cite{kmn},\cite{mir6}) in RQM the harmonic oscillator = $q$ -- oscillator. It is also shown that the relativistic and nonrelativistic QM's are different representations of the same theory. Thetransformation connecting these two representations is found in explicit form. It could be considered as the generalization of the Kontorovich -- Lebedev transformation. 
  We calculate the effective tachyonic potential in closed string field theory up to the quartic term in the tree approximation. This involves an elementary four-tachyon vertex and a sum over the infinite number of Feynman graphs with an intermediate massive state. We show that both the elementary term and the sum can be evaluated as integrals of some measure over different regions in the moduli space of four-punctured spheres. We show that both elementary and effective coupling give negative contributions to the quartic term in the tachyon potential. Numerical calculations show that the fourth order term is big enough to destroy a local minimum which exists in the third order approximation. 
  These days, as high energy particle colliders become unavailable for testing speculative theoretical ideas, physicists are looking to other environments that may provide extreme conditions where theory confronts physical reality. One such circumstance may arise at high temperature $T$, which perhaps can be attained in heavy ion collisions or in astrophysical settings. It is natural therefore to examine the high-temperature behavior of the standard model, and here I shall report on recent progress in constructing the high-$T$ limit of~QCD. 
  We propose a spinon basis for the integrable highest weight modules of $\hsltw$ at levels $k\geq1$, and discuss the underlying Yangian symmetry. Evaluating the characters in this spinon basis provides new quasi-particle type expressions for the characters of these integrable modules, and explicitly exhibits the structure of an RSOS times a Yangian part, known \eg from $S$-matrix results. We briefly discuss generalizations to other groups and more general conformal field theories. 
  Let $V$ be a simple vertex operator algebra and $G$ be a finite nilpotent group of automorphisms of $V.$ We prove the following in this paper: (1) There is a Galois correspondence between subgroups of $G$ and the vertex operator subalgebras of $V$ which contain $V^G$ given by the map $H\mapsto V^H.$ (2) Assume that for every G\in G$ there is unique simple $g$-twisted $V$-module $M(g).$ Then there exists a Hochschild 3-cocycle $\alpha$ on the integral group $Z[G]$ such that there is an equivalence of categories between $V^G$-module category (whose objects are $V^G$-submodules of direct sums of copies of $\oplus_{g\in G}M(g),$ and whose morphisms are $V^G$-module homomorphisms) and the module category for the twisted quantum double $D_{\alpha}(G)$ associated to $\alpha.$ 
  We give a simple diagrammatic algorithm for writing the chiral large $N$ expansion of intersecting Wilson loops in $2D$ $SU(N)$ and $U(N) $Yang Mills theory in terms of symmetric groups, generalizing the result of Gross and Taylor for partition functions. We prove that these expansions compute Euler characters of a space of holomorphic maps from string worldsheets with boundaries. We prove that the Migdal-Makeenko equations hold for the chiral theory and show that they can be expressed as linear constraints on perturbations of the chiral $YM2$ partition functions. We briefly discuss finite $N$ , the non-chiral expansion, and higher dimensional lattice models. 
  There exist certain intrinsic relations between the ultraviolet divergent graphs and the convergent ones at the same loop order in renormalizable quantum field theories. Whereupon we present a new method, the inserter regularization method, to regulate those divergent graphs. In this letter, we demonstrate this method with the $\phi^4$ theory and QED at the one loop order. Some applications to SUSY-models are also made at the one loop order, which shows that supersymmetry is preserved manifestly and consistently. 
  This contribution briefly describes some developments of the use of string symmetries and anomaly cancellation mechanisms to include string loop corrections in the construction of the low-energy effective supergravity of superstrings. (Presented at the 27th International Conference on High Energy Physics, Glasgow, July 1994) 
  A study of two-dimensional QCD on a spatial circle with Majorana fermions in the adjoint representation of the gauge groups SU(2) and SU(3) has been performed. The main emphasis is put on the symmetry properties related to the homotopically non-trivial gauge transformations and the discrete axial symmetry of this model. Within a gauge fixed canonical framework, the delicate interplay of topology on the one hand and Jacobians and boundary conditions arising in the course of resolving Gauss's law on the other hand is exhibited. As a result, a consistent description of the residual $Z_N$ gauge symmetry (for SU(N)) and the ``axial anomaly" emerges. For illustrative purposes, the vacuum of the model is determined analytically in the limit of a small circle. There, the Born-Oppenheimer approximation is justified and reduces the vacuum problem to simple quantum mechanics. The issue of fermion condensates is addressed and residual discrepancies with other approaches are pointed out. 
  We present here the general solution describing generators of \kdef \poin algebra as the functions of classical \poin algebra generators as well as the inverse formulae. Further we present analogous relations for the generators of N=1 D=4 \kdef \poin superalgebra expressed by the classical \poin superalgebra generators. In such a way we obtain the \kdef \poin (super)algebras with all the quantum deformation present only in the coalgebra sector. Using the classical basis of \kdef \poin superalgebra we obtain as a new result the $\k$-deformation of supersymmetric covariant spin square Casimir. 
  We review the structure of the moduli space of particular N = (2,2) superconformal field theories. We restrict attention to those of particular use in superstring compactification, namely those with central charge c = 3d for some integer d and whose NS fields have integer U(1) charge. The cases d = 1, 2 and 3 are analyzed. It is shown that in the case d >= 3 it is important to use techniques of algebraic geometry rather than rely on metric-based ideas. The phase structure of these moduli spaces is discussed in some detail. (Lectures Delivered at Trieste Summer School, 1994.) 
  We clarify the algebraic structure of continuous and discrete quasi-exactly solvable spectral problems by embedding them into the framework of the quantum inverse scattering method. The quasi-exactly solvable hamiltonians in one dimension are identified with traces of quantum monodromy matrices for specific integrable systems with non-periodic boundary conditions. Applications to the Azbel-Hofstadter problem are outlined. 
  Recently two groups have listed all sets of weights (k_1,...,k_5) such that the weighted projective space P_4^{(k_1,...,k_5)} admits a transverse Calabi-Yau hypersurface. It was noticed that the corresponding Calabi-Yau manifolds do not form a mirror symmetric set since some 850 of the 7555 manifolds have Hodge numbers (b_{11},b_{21}) whose mirrors do not occur in the list. By means of Batyrev's construction we have checked that each of the 7555 manifolds does indeed have a mirror. The `missing mirrors' are constructed as hypersurfaces in toric varieties. We show that many of these manifolds may be interpreted as non-transverse hypersurfaces in weighted P_4's, ie, hypersurfaces for which dp vanishes at a point other than the origin. This falls outside the usual range of Landau--Ginzburg theory. Nevertheless Batyrev's procedure provides a way of making sense of these theories. 
  A brief heuristic explanation is given of recent work with Juergen Fuchs, Beatriz Gato-Rivera and Christoph Schweigert on the construction of modular invariant partition functions from Galois symmetry in conformal field theory. A generalization, which we call quasi-Galois symmetry, is also described. As an application of the latter, the invariants of the exceptional algebras at level $g$ (for example $E_8$ level 30) expected from conformal embeddings are presented. [Contribution to the Proceedings of the International Symposium on the Theory of Elementary Particles Wendisch-Rietz, August 30 - September 3, 1994] 
  We solve, by separation of variables, the problem of three anyons with a harmonic oscillator potential. The anyonic symmetry conditions from cyclic permutations are separable in our coordinates. The conditions from two-particle transpositions are not separable, but can be expressed as reflection symmetry conditions on the wave function and its normal derivative on the boundary of a circle. Thus the problem becomes one-dimensional. We solve this problem numerically by discretization. $N$-point discretization with very small $N$ is often a good first approximation, on the other hand convergence as $N\to\infty$ is sometimes very slow. 
  We take the continuum limit of the \sdeq s of the one and two matrix model without expanding them in the length of the loop. The resulting equations agree with those proposed for string field theory in the temporal gauge. We find that the loop operators are required to mix in the two matrix model case and determine the non-constant tadpole terms. 
  Definition of Feynman integrals as solutions of some well defined systems of differential equations is proposed. This definition is equivalent to usual one but needs no regularization and application of $R$-operation. It is argued that proposed renormalization scheme maintains all symmetries that can be maintained in perturbative quantum field theory, and also is convenient for practical calculations. 
  We consider evolution equations of the Lotka-Volterra type, and elucidate especially their formulation as canonical Hamiltonian systems. The general conditions under which these equations admit several conserved quantities (multi-Hamiltonians) are analysed. A special case, which is related to the Liouville model on a lattice, is considered in detail, both as aclassical and as aquantal system 
  In this work we discuss an approach due to F. David to the geometry of world sheets of non-critical strings in quasiclassical approximation. The gravitational dressed conformal dimension is related to the scaling behavior of the two-point function with respect to a distance variable. We show how this approach reproduces the standard gravitational dressing in the next order of perturbation theory. With the same technique we calculate the intrinsic Hausdorff dimension of a world sheet. 
  A new construction using finite dimensional dual grassmannians is developed to study rational and soliton solutions of the KP hierarchy. In the rational case, properties of the tau function which are equivalent to bispectrality of the associated wave function are identified. In particular, it is shown that there exists a bound on the degree of all time variables in tau if and only if the wave function is rank one and bispectral. The action of the bispectral involution, beta, in the generic rational case is determined explicitly in terms of dual grassmannian parameters. Using the correspondence between rational solutions and particle systems, it is demonstrated that beta is a linearizing map of the Calogero-Moser particle system and is essentially the map sigma introduced by Airault, McKean and Moser in 1977. 
  It is shown that the one-loop ultraviolet divergences in renormalizable supersymmetric theories can be regulated by the introduction of heavy Pauli-Villars chiral supermultiplets, provided the generators of the gauge group are traceless in the matter representation. The procedure is extended to include supersymmetric gauged nonlinear sigma models. 
  Dominant weight multiplicities of simple Lie groups are expressed in terms of the modular matrices of Wess-Zumino-Witten conformal field theories, and related objects. Symmetries of the modular matrices give rise to new relations among multiplicities. At least for some Lie groups, these new relations are strong enough to completely fix all multiplicities. 
  We consider the Hamiltonian reduction of the two-loop Wess-Zumino-Novikov-Witten model (WZNW) based on an untwisted affine Kac-Moody algebra $\cgh$. The resulting reduced models, called {\em Generalized Non-Abelian Conformal Affine Toda (G-CAT)}, are conformally invariant and a wide class of them possesses soliton solutions; these models constitute non-abelian generalizations of the Conformal Affine Toda models. Their general solution is constructed by the Leznov-Saveliev method. Moreover, the dressing transformations leading to the solutions in the orbit of the vacuum are considered in detail, as well as the $\tau$-functions, which are defined for any integrable highest weight representation of $\cgh$, irrespectively of its particular realization. When the conformal symmetry is spontaneously broken, the G-CAT model becomes a generalized Affine Toda model, whose soliton solutions are constructed. Their masses are obtained exploring the spontaneous breakdown of the conformal symmetry, and their relation to the fundamental particle masses is discussed. 
  These lectures were prepared to be presented at A.A. Belavin seminar on CFT at Landau Institute for Theoretical Physics. We review bosonization of CFT and show how it can be applied to the studying of representations of Zamolodchikov-Faddeev (ZF) algebras. In the bosonic construction we obtain explicit realization of chiral vertex operators interpolating between irreducible representations of the deformed Virasoro algebra. The commutation relations of these operators are determined by the elliptic matrix of IRF type and their matrix elements are given in the form of the contour integrals of some meromorphic functions. 
  An elementary derivation of the chiral gauge anomaly in all even dimensions is given in terms of noncommutative traces of pseudo-differential operators. 
  We give an algebraic proof of the spin-statistics connection for the parabosonic and parafermionic quantum topological charges of a theory of local observables with a modular PCT-symmetry. The argument avoids the use of the spinor calculus and also works in 1+2 dimensions. It is expected to be a progress towards a general spin-statistics theorem including also (1+2)-dimensional theories with braid group statistics. 
  Trieste Spring School Lectures describing the author's opinions about black hole evaporation and information loss. The remnant, or cornucopion scenario for the endpoint of Hawking evaporation is described in detail. In this picture information can be lost to the original asymptotic observer without violating the rules of quantum mechanics, because a black hole remnant is viewed as a large space connected onto our own by an almost pointlike opening. It does not behave like an elementary particle. Objections to remnants are refuted and the (remote) possibility of testing this scenario experimentally is discussed. Also included is a brief description of Susskind's picture of the stringy origin of Bekenstein-Hawking entropy. An attempt is made to argue that the cornucopion picture and Susskind's model of the states responsible for black hole entropy are compatible with each other. Information is lost to the asymptotic observer in Hawking evaporation, but the information encoded in the BH entropy remains in causal contact with him and is re-emitted with the Hawking radiation. 
  We consider the theory of open bosonic string in massive background fields. The general structure of renormalization is investigated. A general covariant action for a string in background fields of the first massive level is suggested and its symmetries are described. Equations of motion for the background fields are obtained by demanding that the renormalized operator of the energy-momentum tensor trace vanishes. 
  A Symmetry between bosonic coordinates and some Grassmannian-type coordinates is presented. Commuting two of these Grassmannian-type variables results in an arbitrary phase (not just a minus sign). This symmetry is also realised at the level of the field theory. 
  For a wide class of mechanical systems, invariant under gauge transformations with higher (arbitrary) order time derivatives of gauge parameters, the equivalence of Lagrangian and Hamiltonian BRST formalisms is proved. It is shown that the Ostrogradsky formalism establishes the natural rules to relate the BFV ghost canonical pairs with the ghosts and antighosts introduced by the Lagrangian approach. Explicit relation between corresponding gauge-fixing terms is obtained. 
  N=2 supersymmetric field theories in two dimensions have been extensively studied in the last few years. Many of their properties can be determined along the whole renormalization group flow, like their coupling dependence and soliton spectra. We discuss here several models which can be solved completely, when the number of superfields is taken to be large, by studying their topological-antitopological fusion equations. These models are the CPN model, sigma models on Grassmannian manifolds, and certain perturbed $N=2$ Minimal model. 
  We construct a relativistically covariant symmetry of QED. Previous local and nonlocal symmetries are special cases. This generalized symmetry need not be nilpotent, but nilpotency can be arranged with an auxiliary field and a certain condition. The Noether charge generating the symmetry transformation is obtained, and it imposes a constraint on the physical states. 
  The relationship between the Dirac and reduced phase space quantizations is investigated for spin models belonging to the class of Hamiltonian systems having no gauge conditions. It is traced out that the two quantization methods may give similar, or essentially different physical results, and, moreover, it is shown that there is a class of constrained systems, which can be quantized only by the Dirac method. A possible interpretation of the gauge degrees of freedom is given. 
  These lectures give a pedagogical review of dilaton gravity, Hawking radiation, the black hole information problem, and black hole pair creation. (Lectures presented at the 1994 Trieste Summer School in High Energy Physics and Cosmology) 
  The quantum Clebsch-Gordan coefficients and the explicit form of the $\breve{R}_{q}$ matrix related with the minimal representation of the quantum enveloping algebra $U_{q}E_{7}$ are calculated in this paper. 
  We use the $\zeta$-function regularization and an integral representation of the complex power of a pseudo differential operator, to give an unambiguous definition of the determinant of the Dirac operator. We bring this definition to a workable form by making use of an asymmetric Wigner representation. The expression so obtained is amenable to several treatments of which we consider in detail two, the inverse mass expansion and the gradient expansion, with concrete examples. We obtain explicit closed expressions for the corresponding Seeley-DeWitt coefficients to all orders. The determinant is shown to be vector gauge invariant and to posses the correct axial and scale anomalies. The main virtue of our approach is that it is conceptually simple and systematic and can be extended naturally to more general problems (bosonic operators, gravitational fields, etc). In particular, it avoids defining the real and imaginary parts of the effective action separately. In addition, it does not reduce the problem to a bosonic one to apply heat kernel nor performs further analytical rotations of the fields to make the Dirac operator Hermitian. We illustrate the flexibility of the method by studying some interesting cases. 
  The antifield-BRST formalism and the various cohomologies associated with it are surveyed and illustrated in the context of Yang-Mills gauge theory. In particular, the central role played by the Koszul-Tate resolution and its relation to the characteristic cohomology are stressed. 
  Extension of the braid relations to the multiple braided tensor product of algebras that can be used for quantization of nonultralocal models is presented. The Yang--Baxter--type consistency conditions as well as conditions for the existence of the multiple coproduct (monodromy matrix), which can be used for construction of the commuting subalgebra, are given. Homogeneous and local algebras are introduced, and simplification of the Yang--Baxter--type conditions for them is shown. The Yang--Baxter--type equations and multiple coproduct conditions for homogeneous and local of order 2 algebras are solved. 
  A comparative dynamical study of axial gauge QED and QCD is presented. Elementary excitations associated with particular field configurations are investigated. Gluonic excitations analogous to linearly polarized photons are shown to acquire infinite energy. Suppression of this class of excitations in QCD results from quantization of the chromelectric flux and is interpreted as a dual Meissner effect, i.e. as expulsion from the QCD vacuum of chromo-electric fields which are constant over significant distances. This interpretation is supported by a comparative evaluation of the interaction energy of static charges in the axial gauge representation of QED and QCD. 
  Thermodynamics of scalar fields is investigated in three dimensional black hole backgrounds in two approaches. One is mode expansion and direct computation of the partition sum, and the other is the Euclidean path integral approach. We obtain a number of exact results, for example, mode functions, Hartle-Hawking Green functions on the black holes, Green functions on a cone geometry, free energies and entropies. They constitute reliable bases for the thermodynamics of scalar fields. It is shown that thermodynamic quantities largely depend upon the approach to calculate them, boundary conditions for the scalar fields and regularization method. We find that, in general, the entropies are not proportional to the area of the horizon and that their divergent parts are not necessarily due to the existence of the horizon. 
  Gauge fixing and the observable fields for both abelian and non-abelian gauge theories with spontaneous breaking of gauge symmetry are studied. We explicitly show that it is possible to globally fix the gauge in the broken sector and hence construct physical fields even in the non-abelian theory. We predict that any high temperature restoration of gauge symmetry will be accompanied by a confining transition. 
  The Becchi-Rouet-Stora-Tyutin (BRST) method is applied to the quantization of the solitons of the non-linear $O(3)$ model in $2+1$ dimensions. We show that this method allows for a simple and systematic treatment of zero-modes with a non-commuting algebra. We obtain the expression of the BRST hamiltonian and show that the residual interaction can be perturbatively treated in an IR-divergence-free way. As an application of the formalism we explicitly evaluate the two-loop correction to the soliton mass. 
  A family of harmonic superspaces associated with four-dimensional spacetime is described. Some applications to supersymmetric field theories, including supergravity, are given. 
  Although the WKB approximation for multicomponent systems has been intensively studied in the literature, its geometric and global aspects are much less well understood than in the scalar case. In this paper we give a completely geometric derivation of the transport equation, without using local sections and without assuming complete diagonalizability of the matrix valued principal symbol, or triviality of its eigenbundles. The term (called ``no-name term'' in some previous literature) appearing in the transport equation in addition to the covariant derivative with respect to a natural projected connection will be a tensor, independent of the choice of any sections. We give a geometric interpretation of this tensor, involving the contraction of the curvature of the eigenbundle and an analog of the second fundamental form with the Poisson tensor in phase space. In the non-degenerate case this term may be rewritten in an even simpler geometric form. Finally, we discuss obstructions to the existence of WKB states and give a geometric description of the quantization condition for WKB states for a non-degenerate eigenvalue-function. 
  We discuss a phase structure of chiral symmetry breaking in the Gross-Neveu model at finite temperature, density and constant curvature. The effective potential is evaluated in the leading order of the $1/N$-expansion and in a weak curvature approximation. The third order critical line is found on the critical surface in the parameter space of temperature, chemical potential and constant curvature. 
  We present two-parameter solutions of the low-energy four-dimensional heterotic string which in the extremal limit reduce to supersymmetric monopole, string and domain wall solutions. The effective scalar coupling to the Maxwell field, $e^{-\alpha \phi} F_{\mu\nu} F^{\mu\nu}$, gives rise to a new string black hole with $\alpha = \sqrt{3}$, in contrast to the pure dilaton black hole solution which has $\alpha=1$. Implications of string/fivebrane duality in $D=10$ to four-dimensional dualities are discussed. 
  We display continuous families of SU(2) vector potentials $A_i^a(x)$ in 3 space dimensions which generate the same magnetic field $B^{ai}(x)$ (with det $B\neq 0$). These Wu-Yang families are obtained from the Einstein equation $R_{ij}=-2G_{ij}$ derived recently via a local map of the gauge field system into a spatial geometry with $2$-tensor $G_{ij}=B^a{}_i B^a{}_j\det B$ and connection $\Gamma_{jk}^i$ with torsion defined from gauge covariant derivatives of $B$. 
  We present an alternative $N=(4,0)$ superstring theory, with field content different from that of previously-known $N=(4,0)$ superstring theories. This theory is presented as a non-linear $\s$-model on the coset $SU(n,1) / SU(n) \otimes U(1)$ as the target space-time with torsion, which is coupled to $N=(4,0)$ world-sheet superconformal gravity. Our result indicates that the target space-time for $N=4$ superstring theory does not necessarily have to be a hyper-K\"ahler or quaternionic K\"ahler manifold. 
  The differential geometry on a Hopf algebra is constructed, by using the basic axioms of Hopf algebras and noncommutative differential geometry. The space of generalized derivations on a Hopf algebra of functions is presented via the smash product, and used to define and discuss quantum Lie algebras and their properties. The Cartan calculus of the exterior derivative, Lie derivative, and inner derivation is found for both the universal and general differential calculi of an arbitrary Hopf algebra, and, by restricting to the quasitriangular case and using the numerical R-matrix formalism, the aforementioned structures for quantum groups are determined. 
  The Gordon-Andrews identities, which generalize the Rogers-Ramanujan-Schur identities, provide product and fermionic forms for the characters of the minimal conformal field theories (CFTs) M(2,2k+1). We discuss/conjecture identities of a similar type, providing two different fermionic forms for the characters of the models SM(2,4k) in the minimal series of N=1 super-CFTs. These two forms are related to two families of thermodynamic Bethe Ansatz (TBA) systems, which are argued to be associated with the $\hat{\phi}_{1,3}^{\rm top}$- and $\hat{\phi}_{1,5}^{\rm bot}$-perturbations of the models SM(2,4k). Certain other q-series identities and TBA systems are also discussed, as well as a possible representation-theoretical consequence of our results, based on Andrews's generalization of the Gollnitz-Gordon theorem. 
  Recently E.Verlinde and H.Verlinde have suggested an effective two-dimensional theory describing the high-energy scattering in QCD. In this report we attempt to clarify some issues of this suggestion. We consider {\it anisotropic asymptotics} of correlation functions for scalar and gauge theories in four dimensions. Anisotropic asymptotics describe behaviour of correlation functions when some components of coordinates are large as compare with others components. It is occurred that (2+2) anisotropic asymptotics for 4-points functions are related with the well known Regge regime of scattering amplitudes. We study an expansion of correlation functions with respect to the rescaling parameter $\lambda$ over a part of variables (anisotropic $\lambda$-expansion). An effective theory describing the anisotropic limit of free scalar field contains two 2 dim conformal theories. One of them is a conformal theory in configuration space and another one is a conformal theory in momentum space. In some special cases ,in particular for the Wilson line correlators in gauge theories, the leading term of the anisotropic expansion involves only one of the conformal theories and it can be described by an effective theory with an action being a dimensional reduction of the original action. 
  In order to highlight the onset of chaos in the Pullen-Edmonds model a quantal analog of the resonance overlap criterion has been examined. A quite good agreement between analytical and numerical results is obtained. 
  In a series of papers Amati, Ciafaloni and Veneziano and 't Hooft conjectured that black holes occur in the collision of two light particles at planckian energies. In this paper we discuss a possible scenario for such a process by using the Chandrasekhar-Ferrari-Xanthopoulos duality between the Kerr black hole solution and colliding plane gravitational waves. We clarify issues arising in the definition of transition amplitude from a quantum state containing only usual matter without black holes to a state containing black holes. Collision of two plane gravitational waves producing a space-time region which is locally isometric to an interior of black hole solution is considered. The phase of the transition amplitude from plane waves to white and black hole is calculated by using the Fabbrichesi, Pettorino, Veneziano and Vilkovisky approach. An alternative extension beyond the horizon in which the space-time again splits into two separating gravitational waves is also discussed. Such a process is interpreted as the scattering of plane gravitational waves through creation of virtual black and white holes. 
  We review the generalization of the work of Seiberg and Witten on N=2 supersymmetric SU(2) Yang-Mills theory to SU(n) gauge groups. The quantum moduli spaces of the effective low energy theory parametrize a special family of hyperelliptic genus n-1 Riemann surfaces. We discuss the massless spectrum and the monodromies. 
  Unitarity and locality imply a remnant solution to the information problem, and also imply that Reissner-Nordstrom black holes have infinite numbers of internal states. Pair production of such black holes is reexamined including the contribution of these states. It is argued that the rate is proportional to the thermodynamic quantity Tr e^{-beta H}, where the trace is over the internal states of a black hole; this is in agreement with estimates from an effective field theory for black holes. This quantity, and the rate, is apparently infinite due to the infinite number of states. One obvious out is if the number of internal states of a black hole is finite. 
  Exact instanton solutions to $D=11$ spherical supermembranes moving in flat target spacetime backgrounds are construted. Our starting point is Super Yang-Mills theories, based on the infinite dimensional $SU(\infty)$ group, dimensionally reduced to one time dimension. In this fashion the super-Toda molecule equation is recovered preserving only one supersymmetry out of the $N=16$ that one would have obtained otherwise. It is conjectured that the expected critical target spacetime dimensions for the (super) membrane, ($D=11$) $D=27$ is closely related to that of the $noncritical$ (super) $W_{\infty}$ strings. A BRST analysis of these symmetries should yield information about the quantum consistency of the ($D=11$) $D=27$ dimensional (super) membrane. Comments on the role that Skyrmions might play in the two types of Spinning- Membrane actions construted so far is presented at the conclusion. Finally, the importance that integrability on light-lines in complex superspaces has in other types of solutions is emphasized. 
  Attention is paid to the fact that temperature of a classical black hole can be derived from the extremality condition of its free energy with respect to variation of the mass of a hole. For a quantum Schwarzschild black hole evaporating massless particles the same condition is shown to result in the following one-loop temperature $T=(8\pi M)^{-1} (1+\sigma (8\pi M^2)^{-1})$ and entropy $S = 4\pi M^2 - \sigma\log M$ expressed in terms of the effective mass $M$ of a hole together with its radiation and the integral of the conformal anomaly $\sigma$ that depends on the field species. Thus, in the given case quantum corrections to $T$ and $S$ turn out to be completely provided by the anomaly. When it is absent ($\sigma=0$), which happens in a number of supersymmetric models, the one-loop expressions of $T$ and $S$ preserve the classical form. On the other hand, if the anomaly is negative ($\sigma<0$) an evaporating quantum hole seems to cease to heat up when its mass reaches the Planck scales. 
  The chiral null model is a generalization of the fundamental string and gravitational wave background. It is an example of a conformally invariant model in all orders in $\alpha'$ and has unbroken supersymmetries. In a Kaluza--Klein approach we start in 10 dimensions and reduce the model down to 4 dimensions without making any restrictions. The 4-D field content is given by the metric, torsion, dilaton, a moduli field and 6 gauge fields. This model is self-dual and near the singularities asymptotically free. The relation to known IWP, Taub-NUT and rotating black hole solutions is discussed. 
  Rigidly superconformal sigma models in higher than two dimensions are constructed. These models rely on the existence of conformal Killing spinors on the $p+1$ dimensional worldvolume $(p\le 5)$, and homothetic conformal Killing vectors in the $d$--dimensional target space. In the bosonic case, substituting into the action a particular form of the target space metric admitting such Killing vectors, we obtain an action with manifest worldvolume conformal symmetry, which describes the coupling of $d-1$ scalars to a conformally flat metric on the worldvolume. We also construct gauged sigma models with worldvolume conformal supersymmetry. The models considered here are generalizations of the singleton actions on $S^p\times S^1$, constructed sometime ago by Nicolai and these authors. 
  A consistent quantization procedure of anomalous chiral models is discussed. It is based on the modification of the classical action by adding Wess-Zumino terms. The $SO(3)$ invariant WZ action for the $SO(3)$ model is constructed. Quantization of the corresponding modified theory is considered in details. 
  The overlap formula proposed by Narayanan and Neuberger in chiral gauge theories is examined. The free chiral and Dirac Green's functions are constructed in this formalism. Four dimensional anomalies are calculated and the usual anomaly cancellation for one standard family of quarks and leptons is verified. 
  An associative $*$-algebra is introduced (containing a $TTR$-algebra as a subalgebra) that implements the form factor axioms, and hence indirectly the Wightman axioms, in the following sense: Each $T$-invariant linear functional over the algebra automatically satisfies all the form factor axioms. It is argued that this answers the question (posed in the functional Bethe ansatz) how to select the dynamically correct representations of the $TTR$-algebra. Applied to the case of integrable QFTs with diagonal factorized scattering theory a universal formula for the eigenvalues of the conserved charges emerges. 
  The existence of a minimal observable length has long been suggested, in quantum gravity, as well as in string theory. In this context a generalized uncertainty relation has been derived which quantum theoretically describes the minimal length as a minimal uncertainty in position measurements. Here we study in full detail the quantum mechanical structure which underlies this uncertainty relation. 
  We review the origin of anomaly-induced dynamics in theories of $d=2$ gravity from a BRST viewpoint and show how quantum canonical transformations may be used to solve the resulting Liouville or Toda models for the anomalous modes. 
  We compute the exact partition function of the 2D Ising Model at critical temperature but with nonzero magnetic field at the boundary. The model describes a renormalization group flow between the free and fixed conformal boundary conditions in the space of boundary interactions. For this flow the universal ground state degeneracy $g$ and the full boundary state is computed exactly. 
  An exactly integrable symplectic correspondence is derived which in a continuum limit leads to the equations of motion of the relativistic generalization of the Calogero-Moser system, that was introduced for the first time by Ruijsenaars and Schneider. For the discrete-time model the equations of motion take the form of Bethe Ansatz equations for the inhomogeneous spin-1/2 Heisenberg magnet. We present a Lax pair, the symplectic structure and prove the involutivity of the invariants. Exact solutions are investigated in the rational and hyperbolic (trigonometric) limits of the system that is given in terms of elliptic functions. These solutions are connected with discrete soliton equations. The results obtained allow us to consider the Bethe Ansatz equations as ones giving an integrable symplectic correspondence mixing the parameters of the quantum integrable system and the parameters of the corresponding Bethe wavefunction. 
  Reducible off-shell anomalous gauge theories are studied in the framework of an extended Field-Antifield formalism by introducing new variables associated with the anomalous gauge degrees of freedom. The Wess-Zumino term for these theories is constructed and new gauge invariances appear. The quantum effects due to the extra variables are considered. 
  Stability of some solutions of the equations of motion of bosonic p-branes in curved and flat spacetimes is stated. 
  A new effective action for the high energy quark-quark scatterings is obtained by applying a scaling approximation to the QCD action. The propagators are shown to factorize into the transverse and the longitudinal parts so that the scattering amplitudes are given in terms of the products of two dimensional $S$-matrices. we show that our action provides a natural effective field theory for the Lipatov's theory of quark scatterings with quasi-elastic unitarity. The amplitude with quasi-elastic unitarity obtained from this action shows `Regge' behavior and is eikonalized. 
  We consider the $ U(1) $ sigma model in the two dimensional space-time which is a field-theoretical model possessing a nontrivial topology. It is pointed out that its topological structure is characterized by the zero-mode and the winding number. A new type of commutation relations is proposed to quantize the model respecting the topological nature. Hilbert spaces are constructed to be representation spaces of quantum operators. It is shown that there are an infinite number of inequivalent representations as a consequence of the nontrivial topology. The algebra generated by quantum operators is deformed by the central extension. When the central extension is introduced, it is shown that the zero-mode variables and the winding variables obey a new commutation relation, which we call twist relation. In addition, it is shown that the central extension makes momenta operators obey anomalous commutators. We demonstrate that topology enriches the structure of quantum field theories. 
  We examine a few problems of enumerative geometry and present their solutions in the framework of deformed (quantum) cohomology rings. 
  This review contains a summary of work by J.-L. Gervais and the author on the operator approach to 2d gravity. Special emphasis is placed on the construction of local observables -the Liouville exponentials and the Liouville field itself - and the underlying algebra of chiral vertex operators. The double quantum group structure arising from the presence of two screening charges is discussed and the generalized algebra and field operators are derived. In the last part, we show that our construction gives rise to a natural definition of a quantum tau function, which is a noncommutative version of the classical group-theoretic representation of the Liouville fields by Leznov and Saveliev. 
  We investigate the role of the torsion field at the quantum level in the affine-metric theory of gravity. One-loop counterterms are calculated in the theory with terms quadratic in the torsion field. 
  Motivated by the example of the superconducting cosmic string which can be a physical representation of a straight wire carrying a steady current, we derive in this case the explicit expressions of the induced vector potential, current density and magnetic field due to the vacuum polarization at the first order in the fine structure constant. 
  It is proven that up to possible surface terms, the only non-vanishing momentum-dependent amplitudes for the self-dual N=2 string in $R^{2,2}$ are the tree-level two and three-point functions, and the only non-vanishing momentum-independent amplitudes are the one-loop partition function and the tree-level two and four-point functions. The calculations are performed using the topological prescription developed in an earlier paper with Vafa. As in supersymmetric non-renormalization theorems, the vanishing proof is based on a relationship between the zero-momentum dilaton and axion. 
    I derive a general formalism for finding kinetic terms of the effective Lagrangian for slowly moving Chern-Simons vortices. Deformations of fields linear in velocities are taken into account. From the equations they must satisfy I extract the kinetic term in the limit of coincident vortices. For vortices passing one over the other there is locally the right-angle scattering. The method is based on analysis of field equations instead of action functional so it may be useful also for nonvariational equations in nonrelativistic models of Condensed Matter Physics. 
  We consider a scalar-metric gauge theory of gravity with independent metric, connection and dilaton. The role of the dilaton is to provide the scale of all masses, via its vacuum expectation value. In this theory, we study the renormalization group flow of the dilaton potential, taking into account threshold effects at the Planck scale. Due to the running of the VEV of the dilaton all particles that would naively seem to have masses larger than Planck's mass, may actually not propagate. This could solve the problem of unitarity in these theories. 
  Principal chiral models on a d-1 dimensional simplex are introduced and studied analytically in the large $N$ limit. The $d = 0, 2, 4$ and $\infty$ models are explicitly solved. Relationship with standard lattice models and with few-matrix systems in the double scaling limit are discussed. 
  We study the $O(N)$ non-linear $\sigma$ model on three-dimensional manifolds of constant curvature by means of the large $N$ expansion at the critical point. We examine saddle point equations imposing anti-periodic boundary condition in time direction. In the case $S^1 \times S^2$ we find that a solution is inevitably unstable. We briefly refer to the case $S^1 \times S^1 \times S^1$. 
  We review the status of solitons in superstring theory, with a view to understanding the strong coupling regime. These {\it solitonic} solutions are non-singular field configurations which solve the empty-space low-energy field equations (generalized, whenever possible, to all orders in $\alpha'$), carry a non-vanishing topological "magnetic" charge and are stabilized by a topological conservation law. They are compared and contrasted with the {\it elementary} solutions which are singular solutions of the field equations with a $\sigma$-model source term and carry a non-vanishing Noether "electric" charge. In both cases, the solutions of most interest are those which preserve half the spacetime supersymmetries and saturate a Bogomol'nyi bound. They typically arise as the extreme mass=charge limit of more general two-parameter solutions with event horizons. We also describe the theory {\it dual} to the fundamental string for which the roles of elementary and soliton solutions are interchanged. In ten spacetime dimensions, this dual theory is a superfivebrane and this gives rise to a string/fivebrane duality conjecture according to which the fivebrane may be regarded as fundamental in its own right, with the strongly coupled string corresponding to the weakly coupled fivebrane and vice-versa. After compactification to four spacetime dimensions, the fivebrane appears as a magnetic monopole or a dual string according as it wraps around five or four of the compactified dimensions. This gives rise to a four-dimensional string/string duality conjecture which subsumes a Montonen-Olive type duality in that the magnetic monopoles of the fundamental string correspond to the electric winding states of the dual string. This leads to a {\it duality of dualities} whereby under string/string duality the the strong/weak coupling $S$-duality trades places with the minimum/maximum length $T$-duality. Since these magnetic monopoles are extreme black holes, a prediction of $S$-duality is that the corresponding electric massive states of the fundamental string are also extreme black holes. 
  The relations among coupling constants and masses in the standard model \`a la Connes-Lott with general scalar product are computed in detail. We find a relation between the top and the Higgs masses. For $m_t=174\pm22\ GeV$ it yields $m_H=277\pm40\ GeV$. The Connes-Lott theory privileges the masses $m_t=160.4\ GeV$ and $m_H=251.8\ GeV$. 
   Instanton effects in a family of completely massive Higgs models with N=1 supersymmetry are investigated. The models have $N_c=2$ and $N_f\ge 2$. In each model, we show that a certain gauge invariant correlation function depends in a non-trivial way on its coordinates, in spite of the fact that supersymmetry requires its constancy. This means that non-perturbative effects break supersymmetry explicitly in the one instanton sector. We also show that condensates arising in the point-like limit of the above correlation functions can in principle be used to induce the Electro-Weak scale. 
  We describe, in some detail, a number of different Coulomb gas formulations of $N=2$ superconformal coset models. We also give the mappings between these formulations. The ultimate purpose of this is to show how the Landau-Ginzburg structure of these models can be used to extract the $W$-generators, and to show how the computation of the elliptic genus can be refined so as to extract very detailed information about the characters of component parts of the model. 
  We solve the time evolution of the density matrix both for fermions and bosons in the presence of a homogeneous but time dependent external electric field. The number of particles produced by the external field, as well as their distribution in momentum space is found for finite times. Furthermore, we calculate the probability of finding a given number of particles in the ensemble. In all cases, there is a nonvanishing thermal contribution. The bosonic and the fermionic density matrices are expressed in a "functional field basis". This constitutes an extension of the "field basis" concept to fermions. 
  This thesis is a review of black hole evaporation with emphasis on recent results obtained for two dimensional black holes. First, the geometry of the most general stationary black hole in four dimensions is described and some classical quantities are defined. Then, a derivation of the spectrum of the radiation emitted during the evaporation is presented. In section four, a two dimensional model which has black hole solutions is introduced, the so-called CGHS model. These two dimensional black holes are found to evaporate. Unlike the four dimensional case, the evaporation process can be studied analytically as long as the mass of the black hole is well above the two dimensional analog of the Planck mass. Finally, some proposals for resolving the so-called information paradox are reviewed and it is concluded that none of them is fully satisfactory. 
  In this work we generalise previous results connecting (rational) Gaudin magnet models and classical separation of variables. It is shown that the connection persists for the case of linear r-matrix algebra which corresponds to the trigonometric 4x4 r-matrix (of the XXZ type). We comment also on the corresponding problem for the elliptic (XYZ) r-matrix. A prescription for obtaining integrable systems associated with multiple poles of an L-operator is given. 
  We consider the relation between affine Toda field theories (ATFT) and Landau-Ginzburg Lagrangians as alternative descriptions of deformed 2d CFT. First, we show that the two concrete implementations of the deformation are consistent once quantum corrections to the Landau-Ginzburg Lagrangian are taken into account. Second, inspired by Gepner's fusion potentials, we explore the possibility of a direct connection between both types of Lagrangians; namely, whether they can be transformed one into another by a change of variables. This direct connection exists in the one-variable case, namely, for the sine-Gordon model, but cannot be established in general. Nevertheless, we show that both potentials exhibit the same structure of extrema. 
  ( We present complete solutions of $K$-matrix for the quantum Mikhailov-Shabat model. It has been known that there are three diagonal solutions with no free parameters, one being trivial identity solution, the others non-trivial. The most general solutions which we found consist of three families corresponding to each diagonal solutions. One family of solutions depends on two arbitrary parameters. If one of the parameters vanishes, the other must also vanish so that the solutions reduces to trivial identity solution. The other two families for each non-trivial diagonal solutions have only one arbitrary parameter.) 
  The scalar field is quantized in the discretized light-front framework following the {\em standard} Dirac procedure and its infinite volume limit taken. The background field and the nonzero mode variables do not commute for finite volume; they do so only in the continuum limit. A {\em non-local constraint} in the theory relating the two is shown to follow and we must deal with it along with the Hamiltonian. At the tree level the constraint leads to a description of the spontaneous symmetry breaking. The elimination of the constraint would lead to a highly involved light-front Hamiltonian in contrast to the one found when we ignore altogether the background field. The renormalized constraint equation would also account for the instability of the symmetric phase for large enough couping constant. 
  The Yangian symmetry Y(su($n$)) of the Calogero-Sutherland-Moser spin model is reconsidered. The Yangian generators are constructed from two non-commuting su($n$)-loop algebras. The latters generate an infinite dimensional symmetry algebra which is a deformation of the $W_\infty$-algebra. We show that this deformed $W_\infty$-algebra contains an infinite number of Yangian subalgebras with different deformation parameters. 
  By using the exact renormalization group formulation we prove perturbatively the Slavnov-Taylor (ST) identities in SU(2) Yang-Mills theory. This results from two properties: {\it locality}, i.e. the ST identities are valid if their local part is valid; {\it solvability}, i.e. the local part of ST identities is valid if the couplings of the effective action with non-negative dimensions are properly chosen. 
  We investigate the low--energy properties of a Z_12 orbifold with continuous Wilson line moduli. They give rise to a (0,2) superstring compactification. Their Kaehler potentials and Yukawa couplings are calculated. We study the discrete symmetries of the model and their implications on the threshold corrections to the gauge couplings as well as for string unification. 
  The general d-dimensional twisted group lattice is solved. The irreducible representations of the corresponding group are constructed by an explicit procedure. It is proven that they are complete. All matrix representation solutions to the quantum hyperplane equations are obtained. 
  We show that the BRST structure of the topological string is encoded in the ``small'' $N=4$ superconformal algebra, enabling us to obtain, in a non-trivial way, the string theory from hamiltonian reduction of $A(1|1)$. This leads to the important conclusion that not only ordinary string theories, but topological strings as well, can be obtained, or even defined, by hamiltonian reduction from WZW models. Using two different gradations, we find either the standard $N=2$ minimal models coupled to topological gravity, or an embedding of the bosonic string into the topological string. We also comment briefly on the generalization to super Lie algebras $A(n|n)$. 
  In these lectures, we study and compare two different formulations of $SU(2)$, level $k=1$, Wess-Zumino-Witten conformal field theory. The first, conventional, formulation employs the affine symmetry of the model; in this approach correlation functions are derived from the so-called Knizhnik-Zamolodchikov equations. The second formulation is based on an entirely different algebraic structure, the so-called Yangian $Y(sl_2)$. In this approach, the Hilbert space of the theory is obtained by repeated application of modes of the so-called spinon field, which has $SU(2)$ spin $j=\thalf$ and obeys fractional (semionic) statistics. We show how this new formulation, which can be generalized to many other rational conformal field theories, can be used to compute correlation functions and to obtain new expressions for the Virasoro and affine characters in the theory. [Lectures given at the 1994 Trieste Summer School on High Energy Physics and Cosmology, Trieste, July 1994.] 
  We consider duality transformations in N=2 Yang--Mills theory coupled to N=2 supergravity, in a manifestly symplectic and coordinate covariant setting. We give the essential of the geometrical framework which allows one to discuss stringy classical and quantum monodromies, the form of the spectrum of BPS saturated states and the Picard--Fuchs identities encoded in the special geometry of N=2 supergravity theories. 
  Analytic continuation of quantum statistical physics from imaginary to real time is analyzed. Adiabatic vanishing of interactions at real time infinities gives origin to singularities at complex times. This undermines the hypothesis of decoupling of interactions at $t \rightarrow \infty$. Hence an interacting thermal vacuum is a necessary component of the exact real-time formalism. Consequences for Thermo-field dynamics are discussed. 
  I review three different problems occuring in two dimensional field theory: 1) classification of conformal field theories; 2) construction of lattice integrable realizations of the latter; 3) solutions to the WDVV equations of topological field theories. I show that a structure of Coxeter group is hidden behind these three related problems. 
  We study the full set of planar Green's functions for a two-matrix model using the language of functions of non-commuting variables. Both the standard Schwinger-Dyson equations and equations determining connected Green's functions can be efficiently discussed and solved. This solution determines the master field for the model in the `$C$-representation.' 
  The spontaneous symmetry breaking (and Higgs) mechanism in the theory quantized on the light-front ({\it l.f.}), in the {\it discretized formulation}, is discussed. The infinite volume limit is taken to obtain the {\it continuum version}. The hamiltonian formulation is shown to contain a new ingredient in the form of nonlocal {\it constraint eqs.} which lead to a {\it nonlocal l.f. Hamiltonian}. The description of the broken symmetry here has the same physical content as in the conventional formulation though arrived at through a different mechanism. 
  The renormalization of the two dimensional light-front quantized $\phi^{4}$ theory is discussed. The mass renormalization condition and the renormalized constraint equation are shown to contain all the information to describe the phase transition in the theory, which is found to be of the second order in agreement with the conjecture of Simons and Griffith. We argue that the same result is also be obtained in the conventional equal-time formulation. 
  We twist the monopole equations of Seiberg and Witten and show how these equations are realized in topological Yang-Mills theory. A Floer derivative and a Morse functional are found and are used to construct a unitary transformation between the usual Floer cohomologies and those of the monopole equations. Furthermore, these equations are seen to reside in the vanishing self-dual curvature condition of an $OSp(1|2)$-bundle. Alternatively, they may be seen arising directly from a vanishing self-dual curvature condition on an $SU(2)$-bundle in which the fermions are realized as spanning the tangent space for a specific background. 
  This note for the Proceedings of the International Congress of Mathematical Physics gives an account of a construction of an ``elliptic quantum group'' associated with each simple classical Lie algebra. It is closely related to elliptic face models of statistical mechanics, and, in its semiclassical limit, to the Wess-Zumino-Witten model of conformal field theory on tori. 
  A simple solution of Witten's monopole equations is given. 
  We present the computation of threshold functions for Abelian orbifold compactifications. Specifically, starting from the massive, moduli-dependent string spectrum after compactification, we derive the threshold functions as target space duality invariant free energies (sum over massive string states). In particular we work out the dependence on the continuous Wilson line moduli fields. In addition we concentrate on the physically interesting effect that at certain critical points in the orbifold moduli spaces additional massless states appear in the string spectrum leading to logarithmic singularities in the threshold functions. We discuss this effect for the gauge coupling threshold corrections; here the appearance of additional massless states is directly related to the Higgs effect in string theory. In addition the singularities in the threshold functions are relevant for the loop corrections to the gravitational coupling constants. 
  We present a procedure for constructing actions describing propagation of W-strings on group manifolds by using the Hamiltonian canonical formalism and representations of W-algebras in terms of Kac-Moody currents. An explicit construction is given in the case of the $W_3$ string. 
  In the talk different definitions of the black hole entropy are discussed and compared. It is shown that the Bekenstein-Hawking entropy $S^{BH}$ (defined by the response of the free energy of a system containing a black hole on the change of the temperature) differs from the statistical- mechanical entropy $S^{SM}=-\mbox{Tr}(\hat{\rho}\ln \hat{\rho})$ (defined by counting internal degrees of freedom of a black hole). A simple explanation of the universality of the Bekenstein-Hawking entropy (i.e. its independence of the number and properties of the fields which might contribute to $S^{SM}$) is given. 
  A string field theory including open string fields is constructed in the temporal gauge. It consists of string interaction vertices similar to the light-cone gauge string field theory. A slight modification of the definition of the time coordinate is needed because of the existence of the open string end points. 
  Lectures at the CRM-CAP Summer School 'Particles and Fields 94' August 16-24 1994, Banff, Alberta, Canada. 
  In this article, the gap equation for the constituent quark mass in the U(2)*U(2) Nambu-Jona-Lasinio model for the 1/Nc approximation is investigated. It is shown that taking into account scalar isovector mesons plays an important role for the correct description of quark masses in this approximation. The role of the Ward identity in calculations of 1/Nc corrections to the meson vertex functions is shortly discussed. 
  Euclidean invariant Klein-Gordon, Dirac and massive Chern-Simons field theories are constructed in terms of a random walk with a spin factor on a three dimensional lattice. We exactly calculate the free energy and the correlation functions which allow us to obtain the critical diffusion constant and associated critical exponents. It is pointed out that these critical exponents do not satisfy the hyper-scaling relation but the scaling inequalities. We take the continuum limit of this theory on the basis of these analyses. We check the universality of obtained results on other lattice structure such as triclinic lattice and body centered lattice. 
  We discuss the structure of one-loop counterterms for the two-dimensional theory of gravitation in the covariant scheme and study the effect of quantum reparametrizations.Some of them are shown to be equivalent to the introduction of 2+$epsilon$-dimensional terms into the initially 2-dimensional theory. We also argue that the beta-function for the Einstein constant has a non-trivial ultraviolet stable point beyond two dimensions. 
  The lagrangian of the N=1, D=10 dual supergravity interacting with the Yang-Mills matter multiplet is constructed starting immediately from the equations of motion obtained from the Bianchi Identities in the superspace approach. The difference is established in comparison with the Gates-Nishino lagrangian at the fourth order level in fermionic fields. 
  The operator ordering problem due to the quantization or regularization ambiguity in the Chern-Simons theory exists. However, we show that this can be avoided if we require Galilei covariance of the nonrelativistic Abelian Chern-Simons theory even at the quantum level for the extended sources. The covariance can be recovered only by choosing some particular operator orderings for the generators of the Galilei group depending on the quantization ambiguities of the $gauge-matter$ commutation relation. We show that the desired ordering for the unusual prescription is not the same as the well-known normal ordering but still satisfies all the necessary conditions. Furthermore, we show that the equations of motion can be expressed in a similar form regardless of the regularization ambiguity. This suggests that the different regularization prescriptions do not change the physics. On the other hand, for the case of point sources the regularization prescription is uniquely determined, and only the orderings, which are equivalent to the usual one, are allowed. 
  We study the $SU(2)$ WZNW model over a family of elliptic curves. Starting from the formulation developed by Tsuchiya, Ueno and Yamada, we derive a system of differential equations which contains the Knizhnik-Zamolodchikov-Bernard equations. Our system completely determines the $N$-point functions and it is regarded as a natural elliptic analogue of the one developed by Tsuchiya and Kanie for the projective line. We also calculate the system for the 1-point functions explicitly. This gives a generalization of the system for $\hat\frak{sl}(2,\C)$-characters derived by Eguchi and Ooguri. 
  We consider a four-dimensional space-time supplemented by two discrete points assigned to a $Z_2$ algebraic structure and develop the formalism of noncommutative geometry. By setting up a generalised vielbein, we study the metric structure. Metric compatible torsion free connection defines a unique finite field content in the model and leads to a discretized version of Kaluza-Klein theory. We study some special cases of this model that illustrate the rich and complex structure with massive modes and the possible presence of a cosmological constant. 
  We construct a field theory for anyons in the lowest Landau level starting from the $N$-particle description, and discuss the connection to the full field theory of anyons defined using a statistical gauge potential. The theory is transformed to free form, with the fields defined on the circle and satisfying modified commutation relations. The Fock space of the anyons is discussed, and the theory is related to that of edge excitations of an anyon droplet in a harmonic oscillator well. 
  We extend Witten's discussion of actions related to the Landau-Ginzburg description of Calabi-Yau hypersurfaces in weighted projective spaces to cover the mirror class of models that include twisted chiral matter multiplets and a newly discovered 2D, N = 2 twisted vector multiplet. Certain integrability obstructions are observed that constrain the most general constructions containing both matter and twisted matter simultaneously. It is conjectured that knot invariants will ultimately play a role in describing the most general such model. 
   The behavior of small perturbations around the Kaluza-Klein monopole in the five dimensional space-time is investigated. The fact that the odd parity gravitational wave does not propagate in the five dimensional space-time with Kaluza-Klein monopole is found provided that the gravitational wave is constant in the fifth direction. 
  A pedagogical discussion is given of some aspects of \lq\lq quantum black holes", primarily using recently developed two-dimensional models. After a short preliminary concerning classical black holes, we give several motivations for studying such models, especially the so called dilaton gravity models in $1+1$ dimensions. Particularly attractive is the one proposed by Callan,Giddings, Harvey and Strominger (CGHS), which is classically solvable and contains black hole solutions. Its semi-classical as well as classical properties will be reviewed, including how a flux of matter fields produces a black hole with a subsequent emission of Hawking radiation. Breakdown of such an approximation near the horizon, however, calls for exactly solvable variants of this model and some attempts in this direction will then be described. A focus will be placed on a model with 24 matter fields, for which exact quantization can be performed and physical states constructed. A method will then be proposed to extract space-time geometry described by these states in the sense of quantum average and examples containing a black hole will be presented. Finally we give a (partial) list of future problems and discuss the nature of difficulties in resolving them. 
  The Pauli Hamiltonian for a spin $\frac{1}{2}$ charged particle interacting with a point magnetic vortex and $1/r^{2}$ potential exhibits a dynamical supersymmetry $Osp(1,1)$ on the plane except at the origin. Using this symmetry, the spectrum and the wave functions have been obtained. And, the dynamical supersymmetry could be imported to the case when an external harmonic potential is added. 
  Using the associativity relations of the topological Sigma Models with target spaces, $CP^3, CP^4$ and $Gr(2,4)$ , we derive recursion relations of their correlation and evaluate them up to certain order in the expansion over the instantons. The expansion coeffieients are regarded as the number of rational curves in $CP^3, CP^4$ and $Gr(2,4)$ which intersect various types of submanifolds corresponding to the choice of BRST invariant operators in the correlation functions. 
  We study one (or two) matrix models modified by terms of the form $g(\rho(P))^2 + g'(\rho'({\cal{O}}))^2$, where the matrix representation of the puncture operator $P$ and the one of a scaling operator ${\cal{O}}$ are denoted by $\rho(P)$ and $\rho'({\cal{O}})$ respectively. We rewrite the modified models as effective theories of baby universes. We find an upper bound for the gravitational dimension of ${\cal{O}}$ under which we can fine tune the coupling constants to obtain new critical behaviors in the continuum limit. The simultaneous tuning of $g$ and $g'$ is possible if the representations $\rho(P)$ and $\rho'({\cal{O}})$ are chosen so that the non-diagonal elements of the mass matrix of the effective theory vanish. 
  The antibracket formalism for gauge theories, at both the classical and quantum level, is reviewed. Gauge transformations and the associated gauge structure are analyzed in detail. The basic concepts involved in the antibracket formalism are elucidated. Gauge-fixing, quantum effects, and anomalies within the field-antifield formalism are developed. The concepts, issues and constructions are illustrated using eight gauge-theory models. 
  We construct the quantum versions of the monodromy matrices of KdV theory. The traces of these quantum monodromy matrices, which will be called as ``${\bf T}$-operators'', act in highest weight Virasoro modules. The ${\bf T}$-operators depend on the spectral parameter $\lambda$ and their expansion around $\lambda = \infty$ generates an infinite set of commuting Hamiltonians of the quantum KdV system. The ${\bf T}$-operators can be viewed as the continuous field theory versions of the commuting transfer-matrices of integrable lattice theory. In particular, we show that for the values $c=1-3{{(2n+1)^2}\over {2n+3}} , n=1,2,3,... $of the Virasoro central charge the eigenvalues of the ${\bf T}$-operators satisfy a closed system of functional equations sufficient for determining the spectrum. For the ground-state eigenvalue these functional equations are equivalent to those of massless Thermodynamic Bethe Ansatz for the minimal conformal field theory ${\cal M}_{2,2n+3}$; in general they provide a way to generalize the technique of Thermodynamic Bethe Ansatz to the excited states. We discuss a generalization of our approach to the cases of massive field theories obtained by perturbing these Conformal Field Theories with the operator $\Phi_{1,3}$. The relation of these ${\bf T}$-operators to the boundary states is also briefly described. 
  We consider the (smoothed) average correlation between the density of energy levels of a disordered system, in which the Hamiltonian is equal to the sum of a deterministic H0 and of a random potential $\varphi$. Remarkably, this correlation function may be explicitly determined in the limit of large matrices, for any unperturbed H0 and for a class of probability distribution P$(\varphi)$ of the random potential. We find a compact representation of the correlation function. From this representation one obtains readily the short distance behavior, which has been conjectured in various contexts to be universal. Indeed we find that it is totally independent of both H0 and P($\varphi$). 
  In this work we describe the mathematical foundations used in the construction of primary fields of minimal models of conformal field theory. The work contains two parts: In the first part we give a description of Verma and Fock modules for the Virasoro algebra and develop their imbedding patterns. This part is a simplification of the work of Feigin and Fuks (we correct a mistake in their patterns in the case III_+), Rocha-Charidi and some new ideas which yield a simplification of the original papers. In the second part we define (free) vertex operators as unbounded Hilbert space operators, acting on Fock spaces, which are Virasoro modules. We prove several properties of these operators: under appropriate conditions vertex operators are densely defined, not closable operators. Radially ordered products of vertex operators exists on a dense subset. We prove commutation relations between vertex operators and elements of the Virasoro algebra. Next we define, following the (non rigouros) work of G. Felder, integrated vertex operators and prove that these operators resemble the properties of the not integrated vertex operators. Special integrated vertex operators can be identified with conformal fields and a Virasoro invariant subspace of Fock space can be identified with the physical Hilbert space for the conformal theory. 
  We discuss the possibility that quantum black holes have discrete mass spectrum. Different arguments leading to this conclusion are considered, particularly the decoupling between left and right sectors in string theory - the so-called heterotic principle. The case of a $2+1$ dimensional black holes is considered as an argument in favour of this argument. The possible connection between membrane model of the black hole horizon and topological membrane is briefly discussed. 
  A manifestly gauge invariant formulation of chiral theories with fermions on the lattice is developed. It combines SLAC lattice derivative \cite{DWY}, \cite{ACS}, \cite{S} and generalized Pauli-Villars regularization \cite{FS}. The theory is free of fermion doubling, requires only local gauge invariant counterterms and produces correct results when applied to exactly solvable two dimensional models. 
  As the title indicates 
  A string action is considered in four spacetime dimensions which is obtained by dimensionally reducing the ten dimensional effective action. The equations of motion admit string like solutions. The symmetry properties of the four dimensional action is discussed. It is shown that new background configurations can be generated by implementing suitable $O(d,d)$ transformations. 
  We use the gauged linear sigma model introduced by Witten to calculate instanton expansions for correlation functions in topological sigma models with target space a toric variety $V$ or a Calabi--Yau hypersurface $M \subset V$. In the linear model the instanton moduli spaces are relatively simple objects and the correlators are explicitly computable; moreover, the instantons can be summed, leading to explicit solutions for both kinds of models. In the case of smooth $V$, our results reproduce and clarify an algebraic solution of the $V$ model due to Batyrev. In addition, we find an algebraic relation determining the solution for $M$ in terms of that for $V$. Finally, we propose a modification of the linear model which computes instanton expansions about any limiting point in the moduli space. In the smooth case this leads to a (second) algebraic solution of the $M$ model. We use this description to prove some conjectures about mirror symmetry, including the previously conjectured ``monomial-divisor mirror map'' of Aspinwall, Greene, and Morrison. 
  A generalization of the quantum inverse scattering method is proposed replacing the quantum group $RLL$ commutation relations of Lax operators by reflection equation type $RLRL$ commutation relations. Under some natural assumptions the most general algebra of this type allowing to construct the neccessary integrals of motion is found. It serves to describe Lax operators with completely non-ultralocal commutation relations. An example of this new formalism is an integrable model on monodromies of flat connections on a Riemann surface which is related to the XXZ quantum spin chain. 
  A possible generalisation is given to the meaning of maximal symmetry in the presence of torsion 
  Chern-Simons theory coupled to complex scalars is quantized on the light- front in the local light-cone gauge by constructing the self-consistent hamiltonian theory. It is shown that no inconsistency arises on using two local gauge-fixing conditions in the Dirac procedure. The light-front Hamiltonian turns out to be simple and the framework may be useful to construct renormalized field theory of particles with fractional statistics ({\it anyons}). The theory is shown to be relativistic and the extra term in the transformation of the matter field under space rotations, interpreted in previous works as anomaly, is argued to be gauge artefact. 
  The light-front Hamiltonian formulation for the scalar field theory contains a new ingredient in the form of a constraint equation. Renormalization of the two dimensional $\phi^{4}$ theory, described in the continuum, is discussed. The mass renormalization condition and the renormalized constraint equation contain all the information to describe the phase transition in the theory, which is found to be of the second order. We argue that the same result would also be obtained in the conventional equal-time formulation. 
  In this report we discuss some results of non--commutative (quantum) probability theory relating the various notions of statistical independence and the associated quantum central limit theorems to different aspects of mathematics and physics including: $q$--deformed and free central limit theorems; the description of the master (i.e. central limit) field in matrix models along the recent Singer suggestion to relate it to Voiculescu's results on the freeness of the large $N$ limit of random matrices; quantum stochastic differential equations for the gauge master field in QCD; the theory of stochastic limits of quantum fields and its applications to stochastic bosonization of Fermi fields in any dimensions; new structures in QED such as a nonlinear modification of the Wigner semicircle law and the interacting Fock space: a natural explicit example of a self--interacting quantum field which exhibits the non crossing diagrams of the Wigner semicircle law. 
  I review the covariant quantization of the critical $N{=}2$ fermionic string with and without a global ${\bf Z}_2$ twist. The BRST analysis yields massless bosonic and fermionic vertex operators in various ghost and picture number sectors, as well as picture-changers and their inverses, depending on the field basis chosen for bosonization. Two distinct GSO projections exist, one (untwisted) retaining merely the known bosonic scalar and its spectral-flow partner, the other (twisted) yielding two fermions and one boson, on the massless level. The absence of interactions in the latter case rules out standard spacetime supersymmetry. In the untwisted theory, the $U(1,1)$-invariant three-point and vanishing four-point functions are confirmed at tree level. I comment on the $N{=}2$ string field theory, the integration over moduli and the realization of spectral flow. 
  General relativity is applied to the strong interaction; the nexus between the two being arrived at by constructing a line element having the Yukawa form, which is used to describe geometrically the classical dynamics of a particle moving under the influence of the short-range strong interaction. It is shown that, with reasonable assumptions, the theory of general relativity can be made compatible with quantum mechanics by using the general relativistic field equations to construct a Robertson-Walker metric for a quantum particle. The resulting line element of the particle can be transformed entirely to that of a Minkowski spacetime, and the spacetime dynamics of the particle described by a Minkowski observer takes the form of quantum mechanics. It is also discussed the physical aspects of the affine connection in general relativity and its relationship with the field strength of the electromagnetic field and strong interaction. A heuristic geometric formulation of the electromagnetic field as an independent spacetime structure is presented. 
  The Hamiltonization of local symmetries of the form $\delta q^A = \ea{R_a}^A(q,\dot q)$ or $\delta q^A = \dot\ea{R_a}^A (q,\dot q)$ for arbitrary Lagrangean model $L(q^A,\dot q^A)$ is considered. We show as the initial symmetries are transformed in the transition from $L$ to first order action, and then to the Hamiltonian action $S_H=\int{\rm d}\tau(p_A\dot q^A-H_0-v^\alpha\Phi_\alpha)$, where $\Phi_\alpha$ are the all (first and second class) primary constraints. An exact formulae for local symmetries of $S_H$ in terms of the initial generators ${R_a}^A$ and all primary constraints $\Phi_\alpha$ are obtained. 
We study the influence of the anomaly on the physical quantum picture of the chiral Schwinger model (CSM) defined on $S^1$. We show that such phenomena as the total screening of charges and the dynamical mass generation characteristic for the standard Schwinger model do not take place here. Instead of them, the anomaly results in the background linearly rising electric field or, equivalently, in the exotic statistics of the physical matter field. We construct the algebra of the Poincare generators and show that it differs from the Poincare one. For the CSM on $R^1$, the anomaly influences only the mass generation mechanism. 
  In this work the supersymmetric gauge invariant action for the massive Abelian $N$$=$$1$ super-{\QED} in the Atiyah-Ward space-time ({\DDdd}) is formulated. The questions concerning the scheme of the gauge invariance in {\DDdd} by means of gauging the massive $N$$=$$1$ super-{\QED} are investigated. We study how to ensure the gauge invariance at the expenses of the introduction of a complex vector superfield. We discuss the Wess-Zumino gauge and thereupon we conclude that in this gauge, only the imaginary part of the complex vector field, $B_{\m}$, gauges a $U(1)$-symmetry, whereas its real part gauges a Weyl symmetry. We build up the gauge invariant massive term by introducing a pair of chiral and anti-chiral superfields with opposite $U(1)$-charges. We carry out a dimensional reduction {\it{\`a la}} Scherk of the massive $N$$=$$1$ super-{\QED} action from {\DDdd} to {\Ddd}. Truncations are needed in order to suppress non-physical modes and one ends up with a parity-preserving $N$$=$$1$ super-QED$_{1+2}$ (rather than $N$$=$$2$) in $D$$=$$1$$+$$2$. Finally, we show that the $N$$=$$1$ super-QED$_{1+2}$ we have got is the supersymmetric version ofthe ${\tau}_{3}$QED . 
Some general formulas are derived for the solutions of a BRST quantization on inner product spaces of finite dimensional bosonic gauge theories invariant under arbitrary Lie groups. A detailed analysis is then performed of SL(2,R) invariant models and some possible geometries of the Lagrange multipliers are derived together with explicit results for a class of SL(2,R) models. Gauge models invariant under a nonunimodular gauge group are also studied in some detail. 
By means of a generalized quartet mechanism we show in a model independent way that a BRST quantization on an inner product space leads to physical states of the form |ph>=e^{[Q, \psi]} |ph>_0 where Q is the nilpotent BRST operator, \psi a hermitian fermionic gauge fixing operator, and |ph>_0 BRST invariant states determined by a hermitian set of BRST doublets in involution. |ph>_0 does not belong to an inner product space although |ph> does. Since the BRST quartets are split into two sets of hermitian BRST doublets there are two choices for |ph>_0 and the corresponding \psi. When applied to general, both irreducible and reducible, gauge theories of arbitrary rank within the BFV formulation we find that |ph>_0 are trivial BRST invariant states which only depend on the matter variables for one set of solutions, and for the other set |ph>_0 are solutions of a Dirac quantization. This generalizes previous Lie group solutions obtained by means of a bigrading. 
  An explicit construction is presented for the action of the su(1,1) subalgebra of the Virasoro algebra on path spaces for the c(2,q) minimal models. In the case of the Lee-Yang edge singularity, we show how this action already fixes the central charge of the full Virasoro algebra. For this case, we additionally construct a representation in terms of generators of the corresponding Temperley-Lieb algebra. 
The possibility of the application of the free field representation developed by Lukyanov for massive integrable models is investigated in the context of the O(3) sigma model. We use the bootstrap fusion procedure to construct a free field representation for the O(3) Zamolodchikov- Faddeev algebra and to write down a representation for the solutions of the form-factor equations which is similar to the ones obtained previously for the sine-Gordon and SU(2) Thirring models. We discuss also the possibility of developing further this representation for the O(3) model and comment on the extension to other integrable field theories. 
I present here another example of a lattice fibration, a discrete version of the highest dimensional Hopf fibration: $S^{7}\longrightarrow S^{15} \longrightarrow S^{8}$. 
One of the methods used to extend two-dimensional bosonization to four space-time dimensions involves a transformation to new spatial variables so that only one of them appears kinematically. The problem is then reduced to an Abelian version of two-dimensional bosonization with extra ``internal'' coordinates. On a formal level, putting these internal coordinates on a finite lattice seems to provide a well-defined prescription for calculating correlation functions. However, in the infinite-lattice or continuum limits, certain difficulties appear that require very delicate specification of all of the many limiting procedures involved in the construction. 
At the classical level we study open bosonic strings. A generic description of string self-interactions localized at string ends is given. Self-interactions are characterized by two dimensionless coupling constants. The model is rewritten using complex Liouville fields. Using these Lorentz and reparametrization invariant variables, equations of motion get greatly simplified and reduce to some boundary problem for Liouville equation. 
We examine the relativistic Nambu-Goto model with Gauss-Bonnet boundary term added to the action integral. The system is analysed using an invariant representation of dynamical string degrees of freedom by complex Liouville fields. The solutions of classical equations of motion for open strings are described. 
Abstract: We discuss the structure of one-loop counterterms for the two-dimensional theory of gravitation in the covariant scheme and study the effect of quantum reparametrizations. Some of them are shown to be equivalent to the introduction of $2+\epsilon$-dimensional terms into the initially 2-dimensional theory. We also argue that the $\beta$-function for the Einstein constant has a non-trivial ultraviolet stable point beyond two dimensions. 
Abstract In this letter we show that the overlap formulation of chiral gauge theories correctly reproduces the gravitational Lorentz anomaly in 2-dimensions. This formulation has been recently suggested as a solution to the fermion doubling problem on the lattice. The well known response to general coordinate transformations of the effective action of Weyl fermions coupled to gravity in 2-dimensions can also be recovered. 
Inspired by the usefulness of local scaling of time in the path integral formalism, we introduce a new kind of hamiltonian path integral in this paper. A special case of this new type of path integral has been earlier found useful in formulating a scheme of hamiltonian path integral quantization in arbitrary coordinates. This scheme has the unique feature that quantization in arbitrary co-ordinates requires hamiltonian path integral to be set up in terms of the classical hamiltonian only, without addition of any adhoc $ O(\hbar ^2) $terms. In this paper we further study the properties of hamiltonian path integrals in arbitrary co-ordinates with and without local scaling of time and obtain the Schrodinger equation implied by the hamiltonian path integrals. As a simple illustrative example of quantization in arbitrary coordinates and of exact path integration we apply the results obtained to the case of Coulomb problem in two dimensions. 
We briefly review a hamiltonian path integral formalism developed earlier by one of us. An important feature of this formalism is that the path integral quantization in arbitrary co-ordinates is set up making use of only classical hamiltonian without addition of adhoc $\hbar^2$ terms. In this paper we use this hamiltonian formalism and show how exact path integration may be done for several potentials. 
Unlike Einstein gravity, dilaton-Maxwell gravity with matter is renormalizable in $2+\epsilon$ dimensions and has a smooth $\epsilon\rightarrow 0$ limit.By performing a renormalization- group study of this last theory we show that the gravitational coupling constant $G$ has a non-trivial,ultraviolet stable fixed point (asymptotic freedom) and that the dilatonic coupling functions (including the dilatonic potential) exhibit also a real, non-trivial fixed point. At such point the theory represents a standard charged string-inspired model.Stability and the gauge dependence of the fixed-point solution is discussed.It is shown that all these properties remain valid in a dilatonic-Yang-Mills theory with $n$ scalars and $m$ spinors, that has the UF stable fixed point $G^* = 3\epsilon (48+12N-m-2n)^{-1}$.In addition, it is seen that by increasing $N$ (number of gauge fields) the matter central charge $C=n+m/2$ ($0<C<24+6N$) can be increased correspondingly ( in pure dilatonic gravity $0<C<24$). 
Conventional quantization of two-dimensional diffeomorphism and Weyl invariant theories sacrifices the latter symmetry to anomalies, while maintaining the former. When alternatively Weyl invariance is preserved by abandoning diffeomorphism invariance, we find that some invariance against coordinate redefinition remains: one can still perform at will transformations possessing a constant Jacobian. The alternate approach enjoys as much ``gauge'' symmetry as the conventional formulation. 
We find the cross-over behavior for the spin-spin correlation function for the 2D Ising and 3-states Potts model with random bonds at the critical point. The procedure employed is the renormalisation approach of the perturbation series around the conformal field theories representing the pure models. We obtain a crossover in the amplitude for the correlation function for the Ising model which doesn't change the critical exponent, and a shift in the critical exponent produced by randomness in the case of the Potts model. A comparison with numerical data is discussed briefly. 
  We present an unifying description of the graded $SL(p,q)$ KP-KdV hierarchies, including the Wronskian construction of their tau-functions as well as the coefficients of the pertinent Lax operators, obtained via successive applications of special Darboux-B\"{a}cklund transformations. The emerging Darboux-B\"{a}cklund structure is identified as a constrained generalized Toda lattice system relevant for the two-matrix string model. Also, the exact Wronskian solution for the two-matrix model partition function is found. 
Starting from the known expression for the three-point correlation functions for Liouville exponentials with generic real coefficients at we can prove the Liouville equation of motion at the level of three-point functions. Based on the analytical structure of the correlation functions we discuss a possible mass shell condition for excitations of noncritical strings and make some observations concerning correlators of Liouville fields. 
  Infrared regularized versions of 4-D N=1 superstring ground states are constructed by curving the spacetime. A similar regularization can be performed in field theory. For the IR regularized string ground states we derive the exact one-loop effective action for non-zero U(1) or chromo-magnetic fields as well as gravitational and axionic-dilatonic fields. This effective action is IR and UV finite. Thus, the one-loop corrections to all couplings (gravitational, gauge and Yukawas) are unabiguously computed. These corrections are necessary for quantitative string superunification predictions at low energies. The one-loop corrections to the couplings are also found to satisfy Infrared Flow Equations. 
We study the phenomenon of turbulence from the point of view of statistical physics. We discuss what makes the turbulent states different from the thermodynamic equilibrium and give the turbulent analog of the partition function. Then, using the soluble theory of turbulence of waves as an example, we construct the turbulent action and show how one can compute the turbulent correlation functions perturbatively thus developing the turbulent Feynman diagrams. And at last, we discuss which part of what we learnt from the turbulence of waves can be used in other types of turbulence, in particular, the hydrodynamic turbulence of fluids. This paper is based on the talk delivered at SMQFT (1993) conference at the University of Southern California. 
The infrared limit of $D=4,~~N=4$ Yang-Mills theory with compact gauge group $G$ compactified on a two-torus is governed by an effective superconformal field theory. We conjecture that this is a certain orbifold involving the maximal torus of $G$. Yang-Mills $S$-duality makes predictions for all correlators of this effective conformal field theory. These predictions are shown to be implied by the standard $T$-duality of the conformal field theory. Consequently, Montonen-Olive duality between electric and magnetic states reduces to the standard two-dimensional duality between momentum and winding states. 
We relate two integrable models in (1+1) dimensions, namely, multicomponent Calogero-Sutherland model with particles and antiparticles interacting via the hyperbolic potential and the nonrelativistic factorizable $S$-matrix theory with $SU(N)$-invariance. We find complete solutions of the Yang-Baxter equations without implementing the crossing symmetry, and one of them is identified with the scattering amplitudes derived from the Schr\"{o}dinger equation of the Calogero-Sutherland model. This particular solution is of interest in that it cannot be obtained as a nonrelativistic limit of any known relativistic solutions of the $SU(N)$-invariant Yang-Baxter equations. 
We discuss two dimensional Yang -- Mills theories with massless fermions in arbitrary representations of a gauge group $G$. It is shown that the physics (spectrum and interactions) of the massive states in such models is independent of the detailed structure of the model, and only depends on the gauge group $G$ and an integer $k$ measuring the total anomaly. The massless physics, which does depend on the details of the model, decouples (almost) completely from that of the massive one. As an example, we discuss the equivalence of QCD$_2$ coupled to fermions in the adjoint, and fundamental representations. 
  Off-shellness and inhomogeneities are in the non-equilibrium dynamics of the lambda phi^4 model are studied using the closed timepath formalism reformulated as kinetic field theory. We take into account initial correlations up to the 4-point functions. The model is shown to exhibit a SO(1,1) symmetry broken by interactions and initial conditions. The divergence of the corresponding Noethercurrent is calculated and the Ward-Takahashi relations for the broken symmetry are given. They constitute a set of generalized integrated kinetic equations for the general n-point functions. We demonstrate that energy-momentum conservation follows from the transport equations. As an application we discuss inhomogeneities and general non-equilibrium conditions in the free field model. In our solution we identify the casimir-effect. 
  The heat kernel coefficients $H_k$ to the Schr\"odinger operator with a matrix potential are investigated. We present algorithms and explicit expressions for the Taylor coefficients of the $H_k$. Special terms are discussed, and for the one-dimensional case some improved algorithms are derived. 
Dirac equation for the finite dipole potential is solved by the method of the join of the asymptotics. The formulas for the near continuum state energy term of a relativistic electron-dipole system are obtained analytically. Two cases are considered: $Z < 137$ and $Z >137$ 
A method to determine the full structure of the space of local operators of massive integrable field theories, based on the form factor bootstrap approach is presented. This method is applied to the integrable perturbations of the Ising conformal point. It is found that the content of local operators can be expressed in terms of fermionic sum representations of the characters $\c(q)$ of the Virasoro irreducible representations of the minimal model $\M_{3,4}$. The space of operators factorises into chiral components as $Z=\sum \c(q) \c(\q)$, but with the relation $\q=q^{-1}$. 
The space of local operators in massive deformations of conformal field theories is analysed. For several model systems it is shown that one can define chiral sectors in the theory, such that the chiral field content is in a one-to-one correspondence with that of the underlying conformal field theory. The full space of operators consists of the descendent spaces of all scalar fields. If the theory contains asymptotic states which satisfy generalised statistics, the form factor equations admit in addition also solutions corresponding to the descendent spaces of the para-fermionic operators of the same spin as the asymptotic states. The derivation of these results uses $q$-sum expressions for the characters and $q$-difference equations used in proving Rogers-Ramanujan type identities. 
  On compactification to six spacetime dimensions, the fundamental heterotic string admits as a soliton a dual string whose effective worldsheet action couples to the background fields of the dual formulation of six-dimensional supergravity. On further toroidal compactification to four spacetime dimensions, the dual string acquires an $O(2,2;Z)$ target-space duality. This contains as a subgroup the axion-dilaton $SL(2,Z)$ which corresponds to a strong/weak coupling duality for the fundamental string. The dual string also provides a new non-perturbative mechanism for enhancement of the gauge symmetry. 
The method of the calculation of the multi-loop superstring amplitudes is proposed. The amplitudes are calculated from the equations that are none other than Ward identities. They are derived from the requirement that the discussed amplitudes are independent from a choice of gauge of both the vierbein and the gravitino field. The amplitudes are calculated in the terms of the superfields vacuum correlators on the complex (1|1) supermanifolds. The superconformal Schottky groups appropriate for this aim are built for all the spinor structures. The calculation of the multi- loop boson emission amplitudes in the closed, oriented Ramond-Neveu-Schwarz superstring theory is discussed in details. The main problem arises for those spinor structures that correspond to the Ramond fermion loops. Indeed, in this case the superfield vacuum correlators can not be derived by a simple extension of the boson string results. The method of the calculation of the above correlators is proposed. The discussed amplitudes due to all the even spinor structures is given in the explicit form. 
  Following the successful construction of the finite T real-time formalism in temporal axial gauge, we attempt to further study the equivalent new imaginary-time formalism of James and Landshoff based on the same Hamiltonian approach in the hope that it will provide the answer to Debye screening in QCD. It turns out that, unlike in the real-time case, energy conservation does not hold because of the unusual representation of the longitudinal field forced upon by the Hamiltonian formulation. 
In the present contribution, I report on certain {\it non-linear} and {\it non-local} extensions of the conformal (Virasoro) algebra. These so-called $V$-algebras are matrix generalizations of $W$-algebras. First, in the context of two-dimensional field theory, I discuss the non-abelian Toda model which possesses three conserved (chiral) ``currents". The Poisson brackets of these ``currents" give the simplest example of a $V$-algebra. The classical solutions of this model provide a free-field realization of the $V$-algebra. Then I show that this $V$-algebra is identical to the second Gelfand-Dikii symplectic structure on the manifold of $2\times 2$-matrix Schr\"odinger operators $L=-\d^2+U$ (with $\tr\sigma_3 U=0$). This provides a relation with matrix KdV-hierarchies and allows me to obtain an infinite family of conserved charges (Hamiltonians in involution). Finally, I work out the general $V_{n,m}$-algebras as symplectic structures based on $n\times n$-matrix $m^{\rm th}$-order differential operators $L=-\d^m +U_2\d^{m-2}+U_3 \d^{m-3}+\ldots +U_m$. It is the absence of $U_1$, together with the non-commutativity of matrices that leads to the non-local terms in the $V_{n,m}$-algebras. I show that the conformal properties are similar to those of $W_m$-algebras, while the complete $V_{n,m}$-algebras are much more complicated, as is shown on the explicit example of $V_{n,3}$. 
We introduce a new variational characterization of Gaussian diffusion processes as minimum uncertainty states. We then define a variational method constrained by kinematics of diffusions and Schr\"{o}dinger dynamics to seek states of local minimum uncertainty for general non-harmonic potentials. 
The dynamical algebra associated to a family of isospectral oscillator Hamiltonians is studied through the analysis of its representation in the basis of energy eigenstates. It is shown that this representation becomes similar to that of the standard Heisenberg algebra, and it is dependent of a parameter $w\geq 0$. We name it {\it distorted Heisenberg algebra}, where $w$ is the distortion parameter. The corresponding coherent states for an arbitrary $w$ are derived, and some particular examples are discussed in full detail. A prescription to produce the squeezing, by adequately selecting the initial state of the system, is given. 
We derive the $t{\bar t}$-equations for generic $N\!=\!2$ topological field theories as consistency conditions for the contact term algebra of topological strings. A generalization of the holomorphic anomaly equation, known for the critical ${\hat c}\!=\!3$ case, to arbitrary non critical topological strings is presented. The interplay between the non trivial cohomology of the $b$-antighost, gravitational descendants and $\bar t$-dependence is discussed. The physical picture emerging from this study is that the $\bar t$ (background) dependence of topological strings with non trivial cohomology for the $b$-antighost, is determined by gravitational descendants. 
The Kazakov--Migdal (KM) Model is a U(N) Lattice Gauge Theory with a Scalar Field in the adjoint representation but with no kinetic term for the Gauge Field. This model is formally soluble in the limit $N\rightarrow \infty$ though explicit solutions are available for a very limited number of scalar potentials. A ``Double Penner'' Model in which the potential has two logarithmic singularities provides an example of a explicitly soluble model. We begin by reviewing the formal solution to this Double Penner KM Model. We pay special attention to the relationship of this model to an ordinary (one) matrix model whose potential has two logarithmic singularities (the Double Penner Model). We present a detailed analysis of the large N behavior of this Double Penner Model. We describe the various one cut and two cut solutions and we discuss cases in which ``eigenvalue condensation'' occurs at the singular points of the potential. We then describe the consequences of our study for the KM Model described above. We present the phase diagram of the model and describe its critical regions. 
The quantum and classical aspects of a deformed $c=1$ matrix model proposed by Jevicki and Yoneya are studied. String equations are formulated in the framework of Toda lattice hierarchy. The Whittaker functions now play the role of generalized Airy functions in $c<1$ strings. This matrix model has two distinct parameters. Identification of the string coupling constant is thereby not unique, and leads to several different perturbative interpretations of this model as a string theory. Two such possible interpretations are examined. In both cases, the classical limit of the string equations, which turns out to give a formal solution of Polchinski's scattering equations, shows that the classical scattering amplitudes of massless tachyons are insensitive to deformations of the parameters in the matrix model. 
We introduce a new parametrisation for the Fermi sea of the $c = 1$ matrix model. This leads to a simple derivation of the scattering matrix, and a calculation of boundary corrections in the corresponding $1+1$--dimensional string theory. The new parametrisation involves relativistic chiral fields, rather than the non-relativistic fields of the usual formulations. The calculation of the boundary corrections, following recent work of Polchinski, allows us to place restrictions on the boundary conditions in the matrix model. We provide a consistent set of boundary conditions, but believe that they need to be supplemented by some more subtle relationship between the space-time and matrix model. Inspired by these boundary conditions, some thoughts on the black hole in $c=1$ string theory are presented. 
  The statistical mechanical properties of interacting quantum fields in terms of the dynamics of the correlation functions are investigated. We show how the Dyson - Schwinger equations may be derived from a formal action functional, the n-particle irreducible ($nPI, n \to \infty$) or the `master' effective action. It is related to the decoherence functional between histories defined in terms of correlations. Upon truncation of the Dyson - Schwinger hierarchy at a certain order, the master effective action becomes complex, its imaginary part arising from the higher order correlation functions, the fluctuations of which we define as the correlation noises of that order. Decoherence of correlation histories via these noises gives rise to classical stochastic histories %driven by the flucutations of these higher correlation functions. Ordinary quantum field theory corresponds to taking the lowest order functions, usually the mean field and the 2-point functions. As such, our reasoning shows that it is an effective theory which can be intrinsically dissipative. The relation of loop expansion and correlation order as well as the introduction of an arrow of time from the choice of boundary conditions are expounded with regard to the origin of dissipation in quantum fields. Relation with critical phenomena, quantum transport, molecular hydrodynamics and potential applications to quantum gravity, early universe processes and black hole physics are mentioned. 
We show the existence of self-dual semilocal nontopological vortices in a $\Phi^2$ Chern-Simons (C-S) theory. The model of scalar and gauge fields with a $SU(2)_{global} \times U(1)_{local}$ symmetry includes both the C-S term and an anomalous magnetic contribution. It is demonstrated here, that the vortices are stable or unstable according to whether the vector topological mass $\kappa$ is less than or greater than the mass $m$ of the scalar field. At the boundary, $\kappa = m$, there is a two-parameter family of solutions all saturating the self-dual limit. The vortex solutions continuously interpolates between a ring shaped structure and a flux tube configuration. 
We study exact renormalization group equations in the framework of the effective average action. We present analytical approximate solutions for the scale dependence of the potential in a variety of models. These solutions display a rich spectrum of physical behaviour such as fixed points governing the universal behaviour near second order phase transitions, critical exponents, first order transitions (some of which are radiatively induced) and tricritical behaviour. 
We study relativistic self-dual Chern-Simons-Higgs systems in the presence of uniform background fields that explicitly break CTP. A rich, but discrete vacuum structure is found when the gauge symmetry is spontaneously broken, while the symmetric phase can have an infinite vacuum degeneracy at tree level. The latter is due to the proliferation of neutral solitonic states that cost zero energy. Various novel self-dual solitons, such as these, are found in both the symmetric and the asymmetric phases. Also by considering a similar system on a two-sphere and the subsequent large sphere limit, we isolate sensible and finite expressions for the conserved angular and linear momenta, which satisfy anomalous commutation relations. We conclude with a few remarks on unresolved issues. 
  We consider integrable models of the Haldane-Shastry type with open boundary conditions. We define monodromy matrices, obeying the reflection equation, which generate the symmetries of these models. Using a map to the Calogero-Sutherland Hamiltonian of BC type, we derive the spectrum and the highest weight eigenstates. 
  The CGHS model of two-dimensional dilaton gravity coupled to a sine-Gordon matter field is considered. The theory is exactly solvable classically, and the solutions of a kink and two-kink type solitons are studied in connection with black hole formation. 
  A Lagrangian of the topological field theory is found in the twisted osp(2|2)\oplus osp(2|2)conformal super algebra. The reduction on a moduli space is then elaborated through the vanishing Noether current. 
  A superfield method of computing the effective potential in supersymmetric field theories is suggested. Analysis of the structure of the effective potential in the Wess-Zumino model is carried out. It is shown that the superfield effective potential is defined by three objects: kahlerian effective potential, chiral effective potential and auxiliary fields' effective potential. One-loop kahlerian effective potential is calculated using the superfield heat kernel technique. A corresponding superfield kernel is found in explicit form. Procedure for computing higher loop corrections is developed. First (two-loop) correction to chiral effective potential is found. Two-loop kahlerian effective potential is calculated. 
We apply the Mellin-Barnes integral representation to several situations of interest in mathematical-physics. At the purely mathematical level, we derive useful asymptotic expansions of different zeta-functions and partition functions. These results are then employed in different topics of quantum field theory, which include the high-temperature expansion of the free energy of a scalar field in ultrastatic curved spacetime, the asymptotics of the $p$-brane density of states, and an explicit approach to the asymptotics of the determinants that appear in string theory. 
The 2-point function is the natural object in quantum gravity for extracting critical behavior: The exponential fall off of the 2-point function with geodesic distance determines the fractal dimension $d_H$ of space-time. The integral of the 2-point function determines the entropy exponent $\gamma$, i.e. the fractal structure related to baby universes, while the short distance behavior of the 2-point function connects $\gamma$ and $d_H$ by a quantum gravity version of Fisher's scaling relation. We verify this behavior in the case of 2d gravity by explicit calculation. 
We consider the (massive) Gross-Neveu model using the light cone quantization where we solve the constraints explicitly. We show that the vacuum is trivial and that the quantization fails when $m=0$. We discuss how the running coupling constant emerges as a pure normal ordering effect in this context and the bound state equation. 
We compute the Casimir energy for a free scalar field on the spaces $\,R^{m+1}\,\times\,\tilde S^2\,$ where $,\tilde S^2\,$ is two-dimensional deformed two-sphere. 
  Drawing on the parallel between general relativity and Yang-Mills theory we obtain an exact Schwarzschild-like solution for SU(2) gauge fields coupled to a massless scalar field. Pushing the analogy further we speculate that this classical solution to the Yang-Mills equations shows confinement in the same way that particles become confined once they pass the event horizon of the Schwarzschild solution. Two special cases of the solution are considered. 
A survey of solutions to the cosmological moduli problem in string theory. The only extant proposal which may work is Intermediate Scale Inflation as proposed by Randall and Thomas. Supersymmetry preserving dynamics which could give large masses to the moduli is strongly constrained by cosmology and requires the existence of string vacuum states possessing properties different from those of any known vacuuum. Such a mechanism cannot give mass to the dilaton unless there are cancellations between different exponentially small contributions to the superpotential. Our investigation also shows that stationary points of the effective potential with negative vacuum energy do not correspond to stationary solutions of the equations of postinflationary cosmology. This suggests that supersymmetry breaking is a requirement for a successful inflationary cosmology. 
Decoherence in quantum systems which are classically chaotic is studied. It is well-known that a classically chaotic system when quantized loses many prominent chaotic traits. We show that interaction of the quantum system with an environment can under general circumstances quickly diminish quantum coherence and reenact some characteristic classical chaotic behavior. We use the Feynman-Vernon influence functional formalism to study the effect of an ohmic environment at high temperature on two classically-chaotic systems: The linear Arnold cat map (QCM) and the nonlinear quantum kicked rotor (QKR). Features of quantum chaos such as recurrence in QCM and diffusion suppression leading to localization in QKR are destroyed in a short time due to environment-induced decoherence. Decoherence also undermines localization and induces an apparent transition from reversible to irreversible dynamics in quantum chaotic systems. 
  We present exact solutions characterised by Bianchi-type I,II,III,V,VI four-dimensional metric, space-independent dilaton, and vanishing torsion background, for the low energy string effective action with zero central charge deficit. We show that, in such a context, curvature singularities cannot be avoided, except for the trivial case of flat spacetime and constant dilaton. We also provide a further example of the failure of the standard prescription for connecting conformal string backgrounds through duality transformations associated to non-semisimple, non-Abelian isometry group. 
We study two--loop renormalization in $(2+\epsilon)$--dimensional quantum gravity. As a first step towards the full calculation, we concentrate on the divergences which are proportional to the number of matter fields. We calculate the $\beta$ functions and show how the nonlocal divergences as well as the infrared divergences cancel among the diagrams. Although the formalism includes a subtlety concerning the general covariance due to the dynamics of the conformal mode, we find that the renormalization group allows the existence of a fixed point which possesses the general covariance. Our results strongly suggest that we can construct a consistent theory of quantum gravity by the $\epsilon$ expansion around two dimensions. 
We study the algebraic renormalization of $N=2$ Supersymmetric Yang--Mills theories coupled to matter. A regularization procedure preserving both the BRS invariance and the supersymmetry is not known yet, therefore it is necessary to adopt the algebraic method of renormalization, which does not rely on any regularization scheme. The whole analysis is reduced to the solution of cohomology problems arising from the generalized Slavnov operator which summarizes all the symmetries of the model. Besides to unphysical renormalizations of the quantum fields, we find that the only coupling constant of $N=2$ SYMs can get quantum corrections. Moreover we prove that all the symmetries defining the theory are algebraically anomaly--free. 
We calculate correlation functions in matrix models modified by trace-squared terms. First we study scaling operators in modified one-matrix models and find that their correlation functions satisfy modified Virasoro constraints. Then we turn to dressed order parameters in minimal models and show that their correlators satisfy Goulian-Li formulae continued to negative Liouville dressing exponents. Our calculations provide additional support for the idea that the modified matrix models contain operators with the negative branch of gravitational dressing. 
In terms of the form factor bootstrap we describe all the local fields in SG theory and check the agreement with the free fermion case. We discuss the interesting structure responsible for counting the local fields. 
In this talk I discuss both the present status and some recent work on the Kazakov--Migdal Model which was originally proposed as a soluble, large $N$ realization of QCD. After a brief description of the model and a discussion of its solubility in the large $N$ limit I discuss several of the serious problems with the model which lead to the conclusion that it does {\it not} induce QCD. The model is nonetheless a very interesting example of a Gauge Theory and it is related to some very interesting Matrix Models. I then outline a technique \REF\dmsxyz{\dms}\refend which uses ``Loop Equations'' for solving such models. A Penner--like model is then discussed with two logarithmic singularities. This model is distinguished by the fact that it is exactly and explicitly soluble in spite of the fact that it is not Gaussian. It is shown how to analyze this model using both a technical approach and from a more physical point of view. 
The N-extended supersymmetric self-dual Poincar\'e supergravity equations provide a natural local description of supermanifolds possessing hyperk\"ahler structure. These equations admit an economical formulation in chiral superspace. A reformulation in harmonic superspace encodes self-dual supervielbeins and superconnections in a graded skew-symmetric supermatrix superfield prepotential satisfying generalised Cauchy-Riemann conditions. A recipe is presented for extracting explicit self-dual supervielbeins and superconnections from such `analytic' prepotentials. We demonstrate the method by explicitly decoding a simple example of superfield prepotential, analogous to that corresponding to the Taub-NUT solution. The superspace we thus construct is an interesting $N=2$ supersymmetric deformation of flat space, having flat `body' and constant curvature `soul'. 
Using the restricted star-triangle relation, it is shown that the $N$-state spin integrable model on a three-dimensional lattice with spins interacting round each elementary cube of the lattice proposed by Mangazeev, Sergeev and Stroganov is a particular case of the Bazhanov-Baxter model. 
For any given algebra of local observables in Minkowski space an associated scaling algebra is constructed on which renormalization group (scaling) transformations act in a canonical manner. The method can be carried over to arbitrary spacetime manifolds and provides a framework for the systematic analysis of the short distance properties of local quantum field theories. It is shown that every theory has a (possibly non-unique) scaling limit which can be classified according to its classical or quantum nature. Dilation invariant theories are stable under the action of the renormalization group. Within this framework the problem of wedge (Bisognano-Wichmann) duality in the scaling limit is discussed and some of its physical implications are outlined. 
We consider the heat-kernel expansion of the massive Laplace operator on the three dimensional ball with Dirichlet boundary conditions. Using this example, we illustrate a very effective scheme for the calculation of an (in principle) arbitrary number of heat-kernel coefficients for the case where the basis functions are known. New results for the coefficients $B_{\frac 5 2},...,B_5$ are presented. 
We show how to formulate the phenomenon of gaugino condensation in a super-Yang-Mills theory with a field-dependent gauge coupling described with a linear multiplet. We prove the duality equivalence of this approach with the more familiar formulation using a chiral superfield. In so doing, we resolve a longstanding puzzle as to how a linear-multiplet formulation can be consistent with the dynamical breaking of the Peccei-Quinn symmetry which is thought to occur once the gauginos condense. In our approach, the composite gauge degrees of freedom are described by a real vector superfield, $V$, rather than the chiral superfield that is obtained in the traditional dual formulation. Our dualization, when applied to the case of several condensing gauge groups, provides strong evidence that this duality survives strong-coupling effects in string theory. 
  We compute the spectrum of scalar and tensor metric perturbations generated, as amplified vacuum fluctuations, during an epoch of dilaton-driven inflation of the type occurring naturally in string cosmology. In the tensor case the computation is straightforward while, in the scalar case, it is made delicate by the appearance of a growing mode in the familiar longitudinal gauge. In spite of this, a reliable perturbative calculation of perturbations far outside the horizon can be performed by resorting either to appropriate gauge invariant variables, or to a new coordinate system in which the growing mode can be "gauged down". The simple outcome of this complicated analysis is that both scalar and tensor perturbations exhibit nearly Planckian spectra, whose common "temperature" is related to some very basic parameters of the string-cosmology background. 
Two models of dilatonic gravity are investigated: (i) dilaton-Yang-Mills gravity and (ii) higher-derivative dilatonic gravity. Both are renormalizable in $2+\epsilon$ dimensions and have a smooth limit for $\epsilon \rightarrow 0$. The corresponding one-loop effective actions and beta-functions are found. Both theories are shown to possess a non-trivial ultraviolet fixed point ---for all dilatonic couplings--- in which the gravitational constant is asymptotically free. It is shown that in the regime of asymptotic freedom the matter central charge can be significantly increased by two different mechanisms ---as compared with pure dilatonic gravity, where $n < 24$. 
It is argued that the type IIA 10-dimensional superstring theory is actually a compactified 11-dimensional supermembrane theory in which the fundamental supermembrane is identified with the the solitonic membrane of 11-dimensional supergravity. The charged extreme black holes of the 10-dimensional type IIA string theory are interpreted as the Kaluza-Klein modes of 11-dimensional supergravity and the dual sixbranes as the analogue of Kaluza-Klein monopoles. All other p-brane solutions of the type IIA superstring theory are derived from the 11-dimensional membrane and its magnetic dual fivebrane soliton. 
We derive a bound on the energy of the general (p,q)-supersymmetric two-dimensional massive sigma model with torsion, in terms of the topological and Noether charges that appear as central charges in its supersymmetry algebra.The bound is saturated by soliton solutions of first-order Bogomol'nyi-type equations. This generalizes results obtained previously for p=q models without torsion. We give examples of massive (1,1) models with torsion that have a group manifold as a target space. We show that they generically have multiple vacua and find an explicit soliton solution of an SU(2) model. We also construct a new class of zero torsion massive (4,4) models with multiple vacua and soliton solutions. In addition, we compute the metrics on the one-soliton moduli spaces for those cases for which soliton solutions are known explicitly, and discuss their interpretation. 
We examine here discontinuities in the metric, the antisymmetric tensor and the dilaton field which are allowed by conformal invariance. We find that the surfaces of discontinuity must necessarily be null surfaces and shock and impulsive waves are both allowed. We employ our results for the case of colliding plane gravitational waves and we discuss the SL(2,R)XSU(2)/RXR WZW model in the present prespective. In particular, the singularities encountered in this model may be viewed as the result of the mutual focusing of the colliding waves. 
Contents:  1. Introduction  2. Causal Structure and Penrose Diagrams      Minkowski Space; 1+1 Dimensional Minkowski Space; Schwarzchild Black       Holes; Gravitational Collapse and the Vaidya Spacetimes; Event Horizons,       Apparent Horizons, and Trapped Surfaces  3. Black Holes in Two Dimensions      General Relativity in the $S$-Wave Sector; Classical Dilaton Gravity;      Eternal Black Holes; Coupling to Conformal Matter; Hawking Radiation and       the Trace Anomaly; The Quantum State; Including the Back-Reaction;      The Large $N$ Approximation; Conformal Invariance and Generalizations of      Dilaton Gravity; The Soluble $RST$ Model  4. The Information Puzzle in Four Dimensions      Can the Information Come Out Before the Endpoint?;      Low-Energy Effective Descriptions of the Planckian Endpoint; Remnants?;      Information Destruction?; The Superposition Principle; Energy Conservation      The New Rules; Superselection Sectors, $\alpha$-parameters, and the      Restoration of Unitarity  5. Conclusions and Outlook 
We present a dual formulation of the construction of N=2 nonlinear sigma-model type conformal field theories with c=9 which are mainly used as internal sectors of Calabi-Yau heterotic string compactifications. The supercurrents and the higher components of the spectral flow superfields turn out in each case to be tensor products of two simple currents of a c=1 and a non-supersymmetric parafermionic c=8 CFT. The characters of the latter model can be regarded as string functions of the coset sigma/U(1). In particular, for the (1)^9 Gepner model we discuss these string functions in detail, realizing this model contains a broken E_8 gauge symmetry of which only the abelian subalgebra remains a symmetry of the spectrum. As an example, we construct a simple model leading to a string theory with a massless spectrum of 36 E_6 generations and an extended gauge symmetry E_6 x SU(3)^4. 
Relativistic domain walls are studied in the framework of a polynomial approximation to the field interpolating between different vacua and forming the domain wall. In this approach we can calculate evolution of a core and of a width of the domain wall. In the single, cubic polynomial approximation used in this paper, the core obeys Nambu-Goto equation for a relativistic membrane. The width of the domain wall obeys a nonlinear equation which is solved perturbatively. There are two types of corrections to the constant zeroth order width: the ones oscillating in time, and the corrections directly related to curvature of the core. We find that curving a static domain wall is associated with an increase of its width. As an example, evolution of a toroidal domain wall is investigated. 
A generalization of the twistor shift procedure to the case of superparticle interacting with the background D=3 N=1 Maxwell and D=3 N=1 supergravity supermultiplet is considered. We investigate twistor shift effects and discuss the structure of the resulting constraint algebra. 
  We review localization techniques for functional integrals which have recently been used to perform calculations in and gain insight into the structure of certain topological field theories and low-dimensional gauge theories. These are the functional integral counterparts of the Mathai-Quillen formalism, the Duistermaat-Heckman theorem, and the Weyl integral formula respectively. In each case, we first introduce the necessary mathematical background (Euler classes of vector bundles, equivariant cohomology, topology of Lie groups), and describe the finite dimensional integration formulae. We then discuss some applications to path integrals and give an overview of the relevant literature. The applications we deal with include supersymmetric quantum mechanics, cohomological field theories, phase space path integrals, and two-dimensional Yang-Mills theory. 
We present a new application of affine Lie algebras to massive quantum field theory in 2 dimensions, by investigating the $q\to 1$ limit of the q-deformed affine $\hat{sl(2)}$ symmetry of the sine-Gordon theory, this limit occurring at the free fermion point. We describe how radial quantization leads to a quasi-chiral factorization of the space of fields. The conserved charges which generate the affine Lie algebra split into two independent affine algebras on this factorized space, each with level 1, in the anti-periodic sector. The space of fields in the anti-periodic sector can be organized using level-$1$ highest weight representations, if one supplements the $\slh$ algebra with the usual local integrals of motion. Using the integrals of motion, a momentum space bosonization involving vertex operators is formulated. This leads to a new way of computing form-factors, as vacuum expectation values in momentum space. The problem of non-trivial correlation functions in this model is also discussed; in particular it is shown how space-time translational anomalies which arise in radial quantization can be used to compute the short distance expansion of some simple correlation functions. 
We give a detailed account of the computation of the Yang-Mills action for the Connes-Lott model with general coupling constant in the commutant of the $K$-cycle. This leads to tree-approximation results amazingly compatible with experiment, yielding a first indication on the Higgs mass. 
We quantise the reduced theory obtained by substituting the soliton solutions of affine Toda theory into its symplectic form. The semi-classical S-matrix is found to involve the classical Euler dilogarithm. 
  By placing theories with Yangian charges on the lattice in the analogue of the St Petersburg school's approach to the sine-Gordon system, we exhibit the Yangian structure of the auxiliary algebra, and explain how the two Yangians are related. 
We obtain a bilocal classical field theory as the large $N$ limit of the chiral Gross--Neveu (or non--abelian Thirring) model. Exact classical solutions that describe topological solitons are obtained. It is shown that their mass spectrum agrees with the large $N$ limit of the spectrum of the chiral Gross--Neveu model. 
  The chiral boson actions of Floreanini and Jackiw (FJ), and of McClain,Wu and Yu (MWY) have been recently shown to be different representations of the same chiral boson theory. MWY displays manifest covariance and also a (gauge) symmetry that is hidden in the FJ side, which, on the other hand, displays the physical spectrum in a simple manner. We make use of the covariance of the MWY representation for the chiral boson to couple it to background gravity showing explicitly the equivalence with the previous results for the FJ representation 
We demonstrate in detail how the space of two-dimensional quantum field theories can be parametrized by off-shell closed string states. The dynamic equation corresponding to the condition of conformal invariance includes an infinite number of higher order terms, and we give an explicit procedure for their calculation. The symmetries corresponding to equivalence relations of CFT are described. In this framework we show how to perform nonperturbative analysis in the low-energy limit and prove that it corresponds to the Brans-Dicke theory of gravity interacting with a skew symmetric tensor field. (Talk presented at the Gursey memorial conference, Istanbul, Turkey, June, 1994.) 
We calculate the leading quantum and semi-classical corrections to the Newtonian potential energy of two widely separated static masses. In this large-distance, static limit, the quantum behaviour of the sources does not contribute to the quantum corrections of the potential. These arise exclusively from the propagation of massless degrees of freedom. Our one-loop result is based on Modanese's formulation and is in disagreement with Donoghue's recent calculation. Also, we compare and contrast the structural similarities of our approach to scattering at ultra-high energy and large impact parameter. We connect our approach to results from string perturbation theory. 
We study the cohomology arising in the BRST formulation of G/H gauged WZNW models, i.e. in which the states of the gauged theory are projected out from the ungauged one by means of a BRST condition. We will derive for a general simple group $H$ with arbitrary level, conditions for which the cohomology is non-trivial. We show, by introducing a small perturbation due to Jantzen, in the highest weights of the representations, how states in the cohomology, "singlet pairs", arise from unphysical states, "Kugo-Ojima quartets", as the perturbation is set to zero. This will enable us to identify and construct states in the cohomology. The ghost numbers that will occur are $\pm p$, with $p$ uniquely determined by the representations of the algebras involved. Our construction is given in terms of the current modes and relies on the explicit form of highest weight null-states given by Malikov, Feigen and Fuchs. 
New features of a previously introduced Group Approach to Quantization are presented. We show that the construction of the symmetry group associated with the system to be quantized (the ``quantizing group") does not require, in general, the explicit construction of the phase space of the system, i.e., does not require the actual knowledgement of the general solution of the classical equations of motion: in many relevant cases an implicit construction of the group can be given, directly, on configuration space. As an application we construct the symmetry group for the conformally invariant massless scalar and electromagnetic fields and the scalar and Dirac fields evolving in a symmetric curved space- time or interacting with symmetric classical  electromagnetic fields. Further generalizations of the present procedure are also discussed and in particular the conditions under which non-abelian (mainly Kac-Moody) groups can be included. 
The 3+1 dimensional Yang-Mills theory with the Pontryagin term included is studied on manifolds with a boundary. Based on the geometry of the universal bundle for Yang-Mills theory, the symplectic structure of this model is exhibited. The topological type of the quantization line bundles is shown to be determined by the torsion elements in the cohomology of the gauge orbit space. 
The random vector potential model describes massless fermions coupled to a quenched random gauge field. We study its abelian and non-abelian versions. The abelian version can be completely solved using bosonization. We analyse the non-abelian model using its supersymmetric formulation and show, by a perturbative renormalisation group computation, that it is asymptotically free at large distances. We also show that all the quenched chiral current correlation functions can be computed exactly, without using the replica trick or the supersymmetric formulation, but using an exact expression for the effective action for any sample of the random gauge field. These chiral correlation functions are purely algebraic. 
We obtain a bijection between certain lattice paths and partitions. This implies a proof of polynomial identities conjectured by Melzer. In a limit, these identities reduce to Rogers--Ramanujan-type identities for the $\chi_{1,1}^{(p,p+1)}(q)$ Virasoro characters, conjectured by the Stony Brook group. 
We generalize and extend a quantization procedure proposed by the present authors which is designed to quantize SU(N) gauge theories in the continuum without fixing the gauge and thereby avoid the Gribov problem. In particular we discuss the BRS symmetry underlying the effective action. We proceed to use this BRS symmetry to discuss the perturbative renormalization of the theory and show that perturbatively the procedure is equivalent to Landau gauge fixing. This generalizes earlier results obtained in the Abelian case to the non-abelian case and confirms the widely held believe that the Gribov problem manifests itself on the non-perturbative level, while not affecting the perturbative results. A relation between the gluon mass and gluon condensate in QCD is obtained which yields a gluon mass consistent with other estimates for values of the gluon condensate obtained from QCD sum rules. 
We consider the geometric entropy of free nonrelativistic fermions in two dimensions and show that it is ultraviolet finite for finite fermi energies, but divergent in the infrared. In terms of the corresponding collective field theory this is a {\em nonperturbative} effect and is related to the soft behaviour of the usual thermodynamic entropy at high temperatures. We then show that thermodynamic entropy of the singlet sector of the one dimensional matrix model at high temperatures is governed by nonperturbative effects of the underlying string theory. In the high temperature limit the ``exact'' expression for the entropy is regular but leads to a negative specific heat, thus implying an instability. We speculate that in a properly defined two dimensional string theory, the thermodynamic entropy could approach a constant at high temperatures and lead to a geometric entropy which is finite in the ultraviolet. 
A new method of gauging of WZNW models is presented, leading to a class of exact string solutions with a target space metric of Minkowskian signature. The corresponding models may be interpreted as $\sigma$-model analogues of the Toda field theories. 
We discuss a non--commutative integration calculus arising in the mathematical description of anomalies in fermion--Yang--Mills systems. We consider the differential complex of forms $u_0\ccr{\eps}{u_1}\cdots\ccr{\eps}{u_n}$ with $\eps$ a grading operator on a Hilbert space $\cH$ and $u_i$ bounded operators on $\cH$ which naturally contains the compactly supported de Rham forms on $\R^d$ (i.e.\ $\eps$ is the sign of the free Dirac operator on $\R^d$ and $\cH$ a $L^2$--space on $\R^d$). We present an elementary proof that the integral of $d$--forms $\int_{\R^d}\trac{X_0\dd X_1\cdots \dd X_d}$ for $X_i\in\Map(\R^d;\gl_N)$, is equal, up to a constant, to the conditional Hilbert space trace of $\Gamma X_0\ccr{\eps}{X_1}\cdots\ccr{\eps}{X_d}$ where $\Gamma=1$ for $d$ odd and $\Gamma=\gamma_{d+1}$ (`$\gamma_5$--matrix') a spin matrix anticommuting with $\eps$ for $d$ even. This result provides a natural generalization of integration of de Rham forms to the setting of Connes' non--commutative geometry which involves the ordinary Hilbert space trace rather than the Dixmier trace. 
We consider the model of self-interacting complex scalar fields with a rigid gauge invariance under an arbitrary gauge group $G$. In order to analyze the phenomenon of Bose-Einstein condensation finite temperature and the possibility of a finite background charge is included. Different approaches to derive the relevant high-temperature behaviour of the theory are presented. 
I study the time--symmetric initial--data problem in theories with a massless scalar field (dilaton), free or coupled to a Maxwell field in the stringy way, finding different initial--data sets describing an arbitrary number of black holes with arbitrary masses, charges and asymptotic value of the dilaton. The presence of the scalar field gives rise to a number of interesting effects. The mass and charges of a single black hole are different in its two asymptotically flat regions across the Einstein--Rosen bridge. The same happens to the value of the dilaton at infinity. This forbids the identification of these asymptotic regions in order to build (Misner) wormholes in the most naive way. Using different techniques, I find regular initial data for stringy wormholes. The price payed is the existence singularities in the dilaton field. The presence of a single--valued scalar seems to constrain strongly the allowed topologies of the initial space--like surface. Other kinds of scalar fields (taking values on a circle or being defined up to an additive constant) are also briefly considered. 
We study systematically the Lax description of the KdV hierarchy in terms of an operator which is the geometrical recursion operator. We formulate the Lax equation for the $n$-th flow, construct the Hamiltonians which lead to commuting flows. In this formulation, the recursion relation between the conserved quantities follows naturally. We give a simple and compact definition of all the Hamiltonian structures of the theory which are related through a power law. 
  By considering a (partial) topological twisting of supersymmetric Yang-Mills compactified on a 2d space with `t Hooft magnetic flux turned on we obtain a supersymmetric $\sigma$-model in 2 dimensions. For N=2 SYM this maps Donaldson observables on products of two Riemann surfaces to quantum cohomology ring of moduli space of flat connections on a Riemann surface. For N=4 SYM it maps $S$-duality to $T$-duality for $\sigma$-models on moduli space of solutions to Hitchin equations. 
  In this paper we extend our previously discovered exact solution for an SU(2) gauge theory coupled to a massless, non-interacting scalar field, to the general group SU(N+1). Using the first-order formalism of Bogomolny, an exact, spherically symmetric solution for the gauge and scalar fields is found. This solution is similiar to the Schwarzschild solution of general relativity, in that the gauge and scalar fields become infinite at a radius, $r_0 = K$, from the origin. It is speculated that this may be the confinement mechanism that has long been sought for in non-Abelian gauge theories, since any particle which carries the SU(N+1) charge would become permanently trapped once it entered the region $r < r_0$. The energy of the field configuration of this solution is calculated. 
Boundary conditions compatible with classical integrability are studied both directly, using an approach based on the explicit construction of conserved quantities, and indirectly by first developing a generalisation of the Lax pair idea. The latter approach is closer to the spirit of earlier work by Sklyanin and yields a complete set of conjectures for permissible boundary conditions for any affine Toda field theory. 
We show that the large N partition functions and Wilson loop observables of two-dimensional Yang-Mills theories admit a universal functional form irrespective of the gauge group. We demonstrate that U(N) QCD_2 undergoes a large N, third-order phase transition on the projective plane at an area-coupling product of \pi^2/2. We use this as a lemma to provide a direct transcription of the partition functions and phase portraits of Yang-Mills theory from the U(N) on RP^2 at large N to the other classical Lie groups on S^2. We compute the expectation value of the Wilson loops in the fundamental representation for SO(N) and Sp(N) on the two sphere. Finally we compare the strong- and weak-coupling limit of these expressions with those found elswhere in the literature. 
We present the complete structure of the nonlinear $N=2$ super extension of Polyakov-Bershadsky, $W_3^{(2)}$, algebra with the generic central charge, $c$, at the {\it quantum} level. It contains extra two pairs of fermionic currents with integer spins 1 and 2, besides the currents of $N=2$ superconformal and $W_3^{(2)}$ algebras. For $c\rightarrow \infty$ limit, the algebra reduces to the classical one, which has been studied previously. The 'hybrid' field realization of this algebra is also discussed. 
We find new classes of {\it exact} string solutions in a variety of curved backgrounds. They include stationary and dynamical (open, closed, straight, finitely and infinitely long) strings as well as {\it multi-string} solutions, in terms of elliptic functions. The physical properties, string length, energy and pressure are computed and analyzed. In anti de Sitter spacetime, the solutions describe an {\it infinite} number of infinitely long stationary strings of equal energy but different pressures. In de Sitter spacetime, outside the horizon, they describe infinitely many {\it dynamical} strings infalling non-radially, scattering at the horizon and going back to spatial infinity in different directions. For special values of the constants of motion, there are families of solutions with {\it selected finite} numbers of different and independent strings. In black hole spacetimes (without cosmological constant), {\it no} multi-string solutions are found. In the Schwarzschild black hole, inside the horizon, we find one straight string infalling non-radially, with {\it indefinetely} growing size, into the $r=0$ singularity. In the $2+1$ black hole anti de Sitter background, the string stops at $r=0$ with {\it finite} length. 
We study the effects of the spatial curvature on the classical and quantum string dynamics. We find the general solution of the circular string motion in static Robertson-Walker spacetimes with closed or open sections. This is given closely and completely in terms of elliptic functions. The physical properties, string length, energy and pressure are computed and analyzed. We find the {\it back-reaction} effect of these strings on the spacetime: the self-consistent solution to the Einstein equations is a spatially closed $(K>0)$ spacetime with a selected value of the curvature index $K$ (the scale f* is normalized to unity). No self-consistent solutions with $K\leq 0$ exist. We semi-classically quantize the circular strings and find the mass $m$ in each case. For $K>0,$ the very massive strings, oscillating on the full hypersphere, have $m^2\sim K n^2\;\;(n\in N_0)$ {\it independent} of $\alpha'$ and the level spacing {\it grows} with $n,$ while the strings oscillating on one hemisphere (without crossing the equator) have $m^2\alpha'\sim n$ and a {\it finite} number of states $N\sim 1/(K\alpha').$ For $K<0,$ there are infinitely many string states with masses $m\log m\sim n,$ that is, the level spacing grows {\it slower} than $n.$ The stationary string solutions as well as the generic string fluctuations around the center of mass are also found and analyzed in closed form. 
The physical phase space of the relativistic top, as defined by Hanson and Regge, is expressed in terms of canonical coordinates of the Poincar\'e group manifold. The system is described in the Hamiltonian formalism by the mass shell condition and constraints that reduce the number of spin degrees of freedom. The constraints are second class and are modified into a set of first class constraints by adding combinations of gauge fixing functions. The Batalin-Fradkin-Vilkovisky (BFV) method is then applied to quantize the system in the path integral formalism in Hamiltonian form. It is finally shown that different gauge choices produce different equivalent forms of the constraints. 
The one--loop determinant computed around the kink solution in the 3D $\phi^4$ theory, in cylindrical geometry, allows one to obtain the partition function of the interface separating coexisting phases. The quantum fluctuations of the interface around its equilibrium position are described by a $c=1$ two--dimensional conformal field theory, namely a 2D free massless scalar field living on the interface. In this way the capillary wave model conjecture for the interface free energy in its gaussian approximation is proved. 
In this paper we consider affine Toda systems defined on the half-plane and study the issue of integrability, i.e. the construction of higher-spin conserved currents in the presence of a boundary perturbation. First at the classical level we formulate the problem within a Lax pair approach which allows to determine the general structure of the boundary perturbation compatible with integrability. Then we analyze the situation at the quantum level and compute corrections to the classical conservation laws in specific examples. We find that, except for the sinh-Gordon model, the existence of quantum conserved currents requires a finite renormalization of the boundary potential. 
  An argument is presented for the inconsistency of black hole remnants which store the information which falls into black holes. Unlike previous arguments it is not concerned with a possible divergence in the rate of pair production. It is argued that the existence of remnants in the thermal atmosphere of Rindler space will drive the renormalized Newton constant to zero. 
  Quantization of gauge theories on characteristic surfaces and in the light-cone gauge is discussed. Implementation of the Mandelstam-Leibbrandt prescription for the spurious singularity is shown to require two distinct null planes, with independent degrees of freedom initialized on each. The relation of this theory to the usual light-cone formulation of gauge field theory, using a single null plane, is described. A connection is established between this formalism and a recently given operator solution to the Schwinger model in the light-cone gauge. 
A method is developed to construct a non-local massless scalar field theory in a flat quantised space-time generated by an operator algebra. Implicit in the operator algebra is a fundamental length scale of the space-time. The fundamental two-point function of free fields is constructed by assuming that the causal Green functions still have support on the light cone in the operator algebra quantised space-time. In contrast to previous stochastic approaches, the method introduced here requires no explicit averaging over spacetime coordinates. The two- and four-point functions of~$g \varphi^4$ theory are calculated to the one-loop level, and no ultraviolet divergences are encountered. It is also demonstrated that there are no IR divergences in the processes considered. 
We begin with a review of the antiferromagnetic spin 1/2 Heisenberg chain. In particular, we show that the model has particle-like excitations with spin 1/2, and we compute the exact bulk S matrix. We then review our recent work which generalizes these results. We first consider an integrable alternating spin 1/2 - spin 1 chain. In addition to having excitations with spin 1/2, this model also has excitations with spin 0. We compute the bulk S matrix, which has some unusual features. We then consider the open antiferromagnetic spin 1/2 Heisenberg chain with boundary magnetic fields. We give a direct calculation of the boundary S matrix. (Talk presented at the conference on Statistical Mechanics and Quantum Field Theory at USC, 16 - 21 May 1994) 
  We study a supersymmetric 2-dimensional harmonic oscillator which carries a representation of the general graded Lie algebra GL(2$\vert$1), formulate it on the superspace, and discuss its physical spectrum. 
We discuss a duality of (0,2) heterotic string vacua which implies that certain pairs of (0,2) Calabi-Yau compactifications on topologically distinct target manifolds yield identical string theories. Some complex structure moduli in one model are interpreted in the dual model as deforming the holomorphic structure of the vacuum gauge bundle (and vice versa). A better understanding of singularity resolution for (0,2) models may reveal that this duality of compactifications on singular spaces is part of a larger story, involving smooth topology-changing processes which interpolate between the (0,2) models on the resolved spaces. 
A review is given of the presymplectic approach to relativistic physical systems and of the determination of their Dirac's observables. After relativistic mechanics and Nambu string, the Dirac's observables of Yang-Mills theory with fermions are given for the case of massless vector bosons (like in QED). A Dirac-Yukawa-like intrinsic ultraviolet cut-off is identified from the study of the covariantization of Hamiltonian classical field theory in the Dirac-Tomonaga-Schwinger sens. The implications fo the solution of the constraints of tetrad gravity are shown. 
We perform a generalization of the geometrical approach to describing extended objects for studying the doubly supersymmetric twistor--like formulation of super--p--branes. Some basic features of embedding world supersurface into target superspace specified by a geometrodynamical condition are considered. It is shown that the main attributes of the geometrical approach, such as the second fundamental form and extrinsic torsion of the embedded surface, and the Codazzi, Gauss and Ricci equations, have their doubly supersymmetric counterparts. At the same time the embedding of supersurface into target superspace has its particular features. For instance, the embedding may cause more rigid restrictions on the geometrical properties of the supersurface. This is demonstrated with the examples of an N=1 twistor--like supermembrane in D=11 and type II superstrings in D=10, where the geometrodynamical condition causes the embedded supersurface to be minimal and puts the theories on the mass shell. 
  The usual spinor construction from one fermion yields four irreducible representations of the Virasoro algebra with central charge $c = 1/2$. The Neveu-Schwarz (NS) sector is the direct sum of an $h = 0$ and an $h = 1/2$ module, and the Ramond (R) sector is the direct sum of two copies of an $h = 1/16$ module. In addition to the fundamental fermions, which represent a Clifford algebra, and the Virasoro operators, there are infinitely many other vertex operators, in one-to-one correspondence with the vectors (states) in the NS sector. These give the NS sector the structure of a Vertex Operator SuperAlgebra, and the R sector the structure of a ${\bold Z}_2$-twisted module for that VOSA. Keeping both copies of the $h = 1/16$ modules in the R sector, we can define intertwining operators in one-to-one correspondence with the states in the R sector such that the usual Ising fusion rules for just three modules are replaced by a rule given by the group ${\bold Z}_4$. The main objective is to find a generalization of the VOSA Jacobi-Cauchy identity which is satisfied by these intertwining operators. There are several novel features of this new ``Matrix'' Jacobi-Cauchy Identity (MJCI), most of which come from the fact that correlation functions made from two intertwiners are hypergeometric functions. In order to relate and rationalize the correlation functions we use the Kummer quadratic transformation formulas, lifting the functions to a four-sheeted covering, branched over the usual three poles, where the Cauchy residue theorem can be applied. The six possible poles on the cover give six terms in the MJCI. Furthermore, we organize those functions into $2\times 4$ matrices and find the $2\times 2$ (fusion and braiding) matrices which relate them at the six poles. These results for intertwiners 
The Hamilton--Jacobi formalism generalized to 2--dimensional field theories according to Lepage's canonical framework is applied to several covariant real scalar fields, e.g. massless and massive Klein--Gordon, Sine--Gordon, Liouville and $\phi^4$ theories. For simplicity we use the Hamilton--Jacobi equation of DeDonder and Weyl. Unlike mechanics we have to impose certain integrability conditions on the velocity fields to guarantee the transversality relations between Hamilton--Jacobi wave fronts and the corresponding families of extremals embedded therein. B\"acklund Transformations play a crucial role in solving the resulting system of coupled nonlinear PDEs. 
  In this paper we construct the exact representation of the Ising partition function in the form of the $ SL_q(2,R)$-invariant functional integral for the lattice free $(l,q)$-fermion field theory ($l=q=-1$). It is shown that the $(l,q)$-fermionization allows one to re-express the partition function of the eight-vertex model in external field through functional integral with four-fermion interaction. To construct these representations, we define a lattice $(l,q,s)$-deformed Grassmann bispinor field and extend the Berezin integration rules to this field. At $l=q=-1, s=1$ we obtain the lattice $(l,q)$-fermion field which allows us to fermionize the two-dimensional Ising model. We show that the Gaussian integral over $(q,s)$-Grassmann variables is expressed through the $(q,s)$-deformed Pfaffian which is equal to square root of the determinant of some matrix at $q=\pm 1, s=\pm 1$. 
The problem of obtaining a realistic, relativistic description of a quantum system is discussed in the context of a simple (light-cone) lattice field theory. A natural stochastic model is proposed which, although non-local, is relativistic (in the appropriate lattice sense), and which is operationally indistinguishable from the standard quantum theory. The generalization to a broad class of lattice theories is briefly described. 
  The QED(0+1) model describing a quantum mechanical particle on a circle with minimal electromagnetic interaction and with a potential -M cos(phi - theta_M), which mimics the massive Schwinger model, is discussed as a prototype of mechanisms and infrared structures of gauge quantum field theories in positive gauges. The functional integral representation displays a complex measure, with a crucial role of the boundary conditions, and the decomposition into theta sectors takes place already in finite volume. In the infinite volume limit, the standard results are reproduced for M=0 (massless fermions), but one meets substantial differences for M not = 0: for generic boundary conditions, independently of the lagrangean angle of the topological term, the infinite volume limit selects the sector with theta = theta_M, and provides a natural "dynamical" solution of the strong CP problem. In comparison with previous approaches, the strategy discussed here allows to exploit the consequences of the theta-dependence of the free energy density, with a unique minimum at theta = theta_M. 
  We propose a new method for an analytical, non-perturbative computation of effective quark interactions from QCD. It is based on an exact flow equation which describes the scale dependence of the effective average action for quarks in presence of gluons. 
An integral form of the discrete superloop equations for the supereigenvalue model of Alvarez-Gaume, Itoyama, Manes and Zadra is given. By a change of variables from coupling constants to moments we find a compact form of the planar solution for general potentials. In this framework an iterative scheme for the calculation of higher genera contributions to the free energy and the multi-loop correlators is developed. We present explicit results for genus one. 
The fourth derivative models for two dimensional gravity are shown to be equivalent to the special version of the nonlinear sigma models coupled to 2d quantum gravity. The reduction consists in the introduction of the auxiliary scalar fields and can be performed in an explicit way for both metric and general metric-dilaton cases. In view of this we can evaluate the structure of possible counterterms and show that they contains second derivative structures only. The calculations in the theory with an auxiliary fields require some special procedure to be applied. We perform the explicit calculations in a different gauges and explore the features of the auxiliary fields. 
 We study the connection between $\zeta $- and cutoff-regularized Casimir energies for scalar fields. We show that, in general, both regularization schemes lead to divergent contributions, and to finite parts which do not coincide. We determine the relationships among the various coefficients appearing in one approach and the other. As an application, we discuss the case of scalar fields in $d$-dimensional boxes under periodic boundary conditions. 
  By means of general kinematical arguments, the lifetime $\tau$ of a graviton of energy $E$ for decay into gravitons is found to have the form $\tau^{-1} = \frac{1}{EG} \sum_{j=1,2,...} c_j (\Lambda G)^j$. Some recent, preliminary results of non perturbative simplicial quantum gravity are then employed to estimate the effective values of $G$ and $\Lambda G$. It turns out that a short lifetime of the graviton cannot be excluded. 
  Operator cutoff regularization based on the original Schwinger's proper-time formalism is examined. By constructing a regulating smearing function for the proper-time integration, we show how this regularization scheme simulates the usual momentum cutoff prescription yet preserves gauge symmetry even in the presence of the cutoff scales. Similarity between the operator cutoff regularization and the method of higher (covariant) derivatives is also observed. The invariant nature of the operator cutoff regularization makes it a promising tool for exploring the renormalization group flow of gauge theories in the spirit of Wilson-Kadanoff blocking transformation. 
In this letter we develope an operator formalism for the $b-c$ systems with conformal weight $\lambda=1$ defined on a general closed and orientable Riemann surface. The advantage of our approach is that the Riemann surface is represented as an affine algebraic curve. In this way it is possible to perform explicit calculations in string theory at any perturbative order. Besides the obvious applications in string theories and conformal field theories, (the $b-c$ systems at $\lambda=1$ are intimately related to the free scalar field theory), the operator formalism presented here sheds some light also on the quantization of field theories on Riemann surfaces. In fact, we are able to construct explicitly the vacuum state of the $b-c$ systems and to define creation and annihilation operators. All the amplitudes are rigorously computed using simple normal ordering prescriptions as in the flat case. 
We discuss a two-dimensional lagrangian model with $N=2$ supersymmetry described by a K\"{a}hler potential $K(X,\bar{X})$ and superpotential $gX^k$ which explicitly exhibits renormalization group flows to infrared fixed points where the central charge has a value equal that of the $N=2$, $A_{k-1}$ minimal model. We consider the dressing of such models by N=2 supergravity: in contradistinction to bosonic or $N=1$ models, no modification of the $\b$-function takes place. 
A consistent approach to the description of integral coordinate invariant functionals of the metric on manifolds ${\cal M}_{\alpha}$ with conical defects (or singularities) of the topology $C_{\alpha}\times\Sigma$ is developed. According to the proposed prescription ${\cal M}_{\alpha}$ are considered as limits of the converging sequences of smooth spaces. This enables one to give a strict mathematical meaning to a number of invariant integral quantities on ${\cal M}_{\alpha}$ and make use of them in applications. In particular, an explicit representation for the Euler numbers and Hirtzebruch signature in the presence of conical singularities is found. Also, higher dimensional Lovelock gravity on ${\cal M}_{\alpha}$ is shown to be well-defined and the gravitational action in this theory is evaluated. Other series of applications is related to computation of black hole entropy in the higher derivative gravity and in quantum 2-dimensional models. This is based on its direct statistical-mechanical derivation in the Gibbons-Hawking approach, generalized to the singular manifolds ${\cal M}_{\alpha}$, and gives the same results as in the other methods. 
  We represent a classical Maxwell-Bloch equation and related to it positive part of the AKNS hierarchy in geometrical terms. The Maxwell-Bloch evolution is given by an infinitesimal action of a nilpotent subalgebra $n_+$ of affine Lie algebra $\hat {sl}_2$ on a Maxwell-Bloch phase space treated as a homogeneous space of $n_+$. A space of local integrals of motion is described using cohomology methods. We show that hamiltonian flows associated to the Maxwell-Bloch local integrals of motion (i.e. positive AKNS flows) are identified with an infinitesimal action of an abelian subalgebra of the nilpotent subalgebra $n_+$ on a Maxwell- Bloch phase space. Possibilities of quantization and latticization of Maxwell-Bloch equation are discussed. 
The time evolution in a supersymmetric extension of the Kodomtsev-Petviashvilli hierarchy, a classical integrable system, is shown to be Hamiltonian. The canonical bracket associated to the Hamiltonian evolution is the classical analog of the antibracket encountered in the quantization of gauge theories. This provides a new understanding of supersymmetric Hamiltonian systems. 
We study symmetry breaking in the static coordinate-system of de Sitter space. This is done with the help of the functional-Schr\"odinger approach used in previous calculations by Ratra [1]. We consider a massless, minimally coupled scalar field as the parameter of a continuous symmetry (the angular component of an O(2) symmetry). Then we study the correlation function of the massless scalar field, to derive the correlation function of the original field, which finally shows the restoration of the continuous symmetry. 
  It has recently been realized that a large class of Calabi-Yau models in which the VEV of the gauge connection is not set equal to the spin connection of the Calabi-Yau manifold are valid classical solutions of string theory. We provide some examples of three generation models based on such generalized Calabi-Yau compactifications, including models with observable gauge group $SU(3)\times SU(2)\times U(1)$. 
  We argue for the presence of a ${\bf Z}_2$ topological structure in the space of static gauge-Higgs field configurations of $SU(2n)$ and $SO(2n)$ Yang-Mills theories. We rigorously prove the existence of a ${\bf Z}_2$ homotopy group of mappings from the 2-dim. projective sphere ${\bf R}P^2$ into $SU(2n)/{\bf Z}_2$ and $SO(2n)/{\bf Z}_2$ Lie groups respectively. Consequently the symmetric phase of these theories admits infinite surfaces of odd-parity static and unstable gauge field configurations which divide into two disconnected sectors with integer Chern-Simons numbers $n$ and $n+1/2$ respectively. Such a ${\bf Z}_2$ structure persists in the Higgs phase of the above theories and accounts for the existence of $CS=1/2$ odd-parity saddle point solutions to the field equations which correspond to spontaneous symmetry breaking mass scales. 
   The behavior of small perturbations around the Kaluza-Klein monopole in the five dimensional space-time is investigated. The fact that the odd parity gravitational wave does not propagate in the five dimensional space-time with Kaluza-Klein monopole is found provided that the gravitational wave is constant in the fifth direction. 
The problem of computing the one-dimensional configuration sums of the ABF model in regime III is mapped onto the problem of evaluating the grand-canonical partition function of a gas of charged particles obeying certain fermionic exclusion rules. We thus obtain a new {\em fermionic} method to compute the local height probabilities of the model. Combined with the original {\em bosonic} approach of Andrews, Baxter and Forrester, we obtain a new proof of (some of) Melzer's polynomial identities. In the infinite limit these identities yield Rogers--Ramanujan type identities for the Virasoro characters $\chi_{1,1}^{(r-1,r)}(q)$ as conjectured by the Stony Brook group. As a result of our working the corner transfer matrix and thermodynamic Bethe Ansatz approaches to solvable lattice models are unified. 
We construct a string field Hamiltonian describing the dynamics of open and closed strings with effective target-space dimension $c\le 1 $. In order to do so, we first derive the Dyson-Schwinger equations for the underlying large $N$ vector+matrix model and formulate them as a set of constraints satisfying decoupled Virasoro and U(1) current algebras. The Hamiltonian consists of a bulk and a boundary term having different scaling dimensions. The time parameters corresponding to the two terms are interpreted from the the point of view of the fractal geometry of the world surface. 
  Using Hollowood's conjecture for the S-matrix for elementary solitons in complex $a_n^{(1)}$ affine Toda field theories we examine the interactions of bound states of solitons in $a_2^{(1)}$ theory. The elementary solitons can form two different kinds of bound states: scalar bound states (the so-called breathers), and excited solitons, which are bound states with non-zero topological charge. We give explicit expressions of all S-matrix elements involving the scattering of breathers and excited solitons and examine their pole structure in detail. It is shown how the poles can be explained in terms of on-shell diagrams, several of which involve a generalized Coleman-Thun mechanism. 
  Scalar field theory at finite temperature is investigated via an improved renormalization group prescription which provides an effective resummation over all possible non-overlapping higher loop graphs. Explicit analyses for the lambda phi^4 theory are performed in d=4 Euclidean space for both low and high temperature limits. We generate a set of coupled equations for the mass parameter and the coupling constant from the renormalization group flow equation. Dimensional reduction and symmetry restoration are also explored with our improved approach. 
We consider the recently obtained integral representation of quantum Knizhnik-Zamolodchikov equation of level 0. We obtain the condition for the integral kernel such that these solutions satisfy three axioms for form factor \'{a} la Smirnov. We discuss the relation between this integral representation and the form factor of XXZ spin chain. 
We suggest a method of bosonizing any D=2 theory. We demonstrate how it works with the examples of the Thirring and the Schwinger models, known results are reproduced. This method, being applied to the Gross-Neveu model, yields nonlinear boson WZW-type theory with additional constraint in the field space. Relation to the nonlinear sigma - model is also discussed. 
We generalize the standard $N=2$ supersymmetric Kazama-Suzuki coset construction to the $N=4$ case by requiring the {\it non-linear} (Goddard-Schwimmer) $N=4~$ quasi-superconformal algebra to be realized on cosets. The constraints that we find allow very simple geometrical interpretation and have the Wolf spaces as their natural solutions. Our results obtained by using components-level superconformal field theory methods are fully consistent with standard results about $N=4$ supersymmetric two-dimensional non-linear sigma-models and $N=4$ WZNW models on Wolf spaces. We construct the actions for the latter and express the quaternionic structure, appearing in the $N=4$ coset solution, in terms of the symplectic structure associated with the underlying Freudenthal triple system. Next, we gauge the $N=4~$ QSCA and build a quantum BRST charge for the $N=4$ string propagating on a Wolf space. Surprisingly, the BRST charge nilpotency conditions rule out the non-trivial Wolf spaces as consistent string backgrounds. 
It is shown that several Hamiltonian systems possessing dynamical or hidden symmetries can be realized within the framework of Nambu's generalized mechanics. Among such systems are the SU(n)-isotropic harmonic oscillator and the SO(4)-Kepler problem. As required by the formulation of Nambu dynamics, the integrals of motion for these systems necessarily become the so-called generalized Hamiltonians. Furthermore, in most of these problems, the definition of these generalized Hamiltonians is not unique. 
By a suitable choice of variables we show that every Connes-Lott model is a Yang-Mills-Higgs model. The contrary is far from being true. Necessary conditions are given. Our analysis is pedestrian and illustrated by examples. 
It it known that to get the usual Hamiltonian formulation of lattice Yang-Mills theory in the temporal gauge $A_{0}=0$ one should place on every link the cotangent bundle of a Lie group. The cotangent bundle may be considered as a limiting case of a so called Heisenberg double of a Lie group which is one of the basic objects in the theory of Lie-Poisson and quantum groups. It is shown in the paper that there is a generalization of the usual Hamiltonian formulation to the case of the Heisenberg double. 
  This is a review of irrational conformal field theory, which includes rational conformal field theory as a small subspace. Central topics of the review include the Virasoro master equation, its solutions and the dynamics of irrational conformal field theory. Discussion of the dynamics includes the generalized Knizhnik-Zamolodchikov equations on the sphere, the corresponding heat-like systems on the torus and the generic world- sheet action of irrational conformal field theory. 
  Analysis of the WKB exactness in some homogeneous spaces is attempted. $CP^N$ as well as its noncompact counterpart $D_{N,1}$ is studied. $U(N+1)$ or U(N,1) based on the Schwinger bosons leads us to $CP^N$ or $D_{N,1}$ path integral expression for the quantity, ${\rm tr} e^{-iHT}$, with the aid of coherent states. The WKB approximation terminates in the leading order and yields the exact result provided that the Hamiltonian is given by a bilinear form of the creation and the annihilation operators. An argument on the WKB exactness to more general cases is also made. 
The ten dimensional heterotic string effective action with graviton, dilaton and antisymmetric tensor fields is dimensionally reduced to two spacetime dimensions. The resulting theory, with some constraints on backgrounds, admits infinite sequence of conserved nonlocal currents. It is shown that generators of the infinitesimal transformations associated with these currents satisfy Kac-Moody algebra. 
A two dimensional string effective action is obtained by dimensionally reducing the bosonic part of the ten dimensional heterotic string effective action. It is shown that this effective action, with a few restrictions on some backgrounds describes a two dimensional model which admits an infinite sequence of nonlocal conserved currents. 
  We develop various complementary concepts and techniques for handling quantum fluctuations of Goldstone bosons.We emphasise that one of the consequences of the masslessness of Goldstone bosons is that the longitudinal fluctuations also have a diverging susceptibility characterised by an anomalous dimension $(d-2)$ in space-time dimensions $2<d<4$.In $d=4$ these fluctuations diverge logarithmically in the infrared region.We show the generality of this phenomenon by providing three arguments based on i). Renormalization group flows, ii). Ward identities, and iii). Schwinger-Dyson equations.We obtain an explicit form for the generating functional of one-particle irreducible vertices of the O(N) (non)--linear $\sigma$--models in the leading 1/N approximation.We show that this incorporates all infrared behaviour correctly both in linear and non-linear $\sigma$-- models. Our techniques provide an alternative to chiral perturbation theory.Some consequences are discussed briefly. 
A simple integral representation is derived for the quasiclassical Green function of the Dirac equation in an arbitrary spherically-symmetric decreasing external field. The consideration is based on the use of the quasiclassical radial wave functions with the main contribution of large angular momenta taken into account. The Green function obtained is applied to the calculation of the Delbruck scattering amplitudes in a screened Coulomb field. 
The Hamilton-Jacobi formalism generalized to 2-dimensional field theories according to Lepage's canonical framework is applied to several relativistic real scalar fields, e.g. massless and massive Klein-Gordon, Sinh and Sine-Gordon, Liouville and $\phi^4$ theories. The relations between the Euler-Lagrange and the Hamilton-Jacobi equations are discussed in DeDonder and Weyl's and the corresponding wave fronts are calculated in Carath\'eodory's formulation. Unlike mechanics we have to impose certain integrability conditions on the velocity fields to guarantee the transversality relations and especially the dynamical equivalence between Hamilton-Jacobi wave fronts and families of extremals embedded therein. B\"acklund Transformations play a crucial role in solving the resulting system of coupled nonlinear PDEs. 
ABSTRACT: The first quantum corrections to the free energy for an eternal 4-dimensional black hole is investigated at one-loop level, in the large mass limit of the black hole, making use of the conformal techniques related to the optical metric. The quadratic and logarithmic divergences as well as a finite part associated with the first quantum correction to the entropy are obtained at a generic temperature. It is argued that, at the Hawking temperature, the horizon divergences of the internal energy should cancel. Some comments on the divergences of the entropy are also presented. 
The dynamics of the early universe may have been profoundly influenced by spatial anisotropies. A search for such backgrounds in the context of string cosmology has uncovered the existence of an entire class of (spatatially) homogeneous but not necessarily isotropic space-times, analogous to the class of Bianchi-types in general relativity. Configurations with vanishing cosmological constant but non-vanishing dilaton and antisymmetric field are explicitly found for all types. This is a new class of solutions, whose isotropy limits reproduce all known and, further, all possible FRW-type of models in the string-cosmology context considered. There is always an initial singularity and no inflation. Other features of the general solutions, including their behaviour under abelian duality are are also discussed. 
Abstract: We show that a closed Nielsen-Olesen string in presence of a point scalar source exhibits the phenomenon of Fermi-Bose transmutation. This provides physical support to previous claims about transmutation between bosonic and fermionic one-dimensional structures in (3+1) dimensions. In order to render the computations mathematically rigorous we have resorted to an Euclidean lattice regularization. 
  We construct an octonionic instanton solution to the seven dimensional Yang-Mills theory based on the exceptional gauge group $G_2$ which is the automorphism group of the division algebra of octonions. This octonionic instanton has an extension to a solitonic two-brane solution of the low energy effective theory of the heterotic string that preserves two of the sixteen supersymmetries and hence corresponds to $N=1$ space-time supersymmetry in the (2+1) dimensions transverse to the seven dimensions where the Yang-Mills instanton is defined. 
 In Batalin-Vilkovisky formalism a classical mechanical system is specified by means of a solution to the {\sl classical master equation}. Geometrically such a solution can be considered as a $QP$-manifold, i.e. a super\m equipped with an odd vector field $Q$ obeying $\{Q,Q\}=0$ and with $Q$-invariant odd symplectic structure. We study geometry of $QP$-manifolds. In particular, we describe some construction of $QP$-manifolds and prove a classification theorem (under certain conditions).  We apply these geometric constructions to obtain in natural way the action functionals of two-dimensional topological sigma-models and to show that the Chern-Simons theory in BV-formalism arises as a sigma-model with target space $\Pi {\cal G}$. (Here ${\cal G}$ stands for a Lie algebra and $\Pi$ denotes parity inversion.) 
We present a complete study of boundary bound states and related boundary S-matrices for the sine-Gordon model with Dirichlet boundary conditions. Our approach is based partly on the bootstrap procedure, and partly on the explicit solution of the inhomogeneous XXZ model with boundary magnetic field and of the boundary Thirring model. We identify boundary bound states with new ``boundary strings'' in the Bethe ansatz. The boundary energy is also computed. 
  In these lecture notes, I review the ``linear sigma model" approach to (0,2) string vacua. My aim is to provide the reader with a toolkit for studying a very broad class of (0,2) superconformal field theories with the requisite properties to be candidate string vacua. These lectures were delivered at the 1994 Trieste Summer School. 
Recently Seiberg has conjectured a duality symmetry connecting different theories of the supersymmetric QCD type. We provide support for this conjecture by analyzing a flat direction of the theory along which the two dual theories go over to the same theory in the IR. 
We propose a matrix-model derivation of the scaling exponents of the critical and tricritical q-states Potts model coupled to gravity on a sphere. In close analogy with the $O(n)$ model, we reduce the determination of the one-loop-to-vacuum expectation to the resolution of algebraic equations; and find the explicit scaling law for the case q=3. 
Functional determinants for the scalar Laplacian on spherical caps and slices, flat balls, shells and generalised cylinders are evaluated in two, three and four dimensions using conformal techniques. Both Dirichlet and Robin boundary conditions are allowed for. Some effects of non-smooth boundaries are discussed; in particular the 3-hemiball and the 3-hemishell are considered. The edge and vertex contributions to the $C_{3/2}$ coefficient are examined. 
  Representations of the Klein-Gordon and Dirac propagators are determined in a $N$ dimensional conical background for massive fields twisted by an arbitrary angle $2\pi\sigma$. The Dirac propagator is shown to be obtained from the Klein-Gordon propagator twisted by angles $2\pi\sigma\pm {\cal D}/2$ where ${\cal D}$ is the cone deficit angle. Vacuum expectation values are determined by a point-splitting method in the proper time representation of the propagators. Analogies with the Aharonov-Bohm effect are pointed out throughout the paper and a conjecture on an extension to fields of arbitrary spin is given. 
Abstract: A modification of Kaluza-Klein theory is proposed which is general enough to admit an arbitrary finite noncommutative internal geometry. It is shown that the existence of a non-trival extension to the total geometry of a linear connection on space-time places severe restrictions on the structure of the noncommutative factor. A counter-example is given. 
The well-known idea to construct domain wall type solutions of field equations by means of an expansion in the width of the domain wall is reexamined. We observe that the problem involves singular perturbations. Hilbert-Chapman-Enskog method is used to construct a consistent perturbative expansion. We obtain the solutions to the second order in the width without introducing an effective action for the domain wall. We find that zeros of the scalar field in general do not lie on a Nambu-Goto trajectory. As examples we consider cylindrical and spherical domain walls. We find that the spherical domain wall, in contradistinction to the cylindrical one, shows an effective rigidity. 
We present a simple technique that allows to generate Feynman diagrams for vector models with interactions of order $2n$ and similar models (Gross-Neveu, Thirring model), using a bootstrap equation that uses only the free field value of the energy as an input. The method allows to find the diagrams to, in principle, arbitrarily high order and applies to both energy and correlation functions. It automatically generates the correct symmetry factor (as a function of the number of components of the field) and the correct sign for any diagram in the case of fermion loops. We briefly discuss the possibility of treating QED as a Thirring model with non-local interaction. 
  Triviality and Landau poles are often greeted as harbingers of new physics at 1 TeV. After briefly reviewing the ideas behind this, a model of singular quantum mechanics is introduced. Its ultraviolet structure, as well as some features of its vacuum, related to triviality, very much parallel $\lambda\phi^4$. The model is solvable, exactly and perturbatively, in any dimension. From its analysis we learn that Landau poles do not appear in any exactly computed observable, but only in truncated perturbation theory, when perturbation theory is performed with the wrong sign coupling. If these findings apply to the standard model no new physics at 1 TeV should be expected but only challenges for theorists. 
In this paper, we pursue our analysis of the W-infinity symmetry of the low-energy edge excitations of incompressible quantum Hall fluids. These excitations are described by (1+1)-dimensional effective field theories, which are built by representations of the W-infinity algebra. Generic W-infinity theories predict many more fluids than the few, stable ones found in experiments. Here we identify a particular class of W-infinity theories, the minimal models, which are made of degenerate representations only - a familiar construction in conformal field theory. The W-infinity minimal models exist for specific values of the fractional conductivity, which nicely fit the experimental data and match the results of the Jain hierarchy of quantum Hall fluids. We thus obtain a new hierarchical construction, which is based uniquely on the concept of quantum incompressible fluid and is independent of Jain's approach and hypotheses. Furthermore, a surprising non-Abelian structure is found in the W-infinity minimal models: they possess neutral quark-like excitations with SU(m) quantum numbers and non-Abelian fractional statistics. The physical electron is made of anyon and quark excitations. We discuss some properties of these neutral excitations which could be seen in experiments and numerical simulations. 
We study the soliton-type solutions of the system introduced by B. Feigin and the author in in [EF]. We show that it reduces to a top-like system, and we study the behaviour of the solutions at the lattice infinity. We compute the scattering of the solitons and study some periodic solutions of the system. 
Abstract: We discuss the so called gauge invariant quantization of anomalous gauge field theory, originally due to Faddeev and Shatashvili. It is pointed out that the further non invariance of relevant path integral measures poses a problem when one tries to translate it to BRST formalism. The method by which we propose to get around of this problem introduces certain arbitrariness in the model. We speculate on the possibility of using such an arbitrariness to build series of non equivalent models of two dimensional induced gravity. 
  In the presence of Chern-Simons interactions the wave functionals of physical states in 2+1-dimensional gauge theories vanish at anumber of nodal points. We show that those nodes are located at some classical configurations which carry a non-trivial magnetic charge. In abelian gauge theories this fact explains why magnetic monopoles are suppressed by Chern-Simons interactions. In non-abelian theories it suggests a relevant role for nodal gauge field configurations in the confinement mechanism of Yang-Mills theories. We show that the vacuum nodes correspond to the chiral gauge orbits of reducible gauge fields with non-trivial magnetic monopole components. 
  We show that regularization of gauge theories by higher covariant derivatives and gauge invariant Pauli-Villars regulators is a consistent method if the Pauli-Villars vector fields are considered in a covariant in the regulating Pauli-Villars fields is pathological and the original Slavnov proposal in covariant Landau gauge is not correct because of the appearance of massless modes in the regulators which do not decouple when the ultraviolet regulator is removed. In such a case the method does not correspond to the regularization of a pure gauge theory but that of a gauge theory in interaction with massless ghost fields. This explains the problems pointed out by Martin and Ruiz in covariant Landau gauge. However, a minor modification of Slavnov method provides a consistent regularization even for such a case. The regularization that we introduce also solves the problem of overlapping divergences in a way similar to geometric regularization and yields the standard values of the $\beta$ and $\gamma$ functions of the renormalization group equations. 
We discuss a model of a conformal p-brane interacting with the world volume metric and connection. The purpose of the model is to suggest a mechanism by which gravity coupled to p-branes leads to the formation of structure rather than homogeneity in spacetime. Furthermore, we show that the formation of structure is accompanied by the appearance of a multivalued cosmological constant, i.e., one which may take on different values in different domains, or cells, of spacetime. The above results apply to a broad class of non linear gravitational lagrangians as long as metric and connection on the p-brane manifold are treated as independent variables. 
We study the Wess-Zumino theory on ${\bf R}^3 \times S^1$ where a spatial coordinate is compactified. We show that when the bosonic and fermionic fields satisfy the same boundary condition, the theory does not develop a vacuum energy or tadpoles. We work out the two point functions at one loop and show that their forms are consistent with the nonrenormalization theorem. However, the two point functions are nonanalytic and we discuss the structure of this nonanalyticity. 
Resubmitted as hep-ph/9502316. Removed from hep-th. Incorrigibly inept submitter publicly excoriated. 
We show that there is a series of topological string theories whose integrable structure is described by the Toda lattice hierarchy. The monodromy group of the Frobenius manifold for the matter sector is an extension of the affine Weyl group $\widetilde W (A_N^{(1)})$ introduced by Dubrovin. These models are generalizations of the topological $CP^1$ string theory with scaling violation. The logarithmic Hamiltonians generate flows for the puncture operator and its descendants. We derive the string equation from the constraints on the Lax and the Orlov operators. The constraints are of different type from those for the $c=1$ string theory. Higher genus expansion is obtained by considering the Lax operator in matrix form. 
The Sp(2)-symmetric Lagrangian quantization scheme is represented in a completely anticanonical form. Antifields are assigned to all field variables including former "parametric" ones \pi^{Aa}. The antibrackets (F, G)^a as well as the operators \triangle^a and V^a are extended to include the new anticanonical pairs \pi^{Aa}, \bar{\phi}_A. A new version of the gauge fixing mechanism in the Lagrangian effective action is proposed. The corresponding functional integral is shown to be gauge independent. 
A geometric description is given for the Sp(2) covariant version of the field-antifield quantization of general constrained systems in the Lagrangian formalism. We develop differential geometry on manifolds in which a basic set of coordinates (`fields') have two superpartners (`antifields'). The quantization on such a triplectic manifold requires introducing several specific differential-geometric objects, whose properties we study. These objects are then used to impose a set of generalized master-equations that ensure gauge-independence of the path integral. The theory thus quantized is shown to extend to a level-1 theory formulated on a manifold that includes antifields to the Lagrange multipliers. We also observe intriguing relations between triplectic and ordinary symplectic geometry. 
We investigate the non-perturbative structure of two planar $Z_p \times Z_p$ lattice gauge models and discuss their relevance to two-dimensional condensed matter systems and Josephson junction arrays. Both models involve two compact $U(1)$ gauge fields with Chern-Simons interactions, which break the symmetry down to $Z_p \times Z_p$. By identifying the relevant topological excitations (instantons) and their interactions we determine the phase structure of the models. Our results match observed quantum phase transitions in Josephson junction arrays and suggest also the possibility of {\it oblique confining ground states} corresponding to quantum Hall regimes for either charges or vortices. 
Vector supersymmetry is typical of topological field theory. Its role in the construction of gauge invariant quantities is explained, as well as its role in the cancellation of the ultraviolet divergences. The example of the Chern-Simons theory in three dimensions is treated in details. 
We investigate quantum chromodynamics in 2+1 dimensions ($\rm{QCD}_3$) using the Hamiltonian lattice field theory approach. The long wavelength structure of the ground state, which is closely related to the confinement phenomenon, is analyzed and its vacuum wave function is evaluated by means of the recently developed truncated eigenvalue equation method. The third order estimations show nice scaling for the physical quantities. 
Abstract: We present a unified framework for the quantization of a family of discrete dynamical systems of varying degrees of ``chaoticity". The systems to be quantized are piecewise affine maps on the two-torus, viewed as phase space, and include the automorphisms, translations and skew translations. We then treat some discontinuous transformations such as the Baker map and the sawtooth-like maps. Our approach extends some ideas from geometric quantization and it is both conceptually and calculationally simple. 
We consider a model of p independent Ising spins on a dynamical planar phi-cubed graph. Truncating the free energy to two terms yields an exactly solvable model that has a third order phase transition from a pure gravity region (gamma=-1/2) to a tree-like region (gamma=1/2), with gamma=1/3 on the critical line. We are able to make an order of magnitude estimate of the value of p above which there exists a branched polymer (ie tree-like) phase in the full model, that is, p is approximately 13-23, which corresponds to a central charge c of about 6-12. 
There is some experimental evidence that some stars are older than the Universe in General Relativity based cosmology. In TGD based cosmology the paradox has explanation. Photons can be either topologically condensed on background spacetime surface or in 'vapour phase' that is progate in $M^4_+\times CP_2$ as small surfaces. The time for propagation from A to B is in general larger in condensate than in vapour phase. In principle observer detects from a given astrophysical object both vapour phase and condensate photons, vapour phase photons being emitted later than condensate photons. Therefore the erraneous identification of vapour phase photons as condensate photons leads to an over estimate for the age of the star and star can look older than the Universe. The Hubble constant for vapour phase photons is that associated with $M^4_+$ and smaller than the Hubble constant of matter dominated cosmology. This could explain the measured two widely different values of Hubble constant if smaller Hubble constant corresponds to the Hubble constant of the future light cone $M^4_+$. The ratio of propagation velocities of vapour phase and condensate photons equals to ratio of the two Hubble constants, which in turn is depends on the ratio of mass density and critical mass density, only. Anomalously large redshifts are possible since vapour phase photons can come also from region outside the horizon. 
  We consider a new 3-parameter class of exact 4-dimensional solutions in closed string theory and solve the corresponding string model, determining the physical spectrum and the partition function. The background fields (4-metric, antisymmetric tensor, two Kaluza-Klein vector fields, dilaton and modulus) generically describe axially symmetric stationary rotating (electro)magnetic flux-tube type universes. Backgrounds of this class include both the dilatonic (a=1) and Kaluza-Klein (a=\sqrt 3) Melvin solutions and the uniform magnetic field solution, as well as some singular space-times. Solvability of the string sigma model is related to its connection via duality to a simpler model which is a ``twisted" product of a flat 2-space and a space dual to 2-plane. We discuss some physical properties of this model (tachyonic instabilities in the spectrum, gyromagnetic ratios, issue of singularities, etc.). It provides one of the first examples of a consistent solvable conformal string model with explicit D=4 curved space-time interpretation. 
We find all the diagonal $K$-matrices for the $R$-matrix associated with the minimal representation of the exceptional affine algebra $G^{(1)}_2$. The corresponding transfer matrices are diagonalized with a variation of the analytic Bethe ansatz. We find many similarities with the case of the Izergin-Korepin $R$-matrix associated with the affine algebra $A^{(2)}_2$. 
We consider the six-vertex model with anti-periodic boundary conditions across a finite strip. The row-to-row transfer matrix is diagonalised by the `commuting transfer matrices' method. {}From the exact solution we obtain an independent derivation of the interfacial tension of the six-vertex model in the anti-ferroelectric phase. The nature of the corresponding integrable boundary condition on the $XXZ$ spin chain is also discussed. 
We study integrable vertex models and quantum spin chains with toroidal boundary conditions. An interesting class of such boundaries is associated with non-diagonal twist matrices. For such models there are no trivial reference states upon which a Bethe ansatz calculation can be constructed, in contrast to the well-known case of periodic boundary conditions. In this paper we show how the transfer matrix eigenvalue expression for the spin-$s$ XXZ chain twisted by the charge-conjugation matrix can in fact be obtained. The technique used is the generalization to spin-$s$ of the functional relation method based on ``pair-propagation through a vertex''. The Bethe ansatz-type equations obtained reduce, in the case of lattice size $N=1$, to those recently found for the Hofstadter problem of Bloch electrons on a square lattice in a magnetic field. 
The study of hidden symmetries within Dirac's formalism does not possess a systematic procedure due to the lack of first-class constraints to act as symmetry generators. On the other hand, in the Faddeev-Jackiw approach, gauge and reparametrization symmetries are generated by the null eigenvectors of the sympletic matrix and not by constraints, suggesting the possibility of dealing systematically with hidden symmetries throughout this formalism. It is shown in this paper that indeed hidden symmetries of noninvariant or gauge fixed systems are equally well described by null eigenvectors of the sympletic matrix, just as the explicit invariances. The Faddeev-Jackiw approach therefore provides a systematic algorithm for treating all sorts of symmetries in an unified way. This technique is illustrated here by the SL(2,R) affine Lie algebra of the 2-D induced gravity proposed by Polyakov, which is a hidden symmetry in the canonical approach of constrained systems via Dirac's method, after conformal and reparametrization invariances have been fixed. 
Spontaneous breakdown of the continuous symmetry is studied in the framework of discretized light-front quantization. We consider linear sigma model in 3+1 dimensions and show that the careful treatment of zero modes together with the regularization of the theory by introducing NG boson mass leads to the correct description of Nambu-Goldstone phase on the light-front. 
We derive the exact action for a damped mechanical system ( and the special case of the linear oscillator) from the path integral formulation of the quantum Brownian motion problem developed by Schwinger and by Feynman and Vernon. The doubling of the phase-space degrees of freedom for dissipative systems and thermal field theories is discussed and the initial values of the doubled variables are related to quantum noise effects. 
It is proven that there are precisely $n$ odd-parity sphaleron-like unstable modes of the $n$-th Bartnik-McKinnon soliton and the $n$-th non-abelian black hole solution of the Einstein-Yang-Mills theory for the gauge group $SU(2)$. 
An explicit orbifold example of the non-zero correlation functions related to the additional contribution to the induced mass term for Higgs particles at low energies is given. We verify that they form finite dimensional representations of the target space modular transformation $SL_2(Z)$. This action of the modular group is shown to be consistent with its action on the fixed points set defining the twisted fields. 
The nonabelian generalization of a recently proposed abelian axial gauge model for tensor matter fields is obtained. In both cases the model can be derived from a $\vf^{4}-$type theory for antisymmetric fields obeying a complex self-dual condition. 
In the continuum O(3) sigma model in two spatial dimensions, there are topological solitons whose size can be stabilized by adding Skyrme and potential terms. This paper describes a lattice version, namely a natural way of modifying the 2d Heisenberg model to achieve topological stability on the lattice. 
We find exact solutions of the string equations of motion and constraints describing the {\em classical}\ splitting of a string into two. We show that for the same Cauchy data, the strings that split have {\bf smaller} action than the string without splitting. This phenomenon is already present in flat space-time. The mass, energy and momentum carried out by the strings are computed. We show that the splitting solution describes a natural decay process of one string of mass $M$ into two strings with a smaller total mass and some kinetic energy. The standard non-splitting solution is contained as a particular case. We also describe the splitting of a closed string in the background of a singular gravitational plane wave, and show how the presence of the strong gravitational field increases (and amplifies by an overall factor) the negative difference between the action of the splitting and non-splitting solutions. 
We review our recent work on the algebraic characterization of quantum Hall fluids. Specifically, we explain how the incompressible quantum fluid ground states can be classified by effective edge field theories with the W-infinity dynamical symmetry of ``quantum area-preserving diffeomorphisms''. Using the representation theory of W-infinity, we show how all fluids with filling factors $\nu = m/(pm +1)$ and $\nu = m/(pm-1)$ with $m$ and $p$ positive integers, $p$ even, correspond exactly to the W-infinity {\it minimal models}. 
A class of quantum chains possessing a family of local conserved charges with a Catalan tree pattern is studied. Recently, we have identified such a structure in the integrable $SU(N)$-invariant chains. In the present work we find sufficient conditions for the existence of a family of charges with this structure in terms of the underlying algebra. Two additional systems with a Catalan tree structure of conserved charges are found. One is the spin 1/2 XXZ model with $\Delta=-1$. The other is a new octonionic isotropic chain, generalizing the Heisenberg model. This system provides an interesting example of an infinite family of noncommuting local conserved quantities. 
These days, as high energy particle colliders become unavailable for testing speculative theoretical ideas, physicists are looking to other environments that may provide extreme conditions where theory confronts physical reality. One such circumstance may arise at high temperature $T$, which perhaps can be attained in heavy ion collisions or in astrophysical settings. It is natural therefore to examine the high-temperature behavior of the standard model, and here I shall report on recent progress in constructing the high-$T$ limit of QCD. 
  The fused six-vertex models with open boundary conditions are studied. The Bethe ansatz solution given by Sklyanin has been generalized to the transfer matrices of the fused models. We have shown that the eigenvalues of transfer matrices satisfy a group of functional relations, which are the $su$(2) fusion rule held by the transfer matrices of the fused models. The fused transfer matrices form a commuting family and also commute with the quantum group $U_q[sl(2)]$. In the case of the parameter $q^h=-1$ ($h=4,5,\cdots$) the functional relations in the limit of spectral parameter $u\to \i\infty$ are truncated. This shows that the $su$(2) fusion rule with finite level appears for the six vertex model with the open boundary conditions. We have solved the functional relations to obtain the finite-size corrections of the fused transfer matrices for low-lying excitations. From the corrections the central charges and conformal weights of underlying conformal field theory are extracted. To see different boundary conditions we also have studied the six-vertex model with a twisted boundary condition. 
  The properties of static spherically symmetric black holes, which are both electrically and magnetically charged, and which are coupled to the dilaton in the presence of a cosmological constant, Lambda, are considered. It is shown that apart from the Reissner-Nordstrom-de Sitter solution with constant dilaton, such solutions do not exist if Lambda > 0 (in arbitrary spacetime dimension >=4 ). However, asymptotically anti-de Sitter dyonic black hole solutions with a non-trivial dilaton do exist if Lambda < 0. Both these solutions and the asymptotically flat (Lambda = 0) solutions are studied numerically for arbitrary values of the dilaton coupling parameter, g_0, in four dimensions. The asymptotically flat solutions are found to exhibit two horizons if g_0 = 0, 1, \sqrt{3}, \sqrt{6}, ..., \sqrt{n(n+1)/2},..., and one horizon otherwise. For asymptotically anti-de Sitter solutions the result is similar, but the corresponding values of g_0 are altered in a non-linear fashion which depends on Lambda and the mass and charges of the black holes. All dyonic solutions with Lambda <= 0 are found to have zero Hawking temperature in the extreme limit, however, regardless of the value of g_0. 
Periodic classical trajectories are of fundamental importance both in classical and quantum physics. Here we develop path integral techniques to investigate such trajectories in an arbitrary, not necessarily energy conserving hamiltonian system. In particular, we present a simple derivation of a lower bound for the number of periodic classical trajectories. 
 We show that a complete covariantization of the chiral constraint in the Floreanini-Jackiw necessitates an infinite number of auxiliary Wess-Zumino fields otherwise the covariantization is only partial and unable to remove the nonlocality in the chiral boson operator. We comment on recent works that claim to obtain covariantization through the use of Batalin-Fradkin-Tyutin method, that uses just one Wess-Zumino field. 
't Hooft construction of free energy, electric and magnetic fluxes, and of the partition function with twisted boundary conditions, is extended to the case of $N=4$ supersymmetric Yang-Mills theories based on arbitrary compact, simple Lie groups. The transformation of the fluxes and the free energy under S-duality is presented. We consider the partition function of $N=4$ for a particular choice of boundary conditions, and compute exactly its leading infrared divergence. We verify that this partition function obeys the transformation laws required by S-duality. This provides independent evidence in favor of S-duality in $N=4$ theories. 
The topological description of $2D$ string theory at the self-dual radius is studied in the algebro-geometrical formulation of the $A_{k+1}$ topological models at $k=-3$. Genus zero correlators of tachyons and their gravitational descendants are computed as intersection numbers on moduli space and compared to $2D$ string results. The interpretation of negative momentum tachyons as gravitational descendants of the cosmological constant, as well as modifications of this, is shown to imply a disagreement between $2D$ string correlators and the associated intersection numbers. 
In this paper we investigate the canonical quantization of a non-Abelian topologically massive Chern-Simons theory in which the gauge fields are minimally coupled to a multiplet of scalar fields in such a way that the gauge symmetry is spontaneously broken. Such a model produces the Chern-Simons-Higgs mechanism in which the gauge excitations acquire mass both from the Chern-Simons term and from the Higgs-Kibble effect. The symmetry breaking is chosen to be only partially broken, in such a way that in the broken vacuum there remains a residual non-Abelian symmetry. We develop the canonical operator structure of this theory in the broken vacuum, with particular emphasis on the particle-content of the fields involved in the Chern-Simons-Higgs mechanism. We construct the Fock space and express the dynamical generators in terms of creation and annihilation operator modes. The canonical apparatus is used to obtain the propagators for this theory, and we use the Poincar\'e generators to demonstrate the effect of Lorentz boosts on the particle states. 
Correlation functions and form factors in vertex models or spin chains are known to satisfy certain difference equations called the quantum Knizhnik-Zamolodchikov equations. We find similar difference equations for the case of semi-infinite spin chain systems with integrable boundary conditions. We derive these equations using the properties of the vertex operators and the boundary vacuum state, or alternatively through corner transfer matrix arguments for the 8-vertex model with a boundary. The spontaneous boundary magnetization is found by solving such difference equations. The boundary $S$-matrix is also proposed and compared, in the sine-Gordon limit, with Ghoshal--Zamolodchikov's result. The axioms satisfied by the form factors in the boundary theory are formulated. 
In this monograph we prove that the nonlinear Lie algebra representation given by the manifestly covariant Maxwell-Dirac (M-D) equations is integrable to a global nonlinear representation $U$ of the Poincar\'e group ${\cal P}_0$ on a differentiable manifold ${\cal U}_\infty$ of small initial conditions for the M-D equations. This solves, in particular, the Cauchy problem for the M-D equations, namely existence of global solutions for initial data in ${\cal U}_\infty$ at $t=0$. The existence of modified wave operators $\Omega_+$ and $\Omega_-$ and asymptotic completeness is proved. The asymptotic representations $U^{(\varepsilon)}_g = \Omega^{-1}_\varepsilon \circ U_g \circ \Omega_\varepsilon$, $\varepsilon = \pm$, $g \in {\cal P}_0$, turn out to be nonlinear. A cohomological interpretation of the results in the spirit of nonlinear representation theory and its connection to the infrared tail of the electron is given. 
  The two-dimensional model which emerges from low-energy considerations of string theory is written down. Solutions of this classical model are noted, including some examples which have nontrivial tachyon field. One such represents the classical backreaction of the tachyon field on the black hole for a two parameter set of tachyon potentials. Assuming the classical black hole background in the `Eddington-Finkelstein' gauge, the tachyon equation is separable and the radial part is solved by a hypergeometric function, which is in general of complex argument. A semi-classical prescription for including the quantum effects of the tachyon field is described, and the resulting equations of motion are found. Special solutions of these equations are written down. 
  Matrix hierarchies are: multi-component KP, general Zakharov-Shabat (ZS) and its special cases, e.g., AKNS. The ZS comprises all integrable systems having a form of zero-curvature equations with rational dependence of matrices on a spectral parameter. The notion of a $\tau$-function is introduced here in the most general case along with formulas linking $\tau$-functions with wave Baker functions. The method originally invented by Sato et al. for the KP hierarchy is used. This method goes immediately from definitions and does not require any assumption about the character of a solution, being the most general. Applied to the matrix hierarchies, it involves considerable sophistication. The paper is self-contained and does not expect any special prerequisite from a reader. 
We review the construction of extended ( N=2 and N=4 ) superconformal algebras over triple systems and the gauged WZW models invariant under them. The N=2 superconformal algebras (SCA) realized over Freudenthal triple systems (FTS) admit extension to ``maximal'' N=4 SCA's with SU(2)XSU(2)XU(1) symmetry. A detailed study of the construction and classification of N=2 and N=4 SCA's over Freudenthal triple systems is given. We conclude with a study and classification of gauged WZW models with N=4 superconformal symmetry. 
Four-dimensional string backgrounds with local realizations of N = 4 world-sheet supersymmetry have, in the presence of a rotational Killing symmetry, only one complex structure which is an SO(2) singlet, while the other two form an SO(2) doublet. Although N = 2 world-sheet supersymmetry is always preserved under Abelian T-duality transformations, N = 4 breaks down to N = 2 in the rotational case. A non-local realization of N = 4 supersymmetry emerges, instead, with world-sheet parafermions. For SO(3)-invariant metrics of purely rotational type, like the Taub-NUT and the Atiyah-Hitchin metrics, none of the locally realized extended world-sheet supersymmetries can be preserved under non-Abelian duality. 
We discuss bosonization in three dimensions of an $SU(N)$ massive Thirring model in the low-energy regime. We find that the bosonized theory is related (but not equal) to $SU(N)$ Yang-Mills-Chern-Simons gauge theory. For free massive fermions bosonization leads, at low energies, to the pure $SU(N)$ (level $k=1$) Chern-Simons theory. 
Functional equations, in the form of fusion hierarchies, are studied for the transfer matrices of the fused restricted $A_{n-1}^{(1)}$ lattice models of Jimbo, Miwa and Okado. Specifically, these equations are solved analytically for the finite-size scaling spectra, central charges and some conformal weights. The results are obtained in terms of Rogers dilogarithm and correspond to coset conformal field theories based on the affine Lie algebra $A_{n-1}^{(1)}$ with GKO pair $A^{(1)}_{n-1}\; \oplus A^{(1)}_{n-1}\;\supset \; A^{(1)}_{n-1}$. 
We define an $ sl(N) $ analog of Onsager's Algebra through a finite set of relations that generalize the Dolan Grady defining relations for the original Onsager's Algebra. This infinite-dimensional Lie Algebra is shown to be isomorphic to a fixed point subalgebra of $ sl(N) $ Loop Algebra with respect to a certain involution. As the consequence of the generalized Dolan Grady relations a Hamiltonian linear in the generators of $ sl(N) $ Onsager's Algebra is shown to posses an infinite number of mutually commuting integrals of motion. 
  There exists a widely spread notion that gravitational effects can strongly violate global symmetries. It may lead to many important consequences. We will argue, in particular, that nonperturbative gravitational effects in the axion theory lead to a strong violation of CP invariance unless they are suppressed by an extremely small factor 10^{-82}. One could hope that this problem disappears if one represents the global symmetry of a pseudoscalar axion field as a gauge symmetry of the Ogievetsky-Polubarinov-Kalb-Ramond antisymmetric tensor field. We will show, however, that this gauge symmetry does not protect the axion mass from quantum corrections. The amplitude of gravitational effects violating global symmetries could be strongly suppressed by e^{-S}, where S is the action of a wormhole which may eat the global charge. Unfortunately, in a wide variety of theories based on the Einstein theory of gravity the action appears to be fairly small, S = O(10). However, we have found that the existence of wormholes and the value of their action are extremely sensitive to the structure of space on the nearly Planckian scale. We consider several examples (Kaluza-Klein theory, conformal anomaly, R^2 terms) which show that modifications of the Einstein theory on the length scale l ~ 10 M_P^{-1} may strongly suppress violation of global symmetries. We have found also that in string theory there exists an additional suppression of topology change by the factor e^{-{8\pi^2\over g^2}}. This effect is strong enough to save the axion theory for the natural values of the stringy gauge coupling constant. 
  We give another reformulation of the Thirring model (with four-fermion interaction of the current-current type) as a gauge theory and identify it with a gauge-fixed version of the corresponding gauge theory according to the Batalin-Fradkin formalism. Based on this formalism, we study the chiral symmetry breaking of the $D$-dimensional Thirring model ($2<D<4$) with $N$ flavors of 4-component fermions. By constructing the gauge covariant effective potential for the chiral order parameter, up to the leading order of $1/N$ expansion, we show the existence of the second order chiral phase transition and obtain explicitly the critical number of flavors $N_c$ (resp. critical four-fermion coupling $G_c$) as a function of the four-fermion coupling $G$ (resp. $N$), below (resp. above) which the chiral symmetry is spontaneously broken. 
Lectures presented at the VI Mexican School of Particles and Fields, Villahermosa, 3-7 October, 1994.  Contents:  1. Introduction  2. The Symmetry Group Approach to the Quantum Mechanics of Identical Particles  3. How Come Anyons?  4. The Transmutation of Statistics into a Topological Interaction  5. The Chern-Simons Action and Anyon Statistics  6. Nonrelativistic Chern-Simons-(Maxwell) Field Theory  7. Epilogue 
We consider duality transformations in N=2, d=4 Yang--Mills theory coupled to N=2 supergravity. A symplectic and coordinate covariant framework is established, which allows one to discuss stringy `classical and quantum duality symmetries' (monodromies), incorporating T and S dualities. In particular, we shall be able to study theories (like N=2 heterotic strings) which are formulated in symplectic basis where a `holomorphic prepotential' F does not exist, and yet give general expressions for all relevant physical quantities. Duality transformations and symmetries for the N=1 matter coupled Yang--Mills supergravity system are also exhibited. The implications of duality symmetry on all N>2 extended supergravities are briefly mentioned. We finally give the general form of the central charge and the N=2 semiclassical spectrum of the dyonic BPS saturated states (as it comes by truncation of the N=4 spectrum). 
  Starting from the dual action of $(4,4)$ $2D$ twisted multiplets in the harmonic superspace with two independent sets of $SU(2)$ harmonic variables, we present its generalization which hopefully provides an off-shell description of general $(4,4)$ supersymmetric sigma models with torsion. Like the action of the torsionless $(4,4)$ hyper-K\"ahler sigma models in the standard harmonic superspace, it is characterized by a number of superfield potentials. They depend on $n$ copies of a triple of analytic harmonic $(4,4)$ superfields. As distinct from the hyper-K\"ahler case, the potentials prove to be severely constrained by the self-consistency condition which stems from the commutativity of the left and right harmonic derivatives. We show that for $n=1$ these constraints reduce the general action to that of $(4,4)$ twisted multiplet, while for $n\geq 2$ there exists a wide class of new actions which cannot be written only via twisted multiplets. Their most striking feature is the nonabelian and in general nonlinear gauge invariance which substitutes the abelian gauge symmetry of the dual action of twisted multiplets and ensures the correct number of physical degrees of freedom. We show, on a simple example, that these actions describe sigma models with non-commuting left and right complex structures on the bosonic target. 
  We investigate the effect of gravitational back-reaction on the black hole evaporation process. The standard derivation of Hawking radiation is re-examined and extended by including gravitational interactions between the infalling matter and the outgoing radiation. We find that these interactions lead to substantial effects. In particular, as seen by an outside observer, they lead to a fast growing uncertainty in the position of the infalling matter as it approaches the horizon. We argue that this result supports the idea of black hole complementarity, which states that, in the description of the black hole system appropriate to outside observers, the region behind the horizon does not establish itself as a classical region of space-time. We also give a new formulation of this complementarity principle, which does not make any specific reference to the location of the black hole horizon. 
  The standard formulation of a massive Abelian vector field in $2+1$ dimensions involves a Maxwell kinetic term plus a Chern-Simons mass term; in its place we consider a Chern-Simons kinetic term plus a Stuekelberg mass term. In this latter model, we still have a massive vector field, but now the interaction with a charged spinor field is renormalizable (as opposed to super renormalizable). By choosing an appropriate gauge fixing term, the Stuekelberg auxiliary scalar field decouples from the vector field. The one-loop spinor self energy is computed using operator regularization, a technique which respects the three dimensional character of the antisymmetric tensor $\epsilon_{\alpha\beta\gamma}$. This method is used to evaluate the vector self energy to two-loop order; it is found to vanish showing that the beta function is zero to two-loop order. The canonical structure of the model is examined using the Dirac constraint formalism. 
We consider the Hamiltonian theory for the multi-component KP hierarchy. We show that the second Hamiltonian structures constructed by Sidorenko and Strampp[J. Math. Phys. {\bf 34}, 1429(1993)] are not Hamiltonian. A candidate for the second Hamiltonian Structures is proposed and is proved to lead to hereditary operators. 
We investigate the field dependence of the gauge couplings of $N=1$ string vacua from the point of view of the low energy effective quantum field theory. We find that field-theoretical considerations severely constrain the form of the string loop corrections; in particular, the dilaton dependence of the gauge couplings is completely universal at the one-loop level. The moduli dependence of the string threshold corrections is also constrained, and we illustrate the power of such constraints with a detailed discussion of the orbifold vacua and the $(2,2)$ (Calabi-Yau) vacua of the heterotic string. 
We show how a stress-energy pseudotensor can be constructed in two-dimensional dilatonic gravity theories (classical, CGHS and RST) and derive the expression for the ADM mass in these theories from it. We define the Bondi mass for these theories by using the pseudotensor formalism. The resulting expression is the generalization of the expression for the ADM mass. The boundary condition needed for the energy conservation is also investigated. It is shown that under appropriate boundary conditions, our definition of the Bondi mass is exactly the ADM mass minus the matter radiation energy at null infinity. 
  In this Letter, we construct a set of order parameters for non-Abelian gauge theories which probe directly the unbroken group and are free of the deficiencies caused by quantum fluctuations and gauge fixing which have plagued all previous attempts. These operators can be used to map out the phase diagram of a non-Abelian gauge theory. 
  The Aharonov-Bohm effect has been invoked to probe the phase structure of a gauge theory. Yet in the case of non-Abelian gauge theories, it proves difficult to formulate a general procedure that unambiguously specifies the realization of the gauge symmetry, e.g. the unbroken subgroup. In this paper, we propose a set of order parameters that will do the job. We articulate the fact that any useful Aharonov-Bohm experiment necessarily proceeds in two stages: calibration and measurement. World sheets of virtual cosmic string loops can wrap around test charges, thus changing their states relative to other charges in the universe. Consequently, repeated flux measurements with test charges will not necessarily agree. This was the main stumbling block to previous attempts to construct order parameters for non-Abelian gauge theories. In those works, the particles that one uses for calibration and subsequent measurement are stored in {\em separate} ``boxes''. By storing all test particles in the {\em same} ``box'', we show how quantum fluctuations can be overcome. The importance of gauge fixing is also emphasized. 
  We find non-covariant local symmetries in the Abelian gauge theories. The N\"other charges generating these symmetries are nilpotent as BRST charges, and they impose constraints on the physical states. 
  We construct the BRST cohomology under a positive definite inner product and obtain the Hodge decomposition theorem at a non-degenerate state vector space $V$. The harmonic states isomorphic with a BRST cohomology class correspond to the physical Hilbert space with positive norm as long as the completeness of $Q_{BRST}$ is satisfied. We explicitly define ``co-BRST'' operator and analyze the quartet mechanism in QED. 
Color confinement by the mechanism of Kugo and Ojima can treat confinement of any quantized color carrying fields including dynamical quarks. However, the non-perturbative condition for this confinement has been known to be satisfied only in the pure-gauge model (PGM), which is a topological model without physical degrees of freedom. Here we analyze the Yang-Mills theory by adding physical degrees of freedom as perturbation to PGM. We find that quarks and gluons are indeed confined in this perturbation theory. 
  We consider static solutions of two dimensional dilaton gravity models as toy laboratories to address the question of the final fate of black holes. A non perturbative correction to the CGHS potential term is shown to lead classically to an extremal black hole geometry, thus providing a plausible solution to Hawking evaporation paradox. However, the full quantum theory does not admit an extremal solution. 
We compute the $\beta$-function of a YM theory, broken to $U(1)$, by evaluating the coupling constant renormalization in the broken phase. We perform the calculation in the unitary gauge where only physical particles appear and the theory looks like a version of QED containing massive charged spin 1 particles. We consider an on-shell scattering process and after verifying that the non-renormalizable divergences which appear in the Green's functions cancel in the expression of the amplitude, we show that the coupling constant renormalization is entirely due to the photon self-energy as in QED. However we get the expected asymptotic freedom and the physical charge decreases logarithmically as a function of the symmetry breaking scale. 
We present and discuss an euclidean solution of the low-energy effective string action that can be interpreded as a semiclassical decay process of the ground state of the theory. 
  The general BRST-BFV analysis of anomaly in the string theory coupled to background fields is carried out. An exact equation for c-valued symbol of anomaly operator is found and structure of its solutions is studied. 
The total twist number, which represents the first non-trivial Vassiliev knot invariant, is derived from the second order expression of the Wilson loop expectation value in the Chern-Simons theory. Using the well-known fact that the analytical expression is an invariant, a non-recursive formulation of the total twist based on the evaluation of knot diagrams is constructed by an appropriate deformation of the knot line in the three-dimensional Euclidian space. The relation to the original definition of the total twist is elucidated. 
We find a volume form on moduli space of double punctured Riemann surfaces whose integral satisfies the Painlev\'e I recursion relations of the genus expansion of the specific heat of 2D gravity. This allows us to express the asymptotic expansion of the specific heat as an integral on an infinite dimensional moduli space in the spirit of Friedan-Shenker approach. We outline a conjectural derivation of such recursion relations using the Duistermaat-Heckman theorem. 
It is shown that two dimensional ($2d$) topological gravity in the conformal gauge has a larger symmetry than has been hitherto recognized; in the formulation of Labastida, Pernici and Witten it contains a twisted ``small'' $N=4$ superconformal symmetry. There are in fact two distinct twisted $N=2$ structures within this $N=4$, one of which is shown to be isomorphic to the algebra discussed by the Verlindes and the other corresponds, through bosonization, to $c_M\leq 1$ string theory discussed by Bershadsky et.al. As a byproduct, we find a twisted $N=4$ structure in $c_M\leq 1$ string theory. We also study the ``mirror'' of this twisted $N=4$ algebra and find that it corresponds, through another bosonization, to a constrained topological sigma model in complex dimension one. 
The theory of matrix models is reviewed from the point of view of its relation to integrable hierarchies. Determinantal formulas, relation to conformal field models and the theory of Generalized Kontsevich model are discussed in some detail. Attention is also paid to the group-theoretical interpretation of $\tau$-functions which allows to go beyond the restricted set of the (multicomponent) KP and Toda integrable hierarchies. 
The master field for a subclass of planar diagrams, so called rainbow diagrams, for higher dimensional large N theories is considered. An explicit representation for the master field in terms of noncommutative random variables in the modified interaction representation in the Boltzmannian Fock space is given. A natural interaction in the Boltzmannian Fock space is formulated by means of a rational function of the interaction Lagrangian instead of the ordinary exponential function in the standard Fock space. 
We compute the external moduli and dilaton hair to $O(\alpha')$ in the framework of the one-loop corrected superstring effective action for a rotating black hole background. 
We discuss the dressing of one-loop sigma-model beta-functions by induced supergravity, for both N=1 and N=2 supersymmetric theories. We obtain exact results by a superconformal gauge argument, and verify them in the semi-classical limit by explicit perturbative calculations in the light-cone gauge. We find that for N=2 theories there is no dressing of the one-loop beta-functions. 
We use a numerical method to obtain the weak coupling perturbative coefficients of local operators with lattice regularization. Such a method allows us to extend the perturbative expansions obtained so far by analytical Feynman diagrams calculations. In SU(3) lattice gauge theory in four dimensions we compute the first eight coefficients of the expectation value of the Wilson loop on the elementary plaquette which is related to the gluon condensate. The computed eight coefficients grow with the order much faster than predicted by the presence of the infrared renormalon associated to the dimension of the gluon condensate. However the renormalon behaviour for large order is quite well reproduced if one considers the expansion coefficients in a new coupling related to the lattice coupling by large perturbative corrections. This is expected since the lattice and continuum Lambda scales differ by almost two orders of magnitude. 
Recently, sum rules were derived for the inverse eigenvalues of the Dirac operator. They were obtained in two different ways: i) starting from the low-energy effective Lagrangian and ii) starting from a random matrix theory with the symmetries of the Dirac operator. This suggests that the effective theory can be obtained directly from the random matrix theory. Previously, this was shown for three or more colors with fundamental fermions. In this paper we construct the effective theory from a random matrix theory for two colors in the fundamental representation and for an arbitrary number of colors in the adjoint representation. We construct a fermionic partition function for Majorana fermions in Euclidean space time. Their reality condition is formulated in terms of complex conjugation of the second kind. 
We study in detail the algebra of free ghost fields which we realize in a Hilbert-Fock space with positive metric. The investigation of causality clarifies the exact reason for the failure of the spin-statistics theorem and leads to the introduction of the Krein Operator. We study the charge algebra of the ghost fields which gives a representation of ${\rm gl}(2,{\cal C})$. The symmetries of the $S$-matrix in ghost space are pointed out. 
We study a modified two-dimensional dilaton gravity theory which is exactly solvable in the semiclassical approximation including back-reaction. The vacuum solutions of this modified theory are asymptotically flat static space-times. Infalling matter forms a black hole if its energy is above a certain threshold. The black hole singularity is initially hidden behind a timelike apparent horizon. As the black hole evaporates by emitting Hawking radiation, the singularity meets the shrinking horizon in finite retarded time to become naked. A natural boundary condition exists at the naked singularity such that for general infalling matter-configuration the evaporating black hole geometries can be matched continuously to a unique static end-state geometry. This end-state geometry is asymptotically flat at its right spatial infinity, while its left spatial infinity is a semi-infinite throat extending into the strong coupling region. 
Static, four-dimensional (4-d) black holes (BH's) in ($4+n$)-d Kaluza-Klein (KK) theory with Abelian isometry and diagonal internal metric have at most one electric ($Q$) and one magnetic ($P$) charges, which can either come from the same $U(1)$-gauge field (corresponding to BH's in effective 5-d KK theory) or from different ones (corresponding to BH's with $U(1)_M\times U(1)_E$ isometry of an effective 6-d KK theory). In the latter case, explicit non-extreme solutions have the global space-time of Schwarzschild BH's, finite temperature, and non-zero entropy. In the extreme (supersymmetric) limit the singularity becomes null, the temperature saturates the upper bound $T_H=1/4\pi\sqrt{|QP|}$, and entropy is zero. A class of KK BH's with constrained charge configurations, exhibiting a continuous electric-magnetic duality, are generated by global $SO(n)$ transformations on the above classes of the solutions. 
  Starting from a reformulation of the Thirring model as a gauge theory, we consider the bosonization of the $D$-dimensional multiflavor massive Thirring model $(D \ge 2)$ with four-fermion interaction of the current-current type. Our method leads to a novel interpolating Lagrangian written in terms of two gauge fields. Especially we pay attention to the case of very massive fermion $m \gg 1$ in (2+1) and (1+1) dimensions. Up to the next-to-leading order of $1/m$, we show that the (2+1)-dimensional massive Thirring model is mapped to the Maxwell-Chern-Simons theory and that the (1+1)-dimensional massive Thirring model is equivalent to the massive free scalar field theory. In the process of the bosonization of the Thirring model, we point out the importance of the gauge-invariant formulation. Finally we discuss a possibility of extending this method to the non-Abelian case. 
Two dimensional quantum R$^2$-gravity and its phase structure are examined in the semiclassical approach and compared with the results of the numerical simulation. Three phases are succinctly characterized by the effective action. A classical solution of R$^2$-Liouville equation is obtained by use of the solution of the ordinary Liouville equation. The partition function is obtained analytically. A toatal derivative term (surface term) plays an important role there. It is shown that the classical solution can sufficiently account for the cross-over transition of the surface property seen in the numerical simulation. 
  The spectrum of the massive Schwinger model in the strong coupling region is obtained by using the light-front Tamm-Dancoff (LFTD) approximation up to including six-body states. We numerically confirm that the two-meson bound state has a negligibly small six-body component. Emphasis is on the usefulness of the information about states (wave functions). It is used for identifying the three-meson bound state among the states below the three-meson threshold. We also show that the two-meson bound state is well described by the wave function of the relative motion. 
We apply a truncated set of dynamical equations of motion for connected equal-time Green functions up to the 4-point level to the investigation of spontaneous ground state symmetry breaking in $\Phi^4_{2+1}$ quantum field theory. Within our momentum space discretization we obtain a second order phase transition as soon as the connected 3-point function is included. However, an additional inclusion of the connected 4-point function still shows a significant influence on the shape of the effective potential and the critical coupling. 
The massive scalar field theory and the chiral Schwinger model are quantized on a Poincar\'e disk of radius $\rho$. The amplitudes are derived in terms of hypergeometric functions. The behavior at long distances and near the boundary of some of the relevant correlation functions is studied. The exact computation of the chiral determinant appearing in the Schwinger model is obtained exploiting perturbation theory. This calculation poses interesting mathematical problems, as the Poincar\'e disk is a noncompact manifold with a metric tensor which diverges approaching the boundary. The results presented in this paper are very useful in view of possible extensions to general Riemann surfaces. Moreover, they could also shed some light in the quantization of field theories on manifolds with constant curvature scalars in higher dimensions. 
I discuss two techniques that can be used in the investigation of the properties of ``colliding" and ``non-colliding" anyons. 
String amplitudes with an arbitrary number of world-sheet boundaries on which the coordinates satisfy Dirichlet boundary conditions are analyzed in a path integral framework. Special attention is payed to the novel divergences associated with such conditions. Certain helicity amplitudes involving massless closed-string states are free of such divergences to all orders in perturbation theory and their behavior can be analysed unambiguously. The high energy fixed-angle behavior of these amplitudes is discussed in the presence of one or two boundaries and the asymptotic behavior of the amplitudes is shown to be power behaved. 
  Witten has argued that in $2+1$ dimensions local supersymmetry can ensure the vanishing of the cosmological constant without requiring the equality of bose and fermi masses. We find that this mechanism is implemented in a novel fashion in the (2+1)-dimensional supersymmetric abelian Higgs model coupled to supergravity. The vortex solitons are annihilated by half of the supersymmetry transformations. The covariantly constant spinors required to define these supersymmetries exist by virtue of a surprising cancellation between the Aharonov-Bohm phase and the phase associated with the holonomy of the spin connection. However the other half of the supersymmetry transformations, whose actions ordinarily generate the soliton supermultiplet, are not well-defined and bose-fermi degeneracy is consequently absent in the soliton spectrum. 
We investigate new realisations of the $W_3$ algebra with arbitrary central charge, making use of the fact that this algebra can be linearised by the inclusion of a spin-1 current. We use the new realisations with $c=102$ and $c=100$ to build non-critical and critical $W_3$ BRST operators. Both of these can be converted by local canonical transformations into a BRST operator for the Virasoro string with $c=28-2$, together with a Kugo-Ojima topological term. Consequently, these new realisations provide embeddings of the Virasoro string into non-critical and critical $W_3$ strings. 
The geometric construction of the functional integral over coset spaces ${\cal M}/{\cal G}$ is reviewed. The inner product on the cotangent space of infinitesimal deformations of $\cal M$ defines an invariant distance and volume form, or functional integration measure on the full configuration space. Then, by a simple change of coordinates parameterizing the gauge fiber $\cal G$, the functional measure on the coset space ${\cal M}/{\cal G}$ is deduced. This change of integration variables leads to a Jacobian which is entirely equivalent to the Faddeev-Popov determinant of the more traditional gauge fixed approach in non-abelian gauge theory. If the general construction is applied to the case where $\cal G$ is the group of coordinate reparametrizations of spacetime, the continuum functional integral over geometries, {\it i.e.} metrics modulo coordinate reparameterizations may be defined. The invariant functional integration measure is used to derive the trace anomaly and effective action for the conformal part of the metric in two and four dimensional spacetime. In two dimensions this approach generates the Polyakov-Liouville action of closed bosonic non-critical string theory. In four dimensions the corresponding effective action leads to novel conclusions on the importance of quantum effects in gravity in the far infrared, and in particular, a dramatic modification of the classical Einstein theory at cosmological distance scales, signaled first by the quantum instability of classical de Sitter spacetime. Finite volume scaling relations for the functional integral of quantum gravity in two and four dimensions are derived, and comparison with the discretized dynamical triangulation approach to the integration over geometries are discussed. 
  We study the first-order formalism of pure four-dimensional ${\rm SU}(2)$ Yang--Mills theory with theta-term. We describe the Green functions associated to electric and magnetic flux operators \`a la 't~Hooft by means of gauge-invariant non-local operators. These Green functions are related to Witten's invariants of four-manifolds. 
We study the one loop renormalization in the most general metric-dilaton theory with the second derivative terms only. The general theory can be divided into two classes, models of one are equivalent to conformally coupled with gravity scalar field and also to general relativity with cosmological term. The models of second class have one extra degree of freedom which corresponds to dilaton. We calculate the one loop divergences for the models of second class and find that the arbitrary functions of dilaton in the starting action can be fine-tuned in such a manner that all the higher derivative counterterms disappear on shell. The only structures in both classical action and counterterms, which survive on shell, are the potential (cosmological) ones. They can be removed by renormalization of the dilaton field which acquire the nontrivial anomalous dimension, that leads to the effective running of the cosmological constant. For some of the renormalizable solutions of the theory the observable low energy value of the cosmological constant is small as compared with the Newtonian constant. We also discuss another application of our result. 
We show that the effective coarse graining of a two-mode squeezed density matrix, implicit in the Wehrl approaches to a semiclassical phase-space distribution, leads to results in agreement with previous different definitions of entropy for the process of pair production from the vacuum. We also present, in this context, a possible interpretation of the entropy growth as an amplification (due to the squeezing) of our lack of knowledge about the initial conditions, which gives rise to an effective decoherence of the squeezed density matrix. 
QED with N species of massive fermions on a circle of circumference L is analyzed by bosonization. The problem is reduced to the quantum mechanics of the 2N fermionic and one gauge field zero modes on the circle, with nontrivial interactions induced by the chiral anomaly and fermions masses. The solution is given for N=2 and fermion masses (m) much smaller than the mass of the U(1) boson with mass \mu=\sqrt{2e^2/\pi} when all fermions satisfy the same boundary conditions. We show that the two limits m \go 0 and L \go \infty fail to commute and that the behavior of the theory critically depends on the value of mL|\cos\onehalf\theta| where \theta is the vacuum angle parameter. When the volume is large \mu L \gg 1, the fermion condensate <\psibar \psi> is -(e^{4\gamma} m\mu^2 \cos^4\onehalf\theta/4\pi^3)^{1/3} or $-2e^\gamma m\mu L \cos^2 \onehalf\theta /\pi^2 for mL(\mu L)^{1/2} |\cos\onehalf\theta| \gg 1 or \ll 1, respectively. Its correlation function decays algebraically with a critical exponent \eta=1 when m\cos\onehalf\theta=0. 
We explicitly show that the net number of degrees of freedom in the two-dimensional dilaton gravity is zero through the Hamiltonian constraint analysis. This implies that the local space-time dependent physical excitations do not exist. From the linear perturbation around the black hole background, we explicitly prove that the exponentially growing mode with time is in fact eliminated outside the horizon. Therefore, the two-dimensional dilation gravity is essentially stable. 
We define the Bondi energy for two-dimensional dilatonic gravity theories by generalizing the known expression of the ADM energy. We show that our definition of the Bondi energy is exactly the ADM energy minus the radiation energy at null infinity. An explicit calculation is done for the evaporating black hole in the RST model with the Strominger's ghost decoupling term. It is shown that the infalling matter energy is completely recovered through the Hawking radiation and the thunderpop. 
  By casting the Yang-Mills-Higgs equations of an SU(2) theory in the form of the Ernst equations of general relativity, it is shown how the known exact solutions of general relativity can be used to give similiar solutions for Yang-Mills theory. Thus all the known exact solutions of general relativity with axial symmetry (e.g. the Kerr metric, the Tomimatsu-Sato metric) have Yang-Mills equivalents. In this paper we only examine in detail the Kerr-like solution. It will be seen that this solution has surfaces where the gauge and scalar fields become infinite, which correspond to the infinite redshift surfaces of the normal Kerr solution. It is speculated that this feature may be connected with the confinement mechanism since any particle which carries an SU(2) color charge would tend to become trapped once it passes these surfaces. Unlike the Kerr solution, our solution apparently does not have any intrinsic angular momentum, but rather appears to give the non-Abelian field configuration associated with concentric shells of color charge. 
Analogs of ordinary Gaussian coherent states on bosonic Fock spaces are constructed for the case of free Fock spaces, which appear to be natural mathematical structures suitable for description of large N matrix models. 
We present fermionic sum representation for the general Virasoro character of the unitary minimal superconformal series ($N=1$). Example of the corresponding ``finitizated" identities relating corner transfer matrix polynomials with fermionic companions is considered. These identities in the thermodynamic limit lead to the generalized Rogers-Ramanujan identities. 
We point out that in heterotic string theory compactified on a 6-torus, after a consistent truncation of the 10-d gauge fields and the antisymmetric tensor fields, 4-dimensional black holes of Kaluza-Klein theory on a 6-torus constitute a subset of solutions. 
  Massive neutrinos can be accommodated into the noncommutative geometry reinterpretation of the Standard Model. The constrained Standard Model Lagrangian is computed anew under the assumption of nonzero neutrino masses. This gives the ``prediction" of a mass for the Higgs particle somewhat higher than in the vanishing neutrino mass case. 
Hamiltonian lattice gauge models based on the assignment of the Heisenberg double of a Lie group to each link of the lattice are constructed in arbitrary space-time dimensions. It is shown that the corresponding generalization of the gauge-invariant Wilson line observables requires to attach to each vertex of the line a vertex operator which goes to the unity in the continuum limit. 
  The standard notion of the non-Abelian duality in string theory is generalized to the class of $\si$-models admitting `non-commutative conserved charges'. Such $\si$-models can be associated with every Lie bialgebra $(\cg ,\cgt)$ and they possess an isometry group iff the commutant   $[\cgt,\cgt]$ is not equal to $\cgt$. Within the enlarged class of the backgrounds the non-Abelian duality {\it is} a duality transformation in the proper sense of the word. It exchanges the roles of $\cg$ and $\cgt$ and it can be interpreted as a symplectomorphism of the phase spaces of the mutually dual theories. We give explicit formulas for the non-Abelian duality transformation for any $(\cg,\cgt)$. The non-Abelian analogue of the Abelian modular space $O(d,d;{\bf Z})$ consists of all maximally isotropic decompositions of the corresponding Drinfeld double. 
Using thermodynamic arguments we find that the probability that there are no eigenvalues in the interval (-s,\infty) in the double scaling limit of Hermitean matrix models is O(exp(-s^{2m+1})) as s\rightarrow+\infty.Here m=1,2,3.. determine the m^{th} multi-critical point of the level density:\sigma(x)\sim b[1-(x/b)^2]^{m-1/2} and b^2\sim N.Furthermore,the size of the transition zone where the eigenvalue density becomes vanishingly small at the tail of the spectrum is \sim N^{(m-3/2)/(2m+1)} in agreement with earlier work based on the string equation. 
We explore geometrical properties of fermionic vertex operators for a NSR superstring in order to establish connection between worldsheet and target space supersymmetries. The mechanism of picture-changing is obtained as a result of imposing certain constraints on a world-sheet gauge group of the NSR theory. It is found that picture-changing operators of various integer ghost numbers form a polynomial ring. By using properties of the picture-changing formalism we establish connection between the NSR and GS superstring theories. We explore the properties of the $\kappa$-symmetry in the NSR formalism and show that it leads to some new identities between correlation functions. 
  We construct the world-line action for a Dirac particle coupled to a classical scalar or pseudo-scalar background field. This action can be used to compute loop diagrams and the effective action in the Yukawa model using the world-line path-integral formalism for spinning particles. 
A generating functional $F$ is found for a canonical nonabelian dual transformation which maps the supersymmetric chiral O(4) $\sigma$-model to an equivalent supersymmetric extension of the dual $\sigma$-model. This $F$ produces a mapping between the classical phase spaces of the two theories in which the bosonic (coordinate) fields transform nonlocally, the fermions undergo a local tangent space chiral rotation, and all currents (fermionic and bosonic) mix locally. Purely bosonic curvature-free currents of the chiral model become a {\em symphysis} of purely bosonic and fermion bilinear currents of the dual theory. The corresponding transformation functional $T$ which relates wavefunctions in the two quantum theories is argued to be {\em exactly} given by $T=\exp(iF)$. 
Two-dimensional supergravity theory is quantized as an anomalous gauge theory. In the Batalin-Fradkin (BF) formalism, the anomaly-canceling super-Liouville fields are introduced to identify the original second-class constrained system with a gauge-fixed version of a first-class system. The BFV-BRST quantization applies to formulate the theory in the most general class of gauges. A local effective action constructed in the configuration space contains two super-Liouville actions; one is a noncovariant but local functional written only in terms of 2D supergravity fields, and the other contains the super-Liouville fields canceling the super-Weyl anomaly. Auxiliary fields for the Liouville and the gravity super-multiplets are introduced to make the BRST algebra close off-shell. Inclusion of them turns out to be essentially important especially in the super-lightcone gauge-fixing, where the super-curvature equations ($\dl^3_-g_{++} =\dl^2_-\chi_{++}=0$) are obtained as a result of BRST invariance of the theory. Our approach reveals the origin of the graded-SL(2,R) current algebra symmetry in a transparent manner. 
A model for planar phenomena introduced by Jackiw and Pi and described by a Lagrangian including a Chern-Simons term is considered. The associated equations of motion, among which a 2+1 gauged nonlinear Schr\"odinger equation, are rewritten into a gauge independent form involving the modulus of the matter field. Application of a Painlev\'e analysis, as adapted to partial differential equations by Weiss, Tabor and Carnevale, shows up resonance values that are all integer. However, compatibility conditions need be considered which cannot be satisfied consistently in general. Such a result suggests that the examined equations are not integrable, but provides tools for the investigation of the integrability of different reductions. This in particular puts forward the familiar integrable Liouville and 1+1 nonlinear Schr\"odinger equations. 
The reflection equations in a $su(3)$ spin chain with open boundary conditions are analyzed. We find non diagonal solutions to the boundary matrices. The corresponding hamiltonian is given. The solutions are generalized to $su(n)$. 
We discuss the temporal variation of the equation of state of a classical string network, evolving in a background in which the Hubble radius 1/H shrinks to a minimum and then re-expands to infinity. We also present a method to look for self-consistent non-vacuum string backgrounds, correponding to the simultaneous solution of the gravi-dilaton backgrounds equations and of the string equations of motion. 
  We investigate the phase structure and the infrared properties of higher-derivative quantum gravity (QG) with matter, in $4-\varepsilon$ dimensions. The renormalization group (RG) equations in $4-\varepsilon$ dimensions are analysed for the following types of matter: the $f\varphi^4$-theory, the $O(N) \varphi^4$-theory, scalar electrodynamics, and the $SU(2)$ model with scalars. New fixed points for the scalar coupling appear ---one of which is IR stable--- being some of them induced by QG. The IR stable fixed point perturbed by QG leads to a second-order phase transition for the theory at non-zero temperature. The RG improved effective potential in the $SU(2)$ theory (which can be considered as the confining phase of the standard model) is calculated at nonzero temperature and it is shown that its shape is clearly influenced by QG. 
We consider generalized one-matrix models in which external fields allow control over the coordination numbers on both the original and dual lattices. We rederive in a simple fashion a character expansion formula for these models originally due to Itzykson and Di Francesco, and then demonstrate how to take the large N limit of this expansion. The relationship to the usual matrix model resolvent is elucidated. Our methods give as a by-product an extremely simple derivation of the Migdal integral equation describing the large $N$ limit of the Itzykson-Zuber formula. We illustrate and check our methods by analyzing a number of models solvable by traditional means. We then proceed to solve a new model: a sum over planar graphs possessing even coordination numbers on both the original and the dual lattice. We conclude by formulating equations for the case of arbitrary sets of even, self-dual coupling constants. This opens the way for studying the deep problem of phase transitions from random to flat lattices. 
  We have solved exactly the $Osp(1|2)$ spin chain by the Bethe ansatz approach. Our solution is based on an equivalence between the $Osp(1|2)$ chain and certain special limit of the Izergin-Korepin vertex model. The completeness of the Bethe ansatz equations is discussed for a system with four sites and it is noted the appearance of special string structures. The Bethe ansatz presents an important phase-factor which distinguishes the even and odd sectors of the theory. The finite size properties are governed by a conformal field theory with central charge $c=1$. 
We find a new solution of the renormalization group for the Potts model with ferromagnetic random valued coupling constants. The solution exhibits universality and broken replica symmetry. It is argued that the model reaches this universality class if the replica symmetry is broken initially. Otherwise the model stays with the replica symmetric renormalization group flow and reaches the fixed point which has been considered before. 
We have applied the analytical Bethe ansatz approach in order to solve the $Osp(1|2n)$ invariant magnet. By using the Bethe ansatz equations we have calculated the ground state energy and the low-lying dispersion relation. The finite size properties indicate that the model has a central charge $c=n$. 
A classification of all possible spatially homogeneous 4D string backgrounds (HSBs) has been obtained by appropriate ramification of the existing nine Bianchi types of homogeneous 3D spaces. A total of $24^2=576$ HSBs which have been classified as distinct contains a subclass of 192 which includes all possible FRW models as well as those in which SO(3) isotropy is attained asymptotically. A discussion of these results also aims to fascilitate the identification of HSBs which have already appeared in the literature. The basic physical perspective of the parameters of classification is outlined together with certain features relating to deeper aspects of string theory. 
The off-shell nilpotent BRST charge and the BRST invariant effective action for non-abelian BF topological theories over D-dimensional manifolds are explicitly constructed. These theories have the feature of being reducible with exactly D-3 stages of reducibility. The adequate extended phase space including the different levels of ghosts for ghosts is explicitly obtained. Using the structure of the resulting BRST charge we show that for topological BF theories the semi-classical approximation completely describes the quantum theory. The independence of the partition function on the metric also follows from our explicit construction in a straightforward way. 
We investigate orbifold constructions of conformal field theories from lattices by no-fixed-point automorphisms (NFPA's) $Z_p$ for $p$ prime, $p>2$, concentrating on the case $p=3$. Explicit expressions are given for most of the relevant vertex operators, and we consider the locality relations necessary for these to define a consistent conformal field theory. A relation to constructions of lattices from codes, analogous to that found in earlier work in the $p=2$ case which led to a generalisation of the triality structure of the Monster module, is also demonstrated. 
  The problem of the higher-loop contributions to the axial anomaly is reexamined by a new method. We demonstrate that these contributions depend on the order of the calculations. If the divergence of the axial current by nonperturbative Fujikawa method is calculated first and then average it over the photon field in the presence of an external photon source, a nonzero contribution is obtained. However perturbative Feynman diagram method has an uncertainty. Depending on the order of the calculations above mentioned or zero results are obtained. 
  The nonlocal regularization method, recently proposed in ref.\,\ct{emkw91,kw92,kw93}, is extended to general gauge theories by reformulating it along the ideas of the antibracket-antifield formalism. From the interplay of both frameworks a fully regularized version of the field-antifield (FA) formalism arises, being able to deal with higher order loop corrections and to describe higher order loop contributions to the BRST anomaly. The quantum master equation, considered in the FA framework as the quantity parametrizing BRST anomalies, is argued to be incomplete at two and higher order loops and conjectured to reproduce only the one-loop corrections to the $\hbar^p$ anomaly generated by the addition of $O(\hbar^{k})$, $k<p$, counterterms. Chiral $W_3$ gravity is used to exemplify the nonlocally regularized FA formalism. First, the regularized one-loop quantum master equation is used to compute the complete one-loop anomaly. Its two-loop order, however, is shown to reproduce only the modification to the two-loop anomaly produced by the addition of a suitable one-loop counterterm, thereby providing an explicit verification of the previous statement for $p=2$. The well-known universal two-loop anomaly, instead, is alternatively obtained from the BRST variation of the nonlocally regulated effective action. Incompleteness of the quantum master equation is thus concluded to be a consequence of a naive derivation of the FA BRST Ward identity. 
We revise the twistor--like superfield approach to describing super--p--branes by use of the basic principles of the group--manifold approach \cite{rheo}. A super--p--brane action is constructed solely of geometrical objects as the integral over a (p+1)--surface. The Lagrangian is the external product of supervielbein differential forms in world supersurface and target superspace without any use of Lagrange multipliers. This allows one to escape the problem of infinite irreducible symmetries and redundant propagating fields. All the constraints on the geometry of world supersurface and the conditions of its embedding into target superspace arise from the action as differential form equations. 
A new approximation scheme for non-perturbative calculations in a quantum field theory is proposed. The scheme is based on investigation of solutions of the Schwinger equation with its singular character taken into account. As a necessary supplementary boundary condition the Green functions' connected structures correspondence principle is used. Besides the usual perturbation theory expansion which is always available as a particular solution of our scheme some non-perturbative solutions of an equation for the propagator are found in the model of a self-interacting scalar field. 
We consider a formulation of N=1 D=3,4 and 6 superparticle mechanics, which is manifestly supersymmetric on the worldline and in the target superspace. For the construction of the action we use only geometrical objects that characterize the embedding of the worldline superspace into the target superspace, such as target superspace coordinates of the superparticle and twistor components. The action does not contain the Lagrange multipliers which may cause the problem of infinite reducible symmetries, and, in fact, is a worldline superfield generalization of the supertwistor description of superparticle dynamics. 
By employing the symmetries of the underlying conformal field theory, the tree-level K\"ahler potentials for untwisted moduli of the heterotic string compactifications on orbifolds with continuous Wilson lines are derived. These symmetries act linearly on bosonic (toroidal and $E_8\times E_8$ gauge) string coordinates as well as on the untwisted (toroidal and continuous Wilson lines) moduli; they correspond to the scaling of toroidal moduli, the axionic shift of toroidal moduli and the shift of the continuous Wilson line moduli. In turn such symmetries provide sufficient constraints to determine the form of the low-energy effective action associated with the untwisted moduli up to a multiplicative factor. 
  The $SL(2,\R)$ WZNW $\rightarrow$ Liouville reduction leads to a nontrivial phase space on the classical level both in $0+1$ and $1+1$ dimensions. To study the consequences in the quantum theory, the quantum mechanics of the $0+1$ dimensional, point particle version of the constrained WZNW model is investigated. The spectrum and the eigenfunctions of the obtained---rather nontrivial---theory are given, and the physical connection between the pieces of the reduced configuration space is discussed in all the possible cases of the constraint parameters. 
In a recent paper it was suggested that some multi-black hole solutions in five or more dimensions have horizons that are not smooth. These black hole configurations are solutions to $d$-dimensional Einstein gravity (with no dilaton) and are extremally charged with a magnetic type $(d-2)$-form. In this work these solutions will be investigated further. It will be shown that although the curvature is bounded as the horizon of one of the black holes is approached, some derivatives of the curvature are not. This shows that the metric is not $C^{\infty },$ but rather it is only $C^k$ with $k$ finite. These solutions are static so their lack of smoothness cannot be attributed to the presence of radiation. 
We quantize the three-dimensional $BF$-model using axial gauge conditions. Exploiting the rich symmetry-structure of the model we show that the Green-functions correspond to tree graphs and can be obtained as the unique solution of the Ward-Identities. Furthermore, we will show that the theory can be uniquely determined by symmetry considerations without the need of an action principle. 
We describe a Lagrangian quantization of the free massless gauge superfield theories of higher superspins both in the anti-de Sitter and flat global superspaces. 
The topological $\sigma$ model with the black hole metric of the target space is considered. It has been shown before that this model is in the phase with BRST-symmetry broken. In particular, vacuum energy is non-\-zero and correlation functions of observables show the coordinate dependence. However these quantities turned out to be infrared (IR) divergent. It is shown here that IR divergences disappear after the sum over an arbitrary number of additional instanton-\-anti-\-instanton pairs is performed. The model appears to be equivalent to Coulomb gas/Sine Gordon system. 
  A homogeneous anisotropic four dimensional spacetime with Lorentzian signature is constructed from an ungauged WZW model based on a non-semisimple Lie group. The associated non-linear $\sigma $-model describes string propagation in an expanding-contracting universe with antisymmetric tensor and dilaton backgrounds. The current algebra of SL(2,R)$\times $R is constructed in terms of two free boson fields and two generalized parafermions, or four free bosons with background charge. This representation is used to study the string spectrum in the cosmological background. 
The finite-element approach to lattice field theory is both highly accurate (relative errors $\sim 1/N^2$, where $N$ is the number of lattice points) and exactly unitary (in the sense that canonical commutation relations are exactly preserved at the lattice sites). In this paper we construct matrix elements for the time evolution operator for the anharmonic oscillator, for which the continuum Hamiltonian is $H=p^2/2+\lambda q^{2k}/2k$. Construction of such matrix elements does not require solving the implicit equations of motion. Low order approximations turn out to be quite accurate. For example, the matrix element of the time evolution operator in the harmonic oscillator ground state gives a result for the $k=2$ anharmonic oscillator ground state energy accurate to better than 1\%, while a two-state approximation reduces the error to less than 0.1\%. Accurate wavefunctions are also extracted. Analogous results may be obtained in the continuum, but there the computation is more difficult, and not generalizable to field theories in more dimensions. 
  The parametrization and gauge dependencies of the one-loop counterterms on the mass-shell in the Einstein gravity are investigated. The physical meaning of the loop calculation results on the mass shell and the parametrization dependence of the renormgroup functions in the nonrenormalizable theories are discussed. 
In string theory there seems to be an intimate connection between spacetime and world-sheet physics. Following this line of thought we investigate the family problem in a particular class of string solutions, namely the free fermionic string models. We find that the number of generations $N_g$ is related to the index of the supersymmetry generator of the underlying $N=2$ internal superconformal field theory which is always present in any $N=1$ spacetime supersymmetric string vacuum. We also derive a formula for the index and thus for the number of generations which is sensitive to the boundary condition assignments of the internal fermions and to certain coefficients which determine the weight with which each spin-structure of the model contributes to the one-loop partition function. Finally we apply our formula to several realistic string models in order to derive $N_g$ and we verify our results by constructing explicitly the massless spectrum of these string models. 
q-deformed nonlinear field equations are constructed including Klein-Gordon and Maxwell equations. The q-deformation is interpreted as mathematical structure describing specific nonlinearity for which frequency of vibration exponentially depends on the energy 
  We discuss the status of the black hole entropy formula $S_{\rm BH} = A_H /4G$ in low energy effective field theory. The low energy expansion of the black hole entropy is studied in a non-equilibrium situation: the semiclassical decay of hot flat space by black hole nucleation. In this context the entropy can be defined as an enhancement factor in the semiclassical decay rate, which is dominated by a sphaleron-like saddle point. We find that all perturbative divergences appearing in Euclidean calculations of the entropy can be renormalized in low energy couplings. We also discuss some formal aspects of the relation between the Euclidean and Hamiltonian approaches to the one loop corrections to black hole entropy and geometric entropy, and we emphasize the virtues of the use of covariant regularization prescriptions. In fact, the definition of black hole entropy in terms of decay rates {\it requires} the use of covariant measures and accordingly, covariant regularizations in path integrals. Finally, we speculate on the possibility that low energy effective field theory could be sufficient to understand the microscopic degrees of freedom underlying black hole entropy. We propose a qualitative physical picture in which black hole entropy refers to a space of quasi-coherent states of infalling matter, together with its gravitational field. We stress that this scenario might provide a low energy explanation of both the black hole entropy and the information puzzle. 
We present a unified point of view on the different methods available in the literature to extract gauge theory renormalization constants from the low-energy limit of string theory. The Bern-Kosower method, based on an off-shell continuation of string theory amplitudes, and the construction of low-energy string theory effective actions for gauge particles, can both be understood in terms of strings interacting with background gauge fields, and thus reproduce, in the low-energy limit, the field theory results of the background field method. We present in particular a consistent off-shell continuation of the one-loop gluon amplitudes in the open bosonic string that reproduces exactly the results of the background field method in the Feynman gauge. 
  An outline is given of an extended perturbative solution of Euclidean QCD which systematically accounts for a class of nonperturbative effects, while allowing renormalization by the perturbative counterterms. Proper vertices Gamma are approximated by a double sequence Gamma[r,p], with r the degree of rational approximation w.r.t. the QCD mass scale Lambda, nonanalytic in the coupling g, and p the order of perturbative corrections in g-squared, calculated from Gamma[r,0] - rather than from the perturbative Feynman rules Gamma(0)(pert) - as a starting point. The mechanism allowing the nonperturbative terms to reproduce themselves in the Dyson-Schwinger equations preserves perturbative renormalizability and is tied to the divergence structure of the theory. As a result, it restricts the self-consistency problem for the Gamma[r,0] rigorously - i.e. without decoupling approximations - to the superficially divergent vertices. An interesting aspect of the scheme is that rational-function sequences for the propagators allow subsequences describing short-lived excitations. The method is calculational, in that it allows known techniques of loop computation to be used while dealing with integrands of truly nonperturbative content. 
The usual prescription for constructing gauge-invariant Lagrangian is generalized to the case where a Lagrangian contains second derivatives of fields as well as first derivatives. Symmetric tensor fields in addition to the usual vector fields are introduced as gauge fields. Covariant derivatives and gauge-field strengths are determined. 
The abelian sigma model in (1+1) dimensions is a field theoretical model which has a field $ \phi : S^1 \to S^1 $. An algebra of the quantum field is defined respecting the topological aspect of the model. It is shown that the zero-mode has an infinite number of inequivalent quantizations. It is also shown that when a central extension is introduced into the algebra, the winding operator and the momenta operators satisfy anomalous commutators. 
  We introduce the notion of (nondegenerate) strongly-modular fusion algebras. Here strongly-modular means that the fusion algebra is induced via Verlinde's formula by a representation of the modular group whose kernel contains a congruence subgroup. Furthermore, nondegenerate means that the conformal dimensions of possibly underlying rational conformal field theories do not differ by integers. Our first main result is the classification of all strongly-modular fusion algebras of dimension two, three and four and the classification of all nondegenerate strongly-modular fusion algebras of dimension less than 24. Secondly, we show that the conformal characters of various rational models of W-algebras can be determined from the mere knowledge of the central charge and the set of conformal dimensions. We also describe how to actually construct conformal characters by using theta series associated to certain lattices. On our way we develop several tools for studying representations of the modular group on spaces of modular functions. These methods, applied here only to certain rational conformal field theories, are in general useful for the analysis rational models. 
This letter continues the program aimed at analysis of the scalar product of states in the Chern-Simons theory. It treats the elliptic case with group SU(2). The formal scalar product is expressed as a multiple finite dimensional integral which, if convergent for every state, provides the space of states with a Hilbert space structure. The convergence is checked for states with a single Wilson line where the integral expressions encode the Bethe-Ansatz solutions of the Lame equation. In relation to the Wess-Zumino-Witten conformal field theory, the scalar product renders unitary the Knizhnik-Zamolodchikov-Bernard connection and gives a pairing between conformal blocks used to obtain the genus one correlation functions. 
We construct the most general local effective actions for Goldstone boson fields associated with spontaneous symmetry breakdown from a group $G$ to a subgroup $H$. In a preceding paper, it was shown that any $G$-invariant term in the action, which results from a non-invariant Lagrangian density, corresponds to a non-trivial generator of the de Rham cohomology classes of $G/H$. Here, we present an explicit construction of all the generators of this cohomology for any coset space $G/H$ and compact, connected group $G$. Generators contributing to actions in 4-dimensional space-time arise either as products of generators of lower degree such as the Goldstone-Wilczek current, or are of the Wess-Zumino-Witten type. The latter arise if and only if $G$ has a non-zero $G$-invariant symmetric $d$-symbol, which vanishes when restricted to the subgroup $H$, i.e. when $G$ has anomalous representations in which $H$ is embedded in an anomaly free way. Coupling of additional gauge fields leads to actions whose gauge variation coincides with the chiral anomaly, which is carried here by Goldstone boson fields at tree level. Generators contributing to actions in 3-dimensional space-time arise as Chern-Simons terms evaluated on connections that are composites of the Goldstone field. 
We generalize the Gervais-Neveu gauge to four-dimensional N=1 superspace. The model describes an N=2 super Yang-Mills theory. All chiral superfields (N=2 matter and ghost multiplets) exactly cancel to all loops. The remaining hermitian scalar superfield (matrix) has a renormalizable massive propagator and simplified vertices. These properties are associated with N=1 supergraphs describing a superstring theory on a random lattice world-sheet. We also consider all possible finite matrix models, and find they have a universal large-color limit. These could describe gravitational strings if the matrix-model coupling is fixed to unity, for exact electric-magnetic self-duality. 
The structure constants $N_{\lambda, \mu}^{\mu+\nu}$ of the $sl_2$ Verlinde algebra as functions of $\mu$ either vanish or can be expressed after a change of variable as the weight function of an irreducible representation of $sl_2$. We give a similar formula in the $sl_3$ case. 
  We give an integral representation of solutions of the elliptic Knizhnik-Zamolodchikov-Bernard equations for arbitrary simple Lie algebras. If the level is a positive integer, we obtain formulas for conformal blocks of the WZW model on a torus. The asymptotics of our solutions at critical level gives eigenfunctions of Euler-Calogero-Moser integrable $N$-body systems. As a by-product, we obtain some remarkable integral identities involving classical theta functions. 
We construct an approximate solution of field equations in the Abelian Higgs model which describes motion of a curved vortex. The solution is found to the first order in the inverse mass of the Higgs field with the help of the Hilbert-Chapman-Enskog method. Consistency conditions for the approximate solution are obtained with the help of a classical Ward identity. We find that the Higgs field of the curved vortex of the topological charge $n \geq 2$ in general does not have single n-th order zero. There are two zeros: one is of the (n-1)-th order and it follows a Nambu-Goto type trajectory, the other one is of the first order and its trajectory in general is not of the Nambu-Goto type. For $|n|=1$ the single zero in general does not lie on Nambu-Goto type trajectory. 
We review two-dimensional QCD. The report contains:   1. Introduction.   2. QCD$_2$ as a field theory    2.1 The $1/N$ expansion and spectrum, 2.2 Ambiguity in the self-energy of the quark, 2.3 Polyakov--Wiegmann formula and gauge interactions,  2.4 Strong coupling analysis, 2.5 A Lagrangian realization of the coset construction, 2.6 Chiral interactions.   3. Pure QDC$_2$ and string theory,    3.1 Introduction, 3.2 Wilson loop average and large-$N$ limit, 3.3 String interpretation, 3.4 Collective coordinates approach, 3.5 Phase structure of QCD$_2$,   4. Generalized QCD$_2$ and adjoint-matter coupling ,     4.1 Introduction and motivation, 4.2 Scalar and fermionic matter coupling; quantization 4.3 The Hagedorn transition; supersymmetry 4.4 Landau--Ginzburg  description; spectrum and string theory   5. Algebraic aspects of QCD$_2$ and integrability     5.1 $W_\infty$ algebras for colourless bilinears 5.2 Integrability and duality 5.3 Constraint structure of the theory 5.4 Spectrum and comparison with the $1/N$ expansion 5.5 Integrability conditions and Calogero-type models 5.6 QCD at high energies and two-dimensional field theory   6. Conclusions, 3 appendices. 
We propose two generalisations of the Coulomb potential equation of quantum mechanics and investigate the occurence of algebraic eigenfunctions for the corresponding Scrh\"odinger equations. Some relativistic counterparts of these problems are also discussed. 
In this paper, by using an analogy among {\it quantum mechanics}, {\it electromagnetic beam optics in optical fibers}, and {\it charge particle beam dynamics}, we introduce the concept of {\it coherent states} for charged particle beams in the framework of the {\it Thermal Wave Model} (TWM). We give a physical meaning of the Gaussian-like coherent structures of charged particle distribution that are both naturally and artificially produced in an accelerating machine in terms of the concept of coherent states widely used in quantum mechanics and in quantum optics. According to TWM, this can be done by using a Schr\"{o}dinger-like equation for a complex function, the so-called {\it beam wave function} (BWF), whose squared modulus is proportional to the transverse beam density profile, where Planck's constant and the time are replaced by the transverse beam emittance and by the propagation coordinate, respectively. The evolution of the particle beam, whose initial BWF is assumed to be the simplest coherent state (ground-like state) associated with the beam, in an infinite 1-D quadrupole-like device with small sextupole and octupole aberrations, is analytically and numerically investigated. 
 We show that an infinite number of non-unitary minimal models may describe two dimensional turbulent magnetohydrodynamics (MHD), both in the presence and absence of the Alf'ven effect.  We argue that the existence of a critical dynamical index results in the Alf'ven effect or equivelently the equipartition of energy. We show that there are an infinite number of conserved quantities in $2D-MHD$ turbulent systems both in the limit of vanishing the viscocities and in force free case. In the force free case, using the non-unitary minimal model $ M_{2,7} $ we derive the correlation functions for the velocity stream function and magnetic flux function. Generalising this simple model we find the exponents of the energy spectrum in the inertial range for a class of conformal field theories. 
  We discuss scaling relations in four dimensional simplicial quantum gravity. Using numerical results obtained with a new algorithm called ``baby universe surgery'' we study the critical region of the theory. The position of the phase transition is given with high accuracy and some critical exponents are measured. Their values prove that the transition is continuous. We discuss the properties of two distinct phases of the theory. For large values of the bare gravitational coupling constant the internal Hausdorff dimension is {\em two} (the elongated phase), and the continuum theory is that of so called branched polymers. For small values of the bare gravitational coupling constant the internal Hausdorff dimension seems to be {\em infinite} (the crumpled phase). We conjecture that this phase corresponds to a theory of topological gravity. {\em At} the transition point the Hausdorff dimension might be finite and larger than two. This transition point is a potential candidate for a non-perturbative theory of quantum gravity. 
A formula is proposed for continuing physical correlation functions to non-integer numbers of dimensions, expressing them as infinite weighted sums over the same correlation functions in arbitrary integer dimensions. The formula is motivated by studying the strong coupling expansion, but the end result makes no reference to any perturbation theory. It is shown that the formula leads to the correct dimension dependence in weak coupling perturbation theory at one loop. 
We present the exact and explicit solution of the principal chiral field in two dimensions for an infinitely large rank group manifold.  The energy of the ground state is explicitly found for the external Noether's fields of an arbitrary magnitude.  At small field we found an inverse logarithmic singularity in the ground state energy at the mass gap which indicates that at $N=\infty$ the spectrum of the theory contains extended objects rather than pointlike particles. 
In this article we consider an N-brane description of an (N+3)-dimensional black hole horizon. First of all, we start by reviewing a previous work where a string theory is used as describing the dynamics of the event horizon of a four dimensional black hole. Then we consider a particle model defined on one dimensional Euclidean line in a three dimensional black hole as a target spacetime metric. By solving the field equations we find a ``world line instanton'' which connects the past event horizon with the future one. This solution gives us the exact value of the Hawking temperature and to leading order the Bekenstein-Hawking formula of black hole entropy. We also show that this formalism is extensible to an arbitrary spacetime dimension. Finally we make a comment of one-loop quantum correction to the black hole entropy . 
The analysis of $90^{\circ}$ vortex-vortex scattering is extended to $\frac{\pi}{n}$ scattering in all head-on collisions of $n$ vortices in the Abelian Higgs model. A Cauchy problem with initial data that describe the scattering of $n$ vortices is formulated. It is shown that this Cauchy problem has a unique global finite-energy solution. The symmetry of the solution and the form of the local analytic solution then show that $\frac{\pi}{n}$ scattering is realised. 
The $O(n)$ Gross-Neveu model for $n < 2$ presents a massless phase that can be characterized by right-left mover scattering processes. The limit $n \goto 0$ describes the on-shell properties of the random bond Ising model. 
  We examine the Seiberg-Witten treatment of N=2 super Yang-Mills theory, and note that in the strong coupling region of moduli space, some massive particle excitations appear to have negative norm. We discuss the significance of our observation. 
  A bosonized action, that reproduces the structure of the 't Hooft equation for $QCD_2$ in the large-$N$ limit, up to regularization dependent terms, is derived. 
  We study conformal field theories describing two massless one-dimensional fields interacting at a single spatial point. The interactions we include are periodic functions of the bosonized fields separately plus a ``magnetic'' interaction that mixes the two fields. Such models arise in open string theory and dissipative quantum mechanics and perhaps in edge state tunneling in the fractional quantized Hall effect. The partition function for such theories is a Coulomb gas with interchange phases arising from the magnetic field. These ``fractional statistics'' have a profound effect on the phase structure of the Coulomb gas. In this paper we present new exact and approximate results for this type of generalized Coulomb gas. 
The method of differential regularization is applied to calculate explicitly the one-loop effective potential of a bosonized Nambu--Jona-Lasinio model consisting of scalar and pseudoscalar fields. The regularization scheme independent relation for the $\sigma$ mass sum rule is obtained. This method can be readily applied to extended NJL models with gauge fields. 
  We compare the one-loop corrections to the entropy of a black hole, from quantum fields of spin zero, one-half, and one, to the entropy of entanglement of the fields. For fields of spin zero and one-half the black hole entropy is identical to the entropy of entanglement. For spin one the two entropies differ by a contact interaction with the horizon which appears in the black hole entropy but not in the entropy of entanglement. The contact interaction can be expressed as a path integral over particle paths which begin and end on the horizon; it is the field theory limit of the interaction proposed by Susskind and Uglum which couples a closed string to an open string stranded on the horizon. 
In this paper we describe in some detail the representation of the topological $CP^1$ model in terms of a matrix integral which we have introduced in a previous article. We first discuss the integrable structure of the $CP^1$ model and show that it is governed by an extension of the 1-dimensional Toda hierarchy. We then introduce a matrix model which reproduces the sum over holomorphic maps from arbitrary Riemann surfaces onto $CP^1$. We compute intersection numbers on the moduli space of curves using geometrical method and show that the results agree with those predicted by the matrix model. We also develop a Landau-Ginzburg (LG) description of the $CP^1$ model using a superpotential $e^X+e^{t_{0,Q}}e^{-X}$ given by the Lax operator of the Toda hierarchy ($X$ is the LG field and $t_{0,Q}$ is the coupling constant of the K\"ahler class). The form of the superpotential indicates the close connection between $CP^1$ and $N=2$ supersymmetric sine-Gordon theory which was noted some time ago by several authors. We also discuss possible generalizations of our construction to other manifolds and present a LG formulation of the topological $CP^2$ model. 
The third Poisson structure of KdV equation in terms of canonical ``free fields'' and reduced WZNW model is discussed. We prove that it is ``diagonalized'' in the Lagrange variables which were used before in formulation of 2D gravity. We propose a quantum path integral for KdV equation based on this representation. 
We provide explicit realizations for the operators which when exchanged give rise to the scattering matrix. For affine Toda field theory we present two alternative constructions, one related to a free bosonic theory and the other formally to the quantum affine Heisenberg algebra of $U_q(\hat{Sl_2})$. 
We exhibit the gauge-group independence (``universality'') of all normalized non-intersecting Wilson loop expectation values in the large N limit of two-dimensional Yang-Mills theory. This universality is most easily understood via the string theory reformulation of these gauge theories. By constructing an isomorphism between the string maps contributing to normalized Wilson loop expectation values in the different theories, we prove the large N universality of these observables on any surface. The string calculation of the Wilson loop expectation value on the sphere also leads to an indication of the large N phase transition separating strong- and weak-coupling phases. 
  We show that the strong coupling phase of the non-Abelian Thirring model is dual to the weak-coupling phase of a system of two WZNW models coupled to each other through a current-current interaction. This latter system is integrable and is related to a perturbed conformal field theory which, in the large $|k|$ limit, has a nontrivial zero of the perturbation-parameter beta-function. The non-Abelian Thirring model reduces to a free fermion theory plus a topological field theory at this critical point, which should therefore be identified with the isoscalar Dashen-Frishman conformal point. The relationship with the Gross-Neveu model is discussed. 
Proper symmetries act on fields while pseudo-symmetries act on both fields and coupling constants. We identify the pseudo-duality groups that act as symmetries of the equations of motion of general systems of scalar and vector fields and apply our results to $N=2,4$ and $8$ supergravity theories. We present evidence that the pseudo-duality group for both the heterotic and type II strings toroidally compactified to four dimensions is $Sp(56;\Z)\times D$, where $D$ is a certain subgroup of the diffeomorphism group of the scalar field target space. This contains the conjectured heterotic $S\times T$ or type II $U$ proper duality group as a subgroup. 
We present a very quick and powerful method for the calculation of heat-kernel coefficients. It makes use of rather common ideas, as integral representations of the spectral sum, Mellin transforms, non-trivial commutation of series and integrals and skilful analytic continuation of zeta functions on the complex plane. We apply our method to the case of the heat-kernel expansion of the Laplace operator on a $D$-dimensional ball with either Dirichlet, Neumann or, in general, Robin boundary conditions. The final formulas are quite simple. Using this case as an example, we illustrate in detail our scheme ---which serves for the calculation of an (in principle) arbitrary number of heat-kernel coefficients in any situation when the basis functions are known. We provide a complete list of new results for the coefficients $B_3,...,B_{10}$, corresponding to the $D$-dimensional ball with all the mentioned boundary conditions and $D=3,4,5$. 
  It has been argued that any evolution law taking pure states to mixed states in quantum field theory necessarily gives rise to violations of either causality or energy-momentum conservation, in such a way as to have unacceptable consequences for ordinary laboratory physics. We show here that this is not the case by giving a simple class of examples of Markovian evolution laws where rapid evolution from pure states to mixed states occurs for a wide class of states with appropriate properties at the ``Planck scale", suitable locality and causality properties hold for all states, and the deviations from ordinary dynamics (and, in particular, violations of energy-momentum conservation) are unobservably small for all states which one could expect to produce in a laboratory. In addition, we argue (via consideration of other, non-Markovian models) that conservation of energy and momentum for all states is not fundamentally incompatible with causality in dynamical models in which pure states evolve to mixed states. 
A $q$-deformed free scalar relativistic particle is discussed in the framework of the BRST formalism. The $q$-deformed local gauge symmetry and reparametrization invariance of the first-order Lagrangian have been exploited for the BRST quantization of this system on a $GL_{q}(2)$ invariant quantum world-line. The on-shell equivalence of these BRST charges requires the deformation parameter to be $ \pm 1 $ under certain identifications.The same restriction ($ q= \pm 1 $) emerges from the conservation of the $q$-deformed BRST charge on an arbitrary (unconstrained) manifold and the validity of the BRST algebra. The solutions for the equations of motion respect $GL_{q}(2)$ invariance on the mass-shell at any arbitrary value of the evolution parameter characterizing the quantum world-line. 
The Renormalization Group equation describing the evolution of the metric of the nonlinear sigma model poses some nice mathematical problems involving functional analysis, differential geometry and numerical analysis. In this article we briefly report some results obtained from the numerical study of the solutions in the case of a two dimensional target space (deformation of the $O(3)$ sigma model). In particular, our analysis shows that the so-called sausages define an attracting manifold in the $U(1)$-symmetric case, at one-loop level. Moreover, data from two-loop evolution are used to test the association put forward in Nucl. Phys., B406 (1993) 521 between the so-called $SSM_{\nu}$ field theory and a certain $U(1)$-symmetric, factorized scattering theory (FST). 
A conformal field theory representing a four-dimensional classical solution of heterotic string theory is presented. The low-energy limit of this solution has U(1) electric and magnetic charges, and also nontrivial axion and dilaton fields. The low-energy metric contains mass, NUT and rotation parameters. We demonstrate that this solution corresponds to part of an extremal limit of the Kerr-Taub-NUT dyon solution. This limit displays interesting `remnant' behaviour, in that asymptotically far away from the dyon the angular momentum vanishes, but far down the infinite throat in the neighbourhood of the horizon (described by our CFT) there is a non-zero angular velocity. A further natural generalization of the CFT to include an additional parameter is presented, but the full physical interpretation of its role in the resulting low energy solution is unclear. 
We review some recent results on the Calogero-Sutherland model with emphasis upon its algebraic aspects. We give integral formulae for excited states (Jack polynomials) of this model and their relations with W_n singular vectors and generalized matrix models. 
     We classify states saturating a double or a single supersymmetric positivity bound of a four-dimensional N=4 supersymmetry. The massive four-dimensional double-bound states (Bogomolny states) are shown to form a light-like representation of ten-dimensional supersymmetry. The single-bound states form a massive representation (centrino multiplet) of a four-dimensional supersymmetry. The first component of the centrino multiplet is identified with extreme black holes with regular horizon which have one quarter of unbroken supersymmetry. The centrino multiplet includes a massive spin 3/2 state, the centrino, as a highest spin state.      Existence of massive black hole supermultiplets may affect the massless sector of the theory. Assuming that gluino condensate is formed one can study its properties. The bilinear combination of covariantly constant Killing spinors supplies the possible form for a gluino condensate. The condensate has null properties, does not introduce a cosmological constant, and may lead to a spontaneous breaking of local supersymmetry. This suggests that centrino may provide a consistent super-Higgs mechanism. 
I study the spontaneous breakdown of supersymmetry when higher-dimensional Yang-Mills or the type-I $SO(32)$ string theory are compactified on magnetized tori. Because of the universal gyromagnetic ratio $g=2$, the splittings of all multiplets are given by the product of charge times internal helicity operators. As a result such compactifications have two remarkable and robust features: {\it (a)} they can reconcile {\it chirality} with {\it extended} low-energy supersymmetry in the limit of large tori, and {\it (b)} they can trigger gauge-symmetry breaking, via Nielsen-Olesen instabilities, at a scale tied classically to $m_{SUSY}$. I exhibit a compactification of the $SO(32)$ superstring, in which magnetic fields break spontaneously $N=4$ supersymmetry, produce the standard-model gauge group with three chiral families of quarks and leptons, and trigger electroweak symmetry breaking. I discuss supertrace relations and the ensuing ultraviolet softness. As with other known mechanisms of supersymmetry breaking, the one proposed here faces two open problems: the threat to perturbative calculability in the decompactification limit, and the problem of gravitational stability and in particular of the cosmological constant. I explain, however, why a good classical description of the vacuum may require small tadpoles for the dilaton, moduli and metric. 
Starting from string field theory for 2d gravity coupled to c=1 matter we analyze N-point off-shell tree amplitudes of discrete states. The amplitudes exhibit the pole structure and we use the oscillator representation to extract the residues. The residues are generated by a simple effective action. We show that the effective action can be directly deduced from a string field action in a special transversal-like gauge. 
The perturbation theory expansion of the Aharonov-Bohm scattering amplitude has previously been studied in the context of quantum mechanics for spin zero and spin-1/2 particles as well in Galilean covariant field theory. This problem is reconsidered in the framework of the model in which the flux line is considered to have a finite radius which is shrunk to zero at the end of the calculation. General agreement with earlier results is obtained but with the advantage of a treatment which unifies all the various subcases. 
The Schwinger-Dyson equations of the Makeenko-Migdal type, when supplemented with some simple equations as consequence of supersymmetry, form a closed set of equations for Wilson loops and related quantities in the two dimensional super-gauge theory. We solve these equations. It appears that the planar Wilson loops are described by the Nambu string without folds. We also discuss how to put the model on a spatial lattice, where a peculiar gauge is chosen in order to keep one supersymmetry on the lattice. Supersymmetry is unbroken in this theory. We comment on possible generalization of these considerations to other models. 
The stability of a new class of hairy black hole solutions in the coupled system of Einstein-Yang-Mills-Higgs is examined, generalising a method suggested by Brodbeck and Straumann and collaborators, and Volkov and Gal'tsov. The method maps the algebraic system of linearised radial perturbations of the various field modes around the black hole solution into a coupled system of radial equations of Schr\"odinger type. No detailed knowledge of the black hole solution is required, except from the fact that the boundary conditions at the physical space-time boundaries (horizons) must be such so as to guarantee the {\it finiteness} of the various expressions involved. In this way, it is demonstrated that the above Schr\"odinger equations have bound states, which implies the instability of the associated black hole solution. 
  Recent work has made clear that we have far more control over the dynamics of supersymmetric than non-supersymmetric theories. Here, I discuss some issues in dynamical supersymmetry breaking both in ordinary field theory and in string theory. In string theory, in particular, it is possible to show, in some circumstances, that stringy non-perturbative effects are {\it smaller} than effects visible in the low energy field theory. Observations of this sort suggest a general approach to string phenomenology. 
We study the path integrals of the holomorphic Yang-Mills theory on compact K\"{a}hler surface with $b_2^+ = 1$. Based on the results, we examine the correlation functions of the topological Yang-Mills theory and the corresponding Donaldson invariants as well as their transition formulas. 
We find that the Boltzmann weight of the three-dimensional Baxter-Bazhanov model is dependent on four spin variables which are the linear combinations of the spins on the corner sites of the cube and the Wu-Kadanoff duality between the cube and vertex type tetrahedron equations is obtained explicitly for the Baxter-Bazhanov model. Then a three-dimensional vertex model is obtained by considering the symmetry property of the weight function, which is corresponding to the three-dimensional Baxter-Bazhanov model. The vertex type weight function is parametrized as the dihedral angles between the rapidity planes connected with the cube. And we write down the symmetry relations of the weight functions under the actions of the symmetry group $G$ of the cube. The six angles with a constrained condition, appeared in the tetrahedron equation, can be regarded as the six spectrums connected with the six spaces in which the vertex type tetrahedron equation is defined. 
We reduce the dual version of $D=10$, $N=1$ supergravity coupled to $n$ vector fields to four dimensions, and derive the $SL(2,R)\times O(6,6+n)$ transformations which leave the equations of motion invariant. For $n=0$ $SL(2,R)$ is also a symmetry of the action, but for $n>0$ only those $SL(2,R)$ transformations which act linearly on all fields leave the action invariant. The resulting four-dimensional theory is related to the bosonic part of the usual formulation of $N=4$ supergravity coupled to matter by a duality transformation. 
We recast the quaternionic Gursey-Tze solution, which is a fourfold quasi-periodic self-dual Yang-Mills field with a unit instanton number per Euclidean spacetime cell, into an ordinary coordinate formulation. After performing the sum in the Euclidean time direction, we use an observation by Rossi which suggests the solution represents an arrangement with a BPS monopole per space lattice cell. This may provide a concrete realization of a monopole condensate in pure Yang-Mills theory. 
We propose a way of bosonizing free world-sheet fermions for $4$-dimensional heterotic string theory formulated in Minkowski space-time. We discuss the differences as compared to the standard bosonization performed in Euclidean space-time. 
In recent works by Singer, Douglas and Gopakumar and Gross an application of results of Voiculescu from non-commutative probability theory to constructions of the master field for large $N$ matrix field theories have been suggested. In this note we consider interrelations between the master field and quantum groups. We define the master field algebra and observe that it is isomorphic to the algebra of functions on the quantum group $SU_q(2)$ for $q=0$. The master field becomes a central element of the quantum group Hopf algebra. The quantum Haar measure on the $SU_q(2)$ for any $q$ gives the Wigner semicircle distribution for the master field. Coherent states on $SU_q(2)$ become coherent states in the master field theory. 
  Recently, it has been shown how to perform the quantum hamiltonian reduction in the case of general $sl(2)$ embeddings into Lie (super)algebras, and in the case of general $osp(1|2)$ embeddings into Lie superalgebras. In another development it has been shown that when $H$ and $H'$ are both subalgebras of a Lie algebra $G$ with $H'\subset H$, then classically the $W(G,H)$ algebra can be obtained by performing a secondary hamiltonian reduction on $W(G,H')$. In this paper we show that the corresponding statement is true also for quantum hamiltonian reduction when the simple roots of $H'$ can be chosen as a subset of the simple roots of $H$. As an application, we show that the quantum secondary reductions provide a natural framework to study and explain the linearization of the $W$ algebras, as well as a great number of new realizations of $W$ algebras. 
  Using the collective field method, we find a relation between the Jack symmetric polynomials, which describe the excited states of the Calogero-Sutherland model, and the singular vectors of the $W_N$ algebra. Based on this relation, we obtain their integral representations. We also give a direct algebraic method which leads to the same result, and integral representations of the skew-Jack polynomials. 
  In the abelian Higgs model, among other situations, it has recently been realized that the head-on scattering of $n$ solitons distributed symmetrically around the point of scattering is by an angle $\pi/n$, independant of various details of the scattering. In this note, it is first observed that this result is in fact not entirely surprising: the above is one of only two possible outcomes. Then, a generalization of an argument given by Ruback for the case of two gauge theory vortices in the Bogomol'nyi limit is used to show that in the geodesic approximation the above result follows from purely geometric considerations. 
We construct explicit canonical transformations producing non-abelian duals in principal chiral models with arbitrary group G. Some comments concerning the extension to more general $\sigma$-models, like WZW models, are given. 
The Galilean invariance in three dimensional space-time is considered. It appears that the Galilei group in 2+1 dimensions posses a three-parameter family of projective representations. Their physical interpretation is discussed in some detail. 
The higher spin properties of the non-abelian bosonization in the classical theory are investigated. Both the symmetry transformation algebra and the classical current algebra for the non-abelian free fermionic model are linear Gel'fand-Dickey type algebras. However, for the corresponding WZNW model these algebras are different. There exist symmetry transformations which algebra remains the linear Gel'fand-Dickey algebra while in the corresponding current algebra nonlinear terms arised. Moreover, this algebra is closed (in Casimir form) only in an extended current space in which nonlinear currents are included. In the affine sector, it is necessary to be included higher isotopic spin current too. As result we have a triple extended algebra. 
We show that QCD$_2$ on 2D pseudo-manifolds is consistent with the Gross-Taylor string picture. It allows us to introduce a model describing the one-dimensional evolution of the QCD$_2$ string (in the sense that QCD$_2$ itself is regarded as a zero-dimensional system). The model is shown to possess the third order phase transition associated with the $c=1$ Bose string below which it becomes equivalent to the vortex-free sector of the 1-dimensional matrix model. We argue that it could serve as a toy model for the glueball-threshold behavior of multicolor QCD. 
  Some exact relations for the spectral density $\rho(\lambda)$ of the Euclidean Dirac operator in $QCD$ are derived. They follow directly from the chiral symmetry of the $QCD$ lagrangian with massless quarks. New results are obtained both in thermodynamic limit when the Euclidean volume $V$ is sent to infinity and also in the theory defined in finite volume where the spectrum is discrete and a nontrivial information on $\rho(\lambda)$ in the region $\lambda \sim 1/(|<\bar{q}q>_0|V)$ (the characteristic level spacing) can be obtained. These exact results should be confronted with "experimental" numerical simulations on the lattices and in some particular models for $QCD$ vacuum structure and may serve as a nontrivial test of the validity of these simulations. 
Symmetry properties of stochastic dynamical systems described by stochastic differential equation of Stratonovich type and related conserved quantities are discussed, extending previous results by Misawa. New conserved quantities are given by applying symmetry operators to known conserved quantities. Some detailed examples are presented. 
In the present article we analyze the problem of a relativistic Dirac electron in the presence of a combination of a Coulomb field, a $1/r$ scalar potential as well as a Dirac magnetic monopole and an Aharonov-Bohm potential. Using the algebraic method of separation of variables, the Dirac equation expressed in the local rotating diagonal gauge is completely separated in spherical coordinates, and exact solutions are obtained. We compute the energy spectrum and analyze how it depends on the intensity of the Aharonov-Bohm and the magnetic monopole strengths. 
A method to define the complex structure and separate the conformal mode is proposed for a surface constructed by two-dimensional dynamical triangulation. Applications are made for surfaces coupled to matter fields such as $n$ scalar fields ($n=0,1$ and $4$) and $m$ Ising spins ($m=1$ and $3$). We observe a well-defined complex structure for cases when the matter central charges are less than and equal to one, while it becomes unstable beyond $c=1$. This can be regarded as the transition expected in analytic theories. 
The octonion X-product changes the octonion multiplication table, but does not change the role of the identity. The octonion XY-product is very similar, but shifts the identity as well. This will be of interest to those applying th octonions to string theory. 
We derive a new algebraic relation which can be used to find various spinor loop anomalies. We show that this relation includes the Wess-Zumino consistency condition. For an example, we consider the chiral anomaly. With this formalism, the consistent anomaly and the covariant anomaly are determined simultaneously. 
  We demonstrate that string consistency in four spacetime dimensions leads to a spectrum of string states which satisfies the supertrace constraints Str(M^0)=0 and Str(M^2)=\Lambda at tree level, where \Lambda is the one-loop string cosmological constant. This result holds for a large class of string theories, including critical heterotic strings. For strings lacking spacetime supersymmetry, these supertrace constraints will be satisfied as a consequence of a hidden ``misaligned supersymmetry'' in the string spectrum. These results thus severely constrain the possible supersymmetry-breaking scenarios in string theory, and suggest a new intrinsically stringy mechanism whereby such supertrace constraints may be satisfied without phenomenologically unacceptable consequences. 
Allowing for the inclusion of the parity operator, it is possible to construct an oscillator model whose Hamiltonian admits an EXACT square root, which is different from the conventional approach based on creation and annihilation operators. We outline such a model, the method of solution and some generalizations. 
The equations of motion of the massless sector of the two dimensional string theory, obtained by compactifying the heterotic string theory on an eight dimensional torus, is known to have an affine o(8,24) symmetry algebra generating an O(8,24) loop group. In this paper we study how various known discrete S- and T- duality symmetries of the theory are embedded in this loop group. This allows us to identify the generators of the discrete duality symmetry group of the two dimensional string theory. 
  We propose a nonperturbative formulation of chiral gauge theories. The method involves a `pre-regulation' of the gauge fields, which may be implemented on a lattice, followed by a computation of the chiral fermion determinant in the form of a functional integral which is regularized in the continuum. Our result for the chiral determinant is expressed in terms of the vector-like Dirac operator and hence can be realized in lattice simulations. We investigate the local and global anomalies within our regularization scheme. We also compare our result for the chiral determinant to previous exact $\zeta$-function results. Finally, we use a symmetry property of the chiral determinant to show that the partition function for a chiral gauge theory is real. 
We give empirical evidence that the UV-divergences of a renormalizable field theory are knot invariants. 
A general description of string excitations in stationary spacetimes is developed. If a stationary string passes through the ergosphere of a 4-dimensional black hole, its world-sheet describes a 2-dimensional black (or white) hole with horizon coinciding with the static limit of the 4-dimensional black hole. Mathematical results for 2-dimensional black holes can therefore be applied to physical objects (say) cosmic strings in the vicinity of Kerr black holes. An immediate general result is that the string modes are thermally excited. The string excitations are determined by a coupled system of scalar field equations in the world-sheet metric. In the special case of excitations propagating along a stationary string in the equatorial plane of the Kerr-Newman black hole, they reduce to the $s$-wave scalar field equations in the 4-dimensional Reissner-Nordstr\"{o}m black hole. We briefly discuss possible applications of our results to the black hole information puzzle. 
The non-relativistic `Dirac' equation of L\'evy-Leblond is used to describe a spin {\small 1/2} particle interacting with a Chern-Simons gauge field. Static, purely magnetic, self-dual spinor vortices are constructed. The solution can be `exported' to a uniform magnetic background field. 
  The Hamiltonian BRST quantization of the null spinning string for different number of supersymmetries is given. A null spinning string with manifest space-time conformal invariance is constructed. Its Brst quantization gives negative critical dimension for $N\neq 0$ and reproduces previous results for $N=0$. 
We find a new class of (2,0)-supersymmetric two-dimensional sigma models with torsion and target spaces almost complex manifolds extending similar results for models with (2,2) supersymmetry. These models are invariant under a new symmetry which is generated by a Noether charge of Lorentz weight one and it is associated to the Nijenhuis tensor of the almost complex structure of the sigma model target manifold. We compute the Poisson bracket algebra of charges of the above (2,0)-and (2,2)-supersymmetric sigma models and show that it closes but it is not isomorphic to the standard (2,0) and (2,2) supersymmetry algebra, respectively. Examples of such (2,0)- and (2,2)-supersymmetric sigma models with target spaces group manifolds are also given. In addition, we study the quantisation of the (2,0)-supersymmetric sigma models, compute the anomalies of their classical symmetries and examine their cancellation. Furthermore, we examine the massive extension of (2,0)-supersymmetric sigma models with target spaces almost complex manifolds, and study the topological twist of the new supersymmetry algebras. 
   Recently, using the model of $N=2$ supergravity --- vector multiplets interaction with the scalar field geometry $SU(1,m)/SU(m)\otimes U(1)$ as an example, we have shown that even when the scalar field geometry is fixed, one can have a whole family of the Lagrangians, which differ by vector field duality transformation. In this paper we carry out the construction of such families for the case of $N=3$ and $N=4$ supergravities, the scalar field geometry being $SU(3,m)/SU(3)\otimes SU(m)\otimes U(1)$ and $SU(1,1)/U(1)\otimes O(6,m)/O(6)\otimes O(m)$, correspondingly. Moreover, it turns out that these families contain, as a partial case, the models describing the interaction of arbitrary number of vector multiplets with our hidden sectors, admitting spontaneous supersymmetry breaking without a cosmological term. 
We consider 1+1 D theories which are free everywhere except for cosine and magnetic interactions on the boundary. These theories arise in dissipative quantum systems, open string theory, and, in special cases, tunneling in quantum Hall systems. These boundary systems satisfy an approximate SL(2,Z) symmetry as a function of flux per unit cell and dissipation. At special multicritical points, they also satisfy a set of reparametrization Ward identities and have homogeneous, piecewise-linear correlation functions in momentum space. In this paper, we use these symmetries to find exact solutions for some of the correlation functions. We also comment on the form of the correlation functions in general, and verify that the SL(2,Z) duality transformation between different critical points is satisfied exactly in all cases where the full solution is known. 
We study a quite general family of dynamical $r$-matrices for an auxiliary loop algebra ${\cal L}({su(2)})$ related to restricted flows for equations of the KdV type. This underlying $r$-matrix structure allows to reconstruct Lax representations and to find variables of separation for a wide set of the integrable natural Hamiltonian systems. As an example, we discuss the Henon-Heiles system and a quartic system of two degrees of freedom in detail. 
In a recent work, the consequences of quantizing a real scalar field $\Phi$ according to generalized ``quon'' statistics in a dynamically evolving curved spacetime (~which, prior to some initial time and subsequent to some later time, is flat~) were considered. Here a similar calculation is performed; this time we quantize $\Phi$ via the Calogero-Vasiliev oscillator algebra, described by a real parameter $\nu > -1/2$. It is found that both conservation ( $\nu \rightarrow \nu$ ) and anticonservation ( $\nu \rightarrow - \nu$ ) of statistics is allowed. We find that for mathematical consistency the Bogoliubov coefficients associated with the $i$'th field mode must satisfy $|\alpha_i |^2 - | \beta_i |^2 = 1$ with $| \beta_i |^2$ taking an integer value. 
  In a recent paper \cite{[Good1]} Good postulated new rules of quantization, one of the major features of which is that the quantum evolution of the wave function is always given by ordinary differential equations. In this paper we analyse the proposal in some detail and discuss its viability and its relationship with the standard quantum theory. As a byproduct, a simple derivation of the `mass spectrum' for the Klein-Gordon field is presented, but it is also shown that there is a complete additional spectrum of negative `masses'. Finally, two major reasons are presented against the viability of this alternative proposal: a) It does not lead to the correct energy spectrum for the hydrogen atom. b) For field models, the standard quantum theory cannot be recovered from this alternative description. 
The dynamics of an M-dimensional extended object whose M+1 dimensional world volume in M+2 dimensional space-time has vanishing mean curvature is formulated in term of geometrical variables (the first and second fundamental form of the time-dependent surface $\sum_M$), and simple relations involving the rate of change of the total area of $\sum_M$, the enclosed volume as well as the spatial mean -- and intrinsic scalar curvature, integrated over $\sum_M$, are derived. It is shown that the non-linear equations of motion for $\sum_M(t)$ can be viewed as consistency conditions of an associated linear system that gives rise to the existence of non-local conserved quantities (involving the Christoffel-symbols of the flat M+1 dimensional euclidean submanifold swept out in ${\Bbb R}^{M+1}$). For M=1 one can show that all motions are necessarily singular (the curvature of a closed string in the plane can not be everywhere regular at all times) and for M=2, an explicit solution in terms of elliptic functions is exhibited, which is neither rotationally nor axially symmetric. As a by-product, 3-fold-periodic spacelike maximal hypersurfaces in ${\Bbb R}^{1,3}$ are found. 
  We suggest a new family of unitary RSOS scattering models which is obtained by placing the SO(N) critical models in "electric" or "magnetic" field. These fields are associated with two operators from the space of the SO(N) RCFT corresponding to the highest weight of the vector representation of SO(N). A perturbation by the external fields destroys the Weyl group symmetry of an original statistical model. We show that the resulting kinks scattering theories can be viewed as affine imaginary Toda models for non-simply-laced and twisted algebras taken at rational values (roots of unity) of $q$-parameter. We construct the fundamental kink $S$-matrices for these models. At the levels $k=1, 2, \infty$ our answers match the known results for the Sine-Gordon, $Z_{2N}$ - parafermions and free fermions respectively. As a by-product in the SO(4)-case we obtain an RSOS $S$-matrix describing an integrable coupling of two minimal CFT. 
Manin triple construction of N=4 superconformal field theories is considered. The correspondence between quasi Frobenius finite-dimensional Lie algebras and N=4 superconformal field theories is established. 
The emergence of violations of conformal invariance in the form of non-local operators in the two-dimensional action describing solitons inevitably leads to the introduction of collective coordinates as two dimensional ``wormhole parameters''. 
  We discuss minisuperspace aspects a non empty Robertson-Walker universe containing scalar matter field. The requirement that the Wheeler-DeWitt (WDW) operator be self adjoint is a key ingredient in constructing the physical Hilbert space and has non-trivial cosmological implications since it is related with the problem of time in quantum cosmology. Namely, if time is parametrized by matter fields we find two types of domains for the self adjoint WDW operator: a non trivial domain is comprised of zero current (Hartle-Hawking type) wave functions and is parametrized by two new parameters, whereas the domain of a self adjoint WDW operator acting on tunneling (Vilenkin type) wave functions is a {\em single} ray. On the other hand, if time is parametrized by the scale factor both types of wave functions give rise to non trivial domains for the self adjoint WDW operators, and no new parameters appear in them. 
  To comply with recent developments of path integrals in spaces with curvature and torsion we find the correct variational principle for the classical trajectories. Although the action depends only on the length, the trajectories are {\em autoparallels} rather than geodesics due to the effects of a new torsion force. 
  We use a recently developed action principle in spaces with curvature and torsion to derive the Euler equations of motion for a rigid body within the body-fixed coordinate system. This serves as an example that the particle trajectories in a space with curvature and torsion follow the straightest paths (autoparallels), not the shortest paths (geodesics), as commonly believed. 
We consider some differential geometric classes of local and nonlocal Poisson and symplectic structures on loop spaces of smooth manifolds which give natural Hamiltonian and multihamiltonian representations for some important nonlinear equations of mathematical physics and field theory such as nonlinear sigma models with torsion, degenerate Lagrangian systems of field theory, systems of hydrodynamic type, N-component systems of Heisenberg magnet type, Monge-Amp\`ere equations, the Krichever-Novikov equation and others. In addition, we shall prove integrability of some class of nonhomogeneous systems of hydrodynamic type and give a description of nonlinear partial differential equations of associativity in $2D$ topological field theories (for some special type solutions of the Witten-Dijkgraaf-E.Verlinde-H.Verlinde (WDVV) system) as integrable nondiagonalizable weakly nonlinear homogeneous systems of hydrodynamic type. 
A microscopic formulation of Haldane's exclusions statistics is given in terms of a priori occupation probabilities of states. It is shown that negative probabilities are always necessary to reproduce fractional statistics. Based on this formulation, a path-integral realization for systems with exclusion statistics is derived. This has the advantage of being generalizable to interacting systems, and can be used as the starting point for further generalizations of statistics. As a byproduct, the vanishing of the heat capacity at zero temperature for exclusion statistics systems is proved. 
  It is well-known that principal chiral models and symmetric space models in two-dimensional Minkowski space have an infinite-dimensional algebra of hidden symmetries. Because of the relevance of symmetric space models to duality symmetries in string theory, the hidden symmetries of these models are explored in some detail. The string theory application requires including coupling to gravity, supersymmetrization, and quantum effects. However, as a first step, this paper only considers classical bosonic theories in flat space-time. Even though the algebra of hidden symmetries of principal chiral models is confirmed to include a Kac--Moody algebra (or a current algebra on a circle), it is argued that a better interpretation is provided by a doubled current algebra on a semi-circle (or line segment). Neither the circle nor the semi-circle bears any apparent relationship to the physical space. For symmetric space models the line segment viewpoint is shown to be essential, and special boundary conditions need to be imposed at the ends. The algebra of hidden symmetries also includes Virasoro-like generators. For both principal chiral models and symmetric space models, the hidden symmetry stress tensor is singular at the ends of the line segment. 
  We show how to construct the exact factorized S-matrices of 1+1 dimensional quantum field theories whose symmetry charges generate a quantum affine algebra. Quantum affine Toda theories are examples of such theories. We take into account that the Lorentz spins of the symmetry charges determine the gradation of the quantum affine algebras. This gives the S-matrices a non-rigid pole structure. It depends on a kind of ``quantum'' dual Coxeter number which will therefore also determine the quantum mass ratios in these theories. As an example we explicitly construct S-matrices with $U_q(c_n^{(1)})$ symmetry. 
We use time-independent canonical transformation methods to discuss the energy eigenfunctions for the simple linear potential, pedagogically setting the stage for some field theory calculations to follow. We then discuss the Schr\"odinger wave-functional method of calculating correlation functions for Liouville field theory. We compare this approach to earlier treatments, in particular we check against known weak-coupling results for the Liouville field defined on a cylinder. Finally, we further set the stage for future Liouville calculations on curved two-manifolds and briefly discuss simple quantum mechanical systems with time-dependent Hamiltonians. 
  We complete the study of 4-dimensional (4-d), static, spherically symmetric, supersymmetric black holes (BH's) in Abelian $(4+n)$-d Kaluza-Klein theory, by showing that for such solutions $n$ electric charges $\vec{\cal Q} \equiv (Q_1,...,Q_n)$ and $n$ magnetic charges $\vec{\cal P} \equiv (P_1,...,P_n)$ are subject to the constraint $\vec{\cal P}\cdot \vec{\cal Q}=0$. All such solutions can be obtained by performing the SO(n) rotations, which do not affect the 4-d space-time metric and the volume of the internal space, on the supersymmetric $U(1)_M\times U(1)_E$ BH's, {\it i.e.}, supersymmetric BH's with a diagonal internal metric. 
  We present the explicit form for all the four dimensional, static, spherically symmetric solutions in $(4+n)$-d Abelian Kaluza-Klein theory by performing a subset of $SO(2,n)$ transformations corresponding to four $SO(1,1)$ boosts on the Schwarzschild solution, supplemented by $SO(n)/SO(n-2)$ transformations. The solutions are parameterized by the mass $M$, Taub-Nut charge $a$, $n$ electric $\vec{\cal Q}$ and $n$ magnetic $\vec{\cal P}$ charges. Non-extreme black holes (with zero Taub-NUT charge) have either the Reissner-Nordstr\" om or Schwarzschild global space-time. Supersymmetric extreme black holes have a null or naked singularity, while non-supersymmetric extreme ones have a global space-time of extreme Reissner-Nordstr\" om black holes. 
Abstarct: Boundary effects caused by the boundary interactions in various integrable field theories on a half line are discussed at the classical as well as the quantum level. Only the so-called ``integrable" boundary interactions are discussed. They are obtained by the requirement that certain combinations of the lower members of the infinite set of conserved quantities should be preserved. Contrary to the naive expectations, some ``integrable" boundary interactions can drastically change the character of the theory. In some cases, for example, the sinh-Gordon theory, the theory becomes ill-defined because of the instability introduced by ``integrable" boundary interactions for a certain range of the parameter. 
We find U(1)_{E} \times U(1)_{M} non-extremal black hole solutions of 6-dimensional Kaluza-Klein supergravity theories. Extremal solutions were found by Cveti\v{c} and Youm\cite{C-Y}. Multi black hole solutions are also presented. After electro-magnetic duality transformation is performed, these multi black hole solutions are mapped into the the exact solutions found by Horowitz and Tseytlin\cite{H-T} in 5-dimensional string theory compactified into 4-dimensions. The massless fields of this theory can be embedded into the heterotic string theory compactified on a 6-torus. Rotating black hole solutions can be read off those of the heterotic string theory found by Sen\cite{Sen3}. 
The dynamical behaviour of domain boundaries between different realizations of the vacuum of scalar fields with spontaneously broken phases is investigated. They correspond to zero-modes of the Goldstone fields, moving with the speed of light, and turn out to be accompanied by strongly oscillating gravitational fields. In certain space-time topologies this leads to a quantization condition for the symmetry breaking scale in terms of the Planck mass. 
Recently N. Seiberg has shown that certain N=1 supersymmetric Yang -- Mills theories have the property that their infrared physics can be equivalently described by {\it two different} gauge theories, with the strong coupling region of one corresponding to the weak coupling region of the other. This duality leads to interesting results about the infrared dynamics of these theories. We generalize Seiberg's ideas to a new class of models for which a similar duality allows one to obtain rather detailed dynamical information about the long distance physics. 
In the SU(3) Einstein-Skyrme system static spherically symmetric particle-like solutions and black holes exist for both the SU(2) and the SO(3) embedding. The SO(3) embedding leads to new particle-like solutions and black holes, sharing many features with the SU(2) solutions. In particular, there are always two branches of solutions, forming a cusp at a critical coupling constant. The regular SO(3) solutions have even topological charge $B$. The mass of the $B=2$ SO(3) solutions is less than twice the mass of the $B=1$ SU(2) solutions. We conjecture, that the lowest SO(3) branches correspond to stable particle-like solutions and stable black holes. 
  We study bosonization of the sine-Gordon theory in the presence of boundary interactions at the free fermion point. In this way we obtain the boundary S-matrix as a function of physical parameters in the boundary sine-Gordon Lagrangian. The boundary S-matrix can be matched onto the solution of Ghoshal and Zamolodchikov, thereby relating the formal parameters in the latter solution to the physical parameters in the lagrangian. 
Three dimensional black holes in a generalized dilaton gravity action theory are analysed. The theory is specified by two fields, the dilaton and the graviton, and two parameters, the cosmological constant and the Brans-Dicke parameter. It contains seven different cases, of which one distinguishes as special cases, string theory, general relativity and a theory equivalent to four dimensional general relativity with one Killing vector. We study the causal structure and geodesic motion of null and timelike particles in the black hole geometries and find the ADM masses of the different solutions. 
We present a hierarchy of gauged Grassmanian models in $4p$ dimensions, where the gauge field takes its values in the $2^{2p- 1}\times 2^{2p-1}$ chiral representation of SO(4p). The actions of all these models are absolutely minimised by a hierarchy of self-duality equations, all of which reduce to a single pair of coupled ordinary differential equations when subjected to $4p$ dimensional spherical symmetry. 
  Considering bilayer systems as extensions of the planar ones by an internal space of two discrete points, we use the ideas of Noncommutative Geometry to construct the gauge theories for these systems. After integrating over the discrete space we find an effective $2+1$ action involving an extra complex scalar field, which can be interpreted as arising from the tunneling between the layers. The gauge fields are found in different phases corresponding to the different correlations due to the Coulomb interaction between the layers. In a particular phase, when the radial part of the complex scalar field is a constant, we recover the Wen-Zee model of Bilayer Quantum Hall systems. There are some circumstances, where this radial part may become dynamical and cause dissipation in the oscillating supercurrent between the layers. 
The effect of the Gauss--Bonnet term on the SU(2) non--Abelian regular stringy sphaleron solutions is studied within the non--perturbative treatment. It is found that the existence of regular solutions depends crucially on the value of the numerical factor $\beta$ in front of the Gauss--Bonnet term in the four--dimensional effective action. Numerical solutions are constructed in the N=1, 2, 3 cases for different $\beta$ below certain critical values $\beta_N$ which decrease with growing N (N being the number of nodes of the Yang--Mills function). It is proved that for any static spherically symmetric asymptotically flat regular solution the ADM mass is exactly equal to the dilaton charge. No solutions were found for $\beta$ above critical values, in particular, for $\beta=1$. 
We study the Riemannian aspect and the Hilbert-Einstein gravitational action of the non-commutative geometry underlying the Connes-Lott construction of the action functional of the standard model. This geometry involves a two-sheeted, Euclidian space-time. We show that if we require the space of forms to be locally isotropic and the Higgs scalar to be dynamical, then the Riemannian metrics on the two sheets of Euclidian space-time must be identical. We also show that the distance function between the two sheets is determined by a single, real scalar field whose VEV sets the weak scale. 
The Kerr solution is considered as a soliton-like background for spinning elementary particles. Two stringy structures may be found in the Kerr geometry, one string is real and another one is complex. The main attention in this paper is paid to the complex string, which is connected with the Lind-Newman representation of the Kerr metric source as an object with a complex world-line. In a separate note, one new result is announced concerning the real string: in the dilaton-axion gravity the Kerr singular ring may be considered as a heterotic string, the field near the Kerr singularity coincides with the solution obtained by Sen for heterotic string. 
  An extension of the method and results of A. Schwarz for evaluating the partition function of a quadratic functional is presented. This enables the partition functions to be evaluated for a wide class of quadratic functionals of interest in topological quantum field theory, for which no other method has previously been available. In particular it enables the partition functions appearing in the semiclassical approximation for the Witten-invariant to be evaluated in the most general case. The resulting k-dependence is precisely that conjectured by D. Freed and R. Gompf. 
A generalization of the non-Abelian version of the $CP^{N-1}$ models (also known as Grassmannian models) is presented. The generalization helps accommodate a partial breaking of the non-Abelian gauge symmetry. Constituents of the composite gauge fields, in many cases, are naturally constrained to belong to an anomaly free representation which in turn generates a composite scalar simulating Higgs mechanism to break the gauge symmetry dynamically for large $N$. Two cases are studied in detail: one based on the SU(2) gauge group and the other on SO(10). Breakings such as SU(2)$\to$U(1) or SO(10)$\to$SU(5)$\times$U(1) are found feasible. Properties of the composites fields and gauge boson masses are computed by doing a derivative expansion of the large $N$ effective action. 
  We propose a simple version of chaotic inflation scenario which leads to a division of the universe into infinitely many open universes with all possible values of $\Omega$ from 1 to 0. 
  The conformal spectra of the critical dilute A-D-E lattice models are studied numerically. The results strongly indicate that, in branches 1 and 2, these models provide realizations of the complete A-D-E classification of unitary minimal modular invariant partition functions given by Cappelli, Itzykson and Zuber. In branches 3 and 4 the results indicate that the modular invariant partition functions factorize. Similar factorization results are also obtained for two-colour lattice models. 
   Using the topological techniques developed in an earlier paper with Vafa, a field theory action is constructed for any open string with critical N=2 worldsheet superconformal invariance. Instead of the Chern-Simons-like action found by Witten, this action resembles that of a Wess-Zumino-Witten model. For the N=2 string which describes (2,2) self-dual Yang-Mills, the string field generalizes the scalar field of Yang.     As was shown in recent papers, an N=2 string can also be used to describe the Green-Schwarz superstring in a Calabi-Yau background. In this case, one needs three types of string fields which generalize the real superfield of the super-Yang-Mills prepotential, and the chiral and anti-chiral superfields of the Calabi-Yau scalar multiplet. The resulting field theory action for the open superstring in a Calabi-Yau background has the advantages over the standard RNS action that it is manifestly SO(3,1) super-Poincar\'e invariant and does not require contact terms to remove tree-level divergences. 
We show the relationship between a fluid of particles having charge and magnetic moment in $2+1$ dimensional electromagnetism and the Chern-Simons statistical field. The matter current which is minimally coupled to the electromagnetic field has two parts: the global electromagnetic current, and the corresponding topological current. The topological current is associated to the induced electromagnetic current, via a simple constitutive relation between charges and magnetic moments. We also study the edge states, when the region that the currents occupy is bounded. 
We consistently quantize a class of relativistic non-local field equations characterized by a non-local kinetic term in the lagrangian. We solve the classical non-local equations of motion for a scalar field and evaluate the on-shell hamiltonian. The quantization is realized by imposing Heisenberg's equation which leads to the commutator algebra obeyed by the Fourier components of the field. We show that the field operator carries, in general, a reducible representation of the Poincare group. We also consider the Gupta-Bleuler quantization of a non-local gauge field and analyze the propagators and the physical states of the theory. 
A possible ground state of Quantum Gravity is Wheeler's ``space-time foam'', which can be modeled as a ``Planck-lattice'', a space-time cubic lattice of lattice constant $a_P\simeq 10^{33}$cm, the Planck length. I analyse the structure of the Standard Model defined on the Planck Lattice, in the light of the ``no-go'' theorem of Nielsen and Ninomiya, which requires an extension of the continuum model through Nambu-Jona Lasinio terms, quadrilinear in the Fermi-fields. As a result, a theory of masses (of both fermions and gauge bosons) is seen to emerge that, without Higgs excitations, agrees well with observations. 
We discuss two concepts of metric and linear connections in noncommutative geometry, applying them to the case of the product of continuous and discrete (two-point) geometry. 
 First, an algebraic criterion for integrability is discussed -the so-called `superintegrability'- and some results on the classification of superintegrable quantum spin Hamiltonians based on sl(2) are obtained.  Next, the massive phases of the Z_n-chiral Potts quantum spin chain (a model that violates parity) are studied in detail. It is shown that the excitation spectrum of the massive high-temperature phase can be explained in terms of n-1 fundamental quasiparticles. We compute correlation functions from a perturbative and numerical evaluation of the groundstate for the Z_3-chain. In addition to an exponential decay we observe an oscillating contribution. The oscillation length seems to be related to the asymmetry of the dispersion relations. We show that this relation is exact at special values of the parameters for general Z_n using a form factor expansion.  Finally, we discuss several aspects of extended conformal algebras (W-algebras). We observe an analogy between boundary conditions for Z_n-spin chains and W-algebras and then turn to statements about the structure of W-algebras. In particular, we briefly summarize results on unifying structures present in the space of all quantum W-algebras. 
  We present a topological quantum field theory which corresponds to the moduli problem associated to Witten's monopole equations for four-manifolds. The construction of the theory is carried out in purely geometrical terms using the Mathai-Quillen formalism, and the corresponding observables are described. These provide a rich set of new topological quantites. 
  The classic question of a nonabelian Yang-Mills analogue to electromagnetic duality is here examined in a minimalist fashion at the strictly 4-dimensional, classical field and point charge level. A generalisation of the abelian Hodge star duality is found which, though not yet known to give dual symmetry, reproduces analogues to many dual properties of the abelian theory. For example, there is a dual potential, but it is a 2-indexed tensor $T_{\mu\nu}$ of the Freedman-Townsend type. Though not itself functioning as such, $T_{\mu\nu}$ gives rise to a dual parallel transport, $\tilde{A}_\mu$, for the phase of the wave function of the colour magnetic charge, this last being a monopole of the Yang-Mills field but a source of the dual field. The standard colour (electric) charge itself is found to be a monopole of $\tilde{A}_\mu$. At the same time, the gauge symmetry is found doubled from say $SU(N)$ to $SU(N) \times SU(N)$. A novel feature is that all equations of motion, including the standard Yang-Mills and Wong equations, are here derived from a `universal' principle, namely the Wu-Yang (1976) criterion for monopoles, where interactions arise purely as a consequence of the topological definition of the monopole charge. The technique used is the loop space formulation of Polyakov (1980). 
  The structure of block-spin embeddings of the U(1) lattice current algebra is described. For an odd number of lattice sites, the inner realizations of the shift automorphism areclassified. We present a particular inner shift operator which admits a factorization involving quantum dilogarithms analogous to the results of Faddeev and Volkov. 
We show how it is possible to formulate Euclidean two-dimensional quantum gravity as the scaling limit of an ordinary statistical system by means of dynamical triangulations, which can be viewed as a discretization in the space of equivalence classes of metrics. Scaling relations exist and the critical exponents have simple geometric interpretations. Hartle-Hawkings wave functionals as well as reparametrization invariant correlation functions which depend on the geodesic distance can be calculated. The discretized approach makes sense even in higher dimensional space-time. Although analytic solutions are still missing in the higher dimensional case, numerical studies reveal an interesting structure and allow the identification of a fixed point where we can hope to define a genuine non-perturbative theory of four-dimensional quantum gravity. 
We discuss SU(N) gluo-dynamics at finite temperature and on a spatial circle. We show that the effective action for the Polyakov Loop operator is a one dimensional gauged SU(N) principle chiral model with variables in the loop space and loop algebra of the gauge group. We find that the quantum states can be characterized by a discrete $\theta$-angle which appears with a particular 1-dimensional topological term in the effective action. We present an explicit computation of the partition function and obtain the spectrum of the model, together with its dependence on the discrete theta angle. We also present explicit formulae for 2-point correlators of Polyakov loop operators and an algorithm for computing all N-point correlators. 
We obtain a fermionic coset realization of the primaries of minimal unitary models and show how their four-point functions may be calculated by the use of a reduction formula. We illustrate the construction for the Ising model, where we obtain an explicit realization of the energy operator, Onsager fermions, as well as of the order and disorder operators realizing the dual algebra, in terms of constrained Dirac fermions. The four-point correlators of these operators are shown to agree with those obtained by other methods. 
The Higgs-Yukawa model in curved spacetime (renormalizable in the usual sense) is considered near the critical point, employing the $1/N$--expansion and renormalization group techniques. By making use of the equivalence of this model with the standard NJL model, the effective potential in the linear curvature approach is calculated and the dynamically generated fermionic mass is found. A numerical study of chiral symmetry breaking by curvature effects is presented. 
We examine the two-point correlation functions of the fields exp(i$\alpha\Phi$) in the sine-Gordon theory at all values of the coupling constant $\hat\beta$. Using conformal perturbation theory, we write down explicit integral expressions for every order of the short distance expansion. Using a novel technique analagous to dimensional regularisation, we evaluate these integrals for the first few orders finding expressions in terms of generalised hypergeometric functions. From these derived expressions, we examine the limiting forms at the points where the sine-Gordon theory maps onto a doubled Ising and the Gross-Neveu SU(2) models. In this way we recover the known expansions of the spin and disorder fields about criticality in the Ising model and the well known Kosterlitz-Thouless flows in the Gross-Neveu SU(2) model. 
  Moduli fields, which parameterize perturbative flat directions of the potential in supersymmetric theories, are natural candidates to act as inflatons. An inflationary potential on moduli space can result if the scale of dynamical SUSY breaking in some sector of the theory is determined by a moduli dependent coupling. The magnitude of density fluctuations generated during inflation then depends on the scale of SUSY breaking in this sector. This can naturally be hierarchically smaller than the Planck scale in a dynamical model, giving small fluctuations without any fine tuning of parameters. It is also natural for SUSY to be restored at the minimum of the moduli potential, and to leave the universe with zero cosmological constant after inflation. Acceptable reheating can also be achieved in this scenario. 
An exploratory study of the cosmology of moduli in string theory. Moduli are argued to be natural inflaton fields and lead to a robust inflationary cosmology in which inflation takes place at the top of domain walls. The amplitude of microwave background fluctuations constrains the dynamics responsible for inflation to take place at a higher scale than supersymmetry breaking. Models explaining this difference in scales and also preventing the dilaton from running to infinity are proposed. The problem of dilaton domination of the energy density of the universe is not resolved. 
We relate the non-perturbative exact results in supersymmetry to perturbation theory using several different methods: instanton calculations at weak or strong coupling, a method using gaugino condensation and another method relating strong and weak coupling. This allows many precise numerical checks of the consistency of these methods, especially the amplitude of instanton effects, and of the network of exact solutions in supersymmetry. However, there remain difficulties with the instanton computations at strong coupling. 
We have examined a modified dilaton gravity whose action is separable into the kinetic and the cosmological terms for the sake of the quantization. The black hole solutions survive even in the quantized theory, but the ADM mass of the static solution is unbounded from below. Quantum gravitational effect on the interacting matter field and the string susceptibility of the dilaton gravity are also examined. 
After showing that the magnetic translation operators are not the symmetries of the QHE on non-flat surfaces , we show that there exist another set of operators which leads to the quantum group symmetries for some of these surfaces . As a first example we show that the $su(2)$ symmetry of the QHE on sphere leads to $su_q(2)$ algebra in the equator . We explain this result by a contraction of $su(2)$ . Secondly , with the help of the symmetry operators of QHE on the Pioncare upper half plane , we will show that the ground state wave functions form a representation of the $su_q(2)$ algebra . 
  We show that a wide class of $W$-(super)algebras, including $W_N^{(N-1)}$, $U(N)$-superconformal as well as $W_N$ nonlinear algebras, can be linearized by embedding them as subalgebras into some {\em linear} (super)conformal algebras with finite sets of currents. The general construction is illustrated by the example of $W_4$ algebra. 
< \psibarpsi > vanishing above $T_c$ indicates chiral symmetry restoration at high $T$. But is it the old $T=0$ chiral symmetry that is `restored'? In this talk, I report on the spacetime quantization of the BPFTW effective action for quarks in a hot environ. The fermion propagator is known to give a pseudo-Lorentz invariant particle pole as well as new spacelike cuts. Our quantization shows that the spacelike cuts directly lead to a thermal vacuum that is a generalized NJL state, with a curious $90^{o}$ phase. This $90^{o}$ is responsible for < \psibarpsi > vanishing at high $T$. The thermal vacuum is invariant under a new chiral charge, but continues to break the old zero temperature chirality. Our quantization suggests a new class of order parameters that probe the physics of these spacelike cuts. In usual scenario, the pion dissociates in the early alphabet soup. With this new understanding of the thermal vacuum, the pion remains a Nambu-Goldstone particle at high $T$, and will not dissociate. It propagates at the speed of light but with a halo. 
About twenty years ago Hawking made the remarkable suggestion that the black hole evaporation process will inevitably lead to a fundamental loss of quantum coherence. The mechanism by which the quantum radiation is emitted appears to be insensitive to the detailed history of the black hole, and thus it seems that most of the initial information is lost for an outside observer. However, direct examination of Hawking's original derivation (or any later one) of the black hole emission spectrum shows that one inevitably needs to make reference to particle waves that have arbitrarily high frequency near the horizon as measured in the reference frame of the in-falling matter. This exponential red-shift effect associated with the black hole horizon leads to a breakdown of the usual separation of length scales, and effectively works as a magnifying glass that makes the consequences of the short distance, or rather, high energy physics near the horizon visible at larger scales to an asymptotic observer. 
  We show that manifolds of fixed points, which are generated by exactly marginal operators, are common in N=1 supersymmetric gauge theory. We present a unified and simple prescription for identifying these operators, using tools similar to those employed in two-dimensional N=2 supersymmetry. In particular we rely on the work of Shifman and Vainshtein relating the $\bt$-function of the gauge coupling to the anomalous dimensions of the matter fields. Finite N=1 models, which have marginal operators at zero coupling, are easily identified using our approach. The method can also be employed to find manifolds of fixed points which do not include the free theory; these are seen in certain models with product gauge groups and in many non-renormalizable effective theories. For a number of our models, S-duality may have interesting implications. Using the fact that relevant perturbations often cause one manifold of fixed points to flow to another, we propose a specific mechanism through which the N=1 duality discovered by Seiberg could be associated with the duality of finite N=2 models. 
  It is shown that modular invariance provides a natural explanation for the absence of monopoles when assumed to be a discrete gauge symmetry. It follows that monopoles can not be seen because it is always possible to find a suitable gauge-fixing in which they are not present. This result relies upon an easy to prove but non-trivial property of the modular group. A modular-invariant formulation for the hamiltonian of the electromagnetism is given. No monopole arises if independent modular transformations are allowed at each point in space-time where point-like charges are present. 
  The Lee-Weinberg $U(1)$ magnetic monopoles, which have been reinterpreted as topological solitons of a certain non-Abelian gauged Higgs model recently, are considered for some specific choice of Higgs couplings. The model under consideration is shown to admit a Bogomol'nyi-type bound which is saturated by the configurations satisfying the generalized BPS equations. We consider the spherically symmetric monopole solutions in some detail. 
  The strong coupling dynamics of string theories in dimension $d\geq 4$ are studied. It is argued, among other things, that eleven-dimensional supergravity arises as a low energy limit of the ten-dimensional Type IIA superstring, and that a recently conjectured duality between the heterotic string and Type IIA superstrings controls the strong coupling dynamics of the heterotic string in five, six, and seven dimensions and implies $S$ duality for both heterotic and Type II strings. 
  We present a brief overview of 2-dim. string theory and its connection to the theory of non-relativistic fermions in one dimension. We emphasize (i) the role of $W_\infty$ algebra and (ii) the modelling of some aspects of 2-dim. black hole physics using the phase space representation of the fermi fluid. 
  The possibility of excitations with fractional spin and statististics in $1+1$ dimensions is explored. The configuration space of a two-particle system is the half-line. This makes the Hamiltonian self-adjoint for a family of boundary conditions parametrized by one real number $\gamma$. The limit $\gamma \rightarrow 0, (\infty$) reproduces the propagator of non-relativistic particles whose wavefunctions are even (odd) under particle exchange. A relativistic ansatz is also proposed which reproduces the correct Polyakov spin factor for the spinning particle in $1+1$ dimensions. These checks support validity of the interpretation of $\gamma$ as a parameter related to the ``spin'' that interpolates continuously between bosons ($\gamma =0$) and fermions ($\gamma =\infty$). Our approach can thus be useful for obtaining the propagator for one-dimensional anyons. 
  A brief review of the status of duality symmetries in string theory is presented. The evidence is accumulating rapidly that an enormous group of duality symmetries, including perturbative T dualities and non-perturbative S-dualities, underlies string theory. It is my hope that an understanding of these symmetries will suggest the right way to formulate non-perturbative string theory. Whether or not this hope is realized, it has already been demonstrated that this line of inquiry leads to powerful new tools for understanding gauge theories and new evidence for the uniqueness of string theory, as well as deep mathematical results. 
  I propose an approximation scheme for asymptotically free field theories combining both weak coupling and strong coupling series. The weak coupling expansion is used to integrate the high frequency modes and the resulting low energy effective theory is solved using the strong coupling expansion. In some models there exists an intermediate scale at which both expansions make sense. The method is tested on few low dimensional models for which an exact solution is known. 
  We address the question whether so-called m-invariants of the N=2 super Virasoro algebra can be used for the construction of reasonable four-dimensional string models. It turns out that an infinite subset of those are pathological in the sense that - although N=2 supersymmetric - the Ramond sector is not isomorphic to the Neveu-Schwarz sector. Consequently, these two properties are independent and only requiring both guarantees an N=1 space-time supersymmetric string spectrum. However, the remaining 529 consistent spectra - 210 of them are mirrors of Gepner models and 76 real orbifolds - show exact mirror symmetry and are contained in a recent classification of orbifolds of Gepner models. 
  The high temperature behaviour of the open bosonic string free energy in the space $S^1 \otimes H^N$ with vanishingly small curvature is investigated. The leading term of the high temperature expansion of the one-loop free energy, near the Hagedorn instability, is obtained. The problem of infrared regularization of thermodynamical quantities is pointed out. For minimally coupling quantum fields related to the normal modes of strings, the results are similar to the ones valid for Rindler space. In the lower mass string states regime a connection with the quantum corrections to the black hole entropy is outlined. 
  We note that in (2+1)-dimensional gauge theories with even number of massless fermions, there is anomalous $Z_2$ symmetry if theory is regularized in a parity-invariant way. We then consider a parity invariant $U(1)_V\times U(1)_A$ model, which induces a mutual Chern-Simons term in the effective action due to $Z_2$ anomaly. The effect of the discrete anomaly is studied in the induced spin and in the dynamical fermion mass. 
  Using our recently proposed covariant algebraic approach the heat kernel for a Laplace-like differential operator in low-energy approximation is studied. Neglecting all the covariant derivatives of the gauge field strength (Yang-Mills curvature) and the covariant derivatives of the potential term of third order and higher a closed formula for the heat kernel as well as its diagonal is obtained. Explicit formulas for the coefficients of the asymptotic expansion of the heat kernel diagonal in terms of the Yang-Mills curvature, the potential term and its first two covariant derivatives are obtained. 
  It is shown that there exists such collection of variables that the standard QCD Lagrangian can be represented as the sum of usual Palatini Lagrangian for Einstein general relativity and the Lagrangian of matter and some other fields where the tetrad fields and the metric are constructed from initial $SU(3)$ Yang - Mills fields. 
  Using the string field theory recently proposed by the authors and collaborators, we give a background independent formulation of rational noncritical string theories with $c\leq 1$. With a little modification of the string field Hamiltonians previously constructed, we obtain string field theories which include various rational noncritical string theories as classical backgrounds. 
  We discuss dissipative systems in Quantum Field Theory by studying the canonical quantization of the damped harmonic oscillator (dho). We show that the set of states of the system splits into unitarily inequivalent representations of the canonical commutation relations. The irreversibility of time evolution is expressed as tunneling among the unitarily inequivalent representations. Canonical quantization is shown to lead to time dependent SU(1,1) coherent states. We derive the exact action for the dho from the path integral formulation of the quantum Brownian motion developed by Schwinger and by Feynman and Vernon. The doubling of the phase-space degrees of freedom for dissipative systems is related to quantum noise effects. Finally, we express the time evolution generator of the dho in terms of operators of the $q$-deformation of the Weyl-Heisenberg algebra. The $q$-parameter acts as a label for the unitarily inequivalent representations. 
  We discuss the r\^ole of quantum deformation of Weyl-Heisenberg algebra in dissipative systems and finite temperature systems. We express the time evolution generator of the damped harmonic oscillator and the generator of thermal Bogolubov transformations in terms of operators of the quantum Weyl-Heisenberg algebra. The quantum parameter acts as a label for the unitarily inequivalent representations of the canonical commutation relations in which the space of the states splits in the infinite volume limit. 
  QED_2 with mass and N flavors of fermions is constructed using Euclidean path integrals. The fermion masses are treated perturbatively and the convergence of the mass perturbation series is proven for a finite space-time cutoff. The expectation functional is decomposed into clustering theta-vacua and their properties are compared to the theta-vacua of QCD for zero fermion mass. The sector that is created by the N^2 classically conserved vector currents is identified. The currents that correspond to a Cartan subalgebra of U(N) are bosonized together with the chiral densities in terms of a generalized Sine-Gordon model. The solution of the U(1)-problem of QED_2 is discussed and a Witten-Veneziano formula is shown to hold for the mass spectrum of the pseudoscalars. Evaluation of the Fredenhagen-Marcu confinement order parameter clarifies the structure of superselection sectors. 
  The three-flavor Skyrme-'t Hooft-Witten model is interpreted in terms of a quark-like substructure, leading to a new model of explicitly confined color-free ``quarks'' reminiscent of Gell-Mann's original pre-color quarks, but with unexpected and significant differences. 
  A method to construct free field realizations for the form factors of diagonal factorized scattering theories is described. Form factors are constructed from linear functionals over an associative `form factor algebra', which in particular generate solutions of the deformed Knizhnik-Zamolodchikov equations with parameter $2\pi$. We show that there exists a unique deformation off the (`Rindler') value $2\pi$ that preserves the original $S$-matrix and which allows one to realize form factors as vector functionals over an algebra of generalized vertex operators. 
  Consequences of a symmetry, e.g.\ relations amongst Green functions, are renormalization scheme independently expressed in terms of a rigid Ward identity. The corresponding local version yields information on the respective current. In the case of spontaneous breakdown one has to define the theory via the BRS invariance and thus to construct rigid and current Ward identity non-trivially in accordance with it. We performed this construction to all orders of perturbation theory in the abelian Higgs model as a prelude to the standard model. A technical tool of interest in itself is the use of a doublet of external scalar ``background'' fields. The Callan-Symanzik equation has an interesting form and follows easily once the rigid invariance is established. 
  We consider those two-dimensional rational conformal field theories (RCFTs) whose chiral algebras, when maximally extended, are isomorphic to the current algebra formed from some affine non-twisted Kac--Moody algebra at fixed level. In this case the partition function is specified by an automorphism of the fusion ring and corresponding symmetry of the Kac--Peterson modular matrices. We classify all such partition functions when the underlying finite-dimensional Lie algebra is simple. This gives all possible spectra for this class of RCFTs. While accomplishing this, we also find the primary fields with second smallest quantum dimension. 
  A new class of spatially homogeneous 4D string backgrounds, the $X(d\rightarrow)$ according to a recent classification, is presented and shown to contain only five generic types. In contrast to the case of $X(d\uparrow)$ (which contains as a subclass all possible FRW backgrounds), exact $SO(3)$ isotropy is always broken in the $X(d\rightarrow)$ class. This is due to the $H$-field, whose dual is necessarily along a principal direction of anisotropy. Nevertheless, FRW symmetry can be attained asymptotically for Bianchi-types $I$ and $VII_0$ in a rather appealing physical context. Other aspects of the solutions found for types $X=I,II,III,VI_{-1}$, and of the $VII_0$ case are briefly discussed. 
  In the large N limit, conditions for the conformal invariance of the generalized Thirring model are derived, using two different approaches: the background field method and the Hamiltonian method based on an operator algebra, and the agreement between them is established. A free field representation of the relevant algebra is presented, and the structure of the stress tensor in terms of free fields (and free currents) is studied in detail. 
  We show that self-dual Nielsen Olesen (NO) vortices in $3$ dimensions give rise to a class of exact solutions when coupled to Einstein Maxwell Dilaton gravity obeying the Majumdar-Papapetrou(MP) relation between gravitational and Maxwell couplings , provided certain Chern-Simons type interactions are present. The metric may be solved for explicitly in terms of the NO vortex function and becomes degenerate at scales $r_H \sim l_S exp(\frac{l_S}{l_P})$ where $l_S$ is the vortex core size and $l_P$ the Planck length. For typical $l_S\geq 10^4 l_P$ the horizon is thus pushed out to exponentially large scales. In the intermediate asymptotic region (IAR) $l_S<<r<<r_H$ there is a logarithmic deviation of the metric from the flat metric and of the electric field from that of a point charge (which makes it decrease {\it{slower}} than $r^{-1}$ :hence the prefix hyper). In the IAR the ADM energy and charge integrals increase logarithmically with the distance from the core region and finally diverge at the signature change horizon. String solutions in $4+p$ dimensions are obtained by replacing the Maxwell fiel0d by an antisymmetric tensor field (of rank $2+p$) and have essentially similar properties with $r_H \sim l_S exp((\frac{l_S}{l_P})^{2+p})$ and with the antisymmetric charge playing the role of the topological electric charge . 
  A formula is proposed which expresses free fermion fields in 2K dimensions in terms of the Cartan currents of the free fermion current algebra. This leads, in an obvious manner, to a vertex operator construction of nonabelian free fermion current algebras in arbitrary even dimension. It is conjectured that these ideas may generalize to a wide class of conformal field theories. 
  Chiral symmetry breaking due to instanton-produced fermion zero modes in the confining vacuum is considered. Zero modes provide 'tHooft-type determinant quark interaction, which is bosonized by introduction of auxiliary fields. For three flavours this procedure becomes nontrivial and a method is suggested which allows to derive effective Lagrangian for massless Nambu--Goldstone modes. 
  The connection between the Kac-Moody algebras of currents and the chiral symmetries of the two dimensional WZNW model is clarified. It is shown that only the zero modes of the Kac-Moody currents are the first class constraints, and that, consequently, the corresponding gauge symmetries are chiral. 
  Covariant quantization of theories based on nonlinear extensions of Lie algebras in $2d$ is studied by using a generalized Lagrangian BRST formalism. The quantum action is constructed to be invariant under the off--shell nilpotent BRST transformations by using a set of independent antifields as auxiliary, nonpropagating variables in the quantum theory. The general results are applied to the quantization of nonlinear gauge theory based on quadratic Poincar\'e algebra, which is closely related to $2d$ gravity with dynamical torsion. 
  String theory abounds with light scalar fields (the dilaton and various moduli) which create a host of observational problems, and notably some serious cosmological difficulties similar to the ones associated with the Polonyi field in the earliest versions of spontaneously broken supergravity. We show that all these problems are naturally avoided if a recently introduced mechanism for fixing the vacuum expectation values of the dilaton and/or moduli is at work. We study both the classical evolution and the quantum fluctuations of such scalar fields during a primordial inflationary era and find that the results are naturally compatible with observational facts. In this model, dilatons or moduli within a very wide range of masses (which includes the SUSY-breaking favored 1 TeV value and extends up to the Planck scale) qualify to define a novel type of essentially stable ultra-weakly interacting massive particles able to provide enough mass density to close the universe 
  A four-point function of $E_6$ singlets, of interest in elucidating the moduli space of (0,2) deformations of the quintic string vacuum, is computed using analytic and numerical methods. The conformal field theory amplitude satisfies the requisite selection rules and monodromy conditions, but the integrated string amplitude vanishes. Together with selection rules coming from the spacetime R-symmetry \dksecond, this demonstrates the flatness of the gauge-singlet spacetime superpotential through fourth order. Relevance to the more general program of determining the (0,2) moduli space and superpotential is discussed. 
  Nonhamiltonian interaction of hamiltonian systems is considered. Dynamical equations are constructed by use of symmetric designs on Lie algebras. The results of analysis of these equations show that some class of symmetric designs on Lie algebras beyond Jordan ones may be useful for a description of almost periodic, asymptotically periodic, almost asymptotically periodic, and, possibly, more chaotic systems. However, the behaviour of systems related to symmetric designs with additional identities is simpler than for general ones from different points of view. These facts confirm a general thesis that various algebraic structures beyond Lie algebras may be regarded as certain characteristics for a wide class of dynamical systems. 
  We prove a generalized version of the well-known Lichnerowicz formula for the square of the most general Dirac operator $\widetilde{D}$\ on an even-dimensional spin manifold associated to a metric connection $\widetilde{\nabla}$. We use this formula to compute the subleading term $\Phi_1(x,x, \widetilde{D}^2)$\ of the heat-kernel expansion of $\widetilde{D}^2$. The trace of this term plays a key-r$\hat {\petit\rm o}$le in the definition of a (euclidian) gravity action in the context of non-commutative geometry. We show that this gravity action can be interpreted as defining a modified euclidian Einstein-Cartan theory. 
  We study Dirac operators acting on sections of a Clifford module ${\cal E}$\ over a Riemannian manifold $M$. We prove the intrinsic decomposition formula for their square, which is the generalisation of the well-known formula due to Lichnerowicz [L]. This formula enables us to distinguish Dirac operators of simple type. For each Dirac operator of this natural class the local Atiyah-Singer index theorem holds. Furthermore, if $M$\ is compact and ${{\petit \rm dim}\;M=2n\ge 4}$, we derive an expression for the Wodzicki function $W_{\cal E}$, which is defined via the non-commutative residue on the space of all Dirac operators ${\cal D}({\cal E})$. We calculate this function for certain Dirac operators explicitly. From a physical point of view this provides a method to derive gravity, resp. combined gravity/Yang-Mills actions from the Dirac operators in question. 
  We explicitly obtain the generalization of the Ehlers transformation for stationary axisymmetric Einstein equations to string theory. This is accomplished by finding the twist potential corresponding to the moduli fields in the effective two dimensional theory. Twist potential and symmetric moduli are shown to transform under an $O(d,d)$ which is a manifest symmetry of the equations of motion. The non-trivial action of this $O(d,d)$ is given by the Ehlers transformation and belongs to the set $O(d) \times O(d)\over O(d) $. 
  The question whether it is necessary to decompactify the gauge eigenvalue degrees of freedom in QCD$_{1+1} $ is addressed. A careful consideration of the dynamics governing these degrees of freedom leads to the conclusion that eigenvalue decompactification is not necessary due to the curvature on the space of eigenvalues. 
  Classical and quantum solutions for the relativistic straight-line string with arbitrary dependence on the world surface curvature are obtained. They differ from the case of the usual Nambu-Goto interaction by the behaviour of the Regge trajectory which in general can be nonlinear. Regularization of the action is considered and comparison with relativistic point with curvature is made. 
  We show that spin generalization of elliptic Calogero-Moser system, elliptic extension of Gaudin model and their cousins can be treated as a degenerations of Hitchin systems. Applications to the constructions of integrals of motion, angle-action variables and quantum systems are discussed. Explicit formulas for the Lax operator on the higher genus surfaces are obtained in the Shottky parameterization. The constructions are motivated by the Conformal Field Theory, and their quantum counterpart can be treated as a degeneration of the critical level Knizhnik-Zamolodchikov-Bernard equations. 
  In this paper, we construct non-critical BRST operators for matter and Liouville systems whose currents generate two different $W$ algebras. At the classical level, we construct the BRST operators for $W^{\rm M}_{2,s}\otimes W^{\rm L}_{2,s'}$. The construction is possible for $s=s'$ or $s\ge s'+2$. We also obtain the BRST operator for $W^{\rm M}_{2,4}\otimes W^{\rm L}_4$ at the classical level. We use free scalar realisations for the matter currents in the above constructions. At the full quantum level, we obtain the BRST operators for $W^{\rm M}_{2,s}\otimes W^{\rm L}_2$ with $s=4,5, 6$, where $W_2$ denotes the Virasoro algebra. For the first and last cases, the BRST operators are expressed in terms of abstract matter and Liouville currents. As a by-product, we obtain the $W_{2,4}$ algebra at $c=-24$ and the $W_{2,6}$ algebra at $c=-2$ and $-\ft{286}3$, at which values the algebras were previously believed not to exist. 
  It has been shown that certain $W$ algebras can be linearised by the inclusion of a spin--1 current. This provides a way of obtaining new realisations of the $W$ algebras. Recently such new realisations of $W_3$ were used in order to embed the bosonic string in the critical and non-critical $W_3$ strings. In this paper, we consider similar embeddings in $W_{2,4}$ and $W_{2,6}$ strings. The linearisation of $W_{2,4}$ is already known, and can be achieved for all values of central charge. We use this to embed the bosonic string in critical and non-critical $W_{2,4}$ strings. We then derive the linearisation of $W_{2,6}$ using a spin--1 current, which turns out to be possible only at central charge $c=390$. We use this to embed the bosonic string in a non-critical $W_{2,6}$ string. 
  The subject of low energy photon-photon scattering is considered in arbitrary dimensional space-time and the interaction is widened to include scattering events involving an arbitrary number of photons. The effective interaction Lagrangian for these processes in QED has been determined in a manifestly invariant form. This generalisation resolves the structure of the weak-field Euler-Heisenberg Lagrangian and indicates that the component invariant functions have coefficients related, not only to the space-time dimension, but also to the coefficients of the Bernoulli polynomial. 
  In this paper it is stressed that there is no {\em physical} reason for symmetries to be linear and that Lie group theory is therefore too restrictive. We illustrate this with some simple examples. Then we give a readable review on the theory finite $W$-algebras, which is an important class of non-linear symmetries. In particular, we discuss both the classical and quantum theory and elaborate on several aspects of their representation theory. Some new results are presented. These include finite $W$ coadjoint orbits, real forms and unitary representation of finite $W$-algebras and Poincare-Birkhoff-Witt theorems for finite $W$-algebras. Also we present some new finite $W$-algebras that are not related to $sl(2)$ embeddings. At the end of the paper we investigate how one could construct physical theories, for example gauge field theories, that are based on non-linear algebras. 
  We describe a scenario for inflation in the framework of non-critical string theory, which does not employ an inflaton field. There is an exponential expansion of the volume of the Universe, induced by enormous entropy production in the early stages of cosmological evolution. This is associated with the loss of information carried by global string modes that cross the particle horizon. It is the same loss of information that induces irreversible time flow when target time is identified with the world-sheet Liouville mode. The resulting scenario for inflation is described by a string analogue of the Fokker-Planck equation that incorporates diffusion and dissipative effects. Cosmological density perturbations are naturally small. 
  We study the physics of the Seiberg-Witten and Argyres-Faraggi-Klemm-Lerche-Theisen-Yankielowicz solutions of $D=4$, $\CN=2$ and $\CN=1$ $SU(N)$ supersymmetric gauge theory. The $\CN=1$ theory is confining and its effective Lagrangian is a spontaneously broken $U(1)^{N-1}$ abelian gauge theory. We identify some features of its physics which see this internal structure, including a spectrum of different string tensions. We discuss the limit $N\rightarrow\infty$, identify a scaling regime in which instanton and monopole effects survive, and give exact results for the crossover from weak to strong coupling along a scaling trajectory. We find a large hierarchy of mass scales in the scaling regime, including very light $W$ bosons, and the absence of weak coupling. The light $W$'s leave a novel imprint on the effective dual magnetic theory. The effective Lagrangian appears to be inadequate to understand the conventional large $N$ limit of the confining $\CN=1$ theory. 
  We consider the classical \w42 algebra from the integrable system viewpoint. The integrable evolution equations associated with the \w42 algebra are constructed and the Miura maps , consequently modifications, are presented. Modifying the Miura maps, we give a free field realization the classical \w42 algebra. We also construct the Toda type integrable systems for it. 
  We study the BRST-cohomology in the quantum hamiltonian reduction of affine Lie algebras of non-simply laced type. We obtain the free field realization of the $W{\bf g}$-algebra for $\bg=B_{2}$, $B_{3}$, $C_{3}$ and $G_{2}$. The $WC_{3}$ algebra is shown to be equal to the $WB_{3}$ algebra at the quantum level by duality transformation. 
  We comment on relations between the linear W_{2,4}^{linear} algebra and non-linear W(2,4)$ algebra appearing in a Sp(4) particle mechanics model by using Lax equations. The appearance of the non-local V_{2,2} algebra is also studied. 
  The first chapters introduce briefly conformal theories, Moore and Seiberg polynomial equations and Gervais-Neveu quantization of Liouville theory. The next chapters present the original results of this thesis. First, the algebra of the chiral components is completely elucidated, both in the Bloch wave and in the Quantum Group basis. Then, in the strong coupling regime, the proof of the truncation theorem of Gervais to real weight operators for C = 7, 13, 19 is completed, especially including fractionnal spins. In this strong coupling regime, the new cosmological term yields a real string susceptibility equal to the real part of the KPZ formula. And eventually the N-point functions of a strongly coupled topological model are obtained. 
  New boundary conditions for integrable nonlinear lattices of the XXX type, such as the Heisenberg chain and the Toda lattice are presented. These integrable extensions are formulated in terms of a generic XXX Heisenberg magnet interacting with two additional spins at each end of the chain. The construction uses the most general rank 1 ansatz for the 2x2 L-operator satisfying the reflection equation algebra with rational r-matrix. The associated quadratic algebra is shown to be the one of dynamical symmetry for the A1 and BC2 Calogero-Moser problems. Other physical realizations of our quadratic algebra are also considered. 
  A procedure of bosonization of Fermions in an arbitrary dimension is suggested. It is shown that a quadratic expression in the fermionic fields after rescaling time $t\to t/\lambda^2$ and performing the limit $\lambda\to0$ (stochastic limit), gives rise to a bosonic operator satisfying the boson canonical commutation relations. This stochastic bosonization of Fermions is considered first for free fields and then for a model with three--linear couplings. The limiting dynamics of the bosonic theory turns out to be described by means of a quantum stochastic differential equations. 
  The status of the gaugino condensation as the source of supersymmetry breaking is reexamined. It is argued that one cannot have stable minima with broken supersymmetry in models where the dilaton is coupled only linearly to the gaugino condensate. We show that the problems of the gaugino condensate mechanism can be solved by considering nonstandard gauge kinetic functions, created by nonperturbative effects. As an example we use the principle of S-duality to modify the coupling of the gaugino condensate to effective supergravity (superstring) Lagrangians. We show that such an approach can solve the problem of the runaway dilaton and lead to satisfactory supersymmetry breaking in models with a {\em single} gaugino condensate. We exhibit a general property of theories containing a symmetry acting on the dilaton and also shed some light on the question whether it is generically the auxiliary field of the modulus $T$, which dominates supersymmetry breaking. 
  The free propagator for the scalar $\lambda \phi^4$--theory is calculated exactly up to the second derivative of a background field. Using this propagator I compute the one--loop effective action, which then contains all powers of the field but with at most two derivatives acting on each field. The standard derivative expansion, which only has a finite number of derivatives in each term, breaks down for small fields when the mass is zero, while the expression obtained here has a well--defined expansion in $\phi$. In this way the resummation of derivatives cures the naive IR divergence. The extension to finite temperature is also discussed. 
  We derive the $\sigma$-model tachyon $\beta$-function equation of 2-dimensional string theory, in the background of flat space and linear dilaton, working entirely within the $c=1$ matrix model. The tachyon $\beta$-function equation is satisfied by a \underbar{nonlocal} and \underbar{nonlinear} combination of the (massless) scalar field of the matrix model. We discuss the possibility of describing the `discrete states' as well as other possible gravitational and higher tensor backgrounds of 2-dimensional string theory within the $c=1$ matrix model. We also comment on the realization of the $W$-infinity symmetry of the matrix model in the string theory. The present work reinforces the viewpoint that a nonlocal (and nonlinear) transform is required to extract the space-time physics of 2-dimensional string theory from the $c=1$ matrix model. 
  We introduce the Liouville mode into the Green-Schwarz superstring. Like massive supersymmetry without central charges, there is no kappa symmetry. However, the second-class constraints (and corresponding Wess-Zumino term) remain, and can be solved by (twisted) chiral superspace in dimensions D=4 and 6. The matter conformal anomaly is c = 4-D < 1. It thus can be canceled for physical dimensions by the usual Liouville methods, unlike the bosonic string (for which the consistency condition is c = D <= 1). 
  We show that the numbers of generations and anti-generations of a (2,2) string compactification with diagonal internal theory can be expressed in terms of certain specifications of the elliptic genus of the untwisted internal theory which can be computed from the Poincare polynomial. To establish this result we show that there are no cancellations of positive and negative contributions to the Euler characteristic within a fixed twisted sector. For our considerations we recast the orbifolding procedure into an algebraic language using simple currents. Turning the argument around, this allows us to define the `extended Poincare polynomial' P(t,x), which encodes information on the orbits of the spinor current under fusion, for non-diagonal N=2 superconformal field theories. As an application, we derive an explicit formula for P(t,x) for general Landau-Ginzburg orbifolds. 
  We derive several new solutions in three-dimensional stringy gravity. The solutions are obtained with the help of string duality transformations. They represent stationary configurations with horizons, and are surrounded by (quasi) topologically massive Abelian gauge hair, in addition to the dilaton and the Kalb-Ramond axion. Our analysis suggests that there exists a more general family, where our solutions are special limits. Finally, we use the generating technique recently proposed by Garfinkle to construct a traveling wave on the extremal variant of one of our solutions. 
  We show that the Painlev\'{e} test is useful not only for probing (non-)integrability but also for finding the values of spins of conserved currents (W currents) in Toda field theories (TFTs). In the case of the TFTs based on simple Lie algebras the locations of resonances are shown to give precisely the spins of conserved W currents. We apply this test to TFTs based on strictly hyperbolic Kac-Moody algebras and show that there exist no resonances other than that at n=2, which corresponds to the energy-momentum tensor, indicating their non-integrability. We also check by direct calculation that there are no spin-3 nor -4 conserved currents for all the hyperbolic TFTs in agreement with the result of our Painlev\'{e} analysis. 
  A canonical formalism for higher-derivative theories is presented on the basis of Dirac's method for constrained systems. It is shown that this formalism shares a path integral expression with Ostrogradski's canonical formalism. 
  Boundary operators and boundary states in $SU(2)$-invariant Thirring model are considered from the point of view of bosonization and oscillator realizations of bulk and boundary Zamolodchikov-Faddeev algebras. 
  We study supersymmetric $SO(N_c)$ gauge theories with $N_f$ flavors of quarks in the vector representation. Among the phenomena we find are dynamically generated superpotentials with physically inequivalent branches, smooth moduli spaces of vacua, confinement and oblique confinement, confinement without chiral symmetry breaking, massless composites (glueballs, exotics, monopoles and dyons), non-trivial fixed points of the renormalization group and massless magnetic quarks and gluons. Our analysis sheds new light on a recently found duality in $N=1$ supersymmetric theories. The dual forms of some of the theories exhibit ``quantum symmetries'' which involve non-local transformations on the fields. We find that in some cases the duality has both $S$ and $T$ transformations generating $SL(2,Z)$ (only an $S_3$ quotient of which is realized non-trivially). They map the original non-Abelian electric theory to magnetic and dyonic non-Abelian theories. The magnetic theory gives a weak coupling description of confinement while the dyonic theory gives a weak coupling description of oblique confinement. Our analysis also shows that the duality in $N=1$ is a generalization of the Montonen-Olive duality of $N=4$ theories. 
  In this letter we show how the action functional of the standard model and of gravity can be derived from a specific Dirac operator. Far from being exotic this particular Dirac operator turns out to be structurally determined by the Yukawa coupling term. The main feature of our approach is that it naturally unifies the action of the standard model with gravity. 
  By regularizing the singularities appearing in the two dimensional Regge calculus by means of a segment of a sphere or pseudo-sphere and then taking the regulator to zero, we obtain a simple formula for the gauge volume which appears in the functional integral. Such a formula is an analytic function of the opening of the conic singularity in the interval from $\pi$ to $4\pi$ and in the continuum limit it goes over to the correct result. 
  We consider a generalization of a duality symmetric model proposed by Schwarz and Sen. It is based on enlarging the model with a dynamical vector field being a time-like component of a local Lorentz frame. This allows one to preserve the manifest Lorentz invariance of the model in flat space--time. The presence of this field is regarded as a relic of gravitational interaction which respects the general coordinate invariance in curved space--time but breaks the local Lorentz symmetry in tangent space down to its spatial subgroup. 
  Second quantization is revisited and creation and annihilation operators are shown to be related, on the same footing both to the algebra ${\it h}(1)$, ${\underline {and}}$ to the superalgebra ${\it osp}(1|2)$ that are shown to be both compatible with Bose ${\underline {and}}$ Fermi statistics. The two algebras are completely equivalent in the one-mode sector but, because of grading of ${\it osp}(1|2)$, differ in the many-particle case. The possibility of a unorthodox quantum field theory is suggested. 
  We study the quark confinement problem in 2+1 dimensional pure Yang-Mills theory using euclidean instanton methods. The instantons are regularized and dressed Wu-Yang monopoles. The dressing of a monopole is due to the mean field of the rest of the monopoles. We argue that such configurations are stable to small perturbations unlike the case of singular, undressed monopoles. Using exact non-perturbative results for the 3-dim. Coulomb gas, where Debye screening holds for arbitrarily low temperatures, we show in a self-consistent way that a mass gap is dynamically generated in the gauge theory. The mass gap also determines the size of the monopoles. In a sense the pure Yang-Mills theory generates a dynamical Higgs effect. We also identify the disorder operator of the model in terms of the Sine-Gordon field of the Coulomb gas. 
  The loop equation for the complex one-matrix model with a multi-cut structure is derived and solved in the planar limit. An iterative scheme for higher genus contributions to the free energy and the multi-loop correlators is presented for the two-cut model, where explicit results are given up to and including genus two. The double-scaling limit is analyzed and the relation to the one-cut solution of the hermitian and complex one-matrix model is discussed. 
  We show how recent exact results in supersymmetric theories can be extended to models which include {\it explicit} soft supersymmetry breaking terms. We thus derive new exact results for non-supersymmetric models. 
  We show that the trace anomaly for gravitons calculated using the usual effective action formalism depends on the choice of gauge when the background spacetime is not a solution of the classical equation of motion, that is, when off-shell. We then use the gauge independent Vilkovisky-DeWitt effective action to restore gauge independence to the off-shell case. Additionally we explicitly evaluate trace anomalies for some N-sphere background spacetimes. 
  We evaluate zeta-functions $\zeta(s)$ at $s=0$ for invariant non-minimal 2nd-order vector and tensor operators defined on maximally symmetric even dimensional spaces. We decompose the operators into their irreducible parts and obtain their corresponding eigenvalues. Using these eigenvalues, we are able to explicitly calculate $\zeta(0)$ for the cases of Euclidean spaces and $N$-spheres. In the $N$-sphere case, we make use of the Euler-Maclaurin formula to develop asymptotic expansions for the required sums. The resulting $\zeta(0)$ values for dimensions 2 to 10 are given in the Appendix. 
  The nonassociativity of the octonion algebra necessitates a bimodule representation, in which each element is represented by a left and a right multiplier. This representation can then be used to generate gauge transformations for the purpose of constructing a field theory symmetric under a gauged octonion algebra, the nonassociativity of which appears as a failure of the representation to close, and hence produces new interactions in the gauge field kinetic term of the symmetric Lagrangian. 
  We investigate the integrability of the Schwinger-Dyson equations in $c = 1 - \frac{6}{m(m+1)}$ string field theory which were proposed by Ikehara et al as the continuum limit of the Schwinger-Dyson equations of the matrix chain model. We show the continuum Schwinger-Dyson equations generate a closed algebra. This algebra contains Virasoro algebra but does not coincide with $W_{\infty}$ algebra. We include in the Schwinger-Dyson equations a new process of removing from the loop boundaries the operator ${\cal H}(\sigma)$ which locally changes the spin configuration. We also derive the string field Hamiltonian from the continuum Schwinger-Dyson equations. Its form is universal for all $c = 1 - \frac{6}{m(m+1)}$ string theories. 
  The free field realization of irreducible representations of $\os$ is constructed, by a unified and systematic scheme. The $q$-analog of this unified scheme is used to construct $q$-free field realization of irreducible representations of $\uos$. By using these realization, the two point function of $N=1$ superconformal (q-superconformal) model based on $\os$ ($\uos$) symmetry have been calculated. 
  We discuss the Chern-Simons theory in three-dimensional curved space-time in the vielbein formalism. Due to the additional presence of the local Lorentz symmetry, beside the diffeomorphisms, we will include a local gravitational supersymmetry (superdiffeomorphisms and super-Lorentz transformations), which allows us to show the perturbative finiteness at all orders. 
  We clarify the physical origin of the difference between gauge properties of conserved currents in abelian and nonabelian theories. In the latter, but not in the former, such currents can always be written on shell as gauge invariants modulo identically conserved, superpotential, terms. For the ``isotopic" vector and the stress tensor currents of spins 1 and 2 respectively, we explain this difference by the fact that the non-abelian theories are just the self-coupled versions of the abelian ones using these currents as sources. More precisely, we indicate how the self-coupling turns the non-invariantizable abelian conserved currents into (on-shell) superpotentials. The fate of other conserved currents is also discussed. 
  The technique of extended dualization developed in this paper is used to bosonize quantized fermion systems in arbitrary dimension $D$ in the low energy regime. In its original (minimal) form, dualization is restricted to models wherein it is possible to define a dynamical quantized conserved charge. We generalize the usual dualization prescription to include systems with dynamical non--conserved quantum currents. Bosonization based on this extended dualization requires the introduction of an additional rank $0$ (scalar) field together with the usual antisymmetric tensor field of rank $(D-2)$. Our generalized dualization prescription permits one to clearly distinguish the arbitrariness in the bosonization from the arbitrariness in the quantization of the system. We study the bosonization of four--fermion interactions with large mass in arbitrary dimension. First, we observe that dualization permits one to formally bosonize these models by invoking the bosonization of the free massive Dirac fermion and adding some extra model--dependent bosonic terms. Secondly, we explore the potential of extended dualization by considering the particular case of \underbar{chiral} four--fermion interactions. Here minimal dualization is inadequate for calculating the extra bosonic terms. We demonstrate the utility of extended dualization by successfully completing the bosonization of this chiral model. Finally, we consider two examples in two dimensions which illuminate the utility of using extended dualization by showing how quantization ambiguities in a fermionic theory propagate into the bosonized version. An explicit parametrization of the quantization ambiguities of the chiral current in the Chiral Schwinger model is obtained. Similarly, for the sine--Gordon interaction in the massive Thirring model the quantization 
  The one-loop effective action for Einstein gravity in a special one-parameter background gauge is calculated up to first order in a gauge parameter. It is shown that the effective action does not depend upon the gauge parameter on shell. 
  We present a model in which a gauge symmetry of a field theory is intrinsic in the geometry of an extended space time itself. A consequence is that the dimension of our space time is restricted through the BRS cohomology. If the Hilbert space is a dense subspace of the space of all square integrable $C^{\infty}$ functions, the BRS cohomology classes are nontrivial only when the dimension is two or four. 
  Irreducible gauge theories in both the Lagrangian and Hamiltonian versions of the Sp(2)-covariant quantization method are studied. Solutions to generating equations are obtained in the form of expansions in power series of ghost and auxiliary variables up to the 3d order inclusively. 
  A simple connection between the universal $R$ matrix of $U_q(sl(2))$ (for spins $\demi$ and $J$) and the required form of the co-product action of the Hilbert space generators of the quantum group symmetry is put forward. This gives an explicit operator realization of the co-product action on the covariant operators. It allows us to derive the quantum group covariance of the fusion and braiding matrices, although it is of a new type: the generators depend upon worldsheet variables, and obey a new central extension of $U_q(sl(2))$ realized by (what we call) fixed point commutation relations. This is explained by showing that the link between the algebra of field transformations and that of the co-product generators is much weaker than previously thought. The central charges of our extended $U_q(sl(2))$ algebra, which includes the Liouville zero-mode momentum in a nontrivial way are related to Virasoro-descendants of unity. We also show how our approach can be used to derive the Hopf algebra structure of the extended quantum-group symmetry $U_q(sl(2))\odot U_{\qhat}(sl(2))$ related to the presence of both of the screening charges of 2D gravity. 
  We consider a nonrelativistic Chern-Simons theory of planar matter fields interacting with the Chern-Simons gauge field in a $SU(N)_{global} \times U(1)_{local}$ invariant fashion. We find that this model admits static zero-energy self-dual soliton solutions. We also present a set of exact soliton solutions. The exact time-dependent solutions are also obtained, when this model is considered in the background of an external uniform magnetic field. 
  We investigate quantum nucleation of vortex string loops in the relativistic quantum field theory of a complex scalar field by using the Euclidean path integral. Our initial metastable homogeneous field dominated by the $O(3)$ symmetric bounce solution. The nucleation rate and the critical vortex loop size are obtained approximately. Gradually the initial current will be reduced to zero as the induced current inside vortex loops is opposite to the initial current. We also discuss a similar process in Maxwell-Higgs systems and possible physical implications. 
  We study the twisted $N=2$ supersymmetric Yang-Mills theory coupled with the hypermultiplets (TQCD). We suggest that the family of TQCD can be served as a powerful tool for studying the quantum field theoretic properties of the underlying physical theories. 
  The coupling of a Nambu-Goto string to gravity allows for Schwarzschild black holes whose entropy to area relation is $S=(A/4)(1-4\mu)$, where $\mu$ is the string tension. 
  We consider some curious aspects of single-species free Fock spaces, such as novel bosonization and fermionization formulae and relations to various physical properties of bosonic particles. We comment on generalizations of these properties to physically more interesting many-species free Fock spaces. 
  We explicitly construct the adjoint operator of coboundary operator and obtain the Hodge decomposition theorem and the Poincar\'e duality for the Lie algebra cohomology of the infinite-dimensional gauge transformation group. We show that the adjoint of the coboundary operator can be identified with the BRST adjoint generator $Q^{\dagger}$ for the Lie algebra cohomology induced by BRST generator $Q$. We also point out an interesting duality relation - Poincar\'e duality - with respect to gauge anomalies and Wess-Zumino-Witten topological terms. We consider the consistent embedding of the BRST adjoint generator $Q^{\dagger}$ into the relativistic phase space and identify the noncovariant symmetry recently discovered in QED with the BRST adjoint N\"other charge $Q^{\dagger}$. 
  The unitarity problem in curved spacetime is solved for the string described by the SL(2,R) WZW model. The spectrum is computed exactly and demonstratedto be ghost-free. The new features include (i) SL(2,R) left/right symmetrycurrents that have logarithmic cuts on the world sheet but that satisfy theusual local operator products or commutation rules, (ii) physical statesconsistent with the monodromy condition of closed strings despite thelogarithmic singularity in the currents, and (iii) a new free boson realization for these currents which render the SL(2,R) WZW model completely solvable. 
  Point particle scattering at Planckian centre-of-mass (cm) energies and low fixed momentum transfers, occurring due both to electromagnetic and gravitational interactions, is surveyed, with particular emphasis on the novel features occurring in electromagnetic charge-monopole scattering. The issue of possible mixing of the shock waves occurring due to both kinds of interactions is then addressed within the framework of Einsteinian general relativity and the dilatonic extension suggested by string theory. 
  We show how to generalize the $SU(2)$ WZW models to allow for open and unoriented sectors. The construction exhibits some novel patterns of Chan-Paton charge assignments and projected spectra that reflect the underlying current algebra. 
  We study a topological obstruction of a very stringy nature concerned with deforming the target space of an $N=2$ non-linear \sm. This target space has a singularity which may be smoothed away according to the conventional rules of geometry but when one studies the associated conformal field theory one sees that such a deformation is not possible without a discontinuous change in some of the correlation functions. This obstruction appears to come from torsion in the homology of the target space (which is seen by deforming the theory by an irrelevant operator). We discuss the link between this phenomenon and orbifolds with discrete torsion as studied by Vafa and Witten. 
  After giving a pedagogical review of the chiral gauge approach to 2D gravity, with particular emphasis on the derivation of the gravitational Ward identities, we discuss in some detail the interpretation of matter correlation functions coupled to gravity in chiral gauge. We argue that in chiral gauge no {\it explicit} gravitational dressing factor, analogue to the Liouville exponential in conformal gauge, is necessary for left-right symmetric matter operators. In particular, we examine the gravitationally dressed four-point correlation function of products of left and right fermions. We solve the corresponding gravitational Ward identity exactly: in the presence of gravity this four-point function exhibits a logarithmic short-distance singularity, instead of the power-law singularity in the absence of gravity. This rather surprising effect is non-perturbative in the gravitational coupling and is a sign for logarithms in the gravitationally dressed operator product expansions. We also discuss some perturbative evidence that the chiral Gross-Neveu model may remain integrable when coupled to gravity. 
  For the closed superstring, spin fields and bi-spinor states are defined directly in four spacetime dimensions. Explicit operator product expansions are given, including those for the internal superconformal field theory, which are consistent with locality and BRST invariance for the string vertices. The most general BRST picture changing for these fields is computed. A covariant notation for the spin decomposition of these states is developed in which non-vanishing polarizations are selected automatically. The kinematics of the three-gluon dual model amplitude in both the Neveu-Schwarz and Ramond sectors in the Lorentz gauges is calculated and contrasted. Modular invariance and enhanced gauge symmetry of four-dimensional models incorporating these states is described. 
  Some new developments in constrained Lax integrable systems and their applications to physics are reviewed. After summarizing the tau function construction of the KP hierarchy and the basic concepts of the symmetry of nonlinear equations, more recent ideas dealing with constrained KP models are described. A unifying approach to constrained KP hierarchy based on graded $SL(r+n,n)$ algebra is presented and equivalence formulas are obtained for various pseudo-differential Lax operators appearing in this context. It is then shown how the Toda lattice structure emerges from constrained KP models via canonical Darboux-B\"{a}cklund transformations. These transformations enable us to find simple Wronskian solutions for the underlying tau-functions. We also establish a relation between two-matrix models and constrained Toda lattice systems and derive from this relation expressions for the corresponding partition function. 
  It is argued that many linear (0,2) models flow in the infrared to conformally invariant solutions of string theory. The strategy in the argument is to show that the effective space-time superpotential must vanish because there is no place where it can have a pole. This conclusion comes from either of two different analyses, in which the Kahler class or the complex structure of the gauge bundle is varied, while keeping everything else fixed. In the former case, we recover from the linear sigma model the usual simple pole in the ${\bf \bar {27}}^3$ Yukawa coupling but show that an analogous pole does not arise in the couplings of gauge singlet modes. In the latter case, a dimension count shows that the world-sheet instanton sum does not ``see'' the singularities of the gauge bundle and hence cannot have a pole. 
  We are able to perform the duality transformation of the spin system which was found before as a lattice realization of the string with linear action. In four and higher dimensions this spin system can be described in terms of a two-plaquette gauge Hamiltonian. The duality transformation is constructed in geometrical and algebraic language. The dual Hamiltonian represents a new type of spin system with local gauge invariance. At each vertex $\xi$ there are $d(d-1)/2$ Ising spins $\Lambda_{\mu,\nu}= \Lambda_{\nu,\mu}$, $\mu \neq \nu = 1,..,d$ and one Ising spin $\Gamma$ on every link $(\xi,\xi +e_{\mu})$. For the frozen spin $\Gamma \equiv 1$ the dual Hamiltonian factorizes into $d(d-1)/2$ two-dimensional Ising ferromagnets and into antiferromagnets in the case $\Gamma \equiv -1$. For fluctuating $\Gamma$ it is a sort of spin glass system with local gauge invariance. The generalization to $p$-branes is given. 
  Some applications of the odd Poisson bracket to the description of the classical and quantum dynamics are represented. 
  The ambiguity in the calculations of one-loop counterterms by the background field method in nonrenormalizable theories of gravity is discussed. Some examples of such ambiguous calculations are given. The non-equivalence of the first and second order formalism in the quantum gravity is shown. 
  We investigate the tachyon-dilaton-metric system to study the "graceful exit" problem in string theoretic inflation, where tachyon plays the role of the scalar field. From the phase space analysis, we find that the inflationary phase does not smoothly connect to a Friedmann-Robertson-Walker (FRW) expanding universe, thereby providing a simple tachyonic extension of the recently proved stringy no-go theorem. 
  The $p\times p$ matrix version of the $r$-KdV hierarchy has been recently treated as the reduced system arising in a Drinfeld-Sokolov type Hamiltonian symmetry reduction applied to a Poisson submanifold in the dual of the Lie algebra $\widehat{gl}_{pr}\otimes {\Complex}[\lambda, \lambda^{-1}]$. Here a series of extensions of this matrix Gelfand-Dickey system is derived by means of a generalized Drinfeld-Sokolov reduction defined for the Lie algebra $\widehat{gl}_{pr+s}\otimes {\Complex}[\lambda,\lambda^{-1}]$ using the natural embedding $gl_{pr}\subset gl_{pr+s}$ for $s$ any positive integer. The hierarchies obtained admit a description in terms of a $p\times p$ matrix pseudo-differential operator comprising an $r$-KdV type positive part and a non-trivial negative part. This system has been investigated previously in the $p=1$ case as a constrained KP system. In this paper the previous results are considerably extended and a systematic study is presented on the basis of the Drinfeld-Sokolov approach that has the advantage that it leads to local Poisson brackets and makes clear the conformal ($\cal W$-algebra) structures related to the KdV type hierarchies. Discrete reductions and modified versions of the extended $r$-KdV hierarchies are also discussed. 
  Collision of plane waves in dilaton gravity theories and low energy limit of string theory is considered. The formulation of the the problem and some exact solutions are presented. 
  The Poincare algebra of classical electrodynamics in one spatial dimension is studied using light-cone coordinates and ordinary Minkowski coordinates. We show that it is possible to quantize the theory by a canonical quantization procedure in a Poincare invariant manner on the light-cone. We also show that this is not possible when using ordinary coordinates. The physical reason of this anomaly is analysed. 
  The general correlations between massless fermions are calculated in the Schwinger model at arbitrary temperature. The zero temperature calculations on the plane are reviewed and clarified. Then the finite temperature fermionic Green's function is computed and the results on the torus are compared to those on the plane. It is concluded that a simpler way to calculate the finite temperature results is to associate certain terms in the zero temperature structure with their finite temperature counterparts. 
  An evidence for nontriviality of asymptotically non-free (ANF) Yang-Mills theories is found on the basis of optimized perturbation theory. It is argued that these theories with matter couplings can be made nontrivial by means of the reduction of couplings, leading to the idea of dynamical unification of couplings (DUC) The second-order reduction of couplings in the ANF $SU(3)$-gauged Higgs-Yukawa theory, which is assumed to be nontrivial here, is carried out to motivate independent investigations on its nontriviality and DUC. 
  Some of the motivations for as well as the main points of the quantization of the Nappi Witten string in the light cone gauge are reviewed. 
  A cosmological model describing the evolution of n Ricci-flat spaces (n>1) in the presence of 1-component perfect-fluid and minimally coupled scalar field is considered. When the pressures in all spaces are proportional to the density, the Einstein and Wheeler-DeWitt equations are integrated for a large variety of parameters. Classical and quantum wormhole solutions are obtained for negative density. Some special classes of solutions, e.g. solutions with spontaneous and dynamical compactification, exponential and power-law inflations, are singled out. For positive density a third quantized cosmological model is considered and the Planckian spectrum of ``created universes'' is obtained. 
  In this letter we argue that there is no ambiguity between the Pauli-Villars and other methods of regularization in (2+1)-dimensional quantum electrodynamics with respect to dynamical mass generation, provided we properly choose the couplings for the regulators. 
  For all Poincar\'e invariant Lagrangians of the form ${\cal L}\equiv f(F_{\mu\nu})$, in three Euclidean dimensions, where $f$ is any invariant function of a non-compact $U(1)$ field strength $F_{\mu\nu}$, we find that the only continuum limit (described by just such a gauge field) is that of free field theory: First we approximate a gauge invariant version of Wilson's renormalization group by neglecting all higher derivative terms $\sim \partial^nF$ in ${\cal L}$, but allowing for a general non-vanishing anomalous dimension. Then we prove analytically that the resulting flow equation has only one acceptable fixed point: the Gaussian fixed point. The possible relevance to high-$T_c$ superconductivity is briefly discussed. 
  It has been suggested that a possible classical remnant of the phenomenon of target-space duality (T-duality) would be the equivalence of the classical string Hamiltonian systems. Given a simple compact Lie group $G$ with a bi-invariant metric and a generating function $\Gamma$ suggested in the physics literature, we follow the above line of thought and work out the canonical transformation $\Phi$ generated by $\Gamma$ together with an $\Ad$-invariant metric and a B-field on the associated Lie algebra $\frak g$ of $G$ so that $G$ and $\frak g$ form a string target-space dual pair at the classical level under the Hamiltonian formalism. In this article, some general features of this Hamiltonian setting are discussed. We study properties of the canonical transformation $\Phi$ including a careful analysis of its domain and image. The geometry of the T-dual structure on $\frak g$ is lightly touched. 
  We study the ground state energy of integrable $1+1$ quantum field theories with boundaries (the genuine Casimir effect). In the scalar case, this is done by introducing a new, ``R-channel TBA'', where the boundary is represented by a boundary state, and the thermodynamics involves evaluating scalar products of boundary states with all the states of the theory. In the non-scalar, sine-Gordon case, this is done by generalizing the method of Destri and De Vega. The two approaches are compared. Miscellaneous other results are obtained, in particular formulas for the overall normalization and scalar products of boundary states, exact partition functions for the critical Ising model in a boundary magnetic field, and also results for the energy, excited states and boundary S-matrix of $O(n)$ and minimal models. 
  Witten proposed that the low energy physics of strongly coupled D=10 type-IIA superstring may be described by D=11 supergravity. To explore the stringy aspects of the underlying theory we examine the stringy massive states. We propose a systematic formula for identifying non-perturbative states in D=10 type-IIA superstring theory, such that, together with the elementary excited string states, they form D=11 supersymmetric multiplets multiplets in SO(10) representations. This provides hints for the construction of a weakly coupled D=11 theory that is dual to the strongly coupled D=10 type IIA superstring. 
  We construct the effective field theory of the Calogero-Sutherland model in the thermodynamic limit of large number of particles $N$. It is given by a $\winf$ conformal field theory (with central charge $c=1$) that describes {\it exactly} the spatial density fluctuations arising from the low-energy excitations about the Fermi surface. Our approach does not rely on the integrable character of the model, and indicates how to extend previous results to any order in powers of $1/N$. Moreover, the same effective theory can also be used to describe an entire universality class of $(1+1)$-dimensional fermionic systems beyond the Calogero-Sutherland model, that we identify with the class of {\it chiral Luttinger systems}. We also explain how a systematic bosonization procedure can be performed using the $\winf$ generators, and propose this algebraic approach to {\it classify} low-dimensional non-relativistic fermionic systems, given that all representations of $\winf$ are known. This approach has the appeal of being mathematically complete and physically intuitive, encoding the picture suggested by Luttinger's theorem. 
  Motivated by the discovery of errors in six of the 135 diagrams in the published five-loop expansions of the $\beta$-function and the anomalous dimensions of the ${O}(n)$-symmetric $\phi^4$-theory in $D=4-\ep$ dimensions we present the results of a full analytic reevaluation of all diagrams. The divergences are removed by minimal subtraction and $\ep$-expansions are given for the critical exponents $\eta$, $\nu$, and $\omega$ up to order $\epsilon^5$. 
  The genus-dependence of multi-loop superstring amplitudes is bounded at large orders in perturbation theory using the super-Schottky group parametrization of supermoduli space. Partial estimates of supermoduli space integrals suggest an exponential dependence on the genus when the integration region is restricted to a single fundamental domain of the super-modular group in the super-Schottky parameter space. Bounds for N-point superstring scattering amplitudes are obtained for arbitrary N and are shown to be consistent with exact results recently obtained for special type II string amplitudes for orbifold or Calabi-Yau compactifications. It is suggested that the generic estimates, which imply the validity of superstring perturbation theory in the weak-coupling limit, might be used to determine scattering amplitudes at strong coupling because of the S-duality of type II and heterotic string theories. Non-perturbative effects are consistent with these estimates, based on a sum over closed surfaces, because they can be derived from an additional contribution to the sum over surfaces corresponding to the insertion of Dirichlet boundaries. 
  The general structure of N=2 moduli space at arbitrary genus and instanton number is investigated. The N=2 NSR string measure is calculated, yielding picture- and U(1) ghost number-changing operator insertions. An explicit formula for the spectral flow operator acting on vertex operators is given, and its effect on N=2 string amplitudes is discussed. 
  We examine the notion of Haldane's dimension and the corresponding statistics in a probabilistic spirit. Motivated by the example of dimensional-regularization we define the dimension of a space as the trace of a diagonal `unit operator', where the diagonal matrix elements are not, in general, unity but are probabilities to place the system into a given state. These probabilities are uniquely defined by the rules of Haldane's statistics. We calculate the second virial coefficient for our system and demonstrate agreement with Murthy and Shankar's calculation. The partition function for an ideal gas of the particles, a state-counting procedure, the entropy and a distribution function for the particles are investigated using our probabilistic definition. We compare our results with previous calculations of exclusion statistics. 
  A charge-monopole theory is derived from simple and self-evident postulates. Charges and monopoles take an analogous theoretical structure. It is proved that charges interact with free waves emitted from monopoles but not with the corresponding velocity fields. Analogous relations hold for monopole equations of motion. The system's equations of motion can be derived from a regular Lagrangian function. 
  Using the integrability conditions that we recently obtained in QCD$_2$ with massless fermions, we arrive at a sufficient number of conservation laws to be able to fix the scattering amplitudes involving a local version of the Wilson loop operator. 
  We propose a new fractional statistics for arbitrary dimensions, based on an extension of Pauli's exclusion principle, to allow for finite multi-occupancies of a single quantum state. By explicitly constructing the many-body Hilbert space, we obtain a new algebra of operators and a new thermodynamics. The new statistics is different from fractional exclusion statistics; and in a certain limit, it reduces to the case of parafermi statistics. 
  We construct an explicit and manifestly (1,0) heterotic sigma-model where the background fields are those of 10D, N=IIB supergravity. 
  We construct integrable models on flag manifold by using the symplectic structure explicitly given in the Bruhat coordinatization of flag manifold. They are non-commutative integrable and some of the conserved quantities are given by the Casimir invariants. We quantize the systems using the coherent state path integral technique and find the exact expression for the propagator for some special cases. 
  De Witt--Seeley--Gilkey coefficients are calculated for the most general minimal differential fourth--order operator on Riemannian space of an arbitrary dimension. 
  Recently, certain higher dimensional complex manifolds were obtained in [hep-th/9412078] by associating a higher dimensional uniformisation to the generalised Teichm\"uller spaces of Hitchin. The extra dimensions are provided by the ``times'' of the generalised KdV hierarchy. In this paper, we complete the proof that these manifolds provide the analog of superspace for W-gravity and that W-symmetry linearises on these spaces. This is done by explicitly constructing the relationship between the Beltrami differentials which naturally occur in the higher dimensional manifolds and the Beltrami differentials which occur in W-gravity. This also resolves an old puzzle regarding the relationship between KdV flows and W-diffeomorphisms. 
  We analyze multi-instanton sector in two dimensional U(N) Yang-Mills theory on a sphere. We obtain a contour intregrals representation of the multi-instanton amplitude and find ``neutral'' configurations of the even number instantons are dominant in the large N limit. Using this representation, we calculate 1,2,3,4 bodies interactions and the free energies for $N =3,4,5$ numerically and find that in fact the multi-instanton interaction effect essentially contribute to the large N phase transition discovered by Douglas and Kazakov. 
  A topological quantum field theory is introduced which reproduces the Seiberg-Witten invariants of four-manifolds. Dimensional reduction of this topological field theory leads to a new one in three dimensions. Its partition function yields a three-manifold invariant, which can be regarded as the Seiberg-Witten version of Casson's invariant. A Geometrical interpretation of the three dimensional quantum field theory is also given. 
  We study the low-energy effective Lagrangian of $N=2$ heterotic string vacua at the classical and quantum level. The couplings of the vector multiplets are uniquely determined at the tree level, while the loop corrections are severely constrained by the exact discrete symmetries of the string vacuum. We evaluate the general transformation law of the perturbative prepotential and determine its form for the toroidal compactifications of six-dimensional $N=1$ supersymmetric vacua. 
  Recent results on classical and quantum strings in a variety of black hole and cosmological spacetimes, in various dimensions, are presented. The curved backgrounds under consideration include the $2+1$ black hole anti de Sitter spacetime and its dual, the black string, the ordinary $D\geq 4$ black holes with or without a cosmological constant, the de Sitter and anti de Sitter spacetimes and static Robertson-Walker spacetimes. Exact solutions to the string equations of motion and constraints are obtained in these backgrounds and their physical properties (length, energy, pressure) are described. The existence of {\it multi-string} solutions, describing finitely or infinitely many strings, is shown to be a general feature of spacetimes with a positive or negative cosmological constant. Furthermore, using a canonical quantization procedure, we find the string mass spectrum in de Sitter and anti de Sitter spacetimes. New features as compared to the string spectrum in flat Minkowski spacetime appear, for instance the {\it fine-structure effect} at all levels beyond the graviton in both de Sitter and anti de Sitter spacetimes, and the {\it non-existence} of a Hagedorn temperature in anti de Sitter spacetime. Finally, we consider the effect of spatial curvature on the string dynamics in Robertson-Walker spacetimes. 
  We study the relationship among the XXZ Heisenberg model and three models obtained from it by various transformations. In particular, we emphasize the role of a non trivial central element $t^Z$ in the underlying algebra and its relationship with the twisted boundary conditions, $S^{\pm}_{N+1}=t^{\pm N}S^{\pm}_1$. 
  The two-flavor Wess-Zumino model coupled to electromagnetism is treated as a constraint system using the Faddeev-Jackiw method. Expanding into series of powers of the pion fields and keeping terms up to second and third order we obtain Coulomb- gauge Lagrangeans containing non-local terms. 
  We present a non-abelian generalization of Witten monopole equations and we analyze the associated moduli problem, which can be regarded as a generalization of Donaldson theory. The moduli space of solutions for SU(2) monopoles on K\"ahler manifolds is discussed. We also construct, using the Mathai-Quillen formalism, the topological quantum field theory corresponding to the new moduli problem. This theory involves the coupling of topological Yang-Mills theory to topological matter in four dimensions 
  In the example of $R^2+T^2$ gravity on the unit two dimensional disk we demonstrate that in the presence of an independent spin connection it is possible to define local gauge invariant boundary conditions even on boundaries which are not totally geodesic. One-loop partition function and the corresponding heat kernel are calculated. 
  We show how the supersymmetric properties of three dimensional black holes can be obtained algebraically. The black hole solutions are constructed as quotients of the supergroup $OSp(1|\,2;R)$ by a discrete subgroup of its isometry supergroup. The generators of the action of the isometry supergroup which commute with these identifications are found. These yield the supersymmetries for the black hole as found in recent studies as well as the usual geometric isometries. It is also shown that in the limit of vanishing cosmological constant, the black hole vacuum becomes a null orbifold, a solution previously discussed in the context of string theory. 
  The large N limit of the hermitian matrix model in three and four Euclidean space-time dimensions is studied with the help of the approximate Renormalization Group recursion formula. The planar graphs contributing to wave function, mass and coupling constant renormalization are identified and summed in this approximation. In four dimensions the model fails to have an interacting continuum limit, but in three dimensions there is a non trivial fixed point for the approximate RG relations. The critical exponents of the three dimensional model at this fixed point are $\nu = 0.665069$ and $\eta=0.19882$. The existence (or non existence) of the fixed point and the critical exponents display a fairly high degree of universality since they do not seem to depend on the specific (non universal) assumptions made in the approximation. 
  The running coupling constants (in particular, the gravitational one) are studied in asymptotically free GUTs and in finite GUTs in curved spacetime, with explicit examples. The running gravitational coupling is used to calculate the leading quantum GUT corrections to the Newtonian potential, which turn out to be of logarithmic form in asymptotically free GUTs. A comparison with the effective theory for the conformal factor ---where leading quantum corrections to the Newtonian potential are again logarithmic--- is made. A totally asymptotically free $O(N)$ GUT with quantum higher derivative gravity is then constructed, using the technique of introducing renormalization group (RG) potentials in the space of couplings. RG equations for the cosmological and gravitational couplings in this theory are derived, and solved numerically, showing the influence of higher-derivative quantum gravity on the Newtonian potential. The RG-improved effective gravitational Lagrangian for asymptotically free massive GUTs is calculated in the strong (almost constant) curvature regime, and the non-singular De Sitter solution to the quantum corrected gravitational equations is subsequently discussed. Finally, possible extensions of the results here obtained are briefly outlined. 
  We display properties of the general formalism which associates to any given gauge symmetry a topological action and a system of topological BRST and anti-BRST equations. We emphasize the distinction between the antighosts of the geometrical BRST equations and the antighosts occuring in field theory. We propose a transmutation mechanism between these objects. We illustrate our general presentation by examples. 
  Canonical quantization of anomalous SU(N) Yang-Mills models is considered. It is shown that the gauge invariance of the quantum theory can be saved in spite of degeneracy of the Wess-Zumino action.  
  We consider two-dimensional quantum gravity coupled to matter in the temporal gauge, using the Polyakov path integral. We show that the integration over the metric can be explicitly performed under some plausible assumptions. We also discuss that the critical dimensions in string theory may not be determined in the temporal gauge. 
  We study boundary reflection matrix for the quantum field theory defined on a half line using Feynman's perturbation theory. The boundary reflection matrix can be extracted directly from the two-point correlation function. This enables us to determine the boundary reflection matrix for affine Toda field theory with the Neumann boundary condition modulo `a mysterious factor half'. 
  A number of attempts have recently been made to extend the conjectured $S$ duality of Yang Mills theory to gravity. Central to these speculations has been the belief that electrically and magnetically charged black holes, the solitons of quantum gravity, have identical quantum properties. This is not obvious, because although duality is a symmetry of the classical equations of motion, it changes the sign of the Maxwell action. Nevertheless, we show that the chemical potential and charge projection that one has to introduce for electric but not magnetic black holes exactly compensate for the difference in action in the semi-classical approximation. In particular, we show that the pair production of electric black holes is not a runaway process, as one might think if one just went by the action of the relevant instanton. We also comment on the definition of the entropy in cosmological situations, and show that we need to be more careful when defining the entropy than we are in an asymptotically-flat case. 
  A recent result of Gusynin, Miransky and Shovkovy concerning chiral symmetry breaking by a constant external magnetic field in parity-invariant three-dimensional QED is generalised to the case of inhomogeneous fields by relating the phenomenon to the zero modes of the Dirac equation. Virtual photon radiative corrections and four-dimensional QED are briefly discussed. 
  General static solutions for a massless scalar field coupled to a class of effectively 2-d gravity theories continuously connecting spherically symmetric $d$-dimensional Einstein gravity ($d >3$) and the CGHS model are analytically obtained. They include black holes and point scalar charge solutions with naked singularities, and are used to give an analytic proof of no-hair theorem. Exact scattering solutions in $s$-wave 4-d Einstein gravity are constructed as a generalization of corresponding static solutions. They show the existence of black hole formation threshold for square pulse type incoming stress-energy flux, above which trapped surfaces are dynamically formed. The relationship between this behavior and the numerically studied phase transition in this system \cite{choptuik} is discussed. 
  The quantum entanglement entropy of an eternal black hole is studied. We argue that the relevant Euclidean path integral is taken over fields defined on $\alpha$-fold covering of the black hole instanton. The statement that divergences of the entropy are renormalized by renormalization of gravitational couplings in the effective action is proved for non-minimally coupled scalar matter. The relationship of entanglement and thermodynamical entropies is discussed. 
  An exact quantization of the spherical membrane moving in flat target spacetime backgrounds is performed. Crucial ingredients are the exact integrabilty of the $3D~SU(\infty)$ continuous Toda equation and the quasi-finite highest weight irreducible representations of $W_{\infty}$ algebras. Both continuous and discrete energy levels are found. The latter are found for periodic-like solutions. Membrane wavefunctionals solutions are found in terms of Bessel's functions and plausible relations to singleton field theory are outlined. 
  The path integral of the relativistic Coulomb system is solved, and the wave functions are extracted from the resulting amplitude. 
  An explicit construction of theories of spinning particles, both massive and massless, is given with arbitrary extended supersymmetry on the world-line. As an application of our results, we give a universal description of 3D (and via truncation all lower dimensional) supersymmetric scalar multiplets with arbitrary N-extended supersymmetry. 
  Motivated by the work of Kalloniatis, Pauli and Pinsky, we consider the theory of light-cone quantized $QCD_{1+1}$ on a spatial circle with periodic and anti-periodic boundary conditions on the gluon and quark fields respectively. This approach is based on Discretized Light-Cone Quantization (DLCQ). We investigate the canonical structures of the theory. We show that the traditional light-cone gauge $A_- = 0$ is not available and the zero mode (ZM) is a dynamical field, which might contribute to the vacuum structure nontrivially. We construct the full ground state of the system and obtain the Schr\"{o}dinger equation for ZM in a certain approximation. The results obtained here are compared to those of Kalloniatis et al. in a specific coupling region. 
  It has recently been conjectured that the type IIA string theory compactified on K3 and the heterotic string theory compactified on a four dimensional torus describe identical string theories. The fundamental heterotic string can be regarded as a non-singular soliton solution of the type IIA string theory with a semi-infinite throat. We show that this solution admits 24 parameter non-singular deformation describing a fundamental heterotic string carrying electric charge and current. The charge is generated due to the coupling of the gauge fields to the anti-symmetric tensor field, and not to an explicit source term. This clarifies how soliton solutions carrying charge under the Ramond-Ramond fields can be constructed in the type IIA theory, and provides further support to the string string duality conjecture. Similarly, the fundamental type IIA string can be regarded as a non-singular solution of the heterotic string theory with a semi-infinite throat, but this solution does not admit any deformation representing charged string. This is also consistent with the expectation that a fundamental type IIA string does not carry any charge that couples to the fields originating in the Ramond-Ramond sector. 
  We present the solution of the problem of the $1/\Box, \Box \to 0,$ asymptotic terms discovered in the one-loop form factors of the gravitational effective action. Owing to certain constraints among their coefficients, which we establish, these terms cancel in the vacuum stress tensor and do not violate the asymptotic flatness of the expectation value of the metric. They reappear, however, in the Riemann tensor of this metric and stand for a new effect: a radiation of gravitational waves induced by the vacuum stress. This coherent radiation caused by the backreaction adds to the noncoherent radiation caused by the pair creation in the case where the initial state provides the vacuum stress tensor with a quadrupole moment. 
  We study the fifth term in the asymptotic expansion of the heat operator trace for a partial differential operator of Laplace type on a compact Riemannian manifold with Dirichlet or Neumann boundary conditions. 
  We obtain the conserved, nonlocal charges for the supersymmetric two boson hierarchy from fractional powers of its Lax operator. We show that these charges reduce to the ones of the supersymmetric KdV system under appropriate reduction. We study the algebra of the nonlocal, local and supersymmetry charges with respect to the first and the second Hamiltonian structures of the system and discuss how they close as a graded nonlinear cubic algebra. 
  We extend existing treatments of black hole solutions in String Gravity to include moduli fields. We compute the external moduli and dilaton hair, as well as of their associated axions,to $O(\alpha')$ in the framework of the loop corrected superstring effective action for a Kerr-Newman black hole background. 
  We construct $N=1$ supergravity models where the gauge symmetry and supersymmetry are both spontaneously broken, with naturally vanishing classical vacuum energy and unsuppressed Goldstino components along gauge non-singlet directions. We discuss some physically interesting situations where such a mechanism could play a role, and identify the breaking of a grand-unified gauge group as the most likely possibility. We show that, even when the gravitino mass is much smaller than the scale $m_X$ of gauge symmetry breaking, important features can be missed if we first naively integrate out the degrees of freedom of mass ${\cal O} (m_X)$, in the limit of unbroken supersymmetry, and then describe the super-Higgs effect in the resulting effective theory. We also comment on possible connections with extended supergravities and realistic four-dimensional string constructions. 
  We discuss the relation between the micro-canonical and the canonical ensemble for black holes, and highlight some problems associated with extreme black holes already at the classical level. Then we discuss the contribution of quantum fields and demonstrate that the partition functions for scalar and Dirac (Majorana) fields in static space-time backgrounds, can be expressed as functional integrals in the corresponding optical space, and point out that the difference between this and the functional integrals in the original metric is a Liouville-type action. The optical method gives both the correction to the black hole entropy and the bulk contribution to the entropy due to the radiation, while (if the Liouville term is ignored) the conical singularity method just gives the divergent contribution to the black hole entropy. A simple derivation of a general formula for the free energy in the high-temperature approximation is given and applied to various cases. We conclude with a discussion of the second law. 
  We discuss the prepotential describing the effective field theory of N=2 heterotic superstring models. At the one loop-level the prepotential develops logarithmic singularities due to the appearance of charged massless states at particular surfaces in the moduli space of vector multiplets. These singularities modify the classical duality symmetry group which now becomes a representation of the fundamental group of the moduli space minus the singular surfaces. For the simplest two-moduli case, this fundamental group turns out to be a certain braid group and we determine the resulting full duality transformations of the prepotential, which are exact in perturbation theory. 
  In this paper we formulate an integrable model on the simple cubic lattice. The $N$ -- valued spin variables of the model belong to edges of the lattice. The Boltzmann weights of the model obey the vertex type Tetrahedron Equation. In the thermodynamic limit our model is equivalent to the Bazhanov -- Baxter Model. In the case when $N=2$ we reproduce the Korepanov's and Hietarinta's solutions of the Tetrahedron equation as some special cases. 
  We study the BRST cohomology within a local conformal Lagrangian field theory model built on a two dimensional Riemann surface with no boundary. We deal with the case of the complex structure parametrized by Beltrami differential and the scalar matter fields. The computation of {\em all} elements of the BRST cohomology is given. 
  The BRST cohomology group in the space of local functionals of the fields for the two-dimensional conformally invariant gravity is calculated. All classical local actions (ghost number equal to zero) and all candidate anomalies are given and discussed for our model. 
  We show that the exchange statistics have consequences in 1D systems with compact topology, contrary to the common opinion that exchange statistics is arbitrary in 1D. As examples of non-trivial statistical behavior we exactly calculate the partition function and correlators for systems of free q-particles on compactified chains using functional integral techniques and the supersymmetric trick. In particular we consider a spin 1/2 XY-chain with periodic boundary conditions that corresponds to the case of q=-1. 
  The field-antifield quantization method is used to calculate the trace anomaly for a massless scalar field in a curved background, by means of the zeta function regularization procedure. 
  The Leech lattice, $\Lambda_{24}$, is represented on the space of octonionic 3-vectors. It is built from two octonionic representations of $E_{8}$, and is reached via $\Lambda_{16}$. It is invariant under the octonion index cycling and doubling maps. 
  We construct quantum operators solving the quantum versions of the Sturm-Liouville equation and the resolvent equation, and show the existence of conserved currents. The construction depends on the following input data: the basic quantum field $O(k)$ and the regularization . 
  In the paper we study nonlocal functionals whose kernels are homogeneous generalized functions. We also use such functionals to solve the Korteweg-de Vries , the nonlinear Schr\"odinger and the Davey-Stewartson equations. 
  Recently we have shown that a phase transition occurs in the leading and sub-leading approximation of the large N limit in rigid strings coupled to long range Kalb-Ramond interactions. The disordered phase is essentially the Nambu-Goto-Polyakov string theory while the ordered phase is a new theory. In this letter we compute the free energy per unit length of the interacting rigid string at finite temperature. We show that the mass of the winding states solves that of QCD strings in the limit of high temperature. We obtain a precise identification of the QCD coupling constant and those of the interacting rigid string. The relation we obtain is $Ng_{QCD}^2=({4\pi^2(D-2)\over 3})^2{1\over 3\kappa}$ where $\kappa={Dt\alpha\over\pi\mu_{c}}$ is the ratio of the extrinsic curvature coupling constant t, the Kalb-Ramond coupling constant $\alpha$, and the critical string tension $\mu_{c}$. The running beta function of $\kappa$ reproduces correctly the asymptotic behaviour of QCD. 
  Rigid QED is a renormalizable generalization of Feynman's space-time action characterized by the addition of the curvature of the world line (rigidity). We have recently shown that a phase transition occurs in the leading approximation of the large N limit. The disordered phase essentially coincides with ordinary QED, while the ordered phase is a new theory. We have further shown that both phases of the quantum theory are free of ghosts and tachyons. In this letter, we study the first sub-leading quantum corrections leading to the renormalized mass gap equation. Our main result is that the phase transition does indeed survive these quantum fluctuations. 
  Recently we have shown that a phase transition occurs in the leading approximation of the large N limit in rigid strings coupled to long range Kalb-Ramond interactions. The disordered phase is essentially the Nambu-Goto-Polyakov string theory while The ordered phase is a new theory. In this part I letter we study the first sub-leading quantum corrections of the free rigid string and derive the renormalization group equation. We show that the theory is asymptotically free, thus the extrinsic curvature of the string drops out at large distance scales in the disordered phase. In part II we generalize the results of this letter to the interacting theory of rigid strings with the long range Kalb-Ramond interactions. We derive the renormalized mass gap equation and obtain the renormalized critical line. Our main and final result is that the phase transition does indeed survive quantum fluctuations. 
  Recently we have shown that a phase transition occurs in the leading approximation of the large N limit in rigid strings coupled to long range Kalb-Ramond interactions. The disordered phase is essentially the Nambu-Goto-Polyakov string theory while the ordered phase is a new theory. In this part II letter we study the first sub-leading quantum corrections we started in I. We derive the renormalized mass gap equation and obtain the renormalized critical line of the interacting theory. Our main and final result is that the phase transition does indeed survive quantum fluctuations. 
  It is shown that the Type IIA superstring compactified on $K3$ has a smooth string soliton with the same zero mode structure as the heterotic string compactified on a four torus, thus providing new evidence for a conjectured exact duality between the two six-dimensional string theories. The chiral worldsheet bosons arise as zero modes of Ramond-Ramond fields of the IIA string theory and live on a signature $(20,4)$ even, self-dual lattice. Stable, finite loops of soliton string provide the charged Ramond-Ramond states necessary for enhanced gauge symmetries at degeneration points of the $K3$ surface. It is also shown that Type IIB strings toroidally compactified to six dimensions have a multiplet of string solutions with Type II worldsheets. 
  We systematically improve the recent variational calculation of the imaginary part of the ground state energy of the quartic anharmonic oscillator.   The results are extremely accurate as demonstrated by deriving, from the calculated imaginary part, all perturbation coefficients via a dispersion relation and reproducing the exact values with a relative error of less than $10^{-5}$. A comparison is also made with results of a Schr\"{o}dinger calculation based on the complex rotation method. 
  Topological Yang-Mills theory with the Belavin-Polyakov-Schwarz-Tyupkin $SU(2)$ instanton is solved completely, revealing an underlying multi-link intersection theory. Link invariants are also shown to survive the coupling to a certain kind of matter (hyperinstantons). The physical relevance of topological field theory and its invariants is discovered. By embedding topological Yang-Mills theory into pure Yang-Mills theory, it is shown that the topological version TQFT of a quantum field theory QFT allows us to formulate consistently the perturbative expansion of QFT in the topologically nontrivial sectors. In particular, TQFT classifies the set of good measures over the instanton moduli space and solves the inconsistency problems of the previous approaches. The qualitatively new physical implications are pointed out. Link numbers in QCD are related to a non abelian analogoue of the Aharonov-Bohm effect. 
  The exact general solution of circular strings in $2+1$ dimensional de Sitter spacetime is described completely in terms of elliptic functions. The novel feature here is that one single world-sheet generically describes {\it infinitely many} (different and independent) strings. This has no analogue in flat spacetime. The circular strings are either oscillating ("stable") or indefinitely expanding ("unstable"). We then compute the {\it exact} equation of state of circular strings in the $2+1$ dimensional de Sitter (dS) and anti de Sitter (AdS) spacetimes, and analyze its properties for the different (oscillating, contracting and expanding) strings. We finally perform a semi-classical quantization of the oscillating circular strings. We find the mass formula $\alpha'm^2_{\mbox{dS}}\approx 4n-5H^2\alpha'n^2,\;(n\in N_0),$ and a {\it finite} number of states $N_{\mbox{dS}}\approx 0.34/(H^2\alpha')$ in de Sitter spacetime; $m^2_{\mbox{AdS}}\approx H^2n^2$ (large $n\in N$) and $N_{\mbox{AdS}}=\infty$ in anti de Sitter spacetime. The level spacing grows with $n$ in AdS spacetime, while is approximately constant (although smaller than in Minkowski spacetime and slightly decreasing) in dS spacetime. 
  The regularization scheme is proposed for the constrained Hamiltonian formulation of the gauge fields coupled to the chiral or axial fermions. The Schwinger terms in the regularized operator first-class constraint algebra are shown to be consistent with the covariant divergence anomaly of the corresponding current. Regularized quantum master equations are studied, and the Schwinger terms are found out to break down both nilpotency of the BRST-charge and its conservation law. Wess-Zumino consistency conditions are studied for the BRST anomaly and they are shown to contradict to the covariant Schwinger terms in the BRST algebra. 
  The quasi-particle structure of the higher spin XXZ model is studied. We obtained a new description of crystals associated with the level $k$ integrable highest weight $U_q(\widehat{sl_2})$ modules in terms of the creation operators at $q=0$ (the crystaline spinon basis). The fermionic character formulas and the Yangian structure of those integrable modules naturally follow from this description. We have also derived the conjectural formulas for the multi quasi-particle states at $q=0$. 
  In the SU(3) Einstein-Yang-Mills system sequences of static spherically symmetric regular solutions and black hole solutions exist for both the SU(2) and the SO(3) embedding. We construct the lowest regular solutions of the SO(3) embedding, missed previously, and the corresponding black holes. The SO(3) solutions are classified according to their boundary conditions and the number of nodes of the matter functions. Both, the regular and the black hole solutions are unstable. 
  We perform numerical simulations of the two and three-dimensional spin systems with competing interaction. They describe the model of random surfaces with linear-gonihedric action.The degeneracy of the vacuum state of this spin system is equal to $~~d \cdot 2^{N}~~$ for the lattice of the size $~N^{d}~$. We observe the second order phase transition of the three-dimensional system, at temperature $\beta_{c} \simeq 0.43932$ which almost coincides with $\beta_{c}$ of the 2D Ising model. This confirms the earlier analytical result for the case when self-interaction coupling constant $k$ is equal to zero. We suggest the full set of order parameters which characterize the structure of the vacuum states and of the phase transition. 
  We consider Clifford algebras with nonsymmetric bilinear forms, which are isomorphic to the standard symmetric ones, but not equal. Observing, that the content of physical theories is dependent on the injection $\oplus^n\bigwedge \V^{(n)}\rightarrow CL({\cal V},Q)$ one has to transform to the standard construction. The injection is of course dependent on the antisymmetric part of the bilinear form. This process results in the appropriate vertex normalordering terms, which are now obtained from the theory itself and not added ad hoc via a regularization argument. 
  We analyze the action of the spectral flows on N=2 twisted topological theories. We show that they provide a useful mapping between the two twisted topological theories associated to a given N=2 superconformal theory. This mapping can also be viewed as a topological algebra automorphism. In particular null vectors are mapped into null vectors, considerably simplifying their computation. We give the level 2 results. Finally we discuss the spectral flow mapping in the case of the DDK and KM realizations of the topological algebra. 
  Bianchi-type string cosmology involves generalizations of the FRW backgrounds with three transitive spacelike Killing symmetries, but without any a priori assumption of isotropy in the 3D sections of homogeneity. With emphasis on those cases with diagonal metrics and vanishing cosmological constant which which have not been previously examined in the literature, the present findings allow an overview and the classification of all Bianchi-type backgrounds. These string solutions (at least to lowest order in alpha prime) offer prototypes for the study of spatial anisotropy and its impact on the dynamics of the early universe. 
  A non trivial application of a modern computer language ("C") in a highly structured and object-oriented fashion is presented. The contest is that of Lie algebra representations (irreps), specifically the problem of reducing the products of irreps with the weight tree algorithm. The new WBase V2.0 version with table-generation and Young tableaux display capabilities is introduced. 
  We show, in a simple quantum mechanical model, how a theory can become supersymmetric in the presence of interactions even when the free theory is not. This dynamical generation of supersymmetry relaxes the condition on the equality of masses of the superpartners which would be of phenomenological interest. 
  We discuss an approach to compute two-particle scattering amplitudes for spinless particles colliding at Planckian centre-of-mass energies, with increasing momentum transfer away from the eikonal limit. For electrically neutral particles, the amplitude exhibits poles on the imaginary squared cm energy axis at locations that are distinct from those appearing in the eikonal limit. For charged particles, electromagnetic and gravitational effects remain decoupled for the eikonal situation as also the leading order (in momentum transfer, or equivalently, the impact parameter) correction, but mix non-trivially for higher orders. 
  The classical and the quantum massive string model based on a modified BDHP action is analyzed in the range of dimensions $1<d<25$. The discussion concerning classical theory includes a formulation of the geometrical variational principle, a phase-space description of the two-dimensional dynamics, and a detailed analysis of the target space geometry of classical solutions. The model is quantized using "old" covariant method. In particular an appropriate construction of DDF operators is given and the no-ghost theorem is proved. For a critical value of one of free parameters of the model the quantum theory acquires an extra symmetry not present on the classical level. In this case the quantum model is equivalent to the noncritical Polyakov string and to the old Fairlie-Chodos-Thorn massive string. 
  We show that the one-instanton sector moduli-space divergence of the O(3) Sigma Model leads to an unacceptable dependence of Green's functions on the arbitrary way that the field is split into a quantum fluctuation about a classical background. Since the divergence is associated with the degeneration of field configurations to those of the zero-instanton sector this arbitrariness may be cancelled by a `topological counter-term' which we construct. 
  We analyze the gaugino condensation in the effective theory for N=1 SU(N) Supersymmetric QCD with $N_{f}$ flavors. It is known that taking the vacuum expectation value of the matter field to be infinite, we can show that gaugino condensation can occur. At such a limit we should consider only pure supersymmetric Yang-Mills theory. But when we include an interaction term of order $O(\frac{1}{v})$, the situation can change. We analyze the effect of this interaction term and examine the gaugino condensation in the low energy Yang-Mills theory by using the scheme of Nambu-Jona-Lasinio. 
  The first quantum corrections to the entropy for an eternal 4-dimensional extremal Reissner-Nordstr\"om black hole is investigated at one-loop level, in the large mass limit of the black hole, making use of the conformal techniques related to the optical metric. A leading cubic horizon divergence is found and other divergences appear due to the singular nature of the optical manifold. The area law is shown to be violated. 
  We consider a N=2 supersymmetric Yang-Mills-Chern-Simons model, coupled to matter, in the Wess-Zumino gauge. The theory is characterized by a superalgebra which displays two kinds of obstructions to the closure on the translations: field dependent gauge transformations, which give rise to an infinite algebra, and equations of motion. The aim is to put the formalism in a closed form, off-shell, without introducing auxiliary fields. In order to perform that, we collect all the symmetries of the model into a unique nilpotent Slavnov-Taylor operator. Furthermore, we prove the renormalizability of the model through the analysis of the cohomology arising from the generalized Slavnov-Taylor operator. In particular, we show that the model is free of anomaly. 
  We convert the self-dual model of Townsend, Pilch, and Nieuwenhuizen to a first-class system using the generalized canonical formalism of Batalin, Fradkin, and Tyutin and show that gauge-invariant fields in the embedded model can be identified with observables in the Maxwell-Chern-Simons theory as well as with the fundamental fields of the self-dual model. We construct the phase-space partition function of the embedded model and demonstrate how a basic set of gauge-variant fields can play the role of either the vector potentials in the Maxwell-Chern-Simons theory or the fundamental fields of the self-dual model by appropriate choices of gauge. 
  We study the connection between the Green functions of the Maxwell-Chern-Simons theory and a self-dual model by starting from the phase-space path integral representation of the Deser-Jackiw master Lagrangian. Their equivalence is established modulo time-ordering ambiguities. 
  The Higgs mechanism is reconsidered in the canonical Weyl gauge formulation of quantized gauge theories, using an approach in which redundant degrees of freedom are eliminated. As a consequence, its symmetry aspects appear in a different light. All the established physics consequences of the Higgs mechanism are recovered without invoking gauge symmetry breaking. The occurence of massless vector bosons in non-abelian Higgs models is interpreted as signal of spontaneous breakdown of certain global symmetries. Characteristic differences between the relevant ``displacement symmetries'' of QED and the Georgi Glashow model are exhibited. Implications for the symmetry aspects of the electroweak sector of the standard model and the interpretation of the physical photon as Goldstone boson are pointed out. 
  A general fusion method to find solutions to the reflection equation in higher spin representations starting from the fundamental one is shown. The method is illustrated by applying it to obtaining the $K$ diagonal boundary matrices in an alternating spin $1/2$ and spin $1$ chain. The hamiltonian is also given. The applicability of the method to higher rank algebras is shown by obtaining the $K$ diagonal matrices for a spin chain in the $\left\{ 3^* \right\}$ representation of $su(3)$ from the $\left\{ 3\right\}$ representation. 
  Starting from the action of $(4,4)$ $2D$ twisted multiplets in the harmonic superspace with a double set of $SU(2)$ harmonic variables, we present its generalization which provides an off-shell description of a wide class of $(4,4)$ sigma models with torsion and non-commuting left and right complex structures. The distinguishing features of the action constructed are: (i) a nonabelian and in general nonlinear gauge invariance ensuring a correct number of physical degrees of freedom; (ii) an infinite tower of auxiliary fields. For a particular class of such models we explicitly demonstrate the non-commutativity of complex structures on the bosonic target. 
  We prove that particle models whose action is given by the integrated $n$-th curvature function over the world line possess $n+1$ gauge invariances. A geometrical characterization of these symmetries is obtained via Frenet equations by rephrasing the $n$-th curvature model in $\reals^d$ in terms of a standard relativistic particle in $S^{d-n}$. We ``prove by example'' that the algebra of these infinitesimal gauge invariances is nothing but $\W_{n+2}$, thus providing a geometrical picture of the $\W$-symmetry for these models. As a spin-off of our approach we give a new global invariant for four-dimensional curves subject to a curvature constraint. 
  It is observed that, at short range, the field equations of general relativity admit a line element that takes the form of Yukawa potential. The result leads to the possibility that strong interaction may also be described by field equations that have the same form as that of general relativity. It is then shown how such field equations may arise from the coupling of two strong fields. 
  We review the closed time path formalism of Schwinger using a path integral approach. We apply this formalism to the study of pair production from strong external fields as well as the time evolution of a nonequilibrium chiral phase transition. 
  In this note we review the spinon basis for the integrable highest weight modules of sl2^ at levels k\geq1, and give the corresponding character formula. We show that our spinon basis is intimately related to the basis proposed by Foda et al. in the principal gradation of the algebra. This gives rise to new identities for the q-dimensions of the integrable modules. 
  We evaluate the quantum corrections of the Einstein-Hilbert action with boundaries in the $2+\epsilon$ dimensional expansion approach. We find the Einstein-Hilbert action with boundaries to be renormalizable to the one loop order. We compute the geometric entropy beyond the semiclassical approximation. It is found that the exact geometric entropy is related to the string succeptibility by the analytic continuation in the central charge. Our results also show that we can renormalize the divergent quantum corrections for the Bekenstein-Hawking entropy of blackholes by the gravitational coupling constant renormalization beyond two dimensions. 
  We examine the geometrical and topological properties of surfaces surrounding clusters in the 3--$d$ Ising model. For geometrical clusters at the percolation temperature and Fortuin--Kasteleyn clusters at $T_c$, the number of surfaces of genus $g$ and area $A$ behaves as $A^{x(g)}e^{-\mu(g)A}$, with $x$ approximately linear in $g$ and $\mu$ constant. These scaling laws are the same as those we obtain for simulations of 3--$d$ bond percolation. We observe that cross--sections of spin domain boundaries at $T_c$ decompose into a distribution $N(l)$ of loops of length $l$ that scales as $l^{-\tau}$ with $\tau \sim 2.2$. We also present some new numerical results for 2--$d$ self-avoiding loops that we compare with analytic predictions. We address the prospects for a string--theoretic description of cluster boundaries. 
  The structure of on-shell and off-shell 2D, (4,4) supersymmetric scalar multiplets is investigated, in components and in superspace. We reach the surprising result that there exist eight {\underline {distinct}} on-shell versions and an even greater variety of off-shell ones. The off-shell generalised tensor and relaxed N = 4 multiplets are introduced in superspace, and their universal invariant self-interaction is constructed. 
  Using the new variational approach proposed recently for a systematic improvement of the locally harmonic Feynman-Kleinert approximation to path integrals we calculate the partition function of the anharmonic oscillator for all temperatures and coupling strength with high accuracy. 
  Light-cone quantization procedure recently presented is applied to the two-dimensional light-cone theories. By introducing the two distinct null planes it is shown that the modification term in the two-dimensional massless light-cone propagators suggested about twenty years ago vanishs. 
  We present the exact effective superpotentials in $4d$, $N=1$ supersymmetric $SU(2)$ gauge theories with $N_3$ triplets and $N_2$ doublets of matter superfields. For the theories with a single triplet matter superfield we present the exact gauge couplings for arbitrary bare masses and Yukawa couplings. 
  We derive the T-duality transformations that transform a general d=10 solution of the type-IIA string with one isometry to a solution of the type-IIB string with one isometry and vice versa. In contrast to other superstring theories, the T-duality transformations are not related to a non-compact symmetry of a d=9 supergravity theory. We also discuss S-duality in d=9 and d=10 and the relationship with eleven-dimensional supergravity theory. We apply these dualities to generate new solutions of the type IIA and type IIB superstrings and of eleven-dimensional supergravity. 
  Structure constants of Operator Algebras for the SL(2) degenerate conformal field theories are calculated. 
  Sufficiently large seeds for generating the observed (inter)galactic magnetic fields emerge naturally in string cosmology from the amplification of electromagnetic vacuum fluctuations due to a dynamical dilaton background. The success of the mechanism depends crucially on two features of the so-called pre-big-bang scenario, an early epoch of dilaton-driven inflation at very small coupling, and a sufficiently long intermediate stringy era preceding the standard radiation-dominated evolution. 
  We show that the well known $N=1$ NLS equation possesses $N=2$ supersymmetry and thus it is actually the $N=2$ NLS equation. This supersymmetry is hidden in terms of the commonly used $N=1$ superfields but it becomes manifest after passing to the $N=2$ ones. In terms of the new defined variables the second Hamiltonian structure of the supersymmetric NLS equation coincides with the $N=2$ superconformal algebra and the $N=2$ NLS equation belongs to the $N=2$ $a=4$ KdV hierarchy. We propose the KP-like Lax operator in terms of the $N=2$ superfields which reproduces all the conserved currents for the corresponding hierarchy. 
  We determine the eigenvalues of the transfer matrices for integrable open quantum spin chains which are associated with the affine Lie algebras $A^{(2)}_{2n-1}, B^{(1)}_n, C^{(1)}_n, D^{(1)}_n$, and which have the quantum-algebra invariance U_q(C_n), U_q(B_n), U_q(C_n), U_q(D_n)$, respectively. 
  We show that the formation of condensates in the presence of a constant magnetic field in 2+1 dimensions is extremely unstable. It disappears as soon as a heat bath is introduced with or without a chemical potential. We point out some new nonanalytic behavior that develops in this system at finite temperature. 
  We show a surprising connection between known integrable Hamiltonian systems with quartic potential and the stationary flows of some coupled KdV systems related to fourth order Lax operators. In particular, we present a connection between the Hirota-Satsuma coupled KdV system and (a generalisation of) the $1:6:1$ integrable case quartic potential. A generalisation of the $1:6:8$ case is similarly related to a different (but gauge related) fourth order Lax operator. We exploit this connection to derive a Lax representation for each of these integrable systems. In this context a canonical transformation is derived through a gauge transformation. 
  We present a new quantization scheme for $2D$ gravity coupled to an $SU(2)$ principal chiral field and a dilaton; this model represents a slightly simplified version of stationary axisymmetric quantum gravity. The analysis makes use of the separation of variables found in our previous work [1] and is based on a two-time hamiltonian approach. The quantum constraints are shown to reduce to a pair of compatible first order equations, with the dilaton playing the role of a ``clock field''. Exact solutions of the Wheeler-DeWitt equation are constructed via the integral formula for solutions of the Knizhnik-Zamolodchiokov equations. 
  The double-scaling limit of the supereigenvalue model is performed in the moment description. This description proves extremely useful for the identification of the multi-critical points in the space of bosonic and fermionic coupling constants. An iterative procedure for the calculation of higher-genus contributions to the free energy and to the multi-loop correlators in the double-scaling limit is developed. We present the general structure of these quantities at genus g and give explicit results up to and including genus two. 
  Low-energy effective field theories arising from Calabi-Yau string compactifications are generically inconsistent or ill-defined at the classical level because of conifold singularities in the moduli space. It is shown, given a plausible assumption on the degeneracies of black hole states, that for type II theories this inconsistency can be cured by nonperturbative quantum effects: the singularities are resolved by the appearance of massless Ramond-Ramond black holes. The Wilsonian effective action including these light black holes is smooth near the conifold, and the singularity is reproduced when they are integrated out. In order for a quantum effect to cure a classical inconsistency, it can not be suppressed by the usual string coupling $g_s$. It is shown how the required $g_s$ dependence arises as a result of the peculiar couplings of Ramond-Ramond gauge fields to the dilaton. 
  We discuss the origin of the leg factors appearing in 2D string theory. Computing in the world sheet framework we use the semiclassical method to study string amplitudes at high energy. We show that in the case of a simplest 2-point amplitude these factors correspond entirely to the time delay for reflection off the Liouville wall. Our semiclassical calculation reveals that the string longitudinal modes, although nonpropagating in 2D spacetime, have the effect of doubling the phase shift. Particular emphasis is put on comparison with the point particle (center of mass) case. A general method is then given for calculating an arbitrary amplitude semiclassically. 
  We study a scalar field theory coupled to gravity on a flat background, below Planck's energy. Einstein's theory is treated as an effective field theory. Within the context of Wilson's renormalization group, we compute gravitational corrections to the beta functions and the anomalous dimension of the scalar field, taking into account threshold effects. 
  We study a renormalizable, general theory of dilatonic gravity (with a kinetic-like term for the dilaton) interacting with scalar matter near two dimensions. The one-loop effective action and the beta functions for this general theory are written down. It is proven that the theory possesses a non-trivial ultraviolet fixed point which yields an asymptotically free gravitational coupling constant (at $\epsilon \rightarrow 0$) in this regime. Moreover, at the fixed point the theory can be cast under the form of a string-inspired model with free scalar matter. The renormalization of the Jackiw-Teitelboim model and of lineal gravity in $2+\epsilon$ dimensions is also discussed. We show that these two theories are distinguished at the quantum level. Finally, fermion-dilatonic gravity near two dimensions is considered. 
  Through the continuum limit of the one matrix model on the multicritical point the corresponding Schwinger-Dyson equation of temporal-gauge string field theory is derived. It agrees with that of the background independent formulation recently proposed. 
  The conjectured equivalence of the heterotic string to a $K_3$ compactified type IIA superstring is combined with the conjectured equivalence of the latter to a compactified 11-dimensional supermembrane to derive a string membrane duality in seven dimensions; the membrane is a soliton of the string theory and vice versa. A prediction of this duality is that the heterotic string is a $K_3$ compactification of the solitonic 11-dimensional fivebrane. It is verified that the worldsheet action of the D=10 heterotic string is indeed obtainable by $K_3$ compactification of the worldvolume action of the 11-dimensional fivebrane, and it is suggested how the worldvolume action of the D=11 supermebrane may be similarly obtained by $T^3$ compactification of the worldvolume action of a D=10 heterotic fivebrane. Generalizations to $D=8$ string-threebrane and membrane-membrane duality are also discussed. 
  We investigate the question of how the knowledge of sufficiently many local conservation laws for a model can be utilized to solve the model. We show that for models where the conservation laws can be written in one-sided forms, like $\barpartial Q_s = 0$, the problem can always be reduced to solving a closed system of ordinary differential equations. We investigate the $A_1$, $A_2$, and $B_2$ Toda field theories in considerable detail from this viewpoint. One of our findings is that there is in each case a transformation group intrinsic to the model. This group is built on a specific real form of the Lie algebra used to label the Toda field theory. It is the group of field transformations which leaves the conserved densities invariant. 
  By carefully analyzing the relations between operator methods and the discretized and continuum path integral formulations of quantum-mechanical systems, we have found the correct Feynman rules for one-dimensional path integrals in curved spacetime. Although the prescription how to deal with the products of distributions that appear in the computation of Feynman diagrams in configuration space is surprising, this prescription follows unambiguously from the discretized path integral. We check our results by an explicit two-loop calculation. 
  Progress on string theory in curved spacetimes since 1992 are reviewed. After a short introduction on strings in Minkowski and curved spacetimes, we focus on strings in cosmological spacetimes. The classical behaviour of strings in FRW and inflationary spacetimes is now understood in a large extent from various types of explicit string solutions. Three different types of behaviour appear in cosmological spacetimes: {\bf unstable, dual} to unstable and {\bf stable}. For the unstable strings, the energy and size grow for large scale factors $R \to \infty$, proportional to $R$. For the dual to unstable strings, the energy and size blow up for $R\to 0$ as $1/R$. For stable strings, the energy and proper size are bounded. (In Minkowski spacetime, all string solutions are of the stable type).Recent progress on self-consistent solutions to the Einstein equations for string dominated universes is reviewed. The energy-momentum tensor for a gas of strings is then considered as source of the spacetime geometry and from the above string behaviours the string equation of state is determined. The self-consistent string solution exhibits the realistic matter dominated behaviour $R \sim (X^0)^{2/3}\; $ for large times and the radiation dominated behaviour $R \sim (X^0)^{1/2}\;$ for early times. Finally, we report on the exact integrability of the string equations plus the constraints in de Sitter spacetime that allows to systematically find {\bf exact} string solutions by soliton methods and the multistring solutions. {\bf Multistring solutions} are a new feature in curved spacetimes. That is, a single world-sheet simultaneously describes many different and independent strings. This phenomenon has no analogue in flat spacetime and follows to the coupling of the strings with the geometry. 
  It was previously shown that most of the superstrings can be obtained from those with higher world-sheet supersymmetry as spontaneously broken phases. In this paper, we show that the small $N=4$ superstring, which was left out of this hierarchy of the universal string, can be obtained from the large $N=4$ strings. We also show that the $N=2$ string is a special vacuum of small $N=4$ string. Thus all the known superstring theories can be derived from a universal string by spontaneous breakdown of supersymmetry. 
  In the spirit of previous papers, but using more general field configurations, the non-linear O(3) model in (2+1)-D, modified by the addition of both a potential-like term and a Skyrme-like term, is considered. The instanton solutions are numerically evolved in time and some of their stability properties studied. They are found to be stable, and a repulsive force is seen to exist among them. These results, which are restricted to the case of zero speed systems, confirm those obtained in previous investigations, in which a similar problem was studied for a different choice of the potential-like term. 
  Based on the idea of quantum groups and paragrassmann variables, we presenta generalization of supersymmetric classical mechanics with a deformation parameter $q= \exp{\frac{2 \pi i}{k}}$ dealing with the $k =3$ case. The coordinates of the $q$-superspace are a commuting parameter $t$ and a paragrassmann variable $\theta$, where $% \theta^3 = 0$. The generator and covariant derivative are obtained, as well as the action for some possible superfields. 
  We write down the weak-coupling limit of N=2 supersymmetric Yang-Mills theory with arbitrary gauge group \( G \). We find the weak-coupling monodromies represented in terms of \( Sp(2r,\bzeta ) \) matrices depending on paths closed up to Weyl transformations in the Cartan space of complex dimension r, the rank of the group. There is a one to one relation between Weyl orbits of these paths and elements of a generalized braid group defined from \( G \). We check that these weak-coupling monodromies behave correctly in limits of the moduli space corresponding to restrictions to subgroups. In the case of $SO(2r+1)$ we write down the complex curve representing the solution of the theory. We show that the curve has the correct monodromies. 
  Previous path integral treatments of Yang-Mills on a Riemann surface automatically sum over principal fiber bundles of all possible topological types in computing quantum expectations. This paper extends the path integral formulation to treat separately each topological sector. The formulation is sufficiently explicit to calculate Wilson line expectations exactly. Further, it suggests two new measures on the moduli space of flat connections, one of which proves to agree with the small-volume limit of the Yang-Mills measure. \copyright {\em 1996 American Institute of Physics.} 
  We consider the field theory of $N$ massless bosons which are free except for an interaction localized on the boundary of their 1+1 dimensional world. The boundary action is the sum of two pieces: a periodic potential and a coupling to a uniform abelian gauge field. Such models arise in open string theory and dissipative quantum mechanics, and possibly in edge state tunneling in the fractional quantized Hall effect. We explicitly show that conformal invariance is unbroken for certain special choices of the gauge field and the periodic potential. These special cases are naturally indexed by semi-simple, simply laced Lie algebras. For each such algebra, we have a discrete series of conformally invariant theories where the potential and gauge field are conveniently given in terms of the weight lattice of the algebra. We compute the exact boundary state for these theories, which explicitly shows the group structure. The partition function and correlation functions are easily computed using the boundary state result. 
  We argue that, at finite temperature, parity invariant non-compact electrodynamics with massive electrons in 2+1 dimensions can exist in both confined and deconfined phases. We show that an order parameter for the confinement-deconfinement phase transition is the Polyakov loop operator whose average measures the free energy of a test charge that is not an integral multiple of the electron charge. The effective field theory for the Polyakov loop operator is a 2-dimensional Euclidean scalar field theory with a global discrete symmetry $Z$, the additive group of the integers. We argue that the realization of this symmetry governs confinement and that the confinement-deconfinement phase transition is of Berezinskii-Kosterlitz-Thouless type. We compute the effective action to one-loop order and argue that when the electron mass $m$ is much greater than the temperature $T$ and dimensional coupling $e^2$, the effective field theory is the Sine-Gordon model. In this limit, we estimate the critical temperature, $T_{\rm crit.}=e^2/8\pi(1-e^2/12\pi m+\ldots)$. 
  We find a conserved monodromy matrix differential operator T in the quantum Self-Dual Yang-Mills (SDYM) system and show that it satisfies the exchange algebra RTT=TTR. From its two infinitesimal forms, we obtain the infinite conserved quantum nonlocal-charge algebras and the infinite conserved Yangian algebras. It is remarkable that such conserved algebras exist in a four-dimensional nontrivial quantum field theory with interactions. 
  The restricted SOS model of Andrews, Baxter and Forrester has been studied. The finite size corrections to the eigenvalue spectra of the transfer matrix of the model with a more general crossing parameter have been calculated. Therefore the conformal weights and the central charges of the non-unitary or unitary minimal conformal field have been extracted from the finite size corrections. 
  A D-instanton is a space-time event associated with world-sheet boundaries that contributes non-perturbative effects of order $e^{-const/\kappa}$ to closed-string amplitudes. Some properties of a gas of D-instantons are discussed in this paper. 
  Two-dimensional dilaton gravity coupled to a Klein-Gordon matter field with a quartic interaction term is considered. The theory has a classical solution which exhibits black hole formation by a soliton. The geometry of black hole induced by a soliton is investigated. 
  We describe the generators of kappa-conformal transformations, leaving invariant the kappa-deformed d'Alembert equation. In such a way one obtains the conformal extension of the off-shell spin zero realization of kappa-deformed Poincare algebra. Finally the algebraic structure of kappa-deformed conformal algebra is discussed. 
  It is shown that the generators of two discrete Heisenberg-Weyl groups with irrational rotation numbers $\theta$ and $-1/ \theta$ generate the whole algebra $\cal B$ of bounded operators on $L_2(\bf R)$. The natural action of the modular group in $\cal B$ is implied. Applications to dynamical algebras appearing in lattice regularization and some duality principles are discussed. 
  We study deformations of closed string theory by primary fields of conformal weight $(1,1)$, using conformal techniques on the complex plane. A canonical surface integral formalism for computing commutators in a non-holomorphic theory is constructed, and explicit formul\ae for deformations of operators are given. We identify the unique regularization of the arising divergences that respects conformal invariance, and consider the corresponding parallel transport. The associated connection is metric compatible and carries no curvature. 
  We present more examples of dual N=1 SUSY gauge theories. This set of theories is connected by flows to both Seiberg's and Kutasov's dual theories. This provides a unifying picture of the various dual theories. We investigate the dual theories, their flat directions and mass perturbations. 
  Using mirror symmetry in Calabi-Yau manifolds M, three point functions of A(M)-model operators on the genus $0$ Riemann surface in cases of one-parameter families of $d$-folds realized as Fermat type hypersurfaces embedded in weighted projective spaces and a two-parameter family of $d$-fold embedded in a weighted projective space ${\amb}$ are studied. These three point functions ${\corr{\,{{\cal O}^{(1)}_{a}}\, {{\cal O}^{(l-1)}_{b}}\, {{\cal O}^{(d-l)}_{c}}\, }}$ are expanded by indeterminates ${q_l}$=${e^{2\pi i {t_l}}}$ associated with a set of {\kae} coordinates $\{{t_l}\}$ and their expansion coefficients count the number of maps. From these analyses, we can read fusion structure of Calabi-Yau A(M)-model operators. In our cases they constitute a subring of a total quantum cohomology ring of the A(M)-model operators. In fact we switch off all perturbation operators on the topological theories except for marginal ones associated with {\kae} forms of M. For that reason, the charge conservation of operators turns out to be a classical one. 
  For a one-parameter family of Calabi-Yau d-fold M embedded in ${{CP}^{d+1}}$, we consider a new quasi-topological field theory ${A^{\ast}}$(M)-model compared with the $A$(M)-model. The two point correlators on the sigma model moduli space (the hermitian metrics) are analyzed by the $A{A^{\ast}}$-fusion on the world sheet sphere. A set of equations of these correlators turns out to be a non-affine A-type Toda equation system for the d-fold M. This non-affine property originates in the vanishing first Chern class of M. Using the results of the $A{A^{\ast}}$-equation, we obtain a genus one partition function of the sigma model associated to the M in the recipe of the holomorphic anomaly. By taking an asymmetrical limit of the complexified {\kae} parameters ${\bar{t}\rightarrow \infty}$ and $t$ is fixed, the ${A^{\ast}}$(M)-model part is decoupled and we can obtain a partition function (or one point function of the operator ${{\cal O}^{(1)}}$ associated to a {\kae} form of M) of the $A$(M)-matter coupled with the topological gravity at the stringy one loop level. The coefficients of the series expansion with respect to an indeterminate $q:={e^{2\pi i t}}$ are integrals of the top Chern class of the vector bundle {\Large $\nu $} over the moduli space of stable maps with definite degrees. 
  In this paper we investigate the form of induced gauge fields that arises in two types of quantum systems. In the first we consider quantum mechanics on coset spaces G/H, and argue that G-invariance is central to the emergence of the H-connection as induced gauge fields in the different quantum sectors. We then demonstrate why the same connection, now giving rise to the non-abelian generalization of Berry's phase, can also be found in systems which have slow variables taking values in such a coset space. 
  We consider pure $SU(2)$ Yang-Mills theory when the space is compactified to a 3-dimensional sphere with finite radius. The Euclidean classical self-dual solutions of the equations of motion (the instantons) and the static finite energy solutions (the sphalerons) which have been found earlier are rewritten in handy physical variables with the gauge condition $A_0 = 0$. Stationary solutions to the equations of motion in the Minkowski space-time (the standing waves) are discussed. We briefly discuss also the theory defined in a flat finite spherical box with rigid boundary conditions and present the numerical solution describing the sphaleron. 
  Starting from the hypothesis that both physics, in particular space-time and the physical vacuum, and the corresponding mathematics are discrete on the Planck scale we develop a certain framework in form of a '{\it cellular network}' consisting of cells interacting with each other via bonds. Both the internal states of the cells and the "strength" of the bonds are assumed to be dynamical variables. In section 3 the basis is laid for a version of '{\it discrete analysis}' which, starting from different, perhaps more physically oriented principles, manages to make contact with the much more abstract machinery of Connes et al. and may complement the latter approach. In section 4 a, as far as we can see, new concept of '{\it topological dimension}' in form of a '{\it degree of connectivity}' for graphs, networks and the like is developed. It is then indicated how this '{\it dimension}', which for continuous structures or lattices being embedded in a continuous background agrees with the usual notion of dimension, may change dynamically as a result of a '{\it phase transition like}' change in '{\it connectivity}' in the network. A certain speculative argument, along the lines of statistical mechanics, is supplied in favor of the naturalness of dimension 4 of ordinary (classical) space-time. 
  This work is a continuation of paper (hep-th/9407146) where the Boltzmann weights for the N-state integrable spin model on the cubic lattice has been obtained only numerically. In this paper we present the analytical formulae for this model in a particular case. Here the Boltzmann weights depend on six free parameters including the elliptic modulus. The obtained solution allows to construct a two-parametric family of the commuting two-layer transfer matrices. Presented model is expected to be simpler for a further investigation in comparison with a more general model mentioned above. 
  We comment on some aspects of the semiclassical BPS saturated states close to the curve $Im a_{D}/a =0$ 
  In this paper we construct a $(2,2)$ dimensional string theory with manifest $N=1$ spacetime supersymmetry. We use Berkovits' approach of augmenting the spacetime supercoordinates by the conjugate momenta for the fermionic variables. The worldsheet symmetry algebra is a twisted and truncated ``small'' $N=4$ superconformal algebra. The realisation of the symmetry algebra is reducible with an infinite order of reducibility. We study the physical states of the theory by two different methods. In one of them, we identify a subset of irreducible constraints, which is by itself critical. We construct the BRST operator for the irreducible constraints, and study the cohomology and interactions. This method breaks the $SO(2,2)$ spacetime symmetry of the original reducible theory. In another approach, we study the theory in a fully covariant manner, which involves the introduction of infinitely many ghosts for ghosts. 
  There are at present two known string theories in $(2,2)$ dimensions. One of them is the well known $N=2$ string, and the other one is a more recently constructed $N=1$ spacetime supersymmetric string. They are both based on certain twistings and/or truncations of the small $N=4$ superconformal algebra, realised in terms of $(2,2)$ superspace variables. In this paper, we investigate more general possibilities for string theories based on algebras built with the same set of fields. We find that there exists one more string theory, based on an algebra which is not contained within the $N=4$ superconformal algebra. We investigate the spectrum and interactions of this theory. 
  It has been argued by Grigoriev and Rubakov that one can simulate real time processes involving baryon number non-conservation at high temperature using real time evolution of classical equations, and summing over initial conditions with a classical thermal weight. It is known that such a naive algorithm is plagued by ultraviolet divergences. In quantum theory the divergences are regularized, but the corresponding graphs involve the contributions from the hard momentum region and also the new scale $\sim gT$ comes into play. We propose a modified algorithm which involves solving the classical equations of motion for the effective hard thermal loop Hamiltonian with an ultraviolet cutoff $\mu \gg gT$ and integrating over initial conditions with a proper thermal weight. Such an algorithm should provide a determination of the infrared behavior of real time correlation function $<Q(t) Q(0)>_T$ determining the baryon violation rate. Hopefully, the results obtained in this modified algorithm would be cutoff-independent. 
  Witten's linear sigma model for ADHM instantons possesses a natural $(0,4)$ supersymmetry. We study generalizations of the infrared limit of the model that are invariant under $(4,4)$ supersymmetry. In the case of four space-time dimensions a background with a conformally flat metric and torsion is required. The geometry is specified by a single real scalar function satisfying Laplace's equation. It gives rise to 't Hooft instantons for the gauge group $SU(2)$, instead of the general ADHM instantons for an $SO(n)$ gauge group in the case $(0,4)$. 
  Path integral for the $SU(2)$ spin system is reconsidered. We show that the Nielsen-Rohrlich(NR) formula is equivalent to the spin coherent state expression so that the phase space in the NR formalism is not topologically nontrivial. We also perform the WKB approximation in the NR formula and find that it gives the exact result. 
  We study the renormalization of the Ricci curvature as an example of generally covariant operators in quantum gravity near two dimensions. We find that it scales with a definite scaling dimension at short distance. The Ricci curvature singularity at the big bang can be viewed as such a scaling phenomenon. The problem of the spacetime singularity may be resolved by the scale invariance of the spacetime at short distance. 
  We show how to deal with screening charges involving fractional powers of free fields. This enables us to use the free field Wakimoto construction to obtain complete expressions for integral representations of conformal blocks for N-point functions on the sphere, also in the case of non-integrable representations, in particular for admissible representations. We verify several formal properties including the Knizhnik-Zamolodchikov equations. We discuss the fusion rules which result from our treatment, and compare with the literature. 
  Starting from an extension of the Poisson bracket structure and Kubo-Martin-Schwinger-property of classical statistical mechanics of continuous systems to spin systems, defined on a lattice, we derive a series of, as we think, new and interesting bounds on correlation functions for general lattice systems. Our method is expected to yield also useful results in Euclidean Field Theory. Furthermore the approach is applicable in situations where other techniques fail, e.g. in the study of phase transitions without breaking of a {\bf continuous} symmetry like $P(\phi)$-theories with $\phi (x)$ scalar. 
  We derive the moduli dependence of the one--loop gauge couplings for non--vanishing gauge background fields in a four--dimensional heterotic (0,2) string compactification. Remarkably, these functions turn out to have a representation as modular functions on an auxiliary Riemann surface on appropriate truncations of the full moduli space. In particular, a certain kind of one--loop functions is given by the free energy of two--dimensional solitons on this surface. 
  We show in three dimensions, using functional integral techniques, the equivalence between the partition functions of the massive Thirring model and a gauge theory with two gauge fields, to all orders in the inverse fermion mass. Detailed bosonisation identities, also valid to all orders in the inverse mass, are derived. Specialisation to the lowest (and next to lowest) orders reveals that the gauge theory simplifies to the Maxwell-Chern-Simons theory. Some interesting consequences are discussed in this case. 
  In the present paper we obtain some integrable generalisations of the continuous Toda system generated by a flat connection form taking values in higher grading subspaces of the algebra of the area--preserving diffeomorphism of the torus $T^2$, and construct their general solutions. The grading condition which we use here, imposed on the connection, can be realised in terms of some holomorphic distributions on the corresponding homogeneous spaces. 
  In this paper we use the Batalin-Fradkin-Vilkovisky formalism to study a recently proposed nonlocal symmetry of QED. In the BFV extended phase space we show that this symmetry stems from a canonical transformation in the ghost sector. 
  This is the first of three papers on the short-distance properties of the energy-momentum tensor in field theory. We study the energy-momentum tensor for renormalized field theory in curved space. We postulate an exact Ward identity of the energy-momentum tensor. By studying the consistency of the Ward identity with the renormalization group and diffeomorphisms, we determine the short-distance singularities in the product of the energy-momentum tensor and an arbitrary composite field in terms of a connection for the space of composite fields over theory space. We discuss examples from the four-dimensional $\phi^4$ theory. In the forthcoming two papers we plan to discuss the torsion and curvature of the connection. 
  The problems connected with a choice of the spinorial basis in the $(j,0)\oplus (0,j)$ representation space are discussed. It is shown to have profound significance in relativistic quantum theory. From the methodological viewpoint this fact is related with the important dynamical role played by space-time symmetries for all kind of interactions. 
  We study the topologically twisted osp(2|2)+osp(2|2) conformal superalgebra. The algebra includes the Lagrangians which are intrinsic to the topological field theory and composed of fermionic generators. Studying the Lagrangians through a gauge system of osp(2|2)+osp(2|2), geometrical features inherent to the algebra are revealed: a moduli space associated with the algebra is derived and the crucial roles which the fermionic generators play in the moduli space are clarified It is argued that there exists a specific relation between the topological twist and the moduli problem through a geometrical aspect of the algebra. 
  The quantum deformation of the oscillator algebra and its implications on the phase operator are studied from a view point of an index theorem by using an explicit matrix representation. For a positive deformation parameter $q$ or $q=exp(2\pi i\theta)$ with an irrational $\theta$, one obtains an index condition $\dml a - \dml a^{\dagger} = 1$ which allows only a non-hermitian phase operator with $\dml \expon^{i \varphi} - \dml (\expon^{i\varphi})^{\dagger} = 1$. For $q=exp(2\pi i\theta)$ with a rational $\theta$ , one formally obtains the singular situation $\dml a =\infty$ and $ \dml a^{\dagger} = \infty$, which allows a hermitian phase operator with $\dml \expon^{i \Phi} - \dml (\expon^{i\Phi})^{\dagger} = 0$ as well as the non-hermitian one with $\dml \expon^{i \varphi} - \dml (\expon^{i\varphi})^{\dagger} = 1$. Implications of this interpretation of the quantum deformation are discussed. We also show how to overcome the problem of negative norm for $q=exp(2\pi i\theta)$. 
  Odd time was introduced to formulate the Batalin-Vilkovisky method of quantization of gauge theories in a systematic manner. This approach is presented emphasizing the odd time canonical formalism beginning from an odd time Lagrangian. To let the beginners have access to the method essential notions of the gauge theories are briefly discussed, and each step is illustrated with examples. Moreover, the method of solving the master equation in an easy way for a class of gauge theories is reviewed. When this method is applicable some properties of the solutions can easily be extracted as shown in the related examples. 
  A manifestly $N=2$ supersymmetric coset formalism is introduced to describe integrable hierarchies. It is applied to analyze the super-NLS equation. It possesses an $N=2$ symmetry since it can be obtained from a manifest $N=2$ coset algebra construction. A remarkable result is here discussed: the existence of a B\"{a}cklund transformation which connects the super-NLS equation to the equations belonging to the integrable hierarchy of one particular (the $a=4$) $N=2$ super-KdV equation. $N=2$ scalar Lax pair operators are introduced for both super-KdV and super-NLS. 
  The Kerr solution to axidilaton gravity is analyzed in the Debney--Kerr--Schild formalism. It is shown that the Kerr principal null congruence retains its property to be geodesic and shear free, however, the axidilatonic Kerr solution is not algebraically special. A limiting form of this solution is considered near the ring-like Kerr singularity. This limiting solution coincides with the field around a fundamental heterotic string obtained by Sen. 
  We present a geometric formulation of super $p$--brane theories in which the Wess--Zumino term is $(p+1)$--th order in the supersymmetric currents, and hence is manifestly supersymmetric. The currents are constructed using a supergroup manifold corresponding to a generalization of a superalgebra which we found sometime ago. Our results generalize Siegel's analogous reformulation of the Green--Schwarz superstring. The new superalgebras we construct underly the free differential superalgebras introduced by de Azc\'arraga and Townsend a few years ago. 
  We argue that supersymmetric gluodynamics has two phases with equivalent infrared behavior, one of which is asymptotically free and another one is superstrongly coupled in the ultraviolet domain. 
  We present interesting relationship between what are called stationary axisymmetric black hole solutions for the vacuum Einstein equations in the ordinary four-dimensions and exact solutions for self-dual Yang-Mills fields in flat $2+2$ dimensions which are nothing but the consistent backgrounds for $~N=2$~ open superstring. We show that any stationary axisymmetric black hole solution for the former automatically provides an exact solution for the latter. We also give a nice relation between the physical parameters of black holes and an invariant integral analogous to instanton charges for such a self-dual Yang-Mills solution. This result indicates that any general black hole solution in the usual four-dimensions can be the background of $~N=2$~ superstring at the same time. We also give an interesting embedding of stationary axisymmetric solutions into the background of $~N=2$~ closed superstring. Finally we show some indication that the Kerr solution with naked singularity (for $a > m$) is nothing else than the gravitational field generated by a closed string. 
  We analyze gaugino condensation in the presence of a dilaton and an antisymmetric tensor field, with couplings reminiscent of string theories. The degrees of freedom relevant to a supersymmetric description of the effective theory below the scale of condensation are discussed in this context. 
  In this paper a topological theory of gravity is studied on a four-manifold using the formalism of Capovilla {\sl et al}. We show that it is fact equivalent to Anselmi and Fre's topological gravity using the topological symmetries. Using this formalism gives us a new way to study topological gravity and the intersection theory of gravitational instantons if the (3+1) decomposition with respect to local coordinates is performed. 
  It is argued that black hole condensation can occur at conifold singularities in the moduli space of type II Calabi--Yau string vacua. The condensate signals a smooth transition to a new Calabi--Yau space with different Euler characteristic and Hodge numbers. In this manner string theory unifies the moduli spaces of many or possibly all Calabi--Yau vacua. Elementary string states and black holes are smoothly interchanged under the transitions, and therefore cannot be invariantly distinguished. Furthermore, the transitions establish the existence of mirror symmetry for many or possibly all Calabi--Yau manifolds. 
  A topological gauge invariant lagrangian for Seiberg-Witten monopole equations is constructed. The action is invariant under a huge class of gauge transformations which after BRST fixing leads to the BRST invariant action associated to Seiberg-Witten monopole topological theory. The supersymmetric transformation of the fields involved in the construction is obtained from the nilpotent BRST algebra. 
  Some of the extremal black hole solutions in string theory have the same quantum numbers as the Bogomol'nyi saturated elementary string states. We explore the possibility that these black holes can be identified to elementary string excitations. It is shown that stringy effects could correct the Bekenstein-Hawking formula for the black hole entropy in such a way that it correctly reproduces the logarithm of the density of elementary string states. In particular, this entropy has the correct dependence on three independent parameters, the mass and the left-handed charge of the black hole, and the string coupling constant. 
  We study the non-localization of extended worldsheet supersymmetry under T-duality, when the associated complex structure depends on the coordinate with respect to which duality is performed. First, the canonical transformation which implements T-duality is generalized to the supersymmetric non-linear $\sigma$-models. Then, we obtain the non-local object which replaces the complex structure in the dual theory and write down the condition it should satisfy so that the dual action is invariant under the non-local supersymmetry. For the target space, this implies that the supersymmetry transformation parameter is a non-local spinor. The analogue of the Killing equation for this non-local spinor is obtained. It is argued that in the target space, the supersymmetry is no longer realized in the standard way. The string theoretic origin of this phenomenon is briefly discussed. 
  I present arguments to the affect that the topological phase of string theory must be event-symmetric. This motivates a search for a universal string group for discrete strings in event-symmetric space-time which unifies space-time symmetry with internal gauge symmetry. This is partially successful but the results are incomplete and I speculate on the use of quantum groups to define a well behaved theory which would resolve the discrete/continuum dual nature of stringy space-time. 
  Physical mass spectra of supersymmetric Yang-Mills theories in 1+1 dimensions are evaluated in the light-cone gauge with a compact spatial dimension. The supercharges are constructed and the infrared regularization is unambiguously prescribed for supercharges, instead of the light-cone Hamiltonian. This provides a manifestly supersymmetric infrared regularization for the discretized light-cone approach. By an exact diagonalization of the supercharge matrix between up to several hundred color singlet bound states, we find a rapidly increasing density of states as mass increases. 
  We quantize the Chern-Simons-Proca theory in three dimensions by using the Batalin-Tyutin Hamiltonian method, which systematically embeds second class constraint system into first class by introducing new fields in the extended phase space. As results, we obtain simultaneously the St\"uckelberg scalar term, which is needed to cancel the gauge anomaly due to the mass term, and the new type of Wess-Zumino action, which is irrelevant to the gauge symmetry. We also investigate the infrared property of the Chern-Simons-Proca theory by using the Batalin-Tyutin formalism comparing with the symplectic formalism. As a result, we observe that the resulting theory is precisely the gauge invariant Chern-Simons-Proca quantum mechanical version of this theory. 
  The method of constructing of extended phase space for singular theories which permits the consideration of covariant gauges without the introducing of a ghost fields, is proposed. The extension of the phase space is carried out by the identification of the initial theory with an equivalent theory with higher derivatives and applying to it the Ostrogradsky method of Hamiltonian description . 
  The systematic method for the conversion of first class constraints to the equivalent set of Abelian one based on the Dirac equivalence transformation is developed. The representation for the corresponding matrix performing this transformation is proposed. This representation allows one to lead the problem of abelianization to the solution of a certain system of first order {\it linear } differential equations for matrix elements . 
  The {\it {gauge - fixing} } and {\it gaugeless } methods for reducing the phase space in the generalized Hamiltonian dynamics are compared with the aim to define the class of admissible gauges . In the gaugeless approach, the reduced phase space of a Hamiltonian system with the first class constraints is constructed locally, without any gauge fixing, using the following procedure: abelianization of constraints with the subsequent canonical transformation so that some of the new momenta are equal to the new abelian constraints. As a result the corresponding conjugate coordinates are ignorable ( nonphysical ) one while the remaining canonical pairs corresponds to the true dynamical variables. This representation for the phase space prompts us the definition of subclass of admissible gauges -- canonical gauges as functions depending only on the ignorable coordinates. A practical method to recognize the canonical gauge is proposed . 
  Gravity coupled three--dimensional $\sigma$--model describing the stationary Einstein--Maxwell--dilaton system with general dilaton coupling is studied. Killing equations for the corresponding five--dimensional target space are integrated. It is shown that for general coupling constant $\alpha$ the symmetry algebra is isomorphic to the maximal solvable subalgebra of $sl(3,R)$. For two critical values $\alpha =0$ and $\alpha =\sqrt{3}$, Killing algebra enlarges to the full $sl(3,R)$ and $su(2,1)\times R$ algebras respectively, which correspond to five--dimensional Kaluza--Klein and four--dimensional Brans--Dicke--Maxwell theories. These two models are analyzed in terms of the unique real variables. Relation to the description in terms of complex Ernst potentials is discussed. Non--trivial discrete maps between different subspaces of the target space are found and used to generate new arbitrary--$\alpha$ solutions to dilaton gravity. 
  We consider a new action of a two-dimensional field theory interacting with gravitational field. The action is interpreted as the area of a surface imbedded into four-dimensional Mincowski target space. In addition to reparametrization invariance the new action has one extra infinite-dimensional local symmetry with a clear geometrical meaning. The special gauge choice, which includes the gauge condition of tracelessness of the energy-momentum tensor, leads to an effective free scalar field theory. The problem of anomalies in quantum theory and possible connection with matrix quantum mechanics are also discussed. 
  The wave equations for self/anti-self conjugate Majorana-McLennan-Case $j=1/2$ and $j=1$ spinors, proposed by Ahluwalia, are re-written to covariant form. The connection with the Foldy-Nigam-Bargmann-Wightman-Wigner (FNBWW) type quantum field theory is discussed. 
  The first part of this article (Sections I and II) presents oneself an overview of theory and phenomenology of truly neutral particles based on the papers of Majorana, Racah, Furry, McLennan and Case. The recent development of the construct, undertaken by Ahluwalia [{\it Mod. Phys. Lett. A}{\bf 9} (1994) 439; {\it Acta Phys. Polon. B}{\bf 25} (1994) 1267; Preprints LANL LA-UR-94-1252, LA-UR-94-3118], could be relevant for explanation of the present experimental situation in neutrino physics and astrophysics.   In Section III the new fundamental wave equations for self/anti-self conjugate type-II spinors, proposed by Ahluwalia, are re-casted to covariant form. The connection with the Foldy-Nigam-Bargmann-Wightman- Wigner (FNBWW) type quantum field theory is found. The possible applications to the problem of neutrino oscillations are discussed. 
  This paper has rather a pedagogical meaning. Surprising symmetries in the $(j,0)\oplus (0,j)$ Lorentz group representation space are analyzed. The aim is to draw reader's attention to the possibility of describing the particle world on the ground of the Dirac "doubles". Several tune points of the variational principle for this kind of equations are briefly discussed. 
  A representation of a subgroup H of a finite-dimensional group G can be used to induce a nonlinear realization of G. If the nonlinearly realized symmetry is gauged, then the BRST charge can be related by a similarity transformation to the BRST charge for the gauged linear realization of H (plus a cohomologically trivial piece). It is shown that the relation between the two BRST charges is a reflection of the fact that they can be interpreted geometrically as expressions for the exterior derivative on G relative to two different bases, and an explicit expression for the generator of the similarity transformation is obtained. This result is applied in an infinite-dimensional setting, where it yields the similarity transformation used by Ishikawa and Kato to prove the equivalence of the Berkovits-Vafa superstring with the underlying bosonic string theory. 
  Four dimensional heterotic string effective action is known to admit non-rotating electrically and magnetically charged black hole solutions. It is shown that the partition functions and entropies in both the cases are identical when these black hole solutions are related by S-duality transformations. The entropy is computed and is vanishing for each black hole in the extremal limit. 
  The conformal transformation in the Einstein - Hilbert action leads to a new frame where an extra scalar degree of freedom is compensated by the local conformal-like symmetry. We write down a most general action resulting from such transformation and show that it covers both general relativity and conformally coupled to gravity scalar field as the particular cases. On quantum level the equivalence between the different frames is disturbed by the loop corrections. New conformal-like symmetry in anomalous and, as a result, the theory is not finite on shell at the one-loop order. 
  The nature of Mean Field Solutions to the Equations of Motion of the Chern--Simons Landau--Ginsberg (CSLG) description of the Fractional Quantum Hall Effect (FQHE) is studied. Beginning with the conventional description of this model at some chemical potential $\mu_0$ and magnetic field $B$ corresponding to a ``special'' filling fraction $\nu=2\pi\rho/eB=1/n$ ($n=1,3,5\cdot \cdot\cdot$) we show that a deviation of $\mu$ in a finite range around $\mu_0$ does not change the Mean Field solution and thus the mean density of particles in the model. This result holds not only for the lowest energy Mean Field solution but for the vortex excitations as well. The vortex configurations do not depend on $\mu$ in a finite range about $\mu_0$ in this model. However when $\mu-\mu_0 < \mu_{cr}^-$ (or $\mu-\mu_0>\mu_{cr}^+$) the lowest energy Mean Field solution describes a condensate of vortices (or antivortices). We give numerical examples of vortex and antivortex configurations and discuss the range of $\mu$ and $\nu$ over which the system of vortices is dilute. 
  It is argued that the gauge anomalies are only the artefacts of quantum field theory when certain subtleties are not taken into account. With the Berry's phase needed to satisfy certain boundary conditions of the generating path integral, the gauge anomalies associated with homotopically nontrivial gauge transformations are shown explicitly to be eliminated, without any extra quantum fields introduced. This is in contra-distinction to other quantisations of `anomalous' gauge theory where extra, new fields are introduced to explicitly cancel the anomalies. 
  A gauge theory model in which there exists a specific interaction between instantons is considered. An effective action describing this interaction possesses a minimum when the instantons have identical orientation. The considered interaction might provide a phase transition into the state where instantons have a preferred orientation. This phase of the gauge-field theory is important because it can give the description of gravity in the framework of the gauge theory. 
  We complete the formulation of the equations of motion of a non-Abelian gauge field coupled to fermions on a finite-element lattice in four space-time dimensions. This is accomplished by a straightforward iterative approach, in which successive interaction terms are added to the Dirac and Yang-Mills equations of motion, and to the field strength, in order to preserve lattice gauge invariance exactly, yielding a series in powers of $ghA$. Here $g$ is the coupling constant, $h$ is the lattice spacing, and $A$ is the gauge potential. Gauge transformations of the potentials are determined simultaneously. The interaction terms in the equations of motion are nonlocal, and can be expressed either by an iterative formula or by a difference equation. On the other hand, the field strength is locally constructed from the potentials in terms of a path-ordered product of exponentials. 
  We show that if actions more general than the usual simple plaquette action ($\sim F_{\mu\nu}^2$) are considered, then compact $U(1)$ {\sl pure} gauge theory in three Euclidean dimensions can have two phases. Both phases are confining phases, however in one phase the monopole condensate spontaneously `magnetizes'. For a certain range of parameters the phase transition is continuous, allowing the definition of a strong coupling continuum limit. We note that these observations have relevance to the `fictitious' gauge field theories of strongly correlated electron systems, such as those describing high-$T_c$ superconductors. 
  We discuss non-abelian $SU(N_c)$ gauge theory coupled to an adjoint chiral superfield $X$, and a number of fundamental chiral superfields $Q^i$. Using duality, we show that turning on a superpotential $W(X)=\Tr\sum_{l=1}^k g_l X^{l+1}$ leads to non-trivial long distance dynamics, a large number of multicritical IR fixed points and vacua, connected to each other by varying the coefficients $g_l$. 
  We study two-dimensional U($N$) and SU($N$) gauge theories with a topological term on arbitrary surfaces. Starting from a lattice formulation we derive the continuum limit of the action which turns out to be a generalisation of the heat kernel in the presence of a topological term. In the continuum limit we can reconstruct the topological information encoded in the theta term. In the topologically trivial cases the theta term gives only a trivial shift to the ground state energy but in the topologically nontrivial ones it remains to be coupled to the dynamics in the continuum. In particular for the U($N$) gauge group on orientable surfaces it gives rise to a phase transition at $\theta= \pi$, similar to the ones observed in other models. Using the equivalence of 2d QCD and a 1d fermion gas on a circle we rewrite our result in the fermionic language and show that the theta term can be also interpreted as an external magnetic field imposed on the fermions. 
  We study $N=1$ supersymmetric $SP(N_c)$ gauge theories with $N_f$ flavors of quarks in the fundamental representation. Depending on $N_f$ and $N_c$, we find exact, dynamically generated superpotentials, smooth quantum moduli spaces of vacua, quantum moduli spaces of vacua with additional massless composites at strong coupling, confinement without chiral symmetry breaking, non-trivial fixed points of the renormalization group, and massless magnetic quarks and gluons. 
  In this paper, a three-dimensional vertex model is obtained. It is a duality of the three-dimensional integrable lattice model with $N$ states proposed by Boos, Mangazeev, Sergeev and Stroganov. The Boltzmann weight of the model is dependent on four spin variables, which are the linear combinations of the spins on the corner sites of the cube, and obeys the modified vertex type tetrahedron equation. This vertex model can be regard as a deformation of the one related the three-dimensional Baxter-Bazhanov model. The constrained conditions of the spectrums are discussed in detail and the symmetry properties of weight functions of the vertex model are presented. 
  We consider the Hamiltonian of the closed $SU(2)_{q}$ invariant chain. We project a particular class of statistical models belonging to the unitary minimal series. A particular model corresponds to a particular value of the coupling constant. The operator content is derived. This class of models has charge-dependent boundary conditions. In simple cases (Ising, 3-state Potts) corresponding Hamiltonians are constructed. These are non-local as the original spin chain. 
  We give a simple geometrical picture of the basic structures of the covariant $Sp(2)$ symmetric quantization formalism -- triplectic quantization -- recently suggested by Batalin, Marnelius and Semikhatov. In particular, we show that the appearance of an even Poisson bracket is not a particular property of triplectic quantization. Rather, any solution of the classical master equation generates on a Lagrangian surface of the antibracket an even Poisson bracket. Also other features of triplectic quantization can be identified with aspects of conventional Lagrangian BRST quantization without extended BRST symmetry. 
  Anomalous contributions to the energy-momentum commutators are calculated for even dimensions, by using a non-perturbative approach that combines operator product expansion and Bjorken-Johnson-Low limit techniques. We first study the two dimensional case and give the covariant expression for the commutators. The expression in terms of light-cone coordinates is then calculated and found to be in perfect agreement with the results in the literature. The particular scenario of the light-cone frame is revisited using a reformulation of the BJL limit in such a frame. The arguments used for $n=2$ are then generalized to the case of any even dimensional Minkowskian spacetime and it is shown that there are no anomalous contributions to the commutators for $n\not=2$. These results are found to be valid for both fermionic and bosonic free fields. A generalization of the BJL-limit is later used to obtain double commutators of energy-momentum tensors and to study the Jacobi identity. The two dimensional case is studied and we find no existance of 3-cocycles in both the Abelian and non-Abelian case. 
  It is shown that from some solutions of generalized Knizhnik-Zamolodchikov equations one can construct eigenfunctions of the Calogero-Sutherland-Moser Hamiltonians with exchange terms, which are characterized by any given permutational symmetry under particle exchange. This generalizes some results previously derived by Matsuo and Cherednik for the ordinary Calogero-Sutherland-Moser Hamiltonians. 
  We consider the topological gauged WZW model in the generalized momentum representation. The chiral field $g$ is interpreted as a counterpart of the electric field $E$ of conventional gauge theories. The gauge dependence of wave functionals $\Psi(g)$ is governed by a new gauge cocycle $\phi_{GWZW}$. We evaluate this cocycle explicitly using the machinery of Poisson $\sigma$-models. In this approach the GWZW model is reformulated as a Schwarz type topological theory so that the action does not depend on the world-sheet metric. The equivalence of this new formulation to the original one is proved for genus one and conjectured for an arbitrary genus Riemann surface. As a by-product we discover a new way to explain the appearance of Quantum Groups in the WZW model. 
  A gauge invariant quantum field theory with a spacetime dependent Chern-Simons coefficient is studied. Using a constraint formalism together with the Schwinger action principle it is shown that non-zero gradients in the coefficient induce magnetic-moment corrections to the Hall current and transform vortex singularities into non-local objects. The fundamental commutator for the density fluctuations is obtained from the action principle and the Hamiltonian of the Chern-Simons field is shown to vanish only under the restricted class of variations which satisfy the gauge invariance constraint. 
  A superspace formulation for the Batalin Vilkovisky formalism (also called field-antifield quantization ) with extended BRST invariance (BRST and anti-BRST invariance ) for gauge theories with closed algebra is presented. In contrast to a recent formulation, where only BRST invariance holds off shell, two collective sets of fields are introduced and an off shell realization of the extended algebra in a superspace with two Grassmann coordinates is obtained. The example of the Yang Mills theory is also considered. 
  Starting from a nonlinear isospinor-spinor field equation, generalized three-particle Bargmann-Wigner equations are derived. In the strong-coupling limit, a special class of spin 1/2 bound-states are calculated. These solutions which are antisymmetric with respect to all indices, have mixed symmetries in isospin-superspin space and in spin orbit space. As a consequence of this mixed symmetry, we get three solution manifolds. In appendix \ref{b}, table 2, these solution manifolds are interpreted as the three generations of leptons and quarks. This interpretation will be justified in a forthcoming paper. 
  Using the overlap formulation, we calculate the fermionic determinant on the lattice for chiral fermions with twisted boundary conditions in two dimensions. When the lattice spacing tends to zero we recover the results of the usual string-theory continuum calculations. 
  The matrix sine-Gordon theory, a matrix generalization of the well-known sine-Gordon theory, is studied. In particular, the $A_{3}$-generalization where fields take value in $SU(2)$ describes integrable deformations of conformal field theory corresponding to the coset $SU(2) \times SU(2) /SU(2)$. Various classical aspects of the matrix sine-Gordon theory are addressed. We find exact solutions, solitons and breathers which generalize those of the sine-Gordon theory with internal degrees of freedom, by applying the Zakharov-Shabat dressing method and explain their physical properties. Infinite current conservation laws and the B\"{a}cklund transformation of the theory are obtained from the zero curvature formalism of the equation of motion. From the B\"{a}cklund transformation, we also derive exact solutions as well as a nonlinear superposition principle by making use of the Bianchi's permutability theorem. 
  A long-standing problem in string phenomenology has been the fact that the string unification scale disagrees with the GUT scale obtained by extrapolating low-energy data within the framework of the minimal supersymmetric standard model (MSSM). In this paper we examine several effects that may modify the minimal string predictions and thereby bring string-scale unification into agreement with low-energy data. These include heavy string threshold corrections, non-standard hypercharge normalizations, light SUSY thresholds, intermediate gauge structure, and thresholds arising from extra matter beyond the MSSM. We explicitly evaluate these contributions within a variety of realistic free-fermionic string models, including the flipped SU(5), SO(6) x SO(4), and various SU(3) x SU(2) x U(1) models, and find that most of these sources do not substantially alter the minimal string predictions. Indeed, we find that the only way to reconcile string unification with low-energy data is through certain types of extra matter. Remarkably, however, many of the realistic string models contain precisely this required matter in their low-energy spectra. 
  Aspects of the monopole condensation picture of confinement are discussed. First, the nature of the monopole singularities in the abelian projection approach is analysed. Their apparent gauge dependence is shown to have a natural interpretation in terms of 't~Hooft-Polyakov-like monopoles in euclidean SU(2) gauge theory. Next, the results and predictions of a realization of confinement through condensation of such monopoles are summarized and compared with numerical data. 
  We find that the energy spectra of four and five anyons in a harmonic potential exhibit some mirror symmetric (reflection symmetric about the semionic statistics point $\theta=\pi/2$) features analogous to the mirror symmetry in the two and three anyon spectra. However, since the $\ell=0$ sector remains non-mirror symmetric, the fourth and fifth virial coeffients do not reflect this symmetry. 
  We study quantum aspects of the accelerated black holes in some detail. Explicitly shown is the fact that a uniform acceleration stabilizes certain charged black holes against the well-known thermal evaporation. Furthermore, a close inspection of the geometry reveals that this is possible only for near-extremal black holes and that most nonextremal varieties continue to evaporate with a modified spectrum under the acceleration. We also introduce a two-dimensional toy model where the energy-momentum flow is easily obtained for general accelerations, and find the behavior to be in accordance with the four-dimensional results. After a brief comparison to the classical system of a uniformly accelerated charge, we close by pointing out the importance of this result in the WKB expansion of the black hole pair-creation rate. 
  The sine-Gordon Y-systems and those of the minimal $M_{p,q}+\phi_{13}$ models are determined in a compact form and a correspondence between the rational numbers and a new infinite family of multi-parameter functional equations for the Rogers dilogarithm is pointed out. The relation between the TBA-duality and the massless RG fluxes in the minimal models recently conjectured is briefly discussed. 
  By studying string loop corrections to superpotential of type II strings compactified on Calabi-Yau threefolds we find a quantum stringy test and a confirmation of a recent proposal of Strominger on the fate of the conifold singularity. We also propose a connection between the spectrum of Bogomolnyi saturated solitons and one-loop string partition function of $N=2$ topological strings. 
  The role of edge states in phenomena like the quantum Hall effect is well known. In this paper we show how the choice of boundary conditions for a one-particle Schr\"odinger equation can give rise to states localized at the edge of the system. We consider both the example of a free particle and the more involved example of a particle in a magnetic field. In each case, edge states arise from a non-trivial scaling limit involving the boundary conditions. Second quantization of these quantum mechanical systems leads to a multi-particle ground state carrying a persistent current at the edge. We show that the theory quantized with this vacuum displays an ``anomaly'' at the edge which is the mark of a quantized Hall conductivity in the presence of an external magnetic field. We also offer interpretations for the physics of such boundary conditions which may have a bearing on the nature of the excitations of these systems. 
  We analyze the U-duality group for the case of a type II superstring compactified to four dimensions on a K3 surface times a torus. The various limits of this theory are considered which have interpretations as type IIA and IIB superstrings, the heterotic string, and eleven-dimensional supergravity, allowing all these theories to be directly related to each other. The integral structure which appears in the Ramond-Ramond sector of the type II superstring is related to the quantum cohomology of general Calabi-Yau threefolds which allows the moduli space of type II superstring compactifications on Calabi-Yau manifolds to be analyzed. 
  New formulations of the solutions of N=1 and N=2 super Toda field theory are introduced, using Hamiltonian Reduction of the N=1 and N=2 super WZNW Models to the super Toda Models. These parameterisations are then used to present the Hamiltonian formulations of the super Toda theories on the spaces of solutions. 
  In this paper we discuss topological BF theories in 3 and 4 dimensions. Observables are associated to ordinary knots and links (in 3 dimensions) and to 2-knots (in 4 dimensions). The vacuum expectation values of such observables give a wide range of invariants. Here we consider mainly the 3-dimensional case, where these invariants include Alexander polynomials, HOMFLY polynomials and Kontsevich integrals. 
  String theory in 4 dimensions has the unique feature that a topological term, the oriented self-intersection number, can be added to the usual action. It has been suggested that the corresponding theory of random surfaces wold be free from the problem encountered in the scaling of the string tension. Unfortunately, in the usual dynamical triangulation it is not clear how to write such a term. We show that for random surfaces on a hypercubic lattice however, the analogue of the oriented self-intersection number $I[\s]$ can be defined and computed in a straightforward way. Furthermore, $I[\s]$ has a genuine topological meaning in the sense that it is invariant under the discrete analogue of continuous deformations. The resulting random surface model is no longer free and may lead to a non trivial continuum limit. 
  Multistring solutions of the string equations of motion are found for inflationary spacetimes with expansion factor $R(\eta)\propto \eta^k$ for any $k<0$ and $\eta$ conformal time. If $0>k>-1$ only two-string solutions can be found within our ansatz, whereas for $k=-1-1/n$, with $n=1,2,\ldots$, multistring solutions exist with an infinite number of strings [In the special case $k=-2$ we recover de Sitter spacetime where multistring solutions were first found]. 
  Taking advantage of the representation of dilatonic gravity with the  $R^2$-term under the form of low-derivative dilatonic gravity coupled to an  additional scalar, we construct a general renormalizable model motivated by  this theory. Exact black hole solutions are found for some special versions  of the model, and their thermodynamical properties are described in detail.  In particular, their horizons and temperatures are calculated. Finally, the corresponding one-loop effective action is obtained in the conformal gauge, and a number of its properties -including the construction of one-loop finite models-are briefly described. 
  We apply the time-dependent variational principle of Balian and V\'en\'eroni to the $ \Phi^4$ theory. An appropriate parametrization for the variational objects allows us to write coupled dynamical equations from which we derive approximations for the two-time correlation functions involving two, three or four field operators. 
  We construct a new class of $(p,q)$-extended Poincar\'e supergravity theories in 2+1 dimensions as Chern-Simons theories of supersymmetry algebras with both central and automorphism charges. The new theories have the advantage that they are limits of corresponding $(p,q)$ adS supergravity theories and, for not too large a value of $N=p+q$, that they have a natural formulation in terms of off-shell superfields, in which context the distinction between theories having the same value of $N$ but different $(p,q)$ arises because of inequivalent conformal compensator superfields. We also show that, unlike previously constructed N-extended Poincar\'e supergravity theories, the new (2,0) theory admits conical spacetimes with Killing spinors. Many of our results on (2,0) Poincar\'e supergravity continue to apply in the presence of coupling to N=2 supersymmetric sigma-model matter. 
  The transformation of the $SL(2,R)/U(1)$ black hole under a boost of the subgroup U(1) is studied. It is found that the tachyon vertex operators of the black hole go into those of the $c=1$ conformal field theory coupled to gravity. The discrete states of the black hole also tend to the discrete states of the 2-d gravity theory. The fate of the extra discrete states of the black hole under boost are discussed. 
  We Describe a one-parameter family of c=1 CFT's as a continuous conformal deformation of the SL(2)_4/U(1) coset. 
  The exact Seiberg-Witten (SW) description of the light sector in the $N=2$ SUSY $4d$ Yang-Mills theory is reformulated in terms of integrable systems and appears to be a Gurevich-Pitaevsky (GP) solution to the elliptic Whitham equations. We consider this as an implication that dynamical mechanism behind the SW solution is related to integrable systems on the moduli space of instantons. We emphasize the role of the Whitham theory as a possible substitute of the renormalization-group approach to the construction of low-energy effective actions. 
  Using the effective potential, the large-$N$ nonlinear $O(N)$ sigma model with the curvature coupled term is studied on $S^2\times R^1$. We show that, for the conformally coupled case, the dynamical mass generation of the model in the strong-coupled regime on $R^3$ takes place for any finite scalar curvature (or radius of the $S^2$). If the coupling constant is larger than that of the conformally coupled case, there exist a critical curvature (radius) above (below) which the dynamical mass generation does not take place even in the strong-coupled regime. Below the critical curvature, the mass generation occurs as in the model on $R^3$. 
  The Einstein-Maxwell equations with a negative cosmological constant in 2+1 spacetime dimensions discussed by Banados, Teitelboim and Zanelli are solved by assuming a self(anti-self)dual equation. The solution describes an electrically charged extreme black hole with an angular momentum. 
  The most general electrically and magnetically charged rotating black hole solutions of 5 dimensional \KK\ theory are given in an explicit form. Various classical quantities associated with the black holes are derived. In particular, one finds the very surprising result that the gyromagnetic and gyroelectric ratios can become {\tenit arbitrarily large}. The thermodynamic quantities of the black holes are calculated and a Smarr-type formula is obtained leading to a generalized first law of black hole thermodynamics. The properties of the extreme solutions are investigated and it is shown how they naturally separate into two classes. The extreme solutions in one class are found to have two unusual properties: (i). Their event horizons have zero angular velocity and yet they have non-zero ADM angular momentum. (ii). In certain circumstances it is possible to add angular momentum to these extreme solutions without changing the mass or charges and yet still maintain an extreme solution. Regarding the extreme black holes as elementary particles, their stability is discussed and it is found that they are stable provided they have sufficient angular momentum. 
  Action-angle type variables for spin generalizations of the elliptic Ruijsenaars-Schneider system are constructed. The equations of motion of these systems are solved in terms of Riemann theta-functions. It is proved that these systems are isomorphic to special elliptic solutions of the non-abelian 2D Toda chain. A connection between the finite gap solutions of solitonic equations and representations of the Sklyanin algebra is revealed and discrete analogs of the Lame operators are introduced. A simple way to construct representations of the Sklyanin algebra by difference operators is suggested. 
  In the presence of consistent regulators, the standard procedure of BRST gauge fixing (or moving from one gauge to another) can require non-trivial modifications. These modifications occur at the quantum level, and gauges exist which are only well-defined when quantum mechanical modifications are correctly taken into account. We illustrate how this phenomenon manifests itself in the solvable case of two-dimensional bosonization in the path-integral formalism. As a by-product, we show how to derive smooth bosonization in Batalin-Vilkovisky Lagrangian BRST quantization. 
  We show that the supersymmetric nonlinear Schr\"odinger equation is a bi-Hamiltonian integrable system. We obtain the two Hamiltonian structures of the theory from the ones of the supersymmetric two boson hierarchy through a field redefinition. We also show how the two Hamiltonian structures of the supersymmetric KdV equation can be derived from a Hamiltonian reduction of the supersymmetric two boson hierarchy as well. 
  We discuss in detail the parasupersymmetric quantum mechanics of arbitrary order where the parasupersymmetry is between the normal bosons and those corresponding to the truncated harmonic oscillator. We show that even though the parasusy algebra is different from that of the usual parasusy quantum mechanics, still the consequences of the two are identical. We further show that the parasupersymmetric quantum mechanics of arbitrary order p can also be rewritten in terms of p supercharges (i.e. all of which obey $Q_i^{2} = 0$). However, the Hamiltonian cannot be expressed in a simple form in terms of the p supercharges except in a special case. A model of conformal parasupersymmetry is also discussed and it is shown that in this case, the p supercharges, the p conformal supercharges along with Hamiltonian H, conformal generator K and dilatation generator D form a closed algebra. 
  In this article we review some of the recent advances regarding the charged vortex solutions in abelian and nonabelian gauge theories with Chern-Simons (CS) term in two space dimensions. Since these nontrivial results are essentially because of the CS term, hence, we first discuss in some detail the various properties of the CS term in two space dimensions. In particular, it is pointed out that this parity (P) and time reversal (T) violating but gauge invariant term when added to the Maxwell Lagrangian gives a massive gauge quanta and yet the theory is still gauge invariant. Further, the vacuum of such a theory shows the magneto-electric effect. Besides, we show that the CS term can also be generated by spontaneous symmetry breaking as well as by radiative corrections. A detailed discussion about Coleman-Hill theorem is also given which aserts that the parity-odd piece of the vacuum polarization tensor at zero momentum transfer is unaffected by two and multi-loop effects. Topological quantization of the coefficient of the CS term in nonabelian gauge theories is also elaborated in some detail. One of the dramatic effect of the CS term is that the vortices of the abelian (as well as nonabelian) Higgs model now acquire finite quantized charge and angular momentum. The various properties of these vortices are discussed at length with special emphasis on some of the recent developments including the discovery of the self-dual charged vortex solutions. 
  The amplitudes for boson-boson and fermion-boson interactions are calculated in the second order of perturbation theory in the Lobachevsky space. An essential ingredient of the used model is the Weinberg's $2(2j+1)$ component formalism for describing a particle of spin $j$, recently developed substantially. The boson-boson amplitude is then compared with the two-fermion amplitude obtained long ago by Skachkov on the ground of the hamiltonian formulation of quantum field theory on the mass hyperboloid, $p_0^2 -{\bf p}^2=M^2$, proposed by Kadyshevsky. The parametrization of the amplitudes by means of the momentum transfer in the Lobachevsky space leads to same spin structures in the expressions of $T$ matrices for the fermion and the boson cases. However, certain differences are found. Possible physical applications are discussed. 
  A class of 4-dimensional supersymmetric dyonic black hole solutions that arise in an effective 11-dimensional supergravity compactified on a 7-torus is presented. We give the explicit form of dyonic solutions with diagonal internal metric, associated with the Kaluza-Klein sector as well as the three-form field, and relate them to a class of solutions with a general internal metric by imposing a subset of $SO(7)\subset E_7$ transformations. We also give the field transformations which relate the above configurations to 4-dimensional ground state configurations of Ramond-Ramond and Neveu-Schwarz-Neveu-Schwarz sector of type-IIA strings on a 6-torus. 
  We discuss the unification of gauge couplings within the framework of a wide class of realistic free-fermionic string models which have appeared in the literature, including the flipped SU(5), SO(6)xSO(4), and various SU(3)xSU(2)xU(1) models. If the matter spectrum below the string scale is that of the Minimal Supersymmetric Standard Model (MSSM), then string unification is in disagreement with experiment. We therefore examine several effects that may modify the minimal string predictions. First, we develop a systematic procedure for evaluating the one-loop heavy string threshold corrections in free-fermionic string models, and we explicitly evaluate these corrections for each of the realistic models. We find that these string threshold corrections are small, and we provide general arguments explaining why such threshold corrections are suppressed in string theory. Thus heavy thresholds cannot resolve the disagreement with experiment. We also study the effect of non-standard hypercharge normalizations, light SUSY thresholds, and intermediate-scale gauge structure, and similarly conclude that these effects cannot resolve the disagreement with low-energy data. Finally, we examine the effects of additional color triplets and electroweak doublets beyond the MSSM. Although not required in ordinary grand unification scenarios, such states generically appear within the context of certain realistic free-fermionic string models. We show that if these states exist at the appropriate thresholds, then the gauge couplings will indeed unify at the string scale. Thus, within these string models, string unification can be in agreement with low-energy data. 
  In the present paper we obtain some integrable generalisations of the Toda system generated by flat connection forms taking values in higher ${\bf Z}$--grading subspaces of a simple Lie algebra, and construct their general solutions. One may think of our systems as describing some new fields of the matter type coupled to the standard Toda systems. This is of special interest in nonabelian Toda theories where the latter involve black hole target space metrics. We also give a derivation of our conformal system on the base of the Hamiltonian reduction of the WZNW model; and discuss a relation between abelian and nonabelian systems generated by a gauge transformation that maps the first grading description to the second. The latter involves grades larger than one. 
  We present a precise lattice computation of the slope of the effective potential for massless $(\lambda\Phi^4)_4$ theory in the region of bare parameters indicated by the Brahm's analysis of lattice data. Our results confirm the existence on the lattice of a remarkable phase of $(\lambda\Phi^4)_4$ where Spontaneous Symmetry Breaking is generated through ``dimensional transmutation''. The resulting effective potential shows no evidence for residual self-interaction effects of the shifted `Higgs' field $h(x)=\Phi(x)-\langle\Phi\rangle$, as predicted by ``triviality'', and cannot be reproduced in perturbation theory. Accordingly the mass of the Higgs particle, by itself, does not represent a measure of any observable interaction. 
  We discuss recent results on bosonization in $d \geq 2$ space-time dimensions by giving a very simple derivation for the bosonic representation of the original free fermionic model both in the abelian and non-abelian cases. We carefully analyse the issue of symmetries in the resulting bosonic model as well as the recipes for bosonization of fermion currents 
  All possible action functionals on the space of surfaces in ${\bf R}^4$ that depend only on first and second derivatives of the functions, entering the equation of the surface, and satisfy the condition of invariance with respect to rigid motions are described. 
  We present evidence for new, non-trivial RG fixed points with dual magnetic descriptions in $N=1$ supersymmetric $SP(N_c)$ and $SO(N_c)$ gauge theories. The $SP(N_c)$ case involves matter $X$ in the antisymmetric tensor representation and $N_f$ flavors of quarks $Q$ in the fundamental representation. The $SO(N_c)$ case involves matter $X$ in the symmetric tensor representation and $N_f$ flavors of quarks $Q$ in the vector representation of $SO(N_c)$. Perturbing these theories by superpotentials $W(X)$, we find a variety of interesting RG fixed points with dual descriptions. The duality in these theories is similar to that found by Kutasov and by Kutasov and Schwimmer in $SU(N_c)$ with adjoint $X$ and $N_f$ quarks in the fundamental. 
  We review explicitly known exact $D=4$ solutions with Minkowski signature in closed bosonic string theory. Classical string solutions with space-time interpretation are represented by conformal sigma models. Two large (intersecting) classes of solutions are described by gauged WZW models and `chiral null models' (models with conserved chiral null current). The latter class includes plane-wave type backgrounds (admitting a covariantly constant null Killing vector) and backgrounds with two null Killing vectors (e.g., fundamental string solution). $D>4$ chiral null models describe some exact $D=4$ solutions with electromagnetic fields, for example, extreme electric black holes, charged fundamental strings and their generalisations. In addition, there exists a class of conformal models representing axially symmetric stationary magnetic flux tube backgrounds (including, in particular, the dilatonic Melvin solution). In contrast to spherically symmetric chiral null models for which the corresponding conformal field theory is not known explicitly, the magnetic flux tube models (together with some non-semisimple WZW models) are among the first examples of solvable unitary conformal string models with non-trivial $D=4$ curved space-time interpretation. For these models one is able to express the quantum hamiltonian in terms of free fields and to find explicitly the physical spectrum and string partition function. 
  We test type IIA-heterotic string duality in six dimensions by showing that the sigma model anomaly of the heterotic string is generated by a combination of a tree level and a string one-loop correction on the type IIA side. 
  The existence of maximally supersymmetric solutions to heterotic string theory that are not toroidal compactifications of the ten-dimensional superstring is established. We construct an exact fermionic realization of an N=1 supersymmetric string theory in D=8 with non-simply-laced gauge group Sp(20). Toroidal compactification to six and four dimensions gives maximally extended supersymmetric theories with reduced rank (4,12) and (6,14) respectively. 
  We study metastability and nucleation for the Blume-Capel model: a ferromagnetic nearest neighbour two-dimensional lattice system with spin variables taking values in -1,0,+1. We consider large but finite volume, small fixed magnetic field h and chemical potential "lambda" in the limit of zero temperature; we analyze the first excursion from the metastable -1 configuration to the stable +1 configuration. We compute the asymptotic behaviour of the transition time and describe the typical tube of trajectories during the transition. We show that, unexpectedly, the mechanism of transition changes abruptly when the line h=2*lambda is crossed. 
  In this paper, a novel method is presented for the study of the dependence of the functional determinant of the Laplace operator associated to a subbundle $F$ of a hermitian holomorphic vector bundle $E$ over a Riemann surface $\Sigma$ on the hermitian structure $(h,H)$ of $E$. The generalized Weyl anomaly of the effective action is computed and found to be expressible in terms of a suitable generalization of the Liouville and Donaldson actions. The general techniques worked out are then applied to the study of a specific model, the Drinfeld--Sokolov (DS) ghost system arising in $W$--gravity. The expression of generalized Weyl anomaly of the DS ghost effective action is found. It is shown that, by a specific choice of the fiber metric $H_h$ depending on the base metric $h$, the effective action reduces into that of a conformal field theory. Its central charge is computed and found to agree with that obtained by the methods of hamiltonian reduction and conformal field theory. The DS holomorphic gauge group and the DS moduli space are defined and their dimensions are computed. 
  We calculate correlation functions of topological sigma model (A-model) on Calabi-Yau hypersurfaces in $CP^{N-1}$ using torus action method. We also obtain path-integral represention of free energy of the theory coupled to gravity. 
  We consider the theory of four-fermion interactions with N-component fermions in de Sitter space. It is found that the effective potential for a composite operator in the theory is calculable in the leading order of the 1/N expansion. The resulting effective potential is analyzed by varying both the four-fermion coupling constant and the curvature of the space-time. The critical curvature at which the dynamically generated fermion mass disappears is found to exist and is calculated analytically. The dynamical fermion mass is expressed as a function of the space-time curvature. 
  We prove the equality between the statistics phase and the conformal univalence for a superselection sector with finite index in Conformal Quantum Field Theory on $S^1$. A relevant point is the description of the PCT symmetry and the construction of the global conjugate charge. 
  A supersymmetric extension of the color Calogero-Sutherland model is considered based on the Yangian $Y(gl(n|m))$. The algebraic structure of the model is discussed in some details. We show that the commuting conserved quantities can be generated from the super-quantum determinant, thus establishing the integrability of the model. In addition, rational limit of the model is studied where the Yangian symmetry degenerates into a super loop algebra. 
  The heat-kernel expansion and $\zeta$-regularization techniques for quantum field theory and extended objects on curved space-times are reviewed. In particular, ultrastatic space-times with spatial section consisting in manifold with constant curvature are discussed in detail. Several mathematical results, relevant to physical applications are presented, including exact solutions of the heat-kernel equation, a simple exposition of hyperbolic geometry and an elementary derivation of the Selberg trace formula. With regards to the physical applications, the vacuum energy for scalar fields, the one-loop renormalization of a self-interacting scalar field theory on a hyperbolic space-time, with a discussion on the topological symmetry breaking, the finite temperature effects and the Bose-Einstein condensation, are considered. Some attempts to generalize the results to extended objects are also presented, including some remarks on path integral quantization, asymptotic properties of extended objects and a novel representation for the one-loop (super)string free energy. 
  We show that four-dimensional N=2 supersymmetric SU(n) gauge theory for n>2 necessarily contains vacua with mutually non-local massless dyons, using only analyticity of the effective action and the weak coupling limit of the moduli space of vacua. A specific example is the Z_3 point in the exact solution for SU(3), and we study its effective Lagrangian. We propose that the low-energy theory at this point is an N=2 superconformal U(1) gauge theory containing both electrically and magnetically charged massless hypermultiplets. 
  The existence of inequivalent representations in quantum field theory with {\it finitely} many degrees of freedom is shown. Their properties are exemplified and analysed for concrete and simple models. In particular the relations to Bogoliubov--Valatin quasi-particles, to thermo field dynamics, and to $q$--deformed quantum theories are put foreward. The thermal properties of the non-trivial vacuum are given and it is shown that the thermodynamic equilibrium state is uniquely obtained by an irreversible vacuum dynamics. Finally, the theory is applied to a realistic model: the BCS--theory of superconductivity. An exact solution in order $O(N^{-1})$ for the full particle number conserving BCS--Hamiltonian with particle number symmetric ground state is given. 
  The equations of motion for a Lagrangian ${\cal L}(k_1)$, depending on the curvature $k_1$ of the particle worldline, embedded in a space--time of constant curvature, are considered and reformulated in terms of the principal curvatures. It is shown that for arbitrary Lagrangian function ${\cal L}(k_1)$ the general solution of the motion equations can be obtained by integrals. By analogy with the flat space--time case, the constants of integration are interpreted as the particle mass and its spin. As examples, we completely investigate Lagrangians linear and quadratic in $(k_1)$ and the model of relativistic particle with maximal proper acceleration, in a space--time with constant curvature. 
  A number of 2d and 3d four-fermion models which are renormalizable ---in the $1/N$ expansion--- in a maximally symmetric constant curvature space, are investigated. To this purpose, a powerful method for the exact study of spinor heat kernels and propagators on maximally symmetric spaces is reviewed. The renormalized effective potential is found for any value of the curvature and its asymptotic expansion is given explicitly, both for small and for strong curvature. The influence of gravity on the dynamical symmetry breaking pattern of some U(2) flavor-like and discrete symmetries is described in detail. %It is seen explicitly that the effect of a %negative curvature is similar to that of a magnetic field. The phase diagram in $S^2$ is constructed and it is shown that, for any value of the coupling constant, a curvature exists above which chiral symmetry is restored. For the case of $H^2$, chiral symmetry is always broken. In three dimensions, in the case of positive curvature, $S^3$, it is seen that some values of the coupling constants lead to a gap equation which has no solutions. Both for $H^3$ and $S^3$ the configuration given by the auxiliary fields equated to zero is not a solution of the gap equation. 
  In the zero temperature Glauber dynamics of the ferromagnetic Ising or $q$-state Potts model, the size of domains is known to grow like $t^{1/2}$. Recent simulations have shown that the fraction $r(q,t)$ of spins which have never flipped up to time $t$ decays like a power law $r(q,t) \sim t^{-\theta(q)}$ with a non-trivial dependence of the exponent $\theta(q)$ on $q$ and on space dimension. By mapping the problem on an exactly soluble one-species coagulation model ($A+A\rightarrow A$), we obtain the exact expression of $\theta(q)$ in dimension one. 
  We suggest a dual to an $SU(2k)$ Susy gauge theory containing an antisymmetric tensor, $\nf$ fundamentals and $\nfb$ anti-fundamentals. This is done by expanding the theory into an equivalent description with two gauge groups and then performing known duality tranformations on each gauge group separately. Chiral operators, mass perturbations and flat directions are discussed. 
  Starting with the prepotential description of two-dimensional $(2,2)$ supergravity we use local supersymmetry transformations to go to light-cone gauge. We discuss properties of the theory in this gauge and derive Ward identities for correlation functions defined with respect to the induced supergravity action. 
  We describe the programming method for generating the spectrum of bound states for relativistic quantum field theories using the nonperturbative Hamiltonian approach of Discretized Light-Cone Quantization. The method is intended for eventual application to quantum chromodynamics in 3+1 dimensions. Here the fundamental principles are illustrated concretely by treatment of QED in two dimensions. The code is intended as a basis for extensions to include more complicated gauge symmetry groups, such as SU(3) color, and other quantum numbers. The code was written in Fortran 77 and implemented on a DEC 5000-260 workstation with a typical runtime of 0.2s at total momentum 10. 
  Starting from the Abelian Higgs field theory, we construct the theory of quantum Abrikosov--Nielsen--Olesen strings. It is shown that in four space -- time dimensions in the limit of infinitely thin strings, the conformal anomaly is absent, and the quantum theory exists. We also study an analogue of the Aharonov--Bohm effect: the corresponding topological interaction is proportional to the linking number of the string world sheet and the particle world trajectory. The creation operators of the strings are explicitly constructed in the path integral and in the Hamiltonian formulation of the theory. We show that the Aharonov--Bohm effect gives rise to several nontrivial commutation relations. 
  The exchange operator formalism previously introduced for the Calogero problem is extended to the three-body Calogero-Marchioro-Wolfes one. In the absence of oscillator potential, the Hamiltonian of the latter is interpreted as a free particle Hamiltonian, expressed in terms of generalized momenta. In the presence of oscillator potential, it is regarded as a free modified boson Hamiltonian. The modified boson operators are shown to belong to a $D_6$-extended Heisenberg algebra. A proof of complete integrability is also provided. 
  The path integral representation for a system of N non-relativistic particles on the plane, interacting through a Chern-Simons gauge field, is obtained from the operator formalism. An effective interaction between the particles appears, generating the usual Aharonov-Bohm phases. The spin-statistics relation is also considered. 
  Certain four-dimensional $N=4$ supersymmetric theories have special vacua in which massive charged vector supermultiplets become massless, resulting in an enhanced non-abelian gauge symmetry. We show here that any two $N=4$ theories having the same Bogomolnyi spectrum at corresponding points of their moduli spaces have the same enhanced symmetry groups. In particular, the $K_3\times T^2$ compactified type II string is argued to have the same enhanced symmetry groups as the $T^6$-compactified heterotic string, giving further evidence for our conjecture that these two string theories are equivalent. A feature of the enhanced symmetry phase is that for every electrically charged state whose mass tends to zero as an enhanced symmetry point is approached, there are magnetically charged and dyonic states whose masses also tend to zero, a result that applies equally to N=4 super Yang-Mills theory. These extra non-perturbative massless states in the $K_3$ compactification result from $p$-branes wrapping around collapsed homology two-cycles of $K_3$. Finally, we show how membrane `wrapping modes' lead to symmetry enhancement in D=11 supergravity, providing further evidence that the $K_3$-compactified D=11 supergravity is the effective field theory of the strong coupling limit of the $T^3$-compactified heterotic string. 
  It has been argued that the consistency of quantum theory with black hole physics requires nonlocality not present in ordinary effective field theory. We examine the extent to which such nonlocal effects show up in the perturbative S-matrix of string theory. 
  We construct families of hyper-elliptic curves which describe the quantum moduli spaces of vacua of $N=2$ supersymmetric $SU(N_c)$ gauge theories coupled to $N_f$ flavors of quarks in the fundamental representation. The quantum moduli spaces for $N_f < N_c$ are determined completely by imposing $R$-symmetry, instanton corrections and the proper classical singularity structure. These curves are verified by residue and weak coupling monodromy calculations. The quantum moduli spaces for $N_f\geq N_c$ theories are parameterized and their general structure is worked out using residue calculations. Global symmetry considerations suggest a complete description of them. The results are supported by weak coupling monodromy calculations. The exact metrics on the quantum moduli spaces as well as the exact spectrum of stable massive states are derived. We find an example of a novel symmetry of a quantum moduli space: Invariance under the exchange of a moduli parameter and the bare mass. We apply our method for the construction of the quantum moduli spaces of vacua of $N=1$ supersymmetric theories in the coulomb phase. 
  Finite size effects for the Ising Model coupled to two dimensional random surfaces are studied by exploiting the exact results from the 2-matrix models. The fixed area partition function is numerically calculated with arbitrary precision by developing an efficient algorithm for recursively solving the quintic equations so encountered. An analytic method for studying finite size effects is developed based on the behaviour of the free energy near its singular points. The generic form of finite size corrections so obtained are seen to be quite different from the phenomenological parameterisations used in the literature. The method of singularities is also applied to study the magnetic susceptibility. A brief discussion is presented on the implications of these results to the problem of a reliable determination of string susceptibility from numerical simulations. 
  Higher order coefficients of the inverse mass expansion of one--loop effective actions are obtained from a one--dimensional path integral representation. For the evaluation of the path integral with Wick contractions a suitable Green function has to be chosen. We consider the case of a massive scalar loop in the background of both a scalar potential and a (non--abelian) gauge field. For the pure scalar case the method yields the coefficients of the expansion in a minimal set of basis terms whereas complicated ordering problems arise in gauge theory. An appropriate reduction scheme is discussed. 
  A Hamiltonian formulation for the classical problem of electromagnetic interaction of two charged relativistic particles is found. 
  Starting from the generalization of the Itzykson-Zuber integral for $U(m|n)$ we determine the orthogonality relations for this supergroup. 
  Consistent heterotic free fermionic string models are classified in terms of their number of spacetime supersymmetries, N. For each of the six distinct choices of gravitino sector, we determine what number of supersymmetries can survive additional GSO projections. We prove by exhaustive search that only three of the six can yield N = 1, in addition to the N = 4, 2, or 0 that five of the six can yield. One choice of gravitino sector can only produce N = 4 or 0. Relatedly, we find that only Z_2, Z_4, and Z_8 twists of the internal fermions with worldsheet supersymmetry are consistent with N=1 in free fermionic models. Any other twists obviate N=1. 
  A classification of stable singular points on world sheets of open relativistic strings is carried out. 
  There are supersymmetric gauge theories which do not possess any parameters nor flat directions, and hence cannot be studied anywhere in the field space using holomorphy (``non-calculable''). Some of them are believed to break supersymmetry dynamically. We propose a simple technique to analyze these models. Introducing a vector-like field into the model, one finds flat directions where one can study the dynamics. We unambiguously show that the supersymmetry is broken when the mass of the vector-like field is small but finite, and hence Witten index vanishes. If we increase the mass of the vector-like field, it eventually decouples from the dynamics and the models reduce to the original non-calculable models. Assuming the continuity of the Witten index in the parameter space, one can establish the dynamical supersymmetry breaking in the non-calculable models. 
  The crystalline spinon basis for the RSOS models associated with $\widehat{sl_2}$ is studied. This basis gives fermionic type character formulas for the branching coefficients of the coset $(\widehat{sl_2})_l \times (\widehat{sl_2})_N/(\widehat{sl_2})_{l+N}$. In addition the path description of the parafermion characters is found as a limit of the spinon description of the string functions. 
  We improve and generalize in several accounts the recent rigorous proof of convergence of delta expansion - order dependent mappings (variational perturbation expansion) for the energy eigenvalues of anharmonic oscillator. For the single-well anharmonic oscillator the uniformity of convergence in $g\in[0,\infty]$ is proven. The convergence proof is extended also to complex values of $g$ lying on a wide domain of the Riemann surface of $E(g)$. Via the scaling relation \`a la Symanzik, this proves the convergence of delta expansion for the double well in the strong coupling regime (where the standard perturbation series is non Borel summable), as well as for the complex ``energy eigenvalues'' in certain metastable potentials. Sufficient conditions for the convergence of delta expansion are summarized in the form of three theorems, which should apply to a wide class of quantum mechanical and higher dimensional field theoretic systems. 
  We study tensor products of fundamental representations of Yangians and show that the fundamental quotients of such tensor products are given by Dorey's rule. 
  It is known that classical electromagnetic radiation at a frequency in resonance with energy splittings of atoms in a dielectric medium can be described using the classical sine-Gordon equation. In this paper we quantize the electromagnetic field and compute quantum corrections to the classical results by using known results from the sine-Gordon quantum field theory. In particular, we compute the intensity of spontaneously emitted radiation using the thermodynamic Bethe ansatz with boundary interactions. 
  A perturbative technique, the low-temperature expansion, is developed for matrix models of random surfaces. It can be applied to models with arbitrary target spaces, including ones with c>1. As a simple illustration, the series is worked out to 10th order for the surface coupled to a q-state Potts model. Accurate estimates for, e.g., $\gamma_{str}$ are obtained both in the low q (c<1) and high q (branched polymer) regimes, including the logarithmic corrections to scaling. 
  We discuss electric-magnetic duality in two new classes of supersymmetric Yang-Mills theories. The models have gauge group $Sp(2\nc)$ or $SO(\nc)$ with matter in both the adjoint and defining representations. By perturbing these theories with various superpotentials, we find a variety of new infrared fixed points with dual descriptions. This work is complementary to that of Kutasov and Schwimmer on $SU(\nc)$ and of Intriligator on other models involving $Sp(2\nc)$ and $SO(\nc)$. 
  I examine various aspects of event-symmetric physics such as phase changes, symmetry breaking and duality by studying a number of simple toy-models. 
  This is a comment on the article "Integrable Systems in Stringy Gravity" by D. V. Gal'tsov, Phys. Rev. Lett. 74, 2863, (1995). 
  We quantize the spin Calogero-Moser model in the $R$-matrix formalism. The quantum $R$-matrix of the model is dynamical. This $R$-matrix has already appeared in Gervais-Neveu's quantization of Toda field theory and in Felder's quantization of the Knizhnik-Zamolodchikov-Bernard equation. 
  QED_2 with mass and flavor has in common many features with QCD, and thus is an interesting toy model relevant for four dimensional physics. The model is constructed using Euclidean path integrals and mass perturbation series. The vacuum functional is carefully decomposed into clustering states being the analogue of the theta-vacuum of QCD. Finally the clustering theory can be mapped onto a generalized Sine-Gordon model. Having at hand this bosonized version, several lessons on the theta-vacuum, the U(1)-problem and Witten-Veneziano-type formulas will be drawn. This sheds light on the corresponding structures of QCD. 
  In this paper, we review various properties of the supersymmetric Two Boson (sTB) system. We discuss the equation and its nonstandard Lax representation. We construct the local conserved charges as well as the Hamiltoniam structures of the system. We show how this system leads to various other known supersymmetric integrable models under appropriate field redefinition. We discuss the sTB and the supersymmetric nonlinear Schr\"odinger (sNLS) equations as constrained, nonstandard supersymmetric Kadomtsev-Petviashvili (sKP) systems and point out that the nonstandard sKP systems naturally unify all the KP and mKP flows while leading to a new integrable supersymmetrization of the KP equation. We construct the nonlocal conserved charges associated with the sTB system and show that the algebra of charges corresponds to a graded, cubic algebra. We also point out that the sTB system has a hidden supersymmetry making it an $N=2$ extended supersymmetric system. 
  Under special conditions (Meissner-effect levitation in a high frequency magnetic field and rapid rotation) a disk of high-$T_c$ superconducting material has recently been found to produce a weak shielding of the gravitational field. We show that this phenomenon has no explanation in the standard gravity theories, except possibly in the non-perturbative Euclidean quantum theory. 
  We use the BV quantization method for a theory with coupled tensor and vector gauge fields through a topological term. We consider in details the reducibility of the tensorial sector as well as the appearance of a mass term in the effective vectorial theory . 
  We make a comparative study of chiral-boson theories in the Florenani-Jackiw (FJ) and linear constraint formulations. A special attention is given to the case with an improved way of implementing the linear constraint. We show that it has the same spectrum of the FJ formulation. 
  We describe special Ka\"hler geometry, special quaternionic manifolds, and very special real manifolds and analyze the structure of their isometries. The classification of the homogeneous manifolds of these types is presented. 
  I discuss several issues concerning the use of string models as unified theories of all interactions. After a short review of gauge coupling unification in the string context, I discuss possible motivations for the construction of $SU(5)$ and $SO(10)$ String-GUTs. I describe the construction of such String-GUTs using different orbifold techniques and emphasize those properties which could be general. Although $SO(10)$ and $SU(5)$ String-GUTs are relatively easy to build, the spectrum bellow the GUT scale is in general bigger than that of the MSSM and includes colour octets and $SU(2)$ triplets. The phenomenological prospects of these theories are discussed. I then turn to discuss soft SUSY-breaking terms obtained under the assumption of dilaton/moduli dominance in SUSY-breaking string schemes. I underline the unique finiteness pr of the soft terms induced by the dilaton sector. These improved finiteness properties seem to be related to the underlying $SU(1,1)$ structure of the dilaton couplings. I conclude with an outlook and some speculations regarding $N=1$ duality. 
  The one-loop anomalies for chiral $W_{3}$ gravity are derived using the Fujikawa regularisation method. The expected two-loop anomalies are then obtained by imposing the Wess-Zumino consistency conditions on the one-loop results. The anomalies found in this way agree with those already known from explicit Feynman diagram calculations. We then directly verify that the order $\hbar^2$ non-local BRST Ward identity anomalies, arising from the ``dressing'' of the one-loop results, satisfy Lam's theorem. It is also shown that in a rigorous calculation of $Q^2$ anomaly for the BRST charge, one recovers both the non-local as well as the local anomalies. We further verify that, in chiral gravities, the non-local anomalies in the BRST Ward identity can be obtained by the application of the anomalous operator $Q^2$, calculated using operator products, to an appropriately defined gauge fermion. Finally, we give arguments to show why this relation should hold generally in reparametrisation-invariant theories. 
  We present an explicit non-perturbative solution of N=2 supersymmetric SU(N_c) gauge theory with N_f \le 2N_c flavors generalizing results of Seiberg and Witten for N_c=2. 
  In this paper the stability and the renormalizability of Yang-Mills theory in the Background Field Gauge are studied. By means of Ward Identities of Background gauge invariance and Slavnov-Taylor Identities the stability of the classical model is proved and, in a regularization independent way, its renormalizability is verified. A prescription on how to build the counterterms is given and the possible anomalies which may appear for Ward Identities and for Slavnov-Taylor Identities are shown. 
  In the Thermodynamic Bethe Ansatz approach to 2D integrable, ADE-related quantum field theories one derives a set of algebraic functional equations (a Y-system) which play a prominent role. This set of equations is mapped into the problem of finding finite triangulations of certain 3D manifolds. This mapping allows us to find a general explanation of the periodicity of the Y-system. For the $A_N$ related theories and more generally for the various restrictions of the fractionally-supersymmetric sine-Gordon models, we find an explicit, surprisingly simple solution of such functional equations in terms of a single unknown function of the rapidity. The recently-found dilogarithm functional equations associated to the Y-system simply express the invariance of the volume of a manifold for deformations of its triangulations. 
  We study the WZW model based on the centrally extended 2D de Sitter algebra. We obtain the spacetime metric and its explicitly conformally flat expression. The symmetries of the spacetime are found by identifying the Killing vectors with the group generators. The energy-momentum tensor obtained from the affine-Sugawara construction agrees with that from the more conventional approach. The exact center charge agrees to one-loop order with the one-loop beta function equations. We have also studied the representations of the corresponding enveloping Virasoro algebra. 
  A real space renormalization group technique, based on the hierarchical baby-universe structure of a typical dynamically triangulated manifold, is used to study scaling properties of 2d and 4d lattice quantum gravity. In 4d, the $\beta$-function is defined and calculated numerically. An evidence for the existence of an ultraviolet stable fixed point of the theory is presented 
  We search for $N=2$, $d=4$ theories which can be realized both as heterotic string compactifications on $K_{3}\times T^{2}$ and as type II string compactifications on Calabi-Yau threefolds. In such cases, the exact non-perturbative superpotential of one string theory is given in terms of tree level computations in the other string theory. In particular we find concrete examples which provide the stringy realization of the results of Seiberg and Witten on N=2 Yang-Mills theory, corrected by gravitational/stringy effects. We also discuss some examples which shed light on how the moduli spaces of different N=2 heterotic vacua are connected. 
  Multistring vertices and the overlap identities which they satisfy are exploited to understand properties of hyperbolic Kac Moody algebras, and $E_{10}$ in particular. Since any such algebra can be embedded in the larger Lie algebra of physical states of an associated completely compactified subcritical bosonic string, one can in principle determine the root spaces by analyzing which (positive norm) physical states decouple from the $N$-string vertex. Consequently, the Lie algebra of physical states decomposes into a direct sum of the hyperbolic algebra and the space of decoupled states. Both these spaces contain transversal and longitudinal states. Longitudinal decoupling holds generally, and may also be valid for uncompactified strings, with possible consequences for Liouville theory; the identification of the decoupled states simply amounts to finding the zeroes of certain ``decoupling polynomials''. This is not the case for transversal decoupling, which crucially depends on special properties of the root lattice, as we explicitly demonstrate for a non-trivial root space of $E_{10}$. Because the $N$-vertices of the compactified string contain the complete information about decoupling, all the properties of the hyperbolic algebra are encoded into them. In view of the integer grading of hyperbolic algebras such as $E_{10}$ by the level, these algebras can be interpreted as interacting strings moving on the respective group manifolds associated with the underlying finite-dimensional Lie algebras. 
  A general formula for the discrete states (Neveu-Schwarz sector) in $N=1$ $2D$ super-Liouville theory is written down in the world-sheet supersymmetric form. We then derive a set of gauge states at the discrete momenta. These discrete gauge states are shown to carry the $w_\infty$ charges and serve as the symmetry parameters in the old covariant quantization of the theory. 
  We construct exact wavefunctions of two vortices on a plane, a single vortex on the cylinder and a vortex on the torus. In each case, the physics is shown to be equivalent to a particle moving in a covering space, something simple to solve in those examples. We describe how our solutions fit into the general theory of quantum mechanics of $N$ particles on a two-dimensional space and attribute our success to the fact that the fundamental groups are Abelian in those simple cases that we are considering. 
  We review the correspondence between effective actions resulting from non-invariant Lagrangian densities, for Goldstone bosons arising from spontaneous breakdown of a symmetry group G to a subgroup H, and non-trivial generators of the de Rham cohomology of G/H. We summarize the construction of cohomology generators in terms of symmetric tensors with certain invariance and vanishing properties with respect to G and H. The resulting actions in four dimensions arise either from products of generators of lower degree such as the Goldstone-Wilczek current, or are of the Wess-Zumino-Witten type. Actions in three dimensions arise as Chern-Simons terms evaluated on composite gauge fields and may induce fractional spin on solitons. {Contribution to the Proceedings of STRINGS 95, held at University of Southern California, March 13 - 18, 1995.} 
  The spectrum of the anomalous dimensions of the composite operators (with arbitrary number of fields $n$ and derivatives $l$) in the scalar $\phi^4$ - theory in the first order of the $\epsilon$ -expansion is investigated. The exact solution for the operators with number of fields $\leq 4$ is presented. The behaviour of the anomalous dimensions in the large $l$ limit has been analyzed. It is given the qualitative description of the %structure of the spectrum for the arbitrary $n$. 
  The long-time large-distance behaviour of free decaying two dimensional turbulence is studied. Stochastic solutions of the Navier-Stokes equation are explicitly shown to follow renormalisation group trajectories.  It is proven that solutions of the Navier-Stokes equation asymptotically converge to fixed points which are conformal field theories. A particular fixed point is given by the free Gaussian field with a charge at infinity. The stream function is identified with a vertex operator. It happens that this solution also admits constant $n$-enstrophy fluxes in the asymptotic regime, therefore fulfilling all requirements to represent an asymptotic state of two-dimensional turbulence.  The renormalisation basin of attraction of this fixed point consists of a charged Coulomb gas. This Coulomb gas gives an effective description of turbulence. 
  I discuss the global structure of the strongly interacting gauge theory of quarks and gluons as a function of the quark masses and the CP violating parameter $\theta$. I concentrate on whether a first order phase transition occurs at $\theta=\pi.$ I show why this is expected when multiple flavors have a small degenerate mass. This transition can be removed by sufficient flavor-breaking. I speculate on the implications of this structure for Wilson's lattice fermions. 
  In a class of three-dimensional Abelian gauge theories with both light and heavy fermions, heavy chiral fermions can trigger dynamical generation of a magnetic field, leading to the spontaneous breaking of the Lorentz invaiance. Finite masses of light fermions tend to restore the Lorentz invariance. 
  Following a remark advanced by Feynman,we study the connection between the form of the nonlinear vertices involving gauge particles and the Abelian gauge invariance of physical tree amplitudes. We show that this requirement, together with some natural assumptions, fixes uniquely the structure of the Yang-Mills theory. However, the constraints imposed by the above property are not sufficient to single out the gauge theory of gravitation. 
  We study a non-Abelian Chern-Simons gauge theory in $ 2+ 1$ dimensions with the inclusion of an anomalous magnetic interaction. For a particular relation between the Chern-Simons (CS) mass and the anomalous magnetic coupling the equations for the gauge fields reduce from second- to first order differential equations of the pure CS type. We derive the Bogomol'nyi-type or self-dual equations for a $\bphi^2$ scalar potential, when the scalar and topological masses are equal. The corresponding vortex solutions carry magnetic flux that is not quantized due to the non-toplogical nature of the solitons. However, as a consequence of the quantization of the CS term, both the electric charge and angular momentum are quantized. 
  The gravitini zero modes riding on top of the extreme Reissner-Nordstrom black-hole solution of N=2 supergravity are shown to be normalizable. The gravitini and dilatini zero modes of axion-dilaton extreme black-hole solutions of N=4 supergravity are also given and found to have finite norms. These norms are duality invariant. The finiteness and positivity of the norms in both cases are found to be correlated with the Witten-Israel-Nester construction; however, we have replaced the Witten condition by the pure-spin-3/2 constraint on the gravitini. We compare our calculation of the norms with the calculations which provide the moduli space metric for extreme black holes.     The action of the N=2 hypermultiplet with an off-shell central charge describes the solitons of N=2 supergravity. This action, in the Majumdar-Papapetrou multi-black-hole background, is shown to be N=2 rigidly supersymmetric. 
  A model-independent formulation of anyons as spinning particles is presented. The general properties of the classical theory of (2+1)-dimensional relativistic fractional spin particles and some properties of their quantum theory are investigated. The relationship between all the known approaches to anyons as spinning particles is established. Some widespread misleading notions on the general properties of (2+1)-dimensional anyons are removed. 
  The basic concepts underlying our analysis of {\it W-algebras} as extended symmetries of integrable systems are summarized. The construction starts from the second hamiltonian structure of ``Generalized Drinfel'd-Sokolov'' hierarchies, and its correspondence with the $A_1$-embeddings is established, providing a rather simple and general scheme. 
  We explicitly construct the metric and torsion couplings of two-dimensional (4,0)-super\-sym\-metric sigma models with target space a four-manifold that are invariant under a $U(1)$ symmetry generated by a tri-holomorphic Killing vector field that leaves in addition the torsion invariant. We show that the metric couplings arise from magnetic monopoles on the three-sphere which is the space of orbits of the group action generated by the tri-holomorphic Killing vector field on the sigma model target manifold. We also examine the global structure of a subclass of these metrics that are in addition $SO(3)$-invariant and find that the only non-singular one, for models with non-zero torsion, is that of $SU(2)\times U(1)$ WZW model. 
  For a large class of $N=2$ SCFTs, which includes minimal models and many $\s$ models on Calabi-Yau manifolds, the mirror theory can be obtained as an orbifold. We show that in such a situation the construction of the mirror can be extended to the presence of discrete torsions. In the case of the $\ZZ_2\ex\ZZ_2$ torus orbifold, discrete torsion between the two generators directly provides the mirror model. Working at the Gepner point it is, however, possible to understand this mirror pair as a special case of the Berglund--H"ubsch construction. This seems to indicate that the $\ZZ_2\ex\ZZ_2$ example is a mere coincidence, due to special properties of $\ZZ_2$ twists, rather than a hint at a new mechanism for mirror symmetry. 
  We study the entanglement entropy arising from coherent states and one--particle states. We show that it is possible to define a finite entanglement entropy by subtracting the vacuum entropy from that of the considered states, when the unobserved region is the same. 
  Based on the dispersionless KP (dKP) theory, we give a comprehensive study of the topological Landau-Ginzburg (LG) theory characterized by a rational potential. Writing the dKP hierarchy in a general form, we find that the hierarchy naturally includes the dispersionless (continuous) limit of Toda hierarchy and its generalizations having finite number of primaries. Several flat solutions of the topological LG theory are obtained in this formulation, and are identified with those discussed by Dubrovin. We explicitly construct gravitational descendants for all the primary fields. Giving a residue formula for the 3-point functions of the fields, we show that these 3-point functions satisfy the topological recursion relation. The string equation is obtained as the generalized hodograph solutions of the dKP hierarchy, which show that all the gravitational effects to the constitutive equations (2-point functions) can be renormalized into the coupling constants in the small phase space. 
  We discuss R-symmetries in locally supersymmetric N=2 gauge theories coupled to hypermultiplets which can be thought of as effective theories of heterotic superstring models. In this type of supergravities a suitable R-symmetry exists and can be used to topologically twist the theory: the vector multiplet containing the dilaton-axion field has different R-charge assignments with respect to the other vector multiplets. Correspondingly a system of coupled instanton equations emerges, mixing gravitational and Yang--Mills instantons with triholomorphic hyperinstantons and axion-instantons. For the tree-level classical special manifolds $ST(n)=SU(1,1)/U(1)\times SO(2,n)/(SO(2)$ $\times SO(n))$ R-symmetry with the specified properties is a continuous symmetry, but for the quantum corrected manifolds ${\hat {ST}}(n)$ a discrete R--group of electric--magnetic duality rotations is sufficient and we argue that it exists. 
  We examine the reflection matrix acting on the $SO(N)$ vector multiplet and fuse it to obtain that acting on the rank two particle multiplet, and give its decomposition. 
  We review developments in the world-sheet action formulation of the generic irrational conformal field theory, including the non-linear and the linearized forms of the action. These systems form a large class of spin-two gauged WZW actions which exhibit exotic gravitational couplings. Integrating out the gravitational field, we also speculate on a connection with sigma models. 
  Using DLCQ as a nonperturbative method, we test Fock-space truncations in ${\rm QCD}_{1+1}$ by studying the mass spectra of hadrons in colour SU(2) and SU(3) at finite harmonic resolution $K$. We include $q\bar q q\bar q$ states for mesons and up to $qqq q\bar q$ states for baryons. With this truncation, we give `predictions' for the masses of the first five states where finite $K$ effects are minimal. 
  We derive a Kontsevich-type matrix model for the c=1 string directly from the W-infinity solution of the theory. The model that we obtain is different from previous proposals, which are proven to be incorrect. Our matrix model contains the Penner and Kontsevich cases, and we study its quantum effective action. The simplicity of our model leads to an encouraging interpretation in the context of background-independent noncritical string field theory. 
  We present a new form of Quantum Electrodynamics where the photons are composites made out of zero-dimensional scalar ``primitives''. The r\^{o}le of the local gauge symmetry is taken over by an {\em infinite-dimensional global Noether symmetry} -- the group of volume-preserving (symplectic) diffeomorphisms of the target space of the scalar primitives. Similar construction is carried out for higher antisymmetric tensor gauge theories. Solutions of Maxwell's equations are automatically solutions of the new system. However, the latter possesses additional non-Maxwell solutions which display some interesting new effects: (a) a magneto-hydrodynamical analogy, (b) absence of electromagnetic self-energy for electron plane wave solutions, and (c) gauge invariant photon mass generation, where the generated mass is arbitrary. 
  We consider analogs of Yang-Mills theories for non-semisimple real Lie algebras which admit invariant non-degenerate metrics. These 4-dimensional theories have many similarities with corresponding WZW models in 2 dimensions and Chern-Simons theories in 3 dimensions. In particular, the quantum effective action contains only 1-loop term with the divergent part that can be eliminated by a field redefinition. The on-shell scattering amplitudes are thus finite (scale invariant). This is a consequence of the presence of a null direction in the field space metric: one of the field components is a Lagrange multiplier which `freezes out' quantum fluctuations of the `conjugate' field. The non-positivity of the metric implies that these theories are apparently non-unitary. However, the special structure of interaction terms (degenerate compared to non-compact YM theories) suggests that there may exist a unitary `truncation'. We discuss in detail the simplest theory based on 4-dimensional algebra E^c_2. The quantum part of its effective action is expressed in terms of 1-loop effective action of SU(2) gauge theory. The E^c_2 model can be also described as a special limit of SU(2) x U(1) YM theory with decoupled ghost-like U(1) field. 
  For any given algebra of local observables in relativistic quantum field theory there exists an associated scaling algebra which permits one to introduce renormalization group transformations and to construct the scaling (short distance) limit of the theory. On the basis of this result it is discussed how the phase space properties of a theory determine the structure of its scaling limit. Bounds on the number of local degrees of freedom appearing in the scaling limit are given which allow one to distinguish between theories with classical and quantum scaling limits. The results can also be used to establish physically significant algebraic properties of the scaling limit theories, such as the split property. 
  O(N) vector sigma models possessing catastrophes in their action are studied. Coupling the limit N --> infinity with an appropriate scaling behaviour of the coupling constants, the partition function develops a singular factor. This is a generalized Airy function in the case of spacetime dimension zero and the partition function of a scalar field theory for positive spacetime dimension. 
  Quantum theory of dilaton gravity coupled to a nonlinear sigma model with a maximally symmetric target space is studied in $2+\epsilon$ dimensions. The ultraviolet stable fixed point for the curvature of the nonlinear sigma model demands a new fixed point theory for the dilaton coupling function. The fixed point of the dilaton coupling is a saddle point similarly to the previous case of the flat target space. 
  We give a systematic classification and a detailed discussion of the structure, motion and scattering of the recently discovered negaton and positon solutions of the Korteweg-de Vries equation. There are two distinct types of negaton solutions which we label $[S^{n}]$ and $[C^{n}]$, where $(n+1)$ is the order of the Wronskian used in the derivation. For negatons, the number of singularities and zeros is finite and they show very interesting time dependence. The general motion is in the positive $x$ direction, except for certain negatons which exhibit one oscillation around the origin. In contrast, there is just one type of positon solution, which we label $[\tilde C^n]$. For positons, one gets a finite number of singularities for $n$ odd, but an infinite number for even values of $n$. The general motion of positons is in the negative $x$ direction with periodic oscillations. Negatons and positons retain their identities in a scattering process and their phase shifts are discussed. We obtain a simple explanation of all phase shifts by generalizing the notions of ``mass" and ``center of mass" to singular solutions. Finally, it is shown that negaton and positon solutions of the KdV equation can be used to obtain corresponding new solutions of the modified KdV equation. 
  It is known that there is a proportionality factor relating the $\beta$-function and the equations of motion viz. the Zamolodchikov metric. Usually this factor has to be obtained by other methods. The proper time equation, on the other hand, is the full equation of motion. We explain the reasons for this and illustrate it by calculating corrections to Maxwell's equation. The corrections are calculated to cubic order in the field strength, but are exact to all orders in derivatives. We also test the gauge covariance of the proper time method by calculating higher (covariant) derivative corrections to the Yang-Mills equation. 
  For $N\!=\!2$ SUSY theories with non-vanishing $\beta$-function and one-dimensional quantum moduli, we study the representation on the special coordinates of the group of motions on the quantum moduli defined by $\Gamma_W\!=\!Sl(2;Z)\!/\!\Gamma_M$, with $\Gamma_M$ the quantum monodromy group. $\Gamma_W$ contains both the global symmetries and the strong-weak coupling duality. The action of $\Gamma_W$ on the special coordinates is not part of the symplectic group $Sl(2;Z)$. After coupling to gravity, namely in the context of non-rigid special geometry, we can define the action of $\Gamma_W$ as part of $Sp(4;Z)$. To do this requires singular gauge transformations on the "scalar" component of the graviphoton field. In terms of these singular gauge transformations the topological obstruction to strong-weak duality can be interpreted as a $\sigma$-model anomaly, indicating the possible dynamical role of the dilaton field in $S$-duality. 
  We investigate the gauge-independent Hamiltonian formulation and the anomalous Ward identities of a matter-induced 1+1-dimensional gravity theory invariant under Weyl transformations and area-preserving diffeomorphisms, and compare the results to the ones for the conventional diffeomorphism-invariant theory. We find that, in spite of several technical differences encountered in the analysis, the two theories are essentially equivalent. 
  The spontaneous symmetry breaking associated to the tearing of a random surface, where large dynamical holes fill the surface, was recently analized obtaining a non-universal critical exponent on a border phase. Here the issue of universality is explained by an independent analysis. The one hole sector of the model is useful to manifest the origin of the (limited) non-universal behaviour, that is the existence of two inequivalent critical points. 
  We present one loop boundary reflection matrix for $d_4^{(1)}$ Toda field theory defined on a half line with the Neumann boundary condition. This result demonstrates a nontrivial cancellation of non-meromorphic terms which are present when the model has a particle spectrum with more than one mass. Using this result, we determine uniquely the exact boundary reflection matrix which turns out to be \lq non-minimal' if we assume the strong-weak coupling \lq duality'. 
  We consider the semiclassical dynamics of CGHS black holes with a Weyl-invariant effective action for conformal matter. The trace anomaly of Polyakov effective action is converted into the Virasoro anomaly thus leading to the same flux of Hawking radiation. The covariance of semiclassical equations can be restored through a non-local redefinition of the metric-dilaton fields. The resulting theory turns out to be equivalent to the RST model. This provides a mechanism to solve semiclassical equations of 2D dilaton gravity coupled to conformal matter for classically soluble models. 
  We analyze the topological nature of $c=1$ string theory at the self--dual radius. We find that it admits two distinct topological field theory structures characterized by two different puncture operators. We show it first in the unperturbed theory in which the only parameter is the cosmological constant, then in the presence of any infinitesimal tachyonic perturbation. We also discuss in detail a Landau--Ginzburg representation of one of the two topological field theory structures. 
  We present a quasiperiodic self-dual metric of the Gibbons--Hawking type with one gravitational instanton per spacetime cell. The solution, based on an adaptation of Weierstrassian $\zeta$ and $\sigma$ functions to three dimensions, conforms to a definition of spacetime foam given by Hawking. 
  We present a manifestly $N=2$ supersymmetric formulation of $N=2$ super-$W_3^{(2)}$ algebra (its classical version) in terms of the spin 1 unconstrained supercurrent generating a $N=2$ superconformal subalgebra and the spins 1/2, 2 bosonic and spins 1/2, 2 fermionic constrained supercurrents. We consider a superfield reduction of $N=2$ super-$W_3^{(2)}$ to $N=2$ super-$W_3$ and construct a family of evolution equations for which $N=2$ super-$W_3^{(2)}$ provides the second hamiltonian structure. 
  The quantized form of the soft N=8 superconformal algebra is investigated. Its operator product expansions are shown to exhibit a one-parameter-class of (soft) anomalies, which may be arbitrarily shifted by certain suitable quantum corrections of the generators. In particular, the BRST operator can be constructed and made nilpotent in the quantum version of all known realizations of the algebra. This generalizes the results of Cederwall and Preitschopf, who studied the $\widehat{S^7}$-algebra, that is contained as a soft Kac-Moody part in the superconformal algebra. A Fock-space representation is given, that has to be somewhat unusual in certain modes. 
  The Hamiltonian dynamics of \(2 + 1\) dimensional Yang-Mills theory with gauge group SU(2) is reformulated in gauge invariant, geometric variables, as in earlier work on the \(3 + 1\) dimensional case. Physical states in electric field representation have the product form \(\Psi_{\mathrm{phys}} [E^{a i}] = \exp ( i \Omega [ E ] / g ) F [G_{ij}]\), where the phase factor is a simple local functional required to satisfy the Gauss law constraint, and \(G_{ij}\) is a dynamical metric tensor which is bilinear in \(E^{a k}\). The Hamiltonian acting on \(F [ G_{ij} ]\) is local, but the energy density is infinite for degenerate configurations where \(\det G (x)\) vanishes at points in space, so wave functionals must be specially constrained to avoid infinite total energy. Study of this situation leads to the further factorization \(F [G_{ij} ] = F_c [ G_{ij} ] \mathcal R [ G_{ij} ]\), and the product \(\Psi_c [E] \equiv \exp (i \Omega [ E ] / g ) F_c [G_{ij}]\) is shown to be the wave functional of a topological field theory. Further information from topological field theory may illuminate the question of the behavior of physical gauge theory wave functionals for degenerate fields. 
  We consider a class of conformal models describing closed strings in axially symmetric stationary magnetic flux tube backgrounds. These models are closed string analogs of the Landau model of a particle in a magnetic field or the model of an open string in a constant magnetic field. They are interesting examples of solvable unitary conformal string theories with non-trivial 4-dimensional curved space-time interpretation. In particular, their quantum Hamiltonian can be expressed in terms of free fields and the physical spectrum and string partition function can be explicitly determined. In addition to the presence of tachyonic instabilities and existence of critical values of magnetic field the closed string spectrum exhibits also some novel features which were absent in the open string case. (Contribution to the Proceedings of "Strings 95" Conference) 
  It is known that in the 2+1 dimensional quantum electrodynamics with Chern-Simons term, spontaneous magnetic field induces Lorentz symmetry breaking. In this paper, thermodynamical characters, especially the phase structure of this model are discussed. To see the behavior of the spontaneous magnetic field at finite temperature, the effective potential in the finite temperature system is calculated within the weak field approximation and in the fermion massless limit. We found that the spontaneous magnetic field never vanishes at any finite temperature. This result doesn't change even when the chemical potential is introduced. We also investigate the consistency condition and the probability that fermion stays in a lowest Landau level at finite temperature. 
  The nature of duality symmetries is explored in closed bosonic string theory, particularly in the case of a four-dimensional target space admitting a one-parameter isometry. It appears that the S-duality of string theory behaves analogously to the Ehlers' symmetry of General Relativity. Furthermore, it is demonstrated that the O(1,1) target space duality arising from the isometry interchanges the roles of these two symmetries. The inclusion of the tachyon field is shown to be consistent with T-duality but incompatible with S-duality. Finally, extrapolating to dimensions other than four, the effective action is found to be invariant under a larger group of symmetries than the usual O(1,1). 
  In this paper we consider Hamiltonian systems on the quantum plane and we show that the set of Q-meromorphic Hamiltonians is a Virasoro algebra with central charge zero and the set of Hamiltonian derivations of the algebra of $Q$-analytic functions ${\cal A}_q$ with values in the algebra of $Q$-meromorphic functions ${\cal M}_q$ is the Lie algebra $sl(2,A_1(q)).$ Moreover we will show that any motion on a quantum space is associated with a quadratic Hamiltonian. 
  We study the realization of the (super) conformal topological symmetry in two-dimensional field theories. The mirror automorphism of the topological algebra is represented as a reflection in the space of fields. As a consequence, a double BRST structure for topological matter theories is found. It is shown that the implementation of the topological symmetry in non-critical (super)string theories depends on the matter content of the two realizations connected by the mirror transformation. 
  We elaborate on our previous work on N=2 supersymmetric Yang-Mills theory. In particular, we show how to explicitly determine the low energy quantum effective action for $G=SU(3)$ from the underlying hyperelliptic Riemann surface, and calculate the leading instanton corrections. This is done by solving Picard-Fuchs equations and asymptotically evaluating period integrals. We find that the dynamics of the $SU(3)$ theory is governed by an Appell system of type $F_4$, and compute the exact quantum gauge coupling explicitly in terms of Appell functions. 
  Analysing the static, spherically symmetric graviton-dilaton solutions in low energy string and Brans-Dicke theory, we find the following. For a charge neutral point star, these theories cannot predict non trivial PPN parameters, $\beta$ and $\gamma$, without introducing naked singularities. We then couple a cosmological constant $\Lambda$ as in low energy string theory. We find that only in low energy string theory, a non zero $\Lambda$ leads to a curvature singularity, which is much worse than a naked singularity. Requiring its absence upto a distance $r_*$ implies a bound $| \Lambda | < 10^{- 102} (\frac{r_*}{{\rm pc}})^{- 2}$ in natural units. If $r_* \simeq 1 {\rm Mpc}$ then $| \Lambda | < 10^{- 114}$ and, if $r_* \simeq 10^{28} {\rm cm}$ then $| \Lambda | < 10^{- 122}$ in natural units. 
  Even the uninitiated will know that Quantum Field Theory cannot be introduced systematically in just four lectures. I try to give a reasonably connected outline of part of it, from second quantization to the path-integral technique in Euclidean space, where there is an immediate connection with the rules for Feynman diagrams and the partition function of Statistical Mechanics. 
  We construct an integral representation for the momentum space Green's function for a Neutron in interaction with a straight current carrying wire. 
  We calculate the partition function of a harmonic oscillator with quasi-periodic boundary conditions using the zeta-function method. This work generalizes a previous one by Gibbons and contains the usual bosonic and fermionic oscillators as particular cases. We give an alternative prescription for the analytic extension of the generalized Epstein function involved in the calculation of the generalized thermal zeta-functions. We also conjecture about the relation of our calculation to anyonic systems. 
  We consider lattice analogues of some conformal theories, including WZW and Toda models. We describe discrete versions of Drinfeld-Sokolov reduction and Sugawara construction for the WZW model. We formulate perturbation theory in chiral sector. We describe the spaces of Integrals of Motion in the perturbed theories. We interpret the perturbed WZW model in terms of NLS hierarchy and obtain of this model into the lattice KP-hierarchy. 
  Requiring that a Lagrangian path integral leads to certain identities (Ward identities in a broad sense) can be formulated in a general BRST language, if necessary by the use of collective fields. The condition of BRST symmetry can then be expressed with the help of the antibracket, and suitable generalizations thereof. In particular, a new Grassmann-odd bracket, which reduces to the conventional antibracket in a special limit, naturally appears. We illustrate the formalism with various examples. 
  We present a direct approach for the calculation of functional determinants of the Laplace operator on balls. Dirichlet and Robin boundary conditions are considered. Using this approach, formulas for any value of the dimension, $D$, of the ball, can be obtained quite easily. Explicit results are presented here for dimensions $D=2,3,4,5$ and $6$. 
  This is a short review on strings in curved spacetimes. We start by recalling the classical and quantum string behaviour in singular plane waves backgrounds. We then report on the string behaviour in cosmological spacetimes (FRW, de Sitter, power inflation) which is by now largerly understood. Recent progress on self-consistent solutions to the Einstein equations for string dominated universes is reviewed. The energy-momentum tensor for a gas of strings is considered as source of the spacetime geometry. The string equation of state is determined from the behaviour of the explicit string solutions. This yields a self-consistent cosmological solution exhibiting realistic matter dominated behaviour $ R \sim (T)^{2/3}\; $ for large times and radiation dominated behaviour $ R \sim (T)^{1/2}\; $ for early times. Inflation in the string theory context is discussed. 
  We develop a model of $(1+1)$-dimensional parent and baby universes as macroscopic and microscopic fundamental closed strings. We argue, on the basis of understanding of strings from the point of view of target $D$-dimensional space-time, that processes involving baby universes/wormholes not only induce $c$-number "$\alpha$-parameters" in $(1+1)d$ action, but also lead to loss of quantum coherence for a $(1+1)d$ observer in the parent universe. 
  In this paper we present a model of confinement based on an analogy with the confinement mechanism of the Schwarzschild solution of general relativity. Using recently discovered exact, Schwarzschild-like solutions of the SU(2) Yang-Mills-Higgs equations we study the behaviour of a scalar, SU(2) charged test particle placed in the gauge fields of this solution. We find that this test particle is indeed confined inside the color event horizon of our solution. Additionally it is found that this system is a composite fermion even though there are no fundamental fermions in the original Lagrangian. 
  We revisit the quantization of matter-coupled, two-dimensional dilaton gravity. At the classical level and with a cosmological term, a series of field transformations leads to a set of free fields of indefinite signature. Without matter the system is represented by two scalar fields of opposite signature. With a particular quantization for the scalar with negative kinetic energy, the system has zero central charge and we find some physical states satisfying {\it all} the Virasoro conditions. With matter, the constraints cannot be solved because of the Virasoro anomaly. We discuss two avenues for consistent quantization: modification of the constraints, and BRST quantization. The first avenue appears to lead to very few physical states. The second, which roughly corresponds to satisfying half of the Virasoro conditions, results in a rich spectrum of physical states. This spectrum, however, differs significantly from that of free matter fields propagating on flat two-dimensional space-time. 
  We propose and give strong evidence for a duality relating Type II theories on Calabi-Yau spaces and heterotic strings on $K3 \times T^2$, both of which have $N=2$ spacetime supersymmetry. Entries in the dictionary relating the dual theories are derived from an analysis of the soliton string worldsheet in the context of $N=2$ orbifolds of dual $N=4$ compactifications of Type II and heterotic strings. In particular we construct a pairing between Type II string theory on a self-mirror Calabi-Yau space $X$ with $h^{11}= h^{21}= 11$ and a $(4,0)$ background of heterotic string theory on $K3\times T^2$. Under the duality transformation the usual first-quantized mirror symmetry of $X$ becomes a second-quantized mirror symmetry which determines nonperturbative quantum effects. This enables us to compute the exact quantum moduli space. Mirror symmetry of $X$ implies that the low-energy $N=2$ gauge theory is finite, even at enhanced symmetry points. This prediction is verified by direct computation on the heterotic side. Other branches of the moduli space, and corresponding dual pairs which are not finite $N=2$ theories, are connected to this one via black hole condensation. 
  Correlation functions can be calculated on Riemann surfaces using the operator formalism. The state in the Hilbert space of the free field theory on the punctured disc, corresponding to the Riemann surface, is constructed at infinite genus, verifying the inclusion of these surfaces in the Grassmannian. In particular, a subset of the class of $O_{HD}$ surfaces can be identified with a subset of the Grassmannian. The concept of flux through the ideal boundary is used to study the connection between infinite-genus surface and the domain of string perturbation theory. The different roles of effectively closed surfaces with Dirichlet boundaries in a more complete formulation of string theory are identified. 
  We consider the gauge neutral matter in the low--energy effective action for string theory compactification on a \cym\ with $(2,2)$ world--sheet supersymmetry. At the classical level these states (the \sing's of $E_6$) correspond to the cohomology group $H^1(\M,{\rm End}\>T)$. We examine the first order contribution of instantons to the mass matrix of these particles. In principle, these corrections depend on the \K\ parameters $t_i$ through factors of the form $e^{2\p i t_i}$ and also depend on the complex structure parameters. For simplicity we consider in greatest detail the quintic threefold $\cp4[5]$. It follows on general grounds that the total mass is often, and perhaps always, zero. The contribution of individual instantons is however nonzero and the contribution of a given instanton may develop poles associated with instantons coalescing for certain values of the complex structure. This can happen when the underlying \cym\ is smooth. Hence these poles must cancel between the coalescing instantons in order that the superpotential be finite. We examine also the \Y\ couplings involving neutral matter \ysing\ and neutral and charged fields \ymix, which have been little investigated even though they are of phenomenological interest. We study the general conditions under which these couplings vanish classically. We also calculate the first--order world--sheet instanton correction to these couplings and argue that these also vanish. 
  A form of the constraints, specifying a $D$-dimensional manifold embedded in $D+1$ dimensional Euclidean space, is discussed in the path integral formula given by a time discretization. Although the mid-point prescription is privileged in the sphere $S^D$ case, it is more involved in generic cases. An interpretation on the validity of the formula is put in terms of the operator formalism. Operators from this path integral formula are also discussed. 
  On the basis of the classical theory of envelopes, we formulate the renormalization group (RG) method for global analysis, recently proposed by Goldenfeld et al. It is clarified why the RG equation improves things. 
  We look for a graviton-dilaton theory which can predict non trivial values of the PPN parameters $\beta$ and/or $\gamma$ for a charge neutral point star, without any naked singularity. With the potential for dilaton $\phi$ set to zero, it contains one arbitrary function $\psi(\phi)$. Our requirements impose certain constraints on $\psi$, which lead to the following generic and model independent novel results: For a charge neutral point star, the gravitational force becomes repulsive at distances of the order of, but greater than, the Schwarzschild radius $r_0$. There is also no horizon for $r > r_0$. These results suggest that black holes are unlikely to form in a stellar collapse in this theory. 
  The massive multi-flavor Schwinger model on a circle is reduced to a finite dimensional quantum mechanics problem. The model sensitively depends on the parameter $mL|\cos\onehalf\theta|$ where $m$, $L$, and $\theta$ are a typical fermion mass, the volume, and the vacuum angle, respectively. 
  A new interpretation of the causality implementation in the Lienard-Wiechert solution raises new doubts against the validity of the Lorentz-Dirac equation and the limits of validity of the Minkowski structure of spacetime. 
  Many two-dimensional classical field theories have hidden symmetries that form an infinite-dimensional algebra. For those examples that correspond to effective descriptions of compactified superstring theories, the duality group is expected to be a large discrete subgroup of the hidden symmetry group. With this motivation, we explore the hidden symmetries of principal chiral models and symmetric space models. 
  We study in detail gaugino condensation in globally and locally supersymmetric Yang-Mills theories. We focus on models for which gauge-neutral matter couples to the gauge bosons only through nonminimal gauge kinetic terms, for the cases of one and several condensing gauge groups. Using only symmetry arguments, the low-energy expansion, and general properties of supersymmetry, we compute the low energy Wilson action, as well as the (2PI) effective action for the composite {\it classical} superfield $U\equiv\langle \Tr\WW \rangle$, with $W_\alpha$ the supersymmetric gauge field strength. The 2PI effective action provides a firmer foundation for the approach of Veneziano and Yankielowicz, who treated the composite superfield, $U$, as a quantum degree of freedom. We show how to rederive the Wilson action by minimizing the 2PI action with respect to $U$. We determine, in both formulations and for global and local supersymmetry, the effective superpotential, $W$, the non-perturbative contributions to the low-energy K\"ahler potential $K$, and the leading higher supercovariant derivative terms in an expansion in inverse powers of the condensation scale. As an application of our results we include the string moduli dependence of the super- and K\"ahler potentials for simple orbifold models. 
  We discuss the definition of the average effective action in terms of the heat-kernel. As an example we examine a model describing a self-interacting scalar field, both in flat and curved background, and study the renormalization group flow of some of the parameters characterizing its effective potential. Some implications of the running of these parameters for inflationary cosmology are also briefly discussed. 
  We analyse in detail the local BRST cohomology in Einstein-Yang-Mills theory using the antifield formalism. We do not restrict the Lagrangian to be the sum of the standard Hilbert and Yang-Mills Lagrangians, but allow for more general diffeomorphism and gauge invariant actions. The analysis is carried out in all spacetime dimensions larger than 2 and for all ghost numbers. This covers the classification of all candidate anomalies, of all consistent deformations of the action, as well as the computation of the (equivariant) characteristic cohomology, i.e. the cohomology of the spacetime exterior derivative in the space of (gauge invariant) local differential forms modulo forms that vanish on-shell. We show in particular that for a semi-simple Yang-Mills gauge group the antifield dependence can be entirely removed both from the consistent deformations of the Lagrangian and from the candidate anomalies. Thus, the allowed deformations of the action necessarily preserve the gauge structure, while the only candidate anomalies are those provided by previous works not dealing with antifields, and by ``topological" candidate anomalies which are present only in special spacetime dimensions (6,9,10,13,...). This result no longer holds in presence of abelian factors where new candidate anomalies and deformations of the action can be constructed out of the conserved Noether currents (if any). The Noether currents themselves are shown to be covariantizable, with a few exceptions discussed as well. 
  We propose a new BRST operator for the B-twist of $N=2$ Landau-Ginzburg (LG) models. It solves the problem of the fractional ghost numbers of Vafa's old BRST operator and shows how the model is obtained by gauge fixing a zero action. An essential role is played by the anti-BRST operator,which is given by one of the supersymmetries of the $N=2$ algebra. Its presence is needed in proving that the model is indeed a topological field theory. The space of physical observables, defined by taking the anti-BRST cohomology in the BRST cohomology groups, is unchanged. 
  We describe the self-interacting scalar field on the truncated sphere and we perform the quantization using the functional (path) integral approach. The theory posseses a full symmetry with respect to the isometries of the sphere. We explicitely show that the model is finite and the UV-regularization automatically takes place. 
  A model in which the massive dilaton is introduced by minimally extending the two dimensional topological gravity is studied semi-classically. The theory is no longer topological because of the explicit Weyl scale symmetry breaking. Due to the dilaton the semiclassical stress-energy tensor gets renormalized and it is shown how the gravitational background coupled to the the dilaton depends on the dilaton mass as well as the renormalization mass scale, but not on the Newton's constant. 
  We study N=2(4) superstring backgrounds which are four-dimensional non-\Kahlerian with non-trivial dilaton and torsion fields. In particular we consider the case that the backgrounds possess at least one $U(1)$ isometry and are characterized by the continual Toda equation and the Laplace equation. We obtain a string background associated with a non-trivial solution of the continual Toda equation, which is mapped, under the T-duality transformation, to the hyper-\Kahler Taub-NUT instanton background. It is shown that the integrable property of the non-\Kahlerian spaces have the direct origin in the real heavens: real, self-dual, euclidean, Einstein spaces. The Laplace equation and the continual Toda equation imposed on quasi-\Kahler geometry for consistent string propagation are related to the self-duality conditions of the real heavens with ``translational'' and ``rotational''Killing symmetry respectively. 
  The classical spin model in planar condensed media is represented as the U(1) Chern-Simons gauge field theory. When the vorticity of the continuous flow of the media coincides with the statistical magnetic field, which is necessary for the model's integrability, the theory admits zero curvature connection. This allows me to formulate the model in terms of gauge - invariant fields whose evolution is described by the Davey-Stewartson (DS) equations. The Self-dual Chern-Simons solitons described by the Liouville equation are subjected to corresponding integrable dynamics. As a by-product the 2+1-dimensional zero-curvature representation for the DS equation is obtained as well as the new reduction conditions related to the DS-I case. Some possible applications for the statistical transmutation in the anyon superfluid and TQFT are briefly discussed. 
  A new type of semigroups which appears while dealing with $N=1$ superconformal symmetry in superstring theories is considered. The ideal series having unusual abstract properties is constructed. Various idealisers are introduced and studied. The ideal quasicharacter is defined. Green's relations are found and their connection with the ideal quasicharacter is established. 
  Equations of associativity in two-dimensional topological field theory (they are known also as the Witten-Dijkgraaf-H.Verlinde-E.Verlinde (WDVV) system) are represented as an example of the general theory of integrable Hamiltonian nondiagonalizable (i.e., do not possessing Riemann invariants) systems of hydrodynamic type. A corresponding local nondegenerate Hamiltonian structure of hydrodynamic type (Poisson bracket of Dubrovin-Novikov type) is found. For n=3 the equations of associativity are reduced to the integrable three wave system by a chain of explicit transformations. Any solution of the integrable three wave system generates solutions of the equations of associativity. Explicit B\"acklund type transformations connecting solutions of different equations of associativity are found. 
  We give the superdiffeomorphisms transformations of the four-dimensional topological Yang-Mills theory in curved manifold and we discuss the ultraviolet renormalization of the model. The explicit expression of the most general counterterm is given. 
  The leading long-distance quantum correction to the Newtonian potential for heavy spinless particles is computed in quantum gravity. The potential is obtained directly from the sum of all graviton exchange diagrams contributing to lowest non-trivial order to the scattering amplitude. The calculation correctly reproduces the leading classical relativistic post-Newtonian correction. The sign of the perturbative quantum correction would indicate that, in the absence of a cosmological constant, quantum effects lead to a slow increase of the gravitational coupling with distance. 
  Using the N=4 topological reformulation of N=2 strings, we compute all loop partition function for special compactifications of N=2 strings as a function of target moduli. We also reinterpret N=4 topological amplitudes in terms of slightly modified N=2 topological amplitudes. We present some preliminary evidence for the conjecture that N=2 strings is the large N limit of Holomorphic Yang-Mills in 4 dimensions. 
  We present two different families of stationary black strings in three dimensions carrying electric and axion charges. Both solutions contain a singular region, which however is quite harmless for string frame geodesic observers, because the solutions are causally geodesically complete. In addition, we exhibit a variety of extremal limits of our solutions, and argue that there must exist a more general family continuously interpolating between them. 
  It is shown that nonvacuum pseudoparticles can account for quantum tunneling and metastability. In particular the saddle-point nature of the pseudoparticles is demonstrated, and the evaluation of path-integrals in their neighbourhood. Finally the relation between instantons and bounces is used to derive a result conjectured by Bogomolny and Fateyev. 
  U(1) gauge theory on ${\bf R}^4$ is known to possess an electric-magnetic duality symmetry that inverts the coupling constant and extends to an action of $SL(2,{\bf Z})$. In this paper, the duality is studied on a general four-manifold and it is shown that the partition function is not a modular-invariant function but transforms as a modular form. This result plays an essential role in determining a new low-energy interaction that arises when N=2 supersymmetric Yang-Mills theory is formulated on a four-manifold; the determination of this interaction gives a new test of the solution of the model and would enter in computations of the Donaldson invariants of four-manifolds with $b_2^+\leq 1$. Certain other aspects of abelian duality, relevant to matters such as the dependence of Donaldson invariants on the second Stieffel-Whitney class, are also analyzed. 
  We give an argument showing that in N=4 supersymmetric gauge theories there exists at least one bound state saturating the Bogomol'nyi bound with electric charge $p$ and magnetic charge $q$, for each $p$ and $q$ relatively prime, and we comment on the uniqueness of such state. This result is a necessary condition for the existence of an exact S-duality in N=4 supersymmetric theories. 
  A new set of gauge invariant variables is defined to describe the physical Hilbert space of $d = 3 + 1$ $SU(2)$ Yang-Mills theory in the fixed-time canonical formalism. A natural geometric interpretation arises due to the $GL(3)$ covariance found to hold for the basic equations and commutators of the theory in the canonical formalism. We emphasize, however, that we are not interested in and do not consider the coupling of the theory to gravity. We concentrate here on a technical difficulty of this approach, the calculation of the electric field energy. This in turn hinges on the well-definedness of the transformation of variables, an issue which is settled through degenerate perturbation theory arguments. 
  The Feynman-Garrod path integral representation for time evolution is extended to arbitrary one-parameter continuous canonical transformations. One thereupon obtains a generalized Kerner-Sutcliffe formula for the unique quantum representation of the transformation generator, which can be an arbitrary classical dynamical variable. This closed-form quantization procedure is shown to be equivalent to a natural extension of the standard canonical quantization recipe -- an extension that resolves the operator-ordering ambiguity in favor of the Born-Jordan rule. 
  We present a simple argument determining the shape of the curves of marginal stability in the N=2 supersymmetric SU(2) QCD with less than 4 massless flavors. The argument relies only on the modular properties of $a_D/a$ and its weak-coupling behavior. 
  The self-gravitational correction of the ultraviolet-divergent second- order "vacuum polarization" radiative correction insertion Feynman diagram is carried out using full, self-consistent Einstein equation propagation of the intermediate virtual gravitons, which takes into account their important non-linear interactions with each other. (As a by-product, the subsequent perturbative treatment of these non-linearities is avoided, which eliminates the source of the ultraviolet divergences of the second- quantized gravity theory itself.) The corrected diagram is finite, makes no contribution to charge renormalization (as could be expected of a diagram involving but a single transient virtual pair), and its dynamical behaviour accords with the standard quantum electrodynamics result except at inaccessibly extreme (Planck-scale-related) values of the momentum transfer. There, the standard logarithmic rise with momentum transfer which this diagram contributes to the effective coupling strength falls away, as the diagram proceeds instead to decrease strongly toward zero. The same self-gravitational correction is made to the closely related quartically divergent second-order vacuum-to-vacuum amplitude correction Feynman diagram, and it is found that the result vanishes identically. 
  We make a short review of the formalism that describes Higgs and Yang Mills fields as two particular cases of an appropriate generalization of the notion of connection. We also comment about the several variants of this formalism, their interest, the relations with noncommutative geometry, the existence (or lack of existence) of phenomenological predictions, the relation with Lie super-algebras etc. 
  We prove the renormalizability of quantum gravity near two dimensions. The successful strategy is to keep the volume preserving diffeomorphism as the manifest symmetry of the theory. The general covariance is recovered by further imposing the conformal invariance. The proof utilizes BRS formalism in parallel with Yang-Mills theory. The crucial ingredient of the proof is the relation between the conformal anomaly and the $\beta$ functions. 
  We discuss one of the generic spacetime consequences of having (1,0) worldsheet supersymmetry in tachyon-free string theory, namely the appearance of a ``misaligned supersymmetry'' in the corresponding spacetime spectrum. Misaligned supersymmetry is a universal property of (1,0) string vacua which describes how the arrangement of bosonic and fermionic states at all string energy levels conspires to preserve finite string amplitudes, even in the absence of full spacetime supersymmetry. Misaligned supersymmetry also constrains the degree to which spacetime supersymmetry can be broken without breaking modular invariance, and is responsible for the vanishing of various mass supertraces evaluated over the infinite string spectrum.  [Talk delivered at Strings '95, based on material drawn from hep-th/9402006 and hep-th/9409114. To appear in Proceedings.] 
  We demonstrate that the spectrum of any consistent string theory in $D$ dimensions must satisfy a number of supertrace constraints: $ Str~M^{2n}=0 $ for $0 \leq n < D/2-1$, integer $n$. These results hold for a large class of string theories, including critical heterotic strings. For strings lacking spacetime supersymmetry, these supertrace constraints will be satisfied as a consequence of a hidden ``misaligned supersymmetry'' in the string spectrum. [Talk given by R.C.M. at Strings '95; to appear in Proceedings] 
  We present the alternative topological twisting of N=4 Yang-Mills, in which the path integral is dominated not by instantons, but by flat connections of the COMPLEXIFIED gauge group. The theory is nontrivial on compact orientable four-manifolds with nonpositive Euler number, which are necessarily not simply connected. On such manifolds, one finds a single topological invariant, analogous to the Casson invariant of three-manifolds. 
  A recent application of an index relation of the form, $dim\ ker\ M - dim\ ker\ M^{\dagger} = \nu$, to the generation of chiral fermions in a vector-like gauge theory is reviewed. In this scheme the chiral structure arises from a mass term with a non-trivial index.The essence of the generalized Pauli-Villars regularization of chiral gauge theory, which is based on this mechanism, is also clarified. 
  The scale invariance of the $O(3)$ sigma model can be broken by gauging a $U(1)$ subgroup of the $O(3)$ symmetry and including a Maxwell term for the gauge field in the Lagrangian. Adding also a suitable potential one obtains a field theory of Bogomol'nyi type with topological solitons. These solitons are stable against rescaling and carry magnetic flux which can take arbitrary values in some finite interval. The soliton mass is independent of the flux, but the soliton size depends on it. However, dynamically changing the flux requires infinite energy, so the flux, and hence the soliton size, remains constant during time evolution. 
  The geometric description of Yang-Mills theories and their configuration space M is reviewed. The presence of singularities in M is explained and some of their properties are described. The singularity structure is analyzed in detail for structure group SU(2). 
  Report in extended abstract form. Based on talk given at Strings '95, Los Angeles, CA. This is a summary of the findings initially presented in hep-th/9505080. Results of a step (the complete classification of free fermionic solutions to $N=1$ spacetime) in the search for stringy three-generation SO(10) SUSY-GUTs are reviewed. 
  The dispersionless differential Fay identity is shown to be equivalent to a kernel expansion providing a universal algebraic characterization and solution of the dispersionless Hirota equations. Some calculations based on D-bar data of the action are also indicated. 
  We derive an appropriate definition of transpose for quaternionic matrices and give a new panoramic review of the quaternionic groups. We aim to analyse possible quaternionic groups for GUTs. 
  We consider an extension of the ordinary four dimensional Minkowski space by introducing additional dimensions which have their own Lorentz transformation. Particles can transform in a different way under each Lorentz group. We show that only quark interactions are slightly modified and that color confinement is automatic since these degrees of freedom run only in the extra dimensions. No compactification of the extra dimensions is needed. 
  A pseudoclassical model to describe Weyl particle in 10 dimensions is proposed. In course of quantization both the massless Dirac equation and the Weyl condition are reproduced automatically. The construction can be relevant to Ramond-Neveu-Schwarz strings where the Weyl reduction in the Ramond sector has to be made by hand. 
  We show that the partition function of free Maxwell theory on a generic Euclidean four-manifold transforms in a non-trivial way under electric-magnetic duality. The classical part of the partition sum can be mapped onto the genus-one partition function of a 2d toroidal model, without the oscillator contributions. This map relates electric-magnetic duality to modular invariance of the toroidal model and, conversely, the $O(d,d',\Z)$ duality to the invariance of Maxwell theory under the 4d mapping class group. These dualities and the relation between toroidal models and Maxwell theory can be understood by regarding both theories as dimensional reductions of a self-dual 2-form theory in six dimensions. Generalizations to more $U(1)$-gauge fields and reductions from higher dimensions are also discussed. We find indications that the Abelian gauge theories related to 4d string theories with $N=4$ space-time supersymmetry are exactly duality invariant. 
  The spectrum of $H$-monopoles of the heterotic string compactified on a six torus and its relationship to the $S$-duality conjecture is briefly reviewed. It is based on work done in collaboration with J. Harvey and is a contribution to the proceedings of Strings '95, USC, March 1995. 
  The idea of the effective topological theory for high-energy scattering proposed by H. and E. Verlinde is applied to the $(2+1)$ dimensional gravity with Einstein action plus Chern-Simons terms. The calculational steps in the topological description are compared with the eikonal approximation. It is shown that the Lagrangian of the effective topological theory turns out to vanish except for boundary terms. 
  We extend the construction of open descendants to the $SU(2)$ WZW models with non-diagonal left-right pairing, namely $E_7$ and the $D_{odd}$ series in the $ADE$ classification of Cappelli, Itzykson and Zuber. The structure of the resulting models is determined to a large extent by the ``crosscap constraint'', while their Chan-Paton charge sectors may be embedded in a general fashion into those of the corresponding diagonal models. 
  We discuss the statistical mechanics of a two-dimensional gas of non-Abelian Chern-Simons particles which obey the non-Abelian braid statistics. The second virial coefficient is evaluated in the framework of the non-Abelian Chern-Simons quantum mechanics. 
  Starting from a chiral conformal Haag-Kastler net on 2 dimensional Minkowski space we construct associated charged pointlike localized fields which intertwine between arbitrary superselection sectors with finite statistics of the theory. This amounts to a proof of the Spin-Statistics theorem, the PCT theorem and the existence of operator product expansions. This paper generalizes similar results of a recently published paper by Fredenhagen and the author \cite{FrJ} from the neutral vacuum sector to the the full theory with arbitrary charge and finite statistics. 
  We study the theory of systems with constraints from the point of view of the formal theory of partial differential equations. For finite-dimensional systems we show that the Dirac algorithm completes the equations of motion to an involutive system. We discuss the implications of this identification for field theories and argue that the involution analysis is more general and flexible than the Dirac approach. We also derive intrinsic expressions for the number of degrees of freedom. 
  Radiation symmetry is briefly reviewed, along with its historical, experimental, computational, and theoretical relevance. A sketch of the proof of a theorem for radiation zeros is used to highlight the connection between gauge-boson couplings and Poincare transformations. It is emphasized that while mostly bad things happen to good zeros, the weak-boson self-couplings continue to be intimately tied to the best examples of exact or approximate zeros. 
  Wilson's exact renormalization group equations are derived and integrated for the relevant part of the pure Yang-Mills action. We discuss in detail how modified Slavnov-Taylor identities controle the breaking of BRST invariance in the presence of a finite infrared cutoff $k$ through relations among different parameters in the effective action. In particular they imply a nonvanishing gluon mass term for nonvanishing $k$. The requirement of consistency between the renormalization group flow and the modified Slavnov- Taylor identities allows to control the self-consistency of truncations of the effective action. 
  We study homogeneous isotropic universe in a graviton-dilaton theory obtained, in a previous paper, by a simple requirement that the theory be able to predict non trivial values for $\beta$ and/or $\gamma$ for a charge neutral point star, without any naked singularities. We find that in this universe the physical time can be continued indefinitely into the past or future, and that all the physical curvature invariants are always finite, showing the absence of big bang singularity. Adding a dilaton potential, we find again the same features. As a surprising bonus, there emerges naturally a Brans-Dicke function, which has precisely the kind of behaviour needed to make $\omega_{bd} ({\rm today}) > 500$ in hyperextended inflation. 
  Excitation of a vortex in the Abelian Higgs model is investigated with the help of a polynomial approximation. The excitation can be regarded as a longitudinal component of the vector field trapped by the vortex. The energy and profile of the excitation are found. Back-reaction of the excitation on the vortex is calculated in the small $\kappa$ limit. It turns out that in the presence of the excitation the vortex effectively becomes much wider - its radius oscillates in time and for all times it is not smaller than the radius of the unexcited vortex. Moreover, we find that the vector field of the excited vortex has long range radiative component. Bound on the amplitude of the excitation is also found. 
  We derive a minimal set of Feynman rules for the loop amplitudes in unitary models of closed strings, whose target space is a simply laced (extended) Dynkin diagram. The string field Feynman graphs are composed of propagators, vertices (including tadpoles) of all topologies, and leg factors for the macroscopic loops. A vertex of given topology factorizes into a fusion coefficient for the matter fields and an intersection number associated with the corresponding punctured surface. As illustration we obtain explicit expressions for the genus-one tadpole and the genus-zero four-loop amplitude. 
  The quantum theory of the Liouville model with imaginary field is considered using the quantum inverse scattering method. An integrable structure with nontrivial spectral parameter dependence is developed for lattice Liouville theory by scaling the $L$-matrix of lattice sine-Gordon theory. This $L$-matrix yields Bethe Ansatz equations for Liouville theory, by the methods of the algebraic Bethe Ansatz. Using the string picture of exited Bethe states, the lattice Liouville Bethe equations are mapped to the corresponding equations for the spin 1/2 XXZ chain. The well developed theory of finite size corrections in spin chains is used to deduce the conformal properties of the lattice Liouville Bethe states. The unitary series of conformal field theories emerge for Liouville couplings of the form $\gam = \pi\frac{\nu}{\nu+1}$, corresponding to root of unity XXZ anisotropies. The Bethe states give the full spectrum of the corresponding unitary conformal field theory, with the primary states in the \Kac table parameterized by a string length $K$, and the remnant of the chain length mod $(\nu+1)$. 
  A connection between the conifold locus of the type II string on the $W\:P_{11226}^4$ Calabi-Yau manifold and the geometry of the quantum moduli of $N = 2$ $SU(2)$ super Yang-Mills is presented. This relation is obtained from the anomalous behaviour of the $SU(2)$ super Yang-Mills special coordinates under $S$-duality transformation in $Sl(2;Z) / \Gamma_2$. 
  The three-flavour Wess-Zumino model coupled to electromagnetism is treated as a constraint system using the Faddeev-Jackiw method. Expanding into series of powers of the Goldstone boson fields and keeping terms up to second and third order we obtain Coulomb-gauge hamiltonian densities. 
  We show that the instantons induce the abelian monopoles in the Maximal Abelian Projection of $SU(2)$ gluodynamics. As an example we consider the case of one instanton and the case of a set of instantons arranged along a straight line. The abelian monopoles which are induced by instantons may play some role in the confinement scenario. 
  We consider the problem of constructing a non-singular inflationary universe in stringy gravity via branch changing, from a previously superexponentially expanding phase to an FRW-like phase. Our approach is based on the phase space analysis of the dynamics, and we obtain a no-go theorem which rules out the efficient scenario of branch changing catalyzed by dilaton potential and stringy fluid sources. We furthermore consider the effects of string-loop corrections to the gravitational action in the form recently suggested by Damour and Polyakov. These corrections also fail to produce the desired branch change. However, focusing on the possibility that these corrections may decouple the dilaton, we deduce that they may lead to an inflationary expansion in the presence of a cosmological constant, which asymptotically approaches Einstein-deSitter solution. 
  A discussion of the number of degrees of freedom, and their dynamical properties, in higher derivative gravitational theories is presented. It is shown that non-vanishing $(C_{mnpq})^{2}$ terms arise in N=1, D=4 superstring Lagrangians due to one-loop radiative corrections with light field internal lines. 
  In a superspace formulation of Yang-Mills theory previously proposed, we show how gauge-invariant operators and scalars can be incorporated keeping intact the (broken) $Osp(3,1|2)$ symmetry of the superspace action. We show in both cases, that the WT identities can be cast in a simple form $\frac{\partial\bar{W}}{\partial\theta}=0$. 
  The statistics of meander and related problems are studied as particular realizations of compact polymer chain foldings. This paper presents a general discussion of these topics, with a particular emphasis on three points: (i) the use of a direct recursive relation for building (semi) meanders (ii) the equivalence with a random matrix model (iii) the exact solution of simpler related problems, such as arch configurations or irreducible meanders. 
  We present a complete set of conjectures for the exact boundary reflection matrix for $ade$ affine Toda field theory defined on a half line with the Neumann boundary condition. 
  Dyonic classical solutions of Yang-Mills theory are considered and the complete set of fermionic zero modes of these solutions are studied. Representing the QCD vacuum as a gas of dyons, one obtains chiral symmetry breaking due to zero modes similarly to the case of instantonic vacuum. 
  The $\kappa$--anomaly cancellation mechanism in the heterotic superstring determines the superspace constraints for N=1, D=10 Supergravity--Super--Yang--Mills theory. We point out that the constraints found recently in this way appear to disagree with superspace solutions found in the past. We solve this puzzle establishing perfect agreement between the two methods. 
  The dual of the four dimensional non-linear sigma model is constructed using techniques familiar to string theory. This construction necessitates the introduction of a rank two antisymmetric tensor field whose properties are examined. The physics of the dual theory and that of the original model are compared. As an illustration we study in detail the SU(2) chiral model. We find that the scattering amplitudes of the charged Goldstone bosons in the two theories are in complete agreement at the one loop level. 
  We show that there is a function of one variable's worth of Lagrangians for a single Maxwell field coupled to gravity whose equations of motion admit electric-magnetic duality. Such Lagrangians are given by solutions of the Hamilton-Jacobi equation for timelike geodesics in Witten's two-dimensional black hole. Among them are the Born-Infeld Lagrangian which arises in open string theory. We investigate the effect of the axion and the dilaton in the open superstring case and we show that this theory loses its electric-magnetic duality invariance when one considers the higher order electromagnetic field terms. We discuss some implications for black holes in string theory and an extension to $2k$-forms in $4k$ spacetime dimensions. 
  In this talk, the new spacetime-supersymmetric description of the superstring is reviewed and some of its applications are described. These applications include the manifestly spacetime-supersymmetric calculation of scattering amplitudes, the construction of a super-Poincar\'e invariant open superstring field theory, and the beta-function calculation of low-energy equations of motion in superspace. Parts of this work have been done in collaboration with deBoer, van Nieuwenhuizen, Ro\v{c}ek, Sezgin, Skenderis, Stelle, and especially, Siegel and Vafa. 
  We study the symmetries of the two dimensional Heterotic string theory by following the approach of Kinnersley et al for the study of stationary-axially symmetric Einstein-Maxwell equations. We identify the finite dimensional groups $G'$ and $H'$ for the Einstein-Maxwell equations. We also give the constructions for the infinite number of conserved currents and the affine $\hat{o}(8, 24)$ symmetry algebra in this formulation. The generalized Ehlers and Harrison transformations are identified and a parallel between the infinite dimensional symmetry algebra for the heterotic string case with $\hat{sl}(3, R)$ that arise in the case of Einstein-Maxwell equations is pointed out. 
  In the context of $ISO(2,1)$ gauge theory, we consider $(2+1)$-dimensional gravity with the gravitational Chern-Simons term (CST). This formulation allows the `exact' solution for the system coupled to a massive point particle (which is not the case in the conventional Chern-Simons gravity). The solution exhibits locally trivial structure even with the CST, although still shows globally nontrivialness such as the conical space and the helical time structure. Since the solution is exact, we can say the CST induces spin even for noncritical case of $\s+\al m\ne 0$. 
  WZNW models, especially gauged WZNW models, are important in the study of conformal field theories. Karabali and Schnitzer initiated the study of the BRST cohomology of a WZNW model gauged by an anomaly free vector sub-group and results were given for abelian sub-groups. This result was generalized to non-abelian sub-groups for a specific set of representations \cite{HR1}. The subject of this talk is the analysis of arbitrary representations \cite{HR2,Hw}. 
  The strong coupling physics of two dimensional gravity at $C=7$, $13$, $19$ is summarized. It is based on a new set of local fields which do not preserve chirality. Thus this quantum number becomes ``deconfined'' in the strongly coupled regime. This new set leads to a novel definition of the area elements, and hence to a modified expression for the string suceptibility, which the real part of the KPZ formula. It allows to define topological (strongly coupled ) Liouville string theories (without transverse degree of freedom) which are completely solvable, and are natural extensions of the standard matrix models. 
  It is suggested that quantum fluctuations of the light cone are at the origin of what appears at low energy to be a higher-dimensional structure over space-time. A model is presented which has but a finite number of Yang-Mills fields although the supplementary algebraic structure is of infinite dimension. 
  Techniques are presented for calculating directly the scalar functional determinant on the Euclidean d-ball. General formulae are given for Dirichlet and Robin boundary conditions. The method involves a large mass asymptotic limit which is carried out in detail for d=2 and d=4 incidentally producing some specific summations and identities. Extensive use is made of the Watson-Kober summation formula. 
  In this paper we calculate the high--energy quark--quark scattering amplitude, first in the case of scalar QCD, using Fradkin's approach to derive the scalar quark propagator in an external gluon field and computing it in the eikonal approximation. (This approach was also recently used by Fabbrichesi, Pettorino, Veneziano and Vilkovisky to study the four--dimensional Planckian--energy scattering in gravity.) We then extend the results to the case of ``real'' ({\it i.e.} fermion) QCD, thus deriving again, in a rather direct way, the results previously found by Nachtmann. The abelian case (QED) is also discussed in the Appendix. 
  The non-commutative approach of the standard model produces a relation between the top and the Higgs masses. We show that, for a given top mass, the Higgs mass is constrained to lie in an interval. The length of this interval is of the order of $m_\tau^2/m_t$. 
  I present an outline for cosmological evolution in the framework of string theory with emphasis on a phase of dilaton-driven kinetic inflation. It is shown that a typical background of stochastic gravitational radiation is generated, with strength that may allow its detection in future gravity wave experiments. 
  Two-dimensional heavy-quark QCD is studied in the light-cone coordinates with (anti-) periodic field boundary conditions. We carry out the light-cone quantization of this gauge invariant model. To examine the role of the zero modes of the gauge degrees of freedom, we consider the quantization procedure in the zero mode and the nonzero mode sectors separately. In both sectors, we obtain the physical variables and their canonical (anti-) commutation relations. The physical Hamiltonian is constructed via a step-by-step elimination of the unphysical degrees of freedom. It is shown that the zero modes play a crucial role in the self-interaction potential of both the heavy-quarks and gluons, and in the interaction potential between them. It is also shown that the Faddeev-Popov determinant depends on the zero modes of the gauge degrees of freedom. Therefore, one needs to introduce the Faddeev-Popov ghosts in their own nonzero mode sector. 
  Basic quantities related to 2-D gravity, such as Polyakov extrinsic action, Nambu-Goto action, geometrical action, and Euler characteristic are studied using generalized Weierstrass-Enneper (GWE) inducing of surfaces. Connection of the GWE inducing with conformal immersion is made and varius aspects of the theory are shown to be invariant under the modified Veselov-Novikov hierarchy of flows. The geometry of certain surfaces is shown to be connected with the dynamics of infinite and finite dimensional integrable systems. Connections to Liouville-Beltrami gravity are indicated. 
  We discuss an orbifold of the toroidally compactified heterotic string which gives a global reduction of the dimension of the moduli space while preserving the supersymmetry. This construction yields the moduli space of the first of a series of reduced rank theories with maximal supersymmetry discovered recently by Chaudhuri, Hockney, and Lykken. Such moduli spaces contain non-simply-laced enhanced symmetry points in any spacetime dimension D<10. Precisely in D=4 the set of allowed gauge groups is invariant under electric-magnetic duality, providing further evidence for S-duality of the D=4 heterotic string. 
  In several unified field theories the torsion trace is set equal to the electromagnetic potential. Using fibre bundle techniques we show that this is no leading principle but a formal consequence of another geometric relation between space-time and electromagentism. 
  In this article we consider $SU(2)$ Chern-Simons/Higgs theory coupled to gravity in three-dimensions. It is shown that for a cylindrically symmetric vortex both the Einstein equations and the field equations can be reduced to a set of first-order Bogomol'nyi equations provided that we choose a specific eighth-order potential. 
  We show that instantons violate a supersymmetric identity in a classically supersymmetric Higgs model with no massless fermions. This anomalous breaking arises because the correct perturbative expansion in the instanton sector is not supersymmetric. The attempt to construct a manifestly supersymmetric expansion generates infra-red divergences. 
  The Lagrangian for the motion of $n$ well-separated BPS monopoles is calculated, by treating the monopoles as point particles with magnetic, electric and scalar charges. It can be reinterpreted as the Lagrangian for geodesic motion on the asymptotic region of the $n$-monopole moduli space, thereby determining the asymptotic metric on the moduli space. The metric is hyperk\"ahler, and is an explicit example of a type of metric considered previously. 
  Noncritical strings in the "coupling constant" phase space and bosonic string in the affine-metric curved space are dissipative systems. But the quantum descriptions of the dissipative systems have well known ambiguities. We suggest some approach to solve the problems of this description. The generalized Poisson algebra for dissipative systems is considered. 
  We give explicit field theoretical representations for the observables of 2+1 dimensional Chern-Simons theory in terms of gauge invariant composites of 2D WZW fields. To test our identification we compute some basic Wilson loop correlators reobtaining known results. 
  A critically discerning discussion of path integral bosonization is given. Successively evaluating the conventional path integral bosonization of QCD it is shown without any approximations that gluons must be composed of two quarks. This contradicts the fundamentals of QCD, where quarks and gluons are independent fields. Furthermore, bosonizing the Fierz reordered effective four quark interaction term yields gluons, too. Colorless ``mesons'' are shown to be Fierz equivalent to a submanifold of gluons. The results obtained are not specific to QCD, but apply to other models as well. 
  Supersymmetry breaking may be linked to the formation of gaugino condensates in a hidden sector. In this work, the process of formation of the condensate is examined in a cosmological context, using an effective field theory of the gaugino bilinear which provides a reasonable interpolation between the high- and low-temperature phases. The implementation of anomaly requirements generates a large potential barrier between the zero-condensate configuration and that of the true (SUSY-breaking) vacuum. As a consequence, the transition to bubbles of true vacuum may be subject to an enormous exponential suppression. This leads to the same difficulties with inhomogeneity of the universe which occurred in the original inflationary scenarios. 
  We present new classes of string-like soliton solutions in ($N=1$; $D=10$), ($N=2$; $D=6$) and ($N=4$; $D=4$) heterotic string theory. Connections are made between the solution-generating subgroup of the $T$-duality group of the compactification and the number of spacetime supersymmetries broken. Analogous solutions are also noted in ($N=1,2$; $D=4$) compactifications, where a different form of supersymmetry breaking arises. 
  We analyze the properties of mesons in 1+1 dimensional QCD with bosonic and fermionic ``quarks'' in the large $\nc$ limit. We study the spectrum in detail and show that it is impossible to obtain massless mesons including boson constituents in this model. We quantitatively show how the QCD mass inequality is realized in two dimensional QCD. We find that the mass inequality is close to being an equality even when the quarks are light. Methods for obtaining the properties of ``mesons'' formed from boson and/or fermion constituents are formulated in an explicit manner convenient for further study. We also analyze how the physical properties of the mesons such as confinement and asymptotic freedom are realized. 
  We quantize the tachyon field in a static two dimensional dilaton gravity black hole background,and we calculate the Hawking radiation rate. We find that the thermal radiation flux, due to the tachyon field, is larger than the conformal matter one. We also find that massive scalar fields which do not couple to the dilaton, do not give any contribution to the thermal radiation, up to terms quadratic in the scalar curvature. 
  The Weierstrassian $\wp, \zeta$ and $\sigma $ functions are generalized to ${\bf R}^{n}$. The $n=3$ and $n=4$ cases have already been used in gravitational and Yang-Mills instanton solutions which may be interpreted as explicit realizations of spacetime foam and the monopole condensate, respectively. The new functions satisfy higher dimensional versions of the periodicity properties and Legendre's relations obeyed by their familiar complex counterparts. For $n=4$, the construction reproduces functions found earlier by Fueter using quaternionic methods. Integrating over lattice points along all directions but two, one recovers the original Weierstrassian elliptic functions. 
  The coupling, in a non-standard way, of a bosonic string theory with a dilaton and antisymmetric fields is investigated. By integrating over the antisymmetric fields, a Coulomb-like interaction term is generated. The static potential of a theory of this kind is obtained from the corresponding non-local zeta function, in some approximation. An interpretation of the static potential as a type of non-local Casimir effect is given. 
  We analyze a continuous spin Gaussian model on a toroidal triangular lattice with periods $L_0$ and $L_1$ where the spins carry a representation of the fundamental group of the torus labeled by phases $u_0$ and $u_1$. We find the {\it exact finite size and lattice corrections}, to the partition function $Z$, for arbitrary mass $m$ and phases $u_i$. Summing $Z^{-1/2}$ over phases gives the corresponding result for the Ising model. The limits $m\rightarrow0$ and $u_i\rightarrow0$ do not commute. With $m=0$ the model exhibits a {\it vortex critical phase} when at least one of the $u_i$ is non-zero. In the continuum or scaling limit, for arbitrary $m$, the finite size corrections to $-\ln Z$ are {\it modular invariant} and for the critical phase are given by elliptic theta functions. In the cylinder limit $L_1\rightarrow\infty$ the ``cylinder charge'' $c(u_0,m^2L_0^2)$ is a non-monotonic function of $m$ that ranges from $2(1+6u_0(u_0-1))$ for $m=0$ to zero for $m\rightarrow\infty$. 
  By studying the pure Yang-Mills theory on a circle, as well as an adjoint scalar coupled to the gauge field on a circle, we propose a quenching prescription in which the combination of the spatial component of the gauge field and $P$ is treated as a dynamic variable. Averaging over momentum is not necessary, therefore the usual ultraviolet cut-off is eliminated. We then apply this prescription to study the large $N$ two dimensional supersymmetric gauge theory. An one dimensional supersymmetric matrix model is obtained. It is not known whether this model can be solved exactly. However, an extended model with one more complex fermion is exactly solvable, with $N=1$ supersymmetry as Parisi-Sourlas supersymmetry. The exact solvability may have some implications for the $N=1$ quenched model. 
  Quantum integrable models that possess $N=2$ supersymmetry are investigated on the half-space. Conformal perturbation theory is used to identify some $N=2$ supersymmetric boundary integrable models, and the effective boundary Landau-Ginzburg formulations are constructed. It is found that $N=2$ supersymmetry largely determines the boundary action in terms of the bulk, and in particular, the boundary bosonic potential is $|W|^2$, where $W$ is the bulk superpotential. Supersymmetry is also investigated using the affine quantum group symmetry of exact scattering matrices, and the affine quantum group symmetry of boundary reflection matrices is analyzed both for supersymmetric and more general models. Some $N=2$ supersymmetry preserving boundary reflection matrices are given, and their connection with the boundary Landau-Ginzburg actions is discussed. 
  In this review I discuss various aspects of some of the recently constructed black hole and soliton solutions in string theory. I begin with the axionic instanton and related solutions of bosonic and heterotic string theory. The latter ten-dimensional solutions can be compactified to supersymmetric monopole, string and domain wall solutions which break $1/2$ of the spacetime supersymmetries of $N=4, D=4$ heterotic string theory, and which can be generalized to two-parameter charged black hole solutions. The low-energy dynamics of these solutions is also discussed, as well as their connections with strong/weak coupling duality and target space duality in string theory. Finally, new solutions are presented which break $3/4, 7/8$ and all of the spacetime supersymmetries and which also arise in more realistic $N=1$ and $N=2$ compactifications. 
  The quantum corrections to black hole entropy, variously defined, suffer quadratic divergences reminiscent of the ones found in the renormalization of the gravitational coupling constant (Newton constant). We consider the suggestion, due to Susskind and Uglum, that these divergences are proportional, and attempt to clarify its precise meaning. Using a Euclidean formulation the proportionality is a fairly immediate consequence of basic principles -- a low-energy theorem. Thus in this framework renormalizing the Newton constant renders the entropy finite, and equal to its semiclassical value. As a partial check on our formal arguments we compare the one loop determinants, calculated using heat kernel regularization. An alternative definition of black hole entropy relates it to behavior at conical singularities in two dimensions, and thus to a suitable definition of geometric entropy. Geometric entropy permits the same renormalization, but it does not yield an intrinsically positive quantity. For scalar fields geometric entropy is subtly sensitive to curved space couplings, even in the limit of flat space. Fermions and gauge fields are considered as well. Their functional determinants are closely related to the determinants for non-minimally coupled scalar fields with specific values for the curvature coupling, and pose no further difficulties. 
  We study boundary bound states using the Bethe ansatz formalism for the finite $XXZ$ $(\Delta>1)$ chain in a boundary magnetic field $h$. Boundary bound states are represented by the ``boundary strings'' similar to those described in the work of H.Saleur, S.Skorik. We find that for certain values of $h$ the ground state wave function contains boundary strings, and from this infer the existence of two ``critical'' fields in agreement with the results of Jimbo et al. An expression for the vacuum surface energy in the thermodynamic limit is derived and found to be an analytic function of $h$. We argue that boundary excitations appear only in pairs with ``bulk'' excitations or with boundary excitations at the other end of the chain. We mainly discuss the case where the magnetic fields at the left and the right boundaries are antiparallel, but we also comment on the case of the parallel fields. In the Ising ($\Delta=\infty$) and isotropic ($\Delta=1$) limits our results agree with those previously known. 
  We use the methods of group theory to reduce the equations of motion of the $CP^{1}$ model in (2+1) dimensions to sets of two coupled ordinary differential equations. We decouple and solve many of these equations in terms of elementary functions, elliptic functions and Painlev{\'e} transcendents. Some of the reduced equations do not have the Painlev{\'e} property thus indicating that the model is not integrable, while it still posesses many properties of integrable systems (such as stable ``numerical'' solitons). 
  In the first part, the induced vacuum spin around an Aharonov-Bohm flux string in massless three-dimensional QED is computed explicitly and the result is shown to agree with a general index theorem. A previous observation in the literature, that the presence of induced vacuum quantum numbers which are not periodic in the flux make an integral-flux AB string visible, is reinforced. In the second part, a recent discussion of chiral symmetry breaking by external magnetic fields in parity invariant QED$_3$ and its relation to the induced spin in parity non-invariant QED$_3$ is further elaborated. Finally other vacuum polarisation effects around flux tubes in different variants of QED, in three and four dimensions are mentioned. 
  We study a topological sigma-model ($A$-model) in the case when the target space is an ($m_0|m_1$)-dimensional supermanifold. We prove under certain conditions that such a model is equivalent to an $A$-model having an ($m_0-m_1$)-dimensional manifold as a target space. We use this result to prove that in the case when the target space of $A$-model is a complete intersection in a toric manifold, this $A$-model is equivalent to an $A$-model having a toric supermanifold as a target space. 
  An exact conformal model representing a constant magnetic field background in heterotic string theory is explicitly solved in terms of free creation/annihilation operators. The spectrum of physical states is examined for different possible embeddings of the magnetic U(1) subgroup. We find that an arbitrarily small magnetic field gives rise to an infinite number of tachyonic excitations corresponding to charged vector states of the massless level and to higher level states with large spins and charges. 
  A mechanism of generating the metric is proposed, where the Kalb-Ramond symmetry existing in the topological BF theory is broken through the condensation of the string fields which are so introduced as to couple with the anti-symmetric tensor fields $B$, invariantly under the Kalb-Ramond symmetry. In the chiral decomposition of the local Lorentz group, the non-Abelian $B$ fields need to be generalized to the string fields. The mechanism of the condensation is discussed, viewing the confinement problem and the polymer physics. 
  In the framework of the two-form gravity, which is classically equivalent to the Einstein gravity, the one-loop effective potential for the conformal factor of metric is calculated in the finite volume and in the finite temperature by choosing a temporal gauge condition. There appears a quartically divergent term which cannot be removed by the renormalization of the cosmological term and we find there is only one non-trivial minimum in the effective potential. If the cut-off scale has a physical meaning, \eg the Planck scale coming from string theory, this minimum might explain why the space-time is generated, \ie why the classical metric has a non-trivial value. 
  We study the $O(n)$ loop model on the honeycomb lattice with open boundary conditions. Reflection matrices for the underlying Izergin-Korepin $R$-matrix lead to three inequivalent sets of integrable boundary weights. One set, which has previously been considered, gives rise to the ordinary surface transition. The other two sets correspond respectively to the special surface transition and the mixed ordinary-special transition. We analyse the Bethe ansatz equations derived for these integrable cases and obtain the surface energies together with the central charges and scaling dimensions characterizing the corresponding phase transitions. 
  The generalisation of the rigid special geometry of the vector multiplet quantum moduli space to the case of supergravity is discussed through the notion of a dynamical Calabi--Yau threefold. Duality symmetries of this manifold are connected with the analogous dualities associated with the dynamical Riemann surface of the rigid theory. N=2 rigid gauge theories are reviewed in a framework ready for comparison with the local case. As a byproduct we give in general the full duality group (quantum monodromy) for an arbitrary rigid $SU(r+1)$ gauge theory, extending previous explicit constructions for the $r=1,2$ cases. In the coupling to gravity, R--symmetry and monodromy groups of the dynamical Riemann surface, whose structure we discuss in detail, are embedded into the symplectic duality group $\Gamma_D$ associated with the moduli space of the dynamical Calabi--Yau threefold. 
  This paper is a sequel to one in which we examined the affine symmetry algebras of arbitrary classical principal chiral models and symmetric space models in two dimensions. It examines the extension of those results in the presence of gravity. The main result is that even though the symmetry transformations of the fields depend on the gravitational background, the symmetry algebras of these classical theories are completely unchanged by the presence of arbitrary gravitational backgrounds. On the other hand, we are unable to generalize the Virasoro symmetries of the flat-space theories to theories with gravity. 
  Recently the vacuum structure of a large class of four dimensional (supersymmetric) quantum field theories was determined exactly. These theories exhibit a wide range of interesting new physical phenomena. One of the main new insights is the role of ``electric-magnetic duality.'' In its simplest form it describes the long distance behavior of some strongly coupled, and hence complicated, ``electric theories'' in terms of weakly coupled ``magnetic theories.'' This understanding sheds new light on confinement and the Higgs mechanism and uncovers new phases of four dimensional gauge theories. We review these developments and speculate on the outlook. 
  We determine the motions of the roots of the Bethe ansatz equation for the ground state in the XXZ spin chain under a varying twist angle. This is done by analytic as well as numerical study at a finite size system. In the attractive critical regime $ 0< \Delta <1 $, we reveal intriguing motions of strings due to the finite size corrections to the length of the strings: in the case of two-strings, the roots collide into the branch points perpendicularly to the imaginary axis, while in the case of three-strings, they fluctuate around the center of the string. These are successfully generalized to the case of $n$-string. These results are used to determine the final configuration of the momenta as well as that of the phase shift functions. We obtain these as well as the period and the Berry phase also in the regime $ \Delta \leq -1$, establishing the continuity of the previous results at $ -1 < \Delta < 0 $ to this regime. We argue that the Berry phase can be used as a measure of the statistics of the quasiparticle ( or the bound state) involved in the process. 
  In topological field theories determinants of maps with negative as well as positive eigenvalues arise. We give a generalisation of the zeta-regularisation technique to derive expressions for the phase and scaling-dependence of these determinants. For theories on odd-dimensional manifolds a simple formula for the scaling dependence is obtained in terms of the dimensions of certain cohomology spaces. This enables a non-perturbative feature of Chern-Simons gauge theory to be reproduced by path-integral methods. 
  The structure of a previously developed representation of the Leech lattice, $\Lambda_{24}$, is exposed to further light with this unified and very simple construction. 
  Duality between the `axion' field $a$ and the antisymmetric tensor field $B_{\mu\nu}$ is traced after a nonperturbative effect, gaugino condensation, breaks the Peccei-Quinn symmetry $a \rightarrow a +c$. Even though the PQ symmetry was at its origin, duality is nevertheless {\it not} broken by this effect. Below condensation scale, the axion simply gets a mass, but in the `stringy' version, the $ B_{\mu\nu}$ field disappears from the propagating spectrum. Its place is taken by a massive 3-index antisymmetric field $H_{\mu\nu\rho}$ which is the one dual to the massive axion. This is a particular case of a general duality in $D$-dimensions among {\it massive} $p$ and $D-p-1$-index antisymmetric tensor fields. 
  The theoretical foundation of the object moving faster than light in vacuum ({\it tachyon}) is still missing or incomplete. Here we present the classical foundation of the relativistic dynamics including the tachyon. An anomalous sign-factor extracted from the transformation of ${ \sqrt{1-u^{2}/c^{2} } }$ under the Lorentz transformation, which has been always missed in the usual formulation of the tachyon, has a crucial role in the dynamics of the tachyon. Due to this factor the mass of the tachyon transforms in the unusual way although the energy and momentum, which are defined as the conserved quantities in all uniformly moving systems, transform in the usual way as in the case of the object moving slower than light ({\it bradyon}). We show that this result can be also obtained from the least action approach. On the other hand, we show that the ambiguities for the description of the dynamics for the object moving with the velocity of light ({\it luxon}) can be consistently removed only by introducing a new dynamical variable. Furthermore, by using the fundamental definition of the momentum and energy we show that the zero-point energy for any kind of the objects, {\it i.e.}, the tachyon, bradyon, and luxon, which has been known as the undetermined constant, should satisfy some constraints for consistency, and we note that this is essentially another novel relativistic effect. Finally, we remark about the several unsolved problems. 
  The equations describing self/anti-self charge conjugate states, recently proposed by Ahluwalia, are re-written to covariant form. The corresponding Lagrangian for the neutral particle theory is proposed. From a group-theoretical viewpoint the construct is an example of the Nigam-Foldy-Bargmann-Wightman-Wigner-type quantum field theory based on the doubled representations of the extended Lorentz group. Relations with the Sachs-Schwebel and Ziino-Barut concepts of relativistic quantum theory are discussed. 
  We discuss the phases of four dimensional gauge theories and demonstrate them in solvable examples. Some of our simple examples exhibit confinement and oblique confinement. The theory has dual magnetic and dual dyonic descriptions in which these phenomena happen at weak coupling. Combined with the underlying electric theory, which gives a weak coupling description of the Higgs phase, we have electric-magnetic-dyonic triality. In an appendix we clarify some points regarding the use of 1PI superpotentials in these theories. 
  We evaluate the exact ${\rm QED}_{2+1}$ effective energy for charged spin zero and spin half fields in the presence of a family of static magnetic field profiles localized in a strip of width $\lambda$. The exact result yields an infinite set of relations between the terms in the derivative expansion of the effective energy for a general magnetic field. Upon addition of the standard Maxwell magneto-static energy, the minimum energy configuration at fixed flux corresponds to a uniform magnetic field. 
  I discuss the role of spacetime supersymmetry in the interplay between strong/weak coupling duality and target space duality in string theory which arises in string/string duality. This can be seen via the construction of string soliton solutions which in $N=4$ compactifications of heterotic string theory break more than $1/2$ of the spacetime supersymmetries but whose analogs in $N=2$ and $N=1$ compactifications break precisely $1/2$ of the spacetime supersymmetries. As a result, these solutions may be interpreted as stable solitons in the latter two cases, and correspond to Bogomol'nyi-saturated states in their respective spectra. 
  New aspects of the complex sine-Gordon theory are addressed through the reformulation of the theory in terms of the gauged Wess-Zumino-Witten action. A dual transformation between the theory for the coupling constant $\b > 0$ and the theory for $\b < 0$ is given which agrees with the Krammers-Wannier duality in the context of perturbed conformal field theory. The B\"{a}cklund transform and the nonlinear superposition rule for the complex sine-Gordon theory are presented and from which, exact solutions, solitons and breathers with U(1) charge, are derived. We clarify topological and nontopological nature of neutral and charged solitons respectively, and discuss about the duality between the vector and the axial U(1) charges. 
  We provide a description of W_3 transformations in terms of deformations of convex curves in two dimensional Euclidean space. This geometrical interpretation sheds some light on the nature of finite W_3-morphisms. We also comment on how this construction can be extended to the case of W_n and ``nicely curved'' curves in $\reals^{n-1}$. 
  String equations of the $p$-th generalized Kontsevich model and the compactified $c = 1$ string theory are re-examined in the language of the Toda lattice hierarchy. As opposed to a hypothesis postulated in the literature, the generalized Kontsevich model at $p = -1$ does not coincide with the $c = 1$ string theory at self-dual radius. A broader family of solutions of the Toda lattice hierarchy including these models are constructed, and shown to satisfy generalized string equations. The status of a variety of $c \le 1$ string models is discussed in this new framework. 
  We discuss an extension of the quantization method based on the induced representation of the canonical group. 
  The moduli dependence of $(2,2)$ superstring compactifications based on Calabi--Yau hypersurfaces in weighted projective space has so far only been investigated for Fermat-type polynomial constraints. These correspond to Landau-Ginzburg orbifolds with $c=9$ whose potential is a sum of $A$-type singularities. Here we consider the generalization to arbitrary quasi-homogeneous singularities at $c=9$. We use mirror symmetry to derive the dependence of the models on the complexified K\"ahler moduli and check the expansions of some topological correlation functions against explicit genus zero and genus one instanton calculations. As an important application we give examples of how non-algebraic (``twisted'') deformations can be mapped to algebraic ones, hence allowing us to study the full moduli space. We also study how moduli spaces can be nested in each other, thus enabling a (singular) transition from one theory to another. Following the recent work of Greene, Morrison and Strominger we show that this corresponds to black hole condensation in type II string theories compactified on Calabi-Yau manifolds. 
  The principles of finiteness and reduction of couplings can be applied to achieve Gauge-Yukawa Unification. It is found that the observed top-bottom hierarchy and the top quark mass naturally follow if there exists Gauge-Yukawa Unification which is a simple functional relation among the gauge coupling and the Yukawa couplings of the third generation in various susy unified gauge models. We briefly outline the basic idea of these principles and present the main results of the Gauge-Yukawa Unified models that have recently been studied in detail. 
  Following the work of Khare {\it et al}, we show that the generalization to systems with spontaneous symmetry breaking of the Coleman-Hill theorem to one-loop order, can be extended to the case including fermions with the most general interactions. Although the correction to the parity-odd part of the vacuum polarization looks complicated in the Higgs phase, it turns out that the correction to the Chern-Simons term is identical to that in the symmetric phase, with the difference coming only from the contribution of the would be Chern-Simons term. We also discuss the implication of our result to nonabelian systems. 
  The superfield formulation of two - dimensional $N=4$ Extended Supersymmetric Quantum Mechanics (SQM) is described. It is shown that corresponding classical Lagrangian describes the motion in the conformally flat metric with additional potential term. The Bose and Fermi sectors of two- and three-dimensional $N=4$ SQM are analyzed. The structure of the quantum Hamiltonians is such, that the usual Shr\"{o}dinger equation in the flat space arises after some unitary transformation, demonstrating the effect of transmutation of the coupling constant and the energy of the initial model in some special cases. 
  Pure Yang-Mills theories on the $S_1\times R$ cylinder are quantized in light-cone gauge $A_-=0$ by means of ${\bf equal-time}$ commutation relations. Positive and negative frequency components are excluded from the ``physical" Hilbert space by imposing Gauss' law in a weak sense. Zero modes, related to the winding on the cylinder, provide non trivial topological variables of the theory. A Wilson loop with light-like sides is studied: in the abelian case it can be exactly computed obtaining the expected area result, whereas difficulties are pointed out in non abelian cases. 
  We summarize all the known properties of the supersymmetric integrable Two Boson equation. We present its nonstandard Lax formulation and tri-Hamiltonian structure, its reduction to the supersymmetric nonlinear Schr\"odinger equation and the local as well as nonlocal conserved charges. We also present the algebra of the conserved charges and identify its second Hamiltonian structure with the twisted $N=2$ superconformal algebra. 
  The mathematical basis of p-adic Higgs mechanism discussed in papers hep-th@xxx.lanl.gov 9410058-62 is considered in this paper. The basic properties of p-adic numbers, of their algebraic extensions and the so called canonical identification between positive real numbers and p-adic numbers are described. Canonical identification induces p-adic topology and differentiable structure on real axis and allows definition of definite integral with physically desired properties. p-Adic numbers together with canonical identification provide analytic tool to produce fractals. Canonical identification makes it possible to generalize probability concept, Hilbert space concept, Riemannian metric and Lie groups to p-adic context. Conformal invariance generalizes to arbitrary dimensions since p-adic numbers allow algebraic extensions of arbitrary dimension. The central theme of all developments is the existence of square root, which forces unique algebraic extension with dimension $D=4$ and $D=8$ for $p>2$ and $p=2$ respectively. This in turn implies that the dimensions of p-adic Riemann spaces are multiples of $4$ in $p>2$ case and of $8$ in $p=2$ case. 
  We give a simple description of the classical moduli space of vacua for supersymmetric gauge theories with or without a superpotential. The key ingredient in our analysis is the observation that the lagrangian is invariant under the action of the complexified gauge group $\Gc$. From this point of view the usual $D$-flatness conditions are an artifact of Wess--Zumino gauge. By using a gauge that preserves $\Gc$ invariance we show that every constant matter field configuration that extremizes the superpotential is $\Gc$ gauge-equivalent (in a sense that we make precise) to a unique classical vacuum. This result is used to prove that in the absence of a superpotential the classical moduli space is the algebraic variety described by the set of all holomorphic gauge-invariant polynomials. When a superpotential is present, we show that the classical moduli space is a variety defined by imposing additional relations on the holomorphic polynomials. Many of these points are already contained in the existing literature. The main contribution of the present work is that we give a careful and self-contained treatment of limit points and singularities. 
  A gauged (2+1)-dimensional version of the Skyrme model is investigated. The gauge group is $U(1)$ and the dynamics of the associated gauge potential is governed by a Maxwell term. In this model there are topologically stable soliton solutions carrying magnetic flux which is not topologically quantized. The properties of rotationally symmetric solitons of degree one and two are discussed in detail. It is shown that the electric field for such solutions is necessarily zero. The solitons' shape, mass and magnetic flux depend on the $U(1)$ coupling constant, and this dependence is studied numerically from very weak to very strong coupling. 
  We investigate the generalized Gross-Neveu model using the discretized light cone quantization and we find that the vacuum of the bare theory is {\sl non} trivial in presence of vectorial current coupling when the simplest and most natural form of quantum constraints is used. Nevertheless the vacuum of the renormalized theory is trivial. In the thermodynamic the Bethe-Salpiter equations which are obtained contain all the terms needed to make them finite. 
  The vanishing of the cosmological constant and absence of a massless dilaton might be explained by a duality between a supersymmetric string vacuum in three dimensions and a non-supersymmetric string vacuum in four dimensions. 
  We find the transformation properties of the prepotential ${\cal F}$ of $N=2$ SUSY gauge theory with gauge group $SU(2)$. In particular we show that ${\cal G}(a)=\pi i\left({\cal F}(a)-{1\over 2}a\partial_a{\cal F}(a)\right)$ is modular invariant. This function satisfies the non-linear differential equation $\left(1-{\cal G}^2\right){\cal G}''+{1\over 4}a {{\cal G}'}^3=0$, implying that the instanton contribution are determined by recursion relations. Finally, we find $u=u(a)$ and give the explicit expression of ${\cal F}$ as function of $u$. These results can be extended to more general cases. 
  We solve general 1-matrix models without taking the double scaling limit. A method of computing generating functions is presented. We calculate the generating functions for a simple and double torus. Our method is also applicable to more higher genus. Each generating function can be expressed by a ``specific heat'' function for sphere. Universal terms, which are survived in the double scaling limit can be easily picked out from our exact solutions. We also find that the regular part of the spherical generating function is at most bilinear in coupling constants of source terms. 
  We present a construction of modular invariant partition functions for heterotic (0,2) supersymmetric classical string vacua. This generalization of Gepner's construction yields GUT gauge groups E_6, SO(10), SU(5) and SU(3) x SU(2) x U(1)^r, respectively. By calculating the massless spectrum of some of these models we find strong indications that they correspond to (0,2) string vacua discussed recently in the context of CYM/LG phases. 
  Translationally invariant symmetric polynomials as coordinates for $N$-body problems with identical particles are proposed. It is shown that in those coordinates the Calogero and Sutherland $N$-body Hamiltonians, after appropriate gauge transformations, can be presented as a {\it quadratic} polynomial in the generators of the algebra $sl_N$ in finite-dimensional degenerate representation. The exact solvability of these models follows from the existence of the infinite flag of such representation spaces, preserved by the above Hamiltonians. A connection with Jack polynomials is discussed. 
  This model also known as chiral null model is a generalization of the gravitational wave and fundamental string background and is exact in the $\a'$ expansion. The reduction to 4 dimensions yields a stationary IWP solution which couples to 7 gauge fields (one gravi-photon and 6 matter gauge fields) and 4 scalars. Special cases are the Taub-NUT geometry and rotating black holes. These solutions possess a T-self-dual point where the black hole becomes massless. Discussing the S-duality we show that the Taub-NUT geometry allows an S-self-dual point and that the electric black hole corresponds to a magnetic black hole or an H-monopole. We could identify the massless black hole as $N_L=0$ and confirm the H-monopole as an $N_L=1$ string states. 
  The fusion procedure of dilute $A_L$ models is constructed. It has been shown that the fusion rules have two types: $su(2)$ and $su(3)$. This paper is concerned with the $su(2)$ fusion rule mainly and the corresponding functional relations of commuting transfer matrices in the $su(2)$ fusion hierarchy are found. Specially, it has been found that the fusion hierarchy does not close. These two types of fusion generate different solvable models, but, they are not totally irrelevant. The $su(2)$ fusion of level $2$ is equivalent to the $su(3)$ fusion of level $(1,1)$. According to this relationship the Bethe ansatz of fused model of level $(1,1)$ in $su(3)$ hierarchy has been represented by that of level $2$ in $su(2)$ fusion hierarchy. 
  The fusion procedure is implemented for the dilute $A_L$ lattice models and a fusion hierarchy of functional equations with an $su(3)$ structure is derived for the fused transfer matrices. We also present the Bethe ansatz equations for the dilute $A_L$ lattice models and discuss their connection with the fusion hierarchy. The solution of the fusion hierarchy for the eigenvalue spectra of the dilute $A_L$ lattice models will be presented in a subsequent paper. 
  We consider a space-time invariant duality symmetric action for a free Maxwell field and an $SL(2,{\bf R})\times SO(6,22)$ invariant effective action describing a low-energy bosonic sector of the heterotic string compactified on a six-dimensional torus. The manifest Lorentz and general coordinate invariant formulation of the models is achieved by coupling dual gauge fields to an auxiliary vector field from an axionic sector of the theory. 
  We collect further evidence for the proposed duality between $N=2$ heterotic and type II string vacua in a specific model suggested by Kachru and Vafa. In the gauge sector the previous analysis is extended; it is further shown that the duality also holds for the one--loop gravitational couplings to the vector multiplets. 
  We consider the generalization of Haldane's state-counting procedure to describe all possible types of exclusion statistics which are linear in the deformation parameter $g$. The statistics are parametrized by elements of the symmetric group of the particles in question. For several specific cases we determine the form of the distribution functions which generalizes results obtained by Wu. Using them we analyze the low-temperature behavior and thermodynamic properties of these systems and compare our results with previous studies of the thermodynamics of a gas of $g$-ons. Various possible physical applications of these constructions are discussed. 
  We analyze the map between heterotic and type II N=2 supersymmetric string theories for certain two and three moduli examples found by Kachru and Vafa. The appearance of elliptic j-functions can be traced back to specializations of the Picard-Fuchs equations to systems for $K_3$ surfaces. For the three-moduli example we write the mirror maps and Yukawa couplings in the weak coupling limit in terms of j-functions; the expressions agree with those obtained in perturbative calculations in the heterotic string in an impressive way. We also discuss symmetries of the world-sheet instanton numbers in the type II theory, and interpret them in terms of S--duality of the non-perturbative heterotic string. 
  A general covariant quantization of superparticle, Green-Schwarz superstring and a supermembrane with manifest supersymmetry and duality symmetry is proposed. This quantization provides a natural quantum mechanical description of curved BPS-type backgrounds related to the ultra-short supersymmetry multiplets. Half-size commuting and anticommuting Killing spinors admitted by such backgrounds in quantum theory become truncated $\kappa$-symmetry ghosts. The symmetry of Killing spinors under dualities transfers to the symmetry of the spectrum of states.   GS superstring in the generalized semi-light-cone gauge can be quantized consistently in the background of ten-dimensional supersymmetric gravitational waves. Upon compactification they become supersymmetric electrically charged black holes, either massive or massless. However, the generalized light-cone gauge breaks S-duality. We propose a new family of gauges, which we call black hole gauges. These gauges are suitable for quantization both in flat Minkowski space and in the black hole background, and they are duality symmetric. As an example, a manifestly S-duality symmetric black hole gauge is constructed in terms of the axion-dilaton-electric-magnetic black hole hair. We also suggest the U-duality covariant class of gauges for type II superstrings. 
  From the point of view of topology we study the induced representation technique which E. Wigner proposed in 1939. We comment on the gauge structure in the induced representation technique and construc the explicit form of the gauge fields. The topological results ofour study are applied to quantum mechanics on a d-dimensional sphere and its path integral is formulated. 
  It is well known that anomaly cancellation {\it almost} determines the hypercharges in the standard model. A related (and somewhat more stronger) phenomenon takes place in Connes' NCG framework: unimodularity (a technical condition on elements of the algebra) is {\it strictly} equivalent to anomaly cancellation (in the absence of right-handed neutrinos); and this in turn reduces the symmetry group of the theory to the standard $SU(3)\times SU(2) \times U(1)$. 
  Using Operator Product Expansions and a graphical ansatz for the four-point function of the fundamental field \phi^{\alpha}(x) in the conformally invariant O(N) vector model, we calculate the next-to-leading order in 1/N values of the quantities C_{T} and C_{J}. We check the results against what is expected from possible generalisations of the C- and k-theorems in higher dimensions and also against known three-loop calculations in a O(N) invariant \phi^{4} theory for d=4-\epsilon. 
  We discuss the renormalization group improved effective action and running surface couplings in curved spacetime with boundary. Using scalar self-interacting theory as an example, we study the influence of the boundary effects to effective equations of motion in spherical cap and the relevance of surface running couplings to quantum cosmology and symmetry breaking phenomenon. Running surface couplings in the asymptotically free SU(2) gauge theory are found. 
  The renormalization group flow in two-dimensional field theories that are coupled to gravity has unusual features: First, the flow equations are second order in derivatives. Second, in the presence of handles the flow has quantum mechanical properties. Third, the beta functions contain the elementary higher-genus vertices of closed string field theory. This is demonstrated at simple examples and is applied to derive various results about gravitationally dressed beta functions. The possibility of interpreting closed string field theory as the theory of the renormalization group on random surfaces with random topology is considered. 
  We present a Baxterization of a two-colour generalization of the Birman--Wenzl--Murakami (BWM) algebra. Appropriately combining two RSOS-type representations of the ordinary BWM algebra, we construct representations of the two-colour algebra. Using the Baxterization, this provides new RSOS-type solutions to the Yang--Baxter equation. 
  The light--cone lattice approach to the massive Thirring model is reformulated using a local and integrable lattice Hamiltonian written in terms of discrete fermi fields. Several subtle points concerning boundary conditions, normal--ordering, continuum limit, finite renormalizations and decoupling of fermion doublers are elucidated. The relations connecting the six--vertex anisotropy and the various coupling constants of the continuum are analyzed in detail. 
  In a recent work, Unruh showed that Hawking radiation is unaffected by a truncation of free field theory at the Planck scale. His analysis was performed numerically and based on a hydrodynamical model. In this work, by analytical methods, the mathematical and physical origin of Unruh's result is revealed. An alternative truncation scheme which may be more appropriate for black hole physics is proposed and analyzed. In both schemes the thermal Hawking radiation remains unaffected even though transplanckian energies no longer appear. The universality of this result is explained by working in momentum space. In that representation, in the presence of a horizon, the d'Alembertian equation becomes a singular first order equation. In addition, the boundary conditions corresponding to vacuum before the black hole formed are that the in--modes contain positive momenta only. Both properties remain valid when the spectrum is truncated and they suffice to obtain Hawking radiation. 
  We show that the non-critical $c=1$ string at the self-dual radius is equivalent to topological strings based on the deformation of the conifold singularity of Calabi-Yau threefolds. The Penner sum giving the genus expansion of the free energy of the $c=1$ string theory at the self-dual radius therefore gives the universal behaviour of the topological partition function of a Calabi-Yau threefold near a conifold point. 
  We analyze topological objects in pure QCD in the presence of external quarks by calculating the distributions of instanton and monopole densities around static color sources. We find a suppression of the densities close to external sources and the formation of a flux tube between a static quark--antiquark pair. The similarity in the behavior of instantons and monopoles around static sources might be due to a local correlation between these topological objects. On an $8^{3} \times 4$ lattice at $\beta=5.6$, it turns out that topological quantities are correlated approximately two lattice spacings. 
  We analyze multi--matrix chain models. They can be considered as multi--component Toda lattice hierarchies subject to suitable coupling conditions. The extension of such models to include extra discrete states requires a weak form of integrability. The discrete states of the $q$--matrix model are organized in representations of $sl_q$. We solve exactly the Gaussian--type models, of which we compute several all-genus correlators. Among the latter models one can classify also the discretized $c=1$ string theory, which we revisit using Toda lattice hierarchy methods. Finally we analyze the topological field theory content of the $2q$--matrix models: we define primary fields (which are $\infty^q$), metrics and structure constants and prove that they satisfy the axioms of topological field theories. We outline a possible method to extract interesting topological field theories with a finite number of primaries. 
  We extend the model of string as a polymer of string bits to the case of superstring. We mainly concentrate on type II-B superstring, with some discussion of the obstacles presented by not II-B superstring, together with possible strategies for surmounting them. As with previous work on bosonic string we work within the light-cone gauge. The bit model possesses a good deal less symmetry than the continuous string theory. For one thing, the bit model is formulated as a Galilei invariant theory in $(D-2)+1$ dimensional space-time. This means that Poincar\'e invariance is reduced to the Galilei subgroup in $D-2$ space dimensions. Naturally the supersymmetry present in the bit model is likewise dramatically reduced. Continuous string can arise in the bit models with the formation of infinitely long polymers of string bits. Under the right circumstances (at the critical dimension) these polymers can behave as string moving in $D$ dimensional space-time enjoying the full $N=2$ Poincar\'e supersymmetric dynamics of type II-B superstring. 
  Membrane/fivebrane duality in D=11 implies Type IIA string/Type IIA fivebrane duality in D=10, which in turn implies Type IIA string/heterotic string duality in D=6. To test the conjecture, we reproduce the corrections to the 3-form field equations of the D=10 Type IIA string (a mixture of tree-level and one-loop effects) starting from the Chern-Simons corrections to the 7-form Bianchi identities of the D=11 fivebrane (a purely tree-level effect). K3 compactification of the latter then yields the familiar gauge and Lorentz Chern-Simons corrections to 3-form Bianchi identities of the heterotic string. We note that the absence of a dilaton in the D=11 theory allows us to fix both the gravitational constant and the fivebrane tension in terms of the membrane tension. We also comment on an apparent conflict between fundamental and solitonic heterotic strings and on the puzzle of a fivebrane origin of S-duality. 
  We derive anomalous Ward identities in two different approaches to the quantization of massless matter-gravity fields in 1+1 dimensions. 
  In the semiclassical analysis of black hole radiation in matter-coupled dilaton gravity, a one-parameter ``$k$"-family of measures for the path integral quantization of the matter fields is considered. The Weyl anomaly is proportional to the parameter $k$, but the black hole radiation seen by minkowskian observers at future null infinity is $k$-independent. 
  A covariant scalar representation of $iosp(d,2/2)$ is constructed and analysed in comparison with existing methods for the quantization of the scalar relativistic particle. It is found that, with appropriately defined wavefunctions, this $iosp(d,2/2)$ produced representation can be identified with the state space arising from the canonical BFV-BRST quantization of the modular invariant, unoriented scalar particle (or antiparticle) with admissible gauge fixing conditions. For this model, the cohomological determination of physical states can thus be obtained purely from the representation theory of the $iosp(d,2/2)$ algebra. 
  We show that the classical non-abelian pure Chern-Simons action is related to nonrelativistic models in (2+1)-dimensions, via reductions of the gauge connection in Hermitian symmetric spaces. In such models the matter fields are coupled to gauge Chern-Simons fields, which are associated with the isotropy subgroup of the considered symmetric space. Moreover, they can be related to certain (integrable and non-integrable) evolution systems, as the Ishimori and the Heisenberg model. The main classical and quantum properties of these systems are discussed in connection with the topological field theory and the condensed matter physics. 
  We show that the exact static, i.e. `anti-gravitating', magnetic multi monopole solutions of the Einstein/Maxwell/dilaton-YM/Higgs equations found by Kastor, London, Traschen, and the authors, for arbitrary non-zero dilaton coupling constant $a$, are equivalent to the string theory BPS magnetic monopole solutions of Harvey and Liu when $a=\sqrt{3}$. For this value of $a$, the monopole solutions also solve the equations of five-dimensional supergravity/YM theory. We also discuss some features of the dyon solutions obtained by boosting in the fifth dimension and some features of the moduli space of anti-gravitating multi-monopoles. 
  We discuss the equivalence of two dual scalar field theories in 2 dimensions. The models are derived though the elimination of different fields in the same Freedman--Townsend model. It is shown that tree $S$-matrices of these models do not coincide. The 2-loop counterterms are calculated. It turns out that while one of these models is single-charged, the other theory is multi-charged. Thus the dual models considered are non-equivalent on classical and quantum levels. It indicates the possibility of the anomaly leading to non-equivalence of dual models. 
  Constrained KP and super-KP hierarchies of integrable equations (generalized NLS hierarchies) are systematically produced through a Lie algebraic AKS-matrix framework associated to the homogeneous grading. The role played by different regular elements to define the corresponding hierarchies is analyzed as well as the symmetry properties under the Weyl group transformations. The coset structure of higher order hamiltonian densities is proven.\par For a generic Lie algebra the hierarchies here considered are integrable and essentially dependent on continuous free parameters. The bosonic hierarchies studied in \cite{{FK},{AGZ}} are obtained as special limit restrictions on hermitian symmetric-spaces.\par In the supersymmetric case the homogeneous grading is introduced consistently by using alternating sums of bosons and fermions in the spectral parameter power series.\par The bosonic hierarchies obtained from ${\hat {sl(3)}}$ and the supersymmetric ones derived from the $N=1$ affinization of $sl(2)$, $sl(3)$ and $osp(1|2)$ are explicitly constructed. \par An unexpected result is found: only a restricted subclass of the $sl(3)$ bosonic hierarchies can be supersymmetrically extended while preserving integrability. 
  The Schwinger--DeWitt expansion for the evolution operator kernel is used to investigate analytical properties of the Schr\"odinger equation solution in time variable. It is shown, that this expansion, which is in general asymptotic, converges for a number of potentials (widely used, in particular, in one-dimensional many-body problems), and besides, the convergence takes place only for definite discrete values of the coupling constant. For other values of charge the divergent expansion determines the functions having essential singularity at origin (beyond usual $\delta$-function). This does not permit one to fulfil the initial condition. So, the function obtained from the Schr\"odinger equation cannot be the evolution operator kernel. The latter, rigorously speaking, does not exist in this case. Thus, the kernel exists only for definite potentials, and moreover, at the considered examples the charge may have only quantized values. 
  Any automorphism of the Dynkin diagram of a symmetrizable Kac-Moody algebra induces an automorphism of the algebra and a mapping between its highest weight modules. For a large class of such Dynkin diagram automorphisms, we can describe various aspects of these maps in terms of another Kac-Moody algebra, the `orbit Lie algebra'. In particular, the generating function for the trace of the map on modules, the `twining character', is equal to a character of the orbit Lie algebra. Orbit Lie algebras and twining characters constitute a crucial step towards solving the fixed point resolution problem in conformal field theory. 
  An analytic expression is proposed for the three-point function of the exponential fields in the Liouville field theory on a sphere. In the classical limit it coincides with what the classical Liouville theory predicts. Using this function as the structure constant of the operator algebra we construct the four-point function of the exponential fields and verify numerically that it satisfies the conformal bootstrap equations, i.e., that the operator algebra thus defined is associative. We consider also the Liouville reflection amplitude which follows explicitly from the structure constants. 
  Electric-magnetic duality and higher dimensional analogues are obtained as symmetries in generalized coset constructions, similar to the axial-vector duality of two dimensional coset models described by Rocek and Verlinde. We also study global aspects of duality between p-forms and (d-p-2)-forms in d-manifolds. In particular, the modular duality anomaly is governed by the Euler character as in four and two dimensions. Duality transformations of Wilson line operator insertions are also considered. 
  The evaporation of a large mass black hole can be described throughout most of its lifetime by a low-energy effective theory defined on a suitably chosen set of smooth spacelike hypersurfaces. The conventional argument for information loss rests on the assumption that the effective theory is a local quantum field theory. We present evidence that this assumption fails in the context of string theory. The commutator of operators in light-front string theory, corresponding to certain low-energy observers on opposite sides of the event horizon, remains large even when these observers are spacelike separated by a macroscopic distance. This suggests that degrees of freedom inside a black hole should not be viewed as independent from those outside the event horizon. These nonlocal effects are only significant under extreme kinematic circumstances, such as in the high-redshift geometry of a black hole. Commutators of space-like separated operators corresponding to ordinary low-energy observers in Minkowski space are strongly suppressed in string theory. 
  We study the $N=1$ supersymmetric $G_2$ gauge theories with $N_f$ flavors of quarks in the fundamental vector representation. We find dynamically generated superpotentials, smooth quantum moduli space, quantum moduli space with additional mesons, non trivial IR fixed points. 
  Inflationary string cosmology backgrounds can amplify perturbations in a more efficient way than conventional inflationary backgrounds, because the perturbation amplitude may grow - instead of being constant - outside the horizon. If not gauged away, the growing mode can limit the range of validity of a linearized description of perturbations. Even in the restricted linear range, however, this enhanced amplification may lead to phenomenological consequences unexpected in the context of the standard inflationary scenario. In particular, the production of a relic graviton background strong enough to be detected in future by LIGO, and/or the generation of a stochastic electromagnetic background strong enough to seed the cosmic magnetic fields and to be responsible for the observed large scale anisotropy. 
  At large distances and in the low temperature phase, the quenched correlation functions in the 2d random phase sine-Gordon model have been argued to be of the form~: $ \bar {\vev{~[\varphi(x)-\varphi(0)]^2~}}_* = A (\log|x|) + B \ep^2 (\log|x|)^2 $, with $\ep=(T-T_c)$. However, renormalization group computations predict $B\not=0$ while variational approaches (which are supposed to be exact for models with a large number of components) give $B=0$. We introduce a large $N$ version of the random phase sine-Gordon model. Using non-Abelian bosonization and renormalization group techniques, we show that the correlation functions of our models have the above form but with a coefficient $B$ suppressed by a factor $1/N^3$ compared to $A$. 
  A weaker Haag, Narnhofer and Stein prescription as well as a weaker Hessling Quantum Equivalence Principle for the behaviour of thermal Wightman functions on an event horizon are analysed in the case of an extremal Reissner-Nordstr\"{o}m black hole in the limit of a large mass. In order to avoid the degeneracy of the metric in the stationary coordinates on the horizon, a method is introduced which employs the invariant length of geodesics which pass the horizon. First the method is checked for a massless scalar field on the event horizon of the Rindler wedge, extending the original procedure of Haag, Narnhofer and Stein onto the {\em whole horizon} and recovering the same results found by Hessling. Afterwards the HNS prescription and Hessling's prescription for a massless scalar field are analysed on the whole horizon of an extremal Reissner-Nordstr\"{o}m black hole in the limit of a large mass. It is proved that the weak form of the HNS prescription is satisfyed for all the finite values of the temperature of the KMS states, i.e., this principle does not determine any Hawking temperature. It is found that the Reissner-Nordstr\"{o}m vacuum, i.e., $T=0$ does satisfy the weak HNS prescription and it is the only state which satisfies weak Hessling's prescription, too. Finally, it is suggested that all the previously obtained results should be valid dropping the requirements of a massless field and of a large mass black hole. 
  CP-violating phases which contribute to the electric dipole moment(EDM) of the neutron are considered in the context of orbifold compactificationof the heterotic string. In particular, we study the situation where CP is spontaneously broken by moduli fields acquiring, in general, complex expectation values at the minimum of duality invariant low energy effective potentials. We show, by explicit minimization of such a potential in the case of the ${\bf Z}_{6}-{\rm IIb}$ orbifold, that it is the presence of so called Green-Schwarz anomaly coefficients $\delta_{\rm GS}^{i} $, that leads to significant CP violating expectation values of the moduli. By evaluating the soft supersymmetry breaking moduli dependent $A$ and $B$ terms in this model, we find that the experimental bounds $\Phi (A) $, $ \Phi (B) $ $\leq 5 \times 10^{-3} $ are exceeded for a particular range of values of the auxiliary field of the $S$ modulus. 
  We consider generic features of eleven dimensional supergravity compactified down to five dimensions on an arbitrary Calabi-Yau threefold. 
  A supercurrent superfield whose components include a conserved energy-momentum tensor and supersymmetry current as well as a (generally broken) R-symmetry current is constructed for a generic effective N=1 supersymmetric gauge theory. The general form of the R-symmetry breaking is isolated. Included within the various special cases considered is the identification of those models which exhibit an unbroken R-symmetry. One such example corresponds to a non-linearly realized gauge symmetry where the chiral field R-weight is required to vanish. 
  In this letter, we present the Parkes-Siegel formulation for the massive Abelian $N$$=$$1$ super-{\QED} coupled to a self-dual supermultiplet, by introducing a chiral multiplier superfield. We show that after carrying out a suitable dimensional reduction from ($2$$+$$2$) to ($1$$+$$2$) dimensions, and performing some necessary truncations, the simple supersymmetric extension of the ${\tau}_{3}$QED$_{1+2}$ coupled to a Chern-Simons term naturally comes out. 
  We discuss the $2+1$ dimensional Abelian Higgs model coupled to $N=2$ supergravity. We construct the supercharge algebra and, from it, we show that the mass of classical static solutions is bounded from below by the topological charge. As it happens in the global case, half of the supersymmetry is broken when the bound is attained and Bogomol'nyi equations, resulting from the unbroken supersymmetry, hold. These equations, which correspond to gravitating vortices, include a first order self-duality equation whose integrability condition reproduces the Einstein equation. 
  We present evidence for renormalization group fixed points with dual magnetic descriptions in fourteen new classes of four-dimensional $N=1$ supersymmetric models. Nine of these classes are chiral and many involve two or three gauge groups. These theories are generalizations of models presented earlier by Seiberg, by Kutasov and Schwimmer, and by the present authors. The different classes are interrelated; one can flow from one class to another using confinement or symmetry breaking. 
  A procedure for Pauli-Villars regularization of locally and globally supersymmetric theories is described. Implications for specific theories, especially those obtained from superstrings, are discussed with emphasis on the role of field theory anomalies. 
  We investigate the compactification of D=11 supergravity to D=5,4,3, on compact manifolds of holonomy $SU(3)$ (Calabi-Yau), $G_2$, and $Spin(7)$, respectively, making use of examples of the latter two cases found recently by Joyce. In each case the lower dimensional theory is a Maxwell/Einstein supergravity theory. We find evidence for an equivalence, in certain cases, with heterotic string compactifications from D=10 to D=5,4,3, on compact manifolds of holonomy $SU(2)$ ($K_3\times S^1$), $SU(3)$, and $G_2$, respectively. Calabi-Yau manifolds with Hodge numbers $h_{1,1}=h_{1,2}=19$ play a significant role in the proposed equivalences. 
  We study two-dimensional WZW models with target space a nonreductive Lie group. Such models exist whenever the Lie group possesses a bi-invariant metric. We show that such WZW models provide a lagrangian description of the nonreductive (affine) Sugawara construction. We investigate the gauged WZW models and we prove that gauging a diagonal subgroup results in a conformal field theory which can be identified with a coset construction. A large class of exact four-dimensional string backgrounds arise in this fashion. We then study the topological conformal field theory resulting from the $G/G$ coset. We identify the Kazama algebra extending the BRST algebra, and the BV algebra structure in BRST cohomology which it induces. 
  A finite-dimensional Lie algebra is called (symmetric) self-dual, if it possesses an invariant nondegenerate (symmetric) bilinear form. Symmetric self-dual Lie algebras have been studied by Medina and Revoy, who have proven a very useful theorem about their structure. In this paper we prove a refinement of their theorem which has wide applicability in Conformal Field Theory, where symmetric self-dual Lie algebras start to play an important role due to the fact that they are precisely the Lie algebras which admit a Sugawara construction. We also prove a few corollaries which are important in Conformal Field Theory. (This paper provides mathematical details of results used, but only sketched, in the companion paper hep-th/9506151.) 
  Parasupersymmetric quantum mechanics is exploited to introduce a topological invariant associated with a pair of parameter dependent Fredholm (respectively elliptic differential) operators satisfying two compatibility conditions. An explicit algebraic expression for this topological invariant is provided. The latter identifies the parasupersymmetric topological invariant with the sum of the analytic (Atiyah-Singer) indices of the corresponding operators. 
  It is shown that, by defining a suitable energy momentum tensor, the field equations of general relativity admit a line element of Yukawa potential as an exact solution. It is also shown that matter that produces strong force may be negative, in which case there would be no Schwarzschild-like singularity 
  We study the theory of non-relativistic matter with non-Abelian $U(2)$ Chern-Simons gauge interaction in $(2+1)$ dimensions. We adopt the mean field approximation in the current-algebra formulation already applied to the Abelian anyons. We first show that this method is able to describe both ``boson-based'' and ``fermion-based'' anyons and yields consistent results over the whole range of fractional statistics. In the non-Abelian theory, we find a superfluid (and superconductive) phase, which is smoothly connected with the Abelian superfluid phase originally discovered by Laughlin. The characteristic massless excitation is the Goldstone particle of the specific mechanism of spontaneous symmetry breaking. An additional massive mode is found by diagonalizing the non-Abelian, non-local, Hamiltonian in the radial gauge. 
  We study exhaustively the solution-generating transformations (dualities) that occur in the context of the low-energy effective action of superstring theory. We first consider target-space duality (``T duality'') transformations in absence of vector fields. We find that for one isometry the full duality group is (SO^{\uparrow}(1,1))^{3} x D_{4}, the discrete part (D_{4}) being non-Abelian. We, then, include non-Abelian Yang--Mills fields and find the corresponding generalization of the T duality transformations. We study the \alpha^{\prime} corrections to these transformations and show that the T duality rules considerably simplify if the gauge group is embedded in the holonomy group. Next, in the case in which there are Abelian vector fields, we consider the duality group that includes the transformation introduced by Sen that rotates among themselves components of the metric, axion and vector field. Finally we list the duality symmetries of the Type II theories with one isometry. 
  Integrability and supersymmetry of the supersymmetric extension of the sine-Gordon theory on a half-line are examined and the boundary potential which preserves both the integrability and supersymmetry on the bulk is derived. It appears that unlike the boundary bosonic sine-Gordon theory, integrability and supersymmetry strongly restrict the form and parameters of the boundary potential, so that no free parameter in the boundary term is allowed up to a choice of signs. 
  In this talk, we briefly review the basic concepts of anomalous gauge theories. It has been known for some time how theories with local anomalies can be handled. Recently it has been pointed out that global anomalies, which obstruct the quantization of certain gauge theories in the temporal gauge, get bypassed in canonical quantization. 
  String theory is currently the most promising theory to explain the spectrum of the elementary particles and their interactions. One of its most important features is its large symmetry group, which contains the conformal transformations in two dimensions as a subgroup. At quantum level, the symmetry group of a theory gives rise to differential equations between correlation functions of observables. We show that these Ward-identities are equivalent to Operator Product Expansions (OPEs), which encode the short-distance singularities of correlation functions with symmetry generators. The OPEs allow us to determine algebraically many properties of the theory under study. We analyse the calculational rules for OPEs, give an algorithm to compute OPEs, and discuss an implementation in Mathematica. There exist different string theories, based on extensions of the conformal algebra to so-called W-algebras. These algebras are generically nonlinear. We study their OPEs, with as main results an efficient algorithm to compute the beta-coefficients in the OPEs, the first explicit construction of the WB_2-algebra, and criteria for the factorisation of free fields in a W-algebra. An important technique to construct realisations of W-algebras is Drinfel'd- Sokolov reduction. The method consists of imposing certain constraints on the elements of an affine Lie algebra. We quantise this reduction via gauged WZNW-models. This enables us in a theory with a gauged W-symmetry, to compute exactly the correlation functions of the effective theory. Finally, we investigate the critical W-string theories based on an extension of the conformal algebra with one symmetry generator of dimension N. We clarify how the spectrum of this theory forms a minimal model of the W_N-algebra. 
  It is shown that the heterotic string emerges as a soliton in the type I superstring theory in ten dimensions. The collective coordinates of the soliton are described by a smooth, chiral worldsheet theory. There are eight bosonic and eight right-moving fermionic zero modes that arise from the partially broken supertranslations. In addition, there are 496 charged bosonic zero modes of the gauge field that describe a left-moving WZNW model on a $spin(32)/{Z_2}$ group manifold. Small, stable loops of the solitonic string furnish the massive states required by duality that transform as spinors of $spin(32)$. 
  Perturbative renormalization group theory is developed as a unified tool for global asymptotic analysis. With numerous examples, we illustrate its application to ordinary differential equation problems involving multiple scales, boundary layers with technically difficult asymptotic matching, and WKB analysis. In contrast to conventional methods, the renormalization group approach requires neither {\it ad hoc\/} assumptions about the structure of perturbation series nor the use of asymptotic matching. Our renormalization group approach provides approximate solutions which are practically superior to those obtained conventionally, although the latter can be reproduced, if desired, by appropriate expansion of the renormalization group approximant. We show that the renormalization group equation may be interpreted as an amplitude equation, and from this point of view develop reductive perturbation theory for partial differential equations describing spatially-extended systems near bifurcation points, deriving both amplitude equations and the center manifold. 
  The four-dimensional topological Yang-Mills theory with two anticommuting charges is naturally formulated on K\"ahler manifolds. By using a superspace approach we clarify the structure of the Faddeev-Popov sector and determine the total action. This enables us to perform perturbation theory around any given instanton configuration by manifestly maintaining all the symmetries of the topological theory. The superspace formulation is very useful for recognizing a trivial observable (i.e. having vanishing correlation functions only) as the highest component of a gauge invariant superfield. As an example of non-trivial observables we construct the complete solution to the simultaneous cohomology problem of both fermionic charges. We also show how this solution has to be used in order to make Donaldson's interpretation possible. 
  We propose actions for non-linear sigma models on cosets $G/H$ in 2+1 dimensions that include the most general non-linear realizations of Chern-Simons terms. When $G$ is simply connected and $H$ contains $r$ commuting U(1) factors, there are $r$ different topologically conserved charges and generically $r$ different types of topological solitons. Soliton spin fractionalizes as a function of the Chern-Simons couplings, with independent spins associated to each species of soliton charge, as well as to pairs of different charges. This model of soliton spin fractionalization generalizes to arbitrary $G/H$ a model of Wilczek and Zee for one type of soliton. 
  We classify the moduli spaces of the four-dimensional topological half-flat gravity models by using the canonical bundle. For a K3-surface or four-dimensional torus, they describe an equivalent class of a trio of the Einstein-Kahler forms ( the hyperkahler forms ). We calculate the dimensions of these moduli spaces by using the Atiyah-Singer Index theorem. We mention the partition function and the possibility of the observables in the Witten-type topological half-flat gravity model case. 
  We present a theory of 'maximal' super-KP(SKP) hierarchy whose flows are maximally extended to include all those of known SKP hierarchies, including, for example, the MRSKP hierarchy of Manin and Radul and the Jacobian SKP(JSKP) introduced by Mulase and Rabin. It is shown that SKP hierarchies has a natural field theoretic description in terms of the B-C system, in analogous way as the ordinary KP hierarchy. For this SKP hierarchy, we construct the vertex operators by using Kac-van de Leur superbosonization. The vertex operators act on the \(\tau\)-function and then produce the wave function and the dual wave function of the hierarchy. Thereby we achieve the description of the 'maximal' SKP hierarchy in terms of the \(\tau\)-function, which seemed to be lacking till now. Mutual relations among the SKP hierarchies are clarified. The MRSKP and the JSKP hierarchies are obtained as special cases when the time variables are appropriately restricted. 
  We show that the anomaly of nonlocal symmetry can be canceled by the well-known Wess-Zumino acton which cancels the gauge anomaly in the two-dimensional chiral electrodynamics. 
  Analytic Bethe ansatz is executed for a wide class of finite dimensional $U_q(B^{(1)}_r)$ modules. They are labeled by skew-Young diagrams which, in general, contain a fragment corresponding to the spin representation. For the transfer matrix spectra of the relevant vertex models, we establish a number of formulae, which are $U_q(B^{(1)}_r)$ analogues of the classical ones due to Jacobi-Trudi and Giambelli on Schur functions. They yield a full solution to the previously proposed functional relation ($T$-system), which is a Toda equation 
  We present the Higgs mechanism in (0,2) compactifications. The existence of a vector bundle data duality (VBDD) in $(0,2)$ compactifications which is present at the Landau-Ginzburg point allows us to connect in a smooth manner theories with different gauge groups with the same base manifold and same number of effective generations. As we move along the Kahler moduli space of the theories with $E_6$ gauge group, some of the gauginos pick up masses and break the gauge group to $SO(10)$ or $SU(5)$. 
  We conjecture an exact S-matrix for the scattering of solitons in $d_{n+1}^{(2)}$ affine Toda field theory in terms of the R-matrix of the quantum group $U_q(c_n^{(1)})$. From this we construct the scattering amplitudes for all scalar bound states (breathers) of the theory. This S-matrix conjecture is justified by detailed examination of its pole structure. We show that a breather-particle identification holds by comparing the S-matrix elements for the lowest breathers with the S-matrix for the quantum particles in real affine Toda field theory, and discuss the implications for various forms of duality. 
  We study one-loop correction to the Chern-Simons coefficient $\kappa=k/4\pi$ in $N=1,2,3$ supersymmetric Yang-Mills Chern-Simons systems. In the pure bosonic case, the shift of the parameter $k$ is known to be $k\rightarrow k + c_v$, where $c_v$ is the quadratic Casimir of the gauge group. In the $N=1$ case, the fermionic contribution cancels the bosonic contribution by half and the shift is $k \rightarrow k+ c_v/2$, making the theory anomalous if $c_v$ is odd. In the $N=2,3$ cases, the fermionic contribution cancels the bosonic contribution completely and there is no correction. We also calculate the mass corrections, showing the supersymmetry is preserved. As the matter fields decouple from the gauge field in the pure Chern-Simons limit, this work sheds some light on the regularization dependency of the correction in pure Chern-Simons systems. We also discuss the implication to the case when the gauge symmetry is spontaneously broken by the Higgs mechanism. 
  This essay is a tour around many of the lesser known pregeometric models of physics, as well as the mainstream approaches to quantum gravity, in search of common themes which may provide a glimpse of the final theory which must lie behind them. 
  This short article presents a table of new equations which can be regarded as the generalized equations of the dispersionless limit of several nonlinear equations. From the definition expressed in an algebraic formula, one can get an equation for any positive numbers p and q. The equations were calculated by using the computers and were examined by hand-calculation up to p=10. Relations with some dispersionless hierarchies are mentioned. 
  A discussion of the number of degrees of freedom, and their dynamical properties, in higher-derivative gravitational theories is presented. The complete non-linear sigma model for these degrees of freedom is exhibited using the method of auxiliary fields. As a by-product we present a consistent non-linear coupling of a spin-2 tensor to gravitation. It is shown that non-vanishing $(C_{\mu\nu\alpha\beta})^{2}$ terms arise in $N=1$, $D=4$ superstring Lagrangians due to one-loop radiative corrections with light field internal lines. 
  We continue our study of matrix models of dually weighted graphs. Among the attractive features of these models is the possibility to interpolate between ensembles of regular and random two-dimensional lattices, relevant for the study of the crossover from two-dimensional flat space to two-dimensional quantum gravity. We further develop the formalism of large $N$ character expansions. In particular, a general method for determining the large $N$ limit of a character is derived. This method, aside from being potentially useful for a far greater class of problems, allows us to exactly solve the matrix models of dually weighted graphs, reducing them to a well-posed Cauchy-Riemann problem. The power of the method is illustrated by explicitly solving a new model in which only positive curvature defects are permitted on the surface, an arbitrary amount of negative curvature being introduced at a single insertion. 
  In a space of $d=15 $ Grassmann coordinates, two types of generators of the Lorentz transformations, one of spinorial and the other of vectorial character, both linear operators in Grassmann space, forming the group $ SO(1,14) $ which contains as subgroups $ SO(1,4) $ and $ SO(10) $ ${\supset SU(3)} { \times SU(2)} { \times U(1)} $, define the fundamental and the adjoint representations of the group, respectively. The eigenvalues of the commuting operators can be identified with spins of fermionic and bosonic fields $ (SO(1,4)) $, as well as with their Yang-Mills charges $ (SU(3)$, $ SU(2)$, $ U(1)) $, offering the unification of not only all Yang - Mills charges but of all the internal degrees of freedom of fermionic and bosonic fields - Yang - Mills charges and spins - and accordingly of all interactions - gauge fields and gravity. The theory suggests that elementary particles are either in the "spinorial" representations with respect to spins and all charges, or they are in the "vectorial" representations with respect to spins and all charges, which indeed is the case with the quarks, the leptons and the gauge bosons.   The algebras of the two kinds of generators of Lorentz transformations in Grassmann space were studied and the representations are commented on. 
  We develop a formalism to investigate the behavior of quantum field and quantum ground state when the field is coupled to perturbation that periodically oscillates. Working in the Schroedinger picture of quantum field theory, we confirm that the phenomenon of parametric resonance in the classical theory implies an instability of quantum vacuum, and correspondingly it gives rise to catastrophic particle production if the oscillation lasts indefinitely; the produced number of particles exponentially increases without bound as time proceeds. The density matrix describing the limiting stage of the quantum state is determined by a small set of parameters. Moreover, the energy spectrum and the intensity of produced particles are worked out in greatest detail in the limit of weak coupling or small amplitude perturbation. In the case of strong coupling or large amplitude perturbation the leading adiabatic formula is derived. Application to cosmological fate of weakly interacting spinless fields (WISF) such as the invisible axion, the Polonyi, and the modular fields is discussed. Although very little effect is expected on the invisible axion, the Polonyi type field has a chance that it catastrophically decays at an early epoch without much production of entropy, provided that an intrinsic coupling is large enough. 
  Fractional supersymmetry denotes a generalisation of supersymmetry which may be constructed using a single real generalised Grassmann variable, $\theta = \bar{\theta}, \, \theta^n = 0$, for arbitrary integer $n = 2, 3, ...$. An explicit formula is given in the case of general $n$ for the transformations that leave the theory invariant, and it is shown that these transformations possess interesting group properties. It is shown also that the two generalised derivatives that enter the theory have a geometric interpretation as generators of left and right transformations of the fractional supersymmetry group. Careful attention is paid to some technically important issues, including differentiation, that arise as a result of the peculiar nature of quantities such as $\theta$. 
  The string one loop renormalization of the gauge coupling constants is examined in abelian orbifold models. The contributions to string threshold corrections independent of the compactification moduli fields are evaluated numerically for several representative examples of orbifold models. We consider cases with standard and non-standard embeddings as well as cases with discrete Wilson lines background fields which match reasonably well with low energy phenomenology. We examine one loop gauge coupling constants unification in a description incorporating the combined effects of moduli dependent and independent threshold corrections, an adjustable Kac-Moody level for the hypercharge group factor and a large mass threshold associated with an anomalous $U(1)$ mechanism. 
  We show that, in quaternion quantum mechanics with a complex geometry, the minimal four Higgs of the unbroken electroweak theory naturally determine the quaternion invariance group which corresponds to the Glashow group. Consequently, we are able to identify the physical significance of the anomalous Higgs scalar solutions. We introduce and discuss the complex projection of the Lagrangian density. 
  Since the work of Bershadsky and Ooguri and Feigin and Frenkel it is well known that correlators of $SL(2)$ current algebra for admissible representations should reduce to correlators for conformal minimal models. A precise proposal for this relation has been given at the level of correlators: When $SL(2)$ primary fields are expressed as $\phi_j(z_n,x_n)$ with $x_n$ being a variable to keep track of the $SL(2)$ representation multiplet (possibly infinitely dimensional for admissible representations), then the minimal model correlator is supposed to be obtained simply by putting all $x_n=z_n$. Although strong support for this has been presented, to the best of our understanding a direct, simple proof seems to be missing so in this paper we present one based on the free field Wakimoto construction and our previous study of that in the present context. We further verify that the explicit $SL(2)$ correlators we have published in a recent preprint reduce in the above way, up to a constant which we also calculate. We further discuss the relation to more standard formulations of hamiltonian reduction. 
  The critical curve ${\cal C}$ on which ${\rm Im}\,\hat\tau =0$, $\hat\tau=a_D/a$, determines hyperbolic domains whose Poincar\'e metric is constructed in terms of $a_D$ and $a$. We describe ${\cal C}$ in a parametric form related to a Schwarzian equation and prove new relations for $N=2$ Super $SU(2)$ Yang-Mills. In particular, using the Koebe 1/4-theorem and Schwarz's lemma, we obtain inequalities involving $u$, $a_D$ and $a$, which seem related to the Renormalization Group. Furthermore, we obtain a closed form for the prepotential as function of $a$. Finally, we show that $\partial_{\hat\tau} \langle {\rm tr}\,\phi^2\rangle_{\hat \tau}={1\over 8\pi i b_1}\langle \phi\rangle_{\hat\tau}^2$, where $b_1$ is the one-loop coefficient of the beta function. 
  We consider corrections to the entropy of a black hole from an $O(N)$ invariant linear $\s$-model. We obtain the entropy from a $1/N$ expansion of the partition function on a cone. The entropy arises from diagrams which are analogous to those introduced by Susskind and Uglum to explain black hole entropy in string theory. The interpretation of the \sm entropy depends on scale. At short distances, it has a state counting interpretation, as the entropy of entanglement of the $N$ fields $\pa$. In the infrared, the effective theory has a single composite field $\s \sim \pa \pa$, and the state counting interpretation of the entropy is lost. 
  Linear connections are introduced on a series of noncommutative geometries which have commutative limits. Quasicommutative corrections are calculated. 
  We have found that the Regge gravity \cite{regge,sorkin}, can be represented as a $superposition$ of less complicated theory of random surfaces with $Euler~character$ as an action. This extends to Regge gravity our previous result \cite{savvidy}, which allows to represent the gonihedric string \cite{savvidy1} as a superposition of less complicated theory of random paths with $curvature$ action. We propose also an alternative linear action $A(M_{4})$ for the four and high dimensional quantum gravity. From these representations it follows that the corresponding partition functions are equal to the product of Feynman path integrals evaluated on time slices with curvature and length action for the gonihedric string and with Euler character and gonihedric action for the Regge gravity. In both cases the interaction is proportional to the overlapping sizes of the paths or surfaces on the neighboring time slices. On the lattice we constructed spin system with local interaction, which have the same partition function as the quantum gravity. The scaling limit is discussed. 
  Relevant physical models are described by singular Lagrangians, so that their Hamiltonian description is based on the Dirac theory of constraints. The qualitative aspects of this theory are now understood, in particular the role of the Shanmugadhasan canonical transformation in the determination of a canonical basis of Dirac's observables allowing the elimination of gauge degrees of freedom from the classical description of physical systems. This programme was initiated by Dirac for the electromagnetic field with charged fermions. Now Dirac's observables for Yang-Mills theory with fermions (whose typical application is QCD) have been found in suitable function spaces where the Gribov ambiguity is absent. Also the ones for the Abelian Higgs model are known and those for the $SU(2) \times U(1)$ electroweak theory with fermions are going to be found with the same method working for the Abelian case. The main task along these lines will now be the search of Dirac's observables for tetrad gravity in the case of asymptotically flat 3-manifolds. The philosophy behind this approach is ``first reduce, then quantize": this requires a global symplectic separation of the physical variables from the gauge ones so that the role of differential geometry applied to smooth field configurations is dominating, in contrast with the standard approach of ``first quantizing, then reducing", where, in the case of gauge field theory, the reduction process takes place on distributional field configurations, which dominate in quantum measures. This global separation has been accomplished till now, at least at a heuristic level, and one is going to have a classical (pseudoclassical for the fermion) variables basis for the physical description of the $SU(3)\times SU(2)\times U(1)$ standard model; instead, with tetrad gravity one expects to 
  Unitary representations of centrally extended mapping class groups $\tilde M_{g,1}, g\geq 1$ are given in terms of a rational Hopf algebra $H$, and a related generalization of the Verlinde formula is presented. Formulae expressing the traces of mapping class group elements in terms of the fusion rules, quantum dimensions and statistics phases are proposed. 
  The canonical structure of higher dimensional pure Chern-Simons theories is analysed. It is shown that these theories have generically a non-vanishing number of local degrees of freedom, even though they are obtained by means of a topological construction. This number of local degrees of freedom is computed as a function of the spacetime dimension and the dimension of the gauge group. 
  In this first paper we begin the application of variational methods to renormalisable asymptotically free field theories, using the Gross-Neveu model as a laboratory. This variational method has been shown to lead to a numerically convergent sequence of approximations for the anharmonic oscillator. Here we perform a sample calculation in lowest orders, which shows the superficially disastrous situation of variational calculations in quantum field theory, and how in the large-$N$ limit all difficulties go away, as a warm up exercise for the finite-$N$ case and for QCD. 
  We develop exact field theoretic methods to treat turbulence when the effect of pressure is negligible. We find explicit forms of certain probability distributions, demonstrate that the breakdown of Galilean invariance is responsible for intermittency and establish the operator product expansion. We also indicate how the effects of pressure can be turned on perturbatively. 
  We discuss the structure of the vacua in $O(2)$ topologically massive gauge theory on a torus. Since $O(2)$ has two connected components, there are four classical vacua. The different vacua impose different boundary conditions on the gauge potentials. We also discuss the non-perturbative transitions between the vacua induced by vortices of the theory. 
  It is shown that the renormalized finite temperature effective potential for continuum $SU(2)$ Yang-Mills theory develops a non-perturbative minimum for sufficiently strong coupling, i.e. below a critical temperature. The corresponding phase can be the candidate for the confining phase of the continuum theory and becomes energetically favoured basicly due to the decay of the $A^0$ condensate into three gluons. 
  The non-linear sigma model of the dimensionally reduced Einstein (-Maxwell) theory is diagonally embedded into that of the two-dimensional heterotic string theory. Consequently, the embedded string backgrounds satisfy the (electro-magnetic) Ernst equation. In the pure Einstein theory, the Matzner-Misner SL(2,{\bf R}) transformation can be viewed as a change of conformal structure of the compactified flat two-torus, and in particular its integral subgroup SL(2,{\bf Z}) acts as the modular transformation. The Ehlers SL(2,{\bf R}) and SL(2,{\bf Z}) similarly act on another torus whose conformal structure is induced through the Kramer-Neugebauer involution. Either of the Matzner-Misner and the Ehlers SL(2,{\bf Z}) can be embedded to a special T-duality, and if the former is chosen, then the Ehlers SL(2,{\bf Z}) is shown to act as the S-duality on the four-dimensional sector. As an application we obtain some new colliding string wave solutions by using this embedding as well as the inverse scattering method. 
  We present an exact solution of the $O(n)$ model on a random lattice. The coupling constant space of our model is parametrized in terms of a set of moment variables and the same type of universality with respect to the potential as observed for the one-matrix model is found. In addition we find a large degree of universality with respect to $n$; namely for $n\in ]-2,2[$ the solution can be presented in a form which is valid not only for any potential, but also for any $n$ (not necessarily rational). The cases $n=\pm 2$ are treated separately. We give explicit expressions for the genus zero contribution to the one- and two-loop correlators as well as for the genus one contribution to the one-loop correlator and the free energy. It is shown how one can obtain from these results any multi-loop correlator and the free energy to any genus and the structure of the higher genera contributions is described. Furthermore we describe how the calculation of the higher genera contributions can be pursued in the scaling limit. 
  The heterotic string occurs as a soliton of the type I superstring in ten dimensions, supporting the conjecture that these two theories are equivalent.  The conjecture that the type IIB string is self-dual, with the strong coupling dynamics described by a dual type IIB theory, is supported by the occurrence of the dual string as a Ramond-Ramond soliton of the weakly-coupled theory. 
  We construct the family of spin chain Hamiltonians, which have affine $U_q g$ guantum group symmetry. Their eigenvalues coincides with the eigenvalues of the usual spin chain Hamiltonians which have non-affine $U_q g_0$ quantum group symmetry, but have the degeneracy of levels, corresponding to affine $U_q g$. The space of states of these chaines are formed by the tensor product of the fully reducible representations. 
  We begin an investigation of supersymmetric theories based on exceptional groups. The flat directions are most easily parameterized using their correspondence with gauge invariant polynomials. Symmetries and holomorphy tightly constrain the superpotentials, but due to multiple gauge invariants other techniques are needed for their full determination. We give an explicit treatment of $G_2$ and find gaugino condensation for $N_f\leq 2$, and an instanton generated superpotential for $N_f=3$. The analogy with $SU(N_c)$ gauge theories continues with modified and unmodified quantum moduli spaces for $N_f=4$ and $N_f=5$ respectively, and a non-Abelian Coulomb phase for $N_f\geq6$. Electric variables suffice to describe this phase over the full range of $N_f$. The appendix gives a self-contained introduction to $G_2$ and its invariant tensors. 
  In this talk I present recent results on Lorentz covariant correlation functions $\langle q(p_1)\overline{q}(p_2)\rangle$ on the cone $p^2=0$. In particular, chiral symmetry breaking terms are constructed which resemble fermionic 2--point functions of 2--D CFT up to a scalar factor. 
  The quantum affine symmetry of the sine-Gordon theory at q^2 = 1, which occurs at the reflectionless points, is studied. Conserved currents that correspond to the closure of simple root generators are considered, and shown to be local. We argue that they satisfy the affine sl(2) algebra. Examples of these currents are explicitly constructed. 
  Working directly on affine Lie groups, we construct several new formulations of the WZW model. In one formulation WZW is expressed as a one-dimensional mechanical system whose variables are coordinates on the affine Lie group. When written in terms of the affine group element, this formulation exhibits a two-dimensional WZW term. In another formulation WZW is written as a two-dimensional field theory, with a three-dimensional WZW term, whose fields are coordinates on the affine group. On the basis of these equivalent formulations, we develop a translation dictionary in which the new formulations on the affine Lie group are understood as mode formulations of the conventional WZW formulation on the Lie group. Using this dictionary, we also express WZW as a three-dimensional field theory on the Lie group with a four-dimensional WZW term. 
  Following the work of Sen, we consider the correspondence between extremal black holes and string states in the context of the entropy. We obtain and study properties of electrically charged black hole backgrounds of tree level heterotic string theory compactified on a $p$ dimensional torus, for $D=(10-p)=4 \ldots 9$. We study in particular a one--parameter extremal class of these black holes, the members of which are shown to be supersymmetric. We find that the entropy of such an extremal black hole, when calculated at the stringy stretched horizon, scales in such a way that it can be identified with the entropy of the elementary string state with the corresponding quantum numbers. 
  The energy and momentum spectrum of the spin models constructed from the vector representation of the quantized affine algebras of type $\B$ and $\D$ are computed using the approach of Davies et al. \cite{DFJMN92}. The results are for the anti-ferromagnetic (massive) regime, and they agree with the mass spectrum found from the factorized S--matrix theory by Ogievetsky et al. \cite{ORW87}. The other new result is the explicit realization of the fusion construction for the quantized affine algebras of type $\B$ and $\D$.} 
  The effective action for 2d-gravity in Weyl-invariant regularization is extended to supersymmetric case. The super area-preserving invariance and cocyclic properties under general supergravitational transformations of the last action is shown. 
  The Hamiltonian (BFV) and Lagrangian (BV) quantization schemes are proved to be equivalent perturbatively to each other. It is shown in particular that the quantum master equation being treated perturbatively possesses a local formal solution. 
  Functional determinants on various domains of the sphere and flat space are presented for scalar and spinor fields. 
  We examine 4-dimensional string backgrounds compactified over a two torus. There exist two alternative effective Lagrangians containing each two $SL(2)/U(1)$ sigma-models. Two of these sigma-models are the complex and the K\"ahler structures on the torus. The effective Lagrangians are invariant under two different $O(2,2)$ groups and by the successive applications of these groups the affine $\widehat{O}(2,2)$ Kac-Moody is emerged. The latter has also a non-zero central term which generates constant Weyl rescalings of the reduced 2-dimensional background. In addition, there exists a number of discrete symmetries relating the field content of the reduced effective Lagrangians. 
  The one-loop quantum corrections to geometry and thermodynamics of black hole are studied for the two-dimensional RST model. We chose boundary conditions corresponding to the eternal black hole being in the thermal equilibrium with the Hawking radiation. The equations of motion are exactly integrated. The one of the solutions obtained is the constant curvature space-time with dilaton being a constant function. Such a solution is absent in the classical theory. On the other hand, we derive the quantum-corrected metric (\ref{solution}) written in the Schwarzschild like form which is a deformation of the classical black hole solution \cite{5d}. The space-time singularity occurs to be milder than in classics and the solution admits two asymptotically flat black hole space-times lying at "different sides" of the singularity. The thermodynamics of the classical black hole and its quantum counterpart is formulated. The thermodynamical quantities (energy, temperature, entropy) are calculated and occur to be the same for both the classical and quantum-corrected black holes. So, no quantum corrections to thermodynamics are observed. The possible relevance of the results obtained to the four-dimensional case is discussed. 
  We clarify the role of approximate S-duality in effective supergravity theories that are the low energy limits of string theories, and show how this partial symmetry may be used to constrain effective lagrangians for gaugino condensation. 
  We consider the relation between the five-dimensional BF model and a four-dimensional local current algebra from the point of view of perturbative local quantum field theory. We use an axial gauge fixing procedure and show that it allows for a well defined theory which actually can be solved exactly. 
  In this paper we consider the contributions of anomalous commutators to various QCD sum rules. Using a combination of the BJL limit with the operator product expansion the results are presented in terms of the vacuum condensates of gauge invariant operators. It is demonstrated that the anomalous contributions are no negligible and reconcile various apparently contradictory calculations. 
  We give a brief introduction to the study of the algebraic structures -- and their geometrical interpretations -- which arise in the BRST construction of a conformal string background. Starting from the chiral algebra $\cA$ of a string background, we consider a number of elementary but universal operations on the chiral algebra. From these operations we deduce a certain fundamental odd Poisson structure, known as a Gerstenhaber algebra, on the BRST cohomology of $\cA$. For the 2D string background, the correponding G-algebra can be partially described in term of a geometrical G-algebra of the affine plane $\bC^2$. This paper will appear in the proceedings of {\it Strings 95}. 
  We study the effect of nonzero temperature on the induced electric charge around a Dirac monopole. While at zero temperature the charge is known to be proportional to a CP violating $\theta$ parameter, we find that at high temperature the charge is proportional to sin $\theta$. Other features of the charge at nonzero temperature are discussed. We also compute the induced charge at nonzero temperature around an Aharonov-Bohm flux string in $2+1$ dimensions and compare the result with an index theorem, and also with the electron-monopole problem in $3+1$ dimensions. 
  We give an explicit construction of the quantum-group generators ---local, semi-local, and global --- in terms of the group-valued quantum fields $\tilde g$ and $\tilde g^{-1}$ in the Wess-Zumino-Novikov-Witten (WZNW) theory. The algebras among the generators and the fields make concrete and clear the bi-module properties of the $\tilde g$ and the $\tilde g^{-1}$ fields. We show that the correlation functions of the $\tilde g$ and $\tilde g^{-1}$ fields in the vacuum state defined through the semi-local quantum-group generator satisfy a set of quantum-group difference equations. We give the explicit solution for the two point function. A similar formulation can also be done for the quantum Self-dual Yang-Mills (SDYM) theory in four dimensions. 
  Classical solutions for a four-dimensional Minkowskian string effective action and an Euclidean one with cosmological constant term are derived. The former corresponds to electrovac solutions whereas the later solutions are identified as gravitational instanton solutions for Fubini-Study metric. The symmetries of the effective actions are identified and new classical solutions are generated by implementing appropriate noncompact transformations. The S-duality transformations on the equations of motion are discussed and it is found that they are S-duality noninvariant due to the presence of cosmological constant term. 
  The string equations of motion and constraints are solved near the horizon and near the singularity of a Schwarzschild black hole. In a conformal gauge such that $\tau = 0$ ($\tau$ = worldsheet time coordinate) corresponds to the horizon ($r=1$) or to the black hole singularity ($r=0$), the string coordinates express in power series in $\tau$ near the horizon and in power series in $\tau^{1/5}$ around $r=0$. We compute the string invariant size and the string energy-momentum tensor. Near the horizon both are finite and analytic. Near the black hole singularity, the string size, the string energy and the transverse pressures (in the angular directions) tend to infinity as $r^{-1}$. To leading order near $r=0$, the string behaves as two dimensional radiation. This two spatial dimensions are describing the $S^2$ sphere in the Schwarzschild manifold. 
  We consider the (A_n,A_1) Y-system arising in the Thermodynamic Bethe Ansatz. We prove the periodicity of solutions of this Y-system conjectured by Al.B. Zamolodchikov, and the dilogarithm identities conjectured by F. Gliozzi and R. Tateo. 
  Affine Toda field theory with a pure imaginary coupling constant is a non-hermitian theory. Therefore the solutions of the equation of motion are complex. However, in $1+1$ dimensions it has many soliton solutions with remarkable properties, such as real total energy/momentum and mass. Several authors calculated quantum mass corrections of the solitons by claiming these solitons are stable. We show that there exists a large class of classical solutions which develops singularity after a finite lapse of time. Stability claims, in earlier literature, were made ignoring these solutions. Therefore we believe that a formulation of quantum theory on a firmer basis is necessary in general and for the quantum mass corrections of solitons, in particular. 
  In the following paper we continue the work of Bimonte-Lizzi-Sparano on distances on a one dimensional lattice. We succeed in proving analytically the exact formulae for such distances. We find that the distance to an even point on the lattice is the geometrical average of the ``predecessor'' and ``successor'' distances to the neighbouring odd points. 
  In this letter we report on the computation of instanton-dominated correlation functions in supersymmetric YM theories on ALE spaces. Following the approach of Kronheimer and Nakajima, we explicitly construct the self-dual connection on ALE spaces necessary to perform such computations. We restrict our attention to the simplest case of an $SU(2)$ connection with lowest Chern class on the Eguchi-Hanson gravitational background. 
  We study the system of two WZNW models coupled to each other via the current-current interaction. The system is proven to possess the strong/weak coupling duality symmetry. The strong coupling phase of this theory is discussed in detail. It is shown that in this phase the interacting WZNW models approach non-trivial conformal points along the renormalization group flow. The relation between the principal chiral model and interacting WZNW models is investigated. 
  We study a bosonic four--dimensional effective action corresponding to the heterotic string compactified on a 6--torus (dilaton--axion gravity with one vector field) on a curved space--time manifold possessing a time--like Killing vector field. Previously an existence of the $SO(2,3)\sim Sp(4, R)$ global symmetry ($U$--duality) as well as the symmetric space property of the corresponding $\sigma$--model have been established following Neugebauer and Kramer approach. Here we present an explicit form of the $Sp(4, R)$ generators in terms of coset variables and construct a representation of the coset in terms of the physical target space coordinates. Complex symmetric $2\times 2$ matrix $Z$ (``matrix dilaton --axion'') is introduced for which $U$--duality takes the matrix valued $SL(2, R)$ form. In terms of this matrix the theory is further presented as a K\"ahler $\sigma$--model. This leads to a more concise $2\times 2$ formulation which opens new ways to construct exact classical solutions. New solution (corresponding to constant ${\rm Im} Z$ ) is obtained which describes the system of point massless magnetic monopoles endowed with axion charges equal to minus monopole charges. In such a system mutual magnetic repulsion is exactly balanced by axion attraction so that the resulting space time is locally flat but possesses multiple Taub--NUT singularities. 
  We show that the classical null strings generate the Hilbert-Einstein gravity corresponding to D-dimensional Friedmann universes. 
  We extend the Galilei group of space-time transformations by gradation, construct interacting field-theoretic representations of this algebra, and show that non-relativistic Super-Chern-Simons theory is a special case. We also study the generalization to matrix valued fields, which are relevant to the formulation of superstring theory as a $1/N_c$ expansion of a field theory. We find that in the matrix case, the field theory is much more restricted by the supersymmetry. 
  We present families of algebraic curves describing the moduli-space of $N\!=\!2$ supersymmetric Yang-Mills theory with gauge group $SO(2n)$. We test our curves by computing the weak coupling monodromies and the number of $N\!=\!1$ vacua. 
  The standard approach of counting the number of eigenmodes of $N$ scalar fields near the horizon is used as a basis to provide a simple statistical mechanical derivation of the black hole entropy in two and four dimensions. The Bekenstein formula $S={A\ov 4G\h}$ and the two-dimensional entropy $S=2M/\l\h $ are naturally obtained (up to a numerical constant of order 1). This approach provides an explanation on why the black hole entropy is of order $1/\h $ and why it is independent of the number of field-theoretical degrees of freedom. 
  The form factor bootstrap approach is used to compute the exact contributions in the large distance expansion of the correlation function $<\sigma(x) \sigma(0)>$ of the two-dimensional Ising model in a magnetic field at $T=T_c$. The matrix elements of the magnetization operator $\sigma(x)$ present a rich analytic structure induced by the (multi) scattering processes of the eight massive particles of the model. The spectral representation series has a fast rate of convergence and perfectly agrees with the numerical determination of the correlation function. 
  We review some simple group theoretical properties of BPS states, in relation with the singular homology of level surfaces. Primary focus is on classical and quantum N=2 supersymmetric Yang-Mills theory, though the considerations can be applied to string theory as well. 
  String-string duality dictates that type IIA strings compactified on a K3 surface acquire non-abelian gauge groups for certain values of the K3 moduli. We argue that, contrary to expectation, the theories for which such enhanced gauge symmetries appear are not orbifolds in the string sense. For a specific example we show that a theory with enhanced gauge symmetry and an orbifold theory have the same classical K3 surface as a target space but the value of the ``B-field'' differs. This raises the possibility that the conformal field theory associated to a string theory with an enhanced gauge group is badly behaved in some way. 
  We extend Seiberg's qualitative picture of the behavior of supersymmetric QCD to nonsupersymmetric models by adding soft supersymmetry breaking terms. In this way, we recover the standard vacuum of QCD with $N_f$ flavors and $N_c$ colors when $N_f < N_c$. However, for $N_f \geq N_c$, we find new exotic states---new vacua with spontaneously broken baryon number for $N_f = N_c$, and a vacuum state with unbroken chiral symmetry for $N_f > N_c$. These exotic vacua contain massless composite fermions and, in some cases, dynamically generated gauge bosons. In particular Seiberg's electric-magnetic duality seems to persist also in the presence of (small) soft supersymmetry breaking. We argue that certain, specially tailored, lattice simulations may be able to detect the novel phenomena. Most of the exotic behavior does not survive the decoupling limit of large SUSY breaking parameters. 
  We give a geometric characterization of the quasi axial-vector (Kiritsis-Obers) target space duality in the spirit of the bi-algebra (Klimcik-Severa) approach. We show that the sigma-models constructed by taking quotients have non-abelian chiral currents that obey "non-commutative conservation laws" and provide the criterion for a sigma-model to have a dual using the axial-vector procedure. 
  The $O(3)$ nonlinear sigma model with its $U(1)$ subgroup gauged, where the gauge field dynamics is solely governed by a Chern-Simons term, admits both topological as well as nontopological self-dual soliton solutions for a specific choice of the potential. It turns out that the topological solitons are infinitely degenerate in any given sector. 
  The problems arising when quantizing systems with periodic boundary conditions are analysed, in an algebraic (group-) quantization scheme, and the ``failure" of the Ehrenfest theorem is clarified in terms of the already defined notion of {\it good} (and {\it bad}) operators. The analysis of ``constrained" Heisenberg-Weyl groups according to this quantization scheme reveals the possibility for new quantum (fractional) numbers extending those allowed for Chern classes in traditional Geometric Quantization. This study is illustrated with the examples of the free particle on the circumference and the charged particle in a homogeneous magnetic field on the torus, both examples featuring ``anomalous" operators, non-equivalent quantization and the latter, fractional quantum numbers. These provide the rationale behind flux quantization in superconducting rings and Fractional Quantum Hall Effect, respectively. 
  A large class of string-cosmology backgrounds leads to a spectrum of relic stochastic gravitational waves, strongly tilted towards high frequencies, and characterized by two basic parameters of the cosmological model. We estimate the required sensitivity for detection of the predicted gravitational radiation and show that a region of our parameter space is within reach for some of the planned gravitational-wave detectors. 
  We study $N=1$ SUSY gauge theories in four dimensions with gauge group $Spin(7)$ and $N_f$ flavors of quarks in the spinorial representation. We find that in the range $6< N_f < 15$, this theory has a long distance description in terms of an $SU(N_f-4)$ gauge theory with a symmetric tensor and $N_f$ antifundamentals. As a spin-off, we obtain by deforming along a flat direction a dual description of the theories based on the exceptional gauge group $G_2$ with $N_f$ fundamental flavors of quarks. 
  This article presents a formula for some dispersionless equations and a brief review of the operators which have been used for the dispersionless KP hierarchy. 
  The theory of Poisson-$\sigma$-models employs the mathematical notion of Poisson manifolds to formulate and analyze a large class of topological and almost topological two dimensional field theories. As special examples this class of field theories includes pure Yang-Mills and gravity theories, and, to some extent, the G/G gauged WZW-model. The aim of this contribution is to give a pedagogical introduction, explaining many aspects of the general theory by illustrative examples. 
  In the present article we solve the Dirac equation in a de Sitter universe when a constant electric field is present. Using the Bogoliubov transformations, we compute the rate of spin 1/2 created particles by the electric field. We compare our results with the scalar case. We also analyze the behavior of the density of particles created in the limit H=0, when de Sitter background reduces to a flat space-time. 
  We study special points in the moduli space of vacua at which supersymmetric electric solutions of the heterotic string theory become massless. We concentrate on configurations for which supersymmetric non-renormalization theorem is valid. Those are ten-dimensional supersymmetric string waves and generalized fundamental strings with SO(8) holonomy group. From these we find the four dimensional spherically symmetric configurations which saturate the BPS bound, in particular near the points of the vanishing ADM mass. The non-trivial massless supersymmetric states in this class exist only in the presence of non-Abelian vector fields.We also find a new class of supersymmetric massive solutions, closely related to the massless ones. A distinctive property of all these objects, either massless or massive, is the existence of gravitational repulsion. They reflect all particles with nonvanishing mass and/or angular momentum, and therefore they can be called white holes (repulsons), in contrast to black holes which tend to absorb particles of all kinds. If such objects can exist, we will have the first realization of the universal gravitational force which repels all particles with the strength proportional to their mass and therefore can be associated with antigravity. 
  It has previously been shown that a BRST quantization on an inner product space leads to physical states of the form |ph>=e^[Q, \psi] |\phi> where |\phi> is either a trivially BRST invariant state which only depends on the matter variables, |\phi>_1, or a solution of a Dirac quantization, |\phi>_2. \psi is a corresponding fermionic gauge fixing operator, \psi_1 or \psi_2. We show here for abelian and nonabelian models that one may also choose a linear combination of \psi_1 and \psi_2 for both choices of |\phi> except for a discrete set of relations between the coefficients. A general form of the coBRST charge operator is also determined and shown to be equal to such a \psi for an allowed linear combination of \psi_1 and \psi_2. This means that the coBRST charge is always a good gauge fixing fermion. 
  A large class of two-dimensional topological conformal field theories (TCFTs) are obtained by the twisting construction of Witten and Eguchi-Yang. However there seem to exist TCFTs which are not obtained in this way; for instance, TCFTs obtained from the Kazama algebra and critical string theories with generic background. We will show that by embedding the critical bosonic string into the NSR string, its TCFT can indeed be obtained by twisting a N=2 SCFT. A closer look at the construction of the N=2 superconformal algebra will show that the embedding is not essential, and this will tell us how to generalise this to other string theories. We thus conclude with the natural conjecture that _all_ TCFTs have a description as topologically twisted N=2 SCFTs. (Talk given at the Workshop on Strings, Gravity and Related Topics, held at the ICTP (Trieste, Italy) on 29-30 June, 1995.) 
  Zero modes of rotationally symmetric vortices in a hierarchy of generalized Abelian Higgs models are studied. Under the finite-energy and the smoothness condition, it is shown, that in all models, $n$ self-dual vortices superimposed at the origin have $2n$ modes. The relevance of these modes for vortex scattering is discussed, first in the context of the slow-motion approximation. Then a corresponding Cauchy problem for an all head-on collision of $n$ vortices is formulated. It is shown that the solution of this Cauchy problem has a $\frac{\pi}{n}$ symmetry. 
  The correspondence between the BV-formalism and integration theory on supermanifolds is established. An explicit formula for the density on a Lagrangian surface in a superspace provided with an odd symplectic structure and a volume form is proposed. 
  We propose a candidate for the dual (in the weak/strong coupling sense) of the six-dimensional heterotic string compactification constructed recently by Chaudhuri, Hockney and Lykken. It is a type IIA string theory compactified on an orbifold $K3/Z_2$, where the $Z_2$ action involves an involution of $K3$ with fixed points, and also has an embedding in the U(1) gauge group associated with the Ramond-Ramond sector of the type IIA string theory. This introduces flux of the U(1) gauge field concentrated at the orbifold points. This construction provides an explicit example where the dual of a super-conformal field theory background of the heterotic string theory is not a standard super-conformal field theory background of the type IIA string theory. 
  A new method is developed for solving the conformally invariant integrals that arise in conformal field theories with a boundary. The presence of a boundary makes previous techniques for theories without a boundary less suitable. The method makes essential use of an invertible integral transform, related to the radon transform, involving integration over planes parallel to the boundary. For successful application of this method several nontrivial hypergeometric function relations are also derived. 
  The notion of modular covariance is reviewed and the reconstruction of the Poincar\'e group extended to the low-dimensional case. The relations with the PCT symmetry and the Spin and Statistics theorem are described. 
  The action with the Grassmann-odd Lagrangian for the supersymmetric classical Witten mechanics is constructed. It is shown that the exterior differential can be used for the connection between Grassmann-even and Grassmann-odd formulations of the classical dynamics in the superspace in both Hamilton's and Lagrange's approaches. 
  Incorporating the zonal spherical function (zsf) problems on real and $p$-adic hyperbolic planes into a Zakharov-Shabat integrable system setting, we find a wide class of integrable evolutions which respect the number-theoretic properties of the zsf problem. This means that at {\it all} times these real and $p$-adic systems can be unified into an adelic system with an $S$-matrix which involves (Dirichlet, Langlands, Shimura...) L-functions. 
  In this paper we discuss the hyperelliptic curve for $N=2$ $SU(3)$ super Yang-Mills with six flavors of hypermultiplets. We start with a generic genus two surface and construct the curve in terms of genus two theta functions. From this one can construct the curve for $m_i=u=0$. This curve is explicitly dual under a subgroup of $Sp(4,Z)$ which is not isomorphic to $Sp(2,Z)$. We then proceed to construct the curve for the general $SU(3)$ theory and discuss the duality properties of the theory. The results given here differ from those given previously. 
  The equations of motion (e.m.'s) of the N=1, D=10 anomaly free supergravity, obtained in the framework of the superspace approach, are analyzed. The formal equivalence of the usual and dual supergravities is discussed at the level of e.m.'s. The great simplicity of the dual formulation is established. The possibillity of the lagrangian formulation of the dual supergravity is pointed out. The bosonic part of the lagrangian is constructed including anomaly compensating superstring corrections. 
  We develop the concept of trajectories in anyon spectra, i.e., the continuous dependence of energy levels on the kinetic angular momentum. It provides a more economical and unified description, since each trajectory contains an infinite number of points corresponding to the same statistics. For a system of non-interacting anyons in a harmonic potential, each trajectory consists of two infinite straight line segments, in general connected by a nonlinear piece. We give the systematics of the three-anyon trajectories. The trajectories in general cross each other at the bosonic/fermionic points. We use the (semi-empirical) rule that all such crossings are true crossings, i.e.\ the order of the trajectories with respect to energy is opposite to the left and to the right of a crossing. 
  Quantum fluctuations in the background geometry of a black hole are shown to affect the propagation of matter states falling into the black hole in a foliation that corresponds to observations purely outside the horizon. A state that starts as a Minkowski vacuum at past null infinity gets entangled with the gravity sector, so that close to the horizon it can be represented by a statistical ensemble of orthogonal states. We construct an operator connecting the different states and comment on the possible physical meaning of the above construction. The induced energy-momentum tensor of these states is computed in the neighbourhood of the horizon, and it is found that energy-momentum fluctuations become large in the region where the bulk of the Hawking radiation is produced. The background spacetime as seen by an outside observer may be drastically altered in this region, and an outside observer should see significant interactions between the infalling matter and the outgoing Hawking radiation. The boundary of the region of strong quantum gravitational effects is given by a time-like hypersurface of constant Schwarzschild radius $r$ one Planck unit away from the horizon. This boundary hypersurface is an example of a stretched horizon. 
  After adding auxiliary fields and integrating out the original variables, the Yang-Mills action can be expressed in terms of local gauge invariant variables. This method reproduces the known solution of the two dimensional $SU(N)$ theory. In more than two dimensions the action splits into a topological part and a part proportional to $\alpha_s$. We demonstrate the procedure for $SU(2)$ in three dimensions where we reproduce a gravity-like theory. We discuss the four dimensional case as well. We use a cubic expression in the fields as a space-time metric to obtain a covariant Lagrangian. We also show how the four-dimensional $SU(2)$ theory can be expressed in terms of a local action with six degrees of freedom only. 
  Renormalization procedure is generalized to be applicable for non renormalizable theories. It is shown that introduction of an extra expansion parameter allows to get rid of divergences and express physical quantities as series of finite number of interdependent expansion parameters. Suggested method is applied to quantum (Einstein's) gravity. 
  We complete the proof of the ghost-dilaton theorem in string theory by showing that the coupling constant dependence of the vacuum vertices appearing in the closed string action is given correctly by one-point functions of the ghost-dilaton. To prove this at genus one we develop the formalism required to evaluate off-shell amplitudes on tori. 
  In this work we present a number of generalizations of Wick's theorems on integrals with Gaussian weight to a larger class of weights which we call subgaussian. Examples of subgaussian contractions are that of Kac-Moody or Virasoro type, although the concept of a subgaussian weight does not refer a priori to two-dimensional field theory. The generalization was chosen in such a way that the contraction rules become a combinatorical way of solving the Schwinger-Dyson equation. In a still more general setting we prove a relation between solutions of the Schwinger-Dyson equation and a map $N$, which in the Gaussian case reduces to normal ordering. Furthermore, we give a number of results concerning contractions of composite insertions, which do not suffer from the Johnson-Low problem of ``commutation'' relations that do not satisfy the Jacobi identity. 
  A dual action is obtained for a general non-abelian and non-supersymmetric gauge theory at the classical level. The construction follows steps similar to those used in pure abelian gauge theory. As an example we study the spontaneously broken SO(3) gauge theory and show that the electric and the magnetic fields get interchanged in the dual theory. 
  We propose a new formulation of the space-time interpretation of the $c=1$ matrix model. Our formulation uses the well-known leg-pole factor that relates the matrix model amplitudes to that of the 2-dimensional string theory, but includes fluctuations around the fermi vacuum on {\sl both sides} of the inverted harmonic oscillator potential of the double-scaled model, even when the fluctuations are small and confined entirely within the asymptotes in the phase plane. We argue that including fluctuations on both sides of the potential is essential for a consistent interpretation of the leg-pole transformed theory as a theory of space-time gravity. We reproduce the known results for the string theory tree level scattering amplitudes for flat space and linear dilaton background as a special case. We show that the generic case corresponds to more general space-time backgrounds. In particular, we identify the parameter corresponding to background metric perturbation in string theory (black hole mass) in terms of the matrix model variables. Possible implications of our work for a consistent nonperturbative definition of string theory as well as for quantized gravity and black-hole physics are discussed. 
  We use a nonperturbative variational method to investigate the phase transition of the Gross-Neveu model. It is shown that the variational procedure can be generalized to the finite temperature case. The large N result for the phase transition is correctly reproduced. 
  Integrability of Quantum Chromodynamics in 1+1 dimensions has recently been suggested by formulating it as a perturbed conformal Wess-Zumino-Witten Theory. The present paper further elucidates this formulation, by presenting a detailed BRST analysis. 
  The spectrum of critical exponents of the $N$--vector model in $4-\eps$~dimensions is investigated to the second order in~$\eps$. A generic class of one--loop degeneracies that has been reported in a previous work is lifted in two--loop order. One-- and two--loop results lead to the conjecture that the spectrum possesses a remarkable hierarchical structure: The naive sum of any two anomalous dimensions generates a limit point in the spectrum, an anomalous dimension plus a limit point generates a limit point of limit points and so on. An infinite hierarchy of such limit points can be observed in the spectrum. 
  The complete renormalization procedure of a general N=1 supersymmetric gauge theory in the Wess-Zumino gauge is presented, using the regulator free ``algebraic renormalization'' procedure. Both gauge invariance and supersymmetry are included into one single BRS invariance. The form of the general nonabelian anomaly is given. Furthermore, it is explained how the gauge BRS and the supersymmetry functional operators can be extracted from the general BRS operator. It is then shown that the supersymmetry operators actually belong to the closed, finite, Wess-Zumino superalgebra when their action is restricted to the space of the ``gauge invariant operators'', i.e. to the cohomology classes of the gauge BRS operator. An erratum is added at the end of the paper. 
  The non-relativistic dynamics of a spin-1/2 particle in a monopole field possesses a rich supersymmetry structure. One supersymmetry, uncovered by d'Hoker and Vinet, is of the standard type: it squares to the Hamiltonian. In this paper we show the presence of another supersymmetry which squares to the Casimir invariant of the full rotation group. The geometrical origin of this supersymmetry is traced, and its relationship with the constrained dynamics of a spinning particle on a sphere centered at the monopole is described. 
  The highest weight modules of the chiral algebra of orthogonal WZW models at level one possess a realization in fermionic representation spaces; the Kac-Moody and Virasoro generators are represented as unbounded limits of even CAR algebras. It is shown that the representation theory of the underlying even CAR algebras reproduces precisely the sectors of the chiral algebra. This fact allows to develop a theory of local von Neumann algebras on the punctured circle, fitting nicely in the Doplicher-Haag-Roberts framework. The relevant localized endomorphisms which generate the charged sectors are explicitly constructed by means of Bogoliubov transformations. Using CAR theory, the fusion rules in terms of sector equivalence classes are proven. 
  The ten or eleven dimensional origin of central charges in the N=4 or N=8 supersymmetry algebra in four dimensions is reviewed: while some have a standard Kaluza-Klein interpretation as momenta in compact dimensions, most arise from $p$-form charges in the higher-dimensional supersymmetry algebra that are carried by $p$-brane `solitons'. Although $p=1$ is singled out by superstring perturbation theory, U-duality of N=8 superstring compactifications implies a complete `$p$-brane democracy' of the full non-perturbative theory. An `optimally democratic' perturbation theory is defined to be one in which the perturbative spectrum includes all particles with zero magnetic charge. Whereas the heterotic string is optimally democratic in this sense, the type II superstrings are not, although the 11-dimensional supermembrane might be. 
  Some considerations showing that renormalizable theories with consistent perturbative theries can not be nonperturbatively finite (in terms of bare parameters) are provided. Accordingly any fundamental unified theory has to be either non renormalizable or order by order finite. 
  Based on a simple adiabatic argument and by considering the heterotic string counterpart of certain symmetries of Type II superstrings such as $(-1)^{F_L}$ and orientation reversal, we construct orbifold candidates for dual pairs of heterotic and Type II string theories with $N=2$ and $N=1$ supersymmetry. We also analyze from a similar point of view the ${\rm K3}$ fibrations that enter in recently proposed $N=2$ candidates and use this structure together with certain orientation-reversing symmetries to construct $N=1$ dual pairs. These pairs involve generalizations of Type I vacua which can be equivalent to $E_8\times E_8$ heterotic strings, while standard Type I vacua are related to $SO(32)$. 
  Exact superstring solutions are constructed in 4-D space-time, with positive curvature and non-trivial dilaton and antisymmetric tensor fields. The full spectrum of string excitations is derived as a function of moduli fields $T^{i}$ and the scale $\mu^2=1/(k+2)$ which is induced by the non-zero background fields. The spectrum of string excitations has a non-zero mass gap $\mu^2$ and in the weak curvature limit ($\mu$ small) $\mu^2$ plays the role of a well defined infrared regulator, consistent with modular invariance, gauge invariance, supersymmetry and chirality.    The effects of a covariantly constant (chomo)magnetic field $H$ as well as additional curvature can be derived exactly up to one string-loop level. Thus, the one-loop corrections to all couplings (gravitational, gauge and Yukawas) are unambiguously computed and are finite both in the UltraViolet and the InfraRed regime. These corrections are necessary for quantitative string superunification predictions at low energies.  The one-loop corrections to the couplings are also found to satisfy Infrared Flow Equations. Having in our disposal an exact description which goes beyond the leading order in the $\alpha'$-expansion or the linearized approximation in the magnetic field, we find interesting clues about the physics of string theory in strong gravitational and magnetic fields. In particular, the nature of gravitational or magnetic instabilities is studied.   
  The symplectic formalism is fully employed to study the gauge-invariant CP$^1$ model with the Chern-Simons term. We consistently accommodate the CP$^1$ constraint at the Lagrangian level according to this formalism. 
  In this report we compute the boundary states (including the boundary entropy) for the boundary sine-Gordon theory. From the boundary states, we derive both correlation and partition functions. Through the partition function, we show that boundary sine-Gordon maps onto a doubled boundary Ising model. With the current-current correlators, we calculate for finite system size the ac-conductance of tunneling quantum wires with dimensionless free conductance 1/2 (or, alternatively interacting quantum Hall edges at filling fraction 1/2). In the dc limit, the results of C. Kane and M. Fisher, Phys. Rev. B46 (1992) 15233, are reproduced. 
  The full cohomology ring of the Lian-Zuckerman type operators (states) in ${\hat c_M}<1$ Neveu-Schwarz-Ramond (NSR) string theory is argued to be generated by three elements $x$, $y$ and $w$ in analogy with the corresponding results in the bosonic case. The ground ring generators $x$ and $y$ are non-invertible and belong to the Ramond sector whereas the higher ghost number operators are generated by an invertible element $w$ with ghost number one less than that of the ground ring generators and belongs to either Neveu-Schwarz (NS) or Ramond (R) sector depending on whether we consider (even, even) or (odd, odd) series coupled to $2d$ supergravity. We explicitly construct these operators (states) and illustrate our result with an example of pure Liouville supergravity. 
  We give a new proof of the renormalizability of a class of matter field theories on a space-time lattice; in particular we consider $\phi^4$ and massive Yukawa theories with Wilson fermions. We use the Polchinski approach to renormalization, which is based on the Wilson flow equation; this approach is substantially simpler than the BPHZ method, applied to the lattice by Reisz. We discuss matter theories with staggered fermions. In particular we analyse a simple kind of staggered fermions with minimal doubling, using which we prove the renormalizability of a chiral sigma model with exact chiral symmetry on the lattice. 
  The Hamiltonian derived by Bartels, Kwiecinski and Praszalowicz for the study of high-energy QCD in the generalized logarithmic approximation was found to correspond to the Hamiltonian of an integrable $XXX$ spin chain. We study the odderon Hamiltonian corresponding to three sites by means of the Bethe Ansatz approach. We rewrite the Baxter equation, and consequently the Bethe Ansatz equations, as a linear triangular system. We derive a new expression for the eigenvectors and the eigenvalues, and discuss the quantization of the conserved quantities. 
  It is shown that graphs that generalize the ADE Dynkin diagrams and have appeared in various contexts of two-dimensional field theory may be regarded in a natural way as encoding the geometry of a root system. After recalling what are the conditions satisfied by these graphs, we define a bilinear form on a root system in terms of the adjacency matrices of these graphs and undertake the study of the group generated by the reflections in the hyperplanes orthogonal to these roots. Some ``non integrally laced " graphs are shown to be associated with subgroups of these reflection groups. The empirical relevance of these graphs in the classification of conformal field theories or in the construction of integrable lattice models is recalled, and the connections with recent developments in the context of ${\cal N}=2$ supersymmetric theories and topological field theories are discussed. 
  The natural relativistic generalisation of Landau's two constituent superfluid theory can be formulated in terms of a Lagrangian $L$ that is given as a function of the entropy current 4-vector $s^\rho$ and the gradient $\nabla\varphi$ of the superfluid phase scalar. It is shown that in the ``cool" regime, for which the entropy is attributable just to phonons (not rotons), the Lagrangian function $L(\vec s, \nabla\varphi)$ is given by an expression of the form $L=P-3\psi$ where $P$ represents the pressure as a function just of $\nabla\varphi$ in the (isotropic) cold limit. The entropy current dependent contribution $\psi$ represents the generalised pressure of the (non-isotropic) phonon gas, which is obtained as the negative of the corresponding grand potential energy per unit volume, whose explicit form has a simple algebraic dependence on the sound or ``phonon" speed $c_P$ that is determined by the cold pressure function $P$. 
  We give a class of explicit solutions for the stationary and cylindrically symmetric vortex configurations for a ``cool'' two-component superfluid (i.e. superfluid with an ideal gas of phonons). Each solution is characterized only by a set of (true) constants of integration. We then compute the effective asymptotic contribution of the vortex to the stress energy tensor by comparison with a uniform reference state without vortex. 
  We present a string theory that reproduces the large-$N$ expansion of two dimensional Yang-Mills gauge theory on arbitrary surfaces. First, a new class of topological sigma models is introduced, with path integrals localized to the moduli space of harmonic maps. The Lagrangian of these harmonic topological sigma models is of fourth order in worldsheet derivatives. Then we gauge worldsheet diffeomorphisms by introducing the induced worldsheet metric. This leads to a topological string theory, whose Lagrangian coincides in the bose sector with the rigid string Lagrangian discussed some time ago by Polyakov and others as a candidate for QCD string theory. The path integral of this topological rigid string theory is localized to the moduli spaces of minimal-area maps, and calculates their Euler numbers. The dependence of the large-$N$ QCD partition functions on the target area emerges from measuring the volume of the moduli spaces, and can be reproduced by adding a Nambu-Goto term (improved by fermionic terms) to the Lagrangian of the topological rigid string. 
  The coupling of a string to gravity allows for Schwarzschild black holes whose entropy to area relation is $S=(A/4)(1-4\mu)$, where $\mu$ is the string tension. This departure from the A/4 universality results from a string instanton generating a black hole with smaller entropy at a temperature exceeding the Hawking value. The temperature is sensitive to the presence of classical matter outside the black hole horizon but the entropy is not. The horizon materializes at the quantum level. It is conjectured that such a macroscopic non local effect may be operative in retrieving information from incipient black holes. 
  We apply light-cone quantization to a $1+1$ dimensional supersymmetric field theory of large N matrices. We provide some preliminary numerical evidence that when the coupling constant is tuned to a critical value, this model describes a 2+1 dimensional non-critical superstring. 
  The S-duality invariance of the equations of motion of four dimensional string effective action with cosmological constant, $\Lambda $, is studied. It is demonstrated that the S-duality symmetry of the field equations are broken for nonzero $\Lambda$. The ``naturalness'' hypothesis is invoked to argue that $\Lambda $ should remain small since exact S-duality symmetry will force the cosmological constant to vanish in the string effective action. 
  Evidence in favor of $SL(2,Z)$ S-duality in $N=4$ supersymmetric Yang-Mills theories in four dimensions and with general compact, simple gauge groups is presented. (Contribution to the Proceedings of the Strings '95 conference, March 13-18, 1995, USC, and the Proceedings of the Trieste Conference on S-Duality and Mirror Symmetry, June 5-9, 1995.) 
  The affine Toda field theory is studied as a 2+1-dimensional system. The third dimension appears as the discrete space dimension, corresponding to the simple roots in the $A_N$ affine root system, enumerated according to the cyclic order on the $A_N$ affine Dynkin diagram. We show that there exists a natural discretization of the affine Toda theory, where the equations of motion are invariant with respect to permutations of all discrete coordinates. The discrete evolution operator is constructed explicitly. The thermodynamic Bethe ansatz of the affine Toda system is studied in the limit $L,N\to\infty$. Some conjectures about the structure of the spectrum of the corresponding discrete models are stated. 
  The methods of reduced phase space quantization and Dirac quantization are examined in a simple gauge theory. A condition for the possible equivalence of the two methods is discussed. 
  A perturbative approach for non renormalizable theories is developed. It is shown that the introduction of an extra expansion parameter allows one to get rid of divergences and express physical quantities as series with finite coefficients. The method is demonstrated on the example of massive non abelian field coupled to a fermion field. 
  We consider perturbations of the non-unitary minimal model solutions of two-dimensional conformal turbulence proposed by Polyakov. Demanding the absence of non-integrable singularities in the resulting theories leads to constraints on the dimension of the perturbing operator. We give some general solutions of these constraints, illustrating with examples of specific models. We also examine the effect of such perturbations on the Hopf equation and derive the interesting result that the latter is invariant under a certain class of perturbations, to first order in perturbation theory, examples of which are given in specific cases. 
  It is shown that, with some reasonable assumptions, the theory of general relativity can be made compatible with quantum mechanics by using the field equations of general relativity to construct a Robertson-Walker metric for a quantum particle so that the line element of the particle can be transformed entirely to that of the Minkowski spacetime, which is assumed by a quantum observer, and the spacetime dynamics of the particle described by a Minkowski observer takes the form of quantum mechanics. Spacetime structure of a quantum particle may have either positive or negative curvature. However, in order to be describable using the familiar framework of quantum mechanics, the spacetime structure of a quantum particle must be "quantised" by an introduction of the imaginary number $i$. If a particle has a positive curvature then the quantisation is equivalent to turning the pseudo-Riemannian spacetime of the particle into a Riemannian spacetime. This means that it is assumed the particle is capable of measuring its temporal distance like its spatial distances. On the other hand, when a particle has a negative curvature and a negative energy density then quantising the spacetime structure of the particle is equivalent to viewing the particle as if it had a positive curvature and a positive energy density. 
  The definitions of the main notions related to the quantum inverse scattering methods are given. The Yang-Baxter equation and reflection equations are derived as consistency conditions for the factorizable scattering on the whole line and on the half-line using the Zamolodchikov-Faddeev algebra. Due to the vertex-IRF model correspondence the face model analogue of the ZF-algebra and the IRF reflection equation are written down as well as the $Z_2$-graded and colored algebra forms of the YBE and RE. 
  On the example of topologically massive gauge field theory we find the origin of possible inconsistency of working with gauge fixing terms (together with relevant ghost sector) 
  We prove polynomial identities for the N=1 superconformal model SM(2,4\nu) which generalize and extend the known Fermi/Bose character identities. Our proof uses the q-trinomial coefficients of Andrews and Baxter on the bosonic side and a recently introduced very general method of producing recursion relations for q-series on the fermionic side. We use these polynomials to demonstrate a dual relation under q \rightarrow q^{-1} between SM(2,4\nu) and M(2\nu-1,4\nu). We also introduce a generalization of the Witten index which is expressible in terms of the Rogers false theta functions. 
  The wave functions of the Calogero-Sutherland model are known to be expressible in terms of Jack polynomials. A formula which allows to obtain the wave functions of the excited states by acting with a string of creation operators on the wave function of the ground state is presented and derived. The creation operators that enter in this formula of Rodrigues-type for the Jack polynomials involve Dunkl operators. 
  A lattice-type regularization of the supersymmetric field theories on a supersphere is constructed by approximating the ring of scalar superfields by an integer-valued sequence of finite dimensional rings of supermatrices and by using the differencial calculus of non-commutative geometry. The regulated theory involves only finite number of degrees of freedom and is manifestly supersymmetric. 
  New geometrical features of the Landau-Ginzburg orbifolds are presented, for models with a typical type of superpotential. We show the one-to-one correspondence between some of the $(a,c)$ states with $U(1)$ charges $(-1,1)$ and the integral points on the dual polyhedra, which are useful tools for the construction of mirror manifolds. Relying on toric geometry, these states are shown to correspond to the $(1,1)$ forms coming from blowing-up processes. In terms of the above identification, it can be checked that the monomial-divisor mirror map for Landau-Ginzburg orbifolds, proposed by the author, is equivalent to that mirror map for Calabi-Yau manifolds obtained by the mathematicians. 
  We construct a non-perturbative, single-valued solution for the metric and the motion of two interacting particles in ($2+1$)-Gravity, by using a Coulomb gauge of conformal type. The method provides the mapping from multivalued ( minkowskian ) coordinates to single-valued ones, which solves the non-abelian monodromies due to particles's momenta and can be applied also to the general N-body case. 
  By defining a regular gauge which is conformal-like and provides instantaneous field propagation, we investigate classical solutions of (2+1)-Gravity coupled to arbitrarily moving point-like particles. We show how to separate field equations from self-consistent motion and we provide a solution for the metric and the motion in the two-body case with arbitrary speed, up to second order in the mass parameters. 
  A few years ago, Matsuo and Cherednik proved that from some solutions of the Knizhnik-Zamolodchikov (KZ) equations, which first appeared in conformal field theory, one can obtain wave functions for the Calogero integrable system. In the present communication, it is shown that from some solutions of generalized KZ equations, one can construct wave functions, characterized by any given permutational symmetry, for some Calogero-Sutherland-Moser integrable models with exchange terms. Such models include the spin generalizations of the original Calogero and Sutherland ones, as well as that with $\delta$-function interaction. 
  We study the effect of next-to-leading order contributions on the phenomenon of symmetry non-restoration at high temperature in an $O(N_1)\times O(N_2)$ model. 
  We study the nonlinear $O(N)$ sigma model on $S^2$ with the gravitational coupling term, by evaluating the effective potential in the large-$N$ limit. It is shown that there is a critical curvature $R_c$ of $S^2$ for any positive gravitational coupling constant $\xi$, and the dynamical mass generation takes place only when $R<R_c$. The critical curvature is analytically found as a function of $\xi$ $(>0)$, which leads us to define a function looking like a natural generalization of Euler-Mascheroni constant. 
  A method for deriving the Schrodinger equation for Lagrangian path integral with scaling of local time is given. 
  The properties of cosmic strings have been investigated in detail for their implications in early-universe cosmology. Although many variations of the basic structure have been discovered, with implications for both the microscopic and macroscopic properties of cosmic strings, the cylindrical symmetry of the short-distance structure of the string is generally unaffected. In this paper we describe some mechanisms leading to an asymmetric structure of the string core, giving the defects a quasi-two-dimensional character. We also begin to investigate the consequences of this internal structure for the microscopic and macroscopic physics. 
  We consider representations of meromorphic bosonic chiral conformal field theories, and demonstrate that such a representation is completely specified by a state within the theory. The necessary and sufficient conditions upon this state are derived, and, because of their form, we show that we may extend the representation to a representation of a suitable larger conformal field theory. In particular, we apply this procedure to the lattice (FKS) conformal field theories, and deduce that Dong's proof of the uniqueness of the twisted representation for the reflection-twisted projection of the Leech lattice conformal field theory generalises to an arbitrary even (self-dual) lattice. As a consequence, we see that the reflection-twisted lattice theories of Dolan et al are truly self-dual, extending the analogies with the theories of lattices and codes which were being pursued. Some comments are also made on the general concept of the definition of an orbifold of a conformal field theory in relation to this point of view. 
  The generalization of Agranovich-Toshich representation of paulion operators in terms of bosonic ones for the case of truncated oscillators of higher ranks is represented. We use this generalization to introduce a new constraint free bosonic description of truncated oscillator systems. The corresponding functional integral representations for thermodynamic quantities are given and the application to investigations of Long Rang Order in the system is discussed. 
  Following on from recent work describing the representation content of a meromorphic bosonic conformal field theory in terms of a certain state inside the theory corresponding to a fixed state in the representation, and using work of Zhu on a correspondence between the representations of the conformal field theory and representations of a particular associative algebra constructed from it, we construct a general solution for the state defining the representation and identify the further restrictions on it necessary for it to correspond to a ground state in the representation space. We then use this general theory to analyze the representations of the Heisenberg algebra and its $Z_2$-projection. The conjectured uniqueness of the twisted representation is shown explicitly, and we extend our considerations to the reflection-twisted FKS construction of a conformal field theory from a lattice. 
  In this paper the classical limit of relativistic transport theories for spin 1/2 fermions is examined through a comparison with the classical kinetic theory derived from N=1 supersymmetric classical mechanics. The conclusion is that in the classical limit spindensities, i.e. the axial-vector contribution to the relativistic Wigner-function, vanishes and dipole-densities, i.e. the spin-tensor contributions to the relativistic Wigner function, may survive. 
  An algebraic construction more general and intimately connected with that of Faddeev$^1$, along with its application for generating different classes of quantum integrable models are summarised to complement the recent results of ref. 1 ( L.D. Faddeev, {\it Int. J. Mod. Phys. } {\bf A10}, 1845 (1995) ). 
  I discuss examples where basic structures from Connes' noncommutative geometry naturally arise in quantum field theory. The discussion is based on recent work, partly collaboration with J. Mickelsson. 
  We propose a possible resolution for the problem of why the semicircular law is not observed, whilst the random matrix hypothesis describes well the fluctuation of energy spectra. We show in the random 2-matrix model that the interactions between the quantum subsystems alter the semicircular law of level density. We consider also other types of interactions in the chain- and star-multimatrix models. The connection with the Calogero-Sutherland models is briefly discussed. 
  Within effective heterotic superstring theory compactified on a six-torus we derive minimum energy (supersymmetric), static, spherically symmetric solutions, which are manifestly invariant under the target space $O(6,22)$ and the strong-weak coupling $SL(2)$ duality symmetries with 28 electric and 28 magnetic charges subject to one constraint. The class of solutions with a constant axion corresponds to dyonic configurations subject to two charge constraints, with purely electric [or purely magnetic] and dyonic configurations preserving ${1\over 2}$ and ${1\over 4}$ of $N=4$ supersymmetry, respectively. General dyonic configurations in this class have a space-time of extreme Reissner-Nordstr\" om black holes while configurations with more constrained charges have a null or a naked singularity. 
  Following Sutherland's work on one-dimensional integrable systems we formulate and study its two-dimensional version. Physically it expresses the absence of true 3-body forces among an assembly of N particles leaving exclusively effective 2-body interactions. This criterion may be a suitable candidate for an integrability condition. 
  The solution of the classical open-chain n-body Toda problem is derived from an ansatz and is found to have a highly symmetric form. The proof requires an unusual identity involving Vandermonde determinants. The explicit transformation to action-angle variables is exhibited. 
  Non-perturbative exact flow equations describe the scale dependence of the effective average action. We present a numerical solution for an approximate form of the flow equation for the potential in a three-dimensional N-component scalar field theory. The critical behaviour, with associated critical exponents, can be inferred with good accuracy. 
  Is string theory relevant to the black hole information problem? This is an attempt to clarify some of the issues involved. Presented at Strings '95. 
  A non-minimal coupling for spin $3/2$ fields is obtained. We use the fact that the Rarita-Schwinger field equations are the square root of the full linearized Einstein field equations in order to investigate the form of the interaction for the spin $3/2$ field with gauge fields. We deduce the form of the interaction terms for the electromagnetic and non-Abelian Yang-Mills fields by implementing appropiate energy momentum tensors on the linearized Einstein field equations. The interaction found for the electromagnetic case happens to coincide with the dipole term found by Ferrara {\it et al} by a very different procedure, namely by demanding $g=2$ at the tree level for the electromagnetic interaction of arbitrary spin particles. The same interaction is found by using the resource of linearized Supergravity N=2. For the case of the Yang-Mills field Supergravity N=4 is linearized, providing the already foreseen interaction. 
  A simplified direct method is described for obtaining massless scalar functional determinants on the Euclidean ball. The case of odd dimensions is explicitly discussed. 
  Quasifree states of a linear Klein-Gordon quantum field on globally hyperbolic spacetime manifolds are considered. Using techniques from the theory of pseudodifferential operators and wavefront sets on manifolds a criterion for a state to be an Hadamard state is developed. It is shown that ground- and KMS-states on certain static spacetimes and adiabatic vacuum states on Robertson-Walker spaces are Hadamard states. Finally, the problem of constructing Hadamard states on arbitrary curved spacetimes is solved in principle. 
  We study the conditions under which a symmetry is spontaneously broken in the Wilson renormalization group formulation. Both for a global and local symmetry, the result is that in perturbation theory one has to perform a fine tuning of the boundary conditions for the flow of the relevant couplings. We consider in detail the discrete $Z_2$ case and the Abelian Higgs model. 
  The low-momentum structure of the gravitational polarization tensor of an ultrarelativistic plasma of scalar particles with $\lambda\phi^4$ interactions is evaluated in a two-loop calculation up to and including order $\lambda^{3/2}$. This turns out to require an improved perturbation theory which resums a local thermal mass term as well as nonlocal hard-thermal-loop vertices of scalar and gravitational fields. 
  We quantize the chiral Schwinger Model by using the Batalin-Tyutin formalism. We show that one can systematically construct the first class constraints and the desired involutive Hamiltonian, which naturally generates all secondary constraints. For $a>1$, this Hamiltonian gives the gauge invariant Lagrangian including the well-known Wess-Zumino terms, while for $a=1$ the corresponding Lagrangian has the additional new type of the Wess-Zumino terms, which are irrelevant to the gauge symmetry. 
  The two-point Green function of the massive scalar $(3+1)$-quantum field theory with $\lambda\varphi^4$ interaction at finite temperature is evaluated up to the 2nd order of perturbation theory. The averaging on the vacuum fluctuations is separated from the averaging on the thermal fluctuations explicitly. As a result, the temperature dependent part of the propagator is expressed through the scattering amplitudes. The obtained expression is generalized for higher orders of perturbation theory. 
  We study the connection between N=2 supersymmetry and a topological bound in a two-Higgs-doublet system having an $SU(2)\times U(1)_Y\times U(1)_{Y'}$ gauge group. We derive Bogomol'nyi equations from supersymmetry considerations showing that they hold provided certain conditions on the coupling constants, which are a consequence of the huge symmetry of the theory, are satisfied. 
  We derive a simple calculation rule for Aoyama--Tamra's prescription for path integral with degenerated potential minima. Non-perturbative corrections due to the restricted functional space (fundamental region) can systematically be computed with this rule. It becomes manifest that the prescription might give Borel summable series for finite temperature (or volume) system with quantum tunneling, while the advantage is lost at zero temperature (or infinite volume) limit. 
  Lagragian quantization rules for general gauge theories are proposed on a basis of a superfield formulation of the standard BRST symmetry. Independence of the $S$-matrix on a choice of the gauge is proved. The Ward identities in terms of superfields are derived. 
  A superfield version on superspace $(x^\mu,\theta^a)$ is proposed for the $Sp(2)$-- covariant Lagrangian quantization of general gauge theories. The BRST- and antiBRST- transformations are realized on superfields as supertranslations in the $\theta^a$-- directions. A new (geometric) interpretation of the Ward identities in the quantum gauge theory is given. 
  In order to use the Gaussian representation for propagators in Feynman amplitudes, a representation which is useful to relate string theory and field theory, one has to prove first that each $\alpha$- parameter (where $\alpha$ is the parameter associated to each propagator in the $\alpha$-representation of the Feynman amplitudes) can be replaced by a constant instead of being integrated over and second, prove that this constant can be taken equal for all propagators of a given graph. The first proposition has been proven in one recent letter when the number of propagators is infinite. Here we prove the second one. In order to achieve this, we demonstrate that the sum over the weighted spanning trees of a Feynman graph $G$ can be factorized for disjoint parts of $G$. The same can also be done for cuts on $G$, resulting in a rigorous derivation of the Gaussian representation for super-renormalizable scalar field theories. As a by-product spanning trees on Feynman graphs can be used to define a discretized functional space. 
  Hartle's generalized quantum mechanics in the sum-over-histories formalism is used to describe a nonabelian gauge theory. Predictions are made for certain alternatives, with particular attention given to coarse-grainings involving the constraint. In this way, the theory is compared to other quantum-mechanical descriptions of gauge theories in which the constraints are imposed by hand. The vanishing of the momentum space constraint is seen to hold, both through a simple formal argument and via a more careful description of the Lorentzian path integral as defined on a spacetime lattice. (Incidentally, the treatment of the time slicing in the path integral may be of general technical interest.) The configuration space realization of the constraint is shown to behave in a more complicated fashion. For some coarse grainings, we recover the known result from an abelian theory, that coarse grainings by values of the constraint either predict its vanishing or fail to decohere. However, sets of alternatives defined in terms of a more complicated quantity in the abelian case are exhibited where definite predictions can be made which disagree with the assumption that the constraints vanish. Finally, the configuration space sum-over-histories theory is exhibited in a manifestly Lorentz-invariant formulation. 
  The connection between Supergravity and the low-energy world is analyzed. In particular, the soft Supersymmetry-breaking terms arising in Supergravity, the $\mu$ problem and various solutions proposed to solve it are reviewed. The soft terms arising in Supergravity theories coming from Superstring theory are also computed and the solutions proposed to solve the $\mu$ problem, which are naturally present in Superstrings, are also discussed. The $B$ soft terms associated are given for the different solutions. Finally, the low-energy Supersymmetric-spectra, which are very characteristic, are obtained. 
  The recently proposed expression for the general three point function of exponential fields in quantum Liouville theory on the sphere is considered. By exploiting locality or crossing symmetry in the case of those four-point functions, which may be expressed in terms of hypergeometric functions, a set of functional equations is found for the general three point function. It is shown that the expression proposed by the Zamolodchikovs solves these functional equations and that under certain assumptions the solution is unique. 
  We consider a large-N Chern-Simons theory for the attractive bosonic matter (Jackiw-Pi model) in the Hamiltonian, collective-field approach based on the 1/N expansion. We show that the dynamics of density excitations around the ground-state semiclassical configuration is governed by the Calogero or by the Sutherland Hamiltonian, depending on the symmetry of the underlying static-soliton configuration. The relationship between the Chern-Simons coupling constant $\l$ and the Calogero-Sutherland statistical parameter $\l_c$ signalizes some sort of statistical transmutation accompanying the dimensional reduction of the initial problem. 
  The instanton configuration in the SU(2)-gauge system with a Higgs doublet is constructed by using the new valley method. This method defines the configuration by an extension of the field equation and allows the exact conversion of the quasi-zero eigenmode to a collective coordinate. It does not require ad-hoc constraints used in the current constrained instanton method and provides a better mathematical formalism than the constrained instanton method. The resulting instanton, which we call ``valley instanton'', is shown to have desirable behaviors. The result of the numerical investigation is also presented. 
  We analyze in detail the possible breaking of spacetime supersymmetry under T-duality transformations. We find that when appropiate world-sheet effects are taken into account apparent puzzles concerning supersymmetry in spacetime are solved. We study T-duality in general heterotic $\sigma$-models analyzing possible anomalies, and we find some modifications of Buscher's rules. We then work out a simple but representative example which contains most of the difficulties. We also consider the issue of supersymmetry versus duality for marginal deformations of WZW models and present a mechanism that restores supersymmetry dynamically in the effective theory. 
  We address non-perturbative effects and duality symmetries in $N=2$ heterotic string theories in four dimensions. Specifically, we consider how each of the four lines of enhanced gauge symmetries in the perturbative moduli space of $N=2$ $T_2$ compactifications is split into 2 lines where monopoles and dyons become massless. This amounts to considering non-perturbative effects originating from enhanced gauge symmetries at the microscopic string level. We show that the perturbative and non-perturbative monodromies consistently lead to the results of Seiberg-Witten upon identication of a consistent truncation procedure from local to rigid $N=2$ supersymmetry. 
  Using a formulation of QCD_2 as a perturbed conformally invariant theory involving fermions, ghosts, as well as positive and negative level Wess-Zumino-Witten fields, we show that the BRST conditions become restrictions on the conformally invariant sector, as described by a G/G topological theory. By solving the corresponding cohomology problem we are led to a finite set of vacua. For G=SU(2) these vacua are two-fold degenerate. 
  We test the recently conjectured duality between $N=2$ supersymmetric type II and heterotic string models by analysing a class of higher dimensional interactions in the respective low-energy Lagrangians. These are $F$-terms of the form $F_g W^{2g}$ where $W$ is the gravitational superfield. On the type II side these terms are generated at the $g$-loop level and in fact are given by topological partition functions of the twisted Calabi-Yau sigma model. We show that on the heterotic side these terms arise at the one-loop level. We study in detail a rank 3 example and show that the corresponding couplings $F_g$ satisfy the same holomorphic anomaly equations as in the type II case. Moreover we study the leading singularities of $F_g$'s on the heterotic side, near the enhanced symmetry point and show that they are universal poles of order $2g{-}2$ with coefficients that are given by the Euler number of the moduli space of genus-$g$ Riemann surfaces. This confirms a recent conjecture that the physics near conifold singularity is governed by $c{=}1$ string theory at the self-dual point. 
  The algebra of observables of $SO_{q}(3)$-symmetric quantum mechanics is extended to include the inverse $\frac{1}{R}$ of the radial coordinate and used to obtain eigenvalues and eigenfunctions of a \q-deformed Coulomb Hamiltonian. 
  A simple field theoretical approach to Mathai-Quillen topological field theories of maps $X: M_I \to M_T$ from an internal space to a target space is presented. As an example of applications of our formalism we compute by applying our formulas the action and Q-variations of the fields of two well known topological systems: Topological Quantum Mechanics and type-A topological Sigma Model. 
  We use boundary weights and reflection equations to obtain families of commuting double-row transfer matrices for interaction-round-a-face models with fixed boundary conditions. In particular, we consider the fusion hierarchy of the Andrews-Baxter-Forrester models, for which we find that the double-row transfer matrices satisfy functional equations with an su(2) structure. 
  We show that the kinetic approach to statistical mechanics permits an elegant and efficient treatment of fractional exclusion statistics. By using the exclusion-inclusion principle recently proposed [Phys. Rev. E49, 5103 (1994)] as a generalization of the Pauli exclusion principle, which is based on a proper definition of the transition probability between two states, we derive a variety of different statistical distributions interpolating between bosons and fermions. The Haldane exclusion principle and the Haldane-Wu fractional exclusion statistics are obtained in a natural way as particular cases. The thermodynamic properties of the statistical systems obeying the generalized exclusion-inclusion principle are discussed. 
  Following a review of the dual description of the non-linear sigma model we investigate the one-loop quadratic divergences. We use the covariant background field method for the general case and apply the results to the important example of $SU(2)$. 
  Three subjects are considered here: a self-dual non-critical string that appears in Type IIB superstring theory at points in ${\rm K3}$ moduli space where the Type IIA theory has extended gauge symmetry; a conformal field theory singularity at such points which may signal quantum effects that persist even at weak coupling; and the rich dynamics of the real world under compactification, which may be relevant to some attempts to explain the vanishing of the cosmological constant. 
  We analyze the role played by temperature in QCD_3 by means of a dimensional interpolating approach. Pure gauge QCD_3 is defined on a strip of finite width L, which acts as an interpolating parameter between two and three dimensions. A two-dimensional effective theory can be constructed for small enough widths giving the same longitudinal physics as QCD_3. Explicit calculations of T-dependent QCD_3 observables can thus be performed. The generation of a deconfinig phase transition, absent in QCD_2, is proven through an exact calculation of the electric or Debye mass at high T. Low and high T behaviors of relevant thermodynamic functions are also worked out. An accurate estimate of the critical temperature is given and its evolution with L is studied in detail. 
  We give the infinite-dimensional representation for the elliptic $ K $-operator satisfying the boundary Yang-Baxter equation. By restricting the functional space to finite-dimensional space, we construct the elliptic $ K $-matrix associated to Belavin's completely $ \mathbb{Z} $-symmetric $ R $-matrix. 
  We present the extension of the Lagrangian loop gauge invariant representation in such a way to include matter fields. The partition function of lattice compact U(1)-Higgs model is expressed as a sum over closed as much as open surfaces. We have simulated numerically the loop action equivalent to the Villain form of the action and mapped out the beta-gamma phase diagram of this model. 
  We present recent developments in the theory of Nambu mechanics, which include new examples of Nambu-Poisson manifolds with linear Nambu brackets and new representations of Nambu-Heisenberg commutation relations. 
  We reformulate two dimensional string-inspired gravity with point particles as a gauge theory of the extended Poincar\'e group. A non-minimal gauge coupling is necessary for the equivalence of the two descriptions. The classical one-particle problem is analyzed completely. In addition, we obtain the many-particle effective action after eliminating the gravity degrees of freedom. We investigate properties of this effective action, and show how to recover the geometrical description. Quantization of the gauge-theoretic model is carried out and the explicit one-particle solution is found. However, we show that the formulation leads to a quantum mechanical inconsistency in the two-particle case. Possible cures are discussed. 
  The set of Casimir operators associated with the global symmetries of a charged string in a constant magnetic background are found. It is shown that the string rest energy can be expressed as a combination of these invariants. Using this result, the Regge trajectories of the system are derived. The first Regge trajectory is given by a family of infinitely many parallel straight-lines, one for each spin projection along the magnetic field. 
  Explicit expressions for the Temperley-Lieb-Martin algebras, i.e., the quotients of the Hecke algebra that admit only representations corresponding to Young diagrams with a given maximum number of columns (or rows), are obtained, making explicit use of the Hecke algebra representation theory. Similar techniques are used to construct the algebras whose representations do not contain rectangular subdiagrams of a given size. 
  The tracing of cosmological relics from the early string dynamics may enhance the theory and provide new perspectives on the major cosmological problems. This point is illustrated in a leading-order Bianchi-type $VII_0$ string background, wherein spatial isotropy can be claimed as such a relic. A much finer one, descending from a premordial gravitational wave, could be retrieved from its imprint on the small-scale structure of the cosmic microwave background. In spite of the absence of conventional inflation, there is no horizon problem thanks to the presence of an equally fundamental mixmaster dynamics. Implications and certain new perspectives which thus arise for the more general problem of cosmological mixing are briefly discussed. 
  In this paper, we extend previous work on scalar $\phi^4$ theory using the Source Galerkin method. This approach is based on finding solutions $Z[J]$ to the lattice functional equations for field theories in the presence of an external source $J$. Using polynomial expansions for the generating functional $Z$, we calculate propagators and mass-gaps for a number of systems. These calculations are straightforward to perform and are executed rapidly compared to Monte Carlo. The bulk of the computation involves a single matrix inversion. The use of polynomial expansions illustrates in a clear and simple way the ideas of the Source Galerkin method. But at the same time, this choice has serious limitations. Even after exploiting symmetries, the size of calculations become prohibitive except for small systems. The calculations in this paper were made on a workstation of modest power using a fourth order polynomial expansion for lattices of size $8^2$,$4^3$,$2^4$ in $2D$, $3D$, and $4D$. In addition, we present an alternative to the Galerkin procedure that results in sparse matrices to invert. 
  A new deterministic, numerical method to solve fermion field theories is presented. This approach is based on finding solutions $Z[J]$ to the lattice functional equations for field theories in the presence of an external source $J$. Using Grassmann polynomial expansions for the generating functional $Z$, we calculate propagators for systems of interacting fermions. These calculations are straightforward to perform and are executed rapidly compared to Monte Carlo. The bulk of the computation involves a single matrix inversion. Because it is not based on a statistical technique, it does not have many of the difficulties often encountered when simulating fermions. Since no determinant is ever calculated, solutions to problems with dynamical fermions are handled more easily. This approach is very flexible, and can be taylored to specific problems based on convenience and computational constraints. We present simple examples to illustrate the method; more general schemes are desirable for more complicated systems. 
  The formalism of Supersymmetric Quantum Mechanics provides us the eigenfunctions to be used in the variational mathod to obtain the eigenvalues for the Hulth\'en Potential. 
  We investigate with the help of Clifford algebraic methods the Mandelbrot set over arbitrary two-component number systems. The complex numbers are regarded as operator spinors in D\times spin(2) resp. spin(2). The thereby induced (pseudo) normforms and traces are not the usual ones. A multi quadratic set is obtained in the hyperbolic case contrary to [1]. In the hyperbolic case a breakdown of this simple dynamics takes place. 
  The propagation of a localized wave packet in the conical space-time created by a pointlike massive source in 2+1 dimensional gravity is analyzed. The scattering amplitude is determined and shown to be finite along the classical scattering directions due to interference between the scattered and the transmitted wave functions. The analogy with diffraction theory is emphasized. 
  We consider the low-energy effective action for the Coulomb phase of an $N = 2$ supersymmetric gauge theory with a rank one gauge group. The $N = 2$ superspace formalism is naturally invariant under an $SL(2, {\bf Z})$ group of duality transformations, regardless of the form of the action. The leading and next to leading terms in the long distance expansion of the action are given by the holomorphic prepotential and a real analytic function respectively. The latter is shown to be modular invariant with respect to $SL(2, {\bf Z})$. 
  We discuss the behavior of theories of fermions coupled to Chern-Simons gauge fields with a non-abelian gauge group in three dimensions and at finite temperature. Using non-perturbative arguments and gauge invariance, and in contradiction with perturbative results, we show that the coefficient of the Chern-Simons term of the effective actions for the gauge fields at finite temperature can be {\it at most} an integer function of the temperature. This is in a sense a generalized no-renormalization theorem. We also discuss the case of abelian theories and give indications that a similar condition should hold there too. We discuss consequences of our results to the thermodynamics of anyon superfluids and fractional quantum Hall systems. 
  The mean field like gauge invariant variational method formulated recently, is applied to a topologically massive QED in 3 dimensions. We find that the theory has a phase transition in the Chern Simons coefficient $n$. The phase transition is of the Berezinsky-Kosterlitz - Thouless type, and is triggered by the liberation of Polyakov monopoles, which for $n>8$ are tightly bound into pairs. In our Hamiltonian approach this is seen as a similar behaviour of the magnetic vortices, which are present in the ground state wave functional of the compact theory. For $n>8$, the low energy behavior of the theory is the same as in the noncompact case. For $n<8$ there are no propagating degrees of freedom on distance scales larger than the ultraviolet cutoff. The distinguishing property of the $n<8$ phase, is that the magnetic flux symmetry is spontaneoously broken. 
  Under general assumptions, we present a low-energy effective action for the quantum Hall state when edges exist. It is shown that the chiral edge current is necessary to make the effective action to be gauge invariant. However the chiral edge current is irrelevant to the Hall current. The exactly quantized value of $\sigma_{xy}$ is observed only when the Hall current does not flow at the edge region. Our effective theory is applicable to the quantum Hall liquid on a surface with non-trivial topology and physical meanings of the topology are discussed. 
  The Wightman functions in the Rindler portion of Minkowski space-time are presented for any value of the temperature and for massless spin fields up to s=1 and the renormalized stress tensor relative to Minkowski vacuum is discussed. A gauge ambiguity in the vector case is pointed out. 
  We present an explicit expression for the topological invariants associated to $SU(2)$ monopoles in the fundamental representation on spin four-manifolds. The computation of these invariants is based on the analysis of their corresponding topological quantum field theory, and it turns out that they can be expressed in terms of Seiberg-Witten invariants. In this analysis we use recent exact results on the moduli space of vacua of the untwisted $N=1$ and $N=2$ supersymmetric counterparts of the topological quantum field theory under consideration, as well as on electric-magnetic duality for $N=2$ supersymmetric gauge theories. 
  In the 1-dimensional matrix model one identifies the tachyon field in the asymptotic region with a nonlocal transform of the density of fermions. But there is a problem in relating the classical tachyon field with the surface profile of the fermi fluid if a fold forms in the fermi surface. Besides the collective field additional variables $w_j(x)$ are required to describe folds. In the quantum theory we show that the $w_j$ are the quantum dispersions of the collective field. These dispersions become $O(1)$ rather than $O(\hbar)$ precisely after fold formation, thus giving additional `classical' quantities and leading to a rather nontrivial classical limit. A coherent pulse reflecting from the potential wall turns into high energy incoherent quanta (if a fold forms), the frequency amplification being of the order of the square root of the number of quanta in the incident wave. 
  The main result of this paper is the construction of a conformally covariant operator in two dimensions acting on scalar fields and containing fourth order derivatives. In this way it is possible to derive a class of Lagrangians invariant under conformal transformations. They define conformal field theories satisfying equations of the biharmonic type. Two kinds of these biharmonic field theories are distinguished, characterized by the possibility or not of the scalar fields to transform non-trivially under Weyl transformations. Both cases are relevant for string theory and two dimensional gravity. The biharmonic conformal field theories provide higher order corrections to the equations of motion of the metric and give a possibility of adding new terms to the Polyakov action. 
  Magnetic fields in five-dimensional Kaluza-Klein theory compactified on a circle correspond to ``twisted'' identifications of five dimensional Minkowski space. We show that a five dimensional generalisation of the Kerr solution can be analytically continued to construct an instanton that gives rise to two possible decay modes of a magnetic field. One decay mode is the generalisation of the ``bubble decay" of the Kaluza-Klein vacuum described by Witten. The other decay mode, rarer for weak fields, corresponds in four dimensions to the creation of monopole-anti-monopole pairs. An instanton for the latter process is already known and is given by the analytic continuation of the \KK\ Ernst metric, which we show is identical to the five dimensional Kerr solution. We use this fact to illuminate further properties of the decay process. It appears that fundamental fermions can eliminate the bubble decay of the magnetic field, while allowing the pair production of Kaluza-Klein monopoles. 
  We calculate the prepotential of the low-energy effective action for $N=2$ $SU(2)$ supersymmetric Yang-Mills theory with $N_f$ massless hypermultiplets ($N_f=1,\, 2,\, 3$). The precise evaluation of the instanton corrections is performed by making use of the Picard-Fuchs equations associated with elliptic curves. The flavor dependence of the instanton effect is determined explicitly both in the weak- and strong-coupling regimes. 
  The BRST cohomology of any topological conformal field theory admits the structure of a Batalin--Vilkovisky algebra, and string theories are no exception. Let us say that two topological conformal field theories are ``cohomologically equivalent'' if their BRST cohomologies are isomorphic as Batalin--Vilkovisky algebras. What we show in this paper is that any string theory (regardless of the matter background) is cohomologically equivalent to some twisted N=2 superconformal field theory. We discuss three string theories in detail: the bosonic string, the NSR string and the W_3 string. In each case the way the cohomological equivalence is constructed can be understood as coupling the topological conformal field theory to topological gravity. These results lend further supporting evidence to the conjecture that _any_ topological conformal field theory is cohomologically equivalent to some topologically twisted N=2 superconformal field theory. We end the paper with some comments on different notions of equivalence for topological conformal field theories and this leads to an improved conjecture. 
  We study the relation between lattice construction and surgery construction of three-dimensional topological field theories. We show that a class of the Chung-Fukuma-Shapere theory on the lattice has representation theoretic reformulation which is closely related to the Altschuler-Coste theory constructed by surgery. There is a similar relation between the Turaev-Viro theory and the Reshetikhin-Turaev theory. 
  We address the problem of the continuum limit for a system of Hausdorff lattices (namely lattices of isolated points) approximating a topological space $M$. The correct framework is that of projective systems. The projective limit is a universal space from which $M$ can be recovered as a quotient. We dualize the construction to approximate the algebra ${\cal C}(M)$ of continuous functions on $M$. In a companion paper we shall extend this analysis to systems of noncommutative lattices (non Hausdorff lattices). 
  We consider finite approximations of a topological space $M$ by noncommutative lattices of points. These lattices are structure spaces of noncommutative $C^*$-algebras which in turn approximate the algebra $\cc(M)$ of continuous functions on $M$. We show how to recover the space $M$ and the algebra $\cc(M)$ from a projective system of noncommutative lattices and an inductive system of noncommutative $C^*$-algebras, respectively. 
  A recurrence relation of Riccati-type differential equations known in supersymmetric quantum mechanics is investigated to find exactly solvable potentials. Taking some simple {\it ans\"atze}, we find new classes of solvable potentials as well as reproducing the known shape-invariant ones. 
  Recently Connes has proposed a new geometric version of the standard model including a non-commutative charge conjugation. We present a systematic analysis of the relations among masses and coupling constants in this approach. In particular, for a given top mass, the Higgs mass is constrained to lie in an interval. Therefore this constraint is locally stable under renormalization flow. 
  Motivated by the recent work of Kachru-Vafa in string theory, we study in Part A of this paper, certain identities involving modular forms, hypergeometric series, and more generally series solutions to Fuchsian equations. The identity which arises in string theory is the simpliest of its kind. There are nontrivial generalizations of the identity which appear new. We give many such examples -- all of which arise in mirror symmetry for algebraic K3 surfaces.  In Part B, we study the integrality property of certain $q$-series, known as mirror maps, which arise in mirror symmetry. 
  A four dimensional fermion determinant is presented as a path integral of the exponent of a local five dimensional action describing constrained bosonic system. The construction is carried out both in the continuum theory and in the lattice model. 
  As a continuation of \lianyaufour, we study modular properties of the periods, the mirror maps and Yukawa couplings for multi-moduli Calabi-Yau varieties. In Part A of this paper, motivated by the recent work of Kachru-Vafa, we degenerate a three-moduli family of Calabi-Yau toric varieties along a codimension one subfamily which can be described by the vanishing of certain Mori coordinate, corresponding to going to the ``large volume limit'' in a certain direction. Then we see that the deformation space of the subfamily is the same as a certain family of K3 toric surfaces. This family can in turn be studied by further degeneration along a subfamily which in the end is described by a family of elliptic curves. The periods of the K3 family (and hence the original Calabi-Yau family) can be described by the squares of the periods of the elliptic curves. The consequences include: (1) proofs of various conjectural formulas of physicists \vk\lkm~ involving mirror maps and modular functions; (2) new identities involving multi-variable hypergeometric series and modular functions -- generalizing \lianyaufour. In Part B, we study for two-moduli families the perturbation series of the mirror map and the type A Yukawa couplings near certain large volume limits. Our main tool is a new class of polynomial PDEs associated with Fuchsian PDE systems. We derive the first few terms in the perturbation series. For the case of degree 12 hypersurfaces in $\P^4[6,2,2,1,1]$, in one limit the series of the couplings are expressed in terms of the $j$ function. In another limit, they are expressed in terms of rational functions. The latter give explicit formulas for infinite sequences of ``instanton numbers'' $n_d$. 
  Using quantum Hamilton-Jacobi formalism of Leacock and Padgett, we show how to obtain the exact eigenvalues for supersymmetric (SUSY) potentials. 
  We present numerical evidence that a simple variational improvement of the ordinary perturbation theory of the quantum anharmonic oscillator can give a convergent sequence of approximations even in the extreme strong coupling limit, the purely anharmonic case. Some of the new techniques of this paper can be extended to renormalizable field theories. 
  We present an alternative pathway in the application of the variation improvement of ordinary perturbation theory exposed in [1] which can preserve the internal symmetries of a model by means of a time compactification. 
  Factorization of string amplitudes is one way of constructing string interaction vertices. We show that correlation functions in string theory can be conveniently factorized using loop variables representing delta functionals. We illustrate this construction with some examples where one particle is off-shell. These vertices are ``correct'' in the sense that they are guaranteed, by construction, to reproduce S-matrix elements when combined with propagators in a well defined way. 
  Non-perturbative instanton corrections to the moduli space geometry of type IIA string theory compactified on a Calabi-Yau space are derived and found to contain order $e^{-1/g_s}$ contributions, where $g_s$ is the string coupling. The computation reduces to a weighted sum of supersymmetric extremal maps of strings, membranes and fivebranes into the Calabi-Yau space, all three of which enter on equal footing. It is shown that a supersymmetric 3-cycle is one for which the pullback of the K\"ahler form vanishes and the pullback of the holomorphic three-form is a constant multiple of the volume element. Quantum mirror symmetry relates the sum in the IIA theory over supersymmetric, odd-dimensional cycles in the Calabi-Yau space to a sum in the IIB theory over supersymmetric, even-dimensional cycles in the mirror. 
  The scaling form of the critical equation of state is computed for $O(N)$-symmetric models. We employ a method based on an exact flow equation for a coarse grained free energy. A suitable truncation is solved numerically. 
  A class of supersymmetric (BPS saturated), static, spherically symmetric solutions of four-dimensional effective $N=4$ supersymmetric superstring vacua, which become massless at special points of moduli space, is studied in terms of the fields of the effective heterotic string theory compactified on a six-torus. Those are singular four-dimensional solutions corresponding to $O(6,22,Z)$ orbits of dyonic configurations (with zero axion), whose left-moving as well as right-moving electric and magnetic charges are orthogonal (light-like in the $O(6,22,Z)$ sense), while the $O(6,22,Z)$ norms of both the electric and magnetic charges are negative. Purely electric [or purely magnetic] and dyonic configurations preserve $1\over 2$ and $1\over 4$ of $N=4$ supersymmetry, respectively, thus belonging to the vector and the highest spin $3\over 2$ supermultiplets, respectively. Purely electric [or purely magnetic] solutions (along with an infinite tower of $SL(2,Z)$ transformed states) become massless at a point of the corresponding ``one-torus'', thus may contribute to the enhancement of non-Abelian gauge symmetry, while dyonic solutions become simultaneously massless at a point of the corresponding two-torus, and thus may in addition contribute to the enhancement of the local supersymmetry there. 
  A recently introduced model describing the folding of the triangular lattice is generalized allowing for defects in the lattice and written as an Ising model with nearest-neighbor and plaquette interactions on the honeycomb lattice. Its phase diagram is determined in the hexagon approximation of the cluster variation method and the crossover from the pure Ising to the pure folding model is investigated, obtaining a quite rich structure with several multicritical points. Our results are in very good agreement with the available exact ones and extend a previous transfer matrix study. 
  Recent developments about the construction of standard $SO(10)$ and $SU(5)$ grand unified theories from 4-dimensional superstrings are presented. Explicit techniques involving higher level affine Lie algebras, for obtaining such stringGUTs from symmetric orbifolds are discussed. Special emphasis is put on the different constraints and selection rules for model building in this string framework, trying to disentangle those which are generic from those depending on the orbifold construction proposed. Some phenomenological implications from such constraints are briefly discussed. 
  We present a supersymmetric non-linear $\s$-model built up in the $N=1$ superspace of Atiyah-Ward space-time. A manifold of the K\"ahler type comes out that is restricted by a particular decomposition of the K\"ahler potential. The gauging of the $\s$-model isometries is also accomplished in superspace. 
  New and surprisingly simple representation is found for the heterotic string bosonic effective action in three dimensions in terms of complex potentials. The system is presented as a K\"ahler $\sigma$--model using complex symmetric $2\times 2$ matrix (matrix dilaton--axion) which depends linearly on three Ernst--type potentials and transforms under $U$--duality via matrix valued $SL(2,R)$. Two discrete automorphisms relating ten isometries of the target space (U--duality transformations) are found and used to generate the non--trivial Ehlers--Harrison sector by a map from the trivial gauge sector. Finite transformations are obtained in a simple form in terms of complex potentials. New solution generating technique is used to construct EMDA double--Kerr solution. 
  We find the exact N-point generating function in Polyakov's approach to Burgers turbulence. 
  Noting that two-dimensional magnetohydrodynamics can be modeled by conformal field theory, we argue that when the Alf'ven effect is also taken into account one is naturally lead to consider conformal field theories, which have logarithmic terms in their correlation functions. We discuss the implications of such logarithmic terms in the context of magnetohydrodynamics, and derive a relationship between conformal dimensions of the velocity stream function, the magnetic flux function and the Reynolds number. 
  With the perspective of looking for experimentally detectable physical applications of the so-called topological embedding, a procedure recently proposed by the author for quantizing a field theory around a non-discrete space of classical minima (instantons, for example), the physical implications are discussed in a ``theoretical'' framework, the ideas are collected in a simple logical scheme and the topological version of the Ginzburg-Landau theory of superconductivity is solved in the intermediate situation between type I and type II superconductors. 
  We discuss duality between Type IIA string theory, eleven-dimensional supergravity, and heterotic string theory in four spacetime dimensions with $N=1$ supersymmetry. We find theories whose infrared limit is trivial at enhanced symmetry points as well as theories with $N=1$ supersymmetry but the field content of $N=4$ theories which flow to the $N=4$ fixed line in the infrared. 
  We investigate the low-energy dynamics of $SU(N)$ gauge theories with one antisymmetric tensor field, $N - 4 + N_f$ antifundamentals and $N_f$ fundamentals, for $N_f \le 3$. For $N_f = 3$ we construct the quantum moduli space, and for $N_f < 3$ we find the exact quantum superpotentials. We find two large classes of models with dynamical supersymmetry breaking. The odd $N$ theories break supersymmetry once appropriate mass terms are added in the superpotential.  The even $N$ theories break supersymmetry after gauging an extra chiral $U(1)$ symmetry. 
  We discuss duality in $N=1$ SUSY gauge theories in Seiberg's conformal window, $(3N_c/2)<N_f<3N_c$. The 't Hooft consistency conditions -- the basic tool for establishing the infrared duality -- are considered taking into account higher order $\alpha$ corrections. The conserved (anomaly free) $R$ current is built to all orders in $\alpha$. Although this current contains all orders in $\alpha$ the 't Hooft consistency conditions for this current are shown to be one-loop. This observation thus justifies Seiberg's matching procedure. We also briefly discuss the inequivalence of the ``electric" and ``magnetic" theories at short distances. 
  We couple non-linear $\sigma$-models to Liouville gravity, showing that integrability properties of symmetric space models still hold for the matter sector. Using similar arguments for the fermionic counterpart, namely Gross--Neveu-type models, we verify that such conclusions must also hold for them, as recently suggested. 
  We apply the canonical perturbation theory to the semi--quantal hamiltonian of the SU(3) shell model. Then, we use the Einstein--Brillowin--Keller quantization rule to obtain an analytical semi--quantal formula for the energy levels, which is the usual semi--classical one plus quantum corrections. Finally, a test on the numerical accuracy of the semiclassical approximation and of its quantum corrections is performed. 
  The metric on the moduli space of SU(2) charge four BPS monopoles with tetrahedral symmetry is calculated using numerical methods. In the asymptotic region, in which the four monopoles are located on the vertices of a large tetrahedron, the metric is in excellent agreement with the point particle metric. We find that the four monopoles are accelerated through the cubic monopole configuration and compute the time advance. Numerical evidence is presented for a remarkable equivalence between a proper distance in the 4-monopole moduli space and a related proper distance in the point particle moduli space. 
  Given a finite dimensional C-*-Hopf algebra H and its dual H^ we construct the infinite crossed product A=... x H x H^ x H x ... and study its representations. A is the observable algebra of a generalized spin model with H-order and H^-disorder symmetries. By pointing out that A possesses a certain compressibility property we can classify all DHR-sectors of A --- relative to some Haag dual vacuum representation --- and prove that their symmetry is described by the Drinfeld double D(H). Complete, irreducible, translation covariant field algebra extensions F > A are shown to be in one-to-one correspondence with cohomology classes of 2-cocycles u in D(H) @ D(H). 
  Recently proposed mechanism of the black hole condensation at conifold singularity in type II string is an interesting idea from which we can interpret the phase of the universal moduli space of the string vacua. It might also be expected that the true physics is on the conifold singularity after supersymmetry breaking. We derive a mass formula for the extreme black holes caused by the self-dual 5-form field strength, which is stable and supersymmetric. It is shown that the formula can be written by the moduli parameters of Calabi-Yau manifold and can be calculated explicitly. 
  We give an explicit formalism connecting softly broken supersymmetric gauge theories (with QCD as one limit) to $N=2$ and $N=1$ supersymmetric theories possessing exact solutions, using spurion fields to embed these models in an enlarged $N=1$ model. The functional forms of effective Lagrangian terms resulting from soft supersymmetry breaking are constrained by the symmetries of the enlarged model, although not well enough to fully determine the vacuum structure of generic softly broken models. Nevertheless by perturbing the exact $N=1$ model results with sufficiently small soft breaking masses, we show that there exist nonsupersymmetric models that exhibit monopole condensation and confinement in the same modes as the $N=1$ case. 
  Consistent Yang--Mills anomalies $\int\om_{2n-k}^{k-1}$ ($n\in\N$, $ k=1,2, \ldots ,2n$) as described collectively by Zumino's descent equations $\delta\om_{2n-k}^{k-1}+\dd\om_{2n-k-1}^{k}=0$ starting with the Chern character $Ch_{2n}=\dd\om_{2n-1}^{0}$ of a principal $\SU(N)$ bundle over a $2n$ dimensional manifold are considered (i.e.\ $\int\om_{2n-k}^{k-1}$ are the Chern--Simons terms ($k=1$), axial anomalies ($k=2$), Schwinger terms ($k=3$) etc.\ in $(2n-k)$ dimensions). A generalization in the spirit of Connes' noncommutative geometry using a minimum of data is found. For an arbitrary graded differential algebra $\CC=\bigoplus_{k=0}^\infty \CC^{(k)}$ with exterior differentiation $\dd$, form valued functions $Ch_{2n}: \CC^{(1)}\to \CC^{(2n)}$ and $\om_{2n-k}^{k-1}: \underbrace{\CC^{(0)}\times\cdots \times \CC^{(0)}}_{\mbox{{\small $(k-1)$ times}}} \times \CC^{(1)}\to \CC^{(2n-k)}$ are constructed which are connected by generalized descent equations $\delta\om_{2n-k}^{k-1}+\dd\om_{2n-k-1}^{k}=(\cdots)$. Here $Ch_{2n}= (F_A)^n$ where $F_A=\dd(A)+A^2$ for $A\in\CC^{(1)}$, and $(\cdots)$ is not zero but a sum of graded commutators which vanish under integrations (traces). The problem of constructing Yang--Mills anomalies on a given graded differential algebra is thereby reduced to finding an interesting integration $\int$ on it. Examples for graded differential algebras with such integrations are given and thereby noncommutative generalizations of Yang--Mills anomalies are found. 
  We consider the thermal properties of a scalar field theory on curved spacetimes. In particular, we argue for the existence in the de Sitter, Kruskal and Rindler manifolds of a discrete spectrum of allowed temperatures (the odd multiples of a fundamental one). For each temperature we give an explicit construction of the relative two point function in terms of the lowest temperature one. These results are actually valid for a wider class of static metrics with bifurcate Killing horizons, originally studied by Sewell. Some comments on the interpretation of our results are given. 
  We construct N=2 affine current algebras for the superalgebras sl(n|n-1)^{(1)} in terms of N=2 supercurrents subjected to nonlinear constraints and discuss the general procedure of the hamiltonian reduction in N=2 superspace at the classical level. We consider in detail the simplest case of N=2 sl(2|1)^{(1)} and show how N=2 superconformal algebra in N=2 superspace follows via the hamiltonian reduction. Applying the hamiltonian reduction to the case of N=2 sl(3|2)^{(1)}, we find two new extended N=2 superconformal algebras in a manifestly supersymmetric N=2 superfield form. Decoupling of four component currents of dimension 1/2 in them yields, respectively, u(2|1) and u(3) Knizhnik-Bershadsky superconformal algebras. We also discuss how the N=2 superfield formulations of N=2 W_{3} and N=2 W_{3}^{(2)} superconformal algebras come out in this framework, as well as some unusual extended N=2 superconformal algebras containing constrained N=2 stress tensor and/or spin 0 supercurrents. 
  We discuss bosonized two-dimensional QCD with massless fermions in the adjoint and multi-flavor fundamental representations. We evaluate the massive mesonic spectra of several models by using the light-front quantization and diagonalizing the mass operator $M^2=2P^+P^- $. We recover previous results in the case of one flavor adjoint fermions and we find the exact massive spectrum of multi flavor QCD in the limit of large number of flavors. 
  The $(g_2^{(1)}, d_4^{(3)})$\ pair of non simply laced affine Toda theories is studied from the point of view of non perturbative duality. The classical spectrum of each member is composed of two massive scalar particles. The exact S-matrix prediction for the dual behaviour of the coupling dependent mass ratio is found to be in strong agreement with Monte Carlo data. 
  We present a real space renormalization-group map for probabilities of random walks on a hierarchical lattice. From this, we study the asymptotic behavior of the end-to-end distance of a weakly self- avoiding random walk (SARW) that penalizes the (self-)intersection of two random walks in dimension four on the hierarchical lattice. 
  From the computation of three-point singlet correlators in the two-matrix model, we obtain an explicit expression for the macroscopic three-loop amplitudes having boundary lengths $\ell_{i}$ $(i = 1\sim 3)$ in the case of the unitary series $(p,q)= (m+1,m)$ coupled to two-dimensional gravity. The sum appearing in this expression is found to conform to the structure of the CFT fusion rules while the summand factorizes through a product of three modified Bessel functions. We briefly discuss a possible generalization of these features to macroscopic $n$-loop amplitudes. 
  We develop a relativistic free wave equation on the complexified quaternions, linear in the derivatives. Even if the wave functions are only one-component, we show that four independent solutions, corresponding to those of the Dirac equation, exist. A partial set of translations between complex and complexified quaternionic quantum mechanics may be defined. 
  We reformulate Special Relativity by a quaternionic algebra on reals. Using {\em real linear quaternions}, we show that previous difficulties, concerning the appropriate transformations on the $3+1$ space-time, may be overcome. This implies that a complexified quaternionic version of Special Relativity is a choice and not a necessity. 
  We reformulate the self-dual Einstein equation as a trio of differential form equations for simple two-forms. Using them, we can quickly show the equivalence of the theory and 2D sigma models valued in an infinite-dimensional group, which was shown by Park and Husain earlier. We also derive other field theories including the 2D Higgs bundle equation. This formulation elucidates the relation among those field theories. 
  We give a simple geometric description of all formal deformation quantizations on a K\"ahler manifold $M$ which enjoy the following property of separation of variables into holomorphic and antiholomorphic ones. For each open subset $U\subset M$, $\star$-multiplication from the left by a holomorphic function and from the right by an antiholomorphic function on $U$ coincides with the pointwise multiplication by these functions. These quantizations are in 1-1 correspondence with formal deformations of the original K\"ahler metrics on $M$. It has been shown in [Ka] that the formal deformation quantizations obtained from the full asymptotic expansion of Berezin's $*$-product on the orbits of a compact semisimple Lie group in [Mo2] and [CGR1] and on bounded symmetric domains in [Mo1] and [CGR2] are those with separation of variables and correspond to the trivial deformation of the original K\"ahler metrics. 
  The flow of couplings under anisotropic scaling of momenta is computed in $\phi^3$ theory in 6 dimensions. It is shown that the coupling decreases as momenta of two of the particles become large, keeping the third momentum fixed, but at a slower rate than the decrease of the coupling if all three momenta become large simultaneously. This effect serves as a simple test of effective theories of high energy scattering, since such theories should reproduce these deviations from the usual logarithmic scale dependence. 
  The large-N saddle-point equations for the principal chiral models defined on a d-1 dimensional simplex are derived from the external field problem for unitary integrals. The saddle point equation are studied analytically and numerically in many relevant instances, including d=4 and $d\rightarrow\infty$, with special attention to the critical domain, which is found to correspond to $\beta_c=1/d$ for all d. Related models (chiral chains) are discussed and large-N solutions are analyzed. 
  Spin generalization of the relativistic Calogero-Sutherland model is constructed by using the affine Hecke algebra and shown to possess the quantum affine symmetry $\uqglt$. The spin-less model is exactly diagonalized by means of the Macdonald symmetric polynomials. The dynamical density-density correlation function as well as one-particle Green function are evaluated exactly. We also investigate the finite-size scaling of the model and show that the low-energy behavior is described by the $C=1$ Gaussian theory. The results indicate that the excitations obey the fractional exclusion statistics and exhibit the Tomonaga-Luttinger liquid behavior as well. 
  We show how the exact renormalization group for the effective action with a sharp momentum cutoff, may be organised by expanding one-particle irreducible parts in terms of homogeneous functions of momenta of integer degree (Taylor expansions not being possible). A systematic series of approximations -- the $O(p^M)$ approximations -- result from discarding from these parts, all terms of higher than the $M^{\rm th}$ degree. These approximations preserve a field reparametrization invariance, ensuring that the field's anomalous dimension is unambiguously determined. The lowest order approximation coincides with the local potential approximation to the Wegner-Houghton equations. We discuss the practical difficulties with extending the approximation beyond $O(p^0)$. 
  We derive the kinematical constraints which characterize the decay of any massless particle in flat spacetime. We show that in perturbation theory the decay probabilities of photons and Yang-Mills bosons vanish to all orders; the decay probability of the graviton vanishes to one-loop order for graviton loops and to all orders for matter loops. A general power counting argument indicates in which conditions a decay of a massless particle could be possible: the lagrangian should contain a self-coupling without derivatives and with a coupling constant of positive mass dimension. 
  An Introduction to Hopf algebras as a tool for the regularization of relavent quantities in quantum field theory is given. We deform algebraic spaces by introducing q as a regulator of a non-commutative and non-cocommutative Hopf algebra. Relevant quantities are finite provided q\neq 1 and diverge in the limit q\rightarrow 1. We discuss q-regularization on different q-deformed spaces for \lambda\phi^4 theory as example to illustrate the idea. 
  The main result in this paper is the character formula for arbitrary irreducible highest weight modules of W algebras. The key ingredient is the functor provided by quantum Hamiltonian reduction, that constructs the W algebras from affine Kac-Moody algebras and in a similar fashion W modules from KM modules. Assuming certain properties of this functor, the W characters are subsequently derived from the Kazhdan-Lusztig conjecture for KM algebras. The result can be formulated in terms of a double coset of the Weyl group of the KM algebra: the Hasse diagrams give the embedding diagrams of the Verma modules and the Kazhdan-Lusztig polynomials give the multiplicities in the characters. 
  We show that S-duality in four dimensional non-supersymmetric abelian gauge theories can be formulated as a canonical transformation in the phase space of the theory. This transformation is the usual interchange between electric and magnetic degrees of freedom. It is shown that in phase space the modular anomaly emerges as the result of integrating out the momenta degrees of freedom. The generalization to d dimensional abelian gauge theories of p forms is also considered. In the case of non-abelian gauge theories a careful analysis of the constraints implied by the canonical transformation shows that it does not relate Yang-Mills theories with inverted couplings. In fact the dual theory is shown to be of Freedman- Townsend's type, also with ${\tilde \tau}=-1/\tau$, $\tau=\frac{\theta}{2\pi}+\frac{4\pi i}{g^2}$. 
  We describe three different methods for generating quasi-exactly solvable potentials, for which a finite number of eigenstates are analytically known. The three methods are respectively based on (i) a polynomial ansatz for wave functions; (ii) point canonical transformations; (iii) supersymmetric quantum mechanics. The methods are rather general and give considerably richer results than those available in the current literature. 
  In this report it is proposed to generalize the definition of Poisson brackets in order to treat spatial integrals of divergences as Hamiltonians which generate a kind of Hamiltonian equations on the boundary. Nonlinear Schrodinger equation is used as an illustrative example. 
  This is a brief review of the main properties of sphalerons in various theories with a Yang--Mills field. Talk given at the Heat Kernel Techniques and Quantum Gravity, Winnipeg, Canada, August 2-6, 1994. 
  The Regge behaviour of the scattering amplitudes in perturbative QCD is governed in the generalized leading logarithmic approximation by the contribution of the color--singlet compound states of Reggeized gluons. The interaction between Reggeons is described by the effective hamiltonian, which in the multi--color limit turns out to be identical to the hamiltonian of the completely integrable one--dimensional XXX Heisenberg magnet of noncompact spin $s=0$. The spectrum of the color singlet Reggeon compound states - perturbative Pomerons and Odderons, is expressed by means of the Bethe Ansatz in terms of the fundamental $Q-$function, which satisfies the Baxter equation for the XXX Heisenberg magnet. The exact solution of the Baxter equation is known only in the simplest case of the compound state of two Reggeons, the BFKL Pomeron. For higher Reggeon states the method is developed which allows to find its general solution as an asymptotic series in powers of the inverse conformal weight of the Reggeon states. The quantization conditions for the conserved charges for interacting Reggeons are established and an agreement with the results of numerical solutions is observed. The asymptotic approximation of the energy of the Reggeon states is defined based on the properties of the asymptotic series, and the intercept of the three--Reggeon states, perturbative Odderon, is estimated. 
  The Landau-Ginzburg formulation of two-dimensional topological sigma models on the target space with positive first Chern class is considered. The effective Landau-Ginzburg superpotential takes the form of logarithmic type which is characteristic of supersymmetric theories with the mass gap. The equations of motion yield the defining relations of the quantum cohomology ring. Topological correlation functions in the $CP^{n-1}$ and Grassmannian models are explicitly evaluated with the use of the logarithmic superpotential. 
  Quantum canonical transformations of the second kind and the non-Hermitian realizations of the basic canonical commutation relations are investigated with a special interest in the generalization of the conventional ladder operators. The operator ordering problem is shown to be resolved when the non-Hermitian realizations for the canonical variables which can not be measured simultaneously with the energy are chosen for the canonical quantizations. Another merit of the non-Hermitian representations is that it naturally allows us to introduce the generalized ladder operators with which one can solve eigenvalue problems quite neatly. 
  There exist certain intrinsic relations between the ultraviolet divergent graphs and the convergent ones at the same loop order in renormalizable quantum field theories. Whereupon we may establish a new method, the intrinsic regularization method, to regularize those divergent graphs. In this paper, we apply this method to QCD at the one loop order. It turns out to be satisfactory:The gauge invariance is preserved manifestly and the results are the same as those derived by means of other regularization methods. 
  We study both spherically symmetric and rotating black holes with dilaton coupling and discuss the evaporation of these black holes via Hawking's quantum radiation and their fates. We find that the dilaton coupling constant $\alpha$ drastically affects the emission rates, and therefore the fates of the black holes. When the charge is conserved, the emission rate from the non-rotating hole is drastically changed beyond $\alpha = 1$ (a superstring theory) and diverges in the extreme limit. In the rotating cases, we analyze the slowly rotating black hole solution with arbitrary $\alpha$ as well as three exact solutions, the Kerr--Newman ($\alpha = 0$), and Kaluza--Klein ($\alpha = \sqrt{3}$), and Sen black hole ($\alpha = 1$ and with axion field). Beyond the same critical value of $\alpha \sim 1$, the emission rate becomes very large near the maximally charged limit, while for $\alpha<1$ it remains finite. The black hole with $\alpha > 1$ may evolve into a naked singularity due to its large emission rate. We also consider the effects of a discharge process by investigating superradiance for the non-rotating dilatonic black hole. 
  The topological field theories associated with affine Lie superalgebras are constructed. Their BRST symmetry is characterised by a Kazama algebra containing spin 1, 2 and 3 operators and closes linearly. Under this symmetry all operators are grouped into BRST doublets. The relation between the models constructed and non-critical string theories is explored. 
  An algorithm is described to convert Lorentz and gauge invariant expressions in non--Abelian gauge theories with matter into a standard form, consisting of a linear combination of basis invariants. This algorithm is needed for computer calculations of effective actions. The defining properties of the basis invariants are reported. The number of basis invariants up to mass dimension 16 are presented. 
  We consider an internal space of two discrete points in the fifth dimension of the Kaluza-Klein theory by using the formalism of noncommutative geometry developed in a previous paper \cite{VIWA} of a spacetime supplemented by two discrete points. With the nonvanishing internal torsion 2-form there are no constraints implied on the vielbeins. The theory contains a pair of tensor, a pair of vector and a pair of scalar fields. Using the generalized Cartan structure equation we are able not only to determine uniquely the hermitian and metric compatible connection 1-forms, but also the nonvanishing internal torsion 2-form in terms of vielbeins. The resulting action has a rich and complex structure, a particular feature being the existence of massive modes. Thus the nonvanishing internal torsion generates a Kaluza-Klein type model with zero and massive modes. 
  We study in detail the structure of Grand Unified Theories derived as the low-energy limit of orbifold four-dimensional strings. To this aim, new techniques for building level-two symmetric orbifold theories are presented. New classes of GUTs in the context of symmetric orbifolds are then constructed. The method of permutation modding is further explored and SO(10) GUTs with both $45$ or $54$-plets are obtained. SU(5) models are also found through this method. It is shown that, in the context of symmetric orbifold $SO(10)$ GUTs, only a single GUT-Higgs, either a $54$ or a $45$, can be present and it always resides in an order-two untwisted sector. Very restrictive results also hold in the case of $SU(5)$. General properties and selection rules for string GUTs are described. Some of these selection rules forbid the presence of some particular GUT-Higgs couplings which are sometimes used in SUSY-GUT model building. Some semi-realistic string GUT examples are presented and their properties briefly discussed. 
  One Lagrangian BRST quantization principle is that of imposing correct Schwinger-Dyson equations through the BRST Ward identities. In this paper we show how to derive the analogous $Sp(2)$-symmetric quantization condition in flat coordinates from an underlying $Sp(2)$-symmetric Schwinger-Dyson BRST symmetry. We also show under what conditions this can be recast in the language of triplectic quantization. 
  The basic mathematical assumptions for autonomous linear kinetic equations for a classical system are formulated, leading to the conclusion that if they are differential equations on its phase space $M$, they are at most of the 2nd order. For open systems interacting with a bath at canonical equilibrium they have a particular form of an equation of a generalized Fokker-Planck type. We show that it is possible to obtain them as Liouville equations of Hamiltonian dynamics on $M$ with a particular non-commutative differential structure, provided certain geometric in character, conditions are fulfilled. To this end, symplectic geometry on $M$ is developped in this context, and an outline of the required tensor analysis and differential geometry is given. Certain questions for the possible mathematical interpretation of this structure are also discussed. 
  We study several aspects of the canonical quantization of supergravity in terms of the Asthekar variables. We cast the theory in terms of a $GSU(2)$ connection and we introduce a loop representation. The solution space is similar to the loop representation of ordinary gravity, the main difference being the form of the Mandelstam identities. Physical states are in general given by knot invariants that are compatible with the $GSU(2)$ Mandelstam identities. There is an explicit solution to all the quantum constraint equations connected with the Chern-Simons form, which coincides exactly with the Dubrovnik version of the Kauffman Polynomial. This provides for the first time the possibility of finding explicit analytic expressions for the coefficients of that knot polynomial. 
  The ``new fields" or ``superconformal functions" on $N=1$ super Riemann surfaces introduced recently by Rogers and Langer are shown to coincide with the Abelian differentials (plus constants), viewed as a subset of the functions on the associated $N=2$ super Riemann surface. We confirm that, as originally defined, they do not form a super vector space. 
  We study quantized Yang-Mills theory with massive vector fields in the framework of causal perturbation theory. The most general form of the interaction which is invariant under operator gauge transformations is pointed out. The generator of these transformations generally fails to be nilpotent. This defect, however, is easily cured by including scalar fields in the gauge transformations. Due to gauge invariance these scalar gauge fields couple to the Yang-Mills fields with predicted strength. We also show that invariance under ghost charge conjugation fixes the form of the interaction completely. The coupling of the Yang-Mills fields and the scalar gauge fields to matter is investigated. It is proven that gauge invariance implies unitarity of the physical $S$-matrix. We always work in the Fock space of free quantum fields in which all expressions are mathematically well defined. 
  We discuss the quantization of the ADHM sigma model. We show that the only quantum contributions to the effective theory come from the chiral anomalies and compute the first and second order terms. Finally the limit of vanishing instanton size is discussed. 
  The effects of three-dimensional perturbations in two-dimensional turbulence are investigated, through a conformal field theory approach. We compute scaling exponents for the energy spectra of enstrophy and energy cascades, in a strong coupling limit, and compare them to the values found in recent experiments. The extension of unperturbed conformal turbulence to the present situation is performed by means of a simple physical picture in which the existence of small scale random forces is closely related to deviations of the exact two-dimensional fluid motion. 
  The lattice definition of the two-dimensional topological quantum field theory [Fukuma, {\em et al}, Commun.~Math.~Phys.\ {\bf 161}, 157 (1994)] is generalized to arbitrary (not necessarily orientable) compact surfaces. It is shown that there is a one-to-one correspondence between real associative $*$-algebras and the topological state sum invariants defined on such surfaces. The partition and $n$-point functions on all two-dimensional surfaces (connected sums of the Klein bottle or projective plane and $g$-tori) are defined and computed for arbitrary $*$-algebras in general, and for the the group ring $A=\R[G]$ of discrete groups $G$, in particular. 
  The elementary and solitonic supersymmetric $p$-brane solutions to supergravity theories form families related by dimensional reduction, each headed by a maximal (`stainless') member that cannot be isotropically dimensionally oxidized into higher dimensions. We find several new families, headed by stainless solutions in various dimensions $D\le 9$. In some cases, these occur with dimensions $(D,p)$ that coincide with those of descendants of known families, but since the new solutions are stainless, they are necessarily distinct. The new stainless supersymmetric solutions include a 6-brane and a 5-brane in $D=9$, a string in $D=5$, and particles in all dimensions $5\le D\le 9$. 
  By studying an effective action description of the coupling of charged gauge fields in $N=2$ $SU(n)$ supersymmetric Yang-Mills theories, we can describe regions of moduli space where one or more of these fields becomes unphysical. We discuss subtleties in the structure of the moduli space for $SU(3)$. 
  We demonstrate that type II string theory compactified on a singular Calabi-Yau manifold is related to $c=1$ string theory compactified at the self-dual radius. We establish this result in two ways. First we show that complex structure deformations of the conifold correspond, on the mirror manifold, to the problem of maps from two dimensional surfaces to $S^2$. Using two dimensional QCD we show that this problem is identical to $c=1$ string theory. We then give an alternative derivation of this correspondence by mapping the theory of complex structure deformations of the conifold to Chern-Simons theory on $S^3$. These results, in conjunction with similar results obtained for the compactification of the heterotic string on $K_3\times T^2$, provide strong evidence in favour of S-duality between type II strings compactified on a Calabi-Yau manifold and the heterotic string on $K_3\times T^2$. 
  In a recent paper, Duff and Rahmfeld argued that certain massive $N_R=1/2$ states of the four-dimensional heterotic string correspond to extreme black hole solutions. We provide further, dynamical, evidence for this identification by comparing the scattering of these elementary string states with that of the corresponding extreme black holes, in the limit of low velocities. 
  We construct the heterotic dual theory in four dimensions of eleven dimensional supergravity compactified on a particular Joyce manifold, $J$. In particular, $J$ is constructed from resolving fixed point singularities of orbifolds of the seven torus in such a way that one is forced to consider a generalised orbifold on the heterotic side. We conjecture that a heterotic dual exists for all the compact 7-manifolds of $G_{2}$ holonomy constructed by Joyce. 
  We propose a solution to the problem of compatibility of Bose-Fermi statistics with symmetry transformations implemented by compact quantum groups of Drinfel'd type. We use unitary transformations to conjugate multi-particle symmetry postulates, so as to obtain a twisted realization of the symmetric groups S_n. 
  Following an argument advanced by Feynman, we consider a method for obtaining the effective action which generates the sum of tree diagrams with external physical particles. This technique is applied, in the unbroken \lambda \phi^4 theory, to the derivation of the threshold amplitude for the production of $n$ scalar particles by $n$ initial particles. The leading contributions to the tree amplitude, which become singular in the threshold limit, exhibit a factorial growth with n. 
  The supersymmetric generalization of a recently proposed Abelian axial gauge model with antisymmetric tensor matter fields is presented. 
  General fermionic expressions for the branching functions of the rational coset conformal field theories $\widehat{su}(2)_{M}\times \widehat{su}(2)_N/\widehat{su}(2)_{M+N}$ are given. The equality of the bosonic and fermionic representations for the branching functions is proven by introducing polynomial truncations of these branching functions which are the configuration sums of the RSOS models in regime III. The path space interpretation of the RSOS models provides recursion relations for the configuration sums. The proof of the recursion relations for the fermionic expressions is given by using telescopic expansion techniques. The configuration sums of the RSOS model in regime II which correspond to the branching functions of the $Z_{M+N}$-parafermion conformal field theory are obtained by the duality transformation $q\rightarrow q^{-1}$. 
  A physical system composed by a scalar field minimally coupled to gravity and a thermal reservoir, as in thermo field dynamics, all of them in curved space time, is considered. When the formalism of thermo field dynamics is generalized to the above mentioned case, an amplification in the number of created particles is predicted. 
  We consider a $\lambda \phi^4$ theory in Minkowski spacetime. We compute a "coarse grained effective action" by integrating out the field modes with wavelength shorter than a critical value. From this effective action we obtain the evolution equation for the reduced density matrix (master equation). We compute the diffusion coefficients of this equation and analyze the decoherence induced on the long- wavelength modes. We generalize the results to the case of a conformally coupled scalar field in DeSitter spacetime. We show that the decoherence is effective as long as the critical wavelength is taken to be not shorter than the Hubble radius. 
  We discuss some issues related to the definition of different effective actions, in connection with the work on supersymmetric theories by Seiberg and collaborators. We also comment on the possibility of extending this work to broken supersymmetric theories. 
  A lagrangian euclidean model of Drinfeld--Sokolov (DS) reduction leading to general $W$--algebras on a Riemann surface of any genus is presented. The background geometry is given by the DS principal bundle $K$ associated to a complex Lie group $G$ and an $SL(2,\Bbb C)$ subgroup $S$. The basic fields are a hermitian fiber metric $H$ of $K$ and a $(0,1)$ Koszul gauge field $A^*$ of $K$ valued in a certain negative graded subalgebra $\goth x$ of $\goth g$ related to $\goth s$. The action governing the $H$ and $A^*$ dynamics is the effective action of a DS field theory in the geometric background specified by $H$ and $A^*$. Quantization of $H$ and $A^*$ implements on one hand the DS reduction and on the other defines a novel model of $2d$ gravity, DS gravity. The gauge fixing of the DS gauge symmetry yields an integration on a moduli space of DS gauge equivalence classes of $A^*$ configurations, the DS moduli space. The model has a residual gauge symmetry associated to the DS gauge transformations leaving a given field $A^*$ invariant. This is the DS counterpart of conformal symmetry. Conformal invariance and certain non perturbative features of the model are discussed in detail. 
  We examine the occurrence of Bose-Einstein condensation in both nonrelativistic and relativistic systems with no self-interactions in a general setting. A simple condition for the occurrence of Bose-Einstein condensation can be given if we adopt generalized $\zeta$-functions to define the quantum theory. We show that the crucial feature governing Bose-Einstein condensation is the dimension $q$ associated with the continuous part of the eigenvalue spectrum of the Hamiltonian for nonrelativistic systems or the spatial part of the Klein-Gordon operator for relativistic systems. In either case Bose-Einstein condensation can only occur if $q\ge3$. 
  Since T-duality has been proved only perturbatively and most of the heterotic states map into solitonic, non-perturbative, type II states, the 6-dimensional string-string duality between the heterotic string and the type II string is not sufficient to prove the S-duality of the former, in terms of the known T-duality of the latter. We nevertheless show in detail that perturbative T-duality, together with the heterotic-type II duality, does imply the existence of heterotic H-monopoles, with the correct multiplicity and multiplet structure. This construction is valid at a generic point in the moduli space of heterotic toroidal compactifications. 
  We use the weak field approximation to show that information is lost in principle in quantum gravity. 
  We summarize results for all four-dimensional Bogomol'nyi-Sommerfield-Prasat (BPS) saturated and non-extreme solutions of the ($4+n$)-dimensional Abelian Kaluza-Klein theory. Within effective $N=4$ supersymmetric string vacua, parameterized in terms of fields of the heterotic string on a six-torus, we then present a class of BPS saturated states and the corresponding non-extreme solutions, specified by $O(6,22,Z)$ and $SL(2,Z)$ orbits of general dyonic charge configurations with zero axion. The BPS saturated states with non-negative $O(6,22,Z)$ norms for electric and magnetic charge vectors, along with the corresponding set of non-extreme solutions, are regular with non-zero masses. BPS saturated states with the negative charge norms are singular, unaccompanied by non-extreme solutions and become massless at particular points of the moduli space. The role that such massless states may play in the enhancement of non-Abelian gauge symmetry as well as local supersymmetry is addressed. 
  We analyse MINBU distribution of 2 dimensional quantum gravity. New data of R$^2$-gravity by the Monte Carlo simulation and its theoretical analysis by the semiclassical approach are presented. The cross-over phenomenon takes place at some size of the baby universe where the randomness competes with the smoothing force of $R^2$-term. The dependence on the central charge $c_m$\ and on the $R^2$-coupling are explained for the ordinary 2d quantum gravity and for $R^2$-gravity. The $R^2$-Liouville solution plays the central role in the semiclassical analysis. A total derivative term (surface term) and the infrared regularization play important roles . The surface topology is that of a sphere. 
  Two dimensional quantum R$^2$-gravity is formulated in the semiclassical method. The thermodynamic properties,such as the equation of state, the temperature and the entropy, are explained. The topology constraint and the area constraint are properly taken into account. A total derivative term and an infrared regularization play important roles. The classical solutions (vacua) of R$^2$-Liouville equation are obtained by making use of the well-known solution of the ordinary Liouville equation. The positive and negative constant curvature solutions are 'dual' each other. Each solution has two branches($\pm$). We characterize all phases. The topology of a sphere is mainly considered. 
  A convenient formalism is developed to treat classical dynamical systems involving $(p=2)$ parafermionic and parabosonic dynamical variables. This is achieved via the introduction of a parabracket which summarizes the paracommutation relations of the corresponding Green components in a unified manner. Furthermore, it is shown that Peierls quantization scheme may be applied to such systems provided that one uses the above mentioned parabracket to express the quantum paracommutation relations. Application of the Peierls scheme also provides the form of the parafermionic and parabosonic kinetic terms in the Lagrangian. 
  After reviewing the $\beta$-function equations for consistent string backgrounds in the $\sigma$-model approach, including metric and antisymmetric tensor, dilaton and tachyon potential, we apply this formalism to WZW models. We particularly emphasize the case where the WZW model is perturbed by an integrable marginal tachyon potential operator leading to the non-abelian Toda theories. Already in the simplest such theory, there is a large non-linear and non-local chiral algebra that extends the Virasoro algebra. This theory is shown to have two formulations, one being a classical reduction of the other. Only the non-reduced theory is shown to satisfy the $\beta$-function equations. 
  The link between the treatment of singular lagrangians as field systems and the general approch is studied. It is shown that singular Lagrangians as field systems are always in exact agreement with the general approch.   Two examples and the singular Lagrangian with zero rank Hessian matrix are studied. The equations of motion in the field systems are equivalent to the equations which contain accleration, and the constraints are equivalent to the equations which do not contain acceleration in the general approch treatment. 
  Using a $U$-duality symmetry of type II compactification on $T^4$ represented by triality action on the $T$-duality group, and applying the adiabatic argument we construct dual pairs of type II compactifications in lower dimensions. The simplicity of this construction makes it an ideal set up for testing various conjectures about string dualities. In some of these models the type II string has a perturbative non-abelian gauge symmetry. Examples include models with $N=2,4,6$ supersymmetry in four dimensions. There are also self-dual (in the sense of $S-T$ exchange symmetric) $N=2$ and $N=6$ examples. A generalization of the adiabatic argument can be used to construct dual pairs of models with $N=1$ supersymmetry. 
  Definition of the determinant of Euclidean Dirac operator in the nontrivial sector of gauge fields suffers from an inherent ambiguity. The popular Osterwalder-Schrader (OS) recipe for the conjugate Dirac field leads to the option of a vanishing determinant. We propose a novel representation for the conjugate field which depends linearly on the Dirac field and yields a nonvanishing determinant in the nontrivial sector. Physics, it appears, chooses this second option becuase the novel representation leads to a satisfactory resolution of two outstanding problems, the strong CP and U(1) problems, attributed to instanton effects. 
  Using Schwinger's quantum action principle, dispersion relations are obtained for neutral scalar mesons interacting with bi-local sources. These relations are used as the basis of a method for representing the effect of interactions in the Gaussian approximation to field theory, and it is argued that a marked inhomogeneity, in space-time dependence of the sources, forces a discrete spectrum on the field. The development of such a system is characterized by features commonly associated with chaos and self-organization (localization by domain or cell formation). The Green functions play the role of an iterative map in phase space. Stable systems reside at the fixed points of the map. The present work can be applied to self-interacting theories by choosing suitable properties for the sources. Rapid transport leads to a second order phase transition and anomalous dispersion. Finally, it is shown that there is a compact representation of the non-equilibrium dynamics in terms of generalized chemical potentials, or equivalently as a pseudo-gauge theory, with an imaginary charge. This analogy shows, more clearly, how dissipation and entropy production are related to the source picture and transform a flip-flop like behaviour between two reservoirs into the Landau problem in a constant `magnetic field'. A summary of conventions and formalism is provided as a basis for future work. 
  We use path integral methods and topological quantum field theory techniques to investigate a generic classical Hamiltonian system. In particular, we show that Floer's instanton equation is related to a functional Euler character in the quantum cohomology defined by the topological nonlinear $\sigma$--model. This relation is an infinite dimensional analog of the relation between Poincar\'e--Hopf and Gauss--Bonnet--Chern formul\ae$~$ in classical Morse theory, and can also be viewed as a loop space generalization of the Lefschetz fixed point theorem. By applying localization techniques to path integrals we then show that for a K\"ahler manifold our functional Euler character coincides with the Euler character determined by the finite dimensional de Rham cohomology of the phase space. Our results are consistent with the Arnold conjecture which estimates periodic solutions to classical Hamilton's equations in terms of de Rham cohomology of the phase space. 
  Superstring models describing curved 4-dimensional magnetic flux tube backgrounds are exactly solvable in terms of free fields. We first consider the simplest model of this type (corresponding to `Kaluza-Klein' Melvin background). Its 2d action has a flat but topologically non-trivial 10-dimensional target space (there is a mixing of angular coordinate of the 2-plane with an internal compact coordinate). We demonstrate that this theory has broken supersymmetry but is perturbatively stable if the radius R of the internal coordinate is larger than R_0=\sqrt{2\a'}. In the Green-Schwarz formulation the supersymmetry breaking is a consequence of the presence of a flat but non-trivial connection in the fermionic terms in the action. For R < R_0 and the magnetic field strength parameter q > R/2\a' there appear instabilities corresponding to tachyonic winding states. The torus partition function Z(q,R) is finite for R > R_0 (and vanishes for qR=2n, n=integer). At the special points qR=2n (2n+1) the model is equivalent to the free superstring theory compactified on a circle with periodic (antiperiodic) boundary condition for space-time fermions. Analogous results are obtained for a more general class of static magnetic flux tube geometries including the a=1 Melvin model. 
  After analyzing the implication of investigations on the C, P and T transformations since 1956, we propose that there is a basic symmetry in particle physics. The combined space-time inversion is equivalent to particle-antiparticle transformation, denoted by ${\cal PT=C}$. It is shown that the relativistic quantum mechanics and quantum field theory do contain this invariance explicitly or implicitly. In particular, (a) the appearance of negative energy or negative probability density in single particle theory -- corresponding to the fact of existence of antiparticle, (b) spin- statistics connection, (c) CPT theorem, (d) the Feynman propagator are linked together via this symmetry. Furthermore, we try to derive the main results of special relativity, especially, (e) the mass-energy relation, (f) the Lorentz transformation by this one ``relativistic'' postulate and some ``nonrelativistic'' knowledge. 
  We propose an alternative dimensional reduction prescription which in respect with Green functions corresponds to drop the extra spatial coordinate. From this, we construct the dimensionally reduced Lagrangians both for scalars and fermions, discussing bosonization and supersymmetry in the particular 2-dimensional case. We argue that our proposal is in some situations more physical, in the sense that it mantains the form of the interactions between particles, thus preserving the dynamics corresponding to the higher dimensional space. 
  Generically, string models with $N=1$ supersymmetry are not expected to have moduli beyond perturbation theory; stringy non-perturbative effects as well as low energy field-theoretic phenomena such as gluino condensation will lift any flat directions. In this note, we describe models where some subspace of the moduli space survives non-perturbatively. Discrete $R$ symmetries forbid any inherently stringy effects, and dynamical considerations control the field-theoretic effects. The surviving subspace is a space of high symmetry; the system is attracted to this subspace by a potential which we compute. Models of this type may be useful for considerations of duality and raise troubling cosmological questions about string theory. Our considerations also suggest a mechanism for fixing the expectation value of the dilaton. 
  It is shown that extremal magnetic black hole solutions of N = 2 supergravity coupled to vector multiplets $X^\Lambda$ with a generic holomorphic prepotential $F(X^\Lambda)$ can be described as supersymmetric solitons which interpolate between maximally symmetric limiting solutions at spatial infinity and the horizon. A simple exact solution is found for the special case that the ratios of the $X^\Lambda$ are real, and it is seen that the logarithm of the conformal factor of the spatial metric equals the Kahler potential on the vector multiplet moduli space. Several examples are discussed in detail. 
  We examine the problem of constructing three generation free fermionic string models with grand unified gauge groups. We attempt the construction of $G\times G$ models, where $G$ is a grand unified group realized at level 1. This structure allows those Higgs representations to appear which are necessary to break the symmetry down to the standard model gauge group. For $G=SO(10)$, we find only models with an even number of generations. However, for $G=SU(5)$ we find a number of 3 generation models. 
  Symbolic algebra relevant to the renormalization of gauge theories can be efficiently performed by machine using modern packages. We devise a scheme for representing and manipulating the objects involved in perturbative calculations of gauge theories. This scheme is readily implemented using the general purpose package, Mathematica. The techniques discussed are used to calculate renormalization group functions for a non-abelian $SU(m)$ gauge theory with massless fermions in a representation R, in the two-loop approximation, and to simplify some expressions arising in electroweak calculations at the two loop level. 
  A recently proposed path-integral bosonization scheme for massive fermions in $3$ dimensions is extended by keeping the full momentum-dependence of the one-loop vacuum polarization tensor. This makes it possible to discuss both the massive and massless fermion cases on an equal footing, and moreover the results it yields for massless fermions are consistent with the ones of another, seemingly different, canonical quantization approach to the problem of bosonization for a massless fermionic field in $3$ dimensions. 
  We address the issue of when generalized quantum dynamics, which is a classical symplectic dynamics for noncommuting operator phase space variables based on a graded total trace Hamiltonian ${\bf H}$, reduces to Heisenberg picture complex quantum mechanics. We begin by showing that when ${\bf H}={\bf Tr} H$, with $H$ a Weyl ordered operator Hamiltonian, then the generalized quantum dynamics operator equations of motion agree with those obtained from $H$ in the Heisenberg picture by using canonical commutation relations. The remainder of the paper is devoted to a study of how an effective canonical algebra can arise, without this condition simply being imposed by fiat on the operator initial values. We first show that for any total trace Hamiltonian which involves no noncommutative constants, there is a conserved anti--self--adjoint operator $\tilde C$ with a structure which is closely related to the canonical commutator algebra. We study the canonical transformations of generalized quantum dynamics, and show that $\tilde C$ is a canonical invariant, as is the operator phase space volume element. The latter result is a generalization of Liouville's theorem, and permits the application of statistical mechanical methods to determine the canonical ensemble governing the equilibrium distribution of operator initial values. We give arguments based on a Ward identity analogous to the equipartition theorem of classical statistical mechanics, suggesting that statistical ensemble averages of Weyl ordered polynomials in the operator phase space variables correspond to the Wightman functions of a unitary complex quantum mechanics, with a conserved operator Hamiltonian and with the standard canonical commutation relations obeyed by Weyl ordered operator strings. Thus there is a well--defined sense in 
  We adapt a calculation due to Massacand and Schmid to the coordinate independent definition of time and vacuum given by Capri and Roy in order to compute the trace anomaly for a massless scalar field in a curved spacetime in 1+1 dimensions. The computation which requires only a simple regulator and normal ordering yields the well-known result $\frac{R}{24\pi}$ in a straightforward manner. 
  A large class of four-dimensional supersymmetric ground states of closed superstrings with a non-zero mass gap are constructed. For such ground states we turn on chromo-magnetic fields as well as curvature. The exact spectrum as function of the chromo-magnetic fields and curvature is derived. We examine the behavior of the spectrum, and find that there is a maximal value for the magnetic field $H_{\rm max}\sim M_{\rm planck}^2$. At this value all states that couple to the magnetic field become infinitely massive and decouple. We also find tachyonic instabilities for strong background fields of the order ${\cal O}(\mu M_{\rm planck})$ where $\mu$ is the mass gap of the theory. Unlike the field theory case, we find that such ground states become stable again for magnetic fields of the order ${\cal O}(M^2_{\rm planck})$. The implications of these results are discussed. 
  We compute the one-dimensional configuration sums of the ABF model using the fermionic technique introduced in part I of this paper. Combined with the results of Andrews, Baxter and Forrester, we find proof of polynomial identities for finitizations of the Virasoro characters $\chi_{b,a}^{(r-1,r)}(q)$ as conjectured by Melzer. In the thermodynamic limit these identities reproduce Rogers--Ramanujan type identities for the unitary minimal Virasoro characters, conjectured by the Stony Brook group. We also present a list of additional Virasoro character identities which follow from our proof of Melzer's identities and application of Bailey's lemma. 
  We construct and solve the boundary Yang-Baxter equation in the RSOS/SOS representation. We find two classes of trigonometric solutions; diagonal and non-diagonal. As a lattice model, these two classes of solutions correspond to RSOS/SOS models with fixed and free boundary spins respectively. Applied to (1+1)-dimenional quantum field theory, these solutions give the boundary scattering amplitudes of the particles. For the diagonal solution, we propose an algebraic Bethe ansatz method to diagonalize the SOS-type transfer matrix with boundary and obtain the Bethe ansatz equations. 
  A short review of special relativistic dynamics describing a particle acted upon by an arbitrary conservative external force is presented. If the mass of the particle is zero and the force is central then the equations of motion turn out to be completely integrable. A well-known result. Hamiltonian flows on the twistor phase space T are constructed which, for conservative forces and value of the helicity equal to zero, reproduce equations of motion of the classical massless particle. For helicities different from zero the same hamiltonian flows produce equations of motion showing a curious "Zitterbewegung" like behaviour. A canonical Poincare covariant quantization procedure on T is suggested. One simple example describing a spinning and massless 3-D quantum mechanical harmonic oscillator is analysed in some detail. 
  Using known mode properties, the functional determinant for massless spin-half fields on the Euclidean 4-ball is calculated and shown to be different for spectral (nonlocal) and mixed (local) boundary conditions. The local result agrees with that from a conformal argument. Some higher-spin results are also given. 
  Using Lorentz force equation as an input a Hamiltonian mechanics on the non-projective two twistor phase space TxT is formulated. Such a construction automatically reproduces dynamics of the intrinsic classical relativistic spin. The charge appears as a dynamical variable. It is also shown that if the classical relativistic spin function on TxT vanishes, the natural conformally invariant symplectic structure on TxT reduces to the natural symplectic structure on the cotangent bundle of the Kaluza-Klein space. 
  The Poisson bracket algebra corresponding to the second Hamiltonian structure of a large class of generalized KdV and mKdV integrable hierarchies is carefully analysed. These algebras are known to have conformal properties, and their relation to $\cal W$-algebras has been previously investigated in some particular cases. The class of equations that is considered includes practically all the generalizations of the Drinfel'd-Sokolov hierarchies constructed in the literature. In particular, it has been recently shown that it includes matrix generalizations of the Gelfand-Dickey and the constrained KP hierarchies. Therefore, our results provide a unified description of the relation between the Hamiltonian structure of soliton equations and $\cal W$-algebras, and it comprises almost all the results formerly obtained by other authors. The main result of this paper is an explicit general equation showing that the second Poisson bracket algebra is a deformation of the Dirac bracket algebra corresponding to the $\cal W$-algebras obtained through Hamiltonian reduction. 
  The holomorphy of the superpotential along with symmetries gives very strong constraints on any stringy non-perturbative effects. This observation suggests an approach to string phenomenology. (Presented at ``Strings 95'', March 1995. 
  We review the definition of the Casimir energy steming naturally from the concept of functional determinant through the zeta function prescription. This is done by considering the theory at finite temperature and by defining then the Casimir energy as its energy in the limit $T\to 0$. The ambiguity in the coefficient $C_{d/2}$ is understood to be a result of the necessary renormalization of the free energy of the system. Then, as an exact, explicit example never calculated before, the Casimir energy for a massive scalar field living in a general $(1+2)$-dimensional toroidal spacetime (i.e., a general surface of genus one) with flat spatial geometry ---parametrized by the corresponding Teichm\"uller parameters--- and its precise dependence on these parameters and on the mass of the field is obtained under the form of an analytic function. 
  The O(N) symmetric scalar quantum field theory with \lambda\Phi^4 interaction is discussed in the Gaussian approximation. It is shown that the Goldstone theorem is fulfilled for arbitrary N. 
  The zeta function associated with higher-spin fields on the Euclidean $4$-ball is investigated. The leading coefficients of the corresponding heat-kernel expansion are given explicitly and the zeta functional determinant is calculated. For fermionic fields the determinant is shown to differ for local and spectral boundary conditions. 
  This talk, given at several conferences and meetings, explains the background leading to the formulation of the exact electromagnetic duality conjecture believed to be valid in N=4 supersymmetric SU(2) gauge theory. 
  A general homogeneous two dimensional dilaton gravity model considered recently by Lemos and S\` a, is given quantum matter Polyakov corrections and is solved numerically for several static, equilibrium scenarii. Classically the dilaton field ranges the whole real line, whereas in the semi-classical theory, with the usual definition, it is always below a certain critical value at which a singularity appears. We give solutions for both sub- and super-critical dilaton field. The pasting together of the spacetime on both sides of a singularity in semi-classical planar general relativity is discussed. 
  We show how a solitonic ``magnetically'' charged $p$-brane solution of a given supergravity theory, with the magnetic charge carried by an antisymmetric tensor gauge field, can be generalized to a dyonic solution. We discuss the cases of ten-dimensional and eleven-dimensional supergravity in more detail and a new dyonic five-brane solution in ten dimensions is given. Unlike the purely electrically or magnetically charged five-brane solution the dyonic five-brane contains non-zero Ramond--Ramond fields and is therefore an intrinsically type~II solution. The solution preserves half of the type~II spacetime supersymmetries. It is obtained by applying a solution-generating $SL(2,\R) \times SL(2,\R)$ $S$~duality transformation to the purely magnetically charged five-brane solution. One of the $SL(2,\R)$ duality transformations is basically an extension to the type~II case of the six-dimensional $\Z_2$ string/string duality. We also present an action underlying the type IIB supergravity theory. 
  Some recent results on the applications of duality (and related) transformations to general four-dimensional, spherically symmetric, asymptotically flat and time-independent string configurations are summarized. Two classes of results have been obtained. First, these transformations are used to generate the general such solution to the lowest-order field equations in the alpha' expansion. Second, the action and implications of duality (based on time-translation) on the general configuration is determined. It is found to interchange two pairs of the six parameters which label these configurations, namely: (1) the mass with the dilaton charge, and (2) the axion charge with the Taub-NUT parameter. For the special case of the Schwarzshild black hole this implies the relation M -> - k/M, where k is a known, positive, quantity. It is argued that, in some circumstances, dual theories need not be equivalent in the simplest sense. 
  The reduction by symmetry of the linear system of the self-dual Yang-Mills equations in four-dimensions under representatives of the conjugacy classes of subgroups of the connected part to the identity of the corresponding Euclidean group under itself is carried out. Only subgroups leading to systems of differential equations nonequivalent to conditions of zero curvature without parameter, or to systems of uncoupled first order linear O.D.E.'s are considered. Lax pairs for a modified form of the Nahm's equations as well as for systems of partial differential equations in two and three dimensions are written out. 
  In six spacetime dimensions, the heterotic string is dual to a Type $IIA$ string. On further toroidal compactification to four spacetime dimensions, the heterotic string acquires an $SL(2,\BbbZ)_S$ strong/weak coupling duality and an $SL(2,\BbbZ)_T \times SL(2,\BbbZ)_U$ target space duality acting on the dilaton/axion, complex Kahler form and the complex structure fields $S,T,U$ respectively. Strong/weak duality in $D=6$ interchanges the roles of $S$ and $T$ in $D=4$ yielding a Type $IIA$ string with fields $T,S,U$. This suggests the existence of a third string (whose six-dimensional interpretation is more obscure) that interchanges the roles of $S$ and $U$. It corresponds in fact to a Type $IIB$ string with fields $U,T,S$ leading to a four-dimensional string/string/string triality. Since $SL(2,\BbbZ)_S$ is perturbative for the Type $IIB$ string, this $D=4$ triality implies $S$-duality for the heterotic string and thus fills a gap left by $D=6$ duality. For all three strings the total symmetry is $SL(2,\BbbZ)_S \times O(6,22;\BbbZ)_{TU}$. The $O(6,22;\BbbZ)$ is {\it perturbative} for the heterotic string but contains the conjectured {\it non-perturbative} $SL(2,\BbbZ)_X$, where $X$ is the complex scalar of the $D=10$ Type $IIB$ string. Thus four-dimensional triality also provides a (post-compactification) justification for this conjecture. We interpret the $N=4$ Bogomol'nyi spectrum from all three points of view. In particular we generalize the Sen-Schwarz formula for short multiplets to include intermediate multiplets also and discuss the corresponding black hole spectrum both for the $N=4$ theory and for a truncated $S$--$T$--$U$ symmetric $N=2$ theory. Just as the first two strings are described by the four-dimensional {\it elementary} and {\it dual solitonic} solutions, so the 
  Changes of variables giving the dual model are constructed explicitly for sigma-models without isotropy. In particular, the jacobian is calculated to give the known results. The global aspects of the abelian case as well as some of those of the cases where the isometry group is simply connected are considered.         Considering the anomalous case, we infer by a consistency argument that the `multiplicative anomaly' should be replaceable by adequate rules for factorization of composite jacobians. These rules are then generalized in a simple way for composite jacobians defined in spaces of different types. Implimentation of these rules then gives specific formulas for the anomally for semisimple algebras and also for solvable ones. 
  We begin a search for nonsupersymmetric/supersymmetric dual string pairs by constructing candidate critical nonsupersymmetric strings as solitons in supersymmetric string theories. Using orbifold techniques, one can construct cosmic string solutions which lie in supersymmetric vacua but which do not fall in supermultiplets. We discuss two three-dimensional examples in detail. The effective worldsheet actions for the soliton strings have (0,2) and (1,1) supersymmetry and the correct numbers of massless bosons and fermions to be critical heterotic and type II strings, respectively. 
  Owing to its lack of derivability, the dissipative anomaly operator appearing in the theory of turbulence without pressure recently proposed by Polyakov appears to be quite elusive. In particular, we give arguments that seem to lead to the conclusion that an anomaly in the first equation of the sequence of conservation laws cannot be {\it a priori} excluded. 
  We investigate asymmetric orbifold models constructed from non-supersymmetric heterotic strings. We systematically classify the asymmetric orbifold models with standard embeddings and present a list of asymmetric orbifolds which are geometrically interpreted as toroidal compactifications of non-supersymmetric heterotic strings. By studying non-standard embedding models, we also construct examples of the {\em supersymmetric} asymmetric orbifold models based on non-supersymmetric heterotic strings. 
  Exact expressions of the boundary state and the form factors of the Ising model are used to derive differential equations for the one-point functions of the energy and magnetization operators of the model in the presence of a boundary magnetic field. We also obtain explicit formulas for the massless limit of the one-point and two-point functions of the energy operator. 
  We analyze the BRST cohomology of the critical N=2 NSR string using chiral bosonization. Picture-changing and spectral flow is made explicit in a holomorphic field basis. The integration of fermionic and U(1) moduli is performed and yields picture- and U(1) ghost number-changing insertions into the string measure for n-point amplitudes at arbitrary genus and U(1) instanton number. 
  We rederive non-perturbatively the Coleman-Weinberg expression for the effective potential for massless scalar QED. Our result is not restricted to small values of the coupling constants. This shows that the Coleman- Weinberg result can be established beyond the range of validity of perturbation theory. Also, we derive it in a manifestly renormalization group invariant way. It is shown that with the derivation given no Landau ghost singularity arises. The finite temperature case is discussed.   Pacs number: 11.10.Ef,11.10.Gh. 
  We study the behavior of two diferent models at finite temperature in a $D$-dimensional spacetime. The first one is the $\lambda\varphi^{4}$ model and the second one is the Gross-Neveu model. Using the one-loop approximation we show that in the $\lambda\varphi^{4}$ model the thermal mass increase with the temperature while the thermal coupling constant decrese with the temperature. Using this facts we establish that in the $(\lambda\varphi^{4})_{D=3}$ model there is a temperature $\beta^{-1}_{\star}$ above which the system can develop a first order phase transition, where the origin corresponds to a metastable vacuum. In the massless Gross-Neveu model, we demonstrate that for $D=3$ the thermal correction to the coupling constant is zero. For $D\neq 3$ our results are inconclusive.  Pacs numbers: 11.10.Ef, 11.10.Gh 
  We describe a finite analogue of the Poisson algebra of Wilson loops in Yang-Mills theory. It is shown that this algebra arises in an apparently completely different context; as a Lie algebra of vector fields on a non-commutative space. This suggests that non-commutative geometry plays a fundamental role in the manifestly gauge invariant formulation of Yang-Mills theory. We also construct the deformation of the loop algebra induced by quantization, in the large N_c limit. 
  The first quantum corrections to the free energy for massive fields in $D$-dimensional space-times of the form $\R\times\R^+\times\M^{N-1}$, where $D=N+1$ and $\M^{N-1}$ is a constant curvature manifold, is investigated by means of the $\zeta$-function regularization. It is suggested that the nature of the divergences, which are present in the thermodynamical quantities, might be better understood making use of the conformal related optical metric and associated techniques. The general form of the horizon divercences of the free energy is obtained as a function of free energy densities of fields having negative square masses (absence of the gap in the Laplace operator spectrum) on ultrastatic manifolds with hyperbolic spatial section $H^{N-2n}$ and of the Seeley-DeWitt coefficients of the Laplace operator on the manifold $\M^{N-1}$. Furthermore, recurrence relations are found relating higher and lower dimensions. The cases of Rindler space, where $\M^{N-1}=\R^{N-1}$ and very massive $D$-dimensional black holes, where $\M^{N-1}=S^{N-1}$ are treated as examples. The renormalization of the internal energy is also discussed. 
  It is shown that a WZW model corresponding to a general simple group possesses in general different quantisations which are parametrised by $Hom(\pi_1(G),Hom(\pi_1(G),U(1)))$. The quantum theories are generically neither monodromy nor modular invariant, but all the modular invariant theories of Felder et.al. are contained among them.  A formula for the transformation of the Sugawara expression for $L_0$ under conjugation with respect to non-contractible loops in $LG$ is derived. This formula is then used to analyse the monodromy properties of the various quantisations. It turns out that for $\pi_1(G)\cong \Zop_N$, with $N$ even, there are $2$ monodromy invariant theories, one of which is modular invariant, and for $\pi_1(G)\cong \Zop_2\times\Zop_2$ there are $8$ monodromy invariant theories, two of which are modular invariant. A few specific examples are worked out in detail to illustrate the results. 
  The ($p=2$) parabose-parafermi supersymmetry is studied in general terms. It is shown that the algebraic structure of the ($p=2$) parastatistical dynamical variables allows for (symmetry) transformations which mix the parabose and parafermi coordinate variables. The example of a simple parabose-parafermi oscillator is discussed and its symmetries investigated. It turns out that this oscillator possesses two parabose- -parafermi supersymmetries. The combined set of generators of the symmetries forms the algebra of supersymmetric quantum mechanics supplemented with an additional central charge. In this sense there is no relation between the parabose-parafermi supersymmetry and the parasupersymmetric quantum mechanics. A precise definition of a quantum system involving this type of parabose- parafermi supersymmetry is offered, thus introducing ($p=2$) Supersymmetric Paraquantum Mechanics. The spectrum degeneracy structure of general ($p=2$) supersymmetric paraquantum mechanics is analyzed in detail. The energy eigenvalues and eigenvectors for the parabose-parafermi oscillator are then obtained explicitly. The latter confirms the validity of the results obtained for general supersymmetric paraquantum mechanics. 
  Superconformal sigma models with Calabi--Yau target spaces described as complete intersection subvarieties in toric varieties can be obtained as the low-energy limit of certain abelian gauge theories in two dimensions. We formulate mirror symmetry for this class of Calabi--Yau spaces as a duality in the abelian gauge theory, giving the explicit mapping relating the two Lagrangians. The duality relates inequivalent theories which lead to isomorphic theories in the low-energy limit. This formulation suggests that mirror symmetry could be derived using abelian duality. The application of duality in this context is complicated by the presence of nontrivial dynamics and the absence of a global symmetry. We propose a way to overcome these obstacles, leading to a more symmetric Lagrangian. The argument, however, fails to produce a derivation of the conjecture. 
  The problem of how to put interactions in two-dimensional quantum gravity in the strong coupling regime is studied. It shows that the most general interaction consistent with this symmetry is a Liouville term that contain two parameters $(\alpha, \beta)$ satisfying the algebraic relation $2\beta - \alpha =2$ in order to assure the closure of the diffeomorphism algebra. The model is classically soluble and it contains as general solution the temporal singularity. The theory is quantized and we show that the propagation amplitude fall tozero in $\tau =0$. This result shows that the classical singularities are smoothed by quantum effects and the bing-bang concept could be considered as a classical extrapolation instead of a physical concept. 
  Contribution to the proceedings of Schladming 1995. A review of the form factor approach and its utilisation to determine the space of local operators of integrable massive quantum theories is given. A few applications are discussed. 
  We present generalized Rogers-Ramanujan identities which relate the fermi and bose forms of all the characters of the superconformal model $SM(2,4\nu).$ In particular we show that to each bosonic form of the character there is an infinite family of distinct fermionic $q-$ series representations. 
  Motivated by recent findings due to Wiegmann and Zabrodin, Faddeev and Kashaev concerning the appearence of the quantum U_q(sl(2)) symmetry in the problem of a Bloch electron on a two-dimensional magnetic lattice, we introduce a modification of the tight binding Azbel--Hofstadter Hamiltonian that is a specific spin-S Euler top and can be considered as its ``classical'' analog. The eigenvalue problem for the proposed model, in the coherent state representation, is described by the S-gap Lam\'e equation and, thus, is completely solvable. We observe a striking similarity between the shapes of the spectra of the two models for various values of the spin S. 
  We analyze the collective motion of micro-organisms in the fluid and consider the problem of the red tide. The red tide is produced by the condensation of the micro-organisms, which might be a similar phenomenon to the condensation of the strings. We propose a model of the generation of the red tide. By considering the interaction between the micro- organisms mediated by the velocity fields in the fluid, we derive the Van der Waals type equation of state, where the generation of the red tide can be regarded as a phase transition from the gas of micro-organisms to the liquid. (The number density of micro-organisms which generates the red tide is order estimated.) 
  Introducing a chemical potential in the functional method, we construct the effective action of QED$_3$ with a Chern-Simons term. We examine a possibility that charge condensation $\langle\psi^\dagger\psi \rangle$ remains nonzero at the limit of the zero chemical potential. If it happens, spontaneous magnetization occurs due to the Gauss' law constraint which connects the charge condensation to the background magnetic field. It is found that the stable vacuum with nonzero charge condensation is realized only when fermion masses are sent to zero, keeping it lower than the chemical potential. This result suggests that the spontaneous magnetization is closely related to the fermion mass. 
  A numerical study of static spherically symmetric monople solutions of a spontaneously broken SU(2) gauge theory coupled to a dilaton field is presented. Regular solutions seem to exist only up to a maximal value of the dilaton coupling. In addition to the generalization of the 't Hooft-Polyakov monopole a discrete family of regular solutions is found, corresponding to radial excitations, absent in the theory without dilaton. 
  We derive an exact expression for the single particle Green function in the Calogero-Sutherland model for all rational values of the coupling $\beta$. The calculation is based on Jack polynomial techniques and the results are given in the thermodynamical limit. Two type of intermediate states contribute. The firts one consists of a particle propagating out of the Fermi sea and the second one consists of a particle propagating in one direction, q particles in the opposite direction and p holes. 
  The nature of finite temperature transition in QCD is studied on a lattice with Wilson fermion. For massless quarks, the transition is smooth for two flavors, while it is of first order for three and six flavors. Form the study of transition in the case of non-degenerate as well as degenerate massive quark case,it is suggest that the finite temperature transition in the real world is first order. 
  We study the existence of monopole bound states saturating the BPS bound in N=2 supersymmetric Yang-Mills theories. We describe how the existence of such bound states relates to the topology of index bundles over the moduli space of BPS solutions. Using an $L^2$ index theorem, we prove the existence of certain BPS states predicted by Seiberg and Witten based on their study of the vacuum structure of N=2 Yang-Mills theories. 
  Symplectic geometry of the vortex filament in a curved three-manifold is investigated. There appears an infinite sequence of constants of motion in involution in the case of constant curvature. The Duistermaat-Heckman formula is examined perturbatively for the classical partition function in our model and verified up to the 3-loop order. 
  We discuss a cosmological Friedmann model modified by inclusion of off-shell matter which has an equation of state $p,\rho \propto T^5,$ $p=1/4\rho .$ Such matter is shown to have energy density comparable with that of non-interacting radiation at temperatures of the order of the Hagedorn temperature, $\sim 10^{12}$ K, indicating the possibility of a phase transition. It is argued that the $T^5$-phase, or an admixture, lies below the high-temperature $T^4$-phase. 
  We consider 5D Kaluza-Klein type cosmological model with the fifth coordinate being a generalization of the invariant ``historical'' time $\tau $ of the covariant theory of Horwitz and Piron. We distinguish between vacuum-, off-shell matter-, and on-shell matter-dominated eras as the solutions of the corresponding 5D gravitational field equations, and build an inflationary scenario according to which passage from the off-shell matter-dominated era to the on-shell one occurs, probably as a phase transition. We study the effect of this phase transition on the expansion rate in both cases of local $O(4,1)$ and $O(3,2)$ invariance of the extended $(x^\mu ,\tau )$ manifold and show that it does not change in either case. The expansion of the model we consider is not adiabatic; the thermodynamic entropy is a growing function of cosmic time for the closed universe, and can be a growing function of historical time for the open and the flat universe. A complete solution of the 5D gravitational field equations is obtained for the on-shell matter-dominated universe. The open and the closed universe are shown to tend asymptotically to the standard 4D cosmological models, in contrast to the flat universe which does not have the corresponding limit. Finally, possible cosmological implications are briefly discussed. 
  We discuss the properties of an ideal relativistic gas of events possessing Bose-Einstein statistics. We find that the mass spectrum of such a system is bounded by $\mu \leq m\leq 2M/\mu _K,$ where $\mu $ is the usual chemical potential, $M$ is an intrinsic dimensional scale parameter for the motion of an event in space-time, and $\mu _K$ is an additional mass potential of the ensemble. For the system including both particles and antiparticles, with nonzero chemical potential $\mu ,$ the mass spectrum is shown to be bounded by $|\mu |\leq m\leq 2M/\mu _K,$ and a special type of high-temperature Bose-Einstein condensation can occur. We study this Bose-Einstein condensation, and show that it corresponds to a phase transition from the sector of continuous relativistic mass distributions to a sector in which the boson mass distribution becomes sharp at a definite mass $M/\mu _K.$ This phenomenon provides a mechanism for the mass distribution of the particles to be sharp at some definite value. 
  We consider the relativistic statistical mechanics of an ensemble of $N$ events with motion in space-time parametrized by an invariant ``historical time'' $\tau .$ We generalize the approach of Yang and Yao, based on the Wigner distribution functions and the Bogoliubov hypotheses, to find the approximate dynamical equation for the kinetic state of any nonequilibrium system to the relativistic case, and obtain a manifestly covariant Boltzmann-type equation which is a relativistic generalization of the Boltzmann-Uehling-Uhlenbeck (BUU) equation for indistinguishable particles. This equation is then used to prove the $H$-theorem for evolution in $\tau .$ In the equilibrium limit, the covariant forms of the standard statistical mechanical distributions are obtained. We introduce two-body interactions by means of the direct action potential $V(q),$ where $q$ is an invariant distance in the Minkowski space-time. The two-body correlations are taken to have the support in a relative $O( 2,1)$-invariant subregion of the full spacelike region. The expressions for the energy density and pressure are obtained and shown to have the same forms (in terms of an invariant distance parameter) as those of the nonrelativistic theory and to provide the correct nonrelativistic limit. 
  We consider a relativistic strongly interacting Bose gas. The interaction is manifested in the off-shellness of the equilibrium distribution. The equation of state that we obtain for such a gas has the properties of a realistic equation of state of strongly interacting matter, i.e., at low temperature it agrees with the one suggested by Shuryak for hadronic matter, while at high temperature it represents the equation of state of an ideal ultrarelativistic Stefan-Boltzmann gas, implying a phase transition to an effectively weakly interacting phase. 
  We derive, for $N\is2$ super-Yang--Mills with gauge group $SU(2)$ and massless matter, the supersymmetric quantum mechanical models describing the time evolution of multi-monopole configurations in the low energy approximation. This is a first step towards identifying the solitonic states mapped to fundamental excitations by duality in the model with four hypermultiplets in the fundamental representation. 
  By explicit calculation, I determine the structure of the ground ring of the critical $W_3$ gravity and show that there is an $su(3)$ invariant quadratic relation among the six basic elements. By using this result, I also construct some discrete physical states of the critical $W_3$ gravity. 
  The complete structure of the $WG_2$ algebra is obtained from an explicit realization by an abstract Virasoro algebra and a free boson field. We then construct its BRST operator and find a seven-parameter family of nilpotnt BRST operators. These free parameters are related to the canonical transformations of the ghost, antighost fields which leave the total stress-energy tensor and the antighost field $b$ invariant. 
  Different regularizations are studied in localization of path integrals. We discuss the effect of the choice of regularization by evaluating the partition functions for the harmonic oscillator and the Weyl character for SU(2). In particular, we solve the Weyl shift problem that arises in path integral evaluation of the Weyl character by using the Atiyah-Patodi-Singer $\eta$-invariant and the Borel-Weil theory. 
  A relation between the MacDowell-Mansouri theory of gravity and the Pontrjagin toplogical invariant in $(3+1)$ dimensions is discussed. This relation may be of especial interest in the quest of finding a mechanism to go from non-dynamical to dynamical gravity. 
  Using the symmetry reductions of the self-dual Yang-Mills (SDYM) equations in (2+2) dimensions, we introduce new integrable equations which are nonautonomous versions of the chiral model in (2+1) dimensions, generalized nonlinear Schrodinger, Korteweg-de Vries, Toda lattice, Garnier and Euler-Arnold equations. The Lax pairs for all of these equations are derived by the symmetry reductions of the Lax pair for the SDYM equations. 
  We show that the problem of Random Walk with boundary attractive potential may be mapped onto the free massive bosonic Quantum Field Theory with a line of defect. This mapping permits to recover the statistical properties of the Random Walks by using boundary $S$--matrix and Form Factor techniques. 
  A systematic derivation is given of the worldline path integrals for the effective action of a multiplet of Dirac fermions interacting with general matrix-valued classical background scalar, pseudoscalar, and vector gauge fields. The first path integral involves worldline fermions with antiperiodic boundary conditions on the worldline loop and generates the real part of the one loop (Euclidean) effective action. The second path integral involves worldline fermions with periodic boundary conditions and generates the imaginary part of the (Euclidean) effective action, i.e. the phase of the fermion functional determinant. Here we also introduce a new regularization for the phase of functional determinants resembling a heat-kernel regularization. Compared to the known special cases, our worldline Lagrangians have a number of new interaction terms; the validity of some of these terms is checked in perturbation theory. In particular, we obtain the leading order contribution (in the heavy mass expansion) to the Wess-Zumino-Witten term, which generates the chiral anomaly. 
  Different compactifications of six-dimensional string theory on $M_4 \times T^2$ are considered. Particular attention is given to the roles of the reduced modes as the $S$ and $T$ fields. It is shown that there is a discrete group of invariances of an equilateral triangle hidden in the model. This group is realized as the interchanges of the two-form fields present in the intermediate step of dimensional reduction in five dimensions. The key ingredient for the existence of this group is the presence of an additional $U(1)$ gauge field in five dimensions, arising as the dual of the Kalb-Ramond axion field strength. As a consequence, the theory contains more four-dimensional $SL(2,R)$ representations, with the resulting complex scalar axidilaton related to the components of the Kaluza-Klein vector fields of the naive dimensional reduction. An immediate byproduct of this relationship is a triadic correspondence among the fundamental string, the solitonic string, and a singular Brinkmann pp wave. 
  We obtain the Bethe Ansatz equations for the broken ${\bf Z}_{N}$-symmetric model by constructing a functional relation of the transfer matrix of $L$-operators. This model is an elliptic off-critical extension of the Fateev-Zamolodchikov model. We calculate the free energy of this model on the basis of the string hypothesis. 
  Recently, it has been observed that a certain class of classical theories with constraints can be quantized by a mathematical procedure known as Rieffel induction. After a short exposition of this idea, we apply the new quantization theory to the Stueckelberg-Kibble model. We explicitly construct the physical state space ${\cal H}_{phys}$, which carries a massive representation of the Poincar\'e group. The longitudinal one-particle component arises from a particular Bogoliubov transformation of the five (unphysical) degrees of freedom one has started with. Our discussion exhibits the particular features of the proposed constrained quantization theory in great clarity. 
  We consider how to normalize the scattering amplitudes of 4D heterotic superstrings in a Minkowski background. We fix the normalization of the vacuum amplitude (the string partition function) at each genus, and of every vertex operator describing a physical external string state in a way consistent with unitarity of the $S$-matrix. We also provide an explicit expression for the map relating the vertex operator of an incoming physical state to the vertex operator describing the same physical state, but outgoing. This map is related to hermitean conjugation and to the hermiticity properties of the scattering amplitudes. 
  Path-integral expressions for one-particle propagators in scalar and fermionic field theories are derived, for arbitrary mass. This establishes a direct connection between field theory and specific classical point-particle models. The role of world-line reparametrization invariance of the classical action and the implementation of the corresponding BRST-symmetry in the quantum theory are discussed. The presence of classical world-line supersymmetry is shown to lead to an unwanted doubling of states for massive spin-1/2 particles. The origin of this phenomenon is traced to a `hidden' topological fermionic excitation. A different formulation of the pseudo-classical mechanics using a bosonic representation of $\gam_5$ is shown to remove these extra states at the expense of losing manifest supersymmetry. 
  It is shown that $U(1)$--Hamiltonian reduction of a four--dimensional isotropic quantum oscillator results in a bound system of two spinless Schwinger's dyons. Its wavefunctions and spectrum are constructed. 
  "GRAMA" is a Mathematica package for doing symbolic tensor computations and complicated algebraic manipulations in 10-dimensional (D=10) simple (N=1) supergravity. The main new ingredients of this package inside the general Mathematica environment are the computation of complicated products of Dirac matrices and the treatment of covariant derivatives: spinorial and vectorial. In principle, with small modifications, GRAMA can also be used for calculations in 4-dimensional supergravity. With the help of this package we were able to obtain the equations of motion and the Lagrangian for the 10-D supergravity including superstring corrections (see hep-th/9507033) - a calculation that would be otherwise impossible to perform. "GRAMA" is designed as a community- and user-friendly program. 
  In the framework of the prepotential description of superspace two-dimensional $(2,2)$ supergravity, we discuss the construction of invariant integrals. In addition to the full superspace measure, we derive the measure for chiral superspace, and obtain the explicit expressions for going from superspace actions to component actions. We consider both the minimal $U_A(1)$ and the extended $U_V(1) \otimes U_A(1)$ theories. 
  We define a weak-strong coupling transformation based on the Legendre transformation of the effective action. In the case of $N\es 2$ supersymmetric Yang-Mills theory, this coincides with the duality transform on the low energy effective action considered by Seiberg and Witten. This Legendre transform interpretation of duality generalizes directly to the full effective action, and in principle to other theories. 
  An account is given of a technique for testing the equivalence between an exact factorizable S-matrix and an asymptotically-free Lagrangian field theory in two space-time dimensions. The method provides a way of resolving CDD ambiguities in the S-matrix and it also allows for an exact determination of the physical mass in terms of the Lambda parameter of perturbation theory. The results for various specific examples are summarized. (To appear in the Proceedings of the Conference on Recent Developments in Quantum Field Theory and Statistical Mechanics, ICTP, Trieste, Easter 1995). 
  The Thirring model and various generalizations of it are analyzed in detail. The four-Fermi interaction modifies the equation of state. Chemical potentials and twisted boundary conditions both result in complex fermionic determinants which are analyzed. The non-minimal coupling to gravity does deform the conformal algebra which in particular contains the minimal models. We compute the central charges, conformal weights and finite size effects. For the gauged model we derive the partition functions and the explicit expression for the chiral condensate at finite temperature and curvature. The Bosonization in compact curved space-times is also investigated. 
  An SL(2, Z) family of string solutions of type IIB supergravity in ten dimensions is constructed. The solutions are labeled by a pair of relatively prime integers, which characterize charges of the three-form field strengths. The string tensions depend on these charges in an SL(2, Z) covariant way. Compactifying on a circle and identifying with eleven-dimensional supergravity compactified on a torus implies that the modulus of the IIB theory should be equated to the modular parameter of the torus. 
  Using finite abelian automorphism groups of $K3$ we construct orbifold candidates for Type IIA-heterotic dual pairs with maximal supersymmetry in six and lower dimensions. On the heterotic side, these results extend the series of known reduced rank theories with maximal supersymmetry. The corresponding Type IIA theories generalize the Schwarz and Sen proposal for the dual of the simplest of the reduced rank theories constructed as a novel Type IIA $\IZ_2$ orbifold. 
  The hierarchy of Integrable Spin Chain Hamiltonians, which are trigonometric analogs of the Haldane Shastry Model and of the associated higher conserved charges, is derived by a reduction from the trigonometric Dynamical Models of Bernard-Gaudin-Haldane-Pasquier. The Spin Chain Hamiltonians have the property of $U_q(\hat{gl}_2)$-invariance. The spectrum of the Hamiltonians and the $U_q(\hat{gl}_2)$-representation content of their eigenspaces are found by a descent from the Dynamical Models. 
  An alternative approach to perturbative Yang-Mills theories in four (3+1) dimensional space-time based on the causal Epstein-Glaser method in QFT was recently proposed. In this short note we show that the set of identities between C-number distributions expressing nonabelian gauge invariance in the causal approach imply identities which are analogous to the well-known Slavnov-Taylor identities. We explicitly derive the Z-factor relations at one-loop level. 
  An ansatz is presented for a possible non-associative deformation of the standard Yang-Mills type gauge theories. An explicit algebraic structure for the deformed gauge symmetry is put forward and the resulting gauge theory developed. The non-associative deformation is constructed in such a way that an apparently associative Lie algebraic structure is retained modulo a closure problem for the generators. It is this failure to close which leads to new physics in the model as manifest in the gauge field kinetic term in the resulting Lagrangian. A possible connection between this model and quantum group gauge theories is also investigated. 
  For fields that vary slowly on the scale of the lightest mass the logarithm of the vacuum functional can be expanded as a sum of local functionals, however this does not satisfy the obvious form of the Schr\"odinger equation. For $\varphi^4$ theory we construct the appropriate equation that this expansion does satisfy. This reduces the eigenvalue problem for the Hamiltonian to a set of algebraic equations. We suggest two approaches to their solution. The first is equivalent to the usual semi-classical expansion whilst the other is a new scheme that may also be applied to theories that are classically massless but in which mass is generated quantum mechanically. 
  We study the classical chaos--order transition in the spatially homogenous SU(2) Yang--Mills--Higgs system by using a quantal analog of Chirikov's resonance overlap criterion. We obtain an analytical estimation of the range of parameters for which there is chaos suppression. 
  We present a formulation of $N=1,D=10$ Supergravity--Super--Maxwell theory in superspace in which the graviphoton can be described by a 2--form $B_2$ or a 6--form $B_6$, the photon by a 1--form $A_1$ or a 7--form $A_7$ and the dilaton by a scalar $\varphi$ or an 8--form $\varphi_8$, the supercurvatures of these fields being related by duality. Duality interchanges Bianchi identities and equations of motion for each of the three couples of fields. This construction envisages the reformulation of $D=10$ Supergravity, involving 7--forms as gauge fields, conjectured by Schwarz and Sen, which, upon toroidal compactification to four dimensions, gives the manifestly $SL(2,R)_S$ invariant form of the heterotic string effective action. 
  A concise survey of the black hole information paradox and its current status is given. A summary is also given of recent arguments against remnants. The assumptions underlying remnants, namely unitarity and causality, would imply that Reissner Nordstrom black holes have infinite internal states. These can be argued to lead to an unacceptable infinite production rate of such black holes in background fields. (To appear in the proceedings of the PASCOS symposium/Johns Hopkins Workshop, Baltimore, MD, March 22-25, 1995). 
  We present a manifestly supersymmetric off-shell formulation of a wide class of $(4,4)$ $2D$ sigma models with torsion and both commuting and non-commuting left and right complex structures in the harmonic superspace with a double set of $SU(2)$ harmonic variables. The distinguishing features of the relevant superfield action are: (i) in general nonabelian and nonlinear gauge invariance ensuring a correct number of physical degrees of freedom; (ii) an infinite tower of auxiliary fields. This action is derived from the most general one by imposing the integrability condition which follows from the commutativity of the left and right analyticity-preserving harmonic derivatives. For a particular class of such models we explicitly demonstrate the non-commutativity of complex structures on the bosonic target. 
  Within a four dimensional manifestly N = 1 supersymmetric action, we show that Wess-Zumino-Novikov-Witten (WZNW) terms can be embedded in an extraordinarily simple manner into a purely chiral superaction. In order to achieve this result it is necessary to assign spin-0 and spin-1/2 degrees of freedom both to chiral superfields and as well to non-minimal scalar multiplets. We propose a new formulation for the effective low-energy action of 4D, N = 1 supersymmetric QCD that is consistent with holomorphy through fourth order in the pion superfield. After reduction to a 2D, N = 2 theory we find a new class of manifestly supersymmetric non-linear sigma models with torsion. 
  Some relationships between string theories and eleven-dimensional supergravity are discussed and reviewed. We see how some relationships can be derived from others. The cases of N=2 supersymmetry in nine dimensions and N=4 supersymmetry in four dimensions are discussed in some detail. The latter case leads to consideration of quotients of a K3 surface times a torus and to a possible peculiar relationship between eleven-dimensional supergravity and the heterotic strings in ten dimensions. Lecture given at "S-Duality and Mirror Symmetry", Trieste, June 1995. 
  Using heterotic/type II string duality, we obtain exact nonperturbative results for the point particle limit (alpha' -> 0) of some particular four dimensional, N=2 supersymmetric compactifications of heterotic strings. This allows us to recover recent exact nonperturbative results on N=2 gauge theory directly from tree-level type II string theory, which provides a highly non-trivial, quantitative check on the proposed string duality. We also investigate to what extent the relevant singular limits of Calabi-Yau manifolds are related to the Riemann surfaces that underlie rigid N=2 gauge theory. 
  We study the dyon spectrum in $N=2$ Super Yang-Mills theory with gauge group $SU(2)$ coupled to $N_f$ matter multiplets in the fundamental representation. For magnetic charge one and two we determine the spectrum explicitly and show that it is in agreement with the duality predictions of Seiberg and Witten. We briefly discuss the extension to higher charge monopoles for the self-dual $N_f=4$ case and argue that the conjectured spectrum of dyons predicts the existence of certain harmonic spinors on the moduli space of higher charge monopoles. 
  We consider the gravitational coupling of a scalar field, in a reformulation of General Relativity exhibiting local GL(4) invariance at the classical level. We compute the one-loop contribution of the scalar to the quantum effective potential of the vierbein and find that it does not have GL(4) invariance. The minima of the effective potential occur for a vierbein which is proportional to the unit matrix. 
  Lorentz invariance of the current operators implies that they satisfy the well-known commutation relations with the representation operators of the Lorentz group. It is shown that if the standard construction of the current operators in quantum field theory is used then the commutation relations are broken by the Schwinger terms. 
  We examine the electric-magnetic duality for a U(1) gauge theory on a general 4-manifold. The partition function for such a theory transforms as a modular form of specific weight. However, in the canonical approach, we show that S-duality, like T-duality, is generated by a canonical transformation leading to a modular invariant partition function. 
  The adiabatic motion of a charged, spinning, quantum particle in a two - dimensional magnetic field is studied. A suitable set of operators generalizing the cinematical momenta and the guiding center operators of a particle moving in a homogeneous magnetic field is constructed. This allows us to separate the two degrees of freedom of the system into a {\sl fast} and a {\sl slow} one, in the classical limit, the rapid rotation of the particle around the guiding center and the slow guiding center drift. In terms of these operators the Hamiltonian of the system rewrites as a power series in the magnetic length $\lb=\sqrt{\hbar c\over eB}$ and the fast and slow dynamics separates. The effective guiding center Hamiltonian is obtained to the second order in the adiabatic parameter $\lb$ and reproduces correctly the classical limit. 
  The operator ${\bf S}$ in Fock space which describes the scattering and particle production processes in an external time-dependent electromagnetic potential $A$ can be constructed from the one-particle S-matrix up to a physical phase $\lambda [A]$. In this work we determine this phase for $QED$ in (2+1) dimensions, by means of causality, and show that no ultraviolet divergences arise, in contrast to the usual formalism of $QED$. 
  We study the path integral of a twisted $N=2$ supersymmetric Yang-Mills theory coupled with hypermultiplet having the bare mass. We explicitly compute the topological correlation functions for the $SU(2)$ theory on a compact oriented simply connected simple type Riemann manifold with $b_2^+ \geq 3$. As the corollaries, we determine the topological correlation functions of the theory without the bare mass and those of the theory without coupling to the hypermultiplet. This includes a concrete field theoretic proof of the relation between the Donaldson and the Seiberg-Witten invariants. 
  We construct the Picard-Fuchs equations of the $N=2$ supersymmetric $SU(2)$ gauge theories with $N_f=0,1,3$ matter multiplets. For the $N_f=0$ theory from the solutions of the Picard-Fuchs equation the monodromy matrices on the quantum moduli space are determined. We analyze the Seiberg-Witten solutions to compute monodromies exactly and present the instanton expansion of the periods for the $N_f=0,3$ theories. 
  Some recent experiments led to the claim that something can travel faster than light in vacuum. However, such results do not seem to place relativistic causality in jeopardy. Actually, it is possible to solve also the known causal paradoxes, devised for ``faster than $c$" motion: even if this is not widely recognized. Here we want to show, in detail and rigorously, how to solve the oldest causal paradox, originally proposed by Tolman, which is the kernel of so many further tachyon paradoxes. The key to the solution is a careful application of {\em tachyon mechanics}, that can be unambiguously derived from special relativity. 
  The path integral on a homogeneous space $ G/H $ is constructed, based on the guiding principle `first lift to $ G $ and then project to $ G/H $'. It is then shown that this principle admits inequivalent quantizations inducing a gauge field (the canonical connection) on the homogeneous space, and thereby reproduces the result obtained earlier by algebraic approaches. 
  We present a brief overview of the different kinds of electromagnetic radiations expected to come from (or to be induced by) space-like sources (tachyons). New domains of radiation are here considered; and the possibility of experimental observation of tachyons via electromagnetic radiation is discussed. 
  We bring together two apparently disconnected lines of research (of mathematical and of physical nature, respectively) which aim at the definition, through the corresponding zeta function, of the determinant of a differential operator possessing, in general, a complex spectrum. It is shown explicitly how the two lines have in fact converged to a meeting point at which the precise mathematical conditions for the definition of the zeta function and the associated determinant are easy to understand from the considerations coming up from the physical approach, which proceeds by stepwise generalization starting from the most simple cases of physical interest. An explicit formula that establishes the bridge between the two approaches is obtained. 
  In a recent paper, we have investigated the classical theory of Barut and Zanghi (BZ) for the electron spin [which interpreted the Zitterbewegung (zbw) motion as an internal motion along helical paths], and its ``quantum'' version, just by using thelanguage of Clifford algebras. And, in so doing, we have ended with a new non-linear Dirac--like equation (NDE). We want to readdress here the whole subject, and extend it, by translating it however into the ordinary tensorial language, within the frame of a first quantization formalism. In particular, we re-derive here the NDE for the electron field, and show it to be associated with a new conserved probability current (which allows us to work out a ``quantum probabilistic" interpretation of our NDE). Actually, we propose this equation in substitution for the Dirac equation, which is obtained from the former by averaging over a zbw cycle. We then derive a new equation of motion for the 4-velocity field which will allow us to regard the electron as an extended--type object with a classically intelligible internal structure.We carefully study the solutions of the NDE; with special attention to those implying (at the classical limit) light-like helical motions, since they appear to be the most adequate solutions for the electron description from a kinematical and physical point of view, and do cope with the electromagnetic properties of the electron. At last we propose a natural generalization of our approach, for the case in which an external electromagnetic potential $A^\mu$ is present; \ it happens to be based on a new system of {\em five} first--order differential field equations. 
  We report the calculation of the fourth coefficient in an expansion of the heat kernel of a non-minimal, non-abelian kinetic operator in an arbitrary background gauge in arbitrary space-time dimension. The fourth coefficient is shown to bring a nontrivial gauge dependence due to the contribution of the lowest order off-shell gauge invariant structure. 
  This is a self-contained review on the theory of quantum group and its applications to modern physics. A brief introduction is given to the Yang-Baxter equation in integrable quantum field theory and lattice statistical physics. The quantum group is primarily introduced as a systematic method for solving the Yang-Baxter equation. Quantum group theory is presented within the framework of quantum double through quantizing Lie bi-algebra. Both the highest weight and the cyclic representations are investigated for the quantum group and emphasis is laid on the new features of representations for $q$ being a root of unity. Quantum symmetries are explored in selected topics of modern physics. 
  We examine supersymmetric quantum mechanics on $SO(4)$ to realize Witten's idea. We find instanton solutions connecting approximate vacuums. We calculate Hessian matrices for these solutions to determine true vacuums. Our result is in agreement with de Rham cohomology of $SO(4)$. We also give a criterion for cancellation of instanton effects for a pair of instanton paths. 
  For the anisotropic $[u (\sum_{i=1^N {\phi}_i^2)^2+v \sum_{i=1^N \phi_i^4]$-theory with {$N=2,3$} we calculate the imaginary parts of the renormalization-group functions in the form of a series expansion in $v$, i.e., around the isotropic case. Dimensional regularization is used to evaluate the fluctuation determinants for the isotropic instanton near the space dimension 4. The vertex functions in the presence of instantons are renormalized with the help of a nonperturbative procedure introduced for the simple $g{\phi^4$-theory by McKane et al. 
  The mechanism of gaugino condensation has emerged as a prime candidate for supersymmetry breakdown in low energy effective supergravity (string) models. One of the open questions in this approach concerns the size of the gauge coupling constant which is dynamically fixed through the vev of the dilaton. We argue that a nontrivial gauge kinetic function $f(S)$ could solve the potential problem of a runaway dilaton. The actual form of $f(S)$ might be constrained by symmetry arguments. 
  It is a review of some results in Odd symplectic geometry related to the Batalin-Vilkovisky Formalism 
  We present new supersymmetric extensions of Conformal Toda and $A^{(1)}_N$ Affine Toda field theories. These new theories are constructed using methods similar to those that have been developed to find supersymmetric extensions of two-dimensional bosonic sigma models with a scalar potential. In particular, we show that the Conformal Toda field theory admits a (1,1)-supersymmetric extension, and the $A^{(1)}_N$ Affine Toda field admits a (1,0)-supersymmetric extension. 
  We use the large $N$ self consistency method to compute the critical exponents of the fields and coupling of the supersymmetric CP(N) sigma model at leading order in $1/N$ in various dimensions. We verify that the correction to the critical beta-function slope vanishes at O(1/N) which is consistent with supersymmetry. The three dimensional model is investigated explicitly when a Chern Simons term is included supersymmetrically. We determine the modification that this has on the gauge independent quantity \beta^\prime(g_c) as a function of the Chern Simons coupling, \vartheta. For an N = 2 supersymmetric Chern Simons term the exponent is independent of \vartheta at O(1/N), whilst it is invariant under \vartheta \rightarrow 1/\vartheta when an N = 1 supersymmetric Chern Simons term is included. 
  We present dyonic multi-membrane solutions of the N=2 D=8 supergravity theory that serves as the effective field theory of the $T^2$-compactified type II superstring theory. The `electric' charge is fractional for generic asymptotic values of an axion field, as for D=4 dyons. These membrane solutions are supersymmetric, saturate a Bogomolnyi bound, fill out orbits of an $Sl(2;\Z)$ subgroup of the type II D=8 T-duality group, and are non-singular when considered as solutions of $T^3$-compactified D=11 supergravity. On $K_3$ compactification to D=4, the conjectured type II/heterotic equivalence allows the $Sl(2;\Z)$ group to be reinterpreted as the S-duality group of the toroidally compactified heterotic string and the dyonic membranes wrapped around homology two-cycles of $K_3$ as S-duals of perturbative heterotic string states. 
  Threshold corrections to the running of gauge couplings are calculated for superstring models with free complex world sheet fermions. For two N=1 $SU(2)\times U(1)^5$ models, the threshold corrections lead to a small increase in the unification scale. Examples are given to illustrate how a given particle spectrum can be described by models with different boundary conditions on the internal fermions. We also discuss how complex twisted fermions can enhance the symmetry group of an N=4 $SU(3)\times U(1)\times U(1)$ model to the gauge group $SU(3)\times SU(2)\times U(1)$. It is then shown how a mixing angle analogous to the Weinberg angle depends on the boundary conditions of the internal fermions. 
  We demonstrate that the technique of abelian bosonization through duality transformations can be extended to gauging anomalous symmetries. The example of a Dirac fermion theory is first illustrated. This idea is then also applied to bosonize a chiral fermion by gauging its chiral phase symmetry. 
  We give the description of discretized moduli spaces (d.m.s.) $\Mcdisc$ introduced in \cite{Ch1} in terms of discrete de Rham cohomologies for moduli spaces $\Mgn$. The generating function for intersection indices (cohomological classes) of d.m.s. is found. Classes of highest degree coincide with the ones for the continuum moduli space $\Mc$. To show it we use a matrix model technique. The Kontsevich matrix model is the generating function in the continuum case, and the matrix model with the potential $N\alpha \tr {\bigl(- \fr 14 \L X\L X -\fr12\log (1-X)-\fr12X\bigr)}$ is the one for d.m.s. In the latest case the effects of Deligne--Mumford reductions become relevant, and we use the stratification procedure in order to express integrals over open spaces $\Mdisc$ in terms of intersection indices, which are to be calculated on compactified spaces $\Mcdisc$. We find and solve constraint equations on partition function $\cal Z$ of our matrix model expressed in times for d.m.s.: $t^\pm_m=\tr \fr{\d^m}{\d\l^m}\fr1{\e^\l-1}$. It appears that $\cal Z$ depends only on even times and ${\cal Z}[t^\pm_\cdot]=C(\aa N) \e^{\cal A}\e^{F(\{t^{-}_{2n}\}) +F(\{-t^{+}_{2n}\})}$, where $F(\{t^\pm_{2n}\})$ is a logarithm of the partition function of the Kontsevich model, $\cal A$ being a quadratic differential operator in $\dd{t^\pm_{2n}}$. 
  These notes explain recent developments concerning chiral anomalies and hamiltonian quantization, their relation to the theory of gerbes, and extensions of generalized loop algebras using the residue calculus of pseudodifferential operators. The renormalization of the Dirac field, leading to Schwinger terms in equal time commutation relations, is treated in a mathematically rigorous manner. The same renormalization can be used to prove the existence of S-operator in background field problems. 
  A general expression for the scattering amplitude of nonrelativistic spinless particles in the Aharonov-Bohm gauge potential is obtained within the time independent formalism. The result is valid also in the backward and forward directions as well as for any choice of the boundary conditions on the wave function at the flux tube position. 
  We study the perturbation theory for the general non-integrable chiral Potts model depending on two chiral angles and a strength parameter and show how the analyticity of the ground state energy and correlation functions dramatically increases when the angles and the strength parameter satisfy the integrability condition. We further specialize to the superintegrable case and verify that a sum rule is obeyed. 
  The DeWitt expansion of the matrix element $M_{xy} = \left\langle x \right| \exp -[\case{1}{2} (p-A)^2 + V]t \left| y \right\rangle$, $(p=-i\partial)$ in powers of $t$ can be made in a number of ways. For $x=y$ (the case of interest when doing one-loop calculations) numerous approaches have been employed to determine this expansion to very high order; when $x \neq y$ (relevant for doing calculations beyond one-loop) there appear to be but two examples of performing the DeWitt expansion. In this paper we compute the off-diagonal elements of the DeWitt expansion coefficients using the Fock-Schwinger gauge. Our technique is based on representing $M_{xy}$ by a quantum mechanical path integral. We also generalize our method to the case of curved space, allowing us to determine the DeWitt expansion of $\tilde M_{xy} = \langle x| \exp \case{1}{2} [\case{1}{\sqrt {g}} (\partial_\mu - i A_\mu)g^{\mu\nu}{\sqrt{g}}(\partial_\nu - i A_\nu) ] t| y \rangle$ by use of normal coordinates. By comparison with results for the DeWitt expansion of this matrix element obtained by the iterative solution of the diffusion equation, the relative merit of different approaches to the representation of $\tilde M_{xy}$ as a quantum mechanical path integral can be assessed. Furthermore, the exact dependence of $\tilde M_{xy}$ on some geometric scalars can be determined. In two appendices, we discuss boundary effects in the one-dimensional quantum mechanical path integral, and the curved space generalization of the Fock-Schwinger gauge. 
  A discussion of the number of degrees of freedom, and their dynamical properties, in higher derivative gravitational theories is presented. The complete non-linear sigma model for these degrees of freedom is exhibited using the method of auxiliary fields. As a by-product we present a consistent non-linear coupling of a spin-2 tensor to gravitation. It is shown that non-vanishing $(C_{\mu\nu\alpha\beta})^{2}$ terms arise in $N=1$, $D=4$ superstring Lagrangians due to one-loop radiative corrections with light field internal lines. We discuss the general form of quadratic $(1,1)$ supergravity in two dimensions, and show that this theory is equivalent to two scalar supermultiplets coupled to the usual Einstein supergravity. It is demonstrated that the theory possesses stable vacua with vanishing cosmological constant which spontaneously break supersymmetry. 
  We construct new $U_q(a^{(2)}_{2n-1})$ and $U_q(e^{(2)}_6)$ invariant $R$-matrices and comment on the general construction of $R$-matrices for twisted algebras. We use the former to construct $S$-matrices for $b^{(1)}_n$ affine Toda solitons and their bound states, identifying the lowest breathers with the $b^{(1)}_n$ particles. 
  We investigate two methods of obtaining exactly solvable potentials with analytic forms. 
  We describe an $N=2$ heterotic superstring model of rank-3 which is dual to the type-II string compactified on a Calabi-Yau manifold with Betti numbers $b_{1,1}=2$ and $b_{1,2}=86$. We show that the exact duality symmetry found from the type II realization contains the perturbative duality group of the heterotic model, as well as the exact quantum monodromies of the rigid $SU(2)$ super-Yang-Mills theory. Moreover, it contains a non-perturbative monodromy which is stringy in origin and corresponds roughly to an exchange of the string coupling with the compactification radius. 
  New prescription for the singularities of light-cone gauge theories is suggested. The new prescription provides Green's function which is identical with and different from that of Mandelstam-Leibbrandt prescription at $d=4$ and $d=2$ respectively. 
  The evolution of QCD coupling constant at finite temperature is considered by making use of the finite temperature renormalization group equation up to the one-loop order in the background field method with the Feynman gauge and the imaginary time formalism. The results are compared with the ones obtained in the literature. We point out, in particular, the origin of the discrepancies between different calculations, such as the choice of gauge, the break-down of Lorentz invariance, imaginary versus real time formalism and the applicability of the Ward identities at finite temperature. 
  One loop anomalies and their dependence on antifields for general gauge theories are investigated within a Pauli-Villars regularization scheme. For on-shell theories {\it i.e.}, with open algebras or on-shell reducible theories, the antifield dependence is cohomologically non trivial. The associated Wess-Zumino term depends also on antifields. In the classical basis the antifield independent part of the WZ term is expressed in terms of the anomaly and finite gauge transformations by introducing gauge degrees of freedom as the extra dynamical variables. The complete WZ term is reconstructed from the antifield independent part. 
  A family of solvable self-dual Lie algebras that are not double extensions of Abelian algebras and, therefore, cannot be obtained through a Wigner contraction, is presented. We construct WZNW and gauged WZNW models based on the first two algebras in this family. We also analyze some general phenomena arising in such models. 
  Target space duality symmetries, which acts on K\"ahler and continuous Wilson line moduli, of a ${\bf Z}_N$ ($N\not=2$) 2-dimensional subspace of the moduli space of orbifold compactification are modified to include twisted moduli. These spaces described by the cosets $SU(n,1)\over SU(n)\times U(1)$ are $special$ K\"ahler, a fact which is exploited in deriving the extension of tree level duality transformation to include higher orders of the twisted moduli. Also, restrictions on these higher order terms are derived. 
  We develop a new operator quantization scheme for gauge theories in which the dynamics of the ghost sector is described by an N=2 supersymmetry. In this scheme no gauge condition is imposed on the gauge fields. The corresponding path integral is explicitly Lorentz invariant and, in contrast to the BRST-BFV path integral in the Lorentz gauge, it is free of the Gribov ambiguity, i.e., it is also valid in the non-perturbative domain. The formalism can therefore be used to study the non-perturbative properties of gauge theories in the infra-red region (gluon confinement). 
  We show that, when quantized on a curved ``intra-hadronic background'', QCD induces an effective pseudo gravitational interaction with gravitational and cosmological constants in the GeV range. 
  Exact Superstring solutions are constructed moving in 4-D space-time with positive curvature and non-trivial dilaton and antisymmetric tensor fields. The full spectrum of string excitations is derived as a function of moduli fields $T^{i}$ and the scale $\mu^2=1/(k+2)$ which induced by the non-zero background fields. The spectrum of string excitations has a non-zero mass gap $\mu^2$ and in the weak curvature limit ($\mu$ small) $\mu^2$ plays the role of a well defined infrared regulator, consistent with modular invariance, gauge invariance, supersymmetry and chirality.  The effects of a covariantly constant chomo-magnetic field $B$ as well as additional curvature can be derived exactly up to one string-loop level. Thus, the one-loop corrections to all couplings (gravitational, gauge and Yukawas) are unambiguously computed and are finite both in the Ultra-Violet and the Infra-Red regime. These corrections are necessary for quantitative string superunification predictions at low energies. Similar calculations are done in the effective field theory. The one-loop corrections to the couplings are also found to satisfy Infrared Flow Equations. 
  In a previous paper, field theory in curved space was considered, and a formula that expresses the first order variation of correlation functions with respect to the external metric was postulated. The formula is given as an integral of the energy-momentum tensor over space, where the short distance singularities of the product of the energy-momentum tensor and an arbitrary composite field must be subtracted, and finite counterterms must be added. These finite counterterms have been interpreted geometrically as a connection for the linear space of composite fields over theory space. In this paper we will study a second order consistency condition for the variational formula and determine the torsion of the connection. A non-vanishing torsion results from the integrability of the variational formula, and it is related to the Bose symmetry of the product of two energy-momentum tensors. The massive Ising model on a curved two-dimensional surface is discussed as an example, and the short-distance singularities of the product of two energy-momentum tensors are calculated explicitly. 
  In previous work on the quantum Hall effect on an annulus, we used $O(d,d;{\bf Z})$ duality transformations on the action describing edge excitations to generate the Haldane hierarchy of Hall conductivities. Here we generate the corresponding hierarchy of ``bulk actions'' which are associated with Chern-Simons (CS) theories, the connection between the bulk and edge arising from the requirement of anomaly cancellation. We also find a duality transformation for the CS theory exactly analogous to the $R\rightarrow \frac{1}{R}$ duality of the scalar field theory at the edge. 
  We formulate QCD in (d+1) dimensions using Dirac's front form with periodic boundary conditions, that is, within Discretized Light-Cone Quantization. The formalism is worked out in detail for SU(2) pure glue theory in (2+1) dimensions which is approximated by restriction to the lowest {\it transverse} momentum gluons. The dimensionally-reduced theory turns out to be SU(2) gauge theory coupled to adjoint scalar matter in (1+1) dimensions. The scalar field is the remnant of the transverse gluon. This field has modes of both non-zero and zero {\it longitudinal} momentum. We categorize the types of zero modes that occur into three classes, dynamical, topological, and constrained, each well known in separate contexts. The equation for the constrained mode is explicitly worked out. The Gauss law is rather simply resolved to extract physical, namely color singlet states. The topological gauge mode is treated according to two alternative scenarios related to the In the one, a spectrum is found consistent with pure SU(2) gluons in (1+1) dimensions. In the other, the gauge mode excitations are estimated and their role in the spectrum with genuine Fock excitations is explored. A color singlet state is given which satisfies Gauss' law. Its invariant mass is estimated and discussed in the physical limit. 
  We study two-dimensional N=2 supersymmetric actions describing general models of scalar and vector multiplets coupled to supergravity. 
  \(\Un{N}\) coherent states over Grassmann manifold, \(\grsmn{N}{n}\simeq\Un{N}/ (\Un{n}\times \Un{N-n})\), are formulated to be able to argue the WKB-exactness, so called the localization of Duistermaat-Heckman, in the path integral representation of a character formula. The exponent in the path integral formula is proportional to an integer \(k\) that labels the \(\Un{N}\) representation. Thus when \(k \rightarrow\infty\) a usual semiclassical approximation, by regarding \(k \sim 1 / \hbar\), can be performed yielding to a desired conclusion. The mechanism of the localization is uncovered by means of a view from an extended space realized by the Schwinger boson technique. 
  We investigate whether dual strings could be solutions of the magnetohydrodynamics equations in the limit of infinite conductivity. We find that the induction equation is satisfied, and we discuss the Navier-Stokes equation (without viscosity) with the Lorentz force included. We argue that the dual string equations (with a non-universal maximum velocity) should describe the large scale motion of narrow magnetic flux tubes, because of a large reparametrization (gauge) invariance of the magnetic and electric string fields. It is shown that the energy-momentum tensor for the dual string can be reinterpreted as an energy-momentum tensor for magnetohydrodynamics, provided certain conditions are satisfied. We also give a brief discussion of the case when magnetic monopoles are included, and indicate how this can lead to a non-relativistic "electrohydrodynamics" picture of confinement. 
  A quantum theory is developed for a difference-difference system which can serve as a toy-model of the quantum Korteveg-de-Vries equation. 
  The Ward identities of the $W_{\infty}$ symmetry in two dimensional string theory in the tachyon background are studied in the continuum approach. We consider amplitudes different from 2D string ones by the external leg factor and derive the recursion relations among them. The recursion relations have non-linear terms which give relations among the amplitudes defined on different genus. The solutions agree with the matrix model results even in higher genus. We also discuss differences of roles of the external leg factor between the $c_M = 1$ model and the $c_M <1$ model. 
  We show equivalence between the standard weak coupling regime c>25 of the two-dimensional quantum gravity and regime h<1/2 of the original geometric approach of Polyakov [1,2], developed in [3,4,5]. 
  SU(2) Yang-Mills Theory coupled to massive adjoint scalar matter is studied in (1+1) dimensions using Discretised Light-Cone Quantisation. This theory can be obtained from pure Yang-Mills in 2+1 dimensions via dimensional reduction. On the light-cone, the vacuum structure of this theory is encoded in the dynamical zero mode of a gluon and a constrained mode of the scalar field. The latter satisfies a linear constraint, suggesting no nontrivial vacua in the present paradigm for symmetry breaking on the light-cone. I develop a diagrammatic method to solve the constraint equation. In the adiabatic approximation I compute the quantum mechanical potential governing the dynamical gauge mode. Due to a condensation of the lowest omentum modes of the dynamical gluons, a centrifugal barrier is generated in the adiabatic potential. In the present theory however, the barrier height appears too small to make any impact in this odel. Although the theory is superrenormalisable on naive powercounting grounds, the removal of ultraviolet divergences is nontrivial when the constrained mode is taken into account. The open aspects of this problem are discussed in detail. 
  We show that recently found symmetries in QED are just non-local versions of standard BRST symmetry. 
  We classify nonultralocal Poisson brackets for 1-dimensional lattice systems and describe the corresponding regularizations of the Poisson bracket relations for the monodromy matrix . A nonultralocal quantum algebras on the lattices for these systems are constructed.For some class of such algebras an ultralocalization procedure is proposed.The technique of the modified Bethe-Anzatz for these algebras is developed.This technique is applied to the nonlinear sigma model problem. 
  The canonical Poisson structure of nonlinear sigma-model is presented as a Lie-Poisson r-matrix bracket on coadjoint orbits. It is shown that the Poisson structure of this model is determined by some `hidden singularities' of the Lax matrix. 
  The gauging of free differential algebras (FDA's) produces gauge field theories containing antisymmetric tensors. The FDA's extend the Cartan-Maurer equations of ordinary Lie algebras by incorporating p-form potentials ($p > 1$). We study here the algebra of FDA transformations. To every p-form in the FDA we associate an extended Lie derivative $\ell$ generating a corresponding ``gauge" transformation. The field theory based on the FDA is invariant under these new transformations. This gives geometrical meaning to the antisymmetric tensors. The algebra of Lie derivatives is shown to close and provides the dual formulation of FDA's. 
  We review our new method, which might be the most direct and efficient way for approaching the continuum physics from Hamiltonian lattice gauge theory. It consists of solving the eigenvalue equation with a truncation scheme preserving the continuum limit. The efficiency has been confirmed by the observations of the scaling behaviors for the long wavelength vacuum wave functions and mass gaps in (2+1)-dimensional models and (1+1)-dimensional $\sigma$ model even at very low truncation orders. Most of these results show rapid convergence to the available Monte Carlo data, ensuring the reliability of our method. 
  The exact solution for a static spherically symmetric field outside a charged point particle is found in a non-linear $U(1)$ gauge theory with a logarithmic Lagrangian. The electromagnetic self-mass is finite, and for a particular relation between mass, charge, and the value of the non-linearity coupling constant, $\lambda$, the electromagnetic contribution to the Schwarzschild mass is equal to the total mass. If we also require that the singularity at the origin be hidden behind a horizon, the mass is fixed to be slightly less than the charge. This object is a {\em black point.} 
  We discuss several aspects of second quantized scattering operators $\hat S$ for fermions in external time dependent fields. We derive our results on a general, abstract level having in mind as a main application potentials of the Yang--Mills type and in various dimensions. We present a new and powerful method for proving existence of $\hat S$ which is also applicable to other situations like external gravitational fields. We also give two complementary derivations of the change of phase of the scattering matrix under generalized gauge transformations which can be used whenever our method of proving existence of $\hat S$ applies. The first is based on a causality argument i.e.\ $\hat S$ (including phase) is determined from a time evolution, and the second exploits the geometry of certain infinite-dimensional group extensions associated with the second quantization of 1-particle operators. As a special case we obtain a Hamiltonian derivation of the the axial Fermion-Yang-Mills anomaly and the Schwinger terms related to it via the descent equations, which is on the same footing and traces them back to a common root. 
  We compute completely the BRST--antibracket cohomology on local functionals in two-dimensional Weyl invariant gravity for given classical field content (two dimensional metric and scalar matter fields) and gauge symmetries (two dimensional diffeomorphisms and local Weyl transformations). This covers the determination of all classical actions, of all their rigid symmetries, of all background charges and of all candidate gauge anomalies. In particular we show that the antifield dependence can be entirely removed from the anomalies and that, if the target space has isometries, the condition for the absence of matter field dependent Weyl anomalies is more general than the familiar `dilaton equations'. 
  We consider adjoint scalar matter coupled to QCD(1+1) in light-cone quantization on a finite `interval' with periodic boundary conditions. We work with the gauge group SU(2) which is modified to ${\rm{SU(2)/Z_2}}$ by the non-trivial topology. The model is interesting for various nonperturbative approaches because it is the sector of zero transverse momentum gluons of pure glue QCD(2+1), where the scalar field is the remnant of the transverse gluon component. We use the Hamiltonian formalism in the gauge $\partial_- A^+ = 0$. What survives is the dynamical zero mode of $A^+$, which in other theories gives topological structure and degenerate vacua. With a point-splitting regularization designed to preserve symmetry under large gauge transformations, an extra $A^+$ dependent term appears in the current $J^+$. This is reminiscent of an (unwanted) anomaly. In particular, the gauge invariant charge and the similarly regulated $P^+$ no longer commute with the Hamiltonian. We show that nonetheless one can construct physical states of definite momentum which are not {\it invariant} under large gauge transformations but do {\it transform} in a well-defined way. As well, in the physical subspace we recover vanishing {\it expectation values} of the commutators between the gauge invariant charge, momentum and Hamiltonian operators. It is argued that in this theory the vacuum is nonetheless trivial and the spectrum is consistent with the results of others who have treated the large N, SU(N), version of this theory in the continuum limit. 
  A current of the deformed Virasoro algebra is identified with the Zamolodchikov-Faddeev operator for the basic scalar particle in the XYZ model. 
  The problem of the passage of the neutral massless particle with anomalous magnetic moment through the external electromagnetic field is considered both in pseudoclassical and quantum mechanics. The quantum description uses the hamiltonian in the Foldy--Wouthuysen representation, obtained from the pseudoclassical hamiltonian of the massive charged particle with anomalous magnetic moment in interaction with the external electromagnetic field using Weyl quantization scheme. 
  We consider a class of 2 dimensional Toda equations on discrete space-time. It has arisen as functional relations in commuting family of transfer matrices in solvable lattice models associated with any classical simple Lie algebra $X_r$. For $X_r = B_r, C_r$ and $D_r$, we present the solution in terms of Pfaffians and determinants. They may be viewed as Yangian analogues of the classical Jacobi-Trudi formula on Schur functions. 
  We introduce the concept of conformal spin gradation of the untwisted affine Lie superalgebra $\hat A(n-1| n-1)^{(1)}$ to study the $W\hat A (n-1| n-1)^{(1)}$ Miura transformation. We show that the essential of $\hat A(n-1| n-1)^{(1)}$ may be read from the conformal spin gradation of the canonical vector basis of the $SL(n| n)$ vector representation space $V_{2n}$ and a spectral parameter $\mu$. We give the generic formula of their conformal spin weights. Then, we set up the fundamentals of a manifestly $N=2 U(1)$ Lax formalism leading to a manifestly $N=2 W\hat A(n-1| n-1)^{(1)}$ Miura transformation. Its explicit form is obtained and is shown to have a similar structure as in the N=0 case. Both $N=0 W\hat A(n-1)^{(1)}$ and $N=2 W\hat A(n-1| n-1)^{(1)}$ Miura transformations involve $(n-1) N=0$ and N=2 conserved currents with integer conformal spins. The leading cases are discussed. Using the U(1) charge of the N=2 algebra, we develop also a new method of constructing N=2 superfield realizations of the N=2 higher spin supercurrents. Among other results, we find that in general there are three series of $(n-1)$ higher conformal spin N=2 supercurrents. The usual N=2 super $W$ currents are the only hermitian ones. At the $n=3$ level, we find a new Feigin Fuchs type extension of the conformal spin one N=2 supercurrent. Such a feature, which has no analogue at the $n=2$ level, is also present for $n>3$. Finally, we give the N=2 superfield formulation of the N=2 Boussinesq equation and its generalization involving complex N=2 supercurrents. 
  The geometry of self-dual 2-forms in eight dimensions is studied. These 2-forms determine an $n^2-n+1$ dimensional manifold ${\cal S}_{2n}$. We prove that for add $n$, it has only one dimensionallinear subspaces. In eight dimensions, the self-dual forms of Corrigan et al constitue a seven dimensional linear subspace of ${\cal S}_8$, among many other intersting linear subspaces. 
  The evidence for string/string-duality can be extended from the matching of the vector couplings to gravitational couplings. In this note this is shown in the rank three example, the closest stringy analog of the Seiberg/Witten-setup, which is related to the Calabi-Yau $WP^4_{1,1,2,2,6}(12)$. I provide an exact analytical verification of a relation checked by coefficient comparison to fourth order by Kaplunovsky, Louis and Theisen. 
  We construct groundstates of the string with non-zero mass gap and non-trivial chromo-magnetic fields as well as curvature. The exact spectrum as function of the chromo-magnetic fields and curvature is derived. We examine the behavior of the spectrum, and find that there is a maximal value for the magnetic field $H_{\rm max}\sim M_{\rm Plank}^2$. At this value all states that couple to the magnetic field become infinitely massive and decouple. We also find tachyonic instabilities for strong background fields of the order ${\cal O}(\mu M_{\rm Plank})$ where $\mu$ is the mass gap of the theory. Unlike the field theory case, we find that such ground states become stable again for magnetic fields of the order ${\cal O}(M^2_{\rm Plank})$. The implications of these results are discussed. 
  General expressions for the anomalies appearing in pure W_3 gravity are found by requiring that they satisfy a modified version of the Wess-Zumino consistency conditions in which the Ward identities are treated as nonvanishing quantities. 
  In this paper we introduce a unified approach to Toda field theories which allows us to formulate the classes of $A_n$, $B_n$ and $C_n$ models as unique models involving an arbitrary continuous parameter $\nu$. For certain values of $\nu $, the model describes the standard Toda theories. For other values of $\nu$ it defines a class of models that involve infinitely many fields. These models interpolate between the various standard Toda field theories. They are conformally invariant and possess infinitely many conserved higher-spin currents thus making them candidates for a new set of integrable systems. A general construction is performed, which can effectively be used for the derivation of explicit forms of particular higher-spin currents. We also study the currents in a different representation in which they are linear in the dynamical variables having, however, a non-linear Poisson bracket algebra. An explicit formula for this Poisson structure is found. 
  In two - loop effective Lagrangian, the low - temperature expansion of the $QED_{3+1}$ with a constant magnetic field and a finite chemical potential is performed. We then calculate the total fermion density, some components of polarization operator and de Hass -van Alphen oscillations. We find that there is a significant contribution from two-loop expansion to magnetization and fermion density for higher values of chemical potentials. 
  We present here the zero curvature formulation for a wide class of field theory models. This formalism, which relies on the existence of an operator $\d$ which decomposes the exterior space-time derivative as a BRS commutator, turns out to be particularly useful in order to solve the Wess-Zumino consistency condition. The examples of the topological theories and of the $B$-$C$ string ghost system are considered in detail. 
  The Yang-Mills type theories and their BRS cohomology are analysed within the zero curvature formalism. 
  We review the geometrical framework required for understanding the moduli space of $(2,2)$ superconformal-field theories, highlighting various aspects of its phase structure. In particular, we indicate the types of phase diagrams that emerge for ``generic'' Calabi-Yau theories and review an efficient method for their determination. We then focus on some special types of phase diagrams that have bearing on the issues of rigid manifolds, mirror symmetry and geometrical duality. 
  We suggest that for singular rotationally invariant closed string backgrounds which need sources for their support at the origin (in particular, for special plane waves and fundamental strings) certain `trivial' \a'-corrections (which are usually ignored since in the absence of sources they can be eliminated by a field redefinition) may play an important role eliminating the singularities in the exact solutions. These corrections effectively regularize the source delta-function at the \a'-scale. We demonstrate that similar smearing of the singularity of the leading-order point-source solution indeed takes place in the open string theory. 
  We study the differential equations governing mirror symmetry of elliptic curves, and obtain a characterization of the ODEs which give rise to the integral ${\bf q}$-expansion of mirror maps. Through theta function representation of the defining equation, we express the mirror correspondence in terms of theta constants. By investigating the elliptic curves in $X_9$-family, the identification of the Landau-Ginzburg potential with the spectral curve of Ising model is obtained. Through the Jacobi elliptic function parametrization of Boltzmann weights in the statistical model, an exact Jacobi form-like formula of mirror map is described . 
  A way of covariantizing duality symmetric actions is proposed. As examples considered are a manifestly space-time invariant duality--symmetric action for abelian gauge fields coupled to axion-dilaton fields and gravity in D=4, and a Lorentz-invariant action for chiral bosons in D=2. The latter is shown to admit a manifestly supersymmetric generalization for describing chiral superfields in n=(1,0) D=2 superspace. 
  An exact solution for an SU(2) Yang-Mills field coupled to a scalar field is given. This solution has potentials with a linear and Coulomb part. This may have some physical importance since many phenomenological QCD studies assume a linear plus Coulomb potential. Usually the linear potential is motivated with lattice gauge theory arguments. Here the linear potential is an exact result of the field equations. We also show that in the Nielsen-Olesen Abelian model there is an exact solution in the BPS limit which has a Coulomb-like electromagnetic field and a logarithmically rising scalar field. Both of these solutions must be cut off from above to avoid infinite field energy. 
  A discussion of the number of degrees of freedom, and their dynamical properties, in higher derivative gravitational theories is presented. The complete non-linear sigma model for these degrees of freedom is exhibited using the method of auxiliary fields. As a by-product we present a consistent non-linear coupling of a spin-2 tensor to gravitation. It is shown that non-vanishing $(C_{\mu\nu\alpha\beta})^2$ terms arise in $N=1$, $D=4$ superstring Lagrangians due to one-loop radiative corrections with light field internal lines. We discuss the general form of quadratic $(1,1)$ supergravity in two dimensions, and show that this theory is equivalent to two scalar supermultiplets coupled to the usual Einstein supergravity. It is demonstrated that the theory possesses stable vacua with vanishing cosmological constant which spontaneously break supersymmetry. We then generalize this result to $N=1$ supergravity in four dimensions. Specifically, we demonstrate that a class of higher derivative supergravity theories is equivalent to two chiral supermultiplets coupled in a specific way to Einstein supergravity. These theories are shown to possess stable vacuum states with vanishing cosmological constant which spontaneously break the $N=1$ supersymmetry. 
  We investigate 2d gravity quantized in the ADM formulation, where only the loop length $l(z)$ is retained as a dynamical variable of the gravitation, in order to get an intuitive physical insight of the theory. The effective action of $l(z)$ is calculated by adding scalar fields of conformal coupling, and the problems of the critical dimension and the time development of $l$ are addressed. 
  Using the boundary Yang-Baxter equations and exact results on the bulk $S$-matrices, we compute exact boundary scattering amplitudes of the supersymmetric sine-Gordon model with integrable boundary potentials. 
  The algebraic structures underlying quasi-exact solvability for spin 1/2 Hamiltonians in one dimension are studied in detail. Necessary and sufficient conditions for a matrix second-order differential operator preserving a space of wave functions with polynomial components to be equivalent to a \sch\ operator are found. Systematic simplifications of these conditions are analyzed, and are then applied to the construction of several new examples of multi-parameter QES spin 1/2 Hamiltonians in one dimension. 
  We show that the Klein-Gordon equation on the quaternion field is equivalent to a pair of DKP equations. We shall also prove that this pair of DKP equations can be taken back to a pair of new KG equations. We shall emphasize the important difference between the standard and the new KG equations. We also present some qualitative arguments, concerning the possibility of interpreting anomalous solution, within a quaternion quantum field theory. 
  We discuss the use of the variational principle within quaternionic quantum mechanics. This is non-trivial because of the non commutative nature of quaternions. We derive the Dirac Lagrangian density corresponding to the two-component Dirac equation. This Lagrangian is complex projected as anticipated in previous articles and this feature is necessary even for a classical real Lagrangian. 
  We reexamine a recently proposed non-inflationary solution to the monopole problem, based on the possibility that spontaneously broken Grand-Unified symmetries do not get restored at high temperature. We go beyond leading order by studying the self-consistent one-loop equations of the model. We find large next-to-leading corrections that reverse the lowest order results and cause symmetry restoration at high temperature. 
  We obtain a bosonization prescription that allows to represent the energy-momentum tensor and supersymmetry generators of non-critical superstring theories with minimal matter as those of topological supergravity. Superstrings with $N=1$ and $N=2$ world-sheet supersymmetry are considered. The topological symmetry associated with the topological supergravity representation is studied. It is shown, in particular, that the compatibility of this topological structure with the supersymmetry enhances the superconformal symmetry of the models concerned. 
  To describe a massive particle with fixed, but arbitrary, spin on $d=4$ anti-de Sitter space $M^4$, we propose the point-particle model with configuration space ${\cal M}^6 = M^{4}\times S^{2}$, where the sphere $S^2$ corresponds to the spin degrees of freedom. The model possesses two gauge symmetries expressing strong conservation of the phase-space counterparts of the second- and fourth-order Casimir operators for $so(3,2)$. We prove that the requirement of energy to have a global positive minimum $E_o$ over the configuration space is equivalent to the relation $E_o > s$, $s$ being the particle's spin, what presents the classical counterpart of the quantum massive condition. States with the minimal energy are studied in detail. The model is shown to be exactly solvable. It can be straightforwardly generalized to describe a spinning particle on $d$-dimensional anti-de Sitter space $M^d$, with ${\cal M}^{2(d-1)} = M^d \times S^{(d-2)}$ the corresponding configuration space. 
  We report on the status of our search for quasi-topological solitons of various dimensions in realistic field theoretical models of condensed matter and of elementary particle physics. 
  We discuss lowering the order of the two-dimensional scalar-tensor $R^2$ quantum gravity, by mapping the most general version of the model to a multi-dilaton gravity, which is essentially the sigma-model coupled with Jackiw-Teitelboim-like gravity. In the continuation of our previous research, we calculate the divergent part of the one-loop effective action in a 2D scalar-tensor (dilatonic) gravity with Ra2-term, which belongs to a specific degenerate case and cannot be obtained from the general expression. The corresponding finiteness conditions are found. 
  The main part of this presentation is a review of the previous original works on the perturbative covariant approach to the $2$-dimensional quantum gravity. We discuss the renormalization of the two-dimensional dilaton gravity in a harmonic gauge, the form of the quantum corrections to the classical potential, and the conditions of Weyl invariance in a theory of string coupled to $2d$ quantum gravity. 
  We review some of the recent work on the dynamics of four dimensional, supersymmetric gauge theories. The kinematics are largely determined by holomorphy and the dynamics are governed by duality. The results shed light on the phases of gauge theories. Some results and interpretations which have not been published before are also included. 
  I discuss the breaking of space-time supersymmetry when magnetic monopole fields are switched on in compact dimensions. 
  We analyze (2+1)-dimensional Gross-Neveu model with a Thirring interaction, where a vector-vector type four-fermi interaction is on equal terms with a scalar-scalar type one. The Dyson-Schwinger equation for fermion self-energy function is constructed up to next-to-leading order in 1/N expansion. We determine the critical surface which is the boundary between a broken phase and an unbroken one in ($\alpha_c,~ \beta_c,~ N_c$) space. It is observed that the critical behavior is mainly controlled by Gross-Neveu coupling $\alpha_c$ and the region of the broken phase is separated into two parts by the line $\alpha_c=\alpha_c^*(=\frac{8}{\pi^2})$. The mass function is strongly dependent upon the flavor number N for $\alpha > \alpha_c^*$, while weakly for $\alpha < \alpha_c^*$. For $\alpha > \alpha_c^*$, the critical flavor number $N_c$ increases as Thirring coupling $\beta$ decreases. By driving the CJT effective potential, we show that the broken phase is energetically preferred to the symmetric one. We discuss the gauge dependence of the mass function and the ultra-violet property of the composite operators. 
  We prove gauge-independence of one-loop path integral for on-shell quantum gravity obtained in a framework of modified geometric approach. We use projector on pure gauge directions constructed via quadratic form of the action. This enables us to formulate the proof entirely in terms of determinants of non-degenerate elliptic operators without reference to any renormalization procedure. The role of the conformal factor rotation in achieving gauge-independence is discussed. Direct computations on $CP^2$ in a general three-parameter background gauge are presented. We comment on gauge dependence of previous results by Ichinose. 
  Recently a great deal of evidence has been found indicating that type IIA string theory compactified on K3 is equivalent to heterotic string theory compactified on T^4. Under the transformation which relates the two theories, the roles of fundamental and solitonic string solutions are interchanged. In this letter we show that there exists a solitonic membrane solution of the heterotic string theory which becomes a singular solution of the type IIA theory, and should therefore be interpreted as a fundamental membrane in the latter theory. We speculate upon the implications that the complete type IIA theory is a theory of membranes, as well as strings. 
  We compute the exact partition function, the universal ground state degeneracy and boundary state of the 2-D Ising model with boundary magnetic field at off-critical temperatures. The model has a domain that exhibits states localized near the boundaries. We study this domain of boundary bound state and derive exact expressions for the ``$g$ function'' and boundary state for all temperatures and boundary magnetic fields. In the massless limit we recover the boundary renormalization group flow between the conformally invariant free and fixed boundary conditions. 
  We consider the nonlinear algebras $W(sl(4),sl(3))$ and $W(sl(3|1),sl(3))$ and find their realizations in terms of currents spanning conformal linearizing algebras. The specific structure of these algebras, allows us to construct realizations modulo null fields of the $W_3$ algebra that lies in the cosets $W(sl(4),sl(3))/u(1)$ and $W(sl(3|1),sl(3))/u(1)$. Such realizations exist for the following values of the $W_3$ algebra central charge: $c_W=-30,-40/7,-98/5,-2$. The first two values are listed for the first time, whereas for the remaining values we get the new realizations in terms of an arbitrary stress tensor and $u(1)\times sl(2)$ affine currents. 
  A general approach to anomaly in quantum field theory is newly formulated by use of the propagator theory in solving the heat-kernel equation. We regard the heat-kernel as a sort of the point-splitting regularization in the space(-time) manifold. Fujikawa's general standpoint that the anomalies come from the path-integral measure is taken. We obtain some useful formulae which are valid for general anomaly calculation. They turn out to be the same as the 1-loop counter-term formulae except some important total derivative terms. Various anomalies in 2 and 4 dimensional theories are systematically calculated. Some important relations between them are concretely shown. As for the representation of general (global SO(n)) tensors, we introduce a graphical one. It makes the tensor structure of invariants very transparent and makes the tensor calculation so simple. 
  In this paper we examine the classical evolution of a cosmological model derived from the low-energy tree-level limit of a generic string theory. The action contains the metric, dilaton, central charge and an antisymmetric tensor field. We show that with a homogeneous and isotropic metric, allowing spatial curvature, there is a formal equivalence between this system and a scalar field minimally coupled to Einstein gravity in a spatially flat metric. We refer to this system as the shifted frame and using it we describe the full range of cosmological evolution that this model can exhibit. We show that generic solutions begin (or end) with a singularity. As the system approaches a singularity the dilaton becomes becomes large and loop corrections will become important. 
  This paper is an overview on our recent results in the calculation of the heat kernel in quantum field theory and quantum gravity. We introduce a deformation of the background fields (including the metric of a curved spacetime manifold) and study various asymptotic expansions of the heat kernel diagonal associated with this deformation. Especial attention is payed to the low-energy approximation corresponding to the strong slowly varying background fields. We develop a new covariant purely algebraic approach for calculating the heat kernel diagonal in low-energy approximation by taking into account a finite number of low-order covariant derivatives of the background fields, and neglecting all covariant derivatives of higher orders. Then there exist a set of covariant differential operators that together with the background fields and their low-order derivatives generate a finite dimensional Lie algebra. In the zeroth order of the low-energy perturbation theory, determined by covariantly constant background, we use this algebraic structure to present the heat operator in the form of an average over the corresponding Lie group. This simplifies considerably the calculations and allows to obtain closed explicitly covariant formulas for the heat kernel diagonal. These formulas serve as the generating functions for the whole sequence of the Hadamard-Minakshisundaram- De Witt-Seeley coefficients in the low-energy approximation. 
  We classify the spherically symmetric solutions of the Einstein-Maxwell Dilaton field equations in D dimensions and find some exact solutions of the string theory at all orders of the string tension parameter. We also show the uniqueness of the black hole solutions of this theory in static axially symmetric spacetimes. 
  A short informal overview about recent progress in the calculation of the effective action in quantum gravity is given. I describe briefly the standard heat kernel approach to the calculation of the effective action and discuss the applicability of the Schwinger - De Witt asymptotic expansion in the case of strong background fields. I propose a new ansatz for the heat kernel that generalizes the Schwinger - De Witt one and is always valid. Then I discuss the general structure of the asymptotic expansion and put forward some approximate explicitly covariant methods for calculating the heat kernel, namely, the high-energy approximation as well as the low-energy one. In both cases the explicit formulae for the heat kernel are given. 
  The covariant technique for calculating the heat kernel asymptotic expansion for an elliptic differential second order operator is generalized to manifolds with boundary. The first boundary coefficients of the asymptotic expansion which are proportional to $t^{1/2}$ and $t^{3/2}$ are calculated. Our results coincide with completely independent results of previous authors. 
  A new algebraic approach for calculating the heat kernel for the Laplace operator on any Riemannian manifold with covariantly constant curvature is proposed. It is shown that the heat kernel operator can be obtained by an averaging over the Lie group of isometries. The heat kernel diagonal is obtained in form of an integral over the isotropy subgroup. 
  Open-string theories may be related to suitable models of oriented closed strings. The resulting construction of ``open descendants'' is illustrated in a few simple cases that exhibit some of its key features. 
  We present a way for calculating the Lagrangian path integral measure directly from the Hamiltonian Schwinger--Dyson equations. The method agrees with the usual way of deriving the measure, however it may be applied to all theories, even when the corresponding momentum integration is not Gaussian. Of particular interest is the connection that is made between the path integral measure and the measure in the corresponding 0-dimensional model. This allows us to uniquely define the path integral even for the case of Euclidean theories whose action is not bounded from below. 
  The relativistic finite-difference SUSY Quantum Mechanics (QM) is developed. We show that it is connected in a natural way with the q-deformed SUSY Quantum Mechanics. Simple examples are given. 
  This paper has been withdrawn to address an omission. It will be resubmitted in the near future. 
  Curci and Ferrari found a unique BRS-invariant action for non-Abelian gauge theories which includes a mass term for the gauge bosons. I analyze this action. While the BRS operator is not nilpotent, the Zinn-Justin equation generalizes in a simple way so that the renormalization of the theory is consistent with the infrared regularization provided by the mass---infrared singularities and ultraviolet infinities are therefore clearly separated. Relations between renormalization constants are derived in dimensional regularization with minimal subtraction. Additional new symmetries allow a simple characterization of physical operators. A new formula is given for the gauge parameter dependence of physical operators. 
  This paper is being revised to make it intelligible, and to incorporate some corrections. 
  A non-abelian ``self-dual'' massive gauge theory describing a massive spin one physical mode is presented. The action is expressed in terms of two independent connections on a principle bundle over $2+1$ space-time. The kinetic terms are respectively a Chern-Simons and a $BF$ lagrangians, while the gauge invariant interaction is expressed in terms of the difference of the two connections. At the linearized level the theory is equivalent to the Topological Massive gauge theory. The covariant, $BRST$ invariant, effective action of the non-abelian sefl dual gauge theory is constructed. 
  In quantizing gravity based on stochastic quantization method, the stochastic time plays a role of the proper time. We study 2D and 4D Euclidean quantum gravity in this context. By applying stochastic quantization method to real symmetric matrix models, it is shown that the stochastic process defined by the Langevin equation in loop space describes the time evolution of the non-orientable loops which defines non-orientable 2D surfaces. The corresponding Fokker-Planck hamiltonian deduces a non-orientable string field theory at the continuum limit. The strategy, which we have learned in the example of 2D quantum gravity, is applied to 4D case. Especially, the Langevin equation for the stochastic process of 3-geometries is proposed to describe the (Euclidean) time evolution in 4D quantum gravity with Ashtekar's canonical variables. We present it in both lattice regularized version and the naive continuum limit. 
  Relaxing first-class constraint conditions in the usual Drinfeld-Sokolov Hamiltonian reduction leads, after symmetry fixing, to realizations of W algebras expressed in terms of all the J-current components. General results are given for G a non exceptional simple (finite and affine) algebra. Such calculations directly provide the commutant, in the (closure of) G enveloping algebra, of the nilpotent subalgebra $G_-$, where the subscript refers to the chosen gradation in G. In the affine case, explicit expressions are presented for the Virasoro, $W_3$, and Bershadsky algebras at the quantum level. 
  It is shown, under mild assumptions, that classical degrees of freedom dynamically coupled to quantum ones do not inherit their quantum fluctuations. It is further shown that, if the assumptions are strengthen by imposing the existence of a canonical structure, only purely classical or purely quantum dynamics are allowed. 
  For extremal black holes, the thermodynamic entropy is not proportional to the area. The general form allowed by thermodynamics is worked out for three classes of extremal black hole solutions of string theory and shown to be consistent with the entropy calculated from the density of elementary string states. On the other hand, the entanglement entropy does not in general agree with these results. 
  We propose a new method for the study of the chiral properties of the ground state in QFT's based on the computation of the probability distribution function of the chiral condensate. It can be applied directly in the chiral limit and therefore no mass extrapolations are needed. Furthermore this approach allows to write up equations relating the chiral condensate with quantities computable by standard numerical methods, the functional form of these relations depending on the broken symmetry group. As a check, we report some results for the compact Schwinger model. 
  It is demonstrated that the action of SU$(N)$ principal chiral model leads in the limit $N \to {\infty}$ to the action for Husain's heavenly equation. The principal chiral model in the Hilbert space $L^2(\Re^1)$ is considered and it is shown, that in this case the chiral equation is equivalent to the Moyal deformation of Husain's heavenly equation. New method of searching for solutions to this latter equation, via Lie algebra representations in $L^2(\Re^1)$ is given. 
  In this work a deep relation between topology and thermodynamical features of manifolds with boundaries is shown. The expression for the Euler characteristic, through the Gauss- Bonnet integral, and the one for the entropy of gravitational instantons are proposed in a form which makes the relation between them self-evident. A generalization of Bekenstein-Hawking formula, in which entropy and Euler characteristic are related in the form $S=\chi A/8$, is obtained. This formula reproduces the correct result for extreme black hole, where the Bekenstein-Hawking one fails ($S=0$ but $A \neq 0$). In such a way it recovers a unified picture for the black hole entropy law. Moreover, it is proved that such a relation can be generalized to a wide class of manifolds with boundaries which are described by spherically symmetric metrics (e.g. Schwarzschild, Reissner-Nordstr\"{o}m, static de Sitter). 
  Noether's symmetry transformations for higher-order lagrangians are studied. A characterization of these transformations is presented, which is useful to find gauge transformations for higher-order singular lagrangians. The case of second-order lagrangians is studied in detail. Some examples that illustrate our results are given; in particular, for the lagrangian of a relativistic particle with curvature, lagrangian gauge transformations are obtained, though there are no hamiltonian gauge generators for them. 
  A description of dual non-Abelian duality is given, based on the notion of the Drinfeld double. The presentation basically follows the original paper \cite{KS2}, written in collaboration with P. \v Severa, but here the emphasis is put on the algebraic rather than the geometric aspect of the construction and a concrete example of the Borelian double is worked out in detail. 
  We formulate the constrained KP hierarchy (denoted by \cKP$_{K+1,M}$) as an affine ${\widehat {sl}} (M+K+1)$ matrix integrable hierarchy generalizing the Drinfeld-Sokolov hierarchy. Using an algebraic approach, including the graded structure of the generalized Drinfeld-Sokolov hierarchy, we are able to find several new universal results valid for the \cKP hierarchy. In particular, our method yields a closed expression for the second bracket obtained through Dirac reduction of any untwisted affine Kac-Moody current algebra. An explicit example is given for the case ${\widehat {sl}} (M+K+1)$, for which a closed expression for the general recursion operator is also obtained. We show how isospectral flows are characterized and grouped according to the semisimple {\em non-regular} element $E$ of $sl (M+K+1)$ and the content of the center of the kernel of $E$. 
  The connection between the proper time equation and the Zamolodchikov metric is discussed. The connection is two-fold: First, as already known, the proper time equation is the product of the Zamolodchikov metric and the renormalization group beta function. Second, the condition that the two-point function is the Zamolodchikov metric, implies the proper time equation. We study the massless vector of the open string in detail. In the exactly calculable case of a uniform electromgnetic field strength we recover the Born-Infeld equation. We describe the systematics of the perturbative evaluation of the gauge invariant proper time equation for the massless vector field. The method is valid for non-uniform fields and gives results that are exact to all orders in derivatives. As a non trivial check, we show that in the limit of uniform fields it reproduces the lowest order Born-Infeld equation. 
  We construct explicitly the quantization of classical linear maps of $SL(2, R)$ on toroidal phase space, of arbitrary modulus, using the holomorphic (chiral) version of the metaplectic representation. We show that Finite Quantum Mechanics (FQM) on tori of arbitrary integer discretization, is a consistent restriction of the holomorphic quantization of $SL(2, Z)$ to the subgroup $SL(2, Z)/\Gamma_l$, $\Gamma_l$ being the principal congruent subgroup mod l, on a finite dimensional Hilbert space. The generators of the ``rotation group'' mod l, $O_{l}(2)\subset SL(2,l)$, for arbitrary values of l are determined as well as their quantum mechanical eigenvalues and eigenstates. 
  The $N=4$ supersymmetric self-dual Yang-Mills theory in a four- dimensional space with signature $(2,2)$ is formulated in harmonic superspace. The on-shell constraints of the theory are reformulated in the equivalent form of vanishing curvature conditions for three gauge connections (one harmonic and two space-time). The constraints are then obtained as variational equations from a superspace action of the Chern-Simons type. The action is manifestly $SO(2,2)$ invariant. It can be viewed as the Lorentz-covariant form of the light-cone superfield action proposed by Siegel. 
  Given a finite dimensional C^*-Hopf algebra H and its dual H^ we construct the infinite crossed product A=... x H x H^ x H ... and study its superselection sectors in the framework of algebraic quantum field theory. A is the observable algebra of a generalized quantum spin chain with H-order and H^-disorder symmetries, where by a duality transformation the role of order and disorder may also appear interchanged. If H=\CC G is a group algebra then A becomes an ordinary G-spin model. We classify all DHR-sectors of A --- relative to some Haag dual vacuum representation --- and prove that their symmetry is described by the Drinfeld double D(H). To achieve this we construct localized coactions \rho: A \to (A \otimes D(H)) and use a certain compressibility property to prove that they are universal amplimorphisms on A. In this way the double D(H) can be recovered from the observable algebra A as a universal cosymmetrty. 
  A class of exact static spherically symmetric solutions of the Einstein-Maxwell gravity coupled to a massless scalar field has been obtained in harmonic coordinates of the Minkowski space-time. For each value of the coupling constant $a$, these solutions are characterized by a set of three parameters, the physical mass $\mu_0$, the electric charge $Q_0$ and the scalar field parameter $k$. We have found that the solutions for both gravitational and electromagneticfields are not only affected by the scalar field, but also the non-trivial coupling with matter constrains the scalar field itself. In particular, we have found that the constant $k$ generically differs from $\pm 1/2$, falling into the interval $|k|\in [0, {1\over2}\sqrt{1+a^2} \hskip 2pt ]$. It takes these values only for black holes or in the case when a scalar field $\phi$ is totally decoupled from the matter. Our results differ from those previously obtained in that the presence of arbitrary coupling constant $a$ gives an opportunity to rule out the non-physica horizons. In one of the special cases, the obtained solution corresponds to a charged dilatonic black hole with only one horizon $\mu_+$ and hence for the Kaluza-Klein case. The most remarkable property of this result is that the metric, the scalar curvature, and both electromagnetic and scalar fields are all regular on this surface. Moreover, while studying the dilaton charge, we found that the inclusion of the scalar field in the theory result in a contraction of the horizon. The behavior of the scalar curvature was analyzed. 
  We present a static multi-center magnetic solution of toroidally compactified heterotic string theory, which is T-duality covariant. The space-time geometry depends on the mass M and on the O(6,22)-norm N of the magnetic charges. For different range of parameters (M,N)-solution includes 1) two independent positive parameters extremal magnetic black holes with non-singular geometry in stringy frame ($a=1$ black holes included), 2) $a=\sqrt 3$ extremal black holes, 3) singular massive and massless magnetic white holes (repulsons). The electric multi-center solution is also given in an O(6,22)-symmetric form. 
  We describe various aspects of two-dimensional $N=2$ supergravity in superspace. We present the solution to the constraints in terms of unconstrained prepotentials, and the different superspace measures (full and chiral) used in the construction of invariant actions. We discuss aspects of the theory in light-cone gauge, including the Ward identities for correlation functions defined with respect to the induced supergravity action. 
  In order to address the problem of the validity of the "background field approximation", we introduce a dynamical model for a mirror described by a massive quantum field. We then analyze the properties of the scattering of a massless field from this dynamical mirror and compare the results with the corresponding quantities evaluated using the original Davies Fulling model in which the mirror is represented by a boundary condition imposed on the massless field at its surface. We show that in certain circumstances, the recoils of the dynamical mirror induce decoherence effects which subsist even when the mass of the mirror is sent to infinity. In particular we study the case of the uniformly accelerated mirror and prove that, after a certain lapse of proper time, the decoherence effects inevitably dominate the physics of the quanta emitted forward. Then, the vanishing of the mean flux obtained in the Davies Fulling model is no longer found but replaced by a positive incoherent flux. 
  The fixed point resolution problem is solved for diagonal coset theories. The primary fields into which the fixed points are resolved are described by submodules of the branching spaces, obtained as eigenspaces of the automorphisms that implement field identification. To compute the characters and the modular S-matrix we use `orbit Lie algebras' and `twining characters', which were introduced in a previous paper (hep-th/9506135). The characters of the primary fields are expressed in terms of branching functions of twining characters. This allows us to express the modular S-matrix through the S-matrices of the orbit Lie algebras associated to the identification group. Our results can be extended to the larger class of `generalized diagonal cosets'. 
  We review electric/magnetic duality in $N=4$ (and certain $N=2$) globally supersymmetric gauge theories and show how this duality, which relates strong to weak coupling, follows as a consequence of a string/string duality. Black holes, eleven dimensions and supermembranes also have a part to play in the big picture. 
  The operator solution of the anomalous chiral Schwinger model is discussed on the basis of the general principles of Wightman field theory. Some basic structural properties of the model are analyzed taking a careful control on the Hilbert space associated with the Wightman functions. The isomorphism between gauge noninvariant and gauge invariant descriptions of the anomalous theory is established in terms of the corresponding field algebras. We show that (i) the Theta-vacuum representation and (ii) the suggested equivalence of vector Schwinger model and chiral Schwinger model cannot be established in terms of the intrinsic field algebra. 
  We construct and study the electrically charged, rotating black hole solution in heterotic string theory compactified on a $(10-D)$ dimensional torus. This black hole is characterized by its mass, angular momentum, and a $(36-2D)$ dimensional electric charge vector. One of the novel features of this solution is that for $D >5$, its extremal limit saturates the Bogomol'nyi bound. This is in contrast with the $D=4$ case where the rotating black hole solution develops a naked singularity before the Bogomol'nyi bound is reached. The extremal black holes can be superposed, and by taking a periodic array in $D>5$, one obtains effectively four dimensional solutions without naked singularities. 
  The mechanism of dimensional transmutation is discussed in the context of Maxwell-Chern-Simons scalar QED. The method used is non-perturbative. The effective potential describes a broken symmetry state. It is found that the symmetry breaking vacuum is more stable when the Chern-Simons mass is different from zero.   Pacs number: 11.10.Ef, 11.10.Gh. 
  We show that when the Abelian \CS\ theory coupled to matter fields is quantized in a vacuum with non vanishing magnetic flux (or electric charge), the requirement of gauge invariance at finite temperature leads to the quantization of the \CS\ coefficient and its quantum corrections, in a manner similar to the non-Abelian case. 
  We consider BRST-invariant inner product states for quantum electrodynamics constructed from trivial BRST-invariant states and a gauge regulator. The trivial states are products of matter and ghost states and are annihilated by hermitian operators. The co-BRST operator and some further gauge-fixing regulators are found. The relation between gauge fixing and time evolution of both the trivial and the inner product states is discussed. 
  The ADM and Bondi mass for the RST model have been first discussed from Hawking and Horowitz's argument. Expressing the localized RST action in terms of the ADM formulation, the RST Hamiltonian can be derived, meanwhile keeping track of all boundary terms. Then the total boundary terms can be taken as the total energy for the RST model. It has been found that there is a new contribution to the ADM and Bondi mass from the RST boundary due to the existence of the hidden dynamical field. The ADM and Bondi mass have been discussed respectively in detail, and some new properties have been found. 
  The variational properties of the scalar so--called ``Universal'' equations are reviewed and generalised. In particular, we note that contrary to earlier claims, each member of the Euler hierarchy may have an explicit field dependence. The Euler hierarchy itself is given a new interpretation in terms of the formal complex of variational calculus, and is shown to be related to the algebra of distinguished symmetries of the first source form. 
  We show that a (3+1)-dimensional system composed of an open magnetic vortex and an electrical point charge exhibits the phenomenon of Fermi-Bose transmutation. In order to provide the physical realization of this system we focus on the lattice compact scalar electrodynamics $SQED_c$ whose topological excitations are open Nielsen-Olesen strings with magnetic monopoles attached at their ends. 
  A way to break supersymmetry in perturbative superstring theory is the string version of the Scherk-Schwarz mechanism. There, the fermions and bosons have mass splitting due to different compactification boundary conditions. We consider the implementation of this mechanism in abelian orbifold compactifications with Wilson line backgrounds. For $Z_N$ and $Z_N\times Z_M$ orbifolds, we give the possible $U(1)$ R-symmetries which determine the mass splitting, and thus, the supersymmetry breaking at the perturbative level. The phenomenlogical viability of this mechanism implies some dimension(s) to be as large as the TeV scale. We explain how the lighter Kaluza-Klein states associated with the extra-dimension(s) have quantum numbers depending on the Wilson lines used. 
  First, we show that in the $(1,0)\oplus(0,1)$ representation space there exist not one but two theories for charged particles. In the Weinberg construct, the boson and its antiboson carry {\it same} relative intrinsic parity, whereas in our construct the relative intrinsic parities of the boson and its antiboson are {\it opposite}. These results originate from the commutativity of the operations of Charge conjugation and Parity in Weinberg's theory, and from the anti-commutativity of the operations of Charge conjugation and Parity in our theory. We thus claim that we have constructed a first non-trivial quantum theory of fields for the Wigner-type particles. Second, the massless limit of both theories seems formally identical and suggests a fundamental modification of Maxwell equations. At its simplest level, the modification to Maxwell equations enters via additional boundary condition(s). 
  Two-dimensional large-$N$ quantum chromodynamics with scalar quarks is considered with particular emphasis on its strong coupling regime which has not been studied so far. Techniques necessary to deal with the infinitely oscillatory bound state wave functions in the strong coupling regime are developed. I derive an estimate for the ground state mass and show that (1) the lightest hadron in the theory is massless and (2) the ground state mass is continuous across the transition between the weak and the strong coupling. 
  The dilaton in three dimensions does not roll. Witten's conjecture that duality between theories in three and four dimensions solves the cosmological constant problem thus may also solve the dilaton problem in string theory. 
  The noncritical $D=4$ $W_3$ string is a model of $W_3$ gravity coupled to two free scalar fields. In this paper we discuss its BRST quantization in direct analogy with that of the $D=2$ (Virasoro) string. In particular, we calculate the physical spectrum as a problem in BRST cohomology. The corresponding operator cohomology forms a BV-algebra. We model this BV-algebra on that of the polyderivations of a commutative ring on six variables with a quadratic constraint, or, equivalently, on the BV-algebra of (polynomial) polyvector fields on the base affine space of $SL(3,C)$. In this paper we attempt to present a complete summary of the progress made in these studies. [...] 
  It is shown that the McCall-Hahn theory of self-induced transparency in coherent optical pulse propagation can be identified with the complex sine-Gordon theory in the sharp line limit. We reformulate the theory in terms of the deformed gauged Wess-Zumino-Witten sigma model and address various new aspects of self-induced transparency. 
  The noncritical $4D$ $\cW_3$ string is a model of $\cW_3$ gravity coupled to two free scalar fields. In this paper we discuss its BRST quantization in direct analogy with that of the $2D$ (Virasoro) string. The physical operators form a BV-algebra. We model this BV-algebra on that of the polyderivations of a commutative ring on six variables with a quadratic constraint, or, equivalently, on the BV-algebra of (polynomial) polyvector fields on the base affine space of $SL(3,\CC)$. Details have appeared elsewhere. To appear in the proceedings of ``STRINGS '95: Future Perspectives in String Theory,'' USC, March 13--18, 1995 
  A set of integral relations for rotational and translational zero modes in the vicinity of the classical soliton solution are derived from the particle-like properties of the latter. The validity of these all relations is considered for a number of soliton models in 2+1- and 3+1-dimensions. 
  A pair of conformal sigma models related by Poisson-Lie T-duality is constructed by starting with the O(2,2) Drinfeld double. The duality relates the standard SL(2,R) WZNW model to a constrained sigma model defined on SL(2,R) group space. The quantum equivalence of the models is established by using a path integral argument. 
  We give a sequence of equivalent formulations of the $ADE$ and $\hat A\hat D\hat E$ height models defined on a random triangulated surface: random surfaces immersed in Dynkin diagrams, chains of coupled random matrices, Coulomb gases, and multicomponent Bose and Fermi systems representing soliton $\tau$-functions. We also formulate a set of loop-space Feynman rules allowing to calculate easily the partition function on a random surface with arbitrary topology. The formalism allows to describe the critical phenomena on a random surface in a unified fashion and gives a new meaning to the $ADE$ classification. 
  Using canonical quantization we find the Virasoro centre for a class of conformally-invariant interacting Wess-Zumino-Witten theories. The theories have a group structure similar to that of Toda theories (both abelian and non-abelian) but the usual Toda constraints on the coupling constants are relaxed and the theories are not necessarily integrable. The general formula for the Virasoro centre is compared to that derived by BRST methods in the Toda case, and helps to explain the structure of the latter. 
  We summarize some recent results obtained in collaboration with J. McCarthy on the spectrum of physical states in $W_3$ gravity coupled to $c=2$ matter. We show that the space of physical states, defined as a semi-infinite (or BRST) cohomology of the $W_3$ algebra, carries the structure of a BV-algebra. This BV-algebra has a quotient which is isomorphic to the BV-algebra of polyvector fields on the base affine space of $SL(3,C)$. Details have appeared elsewhere. [Published in the proceedings of "Gursey Memorial Conference I: Strings and Symmetries," Istanbul, June 1994, eds. G. Aktas et al., Lect. Notes in Phys. 447, (Springer Verlag, Berlin, 1995)] 
  I report recent studies on the evolution of perturbations in the context of the ``pre-big-bang" scenario typical of string cosmology, with emphasis on the formation of a stochastic background of relic photons and gravitons, and its possible direct/indirect observable consequences. I also discuss the possible generation of a thermal microwave background by using, as example, a simple gravi-axio-dilaton model whose classical evolution connects smoothly inflationary expansion to decelerated contraction. By including the quantum back-reaction of the produced radiation the model eventually approaches the standard radiation-dominated (constant dilaton) regime. 
  An index relation $dim\ ker\ a - dim\ ker\ a^{\dagger} = 1$ is satisfied by the creation and annihilation operators $a^{\dagger}$ and $a$ of a harmonic oscillator. Implications of this analytic index on the possible form of the phase operator are discussed. A close analogy between the present phase operator problem and chiral anomaly in gauge theory, which is associated with Atiyah-Singer index theorem, is emphasized. 
  Generically coupled neutral scalar bosons and chiral fermions are shown, in the eikonal kinematical limit, to be described by a reduced (free field) theory with N=1 {\it on-shell} supersymmetry. {\it Charged} scalars and spinors turn out to be described in the eikonal limit by a reduced interacting theory with a modified and restricted on-shell N=1 supersymmetry. Consequences of such a symmetry for the nontrivial scattering amplitudes in this latter case are discussed. 
  We derive the exact effective superpotential in 4d, N=1 supersymmetric SU(2) gauge theories with $N_A$ triplets and $2N_f$ doublets of matter superfields. We find the quantum vacua of these theories; the equations of motion (for $N_A=1$) can be reorganized into the singularity conditions of an elliptic curve. From the phase transition points to the Coulomb branch, we find the exact Abelian gauge couplings, $\tau$, for arbitrary bare masses and Yukawa couplings. We thus {\em derive} the result that $\tau$ is a section of an $SL(2,\Z)$ bundle over the moduli space and over the parameters space of bare masses and Yukawa couplings. For $N_c>2$, we derive the exact effective superpotential in branches of supersymmetric $SU(N_c)$ gauge theories with one supermultiplet in the adjoint representation ($N_A=1$) and zero or one flavor ($N_f=0,1$). We find the quantum vacua of these theories; the equations of motion can be reorganized into the singularity conditions of a genus $N_c-1$ hyperelliptic curve. Finally, we present the effective superpotential in the $N_A$, $N_f<N_c$ cases. 
  In line with a previous paper, a gauge-invariant regularization is developed for the Weyl determinant of a Euclidean gauged chiral fermion. We restrict ourselves to gauge configurations with the $A$ field going to zero at infinity in Euclidean space; and thus restrict gauge transformations to those with $U$ the identity at infinity. For each finite cutoff one gets a strictly gauge-invariant expression for the Weyl determinant. Full Euclidean invariance is only to be sought in the limit of removing the cutoff. We expect the limit to be Euclidean invariant, but this has not yet been proved. One need not enforce the no-anomaly condition on the representation of the gauge group! We leave to future research relating the present results to conventional physics wisdom. 
  We suggest that some of the remarkable results on stringy dynamics which have been found recently indicate the existence of another dynamical length scale in string theory that, at weak coupling, is much shorter than the string scale. This additional scale corresponds to a mass $\sim m_{\rm s}/g_{\rm s}$ where $m_{\rm s}$ is the square root of the string tension and $g_{\rm s}$ is the string coupling constant. In four dimensions this coincides with the Planck mass. 
  A functional integral approach is developed to discuss the bosonisation of the massive Thirring and the massive Schwinger models in arbitrary D-dimensions. It is found that these models, to {\it all} orders in the inverse fermi mass, bosonise to a theory involving a usual gauge field and a (D-2) rank antisymmetric (Kalb-Ramond) tensor field. Explicit bosonisation identities for the fermion current are deduced. Specialising to the lowest order reveals (for any $D \geq 4$) a mapping between the massive Thirring model and the Proca model. It also establishes an exact duality between the Proca model and the massive (D-2) rank Kalb-Ramond model. Schwinger terms in the current algebra are computed. Conventional bosonisation results in D=2, 3 are reproduced. 
  I summarize some recent results obtained in collaboration with P.~Bouwknegt and K.~Pilch on the spectrum of physical states in $\cW_3$ gravity coupled to $c=2$ matter. In particular, it is shown that the algebra of operators corresponding to physical states -- defined as a semi-infinite (or BRST) cohomology of the $\cW_3$ algebra -- carries the structure of a G-algebra. This G-algebra has a quotient which is isomorphic to the G-algebra of polyvector fields on the base affine space of $SL(3,\CC)$. Details have appeared elsewhere. To appear (with title change) in the proceedings of the ``H.S. Green and A. Hurst Festschrift: Confronting the Infinite'' Adelaide, March 1994. 
  We investigate the classical and quantum properties of a system of $SU(N)$ non-Abelian Chern-Simons (NACS) particles. After a brief introduction to the subject of NACS particles, we first discuss about the symplectic structure of various $SU(N)$ coadjoint orbits which are the reduced phase space of $SU(N)$ internal degrees of freedom or isospins. A complete Dirac's constraint analysis is carried out on each orbit and the Dirac bracket relations among the isospin variables are calculated. Then, the spatial degrees of freedom and interaction with external gauge field are introduced by considering the total reduced phase space which is given by an associated bundle whose fiber is one of the coadjoint orbits. Finally, the theory is quantized by using the coherent state method and various quantum mechanical properties are discussed in this approach. In particular, a coherent state representation of the Knizhnik-Zamolodchikov equation is given and possible solutions in this representation are discussed. 
  We investigate mesons in the bosonized massive Schwinger model in the light-front Tamm-Dancoff approximation in the strong coupling region. We confirm that the three-meson bound state has a few percent fermion six-body component in the strong coupling region when expressed in terms of fermion variables, consistent with our previous calculations. We also discuss some qualitative features of the three-meson bound state based on the information about the wave function. 
  We describe applications of (perturbed) conformal field theories to two-dimensional disordered systems. We present various methods of study~: (i) {\it A direct method} in which we compute the explicit disorder dependence of the correlation functions for any sample of the disorder. This method seems to be specific to two dimensions. The examples we use are disordered versions of the Abelian and non-Abelian WZW models. We show that the disordered WZW model over the Lie group $\CG$ at level $k$ is equivalent at large impurity density to the product of the WZW model over the coset space $\CG^C/\CG$ at level $(-2h^v)$ times an arbitrary number of copies of the original WZW model. (ii) {\it The supersymmetric method} is introduced using the random bond Ising model and the random Dirac theory as examples. In particular, we show that the relevent algebra is the affine $OSp(2N|2N)$ Lie superalgebra, an algebra with zero superdimension. (iii) {\it The replica method} is introduced using the random phase sine-Gordon model as example. We describe particularities of its renormalization group flow. (iv) {\it A variationnal approach} is also presented using the random phase sine-Gordon model as example. Lectures presented at the '95 Cargese Summer School on "Low dimensional application of quantum field theory". 
  We examine the thermal behavior of a relativistic anyon system, dynamically realized by coupling a charged massive spin-1 field to a Chern-Simons gauge field. We calculate the free energy (to the next leading order), from which all thermodynamic quantities can be determined. As examples, the dependence of particle density on the anyon statistics and the anyon anti-anyon interference in the ideal gas are exhibited. We also calculate two and three-point correlation functions, and uncover certain physical features of the system in thermal equilibrium. 
  We apply The Batalin-Tyutin constraint formalism of converting a second class system into a first class system for the rotational quantisation of the SU(2) Skyrme model. We obtain the first class constraint and the Hamiltonian in the extended phase space. The vacuum functional is constructed and evaluated in the unitary gauge and a multiplier dependent gauge. Finally, we discuss the spectrum of the extended theory. The use of the BT formalism on the collective coordinates quantisation of the SU(2) Skyrme model leads an additional term in the usual quantum Hamiltonian that can improve the phenomenology predicted by the Skyrme model. 
  The multiplets that occur in four dimensional rigidly supersymmetric theories can be described either by chiral superfields in Minkowski superspace or analytic superfields in harmonic superspace. The superconformal Ward identities for Green's functions of gauge invariant operators of these types are derived. It is shown that there are no chiral superconformal invariants. It is further shown that the Green's functions of analytic operators are severely restricted by the superconformal Ward when analyticity is taken into account. 
  The most general Lagrangian for non-linear electrodynamics coupled to an axion $a$ and a dilaton $\phi$ with $SL(2,\mbox{\elevenmsb R})$ invariant equations of motion is $$ -\half\left(\nabla\phi\right)^2 - \half e^{2\phi}\left(\nabla a\right)^2 + \fraction{1}{4}aF_{\mu\nu}\star F^{\mu\nu} + L_{\rm inv}(g_{\mu\nu},e^{-\frac{1}{2}\phi}F_{\rho\sigma}) $$ where $L_{\rm inv}(g_{\mu\nu},F_{\rho\sigma})$ is a Lagrangian whose equations of motion are invariant under electric-magnetic duality rotations. In particular there is a unique generalization of Born-Infeld theory admitting $SL(2,\mbox{\elevenmsb R})$ invariant equations of motion. 
  A discussion of the field content of quadratic higher-derivative gravitation is presented, together with a new example of a massless spin-two field consistently coupled to gravity. The full quadratic gravity theory is shown to be equivalent to a canonical second-order theory of a massive scalar field, a massive spin-two symmetric tensor field and gravity. The conditions showing that the tensor field describes only spin-two degrees of freedom are derived. A limit of the second-order theory provides a new example of massless spin-two field consistently coupled to gravity. A restricted set of vacua of the second-order theory is also discussed. It is shown that flat-space is the only stable vacuum of this type, and that the spin-two field around flat space is unfortunately always ghost-like. 
  Vacuum energies are computed in light-cone field theories to obtain effective potentials which determine vacuum condensate. Quantization surfaces interpolating between the light-like surface and the usual spatial one are useful to define the vacuum energies unambiguously. The Gross-Neveu, SU(N) Thirring, and O(N) vector models are worked out in the large $N$ limit. The vacuum energies are found to be independent of the interpolating angle to define the quantization surface. Renormalization of effective potential is explicitly performed. As an example of the case with nonconstant order parameter, two-dimensional QCD is also studied. Vacuum energies are explicitly obtained in the large $N$ limit which give the gap equation as the stationary point. 
  In order to construct a massive tensor theory with a smooth massless limit, we apply the Batalin-Fradkin algorithm to the ordinary massive tensor theory. By introducing an auxiliary vector field all second-class constraints are converted into first-class ones. We find a gauge-fixing condition which produces a massive tensor theory of desirable property. 
  We discuss the duality symmetries of Type II string effective actions in nine, ten and eleven dimensions. As a by-product we give a covariant action underlying the ten--dimensional Type IIB supergravity theory. We apply duality symmetries to construct dyonic Type II string solutions in six dimensions and their reformulation as solutions of the ten--dimensional Type IIB theory in ten dimensions. 
  Computation of the holomorphic $F_1$-function describing one-loop gravitational couplings of vectormultiplets is shown to confirm string-string duality of the proposed dual pair consisting of the heterotic string on $K3\times T^2$ with gauge group $U(1)^4$ and the type IIA string on the Calabi-Yau $WP^4_{1,1,2,8,12}(24)$. 
  A discussion of an extended class of higher-derivative classical theories of gravity is presented. A procedure is given for exhibiting the new propagating degrees of freedom, at the full non-linear level, by transforming the higher-derivative action to a canonical second-order form. For general fourth-order theories, described by actions which are general functions of the scalar curvature, the Ricci tensor and the full Riemann tensor, it is shown that the higher-derivative theories may have multiple stable vacua. The vacua are shown to be, in general, non-trivial, corresponding to deSitter or anti-deSitter solutions of the original theory. It is also shown that around any vacuum the elementary excitations remain the massless graviton, a massive scalar field and a massive ghost-like spin-two field. The analysis is extended to actions which are arbitrary functions of terms of the form $\nabla^{2k}R$, and it is shown that such theories also have a non-trivial vacuum structure. 
  This talk is divided into two parts. The first part reviews some of the duality relationships between superstring theories. These relationships are interpreted as providing evidence for the existence of a unique underlying fundamental theory. The second part describes my recent work on the SL(2,Z) duality group of the type IIB superstring theory in ten dimensions and its interpretation in terms of a possible theory of supermembranes in eleven dimensions. 
  We attempt to understand the fate of spacelike gravitational singularities in string theory via the quantum stress tensor for string matter in a fixed background. We first approximate the singularity with a homogeneous anisotropic background and review the minisuperspace equations describing the evolution of the scale factors and the dilaton. We then review and discuss the behavior of large strings in such models. In a simple model which expands isotropically for a finite period of time we compute the number density of strings produced by quantum pair production and find that this number, and thus the stress tensor, becomes infinite when the Hubble volume of the expansion exceeds the string scale, in a manner reminiscent of the Hagedorn transition. Based on this calculation we argue that either the region near the singularity undergoes a phase transition when the density reaches the order of a string mass per string volume, or that the backreaction of the produced string matter dramatically modifies the geometry. 
  A general formula for the canonical partition function for a system obeying any statistics based on the permutation group is derived. The formula expresses the canonical partition function in terms of sums of Schur functions. The only hitherto known result due to Suranyi [ Phys. Rev. Lett. {\bf 65}, 2329 (1990)] for parasystems of order two is shown to arise as a special case of our general formula. Our results also yield all the relevant information about the structure of the Fock spaces for parasystems. 
  We investigate a 4D analog of 2D WZW theory. The theory turns out to have surprising finiteness properties and an infinite-dimensional current algebra symmetry. Some correlation functions are determined by this symmetry. One way to define the theory systematically proceeds by the quantization of moduli spaces of holomorphic vector bundles over algebraic surfaces. We outline how one can define vertex operators in the theory. Finally, we define four-dimensional ``conformal blocks'' and present an analog of the Verlinde formula. 
  We present a succinct way of obtaining all possible higher dimensional generalization of Quantum Yang-Baxter Equation (QYBE). Using the scheme, we could generate the two popular three-simplex equations, namely: Zamolodchikov's tetrahedron equation (ZTE) and Frenkel and Moore equation (FME). 
  In the context of integrable field theory with boundary, the integrable non-linear sigma models in two dimensions, for example, the $O(N)$, the principal chiral, the ${\rm CP}^{N-1}$ and the complex Grassmannian sigma models are discussed on a half plane. In contrast to the well known cases of sine-Gordon, non-linear Schr\"odinger and affine Toda field theories, these non-linear sigma models in two dimensions are not classically integrable if restricted on a half plane. It is shown that the infinite set of non-local charges characterising the integrability on the whole plane is not conserved for the free (Neumann) boundary condition. If we require that these non-local charges to be conserved, then the solutions become trivial. 
  Using the Schwinger-Keldysh (closed time path or CTP) and Feynman-Vernon influence functional formalisms we obtain a Langevin equation for the description of the charged particle creation in electric field and of backreaction of charged quantum fields and their fluctuations on time evolution of this electric field. We obtain an expression for the influence functional in terms of Bogoliubov coefficients for the case of quantum electrodynamics with spin 0 charged particles. Then we derive a CTP effective action in semiclassical approximation and its cumulant expansion. An intimate connection between CTP effective action and decoherence functional will allow us to analyze how macroscopic electromagnetic fields are ``measured'' through interaction with charges and thereby rendered classical. 
  Recently Borchers has shown that in a theory of local observables, certain unitary and antiunitary operators, which are obtained from an elementary construction suggested by Bisognano and Wichmann, commute with the translation operators like Lorentz boosts and \pct-operators, respectively. We conclude from this that as soon as the operators considered implement {\em any} symmetry, this symmetry can be fixed up to at most some translation. As a symmetry, we admit any unitary or antiunitary operator under whose adjoint action any algebra of local observables is mapped onto an algebra which can be localized somewhere in Minkowski space. 
  The scattering properties of the non-linear $O(3)$ model in (2+1)-D, modified by the addition of both a potential-like term and a Skyrme-like term, are considered. Most of the work is numerical. The skyrmion-scattering is found to be quasi-elastic, the skyrmions' energy density profiles remaining unscathed after collisions. In low-energy processes the skyrmions exhibit back-scattering, while at larger energies they scatter at right angles. These results confirm those obtained in previous investigations, in which a similar problem was studied for a different choice of the potential-like term. 
  In Abelian theories of monopoles the magnetic charge is required to be enormous. Using the electric-magnetic duality of electromagnetism it is argued that the existence of such a large, non-perturbative magnetic coupling should lead to a phase transition where magnetic charge is permanently confined and the photon becomes massive. The apparent masslessness of the photon could then be used as an argument against the existence of such a large, non-perturbative magnetic charge. Finally it is shown that even in the presence of this dynamical mass generation the Cabbibo-Ferrari formulation of magnetic charge gives a consistent theory. 
  We construct the path integral for one-dimensional non-linear sigma models, starting from a given Hamiltonian operator and states in a Hilbert space. By explicit evaluation of the discretized propagators and vertices we find the correct Feynman rules which differ from those often assumed. These rules, which we previously derived in bosonic systems \cite{paper1}, are now extended to fermionic systems. We then generalize the work of Alvarez-Gaum\'e and Witten \cite{alwi} by developing a framework to compute anomalies of an $n$-dimensional quantum field theory by evaluating perturbatively a corresponding quantum mechanical path integral. Finally, we apply this formalism to various chiral and trace anomalies, and solve a series of technical problems: $(i)$ the correct treatment of Majorana fermions in path integrals with coherent states (the methods of fermion doubling and fermion halving yield equivalent results when used in applications to anomalies), $(ii)$ a complete path integral treatment of the ghost sector of chiral Yang-Mills anomalies, $(iii)$ a complete path integral treatment of trace anomalies, $(iv)$ the supersymmetric extension of the Van Vleck determinant, and $(v)$ a derivation of the spin-$3\over 2$ Jacobian of Alvarez-Gaum\'{e} and Witten for Lorentz anomalies. 
  This is a reply to the above comment (hep-th/9509028). We argue that QED displays a class of symmetries which may be used to select out the various velocity dependent superselection sectors. 
  We report on the formulation of $N=2$ $D=4$ supergravity coupled to $n_V$ abelian vector multiplets in presence of electric and magnetic charges. General formulae for the (moduli dependent) electric and magnetic charges for the $n_V+1$ gauge fields are given which reflect the symplectic structure of the underlying special geometry. The specification to Type IIB strings compactified on Calabi-Yau manifolds, with gauge group $U(1)^{h_{21}+1}$ is given. 
  After the work of Seiberg and Witten, it has been seen that the dynamics of N=2 Yang-Mills theory is governed by a Riemann surface $\Sigma$. In particular, the integral of a special differential $\lambda_{SW}$ over (a subset of) the periods of $\Sigma$ gives the mass formula for BPS-saturated states. We show that, for each simple group $G$, the Riemann surface is a spectral curve of the periodic Toda lattice for the dual group, $G^\vee$, whose affine Dynkin diagram is the dual of that of $G$. This curve is not unique, rather it depends on the choice of a representation $\rho$ of $G^\vee$; however, different choices of $\rho$ lead to equivalent constructions. The Seiberg-Witten differential $\lambda_{SW}$ is naturally expressed in Toda variables, and the N=2 Yang-Mills pre-potential is the free energy of a topological field theory defined by the data $\Sigma_{\gg,\rho}$ and $\lambda_{SW}$. 
  The exact solution of $N=2$ supersymmetric $SU(N)$ Yang-Mills theory is studied in the framework of the Whitham hierarchies. The solution is identified with a homogeneous solution of a Whitham hierarchy. This integrable hierarchy (Whitham-Toda hierarchy) describes modulation of a quasi-periodic solution of the (generalized) Toda lattice hierarchy associated with the hyperelliptic curves over the quantum moduli space. The relation between the holomorphic pre-potential of the low energy effective action and the $\tau$ function of the (generalized) Toda lattice hierarchy is also clarified. 
  The restrictions imposed on the strong force in the `non-commutative standard model' are examined. It is concluded that given the framework of non-commutative geometry and assuming the electroweak sector of the standard model many details of the strong force can be explained including its vectorial nature. 
  We introduce an approach for calculating the quantum loop corrections in the $\phi^4$ theory. Differential regularization and background-field method are essential tools and are used to calculate the effective action of the theory to two-loop order. Our approach is considerably simpler than other known methods and can be readily extended to higher-loop calculations and to other models. 
  We analyze the theory of softly broken supersymmetric $QCD$. Exotic behavior like spontaneously broken baryon number, massless composite fermions and Seiberg's duality seems to persist also in the presence of (small) soft supersymmetry breaking. We argue that certain, specially tailored, lattice simulations may be able to detect the novel phenomena. Most of the exotic behavior does not survive the decoupling limit of large SUSY breaking parameters. 
  We extend the definitions of characters and partition functions to the case of conformal field theories which contain operators with logarithmic correlation functions. As an example we consider the theories with central charge c = c(p,1) = 13-6(p+1/p), the ``border'' of the discrete minimal series. We show that there is a slightly generalized form of the property of rationality for such logarithmic theories. In particular, we obtain a classification of theories with c = c(p,1) which is similar to the A-D-E classification of c = 1 models. 
  A summary of the properties of the Wigner Clebsch-Gordan coefficients and isoscalar factors for the group SU3 in the SU2$\otimes$U1 decomposition is presented. The outer degeneracy problem is discussed in detail with a proof of a conjecture (Braunschweig's) which has been the basis of previous work on the SU3 coupling coefficients. Recursion relations obeyed by the SU3 isoscalar factors are produced, along with an algorithm which allows numerical determination of the factors from the recursion relations. The algorithm produces isoscalar factors which share all the symmetry properties under permutation of states and conjugation which are familiar from the SU2 case. The full set of symmetry properties for the SU3 Wigner-Clebsch-Gordan coefficients and isoscalar factors are displayed. 
  We analyze the constraints of general coordinate invariance for quantum theories possessing conformal symmetry in four dimensions. The character of these constraints simplifies enormously on the Einstein universe $R \times S^3$. The $SO(4,2)$ global conformal symmetry algebra of this space determines uniquely a finite shift in the Hamiltonian constraint from its classical value. In other words, the global Wheeler-De Witt equation is {\it modified} at the quantum level in a well-defined way in this case. We argue that the higher moments of $T^{00}$ should not be imposed on the physical states {\it a priori} either, but only the weaker condition $\langle \dot T^{00} \rangle = 0$. We present an explicit example of the quantization and diffeomorphism constraints on $R \times S^3$ for a free conformal scalar field. 
  The conformal factor of the spacetime metric becomes dynamical due to the trace anomaly of matter fields. Its dynamics is described by an effective action which we quantize by canonical methods on the Einstein universe $R\times S^3$. We find an infinite tower of discrete states which satisfy the constraints of quantum diffeomorphism invariance. These physical states are in one-to-one correspondence with operators constructed by integrating integer powers of the Ricci scalar. 
  A Lagrangian definition of a large family of (0,2) supersymmetric conformal field theories may be made by an appropriate gauge invariant combination of a gauged Wess-Zumino-Witten model, right-moving supersymmetry fermions, and left-moving current algebra fermions. Throughout this paper, use is made of the interplay between field theoretic and algebraic techniques (together with supersymmetry) which is facilitated by such a definition. These heterotic coset models are thus studied in some detail, with particular attention paid to the (0,2) analogue of the N=2 minimal models, which coincide with the `monopole' theory of Giddings, Polchinski and Strominger. A family of modular invariant partition functions for these (0,2) minimal models is presented. Some examples of N=1 supersymmetric four dimensional string theories with gauge groups E_6 X G and SO(10) X G are presented, using these minimal models as building blocks. The factor G represents various enhanced symmetry groups made up of products of SU(2) and U(1). 
  The influence of world-sheet boundary condensates on the toroidal compactification of bosonic string theories is considered. At the special points in the moduli space at which the closed-string theory possesses an enhanced unbroken $G\times G$ symmetry (where $G$ is a semi-simple product of simply laced groups) a scalar boundary condensate parameterizes the coset $G\times G/G$. Fluctuations around this background define an open-string generalization of the corresponding chiral nonlinear sigma model. Tree-level scattering amplitudes of on-shell massless states (\lq pions') reduce to the amplitudes of the principal chiral model for the group $G$ in the low energy limit. Furthermore, the condition for the vanishing of the renormalization group beta function at one loop results in the familiar equation of motion for that model. The quantum corrections to the open-string theory generate a mixing of open and closed strings so that the coset-space pions mix with the closed-string $G\times G$ gauge fields, resulting in a Higgs-like breakdown of the symmetry to the diagonal $G$ group. The case of non-oriented strings is also discussed. 
  The stochastic quantization of the fermion field is performed starting from Dirac equations. The statistical properties of stochastic terms in Langevin equations are described by explicit formulae of a Markov process. The interaction of the field is introduced as correlation of the stochastic terms. In the long time limit free fermions disappear and proper combinations of field components propagate as a scalar boson field. The existence and uniqueness of the long time limit is proved in the first order approximation of stochastic Liouville equation. 
  We propose to use a novel master Lagrangian for performing the bosonization of the $D$-dimensional massive Thirring model in $D=d+1 \ge 2$ dimensions. It is shown that our master Lagrangian is able to relate the previous interpolating Lagrangians each other which have been recently used to show the equivalence of the massive Thirring model in (2+1) dimensions with the Maxwell-Chern-Simons theory. Starting from the phase-space path integral representation of the master Lagrangian, we give an alternative proof for this equivalence up to the next-to-leading order in the expansion of the inverse fermion mass. Moreover, in (3+1)-dimensional case, the bosonized theory is shown to be equivalent to the massive antisymmetric tensor gauge theory. As a byproduct, we reproduce the well-known result on bosonization of the (1+1)-dimensional Thirring model following the same strategy. Finally a possibility of extending our strategy to the non-Abelian case is also discussed. 
  We discuss the general form of quadratic (1,1) supergravity in two dimensions, and show that this theory is equivalent to two scalar supermultiplets coupled to non-trivial supergravity. It is demonstrated that the theory possesses stable vacua with vanishing cosmological constant which spontaneously break supersymmetry. 
  We determine the vacuum structure of N=2 supersymmetric QCD with fundamental quarks for gauge groups SO(n) and Sp(2n), extending prior results for SU(n). The solutions are all given in terms of families of hyperelliptic Riemann surfaces of genus equal to the rank of the gauge group. In the scale invariant cases, the solutions all have exact S-dualities which act on the couplings by subgroups of PSL(2,Z) and on the masses by outer automorphisms of the flavor symmetry. They are shown to reproduce the complete pattern of symmetry breaking on the Coulomb branch and predict the correct weak--coupling monodromies. Simple breakings with squark vevs provide further consistency checks involving strong--coupling physics. 
  Families of hyper-elliptic curves which describe the quantum moduli spaces of vacua of $N=2$ supersymmetric $SO(N_c)$ gauge theories coupled to $N_f$ flavors of quarks in the vector representation are constructed. The quantum moduli spaces for $2N_f < N_c-1$ are determined completely by imposing $R$-symmetry, instanton corrections and the proper classical singularity structure. These curves are verified by residue calculations. The quantum moduli spaces for $2N_f\geq N_c-1$ theories are parameterized and their general structure is worked out using residue calculations. The exact metrics on the quantum moduli spaces as well as the exact spectrum of stable massive states are derived. The results presented here together with recent results of Martinec and Warner provide a natural conjecture for the form of the curves for the other gauge groups. 
  The generalized Drinfeld-Sokolov construction of KdV systems is reviewed in the case of an arbitrary affine Lie algebra paying particular attention to Hamiltonian aspects and $\W$-algebras. Some extensions of known results as well as a new interpretation of the construction are also presented. 
  A pseudoclassical theory of Weyl particle in the space--time dimensions $D=2n$ is constructed. The canonical quantization of that pseudoclassical theory is carried out and it results in the theory of the $D=2n$ dimensional Weyl particle in the Foldy--Wouthuysen representation. A quantum mechanics of the neutral Weyl particle in even--dimensional space--time is suggested and the connection of this theory with the theory of Mayorana--Weyl particle in QFT is discussed for $D=10$. 
  We point out that averages of equivariant observables of 2D topological gravity are not globally defined forms on moduli space, when one uses the functional measure corresponding to the formulation of the theory as a 2D superconformal model. This is shown to be a consequence of the existence of the Gribov horizon {\it and} of the dependence of the observables on derivatives of the super-ghost field. By requiring the absence of global BRS anomalies, it is nevertheless possible to associate global forms to correlators of observables by resorting to the \v{C}ech-De Rham notion of form cohomology. To this end, we derive and solve the ``descent'' of local Ward identities which characterize the functional measure. We obtain in this way an explicit expression for the \v{C}ech-De Rham cocycles corresponding to arbitrary correlators of observables. This provides the way to compute and understand contact terms in string theory from first principles. 
  The effective Lagrangian of arbitrary varying in space electromagnetic field in a dense medium is derived. It has been used for investigation of interaction between charged fermions in the medium. It is shown the possibility for the formation of metastable electron bound states in the medium when external magnetic field is applied. 
  Based on our previous studies of the BRST cohomology of the critical N=2 strings, we construct the loop measure and make explicit the role of the spectral flow at arbitrary genus and Chern class, in a holomorphic field basis. The spectral flow operator attributes to the existence of the hidden `small' N=4 superconformal symmetry which is non-linearly realized. We also discuss the symmetry properties of N=2 string amplitudes on locally-flat backgrounds. 
  Some years ago, one of the authors~(MM) revived a concept to which he gave the name of single-particle Dirac oscillator, while another~(CQ) showed that it corresponds to a realization of supersymmetric quantum mechanics. The Dirac oscillator in its one- and many-body versions has had a great number of applications. Recently, it included the analytic expression for the eigenstates and eigenvalues of a two-particle system with a new type of Dirac oscillator interaction of frequency~$\omega$. By considering the latter together with its partner corresponding to the replacement of~$\omega$ by~$-\omega$, we are able to get a supersymmetric formulation of the problem and find the superalgebra that explains its degeneracy. 
  We analyze (3+1)-dimensional black-hole space-times in spontaneously broken Yang-Mills gauge theories that have been recently presented as candidates for an evasion of the scalar-no-hair theorem. Although we show that in principle the conditions for the no-hair theorem do not apply to this case, however we prove that the `spirit' of the theorem is not violated, in the sense that there exist instabilities, in both the sphaleron and gravitational sectors. The instability analysis of the sphaleron sector, which was expected to be unstable for topological reasons, is performed by means of a variational method. As shown, there exist modes in this sector that are unstable against linear perturbations. Instabilities exist also in the gravitational sector. A method for counting the gravitational unstable modes, which utilizes a catastrophe-theoretic approach is presented. The r\^ole of the catastrophe functional is played by the mass functional of the black hole. The Higgs vacuum expectation value (v.e.v.) is used as a control parameter, having a critical value beyond which instabilities are turned on. The (stable) Schwarzschild solution is then understood from this point of view. The catastrophe-theory appproach facilitates enormously a universal stability study of non-Abelian black holes, which goes beyond linearized perturbations. Some elementary entropy considerations are also presented... 
  The Tricritical Ising model perturbed by the subleading energy operator \Phi_(3/5) was known to be an Integrable Scattering Theory of massive kinks and in fact preserves supersymmetry. We consider here the model defined on the half-plane with a boundary and computed the associated factorizable boundary S-matrix. The conformal boundary conditions of this model were identified and the corresponding S-matrices were found. We also show how some of these S-matrices can be perturbed and generate ``flows'' between different boundary conditions. 
  Effective theories with the matter content of the minimal supersymmetric Standard Model below the string scale M_string predict a wrong value for the weak--mixing angle sin^2\theta_W and strong coupling constant \alpha_S at the scale M_Z. To resolve this problem one needs large threshold corrections. At the same time one would like to avoid introducing new intermediate scales that are small compared to M_string. Two requests which seem to be incompatible. We show how both requirements can be satisfied in a class of (0,2) heterotic superstring compactifications with a natural choice of the vevs of the moduli fields entering the moduli dependent string threshold corrections. 
  Whilst many solutions have been found for the Quantum Yang-Baxter Equation (QYBE), there are fewer known solutions available for its higher dimensional generalizations: Zamolodchikov's tetrahedron equation (ZTE) and Frenkel and Moore's simplex equation (FME). In this paper, we present families of solutions to FME which may help us to understand more about higher dimensional generalization of QYBE. 
  A non--commutative analogue of the classical differential forms is constructed on the phase--space of an arbitrary quantum system. The non--commutative forms are universal and are related to the quantum mechanical dynamics in the same way as the classical forms are related to classical dynamics.  They are constructed by applying the Weyl--Wigner symbol map to the differential envelope of the linear operators on the quantum mechanical Hilbert space. This leads to a representation of the non--commutative forms considered by A.~Connes in terms of multiscalar functions on the classical phase--space. In an appropriate coincidence limit they define a quantum deformation of the classical tensor fields and both commutative and non--commutative forms can be studied in a unified framework. We interprete the quantum differential forms in physical terms and comment on possible applications. 
  In this paper the stabilization of 2D quantum Gravity by branching interactions is considered. The perturbative expansion and the first nonperturbative term of the stabilized model are the same than the unbounded matrix model which define pure Gravity, but it has new nonperturbative effects that survives in the continuum limit. 
  The formalism for a non-abelian pure gauge theory in (2+1) dimensions has recently been derived within Discretized Light-Cone Quantization, restricting to the lowest {\it transverse} momentum gluons. It is argued why this model can be a paradigm for full QCD. The physical vacuum becomes non-trivial even in light-cone quantization. The approach is brought here to tractable form by suppressing by hand both the dynamical gauge and the constraint zero mode, and by performing a Tamm-Dancoff type Fock-space truncation. Within that model the Hamiltonian is diagonalized numerically, yielding mass spectra and wavefunctions of the glue-ball states. We find that only color singlets have a stable and discrete bound state spectrum. The connection with confinement is discussed. The structure function of the gluons has a shape like $ [{x(1-x)}] ^{1\over 3} $. The existence of the continuum limit is verified by deriving a coupled set of integral equations. 
  We study a field--theoretical analogue of the Aharonov--Bohm effect in the Abelian Higgs Model: the corresponding topological interaction is proportional to the linking number of the Abrikosov--Nielsen--Olesen string world sheets and the particle world trajectory. The creation operators of the strings are explicitly constructed in the path integral and in the Hamiltonian formulation of the theory. We show that the Aharonov--Bohm effect gives rise to several nontrivial commutation relations. We also study the Aharonov--Bohm effect in the lattice formulation of the Abelian Higgs Model. It occurs that this effect gives rise to a nontrivial interaction of tested charged particles. 
  The realizations of the Lie algebra corresponding to the dynamical symmetry group SO(2,1) of the Schr\"{o}dinger equations for the Morse and the $V=u^2+1/u^2$ potentials were known to be related by a canonical transformation. q-deformed analog of this transformation connecting two different realizations of the sl_q(2) algebra is presented. By the virtue of the q-canonical transformation a q-deformed Schr\"{o}dinger equation for the Morse potential is obtained from the q-deformed $V=u^2+ 1/u^2$ Schr\"{o}dinger equation. Wave functions and eigenvalues of the q-Schr\"{o}dinger equations yielding a new definition of the q-Laguerre polynomials are studied. 
  Some of the properties of string theory defined on world-sheets with boundaries are reviewed. Particular emphasis is put on the possibility of identifying string configurations (\lq D-instantons' and \lq D-branes') that give rise to stringy non-perturbative effects. 
  We show that Dirichlet-branes, extended objects defined by mixed Dirichlet-Neumann boundary conditions in string theory, break half of the supersymmetries of the type~II superstring and carry a complete set of electric and magnetic Ramond-Ramond charges. We also find that the product of the electric and magnetic charges is a single Dirac unit, and that the quantum of charge takes the value required by string duality. This is strong evidence that the Dirchlet-branes are intrinsic to type II string theory and are the Ramond-Ramond sources required by string duality. We also note the existence of a previously overlooked 9-form potential in the IIa string, which gives rise to an effective cosmological constant of undetermined magnitude. 
  Effective superpotentials are considered which include glueball chiral superfields among their arguments in supersymmetric gauge theories. It is seen that the accommodation of glueball superfields is suitable for their complete determination. 
  Starting from the structural similarity between the quantum theory of gauge systems and that of the Kepler problem, an SU(2) gauge description of the five-dimensional Kepler problem is given. This non-abelian gauge system is used as a testing ground for the application of an algebraic constraint quantization scheme which can be formulated entirely in terms of observable quantities. For the quantum mechanical reduction only the quadratic Casimir of the constraint algebra, interpreted as an observable, is needed. 
  A general method of finding functional determinants is presented that depends on the asymptotic behaviour of the resolvent. Its application to the case of a bounded trihedral corner for which the eigenvalues are known only implicitly is outlined and a generalisation of Barnes' product form of multiple gamma functions is given 
  In this paper I review the multiplet calculus of $N = 1$, $D = 1$ local supersymmetry with applications to the construction of models for spinning particles in background fields, and models with space-time supersymmetry. New features include a non-linear realization of the local supersymmetry algebra and the coupling to anti-symmetric tensor fields of both odd and even rank. The non-linear realization allows the construction of a $D = 1$ cosmological-constant term, which provides a mass term in the equations of motion. 
  We discuss non-perturbative aspects of string effective field theories with N=1 supersymmetry in four dimensions. By the use of a scalar potential, which is on-shell invariant under the supersymmetric duality of the dilaton, we study gaugino condensation in (2,2) symmetric Z_{N} orbifold compactifications. The duality under consideration relates a two-form antisymmetric tensor to a pseudoscalar. We show, that our approach is independent of the superfield-representation of the dilaton and preserves the U(1)_{PQ} Peccei-Quinn symmetry exactly. 
  Lectures given by C.G. in the Advanced School on Effective Theories (Almu\~{n}ecar, Granada, 1995) on duality in N=2 supersymmetric Yang-Mills, and the coupling to gravity. 
  The study of factorization in linearized gravity is extended to the graviton scattering processes with an electron for the massive vector boson productions such as $g e \rightarrow Z e$ and $g e \rightarrow W \nu_e$. It is shown that every transition amplitude is completely factorized due to gravitational gauge invariance and Lorentz invariance. Also the explicit values of vector boson polarizations are obtained. 
  I review the k-factorization method to combine the high-energy behaviour in QCD with the renormalization group. Resummation formulas for coefficient functions and anomalous dimensions are derived, and their applications to small-x scaling violations in structure functions are briefly discussed. 
  The aim of these notes is both to review the standard understanding of the Hawking effect, and to discuss the modifications to this understanding that might be required by new physics at short distances. The fundamentals of the Unruh effect are reviewed, and then the Hawking effect is explained as a ``gravitational Unruh effect", with particular attention to the state-dependence of this picture. The order of magnitude of deviations from the thermal spectrum of Hawking radiation is estimated under various hypotheses on physics at short distances. The behavior of black hole radiation in a linear model with altered short distance physics---the Unruh model---is discussed in detail. [Based on lectures given at the First Mexican School on Gravitation and Mathematical Physics, Guanajuato, December 1994.] 
  After recalling a few basic concepts from cosmology and string theory, I will outline the main ideas/assumptions underlying (our own group's approach to) string cosmology and show how these lead to the definition of a two-parameter family of ``minimal" models. I will then briefly explain how to compute, in terms of those parameters, the spectrum of scalar, tensor and electromagnetic perturbations, and mention their most relevant physical consequences. More details on the latter part of this talk can be found in Maurizio Gasperini's contribution to these proceedings. 
  Duality transformations with respect to rotational isometries relate supersymmetric with non-supersymmetric backgrounds in string theory. We find that non-local world-sheet effects have to be taken into account in order to restore supersymmetry at the string level. The underlying superconformal algebra remains the same, but in this case T-duality relates local with non-local realizations of the algebra in terms of parafermions. This is another example where stringy effects resolve paradoxes of the effective field theory. (Contribution to the proceedings of the Trieste conference on S-Duality and Mirror Symmetry; to appear in Nucl Phys B Proc Suppl) 
  One would expect spacetime to have a foam-like structure on the Planck scale with a very high topology. If spacetime is simply connected (which is assumed in this paper), the non-trivial homology occurs in dimension two, and spacetime can be regarded as being essentially the topological sum of $S^2\times S^2$ and $K3$ bubbles. Comparison with the instantons for pair creation of black holes shows that the $S^2\times S^2$ bubbles can be interpreted as closed loops of virtual black holes. It is shown that scattering in such topological fluctuations leads to loss of quantum coherence, or in other words, to a superscattering matrix $\$ $ that does not factorise into an $S$ matrix and its adjoint. This loss of quantum coherence is very small at low energies for everything except scalar fields, leading to the prediction that we may never observe the Higgs particle. Another possible observational consequence may be that the $\theta $ angle of QCD is zero without having to invoke the problematical existence of a light axion. The picture of virtual black holes given here also suggests that macroscopic black holes will evaporate down to the Planck size and then disappear in the sea of virtual black holes. 
  It is shown that the quantum master equation of the Field Antifield quantization method at one loop order can be translated into the requirement of a superfield structure for the action. The Pauli Villars regularization is implemented in this BRST superspace and the case of anomalous gauge theories is investigated. The quantum action, including Wess Zumino terms, shows up as one of the components of a superfield that includes the BRST anomalies in the other component. The example of W2 quantum gravity is also discussed. 
  The fluctuation determinant in the BPS monopole background is calculated in the finite--temperature SU(2) gauge theory. 
  Using the representation of the quantum group $SL_q$(2) by the Weyl ope\-ra\-tors of the canonical commutation relations in quantum mechanics, we construct and solve a new vertex model on a square lattice. Random variables on horizontal bonds are Ising variables, and those on the vertical bonds take half positive integer values. The vertices is subjected to a genera\-li\-zed form of the so-called ``ice-rule'', its property are studied in details and its free energy calculated with the method of quantum inverse scattering. Remarkably in analogy with the usual six-vertex model, there exists a ``Free-Fermion'' limit with a novel rich operator structure. The existing algebraic structure suggests a possible connection with a lattice neutral plasma of charges, via the Fermion-Boson correspondence. 
  We present the results of further analysis of the integrability properties of the $N=4$ supersymmetric KdV equation deduced earlier by two of us (F.D. \& E.I.) as a hamiltonian flow on $N=4$ $SU(2)$ superconformal algebra in the harmonic $N=4$ superspace. To make this equation and the relevant hamiltonian structures more tractable, we reformulate it in the ordinary $N=4$ and further in $N=2$ superspaces. In $N=2$ superspace it is given by a coupled system of evolution equations for a general $N=2$ superfield and two chiral and antichiral superfields, and involves two independent real parameters, $a$ and $b$. We construct a few first bosonic conserved charges in involution, of dimensions from 1 to 6, and show that they exist only for the following choices of the parameters: (i) $a= 4, \;b=0$; (ii) $a= -2,\; b= -6$; (iii) $a= -2, \;b= 6$. The same values are needed for the relevant evolution equations to be bi-hamiltonian. We demonstrate that these three options are related via $SU(2)$ transformations and actually amount to the $SU(2)$ covariant integrability condition found in the harmonic superspace approach. Our results provide a strong evidence that the unique $N=4$ $SU(2)$ super KdV hierarchy exists. Upon reduction to $N=2$ KdV, the above three possibilities cease to be equivalent. They give rise to the $a=4$ and $a=-2$ $N=2$ KdV hierarchies, which thus prove to be different truncations of the single $N=4$ $SU(2)$ KdV one. 
  World-sheet and spacetime supersymmetries that are manifest in some string backgrounds may not be so in their T-duals. Nevertheless, they always remain symmetries of the underlying conformal field theory. In previous work the mechanism by which T-duality destroys manifest supersymmetry and gives rise to non-local realizations was found. We give the general conditions for a 2-dim N=1 supersymmetric sigma-model to have non-local and hence non-manifest extended supersymmetry. We then examine T-duality as a mechanism of restoring manifest supersymmetry. This happens whenever appropriate combinations of non-local parafermions of the underlying conformal field theory become local due to non-trivial world-sheet effects. We present, in detail, an example arising from the model SU(2)/U(1) X SL(2,R)/U(1) and obtain a new exact 4-dim axionic instanton, that generalizes the SU(2) X U(1) semi-wormhole, and has manifest spacetime as well as N=4 world-sheet supersymmetry. In addition, general necessary conditions for abelian T-duality to preserve manifest N=4 world-sheet supersymmetry are derived and applied to WZW models based on quaternionic groups. We also prove some theorems for sigma-models with non-local N=4 world-sheet supersymmetry. 
  Basis states and generator matrix elements are given for the generic representation $(a,b)$ of $G_2$ in an $SU(3)$ basis. 
  We construct a world-line representation for the fermionic one-loop effective action with axial and also vector, scalar, and pseudo-scalar couplings. We use this expression to compute a few selected scattering amplitudes. These allow us to verify that our method yields the same results as standard field theory. In particular, we are able to reproduce the chiral anomaly. Our starting point is the second-order formulation for the Dirac fermion. We translate the second order expressions into a world-line action. 
  We construct a transformation that transforms perturbative states into states that implement Gauss's law for `pure gluonic' Yang-Mills theory and QCD. The fact that this transformation is not and cannot be unitary has special significance. Previous work has shown that only states that are unitarily equivalent to perturbative states necessarily give the same S-matrix elements as are obtained with Feynman rules. 
  Sogami recently proposed the new idea to express Higgs particle as a kind of gauge particle by prescribing the generalized covariant derivative with gauge and Higgs fields operating on quark and lepton fields. The field strengths for both the gauge and Higgs fields are defined by the commutators of the covariant derivative by which he could obtain the Yang-Mills Higgs Lagrangian in the standard model. Inspired by Sogami's work, we present a modification of our previous scheme to formulate the spontaneously broken gauge theory in non-commutative geometry on the discrete space; Minkowski space multiplied by two points space by introducing the generation mixing matrix in operation of the generalized derivative on the more fundamental fields a_i(x,y) which compose the gauge and Higgs fields. The standard model is reconstructed according to the modified scheme, which does not yields not only any special relations between the particle masses but also the special restriction on the Higgs potential. 
  The scheme previously proposed by the present authors is modified to incorporate the strong interaction by affording the direct product internal symmetry. We do not need to prepare the extra discrete space for the color gauge group responsible for the strong interaction to reconstruct the standard model and the left-right symmetric gauge model(LRSM). The approach based on non-commutative geometry leads us to presents many attractive points such as the unified picture of the gauge and Higgs field as the generalized connection on the discrete space; Minkowski space multipied by N-points discrete space. This approach leads us to unified picture of gauge and Higgs fields as the generalized connection. The standard model needs N=2 discrete space for reconstruction in this formalism. \lr is still alive as a model with the intermediate symmetry of the spontaneously broken SO(10) grand unified theory(GUT). N=3 discrete space is needed for the reconstruction of LRSM to include two Higgs bosons $\phi$ and $\xi$ which are as usual transformed as (2,2*,0)$ and (1,3,-2) under left-handed SU(2)x right-handed SU(2)x U(1), respectively. xi is responsible to make the right handed-neutrino Majorana fermion and so well explains the seesaw mechanism. Up and down quarks have the different masses through the vacuum expectation value of phi. 
  By regularizing the conical singularities by means of a segment of a sphere or pseudosphere and then taking the regulator to zero, we compute exactly the Faddeev--Popov determinant related to the conformal gauge fixing for a piece-wise flat surface with the topology of the sphere. The result is analytic in the opening angles of the conical singularities in the interval ($\pi$, $4\pi$) and in the smooth limit goes over to the continuum expression. The Riemann-Roch relation on the dimensions of ker$(L^{\dag}L)$ and ker$(LL^{\dag})$ is satisfied. 
  We consider a 2-parameter class of solvable closed superstring models which `interpolate' between Kaluza-Klein and dilatonic Melvin magnetic flux tube backgrounds. The spectrum of string states has similarities with Landau spectrum for a charged particle in a uniform magnetic field. The presence of spin-dependent `gyromagnetic' interaction implies breaking of supersymmetry and possible existence (for certain values of magnetic parameters) of tachyonic instabilities. We study in detail the simplest example of the Kaluza-Klein Melvin model describing a superstring moving in flat but non-trivial 10-d space containing a 3-d factor which is a `twisted' product of a 2-plane and an internal circle. We also discuss the compact version of this model constructed by `twisting' the product of the two groups in SU(2) x U(1) WZNW theory without changing the local geometry (and thus the central charge). We explain how the supersymmetry is broken by continuous `magnetic' twist parameters and comment on possible implications for internal space compactification models. (Contribution to the Proceedings of the 1995 Erice School "String Gravity and Physics at the Planck Scale") 
  It is sometimes said that there may be a unique algebraic theory independent of space-time topologies which underlies superstring and p-brane theories. In this paper, I construct some algebras using knot relations within the framework of event-symmetric string theory, and ask the question "Is string theory in knots?" 
  For the case of the single-O($N$)-vector linear sigma models the critical behaviour following from any $A_k$ singularity in the action is worked out in the double scaling limit $N \rightarrow \infty$, $f_r \rightarrow f_r^c$, $2 \leq r \leq k$. After an exact elimination of Gaussian degrees of freedom, the critical objects such as coupling constants, indices and susceptibility matrix are derived for all $A_k$ and spacetime dimensions $0 \leq D < 4$. There appear exceptional spacetime dimensions where the degree $k$ of the singularity $A_k$ is more strongly constrained than by the renormalizability requirement. 
  A complete analysis of the consequences of introducing a set of holonomic gauge fixing constraints (to fix the dynamics) into a singular Lagrangian is performed. It is shown in general that the dynamical system originated from the reduced Lagrangian erases all the information regarding the first class constraints of the original theory, but retains its second class. It is proved that even though the reduced Lagrangian can be singular, it never possesses any gauge freedom. As an application, the example of $n \cdot A = 0$ gauges in electromagnetism is treated in full detail. 
  We use the dual description proposed by Seiberg to calculate the pressure in the low temperature confined phase of $N=1$ supersymmetric QCD using perturbation theory to $O(g^3_m)$, where $g_m$ is the gauge coupling in the dual theory. Combining this result with the usual high temperature expansion based on asymptotic freedom, we study how the physics in the intermediate temperature regime depends on the relative size of the scale parameters in the two descriptions. In particular we explore the possibility of having a temperature range where both perturbation expansions are valid. 
  A new approach with BRST invariance is suggested to cure the degeneracy problem of ill defined path integrals in the path-integral calculationof quantum mechanical tunneling effects in which the problem arises due to the occurrence of zero modes. The Faddeev-Popov procedure is avoided and the integral over the zero mode is transformed in a systematic way into a well defined integral over instanton positions. No special procedure has to be adopted as in the Faddeev-Popov method in calculating the Jacobian of the transformation. The quantum mechanical tunneling for the Sine-Gordon potential is used as a test of the method and the width of the lowest energy band is obtained in exact agreement with that of WKB calculations. 
  We propose an alternative description of 2 dimensional Conformal Field Theory in terms of Quantum Inverse Scattering. It is based on the generalized KdV systems attached to $A_2^{(2)}$, yielding the classical limit of Virasoro as Poisson bracket structure. The corresponding T-system is shown to coincide with the one recently proposed by Kuniba and Suzuki. We classify the primary operators of the minimal models that commute with all the Integrals of Motion, and that are therefore candidates to perturb the model by keeping the conservation laws. For our $A_2^{(2)}$ structure these happen to be $\phi_{1,2},\phi_{2,1},\phi_{1,5}$, in contrast to the $A_1^{(1)}$ case, studied by Bazhanov, Lukyanov and Zamolodchikov~\cite{BLZ}, related to $\phi_{1,3}$. 
  We analyze the initial value problem for spinor fields obeying the Dirac equation, with particular attention to the characteristic surfaces. The standard Cauchy initial value problem for first order differential equations is to construct a solution function in a neighborhood of space and time from the values of the function on a selected initial value surface. On the characteristic surfaces the solution function may be discontinuous, so the standard Cauchy construction breaks down. For the Dirac equation the characteristic surfaces are null surfaces. An alternative version of the initial value problem may be formulated using null surfaces; the initial value data needed differs from that of the standard Cauchy problem, and in the case we here discuss the values of separate components of the spinor function on an intersecting pair of null surfaces comprise the necessary initial value data. We present an expression for the construction of a solution from null surface data; two analogues of the quantum mechanical Hamiltonian operator determine the evolution of the system. 
  We analyze the initial value problem for scalar fields obeying the Klein-Gordon equation. The standard Cauchy initial value problem for second order differential equation is to construct a solution function in a neighborhood of space and time form values of the function and its time derivative on a selected initial value surface. On the characteristic surfaces the time derivative of the solution function may be discontinuous, so the standard Cauchy construction breaks down. For the Klein-Gordon equation the characteristic surfaces are null surfaces. An alternative version of the initial data needed differs from that of the standard Cauchy problem, and in the case we discuss here the values of the function on an intersecting pair of null surfaces comprise the necessary initial value data. We also present an expression for the construction of a solution from null surface data; two analogues of the quantum mechanical Hamiltonian operator determine the evolution of the system. 
  For systems with first class constraints the reduction scheme to the gauge invariant variables is considered. The method is based on the analysis of restricted 1-forms in gauge invariant variables. This scheme is applied to the models of electrodynamics and Yang-Mills theory. For the finite dimensional model with $SU(2)$ gauge group of symmetry the possible mechanism of confinement is obtained. 
  Some inadequacy in the traditional description of the phenomenon of electro-magnetic field radiation created by a point charge moving along a straight line with an acceleration is found and discussed in this paper in detail. The possibility of simultaneous coexistence of Newton instantaneous long-range interaction and Faraday-Maxwell short-rang interaction is pointed out. 
  The inadequacy of Li\'{e}nard-Wiechert potentials is demonstrated as one of the examples related to the inconsistency of the conventional classical electrodynamics. The insufficiency of the Faraday-Maxwell concept to describe the whole electromagnetic phenomena and the incompleteness of a set of solutions of Maxwell equations are discussed and mathematically proved. Reasons of the introduction of the so-called ``electrodynamics dualism concept" (simultaneous coexistence of instantaneous Newton long-range and Faraday-Maxwell short-range interactions) have been displayed. It is strictly shown that the new concept presents itself as the direct consequence of the complete set of Maxwell equations and makes it possible to consider classical electrodynamics as a self-consistent and complete theory, devoid of inward contradictions. In the framework of the new approach, all main concepts of classical electrodynamics are reconsidered. In particular, a limited class of motion is revealed when accelerated charges do not radiate electromagnetic field. 
  There exist extremal p-brane solutions of $D\!=\!11$ supergravity for p=2~and~5. In this paper we investigate the zero modes of the membrane and the five-brane solutions as a first step toward understanding the full quantum theory of these objects. It is found that both solutions possess the correct number of normalizable zero modes dictated by supersymmetry. 
  Many new solitary wave solutions of the recently studied Lienard equation are obtained by mapping it to the field equation of the $\phi^6-$field theory. Further, it is shown that the exact solutions of the Lienard equation are also the exact solutions of the various perturbed soliton equations. Besides, we also consider a one parameter family of generalised Lienard equations and obtain exact solitary wave solutions of these equations and show that these are also the exact solutions of the various other generalised nonlinear equations. 
  We identify the exactly solvable theory of the conformal fixed point of (0,2) Calabi-Yau sigma-models and their Landau-Ginzburg phases. To this end we consider a number of (0,2) models constructed from a particular (2,2) exactly solvable theory via the method of simple currents. In order to establish the relation between exactly solvable (0,2) vacua of the heterotic string, (0,2) Landau-Ginzburg orbifolds, and (0,2) Calabi-Yau manifolds, we compute the Yukawa couplings in the exactly solvable model and compare the results with the product structure of the chiral ring which we extract from the structure of the massless spectrum of the exact theory. We find complete agreement between the two up to a finite number of renormalizations. For a particularly simple example we furthermore derive the generating ideal of the chiral ring from a (0,2) linear sigma-model which has both a Landau-Ginzburg and a (0,2) Calabi-Yau phase. 
  The axially symmetric non-local solution in the Heisenberg equation is found. It is regular in the whole space and has the finite energy on the unit of length according to this we may consider the solution as a string. Taking the non-local spherically symmetric solution, which was found by Finkelstein et. al., and our solution in account we suggest to consider the Heisenberg equation as a quantum equation for non-local objects (strings, flux tubes, membranes and so on). The received solution is used for the obtaining the meson model as a rotating string with the quark on its ends. 
  There exists an exact relationship between the quasi-exactly solvable problems of quantum mechanics and models of square and rectangular random complex matrices. This relationship enables one to reduce the problem of constructing topological ($1/N$) expansions in random matrix models to the problem of constructing semiclassical expansions for observables in quasi-exactly solvable problems. Lie algebraic aspects of this relationship are also discussed. 
  We argue that the gauge-fermion interaction in multiflavour quantum electrodynamics in $(2 + 1)$-dimensions is responsible for non-fermi liquid behaviour in the infrared, in the sense of leading to the existence of a non-trivial (quasi) fixed point that lies between the trivial fixed point (at infinite momenta) and the region where dynamical symmetry breaking and mass generation occurs. This quasi-fixed point structure implies slowly varying, rather than fixed, couplings in the intermediate regime of momenta, a situation which resembles that of (four-dimensional) `walking technicolour' models of particle physics. The inclusion of wave-function renormalization yields marginal $O(1/N)$-corrections to the `bulk' non-fermi liquid behaviour caused by the gauge interaction in the limit of infinite flavour number. Such corrections lead to the appearance of modified critical exponents. In particular, at low temperatures there appear to be logarithmic scaling violations of the linear resistivity of the system of order $O(1/N)$. Connection with the anomalous normal-state properties of certain condensed matter systems relevant for high-temperature superconductivity is briefly discussed. The relevance of the large (flavour) $N$ expansion to the fermi-liquid problem is emphasized. As a partial result of our analysis, we point out the absence of Charge-Density-Wave Instabilities from the effective low-energy theory, as a consequence of gauge invariance. 
  A review is presented of the recently obtained expressions for conformal blocks for {\it admissible} representations in $SL(2)$ current algebra based on the Wakimoto free field construction. In this realization one needs to introduce a second screening charge, one which depends on fractional powers of free fields. The techniques necessary to deal with these complications are developed, and explicit general integral representations for conformal blocks on the sphere are provided. The fusion rules are discussed and as a check it is verified that the conformal blocks satisfy the Knizhnik-Zamolodchikov equations. (Talk presented by J. Rasmussen at the Leuven workshop, July 10-14 1995, to appear in the proceedings) 
  In this paper we consider 2+1-dimensional gravity coupled to N point-particles. We introduce a gauge in which the $z$- and $\bar{z}$-components of the dreibein field become holomorphic and anti-holomorphic respectively. As a result we can restrict ourselves to the complex plane. Next we show that solving the dreibein-field: $e^a_z(z)$ is equivalent to solving the Riemann-Hilbert problem for the group $SO(2,1)$. We give the explicit solution for 2 particles in terms of hypergeometric functions. In the N-particle case we give a representation in terms of conformal field theory. The dreibeins are expressed as correlators of 2 free fermion fields and twistoperators at the position of the particles. 
  In a recent paper, the authors have shown that the secondary reduction of W-algebras provides a natural framework for the linearization of W-algebras. In particular, it allows in a very simple way the calculation of the linear algebra $W(G,H)_{\geq0}$ associated to a wide class of W(G,H) algebras, as well as the expression of the W generators of W(G,H) in terms of the generators of $W(G,H)_{\geq0}$. In this paper, we present the extension of the above technique to W-superalgebras, i.e. W-algebras containing fermions and bosons of arbitrary (positive) spins. To be self-contained the paper recall the linearization of W-algebras. We include also examples such as the linearization of W_n algebras; W(sl(3|1),sl(3)) and W(osp(1|4),sp(4)) = WB_2 superalgebras. 
  Motivated by the weak-strong coupling expansion \cite{Rosenstein}, we calculate the spectrum of hadrons using a systematic $1/d$ ($d$ - dimensionality of spacetime) in addition to a strong coupling expansion in $\beta$. The $1/d$ expansion is pushed to the next to leading order in ($1/d$) for mesons and next to next to leading order for baryons. We do the calculation using Wilson fermions with arbitrary $r$ and show that doublers decouple from the spectrum only when $r$ is close to the Wilson's value $r=1$. For these $r$ the spectrum is much closer to the lattice results and the phenomenological values than those obtained by using either the (nonsystematic) "randomwalk" approximation or the hopping parameter expansion. In particular, the value of the nucleon to $\rho$ - meson mass ratio is lowered to $\frac {3 \log d -1/4}{2 {\rm arccosh} 2}+O(1/d)\approx 1.48$. The result holds even for $\beta$ as large as 5, where the weak-strong coupling expansion is applicable and therefore these results are expected to be reasonable. 
  We study the $\lambda \phi^4$ field theory in a flat Robertson-Walker space-time using the functional Sch\"odinger picture. We introduce a simple Gaussian approximation to analyze the time evolution of pure states and we establish the renormalizability of the approximation. We also show that the energy-momentum tensor in this approximation is finite once we consider the usual mass and coupling constant renormalizations. 
  It is known that the partition function and correlators of the two-dimensional topological field theory $G_K(N)/ G_K(N)$ on the Riemann surface $\Sigma_{g,s}$ is given by Verlinde numbers, dim($V_{g,s,K}$) and that the large $K$ limit of dim($V_{g,s,K}$) gives Vol(${\cal M}_s$), the volume of the moduli space of flat connections of gauge group $G(N)$ on $\Sigma_{g,s}$, up to a power of $K$. Given this relationship, we complete the computation of Vol(${\cal M}_s$) using only algebraic results from conformal field theory. The group-level duality of $G(N)_K$ is used to show that if $G(N)$ is a classical group, then $\displaystyle \lim_{N\rightarrow \infty} G_K(N) / G_K(N)$ is a BF theory with gauge group $G(K)$. Therefore this limit computes Vol(${\cal M}^\prime_s$), the volume of the moduli space of flat connections of gauge group $G(K)$. 
  The full ``classical" Dirac-Maxwell equations are considered under various simplifying assumptions. A reduction of the equations is performed in the case when the Dirac field is {\em static} and a further reduction is performed in the case of {\em spherical symmetry}. These static spherically symmetric equations are examined in some detail and a numerical solution presented. Some surprising results emerge:  * Spherical symmetry necessitates the existence of a magnetic monopole.  * There exists a uniquely defined solution, determined only by the demand that the solution be analytic at infinity.  * The equations describe highly compact objects with an inner onion-like shell structure. 
  In this paper the relationship between the problem of constructing the ground state energy for the quantum quartic oscillator and the problem of computing mean eigenvalue of large positively definite random hermitean matrices is established. This relationship enables one to present several more or less closed expressions for the oscillator energy. One of such expressions is given in the form of simple recurrence relations derived by means of the method of orthogonal polynomials which is one of the basic tools in the theory of random matrices. 
  The vacuum polarization due to chiral fermions on a 4--dimensional Euclidean lattice is calculated according to the overlap prescription. The fermions are coupled to weak and slowly varying background gauge and Higgs fields, and the polarization tensor is given by second order perturbation theory. In this order the overlap constitutes a gauge invariant regularization of the fermion vacuum amplitude. Its low energy -- long wavelength behaviour can be computed explicitly and we verify that it coincides with the Feynman graph result obtainable, for example, by dimensional regularization of continuum gauge theory. In particular, the Standard Model Callan--Symanzik RG functions are recovered. Moreover, there are no residual lattice artefacts such as a dependence on Wilson--type mass parameters. 
  A C*-algebra of asymptotic fields which properly describes the infrared structure in quantum electrodynamics is proposed. The algebra is generated by the null asymptotic of electromagnetic field and the time asymptotic of charged matter fields which incorporate the corresponding Coulomb fields. As a consequence Gauss' law is satisfied in the algebraic setting. Within this algebra the observables can be identified by the principle of gauge invariance. A class of representations of the asymptotic algebra is constructed which resembles the Kulish-Faddeev treatment of electrically charged asymptotic fields. 
  SU(3) Einstein-Yang-Mills-dilaton theory possesses sequences of static spherically symmetric sphaleron and black hole solutions for the SU(2) and the SO(3) embedding. The solutions depend on the dilaton coupling constant $\gamma$, approaching the corresponding Einstein-Yang-Mills solutions for $\gamma \rightarrow 0$, and Yang-Mills-dilaton solutions in flat space for $\gamma \rightarrow \infty$. The sequences of solutions tend to Einstein-Maxwell-dilaton solutions with different magnetic charges. The solutions satisfy analogous relations between the dilaton field and the metric for general $\gamma$. Thermodynamic properties of the black hole solutions are discussed. 
  We consider the soliton solutions in 1- and (1+1)-dimensional Toda lattice models with a boundary. We make use of the solutions already known on a full line by means of the Hirota's method. We explicitly construct the solutions satisfying the boundary conditions. The ${\bf Z}_{\infty}$-symmetric boundary condition can be introduced by the two-soliton solutions naturally. 
  Classical integrability is investigated for affine Toda field theories in the presence of a constant background tensor field. This leads to a further set of discrete possibilities for integrable boundary conditions depending upon the time-derivative of the fields at the boundary but containing no free parameters other than the bulk coupling constant. 
  We review the new approach to the theory of nonlinear $W$-algebras which is developed recently and called {\it conformal linearization}. In this approach $W$-algebras are embedded as subalgebras into some {\it linear conformal} algebras with a finite set of currents and most of their properties could be understood in a much simpler way by studing their linear counterpart. The general construction is illustrated by the examples of $u(N)$-superconformal, $W(sl(N),sl(2))$, $W(sl(N),sl(N))$ as well as $W(sl(N),sl(3))$ algebras. Applications to the construction of realizations (included modulo null fields realizations) as well as central charge spectrum for minimal models of nonlinear algebras are discussed. (To appear in ``Geometry and Integrable Models'', Eds.: P.N.Pyatov & S.N.Solodukhin, World Scientific Publ. Co. (in press)). 
  Some general features of locally supersymmetric theories (N=1 in four dimensions) involving Chern-Simons forms and antisymmetric tensors are sketched out. The relevance of the three-form multiplet both for the description of Chern-Simons forms and the supersymmetry properties of the gaugino condensate is pointed out. 
  It is shown that the minimal Higgs sector of a generic N=2 supergravity theory with unbroken N=1 supersymmetry must contain a Higgs hypermultiplet and a vector multiplet. When the multiplets parametrize the quaternionic manifold SO(4,1)/SO(4), and the special Kahler manifold SU(1,1)/U(1), respectively, a vanishing vacuum energy with a sliding massive spin 3/2 multiplet is obtained. Potential applications to N=2 low energy effective actions of superstrings are briefly discussed. 
  We show how a general nonstandard Lax equation (supersymmetric or otherwise) can be expressed as a standard Lax equation. This enables us to define the Gelfand-Dikii brackets for a nonstandard supersymmetric equation. We discuss the Hamiltonian structures for the nonstandard super KP system and work out explicitly the two Hamiltonian structures of the supersymmetric Two Boson system from this point of view. 
  We investigate classical integrable spins defined on the reduced phase spaces of coadjoint orbits of $G= SU(N)$ and study quantum mechanics of them.  After discussions on a complete set of commuting functions on each orbit and construction of integrable spin models on the flag manifolds, we quantize a concrete example of integrable spins on SU(3) flag manifold in the coherent state quantization scheme and solve explicitly the time-dependent Schr\"odinger equation. 
  The Moyal quantization is described as a discretization of the classical phase space by using difference analogue of vector fields. Difference analogue of Lie brackets plays the role of Heisenberg commutators. 
  I present a criterion for all-order finiteness in $N=1$ SYM theories. The structure of the supercurrent anomaly, the Callan-Symanzik equation and the supersymmetric non-renormalization theorem for chiral anomalies are the essential ingredients of the proof. 
  We review the derivation and the basic properties of the perturbative prepotential in N=2 compactifications of the heterotic superstring. We discuss the structure of the perturbative monodromy group and the embedding of rigidly supersymmetric monodromies associated with enhanced gauge groups, at both perturbative and non-perturbative level. 
  We are discussing the $S$ \& $T$ duality for special class of heterotic string configurations. This class of solutions includes various types of black hole solutions and Taub-NUT geometries. It allows a self-dual point for both dualities which corresponds to massless configurations. As string state this point corresponds to $N_R=1/2$ and $N_L=0$. The string/string duality is shortly discussed. 
  A spectrum of relic stochastic gravitational radiation, strongly tilted towards high frequencies, and characterized by two basic parameters is shown to emerge in a class of string theory models. We estimate the required sensitivity for detection of the predicted gravitational radiation and show that a region of our parameter space is within reach for some of the plannedgravitational-wave detectors. 
  We show the OPE formulae for three types of deformed super-Virasoro algebras: Chaichian-Presnajder's deformation, Belov-Chaltikhian's one and its modified version. Fundamental (anti-)commutation relations toward a ghost realization of deformed super-Virasoro algebra are also discussed. 
  In the framework of noncommutative geometry we describe spinor fields with nonvanishing winding number on a truncated (fuzzy) sphere. The corresponding field theory actions conserve all basic symmetries of the standard commutative version (space isometries and global chiral symmetry), but due to the noncommutativity of the space the fields are regularized and they contain only finite number of modes. 
  We study the quantum integrability of nonsimply--laced affine Toda theories defined on the half--plane and explicitly construct the first nontrivial higher--spin charges in specific examples. We find that, in contradistinction to the classical case, addition of total derivative terms to the "bulk" current plays a relevant role for the quantum boundary conservation. 
  A calculation of the Aharonov-Bohm wave function is presented. The result is a series of confluent hypergeometric functions which is finite at the forward direction. 
  A proposed duality between type IIB superstring theory on R^9 X S^1 and a conjectured 11D fundamental theory (``M theory'') on R^9 X T^2 is investigated. Simple heuristic reasoning leads to a consistent picture relating the various p-branes and their tensions in each theory. Identifying the M theory on R^{10} X S^1 with type IIA superstring theory on R^{10}, in a similar fashion, leads to various relations among the p-branes of the IIA theory. 
  We raise the issue whether gauge theories, that are not renormalizable in the usual power-counting sense, are nevertheless renormalizable in the modern sense that all divergences can be cancelled by renormalization of the infinite number of terms in the bare action. We find that a theory is renormalizable in this sense if the {\em a priori} constraints that we impose on the form of the bare action correspond to the cohomology of the BRST transformations generated by the action. Recent cohomology theorems of Barnich, Brandt, and Henneaux are used to show that conventionally nonrenormalizable theories of Yang-Mills fields (such as quantum chromodynamics with heavy quarks integrated out) and/or gravitation are renormalizable in the modern sense. 
  Distler-Kachru models which yield three generations of chiral fermions with gauge group SO(10) are found. These models have mirror partners. 
  The antighost equation valid for usual gauge theories in the Landau gauge, is generalized to the case of $N=1$ supersymmetric gauge theories in a supersymmetric version of the Landau gauge. This equation, which expresses the nonrenormalization of the Faddeev-Popov ghost field, plays an important role in the proof of the nonrenormalization theorems for the chiral anomalies. 
  The Schwinger model (QED_2) with N flavors of massive fermions on a circle of circumference L, or equivalently at finite temperature T, is reduced to a quantum mechanical system of N-1 degrees of freedom. With degenerate fermion masses (m) the chiral condensate develops a cusp singularity at $\theta=\pm \pi$ in the limit L -> $\infty$ or T -> 0, which is removed by a large asymmetry in the fermion masses. Physical quantities sensitively depend on the parameter mL or m/T, and the m -> 0 and L -> $\infty$ (or T -> 0) limits do not commute. A detailed analysis is given for N=3. 
  In the final few years of his life, Julian Schwinger proposed that the ``dynamical Casimir effect'' might provide the driving force behind the puzzling phenomenon of sonoluminescence. Motivated by that exciting suggestion, I have computed the static Casimir energy of a spherical cavity in an otherwise uniform material with dielectric constant $\epsilon$ and permeability $\mu$. As expected the result is divergent; yet a plausible finite answer is extracted, in the leading uniform asymptotic approximation. That result gives far too small an energy to account for the large burst of photons seen in sonoluminescence. If the divergent result is retained (which is different from that guessed by Schwinger), it is of the wrong sign to drive the effect. Dispersion does not resolve this contradiction. However, dynamical effects are not yet included. 
  We discuss the non-abelian duality procedure for groups which do not act freely. As an example we consider Taub-NUT space, which has the local isometry group $SU(2) \otimes U(1)$. We dualise over the entire symmetry group as well as the subgroups $SO(3)$ and $U(1)$, presenting unusual new solutions to low energy string theory. The solutions obtained highlight the relationship between fixed points of an isometry in one solution and singular points in another. We also find the interesting results that, in this case, the $U(1)$ and $SO(3)$ $T$-duality procedures commute with each other, and that the extreme points of the $O(1,1)$ duality group for the time translations have special significance under the $SO(3)$ T-duality. 
  We report on a search for $N=2$ heterotic strings that are dual candidates of type II compactifications on Calabi-Yau threefolds described as $K3$ fibrations. We find many new heterotic duals by using standard orbifold techniques. The associated type II compactifications fall into chains in which the proposed duals are heterotic compactifications related one another by a sequential Higgs mechanism. This breaking in the heterotic side typically involves the sequence $SU(4)\rightarrow SU(3)\rightarrow $ $SU(2)\rightarrow 0$, while in the type II side the weights of the complex hypersurfaces and the structure of the $K3$ quotient singularities also follow specific patterns. 
  We argue that moduli in the adjoint representation of the standard- model gauge group are a natural feature of superstring models, and that they can account for the apparent discrepancy between the string and unification scales. 
  The functional relations of the transfer matrices of fusion hierachies for six- and eight-vertex models with open boundary conditions have been presented in this paper. We have shown the su($2$) fusion rule for the models with more general reflection boundary conditions, which are represented by off-diagonal reflection matrices. Also we have discussed some physics properties which are determined by the functional relations. Finally the intertwining relation between the reflection $K$ matrices for the vertex and SOS models is discussed. 
  Complete energy spectrum is obtained for the quantum mechanical problem of N one dimensional equal mass particles interacting via potential $$V(x_1,x_2,...,x_N) = g\sum^N_{i < j}{1\over (x_i-x_j)^2} - {\alpha\over \sqrt{\sum_{i < j} (x_i-x_j)^2}}$$ Further, it is shown that scattering configuration, characterized by initial momenta $p_i (i=1,2,...,N)$ goes over into a final configuration characterized uniquely by the final momenta $p'_i$ with $p'_i=p_{N+1-i}$. 
  We show that a four-parameter generating solution for a general class of four-dimensional, spherically-symmetric, static, dyonic BPS saturated solutions of leading-order effective equations of toroidally compactified heterotic or type II superstring theory are exact string solutions. The corresponding ten-dimensional background defines a conformal sigma-model which is a particular case of a `chiral null model' with curved `transverse' part. The exact conformal invariance is a consequence of the chiral null structure of the `electric' part of the model and the N=4 world-sheet supersymmetry of its transverse `magnetic' part. The sigma-model action has a remarkable covariance under both target space and the electromagnetic $S$-duality transformations, and it illustrates the relation between string-string duality in six dimensions and $S$-duality in four dimensions. In general, there exists a large class of exact six-dimensional superstring solutions described by chiral null models with four-dimensional transverse parts represented by N=4 supersymmetric sigma-models with metrics conformal to hyper-Kahler ones. 
  We summarize the results for four-dimensional Bogomol'nyi-Prasad- Sommerfield (BPS) saturated dyonic black hole solutions arising in the Kaluza-Klein sector and the three-form field sector of the eleven-dimensional supergravity on a seven-torus. These black hole solutions break $3\over 4$ of $N=8$ supersymmetry, fill out the multipletes of $U$-duality group, and the two classes of solutions are related to each other by a discrete symmetry transformation in $E_7$. Using the field redefinitions of the corresponding effective actions, we present these solutions in terms of fields describing classes of dyonic black holes carrying charges of $U(1)$ gauge fields in Neveu-Schwarz-Neveu-Schwarz and/or Ramond-Ramond sector(s) of the type-IIA superstring on a six-torus. We also summarize the dependence of their ADM masses on the asymptotic values of scalar fields, parameterizing the toroidal moduli space and the string coupling constant. 
  We incorporate Sogami's idea in the standard model into our previous formulation of non-commutative differential geometry by extending the action of the extra exterior derivative operator on spinors defined over the discrete space-time; four dimensinal Minkovski space multiplyed by two point discrete space. The extension consists in making it possible to require that the operator become nilpotent when acting on the spinors. It is shown that the generalized field strength leads to the most general, gauge-invariant Yang-Mills-Higgs Lagrangian even if the extra exterior derivative operator is not nilpotent, while the fermionic part remains intact. The proof is given for a single Higgs model. The method is applied to reformulate the standard model by putting left-handed fermion doublets on the upper sheet and right-handed fermion singlets on the lower sheet with generation mixing among quarks being taken into account. We also present a matrix calculus of the method without referring to the discrete space-time. 
  We show that non-chiral $N=2$ supergravity in ten-dimensions admits a family of dual actions where the one-form, two-form or three-form is replaced by the seven-form, six-form or five-form respectively. The dual actions and supersymmetry transformations are given. 
  The Coulomb branch of $N=2$ supersymmetric gauge theories in four dimensions is described in general by an integrable Hamiltonian system in the holomorphic sense. A natural construction of such systems comes from two-dimensional gauge theory and spectral curves. Starting from this point of view, we propose an integrable system relevant to the $N=2$ $SU(n)$ gauge theory with a hypermultiplet in the adjoint representation, and offer much evidence that it is correct. The model has an $SL(2,{\bf Z})$ $S$-duality group (with the central element $-1$ of $SL(2,{\bf Z})$ acting as charge conjugation); $SL(2,{\bf Z})$ permutes the Higgs, confining, and oblique confining phases in the expected fashion. We also study more exotic phases. 
  Yokoyama's gaugeon formalism is knwon to admit $q$-number gauge transformation. We introduce BRST symmetries into the formalism for the Yang-Mills gauge field. Owing to the BRST symmetry, Yokoyama's physical subsidiary conditions are replaced by a single condition of the Kugo-Ojima type. Our physical subsidiary condition is invariant under the $q$-number gauge transformation. Thus, our physical subspace is gauge invariant. 
  The interplay between T-duality and supersymmetry in string theory is explored. It is shown that T-duality is always compatible with supersymmetry and simply changes a local realization to a non-local one and vice versa. Non-local realizations become natural using classical parafermions of the underlying conformal field theory. Examples presented include hyper-kahler metrics and the backgrounds for the SU(2) X U(1) and SU(2)/U(1) X U(1) X U(1) exact conformal field theories. 
  We propose the tube model as a first step in solving the bound state problem in light-front QCD. In this approach we neglect transverse variations of the fields, producing a model with 1+1 dimensional dynamics. We then solve the two, three, and four particle sectors of the model for the case of pure glue SU(3). We study convergence to the continuum limit and various properties of the spectrum. 
  In this paper some quite simple examples of applications of the zeta-function regularization to superstring theories are presented. It is shown that the Virasoro anomaly in the BRST formulation of (super)strings can be directly computed from the original expressions of the operators as well as normal ordering constants and masses of ground levels. Hawking's zeta regularization is recognized as an efficient tool for direct calculations, bringing no ambiguities.    Possible implications for global GSO operators' phases definitions (maybe ensuring modular invariance) will be discussed elsewhere. 
  Two-dimensional sigma models are defined for the new manifestly spacetime supersymmetric description of four-dimensional compactified superstrings. The resulting target-superspace effective action is constrained by the way the spacetime dilaton couples to the worldsheet curvature: For the heterotic superstring, the worldsheet curvature couples to the real part of a chiral multiplet, and for Type II it couples to the real part of the sum of a vector multiplet and a tensor hypermultiplet.    For the Type II superstring, this contradicts the standard folklore that only a hypermultiplet counts string-loops, explains the peculiar dilaton coupling of Ramond-Ramond fields, and allows the effective action to be easily written in N=2 $4D$ superspace. It also implies that vector multiplet interactions get no quantum corrections, while hypermultiplet interactions can only get corrections if mirror symmetry is non-perturbatively broken. 
  Emission of hard microscopic string (graviton) by an excited macroscopic string may be viewed as a model of branching of a $(1+1)$-dimensional baby universe off large parent one. We show that, apart from a trivial factor, the total emission rate is not suppressed by the size of the macroscopic string. This implies unsuppressed loss of quantum coherence in $(1+1)$-dimensional parent universe. 
  A group theory justification of one dimensional fractional supersymmetry is proposed using an analogue of a coset space, just like the one introduced in $1D$ supersymmetry. This theory is then gauged to obtain a local fractional supersymmetry {\it i.e.} a fractional supergravity which is then quantized {\it \`a la Dirac} to obtain an equation of motion for a particle which is in a representation of the braid group and should describe alternative statistics. A formulation invariant under general reparametrization is given, by means of a curved fractional superline. 
  A one-parameter class of simple models of two-dimensional dilaton gravity, which can be exactly solved including back-reaction effects, is investigated at both classical and quantum levels. This family contains the RST model as a special case, and it continuously interpolates between models having a flat (Rindler) geometry and a constant curvature metric with a non-trivial dilaton field. The processes of formation of black hole singularities from collapsing matter and Hawking evaporation are considered in detail. Various physical aspects of these geometries are discussed, including the cosmological interpretation. 
  The question that guides our discussion is "how did the geometry and particles come into being?" To explore this query we suggest the theory of goyaks, which reveals the primordial deeper structures underlying fundamantal concepts of contemporary physics. It address itself to the question of the prime-cause of origin of geometry and basic concepts of particle physics such as the fundamental fields of quarks and leptons with the spins and various quantum numbers, internal symmetries and so on; also basic principles of Relativity, Quantum, Gauge and Color Confinement, which are, as it was proven, all derivative and come into being simultaneously. The substance out of which the geometry and particles are made is a set of new physical structures-the goyaks involved into reciprocal linkage establishing processes. We elaborated a new mathematical framework, which is a still wider generalization of the familiar methods of secondary quantization with appropriate expansion over the geometric objects. One interesting offshoot of it directly leads to the formalism of operator manifold, which framed our discussion throughout this paper. It yields the quantization of geometry, which differs in principle from all earlier studies. Many of the important anticipated properties, basic concepts and principles of particle physics are appeared quite naturally in the framework of suggested theory. It predicts a class of possible models of internal symmetries, which utilize the whole idea of gauge symmetry and reproduce the known phenomenology of electromagnetic, weak and strong interactions. Here we focused our attention mainly on developing the mathematical foundations for our novel viewpoint. We believe that the more realistic final theory of particles and interactions can be found within the 
  A path integral with BRST symmetry can be formulated by summing the Gribov-type copies in a very specific way if the functional correspondence between $\tau$ and the gauge parameter $\omega$ defined by $\tau (x) = f( A_{\mu}^{\omega})$ is ``globally single valued'', where $f( A_{\mu}^{\omega}) = 0 $ specifies the gauge condition. A soluble gauge model with Gribov-type copies recently analyzed by Friedberg, Lee, Pang and Ren satisfies this criterion. A detailed BRST analysis of the soluble model proposed by the above authors is presented. The BRST symmetry, if it is consistently implemented, ensures the gauge independence of physical quantities. In particular, the vacuum (ground) state and the perturbative corrections to the ground state energy in the above model are analysed from a view point of BRST symmetry and $R_{\xi}$-gauge. Implications of the present analysis on some aspects of the Gribov problem in non-Abelian gauge theory, such as the $1/N$ expansion in QCD and also the dynamical instability of BRST symmetry, are briefly discussed. 
  The non-local one-loop contribution to the gravitational effective action around de Sitter space is computed using the background field method with pure trace external gravitational fields and it is shown to vanish. The calculation is performed in a generic covariant gauge and the result is verified to be gauge invariant. 
  A non-gauge dynamical system depending on parameters is considered. It is shown that these parameters can have such values that corresponding canonically quantized theory will be gauge invariant. The equations allowing to find these values of parameters are derived. The prescription under consideration is applied to obtaining the equation of motion for tachyon background field in closed bosonic string theory. 
  It is shown that the non-relativistic `Dirac' equation of L\'evy-Leblond, we used recently to describe a spin $1/2$ field interacting non-relativistically with a Chern-Simons gauge field, can be obtained by lightlike reduction from $3+1$ dimensions. This allows us to prove that the system is Schr\"odinger symmetric. A spinor representation of the Schr\"odinger group is presented. Static, self-dual solutions, describing spinor vortices are given and shown to be the non-relativistic limits of the fermionic vortices found by Cho et al. The construction is extended to external harmonic and uniform magnetic fields. 
  The semiclassical approximation is studied on hypersurfaces approaching the union of future null infinity and the event horizon on a large class of four dimensional black hole backgrounds. Quantum fluctuations in the background geometry are shown to lead to a breakdown of the semiclassical approximation in these models. The boundary of the region where the semiclassical approximation breaks down is used to define a `stretched horizon'. It is shown that the same effect that brings about the breakdown in semiclassical evolution associates a temperature and an entropy to the region behind the stretched horizon, and identifies the microstates that underlie the thermodynamical properties. The temperature defined in this way is equal to that of the black hole and the entropy is equal to the Bekenstein entropy up to a factor of order unity. 
  A crucial property of the standard antifield-BRST cohomology at non negative ghost number is that any cohomological class is completely determined by its antifield independent part. In particular, a BRST cocycle that vanishes when the antifields are set equal to zero is necessarily exact.\ \ This property, which follows from the standard theorems of homological perturbation theory, holds not only in the algebra of local functions, but also in the space of local functionals. The present paper stresses how important it is that the antifields in question be the usual antifields associated with the gauge invariant description. By means of explicit counterexamples drawn from the free Maxwell-Klein-Gordon system, we show that the property does not hold, in the case of local functionals, if one replaces the antifields of the gauge invariant description by new antifields adapted to the gauge fixation. In terms of these new antifields, it is not true that a local functional weakly annihilated by the gauge-fixed BRST generator determines a BRST cocycle; nor that a BRST cocycle which vanishes when the antifields are set equal to zero is necessarily exact. 
  We reexamine the graceful exit problem in the Pre-Big-Bang inflationary scenario. The dilaton-gravity action is generalized by adding the axion and a general axion/dilaton potential. We provide a phase space analysis of the dynamics which leads us to extend the previous no-go theorem and rule out the branch change necessary for graceful exit in this context. 
  The spacetime singularities play a useful role in gravitational theories by distinguishing physical solutions from non-physical ones. The problem, we studying in this paper is: are these singularities stable? To answer this question, we have analyzed the general problem of stability of the family of the static spherically symmetric solutions of the standard Einstein-Maxwell model coupled to an extra free massless scalar field. We have obtained the equations for the axial and polar perturbations. The stability against axial perturbations has been proven. 
  At the first stage of reheating after inflation, parametric resonance may rapidly transfer most of the energy of an inflaton field $\phi$ to the energy of other bosons. We show that quantum fluctuations of scalar and vector fields produced at this stage are much greater than they would be in a state of thermal equilibrium. This leads to cosmological phase transitions of a new type, which may result in a copious production of topological defects and in a secondary stage of inflation after reheating. 
  We study gravitational quantum mechanics violating (QMV) effects to masses of Nambu-Goldstone bosons, taking majoron as an example. We show a supersymmetric majoron has either mass of O(keV) for the dimension five potential or smaller mass for effective potentials with higher dimensions. We extend the Dashen's formula for pseudo Nambu-Goldstone bosons to include possible effects of QMV. 
  Models with dynamical supersymmetry breaking are interesting because they may provide a solution to both the gauge hierarchy and the fine-tuning problems. However, because of strongly interacting dynamics, it is in general impossible to analyze them quantitatively. One of the few models with calculable dynamical supersymmetry breaking is a model with SU(5) gauge symmetry and two $10$'s and two $\bar 5$'s as the matter content. We determine the ground state of this model, find the vacuum energy, reveal the symmetry breaking pattern and calculate the mass spectrum. The supertrace mass relation is exploited to verify the consistency of the calculated mass spectrum, and an accidental degeneracy is explained. 
  We study the propagation of string fields (metric $G_{\mu\nu}$, Mawxell gauge potential $A_{\mu}$, dilaton $\Phi$, and tachyon $T$) in a two-dimensional (2D) charged black hole. It is shown that the tachyon is a propagating field both inside and outside the black hole. This becomes infinitely blueshifted at the inner horizon. We confirm that the inner horizon is unstable, whereas the outer horizon is stable. 
  We found an infinite number of potentials surrounding 2d black hole. According to the transmission ${\cal T}$ and reflection ${\cal R}$ coefficients for scattering of string fields off 2d black hole, we can classify an infinite number of potentials into three : graviton-dilaton, tachyon and the other types. We suggest that the discrete states from all the Virasoro levels be candidates for new potentials (modes). 
  The charged 2D black hole is visualized as presenting an potential barrier $V^{OUT}(r^*)$ to on-coming tachyon wave. Since this takes the complicated form, an approximate form $V^{APP}(r^*)$ is used for scattering analysis. We calculate the reflection and transmission coefficients for scattering of tachyon off the charged 2D black hole. The Hawking temperature is also derived from the reflection coefficient by Bogoliubov transformation. In the limit of $Q \to 0$, we recover the Hawking temperature of the 2D dilaton black hole. 
  We obtain the $\epsilon<2$ new extremal ground states of a two-dimensional (2D) charged black hole where $\epsilon$ is the dilaton coupling parameter for the Maxwell term. The stability analysis is carried out for all these extremal black holes. It is found that the shape of potentials to an on-coming tachyon (as a spectator) take all barrier-well types. These provide the bound state solutions, which imply that they are unstable. We conclude that the 2D, $\epsilon<2$ extremal black holes should not be considered as a toy model for the stable endpoint of the Hawking evaporation. 
  We discuss the stability of the extremal ground states of a two-dimensional (2D) charged black hole which carries both electric ($Q_E$) and magnetic ($Q_M$) charges. The method is first to find the physical field and then to derive the equation of the Schr\"odinger type. It is found that the presenting potential to an on-coming tachyon (as a spectator) takes a barrier-well type. This provides the bound state solution, which implies an exponentially growing mode with respect to time. The 2D extremal ground states all are classically unstable. We conclude that the 2D extremal charged black holes are not considered as the candidates for the stable endpoint of the Hawking evaporation. 
  We discuss the two-dimensional (2D), $\epsilon<2$ extremal ground states of charged black hole. Here $\epsilon$ is the dilaton coupling parameter for the Maxwell term. The complete analysis of stability is carried out for all these extremal black holes. It is found that they are all unstable. To understand this instability, we study the non-extremal charged black hole with two (inner and outer) horizons. The extremal black holes appear when two horizons coalesce. It conjectures that the instability originates from the inner horizon. 
  We consider the perturbation of tachyon about the extremal ground state of a two-dimensional (2D) electrically charged black hole. It is found that the presenting potential to on-coming tachyonic wave takes a double-humped barrier well. This allows an exponentially growing mode with respect to time. This extremal ground state is classically unstable. We conclude that the 2D extremal electrically charged black hole cannot be a candidate for the stable endpoint of the Hawking evaporation. 
  We give a simple proof of the relation $\Lambda\p_artial{\Lambda}\F= {i\over2\pi}b_1\langle\Tr\phi^2\rangle$, which is valid for $N=2$ supersymmetric QCD with massless quarks. We consider $SU(N_c)$ gauge theories as well as $SO(N_c)$ and $SP(N_c)$. Aa analogous relation which corresponds to massive hypermultiplets is written down. We also discuss the generalizations to $N=1$ models in the Coulomb phase. 
  We calculate the beta-functions of the general massive (p,q) supersymmetric sigma model to two loop order using (1,0) superfields. The conditions for finiteness are discussed in relation to (p,q) supersymmetry. We also calculate the effective potential using component fields to one loop order and consider the possibility of perturbative breaking of supersymmetry. The effect of one loop finite local counter terms and the ultra-violet behaviour of the off-shell (p,q) models to all orders in perturbation theory are also addressed. 
  A lattice regularized Lax operator for the nonultralocal modified Korteweg de Vries (mKdV) equation is proposed at the quantum level with the basic operators satisfying a $q$-deformed braided algebra. Finding further the associated quantum $R$ and $Z$-matrices the exact integrability of the model is proved through the braided quantum Yang--Baxter equation, a suitably generalized equation for the nonultralocal models. Using the algebraic Bethe ansatz the eigenvalue problem of the quantum mKdV model is exactly solved and its connection with the spin-$\ha$ XXZ chain is established, facilitating the investigation of the corresponding conformal properties. 
  According to the experimental data, it is still controversial whether the neutrinos, especially the electron-neutrino and muon-neutrino, can be considered as the fermionic spinorial tachyons, and there is still no reliable report on the existence of the right-handed neutrinos. In this letter, we show that the neutrinos with the single handedness can not be the tachyons, but only those of the both handedness can be. Several implications of this result are discussed. 
  The string quantum kernel is normally written as a functional sum over the string coordinates and the world--sheet metrics. As an alternative to this quantum field--inspired approach, we study the closed bosonic string propagation amplitude in the functional space of loop configurations. This functional theory is based entirely on the Jacobi variational formulation of quantum mechanics, {\it without the use of a lattice approximation}. The corresponding Feynman path integral is weighed by a string action which is a {\it reparametrization invariant} version of the Schild action. We show that this path integral formulation is equivalent to a functional ``Schrodinger'' equation defined in loop--space. Finally, for a free string, we show that the path integral and the functional wave equation are {\it exactly } solvable. 
  We show that polarization dependent string-string scattering provides new evidence for the identification of the Dabholkar-Harvey (DH) string solution with the heterotic string itself. First, we construct excited versions of the DH solution which carry arbitrary left-moving waves yet are annihilated by half the supersymmetries. These solutions correspond in a natural way to Bogomolny-bound-saturating excitations of the ground state of the heterotic string. When the excited string solutions are compactified to four dimensions, they reduce to Sen's family of extremal black hole solutions of the toroidally compactified heterotic string. We then study the large impact parameter scattering of two such string solutions. We develop methods which go beyond the metric on moduli space approximation and allow us to read off the subleading polarization dependent scattering amplitudes. We find perfect agreement with heterotic string tree amplitude predictions for the scattering of corresponding string states. Taken together, these results clearly identify the string states responsible for Sen's extremal black hole entropy. We end with a brief discussion of implications for the black hole information problem. 
  The recent discovery of an explicit conformal field theory description of Type II $p$-branes makes it possible to investigate the existence of bound states of such objects. In particular, it is possible with reasonable precision to verify the prediction that the Type IIB superstring in ten dimensions has a family of soliton and bound state strings permuted by $SL(2,{\bf Z})$. The space-time coordinates enter tantalizingly in the formalism as non-commuting matrices. 
  We present several forms in which the BRST transformations of QCD in covariant gauges can be cast. They can be non-local and even not manifestly covariant. These transformations may be obtained in the path integral formalism by non standard integrations in the ghost sector or by performing changes of ghost variables which leave the action and the path integral measure invariant. For different changes of ghost variables in the BRST and anti-BRST transformations these two transformations no longer anticommute. 
  We present detailed discussions on the stochastic Hamiltonians for non-critical string field theories on the basis of matrix models. Beginning from the simplest $c=0$ case, we derive the explicit forms of the Hamiltonians for the higher critical case $k=3$ (which corresponds to $c=-22/5$) and for the case $c=1/2$, directly from the double-scaled matrix models. In particular, for the two-matrix case, we do not put any restrictions on the spin configurations of the string fields. The properties of the resulting infinite algebras of Schwinger-Dyson operators associated with the Hamiltonians and the derivation of the Virasoro and $W_3$ algebras therefrom are also investigated. Our results suggest certain universal structure of the stochastic Hamiltonians, which might be useful for an attempt towards a background independent string field theory. 
  We consider the spectrum of BPS saturated states in $N = 2$ gauge theories in four dimensions. This spectrum may be discontinuous across real codimension one submanifolds of marginal stability in the moduli space of vacua. An example, which can be treated with semiclassical methods in the weak coupling limit, is the decay of quark-soliton bound states. For a quark and a soliton of electric-magnetic charge vectors $Q$ and $Q^\prime$ respectively, we find that as the manifold of marginal stability is crossed, the number of soliton states changes by a factor of $2^{Q \cdot Q^\prime}$, where the dot denotes the symplectic product. 
  Using the well-known result for the fermionic determinant in terms of a WZW theory, we write QCD$_2$ in bosonized form. After some manipulations we give two versions of the theory, where it is factorized as a product of the conformally invariant WZW models, ghost terms, and a WZW action perturbed off criticality. We first prove that the latter is an integrable model. Furthermore, as a consequence of the BRST analysis, there are several constraints. Second-class constraints also show up. Using the integrability condition we find higher conservation laws, their action on the asymptotic states, and we propose an exact $S$-matrix for the physical excitations. The vacuum structure is analysed, and we prove that a finite number of different vacua exist. An outline of possible generalizations of the procedure is given. 
  The main results are: 1. A manifestly covariant technique for the calculation of De Witt coefficients is elaborated; 2. The coefficients $a_3$ and $a_4$ are calculated; 3. Covariant methods for the study of the nonlocal structure of the effective action are developed. The terms of first and second order in background fields in De Witt coefficients are calculated. The summation of these terms is carried out and nonlocal covariant expression for the Green function, the heat kernel and the effective action are obtained. It is shown that in the conform-invariant case the Green function is finite. A finite effective action in the conform-invariant case of massless scalar field in two-dimensional space is obtained; 4. The off-shell one-loop divergences of the effective action in arbitrary covariant gauge as well as those of the `unique' effective action in higher-derivative quantum gravity are calculated; The ultraviolet asymptotics of coupling constants are found. It is shown that in the `physical' region of coupling constants (no tachyons!) the conformal sector has `zero-charge' behavior contrary to previous authors. This means that the theory goes beyond the limits of weak conformal coupling at higher energies. In other words, the condition of conformal stability of flat background is incompatible with the asymptotic freedom in the conformal sector. There is a stable non-flat ground state but only in the case of positive definite Euclidean action. In this case the theory is asymptotically free both in tensor and conformal sectors. The off-shell one-loop effective action in arbitrary covariant gauge and the `unique' effective action on De Sitter background are calculated. 
  We introduce the self-dual abelian gauged $O(3)$ sigma models where the Maxwell and Chern-Simons terms constitute the kinetic terms for the gauge field. These models have quite rich structures and various limits. Our models are found to exhibit both symmetric and broken phases of the gauge group. We discuss the pure Chern-Simons limit in some detail and study rotationally symmetric solitons. 
  We carefully analyze the N=2 dual pair of string theories in four dimensions introduced by Ferrara, Harvey, Strominger and Vafa. The analysis shows that a second discrete degree of freedom must be switched on in addition to the known `Wilson line' to achieve a non-perturbatively consistent theory. We also identify the phase transition this model undergoes into another dual pair via a process analogous to a conifold transition. This provides the first known example of a phase transition which is understood from both the type II and the heterotic string picture. 
  Knizhnik-Zamolodchikov-Bernard (KZB) equation on an elliptic curve with a marked point is derived by the classical Hamiltonian reduction and further quantization. We consider classical Hamiltonian systems on cotangent bundle to the loop group $L(GL(N,{\bf C}))$ extended by the shift operators, to be related to the elliptic module. After the reduction we obtain the Hamiltonian system on cotangent bundle to the moduli of holomorphic principle bundles and the elliptic module. It is a particular example of generalized Hitchin systems (GHS), which are defined as hamiltonian systems on cotangent bundles to the moduli of holomorphic bundles and to the moduli of curves. They are extensions of the Hitchin systems by the inclusion the moduli of curves. In contrast with the Hitchin systems the algebra of integrals are noncommutative on GHS. We discuss the quantization procedure in our example. The quantization of the quadratic integral leads to the KZB equation. We present the explicite form of higher quantum Hitchin integrals, which upon on reducing from GHS phase space to the Hitchin phase space gives a particular example of the Belinson-Drinfeld commutative algebra of differential operators on the moduli of holomorphic bundles. 
  Small corrections to the uncertainty relations, with effects in the ultraviolet and/or infrared, have been discussed in the context of string theory and quantum gravity. Such corrections lead to small but finite minimal uncertainties in position and/or momentum measurements. It has been shown that these effects could indeed provide natural cutoffs in quantum field theory. The corresponding underlying quantum theoretical framework includes small `noncommutative geometric' corrections to the canonical commutation relations. In order to study the full implications on the concept of locality it is crucial to find the physical states of then maximal localisation. These states and their properties have been calculated for the case with minimal uncertainties in positions only. Here we extend this treatment, though still in one dimension, to the general situation with minimal uncertainties both in positions and in momenta. 
  We discuss some aspects of the two-dimensional scalar field, considering particularly the action for the conformal anomaly as an ``improved'' gravitational coupling, and the possibility of introducing a dual coupling, which provides a ``chiral'' energy-momentum tensor improvement. 
  Symmetry restoration processes during the non-equilibrium stage of ``preheating'' after inflation is studied. It is shown that symmetry restoration is very efficient when the majority of created particles are concentrated at energies much smaller than the temperature $T$ in equilibrium. The strength of symmetry restoration measured in terms of the equivalent temperature can exceed $T$ by many orders of magnitude. In some models the effect can be equivalent to that if the temperature of instant reheating would be close to the Planck scale. This can have an important impact on GUT and axion models. 
  Area preserving diffeomorphisms of a 2-d compact Riemannian manifold with or without boundary are studied. We find two classes of decompositions of a Riemannian metric, namely, h- and g-decomposition, that help to formulate a gravitational theory which is area preserving diffeomorphism (SDiff$M$-) invariant but not necessarily diffeomorphism invariant. The general covariance of equations of motion of such a theory can be achieved by incorporating proper Weyl rescaling. The h-decomposition makes the conformal factor of a metric SDiff$M$-invariant and the rest of the metric invariant under conformal diffeomorphisms, whilst the g-decomposition makes the conformal factor a SDiff$M$ scalar and the rest a SDiff$M$ tensor. Using these, we reformulate Liouville gravity in SDiff$M$ invariant way. In this context we also further clarify the dual formulation of Liouville gravity introduced by the author before, in which the affine spin connection is dual to the Liouville field. 
  We present a dual description for $SU(N)$ supersymmetric gauge theory with an antisymmetric tensor and fundamentals, and no superpotential. This duality is derived from the dualities of Seiberg. Under a perturbation of the superpotential, the dual theory breaks supersymmetry at tree level. 
  Conformal field theory at $c=-2$ provides the simplest example of a theory with ``logarithmic'' operators. We examine in detail the $(\xi,\eta)$ ghost system and Coulomb gas construction at $c=-2$ and show that, in contradistinction to minimal models, they can not be described in terms of conformal families of {\em primary\/} fields alone but necessarily contain reducible but indecomposable representations of the Virasoro algebra. We then present a construction of ``logarithmic'' operators in terms of ``symplectic'' fermions displaying a global $SL(2)$ symmetry. Orbifolds with respect to finite subgroups of $SL(2)$ are reminiscent of the $ADE$ classification of $c=1$ modular invariant partition functions, but are isolated models and not linked by massless flows. 
  The massless superfield content of four-dimensional compactifications of closed superstrings with extended (N=2, 3, or 4) supersymmetry is derived by multiplying two (N=0, 1, or 2) Yang-Mills multiplets. In some cases these superfields are known, and the low-energy actions are determined from the fact that the compensator (dilaton) supermultiplets occur quadratically classically. In the other cases these superfields suggest new formulations of extended superspace theories. 
  Extending our prior investigation, we give a new off-shell construction of theories of spinning particles propagating in Minkowski spaces with arbitrary $N$-extended supersymmetry on the world-line. The basis of the new off-shell formulation is provided by realizations of new algebraic structures ${\cal G}{\cal R}$(${\rm d}, N$) that are certain generalizations of Pauli algebras. 
  Utilizing techniques suggested by the recently obtained construction of off-shell spinning particles, we propose the arbitrary $N$-extension of supersymmetry for the KdV system. It is further suggested that the ${\aleph}_0$ extension for the SKdV system provides a paradigm for {\underline {all}} supersymmetric completely integrable systems. 
  We present mechanisms for generating conical singularities both in three and four-dimensions in the systems with copies of scalar or chiral multiplets coupled to $N=2$ or $N=1$ supergravity. Our mechanisms are useful for supersymmetry breaking, maintaining the zero cosmological constants in three and four-dimensions. A strong coupling duality connecting these two dimensionalities is also studied. 
  The path integral computation of field strength correlation functions for two dimensional Yang-Mills theories over Riemann surfaces is studied. The calculation is carried out by abelianization, which leads to correlators that are topological. They are nontrivial as a result of the topological obstructions to the abelianization. It is shown in the large N limit on the sphere that the correlators undergo second order phase transitions at the critical point. Our results are applied to a computation of contractible Wilson loops. 
  We continue our analysis of the field strength correlation functions of two-dimensional QCD on Riemann surfaces by studying the large $N$ limit of these correlation functions on the sphere for gauge group $U(N)$. Our results allow us to exhibit an explicit master field for the field strength $F_{\mu\nu}$ in a ``topological gauge'', given by a single master matrix in the Lie algebra of the maximal torus of the gauge group. Field correlators are obtained from traces of products of the master field. We also obtain a master field for the gauge potential $A_{\mu}$ on the sphere, consistent with the master field for the field strength. 
  We discuss and compare different definitions of the entropy of a black hole. In particular we show that the thermodynamical entropy defined by the response of the free energy of a black hole to the change of temperature does not coincide with the statistical-mechanical entropy, obtained by counting its dynamical degrees of freedom. The no-boundary wavefunction of a black hole, its relation to the entropy problem and its generalization are discussed. We discuss the relation between on-shell and off-shell calculations of the entropy, and the role of the renormalization of the gravitational constant. 
  The Hamiltonian formulation for a non-Abelian gauge theory in two spatial dimensions is carried out in terms of a gauge-invariant matrix parametrization of the fields. The Jacobian for the relevant transformation of variables is given in terms of the WZW-action for a hermitian matrix field. Some gauge-invariant eigenstates of the kinetic term of the Hamiltonian are given; these have zero charge and exhibit a mass gap. 
  Several physical systems (two identical particles in two dimensions, isotropic oscillator and Kepler system in a 2-dim curved space) and mathematical structures (quadratic algebra QH(3), finite W algebra $\bar {\rm W}_0$) are shown to posses the structure of a generalized deformed su(2) algebra, the representation theory of which is known. Furthermore, the generalized deformed parafermionic oscillator is identified with the algebra of several physical systems (isotropic oscillator and Kepler system in 2-dim curved space, Fokas--Lagerstrom, Smorodinsky--Winternitz and Holt potentials) and mathematical constructions (generalized deformed su(2) algebra, finite W algebras $\bar {\rm W}_0$ and W$_3^{(2)}$). The fact that the Holt potential is characterized by the W$_3^{(2)}$ symmetry is obtained as a by-product. 
  Polchinski has argued that the prediction of Hawking radiation must be independent of the details of unknown high-energy physics because the calculation may be performed using `nice slices', for which the adiabatic theorem may be used. If this is so, then any calculation using a manifestly covariant --- and so slice-independent --- ultraviolet regularization must reproduce the standard Hawking result. We investigate the dependence of the Hawking radiation on such a short-distance regulator by calculating it using a Pauli--Villars regularization scheme. We find that the regulator scale, $\Lambda$, only contributes to the Hawking flux by an amount that is exponentially small in the large variable ${\Lambda}/{T_\ssh} \gg 1$, where $T_\ssh$ is the Hawking temperature; in agreement with Polchinski's arguments. We also solve a technical puzzle concerning the relation between the short-distance singularities of the propagator and the Hawking effect. 
  Classical Electrodynamics is not a consistent theory because of its field inadequate behaviour in the vicinity of their sources. Its problems with the electron equation of motion and with non-integrable singularity of the electron self field and of its stress tensor are well known. These inconsistencies are eliminated if the discrete and localized (classical photons) character of the electromagnetic interaction is anticipatively recognized already in a classical context. This is possible, in a manifestly covariant way, with a new model of spacetime structure, shown in a previous paper $^{1}$, that invalidates the Lorentz-Dirac equation. For a point classical electron there is no field singularity, no causality violation and no conflict with energy conservation in the electron equation of motion. The electromagnetic field must be re-interpreted in terms of average flux of classical photons. Implications of a singularity-free formalism to field theory are discussed. 
  Polchinski's recent construction of Dirichlet-branes of R-R charges, together with Witten's mechanism for forming bound states of both NS-NS charges and R-R charges, provides a rigorous method to treat these dy-branes. We construct the massless sector of boundary states of D-branes, as well as of dy-strings of charge $(p,1)$. As a consequence, the string tension formula predicted by duality in the type IIB theory is obtained. 
  Covariant or invariant functions under a compact linear group can be expressed in terms of functions defined in the orbit space of the group. The semialgebraic relations defining the orbit spaces of all finite coregular real linear groups with at most 4 basic invariants are determined. For each group $G$ acting in $\real^n$, the results are obtained through the computation of a metric matrix $\widehat P(p)$, which is defined only in terms of the scalar products between the gradients of a set of basic polynomial invariants $p_1(x),\dots p_q(x),\x\in\real^n$ of $G$; the semi-positivity conditions $\widehat P(p)\ge 0$ are known to determine all the equalities and inequalities defining the orbit space $\real^n/G$ of $G$ as a semi-algebraic variety in the space $\real^q$ spanned by the variables $p_1,\dots ,p_q$. In a recent paper, the $\widehat P$-matrices, for $q\le 4$, have been determined in an alternative way, as solutions of a universal differential equation;the present paper yields a partial, but significant, check on the correctness and completeness of these solutions. Our results can be widely exploited,e.g. in the determination of patterns of spontaneous symmetry breaking, in the analysis of structural phase transitions (Landau's theory),in covariant bifurcation theory,in crystal field theory and in solid state theory where symmetry adapted functions are used. 
  By applying the Hamiltonian reduction method to the cotangent bundle over loop groups we recover the well-known classical trigonometric $r$-matrices of the periodic Toda lattice. 
  Known mechanisms for breaking of supersymmetry at the level of string theory imply that at least one of the internal dimensions has a very large size. Experimental detection of the associated light Kaluza-Klein (KK) excitations would be a strong hint for the existence of string like elementary objects, as no consistent field theory describing them is known. We restrict the discussion to the Scherk-Schwarz mechanism in orbifold compactifications. For this case we investigate the quantum number of the lightest predicted KK states. 
  A new series of integrable cases of the many-body problem in many-dimensional spaces is found. That series appears as a part of the larger series of integrable problems, which are in 1-1 correspondence with Krichever-Novikov algebras of affine type (that is with pairs each one consisting of some finite root system and some Riemann surface of finite genus with two marked points). 
  We compute the classical $r$-matrix for the relativistic generalization of the Calogero-Moser model, or Ruijsenaars-Schneider model, at all values of the speed-of-light parameter $\lambda$. We connect it with the non-relativistic Calogero-Moser $r$-matrix $(\lambda \rightarrow -1)$ and the $\lambda = 1$ sine-Gordon soliton limit. 
  We prove the renormalizability of the Curci-Ferrari model with and without auxiliary fields using BRST methods. In both cases we find 5 $Z$ factors instead of 3. We verify our results by explicit one loop calculations. We determine a set of generators for the ``physical states'', many of which have negative norm. Supersymmetrization is considered. 
  We give explicit expressions for the q-multinomial generalizations of the q-binomials and Andrews' and Baxter's q-trinomials. We show that the configuration sums for the generalized RSOS models in regime III studied by Date et al. can be expressed in terms of these multinomials. This generalizes the work of ABF and AB where configuration sums of statistical mechanical models have been expressed in terms of binomial and trinomial coefficients. These RSOS configuration sums yield the branching functions for the $\widehat{su}(2)_{M}\times \widehat{su}(2)_{N}/\widehat{su}(2)_{M+N}$ coset models. The representation in terms of multinomials gives Rocha-Caridi like formulas whereas the representation of Date et al. gives a double sum representation for the branching functions. 
  A study is made of the implications of heterotic string $T$-duality and extended gauge symmetry for the conjectured equivalence of heterotic and Type I superstrings. While at first sight heterotic string world-sheet dynamics appears to conflict with Type I perturbation theory, a closer look shows that Type I perturbation theory ``miraculously'' breaks down, in some cases via novel mechanisms, whenever the heterotic string has massless particles not present in Type I perturbation theory. This strongly suggests that the two theories actually are equivalent. As further evidence in the same direction, we show that the Dirichlet one-brane of type I string theory has the same world-sheet structure as the heterotic string. 
  We present an exactly solvable quantum field theory which allows rearrangement collisions. We solve the model in the relevant sectors and demonstrate the orthonormality and completeness of the solutions, and construct the S-matrix. In the light of the exact solutions constructed, we discuss various issues and assumptions in quantum scattering theory, including the isometry of the M\"oller wave matrix, the normalization and completeness of asymptotic states, and the non-orthogonality of basis states. We show that these common assertions do not obtain in this model. We suggest a general formalism for scattering theory which overcomes these, and other, shortcomings and limitations of the existing formalisms in the literature. 
  Form factor bootstrap approach is applied for diagonal scattering theories. We consider the ADE theories and determine the functional equations satisfied by the minimal two-particle form factors. We also determine the parameterization of the singularities in two particle form factors. For $A^{(1)}_{2}$ Affine Toda field theory which is the simplest non-self conjugate theory, form factors are derived up to four-body and identification of operator is done. Generalizing this identification to the $A^{(1)}_N$ Affine Toda cases, we fix the two particle form factors. We also determine the additional pole structure of form factors which comes from the double pole of the $S$-matrices of the $A^{(1)}_N$ theory. For $A_N$ theories, existence of the conserved ${\bf Z}_{N+1}$ charge leads to the division of the set of form factors into $N+1$ decoupled sectors. 
  The system of a closed vortex filament is an integrable Hamiltonian one, namely, a Hamiltonian system with an infinite sequense of constants of motion in involution. An algebraic framework is given for the aim of describing differential geometry of this system. A geometrical structure related to the integrability of this system is revealed. It is not a bi-Hamiltonian structure but similar one. As a related topic, a remark on the inspection of J.Langer and R.Perline, J.Nonlinear Sci.1, 71 (1991), is given. 
  We provide some additional evidence in favour of the strong - weak coupling duality between the SO(32) heterotic and type I superstring theories by comparing terms quartic in the gauge field strength in their low-energy effective actions. We argue that these terms should not receive higher-loop string corrections so that duality should relate the leading-order perturbative coefficients in the two theories. In particular, we demonstrate that the coefficient of the $F^4$-term in the one-loop (torus) part of the SO(32) heterotic string action is exactly the same as the coefficient of the $F^4$-term in the tree-level (disc) part of the type I action. 
  We analyze the symplectic structure on the dressing group in the \shG\, model by calculating explicitly the Poisson bracket $\{g\x g\}$ where $g$ is the \dg\, element which creates a generic one soliton solution from the vacuum. Our result is that this bracket does not coincide with the Semenov--Tian--Shansky one. The last induces a Lie--Poisson structure on the \dg . To get the bracket obtained by us from the Semenov--Tian--Shansky bracket we apply the formalism of the constrained Hamiltonian systems. The constraints on the \dg\, appear since the element which generates one solitons from the vacuum has a specific form. 
  In this paper, we pursue the discussion of the connections between rational conformal field theories (CFT) and graphs. We generalize our recent work on the relations of operator product algebra (OPA) structure constants of $sl(2)\,$ theories with the Pasquier algebra attached to the graph. We show that in a variety of CFT built on $sl(n)\,$ -- typically conformal embeddings and orbifolds, similar considerations enable one to write a linear system satisfied by the matrix elements of the Pasquier algebra in terms of conformal data -- quantum dimensions and fusion coefficients. In some cases, this provides a sufficient information for the determination of all the eigenvectors of an adjacency matrix, and hence of a graph. 
  A recently proposed variation on the usual procedure to perform the topological B-twist in rigid $N=2$ models is applied to the case of the $\sigma $ model on a K\"ahler manifold. This leads to an alternative description of Witten's topological $\sigma $ model, which allows for a proper BRST interpretation and ghost number assignement. We also show that the auxiliary fields, which are responsible for the off shell closure of the $N=2$ algebra, play an important role in our construction. 
  We review recent progress in formulating two-dimensional models over noncommutative manifolds where the space-time coordinates enter in the formalism as non-commuting matrices. We describe the Fuzzy sphere and a way to approximate topological nontrivial configurations using matrix models. We obtain an ultraviolet cut off procedure, which respects the symmetries of the model. The treatment of spinors results from a supersymmetric formulation; our cut off procedure preserves even the supersymmetry. 
  We apply the duality transformation relating the heterotic to the IIA string in 6D to the class of exact string solutions described by the chiral null model and derive explicit formulas for all fields after reduction to 4D. If the model is restricted to asymptotically flat black hole type solutions with well defined mass and charges the purely electric solutions on the heterotic side are mapped to dyonic ones on the IIA side. The mass remains invariant. Before and after the duality transformation the solutions belong to short $N=4$ SUSY multiplets and saturate the corresponding Bogomol'nyi bounds. 
  We consider the harmonic-superspace ($HS$) system of equations that contains superfield $SYM^1_6$ constraints and equations of motion. A dynamical equation in the special $A$-frame is equivalent to the zero-curvature equation corresponding to a covariant conservation of $HS$-analyticity. Properties of a general $SYM^1_6$ solution for the gauge group $SU(2)$ are studied in the simplest harmonic gauge. An analogous approach to the integrability interpretation of $SYM$-$SG$-matter systems in $HS$ is discussed briefly. 
  An action principle for spacetimes with the topology of an Euclidean black-hole is given. The gravitational field is described by the ordinary volume degrees of freedom plus additional surface fields at the horizon. The surface degrees of freedom correspond to diffeomorphisms on the sphere at the horizon and a field of ``opening angles''. General covariance forces the surface modes to be confined to a box of an unusual exponential shape, whose volume must be specified as part of the definition of the statistical ensemble. This gives rise to the Bekenstein-Hawking entropy. 
  We report on a new solution to the Einstein-Maxwell equations in 2+1 dimensions with a negative cosmological constant. The solution is static, rotationally symmetric and has a non-zero magnetic field. The solution can be interpreted as a monopole with an everywhere finite energy density. 
  We clarify the role played by BPS states in the calculation of threshold corrections of D=4, N=2 heterotic string compactifications. We evaluate these corrections for some classes of compactifications and show that they are sums of logarithmic functions over the positive roots of generalized Kac-Moody algebras. Moreover, a certain limit of the formulae suggests a reformulation of heterotic string in terms of a gauge theory based on hyperbolic algebras such as $E_{10}$. We define a generalized Kac-Moody Lie superalgebra associated to the BPS states. Finally we discuss the relation of our results with string duality. 
  Using recently proposed soliton equations we derive a basic identity for the scaling violation of $N=2$ supersymmetric gauge theories $\sum_i a_i\partial F/\partial a_i-2F=8 \pi i b_1 u$. Here $F$ is the prepotential, $a_i$'s are the expectation values of the scalar fields in the vector multiplet, $u=1/2\, {\rm Tr}\, \langle\phi^2\rangle$ and $b_1$ is the coefficient of the one-loop $\beta$-function. This equation holds in the Coulomb branch of all $N=2$ supersymmetric gauge theories coupled with massless matter. 
  An algebra ${\cal G}$ of symmetric {\em one-particle} operators is constructed for the Calogero model. This is an infinite-dimensional Lie-algebra, which is independent of the interaction parameter $\lambda$ of the model. It is constructed in terms of symmetric polynomials of raising and lowering operators which satisfy the commutation relations of the $S_N$-{\em extended} Heisenberg algebra. We interpret ${\cal G}$ as the algebra of observables for a system of identical particles on a line. The parameter $\lambda$, which characterizes (a class of) irreducible representations of the algebra, is interpreted as a statistics parameter for the identical particles. 
  We describe three different approaches to the extended (N=2) supersymmetrization of the multicomponent KP hierarchy. In the first one we utilize only superfermions while in the second only superbosons and in the third superbosons as well as superfermions. It is shown that many soliton equations can be embedded in the supersymmetry theory by using the first approach even if we do not change these equations in the bosonic limit of the supersymmetry. In the second or third approach we obtain a generalization of the soliton equations in the bosonic limit which remains in the class of the usual commuting functions. As the byproduct of our analysis we prove that for the first procedure the bosonic part of the one--component supersym\-me\-tric KP hierarchy coincides with the usual classical two--component KP hierarchy. 
  Special Kahler manifolds are defined by coupling of vector multiplets to $N=2$ supergravity. The coupling in rigid supersymmetry exhibits similar features. These models contain $n$ vectors in rigid supersymmetry and $n+1$ in supergravity, and $n$ complex scalars. Apart from exceptional cases they are defined by a holomorphic function of the scalars. For supergravity this function is homogeneous of second degree in an $(n+1)$-dimensional projective space. Another formulation exists which does not start from this function, but from a symplectic $(2n)$- or $(2n+2)$-dimensional complex space. Symplectic transformations lead either to isometries on the manifold or to symplectic reparametrizations. Finally we touch on the connection with special quaternionic and very special real manifolds, and the classification of homogeneous special manifolds. 
  We study global aspects of N=2 Kazama-Suzuki coset models by investigating topological G/H Kazama-Suzuki models in a Lagrangian framework based on gauged Wess-Zumino-Witten models. We first generalize Witten's analysis of the holomorphic factorization of bosonic G/H models to models with N=1 and N=2 supersymmetry. We also find some new anomaly-free and supersymmetric models based on non-diagonal embeddings of the gauge group. We then explain the basic properties (action, symmetries, metric independence, ...) of the topologically twisted G/H Kazama-Suzuki models. We explain how all of the above generalizes to non-trivial gauge bundles.   We employ the path integral methods of localization and abelianization (shown to be valid also for non-trivial bundles) to establish that the twisted G/H models can be localized to bosonic H/H models (with certain quantum corrections), and can hence be reduced to an Abelian bosonic T/T model, T a maximal torus of H. We also present the action and the symmetries of the coupling of these models to topological gravity. We determine the bosonic observables for all the models based on classical flag manifolds and the bosonic observables and their fermionic descendants for models based on complex Grassmannians. 
  For fields that vary slowly on the scale of the lightest mass the logarithm of the vacuum functional can be expanded as a sum of local functionals. For Yang-Mills theory the leading term in the expansion dominates large distance effects and leads to an area law for the Wilson loop. However, this expansion cannot be expected to converge for fields that vary more rapidly. By studying the analyticity of the vacuum functional under scale transformations we show how to re-sum this series so as to reconstruct the vacuum functional for arbitrary fields. 
  We discuss the $q$-Virasoro algebra based on the arguments of the Noether currents in a two-dimensional massless fermion theory as well as in a three-dimensional nonrelativistic one. Some notes on the $q$-differential operator realization and the central extension are also included. 
  A review is given of the implications of supersymmetric black holes for the non-perturbative formulation of toroidally compactified superstrings, with particular emphasis on symmetry enhancement at special vacua and S-duality of the heterotic string. 
  Bose-Einstein condensation of charged scalar and vector particles may actually occur in presence of a constant homogeneous magnetic field, but there is no critical temperature at which condensation starts. The condensate is described by the statistical distribution. The Meissner effect is possible in the scalar, but not in the vector field case, which exhibits a ferromagnetic behavior. 
  A Chern-Simons action written with Christoffel Symbols has a natural gauge symmetry of diffeomorphisms. This Chern-Simons action will induce a Wess-Zumio-Witten model on the boundary of the manifold. If we restrict the diffeomorphisms to chiral diffeomorphism, the Wess-Zumio-Witten model is equivalent to a quantum Liouville action. 
  The functional quantum field theory, developed by Stumpf, provides the possibility to derive the quantum dynamics of a positronium gas from Coulomb interacting electrons and positrons. By this example, the method will be brought in a Clifford algebraic light, through identifying the functional space with an infinite dimensional Euclidean Clifford algebra. 
  We investigate in detail the topological gauged Wess-Zumino-Witten models describing topological Kazama-Suzuki models based on complex Grassmannians. We show that there is a topological sector in which the ring of observables (constructed from the Grassmann odd scalars of the theory) coincides with the classical cohomology ring of the Grassmannian for all values of the level k. We perform a detailed analysis of the non-trivial topological sectors arising from the adjoint gauging, and investigate the general ring structure of bosonic correlation functions, uncovering a whole hierarchy of level-rank relations (including the standard level-rank duality) among models based on different Grassmannians. Using the previously established localization of the topological Kazama-Suzuki model to an Abelian topological field theory, we reduce the correlators to finite-dimensional purely algebraic expressions. As an application, these are evaluated explicitly for the CP{2} model at level k and shown for all k to coincide with the cohomological intersection numbers of the two-plane Grassmannian G(2,k+2), thus realizing the level-rank duality between this model and the G(2,k+2) model at level one. 
  We construct and discuss all background charges and continuous consistent deformations of standard $2d$ gravity theories with scalar matter fields. It turns out that the background charges and those deformations which change nontrivially both the form of the action and of its gauge symmetries are closely linked and exist only if the target space has at least one special (`covariantly constant') Killing vector which must be a null vector in the case of the deformations. The deformed actions provide interesting novel $2d$ gravity models. We argue that some of them lead to non-critical string theories. 
  Applying the principle of analytic extension for generalized functions we derive causal propagators for algebraic non-covariant gauges. The so generated manifestly causal gluon propagator in the light-cone gauge is used to evaluate two one-loop Feynman integrals which appear in the computation of the three-gluon vertex correction. The result is in agreement with that obtained through the usual prescriptions. 
  We extend work by Callan and Harvey and show how the phase of the chiral fermion determinant in four dimensions is reproduced by zeromodes bound to a domain wall in five dimensions. The analysis could shed light on the applicability of zeromode fermions and the vacuum overlap formulation of Narayanan and Neuberger for chiral gauge theories on the lattice. 
  We study the partition function of N=1 supersymmetric De Rham quantum mechanics on a Riemannian manifold, with a nontrivial chemical potential $\mu$ for the fermions. General arguments suggest that when $\mu \to \infty$ we should get the partition function of a free point particle. We investigate this limit by exact evaluation of the fermionic path integral. In even dimensions we find the De Witt term with a definite numerical factor. However, in odd dimensions our result is pestered by a quantum mechanical anomaly and the numerical factor in the De Witt term remains ambiquous. 
  We present a self-contained analysis of theories of discrete 2D gravity coupled to matter, using geometric methods to derive equations for generating functions in terms of free (noncommuting) variables. For the class of discrete gravity theories which correspond to matrix models, our method is a generalization of the technique of Schwinger-Dyson equations and is closely related to recent work describing the master field in terms of noncommuting variables; the important differences are that we derive a single equation for the generating function using purely graphical arguments, and that the approach is applicable to a broader class of theories than those described by matrix models. Several example applications are given here, including theories of gravity coupled to a single Ising spin ($c = 1/2$), multiple Ising spins ($c = k/2$), a general class of two-matrix models which includes the Ising theory and its dual, the three-state Potts model, and a dually weighted graph model which does not admit a simple description in terms of matrix models. 
  We obtain form factors for scattering of gravitons and anti-symmetric tensor particles off Dirichlet $p$-branes in Type II superstring theory. As expected, the form factor of the -1-brane (the D-instanton) exhibits point-like behavior: in fact it is saturated by a single dilaton tadpole graph. In contrast, $p$-branes with $p>-1$ acquire size of order the string scale due to quantum effects and exhibit Regge behavior. We find their leading form factors in closed form and show that they contain an infinite sequence of poles associated with $p$-brane excitations. Finally, we argue that the $p$-brane form factors for scattering of R-R bosons will have the same stringy features found with NS-NS states. 
  The general structure of the Sp(2) covariant version of the field-antifield quantization of general constrained systems in the Lagrangian formalism, the so called triplectic quantization, as presented in our previous paper with A.M.Semikhatov is further generalized and clarified.  We present new unified expressions for the generating operators which are more invariant and which yield a natural realization of the operator V^a and provide for a geometrical explanation for its presence. This V^a operator provides then for an invariant definition of a degenerate Poisson bracket on the triplectic manifold being nondegenerate on a naturally defined submanifold. We also define inverses to nondegenerate antitriplectic metrics and give a natural generalization of the conventional calculus of exterior differential forms which e g explains the properties of these inverses. Finally we define and give a consistent treatment of second class hyperconstraints. 
  We consider the metrics for cosmic strings and $p$-branes in spacetime dimension $N>4$, that is, we look for solutions to Einstein-Maxwell-Dilaton gravity in $N$-dimensions with boost symmetry in the $p$-directions along the brane. Focussing first in detail on the five dimensional uncharged cosmic string we discuss the solution, which turns out to have a naked singularity on the brane, as well as considering its Kaluza-Klein reduction. We show how singularities may be avoided with particular core models. We then derive the general uncharged $p$-brane solution in arbitrary dimension. Finally, we consider an Einstein-Maxwell-Dilaton action, with arbitrary value of the dilaton coupling parameter, deriving the solutions for electrically and magnetically charged branes, as well as a class of self-dual branes. 
  Lower bounds are placed on the fermionic determinants of Euclidean quantum electrodynamics in two and four dimensions in the presence of a smooth, finite-flux, static, unidirectional magnetic field $B(r) =(0,0,B(r))$, where $B(r) \geq 0$ or $B(r) \leq 0$, and $r$ is a point in the xy-plane. 
  The effective action of N=2 Yang-Mills theory with adjoint matter is shown to be governed by an integrable spin model with spectral parameter on an elliptic curve. We sketch a route to deriving this effective dynamics from the underlying Yang-Mills theory. Natural generalizations of this structure to all N=2 models, and to string theory, are suggested. 
  The vacuum action for the gravitational field admits a known expansion in powers of the Ricci tensor with nonlocal operator coefficients (form factors). We show that going over to a different basis of curvature invariants makes possible a partial summation of this expansion. Only the form factors of the Weyl-tensor invariants need be calculated. The full action is then uniquely recovered to all orders from the knowledge of the trace anomaly. We present an explicit expression for the partially summed action, and point out simplifications resulting in the vertex functions. An application to the effect of the vacuum gravitational waves is discussed. 
  We present a brief overview of several approaches for calculating the local asymptotic expansion of the heat kernel for Laplace-type operators. The different methods developed in the papers of both authors some time ago are described in more detail. 
  This is a non-technical version of a talk presented at the conference, "S-Duality and Mirror Symmetry in String Theory" Trieste, June, 1996 which will appear in the proceedings. 
  We use the method of discrete loop equations to calculate exact correlation functions of spin and disorder operators on the sphere and on the boundary of a disk in the $c = 1/2$ string, both in the Ising and dual Ising matrix model formulations. For both the Ising and dual Ising theories the results on the sphere are in agreement with the KPZ/DDK scaling predictions based on Liouville theory; the results on the disk agree with the scaling predictions of Martinec, Moore, and Seiberg for boundary operators. The calculation of Ising disorder correlations on the sphere requires the use of boundary variables introduced in [hep-th/9510199], which have no matrix model analog. A subtlety in the calculation on the disk arises because the expansions of the correlation functions have leading singular terms which are nonuniversal; we show that this issue may be resolved by using separate cosmological constants for each boundary domain. These results give evidence that the Kramers-Wannier duality symmetry of the $c = 1/2$ conformal field theory survives coupling to quantum gravity, implying a duality symmetry of the $c = 1/2$ string even in the presence of boundary operators. 
  We propose that the ten-dimensional $E_8\times E_8$ heterotic string is related to an eleven-dimensional theory on the orbifold ${\bf R}^{10}\times {\bf S}^1/{\bf Z}_2$ in the same way that the Type IIA string in ten dimensions is related to ${\bf R}^{10}\times {\bf S}^1$. This in particular determines the strong coupling behavior of the ten-dimensional $E_8\times E_8$ theory. It also leads to a plausible scenario whereby duality between $SO(32)$ heterotic and Type I superstrings follows from the classical symmetries of the eleven-dimensional world, just as the $SL(2,{\bf Z})$ duality of the ten-dimensional Type IIB theory follows from eleven-dimensional diffeomorphism invariance. 
  The master fields for the large $N$ limit of matrix models and gauge theory are constructed. The master fields satisfy to standard equations of relativistic field theory but fields are quantized according to a new rule. To define the master field we use the Yang-Feldman equation with a free field quantized in the Boltzmannian Fock space. The master field for gauge theory does not take values in a finite-dimensional Lie algebra however there is a non-Abelian gauge symmetry. For the construction of the master field it is essential to work in Minkowski space-time and to use the Wightman correlation functions. The BRST quantization of the master field for gauge theory and a loop equation are considered. 
  For each r = (r_1, r_2,...,r_N) we construct a highest weight module M_r of the Lie algebra W_{1+infty}. The highest weight vectors are specific tau-functions of the N-th Gelfand--Dickey hierarchy. We show that these modules are quasifinite and we give a complete description of the reducible ones together with a formula for the singular vectors.     This paper is the first of a series of papers (q-alg/9602010, q-alg/9602011, q-alg/9602012) on the bispectral problem. 
  We discuss the limitations on space time measurement in the Schwarzchild metric. We find that near the horizon the limitations on space time measurement are of the order of the black hole radius. We suggest that it indicates that a large mass black hole cannot be described by means of local field theory even at macroscopic distances and that any attempt to describe black hole formation and evaporation by means of an effective local field theory will necessarily lead to information loss. We also present a new interpretation of the black hole entropy which leads to $S=cA$ , where $c$ is a constant of order $1$ which does not depend on the number of fields. 
  The previously proposed generalized action principle approach to supersymmetric extended objects is considered in some details for the case of heterotic string in $D=3, 4, 6 ~and~ 10$ space--time dimensions. The proof of the 'off--shell' superdiffeomorphism invariance of the generalized action is presented. The doubly supersymmetric geometric approach to heterotic string is constructed on the basis of generalized action principle (instead of the geometrodynamic condition, used for this previously). It is demonstrated that $D=3$ heterotic string is described by $n=(1,0)$ supersymmetric generalization of the nonlinear Liouville equation. 
  The relativistic model with two types of planar fermions interacting with the Chern-Simons and Maxwell fields is applied to the study of anyon superconductor. It is demonstrated, that the Meissner effect can be realized in the case of the simultaneous presence of the fermions with a different magnetic moment interactions. Under the certain conditions there occures an extra plateau at the magnetization curve. In the order under consideration the spectrum of the electromagnetic field excitations contains the long-range interaction and one massive "photon" state. 
  We present a derivation of the general form of the scalar potential in Yang-Mills theory of a non-commutative space which is a product of a four-dimensional manifold times a discrete set of points. We show that a non-trivial potential without flat directions is obtained after eliminating the auxiliary fields only if constraints are imposed on the mass matrices utilised in the Dirac operator. The constraints and potential are related to a prepotential function. 
  It is shown that a zero curvature representation for $D$-- dimensional $p$-- brane equations of motion originates naturally in the geometric (Lund- Regge- Omnes) approach. To study the possibility to use this zero curvature representation for investigation of nonlinear equations of $p$-- branes, the simplest case of $D$-- dimensional string ($p=1$) is considered. The connection is found between the $SO(1,1)$ gauge (world--sheet Lorentz) invariance of the string theory with a nontrivial dependence on a spectral parameter of the Lax matrices associated with the nonlinear equations describing the embedding of a string world sheet into flat $D$-- dimensional space -- time. Namely, the spectral parameter can be identified with a parameter of constant $SO(1,1)$ gauge transformations, after the deformation of the Lax matrices has been performed. 
  Lattice discretizations of continuous manifolds are common tools used in a variety of physical contexts. Conventional discrete approximations, however, cannot capture all aspects of the original manifold, notably its topology. In this paper we discuss an approximation scheme due to Sorkin which correctly reproduces important topological aspects of continuum physics. The approximating topological spaces are partially ordered sets (posets), the partial order encoding the topology. Now, the topology of a manifold $M$ can be reconstructed from the commutative C*-algebra $\cc(M)$ of continuous functions defined on it. In turn, this algebra is generated by continuous probability densities in ordinary quantum physics on $M$. The latter also serve to specify the domains of observables like the Hamiltonian. For a poset, the role of this algebra is assumed by a noncommutative C$^*$-algebra $\ca $. This fact makes any poset a genuine `noncommutative' (`quantum') space, in the sense that the algebra of its `continuous functions' is a noncommutative C$^*$-algebra. We therefore also have a remarkable connection between finite approximations to quantum physics and noncommutative geometries. We use this connection to develop various approximation methods for doing quantum physics using $\ca$. 
  Calculations of the number of curves on a Calabi-Yau manifold via an instanton expansion do not always agree with what one would expect naively. It is explained how to account for continuous families of instantons via deformation theory and excess intersection theory. The essential role played by degenerate instantons is also explained. This paper is a slightly expanded version of the author's talk at the June 1995 Trieste Conference on S-Duality and Mirror Symmetry. 
  In this talk we discuss a new approximation scheme for non-perturbative calculations in a quantum field theory which is based on the fact that the Schwinger equation of a quantum field model belongs to the class of singularly perturbed equations. The self-interacting scalar field and the Gross-Neveu model are taken as the examples and some non-perturbative solutions of an equation for the propagator are found for these models. The application to QCD is also discussed. 
  We present new $n=(1,1)$ and $n=(1,0)$ supersymmetric generalization of the Liouville equation, which originate from a geometrical approach to describing the classical dynamics of Green--Schwarz superstrings in $N=2,~D=3$ and $N=1,~D=3$ target superspace. Considered are a zero curvature representation and B\"acklund transformations associated with the supersymmetric non--linear equations. 
  A consistent method for calculating the interquark potential generated by the relativistic string with massive ends is proposed. In this approach the interquark potential in the model of the Nambu--Goto string with point--like masses at its ends is calculated. At first the calculation is done in the one--loop approximation and then the variational estimation is performed. The quark mass correction results in decreasing the critical distance (deconfinement radius). When quark mass decreases the critical distance also decreases. For obtaining a finite result under summation over eigenfrequencies of the Nambu--Goto string with massive ends a suitable mode--by--mode subtraction is proposed. This renormalization procedure proves to be completely unique. In the framework of the developed approach the one--loop interquark potential in the model of the relativistic string with rigidity is also calculated. 
  We study in detail the space of perturbations of a pair of dual $N=1$ supersymmetric theories based on an $SU(N_c)$ gauge theory with an adjoint $X$ and fundamentals with a superpotential which is polynomial in $X$. The equivalence between them depends on non-trivial facts about polynomial equations, i.e.\ singularity theory. The classical chiral rings of the two theories are different. Quantum mechanically there are new relations in the chiral rings which ensure their equivalence. Duality interchanges ``trivial'' classical relations in one theory with quantum relations in the other and vice versa. We also speculate about the behavior of the theory without the superpotential. 
  We explore the extent to which string theories with higher-level gauge symmetries and non-standard hypercharge normalizations can reconcile the discrepancy between the string unification scale and the GUT scale extrapolated from the Minimal Supersymmetric Standard Model (MSSM). We determine the phenomenologically allowed regions of (k_Y,k_2,k_3) parameter space, and investigate the proposal that there might exist string models with exotic hypercharge normalizations k_Y which are less than their usual value k_Y=5/3. For a broad class of heterotic string models (encompassing most realistic string models which have been constructed), we prove that k_Y >= 5/3. Beyond this class, however, we show that there exist consistent MSSM embeddings which lead to k_Y < 5/3. We also consider the constraints imposed on k_Y by demanding charge integrality of all unconfined string states, and show that only a limited set of hypercolor confining groups and corresponding values of k_Y are possible. 
  We discuss a general approach to the nonperturbative treatment of quantum field theories based on existence of effective gauge theory on auxiliary ''spectral" Riemann curve. We propose an effective formulation for the exact solutions to some examples of $2d$ string models and $4d$ supersymmetric Yang-Mills theories and consider their natural generalizations. 
  We study the mechanism for appearance of massless solitons in type II string compactifications. We find that by combining $T$-duality with strong/weak duality of type IIB in 10 dimensions enhanced gauge symmetries and massless solitonic hypermultiplets encountered in Calabi-Yau compactifications can be studied perturbatively using D-strings (the strong/weak dual to type IIB string) compactified on ``D-manifolds''. In particular the nearly massless solitonic states of the type IIB compactifications correspond to elementary states of D-strings. As examples we consider the D-string description of enhanced gauge symmetries for type IIA string compactification on ALE spaces with $A_n$ singularities and type IIB on a class of singular Calabi-Yau threefolds. The class we study includes as a special case the conifold singularity in which case the perturbative spectrum of the D-string includes the expected massless hypermultiplet with degeneracy one. 
  A package of Maple 5.3 commands for doing calculations with anticommutative variables is presented. 
  Lorentz-invariant expectation values for antisymmetric tensor field strengths in Calabi-Yau compactification of IIA string theory are considered. These are found to impart magnetic and/or electric charges to the dilaton hypermultiplet. This results in a potential which can have supersymmetric minima at zero coupling or at conifold points in the moduli space. The latter occurs whenever the dilaton charge is aligned with that of the light black hole at the conifold. It is shown that there is a flat direction extending from the conifold along which there is a black hole condensate whose strength is of order the string coupling $g_s$. It is speculated that these new vacua correspond to string compactification on generalized Calabi-Yau spaces which have $c_1=0$ but are not Kahler. 
  We study supersymmetric $SU(N-4)$ gauge theories with a symmetric tensor and $N$ antifundamental representations. The theory with $W=0$ has a dual description in terms of a non-chiral $Spin(8)$ theory with one spinor and $N$ vectors. This duality flows to the $SO(N)$ duality of Seiberg and to a duality proposed by one of us. It also flows to dualities for a number of $Spin(m)$ theories, $m\le 8$. For $N=6$, when an ${\cal N}=2$ SUSY superpotential is added, the singularities of Seiberg and Witten are recovered. For $N\le 6$, a mass for the spinor generates the branches of $SO(8)$ theories found by Intriligator and Seiberg. Other phenomena include a classical constraint mapped to an anomaly equation under duality and an intricate consistency check on the renormalization group flow. 
  Spectrum of elementary string states in type II string theory contains ultra-short multiplets that are marginally stable. $U$-duality transformation converts these states into bound states at threshold of $p$-branes carrying Ramond-Ramond charges, and wrapped around $p$-cycles of a torus. We propose a test for the existence of these marginally stable bound states. Using the recent results of Polchinski and of Witten, we argue that the spectrum of bound states of $p$-branes is in agreement with the prediction of $U$-duality. 
  A recent development of the studies on classical and quasi-classical properties of supersymmetric quantum mechanics in Witten's version is reviewed. First, classical mechanics of a supersymmetric system is considered. Solutions of the classical equations of motion are given and their properties are discussed in some detail. The corresponding quantum model is constructed by canonical quantization. The quantum model is analyzed by Feynman's path integral within a stationary-phase approximation. A quasi-classical quantization rule is derived, which is applicable when supersymmetry is exact or spontaneously broken. 
  The interaction of a cosmic string with a four-dimensional stationary black hole is considered. If a part of an infinitely long string passes close to a black hole it can be captured. The final stationary configurations of such captured strings are investigated. A uniqueness theorem is proved, namely it is shown that the minimal 2-D surface $\Sigma$ describing a captured stationary string coincides with a {\it principal Killing surface}, i.e. a surface formed by Killing trajectories passing through a principal null ray of the Kerr-Newman geometry. Geometrical properties of principal Killing surfaces are investigated and it is shown that the internal geometry of $\Sigma$ coincides with the geometry of a 2-D black or white hole ({\it string hole}). The equations for propagation of string perturbations are shown to be identical with the equations for a coupled pair of scalar fields 'living' in the spacetime of a 2-D string hole. Some interesting features of physics of 2-D string holes are described. In particular, it is shown that the existence of the extra dimensions of the surrounding spacetime makes interaction possible between the interior and exterior of a string black hole; from the point of view of the 2-D geometry this interaction is acausal. Possible application of this result to the information loss puzzle is briefly discussed. 
  The massive scalar field with $\lambda\varphi^4$ interaction placed in $(3+1)$ dimensional box is considered. The sizes of the box are $V\times \beta$ $(V=L^3$ is the volume, $T=1/\beta$ is the temperature). The free energy is evaluated up to the 2nd order of $PT$. The averaging on the vacuum fluctuations is separated from the averaging on the thermal fluctuations explicitly. As result the free-energy is expressed through the scattering amplitudes. We find that in 3-loop approximation the expression for free energy coincides with the ansatz of Bernstein, Dashen, Ma suggested on the base of $S$-matrix formulation of statistical mechanics. The obtained expressions are generalized for higher order of $PT$. 
  We study some properties of a singular Landau-Ginzburg family characterized by the multi-variable superpotential $W=-X^{-1}(Y_1Y_2)^{n-1} + {1\over n} (Y_1Y_2)^n - Y_3Y_4$. We will argue that (the infra-red limit of) this theory describes the topological degrees of freedom of the $c=1$ string compactified at $n$ times the self-dual radius. We also briefly comment on the possible realization of these line singularities as singularities of Calabi-Yau manifolds. 
  We consider the general case of N=2 dual pairs of type IIA/heterotic string theories in four dimensions. We show that if the type IIA string in this pair can be viewed as having been compactified on a Calabi-Yau manifold in the usual way then this manifold must be of the form of a K3 fibration. We also see how the bound on the rank of the gauge group of the perturbative heterotic string has a natural interpretation on the type IIA side. 
  Self-dual gauge potentials admit supersymmetric couplings to higher-spin fields satisfying interacting forms of the first order Dirac--Fierz equation. The interactions are governed by conserved currents determined by supersymmetry. These super-self-dual Yang-Mills systems provide on-shell supermultiplets of arbitrarily extended super-Poincar\'e algebras; classical consistency not setting any limit on the extension N. We explicitly display equations of motion up to the $N=6$ extension. The stress tensor, which vanishes for the $N\le 3$ self-duality equations, not only gets resurrected when $N=4$, but is then a member of a conserved multiplet of gauge-invariant tensors. 
  The physics of the bare Seiberg-Witten action, without supersymmetric partners, is considered in the framework of standard Quantum Field Theory. The topological analysis related to the solutions of the Seiberg-Witten equations is performed and the phase structure of the model is analysed. 
  Problems connected with a choice of the spinorial basis in the $(j,0)\oplus (0,j)$ representation space are discussed. As shown it has profound significance in the relativistic quantum theory. From the methodological viewpoint this fact is related with the important dynamical role played by space-time symmetries for all kind of interactions. 
  Amplitudes for boson-boson and fermion-boson interactions are calculated in the second order of perturbation theory in the Lobachevsky space. An essential ingredient of the used model is the Weinberg's $2(2j+1)$ component formalism for describing a particle of spin $j$, recently developed substantially. The boson-boson amplitude is then compared with the two-fermion amplitude obtained by Skachkov long ago on the ground of the hamiltonian formulation of quantum field theory on the mass hyperboloid, $p_0^2 -{\vec p}^{2}=M^2$, proposed by Kadyshevsky. The parametrization of the amplitudes by means of the momentum transfer in the Lobachevsky space leads to same spin structures in the expressions of $T$ matrices for the fermion and the boson cases. However, certain differences are found. Possible physical applications are discussed. 
  We study the $W_\infty$ algebra in the Calegero-Sutherland model using the exchange operators. The presence of all the sub-algebras of $W_\infty$ is shown in this model. A simplified proof for this algebra, in the symmetric ordered basics, is given. It is pointed out that the algebra contains in general, nonlinear terms. Possible connection to the nonlinear $W_\infty$ is discussed. 
  We study the eigenvectors of the renormalization-group matrix for scalar fields at the Gaussian fixed point, and find that that there exist ``relevant'' directions in parameter space. They correspond to theories with exponential potentials that are nontrivial and asymptotically free. All other potentials, including polynomial potentials, are ``irrelevant,'' and lead to trivial theories. Away from the Gaussian fixed point, renormalization does not induce derivative couplings, but it generates non-local interactions. 
  We obtain three generation SU(3)_c X SU(2)_L X U(1)_Y string models in all of the exactly solvable (0,2) constructions sampled by fermionization. None of these examples, including those that are symmetric abelian orbifolds, rely on the Z_2 X Z_2 orbifold underlying the NAHE basis. We present the first known three generation models for which the hypercharge normalization, k_1, takes values smaller than that obtained from an SU(5) embedding, thus lowering the effective gauge coupling unification scale. All of the models contain fractional electrically charged and vectorlike exotic matter that could survive in the light spectrum. 
  We study the $SU(N)$, level $1$ Wess-Zumino-Witten model, with affine primary fields as spinon fields of fundamental representation. By evaluating the action of the Yangian generators $Q_{0}^{a}, Q_{1}^{a}$ and the Hamiltonian $H_2$ on two spinon states we get a new connection between this conformal field theory and the Calogero-Sutherland model with $SU(N)$ spin. This connection clearly confirms the need for the $W_3$ generator in $H_2$ and an additional term in the $Q^{a}_{1}$. We also evaluate some energy spectra of $H_2$, by acting it on multi-spinon states. 
  The massless Schwinger model without the kinetic term of gauge field has gauge anomaly. We quantize the model as an anomalous gauge theory in the most general class of gauge conditions. We show that the gauge field becomes a dynamical variable because of gauge anomaly. 
  A general method is presented which allows one to determine from the local gauge invariant observables of a quantum field theory the underlying particle and symmetry structures appearing at the lower (ultraviolet) end of the spatio--temporal scale. Particles which are confined to small scales, i.e., do not appear in the physical spectrum, can be uncovered in this way without taking recourse to gauge fields or indefinite metric spaces. In this way notions such as quark, gluon, colour symmetry and confinement acquire a new and intrinsic meaning which is stable under gauge or duality transformations. The method is illustrated by the example of the Schwinger model. 
  Lavelle-McMullan symmetry of QED is examined at classical and quantum levels. It is shown that Lavelle-McMullan symmetry does not give any new non-trivial information in QED by examining the Ward-Takahashi identities. Being inspired by the examination of Ward-Takahashi identity, we construct the generalized non-local and non-covariant symmetries of QED. 
  We address the problem of finite-size anyons, i.e., composites of charges and finite radius magnetic flux tubes. Making perturbative calculations in this problem meets certain difficulties reminiscent of those in the problem of pointlike anyons. We show how to circumvent these difficulties for anyons of arbitrary spin. The case of spin 1/2 is special because it allows for a direct application of perturbation theory, while for any other spin, a redefinition of the wave function is necessary. We apply the perturbative algorithm to the N-body problem, derive the first-order equation of state and discuss some examples. 
  We use path-integrals to derive a general expression for the semiclassical approximation to the partition function of a one-dimensional quantum-mechanical system. Our expression depends solely on ordinary integrals which involve the potential. For high temperatures, the semiclassical expression is dominated by single closed paths. As we lower the temperature, new closed paths appear, including tunneling paths. The transition from single to multiple-path regime corresponds to well-defined catastrophes. Tunneling sets in whenever they occur. (Our formula fully accounts for this feature.) 
  The general theory of the reduction in the number of coupling parameters is discussed. The method involves renormalization group invariant relations between couplings. It is more general than the imposition of symmetries. There are reduced theories with no known symmetry. The reduction scheme is finding many applications. Discussed in some detail are the construction of gauge theories with ``minimal'' coupling for Yang-Mills and matter fields, and the Gauge-Yukawa Unification within N=1 supersymmetric GUT's. 
  The analytic structure of {\it physical} amplitudes is considered for gauge theories with confinement of excitations corresponding to the elementary fields. Confinement is defined in terms of the BRST algebra. BRST-invariant, local, composite fields are introduced, which interpolate between physical asymptotic states. It is shown that the singularities of physical amplitudes are the same as in an effective theory with only physical fields. In particular, there are no structure singularities (anomalous thresholds) associated with confined constituents, like quarks and gluons. The old proofs of dispersion relations for hadronic amplitudes remain valid in QCD. 
  We study a series of $N\!=\!1$ supersymmetric integrable particle theories in $d=1+1$ dimensions. These theories are represented as integrable perturbations of specific $N\!=\!1$ superconformal field theories. Starting from the conjectured $S$-matrices for these theories, we develop the Thermodynamic Bethe Ansatz (TBA), where we use that the 2-particle $S$-matrices satisfy a free fermion condition. Our analysis proves a conjecture by E.~Melzer, who proposed that these $N\!=\!1$ supersymmetric TBA systems are ``folded'' versions of $N\!=\!2$ supersymmetric TBA systems that were first studied by P.~Fendley and K.~Intriligator. 
  We briefly review results on two-dimensional supersymmetric quantum field theories that exhibit factorizable particle scattering. Our particular focus is on a series of $N\!=\!1$ supersymmetric theories, for which exact $S$-matrices have been obtained. A Thermodynamic Bethe Ansatz (TBA) analysis for these theories has confirmed the validity of the proposed $S$-matrices and has pointed at an interesting `folding' relation with a series of $N\!=\!2$ supersymmetric theories. 
  The general four parameter point interaction in one dimensional quantum mechanics is regulated. It allows the exact solution, but not the perturbative one. We conjecture that this is due to the interaction not being asymptotically free. We then propose a different breakup of unperturbed theory and interaction, which now is asymptotically free but leads to the same physics. The corresponding regulated potential can be solved both exactly and perturbatively, in agreement with the conjecture. 
  We feel that non-commutative geometry is to particle physics what Riemannian geometry is to gravity. We try to explain this feeling. 
  It is shown that for the Calogero-Cohn type upper bounds on the number of bound states of a negative spherically symmetric potential $V(r)$, in each angular momentum state, that is, bounds containing only the integral $\int^\infty_0 |V(r)|^{1/2}dr$, the condition $V'(r) \geq 0$ is not necessary, and can be replaced by the less stringent condition $(d/dr)[r^{1-2p}(-V)^{1-p}] \leq 0, 1/2 \leq p < 1$, which allows oscillations in the potential. The constants in the bounds are accordingly modified, depend on $p$ and $\ell$, and tend to the standard value for $p = 1/2$. 
  We give a realization of quantum affine Lie algebra $U_q(\hat A_{N-1})$ in terms of anyons defined on a two-dimensional lattice, the deformation parameter $q$ being related to the statistical parameter $\nu$ of the anyons by $q = e^{i\pi\nu}$. In the limit of the deformation parameter going to one we recover the Feingold-Frenkel fermionic construction of undeformed affine Lie algebra. 
  Arguments for the confinement of transverse gauge field excitations, which are based upon superconvergence relations of the propagator, and upon the BRST algebra, are reviewed and applied to supersymmetric models. They are shown to be in agreement with recent results obtained as a consequence of holomorphy and duality in certain $N=1$ SUSY models. The significance of the one loop anomalous dimension of the gauge field in the Landau gauge is emphasized. For the models considered, it is shown to be proportional, with negative relative sign, to the one loop coefficient of the renormalization group function for the {\it dual map} of the original theory. 
  Based on a model of the d=3 SU(2) pure gauge theory vacuum as a monopole-vortex condensate, we give a quantitative (if model-dependent) estimate of the relation between the string tension and a gauge-invariant measure of the Chern-Simons susceptibility, due to vortex linkages, in the absence of a Chern-Simons term in the action. We also give relations among these quantities and the vacuum energy and gauge-boson mass. Both the susceptibility and the string tension come from the same physics: The topology of linking, twisting, and writhing of closed vortex strings. The closed-vortex string is described via a complex scalar field theory whose action has a precisely-specified functional form, inferred from previous work giving the exact form of a gauge-theory effective potential at low momentum. Applications to high-T phenomena, including B+L anomalous violation, are mentioned. 
  We examine the one-loop vacuum structure of an effective theory of gaugino condensation coupled to the dilaton for string models in which the gauge coupling constant does not receive string threshold corrections. The new ingredients in our treatment are that we take into account the one-loop correction to the dilaton K\"ahler potential and we use a formulation which includes a chiral field $H$ corresponding to the gaugino bilinear. We find through explicit calculation that supersymmetry in the Yang-Mills sector is broken by gaugino condensation.  The dilaton and $H$ field have masses on the order of the gaugino condensation scale independently of the dilaton VEV. Although the calculation performed here is at best a model of the full gaugino condensation dynamics, the result shows that the one-loop correction to the dilaton K\"ahler potential as well as the detailed dynamics at the gaugino condensation scale may play an important role in solving the dilaton runaway problem. 
  The topological twist of N=2, D=4 matter-coupled supergravities requires a suitable R-symmetry. This symmetry is realised in the effective supergravities arising at tree level from certain heterotic compactifications. The set of instanton equations (topological gauge-fixings) is thus obtained. The conditions that R-symmetry should satisfy also when these theories are replaced by their ``exact'' quantum-corrected counterparts are investigated. 
  It is shown that the physical phase space of $\g$-deformed Hamiltonian lattice Yang-Mills theory, which was recently proposed in refs.[1,2], coincides as a Poisson manifold with the moduli space of flat connections on a Riemann surface with $(L-V+1)$ handles and therefore with the physical phase space of the corresponding $(2+1)$-dimensional Chern-Simons model, where $L$ and $V$ are correspondingly a total number of links and vertices of the lattice. The deformation parameter $\g$ is identified with $\frac {2\pi}{k}$ and $k$ is an integer entering the Chern-Simons action. 
  The requirement of target-space duality and the use of nonrenormalization theorems lead to strong constraints on the perturbative prepotential that encodes the low-energy effective action of $N=2$ heterotic superstring vacua. The analysis is done in the context of special geometry, which governs the couplings of the vector multiplets. The presentation is kept at an introductory level. 
  We give a formal proof of the equivalence of Hamiltonian and Lagrangian BRST quantization. This is done for a generic $Sp(2)$-symmetric theory using flat (Darboux) coordinates. A new quantum master equation is derived in a Hamiltonian setting which contains all the Hamiltonian fields and momenta of a given theory. 
  Starting from the self-dual "triplet" of gravitational instanton solutions in Euclidean gravity, we obtain the corresponding instanton solutions in string theory by making use of the target space duality symmetry. These dual triplet solutions can be obtained from the general dual Taub-NUT de Sitter solution through some limiting procedure as in the pure gravity case. The dual gravitational instanton solutions obtained here are self-dual for some cases, with respect to certain isometries, but not always. 
  A relativistic version of the Kubo--Martin--Schwinger boundary condition is presented which fixes the properties of thermal equilibrium states with respect to arbitrary space--time translations. This novel condition is a natural generalization of the relativistic spectrum condition in the vacuum theory and has similar consequences. In combination with the condition of locality it gives rise to a Kaellen--Lehmann type representation of thermal propagators with specific regularity properties. Possible applications of the results and some open problems are outlined. 
  The mathematical description of stable particle-like systems appearing in relativistic quantum field theory at large, respectively small scales or non-zero temperatures is discussed. 
  This is the written version of lectures presented at Cargese 95. A new formulation for a ``restricted'' type of target space duality in classical two dimensional nonlinear sigma models is presented. The main idea is summarized by the analogy: euclidean geometry is to riemannian geometry as toroidal target space duality is to ``restricted'' target space duality. The target space is not required to possess symmetry. These lectures only discuss the local theory. The restricted target space duality problem is identified with an interesting problem in classical differential geometry. 
  The dynamical system of a point particle constrained on a torus is quantized \`a la Dirac with two kinds of coordinate systems respectively; the Cartesian and toric coordinate systems. In the Cartesian coordinate system, it is difficult to express momentum operators in coordinate representation owing to the complication in structure of the commutation relations between canonical variables. In the toric coordinate system, the commutation relations have a simple form and their solutions in coordinate representation are easily obtained with, furthermore, two quantum Hamiltonians turning up. A problem comes out when the coordinate system is transformed, after quantization, from the Cartesian to the toric coordinate system. 
  Spectrum of elementary string states in type II string theory compactified on a torus contains short multiplets which are invariant under only one quarter of the space-time supersymmetry generators. $U$-duality transformation converts these states into bound states of Dirichlet branes which wrap around intersecting cycles of the internal torus. We study a class of these bound states that are dual to the elementary string states at the first excited level, and argue that the degeneracy of these bound states is in agreement with the $U$-duality prediction. 
  A general theory is presented of quantum mechanics of singular, non-autonomous, higher derivative systems. Within that general theory, $n$-th order and $m$-th order Lagrangians are shown to be quantum mechanically equivalent if their difference is a total derivative. 
  We show that a (2+1)-dimensional $P,T-$invariant free fermion system, relevant to $P,T-$conserving models of high-$T_c$ superconductivity, has a U(1,1) dynamical symmetry as well as an $N=3$ supersymmetry with the even generator being a quadratic function of the spin operator and of the generator of chiral ${\rm U_c(1)}$ transformations. We demonstrate that the hidden supersymmetry leads to a non-standard superextension of the (2+1)-dimensional Poincar\'e group. As a result, the one particle states of the $P,T-$invariant fermion system realize an irreducible representation of the Poincar\'e supergroup labelled by the zero eigenvalue of the superspin operator. 
  Fifth order exact corrections to the non-singlet electron structure function in QED are the leading logarithmic approximation using the ad hoc exponentiation prescription proposed by Jadach and Ward and a recurence formula for the elements of the Jadach-Ward series. A comparison with existing third order solutions is also presented. The three next elements of  the Jadac Ward series were calculated numerically and parametrized with an accuracy better than 5x10^-6 in the range of x between 0.01 and 1. 
  A long-standing puzzle about the heterotic string has been what happens when an instanton shrinks to zero size. It is argued here that the answer at the quantum level is that an extra $SU(2)$ gauge symmetry appears that is supported in the core of the instanton. Thus in particular the quantum heterotic string has vacua with higher rank than is possible in conformal field theory. When $k$ instantons collapse at the same point, the enhanced gauge symmetry is $Sp(k)$. These results, which can be tested by comparison to Dirichlet five-branes of Type I superstrings and to the ADHM construction of instantons, give the first example for the heterotic string of a non-perturbative phenomenon that cannot be turned off by making the coupling smaller. They have applications to several interesting puzzles about string duality. 
  Using the renormalization-group formalism, a sigma model of a special type- in which the metric and the dilaton depend explicitly on one of the string coordinates only-is investigated near two dimensions. It is seen that dilatonic gravity coupled to N scalar fields can be expressed in this form, using a string parametrization, and that it possesses the usual UV fixed point. However,in this stringy parametrization of the theory the fixed point for the scalar-dilaton coupling turns out to be trivial, while the fixed point for the gravitational coupling remains to be the same as in previous studies being, in particular, non-trivial. 
  In the paper we present a different proof of the theorem of B. L. Feigin and D. B. Fuchs about the structure of Verma modules over Virasoro algebra. We state some new results about the structure of Verma modules over Neveu-Schwarz. The proof has thwo advantages: first, it is simplier in the most interesting cases (for example in the so called minimal models), second, it can be generalized for Neveu-Schwarz algebra for some class of Verma modules. 
  The ratio Z_1/Z_3 of vertex and wave-function renormalization factors, which is universal (i.e., matter-independent), is shown to equal 1+u which gives the residue of the scalar pole $\propto p_\mu p_\nu /p^2$ of 2-point function < D_\mu c D_\nu \bar c >. This relation is interesting since 1+u=0 has been known to give a sufficient condition for color confinement. We also give an argument that, when 1+u=0 holds, it will be realized by the disappearance of the massless gauge boson pole and is related with the restoration of a certain ``local gauge symmetry" as was discussed by Hata. (Talk given at International Symposium on BRS Symmetry, Sept.~18 -- 22, 1995, Kyoto.) 
  We critically analyze the introduction of an independent zero momentum mode field renormalization for Phi4. It leads to an infrared divergent effective action. It does not achieve its purpose: triviality still gives massless particles in the broken phase in the continuum limit. It leads to an effective potential which is not the low energy limit of the effective action. 
  We present a new method for calculating the Green functions for a lattice scalar field theory in $D$ dimensions with arbitrary potential $V(\phi)$. The method for non-perturbative evaluation of Green functions for $D \! = \! 1$ is generalized to higher dimensions. We define ``hole functions'' $A^{(i)}~(i=0,1,2,\cdots,N \! -\! 1)$ from which one can construct $N$-point Green functions. We derive characteristic equations of $A^{(i)}$ that form a {\it finite closed} set of coupled local equations. It is shown that the Green functions constructed from the solutions to the characteristic equations satisfy the Dyson-Schwinger equations. To fix the boundary conditions of $A^{(i)}$, a prescription is given for selecting the vacuum state at the boundaries. 
  An effective lagrangian analysis of gaugino condensation is performed in a supersymmetric gauge theory with field-dependent gauge couplings described with a linear multiplet. An original aspect of this effective lagrangian is the use of a real vector superfield to describe composite gauge invariant degrees of freedom. The duality equivalence of this approach with the more familiar formulation using a chiral superfield is demonstrated. These results strongly suggest that chiral-linear duality survives nonperturbative effects in superstrings. 
  A brief introduction to Topological Quantum Field Theory as well as a description of recent progress made in the field is presented. I concentrate mainly on the connection between Chern-Simons gauge theory and Vassiliev invariants, and Donaldson theory and its generalizations and Seiberg-Witten invariants. Emphasis is made on the usefulness of these relations to obtain explicit expressions for topological invariants, and on the universal structure underlying both systems. 
  These are the lecture notes of a set of lectures delivered at the 1995 Trieste summer school in June. I review some recent work on duality in four dimensional Maxwell theory on arbitrary four manifolds, as well as a new set of topological invariants known as the Seiberg-Witten invariants. Much of the necessary background material is given, including a crash course in topological field theory, cohomology of manifolds, topological gauge theory and the rudiments of four manifold theory. My main hope is to wet the readers appetite, so that he or she will wish to read the original works and perhaps to enter this field. 
  We review the recent generalization of the basic structures of classical analytical mechanics to field theory within the framework of the De Donder-Weyl (DW) covariant canonical theory. We start from the Poincar\'e-Cartan form and construct the analogue of the symplectic form -- the polysymplectic form of degree n+1, where n is the dimension of the space-time. The dynamical variables are represented by differential forms and the polysymplectic form leads to the definition of the Poisson brackets on forms. The Poisson brackets equip the exterior algebra of dynamical variables with a structure of a "higher-order" Gerstenhaber algebra. We also briefly outline a possible approach to field quantization which proceeds from the DW Hamiltonian formalism and the Poisson brackets of forms. 
  In this paper we explicitly classify all modular invariant partition functions for su(n) at level 2 and 3. Previously, these were known only for level 1. The level 2 exceptionals exist at n=10, 16, and 28; the level 3 exceptionals exist at n=5, 9, and 21. One of these is new, but the others were all anticipated by the "rank-level duality" relating su(n) level k and su(k) level n. The main recent result which this paper rests on is the classification of "ADE_7-type invariants". 
  For fields that vary slowly on the scale of the lightest mass the logarithm of the vacuum functional of a massive quantum field theory can be expanded in terms of local functionals satisfying a form of the Schr\"odinger equation, the principal ingredient of which is a regulated functional Laplacian. We construct to leading order a Laplacian for the O(N) sigma-model that acts on such local functionals. It is determined by imposing rotational invariance in the internal space together with closure of the Poincar\'e algebra. 
  A generalisation to electrodynamics and Yang-Mills theory is presented that permits computation of the speed of light. The model presented herewithin indicates that the speed of light in vacuo is not a universal constant. This may be relevant to the current debate in astronomy over large values of the Hubble constant obtained through the use of the latest generation of ground and space-based telescopes. An experiment is proposed based on Compton scattering to measure deviations in the speed of light. 
  I calculate the semiclassical phase shift ($\delta$), as function of impact parameter ($b$) and velocity ($v$), when one D-brane moves past another. From its low-velocity expansion I show that, for torroidal compactifications, the moduli space of two identical D-branes stays flat to all orders in $\alpha^\prime$. For K3 compactifications, the calculation of the D-brane moduli-space metric can be mapped to a dual gauge-coupling renormalization problem. In the ultrarelativistic regime, the absorptive part of the phase shift grows as if the D-branes were black disks of area $\sim \alpha^\prime ln{1\over 1-v^2}$. The scattering of large fundamental strings shares all the above qualitative features. A side remark concerns the intriguing duality between limiting electric fields and the speed of light. 
  We investigate the Chung-Fukuma-Shapere theory, or Kuperberg theory, of three-dimensional lattice topological field theory. We construct a functor which satisfies the Atiyah's axioms of topological quantum field theory by reformulating the theory as Turaev-Viro type state-sum theory on a triangulated manifold. The theory can also be extended to give a topological invariant of manifolds with boundary. 
  We report on progress towards evaluation of stringy non-perturbative effects, using a two dimensional effective field theory for matrix models. We briefly discuss the relevance of such effects to models of dynamical supersymmetry breaking. 
  We derive a manifestly gauge invariant low energy blocked action for Yang-Mills theory using operator cutoff regularization, a prescription which renders the theory finite with a regulating smearing function constructed for the proper-time integration. By embedding the momentum cutoff scales in the smearing function, operator cutoff formalism allows for a direct application of Wilson-Kadanoff renormalization group to Yang-Mills theory in a completely gauge symmetry preserving manner. In particular, we obtain a renormalization group flow equation which takes into consideration the contributions of higher dimensional operators and provides a systematic way of exploring the influence of these operators as the strong coupling, infrared limit is approached. 
  It is known that Ashtekar's formulation for pure Einstein gravity can be cast into the form of a topological field theory, namely the $SU(2)$ BF theory, with the B-fields subject to an algebraic constraint. We extend this relation between Ashtekar's formalism and BF theories to $N=1$ and $N=2$ supergravities. The relevant gauge groups in these cases become graded Lie groups of $SU(2)$ which are generated by left-handed local Lorentz transformations and left-supersymmetry transformations. As a corollary of these relations, we provide topological solutions for $N=2$ supergravity with a vanishing cosmological constant. It is also shown that, due to the algebraic constraints, the Kalb-Ramond symmetry which is characteristic of BF theories breaks down to the symmetry under diffeomorphisms and right-supersymmetry transformations. 
  In this paper we summarize our recent work about perturbative and non-perturbative effects in four-dimensional heterotic strings with $N=2$ space-time supersymmetry. 
  We present an algebraic approach to string theory. An embedding of $sl(2|1)$ in a super Lie algebra together with a grading on the Lie algebra determines a nilpotent subalgebra of the super Lie algebra. Chirally gauging this subalgebra in the corresponding Wess-Zumino-Witten model, breaks the affine symmetry of the Wess-Zumino-Witten model to some extension of the $N=2$ superconformal algebra. The extension is completely determined by the $sl(2|1)$ embedding. The realization of the superconformal algebra is determined by the grading. For a particular choice of grading, one obtains in this way, after twisting, the BRST structure of a string theory. We classify all embeddings of $sl(2|1)$ into Lie super algebras and give a detailed account of the branching of the adjoint representation. This provides an exhaustive classification and characterization of both all extended $N=2$ superconformal algebras and all string theories which can be obtained in this way. 
  We review some aspects of gauged WZW models. By choosing a nilpotent subgroup as gauge group, one is lead to three main applications: the construction of field theories with an extended conformal symmetry, the construction of the effective action of (extended) 2D gravities and the systematic construction of string theories with some extended gauge symmetry. 
  We discuss the recently devised one-loop gap equation for the magnetic mass of hot QCD. An alternative, and one would hope equivalent, gap equation is presented, which however shows no mass generation at the one-loop level. 
  The holomorphic prepotential of ultraviolet finite N=2 supersymmetric gauge theories is obtained by a partial twisting of N=1 gauge theory in six dimensions, compactified on $\IR^4\timesT^2$. We show that Ward identities for the conserved chiral $\R$-symmetry in these theories generate a set of constraints on the correlation functions of chiral ring operators. These correlators depend only on the coordinates of the $T^2$, and the constraints are analogs of the Knihnik-Zamolodchikov-Bernard equations at the critical level. 
  Supersymmetric closed string theories contain an infinite tower of BPS-saturated, oscillating, macroscopic strings in the perturbative spectrum. When these theories have dual formulations, this tower of states must exist nonperturbatively as solitons in the dual theories. We present a general class of exact solutions of low-energy supergravity that corresponds to all these states. After dimensional reduction they can be interpreted as supersymmetric black holes with a degeneracy related to the degeneracy of the string states. {}For example, in four dimensions we obtain a point-like solution which is asymptotic to a stationary, rotating, electrically-charged black hole with Regge-bounded angular momentum and with the usual ring-singularity replaced by a string source. This further supports the idea that the entropy of supersymmetric black holes can be understood in terms of counting of string states. We also discuss some applications of these solutions to string duality. 
  I build $N=1$ superstrings in $\Bbb R^4$ out of purely geometric bosonic data. The world-sheet is a bilayer of uniform thickness and the $2D$ supercharge vanishes in a natural way. 
  The vacuum energy density is calculated for the $O(N)$ nonlinear sigma models in two dimensions. To obtain $\varepsilon_{vac}$ we assume that each point of the space in which non-perturbative f\/ields are determined can be replaced by a sphere $S^2$ having a small radius $r$ which approaches zero at the very end of the calculation. This assumption allows to get the classical f\/ields generating v.e.v. of the trace of the energy-momentum tensor. 
  We study non-local realizations of extended worldsheet supersymmetries and the associated space-time supersymmetries which arise under a T-duality transformation. These non-local effects appear when the supersymmetries do not commute with the isometry with respect to which T-duality is performed. 
  We comment on an earlier paper of M. Gleiser, regarding mechanisms of first-order phase transitions. 
  Recent work initiated by Strominger has lead to a consistent physical interpretation of certain types of transitions between different string vacua. These transitions, discovered several years ago, involve singular conifold configurations which connect distinct Calabi-Yau manifolds. In this paper we discuss a number of aspects of conifold transitions pertinent to both worldsheet and spacetime mirror symmetry. It is shown that the mirror transform based on fractional transformations allows an extension of the mirror map to conifold boundary points of the moduli space of weighted Calabi-Yau manifolds. The conifold points encountered in the mirror context are not amenable to an analysis via the original splitting constructions. We describe the first examples of such nonsplitting conifold transitions, which turn out to connect the known web of Calabi-Yau spaces to new regions of the collective moduli space. We then generalize the splitting conifold transition to weighted manifolds and describe a class of connections between the webs of ordinary and weighted projective Calabi-Yau spaces. Combining these two constructions we find evidence for a dual analog of conifold transitions in heterotic N$=$2 compactifications on K3$\times $T$^2$ and in particular describe the first conifold transition of a Calabi-Yau manifold whose heterotic dual has been identified by Kachru and Vafa. We furthermore present a special type of conifold transition which, when applied to certain classes of Calabi-Yau K3 fibrations, preserves the fiber structure. 
  We discuss phase structure of chiral symmetry breaking of the $D$-dimensional ($2\leq D\leq3$) Gross-Neveu model at finite temperature, density and constant curvature. We evaluate the effective potential in a weak background approximation to thermalize the model as well as in the leading order of the $1/N$-expansion. A third order critical line is observed similarly to the $D=2$ case. 
  The mode properties for spectral and mixed boundary conditions for massless spin-half fields are derived for the $d$--ball. The corresponding functional determinants and heat-kernel coefficients are presented, the latter as polynomials in $d$. 
  T-Duality is a poorly understood symmetry of the space-time fields of string theory that interchanges long and short distances. It is best understood in the context of toroidal compactification where, loosely speaking, radii of the torus are inverted. Even in this case, however, conventional techniques permit an understanding of the transformations only in the case where the metric on the torus is endowed with Abelian Killing symmetries. Attempting to apply these techniques to a general metric appears to yield a non-local world-sheet theory that would defy interpretation in terms of space-time fields. However, there is now available a simple but powerful general approach to understanding the symmetry transformations of string theory, which are generated by certain similarity transformations of the stress-tensors of the associated conformal field theories. We apply this method to the particular case of T-Duality and i) rederive the known transformations, ii) prove that the problem of non-locality is illusory, iii) give an explicit example of the transformation of a metric that lacks Killing symmetries and iv) derive a simple transformation rule for arbitrary string fields on tori. 
  Starting from the generating functional of the theory of relativistic spinors in 2+1 dimensions interacting through the pure Chern-Simons gauge field, the S-matrix is constructed and seen to be formally the same as that of spinor quantum electrodynamics in 2+1 dimensions with Feynman diagrams having external photon lines excluded, and with the propagator of the topological Chern-Simons photon substituted for the Maxwell photon propagator. It is shown that the absence of real topological photons in the complete set of vector states of the total Hilbert space leads in a given order of perturbation theory to topological unitarity identities that demand the vanishing of the gauge-invariant sum of the imaginary parts of Feynman diagrams with a given number of internal on-shell free topological photon lines. It is also shown, that these identities can be derived outside the framework of perturbation theory. The identities are verified explicitly for the scattering of a fermion-antifermion pair in one-loop order. 
  We formulate a new model which describes higher-spin gauge interactions for matter fields in two dimensions. This model is a higher-spin generalization of d2 gravity and turns out to be integrable. No vanishing higher-spin current conditions are imposed on the matter fields. 
  We present a number of qualitative arguments which strongly suggest that extremal dyonic black holes in the 4--dimensional low energy, classical field theory limit of toroidally compactified heterotic string theory represent largely degenerate classes of states in the quantum theory. We propose a simple expression for the full non--perturbative degeneracy, which contains no free continuous parameters and reproduces the Bekenstein-Hawking $S={A\over 4G_N}$ in the large-area limit (with the ${1\over 4}$ arising from microphysics). We sketch the elements of a physical picture leading to this expression: the holes support much hair, such that in counting it we are led to an effective string theory, and a matching condition whose solutions give the degeneracy. 
  A proof for a non-perturbative C-theorem in four dimensions, capturing the irreversibility of the renormalization group flow in the space of unitary quantum field theories, has not been accomplished, yet. We test the conjectured C-theorems using the exact results recently obtained in N=1 supersymmetric gauge theories. We find that the flow towards the infrared region is consistent with the main proposals for a C-theorem. 
  The $\Delta$-operator of the Batalin-Vilkovisky formalism is the Hamiltonian BRST charge of Abelian shift transformations in the ghost momentum representation. We generalize this $\Delta$-operator, and its associated hierarchy of antibrackets, to that of an arbitrary non-Abelian and possibly open algebra of any rank. We comment on the possible application of this formalism to closed string field theory. 
  We find exact charged black hole solutions of a string effective action that is invariant under S-duality transformations. These black hole solutions have the same causal structure as the Reissner-Nordstrom (RN) solutions. They reduce to the RN solutions for self-dual configurations of the dilaton and to the Garfinkle-Horowitz-Strominger (GHS) solution in the weak (or strong) coupling regime. Using the purely magnetic solutions of the S-duality model we also generate dyonic black hole solutions of the GHS model, which have the causal structure of the RN solutions. 
  We introduce an infrared regulator in Yang--Mills theories under the form of a mass term for the nonabelian fields. We show that the resulting action, built in a covariant linear gauge, is multiplicatively renormalizable by proving the validity at all orders of the Slavnov identity defining the theory. 
  The interaction of a cosmic string with a four-dimensional stationary black hole is considered. If a part of an infinitely long string passes close to a black hole it can be captured. The final stationary configurations of such captured strings are investigated. A uniqueness theorem is proved, namely it is shown that the minimal 2-D surface $\Sigma$ describing a captured stationary string coincides with a {\it principal Killing surface}, i.e. a surface formed by Killing trajectories passing through a principal null ray of the Kerr-Newman geometry. Geometrical properties of principal Killing surfaces are investigated and it is shown that the internal geometry of $\Sigma$ coincides with the geometry of a 2-D black or white hole ({\it string hole}). The equations for propagation of string perturbations are shown to be identical with the equations for a coupled pair of scalar fields 'living' in the spacetime of a 2-D string hole. Some interesting features of physics of 2-D string holes are described. In particular, it is shown that the existence of the extra dimensions of the surrounding spacetime makes interaction possible between the interior and exterior of a string black hole; from the point of view of the 2-D geometry this interaction is acausal. Possible application of this result to the information loss puzzle is briefly discussed. 
  By applying the Hamiltonian reduction scheme we recover the R-matrix of the trigonometric and elliptic Calogero-Moser system. 
  We give analytical arguments and demonstrate numerically the existence of black hole solutions of the $4D$ Effective Superstring Action in the presence of Gauss-Bonnet quadratic curvature terms. The solutions possess non-trivial dilaton hair. The hair, however, is of ``secondary" type", in the sense that the dilaton charge is expressed in terms of the black hole mass. Our solutions are not covered by the assumptions of existing proofs of the ``no-hair" theorem. We also find some alternative solutions with singular metric behaviour, but finite energy. The absence of naked singularities in this system is pointed out. 
  A pseudoclassical model is proposed for the description of planar $P,T-$invariant massive fermions. The quantization of the model leads to the (2+1)-dimensional $P,T-$invariant fermion model used recently in $P,T-$conserving theories of high-T${}_c$ superconductors. The rich symmetry of the quantum model is elucidated through the analysis of the canonical structure of its pseudoclassical counterpart. We show that both the quantum $P,T-$invariant planar massive fermion model and the proposed pseudoclassical model --- for a particular choice of the parameter appearing in the Lagrangian --- have a U(1,1) dynamical symmetry as well as an $N=3$ supersymmetry. The hidden supersymmetry leads to a non-standard superextension of the (2+1)-dimensional Poincar\'e group. In the quantum theory the one particle states provide an irreducible representation of the extended supergroup labelled by the zero eigenvalue of the superspin. We discuss the gauge modification of the pseudoclassical model and compare our results with those obtained from the standard pseudoclassical model for massive planar fermions. 
  For a large class of field theories there exist portions of parameter space for which the loop expansion predicts increased symmetry breaking at high temperature. Even though this behavior would clearly have far reaching implications for cosmology such theories have not been fully investigated in the literature. This is at least partially due to the counter intuitive nature of the result, which has led to speculations that it is merely an artifact of perturbation theory. To address this issue we study the simplest model displaying high temperature symmetry breaking using a Wilson renormalization group approach. We find that although the critical temperature is not reliably estimated by the loop expansion the total volume of parameter space which leads to the inverse phase structure is not significantly different from the perturbative prediction. We also investigate the temperature dependence of the coupling constants and find that they run approximately according to their one-loop $\beta$-functions at high temperature. Thus, in particular, the quartic coupling of $phi^4$ theory is shown to increase with temperature, in contrast to the behavior obtained in some previous studies. 
  A novel BRST quantization is described, which is applied in generalizing the Jackiw-Nair construction of anyon. We have explicitly shown that the matter states connected to an unconventional ("non-zero") BRST ghost sector are physical. They are identified to the Jackiw-Nair system in a particular gauge. Also for the first time an indepth analysis of the present kind for a reducible constraint system, ( where the constraints are not independent), has been performed. 
  We investigate some classical aspects of fundamental strings via numerical experiments. In particular, we study the thermodynamics of a string network within a toroidal universe, as a function of string energy density and space dimensionality. We find that when the energy density of the system is low, the dominant part of the string is in the form of closed loops of the shortest allowed size, which correspond to the momentum string modes. At a certain critical energy density corresponding to the Hagedorn temperature, the system undergoes a phase transition characterized by the formation of very long loops, winding a number of times around the torus. These loops correspond to the winding string modes. As the energy density is increased, all the extra energy goes into these long strings. We then study the lifetime of winding modes as a function of the space densionality. We find that in the low--energy density regime, long winding strings decay only if the space dimensionality of the toroidal universe is equal to 3. This finding supports the proposed cosmological scenario by Brandenberger and Vafa, which attempts to explain the space dimensionality and to avoid the initial singularity by means of string theory. 
  The star-matrix models are difficult to solve due to the multiple powers of the Vandermonde determinants in the partition function. We apply to these models a modified Q-matrix approach and we get results consistent with those obtained by other methods.As examples we study the inhomogenous gaussian model on Bethe tree and matrix $q$-Potts-like model. For the last model in the special cases $q=2$ and $q=3$, we write down explicit formulas which determinate the critical behaviour of the system.For $q=2$ we argue that the critical behaviour is indeed that of the Ising model on the $\phi^3$ lattice. 
  In general bosonic closed string backgrounds the ghost-dilaton is not the only state in the semi-relative BRST cohomology that can change the dimensionless string coupling. This fact is used to establish complete dilaton theorems in closed string field theory. The ghost-dilaton, however, is the crucial state: for backgrounds where it becomes BRST trivial we prove that the string coupling becomes an unobservable parameter of the string action. For backgrounds where the matter CFT includes free uncompactified bosons we introduce a refined BRST problem by including the zero-modes "x" of the bosons as legal operators on the complex. We argue that string field theory can be defined on this enlarged complex and that its BRST cohomology captures accurately the notion of a string background. In this complex the ghost-dilaton appears to be the only BRST-physical state changing the string coupling. 
  The first and second homology groups are computed for configuration spaces of framed three-dimensional point particles with annihilation included, when up to two particles and an antiparticle are present. 
  We present a version of ten-dimensional IIA supergravity containing a 9-form potential for which the field equations are equivalent to those of the standard, massless, IIA theory for vanishing 10-form field strength, $F_{10}$, and to those of the `massive' IIA theory for non-vanishing $F_{10}$. We exhibit a multi 8-brane solution of this theory which preserves half the supersymmetry. We propose this solution as the effective field theory realization of the Dirichlet 8-brane of type IIA superstring theory. 
  Instanton and seven-brane solutions of type IIB supergravity carrying charges in the Ramond-Ramond sector are constructed. The singular seven-brane has a quantized \RR \ \lq magnetic' charge whereas its dual is the instanton, which is non-singular in the string frame and has an associated {\it global} \lq electric' charge. The product of these charges is constrained by a Dirac quantization condition. The instanton has the form of a space-time wormhole in the string frame, and is responsible for the non-conservation of the Noether current. 
  The Weil algebra of a semisimple Lie group and an exterior algebra of a symplectic manifold possess antibrackets. They are applied to formulate the models of non--abelian equivariant cohomologies. 
  We elaborate further the functional Schr\"{o}dinger-picture approach to the quantum field in curved spacetimes using the generalized invariant method and construct explicitly the Fock space, which we relate with the thermal field theory. We apply the method to a free massive scalar field in the de Sitter spacetime, and find the exact quantum states, construct the Fock space, and evaluate the two-point function and correlation function. 
  A few observations concerning topological string theories at the string-tree level are presented: (1) The tree-level, large phase space solution of an arbitrary model is expressed in terms of a variational problem, with an ``action'' equal, at the solution, to the one-point function of the puncture operator, and found by solving equations of Gauss-Manin type; (2) For $A_k$ Landau-Ginzburg models, an extension to large phase space of the usual residue formula for three-point functions is given. 
  q-oscillator models are considered in two and higher dimensions and their symmetries are explored. New symmetries are found for both isotropic and anisotropic cases. Applications to the spectra of triatomic molecules and superdeformed nuclei are discussed. 
  The Greensite-Halpern method of stabilizing bottomless Euclidean actions is applied to zerodimensional O(N) sigma models with unstable $A_k$ singularities in the $N = \infty$ limit. 
  We review the construction of $N=1$ supergravity models where the Higgs and super-Higgs effects are simultaneously realized, with naturally vanishing classical vacuum energy and goldstino components along gauge-non-singlet directions: this situation is likely to occur in the effective theories of realistic string models. (Invited talk presented at SUSY--95, Palaiseau, France, 15--19 May 1995) 
  The quantization of isomonodromic deformation of a meromorphic connection on the torus is shown to lead directly to the Knizhnik-Zamolodchikov-Bernard equations in the same way as the problem on the sphere leads to the system of Knizhnik-Zamolodchikov equations. The Poisson bracket required for a Hamiltonian formulation of isomonodromic deformations is naturally induced by the Poisson structure of Chern-Simons theory in a holomorphic gauge fixing. This turns out to be the origin of the appearance of twisted quantities on the torus. 
  We test the prediction of a hagedorn density of BPS states which carry RR charge in type II compactifications. We find that in certain cases they correspond to the supersymmetric ground states for a gas of identical 0-branes. 
  This is an elementary review of our recent work on the classification of the spectra of those two-dimensional rational conformal field theories (RCFTs) whose (maximal) chiral algebras are current algebras. We classified all possible partition functions for such theories when the defining finite-dimensional Lie algebra is simple. The concepts underlying this work are emphasized, and are illustrated using simple examples. 
  We consider entropy and relative entropy in Field theory and establish relevant monotonicity properties with respect to the couplings. The relative entropy in a field theory with a hierarchy of renormalization group fixed points ranks the fixed points, the lowest relative entropy being assigned to the highest multicritical point. We argue that as a consequence of a generalized $H$ theorem Wilsonian RG flows induce an increase in entropy and propose the relative entropy as the natural quantity which increases from one fixed point to another in more than two dimensions. 
  A particular dispersive generalization of long water wave equation in $1+1$ dimensions, which is important in the study of matrix models without scaling limit, known as two--Boson (TB) equation, as well as the associated hierarchy has been derived from the zero curvature condition on the gauge group $SL(2,R)\otimes U(1)$. The supersymmetric extension of the two--Boson (sTB) hierarchy has similarly been derived from the zero curvature condition associated with the gauge supergroup $OSp(2|2)$. Topological algebras arise naturally as the second Hamiltonian structure of these classical integrable systems, indicating a close relationship of these models with 2d topological field theories. 
  We introduce an operator version of the BRST-BFV effective action for arbitrary systems with first-class constraints. Using the Schwinger action principle we calculate the propagators corresponding to: (i) the parametrized non-relativistic free particle, (ii) the relativistic free particle and (iii) the spining relativistic free particle. Our calculation correctly imposes the BRST-invariance at the end-points. The precise use of the additional boundary terms required in the description of fermionic variables is also incorporated. 
  The three generation superstring models in the free fermionic models have had remarkable success in describing the real--world. The most explored models use the NAHE set to obtain three generations and to separate the hidden and observable sectors. It is of course well known that the full NAHE set is not required in order to construct three generation free fermionic models. I argue that all the semi--realistic free fermionic models that have been constructed to date correspond to $Z_2\times Z_2$ orbifolds. Thus, the successes of the semi--realistic free fermionic models, if taken seriously, suggest that the true string vacuum is a $Z_2\times Z_2$ orbifold with nontrivial background fields and quantized Wilson lines. 
  A lagrangian for gauge fields coupled to fermions with the Kac-Moody group as its gauge group yields, for the pure fermions sector, an ideal gas of Kac-Moody fermions. The canonical partition function for the $\hat U(1)$ case is shown to have a maximum temperature $kT_{M} = |\lambda| /\pi$, where $\lambda $ is the coupling of the super charge operator $G_0 $ to the fermions. This result is similar to the case of strings but unlike strings the result is obtained from a well-defined lagrangian. 
  In this paper we show how the BRST quantization can be applied to systems possessing only second-class constraints through their conversion to some first-class ones starting with our method exposed in [Nucl.Phys. B456 (1995)473]. Thus, it is proved that i) for a certain class of second-class systems there exists a standard coupling between the variables of the original phase-space and some extravariables such that we can transform the original system into a one-parameter family of first-class systems; ii) the BRST quantization of this family in a standard gauge leads to the same path integral as that of the original system. The analysis is accomplished in both reducible and irreducible cases. In the same time, there is obtained the Lagrangian action of the first-class family and its provenience is clarified. In this context, the Wess-Zumino action is also derived. The results from the theoretical part of the paper are exemplified in detail for the massive Yang-Mills theory and for the massive abelian three-form gauge fields. 
  We study a functional field theory of membranes coupled to a rank--three tensor gauge potential. We show that gauge field radiative corrections lead to membrane condensation which turns the gauge field into a {\it massive spin--0 field}. This is the Coleman--Weinberg mechanism for {\it membranes}. An analogy is also drawn with a type--II superconductor. The ground state of the system consists of a two--phase medium in which the superconducting background condensate is ``pierced'' by four dimensional domains, or ``bags'', of non superconducting vacuum. Bags are bounded by membranes whose physical thickness is of the order of the inverse mass acquired by the gauge field. 
  A new generalization of the vector Schwinger model is considered where gauge symmetry is broken at the quantum mechanical level. By proper extension of the phase space this broken symmetry has been restored. Also an equivalent first class theory is reformulated in the actual phase space using Mitra and Rajaraman's prescription \cite{mr1,mr2}. A BRST invariant effective action is also formulated. The new dynamical fields introduced, turn into Wess-Zumino scalar. 
  We employ an algebraic approach for unifying perturbative and non-perturbative superstring states on an equal footing, in the form of U-duality multiplets, at all excited string levels. In compactified type-IIA supertring theory we present evidence that the multiplet is labelled by two spaces, ``index'' space and ``base'' space, on which U acts without mixing them. Both spaces are non-perturbative extensions of similar spaces that label perturbative T-duality multiplets. Base space consists of all the central charges of the 11D SUSY algebra, while index space corresponds to represetations of the maximal compact subgroup K in U. This structure predicts the quantum numbers of the non-perturbative states. We also discuss whether and how U-multiplets may coexist with 11-dimensional multiplets, that are associated with an additional non-perturbative 11D structure that seems to be lurking behind in the underlying theory. 
  The spectral properties of the Dirac Hamiltonian in the the Aharonov-Bohm potential are discussed. By using the Krein-Friedel formula, the density of states (DOS) for different self-adjoint extensions is calculated. As in the nonrelativistic case, whenever a bound state is present in the spectrum it is always accompanied by a (anti)resonance at the energy. The Aharonov-Casher theorem must be corrected for singular field configurations. There are no zero (threshold) modes in the Aharonov-Bohm potential. For our choice of the 2d Dirac Hamiltonian, the phase-shift flip is shown to occur at only positive energies. This flip gives rise to a surplus of the DOS at the lower threshold coming entirely from the continuous part of the spectrum. The results are applied to several physical quantities: the total energy, induced fermion-number, and the axial anomaly. 
  We show that abelian bosonization of 1+1 dimensional fermion systems can be interpreted as duality transformation and, as a conseguence, it can be generalized to arbitrary dimensions in terms of gauge forms of rank $d-1$, where $d$ is the dimension of the space. This permit to treat condensed matter systems in $d>1$ as gauge theories. Furthermore we show that in the ``scaling" limit the bosonized action is quadratic in a wide class of condensed matter systems. (Talk given at ``Common trends in Condensed Matter and High Energy Physics", September 3--10, 1995 -- Chia). 
  We show how to perform systematically improvable variational calculations in the $O(2N)$ Gross-Neveu model for generic $N$, in such a way that all infinities usually plaguing such calculations are accounted for in a way compatible with the perturbative renormalization group . The final point is a general framework for the calculation of non-perturbative quantities like condensates, masses etc$\ldots$, in an asymptotically free field theory. For the Gross-Neveu model, the numerical results obtained from a ``2-loop'' down to low values of $N$. 
  The two dimensional surface of a sphere can be parametrized by coordinates representing two charged pions acting as Goldstone bosons of a broken $SU_2$ symmetry. We construct in full concrete detail, and in a general class of coordinate systems, all the relevant structure forming a framework for this low energy effective theory. 
  Schild's null (tensionless) strings are discussed in certain flat and curved backgrounds. We find closed, stationary, null strings as natural configurations existing on the horizons of spacetimes which possess such null hypersurfaces. Examples of these are obtained in Schwarzschild and Rindler spacetimes. A dynamic null string is also identified in Rindler spacetime. Furthermore, a general prescription (with explicit examples) is outlined by means of which null string configurations can be obtained in a large class of cosmological backgrounds. 
  We show that, in 1+1 dimensional gauge theories, a heavy probe charge is screened by dynamical massless fermions both in the case when the source and the dynamical fermions belong to the same representation of the gauge group and, unexpectedly, in the case when the representation of the probe charge is smaller than the representation of the massless fermions. Thus, a fractionally charged heavy probe is screened by dynamical fermions of integer charge in the massless Schwinger model, and a colored probe in the fundamental representation is screened in $QCD_2$ with adjoint massless Majorana fermions. The screening disappears and confinement is restored as soon as the dynamical fermions are given a non-zero mass. For small masses, the string tension is given by the product of the light fermion mass and the fermion condensate with a known numerical coefficient. Parallels with 3+1 dimensional $QCD$ and supersymmetric gauge theories are discussed. 
  We employ projection operator techniques in Hilbert space to derive a continuous sequence of effective Hamiltonians which describe the dynamics on successively larger length scales. We show for the case of \phi^4 theory that the masses and couplings in these effective Hamiltonians vary in accordance with 1-loop renormalization group equations. This is evidence for an intimate connection between Wilson's renormalization and the venerable Bloch-Feshbach formalism. 
  A q-deformed two-dimensional phase space is studied as a model for a noncommutative phase space. A lattice structure arises that can be interpreted as a spontaneous breaking of a continuous symmetry. The eigenfunctions of a Hamiltonian that lives on such a lattice are derived as wavefunctions in ordinary $x$-space. 
  These lectures deal with: (1) a brief review of the theory of flexible random manifolds (with fixed intrinsic metric), connected to the physics of polymerized membranes, and of the effect of extrinsic curvature (crumpling transitions); (2) a discussion of the effect of self-avoidance and its renormalization group treatment in term of a non-local field theory (this last part is not much different from cond-mat/9509096). 
  We study quantum effects in five dimensions in heterotic superstring theory compactified on K_3 x S_1 and analyze the conjecture that its dual effective theory is eleven-dimensional supergravity compactified on a Calabi-Yau threefold. This theory is also equivalent to type II superstring theory compactified on the same Calabi-Yau manifold, in an appropriate large volume limit. In this limit the conifold singularity disappears and is replaced by a singularity associated to enhanced gauge symmetries, as naively expected from the heterotic description. Furthermore, we exhibit the existence of additional massless states which appear in the strong coupling regime of the heterotic theory and are related to a different type of singular points on Calabi-Yau threefolds. 
  A quantum effective action for gauge field theories is constructed that is gauge invariant and independent of the choice of gauge breaking terms and the ghost determinant. 
  We compute the effective action and correlators of the Polyakov loop operator in the Schwinger model at finite temperature and discuss the realization of the discrete symmetries that occur there. We show that, due to nonlocal effects of massless fermions in two spacetime dimensions, the discrete symmetry which governs the screening of charges is spontaneously broken even in an effective one-dimensional model, when the volume is infinite. In this limit, the thermal state of the Schwinger model screens an arbitrary external charge; consequently the model is in the deconfined phase, with the charge of the deconfined fermions completely screened. In a finite volume we show that the Schwinger model is always confining. 
  We consider the theory of bosonic closed strings on the flat background R(25,1). We show how the BRST complex can be extended to a complex where the string center of mass operator, x^mu_0, is well defined. We investigate the cohomology of the extended complex. We demonstrate that this cohomology has a number of interesting features. Unlike in the standard BRST cohomology, there is no doubling of physical states in the extended complex. The cohomology of the extended complex is more physical in a number of of aspects related to the zero-momentum states. In particular, we show that the ghost number one zero-momentum cohomology states are in one to one correspondence with the generators of the global symmetries of the background i.e., the Poincare algebra. 
  We study conditions under which an odd symmetry of the integrand leads to localization of the corresponding integral over a (super)manifold. We also show that in many cases these conditions guarantee exactness of the stationary phase approximation of such integrals. 
  We apply the method of \underline{zeta} functions, together with the $n_\mu^*$-prescription for the temporal gauge, to evaluate the thermodynamic pressure in QCD at finite temperature $T$. Working in the imaginary-time formalism and employing a special version of the unified-gauge prescription, we show that the pure-gauge contribution to the pressure at two loops is given by $P_2^{\mbox{{\scriptsize gauge}}} = -(g^2/144)N_cN_gT^4$, where $N_c$ and $N_g$ denote the number of colours and gluons, respectively. This result agrees with the value in the Feynman gauge. 
  We consider a real manifold of dimension 3 or 4 with Minkovsky metric, and with a connection for a trivial GL(n,C) bundle over that manifold. To each light ray on the manifold we assign the data of paralel transport along that light ray. It turns out that these data are not enough to reconstruct the connection, but we can add more data, which depend now not from lines but from 2-planes, and which in some sence are the data of parallel transport in the complex light-like directions, then we can reconstruct the connection up to a gauge transformation. There are some interesting applications of the construction: 1) in 4 dimensions, the self-dual Yang Mills equations can be written as the zero curvature condition for a pair of certain first order differential operators; one of the operators in the pair is the covariant derivative in complex light-like direction we studied. 2) there is a relation of this Radon transform with the supersymmetry. 3)using our Radon transform, we can get a measure on the space of 2 dimensional planes in 4 dimensional real space. Any such measure give rise to a Crofton 2-density. The integrals of this 2-density over surfaces in R^4 give rise to the Lagrangian for maps of real surfaces into R^4, and therefore to some string theory. 4) there are relations with the representation theory. In particular, a closely related transform in 3 dimensions can be used to get the Plancerel formula for representations of SL(2,R). 
  Until now all known static multi black hole solutions described BPS states with charges of the same sign. Such solutions could not be related to flat directions in the space of BPS states. The total number of such states could not spontaneously increase because of the charge conservation. We show that there exist static BPS configurations which remain in equilibrium even if they consist of states with opposite electric (or magnetic) charges from vector multiplets. This is possible because of the exact cancellation between the Coulomb and scalar forces. In particular, in the theories with N=4 or N=2 supersymmetry there exist stable massless multi center configurations with vanishing total charge. Since such configurations have vanishing energy and charge independently of their number, they can be associated with flat directions in the space of all possible BPS states. For N=2 case this provides a realization of the idea that BPS condensates could relate to each other different vacua of the string theory. 
  Consistency of the Bekenstein bound on entropy requires the physical dimensions of particles to grow with momentum as the particle is boosted to transplanckian energies. In this paper the problem of particle growth in heterotic string theory is mapped into a problem involving the properties of BPS saturated black holes as the charge is increased. Explicit calculation based on the black hole solutions of Sen are shown to lead to a growth pattern consistent with the holographic speculation described in earlier work. 
  We consider correlation functions of operators and the operator product expansion in two-dimensional quantum gravity. First we introduce correlation functions with geodesic distances between operators kept fixed. Next by making two of the operators closer, we examine if there exists an analog of the operator product expansion in ordinary field theories. Our results suggest that the operator product expansion holds in quantum gravity as well, though special care should be taken regarding the physical meaning of fixing geodesic distances on a fluctuating geometry. 
  The Toda lattice defined by the Hamiltonian $H={1\over 2} \sum_{i=1}^n p_i^2 + \sum_{i=1}^{n-1} \nu_i e^{q_i-q_{i+1}}$ with $\nu_i\in \{ \pm 1\}$, which exhibits singular (blowing up) solutions if some of the $\nu_i=-1$, can be viewed as the reduced system following from a symmetry reduction of a subsystem of the free particle moving on the group $G=SL(n,\Real )$. The subsystem is $T^*G_e$, where $G_e=N_+ A N_-$ consists of the determinant one matrices with positive principal minors, and the reduction is based on the maximal nilpotent group $N_+ \times N_-$. Using the Bruhat decomposition we show that the full reduced system obtained from $T^*G$, which is perfectly regular, contains $2^{n-1}$ Toda lattices. More precisely, if $n$ is odd the reduced system contains all the possible Toda lattices having different signs for the $\nu_i$. If $n$ is even, there exist two non-isomorphic reduced systems with different constituent Toda lattices. The Toda lattices occupy non-intersecting open submanifolds in the reduced phase space, wherein they are regularized by being glued together. We find a model of the reduced phase space as a hypersurface in ${\Real}^{2n-1}$. If $\nu_i=1$ for all $i$, we prove for $n=2,3,4$ that the Toda phase space associated with $T^*G_e$ is a connected component of this hypersurface. The generalization of the construction for the other simple Lie groups is also presented. 
  We review the global issues associated to gauge fixing ambiguities and their consequence for glueball spectroscopy. To avoid infrared singularities the theory is formulated in a finite volume. The examples of a cubic and spherical geometry will be discussed in some detail. Our methods are not powerful enough to study the infinite volume limit, but the results clearly indicate that for low-lying states, wave functionals are sensitive to global gauge copies which we will argue is equivalent to saying that they are sensitive to the geometric and topological features of configuration space. 
  We give three formulas for meromorphic eigenfunctions (scattering states) of Sutherland's integrable N-body Schroedinger operators and their generalizations. The first is an explicit computation of the Etingof-Kirillov traces of intertwining operators, the second an integral representation of hypergeometric type, and the third is a formula of Bethe ansatz type. The last two formulas are degenerations of elliptic formulas obtained previously in connection with the Knizhnik-Zamolodchikov-Bernard equation. The Bethe ansatz formulas in the elliptic case are reviewed and discussed in more detail here: Eigenfunctions are parametrized by a ``Hermite-Bethe'' variety, a generalization of the spectral variety of the Lame' operator. We also give the q-deformed version of our first formula. In the scalar sl_N case, this gives common eigenfunctions of the commuting Macdonald-Rujsenaars difference operators. 
  The Dirac Hamiltonian with the Aharonov-Bohm potential provides an example of a non-Fredholm operator for which all spectral asymmetry comes entirely from the continuous spectrum. In this case one finds that the use of standard definitions of the resolvent regularized, the heat kernel regularized, and the Witten indices misses the contribution coming from the continuous spectrum and gives vanishing spectral asymmetry and axial anomaly. This behaviour in the case of the continuous spectrum seems to be general and its origin is discussed. 
  In the present article we display a new constructive quantum field theory approach to quantum gauge field theory, utilizing the recent progress in the integration theory on the moduli space of generalized connections modulo gauge transformations. That is, we propose a new set of Osterwalder Schrader like axioms for the characteristic functional of a measure on the space of generalized connections modulo gauge transformations rather than for the associated Schwinger distributions. We show non-triviality of our axioms by demonstrating that they are satisfied for two-dimensional Yang-Mills theory on the plane and the cylinder. As a side result we derive a closed and analytical expression for the vacuum expectation value of an arbitrary product of Wilson loop functionals from which we derive the quantum theory along the Glimm and Jaffe algorithm which agrees exactly with the one as obtained by canonical methods. 
  We determine the dilaton and moduli vacuum expectation values using the one-loop effective potential and topological constraints. A new ingredient of this analysis is that we use a dilaton K\"ahler potential that includes renormalization effects to all loops. We find that the dilaton vacuum expectation value is related to certain topological properties of the compact spacetime. We demonstrate that values of the dilaton vacuum expectation value that are consistent with the weak scale measurements can be dynamically obtained in this fashion. 
  After a brief historical comment on the study of BRS(or BRST) symmetry , we discuss the quantization of gauge theories with Gribov copies. A path integral with BRST symmetry can be formulated by summing the Gribov-type copies in a very specific way if the functional correspondence between $\tau$ and the gauge parameter $\omega$ defined by $\tau (x) = f( A_{\mu}^{\omega}(x))$ is ``globally single valued'', where $f( A_{\mu}^{\omega}(x)) = 0 $ specifies the gauge condition. As an example of the theory which satisfies this criterion, we comment on a soluble gauge model with Gribov-type copies recently analyzed by Friedberg, Lee, Pang and Ren. We also comment on a possible connection of the dynamical instability of BRST symmetry with the Gribov problem on the basis of an index notion. 
  In cohomological field theory we can obtain topological invariants as correlation functions of BRS cohomology classes. A proper understanding of BRS cohomology which gives non-trivial results requires the equivariant cohomology theory. Both topological Yang-Mills theory and topological string theory are typical examples of this fact. After reviewing the role of the equivariant cohomology in topological Yang-Mills theory, we show in purely algebraic framework how the $U(1)$ equivariant cohomology in topological string theory gives the gravitational descendants. The free energy gives a generating function of topological correlation functions and leads us to consider a deformation family of cohomological field theories. In topological strings such a family is controlled by the theory of integrable system. This is most easily seen in the Landau-Ginzburg approach by looking at the contact term interactions between topological observables. 
  Interpretation of exact results on the low-energy limit of $4d$ $N=2$ SUSY YM in the language of $1d$ integrability theory is reviewed. The case of elliptic Calogero system, associated with the flow between $N=4$ and $N=2$ SUSY in $4d$, is considered in some detail. 
  A very elementary model of a single positive hermitian random matrix coupled to an external matrix is defined and studied. Expanding the exact effective action around its classical solution leads to the ``quantum Penner action'', from which a rich structure of correlation functions is obtained. These are shown to be equal to the all-orders perturbative expansion of tachyon amplitudes in the two-dimensional string at self-dual radius. 
  The renormalization of the Chern-Simons parameter is investigated by using an exact and manifestly gauge invariant evolution equation for the scale-dependent effective average action. 
  We study the soldering of two Siegel chiral bosons into one scalar field in a gravitational background. 
  It is shown how to extend the formal variational calculus in order to incorporate integrals of divergences into it. Such a generalization permits to study nontrivial boundary problems in field theory on the base of canonical formalism. 
  The status of gaugino condensation in low-energy string theory is reviewed. Emphasis is given to the determination of the efective action below condensation scale in terms of the 2PI and Wilson actions. We illustrate how the different perturbative duality symmetries survive this simple nonperturbative phenomenon, providing evidence for the believe that these are exact nonperturbative symmetries of string theory. Consistency with T duality lifts the moduli degeneracy. The $B_{\mu\nu}-axion$ duality also survives in a nontrivial way in which the degree of freedom corresponding to $B_{\mu\nu}$ is replaced by a massive $H_{\mu\nu\rho}$ field but duality is preserved. S duality may also be implemented in this process. Some general problems of this mechanism are mentioned and the possible nonperturbative scenarios for supersymmetry breaking in string theory are discussed. 
  We obtain a Quantum Electrodynamics in 2+1 dimensions by applying a Kaluza--Klein type method of dimensional reduction to Quantum Electrodynamics in 3+1 dimensions rendering the model more realistic to application in solid-state systems, invariant under translations in one direction. We show that the model obtained leads to an effective action exhibiting an interesting phase structure and that the generated Chern--Simons term survives only in the broken phase. 
  The radiation from the black holes of a 1+1-dimensional chiral quantum gravity model is studied. Most notably, a non-trivial dependence on a renormalization parameter that characterizes the anomaly relations is uncovered in an improved semiclassical approximation scheme; this dependence is not present in the naive semiclassical approximation. 
  We study the model of (2 + 1)-dimensional relativistic fermions in a random non-Abelian gauge potential at criticality. The exact solution shows that the operator expansion contains a conserved current - a generator of a continuous symmetry. The presence of this operator changes the operator product expansion and gives rise to logarithmic contributions to the correlation functions at the critical point. We calculate the distribution function of the local density of states in this model and find that it follows the famous log-normal law. 
  It is demonstrated that in the (2+1)-dimensional topologically massive gauge theories an agreement of the Pauli-Villars regularization scheme with the other schemes can be achieved by employing pairs of auxiliary fermions with the opposite sign masses. This approach does not introduce additional violation of discrete (P and T) symmetries. Although it breaks the local gauge symmetry only in the regulator fields' sector, its trace disappears completely after removing the regularization as a result of superrenormalizability of the model. It is shown also that analogous extension of the Pauli-Villars regularization in the vector particle sector can be used to agree the arbitrary covariant gauge results with the Landau ones. The source of ambiguities in the covariant gauges is studied in detail. It is demonstrated that in gauges that are softer in the infrared region (e.g. Coulomb or axial) nonphysical ambiguities inherent to the covariant gauges do not arise. 
  We propose a possible internal structure for a Schwarzschild black hole resulting from the creation of multiple de Sitter universes with lightlike boundaries when the curvature reaches Planckian values. The intersection of the boundaries is studied and a scenario leading to disconnected de Sitter universes is proposed. The application to the information loss problem is then discussed. 
  (2+1)-dimensional relativistic fractional spin particles are considered within the framework of the group-theoretical approach to anyons starting from the level of classical mechanics and concluding by the construction of the minimal set of linear differential field equations. 
  This paper shows that there is a correspondence between quasi-exactly solvable models in quantum mechanics and sets of orthogonal polynomials $\{ P_n\}$. The quantum-mechanical wave function is the generating function for the $P_n (E)$, which are polynomials in the energy $E$. The condition of quasi-exact solvability is reflected in the vanishing of the norm of all polynomials whose index $n$ exceeds a critical value $J$. The zeros of the critical polynomial $P_J(E)$ are the quasi-exact energy eigenvalues of the system. 
  We present some generalizations of a recently proposed alternative approach to nonabelian gauge theories based on the causal Epstein-Glaser method in perturbative quantum field theory. Nonabelian gauge invariance is defined by a simple commutator relation in every order of perturbation theory separately using only the linear (abelian) BRS-transformations of the asymptotic fields. This condition is sufficient for the unitarity of the S-matrix in the physical subspace. We derive the most general specific coupling compatible with the conditions of nonabelian gauge invariance and normalizability. We explicitly show that the quadrilinear terms, the four-gluon-coupling and the four-ghost-coupling, are generated by our linear condition of nonabelian gauge invariance. Moreover, we work out the required generalizations for linear gauges. 
  We test the unified-gauge formalism by computing a Wilson loop in Yang-Mills theory to one-loop order. The unified-gauge formalism is characterized by the abritrary, but fixed, four-vector $N_\mu$, which collectively represents the light-cone gauge $(N^2 = 0)$, the temporal gauge $(N^2 > 0)$, the pure axial gauge $(N^2 < 0)$ and the planar gauge $(N^2 < 0)$. A novel feature of the calculation is the use of distinct sets of vectors, $\{ n_{\mu}, n_{\mu}^{\ast} \}$ and $\{N_{\mu}, N_{\mu}^{\ast}\}$, for the path and for the gauge-fixing constraint, respectively. The answer for the Wilson loop is independent of $N_{\mu}$, and agrees numerically with the result obtained in the Feymman gauge. 
  We give background material and some details of calculations for two recent papers [1,2] where we derived a path integral representation of the transition element for supersymmetric and nonsupersymmetric nonlinear sigma models in one dimension (quantum mechanics). Our approach starts from a Hamiltonian $H(\hat{x}, \hat{p}, \hat{\psi}, \hat{\psi}^\dagger)$ with a priori operator ordering. By inserting a finite number of complete sets of $x$ eigenstates, $p$ eigenstates and fermionic coherent states, we obtain the discretized path integral and the discretized propagators and vertices in closed form. Taking the continuum limit we read off the Feynman rules and measure of the continuum theory which differ from those often assumed. In particular, mode regularization of the continuum theory is shown in an example to give incorrect results. As a consequence of time-slicing, the action and Feynman rules, although without any ambiguities, are necessarily noncovariant, but the final results are covariant if $\hat{H}$ is covariant. All our derivations are exact. Two loop calculations confirm our results. 
  We discuss the threshold tree amplitudes in diverse nonintegrable quantum field theories in the framework of integrability. The amplitudes are related to some Baker functions defined on the auxiliary spectral curves and the nullification phenomena are shown to allow a topological interpretation. 
  We consider the supersymmetric WZNW model gauged in a manifestly supersymmetric way. We find the BRST charge and the necessary condition for nilpotency. In the BRST framework the model proves to be a Lagrangian formulation of the supersymmetric coset construction, known as the N=1 Kazama-Suzuki coset construction. 
  The problems of Classical Electrodynamics with the electron equation of motion and with non-integrable singularity of its self-field stress tensor are well known. They are consequences, we show, of neglecting terms that are null off the charge world line but that gives a non null contribution on its world line. The self-field stress tensor of a point classical electron is integrable, there is no causality violation and no conflict with energy conservation in its equation of motion, and there is no need of any kind of renormalization nor of any change in the Maxwell's theory for this.   (This is part of the paper hep-th/9510160, stripped , for simplicity, of its non-Minkowskian geometrization of causality and of its discussion about the physical meaning of the Maxwell-Faraday concept of field). 
  Let $B_{\mu \nu }^a$ ($a=1,...,N$) be a system of $N$ free two-form gauge fields, with field strengths $H_{\mu \nu \rho }^a = 3 \partial _{[\mu }B_{\nu \rho ]}^a$ and free action $S_0$ equal to $(-1/12)\int d^nx\ g_{ab}H_{\mu \nu \rho }^aH^{b\mu \nu \rho }$ ($n\geq 4$). It is shown that in $n>4$ dimensions, the only consistent local interactions that can be added to the free action are given by functions of the field strength components and their derivatives (and the Chern-Simons forms in $5$ mod $3$ dimensions). These interactions do not modify the gauge invariance $B_{\mu \nu }^a\rightarrow B_{\mu \nu }^a+\partial _{[\mu }\Lambda _{\nu ]}$ of the free theory. By contrast, there exist in $n=4$ dimensions consistent interactions that deform the gauge symmetry of the free theory in a non trivial way. These consistent interactions are uniquely given by the well-known Freedman-Townsend vertex. The method of proof uses the cohomological techniques developed recently in the Yang-Mills context to establish theorems on the structure of renormalized gauge-invariant operators. 
  A class class of transformations in a super phase space (we call them D-transformations) is described, which play in theories with second-class constraints the role of ordinary canonical transformations in theories without constraints. 
  We discuss in detail the semiclassical approximation for the CGHS model of two-dimensional dilatonic black holes. This is achieved by a formal expansion of the full Wheeler-DeWitt equation and the momentum constraint in powers of the gravitational constant. In highest order, the classical CGHS solution is recovered. The next order yields a functional Schr\"odinger equation for quantum fields propagating on this background. We show explicitly how the Hawking radiation is recovered from this equation. Although described by a pure quantum state, the expectation value of the number operator exhibits a Planckian distribution with respect to the Hawking temperature.  We then show how this Hawking radiation can lead to the decoherence of black hole superpositions. The cases of a superposition of a black hole with a white hole, as well as of a black hole with no hole, are treated explicitly. 
  We give an explicit demonstration, using the rigorous Feynman rules developed in~$\0^{1}$, that the regularized trace $\tr \gamma_5 e^{-\beta \Dslash^2}$ for the gravitational chiral anomaly expressed as an appropriate quantum mechanical path integral is $\beta$-independent up to two-loop level. Identities and diagrammatic notations are developed to facilitate rapid evaluation of graphs given by these rules. 
  We study quantum corrections for a family of 24 non-supersymmetric heterotic strings in two dimensions. We compute their genus two cosmological constant using the hyperelliptic formalism and the genus one two-point functions for the massless states. From here we get the mass corrections to the states in the massless sector and discuss the role of the infrared divergences that appear in the computation. We also study some tree-level aspects of these theories and find that they are classified not only by the corresponding Niemeier lattice but also by their {\it hidden} right-moving gauge symmetry. 
  The paper is a chapter of the above-mentioned book. It aims to give an expository presentation of author's version of the non-Abelian Stokes theorem in the framework of path-integral formalism. 
  We give an Atiyah-Patodi-Singer index theory construction of the bundle of fermionic Fock spaces parametrized by vector potentials in odd space dimensions and prove that this leads in a simple manner to the known Schwinger terms (Faddeev-Mickelsson cocycle) for the gauge group action. We relate the APS construction to the bundle gerbe approach discussed recently by Carey and Murray, including an explicit computation of the Dixmier-Douady class. An advantage of our method is that it can be applied whenever one has a form of the APS theorem at hand, as in the case of fermions in an external gravitational field. 
  The periods of the three-form on a Calabi-Yau manifold are found as solutions of the Picard-Fuchs equations; however, the toric varietal method leads to a generalized hypergeometric system of equations which has more solutions than just the periods. This same extended set of equations can be derived from symmetry considerations. Semi-periods are solutions of this extended system. They are obtained by integration of the three-form over chains; these chains can be used to construct cycles which, when integrated over, give periods. In simple examples we are able to obtain the complete set of solutions for the extended system. We also conjecture that a certain modification of the method will generate the full space of solutions in general. 
  It is shown that the idea of ``minimal'' coupling to gauge fields can be conveniently implemented in the proper time formalism by identifying the equivalent of a ``covariant derivative''. This captures some of the geometric notion of the gauge field as a connection. The proper time equation is also generalized so that the gauge invariances associated with higher spin massive modes can be made manifest, at the free level, using loop variables. Some explicit examples are worked out illustrating these ideas. 
  New examples of N=2 supersymmetric conformal field theories are found as fixed points of SU(2) N=2 supersymmetric QCD. Relations among the scaling dimensions of their relevant chiral operators, global symmetries, and Higgs branches are understood in terms of the general structure of relevant deformations of non-trivial N=2 conformal field theories. The spectrum of scaling dimensions found are all those compatible with relevant deformations of a y^2 = x^3 singular curve. 
  In the Bargmann-Fock representation the coordinates $z^i$ act as bosonic creation operators while the partial derivatives $\partial_{z^j}$ act as annihilation operators on holomorphic $0$-forms as states of a $D$-dimensional bosonic oscillator. Considering also $p$-forms and further geometrical objects as the exterior derivative and Lie derivatives on a holomorphic ${\bf C}^D$, we end up with an analogous representation for the $D$-dimensional supersymmetric oscillator. In particular, the supersymmetry multiplet structure of the Hilbert space corresponds to the cohomology of the exterior derivative. In addition, a 1-complex parameter group emerges naturally and contains both time evolution and a homotopy related to cohomology. Emphasis is on calculus. 
  We present a very simplified analysis of how one can overcome the Gribov problem in a non-abelian gauge theory. Our formulae, albeit quite simplified, show that possible breakdowns of the Slavnov-Taylor identity could in principle come from singularities in space of gauge orbits. To test these ideas we exhibit the calculation of a very simple correlation function of 2-dimensional topological gravity and we show how in this model the singularities of the moduli space induce a breakdown of the Slavnov-Taylor identity. We comment on the technical relevance of the possibility of including the singularities into a finite number of cells of the moduli space. 
  This is a talk given at YKIS '95, primarily to non-string theorists. I review the evidence for string duality, the principle that any string theory at strong coupling looks like another string theory at weak coupling. A postscript summarizes developments since the conference. 
  We discuss a relation between two-loop bosonic worldline Green functions which are obtained by Schmidt and Schubert in two different parametrizations of a two-loop worldline. These Green functions are transformed into each other by some transformation rules based on reparametrizations of the proper time and worldline modular parameters. 
  Ten dimensional type IIA and IIB theories with p-branes are compactified to 8-dimensions. It is shown that resulting branes can be classified according to the representations of $\bf {SL(3,Z) \times SL(2,Z)}$. These p-branes can also be obtained by compactification of M theory on three torus and various wrappings of membrane and five brane of the eleven dimensional theory. It is argued that there is evidence for bound states of the branes in eight dimensions as is the case in the interpretation of $\bf {SL(2,Z)}$ family of string solutions obtained by Schwarz. 
  An extended free fermionic construction of the internal N=1 world sheet supercurrent for four-dimensional superstring theory is given. We show how it can describe theories with massless fermions, and we discuss the corresponding N=2 superconformal algebra. As an intermediate step, we show that an internal N=2 global superconformal invariance occurs in any superstring theory with massless fermions at tree level. To demonstrate this fact, we give the N=2 supercurrents for a model with N=1 space-time supersymmetry and a model without space-time supersymmetry. 
  We study some low-lying physical states in a superstring theory based on the quadratically non-linear $SO(N)$--extended superconformal algebra. In the realisation of the algebra that we use, all the physical states are discrete, analogous to the situation in a one-scalar bosonic string. The BRST operator for the $N=3$ case needs to be treated separately, and its construction is given here. 
  The usual supermembrane solution of $D=11$ supergravity interpolates between $R^{11}$ and $AdS_4 \times round~S^7$, has symmetry $P_3 \times SO(8)$ and preserves $1/2$ of the spacetime supersymmetries for either orientation of the round $S^7$. Here we show that more general supermembrane solutions may be obtained by replacing the round $S^7$ by any seven-dimensional Einstein space $M^7$. These have symmetry $P_3 \times G$, where $G$ is the isometry group of $M^7$. For example, $G=SO(5) \times SO(3)$ for the squashed $S^7$. For one orientation of $M^7$, they preserve $N/16$ spacetime supersymmetries where $1\leq N \leq 8$ is the number of Killing spinors on $M^7$; for the opposite orientation they preserve no supersymmetries since then $M^7$ has no Killing spinors. For example $N=1$ for the left-squashed $S^7$ owing to its $G_2$ Weyl holonomy, whereas $N=0$ for the right-squashed $S^7$. All these solutions saturate the same Bogomol'nyi bound between the mass and charge. Similar replacements of $S^{D-p-2}$ by Einstein spaces $M^{D-p-2}$ yield new super $p$-brane solutions in other spacetime dimensions $D\leq 11$. In particular, simultaneous dimensional reduction of the above $D=11$ supermembranes on $S^1$ leads to a new class of $D=10$ elementary string solutions which also have fewer supersymmetries. 
  Within the metric structure endowed with two orthogonal space-like Killing vectors a class of solutions of the Einstein-Maxwell-Dilaton field equations is presented. Two explicitly given sub-classes of solutions bear an interpretation as colliding plane waves in the low-energy limit of the heterotic string theory. 
  We study the degenerating limits of superconformal theories for compactifications on singular K3 and Calabi-Yau threefolds. We find that in both cases the degeneration involves creating an Euclidean two-dimensional black hole coupled weakly to the rest of the system. Moreover we find that the conformal theory of A_n singularities of K3 are the same as that of the symmetric fivebrane. We also find intriguing connections between ADE (1,n) non-critical strings and singular limits of superconformal theories on the corresponding ALE space. 
  $ \hat {U}(1)$ Kac-Moody gauge fields have the infinite dimensional $ \hat{U}(1)$ Kac-Moody group as their gauge group. The pure gauge sector, unlike the usual $U(1)$ Maxwell lagrangian, is nonlinear and nonlocal; the Euclidean theory is defined on a $d+1$-dimensional manifold $ {\cal{R}}_d \times {\cal{S}}^1 $ and hence is also asymmetric. We quantize this theory using the background field method and examine its renormalizability at one-loop by analyzing all the relevant diagrams. We find that, for a suitable choice of the gauge field propagators, this theory is one-loop renormalizable in $3+1$ dimensions. This pure abelian Kac-Moody gauge theory in $3+1$ dimensions has only one running coupling constant and the theory is asymptotically free. When fermions are added the number of independent couplings increases and a richer structure is obtained. Finally, we note some features of the theory which suggest its possible relevance to the study of anisotropic condensed matter systems, in particular that of high-temperature superconductors. 
  We present a theorem describing a dual relation between the local geometry of a space admitting a symmetric second-rank Killing tensor, and the local geometry of a space with a metric specified by this Killing tensor. The relation can be generalized to spinning spaces, but only at the expense of introducing torsion. This introduces new supersymmetries in their geometry. Interesting examples in four dimensions include the Kerr-Newman metric of spinning black-holes and self-dual Taub-NUT. 
  The static potential is investigated in string and membrane theories coupled to $U(1)$ gauge fields (specifically, external magnetic fields) and with antisymmetric tensor fields. The explicit dependence of the potential on the shape of the extended objects is obtained, including a careful calculation of the quantum effects. Noting the features which are common to the dynamics of strings and membranes moving in background fields and to the swimming of micro-organisms in a fluid, the latter problem is studied. The Casimir energy of a micro-organism is estimated, taking into account the quantum effects and the backreaction from the outside fluid. The static potential is investigated in string and membrane theories coupled to $U(1)$ gauge fields (specifically, external magnetic fields) and with antisymmetric tensor fields. The explicit dependence of the potential on the shape of the extended objects is obtained, including a careful calculation of the quantum effects. Noting the features which are common to the dynamics of strings and membranes moving in background fields and to the swimming of micro-organisms in a fluid, the latter problem is studied. The Casimir energy of a micro-organism is estimated, taking into account the quantum effects and the backreaction from the outside fluid. 
  We show that the zero-temperature physics of planar Josephson junction arrays in the self-dual approximation is governed by an Abelian gauge theory with periodic mixed Chern-Simons term describing the charge-vortex coupling. The periodicity requires the existence of (Euclidean) topological excitations which determine the quantum phase structure of the model. The electric-magnetic duality leads to a quantum phase transition between a superconductor and a superinsulator at the self-dual point. We also discuss in this framework the recently proposed quantum Hall phases for charges and vortices in presence of external offset charges and magnetic fluxes: we show how the periodicity of the charge-vortex coupling can lead to transitions to anyon superconductivity phases. We finally generalize our results to three dimensions, where the relevant gauge theory is the so-called BF system, with an antisymmetric Kalb-Ramond gauge field. 
  The notion of a higher bundle gerbe is introduced to give a geometric realization of the higher degree integral cohomology of certain manifolds. We consider examples using the infinite dimensional spaces arising in gauge theories. 
  We consider slow motion of a pointlike topological defect (vortex) in the nonlinear Schrodinger equation minimally coupled to Chern-Simons gauge field and subject to external uniform magnetic field. It turns out that a formal expansion of fields in powers of defect velocity yields only the trivial static solution. To obtain a nontrivial solution one has to treat velocities and accelerations as being of the same order. We assume that acceleration is a linear form of velocity. The field equations linearized in velocity uniquely determine the linear relation. It turns out that the only nontrivial solution is the cyclotron motion of the vortex together with the whole condensate. This solution is a perturbative approximation to the center of mass motion known from the theory of magnetic translations. 
  We consider the general case of a type IIA string compactified on a Calabi-Yau manifold which has a heterotic dual description. It is shown that the nonabelian gauge symmetries which can appear nonperturbatively in the type II string but which are understood perturbatively in the heterotic string are purely a result of string-string duality in six dimensions. We illustrate this with some examples. 
  We present a new realization of scalar integrable hierarchies in terms of the Toda lattice hierarchy. In other words, we show on a large number of examples that an integrable hierarchy, defined by a pseudodifferential Lax operator, can be embedded in the Toda lattice hierarchy. Such a realization in terms the Toda lattice hierarchy seems to be as general as the Drinfeld--Sokolov realization. 
  We construct the open string boundary states corresponding to various time-dependent deformations of the D-brane and explore several ways in which they may be used to study stringy soliton collective coordinate quantum dynamics. Among other things, we find that D-strings have exact moduli corresponding to arbitrary chiral excitations of the basic soliton. These are presumably the duals of the BPS-saturated excitations of the fundamental Type IIB string. These first steps in a systematic study of the dynamics and interactions of Dirichlet-brane solitons give further evidence of the consistency of Polchinski's new approach to string soliton physics. 
  The dilute A_3 model is a solvable IRF (interaction-round-a-face) model with three local states and adjacency conditions encoded by the Dynkin diagram of the Lie algebra A_3. It can be regarded as a solvable version of a critical Ising model in a magnetic field. One therefore expects the scaling limit to be governed by Zamolodchikov's integrable perturbation of the c=1/2 conformal field theory. We perform a detailed numerical investigation of the solutions of the Bethe ansatz equation for the off-critical model. Our results agree perfectly with the predicted values for the lowest masses of the stable particles and support the assumptions on the nature of the Bethe ansatz solutions which enter crucially in a recent thermodynamic Bethe ansatz calculation of the factorized scattering matrix. 
  It is shown that the renormalisation group flow in coupling constant space can be interpreted in terms of a dynamical equation for the couplings analogous to viscous fluid flow under the action of a potential. For free scalar field theory the flow is geodesic in two dimensions, while for $D \neq 2$ it is only geodesic in certain limits, e.g. for vanishing external source. For the 1-D Ising model the renormalisation flow is geodesic when the external magnetic field vanishes. 
  We analyze nonabelian massive Higgs-free theories in the causal Epstein-Glaser approach. Recently, there has been renewed interest in these models. In particular we consider the well-known Curci-Ferrari model and the nonabelian St\"uckelberg models. We explicitly show the reason why the considered models fail to be unitary. In our approach only the asymptotic (linear) BRS-symmetry has to be considered. 
  The quark mass dependence of the energy spectrum in the Nambu--Goto string with point--like masses (quarks) at its ends is analyzed. To this end, linearized equations of motion and boundary conditions in this model are considered. It is shown that for sufficiently large quark masses, the first excited state in string spectrum may be arbitrary close to the ground state. Obviously this points to infrared instability in the system under consideration. Possible modifications of string model which could remove this drawback are discussed. 
  We review the status of duality symmetries in superstring theories. These discrete symmetries mark the striking differences between theories of pointlike objects and theories of extended objects. They prove to be very helpful in understanding non-perturbative effects in string theories. We will also briefly discuss the strange role played by open strings and their solitons in the emerging scenario. 
  The Green's Function Monte Carlo method of Chin et al is applied to SU(2) Yang-Mills theory in (2+1)D. Accurate measurements are obtained for the ground-state energy and mean plaquette value, and for various Wilson loops. The results are compared with series expansions and coupled cluster estimates, and with the Euclidean Monte Carlo results of Teper. A striking demonstration of universality between the Hamiltonian and Euclidean formulations is obtained. 
  We find low energy equivalences between $N=2$ supersymmetric gauge theories with different simple gauge groups with and without matter. We give a construction of equivalences based on subgroups and find all examples with maximal simple subgroups. This is used to solve some theories with exceptional gauge groups $G_2$ and $F_4$. We are also able to solve an $E_6$ theory on a codimension one submanifold of its moduli space. 
  In this paper we formulate Maxwell and Dirac theories as an already unified theory (in the sense of Misner and Wheeler). We introduce Dirac spinors as "Dirac square root" of the Faraday bivector, and use this in order to find a spinorial representation of Maxwell equations. Then we show that under certain circunstances this spinor equation reduces to an equation formally identical to Dirac equation. Finally we discuss certain conditions under which this equation can be really interpreted as Dirac equation, and some other possible interpretations of this result. 
  We show that Maxwell equations and Dirac equation (with zero mass term) have both subluminal and superluminal solutions in vacuum. We also discuss the possible fundamental physical consequences of our results. 
  We study the interacting chiral boson and observe that a naive gauging procedure leaves the covariant chiral constraint incompatible with the field equations. Consistency, therefore, rules out most gauging schemes: in a left chiral scalar, only the coupling with the left chiral currents leads to consistent results, in discordance with current literature. 
  We show that the Nonlinear Schr\"odinger Equation and the related Lax pair in 1+1 dimensions can be derived from 2+1 dimensional Chern-Simons Topological Gauge Theory. The spectral parameter, a main object for the Loop algebra structure and the Inverse Spectral Transform, has appear as a homogeneous part (condensate) of the statistical gauge field, connected with the compactified extra space coordinate. In terms of solitons, a natural interpretation for the one-dimensional analog of Chern-Simons Gauss law is given. 
  We present an explicit construction for the central extension of the group $\Map(X, G)$ where $X$ is a compact manifold and $G$ is a Lie group. If $X$ is a complex curve we obtain a simple construction of the extension by the Picard variety $\Pic(X)$. The construction is easily adapted to the extension of $\Aut(E)$, the gauge group of automorphisms of a nontrivial vector bundle $E$. 
  Two-dimensional fermionic string theory is shown to have a structure of topological model, which is isomorphic to a tensor product of two topological ghost systems independent of each other. One of them is identified with $c=1$ bosonic string theory while the other has trivial physical contents. This fact enables us to regard two-dimensional fermionic string theory as an embedding of $c=1$ bosonic string theory in the moduli space of fermionic string theories. Upon this embedding, the discrete states of $c=1$ string theory are mapped to those of fermionic string theory, which is considered to be the origin of the similarity between the physical spectra of these two theories. We also discuss a novel BRST operator associated with this topological structure. 
  The SL(2,R) WZW model, one of the simplest models for strings propagating in curved space time, was believed to be non-unitary in the algebraic treatment involving affine current algebra. It is shown that this was an error that resulted from neglecting a zero mode that must be included to describe the correct physics of non-compact WZW models. In the presence of the zero mode the mass-shell condition is altered and unitarity is restored. The correct currents, including the zero mode, have logarithmic cuts on the worldsheet. This has physical consequences for the spectrum because a combination of zero modes must be quantized in order to impose periodic boundary conditions on mass shell in the physical sector. To arrive at these results and to solve the model completely, the SL(2,R) WZW model is quantized in a free field formalism that differs from previous ones in that the fields and the currents are Hermitean, there are cuts, and there is a new term that could be present more generally, but is excluded in the WZW model. 
  Einstein-Maxwell equations with a cosmological constant are analyzed in a four dimensional stationary spacetime admitting in addition a two dimensional group $G_2$ of spatial isometries. Charged rotating open and closed black string solutions are found. 
  It is shown how to transform the three dimensional BTZ black hole into a four dimensional cylindrical black hole (i.e., black string) in general relativity. This process is identical to the transformation of a point particle in three dimensions into a straight cosmic string in four dimensions. 
  We review some of our recent results concerning the relationship between the Real-Space Renormalization Group method and Quantum Groups. We show this relation by applying real-space RG methods to study two quantum group invariant Hamiltonians, that of the XXZ model and the Ising model in a transverse field (ITF) defined in an open chain with appropriate boundary terms. The quantum group symmetry is preserved under the RG transformation except for the appearence of a quantum group anomalous term which vanishes in the classical case. This is called {\em the quantum group anomaly}. We derive the new qRG equations for the XXZ model and show that the RG-flow diagram obtained in this fashion exhibits the correct line of critical points that the exact model has. In the ITF model the qRG-flow equations coincide with the tensor product decomposition of cyclic irreps of $SU_q(2)$ with $q^4=1$. 
  In this work we study the zero-charge sector of massive two-dimensional Quantum Chromodynamics in the decoupled formulation. We find that some general features of the massless theory, concerning the constraints and the right- and left-moving character of the corresponding BRST currents, survive in the massive case. The implications for the integrability properties previously valid in the massless case, and the structure of the Hilbert space are discussed. 
  We study the formation of vacuum condensates in $2+1$ dimensional QED in the presence of inhomogeneous background magnetic fields. For a large class of magnetic fields, the condensate is shown to be proportional to the inhomogeneous magnetic field, in the large flux limit. This may be viewed as a {\it local} form of the {\it integrated} degeneracy-flux relation of Aharonov and Casher. 
  We discuss string theory vacua which have the wrong number of spacetime dimensions, and give a crude argument that vacua with more than four large dimensions are improbable. We then turn to two dimensional vacua, which naively appear to violate Bekenstein's entropy principle. A classical analysis shows that the naive perturbative counting of states is unjustified. All excited states of the system have strong coupling singularities which prevent us from concluding that they really exist. A speculative interpretation of the classical solutions suggests only a finite number of states will be found in regions bounded by a finite area. We also argue that the vacuum degeneracy of two dimensional classical string theory is removed in quantum mechanics. The system appears to be in a Kosterlitz-Thouless phase. This leads to the conclusion that it is also improbable to have only two large spacetime dimensions in string theory. However, we note that, unlike our argument for high dimensions, our conclusions about the ground state have neglected two dimensional quantum gravitational effects, and are at best incomplete. 
  The force between like sign BPS saturated objects generally vanishes. This is a reflection of the fact that BPS states are really massless uncharged particles with nonvanishing momenta in compactified directions. Two like sign BPS objects with zero relative velocity can be viewed as a boosted state of two neutral massless particles in a state of vanishing relative motion. By contrast two unlike sign BPS particles may be thought of as colliding objects moving in opposite directions in compact space. This leads to complicated interactions which are totally intractable at present. We illustrate this by considering the potential between opposite sign zero-D- branes . 
  This thesis deals with planar gauge theories in which some gauge group G is spontaneously broken to a finite subgroup H. The spectrum consists of magnetic vortices, global H charges and dyonic combinations exhibiting topological Aharonov-Bohm interactions. Among other things, we review the Hopf algebra D(H) related to this residual discrete H gauge theory, which provides an unified description of the spin, braid and fusion properties of the aforementioned particles. The implications of adding a Chern-Simons (CS) term to these models are also addressed. We recall that the CS actions for a compact gauge group G are classified by the cohomology group H^4(BG,Z). For finite groups H this classification boils down to the cohomology group H^3(H,U(1)). Thus the different CS actions for a finite group H are given by the inequivalent 3-cocycles of H. It is argued that adding a CS action for the broken gauge group G leads to additional topological interactions for the vortices governed by a 3-cocycle for the residual finite gauge group H determined by a natural homomorphism from H^4(BG,Z) to H^3(H,U(1)). Accordingly, the related Hopf algebra D(H) is deformed into a quasi-Hopf algebra. These general considerations are illustrated by CS theories in which the direct product of some U(1) gauge groups is broken to a finite subgroup H. It turns out that not all conceivable 3-cocycles for finite abelian gauge groups H can be obtained in this way. Those that are not reached are the most interesting. A Z_2 x Z_2 x Z_2 CS theory given by such a 3-cocycle, for instance, is dual to an ordinary gauge theory with nonabelian gauge group the dihedral group of order eight. Finally, the CS theories with nonabelian finite gauge group a dihedral or double dihedral group are also discussed in full detail. 
  The large $k$ asymptotics (perturbation series) for integrals of the form $\int_{\cal F}\mu e^{i k S}$, where $\mu$ is a smooth top form and $S$ is a smooth function on a manifold ${\cal F}$, both of which are invariant under the action of a symmetry group ${\cal G}$, may be computed using the stationary phase approximation. This perturbation series can be expressed as the integral of a top form on the space $\cM$ of critical points of $S$ mod the action of ${\cal G}$. In this paper we overview a formulation of the ``Feynman rules'' computing this top form and a proof that the perturbation series one obtains is independent of the choice of metric on ${\cal F}$ needed to define it. We also overview how this definition can be adapted to the context of $3$-dimensional Chern--Simons quantum field theory where ${\cal F}$ is infinite dimensional. This results in a construction of new differential invariants depending on a closed, oriented $3$-manifold $M$ together with a choice of smooth component of the moduli space of flat connections on $M$ with compact structure group $G$. To make this paper more accessible we warm up with a trivial example and only give an outline of the proof that one obtains invariants in the Chern--Simons case. Full details will appear elsewhere. 
  Recently it was shown how to formulate the finite-element equations of motion of a non-Abelian gauge theory, by gauging the free lattice difference equations, and simultaneously determining the form of the gauge transformations. In particular, the gauge-covariant field strength was explicitly constructed, locally, in terms of a path ordered product of exponentials (link operators). On the other hand, the Dirac and Yang-Mills equations were nonlocal, involving sums over the entire prior lattice. Earlier, Matsuyama had proposed a local Dirac equation constructed from just the above-mentioned link operators. Here, we show how his scheme, which is closely related to our earlier one, can be implemented for a non-Abelian gauge theory. Although both Dirac and Yang-Mills equations are now local, the field strength is not. The technique is illustrated with a direct calculation of the current anomalies in two and four space-time dimensions. Unfortunately, unlike the original finite-element proposal, this scheme is in general nonunitary. 
  In presence of string solitons, index theorems for the generalised Dirac operators have to be revisited. We show that in supersymmetric configurations the fermionic operators decouple, so that there are no mixing effects between different fermions in the index theorems. We extend the index theorems in presence of torsion to the generic case of manifolds with boundary, which naturally appear in string solutions and apply this result to the soliton solution by Callan, Harvey and Strominger. 
  We derive a geometric integration formula for the partition function of a classical dynamical system and use it to show that corrections to the WKB approximation vanish for any Hamiltonian which generates conformal motions of some Riemannian geometry on the phase space. This generalizes previous cases where the Hamiltonian was taken as an isometry generator. We show that this conformal symmetry is similar to the usual formulations of the Duistermaat-Heckman integration formula in terms of a supersymmetric Ward identity for the dynamical system. We present an explicit example of a localizable Hamiltonian system in this context and use it to demonstrate how the dynamics of such systems differ from previous examples of the Duistermaat-Heckman theorem. 
  We investigate the gauging of conformal algebras with relations between the generators. We treat the $W_{5/2}$--algebra as a specific example. We show that the gauge-algebra is in general reducible with an infinite number of stages. We show how to construct the BV-extended action, and hence the classical BRST charge. An important conclusion is that this can always be done in terms of the generators of the $W$--algebra only, that is, independent of the realisation. The present treatment is still purely classical, but already enables us to learn more about reducible gauge algebras and the BV-formalism. 
  In these lecture notes, we present a self-contained discussion of planar gauge theories broken down to some finite residual gauge group H via the Higgs mechanism. The main focus is on the discrete H gauge theory describing the long distance physics of such a model. The spectrum features global H charges, magnetic vortices and dyonic combinations. Due to the Aharonov-Bohm effect, these particles exhibit topological interactions. Among other things, we review the Hopf algebra related to this discrete H gauge theory, which provides an unified description of the spin, braid and fusion properties of the particles in this model. Exotic phenomena such as flux metamorphosis, Alice fluxes, Cheshire charge, (non)abelian braid statistics, the generalized spin-statistics connection and nonabelian Aharonov-Bohm scattering are explained and illustrated by representative examples.    Preface: Broken symmetry revisited,    1 Basics: 1.1 Introduction, 1.2 Braid groups, 1.3 Z_N gauge theory,        1.3.1 Coulomb screening, 1.3.2 Survival of the Aharonov-Bohm effect,        1.3.3 Braid and fusion properties of the spectrum, 1.4 Nonabelian        discrete gauge theories, 1.4.1 Classification of stable magnetic        vortices, 1.4.2 Flux metamorphosis, 1.4.3 Including matter,    2 Algebraic structure: 2.1 Quantum double, 2.2 Truncated braid groups,       2.3 Fusion, spin, braid statistics and all that...,    3 \bar{D}_2 gauge theory: 3.1 Alice in physics, 3.2 Scattering doublet        charges off Alice fluxes, 3.3 Nonabelian braid statistics,        3.A Aharonov-Bohm scattering, 3.B B(3,4) and P(3,4),    Concluding remarks and outlook 
  Standard lattice-space formulations of quartic self-coupled Euclidean scalar quantum fields become trivial in the continuum limit for sufficiently high space-time dimensions, and in particular the moment generating functional for space-time smeared fields becomes a Gaussian appropriate to that of a (possibly generalized) free field. For sharp-time fields this fact implies that the ground-state expectation functional also becomes Gaussian in the continuum limit. To overcome these consequences of the central limit theorem, an auxiliary, nonclassical potential is appended to the original lattice form of the model and parameters are tuned so that a generalized Poisson field distribution emerges in the continuum limit for the ground-state probability distribution. As a consequence, the sharp-time expectation functional is infinitely divisible, but the Hamiltonian operator is such, in the general case, that the generating functional for the space-time smeared field is not infinitely divisible in Minkowski space. This feature permits the models in question to escape a manifestly trivial scattering matrix imposed on all infinitely divisible covariant Minkowski fields. Two sequentially related proposals for an alternative lattice formulation of interacting covariant models in four and more space-time dimensions are analyzed in some detail. 
  We find new 4-brane and 5-brane solitons in massive gauged $D=6$, $N=2$ and $D=7$, $N=1$ supergravities. In each case, the solutions preserve half of the original supersymmetry. These solutions make use of the metric and dilaton fields only. We also present more general dilatonic $(D-2)$-branes in $D$ dimensions. 
  We review recent work concerning topology changing phase transitions through black hole condensation in Type II string theory. We then also briefly describe a present study aimed at extending the known web of interconnections between Calabi-Yau manifolds. We show, for instance, that all 7555 Calabi-Yau hypersurfaces in weighted projective four space are mathematically connected by extremal transitions. 
  A new four-dimensional $N=1$ superfield model is suggested. The model is induced by supertrace anomaly of matter superfields in curved superspace and leads to effective theory of supergravity chiral compensator. A renormalization structure of this model is studied, one-loop counterterms are calculated and renormalization group equations are investigated. It is shown that the theory under consideration is infrared free. 
  Using the torus action method, we construct one variable polynomial representation of quantum cohomology ring for degree $k$ hypersurface in $CP^{N-1}$ . The results interpolate the well-known result of $CP^{N-2}$ model and the one of Calabi-Yau hypersuface in $CP^{N-1}$. We find in $k\leq N-2$ case, principal relation of this ring have very simple form compatible with toric compactification of moduli space of holomorphic maps from $CP^{1}$ to $CP^{N-1}$. 
  We construct a non-perturbative, single-valued solution for the metric and the motion of $N$ interacting particles in $2+1$-Gravity. The solution is explicit for two particles with any speed and for any number of particles with small speed. It is based on a mapping from multivalued Minkowskian coordinates to single-valued ones, which solves the non-abelian monodromies due to particles' momenta. The two and three-body cases are treated in detail. 
  Recently we showed that the spectral flow acting on the N=2 twisted topological theories gives rise to a topological algebra automorphism. Here we point out that the untwisting of that automorphism leads to a spectral flow on the untwisted N=2 superconformal algebra which is different from the usual one. This "other" spectral flow does not interpolate between the chiral ring and the antichiral ring. In particular, it maps the chiral ring into the chiral ring and the antichiral ring into the antichiral ring. We discuss the similarities and differences between both spectral flows. We also analyze their action on null states. 
  We consider here renormalizable theories without relevant couplings and present an I.R. consistent technique to study corrections to short distance behavior (Wilson O.P.E. coefficients) due to a relevant perturbation. Our method is the result of a complete reformulation of recent works on the field, and is characterized by a more orthodox treatment of U.V. divergences that allows for simpler formulae and consequently an explicit all order (regularization invariant) I.R. finitess proof. Underlying hypotheses are discussed in detail and found to be satisfied in conformal theories that constitute a natural field of application of this approach. 
  We identify a new symmetry for the equations governing odderon amplitudes, corresponding in the Regge limit of QCD to the exchange of 3 reggeized gluons. The symmetry is a modular invariance with respect to the unique normal subgroup of sl(2,Z) {\,} of index 2.  This leads to a natural description of the Hamiltonian and conservation-law operators as acting on the moduli space of elliptic curves with a fixed ``sign'': elliptic curves are identified if they can be transformed into each other by an {\em even} number of Dehn twists. 
  In these notes we will review some approaches to 2+1 dimensional gravity and the way it is coupled to point-particles. First we look into some exact static and stationary solutions with and without cosmological constant. Next we study the polygon approach invented by 't Hooft. The third section treats the Chern-Simonons formulation of 2+1-gravity. In the last part we map the problem of finding the gravitational field around point-particles to the Riemann-Hilbert problem. 
  We convert the second class Proca model into a first class theory by using the generalised prescription of Batalin, Fradkin and Tyutin. We then show how a basic set of gauge invariant fields in the embedded model can be identified with the fundamental fields in the proca model as well as with the observables in the St\"uckelberg model or in the model involving the interaction of an abelian 2-form field with the Maxwell field. The connection of these models with the massive Kalb-Ramond model is also elucidated within a path integral approach. 
  We consider an eleven dimensional supergravity compactified on $K3\times T^2$ and show that the resulting five dimensional theory has identical massless states as that of heterotic string compactified on a specific five torus $T^5$. The strong-weak coupling duality of the five dimensional theory is argued to represent a ten dimensional Type $IIA$ string compactified on $K3 \times S^1$, supporting the conjecture of string-string duality in six dimensions. In this perspective, we present magnetically charged solution of the low energy heterotic string effective action in five dimensions with a charge defined on a three sphere $S^3$ due to the two form potential. We use the Poincare duality to replace the antisymmetric two form with a gauge field in the effective action and obtain a string solution with charge on a two sphere $S^2$ instead of that on a three sphere $S^3$ in the five dimensional spacetime. We note that the string-particle duality is accompanied by a change of topology from $S^3$ to $S^2$ and viceversa. 
  We discuss two issues regarding the question of degrees of freedom in two dimensional string theory. The first issue relates to the classical limit of quantum string theory. In the classical theory one requires an infinite number of fields in addition to the collective field to describe ``folds'' on the fermi surface. We argue that in the quantum theory these are not additional degrees of freedom. Rather they represent quantum dispersions of the collective field which are {\em not} suppressed when $\hbar \rightarrow 0$ whenever a fold is present, thus leading to a nontrivial classical limit. The second issue relates to the ultraviolet properties of the geometric entropy. We argue that the geometric entropy is finite in the ultraviolet due to {\em nonperturbative} effects. This indicates that the true degrees of freedom of the two dimensional string at high energies is much smaller than what one naively expects.  (Based on talks at Spring Workshop on String theory and Quantum Gravity, ICTP, Trieste, March 1995 and VIIth Regional Conference on Mathematical Physics, Bandar-Anzali, October 1995.) 
  By an extension of the methods used for the reduction of the two body problem in 2+1 dimensional gravity, we show that the two body problem in N=2 Chern Simons supergravity can be reduced exactly to an equivalent on body formalism. We give exact expressions for the invariants of the reduced one body problem. 
  In the theory of nets of observable algebras, the modular operators associated with wedge regions are expected to have a natural geometric action, a generalization of the Bisognano-Wichmann condition for nets associated with Poincare-covariant fields. Here many possible such modular covariance conditions are discussed (in spacetime of at least three dimensions), including several conditions previously proposed and known to imply versions of the PCT and spin-statistics theorems. The logical relations between these conditions are explored: for example, it is shown that most of them are equivalent, and that all of them follow from appropriate commutation relations for the modular automorphisms alone. These results allow us to reduce the study of modular covariance to the case of systems describing non-interacting particles. Given finitely many Poincare-covariant non-interacting particles of any given mass, it is shown that modular covariance and wedge duality must hold, and the modular operators for wedge regions must have the Bisognano-Wichmann form, so that the usual free fields are the only possibility. For models describing interacting particles, it is shown that if they have a complete scattering interpretation in terms of such non-interacting particles, then again modular covariance and wedge duality must hold, and the modular operators for wedge regions must have the Bisognano-Wichmann form, so that wedge duality and the PCT and spin-statistics theorems must hold. 
  We study the renormalizability of quantum gravity near two dimensions. Our formalism starts with the tree action which is invariant under the volume preserving diffeomorphism. We identify the BRS invariance which originates from the full diffeomorphism invariance. We study the Ward-Takahashi identities to determine the general structure of the counter terms. We prove to all orders that the counter terms can be supplied by the coupling and the wave function renormalization of the tree action. The bare action can be constructed to be the Einstein action form which ensures the full diffeomorphism invariance. 
  We probe the geometry around an elementary BPS (EBPS) state in heterotic string theory compactified on a six-torus by scattering a massless scalar off it and comparing with the corresponding experiment in which the EBPS state is replaced by a classical extremal black hole background satisfying the BPS condition. We find that the low energy limit of the scattering amplitudes precisely agree if one takes the limit $m_{\rm bh} >> m_P$. In the classical experiment, beyond a certain frequency of the incident wave, part of the wave is found to be absorbed by the black hole, whereas in case of the string scattering there is a critical frequency (inelastic threshold) of the probe beyond which the EBPS state gets excited to a higher mass non-BPS elementary state. The classical absorption threshold matches exactly with the inelastic threshold in the limit of maximum degeneracy of the EBPS state of a given mass. In that limit we can therefore identify absorption by the black hole as excitation of the elementary string state to the next vibrational state of the string and consequently also identify the non-BPS string state as a non-extremal black hole. 
  Path integral expressions for three canonical formalisms -- Ostrogradski's one, constrained one and generalized one -- of higher-derivative theories are given. For each fomalism we consider both nonsingular and singular cases. It is shown that three formalisms share the same path integral expressions. In paticular it is pointed out that the generalized canonical formalism is connected with the constrained one by a canonical transformation. 
  We investigate the structure of the macroscopic $n$-loop amplitude obtained from the two-matrix model at the unitary minimal critical point $(m+1,m)$. We derive a general formula for the $n$-resolvent correlator at the continuum planar limit whose inverse Laplace transform provides the amplitude in terms of the boundary lengths $\ell_{i}$ and the renormalized cosmological constant $t$. The amplitude is found to contain a term consisting of $\left( \frac{\partial} {\partial t} \right)^{n-3}$ multiplied by the product of modified Bessel functions summed over their degrees which conform to the fusion rules and the crossing symmetry. This is found to be supplemented by an increasing number of other terms with $n$ which represent residual interactions of loops. We reveal the nature of these interactions by explicitly determining them as the convolution of modified Bessel functions and their derivatives for the case $n=4$ and the case $n=5$. We derive a set of recursion relations which relate the terms in the $n$-resolvents to those in the $(n-1)$-resolvents. 
  The overlap formulation is applied to calculate the chiral determinant on a two-dimensional torus with twisted boundary conditions. We evaluate first the continuum overlap, which is convergent and well-defined, and yields the correct string theory result for both the real and imaginary parts of the effective action. We then show that the lattice version of the overlap gives the continuum overlap results in the limit when the lattice spacing tends to zero, and that the subleading terms in that limit are irrelevant. 
  In the presence of a D-brane a string theory develops a new subsector. We show that for curved D-branes the corresponding sector is a (partially twisted) topological field theory. We use this result to compute the degeneracy of 2-branes wrapped around $K3$ cycles as well as 3-branes wrapped around CY threefold vanishing 3-cycles. In both cases we find the degeneracy is in accord with expectation. The counting of BPS states of a gas of 0-branes in the presence of a 4-brane in $K3$ is considered and it is noted that the effective 0-brane charge is shifted by 1, due to a quantum correction. This is in accord with string duality and the fact that left-moving ground state energy of heterotic string starts at $-1$. We also show that all the three different topological twistings of four dimensional $N=4$ Yang-Mills theory do arise from curved three-branes embedded in different spaces (Calabi-Yau manifolds and manifolds with exceptional holonomy groups). 
  We construct two classes of higher-derivative supergravity theories generalizing Einstein supergravity. We explore their dynamical content as well as their vacuum structure. The first class is found to be equivalent to Einstein supergravity coupled to a single chiral superfield. It has a unique stable vacuum solution except in a special case, when it becomes identical to a simple no-scale theory. The second class is found to be equivalent to Einstein supergravity coupled to two chiral superfields and has a richer vacuum structure. It is demonstrated that theories of the second class can possess a stable vacuum with vanishing cosmological constant that spontaneously breaks supersymmetry. We present an explicit example of this phenomenon and compare the result with the Polonyi model. 
  A constant magnetic field in 3+1 and 2+1 dimensions is a strong catalyst of dynamical chiral symmetry breaking, leading to the generation of a fermion mass even at the weakest attractive interaction between fermions. The essence of this effect is the dimensional reduction $D/rightarrow D-2$ in the dynamics of fermion pairing in a magnetic field. The effect is illustrated in the Nambu-Jona-Lasinio model and QED. Possible applications of this effect and its extension to inhomogeneous field configurations are discussed. 
  In this note we show that the single soliton solutions known previously in the $1+1$ dimensional affine Toda field theories from a variety of different methods \cite{H1,MM,OTUa,OTUb}, are in fact not the most general single soliton solutions. We exhibit single soliton solutions with additional small parameters which reduce to the previously known solutions when these extra parameters are set to zero. The new solution has the same mass and topological charges as the standard solution when these parameters are set to zero. However we cannot yet completely rule out the possibility that other solutions with larger values of these extra parameters are non-singular, in the cases where the number of extra parameters is greater than one, and if so their topological charges would most likely be different. 
  In a previous publication [1], local gauge invariant geometric variables were introduced to describe the physical Hilbert space of Yang-Mills theory. In these variables, the electric energy involves the inverse of an operator which can generically have zero modes, and thus its calculation is subtle. In the present work, we resolve these subtleties by considering a small deformation in the definition of these variables, which in the end is removed. The case of spherical configurations of the gauge invariant variables is treated in detail, as well as the inclusion of infinitely heavy point color sources, and the expression for the associated electric field is found explicitly. These spherical geometries are seen to correspond to the spatial components of instanton configurations. The related geometries corresponding to Wu-Yang monopoles and merons are also identified. 
  In these lectures we review the quantum physics of large Schwarzschild black holes. Hawking's information paradox, the theory of the stretched horizon and the principle of black hole complementarity are covered. We then discuss how the ideas of black hole complementarity may be realized in string theory. Finally, arguments are given that the world may be a hologram. Lectures delivered at ICTP Spring School on String Theory, Gauge Theory, and Quantum Gravity, 1995. 
  We study a class of four-dimensional N=1 heterotic string theories which have nontrivial quantum dynamics arising from asymptotically free gauge groups. These models are obtained by orbifolding 4d N=2 heterotic/type II dual pairs by symmetries which leave unbroken products of nonabelian gauge groups (without charged matter) in a ``hidden sector'' on the heterotic side. Such models are expected to break supersymmetry through gaugino condensation in the hidden sector. We find a dual description of the effects of gaugino condensation on the type II side, where the corresponding superpotential arises at tree level. We speculate that the conformal field theory underlying the type II description may be related to a class of geometrical nonsupersymmetric string compactifications. 
  We present a derivation of the N=1 and N=2 superconformal coset constructions starting from a supersymmetric WZW model where a diagonal subgroup has been gauged. We work in the general framework of self-dual (not necessarily reductive) Lie algebras; but even in the reductive case these results are new. We show that the BRST cohomology of the gauged supersymmetric WZW model contains the N=1 (and if it exists also the N=2) coset generators. We also extend the BRST analysis to show that the BRST cohomology of the supersymmetric WZW model agrees with that of an ordinary bosonic WZW model (in a representation twisted by the presence of the coset fermions). In particular, in the case of the topological G/G coset, the supersymmetric and nonsupersymmetric theories agree. 
  We show that the moduli space of all Calabi-Yau manifolds that can be realized as hypersurfaces described by a transverse polynomial in a four dimensional weighted projective space, is connected. This is achieved by exploiting techniques of toric geometry and the construction of Batyrev that relate Calabi-Yau manifolds to reflexive polyhedra. Taken together with the previously known fact that the moduli space of all CICY's is connected, and is moreover connected to the moduli space of the present class of Calabi-Yau manifolds (since the quintic threefold P_4[5] is both CICY and a hypersurface in a weighted P_4, this strongly suggests that the moduli space of all simply connected Calabi-Yau manifolds is connected. It is of interest that singular Calabi-Yau manifolds corresponding to the points in which the moduli spaces meet are often, for the present class, more singular than the conifolds that connect the moduli spaces of CICY's. 
  We discuss hypersurface motions in Riemannian manifolds whose normal velocity is a function of the induced hypersurface volume element and derive a second order partial differential equation for the corresponding time function $\tau(x)$ at which the hypersurface passes the point $x$. Equivalently, these motions may be described in a Hamiltonian formulation as the singlet sector of certain diffeomorphism invariant field theories. At least in some (infinite class of) cases, which could be viewed as a large-volume limit of Euclidean $M$-branesmoving in an arbitrary $M+1$-dimensional Riemannian manifold, the models are integrable: In the time-function formulation the equation becomes linear (with $\tau(x)$ a harmonic function on the embedding Riemannian manifold). We explicitly compute solutions to the large volume limit of Euclidean membrane dynamics in $\Real^3$ by methods used in electrostatics and point out an additional gradient flow structure in $\Real^n$. In the Hamiltonian formulation we discover infinitely many hierarchies of integrable, multidimensional, $N$-component theories possessing infinitely many diffeomorphism invariant, Poisson commuting, conserved charges. 
  It has been suggested that the chiral symmetry can be implemented only in classical Lagrangians containing higher covariant derivatives of odd order. Contrary to this belief, it is shown that one can construct an exactly soluble two-dimensional higher-derivative fermionic quantum field theory containing only derivatives of even order whose classical Lagrangian exhibits chiral-gauge invariance. The original field solution is expressed in terms of usual Dirac spinors through a canonical transformation, whose generating function allows the determination of the new Hamiltonian. It is emphasized that the original and transformed Hamiltonians are different because the mapping from the old to the new canonical variables depends explicitly on time. The violation of cluster decomposition is discussed and the general Wightman functions satisfying the positive-definiteness condition are obtained. 
  We show that the equations of motion defined over a specific field space are realizable as operator conditions in the physical sector of a generalized Floer theory defined over that field space. The ghosts associated with such a construction are found not to be dynamical. This construction is applied to gravity on a four dimensional manifold, $M$; whereupon, we obtain Einstein's equations via surgery, along $M$, in a five-dimensional topological quantum field theory. 
  New recent results in supersymmetric gauge theories based on holomorphy and symmetry considerations are extended to the case where the gauge coupling constant is given by the real part of a chiral superfield. We assume here that its dynamics can be described by an effective quantum field theory. Then its vacuum expectation value is a function of the other coupling constants, viewed as chiral background superfields. This functional dependence can be determined exactly and satisfies highly non-trivial consistency checks. 
  We present the hyper-elliptic curve describing the moduli space of the N=2 supersymmetric Yang-Mills theory with the $G_2$ gauge group. The exact monodromies and the dyon spectrum of the theory are determined. It is verified that the recently proposed solitonic equation is also satisfied by our solution. 
  We study spontaneous supersymmetry breaking in N=2 globally supersymmetric theories describing a system of abelian vector multiplets. We find that the most general form of the action admits, in addition to the usual Fayet-Iliopoulos term, a magnetic Fayet-Iliopoulos term for the auxiliary components of dual vector multiplets. In a generic case, N=2 supersymmetry is broken down spontaneously to N=1. In some cases however, the scalar potential can drive the theory towards a N=2 supersymmetric ground state where massless dyons condense in the vacuum. 
  A construction of conservation laws for chiral models (generalized sigma-models on a two-dimensional space-time continuum using differential forms is extended in such a way that it also comprises corresponding discrete versions. This is achieved via a deformation of the ordinary differential calculus. In particular, the nonlinear Toda lattice results in this way from the linear (continuum) wave equation. The method is applied to several further examples. We also construct Lax pairs and B\"acklund transformations for the class of models considered in this work. 
  We present a formulation for the construction of first order equations which describe particles with spin, in the context of a manifestly covariant relativistic theory governed by an invariant evolution parameter; one obtains a consistent quantized formalism dealing with off-shell particles with spin. Our basic requirement is that the second order equation in the theory is of the Schr\"{o}dinger-Stueckelberg type, which exhibits features of both the Klein-Gordon and Schr\"{o}dinger equations. This requirement restricts the structure of the first order equation, in particular, to a chiral form. One thus obtains, in a natural way, a theory of chiral form for massive particles, which may contain both left and right chiralities, or just one of them. We observe that by iterating the first order system, we are able to obtain second order forms containing the transverse and longitudinal momentum relative to a time-like vector $t_{\mu}t^{\mu}=-1$ used to maintain covariance of the theory. This time-like vector coincides with the one used by Horwitz, Piron, and Reuse to obtain an invariant positive definite space-time scalar product, which permits the construction of an induced representation for states of a particle with spin. We discuss the currents and continuity equations, and show that these equations of motion and their currents are closely related to the spin and convection parts of the Gordon decomposition of the Dirac current. The transverse and longitudinal aspects of the particle are complementary, and can be treated in a unified manner using a tensor product Hilbert space. Introducing the electromagnetic field we find an equation which gives rise to the correct gyromagnetic ratio, and is fully Hermitian under the proposed scalar product. Finally, we show that the original structure of Dirac's 
  Using a contour integral representation we analyze the multi-instanton sector in two dimensional $U(N)$ Yang-Mills theory on a sphere and argue the role of multi-instanton in the large $N$ phase transition. In the strong coupling region at the large $N$ , we encounter ``singular saddle point''. Because of this situation, ``neutral'' configurations of the multi-instanton are dominant in this region. Based on the ``neutral'' multi-instanton approximation we numerically calculate the multi-instanton amplitude , the free energies and the Wilson loops for finite $N$ . We also compare our results with the large $N$ exact solution of the free energy and the Wilson loop and argue some problems. We find the ``neutral'' multi-instanton contribution bridges the gap between weak and strong coupling phase. 
  It is shown that in the general case the canonical construction of the current operators in quantum field theory does not render a bona fide vector field since Lorentz invariance is violated by Schwinger terms. We argue that the nonexistence of the canonical current operators for spinor fields follows from a very simple algebraic consideration. As a result, the well-known sum rules in deep inelastic scattering are not substantiated. 
  In this paper, the relationship between the sine-Gordon model with an integrable boundary condition and the Thirring model with boundary is discussed and the reflection $R$-matrix for the massive Thirring model, which is related to the physical boundary parameters of the sine-Gordon model, is given. The relationship between the the boundary parameters and the two formal parameters appearing in the work of Ghoshal and Zamolodchikov is discussed. 
  In this paper, we give a construction of $p$-brane solitons in all maximal supergravity theories in $4\le D \le 11$ dimensions that are obtainable from $D=11$ supergravity by dimensional reduction. We first obtain the full bosonic Lagrangians for all these theories in a formalism adapted to the $p$-brane soliton construction. The solutions that we consider involve one dilaton field and one antisymmetric tensor field strength, which are in general linear combinations of the basic fields of the supergravity theories. We also study the supersymmetry properties of the solutions by calculating the eigenvalues of the Bogomol'nyi matrices, which are derived from the commutators of the supercharges. We give an exhaustive list of the supersymmetric $p$-brane solutions using field strengths of all degrees $n=4,3,2,1$, and the non-supersymmetric solutions for $n=4,3,2$. As well as studying elementary and solitonic solutions, we also discuss dyonic solutions in $D=6$ and $D=4$. In particular, we find that the Bogomol'nyi matrices for the supersymmetric massless dyonic solutions have indefinite signature. 
  It is shown how some results on harmonic maps within the chiral model can be carried over to self-dual gravity. The WZW-like action for self-dual gravity is found. 
  It is shown that there exists a truly marginal deformation of the direct sum of two $G_k$ WZNW models at $k=-2c_V(G)$ (where $c_V(G)$ is the eigenvalue of the quadratic Casimir operator in the adjoint representation of the group $G$) which does not seem to fit the Chaudhuri-Schwartz criterion for truly marginal perturbations. In addition, a continuous family of WZNW models is constructed. 
  Considering the conformal anomaly in an effective action, the critical dimension of string theory can be decided in the harmonic gauge, in which it had been reported before to be indefinite. In this gauge, there is no anomaly for the ghost number symmetry. This can be naturally understood in terms of Faddeev-Popov conjugation in the theory. 
  If $X$ and $Y$ are a mirror pair of Calabi--Yau threefolds, mirror symmetry should extend to an isomorphism between the type IIA string theory compactified on $X$ and the type IIB string theory compactified on $Y$, with all nonperturbative effects included. We study the implications which this proposal has for the structure of the semiclassical moduli spaces of the compactified type II theories. For the type IIB theory, the form taken by discrete shifts in the Ramond-Ramond scalars exhibits an unexpected dependence on the $B$-field. (Based on a talk at the Trieste Workshop on S-Duality and Mirror Symmetry.) 
  The action of the $S$-duality $Sl(2,Z)$ group on the moduli of the Calabi-Yau manifold $W\IP^{12}_{11226}$ appearing in the rank two dual pair $(K^{3}\times T^{2}/W\IP^{12}_{11226})$ is defined by interpreting the $N\!=\!4$ to $N\!=\!2$ flow, for $SU(2)$ supersymmetric Yang-Mills, in terms of the Calabi-Yau moduli. The different singularity loci are mapped in a one to one way, and the ($N\!=\!2$ limit/point particle limit) is obtained in both cases by the same type of blow up. Moreover, it is shown that the $S$-duality group permutes the different singularity loci of the moduli of $W\IP^{12}_{11226}$. We study the transformation under $S$-duality of the Calabi-Yau Yukawa couplings. 
  It is shown that the new version of nonsymmetric gravitational theory (NGT) corresponds in the linear approximation to linear Einstein gravity theory and antisymmetric field equations with a non-conserved string source current. The Hamiltonian for the antisymmetric field equations is bounded from below and describes the exchange of a spin $1^+$ massive vector boson between open strings. The non-Riemannian geometrical theory is formulated in terms of a nonsymmetric fundamental tensor $g_{\mu\nu}$. The weak field limit, $g_{[\mu\nu]}\rightarrow 0$, associated with large distance scales, corresponds to the limit to a confinement region at low energies described by an effective Yukawa potential at galactic distance scales. The limit to this low-energy confinement region is expected to be singular and non-perturbative. The NGT string theory predicts that there are no black hole event horizons associated with infinite red shift null surfaces. 
  We study quantum aspects of the Einstein gravity with one time-like and one space-like Killing vector commuting with each other. The theory is formulated as a $\coset$ nonlinear $\sigma$-model coupled to gravity. The quantum analysis of the nonlinear $\sigma$-model part, which includes all the dynamical degrees of freedom, can be carried out in a parallel way to ordinary nonlinear $\sigma$-models in spite of the existence of an unusual coupling. This means that we can investigate consistently the quantum properties of the Einstein gravity, though we are limited to the fluctuations depending only on two coordinates. We find the forms of the beta functions to all orders up to numerical coefficients. Finally we consider the quantum effects of the renormalization on the Kerr black hole as an example. It turns out that the asymptotically flat region remains intact and stable, while, in a certain approximation, it is shown that the inner geometry changes considerably however small the quantum effects may be. 
  This talk is devoted mainly to the concept of higher-order polarization on a group, which is introduced in the framework of a Group Approach to Quantization, as a powerful tool to guarantee the irreducibility of quantizations and/or representations of Lie groups in those anomalous cases where the Kostant-Kirilov co-adjoint method or the Borel-Weyl-Bott representation algorithm do not succeed. 
  Using the Mathai-Quillen formalism we reexamine the twisted N=4 supersymmetric model of Vafa-Witten theory. Smooth out the relation between the supersymmetric action and the path integral representation of the Thom class. 
  Following the previous work of Ferretti and Yang on the role of magnetic fields in the theory of conformal turbulence, we show that non-unitary minimal model solutions to 2-dimensional magnetohydrodynamics (MHD) obtained by dimensional reduction from 3-dimensions exist under different (and more restrictive) conditions. From a 3-dimensional point of view, these conditions are equivalent to perpendicular flow, in which the magnetic and velocity fields are orthogonal. We also extend the analysis to the finite conductivity case and present some approximate solutions, whose connection to the exact ones of the infinite conductivity case is also discussed. 
  It is shown that for 2D field theories only the first order coefficient of the gravitationally dressed RG $\beta$-function is scheme independent. This is valid even for matter theories with one dimensionless coupling, where the first two coefficients of the original $\beta$-function are scheme independent. 
  We match a few non chiral operators in the electric and magnetic descriptions of SQCD, suggesting the first evidence of electric-magnetic duality outside the chiral ring. Algebraically, these non chiral operators are a module of the chiral ring. Physically, the suggested correspondence essentially transforms certain electric gauge invariant composites containing the electric field strength into magnetic matter composites. 
  We formulate Poisson-Lie T-duality in a path-integral manner that allows us to analyze the quantum corrections. Using the path-integral, we rederive the most general form of a Poisson-Lie dualizeable background and the generalized Buscher transformation rules it has to satisfy. 
  We construct a framework which unifyies in dual pairs the fields and anti-fields of the Batalin and Vilkovisky quantization method. We consider gauge theories of p-forms coupled to Yang-Mills fields. Our algorithm generates many topological models of the Chern-Simon type or of the Donaldson-Witten type. Some of these models can undergo a partial breaking of their topological symmetries. 
  We investigate the properties of 2-D gravity in the Batalin and Vilkovisky quantization scheme. We find a factorized structure which exhibits duality properties analogous to those existing in the topological theories of forms. New conformal field are introduced with their invariant action. 
  We analyze the interaction between the massless RR states with a dilaton in a type IIB superstring. By constructing vertex operators for massless and massive RR states and computing their correlation functions with a dilaton we find the Ramond-Ramond part of the superstring low-energy effective action. The RR terms appearing in the action do not contain the standard dilatonic factor (string coupling constant). The geometrical interpretation of this fact is presented. Namely we argue that the spin operators in the RR vertices effectively decrease the Euler character of the worldsheet by 1 unit. As a result, the additional dilatonic term proportional to string coupling constant appears in the worldsheet string action. 
  We describe a general purpose Mathematica package for computing Superfield Operator Product Expansions in meromorphic $N=2$ superconformal field theory. Given the SOPEs for a set of ``basic" superfields, SOPEs of arbitrarily complicated composites can be computed automatically. Normal ordered products are always reduced to a standard form. It is possible to check the Jacobi identities, and to compute Poisson brackets (``classical SOPEs''). We present two explicit examples: a construction of the ``small'' $N=4$ superconformal algebra in terms of $N=2$ superfields, and a realisation of the $N=2$ superconformal algebra in terms of chiral and antichiral fermionic superfields. 
  The symmetric space sine-Gordon models arise by conformal reduction of ordinary 2-dim $\sigma$-models, and they are integrable exhibiting a black-hole type metric in target space. We provide a Lagrangian formulation of these systems by considering a triplet of Lie groups $F \supset G \supset H$. We show that for every symmetric space $F/G$, the generalized sine-Gordon models can be derived from the $G/H$ WZW action, plus a potential term that is algebraically specified. Thus, the symmetric space sine-Gordon models describe certain integrable perturbations of coset conformal field theories at the classical level. We also briefly discuss their vacuum structure, Backlund transformations, and soliton solutions. 
  We consider a six-dimensional solitonic string solution described by a conformal chiral null model with non-trivial $N=4$ superconformal transverse part. It can be interpreted as a five-dimensional dyonic solitonic string wound around a compact fifth dimension. The conformal model is regular with the short-distance (`throat') region equivalent to a WZW theory. At distances larger than the compactification scale the solitonic string reduces to a dyonic static spherically-symmetric black hole of toroidally compactified heterotic string. The new four-dimensional solution is parameterised by five charges, saturates the Bogomol'nyi bound and has nontrivial dilaton-axion field and moduli fields of two-torus. When acted by combined T- and S-duality transformations it serves as a generating solution for all the static spherically-symmetric BPS-saturated configurations of the low-energy heterotic string theory compactified on six-torus. Solutions with regular horizons have the global space-time structure of extreme Reissner-Nordstrom black holes with the non-zero thermodynamic entropy which depends only on conserved (quantised) charge vectors. The independence of the thermodynamic entropy on moduli and axion-dilaton couplings strongly suggests that it should have a microscopic interpretation as counting degeneracy of underlying string configurations. This interpretation is supported by arguments based on the corresponding six-dimensional conformal field theory. The expression for the level of the WZW theory describing the throat region implies a renormalisation of the string tension by a product of magnetic charges, thus relating the entropy and the number of oscillations of the solitonic string in compact directions. 
  Given a simple, simply laced, complex Lie algebra $\bfg$ corresponding to the Lie group $G$, let $\bfnp$ be the subalgebra generated by the positive roots. In this paper we construct a BV-algebra $\fA[\bfg]$ whose underlying graded commutative algebra is given by the cohomology, with respect to $\bfnp$, of the algebra of regular functions on $G$ with values in $\mywedge (\bfnp\backslash\bfg)$. We conjecture that $\fA[\bfg]$ describes the algebra of {\it all} physical (i.e., BRST invariant) operators of the noncritical $\cW[\bfg]$ string. The conjecture is verified in the two explicitly known cases, $\bfg=\sltw$ (the Virasoro string) and $\bfg=\slth$ (the $\cW_3$ string). 
  We consider a certain local generalization of BRS transformations of Yang-Mills theory in which the anti-commuting parameter is space time dependent. While these are not exact symmetries, they do lead to a new nontrivial WT identity. We make a precise connection between the ``local BRS "and the broken orthosymplectic symmetry recently found in superspace formulation of Yang-Mills theory by showing that the local BRS WT identity is precisely the WT identity obtained in the superspace formulation via a superrotation. This ``local BRS " WT identity could lead to new consequences not contained in the usual BRS WT identity. 
  Some of the four-dimensional Superstring solutions provide a consistent framework for a Supersymmetric Unification of all interactions including gravity. A class of them extends successfully the validity of the standard model up to the string scale ${\cal O}(10^{17})~GeV$. We stress the importance of string corrections which are relevant for low energy ${\cal O}(1)~TeV$ predictions of gauge and Yukawa couplings as well as the spectrum of the supersymmetric particles after supersymmetry breaking. 
  We consider the $SYM^1_6$ harmonic-superspace system of equations that contains superfield constraints and equations of motion for the simplest six-dimensional supersymmetric gauge theory. A special $A$-frame of the analytic basis is introduced where a kinematic equation for the harmonic connection $A^{\s--}$ can be solved . A dynamical equation in this frame is equivalent to the zero-curvature equation corresponding to the covariant conservation of analyticity. Using a simple harmonic gauge condition for the gauge group $SU(2)$ we derive the superfield equations that produce the general $SYM^1_6$ solution . An analogous approach for the analysis of integrability conditions for the $SYM^2_4$-theory and $SYM$-supergravity-matter systems in harmonic superspace is discussed briefly. 
  The general procedure of constructing a consistent covariant Dirac-type bracket for models with mixed first and second class constraints is presented. The proposed scheme essentially relies upon explicit separation of the initial constraints into infinitely reducible first and second class ones (by making use of some appropriately constructed covariant projectors). Reducibility of the second class constraints involved manifests itself in weakening some properties of the bracket as compared to the standard Dirac one. In particular, a commutation of any quantity with the second class constraints and the Jacobi identity take place on the second class constraints surface only. The developed procedure is realized for N=1 Brink--Schwarz superparticle in arbitrary dimension and for N=1, D=9 massive superparticle with Wess--Zumino term. A possibility to apply the bracket for quantizing the superparticles within the framework of the recent unified algebra approach by Batalin and Tyutin [20--22] is examined. In particular, it is shown that for D=9 massive superparticle it is impossible to construct Dirac-type bracket possessing (strong) Jacobi identity in a full phase space. 
  A variety of unitary gauges for perturbation theory in a background field is considered in order to find those most suitable for a Hamiltonian treatment of the system. We select two convenient gauges and derive the propagators $D_{\mu\nu}$ for gluonic quantum fluctuations immersed in background configurations. The first one is a unitary generalization of the usual Coulomb gauge in QED which preserves the decoupling of two propagating polarizations from the instantaneous one. The second possibility is the axial light cone gauge which remains ghost free also in the presence of a background. Applications of the formalism to the spectrum and dynamics of QCD at the confinement scale, such as hybrid states, are briefly discussed. 
  We define a complete set of supertraces on the algebra $SH_N(\nu)$, the algebra of observables of the $N$-body rational Calogero model with harmonic interaction. This result extends the previously known results for the simplest cases of $N=1$ and $N=2$ to arbitrary $N$. It is shown that $SH_N(\nu)$ admits $q(N)$ independent supertraces where $q(N)$ is a number of partitions of $N$ into a sum of odd positive integers, so that $q(N)>1$ for $N\ge 3$. Some consequences of the existence of several independent supertraces of $SH_N (\nu )$ are discussed such as the existence of ideals in associated $W_{\infty}$ - type Lie superalgebras. 
  The BRST formalism has played a fundamental role in the construction of bosonic closed string backgrounds, ie. the stringy analogs of classical solutions to the field equations of general relativity. The concept of a string background has been extended to the notion of $W$-strings, where the BRST symmetry is still largely conjectural. More recently, the BRST formalism has entered the construction of two dimensional topological conformal quantum field theories, such as those that arise from Calabi-Yau varieties. In this lecture, we focus on common features of the BRST cohomology algebras of string backgrounds and topological field theories. In this context, we present some new evidence for a remarkable relationship that transports us from bosonic and $W$-string backgrounds to the B-model topological conformal field theories associated to certain noncompact Calabi-Yau varieties. This paper will appear in the proceedings of the {\it Symposium on BRS Symmetry} held at RIMS, September 18-22, 1995. 
  A duality invariant first order action is constructed on the loop group of a Drinfeld double. It gives at the same time the description of both of the pair of $\sigma$-models related by Poisson-Lie T-duality. Remarkably, the action contains a WZW-term on the Drinfeld double not only for conformally invariant $\si$-models. The resulting actions of the models from the dual pair differ just by a total derivative corresponding to an ambiguity in specifying a two-form whose exterior derivative is the WZW three-form. This total derivative is nothing but the Semenov-Tian-Shansky symplectic form on the Drinfeld double and it gives directly a generating function of the canonical transformation relating the $\si$-models from the dual pair. 
  A model describing the $N=2$ supergravity interaction with vector and linear multiplets is constructed. It admits the introduction of the spontaneous breaking of supersymmetry with arbitrary scales, one of which may be equal to zero, which corresponds to partial super-Higgs effect ($N=2 \rightarrow N=1$). A cosmological term is automatically equal to zero. 
  Dirichlet-branes have emerged as important objects in studying nonperturbative string theory. It is important to generalize these objects to more general backgrounds other than the usual flat background. The simplest case is the linear dilaton condensate. The usual Dirichlet boundary condition violates conformal invariance in such a background. We show that by switching on a certain boundary interaction, conformal invariance is restored. An immediate application of this result is to two dimensional string theory. 
  In these lectures I review the general structure of electric--magnetic duality rotations in every even space--time dimension. In four dimensions, which is my main concern, I discuss the general issue of symplectic covariance and how it relates to the typical geometric structures involved by N=2 supersymmetry, namely Special K\"ahler geometry for the vector multiplets and either HyperK\"ahler or Quaternionic geometry for the hypermultiplets. I discuss classical continuous dualities versus non--perturbative discrete dualities. How the moduli space geometry of an auxiliary dynamical Riemann surface (or Calabi--Yau threefold) relates to exact space--time dualities is exemplified in detail for the Seiberg Witten model of an $SU(2)$ gauge theory. 
  Extensions (modifications) of the Heisenberg Uncertainty principle are derived within the framework of the theory of Special Scale-Relativity proposed by Nottale. In particular, generalizations of the Stringy Uncertainty Principle are obtained where the size of the strings is bounded by the Planck scale and the size of the Universe. Based on the fractal structures inherent with two dimensional Quantum Gravity, which has attracted considerable interest recently, we conjecture that the underlying fundamental principle behind String theory should be based on an extension of the Scale Relativity principle where both dynamics as well as scales are incorporated in the same footing. 
  The super-Weyl cocycle (effective action for supertrace anomaly) and corresponding invariant operator in nonminimal formulation of $d=4$,$N=1$ supergravity are obtained. 
  We study a standard-embedding $N=2$ heterotic string compactification on $K3\times T^2$ with a Wilson line turned on and perform a world-sheet calculation of string threshold correction. The result can be expressed in terms of the quantities appearing in the two-loop calculation of bosonic string. We also comment and speculate on the relevance of our result to generalized Kac-Moody superalgebra and $N=2$ heterotic-type IIA duality. 
  It is known that gauge fields defined on manifolds with spatial boundaries support states localized at the boundaries. In this paper, we demonstrate how coarse-graining over these states can lead to an entanglement entropy. In particular, we show that the entanglement entropy of the ground state for the quantum Hall effect on a disk exhibits an approximate ``area " law. 
  We examine the possibility of a confinement-deconfinement phase transition at finite temperature in both parity invariant and topologically massive three-dimensional quantum electrodynamics. We review an argument showing that the Abelian version of the Polyakov loop operator is an order parameter for confinement, even in the presence of dynamical electrons. We show that, in the parity invariant case, where the tree-level Coulomb potential is logarithmic, there is a Berezinskii-Kosterlitz-Thouless transition at a critical temperature ($T_c=e^2/8\pi+{\cal O}(e^4/m)$, when the ratio of the electromagnetic coupling and the temperature to the electron mass is small). Above $T_c$ the electric charge is not confined and the system is in a Debye plasma phase, whereas below $T_c$ the electric charges are confined by a logarithmic Coulomb potential, qualitatively described by the tree-level interaction. When there is a topological mass, no matter how small, in a strict sense the theory is not confining at any temperature; the model exhibits a screening phase, analogous to that found in the Schwinger model and two-dimensional QCD with massless adjoint matter. However, if the topological mass is much smaller than the other dimensional parameters, there is a temperature for which the range of the Coulomb interaction changes from the inverse topological mass to the inverse electron mass. We speculate that this is a vestige of the BKT transition of the parity-invariant system, separating regions with screening and deconfining behavior. 
  The charge screening, confinement of fermion quantum numbers and the chiral condensate formation in two-dimensional QED is studied in details. It is shown that charge screening and confinement of fermion number in two-dimensional QED is due to an appearance of gauge fields which nullify the Dirac determinant. An appearance of the fields of another type but with the same property yield the chiral condensate formation. In addition, these second type fields ensure the "softness" of the charge screening in a process which is analogous to the $e^+e^-$ annihilation. 
  In a space of $d $ Grassmann coordinates two types of generators of Lorentz transformations can be defined, one of spinorial and the other of vectorial character. Both kinds of operators appear as linear operators in Grassmann space, definig the fundamental and the adjoint representations of the group $ SO(1,d$$-$$1) $, respectively. The eigenvalues of commuting operators belonging to the subgroup $(SO(1,4)) $ can be identified with spins of either fermionic or bosonic fields, while the operators belonging to subgroups of $ SO(d$$-$$5) $ ${\supset SU(3)}$ $ { \times SU(2)}$ $ { \times U(1)} $, determine the Yang-Mills charges. The theory offers unification of all the internal degrees of freedom of fermionic and bosonic fields - spins and all Yang-Mills charges. When accordingly all interactions are unified, Yang-Mills fields appear as part of the gravitational field. The theory suggests that elementary particles are either in the fundamental representations with respect to the groups determining the spin and the charges, or they are in the adjoint representations with respect to the groups, which determine the spin and the charges. 
  We present new family of exact analytic solutions for three anyons in a harmonic potential (or in free space) in terms of generalized harmonics on $S^3$, which supplement the known solutions. The new solutions satisfy the hard-core condition when $\alpha={1\over 3},1$ ($\alpha$ being the statistical parameter) but otherwise, have finite non-vanishing two-particle colliding probability density, which is consistent with self-adjointness of the Hamiltonian. These solutions, however, do not have one-to-one mapping property between bosonic and fermionic spectra. 
  For $n\in [-2,2]$ the $O(n)$ model on a random lattice has critical points to which a scaling behaviour characteristic of 2D gravity interacting with conformal matter fields with $c\in [-\infty,1]$ can be associated. Previously we have written down an exact solution of this model valid at any point in the coupling constant space and for any $n$. The solution was parametrized in terms of an auxiliary function. Here we determine the auxiliary function explicitly as a combination of $\theta$-functions, thereby completing the solution of the model. Using our solution we investigate, for the simplest version of the model, hitherto unexplored regions of the parameter space. For example we determine in a closed form the eigenvalue density without any assumption of being close to or at a critical point. This gives a generalization of the Wigner semi-circle law to $n\neq 0$. We also study the model for $|n|>2$. Both for $n<-2$ and $n>2$ we find that the model is well defined in a certain region of the coupling constant space. For $n<-2$ we find no new critical points while for $n>2$ we find new critical points at which the string susceptibility exponent $\gamma_{str}$ takes the value $+\frac{1}{2}$. 
  The conditions for the cancellation of all gauge, gravitational, and mixed anomalies of $N=1$ supersymmetric models in six dimensions are reviewed and illustrated by a number of examples. Of particular interest are models that cannot be realized perturbatively in string theory. An example of this type, which we verify satisfies the anomaly cancellation conditions, is the K3 compactification of the $SO(32)$ theory with small instantons recently proposed by Witten. When the instantons coincide it has gauge group $SO(32) \times Sp(24)$. Two new classes of models, for which non-perturbative string constructions are not yet known, are also presented. They have gauge groups $SO(2n+8)\times Sp(n)$ and $SU(n)\times SU(n)$, where $n$ is an arbitrary positive integer. 
  We calculate the anomalous magnetic moment of anyons in three dimensional $CP^{N-1}$ model with a Chern-Simons term in various limits in $1/N$ expansion. We have found that for anyons of infinite mass the gyromagnetic ratio ($g$-factor) is 2 up to the next-to-leading order in $1/N$. Our result supports a recent claim that the $g$-factor of nonrelativistic anyons is exactly two. We also found that for $-{8\over\pi }<\theta<0$, the electromagnetic interation between two identical aynons of large mass are attractive. 
  The BRST anomaly which may be present in the induced $W_n$ gravity quantized on the light-cone is evaluated in the geometrical framework of Zucchini. The cocycles linked by the cohomology of the BRST operator to the anomaly are straightforwardly calculated thanks to the analogy between this formulation and the Yang-Mills theory. We give also a conformally covariant formulation of these quantities including the anomaly, which is valid on arbitrary Riemann surfaces. The example of the $W_3$ theory is discussed and a comparison with other candidates for the anomaly available in the literature is presented. 
  Starting from the concept of the universal exterior algebra in non-commutative differential geometry we construct differential forms on the quantum phase-space of an arbitrary system. They bear the same natural relationship to quantum dynamics which ordinary tensor fields have with respect to classical hamiltonian dynamics. 
  The modular transformations of the $(1|1)$ complex supermanifolds in the like-Schottky modular parameterization are discussed. It is shown that these "supermodular" transformations depend on the spinor structure of the supermanifold by terms proportional to the odd modular parameters. The above terms are calculated in the explicit form. They are urgent for the divergency problem in the Ramond-Neveu-Schwarz superstring theory and for calculating the fundamental domain in the modular space. The supermodular transformations of the multi-loop superstring partition functions calculated by the solution of the Ward identities are studied. The above Ward identities are shown to be covariant under the supermodular transformations. So the partition functions necessarily possess the covariance under the transformations discussed. It is demonstrated explicitly the covariance of the above partition functions at zero odd moduli under those supermodular transformations, which turn a pair of even genus-1 spinor structures to a pair of the odd genus-1 spinor ones. The brief consideration of the cancellation of divergences is given. 
  We discuss the role of additional local symmetries related to the transformations of connection fields in the affine-metric theory of gravity. The corresponding BRST transformations connected with all symmetries (general coordinate, local Lorentz and extra) are constructed. It is shown, that extra symmetries give the additional contribution to effective action which is proportional to the corresponding Nielsen-Kallosh ghost one. Some arguments are given, that there is no anomaly associated with extra local symmetries. 
  It is shown that many of the $p$-branes of type II string theory and $d=11$ supergravity can have boundaries on other $p$-branes. The rules for when this can and cannot occur are derived from charge conservation. For example it is found that membranes in $d=11$ supergravity and IIA string theory can have boundaries on fivebranes. The boundary dynamics are governed by the self-dual $d=6$ string. A collection of $N$ parallel fivebranes contains $\half N(N-1)$ self-dual strings which become tensionless as the fivebranes approach one another. 
  We construct more dual pairs of type II-heterotic strings in four dimensions with $N=2,1$ spacetime supersymmetry. On the type II side the construction utilizes the various possible choices of K3 automorphisms with fixed points which transform the holomorphic two-form nontrivially, and rotation plus translation on $T^2$. The Calabi-Yau orbifolds so obtained have non-zero Euler numbers, so quantum corrections exist on the type IIA strings. The heterotic string (asymmetric) orbifold duals are found which depend on going to the enhanced symmetry points. Some aspects of the construction are discussed including the role of the singularity and the possibility of going beyond the adiabatic argument. Many of these examples have also orientifold analogs. 
  We derive double dimensional reduction/oxidation in a framework where it is applicable to describe general non-static (and anisotropic) $p$-brane solutions. Given this procedure, we are able to relate the dynamical interaction potential for parallel extremal $p$-branes in $D$ dimensions to that for extremal black holes in $D-p$ dimensions. In particular, we find that to leading order the potential vanishes for all $\kappa$-symmetric $p$-branes. 
  The 2-brane and 4-brane solutions of ten dimensional IIA supergravity have a dual interpretation as Dirichlet-branes, or `D-branes', of type IIA superstring theory and as `M-branes' of an $S^1$-compactified eleven dimensional supermembrane theory, or M-theory. This eleven-dimensional connection is used to determine the ten-dimensional Lorentz covariant worldvolume action for the Dirichlet super 2-brane, and its coupling to background spacetime fields. It is further used to show that the 2-brane can carry the Ramond-Ramond charge of the Dirichlet 0-brane as a topological charge, and an interpretation of the 2-brane as a 0-brane condensate is suggested. Similar results are found for the Dirichlet 4-brane via its interpretation as a double-dimensional reduction of the eleven-dimensional fivebrane. It is suggested that the latter be interpreted as a D-brane of an open eleven-dimensional supermembrane. 
  We study embeddings of the Prasad-Sommerfield monopole solution in SU(N) Super-QCD (N>2), where the role of the Higgs field is played by the squarks in the fundamental representation. Classically, the resulting configurations live in a phase with unbroken SU(k) subgroups of SU(N) (as a result they are not topologically stable). The structure of zero modes is such that they can be naturally interpreted as massive chiral superfields with R charge one and baryon number zero, transforming in the adjoint representation of a dual gauge group defined using the Goddard-Nuyts-Olive (GNO) framework. We discuss the possible applications of these monopoles to N=1 duality, and more generally the possibility of relating GNO-type dual gauge groups to those appearing in N=1 duality. 
  Based on the new valley equation, we propose the most plausible method for constructing instanton-like configurations in the theory where the presence of a mass scale prevents the existence of the classical solution with a finite radius. We call the resulting instanton-like configuration as valley instanton. The detail comparison between the valley instanton and the constrained instanton in $\phi^4$ theory and the gauge-Higgs system are carried out. For instanton-like configurations with large radii, there appear remarkable differences between them. These differences are essential in calculating the baryon number violating processes with multi bosons. 
  Using the collective field technique, we give the description of the spin Calogero-Sutherland Model (CSM) in terms of free bosons. This approach can be applicable for arbitrary coupling constant and provides the bosonized Hamiltonian of the spin CSM. The boson Fock space can be identified with the Hilbert space of the spin CSM in the large $N$ limit. We show that the eigenstates corresponding to the Young diagram with a single row or column are represented by the vertex operators. We also derive a dual description of the Hamiltonian and comment on the construction of the general eigenstates. 
  An overview is given of the application of twistor geometric ideas to supersymmetry with particular emphasis on the construction of superspaces associated with four-dimensional spacetime. Talk given at Leuven conference, July 1995. 
  All four dimensional orbit spaces of compact coregular linear groups have been determined. The results are obtained through the integration of a universal differential equation, that only requires as input the number of elements of an integrity basis of the ideal of polynomial invariants of the linear group. Our results are relevant and lead to universality properties in the physics of spontaneous symmetry breaking at the classical level. 
  The world-line path-integral representation of fermion propagators is discussed. Particular attention is paid to the representation of $\gamma_5$, which is connected to the realization of manifest world-line supersymmetry. 
  A vertex model introduced by M. Bowick, P. Di Francesco, O. Golinelli, and E. Guitter (cond-mat/9502063) describing the folding of the triangular lattice onto the face centered cubic lattice has been studied in the hexagon approximation of the cluster variation method. The model describes the behaviour of a polymerized membrane in a discrete three--dimensional space. We have introduced a curvature energy and a symmetry breaking field and studied the phase diagram of the resulting model. By varying the curvature energy parameter, a first-order transition has been found between a flat and a folded phase for any value of the symmetry breaking field. 
  The intrinsic covariant 1-time description (rest-frame instant form) for N relativistic scalar particles is defined. The system of N charged scalar particles plus the electromagnetic field is described in this way: the study of its Dirac observables allows the extraction of the Coulomb potential from field theory and the regularization of the classical self-energy by using Grassmann-valued electric charges. The 1-time covariant relativistic statistical mechanics is defined. 
  A complete, straightforward and natural Lagrangian description is given for the classical non-relativistic dynamics of a particle with colour or internal symmetry degrees of freedom moving in a background Yang-Mills field. This provides a new simple Lagrangian formalism for Wong's equations for spinless particles, and presents also their generalisation, in gauge covariant form, for spin-$\frack$ particles, within a complete Lagrangian formalism. 
  The phase structure of a $(2+1)$ - dimensional model of relativistic fermions with a four fermi interaction is analyzed in the strong coupling regime using the large $N$ perturbation theory. It is shown that, this model exhibits a low temperature superconducting phase due to the vortex-anti vortex binding via Kosterlitz-Thouless mechanism. Above a critical temperature, vortices unbind and superconductivity is destroyed; at a still higher temperature the vacuum expectation value of a neutral order parameter vanishes. The ground state respects parity and time reversal symmetries. 
  We study exact renormalization group equations in the framework of the effective average action. We present analytical solutions for the scale dependence of the potential in a variety of models. These solutions display a rich spectrum of physical behaviour such as fixed points governing the universal behaviour near second order phase transitions, critical exponents, first order transitions (some of which are radiatively induced) and tricritical behaviour. 
  Recent progress on string theory in curved spacetimes is reviewed. The string dynamics in cosmological and black hole spacetimes is investigated.The methods to solve the string equations of motion in curved spacetimes are described.That is, the perturbation approach, the null approach, the $\tau$-expansion, and the construction of global solutions.The behaviour of strings in FRW and inflatio- nary spacetimes is obtained from the various types of string solutions. Three different types of behaviour appear:{\bf unstable, dual} to unstable and {\bf stable}.For the unstable strings, the energy and size grow proportional to $R$ for large scale factors $R \to \infty$. For the dual to unstable strings, the energy and size blow up for R\to 0 as 1/R. For stable strings, the energy and size are bounded. (In Minkowski, all solutions are of the stable type). The self-consistent solution to the Einstein equations for string dominated universes is reviewed. The energy-momentum tensor for a gas of strings is taken as source and from the above behaviours the string equation of state is {\bf derived}. The self-consistent string solution exhibits realistic matter dominated behaviour for large times and radiation dominated behaviour for early times. We report on the {\bf exact integrability} of the string dynamics in de Sitter spacetime that allows to systematically find {\bf exact} string solutions by soliton methods. {\bf Multistring solutions} are a new feature in curved spacetimes. That is, a single world-sheet simultaneously describes many different and independent strings. This phenomenon has no analogue in flat spacetime and follows from the coupling of the strings with the geometry. Finally, the string dynamics next and inside a Schwarzschild black hole is analyzed and their physical properties discussed. 
  The Legendre transform and its generalizations, originally found in supersymmetric sigma-models, are techniques that can be used to give constructions of hyperkahler metrics. We give a twistor space interpretation to the generalizations of the Legendre transform construction. The Atiyah-Hitchin metric on the moduli space of two monopoles is used as a detailed example. 
  The second virial coefficient for non-Abelian Chern-Simons particles is recalculated. It is shown that the result is periodic in the flux parameter just as in the Abelian theory. 
  We discuss a set of universal couplings between superstring Ramond-Ramond gauge fields and the gauge fields internal to D-branes, with emphasis on their topological consequences, and argue that instanton solutions in these internal theories are equivalent to D-branes. A particular example is the Dirichlet 5-brane in type I theory, which Witten recently showed is the zero size limit of an SO(32) instanton. Its effective world-volume theory is an Sp(1) gauge theory, unbroken in the zero size limit. We show that the zero size limit of an instanton in this theory is a 1-brane, which can be described as a bound state of the Dirichlet 1-brane with the 5-brane. Considering several 1 and 5-branes provides a description of moduli spaces of Sp(N) instantons, and a type II generalization is given which should describe U(N) instantons. 
  We consider type IIA compactification on $K3$. We show that the instanton subsectors of the worldvolume of $N$ 4-branes wrapped around $K3$ lead to a Hagedorn density of BPS states in accord with heterotic-type IIA duality in 6 dimensions. We also find evidence that the correct framework to understand the results of Nakajima on the appearance of affine Kac-Moody algebras on the cohomology of moduli space of instantons on ALE spaces is in the context of heterotic-type IIA string duality. 
  A general construction is found for `topological' singular vectors of the twisted N=2 superconformal algebra. It demonstrates many parallels with the known construction for sl(2) singular vectors due to Malikov--Feigin--Fuchs, but is formulated independently of the latter. The two constructions taken together provide an isomorphism between topological and sl(2)- singular vectors. The general formula for topological singular vectors can be reformulated as a chain of direct recursion relations that allow one to derive a given singular vector |S(r,s)> from the lower ones |S(r,s'<s)>. We also introduce generalized Verma modules over the twisted N=2 algebra and show that they provide a natural setup for the new construction for topological singular vectors. 
  We derive a worldline path integral representation for the effective action of a multiplet of Dirac fermions coupled to the most general set of matrix-valued scalar, pseudoscalar, vector, axial vector and antisymmetric tensor background fields. By representing internal degrees of freedom in terms of worldline fermions as well, we obtain a formulation which manifestly exhibits chiral gauge invariance. 
  We compare order $R^4$ terms in the 10-dimensional effective actions of SO(32) heterotic and type I superstrings from the point of view of duality between the two theories. Some of these terms do not receive higher-loop corrections being related by supersymmetry to `anomaly-cancelling' terms which depend on the antisymmetric 2-tensor. At the same time, the consistency of duality relation implies that the `tree-level' $R^4$ super-invariant (the one which has $\zeta(3)$-coefficient in the sphere part of the action) should appear also at higher orders of loop expansion, i.e. should be multiplied by a non-trivial function of the dilaton. 
  We discuss an extension of the super-Poincar\'e algebra in $D=11$ which includes an extra fermionic charge and super two-form charges. We give a geometrical reformulation of the $D=11$ supermembrane action which is manifestly invariant under the extended super-Poincar\'e transformations. Using the same set of transformations, we also reformulate a superstring action in $D=11$, considered sometime ago by Curtright. While this paper is primarily a review of a recent work by Bergshoeff and the author, it does contain some new results. 
  The generalized deformed oscillator schemes introduced as unified frameworks of various deformed oscillators are proved to be equivalent, their unified representation leading to a correspondence between the deformed oscillator and the N=2 supersymmetric quantum mechanics (SUSY-QM) scheme. In addition, several physical systems (two identical particles in two dimensions, isotropic oscillator and Kepler system in a 2-dim curved space) and mathematical structures (quadratic algebra QH(3), finite W algebra $\bar {\rm W}_0$) are shown to possess the structure of a generalized deformed su(2) algebra, the representation theory of which is known. Furthermore, the generalized deformed parafermionic oscillator is identified with the algebra of several physical systems (isotropic oscillator and Kepler system in 2-dim curved space, Fokas--Lagerstrom, Smorodinsky--Winternitz and Holt potentials) and mathematical constructions (generalized deformed su(2) algebra, finite W algebras $\bar {\rm W}_0$ and W$_3^{(2)}$). The fact that the Holt potential is characterized by the W$_3^{(2)}$ symmetry is obtained as a by-product. 
  We present unitarity as a method for determining the infinities present in graviton scattering amplitudes. The infinities are a combination of IR and UV. By understanding the soft singularities we may extract the UV infinities and relate these to counter-terms in the effective action. As an demonstration of this method we rederive the UV infinities present at one-loop when gravity is coupled to matter. 
  Construction of integrable field theories in space with a boundary is extended to fermionic models. We obtain general forms of boundary interactions consistent with integrability of the massive Thirring model and study the duality equivalence of the MT model and the sine-Gordon model with boundary terms. We find a variety of integrable boundary interactions in the $O(3)$ Gross-Neveu model from the boundary supersymmetric sine-Gordon theory by using boson-fermion duality. 
  The exact renormalization group approach (ERG) is developed for the case of pure fermionic theories by deriving a Grassmann version of the ERG equation and applying it to the study of fixed point solutions and critical exponents of the two-dimensional chiral Gross-Neveu model. An approximation based on the derivative expansion and a further truncation in the number of fields is used. Two solutions are obtained analytically in the limit $N\to \infty $, with N being the number of fermionic species. For finite N some fixed point solutions, with their anomalous dimensions and critical exponents, are computed numerically. The issue of separation of physical results from the numerous spurious ones is discussed. We argue that one of the solutions we find can be identified with that of Dashen and Frishman, whereas the others seem to be new ones. 
  Using 1-loop renormalisation group equations, we analyze the effect of randomness on multi-critical unitary minimal conformal models. We study the case of two randomly coupled $M_p$ models and found that they flow in two decoupled $M_{p-1}$ models, in the infra-red limit. This result is then extend to the case with $M$ randomly coupled $M_p$ models, which will flow toward $M$ decoupled $M_{p-1}$. 
  Four $SL(2,C)$ spinors are considered within the framework of Wigner's little groups which dictate internal space-time symmetries of relativistic particles. It is indicated that the little group for a massive particle at rest is $O(3)$, while it is $O(3)$-like for a moving massive particle. The little group becomes like $E(2)$ in the infinite-momentum/zero-mass limit. Spin-$\frac{1}{2}$ particles are studied in detail, and the origin of the gauge degrees of freedom for massless particles is clarified. There are sixteen different combinations of direct products of two $SL(2,C)$ spinors for spin-1 and spin-0 particles. The state vectors for the $O(3)$ and $O(3)$-like little groups are constructed. It is shown that in the infinite-momentum/zero-mass limit, these state vectors become scalars, four-potentials and the Maxwell field tensor. It is revealed that the Maxwell field tensor so obtained corresponds to some of the state vectors constructed by Weinberg in 1964. 
  We consider a simple static extremal multi-black hole solution with constituents charged under different $U(1)$ fields. Each of the constituents by itself is an extremal dilatonic black hole of coupling $a=\srt$. For a special case with two electrically and two magnetically charged black holes the multi-black hole solution interpolates between the familiar $a=\sqrt{3},1,\frac{1}{\sqrt{3}}$ and $0$ solutions, depending on how many black holes are placed at infinity. This proves the hypothesis that black holes with the above dilaton couplings arise in string theory as bound states of fundamental $a=\sqrt{3}$ states with zero binding energy. We also generalize the result to states where the action does not admit a single scalar truncation and show that a wide class of dyonic black holes in toroidally compactified string theory can be viewed as bound states of fundamental $a=\srt$ black holes. 
  We consider a completely integrable lattice regularization of the sine-Gordon model with discrete space and continuous time. We derive a determinant representation for a correlation function which in the continuum limit turns into the correlation function of local fields. The determinant is then embedded into a system of integrable integro-differential equations. The leading asymptotic behaviour of the correlation function is described in terms of the solution of a Riemann Hilbert Problem (RHP) related to the system of integro-differential equations. The leading term in the asymptotical decomposition of the solution of the RHP is obtained. 
  After recalling a few basic concepts from cosmology and string theory, I will discuss the main ideas/assumptions underlying string cosmology and show how these lead to a two-parameter family of ``minimal" models. I will then explain how to compute, in terms of those parameters, the spectrum of scalar, tensor and electromagnetic perturbations, point at their ($T$ and $S$-type) duality symmetries, and mention their most relevant physical consequences. 
  We consider the sl(2) current algebra at level k=-4 when the sl(2) BRST operator is nilpotent. We formulate a spectral sequence converging to the cohomology of this BRST operator. At the second term of the spectral sequence, we observe an N=4 algebra. This algebra is generated in a c=-2 bosonic string whose BRST operator Q_{string} represents the next term in the spectral sequence. We realize the cohomology of the irreducible modules as Q_{string}-primitives of the N=4 singular vectors and point out their relation to Lian--Zuckerman states of c=-2 matter. The relation between sl(2)_{-4} WZW model and c=-2 bosonic string is established both at the level of BRST cohomology and at the level of an explicit operator construction. The relation of the N=4 algebra to the known symmetries of matter+gravity systems is also investigated. 
  We consider Diagram algebras, $\Dg(G)$ (generalized Temperley-Lieb algebras) defined for a large class of graphs $G$, including those of relevance for cubic lattice Potts models, and study their structure for generic $Q$. We find that these algebras are too large to play the precisely analogous role in three dimensions to that played by the Temperley-Lieb algebras for generic $Q$ in the planar case. We outline measures to extract the quotient algebra that would illuminate the physics of three dimensional Potts models. 
  We present a brief survey of the results of the Theory of Solitons from the viewpoint of the periodic theory including some new results in the theory of 2-dimensional periodic Schrodinger Operators. The main subjects are: Periodic Solitons and Algebraic Geometry, The Theory of Solitons and the String equation, Topologically trivial and nontrivial periodic two-dimensional Schrodinger operators and Riemann surfaces and Cyclic and semicyclic chains of Laplace transformations (new results of the present author and A.Veselov). 
  We establish a weight-preserving bijection between the index sets of the spectral data of row-to-row and corner transfer matrices for $U_q\widehat{sl(n)}$ restricted interaction round a face (IRF) models. The evaluation of momenta by adding Takahashi integers in the spin chain language is shown to directly correspond to the computation of the energy of a path on the weight lattice in the two-dimensional model. As a consequence we derive fermionic forms of polynomial analogues of branching functions for the cosets ${(A^{(1)}_{n-1})_{\ell -1}\otimes (A^{(1)}_{n-1})_{1}} \over (A^{(1)}_{n-1})_{\ell}$, and establish a bosonic-fermionic polynomial identity. 
  The chiral vertex operators for the minimal models are constructed and used to define a fusion product of representations. The existence of commutativity and associativity operations is proved. The matrix elements of the associativity operations are shown to be given in terms of the 6-j symbols of the weak quasitriangular quasi-Hopf algebra obtained by truncating $\usl$ at roots of unity. 
  One-dimensional $\delta^{'}$-function potential is discussed in the framework of Green's function formalism without invoking perturbation expansion. It is shown that the energy-dependent Green's function for this case is crucially dependent on the boundary conditions which are provided by self-adjoint extension method. The most general Green's function which contains four real self-adjoint extension parameters is constructed. Also the relation between the bare coupling constant and self-adjoint extension parameter is derived. 
  We obtain an explicit expression for the supersymmetric Wilson loop in terms of chiral superfields and supercurrents in superspace. The result turns out to be different from what one would expect from the simple replacement of Lie algebra valued connection in the exponent with the corresponding super-Lie algebraic one. Genralizing the super particle coupling represented by the supersymmetric Wilson loop, we show that there exists a unique dimensionless coupling of the superstring to abelian supersymmetric gauge theories that respects all the known symmetries. The coupling is expressed in terms of chiral currents in superspace. The natural superstring coupling gives rise to a new observable that is "stringy" in nature and has no analogue in non-supersymmetric gauge theories. 
  Recently the Hamilton-Jacobi formulation for first order constrained systems has been developed. In such formalism the equations of motion are written as total differential equations in many variables. We generalize the Hamilton-Jacobi formulation for singular systems with second order Lagrangians and apply this new formulation to Podolsky electrodynamics, comparing with the results obtained through Dirac's method. 
  There are several two dimensional quantum field theory models which are equipped with different vacuum states. For example the Sine-Gordon- and the $\phi^4_2$-model. It is known that in these models there are also states, called soliton- or kink-states, which interpolate different vacua. We consider the following question: Which are the properties a pair of vacuum sates must have, such that an interpolating kink-state can be constructed? Since we are interested in structural aspects and not in specific details of a given model, we are going to discuss this question in the framework of algebraic quantum field theory which includes, for example, the $P(\phi)_2$-models. We have shown that for a large class of vacuum states, including the vacua of the $P(\phi)_2$-models, there is a natural way to construct an interpolating kink-state. 
  Representation of a $D$-dimensional fermion determinant as a path integral of exponent of a $(D+1)$-dimensional Hermitean bosonic action is constructed. 
  In this review I discuss some recent results concerning D=4 doubly supersymmetric membranes within the framework of geometrical approach obtained in the collaboration with Igor Bandos, Dmitrij Sorokin and Dmitrij Volkov. 
  Two approaches concerning the connection of the fermionic kappa -- symmetry with the superstring world -- sheet superdiffeomorphism transformations are discussed. The first approach is based on the twistor -- like formulation of the superstring action and the second one on a reformulation of the superstring and super~-~p~-~brane actions according to the Generalized Action Principle. 
  It is known that any minimal model M_p receives along its phi_31 irrelevant direction *two* massless integrable flows: one from M_{p+1} perturbed by phi_{13}, the other from Z_{p-1} parafermionic model perturbed by its generating parafermion field. By comparing Thermodynamic Bethe Ansatz data and ``predictions'' of infrared Conformal Perturbation Theory we show that these two flows are received by M_p with opposite coupling constants of the phi_31 irrelevant perturbation. Some comments on the massless S matrices of these two flows are added. 
  We investigate higher grading integrable generalizations of the affine Toda systems. The extra fields, associated to non zero grade generators, obey field equations of the Dirac type and are regarded as matter fields. The models possess soliton configurations, which can be interpreted as particles of the theory, on the same footing as those associated to fundamental fields. A special subclass of these models is remarkable. They possess a $U(1)$ Noether current which, after a special gauge fixing of the conformal symmetry, is proportional to a topological current. This leads to the confinement of the matter field inside the solitons, which can be regarded as a one dimensional bag model for QCD. These models are also relevent to the study of electron self--localization in (quasi)-one-dimensional electron--phonon systems. 
  We propose an extension of 't Hooft's large-$N_c$ light-front QCD in two dimensions to include helicity and physical gluon degrees of freedom, modelled on a classical dimensional reduction of four dimensional QCD. A non-perturbative renormalisation of the infinite set of coupled integral equations describing boundstates is performed. These equations are then solved, both analytically in a phase space wavefunction approximation and numerically by discretising momenta, for (hybrid) meson masses and (polarized) parton structure functions. 
  We construct a 1+1 dimensional superstring-bit model for D=3 Type IIB superstring. This low dimension model escapes the problems encountered in higher dimension models: (1) It possesses full Galilean supersymmetry; (2) For noninteracting polymers of bits, the exactly soluble linear superpotential describing bit interactions is in a large universality class of superpotentials which includes ones bounded at spatial infinity; (3) The latter are used to construct a superstring-bit model with the clustering properties needed to define an $S$-matrix for closed polymers of superstring-bits. 
  Halpern and Huang recently showed that there are relevant directions in the space of interactions at the Gaussian fixed point. I show that their result can be derived from Polchinski's form of the Wilson renormalization group. The derivation shows that the existence of these directions is independent of the cutoff function used. 
  This paper has been withdraw by the authors. An enlarged and complete analysis of the constraints is presented in hep-th/9910163. It includes also relativistic particles and strings. The non-relativistic particle quantization has been reanalyzed. 
  We apply the Batalin-Tyutin Hamiltonian method to the Abelian Proca model in order to convert a second class constraint system into a first class one systematically by introducing the new fields. Then, according to the BFV formalism we obtain that the desired resulting Lagrangian preserving standard BRST symmetry naturally includes the well-known St\"ukelberg scalar related to the explicit gauge-breaking effect due to the presence of the mass term. Furthermore, we also discuss the nonlocal symmetry structure of this model in the context of the nonstandard BRST symmetry. 
  Within the framework of algebraic quantum field theory, we construct explicitly localized morphisms of a Haag-Kastler net in 1+1-dimensional Minkowski space showing abelian braid group statistics. Moreover, we investigate the scattering theory of the corresponding quantum fields. 
  Feynman rules for the vacuum amplitude of fermions coupled to external gauge and Higgs fields in a domain wall lattice model are derived using time--dependent perturbation theory. They have a clear and simple structure corresponding to 1--loop vacuum graphs. Their continuum approximations are extracted by isolating the infrared singularities and it is shown that, in each order, they reduce to vacuum contributions for chiral fermions. In this sense the lattice model is seen to constitute a valid regularization of the continuum theory of chiral fermions coupled to weak and slowly varying gauge and Higgs fields. The overlap amplitude, while not gauge invariant, exhibits a well defined (modulo phase conventions) response to gauge transformations of the background fields. This response reduces in the continuum limit to the expected chiral anomaly, independently of the phase conventions. 
  The effective action for the local composite operator $\Phi^2(x)$ in the scalar quantum field theory with $\lambda\Phi^4$ interaction is obtained in the expansion in two-particle-point-irreducible (2PPI) diagrams up to five-loops. The effective potential and 2-point Green's functions for elementary and composite fields are derived. The ground state energy as well as one- and two-particle excitations are calculated for space-time dimension $n=1$, when the theory is equivalent to the quantum mechanics of an anharmonic oscillator. The agreement with the exact spectrum of the oscillator is much better than that obtained within the perturbation theory. 
  Using the nonperturbative Schwinger-Dyson equation, we show that chiral symmetry in weak-coupling massless QED is dynamically broken by a constant but arbitrarily strong external magnetic field. 
  Starting with a manifestly conformal ($O(d,2)$ invariant) mechanics model in $d$ space and 2 time dimensions, we derive the action for a massless spinning particle in $d$-dimensional anti-de Sitter space. The action obtained possesses both gauge $N$-extended worldline supersymmetry and local $O(N)$ invarince. Thus we improve the old statement by Howe et al. that the spinning particle model with extended worldline supersymmetry admits only flat space-time background for $N > 2$ (spin greater one). The original $(d+2)$-dimensional model is characterized by rather unusual property that the corresponding supersymmetry transformations do not commute with the conformal ones, in spite of the explicit $O(d,2)$ invariance of the action. 
  A semi-classical check of the Goddard-Nuyts-Olive (GNO) generalized duality conjecture for gauge theories with adjoint Higgs fields is performed for the case where the unbroken gauge group is non-abelian. The monopole solutions of the theory transform under the non-abelian part of the unbroken global symmetry and the associated component of the moduli space is a Lie group coset space. The well-known problems in introducing collective coordinates for these degrees-of-freedom are solved by considering suitable multi-monopole configurations in which the long-range non-abelian fields cancel. In the context of an $N=4$ supersymmetric gauge theory, the multiplicity of BPS saturated states is given by the number of ground-states of a supersymmetric quantum mechanics on the compact internal moduli space. The resulting degeneracy is expressed as the Euler character of the coset space. In all cases the number of states is consistent with the dimensions of the multiplets of the unbroken dual gauge group, and hence the results provide strong support for the GNO conjecture. 
  We discuss the connection between the construction of Bogomol'nyi bounds and equations in three dimensional gravitational theories and the existence of an underlying $N=2$ local supersymmetric structure. We show that, appart from matter self duality equations, a first order equation for the gravitational field (whose consistency condition gives the Einstein equation) can be written as a consequence of the local supersymmetry. Its solvability makes possible the evasion of the no-go scenario for the construction of Killing spinors in asymptotically conical spacetimes. In particular we show that the existence of non-trivial supercovariantly constant spinors is guaranteed whenever field configurations saturate the topological bound. 
  We consider an electron which is electromagnetically dressed in such a way that it is both gauge invariant and that it has the associated electric and magnetic fields expected of a moving charge. We study the propagator of this dressed electron and, for small velocities, show explicitly at one loop that at the natural (on-shell) renormalisation point, $p_0=m$, ${\bold p}= m{\bold v}$, one can renormalise the propagator multiplicatively. Furthermore the renormalisation constants are infra-red finite. This shows that the dressing we use corresponds to a slowly moving, physical asymptotic field. 
  A unified description of the relationship between the Hamiltonian structure of a large class of integrable hierarchies of equations and W-algebras is discussed. The main result is an explicit formula showing that the former can be understood as a deformation of the latter. 
  The dynamical algebra associated to a family of Isospectral Oscillator Hamiltonians, named {\it Distorted Heisenberg Algebra} because its dependence on a distortion parameter $W \geq 0$, has been recently studied. The connection of this algebra with the Hilbert space of entire functions of growth (1/2, 2) is analized. 
  We describe the counting of BPS states of Type II strings on K3 by relating the supersymmetric cycles of genus $g$ to the number of rational curves with $g$ double points on K3. The generating function for the number of such curves is the left-moving partition function of the bosonic string. 
  From the time-independent current $\tcj(\bar y,\bar k)$ in the quantum self-dual Yang-Mills (SDYM) theory, we construct new group-valued quantum fields $\tilde U(\bar y,\bar k)$ and $\bar U^{-1}(\bar y,\bar k)$ which satisfy a set of exchange algebras such that fields of $\tcj(\bar y,\bar k)\sim\tilde U(\bar y,\bar k)~\partial\bar y~\tilde U^{-1}(\bar y,\bar k)$ satisfy the original time-independent current algebras. For the correlation functions of the products of the $\tilde U(\bar y,\bar k)$ and $\tilde U^{-1}(\bar y,\bar k)$ fields defined in the invariant state constructed through the current $\tcj(\bar y,\bar k)$ we can derive the Knizhnik-Zamolodchikov (K-Z) equations with an additional spatial dependence on $\bar k$. From the $\tilde U(\bar y,\bar k)$ and $\tilde U^{-1}(\bar y,\bar k)$ fields we construct the quantum-group generators --- local, global, and semi-local --- and their algebraic relations. For the correlation functions of the products of the $\tilde U$ and $\tilde U^{-1}$ fields defined in the invariant state constructed through the semi-local quantum-group generators we obtain the quantum-group difference equations. We give the explicit solution to the two point function. 
  We consider the $N\!=\!1$ supersymmetric $\sigma$-model and we examine the transformation properties of the partition function under target-space duality. Contrary to what one would expect, we find that it is not, in general, invariant. In fact, besides the dilaton shift emerging from the Jacobian of the duality transformation of the bosonic part, there also exist a Jacobian for the fermionic part since fermions are also transform under the duality process. The latter is just the parity of the spin structure of the word sheet and since it cannot be compensated, the dual theory is not equvalent to the original one. 
  Global issues of the Poisson-Lie T-duality are addressed. It is shown that oriented open strings propagating on a group manifold $G$ are dual to $D$-brane - anti-$D$-brane pairs propagating on the dual group manifold $\ti G$. The $D$-branes coincide with the symplectic leaves of the standard Poisson structure induced on the dual group $\ti G$ by the dressing action of the group $G$. T-duality maps the momentum of the open string into the mutual distance of the $D$-branes in the pair. The whole picture is then extended to the full modular space $M(D)$ of the Poisson-Lie equivalent $\si$-models which is the space of all Manin triples of a given Drinfeld double.T-duality rotates the zero modes of pairs of $D$-branes living on targets belonging to $M(D)$. In this more general case the $D$-branes are preimages of symplectic leaves in certain Poisson homogeneous spaces of their targets and, as such, they are either all even or all odd dimensional. 
  We demonstrate how negative powers of screenings arise as a nonperturbative effect within the operator approach to Liouville theory. This explains the origin of the corresponding poles in the exact Liouville three point function proposed by Dorn/Otto and $(\hbox{Zamolodchikov})^2$ (DOZZ) and leads to a consistent extension of the operator approach to arbitrary integer numbers of screenings of both types. The general Liouville three point function in this setting is computed without any analytic continuation procedure, and found to support the DOZZ conjecture. We point out the importance of the concept of free field expansions with adjustable monodromies - recently advocated by Petersen, Rasmussen and Yu - in the present context, and show that it provides a unifying interpretation for two types of previously constructed local observables. 
  A method for implementing non-Abelian duality on string backgrounds is given. It is shown that a direct generalisation of the familiar Abelian duality induces an extra local symmetry in the gauge invariant theory. The non-Abelian isometry group is shown to be enlarged to a non-semi-simple group. However, upon eliminating the gauge fields to obtain the dual theory the new algebra does not close. Therefore the gauge fixing procedure becomes problematic. The new method proposed here avoids these issues and leads to a dual theory in the proper sense of duality. 
  We present the most general static, spherically symmetric solutions of heterotic string compactified on a six-torus that conforms to the conjectured ``no-hair theorem'', by performing a subset of O(8,24) transformations, i.e., symmetry transformations of the effective three-dimensional action for stationary solutions, on the Schwarzschild solution. The explicit form of the generating solution is determined by six $SO(1,1)\subset O(8,24)$ boosts, with the zero Taub-NUT charge constraint imposing one constraint among two boost parameters. The non-nontrivial scalar fields are the axion-dilaton field and the moduli of the two-torus. The general solution, parameterized by {\it unconstrained} 28 magnetic and 28 electric charges and the ADM mass compatible with the Bogomol'nyi bound, is obtained by imposing on the generating solution $[SO(6)\times SO(22)]/[SO(4)\times SO(20)] \subset O(6,22)$ (T-duality) transformation and $SO(2)\subset SL(2,R)$ (S-duality) transformation, which do not affect the four-dimensional space-time. Depending on the range of boost parameters, the non-extreme solutions have the space-time of either Schwarzschild or Reissner-Nordstr\" om black hole, while extreme ones have either null (or naked) singularity, or the space-time of extreme Reissner-Nordstr\" om black hole. 
  The instanton solution for the forced Burgers equation is found. This solution describes the exponential tail of the probability distribution function of velocity differences in the region where shock waves are absent. The results agree with the one found recently by Polyakov, who used the operator product conjecture. If this conjecture is true, then our WKB asymptotics of the Wyld functional integral is exact to all orders of the perturbative expansion around the instanton solution. We explicitly checked this in the first order. We also generalized our solution for the arbitrary dimension of Burgers (=KPZ) equation. As a result we found the angular dependence of the velocity difference PDF. 
  We study the BPS spectrum in $D=4, N=4$ heterotic string compactifications, with some emphasis on intermediate $N=4$ BPS states. These intermediate states, which can become short in $N=2$ compactifications, are crucial for establishing an $S-T$ exchange symmetry in $N=2$ compactifications. We discuss the implications of a possible $S-T$ exchange symmetry for the $N=2$ BPS spectrum. Then we present the exact result for the 1-loop corrections to gravitational couplings in one of the heterotic $N=2$ models recently discussed by Harvey and Moore. We conjecture this model to have an $S-T$ exchange symmetry. This exchange symmetry can then be used to evaluate non-perturbative corrections to gravitational couplings in some of the non-perturbative regions (chambers) in this particular model and also in other heterotic models. 
  We describe an infinite-dimensional Kac-Moody-Virasoro algebra of new hidden symmetries for the self-dual Yang-Mills equations related to conformal transformations of the 4-dimensional base space. 
  At the very early Universe the matter fields are described by the GUT models in curved space-time. At high energies these fields are asymptotically free and conformally coupled to external metric. The only possible quantum effect is the appearance of the conformal anomaly, which leads to the propagation of the new degree of freedom - conformal factor. Simultaneously with the expansion of the Universe, the scale of energies decreases and the propagating conformal factor starts to interact with the Higgs field due to the violation of conformal invariance in the matter fields sector. In a previous paper \cite{foo} we have shown that this interaction can lead to special physical effects like the renormalization group flow, which ends in some fixed point. Furthermore in the vicinity of this fixed point there occur the first order phase transitions. In the present paper we consider the same theory of conformal factor coupled to Higgs field and incorporate the temperature effects. We reduce the complicated higher-derivative operator to several ones of the standard second-derivative form and calculate an exact effective potential with temperature on the anti de Sitter (AdS) background. 
  We consider theories with gauged chiral fermions in which there are abelian anomalies, and no nonabelian anomalies (but there may be nonabelian gauge fields present). We construct an associated theory that is gauge invariant, renormalizable, and with the same particle content, by adding a finite number of terms to the action. Alternatively one can view the new theory as arising from the original theory by using another regularization, one that is gauge invariant. The situation is reminiscent of the mechanism of adding Fadeev-Popov ghosts to an unsatisfactory gauge theory, to arrive at the usual quantization procedure. The models developed herein are much like the abelian Wess-Zumino model (an abelian effective theory with a Wess-Zumino counter term), but unlike the W-Z model are renormalizable! 
  This paper is based on lectures presented to mathematical physicists and attempts to provide an overview of the present status of the Standard Model, its experimental tests, phenomenological and experimental motivations for going beyond the Standard Model via supersymmetry and grand unification, and ways to test these ideas with particle accelerators. 
  We apply the techniques of $S^7$-algebras to the construction of N=5-8 superconformal algebras and of S{\bf O}(1,9), a modification of SO(1,9) which commutes with $S^7$-transformations. We discuss the relevance of S{\bf O}(1,9) for off-shell super-Maxwell theory in D=(1,9). 
  We consider SU(N) gauge theory in 1+1 dimensions coupled to chiral fermions in the adjoint representation of the gauge group. With all fields in the adjoint representation the gauge group is actually SU(N)/Z_N, which possesses nontrivial topology. In particular, there are N distinct topological sectors and the physical vacuum state has a structure analogous to a \theta vacuum. We show how this feature is realized in light-front quantization for the case N=2, using discretization as an infrared regulator. In the discretized form of the theory the nontrivial vacuum structure is associated with the zero momentum mode of the gauge field A^+. We find exact expressions for the degenerate vacuum states and the analog of the \theta vacuum. The model also possess a condensate which we calculate. We discuss the difference between this chiral light-front theory and the theories that have previously been considered in the equal-time approach. 
  We start by giving a brief introduction to string theory with emphasis on the example of the bosonic string. In order to fully appreciate string theory it is necessary to study the dynamics of the surface that the string traces out when moving in space-time. This is described by what is known as conformal field theory, which is the subject of chapter two. After this digression we return to string theory to give some insight into how the results of the articles contained in this thesis could provide with interesting ingredients for constructions of realistic string theories. This is also continued on to some extent in chapter seven and eight. The articles in this thesis are discussing mainly two subjects, gauged WZNW models and affine branching functions. The background for those issues is given in chapter seven and eight. In the articles contained in this thesis we use two major tools. First affine Lie algebras and its representation theory, and secondly BRST quantization. In chapter four and five we deal with affine Lie algebras, and in chapter six we try to illuminate some issues on BRST quantization. 
  We discuss the low-energy dynamics of generalized extremal higher membrane black hole solutions of string theory and higher membrane theories following Manton's prescription for multi-soliton solutions. A flat metric is found for those solutions which possess $\kappa$-symmetry on the worldvolume. 
  We show that certain classes of K3 fibered Calabi-Yau manifolds derive from orbifolds of global products of K3 surfaces and particular types of curves. This observation explains why the gauge groups of the heterotic duals are determined by the structure of a single K3 surface and provides the dual heterotic picture of conifold transitions between K3 fibrations. Abstracting our construction from the special case of K3 hypersurfaces to general K3 manifolds with an appropriate automorphism, we show how to construct Calabi-Yau threefold duals for heterotic theories with arbitrary gauge groups. This generalization reveals that the previous limit on the Euler number of Calabi-Yau manifolds is an artifact of the restriction to the framework of hypersurfaces. 
  An introduction to $N=2$ rigid and local supersymmetry is given. The construction of the actions of vector multiplets is reviewed, defining special K\"ahler manifolds. Symplectic transformations lead to either isometries or symplectic reparametrizations. Writing down a symplectic formulation of special geometry clarifies the relation to the moduli spaces of a Riemann surface or a Calabi-Yau 3-fold. The scheme for obtaining perturbative and non-perturbative corrections to a supersymmetry model is explained. The Seiberg-Witten model is reviewed as an example of the identification of duality symmetries with monodromies and symmetries of the associated moduli space of a Riemann surface. 
  We summarize some results in 4d, N=1 supersymmetric SU(2) gauge theories: the exact effective superpotentials, the vacuum structure, and the exact effective Abelian couplings for arbitrary bare masses and Yukawa couplings. 
  Correlation between instantons and QCD-monopoles is studied in the abelian-gauge-fixed QCD. From a simple topological consideration, instantons are expected to appear only around the QCD-monopole trajectory in the abelian-dominating system. The QCD-monopole in the multi-instanton solution is studied in the Polyakov-like gauge, where $A_4(x)$ is diagonalized. The world line of the QCD-monopole is found to be penetrate the center of each instanton. For the single-instanton solution, the QCD-monopole trajectory becomes a simple straight line. On the other hand, in the multi-instanton system, the QCD-monopole trajectory often has complicated topology including a loop or a folded structure, and is unstable against a small fluctuation of the location and the size of instantons. We also study the thermal instanton system in the Polyakov-like gauge. At the high-temperature limit, the monopole trajectory becomes straight lines in the temporal direction. The topology of the QCD-monopole trajectory is drastically changed at a high temperature. 
  Strassler's formulation of the string-derived Bern-Kosower formalism is extended to consider QED processes in homogeneous constant external field. A compact expression for the contribution of the one-loop with arbitrary number of external photon lines is given for scalar QED. Extension to spinor QED is shortly discussed. 
  The vector-tensor multiplet is coupled off-shell to an N=2 vector multiplet such that its central charge transformations are realized locally. A gauged central charge is a necessary prerequisite for a coupling to supergravity and the strategy underlying our construction uses the potential for such a coupling as a guiding principle. The results for the action and transformation rules take a nonlinear form and necessarily include a Chern-Simons term. After a duality transformation the action is encoded in a homogeneous holomorphic function consistent with special geometry. 
  Using the Schwinger-Keldysh (closed time path or CTP) and Feynman-Vernon influence functional formalisms we obtain an expression for the influence functional in terms of Bogoliubov coefficients for the case of spinor quantum electrodynamics. Then we derive a CTP effective action in semiclassical approximation and its cumulant expansion. Using it we obtain a equation for the description of the charged particle creation in electric field and of backreaction of charged quantum fields and their fluctuations on time evolution of this electric field. Also an intimate connection between CTP effective action and decoherence functional will allow us to analyze how macroscopic electromagnetic fields are ``measured'' through interaction with charges and thereby rendered classical. 
  We propose a conformal field theory description of a solitonic heterotic string in type $IIA$ superstring theory compactified on $K3$, generalizing previous work by J. Harvey, A. Strominger and A. Sen. In ten dimensions the construction gives a fivebrane which is related to the fundamental type $II$ string by electric -- magnetic duality, and to the Dirichlet fivebrane of type $IIB$ string theory by $SL(2, Z)$. 
  It is well known that --differing from ordinary gauge systems-- canonical gauges are not admissible in the path integral for parametrized systems. This is the case for the relativistic particle and gravitation. However, a time dependent canonical transformation can turn a parametrized system into an ordinary gauge system. It is shown how to build a canonical transformation such that the fixation of the new coordinates is equivalent to the fixation of the original ones; this aim can be achieved only if the Hamiltonian constraint allows for an intrinsic global time. Thus the resulting action, describing an ordinary gauge system and allowing for canonical gauges, can be used in the path integral for the quantum propagator associated with the original variables. 
  We construct collective field theories associated with one-matrix plus $r$ vector models. Such field theories describe the continuum limit of spin Calogero Moser models. The invariant collective fields consist of a scalar density coupled to a set of fields in the adjoint representation of $U(r)$. Hermiticity conditions for the general quadratic Hamiltonians lead to a new type of extended non-linear algebra of differential operators acting on the Jacobian. It includes both Virasoro and $SU(r)$ (included in $sl(r, {\bf C}) \times sl(r, {\bf C})$) current algebras. A systematic construction of exact eigenstates for the coupled field theory is given and exemplified. 
  We study radiation from black holes in the effective theory produced by integrating gravity and the dilaton out of $1+1$ dilaton gravity. The semiclassical wavefunctions for the dressed particles show that the self-interactions produce an unusual renormalization of the frequencies of outgoing states. Modes propagating in the dynamical background of an incoming quantum state are seen to acquire large scattering phases that nevertheless conspire, in the absence of self-interactions, to preserve the thermality of the Hawking radiation. However, the in-out scattering matrix does not commute with the self-interactions and this could lead to observable corrections to the final state. Finally, our calculations explicitly display the limited validity of the semiclassical theory of Hawking radiation and provide support for a formulation of black hole complementarity. 
  In this letter we study supersymmetric sigma models on toric varieties. These manifolds are generalizations of CP^n manifolds. We examine here sigma models, viewed as gauged linear sigma models, on one of the simplest such manifold, the blow-up of P^2_(2,1,1), and determine their properties using the techniques of topological- antitopological fusion. We find that the model contains solitons which become massless at the singular point of the theory where a gauge symmetry remains unbroken. 
  In our paper~\cite{KR} we began a systematic study of representations of the universal central extension $\widehat{\Cal D}\/$ of the Lie algebra of differential operators on the circle. This study was continued in the paper~\cite{FKRW} in the framework of vertex algebra theory. It was shown that the associated to $\widehat {\Cal D}\/$ simple vertex algebra $W_{1+ \infty, N}\/$ with positive integral central charge $N\/$ is isomorphic to the classical vertex algebra $W (gl_N)$, which led to a classification of modules over $W_{1 + \infty, N}$. In the present paper we study the remaining non-trivial case, that of a negative central charge $-N$. The basic tool is the decomposition of $N\/$ pairs of free charged bosons with respect to $gl_N\/$ and the commuting with $gl_N\/$ Lie algebra of infinite matrices $\widehat{gl}$. 
  Wigner's 1939 paper on representations of the inhomogeneous Lorentz group is one of the most fundamental papers in physics. Wigner maintained his passion for this subject throughout his life. In this report, I will review the papers which he published with me on this subject. These papers deal with the question of unifying the internal space-time symmetries of massive and massless particles. 
  We discuss the different discrete duality symmetries in six dimensions that act within and between (i) the 10-dimensional heterotic string compactified on $T^4$, (ii) the 10-dimensional Type IIA string compactified on $K3$ and (iii) the 10-dimensional Type IIB string compactified on $K3$. In particular we show that the underlying group-theoretical structure of these discrete duality symmetries is determined by the proper cubic group ${\cal C}/\Z_2$. Our group theoretical interpretation leads to simple rules for constructing the explicit form of the different discrete Type II duality symmetries in an arbitrary background. The explicit duality rules we obtain are applied to construct dual versions of the 6-dimensional chiral null model. 
  In a previous paper \cite{lp}, supersymmetric $p$-brane solutions involving one dilatonic scalar field in maximal supergravity theories were classified. Although these solutions involve a number of participating field strengths, they are all equal and thus they carry equal electric or magnetic charges. In this paper, we generalise all these solutions to multi-scalar solutions in which the charges become independent free parameters. The mass per unit $p$-volume is equal to the sum of these Page charges. We find that for generic values of the Page charges, they preserve the same fraction of the supersymmetry as in their single-scalar limits. However, for special values of the Page charges, the supersymmetry can be enhanced. 
  We construct a solution to the low-energy string equations of motion in five dimensions that describes a circular loop of fundamental string exponentially expanding in a background electric $H$-field. Euclideanising this gives an instanton for the creation of a loop of fundamental string in a background $H$-field, and we calculate the rate of nucleation. Solutions describing magnetically charged strings and $p$-branes, where the gauge field comes from Kaluza-Klein reduction on a circle, are also constructed. It is known that a magnetic flux tube in four (reduced) spacetime dimensions is unstable to the pair creation of Kaluza-Klein monopoles. We show that in $(4+p)$ dimensions, magnetic $(p+1)$ ``fluxbranes" are unstable to the nucleation of a magnetically charged spherical $p$-brane. In ten dimensions the instanton describes the nucleation of a Ramond-Ramond magnetically charged six-brane in type IIA string theory. We also find static solutions describing spherical charged $p$-branes or fundamental strings held in unstable equilibrium in appropriate background fields. Instabilities of intersecting magnetic fluxbranes are also discussed. 
  The present status of superstring phenomenology is briefly discussed. 
  We generalize the rules for the free fermionic string construction to include other asymmetric orbifolds. Examples are given to illustrate the use of these rules. 
  We study some features of dyonic Black hole solution in heterotic string theory on a six torus. This solution has 58 parameters. Of these, 28 parameters denote the electric charge of the black hole, another 28 correspond to the magnetic charge, and the other two parameters being the mass and the angular momentum of the black hole. We discuss the extremal limit and show that in various limits, it reduces to the known black hole solutions. The solutions saturating the Bogomolnyi bound are identified. Explicit solution is presented for the non-rotating dyonic black hole. 
  The equation of motion of affine Toda field theory is a coupled equation for $r$ fields, $r$ is the rank of the underlying Lie algebra. Most of the theories admit reduction, in which the equation is satisfied by fewer than $r$ fields. The reductions in the existing literature are achieved by identifying (folding) the points in the Dynkin diagrams which are connected by symmetry (automorphism). In this paper we present many new reductions. In other words the symmetry of affine Dynkin diagrams could be extended and it leads to non-canonical foldings. We investigate these reductions in detail and formulate general rules for possible reductions. We will show that eventually most of the theories end up in $a_{2n}^{(2)}$ that is the theory cannot have a further dimension $m$ reduction where $m<n$. 
  We investigate the internal space of Bessel functions which is associated to the group Z of positive and negative integers defining their orders. As a result we propose and prove a new unifying formula (to be added to the huge literature on Bessel functions) generating Bessel functions of real orders out of integer order one's. The unifying formula is expected to be of great use in applied mathematics. Some applications of the formula are given for illustration. 
  W_{1+infty} is defined as an infinite dimensional Lie algebra spanned by the unit operator and the Laurent modes of a series of local quasiprimary chiral fields V^l(z) of dimension l+1 (l=0,1,2,...). These fields are neutral with respect to the u(1) current J(z)=V^0(z); as a result the (l+2)-fold commutator of J with V^l vanishes. We outline a construction of rational conformal field theories with stress energy tensor T(z)=V^1(z) whose chiral algebras include all V^l's. It is pointed out that earlier work on local extensions of the u(1) current algebra solves the problem of classifying all such theories for Virasoro central charge c=1. 
  Some basic facts about the prepotential in the SW/Whitham theory are presented. Consideration begins from the abstract theory of quasiclassical $\tau$-functions , which uses as input a family of complex spectral curves with a meromorphic differential $dS$, subject to the constraint $\partial dS/\partial(moduli)= \ holomorphic$, and gives as an output a homogeneous prepotential on extended moduli space. Then reversed construction is discussed, which is straightforwardly generalizable from spectral {\it curves} to certain complex manifolds of dimension $d >1$ (like $K3$ and $CY$ families). Finally, examples of particular $N=2$ SUSY gauge models are considered from the point of view of this formalism. At the end we discuss similarity between the $WP^{12}_{1,1,2,2,6}$ -\-Calabi-\-Yau model with $h_{21}=2$ and the $1d$ $SL(2)$ Calogero/Ruijsenaars model, but stop short of the claim that they belong to the same Whitham universality class beyond the conifold limit. 
  We argue that the random-matrix like energy spectra found in pseudointegrable billiards with pointlike scatterers are related to the quantum violation of scale invariance of classical analogue system. It is shown that the behavior of the running coupling constant explains the key characteristics of the level statistics of pseudointegrable billiards. 
  A reexamination of the semiclassical approach of the relativistic electron indicates a possible variation of its helicity for electric and magnetic static fields applied along its global motion due to zitterbewegung effects, proportional to the anomalous part of the magnetic moment. 
  We examine the quantum energy levels of rectangular billiards with a pointlike scatterer in one and two dimensions. By varying the location and the strength of the scatterer, we systematically find diabolical degeneracies among various levels. The associated Berry phase is illustrated, and the existence of localized wave functions is pointed out. In one dimension, even the ground state is shown to display the sign reversal with a mechanism to circumvent the Sturm-Liouville theorem. 
  Global N=2 supersymmetry in four dimensions with a gauged central charge is formulated in superspace. To find an irreducible representation of supersymmetry for the gauge connections a set of constraints is given. Then the Bianchi identities are solved subject to this set of constraints. It is shown that the gauge connection of the central charge is a N=2 vector multiplet. Moreover the Bogomol'nyi bound of the massive particle states is studied. 
  In this talk results of study in various dimensions of the Boltzmann master field for a subclass of planar diagrams, so called half-planar diagrams, found in the recent work by Accardi, Volovich and one of us (I.A.) are presented. 
  The relationship between various methods to calculate the physical degrees of freedom for gauge invariant systems of a general form is established. The set of hidden parameters caused for the superfluous degrees of freedom is revealed. 
  The $e^-$-$e^+$ bound state spectrum of QED3 is investigated in the quenched ladder approximation to the homogeneous Bethe-Salpeter equation with fermion propagators from a rainbow approximation Schwinger-Dyson equation. A detailed analysis of the analytic structure of the fermion propagator is performed so as to test the appropriateness of the methods employed. The large fermion mass limit of the Bethe-Salpeter equation is also considered, including a derivation of the Schr\"{o}dinger equation, and comparisons made with existing non-relativistic calculations. 
  In this talk the Schwarz hypothesis that the duality symmetries should be pieces of the hidden gauge symmetry in a string theory is discussed. Using auxiliary linear system special dual transformations for $N=4$ SYM generalizing the Schwarz dual transformations for a principal chiral model are constructed. These transformations are related with a continuous group of hidden symmetry of a new model involving more fields as compare with $N=4$ SYM. We conjecture that the $Z_{2}$ discrete subgroup of this hidden symmetry group has a stable set of N=4 YM fields and transforms a self-dual configuration to an anti-self-dual and via versa. 
  In a series of papers Amati, Ciafaloni and Veneziano and 't Hooft conjectured that black holes occur in the collision of two light particles at planckian energies. In this talk based on \cite {AVV} we discuss a possible scenario for such a process by using the Chandrasekhar-Ferrari-Xanthopoulos duality between the Kerr black hole solution and colliding plane gravitational waves. 
  The standard model is reconstructed by new method to incorporate strong interaction into our previous scheme based on the non-commutative geometry. The generation mixing is also taken into account. Our characteristic point is to take the fermion field so as to contain quarks and leptons all together which is almost equal to that of SO(10) grand unified theory(GUT). The space-time $M_4\times Z_2$; Minkowski space multiplied by two point discrete space is prepared to express the left-handed and right-handed fermion fields. The generalized gauge field $A(x,y)$ written in one-differential form extended on $M_4\times Z_2$ is well built to give the correct Dirac Lagrangian for fermion sector. The fermion field is a vector in 24-dimensional space and gauge and Higgs fields are written in $24\times24$ matrices. At the energy of the equal coupling constants for both sheets $y=\pm$ expected to be amount to the energy of GUT scale, we can get $\sin^2\theta_{_{W}}=3/8$ and $m_{_{H}}=\sqrt{2}m_{_{W}}$. In general, the equation $m\ma{H}=(4/\sqrt {3})m\ma{W}\sin\theta\ma{W}$ is followed. Then, it should be noticed that the same result as that of the grand unified theory such as SU(5) or SO(10) GUT is obtained without GUT but with the approach based on the non-commutative geometry and in addition the Higgs mass is related to other physical quantities as stated above. 
  The four-point Green's function of static QED, where a fermion and an antifermion are located at fixed space positions, is calculated in covariant gauges. The bound state spectrum does not display any abnormal state corresponding to excitations of the relative time. The equation that was established by Mugibayashi in this model and which has abnormal solutions does not coincide with the Bethe$-$Salpeter equation. Gauge transformation from the Coulomb gauge also confirms the absence of abnormal solutions in the Bethe$-$Salpeter equation. 
  It is shown that classical nonsupersymmetric Yang-Mills theory in 4 dimensions is symmetric under a generalized dual transform which reduces to the usual dual *-operation for electromagnetism. The parallel phase transport $\tilde{A}_\mu(x)$ constructed earlier for monopoles is seen to function also as a potential in giving full description of the gauge field, playing thus an entirely dual symmetric role to the usual potential $A_\mu(x)$. Sources of $A$ are monopoles of $\tilde{A}$ and vice versa, and the Wu-Yang criterion for monopoles is found to yield as equations of motion the standard Wong and Yang-Mills equations for respectively the classical and Dirac point charge; this applies whether the charge is electric or magnetic, the two cases being related just by a dual transform. The dual transformation itself is explicit, though somewhat complicated, being given in terms of loop space variables of the Polyakov type. 
  In this paper we present a detailed study of the quantum conservation laws for Toda field theories defined on the half plane in the presence of a boundary perturbation. We show that total derivative terms added to the currents, while irrelevant at the classical level, become important at the quantum level and in general modify significantly the quantum boundary conservation. We consider the first nontrivial higher--spin currents for the simply laced $a^{(1)}_n $ Toda theories: we find that the spin--three current leads to a quantum conserved charge only if the boundary potential is appropriately redefined through a finite renormalization. Contrary to the expectation we demonstrate instead that at spin four the classical symmetry does not survive quantization and we suspect that this feature will persist at higher--spin levels. Finally we examine the first nontrivial conservations at spin four for the $d^{(2)}_3$ and $c^{(1)}_2$ nonsimply laced Toda theories. In these cases the addition of total derivative terms to the bulk currents is necessary but sufficient to ensure the existence of corresponding quantum exact conserved charges. 
  Talk presented by J.L. Petersen at the 29th Symposium Ahrenshop, Buckow August 29-September 2, 1995. A presentation is given of the free field realization relevant to SL(2) WZW theories with a Hilbert space based on admissible representations. It is known that this implies the presence of two screening charges, one involving a fractional power of a free field. We develop the use of fractional calculus for treating in general such cases. We derive explicit integral representations of $N$-point conformal blocks. We show that they satisfy the Knizhnik-Zamolodchikov equations and we prove how they are related to minimal conformal blocks via a formulation of hamiltonian reduction advocated by Furlan, Ganchev, Paunov and Petkova. 
  We continue the study of symmetries in the Lagrangian formalism of arbitrary order with the help of the so-called Anderson-Duchamp-Krupka equations. For the case of second-order equations and arbitrary vector fields we are able to establish a polynomial structure in the second-order derivatives. This structure is based on the some linear combinations of Olver hyper-Jacobians. We use as the main tools Fock space techniques and induction. This structure can be used to analyze Lagrangian systems with groups of Noetherian symmetries. As an illustration we analyze the case of Lagrangian equations with Abelian gauge invariance. 
  We study the embedding of extreme (multi-) dilaton black hole solutions for the values of the parameter $a=\sqrt{3},1,1/\sqrt{3},0$ in $N=4$ and $N=8$ four-dimensional supergravity. For each black hole solution we find different embeddings in $N=4$ supergravity which have different numbers of unbroken supersymmetries. When embedded in $N=8$ supergravity, all different embeddings of the same solution have the same number of unbroken supersymmetries. Thus, there is a relation between the value of the parameter $a$ and the number of unbroken supersymmetries in $N=8$ supergravity, but not in $N=4$, and the different embeddings must be related by dualities of the $N=8$ theory which are not dualities of the $N=4$ theory. The only exception in this scheme is a {\it dyonic} embedding of the $a=0$ black-hole solution which seems to break all supersymmetries both in the $N=4$ and in the $N=8$ theories. 
  We present a dyonic embedding of the extreme Reissner-Nordstr\"om black hole in $N=4$ and $N=8$ supergravity that breaks all supersymmetries. 
  We discuss the vacuum structure of $\phi^4$-theory in 1+1 dimensions quantised on the light-front $x^+ =0$. To this end, one has to solve a non-linear, operator-valued constraint equation. It expresses that mode of the field operator having longitudinal light-front momentum equal to zero, as a function of all the other modes in the theory. We analyse whether this zero mode can lead to a non-vanishing vacuum expectation value of the field $\phi$ and thus to spontaneous symmetry breaking. In perturbation theory, we get no symmetry breaking. If we solve the constraint, however, non-perturbatively, within a mean-field type Fock ansatz, the situation changes: while the vacuum state itself remains trivial, we find a non-vanishing vacuum expectation value above a critical coupling. Exactly the same result is obtained within a light-front Tamm-Dancoff approximation, if the renormalisation is done in the correct way. 
  We analyze the relation between rigid and local supersymmetric N=2 field theories, when half of the supersymmetries are spontaneously broken. In particular, we show that the recently found partial supersymmety breaking induced by electric and magnetic Fayet-Iliopoulos terms in rigid theories can be obtained by a suitable flat limit of previously constructed N=2 supergravity models with partial super-Higgs in the observable sector. 
  The dynamics of superstring, supergravity and M theories and their compactifications are probed by studying the various perturbation theories that emerge in the strong and weak coupling limits for various directions in coupling constant space. The results support the picture of an underlying non-perturbative theory that, when expanded perturbatively in different coupling constants, gives different perturbation theories, which can be perturbative superstring theories or superparticle theories. The $p$-brane spectrum is considered in detail and a criterion found to establish which $p$-branes govern the strong coupling dynamics. In many cases there are competing conjectures in the literature, and this analysis decides between them. In other cases, new results are found. The chiral six-dimensional theory resulting from compactifying the type IIB string on $K_3$ is studied in detail and it is found that certain strong coupling limits appear to give new theories, some of which hint at the possibility of a twelve-dimensional origin. 
  We demonstrate that the Bailey pair formulation of Rogers-Ramanujan identities unifies the calculations of the characters of $N=1$ and $N=2$ supersymmetric conformal field theories with the counterpart theory with no supersymmetry. We illustrate this construction for the $M(3,4)$ (Ising) model where the Bailey pairs have been given by Slater. We then present the general unitary case. We demonstrate that the model $M(p,p+1)$ is derived from $M(p-1,p)$ by a Bailey renormalization flow and conclude by obtaining the $N=1$ model $SM(p,p+2)$ and the unitary $N=2$ model with central charge $c=3(1-2/p).$ 
  Quantum integrable models that possess $N=2$ supersymmetry are investigated on the half-space. Conformal perturbation theory is used to identify some $N=2$ supersymmetric boundary integrable models, and the effective boundary Landau-Ginzburg actions are constructed. It is found that $N=2$ supersymmetry largely determines the boundary action in terms of the bulk, and in particular, the boundary bosonic potential is $|W|^2$, where $W$ is the bulk superpotential. Supersymmetry is also discussed from the perspective of the affine quantum group symmetry of exact scattering matrices, and exact $N=2$ supersymmetry preserving boundary reflection matrices are described. 
  Different methods of calculation of quantum corrections to the thermodynamical characteristics of a black hole are discussed and compared. The relation between on-shell and off-shell approaches is established. The off-shell methods are used to explicitly demonstrate that the thermodynamical entropy $S^{TD}$ of a black hole, defined by the first thermodynamical law, differs from the statistical-mechanical entropy $S^{SM}$, determined as $S^{SM}=-\mbox{Tr}(\hat{\rho}^H\ln\hat{\rho}^H)$ for the density matrix $\hat{\rho}^H$ of a black hole. It is shown that the observable thermodynamical black hole entropy can be presented in the form $S^{TD}=\pi {\bar r}_+^2+S^{SM}-S^{SM}_{Rindler}$. Here ${\bar r}_+$ is the radius of the horizon shifted because of the quantum backreaction effect, and $S^{SM}_{Rindler}$ is the statistical-mechanical entropy calculated in the Rindler space. 
  The equation for the gluon propagator in the approach of Baker-Ball-Zachariasen is considered. The possibility of non-integer power infrared behaviour is studied, $D(q) \sim (q^2)^{-c}$, $q^2 \rightarrow 0$. It is shown that the characteristic equation for the exponent has no solutions at $-1\leq c\leq 3$. The approximations made to obtain the closed integral equation are analysed and the conclusion on the infrared behaviour of the gluon propagator $D(q) \sim 1/(q^2)^2$, $q^2 \rightarrow 0$ is made when the transverse part of the triple gluon vertex is taken into account. 
  A class of exact solutions are obtained for the problem of N-anyons interacting via the N-body potential $V (\vec x_1,\vec x_2,...,\vec x_N)$ = $-{e^2\over\sqrt{{1\over N}\sum_{i<j} (\vec x_i-\vec x_j)^2}}$ Unlike the oscillator case the resulting spectrum is not linear in the anyon parameter $\alpha (0\leq \alpha\leq 1)$. However, a la oscillator case, cross-over between the ground states is shown to occur for N-anyons $(N\geq 3)$ experiencing the above potential. 
  Solvable theories of 2D dilaton gravity can be obtained from a Liouville theory by suitable field redefinitions. In this paper we propose a new framework to generate 2D dilaton gravity models which can also be exactly solved in the semiclassical approximation. Our approach is based on the recently introduced scheme to quantize massless scalar fields coupled to 2D gravity maintaining invariance under area-preserving diffeomorphisms and Weyl transformations. Starting from the CGHS model with the new effective action we reestablish the full diffeomorphism invariance by means of an adequate family of field redefinitions. The original theory is therefore mapped into a large family of solvable models. We focus our analysis on the one-parameter class of models interpolating between the Russo-Susskind-Thorlacius model and the Bose-Parker-Peleg model. Finally we shall briefly indicate how can we extend our approach to spherically symmetric Einstein gravity coupled to 2D conformal matter. 
  Within a four-dimensional, toroidally compactified heterotic string, we identify (quantized) charge vectors of electrically charged BPS-saturated states (along with the tower of SL(2,Z) related dyonic states), which preserve 1/2 of N=4 supersymmetry and become massless along the hyper-surfaces of enhanced gauge symmetry of the two-torus moduli sub-space. In addition, we identify charge vectors of the dyonic BPS-saturated states (along with the tower of SL(2,Z) related states), which preserve 1/4 of N=4 supersymmetry, and become massless at two points with the maximal gauge symmetry enhancement. 
  I present the moduli space of the (2+2)-dimensional critical closed fermionic string with two world-sheet supersymmetries. The integration of fermionic and Maxwell moduli in the presence of punctures yields the string measure for n-point amplitudes at arbitrary genus and instanton number. Generalized picture-changing and spectral-flow operators emerge, connecting different instanton sectors. Tree and loop amplitudes are computed. 
  We present a novel way to compute the one-loop ring-improved effective potential numerically, which avoids the spurious appearence of complex expressions and at the same time is free from the renormalization ambiguities of the self-consistent approaches, based on the direct application of Schwinger-Dyson type equations to the masses. 
  Recently, Witten has proposed a mechanism for symmetry enhancement in $SO(32)$ heterotic string theory, where the singularity obtained by shrinking an instanton to zero size is resolved by the appearance of an $Sp(1)$ gauge symmetry. In this short letter, we consider spacetime constraints from anomaly cancellation in six dimensions and D-flatness and demonstrate a subtlety which arises in the moduli space when many instantons are shrunk to zero size. 
  We discuss R-symmetry in locally supersymmetric $N=2$ gauge theories coupled to hypermultiplets, which can be viewed as effective theories of heterotic string models. In this type of supergravities a suitable R-symmetry exists and can be used to topologically twist the theory. The vector multiplet of the dilaton-axion field has a different R-charge assignment with respect to the other vector multiplets. 
  Generalizing the Bogoliubov model of a weakly non-ideal Bose gas to massless $\lambda\varphi^4$ theory we show that spontaneous breaking of symmetry occurs due to condensation in a coherent vacuum % which is energetically favoured compared to the perturbative one. and leads to a vacuum energy density which is lower than that obtained by Coleman and Weinberg using the one-loop effective potential method. We discuss the alternative of a squeezed condensate and find that for the massless $\lambda \varphi^4$ theory spontaneous symmetry breaking to a squeezed vacuum does not occur. 
  New types "extended" (super)conformal algebras $G^{(\frac n2)}$ are presented. (Su\-per)twistor spaces $T$ are subspaces in cosets $G^{(\frac n2)}/H$. The (super)twistor correspondence has a cleary defined geometrical meaning. 
  We prove the Mirror Conjecture for Calabi-Yau manifolds equipped with a holomorphic symplectic form. Such manifolda are also known as complex manifolds of hyperkaehler type. We obtain that a complex manifold of hyperkaehler type is Mirror dual to itself. The Mirror Conjecture is stated (following Kontsevich, ICM talk) as the equivalence of certain algebraic structures related to variations of Hodge structures. We compute the canonical flat coordinates on the moduli space of Calabi-Yau manifolds of hyperkaehler type, introduced to Mirror Symmetry by Bershadsky, Cecotti, Ooguri and Vafa. 
  We study Z2-orbifolds of 11-dimensional M-theory on tori of various dimensions. The most interesting model (besides the known S1/Z2 case) corresponds to T5/Z2, for which we argue that the resulting six-dimensional theory is equivalent to the type IIB string compactified on K3. Gravitational anomaly cancellation plays a crucial role in determining what states appear in the twisted sector. Most of the other models appear to break spacetime supersymmetry. We observe that M-theory tends to produce chiral compactifications on orbifolds, and that our results may provide an insight into the mechanism by which twisted-sector states arise in this hypothetical theory. 
  The order dependent mapping method, its convergence has recently been proven for the energy eigenvalue of the anharmonic oscillator, is applied to re-sum the standard perturbation series for Stark effect of the hydrogen atom. We perform a numerical experiment up to the fiftieth order of the perturbation expansion. A simple mapping suggested by the analytic structure and the strong field behavior gives an excellent agreement with the exact value for an intermediate range of the electric field, $0.03\leq E\leq0.25$. The imaginary part of the energy (the decay width) as well as the real part of the energy is reproduced from the standard perturbation series. 
  This report is consisted of six independent chapters, each chapter (except chapter 1) is a paper carried out in colabouration with others, who's names are indicated in chapter1. The topics included are (1)Overview of general properties of Toda-like systems (chapter1) (2)Free field representations of some particular example of entended Toda like system via Drinfeld-Sokolov construction (chapter2) (3)Heterotic conformal Toda system (chapter3) (4)Heterotic Liouville systems related to Bernoulli equation (chapter4) (5)Two-extended Toda field in three-dimensions--a representative example of extended Toda systems in higher dimensions (chapter5) and (6) $W_N^{(l)}$ algebras in the exchange algebra's framework (chapter6) 
  The fine-tuning principles are examined to predict the top-quark and Higgs-boson masses. The modification of the Veltman condition based on the compensation of vacuum energies is developed. It is implemented in the Standard Model and in its minimal extension with two Higgs doublets and Left-Right symmetric Model. The top-quark and Higgs-boson couplings are fitted in the SM for the lowest ultraviolet scale where the fine-tuning can be stable under rescaling. It yields the low-energy values $m_t \simeq 175 GeV;\quad m_H \simeq 210 GeV$. For the Two-Higgs and Left-Right Symmetric Models the fine-tuning principles yield the interval for top-quark mass, compatible with the modern experimental data. For the Left-Right Model the FT principles demand the existence of the right-handed Majorana neitrinos with masses of order of right-handed gauge bosons. 
  Curvature induced phase transition is thoroughly investigated in a four- fermion theory with $N$ components of fermions for arbitrary space-time dimensions $(2 \leq D < 4)$. We adopt the $1/N$ expansion method and calculate the effective potential for a composite operator $\bar{\psi}\psi$. The resulting effective potential is expanded asymptotically in terms of the space-time curvature $R$ by using the Riemann normal coordinate. We assume that the space-time curves slowly and keep only terms independent of $R$ and terms linear in $R$. Evaluating the effective potential it is found that the first-order phase transition is caused and the broken chiral symmetry is restored for a large positive curvature. In the space-time with a negative curvature the chiral symmetry is broken down even if the coupling constant of the four-fermion interaction is sufficiently small. We present the behavior of the dynamically generated fermion mass. The critical curvature, $R_{cr}$, which divides the symmetric and asymmetric phases is obtained analytically as a function of the space-time dimension $D$. At the four-dimensional limit our result $R_{cr}$ agrees with the exact results known in de Sitter space and Einstein universe. 
  Several examples of similarity transformations connecting two string theories with different backgrounds are reviewed. We also discuss general structure behind the similarity transformations from the point of view of the topological conformal algebra and of the non-linear realization of gauge symmetry. 
  We investigate four-fermion interactions with $N$-component fermion in Einstein universe for arbitrary space-time dimensions ($2 \leq D<4$). It is found that the effective potential for composite operator $\overline{\psi}\psi$ is calculable in the leading order of the $1/N$ expansion. The resulting effective potential is analyzed by varying the curvature of the space-time and is found to exhibit the symmetry restoration through the second-order phase transition. The critical curvature at which the dynamical fermion mass disappears is analytically calculated. 
  We investigate possible existence of duality symmetries which exchange the Kaluza-Klein modes with the wrapping modes of a BPS saturated $p$-brane on a torus. Assuming the validity of the conjectured $U$-duality symmetries of type II and heterotic string theories and $M$-theory, we show that for a BPS saturated $p$-brane there is an SL(2,Z) symmetry that mixes the Kaluza-Klein modes on a $(p+1)$ dimensional torus $T^{(p+1)}$ with the wrapping modes of the $p$-brane on $T^{(p+1)}$. The field that transforms as a modular parameter under this SL(2,Z) transformation has as its real part the component of the $(p+1)$-form gauge field on $T^{(p+1)}$, and as its imaginary part the volume of $T^{(p+1)}$, measured in the metric that couples naturally to the $p$-brane. 
  Reflexive polyhedra encode the combinatorial data for mirror pairs of Calabi-Yau hypersurfaces in toric varieties. We investigate the geometrical structures of circumscribed polytopes with a minimal number of facets and of inscribed polytopes with a minimal number of vertices. These objects, which constrain reflexive pairs of polyhedra from the interior and the exterior, can be described in terms of certain non-negative integral matrices. A major tool in the classification of these matrices is the existence of a pair of weight systems, indicating a relation to weighted projective spaces. This is the corner stone for an algorithm for the construction of all dual pairs of reflexive polyhedra that we expect to be efficient enough for an enumerative classification in up to 4 dimensions, which is the relevant case for Calabi-Yau compactifications in string theory. 
  Two-point functions for scalar and spinor fields are investigated in Einstein universe ($R \otimes S^{\sN-1}$). Equations for massive scalar and spinor two-point functions are solved and the explicit expressions for the two-point functions are given. The simpler expressions for massless cases are obtained both for the scalar and spinor cases. 
  Quantization of free string field theory in the Rindler space-time is studied by using the covariant formulation and taking the center-of-mass value of the Rindler string time-coordinate $\eta(\sigma)$ as the time variable for quantization. We construct the string Rindler modes which vanish in either of the Rindler wedges $\pm$ defined by the Minkowski center-of-mass coordinate of the string. We then evaluate the Bogoliubov coefficients between the Rindler string creation/annihilation operators and the Minkowski ones, and analyze the string thermalization. An approach to the construction of the string Rindler modes corresponding to different definitions of the wedges is also presented toward a thorough understanding of the structure of the Hilbert space of the string field theory on the Rindler space-time. 
  We study the post-Newtonian limit of a generalized dilaton gravity in which gravity is coupled to dilaton and eletromagnetic fields. The field equations are derived using the post-Newtonian scheme, and the approximate solution is presented for a point mass with electric and dilaton charges. The result indicates that the dilaton effect can be detected, in post-Newtonian level, using a charged test particle but not a neutral one. We have also checked that the approximate solution is indeed consistent with the weak field expansion of charged dilaton black hole solution in the harmonic coordinate. 
  A perturbative renormalization group (RG) scheme for light-front Hamiltonian is formulated on the basis of the Bloch-Horowitz effective Hamiltonian, and applied to the simplest $\phi^4$ model with spontaneous breaking of the $Z_2$ symmetry. RG equations are derived at one-loop order for both symmetric and broken phases. The equations are consistent with those calculated in the covariant perturbation theory. For the symmetric phase, an initial cutoff Hamiltonian in the RG procedure is made by excluding the zero mode from the canonical Hamiltonian with an appropriate regularization. An initial cutoff Hamiltonian for the broken phase is constructed by shifting $\phi$ as $\phi \rightarrow\phi-v$ in the initial Hamiltonian for the symmetric phase. The shifted value $v$ is determined on a renormalization trajectory. The minimum of the effective potential occurs on the trajectory. 
  Approximating light charged point-like particles in terms of (nonextremal) dilatonic black holes is shown to lead to certain pathologies in Planckian scattering in the eikonal approximation, which are traced to the presence of a (naked) curvature singularity in the metric of these black holes. The existence of such pathologies is confirmed by analyzing the problem in an `external metric' formulation where an ultrarelativistic point particle scatters off a dilatonic black hole geometry at large impact parameters. The maladies disappear almost trivially upon imposing the extremal limit. Attempts to derive an effective three dimensional `boundary' field theory in the eikonal limit are stymied by four dimensional (bulk) terms proportional to the light-cone derivatives of the dilaton field, leading to nontrivial mixing of electromagnetic and gravitational effects, in contrast to the case of general relativity. An eikonal scattering amplitude, showing decoupling of these effects, is shown to be derivable by resummation of graviton, dilaton and photon exchange ladder diagrams in a linearized version of the theory, for an asymptotic value of the dilaton field which makes the string coupling constant non-perturbative. 
  We discuss the target-space interpretation of the world-sheet logarithmic operators in string theory. These operators generate the normalizable zero modes (discrete states) in target space, which restore the symmetries of the theory broken by the background. The problem of the recoil in string theory is considered, as well as some general properties of string amplitudes containing logarithmic operators. 
  I briefly review the present status of bosonic strings and discretized random surfaces in D>1 which seem to be in a polymer rather than stringy phase. As an explicit example of what happens, I consider the Kazakov-Migdal model with a logarithmic potential which is exactly solvable for any D (at large D for an arbitrary potential). I discuss also the meander problem and report some new results on its representation via matrix models and the relation to the Kazakov-Migdal model. A supersymmetric matrix model is especially useful for describing the principal meanders. 
  We study abelian lattice gauge theory defined on a simplicial complex with arbitrary topology. The use of dual objects allows one to reformulate the theory in terms of new dynamical variables; however, we avoid the use of the dual lattice entirely. Topological modes which are present in the transformation now appear as homology classes, in contrast to the cohomology modes found in the dual cell picture. Irregularities of dual cell complexes do not arise in this approach. We treat the two and three dimensional cases in detail. 
  We study the first-order phase transition in a model of scalar field with $O(3)$ symmetry coupled to gravity, and, in high temperature limit, discuss the existence of new bubble solution with a global monopole at the center of the bubble. 
  We discuss the N=2 extension of Polyakov-Bershadsky $W_3^{(2)}$ algebra with the generic central charge, $c$, at the quantum level in superspace. It contains, in addition to the spin 1 N=2 stress tensor, the spins $1/2, 2$ bosonic and spins $1/2, 2$ fermionic supercurrents satisfying the first class nonlinear chiral constraints. In the $c \to \infty $ limit, the ``classical'' N=2 $W_3^{(2)}$ algebra is recovered. 
  The BRST quantization of a gauge theory in noncommutative geometry is carried out in the ``matrix derivative" approach. BRST/anti-BRST transformation rules are obtained by applying the horizontality condition, in the superconnection formalism. A BRST/anti-BRST invariant quantum action is then constructed, using an adaptation of the method devised by Baulieu and Thierry-Mieg for the Yang-Mills case. The resulting quantum action turns out to be the same as that of a gauge theory in the 't Hooft gauge with spontaneously broken symmetry. Our result shows that only the even part of the supergroup acts as a gauge symmetry, while the odd part effectively provides a global symmetry. We treat the general formalism first, then work out the $SU(2/1)$ and $SU(2/2)$ cases explicitly. 
  We study the topological mass generation in the 4 dimensional nonabelian gauge theory, which is the extension of the Allen $et$ $al.$'s work in the abelian theory. It is crucial to introduce a one form auxiliary field in constructing the gauge invariant nonabelian action which contains both the one form vector gauge field $A$ and the two form antisymmetric tensor field $B$. As in the abelian case, the topological coupling $m B\wedge F$, where $F$ is the field strength of $A$, makes the transmutation among $A$ and $B$ possible, and consequently we see that the gauge field becomes massive. We find the BRST/anti-BRST transformation rule using the horizontality condition, and construct a BRST/anti-BRST invariant quantum action. 
  A sequence of solutions to the equation of symmetry for the continuous Toda chain in $1+2$ dimensions is represented in explicit form. This fact leads to the supposition that this equation is completely integrable. 
  We present a procedure in which known solutions to reflection equations for interaction-round-a-face lattice models are used to construct new solutions. The procedure is particularly well-suited to models which have a known fusion hierarchy and which are based on graphs containing a node of valency $1$. Among such models are the Andrews-Baxter-Forrester models, for which we construct reflection equation solutions for fixed and free boundary conditions. 
  We relate Type IIB superstrings compactified to six dimensions on K3 to an eleven-dimensional theory compactified on $({\bf S}^1)^5/{\bf Z}_2$. Eleven-dimensional five-branes enter the story in an interesting way. 
  We give a gauge invariant formulation of $N=2$ supersymmetric abelian Toda field equations in \n2 superspace. Superconformal invariance is studied. The conserved currents are shown to be associated with Drinfeld-Sokolov type gauges. The extension to non-abelian \n2 Toda equations is discussed. Very similar methods are then applied to a matrix formulation in \n2 superspace of one of the \n2 KdV hierarchies. 
  We study the Poisson bracket algebra of the $N=2$ supersymmetric chiral WZNW model in superspace. It involves two classical r-matrices, one of which comes from the geometrical constraints implied by $N=2$ supersymmetry. The phase space itself consists of superfields satisfying constraints involving this r-matrix. An attempt is made to relax these constraints. The symmetries of the model are investigated. 
  A sequence of zero-temperature black-hole spacetimes with angular momentum and electric and magnetic charges is shown to exist in gauged $N=2$ supergravity. Stability of a subset of these spacetimes is demonstrated by saturation of the Bogomol'nyi bound arising from the supersymmetry algebra. The mass of the resulting solitonic black holes is given in terms of the cosmological constant and the angular momentum. We conjecture that at the quantum level these solitons are dyons with angular momentum determined by the electric and magnetic charges. 
  We review the specific problems that arise when studying instantons on a torus. We discuss how the Nahm transformation shows that no exact charge one instanton on T**4 can exist. However, taking one of the directions (the time) to infinity, it can be shown that vacuum to vacuum tunnelling solutions exist. A precise description of the moduli space for T**3xR, studied numerically using lattice techniques, remains an interesting open problem. New is an explicit application of the Nahm transformation to (anti-)selfdual constant curvature solutions on T**4 and a discussion of its properties relevant to instantons on T**3xR. 
  The loop variable approach used earlier to obtain free equations of motion for the massive modes of the open string, is generalized to include interaction terms. These terms, which are polynomial, involve only modes of strictly lower mass. Considerations based on operator product expansions suggest that these equations are particular truncations of the full string equations. The method involves broadening the loop to a band of finite thickness that describes all the different interacting strings. Interestingly, in terms of these variables, the theory appears non-interacting. 
  As an illustration of the formalism of the master field we consider generalised $QCD_2$. We show how Wilson Loop averages for an arbitrary contour can be computed explicitly and with some ease. A generalised Hopf equation is shown to govern the behaviour of the eigenvalue density of Wilson loops. The collective field description of the theory is therefore deduced. Finally, the non-trivial master gauge field and field strengths are obtained. These results do not seem easily accessible with conventional means. 
  We analyze the general class of supersymmetry preserving orbifolds of strong/weak Type IIA/heterotic dual pairs in six dimensions and below. A unified treatment is given by considering compactification to two spacetime dimensions and constructing orbifolds by subgroups of the Fischer-Greiss monster, utilizing the moonshine results of Conway and Norton. Duality requires nontrivial Ramond-Ramond fluxes on the Type IIA side which are localized at the fixed points. Further orbifolding by (-1)^{F_L} gives examples of new four dimensional N=2 Type IIA vacua which are not conformal field theory backgrounds. 
  The McCoy-Roan integral representation for gaps of the integrable Z_n- symmetric Chiral Potts quantum chain is used to calculate the boundary of the incommensurable phase for various n. In the limit n -> \infty an analytic formula for this phase boundary is obtained. The McCoy-Roan formula gives the gaps in terms of a rapidity. For the lowest gap we conjecture the relation of this rapidity to the physical momentum in the high-temperature limit using symmetry properties and comparing the McCoy-Roan formula to high-temperature expansions and finite-size data. 
  We introduce a random lattice corresponding to ordinary Feynman diagrams, with 1/p-squared propagators instead of the Gaussians used in the usual strings. The continuum limit defines a new type of string action with two worldsheet metrics, one Minkowskian and one Euclidean. The propagators correspond to curved lightlike paths with respect to the Minkowskian worldsheet metric. Spacetime dimensionality of four is implied not only as the usual critical dimension of renormalizable quantum field theory, but also from T-duality. 
  Effective world-brane actions for solitons of ten-dimensional type IIA and IIB superstring theory are derived using the formulation of solitons as Dirichlet branes. The one-brane actions are used to recover predictions of SL(2,Z) strong-weak coupling duality. The two-brane action, which contains a hidden eleventh target space coordinate, is shown to be the eleven-dimensional supermembrane action. It can be thought of as the membrane action of `M-theory'. 
  We construct a new extreme black hole solution in toroidally compactified heterotic string theory. The black hole saturates the Bogomol'nyi bound, has zero angular momentum, but nonzero electric dipole moment. It is obtained by starting with a higher dimensional rotating charged black hole, and compactifying one direction in the plane of rotation. 
  We show that a considerable part of the theory of (ultra)distributions and hyperfunctions can be extended to more singular generalized functions, starting from an angular localizability notion introduced previously. Such an extension is needed to treat gauge quantum field theories with indefinite metric in a generic covariant gauge. Prime attention is paid to the generalized functions defined on the Gelfand-Shilov spaces $S_\alpha^0$ which gives the widest framework for construction of gauge-like models. We associate a similar test function space with every open and every closed cone, show that these spaces are nuclear and obtain the required formulas for their tensor products. The main results include the generalization of the Paley--Wiener--Schwartz theorem to the case of arbitrary singularity and the derivation of the relevant theorem on holomorphic approximation. 
  This paper applies $\zeta$-function regularization to evaluate the 1-loop effective action for scalar field theories and Euclidean Maxwell theory in the presence of boundaries. After a comparison of two techniques developed in the recent literature, vacuum Maxwell theory is studied and the contribution of all perturbative modes to $\zeta'(0)$ is derived: transverse, longitudinal and normal modes of the electromagnetic potential, jointly with ghost modes. The analysis is performed on imposing magnetic boundary conditions, when the Faddeev-Popov Euclidean action contains the particular gauge-averaging term which leads to a complete decoupling of all perturbative modes. It is shown that there is no cancellation of the contributions to $\zeta'(0)$ resulting from longitudinal, normal and ghost modes. 
  These are notes from introductory lectures given at the Ecole Normale in Paris and at the Strasbourg meeting dedicated to the memory of Claude Itzykson. I review in considerable detail and in a hopefully pedagogical way the work of Seiberg and Witten on $N=2$ supersymmetric $SU(2)$ gauge theory without extra matter. This presentation basically follows their original work, except in the last section where the low-energy effective action is obtained emphasizing more the relation between monodromies and differential equations rather than using elliptic curves. 
  We consider a critical composite superconformal string model to desribe hadronic interactions. We present a new approach of introducing hadronic quantum numbers in the scattering amplitudes. The physical states carry the quantum numbers and form a common system of eigenfunctions of the operators in this string model. We give explicit constructions of the quantum number operators. 
  It is shown that a gauged nonlinear $O(3)$ sigma model with anomalous magnetic moment interaction in $2+1$ dimensions is exactly integrable for static, self-dual field configurations. The matter fields are exactly equivalent to those of the usual ungauged nonlinear $O(3)$ sigma model. These static soliton solutions can be mapped into an Abelian purely magnetic vortex solutions through a suitable reduction of the non-Abelian gauge group. A relativistic Abelian model in $2+1$ dimensions is also presented where these purely magnetic vortices can be realized. 
  Supersymmetric monopoles of the heterotic string theory associated with arbitrary non-negative number of the left moving modes of the string states are presented. They include H monopoles and their T dual partners F monopoles (ALE instantons). Massive F = H monopoles are T self-dual. Solutions include also an infinite tower of generic T duality covariant non-singular in stringy frame F\&H monopoles with the bottomless throat geometry. The massless F = -H monopoles are invariant under combined T duality and charge conjugation converting a monopole into anti monopole. All F\&H monopoles can be promoted to the exact supersymmetric solutions of the heterotic string theory since the holonomy group is the compact $SO(9)$. The sigma models for ${\bf M}^8$ monopoles, which admit constant complex structures, have enhanced world-sheet supersymmetry: (4,1) in general and (4,4) for the left-right symmetric monopoles. The space-time supersymmetric GS light-cone action in monopole background is directly convertible into the world-sheet supersymmetric NSR action. 
  We analyze the Coulomb phase of theories of $N=2$ SQCD with $SU(N_c)$ gauge groups which are conjectured to have exact electric-magnetic duality. We discuss the duality transformation of the particle spectrum, emphasizing the differences between the general case and the $SU(2)$ case. Some difficulties associated with the definition of the duality transformation for a general gauge group are discussed. We compute the classical monopole spectrum of these theories, and when it is possible we use it to check the consistency of the duality. Generally these theories may have phase transitions between strong and weak coupling, which prevent the semi-classical computation from being useful for checking the duality. 
  The quantum energy-momentum tensor ${\hat T}^{\mu\nu}(x)$ is computed for strings in Minkowski space-time. We compute its expectation value for different physical string states both for open and closed bosonic strings. The states considered are described by normalizable wave-packets in the center of mass coordinates. We find in particular that ${\hat T}^{\mu\nu}(x)$ is {\bf finite} which could imply that the classical divergence that occurs in string theory as we approach the string position is removed at the quantum level as the string position is smeared out by quantum fluctuations. For massive string states the expectation value of ${\hat T}^{\mu\nu}(x)$ vanishes at leading order (genus zero). For massless string states it has a non-vanishing value which we explicitly compute and analyze for both spherically and cylindrically symmetric wave packets. The energy-momentum tensor components propagate outwards as a massless lump peaked at $ r = t $. 
  In this thesis we give an overview of the antifield formalism and show how it must be used to quantise arbitrary gauge theories. The formalism is further developed and illustrated in several examples, including Yang-Mills theory, chiral $W_3$ and $W_{2,5/2}$ gravity, strings in curved backgrounds and topological field theories. All these models are characterised by their gauge algebra, which can be open, reducible, or even infinitly reducible. We show in detail how to perform the gauge fixing and how to compute the anomalies using Pauli-Villars regularisation and the heat kernel method. Finally, we discuss the geometrical structure of the antifield formalism. 
  We study a class of extremal transitions between topological distinct Calabi--Yau manifolds which have an interpretation in terms of the special massless states of a type II string compactification. In those cases where a dual heterotic description exists the exceptional massless states are due to genuine strong (string-) coupling effects. A new feature is the appearance of enhanced non-abelian gauge symmetries in the exact nonperturbative theory. 
  We consider the properties of massive one particle states on a translation covariant Haag-Kastler net in Minkowski space. In two dimensional theories, these states can be interpreted as soliton states and we are interested in the existence of antisolitons. It is shown that for each soliton state there are three different possibilities for the construction of an antisoliton sector which are equivalent if the (statistical) dimension of the corresponding soliton sector is finite. 
  We analyse abelian T-duality for WZW models of simply-connected groups. We demonstrate that the dual theory is a certain orbifold of the original theory, and check that it is conformally invariant. We determine the spectrum of the dual theory, and show that it agrees with the spectrum of the original theory. 
  Twistor phase spaces are used to provide a general description of the dynamics of a finite number of directly interacting massless spinning particles forming a closed relativistic massive and spinning system with an internal structure. A Poincare invariant canonical quantization of the so obtained twistor phase space dynamics is performed. 
  Lee replies to the comment on "Statistical Mechanics of Non-Abelian Chern-Simons Particles" by C. R. Hagen 
  We point out that in infinite spacetime dimensions, the singularity in the interquark potential at small distances disappears if the string is anchored at one end to a heavy quark, at the other end to a light quark. This suggests that if quarks are placed at the end of strings some unphysical features such as tachyon states may be absent also in in finite dimensions. 
  We briefly review models of relativistic particles with spin. Departing from the oldest attempts to describe the spin within the lagrangian framework we pass through various non supersymmetric models. Then the component and superfield formulations of the spinning particle and superparticle models are reviewed. Our focus is mainly on the classical side of the problem, but some quantization questions are mentioned as well. 
  More than twenty years have passed since the threads of the `proper time formalism' in covariant classical and quantum mechanics were brought together to construct a canonical formalism for the relativistic mechanics of many particles. Drawing on the work of Fock, Stueckelberg, Nambu, Schwinger, and Feynman, the formalism was raised from the status of a purely formal mathematical technique to a covariant evolution theory for interacting particles. In the context of this theory, solutions have been found for the relativistic bound state problem, classical and quantum scattering in relativistic potentials, as well as applications in statistical mechanics.  It has been shown that a generalization of the Maxwell theory is required in order that the electromagnetic interaction be well-posed in the theory. The resulting theory of electromagnetism involves a fifth gauge field introduced to compensate for the dependence of the gauge transformation on the invariant time parameter; permitting such dependence relaxes the requirement that individual particles be on fixed mass shells and allows exchange of mass during scattering. In this paper, we develop the quantum field theory of off-shell electromagnetism, and use it to calculate certain elementary processes, including Compton scattering and M{\o}ller scattering. These calculations lead to {\em qualitative} deviations from the usual scattering cross-sections, which are, however, small effects, but may be visible at small angles near the forward direction. The familiar IR divergence of the M{\o}ller scattering is, moreover, completely regularized. 
  Recently it was shown that an asymptotic behaviour of $SU(N)$ gauge theory for large $N$ is described by q-deformed quantum field. The master fields for large N theories satisfy to standard equations of relativistic field theory but fields satisfy $q$-deformed commutation relations with $q=0$. These commutation relations are realized in the Boltzmannian Fock space. The master field for gauge theory does not take values in a finite-dimensional Lie algebra however there is a non-Abelian gauge symmetry. The gauge master field for a subclass of planar diagrams, so called half-planar diagrams, is also considered. A recursive set of master fields summing up a matreoshka of 2-particles reducible planar diagrams is briefly described. 
  We discuss several aspects of superconformal field theories in four dimensions (CFT_4), in the context of electric-magnetic duality. We analyse the behaviour of anomalous currents under RG flow to a conformal fixed point in N=1, D=4 supersymmetric gauge theories. We prove that the anomalous dimension of the Konishi current is related to the slope of the beta function at the critical point. We extend the duality map to the (nonchiral) Konishi current. As a byproduct we compute the slope of the beta function in the strong coupling regime. We note that the OPE of $T_{\mu\nu}$ with itself does not close, but mixes with a special additional operator $\Sigma$ which in general is the Konishi current. We discuss the implications of this fact in generic interacting conformal theories. In particular, a SCFT_4 seems to be naturally equipped with a privileged off-critical deformation $\Sigma$ and this allows us to argue that electric-magnetic duality can be extended to a neighborhood of the critical point. We also stress that in SCFT_4 there are two central charges, c and c', associated with the stress tensor and $\Sigma$, respectively; c and c' allow us to count both the vector multiplet and the matter multiplet effective degrees of freedom of the theory. 
  A construction of conservation laws for $\sigma$-models in two dimensions is generalized in the framework of noncommutative geometry of commutative algebras. This is done by replacing the ordinary calculus of differential forms with other differential calculi and introducing an analogue of the Hodge operator on the latter. The general method is illustrated with several examples. 
  It is demonstrated that quasi-exactly solvable models of quantum mechanics admit an interesting duality transformation which changes the form of their potentials and inverts the sign of all the exactly calculable energy levels. This transformation helps one to reveal some new features of quasi-exactly solvable models and associated orthogonal polynomials. 
  A new formulation of $c\leq 1$ matter coupled to 2D gravity is proposed. This model, being closely analogous to one in the Polyakov light-cone gauge, possesses well defined global properties which allow to calculate correlation functions. As an example, the three point correlation functions of discrete states are found. 
  A supersymmetric gauge invariant action is constructed over any 4-dimensional Riemannian manifold describing Witten's theory of 4-monopoles. The topological supersymmetric algebra closes off-shell. The multiplets include the auxiliary fields and the Wess-Zumino fields in an unusual way, arising naturally from BRST gauge fixing. A new canonical approach over Riemann manifolds is followed, using a Morse function as an euclidean time and taking into account the BRST boundary conditions that come from the BFV formulation. This allows a construction of the effective action starting from gauge principles. 
  We present a new type of spherically symmetric monopole and dyon solutions with the magnetic charge $ 4\pi/e$ in the standard Weinberg-Salam model. The monopole (and dyon) could be interpreted as a non-trivial hybrid between the abelian Dirac monopole and non-abelian 't Hooft-Polyakov monopole (with an electric charge). We discuss the possible physical implications of the electroweak dyon. 
  The Bekenstein-Hawking area-entropy relation $S_{BH}=A/4$ is derived for a class of five-dimensional extremal black holes in string theory by counting the degeneracy of BPS soliton bound states. 
  QCD strings are color-electric flux tubes between quarks with a finite thickness and thus a finite curvature stiffness. Contrary to an earlier rigid-string by Polyakov and Kleinert, and motivated by the properties of a magnetic flux tubes in type-II superconductors we put forward the hypothesis that QCD strings have a {\em negative\/ stiffness. We set up a new string model with this property and show that it is free of the three principal problems of rigid-strings --- particle states with negative norm, nonexistence of a lowest-energy state, and wrong high-temperature behavior of string tension --- thus making it a better candidate for a string description of quark forces than previous models. 
  A formulation for a non-trivial composition of two classical gauge structures is given: Two parent gauge structures of a common base space are synthesized so as to obtain a daughter structure which is fundamental by itself. The model is based on a pair of related connections that take their values in the product space of the corresponding Lie algebras. The curvature, the covariant exterior derivatives and the associated structural identities, all get contributions from both gauge groups. The various induced structures are classified into those whose composition is given just by trivial means, and those which possess an irreducible nature. The pure irreducible piece, in particular, generates a complete super-space of ghosts with an attendant set of super-BRST variation laws, both of which are purely of a geometrical origin. 
  The general solution of the graded contraction equations for a $\zz_2^{\otimes N}$ grading of the real compact simple Lie algebra $so(N+1)$ is presented in an explicit way. It turns out to depend on $2^N-1$ independent real parameters. The structure of the general graded contractions is displayed for the low dimensional cases, and kinematical algebras are shown to appear straightforwardly. The geometrical (or physical) meaning of the contraction parameters as curvatures is also analysed; in particular, for kinematical algebras these curvatures are directly linked to geometrical properties of possible homogeneous space-times. 
  A field theory with local transformations belonging to the quantum group SU_q(n) is defined on a classical spacetime, with gauge potentials belonging to a quantum Lie algebra. Gauge transformations are defined for the potentials which lead to the appropriate quantum-group transformations for field strengths and covariant derivatives, defined for all elements of SU_q(n) by means of the adjoint action. This guarantees a non-trivial deformation. Gauge-invariant commutation relations are identified. 
  For the general operator product algebra coefficients derived by Cremmer Roussel Schnittger and the present author with (positive integer) screening numbers, the coupling constants determine the factors additional to the quantum group 6j symbols.   They are given by path independent products over a two dimensional lattice in the zero mode space. It is shown that the ansatz for the three point function of Dorn-Otto and Zamolodchikov-Zamolodchikov precisely defines the corresponding flat lattice connection, so that it does give a natural generalization of these coupling constants to continuous screening numbers. The consistency of the restriction to integer screening charges is reviewed, and shown to be linked with the orthogonality of the (generalized) 6j symbols. Thus extending this last relation is the key to general screening numbers. 
  We show how to obtain all covariant field equations for massless particles of arbitrary integer, or half-integer, helicity in four dimensions from the quantization of the rigid particle, whose action is given by the integrated extrinsic curvature of its worldline, {\ie} $S=\alpha\int ds \kappa$. This geometrical particle system possesses one extra gauge invariance besides reparametrizations, and the full gauge algebra has been previously identified as classical $\W_3$. The key observation is that the covariantly reduced phase space of this model can be naturally identified with the spinor and twistor descriptions of the covariant phase spaces associated with massless particles of helicity $s=\alpha$. Then, standard quantization techniques require $\alpha$ to be quantized and show how the associated Hilbert spaces are solution spaces of the standard relativistic massless wave equations with $s=\alpha$. Therefore, providing us with a simple particle model for Weyl fermions ($\alpha=1/2$), Maxwell fields ($\alpha=1$), and higher spin fields. Moreover, one can go a little further and in the Maxwell case show that, after a suitable redefinition of constraints, the standard Dirac quantization procedure for first-class constraints leads to a wave-function which can be identified with the gauge potential $A_\mu$. Gauge symmetry appears in the formalism as a consequence of the invariance under $\W_3$-morphisms, that is, exclusively in terms of the extrinsic geometry of paths in Minkowski space. When all gauge freedom is fixed one naturally obtains the standard Lorenz gauge condition on $A_{\mu}$, and Maxwell equations in that gauge. This construction has a direct generalization to arbitrary integer values of $\alpha$, and we comment on the physically interesting case of linearized Einstein gravity ($\alpha =2$). 
  We re-examine the question of heterotic - heterotic string duality in six dimensions and argue that the $E_8\times E_8$ heterotic string, compactified on $K3$ with equal instanton numbers in the two $E_8$'s, has a self-duality that inverts the coupling, dualizes the antisymmetric tensor, acts non-trivially on the hypermultiplets, and exchanges gauge fields that can be seen in perturbation theory with gauge fields of a non-perturbative origin. The special role of the symmetric embedding of the anomaly in the two $E_8$'s can be seen from field theory considerations or from an eleven-dimensional point of view. The duality can be deduced by looking in two different ways at eleven-dimensional $M$-theory compactified on $K3\times {\bf S}^1/\Z_2$. 
  In the context of a recently proposed method for computing exactly string loop corrections regularized in the infra-red, we determine and calculate the universal moduli-dependent part of the threshold corrections to the gauge couplings for the symmetric $Z_2\times Z_2$ orbifold model. We show that these corrections decrease the unification scale of the underlying effective field theory. We also comment on the relation between this infra-red regularization scheme and other proposed methods. 
  We study superstrings with orientifold projections and with generalized open string boundary conditions (D-branes). We find two types of consistency condition, one related to the algebra of Chan-Paton factors and the other to cancellation of divergences. One consequence is that the Dirichlet 5-branes of the Type I theory carry a symplectic gauge group, as required by string duality. As another application we study the Type I theory on a $K3$ $Z_2$ orbifold, finding a family of consistent theories with various unitary and symplectic subgroups of $U(16) \times U(16)$. We argue that the $K3$ orbifold with spin connection embedded in gauge connection corresponds to an interacting conformal field theory in the Type I theory. 
  We reinterpret the Faddeev-Popov gauge-fixing procedure of Yang-Mills theories as the definition of a topological quantum field theory for gauge group elements depending on a background connection. This has the advantage of relating topological gauge-fixing ambiguities to the global breaking of a supersymmetry. The global zero modes of the Faddeev-Popov ghosts are handled in the context of an equivariant cohomology without breaking translational invariance. The gauge-fixing involves constant fields which play the role of moduli and modify the behavior of Green functions at subasymptotic scales. At the one loop level physical implications from these power corrections are gauge invariant. 
  We continue the investigation of massive integrable models by means of the bootstrap fusion procedure, started in our previous work on O(3) nonlinear sigma model. Using the analogy with SU(2) Thirring model and the O(3) nonlinear sigma model we prove a similar relation between sine-Gordon theory and a one-parameter deformation of the O(3) sigma model, the sausage model. This allows us to write down a free field representation for the Zamolodchikov-Faddeev algebra of the sausage model and to construct an integral representation for the generating functions of form-factors in this theory. We also clear up the origin of the singularities in the bootstrap construction and the reason for the problem with the kinematical poles. 
  In this thesis generalizations of matrix and eigenvalue models involving supersymmetry are discussed. Following a brief review of the Hermitian one matrix model, the c=-2 matrix model is considered. Built from a matrix valued superfield this model displays supersymmetry on the matrix level. We stress the emergence of a Nicolai-map of this model to a free Hermitian matrix model and study its diagrammatic expansion in detail. Correlation functions for quartic potentials on arbitrary genus are computed, reproducing the string susceptibility of c=-2 Liouville theory in the scaling limit. The results may be used to perform a counting of supersymmetric graphs.  We then turn to the supereigenvalue model, today's only successful discrete approach to 2d quantum supergravity. The model is constructed in a superconformal field theory formulation by imposing the super-Virasoro constraints. The complete solution of the model is given in the moment description, allowing the calculation of the free energy and the multi-loop correlators on arbitrary genus and for general potentials. The solution is presented in the discrete case and in the double scaling limit. Explicit results up to genus two are stated.  Finally the supersymmetric generalization of the external field problem is addressed. We state the discrete super-Miwa transformations of the supereigenvalue model on the eigenvalue and matrix level. Properties of external supereigenvalue models are discussed, although the model corresponding to the ordinary supereigenvalue model could not be identified so far. 
  I try to argue that the only way out of the black hole information paradox is through a unified quantum field theory of gravity and other interactions. Superstring theory is especially interesting, since in a special limit, the classical picture of 't Hooft emerges. 
  Wilson's approximation scheme of RG recursion formula dropping momentum dependence of the propagators is applied to large-$N$ vector and matrix models in dimensions $2<d<4$ by making use of their exact solutions in zero dimension. In spite of apparent dependence of critical exponents upon the dilatational parameter $\rho$ involved by the approximation, the exact exponents are reproduced for vector models in the limit $\rho\rightarrow 0$. Application to matrix models is then reexamined after the same fashion. It predicts critical exponents $\nu=2/d$ and $\eta=2-d/2$ for the $\tr \Phi^4$ matrix model. 
  We review characteristic features of N=2 supersymmetric vector multiplets and discuss symplectic reparametrizations and their relevance for monopoles and dyons. We close with an analysis of perturbative corrections to the low-energy effective action of N=2 heterotic superstring vacua. 
  We explore quantum electrodynamics in (1+1) dimensions at finite temperature using the method of Discretized Light-Cone Quantisation. The partition function, energy and specific heat are computed in the canonical ensemble using the spectrum of invariant masses computed with a standard DLCQ numerical routine. In particular, the specific heat exhibits a peak which grows as the continuum limit is numerically approached. A critical exponent is tentatively extracted. The surprising result is that the density of states contains significant finite size artifacts even for a relatively high harmonic resolution. These and the other outstanding problems in the present calculation are discussed. 
  A new procedure for regularizing Feynman integrals in the noncovariant Coulomb gauge is proposed for Yang-Mills theory. The procedure is based on a variant of dimensional regularization, called split dimensional regularization, which leads to internally consistent, ambiguity-free integrals. It is demonstrated that split dimensional regularization yields a one-loop Yang-Mills self-energy that is nontransverse, but local. Despite the noncovariant nature of the Coulomb gauge, ghosts are necessary in order to satisfy the appropriate Ward/BRS identity. The computed Coulomb-gauge Feynman integrals are applicable to both Abelian and non-Abelian gauge models.   PACS: 11.15, 12.38.C 
  We extend the discussion of projective group representations in quaternionic Hilbert space which was given in our recent book. The associativity condition for quaternionic projective representations is formulated in terms of unitary operators and then analyzed in terms of their generator structure. The multi--centrality and centrality assumptions are also analyzed in generator terms, and implications of this analysis are discussed. 
  We derive finite temperature expansions for relativistic fermion systems in the presence of background magnetic fields, and with nonzero chemical potential. We use the imaginary-time formalism for the finite temperature effects, the proper-time method for the background field effects, and zeta function regularization for developing the expansions. We emphasize the essential difference between even and odd dimensions, focusing on $2+1$ and $3+1$ dimensions. We concentrate on the high temperature limit, but we also discuss the $T=0$ limit with nonzero chemical potential. 
  We derive and classify all solutions of the boundary Yang-Baxter equation (or the reflection equation) for the 19-vertex model associated with $U_q(\widehat{sl_2})$. Integrable $XXZ$ spin-1 chain hamiltonian with general boundary interactions is also obtained. 
  We investigate the subset of exactly solvable (0,4) world sheet supersymmetric string vacua contained in a recent class of Gepner-like (0,2) superconformal models. The identification of these models with certain points of enhanced gauge symmetry on K_3 x T_2 can be achieved completely. Furthermore, we extend the construction of in general (0,2) supersymmetric exactly solvable models to the case where also a nontrivial part of the vector bundle is embedded into the hidden E_8 gauge group. For some examples we explicitly calculate the enhanced gauge symmetries and show that they open up the way to interesting branches of the N=2 moduli space. For some of these models candidates of typeII dual descriptions exist. 
  We present new diagonal solutions of the reflection equation for elliptic solutions of the star-triangle relation. The models considered are related to the affine Lie algebras $A_n^{(1)},B_n^{(1)},C_n^{(1)},D_n^{(1)}$ and $A_n^{(2)}$. We recover all known diagonal solutions associated with these algebras and find how these solutions are related in the elliptic regime. Furthermore, new solutions of the reflection equation follow for the associated vertex models in the trigonometric limit. 
  The 3-anyon problem is studied using a set of variables recently proposed in an anyon gauge analysis by Mashkevich, Myrheim, Olaussen, and Rietman (MMOR). Boundary conditions to be satisfied by the wave functions in order to render the Hamiltonian self-adjoint are derived, and it is found that the boundary conditions adopted by MMOR are one of the ways to satisfy these general self-adjointness requirements. The possibility of scale-dependent boundary conditions is also investigated, in analogy with the corresponding analyses of the 2-anyon case. The structure of the known solutions of the 3-anyon in harmonic potential problem is discussed in terms of the MMOR variables. Within a series expansion in a boson gauge framework the problem of finding any anyon wavefunction is reduced to a (possibly infinite) set of algebraic equations, whose numerical analysis is proposed as an efficient way to study anyon physics. 
  Vector $SO(3)$ gauged $O(4)$ sigma models on $\R_3$ are presented. The topological charge supplying the lower bound on the energy and rendering the soliton stable coincides with the Baryon number of the Skyrmion. These solitons have vanishing magnetic monopole flux. To exhibit the existence of such solitons, the equations of motion of one of these models is integrated numerically. The structure of the conserved Baryon current is briefly discussed.  
  Over the last two years, the canonical approach to quantum gravity based on connections and triads has been put on a firm mathematical footing through the development and application of a new functional calculus on the space of gauge equivalent connections. This calculus does not use any background fields (such as a metric) and is thus well-suited to a fully non-perturbative treatment of quantum gravity. Using this framework, quantum geometry is examined. Fundamental excitations turn out to be one-dimensional, rather like polymers. Geometrical observables such as areas of surfaces and volumes of regions have purely discrete spectra. Continuum picture arises only upon coarse graining of suitable semi-classical states. Next, regulated quantum diffeomorphism constraints can be imposed in an anomaly-free fashion and the space of solutions can be given a natural Hilbert space structure. Progress has also been made on the quantum Hamiltonian constraint in a number of directions. In particular, there is a recent approach based on a generalized Wick transformation which maps solutions to the Euclidean quantum constraints to those of the Lorentzian theory. These developments are summarized. Emphasis is on conveying the underlying ideas and overall pictures rather than technical details. 
  The purpose of this talk is to address a couple of simple-sounding questions: what boundary conditions are compatible with  (a) Classical integrability?  (b) Quantum integrability? 
  Using explicit expressions for a class of singular vectors of the $N=2$ (untwisted) algebra and following the approach of Malikov-Feigin-Fuchs and Kent, we show that the analytically extended Verma modules contain two linearly independent neutral singular vectors at the same grade. We construct this two dimensional space and we identify the singular vectors of the original Verma modules. We show that in some Verma modules these expressions lead to two linearly independent singular vectors which are at the same grade and have the same charge. 
  The scattering of R-R gauge bosons off of Dirichlet p-branes is computed to leading order in the string coupling. The results are qualitatively similar to those found in the scattering of massless NS-NS bosons: all p-branes with p >= 0 exhibit stringy properties, in particular the Regge behavior. Both the R-R and NS-NS scattering amplitudes agree in the limit of small momentum transfer with scattering off the extremal R-R charged p-brane solutions found in the low-energy supergravities. We interpret this as evidence that Dirichlet-branes are an exact world-sheet description of the extremal p-branes. The -1-brane (D-instanton) is a special object which, unlike all other Dirichlet-branes, exhibits point-like behavior. We find the R-R charged instanton solution to type IIB supergravity and confirm that the field theoretic scattering off of this solution miraculously reproduces the full stringy calculation. As an aside, we include a discussion of the entropy of non-extremal black holes in ten dimensions, produced by exciting the 0-brane. We show that, for large black holes, the entropy grows linearly with the black hole mass. 
  We present a simple physical representation for states of the two-dimensional string theory. In order to incorporate a fundamental cutoff of the order 1/g we use a picture consisting of q-oscillators at the first-quantized level. In this framework we also find a representation for the (singular) negatively dressed states representing nontrivial string backgrounds. 
  We discuss $SL(2,Z)$ subgroups appropriate for the study of $N=2$ Super Yang-Mills with $N_f=2n$ flavors. Hyperelliptic curves describing such theories should have coefficients that are modular forms of these subgroups. In particular, uniqueness arguments are sufficient to construct the $SU(3)$ curve, up to two numerical constants, which can be fixed by making some assumptions about strong coupling behavior. We also discuss the situation for higher groups. We also include a derivation of the closed form $\beta$-function for the $SU(2)$ and $SU(3)$ theories without matter, and the massless theories with $N_f=n$. 
  A free lattice fermion field theory in 1+1 dimensions can be interpreted as SOS-type model, whose spins are integer-valued. We point out that the relation between these spins and the fermion field is similar to the abelian bosonization relation between bosons and fermions in the continuum. Though on the lattice the connected $2n$-point correlation functions of the integer-valued spins are not zero for any $n \ge 1$, the two-point correlation function of these spins is that of free bosons in the infrared. We also conjecture the form of the Wess-Zumino-Witten chiral field operator in a nonabelian lattice fermion model. These constructions are similar in spirit to the ``twistable string" idea of Krammer and Nielsen. 
  Using a simple solvable model, i.e., Higgs--Yukawa system with an infinite number of flavors, we explicitly demonstrate how a dimensional continuation of the $\beta$ function in two dimensional MS scheme {\it fails\/} to reproduce the correct behavior of the $\beta$ function in four dimensions. The mapping between coupling constants in two dimensional MS scheme and a conventional scheme in the cutoff regularization, in which the dimensional continuation of the $\beta$ function is smooth, becomes singular when the dimension of spacetime approaches to four. The existence of a non-trivial fixed point in $2+\epsilon$ dimensions continued to four dimensions $\epsilon\to2$ in the two dimensional MS scheme is spurious and the asymptotic safety cannot be imposed to this model in four dimensions. 
  After carefully regularizing the Wheeler -- De Witt operator, which is the Hamiltonian operator of canonical quantum gravity, we find a class of exact solutions of the Wheeler -- De Witt equation. 
  Quantum mechanics of Hamiltonian (non-dissipative) systems uses Lie algebra and analytic group (Lie group). In order to describe non-Hamiltonian (dissipative) systems in quantum theory we need to use non-Lie algebra and analytic quasigroup (loop).   The author derives that analog of Lie algebra for quantum non-Hamiltonian systems is commutant Lie algebra and analog of Lie group for these systems is analytic commutant associative loop (Valya loop). A commutant Lie algebra is an algebra such that commutant (a subspace which is generated by all commutators) is a Lie subalgebra. Valya loop is a non-associative loop such that the commutant of this loop is associative subloop (group). We prove that a tangent algebra of Valya loop is a commutant Lie algebra. It is shown that generalized Heisenberg-Weyl algebra, suggested by the author to describe quantum non-Hamiltonian (dissipative) systems, is a commutant Lie algebra. As the other example of commutant Lie algebra, it is considered a generalized Poisson algebra for differential 1-forms.   Note that non-Hamiltonian (dissipative) quantum theory has a broad range of application for non-critical strings in "coupling constant" phase space and bosonic string in non-Riemannian (for example, affine-metric) curved space which are non-Hamiltonian (dissipative) systems. 
  The question of how to account for the outgoing black hole modes without drawing upon a transplanckian reservoir at the horizon is addressed. It is argued that the outgoing modes must arise via conversion from ingoing modes. It is further argued that the back-reaction must be included to avoid the conclusion that particle creation cannot occur in a strictly stationary background. The process of ``mode conversion" is known in plasma physics by this name and in condensed matter physics as ``Andreev reflection" or ``branch conversion". It is illustrated here in a linear Lorentz non-invariant model introduced by Unruh. The role of interactions and a physical short distance cutoff is then examined in the sonic black hole formed with Helium-II. 
  A new pseudoclassical supersymmetrical model of a spinning particle in $2+1$ dimensions is proposed. Different ways of its quantization are discussed. They all reproduce the minimal quantum theory of the particle. 
  Possible geometric frameworks for a unified theory of gravity and electromagnetism are investigated: General relativity is enlarged by allowing for an arbitrary complex linear connection and by constructing an extended spinor derivative based on the complex connection. Thereby the spacetime torsion not only is coupled to the spin of fermions and causes a four-fermion contact interaction, but the non-metric vector-part of torsion is also related to the electromagnetic potential. However, this long-standing relation is shown to be valid only in a special U(1) gauge, and it is a formal consequence of the underlying extended geometry. 
  In this paper we present a symmetry of a toroidally compactified type II string theory. This symmetry has the interpretaion that it interchanges the left and the right-moving worldsheet coordinates and reverses the orientations of some of the spatial coordinates. We also identify another discrete symmetry of the type II theory which is related to the above one by a nontrivial U-duality element of string theory. This symmetry, however, has trivial action on the worldsheet coordinates and corresponds to an improper T-duality rotation. We then construct examples of type II dual pairs in four dimensions by modding out the known type II dual pairs by the above symmetries. We show the explicit matching of the spectrum and supersymmetries in these examples. 
  We study the WZNW models based on nonstandard bilinear forms. We approach the problem from algebraic, perturbative and functional exact methods. It is shown that even in the case of integer $k$ we can find irrational CFT's. We prove that when the base group is noncompact with nonabelian maximal compact subgroup, the Kac-Moody representations are nonunitary. 
  We exactly solve a special matrix model of dually weighted planar graphs describing pure two-dimensional quantum gravity with an R^2 interaction. It permits us to study the intermediate regimes between the gravitating and flat metric. Flat space is modeled by a regular square lattice, while localised curvature is introduced through lattice defects. No ``flattening'' phase transition is found with respect to the R^2 coupling: the infrared behaviour of the system is that of pure gravity for any finite R^2 coupling. In the limit of infinite coupling, we are able to extract a scaling function interpolating between pure gravity and a dilute gas of curvature defects on a flat background. We introduce and explain some novel techniques concerning our method of large N character expansions and the calculation of Schur characters on big Young tableaux. 
  The generalized Knizhnik-Zamolodchikov equations of irrational conformal field theory provide a uniform description of rational and irrational conformal field theory. Starting from the known high-level solution of these equations, we first construct the high-level conformal blocks and correlators of all the affine-Sugawara and coset constructions on simple g. Using intuition gained from these cases, we then identify a simple class of irrational processes whose high-level blocks and correlators we are also able to construct. 
  We reconsider quantum mechanical systems based on the classical action being the period of a one form over a cycle and elucidate three main points. First we show that the prepotenial V is no longer completely arbitrary but obeys a consistency integral equation. That is the one form dV defines the same period as the classical action. We then apply this to the case of the punctured plane for which the prepotential is of the form $V= \alpha \theta + \Phi ( \theta )$. The function $ \Phi $ is any but a periodic function of the polar angle. For the topological information to be preserved, we further require that $ \Phi $ be even. Second we point out the existence of a hidden scale which comes from the regularization of the infrared behaviour of the solutions. This will then be used to eliminate certain invariants preselected on dimensional counting grounds. Then provided we discard nonperiodic solutions as being non physical we compute the expectation values of the BRST- exact observables with the general form of the prepotential using only the orthonormality of the solutions (periodic). Third we give topological interpretations of the invariants in terms of the topological invariants wich live naturally on the punctured plane as the winding number and the fundamental group of homotopy,but this requires a prior twisting of the homotopy structure. 
  We study realizations of the exceptional non-linear (quadratically generated, or W-type) N=8 and N=7 superconformal algebras with Spin(7) and G_2 affine symmetry currents, respectively. Both the N=8 and N=7 algebras admit unitary highest-weight representations in terms of a single boson and free fermions in 8 of Spin(7) and 7 of G_2, with the central charges c_8=26/5 and c_7=5, respectively. Furthermore, we show that the general coset Ans"atze for the N=8 and N=7 algebras naturally lead to the coset spaces SO(8)xU(1)/SO(7) and SO(7)xU(1)/G_2, respectively, as the additional consistent solutions for certain values of the central charge. The coset space SO(8)/SO(7) is the seven-sphere S^7, whereas the space SO(7)/G_2 represents the seven-sphere with torsion, S^7_T. The division algebra of octonions and the associated triality properties of SO(8) play an essential role in all these realizations. We also comment on some possible applications of our results to string theory. 
  We study the spectrum of created particles in two-dimensional black hole geometries for a linear, hermitian scalar field satisfying a Lorentz non-invariant field equation with higher spatial derivative terms that are suppressed by powers of a fundamental momentum scale $k_0$. The preferred frame is the ``free-fall frame" of the black hole. This model is a variation of Unruh's sonic black hole analogy. We find that there are two qualitatively different types of particle production in this model: a thermal Hawking flux generated by ``mode conversion" at the black hole horizon, and a non-thermal spectrum generated via scattering off the background into negative free-fall frequency modes. This second process has nothing to do with black holes and does not occur for the ordinary wave equation because such modes do not propagate outside the horizon with positive Killing frequency. The horizon component of the radiation is astonishingly close to a perfect thermal spectrum: for the smoothest metric studied, with Hawking temperature $T_H\simeq0.0008k_0$, agreement is of order $(T_H/k_0)^3$ at frequency $\omega=T_H$, and agreement to order $T_H/k_0$ persists out to $\omega/T_H\simeq 45$ where the thermal number flux is $O(10^{-20}$). The flux from scattering dominates at large $\omega$ and becomes many orders of magnitude larger than the horizon component for metrics with a ``kink", i.e. a region of high curvature localized on a static worldline outside the horizon. This non-thermal flux amounts to roughly 10\% of the total luminosity for the kinkier metrics considered. The flux exhibits oscillations as a function of frequency which can be explained by interference between the various contributions to the flux. 
  We find black hole solutions to Euclidean 2+1 gravity coupled to a relativistic particle which have a dynamical conical singularity at the horizon. These solutions mimic the tree level contribution to the partition function of gravity coupled to a quantum field theory. They are found to violate the standard area law for black hole entropy, their entropy being proportional to the total opening angle. Since each solution depends on the number of windings of the particle path around the horizon, the significance of their summation in the path integral is considered. 
  The point of this paper is see what light new results in hyperbolic geometry may throw on gravitational entropy and whether gravitational entropy is relevant for the quantum origin of the univeres. We introduce some new gravitational instantons which mediate the birth from nothing of closed universes containing wormholes and suggest that they may contribute to the density matrix of the universe. We also discuss the connection between their gravitational action and the topological and volumetric entropies introduced in hyperbolic geometry. These coincide for hyperbolic 4-manifolds, and increase with increasing topological complexity of the four manifold. We raise the questions of whether the action also increases with the topological complexity of the initial 3-geometry, measured either by its three volume or its Matveev complexity. We point out, in distinction to the non-supergravity case, that universes with domains of negative cosmological constant separated by supergravity domain walls cannot be born from nothing. Finally we point out that our wormholes provide examples of the type of Perpetual Motion machines envisaged by Frolov and Novikov. 
  We consider the Salam-Weinberg theory by introducing tensor gauge fields. When these fields are coupled in a topological way with the vector ones, the resulting system constitutes an alternative mechanism of mass generation for vector fields without the presence of Higgs bosons. We show that these masses are in agreement with the ones obtained by means of the spontaneous symmetry breaking. 
  T duality expresses the equivalence of a superstring theory compactified on a manifold K to another (possibly the same) superstring theory compactified on a dual manifold K'. The volumes of K and K' are inversely proportional. In this talk we consider two M theory generalizations of T duality: (i) M theory compactified on a torus is equivalent to type IIB superstring theory compactified on a circle and (ii) M theory compactified on a cylinder is equivalent to SO(32) superstring theory compactified on a circle. In both cases the size of the circle is proportional to the -3/4 power of the area of the dual manifold. 
  A formula is presented for the modular transformation matrix S for any simple current extension of the chiral algebra of a conformal field theory. This provides in particular an algorithm for resolving arbitrary simple current fixed points, in such a way that the matrix S we obtain is unitary and symmetric and furnishes a modular group representation. The formalism works in principle for any conformal field theory. A crucial ingredient is a set of matrices S^J_{ab}, where J is a simple current and a and b are fixed points of J. We expect that these input matrices realize the modular group for the torus one-point functions of the simple currents. In the case of WZW-models these matrices can be identified with the S-matrices of the orbit Lie algebras that we introduced in a previous paper. As a special case of our conjecture we obtain the modular matrix S for WZW-theories based on group manifolds that are not simply connected, as well as for most coset models. 
  Target space duality is reconsidered from the viewpoint of quantization in a space with nontrivial topology. An algebra of operators for the toroidal bosonic string is defined and its representations are constructed. It is shown that there exist an infinite number of inequivalent quantizations, which are parametrized by two parameters $ 0 \le s, t < 1 $. The spectrum exhibits the duality only when $ s = t $ or $ -t $ (mod 1). A deformation of the algebra by a central extension is also introduced. It leads to a kind of twisted relation between the zero mode quantum number and the topological winding number. 
  The multicritical points of the $O(N)$ invariant $N$ vector model in the large $N$ limit are reexamined. Of particular interest are the subtleties involved in the stability of the phase structure at critical dimensions. In the limit $N \to \infty$ while the coupling $g \to g_c$ in a correlated manner (the double scaling limit) a massless bound state $O(N)$ singlet is formed and powers of $1/N$ are compensated by IR singularities. The persistence of the $N \to \infty$ results beyond the leading order is then studied with particular interest in the possible existence of a phase with propagating small mass vector fields and a massless singlet bound state. We point out that under certain conditions the double scaled theory of the singlet field is non-interacting in critical dimensions. 
  The dependence of the effective action for gauge theories on the background field obeys an exact identity. We argue that for Abelian theories the Ward identity follows from the more general background field identity. This observation is particularly relevant for the anomalous Ward identity valid for gauge theories with an effective infrared cutoff as used for flow equations. 
  We show the exact equality of the path integral of the general renormalizable fourth order gravitational action to the path integral of the Einstein action coupled to a massive spin-0 field and a massive spin-2 ghost-like field with non-polynomial interactions. The metric in the Einstein version is a highly nonlinear function of the metric in the quadratic version. Both massive excitations are unstable. The respective cosmological constant terms in the two versions can be very different. Some implications are briefly discussed. 
  Typically the moduli fields acquire mass m =C H in the early universe, which shifts the position of the minimum of their effective potential and leads to an excessively large energy density of the oscillating moduli fields at the later stages of the evolution of the universe. This constitutes the cosmological moduli problem, or Polonyi field problem. We show that the cosmological moduli problem can be solved or at least significantly relaxed in the theories in which C >> 1, as well as in some models with C << 1. 
  Chiral perturbation lagrangian in the framework of non-commutative geometry is considered in full detail. It is found that the explicit symmetry breaking terms appear and some relations between the coupling constants of the theory come out naturally. The WZW term also turns up on the same footing as the other terms of the chiral lagrangian. 
  We analyze the spectrum of dyons in N=4 supersymmetric Yang-Mills theory with gauge group SU(3) spontaneously broken down to U(1)xU(1). The Higgs fields select a natural basis of simple roots. Acting with S-duality on the W-boson states corresponding to simple roots leads to an orbit of BPS dyon states that are magnetically charged with respect to one of the U(1)'s. The corresponding monopole solutions can be obtained by embedding SU(2) monopoles into SU(3) and the S-duality predictions reduce to the SU(2) case. Acting with S-duality on the W-boson corresponding to a non-simple root leads to an infinite set of new S-duality predictions. The simplest of these corresponds to the existence of a harmonic form on the moduli space of SU(3) monopoles that have magnetic charge (1,1) with respect to the two U(1)'s. We argue that the moduli space is given by R^3x(R^1xM)/Z_2, where M is Euclidean Taub-NUT space, and that the latter admits the appropriate normalizable harmonic two form. We briefly discuss the generalizations to other gauge groups. 
  Multi-instanton contributions to QCD sum rules for the pion are investigated within a framework which models the QCD vacuum as an instanton liquid. It is shown that in singular gauge the sum of planar diagrams in leading order of the $1/N_{c}$ expansion provides similar results as the effective single-instanton contribution. These effects are also analysed in regular gauge. Our findings confirm that at large distances the correlator functions are more adequately described in the singular gauge rather than in the regular one. 
  We examine the Toda frame formulation of the SO(3)-invariant hyper-Kahler 4-metrics, namely Eguchi-Hanson, Taub-NUT and Atiyah-Hitchin. Our method exploits the presence of a rotational SO(2) isometry, leading to the explicit construction of all three complex structures as a singlet plus a doublet. The Atiyah-Hitchin metric on the moduli space of BPS SU(2) monopoles with magnetic charge 2 is purely rotational. 
  The paper is devoted to the further study of the remarkable classes of orthogonal polynomials recently discovered by Bender and Dunne. We show that these polynomials can be generated by solutions of arbitrary quasi - exactly solvable problems of quantum mechanichs both one-dimensional and multi-dimensional. A general and model-independent method for building and studying Bender-Dunne polynomials is proposed. The method enables one to compute the weight functions for the polynomials and shows that they are the orthogonal polynomials in a discrete variable $E_k$ which takes its values in the set of exactly computable energy levels of the corresponding quasi-exactly solvable model. It is also demonstrated that in an important particular case, the Bender-Dunne polynomials exactly coincide with orthogonal polynomials appearing in Lanczos tridiagonalization procedure. 
  We give a review of some recent work on the construction and classification of $p$-brane solutions in maximal supergravity theories in all dimensions $4\le D\le 11$. These solutions include isotropic elementary and solitonic $p$-branes, dyonic $p$-branes, and multi-scalar $p$-branes. These latter two categories include massless strings and black holes as special cases. For all the solutions, we analyse their residual unbroken supersymmetry by means of an explicit construction of the eigenvalues of the \bog matrix, defined as the anticommutator of the conserved supercharges. 
  The heat kernel expansion for a general non--minimal operator on the spaces $C^\infty (\Lambda^k)$ and $C^\infty (\Lambda^{p,q})$ is studied. The coefficients of the heat kernel asymptotics for this operator are expressed in terms of the Seeley coefficients for the Hodge--de Rham Laplacian. 
  Various examples of target space duality transformations are investigated up to two loop order in perturbation theory. Our results show that when using the tree level (`naive') transformation rules the dual theories are in general {\it inequivalent} at two loops to the original ones, (both for the Abelian and the non Abelian duality). 
  The method of higher covariant derivative regularization of gauge theories is reviewed. The objections raised in the literature last years are discussed and the consistency of the method is proven. New approach to regularization of overlapping divergencies is developped. 
  A non--Abelian $SU(2)$ model is constructed for a five--dimensional bound system "charge--dyon" on the basis of the Hurwitz--transformed eight--dimensional isotropic quantum oscillator. The principle of dyon--oscillator duality is formulated; the energy spectrum and wave functions of the system "charge--dyon" are calculated. 
  We examine QED(3+1) quantised in the `front form' with finite `volume' regularisation, namely in Discretised Light-Cone Quantisation. Instead of the light-cone or Coulomb gauges, we impose the light-front Weyl gauge $A^-=0$. The Dirac method is used to arrive at the quantum commutation relations for the independent variables. We apply `quantum mechanical gauge fixing' to implement Gau{\ss}' law, and derive the physical Hamiltonian in terms of unconstrained variables. As in the instant form, this Hamiltonian is invariant under global residual gauge transformations, namely displacements. On the light-cone the symmetry manifests itself quite differently. 
  The combined method of Higher Covariant Derivatives and Pauli-Villars regularization to regularize pure Yang-Mills theories is formulated in the framework of Batalin and Vilkovisky. However, BRS invariance is broken by this method and suitable counterterms should be added to restore it. The one loop counterterm is presented. Contrary to the scheme of Slavnov, this method is regularizing and leads to consistent renormalization group functions, which are the same as those found by other regularization schemes. 
  The Supersymmetric Dual Sigma Model (SDSM) is a local field theory introduced to be nonlocally equivalent to the Supersymmetric Chiral nonlinear sigma-Model (SCM), this dual equivalence being proven by explicit canonical transformation in tangent space. This model is here reconstructed in superspace and identified as a chiral-entwined supersymmetrization of the Dual Sigma Model (DSM). This analysis sheds light on the Boson-Fermion Symphysis of the dual transition, and on the new geometry of the DSM. 
  We consider the low-energy dynamics of a pair of distinct fundamental monopoles that arise in the $N=4$ supersymmetric $SU(3)$ Yang-Mills theory broken to $U(1)\times U(1)$. Both the long distance interactions and the short distance behavior indicate that the moduli space is $R^3\times(R^1 \times {\cal M}_0)/Z$ where ${\cal M}_0$ is the smooth Taub-NUT manifold, and we confirm this rigorously. By examining harmonic forms on the moduli space, we find a threshold bound state of two monopoles with a tower of BPS dyonic states built on it, as required by Montonen-Olive duality. We also present a conjecture for the metric of the moduli space for any number of distinct fundamental monopoles for an arbitrary gauge group. 
  We study the high energy, fixed angle, asymptotics of D-brane form factors to all orders in string perturbation theory, using the Gross-Mende saddle point techniques. The effective interaction size of all D-branes grows linearly with the energy as (alpha') E/n, where n is the order of perturbation theory, except for the D-instanton, whose form factor is dominated by end-point contributions, and remains point-like at high orders. The qualitative features are independent of the R-R or NS-NS character of the states used to probe the D-brane. 
  We study the structures of partition functions of the large $N$ generalized two-dimensional Yang-Mills theories ($gYM_2$) by recasting the higher Casimirs. We clarify the appropriate interpretations of them and try to extend the Cordes-Moore-Ramgoolam's topological string model describing the ordinary $YM_2$ \cite{CMR} to those describing $gYM_2$. We present the expressions of the appropriate operators to reproduce the higher Casimir terms in $gYM_2$. The concept of ''deformed gravitational descendants'' will be introduced for this purpose. 
  For the chiral QCD_2 on a cylinder, we give a construction of a quantum theory consistent with anomaly. We construct the algebra of the Poincare generators and show that it differs from the Poincare one. 
  The renormalization group (RG) is used to study the asymptotically free $\phi_6^3$-theory in curved spacetime. Several forms of the RG equations for the effective potential are formulated. By solving these equations we obtain the one-loop effective potential as well as its explicit forms in the case of strong gravitational fields and strong scalar fields. Using zeta function techniques, the one-loop and corresponding RG improved vacuum energies are found for the Kaluza-Klein backgrounds $R^4\times S^1\times S^1$ and $R^4\times S^2$. They are given in terms of exponentially convergent series, appropriate for numerical calculations. A study of these vacuum energies as a function of compactification lengths and other couplings shows that spontaneous compactification can be qualitatively different when the RG improved energy is used. 
  In this article we examine the compatibility of some recent results, results relating M-Theory to String Theory, with the string-string duality conjecture in six-dimensions. In particular, we rederive the relation between M-Theory and Type IIA strings. We then go on to examine in detail M-Theory on $K3 \times S^{1}$ and its relation to the Heterotic theory on $T^{4}$. We conclude with some remarks on M-Theory on $T^{4}\times (S^{1}/{\bf Z}_{2})$ and its relation to the Type II theory on $K3$. 
  The conditions that allow us to consider the vacuum expectation value of the energy-momentum tensor as a statistical average, at some particular temperature, are given. When the mean value of created particles is stationary, a planckian distribution for the field modes is obtained. In the massless approximation, the temperature dependence is as that corresponding to a radiation dominated Friedmann-like model. 
  The free energy in the weak-coupling phase of two-dimensional Yang-Mills theory on a sphere for SO(N) and Sp(N) is evaluated in the 1/N expansion using the techniques of Gross and Matytsin. Many features of Yang-Mills theory are universal among different gauge groups in the large N limit, but significant differences arise in subleading order in 1/N. 
  The loop transform in quantum gauge field theory can be recognized as the Fourier transform (or characteristic functional) of a measure on the space of generalized connections modulo gauge transformations. Since this space is a compact Hausdorff space, conversely, we know from the Riesz-Markov theorem that every positive linear functional on the space of continuous functions thereon qualifies as the loop transform of a regular Borel measure on the moduli space. In the present article we show how one can compute the finite joint distributions of a given characteristic functional, that is, we derive the inverse loop transform. 
  It is shown that the partition function of the 2d Ising model on the dual finite lattice with periodical boundary conditions is expressed through some specific combination of the partition functions of the model on the torus with corresponding boundary conditions. The generalization of the duality relations for the nonhomogeneous case is given. These relations are proved for the weakly-nonhomogeneous distribution of the coupling constants for the finite lattice of arbitrary sizes. Using the duality relations for the nonhomogeneous Ising model, we obtain the duality relations for the two-point correlation function on the torus, the 2d Ising model with magnetic fields applied to the boundaries and the 2d Ising model with free, fixed and mixed boundary conditions. 
  It is described how the standard Poisson bracket formulas should be modified in order to incorporate integrals of divergences into the Hamiltonian formalism and why this is necessary. Examples from Einstein gravity and Yang-Mills gauge field theory are given. 
  We show how enhanced gauge symmetry in type II string theory compactified on a Calabi--Yau threefold arises from singularities in the geometry of the target space. When the target space of the type IIA string acquires a genus $g$ curve $C$ of $A_{N-1}$ singularities, we find that an $SU(N)$ gauge theory with $g$ adjoint hypermultiplets appears at the singularity. The new massless states correspond to solitons wrapped about the collapsing cycles, and their dynamics is described by a twisted supersymmetric gauge theory on $C\times \R^4$. We reproduce this result from an analysis of the $S$-dual $D$-manifold. We check that the predictions made by this model about the nature of the Higgs branch, the monodromy of period integrals, and the asymptotics of the one-loop topological amplitude are in agreement with geometrical computations. In one of our examples we find that the singularity occurs at strong coupling in the heterotic dual proposed by Kachru and Vafa. 
  The supersymmetric action of type IIA D=10 superstring in N=2a, D=10 supergravity background can be derived by double dimensional reduction of the action of supermembrane coupled to D=11 supergravity. We demonstrate that the background Ramond-Ramond fields appear in the resulting superstring action with an extra factor of exponential of the dilaton. 
  We have examined the coupled system of the dilaton gravity and the $CP^{N-1}$ theory known as a model of the confinement of massive scalar quarks. After the quantization of the system, we could see the quantum effect of the gravitation on the coupling constant of $CP^{N-1}$ model and how the coupling constant of dynamically induced gauge field changes near the black hole configuration. 
  We study the quantum mechanical Liouville model with attractive potential which is obtained by Hamiltonian symmetry reduction from the system of a free particle on $SL(2, \Real)$. The classical reduced system consists of a pair of Liouville subsystems which are `glued together' in such a way that the singularity of the Hamiltonian flow is regularized. It is shown that the quantum theory of this reduced system is labelled by an angle parameter $\theta \in [\,0,2\pi)$ characterizing the self-adjoint extensions of the Hamiltonian and hence the energy spectrum. There exists a probability flow between the two Liouville subsystems, demonstrating that the two subsystems are also `connected' quantum mechanically, even though all the wave functions in the Hilbert space vanish at the junction. 
  We give a review of the exact renormalization group (ERG) approach and illustrate its applications in scalar and fermionic theories. The derivative expansion and approximations based on the derivative expansion with further truncation in the number of fields (mixed approximation) are discussed. We analyse the mixed approximation for a three-dimensional scalar theory and show that it is less effective than the pure derivative expansion. For pure fermionic theories analytical solutions for the pure derivative expansion and mixed approximation in the limit $N \to \infty $, where $N$ is the number of fermionic species, are found. For finite $N$ a few series of fixed point solutions with their anomalous dimensions and critical exponents are computed numerically. We argue that one of the fermionic solutions can be identified with that of Dashen and Frishman, whereas the others seem to be new ones. The issues of spurious solutions and scheme dependence of the results are discussed. 
  The magnetic deformation of the Ising Model, the thermal deformations of both the Tricritical Ising Model and the Tricritical Potts Model are governed by an algebraic structure based on the Dynkin diagram associated to the exceptional algebras $E_n$ (respectively for $n=8,7,6$). We make use of these underlying structures as well as of the discrete symmetries of the models to compute the matrix elements of the stress--energy tensor and its two--point correlation function by means of the spectral representation method. 
  The initial classification of fusion rules have shown that rational conformal field theory is very limited. In this paper we study the fusion rules of extend ed current algebras. Explicit formulas are given for the S matrix and the fusion rules, based on the full splitting of the fixed point fields. We find that in s ome cases sensible fusion rules are obtained, while in others this procedure lea ds to fractional fusion constants. 
  In addition to the familiar contribution from a holomorphic function $\FF$, the K\"ahler potential of the scalars in the nonabelian $N=2$ vector multiplet receives contributions from a real function $\HH$. We determine the latter at the one-loop level, taking into account both supersymmetric matter and gauge loops. The function $\HH$ characterizes the four-point coupling of the $N=2$ vector-multiplet vectors for constant values of their scalar superpartners. We discuss the consequences of our results. 
  Within a group-theoretical approach to the description of (2+1)-dimensional anyons, the minimal covariant set of linear differential equations is constructed for the fractional spin fields with the help of the deformed Heisenberg algebra (DHA), $[a^{-},a^{+}]=1+\nu K$, involving the Klein operator $K$, $\{K,a^{\pm}\}=0$, $K^{2}=1$. The connection of the minimal set of equations with the earlier proposed `universal' vector set of anyon equations is established. On the basis of this algebra, a bosonization of supersymmetric quantum mechanics is carried out. The construction comprises the cases of exact and spontaneously broken $N=2$ supersymmetry allowing us to realize a Bose-Fermi transformation and spin-1/2 representation of SU(2) group in terms of one bosonic oscillator. The construction admits an extension to the case of OSp(2$\vert$2) supersymmetry, and, as a consequence, both applications of the DHA turn out to be related. A possibility of `superimposing' the two applications of the DHA for constructing a supersymmetric (2+1)-dimensional anyon system is discussed. As a consequential result we point out that $osp(2|2)$ superalgebra is realizable as an operator algebra for a quantum mechanical 2-body (nonsupersymmetric) Calogero model. 
  The role of Kac-Moody algebras in exploiting symmetries of particle physics and string theory is described. 
  We study a class of rotating dyonic black holes in the heterotic string theory in four dimension which have left, right independent electric charges but have same magnitude for the left and right magnetic charges. In both left and right sector the electric and the magnetic vectors are orthogonal to each other. The gyromagnetic(electric) ratios are in general found not to have an upper bound. 
  The Hamiltonian formulation of the motion of a spinning relativistic particle in an external electromagnetic field is considered. The approach is based on the introduction of new coordinates and their conjugated momenta to describe the spin degrees of freedom together with an appropriate set of constraints in the Dirac formulation. For particles with gyromagnetic ratio $g=2$, the equations of motion do not predict any deviation from the standard Lorentz force, while for $g \neq 2$ an additional force, which corresponds to the magnetic dipole force, is obtained. 
  It has been suggested that quantum fluctuations of the gravitational field could give rise in the lowest approximation to an effective noncommutative version of Kaluza-Klein theory which has as extra hidden structure a noncommutative geometry. It would seem however from the Standard Model, at least as far as the weak interactions are concerned, that a double-sheeted structure is the phenomenologically appropriate one at present accelerator energies. We examine here to what extent this latter structure can be considered as a singular limit of the former. 
  A tensor product generalisation of $B\wedge F$ theories is proposed to give a Bogomol'nyi structure. Non-singular, stable, finite-energy particle-like solutions to the Bogomol'nyi equations are studied. Unlike Yang-Mills(-Higgs) theory, the Bogomol'nyi structure does not appear as a perfect square in the Lagrangian. Consequently, the Bogomol'nyi energy can be obtained in more than one way. The added flexibility permits electric monopole solutions to the field equations. 
  We make some comments on the renormalization of Wilson operators (not just vacuum -expectation values of Wilson operators), and the features which arise in Minkowski space. If the Wilson loop contains a straight light-like segment, charge renormalization does not work in a simple graph-by-graph way; but does work when certain graphs are added together. We also verify that, in a simple example of a smooth loop in Minkowski space, the existence of pairs of points which are light-like separated does not cause any extra divergences. 
  The generally accepted phase diagrams for the discrete $Z_N$ spin models in two dimensions imply the existence of certain renormalisation group flows, both between conformal field theories and into a massive phase. Integral equations are proposed to describe these flows, and some properties of their solutions are discussed. The infrared behaviour in massless and massive directions is analysed in detail, and the techniques used are applied to a number of other models. 
  One may introduce at least three different Lie algebras in any Lagrangian field theory : (i) the Lie algebra of local BRST cohomology classes equipped with the odd Batalin-Vilkovisky antibracket, which has attracted considerable interest recently~; (ii) the Lie algebra of local conserved currents equipped with the Dickey bracket~; and (iii) the Lie algebra of conserved, integrated charges equipped with the Poisson bracket. We show in this paper that the subalgebra of (i) in ghost number $-1$ and the other two algebras are isomorphic for a field theory without gauge invariance. We also prove that, in the presence of a gauge freedom, (ii) is still isomorphic to the subalgebra of (i) in ghost number $-1$, while (iii) is isomorphic to the quotient of (ii) by the ideal of currents without charge. In ghost number different from $-1$, a more detailed analysis of the local BRST cohomology classes in the Hamiltonian formalism allows one to prove an isomorphism theorem between the antibracket and the extended Poisson bracket of Batalin, Fradkin and Vilkovisky. 
  We construct a (1,2) heterotic string with gauge symmetry and determine its particle spectrum. This theory has a local N=1 worldsheet supersymmetry for left movers and a local N=2 worldsheet supersymmetry for right movers and describes particles in either two or three space-time dimensions. We show that fermionizing the bosons of the compactified N=1 space leads to a particle spectrum which has nonabelian gauge symmetry. The fermionic formulation of the theory corresponds to a dimensional reduction of self dual Yang Mills. We also give a worldsheet action for the theory and calculate the one-loop path integral. 
  The solutions of Heisenberg equations and two-particles eigenvalue problems for nonrelativistic models of current-current fermion interaction and $N, \Theta $ model are obtained in the frameworks of dynamical mapping method. The equivalence of different types of dynamical mapping is shown. The connection between renormalization procedure and theory of selfadjoint extensions is elucidated. 
  I show that a usual propagator cannot be defined for the pseudo-diffusion equation of the Q-functions. Instead, a forward-backward propagator is defined, which motivated a generalization of Cahill-Glauber interpolating operator. Our generalized operator ${\bf Q}(p,q;\sigma_p^{-1},\sigma_q)$ depends on two squeezing parameters $\sigma_p$ and $\sigma_q$, and is shown to obey a generalized pseudo-diffusion equation or a diffusion equation, depending on the curve $(\sigma_p(\mu),\sigma_q(\mu))$ along which one moves in the $(\sigma_p,\sigma_q)$ plane. An algorithm is also given for squeezing Q functions directly, using one-dimensional diffusion propagators. 
  In a recent Letter (K.Halpern and K.Huang, Phys. Rev. Lett. 74 (1995) 3526), certain properties of the Local Potential Approximation (LPA) to the Wilson renormalization group were uncovered, which led the authors to conclude that $D>2$ dimensional scalar field theories endowed with {\sl non-polynomial} interactions allow for a continuum of renormalization group fixed points, and that around the Gaussian fixed point, asymptotically free interactions exist. If true, this could herald very important new physics, particularly for the Higgs sector of the Standard Model. Continuing work in support of these ideas, has motivated us to point out that we previously studied the same properties and showed that they lead to very different conclusions. Indeed, in as much as the statements in hep-th/9406199 are correct, they point to some deep and beautiful facts about the LPA and its generalisations, but however no new physics. 
  These are notes based on a lecture given at the Cargese summer school 1995. I describe evidence that the (two-dimensional) integrable chiral Gross-Neveu model might remain integrable when coupled to gravity. The results presented here were obtained in collaboration with Ian Kogan. 
  The sequence of intertwined T-S-T duality transformations acting on the 4D static uncharged black hole leads to a new black hole background with horizon and singularity exchanged. It is shown that this space-time is extendible too. In particular we will see that a string moving into a black hole is dual to a string leaving a white hole. That offers the possibility that a test-string does not see the singularity. 
  The problem of a harmonic oscillator coupling to an electromagnetic potential plus a topological-like (Chern-Simons) massive term, in two-dimensional space, is studied in the light of the symplectic formalism proposed by Faddeev and Jackiw for constrained systems. 
  It was recently conjectured by D. Page that if a quantum system of Hilbert space dimension $nm$ is in a random pure state then the average entropy of a subsystem of dimension $m$ where $m \leq n$ is $ S_{mn} = \sum^{mn}_{k=n+1}(1/k) - (m-1)/2n$. In this letter this conjecture is proved. 
  The effective Lagrangian of QED coupled to dyons is calculated. The resulting generalization of the Euler-Heisenberg Lagrangian contains non-linear $P$- and $T$-nonivariant (but $C$ invariant) terms corresponding to the virtual pair creation of dyons. As examples, the amplitudes for photon splitting and photon coalescence are calculated. 
  An energetic justification of a thermal component during inflation is given. The thermal component can act as a heat reservoir which induces thermal fluctuations on the inflaton field system. We showed previously that such thermal fluctuations could dominate quantum fluctuations in producing the initial seeds of density perturbations. A Langevin-like rate equation is derived from quantum field theory which describes the production of fluctuations in the inflaton field when acted upon by a simple modeled heat reservoir. In a certain limit this equation is shown to reduce to the standard Langevin equation, which we used to construct "Warm Inflation" scenarios in previous work. A particle physics interpretation of our system-reservoir model is offered. 
  An integral solution to the quantum Knizhnik-Zamolodchikov ($q$KZ) equation with $|q|=1$ is presented. Upon specialization, it leads to a conjectural formula for correlation functions of the XXZ model in the gapless regime. The validity of this conjecture is verified in special cases, including the nearest neighbor correlator with an arbitrary coupling constant, and general correlators in the XXX and XY limits. 
  We solve the Schwinger model on a circle by first finding the explicit groundstate functional(s). Having done this, we give the structure of the Hilbert space and derive bosonization formulae in this formalism. 
  We clarifies the group theoretical structure of $N=1$ and $N=2$ two-form supergravity, which is classically equivalent to the Einstein supergravity. $N=1$ and $N=2$ two-form supergravity theories can be formulated as gauge theories. By introducing two Grassmann variables $\theta^A$ ($A=1,2$), we construct the explicit representations of the generators $Q^i$ of the gauge group, which makes to express any product of the generators as a linear combination of the generators $Q^iQ^j=\sum_k f^{ij}_k Q^k$. By using the expression and the tensor product representation, we explain how to construct finite-dimensional representations of the gauge groups. Based on these representations, we construct the Lagrangeans of $N=1$ and $N=2$ two-form supergravity theories. 
  Hamiltonians whose symbols are not simply real valued, but matrix or, more generally, endomorphism valued functions appear in many places in physics, examples being the Dirac equation, multicomponent wave equations like electrodynamics in media, and Yang-Mills theories, and the Born-Oppenheimer approximation in molecular physics. The aim of this paper is to give a completely geometric approach to the WKB approximation od such systems, and to reduce the problem ``as far as possible'' to the scalar case. A star-product formulation of quantum mechanics proves to be particularly useful in this context. As opposed to other approaches in the literature which restrict themselves to the use of the Moyal product and thus to the study of trivial bundles (or local trivializations) over $\real^{2n}$, we will consider general bundles over arbitrary symplectic manifolds. Here, Fedosov's construction \cite{fedosov} will be the adequate tool, since it gives an explicit construction for star products in this general setting. 
  We consider matrix-model representations of the meander problem which describes, in particular, combinatorics for foldings of closed polymer chains. We introduce a supersymmetric matrix model for describing the principal meander numbers. This model is of the type proposed by Marinari and Parisi for discretizing a superstring in D=1 while the supersymmetry is realized in D=0 as a rotational symmetry between bosonic and fermionic matrices. Using non-commutative sources, we reformulate the meander problem in a Boltzmannian Fock space whose annihilation and creation operators obey the Cuntz algebra. We discuss also the relation between the matrix models describing the meander problem and the Kazakov-Migdal model on a D-dimensional lattice. 
  We establish the asymptotic behaviour of the ratio $h^\prime(0)/h(0)$ for $\lambda\rightarrow\infty$, where $h(r)$ is a solution, vanishing at infinity, of the differential equation $h^{\prime\prime}(r) = i\lambda \omega (r) h(r)$ on the domain $0 \leq r <\infty$ and $\omega (r) = (1-\sqrt{r} K_1(\sqrt{r}))/r$. Some results are valid for more general $\omega$'s. 
  The minimal bosonization of supersymmetry in terms of one bosonic degree of freedom is considered. A nontrivial relationship of the construction to the Witten supersymmetric quantum mechanics is illustrated with the help of the simplest $N=2$ SUSY system realized on the basis of the ordinary (undeformed) bosonic oscillator. It is shown that the generalization of such a construction to the case of Vasiliev deformed bosonic oscillator gives a supersymmetric extension of the 2-body Calogero model in the phase of exact or spontaneously broken $N=2$ SUSY. The construction admits an extension to the case of the OSp(2$\vert$2) supersymmetry, and, as a consequence, $osp(2\vert 2)$ superalgebra is revealed as a dynamical symmetry algebra for the bosonized supersymmetric Calogero model. Realizing the Klein operator as a parity operator, we construct the bosonized Witten supersymmetric quantum mechanics. Here the general case of the corresponding bosonized $N=2$ SUSY is given by an odd function being a superpotential. 
  We compute the renormalized trajectory of $\phi^4_4$-theory by perturbation theory in a running coupling. We introduce an iterative scheme without reference to a bare action. The expansion is proved to be finite to every order of perturbation theory. 
  We describe a set of methods to calculate gauge theory renormalization constants from string theory, all based on a consistent prescription to continue off shell open bosonic string amplitudes. We prove the consistency of our prescription by explicitly evaluating the renormalizations of the two, three and four-gluon amplitudes, and showing that they obey the appropriate Ward identities. The field theory limit thus performed corresponds to the background field method in Feynman gauge. We identify precisely the regions in string moduli space that correspond to different classes of Feynman diagrams, and in particular we show how to isolate contributions to the effective action. Ultraviolet divergent terms are then encoded in a single string integral over the modular parameter $\tau$. Finally, we derive a multiloop expression for the effective action by computing the partition function of an open bosonic string interacting with an external non-abelian background gauge field. 
  We discuss a canonical formalism method for constructing actions describing propagation of W-strings on curved backgrounds. The method is based on the construction of a representation of the W-algebra in terms of currents made from the string coordinates and the canonically conjugate momenta. We construct such a representation for a W_3-string propagating in the background metric with one flat direction by using a simple ansatz for the W-generators where each generator is a polynomial of the canonical currents and the veilbeins. In the case of a general background we show that the simple polynomial ansatz fails, and terms containing the veilbein derivatives must be added. 
  In this talk we summarise our recent results on perturbative and non-perturbative monodromies in four-dimensional heterotic strings with $N=2$ space-time supersymmetry, and we compare our results with the rigid $SU(2)$ $N=2$ Yang-Mills monodromies. 
  Using a numerical implementation of the ADHMN construction, we compute the fields and energy densities of a charge three monopole with tetrahedral symmetry and a charge four monopole with octahedral symmetry. We then construct a one parameter family of spectral curves and Nahm data which represent charge four monopoles with tetrahedral symmetry, which includes the monopole with octahedral symmetry as a special case. In the moduli space approximation, this family describes a novel kind of four monopole scattering and we use our numerical scheme to construct the energy density at various times during the motion. 
  It is shown that there exists a charge five monopole with octahedral symmetry and a charge seven monopole with icosahedral symmetry. A numerical implementation of the ADHMN construction is used to calculate the energy density of these monopoles and surfaces of constant energy density are displayed. The charge five and charge seven monopoles look like an octahedron and a dodecahedron respectively. A scattering geodesic for each of these monopoles is presented and discussed using rational maps. This is done with the aid of a new formula for the cluster decomposition of monopoles when the poles of the rational map are close together. 
  By imposing certain combined inversion and rotation symmetries on the rational maps for SU(2) BPS monopoles we construct geodesics in the monopole moduli space. In the moduli space approximation these geodesics describe a novel kind of monopole scattering. During these scattering processes axial symmetry is instantaneously attained and, in some, monopoles with the symmetries of the regular solids are formed. The simplest example corresponds to a charge three monopole invariant under a combined inversion and 90 degree rotation symmetry. In this example three well-separated collinear unit charge monopoles coalesce to form first a tetrahedron, then a torus, then the dual tetrahedron and finally separate again along the same axis of motion. We explicitly construct the spectral curves in this case and use a numerical ADHMN construction to compute the energy density at various times during the motion. We find that the dynamics of the zeros of the Higgs field is extremely rich and we discover a new phenomenon; there exist charge k SU(2) BPS monopoles with more than k zeros of the Higgs field. 
  We propose an alternative to Dirac quantization for a quadratic constrained system. We show that this solves the Jacobi identity violation problem occuring in the Dirac quantization case and yields a well defined Fock space. By requiring the uniqueness of the ground state, we show that for non-constrained systems this approach gives the same results as Dirac quantization. 
  We present a version of ten-dimensional IIA supergravity containing a 9-form potential for which the field equations are equivalent to those of the standard, massless, IIA theory for vanishing 10-form field strength, $F_{10}$, and to those of the `massive' IIA theory for non-vanishing $F_{10}$. We present a multi 8-brane solution of these equations that generalizes the 8-brane of Polchinski and Witten. We show that this solution is T-dual to a new multi 7-brane solution of $S^1$ compactified IIB supergravity, and that the latter is T-dual to the IIA 6-brane. When combined with the $Sl(2;\Z)$ U-duality of the type IIB superstring, the T-duality between type II 7-branes and 8-branes implies a quantization of the cosmological constant of type IIA superstring theory. These results are made possible by the construction of a new {\it massive} N=2 D=9 supergravity theory. We also discuss the 11-dimensional interpretation of these type II p-branes. 
  This is the written version of a talk given at the Santa Barbara Workshop on Supersymmetry in December of 1995. It summarizes a collection of results on superstring cosmology obtained by the author and various collaborators, and contains some speculations about the resolution of the cosmological constant and vacuum selection problems in string theory. 
  We examine the BPS and low energy non-BPS excitations of the D-string, in terms of open strings that travel on the D-string. We use this to study the energy thresholds for exciting a long D-string, for arbitrary winding number. We also compute the leading correction to the entropy from non-BPS states for a long D-string, and observe the relation of all these quantities with the corresponding quantities for the elementary string. 
  We review the recent exact solution of a matrix model which interpolates between flat and random lattices. The importance of the results is twofold: Firstly, we have developed a new large N technique capable of treating a class of matrix models previously thought to be unsolvable. Secondly, we are able to make a first precise statement about two-dimensional R^2 gravity. These notes are based on a lecture given at the Cargese summer school 1995. They contain some previously unpublished results. 
  We discuss the connection between different entropies introduced for black hole. It is demonstrated on the two-dimensional example that the (quantum) thermodynamical entropy of a hole coincides (including UV-finite terms) with its statistical-mechanical entropy calculated according to 't Hooft and regularized by Pauli-Villars. 
  We write down a general action principle for spinning strings in 2+1 dimensional space-time without introducing Grassmann variables. The action is written solely in terms of coordinates taking values in the 2+1 Poincare group, and it has the usual string symmetries, i.e. it is invariant under a) diffeomorphisms of the world sheet and b) Poincare transformations. The system can be generalized to an arbitrary number of space-time dimensions, and also to spinning membranes and p-branes. 
  Micro-organisms can be classified into three different types according to their size. We study the efficiency of the swimming of micro-organism in two dimensional fluid as a device for helping the explanation of this hierarchy in the size. We show that the efficiency of flagellate becomes unboundedly large, whereas that of ciliate has the upper bound. The unboundedness is related to the curious feature of the shape space, that is, a singularity at the basic shape of flagellate. 
  (Anti)self-dual solutions of the scale invariant SU(2) gauged Grassmanian model are sought. A stronger (anti)selfduality condition for this system is defined, referred to as strong self-duality, and spherically symmetric solutions of this {\it strong} (anti)self-duality equations are found in closed form. It is verified that these are the only solutions of the strong (anti)self-duality equations. The usual (anti)self-duality equations for the axially symmetric fields are derived and seen to be not overdetermined. 
  We investigate field theory puzzles occuring in the interplay between supersymmetry and duality in the presense of rotational isometries (also known as non-triholomorphic in hyper-Kahler geometry). We show that T-duality is always compatible with supersymmetry, provided that non-local world-sheet effects are properly taken into account. The underlying superconformal algebra remains the same, and T-duality simply relates local with non-local realizations of it. The non-local realizations have a natural description using parafermion variables of the corresponding conformal field theory. We also comment on the relevance of these ideas to a possible resolution of long standing problems in the quantum theory of black holes. 
  I discuss the indefinite metrical structure of the time-space translations as realized in the indefinite inner products for relativistic quantum fields, familiar in the example of quantum gauge fields. The arising indefinite unitary nondiagonalizable representations of the translations suggest as the positive unitarity condition for the probability interpretable positive definite asymptotic particle state space the requirement of a vanishing nilpotent part in the time-space translations realization. A trivial Becchi-Rouet-Stora charge (classical gauge invariance) for the asymptotics in quantum gauge theories can be interpreted as one special case of this general principle - the asymptotic projection to the eigenstates of the time-space translations. 
  In 4D non-perturbative Regge calculus a positive value of the effective cosmological constant characterizes the collapsed phase of the system. If a local term of the form $S'=\sum_{h \epsilon \{h_1,h_2,...\} } \lambda_h V_h$ is added to the gravitational action, where $\{h_1,h_2,...\}$ is a subset of the hinges and $\{\lambda_h\}$ are positive constants, one expects that the volumes $V_{h_1}$, $V_{h_2}$, ... tend to collapse and that the excitations of the lattice propagating through the hinges $\{h_1,h_2,...\}$ are damped. We study the continuum analogue of this effect. The additional term $S'$ may represent the coupling of the gravitational field to an external Bose condensate. 
  In the series of papers we represent the ``Whittaker'' wave functional of $d+1$-dimensional Liouville model as a correlator in $d+0$-dimensional theory of the sine-Gordon type (for $d=0$ and $1$). Asypmtotics of this wave function is characterized by the Harish-Chandra function, which is shown to be a product of simple $\Gamma$-function factors over all positive roots of the corresponding algebras (finite-dimensional for $d=0$ and affine for $d=1$). This is in nice correspondence with the recent results on 2- and 3-point correlators in $1+1$ Liouville model, where emergence of peculiar double-periodicity is observed. The Whittaker wave functions of $d+1$-dimensional non-affine ("conformal") Toda type models are given by simple averages in the $d+0$ dimensional theories of the affine Toda type. This phenomenon is in obvious parallel with representation of the free-field wave functional, which is originally a Gaussian integral over interior of a $d+1$-dimensional disk with given boundary conditions, as a (non-local) quadratic integral over the $d$-dimensional boundary itself. In the present paper we mostly concentrate on the finite-dimensional case. The results for finite-dimensional "Iwasawa" Whittaker functions were known, and we present their survey. We also construct new "Gauss" Whittaker functions. 
  We describe the explicit construction of Yang-Mills instantons on ALE spaces, following the work of Kronheimer and Nakajima. For multicenter ALE metrics, we determine the abelian instanton connections which are needed for the construction in the non-abelian case. We compute the partition function of Maxwell theories on ALE manifolds and comment on the issue of electromagnetic duality. We discuss the topological characterization of the instanton bundles as well as the identification of their moduli spaces. We generalize the 't Hooft ansatz to SU(2) instantons on ALE spaces and on other hyper-Kahler manifolds. Specializing to the Eguchi-Hanson gravitational background, we explicitly solve the ADHM equations for SU(2) gauge bundles with second Chern class 1/2, 1 and 3/2. 
  We demonstrate that all rational models of the N=2 super Virasoro algebra are unitary. Our arguments are based on three different methods: we determine Zhu's algebra (for which we give a physically motivated derivation) explicitly for certain theories, we analyse the modular properties of some of the vacuum characters, and we use the coset realisation of the algebra in terms of su_2 and two free fermions.   Some of our arguments generalise to the Kazama-Suzuki models indicating that all rational N=2 supersymmetric models might be unitary. 
  U duality transformations must act on a basis of states that form complete multiplets of the U group, at any coupling, even though the states may not be mass degenerate, as for a broken symmetry. Similarly, if superstring theory is related to a non-perturbative 11D M-theory, then an 11D supermultiplet structure is expected, even though the multiplet may contain states of different masses. We analyse the consistency between these two multiplet schemes at the higher excited string levels for various compactifications of the type IIA superstring. While we find complete consistency for a number of compactifications, there remain some unanswered questions in others. The relation to D-branes also needs further clarification. 
  $\lambda\varphi^4$ theory at finite temperature suffers from infrared divergences near the temperature at which the symmetry is restored. These divergences are handled using renormalization group methods. Flow equations which use a fiducial mass as flow parameter are well adapted to predicting the non-trivial critical exponents whose presence is reflected in these divergences. Using a fiducial temperature as flow parameter, we predict the critical temperature, at which the mass vanishes, in terms of the zero-temperature mass and coupling. We find some universal amplitude ratios which connect the broken and symmetric phases of the theory which agree well with those of the three-dimensional Ising model obtained from high- and low-temperature series expansions. 
  General static solutions of effectively 2-dimensional Einstein-Dilaton-Maxwell-Scalar theories are obtained. Our model action includes a class of 2-d dilaton gravity theories coupled with a $U(1)$ gauge field and a massless scalar field. Therefore it also describes the spherically symmetric reduction of $d$-dimensional Einstein-Scalar-Maxwell theories. The properties of the analytic solutions are briefly discussed. 
  Since the background fields of the string low energy action are supposed to be the long range manifestation of a condensate of strings, the addition of world sheet actions to the low energy effective action needs some string theoretic explanation. In this paper we suggest that this may be understood, as being due to string loop effects. We first present arguments using an equation due to Tseytlin and then more rigorously in the particular case of type IIB theory by invoking the Fischler-Susskind effect. The argument provides further justification for ${\rm SL}(2,Z)$ duality between D-strings and F(fundamental)-strings. In an appendix we comment on recent attempts to relate the type IIA membrane to the 11-dimensional membrane. 
  A summary of results is presented, which provide exact description of the low-energy $4d$ $N=2$ and $N=4$ SUSY gauge theories in terms of $1d$ integrable systems. 
  Efforts have been made recently to reformulate traditional Kaluza-Klein theory by using a generalized definition of a higher-dimensional extended space-time. Both electromagnetism and gravity have been studied in this context. We review some of the models which have been proposed, with a special effort to keep the mathematical formalism to a very minimum. 
  The conformal limit over an anti-ferromagnetic vacuum of the fermionic spin  $\frac{1}{2}$ Calogero-Sutherland Model is derived by using the wedge product formalism. The space of states in the conformal limit is identified with the Fock space of two complex fermions, or, equivalently, with a tensor product of an irreducible level-1 module of $\slt$ and a Fock space module of the Heisenberg algebra.The Hamiltonian and the Yangian generators of the Calogero-Sutherland Model are represented in terms of $\slt$ currents and bosons. At special values of the coupling constant they give rise to the Hamiltonian and the Yangian generators of the conformal limit of the Haldane-Shastry Model acting in an irreducible level-1 module of $\slt$. At generic values of the coupling constant the space of states is decomposed into irreducible representations of the Yangian. 
  We obtain a family of type IIB superstring backgrounds involving Ramond-Ramond fields in ten dimensions starting from a heterotic string background with vanishing gauge fields. To this end the global $SL(2,R)$ symmetry of the type IIB equations of motion is implemented as a solution generating transformation. Using a geometrical analysis we show that the type IIB backgrounds obtained are solutions to all orders in $\alpha^{\prime}$. 
  The motion of a vortex-(anti)vortex pair is studied numerically in the framework of a dynamical Ginzburg-Landau model, relevant to the description of a superconductor or of an idealized bosonic plasma. It is shown that up to a fine "cyclotron" internal motion, also studied in detail, two vortices brought together, rotate around each other, while a vortex and an antivortex move in formation parallel to each other. The velocities of the vortices in both cases are measured to be in remarkable agreement with recent theoretical predictions, down to intervortex distances as small as their characteristic diameter. 
  We review a manifestly supersymmetric off-shell formulation of a wide class of torsionful $(4,4)$ $2D$ sigma models and their massive deformations in the harmonic superspace with a double set of $SU(2)$ harmonic variables. Sigma models with both commuting and non-commuting left and right complex structures are treated. 
  The structure of the asymptotic symmetry in the Poincar\'e gauge theory of gravity in 2d is clarified by using the Hamiltonian formalism. The improved form of the generator of the asymptotic symmetry is found for very general asymptotic behaviour of phase space variables, and the related conserved quantities are explicitly constructed. 
  String theory, if it describes nature, is probably strongly coupled. In light of recent developments in string duality, this means that the ``real world'' should correspond to a region of the classical moduli space which admits no weak coupling description. We exhibit, in the heterotic string, one such region of the moduli space, in which the coupling, $\lambda$, is large and the ``compactification radius'' scales as $\lambda^{1/3}$. We discuss some of the issues raised by the conjecture that the true vacuum lies in such a region. These include the question of coupling constant unification, and more generally the problem of what quantities one might hope to calculate and compare with experiment in such a picture. 
  The effective action of a Higgs theory should be gauge-invariant. However, the quantum and/or thermal contributions to the effective potential seem to be gauge-dependent, posing a problem for its physical interpretation. In this paper, we identify the source of the problem and argue that in a Higgs theory, perturbative contributions should be evaluated with the Higgs fields in the polar basis, not in the Cartesian basis. Formally, this observation can be made from the derivation of the Higgs theorem, which we provide. We show explicitly that, properly defined, the effective action for the Abelian Higgs theory is gauge invariant to all orders in perturbation expansion when evaluated in the covariant gauge in the polar basis. In particular, the effective potential is gauge invariant. We also show the equivalence between the calculations in the covariant gauge in the polar basis and the unitary gauge. These points are illustrated explicitly with the one-loop calculations of the effective action. With a field redefinition, we obtain the physical effective potential. The SU(2) non-Abelian case is also discussed. 
  Supersymmetric extreme dyonic black holes of toroidally compactified heterotic or type II string theory can be viewed as lower-dimensional images of solitonic strings wound around a compact dimension. We consider conformal sigma models which describe string configurations corresponding to various extreme dyonic black holes in four and five dimensions. These conformal models have regular short-distance region equivalent to a WZW theory with level proportional to magnetic charges. Arguments are presented suggesting a universal relation between the black hole entropy (area) and the statistical entropy of BPS-saturated oscillation states of solitonic string.[Extended version of a talk at the Workshop "Frontiers in Quantum Field Theory", in honor of the 60th birthday of K. Kikkawa (Osaka, Japan, 14-17 December 1995)] 
  Elimination of the fibre coordinate dependence from the connection form transformation rule for a bundle with a coset manifold standard fibre reduces the structure group. The nonlinear SU(4) action on an $S^7$ bundle is applied to the dimensional reduction of eleven-dimensional supergravity and ten-dimensional superstring theory to four dimensions. A principle, consistent with higher-dimensional superstring theory, is suggested to explain the types of gauge interactions that arise in the standard model based on the geometry of the internal symmetry spaces. It is shown why a Lie group structure is required for vector bosons in pure gauge theories and that the application of division algebras to force unification must begin with the fermions comprising the elementary particle multiplets of the standard model. A suggestion is made for establishing a mechanism for the cancellation of anomalies within this approach to superstring theory. 
  In the vicinity of points in Calabi-Yau moduli space where there are degenerating three-cycles the low energy effective action of type IIA string theory will contain significant contributions arising from membrane instantons that wrap around these three-cycles. We show that the world-volume description of these instantons is Chern-Simons theory. 
  We construct Landau-Ginzburg Lagrangians for minimal bosonic ($N=0$) $W$-models perturbed with the least relevant field, inspired by the theory of $N=2$ supersymmetric Landau-Ginzburg Lagrangians. They agree with the Lagrangians for unperturbed models previously found with Zamolodchikov's method. We briefly study their properties, e.g. the perturbation algebra and the soliton structure. We conclude that the known properties of $N=2$ solitons (BPS, lines in $W$ plane, etc.) hold as well. Hence, a connection with a generalized supersymmetric structure of minimal $W$-models is conjectured. 
  By adopting the standard definition of diffeomorphisms for a Regge surface we give an exact expression of the Liouville action both for the sphere and the torus topology in the discretized case. The results are obtained in a general way by choosing the unique self--adjoint extension of the Lichnerowicz operator satisfying the Riemann--Roch relation. We also give the explicit form of the integration measure for the conformal factor. For the sphere topology the theory is exactly invariant under the $SL(2,C)$ transformations, while for the torus topology we have exact translational and modular invariance. In the continuum limit the results flow into the well known expressions. 
  We construct the solution $\phi(t,{\bf x})$ of the quantum wave equation $\Box\phi + m^2\phi + \lambda:\!\!\phi^3\!\!: = 0$ as a bilinear form which can be expanded over Wick polynomials of the free $in$-field, and where $:\!\phi^3(t,{\bf x})\!: $ is defined as the normal ordered product with respect to the free $in$-field. The constructed solution is correctly defined as a bilinear form on $D_{\theta}\times D_{\theta}$, where $D_{\theta}$ is a dense linear subspace in the Fock space of the free $in$-field. On $D_{\theta}\times D_{\theta}$ the diagonal Wick symbol of this bilinear form satisfies the nonlinear classical wave equation. 
  The internal space-time symmetries of relativistic particles are dictated by Wigner's little groups. The $O(3)$-like little group for a massive particle at rest and the $E(2)$-like little group of a massless particle are two different manifestations of the same covariant little group. Likewise, the quark model and parton pictures are two different manifestations of the one covariant entity. 
  The exact solutions to quantum string and gauge field theories are discussed and their formulation in the framework of integrable systems is presented. In particular I consider in detail several examples of appearence of solutions to the first-order integrable equations of hydrodynamical type and stress that all known examples can be treated as partial solutions to the same problem in the theory of integrable systems. 
  We discuss the bosonized Schwinger model in light-cone quantization, using discretization as an infrared regulator. We consider both the light-cone Coulomb gauge, in which all gauge freedom can be removed and a physical Hilbert space employed, and the light-cone Weyl (temporal) gauge, in which the Hilbert space is unphysical and a Gauss law operator is used to select a physical subspace. We describe the different ways in which the theta vacuum is manifested depending on this choice of gauge, and compute the theta-dependence of the chiral condensate in each case. 
  We consider a variation of $O(N)$-symmetric vector models in which the vector components are Grassmann numbers. We show that these theories generate the same sort of random polymer models as the $O(N)$ vector models and that they lie in the same universality class in the large-$N$ limit. We explicitly construct the double-scaling limit of the theory and show that the genus expansion is an alternating Borel summable series that otherwise coincides with the topological expansion of the bosonic models. We also show how the fermionic nature of these models leads to an explicit solution even at finite-$N$ for the generating functions of the number of random polymer configurations. 
  Polynomial relations for generators of $su(2)$ Lie algebra in arbitrary representations are found. They generalize usual relation for Pauli operators in spin 1/2 case and permit to construct modified Holstein-Primakoff transformations in finite dimensional Fock spaces. The connection between $su(2)$ Lie algebra and q-oscillators with a root of unity q-parameter is considered. The meaning of the polynomial relations from the point of view of quantum mechanics on a sphere are discussed. 
  A dynamical non-abelian two-form potential gives masses to vector bosons via a topological coupling. Unlike in the abelian case, the two-form cannot be dualized to Goldstone bosons. Duality is restored by coupling a flat connection to the theory in a particular way, and the new action is then dualized to a non-linear sigma model. The presence of the flat connection is crucial, which saves the original mechanism of Higgs-free topological mass generation from being dualized to a sigma model. 
  We analyze $M$-theory compactified on $(K3\times S^1)/Z_2$ where the $Z_2$ changes the sign of the three form gauge field, acts on $S^1$ as a parity transformation and on K3 as an involution with eight fixed points preserving SU(2) holonomy. At a generic point in the moduli space the resulting theory has as its low energy limit N=1 supergravity theory in six dimensions with eight vector, nine tensor and twenty hypermultiplets. The gauge symmetry can be enhanced (e.g. to $E_8$) at special points in the moduli space. At other special points in the moduli space tensionless strings appear in the theory. 
  We demonstrate that in addition to the usual fourth-rank superfield $(W_{a b c d})$ which describes the on-shell theory, a spinor superfield $(J_\a )$ can be introduced into the 11D geometrical tensors with engineering dimensions less or equal to one in such a way to satisfy the Bianchi identities in superspace. The components arising from $J_\a$ are identified as some of the auxiliary fields required for a full off-shell formulation. Our result indicates that eleven dimensional supergravity does not have to be completely on-shell. The $\k\-$symmetry of the supermembrane action in the presence of our partial off-shell supergravity background is also confirmed. Our modifications to eleven-dimensional supergravity theory are thus likely relevant for M-theory. We suggest our proposal as a significant systematic off-shell generalization of eleven-dimensional supergravity theory. 
  The effective average actions for gauge theories and the associated nonperturbative evolution equations which govern their renormalization group flow are reviewed and various applications are described. As an example of a topological field theory, Chern-Simons theory is discussed in detail. 
  We find the Laughlin states of the electrons on the Poincare half-plane in different representations. In each case we show that there exist a quantum group $su_q(2)$ symmetry such that the Laughlin states are a representation of it. We calculate the corresponding filling factor by using the plasma analogy of the FQHE. 
  Extreme black holes with 1/8 of unbroken N=8 supersymmetry are characterized by the non-vanishing area of the horizon. The central charge matrix has four generic eigenvalues. The area is proportional to the square root of the invariant quartic form of $E_{7(7)}$. It vanishes in all cases when 1/4 or 1/2 of supersymmetry is unbroken. The supergravity non-renormalization theorem for the area of the horizon in N=8 case protects the unique U-duality invariant. 
  The generalized Killing equations for the configuration space of spinning particles (spinning space) are analysed. Simple solutions of the homogeneous part of these equations are expressed in terms of Killing-Yano tensors. The general results are applied to the case of the four-dimensional euclidean Taub-NUT manifold. 
  Starting from deformation quantization (star-products), the quantization problem of Nambu Mechanics is investigated. After considering some impossibilities and pushing some analogies with field quantization, a solution to the quantization problem is presented in the novel approach of Zariski quantization of fields (observables, functions, in this case polynomials). This quantization is based on the factorization over ${\Bbb R}$ of polynomials in several real variables. We quantize the infinite-dimensional algebra of fields generated by the polynomials by defining a deformation of this algebra which is Abelian, associative and distributive. This procedure is then adapted to derivatives (needed for the Nambu brackets), which ensures the validity of the Fundamental Identity of Nambu Mechanics also at the quantum level. Our construction is in fact more general than the particular case considered here: it can be utilized for quite general defining identities and for much more general star-products. 
  We consider the fermion-boson mapping in three dimensional space-time, in the Abelian case, from the current algebra point of view. We show that in a path-integral framework one can derive a general bosonization recipe leading, in the bosonic language, to the correct equal-time current commutators of the original free fermionic theory. 
  A method is suggested for the calculation of the DeWitt-Seeley-Gilkey (DWSG) coefficients for the operator $\sqrt{-\nabla^2 + V(x)}$ basing on a generalization of the pseudodifferential operator technique. The lowest DWSG coefficients for the operator $\sqrt{-\nabla^2} + V(x)$ are calculated by using the method proposed. It is shown that the method admits a generalization to the case of operators of the type $(-\nabla^2 + V(X))^{1/{\rm m}}$, where m is an arbitrary rational number. A more simple method is proposed for the calculation of the DWSG coefficients for the case of strictly positive operators under the sign of root. By using this method, it is shown that the problem of the calculation of the DWSG coefficients for such operators is exactly solvable. Namely, an explicit formula expressing the DWSG coefficients for operators with root through the DWSG coefficients for operators without root is deduced. 
  Lorentz boosts are squeeze transformations. While these transformations are similar to those in squeezed states of light, they are fundamentally different from both physical and mathematical points of view. The difference is illustrated in terms of two coupled harmonic oscillators, and in terms of the covariant harmonic oscillator formalism. 
  M-branes are related to theories on function spaces $\cal{A}$ involving M-linear non-commutative maps from $\cal{A} \times \cdots \times \cal{A}$ to $\cal{A}$. While the Lie-symmetry-algebra of volume preserving diffeomorphisms of $T^M$ cannot be deformed when M>2, the arising M-algebras naturally relate to Nambu's generalisation of Hamiltonian mechanics, e.g. by providing a representation of the canonical M-commutation relations, $[J_1,\cdots, J_M]=i\hbar$. Concerning multidimensional integrability, an important generalisation of Lax-pairs is given. 
  There are many physically interesting superconformal gauge theories in four dimensions. In this talk I discuss a common phenomenon in these theories: the existence of continuous families of infrared fixed points. Well-known examples include finite ${\cal N}=4$ and ${\cal N}=2$ supersymmetric theories; many finite ${\cal N}=1$ examples are known also. These theories are a subset of a much larger class, whose existence can easily be established and understood using the algebraic methods explained here. A relation between the ${\cal N}=1$ duality of Seiberg and duality in finite ${\cal N}=2$ theories is found using this approach, giving further evidence for the former. This talk is based on work with Robert Leigh (hep-th/9503121). 
  We construct compact examples of D-manifolds for type IIB strings. The construction has a natural interpretation in terms of compactification of a 12 dimensional `F-theory'. We provide evidence for a more natural reformulation of type IIB theory in terms of F-theory. Compactification of M-theory on a manifold $K$ which admits elliptic fibration is equivalent to compactification of F-theory on $K\times S^1$. A large class of $N=1$ theories in 6 dimensions are obtained by compactification of F-theory on Calabi-Yau threefolds. A class of phenomenologically promising compactifications of F-theory is on $Spin(7)$ holonomy manifolds down to 4 dimensions. This may provide a concrete realization of Witten's proposal for solving the cosmological constant problem in four dimensions. 
  An expression for energy in string-inspired dilaton gravity is obtained in canonical approach. 
  We begin with personal notes describing the atmosphere of "Bogoliubov renormalization group" birth. Then we expose the history of RG discovery in the QFT and of the RG method devising in the mid-fifties. The third part is devoted to proliferation of RG ideas into diverse parts of theoretical physics. We conclude with discussing the perspective of RG method further development and its application in mathematical physics. 
  The statistics of meanders is studied in connection with the Temperley-Lieb algebra. Each (multi-component) meander corresponds to a pair of reduced elements of the algebra. The assignment of a weight $q$ per connected component of meander translates into a bilinear form on the algebra, with a Gram matrix encoding the fine structure of meander numbers. Here, we calculate the associated Gram determinant as a function of $q$, and make use of the orthogonalization process to derive alternative expressions for meander numbers as sums over correlated random walks. 
  We study the space-time CPT properties of string theories formulated in a flat Minkowski background of even dimension. We define CPT as a world-sheet transformation acting on the vertex operators and we prove the CPT invariance of the string $S$-matrix elements. Some related issues, including the connection between spin and statistics of physical string states, are also considered. 
  We construct QED_2 with mass and flavor and an extra Thirring term. The vacuum expectation values are carefully decomposed into clustering states using the U(1)-axial symmetry of the considered operators and a limiting procedure. The properties of the emerging expectation functional are compared to the proposed theta-vacuum of QCD. The massive theory is bosonized to a generalized Sine-Gordon model (GSG). The structure of the vacuum of QED_2 manifests itself in symmetry properties of the GSG. We study the U(1)-problem and derive a Witten-Veneziano-type formula for the masses of the pseudoscalars determined from a semiclassical approximation. 
  The model of p Ising spins coupled to 2d gravity, in the form of a sum over planar phi-cubed graphs, is studied and in particular the two-point and spin-spin correlation functions are considered. We first solve a toy model in which only a partial summation over spin configurations is performed and, using a modified geodesic distance, various correlation functions are determined. The two-point function has a diverging length scale associated with it. The critical exponents are calculated and it is shown that all the standard scaling relations apply. Next the full model is studied, in which all spin configurations are included. Many of the considerations for the toy model apply for the full model, which also has a diverging geometric correlation length associated with the transition to a branched polymer phase. Using a transfer function we show that the two-point and spin-spin correlation functions decay exponentially with distance. Finally, by assuming various scaling relations, we make a prediction for the critical exponents at the transition between the magnetized and branched polymer phases in the full model. 
  We evaluate Polyakov loops and string tension in two-dimensional QED with both massless and massive $N$-flavor fermions at zero and finite temperature. External charges, or external electric fields, induce phases in fermion masses and shift the value of the vacuum angle parameter $\theta$, which in turn alters the chiral condensate. In particular, in the presence of two sources of opposite charges, $q$ and $-q$, the shift in $\theta$ is $2\pi(q/e)$ independent of $N$. The string tension has a cusp singularity at $\theta=\pm\pi$ for $N\ge 2$ and is proportional to $m^{2N/(N+1)}$ at $T=0$. 
  A new orientifold of Type-IIB theory on $K3$ is constructed that has $N=1$ supersymmetry in six dimensions. The orientifold symmetry consists of a $Z_2$ involution of $K3$ combined with orientation-reversal on the worldsheet. The closed-string sector in the resulting theory contains nine tensor multiplets and twelve neutral hypermultiplets in addition to the gravity multiplet, and is anomaly-free by itself. The open-string sector contains only 5-branes and gives rise to maximal gauge groups $SO(16)$ or $U(8)\times U(8)$ at different points in the moduli space. Anomalies are canceled by a generalization of the Green-Schwarz mechanism that involves more than one tensor multiplets. 
  We study ${\cal N}=1$ supersymmetric $Spin(10)$ chiral gauge theories with a single spinor representation and $N$ vector representations. We present a dual description in terms of an ${\cal N}=1$ supersymmetric $SU(N-5)$ chiral gauge theory with a symmetric tensor, one fundamental and $N$ antifundamental representations. The $Spin(10)$ theory with $N=0$ breaks supersymmetry at strong coupling; we study how this arises at weak coupling in the dual theory, which is a spontaneously broken gauge theory. Also, we recover various known dualities, find new dual pairs and generate new examples of dynamical supersymmetry breaking. 
  I introduce a class of string constructions based on asymmetric orbifolds leading to level two models. In particular, I derive in detail various models with gauge groups $E_6$ and SO(10), including a four generation $E_6$ model with two adjoint representations. The occurrence of multiple adjoint representations is a generic feature of the construction. In the course of describing this approach, I will address the problem of twist phases in higher twisted sectors of asymmetric orbifolds. 
  A new tool for the investigation of $2+1$ dimensional gravity is proposed. It is shown that in a stationary $2+1$ dimensional spacetime, the eigenvectors of the covariant derivative of the timelike Killing vector form a rigid structure, the {\it principal Killing triad}. Two of the triad vectors are null, and in many respects they play the role similar to the principal null directions in the algebraically special 4-D spacetimes. It is demonstrated that the principal Killing triad can be efficiently used for classification and study of stationary $2+1$ spacetimes.    One of the most interesting applications is a study of minimal surfaces in a stationary spacetime. A {\it principal Killing surface} is defined as a surface formed by Killing trajectories passing through a null ray, which is tangent to one of the null vectors of the principal Killing triad. We prove that a principal Killing surface is minimal if and only if the corresponding null vector is geodesic. Furthermore, we prove that if the $2+1$ dimensional spacetime contains a static limit, then the only regular stationary timelike minimal 2-surfaces that cross the static limit, are the minimal principal Killing surfaces.    A timelike minimal surface is a solution to the Nambu-Goto equations of motion and hence it describes a cosmic string configuration. A stationary string interacting with a $2+1$ dimensional rotating black hole is discussed in detail. 
  We review how the continuous symmetry can support a soliton inside a high-temperature bubble at the time of its nucleation. This solitonic island in disoriented phase remains stable during the growth of bubbles before their collision. 
  We study $N=1$ supersymmetric $SO(10)$ gauge theory with a field in the spinorial representaition and $N_f$ ($\leq8$) fields in the defining representation. It is shown that this theory for $N_f=7,8$ has a dual description, which is $N=1$ supersymmetric $SU(N_f-5)$ gauge theory. Its matter content for $N_f=7$ is different {}from the one for $N_f=8$; for $N_f=7$, it contains $8$ fields in the anti-fundamental representation. For $N_f=8$, a rank-$2$ symmetric tensor and one field in the fundamental representation appears in addition to them. This duality connects along the flat direction to the duality between chiral and vector gauge theory found by Pouliot. 
  Geometric optics is analysed using the techniques of Presymplectic Geometry. We obtain the symplectic structure of the space of light rays in a medium of a non constant refractive index by reduction from a presymplectic structure, and using adapted coordinates, we find Darboux coordinates. The theory is illustrated with some examples and we point out some simple physical applications 
  The tools of presymplectic geometry are used to study light rays trajectories in anisotropic media. 
  The two-dimensional manifestly locally supersymmetric actions describing the N=2 and N=4 extended super-Liouville theory coupled to the N=2 and N=4 conformal supergravity, respectively, are constructed in superspace. It is shown that the N=4 super-Liouville multiplet is described by the improved twisted-II scalar multiplet TM-II, whose kinetic terms are given by the SU(2)xU(1) WZNW model. 
  A systematic loop expansion is formulated in terms of full propagators and vertices. It is based on an expansion of the general solution of an exact non-perturbative flow equation. 
  The Schr\"odinger-type formalism of the Klein-Gordon quantum mechanics is adapted for the case of the $SL(2,\R)$ spacetime. The free particle case is solved, the results of a recent work are reproduced while all the other, topologically nontrivial solutions and the antiparticle modes are also found, and a deeper insight into the physical content of the theory is given. 
  We show that gauge invariant composites in the fermionic realization of $SU(N)_1$ conformal field theory explicitly exhibit the holomorphic factorization of the corresponding WZW primaries. In the $SU(2)_1$ case we show that the holomorphic sector realizes the spinon $Y(sl_2)$ algebra, thus allowing the classification of the chiral Fock space in terms of semionic quasi-particle excitations created by the composite fermions. 
  The screening of an applied magnetic field in the charged anyon fluid at finite density and temperature is investigated. For densities typical of high-temperature superconducting materials we find that the anyon fluid exhibits a superconducting behavior at any temperature. The total Meissner screening is characterized by two penetration lengths corresponding to two short-range eigenmodes of propagation within the anyon fluid. 
  Strominger and Vafa have used D-brane technology to identify and precisely count the degenerate quantum states responsible for the entropy of certain extremal, BPS-saturated black holes. Here we give a Type-II D-brane description of a class of extremal {\it and} non-extremal five-dimensional Reissner-Nordstr\"om solutions and identify a corresponding set of degenerate D-brane configurations. We use this information to do a string theory calculation of the entropy, radiation rate and ``Hawking'' temperature. The results agree perfectly with standard Hawking results for the corresponding nearly extremal Reissner-Nordstr\"om black holes. Although these calculations suffer from open-string strong coupling problems, we give some reasons to believe that they are nonetheless qualitatively reliable. In this optimistic scenario there would be no ``information loss'' in black hole quantum evolution. 
  The present paper proposes a basis for new gravitational mechanics. The problem of finding the spectrum of mass-energy is reduced to a new kind of eigenvalue problem which intrinsically contains the fundamental length ${\it l} = \sqrt{Gh\over c^3}$.The new wave equations are difference wave equations in one representation of the theory. 
  This is a pedagogical review article surveying the various approaches towards understanding gauge coupling unification within string theory. As is well known, one of the major problems confronting string phenomenology has been an apparent discrepancy between the scale of gauge coupling unification predicted within string theory, and the unification scale expected within the framework of the Minimal Supersymmetric Standard Model (MSSM). In this article, I provide an overview of the different approaches that have been taken in recent years towards reconciling these two scales, and outline some of the major recent developments in each. These approaches include   1) string GUT models;   2) higher affine levels and non-standard hypercharge normalizations;   3) heavy string threshold corrections;   4) light supersymmetric thresholds;   5) effects from intermediate-scale gauge and matter structure beyond the MSSM;   6) strings without supersymmetry; and   7) strings at strong coupling. 
  We examine Bose-Einstein condensation as a form of symmetry breaking in the specific model of the Einstein static universe. We show that symmetry breaking never occursin the sense that the chemical potential $\mu$ never reaches its critical value.This leads us to some statements about spaces of finite volume in general. In an appendix we clarify the relationship between the standard statistical mechanical approaches and the field theory method using zeta functions. 
  The conventional path integral expression for the Yang-Mills transition amplitude with flat measure and gauge-fixing built in via the Faddeev-Popov method has been claimed to fall short of guaranteeing gauge invariance in the non-perturbative regime. We show, however, that it yields the gauge invariant partition function where the projection onto gauge invariant wave functions is explicitly performed by integrating over the compact gauge group. In a variant of maximal Abelian gauge the Haar measure arises in the conventional Yang-Mills path integral from the Faddeev-Popov determinant. 
  Pseudoclassical supersymmetric model to describe massive particles with higher spins (integer and half-integer) in $2+1$ dimensions is proposed. The quantization of the model leads to the minimal (with only one polarization state) quantum theory. In particular, the Bargmann-Wigner type equations for higher spins arise in course of the canonical quantization. The cases of spin one-half and one are considered in detail. Here one gets Dirac particles and Chern-Simons particles respectively. A relation with the field theory is discussed. On the basis of the model proposed, and using dimensional reduction considerations, a model to describe Weyl particles with higher spins in $3+1$ dimensions is constructed. 
  The target space theory of the N=(2,1) heterotic string may be interpreted as a theory of gravity coupled to matter in either $1+1$ or $2+1$ dimensions. Among the target space theories in $1+1$ dimensions are the bosonic, type II, and heterotic string world sheet field theories in a physical gauge. The $2+1$ dimensional version describes a consistent quantum theory of supermembranes in $10+1$ dimensions. The unifying framework for all of these vacua is a theory of $2+2$ dimensional self-dual geometries embedded in $10+2$ dimensions. There are also indications that the N=(2,1) string describes the strong coupling dynamics of compactifications of critical string theories to two dimensions, and may lead to insights about the fundamental degrees of freedom of the theory. 
  It has been shown how on-shell forward scattering amplitudes (the ``Barton expansion'') and quantum mechanical path integral (QMPI) can both be used to compute temperature dependent effects in thermal field theory. We demonstrate the equivalence of these two approaches and then apply the QMPI to compute the high temperature expansion for the four-point function in QED, obtaining results consistent with those previously obtained from the Barton expansion. 
  A six-dimensional black string is considered and its Bekenstein-Hawking entropy computed. It is shown that to leading order above extremality, this entropy precisely counts the number of string states with the given energy and charges. This identification implies that Hawking decay of the near-extremal black string can be analyzed in string perturbation theory and is perturbatively unitary. 
  This is a series of remedial lectures on open and unoriented strings for the heterotic string generation. The particular focus is on the interesting features that arise under T-duality---D-branes and orientifolds. The final lecture discusses the application to string duality. There will be no puns. Lectures presented by J. P. at the ITP from Nov. 16 to Dec. 5, 1995. References updated through Jan. 25, 1996. 
  The influence of spatial geometry on the vacuum polarization in 2+1-dimensional spinor electrodynamics is investigated. The vacuum angular momentum induced by an external static magnetic field is found to depend on global geometric surface characteristics connected with curvature. The relevance of the results obtained for the fermion number fractionization is discussed. 
  A conformally invariant theory of Majorana fermions in 2<d<4 with O(N) symmetry is studied using Operator Product Expansions and consistency relations based on the cancellation of shadow singularities. The critical coupling G_{*} of the theory is calculated to leading order in 1/N. This value is then used to reproduce the O(1/N) correction for the anomalous dimension of the fermion field as evidence for the validity of our approach to conformal field theory in d>2. 
  We review some recent results on the calculation of renormalization constants in Yang-Mills theory using open bosonic strings. The technology of string amplitudes, supplemented with an appropriate continuation off the mass shell, can be used to compute the ultraviolet divergences of dimensionally regularized gauge theories. The results show that the infinite tension limit of string amplitudes corresponds to the background field method in field theory. (Proceedings of the Workshop ``Strings, Gravity and Physics at the Planck scale'', Erice (Italy), September 1995. Preprint DFTT 82/95) 
  We review the application of bosonic string techniques to the calculation of renormalization constants and effective actions in Yang-Mills theory. We display the multiloop string formulas needed to compute Yang-Mills amplitudes, and we discuss how the renormalizations of proper vertices can be extracted in the field theory limit. We show how string techniques lead to the background field method in field theory, and indicate how the gauge invariance of the multiloop effective action can be inferred form the string formalism. (Proceedings of the 29th International Symposium on the Theory of Elementary Particles, Buckow (Germany), Aug.-Sept. 1995. Preprint DFTT 04/96) 
  For extremal charged black holes, the thermodynamic entropy is proportional not to the area but to the mass or charges. This is demonstrated here for dyonic extremal black hole solutions of string theory. It is pointed out that these solutions have zero classical action although the area is nonzero. By combining the general form of the entropy allowed by thermodynamics with recent observations in the literature it is possible to fix the entropy almost completely. 
  We review a construction, using the harmonic space method, of solutions to the superfield equations of motion for N-extended self-dual supergravity theories. A superspace gauge condition suitable for the performance of a component analysis is discussed. 
  We examine T-duality transformations for supersymmetric strings with target space geometry with compact abelian isometries. We consider the partition function of these models and we show that although T-duality is not a symmetry, due to an anomaly, it relates type IIA to type IIB strings. In this way we extend the corresponding result for toroidal compactification to the general case of non-trivial backgrounds with abelian isometries and for world sheets of any genera. 
  We establish the consistency of duality transformations for generic systems of $N=2$ vector supermultiplets in the presence of a chiral background field. This is relevant, for instance, when dealing with spurion fields or when considering higher-derivative couplings of vector multiplets to supergravity. We point out that under duality most quantities do not transform as functions. With few exceptions, true functions are nonholomorphic, even though the duality transformations themselves are holomorphic in nature. 
  String duality suggests a fascinating juxtoposition of world-volume and target-space dynamics. This is particularly apparent in the $D$-brane description of stringy solitons that forms a major focus of this article (which is {\it not} intended to be a comprehensive review of this extensive subject). The article is divided into four sections:     1. The oligarchy of string world-sheets     2. $p$-branes and world-volumes     3. World-sheets for world-volumes     4. Boundary states, $D$-branes and space-time supersymmetry   [This article is based on a talk presented at the CERN Workshop on String Duality (December, 1995) and the published version of a talk at the Buckow Symposium (September, 1995).] 
  Using the topological membrane approach to string theory, we suggest a geometric origin for the heterotic string. We show how different membrane boundary conditions lead to different string theories. We discuss the construction of closed oriented strings and superstrings, and demonstrate how the heterotic construction naturally arises from a specific choice of boundary conditions on the left and right boundaries of a cylindrical topological membrane. 
  Vacuum expectation values of products of local bilinears $\bar\psi\psi$ are computed in massless $QED_2$ at finite density. It is shown that chiral condensates exhibit an oscillatory inhomogeneous behaviour depending on the chemical potential. The use of a path-integral approach clarifies the connection of this phenomenon with the topological structure of the theory. 
  D-brane actions depend on a world-volume abelian vector field and are described by Born-Infeld-type actions. We consider the vector field duality transformations of these actions. Like the usual 2d scalar duality rotations of isometric string coordinates imply target space T-duality, this vector duality is intimately connected with SL(2,Z)-symmetry of type IIB superstring theory. We find that in parallel with generalised 4-dimensional Born-Infeld action, the action of 3-brane of type IIB theory is SL(2,Z) self-dual. This indicates that 3-brane should play a special role in type IIB theory and also suggests a possibility of its 12-dimensional reformulation. 
  We obtain a new class of spinning charged extremal black holes in five dimensions, considered both as classical configurations and in the Dirichlet(D)--brane representation. The degeneracy of states is computed from the D--brane side and the entropy agrees perfectly with that obtained from the black hole side. 
  The classical equation of motion of a charged point particle, including its radiation reaction, is described by the Lorentz-Dirac equation. We found a new class of solutions that describe tunneling (in a completely classical context!). For nonrelativistic electrons and a square barrier, the solution is elementary and explicit. We show the persistance of the solution for smoother potentials. For a large range of initial velocities, initial conditions may leave a (discrete) ambiguity on the resulting motion. 
  Massless black holes can be understood as bound states of a (positive mass) extreme a=\sqrt{3} black hole and a singular object with opposite (i.e. negative) mass with vanishing ADM (total) mass but non-vanishing gravitational field. Supersymmetric balance of forces is crucial for the existence of this kind of bound states and explains why the system does not move at the speed of light. We also explain how supersymmetry allows for negative mass as long as it is never isolated but in bound states of total non-negative mass. 
  Additional non-isospectral symmetries are formulated for the constrained Kadomtsev-Petviashvili (\cKP) integrable hierarchies. The problem of compatibility of additional symmetries with the underlying constraints is solved explicitly for the Virasoro part of the additional symmetry through appropriate modification of the standard additional-symmetry flows for the general (unconstrained) KP hierarchy. We also discuss the special case of \cKP --truncated KP hierarchies, obtained as Darboux-B\"{a}cklund orbits of initial purely differential Lax operators. The latter give rise to Toda-lattice-like structures relevant for discrete (multi-)matrix models. Our construction establishes the condition for commutativity of the additional-symmetry flows with the discrete Darboux-B\"{a}cklund transformations of \cKP hierarchies leading to a new derivation of the string-equation constraint in matrix models. 
  A calculation of the renormalization group improved effective potential for the gauged U(N) vector model, coupled to $N_f$ fermions in the fundamental representation, computed to leading order in 1/N, all orders in the scalar self-coupling $\lambda$, and lowest order in gauge coupling $g^2$, with $N_f$ of order $N$, is presented. It is shown that the theory has two phases, one of which is asymptotically free, and the other not, where the asymptotically free phase occurs if $0 < \lambda /g^2 < {4/3} (\frac{N_f}{N} - 1)$, and $\frac{N_f}{N} < {11/2}$. In the asymptotically free phase, the effective potential behaves qualitatively like the tree-level potential. In the other phase, the theory exhibits all the difficulties of the ungauged $(g^2 = 0)$ vector model. Therefore the theory appears to be consistent (only) in the asymptotically free phase. 
  In a certain strong coupling limit, compactification of the $E_8\times E_8$ heterotic string on a Calabi-Yau manifold $X$ can be described by an eleven-dimensional theory compactified on $X\times \S^1/\Z_2$. In this limit, the usual relations among low energy gauge couplings hold, but the usual (problematic) prediction for Newton's constant does not. In this paper, the equations for unbroken supersymmetry are expanded to the first non-trivial order, near this limit, verifying the consistency of the description and showing how, in some cases, if one tries to make Newton's constant too small, strong coupling develops in one of the two $E_8$'s. The lower bound on Newton's constant (beyond which strong coupling develops) is estimated and is relatively close to the actual value. 
  We formulate boundary conditions for an open membrane that ends on the fivebrane of {\cal M}-theory. We show that the dynamics of the eleven-dimensional fivebrane can be obtained from the quantization of a ``small membrane'' that is confined to a single fivebrane and which moves with the speed of light. This shows that the eleven-dimensional fivebrane has an interpretation as a $D$-brane of an open supermembrane as has recently been proposed by Strominger and Townsend. We briefly discuss the boundary dynamics of an infinitely extended planar membrane that is stretched between two parallel fivebranes 
  We show via use of the RADIO technique that an off-shell (4,0) version of the hypermultiplet, in the form first proposed by Fayet, exists and contains 28 - 28 component fields. The off-shell structure uncovered is found to include a chiral truncation of the ``generalized 2D, N = 4 tensor multiplet formalism'' proposed by Ketov. The (4,0) theory is extended to an off-shell 56 - 56 component field (4,4) theory with the addition of a minimal (4,0) minus spinor multiplet together with (4,0) auxiliary multiplets. We propose that our final result gives a solution to a twenty year-old 2D supersymmetry problem in the physics literature. 
  We calculate the one-instanton contribution to the prepotential in $N=2$ supersymmetric $SU(N_c)$ Yang-Mills theory from the microscopic viewpoint. We find that the holomorphy argument simplifies the group integrations of the instanton configurations. For $N_{c}=3$, the result agrees with the exact solution. 
  By using the bosonization technique, we derive an integral representation for multi-point Local Hight Probabilities for the Andrews-Baxter-Forrester model in the regime III. We argue that the dynamical symmetry of the model is provided by the deformed Virasoro algebra. 
  We show how spontaneous supersymmetry breaking in the vacuum state of higher-derivative supergravity is transmitted, as explicit soft supersymmetry-breaking terms, to the effective Lagrangian of the standard electroweak model. The general structure of the soft supersymmetry breaking terms is presented and a new scenario for understanding the gauge hierarchy problem, based on the functional form of these terms, is discussed. 
  The relationship between the quasi-exactly solvable problems and W-algebras is revealed. This relationship enabled one to formulate a new general method for building multi-dimensional and multi-channel exactly and quasi-exactly solvable models with hermitian hamiltonians. The method is based on the use of multi-parameter spectral differential equations constructable from generators of finite-dimensional representations of simple Lie algebras and from generators of the associated W-algebras. It is shown that algebras B(n), C(n+1), D(2n+2) with n>1 and also algebras A(1), G(2), F(4), E(7) and E(8) always lead to models with hermitian hamiltonians. The situation with the remaining algebras A(n), D(2n+3) with n>1 and E(6) is still unclear. 
  The Born--Infeld-like effective world-volume theory of a single 3-brane is deduced from a manifestly space-time supersymmetric description of the corresponding $D$-brane.  This is shown to be invariant under $SL(2,R)$ transformations that act on the abelian gauge field as well as the bulk fields. The effective theory of two nearby parallel three-branes involves massive world-volume supermultiplets which transform under $SL(2,Z)$ into the dyonic solitons of four-dimensional $N=4$ spontaneously broken $SU(2)$ Yang--Mills theory. 
  A geometric formal method for perturbatively expanding functional integrals arising in quantum gauge theories is described when the spacetime is a compact riemannian manifold without boundary. This involves a refined version of the Faddeev-Popov procedure using the covariant background field gauge-fixing condition with background gauge field chosen to be a general critical point for the action functional (i.e. a classical solution). The refinement takes into account the gauge-fixing ambiguities coming from gauge transformations which leave the critical point unchanged, resulting in the absence of infrared divergences when the critical point is isolated modulo gauge transformations. The procedure can be carried out using only the subgroup of gauge transformations which are topologically trivial, possibly avoiding the usual problems which arise due to gauge-fixing ambiguities. For Chern-Simons gauge theory the method enables the partition function to be perturbatively expanded for a number of simple spacetime manifolds such as $S^3$ and lens spaces, and the expansions are shown to be formally independent of the metric used in the gauge-fixing. 
  We give a brief account of two recent applications of the harmonic superspace method: (i) an off-shell description of torsionful $(4,4)$ supersymmetric $2D$ sigma models in the framework of $SU(2)\times SU(2)$ harmonic superspace and (ii) the harmonic superspace formulation of ``small'' $N=4$ $SU(2)$ superconformal algebra and the related super KdV hierarchy. 
  A path integral evaluation of the Green's function for the hydrogen atom initiated by Duru and Kleinert is studied by recognizing it as a special case of the general treatment of the separable Hamiltonian of Liouville-type. The basic dynamical principle involved is identified as the Jacobi's principle of least action for given energy which is reparametrization invariant, and thus the appearance of a gauge freedom is naturally understood. The separation of variables in operator formalism corresponds to a choice of gauge in path integral, and the Green's function is shown to be gauge independent if the operator ordering is properly taken into account. Unlike the conventional Feynman path integral,which deals with a space-time picture of particle motion, the path integral on the basis of the Jacobi's principle sums over orbits in space. We illustrate these properties by evaluating an exact path integral of the Green's function for the hydrogen atom in parabolic coordinates, and thus avoiding the use of the Kustaanheimo-Stiefel transformation. In the present formulation , the Hamiltonian for Stark effect is converted to the one for anharmonic oscillators with an unstable quartic coupling. We also study the hydrogen atom path integral from a view point of one-dimensional quantum gravity coupled to matter fields representing the electron coordinates. A simple BRST analysis of the problem with an evaluation of Weyl anomaly is presented . 
  In this note, we propose an exegesis of the Maxwell equations for electromagnetism. We begin with an analogy between the homogeneous Maxwell equations and the equations needed to describe the vorticity field of an incompressible inviscid fluid. We suggest that the inhomogeneous equations are analogous to two equations valid in turbulent hydrodynamics. Once the analogy is completed we give the mechanical analogue of the Poynting vector and we explain the influence of a long solenoid on the motion of a charged particle. 
  We carefully study the global structure of the solution of the $N=2$ supersymmetric pure Yang-Mills theory with gauge group $SU(2)$ obtained by Seiberg and Witten. We exploit its ${\bf Z}_2$-symmetry and describe the curve in moduli space where BPS states can become unstable, separating the strong-coupling from the weak-coupling region. This allows us to obtain the spectrum of stable BPS states in the strong-coupling region: we prove that only the two particles responsible for the singularities of the solution (the magnetic monopole and the dyon of unit electric charge) are present in this region. Our method also permits us to very easily obtain the well-known weak-coupling spectrum, without using semi-classical methods. We discuss how the BPS states disintegrate when crossing the border from the weak to the strong-coupling region. 
  We investigate the properties of a dressed electron which reduces, in a particular class of gauges, to the usual fermion. A one loop calculation of the propagator is presented. We show explicitly that an infra-red finite, multiplicative, mass shell renormalisation is possible for this dressed electron, or, equivalently, for the usual fermion in the abovementioned gauges. The results are in complete accord with previous conjectures. 
  We analyze the quantum cosmology of one--loop string effective models which exhibit an $O(d,d)$ symmetry. It is shown that due to the large symmetry of these models the Wheeler--de Witt equation can completely be solved. As a result, we find a basis of solutions with well defined transformation properties under $O(d,d)$ and under scale factor duality in particular. The general results are explicitly applied to 2--dimensional target spaces while some aspects of higher dimensional cases are also discussed. Moreover, a semiclassical wave function for the 2-dimensional black hole is constructed as a superposition of our basis. 
  We continue studies on quantum field theories on noncommutative geometric spaces, focusing on classes of noncommutative geometries which imply ultraviolet and infrared modifications in the form of nonzero minimal uncertainties in positions and momenta. The case of the ultraviolet modified uncertainty relation which has appeared from string theory and quantum gravity is covered. The example of euclidean $\phi^4$-theory is studied in detail and in this example we can now show ultraviolet and infrared regularisation of all graphs. 
  We obtain effective potential of $O(N)$-symmetric $\phi^4$ theory for large $N$ starting with a finite lattice system and taking the thermodynamic limit with great care. In the thermodynamic limit, it is globally real-valued and convex in both the symmetric and the broken phases. In particular, it has a flat bottom in the broken phase. Taking the continuum limit, we discuss renormalization effects to the flat bottom and exhibit the effective potential of the continuum theory in three and four dimensions.On the other hand the effective potential is nonconvex in a finite lattice system. Our numerical study shows that the barrier height of the effective potential flattens as a linear size of the system becomes large. It decreases obeying power law and the exponent is about $-2$. The result is clearly understood from dominance of configurations with slowly-rotating field in one direction. 
  We write an O(d,d)-covariant Wheeler-De Witt equation in the ($d^2+1$)-dimensional minisuperspace of low-energy cosmological string backgrounds. We discuss explicit examples of transitions between two duality-related cosmological phases, and we find a finite quantum transition probability even when the two phases are classically separated by a curvature singularity. This quantum approach is completely free from operator ordering ambiguities as a consequence of the duality symmetries of the string effective action. 
  The Seiberg--Witten equations, when dimensionally reduced to $\bf R^{2}\mit$, naturally yield the Liouville equation, whose solutions are parametrized by an arbitrary analytic function $g(z)$. The magnetic flux $\Phi$ is the integral of a singular Kaehler form involving $g(z)$; for an appropriate choice of $g(z)$ , $N$ coaxial or separated vortex configurations with $\Phi=\frac{2\pi N}{e}$ are obtained when the integral is regularized. The regularized connection in the $\bf R^{1}\mit$ case coincides with the kink solution of $\varphi^{4}$ theory. 
  We consider zeta functions and heat-kernel expansions on the bounded, generalized cone in arbitrary dimensions using an improved calculational technique. The specific case of a global monopole is analysed in detail and some restrictions thereby placed on the $A_{5/2}$ coefficient. The computation of functional determinants is also addressed. General formulas are given and known results are incidentally, and rapidly, reproduced. 
  A B\"acklund transformation yielding the static non-relativistic Chern-Simons vortices of Jackiw and Pi is presented. 
  The properties of a generalized version of the Borel Transform in infrared unstable theories with dynamical mass generation are studied. The reconstruction of the nonperturbative structure is unambiguous in this version. Various methods for extracting the singularity structure of the Borel Transform for lattice formulations of such theories are explored, and illustrated explicitly with the O(N) sigma model. The status of the first infrared renormalon in QCD is discussed. The feasibility of a proposed technique for analytically continuing from the left hand Borel plane (where nonperturbative information is available via simulation of lattice field theory) to the positive real axis is examined using the sigma model. 
  We study bosonization ambiguities in two dimensional quantum electrodynamics in the presence and in the absence of topologically charged gauge fields. The computation of fermionic correlation functions suggests that ambiguities may be absent in nontrivial topologies, provided that we do not allow changes of sector as we evaluate functional integrals. This would remove an infinite arbitrariness from the theory. In the case of trivial topologies, we find upper and lower bounds for the Jackiw-Rajaraman parameter, corresponding to the limiting cases of regularizations which preserve gauge or chiral symmetry. 
  We consider the problem of constructing the fully gauged effective action in $2n$-dimensional space-time for Nambu-Goldstone bosons valued in a homogeneous space $G/H$, with the requirement that the action be a solution of the anomalous Ward identity and be invariant under the gauge transformations of $H$. We show that this can be done whenever the homotopy group $\pi_{2n}(G/H)$ is trivial, $G/H$ is reductive and $H$ is embedded in $G$ so as to be anomaly free, in particular if $H$ is an anomaly safe group. We construct the necessary generalization of the Bardeen counterterm and give explicit forms for the anomaly and the effective action. When $G/H$ is a symmetric space the counterterm and the anomaly decompose into a parity even and a parity odd part. In this case, for the parity even part of the action, one does not need the anomaly free embedding of $H$. 
  We reformulate the Hamiltonian approach to lattice gauge theories such that, at the classical level, the gauge group does not act canonically, but instead as a Poisson-Lie group. At the quantum level, it then gets promoted to a quantum group gauge symmetry. The theory depends on two parameters - the deformation parameter $\lambda$ and the lattice spacing $a$. We show that the system of Kogut and Susskind is recovered when $\lambda \rightarrow 0$, while QCD is recovered in the continuum limit (for any $\lambda$). We thus have the possibility of having a two parameter regularization of QCD. 
  The scattering of free particles constrained to move on a cylindrically symmetric curved surface is studied. The nontrivial geometry of the space contributes to the scattering cross section through the kinetic as well as a possible scalar curvature term in the quantum Hamiltonian. The coefficient of the latter term is known to be related to the factor ordering problem in curved space quantization. Hence, in principle, the scattering data may be used to provide an experimental resolution of the theoretical factor ordering ambiguity. To demonstrate the sensitivity required of such an experimental setup, the effect of a localized magnetic field in the scattering process is also analyzed. 
  In a Wheeler-de Witt approach to quantum string cosmology, the present state of the Universe arises from the scattering and reflection of the wave function representing the initial string vacuum in superspace. This scenario is described and compared with the more conventional quantum cosmology picture, in which the birth of the Universe is represented as a process of tunnelling "from nothing" in superspace. 
  We consider $E_8\times E_8$ heterotic compactifications on $K3$ and $K3\times T^2$. The idea of heterotic/heterotic duality in $D=6$ has difficulties for generic compactifications since for large dilaton values some gauge groups acquire negative kinetic terms. Recently Duff, Minasian and Witten (DMW) suggested a solution to this problem which only works if the compactification is performed assuming the presence of symmetric gauge embeddings on both $E_8$'s. We consider an alternative in which asymmetric embeddings are possible and the wrong sign of kinetic terms for large dilaton value is a signal of spontaneous symmetry breaking. Upon further toroidal compactification to $D=4$, we find that the duals in the DMW case correspond to $N=2$ models in which the $\beta$-function of the different group factors verify ${\beta }_\alpha=12$, whereas the asymmetric solutions that we propose have ${\beta }_\alpha=24$. We check the consistency of these dualities by studying the different large $T,S$ limits of the gauge kinetic function. Dual $N=1$, $D=4$ models can also be obtained by the operation of appropriate freely acting twists, as shown in specific examples. 
  Three dyon solutions to the SU(2) Yang-Mills-Higgs system are presented. These solutions are obtained from the BPS dyon by allowing the gauge fields to be complex, or by letting the free parameter of the BPS solution become imaginary. In all cases however the physically measurable quantities connected with these new solutions are entirely real. Although the new solutions are mathematically simple variations of the BPS solution, they have one or more physically distinct characteristics. 
  The well known N=2 string theory describes self-dual gravity, as was shown by Ooguri and Vafa sometime ago. In search of a variant of this theory which would describe self-dual supergravity in 2+2 dimensions, we have constructed two new N=2 strings theories in which the target space is a superspace. Both theories contain massless scalar and spinor fields in their spectrum, and one of them has spacetime supersymmetry. However, we find that the interactions of these fields do not correspond to those of self-dual supergravity. In our construction, we have used the basic (2,2) superspace variables, and considered quadratic constraints in these variables. A more general construction may be needed for a stringy description of self-dual supergravity. 
  The new phase of a gauge theory in which the instantons are ``polarized'', i.e. have the preferred orientation is discussed. A class of gauge theories with the specific condensates of the scalar fields is considered. In these models there exists an interaction between instantons resulting from one-fermion loop correction. The interaction makes the identical orientation of instantons to be the most probable, permitting one to expect the system to undergo the phase transition into the state with polarized instantons. The existence of this phase is confirmed in the mean-field approximation in which there is the first order phase transition separating the ``polarized phase'' from the usual non-polarized one. The considered phase can be important for the description of gravity in the framework of the gauge field theory. 
  Decoupling the chiral dynamics in the canonical approach to the WZNW model requires an extended phase space that includes left and right monodromy variables. Earlier work on the subject, which traced back the quantum qroup symmetry of the model to the Lie-Poisson symmetry of the chiral symplectic form, left some open questions: - How to reconcile the monodromy invariance of the local 2D group valued field (i.e., equality of the left and right monodromies) with the fact that the latter obey different exchange relations? - What is the status of the quantum group symmetry in the 2D theory in which the chiral fields commute? - Is there a consistent operator formalism in the chiral and in the extended 2D theory in the continuum limit? We propose a constructive affirmative answer to these questions for G=SU(2) by presenting the chiral quantum fields as sums of chiral vertex operators and q-Bose creation and annihilation operators. 
  We compactify $M$-theory on a Calabi-Yau manifold to five dimensions by wrapping the membrane and fivebrane solitons of the eleven-dimensional supergravity limit around Calabi-Yau two-cycles and four-cycles respectively. We identify the perturbative and non-perturbative BPS states thus obtained with those of heterotic string theory compactified on $K3\times S^1$. Quantum aspects of the five-dimensional theory are discussed. 
  We analyze the problems with the so called gauge invariant quantization of the anomalous gauge field theories originary due to Faddeev and Shatashvili (FS). Our analysis bring to a generalization of FS method which allows to construct a series of classically equivalent theories which are non equivalent at quantum level. We prove that these classical theories are all consistent with the BRST invariance of the original gauge symmetry with suitably augmented field content. As an example of such a scenario, we discuss the class of physically distinct models of two dimensional induced gravity which are a generalization of the David-Distler-Kawai model. 
  In this paper we consider some analytic properties of the high--energy quark--quark scattering amplitude, which, as is well known, can be described by the expectation value of two lightlike Wilson lines, running along the classical trajectories of the two colliding particles. We shall prove that the expectation value of two infinite Wilson lines, forming a certain hyperbolic angle in Minkowski space--time, and the expectation value of two infinite Euclidean Wilson lines, forming a certain angle in Euclidean four--space, are connected by an analytic continuation in the angular variables. This could open the possibility of evaluating the high--energy scattering amplitude directly on the lattice or using the stochastic vacuum model. The Abelian case (QED) is also discussed. 
  Quantum corrections are studied for a charged black hole in a two-dimensional model obtained by spherisymmetric reduction of the 4D Einstein-Maxwell theory. The classical (tree-level) thermodynamics is re-formulated in the framework of the off-shell approach, considering systems at arbitrary temperature. This implies a conical singularity at the horizon and modifies the gravitational action by terms defined on the horizon. A consistent variational procedure for the action functional is formulated. It is shown that the free energy reaches an extremum on the regular manifold with $T=T_H$. The one-loop contribution to the action in the Liouville-Polyakov form is re-examined. All the boundary terms are taken into account and the dependence on the state of the quantum field is established. The modification of the Liouville-Polyakov term for a 2D space with a conical defect is derived. The backreaction of the Hawking radiation on the geometry is studied and the quantum-corrected black hole metric is calculated perturbatively. Within the off-shell approach the one-loop thermodynamical quantities, energy and entropy, are found. They are shown to contain a part due to hot gas surrounding he black hole and a part due to the hole itself. It is noted that the contribution of the hot gas can be eliminated by appropriate choice of the (generally, non-flat) reference geometry. The deviation of the {\it `` entropy - horizon area''} relation for the quantum-corrected black hole from the classical law is discovered and possible physical consequences are discussed. 
  It is demonstrated that the recently proposed pseudoclassical model for Weyl particles [1] (D.M. Gitman, A.E. Goncalves and I.V. Tyutin, Phys. Rev. D 50 (1994) 5439) is classically inconsistent. A possible way of removing the classical inconsistency of the model is proposed. 
  We consider a $A_{N-1}$ type of spin dependent Calogero-Sutherland model, containing an arbitrary representation of the permutation operators on the combined internal space of all particles, and find that such a model can be solved as easily as its standard $su(M)$ invariant counterpart through the diagonalisation of Dunkl operators. A class of novel representations of the permutation operator $P_{ij}$, which pick up nontrivial phase factors along with interchanging the spins of $i$-th and $j$-th particles, are subsequently constructed. These `anyon like' representations interestingly lead to different variants of spin Calogero-Sutherland model with highly nonlocal interactions. We also explicitly derive some exact eigenfunctions as well as energy eigenvalues of these models and observe that the related degeneracy factors crucially depend on the choice of a few discrete parameters which characterise such anyon like representations. 
  We construct the twistor space associated with an HKT manifold, that is, a hyper-K\"ahler manifold with torsion, a type of geometry that arises as the target space geometry in two-dimensional sigma models with (4,0) supersymmetry. We show that this twistor space has a natural complex structure and is a holomorphic fibre bundle over the complex projective line with fibre the associated HKT manifold. We also show how the metric and torsion of the HKT manifold can be determined from data on the twistor space by a reconstruction theorem. We give a geometric description of the sigma model (4,0) superfields as holomorphic maps (suitably understood) from a twistorial extension of (4,0) superspace (harmonic superspace) into the twistor space of the sigma model target manifold and write an action for the sigma model in terms of these (4,0) superfields. 
  Aiming towards understanding the question of the discrete states in the light-cone gauge in the theory of two-dimensional strings with a linear background charge term, we study the path-integral formulation of the theory. In particular, by gauge fixing Polyakov's path-integral expression for the 2-d strings, we show that the light-cone gauge-fixed generating functional is the same as the conformal gauge-fixed one and is critical for the same value of the background charge (Q=2 $\sqrt 2 $). Since the equivalence is shown at the generating functional level, one expects that the spectra of the two theories are the same. The zero modes of the ratio of the determinants are briefly analyzed and it is shown that only the constant mode survives in this formulation. This is an indication that the discrete states may lie in these zero modes. This result is not particular to the light-cone gauge, but it holds for the conformal gauge as well. 
  Using the renormalization group (RG) approach and the equivalency between the class of gauge-Higgs-Yukawa models and the gauged Nambu-Jona-Lasinio (NJL) model, we study the gauged NJL model in curved space-time. The behaviour of the scalar-gravitational coupling constant $\xi(t)$ in both theories is discussed. The RG improved effective potential of gauged NJL model in curved spacetime is found. The curvature at which chiral symmetry in the gauged NJL model is broken is obtained explicitly in a remarkably simple form. The powerful RG improved effective potential formalizm leads to the same results as ladder Schwinger-Dyson equations which have not been formulated yet in curved spacetime what opens new possibilities in the study of GUTs and NJL-like models in curved spacetime. 
  Extremal BPS-saturated black holes in $N=2$, $d=4$ supergravity can carry electric and magnetic charges $(q^\Lambda_{(m)},q_\Lambda^{(e)})$. It is shown that in smooth cases the moduli fields at the horizon take a fixed "rational" value $X^\Lambda(q_{(m)},q^{(e)})$ which is determined by the charges and is independent of the asymptotic values of the moduli fields. A universal formula for the Bekenstein-Hawking entropy is derived in terms of the charges and the moduli space geometry at $X^\Lambda(q_{(m)},q^{(e)})$. This work extends previous results of Ferrara, Kallosh and the author for the pure magnetic case. 
  In a recent paper Townsend suggested to associate to the D=11 solitonic membrane of N=1 supergravity a certain thickness, and then to identify this membrane with the fundamental supermembrane. By integrating out the 8 transverse dimensions of the "thick" solitonic membrane, we show that the resulting world-volume action indeed contains all the usual supermembrane terms, as well as background curvature terms and extrinsic curvature terms, which are believed to render the membrane spectrum discrete. We also outline the analog derivation for the "thick" D=10 solitonic string solution of N=IIA supergravity. The resulting world-sheet action contains the usual type IIA superstring terms, as well as extra terms whose presence can be interpreted as a rescaling of the background metric, thus preserving kappa-symmetry and conformal invariance. 
  The Hodge--de Rham Laplacian on spheres acting on antisymmetric tensor fields is considered. Explicit expressions for the spectrum are derived in a quite direct way, confirming previous results. Associated functional determinants and the heat kernel expansion are evaluated. Using this method, new non--local counterterms in the quantum effective action are obtained, which can be expressed in terms of Betti numbers. 
  We study compactifications of F-theory on certain Calabi--Yau threefolds. We find that $N=2$ dualities of type II/heterotic strings in 4 dimensions get promoted to $N=1$ dualities between heterotic string and F-theory in 6 dimensions. The six dimensional heterotic/heterotic duality becomes a classical geometric symmetry of the Calabi--Yau in the F-theory setup. Moreover the F-theory compactification sheds light on the nature of the strong coupling transition and what lies beyond the transition at finite values of heterotic string coupling constant. 
  The truncated 4-dimensional sphere $S^4$ and the action of the self-interacting scalar field on it are constructed. The path integral quantization is performed while simultaneously keeping the SO(5) symmetry and the finite number of degrees of freedom. The usual field theory UV-divergences are manifestly absent. 
  We introduce a gauge group of internal symmetries of an ambient algebra as a new tool for investigating the superselection structure of WZW theories and the representation theory of the corresponding affine Lie algebras. The relevant ambient algebra arises from the description of these conformal field theories in terms of free fermions. As an illustration we analyze in detail the \son\ WZW theories at level two. In this case there is actually a homomorphism from the representation ring of the gauge group to the WZW fusion ring, even though the level-two observable algebra is smaller than the gauge invariant subalgebra of the field algebra. 
  The microscopic theories of quantum gravity related to integrable lattice models can be constructed as special deformations of pure gravity. Each such deformation is defined by a second order differential operator acting on the coupling constants. As a consequence, the theories with matter fields satisfy a set of constraints inherited from the integrable structure of pure gravity. In particular, a set of bilinear functional equations for each theory with matter fields follows from the Hirota equations defining the KP (KdV) structure of pure gravity. 
  A type IIA string compactified on a Calabi-Yau manifold which admits a K3 fibration is believed to be equivalent to a heterotic string in four dimensions. We study cases where a Calabi-Yau manifold can have more than one such fibration leading to equivalences between perturbatively inequivalent heterotic strings. This allows an analysis of an example in six dimensions due to Duff, Minasian and Witten and enables us to go some way to prove a conjecture by Kachru and Vafa. The interplay between gauge groups which arise perturbatively and nonperturbatively is seen clearly in this example. As an extreme case we discuss a Calabi-Yau manifold which admits an infinite number of K3 fibrations leading to infinite set of equivalent heterotic strings. 
  Studies in string theory and in quantum gravity suggest the existence of a finite lower bound to the possible resolution of lengths which, quantum theoretically, takes the form of a minimal uncertainty in positions $\Delta x_0$. A finite minimal uncertainty in momenta $\Delta p_0$ has been motivated from the absence of plane waves on generic curved spaces. Both effects can be described as small noncommutative geometric features of space-time. In a path integral approach to the formulation of field theories on noncommutative geometries, we can now generally prove IR regularisation for the case of noncommutative geometries which imply minimal uncertainties $\Delta p_0$ in momenta. 
  T-duality is used to extract information on an instanton of zero size in the $E_8\times E_8$ heterotic string. We discuss the possibility of the appearance of a tensionless anti-self-dual non-critical string through an implementation of the mechanism suggested by Strominger of two coincident 5-branes. It is argued that when an instanton shrinks to zero size a tensionless non-critical string appears at the core of the instanton. It is further conjectured that appearance of tensionless strings in the spectrum leads to new phase transitions in six dimensions in much the same way as massless particles do in four dimensions. 
  A class of loop diagrams in general relativity appears to have a behavior which would upset the utility of the energy expansion for quantum effects. We show through the study of specific diagrams that cancellations occur which restore the expected behaviour of the energy expansion. By considering the power counting in a physical gauge we show that the apparent bad behavior is a gauge artifact, and that the quantum loops enter with a well behaved energy expansion. 
  Quantum field theory offers physicists a tremendously wide range of application; it is both a language with which a vast variety of physical processes can be discussed and also it provides a model for fundamental physics, the so-called ``standard-model,'' which thus far has passed every experimental test. No other framework exists in which one can calculate so many phenomena with such ease and accuracy. Nevertheless, today some physicists have doubts about quantum field theory, and here I want to examine these reservations. 
  We present a line of reasoning based on the analysis of scale variations of the Wilsonian partition function and the trace of the stress tensor in a curved manifold which results in a statement of irreversibility of Wilsonian renormalization group flow for unitary theories. We also analyze subtleties related to subtractions in the case of the 1PI effective action flow. 
  From the postulate that a black hole can be replaced by a boundary on the apparent horizon with suitable boundary conditions, an unconventional scenario for the evolution emerges. Only an insignificant fraction of energy of order $(mG)^{-1}$ is radiated out. The outgoing wave carries a very small part of the quantum mechanical information of the collapsed body, the bulk of the information remaining in the final stable black hole geometry. 
  In the context of a two-dimensional exactly solvable model, the dynamics of quantum black holes is obtained by analytically continuing the description of the regime where no black hole is formed. The resulting spectrum of outgoing radiation departs from the one predicted by the Hawking model in the region where the outgoing modes arise from the horizon with Planck-order frequencies. This occurs early in the evaporation process, and the resulting physical picture is unconventional. The theory predicts that black holes will only radiate out an energy of Planck mass order, stabilizing after a transitory period. The continuation from a regime without black hole formation --accessible in the 1+1 gravity theory considered-- is implicit in an S-matrix approach and suggests in this way a possible solution to the problem of information loss. 
  In this paper we study the production of pairs of neutral and charged black holes by domain walls, finding classical solutions and calculating their classical actions. We find that neutral black holes whose creation is mediated by Euclidean instantons must be produced mutually at rest with respect to one another, but for charged black holes a new type of instanton is possible in which after formation the two black holes accelerate away from one another. These new types of instantons are not possible in Einstein-Maxwell theory with a cosmological constant. We also find that the creation of non-orientable black hole solutions can be mediated by Euclidean instantons and that in addition if one is prepared to consider entirely Lorentzian no-boundary type contributions to the path integral then mutually accelerating pairs may be created even in the neutral case. Finally we consider the production of Kaluza-Klein monopoles both by a standard cosmological term and in the presence of a domain wall. We find that compactification is accompanied by the production of pairs of Kaluza-Klein monopoles. 
  The free quantum states of topologically massive electrodynamics and gravity in 2+1 dimensions, are explicitly found. It is shown that in both theories the states are described by infrared-regular polarization tensors containing a regularization phase which depends on the spin. This is done by explicitly realizing the quantum algebra on a functional Hilbert space and by finding the Wightman function to define the scalar product on such a Hilbert space. The physical properties of the states are analyzed defining creation and annihilation operators.  For both theories, a canonical and covariant quantization procedure is developed. The higher order derivatives in the gravitational lagrangian are treated by means of a preliminary Dirac procedure.  The closure of the Poincar\'e algebra is guaranteed by the infrared-finiteness of the states which is related to the spin of the excitations through the regularization phase. Such a phase may have interesting physical consequences. 
  A supersymmetric formulation of the classical action of interacting charged and neutral fermions with arbitrary anomalous magnetic moment is considered. This formulation generalizes the known action for scalar charged particles investigated in papers by Fokker, Schwarzschild, Tetrode, Wheeler and Feynman. The superfield formulation of the electrodynamics of the Maxwell supermultiplet, constructed from the world coordinates of charged or neutral fermions is carried out basing on the proposed action. 
  We calculate the entropies of the system of classical particles and a quantum scalar field by using the brick wall method in thermal bath in a charged Kerr black hole spacetime. Their leading terms at Hartle-Hawking temperature $T_H = \kappa/(2 \pi) $ are given by $ S_{cl} \approx N \ \ln \left( \frac{A_b}{\epsilon^2} \right)$, and $S \approx N' \frac{A_H}{\epsilon^2}$, where $A_b$ and $A_H$ are the area of the box and the horizon respectively. 
  We study the Schwinger-Dyson equation for the fermion self-energy in massless and massive $QED_2$, in the ladder approximation. When the fermion is massless (and the photon massless or massive), we check the reliability of this approximation by comparing its solutions with the exact ones. They agree only when the photon is massless. For a massive fermion and massless photon, we show that there is no consistent solution at all, the infrared divergences introduced by the approximation forbidding even the trivial solution. When both fermion and photon are massive, we find a non-perturbative (extra) fermion mass generation (which survives in the limit when the bare mass of the fermion tends to zero). We argue that, in this case, the ladder approximation will provide reliable results if the bare masses of the fields are large compared with the (dimensionful) coupling constant. 
  We investigate some properties of a first-order polynomial formulation of the U(1) non-linear sigma-model in two Euclidean dimensions. The variables in this description are a 1-form field plus a 0-form Lagrange multiplier field. The usual spin variables are non-local functions of the new fields. As this construction incorporates O(2) invariance ab initio, only O(2)-invariant correlation functions (the only non-vanishing ones in the model) can be constructed. We show that the vortices play a dual role to the spin variables in the partition function. The equivalent Sine-Gordon description is obtained in a natural way, when one integrates out the 1-form field to get an effective partition function for the Lagrange multiplier. We also show how to introduce strings of vortices within this formulation. 
  Convergence of the standard model gauge coupling constants to a common value at around $2\times 10^{16}$ GeV is studied in the context of orbifold theories where the modular symmetry groups for $T$ and $U$ moduli are broken to subgroups of $PSL(2, Z)$. The values of the moduli required for this unification of coupling constants are studied for this case and also for the case where string unification is accompanied by unification to a gauge group larger then $SU(3)\times SU(2)\times U(1).$ 
  We perform consistently the Gupta-Bleuler quantization combined with Dirac procedure for a chiral boson with the parameter ($\alpha$) on the circle, the boundary of the circular droplet. For $\alpha =1$, we obtain the holomorphic constraints. Using the representation of Bargmann-Fock space and the Schr\"odinger equation, we construct the holomorphic wave functions. In order to interpret these functions, we introduce the $W_\infty$-coherent state to account for the infinite-dimensional translation symmetry for the Fourier (edge) modes. The $\alpha=1$ wave functions explain the neutral edge states for $\nu =1$ quantum Hall fluid very well. In the case of $\alpha = -1$,  we obtain the new wave functions which may describe the higher modes (radial excitations) of edge states. Finally, the charged edge states are described by the $|\alpha| \not=1$ wave functions. 
  We briefly sketch the noncommutative geometry approach to the Standard Model, with attention to what can be inferred about particle masses. 
  We consider slightly non-extremal black 3-branes of type IIB supergravity and show that their Bekenstein-Hawking entropy agrees, up to a mysterious factor, with an entropy derived by counting non-BPS excitations of the Dirichlet 3-brane. These excitations are described in terms of the statistical mechanics of a 3+1 dimensional gas of massless open string states. This is essentially the classic problem of blackbody radiation. The blackbody temperature is related to the temperature of the Hawking radiation. We also construct a solution of type IIB supergravity describing a 3-brane with a finite density of longitudinal momentum. For extremal momentum-carrying 3-branes the horizon area vanishes. This is in agreement with the fact that the BPS entropy of the momentum-carrying Dirichlet 3-branes is not an extensive quantity. 
  We find a general principle which allows one to compute the area of the horizon of N=2 extremal black holes as an extremum of the central charge. One considers the ADM mass equal to the central charge as a function of electric and magnetic charges and moduli and extremizes this function in the moduli space (a minimum corresponds to a fixed point of attraction). The extremal value of the square of the central charge provides the area of the horizon, which depends only on electric and magnetic charges. The doubling of unbroken supersymmetry at the fixed point of attraction for N=2 black holes near the horizon is derived via conformal flatness of the Bertotti-Robinson-type geometry. These results provide an explicit model independent expression for the macroscopic Bekenstein-Hawking entropy of N=2 black holes which is manifestly duality invariant. The presence of hypermultiplets in the solution does not affect the area formula. Various examples of the general formula are displayed. We outline the attractor mechanism in N=4,8 supersymmetries and the relation to the N=2 case. The entropy-area formula in five dimensions, recently discussed in the literature, is also seen to be obtained by extremizing the 5d central charge. 
  By generalizing the Fujikawa approach, we show in the path-integral formalism: (1) how the infinitesimal variation of the fermion measure can be integrated to obtain the full anomalous chiral action; (2) how the action derived in this way can be identified as the Chern-Simons term in five dimensions, if the anomaly is consistent; (3) how the regularization can be carried out, so as to lead to the consistent anomaly and not to the covariant anomaly. Our method uses Schwinger's ``proper-time'' representation of the Green's function and the gauge invariant point-splitting technique. We find that the consistency requirement and the point-splitting technique allow both an anomalous and a non-anomalous action. In the end, the nature of the vacuum determines whether we have an anomalous theory, or, a non-anomalous theory 
  A multilinear M-dimensional generalization of Lax pairs is introduced and its explicit form is given for the recently discovered class of time-harmonic, integrable, hypersurface motions. 
  The exact parity model (EPM) is a simple extension of the Standard Model which reinstates parity invariance as an unbroken symmetry of nature. The mirror matter sector of the model can interact with ordinary matter through gauge boson mixing, Higgs boson mixing and, if neutrinos are massive, through neutrino mixing. The last effect has experimental support through the observed solar and atmospheric neutrino anomalies. In this paper we show that the exact parity model can be formulated in a quaternionic framework. This suggests that the idea of mirror matter and exact parity may have profound implications for the mathematical formulation of quantum theory. 
  In this paper, we study the actions of the Weyl groups of the U duality groups for type IIA string theory toroidally compactified to all dimensions $D\ge 3$. We show how these Weyl groups implement permutations of the field strengths, and we discuss the Weyl group multiplets of all supersymmetric $p$-brane solitons. 
  The Ward-Takahashi identities in large N field theories are expressed in a simple form using master fields. Operators appearing in these expressions are found to be generators of symmetry transformations acting on the master fields. 
  We construct a perturbation theory for the SU(2) non-linear Sigma-model in 2+1 dimensions using a polynomial, first-order formulation, where the variables are a non-Abelian vector field L_mu (the left SU(2) current), and a non-Abelian pseudovector field $\theta_{\mu}$, which imposes the condition F_{mu nu}(L) = 0. The coordinates on the group do not appear in the Feynman rules, but their scattering amplitudes are easily related to those of the currents. We show that all the infinities affecting physical amplitudes at one-loop order can be cured by normal ordering, presenting the calculation of the full propagator as an example of an application. 
  We study the canonical structure of the $SU(N)$ non-linear Sigma-model in a polynomial, first-order representation. The fundamental variables in this description are a non-Abelian vector field L_mu and a non-Abelian antisymmetric tensor field theta_{mu nu}, which constrains L_{mu} to be a `pure gauge' (F_{mu nu}(L) = 0) field. The second-class constraints that appear as a consequence of the first-order nature of the Lagrangian are solved, and the reduced phase-space variables explicitly found. We also treat the first-class constraints due to the gauge-invariance under transformations of the antisymmetric tensor field, constructing the corresponding most general gauge-invariant functionals, which are used to describe the dynamics of the physical degrees of freedom. We present these results in 1+1, 2+1 and 3+1 dimensions, mentioning some properties of the (d+1)-dimensional case. We show that there is a kind of duality between this description of the non-linear $\sigma$-model and the massless Yang-Mills theory. This duality is further extended to more general first-class systems. 
  We present several Orientifolds of M-Theory on $K_3\times S^1$ by additional projections with respect to the finite abelian automorphism groups of $K_3$. The resulting models correspond to anomaly free theories in six dimensions. We construct explicit examples which can be interpreted as models with eight, four, two and one vector multiplets and $N=1$ supersymmetry in six dimensions. 
  A family of type IIB superstring backgrounds involving Ramond-Ramond fields are obtained in ten dimensions starting from a K-model through a generalization of our recent results. The unbroken global $SL(2,R)$ symmetry of the type IIB equations of motion are implemented in this context as a solution generating transfromation. A geometrical analysis, based on the tensor structure of the higher order $\alpha^{\prime}$ terms in the equations of motion, is employed to show that these backgrounds are exact. 
  We derive the $N=2$ and $4$ super Yang-Mills theories from the viewpoint of the $M_4\times Z_2\times Z_2$ gauge theory. Scalars and pseudoscalars appearing in the theories are regarded as gauge fields along the directions on $Z_2\times Z_2$ discrete space. 
  We analyze GKZ(Gel'fand, Kapranov and Zelevinski) hypergeometric systems and apply them to study the quantum cohomology rings of Calabi-Yau manifolds. We will relate properties of the local solutions near the large radius limit to the intersection rings of a toric variety and of a Calabi-Yau hypersurface. (Talk presented at "Frontiers in Quantum Field Theory", Osaka, Japan, Dec.1995) 
  In this letter we propose exact three-point correlation functions for $N=1$ supersymmetric Liouville theory. Along the lines of Zamolodchikov and Zamolodchikov paper (hep-th/9506136) we propose a generalized special function which describe the three-point amplitudes. We consider briefly the so called reflection amplitudes in the supersymmetric case. 
  We consider the 2--dimensional Wess--Zumino--Witten (WZW) model in the canonical formalism introduced in a previous paper by two of us. Using an $r$--$s$ matrix approach to non--ultralocal field theories we find the Poisson algebra of monodromy matrices and of conserved quantities with a new, non--dynamical, $r$ matrix. 
  The Ward identities of the $W_{\infty}$ symmetry in 2D string theory in the tachyon background are studied in the continuum approach. Comparing the solutions with the matrix model results, it is verified that 2D string amplitudes are different from the matrix model amplitudes only by the external leg factors even in higher genus. This talk is based on the recent work [1] and also [2] for the $c_M <1$ model. (Talk given at the workshop on ``Frontiers in Quantum Field Theory'', Osaka, Japan, December 1995.) 
  We present here our considerations concerning the problem of classical consistency of pseudoclassical models touched upon in a recent comment on our paper "New pseudoclassical model for Weyl particle". 
  We discuss the algebraic renormalization of the Yang--Mills gauge field theory in the presence of translations. Due to the translations the algebra between Sorella's $\d$--operator, the exterior derivative and the BRST--operator closes. Therefore, we are able to derive an integrated parameter formula collecting in an elegant and compact way all nontrivial solutions of the descent equations. 
  Under the six-dimensional heterotic/type $IIA$ duality map, a solitonic membrane solution of heterotic string theory transforms into a singular solution of type $IIA$ theory, and should therefore be interpreted as a fundamental membrane in the latter theory. This finding pointed to a gap in the formulation of string theory that was subsequently filled by the discovery of the role of $D$-branes as the carriers of Ramond-Ramond charge in type $II$ string theory. The roles of compactified eleven-dimensional membranes and fivebranes in five-dimensional string theory are also discussed. 
  We consider the type IIA string compactified on the Calabi-Yau space given by a degree 12 hypersurface in the weighted projective space ${\bf P}^4_{(1, 1, 2,2, 6)}$. We express the prepotential of the low-energy effective supergravity theory in terms of a set of functions that transform covariantly under $PSL(2, \IZ)$ modular transformations. These functions are then determined by monodromy properties, by singularities at the massless monopole point of the moduli space, and by $S \leftrightarrow T$ exchange symmetry. 
  An analysis of how the mass gap could arise in pure Yang-Mills theories in two spatial dimensions is given 
  If the Wilsonian renormalization group (RG) is formulated with a cutoff that breaks gauge invariance, then gauge invariance may be recovered only once the cutoff is removed and only once a set of effective Ward identities is imposed. We show that an effective Quantum Action Principle can be formulated in perturbation theory which enables the effective Ward identities to be solved order by order, even if the theory requires non-vanishing subtraction points. The difficulties encountered with non-perturbative approximations are briefly discussed. 
  $SU(N)$ Yang-Mills theory in three dimensions, with a Chern-Simons term of level $k$ (an integer) added, has two dimensionful coupling constants, $g^2 k$ and $g^2 N$; its possible phases depend on the size of $k$ relative to $N$. For $k \gg N$, this theory approaches topological Chern-Simons theory with no Yang-Mills term, and expectation values of multiple Wilson loops yield Jones polynomials, as Witten has shown; it can be treated semiclassically. For $k=0$, the theory is badly infrared singular in perturbation theory, a non-perturbative mass and subsequent quantum solitons are generated, and Wilson loops show an area law. We argue that there is a phase transition between these two behaviors at a critical value of $k$, called $k_c$, with $k_c/N \approx 2 \pm .7$. Three lines of evidence are given: First, a gauge-invariant one-loop calculation shows that the perturbative theory has tachyonic problems if $k \leq 29N/12$.The theory becomes sensible only if there is an additional dynamic source of gauge-boson mass, just as in the $k=0$ case. Second, we study in a rough approximation the free energy and show that for $k \leq k_c$ there is a non-trivial vacuum condensate driven by soliton entropy and driving a gauge-boson dynamical mass $M$, while both the condensate and $M$ vanish for $k \geq k_c$. Third, we study possible quantum solitons stemming from an effective action having both a Chern-Simons mass $m$ and a (gauge-invariant) dynamical mass $M$. We show that if $M \gsim 0.5 m$, there are finite-action quantum sphalerons, while none survive in the classical limit $M=0$, as shown earlier by D'Hoker and Vinet. There are also quantum topological vortices smoothly vanishing as $M \rightarrow 0$. 
  In these lectures several aspects of anyon in one and two dimensions are considered from the path integral formalism. This paper is based in a set of four lectures given by the author in the "V Latinoamerican Workshop of Particles and Fields, hel in Puebla, Mexico. 
  Noncommutative geometric gauge theory is reconstructed based on the superconnection concept. The bosonic action of the Connes-Lott model including the symmetry breaking Higgs sector is obtained by using a new generalized derivative, which consists of the usual 1-form exterior derivative plus an extra element called the matrix derivative, for the curvatures. We first derive the matrix derivative based on superconnections and then show how the matrix derivative can give rise to spontaneous symmetry breaking. We comment on the correspondence between the generalized derivative and the generalized Dirac operator of the Connes-Lott model. 
  We demonstrate that in a certain gauge the Lax matrices of the rational and hyperbolic Ruijsenaars--Schneider models have a quadratic $r$-matrix Poisson bracket which is an exact quadratization of the linear $r$--matrix Poisson bracket of the Calogero--Moser models. This phenomenon is explained by a geometric derivation of Lax equations for arbitrary flows of both hierarchies, which turn out to be governed by the same dynamical $R$--operator. 
  We show that the integrability of the dynamical system recently proposed by Calogero and characterized by the Hamiltonian $ H = \sum_{j,k}^{N} p_j p_k \{\lambda + \mu cos [ \nu ( q_j - q_k)] \} $ is due to a simple algebraic structure . It is shown that the integrals of motion are related to the Casimiar invariants of of the $su(1,1)$ algebra. Our method shows clearly how these types of systems can be generalized . 
  The account of the Poisson-Lie T-duality is presented for the case when the action of the duality group on a target is not free. At the same time a generalization of the picture is given when the duality group does not even act on $\si$-model targets but only on their phase spaces. The outcome is a huge class of dualizable targets generically having no local isometries or Poisson-Lie symmetries whatsoever. 
  The BRS symmetry determines physical states, Lagrange densities and candidate anomalies. It renders gauge fixing unobservable in physical states and is required if negative norm states are to decouple also in interacting models. The relevant mathematical structures and the elementary cohomological investigations are presented. 
  The four dimensional SU(2) WZW model coupled to elecromagnetism is treated as a constraint system in the context of the BFV approach. We show that the Darboux's transformations which are used to diagonalize the canonical one-form in the Faddeev-Jackiw formalism, transform the fields of the model into BRST invariant ones. The same analysis is also carried out in the case of spinor electrodynamics. 
  The expectation value of a Wilson loop in a Chern--Simons theory is a knot invariant. Its skein relations have been derived in a variety of ways, including variational methods in which small deformations of the loop are made and the changes evaluated. The latter method only allowed to obtain approximate expressions for the skein relations. We present a generalization of this idea that allows to compute the exact form of the skein relations. Moreover, it requires to generalize the resulting knot invariants to intersecting knots and links in a manner consistent with the Mandelstam identities satisfied by the Wilson loops. This allows for the first time to derive the full expression for knot invariants that are suitable candidates for quantum states of gravity (and supergravity) in the loop representation. The new approach leads to several new insights in intersecting knot theory, in particular the role of non-planar intersections and intersections with kinks. 
  We show that the N=2 determinant formulae of the Aperiodic NS algebra and the Periodic R algebra can be applied directly to incomplete Verma modules built on chiral primary states and on Ramond ground states, respectively, provided one modifies the interpretation of the zeroes in an appropriate way. That is, the zeroes of the determinat formulae account for the highest weight singular states built on chiral primaries and on Ramond ground states, but the identification of the levels and relative U(1) charges of the singular states is different than for complete Verma modules. In particular, half of the zeroes of the quadratic vanishing surfaces $f^A_{r,s}=0$ and $f^P_{r,s}=0$ correspond to uncharged singular states, and the other half correspond to charged singular states. We derive the spectrum of singular states built on chiral primaries, including the singular states of the Twisted Topological algebra, and the spectrum of singular states built on the Ramond ground states. We also uncover the existence of non-highest weight singular states which are not secondary of any highest weight singular state. 
  We study the moduli space for an arbitrary number of BPS monopoles in a gauge theory with an arbitrary gauge group that is maximally broken to $U(1)^k$. From the low energy dynamics of well-separated dyons we infer the asymptotic form of the metric for the moduli space. For a pair of distinct fundamental monopoles, the space thus obtained is $R^3 \times(R^1\times {\cal M}_0)/Z$ where ${\cal M}_0$ is the Euclidean Taub-NUT manifold. Following the methods of Atiyah and Hitchin, we demonstrate that this is actually the exact moduli space for this case. For any number of such objects, we show that the asymptotic form remains nonsingular for all values of the intermonopole distances and that it has the symmetries and other characteristics required of the exact metric. We therefore conjecture that the asymptotic form is exact for these cases also. 
  We evaluate the Wilson loop at second order in general non-covariant gauges by means of the causal principal-value prescription for the gauge- dependent poles in the gauge-boson propagator and show that the result agrees with the usual causal prescriptions. 
  We study the low-energy approximation for calculation of the heat kernel which is determined by the strong slowly varying background fields in strongly curved quasi-homogeneous manifolds. A new covariant algebraic approach, based on taking into account a finite number of low-order covariant derivatives of the background fields and neglecting all covariant derivatives of higher orders, is proposed. It is shown that a set of covariant differential operators together with the background fields and their low-order derivatives generates a finite dimensional Lie algebra. This algebraic structure can be used to present the heat semigroup operator in the form of an average over the corresponding Lie group. Closed covariant formulas for the heat kernel diagonal are obtained. These formulas serve, in particular, as the generating functions for the whole sequence of the Hadamard-\-Minakshisundaram-\-De~Witt-\-Seeley coefficients in all symmetric spaces. 
  We consider the universal central extension of the Lie algebra $\Vect (S^1)${\math \s}$ C^{\infty}(S^1)$. The coadjoint representation of this Lie algebra has a natural geometric interpretation by matrix analogues of the Sturm-Liouville operators. This approach leads to new Lie superalgebras generalizing the well-known Neveu-Schwartz algebra. 
  We use the method of solving the three-anyon problem developed in our earlier publication to evaluate numerically the third virial coefficient of free anyons. In order to improve precision, we explicitly correct for truncation effects. The present calculation is about three orders of magnitude more precise than the previous Monte Carlo calculation and indicates the presence of a term $a sin^4 \pi\nu$ with a very small coefficient $a \simeq -1.65 10^{-5}$. 
  This is an expanded version of a talk given at ``{\em IInd Recontre du Vietnam}'' held at Ho Chi Minh City in October, 1995. We discuss several aspects of black hole entropy in string theory. We first explain why the geometric entropy in two dimensional noncritical string theory is nonperturbatively finite. We then explain the philosophy of regarding massive string states as black branes and how the Beckenstein-Hawking entropy for extremal BPS black holes may be understood as coming from degeneracy of string states. This is then discussed in the context of D-strings in Type IIB superstrings. We then describe non-BPS excitations of D-strings and their entropy and explore the possibility that their decay describes Hawking radiation. For these D-strings and other D-branes the entropy and temperature are consequences of the physical motion of stuck open strings along the D-brane and this leads to a simple space-time interpretation. Finally we speculate that the horizon may be itself regarded as a D-brane. 
  A geometrical study of supergravity defined on (1|1) complex superspace is presented. This approach is based on the introduction of generalized superprojective structures extending the notions of super Riemann geometry to a kind of super W-Riemann surfaces. On these surfaces a connection is constructed. The zero curvature condition leads to the super Ward identities of the underlying supergravity. This is accomplished through the symplectic form linked to the (super)symplectic manifold of all super gauge connections. The BRST algebra is also derived from the knowledge of the super W-symmetries which are the gauge transformations of the vector bundle canonically associated to the generalized superprojective structures. We obtain the possible consistent BRST (super)anomalies and their cocycles related by the descent equations. Finally we apply our considerations to the case of supergravity. 
  We obtain the exact beta function for $N=2$ SUSY $SU(2)$ Yang-Mills theory and prove the nonperturbative Renormalization Group Equation $$ \partial_\Lambda{\cal F}(a,\Lambda)= {\Lambda\over \Lambda_0}\partial_{\Lambda_0}{\cal F}(a_0,\Lambda_0) e^{-2\int_{\tau_0}^\tau {dx \beta^{-1}(x)}}. $$ 
  Within the Euclidean path integral and mass perturbation theory we derive, from the Dyson-Schwinger equations of the massive Schwinger model, a general formula that incorporates, for sufficiently small fermion mass, all the bound-state mass poles of the massive Schwinger model. As an illustration we perturbatively compute the masses of the three lowest bound states. 
  An overlap method for regularizing Majorana--Weyl fermions interacting with gauge fields is presented. A mod(2) index is introduced in relation to the anomalous violation of a discrete global chiral symmetry. Most of the paper is restricted to 2 dimensions but generalizations to 2+8k dimensions should be straightforward. 
  The Coulomb-gas description of minimal models is considered on the half plane. Screening prescriptions are developed by the perturbative expansion of the Liouville theory with imaginary coupling and with Neumann boundary condition on the bosonic field. To generate the conformal blocks of more general boundary conditions, we propose the insertion of boundary operations. 
  We consider a theory with gauge group $G \times U(1)_A$ containing: i) an abelian factor for which the chiral matter content of the theory is anomalous $\sum_{f} q^f_A \neq 0 \neq \sum_{f} (q^f_A)^3$ ; ii) a nonanomalous factor $G$. In these models, the calculation of consistent gauge anomalies usually found in the literature as a solution to the Zumino-Stora descent equations is reconsidered. Another solution of the descent equations that differs on the terms involving mixed gauge anomalies is presented on this paper. The origin of their difference is analysed, and using Fujikawa's formalism the second result is argued to be the divergence of the usual chiral current. Invoking topological arguments the physical equivalence of both solutions is explained, but only the second one can be technically called the consistent anomaly of a classically invariant theory. The first one corresponds to the addition of noninvariant local counterterms to the action. A consistency check of their physical equivalence is performed by implementing the four dimensional string inspired Green-Schwarz mechanism for both expressions. This is achieved adding slightly different anomaly cancelling terms to the original action, whose difference is precisely the local counterterms mentioned before. The complete anomaly free action is therefore uniquely defined, and the resulting constraints on the spectrum of fermion charges are the same. The Lorentz invariance of the fermion measure in four dimensions forces the Lorentz variation of the Green-Schwarz terms to cancel by itself, producing an additional constraint usually overlooked in the literature. This often happens when a dual description of the theory is used without including all local counterterms. 
  Non--Abelian duality in relation to supersymmetry is examined. When the action of the isometry group on the complex structures is non--trivial, extended supersymmetry is realized non--locally after duality, using path ordered Wilson lines. Prototype examples considered in detail are, hyper--Kahler metrics with SO(3) isometry and supersymmetric WZW models. For the latter, the natural objects in the non--local realizations of supersymmetry arising after duality are the classical non--Abelian parafermions. The canonical equivalence of WZW models and their non--Abelian duals with respect to a vector subgroup is also established. 
  We provide vector-like gauge theories which break supersymmetry dynamically. 
  Strongly self-dual Yang-Mills fields in even dimensional spaces are characterised by a set of constraints on the eigenvalues of the Yang-Mills fields $F_{\mu \nu}$. We derive a topological bound on ${\bf R}^8$, $\int_{M} ( F,F )^2 \geq k \int_{M} p_1^2$ where $p_1$ is the first Pontrjagin class of the SO(n) Yang-Mills bundle and $k$ is a constant. Strongly self-dual Yang-Mills fields realise the lower bound. 
  Making use of the exact solutions of the $N=2$ supersymmetric gauge theories we construct new classes of superconformal field theories (SCFTs) by fine-tuning the moduli parameters and bringing the theories to critical points. SCFTs we have constructed represent universality classes of the 4-dimensional $N=2$ SCFTs. 
  We discuss the singularities in the moduli space of string compactifications to six dimensions with $N=1$ supersymmetry. Such singularities arise from either massless particles or non-critical tensionless strings. The points with tensionless strings are sometimes phase transition points between different phases of the theory. These results appear to connect all known $N=1$ supersymmetric six-dimensional vacua. 
  The general form of N=2 supergravity coupled to an arbitrary number of vector multiplets and hypermultiplets, with a generic gauging of the scalar manifold isometries is given. This extends the results already available in the literature in that we use a coordinate independent and manifestly symplectic covariant formalism which allows to cover theories difficult to formulate within superspace or tensor calculus approach. We provide the complete lagrangian and supersymmetry variations with all fermionic terms, and the form of the scalar potential for arbitrary quaternionic manifolds and special geometry, not necessarily in special coordinates. Our results can be used to explore properties of theories admitting $N=2$ supergravity as low energy limit. 
  We compute the renormalized trajectory of $\phi^4_4$-theory by perturbation theory in a running coupling. We use an exact infinitesimal renormalization group. The expansion is put into a form which is manifestly independent of the scale parameter. 
  Abelian Gauge Theories are quantized in a geometric representation that generalizes the Loop Representation and treates electric and magnetic operators on the same footing. The usual canonical algebra is turned into a topological algebra of non local operators that resembles the order-disorder dual algebra of 't Hooft. These dual operators provide a complete description of the physical phase space of the theories. 
  Planckian scattering of particles with angular momenta is studied by describing them as sources of Kerr metric. In the shock wave formalism, it is found that the angular momenta do not contribute to the scattering amplitude in the eikonal limit. This is confirmed by using the wave equation of the test particle in the Kerr background. 
  One-string-loop (torus topology) corrections to black-string backgrounds corresponding to gauged ${SL(2,R)\times R}/R$ WZW model are calculated using $\beta$-function equations derived from string-loop-corrected effective action. Loop-corrected backgrounds are used to calculate ADM mass of the black string. 
  We discuss the relation between M theory and type II string theories. We show that, assuming ``natural'' interactions between membranes and fivebranes in M theory, the known interactions between strings and D-branes in type II string theories arise in appropriate limits. Our discussion of the interactions is purely at the classical level. We remark on issues associated with the M theory approach to enhanced gauge symmetries, which deserve further investigation. 
  We show how to compute form factors, matrix elements of local fields, in the restricted sine-Gordon model, at the reflectionless points, by quantizing solitons. We introduce (quantum) separated variables in which the Hamiltonians are expressed in terms of (quantum) tau-functions. We explicitly describe the soliton wave functions, and we explain how the restriction is related to an unusual hermitian structure. We also present a semi-classical analysis which enlightens the fact that the restricted sine-Gordon model corresponds to an analytical continuation of the sine-Gordon model, intermediate between sine-Gordon and KdV. 
  We approach the study of non--integrable models of two--dimensional quantum field theory as perturbations of the integrable ones. By exploiting the knowledge of the exact $S$-matrix and Form Factors of the integrable field theories we obtain the first order corrections to the mass ratios, the vacuum energy density and the $S$-matrix of the non-integrable theories. As interesting applications of the formalism, we study the scaling region of the Ising model in an external magnetic field at $T \sim T_c$ and the scaling region around the minimal model $M_{2,7}$. For these models, a remarkable agreement is observed between the theoretical predictions and the data extracted by a numerical diagonalization of their Hamiltonian. 
  Anomalies and BRST invariance are governed, in the context of Lagrangian Batalin-Vilkovisky quantization, by the master equation, whose classical limit is $(S, S)=0$. Using Zimmerman's normal products and the BPHZ renormalisation method, we obtain a corresponding local quantum operator equation, which is valid to all orders in perturbation theory. The formulation implies a calculational method for anomalies to all orders that is useful also outside the BV context and that remains completely within regularised perturbation theory. It makes no difference in principle whether the anomaly appears at one loop or at higher loops. The method is illustrated by computing the one- and two-loop anomalies in chiral $W_3$ gravity. 
  We study the interaction of non-Abelian topological $BF$ theories defined on two dimensional manifolds with point sources carrying non-Abelian charges. We identify the most general solution for the field equations on simply and multiply connected two-manifolds. Taking the particular choice of the so-called extended Poincar\'e group as the gauge group we discuss how recent discussions of two dimensional gravity models do fit in this formalism. 
  The arguments of statistical nature for the existence of constituents of active gravitational masses are presented. The present paper proposes a basis for microscopic theory of universal gravitation. Questions like the relation of cosmological constant and quantum theory, black hole radiance and the nature of inertia are addressed. This paper is the second in the series of papers published in Acta Physica Polonica {\bf B}. 
  We investigate non-abelian gaugings of WZNW models. When the gauged group is semisimple we are able to present exact formulae for the dual conformal field theory, for all values of the level $k$. The results are then applied to non-abelian target space duality in string theory, showing that the standard formulae are quantum mechanically well defined in the low energy limit if the gauged group is semisimple. 
  We develop the general formalism for performing perturbative diagrammatic expansions in the lattice theory of quantum gravity. The results help establish a precise correspondence between continuum and lattice quantities, and should be a useful guide for non-perturbative studies of gravity. The Feynman rules for Regge's simplicial lattice formulation of gravity are then discussed in detail in two dimensions. As an application, the two-dimensional conformal anomaly due to a $D$-component scalar field is explicitly computed in perturbation theory. 
  The RST model is augmented by the addition of a scalar field and a boundary term so that it is well-posed and local. Expressing the RST action in terms of the ADM formulation, the constraint structure can be analysed completely. It is shown that from the view point of local field theories, there exists a hidden dynamical field in the RST model. Thanks to the presence of this hidden dynamical field, we can reconstruct the closed algebra of the constraints which guarantee the general invariance of the RST action. The resulting stress tensors are recovered to be true tensor quantities. At the quantum level, the cancellation condition for the total central charge is reexamined. Finally, with the help of the hidden dynamical field, the fact that the semi-classical static solution of the RST model has two independent parameters (P,M), whereas for the classical CGHS model there is only one, can be explained. 
  The significance of zero modes in the path-integral quantization of some solitonic models is investigated. In particular a Skyrme-like theory with topological vortices in (1+2) dimensions is studied, and with a BRST invariant gauge fixing a well defined transition amplitude is obtained in the one loop approximation. We also present an alternative method which does not necessitate evoking the time-dependence in the functional integral, but is equivalent to the original one in dealing with the quantization in the background of the static classical solution of the non-linear field equations. The considerations given here are particularly useful in - but also limited to - the one-loop approximation. 
  We present a detailed discussion of Spontaneous Symmetry Breaking (SSB) in $(\lambda\Phi^4)_4$. In the usual approach, inspired by perturbation theory, one predicts a second-order phase transition, the Higgs mass $m_h$, related to the value of the renormalized 4-point coupling, gets smaller when increasing the ultraviolet cutoff and this leads to the generally quoted upper bounds $m_h<$700-900 GeV. On the other hand, by exploring the structure of the effective potential in those approximation consistent with `triviality', where the Higgs mass does not represent a measure of any observable interaction, SSB does not require an ultraviolet cutoff, the phase transition is first-order, such that the massless `Coleman-Weinberg' regime lies in the broken phase, and one gets only $m_h<$3 TeV from vacuum stability. To separate out the two alternatives, we present a precise lattice computation of the slope of the effective potential in the region of bare parameters indicated by the Luscher~\&~Weisz and Brahm's analysis of the critical line. Our lattice data strongly support the latter description of SSB. Indeed, our data cannot be reproduced in perturbation theory, and then they confirm the existence on the lattice of a remarkable phase of $(\lambda\Phi^4)_4$ where SSB is generated through ``dimensional transmutation'', and show no evidence for residual self-interaction effects of the shifted ``Higgs'' field $h(x)=\Phi(x)-\langle\Phi\rangle$, in agreement with ``triviality''. 
  Charge and magnetic flux bearing operators are introduced in Chern-Simons theory both in its pure form and when it is coupled to fermions. The magnetic flux creation operator turns out to be the Wilson line. The euclidean correlation functions of these operators are shown to be local and are evaluated exactly in the pure case and through an expansion in the inverse fermion mass whenever these are present. Physical states only occur in the presence of fermions and consist of composite charge-magnetic flux carrying states which are in general anyonic. The large distance behavior of the correlation functions indicates the condensation of charge and magnetic flux. 
  Gauge-invariant boundary conditions in Euclidean quantum gravity can be obtained by setting to zero at the boundary the spatial components of metric perturbations, and a suitable class of gauge-averaging functionals. This paper shows that, on choosing the de Donder functional, the resulting boundary operator involves projection operators jointly with a nilpotent operator. Moreover, the elliptic operator acting on metric perturbations is symmetric. Other choices of mixed boundary conditions, for which the normal components of metric perturbations can be set to zero at the boundary, are then analyzed in detail. Last, the evaluation of the 1-loop divergence in the axial gauge for gravity is obtained. Interestingly, such a divergence turns out to coincide with the one resulting from transverse-traceless perturbations. 
  This paper analyses quantum mechanics in multiply connected spaces. It is shown that the multiple connectedness of the configuration space of a physical system can determine the quantum nature of physical observables, such as the angular momentum. In particular, quantum mechanics in compactified Kaluza-Klein spaces is examined. These compactified spaces give rise to an additional angular momentum which can adopt half-integer values and, therefore, may be identified with the intrinsic spin of a quantum particle. 
  We describe the reduction from four to two dimensions of the SU(2) Donaldson-Witten theory and the dual twisted Seiberg-Witten theory, i.e. the Abelian topological field theory corresponding to the Seiberg--Witten monopole equations. 
  We present a general method of constructing Winf and winf gauge theories in terms of d+2 dimensional local fields. In this formulation the \Winf gauge theory Lagrangians involve non-local interactions, but the winf theories are entirely local. We discuss the so-called classical contraction procedure by which we derive the Lagrangian of winf gauge theory from that of the corresponding Winf gauge theory. In order to discuss the relationship between quantum Winf and quantum winf gauge theory we solve d=1 gauge theory models of a Higgs field exactly by using the collective field method. Based on this we conclude that the Winf gauge theory can be regarded as the large N limit of the corresponding SU(N) gauge theory once an appropriate coupling constant renormalization is made, while the winf gauge theory cannot be. 
  We propose using the method of subtraction to renormalize quantum gauge theories with chiral fermions and with spontaneous symmetry breaking. The Ward-Takahashi identities derived from the BRST invariance in these theories are complex and rich in content. We demonstrate how to use these identities to determine relationships among renormalization constants of the theory and obtain the subtraction constants needed for the renormalization procedure. We have found it particularly convenient to adopt the Landau gauge throughout the scheme. The method of renormalization by subtraction enables one to calculate physical quantities in the theory in the form of a renormalized perturbation series which is unique and definite. There is no ambiguity in handling the $\gamma_5$ matrix associated with chiral fermions. 
  We calculate the triangular anomaly in the next order in the Abelian-Higgs theory with an appropriate number of families of chiral fermions and with Yukawa couplings. The calculation is performed with the method of subtraction aided by the Ward- Takahashi identities. This anomaly amplitude is performed without any regularization and is found to vanish. 
  We describe curved-space BPS dyon solutions, the ADM mass of which saturates the gravitational version of the Bogomol'nyi bound. This generalizes self-gravitating BPS monopole solutions of Gibbons et al. when there is no dilaton. 
  We study the quantum moduli space of vacua of $N=2$ supersymmetric $SU(N_c)$ gauge theories coupled to $N_f$ flavors of quarks in the fundamental representation. We identify the moduli space of the $N_c = 3$ and $N_f=2$ massless case with the full spectral curve obtained from the Lax representation of the Goryachev-Chaplygin top. For the case with {\it massive} quarks, we present an integrable system where the corresponding hyperelliptic curve parametrizing the Laurent solution coincides with that of the moduli space of $N_{c}=3$ with $N_{f}=0, 1, 2$. We discuss possible generalizations of the integrable systems relevant to gauge theories with $N_c \neq 3 $ and general $N_f$. 
  The dual superconductivity of QCD vacuum as a mechanism for colour confinement is reviewed. Recent evidence from lattice of monopole condensation is presented. 
  Preserving the T-duality invariance of the continuum string in its random lattice regularization uniquely determines the random matrix model potential. For D=0 the duality transformation can be performed explicitly on the matrix action, and replaces color with flavor; invariance thus requires that the color and flavor groups be the same. 
  The covariant path integral for chiral bosons obtained by McClain, Wu and Yu is generalized to chiral p-forms. In order to handle the reducibility of the gauge transformations associated with the chiral p-forms and with the new variables (in infinite number) that must be added to eliminate the second class constraints, the field-antifield formalism is used. 
  In this article we show that if the electrons in a quantum Hall sample are subjected to a constant electric field in the plane of the material, comparable in magnitude to the background magnetic field on the system of electrons, a multiplicity of edge states localised in different regions of space is produced in the sample. The actions governing the dynamics of these edge states are obtained starting from the well-known Schr\"odinger field theory for a system of non-relativistic electrons, where on top of the constant background electric and magnetic fields, the electrons are further subject to slowly varying weak electromagnetic fields. In the regions between the edges, dubbed as the "bulk", the fermions can be integrated out entirely and the dynamics expressed in terms of a local effective action involving the slowly varying electromagnetic potentials. It is further shown how the "bulk" action is gauge non-invariant in a particular way and how the edge states conspire to restore the U(1) electromagnetic gauge invariance of the system. 
  We present an ansatz which enables us to construct heterotic/M-theory dual pairs in four dimensions. It is checked that this ansatz reproduces previous results and that the massless spectra of the proposed dual pairs agree. The new dual pairs consist of M-theory compactifications on Joyce manifolds of $G_2$ holonomy and Calabi-Yau compactifications of heterotic strings. These results are further evidence that M-theory is consistent on orbifolds. Finally, we interpret these results in terms of M-theory geometries which are K3 fibrations and heterotic geometries which are conjectured to be $T^3$ fibrations. Even though the new dual pairs are constructed as non-freely acting orbifolds of existing dual pairs, the adiabatic argument is apparently not violated. 
  We consider supersymmetric Sp(2N) gauge theories with F matter fields in the defining representation, one matter field in the adjoint representation, and no superpotential. We construct a sequence of dual descriptions of this theory using the dualities of Seiberg combined with the ``deconfinement'' method introduced by Berkooz. Our duals hint at a new non-perturbative phenomenon that seems to be taking place at asymptotically low energies in these theories: for small F some of the degrees of freedom form massless, non-interacting bound states while the theory remains in an interacting non-Abelian Coulomb phase. This phenomenon is the result of strong coupling gauge dynamics in the original description, but has a simple classical origin in the dual descriptions. The methods used for constructing these duals can be generalized to any model involving arbitrary 2-index tensor representations of Sp(2N), SO(N), or SU(N) groups. 
  An explicit canonical construction of monopole connections on non trivial U(1) bundles over Riemann surfaces of any genus is given. The class of monopole solutions depend on the conformal class of the given Riemann surface and a set of integer weights. The reduction of Seiberg-Witten 4-monopole equations to Riemann surfaces is performed. It is shown then that the monopole connections constructed are solutions to these equations. 
  By using the Ward-Takahashi identities in the Landau gauge, we derive exact relations between particle masses and the vacuum expectation value of the Higgs field in the Abelian gauge field theory with a Higgs meson. 
  Heterotic strings on $R^6 \times K3$ generically appear to undergo some interesting new phase transition at that value of the string coupling for which the one of the six-dimensional gauge field kinetic energies changes sign. An exception is the $E_8 \times E_8$ string with equal instanton numbers in the two $E_8$'s, which admits a heterotic/heterotic self-duality. In this paper, we generalize the dyonic string solution of the six-dimensional heterotic string to include non-trivial gauge field configurations corresponding to self-dual Yang-Mills instantons in the four transverse dimensions. We find that vacua which undergo a phase transition always admit a string solution exhibiting a naked singularity, whereas for vacua admitting a self-duality the solution is always regular. When there is a phase transition, there exists a choice of instanton numbers for which the dyonic string is tensionless and quasi-anti-self-dual at that critical value of the coupling. For an infinite subset of the other choices of instanton number, the string will also be tensionless, but all at larger values of the coupling. 
  I review my new method for solving general 1-matrix models by expanding in $N^{-1}$ without taking a physical continuum limit. Using my method, each coefficient of the free energy in the genus expansion is exactly computable. One can include physical information in a function which is uniquely specified by the action of the model. My method gives completely the same result with the usual one if the physical fine tuning is done and the leading singular terms are extracted. I also calculate in the genus three case and confirm the validity of my method. 
  We extend our previous analysis of the classical integrable models of Calogero in several respects. Firstly we provide the algebraic resaons of their quantum integrability.Secondly we show why these systems allow their initial value problem to be solved in closed form . Furthermore we show that due to their similarity with the above models the classical and quantum Heisenberg magnets with long rang interactions in a magnetic field are also intergrable. Explicit expressions are given for the integrals of motion in involution in the classical case and for the commuting operators in the quantum case. 
  We reconsider the, by Brink and Vasiliev, recently proposed generalized Toda field theories using the framework of WZNW$\rightarrow$Toda reduction. The reduced theory has a gauge symmetry which can be fixed in various ways. We discuss some different gauge choices. In particular we study the ${\cal W}$ algebra associated with the generalized model in some different realizations, corresponding to different gauge choices. We also investigate the mapping between the Toda field and a free field and show the relation between the ${\cal W}$ algebra generators expressed in terms of the two different fields. All results apply also to the case of ordinary Toda theories. 
  The Becchi-Rouet-Stora-Tyutin (BRST) treatment for the quantization of collective coordinates is considered in the Lagrangian formalism. The motion of a particle in a Riemannian manifold is studied in the case when the classical solutions break a non-abelian global invariance of the action. Collective coordinates are introduced, and the resulting gauge theory is quantized in the BRST antifield formalism. The partition function is computed perturbatively to two-loops, and it is shown that the results are independent of gauge-fixing parameters. 
  We analyze in detail the moduli space of vacua of N=2 SUSY QCD with n_c colors and n_f flavors. The Coulomb branch has submanifolds with non-Abelian gauge symmetry. The massless quarks and gluons at these vacua are smoothly connected to the underlying elementary quarks and gluons. Upon breaking N=2 by an N=1 preserving mass term for the adjoint field the theory flows to N=1 SUSY QCD. Some of the massless quarks and gluons on the moduli space of the N=2 theory become the magnetic quarks and gluons of the N=1 theory. In this way we derive the duality in N=1 SUSY QCD by identifying its crucial building blocks---the magnetic degrees of freedom---using only semiclassical physics and the non-renormalization theorem. 
  We semi-classically calculate the entropy of a scalar field in the background of the BTZ black hole, and derive the perimeter law of the entropy. The proper length from the horizon to the ultraviolet cutoff is independent of both the mass and the angular momentum of the black hole. It is shown that the superradiant scattering modes give the sub-leading order contribution to the entropy while the non-superradiant modes give the leading order one, and thus superradiant effect is minor. 
  We quantize the tachyon field in the two-dimensional (2D), $\epsilon<2$ charged black holes where $\epsilon$ is the dilaton coupling parameter for the Maxwell term. Especially the expectation value of the stress-energy tensor $\langle T_{ab}\rangle$, observed by a freely falling observer, is computed. This shows that new divergences such as $\ln f$ and ${1 \over f}$ arises near the horizon ($f \to 0$), compared with conformal matter case. 
  The quantization of spontaneously broken gauge theories in noncommutative geometry(NCG) has been sought for some time, because quantization is crucial for making the NCG approach a reliable and physically acceptable theory. Lee, Hwang and Ne'eman recently succeeded in realizing the BRST quantization of gauge theories in NCG in the matrix derivative approach proposed by Coquereaux et al. The present author has proposed a characteristic formulation to reconstruct a gauge theory in NCG on the discrete space $M_4\times Z_{_N}$. Since this formulation is a generalization of the differential geometry on the ordinary manifold to that on the discrete manifold, it is more familiar than other approaches. In this paper, we show that within our formulation we can obtain the BRST invariant Lagrangian in the same way as Lee, Hwang and Ne'eman and apply it to the SU(2)$\times$U(1) gauge theory. 
  A non homogeneous spin chain in the representations $ \{3 \}$ and $ \{3^*\}$ of $A_2$ is analyzed. We find that the naive nested Bethe ansatz is not applicable to this case. A method inspired in the nested Bethe ansatz, that can be applied to more general cases, is developed for that chain. The solution for the eigenvalues of the trace of the monodromy matrix is given as two coupled Bethe equations different from that for a homogeneous chain. A conjecture about the form of the solutions for more general chains is presented.   PACS: 75.10.Jm, 05.50+q 02.20 Sv 
  The $SO(4)\times U(1)$ Higgs model on $\R_4$ is extended by a $F^3$ term so that the action receives a nonvanishing contribution from the interactions of 2-instantons and 3-instantons, and can be expressed as the inverse of the Laplacian on $\R_4$ in terms of the mutual distances of the instantons. The one-instanton solutions of both the basic and the extended models have been studied in detail numerically. 
  We show that the classical non-abelian pure Chern-Simons action is related in a natural way to completely integrable systems of the Davey-Stewartson hyerarchy, via reductions of the gauge connection in Hermitian spaces and by performing certain gauge choices. The B\"acklund Transformations are interpreted in terms of Chern-Simons equations of motion or, on the other hand, as a consistency condition on the gauge. A mapping with a nonlinear $\sigma$-model is discussed. 
  We find that the fundamental quadratic form of classical string propagation in $2+1$ dimensional constant curvature spacetimes solves the Sinh-Gordon equation, the Cosh-Gordon equation or the Liouville equation. We show that in both de Sitter and anti de Sitter spacetimes (as well as in the $2+1$ black hole anti de Sitter spacetime), {\it all} three equations must be included to cover the generic string dynamics. The generic properties of the string dynamics are directly extracted from the properties of these three equations and their associated potentials (irrespective of any solution). These results complete and generalize earlier discussions on this topic (until now, only the Sinh-Gordon sector in de Sitter spacetime was known). We also construct new classes of multi-string solutions, in terms of elliptic functions, to all three equations in both de Sitter and anti de Sitter spacetimes. Our results can be straightforwardly generalized to constant curvature spacetimes of arbitrary dimension, by replacing the Sinh-Gordon equation, the Cosh-Gordon equation and the Liouville equation by higher dimensional generalizations. 
  The 3-body Calogero problem is solved by separation of variables for arbitrary exchange statistics. A numerical computation of the 4-body spectrum is also presented. The results display new features in comparison with the standard case of bosons and fermions, for instance the energies are not linear with the interaction parameter $\nu$ and Bethe ansatz as well as Haldane's statistics are not verified. 
  We discuss how the structure of massless monopoles in supersymmetric theories with a Coulomb phase can be obtained from effective superpotentials for a phase with a confined photon. To illustrate the technique, we derive effective superpotentials which can be used to derive the curves which describe the Coulomb phase of $N=2$, $SU(N_c)$ gauge theory with $N_f<N_c$ flavors. 
  A pseudoclassical model is proposed to describe massive Dirac (spin one-half) particles in arbitrary odd dimensions. The quantization of the model reproduces the minimal quantum theory of spinning particles in such dimensions. A dimensional duality between the model proposed and the pseudoclassical description of Weyl particles in even dimensions is discussed. 
  We first exhibit in the commutative case the simple algebraic relations between the algebra of functions on a manifold and its infinitesimal length element $ds$. Its unitary representations correspond to Riemannian metrics and Spin structure while $ds$ is the Dirac propagator $ds = \ts \!\!$---$\!\! \ts = D^{-1}$ where $D$ is the Dirac operator. We extend these simple relations to the non commutative case using Tomita's involution $J$. We then write a spectral action, the trace of a function of the length element in Planck units, which when applied to the non commutative geometry of the Standard Model will be shown (in a joint work with Ali Chamseddine) to give the SM Lagrangian coupled to gravity. The internal fluctuations of the non commutative geometry are trivial in the commutative case but yield the full bosonic sector of SM with all correct quantum numbers in the slightly non commutative case. The group of local gauge transformations appears spontaneously as a normal subgroup of the diffeomorphism group. 
  In this paper, we present a parity-preserving QED3 with spontaneous breaking of a local U(1)-symmetry. The breaking is accomplished by a potential of the \vf^6-type. It is shown that a net attractive interaction appears in the M{\o}ller scattering (s and p-wave scattering between two electrons) as mediated by the gauge field and a Higgs scalar. This might favour a pair-condensation mechanism. 
  By using the brick wall method we calculate the free energy and the entropy of the scalar field in the rotating black holes. As one approaches the stationary limit surface rather than the event horizon in comoving frame, those become divergent. Only when the field is comoving with the black hole (i.e. $\Omega_0 = \Omega_H$) those become divergent at the event horizon. In the Hartle-Hawking state the leading terms of the entropy are $ A \frac{1}{h} + B \ln(h) + finite$, where $h$ is the cut-off in the radial coordnate near the horizon. In term of the proper distance cut-off $\epsilon$ it is written as $ S = N A_H/\epsilon^2$. The origin of the divergence is that the density of state on the stationary surface and beyond it diverges. 
  We present a (2+1)-dimensional gauged $O(3) \sigma$-model with an Abelian Chern--Simons term. It shows topologically stable, anyonic vortices as classical solutions. The fields are studied in the case of rotational symmetry and analytic approximations are found for their asymptotic behaviour. The static Euler--Lagrange equations are solved numerically, where particular attention is paid to the dependence of the vortex' properties on the coupling to the gauge field. We compute the vortex mass and charge as a function of this coupling and obtain bound states for two--vortices as well as two--vortices with masses above the stability threshold. 
  We compute the corrections to the transition amplitudes of an accelerated Unruh ``box'' that arise when the accelerated box is replaced by a ``two level ion'' immersed in a constant electric field and treated in second quantization. There are two kinds of corrections, those due to recoil effects induced by the momentum transfers and those due to pair creation. Taken together, these corrections show that there is a direct relationship between pair creation amplitudes described by the Heisenberg-Euler-Schwinger mechanism and the Unruh effect, i.e. the thermalisation of accelerated systems at temperature $a/ 2 \pi$ where $a$ is the acceleration. In particular, there is a thermodynamical consistency between both effects whose origin is that the euclidean action governing pair creation rates acts as an entropy in delivering the Unruh temperature. Upon considering pair creation of charged black holes in an electric field, these relationships explain why black holes are created from vacuum in thermal equilibrium, i.e. with their Hawking temperature equal to their Unruh temperature. 
  I characterize the structure of the master field for $F^{0}_{z \bar z}$ in $SU(\infty)$-$YM_4$ on a product of two Riemann surfaces $Z \times W$ in the gauge $F^{ch}_{z \bar z}=0$ as the sum of a `bulk' constant term and of delta-like `contact' terms.\\ The contact terms may occur because the localization of the functional integral at $N=\infty$ on a master orbit of a constant connection under the action of singular gauge transformations is still compatible with the large-$N$ factorization and translational invariance.\\ In addition I argue that if the gauge group is unbroken and there is a mass gap, that is if the theory confines, the functional measure at $N=\infty$, in the gauge $F^{ch}_{z \bar z}=0$, must be localized on the moduli space of flat connections with punctures on $Z \times W$. 
  We review some examples of heterotic/type II string duality which shed light on the infrared dynamics of string compactifications with N=2 and N=1 supersymmetry in four dimensions. 
  String theory is used to count microstates of four-dimensional extremal black holes in compactifications with $N=4$ and $N=8$ supersymmetry. The result agrees for large charges with the Bekenstein-Hawking entropy. 
  We derive the Bekenstein-Hawking entropy formula for four-dimensional Reissner-Nordstrom extremal black holes in type II string theory. The derivation is performed in two separate (T-dual) weak coupling pictures. One uses a type IIB bound state problem of D5- and D1-branes, while the other uses a bound state problem of D0- and D4-branes with macroscopic fundamental type IIA strings. In both cases, the D-brane systems are also bound to a Kaluza-Klein monopole, which then yields the four-dimensional black hole at strong coupling. 
  We extend the idea of the generalized Pauli-Villars regularization of Frolov and Slavnov and analyze the general structure of the regularization scheme. The gauge anomaly-free condition emerges in a simple way in the scheme, and, under the standard prescription for the momentum assignment, the Pauli-Villars Lagrangian provides a gauge invariant regularization of chiral fermions in arbitrary anomaly-free representations. The vacuum polarization tensor is transverse, and the fermion number and the conformal anomalies have gauge invariant forms. We also point out that the real representation can be treated in a straightforward manner and the covariant regularization scheme is directly implemented. 
  $SO(10)$ grand unified models have an intermediate symmetry group in between $SO(10)$ and $SU(3)_{C} \otimes SU(2)_{L} \otimes U(1)_{Y}$. Hence they lead to a prediction for proton lifetime in agreement with the experimental lower limit. This paper reviews the recent work on the tree-level potential and the one-loop effective potential for such models, with application to inflationary cosmology. The open problems are the use of the most general form of tree-level potential for $SO(10)$ models in the reheating stage of the early universe, and the analysis of non-local effects in the semiclassical field equations for such models in Friedmann-Robertson-Walker backgrounds. 
  It is shown that a particular $q$-deformation of the Virasoro algebra can be interpreted in terms of the $q$-local field $\Phi (x)$ and the Schwinger-like point-splitted Virasoro currents, quadratic in $\Phi (x)$. The $q$-deformed Virasoro algebra possesses an additional index $\alpha$, which is directly related to point-splitting of the currents. The generators in the $q$-deformed case are found to exactly reproduce the results obtained by probing the fields $X(z)$ (string coordinate) and $\Phi (z)$ (string momentum) with the non-splitted Virasoro generators and lead to a particular representation of the $SU_q (1,1)$ algebra characterized by the standard conformal dimension $J$ of the field. Some remarks concerning the $q$-vertex operator for the interacting $q$-string theory are made. 
  Recently the existence of certain SU(2) BPS monopoles with the symmetries of the Platonic solids has been proved. Numerical results in an earlier paper suggest that one of these new monopoles, the tetrahedral 3-monopole, has a remarkable new property, in that the number of zeros of the Higgs field is greater than the topological charge (number of monopoles). As a consequence, zeros of the Higgs field exist (called anti-zeros) around which the local winding number has opposite sign to that of the total winding. In this letter we investigate the presence of anti-zeros for the other Platonic monopoles. Other aspects of anti-zeros are also discussed. 
  We examine the Kac-Schwarz problem of specification of point in Grassmannian in the restricted case of gap-one first-order differential Kac-Schwarz operators. While the pair of constraints satisfying $[{\cal K}_1,W] = 1$ always leads to Kontsevich type models, in the case of $[{\cal K}_1,W] = W$ the corresponding KP $\tau$-functions are represented as more sophisticated matrix integrals. 
  We discuss the limitations of 't Hooft's proposal for the black hole S-matrix. We find that the validity of the S-matrix implies violation of the semi-classical approximation at scales large compared to the Planck scale. We also show that the effect of the centrifugal barrier on the S-matrix is crucial even for large transverse distances. 
  We examine the non-equilibrium time evolution of the hadronic plasma produced in a relativistic heavy ion collision, assuming a spherical expansion into the vacuum. We study the $O(4)$ linear sigma model to leading order in a large-$N$ expansion. Starting at a temperature above the phase transition, the system expands and cools, finally settling into the broken symmetry vacuum state. We consider the proper time evolution of the effective pion mass, the order parameter $\langle \sigma \rangle$, and the particle number distribution. We examine several different initial conditions and look for instabilities (exponentially growing long wavelength modes) which can lead to the formation of disoriented chiral condensates (DCCs). We find that instabilities exist for proper times which are less than 3 fm/c. We also show that an experimental signature of domain growth is an increase in the low momentum spectrum of outgoing pions when compared to an expansion in thermal equilibrium. In comparison to particle production during a longitudinal expansion, we find that in a spherical expansion the system reaches the ``out'' regime much faster and more particles get produced. However the size of the unstable region, which is related to the domain size of DCCs, is not enhanced. 
  Seiberg-Witten solutions of four-dimensional supersymmetric gauge theories possess rich but involved integrable structures. The goal of this paper is to show that an isomonodromy problem provides a unified framework for understanding those various features of integrability. The Seiberg-Witten solution itself can be interpreted as a WKB limit of this isomonodromy problem. The origin of underlying Whitham dynamics (adiabatic deformation of an isospectral problem), too, can be similarly explained by a more refined asymptotic method (multiscale analysis). The case of $N=2$ SU($s$) supersymmetric Yang-Mills theory without matter is considered in detail for illustration. The isomonodromy problem in this case is closely related to the third Painlev\'e equation and its multicomponent analogues. An implicit relation to $t\tbar$ fusion of topological sigma models is thereby expected. 
  Using a collective field method, we obtain explicit solutions of the generalized Calogero-Sutherland models that are characterized by the roots of the classical groups $B_N$ and $C_N$. Starting from the explicit wave functions for $A_{N-1}$ type expressed in terms of the singular vectors of the $W_N$ algebra, we give a systematic method to construct wave functions and derive energy eigenvalues for other types of theories. 
  A construction of supersymmetric field-theoretical models in non-commutative geometry is reviewed. The underlying superstructure of the models is encoded in $osp(2,2)$ superalgebra. 
  We calculate the entropy of a scalar field in a rotating black hole in 2 + 1 dimension. In the Hartle-Hawking state the entropy is proportional to the horizon area, but diverges linearly in $\sqrt{h}$, where $h$ is the radial cut-off. In WKB approximation the superradiant modes do not contribute to the entropy. 
  The Picard-Fuchs equations for $N=2$ supersymmetric $SU(N_{c})$ Yang-Mills theories with massless hypermultiplets are obtained for $N_{c}=2$ and $3$. For $SU(2)$ we derive the non-linear differential equations for the prepotentials and calculate full non-perturbative corrections to the effective gauge coupling constant in the weak and strong coupling regions. 
  The phenomenological aspects of string theory are briefly reviewed. Emphasis is given to the status of 4D string model building, effective Lagrangians, model independent results, supersymmetry breaking and duality symmetries. 
  The infrared behaviour of the ${\phi}^3$-theory is discussed stressing analogies with the Witten-Seiberg story about $N=2$ $QCD$. Though the microscopic theory is apparently not integrable, the effective theory is shown to be integrable at classical level, and a general solution of it in terms of hypergeometric functions is obtained. An effective theory for the multiparticle soft scattering is sketched. 
  It is found that the recently published first coefficient of nonzero $\beta$-function for the Chern-Simons term in the $1/N$ expansion of the $CP^{N-1}$ model is untrue numerically. The correct result is given. The main conclusions of Park's paper are not changed. 
  The gravitational analogue of the electromagnetic Meissner effect is investigated. Starting from the post-Newtonian approximation to general relativity we arrive at gravitational London equations, predicting a gravitational Meissner effect. Applied to neutron stars we arrive at a London penetration depth of 12km, which is about the size of a neutron star. 
  A seven parameter family of five-dimensional black hole solutions depending on mass, two angular momenta, three charges and the asymptotic value of a scalar field is constructed. The entropy is computed as a function of these parameters both from the Bekenstein-Hawking formula and from the degeneracies of the corresponding D-brane states in string theory. The expressions agree at and to leading order away from extremality. 
  The Rubakov-Callan effect is reexamined by considering the gravitational effects caused by the heavy monopole mass. Assuming that the Higgs vacuum expectation value is as large as (or larger than) the Planck mass, we show that the calculational scheme of Rubakov and Callan may be extended in the presence of curved background field. It is argued that the density of the fermion condensate around a magnetically charged black hole is modified in an intricate way. 
  The spin - 3/2 Ising model on a square lattice is investigated. It is shown that this model is reducible to an eight - vertex model on a surface in the parameter space spanned by coupling constants J, K, L and M. It is shown that this model is equivalent to an exactly solvable free fermion model along two lines in the parameter space. 
  We study the low energy effective theory describing the dynamics of D-particles. This corresponds to the quantum mechanical system obtained by dimensional reduction of $9+1$ dimensional supersymmetric Yang-Mills theory to $0+1$ dimensions and can be interpreted as the non relativistic limit of the Born-Infeld action. We study the system of two like-charged D-particles and find evidence for the existence of non-BPS states whose mass grows like $\lambda^{1/3}$ over the BPS mass. We give a string interpretation of this phenomenon in terms of a linear potential generated by strings stretching from the two D-particles. Some comments on the possible relations to black hole entropy and eleven dimensional supergravity are also given. 
  In this paper we investigate how various equivalences between effective field theories of $N=2$ SUSY Yang-Mills theory with matter can be understood through Higgs breaking, i.e. by giving expectation values to squarks. We give explicit expressions for the flat directions for a wide class of examples. 
  Over the past five years, there has been significant progress on the problem of quantization of diffeomorphism covariant field theories with {\it local} degrees of freedom. The absence of a background space-time metric in these theories gives rise to a host of conceptual and technical difficulties because most of the familiar methods from axiomatic, constructive and perturbative quantum field theory are no longer applicable. Perhaps the most striking examples of these problems arise in the construction of a quantum field theory of geometry. We show that these problems can be tackled using new non-perturbative methods. In particular, one can rigorously define certain geometric operators and show that their spectrum is discrete. Thus, there is a precise sense in which the geometry is quantized at the Planck scale and the continuum picture is only a coarse-grained approximation. 
  We show that the replacement of the ``instantaneous'' 't Hooft's potential with the causal form suggested by equal time canonical quantization in light-cone gauge, which entails the occurrence of negative probability states, does not change the bound state spectrum when the difference is treated as a single insertion in the kernel. 
  The duality symmetry group of the cosets ${\textstyle SU(n,1)\over \textstyle SU(n)\otimes U(1)}$, which describe the moduli space of a two-dimensional subspace of an orbifold model with $(n-1)$ complex Wilson lines moduli, is discussed. The full duality group and its explicit action on the moduli fields are derived. 
  These notes present a pedagogical introduction to magnetic monopoles and exact electromagnetic duality in supersymmetric gauge theories. They are based on lectures given at the 1995 Trieste Summer School in High Energy Physics and Cosmology and at the 1995 Busstepp Summer School at Cosener's House. 
  We present the magnetic duals of G\"uven's electric-type solutions of D=11 supergravity preserving $1/4$ or $1/8$ of the D=11 supersymmetry. We interpret the electric solutions as $n$ orthogonal intersecting membranes and the magnetic solutions as $n$ orthogonal intersecting 5-branes, with $n=2,3$; these cases obey the general rule that $p$-branes can self-intersect on $(p-2)$-branes. On reduction to $D=4$ these solutions become electric or magnetic dilaton black holes with dilaton coupling constant $a=1$ (for $n=2$) or $a=1/\sqrt{3}$ (for $n=3$). We also discuss the reduction to D=10. 
  It has been known for some time that the (1,3) perturbations of the (2k+1,2) Virasoro minimal models have conserved currents which are also singular vectors of the Virasoro algebra. This also turns out to hold for the (1,2) perturbation of the (3k+-1,3) models. In this paper we investigate the requirement that a perturbation of an extended conformal field theory has conserved currents which are also singular vectors. We consider conformal field theories with W3 and (bosonic) WBC2 = W(2,4) extended symmetries. Our analysis relies heavily on the general conjecture of de Vos and van Driel relating the multiplicities of W-algebra irreducible modules to the Kazhdan-Lusztig polynomials of a certain double coset. Granting this conjecture, the singular-vector argument provides a direct way of recovering all known integrable perturbations. However, W models bring a slight complication in that the conserved densities of some (1,2)-type perturbations are actually subsingular vectors, that is, they become singular vectors only in a quotient module. 
  The T-duality transformations between open and closed superstrings in different D-manifolds are generalized to curved backgrounds with commuting isometries. We address some global aspects like the occurrence of orientifold boundaries in general sigma models, higher genus world sheets, and the case of non-compact isometries. The various world volume effective actions are shown to transform properly under T-duality. We also include a brief discussion of the canonical transformations of boundary states in the operator formalism. 
  The macroscopic entropy-area formula for supersymmetric black holes in N=2,4,8 theories is found to be universal: in d=4 it is always given by the square of the largest of the central charges extremized in the moduli space. The proof of universality is based on the fact that the doubling of unbroken supersymmetry near the black hole horizon requires that all central charges other than Z=M vanish at the attractor point for N=4,8. The ADM mass at the extremum can be computed in terms of duality symmetric quartic invariants which are moduli independent. The extension of these results for d=5, N=1,2,4 is also reported. A duality symmetric expression for the energy of the ground state with spontaneous breaking of supersymmetry is provided by the power 1/2 (2/3) of the black hole area of the horizon in d=4 (d=5). It is suggested that the universal duality symmetric formula for the energy of the ground state in supersymmetric gravity is given by the modulus of the maximal central charge at the attractor point in any supersymmetric theory in any dimension. 
  We examine the quantum motion of two particles interacting through a contact force which are confined in a rectangular domain in two and three dimensions. When there is a difference in the mass scale of two particles, adiabatic separation of the fast and slow variables can be performed. Appearance of the Berry phase and magnetic flux is pointed out. The system is reduced to a one-particle Aharonov-Bohm billiard in two-dimensional case. In three dimension, the problem effectively becomes the motion of a particle in the presence of closed flux string in a box billiard. 
  In the pseudo-euclidean metrics Chern-Simons gauge theory in the infrared region is found to be associated with dissipative dynamics. In the infrared limit the Lagrangian of 2+1 dimensional pseudo-euclidean topologically massive electrodynamics has indeed the same form of the Lagrangian of the damped harmonic oscillator. On the hyperbolic plane a set of two damped harmonic oscillators, each other time-reversed, is shown to be equivalent to a single undamped harmonic oscillator. The equations for the damped oscillators are proven to be the same as the ones for the Lorentz force acting on two particles carrying opposite charge in a constant magnetic field and in the electric harmonic potential. This provides an immediate link with Chern-Simons-like dynamics of Bloch electrons in solids propagating along the lattice plane with hyperbolic energy surface. The symplectic structure of the reduced theory is finally discussed in the Dirac constrained canonical formalism. 
  We determine the two--dimensional Weyl, Lorentz and $\kappa$--anomalies in the $D=10$ Green--Schwarz heterotic string sigma--model, in an $SO(1,9)$-Lorentz covariant background gauge, and prove their cancellation. 
  The fields of the conjectured ``heterotic" super--fivebrane sigma--model in ten dimensions are made out of a well known gravitational sector, the $X$ and the $\vartheta$, and of a still unknown heterotic sector which should be coupled to the Yang--Mills fields. We compute the one--loop $d=6$ worldvolume and $D=10$ target space Lorentz--anomalies which arise from the gravitational sector of the heterotic super--fivebrane sigma--model, using a method which we developed previously for the Green--Schwarz heterotic superstring. These anomalies turn out to carry an overall coefficient which is $1/2$ of that required by the string/fivebrane duality conjecture. As a consequence the worldvolume anomaly vanishes if the heterotic fields consist of 16 (rather than 32) complex Weyl fermions on the worldvolume. This implies that the string/fivebrane duality conjecture can not be based on a ``heterotic" super--fivebrane sigma--model with only fermions in the heterotic sector. Possible implications of this result are discussed. 
  The Connes and Lott reformulation of the strong and electroweak model represents a promising application of noncommutative geometry. In this scheme the Higgs field naturally appears in the theory as a particular `gauge boson', connected to the discrete internal space, and its quartic potential, fixed by the model, is not vanishing only when more than one fermion generation is present. Moreover, the exact hypercharge assignments and relations among the masses of particles have been obtained. This paper analyzes the possibility of extensions of this model to larger unified gauge groups. Noncommutative geometry imposes very stringent constraints on the possible theories, and remarkably, the analysis seems to suggest that no larger gauge groups are compatible with the noncommutative structure, unless one enlarges the fermionic degrees of freedom, namely the number of particles. 
  It is generally accepted that the double-scaled 1D matrix model is equivalent to the $c=1$ string theory with tachyon condensation. There remain however puzzles that are to be clarified in order to utilize this connection for our quest towards possible non-perturbative formulation of string theory. We discuss some of the issues that are related to the space-time interpretation of matrix models, in particular, the questions of leg poles, causality, and black hole background. Finally, a speculation about a possible connection of a deformed matrix model with the idea of Dirichret brane is presented. 
  In non-diagonal conformal models, the boundary fields are not directly related to the bulk spectrum. We illustrate some of their features by completing previous work of Lewellen on sewing constraints for conformal theories in the presence of boundaries. As a result, we include additional open sectors in the descendants of $D_{odd}$ $SU(2)$ WZW models. A new phenomenon emerges, the appearance of multiplicities and fixed-point ambiguities in the boundary algebra not inherited from the closed sector. We conclude by deriving a set of polynomial equations, similar to those satisfied by the fusion-rule coefficients $N_{ij}^k$, for a new tensor $A_{a b}^i$ that determines the open spectrum. 
  We study the majority rule transformation applied to the Gibbs measure for the 2--D Ising model at the critical point. The aim is to show that the renormalized hamiltonian is well defined in the sense that the renormalized measure is Gibbsian. We analyze the validity of Dobrushin-Shlosman Uniqueness (DSU) finite-size condition for the "constrained models" corresponding to different configurations of the "image" system. It is known that DSU implies, in our 2--D case, complete analyticity from which, as it has been recently shown by Haller and Kennedy, Gibbsianness follows. We introduce a Monte Carlo algorithm to compute an upper bound to Vasserstein distance (appearing in DSU) between finite volume Gibbs measures with different boundary conditions. We get strong numerical evidence that indeed DSU condition is verified for a large enough volume $V$ for all constrained models. 
  We consider an extension of a special class of conformal sigma models (`chiral null models') which describe extreme supersymmetric string solutions. The new models contain both `left' and `right' vector couplings and should correspond to non-BPS backgrounds. In particular, we discuss a conformal six-dimensional model which is a combination of fundamental string and 5-brane models with two extra couplings representing rotations in orthogonal planes. If the two rotational parameters are independent the resulting background is found to be either singular or not asymptotically flat. The non asymptotically flat solution has a regular short distance limit described by a `twisted' product of SL(2,R) and SU(2) WZW theories with two twist parameters mixing the isometric Euler angles of SU(2) with a null direction of SL(2,R). 
  We present the most general rotating black hole solution of five-dimensional N=4 superstring vacua that conforms to the ``no hair theorem''. It is chosen to be parameterized in terms of massless fields of the toroidally compactified heterotic string. The solutions are obtained by performing a subset of O(8,24) transformations, i.e., symmetry transformations of the effective three-dimensional action for stationary solutions, on the five-dimensional (neutral) rotating solution parameterized by the mass m and two rotational parameters $l_1$ and $l_2$. The explicit form of the generating solution is determined by three $SO(1,1)\subset O(8,24)$ boosts, which specify two electric charges $Q_1^{(1)}, Q_{2}^{(2)}$ of the Kaluza-Klein and two-form U(1) gauge fields associated with the same compactified direction, and the charge Q (electric charge of the vector field, whose field strength is dual to the field strength of the five-dimensional two-form field). The general solution, parameterized by 27 charges, two rotational parameters and the ADM mass compatible with the Bogomol'nyi bound, is obtained by imposing $[SO(5)\times SO(21)]/[SO(4)\times SO(20)]\subset O(5,21)$ transformations, which do not affect the five-dimensional space-time. We also analyze the deviations from the BPS-saturated limit. 
  Gauged linear sigma models with C^m-valued scalar fields and gauge group U(1)^d, d \leq m, have soliton solutions of Bogomol'nyi type if a suitably chosen potential for the scalar fields is also included in the Lagrangian. Here such models are studied on (2+1)-dimensional Minkowski space. If the dynamics of the gauge fields is governed by a Maxwell term the appropriate potential is a sum of generalised Higgs potentials known as Fayet-Iliopoulos D-terms. Many interesting topological solitons of Bogomol'nyi type arise in models of this kind, including various types of vortices (e.g. Nielsen-Olesen, semilocal and superconducting vortices) as well as, in certain limits, textures (e.g. CP^(m-1) textures and gauged CP^(m-1) textures). This is explained and general results about the spectrum of topological defects both for broken and partially broken gauge symmetry are proven. When the dynamics of the gauge fields is governed by a Chern-Simons term instead of a Maxwell term a different scalar potential is required for the theory to be of Bogomol'nyi type. The general form of that potential is given and a particular example is discussed. 
  We calculate the electric charge at finite temperature $T$ for non-Abelian monopoles in spontaneously broken gauge theories with a CP violating $\theta$-term. A careful treatment of dyon's gauge degrees of freedom shows that Witten formula for the dyon charge at $T=0$, $ Q = e(n - \theta/2\pi) $, remains valid at $T \ne 0$. 
  In this paper we show that a quasi-exactly solvable (normalizable or periodic) one-dimensional Hamiltonian satisfying very mild conditions defines a family of weakly orthogonal polynomials which obey a three-term recursion relation. In particular, we prove that (normalizable) exactly-solvable one-dimensional systems are characterized by the fact that their associated polynomials satisfy a two-term recursion relation. We study the properties of the family of weakly orthogonal polynomials defined by an arbitrary one-dimensional quasi-exactly solvable Hamiltonian, showing in particular that its associated Stieltjes measure is supported on a finite set. From this we deduce that the corresponding moment problem is determined, and that the $k$-th moment grows like the $k$-th power of a constant as $k$ tends to infinity. We also show that the moments satisfy a constant coefficient linear difference equation, and that this property actually characterizes weakly orthogonal polynomial systems. 
  We review the basic principles of the construction of open and unoriented superstring models and analyze some representative examples. 
  In this paper we propose algebraic universal procedure for deriving "fusion rules" and Baxter equation for any integrable model with $U_q(\widehat{sl}_2)$ symmetry of Quantum Inverse Scattering Method. Universal Baxter Q- operator is got from the certain infinite dimensional representation called q-oscillator one of the Universal R- matrix for $U_q(\widehat{sl}_2)$ affine algebra (first proposed by V. Bazhanov, S.Lukyanov and A.Zamolodchikov for quantum KdV case). We also examine the algebraic properties of Q-operator. 
  The off-shell entropy for a massless scalar field in a D-dimensional Rindler-like space-time is investigated within the conical Euclidean approach in the manifold $C_\be\times\M^N$, $C_\be$ being the 2-dimensional cone, making use of the zeta-function regularisation. Due to the presence of conical singularities, it is shown that the relation between the zeta-function and the heat kernel is non trivial and, as first pointed out by Cheeger, requires a separation between small and large eigenvalues of the Laplace operator. As a consequence, in the massless case, the (naive) non existence of the Mellin transform is by-passed by the Cheeger's analytical continuation of the zeta-function on manifold with conical singularities. Furthermore, the continuous spectrum leads to the introduction of smeared traces. In general, it is pointed out that the presence of the divergences may depend on the smearing function and they arise in removing the smearing cutoff. With a simple choice of the smearing function, horizon divergences in the thermodynamical quantities are recovered and these are similar to the divergences found by means of off-shell methods like the brick wall model, the optical conformal transformation techniques or the canonical path integral method. 
  A string-like model with the "cosmological constant" $\Lambda$ is considered. The Maki-Shiraishi multi-black-hole solution \cite{MS1} is generalized to space-times with a Ricci-flat internal space. For $\Lambda = 0$ the obtained solution in the one-black-hole case is shown to coincide with the extreme limit of the charged dilatonic black hole solution \cite{BI,BM}. The Hawking temperature $T_H$ for the solution \cite{BI,BM} is presented and its extreme limit is considered. For the string value of dilatonic coupling the temperature $T_H$ does not depend upon the internal space dimension. 
  In this note, we comment on Calabi-Yau spaces with Hodge numbers $h_{1,1}=3$ and $h_{2,1}=243$. We focus on the Calabi-Yau space $WP_{1,1,2,8,12}(24)$ and show how some of its instanton numbers are related to coefficients of certain modular forms. We also comment on the relation of four dimensional exchange symmetries in certain $N=2$ dual models to six dimensional heterotic/heterotic string duality. 
  A six-parameter family of five-dimensional black hole solutions is constructed which are labeled by their mass, two asymptotic scalar fields and three charges. It is shown that the Bekenstein-Hawking entropy is exactly matched, arbitrarily far from extremality, by a simple but mysterious duality-invariant extension of previously derived formulae for the number of D-brane states in string theory. 
  The symmetries of the tree level string effective action are discussed. An appropriate effective action is constructed starting from the manifestly SL(2,R) invarint form of string effective action introduced by Schwarz and Sen. The conserved charges are derived and generators of infinitesimal transformations are obtained in the Hamiltonian formalism. Some interesting consequences of the canonical transformations are explored. 
  The principle of boundary bootstrap plays a significant role in the algebraic study of the purely elastic boundary reflection matrix $K_a(\theta)$ for integrable quantum field theory defined on a space-time with a boundary. However, general structure of that principle in the form as was originally introduced by Fring and K\"oberle has remained unclear. In terms of a new matrix $J_a(\theta)=\sqrt{K_a(\theta)/K_{\bar{a}}(i\pi +\theta)}$, the boundary bootstrap takes a simple form. Incidentally, a hypothesised expression of the boundary reflection matrix for simply-laced $ADE$ affine Toda field theory defined on a half line with the Neumann boundary condition is obtained in terms of geometrical quantities of root systems \`a la Dorey. 
  We propose an approach to treat (1+1)--dimensional fermionic systems based on the idea of algebraic bosonization. This amounts to decompose the elementary low-lying excitations around the Fermi surface in terms of basic building blocks which carry a representation of the W_{1+\infty} \times {\overline W_{1+\infty}} algebra, which is the dynamical symmetry of the Fermi quantum incompressible fluid. This symmetry simply expresses the local particle-number current conservation at the Fermi surface. The general approach is illustrated in detail in two examples: the Heisenberg and Calogero-Sutherland models, which allow for a comparison with the exact Bethe Ansatz solution. 
  It is shown that many of the conjectured dualities involving orbifold compactification of M-theory follow from the known dualities involving M-theory and string theory in ten dimensions, and the ansatz that orbifolding procedure commutes with the duality transformation. This ansatz also leads to a new duality conjecture, namely that M-theory compactified on $T^8/Z_2$ is dual to type I string theory on $T^7$. In this case the `twisted sector states' in M-theory live on sixteen membranes transverse to the internal manifold. 
  Poisson algebraic structures on current manifolds (of maps from a finite dimensional Riemannian manifold into a 2-dimensional manifold) are investigated in terms of symplectic geometry. It is shown that there is a one to one correspondence between such current manifolds and Poisson current algebras with three generators. A geometric meaning is given to q-deformations of current algebras. The geometric quantization of current algebras and quantum current algebraic maps is also studied. 
  We discuss a recent approach to quantum field theoretical path integration on noncommutative geometries which imply UV/IR regularising finite minimal uncertainties in positions and/or momenta. One class of such noncommutative geometries arise as `momentum spaces' over curved spaces, for which we can now give the full set of commutation relations in coordinate free form, based on the Synge world function. 
  String and membrane dynamics may be unified into a theory of 2+2 dimensional self-dual world-volumes living in a 10+2 dimensional target space. Some of the vacua of this M-theory are described by the N=(2,1) heterotic string, whose target space theory describes the world-volume dynamics of 2+2 dimensional `M-branes'. All classes of string and membrane theories are realized as particular vacua of the N=(2,1) string: Type IIA/B strings and supermembranes arise in the standard moduli space of toroidal compactifications, while type ${\rm I}'$ and heterotic strings arise from a $\bf Z_2$ orbifold of the N=2 algebra. Yet another vacuum describes M-theory on a ${\bf T}^5/{\bf Z}_2$ orientifold, the type I string on $ {\bf T}^4$, and the six-dimensional self-dual string. We find that open membranes carry `Chan-Paton fields' on their boundaries, providing a common origin for gauge symmetries in M-theory. The world-volume interactions of M-brane fluctuations agree with those of Born-Infeld effective dynamics of the Dirichlet two-brane in the presence of a non-vanishing electromagnetic field on the brane. 
  $p'$-brane solutions to rank $p+1$ composite antisymmetric tensor field theories of the kind developed by Guendelman, Nissimov and Pacheva are found when the dimensionality of spacetime is $D=(p+1)+(p'+1)$. These field theories posses an infinite dimensional group of global Noether symmetries, that of volume-preserving diffeomorphisms of the target space of the scalar primitive field constituents. Crucial in the construction of $p'$ brane solutions are the duality transformations of the fields and the local gauge field theory formulation of extended objects given by Aurilia, Spallucci and Smailagic. Field equations are rotated into Bianchi identities after the duality transformation is performed and the Clebsch potentials associated with the Hamilton-Jacobi formulation of the $p'$ brane can be identified with the $duals$ of the original scalar primitive constituents. Different types of Kalb-Ramond actions are discussed and a particular covariant action is presented which bears a direct relation to the light-cone gauge $p$-brane action. A simple derivation of $S$ and $T$ duality is also given. \medskip 
  We realize the string/$(D-5)$-brane duality on the action level between the $T^{10-D}$-compactified heterotic string effective action and the $(D-5)$- brane effective action in $D$ dimensions by managing a Lagrange multiplier field. A dual dictionary is composed to be available for the translation between the elementary or solitonic solutions of the dual pair of actions. In the same way the $S$ duality is also reconstructed on the action level as a double dualization for the $T^6$-compactified heterotic string effective action. 
  We propose the use of Heaviside transform with respect to the quark mass to investigate dynamical aspects of QCD. We show that at large momentum transfer the transformed propagator of massive quarks behaves softly and thus the dominant effect of explicit chiral symmetry breaking disappears through Heaviside transform. This suggests that the massless approximation would be more convenient to do in the transformed quantity than in the original one. As an example of explicit approximation, we estimate the massless value of the quark condensate. 
  It is shown that the string equation can be obtain from field equations. Such work is performed to scalar field. The equation obtained in nonrelativistic limit describes the nonlinear string. Such string has the effective elasticity connencted with the local string curvature. Some examples of the movement such nonlinear elastic string are considered. 
  For a simplicial manifold we construct the differential geometry structure and use it to investigate linear connections, metric and gravity. We discuss and compare three main approaches and calculate the resulting gravity action functionals. 
  We consider a $CP^N$ model with the subgroup $SU(r)$ completely gauged, where $1 < r < N+1$. The gauge field dynamics is solely governed by a nonabelian Chern-Simons term and the global $SU(N+1)$ symmetry is broken explicitly by introducing a $SU(r)$ invariant scalar potential. We obtain self-dual equations of this gauged $CP^N$ model and find that the energy is bounded from below by a linear combination of the topological charge and a global $U(1)$ charge present in the theory. We also discuss on the self-dual soliton solutions of this model. 
  We show how the $T$--duality between $D$--branes is realized (i) on $p$--brane solutions $(p=0,\cdots ,9)$ of IIA/IIB supergravity and (ii) on the $D$--brane actions ($p=0,\cdots ,3)$ that act as source terms for the $p$--brane solutions. We point out that the presence of a cosmological constant in the IIA theory leads, by the requirement of gauge invariance, to a topological mass term for the worldvolume gauge field in the 2--brane case. 
  The discrete symmetries of the Lorentz group are on the one hand a `complex' interplay between linear and anti-linear operations on spinor fields and on the other hand simple linear reflections of the Minkowski space. We define operations for T, CP and CPT leading to both kinds of actions. These operations extend the action of SL(2,C), representing the action of the proper orthochronous Lorentz group SO^+(1,3) on the Weyl spinors, to an action of the full group O(1,3). But it is more instructive to reverse the arguments. The action of O(1,3) is the natural way how SL(2,C) together with its conjugation structure acts on Minkowski space.  Focusing on the symmetries of these (anti-)linear operations we can for example distinguish between CP-invariant and CP-violating symmetries. This is important if gauge symmetries are included. It turns out that, contrary to the general belief, CP and T are not compatible with SU(n) for n > 2, especially with colour-SU(3) or with the U(3)-Cabibbo-Kobayashi-Maskawa matrix. 
  The relations between integrable Poisson algebras with three generators and two-dimensional manifolds are investigated. Poisson algebraic maps are also discussed. 
  We propose a formulation of 11-dimensional M-theory in terms of five-branes with closed strings on their world-volume. We use this description to construct the complete spectrum of BPS states in compactifications to six and five dimensions. We compute the degeneracy for fixed charge and find it to be in accordance with U-duality (which in our formulation is manifest in six dimensions) and the statistical entropy formula of the corresponding black hole. We also briefly comment on the four-dimensional case. 
  We consider low energy, non-relativistic scattering of two Dirichlet zero-branes as an exercise in quantum mechanics. For weak string coupling and sufficiently small velocity, the dynamics is governed by an effective U(2) gauge theory in 0+1 dimensions. At low energies, D-brane scattering can reliably probe distances much shorter than the string scale. The only length scale in the quantum mechanics problem is the eleven dimensional Planck length. This provides evidence for the role of scales shorter than the string length in the weakly coupled dynamics of type IIA strings. 
  We study the Chern-Simons $CP(N)$ models with a global $U(1)$ symmetry and found the self-dual models among them. The Bogomolnyi-type bound in these self-dual models is a nontrivial generalization of that in the pure $CP(N)$ models. Our models have quite a rich vacuum and soliton structure and approach the many known gauged self-dual models in some limit. 
  The exact solutions (Seiberg-Witten type) of $N=2$ supersymmetric Yang-Mills theory are discussed from the view of Whitham-Toda hierarchy. 
  The notion of index is applied to analyze the phase operator problem associated with the photon. We clarify the absence of the hermitian phase operator on the basis of an index consideration. We point out an interesting analogy between the phase operator problem and the chiral anomaly in gauge theory, and an appearance of a new class of quantum anomaly is noted. The notion of index, which is invariant under a wide class of continuous deformation, is also shown to be useful to characterize the representations of Q-deformed oscillator algebra. 
  We introduce new variables in four dimensional SU(N) Yang-Mills theory. These variables emerge when we sum the path integral over classical solutions and represent the summation as an integral over appropriate degrees of freedom. In this way we get an effective field theory with SU(N)$\times$SU(N) gauge symmetry. In the instanton approximation our effective theory has in addition a N=2 supersymmetry, and when we sum over all possible solutions we find a Parisi-Sourlas supersymmetry. These extra symmetries can then be broken explicitly by a SU(N) invariant and power counting renormalizable mass term. Our results suggest that the confinement mechanism which has been recently identified in the N=2 supersymmetric Yang-Mills theory might also help to understand color confinement in ordinary, pure Yang-Mills theory. In particular, there appears to be an intimate relationship between the N=2 supersymmetry approach to confinement and the Parisi-Sourlas dimensional reduction. 
  We link, by means of a semiclassical approach, the fractional statistics of particles obeying the Haldane exclusion principle to the Tsallis statistics and derive a generalized quantum entropy and its associated statistics. 
  Recent developments in string duality suggest that the string scale may not be irrevocably tied to the Planck scale. Two explicit but unrealistic examples are described where the ratio of the string scale to the Planck scale is arbitrarily small. Solutions which are more realistic may exist in the intermediate coupling or ``truly strong coupling'' region of the heterotic string. Weak scale superstrings have dramatic experimental consequences for both collider physics and cosmology. 
  We derive parts of the monopole and dyon spectra for N=2 super-Yang--Mills theories in four dimensions with gauge groups G of rank r>1 and matter multiplets. Special emphasis is put on G=SU(3) and those matter contents that yield perturbatively finite theories. There is no direct interpretation of the soliton spectra in terms of naive selfduality under strong--weak coupling and exchange of electric and magnetic charges. We argue that, in general, the standard procedure of finding the dyon spectrum will not give results that support a conventional selfduality hypothesis --- the SU(2) theory with four fundamental hypermultiplets seems to be an exception. Possible interpretations of the results are discussed. 
  The issues that arise when using the Cornwall-Jackiw-Tomboulis formalism in multi-field theories are investigated. Particular attention is devoted to the interplay between temperature effects, ultraviolet structure, and the interdependence of the gap equations. Results are presented explicitly in the case of the evaluation of the finite temperature effective potential of a theory with two scalar fields which has attracted interest as a toy model for symmetry nonrestoration at high temperatures. The lowest nontrivial order of approximation of the Cornwall-Jackiw-Tomboulis effective potential is shown to lead to consistent results, which are relevant for recent studies of symmetry nonrestoration by Bimonte and Lozano. 
  The Seiberg-Witten solution of N=2 supersymmetric SU(2) gauge theory may be viewed as a prediction for the infinite family of constants F_n measuring the n-instanton contribution to the prepotential F. Here we examine the instanton physics directly, in particular the contribution of the general self-dual solution of topological charge n constructed by Atiyah, Drinfeld, Hitchin and Manin (ADHM). In both the bosonic and supersymmetric cases, we determine both the large- and short-distance behavior of all the fields in this background. This allows us to construct the exact classical interaction between n ADHM (super-)instantons mediated by the adjoint Higgs. We calculate the one- and two-instanton contributions to the low-energy Seiberg-Witten effective action, and find precise agreement with their predicted values of F_1 and F_2. 
  We investigate the non-abelian $T$-duality of Wess-Zumino-Witten model. The obtained dual model is equivalent to the model dual to the $SU(2)$ chiral model found by Curtright-Zachos. This might tell that the Wess-Zumino term would be induced when the chiral model couples with gravity. 
  We study the $SU(2)_k/U(1)$-parafermion model perturbed by its first thermal operator. By formulating the theory in terms of a (perturbed) fermionic coset model we show that the model is equivalent to interacting WZW fields modulo free fields. In this scheme, the order and disorder operators of the $Z_k$ parafermion theory are constructed as gauge invariant composites. We find that the theory presents a duality symmetry that interchanges the roles of the spin and dual spin operators. For two particular values of the coupling constant we find that the theory recovers conformal invariance and the gauge symmetry is enlarged. We also find a novel self-dual point. 
  This paper surveys recent work on Lie algebras of differential operators and their application to the construction of quasi-exactly solvable Schroedinger operators. 
  The form of the spectral curve for $4d$ $N=2$ supersymmetric Yang-Mills theory with matter fields in the fundamental representation of the gauge group suggests that its $1d$ integrable counterpart should be looked for among (inhomogeneous) $sl(2)$ spin chains with the length of the chain being equal to the number of colours $N_c$. For $N_f < 2N_c$ the relevant spin chain is the simplest $XXX$-model, and this identification is in agreement with the known results in Seiberg-Witten theory. 
  When the swimming of micro-organisms is viewed from the string and membrane theories coupled to the velocity field of the fluid, a number of interesting results are derived; 1) importance of the area (or volume) preserving algebra, 2) usefulness of the $N$-point Reggeon (membranic) amplitudes, and of the gas to liquid transition in case of the red tide issues, 3) close relation between the red tide issue and the generation of Einstein gravity, and 4) possible understanding of the three different swimming ways of micro-organisms from the singularity structure of the shape space. 
  In this paper, we present a systematic analysis of eleven-dimensional supergravity on a manifold with boundary, which is believed to be relevant to the strong coupling limit of the $E_8\times E_8$ heterotic string. Gauge and gravitational anomalies enter at a very early stage, and require a refinement of the standard Green-Schwarz mechanism for their cancellation. This uniquely determines the gauge group to be a copy of $E_8$ for each boundary component, fixes the gauge coupling constant in terms of the gravitational constant, and leads to several striking new tests of the hypothesis that there is a consistent quantum $M$-theory with eleven-dimensional supergravity as its low energy limit. 
  A gauge theory with an underlying SU_q(2) quantum group symmetry is introduced, and its properties examined. With suitable assumptions, this model is found to have many similarities with the usual SU(2)\times U(1) Standard Model, specifically, the existence of four generators and thus four gauge fields. However, the two classical symmetries are unified into one quantum symmetry, and therefore there is only a single coupling constant, rather than two. By incorporating a Higgs sector into the model, one obtains several explicit tree-level predictions in the undeformed limit, such as the Weinberg angle: $sin^2\Theta_{W} = 3/11$. With the Z-boson mass m_Z and fine structure constant alpha as inputs, one can also obtain predictions for the weak coupling constant, the mass of the W, and the Higgs VEV. The breaking of the quantum invariance also results in a remaining undeformed U(1) gauge symmetry. 
  The Wess-Zumino term of the spinning string is constructed in terms of their anomalies using an extended field-antifield formalism. A new feature appears from a fact that the non-anomalous transformations do not form a sub-group. The algebra of the extended variables closes only using the equations of motion derived from the WZ term. 
  In quantum field theory with three-point and four-point couplings the Feynman diagrams of perturbation theory contain momentum independent subdiagrams, the ``tadpoles'' and ``snails''. With the help of Dyson-Schwinger equations we show how these can be summed up completely by a suitable modification of the mass and coupling parameters. This reduces the number of diagrams significantly. The method is useful for the organisation of perturbative calculations in higher orders. 
  On the occasion of the 60th birthday of Professor Keiji Kikkawa, Kikkawa-type physics performed at Ochanomizu was personally reviewed, and the generation of the metric is discussed with the condensation of the string fields. 
  We give the explicit expression for four-dimensional rotating charged black hole solutions of N=4 (or N=8) superstring vacua, parameterized by the ADM mass, four charges (two electric and two magnetic charges, each arising from a different U(1) gauge factors), and the angular momentum (as well as the asymptotic values of four toroidal moduli of two-torus and the dilaton-axion field). The explicit form of the thermodynamic entropy is parameterized in a suggestive way as a sum of the product of the `left-moving' and the `right-moving' terms, which may have an interpretation in terms of the microscopic degrees of freedom of the corresponding D-brane configuration. We also give an analogous parameterization of the thermodynamic entropy for the recently obtained five-dimensional rotating charged black holes parameterized by the ADM mass, three U(1) charges and two rotational parameters (as well as the asymptotic values of one toroidal modulus and the dilaton). 
  By introducing a nonlinearly transforming goldstino field non-super\-sym\-metric matter can be coupled to supergravity. This implies the possibility of coupling a standard model with one Higgs to supergravity. 
  A 2D- fractional supersymmetry theory is algebraically constructed. The Lagrangian is derived using an adapted superspace including, in addition to a scalar field, two fields with spins 1/3,2/3. This theory turns out to be a rational conformal field theory. The symmetry of this model goes beyond the super Virasoro algebra and connects these third-integer spin states. Besides the stress-momentum tensor, we obtain a supercurrent of spin 4/3. Cubic relations are involved in order to close the algebra; the basic algebra is no longer a Lie or a super-Lie algebra. The central charge of this model is found to be 5/3. Finally, we analyse the form that a local invariant action should take. 
  Phase transitions are studied in $M$-theory and $F$-theory. In $M$-theory compactification to five dimensions on a Calabi-Yau, there are topology-changing transitions similar to those seen in conformal field theory, but the non-geometrical phases known in conformal field theory are absent. At boundaries of moduli space where such phases might have been expected, the moduli space ends, by a conventional or unconventional physical mechanism. The unconventional mechanisms, which roughly involve the appearance of tensionless strings, can sometimes be better understood in $F$-theory. 
  We propose the gauged Thirring model as a natural gauge-invariant generalization of the Thirring model, four-fermion interaction of current-current type. In the strong gauge-coupling limit, the gauged Thirring model reduces to the recently proposed reformulation of the Thirring model as a gauge theory. Especially, we pay attention to the effect coming from the kinetic term for the gauge boson field, which was originally the auxiliary field without the kinetic term. In 3 + 1 dimensions, we find the nontrivial phase structure for the gauged Thirring model, based on the Schwinger-Dyson equation for the fermion propagator as well as the gauge-invariant effective potential for the chiral order parameter. Within this approximation, we study the renormalization group flows (lines of constant physics) and find a signal for nontrivial continuum limit with nonvanishing renormalized coupling constant and large anomalous dimension for the gauged Thirring model in 3+1 dimensions, at least for small number of flavors $N_f$. Finally we discuss the (perturbatively) renormalizable extension of the gauged Thirring model. 
  We study particles creation in arbitrary space-time dimensions by external electric fields, in particular, by fields, which are acting for a finite time. The time and dimensional analysis of the vacuum instability is presented. It is shown that the distributions of particles created by quasiconstant electric fields can be written in a form which has a thermal character and seems to be universal. Its application, for example, to the particles creation in external constant gravitational field reproduces the Hawking temperature exactly. 
  We describe a method for simulating the dynamics of an $S_3$ cosmic string network. We use a lattice Monte Carlo to generate initial conditions for the network, which subsequently is allowed to relax continuously according to a simplified model of string dynamics. The dynamics incorporates some novel features which, to our knowledge, have not been studied in previous numerical simulations: The existence of two types of string which may have different tensions, and the possibility that two non-commuting strings may intersect. Simulation of the non-commuting fluxes presents a computational challenge as it requires a rather complex gauge-fixing procedure. The flux definitions change as strings change their positions and orientations relative to each other and must be carefully updated as the network evolves. The method is described here in some detail, with results to be presented elsewhere. 
  In this paper, we present a new solution for the effective theory of Maxwell--Einstein--Dilaton, Low energy string and Kaluza--Klein theories, which contains among other solutions the well known Kaluza--Klein monopole solution of Gross--Perry--Sorkin as special case. We show also the magnetic and electric dipole solutions contained in the general one. 
  A point particle approximation to the classical dynamics of well separated vortices of the abelian Higgs model is developed. A static vortex is asymptotically identical to a solution of the linearized field theory (a Klein-Gordon/Proca theory) in the presence of a singular point source at the vortex centre. It is shown that this source is a composite scalar monopole and magnetic dipole, and the respective charges are determined numerically for various values of the coupling constant. The interaction potential of two well separated vortices is computed by calculating the interaction Lagrangian of two such point sources in the linear theory. The potential is used to model type II vortex scattering. 
  It is shown that there are logarithmic operators in D-brane backgrounds that lead to infrared divergences in open string loop amplitudes. These divergences can be cancelled by changing the closed string background by operators that correspond to the D-brane moving with constant velocity after some instant in time, since it is precisely such operators that give rise to the appropriate ultraviolet divergences in the closed string channel. 
  We perform the renormalization group analysis on the dynamical symmetry breaking under strong external magnetic field, studied recently by Gusynin, Miransky and Shovkovy. We find that any attractive four-Fermi interaction becomes strong in the low energy, thus leading to dynamical symmetry breaking. When the four-Fermi interaction is absent, the $\beta$-function for the electromagnetic coupling vanishes in the leading order in $1/N$. By solving the Schwinger-Dyson equation for the fermion propagator, we show that in $1/N$ expansion, for any electromagnetic coupling, dynamical symmetry breaking occurs due to the presence of Landau energy gap by the external magnetic field. 
  Supersymmetry breaking by the quantum deformation of a classical moduli space is considered. A simple, non-chiral, renormalizable model is presented to illustrate this mechanism. The well known, chiral, $SU(3) \times SU(2)$ model and its generalizations are shown to break supersymmetry by this mechanism in the limit $\Lambda_2 \gg \Lambda_3$. Other supersymmetry breaking models, with classical flat directions that are only lifted quantum mechanically, are presented. Finally, by integrating in vector matter, the strongly coupled region of chiral models with a dynamically generated superpotential is shown to be continuously connected to a weakly coupled description in terms of confined degrees of freedom, with supersymmetry broken at tree level. 
  When the path integral method of anomaly evaluation is applied to chiral gauge theories, two different types of gauge anomaly, i.e., the consistent form and the covariant form, appear depending on the regularization scheme for the Jacobian factor. We clarify the relation between the regularization scheme and the Pauli--Villars--Gupta (PVG) type Lagrangian level regularization. The conventional PVG, being non-gauge invariant for chiral gauge theories, in general corresponds to the consistent regularization scheme. The covariant regularization scheme, on the other hand, is realized by the generalized PVG Lagrangian recently proposed by Frolov and Slavnov. These correspondences are clarified by reformulating the PVG method as a regularization of the composite gauge current operator. 
  It is known that if gauge conditions have Gribov zero modes, then topological symmetry is broken. In this paper we apply it to topological gravity in dimension $n \geq 3$. Our choice of the gauge condition for conformal invariance is $R+{\alpha}=0$ , where $R$ is the Ricci scalar curvature. We find when $\alpha \neq 0$, topological symmetry is not broken, but when $\alpha =0$ and solutions of the Einstein equations exist then topological symmetry is broken. This conditions connect to the Yamabe conjecture. Namely negative constant scalar curvature exist on manifolds of any topology, but existence of nonnegative constant scalar curvature is restricted by topology. This fact is easily seen in this theory. Topological symmetry breaking means that BRS symmetry breaking in cohomological field theory. But it is found that another BRS symmetry can be defined and physical states are redefined. The divergence due to the Gribov zero modes is regularized, and the theory after topological symmetry breaking become semiclassical Einstein gravitational theory under a special definition of observables. 
  We continue our study of compactifications of F-theory on Calabi--Yau threefolds. We gain more insight into F-theory duals of heterotic strings and provide a recipe for building F-theory duals for arbitrary heterotic compactifications on elliptically fibered manifolds. As a byproduct we find that string/string duality in six dimensions gets mapped to fiber/base exchange in F-theory. We also construct a number of new $N=1$, $d=6$ examples of F-theory vacua and study transitions among them. We find that some of these transition points correspond upon further compactification to 4 dimensions to transitions through analogues of Argyres--Douglas points of $N=2$ moduli. A key idea in these transitions is the notion of classifying $(0,4)$ fivebranes of heterotic strings. 
  We analyze free-fermion conditions on vertex models. We show --by examining examples of vertex models on square, triangular, and cubic lattices-- how they amount to degeneration conditions for known symmetries of the Boltzmann weights, and propose a general scheme for such a process in two and more dimensions. 
  The general features of the degeneracy structure of ($p=2$) parasupersymmetric quantum mechanics are employed to yield a classification scheme for the form of the parasupersymmetric Hamiltonians. The method is applied to parasupersymmetric systems whose Hamiltonian is the square root of a forth order polynomial in the generators of the parasupersymmetry. These systems are interesting to study for they lead to the introduction of a set of topological invariants very similar to the Witten indices of ordinary supersymmetric quantum mechanics. The topological invariants associated with parasupersymmetry are shown to be related to a pair of Fredholm operators satisfying two compatibility conditions. An explicit algebraic expression for the topological invariants of a class of parasupersymmetric systems is provided. 
  We explore how the existence of a field with a heavy mass influences the low energy dynamics of a quantum field with a light mass by expounding the stochastic characters of their interactions which take on the form of fluctuations in the number of (heavy field) particles created at the threshold, and dissipation in the dynamics of the light fields, arising from the backreaction of produced heavy particles. We claim that the stochastic nature of effective field theories is intrinsic, in that dissipation and fluctuations are present both above and below the threshold. Stochasticity builds up exponentially quickly as the heavy threshold is approached from below, becoming dominant once the threshold is crossed. But it also exists below the threshold and is in principle detectable, albeit strongly suppressed at low energies. The results derived here can be used to give a quantitative definition of the `effectiveness' of a theory in terms of the relative weight of the deterministic versus the stochastic behavior at different energy scales. 
  In this sequel to my previous paper, "Is String Theory in Knots?" I explore ways of constructing symmetries through an algebraic stepping process using knotted graphs. The hope is that this may lead to an algebraic formulation of string theory. In the conclusion I speculate that the stepping process is a form of quantisation for which the most general form must be sought. By applying the quantisation step a sufficient number (possibly infinite) of times we may construct an algebra which is equivalent to its own quantisation. 
  Complex frequency modes occur for a scalar field near a rapidly rotating star {\it with ergoregion but no event horizon}. Such complex frequency modes must be included in the quantization of the field. As a model for this system, we have investigated a real scalar field with mass $\mu $ in a one-dimensional square-well potential. If the depth of the potential is greater than $\mu^2$, then there exist imaginary frequency modes. It is possible to quantize this simple system, but the mode operators for imaginary frequencies satisfy unusual commutation relations and do not admit a Fock-like representation or a ground state. Similar properties have been discussed already by Fulling for a complex scalar field interacting with an external electrostatic potential. We are interested in the field dynamics in the physical case where the initial state of the quantum field is specified before the complex frequency modes develop. As a model for this, we investigated a free scalar field whose ``mass" is normal in the past and becomes ``tachyonic" in the future. A particle detector in the far future placed in the in-vacuum state shows non-vanishing excitations related to the imaginary frequency modes as well. Implications of these results for the question of vacuum stability near rapidly rotating stars and possible applications to other fields in physics are discussed briefly. 
  Effective field theories in type I and II superstring theories for D-branes located at points in the orbifold C^2/Z_n are supersymmetric gauge theories whose field content is conveniently summarized by a `quiver diagram,' and whose Lagrangian includes non-metric couplings to the orbifold moduli: in particular, twisted sector moduli couple as Fayet-Iliopoulos terms in the gauge theory. These theories describe D-branes on resolved ALE spaces. Their spaces of vacua are moduli spaces of smooth ALE metrics and Yang-Mills instantons, whose metrics are explicitly computable. For U(N) instantons, the construction exactly reproduces results of Kronheimer and Nakajima. 
  We implement the O(d,d,Z) transformations of T-duality as automorphisms of the operator algebras of Conformal Field Theories. This extends these transformations to arbitrary field configurations in the deformation class. 
  We present an equivariant extension of the Thom form with respect to a vector field action, in the framework of the Mathai-Quillen formalism. The associated Topological Quantum Field Theories correspond to twisted $N=2$ supersymmetric theories with a central charge. We analyze in detail two different cases: topological sigma models and non-abelian monopoles on four-manifolds. 
  We discuss how transitions in the space of heterotic K3*T^2 compactifications are mapped by duality into transitions in the space of Type II compactifications on Calabi-Yau manifolds. We observe that perturbative symmetry restoration, as well as non-perturbative processes such as changes in the number of tensor multiplets, have at least in many cases a simple description in terms of the reflexive polyhedra of the Calabi-Yau manifolds. Our results suggest that to many, perhaps all, four-dimensional N=2 heterotic vacua there are corresponding type II vacua. 
  The reduction from N=1, D=10 to N=4, D=4 supergravity with the Yang-Mills matter is considered. For this purpose a set of constraints is imposed in order to exclude six additional abelian matter multiplets and conserve the supersymmetry. We consider both the cases of usual and dual N=1, D=10 supergravity using the superspace approach. Also the effective potential of the deriving theory is written. 
  We investigate the gauged NJL--model in curved spacetime using the RG formulation and the equivalency with the gauge Higgs--Yukawa model in a modified 1/N_c -expansion. The strong curvature induced chiral symmetry breaking is found in the non-perturbative RG approach (presumably equivalent to the ladder Schwinger--Dyson equations). Dynamically generated fermion mass is explicitly calculated and inducing of Einstein gravity is briefly discussed. This approach shows the way to the non-perturbative study of the dynamical symmetry breaking at external fields. 
  The $SL(2, R)/Z$ WZW orbifold describes the (2+1)-dimensional black hole which approaches anti-de~Sitter space asymptotically. We study the $1 \rightarrow 1$ tachyon scattering off the rotating black hole background and calculate the Hawking temperature using the Bogoliubov transformation. 
  A universal model for D=4 spinning particle is constructed with the configuration space chosen as ${\bf R}^{3,1}\times S^2$, where the sphere corresponds to the spinning degrees of freedom. The Lagrangian includes all the possible world--line first order invariants of the manifold. Each combination of the four constant parameters entering the Lagrangian gives the model, which describes the proper irreducible Poincar\'e dynamics both at the classical and quantum levels, and thereby the construction uniformly embodies the massive, massless and continuous helicity cases depending upon the special choice of the parameters. For the massive case, the connection with the Souriau approach to elementary systems is established. Constrained Hamiltonian formulation is built and Dirac canonical quantization is performed for the model in the covariant form. An explicit realization is given for the physical wave functions in terms of smooth tensor fields on ${\bf R}^{3,1}\times S^2$. One-parametric family of consistent interactions with general electromagnetic and gravitational fields is introduced in the massive case. The spin tensor is shown to satisfy the Frenkel-Nyborg equation with arbitrary fixed gyromagnetic ratio in a limit of weakly varying electromagnetic field. 
  The thermodynamical one-loop entropy $S^{TD}$ of a two-dimensional black hole in thermal equilibrium with the massless quantum gas is calculated. It is shown that $S^{TD}$ includes the Bekenstein-Hawking entropy, evaluated for the quantum corrected geometry, and the finite difference of statistical mechanical entropies $-Tr\hat{\rho}\ln\hat{\rho}$ for the gas on the black hole and Rindler spaces. This result demonstrates in an explicit form that the relation between thermodynamical and statistical-mechanical entropies of a black hole is non-trivial and requires special subtraction procedure. 
  I exhibit a middle-dimensional square integrable harmonic form on the moduli space of distinct fundamental BPS monopoles of an arbitrary Lie group. This is in accord with Sen's S-duality conjecture. I also show that the moduli space has no closed or bound geodesics. 
  The massive Schwinger model is quantized on the light cone with great care on the bosonic zero modes by putting the system in a finite (light-cone) spatial box. The zero mode of $A_{-}$ survives Dirac's procedure for the constrained system as a dynamical degree of freedom. After regularization and quantization, we show that the physical space condition is consistently imposed and relates the fermion Fock states to the zero mode of the gauge field. The vacuum is obtained by solving a Schr\"odinger equation in a periodic potential, so that the theta is understood as the Bloch momentum. We also construct a one-meson state in the fermion-antifermion sector and obtained the Schr\"odinger equation for it. 
  The solitons of affine Toda field theory are related to the spin-generalised Ruijsenaars-Schneider (or relativistic Calogero-Moser) models. This provides the sought after extension of the correspondence between the sine-Gordon solitons and the Ruijsenaars-Schneider model. 
  The propagator and corresponding path integral for a system of identical particles obeying parastatistics are derived. It is found that the statistical weights of topological sectors of the path integral for parafermions and parabosons are simply related through multiplication by the parity of the permutation of the final positions of the particles. Appropriate generalizations of statistics are proposed obeying unitarity and factorizability (strong cluster decomposition). The realization of simple maximal occupancy (Gentile) statistics is shown to require ghost states. 
  Gauge invariant local creation operators of charged states are introduced and studied in pure gauge theories of the Maxwell type in 2+1D. These states are usually unphysical because of the subsidiary condition imposed on the physical subspace by Gauss' law. A dual Maxwell theory which possesses a topological electric charge is introduced. Pure Electrodynamics lies in the sector where the topological charge identically vanishes. Charge bearing operators fully expressed in terms of the gauge field, however, can create physical states in the nontrivial topological sectors which thereby generalize QED. An order disorder structure exists relating the charged operators and the magnetic flux creating (vortex) operators, both through commutation rules and correlation functions. The relevance of this structure for bosonization in 2+1D is discussed. 
  We derive the couplings of the 3-form supermultiplet to the general supergravity-matter-Yang-Mills system. Based on the methods of superspace geometry, we identify component fields, establish their supergravity transformations and construct invariant component field actions. Two specific applications are adressed: the appearance of fundamental 3-forms in the context of strong-weak duality and the use of the 3-form supermultiplets to describe effective degrees of freedom relevant to the mechanism of gaugino condensation. 
  We compute the exact string vacuum backgrounds corresponding to the non-compact coset theory $SU(2,1)/SU(2)$. The conformal field theory defined by the level $k= 4$ results in a five dimensional singular solution that factorizes in an asymptotic region as the linear dilaton solution and a $S^3$ model. It presents two abelian compact isometries that allow to reinterpreting it from a four dimensional point of view as a stationary and magnetically charged space-time resembling in some aspects the Kerr-Newman solution of general relativity. The $k=\frac{13}{7}$ theory on the other hand describes a cosmological solution that interpolates between a singular phase at short times and a $S^1 \times S^2$ universe after some planckian times. 
  As shown by Ho\v{r}ava and Witten, there are gravitational anomalies at the boundaries of $M^{10}\times S^1/Z_2$ of 11 dimensional supergravity. They showed that only 10 dimensional vector multiplets belonging to $E_8$ gauge group can be consistently coupled to this theory. Thus, the dimensional reduction of this theory should be the low energy limit of the $E_8\times E_8$ heterotic string. Here we assume that M-theory is a theory of supermembranes which includes twisted supermembranes. We show that for a target space $M^{10}\times S^1/Z_2$, in the limit in which $S^1/Z_2$ is small, the effective action is the $E_8\times E_8$ heterotic string. We also consider supermembranes on $M^{9}\times S^1\times S^1/Z_2$ and find the dualities expected from 11 dimensional supergravity on this manifold. We show that the requirements for worldsheet anomaly cancellations at the boundaries of the worldvolume action are the same requirements imposed on the Ho\v{r}ava-Witten action. 
  Major developments in the history of the subject are critically reviewed in this talk. 
  The scattering problem under the influence of the Aharonov-Bohm (AB) potential is reconsidered. By solving the Lippmann-Schwinger (LS) equation we obtain the wave function of the scattering state in this system. In spite of working with a plane wave as an incident wave we obtain the same wave function as was given by Aharonov and Bohm. Another method to solve the scattering problem is given by making use of a modified version of Gordon's idea which was invented to consider the scattering by the Coulomb potential. These two methods give the same result, which guarantees the validity of taking an incident plane wave as usual to make an analysis of this scattering problem. The scattering problem by a solenoid of finite radius is also discussed, and we find that the vector potential of the solenoid affects the charged particles even when the magnitude of the flux is an odd integer as well as noninteger. It is shown that the unitarity of the $S$ matrix holds provided that a plane wave is taken to be an incident one. 
  We discuss T-duality for open strings in general background fields both in the functional integral formulation as well as in the language of canonical transformations. The Dirichlet boundary condition in the dual theory has to be treated as a constraint on the functional integration. Furthermore, we give meaning to the notion of matrix valued string end point position in the presence of nonabelian gauge field background. 
  It is shown that an alternative supersymmetric version of the Liouville equation extracted from D=3 Green-Schwarz superstring equations naturally arises as a super-Toda model obtained from a properly constrained supersymmetric WZNW theory based on the $sl(2, R)$ algebra. Hamiltonian reduction is performed by imposing a nonlinear superfield constraint which turns out to be a mixture of a first- and second-class constraint on supercurrent components. Supersymmetry of the model is realized nonlinearly and is spontaneously broken. The set of independent current fields which survive the Hamiltonian reduction contains (in the holomorphic sector) one bosonic current of spin 2 (the stress--tensor of the spin 0 Liouville mode) and two fermionic fields of spin ${3/2}$ and $-1/2$. The $n=1$ superconformal system thus obtained is of the same kind as one describing noncritical fermionic strings in a universal string theory. The generalization of this procedure allows one to produce from any bosonic Lie algebra super--Toda models and associated super-W algebras together with their nonstandard realizations. 
  Dual Feynman rules for Dirac monopoles in Yang-Mills fields are obtained by the Wu-Yang (1976) criterion in which dynamics result as a consequence of the constraint defining the monopole as a topological obstruction in the field. The usual path-integral approach is adopted, but using loop space variables of the type introduced by Polyakov (1980). An anti-symmetric tensor potential $L_{\mu\nu}[\xi|s]$ appears as the Lagrange multiplier for the Wu-Yang constraint which has to be gauge-fixed because of the ``magnetic'' $\widetilde{U}$-symmetry of the theory. Two sets of ghosts are thus introduced, which subsequently integrate out and decouple. The generating functional is then calculated to order $g^0$ and expanded in a series in $\widetilde{g}$. It is shown to be expressible in terms of a local ``dual potential'' $\widetilde{A}_\mu (x)$ found earlier, which has the same progagator and the same interaction vertex with the monopole field as those of the ordinary Yang-Mills potential $A_\mu$ with a colour charge, indicating thus a certain degree of dual symmetry in the theory. For the abelian case the Feynman rules obtained here are the same as in QED to all orders in $g$, as expected by dual symmetry. 
  On the example of the quantized spinor field, interacting with arbitrary external electromagnetic field, the commutation function is studied. It is shown that a proper time representation is available in any dimensions. Using it, all the light cone singularities of the function are found explicitly, generalizing the Fock formula in four dimensions, and a path integral representation is constructed. 
  The complete spectrum of states in the supersymmetric principal chiral model based on SU(n) is conjectured, and an exact factorizable S-matrix is proposed to describe scattering amongst these states. The SU(n)_L*SU(n)_R symmetry of the lagrangian is manifest in the S-matrix construction. The supersymmetries, on the other hand, are incorporated in the guise of spin-1/2 charges acting on a set of RSOS kinks associated with su(n) at level n. To test the proposed S-matrix, calculations of the change in the ground-state energy in the presence of a coupling to a background charge are carried out. The results derived from the lagrangian using perturbation theory and from the S-matrix using the TBA are found to be in complete agreement for a variety of background charges which pick out, in turn, the highest weight states in each of the fundamental representations of SU(n). In particular, these methods rule out the possibility of additional CDD factors in the S-matrix. Comparison of the expressions found for the free-energy also yields an exact result for the mass-gap in these models: m/Lambda_{MS-bar}=(n/pi)sin(pi/n). 
  We study the symplectic reparametrizations that are possible for theories of N=2 supersymmetric vector multiplets in the presence of a chiral background and discuss some of their consequences. One of them concerns an anomaly arising from a conflict between symplectic covariance and holomorphy. 
  The BRST generator is realized as a Hermitian nilpotent operator for a finite-dimensional gauge system featuring a quadratic super-Hamiltonian and linear supermomentum constraints. As a result, the emerging ordering for the Hamiltonian constraint is not trivial, because the potential must enter the kinetic term in order to obtain a quantization invariant under scaling. Namely, BRST quantization does not lead to the curvature term used in the literature as a means to get that invariance. The inclusion of the potential in the kinetic term, far from being unnatural, is beautifully justified in light of the Jacobi's principle. 
  We derive closed analytical expressions for the complex Berry phase of an open quantum system in a state which is a superposition of resonant states and evolves irreversibly due to the spontaneous decay of the metastable states. The codimension of an accidental degeneracy of resonances and the geometry of the energy hypersurfaces close to a crossing of resonances differ significantly from those of bound states. We discuss some of the consequences of these differences for the geometric phase factors, such as: Instead of a diabolical point singularity there is a continuous closed line of singularities formally equivalent to a continuous distribution of `magnetic' charge on a diabolical circle; different classes of topologically inequivalent non-trivial closed paths in parameter space, the topological invariant associated to the sum of the geometric phases, dilations of the wave function due to the imaginary part of the Berry phase and others. 
  We derive fully covariant expressions for all two-point scattering amplitudes of two massless closed strings from a Dirichlet $p$-brane. This construction relies on the observation that there is a simple relation between these D-brane amplitudes in type II superstring theory and four-point scattering amplitudes for type I open superstrings. From the two-point amplitudes, we derive the long range background fields for the D-branes, and verify that as expected they correspond to those of extremally charged $p$-brane solutions of the low energy effective action. 
  We identify the states in string theory which are responsible for the entropy of near-extremal rotating four-dimensional black holes in $N=8$ supergravity. For black holes far from extremality (with no rotation), the Bekenstein-Hawking entropy is exactly matched by a mysterious duality invariant extension of the formulas derived for near-extremal black holes states. 
  The four dimensional SU(3) WZW model coupled to electromagnetism is treated as a constrained system in the context of Batalin-Fradkin- Vilkovisky formalism. It is shown that this treatment is equivalent to the Faddeev-Jackiw (FJ) approach. It is also shown that the field redefinitions that transform the fields of the model into BRST and $\sigma$ closed are actually the Darboux's transformations used in the FJ formalism. 
  We study topological Yang-Mills-Higgs theories in two and three dimensions and topological Yang-Mills theory in four dimensions in a unified framework of superconnections. In this framework, we first show that a classical action of topological Yang-Mills type can provide all three classical actions of these theories via appropriate projections. Then we obtain the BRST and anti-BRST transformation rules encompassing these three topological theories from an extended definition of curvature and a geometrical requirement of Bianchi identity. This is an extension of Perry and Teo's work in the topological Yang-Mills case. Finally, comparing this result with our previous treatment in which we used the ``modified horizontality condition", we provide a meaning of Bianchi identity from the BRST symmetry viewpoint and thus interpret the BRST symmetry in a geometrical setting. 
  The interest in string Hamiltonian system has recently been rekindled due to its application to target-space duality. In this article, we explore another direction it motivates. In Sec.\ 1, conformal symmetry and some algebraic structures of the system that are related to interacting strings are discussed. These lead one naturally to the study of Lorentz surfaces in Sec.\ 2. In contrast to the case of Riemann surfaces, we show in Sec.\ 3 that there are Lorentz surfaces that cannot be conformally deformed into Mandelstam diagrams. Lastly in Sec.\ 4, we discuss speculatively the prospect of Lorentzian conformal field theory.   Additionally, to have a view of what quantum picture a string Hamiltonian system may lead to, we discuss independently in the Appendix a formal geometric quantization of the string phase space. 
  In this paper we find an explicit formula for the most general vector evolution of curves on $RP^{n-1}$ invariant under the projective action of $SL(n,R)$. When this formula is applied to the projectivization of solution curves of scalar Lax operators with periodic coefficients, one obtains a corresponding evolution in the space of such operators. We conjecture that this evolution is identical to the second KdV Hamiltonian evolution under appropriate conditions. These conditions give a Hamiltonian interpretation of general vector differential invariants for the projective action of $SL(n,R)$, namely, the $SL(n,R)$ invariant evolution can be written so that a general vector differential invariant corresponds to the Hamiltonian pseudo-differential operator. We find common coordinates and simplify both evolutions so that one can attempt to prove the equivalence for arbitrary $n$. 
  The theory of magnetohydrodynamics is extended to the cases of a plasma of separate magnetic and electric charges, as well as to a plasma of dyons respectively. In both these cases the system possesses electric-magnetic duality symmetry. In the former case we find that because of the existence of two independent generalized Ohm's law equations, the limit of infinite electric and magnetic conductivity results in the vanishing of both electric and magnetic fields, as well as the corresponding currents. In the dyonic case, we find that the resulting duality-invariant system of equations are equivalent to those of ordinary MHD, after suitable field redefinitions. 
  We study the finite temperature symmetry behaviour of O(N_1) \times O(N_2) scalar models on the lattice and we prove that at sufficiently high temperatures and in arbitrary dimensions their full symmetry is always restored or, equivalently, that the phenomenon of Symmetry Non Restoration which, according to lowest order perturbation theory, takes place in the continuum version of these models, does not occur on the lattice. 
  We rewrite the 1+1 Dirac equation in light cone coordinates in two significant forms, and solve them exactly using the classical calculus of finite differences. The complex form yields ``Feynman's Checkerboard''---a weighted sum over lattice paths. The rational, real form can also be interpreted in terms of bit-strings. 
  The lattice Coulomb-gauge hamiltonian is derived from the transfer matrix of Wilson's Euclidean lattice gauge theory, wherein the lattice form of Gauss's law is satisfied identically. The restriction to a fundamental modular region (no Gribov copies) is implemented in an effective hamiltonian by the addition of a "horizon function" $G$ to the lattice Coulomb-gauge hamiltonian. Its coefficient $\gamma_0$ is a thermodynamic parameter that ultimately sets the scale for hadronic mass, and which is related to the bare coupling constant $g_0$ by a "horizon condition". This condition determines the low-momentum behavior of the (ghost) propagator that transmits the instantaneous longitudinal color-electric field, and thereby provides for a confinement-like feature in leading order in a new weak-coupling expansion. 
  We consider anomaly free combinations of chiral fermions coupled to $U(1)$ gauge fields on a 2D torus first in the continuum and then on the lattice in the overlap formulation. Both in the continuum and on the lattice, when the background consists of sufficiently large constant gauge potentials the action induced by the fermions varies significantly under certain singular gauge transformations. ``Ruling away'' such discontinuities cannot be justified in the continuum framework and does not naturally fit on the lattice. Complete gauge invariance in the continuum can be restored in some models by choosing special boundary conditions for the fermions. Evidence is presented that gauge averaging the overlap phases in these models produces correct continuum results. 
  This is an overview of the problem of the vacuum in light-cone field theory, stressing its close connection to other puzzles regarding light-cone quantization. I explain the sense in which the light-cone vacuum is ``trivial,'' and describe a way of setting up a quantum field theory on null planes so that it is equivalent to the usual equal-time formulation. This construction is quite helpful in resolving the puzzling aspects of the light-cone formalism. It furthermore allows the extraction of effective Hamiltonians that incorporate vacuum physics, but that act in a Hilbert space in which the vacuum state is simple. The discussion is fairly informal, and focuses mainly on the conceptual issues. [Talk presented at {\sc Orbis Scientiae 1996}, Miami Beach, FL, January 25--28, 1996. To appear in the proceedings.] 
  We analyze the effects of soft supersymmetry breaking terms on N=1 supersymmetric QCD with $N_f$ flavors and color gauge group $SU(N_c)$. The mass squared of some squarks may be negative, as long as vacuum stability is ensured by a simple mass inequality. For $N_f<N_c$, we include the dynamics of the non-perturbative superpotential and use the original (s)quark and gauge fields, while for $N_f>N_c+1$, we formulate the dynamics in terms of dual (s)quarks and a dual gauge group $SU(N_f-N_c)$. The presence of negative squark mass squared terms leads to spontaneous breakdown of flavor and color symmetry. We determine this breaking pattern, derive the spectrum, and argue that the masses vary smoothly as one crosses from the Higgs phase into the confining phase. 
  We discuss an extension of the $C$-theorem to chiral theories. We show that two monotonically decreasing $C$-functions can be introduced. However, their difference is a constant of the renormalization group flow. This constant reproduces the 't Hooft anomaly matching conditions. 
  We provide N=1 Super Yang-Mills theory in the Wess-Zumino gauge with mass terms for the supersymmetric partners of the gauge fields and of the matter fields, together with a supersymmetric mass term for the fermionic matter fields. All mass terms are chosen in such a way to induce soft supersymmetry breakings at most, while preserving gauge invariance to all orders of perturbation theory. The breakings are controlled through an extended Slavnov-Taylor identity. The renormalization analysis, both in the ultraviolet and in the infrared region, is performed. 
  The Eguchi-Hanson, Taub-NUT and Atiyah-Hitchin metrics are the only complete non-singular SO(3)-invariant hyper-Kahler metrics in four dimensions. The presence of a rotational SO(2) isometry allows for their unified treatment based on solutions of the 3-dim continual Toda equation. We determine the Toda potential in each case and examine the free field realization of the corresponding solutions, using infinite power series expansions. The Atiyah-Hitchin metric exhibits some unusual features attributed to topological properties of the group of area preserving diffeomorphisms. The construction of a descending series of SO(2)-invariant 4-dim regular hyper-Kahler metrics remains an interesting question. 
  We analyze the possible soft breaking of $N=2$ supersymmetric Yang-Mills theory with and without matter flavour preserving the analyticity properties of the Seiberg-Witten solution. For small supersymmetry breaking parameter with respect to the dynamical scale of the theory we obtain an exact expression for the effective potential. We describe in detail the onset of the confinement transition and some of the patterns of chiral symmetry breaking. If we extrapolate the results to the limit where supersymmetry decouples, we obtain hints indicating that perhaps a description of the QCD vacuum will require the use of Lagrangians containing simultaneously mutually non-local degrees of freedom (monopoles and dyons). 
  It is well known that by using the infinite dimensional symmetries that issue from string theories, one can build 2D geometric field theories. These 2D field theories can be identified with gravitational and gauge anomalies that arise in the presence of background gauge and gravitational anomalies. In this work we consider the background fields as residuum from reducing higher dimensional field theories to two dimensions. This implies a new relationship between string theory and field theories. We identify the isotropy equations of the distinct orbits as the Gau\ss 's law constraints of a Yang-Mills theory coupled to a gravitational theory that has been evaluated on a two-dimensional manifold. We show explicitly how one may recover the higher dimensional theories and extract this new theory of gravity and its coupling to Yang-Mills theory. This gravitational theory is able to couple to Yang-Mills via a torsion-like term and yet maintain gauge invariance. Also this new theory of gravity suggest a natural distinction between cosmology and local gravitation. We comment on the analogue of Chern-Simons theory for diffeomorphism, the vacuum structure of gravity, and also the possibility of extracting explicit realizations of distinct differentiable structures in four dimensions. 
  The Green--Schwarz superstring action is modified to include some set of additional (on-shell trivial) variables. A complete constraints system of the theory turns out to be reducible both in the original and in additional variable sectors. The initial $8s$ first class constraints and $8c$ second class ones are shown to be unified with $8c$ first and $8s$ second class constraints from the additional variables sector, resulting with $SO(1,9)$-covariant and linearly independent constraint sets. Residual reducibility proves to fall on second class constraints only. 
  It is shown explicitly that the correlation functions of Conformal Field Theories (CFT) with the logarithmic operators are invariant under the differential realization of Borel subalgebra of $\W_\infty$-algebra. This algebra is constructed by tensor-operator algebra of differential representation of ordinary $sl(2,C)$. This method allows us to write differential equations which can be used to find general expression for three and four-point correlation functions possessing logarithmic operators. The operator product expansion (OPE) coefficients of general logarithmic CFT are given up to third level. 
  A complete Fock space representation of the covariant differential calculus on quantum space is constructed. The consistency criteria for the ensuing algebraic structure, mapping to the canonical fermions and bosons and the consequences of the new algebra for the statistics of quanta are analyzed and discussed. The concept of statistical transmutation between bosons and fermions is introduced. 
  We describe the recently introduced method of Algebraic Bosonization of (1+1)-dimensional fermionic systems by discussing the specific case of the Calogero-Sutherland model. A comparison with the Bethe Ansatz results is also presented. 
  We study some natural connections on spaces of conformal field theories using an analytical regularization method. The connections are based on marginal conformal field theory deformations. We show that the analytical regularization preserves conformal invariance and leads to integrability of the marginal deformations. The connections are shown to be flat and to generate well-defined finite parallel transport. These finite parallel transports yield formulations of the deformed theories in the state space of an undeformed theory. The restrictions of the connections to the tangent space are curved but free of torsion. 
  We derive a set of bilinear functional equations of Hirota type for the partition functions of the $sl(2)$ related integrable statistical models defined on a random lattice. These equations are obtained as deformations of the Hirota equations for the KP integrable hierarchy, which are satisfied by the partition function of the ensemble of planar graphs. 
  The problem of a nonrelativistic particle with an internal color degree of freedom, with and without spin, moving in a free random gauge background is discussed. Freeness is a concept developed recently in the mathematical literature connected with noncommuting random variables. In the context of large-N hermitian matrices, it means that the the multi-matrix model considered contains no bias with respect to the relative orientations of the matrices. In such a gauge background, the spectrum of a colored particle can be solved for analytically. In three dimensions, near zero momentum, the energy distribution for the spinless particle displays a gap, while the energy distribution for the particle with spin does not. 
  We examine non-abelian duality transformations in the open string case. After gauging the isometries of the target space and developing the general formalism, we study in details the duals oftarget spaces with SO(N) isometries which, for the SO(2) case, reduces to the known abelian T-duals. We apply the formalism to electrically and magnetically charged 4D black hole solutions and, as in the abelian case, dual coordinates satisfy Dirichlet conditions. 
  We develope a formalism for the scattering off D-branes that includes collective coordinates. This allows a systematic expansion in the string coupling constant for such processes, including a worldsheet calculation for the D-brane's mass. 
  We present a relation which connects the propagator in the radial (Fock-Schwinger) gauge with a gauge invariant Wilson loop. It is closely related to the well-known field strength formula and can be used to calculate the radial gauge propagator. The result is shown to diverge in four-dimensional space even for free fields, its singular nature is however naturally explained using the renormalization properties of Wilson loops with cusps and self-intersections. Using this observation we provide a consistent regularization scheme to facilitate loop calculations. Finally we compare our results with previous approaches to derive a propagator in Fock-Schwinger gauge. 
  By using the intertwiner and face-vertex correpondence relation, we obtain the Bethe ansatz equation of eight vertex model with open boundary condtitions in the framework of algebraic Bethe ansatz method. The open boundary condition under consideration is the general solution of the reflection equation for eight vertex model with only one restriction on the free parameters of the right side reflecting boundary matrix. The reflecting boundary matrices used in this paper thus may have off-diagonal elements. Our construction can also be used for the Bethe ansatz of SOS model with reflection boundaries. 
  We show that certain classical SU(2) pure gauge configurations give rise to a non-zero string tension. We then investigate cooled configurations generated by Monte Carlo simulations on the lattice and find similar properties. We infer evidence in favour of a classical model of Confinement. 
  We review the status of quantising (non-abelian) gauge theories using different versions of a Hamiltonian formulation corresponding to Dirac's instant and front form of dynamics, respectively. In order to control infrared divergences we work in a finite spatial volume, chosing a torus geometry for convenience. We focus on the determination of the physical configuration space of gauge invariant variables via gauge fixing. This naturally leads us to the issue of the Gribov problem. We discuss it for different gauge choices, in particular finite volume modifications of the axial gauge. Conventional and light-front quantisation are compared and the differences pointed out. 
  Studies of models of current flow behaviour in Electrical Impedance Tomography (EIT) have shown that the current density distribution varies extremely rapidly near the edge of the electrodes used in the technique. This behaviour imposes severe restrictions on the numerical techniques used in image reconstruction algorithms. In this paper we have considered a simple two dimensional case and we have shown how the theory of end point/pinch singularities which was developed for studying the anomalous thresholds encountered in elementary particle physics can be used to give a complete description of the analytic structure of the current density near to the edge of the electrodes. As a byproduct of this study it was possible to give a complete description of the Riemann sheet manifold of the eigenfunctions of the logarithmic kernel. These methods can be readily extended to other weakly singular kernels. 
  I outline various derivations of the non-Abelian Kubo equation, which governs the response of a quark-gluon plasma to hard thermal perturbations. In the static case, it is proven that gauge theories do not support hard thermal solitons. Explicit solutions are constructed within an SU(2) Ansatz and they are shown to support the general result. The time-dependent problem, i.e., non-Abelian plasma waves, has not been completely solved. We express and motivate the hope that the intimate relations linking the gauge-invariance condition for hard thermal loops to the equation of motion for T=0, topological Chern-Simons theory may yield new insight into this field. 
  For the structure functions of the quark propagator, the asymptotic behavior is obtained for general, linear, covariant gauges, and in all directions of the complex $k^2$-plane. Asymptotic freedom is assumed. Corresponding previous results for the gauge field propagator are important in the derivation. Except for coefficients, the leading asymptotic terms are determined by one-loop or by two-loop information, and are gauge independent. Various sum rules are derived. 
  A residue formula which evaluates any correlation function of topological $SU_n$ Yang-Mills theory with arbitrary magnetic flux insertion in two dimensions are obtained. Deformations of the system by two form operators are investigated in some detail. The method of the diagonalization of a matrix valued field turns out to be useful to compute various physical quantities. As an application we find the operator that contracts a handle of a Riemann surface and a genus recursion relation. 
  We derive an infinite sequence of Schwinger-Dyson equations for $N=1$ supersymmetric Yang-Mills theory. The fundamental and the only variable employed is the Wilson-loop geometrically represented in $N=1$ superspace: it organizes an infinite number of supersymmetrizing insertions into the ordinary Wilson-loop as a single entity. In the large $N_{c}$ limit, our equation becomes a closed loop equation for the one-point function of the Wilson-loop average. 
  An oscillator algebra and the associated Fock space with reflecting boundary and generalized statistics are constructed and is generalized to the multicomponent case. The oscillator algebra depends manifestly on the reflection factor and the statistical (exchange) factor, and the corresponding Fock space can be obtained from that of the usual bosonic oscillator without reflection condition by certain projection operation. 
  The paper provides a framework for a systematic analysis of the local BRST cohomology in a large class of gauge theories. The approach is based on the cohomology of s+d in the jet space of fields and antifields, s and d being the BRST operator and exterior derivative respectively. It relates the BRST cohomology to an underlying gauge covariant algebra and reduces its computation to a compactly formulated problem involving only suitably defined generalized connections and tensor fields. The latter are shown to provide the building blocks of physically relevant quantities such as gauge invariant actions, Noether currents and gauge anomalies, as well as of the equations of motion. 
  We analyse the fusion products of certain representations of the Virasoro algebra for c=-2 and c=-7 which are not completely reducible. We introduce a new algorithm which allows us to study the fusion product level by level, and we use this algorithm to analyse the indecomposable components of these fusion products. They form novel representations of the Virasoro algebra which we describe in detail.   We also show that a suitably extended set of representations closes under fusion, and indicate how our results generalise to all (1,q) models. 
  We present a simplified description of higher antibrackets, generalizations of the conventional antibracket of the Batalin-Vilkovisky formalism. We show that these higher antibrackets satisfy relations that are identical to those of higher string products in non-polynomial closed string field theory. Generalization to the case of Sp(2)-symmetry is also formulated. 
  Photon splitting in a very strong magnetic field is analyzed for energy $\omega < 2m$. The amplitude obtained on the base of operator-diagram technique is used. It is shown that in a magnetic field much higher than critical one the splitting amplitude is independent on the field. Our calculation is in a good agreement with previous results of Adler and in a strong contradiction with recent paper of Mentzel et al. 
  Using a generating function for the Wigner's $D$-matrix elements of $SU(3)$ Weyl's character formula for $SU(3)$ is derived using Schwinger's technique. 
  The non-perturbative superpotential can be effectively calculated in $M$-theory compactification to three dimensions on a Calabi-Yau four-fold $X$. For certain $X$, the superpotential is identically zero, while for other $X$, a non-perturbative superpotential is generated. Using $F$-theory, these results carry over to certain Type IIB and heterotic string compactifications to four dimensions with $N=1$ supersymmetry. In the heterotic string case, the non-perturbative superpotential can be interpreted as coming from space-time and world-sheet instantons; in many simple cases contributions come only from finitely many values of the instanton numbers. 
  We generalize the integrable Heisenberg ferromagnet model according to each Hermitian symmetric spaces and address various new aspects of the generalized model. Using the first order formalism of generalized spins which are defined on the coadjoint orbits of arbitrary groups, we construct a Lagrangian of the generalized model from which we obtain the Hamiltonian structure explicitly in the case of $CP(N-1)$ orbit. The gauge equivalence between the generalized Heisenberg ferromagnet and the nonlinear Schr\"{o}dinger models is given. Using the equivalence, we find infinitely many conserved integrals of both models. 
  We derive a bosonic Hamiltonian from two dimensional QCD on the light-front. To obtain the bosonic theory we find that it is useful to apply the boson expansion method which is the standard technique in quantum many-body physics. We introduce bilocal boson operators to represent the gauge-invariant quark bilinears and then local boson operators as the collective states of the bilocal bosons. If we adopt the Holstein-Primakoff type among various representations, we obtain a theory of infinitely many interacting bosons, whose masses are the eigenvalues of the 't Hooft equation. In the large $N$ limit, since the interaction disappears and the bosons are identified with mesons, we obtain a free Hamiltonian with infinite kinds of mesons. 
  It is shown that time-harmonic hypersurface motions in various, conformally flat, N-dimensional manifolds admit a multilinear description, dL/dt={ L, M_1, ... , M_{N-2} }, automatically generating infinitely many conserved quantities, as well as leading to new (integrable) matrix equations. Interestingly, the conformal factor can be changed without changing L. 
  We show how the Riemann surface $\Sigma$ of $N=2$ Yang-Mills field theory arises in type II string compactifications on Calabi-Yau threefolds. The relevant local geometry is given by fibrations of ALE spaces. The $3$-branes that give rise to BPS multiplets in the string descend to self-dual strings on the Riemann surface, with tension determined by a canonically fixed Seiberg-Witten differential $\lambda$. This gives, effectively, a dual formulation of Yang-Mills theory in which gauge bosons and monopoles are treated on equal footing, and represents the rigid analog of type II-heterotic string duality. The existence of BPS states is essentially reduced to a geodesic problem on the Riemann surface with metric $|\lambda|^2$. This allows us, in particular, to easily determine the spectrum of {\it stable} BPS states in field theory. Moreover, we identify the six-dimensional space $\IR^4\times \Sigma$ as the world-volume of a five-brane and show that BPS states correspond to two-branes ending on this five-brane. 
  We present solutions describing supersymmetric configurations of 2 or 3 orthogonally intersecting 2-branes and 5-branes of D=11 supergravity. The configurations which preserve 1/4 or 1/8 of maximal supersymmetry are 2+2, 5+5, 2+5, 2+2+2, 5+5+5, 2+2+5 and 2+5+5 (2+2 stands for orthogonal intersection of two 2-branes over a point, etc.; p-branes of the same type intersect over (p-2)-branes). There exists a simple rule which governs the construction of composite supersymmetric p-brane solutions in D=10 and 11 with a separate harmonic function assigned to each constituent 1/2-supersymmetric p-brane. The resulting picture of intersecting p-brane solutions complements their D-brane interpretation in D=10 and seems to support possible existence of a D=11 analogue of D-brane description. The D=11 solution describing intersecting 2-brane and 5-brane reduces in D=10 to a type II string solution corresponding to a fundamental string lying within a solitonic 5-brane (which further reduces to an extremal D=5 black hole). We also discuss a particular D=11 embedding of the extremal D=4 dyonic black hole solution with finite area of horizon. 
  A generating function for the Wigner's $D$-matrix elements of $SU(3)$ is derived. From this an explicit expression for the individual matrix elements is obtained in a closed form. 
  Collective coordinate quantization of Dirichlet branes is discussed. Utilizing Polchinski's combinatoric rule, semiclassical D-brane wave functional is given in proper-time formalism. D-brane equation of motion is then identified with renormalization group equation of defining Dirichlet open string theory. Quantum mechanical size of macroscopically charged D-brane is illustrated and striking difference from ordinary field theory BPS particle is emphasized. 
  We present master formulas for the divergent part of the one-loop effective action for a minimal operator of any order in the 4-dimensional curved space and for an arbitrary nonminimal operator in the flat space. 
  We discuss the properties of an ideal relativistic gas of events possessing Bose-Einstein statistics. We find that the mass spectrum of such a system is bounded by $\mu \leq m\leq 2M/\mu _K,$ where $\mu $ is the usual chemical potential, $M$ is an intrinsic dimensional scale parameter for the motion of an event in space-time, and $\mu _K$ is an additional mass potential of the ensemble. For the system including both particles and antiparticles, with nonzero chemical potential $\mu ,$ the mass spectrum is shown to be bounded by $|\mu |\leq m\leq 2M/\mu _K,$ and a special type of high-temperature Bose-Einstein condensation can occur. We study this Bose-Einstein condensation, and show that it corresponds to a phase transition from the sector of continuous relativistic mass distributions to a sector in which the boson mass distribution becomes sharp at a definite mass $M/\mu _K.$ This phenomenon provides a mechanism for the mass distribution of the particles to be sharp at some definite value. 
  The gauge variance of wave functionals for a gauge theory quantized in the momentum (curvature) representation is described. It is shown that a gauge transformation gives rise to a cocycle, which for theories in two space-time dimensions is related to the Kirillov-Kostant form. Various derivations are presented, including one based on geometric (pre-) quantization. The formalism is applied to two dimensional gravity models that are equivalently described by "B-F" gauge theories. 
  We develop and study a D-brane realization of 4D N=2 super Yang-Mills theory. It is a type IIB string theory compactified on R^6\times K3 and containing parallel 7-branes. It can also be regarded as a subsector of Vafa's F-theory compactified on K3\times K3 and is thus dual to the heterotic string on K3\times T^2. We show that the one-loop prepotential in this gauge theory is exactly equal to the interaction produced by classical closed string exchange. A monopole configuration corresponds to an open Dirichlet 5-brane wrapping around K3 with ends attached to two 7-branes. 
  The application of D-brane methods to large black holes whose Schwarzschild radius is larger than the compactification scale is problematic. Callan and Maldacena have suggested that despite apparent problems of strong interactions when the number of branes becomes large, the open string degrees of freedom may remain very dilute due to the growth of the horizon area which they claim grows more rapidly than the average number of open strings. Such a picture of a dilute weakly coupled string system conflicts with the picture of a dense string-soup that saturates the bound of one string per planck area. A more careful analysis shows that Callan and Maldacena were not fully consistent in their estimates. In the form that their model was studied it can not be used to extrapolate to large mass without being in conflict with the Hawking Bekenstein entropy formula. A somewhat modified model can reproduce the correct entropy formula. In this ``improved model" the number of string bits on the horizon scales like the entropy in agreement with earlier speculations of Susskind. 
  In these notes evidence is presented for intepreting the moduli space of the integrable model associated to $N\!=\!2$ gauge theories with $N\!=\!4$ matter content, in terms of Calabi-Yau manifolds. We restrict to the case of gauge group $SU(2)$, which is compared with the moduli space of the Calabi-Yau manifold $WP_{11226}^{12}$ appearing in the rank three dual pair $(K^{3}\times T^{2} / WP_{11226}^{12})$. The singularity loci of both spaces are maped in a one to one way and, in the weak coupling limit, $N\!=\!2$ $SU(2)$ pure Yang-Mills is obtained in both cases by the same type of blow up. Comments on the interpretation of the strong coupling locus from the perspective of the integrable system are done. 
  This paper is a direct continuation of\ \BLZ\ where we begun the study of the integrable structures in Conformal Field Theory. We show here how to construct the operators ${\bf Q}_{\pm}(\lambda)$ which act in highest weight Virasoro module and commute for different values of the parameter $\lambda$. These operators appear to be the CFT analogs of the $Q$ - matrix of Baxter\ \Baxn, in particular they satisfy famous Baxter's ${\bf T}-{\bf Q}$ equation. We also show that under natural assumptions about analytic properties of the operators ${\bf Q}(\lambda)$ as the functions of $\lambda$ the Baxter's relation allows one to derive the nonlinear integral equations of Destri-de Vega (DDV)\ \dVega\ for the eigenvalues of the ${\bf Q}$-operators. We then use the DDV equation to obtain the asymptotic expansions of the ${\bf Q}$ - operators at large $\lambda$; it is remarkable that unlike the expansions of the ${\bf T}$ operators of \ \BLZ, the asymptotic series for ${\bf Q}(\lambda)$ contains the ``dual'' nonlocal Integrals of Motion along with the local ones. We also discuss an intriguing relation between the vacuum eigenvalues of the ${\bf Q}$ - operators and the stationary transport properties in boundary sine-Gordon model. On this basis we propose a number of new exact results about finite voltage charge transport through the point contact in quantum Hall system. 
  Quantum mechanics ordinarily describes particles as being pointlike, in the sense that the uncertainty $\Delta x$ can, in principle, be made arbitrarily small. It has been shown that suitable correction terms to the canonical commutation relations induce a finite lower bound to spatial localisation. Here, we perturbatively calculate the corrections to the energy levels of an in this sense nonpointlike particle in isotropic harmonic oscillators. Apart from a special case the degeneracy of the energy levels is removed. 
  A recent generalisation of the Raychaudhuri equations for timelike geodesic congruences to families of $D$ dimensional extremal, timelike, Nambu--Goto surfaces embedded in an $N$ dimensional Lorentzian background is reviewed. Specialising to $D=2$ (i.e the case of string worldsheets) we reduce the equation for the generalised expansion $\theta _{a}, (a =\sigma,\tau)$ to a second order, linear, hyperbolic partial differential equation which resembles a variable--mass wave equation in $1+1$ dimensions. Consequences, such as a generalisation of geodesic focussing to families of worldsheets as well as exactly solvable cases are explored and analysed in some detail. Several possible directions of future research are also pointed out. 
  The energy condition inequalities for the matter stress energy comprised out of the dilaton and Maxwell fields in the dilaton-Maxwell gravity theories emerging out of string theory are examined in detail. In the simplistic 1+1 dimensional models, $R\le 0$ (where $R$ is the Ricci scalar), turns out to be the requirement for ensuring focusing of timelike geodesics. In 3+1 dimensions, we outline the requirements on matter for pure dilaton theories-these in turn constrain the functional forms of the dilaton. Furthermore, in charged dilaton gravity a curious opposite behaviour of the matter stress energy w.r.t the violation/conservation of the Weak Energy Condition is noted for the electric and magnetic black hole metrics written in the string frame of reference. We also investigate the matter that is necessary for creating certain specific non-asymptotically flat black holes. For the electric and magnetic black hole metrics, strangely, matter satisfies the weak energy condition in the string frame. Finally, the Averaged Null Energy Condition is evaluated along radial null geodesics for each of these black hole spacetimes. 
  The generalised Raychaudhuri equations derived by Capovilla and Guven are exclusively for extremal, timelike Nambu--Goto membranes. In this article, we construct the corresponding equations for string world--sheets in the presence of a background Kalb--Ramond field. We analyse the full set of equations by concentrating on special cases in which the generalised shear or the generalised rotation or both are set to zero. If only the generalised shear is set to zero then it is possible to identify the components of the generalised rotation with the projections of the field strength of the Kalb--Ramond potential. 
  A finite expansion of the exponential map for a $N\times N$ matrix is presented. The method uses the Cayley-Hamilton theorem for writing the higher matrix powers in terms of the first N-1 ones. The resulting sums over the corresponding coefficients are rational functions of the eigenvalues of the matrix. 
  The probability amplitude for tunneling between the Dirac vacua corresponding to different signs of a parity breaking fermionic mass $M$ in $2+1$ dimensions is studied, making contact with the continuum overlap formulation for chiral determinants. It is shown that the transition probability in the limit when $M \to \infty$ corresponds, via the overlap formalism, to the squared modulus of a chiral determinant in two Euclidean dimensions. The transition probabilities corresponding to two particular examples: fermions on a torus with twisted boundary conditions, and fermions on a disk in the presence of an external constant magnetic field are evaluated. 
  We discuss the $N=2$ $SU(2)$ Yang-Mills theory coupled with a massive matter  in the weak coupling. In particular, we obtain the instanton expansion of its prepotential. Instanton contributions in the mass-less limit are completely reproduced. We study also the double scaling limit of this massive theory and find that the prepotential with instanton corrections in the double scaling limit coincides with that of $N=2$ $SU(2)$ Yang-Mills theory without matter. 
  We present a class of black $p$-brane solutions of M-theory which were hitherto known only in the extremal supersymmetric limit, and calculate their macroscopic entropy and temperature. 
  A two-dimensional string model with dynamical cancellation of folds is considered. The action of the model contains the self-intersection number which is defined for surfaces immersed into 4D targets. The two additional variables are not dynamical and live on a compact manifold. In this sense the model is a compactification of a 4D theory. The cancellation forces the string $\th$ angle to be equal $\pi$. Candidates for string states are constructed. Some mathematical background is given. 
  A detailed quantitative analysis of the transition process mediated by a sphaleron type non-Abelian gauge field configuration in a static Einstein universe is carried out. By examining spectra of the fluctuation operators and applying the zeta function regularization scheme, a closed analytical expression for the transition rate at the one-loop level is derived. This is a unique example of an exact solution for a sphaleron model in $3+1$ spacetime dimensions. 
  We give a unified description of all BPS states of M-theory compactified on $T^5$ in terms of the five-brane. We compute the mass spectrum and degeneracies and find that the $SO(5,5,Z)$ U-duality symmetry naturally arises as a T-duality by assuming that the world-volume theory of the five-brane itself is described by a string theory. We also consider the compactification on $S^1/Z_2 \times T^4$, and give a new explanation for its correspondence with heterotic string theory by exhibiting its dual equivalence to M-theory on $K3\times S^1$. 
  Here we discuss the two dimensional quantum electrodynamics in curved space-time, especially in the background of some black holes. We first show the existence of some new quantum mechanical solution which has interesting properties. Then for some special black holes we discuss the fermion-black hole scattering problem. The issue of confinement is intimately connected with these solutions and we also comment on this in this background. Finally, the entanglement entropy and the Hawking radiation are also discussed in this background from a slightly different viewpoint. 
  The integrable systems associated with Seiberg-Witten geometry are considered both from the Hitchin-Donagi-Witten gauge model and in terms of intermediate Jacobians of Calabi-Yau threefolds. Dual pairs and enhancement of gauge symmetries are discussed on the basis of a map from the Donagi-Witten ``moduli'' into the moduli of complex structures of the Calabi-Yau threefold. 
  We study the general form of the equations for isotropic single-scalar, multi-scalar and dyonic $p$-branes in superstring theory and M-theory, and show that they can be cast into the form of Liouville, Toda (or Toda-like) equations. The general solutions describe non-extremal isotropic $p$-branes, reducing to the previously-known extremal solutions in limiting cases. In the non-extremal case, the dilatonic scalar fields are finite at the outer event horizon. 
  We discuss N=2 SU(2) Yang-Mills gauge theories coupled with N_f (=2,3) massive hypermultiplets in the weak coupling limit. We determine the exact massive prepotentials and the monodromy matrices around the weak coupling limit. We also study that the double scaling limit of these massive theories and find that the massive N_f -1 theory can be obtained from the massive N_f theory. New formulae for the massive prepotentials and the monodromy matrices are proposed. In these formulae, N_f dependences are clarified. 
  A new version of application Pauli-Villars regularized Green functions in the quantum field theory using higher derivatives is proposed. In this version the regularizing mass $M$ is large but finite. Our approach is demonstrated and discussed on the example of QED. It is shown that in our case there are no ultraviolet divergences and - on the example of the selfenergy spinor Feynman diagram - no infrared ones. 
  The gauge dependence of the effective action of composite fields for general gauge theories in the framework of the quantization method by Batalin, Lavrov and Tyutin is studied. The corresponding Ward identities are obtained. The variation of composite fields effective action is found in terms of new set of operators depending on composite field. The theorem of the on-shell gauge fixing independence for the effective action of composite fields in such formalism is proved. brief discussion of gravitational-vector induced interaction for Maxwell theory with composite fields is given. 
  We discuss spontaneous supersymmetry breaking in $N=2$ globally supersymmetric theories describing abelian vector multiplets. The most general form of the action admits, in addition to the usual Fayet-Iliopoulos term, a magnetic Fayet-Iliopoulos term for the auxiliary components of dual vector multiplets. In a generic case, this leads to a spontaneous breakdown of one of the two supersymmetries. In some cases however, dyon condensation restores $N=2$ SUSY vacuum. This talk is based on the work done in collaboration with H. Partouche \cite{apt}. 
  We discuss the issue of screening and confinement of external colour charges in bosonised two-dimensional quantum chromodynamics. Our computation relies on the static solutions of the semi-classical equations of motion. The significance of the different representations of the matter field is explicitly studied. We arrive at the conclusion that the screening phase prevails, even in the presence of a small mass term for the fermions. To confirm this result further, we outline the construction of operators corresponding to screened quarks. 
  Roughly speaking, naked singularities are singularities that may be seen by timelike observers. The Cosmic Censorship conjecture forbids their existence by stating that a reasonable system of energy will not, under reasonable conditions, collapse into a naked singularity. There are however many (classical) counter-examples to this conjecture in the literature. We propose a defense of the conjecture through the quantum theory. We will show that the Hawking effect and the accompanying back reaction, when consistently applied to naked singularities in two dimensional models of dilaton gravity with matter and a cosmological constant, prevent their formation by causing them to explode or emit radiation catastrophically. This contrasts with black holes which radiate slowly. If this phenomenon is reproduced in the four dimensional world, the explosion of naked singularities should have observable consequences. 
  We calculate the leading order interactions of massless D-brane excitations. Their 4-point functions are found to be identical to those found in type I theory. The amplitude for two massless D-brane fluctuations to produce a massless closed string is found to possess interesting new structure. As a function of its single kinematic invariant, it displays an infinite sequence of alternating zeros and poles. At high transverse momenta, this amplitude decays exponentially, indicating a growing effective thickness of the D-brane. This amplitude is the leading process by which non-extremal D-branes produce Hawking radiation. 
  We obtain a four dimensional exploding universe solution in string theory. The solution is obtained from the string theory in the flat background by using non-abelian $T$-duality and the analytic continuation. In the solution, the radius of the universe is finite for fixed time and the universe is surrounded with the boundary which is composed of the singularity. The boundary runs away with the speed of light and the flat space-time is left behind. 
  We propose a non-critical string field theory for $2d$ quantum gravity coupled to ($p$,$q$) conformal fields. The Hamiltonian is described by the generators of the $W_p$ algebra, and the Schwinger-Dyson equation is equivalent to a vacuum condition imposed on the generators of $W_p$ algebra. 
  We describe the qualitative properties of p-brane solutions of supergravity theories and present two examples of p-brane solutions, first the dyonic membrane solutions of N=2 D=8 supergravity and second the intersecting M-brane solutions of D=11 supergravity. 
  We consider a higher derivative effective theory for an Abelian gauge field in three dimensions, which represents the result of integrating out heavy matter fields interacting with a classical gauge field in a parity-conserving way. We retain terms containing up to two derivatives of $F_{\mu\nu}$, but make no assumption about the strength of this field. We then quantize the gauge field, and compute the one-loop effective action for a constant $\fmn$. The result is explicitly evaluated for the case of a constant magnetic field. 
  We construct several examples where duality transformation commutes with the orbifolding procedure even when the orbifolding group does not act freely, and there are massless states from the twisted sector at a generic point in the moduli space. Often the matching of spectrum in the dual theories is a result of non-trivial identities satisfied by the coefficients of one loop tadpoles in the heterotic, type II and type I string theories. 
  The constructed $Sp(4,R)/GL(2,R)$ matrix operator defines the family of isotropic geodesic containing vacuum point lines in the target space of the stationary D=4 Einstein--Maxwell--dilaton--axion theory. This operator is used to derive a class of solutions which describes a point center system with nontrivial values of mass, parameter NUT, as well as electric, magnetic, dilaton and axion charges. It is shown that this class contains both particular solutions Majumdar--Papapetrou--like black holes and massless asymptotically nonflat naked singularities. 
  We formulate in terms of the quantum inverse scattering method the exact solution of a $spl(2|1)$ invariant vertex model recently introduced in the literature. The corresponding transfer matrix is diagonalized by using the algebraic (nested) Bethe ansatz approach. The ground state structure is investigated and we argue that a Pokrovsky-Talapov transition is favored for certain value of the 4-dimensional $spl(2|1)$ parameter. 
  New loop equations for all genera in $c = \frac{1}{2}$ non-critical string theory are constructed. Our loop equations include two types of loops, loops with all Ising spins up (+ loops) and those with all spins down ( $-$ loops). The loop equations generate an algebra which is a certain extension of $W_3$ algebra and are equivalent to the $W_3$ constraints derived before in the matrix-model formulation of 2d gravity. Application of these loop equations to construction of Hamiltonian for $c = \frac{1}{2}$ string field theory is considered. 
  For a rather broad class of dynamical systems subject to mixed fermionic first and second class constraints or infinitely reducible first class constraints (IR1C), a manifestly covariant scheme of supplementation of IR1C to irreducible ones is proposed. For a model with IR1C only, an application of the scheme leads to a system with covariantly splitted and irreducible first and second class constraints. Modified Lagrangian formulations for the Green--Schwarz superstring, Casalbuoni--Brink--Schwarz superparticle and Siegel superparticle, which reproduce the supplementation scheme, are suggested. 
  Several two-dimensional quantum field theory models have more than one vacuum state. Familiar examples are the Sine-Gordon and the $\phi^4_2$-model. It is known that these models possess states, called kink states, which interpolate different vacua. A general construction scheme for kink states in the framework of algebraic quantum field theory is developed in a previous paper. However, for the application of this method, the crucial condition is the split property for wedge algebras in the vacuum representations of the considered models. It is believed that the vacuum representations of $P(\phi)_2$-models fulfill this condition, but a rigorous proof is only known for the massive free scalar field. Therefore, we investigate in a construction of kink states which can directly be applied to $P(\phi)_2$-model, by making use of the properties of the dynamic of a $P(\phi)_2$-model. 
  The general structure of the renormalization group equations for the low energy effective field theory formulation of pure gravity is presented. The solution of these equations takes a particular simple form if the mass scale of the effective theory is much smaller than the Planck mass (a possibility compatible with the renormalization of the effective theory). A theory with just one free renormalized parameter is obtained when contributions suppressed by inverse powers of the Planck mass are neglected. 
  We show that certain heterotic string amplitudes are given in terms of correlators of the twisted topological (2,0) SCFT, corresponding to the internal sector of the N=1 spacetime supersymmetric background. The genus g topological partition function $F^g$ corresponds to a term in the effective action of the form $W^{2g}$, where W is the gauge or gravitational superfield. We study also recursion relations related to holomorphic anomalies, showing that, contrary to the type II case, they involve correlators of anti-chiral superfields. The corresponding terms in the effective action are of the form $W^{2g}\Pi^n$, where $\Pi$ is a chiral superfield obtained by chiral projection of a general superfield. We observe that the structure of the recursion relations is that of N=1 spacetime supersymmetry Ward identity. We give also a solution of the tree level recursion relations and discuss orbifold examples. 
  We discuss the construction of the spectral curve and the action integrals for the ``elliptic" $XYZ$ spin chain of the length $N_c$. Our analysis can reflect the integrable structure behind the ``elliptic" ${\cal N}=2$ supersymmetric QCD with $N_f=2N_c$. 
  We study the local conserved charges in integrable spin chains of the XYZ type with nontrivial boundary conditions. The general structure of these charges consists of a bulk part, whose density is identical to that of a periodic chain, and a boundary part. In contrast with the periodic case, only charges corresponding to interactions of even number of spins exist for the open chain. Hence, there are half as many charges in the open case as in the closed case. For the open spin-1/2 XY chain, we derive the explicit expressions of all the charges. For the open spin-1/2 XXX chain, several lowest order charges are presented and a general method of obtaining the boundary terms is indicated. In contrast with the closed case, the XXX charges cannot be described in terms of a Catalan tree pattern. 
  Functional relation for commuting quantum transfer matrices of quantum integrable models is identified with classical Hirota's bilinear difference equation. This equation is equivalent to the completely discretized classical 2D Toda lattice with open boundaries. The standard objects of quantum integrable models are identified with elements of classical nonlinear integrable difference equation. In particular, elliptic solutions of Hirota's equation give complete set of eigenvalues of the quantum transfer matrices. Eigenvalues of Baxter's $Q$-operator are solutions to the auxiliary linear problems for classical Hirota's equation. The elliptic solutions relevant to Bethe ansatz are studied. The nested Bethe ansatz equations for $A_{k-1}$-type models appear as discrete time equations of motions for zeros of classical $\tau$-functions and Baker-Akhiezer functions. Determinant representations of the general solution to bilinear discrete Hirota's equation and a new determinant formula for eigenvalues of the quantum transfer matrices are obtained. 
  Unconventional way of handling the perturbative series is presented with the help of Heaviside transformation with respect to the mass. We apply Heaviside transform to the effective potential in the massive Gross-Neveu model and carry out perturbative approximation of the massless potential by dealing with the resulting Heaviside function. We find that accurate values of the dynamical mass can be obtained from the Heaviside function already at finite orders where just the several of diagrams are incorporated. We prove that our approximants converges to the exact massless potential in the infinite order. Small mass expansion of the effective potential can be also obtained in our approach. 
  By using the spherical coordinates in 3+1 dimensions we study the self-adjointness of the Dirac Hamiltonian in an Aharonov-Bohm gauge field of an infinitely thin magnetic flux tube. It is shown that the angular part of the Dirac Hamiltonian requires self-adjoint extensions as well as its radial one. The self-adjoint extensions of the angular part are parametrized by 2x2 unitary matrix. 
  We show that the two sets of self-dual Yang-Mills equations in 8-dimensions proposed in (E.Corrigan, C.Devchand, D.B.Fairlie and J.Nuyts, {\it Nuclear Physics} {\bf B214}, 452-464, (1983)) form respectively elliptic and overdetermined elliptic systems under the Coulomb gauge condition. In the overdetermined case, the Yang-Mills fields can depend at most on $N$ arbitrary constants, where $N$ is the dimension of the gauge group. We describe a subvariety ${\cal P}_8 $ of the skew-symmetric $8\times 8$ matrices by an eigenvalue criterion and we show that the solutions of the elliptic equations of Corrigan et al. are among the maximal linear submanifolds of ${\cal P}_8$. We propose an eight order action for which the elliptic set is a maximum. 
  Using general arguments we determine the allowed region for the end point frequency and the peak energy density of the stochastic background of gravity waves expected in string cosmology. We provide an accurate estimate of the minimal experimental sensitivity required to detect a signal in the Hz to GHz range. 
  We show that excitations of physical interest of the heavenly equation are generated by symmetry operators which yields two reduced equations with different characteristics. One equation is of the Liouville type and gives rise to gravitational instantons, including those found by Eguchi-Hanson and Gibbons-Hawking. The second equation appears for the first time in the theory of heavenly spaces and provides meron-like configurations endowed with a fractional topological charge. A link is also established between the heavenly equation and the socalled Schr{\"o}der equation, which plays a crucial role in the bootstrap model and in the renormalization theory. 
  In a one-dimensional lattice, the induced metric (from a noncommutative geometry calculation) breaks translation invariance. This leads to some inconsistencies among different spectator frames, in the observation of the hoppings of a test particle between lattice sites. To resolve the inconsistencies between the different spectator frames, we replace the test particle's bare mass by an effective locally dependent mass. This effective mass also depends on the lattice constant - i.e. it is a scale dependent variable (a "running" mass). We also develop an alternative approach based on a compensating potential. The induced potential between a spectator frame and the test particle is attractive on the average. We then show that the entire formalism holds for a quantum particle represented by a wave function, just as it applies to the mechanics of a classical point particle. 
  The cvariant path integral quantization of the theory of the scalar and spinor particles interacting through the abelian and non-Abelian Chern-Simons gauge fields is carried out and is shown to be mathematically ill defined due to the absence of the transverse components of these gauge fields. This is remedied by the introduction of the Maxwell or the Maxwell-type (in the non-Abelian case)term which makes the theory superrenormalizable and guarantees its gauge-invariant regularization and renormalization. The generating functionals are constructed and shown to be formally the same as those of QED (or QCD) in 2+1 dimensions with the substitution of the Chern-Simons propagator for the photon (gluon) propagator. By constructing the propagator in the general case, the existence of two limits; pure Chern-Simons and QED (QCD) after renormalization is demonstrated.  By carrying out carefully the path integral quantization of the non-Abelian Chern-Simons theories using the De Witt-Fadeev-Popov and the Batalin-Fradkin- Vilkovisky methods it is demonstrated that there is no need to quantize the dimensionless charge of the theory. The main reason is that the action in the exponent of the path integral is BRST-invariant which acquires a zero winding number and guarantees the BRST renormalizability of the model.  The S-matrix operator is constructed, and starting from this S-matrix operator novel topological unitarity identities are derived that demand the vanishing of the gauge-invariant sum of the imaginary parts of the Feynman diagrams with a given number of intermediate on-shell topological photon lines in each order of perturbation theory. These identities are illustrated by an explicit example. 
  A method for the calculation of the BRST cohomology, recently developed for 2D gravity theory and the bosonic string in the Beltrami parametrization,is generalised to the superstring theories quantized in super-Beltrami parametrization. 
  We carry out a thorough survey of entropy for a large class of $p$-branes in various dimensions. We find that the Bekenstein-Hawking entropy may be given a simple world volume interpretation only for the non-dilatonic $p$-branes, those with the dilaton constant throughout spacetime. The entropy of extremal non-dilatonic $p$-branes is non-vanishing only for the solutions preserving 1/8 of the original supersymmetries. Upon toroidal compactification these reduce to dyonic black holes in 4 and 5 dimensions. For the self-dual string in 6 dimensions, which preserves 1/4 of the original supersymmetries, the near-extremal entropy is found to agree with a world sheet calculation, in support of the existing literature. The remaining 3 interesting cases preserve 1/2 of the original supersymmetries. These are the self-dual 3-brane in 10 dimensions, and the 2- and 5-branes in 11 dimensions. For all of them the scaling of the near-extremal Bekenstein-Hawking entropy with the Hawking temperature is in agreement with a statistical description in terms of free massless fields on the world volume. 
  We review the status of domain walls in $N=1$ supergravity theories for both the vacuum domain walls as well as dilatonic domain walls. We concentrate on a systematic analysis of the nature of the space-time in such domain wall backgrounds and the special role that supersymmetry is playing in determining the nature of such configurations. 
  $D$-brane boundary states for type II superstrings are constructed by enforcing the conditions that preserve half of the space-time supersymmetry. A light-cone coordinate frame is used where time is identified as one of the coordinates transverse to the brane's (euclidean) world-volume so that the $p$-brane is treated as a $(p+1)$-instanton. The boundary states have the superspace interpretation of top or bottom states in a light-cone string superfield. The presence of a non-trivial open-string boundary condensate give rise to the familiar $D$-brane source terms that determine the (linearized) Born--Infeld-like effective actions for $p$-branes and the (linearized) equations of motion for the massless fields implied by the usual $p$-brane ansatze. The `energy' due to closed string exchange between separate $D$-branes is calculated (to lowest order in the string coupling) in situations with pairs of parallel, intersecting as well as orthogonal branes -- in which case the unbroken supersymmetry may be reduced. Configurations of more than two branes are also considered in situations in which the supersymmetry is reduced to $1/8$ or $1/16$ of the full amount. The Ward identities resulting from the non-linearly realized broken space-time supersymmetry in the presence of a $D$-brane are also discussed. 
  We prove that the dynamical system charaterized by the Hamiltonian $ H = \lambda N \sum_{j}^{N} p_j + \mu \sum_{j,k}^{N} {{(p_j p_k)}^{1\over 2}} \{ cos [ \nu ( q_j - q_k)] \} $ proposed and studied by Calogero [1,2] is equivalent to a system of {\it non-interacting} harmonic oscillators. We find the explicit form of the conserved currents which are in involution. We also find the action-angle variables and solve the initial value problem in simple form. 
  The constraint structure of 2D-gravity with the Weyl and area-preserving diffeomorphism invariances is analysed in the ADM formulation. It is found that when the area-preserving diffeomorphism constraints are kept, the usual conformal gauge does not exist, whereas there is the possibility to choose the so-called ``quasi-light-cone'' gauge, in which besides the area-preserving diffeomorphism invariance, the reduced Lagrangian also possesses the SL(2,R) residual symmetry. The string-like approach is applied to quantise this model, but a fictitious non-zero central charge in the Virasoro algebra appears. When a set of gauge-independent SL(2,R) current-like fields is introduced instead of the string-like variables, a consistent quantum theory is obtained. 
  We work out finite-dimensional integral formulae for the scalar product of genus one states of the group $G$ Chern-Simons theory with insertions of Wilson lines. Assuming convergence of the integrals, we show that unitarity of the elliptic Knizhnik-Zamolodchikov-Bernard connection with respect to the scalar product of CS states is closely related to the Bethe Ansatz for the commuting Hamiltonians building up the connection and quantizing the quadratic Hamiltonians of the elliptic Hitchin system. 
  In this paper we derive in a coordinate-free manner the field equations for a lagrangean consisting of Yang-Mills kinetical term plus Chern-Simons self-coupling term. This equation turns out to be an eigenvalue equation for the covariant laplacian. 
  We analyze the behaviour of moduli fields in string effective models between the end of inflation and reheating. The effective moduli potential during this era is derived for a class of simple models. We argue that this potential significantly stabilizes the modulus at its high energy minimum, if some restrictions on modular weights are met. Two mechanisms to further stabilize the moduli to their low energy minima are discussed explicitly: coinciding minima at a point of enhanced symmetry, and the smooth transition from high to low energy minimum by an effective mass term C^2 H^2. For both cases we present explicit examples, and C^2 is found to be O(10) at most. In addition, we show that during a smooth transition the reduction of the modulus amplitude strongly depends on the shape of the low energy potential. 
  We elucidate the interplay between gauge and supersymmetry anomalies in six-dimensional $N=1$ supergravity with generalized couplings between tensor and vector multiplets. We derive the structure of the five-dimensional supergravity resulting from the $S_1$ reduction of these models and give the constraints on Chern-Simons couplings that follow from duality to $M$ theory compactified on a Calabi-Yau threefold. The duality is supported only on a restricted class of Calabi-Yau threefolds and requires a special type of intersection form. We derive five-dimensional central-charge formulas and discuss briefly the associated phase transitions. Finally, we exhibit connections with $F$-theory compactifications on Calabi-Yau manifolds that admit elliptic fibrations. This analysis suggests that $F$ theory unifies Type-$IIb$ three-branes and $M$-theory five-branes. 
  The RSOS restriction of the Zhiber-Mikhailov-Shabat (ZMS) model is investigated. It is shown that in addition to the usual RSOS restriction, corresponding to $\Phi_{(1,2)}$ and $\Phi_{(2,1)}$ perturbations of minimal CFT, there is another one which yields $\Phi_{(1,5)}$ perturbations of non-unitary minimal models. The new RSOS restriction is carried out and the particular case of the minimal models ${\cal M}_{(3,10)}$, ${\cal M}_{(3,14)}$ and ${\cal M}_{(3,16)}$ is discussed in detail. In the first two cases, while the mass spectra of the two RSOS restrictions are the same, the bootstrap systems and the detailed amplitudes are different. In the third case, even the spectra of the two RSOS restrictions are different. In addition, for ${\cal M}_{(3,10)}$ an interpretation in terms of the tensor product of two copies of ${\cal M}_{(2,5)}$ is given. 
  We characterise a class of SU(2) gluonic field configurations in the modified axial gauge where a zero mode component vanishes at some space point but the global Haar measure remains non-zero. The consequence of this is that gluonic wavefunctionals need not vanish at the boundary of the fundamental modular domain, which itself permits $\theta$ dependence in QCD(3+1). 
  We consider a fibrillar medium with a continuous distribution of two-level atoms coupled to quantized electromagnetic fields. Perturbation theory is developed based on the current algebra satisfied by the atomic operators. The one-loop corrections to the dispersion relation for the polaritons and the dielectric constant are computed. Renormalization group equations are derived which demonstrate a screening of the two-level splitting at higher energies. Our results are compared with known results in the slowly varying envelope and rotating wave approximations. We also discuss the quantum sine-Gordon theory as an approximate theory. 
  We show how the method of separation of variables can be used to construct integrable models corresponding to curves describing vacuum structure of four-dimensional ${\cal N} = 2$ SUSY Yang-Mills theories. We use this technique to construct models corresponding to $SU(N)$ Yang-Mills theory with $N_f<2N$ matter hypermultiplets by generalising the periodic Toda lattice. We also show that some special cases of massive $SU(3)$ gauge theory can be equivalently described by the generalisations of the Goryachev-Chaplygin top obtained via separation of variables. 
  We first establish some general results connecting real and complex Lie algebras of first-order differential operators. These are applied to completely classify all finite-dimensional real Lie algebras of first-order differential operators in $R^2$. Furthermore, we find all algebras which are quasi-exactly solvable, along with the associated finite-dimensional modules of analytic functions. The resulting real Lie algebras are used to construct new quasi-exactly solvable Schroedinger operators on $R^2$. 
  We analyze how string theory dualities may be described in M theory. T dualities arise from scalar-vector dualities in the worldvolume of the membrane of M theory. ``Electric-magnetic'' dualities arise from a duality transformation in M theory compactified on a 3-torus, which takes the membrane into a fivebrane wrapped around the 3-torus. 
  We examine the proposal made recently that the su(3) modular invariant partition functions could be related to the geometry of the complex Fermat curves. Although a number of coincidences and similarities emerge between them and certain algebraic curves related to triangular billiards, their meaning remains obscure. In an attempt to go beyond the su(3) case, we show that any rational conformal field theory determines canonically a Riemann surface. 
  Any non-critical N=2 string is shown to have an affine sl(2|1) worldsheet symmetry in the conformal gauge. 
  We prove the uncertainty relation $T_{\triangle V}\triangle m\stackrel{>}{\sim }2\pi \hbar /c^2,$ which is realized on a statistical mechanical level for an ensemble of events in $(1+D)$-dimensional spacetime with motion parametrized by an invariant ``proper time'' $\tau ,$ where $T_{\triangle V}$ is the average passage interval in $\tau $ for the events which pass through a small (typical) $(1+D)$-volume $\triangle V,$ and $\triangle m$ is the dispersion of mass around its on-shell value in such an ensemble. We show that a linear mass spectrum is a completely general property of a $(1+D)$-dimensional off-shell theory. 
  We show that the open membrane action on $T^3\times S^1/Z^2$ is equivalent to the closed membrane action on K3. The main difference between the two actions is that one generates the KK modes in the worldvolume action which is the strong coupling limit of \IIA\ while the other action generates the KK modes in a worldsheet action. Thus explaining membrane-string duality in D=7, which naturally leads to string-string duality in D=6. 
  I give a relatively elementary proof of the symmetric space theorem, due to Goddard, Nahm and Olive \cite{GNO}. Unlike their original proof, which involves the quark-model construction, I only use elementary algebraic techniques. 
  We study a class of Calogero-Sutherland type one dimensional N-body quantum mechanical systems, with potentials given by $$ V( x_1, x_2, \cdots x_N) = \sum_{i <j} {g \over {(x_i - x_j)^2}} - \frac{g^{\prime}}{\sum_{i<j}(x_i - x_j)^2} + U(\sqrt{\sum_{i<j}(x_i - x_j)^2}),$$ where $U(\sqrt{\sum_{i<j}(x_i - x_j)^2})$'s are of specific form. It is shown that, only for a few choices of $U$, the eigenvalue problems can be solved {\it exactly}, for arbitrary $g^{\prime}$. The eigen spectra of these Hamiltonians, when $g^{\prime} \ne 0$, are non-degenerate and the scattering phase shifts are found to be energy dependent. It is further pointed out that, the eigenvalue problems are amenable to solution for wider choices of $U$, if $g^{\prime}$ is conveniently fixed. These conditionally exactly solvable problems also do not exhibit energy degeneracy and the scattering phase shifts can be computed {\it only} for a specific partial wave. 
  T-duality has been shown to arise from a gauge symmetry in some string theories. However, in the case of \IIA\ compactified on a circle, it is not possible to show that this is the case. This situation is rather uncomfortable since string string duality suggests that all T-dualities should arise from a gauge symmetry. Here we show that the T-duality of \IIA\ compactified on a circle arises from the reparametrization of the supermembrane. Then we show how this reparametrization can be understood in terms of a gauge symmetry of the supermembrane, thus allowing to connect T-duality in \II\ to a gauge symmetry of the supermembrane. 
  In this paper, we probe the validity of the tunnelling interpretation that is usually called forth in literature to explain the phenomenon of particle production by time independent classical electromagnetic backgrounds. We show that the imaginary part of the effective lagrangian is zero for a complex scalar field quantized in a time independent, but otherwise arbitrary, magnetic field. This result implies that no pair creation takes place in such a background. But we find that when the quantum field is decomposed into its normal modes in the presence of a spatially confined and time independent magnetic field, there exists a non-zero tunnelling probability for the effective Schr{\" o}dinger equation. According to the tunnelling interpretation, this result would imply that spatially confined magnetic fields can produce particles, thereby contradicting the result obtained from the effective lagrangian. This lack of consistency between these two approaches calls into question the validity of attributing a non-zero tunnelling probability for the effective Schr{\" o}dinger equation to the production of particles by the time independent electromagnetic backgrounds. The implications of our analysis are discussed. 
  We consider the methods by which higher-level and non-simply laced gauge symmetries can be realized in free-field heterotic string theory. We show that all such realizations have a common underlying feature, namely a dimensional truncation of the charge lattice, and we identify such dimensional truncations with certain irregular embeddings of higher-level and non-simply laced gauge groups within level-one simply-laced gauge groups. This identification allows us to formulate a direct mapping between a given subgroup embedding, and the sorts of GSO constraints that are necessary in order to realize the embedding in string theory. This also allows us to determine a number of useful constraints that generally affect string GUT model-building. For example, most string GUT realizations of higher-level gauge symmetries G_k employ the so-called diagonal embeddings G_k\subset G\times G \times...\times G. We find that there exist interesting alternative embeddings by which such groups can be realized at higher levels, and we derive a complete list of all possibilities for the GUT groups SU(5), SU(6), SO(10), and E_6 at levels k=2,3,4 (and in some cases up to k=7). We find that these new embeddings are always more efficient and require less central charge than the diagonal embeddings which have traditionally been employed. As a byproduct, we also prove that it is impossible to realize SO(10) at levels k>4. This implies, in particular, that free-field heterotic string models can never give a massless 126 representation of SO(10). 
  The structure of state vector space for a general (non-anomalous) gauge theory is studied within the Lagrangian version of the $Sp(2)$-symmetric quantization method. The physical {\it S}-matrix unitarity conditions are formulated. The general results are illustrated on the basis of simple gauge theory models. 
  We discuss the relation between the Gell-Mann-Low beta function and the ``flowing couplings'' of the Wilsonian action $S_\L[\phi]$ of the exact renormalization group (RG) at the scale $\L$. This relation involves the ultraviolet region of $\L$ so that the condition of renormalizability is equivalent to the Callan-Symanzik equation. As an illustration, by using the exact RG formulation, we compute the beta function in Yang-Mills theory to one loop (and to two loops for the scalar case). We also study the infrared (IR) renormalons. This formulation is particularly suited for this study since: $i$) $\L$ plays the r\^ole of a IR cutoff in Feynman diagrams and non-perturbative effects could be generated as soon as $\L$ becomes small; $ii$) by a systematical resummation of higher order corrections the Wilsonian flowing couplings enter directly into the Feynman diagrams with a scale given by the internal loop momenta; $iii$) these couplings tend to the running coupling at high frequency, they differ at low frequency and remain finite all the way down to zero frequency. 
  A natural extension of the standard model within non-commutative geometry is presented. The geometry determines its Higgs sector. This determination is fuzzy, but precise enough to be incompatible with experiment. 
  The automorphism group $G_{2}$ of the octonions changes when octonion X,Y-product variants are used. I present here a general solution for how to go from $G_{2}$ to its X,Y-product variant. 
  A quantum deformation of 3-dimensional lattice gauge theory is defined by applying the Reshetikhin-Turaev functor to a Heegaard diagram associated to a given cell complex. In the root-of-unity case, the construction is carried out with a modular Hopf algebra. In the topological (weak-coupling) limit, the gauge theory partition function gives a 3-fold invariant, coinciding in the simplicial case with the Turaev-Viro one. We discuss bounded manifolds as well as links in manifolds. By a dimensional reduction, we obtain a q-deformed gauge theory on Riemann surfaces and find a connection with the algebraic Alekseev-Grosse-Schomerus approach. 
  We apply the method of conical singularities to calculate the tree-level entropy and its one-loop quantum corrections for a charged Kerr black hole. The Euclidean geometry for the Kerr-Newman metric is considered. We show that for an arbitrary periodization in Euclidean space there exists a conical singularity at the horizon. Its $\delta$-function like curvatures are calculated and are shown to behave similar to the static case. The heat kernel expansion for a scalar field on this conical space background is derived and the (divergent) quantum correction to the entropy is obtained. It is argued that these divergences can be removed by renormalization of couplings in the tree-level gravitational action in a manner similar to that for a static black hole. 
  A geometrical formulation of the T-duality rules for type II superstring Ramond--Ramond fields is presented. This is used to derive the Wess-Zumino terms in superstring D-brane actions, including terms proportional to the mass parameter of the IIA theory, thereby completing partial results in the literature. For non-abelian world-volume gauge groups the massive type IIA D-brane actions contain non-abelian Chern--Simons terms for the Born--Infeld gauge potential, implying a quantization of the IIA cosmological constant. 
  The new spacetime-supersymmetric description of the superstring is used to compute tree-level scattering amplitudes for an arbitrary number of massless four-dimensional states. The resulting Koba-Nielsen formula is manifestly SO(3,1) super-Poincare invariant and is easily generalized to scattering in the presence of a D-brane. 
  We point out that off-shell four-dimensional supersymmetry implies strange hermiticity properties for the N=1 RNS superstring. However, these hermiticity properties become natural when the N=1 superstring is embedded into an N=2 superstring. 
  Exact stationary soliton solutions of the fifth order KdV type equation $$ u_t +\alpha u^p u_x +\beta u_{3x}+\gamma u_{5x} = 0$$ are obtained for any p ($>0$) in case $\alpha\beta>0$, $D\beta>0$, $\beta\gamma<0$ (where D is the soliton velocity), and it is shown that these solutions are unstable with respect to small perturbations in case $p\geq 5$. Various properties of these solutions are discussed. In particular, it is shown that for any p, these solitons are lower and narrower than the corresponding $\gamma = 0$ solitons. Finally, for p = 2 we obtain an exact stationary soliton solution even when $D,\alpha,\beta,\gamma$ are all $>0$ and discuss its various properties. 
  This is a review of the new manifestly spacetime-supersymmetric description of the superstring. The new description contains N=2 worldsheet supersymmetry, and is related by a field redefinition to the standard RNS description. It is especially convenient for four-dimensional compactifications since SO(3,1) super-Poincar\'e invariance can be made manifest. Parts of this work have been done in collaboration with Warren Siegel and Cumrun Vafa.   This review is based on lectures given at the VIII J.A. Swieca summer school and should be easily accesible to anyone familiar with the RNS superstring description. 
  The conditions leading to a nontrivial renormalization of the topological charge in four--dimensional Yang--Mills theory are discussed. It is shown that if the topological term is regarded as the limit of a certain nontopological interaction, quantum effects due to the gauge bosons lead to a finite multiplicative renormalization of the theta--parameter while fermions give rise to an additional shift of theta. A truncated form of an exact renormalization group equation is used to study the scale dependence of the theta--parameter. Possible implications for the strong CP--problem of QCD are discussed. 
  We investigate the critical behavior of the lambda phi^4 theory defined on S^1 x R^d having two finite length scales beta, the circumference of S^1, and k^{-1}, the blocking scale introduced by the renormalization group transformation. By numerically solving the coupled differential RG equations for the finite-temperature blocked potential U_{beta,k}(Phi) and the wavefunction renormalization constant Z_{beta,k}(Phi), we demonstrate how the finite-size scaling variable betabar = beta k determines whether the phase transition is (d+1)- or d-dimensional in the limits betabar >> 1 and betabar << 1, respectively. For the intermediate values of betabar, finite-size effects play an important role. We also discuss the failure of the polynomial expansion of the effective potential near criticality. 
  The behaviour of solitons in integrable theories is strongly constrained by the integrability of the theory; i.e. by the existence of an infinite number of conserved quantities which these theories are known to possess. One usually expects the scattering of solitons in such theories to be rather simple, i.e. trivial. By contrast, in this paper we generate new soliton solutions for the planar integrable chiral model whose scattering properties are highly nontrivial; more precisely, in head-on collisions of $N$ indistinguishable solitons the scattering angle (of the emerging structures relative to the incoming ones) is $\pi/N$. We also generate soliton-antisoliton solutions with elastic scattering; in particular, a head-on collision of a soliton and an antisoliton resulting in $90^0$ scattering. 
  It has recently been shown that for certain classical non-dilatonic $p$-branes, the entropy and temperature satisfy the ideal-gas relation $S\sim T^{p}$ in the near-extremal regime. We extend these results to cases where the dilaton is non-vanishing, but nevertheless remains finite on the horizon in the extremal limit, showing that the ideal-gas relation is again satisfied. At the classical level, however, this relation does break down if the dilaton diverges on the horizon. We argue that such a divergence indicates the breakdown of the validity of the classical approximation, and that by taking string and worldsheet loop corrections into account, the validity of the entropy/temperature relation may be extended to include these cases. This opens up the possibility of giving a microscopic interpretation of the entropy for all near-extremal $p$-branes. 
  A generating function is given for the number, $E(l,k)$, of irreducible $k$-fold Euler sums, with all possible alternations of sign, and exponents summing to $l$. Its form is remarkably simple: $\sum_n E(k+2n,k) x^n = \sum_{d|k}\mu(d) (1-x^d)^{-k/d}/k$, where $\mu$ is the M\"obius function. Equivalently, the size of the search space in which $k$-fold Euler sums of level $l$ are reducible to rational linear combinations of irreducible basis terms is $S(l,k) = \sum_{n<k}{\lfloor(l+n-1)/2\rfloor\choose n}$. Analytical methods, using Tony Hearn's REDUCE, achieve this reduction for the 3698 convergent double Euler sums with $l\leq44$; numerical methods, using David Bailey's MPPSLQ, achieve it for the 1457 convergent $k$-fold sums with $l\leq7$; combined methods yield bases for all remaining search spaces with $S(l,k)\leq34$. These findings confirm expectations based on Dirk Kreimer's connection of knot theory with quantum field theory. The occurrence in perturbative quantum electrodynamics of all 12 irreducible Euler sums with $l\leq 7$ is demonstrated. It is suggested that no further transcendental occurs in the four-loop contributions to the electron's magnetic moment. Irreducible Euler sums are found to occur in explicit analytical results, for counterterms with up to 13 loops, yielding transcendental knot-numbers, up to 23 crossings. 
  We study string theory propagating on R^6 times K3 by constructing orientifolds of Type IIB string theory compactified on the T^4/Z_N orbifold limits of the K3 surface. This generalises the Z_2 case studied previously. The orientifold models studied may be divided into two broad categories, sometimes related by T-duality. Models in category A require either both D5- and D9-branes, or only D9-branes, for consistency. Models in category B require either only D5-branes, or no D-branes at all. This latter case is an example of a consistent purely closed unoriented string theory. The spectra of the resulting six dimensional N=1 supergravity theories are presented. Precise statements are made about the relation of the Z_N ALE spaces and D5-branes to instantons in the dual heterotic string theory. 
  By using the brick wall method we calculate the thermodynamic potential of the complex scalar field in a charged Kerr black hole. Using it we show that in the Hartle-Hawking state the leading term of the entropy is proportional to $\frac{ A _H}{\epsilon^2}$, which becomes divergent as the system approaches the black hole horizon. The origin of the divergence is that the density of states diverges at the horizon. 
  In a previous paper, we proposed a construction of $U_q(sl(2))$ quantum group symmetry generators for 2d gravity, where we took the chiral vertex operators of the theory to be the quantum group covariant ones established in earlier works. The basic idea was that the covariant fields in the spin $1/2$ representation themselves can be viewed as generators, as they act, by braiding, on the other fields exactly in the required way. Here we transform this construction to the more conventional description of 2d gravity in terms of Bloch wave/Coulomb gas vertex operators, thereby establishing for the first time its quantum group symmetry properties. A $U_q(sl(2))\otimes U_q(sl(2))$ symmetry of a novel type emerges: The two Cartan-generator eigenvalues are specified by the choice of matrix element (bra/ket Verma-modules); the two Casimir eigenvalues are equal and specified by the Virasoro weight of the vertex operator considered; the co-product is defined with a matching condition dictated by the Hilbert space structure of the operator product. This hidden symmetry possesses a novel Hopf like structure compatible with these conditions.  At roots of unity it gives the right truncation. Its (non linear) connection with the $U_q(sl(2))$ previously discussed is disentangled. 
  A superposition of bosons and generalized deformed parafermions corresponding to an arbitrary paraquantization order $p$ is considered to provide deformations of parasupersymmetric quantum mechanics. New families of parasupersymmetric Hamiltonians are constructed in connection with two examples of su(2) nonlinear deformations such as introduced by Polychronakos and Ro\v cek. 
  It is argued that $M$-theory compactified on {\it any} of Joyce's $Spin(7)$ holonomy 8-manifolds are dual to compactifications of heterotic string theory on Joyce 7-manifolds of $G_2$ holonomy. 
  We discuss why classical hair is desirable for the description of black holes, and show that it arises generically in a wide class of field theories involving extra dimensions. We develop the canonical formalism for theories with the matter content that arises in string theory. General covariance and duality are used to determine the form of surface terms. We derive an effective theory (reduced Hamiltonian) for the hair in terms of horizon variables. % accessible to an observer at infinity. Solution of the constraints expresses these variables in terms of hair accessible to an observer at infinity. We exhibit some general properties of the resulting theory, including a formal identification of the temperature and entropy. The Cveti\v{c}-Youm dyon is described in some detail, as an important example. 
  We consider general properties of charged circular cosmic strings in a general family of world-sheet string models. We then specialize to a model recently proposed by Carter and Peter. This model was shown to give a good description of the features of the superconducting cosmic strings originally discovered by Witten. We derive an explicit expression for the potential determining the dynamics of the string and we present explicit expressions for the string tension and energy density as a function of string-loop radius. We also obtain explicit expressions for the wiggle and woggle speeds (speeds of transverse and longitudinal perturbations, respectively). We show that the contraction of the uniformly charged string is essentially governed by the string tension (for large loop radius) and by a {\it finite} Coulomb barrier (for small loop radius). We argue for the unobstructed contraction of a uniformly charged loop over the Coulombic barrier and its eventual collapse to a charged point. The implication of such an effect to the possible formation of naked singularities, in violation of the cosmic censorship hypothesis, is finally discussed. 
  Target space duality transformations are considered for bosonic sigma models and strings away from RG fixed points. A set of consistency conditions are derived, and are seen to be nontrivially satisfied at one-loop order for arbitrary running metric, antisymmetric tensor and dilaton backgrounds. Such conditions are sufficiently stringent to enable an independent determination of the sigma model beta functions at this order. 
  Short introduction to exotic differential structures on manifolds is given. The possible physical context of this mathematical curiosity is discussed. The topic is very interesting although speculative. 
  We find the spectrum of magnetic monopoles produced in the symmetry breaking SU(5) to [SU(3)\times SU(2)\times U(1)']/Z_6 by constructing classical bound states of the fundamental monopoles. The spectrum of monopoles is found to correspond to the spectrum of one family of standard model fermions and hence, is a starting point for constructing the dual standard model. At this level, however, there is an extra monopole state - the ``diquark'' monopole - with no corresponding standard model fermion. If the SU(3) factor now breaks down to Z_3, the monopoles with non-trivial SU(3) charge get confined by strings in SU(3) singlets. Another outcome of this symmetry breaking is that the diquark monopole becomes unstable (metastable) to fragmentation into fundamental monopoles and the one-one correspondence with the standard model fermions is restored. We discuss the fate of the monopoles if the [SU(2)\times U(1)']/Z_2 factor breaks down to U(1)_Q by a Higgs mechanism as in the electroweak model. Here we find that monopoles that are misaligned with the vacuum get connected by strings even though the electroweak symmetry breaking does not admit topological strings. We discuss the lowest order quantum corrections to the mass spectrum of monopoles. 
  A global superalgebra with 32 supercharges and all possible central extensions is studied in order to extract some general properties of duality and hidden dimensions in a theory that treats $p$-branes democratically. The maximal number of dimensions is 12, with signature (10,2), containing one space and one time dimensions that are hidden from the point of view of perturbative 10-dimensional string theory or its compactifications. When the theory is compactified on $R^{d-1,1}\otimes T^{c+1,1}$ with $d+c+2=12,$ there are isometry groups that relate to the hidden dimensions as well as to duality. Their combined classification schemes provide some properties of non-perturbative states and their couplings. 
  We use an exactly solvable (0,2) supersymmetric conformal field theory with gauge group SO(10) to investigate the superpotential of the corresponding classical string vacuum. We provide evidence that the rational point lies in the Landau-Ginzburg phase of the linear sigma-model and calculate exactly all three- and four-point functions of the gauge singlets. These couplings already put severe constraints on the possible flat directions of the superpotential. Finally, we contemplate about the flat direction related to Kahler deformations of the underlying linear sigma-model. 
  The most general four-dimensional non-linear sigma-model, having the second-order derivatives only and interacting with a background metric and an antisymmetric tensor field, is constructed. Despite its apparent non-renormalizability, just imposing the one-loop UV-finiteness conditions determines the unique model, which may be finite to all orders of the quantum perturbation theory. This model is known as the four-dimensional Donaldson-Nair-Schiff theory, which is a four-dimensional analogue of the standard two-dimensional Wess-Zumino-Novikov-Witten model, and whose unique finiteness properties and an infinite-dimensional current algebra have long been suspected. 
  A deformed differential calculus is developed based on an associative star-product. In two dimensions the Hamiltonian vector fields model the algebra of pseudo-differential operator, as used in the theory of integrable systems. Thus one obtains a geometric description of the operators. A dual theory is also possible, based on a deformation of differential forms. This calculus is applied to a number of multidimensional integrable systems, such as the KP hierarchy, thus obtaining a geometrical description of these systems. The limit in which the deformation disappears corresponds to taking the dispersionless limit in these hierarchies. 
  The method of self-adjoint extensions is employed to determine the vacuum quantum numbers induced by a singular static magnetic vortex in $2+1$-dimensional spinor electrodynamics. The results obtained are gauge-invariant and, for certain values of the extension parameter, both periodic in the value of the vortex flux and possessing definite parity with respect to the charge conjugation. 
  In this work we give a deformation theoretical approach to the problem of quantization. First the notion of a deformation of a noncommutative ringed space over a commutative locally ringed space is introduced within a language coming from Algebraic Geometry and Complex Analysis. Then we define what a Dirac quantization of a commutative ringed space with a Poisson structure, the space of classical observables, is. Afterwards the normal order quantization of the Poisson space of classical polynomial observables on a cotangent bundle is constructed. By using a complete symbol calculus on manifolds we succeed in extending the normal order quantization of polynomial observables to a quantization of a Poisson space of symbols on a cotangent bundle. Furthermore we consider functorial properties of these quantizations. Altogether it is shown that a deformation theoretical approach to quantization is possible not only in a formal sense but also such that the deformation parameter $\hbar$ can attain any real value. 
  Representations of the quantum q-oscillator algebra are studied with particular attention to local Hamiltonian representations of the Schroedinger type. In contrast to the standard harmonic oscillators such systems exhibit a continuous spectrum. The general scheme of realization of the q-oscillator algebra on the space of wave functions for a one-dimensional Schroedinger Hamiltonian shows the existence of non-Fock irreducible representations associated to the continuous part of the spectrum and directly related to the deformation. An algorithm for the mapping of energy levels is described. 
  The purpose of this article is to apply the concept of the spectral triple, the starting point for the analysis of noncommutative spaces in the sense of A.~Connes, to the case where the algebra $\cA$ contains both bosonic and fermionic degrees of freedom. The operator $\cD$ of the spectral triple under consideration is the square root of the Dirac operator und thus the forms of the generalized differential algebra constructed out of the spectral triple are in a representation of the Lorentz group with integer spin if the form degree is even and they are in a representation with half-integer spin if the form degree is odd. However, we find that the 2-forms, obtained by squaring the connection, contains exactly the components of the vector multiplet representation of the supersymmetry algebra. This allows to construct an action for supersymmetric Yang-Mills theory in the framework of noncommutative geometry. 
  We calculate the Kahlerian and the lowest order non-Kahlerian contributions to the one loop effective superpotential using super-Feynman graphs in the massless Wess-Zumino Model, the massive Wess-Zumino Model and N=1, U(1) gauge theory. We also calculate the Kahlerian term in Yang-Mills Theory for a general gauge group. Using this latter result we find the one loop Kahlerian contribution for N=2 Yang-Mills Theory in terms of N=1 superfields and we show that it can only come from non-holomorphic contributions to the N=2 effective potential. 
  It has long been claimed that antisymmetric tensor field of the second rank is longitudinal after quantization. Such a situation is quite unacceptable from a viewpoint of the Correspondence Principle. On the basis of the Lagrangian formalism we calculate the Pauli-Lyuban'sky vector of relativistic spin for this field. Even at the classical level it can be equal to zero after application of the well-known constraints. The correct quantization procedure permits us to propose solution of this puzzle in the modern field theory. Obtained results develop the previous consideration of Evans [{\it Physica A}214 (1995) 605-618]. 
  For any worldline reformulation of a quantum field theory for Dirac fermions, this paper shows that worldline supersymmetry may generally be enforced by the vanishing of the commutator of the Dirac operator with the worldline Hamiltonian. The action of supersymmetry on the worldline Lagrangian may not, however, be written in terms of the variations on the fields in the usual way, except when the spinning particle couples just to a one-form. By reduction from six to four dimensions of the worldline reformulation for a spinning particle coupled to a three-form, corrections to the superworldline Lagrangian are presented which are needed in order to reproduce correct field theory results from worldline perturbation theory in an unambiguous way. 
  We propose a renormalization prescription for the Wheeler-DeWitt equation of (3+1)-dimensional Einstein gravity and also propose a strong coupling expansion as an approximation scheme to probe quantum geometry at length scales much smaller than the Planck length. We solve the Wheeler-DeWitt equation to the second order in the expansion in a class of local solutions and discuss problems arising in our approach. 
  A standard approach to analyzing tunneling processes in various physical contexts is to use instanton or imaginary time path techniques. For systems in which the tunneling takes place in a time dependent setting, the standard methods are often applicable only in special cases, e.g. due to some additional symmetries. We consider a collection of time dependent tunneling problems to which the standard methods cannot be applied directly, and present an algorithm, based on the WKB approximation combined with complex time path methods, which can be used to calculate the relevant tunneling probabilities. This collection of problems contains, among others, the spontaneous nucleation of topological defects in an expanding universe, the production of particle - antiparticle pairs in a time dependent electric field, and false vacuum decay in field theory from a coherently oscillating initial state. To demonstrate the method, we present detailed calculations of the time dependent decay rates for the last two examples. 
  We show how the multiloop bosonic Green function of closed string theory reduces to the world-line Green function as defined by Schmidt and Schubert in the limit where the string world-sheet degenerates into a $\Phi^3$ particle diagram. To obtain this correspondence we have to make an appropriate choice of the local coordinates defined on the degenerate string world sheet. We also present a set of simple rules that specify, in the explicit setting of the Schottky parametrization, which is the corner of moduli space corresponding to a given multiloop $\Phi^3$ diagram. 
  Vertex operators for the deformed Virasoro algebra are defined, their bosonic representation is constructed and difference equation for the simplest vertex operators is described. 
  A new $N=1$ superfield model in $D=4$ flat superspace is suggested. This model describes dynamics of chiral compensator and can be treated as a low-energy limit of $D=4$, $N=1$ quantum superfield supergravity. Renormalization structure of this model is studied and one-loop counterterms are calculated. It is shown that the theory is infrared free. An effective action for the model under consideration is investigated in infrared domain. The lower contributions to the one-loop effective action are computed in explicit form. 
  We construct an explicit representation of the Sugawara generators for arbitrary level in terms of the homogeneous Heisenberg subalgebra, which generalizes the well-known expression at level 1. This is achieved by employing a physical vertex operator realization of the affine algebra at arbitrary level, in contrast to the Frenkel--Kac--Segal construction which uses unphysical oscillators and is restricted to level 1. At higher level, the new operators are transcendental functions of DDF ``oscillators'' unlike the quadratic expressions for the level-1 generators. An essential new feature of our construction is the appearance, beyond level 1, of new types of poles in the operator product expansions in addition to the ones at coincident points, which entail (controllable) non-localities in our formulas. We demonstrate the utility of the new formalism by explicitly working out some higher-level examples. Our results have important implications for the problem of constructing explicit representations for higher-level root spaces of hyperbolic Kac--Moody algebras, and $E_{10}$ in particular. 
  We calculate the potential between two different stationary D-branes and the velocity dependent potential between two different moving D-branes. We identify configurations with some unbroken supersymmetry, using a zero force condition. The potentials are compared with an eleven dimensional calculation of the scattering of a zero black-brane from the $0,2,4$ and 6 black-brane of type IIA supergravity. The agreement of these calculations provide further evidence for the D-brane description of black-branes, and for the eleven dimensional origin of type IIA string theory. 
  We consider possible conformal field theory (CFT) descriptions of the various inertial ranges that exist in $2d$ duality invariant Magnetohydrodynamics. Such models arise as effective theories of dyonic plasmas in 3 dimensions in which all fields are independent of the third coordinate. We find new constraints on the allowed CFT's compared to those that may describe turbulence in $2d$ plasmas of electric charges only. The predictions from CFT concerning equipartition of energy amongst the electric and magnetic fields are discussed, and quantities exhibiting universal scaling are derived. 
  The correlation function of the product of N generalized vertex operators satisfies an infinite set of Ward identities, related to a W_{\infty} algebra, whose extention out of the mass shell gives rise to equations which can be considered as a generalization of the compactified Calogero-Sutherland (CS) hamiltonians. In particular the wave function of the ground state of the compactified CS model is shown to be given by the value of the product of N vertex operators between the vacuum and exitated state. The role of vertex algebra as underlying unifying structure is pointed out. 
  Models of string cosmology predict a stochastic background of gravitational waves with a spectrum that is strongly tilted towards high frequencies. I give simple approximate expressions for spectral densities of the cosmic background which can be directly compared with sensitivities of gravitational wave detectors. 
  We study the effective action in Euclidean Yang-Mills theory with a compact simple gauge group in one-loop approximation assuming a covariantly constant gauge field strength as a background. For groups of higher rank and spacetimes of higher dimensions such field configurations have many independent color components taking values in Cartan subalgebra and many ``magnetic fields'' in each color component. In our previous investigation it was shown that such background is stable in dimensions higher than four provided the amplitudes of ``magnetic fields'' do not differ much from each other. In present paper we calculate exactly the relevant zeta-functions in the case of equal amplitudes of ``magnetic fields''. For the case of two ``magnetic fields'' with equal amplitudes the behavior of the effective action is studied in detail. It is shown that in dimensions $d=4,5,6,7$ $({\rm mod}\, 8)$, the perturbative vacuum is metastable, i.e., it is stable in perturbation theory but the effective action is not bounded from below, whereas in dimensions $d=9,10,11$ $({\rm mod}\, 8)$ the perturbative vacuum is absolutely stable. In dimensions $d=8$ $({\rm mod}\, 8)$ the perturbative vacuum is stable for small values of coupling constant but becomes unstable for large coupling constant leading to the formation of a non-perturbative stable vacuum with nonvanishing ``magnetic fields''. The critical value of the coupling constant and the amplitudes of the vacuum ``magnetic fields'' are evaluated exactly. 
  The two-point functions of the energy-momentum tensor and the Noether current are used to probe the O(3) nonlinear sigma model in an energy range below 10^4 in units of the mass gap $m$. We argue that the form factor approach, with the form factor series trunctated at the 6-particle level, provides an almost exact solution of the model in this energy range. The onset of the (2-loop) perturbative regime is found to occur only at energies around $100m$. 
  In this talk we present some links of the theory of the odderon with elliptic curves. These results were obtained in an earlier work \cite{RJ}. The natural degrees of freedom of the odderon turn out to coincide with conformal invariants of elliptic curves with a fixed `sign'. This leads to a formulation of the odderon which is modular invariant with respect to $\Gamma^2$ --- the unique normal subgroup of \slz {\,} of index 2. 
  The phase structure of $d=3$ Nambu-Jona-Lasinio model in curved spacetime with magnetic field is investigated in the leading order of the $1/N$-expansion and in linear curvature approximation (an external magnetic field is treated exactly). The possibility of the chiral symmetry breaking under the combined action of the external gravitational and magnetic fields is shown explicitly. At some circumstances the chiral symmetry may be restored due to the compensation of the magnetic field by the gravitational field. 
  Compactifications of M-theory to two dimensional space-time on ${(K3\times \T^5)}/ \Z_2$ and ${(K3\times K3\times \S^1)}/ \Z_2$ orientifolds are presented. These orientifolds provide examples of anomaly free chiral supergravity models in two dimensions with (8, 0) and (4, 0) supersymmetries. Anomaly free spectra at the enhanced symmetry points are also obtained. The results confirm the twisted sector contribution to the spectrum in the case of $\T^9/ \Z_2$ discussed earlier.  
  We formulate a conjecture for the three different Lax operators that describe the bosonic sectors of the three possible $N=2$ supersymmetric integrable hierarchies with $N=2$ super $W_n$ second hamiltonian structure. We check this conjecture in the simplest cases, then we verify it in general in one of the three possible supersymmetric extensions. To this end we construct the $N=2$ supersymmetric extensions of the Generalized Non-Linear Schr\"{o}dinger hierarchy by exhibiting the corresponding super Lax operator. To find the correct hamiltonians we are led to a new definition of super-residues for degenerate N=2 supersymmetric pseudodifferential operators. We have found a new non-polinomial Miura-like realization for $N=2$ superconformal algebra in terms of two bosonic chiral--anti--chiral free superfields. 
  We present two 1/8 supersymmetric intersecting p-brane solutions of 11-dimensional supergravity which upon compactification to four dimensions reduce to extremal dyonic black holes with finite area of horizon. The first solution is a configuration of three intersecting 5-branes with an extra momentum flow along the common string. The second describes a system of two 2-branes and two 5-branes. Related (by compactification and T-duality) solution of type IIB theory corresponds to a completely symmetric configuration of four intersecting 3-branes. We suggest methods for counting the BPS degeneracy of three intersecting 5-branes which, in the macroscopic limit, reproduce the Bekenstein-Hawking entropy. 
  The interactions which preserve the structure of the gauge interactions of the free theory are introduced in terms of the generalized fields method of solving the Batalin-Vilkovisky master equation. It is shown that by virtue of this method the solution of the descent equations resulting from the cohomological analysis is provided straightforwardly. The general scheme is illustrated by applying it to spin-1 gauge field in 3 and 4 dimensions, to free BF theory in 2-d and to the antisymmetric tensor field in any dimension. It is shown that it reproduces the results obtained by cohomological techniques. 
  We show how, via $T$-duality, intersecting $D$-Brane configurations in ten (six) dimensions can be obtained from the elementary $D$-Brane configurations by embedding a Type IIB $D$-Brane into a Type IIB Nine-Brane (Five-Brane) and give a classification of such configurations. We show that only a very specific subclass of these configurations can be realized as (supersymmetric) solutions to the equations of motion of IIA/IIB supergravity. Whereas the elementary $D$-brane solutions in $d=10$ are characterized by a single harmonic function, those in $d=6$ contain two independent harmonic functions and may be viewed as the intersection of two $d=10$ elementary $D$-branes. Using string/string/string triality in six dimensions we show that the heterotic version of the elementary $d=6$ $D$-Brane solutions correspond in ten dimensions to intersecting Neveu-Schwarz/Neveu-Schwarz (NS/NS) strings or five-branes and their $T$-duals. We comment on the implications of our results in other than ten and six dimensions. 
  Noncommutative geometry applied to the standard model of electroweak and strong interactions was shown to produce fuzzy relations among masses and gauge couplings. We refine these relations and show then that they are exhaustive. 
  We consider 4D quantum-dilaton gravity with the most general coupling in a homogeneous and isotropic universe, especially an inflationary one, which is essentially characterized by an exponentially expanding scale factor with time. We show that on the inflationary background this theory can be miraculously renormalized, at least at the one-loop level, which must be an effective theory during the inflation of the un-constructed complete quantum theory of gravity. 
  A free superstring with chiral N=2 supersymmetry in six dimensions is proposed. It couples to a two-form gauge field with a self-dual field strength. Compactification to four dimensions on a two-torus gives a strongly coupled N=4 four-dimensional gauge theory with SL(2, Z) duality and an infinite tower of dyons. Various authors have suggested that this string theory should be also the world-volume theory of M theory five-branes. Accepting this proposal, we find a puzzling factor of two in the application to black-hole entropy computations. 
  We present a derivation of the Schr\"odinger equation for a path integral of a point particle in a space with curvature and torsion which is considerably shorter and more elegant than what is commonly found in the literature. 
  The results of analysis of the one--loop spectrum of anomalous dimensions of composite operators in the scalar $ \phi^{4} $ model are presented. We give the rigorous constructive proof of the hypothesis on the hierarchical structure of the spectrum of anomalous dimensions -- the naive sum of any two anomalous dimensions generates a limit point in the spectrum. Arguments in favor of the nonperturbative character of this result and the possible ways of a generalization to other field theories are briefly discussed. 
  We study different aspects of integrable boundary quantum field theories, focusing mostly on the ``boundary sine-Gordon model'' and its applications to condensed matter physics. The first part of the review deals with formal problems. We analyze the classical limit and perform semi-classical quantization. We show that the non-relativistic limit corresponds to the Calogero-Moser model with a boundary potential. We construct a lattice regularization of the problem via the XXZ chain. We classify boundary bound states. We generalize the Destri de Vega method to compute the ground state energy of the theory on a finite interval. The second part deals with some applications to condensed matter physics. We show how to compute analytically time and space dependent correlations in one-dimensional quantum integrable systems with an impurity. Our approach is based on a description of these systems in terms of massless scattering of quasiparticles. Correlators follow then from matrix elements of local operators between multiparticle states -- the massless form-factors. Although, in general an infinite sum of these form-factors has to be considered, we find that for the current, spin and energy operators only a few (two or three) are necessary to obtain an accuracy of more than 1\%. Our results hold for arbitrary impurity strength, in contrary to the perturbative expansions in the coupling constants. As an example, we compute the frequency dependent condunctance, at zero temperature, in a Luttinger liquid with an impurity, and also discuss the susceptibility in the Kondo model and the time-dependent properties of the two-state problem with dissipation. 
  The Kramer--Neugebauer--like transformation is constructed for the stationary axisymmetric D=4 Einstein--Maxwell--dilaton--axion system. This transformation directly maps the dualized sigma--model equations of the theory into the nondualized ones. Also the new chiral $4 \times 4$ matrix representation of the problem is presented. 
  We give general expressions for singular vectors of the N=2 superconformal algebra in the form of {\it monomials} in the continued operators by which the universal enveloping algebra of N=2 is extended. We then show how the algebraic relations satisfied by the continued operators can be used to transform the monomials into the standard Verma-module expressions. Our construction is based on continuing the extremal diagrams of N=2 Verma modules to the states satisfying the twisted \hw{} conditions with complex twists. It allows us to establish recursion relations between singular vectors of different series and at different levels. Thus, the N=2 singular vectors can be generated from a smaller set of the so-called topological singular vectors, which are distinguished by being in a 1:1 correspondence with singular vectors in affine sl(2) Verma modules. The method of `continued products' of fermions is a counterpart of the method of complex powers used in the constructions of singular vectors for affine Lie algebras. 
  It has been argued by Dyson in the context of QED in flat spacetime that perturbative expansions in powers of the electric charge e cannot be convergent because if e is purely imaginary then the vacuum should be unstable to the production of charged pairs. We investigate the spontaneous production of such Dyson pairs in electrodynamics coupled to gravity. They are found to consist of pairs of zero-rest mass black holes with regular horizons. The properties of these zero rest mass black holes are discussed. We also consider ways in which a dilaton may be included and the relevance of this to recent ideas in string theory. We discuss accelerating solutions and find that, in certain circumstances, the `no strut' condition may be satisfied giving a regular solution describing a pair of zero rest mass black holes accelerating away from one another. We also study wormhole and tachyonic solutions and how they affect the stability of the vacuum. 
  We construct several examples of compactification of Type IIB theory on orientifolds and discuss their duals. In six dimensions we obtain models with $N=1$ supersymmetry, multiple tensor multiplets, and different gauge groups. In nine dimensions we obtain a model that is dual to M-theory compactified on a Klein bottle. 
  We construct new supersymmetric solutions of $D$=11 supergravity describing $n$ orthogonally ``overlapping" membranes and fivebranes for $n$=2,\dots,8. Overlapping branes arise after separating intersecting branes in a direction transverse to all of the branes. The solutions, which generalize known intersecting brane solutions, preserve at least $2^{-n}$ of the supersymmetry. Each pairwise overlap involves a membrane overlapping a membrane in a 0-brane, a fivebrane overlapping a fivebrane in a 3-brane or a membrane overlapping a fivebrane in a string. After reducing $n$ overlapping membranes to obtain $n$ overlapping $D$-2-branes in $D$=10, $T$-duality generates new overlapping $D$-brane solutions in type IIA and type IIB string theory. Uplifting certain type IIA solutions leads to the $D$=11 solutions. Some of the new solutions reduce to dilaton black holes in $D$=4. Additionally, we present a $D$=10 solution that describes two $D$-5-branes overlapping in a string. $T$-duality then generates further $D$=10 solutions and uplifting one of the type IIA solutions gives a new $D$=11 solution describing two fivebranes overlapping in a string. 
  We consider the evolution of quantum fields on a classical background space-time, formulated in the language of differential geometry. Time evolution along the worldlines of observers is described by parallel transport operators in an infinite-dimensional vector bundle over the space-time manifold. The time evolution equation and the dynamical equations for the matter fields are invariant under an arbitrary local change of frames along the restriction of the bundle to the worldline of an observer, thus implementing a ``quantum gauge principle''. We derive dynamical equations for the connection and a complex scalar quantum field based on a gauge field action. In the limit of vanishing curvature of the vector bundle, we recover the standard equation of motion of a scalar field in a curved background space-time. 
  Two different massive gauge invariant spin-one theories in $3+1$ dimensions, one Stuckelberg formulation and the other `$B^{\wedge}F$' theory, with Kalb-Ramond field are shown to be related by duality. This is demonstrated by gauging the global symmetry in the model and constraining the corresponding dual field strength to be zero by a Lagrange multiplier, which becomes a field in the dual theory. Implication of this equivalence to the $5$ dimensional theories from which these theories can be obtained is discussed. The self-dual Deser-Jackiw model in $2+1$ dimensions, is also shown to result by applying this procedure to Maxwell-Chern-Simon theory. 
  This paper studies the semiclassical approximation of simple supergravity in Riemannian four-manifolds with boundary, within the framework of $\zeta$-function regularization. The massless nature of gravitinos, jointly with the presence of a boundary and a local description in terms of potentials for spin ${3\over 2}$, force the background to be totally flat. First, nonlocal boundary conditions of the spectral type are imposed on spin-${3\over 2}$ potentials, jointly with boundary conditions on metric perturbations which are completely invariant under infinitesimal diffeomorphisms. The axial gauge-averaging functional is used, which is then sufficient to ensure self-adjointness. One thus finds that the contributions of ghost and gauge modes vanish separately. Hence the contributions to the one-loop wave function of the universe reduce to those $\zeta(0)$ values resulting from physical modes only. Another set of mixed boundary conditions, motivated instead by local supersymmetry and first proposed by Luckock, Moss and Poletti, is also analyzed. In this case the contributions of gauge and ghost modes do not cancel each other. Both sets of boundary conditions lead to a nonvanishing $\zeta(0)$ value, and spectral boundary conditions are also studied when two concentric three-sphere boundaries occur. These results seem to point out that simple supergravity is not even one-loop finite in the presence of boundaries. 
  I review how traditional grand unified theories, which require adjoint (or higher representation) Higgs fields for breaking to the standard model, can be contained within string theory. The status (as of January 1996) of the search for stringy free fermionic three generation SO(10) SUSY--GUT models is discussed. Progress in free fermionic classification of both SO(10)$_2$ charged and uncharged embeddings and in $N=1$ spacetime solutions is presented. Based on talks presented at the Workshop on SUSY Phenomena and SUSY GUTs, Santa Barbara, California, Dec. 7-11, 1995, and at the Orbis Scientiae, Coral Gables, Florida, January 25-28, 1996. Appearing in the Proceedings of Orbis Scientiae, 1996. 
  We show that the partition function of the super eigenvalue model satisfies an infinite set of constraints with even spins $s=4,6,\cdots,\infty$. These constraints are associated with half of the bosonic generators of the super $\left( W_{\infty \over 2}\oplus W_{{1+\infty}\over 2}\right)$ algebra. The simplest constraint $(s=4)$ is shown to be reducible to the super Virasoro constraints, previously used to construct the model. All results hold for finite $N$. 
  We provide an alternative interpretation for the topological terms in physics by investigating the low-energy gauge interacting system. The asymptotic behavior of the gauge field at infinity indicates that it traces out a closed loop in the infinite time interval: -infinity, + infinity. Adopting Berry's argument of geometric phase, we show that the adiabatic evolution of the gauge system around the loop results in an additional term to the effective action: the Chern-Simons term for three-dimensional spacetime, and the Pontrjagin term for the four-dimensional spacetime. 
  We define the $SU(2)\times SU(2)$ harmonic superspace analogs of tensor and nonlinear $(4,4)$, $2D$ supermultiplets. They are described by constrained analytic superfields and provide an off-shell formulation of a class of torsionful $(4,4)$ supersymmetric sigma models with abelian translational isometries on the bosonic target. We examine their relation to $(4,4)$ twisted multiplets and discuss different types of $(4,4)$ dualities associated with them. One of these dualities implies the standard abelian $T$-duality relations between the bosonic targets in the initial and dual sigma model actions. We show that $N=4$, $2D$ superconformal group admits a simple realization on the superfields introduced, and present a new superfield form of the $(4,4)$ $SU(2)\times U(1)$ WZNW action. 
  We introduce topological magnetic field in two-dimensional flat space, which admits a solution of scalar monopole that describes the nontrivial topology. In the Chern-Simons gauge field theory of anyons, we interpret the anyons as the quasi-particles composed of fermions and scalar monopoles in such a form that each fermion is surrounded by infinite number of scalar monopoles. It is the monopole charge that determines the statistics of anyons. We re-analyze the conventional arguments of the connection between topology and statistics in three-dimensional space, and find that those arguments are based on the global topology, which is relatively trivial compared with the monopole structure. Through a simple model, we formulate the three-dimensional anyon field  using infinite number of Dirac's magnetic monopoles that change the ordinary spacetime topology. The quasi-particle picture of the three-dimensional anyons is quite similar to that of the usual two-dimensional anyons. However, the exotic statistics there is not restricted to the usual fractional statistics, but a functional statistics. 
  We present a rigurous disscusion for abelian $BF$ theories in which the base manifold of the $U(1)$ bundle is homeomorphic to a Hilbert space. The theory has an infinte number of stages of reducibility. We specify conditions on the base manifold under which the covarinat quantization of the system can be performed unambiguously. Applications of the formulation to the superparticle and the supertstring are also discussed. 
  We find extremal four dimensional black holes with finite area constructed entirely from intersecting D-branes. We argue that the microscopic degeneracy of these configurations agrees with the Bekenstein-Hawking entropy formula. The absence of solitonic objects in these configurations may make them useful for dynamical studies of black holes. 
  The bound states of two particles are studied in frames of non-relativistic quantum field model with current $\times$ current type interaction by analyzing the Bethe-Salpeter amplitudes. The Bethe-Salpeter equations are obtained in closed form. The existence of Goldstone mode corresponding to the spontaneous breaking of additional SU(2) symmetry of the model is revealed. 
  The combinatorics of the BPHZ subtraction scheme for a class of ladder graphs for the three point vertex in $\phi^3$ theory is transcribed into certain connectivity relations for marked chord diagrams (knots with transversal intersections). The resolution of the singular crossings using the equivalence relations in these examples provides confirmation of a proposed fundamental relationship between knot theory and renormalization in perturbative quantum field theory. 
  Effective critical exponents for the \lambda \phi^4 scalar field theory are calculated as a function of the renormalization group block size k_o^{-1} and inverse critical temperature \beta_c. Exact renormalization group equations are presented up to first order in the derivative expansion and numerical solutions are obtained with and without polynomial expansion of the blocked potential. For a finite temperature system in d dimensions, it is shown that \bar\beta_c = \beta_c k_o determines whether the d-dimensional (\bar\beta_c << 1) or (d+1)-dimensional (\bar\beta_c >> 1) fixed point governs the phase transition. The validity of a polynomial expansion of the blocked potential near criticality is also addressed. 
  We suggest a duality invariant formula for the entropy and temperature of non-extreme black holes in supersymmetric string theory. The entropy is given in terms of the duality invariant parameter of the deviation from extremality and 56 SU(8) covariant central charges. It interpolates between the entropies of Schwarzschild solution and extremal solutions with various amount of unbroken supersymmetries and therefore serves for classification of black holes in supersymmetric string theories. We introduce the second auxiliary 56 via E(7) symmetric constraint. The symmetric and antisymmetric combinations of these two multiplets are related via moduli to the corresponding two fundamental representations of E(7): brane and anti-brane "numbers." Using the CPT as well as C symmetry of the entropy formula and duality one can explain the mysterious simplicity of the non-extreme black hole area formula in terms of branes and anti-branes. 
  This paper studies the role of the axial gauge in the semiclassical analysis of simple supergravity about the Euclidean four-ball, when non-local boundary conditions of the spectral type are imposed on gravitino perturbations at the bounding three-sphere. Metric perturbations are instead subject to boundary conditions completely invariant under infinitesimal diffeomorphisms. It is shown that the axial gauge leads to a non-trivial cancellation of ghost-modes contributions to the one-loop divergence. The analysis, which is based on zeta-function regularization, provides a full $\zeta(0)$ value which coincides with the one obtained from transverse-traceless perturbations for gravitons and gravitinos. The resulting one-loop divergence does not vanish. This property seems to imply that simple supergravity is not even one-loop finite in the presence of boundaries. 
  String propagation on D-dimensional curved backgrounds with Lorentzian signature is formulated as a geometrical problem of embedding surfaces. When the spatial part of the background corresponds to a general WZW model for a compact group, the classical dynamics of the physical degrees of freedom is governed by the coset conformal field theory SO(D-1)/SO(D-2), which is universal irrespective of the particular WZW model. The same holds for string propagation on D-dimensional flat space. The integration of the corresponding Gauss-Codazzi equations requires the introduction of (non-Abelian) parafermions in differential geometry. 
  We study the different phases of field theories of compact antisymmetric tensors of rank $h-1$ in arbitrary space-time dimensions $D=d+1$. Starting in a `Coulomb' phase, topological defects of dimension $d-h-1$ ($(d-h-1)$-branes) may condense leading to a generalized `confinement' phase. If the dual theory is also compact the model may also have a third, generalized `Higgs' phase, driven by the condensation of the dual $(h-2)$-branes. Developing on the work of Julia and Toulouse for ordered solid-state media, we obtain the low energy effective action for these phases. Each phase has two dual descriptions in terms of antisymmetric tensors of different ranks, which are massless for the Coulomb phase but massive for the Higgs and confinement phases. We illustrate our prescription in detail for compact QED in 4D. Compact QED and $O(2)$ models in 3D, as well as a periodic scalar field in 2D (strings on a circle), are also discussed. In this last case we show how $T$-duality is maintained if one considers both worldsheet instantons and their duals. We also unify various approaches to the problem of the axion mass in 4D string models. Finally we discuss possible implications of our results for non-perturbative issues in string theory. 
  It is shown, that the reduction of the circular quantum oscillator by the $Z_2$-group action results to the two systems: a two-dimensional hydrogen atom, and a ``charge - charged magnetic vortex" one, with the spin $\frac 12$. Analogously, the $Z_N$-reduction of the two-dimensional system with the central potential $r^{2(N-1)}$ results into $N$ bound ``charge - magnetic vertex" systems with the interaction potential $r^{2(1/N-1)}$ and spins $\sigma=\frac kN$, $k =0,1,..., (N-1)$. 
  We prove that self-dual gauge fields in type I superstring theory are equivalent to configurations of Dirichlet 5-branes, by showing that the world-sheet theory of a Dirichlet 1-brane moving in a background of 5-branes includes an ``ADHM sigma model.'' This provides an explicit construction of the equivalent self-dual gauge field. We also discuss type II. 
  We provide a unified construction of the symplectic forms which arise in the solution of both N=2 supersymmetric Yang-Mills theories and soliton equations. Their phase spaces are Jacobian-type bundles over the leaves of a foliation in a universal configuration space. On one hand, imbedded into finite-gap solutions of soliton equations, these symplectic forms assume explicit expressions in terms of the auxiliary Lax pair, expressions which generalize the well-known Gardner-Faddeev-Zakharov bracket for KdV to a vast class of 2D integrable models; on the other hand, they determine completely the effective Lagrangian and BPS spectrum when the leaves are identified with the moduli space of vacua of an N=2 supersymmetric gauge theory. For SU($N_c$) with $N_f\leq N_c+1$ flavors, the spectral curves we obtain this way agree with the ones derived by Hanany and Oz and others from physical considerations. 
  Using a global superalgebra with 32 fermionic and 528 bosonic charges, many features of p-brane dualities and hidden dimensions are discussed. 
  The violation of the Jacobi identity by the presence of magnetic charge is accomodated by using an explicitly nonassociative theory of octonionic fields. It is found that the dynamics of this theory is simplified if the Lagrangian contains only dyonic charges, but certain problems in the constrained quantisation remain. The extension of these concepts to string theory may however resolve these difficulties. 
  We render a thorough, physicist's account of the formulation of the Standard Model (SM) of particle physics within the framework of noncommutative differential geometry (NCG). We work in Minkowski spacetime rather than in Euclidean space. We lay the stress on the physical ideas both underlying and coming out of the noncommutative derivation of the SM, while we provide the necessary mathematical tools. Postdiction of most of the main characteristics of the SM is shown within the NCG framework. This framework, plus standard renormalization technique at the one-loop level, suggest that the Higgs and top masses should verify 1.3 m_top \lesssim m_H \lesssim 1.73 m_top. 
  We construct the path integral formula in terms of ``multi-periodic'' coherent state as an extension of the Nielsen-Rohrlich formula for spin. We make an exact calculation of the formula and show that, when a parameter corresponding to the magnitude of spin becomes large, the leading order term of the expansion coincides with the exact result. We also give an explicit correspondence between the trace formula in the multi-periodic coherent state and the one in the ``generalized'' coherent state. 
  We give a derivation of the Dirac operator on the noncommutative $2$-sphere within the framework of the bosonic fuzzy sphere and define Connes' triple. It turns out that there are two different types of spectra of the Dirac operator and correspondingly there are two classes of quantized algebras. As a result we obtain a new restriction on the Planck constant in Berezin's quantization. The map to the local frame in noncommutative geometry is also discussed. 
  The Schrodinger picture description of vacuum states in curved spacetime is further developed. General solutions for the vacuum wave functional are given for both static and dynamic (Bianchi type I) spacetimes and for conformally static spacetimes of Robertson-Walker type. The formalism is illustrated for simple cosmological models with time-dependent metrics and the phenomenon of particle creation is related to a special form of the kernel in the vacuum wave functional. 
  A parquet approximation (generalized ladder diagrams) in matrix models is considered. By means of numerical calculations we demonstrate that in the large $N$ limit the parquet approximation gives an excellent agreement with exact results. 
  We investigate restrictions to be imposed over the non commutative geometry model C+M_2+M_3 to make it to fit with experimental data. By constraining the action over quarks, a leptophobic Z' is got 
  Intrinsic and extrinsic geometric properties of string world sheets in curved space-time background are explored. In our formulation, the only dynamical degrees of freedom of the string are its immersion coordinates. Classical equation of motion and the space-time energy-momentum tensor of the string are obtained. The equations of motion for the extrinsic curvature action are second order for the scalar mean curvature of the world sheet. 1-loop divergent terms are calculated using the background field method. Asymptotic freedom of the extrinsic curvature coupling is established. 
  A class of integrable models of 1+1 dimensional dilaton gravity coupled to scalar and electromagnetic fields is obtained and explicitly solved. More general models are reduced to 0+1 dimensional Hamiltonian systems, for which two integrable classes (called s-integrable) are found and explicitly solved. As a special case, static spherical solutions of the Einstein gravity coupled to electromagnetic and scalar fields in any real space-time dimension are derived. A generalization of the `no-hair' theorem is pointed out and the Hamiltonian formulation that enables quantizing the s-integrable systems is outlined. 
  The implications of conformal invariance, as relevant in quantum field theories at a renormalisation group fixed point, are analysed with particular reference to results for correlation functions involving conserved currents and the energy momentum tensor. Ward identities resulting from conformal invariance are discussed. Explicit expressions for two and three point functions, which are essentially determined by conformal invariance, are obtained. As special cases we consider the three point functions for two vector and an axial current in four dimensions, which realises the usual anomaly simply and unambiguously, and also for the energy momentum tensor in general dimension $d$. The latter is shown to have two linearly independent forms in which the Ward identities are realised trivially, except if $d = 4$, when the two forms become degenerate. This is necessary in order to accommodate the two independent forms present in the trace of the energy momentum tensor on curved space backgrounds for conformal field theories in four dimensions. The coefficients of the two trace anomaly terms are related to the three parameters describing the general energy momentum tensor three point function. The connections with gravitational effective actions depending on a background metric are described. A particular form due to Riegert is shown to be unacceptable. Conformally invariant expressions for the effective action in four dimensions are obtained using the Green function for a differential operator which has simple properties under local rescalings of the metric. 
  We explicitly construct massive (0,4) supersymmetric ADHM sigma models which have heterotic p-brane solitons as their conformal fixed points. These yield the familiar gauge 5-brane and a new 1-brane solution which preserve 1/2 and 1/4 of the spacetime supersymmetry respectively. We also discuss an analogous construction for the type II NS-NS p-branes using (4,4) supersymmetric models. 
  We present in detail the procedure for calculating the heterotic one-loop effective action. We focus on gravitational and gauge couplings. We show that the two-derivative couplings of the gravitational sector are not renormalized at one loop when the ground state is supersymmetric. Arguments are presented that this non-renormalization theorem persists to all orders in perturbation theory. We also derive the full one-loop correction to the gauge coupling. For a class of $N=2$ ground states, namely those that are obtained by toroidal compactification to four dimensions of generic six-dimensional $N=1$ models, we give an explicit formula for the gauge-group independent thresholds, and show that these are equal within the whole family. 
  In the framework of a recently proposed method for computing exactly string amplitudes regularized in the infra-red, I determine the one-loop correlators for auxiliary fields in the symmetric $Z_2\times Z_2$ orbifold model. The $D$-field correlation function turns out to give the one-loop corrections for the gauge couplings, which amounts to a string-theory supersymmetry Ward identity. The two-point function for uncharged $F$ fields leads to the one-loop renormalization of the moduli K\"ahler metric, and eventually to the corrections for the Yukawa couplings. 
  We study a hermitian $(n+1)$-matrix model with plaquette interaction, $\sum_{i=1}^n MA_iMA_i$. By means of a conformal transformation we rewrite the model as an $O(n)$ model on a random lattice with a non polynomial potential. This allows us to solve the model exactly. We investigate the critical properties of the plaquette model and find that for $n\in]-2,2]$ the model belongs to the same universality class as the $O(n)$ model on a random lattice. 
  We present a pedagogical review of old inconsistencies of Classical Electrodynamics and of some new ideas that solve them. Problems with the electron equation of motion and with the non-integrable singularity of its self-field energy tensor are well known. They are consequences, we show, of neglecting terms that are null off the charge world-line but that give a non null contribution on its world-line. The electron self-field energy tensor is integrable without the use of any kind of renormalization; there is no causality violation and no conflict with energy conservation in the electron equation of motion, when its meaning is properly considered. 
  We examine a criterion for the anyonic superconductivity at zero temperature in Abelian matter-coupled Chern-Simons gauge field theories in three dimensions. By solving the Dyson-Schwinger equations, we obtain a critical value of the statistical charge for the superconducting phase in a massless fermion-Chern-Simons model. 
  The Hawking Beckenstein entropy of near extremal fivebranes is calculated in terms of a gas of strings living on the fivebrane. These fivebranes can also be viewed as near extremal black holes in five dimensions. 
  Thermodynamic properties of a class of black $p$-branes in $D$-dimensions considered by Duff and Lu are investigated semi-classically. For black $(d-1)$-brane, thermodynamic quantities depend on $D$ and $d$ only through the combination $\tilde d \equiv D-d-2$. The behavior of the Hawking temperature and the lifetime vary with $\tilde d$, with a critical value $\tilde d=2$. For $\tilde d>2$, there remains a remnant, in which non-zero entropy is stored. Implications of the fact that the Bekenstein-Hawking entropy of the black $(d-1)$-brane depend only on $\tilde d=D-d-2$ is discussed from the point of view of duality. 
  Using the BF version of pure Yang-Mills, it is possible to find a covariant representation of the 't Hooft magnetic flux operator. In this framework, 't Hooft's pioneering work on confinement finds an explicit realization in the continuum. Employing the Abelian projection gauge we compute the expectation value of the magnetic variable and find the expected perimeter law. We also check the area law behaviour for the Wilson loop average and compute the string tension which turns out to be of the right order of magnitude. 
  We explicitly develop a quaternionic version of the electroweak theory, based on the local gauge group $U(1, \; q)_{L}\mid U(1, \; c)_{Y}$. The need of a complex projection for our Lagrangian and the physical significance of the anomalous scalar solutions are also discussed. 
  We describe an approach to classify (meromorphic) representations of a given vertex operator algebra by calculating Zhu's algebra explicitly. We demonstrate this for FKS lattice theories and subtheories corresponding to the Z_2 reflection twist and the Z_3 twist. Our work is mainly offering a novel uniqueness tool, but, as shown in the Z_3 case, it can also be used to extract enough information to construct new representations. We prove the existence and some properties of a new non-unitary representation of the Z_3-invariant subtheory of the (two dimensional) Heisenberg algebra. 
  We complete the rules of translation between standard complex quantum mechanics (CQM) and quaternionic quantum mechanics (QQM) with a complex geometry. In particular we describe how to reduce ($2n$+$1$)-dimensional complex matrices to {\em overlapping\/} ($n$+$1$)-dimensional quaternionic matrices with generalized quaternionic elements. This step resolves an outstanding difficulty with reduction of purely complex matrix groups within quaternionic QM and avoids {\em anomalous} eigenstates. As a result we present a more complete translation from CQM to QQM and viceversa. 
  A technique for evaluating the electromagnetic Casimir energy in situations involving spherical or circular boundaries is presented. Zeta function regularization is unambiguously used from the start and the properties of Bessel and related zeta functions are central. Nontrivial results concerning these functions are given. While part of their application agrees with previous knowledge, new results like the zeta-regularized electromagnetic Casimir energy for a circular wire are included. 
  We derive the BPS mass formulae of the Dirichlet branes from the Born-Infeld type action. BPS saturation is realized when the brane has the minimal volume while keeping the appropriate winding numbers. We apply the idea to two cases, type IIA superstring compactified on $T^4$ and $K3$. The result is consistent with the string duality, and the expected spectrum for the BPS states is reproduced. 
  We consider a scalar field theory in Minkowski spacetime and define a coarse grained Closed Time Path (CTP) effective action by integrating quantum fluctuations of wavelengths shorter than a critical value. We derive an exact CTP renormalization group equation and solve it using a derivative expansion approach. Explicit calculation is performed for the $\lambda \phi^4$ theory. We discuss the relevance of the CTP average action in the study of non-equilibrium aspects of phase transitions in quantum field theory. 
  We consider 4D quantum gravity with N-dilatons with the most general couplings. Especially, on constant dilaton and arbitrary metric background, we show the structure of the divergent terms. We show the constraint between the couplings necessary to cancel the coefficient of the square of the Wyle tensor. Next we show the N dependence of a non-renormalizable divergent term, and found that it cannot be canceled in the case of $N \geq 1$ with any fine-tuning of the couplings. 
  The Laplace operator acting on antisymmetric tensor fields in a $D$--dimensional Euclidean ball is studied. Gauge-invariant local boundary conditions (absolute and relative ones, in the language of Gilkey) are considered. The eigenfuctions of the operator are found explicitly for all values of $D$. Using in a row a number of basic techniques, as Mellin transforms, deformation and shifting of the complex integration contour, and pole compensation, the zeta function of the operator is obtained. From its expression, in particular, $\zeta (0)$ and $\zeta'(0)$ are evaluated exactly. A table is given in the paper for $D=3, 4, ...,8$. The functional determinants and Casimir energies are obtained for $D=3, 4, ...,6$. 
  The infrared dynamics in the (3+1)-dimensional supersymmetric and non-supersymmetric Nambu-Jona-Lasinio model in a constant magnetic field is studied. While at strong coupling the dynamics in these two models is essentially different, it is shown that the models become equivalent at weak coupling. In particular, at weak coupling, as the strength of the magnetic field goes to infinity, both the supersymmetric and non-supersymmetric Nambu-Jona-Lasinio models are reduced to a continuum set of independent (1+1)-dimensional Gross-Neveu models, labeled by the coordinates in the plane perpendicular to the magnetic field. The relevance of these results for cosmological models based on supersymmetric dynamics is pointed out. 
  We analyze the one-loop effective gauge-field action in $Z_2$-orbifold compactifications of type-I theory. We show how, for non-abelian group factors, the threshold effects are ultraviolet finite though given entirely by a six-dimensional field theory expression. 
  The exact quantum integrability problem of the membrane is investigated. It is found that the spherical membrane moving in flat target spacetime backgrounds is an exact quantum integrable system for a particular class of solutions of the light-cone gauge equations of motion : a dimensionally-reduced $SU(\infty)$ Yang-Mills theory to one temporal dimension. Crucial ingredients are the exact integrability property of the $3D~SU(\infty)$ continuous Toda theory and its associated dimensionally-reduced $SU(\infty)$ Toda $molecule$ equation whose symmetry algebra is the $U_\infty$ algebra obtained from a dimensional-reducion of the $W_\infty \oplus {\bar W}_\infty$ algebras that act naturally on the original $3D$ continuous Toda theory. The $U_\infty$ algebra is explicitly constructed in terms of exact quantum solutions of the quantized continuous Toda equation. Highest weight irreducible representations of the $W_\infty$ algebras are also studied in detail. Continuous and discrete energy levels are both found in the spectrum . Other relevant topics are discussed in the conclusion. 
  A scale--dependent effective action for gravity is introduced and an exact nonperturbative evolution equation is derived which governs its renormalization group flow. It is invariant under general coordinate transformations and satisfies modified BRS Ward--Identities. The evolution equation is solved for a simple truncation of the space of actions. In 2+epsilon dimensions, nonperturbative corrections to the beta--function of Newton's constant are derived and its dependence on the cosmological constant is investigated. In 4 dimensions, Einstein gravity is found to be ``antiscreening'', i.e., Newton's constant increases at large distances. 
  Using stochastic quantization method we derive equations for correlators of quantum fluctuations around the classical solution in the massless phi^4 theory. The obtained equations are then solved in the lowest orders of perturbation theory, and the first correction to the free propagator of a quantum fluctuation is calculated. 
  The general form of N=2 supergravity coupled to an arbitrary number of vector multiplets and hypermultiplets, with a generic gauging of the scalar manifold isometries is given. This extends the results already available in the literature in that we use a coordinate independent and manifestly symplectic covariant formalism which allows to cover theories difficult to formulate within superspace or tensor calculus approach. We provide the complete lagrangian and supersymmetry variations with all fermionic terms, and the form of the scalar potential for arbitrary quaternionic manifolds and special geometry, not necessarily in special coordinates. Lagrangians for rigid theories are also written in this general setting and the connection with local theories elucidated. The derivation of these results using geometrical techniques is briefly summarized. 
  We show that the anomalous couplings of $D$-brane gauge and gravitational fields to Ramond-Ramond tensor potentials can be deduced by a simple anomaly inflow argument applied to intersecting $D$-branes and use this to determine the eight-form gravitational coupling. 
  We consider an 8--dimensional gravitational theory, which possesses a principle fiber bundle structure, with Lorentz--scalar fields coupled to the metric. One of them plays the role of a Higgs field and the other one that of a dilaton field. The effective cosmological constant is interpreted as a Higgs potential. The Yukawa couplings are introduce by hand. The extra dimensions constitute a $SU(2)_{L} \times U(1)_{Y} \times SU(2)_{R}$ group manifold. Dirac fields are coupled to the potentials derived from the metric. As result, we obtain an effective four--dimensional theory which contains all couplings of a Weinberg--Salam--Glashow theory in a curved space-time. The masses of the gauge bosons and of the first two fermion families are given by the theory. 
  We recalculate the amplitude for photon splitting in a strong magnetic field below the pair production threshold, using the worldline path integral variant of the Bern--Kosower formalism. Numerical comparison (using programs that we have made available for public access on the Internet) shows that the results of the recalculation are identical to the earlier calculations of Adler and later of Stoneham, and to the recent recalculation by Baier, Milstein, and Shaisultanov. 
  Many N=1 heterotic string compactifications exhibit physically mysterious singularities at codimension one in the moduli space of vacua. At these singularities, Yukawa couplings of charged fields develop poles as a function of the moduli. We explain these conformal field theory singularities, in a class of examples, as arising from non-perturbative gauge dynamics of non-perturbative gauge bosons (whose gauge coupling is the sigma model coupling) in the string theory. 
  The effective action of a SU(N)-gauge theory coupled to fermions is evaluated at a large infrared cut-off scale k within the path integral approach. The gauge field measure includes topologically non-trivial configurations (instantons). Due to the explicit infrared regularisation there are no gauge field zero modes. The Dirac operator of instanton configurations shows a zero mode even after the infrared regularisation, which leads to U_A(1)-violating terms in the effective action. These terms are calculated in the limit of large scales k. 
  The symmetric dynamics of two kinks and one antikink in classical (1+1)-dimensional $\phi^4$ theory is investigated. Gradient flow is used to construct a collective coordinate model of the system. The relationship between the discrete vibrational mode of a single kink, and the process of kink-antikink pair production is explored. 
  A general framework for the Weyl invariant quantization of Liouville field theory by means of an exact renormalization group equation is proposed. This flow equation describes the scale dependence of the effective average action which has a built-in infrared cutoff. For c<1 it is solved approximately by a truncation of the space of action functionals. We derive the Ward identities associated to Weyl transformations in presence of the infrared cutoff. They are used to select a specific universality class for the renormalization group trajectory which is found to connect two conformal field theories with central charges 25-c and 26-c, respectively. 
  An anisotropic integrable spin chain, consisting of spins $s=1$ and $s=\frac{1}{2}$, is investigated \cite{devega}. It is characterized by two real parameters $\bar{c}$ and $\tilde{c}$, the coupling constants of the spin interactions. For the case $\bar{c}<0$ and $\tilde{c}<0$ the ground state configuration is obtained by means of thermodynamic Bethe ansatz. Furthermore the low excitations are calculated. It turns out, that apart from free magnon states being the holes in the ground state rapidity distribution, there exist bound states given by special string solutions of Bethe ansatz equations (BAE) in analogy to \cite{babelon}. The dispersion law of these excitations is calculated numerically. 
  In this paper we continue the investigation of an anisotropic integrable spin chain, consisting of spins $s=1$ and $s=\frac{1}{2}$, started in our paper \cite{meissner}. The thermodynamic Bethe ansatz is analysed especially for the case, when the signs of the two couplings $\bar{c}$ and $\tilde{c}$ differ. For the conformally invariant model ($\bar{c}=\tilde{c}$) we have calculated heat capacity and magnetic susceptibility at low temperature. In the isotropic limit our analysis is carried out further and susceptibilities are calculated near phase transition lines (at $T=0$). 
  Open descendants extend Conformal Field Theory to unoriented surfaces with boundaries. The construction rests on two types of generalizations of the fusion algebra. The first is needed even in the relatively simple case of diagonal models. It leads to a new tensor that satisfies the fusion algebra, but whose entries are signed integers. The second is needed when dealing with non-diagonal models, where Cardy's ansatz does not apply. It leads to a new tensor with positive integer entries, that satisfies a set of polynomial equations and encodes the classification of the allowed boundary operators. 
  We use soliton techniques of the two-dimensional reduced beta-function equations to obtain non-trivial string backgrounds from flat space. These solutions are characterized by two integers (n, m) referring to the soliton numbers of the metric and axion-dilaton sectors respectively. We show that the Nappi-Witten universe associated with the SL(2) x SU(2) / SO(1, 1) x U(1) CFT coset arises as an (1, 1) soliton in this fashion for certain values of the moduli parameters, while for other values of the soliton moduli we arrive at the SL(2)/SO(1, 1) x SO(1, 1)^2 background. Ordinary 4-dim black-holes arise as 2-dim (2, 0) solitons, while the Euclidean worm-hole background is described as a (0, 2) soliton on flat space. The soliton transformations correspond to specific elements of the string Geroch group. These could be used as starting point for exploring the role of U-dualities in string compactifications to two dimensions. 
  Using stochastic quantization method we derive gauge-invariant equations, connecting multilocal vacuum correlators of nonperturbative field configurations immersed into the quantum background. Three alternative methods of stochastic regularization of these equations are suggested, and the corresponding regularized propagators of a background field are obtained in the lowest order of perturbation theory. 
  Stochastic quantization is applied to derivation of equations connecting multilocal gauge-invariant correlators in different field theories. They include Abelian Higgs Model, QCD with spinless quarks at T=0 and T>0 and QED, where spin effects are taken into account exactly. 
  We develop quantization aspects of our Liouville approach to non-critical strings, proposing a path-integral formulation of a second quantization of string theory, that incorporates naturally the couplings of string sources to background fields. Such couplings are characteristic of macroscopic string solutions and/or $D$-brane theories. Resummation over world-sheet genera in the presence of stringy ($\sigma$-model) soliton backgrounds, and recoil effects associated with logarithmic operators on the world sheet, play a crucial r\^ole in inducing such sources as well-defined renormalization-group counterterms. Using our Liouville renormalization group approach, we derive the appropriate second-order equation of motion for the $D$ brane. We discuss within this approach the appearance of open strings, whose ends carry non-trivial Chan-Paton-like quantum numbers related to the $W_\infty$ charges of two-dimensional string black holes. 
  We discuss various dual {\it pairs} of $M$-theory compactifications. Each pair consists of a compactification in which the two-brane plays the crucial role in relation to a string theory and a compactification in which the five-brane takes center stage. We show that in many examples such dual pairs are interchanged by the {\it same} duality transformation in each case. 
  We discuss the classical limit for the long-distance (``soft'') modes of a quantum field when the hard modes of the field are in thermal equilibrium. We address the question of the correct semiclassical dynamics when a momentum cut-off is introduced. Higher order contributions leads to a stochastic interpretation for the effective action in analogy to Quantum Brownian Motion, resulting in dissipation and decoherence for the evolution of the soft modes. Particular emphasis is put on the understanding of dissipation. Our discussion focuses mostly on scalar fields, but we make some remarks on the extension to gauge theories. 
  We consider the propagation of Type I open superstrings on orbifolds with four non-compact dimensions and $N=1$ supersymmetry. In this paper, we concentrate on a non-trivial Z_2xZ_2 example. We show that consistency conditions, arising from tadpole cancellation and algebraic sources, require the existence of three sets of Dirichlet 5-branes. We discuss fully the enhancements of the spectrum when these 5-branes intersect. An amusing attribute of these models is the importance of the tree-level (in Type I language) superpotential to the consistent relationship between Higgsing and the motions of 5-branes. 
  We study some properties of target space strings constructed from (2,1) heterotic strings. We argue that world-sheet complexification is a general property of the bosonic sector of such target world-sheets. We give a target space interpretation of this fact and relate it to the non-gaussian nature of free String Field Theory. We provide several one loop calculations supporting the stringy construction of critical world-sheets in terms of (2,1) models. Using finite temperature boundary conditions in the underlying (2,1) string we obtain non-chiral target space spin structures, and point out some of the problems arising for chiral spin structures, such as the heterotic world-sheet. To this end, we study the torus partition function of the corresponding asymmetric orbifold of the (2,1) string. 
  We construct generating solutions for general D-dimensional ($4 \le D \le 9$) rotating, electrically charged, black holes in the effective action of toroidally compactified heterotic (or Type IIA) string. The generating solution is parameterized by the ADM mass, two electric charges and $[{D-1}\over 2]$ angular momenta (as well as the asymptotic values of one toroidal modulus and the dilaton field). For $D \ge 6$, those are generating solutions for {\it general} black holes in toroidally compactified heterotic (or type IIA) string. Since in the BPS-limit (extreme limit) these solutions have singular horizons or naked singularities, we address the near extreme solutions with all the angular momenta small enough. In this limit, the thermodynamic entropy can be cast in a suggestive form, which has a qualitative interpretation as microscopic entropy of (near)-BPS-saturated charged string states of toroidally compactified heterotic string, whose target-space angular momenta are identified as $[{{D-1}\over 2}]$ U(1) left-moving world-sheet currents. 
  Hidden nonabelian symmetries in nonlinear interactions of radiation with matter are clarified. In terms of a nonabelian potential variable, we construct an effective field theory of self-induced transparency, a phenomenon of lossless coherent pulse propagation, in association with Hermitian symmetric spaces $G/H$. Various new properties of self-induced transparency, e.g. soliton numbers, effective potential energy, gauge symmetry and discrete symmetries, modified pulse area, conserved $U(1)$-charge etc. are addressed and elaborated in the nondegenerate two-level case where $G/H = SU(2)/U(1)$. Using the $U(1)$-charge conservation, a new type of analysis on pulse stability is given which agrees with earlier numerical results. 
  We show that in certain compactifications of ${\cal M}$-theory on eight-manifolds to three-dimensional Minkowski space-time the four-form field strength can have a non-vanishing expectation value, while an $N=2$ supersymmetry is preserved. For these compactifications a warp factor for the metric has to be taken into account. This warp factor is non-trivial in three space-time dimensions due to Chern-Simons corrections to the fivebrane Bianchi identity. While the original metric on the internal space is not K\"ahler, it can be conformally transformed to a metric that is K\"ahler and Ricci flat, so that the internal manifold has $SU(4)$ holonomy. 
  Recently K. Lee, E.J. Weinberg and P. Yi in CU-TP-739, hep-th/9602167, calculated the asymptotic metric on the moduli space of (1, 1, ..., 1) BPS monopoles and conjectured that it was globally exact. I lend support to this conjecture by showing that the metric on the corresponding space of Nahm data is the same as the metric they calculate. 
  We introduce {\em half-whole} dimensions for quaternionic matrices and propose a quaternionic version of the Frobenius-Schur theorem which allows us to obtain the proper quaternionic dimensionality for the representations of the Dirac and Duffin-Kemmer-Petiau (DKP) algebras. 
  We present the results of a numerical calculation of the photon splitting rate below the electron-pair creation threshold ($\omega \le 2m$) in magnetic fields $B ~{> \atop \sim}~ B_{\rm cr} = m^2 /e = 4.414 \times 10^{9}$ T. Our results confirm asymptotic approximations derived in the low-field ($B < B_{\rm cr}$) and high-field ($B \gg B_{\rm cr}$) limit, and allow interpolating between the two asymptotic regions. Our expression for the photon splitting rate is a simplified version of a formula given by Mentzel et al. We also point out that, although the analytical formula is correct, the splitting rates calculated there are wrong due to an error in the numerical calculations. 
  We study the one loop renormalization in the most general metric-dilaton theory with second derivative only. In constant background dilaton theory, there are two types of gravity background which enable the theory renormalizable at one-loop level. We show this concretely to discuss on the spherical symmetric background. 
  We show that the most general stationary electrically charged black-hole solutions of the heterotic string compactified on a (10-D)-torus (where D > 3) can be obtained by using the solution generating transformations of Sen acting on the Myers and Perry metric. The conserved charges labeling these black-hole solutions are the mass, the angular momentum in all allowed commuting planes, and 36-2D electric charges. General properties of these black-holes are also studied. 
  We present a new set of supersymmetric stationary solutions of pure N=4,d=4 supergravity (and, hence, of low-energy effective string theory) that generalize (and include) the Israel-Wilson-Perj\'es solutions of Einstein-Maxwell theory. All solutions have 1/4 of the supersymmetries unbroken and some have 1/2. The full solution is determined by two arbitrary complex harmonic functions {\cal H}_{1,2} which transform as a doublet under SL(2,\R) S duality and N complex constants k^{(n)} that transform as an SO(N) vector. This set of solutions is, then, manifestly duality invariant. When the harmonic functions are chosen to have only one pole, all the general resulting point-like objects have supersymmetric rotating asymptotically Taub-NUT metrics with 1/2 or 1/4 of the supersymmetries unbroken. The static, asymptotically flat metrics describe supersymmetric extreme black holes. Only those breaking 3/4 of the supersymmetries have regular horizons. The stationary asymptotically flat metrics do not describe black holes when the angular momentum does not vanish, even in the case in which 3/4 of the supersymmetries are broken. 
  We show that self-dual 2-forms in 2n dimensional spaces determine a $n^2-n+1$ dimensional manifold ${\cal S}_{2n}$ and the dimension of the maximal linear subspaces of ${\cal S}_{2n}$ is equal to the (Radon-Hurwitz) number of linearly independent vector fields on the sphere $S^{2n-1}$. We provide a direct proof that for $n$ odd ${\cal S}_{2n}$ has only one-dimensional linear submanifolds. We exhibit $2^c-1$ dimensional subspaces in dimensions which are multiples of $2^c$, for $c=1,2,3$. In particular, we demonstrate that the seven dimensional linear subspaces of ${\cal S}_{8}$ also include among many other interesting classes of self-dual 2-forms, the self-dual 2-forms of Corrigan, Devchand, Fairlie and Nuyts and a representation of ${\cal C}l_7$  given by octonionic multiplication. We discuss the relation of the linear subspaces with the representations of Clifford algebras. 
  Two-particle scattering amplitudes for integrable relativistic quantum field theory in 1+1 dimensions can normally have at most singularities of poles and zeros along the imaginary axis in the complex rapidity plane. It has been supposed that single particle amplitudes of the exact boundary reflection matrix exhibit the same structure. In this paper, single particle amplitudes of the exact boundary reflection matrix corresponding to the Neumann boundary condition for affine Toda field theory associated with twisted affine algebras $a_{2n}^{(2)}$ are conjectured, based on one-loop result, as having a new kind of square root singularity. 
  We consider the supersymmetric approach to gaussian disordered systems like the random bond Ising model and Dirac model with random mass and random potential. These models appeared in particular in the study of the integer quantum Hall transition. The supersymmetric approach reveals an osp(2/2)_1 affine symmetry at the pure critical point. A similar symmetry should hold at other fixed points. We apply methods of conformal field theory to determine the conformal weights at all levels. These weights can generically be negative because of non-unitarity. Constraints such as locality allow us to quantize the level k and the conformal dimensions. This provides a class of (possibly disordered) critical points in two spatial dimensions. Solving the Knizhnik-Zamolodchikov equations we obtain a set of four-point functions which exhibit a logarithmic dependence. These functions are related to logarithmic operators. We show how all such features have a natural setting in the superalgebra approach as long as gaussian disorder is concerned. 
  We compute the effect of quantum mechanical backreaction on the spectrum of radiation in a dynamical moving mirror model, mimicking the effect of a gravitational collapse geometry. Our method is based on the use of a combined WKB and saddle-point approximation to implement energy conservation in the calculation of the Bogolyubov coefficients, in which we assume that the mirror particle has finite mass m. We compute the temperature of the produced radiation as a function of time and find that after a relatively short time, the temperature is reduced by a factor 1/2 relative to the standard result. We comment on the application of this method to two-dimensional dilaton gravity with a reflecting boundary, and conclude that the WKB approximation quickly breaks down due to the appearance of naked singularities and/or white hole space-times for the relevant WKB-trajectories. 
  We describe a unified approach to calculating the partition functions of a general multi-level system with a free Hamiltonian. Particularly, we present new results for parastatistical systems of any order in the second quantized approach. Anyonic- like systems are briefly discussed. 
  The one and two-particle form factors of the energy operator in the two-dimensional Ising model in a magnetic field at $T=T_c$ are exactly computed within the form factor bootstrap approach. Together with the matrix elements of the magnetisation operator already computed in ref.\,\cite{immf}, they are used to write down the large distance expansion for the correlators of the two relevant fields of the model. 
  We study the connection between $N=2$ supersymmetry and a topological bound in a two-Higgs-doublet system with an $SU(2)\times U(1)_Y\times U(1)_{Y^{\prime}}$ gauge group. We derive the Bogomol'nyi equations from supersymmetry considerations showing that they hold provided certain conditions on the coupling constants, which are a consequence of the huge symmetry of the theory, are satisfied. Their solutions, which can be interpreted as electroweak cosmic strings breaking one half of the supersymmetries of the theory, are studied. Certain interesting limiting cases of our model which have recently been considered in the literature are finally analyzed. 
  Newly introduced generalized Poisson structures based on suitable skew-symmetric contravariant tensors of even order are discussed in terms of the Schouten-Nijenhuis bracket. The associated `Jacobi identities' are expressed as conditions on these tensors, the cohomological contents of which is given. In particular, we determine the linear generalized Poisson structures which can be constructed on the dual spaces of simple Lie algebras. 
  We show how the Hamiltonian lattice loop representation can be cast straightforwardly in the path integral formalism. The procedure is general for any gauge theory. Here we present in detail the simplest case: pure compact QED. We also analyze the non-Abelian Yang-Mills theory. The lattice loop path integral approach allows to knit together the power of statistical algorithms with the transparency of the gauge invariant loop description. The results produced by numerical simulations with the loop classical action for different lattice models are discused. 
  The Goddard, Nuyts and Olive conjecture for electric-magnetic duality in Yang-Mills theory with an arbitrary gauge group G is extended by including a non-vanishing vacuum angle $\theta$. This extended S-duality conjecture includes the case when the unbroken gauge group is non-abelian and a definite prediction for the spectrum of dyons results. 
  A class of non-linear sigma models possessing a new symmetry is identified. The same symmetry is also present in Chern-Simons theories. This hints at a possible topological origin for this class of sigma models. The non-linear sigma models obtained by non-Abelian duality are a particular case in this class. 
  We consider the notion of a connection on a principal bundle $P(M,G)$ from the point of view of the action of the tangential group $TG$. This leads to a new definition of a connection and allows to interpret gauge transformations as effect of the $TG$ action. 
  An extension of the Connes--Lott model is proposed. It is also within the framework of the A.Connes construction based on a generalized Dirac--Yukawa operator and the K--cycle $(H,D)$, with $H$ a fermionic Hilbert space. The basic algebra $A$ which may be considered as representing the non--commutative extension, plays a less important role in our approach. This allows a new class of natural extensions. The proposed extension lies in a sense between the Connes--Lott and the Marseille--Mainz model. It leads exactly to the standard model of electroweak interactions. 
  We study the ``ordinary'' Scherk-Schwarz dimensional reduction of the bosonic sector of the low energy effective action of a hypothetical M-theory on $S^1 \times S^1 \cong T^2$. We thus obtain the low energy effective actions of type IIA string theory in both ten and nine space-time dimensions. We point out how to obtain the O(1, 1) invariance of the NS-NS sector of the string effective action correctly in nine dimensions. We dimensionally reduce the type IIB string effective action on $S^1$ and show that the resulting nine dimensional theory can be mapped, purely from the bosonic consideration, exactly to the type IIA theory by an O(1, 1) or Buscher's T-duality transformations. We then give a dynamical argument, in analogy with that for the type IIB theory in ten dimensions, to show how an S-duality in the type IIA theory can be understood from the underlying nine dimensional theory by compactifying M-theory on a T-dual torus $\tilde {T}^2$. 
  Topologically non trivial effects appearing in the discussion of duality transformations in higher genus manifolds are discussed in a simple example, and their relation with the properties of Topological Field Theories is established. 
  In string field theory an infinitesimal background deformation is implemented as a canonical transformation whose hamiltonian function is defined by moduli spaces of punctured Riemann surfaces having one special puncture. We show that the consistency conditions associated to the commutator of two deformations are implemented by virtue of the existence of moduli spaces of punctured surfaces with two special punctures. The spaces are antisymmetric under the exchange of the special punctures, and satisfy recursion relations relating them to moduli spaces with one special puncture and to string vertices. We develop the theory of moduli spaces of surfaces with arbitrary number of special punctures and indicate their relevance to the construction of a string field theory that makes no reference to a conformal background. Our results also imply a partial antibracket cohomology theorem for the string action. 
  We provide a new proof of a important theorem in the Lagrangian formalism about necessary and sufficient conditions for a second-order variational system of equations to follow from a first-order Lagrangian. 
  In this paper, we construct multi-scalar, multi-center $p$-brane solutions in toroidally compactified M-theory. We use these solutions to show that all supersymmetric $p$-branes can be viewed as bound states of certain basic building blocks, namely $p$-branes that preserve $1/2$ of the supersymmetry. We also explore the M-theory interpretation of $p$-branes in lower dimensions. We show that all the supersymmetric $p$-branes can be viewed as intersections of M-branes or boosted M-branes in $D=11$. 
  The possibility of composite systems arising out of a point charge interacting with a Nielsen-Olesen vortex in 2+1-dimensions is investigated. It is shown that classical bounded orbits are possible for certain ranges of parameters. Long lived metastable states are shown to exist, in a semi-classical approach, from the study of the effective potential. Loss of self-adjointness of the Hamiltonian and its subsequent self-adjoint extension in some cases leads to bound states. 
  In this paper, colorless bilocal fields are employed to study the large $N$ limit of both fermionic and bosonic vector models. The Jacobian associated with the change of variables from the original fields to the bilocals is computed exactly, thereby providing an exact effective action. This effective action is shown to reproduce the familiar perturbative expansion for the two and four point functions. In particular, in the case of fermionic vector models, the effective action correctly accounts for the Fermi statistics. The theory is also studied non-perturbatively. The stationary points of the effective action are shown to provide the usual large $N$ gap equations. The homogeneous equation associated with the quadratic (in the bilocals) action is simply the two particle Bethe Salpeter equation. Finally, the leading correction in $1\over N$ is shown to be in agreement with the exact $S$ matrix of the model. 
  A non-abelian coupling between antisymmetric fields and Yang-Mills fields proposed by Freedman and Townsend several years ago is derived using the self-interaction mechanism. 
  Applying a technique developed recently [1,2] for an harmonic oscillator coupled to a bath of harmonic oscillators, we present an exact solution for the tunneling problem in an Ohmic dissipative system with inverted harmonic potential. The result shows that while the dissipation tends to suppress the tunneling, the Brownian motion tends to enhance the tunneling. Whether the tunneling rate increases or not would then depend on the initial conditions. We give a specific formula to calculate the tunneling probability determined by various parameters and the initial conditions. 
  In addition to the double-dimensional reduction procedure that employs world-volume Killing symmetries of $p$-brane supergravity solutions and acts diagonally on a plot of $p$ versus spacetime dimension $D$, there exists a second procedure of ``vertical'' reduction. This reduces the transverse-space dimension via an integral that superposes solutions to the underlying Laplace equation. We show that vertical reduction is also closely related to the recently-introduced notion of intersecting $p$-branes. We illustrate this with examples, and also construct a new $D=11$ solution describing four intersecting membranes, which preserves $1/16$ of the supersymmetry. Given the two reduction schemes plus duality transformations at special points of the scalar modulus space, one may relate most of the $p$-brane solutions of relevance to superstring theory. We argue that the maximum classifying duality symmetry for this purpose is the Weyl group of the corresponding Cremmer-Julia supergravity symmetry $E_{r(+r)}$. We also discuss a separate class of duality-invariant $p$-branes with $p=D-3$. 
  A collective field method is extended to obtain all the explicit solutions of the generalized Calogero-Sutherland models that are characterized by the roots of all the classical groups, including the solutions corresponding to spinor representations for $B_N$ and $D_N$ cases. 
  In order to construct a massive tensor theory with a smooth massless limit, we apply two kinds of gauge-fixing procedures, Nakanishi's one and the BRS one, to two models of massive tensor field. The first is of the Fierz-Pauli (FP) type, which describes a pure massive tensor field; the other is of the additional-scalar-ghost (ASG) type, which includes a scalar ghost in addition to an ordinary tensor field. It is shown that Nakanishi's procedure can eliminate massless singularities in both two models, while the BRS procedure regularizes the ASG model only. The BRS-regularized ASG model is most promising in constructing a complete nonlinear theory. 
  In the process of identifying heterotic and Type $II$ BPS string states with extremal dilaton black holes, it has been suggested that solutions with scalar/Maxwell parameters $a=\sqrt{3}$, $1$, $1/\sqrt{3}$ and $0$ correspond to $1-$, $2-$, $3-$ and $4$-particle bound states at threshold. (For example, the Reissner-Nordstrom black hole is just a superposition of four Kaluza-Klein black holes). Here we show that not only the masses, electric charges and magnetic charges but also the spins and supermultiplet structures of the string states are consistent with this interpretation. Their superspin $L$ corresponds to the Kerr-type angular momentum and hence only the $L=0$ elementary BPS states are black holes. Moreover, these results generalize to super $p$-branes in $D$-dimensions. By constructing multi-centered $p$-brane solitons, the new super $p$-branes found recently with various values of $a^2=\Delta-2(p+1)(D-p-3)/(D-2)$ are seen to be bound states of the fundamental ones with $\Delta=4$. 
  Using cumulant expansion for an averaged Wilson loop we derive an action of the gluodynamics string in the form of a series in powers of the correlation length of the vacuum. In the lowest orders it contains the Nambu-Goto term and the rigidity term with the coupling constants computed from the bilocal correlator of gluonic fields. Some higher derivative corrections are calculated. 
  The (1,0) supersymmetry in six dimensions admits a tensor multiplet which contains a second-rank antisymmetric tensor field with a self-dual field strength and a dilaton. We describe the fully supersymmetric coupling of this multiplet to Yang-Mills multiplet, in the absence of supergravity. The self-duality equation for the tensor field involves a Chern-Simons modified field strength, the gauge fermions, and an arbitrary dimensionful parameter. 
  The issue of spontaneous breaking of Lorentz and CPT invariance is studied in the open bosonic string using a truncation scheme to saturate the string-field action at successively higher levels. We find strong evidence for the existence of extrema of the action with nonzero expectation values for certain fields. The Lorentz- and CPT-preserving solution previously suggested in the literature is confirmed through level 12 in the action. A family of Lorentz-breaking, CPT-preserving solutions of the equations of motion is found to persist and converge through level 18 in the action. Two sequences of solutions spontaneously breaking both Lorentz invariance and CPT are discussed. The analysis at this level involves the analytical form of over 20,000 terms in the static potential. 
  We present a numerical classification of the spherically symmetric, static solutions to the Einstein--Yang--Mills equations with cosmological constant $\Lambda$. We find three qualitatively different classes of configurations, where the solutions in each class are characterized by the value of $\Lambda$ and the number of nodes, $n$, of the Yang--Mills amplitude. For sufficiently small, positive values of the cosmological constant, $\Lambda < \Llow(n)$, the solutions generalize the Bartnik--McKinnon solitons, which are now surrounded by a cosmological horizon and approach the deSitter geometry in the asymptotic region. For a discrete set of values $\Lambda_{\rm reg}(n) > \Lambda_{\rm crit}(n)$, the solutions are topologically $3$--spheres, the ground state $(n=1)$ being the Einstein Universe. In the intermediate region, that is for $\Llow(n) < \Lambda < \Lhig(n)$, there exists a discrete family of global solutions with horizon and ``finite size''. 
  We find the nonperturbative relation between $\langle {\rm tr} \phi^2 \rangle$, $\langle {\rm tr} \phi^3\rangle$ the prepotential ${\cal F}$ and the vevs $\langle \phi_i\rangle$ in $N=2$ supersymmetric Yang-Mills theories with gauge group $SU(3)$. Nonlinear differential equations for ${\cal F}$ including the Witten -- Dijkgraaf -- Verlinde -- Verlinde equation are obtained. This indicates that $N=2$ SYM theories are essentially topological field theories and that should be seen as low-energy limit of some topological string theory. Furthermore, we construct relevant modular invariant quantities, derive canonical relations between the periods and investigate the structure of the beta function by giving its explicit form in the moduli coordinates. In doing this we discuss the uniformization problem for the quantum moduli space.  The method we propose can be generalized to $N=2$ supersymmetric Yang-Mills theories with higher rank gauge groups. 
  We present string-theory derivation of the semiclassical entropy of extremal dyonic black holes in the approach based on conformal sigma model (NS-NS embedding of the classical solution). We demonstrate (resolving some puzzles existed in previous related discussions) that the degeneracy responsible for the entropy is due to string oscillations in four transverse dimensions `intrinsic' to black hole: four non-compact directions of the D=5 black hole and three non-compact and one compact (responsible for embedding of magnetic charges) dimensions in the D=4 black hole case. Oscillations in other compact internal dimensions give subleading contributions to statistical entropy in the limit when all charges are large. The dominant term in the statistical entropy is thus universal (i.e. is the same in type II and heterotic string theory) and agrees with Bekenstein-Hawking expression. 
  Problem of asymptotic description for global solutions to the six Painleve equations was investigated. Elliptic anzatzes and appropriate modulation equations were written out. 
  A charge $q$ moving in a reference laboratory system with constant velocity {\bf V} in the $X$-axis produces in the $Z$-axis a longitudinal, phase free, vacuum magnetic field which is identified as the radiated ${\bf B}^{(3)}$ field of Evans, Vigier and others. 
  We introduce a new type of cluster expansion which generalizes a previous formula of Brydges and Kennedy. The method is especially suited for performing a phase-space multiscale expansion in a just renormalizable theory, and allows the writing of explicit non-perturbative formulas for the Schwinger functions. The procedure is quite model independent, but for simplicity we chose the infrared $\ph^4_4$ model as a testing ground. We used also a large field versus small field expansion. The polymer amplitudes, corresponding to graphs without almost local two and for point functions, are shown to satisfy the polymer bound. 
  Using gauge theory and functional integral methods, we derive concrete expressions for the partition functions of BF theory and the U(1|1) model of Rozansky and Saleur on $\Sigma x S^{1}$, both directly and using equivalent two-dimensional theories. We also derive the partition function of a certain non-abelian generalization of the U(1|1) model on mapping tori and hence obtain explicit expressions for the Ray-Singer torsion on these manifolds. Extensions of these results to BF and Chern-Simons theories on mapping tori are also discussed. The topological field theory actions of the equivalent two-dimensional theories we find have the interesting property of depending explicitly on the diffeomorphism defining the mapping torus while the quantum field theory is sensitive only to its isomorphism class defining the mapping torus as a smooth manifold. 
  The N=2 supersymmetric gauge theory with gauge group SU(2) is considered. The instanton field is calculated explicitly using the superfield formalism. The instanton-induced effects are encoded in the effective vertex in the Lagrangian. This vertex produces the large distance expansion of the low energy effective Lagrangian in derivatives of fields. The leading term in this expansion coincides with the one-instanton-induced term of the Seiberg-Witten exact solution of the model. All orders of corrections in derivatives of fields are also calculated. 
  We analyse boundary conformal field theories on random surfaces using the conformal gauge approach of David, Distler and Kawai. The crucial point is the choice of boundary conditions on the Liouville field. We discuss the Weyl anomaly cancellation for Polyakov`s non-critical open bosonic string with Neumann, Dirichlet and free boundary conditions. Dirichlet boundary conditions on the Liouville field imply that the metric is discontinuous as the boundary is approached. We consider the semi-classical limit and argue how it singles out the free boundary conditions on the Liouville field. We define the open string susceptibility, the anomalous gravitational scaling dimensions and a new Yang-Mills Feynman mass critical exponent. 
  A recently introduced model of dually weighted planar graphs is solved in terms of an elliptic parametrization for some particular collection of planar graphs describing the 2D $R^2$ quantum gravity. Along with the cosmological constant $\lambda$ one has a coupling $\beta$ in the model corresponding to the $R^2$-coupling constant. It is shown that for any value of $\beta$ the large scale behavior of the model corresponds to that of the standard pure 2D quantum gravity. On small distances it describes the dynamics of point-like curvature defects introduced into the flat 2D space. The scaling function in the vicinity of almost flat metric is obtained. The major steps of the exact solution are given. 
  We quantize the $SU(n)$ Wess-Zumino-Witten model in terms of left and right chiral variables choosing an appropriate gauge and we compare our results with the results that have been previously obtained in the algebraic treatment of the problem. The algebra of the chiral vertex operators in the fundamental representation is verified by solving appropriate Knizhnik-Zamolodchikov equations. 
  The possibility of non-causal signal propagation is examined for various theories of dense matter. This investigation requires a discussion of definitions of causality, together with interpretations of spacetime position. Specific examples are used to illustrate the satisfaction or violation of causality in realistic calculations. 
  We explicitly determine the global structure of the $SL(2,Z)$ bundle over the Coulomb branch of the moduli space of asymptotically free $N=2$ supersymmetric Yang-Mills theories with gauge group $SU(2)$ when massless hypermultiplets are present. For each relevant number of flavours, we show that there is a curve of marginal stability on the Coulomb branch, diffeomorphic to a circle, across which the BPS spectrum is discontinuous. We determine rigorously and completely the BPS spectra inside and outside the curve. In all cases, the spectrum inside the curve consists of only those BPS states that are responsible for the singularities of the low energy effective action (in addition to the massless abelian gauge multiplet which is always present). The predicted decay patterns across the curve of marginal stability are perfectly consistent with all quantum numbers carried by the BPS states. As a byproduct, we also show that the electric and magnetic quantum numbers of the massless states at the singularities proposed by Seiberg and Witten are the only possible ones. 
  It is known from a work of Feigin and Frenkel that a Wakimoto type, generalized free field realization of the current algebra $\widehat{\cal G}_k$ can be associated with each parabolic subalgebra ${\cal P}=({\cal G}_0+{\cal G}_+)$ of the Lie algebra ${\cal G}$, where in the standard case ${\cal G}_0$ is the Cartan and ${\cal P}$ is the Borel subalgebra. In this letter we obtain an explicit formula for the Wakimoto realization in the general case. Using Hamiltonian reduction of the WZNW model, we first derive a Poisson bracket realization of the ${\cal G}$-valued current in terms of symplectic bosons belonging to ${\cal G}_+$ and a current belonging to ${\cal G}_0$. We then quantize the formula by determining the correct normal ordering. We also show that the affine-Sugawara stress-energy tensor takes the expected quadratic form in the constituents. 
  Starting from the hypothesis that both physics, in particular space-time and the physical vacuum, and the corresponding mathematics are discrete on the Planck scale we develop a certain framework in form of a class of ' cellular networks' consisting of cells (nodes) interacting with each other via bonds according to a certain 'local law' which governs their evolution. Both the internal states of the cells and the strength/orientation of the bonds are assumed to be dynamical variables. We introduce a couple of candidates of such local laws which, we think, are capable of catalyzing the unfolding of the network towards increasing complexity and pattern formation. In section 3 the basis is laid for a version of 'discrete analysis' on 'graphs' and 'networks' which, starting from different, perhaps more physically oriented principles, manages to make contact with the much more abstract machinery of Connes et al. and may complement the latter approach. In section 4 several more advanced geometric/topological concepts and tools are introduced which allow to study and classify such irregular structures as (random)graphs and networks. We show in particular that the systems under study carry in a natural way a 'groupoid structure'. In section 5 a, as far as we can see, promising concept of 'topological dimension' (or rather: ' fractal dimension') in form of a 'degree of connectivity' for graphs, networks and the like is developed. The possibility of dimensional phase transitions is discussed. 
  We consider the RSOS S-matrices of the Phi(1,5) perturbed minimal models which have recently been found in the companion paper [hep-th/9604098]. These S-matrices have some interesting properties, in particular, unitarity may be broken in a stronger sense than seen before, while one of the three classes of Phi(1,5) perturbations (to be described) shares the same Thermodynamic Bethe Ansatz as a related Phi(1,2) perturbation. We test these new S-matrices by the standard Truncated Conformal Space method, and further observe that in some cases the BA equations for two particle energy levels may be continued to complex rapidity to describe (a) single particle excitations and (b) complex eigenvalues of the Hamiltonian corresponding to non-unitary S-matrix elements. We make some comments on identities between characters in the two related models following from the fact that the two perturbed theories share the same breather sector. 
  We spell out the derivation of novel features, put forward earlier in a letter, of two dimensional gravity in the strong coupling regime, at $C_L=7$, $13$, $19$. Within the operator approach previously developed, they neatly follow from the appearence of a new cosmological term/marginal operator, different from the standard weak-coupling one, that determines the world sheet interaction. The corresponding string susceptibility is obtained and found real contrary to the continuation of the KPZ formula. Strongly coupled (topological like) models---only involving zero-mode degrees of freedom---are solved up to sixth order, using the ward identities which follow from the dependence upon the new cosmological constant. They are technically similar to the weakly coupled ones, which reproduce the matrix model results, but gravity and matter quantum numbers are entangled differently. 
  A new numerical integration method for examining a black hole structure was realized. Black hole solutions with dilatonic hair of 4D low energy effective SuperString Theory action with Gauss-Bonnet quadratic curvature contribution were studied, using this method, inside and outside the event horizon. Thermodynamical properties of this solution were also studied. 
  Renormalization factors are most easily extracted by going to the massless limit of the quantum field theory and retaining only a single momentum scale. We derive factors and renormalized Green functions to all orders in perturbation theory for rainbow graphs and vertex (or scattering diagrams) at zero momentum transfer, in the context of dimensional renormalization, and we prove that the correct anomalous dimensions for those processes emerge in the limit D -> 4. 
  In this paper we introduce a new class of theories which dynamically break supersymmetry based on the gauge group SU(n)xSU(3)xU(1) for even n. These theories are interesting in that no dynamical superpotential is generated in the absence of perturbations. For the example SU(4)xSU(3)xU(1) we explicitly demonstrate that all flat directions can be lifted through a renormalizable superpotential and that supersymmetry is dynamically broken. We derive the exact superpotential for this theory, which exhibits new and interesting dynamical phenomena. For example, modifications to classical constraints can be field dependent. We also consider the generalization to SU(n)xSU(3)xU(1) models (with even n>4). We present a renormalizable superpotential which lifts all flat directions. Because SU(3) is not confining in the absence of perturbations, the analysis of supersymmetry breaking is very different in these theories from the n=4 example. When the SU(n) gauge group confines, the Yukawa couplings drive the SU(3) theory into a regime with a dynamically generated superpotential. By considering a simplified version of these theories we argue that supersymmetry is probably broken. 
  Einstein-Yang-Mills-dilaton theory possesses sequences of neutral static spherically symmetric black hole solutions. The solutions depend on the dilaton coupling constant $\gamma$ and on the horizon. The SU(2) solutions are labelled by the number of nodes $n$ of the single gauge field function, whereas the SO(3) solutions are labelled by the nodes $(n_1,n_2)$ of both gauge field functions. The SO(3) solutions form sequences characterized by the node structure $(j,j+n)$, where $j$ is fixed. The sequences of magnetically neutral solutions tend to magnetically charged limiting solutions. For finite $j$ the SO(3) sequences tend to magnetically charged Einstein-Yang-Mills-dilaton solutions with $j$ nodes and charge $P=\sqrt{3}$. For $j=0$ and $j \rightarrow \infty$ the SO(3) sequences tend to Einstein-Maxwell-dilaton solutions with magnetic charges $P=\sqrt{3}$ and $P=2$, respectively. The latter also represent the scaled limiting solutions of the SU(2) sequence. The convergence of the global properties of the black hole solutions, such as mass, dilaton charge and Hawking temperature, is exponential. The degree of convergence of the matter and metric functions of the black hole solutions is related to the relative location of the horizon to the nodes of the corresponding regular solutions. 
  We compute an analogue of the Itzykson-Zuber integral for the case of arbitrary complex matrices. The calculation is done for both ordinary and supermatrices by transferring the Itzykson-Zuber diffusion equation method to the space of arbitrary complex matrices. The integral is of interest for applications in Quantum Chromodynamics and the theory of two-dimensional Quantum Gravity. 
  In hep-th/9506151 we started a programme devoted to the systematic study of the conformal field theories derived from WZW models based on nonreductive Lie groups. In this, the second part, we continue this programme with a look at the N=1 and N=2 superconformal field theories which arise from both gauged and ungauged supersymmetric WZW models. We extend the supersymmetric (affine) Sugawara and coset constructions, as well as the N=2 Kazama--Suzuki construction to general self-dual Lie algebras. 
  It is known that the naive version of D-brane theory is inadequate to explain the black hole entropy in the limit in which the Schwarzschild radius becomes larger than all compactification radii. We present evidence that a more consistent description can be given in terms of strings with rescaled tensions. We show that the rescaling can be interpreted as a redshift of the tension of a fundamental string in the gravitational field of the black hole. An interesting connection is found between the string level number and the Rindler energy. Using this connection, we reproduce the entropies of Schwarzschild black holes in arbitrary dimensions in terms of the entropy of a single string at the Hagedorn temperature. 
  We study the non-perturbative behavior of some N=1 supersymmetric product-group gauge theories with the help of duality. As a test case we investigate an SU(2)xSU(2) theory in detail. Various dual theories are constructed using known simple-group duality for one group or both groups in succession. Several stringent tests show that the low-energy behavior of the dual theories agrees with that of the electric theory. When the theory is in the confining phase we calculate the exact superpotential. Our results strongly suggest that, in general, dual theories for product groups can be constructed in this manner, by using simple-group duality for both groups. Turning to a class of theories with SU(N)xSU(M) gauge symmetry we study the renormalization group flows in the space of the two gauge couplings and show that they are consistent with the absence of phase transitions. Finally, we show that a subset of these theories, with SU(N)xSU(N-1) symmetry break supersymmetry dynamically. 
  We discuss the exact electrically charged BTZ black hole solutions to the Einstein-Maxwell equations with a negative cosmological constant in 2+1 spacetime dimensions assuming a (anti-)self dual condition between the electromagnetic fields. In a coordinate condition there appears a logarithmic divergence in the angular momentum at spatial infinity. We show how it is to be regularized by taking the contribution from the boundary into account. We show another coordinate condition which leads to a finite angular momentum though it brings about a peculiar spacetime topology. 
  We perform a dimensional reduction of the $U(1)\times SU(2)$ Chern--Simons bosonization and apply it to the $t-J$ model, relevant for high $T_c$ superconductors. This procedure yields a decomposition of the electron field into a product of two ``semionic" fields, i.e. fields obeying abelian braid statistics with statistics parameter $\theta={1\over 4}$, one carrying the charge and the other the spin degrees of freedom. A mean field theory is then shown to reproduce correctly the large distance behaviour of the correlation functions of the 1D $t-J$ model at $t>>J$. This result shows that to capture the essential physical properties of the model one needs a specific ``semionic" form of spin--charge separation. 
  There is still confusion about the correct form of the area law for the baryonic Wilson loop (BWL) of QCD. Strong-coupling (i.e., finite lattice spacing in lattice gauge theory) approximations suggest the form $\exp [-KA_Y]$, where $K$ is the $q\bar{q}$ string tension and $A_Y$ is the global minimum area, generically a three-bladed area with the blades joined along a Steiner line ($Y$ configuration). However, the correct answer is $\exp[-(K/2)(A_{12}+A_{13}+A_{23})]$, where, e.g., $A_{12}$ is the minimal area between quark lines 1 and 2 ($\Delta$ configuration). This second answer was given long ago, based on certain approximations, and is also strongly favored in lattice computations. In the present work, we derive the $\Delta$ law from the usual vortex-monopole picture of confine- ment, and show that in any case because of the 1/2 in the $\Delta$ law, this law leads to a larger value for the BWL (smaller exponent) than does the $Y$ law. We show that the three-bladed strong-coupling surfaces, which are infinitesimally thick in the limit of zero lattice spacing, survive as surfaces to be used in the non-Abelian Stokes' theorem for the BWL, which we derive, and lead via this Stokes' theorem to the correct $\Delta$ law. Finally, we extend these considerations, including perturbative contributions, to gauge groups $SU(N)$, with $N>3$. 
  We are presenting a general solution to the classical Einstein--Maxwell--dilaton--axion equations starting from a metric of type--D. Namely, this stringy solution is the result of a transformation on a general vacuum type--D solution to the Einstein's equations which was studied in detail some years ago. 
  Duality groups of Abelian gauge theories on four manifolds and their reduction to two dimensions are considered. The duality groups include elements that relate different space-times in addition to relating different gauge-coupling matrices. We interpret (some of) such dualities as the geometrical symmetries of compactified theories in higher dimensions. In particular, we consider compactifications of a (self-dual) 2-form in 6-D, and compactifications of a self-dual 4-form in 10-D. Relations with a self-dual superstring in 6-D and with the type IIB superstring are discussed. 
  We investigate three classes of supersymmetric models which can be obtained by breaking the chiral SU(2k+3) gauge theories with one antisymmetric tensor and 2k-1 antifundamentals. For N=3, the chiral SU(2k)$\times$SU(3)$\times$U(1) theories break supersym metry by the quantum deformations of the moduli spaces in the strong SU(2k) gauge coupling limit. For N=2, it is the generalization of the SU(5)$\times$U(2)$\times$U(1) model mentioned in the literature. Supersymmetry is broken by carefully choosing the q uark-antiquark-doublet Yukawa couplings in this model. For N=1, this becomes the well-known model discussed in the literature. 
  The non-Abelian Kubo formula is derived from the kinetic theory. That expression is compared with the one obtained using the eikonal for a Chern-Simons theory. The multiple time-scale method is used to solve the non-Abelian Kubo formula, and the damping rate for longitudinal color waves is computed. 
  We apply differential renormalization method to the study of three-dimensional topologically massive Yang-Mills and Chern-Simons theories. The method is especially suitable for such theories as it avoids the need for dimensional continuation of three-dimensional antisymmetric tensor and the Feynman rules for three-dimensional theories in coordinate space are relatively simple. The calculus involved is still lengthy but not as difficult as other existing methods of calculation. We compute one-loop propagators and vertices and derive the one-loop local effective action for topologically massive Yang-Mills theory. We then consider Chern-Simons field theory as the large mass limit of topologically massive Yang-Mills theory and show that this leads to the famous shift in the parameter $k$. Some useful formulas for the calculus of differential renormalization of three-dimensional field theories are given in an Appendix. 
  We respond to Tarrach's criticisms (hep-th/9511034) of our work on lambda Phi^4 theory.  Tarrach does not discuss the same renormalization procedure that we do. He also relies on results from perturbation theory that are not valid. There is no "infrared divergence" or unphysical behaviour associated with the zero-momentum limit of our effective action. 
  Presented is an integral formula for solutions to the quantum Knizhnik--Zamolodchikov equation of level $0$ associated with the vector representation of $U_q (\widehat{ sl_n})$. This formula gives a generalization of both our previous work for $U_q (\widehat{ sl_2})$ and Smirnov's formula for form factors of $SU(n)$ chiral Gross-Neveu model. 
  We consider a bosonic $\s$--model coupled to two--dimensional gravity. In the semiclassical limit, $c\rightarrow -\infty$, we compute the gravity dressing of the $\b$--functions at two--loop order in the matter fields. We find that the corrections due to the presence of dynamical gravity are {\em not} expressible simply in terms of a multiplicative factor as previously obtained at the one--loop level. Our result indicates that the critical points of the theory are nontrivially influenced and modified by the induced gravity. 
  We study density correlation functions for an impenetrable Bose gas in a finite box, with Neumann or Dirichlet boundary conditions in the ground state. We derive the Fredholm minor determinant formulas for the correlation functions. In the thermodynamic limit, we express the correlation functions in terms of solutions of non-linear differential equations which were introduced by Jimbo, Miwa, M\^ori and Sato as a generalization of the fifth Painlev\'e equations. 
  We consider real forms of Lie algebras and embeddings of sl(2) which are consistent with the construction of integrable models via Hamiltonian reduction. In other words: we examine possible non-standard reality conditions for non-abelian Toda theories. We point out in particular that the usual restriction to the maximally non-compact form of the algebra is unnecessary, and we show how relaxing this condition can lead to new real forms of the resulting W-algebras. Previous results for abelian Toda theories are recovered as special cases. The construction can be extended straightforwardly to deal with osp(1|2) embeddings in Lie superalgebras. Two examples are worked out in detail, one based on a bosonic Lie algebra, the other based on a Lie superalgebra leading to an action which realizes the N=4 superconformal algebra. 
  We study the partition function for the low-energy edge excitations of the incompressible electron fluid. On an annular geometry, these excitations have opposite chiralities on the two edges; thus, the partition function takes the standard form of rational conformal field theories. In particular, it is invariant under modular transformations of the toroidal geometry made by the angular variable and the compact Euclidean time. The Jain series of plateaus have been described by two types of edge theories: the minimal models of the W-infinity algebra of quantum area-preserving diffeomorphisms, and their non-minimal version, the theories with U(1)xSU(m) affine algebra. We find modular invariant partition functions for the latter models. Moreover, we relate the Wen topological order to the modular transformations and the Verlinde fusion algebra. We find new, non-diagonal modular invariants which describe edge theories with extended symmetry algebra; their Hall conductivities match the experimental values beyond the Jain series. 
  A closed expression of the Euclidean Wilson-loop functionals is derived for pure Yang-Mills continuum theories with gauge groups $SU(N)$ and $U(1)$ and space-time topologies $\Rl^1\times\Rl^1$ and $\Rl^1\times S^1$. (For the $U(1)$ theory, we also consider the $S^1\times S^1$ topology.) The treatment is rigorous, manifestly gauge invariant, manifestly invariant under area preserving diffeomorphisms and handles all (piecewise analytic) loops in one stroke. Equivalence between the resulting Euclidean theory and and the Hamiltonian framework is then established. Finally, an extension of the Osterwalder-Schrader axioms for gauge theories is proposed. These axioms are satisfied for the present model. 
  In this lecture I will report on some recent progress in understanding the relation of Dirac operators on Clifford modules over an even-dimensional closed Riemannian manifold $M$\ and (euclidean) Einstein-Yang-Mills-Higgs models. 
  A general way to construct chain models with certain Lie algebraic or quantum Lie algebraic symmetries is presented. These symmetric models give rise to series of integrable systems. As an example the chain models with $A_n$ symmetry and the related Temperley-Lieb algebraic structures and representations are discussed. It is shown that corresponding to these $A_n$ symmetric integrable chain models there are exactly solvable stationary discrete-time (resp. continuous-time) Markov chains whose spectra of the transition matrices (resp. intensity matrices) are the same as the ones of the corresponding integrable models. 
  The SO(32) heterotic string on a K3 surface is analyzed in terms of the dual theory of a type II string (or F-theory) on an elliptically fibred Calabi-Yau manifold. The results are in beautiful agreement with earlier work by Witten using very different methods. In particular, we find gauge groups of SO(32) x Sp(k) appearing at points in the moduli space identified with point-like instantons and see hypermultiplets in the (32,2k) representation becoming massless at the same time. We also discuss some aspects of the E8 x E8 case. 
  Costa and Matsas (gr-qc/9412030) claimed in their recent paper that a thermal bath does not affect substantially the transition probability for fast moving inertial Unruh detector. It is shown that their claim holds only for small $\beta \Delta E$.  We show that, for large enough $\beta \Delta E$, the transition probability is not monotonically decreasing function of the detector's speed contrary to their claim. 
  Using twisted realizations of the symmetric groups, we show that Bose and Fermi statistics are compatible with transformations generated by compact quantum groups of Drinfel'd type. 
  In this paper we consider the N = 2 supergravity models in which the hypermultiplets realize the nonlinear sigma-models, corresponding to the nonsymmetric (but homogeneous) quaternionic manifolds. By exploiting the isometries of appropriate manifolds we give an explicit construction for the Lagrangians and supertransformation laws in terms of usual hypermultiplets in the form suitable for the investigation of general properties of such models as well as for the studying of concrete models. 
  We suggest the quasiparticle picture behind the integrable structure of N=2 SYM theory,which arises if the Lax operator is considered as a Hamiltonian for the fermionic system. We compare the meaning of BPS states with the one coming from the D-brane interpretation and give some evidence for the compositeness of the selfdual strings. The temperature phase transition with the disappearance of the mass gap is conjectured. 
  If nature is described by string theory, and if the compactification radius is large (as suggested by the unification of couplings), then the theory is in a regime best described by the low energy limit of $M$-theory. We discuss some phenomenological aspects of this view. The scale at which conventional quantum field theory breaks down is of order the unification scale and consequently (approximate) discrete symmetries are essential to prevent proton decay. There are one or more light axions, one of which solves the strong CP problem. Modular cosmology is still problematic but much more complex than in perturbative string vacua. We also consider a range of more theoretical issues, focusing particularly on the question of stabilizing the moduli. We give a simple, weak coupling derivation of Witten's expression for the dependence of the coupling constants on the eleven dimensional radius. We discuss the criteria for the validity of the long wavelength analysis and find that the "real world" seems to sit just where this analysis is breaking down. On the other hand, residual constraints from N=2 supersymmetry make it difficult to see how the moduli can be stabilized while at the same time yielding a large hierarchy. 
  We show within the Wilson renormalization group framework how the flow equation method can be used to prove the perturbative renormalizability of a relativistic massive selfinteracting scalar field. Furthermore we prove the regularity of the renormalized relativistic one-particle irreducible n-point Green functions in the region predicted by axiomatic quantum field theory which ensures that physical renormalization conditions for the two-point function can be imposed. 
  The relation between super-Virasoro anomaly and super-Weyl anomaly in $N=1$ NSR superstring coupled with 2D supergravity is investigated from canonical theoretical view point. The WZW action canceling the super-Virasoro anomaly is explicitly constructed. It is super-Weyl invariant but nonlocal functional of 2D supergravity. The nonlocality can be remedied by the super-Liouvlle action, which in turn recovers the super-Weyl anomaly. The final gravitational effective action turns out to be local but noncovariant super-Liouville action, describing the dynamical behavior of the super-Liouville fields. The BRST invariance of this approach is examined in the superconformal gauge and in the light-cone gauge. 
  A short outline is given on the application of differential regularization to QCD in the background-field method. The derivation of the propagators in the background gluon field as short-distance expansions is described and the renormalization of the theory is mentioned. 
  We review a class of matrix models whose degrees of freedom are matrices with anticommuting elements. We discuss the properties of the adjoint fermion one-, two- and gauge invariant D-dimensional matrix models at large-N and compare them with their bosonic counterparts which are the more familiar Hermitian matrix models. We derive and solve the complete sets of loop equations for the correlators of these models and use these equations to examine critical behaviour. The topological large-N expansions are also constructed and their relation to other aspects of modern string theory such as integrable hierarchies is discussed. We use these connections to discuss the applications of these matrix models to string theory and induced gauge theories. We argue that as such the fermionic matrix models may provide a novel generalization of the discretized random surface representation of quantum gravity in which the genus sum alternates and the sums over genera for correlators have better convergence properties than their Hermitian counterparts. We discuss the use of adjoint fermions instead of adjoint scalars to study induced gauge theories. We also discuss two classes of dimensionally reduced models, a fermionic vector model and a supersymmetric matrix model, and discuss their applications to the branched polymer phase of string theories in target space dimensions D>1 and also to the meander problem. 
  We formulate in terms of the quantum inverse scattering method the algebraic Bethe ansatz solution of the one-dimensional Hubbard model. The method developed is based on a new set of commutation relations which encodes a hidden symmetry of 6-vertex type. 
  The physical state condition in the BRST quantization of Chern-Simons field theory is used to derive Gauss law constraints in the presence of Wilson loops, which play an important role in explicitly establishing the connection of Chern-Simons field theory with 2-dimensional conformal field theory. 
  In this paper we continue our investigation of the N = 2 supergravity models, where scalar fields of hypermultiplets parameterize the nonsymmetric quaternionic manifolds. Using the results of our previous paper, where we have given an explicit construction for the Lagrangians and the supertransformations and, in-particular, the known global symmetries of the Lagrangians, we consider here the switching on the gauge interaction. We show that in this type of models there appears to be possible to have spontaneous supersymmetry breaking with two different scales and without a cosmological term. Moreover, such a breaking could lead to the generation of the Yukawa interactions of the scalar and spinor fields from the hypermultiplets which are absent in other known models. 
  We present a detailed account of the isomonodromic quantization of dimensionally reduced Einstein gravity with two commuting Killing vectors. This theory constitutes an integrable ``midi-superspace" version of quantum gravity with infinitely many interacting physical degrees of freedom. The canonical treatment is based on the complete separation of variables in the isomonodromic sectors of the model. The Wheeler-DeWitt and diffeomorphism constraints are thereby reduced to the Knizhnik-Zamolodchikov equations for $SL(2,R)$. The physical states are shown to live in a well defined Hilbert space and are manifestly invariant under the full diffeomorphism group. An infinite set of independent observables \`a la Dirac exists both at the classical and the quantum level. Using the discrete unitary representations of $SL(2,R)$, we construct explicit quantum states. However, satisfying the additional constraints associated with the coset space $SL(2,R)/SO(2)$ requires solutions based on the principal series representations, which are not yet known. We briefly discuss the possible implications of our results for string theory. 
  We explicitly extract the structure of higher-derivative curvature-squared terms at genus 0 and 1 in the d=4 heterotic string effective action compactified on symmetric orbifolds by computing on-shell S-matrix superstring amplitudes. In particular, this is done within the context of calculating the graviton 4-point amplitude. We also discuss the moduli-dependent gravitational threshold corrections to the coupling associated with the CP even quadratic curvature terms. 
  The `electromagnetic' $Sl(2;\bZ)$ duality group in spacetime dimension $D=4k$ can be given a Kaluza-Klein interpretation in $D=4k+2$ as the modular group of a compactifying torus. We show how dyonic $2(k-1)$-branes in $D=4k$ can be interpreted as self-dual $(2k-1)$-branes in $D=4k+2$ wound around the homology cycles of the torus. In particular, dyons of the D=4 N=4 heterotic string theory are interpreted as winding modes of a D=6 self-dual string, while D=8 dyonic membranes are interpreted as wound 3-branes of D=10 IIB superstring theory. We also discuss the T-dual IIA interpretations of D=8 dyonic membranes. 
  The Coleman-Mandula theorem, which states that space-time and internal symmetries cannot be combined in any but a trivial way, is generalized to an arbitrarily higher spacelike dimension. Prospects for further generalizations of the theorem (space-like representations, larger time-like dimension, infinite number of particle types) are also discussed. The original proof relied heavily on the Dirac formalism, which was not well defined mathematically at that time. The proof given here is based on the rigorous version of the Dirac formalism, based on the theory of distributions. This work serves also to demonstrate the suitability of this formalism for practical applications. 
  The space of physical states in relativistic scattering theory is constructed, using a rigorous version of the Dirac formalism, where the Hilbert space structure is extended to a Gel'fand triple. This extension enables the construction of ``a complete set of states'', the basic concept of the original Dirac formalism, also in the cases of unbounded operators and continuous spectra. We construct explicitly the Gel'fand triple and a complete set of ``plane waves'' -- momentum eigenstates -- using the group of space-time symmetries. This construction is used (in a separate article) to prove a generalization of the Coleman-Mandula theorem to higher dimension. 
  We compute the $1$-loop effective K\"ahler potential in the most general renormalizable $N=1$ $d=4$ supersymmetric quantum field theory. 
  By analyzing $F$-theory on $K3$ near the orbifold limit of $K3$ we establish the equivalence between $F$-theory on $K3$ and an orientifold of type IIB on $T^2$, which in turn, is related by a T-duality transformation to type I theory on $T^2$. By analyzing the $F$-theory background away from the orbifold limit, we show that non-perturbative effects in the orientifold theory splits an orientifold plane into two planes, with non-trivial SL(2,Z) monodromy around each of them. The mathematical description of this phenomenon is identical to the Seiberg-Witten result for N=2 supersymmetric $SU(2)$ gauge theory with four quark flavors. Points of enhanced gauge symmetry in the orientifold / $F$-theory are in one to one correspondence with the points of enhanced global symmetry in the Seiberg-Witten theory. 
  We find the fusion rules for the c_{p,1} series of logarithmic conformal field theories. This completes our attempts to generalize the concept of rationality for conformal field theories to the logarithmic case. A novelty is the appearance of negative fusion coefficients which can be understood in terms of exceptional quantum group representations. The effective fusion rules (i.e. without signs and multiplicities) resemble the BPZ fusion rules for the virtual minimal models with conformal grid given via c = c_{3p,3}. This leads to the conjecture that (almost) all minimal models with c = c_{p,q}, gcd(p,q) > 1, belong to the class of rational logarithmic conformal field theories. 
  We propose a new way for describing the transition between two quantum Hall effect states with different filling factors within the framework of rational conformal field theory. Using a particular class of non-unitary theories, we explicitly recover Jain's picture of attaching flux quanta by the fusion rules of primary fields. Filling higher Landau levels of composite fermions can be described by taking tensor products of conformal theories. The usual projection to the lowest Landau level corresponds then to a simple coset of these tensor products with several U(1)-theories divided out. These two operations -- the fusion map and the tensor map -- can explain the Jain series and all other observed fractions as exceptional cases. Within our scheme of transitions we naturally find a field with the experimentally observed universal critical exponent 7/3. 
  The heat kernels of Laplacians for spin 1/2, 1, 3/2 and 2 fields, and the asymptotic expansion of their traces are studied on manifolds with conical singularities. The exact mode-by-mode analysis is carried out for 2-dimensional domains and then extended to arbitrary dimensions. The corrections to the first Schwinger-DeWitt coefficients in the trace expansion, due to conical singularities, are found for all the above spins. The results for spins 1/2 and 1 resemble the scalar case. However, the heat kernels of the Lichnerowicz spin 2 operator and the spin 3/2 Laplacian show a new feature. When the conical angle deficit vanishes the limiting values of these traces differ from the corresponding values computed on the smooth manifold. The reason for the discrepancy is breaking of the local translational isometries near a conical singularity. As an application, the results are used to find the ultraviolet divergences in the quantum corrections to the black hole entropy for all these spins. 
  We describe a new kind of transition between topologically distinct $N=2$ type II Calabi--Yau vacua through points with enhanced non-abelian gauge symmetries together with fundamental charged matter hypermultiplets. We connect the appearance of matter to the local geometry of the singularity and discuss the relation between the instanton numbers of the Calabi--Yau manifolds taking part in the transition. In a dual heterotic string theory on $K3\times T^2$ the process corresponds to Higgsing a semi-classical gauge group or equivalently to a variation of the gauge bundle. In special cases the situation reduces to simple conifold transitions in the Coulomb phase of the non-abelian gauge symmetries. 
  A model for lattice fermion is proposed which is, (i) free from doublers, (ii) hermitian, and (iii) chirally invariant. The price paid is the loss of hypercubic and reflection symmetries in the lattice action. Thanks to the $\epsilon$-prescription, correlation functions are free from the ill effects due to the loss of these symmetries. In weak coupling approximation, the U(1) vector current of a gauge theory of lattice fermion in this model is conserved in the continuum limit. As for the U(1) axial vector current, one obtains the ABJ anomaly if the continuum limit is implemented before the chiral limit $m = 0$. The anomaly disappears, as in the Wilson model, if the order of the two limits is reversed. 
  Examples for bounded Bose fields in two dimensions are presented. 
  Starting from a field theoretical description of multicomponent anyons with mutual statistical interactions in the lowest Landau level, we construct a model of interacting chiral fields on the circle, with the energy spectrum characterized by a symmetric matrix $g_{\alpha\beta}$ with nonnegative entries. Being represented in a free form, the model provides a field theoretical realization of (ideal) fractional exclusion statistics for particles with linear dispersion, with $g_{\alpha\beta}$ as a statistics matrix. We derive the bosonized form of the model and discuss the relation to the effective low-energy description of the edge excitations for abelian fractional quantum Hall states in multilayer systems. 
  The cohomology of the BRS operator corresponding to a group of rigid symmetries is studied in a space of local field functionals subjected to a condition of gauge invariance. We propose a procedure based on a filtration operator counting the degree in the infinitesimal parameters of the rigid symmetry transformations. An application to Witten's topological Yang-Mills theory is given. 
  Higher dimensional Chern-Simons theories, even though constructed along the same topological pattern as in 2+1 dimensions, have been shown recently to have generically a non-vanishing number of degrees of freedom. In this paper, we carry out the complete Dirac Hamiltonian analysis (separation of first and second class constraints and calculation of the Dirac bracket) for a group GxU(1). We also study the algebra of surface charges that arise in the presence of boundaries and show that it is isomorphic to the WZW4 discussed in the literature. Some applications are then considered. It is shown, in particular, that Chern-Simons gravity in dimensions greater than or equal to five has a propagating torsion. 
  Chern-Simons-Matter Lagrangian with noncompact gauge symmetry group is considered. The theory is quantized in the holomorphic gauge with a complex gauge fixing condition. The model is discussed, in which the the gauge and matter fields are accompanied by the complex conjugate counterparts. It is argued, that such a theory represents an adequate framework for the description of the quantum Hall states. 
  The q-deformed harmonic oscillator is studied in the light of q-deformed phase space variables. This allows a formulation of the corresponding Hamiltonian in terms of the ordinary canonical variables $x$ and $p$. The spectrum shows unexpected features such as degeneracy and an additional part that cannot be reached from the ground state by creation operators. The eigenfunctions show lattice structure, as expected. 
  It is shown how strictly four-dimensional integration by parts combined with differential renormalization and its infrared analogue can be applied for calculation of Feynman diagrams. 
  We extend recent discussions about the effect of nonzero temperature on the induced electric charge, due to CP violation, of a Dirac or an 't Hooft-Polyakov monopole. In particular, we determine the fractional electric charge of a very small 't Hooft-Polyakov monopole coupled to light fermions at nonzero temperature. If dyons with fractional electric charge exist in the Weinberg-Salam model, as recently suggested in the literature, then their charge too should be temperature dependent. 
  A new version of differential renormalization is presented. It is based on pulling out certain differential operators and introducing a logarithmic dependence into diagrams. It can be defined either in coordinate or momentum space, the latter being more flexible for treating tadpoles and diagrams where insertion of counterterms generates tadpoles. Within this version, gauge invariance is automatically preserved to all orders in Abelian case. Since differential renormalization is a strictly four-dimensional renormalization scheme it looks preferable for application in each situation when dimensional renormalization meets difficulties, especially, in theories with chiral and super symmetries. The calculation of the ABJ triangle anomaly is given as an example to demonstrate simplicity of calculations within the presented version of differential renormalization. 
  We announce results about the nonperturbative mathematically rigorous construction of the $:\!\phi^4_4\!:$ quantum field theory in four-dimensional space-time. The complex structure of solutions of the classical nonlinear (real-valued) wave equation and quantization are closely connected among themselves and allow to construct non-perturbatively the quantum field theory with interaction $:\!\phi^4_4\!:$ in four-dimensional space-time. We consider vacuum averages, in particular, we construct Wightman functions and matrix elements of the scattering operator as generalized functions for finite energies. The constructed theory is obviously nontrivial. 
  We analyze the stability properties of the purely magnetic, static solutions to the Einstein--Yang--Mills equations with cosmological constant. It is shown that all three classes of solutions found in a recent study are unstable under spherical perturbations. Specifically, we argue that the configurations have $n$ unstable modes in each parity sector, where $n$ is the number of nodes of the magnetic Yang--Mills amplitude of the background solution. The ``sphaleron--like'' instabilities (odd parity modes) decouple from the gravitational perturbations. They are obtained from a regular Schr\"odinger equation after a supersymmetric transformation. The body of the work is devoted to the fluctuations with even parity. The main difficulty arises because the Schwarzschild gauge -- which is usually imposed to eliminate the gravitational perturbations from the Yang--Mills equation -- is not regular for solutions with compact spatial topology. In order to overcome this problem, we derive a gauge invariant formalism by virtue of which the unphysical (gauge) modes can be isolated. 
  We introduce and investigate numerically a minimal class of dynamical triangulations of two-dimensional gravity on the sphere in which only vertices of order five, six or seven are permitted. We show firstly that this restriction of the local coordination number, or equivalently intrinsic scalar curvature, leaves intact the fractal structure characteristic of generic dynamically triangulated random surfaces. Furthermore the Ising model coupled to minimal two-dimensional gravity still possesses a continuous phase transition. The critical exponents of this transition correspond to the usual KPZ exponents associated with coupling a central charge c=1/2 model to two-dimensional gravity. 
  Using a formalism developed to include collective coordinates, we calculate the contributions to S-matrix elements due to off-shell D-branes in a string coupling expansion. The formalism is further used to establish a two-dimensional computation of higher order corrections to the D-brane tension, both for the bosonic and the supersymmetric cases. 
  The bulk and boundary magnetizations are calculated for the critical Ising model on a randomly triangulated disk in the presence of a boundary magnetic field h. In the continuum limit this model corresponds to a c = 1/2 conformal field theory coupled to 2D quantum gravity, with a boundary term breaking conformal invariance. It is found that as h increases, the average magnetization of a bulk spin decreases, an effect that is explained in terms of fluctuations of the geometry. By introducing an $h$-dependent rescaling factor, the disk partition function and bulk magnetization can be expressed as functions of an effective boundary length and bulk area with no further dependence on h, except that the bulk magnetization is discontinuous and vanishes at h = 0. These results suggest that just as in flat space, the boundary field generates a renormalization group flow towards h = \infty. An exact analytic expression for the boundary magnetization as a function of $h$ is linear near h = 0, leading to a finite nonzero magnetic susceptibility at the critical temperature. 
  We study gaugino condensation in the context of superstring effective theories using the linear multiplet formulation for the dilaton superfield. Including nonperturbative corrections to the K\"ahler potential for the dilaton may naturally achieve dilaton stabilization, with supersymmetry breaking and gaugino condensation; these three issues are interrelated in a very simple way. In a toy model with a single static condensate, a dilaton $vev$ is found within a phenomenologically interesting range. The effective theory differs significantly from condensate models studied previously in the chiral formulation. 
  We consider the analog in one spatial dimension of the Bose-Fermi transmutation for planar systems. A quantum mechanical system of a spin 1/2 particle coupled to an abelian gauge field, which is classically invariant under gauge transformations and charge conjugation is studied. It is found that unless the flux enclosed by the particle orbits is quantized, and the spin takes a value $n+ 1/2$, at least one of the two symmetries would be anomalous. Thus, charge conjugation invariance and the existence of abelian instantons simultaneously force the particles to be either bosons or fermions, but not anyons. 
  The problem of computing the thermodynamic properties of a one-dimensional gas of particles which transform in the adjoint representation of the gauge group and interact through non-Abelian electric fields is formulated and solved in the large $N$ limit. The explicit solution exhibits a first order confinement-deconfinement phase transition with computable properties and describes two dimensional adjoint QCD in the limit where matter field masses are large. 
  We study the cosmological implications of the one-loop terms in the string expansion. In particular, we find non-singular solutions which interpolate between a contracting universe and an expanding universe, and show that these solutions provide a mechanism for removing the initial conditions problem peculiar to spatially closed FRW cosmologies. In addition, we perform numerical calculations to show that the non-singular cosmologies do not require a careful choice of initial conditions, and estimate the likely magnitude of higher order terms in the string expansion. 
  We study the internal dynamics of Ramond-Ramond solitons excited far from the BPS limit by leading Regge trajectory open strings. The simplest world volume process for such strings is splitting into two smaller pieces, and we calculate the corresponding decay rates. Compared to the conventional open superstring, the splitting of states polarized parallel to the brane is suppressed by powers of logarithms of the energy. The rate for states polarized transverse to the brane {\em decreases} with increasing energy. We also calculate the static force between a D-brane excited by a massive open string and an unexcited D-brane parallel to it. The result shows that transversely polarized massive open strings endow D-branes with a size of order the string scale. 
  Recursion relations for orthogonal polynominals, arising in the study of the one-matrix model of two-dimensional gravity, are shown to be equvalent to the equations of the Toda-chain hierarchy supplemented by additional Virasoro constraints. This is the case without the double scaling limit. A discrete time variable to the matrix model is introduced. The discrete time dependent partition functions are given by $\tau$ functions which satisfy the discrete Toda molecule equation. Further the relations between the matrix model and the discrete time Toda theory are discussed. 
  Classical string cosmology consists of two branches related to each other by scale-factor duality: a super-inflation branch and a Friedmann-Robertson-Walker (FRW) branch. Curvature and string coupling singularity separates the two branches, hence posing `graceful exit problem' to super-inflationary string cosmology. In an exactly soluble two-dimensional compactification model it is shown that quantum back reaction retards curvature and string coupling growth and connects the super-inflation branch to the FRW branch without encountering a singularity. This may offer an attractive solution to the `graceful exit problem' in string inflationary cosmology. 
  The purpose of this Letter is to present a computation of the interaction energy of gauged O(3) Chern-Simons vortices which are infinitely separated. The results will show the behaviour of the interaction energy as a function of the constant coupling the potential, which measures the relative strength of the matter self-coupling and the electromagnetic coupling. We find that vortices attract each other for $\lambda > 1 $ and repel when $\lambda < 1 $. When $\lambda =1 $ there is a topological lower bound on the energy. It is possible to saturate the bound if the fields satisfy a set of first order partial differential equations. 
  Several physics aspects of the Seiberg-Witten solution of N=2 supersymmetric Yang-Mills theory with SU(2) gauge group, supplemented with a small mass term for the "matter" fields which leads to an $N=1$ theory with confinement, are discussed. The light spectrum of the theory is understood on the basis of current algebra relations, and CP invariance of the massless and massive theories is studied. We find that in the massive (confining) theory the low energy physics has an exact CP symmetry, while in a generic vacuum in the massless theory CP invarince is spontaneously broken. 
  We calculate the partition function, average occupation number and internal energy for a $SU_q(2)$ fermionic system and compare this model at $T=0$ with the ordinary fermionic, $q=1$, case. At low temperatures and $q\gg 1$ we find the chemical potential $\mu$ to have the same temperature dependence than the Fermi case. For $q\ll 1$, the function $\mu(T)$ has in addition a linear dependence on $T$. 
  We show that the usual dilaton dominance scenario, derived from the tree level K\"ahler potential, can never correspond to a global minimum of the potential at $V=0$. Similarly, it cannot correspond to a local minimum either, unless a really big conspiracy of different contributions to the superpotential $W(S)$ takes place. These results, plus the fact that the K\"ahler potential is likely to receive sizeable string non-perturbative contributions, strongly suggest to consider a more general scenario, leaving the K\"ahler potential arbitrary. In this way we obtain generalized expressions for the soft breaking terms but a predictive scenario still arises. Finally, we explore the phenomenological capability of some theoretically motivated forms for non-perturbative K\"ahler potentials, showing that it is easy to stabilize the dilaton at the realistic value $S\sim 2$ with just one condensate and no fine-tuning. 
  Conventional weak-coupling perturbation theory suffers from problems that arise from resonant coupling of successive orders in the perturbation series. Multiple-scale perturbation theory avoids such problems by implicitly performing an infinite reordering and resummation of the conventional perturbation series. Multiple-scale analysis provides a good description of the classical anharmonic oscillator. Here, it is extended to study the Heisenberg operator equations of motion for the quantum anharmonic oscillator. The analysis yields a system of nonlinear operator differential equations, which is solved exactly. The solution provides an operator mass renormalization of the theory. 
  The moduli space describing the low-energy dynamics of BPS multi-monopoles for several charge configurations is presented. We first prove the conjectured form of the moduli space of $n-1$ distinct monopoles in a spontaneously broken SU(n) gauge theory. We further propose the solution where one of the charge components has two units, hence asymptotically corresponds to embeddings of two monopoles of one charge type and the rest different. The latter hyperk\"ahler metrics possess features of the two-monopole Atiyah-Hitchin metric. We also conjecture classes of solutions to multi-monopole moduli spaces with arbitrary charge and no more than two units in each component, which models the gluing together of Atiyah-Hitchin metrics. Our approach here uses the generalized Legendre transform technique to find the new hyperk\"ahler manifolds and rederive previously conjectured ones. 
  We find the finite volume QCD partition function for arbitrary quark masses. This is a generalization of a result obtained by Leutwyler and Smilga for equal quark masses. Our result is derived in the sector of zero topological charge using a generalization of the Itzykson-Zuber integral appropriate for arbitrary complex matrices. We present a conjecture regarding the result for arbitrary topological charge which reproduces the Leutwyler-Smilga result in the limit of equal quark masses. We derive a formula of the Itzykson-Zuber type for arbitrary {\em rectangular\/} complex matrices, extending the result of Guhr and Wettig obtained for {\em square\/} matrices. 
  We consider various aspects of compactifications of the Type I/heterotic $Spin(32)/\Z_2$ theory on K3. One family of such compactifications includes the standard embedding of the spin connection in the gauge group, and is on the same moduli space as the compactification of the heterotic $E_8\times E_8$ theory on K3 with instanton numbers (8,16). Another class, which includes an orbifold of the Type I theory recently constructed by Gimon and Polchinski and whose field theory limit involves some topological novelties, is on the moduli space of the heterotic $E_8\times E_8$ theory on K3 with instanton numbers (12,12). These connections between $Spin(32)/\Z_2$ and $E_8\times E_8$ models can be demonstrated by T duality, and permit a better understanding of non-perturbative gauge fields in the (12,12) model. In the transformation between $Spin(32)/\Z_2$ and $E_8\times E_8$ models, the strong/weak coupling duality of the (12,12) $E_8\times E_8$ model is mapped to T duality in the Type I theory. The gauge and gravitational anomalies in the Type I theory are canceled by an extension of the Green-Schwarz mechanism. 
  We show that universal functions play an important role in the observation of the fractal structure of space--time in the numerical simulation of quantum gravity. 
  The massive spinning particle in six-dimensional Minkowski space is described as a mechanical system with the configuration space ${\ R}% ^{5,1}\times {\ CP}^3$. The action functional of the model is unambigiously determined by the requirement of identical (off-shell) conservation for the phase-space counterparts of three Casimir operators of Poincar\'e group. The model is shown to be exactly solvable. Canonical quantization of the model leads to the equations on wave functions which prove to be equivalent to the relativistic wave equations for the irreducible $6d$ fields. 
  I study the technique of Algebraic Bethe Ansatz for solving integrable models and show how it works in detail on the simplest example of spin 1/2 XXX magnetic chain. Several other models are treated more superficially, only the specific details are given. Several parameters, appearing in these generalizations: spin $s$, anisotropy parameter $\ga$, shift $\om$ in the alternating chain, allow to include in our treatment most known examples of soliton theory, including relativistic model of Quantum Field Theory. 
  With a help of the Schwinger --- DeWitt expansion analytical properties of the evolution operator kernel for the Schr\"odinger equation in time variable $t$ are studied for the Coulomb and Coulomb-like (which behaves themselves as $1/|\vec q|$ when $|\vec q| \to 0$) potentials. It turned out to be that the Schwinger --- DeWitt expansion for them is divergent. So, the kernels for these potentials have additional (beyond $\delta$-like) singularity at $t=0$. Hence, the initial condition is fulfilled only in asymptotic sense. It is established that the potentials considered do not belong to the class of potentials, which have at $t=0$ exactly $\delta$-like singularity and for which the initial condition is fulfilled in rigorous sense (such as $V(q) = -\frac{\lambda (\lambda-1)}{2} \frac {1}{\cosh^2 q}$ for integer $\lambda$). 
  A 'differential measure' is used to cast our calculus for the group $SU(3)$ into a form similar to Schwinger's boson operator calculus for the group $SU(2)$. It is then applied to compute (i) the inner product between the basis states and (ii) an algebraic formula for the Clebsch-Gordan coefficients. These were obtained earlier by us using Gaussian integration techniques. 
  Park et al's recent comment (hep-th/9605132) that for detectors with large energy gap in comparison with the temperature of the background thermal bath, the maximum excitation rate is obtained for some non-zero detector's velocity is correct but was previously discussed by ourselves elsewhere, and moreover does not affect in our paper above any mathematical formula, numerical result, and consequently our final conclusion that the background thermal bath does not contribute substantially in the depolarization of electrons at LEP. 
  Canonical coordinates for the Schr\"odinger equation are introduced, making more transparent its Hamiltonian structure. It is shown that the Schr\"odinger equation, considered as a classical field theory, shares with Liouville completely integrable field theories the existence of a {\sl recursion operator} which allows for the infinitely many conserved functionals pairwise commuting with respect to the corresponding Poisson bracket. The approach may provide a good starting point to get a clear interpretation of Quantum Mechanics in the general setting, provided by Stone-von Neumann theorem, of Symplectic Mechanics. It may give new tools to solve in the general case the inverse problem of quantum mechanics whose solution is given up to now only for one-dimensional systems by the Gel'fand-Levitan-Marchenko formula. 
  Using Seiberg-Witten theory it is known that the dynamics of N=2 supersymmetric SU(n) Yang-Mills theory is determined by a Riemann surface. In particular the mass formula for BPS states is given by the periods of a special differential on this surface. In this note we point out that the surface can be obtained from the quotient of a symmetric n-monopole spectral curve by its symmetry group. Known results about the Seiberg-Witten curves then implies that these monopoles are related to the Toda lattice. We make this relation explicit via the ADHMN construction. Furthermore, in the simplest case, that of two SU(2) monopoles, we find that the general two monopole solution is generated by an affine Toda soliton solution of the imaginary coupled theory. 
  In analogy with the Liouville case we study the $sl_3$ Toda theory on the lattice and define the relevant quadratic algebra and out of it we recover the discrete $W_3$ algebra. We define an integrable system with respect to the latter and establish the relation with the Toda lattice hierarchy. We compute the the relevant continuum limits. Finally we find the quantum version of the quadratic algebra. 
  Spontaneous magnetization in the Maxwell QED$_{2+1}$ at finite fermion density is studied. It is shown that at low fermion densities the one-loop free energy has its minimum at some nonvanishing value of the magnetic field. The magnetization is due to the asymmetry of the fermion spectrum of the massive QED$_{2+1}$ in an external magnetic field. 
  We discuss the phase structure of the NJL model in curved spacetime with magnetic field using $1/N$-expansion and linear curvature approximation. The effective potential for composite fields $\bar\psi \psi$ is calculated using the proper-time cut-off in the following cases: a) at non-zero curvature, b) at non-zero curvature and non-zero magnetic field, and c) at non-zero curvature and non-zero covariantly constant gauge field. Chiral symmetry breaking is studied numerically. We show that the gravitational field may compensate the effect of the magnetic field what leads to restoration of chiral symmetry. 
  We consider the large-$N$ Sutherland model in the Hamiltonian collective-field approach based on the $1/N$ expansion. The Bogomol'nyi limit appears and the corresponding solutions are given by static-soliton configurations. They exist only for $\l<1$, i.e. for the negative coupling constant of the Sutherland interaction. We determine their creation energies and show that they are unaffected by higher-order corrections. For $\l=1$, the Sutherland model reduces to the free one-plaquette Kogut-Susskind model. 
  The Lagrangian field-antifield formalism of Batalin and Vilkovisky (BV) is used to investigate the application of the collective coordinate method to soliton quantisation. It is shown how Noether identities and local symmetries of the Lagrangian arise when collective coordinates are introduced in order to avoid divergences related to zero modes. This transformation to collective and fluctuation degrees of freedom is interpreted as a canonical transformation in the symplectic field-antifield space which induces a time-local gauge symmetry. Separating the corresponding Lagrangian path integral of the BV scheme in lowest order into harmonic quantum fluctuations and a free motion of the collective coordinate with the classical mass of the soliton, we show how the BV approach clarifies the relation between zero modes, collective coordinates, gauge invariance and the center-of-mass motion of classical solutions in quantum fields. Finally, we apply the procedure to the reduced nonlinear O(3) sigma-model. 
  Based on the covariant background field method, we calculate the ultraviolet counter\-terms up to two-loop order and discuss the renormalizability of the three-dimensional non-linear sigma models with arbitrary Riemannian manifolds as target spaces. We investigate the bosonic model and its supersymmetric extension. We show that at the one-loop level these models are renormalizable and even finite when the manifolds are Ricci-flat. However, at the two-loop order, we find non-renormalizable counterterms in all cases considered, so the renormalizability and finiteness of such models are completely lost in this order. 
  Last week, A. Sen found an explicit type I string compactification dual to the eight-dimensional F-theory construction with SO(8)^4 nonabelian gauge symmetry. He found that the perturbations around the enhanced symmetry point were described by the mathematics of the solution of N=2, d=4 SU(2) gauge theory with four flavors, and argued more generally that global symmetry enhancement in CN=2, d=4 gauge theories corresponded to gauge symmetry enhancement in F-theory. We show that these N=2, d=4 gauge theories have a physical interpretation in the theory. They are the world-volume theories of 3-branes parallel to the 7-branes. They can be used to probe the structure of the exact quantum F-theory solutions. On the Higgs branch of the moduli space, the objects are equivalent to finite size instantons in the 7-brane gauge theory. 
  Using ``Tate's algorithm,'' we identify loci in the moduli of F-theory compactifications corresponding to enhanced gauge symmetry. We apply this to test the proposed F-theory/heterotic dualities in six dimensions. We recover the perturbative gauge symmetry enhancements of the heterotic side and the physics of small $SO(32)$ instantons, and discover new mixed perturbative/non-perturbative gauge symmetry enhancements. Upon further toroidal compactification to 4 dimensions, we derive the chain of Calabi-Yau threefolds dual to various Coulomb branches of heterotic strings. 
  When $N$ five-branes of M-theory coincide the world-volume theory contains tensionless strings, according to Strominger's construction. This suggests a large $N$ limit of tensionless string theories. For the small $E_8$ instanton theories, the definition would be a large instanton number. An adiabatic argument suggests that in the large $N$ limit an effective extra uncompactified dimension might be observed. We also propose ``surface-equations'', which are an analog of Makeenko-Migdal loop-equations, and might describe correlators in the tensionless string theories. In these equations, the anti-self-dual two forms of 6D and the tensionless strings enter on an equal footing. Addition of strings with CFTs on their world-sheet is analogous to addition of matter in 4D QCD. 
  Using the framework of Nambu's generalised mechanics, we obtain a new description of constrained Hamiltonian dynamics, involving the introduction of another degree of freedom in phase space, and the necessity of defining the action integral on a world sheet. We also discuss the problem of quantising Nambu mechanics. 
  Weak first-order phase transitions proceed with percolation of new phase. The kinematics of this process is clarified from the point of view of subcritical bubbles. We examine the effect of small subcritical bubbles around a large domain of asymmetric phase by introducing an effective geometry. The percolation process can be understood as a perpetual growth of the large domain aided by the small subcritical bubbles. 
  We formulate a theory of generalized Fock spaces which underlies the different forms of quantum statistics such as ``infinite'', Bose-Einstein and Fermi-Dirac statistics. Single-indexed systems as well as multi-indexed systems that cannot be mapped into single-indexed systems are studied. Our theory is based on a three-tiered structure consisting of Fock space, statistics and algebra. This general formalism not only unifies the various forms of statistics and algebras, but also allows us to construct many new forms of quantum statistics as well as many algebras of creation and destruction operators. Some of these are : new algebras for infinite statistics, q-statistics and its many avatars, a consistent algebra for fractional statistics, null statistics or statistics of frozen order, ``doubly-infinite'' statistics, many representations of orthostatistics, Hubbard statistics and its variations. 
  A previously unnoticed connection between the Lax descriptions and the superextensions of the KdV hierarchy is presented. It is shown that the two different Lax descriptions of the KdV hierarchy come out naturally from two different bihamiltonian superextensions of the KdV hierarchy. Some implications of this observation are briefly mentioned. 
  We present master formulas for the divergent part of the one-loop effective action for an arbitrary (both minimal and nonminimal) operators of any order in the 4-dimensional curved space. They can be considered as computer algorithms, because the one-loop calculations are then reduced to the simplest algebraic operations. Some test applications are considered by REDUCE analytical calculation system. 
  Discussions are made on the relationship between physical states and gauge independence in QED. As the first candidate take the LSZ-asymptotic states in a covariant canonical formalism to investigate gauge independence of the (Belinfante's) symmetric energy-momentum tensor. It is shown that expectation values of the energy-momentum tensor in terms of those asymptotic states are gauge independent to all orders. Second, consider gauge invariant operators of electron or photon, such as the Dirac's electron or Steinmann's covariant approach, expecting a gauge invariant result without any restriction. It is, however, demonstrated that to single out gauge invariant quantities is merely synonymous to a gauge fixing, resulting again in use of the asymptotic condition when proving gauge independence. Nevertheless, it is commented that these invariant approaches is helpful to understand the mechanism of the LSZ-mapping and furthermore of quark confinement in QCD. As the final candidate, it is shown that gauge transformations are freely performed under the functional representation or the path integral expression on account of the fact that the functional space is equivalent to a collection of infinitely many inequivalent Fock spaces. The covariant LSZ formalism is shortly reviewed and the basic facts on the energy-momentum tensor are also illustrated. 
  A scalar cubic action that classically reproduces the self-dual Yang--Mills equations is shown to generate one-loop QCD amplitudes for external gluon all with the same helicity. This result is related to the symmetries of the self-dual Yang--Mills equations. 
  The one-loop effective action for a massive self-interacting scalar field is investigated in $4$-dimensional ultrastatic space-time $ R \times H^3/\Gamma$, $H^3/\Gamma$ being a non-compact hyperbolic manifold with finite volume. Making use of the Selberg trace formula, the $\zeta$-function related to the small disturbance operator is constructed. For an arbitrary gravitational coupling, it is found that $\zeta(s)$ has a simple pole at $s=0$. The one-loop effective action is analysed by means of proper-time regularisations and the one-loop divergences are explicitly found. It is pointed out that, in this special case, also $\zeta$-function regularisation requires a divergent counterterm, which however is not necessary in the free massless conformal invariant coupling case. Finite temperature effects are studied and the high-temperature expansion is presented. A possible application to the problem of the divergences of the entanglement entropy for a free massless scalar field in a Rindler-like space-time is briefly discussed. 
  In recent years light-cone quantization of quantum field theory has emerged as a promising method for solving problems in the strong coupling regime. This approach has a number of unique features that make it particularly appealing, most notably, the ground state of the free theory is also a ground state of the full theory. 
  Within a Liouville approach to non-critical string theory, we discuss space-time foam effects on the propagation of low-energy particles. We find an induced frequency-dependent dispersion in the propagation of a wave packet, and observe that this would affect the outcome of measurements involving low-energy particles as probes. In particular, the maximum possible order of magnitude of the space-time foam effects would give rise to an error in the measurement of distance comparable to that independently obtained in some recent heuristic quantum-gravity analyses. We also briefly compare these error estimates with the precision of astrophysical measurements. 
  A non-Abelian analogue of the Abelian T-duality momentum-winding exchange is described. The non-Abelian T-duality relates $\sigma$-models living on the cosets of a Drinfeld double with respect to its isotropic subgroups. The role of the Abelian momentum-winding lattice is in general played by the fundamental group of the Drinfeld double. 
  It is shown that the non-trivial cocycles on simple Lie algebras may be used to introduce antisymmetric multibrackets which lead to higher-order Lie algebras, the definition of which is given. Their generalised Jacobi identities turn out to be satisfied by the antisymmetric tensors (or higher-order `structure constants') which characterise the Lie algebra cocycles. This analysis allows us to present a classification of the higher-order simple Lie algebras as well as a constructive procedure for them. Our results are synthesised by the introduction of a single, complete BRST operator associated with each simple algebra. 
  The four dimensional Abelian Higgs model with monopoles and $\Theta$-term is considered in the limit of the large mass of the higgs boson. We show that for $\Theta=2 \pi$ the theory is equivalent, at large distances, to summation over all possible world-sheets of fermionic strings with Dirichlet type boundary conditions on string coordinates. 
  Using instanton calculus we check, in the weak coupling region, the nonperturbative relation $$ <\Tr\phi^2>=i\pi\left(\cf-{a\over 2} {\partial\cf\over\partial a}\right)$$ obtained for a N=2 globally supersymmetric gauge theory. Our computations are performed for instantons of winding number k, up to k=2 and turn out to agree with previous nonperturbative results. 
  In four dimensions there are 4 different types of extremal Maxwell/scalar black holes characterized by a scalar coupling parameter $a$ with $a=0,1/\sqrt{3} , 1 , \sqrt{3}$. These black holes can be described as intersections of ten--dimensional non-singular Ramond-Ramond objects, i.e.~$D$-branes, waves and Taub-NUT solitons. Using this description it can be shown that the four--dimensional black holes decompactify near the core to higher--dimensional {\em non-singular} solutions. In terms of these higher--dimensional non-singular solutions we define a non-vanishing entropy for all four black hole types from a four--dimensional point of view. 
  The complexified gauging of the de Sitter group gives a unified theory for the electroweak and gravitational interactions. The standard spectrum for the electroweak gauge bosons is recovered with the correct mass assignments, following a spontaneous breaking of the gauge symmetry imposed by the geometry. There is no conventional Higgs sector. New physics is predicted with gravity-induced electroweak processes (at the electroweak and at an intermediate scale of about $10^{10}Gev$) as well as with novel-type of effects (such as gravitational Aharonov-Bohm and violations of the Principle of equivalence to 1 part in $10^{17}$). The new theoretical perspectives emerging from this geometric unification are briefly discussed. 
  We develop an efficient technique to compute anomalies in supersymmetric theories by combining the so-called nonlocal regularization method and superspace techniques. To illustrate the method we apply it to a four dimensional toy model with potentially anomalous N=1 supersymmetry and prove explicitly that in this model all the candidate supersymmetry anomalies have vanishing coefficients at the one-loop level. 
  We study two-dimensional integrable $N=1$ supersymmetric theories (without topological charges) in the presence of a boundary. We find a universal ratio between the reflection amplitudes for particles that are related by supersymmetry and we propose exact reflection matrices for the supersymmetric extensions of the multi-component Yang-Lee models and for the breather multiplets of the supersymmetric sine-Gordon theory. We point out the connection between our reflection matrices and the classical boundary actions for the supersymmetric sine-Gordon theory as constructed by Inami, Odake and Zhang \cite{IOZ}. 
  N=2 noncritical strings are closely related to the $\Slr/\Slr$ Wess-Zumino- Novikov-Witten model, and there is much hope to further probe the former by using the algebraic apparatus provided by the latter. An important ingredient is the precise knowledge of the $\hslc$ representation theory at fractional level. In this paper, the embedding diagrams of singular vectors appearing in $\hslc$ Verma modules for fractional values of the level ($k=p/q-1$, p and q coprime) are derived analytically. The nilpotency of the fermionic generators in $\hslc$ requires the introduction of a nontrivial generalisation of the MFF construction to relate singular vectors among themselves. The diagrams reveal a striking similarity with the degenerate representations of the $N=2$ superconformal algebra. 
  The construction of a supersymmetric $SO(10)$ grand unification with 5 left-handed and 2 right-handed families in the four-dimensional heterotic string theory is presented. The model has one $SO(10)$ adjoint Higgs field. The $SO(10)$ current algebra is realized at level 3. 
  In contrast to the familiar (2,2) case, the singularities which arise in the (0,2) setting can be associated with degeneration of the base Calabi-Yau manifold {\it and/or}\/ with degenerations of the gauge bundle. We study a variety of such singularities and give a procedure for resolving those which can be cured perturbatively. Among the novel features which emerge are models in which smoothing singularities in the base yields a gauge {\it sheaf}\/ as opposed to a gauge bundle as the structure to which left moving fermions couple. Supersymmetric $\sigma$-models with target data being an appropriate sheaf on a Calabi-Yau space therefore appear to be the natural arena for N=1 string models in four dimensions. We also indicate a variety of singularities which would require a nonperturbative treatment for their resolution and briefly discuss applications to heterotic models on K3. 
  We describe the ${\cal{O}}({\alpha'}^0)$ constraints on the target space geometry of the $N=(2,1)$ heterotic superstring due to the left-moving $N=1$ supersymmetry and $U(1)$ currents. In the fermionic description of the internal sector supersymmetry is realized quantum mechanically, so that both tree-level and one-loop effects contribute to the order ${\cal{O}}({\alpha'}^0)$ constraints. We also discuss the physical interpretation of the resulting target space geometry in terms of configurations of a $2+2$-dimensional object propagating in a $10+2$-dimensional spacetime with a null isometry, which has recently been suggested as a unified description of string and M theory. 
  We consider a family of solutions to string theory which depend on arbitrary functions and contain regular event horizons. They describe six dimensional extremal black strings with traveling waves and have an inhomogeneous distribution of momentum along the string. The structure of these solutions near the horizon is studied and the horizon area computed. We also count the number of BPS string states at weak coupling whose macroscopic momentum distribution agrees with that of the black string. It is shown that the number of such states is given by the Bekenstein-Hawking entropy of the black string with traveling waves. 
  We discuss how the theory of quantum cohomology may be generalized to ``gravitational quantum cohomology'' by studying topological sigma models coupled to two-dimensional gravity. We first consider sigma models defined on a general Fano manifold $M$ (manifold with a positive first Chern class) and derive new recursion relations for its two point functions. We then derive bi-Hamiltonian structures of the theories and show that they are completely integrable at least at the level of genus $0$. We next consider the subspace of the phase space where only a marginal perturbation (with a parameter $t$) is turned on and construct Lax operators (superpotentials) $L$ whose residue integrals reproduce correlation functions. In the case of $M=CP^N$ the Lax operator is given by $L= Z_1+Z_2+\cdots +Z_N+e^tZ_1^{-1}Z_2^{-1}\cdots Z_N^{-1}$ and agrees with the potential of the affine Toda theory of the $A_N$ type. We also obtain Lax operators for various Fano manifolds; Grassmannians, rational surfaces etc. In these examples the number of variables of the Lax operators is the same as the dimension of the original manifold. Our result shows that Fano manifolds exhibit a new type of mirror phenomenon where mirror partner is a non-compact Calabi-Yau manifold of the type of an algebraic torus $C^{*N}$ equipped with a specific superpotential. 
  We discuss a novel method of obtaining the fractional spin of abelian and nonabelian Chern-Simons vortices. This spin is interpreted as the difference between the angular momentum obtained by modifying Schwinger's energy momentum tensor by the Gauss constraint, and the canonical (Noether) angular momentum. It is found to be a boundary term depending only on the gauge field and, hence, is independent of the matter sector to which the Chern-Simons term couples. Addition of the Maxwell term does not alter the fractional spin. 
  We show that the measure of the three dimensional Nambu-Goto string theory has a simple decomposition as a measure on two parameter group of induced area-preserving transformations of the immersed surface and a trivial measure for the area of the surface. 
  We present a quantization of previously proposed generalized Chern-Simons theory with $gl(1,{\bf R})$ algebra in 1+1 dimensions. This simplest model shares the common features of generalized CS theories: on-shell reducibility and violations of regularity. On-shell reducibility of the theory requires us to use the Lagrangian Batalin-Vilkovisky and/or Hamiltonian Batalin-Fradkin-Vilkovisky formulation. Since the regularity condition is violated, their quantization is not straightforward. In the present case we can show that both formulations give an equivalent result. It leads to an interpretation that a physical degree of freedom which does not exist at the classical level appears at the quantum level. 
  We use the multimonopole moduli space as a tool for studying the properties of BPS monopoles carrying nonabelian magnetic charges. For configurations whose total magnetic charge is purely abelian, the moduli space for nonabelian breaking can be obtained as a smooth limit of that for a purely abelian breaking. As the asymptotic Higgs field is varied toward one of the special values for which the unbroken symmetry is enlarged to a nonabelian group, some of the fundamental monopoles of unit topological charge remain massive but acquire nonabelian magnetic charges. The BPS mass formula indicates that others should become massless in this limit. We find that these do not correspond to distinct solitons but instead manifest themselves as ``nonabelian clouds'' surrounding the massive monopoles. The moduli space coordinates describing the position and $U(1)$ phase of these massless monopoles are transformed into an equal number of nonabelian global gauge orientation and gauge-invariant structure parameters characterizing the nonabelian cloud. We illustrate this explicitly in a class of $Sp(2N)$ examples for which the full family of monopole solutions is known. We show in detail how the unbroken symmetries of the theory are manifested as isometries of the moduli space metric. We discuss the connection of these results to the Montonen-Olive duality conjecture, arguing in particular that the massless monopoles should be understood as the duals to the massless gauge bosons that appear as the mediators of the nonabelian forces in the perturbative sector. 
  An explicit and simple correspondence, between the basis of the model space of $SU(3)$ on one hand and that of $SU(2)\otimes SU(2)$ or $SO(1,3)$ on the other, is exhibited for the first time. This is done by considering the generating functions for the basis vectors of these model spaces. 
  We study the QED bound-state problem in a light-front hamiltonian approach. Starting with a bare cutoff QED Hamiltonian, $H_{_{B}}$, with matrix elements between free states of drastically different energies removed, we perform a similarity transformation that removes the matrix elements between free states with energy differences between the bare cutoff, $\Lambda$, and effective cutoff, $\lam$ ($\lam < \Lam$). This generates effective interactions in the renormalized Hamiltonian, $H_{_{R}}$. These effective interactions are derived to order $\alpha$ in this work, with $\alpha \ll 1$. $H_{_{R}}$ is renormalized by requiring it to satisfy coupling coherence. A nonrelativistic limit of the theory is taken, and the resulting Hamiltonian is studied using bound-state perturbation theory (BSPT). The effective cutoff, $\lam^2$, is fixed, and the limit, $0 \longleftarrow m^2 \alpha^2\ll \lam^2 \ll m^2 \alpha \longrightarrow \infty$, is taken. This upper bound on $\lam^2$ places the effects of low-energy (energy transfer below $\lam$) emission in the effective interactions in the $| e {\overline e} > $ sector. This lower bound on $\lam^2$ insures that the nonperturbative scale of interest is not removed by the similarity transformation. As an explicit example of the general formalism introduced, we show that the Hamiltonian renormalized to $O(\alpha)$ reproduces the exact spectrum of spin splittings, with degeneracies dictated by rotational symmetry, for the ground state through $O(\alpha^4)$. The entire calculation is performed analytically, and gives the well known singlet-triplet ground state spin splitting of positronium, $7/6 \alpha^2 Ryd$. We discuss remaining corrections other than the spin splittings and how they can be treated in calculating the spectrum with higher precision. 
  We discuss $SU(N_c)$ gauge theory coupled to two adjoint chiral superfields $X$ and $Y$, and a number of fundamental chiral superfields $Q^i$. We add a superpotential that has the form of Arnold's $D$ series $W = \Tr X^{k+1} + \Tr XY^2$. We present a dual description in terms of an $SU(3kN_f - N_c)$ gauge theory, and we show that the duality passes many tests. At the end of the paper, we show how a deformation of this superpotential flows to another duality having a product gauge group $SU(N_c)\times SU(N_c')$, with an adjoint field charged under $SU(N_c)$, an adjoint field charged under $SU(N_c')$, fields in the $(N_c,N_c')$ and $(\overline N_c,\overline N_c')$ representation, and a number of fundamentals. The dual description is an $SU(2kN_f' + kN_f - N_c')\times SU(2kN_f + kN_f' - N_c)$ gauge theory. 
  The action for the su(N) SDYM equations is shown to give in the limit $N \to \infty$ the action for the six-dimensional version of the second heavenly equation. The symmetry reductions of this latter equation to the well known equations of self-dual gravity are given. The Moyal deformation of the heavenly equations are also considered. 
  Classically a black hole can absorb but not emit energy. We discuss how this T-asymmetric property of black holes arises in the recently proposed (T-symmetric) microscopic models of black holes based on bound states of D-branes. In these string theory based models, the nonvanishing classical absorption is made possible essentially by the exponentially increasing degeneracy of quantum states with mass of the black hole. The classical limit of the absorption crosssection computed in the microscopic model agrees with the result obtained from a classical analysis of a wave propagating in the background metric of the corresponding black hole (upto a numerical factor). 
  Using the vertex operator approach we show that fusion of the RSOS models can be considered as a kind of coset construction which is very similar to the coset construction of minimal models in conformal field theory. We reproduce the excitation spectrum and $S$-matrix of the fusion RSOS models in the regime III and show that their correlation functions and form factors can be expressed in terms of those of the ordinary (ABF) RSOS models. 
  We consider an embedding of the extremal four-dimensional Reissner-Nordstr\"om black hole into type $IIB$ string theory. The equivalent type $IIB$ configuration, in the D-brane weak-coupling picture, is a bound state of D1- and D5-branes threaded by fundamental type $IIB$ strings. The bound state involves also a NSNS solitonic 5-brane, mimicking the role of the Kaluza-Klein magnetic monopole. The statistical entropy derived by counting the degeneracy of the BPS-saturated excitations of this bound state agrees perfectly with the (semiclassical) Bekenstein-Hawking formula. 
  We propose a new action principle to be associated with a noncommutative space $(\Ac ,\Hc ,D)$. The universal formula for the spectral action is $(\psi ,D\psi) + \Trace (\chi (D /$ $\Lb))$ where $\psi$ is a spinor on the Hilbert space, $\Lb$ is a scale and $\chi$ a positive function. When this principle is applied to the noncommutative space defined by the spectrum of the standard model one obtains the standard model action coupled to Einstein plus Weyl gravity. There are relations between the gauge coupling constants identical to those of $SU(5)$ as well as the Higgs self-coupling, to be taken at a fixed high energy scale. 
  We derive the partition functions of the Schwarz-type four-dimensional topological half-flat 2-form gravity model on K3-surface or T^4 up to on-shell one-loop corrections. In this model the bosonic moduli spaces describe an equivalent class of a trio of the Einstein-K\"ahler forms (the hyperk\"ahler forms). The integrand of the partition function is represented by the product of some $\bar \partial$-torsions. 
  We compute the boundary scattering amplitudes of the breathers of the supersymmetric sine-Gordon model using the fusion of the soliton-antisoliton pair scattering with the boundary with a known result of the soliton boundary scattering amplitudes. We also solve the boundary Yang-Baxter equation of the eight-vertex free fermion models to find the boundary reflection matrices. The former result is confirmed by the latter since the bulk $S$-matrices of the breathers can be identified with the trigonometric limit of the Boltzmann weights of the free fermion models. Our dual approach can answer a few quesions on the relationships between the free parameters in the boundary potential and those in the scattering amplitudes. 
  An iterative scheme is set up for solving the loop equation of the hermitian one-matrix model with a multi-cut structure. Explicit results are presented for genus one for an arbitrary but finite number of cuts. Due to the complicated form of the boundary conditions, the loop correlators now contain elliptic integrals. This demonstrates the existence of new universality classes for the hermitian matrix model. The two-cut solution is investigated in more detail, including the double-scaling limit. It is shown, that in special cases it differs from the known continuum solution with one cut. 
  We introduce and study a pure gauge abelian theory in 2+1D in which massive quantum vortex states do exist in the spectrum of excitations. This theory can be mapped in a three dimensional gas of point particles with a logarithmic interaction, in the grand-canonical ensemble. We claim that this theory is the 2+1D analog of the Sine-Gordon, the massive vortices being the counterparts of Sine-Gordon solitons. We show that a symmetry breaking, order parameter, similar to the vacuum expectation value of a Higgs field does exist. 
  It is argued that quantum gravity has an interpretation as a topological field theory provided a certain constraint from the path intergral measure is respected. The constraint forces us to couple gauge and matter fields to gravity for space - time dimensions different from 3. We then discuss possible models which may be relevant to our universe. 
  Recently, within the context of the phase space coherent state path integral quantisation of constrained systems, John Klauder introduced a reproducing kernel for gauge invariant physical states, which involves a projection operator onto the reduced Hilbert space of physical states, avoids any gauge fixing conditions, and leads to a specific measure for the integration over Lagrange multipliers. Here, it is pointed out that this approach is also devoid of any Gribov problems and always provides for an effectively admissible integration over all gauge orbits of gauge invariant systems. This important aspect of Klauder's proposal is explicitly confirmed by two simple examples. 
  We derive the anomaly 8-form of 6-dimensional gauge theories arising in F theory compactifications on elliptic Calabi-Yau threefolds. The result allows to determine the matter content of certain such theories in terms of intersection numbers on the base of elliptic fibration. We also discuss gauge theories on 7-branes with double point singularities on the worldvolume. Applications to Type II compactifications on Hirzebruch surfaces and $P^2$ are outlined. 
  Starting from the type IIB Dirichlet 3-brane action, we obtain a Nambu-Goto action. It is interpreted as the world volume action of a fundamental 3-brane, and its target space theory as F-theory. The target space is twelve dimensional, with signature $(11, 1)$. It is an elliptic fibration over a ten dimensional base space. The $SL(2, Z)$ symmetry of type IIB string has now an explicit geometric interpretation. Also, one gets a glimpse of the conjectured self-duality of M-theory. 
  Type II superstring theory with mixed Dirichlet and Neumann boundary conditions admit antisymmetric tensors with varying degrees in the spectrum. We show that there exists a family of dual supergravity lagrangians to the $N=2$ type IIA action in ten dimensions. The duality transformations and the resulting actions are constructed explicitely. 
  We present examples of many-body Wigner quantum systems. The position and the momentum operators ${\bf R}_A$ and ${\bf P}_A,\; A=1,\ldots,n+1$, of the particles are noncanonical and are chosen so that the Heisenberg and the Hamiltonian equations are identical. The spectrum of the energy with respect to the centre of mass is equidistant and has finite number of energy levels. The composite system is spread in a small volume around the centre of mass and within it the geometry is noncommutative. The underlying statistics is an exclusion statistics. 
  The trigonometric Ruijsenaars-Schneider model is diagonalized by means of the Macdonald symmetric functions. We evaluate the dynamical density-density correlation function and the one-particle retarded Green function as well as their thermodynamic limit. Based on these results and finite-size scaling analysis, we show that the low-energy behavior of the model is described by the $C=1$ Gaussian conformal field theory under a new fractional selection rule for the quantum numbers labeling the critical exponents. 
  A perturbation theory with respect to tension parameter $\gamma/\alpha^\prime$ for the non--linear equations of string, moving in curved space--time is considered. Obtained are linearized motion equations for the functions of the $n-$th degree of approximation ($n=0,1,2$) 
  The property of some finite W-algebras to appear as the commutant of a particular subalgebra in a simple Lie algebra G is exploited for the obtention of new G-realizations from a "canonical" differential one.   The method is applied to the conformal algebra so(4,2) and therefore yields also results for its Poincare subalgebra. Unitary irreducible representations of these algebras are recognized in this approach, which is naturally compared -or associated- to the induced representation technic. 
  Free field representations of the affine superalgebra $A(1,0)^{(1)}$ at level $k$ are needed in the description of the noncritical $N=2$ string. The superalgebra admits two inequivalent choices of simple roots. We give the Wakimoto representations corresponding to each of these and derive the relation between the two at the quantum level. 
  The Bekenstein-Hawking entropy for certain BPS-saturated black holes in string theory has recently been derived by counting internal black hole microstates at weak coupling. We argue that the black hole microstate can be measured by interference experiments even in the strong coupling region where there is clearly an event horizon. Extracting information which is naively behind the event horizon is possible due to the existence of statistical quantum hair carried by the black hole. This quantum hair arises from the arbitrarily large number of discrete gauge symmetries present in string theory. 
  We consider the type I theory compactified on $T^3$. When the D5-brane wraps the $T^3$ it yields a D2-brane in seven dimensions. In the leading approximation the moduli space of vacua of the three dimensional field theory on the brane is $T^4/\ZZ_2$. The dual M theory description of this theory is a compactification on K3 and our 2-brane is the eleven dimensional 2-brane at a point in K3. We use this fact to conclude that strong coupling IR effects in the three dimensional theory on the brane turn its moduli space into a K3. This interpretation allows us to solve various strongly coupled gauge theories in three dimensions by identifying their Coulomb branch with a piece of a (sometime singular) K3. 
  A systematic algorithm for obtaining recurrence relations for dimensionally regularized Feynman integrals w.r.t. the space-time dimension $d$ is proposed. The relation between $d$ and $d-2$ dimensional integrals is given in terms of a differential operator for which an explicit formula can be obtained for each Feynman diagram. We show how the method works for one-, two- and three-loop integrals. The new recurrence relations w.r.t. $d$ are complementary to the recurrence relations which derive from the method of integration by parts. We find that the problem of the irreducible numerators in Feynman integrals can be naturally solved in the framework of the proposed generalized recurrence relations. 
  We study here the algebraic structure of the conserved generators from which the microcanonical and canonical ensembles are constructed on an underlying generalized quantum dynamics, and the flows they induce on the phase space. We also discuss briefly the structure of the microcanonical and canonical ensembles. 
  We apply an earlier formulated programme for quantization of nonabelian gauge theories to one-flavour chromodynamics. This programme consists in a complete reformulation of the functional integral in terms of gauge invariant quantities. For the model under consideration two types of gauge invariants occur -- quantities, which are bilinear in quarks and antiquarks (mesons) and a matrix-valued covector field, which is bilinear in quarks, antiquarks and their covariant derivatives. This covector field is linear in the original gauge potential, and can be, therefore, considered as the gauge potential ``dressed'' in a gauge invariant way with matter. Thus, we get a complete bosonization of the theory. The strong interaction is described by a highly non-linear effective action obtained after integrating out quarks and gluons from the functional integral. All constructions are done consequently on the quantum level, where quarks and antiquarks are anticommuting objects. Our quantization procedure circumvents the Gribov ambiguity. 
  We study the Galilean symmetry in a nonrelativistic model, recently advanced by Bak, Jackiw and Pi, involving the coupling of a nonabelian Chern-Simons term with matter fields. The validity of the Galilean algebra on the constraint surface is demonstrated in the gauge independent formalism. Then the reduced space formulation is discussed in the axial gauge using the symplectic method. An anomalous term in the Galilean algebra is obtained which can be eliminated by demanding conditions on the Green function. Finally, the axial gauge is also treated by Dirac's method.  Galilean symmetry is preserved in this method. Comparisions with the symplectic approach reveal some interesting features. 
  The algebraic and Hamiltonian structures of the multicomponent dispersionless Benney and Toda hierarchies are studied. This is achieved by using a modified set of variables for which there is a symmetry between the basic fields. This symmetry enables formulae normally given implicitly in terms of residues, such as conserved charges and fluxes, to be calculated explicitly. As a corollary of these results the equivalence of the Benney and Toda hierarchies is established. It is further shown that such quantities may be expressed in terms of generalized hypergeometric functions, the simplest example involving Legendre polynomials. These results are then extended to systems derived from a rational Lax function and a logarithmic function. Various reductions are also studied. 
  It has recently been shown, by application of statistical mechanical methods to determine the canonical ensemble governing the equilibrium distribution of operator initial values, that complex quantum field theory can emerge as a statistical approximation to an underlying generalized quantum dynamics. This result was obtained by an argument based on a Ward identity analogous to the equipartition theorem of classical statistical mechanics. We construct here a microcanonical ensemble which forms the basis of this canonical ensemble. This construction enables us to define the microcanonical entropy and free energy of the field configuration of the equilibrium distribution and to study the stability of the canonical ensemble. We also study the algebraic structure of the conserved generators from which the microcanonical and canonical ensembles are constructed, and the flows they induce on the phase space. 
  We formulate the WT identity for proper vertices in a simple and compact form $\partial \Gamma / {\partial \theta } =0 $ in a superspace formulation of gauge theories proposed earlier. We show this WT identity (together with a subsidiary constraint) lead, in transparent way, the superfield superspace multiplet renormalizations formulated earlier (and shown to explain symmetries of Yang-Mills theory renormalization). 
  The Virasoro master equation (VME) describes the general affine-Virasoro construction $T=L^{ab}J_aJ_b+iD^a \dif J_a$ in the operator algebra of the WZW model, where $L^{ab}$ is the inverse inertia tensor and $D^a $ is the improvement vector. In this paper, we generalize this construction to find the general (one-loop) Virasoro construction in the operator algebra of the general non-linear sigma model. The result is a unified Einstein-Virasoro master equation which couples the spacetime spin-two field $L^{ab}$ to the background fields of the sigma model. For a particular solution $L_G^{ab}$, the unified system reduces to the canonical stress tensors and conventional Einstein equations of the sigma model, and the system reduces to the general affine-Virasoro construction and the VME when the sigma model is taken to be the WZW action. More generally, the unified system describes a space of conformal field theories which is presumably much larger than the sum of the general affine-Virasoro construction and the sigma model with its canonical stress tensors. We also discuss a number of algebraic and geometrical properties of the system, including its relation to an unsolved problem in the theory of $G$-structures on manifolds with torsion. 
  We introduce a notion of quantum hair which completely characterizes the state of a D-brane in perturbative string theory. The hair manifests itself as a phase (more generally a unitary matrix in subspaces of degenerate string eigenstates) in the scattering amplitudes of elementary strings on the D brane. As the separation of the D-brane and the string center of mass becomes large, the phase goes to zero if we keep the string excitation level fixed. However, by letting the level number increase with the distance, we can keep the phase constant. We argue that this implies that scattering experiments with highly excited strings can detect the state of a D-brane long after it has ``fallen into a black hole'' . 
  We show that the self-dual Yang-Mills equations afford supersymmetrisation to systems of equations invariant under global N-extended super-Poincar\'e transformations for arbitrary values of N, without the limitation (N\le 4) applicable to standard non-self-dual Yang-Mills theories. These systems of equations provide novel classically consistent interactions for vector supermultiplets containing fields of spin up to (N-2)/2. The equations of motion for the component fields of spin greater than 1/2 are interacting variants of the first-order Dirac--Fierz equations for zero rest-mass fields of arbitrary spin. The interactions are governed by conserved currents which are constructed by an iterative procedure. In (arbitrarily extended) chiral superspace, the equations of motion for the (arbitrarily large) self-dual supermultiplet are shown to be completely equivalent to the set of algebraic supercurvature constraints defining the self-dual superconnection. 
  We develop an algebraic approach to studying the spectral properties of the stationary Schr\"odinger equation in one dimension based on its high order conditional symmetries. This approach makes it possible to obtain in explicit form representations of the Schr\"odinger operator by $n\times n$ matrices for any $n\in {\bf N}$ and, thus, to reduce a spectral problem to a purely algebraic one of finding eigenvalues of constant $n\times n$ matrices. The connection to so called quasi exactly solvable models is discussed. It is established, in particular, that the case, when conditional symmetries reduce to high order Lie symmetries, corresponds to exactly solvable Schr\"odinger equations. A symmetry classification of Sch\"odinger equation admitting non-trivial high order Lie symmetries is carried out, which yields a hierarchy of exactly solvable Schr\"odinger equations. Exact solutions of these are constructed in explicit form. Possible applications of the technique developed to multi-dimensional linear and one-dimensional nonlinear Schr\"odinger equations is briefly discussed. 
  A detailed analysis of Chern-Simons (CS) theories in which a compact abelian direct product gauge group U(1)^k is spontaneously broken down to a direct product H of (finite) cyclic groups is presented. The spectrum features global H charges, vortices carrying flux labeled by the elements of H and dyonic combinations. Due to the Aharonov-Bohm effect these particles exhibit toplogical interactions. The remnant of the U(1)^k CS term in the discrete H gauge theory describing the effective long distance physics of such a model is shown to be a 3-cocycle for H summarizing the nontrivial topological interactions cast upon the magnetic vortices by the U(1)^k CS term. It is noted that there are in general three types of 3-cocycles for a finite abelian gauge group H: one type describes topological interactions among vortices carrying flux w.r.t. the same cyclic group in the direct product H, another type gives rise to topological interactions between vortices carrying flux w.r.t. two different cyclic factors of H and a third type leading to topological interactions between vortices carrying flux w.r.t. three different cyclic factors. Among other things, it is demonstrated that only the first two types can be obtained from a spontaneously broken U(1)^k CS theory. The 3-cocycles that can not be reached in this way turn out to be the most interesting. They render the theory nonabelian and in general lead to dualities with planar theories with a nonabelian finite gauge group. In particular, the CS theory with finite gauge group H = Z_2 x Z_2 x Z_2 defined by such a 3-cocycle is shown to be dual to the planar discrete D_4 gauge theory with D_4 the dihedral group of order 8. 
  Several gravitational string backgrounds can be interpreted as 2-dim soliton solutions of reduced axion-dilaton gravity. They include black-hole and worm-hole solutions as well as cosmological models with an exact conformal field theory description. We illustrate the use of gravisolitons for the particular example of Nappi-Witten universe which is thus "created" from flat space by soliton dressing. We also make some general comments about the status of gravisolitons in comparison to soliton solutions of other 2-dim integrable systems without gravity. (Contribution to the proceedings of the 2nd International Sakharov Conference, Moscow) 
  We investigate the conformal string $\sigma $-model corresponding to a general five-dimensional non-extremal black hole solution. In the horizon region the theory reduces to an exactly solvable conformal field theory. We determine the modular invariant spectrum of physical string states, which expresses the Rindler momentum operator in terms of three charges and string oscillators. For black holes with winding and Kaluza-Klein charges, we find that states made with only right-moving excitations have ADM mass equal to the black hole ADM mass, and thus they can be used as sources of the gravitational field. A discussion on statistical entropy is included. 
  Two series of integrable theories are constructed which have soliton solutions and can be thought of as generalizations of the sine-Gordon theory. They exhibit internal symmetries and can be described as gauged WZW theories with a potential term. The spectrum of massive states is determined. 
  We present non-extreme generalisations of intersecting p-brane solutions of eleven-dimensional supergravity which upon toroidal compactification reduce to non-extreme static black holes in dimensions D=4, D=5 and 5<D<10, parameterized by four, three and two charges, respectively. The D=4 black holes are obtained either from non-extreme configuration of three intersecting five-branes with a boost along the common string or from non-extreme intersecting system of two two-branes and two five-branes. The D=5 black holes arise from three intersecting two-branes or from a system of intersecting two-brane and five-brane with a boost along the common string. Five-brane and two-brane with a boost along one direction reduce to black holes in D=6 and D=9, respectively, while D=7 black hole can be interpreted in terms of non-extreme configuration of two intersecting two-branes. We discuss the expressions for the corresponding masses and entropies. 
  We construct new superstring actions which are distinguished from standard superstrings by being space-time scale invariant. Like standard superstrings, they are also reparametrization invariant, space-time supersymmetric, and invariant under local scale transformations of the world sheet. We discuss scenarios in which these actions could play a significant role, in particular one which involves their coupling to supersymmetric gauge theories. 
  In this article we derive the Bekenstein-Hawking formula of black hole entropy from a single string. We consider a open string in the Rindler metric which can be obtained in the large mass limit from the Schwarzschild black hole metric. By solving the field equations we find a nontrivial solution with the exact value of the Hawking temperature. We see that this solution gives us the Bekenstein-Hawking formula of black hole entropy to leading order of approximation. This string has effectively a rescaled string tension having a relation with a redshift factor. It is also pointed out that this formalism is extensible to other black holes in an arbitrary spacetime dimension. The present work might lead us to a surprising idea that the recent picture which the black hole entropy arises from D-brane excitations has the root of the picture that the black hole entropy is stocked in a single long string with a rescaled string tension. 
  We reconsider the issue of large-volume compactifications of the heterotic string in light of the recent discoveries about strongly-coupled string theories. Our conclusion remains firmly negative with respect to classical compactifications of the ten-dimensional field theory, albeit for a new reason: When the internal sixfold becomes large in heterotic units, the theory acquires an additional threshold at energies much less then the naive Kaluza-Klein scale. It is this additional threshold that imposes the ultimate limit on the compactification scale: Any compactification must have M_{Kaluza Klein} > 4*10^7 Gev; for most compactifications, the actual limit is much higher. (Generically, M_{Kaluza Klein} > alpha_{GUT} M_{Planck} in either SO(32) or E_8*E_8 heterotic string.) 
  A lower bound is placed on the fermionic determinant of Euclidean quantum electrodynamics in three dimensions in the presence of a smooth, finite--flux, static, unidirectional magnetic field $\mathbf{B}(\mathbf{r})=(0,0,B(\mathbf{r}))$, where $B(\mathbf{r})\geq 0$ or $B(\mathbf{r})\leq 0$ and $\mathbf r$ is a point in the $xy\mbox{-plane}$. Bounds are also obtained for the induced spin for $2+1$ dimensional QED in the presence of $\mathbf{B}(\mathbf{r})$. An upper bound is placed on the fermionic determinant of Euclidean QED in four dimensions in the presence of a strong, static, directionally-varying, square-integrable magnetic field $\mathbf{B}(\mathbf{r})$ on $\R^3 $. 
  The energy density is computed for a U(2) Chern-Simons theory coupled to a non-relativistic fermion field (a theory of ``non-Abelian anyons'') under the assumptions of uniform charge and matter density. When the matter field is a spinless fermion, we find that this energy is independent of the two Chern-Simons coupling constants and is minimized when the non-Abelian charge density is zero. This suggests that there is no spontaneous breaking of the SU(2) subgroup of the symmetry, at least in this mean-field approximation. For spin-1/2 fermions, we find self-consistent mean-field states with a small non-Abelian charge density, which vanishes as the theory of free fermions is approached. 
  We argue that the behavior of string theory at high temperature and high longitudinal boosts, combined with the emergence of p-branes as necessary ingredients in various string dualities, point to a possible reformulation of strings, as well as p-branes, as composites of bits. We review the string-bit models, and suggest generalizations to incorporate p-branes. 
  It is argued that every Calabi-Yau manifold $X$ with a mirror $Y$ admits a family of supersymmetric toroidal 3-cycles. Moreover the moduli space of such cycles together with their flat connections is precisely the space $Y$. The mirror transformation is equivalent to T-duality on the 3-cycles. The geometry of moduli space is addressed in a general framework. Several examples are discussed. 
  Kinnersley--type representation is constructed for the four--dimensional Einstein--Maxwell--dilaton--axion system restricted to space--times possessing two non--null commuting Killing symmetries. New representation essentially uses the matrix--valued $SL(2,R)$ formulation and effectively reduces the construction of the Geroch group to the corresponding problem for the vacuum Einstein equations. An infinite hierarchy of potentials is introduced in terms of $2\times 2$ real symmetric matrices generalizing the scalar hierarchy of Kinnersley--Chitre known for the vacuum Einstein equations. 
  The four-dimensional bosonic effective action of the toroidally compactified heterotic string incorporating a dilaton, an axion and one $U(1)$ vector field is studied on curved space-time manifolds with one and two commuting Killing vectors. In the first case the theory is reduced to a three-dimensional sigma model possessing a symmetric pseudoriemannian target space isomorphic to the coset $SO(2,3)/(SO(3)\times SO(2))$. The ten-parameter group $SO(2,3)$ of target space isometries contains embedded both $S$ and $T$ classical duality symmetries of the heterotic string. With one more ignorable coordinate, the theory reduces to a two-dimensional chiral model built on the above coset, and therefore belongs to the class of completely integrable systems. This entails infinite-dimensional symmetries of the Geroch--Kinnersley--Chitre type. Purely dilatonic theory is shown to be two-dimensionally integrable only for two particular values of the dilaton coupling constant. In the static case (diagonal metrics) both theories essentially coincide; in this case the integrability property holds for all values of the dilaton coupling. 
  We propose a unified scheme for finding the hyperelliptic curve of $N=2$ SUSY YM theory with any Lie gauge groups. Our general scheme gives the well known results for classical gauge groups and exceptional $G_2$ group. In particular, we present the curve for the exceptional gauge groups $F_4, E_{6,7,8}$ and check consistency condition for them. The exact monodromies and the dyon spectrum of these theories are determined. We note that for any Lie gauge groups, the exact monodromies could be obtained only from the Cartan matrix. 
  The subspace of the moduli space of F-theory on K3 over which the coupling remains constant develops new branches at special values of this coupling. These values correspond to fixed points under the SL(2,Z) duality group of the type IIB string. The branches contain points where K3 degenerates to orbifolds of the four-torus by Z_3,Z_4 and Z_6. A singularity analysis shows that exceptional group symmetries appear on these branches, including pure E_8 xE_8, although SO(32) cannot be realised in this way. The orbifold points can be mapped to a kind of non-perturbative generalization of a IIB orientifold, and to M-theory orbifolds with non-trivial action on 2-brane wrapping modes. 
  A general criterion is given for the vanishing of the beta-functions in N=1 supersymmetric gauge theories. 
  We provide an axiomatic framework for Quantum Field Theory at finite temperature which implies the existence of general analyticity properties of the $ n $-point functions; the latter parallel the properties derived from the usual Wightman axioms in the vacuum representation of Quantum Field Theory. Complete results are given for the propagators, including a generalization of the K\"all\'en-Lehmann representation. Some known examples of ``hard-thermal-loop calculations'' and the representation of ``quasiparticles'' are discussed in this general framework. 
  Supersymmetric $p$-branes that carry a single electric or magnetic charge and preserve $1/2$ of the supersymmetry have been interpreted as the constituents from which all supersymmetric $p$-branes can be constructed as bound states, albeit with zero binding energy. Here we extend the discussion to non-supersymmetric $p$-branes, and argue that they also can be interpreted as bound states of the same basic supersymmetric constituents. In general, the binding energy is non-zero, and can be either positive or negative depending on the specific choice of constituents. In particular, we find that the $a=0$ Reissner-Nordstr\o{m} black hole in $D=4$ can be built from different sets of constituents such that it has zero, positive or negative binding energy. 
  Several complications arise in quantum field theory because of the infinite many degrees of freedom. However, the distinction between one-particle and many-particle effects -- mainly induced by the vacuum -- is not clear up to now. A field theoretic picture of the one-particle Dirac theory is developed in order to explore such questions. Main emphasis is laid on the injection of Grassmann's algebra into the endomorphism Clifford algebra built over it. The obtained ``field theoretic'' functional equation behaves in a very unusual way. New methods to handle Dirac and QFT are given. 
  Using the heterotic--type II duality of $N=2$ string vacua in four space-time dimensions we study non-perturbative couplings of toroidally compactified six-dimensional heterotic vacua. In particular, the heterotic--heterotic $S$-duality and the Coulomb branch of tensor multiplets observed in six dimensions are studied from a four-dimensional point of view. We explicitly compute the couplings of the vector multiplets of several type II vacua and investigate the implications for their heterotic duals. 
  We analyse the fusion of representations of the triplet algebra, the maximally extended symmetry algebra of the Virasoro algebra at c=-2. It is shown that there exists a finite number of representations which are closed under fusion. These include all irreducible representations, but also some reducible representations which appear as indecomposable components in fusion products. 
  We study, for the first time, the phase structure of the Gross--Neveu model with a combination of a (constant) gravitational and a magnetic field. This has been made possible by our finding of an exact solution to the problem, namely the effective potential for the composite fermions. Then, from the corresponding implicit equation the phase diagram for the dynamical fermion mass is calculated numerically for some values of the magnetic field. %(what can be done with arbitrary precision). For a small magnetic field the phase diagram hints to the possibility of a second order phase transition at some critical curvature. With growing magnetic field only the phase with broken chiral symmetry survives, because the magnetic field prevents the decay of the chiral condensate. This result is bound to have important consequences in early universe cosmology. 
  We present the full calculation of the divergent one-loop contribution to the effective boson Lagrangian for supergravity, including the Yang-Mills sector and the helicity-odd operators that arise from integration over fermion fields. The only restriction is on the Yang-Mills kinetic energy normalization function, which is taken diagonal in gauge indices, as in models obtained from superstrings. 
  A precise definition of an adiabaticity parameter $\nu$ of a time-dependent Hamiltonian is proposed. A variation of the time-dependent perturbation theory is presented which yields a series expansion of the evolution operator $U(\tau)=\sum_\ell U^{(\ell)}(\tau)$ with $U^{(\ell)}(\tau)$ being at least of the order $\nu^\ell$. In particular $U^{(0)}(\tau)$ corresponds to the adiabatic approximation and yields Berry's adiabatic phase. It is shown that this series expansion has nothing to do with the $1/\tau$-expansion of $U(\tau)$. It is also shown that the non-adiabatic part of the evolution operator is generated by a transformed Hamiltonian which is off-diagonal in the eigenbasis of the initial Hamiltonian. Some related issues concerning the geometric phase are also discussed. 
  We make an all-order analysis to establish the precise correspondence between nonrelativistic Chern-Simons quantum field theory and an appropriate first-quantized description. Physical role of the field-theoretic contact term in the context of renormalized perturbation theory is clarifed through their connection to self-adjoint extension of the Hamiltonian in the first-quantized approach. Our analysis provides a firm theoretical foundation on quantum field theories of nonrelativistic anyons. 
  We study 3-point functions at finite temperature in the closed time path formalism. We give a general decomposition of the eight component tensor in terms of seven vertex functions. We derive a spectral representation for these seven functions in terms of two independent real spectral functions. We derive relationships between the seven functions and obtain a representation of the vertex tensor that greatly simplifies calculations in real time. 
  A universal formula for an action associated with a noncommutative geometry, defined by a spectal triple $(\Ac ,\Hc ,D)$, is proposed. It is based on the spectrum of the Dirac operator and is a geometric invariant. The new symmetry principle is the automorphism of the algebra $\Ac $ which combines both diffeomorphisms and internal symmetries. Applying this to the geometry defined by the spectrum of the standard model gives an action that unifies gravity with the standard model at a very high energy scale. 
  String duality requires the presence of solitonic $p$-branes. By contrast, the existence of fundamental supermembranes is problematic, since they are probably unstable. In this paper, we re-examine the quantum stability of fundamental supermembranes in 11 dimensions. Previously, supermembranes were shown to be unstable by approximating them with SU(n) super Yang-Mills fields as $n \rightarrow \infty$. We show that this instability persists even if we quantize the continuum theory from the very beginning. Second, we speculate how a theory of decaying membranes may still be physically relevant. We present some heuristic arguments showing that light membranes may be too short-lived to be detected, while very massive membranes, with longer lifetimes, may have only very weak coupling to the particles we see in nature. Either way, decaying membranes would not be detectable in the lab. (This article is to be published in Frontiers in Quantum Field Theory (World Scientific) dedicated to Keiji Kikkawa's 60th birthday.) 
  We find a general p-1-brane solution to supergravity coupled to a p+1-form field strength using the ``standard ansatz'' for the fields. In addition to the well-known elementary and solitonic p-1-brane solutions, which are the only ones preserving half of the supersymmetry (for the cases where the supersymmetry transformations are known), there are other possibilities, the most interesting of which is an elementary type~I string solution with a non-dynamical string source and no conserved charge. 
  We describe the recently introduced method of algebraic bosonization of the $(1+1)$-dimensional Luttinger systems by discussing in detail the specific case of the Calogero-Sutherland model, and mentioning the hard-core Bose gas. We also compare our findings with the exact Bethe Ansatz results. 
  Considering the linearized gravity with matter fields, the effective potential of the ``conformal dilaton'' in the string frame is generated semiclassically by one-loop contribution of heavy matter fields. This in turn generates a nontrivial potential for the physical dilaton in the Einstein frame with the trace of the graviton in the Einstein frame gauged away. The remaining manifest local spacetime symmetry is only the volume preserving diffeomorphism symmetry. The consistency of this procedure is examined and the possibility of spontaneous diffeomorphism symmetry breaking is suggested. 
  We provide an action for self-dual Yang-Mills theory which is a simple truncation of the usual Yang-Mills action. Only vertices that violate helicity conservation maximally are included. One-loop amplitudes in the self-dual theory then follow as a subset of the Yang-Mills ones. In light-cone gauges this action is almost identical to previously proposed actions, but in this formulation the vanishing of all higher-loop amplitudes is obvious; the explicit perturbative S-matrix is known. Similar results apply to gravity. 
  By promoting an invariant subgroup $H$ of $ISO(2,1)$ to a gauge symmetry of a WZWN action, we obtain the description of a bosonic string moving either in a curved 4-dimensional space--time with an axion field and curvature singularities or in 3-dimensional Minkowski space--time. 
  By defining a prepotential function for the stationary Schr\"odinger equation we derive an inversion formula for the space variable $x$ as a function of the wave-function $\psi$. The resulting equation is a Legendre transform that relates $x$, the prepotential ${\cal F}$, and the probability density. We invert the Schr\"odinger equation to a third-order differential equation for ${\cal F}$ and observe that the inversion procedure implies a $x$-$\psi$ duality. This phenomenon is related to a modular symmetry due to the superposition of the solutions of the Schr\"odinger equation. We propose that in quantum mechanics the space coordinate can be interpreted as a macroscopic variable of a statistical system with $\hbar$ playing the role of a scaling parameter. We show that the scaling property of the space coordinate with respect to $\tau=\partial_{\psi}^2{\cal F}$ is determined by the ``beta-function''. We propose that the quantization of the inversion formula is a natural way to quantize geometry. The formalism is extended to higher dimensions and to the Klein-Gordon equation. 
  A quantum mechanical system describing bosons in one space dimension with a kinetic energy of arbitrary order in derivatives and a delta function interaction is studied. Exact wavefunctions for an arbitrary number of particles and order of derivative are constructed. Also, equations determining the spectrum of eigenvalues are found. 
  There exists a simple rule by which path integrals for the motion of a point particle in a flat space can be transformed correctly into those in curved space. This rule arose from well-established methods in the theory of plastic deformations, where crystals with defects are described mathematically by applying nonholonomic coordinate transformations to ideal crystals. In the context of time-sliced path integrals, this has given rise to a {\em quantum equivalence principle\/} which determines the measure of fluctating orbits in spaces with curvature and torsion. The nonholonomic transformations are accompanied by a nontrivial Jacobian which in curved spaces produces an additional energy proportional to the curvature scalar, thereby canceling an equal term found earlier by DeWitt from a naive formulation of Feynman's time-sliced path integral in curved space. The importance of this cancelation has been documented in various systems (H-atom, particle on the surface of a sphere, spinning top). Here we point out its relevance in the process of bosonizing a nonabelian one-dimensional quantum field theory, whose fields live in a flat field space. Its bosonized version is a quantum-mechanical path integral of a point particle moving in a space with constant curvature. The additional term introduced by the Jacobian is crucial for the identity between original and bosonized theory.   A useful bozonization tool is the so-called Hubbard-Stratonovich formula for which we find a nonabelian version. 
  The solutions of a large class of hierarchies of zero-curvature equations that includes Toda and KdV type hierarchies are investigated. All these hierarchies are constructed from affine (twisted or untwisted) Kac-Moody algebras~$\ggg$. Their common feature is that they have some special ``vacuum solutions'' corresponding to Lax operators lying in some abelian (up to the central term) subalgebra of~$\ggg$; in some interesting cases such subalgebras are of the Heisenberg type. Using the dressing transformation method, the solutions in the orbit of those vacuum solutions are constructed in a uniform way. Then, the generalized tau-functions for those hierarchies are defined as an alternative set of variables corresponding to certain matrix elements evaluated in the integrable highest-weight representations of~$\ggg$. Such definition of tau-functions applies for any level of the representation, and it is independent of its realization (vertex operator or not). The particular important cases of generalized mKdV and KdV hierarchies as well as the abelian and non abelian affine Toda theories are discussed in detail. 
  We use the Cutkosky rules as a tool for determining the infinities present in graviton scattering amplitudes. We are able to confirm theoretical derivations of counterterms in Einstein-Maxwell theory and to determine new results in the Dirac-Einstein counter-Lagrangian. 
  The form factor bootstrap approach allows to construct the space of local fields in the massive restricted sine-Gordon model. This space has to be isomorphic to that of the corresponding minimal model of conformal field theory. We describe the subspaces which correspond to the Verma modules of primary fields in terms of the commutative algebra of local integrals of motion and of a fermion (Neveu-Schwarz or Ramond depending on the particular primary field). The description of null-vectors relies on the relation between form factors and deformed hyper-elliptic integrals. The null-vectors correspond to the deformed exact forms and to the deformed Riemann bilinear identity. In the operator language, the null-vectors are created by the action of two operators $\CQ$ (linear in the fermion) and $\CC$ (quadratic in the fermion). We show that by factorizing out the null-vectors one gets the space of operators with the correct character. In the classical limit, using the operators $\CQ$ and $\CC$ we obtain a new, very compact, description of the KdV hierarchy. We also discuss a beautiful relation with the method of Whitham. 
  This work describes new perturbative solutions to the classical, four-dimensional Kerr--Newman dilaton black hole field equations. Our solutions do not require the black hole to be slowly rotating. The unperturbed solution is taken to be the ordinary Kerr solution, and the perturbation parameter is effectively the square of the charge-to-mass ratio $(Q/M)^2$ of the Kerr--Newman black hole. We have uncovered a new, exact conjugation (mirror) symmetry for the theory, which maps the small coupling sector to the strong coupling sector ($\phi \to -\phi$). We also calculate the gyromagnetic ratio of the black hole. 
  We analyse the critical behaviour of anomalous currents in N=1 four-dimensional supersymmetric gauge theories in the context of electric-magnetic duality. We show that the anomalous dimension of the Konishi superfield is related to the slope of the beta function at the critical point. We construct a duality map for the Konishi current in the minimal SQCD. As a byproduct we compute the slope of the beta function in the strong coupling regime. We note that the OPE of the stress tensor with itself does not close, but mixes with the Konishi operator. As a result in superconformal theories in four dimensions (SCFT$_4$) there are {\sl two} central charges; they allow us to count both the vector multiplet and the matter multiplet effective degrees of freedom. Some applications to N=4 SYM are discussed. 
  Using the simple path integral method we calculate the $n$-point functions of field strength of Yang-Mills theories on arbitrary two-dimensional Riemann surfaces. In $U(1)$ case we show that the correlators consist of two parts , a free and an $x$-independent part. In the case of non-abelian semisimple compact gauge groups we find the non-gauge invariant correlators in Schwinger-Fock gauge and show that it is also divided to a free and an almost $x$-independent part. We also find the gauge-invariant Green functions and show that they correspond to a free field theory. 
  We apply here a recently developed approach to compute the short distance corrections to scaling for the correlators of all primary operators of the critical two dimensional Ising model in a magnetic field. The essence of the method is the fact that if one deals with O.P.E. Wilson coefficients instead of correlators, all order I.R. safe formulas can be obtained for the perturbative expansion with respect to magnetic field. This approach yields in a natural way the expected fractional powers of the magnetic field, that are clearly absent in the naive perturbative expression for correlators. The technique of the Mellin transform have been used to compute the I.R. behavior of the regularized integrals. As a corollary of our results, by comparing the existing numerical data for the lattice model we give an estimate of the Vacuum Expectation Value of the energy operator, left unfixed by usual nonperturbative approaches (Thermodynamic Bethe Ansatz). 
  The scalars in vector multiplets of N=2 supersymmetric theories in 4 dimensions exhibit `special Kaehler geometry', related to duality symmetries, due to their coupling to the vectors. In the literature there is some confusion on the definition of special geometry, which we want to clear up, for rigid as well as for local N=2. 
  We develop a new operator quantization scheme for gauge theories where no gauge fixing for gauge fields is needed. The scheme allows one to avoid the Gribov problem and construct a manifestly Lorentz invariant path integral that can be used in the non-perturbative domain. We discuss briefly an application of the method to Abelian projections of QCD. 
  The generating functionals for the local composite operators, $\Phi^2(x)$ and $\Phi^4(x)$, are used to study excitations in the scalar quantum field theory with $\lambda \Phi^4$ interaction. The effective action for the composite operators is obtained as a series in the Planck constant $\hbar$, and the two- and four-particle propagators are derived. The numerical results are studied in the space-time of one dimension, when the theory is equivalent to the quantum mechanics of an anharmonic oscillator. The effective potential and the poles of the composite propagators are obtained as series in $\hbar$, with effective mass and coupling determined by non-perturbative gap equations. This provides a systematic approximation method for the ground state energy, and for the second and fourth excitations. The results show quick convergence to the exact values, better than that obtained without including the operator $\Phi^4$. 
  We present an N=2 multiplet including a three-index antisymmetric tensor gauge potential, and describe it as a solution to the Bianchi identities for the associated fieldstrength superform, subject to some covariant constraints, in extended central charge superspace. We find that this solution is given in terms of an 8+8 tensor multiplet subject to an additional constraint. We give the transformation laws for the multiplet as well as invariant superfield and component field lagrangians, and mention possible couplings to other multiplets. We also allude to the relevance of the 3--form geometry for generic invariant supergravity actions. 
  We describe an infinite-dimensional algebra of hidden symmetries for the self-dual gravity equations. Besides the known diffeomorphism-type symmetries (affine extension of w(infinity) algebra), this algebra contains new hidden symmetries, which are an affine extension of the Lorentz rotations. The full symmetry algebra has both Kac-Moody and Virasoro-like generators, whose exponentiation maps solutions of the field equations to other solutions. Relations to problems of string theories are briefly discussed. 
  We search a canonical basis of Dirac's observables for the classical Abelian Higgs model with fermions in the case of a trivial U(1) principal bundle. The study of the Gauss law first class constraint shows that the model has two disjoint sectors of solutions associated with two physically different phases. In the electromagnetic phase, the electromagnetic field remains massless: after the determination of the Dirac's observables we get that both the reduced physical Hamiltonian and Lagrangian are nonlocal. In the Higgs phase, the electromagnetic field becomes massive and in terms of Dirac's observables we get a local, but nonanalytic in the electric charge (or equivalently in the sum of the electromagnetic mass and of the residual Higgs field), physical Hamiltonian; however the associated Lagrangian is nonlocal. Some comments on the R-gauge-fixing, the possible elimination of the residual Higgs field and on the Nielsen-Olesen vortex solution close the paper. 
  We search a canonical basis of Dirac's observables for the classical non-Abelian Higgs model with fermions in the case of a trivial SU(2) principal bundle with a complex doublet of Higgs fields and with the fermions in a given representation of SU(2). Since each one of the three Gauss law first class constraints can be solved either in the corresponding longitudinal electric field or in the corresponding Higgs momentum, we get a priori eight disjoint phases of solutions of the model. The only two phases with SU(2) covariance are the SU(2) phase with massless SU(2) fields and the Higgs phase with massive SU(2) fields. The Dirac observables and the reduced physical (local) Hamiltonian and (nonlocal) Lagrangian of the Higgs phase are evaluated: the main result is the nonanalyticity in the SU(2) coupling constant, or equivalently in the sum of the residual Higgs field and of the mass of the SU(2) fields. Some comments on the function spaces needed for the gauge fields are made. 
  The heterotic and type II superstring actions are identified in different anomaly-free decompositions of a single topological sigma-model action depending on bosonic and fermionic coordinates, $X^\mu$ and $\r^A$ respectively, and of their topological ghosts. This model results from gauge-fixing the topological gauge symmetry $\delta X^\mu = \epsilon^\mu (z,\bar z)$ ($\mu =1,2,\dots, 10$) and $\delta \r^\alpha= \epsilon^\alpha (z,\bar z)$. ($\alpha=1,2.\dots, 16$). From another viewpoint the heterotic and type II superstring actions emerge as two different gauge-fixings of the same closed two-form. Comments are also made concerning the possibility of relating $\rho^\alpha$ to a Majorana-Weyl space-time spinor superpartner of $X^\mu$. 
  We compute the contribution to the modulus of the one-loop effective action in pure non-Abelian Chern-Simons theory in an arbitrary covariant gauge. We find that the results are dependent on both the gauge parameter ($\alpha$) and the metric required in the gauge fixing. A contribution arises that has not been previously encountered; it is of the form $(\alpha / \sqrt{p^2}) \epsilon _{\mu \lambda \nu} p^\lambda$. This is possible as in three dimensions $\alpha$ is dimensionful. A variant of proper time regularization is used to render these integrals well behaved (although no divergences occur when the regularization is turned off at the end of the calculation). Since the original Lagrangian is unaltered in this approach, no symmetries of the classical theory are explicitly broken and $\epsilon_{\mu \lambda \nu}$ is handled unambiguously since the system is three dimensional at all stages of the calculation. The results are shown to be consistent with the so-called Nielsen identities which predict the explicit gauge parameter dependence using an extension of BRS symmetry. We demonstrate that this $\alpha$ dependence may potentially contribute to the vacuum expectation values of products of Wilson loops. 
  We investigate higher-dimensional analogues of the $bc$ systems of 2D RCFT. When coupled to gauge fields and Beltrami differentials defining integrable holomorphic structures the $bc$ partition functions can be explicitly evaluated using anomalies and holomorphy. The resulting induced actions generalize the chiral algebras of 2D RCFT to $2n$ dimensions. Moreover, $bc$ systems in four and six dimensions are closely related to supersymmetric matter. In particular, we show that $d=4, \CN=2$ hypermultiplets induce a theory of self-dual Yang-Mills fields coupled to self-dual gravity. In this way the $bc$ systems fermionize both the algebraic sector of the $WZW_4$ theory, as defined by Losev et. al., and the classical open $\CN_{ws}=2$ string. 
  We consider supersymmetrization of Hamiltonian dynamics via antibrackets for systems whose Hamiltonian generates an isometry of the phase space. We find that the models are closely related to the supersymmetric non-linear $\sigma$-model. We interpret the corres\-ponding path integrals in terms of super loop space equivariant cohomology. It turns out that they can be evaluated exactly using localizations techniques. 
  A general definition of the curves and geodesics associated with a given connection on a quantized manifold is given. In the particular case of the functional quantization we define geodesics in the same way as in the classical case and we will show that the two definitions are compatible. As an example we examine our results for the quantum plane. 
  After summarising the physical approach leading to twisted homotopy and after developing the cohomological approach further with respect to our previous work we propose a third alternative approach to twisted homotopy based on group theoretic considerations. In this approach the fundamental group $\Pi (m) $ isomorphic to Z which describes homotopic loops on the punctured plane$ R^2/(0) $ is enhanced in a special way to the continuous SO(2) group . This is performed by letting the parameter of the group $ m \rightarrow \lambda $ while keeping its generator unchanged .It is shown that such non-trivial procedure has the effect of introducing well defined self-interactions among loops which are at the basis of twisted homotopy where the angle $ \lambda $ plays the role of the self coupling constant.  KEYWORDS: Homotopy, Group Theory, Quantum Mechanics  MSC:55Q35;  PACS:02.20.Fh ; 03.65.Fd 
  We provide a local geometric description of how charged matter arises in type IIA, M-theory, or F-theory compactifications on Calabi-Yau manifolds. The basic idea is to deform a higher singularity into a lower one through Cartan deformations which vary over space. The results agree with expectations based on string dualities. 
  We investigate heterotic ground states in four dimensions in which N=4 supersymmetry is spontaneously broken to N=2. N=4 supersymmetry is restored at a decompactification limit corresponding to $m_{3/2}\to 0$. We calculate the full moduli dependent threshold corrections and confirm that they are supressed in the decompactification limit $m_{3/2}\to 0$ as expected from the restoration of N=4 supersymmetry. This should be contrasted with the behavior of the standard N=2 groundstates where the coupling blow up linearly with the volume of the decompactifying manifold. This mechanism provides a solution to the decompactification problem for the gauge coupling constants.  We also discuss how the mechanism can be implemented in ground states with lower supersymmetry. 
  The shift operator for a quantum lattice current algebra associated with sl(2) is produced in the form of product of local factors. This gives a natural deformation of the Sugawara construction for discrete space-time. 
  We complete the program outlined in the paper of the author with A. Migdal and sum up exactly all the fluctuations around the instanton solution of the randomly large scale driven Burgers equation. We choose the force correlation function $\kappa$ to be exactly quadratic function of the coordinate difference. The resulting probability distribution satisfy the differential equation proposed by Polyakov without an anomaly term. The result shows that unless the anomaly term is indeed absent it must come from other possible instanton solutions, and not from the fluctuations. 
  We give generalizations of extended Poincar\'e supergravity with {\it arbitrarily many} supersymmetries in the absence of central charges in three-dimensions by gauging its intrinsic global $~SO(N)$~ symmetry. We call these \alephnull (Aleph-Null) supergravity theories. We further couple a non-Abelian supersymmetric Chern-Simons theory and an Abelian topological BF theory to \alephnull supergravity. Our result overcomes the previous difficulty for supersymmetrization of Chern-Simons theories beyond $~N=4$. This feature is peculiar to the Chern-Simons and BF theories including supergravity in three-dimensions. We also show that dimensional reduction schemes for four-dimensional theories such as $~N=1$~ self-dual supersymmetric Yang-Mills theory or $~N=1$~ supergravity theory that can generate \alephnull globally and locally supersymmetric theories in three-dimensions. As an interesting application, we present \alephnull supergravity Liouville theory in two-dimensions after appropriate dimensional reduction from three-dimensions. 
  Coupling of bilinears of gauginos on D-branes to anti-symmetric tensor fields can be deduced by comparing F-theory compactified to 8 dimensions with the heterotic string on $R^8\times T^2$. Application to SUSY breaking in F-theory triggered by gaugino condensation is discussed. Compactification to four dimensions with a warp factor is briefly discussed. In this case some 3-branes whose world-volume coincides with the open spacetime must be introduced, in addition to 7-branes. 
  Explicit examples of quasi-exactly-solvable $N$-body problems on the line are presented. These are related to the hidden algebra $sl_N$, and they are of two types -- containing up to $N$ (infinitely-many eigenstates are known, but not all) and up to 6 body interactions only (a finite number of eigenstates is known). Both types degenerate to the Calogero model. 
  BPS monopoles in N=2 SUSY theories may lead to monopole condensation and confinement. We have found that supersymmetric black holes with non-vanishing area of the horizon may stabilize the moduli in theories where the potential is proportional to the square of the graviphoton central charge. In particular, in known models of spontaneous breaking of N=2 to N=1 SUSY theories, the parameters of the electric and magnetic Fayet--Iliopoulos terms can be considered proportional to electric and magnetic charges of the dyonic black holes. Upon such identification the potential is found to be proportional to the square of the black hole mass. The fixed values of the moduli near the black hole horizon correspond exactly to the minimum of this potential. The value of the potential at the minimum is proportional to the black hole entropy. 
  Using a field theory generalization of the spinning top motion, we construct nonabelian generalizations of the sine-Gordon theory according to each symmetric spaces. A Lagrangian formulation of these generalized sine-Gordon theories is given in terms of a deformed gauged Wess-Zumino-Witten action which also accounts for integrably perturbed coset conformal field theories. As for physical applications, we show that they become precisely the effective field theories of self-induced transparency in nonlinear optics. This provides a dictionary between field theory and nonlinear optics. 
  We present a free boson realization of the vertex operators and their duals for the solvable SOS lattice model of $A^{(1)}_{n-1}$ type. We discuss a possible connection to the calculation of the correlation functions. 
  Actual calculations of monopole and dyon spectra have previously been performed in N=4 SYM and in N=2 SYM with gauge group SU(2), and are in total agreement with duality conjectures for the finite theories. These calculations are extended to N=2 SYM with higher rank gauge groups, and it turns out that the SU(2) model with four fundamental hypermultiplet is an exception in that its soliton spectrum supports duality. This may be an indication that the other perturbatively finite N=2 theories have non-perturbative contributions to the beta-function. This talk contains a short summary of recent results. 
  We study finite-dimensional extra symmetries of generic 2D dilaton gravity models. Using a non-linear sigma model formulation we show that the unique theories admitting an extra (conformal) symmetry are the models with an exponential potential $V \propto e^{\beta\phi}$ ($ S ={1\over2\pi} \int d^2 x \sqrt{-g} [ R \phi + 4 \lambda^2 e^{\beta\phi} ]$), which include the CGHS model as a particular though limiting ($\beta=0$) case. These models give rise to black hole solutions with a mass-dependent temperature. The underlying extra symmetry can be maintained in a natural way in the one-loop effective action, thus implying the exact solubility of the semiclassical theory including back-reaction. Moreover, we also introduce three different classes of (non-conformal) transformations which are extra symmetries for generic 2D dilaton gravity models. Special linear combinations of these transformations turn out to be the (conformal) symmetries of the CGHS and $V \propto e^{\beta\phi}$ models. We show that one of the non-conformal extra symmetries can be converted into a conformal one by means of adequate field redefinitions involving the metric and the derivatives of the dilaton. Finally, by expressing the Polyakov-Liouville effective action in terms of an invariant metric, we are able to provide semiclassical models which are also invariant. This generalizes the solvable semiclassical model of Bose, Parker and Peleg (BPP) for a generic 2D dilaton gravity model. 
  We calculate the two-loop quantum corrections, including the back-reaction of the Hawking radiation, to the one-loop effective metric in a unitary gauge quantization of the CGHS model of 2d dilaton gravity. The corresponding evaporating black hole solutions are analysed, and consistent semi-classical geometries appear in the weak-coupling region of the spacetime when the width of the matter pulse is larger then the short-distance cutoff. A consistent semi-classical geometry also appears in the limit of a shock-wave matter. The Hawking radiation flux receives non-thermal corrections such that it vanishes for late times and the total radiated mass is finite. There are no static remnants for matter pulses of finite width, although a BPP type static remnant appears in the shock-wave limit. Semi-classical geometries without curvature singularities can be obtained as well. Our results indicate that higher-order loop corrections can remove the singularities encountered in the one-loop solutions. 
  We prove the universality of correlation functions of chiral complex matrix models in the microscopic limit (N->\infty, z->0, N z=fixed) which magnifies the crossover region around the origin of the eigenvalue distribution. The proof exploits the fact that the three-term difference equation for orthogonal polynomials reduces into a universal second-order differential (Bessel) equation in the microscopic limit. 
  We consider quantized Yang-Mills theories in the framework of causal perturbation theory which goes back to Epstein and Glaser. In this approach gauge invariance is expressed by a simple commutator relation for the S-matrix. The most general coupling which is gauge invariant in first order contains a two-parametric ambiguity in the ghost sector - a divergence- and a coboundary-coupling may be added. We prove (not completely) that the higher orders with these two additional couplings are gauge invariant, too. Moreover we show that the ambiguities of the n-point distributions restricted to the physical subspace are only a sum of divergences (in the sense of vector analysis). It turns out that the theory without divergence- and coboundary-coupling is the most simple one in a quite technical sense. The proofs for the n-point distributions containing coboundary-couplings are given up to third or fourth order only, whereas the statements about the divergence-coupling are proven in all orders. 
  The Boyer-Finley equation, or $SU(\infty)$-Toda equation is both a reduction of the self-dual Einstein equations and the dispersionlesslimit of the $2d$-Toda lattice equation. This suggests that there should be a dispersive version of the self-dual Einstein equation which both contains the Toda lattice equation and whose dispersionless limit is the familiar self-dual Einstein equation. Such a system is studied in this paper. The results are achieved by using a deformation, based on an associative $\star$-product, of the algebra $sdiff(\Sigma^2)$ used in the study of the undeformed, or dispersionless, equations. 
  We construct the pair of logarithmic operators associated with the recoil of a $D$-brane. This construction establishes a connection between a translation in time and a world-sheet rescaling. The problem of measuring the centre of mass coordinate of the $D$-brane is considered and the relation between the string uncertainty principle and the logarithmic operators is discussed. 
  We discuss the renormalization group flow, duality, and supersymmetry breaking in N = 1 supersymmetric SU(N)xSU(M) gauge theories. 
  We derive a new non-abelian Stokes theorem by rewriting the Wilson loop as a gauge-invariant area integral, at the price of integrating over an auxiliary field from the coset SU(N) / [U(1)]^{N-1} space. We then introduce the relativistic quark--monopole interaction as a Wess--Zumino-type action, and extend it to the non-abelian case. We show that condensation of monopoles and confinement can be investigated in terms of the behaviour of the monopole world lines. One can thus avoid hard problems of how to introduce monopole fields and dual Yang--Mills potentials. 
  We continue the investigation of quantized Yang-Mills theories coupled to matter fields in the framework of causal perturbation theory which goes back to Epstein and Glaser. In this approach gauge invariance is expressed by a simple commutator relation for the S-matrix and the corresponding gauge transformations are simple transformations of the free fields only. In spite of this simplicity, gauge invariance implies the usual Slavnov-Taylor identities. The main purpose of this paper is to prove the latter statement. Since the Slavnov-Taylor identities are formulated in terms of Green's functions, we investigate the agreement of two perturbative definitions of Green's functions, namely of Epstein and Glaser's definition with the Gell-Mann Low series. 
  It is generally admitted that gravitational interactions become large at an invariant distance of order $1$ from the black hole horizon. We show that due to the ``atmosphere'' of high angular particles near the horizon strong gravitational interactions already occur at an invariant distance of the order of $\sqrt[3]{M}$. The implications of these results for the origin of black hole radiation, the meaning of black hole entropy and the information puzzle are discussed. 
  Utilizing (2,0) superfields, we write a supersymmetry-squared action and partially relate it to the new formulation of the Green-Schwarz action given by Berkovits and Siegel. Recent results derived from this new formulation are discussed within the context of some prior proposals in the literature. Among these, we note that 4D, N = 1 beta-FFC superspace geometry with a composite connection for R-symmetry has now been confirmed as the only presently known limit of 4D, N = 1 heterotic string theory that is derivable in a completely rigorous manner. 
  Extending prior investigations, we study three of the the four distinct minimal (4,0) scalar multiplets coupled to (4,0) supergravity. It is found that the scalar multiplets manifest their differences at the component level by possessing totally different couplings to the supergravity fields. Only the SM-I multiplet possesses a conformal coupling. For the remaining multiplets, terms linear in the world sheet curvature and/or SU(2) gauge field strengths are required to appear in the action by local supersymmetry. 
  Within a four dimensional N = 1 superspace, we present a new ansatz for the Skyrme term and for the gauged WZNW term embedded into a superaction. We use the new chiral-nonminimal (CNM) formulation for the effective low-energy action of 4D, N = 1 supersymmetric QCD constructed by assigning right-handed components of Dirac fields to chiral multiplets and left-handed components of Dirac fields to nonminimal multiplets. It is noted that such a construction likely allows for a new type of parity violation in low-energy 4D, N = 1 supersymmetric QCD. 
  We consider theories with gauged chiral fermions in which there are abelian anomalies, and no nonabelian anomalies (but there may be nonabelian gauge fields present). We construct an associated theory that is gauge-invariant, renormalizable, and with the same particle content, by adding a finite number of terms to the action. Alternatively one can view the new theory as arising from the original theory by using another regularization, one that is gauge-invariant. The situation is reminiscent of the mechanism of adding Fadeev-Popov ghosts to an unsatisfactory gauge theory, to arrive at the usual quantization procedure. The models developed herein are much like the abelian Wess-Zumino model (an abelian effective theory with a Wess-Zumino counterterm), but unlike the W-Z model are renormalizable!   Details of the approach are worked out explicitly for the special case of a single massless Dirac fermion, for which we couple one abelian gauge field to the vector current and another abelian gauge field to the axial current. 
  In these notes, we emphasize the r\^ole of spontaneous broken global discrete symmetries acting on the moduli space of $N=2$ susy Yang-Mills theories and show how they can be used, together with the BPS condition, as a spectrum generating symmetry. In particular, in the strong-coupling region, all BPS states come in multiplets of this broken symmetry. This played a key r\^ole in the determination of the strong-coupling spectra. 
  We study the boundary states of D-branes wrapped around supersymmetric cycles in a general Calabi-Yau manifold. In particular, we show how the geometric data on the cycles are encoded in the boundary states. As an application, we analyze how the mirror symmetry transforms D-branes, and we verify that it is consistent with the conjectured periodicity and the monodromy of the Ramond-Ramond field configuration on a Calabi-Yau manifold. This also enables us to study open string worldsheet instanton corrections and relate them to closed string instanton counting. The cases when the mirror symmetry is realized as T-duality are also discussed. 
  We extend our recent analysis of the entropy of extremal black strings with traveling waves. We previously considered waves carrying linear momentum on black strings in six dimensions. Here we study waves carrying angular momentum on these strings, and also waves carrying linear momentum on black strings in five dimensions. In both cases, we show that the horizon remains homogeneous and compute its area. We also count the number of BPS states at weak string coupling with the same distribution of linear and angular momentum, and find complete agreement with the black string entropy. 
  The algebra of non-commutative differential geometry (NCG) on the discrete space $M_4\times Z\ma{N}$ previously proposed by the present author is improved to give the consistent explanation of the generalized gauge field as the generalized connection on $M_4\times Z\ma{N}$. The nilpotency of the generalized exterior derivative ${\mbf d}$ is easily proved. The matrix formulation where the generalized gauge field is denoted in matrix form is shown to have the same content with the ordinary formulation using {\mbf d}, which helps us understand the implications of the algebraic rules of NCG on $M_4\times Z\ma{N}$. The Lagrangian of spontaneously broken gauge theory which has the extra restriction on the coupling constant of the Higgs potential is obtained by taking the inner product of the generalized field strength. The covariant derivative operating on the fermion field determines the parallel transformation on $M_4\times Z\ma{N}$, which confirms that the Higgs field is the connection on the discrete space. This implies that the Higgs particle is a gauge particle on the same footing as the weak bosons. Thus, it is expected that the mass relation $m\ma{H}=\frac{4}{\sqrt{3}}m\ma{W}\sin\theta\ma{W}$ proposed by the present author holds without any correction in the same way as the mass relation $m\ma{W}=m\ma{Z}\cos\theta\ma{W}$. The Higgs kinetic and potential terms are regarded as the curvatures on $M_4\times Z\ma{N}$. 
  Using a Dirac-matrix substitution rule, applied to the electric charge, the anomalous magnetic moments of fermions are incorporated in local form in the two-body relativistic wave equations of constraint theory. The structure of the resulting potential is entirely determined, up to magnetic type form factors, from that of the initial potential descibing the mutual interaction in the absence of anomalous magnetic moments. The wave equations are reduced to a single eigenvalue equation in the sectors of pseudoscalar and scalar states ($j=0$). The requirement of a smooth introduction of the anomalous magnetic moments imposes restrictions on the behavior of the form factors near the origin, in $x$-space. The nonrelativistic limit of the eigenvalue equation is also studied. 
  The bootstrap programme for finding exact S-matrices of integrable quantum field theories with N=1 supersymmetry is investigated. New solutions are found which have the same fusing data as bosonic theories related to the classical affine Lie algebras. When the states correspond to a spinor spot of the Dynkin diagram they are kinks which carry a non-zero topological charge. Using these results, the S-matrices of the supersymmetric O($2n$) sigma model and sine-Gordon model can be shown to close under the bootstrap. 
  The non-Abelian analog of the classical Coulomb gas is discussed. The statistical mechanics of arrays of classical particles which transform under various representations of a non-Abelian gauge group and which interact through non-Abelian electric fields are considered. The problem is formulated on the lattice and, for the case of adjoint charges, it is solved in the large N limit. The explicit solution exhibits a first order confinement-de-confinement phase transition with computable properties. In one dimension, the solution has a continuum limit which describes 1+1-dimensional quantum chromodynamics (QCD) with heavy adjoint matter. 
  We consider the action, in arbitrary curved background, of the eleven-dimensional five-brane to second order in the curvature of the worldvolume tensor field. We show that this action gives upon double dimensional reduction the action of the Dirichlet four-brane up to the same order. We use this result as a starting point to discuss the structure of the action including terms of higher order in the worldvolume curvature. 
  String theory counterparts to Einstein's gravity, cosmology and inflation are described. A very tight upper bound on the Cosmic Gravitational Radiation Background (CGRB) of standard inflation is shown to be evaded in string cosmology, while an interesting signal in the phenomenologically interesting frequency range is all but excluded. The generic features of such a stringy CGRB are presented. 
  A family of locally equivalent models is considered. They can be taken as a generalization to $d+1$ dimensions of the Topological Massive and ``Self-dual'' models in 2+1 dimensions. The corresponding 3+1 models are analized in detail. It is shown that one model can be seen as a gauge fixed version of the other, and their space of classical solutions differs in a topological sector represented by the classical solutions of a pure BF model. The topological sector can be gauged out on cohomologically trivial base manifolds but on general settings it may be responsible of the difference in the long distance behaviour of the models. The presence of this topological sector appears explicitly in the partition function of the theories. The generalization of this models to higher dimensions is shown to be straightfoward. 
  The Proca model is quantized in an open-path dependent representation that generalizes the Loop Representation of gauge theories. The starting point is a gauge invariant Lagrangian that reduces to the Proca Lagrangian when certain gauge is selected. 
  We study the restrictions imposed by cancellation of the tadpoles for two, three, and four-form gauge fields in string theory, M-theory and F-theory compactified to two, three and four dimensions, respectively. For a large class of supersymmetric vacua, turning on a sufficient number of strings, membranes and three-branes, respectively, can cancel the tadpoles, and preserve supersymmetry. However, there are cases where the tadpole cannot be removed in this way, either because the tadpole is fractional, or because of its sign. For M-theory and F-theory compactifications, we also explore the relation of the membranes and three-branes to the nonperturbative space-time superpotential. 
  We look for soliton solutions in $c=0$ noncritical string field theory constructed by the authors and collaborators. It is shown that the string field action itself is very complicated in our formalism but it satisfies a very simple equation. We derive an equation which a solution to the equation of motion should satisfy. Using this equation, we conjecture the form of a soliton solution which is responsible for the nonperturbative effects of order $e^{-A/\kappa}$. (Talk given by N.I. at ``Inauguration Conference of APCTP'', 4-10 June, 1996) 
  The tube solutions in Yang - Mills - Higgs theory are received, in which the Higgs field has the negative energy density. This solutions make up the discrete spectrum numered by two integer and have the finite linear energy density. Ignoring its transverse size, such field configuration is the rest infinity straight string. 
  A model of quantum field theory in which the field operators form a nonassociative algebra is proposed. In such a case, the n-point Green's functions become functionally independent of each other. It is shown that particle interaction in such a theory can be realized by nonlocal virtual objects. 
  The Kowalewski top on Lie algebras $o(4)$, $e(3)$ and $o(3,1)$ is embedded in the SUSY quantum mechanics. In two dimensions we give the new prescription for construction of the pairs of integrable systems by using a standard SUSY algebra. At the proposed scheme the Goryachev-Chapligin top is shown to be a natural partner of the Kowalewski top. 
  We study the infra-red limit of non-abelian Chern-Simons gauge theory perturbed by a non-topological, albeit gauge invariant, mass term. It is shown that, in this limit, we may construct an infinite class of integrable quantum mechanical models which, for the case of SU(2) group, are labelled by the angular momentum eigenvalue. The first non-trivial example in this class is obtained for the triplet representation and it physically describes the gauge invariant coupling of a non-abelian Chern-Simons particle with a particle moving on $S^3$ - the SU(2) group manifold. In addition to this, the model has a fascinating resemblance to the Landau problem and may be regarded as a non-abelian and a non-linear generalisation of the same defined on the three-sphere with the uniform magnetic field replaced by an angular momentum field. We explicitly solve for some eigenstates of this model in a closed form in terms of some generalised orthogonal polynomials. In the process, we unravel some startling connections with Anderson's chain models which are important in the study of disordered systems in condensed matter physics. We also sketch a method which allows us, in principle, to find the energy eigenvalues corresponding to the above eigenstates of the theory if the Lyapunov exponents of the transfer matrix of the infinite chain model involved are known. 
  We show that when correctly formulated the equation $\nabla \times \mbox{\boldmath $a$} = \kappa \mbox{\boldmath $a$}$ does not exhibit some inconsistencies atributed to it, so that its solutions can represent physical fields. 
  Two-dimensional QED with $N$ flavor fermions is solved at zero and finite temperature with arbitrary fermion masses to explore QCD physics such as chiral condensate and string tension. The problem is reduced to solving a Schr\"odinger equation for $N$ degrees of freedom with a specific potential determined by the ground state of the Schr\"odinger problem itself. 
  A new conformal field theory description of two-dimensional turbulence is proposed. The recently established class of rational logarithmic conformal field theories provides a unique candidate solution which resolves many of the drawbacks of former approaches via minimal models. This new model automatically includes magneto-hydrodynamic turbulence and the Alf'ven effect. 
  We present new regular static isolated cylindrically symmetric solutions for the Ginzburg-Landau model which have finite Gibbs free energy. These configurations (which we call the {\it flux tube} and {\it type B} solutions) are energetically favorable in the interval of the external magnetic fields between the thermodynamic critical value $H_{c}$ and the upper critical field $H_{c_2}$ which indicates that they are important new elements of the mixed state describing a transition from vortices to the normal state. 
  This paper studies the one-loop effective action for Euclidean Maxwell theory about flat four-space bounded by one three-sphere, or two concentric three-spheres. The analysis relies on Faddeev-Popov formalism and $\zeta$-function regularization, and the Lorentz gauge-averaging term is used with magnetic boundary conditions. The contributions of transverse, longitudinal and normal modes of the electromagnetic potential, jointly with ghost modes, are derived in detail. The most difficult part of the analysis consists in the eigenvalue condition given by the determinant of a $2 \times 2$ or $4 \times 4$ matrix for longitudinal and normal modes. It is shown that the former splits into a sum of Dirichlet and Robin contributions, plus a simpler term. This is the quantum cosmological case. In the latter case, however, when magnetic boundary conditions are imposed on two bounding three-spheres, the determinant is more involved. Nevertheless, it is evaluated explicitly as well. The whole analysis provides the building block for studying the one-loop effective action in covariant gauges, on manifolds with boundary. The final result differs from the value obtained when only transverse modes are quantized, or when noncovariant gauges are used. 
  We present a systematic way of generating F-theory models dual to nonperturbative vacua (i.e., vacua with extra tensor multiplets) of heterotic E8xE8 strings compactified on K3, using hypersurfaces in toric varieties. In all cases, the Calabi-Yau is an elliptic fibration over a blow up of the Hirzebruch surface F_n. We find that in most cases the fan of the base of the elliptic fibration is visible in the dual polyhedron of the Calabi-Yau, and that the extra tensor multiplets are represented as points corresponding to the blow-ups of the F_n. 
  The quantum mechanics of black holes in generic 2-D dilaton gravity is considered. The Hamiltonian surface terms are derived for boundary conditions corresponding to an eternal black hole with slices on the interior ending on the horizon bifurcation point. The quantum Dirac constraints are solved exactly for these boundary conditions to yield physical eigenstates of the energy operator. The solutions are obtained in terms of geometrical phase space variables that were originally used by Cangemi, Jackiw and Zwiebach in the context of string inspired dilaton gravity. The spectrum is continuous in the Lorentzian sector, but in the Euclidean sector the thermodynamic entropy must be $2\pi n/G$ where $n$ is an integer. The general class of models considered contains as special cases string inspired dilaton gravity, Jackiw-Teitelboim gravity and spherically symmetry gravity. 
  We present the full result for the divergent one-loop contribution to the effective boson Lagrangian for supergravity coupled to chiral and Yang-Mills supermultiplets. We also consider the specific case of dilaton couplings in effective supergravity Lagrangians from superstrings, for which the one-loop result is considerably simplified. 
  We argue for the existence of phase transitions in $3+1$ dimensions associated with the appearance of tensionless strings. The massless spectrum of this theory does not contain a graviton: it consists of one $N=2$ vector multiplet and one linear multiplet, in agreement with the light-cone analysis of the Green-Schwarz string in $3+1$ dimensions. In M-theory the string decoupled from gravity arises when two 5-branes intersect over a three-dimensional hyperplane. The two 5-branes may be connected by a 2-brane, whose boundary becomes a tensionless string with $N=2$ supersymmetry in $3+1$ dimensions. Non-critical strings on the intersection may also come from dynamical 5-branes intersecting the two 5-branes over a string and wrapped over a four-torus. The near-extremal entropy of the intersecting 5-branes is explained by the non-critical strings originating from the wrapped 5-branes. 
  We give a lattice theory treatment of certain one and two dimensional quantum field theories. In one dimension we construct a combinatorial version of a non-trivial field theory on the circle which is of some independent interest in itself while in two dimensions we consider a field theory on a toroidal triangular lattice. We take a continuous spin Gaussian model on a toroidal triangular lattice with periods $L_0$ and $L_1$ where the spins carry a representation of the fundamental group of the torus labeled by phases $u_0$ and $u_1$. We compute the {\it exact finite size and lattice corrections}, to the partition function $Z$, for arbitrary mass $m$ and phases $u_i$. Summing $Z^{-1/2}$ over a specified set of phases gives the corresponding result for the Ising model on a torus. An interesting property of the model is that the limits $m\rightarrow0$ and $u_i\rightarrow0$ do not commute. Also when $m=0$ the model exhibits a {\it vortex critical phase} when at least one of the $u_i$ is non-zero. In the continuum or scaling limit, for arbitrary $m$, the finite size corrections to $-\ln Z$ are {\it modular invariant} and for the critical phase are given by elliptic theta functions. In the cylinder limit $L_1\rightarrow\infty$ the ``cylinder charge'' $c(u_0,m^2L_0^2)$ is a non-monotonic function of $m$ that ranges from $2(1+6u_0(u_0-1))$ for $m=0$ to zero for $m\rightarrow\infty$ but from which one can determine the central charge $c$. The study of the continuum limit of these field theories provides a kind of quantum theoretic analog of the link between certain combinatorial and analytic topological quantities. 
  We study the operator product expansion in the bosonic $SU(2)_0$ and SUSY $SU(2)_2$ WZNW models. We find that these OPEs contain both logarithmic operators and new conserved currents, leading to an extension of the symmetry group. 
  We show that configurations of multiple D-branes related by SU(N) rotations will preserve unbroken supersymmetry. This includes cases in which two D-branes are related by a rotation of arbitrarily small angle, and we discuss some of the physics of this. In particular, we discuss a way of obtaining 4D chiral fermions on the intersection of D-branes.   We also rephrase the condition for unbroken supersymmety as the condition that a `generalized holonomy group' associated with the brane configuration and manifold is reduced, and relate this condition (in Type IIA string theory) to a condition in eleven dimensions. 
  Using the Steiner-Weyl expansion formula for parallel manifolds and the so called gonihedric principle we find a large class of discrete integral invariants which are defined on simplicial manifolds of various dimensions. These integral invariants include the discrete version of the Hilbert-Einstein action found by Regge and alternative actions which are linear with respect to the size of the manifold. In addition the concept of generalized deficit angles appear in a natural way and is related to higher order curvature terms. These angles may be used to introduce various aspects of rigidity in simplicial quantum gravity. 
  We show that excitations in a recently proposed gauge theory for anyons on a line in fact do not obey anomalous statistics. On the other hand, the theory supports novel chiral solitons. Also we construct a field-theoretic description of lineal anyons, but gauge fields play no role. 
  Taking the N=2 strings as the starting point, we discuss the equivalent self-dual field theories and analyse their symmetry structure in 2+2 dimensions. Restoring the full `Lorentz' invariance in the target space necessarily leads to an extension of the N=2 string theory to a theory of 2+2 dimensional supermembranes propagating in 2+10 dimensional target space. The supermembrane requires maximal conformal supersymmetry in 2+2 dimensions, in the way advocated by Siegel. The corresponding self-dual N=4 Yang-Mills theory and the self-dual N=8 (gauged) supergravity in 2+2 dimensions thus appear to be naturally associated to the membrane theory, not a string. Since the same theory of membranes seems to represent the M-theory which is apparently underlying the all known N=1 string theories, the N=2 strings now appear on equal footing with the other string models as particular limits of the unique fundamental theory. Unlike the standard 10-dimensional superstrings, the N=2 strings seem to be much closer to a membrane description of F&M theory. 
  When Alf`ven effect is peresent in magnetohydrodynamics one is naturally lead to consider conformal field theories, which have logarithmic terms in their correlation functions. We discuss the implications of such logarithmic terms and find a unique conformal field theory with centeral charge $c=-\frac{209}{7}$, within the border of the minimal series, which satisfies all the constraints. The energy espectrum is found to be \newline $E(k)\sim k^{-\frac{13}{7}} \log{k}$. 
  We demonstrate that the generalization of the relativistic Toda chain (RTC) is a special reduction of two-dimensional Toda Lattice hierarchy (2DTL). This reduction implies that the RTC is gauge equivalent to the discrete AKNS hierarchy and, which is the same, to the two-component Volterra hierarchy while its forced (semi-infinite) variant is described by the unitary matrix integral. The integrable properties of the RTC hierarchy are revealed in different frameworks of: Lax representation, orthogonal polynomial systems, and $\tau$-function approach. Relativistic Toda molecule hierarchy is also considered, along with the forced RTC. Some applications to biorthogonal polynomial systems are discussed. 
  The Narain lattice construction of string compactifications is generalized to include spontaneously broken supersymmetry. Consistency conditions from modular invariance and Lorentz symmetry are solved in full generality. This framework incorporates models where supersymmetry breaking is inversely proportional to the radii of compact dimensions. The enhanced lattice description, however, might allow for models with a different geometrical or even non-geometrical interpretation. 
  Using D-brane techniques, we compute the spectrum of stable BPS states in N=4 supersymmetric gauge theory, and find it is consistent with Montonen-Olive duality. 
  By use of geometrical methods of surface theory we demonstrate links of Green-Schwarz superstring dynamics with supersymmetric exactly-solvable nonlinear systems and super-WZNW models reduced in an appropriate way. 
  We discuss some aspects of F-theory in four dimensions on elliptically fibered Calabi-Yau fourfolds which are Calabi-Yau threefold fibrations. A particularly simple class of such manifolds emerges for fourfolds in which the generic Calabi-Yau threefold fiber is itself an elliptic fibration and is K3 fibered. Duality between F-theory compactified on Calabi-Yau fourfolds and heterotic strings on Calabi-Yau threefolds puts constraints on the cohomology of the fourfold. By computing the Hodge diamond of Calabi-Yau fourfolds we provide first numerical evidence for F-theory dualities in four dimensions. 
  I investigate three dimensional abelian and non-abelian gauge theories interacting with Dirac fermions. Using a variational method I evaluate the vacuum energy density in the one-loop approximation. It turns out that the states with a constant magnetic condensate lie below the perturbative ground state only in the case of three dimensional quantum electrodynamics with massive fermions. 
  The superspace formulation of eleven-dimensional supergravity with a seven-form field strength is given. 
  The characteristic novel features of strongly coupled gravity at the special values ($C_L=7, 13, 19$) are reviewed in a simple manner using pictures as much as possible. (Notes of lectures at the 1995 Cargese Meeting Low dimensional applications of quantum field theory) 
  We combine analytical and numerical techniques to study the collapse of conformally coupled massless scalar fields in semiclassical 2D dilaton gravity, with emphasis on solutions just below criticality when a black hole almost forms. We study classical information and quantum correlations. We show explicitly how recovery of information encoded in the classical initial data from the outgoing classical radiation becomes more difficult as criticality is approached. The outgoing quantum radiation consists of a positive-energy flux, which is essentially the standard Hawking radiation, followed by a negative-energy flux which ensures energy conservation and guarantees unitary evolution through strong correlations with the positive-energy Hawking radiation. As one reaches the critical solution there is a breakdown of unitarity. We show that this breakdown of predictability is intimately related to a breakdown of the semiclassical approximation. 
  The main limitations of string field theory arise because its present formulation requires a background representing a classical solution, a background defined by a strictly conformally invariant theory. Here we sketch a construction for a gauge-invariant string field action around non-conformal backgrounds. The construction makes no reference to any conformal theory. Its two-dimensional field-theoretic aspect is based on a generalized BRST operator satisfying a set of Weyl descent equations. Its geometric aspect uses a complex of moduli spaces of two-dimensional Riemannian manifolds having ordinary punctures, and organized by the number of special punctures which goes from zero to infinity. In this complex there is a Batalin-Vilkovisky algebra that includes naturally the operator which adds one special puncture. We obtain a classical field equation that appears to relax the condition of conformal invariance usually taken to define classical string backgrounds. 
  We consider the correlation functions of two-dimensional turbulence in the presence and absence of a three-dimensional perturbation, by means of conformal field theory. In the persence of three dimensional perturbation, we show that in the strong coupling limit of a small scale random force, there is some logarithmic factor in the correlation functions of velocity stream functions. We show that the logarithmic conformal field theory $c_{8,1}$ describes the 2D- turbulence both in the absence and the presence of the perturbation. We obtain the following energy spectrum $E(k) \sim k^{-5.125 } \ln(k )$ for perturbed 2D - turbulence and $E(k) \sim k^{-5 } \ln(k )$ for unperturbed turbulence. Recent numerical simulation and experimental results confirm our prediction. 
  A new method for non-perturbative calculation of Green functions in quantum mechanics and quantum field theory is proposed. The method is based on an approximation of Schwinger-Dyson equation for the generating functional by exactly soluble equation in functional derivatives. Equations of the leading approximation and the first step are solved for $\phi^4_d$-model. At $d=1$ (anharmonic oscillator) the ground state energy is calculated. The renormalization program is performed for the field theory at $d=2,3$. At $d=4$ the renormalization of the coupling involves a trivialization of the theory. 
  We study two dimensional freely decaying magnetohydrodynamic turbulence. We investigate the time evolution of the probability law of the gauge field and the stream function. Assuming that this probability law is initially defined by a statistical field theory in the basin of attraction of a renormalisation group fixed point, we show that its time evolution is generated by renormalisation transformations. In the long time regime, the probability law is described by non-unitary conformal field theories. In that case, we prove that the kinetic and magnetic energy spectra are proportional. We then construct a family of fixed points using the $(p,p+2)$ non-unitary minimal models of conformal field theories. 
  Vacuum structures of supersymmetric (SUSY) Yang-Mills theories in $1+1$ dimensions are studied with the spatial direction compactified. SUSY allows only periodic boundary conditions for both fermions and bosons. By using the Born-Oppenheimer approximation for the weak coupling limit, we find that the vacuum energy vanishes, and hence the SUSY is unbroken. Other boundary conditions are also studied, especially the antiperiodic boundary condition for fermions which is related to the system in finite temperatures. In that case we find for gaugino bilinears a nonvanishing vacuum condensation which indicates instanton contributions. 
  Liouville field theory on hyperelliptic surface is considered. The partition function of the Liouville field theory on the hyperelliptic surface are expressed as a correlation function of the Liouville vertex operators on a sphere and the twist fields. 
  We study the path-integral formalism in the imaginary-time to show its validity in a case with a metastable ground state. The well-known method based on the bounce solution leads to the imaginary part of the energy even for a state that is only metastable and has a simple oscillating behavior instead of decaying. Although this has been argued to be the failure of the Euclidean formalism, we show that proper account of the global structure of the path-space leads to a valid expression for the energy spectrum, without the imaginary part. For this purpose we use the proper valley method to find a new type of instanton-like configuration, the ``valley instantons''. Although valley instantons are not the solutions of equation of motion, they have dominant contribution to the functional integration. A dilute-gas approximation for the valley instantons is shown to lead to the energy formula. This method extends the well-known imaginary-time formalism so that it can take into account the global behavior of the theory. 
  The effects of instantons close to the cut-off is studied in four dimensional SU(2) gauge theory with higher order derivative terms in the action. It is found in the framework of the dilute instanton gas approximation that the convergence of the topological observables requires non-universal beta function. 
  The modular symmetries of string loop threshold corrections for gauge coupling constants are studied in the presence of discrete Wilson lines for all examples of abelian orbifolds, where the point group is realised by the action of Coxeter elements or generalised Coxeter elements on the root lattices of the Lie groups. 
  A quantization scheme based on the extension of phase space with application of constrained quantization technic is considered. The obtained method is similar to the geometric quantization. For constrained systems the problem of scalar product on the reduced Hilbert space is investigated and possible solution of this problem is done. Generalization of the Gupta-Bleuler like conditions is done by the minimization of quadratic fluctuations of quantum constraints. The scheme for the construction of generalized coherent states is considered and relation with Berezin quantization is found. The quantum distribution functions are introduced and their physical interpretation is discussed. 
  We formulate and solve the analog of the universal Conformal Ward Identity for the stress-energy tensor on a compact Riemann surface of genus $g>1$, and present a rigorous invariant formulation of the chiral sector in the induced two-dimensional gravity on higher genus Riemann surfaces. Our construction of the action functional uses various double complexes naturally associated with a Riemann surface, with computations that are quite similar to descent calculations in BRST cohomology theory. We also provide an interpretation for the action functional in terms of the geometry of different fiber spaces over the Teichm\"{u}ller space of compact Riemann surfaces of genus $g>1$. 
  A carefully motivated symmetric variant of the Poisson bracket in ordinary (not Grassmann) phase space variables is shown to satisfy identities which are in algebraic correspondence with the anticommutation postulates for quantized Fermion systems. "Symplecticity" in terms of this symmetric Poisson bracket implies generalized Hamilton's equations that can only be of Schroedinger type (e.g., the Dirac equation but not the Klein-Gordon or Maxwell equations). This restriction also excludes the old "four-Fermion" theory of beta decay. 
  Recently Gimon and Johnson (hep-th/9604129) and Dabholkar and Park (hep-th/9604178) have constructed Type I theories on K3 orbifolds. The spectra differ from that of Type I on a smooth K3, having extra tensors. We show that the orbifold theories cannot be blown up to smooth K3's, but rather $Z_2$ orbifold singularities always remain. Douglas's recent proposal to use D-branes as probes is useful in understanding the geometry. The $Z_2$ singularities are of a new type, with a different orientifold projection from those previously considered. We also find a new world-sheet consistency condition that must be satisfied by orientifold models. 
  An analysis of the generalised Raychaudhuri equations for string world sheets is shown to lead to the notion of focusing of timelike worldsheets in the classical Nambu-Goto theory of strings. The conditions under which such effects can occur are obtained . Explicit solutions as well as the Cauchy initial value problem are discussed. The results closely resemble their counterparts in the theory of point particles which were obtained in the context of the analysis of spacetime singularities in General Relativity many years ago. 
  Two-dimensional QED with N-flavor fermions serves as a model of quark dynamics in QCD as well as an effective theory of an anti-ferromagnetic spin chain. It is reduced to N-degree quantum mechanics in which a potential is self-consistently determined by the Schr\"odinger equation itself. 
  We analyze the possible soft breaking of $N=2$ supersymmetric Yang-Mills theory with and without matter flavour preserving the analyticity properties of the Seiberg-Witten solution. We present the formalism for an arbitrary gauge group and obtain an exact expression for the effective potential. We describe in detail the onset of the confinement description and the vacuum structure for the pure $SU(2)$ Yang-Mills case and also some general features in the $SU(N)$ case. A general mass formula is obtained, as well as explicit results for the mass spectrum in the $SU(2)$ case. 
  Starting from the type IIB string on the Z orbifold, we construct some chiral open-string vacua with N=1 supersymmetry in four dimensions. The Chan-Paton group depends on the (quantized) NS-NS antisymmetric tensor. The largest choice, SO(8)xSU(12)xU(1), has an anomalous U(1) factor whose gauge boson acquires a mass of the order of the string scale. The corresponding open-string spectrum comprises only Neumann strings and includes three families of chiral multiplets in the (8,12*) + (1,66) representation. A comparison is drawn with a heterotic vacuum with non-standard embedding, and some properties of the low-energy effective field theory are discussed. 
  We discuss quantum deformations of Lie algebra as described by the noncoassociative modification of its coalgebra structure. We consider for simplicity the quantum $D=1$ Galilei algebra with four generators: energy $H$, boost $B$, momentum $P$ and central generator $M$ (mass generator). We describe the nonprimitive coproducts for $H$ and $B$ and show that their noncocommutative and noncoassociative structure is determined by the two-body interaction terms. Further we consider the case of physical Galilei symmetry in three dimensions. Finally we discuss the noninteraction theorem for manifestly covariant two-body systems in the framework of quantum deformations of $D=4$ \poin algebra and a possible way out. 
  We present the theory, the experimental evidence, and fundamental physical consequences concerning the existence of families of undistorted progressive waves (UPWs) of arbitrary speeds $0\leq v < \infty$, which are solutions of the homogeneous wave equation, Maxwell equations, and Dirac and Weyl equations. 
  In this paper, two things are done. (i) First, it is shown that any global symmetry of a gauge-invariant theory can be extended to the ghosts and the antifields so as to leave invariant the solution of the master-equation (before gauge fixing). (ii) Second, it is proved that the incorporation of the rigid symmetries to the solution of the master-equation through the introduction of a constant ghost for each global symmetry can be obstructed already at the classical level whenever the theory possesses higher order conservation laws. Explicit examples are given. 
  In this note, we consider the reformulation of the Dirac-Born-Infeld action for a Dirichlet p-brane in Brink-Di Vecchia-Howe-Tucker form, i.e., including an independent non-propagating world-volume metric. When p>2, the action becomes non-polynomial. A closed expression is derived for p=3. For selfdual field-strengths, the DBI action is reproduced by an action with a simple F^2 term. We speculate on supersymmetrization of the D_3-brane action. We also give the governing equations for arbitrary p, and derive an implicit expression for the D_4-brane lagrangian. 
  We present a generalization of the Frolov-Slavnov invariant regularization scheme for chiral fermion theories in curved spacetimes. local gauge symmetries of the theory, including local Lorentz invariance. The perturbative scheme works for arbitrary representations which satisfy the chiral gauge anomaly and the mixed Lorentz-gauge anomaly cancellation conditions. Anomalous theories on the other hand manifest themselves by having divergent fermion loops which remain unregularized by the scheme. Since the invariant scheme is promoted to also include local Lorentz invariance, spectator fields which do not couple to gravity cannot be, and are not, introduced. Furthermore, the scheme is truly chiral (Weyl) in that all fields, including the regulators, are left-handed; and only the left-handed spin connection is needed. The scheme is, therefore, well suited for the study of the interaction of matter with all four known forces in a completely chiral fashion. In contrast with the vectorlike formulation, the degeneracy between the Adler-Bell-Jackiw current and the fermion number current in the bare action is preserved by the chiral regularization scheme. 
  Starting from a local quantum field theory with an unbroken compact symmetry group $G$ in 1+1-dimensional spacetime we construct disorder fields implementing gauge transformations on the fields (order variables) localized in a wedge region. Enlarging the local algebras by these disorder fields we obtain a nonlocal field theory, the fixpoint algebras of which under the appropriately extended action of the group $G$ are shown to satisfy Haag duality in every simple sector. The specifically 1+1 dimensional phenomenon of violation of Haag duality of fixpoint nets is thereby clarified. In the case of a finite group $G$ the extended theory is acted upon in a completely canonical way by the quantum double $D(G)$ and satisfies R-matrix commutation relations as well as a Verlinde algebra. Furthermore, our methods are suitable for a concise and transparent approach to bosonization. The main technical ingredient is a strengthened version of the split property which should hold in all reasonable massive theories. In the appendices (part of) the results are extended to arbitrary locally compact groups and our methods are adapted to chiral theories on the circle. 
  A while ago, examples of N=1 vacua in D=6 were constructed as orientifolds of Type IIB string theory compactified on the K3 surface. Among the interesting features of those models was the presence of D5-branes behaving like small instantons, and the appearance of extra tensor multiplets. These are both non-perturbative phenomena from the point of view of heterotic string theory. Although the orientifold models are a natural setting in which to study these non-perturbative Heterotic string phenomena, it is interesting and instructive to explore how such vacua are realised in Heterotic string theory, M-theory and F-theory, and consider the relations between them. In particular, we consider models of M-theory compactified on K3 x S^1/Z_2 with fivebranes present on the interval. There is a family of such models which yields the same spectra as a subfamily of the orientifold models. By further compactifying on T^2 to four dimensions we relate them to Heterotic string spectra. We then use Heterotic/Type IIA duality to deduce the existence of Calabi-Yau 3-folds which should yield the original six dimensional orientifold spectra if we use them to compactify F-theory. Finally, we show in detail how to take a limit of such an F-theory compactification which returns us to the Type IIB orientifold models. 
  Critical phenomena in the two-dimensional Ising model with a defect line are studied using boundary conformal field theory on the $c=1$ orbifold. Novel features of the boundary states arising from the orbifold structure, including continuously varying boundary critical exponents, are elucidated. New features of the Ising defect problem are obtained including a novel universality class of defect lines and the universal boundary to bulk crossover of the spin correlation function. 
  We consider a Yang--Mills theory in loop space whose gauge group is a Kac--Moody group with the central extension. From this theory, we derive a local field theory constructed of Yang--Mills fields and abelian antisymmetric and symmetric tensor fields of the second rank. The Chapline--Manton coupling, that is, coupling of Yang--Mills fields and a second-rank antisymmetric tensor field via the Chern--Simons 3-form is obtained in a systematic manner. 
  The coupling of a Nambu-Goto string to gravity allows for Schwarzschild black holes whose entropy to area relation is $S=(A/4)(1-4\mu)$, where $\mu$ is the string tension. The departure from $A/4$ universality results from a string instanton which leads to a materialisation of the horizon at the quantum level. 
  The free field representation of the osp(1|2) current algebra is analyzed. The four point conformal blocks of the theory are studied. The structure constants for the product of an arbitrary primary operator and a primary field that transforms according to the fundamental representation of osp(1|2) are explicitly calculated. 
  The characteristic cohomology $H^k_{char}(d)$ for an arbitrary set of free $p$-form gauge fields is explicitly worked out in all form degrees $k<n-1$, where $n$ is the spacetime dimension. It is shown that this cohomology is finite-dimensional and completely generated by the forms dual to the field strengths. The gauge invariant characteristic cohomology is also computed. The results are extended to interacting $p$-form gauge theories with gauge invariant interactions. Implications for the BRST cohomology are mentioned. 
  The quantum-mechanical problem of $N$ fermions with $\delta$-function interaction in a one-dimensional potential well of finite depth is solved. It is shown that there exists exact wave function of Bethe-ansatz form in the case that a single particle tunnels outside of the well. The Bethe-ansatz like secular equations for the spectrum are obtained. 
  We define, in a consistent way, non-local pseudo-differential operators acting on a space of analytic functionals. These operators include the fractional derivative case. In this context we show how to solve homogeneous and inhomogeneous equations associated with these operators. We also extend the formalism to d-dimensional space-time solving, in particular, the fractional Wave and Klein-Gordon equations. 
  We consider a class of N=1 supersymmetric Yang-Mills theories, with gauge group SU(N)xSU(N - M) and fundamental matter content. Duality plays an essential role in analyzing the nonperturbative infrared dynamics of these models. We find that Yukawa couplings drive these theories into the confining phase, and show how the nonperturbative superpotentials arise in the dual picture. We show that the odd-N, M = 2 models with an appropriate tree-level superpotential break supersymmetry. 
  We compute the leading order (in coupling) rate of emission of low energy quanta from a slightly nonextremal system of 1 and 5 D-branes. We also compute the classical cross-section, and hence the Hawking emission rate, for low energy scalar quanta for the black hole geometry that corresponds to these branes (at sufficiently strong coupling). These rates are found to agree with each other. 
  By using Heterotic SUSY path integral, one version and generalization of an integrability theorem due to Mayer is obtained. 
  A modification of the Abelian Duality transformations is proposed guaranteeing that a (not necessarily conformally invariant) $\sigma$-model be quantum equivalent (at least up to two loops in perturbation theory) to its dual. This requires a somewhat non standard perturbative treatment of the {\sl dual} $\sigma$-model. Explicit formulae of the modified duality transformation are presented for a special class of block diagonal purely metric $\sigma$-models. 
  Supersymmetry can be consistently generalized in one and two dimensional spaces, fractional supersymmetry being one of the possible extension. 2D fractional supersymmetry of arbitrary order $F$ is explicitly constructed using an adapted superspace formalism. This symmetry connects the fractional spin states ($0,{1 \over F}, ...,{F-1 \over F}$). Besides the stress momentum tensor, we obtain a conserved current of spin($1 + {1 \over F})$. The coherence of the theory imposes strong constraints upon the commutation relations of the modes of the fields. The creation and annihilation operators turn out to generate alternative statistics, currently referred as quons in the literature. We consider, with a special attention, the consistence of the algebra, on the level of the Hilbert space and the Green functions. The central charges are generally irrational numbers except for the particular cases $F=2,3,4$. A natural classification emerges according to the decomposition of $F$ into its product of prime numbers leading to sub-systems with smaller symmetries. 
  We study generic features related to matter contents and flat directions in $Z_{2n}$ orbifold models. It is shown that $Z_{2n}$ orbifold models have massless conjugate pairs, $R$ and $\overline R$, in certain twisted sectors as well as one of untwisted subsectors. Using these twisted sectors, $Z_{2n}$ orbifold models are classified into two types. Conjugate pairs, $R$ and $\overline R$, lead to $D$-flatness as $\langle R \rangle = \langle \overline R \rangle \neq 0$. We investigate generic superpotentials derived from orbifold models so as to show that this direction is indeed a flat direction. 
  $N=2$ string theories are formulated in space-times with 2 space and 2 time dimensions. If the world-sheet matter system consists of 2 chiral superfields, the space-time is Kahler and the dynamics are those of anti-self-dual gravity. If instead one chiral superfield and one twisted chiral superfield are used, the space-time is a hermitian manifold with torsion and a dilaton. The string spectrum consists of a scalar, which is a potential $K$ determining the metric, torsion and dilaton. The dynamics imply that the curvature with torsion is anti-self-dual, and an action is found for the potential $K$. It is argued that any $N=2$ sigma-model with twisted chiral multiplets in any dimension can be deformed to a conformally invariant theory if the lowest order contribution to the conformal anomaly vanishes. If there are isometries, more general geometries are possible in which the dilaton is the Killing potential for a holomorphic Killing vector. 
  We extend previous work on the soft breaking of $N=2$ supersymmetric QCD. We present the formalism for the breaking due to a dilaton spurion for a general gauge group and obtain the exact effective potential. We obtain some general features of the vacuum structure in the pure $SU(N)$ Yang-Mills theory and we also derive a general mass formula for this class of theories, in particular we present explicit results for the mass spectrum in the $SU(2)$ case. Finally we analyze the vacuum structure of the $SU(2)$ theory with one massless hypermultiplet. This theory presents dyon condensation and a first order phase transition in the supersymmetry breaking parameter driven by non-mutually local BPS states. This could be a hint of Argyres-Douglas-like phases in non-supersymmetric gauge theories. 
  These notes are based on work done in collaboration with Frank Ferrari. We show how to determine the spectra of stable BPS states in $N=2$ supersymmetric $SU(2)$ Yang-Mills theories that are asymptotically free, i.e. without and with $N_f=1,2,3$ quark hypermultiplets. In all cases: $\bullet$ There is a curve of marginal stability diffeomorphic to a circle and going through all (finite) singular points of moduli space. $\bullet$ The BPS spectra are discontinuous across these curves. $\bullet$ The strong-coupling spectra (inside the curves) contain only those BPS states that can become massless and are responsible for the singularities. Except for $N_f=3$, they form a multiplet (with different masses) of the broken global discrete symmetry. $\bullet$ All other semi-classical BPS states must and do decay consistently when crossing the curves. $\bullet$ The weak-coupling, i.e. semi-classical BPS spectra, contain no magnetic charges larger than one for $N_f=0,1,2$ and no magnetic charges larger than two for $N_f=3$. 
  We find the general charged rotating black hole solutions of the maximal supergravities in dimensions $4\le D\le 9$ arising from toroidally compactified Type II string or M-theories. In each dimension, these are obtained by acting on a generating solution with classical duality symmetries. In D=4, D=5 and $6\le D \le 9$ the generating solution is specified by the ADM mass, $[{D-1}/2]$-angular momentum components and five, three and two charges, respectively. We discuss the BPS-saturated (static) black holes and derive the U-duality invariant form of the area of the horizon. We also comment on the U-duality invariant form of the BPS mass formulae. 
  We address the issue of topological angles in the context of two dimensional SU(N) Yang-Mills theory coupled to massive fermions in the adjoint representation. Classification of the resulting multiplicity of vacua is carried outin terms of asymptotic fundamental Wilson loops, or equivalently, charges at the boundary of the world. We explicitly demonstrate that the multiplicity of vacuum states is equal to N for SU(N) gauge group. Different worlds of the theory are classified by the integer number k=0,1,...N-1 (superselection rules) which plays an analogous role to the $\theta$ parameter in QCD. Via two completely independent approaches we study the physical properties of these unconnected worlds as a function of k. First, we apply the well known machinery of the loop calculus in order to calculate the effective string tensions in the theory as function of $k$. The second way of doing the same physics is the standard particle/field theoretic calculation for the binding potential of a pair of infinitely massive fermions. We also calculate the vacuum energy as function of k. 
  Stochastic quantization is applied to derivation of the equations for the Wilson loops and generating functionals of the Wilson loops in the large-N limit. These equations are treated both in the coordinate and momentum representations. In the first case the connection of the suggested approach with the problem of random closed contours and supersymmetric quantum mechanics is established, and the equation for the Quenched Master Field Wilson loop is derived. The regularized version of one of the obtained equations is presented and applied to derivation of the equation for the bilocal field correlator. The momentum loop dynamics is also investigated. 
  A brief review of some of the central ideas, terminology and techniques of the technology of orientifolds and D-branes is presented. Some applications are reviewed, including the construction of dual solitonic strings in the context of string/string duality, the computation of the Bekenstein-Hawking entropy/area law for extremal black holes, and the construction of N=1 string vacua in dimensions lower than ten. (Contribution to the proceedings of the fourth International Workshop on Supersymmetry and Unification of Fundamental Interactions (SUSY '96), held at the University of Maryland at College Park, May 29 - June 1, 1996.) 
  The effective action of (2+1)-dimensional QED with finite fermion density is calculated in a uniform electromagnetic field. It is shown that the integer quantum Hall effect and de Haas-van Alphen like phenomena in condensed matter physics are derived directly from the effective action. 
  Using renormalization group methods we calculate the derivative expansion of the effective Lagrangian for a covariantly constant gauge field in curved spacetime. Curvature affects the vacuum, in particular it could induce phase transitions between different vacua. We also consider the effect of quantum fluctuations of the metric , in the context of a renormalizable $R^2$ theory. In this case the critical curvature depends on the gravi tational coupling constants. 
  By examining multi-instantons in N=2 supersymmetric SU(2) gauge theory, we derive, on very general grounds, and to all orders in the instanton number, a relationship between the prepotential F(Phi), and the coordinate on the quantum moduli space u=<Tr Phi^2>. This relation was previously obtained by Matone in the context of the explicit Seiberg-Witten low-energy solution of the model. Our findings can be viewed as a multi-instanton check of the proposed exact results in supersymmetric gauge theory. 
  We investigate the black string in the context of the string theories. It is shown that the graviton is the only propagating mode in the (2+1)--dimensional extremal black string background. Both the dilation and axion turn out to be non-propagating modes. 
  The low energy effective theory of the N=1 supersymmetric SU(5) gauge theory with chiral superfields in the 5* and 10 representations is constructed. Instead of postulating the confinement of SU(5) (confining picture), only the confinement of its subgroup SU(4) is postulated (Higgs picture), and the effective fields are SU(4)-singlet but SU(5)-variant. The classical scalar potential which ensures unique supersymmetric vacuum at the classical level is incorporated into the Kaehler potential of the effective fields. We show that supersymmetry and all other global symmetry are spontaneously broken. The scales of these symmetry breaking and the particle spectrum including Nambu-Goldstone particles are explicitly calculated, and no large scale hierarchy is found. 
  In spite of its simplicity and beauty, the Mathai-Quillen formulation of cohomological topological quantum field theory with gauge symmetry suffers two basic problems: $i$) the existence of reducible field configurations on which the action of the gauge group is not free and $ii$) the Gribov ambiguity associated with gauge fixing, i. e. the lack of global definition on the space of gauge orbits of gauge fixed functional integrals. In this paper, we show that such problems are in fact related and we propose a general completely geometrical recipe for their treatment. The space of field configurations is augmented in such a way to render the action of the gauge group free and localization is suitably modified. In this way, the standard Mathai--Quillen formalism can be rigorously applied. The resulting topological action contains the ordinary action as a subsector and can be shown to yield a local quantum field theory, which is argued to be renormalizable as well. The salient feature of our method is that the Gribov problem is inherent in localization, and thus can be dealt with in a completely equivariant setting, whereas gauge fixing is free of Gribov ambiguities. For the stratum of irreducible gauge orbits, the case of main interest in applications, the Gribov problem is solvable. Conversely, for the the strata of reducible gauge orbits, the Gribov problem cannot be solved in general and the obstruction may be described in the language of sheaf theory. The formalism is applied to the Donaldson--Witten model. 
  We built up a explicit realization of (0+1)-dimensional q-deformed superspace coordinates as operators on standard superspace. A q-generalization of supersymmetric transformations is obtained, enabling us to introduce scalar superfields and a q-supersymmetric action. We consider a functional integral based on this action. Integration is implemented, at the level of the coordinates and at the level of the fields, as traces over the corresponding representation spaces. Evaluation of these traces lead us to standard functional integrals. The generation of a mass term for the fermion field leads, at this level, to an explicitely broken version of supersymmetric quantum mechanics. 
  We study all possible $U(1)$-extensions of the standard model (SM) in the framework of noncommutative geometry (NCG) with the algebra $\hhh\op\cc\op\cc\op M_3(\cc)$. Comparison to experimental data about the mass of a hypothetical $Z'$ gauge boson leads to the necessity of introducing at least one new family of heavy fermions. 
  We consider gravitational scattering of point particles with Planckian centre-of-mass energy and fixed low momentum transfers in the framework of general relativity and dilaton gravity. The geometry around the particles are modelled by arbitrary black hole metrics of general relativity to calculate the scattering amplitudes. However, for dilaton gravity, this modelling can be done {\it only} by extremal black hole metrics. This is consistent with the conjecture that extremal black holes are elementary particles. 
  We evaluate partition functions $Z_I$ in topologically nontrivial (instanton) gauge sectors in the bosonized version of the Schwinger model and in a gauged WZNW model corresponding to $QCD_2$ with adjoint fermions. We show that the bosonized model is equivalent to the fermion model only if a particular form of the WZNW action with gauge-invariant integrand is chosen. For the exact correspondence, it is necessary to integrate over the ways the gauge group $SU(N)/Z_N$ is embedded into the full $O(N^2 - 1)$ group for the bosonized matter field. For even $N$, one should also take into account the contributions of both disconnected components in $O(N^2 - 1)$. In that case, $Z_I \propto m^{n_0}$ for small fermion masses where $2n_0$ coincides with the number of fermion zero modes in a particular instanton background. The Taylor expansion of $Z_I/m^{n_0}$ in mass involves only even powers of $m$ as it should. The physics of adjoint $QCD_2$ is discussed. We argue that, for odd $N$, the discrete chiral symmetry $Z_2 \otimes Z_2$ present in the action is broken spontaneously down to $Z_2$ and the fermion condensate $<\bar{\lambda} \lambda>_0$ is formed. The system undergoes a first order phase transition at $T_c = 0$ so that the condensate is zero at an arbitrary small temperature. It is not yet quite clear what happens for even $N \geq 4$. 
  In this talk I shall first make some brief remarks on quaternionic quantum mechanics, and then describe recent work with A.C. Millard in which we show that standard complex quantum field theory can arise as the statistical mechanics of an underlying noncommutative dynamics. 
  We study 2D non-linear sigma models on a group manifold with a special form of the metric. We address the question of integrability for this special class of sigma models. We derive two algebraic conditions for the metric on the group manifold. Each solution of these conditions defines an integrable model. Although the algebraic system is overdetermined in general, we give two examples of solutions. We find the Lax field for these models and calculate their Poisson brackets. We also obtain the renormalization group (RG) equations, to first order, for the generic model. We solve the RG equations for the examples we have and show that they are integrable along the RG flow. 
  We consider the wave equation for spinors in ${\cal D}$-dimensional Weyl geometry. By appropriately coupling the Weyl vector $\phi _{\mu}$ as well as the spin connection $\omega _{\mu a b } $ to the spinor field, conformal invariance can be maintained. The one loop effective action generated by the coupling of the spinor field to an external gravitational field is computed in two dimensions. It is found to be identical to the effective action for the case of a scalar field propagating in two dimensions. 
  We point out that some M-theory results for brane tension can be derived from Polchinski's formula for D-brane tension. We also argue that this formula determines gravitational and gauge couplings in the low energy but quantum exact effective action. 
  For any rational number $p_0\ge 2$ we prove an identity of Rogers-Ramanujan's type. Bijection between the space od states for $XXZ$ model and that of $XXX$ model is constructed 
  We solve a hot twisted Eguchi-Kawai model with only timelike plaquettes in the deconfined phase, by computing the quadratic quantum fluctuations around the classical vacuum. The solution of the model has some novel features: the eigenvalues of the time-like link variable are separated in L bunches, if L is the number of links of the original lattice in the time direction, and each bunch obeys a Wigner semicircular distribution of eigenvalues. This solution becomes unstable at a critical value of the coupling constant, where it is argued that a condensation of classical solutions takes place. This can be inferred by comparison with the heat-kernel model in the hamiltonian limit, and the related Douglas-Kazakov phase transition in QCD2. As a byproduct of our solution, we can reproduce the dependence of the coupling constant from the parameter describing the asymmetry of the lattice, in agreement with previous results by Karsch. 
  The prolongation structure of Zhiber-Mikhailov-Shabat (ZMS) equation is studied by using Wahlquist-Estabrook's method. The Lax-pair for ZMS equation and Riccati equations for pseudopotentials are formulated respectively from linear and nonlinear realizations of the prolongation structure. Based on nonlinear realization of the prolongation structure, an auto-B$\ddot{a}$cklund transformation of ZMS equation is obtained. 
  The thermal partition function of photons in any covariant gauge and gravitons in the harmonic gauge, propagating in a Rindler wedge, are computed using a local $\zeta$-function regularization approach. The correct Planckian leading order temperature dependence $T^4$ is obtained in both cases. For the photons, the existence of a surface term giving a negative contribution to the entropy is confirmed, as earlier obtained by Kabat, but this term is shown to be gauge dependent in the four-dimensional case and, therefore is discarded. It is argued that similar terms could appear dealing with any integer spin $s\geq 1$ in the massless case and in more general manifolds. Our conjecture is checked in the case of a graviton in the harmonic gauge, where different surface terms also appear, and physically consistent results arise dropping these terms. The results are discussed in relation to the quantum corrections to the black hole entropy. 
  The superselection structure of $\son$ WZW models is investigated from the point of view of algebraic quantum field theory. At level $1$ it turns out that the observable algebras of the WZW theory can be constructed in terms of even CAR algebras. This fact allows to give a formulation of these models close to the DHR framework. Localized endomorphisms are constructed explicitly in terms of Bogoliubov transformations, and the WZW fusion rules are proven using the DHR sector product.   At level $2$ it is shown that most of the sectors are realized in $\HNSh=\HNS\otimes\HNS$ where $\HNS$ is the Neveu-Schwarz sector of the level $1$ theory. The level $2$ characters are derived and $\HNSh$ is decomposed completely into tensor products of the sectors of the WZW chiral algebra and irreducible representation spaces of the coset Virasoro algebra. Crucial for this analysis is the DHR decomposition of $\HNSh$ into sectors of a gauge invariant fermion algebra since the WZW chiral algebra as well as the coset Virasoro algebra are invariant under the gauge group $\Oz$. 
  On the base of Lie algebraic and differential geometry methods, a wide class of multidimensional nonlinear integrable systems is obtained, and the integration scheme for such equations is proposed. 
  We derive the conditions that the coupling constants of the Generalized Thirring Model have to satisfy in order for the model to admit an infinite number of commuting classical conserved quantities. Our treatment uses the bosonized version of the model, with periodic boundary conditions imposed on the space coordinate. Some explicit examples that satisfy these conditions are discussed. We show that, with a different set of boundary conditions, there exist additional conserved quantities, and we find the Poisson Bracket algebra satisfied by them. 
  The massless spectrum of an orientifold of the IIB string theory is computed and shown to be identical to F theory on the Calabi-Yau threefold with $h_{11}=51$ and $h_{21}=3$. Target space duality is also considered in this model. 
  Compactifying the $E_8$ non-critical string in 6D down to 5D the 6D strings give rise to particles and strings in 5D. Using the dual M-theory description compactified on an elliptically fibered Calabi-Yau we compare some of the 5D BPS states to what one expects from non-critical strings with an $E_8$ chiral current algebra. The $E_8$ multiplets of particle states comprise of 2-branes wrapping on irreducible curves together with bound states of several 2-branes. 
  We show that theories in the confining, free magnetic, and conformal phases can break supersymmetry through dynamical effects. To illustrate this, we present theories based on the gauge groups $SU(n)\times SU(4)\times U(1)$ and $SU(n) \times SU(5) \times U(1)$ with the field content obtained by decomposing an $SU(m)$ theory with an antisymmetric tensor and $m-4$ antifundamentals. 
  We suggest that the Higgs might be unobservable as a free particle, due to its origin at a symmetry breaking mechanism. The standard model is kept intact, only the definition of the vacuum for the Higgs is changed. With the new (natural) definition, the Higgs propagator is half advanced and half retarded. This Green function is compatible with the absence of free particles. 
  We show that the anomalies of the Virasoro algebra are due to the asymmetric behavior of raising and lowering operators with respect to the ground state of the string. With the adoption of a symmetric vacuum we obtain a non-anomalous theory in any number of dimensions. In particular for D=4. 
  A scalar field obeying a Lorentz invariant higher order wave equation, is minimally coupled to the electromagnetic field. The propagator and vertex factors for the Feynman diagrams, are determined. As an example we write down the matrix element for the Compton effect. This matrix element is algebraically reduced to the usual one for a charged Klein-Gordon particle. It is proved that the $ n^{th} $ order theory is equivalent to n independent second order theories. It is also shown that the higher order theory is both renormalizable and unitary for arbitrary n. 
  A (globally) neutral two-body system is supposed to obey a pair of coupled Klein-Gordon equations in a constant homogeneous magnetic field. Considering eigenstates of the pseudomomentum four-vector, we reduce these equations to a three-dimensional eigenvalue problem. The frame adapted to pseudomomentum has in general a nonvanishing velocity with respect to the frames where the field is purely magnetic. This velocity plays a crucial role in the occurance of motional terms; these terms are taken into account within a manifestly covariant framework. Perturbation theory is available when the mutual interaction doesnot depend on the total energy; a weak-field-slow-motion approximation is more specially tractable. 
  We present a microscopic index formula for the degeneracy of dyons in four-dimensional N=4 string theory. This counting formula is manifestly symmetric under the duality group, and its asymptotic growth reproduces the macroscopic Bekenstein-Hawking entropy. We give a derivation of this result in terms of the type II five-brane compactified on K3, by assuming that its fluctuations are described by a closed string theory on its world-volume. We find that the degeneracies are given in terms of the denominator of a generalized super Kac-Moody algebra. We also discuss the correspondence of this result with the counting of D-brane states. 
  We construct $(D-3)$-brane and instanton solutions using $N \le 10-D$ one-form field strengths in $D$ dimensions, and show that the equations of motion can be cast into the form of the $SL(N+1,R)$ Toda equations. For generic values of the charges, the solutions are non-supersymmetric; however, they reduce to the previously-known multiply-charged supersymmetric solutions when appropriate charges vanish. 
  We show that the S-matrix ansatz implies a semi-classical metric such that a freely falling test particle will not cross the horizon in its proper time. Instead of reaching the singularity it will reach ${\cal I^{+}}$. 
  In this paper the elliptic genus for a general Calabi-Yau fourfold is derived. The recent work of Kawai calculating N=2 heterotic string one-loop threshold corrections with a Wilson line turned on is extended to a similar computation where K3 is replaced by a general Calabi-Yau 3- or 4-fold. In all cases there seems to be a generalized Kac-Moody algebra involved, whose denominator formula appears in the result. 
  The general structure of the perturbative expansion of the vacuum expectation value of a product of Wilson-loop operators is analyzed in the context of Chern-Simons gauge theory. Wilson loops are opened into Wilson lines in order to unravel the algebraic structure encoded in the group factors of the perturbative series expansion. In the process a factorization theorem is proved for Wilson lines. Wilson lines are then closed back into Wilson loops and new link invariants of finite type are defined. Integral expressions for these invariants are presented for the first three primitive ones of lower degree in the case of two-component links. In addition, explicit numerical results are obtained for all two-component links of no more than six crossings up to degree four. 
  In this paper we explicitly prove that Integrable System solved by Quantum Inverse Scattering Method can be described with the pure algebraic object (Universal R-matrix) and proper algebraic representations. Namely, on the example of the Quantum Volterra model we construct L-operator and fundamental R--matrix from universal R--matrix for Quantum Affine $U_q(\widehat{sl}_2)$ Algebra and q-oscillator representation for it. In this way there exists an equivalence between the Integrable System with symmetry algebra A and the representation of this algebra. 
  Generic partial supersymmetry breaking of $N=2$ supergravity with zero vacuum energy and with surviving unbroken arbitrary gauge groups is exhibited. Specific examples are given. 
  The theory including interaction between Siegel and gauge multiplets leads to the model of nonbreaking supersymmetry which contains massive scalar, four component fermion and gauge fields. The upper bound of Higgs boson mass is estimated as heaviest fermion mass. The idea to replace Higgs field by scalar superpartner or by auxiliary fields of corresponding supermultiplet is discussed. 
  In this paper we consider a class of models for vector and hypermultiplets, interacting with $N=2$ supergravity, with gauge groups being an infinite-dimensional Kac-Moody groups. It is shown that specific properties of Kac-Moody groups, allowing the introduction of the vector fields masses without the usual Higgs mechanism, make it possible to break simultaneously both the supersymmetry and the gauge symmetry. Also, a kind of inverse Higgs mechanism can be realized, that is, in the considered model there exists a possibility to lower masses of the scalar fields, which usually acquire huge masses as a result of supersymmetry breaking. That allows one to use them, for example, as Higgs fields at the second step of the gauge symmetry breaking in the unified models. 
  A generalized spin Sutherland model including a three-body potential is proposed. The problem is analyzed in terms of three first-order differential-difference operators, obtained by combining SUSYQM supercharges with the elements of the dihedral group~$D_6$. Three alternative commuting operators are also introduced. 
  The comultiplication formula for fusion products of untwisted representations of the chiral algebra is generalised to include arbitrary twisted representations. We show that the formulae define a tensor product with suitable properties, and determine the analogue of Zhu's algebra for arbitrary twisted representations.   As an example we study the fusion of representations of the Ramond sector of the N=1 and N=2 superconformal algebra. In the latter case, certain subtleties arise which we describe in detail. 
  We discuss the evaluation of observables in two-dimensional conformal field theory using the topological membrane description. We show that the spectrum of anomalous dimensions can be obtained perturbatively from the topologically massive quantum field theories by computing radiative corrections to Aharonov-Bohm scattering amplitudes for dynamical charged matter fields. The one-loop corrections in the case of topologically massive Yang-Mills theory are shown to coincide with the scaling dimensions of the induced ordinary and supersymmetric WZNW models. We examine the effects of the dressing of a topologically massive gauge theory by topologically massive gravity and show that the one-loop contributions to the Aharonov-Bohm amplitudes coincide with the leading orders of the KPZ scaling relations for two-dimensional quantum gravity. Some general features of the description of conformal field theories via perturbative techniques in the three-dimensional approach are also discussed. 
  Following Polchinski's approach we calculate the one-loop vacuum amplitude for two parallel D-branes connected by open bosonic (neutral or charged)string in a constant uniform electromagnetic (EM) field. For neutral string, external EM field contribution appears as multiplier (Born-Infeld type action) of vacuum amplitude without external EM field. Hence,it gives the alternative way to see the inducing of Born-Infeld type action for description of D-branes. For charged string the situation is more complicated, it may indicate the necessity to modify the induced D-branes action in this case. 
  We study the statistics of semi-meanders, i.e. configurations of a set of roads crossing a river through n bridges, and possibly winding around its source, as a toy model for compact folding of polymers. By analyzing the results of a direct enumeration up to n=29, we perform on the one hand a large n extrapolation and on the other hand we reformulate the available data into a large q expansion, where q is a weight attached to each road. We predict a transition at q=2 between a low-q regime with irrelevant winding, and a large-q regime with relevant winding. 
  We bosonize the Massive Thirring Model in 3+1D for small coupling constant and arbitrary mass. The bosonized action is explicitly obtained both in terms of a Kalb-Ramond tensor field as well as in terms of a dual vector field. An exact bosonization formula for the current is derived. The small and large mass limits of the bosonized theory are examined in both the direct and dual forms. We finally obtain the exact bosonization of the free fermion with an arbitrary mass. 
  An orientifold of Type-IIB theory on a $K3$ realized as a $Z_2$ orbifold is constructed which corresponds to F-theory compactification on a Calabi-Yau orbifold with Hodge numbers $(51, 3)$. The T-dual of this model is analogous to an orbifold with discrete torsion in that the action of orientation reversal has an additional phase on the twisted sectors, and both 9-branes and 5-branes carry orthogonal gauge groups. An orientifold of the $Z_3$ orbifold and its relation to F-theory is briefly discussed. 
  We compute the non--trivial infrared $\phi^4_3$--fixed point by means of an interpolation expansion in fixed dimension. The expansion is formulated for an infinitesimal momentum space renormalization group. We choose a coordinate representation for the fixed point interaction in derivative expansion, and compute its coordinates to high orders by means of computer algebra. We compute the series for the critical exponent $\nu$ up to order twenty five of interpolation expansion in this representation, and evaluate it using \pade, Borel--\pade, Borel--conformal--\pade, and Dlog--\pade resummation. The resummation returns $0.6262(13)$ as the value of $\nu$. 
  Stationary four-dimensional BPS solutions to gravity coupled bosonic theories admitting a three-dimensional sigma-model representation on coset spaces are interpreted as null geodesics of the target manifold equipped with a certain number of harmonic maps. For asymptotically flat (or Taub-NUT) space-times such geodesics can be directly parametrized in terms of charges saturating the Bogomol'nyi-Gibbons-Hull bound, and classified according to the structure of related coset matrices. We investigate in detail the ``dilaton-axion gravity'' with one vector field, and show that in the space of BPS solutions an $SO(1,2) \times SO(2)$ classical symmetry is acting. Within the present formalism the most general multicenter (IWP/Taub-NUT dyon) solutions are derived in a simple way. We also discover a large new class of asymptotically flat solutions for which the dilaton and axion charges are constrained only by the BPS bound. The string metrics for these solutions are generically regular. Both the IWP class and the new class contain massless solutions. 
  Yang-Mills theory in the first order formalism appears as the deformation of a topological field theory, the pure BF theory. In this approach new non local observables are inherited from the topological theory and the operators entering the t'Hooft algebra find an explicit realization. A calculation of the {\it vev}'s of these operators is performed in the Abelian Projection gauge. 
  In gauge field theories with asymptotic freedom, the short distance properties of Green's functions can be obtained on the basis of weak coupling perturbation expansions. Within this framework, the large momentum behavior of the structure functions for gluon, quark and ghost propagators is derived. The limits are found for general, covariant, linear gauges, and in all directions of the complex $k^2-$plane. Except for the coefficients, the functional forms of the leading asymptotic terms for the various structure functions are independent of the gauge parameter. They are determined exactly in terms of one-loop expressions (two-loop expressions in cases where one-loop terms vanish). With the exception of the Landau gauge, the asymptotic expressions for the gauge field propagator play an important r\^{o}le for the corresponding limits of quark and ghost propagators. For {\it all} gauges considered, it is the sign of the one-loop anomalous dimension coefficient of the gluon field in Landau gauge (as a fixed point of the gauge parameter) which is of considerable relevance for the asymptotics of the various propagators. The bounds obtained from the asymptotic expressions, together with the analytic properties of the structure functions, generally lead to un-subtracted dispersion representations. In special cases, for a limited number of flavors, sum rules are obtained for the discontinuities along the real axis. The sum rule for the gluon propagator is a generalization of the superconvergence relation derived previously in the Landau gauge. 
  It is shown that the scaling operators in the conformal limit of a two-dimensional field theory have massive form factors which obey a simple factorisation property in rapidity space. This has been used to identify such operators within the form factor bootstrap approach. A sum rule which yields the scaling dimension of such operators is also derived. 
  We first present a general method for extracting collective variables out of non-relativistic fermions by extending the gauge theory of collective coordinates to fermionic systems. We then apply the method to a system of non-interacting flavored fermions confined in a one-dimensional flavor-independent potential. In the limit of a large number of particles we obtain a Lagrangian with the Wess-Zumino-Witten term, which is the well-known Lagrangian describing the non-Abelian bosonization of chiral fermions on a circle. The result is universal and does not depend on the details of the confining potential. 
  By using the gauge Ward identities, we study correlation functions of gauged WZNW models. We show that the gauge dressing of the correlation functions can be taken into account as a solution of the Knizhnik-Zamolodchikov equation. Our method is analogous to the analysis of the gravitational dressing of 2D field theories. 
  We propose a hypothesis that all gauge theories are equivalent to a certain non-standard string theory. Different gauge groups are accounted for by weights ascribed to the world sheets of different topologies. The hypothesis is checked in the case of the compact abelian theories, where we show how condensing monopole -instanton fields are reproduced by the summation over surfaces. In the non-abelian case we prove that the loop equations are satisfied modulo contact terms. The structure of these terms unfortunately remains undetermined. 
  The strong coupling limit of a quantum system is in general quite complicated, but in some cases a great simplification occurs: the strongly coupled limit is equivalent to the weakly coupled limit of some other system. In string theory conjectures of this type go back several years, but only in the past year and a half has it been understood to be a general principle applying to all string theories. This has improved our understanding of string dynamics, including quantum gravity, in many new and sometimes surprising ways. I describe these developments and put them in the context of the search for the unified theory of particle physics and gravity. 
  We study T-duality for open strings in various $D$-manifolds in the approach of canonical transformations. We show that this approach is particularly useful to study the mapping of the boundary conditions since it provides an explicit relation between initial and dual variables. We consider non-abelian duality transformations and show that under some restrictions the dual is a curved $(d-{\rm dim}G-1)$ D-brane, where $d$ is the dimension of the space-time and $G$ the non-abelian symmetry group. The generalization to $N=1$ supersymmetric sigma models with abelian and non-abelian isometries is also considered. 
  We study the thermodynamics of open superstrings in the presence of $p$-dimensional D-branes. We get some finite temperature dualities relating the one-loop canonical free energy of open strings to the self-energy of D-branes at dual temperature. For the open bosonic string the inverse dual temperature is, as expected, the dual length under T-duality, $4\pi^{2}\alpha^{'}/\beta$. On the contrary, for the $SO(N)$, type-I superstring the dual temperature is given by $\beta$-duality, $2\pi^{2}\alpha^{'}/\beta$. We also study the emergence of the Hagedorn singularity in the dual description as triggered by the coupling of the D-brane to unphysical tachyons as well as the high temperature limit. 
  We review the applications of the integral over anticommuting Grassmann variables (nonquantum fermionic fields) to the analytic solutions and the field-theoretical formulations for the 2D Ising models. The 2D Ising model partition function $Q$ is presentable as the fermionic Gaussian integral. The use of the spin-polynomial interpretation of the 2D Ising problem is stressed, in particular. Starting with the spin-polynomial interpretation of the local Boltzmann weights, the Gaussian integral for $Q$ appears in the universal form for a variety of lattices, including the standard rectangular, triangular, and hexagonal lattices, and with the minimal number of fermionic variables (two per site). The analytic solutions for the correspondent 2D Ising models then follow by passing to the momentum space on a lattice. The symmetries and the question on the location of critical point have an interesting interpretation within this spin-polynomial formulation of the problem. From the exact lattice theory we then pass to the continuum-limit field-theoretical interpretation of the 2D Ising models. The continuum theory captures all relevant features of the original models near $T_c$. The continuum limit corresponds to the low-momentum sector of the exact theory responsible for the critical-point singularities and the large-distance behaviour of correlations. The resulting field theory is the massive two-component Majorana theory, with mass vanishing at $T_c$. By doubling of fermions in the Majorana representation, we obtain as well the 2D Dirac field theory of charged fermions for 2D Ising models. The differences between particular 2D Ising lattices are merely adsorbed, in the field-theoretical formulation, in the definition of the effective mass. 
  We propose a configuration of D-branes welded by analogous orbifold operation to be responsible for the enhancement of $SO(2N)$ gauge symmetry in type II string compactified on the $D_n$-type singular limit of K3. Evidences are discussed from the $D_n$-type ALE and D-manifold point of view. A subtlety regarding the ability of seeing the enhanced $SO(2N)$ gauge symmetry perturbatively is briefly addressed. 
  We build the trigonometric solutions of the Yang-Baxter equation that can not be obtained from quantum groups in any direct way. The solution is obtained using the construction suggested recently from the rational conformal field theory corresponding to the WZW model on $SO(3)_{4 R}=SU(2)_{4 R} / Z_{2}$. We also discuss the full elliptic solution to the Yang-Baxter equation whose critical limit corresponds to the trigonometric solution found below. 
  The string propagation equations in axisymmetric spacetimes are exactly solved by quadratures for a planetoid Ansatz. This is a straight non-oscillating string, radially disposed, which rotates uniformly around the symmetry axis of the spacetime. In Schwarzschild black holes, the string stays outside the horizon pointing towards the origin. In de Sitter spacetime the planetoid rotates around its center. We quantize semiclassically these solutions and analyze the spin/(mass$^2$) (Regge) relation for the planetoids, which turns out to be non-linear. 
  We study orbifold compactifications of F-theory which lead to $N=1$ supersymmetry in 6 and 4 spacetime dimensions. These are dual to specific orientifolds of M-theory, and in many cases to orientifolds of type IIB string theory. The equivalences are demonstrated by mapping the orbifolding transformations in the F, M and string theories to each other using dualities. We observe that M and F-theory appear to possess a property similar to discrete torsion in string theory. This is related to an ambiguity recently noted by Polchinski in the orientifold projection for 6-dimensional models. The 4-dimensional compactifications exhibit similar features, from which we predict the existence of certain new orientifolds of type IIB. Some orbifolds with higher supersymmetry are also examined. 
  Low--energy limits of N=2 supersymmetric field theories in the Higgs branch are described in terms of a non--linear 4--dimensional sigma--model on a \hk target space, classically obtained as a \hk quotient of the original flat hypermultiplet space by the gauge group. We review in a pedagogical way this construction, and illustrate it in various examples, with special attention given to the singularities emerging in the low--energy theory. In particular, we thoroughly study the Higgs branch singularity of Seiberg--Witten $SU(2)$ theory with $N_f$ flavors, interpreted by Witten as a small instanton singularity in the moduli space of one instanton on $\RR^4$. By explicitly evaluating the metric, we show that this Higgs branch coincides with the Higgs branch of a $U(1)$ N=2 SUSY theory with the number of flavors predicted by the singularity structure of Seiberg--Witten's theory in the Coulomb phase. We find another example of Higgs phase duality, namely between the Higgs phases of $U(N_c)\; N_f$ flavors and $U(N_f-N_c)\; N_f$ flavors theories, by using a geometric interpretation due to Biquard et al. This duality may be relevant for understanding Seiberg's conjectured duality $N_c \leftrightarrow N_f-N_c$ in N=1 SUSY $SU(N_c)$ gauge theories. 
  Models that provide experimentally testable violations of ordinary Quantum Mechanics have been recently proposed. These models are based on non-unitary time evolutions of density matrices that are generated by linear positive maps. We discuss the consequences of imposing a stronger condition on those maps, known as complete positivity. It turns out that experimental data on the neutral kaon system giving upper bounds to the parameters characterizing positive maps, also give bounds to those determining completely positive ones. 
  The operator product expansion in four-dimensional superconformal field theory is discussed. The OPE takes a particularly simple form for chiral operators, in $N=1$ and $N=2$, and for analytic operators, in $N=2$ and $N=4$. It is argued that the Green's functions of such operators can be determined up to constants. 
  Configuration space heat-kernel methods are used to evaluate the determinant and hence the effective action for an SU(2) doublet of fermions in interaction with a {\it covariantly constant} SU(2) background field. Exact results are exhibited which are applicable to {\it any} Abelian background on which the only restriction is that $(B^{2}-E^{2})$ and $E\cdot B$ are constant. Such fields include the uniform field and the plane wave field. The fermion propagator is also given in terms of gauge covariant objects. An extension to include finite temperature effects is given and the probability for creation of fermions from the vacuum at finite temperature in the presence of an electric field is discussed. 
  We discuss the quantization of pure string--inspired dilaton--gravity in $(1+1)$--dimensions, and of the same theory coupled to scalar matter. We perform the quantization using the functional Schroedinger and BRST formalisms. We find, both for pure gravity and the matter--coupled theory, that the two quantization procedures give inequivalent ``physical'' results. 
  The infrared regime of fermionic Green and vertex functions is studied analytically within a geometric approach which simulates soft interactions by an {\it effective} theory of contours. Expanding the particle path integral in terms of dominant contours at large distances, all-order results in the coupling constant are obtained for the renormalized fermion propagator and a universal vertex function with physical characteristics close to those associated with the Isgur-Wise function in the weak decays of heavy mesons. The extension to the ultraviolet regime is scetched. 
  We present a `spinon formulation' of the $SU(n)_1$ Wess-Zumino-Witten models. Central to this approach are a set of massless quasi-particles, called `spinons', which transform in the representation ${\bf \bar{n}}$ of $su(n)$ and carry fractional statistics of angle $\theta = \pi/n$. Multi-spinon states are grouped into irreducible representations of the yangian $Y(sl_n)$. We give explicit results for the $su(n)$ content of these yangian representations and present $N$-spinon cuts of the WZW character formulas. As a by-product, we obtain closed expressions for characters of the $su(n)$ Haldane-Shastry spin chains. 
  We develop a method based on the generalised St\"uckelberg prescription for discussing bosonisation in the low energy regime of the SU(2) massive Thirring model in 2+1 dimensions. For arbitrary values of the coupling parameter the bosonised theory is found to be a nonabelian gauge theory whose physical sector is explicitly obtained. In the case of vanishing coupling this gauge theory can be identified with the SU(2) Yang-Mills Chern-Simons theory in the limit when the Yang-Mills term vanishes. Bosonisation identities for the fermionic current are derived. 
  The exact solution of N=2 supersymmetric QCD found by Seiberg and Witten includes a prediction for all multi-instanton contributions to the low-energy dynamics. In the case where the matter hypermultiplets are massless, the leading non-perturbative contribution at weak coupling comes from the two-instanton sector. We calculate this contribution from first principles using the exact two-instanton solution of the self-dual Yang-Mills equation found by Atiyah, Drinfeld, Hitchin and Manin. We find exact agreement with the predictions of Seiberg and Witten for all numbers of flavours. We also confirm their predictions for the case of massive hypermultiplets. 
  Recent developments in superstring theory have led to major advances in understanding. After reviewing where things stood earlier, we sketch the impact of recently identified dualities and D-branes, as well as a hidden eleventh dimension. 
  The usual concept of shape invariance is discussed and one extension of this concept is suggested. 
  We apply the gauge theory on $M_4\times Z_2$ geometry previously proposed by Konisi and Saito to the Weinberg-Salam model for electroweak interactions, especially in order to clarify the geometrical meaning of curvatures in this geometry. Considering the Higgs field to be a gauge field along $Z_2$ direction, we also discuss the BRST invariant gauge fixing in this theory. 
  The free action for massless Ramond-Ramond fields is derived from closed superstring field theory using the techniques of Siegel and Zwiebach. For the uncompactified Type IIB superstring, this gives a manifestly Lorentz-covariant action for a self-dual five-form field strength. Upon compactification to four dimensions, the action depends on a U(1) field strength from 4D N=2 supergravity. However, unlike the standard Maxwell action, this action is manifestly invariant under the electromagnetic duality transformation which rotates $F_{mn}$ into $\epsilon_{mnpq} F^{pq}$. 
  We consider families of geometries of D--dimensional space, described by a finite number of parameters. Starting from the De Witt metric we extract a unique integration measure which turns out to be a geometric invariant, i.e. independent of the gauge fixed metric used for describing the geometries. The measure is also invariant in form under an arbitrary change of parameters describing the geometries. We prove the existence of geometries for which there are no related gauge fixing surfaces orthogonal to the gauge fibers. The additional functional integration on the conformal factor makes the measure independent of the free parameter intervening in the De Witt metric. The determinants appearing in the measure are mathematically well defined even though technically difficult to compute. 
  We study the large N limit of the MATRIX valued Gross-Neveu model in 2<d<4 dimensions. The method employed is a combination of the approximate recursion formula of Polyakov and Wilson with the solution to the zero dimensional large N counting problem of Makeenko and Zarembo. The model is found to have a phase transition at a finite value for the critical temperature and the critical exponents are approximated by nu = 1/(2(d-2)) and eta=d-2. We test the validity of the approximation by applying it to the usual vector models where it is found to yield exact results to leading order in 1/N. 
  In this paper we study Dirac-Hestenes spinor fields (DHSF) on a four-dimensional Riemann-Cartan spacetime (RCST). We prove that these fields must be defined as certain equivalence classes of even sections of the Clifford bundle (over the RCST), thereby being certain particular sections of a new bundle named Spin-Clifford bundle (SCB). The conditions for the existence of the SCB are studied and are shown to be equivalent to the famous Geroch's theorem concerning to the existence of spinor structures in a Lorentzian spacetime. We introduce also the covariant and algebraic Dirac spinor fields and compare these with DHSF, showing that all the three kinds of spinor fields contain the same mathematical and physical information. We clarify also the notion of (Crumeyrolle's) amorphous spinors (Dirac-K\"ahler spinor fields are of this type), showing that they cannot be used to describe fermionic fields. We develop a rigorous theory for the covariant derivatives of Clifford fields (sections of the Clifford bundle (CB)) and of Dirac-Hestenes spinor fields. We show how to generalize the original Dirac-Hestenes equation in Minkowski spacetime for the case of a RCST. Our results are obtained from a variational principle formulated through the multiform derivative approach to Lagrangian field theory in the Clifford bundle. 
  Conventional weak-coupling Rayleigh-Schr\"odinger perturbation theory suffers from problems that arise from resonant coupling of successive orders in the perturbation series. Multiple-scale analysis, a powerful and sophisticated perturbative method that quantitatively analyzes characteristic physical behaviors occurring on various length or time scales, avoids such problems by implicitly performing an infinite resummation of the conventional perturbation series. Multiple-scale perturbation theory provides a good description of the classical anharmonic oscillator. Here, it is extended to study (1) the Heisenberg operator equations of motion and (2) the Schr\"odinger equation for the quantum anharmonic oscillator. In the former case, it leads to a system of coupled operator differential equations, which is solved exactly. The solution provides an operator mass renormalization of the theory. In the latter case, multiple-scale analysis elucidates the connection between weak-coupling perturbative and semiclassical nonperturbative aspects of the wave function. 
  We solve the Dyson--Schwinger equation for the quark propagator in a model with singular infrared behavior for the gluon propagator. We require that the solutions, easily found in configuration space, be tempered distributions and thus have Fourier transforms. This severely limits the boundary conditions that the solutions may satisify. The sign of the dimensionful parameter that characterizes the model gluon propagator can be either positive or negative. If the sign is negative, we find a unique solution. It is singular at the origin in momentum space, falls off like $1/p^2$ as $p^2\rightarrow +/-\infty$, and it is truly nonperturbative in that it is singular in the limit that the gluon--quark interaction approaches zero. If the sign of the gluon propagator coefficient is positive, we find solutions that are, in a sense that we exhibit, unconstrained linear combinations of advanced and retarded propagators. These solutions are singular at the origin in momentum space, fall off like $1/p^2$ asympotically, exhibit ``resonant--like" behavior at the position of the bare mass of the quark when the mass is large compared to the dimensionful interaction parameter in the gluon propagator model, and smoothly approach a linear combination of free--quark, advanced and retarded two--point functions in the limit that the interaction approaches zero. In this sense, these solutions behave in an increasingly ``particle--like" manner as the quark becomes heavy. The Feynman propagator and the Wightman function are not tempered distributions and therefore are not acceptable solutions to the Schwinger--Dyson equation in our model. On this basis we advance several arguments to show that the Fourier--transformable solutions we find are consistent with quark confinement, even though they have singularities on the 
  Microscopic tests of the exact results are performed in N=2 SU(2) supersymmetric QCD. We construct the multi-instanton solution in N=2 supersymmetric QCD and calculate the two-instanton contribution ${\cal F}_2$ to the prepotential ${\cal F}$ explicitly. For $N_f=1,2$, instanton calculus agrees with the prediction of the exact results, however, for $N_f=3$, we find a discrepancy between them. 
  In N=2 ungauged supergravity we have found the most general double-extreme dyonic black holes with arbitrary number n_v of constant vector multiplets and n_h of constant hypermultiplets. They are double-extreme: 1) supersymmetric with coinciding horizons, 2) the mass for a given set of quantized charges is extremal. The spacetime is of the Reissner-Nordstrom form and the vector multiplet moduli depend on dyon charges. As an example we display n_v complex moduli as functions of 2(n_v+1) electric and magnetic charges in a model related to a classical Calabi-Yau moduli space. A specific case includes the complex S, T, U moduli depending on 4 electric and 4 magnetic charges of 4 U(1) gauge groups. 
  Tests of duality between heterotic strings on $K3\times T^2$ (restricted on certain Narain moduli subspaces) and type IIA strings on K3-fibered Calabi-Yau threefolds are attempted in the weak coupling regime on the heterotic side by identifying pertinent modular forms related to the computations of string threshold corrections. Concretely we discuss in parallel the three cases associated with Calabi-Yau manifolds $(A):X(6,2,2,1,1)_{2}^{-252}$, $(B):X(12,8,2,1,1)_{3}^{-480}$ and $(C):X(10,3,3,2,2)_{4}^{-132}$ on the type IIA side. 
  Equations of motion and the lagrangian are derived explicitely for Dual D=10, N=1 Supergravity considered as a field theory limit of a Fivebrane. It is used the mass-shell solution of Heterotic String Bianchi Identites obtained in the 2-dimensional $\sigma$-model two-loop approximation and in the tree-level Heterotic String approximation. As a result the Dual Supergravity lagrangian is derived in the one-loop Five-Brane approximation and in the lowest 6-dimensional $\sigma$-model approximaton. 
  The quantum mechanical concept of quasi-exact solvability is based on the idea of partial algebraizability of spectral problem. This concept is not directly extendable to the systems with infinite number of degrees of freedom. For such systems a new concept based on the partial Bethe Ansatz solvability is proposed. In present paper we demonstrate the constructivity of this concept and formulate a simple method for building quasi-exactly solvable field theoretical models on a one-dimensional lattice. The method automatically leads to local models described by hermitian hamiltonians. 
  We study an ensemble of branched polymers which are embedded on other branched polymers. This is a toy model which allows us to study explicitly the reaction of a statistical system on an underlying geometrical structure, a problem of interest in the study of the interaction of matter and quantized gravity. We find a phase transition at which the embedded polymers begin to cover the basis polymers. At the phase transition point the susceptibility exponent $\gamma$ takes the value 3/4 and the two-point function develops an anomalous dimension 1/2. 
  We prove in two different ways that brane dynamics gives rise to all H-monopoles predicted by S-duality of N=4, 4-dimensional superstring compactifications. One method uses heterotic-type II duality and the self-dual 3-brane dynamics, the other uses the 5-brane description of small instantons in SO(32) superstrings. 
  We extend our study of the field-theoretic description of matrix-vector models and the associated many-body problems of one dimensional particles with spin. We construct their Yangian-su(R) invariant Hamiltonian. It describes an interacting theory of a c=1 collective boson and a k=1 su(R) current algebra. When $R \geq 3$ cubic-current terms arise. Their coupling is determined by the requirement of the Yangian symmetry. The Hamiltonian can be consistently reduced to finite-dimensional subspaces of states, enabling an explicit computation of the spectrum which we illustrate in the simplest case. 
  We consider the standard vector gauging of Lorentz group $ SO(3,1) $ WZW model by its non-semisimple null Euclidean subgroup in two dimensions $ E(2) $. The resultant effective action of the theory is seen to describe a one dimensional bosonic field in the presence of external charge that we interpret it as a Liouville field. Gauging a boosted $ SO(3) $ subgroup, we find that in the limit of the large boost, the theory can be interpreted as an interacting Toda theory. We also take the generalized non-standard bilinear form for $SO(3,1) $ and gauge both $ SO(3) $ and $E(2)$ subgroups and discuss the resultant theories. 
  Some algebraic aspects of field quantization in space-time with boundaries are discussed. We introduce an associative algebra, whose exchange properties are inferred from the scattering processes in integrable models with reflecting boundary conditions on the half line. The basic properties of this algebra are established and the Fock representations associated with certain involutions are derived. We apply these results for the construction of quantum fields and for the study of scattering on the half line. 
  We develop a mathematical concept towards gauge field theories based upon a Hilbert space endowed with a representation of a skew-adjoint Lie algebra and an action of a generalized Dirac operator. This concept shares common features with the non-commutative geometry a la Connes/Lott, differs from that, however, by the implementation of skew-adjoint Lie algebras instead of unital associative *-algebras. We present the physical motivation for our approach and sketch its mathematical strategy. Moreover, we comment on the application of our method to the standard model and the flipped SU(5) x U(1)-grand unification model. 
  We calculate the self-energy at finite temperature in scalar $\lambda\phi ^4$ theory to second order in a modified perturbation expansion. Using the renormalisation group equation to tame the logarithms in momentum, it gives an equation to determine the critical temperature. Due to the infrared freedom of the theory, this equation is satisfied, irrespective of the value of the temperature. We conclude that there is no second order phase transition in this theory. 
  The two flavour, four dimensional WZW model coupled to electromagnetism, is treated as a constraint system in the context of the Faddeev-Jackiw approach. No approximation is made. Detailed exposition of the calculations is given. Solution of the constraints followed by proper Darboux's transformations leads to an unconstrained Coulomb-gauge Lagrangian density. 
  We propose a Wilsonian action compatible with special geometry and higher dimension N=2 corrections, and show that the holomorphic contribution F to the low energy effective action is independent of the infrared cutoff. We further show that for asymptotically free SU(2) super Yang-Mills theories, the infrared cutoff can be tuned to cancel leading corrections to F. We also classify all local higher-dimensional contributions to the N=2 superspace effective action that produce corrections to the Kahler potential when reduced to N=1 superspace. 
  We study phase transitions induced by topological defects in Abelian gauge theories of open p-branes in (d+1) space-time dimensions. Starting from a massive antisymmetric tensor theory for open p-branes we show how the condensation of topological defects can lead to a decoupled phase with a massless tensor coupled to closed (p-1)-branes and a massive tensor coupled to open (p+1)-branes. We also consider the case, relevant in string theory, in which the boundaries of the p-branes are constrained to live on a Dirichlet n-branes. 
  We compute the non-perturbative superpotential in $F$-theory compactification to four dimensions on a complex three-fold $\P^1\times S$, where $S$ is a rational elliptic surface. In contrast to examples considered previously, the superpotential in this case has interesting modular properties; it is essentially an $E_8$ theta function. 
  We study compactifications of the $N=2$ 6D tensionless string on various complex two-folds down to two-dimensions. In the IR limit they become non-trivial conformal field theories in 2D. Using results of Vafa and Witten on the partition functions of twisted Super-Yang-Mills theories, we can study the resulting CFT. We also discuss the contribution of instantons made by wrapping strings on 2-cycles of the complex two-fold. 
  A new $N=1$ superfield theory in $D=4$ flat superspace is suggested. It describes dynamics of supergravity compensator and can be considered as a low-energy limit for $N=1$, $D=4$ superfield supergravity. The theory is shown to be renormalizable in infrared limit and infrared free. A quantum effective action is investigated in infrared domain. 
  Starting with a Hilbert space endowed with a representation of a unitary Lie algebra and an action of a generalized Dirac operator, we develop a mathematical concept towards gauge field theories. This concept shares common features with the non--commutative geometry a la Connes/Lott, differs from that, however, by the implementation of unitary Lie algebras instead of associative *-algebras. The general scheme is presented in detail and is applied to functions $\otimes$ matrices. 
  The mathematical framework for an exact quantization of the two-dimensional coset space sigma-models coupled to dilaton gravity, that arise from dimensional reduction of gravity and supergravity theories, is presented. Extending previous results the two-time Hamiltonian formulation is obtained, which describes the complete phase space of the model in the isomonodromic sector. The Dirac brackets arising from the coset constraints are calculated. Their quantization allows to relate exact solutions of the corresponding Wheeler-DeWitt equations to solutions of a modified (Coset) Knizhnik-Zamolodchikov system. On the classical level, a set of observables is identified, that is complete for essential sectors of the theory. Quantum counterparts of these observables and their algebraic structure are investigated. Their status in alternative quantization procedures is discussed, employing the link with Hamiltonian Chern-Simons theory. 
  We present the construction of the standard model within the framework of non--associative geometry. For the simplest scalar product we get the tree--level predictions $m_W=\frac{1}{2} m_t\,,$ $m_H=\frac{3}{2} m_t$ and $\sin^2 \theta_W= \frac{3}{8}.$ These relations differ slightly from predictions derived in non--commutative geometry. 
  We propose a method of constructing a gauge invariant canonical formulation for non-gauge classical theory which depends on a set of parameters. Requirement of closure for algebra of operators generating quantum gauge transformations leads to restrictions on parameters of the theory. This approach is then applied for illustration to bosonic string theory coupled to background tachyonic field. It is shown that within the proposed canonical formulation the known mass-shell condition for tachyon is produced. 
  We propose a method of constructing a gauge invariant canonical formulation for non-gauge classical theory which depends on a set of parameters. Requirement of closure for algebra of operators generating quantum gauge transformations leads to restrictions on parameters of the theory. This approach is then applied to bosonic string theory coupled to massive background fields. It is shown that within the proposed canonical formulation the correct linear equations of motion for background fields arise. 
  We develop a method of computing the excited state energies in Integrable Quantum Field Theories (IQFT) in finite geometry, with spatial coordinate compactified on a circle of circumference R. The IQFT ``commuting transfer-matrices'' introduced by us (BLZ) for Conformal Field Theories (CFT) are generalized to non-conformal IQFT obtained by perturbing CFT with the operator $\Phi_{1,3}$. We study the models in which the fusion relations for these ``transfer-matrices'' truncate and provide closed integral equations which generalize the equations of Thermodynamic Bethe Ansatz to excited states. The explicit calculations are done for the first excited state in the ``Scaling Lee-Yang Model''. 
  We show, using the large $N$ limit, that there is a non--trivial scale invariant action for four dimensional scalar field theory. We investigate the possibility that the scalar sector of the standard model of particle physics has such a scale invariant action, with scale invariance being spontaneously broken by the vacuum expectation value of the scalar. This leads to a prediction for the mass of the lightest massive scalar particle (the Higgs particle) to be $5.4$ Tev. 
  We develop a simple computational tool for $SU(3)$ analogous to Bargmann's calculus for $SU(2)$. Crucial new inputs are, (i) explicit representation of the Gelfand-Zetlin basis in terms of polynomials in four variables and positive or negative integral powers of a fifth variable (ii) an auxiliary Gaussian measure with respect to which the Gelfand-Zetlin states are orthogonal but not normalized (iii) simple generating functions for generating all basis states and also all invariants. As an illustration of our techniques, an algebraic formula for the Clebsch-Gordan coefficients is obtained for the first time. This involves only Gaussian integrations. Thus $SU(3)$ is made as accessible for computations as $SU(2)$ is. 
  We analyze the zero mass black holes that arise as classical solutions to low energy heterotic string theory. Though these solutions contain naked singularities, it has been conjectured that they should be allowed in the theory. We find a solution describing a pair of oppositely charged massless black holes in uniformly accelerated motion under no external force. By analytically continuing the solution to Euclidean time, we find an instanton mediating the pair production of these objects in Minkowski space. We analyze the creation rate, and discuss some consequences of the result. 
  We introduce Z_3-graded objects which are the generalization of the more familiar Z_2-graded objects that are used in supersymmetric theories and in many models of non-commutative geometry. First, we introduce the Z_3-graded Grassmann algebra, and we use this object to construct the Z_3-matrices, which are the generalizations of the supermatrices. Then, we generalize the concepts of supertrace and superdeterminant. 
  The statistical-mechanical origin of the Bekenstein-Hawking entropy $S^{BH}$ in the induced gravity is discussed. In the framework of the induced gravity models the Einstein action arises as the low energy limit of the effective action of quantum fields. The induced gravitational constant is determined by the masses of the heavy constituents. We established the explicit relation between statistical entropy of constituent fields and black hole entropy $S^{BH}$. 
  Four-dimensional supergravity theories are reinterpreted in a 12-dimensional F-theory framework. The O(8) symmetry of N=8 supergravity is related to a reduction of F-theory on T_8, with the seventy scalars formally associated, by O(8) triality, to a fully compactified four-form A_4. For the N=1 type I model recently obtained from the type IIB string on the Z orbifold, we identify the K\"ahler manifold of the untwisted scalars in the unoriented closed sector with the generalized Siegel upper-half plane Sp(8,R)/(SU(4) \times U(1)). The SU(4) factor reflects the holonomy group of Calabi-Yau fourfolds. 
  I review the covariant quantization of the closed fermionic string with (2,2) extended world-sheet supersymmetry on R^{2,2}. Results on n-point scattering amplitudes are presented, for tree- and one-loop world-sheets with arbitrary Maxwell instanton number. I elaborate the connection between Maxwell moduli, spectral flow, and instantons. It is argued that the latter serve to extend the Lorentz symmetry from U(1,1) to SO(2,2) by undoing the choice of spacetime complex structure. 
  We discuss the known microscopic interpretations of the Bekenstein-Hawking entropy for configurations of intersecting M-branes. In some cases the entropy scales as that of a massless field theory on the intersection. A different situation, found for configurations which reduce to 1-charge D=5 black holes or 2-charge D=4 black holes, is explained by a gas of non-critical strings at their Hagedorn temperature. We further suggest that the entropy of configurations reducing to 1-charge D=4 black holes is due to 3-branes moving within 5-branes. 
  We show that under variation of moduli fields $\phi$ the first law of black hole thermodynamics becomes $dM = {\kappa dA\over 8\pi} + \Omega dJ + \psi dq + \chi dp - \Sigma d\phi$, where $\Sigma$ are the scalar charges. We also show that the ADM mass is extremized at fixed $A$, $J$, $(p,q)$ when the moduli fields take the fixed value $\phi_{\rm fix}(p,q)$ which depend only on electric and magnetic charges. It follows that the least mass of any black hole with fixed conserved electric and magnetic charges is given by the mass of the double-extreme black hole with these charges. Our work allows us to interpret the previously established result that for all extreme black holes the moduli fields at the horizon take a value $\phi= \phi_{\rm fix}(p,q)$ depending only on the electric and magnetic conserved charges: $ \phi_{\rm fix}(p,q)$ is such that the scalar charges $\Sigma ( \phi_{\rm fix}, (p,q))=0$. 
  The prepotential $F(a_i)$, defining the low-energy effective action of the $SU(N)$ ${\cal N}=2$ SUSY Yang-Mills theories, satisfies an enlarged set of the WDVV-like equations $F_iF_k^{-1}F_j = F_jF_k^{-1}F_i$ for any triple $i,j,k = 1,\ldots,N-1$, where matrix $F_i$ is equal to $(F_i)_{mn} = \partial^3 F/\partial a_i\partial a_m\partial a_n$. The same equations are actually true for generic topological theories. In contrast to the conventional formulation, when $k$ is restricted to $k=0$, in the proposed system there is no distinguished ``first'' time-variable, and indices can be raised with the help of any ``metric'' $\eta_{mn}^{(k)} = (F_k)_{mn}$, not obligatory flat. All the equations (for all $i,j,k$) are true simultaneously. This result provides a new parallel between the Seiberg-Witten theory of low-energy gauge models in $4d$ and topological theories. 
  Scale-invariant actions in arbitrary dimensions are investigated in curved space to clarify the relation between scale-, Weyl- and conformal invariance on the classical level. The global Weyl-group is gauged. Then the class of actions is determined for which Weyl-gauging may be replaced by a suitable coupling to the curvature (Ricci gauging). It is shown that this class is exactly the class of actions which are conformally invariant in flat space. The procedure yields a simple algebraic criterion for conformal invariance and produces the improved energy-momentum tensor in conformally invariant theories in a systematic way. It also provides a simple and fundamental connection between Weyl-anomalies and central extensions in two dimensions. In particular, the subset of scale-invariant Lagrangians for fields of arbitrary spin, in any dimension, which are conformally invariant is given. An example of a quadratic action for which scale-invariance does not imply conformal invariance is constructed. 
  We re-examine the question of the stability of quantum supermembranes. In the past, the instability of supermembranes was established by using a regulator, i.e. approximating the membrane by SU(N) super Yang-Mills theory and letting $N \rightarrow \infty$. In this paper, we (a) show that the instability persists even if we directly examine the continuum theory (b) give heuristic arguments that even a theory of unstable membranes at the Planck length may still be compatible with experiment (c) resolve a certain puzzling discrepancy between earlier works on the stability of supermembranes. Presented at the 2nd International Sakharov Conference in Moscow, May 1996. 
  The representation theory of the maximally extended superalgebra with 32 fermionic and 528 bosonic generators is developed in order to investigate non-perturbative properties of the democratic secret theory behind strings and other p-branes. The presence of Lorentz non-singlet central extensions is emphasized, their role for understanding up to 13 hidden dimensions and their physical interpretation as boundaries of p-branes is elucidated. The criteria for a new larger set of BPS-like non-perturbative states is given and the methods of investigation are illustrated with several explicit examples. 
  States obtained by projecting boundary states, associated with D-branes, to fixed mass-level and momentum generically define non-trivial cohomology classes. For on-shell states the cohomology is the standard one, but when the states are off-shell the relevant cohomology is defined using a BRST operator with ghost zero modes removed. The zero momentum cohomology falls naturally into multiplets of $SO(D-1,1)$. At the massless level, a simple set of D-brane configurations generates the full set of zero-momentum states of standard ghost number, including the discrete states. We give a general construction of off-shell cohomology classes, which exhibits a non-trivial interaction between left and right movers that is not seen in on-shell cohomology. This includes, at higher mass levels, states obtained from typical D-brane boundary states as well as states with more intricate ghost dependence. 
  We present new geometric formulations for the fractional spin particle models on the minimal phase spaces. New consistent couplings of the anyon to background fields are constructed. The relationship between our approach and previously developed anyon models is discussed. 
  We reexamine Bethe ansatz solutions of the massive Thirring model. We solve equations of periodic boundary conditions numerically without referring to the density of states. It is found that there is only one bound state in the massive Thirring model. The bound state spectrum obtained here is consistent with Fujita-Ogura's solutions of the infinite momentum frame prescription. Further, it turns out that there exist no solutions for string-like configurations. Instead, we find boson boson scattering states in 2-particle 2-hole configurations where all the rapidity variables turn out to be real. 
  Excitations of a vortex are usually considered in a linear approximation neglecting their backreaction on the vortex. In the present paper we investigate backreaction of Proca type excitations on a straightlinear vortex in the Abelian Higgs model. We propose exact Ansatz for fields of the excited vortex. From initial set of six nonlinear field equations we obtain (in a limit of weak excitations) two linear wave equations for the backreaction corrections. Their approximate solutions are found in the cases of plane wave and wave packet type excitations. We find that the excited vortex radiates vector field and that the Higgs field has a very broad oscillating component. 
  In this work we find static black hole solutions in the context of the two-dimensional dilaton gravity, which is modified by the addition of an $R^2$ term. This term arises from the one-loop effective action of a massive scalar field in its large mass expansion. The basic feature of this term is that it does not contribute to the Hawking radiation of the classical black hole backgrounds of the model. From this point of view a class of the solutions derived are interpreted as describing backreaction effects. In particular it is argued that evolution of a black hole via non-thermal signals is possible. Nevertheless this evolution seems to be 'soft', in the sense that it does not lead to the evaporation of a black hole, leaving the Hawking radiation as the dominant mechanism for this process. 
  The Yang-Mills functional integral is studied in an axial variant of 't Hooft's maximal Abelian gauge. In this gauge Gau\ss ' law can be completely resolved resulting in a description in terms of unconstrained variables. Compared to previous work along this line starting with work of Goldstone and Jackiw one ends up here with half as many integration variables, besides a field living in the Cartan subgroup of the gauge group and in D-1 dimension. The latter is of particular relevance for the infrared behaviour of the theory. Keeping only this variable we calculate the Wilson loop and find an area law. 
  Tensionless null p-branes in arbitrary cosmological backgrounds are considered and their motion equations are solved. It is shown that an ideal fluid of null p-branes may be considered as a source of gravity for D-dimensional Friedmann- Robertson-Walker universes. 
  The symmetry algebra of $N=1$ Super-Liouville field theory in two dimensions is the infinite dimensional $N=1$ superconformal algebra, which allows one to prove, that correlation functions, containing degenerated fields obey some partial linear differential equations. In the special case of four point function, including a primary field degenerated at the first level, this differential equations can be solved via hypergeometric functions. Taking into account mutual locality properties of fields and investigating s- and t- channel singularities we obtain some functional relations for three- point correlation functions. Solving this functional equations we obtain three-point functions in both Neveu-Schwarz and Ramond sectors. 
  We study heterotic $E_8\times E_8$ models that are dual to compactifications of F-theory and type IIA string on certain classes of elliptically fibered Calabi-Yau manifolds. Different choices for the specific torus in the fibration have heterotic duals that are most easily understood in terms of $E_8\times E_8$ models with gauge backgrounds of type $H\times U(1)^{8-d}$, where $H$ is a non-Abelian factor. The case with $d=8$ corresponds to the well known $E_8\times E_8$ compactifications with non-Abelian instanton backgrounds $(k_1,k_2)$ whose F-theory duals are built through compactifications on fibrations of the torus $\IP_2^{(1,2,3)}[6]$ over $\IF_n$. The new cases with $d < 8$ correspond to other choices for the elliptic fiber over the same base and yield unbroken $U(1)$'s, some of which are anomalous and acquire a mass by swallowing zero modes of the antisymmetric $B_{MN}$ field. We also study transitions to models with no tensor multiplets in $D=6$ and find evidence of $E_d$ instanton dynamics. We also consider the possibility of conifold transitions among spaces with different realization of the elliptic fiber. 
  We construct higher-spin N=1 super algebras as extensions of the super Virasoro algebra containing generators for all spins $s\ge 3/2$. We find two distinct classical (Poisson) algebras on the phase super space. Our results indicate that only one of them can be consistently quantized. 
  Different group structures which underline the integrable systems are considered. In some cases, the quantization of the integrable system can be provided with substituting groups by their quantum counterparts. However, some other group structures keep non-deformed in the course of quantizing the integrable system although their treatment is to be changed. Manifest examples of the KP/Toda hierarchy and the Liouville theory are considered. 
  We derive an analytic form for the Heisenberg-Euler Lagrangian in the limit where the component of the electric field parallel to the magnetic field is small. We expand these analytic functions to all orders in the field strength ($F_{\mu\nu}F^{\mu\nu}$) in the limits of weak and strong fields, and use these functions to estimate the pair-production rate in arbitrarily strong electric fields and the photon-splitting rate in arbitrarily strong magnetic fields. 
  Making use of the exact solutions of the $N=2$ supersymmetric gauge theories we construct new classes of superconformal field theories (SCFTs) by fine-tuning the moduli parameters and bringing the theories to critical points. In the case of SCFTs constructed from pure gauge theories without matter $N=2$ critical points seem to be classified according to the A-D-E classification as in the two-dimensional SCFTs. 
  A broad class of two-dimensional loop-corrected dilaton gravity models exhibit cosmological solutions that interpolate between the string perturbative vacuum and a background with asymptotically flat metric and linearly growing dilaton. The curvature singularities of the corresponding tree-level solutions are smoothed out, but no branch-change occurs. Thus, even in the presence of a non-perturbative potential, the system is not attracted by physically interesting fixed points with constant dilaton, and the exit problem of string cosmology persists. 
  We argue that correct account of the quantum properties of macroscopic objects which form reference frames (RF) demand the change of the standard space-time picture accepted in Quantum Mechanics. Galilean or Lorentz space-time transformations are shown to become incorrect in this case and for the description of transformations between different RF the special quantum space-time transformations are introduced. Consequently it results in the generalised Schrodinger equation which depends on the observer mass. The experiments with macroscopic coherent states are proposed in which this effects can be tested. 
  The regular solutions for the Ginzburg-Landau (-Nielsen-Olesen) Abelian gauge model are studied numerically. We consider the static isolated cylindrically symmetric configurations. The well known (Abrikosov) vortices, which present a particular example of such solutions, play an important role in the theory of type II superconductors and in the models of structure formation in the early universe. We find new regular static isolated cylindrically symmetric solutions which we call the type B and the flux tube solutions. In contrast to the pure vortex configurations which have finite energy, the new regular solutions possess a finite Gibbs free energy. The flux tubes appear to be energetically the most preferable configurations in the interval of external magnetic fields between the thermodynamic critical value $H_{c}$ and the upper critical field $H_{c_2}$, while the pure vortex dominate only between the lower critical field $H_{c_1}$ and $H_{c}$. Our conclusion is thus that type B and flux tube solutions are important new elements necessary for the correct understanding of a transition from the vortex state to the completely normal state. 
  Based on our earlier work on free field realizations of conformal blocks for conformal field theories with $SL(2)$ current algebra and with fractional level and spins, we discuss in some detail the fusion rules which arise. By a careful analysis of the 4-point functions, we find that both the fusion rules previously found in the literature are realized in our formulation. Since this is somewhat contrary to our expectations in our first work based on 3-point functions, we reanalyse the 3-point functions and come to the same conclusion. We compare our results on 4-point conformal blocks in particular with a different realization of these found by O. Andreev, and we argue for the equivalence. We describe in detail how integration contours have to be chosen to obtain convenient bases for conformal blocks, both in his and in our own formulation. We then carry out the rather lengthy calculation to obtain the crossing matrix between s- and t-channel blocks, and we use that to determine the monodromy invariant 4-point greens functions. We use the monodromy coefficients to obtain the operator algebra coefficients for theories based on admissible representations. 
  We outline a method of calculation of partition functions of orientable manifolds with fluctuating metric and perform the calculation for the specific case of the unit interval. 
  We analyze in detail the description of type IIB theory on a Calabi-Yau three-fold near a conifold singularity in terms of intersecting D-branes. In particular we study the singularity structure of higher derivative $F$-terms of the form $F_g W^{2g}$ where $W$ is the gravitational superfield. This singularity is expected to be due to a one -loop contribution from a charged soliton hypermultiplet becoming massless at the conifold point. In the intersecting D-brane description this soliton is described by an open string stretched between the two D-branes. After identifying the graviphoton vertex as a closed string operator we show that $F_g$'s have the expected singularity structure in the limit of vanishing soliton mass. 
  We discuss of the conceptual difficulties connected with the anticommutativity of classical fermion fields, and we argue that the "space" of all classical configurations of a model with such fields should be described as an infinite-dimensional supermanifold M.    We discuss the two main approaches to supermanifolds, and we examine the reasons why many physicists tend to prefer the Rogers approach although the Berezin-Kostant-Leites approach is the more fundamental one. We develop the infinite-dimensional variant of the latter, and we show that the functionals on classical configurations considered in a previous paper are nothing but superfunctions on M. We present a programme for future mathematical work, which applies to any classical field model with fermion fields. This programme is (partially) implemented in successor papers. 
  Using a supergeometric interpretation of field functionals developed in previous papers, we show that for quite a large class of systems of nonlinear field equations with anticommuting fields, infinite-dimensional supermanifolds (smf) of classical solutions can be constructed. Such systems arise in classical field models used for realistic quantum field theoretic models. In particular, we show that under suitable conditions, the smf of smooth Cauchy data with compact support is isomorphic with an smf of corresponding classical solutions of the model. 
  A distance function on the set of physical equivalence classes of Yang-Mills configurations considered by Feynman and by Atiyah, Hitchin and Singer is studied for both the $2+1$ and $3+1$-dimensional Hamiltonians. This set equipped with this distance function is a metric space, and in fact a Riemannian manifold as Singer observed. Furthermore, this manifold is complete. Gauge configurations can be used to parametrize the manifold. The metric tensor without gauge fixing has zero eigenvalues, but is free of ambiguities on the entire manifold. In $2+1$ dimensions the problem of finding the distance from any configuration to a pure gauge configuration is an integrable system of two-dimensional differential equations. A calculus of manifolds with singular metric tensors is developed and the Riemann curvature is calculated using this calculus. The Laplacian on Yang-Mills wave functionals has a slightly different form from that claimed earlier. In $3+1$-dimensions there are field configurations an arbitrarily large distance from a pure gauge configuration with arbitrarily small potential energy. These configurations resemble long-wavelength gluons. Reasons why there nevertheless can be a mass gap in the quantum theory are proposed. 
  We describe a model of massive matter fields interacting through higher-spin gauge fields in 2+1 dimensional space-time. The two main conclusions are that the parameter of mass $M$ appears as a module characterizing an appropriate vacuum solution of the full non-linear model and that $M$ affects a structure of a global vacuum higher-spin symmetry which leaves invariant the chosen vacuum solution. 
  The branching ratio is calculated for three different models of 2d gravity, using dynamical planar phi-cubed graphs. These models are pure gravity, the D=-2 Gaussian model coupled to gravity and the single spin Ising model coupled to gravity. The ratio gives a measure of how branched the graphs dominating the partition function are. Hence it can be used to estimate the location of the branched polymer phase for the multiple Ising model coupled to 2d gravity. 
  The rules for the free fermionic string model construction are extended to include general non-abelian orbifold constructions that go beyond the real fermionic approach. This generalization is also applied to the asymmetric orbifold rules recently introduced. These non-abelian orbifold rules are quite easy to use. Examples are given to illustrate their applications. 
  We generalize the rules for the free fermionic string construction to include other asymmetric orbifolds with world-sheet bosons. We use these rules to construct various grand unified string models that involve level-3 current algebras. We present the explicit construction of three classes of 3-chiral-family grand unified models in the heterotic string theory. Each model has 5 left-handed, and two right-handed families, and an adjoint Higgs. Two of them are SO(10) and the third is E_6. With Wilson lines and/or varying moduli, we show how other 3-family grand unified models, such as SO(10) and SU(5), may be constructed from them. 
  We study the BPS states of non-critical strings which arise for zero size instantons of exceptional groups. This is accomplished by using F-theory and M-theory duals and by employing mirror symmetry to compute the degeneracy of membranes wrapped around 2-cycles of the Calabi-Yau threefold. We find evidence for a number of novel physical phenomena, including having infinitely many light states with the first lightest state including a nearly massless gravitino. 
  Octonionic algebra being nonassociative is difficult to manipulate. We introduce left-right octonionic barred operators which enable us to reproduce the associative GL(8,R) group. Extracting the basis of GL(4,C), we establish an interesting connection between the structure of left-right octonionic barred operators and generic 4x4 complex matrices. As an application we give an octonionic representation of the 4-dimensional Clifford algebra. 
  We show how to obtain correctly normalized expressions for the Feynman diagrams of $\Phi^3$ theory with an internal $U(N)$ symmetry group, starting from tachyon amplitudes of the open bosonic string, and suitably performing the zero--slope limit by giving an arbitrary mass $m$ to the tachyon. In particular we present explicit results for the two--loop amplitudes of $\Phi^3$ theory, in preparation for the more interesting case of the multiloop amplitudes of non--abelian gauge theories. 
  Formulating quantum integrability for nonultralocal models (NM) parallel to the familiar approach of inverse scattering method is a long standing problem. After reviewing our result regarding algebraic structures of ultralocal models, we look for the algebra underlying NM. We propose an universal equation represented by braided Yang-Baxter equation and able to derive all basic equations of the known models like WZWN model, nonabelian Toda chain, quantum mapping etc. As further useful application we discover new integrable quantum NM, e.g. mKdV model, anyonic model, Kundu-Eckhaus equation and derive SUSY models and reflection equation from the nonultralocal view point. 
  We propose a generalization of non-commutative geometry and gauge theories based on ternary Z_3-graded structures. In the new algebraic structures we define, we leave all products of two entities free, imposing relations on ternary products only. These relations reflect the action of the Z_3-group, which may be either trivial, i.e. abc=bca=cab, generalizing the usual commutativity, or non-trivial, i.e. abc=jbca, with j=e^{(2\pi i)/3}. The usual Z_2-graded structures such as Grassmann, Lie and Clifford algebras are generalized to the Z_3-graded case. Certain suggestions concerning the eventual use of these new structures in physics of elementary particles are exposed. 
  The string duality revolution calls into question virtually all of the working assumptions of string model builders. A number of difficult questions arise. I use fractional charge as an example of a criterion which one would hope is robust beyond the weak coupling heterotic limit. Talk given at the 5th International Workshop on Supersymmetry and Unification of Fundamental Interactions (SUSY-96), University of Maryland, College Park, May 29 - June 1, 1996. 
  We consider the question of ground states in the supersymmetric system that arises in the search for the missing H-monopole states. By studying the effective theory near certain singularities in the five-brane moduli space, we find the remaining BPS states required by the conjectured S-duality of the toroidally compactified heterotic string. 
  I discuss the status of present knowledge about a possible background of relic gravitons left by an early, pre-big bang cosmological epoch, whose existence in the past of our Universe is suggested by the duality symmetries of string theory. 
  For an $S_4$ space-time manifold global aspects of gauge-fixing are investigated using the relation to Topological Quantum Field Theory on the gauge group. The partition function of this TQFT is shown to compute the regularized Euler character of a suitably defined space of gauge transformations. Topological properties of the space of solutions to a covariant gauge conditon on the orbit of a particular instanton are found using the $SO(5)$ isometry group of the $S_4$ base manifold. We obtain that the Euler character of this space differs from that of an orbit in the topologically trivial sector. This result implies that an orbit with Pontryagin number $\k=\pm1$ in covariant gauges on $S_4$ contributes to physical correlation functions with a different multiplicity factor due to the Gribov copies, than an orbit in the trivial $\k=0$ sector. Similar topological arguments show that there is no contribution from the topologically trivial sector to physical correlation functions in gauges defined by a nondegenerate background connection. We discuss possible physical implications of the global gauge dependence of Yang-Mills theory. 
  Implications of string-string dualities to cosmological string vacua are discussed. Cosmological vacua of classical string theories comprise of disjoint classses mapped one another by scale-factor T-duality. Each classes are, however, afflicted with initial/final cosmological singularities. It is argued that quantum string theories and string-string dualities dramatically resolve these cosmological singularities out so that disjoint classical cosmological vacua are continuously connected in a unified manner. A natural inflationary cosmology follows from this. 
  We investigate some aspects of the spectrum of D-branes and their interactions with closed strings. As argued earlier, a collection of many D-strings behaves, at large dilaton values, as a single multiply wound string. We use this result and T-duality transformations to show that a similar phenomenon occurs for effective strings produced by wrapping p-branes on a small (p-1)-dimensional torus, for suitable coupling. To understand the decay of an excited D-string at large dilaton values, we study the decay of an elementary string at small dilaton values. A long string, multiply wound on a circle, with a small excitation energy is found to predominantly decay into another string with the same winding number and an unwound closed string (rather than two wound strings). This decay amplitude agrees, under duality, with the decay amplitude computed using the Born-Infeld action for the D-string. We compute the absorption cross section for the D-brane model studied by Callan and Maldacena. The absorption cross section for the dilaton equals that for the scalars obtained by reduction of the graviton, and both agree with the cross section expected from a classical hole with the same charges. 
  The production rate of a soft photon from a hot quark-gluon plasma is computed to leading order at logarithmic accuracy. The canonical hard-thermal-loop resummation scheme leads to logarithmically divergent production rate due to mass singularities. We show that these mass singularities are screened by employing the effective hard-quark propagator, which is obtained through resummation of one-loop self-energy part in a self-consistent manner. The damping-rate part of the effective hard-quark propagator, rather than the thermal-mass part, plays the dominant role of screening mass singularities. Diagrams including photon--(hard-)quark vertex corrections also yield leading contribution to the production rate. 
  Effective superpotentials for the phase with a confined photon are obtained in $N=1$ supersymmetric gauge theories. We use the results to derive the hyperelliptic curves which describe the Coulomb phase of $N=2$ theories with classical gauge groups, and thus extending the prior result for $SU(N_c)$ gauge theory by Elitzur et al. Moreover, adjusting the coupling constants in $N=1$ effective superpotentials to the values of $N=2$ non-trivial critical points we find new classes of $N=1$ superconformal field theories with an adjoint matter with a superpotential. 
  Quaternion analysis is considered in full details where a new analyticity condition in complete analogy to complex analysis is found. The extension to octonions is also worked out. 
  The issue of local gauge invariance in the simplicial lattice formulation of gravity is examined. We exhibit explicitly, both in the weak field expansion about flat space, and subsequently for arbitrarily triangulated background manifolds, the exact local gauge invariance of the gravitational action, which includes in general both cosmological constant and curvature squared terms. We show that the local invariance of the discrete action and the ensuing zero modes correspond precisely to the diffeomorphism invariance in the continuum, by carefully relating the fundamental variables in the discrete theory (the edge lengths) to the induced metric components in the continuum. We discuss mostly the two dimensional case, but argue that our results have general validity. The previous analysis is then extended to the coupling with a scalar field, and the invariance properties of the scalar field action under lattice diffeomorphisms are exhibited. The construction of the lattice conformal gauge is then described, as well as the separation of lattice metric perturbations into orthogonal conformal and diffeomorphism part. The local gauge invariance properties of the lattice action show that no Fadeev-Popov determinant is required in the gravitational measure, unless lattice perturbation theory is performed with a gauge-fixed action, such as the one arising in the lattice analog of the conformal or harmonic gauges. 
  We derive exact analytical expressions for the critical amplitudes $A_\psi$, $A_{gap}$ in the scaling laws for the fermion condensate $<\bar \psi \psi> = A_\psi m^{1/3} g^{2/3}$ and for the mass of the lightest state $M_{gap} = A_{gap} m^{2/3} g^{1/3}$ in the Schwinger model with two light flavors, $m \ll g$. $A_\psi$ and $A_{gap}$ are expressed via certain universal amplitude ratios being calculated recently in TBA technique and the known coefficient $A_{\psi\psi}$ in the scaling law $<\bar \psi \psi (x) \bar \psi \psi (0)> = A_{\psi\psi} (g/x)$ at the critical point. Numerically, $A_\psi = -0.388..., A_{gap} = 2.008...$ . The same is done for the standard square lattice Ising model at $T = T_c$. Using recent Fateev's results, we get $<\sigma_{lat}> = 1.058... (H_{lat}/T_c)^{1/15}$ for the magnetization and $M_{gap} = a/\xi = 4.010... (H_{lat}/T_c)^{8/15}$ for the inverse correlation length ($a$ is the lattice spacing). The theoretical prediction for $<\sigma^{lat}>$ is in a perfect agreement with numerical data. Two available numerical papers give the values of $M_{gap}$ which differ from each other by a factor $\approx \sqrt{2}$ . The theoretical result for $M_{gap}$ agrees with one of them. 
  We study the evaporation process of 2D black holes in thermal equilibrium when the incoming radiation is turned off. Our analysis is based on two different classes of 2D dilaton gravity models which are exactly solvable in the semiclassical aproximation including back-reaction. We consider a one parameter family of models interpolating between the Russo-Susskind-Thorlacius and Bose-Parker-Peleg models. We find that the end-state geometry is the same as the one coming from an evaporating black hole formed by gravitational collapse. We also study the quantum evolution of black holes arising in a model with classical action $S = {1\over2\pi} \int d^2x \sqrt{-g} (R\phi + 4\lambda^2e^{\beta\phi})$. The black hole temperature is proportional to the mass and the exact semiclassical solution indicates that these black holes never disappear completely. 
  The perturbation theory with a variational basis is constructed and analyzed.The generalized Gaussian effective potential is introduced and evaluated up to the second order for selfinteracting scalar fields in one and two spatial dimensions. The problem of the renormalization of the mass is discussed in details. Thermal corrections are incorporated. The comparison between the finite temperature generalized Gaussian effective potential and the finite temperature effective potential is critically analyzed. The phenomenon of the restoration at high temperature of the symmetry broken at zero temperature is discussed. 
  We consider the compactification of type IIB superstring theory on K3 $\times$ K3. We obtain the massless spectrum of the resulting two dimensional theory and show that the model is free of gravitational anomaly. We then consider an orbifold and an orientifold projection of the above model and find that their spectrum match identically and are anomaly-free as well. This gives a dual pair of type IIB theory in two dimensions and can be understood as a consequence of SL(2, Z) symmetry of the ten dimensional theory. We also point out the M-theory duals of the type IIB compactifications considered here. 
  We give the details of the computation of the Chamseddine-Connes action by combination of a Lichnerowicz formula with the heat kernel expension. 
  I review the appearence of integrable structures in the formulation of exact nonperturbative solutions to $4d$ supersymmetric quantum gauge theories. Various examples of ${\cal N}\geq 2$ SUSY Yang-Mills nonperturbative solutions are adequately described in terms of the (deformations of the) finite-gap solutions to integrable models: through the generating differential and the $\tau$-function. One of the basic definitions of generating differential is discussed and its role in the theory of integrable systems is demonstrated. 
  A new approach to bosonization in relativistic field theories and many-body systems, based on the use of fermionic composites as integration variables in the Berezin integral defining the partition function of the system, is tested. The method is applied to the study of a simplified version of the BCS model. 
  The main definitions and properties of Lie superalgebras are proposed a la facon de a short dictionary, the different items following the alphabetical order. The main topics deal with the structure of simple Lie superalgebras and their finite dimensional representations; rather naturally, a few pages are devoted to supersymmetry. This modest booklet has two ambitious goals: to be elementary and easy to use. The beginner is supposed to find out here the main concepts on superalgebras, while a more experimented theorist should recognize the necessary tools and informations for a specific use. 
  A brief non-technical review of the recent study of classical integrable structures in quantum integrable systems is given. It is explained how to identify the standard objects of quantum integrable systems (transfer matrices, Baxter's $Q$-operators, etc) with elements of classical non-linear integrable difference equations ($\tau$-functions, Baker-Akhiezer functions, etc). The nested Bethe ansatz equations for $A_{k-1}$-type models emerge as discrete time equations of motion for zeros of classical $\tau$-functions and Baker-Akhiezer functions. The connection with discrete time Ruijsenaars-Schneider system of particles is discussed. 
  We study four dimensional $N=2$ supersymmetric gauge theories on $R^3 \times S^1$ with a circle of radius $R$. They interpolate between four dimensional gauge theories ($R=\infty$) and $N=4$ supersymmetric gauge theories in three dimensions ($R=0$). The vacuum structure can be determined quite precisely as a function of $R$, agreeing with three and four-dimensional results in the two limits. 
  We examine $(D-2)$-brane solutions in supergravities, showing that they fall into four categories depending on the details of the dilaton coupling. In general they describe domain walls, although in one of the four categories the metric describes anti-de Sitter spacetime. We study this case, and its $S^1$ dimensional reduction to a more conventional domain wall in detail, focussing in particular on the manner in which the unbroken supersymmetry of the anti-de Sitter solution is partially broken by the dimensional reduction to the domain wall. 
  We describe several infinite series of rational conformal field theories whose conformal characters are modular units, i.e. which are modular functions having no zeros or poles in the upper complex half plane, and which thus possess simple product expansions. We conjecture that certain infinite series of rational models of Casimir W-algebras always have this property. Furthermore, we describe an algorithm which can be used to prove whether a modular function is a modular unit or not. 
  We discuss the statistical-mechanical entropy of black hole calculated according to 't Hooft. It is argued that in presence of horizon the statistical mechanics of quantum fields depends on their UV behavior. The ``brick wall'' model was shown to provide a correct description when the ``brick wall'' parameter is less than any UV cut-off. 
  We suggest an approach to the problem of finding integral equations for the excited states of an integrable model, starting from the Thermodynamic Bethe Ansatz equations for its ground state. The idea relies on analytic continuation through complex values of the coupling constant, and an analysis of the monodromies that the equations and their solutions undergo. For the scaling Lee-Yang model, we find equations in this way for the one- and two- particle states in the spin-zero sector, and suggest various generalisations. Numerical results show excellent agreement with the truncated conformal space approach, and we also treat some of the ultraviolet and infrared asymptotics analytically. 
  From the equivalence of the bosonic and fermionic representations of finitized characters in conformal field theory, one can extract mathematical objects known as Bailey pairs. Recently Berkovich, McCoy and Schilling have constructed a `generalized' character formula depending on two parameters $\ra$ and $\r2$, using the Bailey pairs of the unitary model $M(p-1,p)$. By taking appropriate limits of these parameters, they were able to obtain the characters of model $M(p,p+1)$, $N=1$ model $SM(p,p+2)$, and the unitary $N=2$ model with central charge $c=3(1-{\frac{2}{p}})$. In this letter we computed the effective central charge associated with this `generalized' character formula using a saddle point method. The result is a simple expression in dilogarithms which interpolates between the central charges of these unitary models. 
  The low energy effective actions of the $N=2$ SUSY $SU(N_c)$ QCD are considered at the symmetric point on the moduli space. The classes of such theories have similar spectral curves. This fact allows us to show that all these models have the same structure of the coupling matrix and to show that the $N_f=2N_c$ spectral curve can not be presented as a double covering of the sphere. We calculate first instanton contributions to the coupling matrix and get nonperturbative $\beta$-functions in the $SU(2)$ gauge theory with non-zero bare masses of the matter hypermultiplets. 
  It is shown that the elliptic Ruijsenaars-Schneider model can be obtained from the affine Heisenberg Double by means of the Poisson reduction procedure. The dynamical $r$-matrix naturally appears in the construction. 
  We demonstrate that an action proposed by A. Khoudeir and N. R. Pantoja in {\sl Phys. Rev.} {\bf D53}, 5974 (1996) for endowing Maxwell theory with manifest electric--magnetic duality symmetry contains, besides the Maxwell field, additional propagating vector degrees of freedom. Hence it cannot be considered as a duality symmetric action for a {\it single} abelian gauge field. 
  We study gaugino condensation in presence of an anomalous $U(1)$ gauge group and find that global supersymmetry is dynamically broken. An example of particular interest is provided by effective string models with 4-dimensional Green-Schwarz anomaly cancellation mechanism. The structure of the hidden sector is constrained by the anomaly cancellation conditions and the scale of gaugino condensation is shifted compared with the usual case. We explicitly compute the resulting soft supersymmetry breaking terms. 
  We consider the simplest $SU_{q}(2)$ invariant fermionic hamiltonian and calculate the low and high temperature behavior for the two distinct cases $q>1$ and $q<1$. For low temperatures we find that entropy values for the Fermi case are an upper bound for those corresponding to $q\neq 1$. At high temperatures we find that the sign of the second virial coefficient depends on $q$, and vanishes at $q=1.96$.   An important consequence of this fact is that the parameter $q$ connects the fermionic and bosonic regions, showing therefore that $SU_{q}(2)$ fermions exhibit fractional statistics in three spatial dimensions. 
  Counterterms that are not reducible to $\zeta_{n}$ are generated by ${}_3F_2$ hypergeometric series arising from diagrams for which triangle and uniqueness relations furnish insufficient data. Irreducible double sums, corresponding to the torus knots $(4,3)=8_{19}$ and $(5,3)=10_{124}$, are found in anomalous dimensions at ${\rm O}(1/N^3)$ in the large-$N$ limit, which we compute analytically up to terms of level 11, corresponding to 11 loops for 4-dimensional field theories and 12 loops for 2-dimensional theories. High-precision numerical results are obtained up to 24 loops and used in Pad\'e resummations of $\varepsilon$-expansions, which are compared with analytical results in 3 dimensions. The ${\rm O}(1/N^3)$ results entail knots generated by three dressed propagators in the master two-loop two-point diagram. At higher orders in $1/N$ one encounters the uniquely positive hyperbolic 11-crossing knot, associated with an irreducible triple sum. At 12 crossings, a pair of 3-braid knots is generated, corresponding to a pair of irreducible double sums with alternating signs. The hyperbolic positive knots $10_{139}$ and $10_{152}$ are not generated by such self-energy insertions. 
  The existence of an electromagnectic field with parallel electric and magnetic components is readdressed in the presence of a gravitational field. A non-parallel solution is shown to exist. Next, we analyse the possibility of finding stationary gravitational waves in de nature. Finaly, We construct a D=4 effective quantum gravity model. Tree-level unitarity is verified. 
  Working in a Hamiltonian formulation with $A_0 = 0$ gauge and also in a path integral formulation, we show that the vacuum wave functional of four-dimensional pure Yang-Mills theory has the form of the exponential of a {\it three}-dimensional Yang-Mills action. This result implies that vacuum expectation values can be calculated in Yang-Mills theory but one dimension lower. Our analysis reveals that this dimensional reduction results from a stochastic nature of the theory. 
  The BRST property of Lukyanov's screening operators in the bosonic representation of the deformed Virasoro algebra is proven. 
  The thermal Euclidean Green functions for Photons propagating in the Rindler wedge are computed employing an Euclidean approach within any covariant Feynman-like gauge. This is done by generalizing a formula which holds in the Minkowskian case. The coincidence of the found $(\be=2\pi)$-Green functions and the corresponding Minkowskian vacuum Green functions is discussed in relation to the remaining static gauge ambiguity already found in previous papers. Further generalizations to more complicated manifolds are discussed. Ward identities are verified in the general case. 
  Physical aspects of the thermofield dynamics of the D=10 heterotic string theory are exemplified through the infrared behaviour of the one-loop dual symmetric cosmological constant in association with the global phase structure of the thermal string ensemble. 
  Taking into account the recent dualities we rederive the super p-brane scan. Our main results are the importance of the metric's signature and the existence of an S self-dual super 5-brane at D=14 with signature (7,7) or (11,3). 
  This paper contains the lecture notes of a short course on the quantization of gauge theories. Starting from a sketchy review of scattering theory, the paper describes the lines of BRST-Faddeev-Popov quantization considering the problem of a non-perturbative extension of this method. The connection between Slavnov-Taylor identity and S-matrix unitarity is also discussed. 
  We generalize duality invariance for the free Maxwell action in an arbitrary background geometry to include the presence of electric and magnetic charges. In particular, it follows that the actions of equally charged electric and magnetic black holes are equal. 
  String-bit models are both an efficient way of organizing string perturbation theory, and a possible non-perturbative composite description of string theory. This is a summary of ideas and results of string-bit and superstring-bit models, as presented in the Strings '96 conference. 
  The restrictions of target--space duality are imposed at the perturbative level on the holomorphic Wilsonian couplings that encode certain higher-order gravitational interactions in $N=2, D=4$ heterotic string compactifications. A crucial role is played by non-holomorphic corrections. The requirement of symplectic covariance and an associated symplectic anomaly equation play an important role in determining their form. For models which also admit a type-II description, this equation coincides with the holomorphic anomaly equation for type-II compactifications in the limit that a specific K\"ahler-class modulus grows large. We explicitly evaluate some of the higher-order couplings for a toroidal compactification with two moduli $T$ and $U$, and we express them in terms of modular forms. 
  We present a model for supersymmetric Yang-Mills theory in 10+2 dimensions. Our construction uses a constant null vector, and leads to a consistent set of field equations and constraints. The model is invariant under generalized translations and an extra gauge transformation. Ordinary dimensional reduction to ten dimensions yields the usual supersymmetric Yang-Mills equations, while dimensional reduction to 2+2 yields supersymmetric Yang-Mills equations in which the Poincar\'e supersymmetry is reduced by a null vector. We also give the corresponding formulation in superspace. 
  In the final few years of his life, Julian Schwinger proposed that the ``dynamical Casimir effect'' might provide the driving force behind the puzzling phenomenon of sonoluminescence. Motivated by that exciting suggestion, we have computed the static Casimir energy of a spherical cavity in an otherwise uniform material. As expected the result is divergent; yet a plausible finite answer is extracted, in the leading uniform asymptotic approximation. This result agrees with that found using zeta-function regularization. Numerically, we find far too small an energy to account for the large burst of photons seen in sonoluminescence. If the divergent result is retained, it is of the wrong sign to drive the effect. Dispersion does not resolve this contradiction. In the static approximation, the Fresnel drag term is zero; on the mother hand, electrostriction could be comparable to the Casimir term. It is argued that this adiabatic approximation to the dynamical Casimir effect should be quite accurate. 
  We present an approach to membrane quantization using matrix quantum mechanics at large N. We show that this leads (through a simple field theory of two-dimensional open strings and the associated SU(\infty) current algebra) to a 4-D dynamics of self-dual gravity plus matter. 
  The aim of these lectures is to describe a construction, as self-contained as possible, of renormalized gauge theories. Following a suggestion of Polchinski, we base our analysis on the Wilson renormalization group method. After a discussion of the infinite cut-off limit, we study the short distance properties of the Green functions verifying the validity of Wilson short distance expansion. We also consider the problem of the extension to the quantum level of the classical symmetries of the theory. With this purpose we analyze in details the breakings induced by the cut-off in a $SU(2)$ gauge symmetry and we prove the possibility of compensating these breakings by a suitable choice of non-gauge invariant counter terms. 
  We discuss the influence of gravitational effects on the stabilization of the chromomagnetic vacuum. The one-loop effective potential for a covariantly constant SU(2) gauge field in ${\bf S}^2 \times {\bf R}^2$ and ${\bf T}^2 \times {\bf R}^2$ is calculated. A possibility of curvature-induced phase transitions between zero and nonzero chromomagnetic vacua is found ---what is also confirmed through the calculation of the renormalization group (RG) improved effective potential on constant-curvature spaces with small curvature. Numerical evaluation indicates that for some curvatures the imaginary part of the effective potential disappears (gravitational stabilization of the chromomagnetic vacuum occurs). 
  Despite recent evidence indicating the existence of a new kind of self-dual six-dimensional superstring, no satisfactory worldsheet formulation of such a string has been proposed. In this note we point out that a theory built from Z_4 parafermions may have the right properties to describe the light-cone conformal field theory of this string. This indicates a possible worldsheet formulation of this theory. 
  A review is given of the canonical reduction of gauge and relativistic particle theories and of a new covariant rest-frame instant form of dynamics according to Dirac's theory of constraints 
  We show that the gauge-fermion interaction in multiflavour $(2+1)$-dimensional quantum electrodynamics with a finite infrared cut-off is responsible for non-fermi liquid behaviour in the infrared, in the sense of leading to the existence of a non-trivial fixed point at zero momentum, as well as to a significant slowing down of the running of the coupling at intermediate scales as compared with previous analyses on the subject. Both these features constitute deviations from fermi-liquid theory. Our discussion is based on the leading- $1/N$ resummed solution for the wave-function renormalization of the Schwinger-Dyson equations . The present work completes and confirms the expectations of an earlier work by two of the authors (I.J.R.A. and N.E.M.) on the non-trivial infrared structure of the theory. 
  We study a special class of higher derivative F-terms of the form $F_{g,n}W^{2g}(\Pi f)^{n}$ where W is the N=2 gravitational superfield and $\Pi$ is the chiral projector applied to a non-holomorphic function $f$ of the heterotic dilaton vector superfield. We analyze these couplings in the heterotic theory on $K3\times T^2$ , where it is found they satisfy an anomaly equation as the well studied $F_{g,0}$ terms. We recognize that, near a point of SU(2) enhancement, a given generating function of the leading singularity of the $F_{g,n}$ reproduces the free energy of a c=1 string at an arbitrary radius R. According to the N=2 heterotic-type II duality in 4d, we then study these couplings near a conifold singularity, using its local description in terms of intersecting D-5-branes. In this context, it turns out that there exists, among the other states involved, a vector gauge field reproducing the heterotic leading singularity structure. 
  Higher order anisotropic superspaces are constructed as generalized vector superbundles provided with compatible nonlinear connection, distinguished connection and metric structures. 
  Torsions, curvatures, structure equations and Bianchi identities for locally anisotropic superspaces (containing as particular cases different supersymmetric extensions and prolongations of Riemann, Finsler, Lagrange and Kaluza--Klein spaces) are investigated. 
  A general approach to formulation of supergravity in higher order anisotropic superspaces (containing as particular cases different supersymmetric extensions and prolongations of Riemann, Finsler, Lagrange and Kaluza--Klein spaces) is given. We analyze three models of locally anisotropic supergravity. 
  This paper studies the 1-loop approximation for a massless spin-1/2 field on a flat four-dimensional Euclidean background bounded by two concentric 3-spheres, when non-local boundary conditions of the spectral type are imposed. The use of $\zeta$-function regularization shows that the conformal anomaly vanishes, as in the case when the same field is subject to local boundary conditions involving projectors. A similar analysis of non-local boundary conditions can be performed for massless supergravity models on manifolds with boundary, to study their 1-loop properties. 
  New rigid string instanton equations are derived. Contrary to standard case, the equations split into three families. Their solutions in $R^4$ are discussed and explicitly presented in some cases. 
  A gauge independent method of obtaining the reduced space of constrained dynamical systems is discussed in a purely lagrangian formalism. Implications of gauge fixing are also considered. 
  We investigate the confining phase vacua of supersymmetric $Sp(2\NC)$ gauge theories that contain matter in both fundamental and antisymmetric representations. The moduli spaces of such models with $\NF=3$ quark flavors and $\NA=1$ antisymmetric field are analogous to that of SUSY QCD with $\NF=\NC+1$ flavors. In particular, the forms of their quantum superpotentials are fixed by classical constraints. When mass terms are coupled to $W_{(\NF=3,\NA=1)}$ and heavy fields are integrated out, complete towers of dynamically generated superpotentials for low energy theories with fewer numbers of matter fields can be derived. Following this approach, we deduce exact superpotentials in $Sp(4)$ and $Sp(6)$ theories which cannot be determined by symmetry considerations or integrating in techniques. Building upon these simple symplectic group results, we also examine the ground state structures of several $Sp(4) \times Sp(4)$ and $Sp(6) \times Sp(2)$ models. We emphasize that the top-down approach may be used to methodically find dynamical superpotentials in many other confining supersymmetric gauge theories. 
  These lectures begin by reviewing the evidence for S duality of the toroidally compactified heterotic string in 4d that was obtained in the period 1992--94. Next they review recently discovered dualities that relate all five of the 10d superstring theories and a quantum extension of 11d supergravity called M theory. The study of p-branes of various dimensions (some of which are D-branes) plays a central role. The final sections survey supersymmetric string vacua in 6d and some of the dual constructions by which they can be obtained. Special emphasis is given to a class of N=1 models that exhibit ``heterotic-heterotic duality.'' 
  We further discuss the N=2 superinstantons in SU(2) gauge theory, obtained from the general self-dual solutions of topological charge n constructed by Atiyah, Drinfeld, Hitchin and Manin (ADHM). We realize the N=2 supersymmetry algebra as actions on the superinstanton moduli. This allows us to recast in concise superfield notation our previously obtained expression for the exact classical interaction between n ADHM superinstantons mediated by the adjoint Higgs bosons, and moreover, to incorporate N_F flavors of hypermultiplets. We perform explicit 1- and 2-instanton checks of the Seiberg-Witten prepotentials for all N_F and arbitrary hypermultiplet masses. Our results for the low-energy couplings are all in precise agreement with the predictions of Seiberg and Witten except for N_F=4, where we find a finite renormalization of the coupling which is absent in the proposed solution. 
  The presence of Killing-Yano tensors implies the existence of non-standard supersymmetries in point particle theories on curved backgrounds. In a string theoretical context these are symmetries of the modes describing the particle-like behavior of the string. In the presence of isometries we show that, in addition to these, one can also define a new type of non-standard supersymmetry among a mixture of particle and winding modes. The interplay with T-duality is also examined and illustrated by explicit examples. 
  I review work developing the idea that string is a composite of point-like entities called string bits. Old and new insights this picture brings into the nature of string theory are discussed. This paper summarizes my talk presented to the Strings96 conference at Santa Barbara, CA, 14-20 July 1996. 
  We dicuss the resolution of initial value problems of electrodynamics in the Lorentz gauge. 
  Certain topological invariants of the moduli space of gravitational instantons are defined and studied. Several amplitudes of two and four dimensional topological gravity are computed. A notion of puncture in four dimensions, that is particularly meaningful in the class of Weyl instantons, is introduced. The topological embedding, a theoretical framework for constructing physical amplitudes that are well-defined order by order in perturbation theory around instantons, is explicitly applied to the computation of the correlation functions of Dirac fermions in a punctured gravitational background, as well as to the most general QED and QCD amplitude. Various alternatives are worked out, discussed and compared. The quantum background affects the propagation by generating a certain effective ``quantum'' metric. The topological embedding could represent a new chapter of quantum field theory. 
  We discuss non-trivial fixed points of the renormalization group with dual descriptions in $N=4$ gauge theories in three dimensions. This new duality acts as mirror symmetry, exchanging the Higgs and Coulomb branches of the theories. Quantum effects on the Coulomb branch arise classically on the Higgs branch of the dual theory. We present examples of dual theories whose Higgs/Coulomb branch are the ALE spaces and whose Coulomb/Higgs branches are the moduli space of instantons of the corresponding $ADE$ gauge group. In particular, we show that in three dimensions small $E_8$ instantons in string theory are described by a local quantum field theory. 
  We perform two-loop calculation of Chern-Simons in background field method using the hybrid regularization of higher-covariant derivative and dimensional regularization. It is explicitly shown that Chern-Simons field theory is finite at the two-loop level. This finiteness plays an important role in the relation of Chern-Simons theory with two-dimensional conformal field theory and the description of link invariant. 
  Laplace operators perturbed by meromorphic potential on the Riemann and separated type Klein surfaces are constructed and their indices are calculated by two different ways. The topological expressions for the indices are obtained from the study of spectral properties of the operators. Analytical expressions are provided by the Heat Kernel approach in terms of the functional integrals. As a result two formulae connecting characteristics of meromorphic (real meromorphic) functions and topological properties of Riemann (separated type Klein) surfaces are derived. 
  We study supersymmetric Sp(2N) gauge theories with matter in the antisymmetric tensor representation and F fundamentals. For F=6 we solve the theory exactly in terms of confined degrees of freedom and a superpotential. By adding mass terms we obtain the theories with F<6 which we find to exhibit a host of interesting non-perturbative phenomena: quantum deformed moduli spaces with N constraints, instanton-induced superpotentials and non-equivalent disjoint branches of moduli spaces. We find a simple dual for F=8 and no superpotential. We show how the F=4 and F=2 theories can be modified to break supersymmetry spontaneously and point out that the Sp(6) theory with F=6 may be very interesting for model builders. 
  We present here a field theory of the spinning electron, by writing down a new equation for the 4-velocity field v^mu (different from that of Dirac theory), which allows a classically intelligible description of the electron. Moreover, we make explicit the noticeable kinematical properties of such velocity field (which also result different from the ordinary ones). At last, we analyze the internal zitterbewegung (zbw) motions, for both time-like and light-like speeds. We adopt in this paper the ordinary tensorial language. Our starting point is the Barut-Zanghi classical theory for the relativistic electron, which related spin with zbw. This paper is dedicated to the memory of Asim O. Barut, who so much contributed to clarifying very many fundamental issues of physics, and whose work constitutes a starting point of these articles. 
  One of the most satisfactory pictures for spinning particles is the Barut-Zanghi (BZ) classical theory for the relativistic extended-like electron, that relates spin to zitterbewegung (zbw). The BZ motion equations constituted the starting point for recent works about spin and electron structure, co-authored by us, which adopted the Clifford algebra language. This language results to be actually suited and fruitful for a hydrodynamical re-formulation of the BZ theory. Working out, in such a way, a ``probabilistic fluid'', we are allowed to re-interpret the original classical spinors as quantum wave-functions for the electron. Thus, we can pass to ``quantize" the BZ theory employing this time the tensorial language, more popular in first-quantization. ``Quantizing'' the BZ theory, however, does not lead to the Dirac equation, but rather to a non-linear, Dirac--like equation, which can be regarded as the actual ``quantum limit'' of the BZ classical theory. Moreover, an original variational approach to the the BZ probabilistic fluid shows that it is a typical ``Weyssenhoff fluid'', while the Hamilton-Jacobi equation (linking together mass, spin and zbw frequency) appears to be nothing but a special case of de Broglie's famous energy-frequency relation. Finally, after having discussed the remarkable correlation between the gauge transformation U(1) and a general rotation on the spin plan, we clarify and comment on the two-valuedness nature of the fermionic wave-function, and on the parity and charge conjugation transformations. 
  Inserting the correct Lorentz factor into the definition of the 4-velocity v^mu for spinning particles entails new kinematical properties for v^2. The well-known constraint (identically true for scalar particles, but entering also the Dirac theory, and assumed a priori in all spinning particle models) p_mu v^mu = m is here derived in a self-consistent way. 
  Starting from the formal expressions of the hydrodynamical (or ``local'') quantities employed in the applications of Clifford Algebras to quantum mechanics, we introduce --in terms of the ordinary tensorial framework-- a new definition for the field of a generic quantity. By translating from Clifford into tensor algebra, we also propose a new (non-relativistic) velocity operator for a spin 1/2 particle. This operator is the sum of the ordinary part p/m describing the mean motion (the motion of the center-of-mass), and of a second part associated with the so-called zitterbewegung, which is the spin ``internal'' motion observed in the center-of-mass frame. This spin component of the velocity operator is non-zero not only in the Pauli theoretical framework, i.e. in presence of external magnetic fields and spin precession, but also in the Schroedinger case, when the wave-function is a spin eigenstate. In the latter case, one gets a decomposition of the velocity field for the Madelung fluid into two distinct parts: which constitutes the non-relativistic analogue of the Gordon decomposition for the Dirac current. We find furthermore that the zitterbewegung motion involves a velocity field which is solenoidal, and that the local angular velocity is parallel to the spin vector. In presence of a non-constant spin vector (Pauli case) we have, besides the component normal to spin present even in the Schroedinger theory, also a component of the local velocity which is parallel to the rotor of the spin vector. 
  An algebraic program of computation and characterization of higher loop BRST anomalies is presented. We propose a procedure for disentangling a genuine{\it local} higher loop anomaly from the quantum dressings of lower loop anomalies. For such higher loop anomalies we derive a local consistency condition, which is the generalisation of the Wess-Zumino condition for the one-loop anomaly. The development is presented in the framework of the field-antifield formalism, making use of a nonlocal regularization method. The theoretical construction is exemplified by explicitly computing the two-loop anomaly of chiral $W_3$ gravity. We also give, for the first time, an explicit check of the local two--loop consistency condition that is associated with this anomaly. 
  Global N=2 supersymmetry in four dimensions with a Fayet-Sohnius hypermultiplet and a complex central charge is studied in N = 2 superspace. It is shown how to construct the complete expansion of the hypermultiplet with respect to the central charge. In addition the low-energy effective action is discussed and it is shown that the `kernel' of the Lagrangian only needs an integration over a `small' superspace to construct a supersymmetric action. 
  A general structure of effective action in new chiral superfield model associated with $N=1$, $D=4$ supergravity is investigated. This model corresponds to finite quantum field theory and does not demand the regularization and renormalization at effective action calculation. It is shown that in local approximation the effective action is defined by two objects called general superfield effective lagrangian and chiral superfield effective lagrangian. A proper-time method is generalized for calculation of these two effective lagrangians in superfield manner. Power expansion of the effective action in supercovariant derivatives is formulated and the lower terms of such an expansion are calculated in explicit superfield form. 
  We discuss a T-duality transformation for the c=1/2 matrix model for the purpose of studying duality transformations in a possible toy example of nonperturbative frameworks of string theory. Our approach is to first investigate the scaling limit of the Schwinger-Dyson equations and the stochastic Hamiltonian in terms of the dual variables and then compare the results with those using the original spin variables. It is shown that the c=1/2 model in the scaling limit is T-duality symmetric in the sphere approximation. The duality symmetry is however violated when the higher-genus effects are taken into account, owing to the existence of global Z_2 vector fields corresponding to nontrivial homology cycles. Some universal properties of the stochastic Hamiltonians which play an important role in discussing the scaling limit and have been discussed in a previous work by the last two authors are refined in both the original and dual formulations. We also report a number of new explicit results for various amplitudes containing macroscopic loop operators. 
  Some formulas and speculations are presented relative to integrable systems and quantum mechanics. 
  We show that one-dimensional superspace is isomorphic to a non-trivial but consistent limit as $q\to-1$ of the braided line. Supersymmetry is identified as translational invariance along this line. The supertranslation generator and covariant derivative are obtained in the limit in question as the left and right derivatives of the calculus on the braided line. 
  An analogue of the Holstein-Primakoff and of the Dyson realization for the Lie superalgebra $sl(1/n)$ is written down. The expressions are formally the same as for the Lie algebra $sl(n+1)$, however in the latter the Bose operators have to be replaced with Fermi operators. 
  The known Holstein-Primakoff and Dyson realizations for the Lie algebras $gl(n+1),\;n=1,2,\ldots$ in terms of Bose operators are generalized to the class of the Lie superalgebras $gl(m/n+1)$ for any $n$ and $m$. Formally the expressions are the same as for $gl(m+n+1)$, however both Bose and Fermi operators are involved. 
  There is a close relation between duality in $N=2$ SUSY gauge theories and integrable models. In particular, the quantum moduli space of vacua of $N=2$ SUSY $SU(3)$ gauge theories coupled to two flavors of massless quarks in the fundamental representation can be related to the spectral curve of the Goryachev-Chaplygin top. Generalizing this to the cases with {\it massive} quarks, and $N_f = 0,1,2$, we find a corresponding integrable system in seven dimensional phase space where a hyperelliptic curve appears in the Painlev\'e test. To understand the stringy origin of the integrability of these theories we obtain exact nonperturbative point particle limit of type II string compactified on a Calabi-Yau manifold, which gives the hyperelliptic curve of $SU(2)$ QCD with $N_f =1$ hypermultiplet. 
  The static color-Coulomb interaction potential is calculated as the solution of a non-linear integral equation which arises in the Hamiltonian Coulomb gauge when the restriction to the interior of the Gribov horizon is implemented. The potential obtained is in qualitative agreement with expectations, being Coulombic with logarithmic corrections at short range, and confining at long range. The values obtained for the string tension and $\Lambda_{\overline{MS}}$ are in semi-quantitative agreement with lattice Monte Carlo and phenomenological determinations. 
  We study chiral induced gravity in the light-cone gauge and show that the theory is consistent for a particular choice of chiralities. The corresponding Kac--Moody central charge has no forbidden region of complex values. Generalized analysis of the critical exponents is given and their relation to the $SL(2,R)$ vacuum states is elucidated. All the parameters containing information about the theory can be traced back to the characteristics of the group of residual symmetry in the light--cone gauge. 
  All global solutions of arbitrary topology of the most general 1+1 dimensional dilaton gravity models are obtained. We show that for a generic model there are globally smooth solutions on any non-compact 2-surface. The solution space is parametrized explicitly and the geometrical significance of continuous and discrete labels is elucidated. As a corollary we gain insight into the (in general non-trivial) topology of the reduced phase space.  The classification covers basically all 2D metrics of Lorentzian signature with a (local) Killing symmetry. 
  A large class of supersymmetric extended objects is considered from the viewpoint of embeddings of super worldsurfaces into target superspaces. It is shown that a simple geometrical condition leads to the equations of motion for the brane in many cases. 
  We analyse the curvature representation of the gonihedric action $A(M)$ for the cases when the dependence on the dihedral angle is arbitrary. 
  We construct open descendants of Gepner models, concentrating mainly on the six-dimensional case, where they give type I vacua with rich patterns of Chan-Paton symmetry breaking and various numbers of tensor multiplets, including zero. We also relate the models in $D < 10$ without open sectors, recently found by other authors, to the generalized Klein-bottle projections allowed by the crosscap constraint. 
  We study the three dimensional Navier-Stokes equation with a random Gaussian force acting on large wavelengths. Our work has been inspired by Polyakov's analysis of steady states of two dimensional turbulence. We investigate the time evolution of the probability law of the velocity potential. Assuming that this probability law is initially defined by a statistical field theory in the basin of attraction of a renormalisation fixed point, we show that its time evolution is obtained by averaging over small scale features of the velocity potential. The probability law of the velocity potential converges to the fixed point in the long time regime. At the fixed point, the scaling dimension of the velocity potential is determined to be ${-{4\over 3}}$. We give conditions for the existence of such a fixed point of the renormalisation group describing the long time behaviour of the velocity potential. At this fixed point, the energy spectrum of three dimensional turbulence coincides with a Kolmogorov spectrum. 
  Maxwell equation $\dirac F = 0$ for $F \in \sec \bwe^2 M \subset \sec \clif (M)$, where $\clif (M)$ is the Clifford bundle of differential forms, have subluminal and superluminal solutions characterized by $F^2 \neq 0$. We can write $F = \psi \gamma_{21} \tilde \psi$ where $\psi \in \sec \clif^+(M)$. We can show that $\psi$ satisfies a non linear Dirac-Hestenes Equation (NLDHE). Under reasonable assumptions we can reduce the NLDHE to the linear Dirac-Hestenes Equation (DHE). This happens for constant values of the Takabayasi angle ($0$ or $\pi$). The massless Dirac equation $\dirac \psi =0$, $\psi \in \sec \clif^+ (M)$, is equivalent to a generalized Maxwell equation $\dirac F = J_{e} - \gamma_5 J_{m} = {\cal J}$. For $\psi = \psi^\uparrow$ a positive parity eigenstate, $j_e = 0$. Calling $\psi_e$ the solution corresponding to the electron, coming from $\dirac F_e =0$, we show that the NLDHE for $\psi$ such that $\psi \gamma_{21} \tilde{\psi} = F_e + F^{\uparrow}$ gives a linear DHE for Takabayasi angles $\pi/2$ and $3\pi/2$ with the muon mass. The Tau mass can also be obtained with additional hypothesis. 
  Quantum field theory of space-like particles is investigated in the framework of absolute causality scheme preserving Lorentz symmetry. It is related to an appropriate choice of the synchronization procedure (definition of time). In this formulation existence of field excitations (tachyons) distinguishes an inertial frame (privileged frame of reference) via spontaneous breaking of the so called synchronization group. In this scheme relativity principle is broken but Lorentz symmetry is exactly preserved in agreement with local properties of the observed world. It is shown that tachyons are associated with unitary orbits of Poincar\'e mappings induced from $SO(2)$ little group instead of $SO(2,1)$ one. Therefore the corresponding elementary states are labelled by helicity. The cases of the helicity $\lambda = 0$ and $\lambda = \pm\frac{1}{2}$ are investigated in detail and a corresponding consistent field theory is proposed. In particular, it is shown that the Dirac-like equation proposed by Chodos et al., inconsistent in the standard formulation of QFT, can be consistently quantized in the presented framework. This allows us to treat more seriously possibility that neutrinos might be fermionic tachyons as it is suggested by experimental data about neutrino masses. 
  In three dimensional Maxwell-Chern-Simons massless scalar electrodynamics with $ \phi^6$ coupling, the $U(1)$ symmetry is spontaneously broken at two loop order regardless of the presence or absence of the Maxwell term. Dimensional transmutation takes place in pure Chern-Simons scalar electrodynamics. The beta function for the $\phi^6$ coupling is independent of gauge couplings. 
  This paper provides a systematic description of the interplay between a specific class of reductions denoted as \cKPrm ($r,m \geq 1$) of the primary continuum integrable system -- the Kadomtsev-Petviashvili ({\sf KP}) hierarchy and discrete multi-matrix models. The relevant integrable \cKPrm structure is a generalization of the familiar $r$-reduction of the full {\sf KP} hierarchy to the $SL(r)$ generalized KdV hierarchy ${\sf cKP}_{r,0}$. The important feature of \cKPrm hierarchies is the presence of a discrete symmetry structure generated by successive Darboux-B\"{a}cklund (DB) transformations. This symmetry allows for expressing the relevant tau-functions as Wronskians within a formalism which realizes the tau-functions as DB orbits of simple initial solutions. In particular, it is shown that any DB orbit of a ${\sf cKP}_{r,1}$ defines a generalized 2-dimensional Toda lattice structure. Furthermore, we consider the class of truncated {\sf KP} hierarchies ({\sl i.e.}, those defined via Wilson-Sato dressing operator with a finite truncated pseudo-differential series) and establish explicitly their close relationship with DB orbits of \cKPrm hierarchies. This construction is relevant for finding partition functions of the discrete multi-matrix models.   The next important step involves the reformulation of the familiar non-isospectral additional symmetries of the full {\sf KP} hierarchy so that their action on \cKPrm hierarchies becomes consistent with the constraints of the reduction. Moreover, we show that the correct modified additional symmetries are compatible with the discrete DB symmetry on the \cKPrm DB orbits.   The above technical arsenal is subsequently applied to obtain complete 
  This thesis is devoted to trying to find a microscopic quantum description of black holes. We consider black holes in string theory which is a quantum theory of gravity. We find that the ``area law'' black hole entropy for extremal and near-extremal charged black holes arises from counting microscopic configurations. We study black holes in five and four spacetime dimensions. We calculate the Hawking temperature and give a physical picture of the Hawking decay process. Hopefully, the reader will find here a moderately self contained review of D-branes and string theory applied to black hole physics. 
  Bubbles are point-like regular solutions of the higher-dimensional Kaluza-Klein equations that appear as naked singularities in four dimensions. We analyze all such possible solutions in 5D Kaluza-Klein theory that are static and spherically symmetric. We show that they can be obtained by taking unusual choices of the parameters in the dyonic black hole solutions, and find that regularity can only be achieved if their electric charge is zero. However, they can be neutral or possess magnetic charge. We study some of their properties, both in theories where the internal dimension is space-like as well as time-like. Since bubbles do not have horizons, they have no entropy, nor do they emit any thermal radiation, but they are, in general, non-extremal objects. In the two-timing case, it is remarkable that non-singular massless monopoles are possible, probably signaling a new pathology of these theories. These two-timing monopoles connect two asymptotically flat regions, and matter can flow from one region to the other. We also present a C-type solution that describes neutral bubbles in uniform acceleration, and we use it to construct an instanton that mediates the breaking of a cosmic string by forming bubbles at its ends. The rate for this process is also calculated. Finally, we argue that a similar solution can be constructed for magnetic bubbles, and that it can be used to describe a semiclassical instability of the two-timing vacuum against production of massless monopole pairs. 
  We formulate the flipped SU(5) x U(1)-GUT within the framework of non-associative geometry. It suffices to take the matrix Lie algebra su(5) as the input; the u(1)-part with its representation on the fermions is an algebraic consequence. The occurring Higgs multiplets (24,5,45,50-representations of su(5)) are uniquely determined by the fermionic mass matrix and the spontaneous symmetry breaking pattern to SU(3) x U(1). We find the most general gauge invariant Higgs potential that is compatible with the given Higgs vacuum. Our formalism yields tree-level predictions for the masses of all gauge and Higgs bosons. It turns out that the low-energy sector is identical with the standard model. In particular, there exists precisely one light Higgs field, whose upper bound for the mass is 1.45 m_t. All remaining 207 Higgs fields are extremely heavy. 
  We consider the actions for ten--dimensional $p$--branes and $D$--branes in an arbitrary curved background and discuss some of their properties. We comment on how the $SL(2,R)$ duality symmetry acts on the five--brane actions. 
  A reformulation of the superconformal Ward identities that combines all the superconformal currents and the associated parameters in one multiplet is given for theories with rigid N=1 or N=2 supersymmetry. This form of the Ward Identities is applied to spontaneously broken N=2 Yang-Mills theory and used to derive a condition on the low energy effective action. This condition is satisfied by the solution proposed by Seiberg and Witten. 
  We propose the ternary generalization of the classical anti-commutativity and study the algebras whose generators are ternary anti-commutative. The integral over an algebra with an arbitrary number of generators N is defined and the formula of a change of variables is proved. In analogy with the fermion integral we define an analogue of the Pfaffian for a cubic matrix by means of Gaussian type integral and calculate its explicit form in the case of N=3. 
  We define and study the ternary analogues of Clifford algebras. It is proved that the ternary Clifford algebra with $N$ generators is isomorphic to the subalgebra of the elements of grade zero of the ternary Clifford algebra with $N+1$ generators. In the case $N=3$ the ternary commutator of cubic matrices induced by the ternary commutator of the elements of grade zero is derived. We apply the ternary Clifford algebra with one generator to construct the $Z_3$-graded generalization of the simplest algebra of supersymmetries. 
  We show that the chiral soliton model recently introduced by Aglietti et al. can be made integrable by adding an attractive potential with a fixed coefficient. The modified model is equivalent to the derivative nonlinear Schr\"{o}dinger model which does not possess parity and Galilean invariance. We obtain explicit one and two classical soliton solutions and show that in the weak coupling limit, they correctly reproduce the bound state energy as well as the time delay of two-body quantum mechanics of the model. 
  The massive soliton theories describe integrable perturbations of WZW cosets as generalized multi-component sine-Gordon models. We study their coupling to 2-dim gravity in the conformal gauge and show that the resulting models can be interpreted as conformal non-Abelian Toda theories when a certain algebraic condition is satisfied. These models, however, do not provide quantum mechanically consistent string backgrounds in the case the underlying WZW constraints are first solved classically. 
  The superconformal algebras of Ademollo et al are generalised to a multi-index form. The structure obtained is similar to the Moyal Bracket analogue of the Neveu-Schwarz Algebra. 
  Commuting transfer matrices of $U_{q}(X_{r}^{(1)})$ vertex models obey the functional relations which can be viewed as an $X_{r}$ type Toda field equation on discrete space time. Based on analytic Bethe ansatz we present, for $X_{r}=D_{r}$, a new expression of its solution in terms of determinants and Pfaffians. 
  I review results from recent investigations of anomalies in fermion--Yang Mills systems in which basic notions from noncommutative geometry (NCG) where found to naturally appear. The general theme is that derivations of anomalies from quantum field theory lead to objects which have a natural interpretation as generalization of de Rham forms to NCG, and that this allows a geometric interpretation of anomaly derivations which is useful e.g. for making these calculations efficient. This paper is intended as selfcontained introduction to this line of ideas, including a review of some basic facts about anomalies. I first explain the notions from NCG needed and then discuss several different anomaly calculations: Schwinger terms in 1+1 and 3+1 dimensional current algebras, Chern--Simons terms from effective fermion actions in arbitrary odd dimensions. I also discuss the descent equations which summarize much of the geometric structure of anomalies, and I describe that these have a natural generalization to NCG which summarize the corresponding structures on the level of quantum field theory.   Contribution to Proceedings of workshop `New Ideas in the Theory of Fundamental Interactions', Szczyrk, Poland 1995; to appear in Acta Physica Polonica B. 
  The quantization of circular strings in an anti-de Sitter background spacetime is performed, obtaining a discrete spectrum for the string mass. A comparison with a four-dimensional homogeneous and isotropic spacetime coupled to a conformal scalar field shows that the string radius and the scale factor have the same classical solutions and that the quantum theories of these two models are formally equivalent. However, the physically relevant observables of these two systems have different spectra, although they are related to each other by a specific one-to-one transformation. We finally obtain a discrete spectrum for the spacetime size of both systems, which presents a nonvanishing lower bound. 
  Recently Banks, Douglas and Seiberg have shown that the world volume theory of a three brane of the type IIB theory in the presence of a configuration of four Dirichlet seven branes and an orientifold plane is described by an N=2 supersymmetric SU(2) gauge theory with four quark flavours in 3+1 dimensions. In this note we show how the BPS mass formula for N=2 supersymmetric gauge theory arises from masses of open strings stretched between the three brane and the seven brane along appropriate geodesics. 
  We present a regularized and renormalized version of the one-loop nonlinear relaxation equations that determine the non-equilibrium time evolution of a classical (constant) field coupled to its quantum fluctuations. We obtain a computational method in which the evaluation of divergent fluctuation integrals and the evaluation of the exact finite parts are cleanly separated so as to allow for a wide freedom in the choice of regularization and renormalization schemes. We use dimensional regularization here. Within the same formalism we analyze also the regularization and renormalization of the energy-momentum tensor. The energy density serves to monitor the reliability of our numerical computation. The method is applied to the simple case of a scalar phi^4 theory; the results are similar to the ones found previously by other groups. 
  A new two-dimensional black hole model, based on the "R=T" relativistic theory, is introduced, and the quantum massless scalar field is studied in its classical gravitational field. In particular infrared questions are discussed. The two-point function, energy-momentum tensor, current, Bogoliubov transformations and the mean number of created particles for a given test function are computed. I show that this black hole emits massless scalar particles spontaneously. Comparison with the corresponding field theory in a thermal bath shows that the spontaneous emission is everywhere thermal, i.e. not only near the horizon. 
  A new, general, field theoretic approach to the derivation of asymptotic conservation laws is presented. In this approach asymptotic conservation laws are constructed directly from the field equations according to a universal prescription which does not rely upon the existence of Noether identities or any Lagrangian or Hamiltonian formalisms. The resulting general expressions of the conservation laws enjoy important invariance properties and synthesize all known asymptotic conservation laws, such as the ADM energy in general relativity. 
  Familiar nonlinear and in particular soliton equations arise as zero curvature conditions for GL(1,R) connections with noncommutative differential calculi. The Burgers equation is formulated in this way and the Cole-Hopf transformation for it attains the interpretation of a transformation of the connection to a pure gauge in this mathematical framework. The KdV, modified KdV equation and the Miura transformation are obtained jointly in a similar setting and a rather straightforward generalization leads to the KP and a modified KP equation.  Furthermore, a differential calculus associated with the Boussinesq equation is derived from the KP calculus. 
  The crucial problem of how the dilaton field is stabilized at a phenomenologically acceptable value in string theories remains essentially unsolved. We show that the usual scenario of assuming that the dilaton is fixed by the (SUSY breaking) dynamics of just the dilaton itself (dilaton dominance scenario) is {\em inconsistent} unless the K\"ahler potential receives very important perturbative or non-perturbative contributions. Then, the usual predictions about soft breaking terms are lost, but still is possible to derive model-independent predictions for them. 
  The classical Yang--Mills equations are analyzed within the geometrical framework of an effective gravity theory. Exact analytical solutions are derived for the cylindrically symmetric configurations of the coupled gauge and isoscalar fields. It turns out that there is an infinite family of solutions parametrized by two real parameters, one of which determines the asymptotic behavior of fields near the symmetry axis and in infinity, while the second locates the singularity. These configurations have a simple pole at a finite value of the radial coordinate, and physically they represent ``thick string''-like objects which possess the confinement properties. It is demonstrated that the particles with gauge charge cannot move classically and quantum mechanically out of the interior region. Such an objects are thus direct analogues of the ``black string'' gravitational configurations reported recently in the literature. 
  We discuss the equivalence between Type I, Type II and Heterotic N=2 superstring theories in four dimensions. We study the effective field theory of Type I models obtained by orientifold reductions of Type IIB compactifications on $K_3\times T^2$. We show that the perturbative prepotential is determined by the one-loop corrections to the Planck mass and is associated to an index. As is the case for threshold corrections to gauge couplings, this renormalization is entirely due to N=2 BPS states that originate from D=6 massless string modes. We apply our result to the so-called S-T-U model which admits simultaneous Type II and Heterotic descriptions, and show that all three prepotentials agree in the appropriate limits as expected from the superstring triality conjecture. 
  It is shown that the elliptic Ruijsenaars-Schneider model can be obtained from the cotangent bundle over the two-dimensional current group by means of the Hamiltonian reduction procedure. 
  An affine vertex operator construction at arbitrary level is presented which is based on a completely compactified chiral bosonic string whose momentum lattice is taken to be the (Minkowskian) affine weight lattice. This construction is manifestly physical in the sense of string theory, i.e., the vertex operators are functions of DDF ``oscillators'' and the Lorentz generators, both of which commute with the Virasoro constraints. We therefore obtain explicit representations of affine highest weight modules in terms of physical (DDF) string states. This opens new perspectives on the representation theory of affine Kac-Moody algebras, especially in view of the simultaneous treatment of infinitely many affine highest weight representations of arbitrary level within a single state space as required for the study of hyperbolic Kac-Moody algebras. A novel interpretation of the affine Weyl group as the ``dimensional null reduction'' of the corresponding hyperbolic Weyl group is given, which follows upon re-expression of the affine Weyl translations as Lorentz boosts. 
  We generalize central-charge relations and differential identities of N=2 Special Geometry to N extended supergravity in any dimension 4 \leq D <10, and p-extended objects. We study the extremization of the ADM mass per unit of p-volume of BPS extended objects. Runaway solutions for a ``dilaton'' degree of freedom leading to a vanishing result are interpreted as BPS extremal states having vanishing Bekenstein-Hawking Entropy. 
  We discuss a general quantum theoretical example of quantum cohomology and show that various mathematical aspects of quantum cohomology have quantum mechanical and also observable significance. 
  N=(2,1) heterotic string theory provides clues about hidden structure in M-theory related to string duality; in effect it geometrizes some aspects of duality. The program whereby one may deduce this hidden structure is outlined, together with the results obtained to date. Speculations are made as to the eventual shape of the theory. Talk presented at Strings '96 (Santa Barbara, July 20-25, 1996). 
  We reconsider the earlier found solutions of the Knizhnik-Zamolodchikov (KZ) equations describing correlators based on the admissible representations of $A_1^{(1)}$. Exploiting a symmetry of the KZ equations we show that the original infinite sums representing the 4-point chiral correlators can be effectively summed up. Using these simplified expressions with proper choices of the contours we determine the duality (braid and fusion) transformations and show that they are consistent with the fusion rules of Awata and Yamada. The requirement of locality leads to a 1-parameter family of monodromy (braid) invariants. These analogs of the ``diagonal'' 2-dimensional local 4-point functions in the minimal Virasoro theory contain in general non-diagonal terms. They correspond to pairs of fields of identical monodromy, having one and the same counterpart in the limit to the Virasoro minimal correlators. 
  Strongly coupled heterotic $E_8\times E_8$ string theory, compactified to four dimensions on a large Calabi-Yau manifold ${\bf X}$, may represent a viable candidate for the description of low-energy particle phenomenology. In this regime, heterotic string theory is adequately described by low-energy $M$-theory on ${\bf R}^4\times{\bf S}^1/{\bf Z}_2\times{\bf X}$, with the two $E_8$'s supported at the two boundaries of the world. In this paper we study the effects of gluino condensation, as a mechanism for supersymmetry breaking in this $M$-theory regime. We show that when a gluino condensate forms in $M$-theory, the conditions for unbroken supersymmetry can still be satisfied locally in the orbifold dimension ${\bf S}^1/{\bf Z}_2$. Supersymmetry is then only broken by the global topology of the orbifold dimension, in a mechanism similar to the Casimir effect. This mechanism leads to a natural hierarchy of scales, and elucidates some aspects of heterotic string theory that might be relevant to the stabilization of moduli and the smallness of the cosmological constant. 
  Quantum theory of 2d gravity is examined by including a special quantum correction, which corresponds to the open string loop corrections and provides a new conformal anomaly for the corresponding $\sigma$ model. This anomaly leads to the condensation of the tachyon, and the resultant effective theory implies a possibility of extending the 2d gravity to the case of $c>1$. 
  We consider a simple model of d families of scalar field interacting with geometry in two dimensions. The geometry is locally flat and has only global degrees of freedom. When d<0 the universe is locally two dimensional but for d>0 it collapses to a one dimensional manifold. The model has some, but not all, of the characteristics believed to be features of the full theory of conformal matter interacting with quantum gravity which has local geometric degrees of freedom. 
  We study the CPT theorem for a two-dimensional conformal field theory on an arbitrary Riemann surface. On the sphere the theorem follows from the assumption that the correlation functions have standard hermiticity properties and are invariant under the transformation $z \rightarrow 1/z$. The theorem can then be extended to higher genus surfaces by sewing. We show that, as a consequence of the CPT theorem on the world-sheet, the scattering $T$-matrix in string theory is {\sl formally\/} hermitean at any loop order. 
  The act of implementing non-Abelian duality in two dimensional sigma models results unavoidably in an additional reducible symmetry. The Batalin-Vilkovisky formalism is employed to handle this new symmetry. Valuable lessons are learnt here with respect to non-Abelian duality. We emphasise, in particular, the effects of the ghost sector corresponding to this symmetry on non-Abelian duality. 
  We study the behavior of D-branes at distances far shorter than the string length scale~$l_s$. We argue that short-distance phenomena are described by the IR behavior of the D-brane world-volume quantum theory. This description is valid until the brane motion becomes relativistic. At weak string coupling $\gs$ this corresponds to momenta and energies far above string scale. We use 0-brane quantum mechanics to study 0-brane collisions and find structure at length scales corresponding to the eleven-dimensional Planck length ($\lp11 \sim \gs^{1/3} l_s$) and to the radius of the eleventh dimension in M-theory ($\R11 \sim \gs l_s$). We use 0-branes to probe non-trivial geometries and topologies at sub-stringy scales. We study the 0-brane 4-brane system, calculating the 0-brane moduli space metric, and find the bound state at threshold, which has characteristic size $\lp11$. We examine the blowup of an orbifold and are able to resolve the resulting $S^2$ down to size $\lp11$. A 0-brane with momentum approaching $1/\R11$ is able to explore a larger configuration space in which the blowup is embedded. Analogous phenomena occur for small instantons. We finally turn to 1-branes and calculate the size of a bound state to be $\sim \gs^{1/2} l_s$, the 1-brane tension scale. 
  It is interesting to superimpose the Pauli-Villars regularization on the lattice regularization. We illustrate how this scheme works by evaluating the axial anomaly in a simple lattice fermion model, the Pauli-Villars Lagrangian with a gauge non-invariant Wilson term. The gauge non-invariance of the axial anomaly, caused by the Wilson term, is remedied by a compensation among Pauli-Villars regulators in the continuum limit. A subtlety in Frolov-Slavnov's scheme for an odd number of chiral fermions in an anomaly free complex gauge representation, which requires an infinite number of regulators, is briefly mentioned. 
  We present a class of graviton-dilaton models in which a homogeneous isotropic universe, such as our observed one, evolves with no singularity at any time. Such models may stand on their own as interesting models for singularity free cosmology, and may be studied further accordingly. They may also arise from string theory. We discuss critically a few such possibilities. 
  The coordinate-free formulation of canonical quantization, achieved by a flat-space Brownian motion regularization of phase-space path integrals, is extended to a special class of closed first-class constrained systems that is broad enough to include Yang-Mills type theories with an arbitrary compact gauge group. Central to this extension are the use of coherent state path integrals and of Lagrange multiplier integrations that engender projection operators onto the subspace of gauge invariant states. 
  We study string propagation in an exact, four-dimensional dyonic black hole background. The general solutions describing string configurations are obtained by solving the string equations of motion and constraints. By using the covariant formalism, we also investigate the propagation of physical perturbations along the string in the given curved background. 
  We study the discrete and gauge symmetries of Quantum Electrodynamics at finite temperature within the real-time formalism.   The gauge invariance of the complete generating functional leads to the finite temperature Ward identities. These Ward identities relate the eight vertex functions to the elements of the self-energy matrix. Combining the relations obtained from the $Z_2$ and the gauge symmetries of the theory we find that only one out of eight longitudinal vertex functions is independent. As a consequence of the Ward identities it is shown that some elements of the vertex function are singular when the photon momentum goes to zero. 
  Within the framework of string field theory the intrinsic Hausdorff dimension d_H of the ensemble of surfaces in two-dimensional quantum gravity has recently been claimed to be 2m for the class of unitary minimal models (p = m+1,q = m). This contradicts recent results from numerical simulations, which consistently find d_H approximatly 4 in the cases m = 2, 3, 5 and infinity. The string field calculations rely on identifying the scaling behavior of geodesic distance and area with respect to a common length scale l. This length scale is introduced by formulating the models on a disk with fixed boundary length l. In this paper we study the relationship between the mean area and the boundary length for pure gravity and the Ising model coupled to gravity. We discuss how this relationship is modified by relevant perturbations in the Ising model. We discuss how this leads to a modified value for the Hausdorff dimension. 
  We discuss the $2+1$ dimensional description of the $\Phi_{1,3}$ deformation of the minimal model $M_p$ leading to a transition $M_p \rightarrow M_{p-1}$. The deformation can be considered as an addition of the charged matter to the Chern-Simons theory describing a minimal model. The $N=1$ superconformal case is also considered. 
  The dynamical properties of the gauge theory of Born-Infeld type action, which is expected as the high-energy effective theory, are investigated by adding a complex scalar field to this gauge system. Especially the Coleman-Weinberg mechanism is addressed in this theory. 
  The O(3) sigma model and abelian Higgs model in two space dimensions admit topological (Bogomol'nyi) lower bounds on their energy. This paper proposes lattice versions of these systems which maintain the Bogomol'nyi bounds. One consequence is that instantons/solitons/vortices on the lattice then have a high degree of stability. 
  In the framework of heterotic compactifications, we consider the one-loop corrections to the gauge couplings, which were shown to be free of any infra-red ambiguity. For a class of N=2 models, namely those that are obtained by toroidal compactification to four dimensions of generic six-dimensional N=1 ground states, we give an explicit formula for the gauge-group independent thresholds, and show that these are equal within this class, as a consequence of an anomaly-cancellation constraint in six dimensions. We further use these results to compute the (N=2)-sector contributions to the thresholds of N=1 orbifolds. We then consider the full contribution of N=1 sectors to the gauge couplings which generically are expected to modify the unification picture. We compute such corrections in several models. We finally comment on the effect of such contributions to the issue of string unification. 
  The canonical front form Hamiltonian for non-Abelian SU(N) gauge theory in 3+1 dimensions is mapped on an effective Hamiltonian which acts only in the Fock space of one quark and one antiquark. The approach is non-perturbative and exact. It is based on Discretized Light-Cone Quantization and the Method of Iterated Resolvents. The method resums the diagrams of perturbation theory to all orders in the coupling constant and is free of Tamm-Dancoff truncations in the Fock-space. Emphasis is put on dealing accurately with the many-body aspects of gauge field theory. Pending future renormalization group analysis the running coupling is derived to all orders in the bare coupling constant.~--- The derived effective interaction has an amazingly simple structure and is gauge invariant and frame independent. It is solvable on a small computer like a work station. The many-body amplitudes can be retrieved self-consistently from these solutions, by quadratures without solving another eigenvalue problem. The structures found allow also for developing simple phenomenological models consistent with non-Abelian gauge field theory. 
  It is shown that supersymmetric integrable models in two dimensions, both relativistic (i.e. super-Toda type theories) and non-relativistic (reductions of super-KP hierarchies) can be associated to general Poisson-brackets structures given by superaffinizations of any bosonic Lie or any super-Lie algebra. This result allows enlarging the set of supersymmetric integrable models, which are no longer restricted to the subclass of superaffinizations of purely fermionic super-Lie algebras (that is admitting fermionic simple roots only). 
  The sphaleron type solution in the electroweak theory, generalized to include the dilaton field, is examined. The solutions describe both the variarions of Higgs and Gauge fields inside the sphaleron and the shape of the dilaton cloud surrounding the sphaleron. Such a cloud is large and extends far outside. These phenomena may play an important role during the baryogenesis which probably took place in the Early Universe. 
  Some mathematical inconsistencies in the conventional form of Maxwell's equations extended by Lorentz for a single charge system are discussed. To surmount these in framework of Maxwellian theory, a novel convection displacement current is considered as additional and complementary to the famous Maxwell displacement current. It is shown that this form of the Maxwell-Lorentz equations is similar to that proposed by Hertz for electrodynamics of bodies in motion. Original Maxwell's equations can be considered as a valid approximation for a continuous and closed (or going to infinity) conduction current. It is also proved that our novel form of the Maxwell-Lorentz equations is relativistically invariant. In particular, a relativistically invariant gauge for quasistatic fields has been found to replace the non-invariant Coulomb gauge. The new gauge condition contains the famous relationship between electric and magnetic potentials for one uniformly moving charge that is usually attributed to the Lorentz transformations. Thus, for the first time, using the convection displacement current, a physical interpretation is given to the relationship between the components of the four-vector of quasistatic potentials. A rigorous application of the new gauge transformation with the Lorentz gauge transforms the basic field equations into an independent pair of differential equations responsible for longitudinal and transverse fields, respectively. The longitudinal components can be interpreted exclusively from the standpoint of the instantaneous "action at a distance" concept and leads to necessary conceptual revision of the conventional Faraday-Maxwell field. The concept of electrodynamic dualism is proposed for self-consistent classical electrodynamics. It implies simultaneous coexistence 
  We study the geometric realization of the Higgs phenomenon in type II string compactifications on Calabi--Yau manifolds. The string description is most directly phrased in terms of confinement of magnetic flux, with magnetic charged states arising from D-branes wrapped around chains as opposed to cycles. The rest of the closed cycle of the D-brane worldvolume is manifested as a confining flux tube emanating from the magnetic charges, in the uncompactified space. We also study corrections to hypermultiplet moduli for type II compactifications, in particular for type IIA near the conifold point. 
  Exact operator solution for quantum Liouville theory is investigated based on the canonical free field. Locality, the field equation and the canonical commutation relations are examined based on the exchange algebra hidden in the theory. The exact solution proposed by Otto and Weigt is shown to be correct to all order in the cosmological constant. 
  We calculate the density and gravitational wave spectrums generated in a version of string cosmology termed pre-big bang scenario. The large scale structures are originated from quantum fluctuations of the metric and dilaton field during a pole-like inflation stage driven by a potential-less dilaton field realized in the low-energy effective action of string theory. The generated classical density field and the gravitational wave in the second horizon crossing epoch show tilted spectrums with n ~ 4 and n_T ~ 3, respectively. These differ from the observed spectrum of the large angular scale anisotropy of the cosmic microwave background radiation which supports scale invariant ones with n ~ 1 and n_T ~ 0. This suggests that the pre-big bang stage is not suitable for generating the present day observable large scale structures, and suggests the importance of investigating the quantum generation processes during stringy era with higher order quantum correction terms. 
  The numerical input for the quantitative consequences of the electroweak standard model, the hypercharge and isospin coupling constants and the Higgs field ground state mass value, are interpreted as normalizations of the symmetries involved. Using an additional statistical argument, a first order quantitative determination of the Weinberg angle as a normalization ratio gives the experimentally acceptable value $\tan^2\th={1\over3}$. 
  We discuss the definition of condensates within light-cone quantum field theory. As the vacuum state in this formulation is trivial, we suggest to abstract vacuum properties from the particle spectrum. The latter can in principle be calculated by solving the eigenvalue problem of the light-cone Hamiltonian. We focus on fermionic condensates which are order parameters of chiral symmetry breaking. As a paradigm identity we use the Gell-Mann-Oakes-Renner relation between the quark condensate and the observable pion mass. We examine the analogues of this relation in the `t~Hooft and Schwinger model, respectively. A brief discussion of the Nambu-Jona-Lasinio model is added. 
  We review some recent work on the existence and classification of extreme black-hole-type solutions in N=8 supergravity. For the black holes considered (those that are also solutions of N=4 supergravity and of the Einstein-Maxwell-dilaton theory with coupling a) a complete classification is achieved: the only possible values of a are \sqrt{3},1,1/\sqrt{3},0. Up to U duality transformations there is only one solution for each of those values. The exception is a=0 for which an additional extreme but non-supersymmetric Reissner-Nordstr\"om black hole solution exists. We also study the so-called massless black-hole solutions. We argue that they can be understood as composite objects. At least one of the components would have ``negative mass''. We also argue that these states, being annihilated by all the generators in the supersymmetry algebra, could also constitute alternative vacua of the supergravity theory. 
  We determine the most general form of the antisymmetric $H$-field tensor derived from a purely time-dependent potential that is admitted by all possible spatially homogeneous cosmological models in 3+1-dimensional low-energy bosonic string theory. The maximum number of components of the $H$ field that are left arbitrary is found for each homogeneous cosmology defined by the Bianchi group classification. The relative generality of these string cosmologies is found by counting the number of independent pieces of Cauchy data needed to specify the general solution of Einstein's equations. The hierarchy of generality differs significantly from that characteristic of vacuum and perfect-fluid cosmologies. The degree of generality of homogeneous string cosmologies is compared to that of the generic inhomogenous solutions of the string field equations. 
  Dynamical supersymmetry breaking is considered in models which admit descriptions in terms of electric, confined, or magnetic degrees of freedom in various limits. In this way, a variety of seemingly different theories which break supersymmetry are actually inter-related by confinement or duality. Specific examples are given in which there are two dual descriptions of the supersymmetry breaking ground state. 
  We obtain the elliptic curve corresponding to an $N=2$ superconformal field theory which has an $E_6$ global symmetry at the strong coupling point $\tau=e^{\pi i/3}$. We also find the Seiberg-Witten differential $\lambda_{SW}$ for this theory. This differential has 27 poles corresponding to the fundamental representation of $E_6$. The complex conjugate representation has its poles on the other sheet. We also show that the $E_6$ curve reduces to the $D_4$ curve of Seiberg and Witten. Finally, we compute the monodromies and use these to compute BPS masses in an $F$-Theory compactification. 
  A new version of the delta expansion is presented, which, unlike the conventional delta expansion, can be used to do nonperturbative calculations in a self-interacting scalar quantum field theory having broken symmetry. We calculate the expectation value of the scalar field to first order in delta, where delta is a measure of the degree of nonlinearity in the interaction term. 
  We show that two-dimensional topological BF theories coupled to particles carrying non-Abelian charge admit a new coupling involving the Lagrange multiplier field. When applied to the gauge theoretic formulation of dilatonic gravity it gives rise to a source term for the gravitational field. We show that the system admits black hole solutions. 
  By completing the old discussion of K.~Wilson, we express the chiral anomaly in terms of a double integral of a three-point function of chiral currents over an arbitrarily small region in the coordinate space. An integrability condition provides an important finite local counterterm to the integral. 
  A convenient framework is developed to generalize Berry's investigation of the adiabatic geometrical phase for a classical relativistic charged scalar field in a curved background spacetime which is minimally coupled to electromagnetism and an arbitrary (non-electromagnetic) scalar potential. It involves a two-component formulation of the corresponding Klein-Gordon equation. A precise definition of the adiabatic approximation is offered and conditions of its validity are discussed. It is shown that the adiabatic geometric phase can be computed without making a particular choice for an inner product on the space of solutions of the field equations. What is needed is just an inner product on the Hilbert space of the square integrable functions defined on the spatial hypersurfaces. The two-component formalism is applied in the investigation of the adiabatic geometric phases for several specific examples, namely, a rotating magnetic field in Minkowski space, a rotating cosmic string, and an arbitrary spatially homogeneous cosmological background. It is shown that the two-component formalism reproduces the known results for the first two examples. It also leads to several interesting results for the case of spatially homogeneous cosmological models. In particular, it is shown that the adiabatic geometric phase angles vanish for Bianchi type I models. The situation is completely different for Bianchi type IX models where a variety of nontrivial non-Abelian adiabatic geometrical phases can occur. The analogy between the adiabatic geometric phases induced by the Bianchi type IX backgrounds and those associated with the well-known time-dependent nuclear quadrupole Hamiltonians is also pointed out. 
  The general treatment of a separable Hamiltonian of Liouville-type is well-known in operator formalism. A path integral counterpart is formulated if one starts with the Jacobi's principle of least action, and a path integral evaluation of the Green's function for the hydrogen atom by Duru and Kleinert is recognized as a special case. The Jacobi's principle of least action for given energy is reparametrization invariant, and the separation of variables in operator formalism corresponds to a choice of gauge in path integral. The Green's function is shown to be gauge independent,if the operator ordering is properly taken into account. These properties are illustrated by evaluating an exact path integral of the Green's function for the hydrogen atom in parabolic coordinates. 
  Orientifolds of the type IIB superstring that descend from F theory and M theory orbifolds are studied perturbatively. One finds strong evidence that a previously ignored twisted open string is required in these models. An attempt is made to interpret the $J$ type torsion in F theory where one finds a realization of Gimon and Johnson's models which does not require these twisted strings. S-duality also provides evidence for these strings. 
  We study the low-energy effective theory in N=2 super Yang-Mills theories by microscopic and exact approaches. We calculate the one-instanton correction to the prepotential for any simple Lie group from the microscopic approach. We also study the Picard-Fuchs equations and their solutions in the semi- classical regime for classical gauge groups with rank r \leq 3. We find that for gauge groups G=A_r, B_r, C_r (r \leq 3) the microscopic results agree with those from the exact solutions. 
  The identification in affine Toda field theory of the quantum particle with the lowest breather allows us to re-interpret discrete modes of excitation of solitons as breathers bound to solitons, and thus to investigate them through the proposed soliton-breather S-matrices. There are implications for the physical spectrum and for the semiclassical soliton mass corrections. 
  Series of extended Epstein type provide examples of non-trivial zeta functions with important physical applications. The regular part of their analytic continuation is seen to be a convergent or an asymptotic series. Their singularity structure is completely determined in terms of the Wodzicki residue in its generalized form, which is proven to yield the residua of all the poles of the zeta function, and not just that of the rightmost pole (obtainable from the Dixmier trace). The calculation is a very down-to-earth application of these powerful functional analytical methods in physics. 
  The bosonization of a massless fermionic field coupled to both vector and axial-vector external sources is developed, following a path-integral approach. The resulting bosonized theory contains two antisymmetric tensor fields whose actions consist of non-local Kalb-Ramond-like terms plus interactions. Exact bosonization rules that take the axial anomaly for the axial current into account are derived, and an approximated bosonized action is constructed. 
  We present a method to perform renormalized perturbation calculation in gauge theories with chiral fermions. We find it proper to focus directly on the Ward-Takahashi identities, relegating dimensional regularization into a supplementary and secondary role. We show with the example of the Abelian-Higgs theory how to handle amplitudes involving fermions, particularly how to handle the matrix $\gamma_5$. As a demonstration of our method of renormalization, we evaluate the radiative corrections of the triangular anomaly in the Abelian-Higgs theory with appropriate number of chiral fermions. This anomaly amplitude is calculated without any regularization and is found to vanish. 
  We find double-extreme black holes associated with the special geometry of the Calabi-Yau moduli space with the prepotential F=STU. The area formula is STU-moduli independent and has ${[SL(2,Z)]}^3$ symmetry in space of charges. The dual version of this theory without prepotential treats the dilaton S asymmetric versus T,U-moduli. We display the dual relation between new (STU) black holes and stringy (S|TU) black holes using particular Sp(8, Z) transformation. The area formula of one theory equals that of the dual theory when expressed in terms of dual charges. We analyse the relation between (STU) black holes to string triality of black holes: (S|TU), (T|US), (U|ST) solutions. In the democratic STU-symmetric version we find that all three S and T and U duality symmetries are non-perturbative and mix electric and magnetic charges. 
  Microscopic tests of the exact results are performed in $N=2$ supersymmetric $SU(2)$ QCD. We present the complete construction of the multi-instanton in $N=2$ supersymmetric QCD. All the defining equations of the super instanton are reduced to the algebraic equations. Using this result, we calculate the two-instanton contribution ${\cal F}_2$ to the prepotential ${\cal F}$ for the arbitrary $N_f$ theories. For $N_f=0,1,2$, instanton calculus agrees with the prediction of the exact results, however, for $N_f=3,4$, we find discrepancies between them. We propose improved curves of the exact results for the massive $N_f=3$ and massless $N_f=4$ theories. 
  The algebraic structure of S-Theory and its representations are described. This structure includes up to 13 hidden dimensions. It implies the existence of an SO(10,2) covariant supergravity theory as a limit of the secret theory behind string theory. The black hole entropy is invariant under transformations including the 11th and 12th dimensions. The discussion includes generalization into curved spacetime via various contractions of the superalgebras OSp(1/32), OSp(1/64) etc.. 
  We investigate dynamical symmetry breaking of the Gross-Neveu model in the light-front formalism without introducing auxiliary fields. While this system cannot have zero-mode constraints, we find that a nontrivial solution to the constraint on nondynamical spinor fields is responsible for symmetry breaking. The fermionic constraint is solved by systematic 1/$N$ expansion using the boson expansion method as a technique. Carefully treating the infrared divergence, we obtain a nonzero vacuum expectation value for fermion condensate in the leading order. We derive the 't Hooft equation including the effect of condensation, and determine the Hamiltonian consistently with the equation of motion. 
  The problem of finding all possible effective field theories for the quantum Hall effect is closely related to the problem of classifying all possible modular invariant partition functions for the algebra $u(1)^m$, as was argued recently by Cappelli and Zemba. This latter problem is also a natural one from the perspective of conformal field theory. In this paper we completely solve this problem, expressing the answer in terms of self-dual lattices, or equivalently, rational points on the dual Grassmannian $G_{m,m}(R)^*$. We also find all modular invariant partition functions for $su(2)\oplus u(1)^m$, from which we obtain the classification of all N=2 superconformal minimal models. The `A-D-E classification' of these, though often quoted in the literature, turns out to be a very coarse-grained one: e.g. associated with the names $E_6,E_7,E_8$, respectively, are precisely 20,30,24 different partition functions. As a by-product of our analysis, we find that the list of modular invariants for su(2) lengthens surprisingly little when commutation with T -- i.e. invariance under $\tau \mapsto \tau+1$ -- is ignored: the other conditions are far more essential. 
  We consider the classical self-dual Yang-Mills equation in 3+1-dimensional Minkowski space. We have found an exact solution, which describes scattering of $n$ plane waves. In order to write the solution in a compact form, it is convenient to introduce a scattering operator $\hat{T}$. It acts in the direct product of three linear spaces: 1) universal enveloping of $su(N)$ Lie algebra, 2) $n$-dimensional vector space and 3) space of functions defined on the unit interval. 
  We construct a locally supersymmetric action for the scalar particle, and study its relation with the usual reparametrization invariant action. The mechanisms at work are similar to those employed in the embedding of the bosonic string into the fermionic one, originally due to Berkovits and Vafa in their search for a universal string. The simpler algebraic structure present in the particle case provides us with a guide on how to prove in a simple way, without the need of fixing the superconformal gauge, that the supersymmetric formulation of the bosonic string is equivalent to the usual one, where reparametrization invariance is the only world-sheet gauge symmetry. 
  We consider the quantum nonlinear Schr\"odinger equation in one space and one time dimension. We are interested in the non-free-fermionic case. We consider static temperature-dependent correlation functions. The determinant representation for correlation functions simplifies in the small mass limit of the Bose particle. In this limit we describe the correlation functions by the vacuum expectation value of a boson-valued solution for Maxwell-Bloch differential equation. We evaluate long-distance asymptotics of correlation functions in the small mass limit. 
  S. Axelrod and I.M. Singer constructed a compactification of the configuration space of distinct points in a Riemannian manifold V. A similar compactification for the moduli space of configurations of distinct points in the plane (mod the affine group action) was considered by E. Getzler and J.D.S. Jones. They observed that this compactification carries a natural structure of an operad. In the present note we show that (non-compactified) configuration spaces form a partial operad (or a partial module over a partial operad) and that the compactification can be described as an operadic (or modular) completion. This approach immediately gives the operad (or module) structure on the compactification. We also discuss the spectral sequence of the stratification and identify the second term of this spectral sequence to the bar resolution of an operadic module. Our results generalize the work of Getzler, Kimura, Jones, Stasheff, Voronov and others to the case of configurations in a general Riemannian manifold. 
  We review equivariant localization techniques for the evaluation of Feynman path integrals. We develop systematic geometric methods for studying the semi-classical properties of phase space path integrals for dynamical systems, emphasizing the relations with integrable and topological quantum field theories. Beginning with a detailed review of the relevant mathematical background -- equivariant cohomology and the Duistermaat-Heckman theorem, we demonstrate how the localization ideas are related to classical integrability and how they can be formally extended to derive explicit localization formulas for path integrals in special instances using BRST quantization techniques. Various loop space localizations are presented and related to notions in quantum integrability and topological field theory. We emphasize the common symmetries that such localizable models always possess and use these symmetries to discuss the range of applicability of the localization formulas. A number of physical and mathematical applications are presented in connection with elementary quantum mechanics, Morse theory, index theorems, character formulas for semi-simple Lie groups, quantization of spin systems, unitary integrations in matrix models, modular invariants of Riemann surfaces, supersymmetric quantum field theories, two-dimensional Yang-Mills theory, conformal field theory, cohomological field theories and the loop expansion in quantum field theory. Some modern techniques of path integral quantization, such as coherent state methods, are also discussed. The relations between equivariant localization and other ideas in topological field theory, such as the Batalin-Fradkin-Vilkovisky and Mathai-Quillen formalisms, are presented. 
  We study the solutions of the equations of motion in the gauged (2+1)-dimensional nonlinear Schr\"odinger model. The contribution of Chern-Simons gauge fields leads to a significant decrease of the critical power of self-focusing. We also show that at appropriate boundary conditions in the considered model there exists a regime of turbulent motions of hydrodynamic type. 
  The vacuum energy of a scalar field in a spherically symmetric background field is considered. It is expressed through the Jost function of the corresponding scattering problem. The renormalization is discussed in detail and performed using the uniform asymptotic expansion of the Jost function. The method is demonstrated in a simple explicit example. 
  The Casimir energy corresponding to a massive scalar field with Dirichlet boundary conditions on a spherical bag is obtained. The field is considered, separately, inside and outside the bag. The renormalization procedure that is necessary to apply in each situation is studied in detail, in particular the differences occurring with respect to the case when the field occupies the whole space. The final result contains several constants that experience renormalization and can be determined only experimentally. The non-trivial finite parts that appear in the massive case are found exactly, providing a precise determination of the complete, renormalized zero-point energy for the first time. 
  The nonabelian Stokes theorem, representing a Wilson loop as an integral over all the orientations in colour space, and the cumulant expansion are used for derivation of the string effective action in SU(2) gluodynamics. The obtained theory is the theory of the rigid string interacting with the rank two antisymmetric Kalb-Ramond fields. In this model there exists a phase where there are no problems of crumpling and wrong high temperature behaviour of the string tension, which are present in the free rigid string theory. The Langevin approach to stochastic quantization of the obtained theory is applied. 
  Recently Olesen has shown the existence of dual string solutions to the equations of ideal Magnetohydrodynamics that describe the long wavelength properties of electrically charged plasmas. Here, we extend these solutions to include the case of plasmas consisting of point like dyons, which carry both electric and magnetic charge. Such strings are dyonic in that they consist of both magnetic and electric flux. We contrast some physical features of dyonic plasmas with those of the purely electric or magnetic type, particularly in relation to the validity of the ideal approximation. 
  We build generalizations of the Grassmann algebras from a few simple assumptions which are that they are graded, maximally symmetric and contain an ordinary Grassmann algebra as a subalgebra. These algebras are graded by Z_{n}^{3} and display surprising properties that indicate their possible application to the modelization of quark fields. We build the generalized supersymmetry generators based on these algebras and their derivation operators. These generators are cubic roots of the usual supersymmetry generators. 
  The holomorphic homogeneous prepotential encoding the special geometry of the special K\"ahler manifolds ${\textstyle SU(1,n)\over \textstyle U(1)\otimes SU(n)}$ is constructed using the symplectic embedding of the isometry group $SU(1,n)$ into $Sp(2n+2,{\hbox{\bf R}})$. Also the automorphic functions of these manifolds are discussed. 
  When quantum supergravity is studied on manifolds with boundary, one may consider local boundary conditions which fix on the initial surface the whole primed part of tangential components of gravitino perturbations, and fix on the final surface the whole unprimed part of tangential components of gravitino perturbations. This paper studies such local boundary conditions in a flat Euclidean background bounded by two concentric 3-spheres. It is shown that, as far as transverse-traceless perturbations are concerned, the resulting contribution to $\zeta(0)$ vanishes when such boundary data are set to zero, exactly as in the case when non-local boundary conditions of the spectral type are imposed. These properties may be used to show that one-loop finiteness of massless supergravity models is only achieved when two boundary 3-surfaces occur, and there is no exact cancellation of the contributions of gauge and ghost modes in the Faddeev-Popov path integral. In these particular cases, which rely on the use of covariant gauge-averaging functionals, pure gravity is one-loop finite as well. 
  $N=1$ supersymmetric gauge theories with global flavor symmetries contain a gauge invariant W-superalgebra which acts on its moduli space of gauge invariants. With adjoint matter, this superalgebra reduces to a graded Lie algebra. When the gauge group is $SO(n_c)$, with vector matter, it is a W-algebra, and the primary invariants form one of its representation. The same superalgebra exists in the dual theory, but its construction in terms of the dual fields suggests that duality may be understood in terms of a charge conjugation within the algebra. We extend the analysis to the gauge group $E_6$. 
  We derive the low energy effective action of the heterotic superstring in superspace. This is achieved by coupling the covariantly quantized Green-Schwarz superstring of Berkovits to a curved background and requiring that the sigma model has superconformal invariance at tree level and at one loop in $\a'$. Tree level superconformal invariance yields the complete supergravity algebra, and one-loop superconformal invariance the equations of motion of the low energy theory. The resulting low energy theory is old-minimal supergravity coupled to a tensor multiplet. The dilaton is part of the compensator multiplet. 
  We investigate quantum corrections to the moduli space for hypermultiplets for type IIA near a conifold singularity. We find a unique quantum deformation based on symmetry arguments which is consistent with a recent conjecture. The correction can be interpreted as an infinite sum coming from multiple wrappings of the Euclidean D-branes around the vanishing cycle. 
  We present a study of the role of fermions in the decay of metastable states of a scalar field via bubble nucleation. We analyze both one and three-dimensional systems by using a gradient expansion for the calculation of the fermionic determinant. The results of the one-dimensional case are compared to the exact results of previous work. 
  It is pointed out that there are gauge-dependent and gauge-independent spinors within the little-group framework for internal space-time symmetries of massless particles. It is shown that two of the $SL(2,c)$ spinors are invariant under gauge transformations while the remaining two are not. The Dirac equation contains only the gauge-invariant spinors leading to polarized neutrinos. It is shown that the gauge-dependent $SL(2,c)$ spinor is the origin of the gauge dependence of electromagnetic four-potentials. 
  General Relativity reduced to two dimensions possesses a large group of symmetries that exchange classical solutions. The associated Lie algebra is known to contain the affine Kac-Moody algebra $A_1^{(1)}$ and half of a real Witt algebra. In this paper we exhibit the full symmetry under the semi-direct product of $\Lie{A_1^{(1)}}$ by the Witt algebra $\Lie{\Wir}$. Furthermore we exhibit the corresponding hidden gauge symmetries. We show that the theory can be understood in terms of an infinite dimensional potential space involving all degrees of freedom: the dilaton as well as matter and gravitation. In the dilaton sector the linear system that extends the previously known Lax pair has the form of a twisted self-duality constraint that is the analog of the self-duality constraint arising in extended supergravities in higher spacetime dimensions. Our results furnish a group theoretical explanation for the simultaneous occurrence of two spectral parameters, a constant one ($=y$) and a variable one ($=t$). They hold for all $2d$ non-linear $\sigma$-models that are obtained by dimensional reduction of $G/H$ models in three dimensions coupled to pure gravity. In that case the Lie algebra is $\Lie{\Wir \semi G^{(1)}}$; this symmetry acts on a set of off shell fields (in a fixed gauge) and preserves the equations of motion. 
  We present illustrations which show the usefulness of algebraic QFT. In particular in low-dimensional QFT, when Lagrangian quantization does not exist or is useless (e.g. in chiral conformal theories), the algebraic method is beginning to reveal its strength. 
  The Wilson (exact) renormalization group equations are used to determine the evolution of a general low energy N=1 supersymmetric action containing a U(1) gauge vector multiplet and a neutral chiral multiplet. The effective theory evolves towards satisfying a fixed relation where the K\"ahler potential and effective gauge coupling are obtained from a N=2 supersymmetric holomorphic prepotential. 
  We use the HyperK\"{a}hler quotient of flat space to obtain some monopole moduli space metrics in explicit form. Using this new description, we discuss their topology, completeness and isometries. We construct the moduli space metrics in the limit when some monopoles become massless, which corresponds to non-maximal symmetry breaking of the gauge group. We also introduce a new family of HyperK"{a}hler metrics which, depending on the ``mass parameter'' being positive or negative, give rise to either the asymptotic metric on the moduli space of many SU(2) monopoles, or to previously unknown metrics. These new metrics are complete if one carries out the quotient of a non-zero level set of the moment map, but develop singularities when the zero-set is considered. These latter metrics are of relevance to the moduli spaces of vacua of three dimensional gauge theories for higher rank gauge groups. Finally, we make a few comments concerning the existence of closed or bound orbits on some of these manifolds and the integrability of the geodesic flow. 
  We study the hypermultiplet moduli space of the type II string compactified on a Calabi-Yau space $\bX$. We do this by using IIA/IIB duality in a compactification of the same theory on $\bX\times \bS ^1$ and by using recent results on three dimensional field theory. 
  Recently suggested causal principal value and causal prescriptions for the "spurious singularity" in light-cone gauge theories are nothing but the different guises of usual Mandelstam-Leibbrandt prescription. 
  A pseudoclassical theories of Majorana, Weyl and Majorana--Weyl particles in the space--time dimensions $D=2n$ are constructed. The canonical quantization of these theories is carried out and as a result we obtain the quantum mechanical description of neutral particle in $D=2n$ , Weyl particle in $D=2n$ and neutral Weyl particle in $D=4n+2$. In $D=2,4({\rm mod}8)$ dimensional space--time the description of the neutral particle coincides with the field theoretical description of the Majorana particle in the Foldy--Wouthuysen representation. In $D=8k+2$ dimensions the neutral Weyl particle coincides with the Majorana--Weyl particle in the Foldy--Wouthuysen representation. 
  The functional determinant of Laplace-type operators on the 3-dimensional non-compact hyperbolic manifold with invariant fundamental domain of finite volume is computed by quadratures and making use of the related terms of the Selberg trace formula. 
  In these lectures we discuss various aspects of gauge theories with extended $N=2$ and $N=4$ supersymmetry that are at the basis of recently found exact results. These results include the exact calculation of the low energy effective action for the light degrees of freedom in the $N=2$ super Yang-Mills theory and the conjecture, supported by some checks, that the $N=4$ super Yang-Mills theory is dual in the sense of Montonen-Olive. 
  We derive a generalization of the Destri - De Vega equation governing the scaling functions of some excited states in the Sine-Gordon theory. In particular configurations with an even number of holes and no strings are analyzed and their UV limits found to match some of the conformal dimensions of the corresponding compactified massless free boson. Quantum group reduction allows to interpret some of our results as scaling functions of excited states of Restricted Sine-Gordon theory, i.e. minimal models perturbed by phi_13 in their massive regime. In particular we are able to reconstruct the scaling functions of the off-critical deformations of all the scalar primary states on the diagonal of the Kac-table. 
  It has been known that the Wigner representation theory for positive energy orbits permits a useful localization concept in terms of certain lattices of real subspaces of the complex Hilbert -space. This ''modular localization'' is not only useful in order to construct interaction-free nets of local algebras without using non-unique ''free field coordinates'', but also permits the study of properties of localization and braid-group statistics in low-dimensional QFT. It also sheds some light on the string-like localization properties of the 1939 Wigner's ''continuous spin'' representations.We formulate a constructive nonperturbative program to introduce interactions into such an approach based on the Tomita-Takesaki modular theory. The new aspect is the deep relation of the latter with the scattering operator. 
  The Lorentz harmonic formulation of D-dimensional bosonic p-brane theory with $D\geq (p+1)(p+2)/2$ coupled to an antisymmetric tensor field of rank d=(p+1) provides the dynamical ground for the description of d=(p+1) dimensional Gravity. It hence realizes the idea of Regge and Teitelboim on a 'string-like' description of gravity. The simplest nontrivial models of such a kind are provided by free D-dimensional p-branes in which world volumes are embedded as minimal surfaces. Possible applications of such a model with d=2+2 and D=2+10 for studying a geometry of bosonic sector of F-theory are considered. Some speculations inspired by the proposed model are presented. 
  In this short printed version of my talk I consider the generalized action for super-p-branes proposed previously in collaboration with D. Sorokin and Dmitrij V. Volkov. I describe briefly, the derivation doubly supersymmetric geometric approach from the generalized action and discuss one of the direction of its possible application. 
  It is shown that a Lagrangian, describing the interaction of the gravitation field with the dilaton and the antisymmetric tensor in arbitrary dimension spacetime, admits an isotropic p-brane solution consisting of three blocks. Relations with known p-brane solutions are discussed. In particular, in ten-dimensional spacetime the three-block p-brane solution is reduced to the known solution, which recently has been used in the D-brane derivation of the black hole entropy. 
  In this note we prove an identity that equates the elliptic genus partition function of a supersymmetric sigma model on the N-fold symmetric product $M^N/S_N$ of a manifold M to the partition function of a second quantized string theory on the space $M \times S^1$. The generating function of these elliptic genera is shown to be (almost) an automorphic form for O(3,2,Z). In the context of D-brane dynamics, this result gives a precise computation of the free energy of a gas of D-strings inside a higher-dimensional brane. 
  Using analyticity of the vacuum wave-functional under complex scalings, the vacuum of a quantum field theory may be reconstructed from a derivative expansion valid for slowly varying fields. This enables the eigenvalue problem for the Hamiltonian to be reduced to algebraic equations. Applied to Yang-Mills theory this expansion leads to a confining force between quarks. 
  The idea of biased symmetries to avoid or alleviate cosmological problems caused by the appearance of some topological defects is familiar in the context of domain walls, where the defect statistics lend themselves naturally to a percolation theory description, and for cosmic strings, where the proportion of infinite strings can be varied or disappear entirely depending on the bias in the symmetry. In this paper we measure the initial configurational statistics of a network of string defects after a symmetry-breaking phase transition with initial bias in the symmetry of the ground state. Using an improved algorithm, which is useful for a more general class of self-interacting walks on an infinite lattice, we extend the work in \cite{MHKS} to better statistics and a different ground state manifold, namely $\R P^2$, and explore various different discretisations. Within the statistical errors, the critical exponents of the Hagedorn transition are found to be quite possibly universal and identical to the critical exponents of three-dimensional bond or site percolation. This improves our understanding of the percolation theory description of defect statistics after a biased phase transition, as proposed in \cite{MHKS}. We also find strong evidence that the existence of infinite strings in the Vachaspati Vilenkin algorithm is generic to all (string-bearing) vacuum manifolds, all discretisations thereof, and all regular three-dimensional lattices. 
  We discuss the transformation properties of classical extremal N=2 black hole solutions in S-T-U like models under S and T duality. Using invariants of (subgroups of) the triality group, which is the symmetry group of the classical BPS mass formula, the transformation properties of the moduli on the event horizon and of the entropy under these transformations become manifest. We also comment on quantum corrections and we make a conjecture for the one-loop corrected entropy. 
  Based on the path integral formalism, we rederive and extend the transverse Ward-Takahashi identities (which were first derived by Yasushi Takahashi) for the vector and the axial vector currents and simultaneously discuss the possible anomaly for them. Subsequently, we propose a new scheme for writing down and solving the Schwinger-Dyson equation in which the the transverse Ward-Takahashi identity together with the usual (longitudinal) Ward-Takahashi identity are applied to specify the fermion-boson vertex function. Especially, in two dimensional Abelian gauge theory, we show that this scheme leads to the exact and closed Schwinger-Dyson equation for the fermion propagator in the chiral limit (when the bare fermion mass is zero) and that the Schwinger-Dyson equation can be exactly solved. 
  We study functional determinants for Dirac operators on manifolds with boundary. We give, for local boundary conditions, an explicit formula relating these determinants to the corresponding Green functions. We finally apply this result to the case of a bidimensional disk under bag-like conditions. 
  A formula relating quotients of determinants of elliptic differential operators sharing their principal symbol, with local boundary conditions, to the corresponding Green function is given. 
  The classical and quantum aspects of the Schwinger model on the torus are considered. First we find explicitly all zero modes of the Dirac operator in the topological sectors with nontrivial Chern index and is spectrum. In the second part we determine the regularized effective action and discuss the propagators related to it.   Finally we calculate the gauge invariant averages of the fermion bilinears and correlation functions of currents and densities. We show that in the infinite volume limit the well-known result for the chiral condensate can be obtained and the clustering property can be established. 
  Global geometry of $K3$-fibration Calabi-Yau threefolds, with Hodge number $h_{2,1}=r+1$, is used to define $N=4$ softly broken $SU(r+1)$ gauge theories, with the bare coupling constant given by the dual heterotic dilaton, and the mass of the adjoint hypermultiplet given by the heterotic string tension. The $U(r+1)$ Donagi-Witten integrable model is also derived from the $K3$-fibration structure, with the extra $U(1)$ associated to the heterotic dilaton. The case of $SU(2)$ gauge group is analyzed in detail. String physics beyond the heterotic point particle limit is partially described by the $N=4$ softly broken theory. 
  The coulomb branch of $N=4$ supersymmetric Yang-Mills gauge theories in $d=2+1$ is studied. A direct connection between gauge theories and monopole moduli spaces is presented. It is proposed that the hyper-K\"ahler metric of supersymmetric $N=4$ $SU(N)$ Yang-Mills theory is given by the charge $N$ centered moduli space of BPS monopoles in $SU(2)$. The theory is compared to $N=2$ supersymmetric Yang-Mills theory in four dimensions through compactification on a circle of the latter. It is found that rational maps are appropriate to this comparison. A BPS mass formula is also written for particles in three dimensions and strings in four dimensions. 
  The embedding of the isometry group of the coset spaces SU(1,n)/ U(1)xSU(n) in Sp(2n+2,R) is discussed. The knowledge of such embedding provides a tool for the determination of the holomorphic prepotential characterizing the special geometry of these manifolds and necessary in the superconformal tensor calculus of N=2 supergravity. It is demonstrated that there exists certain embeddings for which the homogeneous prepotential does not exist. Whether a holomorphic function exists or not, the dependence of the gauge kinetic terms on the scalars characterizing these coset in N=2 supergravity theory can be determined from the knowledge of the corresponding embedding, \`a la Gaillard and Zumino. Our results are used to study some of the duality symmetries of heterotic compactifications of orbifolds with Wilson lines. 
  A systematic method of constructing manifestly supersymmetric $1+1$-dimensional KP Lax hierarchies is presented. Closed expressions for the Lax operators in terms of superfield eigenfunctions are obtained. All hierarchy equations being eigenfunction equations are shown to be automatically invariant under the (extended) supersymmetry. The supersymmetric Lax models existing in the literature are found to be contained (up to a gauge equivalence) in our formalism. 
  Recently Das and Mathur found that the leading order Hawking emission rate of neutral scalars by near-extremal $D=5$ black holes is exactly reproduced by a string theoretic model involving intersecting D-branes. We show that the agreement continues to hold for charged scalar emission. We further show that similar agreement can be obtained for a class of near-extremal $D=4$ black holes using a model inspired by M-theory. In this model, BPS saturated $D=4$ black holes with four charges are realized in M-theory as 5-branes triply intersecting over a string. The low-energy excitations are signals traveling on the intersection string. 
  The low-energy limit of the 6D non-critical string theory with $N=1$ SUSY and $E_8$ chiral current algebra compactified on $T^2$ is generically an $N=2$ $U(1)$ vector multiplet. We study the analog of the Seiberg-Witten solution for the low-energy effective action as a function of $E_8$ Wilson lines on the compactified torus and the complex modulus of that torus. The moduli space includes regions where the Seiberg-Witten curves for $SU(2)$ QCD are recovered as well as regions where the newly discovered 4D theories with enhanced $E_{6,7,8}$ global symmetries appear. 
  As previously shown BRST singlets |s> in a BRST quantization of general gauge theories on inner product spaces may be represented in the form |s>=e^{[Q, \psi]} |\phi> where |\phi> is either a trivially BRST invariant state which only depends on the matter variables, or a solution of a Dirac quantization. \psi is a corresponding fermionic gauge fixing operator. In this paper it is shown that the time evolution is determined by the singlet states of the corresponding reparametrization invariant theory. The general case when the constraints and Hamiltonians may have explicit time dependence is treated. 
  We study (non-renormalizable) five dimensional supersymmetric field theories. The theories are parametrized by quark masses and a gauge coupling. We derive the metric on the Coulomb branch exactly. We use stringy considerations to learn about new non-trivial interacting field theories with exceptional global symmetry $E_n$ ($E_8$, $E_7$, $E_6$, $E_5=Spin(10)$, $E_4=SU(5)$, $E_3=SU(3)\times SU(2)$, $E_2=SU(2)\times U(1)$ and $E_1=SU(2)$). Their Coulomb branch is ${\bf R}^+$ and their Higgs branch is isomorphic to the moduli space of $E_n$ instantons. One of the relevant operators of these theories leads to a flow to $SU(2)$ gauge theories with $N_f=n-1$ flavors. In terms of these $SU(2)$ IR theories this relevant parameter is the inverse gauge coupling constant. Other relevant operators (which become quark masses after flowing to the $SU(2)$ theories) lead to flows between them. Upon further compactifications to four and three dimensions we find new fixed points with exceptional symmetries. 
  We consider theories with degenerate kinetic terms such as those that arise at strong coupling in $N=2$ super Yang-Mills theory. We compute the components of generalized $N=1,2$ supersymmetric sigma model actions in two dimensions. The target space coordinates may be matter and/or Yang-Mills superfield strengths. 
  In this article we examine some points in the moduli space of M-Theory at which there arise enhanced gauge symmetries. In particular, we examine the ``trivial" points of enhanced gauge symmetry in the moduli space of M-Theory on $S^{ 1 } \times S^{ 1 } / { \bf Z }_{ 2 }$ as well as the points of enhanced gauge symmetry in the moduli space of M-Theory on $ K3 $ and those in the moduli space of M-Theory on $T^{5}/{\bf Z}_{2} \times S^{1}$. Also, we employ the above enhanced gauge symmetries to derive the existence of some points of enhanced gauge symmetry in the moduli space of the Type IIA string theory. 
  Dynamics of many supersymmetric monopoles are studied in the low energy approximation. A conjecture for the exact moduli space metric is given for all collections of fundamental monopoles of distinct type, and various partial confirmations of the conjecture are discussed. Upon the quantization of the resulting multi-monopole dynamics in the context of $N=4$ supersymmetric Yang-Mills-Higgs theories, one recovers the missing magnetic states that are dual to some of the massive vector mesons. A generalization to monopoles with nonabelian charges is also discussed. 
  We continue the construction of the $:\phi^4_4:$ quantum field theory. In this paper we consider the Wick kernel of the interacting quantum field. Using the complex structure and the Fock-Bargmann-Berezin-Segal integral representation we prove that this kernel defines a unique operator--valued generalized function on the space $\Sc^\alpha(\R^4)$ for any $\alpha<6/5,$ i.e. the constructed quantum field is the generalized operator-valued function of localizable Jaffe class. The same assertion is valid for the outgoing quantum field. These assertions about the quantum field allow to construct the Wightman functions, the matrix elements of the quantum scattering operator and to consider their properties (positivity, spectrality, Poincare invariance, locality, asymptotic completeness, and unitarity of the quantum scattering). 
  We derive in the SCFT and low energy effective action frameworks the necessary and sufficient conditions for supersymmetric cycles in exceptional holonomy manifolds and Calabi-Yau 4-folds. We show that the Cayley cycles in $Spin(7)$ holonomy eight-manifolds and the associative and coassociative cycles in $G_2$ holonomy seven-manifolds preserve half of the space-time supersymmetry. We find that while the holomorphic and special Lagrangian cycles in Calabi-Yau 4-folds preserve half of the space-time supersymmetry, the Cayley submanifolds are novel as they preserve only one quarter of it. We present some simple examples. Finally, we discuss the implications of these supersymmetric cycles on mirror symmetry in higher dimensions. 
  Superunification underwent a major paradigm shift in 1984 when eleven-dimensional supergravity was knocked off its pedestal by ten-dimensional superstrings. This last year has witnessed a new shift of equal proportions: perturbative ten-dimensional superstrings have in their turn been superseded by a new non-perturbative theory called {\it $M$-theory}, which describes supermembranes and superfivebranes, which subsumes all five consistent string theories and whose low energy limit is, ironically, eleven-dimensional supergravity. In particular, six-dimensional string/string duality follows from membrane/fivebrane duality by compactifying $M$-theory on $S^1/Z_2 \times K3$ (heterotic/heterotic duality) or $S^1 \times K3$ (Type $IIA$/heterotic duality) or $S^1/Z_2 \times T^4$ (heterotic/Type $IIA$ duality) or $S^1 \times T^4$ (Type $IIA$/Type $IIA$ duality). 
  The self-duality of Dirichlet $3$-brane action under the $SL(2,R)$ duality transformation of type IIB superstring theory is shown in the Hamiltonian form of the path integral for the partition function by performing the direct integration with respect to the boundary gauge field. Through the integration in the phase space the canonical momentum conjugate to the boundary gauge field can be effectively replaced by the dual gauge field. 
  The vacuun configuration of dual supergravity in ten dimensions with one-loop fivebrane corrections is analyzed. It is shown that the compactification of this theory with rather general conditions to six-dimensional space leads to zero value of cosmological constant. 
  In this article we note that in a number of situations the operator product and the classical action satisfy a natural compatibility condition. We consider the interest of this condition to be twofold: First, the naturality (functoriality) of the compatibility condition suggests that it be used for geometrical applications of renormalized functional integration. Second, the compatibility can be used as the definition of a category; consideration of this category as the central object of study in quantum field theory seems to have quite some advantages over previously introduced theories of the type ``S-matrix theory'', ``Vertex operator algebras'', since this seems to be the only category in which both the action and the expectation values enter, the two being linked roughly speaking by a combination of the Frobenius property and the renormalized Schwinger-Dyson equation. 
  Electric-magnetic duality allows to calculate the central charges of N=2 supersymmetric theories with massless hypermultiplets as derivatives of simple modular forms. The procedure reproduces the Seiberg-Witten results for N_f=0,2,3 in a uniform way, but indicates open problems for N_f=1. 
  In this paper we numerically demonstrate that massless two-dimensional QCD is not integrable. To this aim, we explicitly solve the 't Hooft integral equation for bound states by an adaptive spline procedure, and compute the decay amplitudes. These amplitudes significantly differ from zero except in all cases in which the decay also produces a pion. 
  We evaluate the Polyakov loop and string tension at zero and finite temperature in $QED_2.$ Using bozonization the problem is reduced to solving the Schr\"odinger equation with a particular potential determined by the ground state. In the presence of two sources of opposite charges the vacuum angle parameter $\theta $ changes by $2\pi (q/e)$, independent of the number of flavors. This, in turn, alters the chiral condensate. Particularly, in the one flavor case through a simple computer algorithm, we explore the chiral dynamics of a heavy fermion. 
  A new approach to the quantization of the relativistic kink - model around the solitonic solution is developed on the ground of the collective coordinates method. The corresponding effective action is proved to be the action of the nonminimal $d=1+1$ point - particle with curvature. It is shown that upon canonical quantization this action yields the spectrum of kink - solution obtained firstly with the help of the WKB - quantization. 
  We study the operator product algebra of the supercurrent J and Konishi superfield K in four-dimensional supersymmetric gauge theories. The Konishi superfield appears in the JJ OPE and the algebra is characterized by two central charges c and c' and an anomalous dimension h for K. In free field (one-loop) approximation, c~3N_v+N_\chi and c'~N_\chi, where N_v and N_\chi are, respectively, the number of vector and chiral multiplets in the theory. In higher order c, c' and h depend on the gauge and Yukawa couplings and we obtain the two-loop contributions by combining earlier work on c with our own calculations of c'. The major result is that the radiative corrections to the central charges cancel when the one-loop beta-functions vanish, suggesting that c and c' (but not h) are invariant under continuous deformations of superconformal theories. The behavior of c and c' along renormalization group flows is studied from the viewpoint of a c-theorem. 
  For an extreme charged black hole some scalars take on a fixed value at the horizon determined by the charges alone. We call them fixed scalars. We find the absorption cross section for a low frequency wave of a fixed scalar to be proportional to the square of the frequency. This implies a strong suppression of the Hawking radiation near extremality. We compute the coefficient of proportionality for a specific model. 
  Method for computing scattering amplitudes of open strings with Dirichlet boundary on one end and Neumann boundary condition on the other is described. Vertex operator for these states are constructed using twist fields which have been studied previously in the context of Ashkin-Teller model and strings on orbifolds. Using these vertex operators, we compute the three- and four-point scattering amplitudes for (5,9) strings on 5-branes and 9-branes. In the low energy limit, these amplitudes are found to be in exact agreement with the field theory amplitudes for supersymmetric Yang-Mills coupled to hypermultiplets in 6-dimensions. We also consider the 1-brane 5-brane system and compute the amplitude for a pair of (1,5) strings to collide and to escape the brane as a closed string. (1,5) strings are found to be remarkably selective in their coupling to massless closed strings in NS-NS sector; they only couple to the dilaton. 
  Causality is studied in the covariant formulation of free string field theory (SFT). We find that, though the string field in the covariant formulation is a functional of the ghost coordinates as well as the space-time coordinate and the latter contains the time-like oscillators with negative norm, the condition for the commutator of two open string fields to vanish is simply given by $\int_0^\pi d\sigma\left(\Delta X^\mu(\sigma)\right)^2 >0$, which is the same condition as in the light-cone gauge SFT. For closed SFT, the corresponding condition is given in a form which is manifestly invariant under the rigid shifts of the $\sigma$ parameters of the two string fields. 
  We determine the exact global structure of the moduli space of $N{=}2$ supersymmetric $SO(n)$ and $\USp(2n)$ gauge theories with matter hypermultiplets in the fundamental representations, using the non-renormalization theorem for the Higgs branches and the exact solutions for the Coulomb branches. By adding an $(N{=}2)$--breaking mass term for the adjoint chiral field and varying the mass, the $N{=}2$ theories can be made to flow to either an ``electric'' $N{=}1$ supersymmetric QCD or its $N{=}1$ dual ``magnetic'' version. We thus obtain a derivation of the $N{=}1$ dualities of Seiberg. 
  We introduce and study a new discrete basis of gravity constraints by making use of harmonic expansion for closed cosmological models. The full set of constraints is splitted into area-preserving spatial diffeomorphisms, forming closed subalgebra, and Virasoro-like generators. Operatorial Hamiltonian BFV-BRST quantization is performed in the framework of perturbative expansion in the dimensionless parameter which is a positive power of the ratio of Planckian volume to the volume of the Universe. For the (N+1) - dimensional generalization of stationary closed Bianchi-I cosmology the nilpotency condition for the BRST operator is examined in the first quantum approximation. It turns out, that certain relationship between dimensionality of the space and the spectrum of matter fields emerges from the requirement of quantum consistency of the model. 
  We present the N=4 superspace constraints for the two-dimensional (2d) off-shell (4,4) supergravity with the superfield strengths expressed in terms of a (4,4) twisted (scalar) multiplet TM-I, as well as the corresponding component results, in a form suitable for applications. The constraints are shown to be invariant under the N=4 super-Weyl transformations, whose N=4 superfield parameters form another twisted (scalar) multiplet TM-II. To solve the constraints, we propose the Ansatz which makes the N=4 superconformal flatness of the N=4 supergravity curved superspace manifest. The locally (4,4) supersymmetric TM-I matter couplings, with the potential terms resulting from spontaneous supersymmetry breaking, are constructed. We also find the full (4,4) superconformally invariant (improved) TM-II matter action. The latter can be extended to the (4,4) locally supersymmetric Liouville action which is suitable for describing (4,4) supersymmetric non-critical strings. 
  A construction of master field describing multicolour QCD is presented. The master fields for large N matrix theories satisfy to standard equations of relativistic field theory but fields are quantized according $q$-deformed commutation relations with $q=0$. These commutation relations are realized in the Boltzmannian Fock space. The master field for gauge theory does not take values in a finite-dimensional Lie algebra, however, there is a non-Abelian gauge symmetry and BRST-invariance. 
  In two space-time dimensions, we write down the exact and closed Schwinger-Dyson equation for the gauged Thirring model which has been proposed recently by the author. The gauged Thirring model is a natural gauge-invariant extension of the Thirring model and reduces to the Schwinger model (in the Abelian case) in the strong four-fermion coupling limit. The exact SD equation is derived by making use of the transverse Ward-Takahashi identity as well as the usual (longitudinal) Ward-Takahashi identity. Moreover the exact solution of the SD equation for the fermion propagator is obtained together with the vertex function in the Abelian gauged case. Finally we discuss the dynamical fermion mass generation based on the solution of the SD equation. 
  It is pointed out that any conformally transformed of a flat space-time metric $\ball g_{ij} = f(x)\;\eta_{ij}$ is a solution to Witten's equation of Chern-Simons gravity, which holds outside matter in (2+1) dimensions. It is also shown that a simultaneous exterior solution to both Witten's and Einstein's equations, yields the lensing effect of an isolated cosmic string, if $f(x)$ is reduced to an arbitrary dimensionless constant. However, the solution to Witten's equation with $f(x)$ being an arbitrary and continuous function of space-time coordinates, also leads to an open circular path for a light ray near the string. 
  We investigate the behavior of softly broken $N=2$ SQCD at non-zero bare theta angle $\theta_0$, using superfield spurions to implement the SUSY breaking. We find a first-order phase transition as $\theta_0$ is varied from zero to $2 \pi$, in agreement with a prediction of `t Hooft. The low-energy theta angle $\theta_{eff}$, which determines the effective charges of dyonic excitations, has a complicated dependence on $\theta_0$ and breaking parameters. 
  We present new exact solutions of the low-energy-effective-action string equations with both dilaton $\phi $ and axion $H$ fields non-zero. The background universe is of Kantowski-Sachs type. We consider the possibility of a pseudoscalar axion field $h$ ($H=e^\phi (dh)^{*}$) that can be either time or space dependent. The case of time-dependent $h$ reduces to that of a stiff perfect-fluid cosmology. For space-dependent $h$ there is just one non-zero time-space-space component of the axion field $H$, and this corresponds to a distinguished direction in space which prevents the models from isotropising. Also, in the latter case, both the axion field $H$ and its tensor potential $B$ ($H=dB$) are dependent on time and space yet the energy-momentum tensor remains time-dependent as required by the homogeneity of the cosmological model. 
  We discuss an application of the known in QCD large $N$ expansion to strings and supermembranes in the strong coupling. In particular we use the recently obtained master field describing $ SU(\infty)$ gauge theory to argue that quantum extreme black holes obey quantum Boltzmann (infinite) statistics. This supports a topological argument by Strominger that black holes obey infinite statistics. We also speculate on a formulation of $X$-theory of strings and p-branes as theory of Grothendieck's motives. The partition function is expressed in terms of $L$-function of a motive. The Beilinson conjectures on the values of $L$-functions are interpreted as dealing with the cosmological constant problem. 
  We apply a recently proposed path-integral approach to non-local bosonization to a Thirring-like system modeling non-relativistic massless particles interacting with localized fermionic impurities. We consider forward scattering processes described by symmetric potentials including interactions between charge, current, spin and spin-current densities. In the general (spin-flipping) problem we obtain an effective action for the collective modes of the model at T = 0, containing WZW-type terms. When spin-flipping processes are disregarded the structure of the action is considerably simplified, allowing us to derive exact expressions for the dispersion relations of collective modes and two point fermionic correlation functions as functionals of the potentials. Finally, as an example, we compute the momentum distribution for the case in which electrons and impurities are coupled through spin and spin-current densities only. The formulae we get suggest that our formalism could be useful in order to seek for a mechanism able to restore Fermi liquid behavior. 
  The problem of extracting the modes of Goldstone bosons from a thermal background is reconsidered in the framework of relativistic quantum field theory. It is shown that in the case of spontaneous breakdown of an internal bosonic symmetry a recently established decomposition of thermal correlation functions contains certain specific contributions which can be attributed to a particle of zero mass. 
  The action of the isometry subgroup which preserves the trivial values of the fields is studied for the stationary D=4 Einstein--Maxwell--Dilaton--Axion theory. The technique for generation of charges and the corresponding procedure for construction of new solutions is formulated. A solution describing the double rotating dyon with independent values of all physical charges is presented. 
  We demonstrate how to obtain explicitly the propagators for quantum fields residing in curved space-time using the heat kernel for which a new construction procedure exists. Propagators are determined for the case of Rindler, Friedman-Robertson-Walker, Schwarzschild and general conformally flat metrics, both for scalar, Dirac and Yang-Mills fields. The calculations are based on an improved formula for the heat kernel in a general curved space. All the calculations are done in $d=4$ dimensions for concreteness, but are easily generalizable to arbitrary $d$. The new method advocated here does not assume that the fields are massive, nor is it based on an aymptotic expansion as such.   Whenever possible, the results are compared to that of other authors. 
  The free energy due to the vacuum fluctuations of matter fields on a classical gravitational background is discussed. It is shown explicitly how this energy is calculated for a non-minimally coupled scalar field in an arbitrary gravitational background, using the heat kernel method. The treatment of (self-)interacting fields of higher spin is outlined, using a meanfield approximation to the gaugefield when treating the gauge boson self interaction and the fermion-gauge boson interaction. 
  We outline the renormalization of the standard model to all orders of perturbation theory in a way which does not make essential use of a specific subtraction scheme but is based on the Slavnov-Taylor identity. Physical fields and parameters are used throughout the paper. The Ward-identity for the global gauge invariance of the vertex functions is formulated. As an application the Callan-Symanzik equation is derived. 
  By using the path integral method , we calculate the Green functions of field strength of Yang-Mills theories on arbitrary nonorientable surfaces in Schwinger-Fock gauge. We show that the non-gauge invariant correlators consist of a free part and an almost $x$-independent part. We also show that the gauge invariant $n$-point functions are those corresponding to the free part , as in the case of orientable surfaces. 
  We consider compactifications of the heterotic string on $K3 \times T2$ so that the resulting theory in $d = 4$ space-time dimensions has $N = 2$ supersymmetry. The gravitational and gauge coupling constants of the low-energy effective theory receive threshold corrections from loops of super-heavy string states. We calculate these corrections for the case when the $K3$-surface is a ${\bf Z}_n$ orbifold of a four torus $T4$. The results are used to determine the one-loop prepotential ${\cal F}_0^{(1)}$ for the vector multiplets and the gravitational coupling ${\cal F}_1^{(1)}$. 
  We investigate average gradient degree of normal random polynomials of fixed algebraic degree n. In particular, for polynomials of two variables, asymptotics of the average gradient degree for large values of n is determined. 
  We consider Dynamical Supersymmetry Breaking (DSB) in models with classical flat directions. We analyze a number of examples, and develop a systematic approach to determine if classical flat directions are stabilized in the full quantum theory, or lead to run-away behavior. In some cases pseudo-flat directions remain even at the quantum level before taking into account corrections to the K\"ahler potential. We show that in certain limits these corrections are calculable. In particular, we find that in the Intriligator-Thomas $SU(2)$ and its generalizations, a potential for moduli is generated. Moreover, there is a region of the parameter space where K\"ahler potential corrections lead to calculable (local) minima at large but finite distance from the origin. 
  Dilatation, i.e. scale, symmetry in the presence of the dilaton in Minkowski space is derived from diffeomorphism symmetry in curved spacetime, incorporating the volume-preserving diffeomorphisms. The conditions for scale invariance are derived and their relation to conformal invariance is examined. In the presence of the dilaton scale invariance automatically guarantees conformal invariance due to diffeomorphism symmetry. Low energy scale-invariant phenomenological Lagrangians are derived in terms of dilaton-dressed fields, which are identified as the fields satisfying the usual scaling properties. The notion of spontaneous scale symmetry breaking is defined in the presence of the dilaton. In this context, possible phenomenological implications are advocated and by computing the dilaton mass the idea of PCDC (partially conserved dilatation current) is further explored. 
  We study the non-abelian extension of the soldering process of two chiral WZW models of opposite chiralities, resulting in a (non-chiral) WZW model living in a 2D space-time with non trivial Riemanian curvature. 
  A general formula for the grand canonical partition function for a para Fermi system of any order and of any number of levels is derived. 
  Explicit formulae for asymptotic expansions of Feynman diagrams in typical limits of momenta and masses with external legs on the mass shell are presented. 
  The front form Hamiltonian for quantum chromodynamics, reduced to an effective Hamiltonian acting only in the $q\bar q$ space, is solved approximately. After coordinate transformation to usual momentum space and Fourier transformation to configuration space a second order differential equation is derived. This retarded Schr\"odinger equation is solved by variational methods and semi-analytical expressions for the masses of all 30 pseudoscalar and vector mesons are derived. In view of the direct relation to quantum chromdynamics without free parameter, the agreement with experiment is remarkable, but the approximation scheme is not adequate for the mesons with one up or down quark. The crucial point is the use of a running coupling constant $\alpha_s(Q^2)$, in a manner similar but not equal to the one of Richardson in the equal usual-time quantization. Its value is fixed at the Z mass and the 5 flavor quark masses are determined by a fit to the vector meson quarkonia. 
  An interpolation between the canonical partition functions of Bose, Fermi and Maxwell-Boltzmann statistics is proposed. This interpolation makes use of the properties of Jack polynomials and leads to a physically appealing interpolation between the statistical weights of the three statistics. This, in turn, can be used to define a new exclusion statistics in the spirit of the work of Haldane. 
  We consider a class of four parameter D=4, N=2 string models, namely heterotic strings compactified on K3 times T2 together with their dual type II partners on Calabi-Yau three-folds. With the help of generalized modular forms (such as Siegel and Jacobi forms), we compute the perturbative prepotential and the perturbative Wilsonian gravitational coupling F1 for each of the models in this class. We check heterotic/type II duality for one of the models by relating the modular forms in the heterotic description to the known instanton numbers in the type II description. We comment on the relation of our results to recent proposals for closely related models. 
  We consider the QCD$_2$ partition function in the non-local, decoupled formulation and systematically establish which subset of the nilpotent Noether charges is required to vanish on the physical states. The implications for the Hilbert space structure are also examined. 
  We present a functional derivation of recursion rules for scattering amplitudes in a non-Abelian gauge theory in a form valid to arbitrary loop order. The tree-level and one-loop recursion rules are explicitly displayed. 
  It has been argued that the superpotential can be renormalized in the presence of massless particles. Possible implications which have been considered include the restoration of supersymmetry at higher loops or a shift to a supersymmetric vacuum state. We argue that even in the presence of massless particles, there are no new contributions to the superpotential at any order in perturbation theory. This confirms the utility of the Wilsonian superpotential for analyzing the moduli space of the low energy theory. 
  In asymptotic free field theories we show that part of the OPE of the trace of the stress-energy tensor and an arbitrary composite field is determined by the anomalous dimension of the composite field. We take examples from the two-dimensional O(N) non-linear sigma model. 
  The Chern-Simons topological term dynamical generation in the effective action is obtained at arbitrary finite density. By using the proper time method and perturbation theory it is shown that $\mu^2 = m^2$ is the crucial point for Chern-Simons. So when $\mu^2 < m^2 \mu$--influence disappears and we get the usual Chern-Simons term. On the other hand when $\mu^2 > m^2$ the Chern-Simons term vanishes because of non-zero density of background fermions. In particular for massless case parity anomaly is absent at any finite density. This result holds in any odd dimension as in abelian so as in nonabelian cases. 
  We analyze gauge parameter dependence by using an algebraic method which relates the gauge parameter dependence of Green functions to an enlarged Slavnov-Taylor identity. In the course of the renormalization it turns out that gauge parameter dependence of physical parameters is already restricted at the level of Green functions. In a first step we consider the on-shell conditions which we find to be in complete agreement with these restrictions to all orders of perturbation theory. The fixing of the coupling, however, is much more involved outside the complete on-shell scheme. In the Abelian Higgs model we prove that this fixing can be properly chosen by requiring the Ward identity of gauge invariance to hold in its tree form to all orders of perturbation theory. 
  One of the perspectives in modern quantum field and string theory is related with the attempts to go beyond the perturbation theory. It turns out that a key principle in the formulation of all known non-perturbative results is {\it integrability}, i.e. arising of the structures of completely integrable systems. I discuss several important steps in this direction and speculate on its further possible development. 
  We give several pieces of evidence to show that extremal black holes cannot be obtained as limits of non-extremal black holes. We review arguments in the literature showing that the entropy of extremal black holes is zero, while that of near-extremal ones obey the Bekenstein-Hawking formula. However, from the counting of degeneracy of quantum (BPS) states of string theory the entropy of extremal stringy black holes obeys the area law. An attempt is made to reconcile these arguments. 
  We study the correspondence between IIb solitonic 1-branes and monopoles in the context of the 3-brane realization of $D=4$ $N=4$ super Yang-Mills theory. We show that a bound state of 1-branes stretching between two separated 3-branes exhibits a family of super-symmetric ground states that can be identified with the ADHMN construction of the moduli space of $SU(2)$ monopoles.. This identification is supported by the construction of the monopole gauge field as a space-time coupling in the quantum mechanical effective action of a 1-brane used as a probe. The analysis also reveals an intriguing aspect of the 1-brane theory:the transverse oscillations of the 1-branes in the ground states are described by non-commuting matrix valued fields which develop poles at the boundary. Finally, the construction is generalized to $SU(n)$ monopoles with arbitrary $n>2$. 
  We review the unified description of massless spinning particles, living in spaces of constant curvature, in the framework of the pseudoclassical approach with a gauged $N$-extended worldline supersymmetry and a local $O(N)$ invariance. 
  We discuss the relationship between the boundary conditions of the Schwinger-Dyson equations and the phase diagram of a bosonic field theory or matrix model. In the thermodynamic limit, many boundary conditions lead to the same solution, while other boundary conditions have no such limit. The list of boundary conditions for which a thermodynamic limit exists depends on the parameters of the theory. The boundary conditions of a physical solution may be quite exotic, corresponding to path integration over various inequivalent complex contours. 
  The formalism of integrable mappings is applied to the problem of constructing hierarchies of $(1+2)$ dimensional integrable systems in the $(2|2)$ superspace. We find new supersymmetric integrable mappings and corresponding to them new hierarchies of integrable systems which, at the reduction to the $(1|2)$ superspace, possess $N=2$ supersymmetry. The general formulae obtained for the hierarchies are used to explicitly derive their first nontrivial equations possessing a manifest $N=2$ supersymmetry. New bosonic substitutions and hierarchies are obtained from the supersymmetric counterparts in the bosonic limit. 
  We use rotation of one D-brane with respect to the other to reveal the hidden structure of D-branes in type-II theories. This is done by calculation of the interaction amplitude for two different parallel and angled branes. The analysis of strings with different boundary conditions at the ends is also given. The stable configuration for two similar branes occurs when they are anti-parallel. For branes of different dimensions stability is attained for either parallel or anti-parallel configurations and when dimensions differ by four the amplitude vanishes at the stable point. The results serve as more evidence that D-branes are stringy descriptions of non-perturbative extended solutions of SUGRA theories, as low energy approximation of superstrings. 
  The gauge theory on $M_4\times Z_2$ geometry is applied to the Brans-Dicke(BD) theory, where $M_4$ is the four dimensional space-time and $Z_2$ is a discrete space with two points. This approach had been previously proposed by Konisi and Saito without recourse to noncommutative geometry(NCG). Since our approach is geometrically simpler and clearer than NCG, one can see more directly the effect of the $Z_2$ space in obtaining the BD theory. 
  We describe a class of topological field theories called ``balanced topological field theories.'' These theories are associated to moduli problems with vanishing virtual dimension and calculate the Euler character of various moduli spaces. We show that these theories are closely related to the geometry and equivariant cohomology of ``iterated superspaces'' that carry two differentials. We find the most general action for these theories, which turns out to define Morse theory on field space. We illustrate the constructions with numerous examples. Finally, we relate these theories to topological sigma-models twisted using an isometry of the target space. 
  The leading order correction to the metric of a Schwarzschild black hole, due to the backreaction of infalling fermionic matter fields, is shown to produce a shift of the event horizon such that particles that would constitute Hawking radiation at late retarded times are now trapped. Fermionic field operators associated with infalling and outgoing modes at the horizon behave canonically in the semiclassical approximation. They are, however, shown to satisfy a nontrivial exchange algebra given in terms of the backreaction, when the shift is `quantized' by means of correspondence. The consequent exchange algebra for bilinear fermionic densities is also obtained. 
  We review our recent work on the low-energy actions and the realizations of strong-weak coupling dualities in non-perturbative phases of compact antisymmetric tensor field theories due to p-brane condensation. As examples we derive and discuss the confining string and confining membrane actions obtained from compact vector and tensor theories in 4D. We also mention the relevance of our results for the description of the Hagedorn phase transition of finite temperature strings. 
  Matrix models have wide applications in nuclear theory, condensed matter theory and quantum field theory. I discuss supersymmetric extensions of matrix models and their applications to branched polymers, the meander problem, and superstrings in lower dimensions. 
  We show how toroidally-compactified eleven-dimensional supergravity can be consistently truncated to yield a variety of maximally-supersymmetric ``massive'' supergravities in spacetime dimensions $D\le 8$. The mass terms arise as a consequence of making a more general ansatz than that in usual Kaluza-Klein dimensional reduction, in which one or more axions are given an additional linear dependence on one of the compactification coordinates. The lower-dimensional theories are nevertheless consistent truncations of eleven-dimensional supergravity. Owing to the fact that the generalised reduction commutes neither with U-duality nor with ordinary dimensional reduction, many different massive theories can result. The simplest examples arise when just a single axion has the additional linear coordinate dependence. We find five inequivalent such theories in D=7, and 71 inequivalent ones in D=4. The massive theories admit no maximally-symmetric vacuum solution, but they do admit $(D-2)$-brane solutions, i.e. domain walls, which preserve half the supersymmetry. We present examples of these solutions, and their oxidations to D=11. Some of the latter are new solutions of D=11 supergravity. 
  We propose a continuous Wick rotation for Dirac, Majorana and Weyl spinors from Minkowski spacetime to Euclidean space which treats fermions on the same footing as bosons. The result is a recipe to construct a supersymmetric Euclidean theory from any supersymmetric Minkowski theory. This Wick rotation is identified as a complex Lorentz boost in a five-dimensional space and acts uniformly on bosons and fermions. For Majorana and Weyl spinors our approach is reminiscent of the traditional Osterwalder Schrader approach in which spinors are ``doubled'' but the action is not hermitean. However, for Dirac spinors our work provides a link to the work of Schwinger and Zumino in which hermiticity is maintained but spinors are not doubled. Our work differs from recent work by Mehta since we introduce no external metric and transform only the basic fields. 
  We discuss the following recent applications of the ``string-inspired'' worldline technique to calculations in quantum electrodynamics: i) Photon splitting in a constant magnetic field, ii) The two-loop Euler-Heisenberg Lagrangian, iii) A progress report on a recalculation of the three-loop QED beta -- function. 
  We evaluate the chiral condensate and Polyakov loop in two-dimensional QED with a fermion of an arbitrary mass ($m$). We find discontinuous $m$ dependence in the chiral condensate and anomalous temperature dependence in Polyakov loops when the vacuum angle $\theta$$\sim$$\pi$ and $m$=O($e$). These nonperturbative phenomena are due to the bifurcation process in the solutions to the vacuum eigenvalue equation. 
  The partial spontaneous breaking of rigid N=2 supersymmetry implies the existence of a massless N=1 Goldstone multiplet. In this paper we show that the spin-(1/2,1) Maxwell multiplet can play this role. We construct its full nonlinear transformation law and find the invariant Goldstone action. The spin-1 piece of the action turns out to be of Born-Infeld type, and the full superfield action is duality invariant. This leads us to conclude that the Goldstone multiplet can be associated with a D-brane solution of superstring theory for p=3. In addition, we find that N=1 chirality is preserved in the presence of the Goldstone-Maxwell multiplet. This allows us to couple it to N=1 chiral and gauge field multiplets. We find that arbitrary Kahler and superpotentials are consistent with partially broken N=2 supersymmetry. 
  Starting from an associated reparametrization-invariant action, the generalization of the BRST-BFV method for the case of nonstationary systems is constructed. The extension of the Batalin-Tyutin conversional approach is also considered in the nonstationary case. In order to illustrate these ideas, the propagator for the time-dependent two-dimensional rotor is calculated by reformulating the problem as a system with only first class constraints and subsequently using the BRST-BFV prescription previously obtained. 
  We study the spin factor problem both in $3+1$ and $2+1$ dimensions which are essentially different for spin factor construction. Doing all Grassmann integrations in the corresponding path integral representations for Dirac propagator we get representations with spin factor in arbitrary external field. Thus, the propagator appears to be presented by means of bosonic path integral only. In $3+1$ dimensions we present a simple derivation of spin factor avoiding some unnecessary steps in the original brief letter (Gitman, Shvartsman, Phys. Lett. {\bf B318} (1993) 122) which themselves need some additional justification. In this way the meaning of the surprising possibility of complete integration over Grassmann variables gets clear. In $2+1$ dimensions the derivation of the spin factor is completely original. Then we use the representations with spin factor for calculations of the propagator in some configurations of external fields. Namely, in constant uniform electromagnetic field and in its combination with a plane wave field. 
  The propagator of a spinning particle in external Abelian field and in arbitrary dimensions is presented by means of a path integral. The problem has different solutions in even and odd dimensions. In even dimensions the representation is just a generalization of one in four dimensions (it has been known before). In this case a gauge invariant part of the effective action in the path integral has a form of the standard (Berezin-Marinov) pseudoclassical action. In odd dimensions the solution is presented for the first time and, in particular, it turns out that the gauge invariant part of the effective action differs from the standard one. We propose this new action as a candidate to describe spinning particles in odd dimensions. Studying the hamiltonization of the pseudoclassical theory with this action, we show that the operator quantization leads to adequate minimal quantum theory of spinning particles in odd dimensions. In contrast with the models proposed formerly in this case the new one admits both the operator and the path integral quantization. Finally the consideration is generalized for the case of the particle with anomalous magnetic moment. 
  We investigate thermalization processes occurring at different time scales in the Yang-Mills-Higgs system at high temperatures. We determine the largest Lyapunov exponent associated with the gauge fields and verify its relation to the perturbatively calculated damping rate of a static gauge boson. 
  The action for a relativistic free particle of mass $m$ receives a contribution $-mds$ from a path segment of infinitesimal length $ds$. Using this action in a path integral, one can obtain the Feynman propagator for a spinless particle of mass $m$. If one of the effects of quantizing gravity is to introduce a minimum length scale $L_P$ in the spacetime, then one would expect the segments of paths with lengths less than $L_P$ to be suppressed in the path integral. Assuming that the path integral amplitude is invariant under the `duality' transformation $ds\to L_P^2/ds$, one can calculate the modified Feynman propagator. I show that this propagator is the same as the one obtained by assuming that: quantum effects of gravity leads to modification of the spacetime interval $(x-y)^2$ to $(x-y)^2+L_P^2$. This equivalence suggests a deep relationship between introducing a `zero-point-length' to the spacetime and postulating invariance of path integral amplitudes under duality transformations. 
  We show that the genus 34 Seiberg-Witten curve underlying $N=2$ Yang-Mills theory with gauge group $E_6$ yields physically equivalent results to the manifold obtained by fibration of the $E_6$ ALE singularity. This reconciles a puzzle raised by $N=2$ string duality. 
  It is shown that the model X_{14}(7,3,2,1,1) has two Calabi--Yau phases. 
  We review a progress in our understanding of the moduli space for an arbitrary number of BPS monopoles in a gauge theory with a group $G$ of rank $r$ that is maximally broken to $U(1)^r$. The derivation of the moduli space metric has been obtained from studying the low energy dynamics of well-separated dyons. 
  We study some aspects of enhanced gauge symmetries in F-theory compactified on K3. We find open string configurations connecting various 7-branes which represent stable BPS states. In this approach we recover $D_n$ and $E_n$ gauge groups previously found from an analysis of sigularities of the moduli space of elliptically fibered K3 manifolds as well as examples of non-perturbative realizations of $A_n$ groups. 
  We study the non-abelian extension for the splitting of a scalar field into chiral components. Using this procedure we find a non ambiguous way of coupling a non abelian chiral scalar field to gravity. We start with a (non-chiral) WZW model covariantly coupled to a background metric and, after the splitting, arrive at two chiral Wess-Zumino-Witten (WZW) models coupled to gravity. 
  After recalling briefly some basic properties of the quantum group $GL_q(2)$, we study the quantum sphere $S_q^2$, quantum projective space $CP_q(N)$ and quantum Grassmannians as examples of complex (K\"{a}hler) quantum manifolds. The differential and integral calculus on these manifolds are discussed. It is shown that many relations of classical projective geometry generalize to the quantum case. For the case of the quantum sphere a comparison is made with A. Connes' method. 
  A method is presented, and used, for determining any heat-kernel coefficient for the form-valued Laplacian on the $D$-ball as an explicit function of dimension and form order. The calculation is offerred as a particular application of a general technique developed earlier for obtaining heat-kernel coefficients on a bounded generalised cone which involves writing the sphere and ball \zfs, and coefficients, in terms of Barnes \zfs\ and generalised Bernoulli polynomials. Functional determinants are computed. Spinors are also treated by the general method. 
  We investigate (1,0)-superconformal Toda theories based on simple Lie algebras and find that the classical integrability properties of the underlying bosonic theories do not survive. For several models based on algebras of low rank, we show explicitly that none of the conserved W-algebra generators can be generalized to the supersymmetric case. Using these results we deduce that at least one W-algebra generator fails to generalize in any model based on a classical Lie algebra. This argument involves a method for relating the bosonic Toda theories and their conserved currents within each classical series. We also scrutinize claims that the (1,0)-superconformal models actually admit (1,1) supersymmetry and find that they do not. Our results are consistent with the belief that all integrable Toda models with fermions arise from Lie superalgebras. 
  It is demonstrated that the field strength approach to Yang Mills theories has essential features of the dual description. In D=3 this approach is formulated in terms of gauge invariant variables. 
  We introduce the chemical potential in a system of massless fermions in a bag by impossing boundary conditions in the Euclidean time direction. We express the fermionic mean number in terms of a functional trace involving the Green's function of the boundary value problem, which we study analytically. Numerical evaluations are made, and an application to a simple hadron model is discussed. 
  The scalar two-loop master diagram is revisited in the massive cases needed for the computation of boson and fermion propagators in QED and QCD. By means of the causal method it is possible in a straightforward manner to express the propagators as double integrals. In the case of vacuum polarization both integrations can be carried out in terms of polylogarithms, whereas the last integral in the fermion propagator cannot be expressed by known special functions. The advantage of the method in comparison with Feynman integral calculations is indicated. 
  We describe a class of 4d N=1 compactifications of the $SO(32)$ heterotic/type I string theory which are destabilized by nonperturbatively generated superpotentials. In the type I description, the destabilizing superpotential is generated by a one instanton effect or gaugino condensation in a nonperturbative $SU(2)$ gauge group. The dual, heterotic description involves destabilization due to worldsheet instanton or $\it half$ worldsheet instanton effects in the two cases. A genericity argument suggests that a (global) supersymmetry-breaking model of Intriligator and Thomas might be typical in a class of string theory models. Our analysis also suggests that the tensionless strings which arise in the $E_8 \times E_8$ theory in six dimensions when an instanton shrinks to zero size should, in some cases, have supersymmetry breaking dynamics upon further compactification to four dimensions. We provide explicit examples, constructed using F-theory, of the two cases of dynamically generated superpotentials. 
  We study cosmological solutions of type II string theory with a metric of the Kaluza--Klein type and nontrivial Ramond--Ramond forms. It is shown that models with only one form excited can be integrated in general. Moreover, some interesting cases with two nontrivial forms can be solved completely since they correspond to Toda models. We find two types of solutions corresponding to a negative time superinflating phase and a positive time subluminal expanding phase. The two branches are separated by a curvature singularity. Within each branch the effect of the forms is to interpolate between different solutions of pure Kaluza--Klein theory. 
  It is known that the critical N=(2,2) string describes 2+2 dimensional self-dual gravity in a non-covariant form, since it requires the choice of a complex structure in the target, which leaves only U(1,1) Lorentz symmetry. We briefly review picture-changing and spectral flow and show that the world-sheet Maxwell instantons individually break the Lorentz group further to SU(1,1). However, their contributions conspire to restore full SO(2,2) global symmetry if dilaton and axion fields are assembled in a null anti-self-dual two-form, denying them the status of Lorentz scalars. We present the fully SO(2,2) invariant tree-level three-point amplitude and the corresponding extension of the Plebanski action for self-dual gravity. 
  String theory posesses numerous axion candidates. The recent realization that the compactification radius in string theory might be large means that these states can solve the strong CP problem. This still leaves the question of the cosmological bound on the axion mass. Here we explore two schemes for accommodating such light axions in cosmology. In the first, we note that in string theory the universe is likely to be dominated early on by the coherent oscillations of some moduli. The usual moduli problem assumes that these fields have masses comparable to the gravitino. We argue that string moduli are likely to be substantially more massive, eliminating this problem. In such cosmologies the axion bound is significantly weakened. Plausible mechanisms for generating the baryon number density are described. In the second, we point out that in string theory, the axion potentials might be much larger at early times than at present. In string theory, if CP violation is described by a small parameter, the axion may sit sufficiently close to its true minimum to invalidate the bounds. 
  The purpose of this contribution is to review some aspects of the loop space formulation of pure gauge theories having the connection defined over a Lie algebra. The emphasis is focused on the discussion of the Mandelstam identities, which provide the basic constraints upon both the classical and the quantum degrees of freedom of the theory. In the case where the connection is extended to be valued on a super Lie algebra, some new results are presented which can be considered as first steps towards the construction of the Mandelstam identities in this situation, which encompasses such interesting cases as supergravity in $3+1$ dimensions together with $2+1$ super Chern-Simons theories, for example. Also, these ideas could be useful in the loop space formulation of fully supersymmetric theories. 
  It is shown that fractional $Z_3$-superspace is isomorphic to the $q\to\exp(2\pi i/3)$ limit of the braided line. $Z_3$-supersymmetry is identified as translational invariance along this line. The fractional translation generator and its associated covariant derivative emerge as the $q\to\exp(2\pi i/3)$ limits of the left and right derivatives from the calculus on the braided line 
  We consider the combined Yang Mills-Dilaton-Gravity system in the presence of a Gauss-Bonnet term as it appears in the $4D$ Effective Superstring Action. We give analytical arguments and demonstrate numerically the existence of black hole solutions with non-trivial dilaton and Yang Mills hair for the particular case of SU(2) gauge fields. The thermodynamical properties of the solutions are also discussed. 
  Taking the (2,2) strings as a starting point, we discuss the equivalent integrable field theories and analyze their symmetry structure in 2+2 dimensions from the viewpoint of string/membrane unification. Requiring the Lorentz invariance and supersymmetry in the (2,2) string target space leads to an extension of the (2,2) string theory to a theory of 2+2 dimensional supermembranes (M-branes) propagating in a higher dimensional target space. The origin of the hidden target space dimensions of the M-brane is related to the maximally extended supersymmetry implied by the Lorentz covariance and dimensional reasons. The K"ahler-Chern-Simons-type action describing the self-dual gravity in 2+2 dimensions is proposed. Its maximal supersymmetric extension (of the Green-Schwarz-type) naturally leads to the 2+10 (or higher) dimensions for the M-brane target space. The proposed OSp(32|1) supersymmetric action gives the pre-geometrical description of M-branes, which may be useful for a fundamental formulation of F&M theory. 
  In this article we examine the appearance of tensionless strings in M-Theory. We subsequently interpret these tensionless strings in a String Theory context. In particular, we examine tensionless strings appearing in M-Theory on $S^{1}$, M-Theory on $S^{1} / {\bf Z}_{2}$, and M-Theory on $T^{2}$; we then interpret the appearance of such strings in a String Theory context. Then we reverse this process and examine the appearance of some tensionless strings in String Theory. Subsequently we interpret these tensionless strings in a M-Theory context. 
  Whereas the usual understanding is that the entropy of only a non-extremal black hole is given by the area of the horizon, there are derivations of an area law for extremal black holes in some model calculations. It is explained here how such results can arise in an approach where one sums over topologies and imposes the extremality condition after quantization. 
  We consider an interaction representation in the Boltzmann field theory. It describes the master field for a subclass of planar diagrams in matrix models, so called half-planar diagrams. This interaction representation was found in the previous paper by Accardi, Volovich and one of us (I.A.) and it has an unusual property that one deals with a rational function of the interaction Lagrangian instead of the ordinary exponential function. Here we study the interaction representation in more details and show that under natural assumptions this representation is in fact unique. We demonstrate that corresponding Schwinger-Dyson equations lead to a closed set of integral equations for two- and four-point correlation functions. Renormalization of the model is performed and renormalization group equations are obtained. Some model examples with discrete number of degrees of freedom are solved numerically. The solution for one degree of freedom is compared with the planar approximation for one matrix model. For large variety of coupling constant it reproduces the planar approximation with good accuracy. 
  We show that coarse graining arguments invented for the analysis of multi-spin systems on a randomly triangulated surface apply also to the O(n) model on a random lattice. These arguments imply that if the model has a critical point with diverging string susceptibility, then either \g=+1/2 or there exists a dual critical point with negative string susceptibility exponent, \g', related to \g by \g=\g'/(\g'-1). Exploiting the exact solution of the O(n) model on a random lattice we show that both situations are realized for n>2 and that the possible dual pairs of string susceptibility exponents are given by (\g',\g)=(-1/m,1/(m+1)), m=2,3,.... We also show that at the critical points with positive string susceptibility exponent the average number of loops on the surface diverges while the average length of a single loop stays finite. 
  An unusual four-dimensional generally covariant and supersymmetric SU(2) gauge theory is described. The theory has propagating degrees of freedom, and is invariant under a local (left-handed) chiral supersymmetry, which is half the supersymmetry of supergravity. The Hamiltonian 3+1 decomposition of the theory reveals the remarkable feature that the local supersymmetry is a consequence of Yang-Mills symmetry, in a manner reminiscent of how general coordinate invariance in Chern-Simons theory is a consequence of Yang-Mills symmetry. It is possible to write down an infinite number of conserved currents, which strongly suggests that the theory is classically integrable. A possible scheme for non-perturbative quantization is outlined. This utilizes ideas that have been developed and applied recently to the problem of quantizing gravity. 
  Starting from the expression for the superdeterminant of $ (xI-M)$, where $M$ is an arbitrary supermatrix , we propose a definition for the corresponding characteristic polynomial and we prove that each supermatrix satisfies its characteristic equation. Depending upon the factorization properties of the basic polynomials whose ratio defines the above mentioned superdeterminant we are able to construct polynomials of lower degree which are also shown to be annihilated by the supermatrix. 
  The Thomas-Fermi screening of non-Abelian gauge fields by fermions or screening of gluon fields in quark matter is discussed. It is described by an effective mass term which is, as with hard thermal loops, related to the eikonal for a Chern-Simons theory and the Wess-Zumino-Witten action. 
  We discuss the differing definitions of complex and quaternionic projective group representations employed by us and by Emch. The definition of Emch (termed here a strong projective representation) is too restrictive to accommodate quaternionic Hilbert space embeddings of complex projective representations. Our definition (termed here a weak projective representation) encompasses such embeddings, and leads to a detailed theory of quaternionic, as well as complex, projective group representations. 
  We introduce a graphical representation for a global SO(n) tensor $\pl_\m\pl_\n h_\ab$, which generally appears in the perturbative approach of gravity around the flat space: $g_\mn=\del_\mn+h_\mn$. We systematically construct global SO(n) invariants. Independence and completeness of those invariants are shown by taking examples of $\pl\pl h$-, and $ (\pl\pl h)^2$- invariants. They are classified graphically. Indices which characterize all independent invariants (or graphs) are given. We apply the results to general invariants with dimension $(Mass)^4$ and the Gauss-Bonnet identity in 4-dim gravity. 
  Tensor calculation of suffix-contraction is carried out by a C-program. Tensors are represented graphically, and the algorithm makes use of the topology of graphs. Classical and quantum gravity, in the weak-field perturbative approach, is a special interest. Examples of the leading order calculation of some general invariants such as $R_{\mn\ls}R^{\mn\ls}$ are given. Application to Weyl anomaly calculation is commented. 
  In this paper we analyze the earlier constructions of the type IIA dual pairs through orientifolding. By an appropriate choice of $\Gamma$-matrix basis for the spinor representations of the $U$-duality group, we give an explicit relationship between the orientifold models and their dual pairs. 
  We use the four-dimensional N=2 central charge superspace to give a geometrical construction of the Abelian vector-tensor multiplet consisting, under N=1 supersymmetry, of one vector and one linear multiplet. We derive the component field supersymmetry and central charge transformations, and show that there is a super-Lagrangian, the higher components of which are all total derivatives, allowing us to construct superfield and component actions. 
  We define an algebra on the space of BPS states in theories with extended supersymmetry. We show that the algebra of perturbative BPS states in toroidal compactification of the heterotic string is closely related to a generalized Kac-Moody algebra. We use D-brane theory to compare the formulation of RR-charged BPS algebras in type II compactification with the requirements of string/string duality and find that the RR charged BPS states should be regarded as cohomology classes on moduli spaces of coherent sheaves. The equivalence of the algebra of BPS states in heterotic/IIA dual pairs elucidates certain results and conjectures of Nakajima and Gritsenko & Nikulin, on geometrically defined algebras and furthermore suggests nontrivial generalizations of these algebras. In particular, to any Calabi-Yau 3-fold there are two canonically associated algebras exchanged by mirror symmetry. 
  We consider a four-fermion theory as a simple model of dynamical symmetry breaking in flat space with non-trivial topology, motivated from recent studies in similar considerations in curved space. The phase structure is investigated, by developing a useful formalism to evaluate the effective potential in arbitrary compactified flat space in 3- and 4-dimensional spacetime. The phase structure is significantly altered due to the finite volume effect in the compactified space. Interestingly, the effect works in different way depending on the boundary condition of the fermion fields. The physical interpretation of the results and its implication on the dynamical symmetry breaking phenomenon in curved space are discussed. 
  We investigate phase transitions in three dimensional scalar matrix models, with special emphasis on complex $2 \times 2$ matrices. The universal equation of state for weak first order phase transitions is computed. We also study the coarse grained free energy. Its dependence on the coarse graining scale gives a quantitative criterion for the validity of the standard treatment of bubble nucleation. 
  Starting from a conformal Haag-Kastler net in 1+1 dimensions, Wightman functions are constructed. 
  By using the standard perturbation theory we study the mass as well as $\theta$ parameter dependence of the Seiberg-Witten theory with $SU(2)$ gauge group, supplemented with a $N=1$ supersymmetric as well as a smaller nonsupersymmetric (e.g.) gaugino mass. Confinement persists at all values of the bare vacuum parameter $\theta_{ph}$; as it is varied there is however a phase transition at $\theta_{ph} =-\pi, \pi, 3\pi, \ldots$. At these values of $\theta_{ph}$ the vacuum is doubly degenerate and CP invariance is spontaneously broken \`a la Dashen. Due to the instanton-induced renormalization effect the low energy effective $\theta$ parameter remains small irrespectively of $\theta_{ph}$ as long as the gaugino mass is sufficiently small. 
  We study general perturbations of two-dimensional conformal field theories by holomorphic fields. It is shown that the genus one partition function is controlled by a contact term (pre-Lie) algebra given in terms of the operator product expansion. These models have applications to vertex operator algebras, two-dimensional QCD, topological strings, holomorphic anomaly equations and modular properties of generalized characters of chiral algebras such as the $W_{1+\infty}$ algebra, that is treated in detail. 
  In this talk we briefly review the concept of supersymmetric quantum mechanics using a model introduced by Witten. A quasi-classical path-integral evaluation for this model is performed, leading to a so-called supersymmetric quasi-classical quantization condition. Properties of this quantization condition are compared with those derived from the standard WKB approach. 
  The integrable mappings formalism is generalized on non--commutative case. Arising hierarchies of integrable systems are invariant with respect to this "quantum" discrete transformations without any assumption about commutative properties of unknown operators they include. Partially, in the scope of this construction are the equations for Heisenberg operators of quantum (integrable) systems. 
  We obtain the exact ground state for the Calogero-Sutherland problem in arbitrary dimensions. In the special case of two dimensions, we show that the problem is connected to the random matrix problem for complex matrices, provided the strength of the inverse-square interaction $g = 2$. In the thermodynamic limit, we obtain the ground state energy and the pair-correlation function and show that in this case there is no long-range order. 
  Black holes do not Hawking radiate strictly blackbody radiation due to well-known frequency-dependent greybody factors. These factors arise from frequency-dependent potential barriers outside the horizon which filter the initially blackbody spectrum emanating from the horizon. D-brane bound states, in a thermally excited state corresponding to near-extremal black holes, also do not emit blackbody radiation: The bound state radiation spectrum encodes the energy spectrum of its excitations. We study a near-extremal five-dimensional black hole. We show that, in a wide variety of circumstances including both neutral and charged emission, the effect of the greybody filter is to transform the blackbody radiation spectrum precisely into the bound state radiation spectrum. Implications of this result for the information puzzle in the context of near-extremal black hole dynamics are discussed. 
  We present two 3-family SU(5) grand unified models in the heterotic string theory. One model has 3 chiral families and 9 pairs of $5+{\overline 5}$ Higgs fields, and an asymptotically-free SU(2) X SU(2) hidden sector, where the two SU(2)s have different matter contents. The other model has 6 left-handed and 3 right-handed 10s, 12 left-handed and 9 right-handed ${\overline 5}$s, and an asymptotically-free SU(3) hidden sector. At the string scale, the gauge couplings $g^2$ of the hidden sector are three times as big as that of SU(5). In addition, both models have an anomalous U(1). 
  We shall discuss various kinds of geometric bremsstrahlung processes in the spatially flat Robertson-Walker universe. Despite that the temperature of the universe is much higher than particle masses and the Hubble parameter, the transition probability of these processes do not vanish. It is also pointed out that explicit forms of the probability possess a new duality with respect to scale factor of background geometry. 
  The path integral for the hydrogen atom, for example, is formulated as a one-dimensional quantum gravity coupled to matter fields representing the electron coordinates. The (renormalized) cosmological constant, which corresponds to the energy eigenvalue, is thus quantized in this model. Possible implications of the quantized cosmological constant are briefly discussed. 
  The structure of the reduced phase space arising in the Hamiltonian reduction of the phase space corresponding to a free particle motion on the group ${\rm SL}(2, {\Bbb R})$ is investigated. The considered reduction is based on the constraints similar to those used in the Hamiltonian reduction of the Wess--Zumino--Novikov--Witten model to Toda systems. It is shown that the reduced phase space is diffeomorphic either to the union of two two--dimensional planes, or to the cylinder $S^1 \times {\Bbb R}$. Canonical coordinates are constructed for the both cases, and it is shown that in the first case the reduced phase space is symplectomorphic to the union of two cotangent bundles $T^*({\Bbb R})$ endowed with the canonical symplectic structure, while in the second case it is symplectomorphic to the cotangent bundle $T^*(S^1)$ also endowed with the canonical symplectic structure. 
  On the base of Lie algebraic and differential geometry methods, a wide class of multidimensional nonlinear systems is obtained, and the integration scheme for such equations is proposed. 
  The use of complex geometry allows us to obtain a consistent formulation of octonionic quantum mechanics (OQM). In our octonionic formulation we solve the hermiticity problem and define an appropriate momentum operator within OQM. The nonextendability of the completeness relation and the norm conservation is also discussed in details. 
  In order to obtain a consistent formulation of octonionic quantum mechanics (OQM), we introduce left-right barred operators. Such operators enable us to find the translation rules between octonionic numbers and $8\times 8$ real matrices (a translation is also given for $4\times 4$ complex matrices). We develop an octonionic relativistic free wave equation, linear in the derivatives. Even if the wave functions are only one-component we show that four independent solutions, corresponding to those of the Dirac equation, exist. 
  We reformulate free equations of motion for massive spin 0 and spin 1/2 matter fields in 2+1 dimensional anti-de Sitter space in the form of some covariant constantness conditions. The infinite-dimensional representation of the anti-de Sitter algebra underlying this formulation is shown to admit a natural realization in terms of the algebra of deformed oscillators with a deformation parameter related to the parameter of mass. 
  We investigate a generalized non-linear O(3) $\sigma$-model in three space dimensions where the fields are maps $S^3 \mapsto S^2$. Such maps are classified by a homotopy invariant called the Hopf number which takes integer values. The model exhibits soliton solutions of closed vortex type which have a lower topological bound on their energies. We explicitly compute the fields for topological charge 1 and 2 and discuss their shapes and binding energies. The effect of an additional potential term is considered and an approximation is given for the spectrum of slowly rotating solitons. 
  We construct representations of the Heisenberg algebra by pushing the perturbation expansion to high orders. If the multiplication operators $B_{1,2}$ tend to differential operators of order $l_{2,1}$, respectively, the singularity is characterized by $(l _{1},l_{2})$. Let $l_{1} \geq l_{2}$. Then the two cases A : ``$l_{2}$ does not divide $l_{1}$'' and B : ``$l_{2}$ divides $l_{1}$'' need a different treatment. The universality classes are labelled $[p,q]$ where $[p,q]$=[$l_{1}$,$l_{2}$] in case A and $[p,q]$=[$l_{1}+1$,$l_{2}$] in case B. 
  We propose a Lagrangian path integral based on gauge symmetries generated by a symmetric higher-order $\Delta$-operator, and demonstrate that this path integral is independent of the chosen gauge-fixing function. No explicit change of variables in the functional integral is required to show this. 
  We construct the nonlinear $W(sl(N+3),sl(3))$ algebras and find the spectrum of values of the central charge that gives rise, by contracting the $W(sl(N+3),sl(3))$ algebras, to a $W_3$ algebra belonging to the coset $W((sl(N+3),sl(3))/(u(1)\oplus sl(N))$. Part of the spectrum was conjectured before, but part of it is given here for the first time. Using the tool of embedding the $W(sl(N+3),sl(3))$ algebras into linearizing algebras, we construct new realizations of $W_3$ modulo null fields. The possibility to predict, within the conformal linearization framework, the central charge spectrum for minimal models of the nonlinear $W(sl(N+3),sl(3))$ algebras is discussed at the end. 
  The thermofield dynamics of the D=26 closed bosonic thermal string theory is described in proper reference to the thermal duality symmetry as well as the thermal stability of modular invariance in association with the global phase structure of the bosonic thermal string ensemble. 
  We analyse approximate solutions to an exact renormalisation group equation with particular emphasis on their dependence on the regularisation scheme, which is kept arbitrary. Physical quantities related to the coarse-grained potential of scalar QED display universal behaviour for strongly first-order phase transitions. Only subleading corrections depend on the regularisation scheme and are suppressed by a sufficiently large UV scale. We calculate the relevant coarse-graining scale and give a condition for the applicability of Langer's theory of bubble nucleation. 
  We determine the effective prepotential for N=2 supersymmetric SU(N_c) gauge theories with an arbitrary number of flavors N_f < 2N_c, from the exact solution constructed out of spectral curves. The prepotential is the same for the several models of spectral curves proposed in the literature. It has to all orders the logarithmic singularities of the one-loop perturbative corrections, thus confirming the non-renormalization theorems from supersymmetry. In particular, the renormalized order parameters and their duals have all the correct monodromy transformations prescribed at weak coupling. We evaluate explicitly the contributions of one- and two-instanton processes. 
  A quantal guiding center theory allowing to systematically study the separation of the different time scale behaviours of a quantum charged spinning particle moving in an external inhomogeneous magnetic filed is presented. A suitable set of operators adapting to the canonical structure of the problem and generalizing the kinematical momenta and guiding center operators of a particle coupled to a homogenous magnetic filed is constructed. The Pauli Hamiltonian rewrites in this way as a power series in the magnetic length $l_B= \sqrt{\hbar c/eB}$ making the problem amenable to a perturbative analysis. The first two terms of the series are explicitly constructed. The effective adiabatic dynamics turns to be in coupling with a gauge filed and a scalar potential. The mechanism producing such magnetic-induced geometric-magnetism is investigated in some detail. 
  We point out an effect which may stabilize a supersymmetric membrane moving on a manifold with boundary, and lead to a light-cone Hamiltonian with a discrete spectrum of eigenvalues. The analysis is carried out explicitly for a closed supermembrane in the regularized $SU(N)$ matrix model version. 
  The notion of the integral over the anticommuting Grassmann variables (nonquantum fermionic fields) seems to be the most powerful tool in order to extract the exact analytic solutions for the 2D Ising models on simple and more complicated lattices, which is the subject of a discussion in this report. 
  The renormalization group method of Goldenfeld, Oono and their collaborators is applied to asymptotic analysis of vector fields. The method is formulated on the basis of the theory of envelopes, as was done for scalar fields. This formulation actually completes the discussion of the previous work for scalar equations. It is shown in a generic way that the method applied to equations with a bifurcation leads to the Landau-Stuart and the (time-dependent) Ginzburg-Landau equations. It is confirmed that this method is actually a powerful theory for the reduction of the dynamics as the reductive perturbation method is. Some examples for ordinary diferential equations, such as the forced Duffing, the Lotka-Volterra and the Lorenz equations, are worked out in this method: The time evolution of the solution of the Lotka-Volterra equation is explicitly given, while the center manifolds of the Lorenz equation are constructed in a simple way in the RG method. 
  This is the text of a talk given at the Inaugural Conference of the Asia Pacific Center for Theoretical Physics, Seoul, Korea, June 9, 1996. If nature is described by string theory, and if the compactification radius is large (as suggested by the unification of couplings), then the theory is in a regime best described by the low energy limit of $M$-theory. We discuss some phenomenological aspects of this view. The scale at which conventional quantum field theory breaks down is of order the unification scale and consequently (approximate) discrete symmetries are essential to prevent proton decay. There are one or more light axions, one of which solves the strong CP problem. Modular cosmology is still problematic but much more complex than in perturbative string vacua. 
  The low-density expansion for a homogeneous interacting Bose gas at zero temperature can be formulated as an expansion in powers of $\sqrt{\rho a^3}$, where $\rho$ is the number density and $a$ is the S-wave scattering length. Logarithms of $\rho a^3$ appear in the coefficients of the expansion. We show that these logarithms are determined by the renormalization properties of the effective field theory that describes the scattering of atoms at zero density. The leading logarithm is determined by the renormalization of the pointlike $3 \to 3$ scattering amplitude. 
  We study the Brans-Dicke model in the presence of an axion. The dynamical equations are solved when the fields are space independent and the metric is spatially flat. It is found that at late time the scale factor undergoes decelerated expansion but the dilaton grows large. At early time, scale factor and the dilaton approach constants. 
  In this talk I describe recent work (hep-th/9606029) in which I classified all conceivable 2+1 dimensional Chern-Simons (CS) theories with continuous compact abelian gauge group or finite abelian gauge group. The CS theories with finite abelian gauge group that can be obtained from the spontaneous breakdown of a CS theory with gauge group the direct product of various compact U(1) gauge groups were also identified. Those that can not be reached in this way are actually the most interesting since they lead to nonabelian phenomena such as nonabelian braid statistics, Alice fluxes and Cheshire charges and quite generally lead to dualities with 2+1 dimensional theories with a nonabelian finite gauge group. 
  This thesis uses Path Integrals and Green's Functions to study Gravity, Quantum Field Theory and Statistical Mechanics, particularly with respect to: finite temperature quantum systems of different spin in gravitational fields; finite temperature interacting quantum systems in perturbative regime; self-interacting fermi models in non-trivial space-time of different dimensions; non-linear quantum models at finite temperatures in a background curved space-time; 3-D topological field models in non-trivial space-time and at finite temperatures; thermal quantum systems in a background curved space-time. Results include: Non-Equivalence of Inertial and Gravitational Mass. 
  The past year has seen enormous progress in string theory. It has become clear that all of the different string theories are different limits of a single theory. Moreover, in certain limits, one obtains a new, eleven-dimensional structure known as $M$-theory. Strings with unusual boundary conditions, known as D-branes, turn out to be soliton solutions of string theory. These have provided a powerful tool to probe the structure of these theories. Most dramatically, they have yielded a partial understanding of the thermodynamics of black holes in a consistent quantum mechanical framework. In this brief talk, I attempt to give some flavor of these developments. 
  In this paper we compute the low energy absorption cross-section for minimally coupled massles scalars and spin-$1/2$ particles, into a general spherically symmetric black hole in arbitrary dimensions. The scalars have a cross section equal to the area of the black hole, while the spin-$1/2$ particles give the area measured in a flat spatial metric conformally related to the true metric. 
  This note discusses some results on the non-BPS excitations of D-branes. We show that the excitation spectrum of a bound state of D-strings changes character when the length of the wrapping circle becomes less than $\sim g^{-1}\LS$. We review the observed relation between the low energy absorption cross-section of D-branes and the low energy absorption cross-section for black holes. We discuss various issues related to the information question for black holes. 
  We propose a lattice formulation of the chiral fermion which maximally respects the gauge symmetry and simultaneously is free of the unwanted species doublers. The formulation is based on the lattice fermion propagator and composite operators, rather than on the lattice fermion action. The fermionic determinant is defined as a functional integral of an expectation value of the gauge current operator with respect to the background gauge field: The gauge anomaly is characterized as the non-integrability. We perform some perturbative test to confirm the gauge covariance and an absence of the doublers. The formulation can be applied rather straightforwardly to numerical simulations in the quenched approximation. 
  We study the Gribov problem within a Hamiltonian formulation of pure Yang-Mills theory. For a particular gauge fixing, a finite volume modification of the axial gauge, we find an exact characterization of the space of gauge-inequivalent gauge configurations. 
  We show that a class of extremal four-dimensional supersymmetric ``high-branes'', i.e. string and domain wall solutions, can be interpreted as intersections of four ten-dimensional Dirichlet branes. These $d=4$ solutions are related, via $T$-duality in ten dimensions, to the four-dimensional extremal Maxwell/scalar black holes that are characterized by a scalar coupling parameter $a$ with $a=0,1/\sqrt{3}, 1, \sqrt{3}$. 
  Using the hierarchical approximation, we discuss the cut-off dependence of the renormalized quantities of a scalar field theory. The naturalness problem and questions related to triviality bounds are briefly discussed. We discuss unphysical features associated with the hierarchical approximation such as the recently observed oscillatory corrections to the scaling laws. We mention a two-parameter family of recursion formulas which allows one to continuously extrapolate between Wilson's approximate recursion formula and the recursion formula of Dyson's hierarchical model. The parameters are the dimension D and 2^zeta, the number of sites integrated in one RG transformation. We show numerically that at fixed D, the critical exponent gamma depends continuously on zeta. We suggest the requirement of zeta -independence as a guide for constructing improved recursion formulas. 
  We employ the method of differential regularization to calculate explicitly the one-loop effective action of a bosonized $U_L(3)\times U_R(3)$ extended Nambu--Jona-Lasinio model consisting of scalar, pseudoscalar, vector and axial vector fields. 
  We apply the method of confining phase superpotentials to N = 2 supersymmetric Yang-Mills theory with the exceptional gauge group G_2. Our findings are consistent with the spectral curve of the periodic Toda lattice, but do not agree with the hyperelliptic curve suggested previously in the literature. We also apply the method to theories with fundamental matter, treating both the example of SO(5) and G_2. 
  We give a general method to construct a complete set of linearly independent Casimir operators of a Lie algebra with rank N. For a Casimir operator of degree p, this will be provided by an explicit calculation of its symmetric coefficients $ g^{A_1,A_2,.. A_p}$. It is seen that these coefficients can be descibed by some rational polinomials of rank N. These polinomials are also multilinear in Cartan sub-algebra indices taking values from the set $I_0 = {1,2,.. N}$. The crucial point here is that for each degree one needs, in general, more than one polinomials. This in fact is related with an observation that the whole set of symmetric coefficients $ g^{A_1,A_2,.. A_p} $ is decomposed into sum subsets which are in one to one correspondence with these polinomials. We call these subsets clusters and introduce some indicators with which we specify different clusters. These indicators determine all the clusters whatever the numerical values of coefficients $g^{A_1,A_2,.. A_p}$ are. For any degree p, the number of clusters is independent of rank N. This hence allows us to generalize our results to any value of rank N.   To specify the general framework explicit constructions of 4th and 5th order Casimir operators of $A_N$ Lie algebras are studied and all the polinomials which specify the numerical value of their coefficients are given explicitly. 
  We compute vacuum expectation values of products of fermion bilinears for two-dimensional Quantum Chromodynamics at finite flavored fermion densities. We introduce the chemical potential as an external charge distribution within the path-integral approach and carefully analyse the contribution of different topological sectors to fermion correlators. We show the existence of chiral condensates exhibiting an oscillatory inhomogeneous behavior as a function of a chemical potential matrix. This result is exact and goes in the same direction as the behavior found in QCD_4 within the large N approximation. 
  We explicitly determine the instanton corrections to the prepotential for N=2 supersymmetric SU(3) Yang-Mills theory with massless hypermultiplets in the weak coupling regions $u\to\infty$ and $v\to\infty$. We construct the Picard-Fuchs equations for $N_f<6$ and calculate the monodromies using Picard-Lefschetz theorem for $N_f=2,4$. For all $N_f<6$ the instanton corrections to the prepotential are determined using the relation between $Tr <\phi^2>$ and the prepotential. 
  The ultraviolet structure of the Calogero-Sutherland models is examined, and, in particular, semions result to have special properties. An analogy with ultraviolet structures known in anyon quantum mechanics is drawn, and it is used to suggest possible physical consequences of the observed semionic properties. 
  The general structure of the partition function of an anyon gas is discussed, especially in relation to statements made in Phys. Rev. Lett. 68 (1992) 1621 and Phys. Rev. Lett. 69( 1992) 2877. 
  We show how to obtain the explicite form of the low energy quantum effective action for $N=2$ supersymmetric Yang-Mills theory in the weak coupling region from the underlying hyperelliptic Riemann surface. This is achieved by evaluating the integral representation of the fields explicitly. We calculate the leading instanton corrections for the group $SU(\nc), SO(N)$ and $SP(2N)$ and find that the one-instanton contribution of the prepotentials for the these group coincide with the one obtained recently by using the direct instanton caluculation. 
  We derive a simple formula for the periods associated with the low energy effective action of $N=2$ supersymmetric $SU(2)$ Yang-Mills theory with massive $N_f\le 3$ hypermultiplets.  This is given by evaluating explicitly the integral associated to the elliptic curve using various identities of hypergeometric functions. Following this formalism, we can calculate the prepotential with massive hypermultiplets both in the weak coupling region and in the strong coupling region. In particular, we show how the Higgs field and its dual field are expressed as generalized hypergeometric functions when the theory has a conformal point. 
  The three-dimensional real scalar model, in which the $Z_2$ symmetry spontaneously breaks, is renormalized in a nonperturbative manner based on the Tamm-Dancoff truncation of the Fock space. A critical line is calculated by diagonalizing the Hamiltonian regularized with basis functions. The marginal ($\phi^6$) coupling dependence of the critical line is weak. In the broken phase the canonical Hamiltonian is tachyonic, so the field is shifted as $\phi(x)\to\varphi(x)+v$. The shifted value $v$ is determined as a function of running mass and coupling so that the mass of the ground state vanishes. 
  We construct the form factors of the trace of energy-momentum tensor for the massless model described by $SU(2)$ principal chiral field model with WZNW tern on level 1. We explain how this construction can be generalized to a class of integrable massless models including the flow from tricritical to critical Ising model. 
  We show that it is possible to formulate a gauge theory starting from a local action at the ultraviolet (UV) momentum cutoff which is BRS invariant. One has to require that fields in the UV action and the fields in the effective action are not the same but related by a local field transformation. The few relevant parameters involved in this transformation (six for the $SU(2)$ gauge theory), are perturbatively fixed by the gauge symmetry. 
  We study five-dimensional supersymmetric field theories with one-dimensional Coulomb branch. We extend a previous analysis which led to non-trivial fixed points with $E_n$ symmetry ($E_8$, $E_7$, $E_6$, $E_5=Spin(10)$, $E_4=SU(5)$, $E_3=SU(3)\times SU(2)$, $E_2=SU(2)\times U(1)$ and $E_1=SU(2)$) by finding two new theories: $\tilde E_1$ with $U(1)$ symmetry and $E_0$ with no symmetry. The latter is a non-trivial theory with no relevant operators preserving the super-Poincar\'e symmetry. In terms of string theory these new field theories enable us to describe compactifications of the type I' theory on $S^1/Z_2$ with 16, 17 or 18 background D8-branes. These theories also play a crucial role in compactifications of M-theory on Calabi--Yau spaces, providing physical models for the contractions of del Pezzo surfaces to points (thereby completing the classification of singularities which can occur at codimension one in K\"ahler moduli). The structure of the Higgs branch yields a prediction which unifies the known mathematical facts about del Pezzo transitions in a quite remarkable way. 
  Samll instantons of exceptional groups arise geometrically by a collapsing del Pezzo surface in a CY. We use this to explain the physics of a 4-brane probe in Type I' compactification to 9 dimensions. 
  We study an integral representation for the zeta function of the one-loop effective potential for a minimally coupled massive scalar field in D-dimensional de Sitter spacetime. By deforming the contour of integration we present it in a form suitable for letting the de Sitter radius tend to infinity, and we demonstrate explicitly for the case D=2 that the quantities $\zeta( 0 )$ and $\zeta'( 0 )$ have the appropriate Minkowski limits. 
  We review the "no-go" theorems that severely constrain the breaking of N=2 supersymmetry to N=1 (both in rigid supersymmetry and supergravity), and we exhibit some models that evade them. 
  We prove that three-dimensional $N=1$ supersymmetric Yang-Mills-Chern-Simons theory is finite to all loop orders. In general this leaves open the possibility that different regularization methods lead to different finite effective actions. We show that in this model dimensional regularization and regularization by dimensional reduction yield the same effective action. Consequently, the superfield approach preserves BRS invariance for this model. 
  We review the arguments that fundamental string states are in one to one correspondence with black hole states. We demonstrate the power of the assumption by showing that it implies that the statistical entropy of a wide class of nonextreme black holes occurring in string theory is proportional to the horizon area. However, the numerical coefficient relating the area and entropy only agrees with the Bekenstein--Hawking formula if the central charge of the string is six which does not correspond to any known string theory. Unlike the current D-brane methods the method used in this paper is applicable for the case of Schwarzschild and highly non-extreme charged black holes. 
  Recently Maldacena and Strominger found that the calculation of greybody factors for $D=5$ black holes carrying three U(1) charges gives striking new evidence for their description as multiply wound effective strings. Here we show that a similar result holds for $D=4$ black holes with four $U(1)$ charges. In this case the effective string may be thought of as the triple intersection of the 5-branes in M-theory compactified on $T^7$. 
  We perform the two loop level renormalization of quantum gravity in $2+\epsilon$ dimensions. We work in the background gauge whose manifest covariance enables us to use the short distance expansion of the Green's functions. We explicitly show that the theory is renormalizable to the two loop level in our formalism. We further make a physical prediction for the scaling relation between the gravitational coupling constant and the cosmological constant which is expected to hold at the short distance fixed point of the renormalization group. It is found that the two loop level calculation is necessary to determine the scaling exponent to the leading order in $\epsilon$. 
  A class of solutions of the low energy string theory in four dimensions is studied. This class admits a geodesic, shear-free null congruence which is non-twisting but in general diverging and the corresponding solutions in Einstein's theory form the Robinson-Trautman family together with a subset of the Kundt's class. The Robinson-Trautman conditions are found to be frame invariant in string theory. The Lorentz Chern-Simons three form of the stringy Robinson-Trautman solutions is shown to be always closed. The stringy generalizations of the vacuum Robinson-Trautman equation are obtained and three subclasses of solutions are identified. One of these subclasses exists, among all the dilatonic theories, only in Einstein's theory and in string theory. Several known solutions including the dilatonic black holes, the pp- waves, the stringy C-metric and certain solutions which correspond to exact conformal field theories are shown to be particular members of the stringy Robinson-Trautman family. Some new solutions which are static or asymptotically flat and radiating are also presented. The radiating solutions have a positive Bondi mass. One of these radiating solutions has the property that it settles down smoothly to a black hole state at late retarded times. 
  We consider scattering amplitudes in non-critical string theory of $N$ external states in the limit where the energy of all external states is large compared to the string tension. We argue that the amplitudes are naturally complex analytic in the matter central charge $c$ and we propose to define the amplitudes for arbitrary value of $c$ by analytic continuation. We show that the high energy limit is dominated by a saddle point that can be mapped onto an equilibrium electro-static energy configuration of an assembly of $N$ pointlike (Minkowskian) charges, together with a density of charges arising from the Liouville field. We argue that the Liouville charges accumulate on segments of curves, and produce quadratic branch cuts on the worldsheet. The electro-statics problem is solved for string tree level in terms of hyper-elliptic integrals and is given explicitly for 3- and 4-point functions. We show that the high energy limit should behave in a string-like fashion with exponential dependence on the energy scale for generic values of $c$. 
  We compute the Form Factors of the relevant scaling operators in a class of integrable models without internal symmetries by exploiting their cluster properties. Their identification is established by computing the corresponding anomalous dimensions by means of Delfino--Simonetti--Cardy sum--rule and further confirmed by comparing some universal ratios of the nearby non--integrable quantum field theories with their independent numerical determination. 
  We give a new description of N=1 super Yang-Mills theory in curved superspace. It is based on the induced geometry approach to a curved superspace in which it is viewed as a surface embedded into C(4|2). The complex structure on C(4|2) supplied with a standard volume element induces a special Cauchy-Riemann (SCR)-structure on the embedded surface. We give an explicit construction of SYM theory in terms of intrinsic geometry of the superspace defined by this SCR-structure and a CR-bundle over the superspace. We write a manifestly SCR-covariant Lagrangian for SYM coupled with matter. We also show that in a special gauge our formulation coincides with the standard one which uses Lorentz connections. Some useful auxiliary results about the integration over surfaces in superspace are obtained. 
  We discuss the structure of heterotic/type II duality in four dimensions as a consequence of string-string duality in six dimensions. We emphasize the new features in four dimensions which go beyond the six dimensional vacuum structure and pertain to the way particular K3 fibers can be embedded in Calabi-Yau threefolds. Our focus is on hypersurfaces as well as complete intersections of codimension two which arise via conifold transitions. 
  Quantizing the electromagnetic field with a group formalism faces the difficulty of how to turn the traditional gauge transformation of the vector potential, $A_{\mu}(x)\rightarrow A_{\mu}(x)+\partial_{\mu}\varphi(x)$, into a group law. In this paper it is shown that the problem can be solved by looking at gauge transformations in a slightly different manner which, in addition, does not require introducing any BRST-like parameter. This gauge transformation does not appear explicitly in the group law of the symmetry but rather as the trajectories associated with generalized equations of motion generated by vector fields with null Noether invariants. In the new approach the parameters of the local group, $U(1)(\vec{x},t)$, acquire dynamical content outside the photon mass shell, a fact which also allows a unified quantization of both the electromagnetic and Proca fields. 
  After emphasizing the importance of obtaining a space-time understanding of black hole entropy, we further elaborate our program to identify the degrees of freedom of black holes with classical space-time degrees of freedom. The Cveti\v{c}-Youm dyonic black holes are discussed in some detail as an example. In this example hair degrees of freedom transforming as an effective string can be identified explicitly. We discuss issues concerning charge quantization, identification of winding, and tension renormalization that arise in counting the associated degrees of freedom. The possibility of other forms of hair in this example, and the prospects for making contact with D-brane ideas, are briefly considered. 
  We consider the behaviour of a quantum scalar field on three-dimensional Euclidean backgrounds: Anti-de Sitter space, the regular BTZ black hole instanton and the BTZ instanton with a conical singularity at the horizon. The corresponding heat kernel and effective action are calculated explicitly for both rotating and non-rotating holes. The quantum entropy of the BTZ black hole is calculated by differentiating the effective action with respect to the angular deficit at the conical singularity. The renormalization of the UV-divergent terms in the action and entropy is considered. The structure of the UV-finite term in the quantum entropy is of particular interest. Being negligible for large outer horizon area $A_+$ it behaves logarithmically for small $A_+$. Such behaviour might be important at late stages of black hole evaporation. 
  We construct a new extension of the Poincar\'e superalgebra in eleven dimensions which contains super one-, two- and five-form charges. The latter two are associated with the supermembrane and the superfivebrane of M-theory. Using the Maurer-Cartan equations of this algebra, we construct closed super seven-forms in a number of ways. The pull-back of the corresponding super six-forms are candidate superfivebrane Wess-Zumino terms, which are manifestly supersymmetric, and contain coordinates associated with the new charges. 
  We discuss a particular stringy modular cosmology with two axion fields in seven space-time dimensions, decomposable as a time and two flat three-spaces. The effective equations of motion for the problem are those of the $SU(3)$ Toda molecule, and hence are integrable. We write down the solutions, and show that all of them are singular. They can be thought of as a generalization of the Pre-Big-Bang cosmology with excited internal degrees of freedom, and still suffering from the graceful exit problem. Some of the solutions however show a rather unexpected property: some of their spatial sections shrink to a point in spite of winding modes wrapped around them. We also comment how more general, anisotropic, solutions, with fewer Killing symmetries can be obtained with the help of STU dualities. 
  We consider the quantum correction to the Lagrangean by the massless free boson in the curved background in three dimensions where one of the coordinates is periodic. The correction term is given by an expansion of the metric with respect to the derivative and the first term expresses to the usual Casimir energy. As an application, we investigate the change of the geometry in three dimensional black hole due to the quantum effect and we show that the geometry becomes like that of the Reissner-Nordstr\o m solution. 
  A path-integral quantization on a homogeneous space G/H is proposed based on the guiding principle `first lift to G and then project to G/H'. It is then shown that this principle gives a simple procedure to obtain the inequivalent quantizations (superselection sectors) along with the holonomy factor (induced gauge field) found earlier by algebraic approaches. We also prove that the resulting matrix-valued path-integral is physically equivalent to the scalar-valued path-integral derived in the Dirac approach, and thereby present a unified viewpoint to discuss the basic features of quantizing on $G/H$ obtained in various approaches so far. 
  We sketch recent applications of the harmonic superspace approach for off-shell formulations of $(4,4)$, $2D$ sigma models with torsion and for constructing super KdV hierarchies associated with "small" and "large" $N=4$ superconformal algebras. 
  Quantum Field Theory (QFT) developed in Rindler space-time and its thermal properties are analyzed by means of quantum groups approach. The quantum deformation parameter, labelling the unitarily inequivalent representations, turns out to be related to the acceleration of the Rindler frame. 
  The paper analyzes the problem fixing the local fermionic gauge symmetry in the Green-Schwarz superstring theory,related to the covariant quantization of this theory.The gauge-fixing procedure reveals the connection between GS and NSR formalisms, conjectured earlier. The analysis relates the kappa-invariance to the Bogomol'ny-type condition,well-known in connection with D-brane theories. The BPS saturation condition appears to be a target space superpartner of the kappa-invariance condition. Relation to RR charges of p-brane solutions of the ten-dimensional supergravity is discussed. 
  A quantum time topological space is developed and applied to solve some problems about quantum theory. It is disconnected and satifies specific separation axioms. The degree of disconnectedness of the time-space is a decreasing function of the number of simultaneous or almost simultaneous fundamental interactions. In this topology the U+R Penrose dynamics is implemented by means of a time evolution operator in QFT. This operator is unitary or non-unitary, depending on the type of quantization of the field action-integral. The time evolution operator allows to find the Boltzmann factor in QFT in the above space-time. From an elementary solution of the Liouville equation the quantization of the time follows and the Planck constant has been calculated. Compatibility between time-reversal and irreversibility is spontaneously obtained. The renormalization of the field action-integral follows from quantization. The solution of the measurement problem and the wave function reduction have been deduced in the framework of the Schroedinger theory. The Schroedinger cat's paradoxon and the paradoxon of the wave packet decay have been resolved. 
  $P$-brane solutions of low-energy string actions have traditionally provided the first evidence for the existence of string dualities, in which fundamental and solitonic $p$-branes are identified with perturbative and non-perturbative BPS states. In this talk we discuss the composite nature of solutions, which allows for the interpretation of general solutions as bound states or intersections of maximally supersymmetric fundamental constituents. This feature lies at the heart of the recent success of string theory in reproducing the Beckenstein-Hawking black hole entropy formula. 
  The M-theory interpretation of certain D=10 IIA p-branes implies the existence of worldvolume Kaluza-Klein modes which are expected to appear as 0-brane/p-brane bound states preserving 1/4 of the spacetime supersymmetry. We construct the corresponding solutions of the effective supergravity theory for $p=1,4$, and show that no such solution exists for $p=8$. 
  The boundary effects in the screening of an applied magnetic field in a charged anyon fluid at finite temperature and density are investigated. By analytically solving the extremum equations of the sytem and minimizing the free energy density, we find that in a sample with only one boundary (the half plane), a total Meissner effect takes place; while the sample with two boundaries (the infinite strip) exhibits a partial Meissner effect. The short-ranges modes of propagation of the magnetic field inside the fluid are characterized by two temp erature dependent penetration lengths. 
  The boundary effects in the screening of an applied magnetic field in a finite temperature 2+1 dimensional model of charged fermions minimally coupled to Maxwell and Chern-Simons fields are investigated. It is found that in a sample with only one boundary -a half-plane- a total Meissner effect takes place, while in a sample with two boundaries -an infinite strip- the external magnetic field partially penetrates the material. 
  We review work done in collaboration with C.B. Thorn on Super-Galilei invariant field theory and its application to the reformulation of superstring theory in terms of constituent superstring-bits. 
  I discuss how the basic phenomenon of asymptotic freedom in QCD can be understood in elementary physical terms. Similarly, I discuss how the long-predicted phenomenon of ``gluonization of the proton'' -- recently spectacularly confirmed at HERA -- is a rather direct manifestation of the physics of asymptotic freedom. I review the broader significance of asymptotic freedom in QCD in fundamental physics: how on the one hand it guides the interpretation and now even the design of experiments, and how on the other it makes possible a rational, quantitative theoretical approach to problems of unification and early universe cosmology. 
  We study the effects of internal symmetries on the decay by bubble nucleation of a metastable false vacuum. The zero modes about the bounce solution that are associated with the breaking of continuous internal symmetries result in an enhancement of the tunneling rate into vacua in which some of the symmetries of the initial state are spontaneously broken. We develop a general formalism for evaluating the effects of these zero modes on the bubble nucleation rate in both flat and curved space-times. 
  It is shown that the physical ``quark number'' charges which appear in the central charge of the supersymmetry algebra of $N=2$ supersymmetric QCD can take irrational values and depend non trivially on the Higgs expectation value. This gives a physical interpretation of the constant shifts which the ``electric'' and ``magnetic'' variables $a_D$ and $a$ undergo when encircling a singularity, and show that duality in this model is truly an electric-magnetic-quark number duality. Also included is a computation of the monodromy matrices directly in the microscopic theory. 
  Some aspects of chiral p-forms, in particular the obstruction that makes it hard to define covariant Green functions, are discussed. It is shown that a proposed resolution involving an infinite set of gauge fields can be made mathematically rigourous in the classical case. We also give a brief demonstration of species doubling for chiral bosons. 
  A derivation is given of the Feynman rules to be used in the perturbative computation of the Green's functions of a generic quantum many-body theory when the action which is being perturbed is not necessarily quadratic. Some applications are discussed. 
  We study the low-energy effective theory in N=2 SU(Nc) supersymmetric QCD with Nf =< 2Nc fundamental hypermultiplets in the Coulomb branch by microscopic and exact approaches. We calculate the one-instanton correction to the modulus u=< 1/2 Tr A^2 > from microscopic instanton calculation. We also study the one-instanton corrections from the exact solutions for Nc=3 with massless hypermultiplets. They agree with each other except for Nf=2Nc-2 and 2Nc cases. These differences come from possible ambiguities in the constructions of the exact solutions or the definitions of the operators in the microscopic theories. 
  We derive, in 2+1 dimensions, classical solutions for metric and motion of two or more spinning particles, in the conformal Coulomb gauge introduced previously. The solutions are exact in the $N$-body static case, and are perturbative in the particles' velocities in the dynamic two-body case. A natural boundary for the existence of our gauge choice is provided by some ``CTC horizons'' encircling the particles, within which closed timelike curves occur. 
  A discretized massless wave equation in two dimensions, on an appropriately chosen square lattice, exactly reproduces the solutions of the corresponding continuous equations. We show that the reason for this exact solution property is the discrete analog of conformal invariance present in the model, and find more general field theories on a two-dimensional lattice that exactly solve their continuous limit equations. These theories describe in general non-linearly coupled bosonic and fermionic fields and are similar to the Wess-Zumino-Witten model. 
  We study some features of bosonic particle path-integral quantization in a twistor-like approach by use of the BRST-BFV quantization prescription. In the course of the Hamiltonian analysis we observe links between various formulations of the twistor-like particle by performing a conversion of the Hamiltonian constraints of one formulation to another. A particular feature of the conversion procedure applied to turn the second-class constraints into the first-class constraints is that the simplest Lorentz-covariant way to do this is to convert a full mixed set of the initial first- and second-class constraints rather than explicitly extracting and converting only the second-class constraints. Another novel feature of the conversion procedure applied below is that in the case of the D=4 and D=6 twistor-like particle the number of new auxiliary Lorentz-covariant coordinates, which one introduces to get a system of first-class constraints in an extended phase space, exceeds the number of independent second-class constraints of the original dynamical system. We calculate the twistor-like particle propagator in D=3, 4 and 6 space-time dimensions and show, that it coincides with that of a conventional massless bosonic particle. 
  The supersymmetric version of a topological quantum field theory describing flat connections, the super BF-theory, is studied in the superspace formalism. A set of observables related to topological invariants is derived from the curvature of the superspace. Analogously to the non-supersymmetric versions, the theory exhibits a vector-like supersymmetry. The role of the vector supersymmetry and an additional new symmetry of the action in the construction of observables is explained. 
  We obtain direct, finite, descriptions of a renormalized quantum mechanical system with no reference to ultraviolet cutoffs and running coupling constants, in both the Hamiltonian and path integral pictures. The path integral description requires a modification to the Wiener measure on continuous paths that describes an unusual diffusion process wherein colliding particles occasionally stick together for a random interval of time before going their separate ways. 
  We consider the generalized chiral $QED_2$ on $S^1$ with a $U(1)$ gauge field coupled with different charges to both chiral components of a fermionic field. Using the adiabatic approximation we calculate the Berry phase and the corresponding ${\rm U}(1)$ connection and curvature for the vacuum and many particle Fock states. We show that the nonvanishing vacuum Berry phase is associated with a projective representation of the local gauge symmetry group and contributes to the effective action of the model. 
  In this note we compare the moduli spaces of the heterotic string compactified on a two-torus and F-Theory compactified on an elliptic K3 surface for the case of an unbroken E8 x E8 gauge group. The explicit map relating the deformation parameters alpha and beta of the F-Theory K3 surface to the moduli T and U of the heterotic torus is found using the close relationship between the K3 discriminant and the discriminant of the Calabi-Yau-threefold X(1,1,2,8,12)[24] in the limit of a large base P1. 
  An abundance of the Poisson-Lie symmetries of the WZNW models is uncovered. They give rise, via the Poisson-Lie $T$-duality, to a rich structure of the dual pairs of $D$-branes configurations in group manifolds. The $D$-branes are characterized by their shapes and certain two-forms living on them. The WZNW path integral for the interacting $D$-branes diagrams is unambiguously defined if the two-form on the $D$-brane and the WZNW three-form on the group form an integer-valued cocycle in the relative singular cohomology of the group manifold with respect to its $D$-brane submanifold. An example of the $SU(N)$ WZNW model is studied in some detail. 
  It was observed some time ago by Shatashvili and Vafa that superstring compactification on manifolds of exceptional holonomy gives rise to superconformal field theories with extended chiral algebras. In their paper, free field realisations are given of these extended superconformal algebras inspired by Joyce's constructions of such manifolds as desingularised toroidal orbifolds. The purpose of this note is to give another realisation of these algebras starting not from free fields, but from the superconformal algebras associated to Calabi--Yau manifolds. These superconformal algebras, originally studied by Odake, are extensions of the N=2 Virasoro algebra. For the case of G_2 holonomy, our realisation is inspired in the conjectured construction of such manifolds as a desingularisation of (K x S^1)/Z_2, where K is a Calabi--Yau 3-fold admitting an antiholomorphic involution. Similarly, for the case of Spin(7) holonomy our realisation suggests a construction of such manifolds as desingularisations of K'/Z_2, where K' is a Calabi-Yau 4-fold admitting an antiholomorphic involution. 
  We study Abelian generalized deformations of the usual product of polynomials introduced in hep-th/9602016. We construct an explicit example for the case of $su/2$ which provides a tentative of a quantum-mechanical description of Nambu Mechanics on $R^3$. By introducing the notions of strong and weak triviality of generalized deformations, we show that the Zariski product is never trivial in either sense, while the example constructed here in a quantum-mechanical context is only strongly non-trivial. 
  Super--inflation driven by dilaton/moduli kinetic energy is naturally realized in compactified string theory. Discussed are selected topics of recent development in string inflationary cosmology: kinematics of super-inflation, graceful exit triggered by quantum back reaction, and clasical and quantum power spectra of density and metric perturbations. 
  A new method to obtain the Picard-Fuchs equations of effective $N = 2$ supersymmetric gauge theories in 4 dimensions is developed. It includes both pure super Yang-Mills and supersymmetric gauge theories with massless matter hypermultiplets. It applies to all classical gauge groups, and directly produces a decoupled set of second-order, partial differential equations satisfied by the period integrals of the Seiberg-Witten differential along the 1-cycles of the algebraic curves describing the vacuum structure of the corresponding $N = 2$ theory. 
  Using connection with quantum field theory, the infinitesimal covariant abelian gauge transformation laws of relativistic two-particle constraint theory wave functions and potentials are established and weak invariance of the corresponding wave equations shown. Because of the three-dimensional projection operation, these transformation laws are interaction dependent. Simplifications occur for local potentials, which result, in each formal order of perturbation theory, from the infra-red leading effects of multiphoton exchange diagrams. In this case, the finite gauge transformation can explicitly be represented, with a suitable approximation and up to a multiplicative factor, by a momentum dependent unitary operator that acts in $x$-space as a local dilatation operator. The latter is utilized to reconstruct from the Feynman gauge the potentials in other linear covariant gauges. The resulting effective potential of the final Pauli-Schr\"odinger type eigenvalue equation has the gauge invariant attractive singularity $\alpha^2/r^2$, leading to a gauge invariant critical coupling constant $\alpha_c =1/2$. 
  I consider an algebraic construction of creation and annihilation operators for superstring and p-brane parton models. The result can be interpreted as a realisation of multiple quantisation and suggests a relationship between quantisation and dimension. The most general algebraic form of quantisation may eventually be expressed in the language of category theory. 
  It has recently been shown that the Field Antifield quantization of anomalous irreducible gauge theories with closed algebra can be represented in a BRST superspace where the quantum action at one loop order, including the Wess Zumino term, and the anomalies show up as components of the same superfield. We show here how the Chiral Schwinger model can be represented in this formulation. 
  Casimir W-algebras are shown to exist in such a way that the conformal spins of primary(generating) fields coincide with the orders of independent Casimir operators. We show here that this coincidence can be extended further to the case that these generating fields have the same eiginvalues with the Casimir operators. 
  Within the framework of semiclassical QCD approximations the short distance behavior of two static color charges in (2+1)-dimensional QCD is discussed. A classical linearization of the field equations is exhibited and leads to analytical results producing the static potential. Beyond the dominant classical part proportional to ln lambda R, QCD contributions of order R^1/2 and R are found. 
  The quantization law for the antisymmetric tensor field of $M$-theory contains a gravitational contribution not known previously. When it is included, the low energy effective action of $M$-theory, including one-loop and Chern-Simons contributions, is well-defined. The relation of $M$-theory to the $E_8\times E_8$ heterotic string greatly facilitates the analysis. 
  We study the properties of color-singlet Reggeon compound states in multicolor high-energy QCD in four dimensions. Their spectrum is governed by completely integrable (1+1)-dimensional effective QCD Hamiltonian whose diagonalization within the Bethe Ansatz leads to the Baxter equation for the Heisenberg spin magnet. We show that nonlinear WKB solution of the Baxter equation gives rise to the same integrable structures as appeared in the Seiberg-Witten solution for $N=2$ SUSY QCD and in the finite-gap solutions of the soliton equations. We explain the origin of hyperelliptic Riemann surfaces out of QCD in the Regge limit and discuss the meaning of the Whitham dynamics on the moduli space of quantum numbers of the Reggeon compound states, QCD Pomerons and Odderons. 
  We show that the WZW fusion rings at finite levels form a projective system with respect to the partial ordering provided by divisibility of the height, i.e. the level shifted by a constant. From this projective system we obtain WZW fusion rings in the limit of infinite level. This projective limit constitutes a mathematically well-defined prescription for the `classical limit' of WZW theories which replaces the naive idea of `sending the level to infinity'. The projective limit can be endowed with a natural topology, which plays an important role for studying its structure. The representation theory of the limit can be worked out by considering the associated fusion algebra; this way we obtain in particular an analogue of the Verlinde formula. 
  In this paper we introduce a class of generalized supersymmetric Toda field theories. The theories are labeled by a continuous parameter and have $N=2$ supersymmetry. They include previously known $N=2$ Toda theories as special cases. Using the WZNW -->Toda reduction approach we obtain a closed expression for the bracket of the associated ${\cal W}$ algebras. We also derive an expression for the generators of the ${\cal W}$ algebra in a free superfield realization. 
  We construct multi-center solutions for charged, dilatonic, non-extremal black holes in D=4. When an infinite array of such non-extremal black holes are aligned periodically along an axis, the configuration becomes independent of this coordinate, which can therefore be used for Kaluza-Klein compactification. This generalises the vertical dimensional reduction procedure to include {\it non-extremal} black holes. We then extend the construction to multi-center non-extremal $(D-4)$-branes in $D$ dimensions, and discuss their vertical dimensional reduction. 
  The standard MS renormalization prescription is inadequate for dealing with multiscale problems. To illustrate this, we consider the computation of the effective potential in the Higgs-Yukawa model. It is argued that the most natural way to deal with this problem is to introduce a 2-scale renormalization group. We review various ways of implementing this idea and consider to what extent they fit in with the notion of heavy particle decoupling. 
  It is found that the number, $M_n$, of irreducible multiple zeta values (MZVs) of weight $n$, is generated by $1-x^2-x^3=\prod_n (1-x^n)^{M_n}$. For $9\ge n\ge3$, $M_n$ enumerates positive knots with $n$ crossings. Positive knots to which field theory assigns knot-numbers that are not MZVs first appear at 10 crossings. We identify all the positive knots, up to 15 crossings, that are in correspondence with irreducible MZVs, by virtue of the connection between knots and numbers realized by Feynman diagrams with up to 9 loops. 
  The loop-expansion of the effective potential in the $O(N)$-symmetric $\phi^4$-model contains generically two types of large logarithms. To resum those systematically a new minimal two-scale subtraction scheme $\tMS$ is introduced in an $O(N)$-invariant generalization of $\MS$. As the $\tMS$ beta functions depend on the renormalization scale-ratio a large logarithms resummation is performed on them. Two partial $\tMS$ renormalization group equations are derived to turn the beta functions into $\tMS$ running parameters. With the use of standard perturbative boundary conditions, which become applicable in $\tMS$, the leading logarithmic $\tMS$ effective potential is computed. The calculation indicates that there is no stable vacuum in the broken phase of the theory for $1<N\leq 4$. 
  We determine the mass dependence of the coupling constant for N=2 SYM with N_f=1,2,3 and 4 flavours. All these cases can be unified in one analytic expression, given by a Schwarzian triangle function. Moreover we work out the connection to modular functions which enables us to give explicit formulas for the periods. Using the form of the J-functions we are able to determine in an elegant way the couplings and monodromies at the superconformal points. 
  The existence of several nilpotent Noether charges in the decoupled formulation of two-dimensional gauge theories does not imply that all of these are required to annihilate the physical states. We elucidate this matter in the context of simple quantum mechanical and field theoretical models, where the structure of the Hilbert space is known. We provide a systematic procedure for deciding which of the BRST conditions is to be imposed on the physical states in order to ensure the equivalence of the decoupled formulation to the original, coupled one. 
  As is well known, one can arrange the parameters of the O(N) non-linear sigma model to reproduce the low energy S-matrix elements of the renormalizable O(N) linear sigma model. In this note we provide details which are necessary in order to obtain the off-shell equivalence between the two theories. 
  Massive spinning particle in $6d$-Minkowski space is described as a mechanical system with the configuration space $R^{5,1} \times CP^3$. The action functional of the model is unambiguously determined by the requirement of identical (off-shell) conservation for the phase-space counterparts of three Casimir operators of Poincar\'e group. The model proves to be completely solvable. Its generalization to the constant curvature background is presented. Canonical quantization of the theory leads to the relativistic wave equations for the irreducible $6d$ fields. 
  In this paper we calculate the finite-size corrections of an anisotropic integrable spin chain, consisting of spins s=1 and s=1/2. The calculations are done in two regions of the phase diagram with respect to the two couplings $\bar{c}$ and $\tilde{c}$. In case of conformal invariance we obtain the final answer for the ground state and its lowest excitations, which generalizes earlier results. 
  The correlation functions of the Z-invariant Ising model are calculated explicitly using the Vertex Operators language developed by the Kyoto school. 
  We consider 3-monopoles symmetric under inversion symmetry. We show that the moduli space of these monopoles is an Atiyah-Hitchin submanifold of the 3-monopole moduli space. This allows what is known about 2-monopole dynamics to be translated into results about the dynamics of 3-monopoles. Using a numerical ADHMN construction we compute the monopole energy density at various points on two interesting geodesics. The first is a geodesic over the two-dimensional rounded cone submanifold corresponding to right angle scattering and the second is a closed geodesic for three orbiting monopoles. 
  The self-dual systems are constrained and so are simpler to understand. In recent years there have been several studies on the self-dual Chern-Simons systems. Here I present a brief survey of works done by my collaborators and myself. I also discuss several questions related to these self-dual models. 
  We present a brief review of some recent results on conformal anomalies in four and more dimensions. The discussion is intended for relativists, so some background on the quantum origin of anomalies and of their simple properties in D=2 is also provided. Topics treated include a critical review of the effective gravitational action uniqueness problem and the derivation of beta functions, independent of ultraviolet behavior, from the type B anomaly. 
  The mechanism underlying the divergence of perturbation theory is exposed. This is done through a detailed study of the violation of the hypothesis of the Dominated Convergence Theorem of Lebesgue using familiar techniques of Quantum Field Theory. That theorem governs the validity (or lack of it) of the formal manipulations done to generate the perturbative series in the functional integral formalism. The aspects of the perturbative series that need to be modified to obtain a convergent series are presented. Useful tools for a practical implementation of these modifications are developed. Some resummation methods are analyzed in the light of the above mentioned mechanism. 
  I shall recall a number of solutions to the Schwinger model in different gauges, having different boundary conditions and using different quantization surfaces. I shall discuss various properties of these solutions emphasizing the degrees of freedom necessary to represent the solution, the way the operator products are defined and the effects these features have on the chiral condensate. 
  We present N=2 supersymmetry transformations, both in N=1, D=4 Minkowski and anti-de Sitter superspaces, for higher superspin massless theories. It is noted that the existence of dual versions of massless supermultiplets with arbitrary superspin s may provide a basis for understanding duality in N=1, D=4 superstring theory. We further conjecture that the N=1, D=4 supergravity pre-potential together with all higher superspin s pre-potentials are components of an N=1, D=4 superstring pre-potential. 
  We present another concrete realization of a quantum field theory, envisaged many years ago by Bargmann, Wightman and Wigner. Considering the special case of the $(1/2,0)\oplus (0,1/2)$ field and developing the Majorana-McLennan-Case-Ahluwalia construct for neutrino we show that fermion and its antifermion can have same intrinsic parities. The construct can be applied to explanation of the present situation in neutrino physics. 
  Results of the work of S. Bruce [{\it Nuovo Cimento} {\bf 110}B (1995) 115] are compared with those of recent papers of D. V. Ahluwalia and myself, devoted to describing neutral particles of spin $j=1/2$ and $j=1$. 
  In the first part of this paper we review several formalisms which give alternative ways for describing the light. They are: the formalism `baroque' and the Majorana-Oppenheimer form of electrodynamics, the Sachs' theory of Elementary Matter, the Dirac-Fock-Podol'sky model, its development by Staruszkiewicz, the Evans-Vigier ${\bf B}^{(3)}$ field, the theory with an invariant evolution parameter of Horwitz, the analysis of the action-at-a-distance concept, presented recently by Chubykalo and Smirnov-Rueda, and the analysis of the claimed `longitudity' of the antisymmetric tensor field after quantization. The second part is devoted to the discussion of the Weinberg formalism and its recent development by Ahluwalia and myself. 
  We calculate the effective prepotentials for N=2 supersymmetric SO(N_c) and Sp(N_c) gauge theories, with an arbitrary number of hypermultiplets in the defining representation, from restrictions of the prepotentials for suitable N=2 supersymmetric gauge theories with unitary gauge groups. (This extends previous work in which the prepotential for N=2 supersymmetric SU(N_c) gauge theories was evaluated from the exact solution constructed out of spectral curves.) The prepotentials have to all orders the logarithmic singularities of the one-loop perturbative corrections, as expected from non-renormalization theorems. We evaluate explicitly the contributions of one- and two-instanton processes. 
  We analyze dispersion relations of the equations recently proposed by Ahluwalia for describing neutrino. Equations for type-II spinors are deduced on the basis of the Wigner rules for left- and right- 2-spinors and the Ryder-Burgard relation. It is shown that equations contain acausal solutions which are similar to those of the Dirac-like second-order equation. The latter is obtained in a similar way, provided that we do not apply to any constraints in the process of its deriving. 
  The Majorana discernment of neutrality is applied to the solutions of $j=1$ Weinberg equations in the $(j,0)\oplus (0,j)$ representation of the Poincar\`e group. 
  This is a brief introduction on the graduate level to mechanics of various spin relativistic particles with oscillatorlike interaction. This mathematical model proposed by M. Moshinsky could have considerable physical applications for describing processes mediated by tensor fields and in bound state theory. 
  This is a brief introduction on the graduate level to recent ideas in the Weinberg $(j,0)\oplus (0,j)$ formalism, appearing after presentation of the Bargamann-Wightman-Wigner-type quantum field theory by D. V. Ahluwalia {\it et al.} 
  The homogeneous Bethe-Salpeter equation is solved in the quenched ladder approximation for the vector positronium states of 4-component quantum electrodynamics in 2 space and 1 time dimensions. Fermion propagator input is from a Rainbow approximation Dyson-Schwinger solution, with a broad range of fermion masses considered. This work is an extension of earlier work on the scalar spectrum of the same model. The non-relativistic limit is also considered via the large fermion mass limit. Classification of states via their transformation properties under discrete parity transformations allows analogies to be drawn with the meson spectrum of QCD. 
  Nonvanishing tadpoles and possible infinities associated in the multiparticle amplitudes are discussed with regard to the disk and $RP^{2}$ diagrams of the Type I' compactification. We find that the infinity cancellation of $SO(32)$ type $I$ theory extends to this case as well despite the presence of tadpoles localized in the D-brane world-volume and the orientifold surfaces. Formalism of string S-matrix generating functional is presented to find a consistent string background as c-number source function: we find this only treats the cancellation of the tadpoles in the linearized approximation. Our formalism automatically provides representation of the string amplitudes on this background to all orders in $\alpha'$. 
  A geometrical approach in the non-symmetric connection framework is employed to examine the issue of higher order $\alpha'$ corrections to D=10 type IIB superstring backgrounds with a covariantly constant null Killing isometry and non-zero Ramond-Ramond field content. These describe generalized supersymmetric string waves and were obtained recently by us through the S-duality transformation of purely NS-NS plane wave backgrounds. We find that the backgrounds are exact subject to the existence of certain field redefinitions and provided certain restrictive conditions are satisfied. 
  In this paper, based on the author's lectures at the 1995 les Houches Summer school, explicit expressions for the Friedan--Shenker connection on the vector bundle of WZW conformal blocks on the moduli space of curves with tangent vectors at $n$ marked points are given. The covariant derivatives are expressed in terms of ``dynamical $r$-matrices'', a notion borrowed from integrable systems. The case of marked points moving on a fixed Riemann surface is studied more closely. We prove a universal form of the (projective) flatness of the connection: the covariant derivatives commute as differential operators with coefficients in the universal enveloping algebra -- not just when acting on conformal blocks. 
  By a closer inspection of the massive Schwinger model within mass perturbation theory we find that, in addition to the $n$-boson bound states, a further type of hybrid bound states has to be included into the model. Further we explicitly compute the decay widths of the three-boson bound state and of the lightest hybrid bound state. 
  Within the framework of Euclidean path integral and mass perturbation theory we compute the Wilson loop of widely separated external charges for the massive Schwinger model. From this result we show for arbitrary order mass perturbation theory that integer external charges are completely screened, whereas for noninteger charges a constant long-range force remains. 
  We present a strong-weak coupling duality for quantum mechanical potentials. Similarly to what happens in quantum field theory, it relates two problems with inverse couplings, leading to a mapping of the strong coupling regime into the weak one, giving information from the nonperturbative region of the parameters space. It can be used to solve exactly power-type potentials and to extract deep information about the energy spectra of polynomial ones. 
  We obtain the closed form of the Picard-Fuchs equations for $N=2$ supersymmetric Yang-Mills theories with classical Lie gauge groups. For a gauge group of rank $r$, there are $r-1$ regular and an exceptional differential equations. We describe the series solutions of the Picard-Fuchs equations in the semi-classical regime. 
  We regularize in a continuous manner the path integral of QED by construction of a non-local version of its action by means of a regularized form of Dirac's $\delta$ functions. Since the action and the measure are both invariant under the gauge group, this regularization scheme is intrinsically non-perturbative. Despite the fact that the non-local action converges formally to the local one as the cutoff goes to infinity, the regularized theory keeps trace of the non-locality through the appearance of a quadratic divergence in the transverse part of the polarization operator. This term which is uniquely defined by the choice of the cutoff functions can be removed by a redefinition of the regularized action. We notice that as for chiral fermions on the lattice, there is an obstruction to construct a continuous and non ambiguous regularization in four dimensions. With the help of the regularized equations of motion, we calculate the one particle irreducible functions which are known to be divergent by naive power counting at the one loop order. 
  We interpret certain strong coupling singularities of the $E_8\times E_8$ heterotic string on K3 in terms of exotic six-dimensional theories in which $E_8$ is a gauge symmetry. These theories are closely related to theories obtained at small instanton singularities, which have $E_8$ as a global symmetry. 
  Metaphysics is the science of being and asks the question ``What really exists?'' The answer to this question has been sought for by mankind since the beginning of recorded time. In the past 2500 years there have been many answers to this question and these answers dominate our view of how physics is done. Examples of questions which were originally metaphysical are the shape of the earth, the motion of the earth, the existence of atoms, the relativity of space and time, the uncertainty principle, the renormalization of field theory and the existence of quarks and strings. I will explore our changing conception of what constitutes reality by examining the views of Aristotle, Ptolemy, St. Thomas Aquinas, Copernicus, Galileo, Bacon, Descartes, Newton, Leibnitz, Compte, Einstein, Bohr, Feynman, Schwinger, Yang, Gell-Mann, Wilson and Witten. 
  We start a systematic analysis of supersymmetric field theories in six dimensions. We find necessary conditions for the existence of non-trivial interacting fixed points. String theory provides us with examples of such theories. We conjecture that there are many other examples. 
  The pattern of duality symmetries acting on the states of compactified superstring models reinforces an earlier suggestion that the Monster sporadic group is a hidden symmetry for superstring models. This in turn points to a supersymmetric theory of self-dual and anti-self-dual K3 manifolds joined by Dirac strings and evolving in a 13 dimensional spacetime as the fundamental theory. In addition to the usual graviton and dilaton this theory contains matter-like degrees of freedom resembling the massless states of the heterotic string, thus providing a completely geometric interpretation for ordinary matter. 
  We study mirror symmetry of Calabi-Yau manifolds within the framework of the Gauss-Manin system. Applying the flat coordinates to the Gauss-Manin system for the periods, we derive differential equations for the mirror map in addition to the ordinary Picard-Fuchs equations for the periods. These equations are obtained for a class of one-parameter models and a two-parameter model of Fermat type Calabi-Yau manifolds. 
  We review the application of the critical point large N_f self-consistency method to QCD. In particular we derive the O(1/N_f) d-dimensional critical exponents whose epsilon-expansion determines the perturbative coefficients in MSbar of the field dimensions, beta-function and various twist-2 operators which occur in the operator product expansion of deep inelastic scattering. 
  String propagation on a curved background defines an embedding problem of surfaces in differential geometry. Using this, we show that in a wide class of backgrounds the classical dynamics of the physical degrees of freedom of the string involves 2-dim sigma-models corresponding to coset conformal field theories. 
  Simple bosonic path integral representation for path ordered exponent is derived. This representation is used, at first, to obtain new variant of non-Abelian Stokes theorem. Then new pure bosonic worldline path integral representations for fermionic determinant and Green functions are presented. Finally, applying stationary phase method, we get quasiclassical equations of motion in QCD. 
  We generalize the previously established (0,2) triality of exactly solvable models, Landau-Ginzburg theories and Calabi-Yau manifolds to a number of different classes of (0,2) compactifications derived from (2,2) vacua. For the resulting models we show that the known (2,2) mirror constructions induce mirror symmetry in the (0,2) context. 
  The quantization condition derived previously for SU(2) solitons quantized with SU(3)-collective coordinates is generalized for SU(3) skyrmions with strangeness content different from zero. Quantization of the dipole-type configuration with large strangeness content found recently is considered as an example. 
  We present the mass spectrum of the tensionless string in 2 dimensions where it has been found that the space time conformal symmetry survives quantization. A BRST treatment of the physical states reveals that the string collapses into a massless particle, a result which agrees with the classical treatment. 
  We consider quantum corrections to classical real time correlation functions at finite temperature. We derive a semi-classical expansion in powers of $\hbar$ with coefficients including all orders in the coupling constant. We give explicit expressions up to order $\hbar^2$. We restrict ourselves to a scalar theory. This method, if extended to gauge theories, might be used to compute quantum corrections to the high temperature baryon number violation rate in the Standard Model. 
  Boundary equations for the relativistic string with masses at ends are formulated in terms of geometrical invariants of world trajectories of masses at the string ends. In the three--dimensional Minkowski space $E^1_2$, there are two invariants of that sort, the curvature $K$ and torsion $\kappa$. Curvatures of trajectories of the string ends with masses are always constant, $K_i = \gamma/m_i (i =1,2,)$, whereas torsions $\kappa_i(\tau)$ obey a system of differential equations with deviating arguments. For these equations with periodic $\kappa_i(\tau+n l)=\kappa(\tau)$, constants of motion are obtained (part I) and exact solutions are presented (part II) for periods $l$ and $2l$ where $l$ is the string length in the plane of parameters $\tau$ and $\sigma \ (\sigma_1 = 0, \sigma_2 =l)$. 
  We present a new real space renormalization-group map, on the space of probabilities, to study the renormalization of the SUSY \phi^4. In our approach we use the random walk representation on a lattice labeled by an ultrametric space. Our method can be extended to any \phi^n. New stochastic meaning is given to the parameters involved in the flow of the map and results are provided. 
  We study a theory of particles interacting with strings. Considering such a theory for Type IIA superstring will give some clue about M-theory. As a first step toward such a theory, we construct the particle-particle-string interaction vertex generalizing the D-particle boundary state. 
  We prove the universality of correlation functions of chiral unitary and unitary ensembles of random matrices in the microscopic limit. The essence of the proof consists in reducing the three-term recursion relation for the relevant orthogonal polynomials into a Bessel equation governing the local asymptotics around the origin. The possible physical interpretation as the universality of soft spectrum of the Dirac operator is briefly discussed. 
  In the present paper we investigate classical dynamics of the Nambu-Goto string with the Gauss-Bonnet term in the action and point-like masses at the ends in the context of effective QCD string. The configurations of rigidly rotating string is studied and its application to phenomenological description of meson is discussed. 
  We argue that all conjectured dualities involving various string, M- and F- theory compactifications can be `derived' from the conjectured duality between type I and SO(32) heterotic string theory, T-dualities, and the definition of M- and F- theories. (Based on a talk given at the conference on `Advanced Quantum Field Theory', Le Londe-les-Maures, France, Aug.31-Sept.5, 1996, in memory of Claude Itzykson) 
  Unitary irreducible representation of the group SO(1,2) is obtained in the mixed basis, i.e. between the compact and noncompact basis and the new addition theorems are derived which are required in path integral applications involving the positively signed potential. The Green function for the potential barrier $V=cosh^{-2}\omega x$ is evaluated from the path integration over the coset space SO(1,2)/K where K is the compact subgroup. The transition and the reflection coefficients are given. Results for the moving barrier $V=\cosh^{-2}\omega (x-g_0t)$ are also presented. 
  This paper studies the normalizability criterion for the one-loop wave function of the universe in a de Sitter background, when various unified gauge models are considered. It turns out that, in the absence of interaction between inflaton field and other matter fields, the supersymmetric version of such unified models is preferred. By contrast, the interaction of inflaton and matter fields, jointly with the request of normalizability at one-loop order, picks out non-supersymmetric versions of unified gauge models. 
  We found a quantum cohomology/homology of quantum Hall effect which arises as the invariant property of the Chern-Simons theory of quantum Hall effect and showed that it should be equivalent to the quantum cohomology which arose as the invariant property of topological sigma models. This isomorphism should be related with an equivalence between the supersymmetric- and quantization structures in two dimensional models and/or with an equivalence between topological sigma models and the Chern-Simons theory by the methode of master equation. 
  We construct sequences of axially symmetric multisphaleron solutions in SU(2) Yang-Mills-dilaton theory. The sequences are labelled by a winding number $n>1$. For $n=1$ the known sequence of spherically symmetric sphaleron solutions is obtained. The solutions within each sequence are labelled by the number of nodes $k$ of the gauge field functions. The limiting solutions of the sequences correspond to abelian magnetic monopoles with $n$ units of charge and energy $E \propto n$. 
  We present new supersymmetric solutions of D=11 supergravity obtained by intersecting the brane configuration interpreted as a 2-brane lying within a 5-brane. Some of these solutions can be boosted along a common string and/or superposed with a Kaluza-Klein monopole. We also present a new embedding of the extreme four dimensional dyonic black hole with finite horizon area. These solutions are a consequence of a rather simple set of rules that allow us to construct the composite M-branes. 
  We discuss the following aspects of two-dimensional N=2 supersymmetric theories defined on compact super Riemann surfaces: parametrization of (2,0) and (2,2) superconformal structures in terms of Beltrami coefficients and formulation of superconformal models on such surfaces (invariant actions, anomalies and compensating actions, Ward identities). 
  We introduce a diagramatic notation for supersymmetric gauge theories. The notation is a tool for exploring duality and helps to present the field content of more complicated models in a simple visual way. We introduce the notation with a few examples from the literature. The power of the formalism allows us to study new models with gauge group $(SU(N))^k$ and their duals. Amongst these are models which, contrary to a naive analysis, possess no conformal phase. 
  The $N=4$ SU(2)$_k$ superconformal algebra has the global automorphism of SO(4) $\approx$ SU(2)$\times$SU(2) with the {\it left} factor as the Kac-Moody gauge symmetry. As a consequence, an infinite set of independent algebras labeled by $\rho$ corresponding to the conjugate classes of the {\it outer} automorphism group SO(4)/SU(2)=SU(2) are obtained \`a la Schwimmer and Seiberg. We construct Feigin-Fuchs representations with the $\rho$ parameter embedded for the infinite set of the $N=4$ nonequivalent algebras. In our construction the extended global SU(2) algebras labeled by $\rho$ are self-consistently represented by fermion fields with appropriate boundary conditions. 
  In 1+1 dimensions two different formulations exist of SU(N) Yang Mills theories in light-cone gauge; only one of them gives results which comply with the ones obtained in Feynman gauge. Moreover the theory, when considered in 1+(D-1) dimensions, looks discontinuous in the limit D=2. All those features are proven in Wilson loop calculations as well as in the study of the $q\bar q$ bound state integral equation in the large N limit. 
  A general action is proposed for the fields of $q$-dimensional differential form over the compact Riemannian manifold of arbitrary dimensions. Mathematical tools come from the well-known de Rham-Kodaira decomposing theorem on the harmonic integral. We have a field-theoretic action suitable for strings and $p$-branes with or without spin degrees of freedom. In a completely-kinematical way is derived the generalized Maxwell theory with a magnetic monopole over a curved space-time, where we have a new type of gauge transformations. 
  A general system constrained with {\it several} initial constraint conditions is quantized based on the Dirac formalism and the Schr\"{o}dinger equation for this system is obtained. These constraint conditions are now allowed to depend not only on the coordinates but also on the velocities. It is shown that the hermiticity for the observables of the system restricts the geometrical structure of our world. 
  We construct asymptotic solutions of the functional Schroedinger equation for a scalar field in the Gaussian approximation at large proper time. These solutions describe the late proper time stages of the expansion of a meson gas with boost invariant boundary conditions. The relevance of these solutions for the formation of a disoriented chiral condensate in ultra relativistic collisions is discussed. 
  We present analytical methods for investigating the interaction of two heavy quarks in QCD_3 using the effective action approach. Our findings result in explicit expressions for the static potentials in QCD_3 for long and short distances. With regard to confinement, our conclusion reflects many features found in the more realistic world of QCD_4. 
  We examine consequences of the stabilization of the dilaton through the axion. An estimate of the resulting dilaton potential yields a relation between the axion parameter $m_a f_{PQ}$ and the average instanton radius, and predicts the ratio between the dilaton mass $m_\phi$ and the axion mass $m_a$. If we identify the string axion with a Peccei--Quinn axion, then $m_\phi m_{Pl} \sim m_a f_{PQ}$, and the dilaton should be strongly aligned $\sqrt{<\phi^2>}\leq 10^{-4}m_{Pl}$ at the QCD scale, in order not to overclose the universe. 
  We construct a new variety of $N=2$ supersymmetric integrable systems by junction of pseudo-differential superspace Lax operators for $a=4$, $N=2$ KdV and multi-component $N=2$ NLS hierarchies. As an important particular case, we obtain Lax operator for $N=4$ super KdV system. A similar extension of one of $N=2$ super Boussinesq hierarchies is given. We also present a minimal $N=4$ supersymmetric extension of the second flow of $N=4$ KdV hierarchy and comment on its possible integrability. 
  The local BRST cohomology is computed in old and new minimal supergravity, including the coupling to Yang-Mills gauge multiplets. This covers the determination of all gauge invariant local actions for these models, the classification of all the possible counterterms that are invariant on-shell, of all candidate gauge anomalies, and of the possible nontrivial (continuous) deformations of the standard actions and gauge transformations. Among others it is proved that in old minimal supergravity the most general gauge invariant action can indeed be constructed from well-known superspace integrals, whereas in new minimal supergravity there are only a few additional (but important) contributions which cannot be constructed in this way without further ado. Furthermore the results indicate that supersymmetry itself is not anomalous in minimal supergravity and show that the gauge transformations are extremely stable under consistent deformations of the models. There is however an unusual deformation converting new into old minimal supergravity with local R-invariance which is reminiscent of a duality transformation. 
  Higher-order WKB methods are used to investigate the border between the solvable and insolvable portions of the spectrum of quasi-exactly solvable quantum-mechanical potentials. The analysis reveals scaling and factorization properties that are central to quasi-exact solvability. These two properties define a new class of semiclassically quasi-exactly solvable potentials. 
  We compute the functional determinant for a Dirac operator in the presence of an Abelian gauge field on a bidimensional disk, under global boundary conditions of the type introduced by Atiyah-Patodi-Singer. We also discuss the connection between our result and the index theorem. 
  Schwinger's Dynamical Casimir Effect is one of several candidate explanations for sonoluminescence. Recently, several papers have claimed that Schwinger's estimate of the Casimir energy involved is grossly inaccurate. In this letter, we show that these calculations omit the crucial volume term. When the missing term is correctly included one finds full agreement with Schwinger's result for the Dynamical Casimir Effect. We have nothing new to say about sonoluminescence itself except to affirm that the Casimir effect is energetically adequate as a candidate explanation. 
  We show how to find explicit expressions for rigid string instantons for general 4-manifold $M$. It appears that they are pseudo-holomorphic curves in the twistor space of $M$. We present explicit formulae for $M=R^4, S^4$. We discuss their properties and speculate on relations to topology of 4-manifolds and the theory of Yang-Mills fields. 
  In this letter we study (2+1)-dimensional QED. The first part contains the computation of the flavor symmetry-breaking condensate and its relation to the trace of the energy-momentum tensor, while the second part is concerned with the computation of the effective action allowing for non-constant static external magnetic fields. We do not find that space derivatives in the magnetic field lower the energy of the ground state as compared to a constant field configuration. 
  The equivalence of several $SL(3)$ sigma models and their special Abelian duals is investigated in the two loop order of perturbation theory. The investigation is based on extracting and comparing various $\beta$ functions of the original and dual models. The role of the discrete global symmetries is emphasized. 
  Trivial second-order Lagrangians are studied and a complete description of the dependence on the second-order derivatives is given. This extends previous work of Olver and others. In particular, this description involves some polynomial expressions called hyper-Jacobians. There exists some linear dependencies between these polynomials which are elucidated for the (second-order) hyper-Jacobians. 
  An exactly solvable sphaleron model in $3+1$ spacetime dimensions is described 
  The dynamics of small perturbations around sphaleron and black hole solutions in the Einstein-Yang-Mills theory for the gauge group $SU(2)$ is investigated. The perturbations can be split into the two independent sectors in accordance with their parity; each sector contains negative modes. The even-parity negative modes are shown to correspond to the negative second variations of the height of the potential energy barrier near the sphaleron. For the odd-parity sector, the existence of precisely $n$ (the number of nodes of the sphaleron solution gauge field function) negative modes is rigorously proven. The same results hold for the Einstein-Yang-Mills black holes as well. 
  We discuss non-Abelian bosonization of two and three dimensional fermions using a path-integral framework in which the bosonic action follows from the evaluation of the fermion determinant for the Dirac operator in the presence of a vector field. This naturally leads to the Wess-Zumino-Witten action for massless two-dimensional fermions and to a Chern-Simons action for very massive three dimensional fermions. One advantage of our approach is that it allows to derive the exact bosonization recipe for fermion currents in a systematic way. 
  Non-trivial extensions of the three dimensional Poincar\'e algebra, beyond the supersymmetric one, are explicitly constructed. These algebraic structures are the natural three dimensional generalizations of fractional supersymmetry of order $F$ already considered in one and two dimensions. Representations of these algebras are exhibited, and unitarity is explicitly checked. It is then shown that these extensions generate symmetries which connect fractional spin states or anyons. Finally, a natural classification arises according to the decomposition of $F$ into its product of prime numbers leading to sub-systems with smaller symmetries. 
  String theory is used to compute the microscopic entropy for several examples of black holes in compactifications with $N=2$ supersymmetry. Agreement with the Bekenstein-Hawking entropy and the moduli-independent $N=2$ area formula is found in all cases. 
  In the context of the Batalin-Vilkovisky formalism, a new observable for the Abelian BF theory is proposed whose vacuum expectation value is related to the Alexander-Conway polynomial. The three-dimensional case is analyzed explicitly, and it is proved to be anomaly free. Moreover, at the second order in perturbation theory, a new formula for the second coefficient of the Alexander-Conway polynomial is obtained. An account on the higher-dimensional generalizations is also given. 
  We study the Luttinger-Schwinger model, i.e. the (1+1) dimensional model of massless Dirac fermions with a non-local 4-point interaction coupled to a U(1)-gauge field. The complete solution of the model is found using the boson-fermion correspondence, and the formalism for calculating all gauge invariant Green functions is provided. We discuss the role of anomalies and show how the existence of large gauge transformations implies a fermion condensate in all physical states. The meaning of regularization and renormalization in our well-defined Hilbert space setting is discussed. We illustrate the latter by performing the limit to the Thirring-Schwinger model where the interaction becomes local. 
  We determine the equations which govern the gauge symmetries of worldsheets with local supersymmetry of arbitrary rank $(N,N')$, and their possible anomalies. Both classical and ghost conformally invariant multiplets of the left or right sector are assembled into the components of a single $O(N)$-superfield. The component with ghost number zero of this superfield is the $N$-supersymmetric generalization of the Beltrami differential. In a Lagrangian approach, and after gauge-fixing, it becomes the super-moduli of Riemann surfaces coupled to local supersymmetry of rank $N$. It is also the source of all linear superconformal currents derived from ordinary operator product techniques. The interconnection between BRST invariant actions with different values of $N\geq 3$, and their possible link to topological 2D-gravity coupled to topological sigma models, are shown by straightforward algebraic considerations. 
  An essentially unique deformation of the product of quantum fields at the same spacetime point is obtained. It is proposed to replace local quantum field theory with another structure which uses a *-product. The resulting theory contains a fundamental length and is free from divergences. This provides the third deformation suggested by Faddeev. 
  We study strong coupling effects in four-dimensional heterotic string models where supersymmetry is spontaneously broken with large internal dimensions, consistently with perturbative unification of gauge couplings. These effects give rise to thresholds associated to the dual theories: type I superstring or M-theory. In the case of one large dimension, we find that these thresholds appear close to the field-theoretical unification scale $\sim 10^{16}$ GeV, offering an appealing scenario for unification of gravitational and gauge interactions. We also identify the inverse size of the eleventh dimension of M-theory with the energy at which four-fermion effective operators become important. 
  We explicitly construct soliton operators in $D<2$ (or $c<1$) string theory, and show that the Schwinger-Dyson equations allow solutions with these solitons as backgrounds. The dominant contributions from 1-soliton background are explicitly evaluated in the weak coupling limit, and shown to agree with the nonperturbative analysis of string equations. We suggest that fermions should be treated as fundamental dynamical variables since both macroscopic loops and solitons are constructed in their bilinear forms. 
  We show the complete cancellation of gauge and gravitational anomalies in the M-theory of Horava and Witten using their boundary contribution, and a term coming from the existence of two and five-branes. A factor of three discrepancy noted in an earlier work is resolved. We end with a comment on flux quantization. 
  The condition of vanishing of static force on a q-brane probe in the gravitational background produced by another p-brane is used to give a simple derivation of the pair-wise intersection rules which govern the construction of BPS combinations of branes. These rules, while implied also by supersymmetry considerations, thus have purely bosonic origin. Imposing the no-force requirement makes possible to add branes `one by one' to construct composite BPS configurations (with zero binding energy) of 2-branes and 5-branes in D=11 and of various p-branes in D=10. The advantage of this elementary approach is its universality, i.e. the cases of different dimensions and different types of branes (e.g., NS-NS, R-R and `mixed' combinations of NS-NS and R-R branes in D=10) are all treated in the same way. 
  In this paper, an explicit expression for the Casimir operator (or the Casimir invariant) of the inhomogeneous group ISL(n,R) in its enveloping algebra is proposed, which using contractions of the tenso- rial indices of the generating operators P^{rho} and E^{nu}{mu} may be presented in the following (slightly more comprehensible as equation (1)) form. The Casimir is obtained by symmetrizing this expression. This tensor form is useful in the classification of particles in affine gravitational gauge theories; such as that based on ISL(4,R). It is also proven that the Casimir of ISL(n,R) can be decomposed in terms of the Casimirs of its little groups, a key point in the posterior construction of its irreducible representations. 
  We discussed the full unitary matrix models from the view points of integrable equations and string equations. Coupling the Toda equations and the string equations, we derive a special case of the Painlev\'{e} III equation. From the Virasoro constrains, we can use the radial coordinate. The relation between $t_{1}$ and $t_{-1}$ is like the complex conjugate. %An implicit to the $t\bar{t}$ fusion of topological sigma model %is therby expected. 
  We give a representation, in terms of iterated Mellin-Barnes integrals, of periods on multi-moduli Calabi-Yau manifolds arising in superstring theory. Using this representation and the theory of multidimensional residues, we present a method for analytic continuation of the fundamental period in the form of Horn series. 
  Matrices are said to behave as free non-commuting random variables if the action which governs their dynamics constrains only their eigenvalues, i.e. depends on traces of powers of individual matrices. The authors use recently developed mathematical techniques in combination with a standard variational principle to formulate a new variational approach for matrix models. Approximate variational solutions of interacting large-N matrix models are found using the free random matrices as the variational space. Several classes of classical and quantum mechanical matrix models with different types of interactions are considered and the variational solutions compared with exact Monte Carlo and analytical results. Impressive agreement is found in a majority of cases. 
  Some aspects of the role of p-branes in non-perturbative superstring theory and M-theory are reviewed. It is then shown how the Chern-Simons terms in D=10 and D=11 supergravity theories determine which branes can end on which, i.e. the `brane boundary rules'. 
  The purpose of this paper is to formulate an action principle which allows for the construction of a classical lagrangean including both electric and magnetic currents. The lagrangean is non-local and shown to yield all the expected (local) equations for dual electrodynamics. 
  We propose a non-perturbative solution of N=2 supersymmetric gauge theory in five dimensions compactified on circle of a radius $R$. We consider the cases of the pure gauge theory as well as the theories with matter in the fundamental and in the adjoint representations. The pure theory as well as the one with adjoint hypermultiplet give rise to the known relativistic integrable systems with ${1\over{R}}$ playing the r\^ole of the speed of light. The theory with adjoint hypermultiplet exhibits some interesting finiteness properties.   Talk given at the III International Conference ``Conformal Field Theories and Integrable Models'', Chernogolovka, June 23-30 1996 
  We show that the BRST operator of Neveu-Schwarz-Ramond superstring is closely related to de Rham differential on the moduli space of decorated super-Riemann surfaces P. We develop formalism where superstring amplitudes are computed via integration of some differential forms over a section of P over the super moduli space M. We show that the result of integration does not depend on the choice of section when all the states are BRST physical. Our approach is based on the geometrical theory of integration on supermanifolds of which we give a short review. 
  We summarize recent work, in which we consider scattering amplitudes of non-critical strings in the limit where the energy of all external states is large compared to the string tension. We show that the high energy limit is dominated by a saddle point that can be mapped onto an electrostatic equilibrium configuration of an assembly of charges associated with the external states, together with a density of charges arising from the Liouville field. The Liouville charges accumulate on line segments, which produce quadratic branch cuts on the worldsheet. The electrostatics problem is solved for string tree level in terms of hyperelliptic integrals and is given explicitly for the 3- and 4-point functions. For generic values of the central charge, the high energy limit behaves in a string-like fashion, with exponential energy dependence. 
  We present a complete set of Feynman rules for non-Abelian gauge fields obeying the radial (Fock-Schwinger) gauge condition and prove the consistency with covariant gauge Feynman rules. 
  Quantum dissipation in thermal environment is investigated, using the path integral approach. The reduced density matrix of the harmonic oscillator system coupled to thermal bath of oscillators is derived for arbitrary spectrum of bath oscillators. Time evolution and the end point of two-body decay of unstable particles is then elucidated: After early transient times unstable particles undergo the exponential decay, followed by the power law decay and finally ending in a mixed state of residual particles containing contributions from both on and off the mass shell, whose abundance does not suffer from the Boltzmann suppression. 
  We discuss the bound states of the massive Thirring model. Here, the periodic boundary condition equations for the Bethe ansatz solutions are numerically solved. It is found that the massive Thirring model has only one bound state and the bound state spectrum as the function of the coupling constant agrees with that of infinite momentum frame prescription by Fujita and Ogura. Boson boson states (2p$-$2h states) appear only as the continuum spectrum without making any bound states.   Further, the finite size correction to the vacuum energy is estimated. The evaluated central charge is found to be close to unity. 
  A generic Lagrangian, in arbitrary spacetime dimension, describing the interaction of a graviton, a dilaton and two antisymmetric tensors is considered. An isotropic p-brane solution consisting of three blocks and depending on four parameters in the Lagrangian and two arbitrary harmonic functions is obtained. For specific values of parameters in the Lagrangian the solution may be identified with previously known superstring solutions. 
  We discuss BF theories defined on manifolds with spatial boundaries. Variational arguments show that one needs to augment the usual action with a boundary term for specific types of boundary conditions. We also show how to use this procedure to find the boundary actions for theories of gravity with first order formulations. Possible connection with the membrane approach is also discussed. 
  Field equations with general covariance are interpreted as equations for a target space describing physical space time co-ordinates, in terms of an underlying base space with conformal invariance. These equations admit an infinite number of inequivalent Lagrangian descriptions. A model for reparametrisation invariant membranes is obtained by reversing the roles of base and target space variables in these considerations. 
  We show that the problem of computing the vacuum expectation values of gauge invariant operators in the strong coupling limit of topologically massive gauge theory is equivalent to the problem of computing similar operators in the $G_k/G$ model where $k$ is the integer coefficient of the Chern-Simons term. The $G_k/G$ model is a topological field theory and many correlators can be computed analytically. We also show that the effective action for the Polyakov loop operator and static modes of the gauge fields of the strongly coupled theory at finite temperature is a perturbed, non-topological $G_k/G$ model. In this model, we compute the one loop effective potential for the Polyakov loop operators and explicitly construct the low-lying excited states. In the strong coupling limit the theory is in a deconfined phase. 
  Minimal N=1/2 supersymmetric extension of bosonic Polyakov's string is constructed. This model is natural generalization of Di Vecchia-Ravndal superparticle. The classical sector of the model is investigated, Noether currents and Virosoro supercondition are found. Minimal spinning string is more simple, than the standard N=1 spinning string of Neveu-Schwarz-Ramond and has a number of unusial properties such as a chiral symmetry, parabolic type of equations of movement, non-triviality fermionic sectors for closed strings only and e.t.c. 
  We point out that the similarities in N = 1 supersymmetric SO, SP gauge theories can be explained by using the trick of extrapolating the groups to the negative dimensions. One of the advantages of this trick is that anomaly matching is automatically satisfied. 
  We consider the Maxwell-Higgs system in the broken phase, described in terms of a Kalb-Ramond field interacting with the electromagnetic field through a topological coupling. We then study the creation operators of states which respectively carry a point charge and a closed magnetic string in the electromagnetic language or a point topological charge and a closed Kalb-Ramond charged string in the Kalb-Ramond dual language. Their commutation relation is evaluated, implying they satisfy a dual algebra and their composite possesses generalized statistics. In the local limit where the radius of the string vanishes, only Fermi or Bose statistics are allowed. This provides an explicit operator realization for statistical transmutation in 3+1D. 
  For the simplest case of a supermembrane matrix model, various symmetry reductions are given, with the fermionic contribution(s) (to an effective Schr\"odinger equation) corresponding to an attractive $\delta$-function potential (towards zero-area configurations). The differential equations are real, and are shown not to admit square-integrable real solutions (even when allowing non-vanishing boundary conditions at infinity). Complex solutions, however, are not excluded by this argument. 
  We find classical solutions to the simply-laced affine Toda equations which satisfy integrable boundary conditions using solitons which are analytically continued from imaginary coupling theories. Both static `vacuum' configurations and the time-dependent perturbations about them which correspond to classical vacua and particle scattering solutions respectively are considered. A large class of classical scattering matrices are calculated and found to satisfy the reflection bootstrap equation. 
  The existence of instantonic decay modes would indicate a semi-classical instability of the vacua of ten and eleven dimensional supergravity theories. Decay modes whose spin structures are incompatible with those of supersymmetric vacua have previously been constructed, and we present generalisations including those involving non trivial dilaton and antisymmetric tensor fields. We then show that the requirement that any instanton describing supersymmetric vacuum decay should admit both a zero momentum hypersurface from which we describe the subsequent Lorentzian evolution and a spin structure at infinity compatible with the putative vacuum excludes all such decay modes, except those with unphysical energy momentum tensors which violate the dominant energy condition. 
  We develop the theory of the nonadiabatic geometric phase, in both the Abelian and non-Abelian cases, in quaternionic Hilbert space. 
  The development of the Exact Renormalization Group for fermionic theories is presented, together with its application to the chiral Gross-Neveu model. We focus on the reliability of various approximations, specifically the derivative expansion and further truncations in the number of fields. The main differences with bosonic theories are discussed. 
  We study the large-N limit of a class of matrix models for dually weighted triangulated random surfaces using character expansion techniques. We show that for various choices of the weights of vertices of the dynamical triangulation the model can be solved by resumming the Itzykson-Di Francesco formula over congruence classes of Young tableau weights modulo three. From this we show that the large-N limit implies a non-trivial correspondence with models of random surfaces weighted with only even coordination number vertices. We examine the critical behaviour and evaluation of observables and discuss their interrelationships in all models. We obtain explicit solutions of the model for simple choices of vertex weightings and use them to show how the matrix model reproduces features of the random surface sum. We also discuss some general properties of the large-N character expansion approach as well as potential physical applications of our results. 
  We discuss the scattering of a light closed-string state off a $D$ brane, taking into account quantum recoil effects on the latter, which are described by a pair of logarithmic operators. The light-particle and $D$-brane subsystems may each be described by a world-sheet with an external source due to the interaction between them. This perturbs each subsystem away from criticality, which is compensated by dressing with a Liouville field whose zero mode we interpret as time. The resulting evolution equations for the $D$ brane and the closed string are of Fokker-Planck and modified quantum Liouville type, respectively. The apparent entropy of each subsystem increases as a result of the interaction between them, which we interpret as the loss of information resulting from non-observation of the other entangled subsystem. We speculate on the possible implications of these results for the propagation of closed strings through a dilute gas of virtual $D$ branes. 
  Using the recent advances in our understanding of non-perturbative aspects of type II strings we show how non-trivial exact results for $N=2$ quantum field theories can be reduced to T-dualities of string theory. This is done by constructing a local geometric realization of quantum field theories together with a local application of mirror symmetry. This construction is not based on any duality conjecture and thus reduces non-trivial quantum field theory results to much better understood T-dualities of type II strings. Moreover it can be used in principle to construct in a systematic way the vacuum structure for arbitrary $N=2$ gauge theories with matter representations. 
  The two--dimensional topological BF model is quantized in the axial gauge. We show that this theory is trivially ultraviolet finite and that the usual infrared problem of the propagator of the scalar field in two dimensions is replaced by an easily solvable long distances problem inherent to the axial gauge. It will also be shown that contrarily to the 3--dimensional case, the action principle cannot be completely replaced by the various Ward identities expressing the symmetries of the model; some of the equation of motion are needed. 
  We consider $N=2$ supergravity coupled to $N=2$ Yang--Mills matter and discuss the nature of one--loop divergences. Using $N=1$ superfields and superspace methods, we describe the quantization of the system in the abelian case. 
  In terms of a gauge-invariant matrix parametrization of the fields, we give an analysis of how the mass gap could arise in non-Abelian gauge theories in two spatial dimensions. 
  The dilute A_3 model is a solvable IRF (interaction round a face) model with three local states and adjacency conditions encoded by the Dynkin diagram of the Lie algebra A_3. It can be regarded as a solvable version of an Ising model at the critical temperature in a magnetic field. One therefore expects the scaling limit to be governed by Zamolodchikov's integrable perturbation of the c=1/2 conformal field theory. Indeed, a recent thermodynamic Bethe Ansatz approach succeeded to unveil the corresponding E_8 structure under certain assumptions on the nature of the Bethe Ansatz solutions. In order to check these conjectures, we perform a detailed numerical investigation of the solutions of the Bethe Ansatz equations for the critical and off-critical model. Scaling functions for the ground-state corrections and for the lowest spectral gaps are obtained, which give very precise numerical results for the lowest mass ratios in the massive scaling limit. While these agree perfectly with the E_8 mass ratios, we observe one state which seems to violate the assumptions underlying the thermodynamic Bethe Ansatz calculation. We also analyze the critical spectrum of the dilute A_3 model, which exhibits massive excitations on top of the massless states of the Ising conformal field theory. 
  In space of d ordinary and d Grassmann coordinates, with d \ge 15, the charges unify with the spin: the Lorentz group SO(1, d-1) in Grassmann space manifests under certain conditions as SO(1,3) (in d=4 subspace) times SO(10) \supset SU(3) \times SU(2) \times U(1) (in the rest of the space), accordingly the symmetry group of the S- matrix, which is approximatelly unitary in d=4 ordinary subspace, manifests as the direct product of the Poincar\'e group in d=4 subspace and the group describing charges. (Talk presented at XIX Triangular Meeting on Recent Development in Quantum Theories, Rome, March 1996 and at IWCQIS 96, Dubna, July 1996.) 
  A class of four-dimensional static supersymmetric black hole solutions of effective supergravity Lagrangian of IIA superstring compactified on $T^6$ is constructed by explicitly solving Killing spinor equations (KSEs). These solutions are dyonic black holes parametrized by four charges, with dilaton and diagonal internal metric components as the only non-zero scalar fields, and preserve $1 \over 8$ of $N=8$ supersymmetry. The KSEs with only Neveu-Schwarz-Neveu-Schwarz charges relate spinors with opposite chirality from ten-dimensional view point, and have identical structures with KSEs of toroidally compactified heterotic string. We also find a solution with four Ramond-Ramond charges which is U-dual to the solution with four Neveu-Schwarz-Neveu-Schwarz charges, and corresponds to the intersecting D-brane configuration with two 2-branes and two 4-branes. A configuration with both Neveu-Schwarz-Neveu-Schwarz charges and Ramond-Ramond charges is also found. We show that the configurations T-dual to the above solutions are also solutions of the KSEs. The patterns of supersymmetry breaking are studied in detail. 
  We analyze the perturbative implications of the most general high derivative approach to quantum gravity based on a diffeomorphism invariant local action. In particular, we consider the super-renormalizable case with a large number of metric derivatives in the action. The structure of ultraviolet divergences is analyzed in some detail. We show that they are independent of the gauge fixing condition and the choice of field reparametrization. The cosmological counterterm is shown to vanish under certain parameter conditions. We elaborate on the unitarity problem of high derivative approaches and the distribution of masses of unphysical ghosts. We also discuss the properties of the low energy regime and explore the possibility of having a multi-scale gravity with different scaling regimes compatible with Einstein gravity at low energies. Finally, we show that the ultraviolet scaling of matter theories is not affected by the quantum corrections of high derivative gravity. As a consequence, asymptotic freedom is stable under those quantum gravity corrections. 
  We consider SU(2) Yang-Mills theory in 1+1 dimensions coupled to massless adjoint fermions. With all fields in the adjoint representation the gauge group is actually SU(2)/Z_2, which possesses nontrivial topology. In particular, there are two distinct topological sectors and the physical vacuum state has a structure analogous to a \theta vacuum. We show how this feature is realized in light-front quantization, using discretization of x^- as an infrared regulator. We find exact expressions for the vacuum states and construct the analog of the \theta vacuum. We calculate the bilinear condensate of the model. We argue that this condensate does not effect the spectrum of the massless theory but gives the string tenson of the massive theory. 
  We find an analog of the electric-magnetic duality, which is a $Z_2$ transformation between magnetic and electric sectors of the static and rotationally symmetric solutions in a class of (2+1)-dimensional Einstein-Maxwell-Dilaton gravity theories. The theories in our consideration include, in particular, one parameter class of theories continuously connecting the Banados-Teitelboim-Zanelli (BTZ) gravity and the low energy string effective theory. When there is no $U(1)$ charge, we have $O(2)$ or $O(1,1)$ symmetry, depending on a parameter that specifies each theory. Via the $Z_2$ transformation, we obtain exact magnetically charged solutions from the known electrically charged solutions. We explain the relationship between the $Z_2$ transformation and $O(2,Z)$ symmetry, and comment on the $T$-duality of the string theory. 
  As has long been known, it is energetically favorable for massive fermions to deform the homogeneous vacuum around them, giving rise to extended bag-like objects. We study this phenomenon non-perturbatively in a model field theory, the $1+1$ dimensional Massive Gross-Neveu model, in the large $N$ limit. We prove that the bags in this model are necessarily time dependent. We calculate their masses variationally and demonstrate their stability. We find a non-analytic behavior in these masses as we approach the standard massless Gross-Neveu model and argue that this behavior is caused by the kink-antikink threshold. This work extends our previous work to a non-integrable field theory. 
  We determine self-consistently the hard-quark and hard-gluon modes in hot QCD. The damping-rate part in resummed hard-quark or hard-gluon propagators, rather than the thermal-mass part, plays the dominant role. 
  The spherical domains $S^d_\beta$ with conical singularities are a convenient arena for studying the properties of tensor Laplacians on arbitrary manifolds with such a kind of singular points. In this paper the vector Laplacian on $S^d_\beta$ is considered and its spectrum is calculated exactly for any dimension $d$. This enables one to find the Schwinger-DeWitt coefficients of this operator by using the residues of the $\zeta$-function. In particular, the second coefficient, defining the conformal anomaly, is explicitly calculated on $S^d_\beta$ and its generalization to arbitrary manifolds is found. As an application of this result, the standard renormalization of the one-loop effective action of gauge fields is demonstrated to be sufficient to remove the ultraviolet divergences up to the first order in the conical deficit angle. 
  Approximation only by derivative (or more generally momentum) expansions, combined with reparametrization invariance, turns the continuous renormalization group for quantum field theory into a set of partial differential equations which at fixed points become non-linear eigenvalue equations for the anomalous scaling dimension $\eta$. We review how these equations provide a powerful and robust means of discovering and approximating non-perturbative continuum limits. Gauge fields are briefly discussed. Particular emphasis is placed on the r\^ole of reparametrization invariance, and the convergence of the derivative expansion is addressed. 
  We present two new classes of axisymmetric stationary solutions of the Einstein-Maxwell-Dilaton equations with coupling constant $\alpha^2=3$. Both classes are written in terms of two harmonic maps $\lambda$ and $\tau$. $\lambda$ determines the gravitational potential and $\tau$ the electromagnetic one in such a form that we can have an arbitrary electromagnetic field. As examples we generate two solutions with mass ($M$), rotation ($s$) and scalar ($\delta$) parameters, one with electric charge ($q$) another one with magnetic dipole ($Q$) parameter. The first solution contains the Kerr metric for $q=\delta=0$. 
  We write down a general geometric action principle for spinning strings in $d$-dimensional Minkowski space, which is formulated without the use of Grassmann coordinates. Instead, it is constructed in terms of the pull-back of a left invariant Maurer-Cartan form on the $d$-dimensional Poincar\'e group to the world sheet. The system contains some interesting special cases. Among them are the Nambu string (as well as, null and tachyionic strings) where the spin vanishes, and also the case of a string with a spin current - but no momentum current. We find the general form for the Virasoro generators, and show that they are first class constraints in the Hamiltonian formulation of the theory. The current algebra associated with the momentum and angular momentum densities are shown, in general, to contain rather complicated anomaly terms which obstruct quantization. As expected, the anomalies vanish when one specializes to the case of the Nambu string, and there one simply recovers the algebra associated with the Poincar\'e loop group. We speculate that there exist other cases where the anomalies vanish, and that these cases give the bosonization of the known pseudoclassical formulations of spinning strings. 
  Using a contraction procedure, we obtain Toda theories and their structures, from affine Toda theories and their corresponding structures. By structures, we mean the equation of motion, the classical Lax pair, the boundary term for half line theories, and the quantum transfer matrix. The Lax pair and the transfer matrix so obtained, depend nontrivially on the spectral parameter. 
  We construct the six-dimensional topological field theory appropriate to describe the ground-state configurations of D5-branes. A close examination on the degenerations of D5-branes gives us the physical observables which can be regarded as the Poincar\'e duals of the cycles of the moduli space. These observables are identified with the creation opeartors of the bound states of D5-branes and lead to the second quantization of five-branes. This identification of the bound states with the cycles also provides their topological stability and suggests that the bound states of five-branes have internal structures. The partition function of the second-quantized five-branes is also discussed. 
  Maxwell theory can be studied in a gauge which is invariant under conformal rescalings of the metric, and first proposed by Eastwood and Singer. This paper studies the corresponding quantization in flat Euclidean 4-space. The resulting ghost operator is a fourth-order elliptic operator, while the operator P on perturbations of the potential is a sixth-order elliptic operator. The operator P may be reduced to a second-order non-minimal operator if a dimensionless gauge parameter tends to infinity. Gauge-invariant boundary conditions are obtained by setting to zero at the boundary the whole set of perturbations of the potential, jointly with ghost perturbations and their normal derivative. This is made possible by the fourth-order nature of the ghost operator. An analytic representation of the ghost basis functions is also obtained. 
  The supermembrane theory on $R^{10}x S^1$ is investigated, for membranes that wrap once around the compact dimension. The Hamiltonian can be organized as describing $N_s$ interacting strings, the exact supermembrane corresponding to $N_s\to \infty$. The zero-mode part of $N_s-1$ strings turn out to be precisely the modes which are responsible of instabilities. For sufficiently large compactification radius $R_0$, interactions are negligible and the lowest-energy excitations are described by a set of harmonic oscillators. We compute the physical spectrum to leading order, which becomes exact in the limit $ g^2 \to \infty $, where $g^2\equiv 4\pi^2 T_3 R_0^3$ and $T_3$ is the membrane tension. As the radius is decreased, more strings become strongly interacting and their oscillation modes get frozen. In the zero-radius limit, the spectrum is constituted of the type IIA superstring spectrum, plus an infinite number of extra states associated with flat directions of the quartic potential. 
  We briefly review these low-energy effective theories for the quantum Hall effect, with emphasis and language familiar to field theorists. Two models have been proposed for describing the most stable Hall plateaus (the Jain series): the multi-component Abelian theories and the minimal W-infinity models. They both lead to a-priori classifications of quantum Hall universality classes. Some experiments already confirmed the basic predictions common to both effective theories, while other experiments will soon pin down their detailed properties and differences. Based on the study of partition functions, we show that the Abelian theories are rational conformal field theories while the minimal W-infinity models are not. 
  A new two-step renormalization procedure is proposed. In the first step, the effects of high-energy states are considered in the conventional (Feynman) perturbation theory. In the second step, the coupling to many-body states is eliminated by a similarity transformation. The resultant effective Hamiltonian contains only interactions which do not change particle number. It is subject to numerical diagonalization. We apply the general procedure to a simple example for the purpose of illustration. 
  I review some recent work that describes the close analogy between self-dual Yang--Mills amplitudes and QCD amplitudes with external gluons of positive helicity. This analogy is carried at tree level for amplitudes with two external quarks and up to one-loop for amplitudes involving only external gluons. 
  In this talk we use nonlinear realizations to study the spontaneous partial breaking of rigid and local supersymmetry. 
  In this work we use constructs from the dual space of the semi-direct product of the Virasoro algebra and the affine Lie algebra of a circle to write a theory of gravitation which is a natural analogue of Yang-Mills theory. The theory provides a relation between quadratic differentials in 1+1 dimensions and rank two symmetric tensors in higher dimensions as well as a covariant local Lagrangian for two dimensional gravity. The isotropy equations of coadjoint orbits are interpreted as Gauss law constraints for a field theory in two dimensions, which enables us to extend to higher dimensions. The theory has a Newtonian limit in any space-time dimension. Our approach introduces a novel relationship between string theories and 2D field theories that might be useful in defining dual theories. We briefly discuss how this gravitational field couples to fermions. 
  We present a brief review on the canonical transformation description of some duality symmetries in string and gauge theories. In particular, we consider abelian and non-abelian T-dualities in closed and open string theories as well as S-duality in abelian and non-abelian non-supersymmetric gauge theories. 
  A careful analysis of differential renormalization shows that a distinguished choice of renormalization constants allows for a mathematically more fundamental interpretation of the scheme. With this set of a priori fixed integration constants differential renormalization is most closely related to the theory of generalized functions. The special properties of this scheme are illustrated by application to the toy example of a free massive bosonic theory. Then we apply the scheme to the phi^4-theory. The two-point function is calculated up to five loops. The renormalization group is analyzed, the beta-function and the anomalous dimension are calculated up to fourth and fifth order, respectively. 
  The recently rigorously proved nonperturbative relation between u and the prepotential, underlying N=2 SYM with gauge group SU(2), implies both the reflection symmetry $\overline{u(\tau)}=u(-\bar\tau)$ and $u(\tau+1)=-u(\tau)$ which hold exactly. The relation also implies that $\tau$ is the inverse of the uniformizing coordinate u of the moduli space of quantum vacua. In this context, the above quantum symmetries are the key points to determine the structure of the moduli space. It turns out that the functions a(u) and a_D(u), which we derive from first principles, actually coincide with the solution proposed by Seiberg and Witten. We also consider some relevant generalizations. 
  We investigate a possible difference between the effective potential and zero-point energy. We define the zero-point ambiguity (ZPA) as the difference between these two definitions of vacuum energy. Using the zeta function technique, in order to obtain renormalized quantities, we show that ZPA vanishes, implying that both of the above definitions of vacuum energy coincide for a large class of geometries and a very general potential. In addition, we show explicitly that an extra term, obtained by E. Myers some years ago for the ZPA, disappears when a scale parameter $\mu$ is consistently introduced in all zeta functions in order to keep them dimensionless. 
  A review of old inconsistencies of Classical Electrodynamics (CED) and of some new ideas that solve them is presented. Problems with causality violating solutions of the wave equation and of the electron equation of motion, and problems with the non-integrable singularity of its self-field energy tensor are well known. The correct interpretation of the two (advanced and retarded) Lienard-Wiechert solutions are in terms of creation and annihilation of particles in classical physics. They are both retarded solutions. Previous work on the short distance limit of CED of a spinless point electron are based on a faulty assumption which causes the well known inconsistencies of the theory: a diverging self-energy (the non-integrable singularity of its self-field energy tensor) and a causality-violating third order equation of motion (the Lorentz-Dirac equation). The correct assumption fixes these problems without any change in the Maxwell's equations and let exposed, in the zero-distance limit, the discrete nature of light: the flux of energy from a point charge is discrete in time. CED cannot have a true equation of motion but only an effective one, as a consequence of the intrinsic meaning of the Faraday-Maxwell concept of field that does not correspond to the classical description of photon exchange, but only to the smearing of its effects in the space around the charge. This, in varied degrees, is transferred to QED and to other field theories that are based on the same concept of fields as space-smeared interactions. 
  A modified interaction representation for the master field describing connected $SU(N)$-invariant Wightman's functions in the large $N$ limit of matrix fields is constructed. This construction is based on the representation of the master field in terms of Boltzmannian field theory found before. In the modified interaction representation we deal with two scalar Boltzmann fields ({\it up} and {\it down} fields). For up and down fields only half-planar diagrams contribute and this could help to write down a recursive set of non-linear integral-differential equations summing up planar diagrams. 
  We study charge k SU(2) BPS monopoles which are symmetric under the cyclic group of order k. Approximate twistor data (spectral curves and Nahm data) is constructed using a new technique based upon a Painleve analysis of Nahm's equation around a pole. With this data both analytical and numerical approximate ADHMN constructions are performed to study the zeros of the Higgs field and the monopole energy densities. The results describe, via the moduli space approximation, a novel type of low energy k monopole scattering. 
  We investigate the five and six-dimensional black strings within Einstein-Maxwell theory. The extremal black strings are endowed with the null Killing symmetry. We study the propagation of Einstein-Maxwell modes in the extremal black string background by using this symmetry. It turns out that one graviton is a propagating mode, while both the Maxwell $F$ and three-form $H$ fields are non-propagating modes. Further we discuss the stability and classical hair of the extremal black strings. 
  It is well known that under T-duality the sigma model dilaton (which is normally thought to be related to the string coupling constant through the simple formula $\kappa = exp <\phi >$), undergoes an additive shift. On the other hand, Kugo and Zwiebach, using a simplified form of string field theory, claim that the string coupling constant does not change under the T-duality. Obviously, what seems to happen is that two different coupling constants, associated to different dilatons, are used. In this contribution we shall try to clarify this, and related issues. 
  We present some striking global consequences of a model quaternionic quantum field theory which is locally complex. We show how making the quaternionic structure a dynamical quantity naturally leads to the prediction of cosmic strings and non-baryonic hot dark matter candidates. 
  A new method for integrating anomalous Ward identities and finding the effective action is proposed. Two-dimensional supergravity and $W_3$-gravity are used as examples to demonstrate its potential. An operator is introduced that associates each physical quantity with a Ward identity, i.e., a quantity that is transformed without anomalous terms and can be nullified in a consistent manner. A covariant form of the action for matter field interacting with a gravitational and $W_3$-gravitational background is proposed. 
  In this paper we study the structure of the Hilbert space for the recent noncommutative geometry models of gauge theories. We point out the presence of unphysical degrees of freedom similar to the ones appearing in lattice gauge theories (fermion doubling). We investigate the possibility of projecting out these states at the various levels in the construction, but we find that the results of these attempts are either physically unacceptable or geometrically unappealing. 
  Threshold amplitudes are considered for $n$-particle production in arbitrary scalar theory. It is found that, like in $\phi ^4$, leading-$n$ corrections to the tree level amplitudes, being summed over all loops, exponentiate. This result provides more evidence in favor of the conjecture on the exponential behavior of the multiparticle amplitudes. 
  The c<1 and c>1 matrix models are analyzed within large N renormalization group, taking into account touching (or branching) interactions. The c<1 modified matrix model with string exponent gamma>0 is naturally associated with an unstable fixed point, separating the Liouville phase (gamma<0) from the branched polymer phase (gamma=1/2). It is argued that at c=1 this multicritical fixed point and the Liouville fixed point coalesce, and that both fixed points disappear for c>1. In this picture, the critical behavior of c>1 matrix models is generically that of branched polymers, but only within a scaling region which is exponentially small when c -> 1. It also explains the behavior of multiple Ising spins coupled to gravity. Large crossover effects occur for c-1 small enough, with a c ~ 1 pseudo-scaling which explains numerical results. 
  We show that a Hamiltonian reduction of affine Lie superalgebras having bosonic simple roots (such as $OSp(1|4)$) ``does'' produce supersymmetric Toda models, with superconformal symmetry being nonlinearly realised for those fields of the Toda system which are related to the bosonic simple roots of the superalgebra. A fermionic $b-c$ system of conformal spin $(3/2,-1/2)$ is a natural ingredient of such models. 
  The recent progress in revealing classical integrable structures in quantum models solved by Bethe ansatz is reviewed. Fusion relations for eigenvalues of quantum transfer matrices can be written in the form of classical Hirota's bilinear difference equation. This equation is also known as the completely discretized version of the 2D Toda lattice. We explain how one obtains the specific quantum results by solving the classical equation. The auxiliary linear problem for the Hirota equation is shown to generalize Baxter's T-Q relation. 
  We consider relative entropy in Field Theory as a well defined (non-divergent) quantity of interest. We establish a monotonicity property with respect to the couplings in the theory. As a consequence, the relative entropy in a field theory with a hierarchy of renormalization group fixed points ranks the fixed points in decreasing order of criticality. We argue from a generalized $H$ theorem that Wilsonian RG flows induce an increase in entropy and propose the relative entropy as the natural quantity which increases from one fixed point to another in more than two dimensions. 
  We give a broad overview of superstring duality, Dirichlet branes, and some implications of both for questions about the structure of space-time at short distances. 
  Two topological issues on membranes in M-theory are studied:     (1) Soliton is an important subject in M-theory. Under the framework of obstruction theory with the help from framed links in $S^3$, we give a complete enumeration of topological membrane solitons in a string-admissible target-space of the form a product of Minkowskian space-times, tori, and K3-surfaces. Patching of these solitons and their topological charges are also defined and discussed.      (2) Loop order of membrane scatterings is the basis for a perturbative M-theory. We explore this concept with emphases on its distinct features from pointlike and stringlike particles.   For completeness, a light exposition on homologies of compact oriented 3-manifolds is given in the Appendix. 
  We suggest and motivate a precise equivalence between uncompactified eleven dimensional M-theory and the N = infinity limit of the supersymmetric matrix quantum mechanics describing D0-branes. The evidence for the conjecture consists of several correspondences between the two theories. As a consequence of supersymmetry the simple matrix model is rich enough to describe the properties of the entire Fock space of massless well separated particles of the supergravity theory. In one particular kinematic situation the leading large distance interaction of these particles is exactly described by supergravity . The model appears to be a nonperturbative realization of the holographic principle. The membrane states required by M-theory are contained as excitations of the matrix model. The membrane world volume is a noncommutative geometry embedded in a noncommutative spacetime. 
  Some basic topics in Light-Front (LF) quantized field theory are reviewed. Poincar\`e algebra and the LF Spin operator are discussed. The local scalar field theory of the conventional framework is shown to correspond to a non-local Hamiltonian theory on the LF in view of the constraint equations on the phase space, which relate the bosonic condensates to the non-zero modes. This new ingredient is useful to describe the spontaneous symmetry breaking on the LF. The instability of the symmetric phase in two dimensional scalar theory when the coupling constant grows is shown in the LF theory renormalized to one loop order. Chern-Simons gauge theory, regarded to describe excitations with fractional statistics, is quantized in the light-cone gauge and a simple LF Hamiltonian obtained which may allow us to construct renormalized theory of anyons. 
  We compare the emission rates from excited D-branes in the microcanonical ensemble with back reaction corrected emission rates from black holes in field theory. In both cases, the rates in the high energy tail of the spectrum differ markedly from what a canonical ensemble or free field theory approach would yield. Instead of being proportional to a Bose-Einstein distribution function, the rates in the high energy tail are proportional to $\exp (-\Delta S_{BH})$ where $\Delta S_{BH}$ is the difference in black hole entropies before and after emission. After including the new effects, we find agreement, at leading order, between the D-brane and field theory rates over the entire range of the spectrum. 
  The known BPS dyon black hole solutions of the N=4 heterotic string in four dimensions with non-zero angular momentum all have naked singularities. We show that it is possible to modify a certain class of these solutions by the addition of massive Kaluza-Klein fields in such a way that the solutions decompactify near the core to five-dimensional black hole solutions with regular event horizons. We argue that the degeneracy of the four-dimensional BPS dyon states is given, for large charges, by the five-dimensional geometric entropy. 
  Geometric quantization on a coset space $G/H$ is considered, intending to recover Mackey's inequivalent quantizations. It is found that the inequivalent quantizations can be obtained by adopting the symplectic 2-form which leads to Wong's equation. The irreducible representations of $H$ which label the inequivalent quantizations arise from Weil's theorem, which ensures a Hermitian bundle over $G/H$ to exist. 
  Within the framework of generalized functions a general consistent definition of double commutators is given. This definition respects the Jacobi identity even if the regularization is removed. The double commutator of fermionic currents is calculated in this limit. We show that BJL--type prescriptions and point--splitting prescriptions for calculating double commutators fail to give correct results in free field theory. 
  We give an explicit derivation of the Picard-Fuchs equations for N=2 supersymmetric SU(3) Yang-Mills theory with $N_f<6$ massive hypermultiplets in the fundamental representation. We determine the instanton corrections to the prepotential in the weak coupling region using the relation between $\tr<\phi^2>$ and the prepotential. This method can be generalized to other gauge groups. 
  We derive the (matrix-valued) Feynman rules of mass perturbation theory of the massive Schwinger model for non-zero vacuum angle theta. Further, we discuss the properties of the three-boson bound state and compute -- by a partial resummation of the mass perturbation series -- its mass and its partial decay widths. 
  The conformal field theory on a Z_N-surface is studied by mapping it on the branched sphere. Using a coulomb gas formalism we construct the minimal models of the theory. 
  We propose a classification of critical behaviours of branched polymers for arbitrary topology. We show that in an appropriately defined double scaling limit the singular part of the partition function is universal. We calculate this partition function exactly in the generic case and perturbatively otherwise. In the discussion section we comment on the relation between branched polymer theory and Euclidean quantum gravity. 
  We show how BPS states of supersymmetric SU(2) Yang-Mills with matter -both massless and massive- are described as self-dual strings on a Riemann surface. This connection enables us to prove the stability and the strong coupling behaviour of these states. The Riemann surface naturally arises from type-IIB Calabi-Yau compactifications whose three-branes wrapped around vanishing two-cycles correspond to one-cycles on this surface. 
  We analyze the 1+1 dimensional Nambu-Jona-Lasinio model non-perturbatively. We study non-trivial saddle points of the effective action in which the composite fields $\sigx=<\bar\psi\psi>$ and $\pix=<\bar\psii\gam_5\psi>$ form static space dependent configurations. These configurations may be viewed as one dimensional chiral bags that trap the original fermions (``quarks'') into stable extended entities (``hadrons''). We provide explicit expressions for the profiles of some of these objects and calculate their masses. Our analysis of these saddle points, and in particular, the proof that the $\sigx, \pix$ condensations must give rise to a reflectionless Dirac operator, appear to us simpler and more direct than the calculations previously done by Shei, using the inverse scattering method following Dashen, Hasslacher, and Neveu. 
  As in an earlier paper we start from the hypothesis that physics on the Planck scale should be described by means of concepts taken from ``discrete mathematics''. This goal is realized by developing a scheme being based on the dynamical evolution of a particular class of ``cellular networks'' being capable of performing an ``unfolding phase transition'' from a (presumed) chaotic initial phase towards a new phase which acts as an ``attractor'' in total phase space and which carries a fine or super structure which is identified as the discrete substratum underlying ordinary continuous space-time (or rather, the physical vacuum). Among other things we analyze the internal structure of certain particular subclusters of nodes/bonds (maximal connected subsimplices, $mss$) which are the fundamental building blocks of this new phase and which are conjectured to correspond to the ``physical points'' of ordinary space-time. Their mutual entanglement generates a certain near- and far-order, viz. a causal structure within the network which is again set into relation with the topological/metrical and causal/geometrical structure of continuous space-time. The mathematical techniques to be employed consist mainly of a blend of a fair amount of ``stochastic mathematics'' with several relatively advanced topics of discrete mathematics like the ``theory of random graphs'' or ``combinatorial graph theory''. Our working philosophy is it to create a scenario in which it becomes possible to identify both gravity and quantum theory as the two dominant but derived(!) aspects of an underlying discrete and more primordial theory (dynamical cellular network) on a much coarser level of resolution, viz. continuous space-time. 
  We derive a path integral expression for the transition amplitude in 1+1-dimensional QCD starting from canonically quantized QCD. Gauge fixing after quantization leads to a formulation in terms of gauge invariant but curvilinear variables. Remainders of the curved space are Jacobians, an effective potential, and sign factors just as for the problem of a particle in a box. Based on this result we derive a Faddeev-Popov like expression for the transition amplitude avoiding standard infinities that are caused by integrations over gauge equivalent configurations. 
  We investigate Aharonov-Bohm scattering in a theory in which charged bosonic matter fields are coupled to topologically massive electrodynamics and topologically massive gravity. We demonstrate that, at one-loop order, the transmuted spins in this theory are related to the ones of ordinary Chern-Simons gauge theory in the same way that the Knizhnik-Polyakov-Zamolodchikov formula relates the Liouville-dressed conformal weights of primary operators to the bare weights in two-dimensional conformal field theories. We remark on the implications of this connection between two-dimensional conformal field theories and three-dimensional gauge and gravity theories for a topological membrane reformulation of strings. We also discuss some features of the gravitational analog of the Aharonov-Bohm effect. 
  We prove Zuber's conjecture establishing connections of the fusion rules of the $su(N)_k$ WZW model of conformal field theory and the intersection form on vanishing cycles of the associated fusion potential. 
  Decay amplitudes for mesons in two-dimensional QCD are discussed. We show that in spite of an infinite number of conserved charges, particle production is not entirely suppressed. This phenomenon is explained in terms of quantum corrections to the combined algebra of higher-conserved and spectrum-generating currents. We predict the qualitative form of particle production probabilities and verify that they are in agreement with numerical data. We also discuss four-dimensional self-dual Yang-Mills theory in the light of our results. 
  We discuss zero-brane solutions in three-dimensional Minkowski space-time annihilated by half of the supersymmetries. The other half of the supersymmetries which should generate the supersymmetric multiplet are ill-defined and lead to a non bose-fermi degenerate solitonic spectrum. 
  The stimulated emission of massless bosons by a relativistic and the CGHS black hole are studied for real and complex scalar fields. The radiations induced by one-particle and thermal states are considered and their thermal properties investigated near the horizon. These exhibit both thermal and non-thermal properties for the two black-hole models. 
  In the first part of the talk we discuss T-duality for a free boson on a world sheet with boundary in a setting suitable for the generalization to non-trivial backgrounds. The gauging method as well as the canonical transformation are considered. In both cases Dirichlet strings as T-duals of Neumann strings arise in a generic way. In the second part the gauging method is employed to construct the T-dual of a model with non-Abelian isometries. 
  In the first part of this talk, I consider some exact string solutions in curved spacetimes. In curved spacetimes with a Killing vector (timelike or spacelike), the string equations of motion and constraints are reduced to the Hamilton equations of a relativistic point-particle in a scalar potential, by imposing a particular ansatz. As special examples I consider circular strings in axially symmetric spacetimes, as well as stationary strings in stationary spacetimes. In the second part of the talk, I then consider in more detail the stationary strings in the Kerr -Newman geometry. It is shown that the world-sheet of a stationary string, that passes the static limit of the 4-D Kerr-Newman black hole, describes a 2-D black hole. Mathematical results for 2-D black holes can therefore be applied to physical objects; (say) cosmic strings in the vicinity of Kerr black holes. As an immediate general result, it follows that the string modes are thermally excited. 
  We reformulate, using super worldline formalism, the pinched gluon vertex operator proposed by Strassler. The pinched vertex operator turns out to be the product of two gluon vertex operators with the insertion of $\delta$-function which makes the super distances between them zero. Thus the pinch procedures turn out to be nothing but the insertions of $\delta$-function. Applying our formulation to two-loop diagrams which are the QED correction to gluon scatterings via a single spinor loop, with the QED charge e being replaced by the strong coupling g, we show various formulae on pinched N-point functions. 
  The properties of BPS monopoles carrying nonabelian magnetic charges are investigated by following the behavior of the moduli space of solutions as the Higgs field is varied from a value giving a purely abelian symmetry breaking to one that leaves a nonabelian subgroup of the gauge symmetry unbroken. As the limit of nonabelian unbroken symmetry is reached, some of the fundamental abelian monopoles remain massive but acquire nonabelian magnetic charges. The BPS mass formula indicates that others should because massless in this limit. These do not correspond to distinct solitons, but instead are manifested as a ``nonabelian cloud'' surrounding the massive monopoles, with their position and phase degrees of freedom being transformed into parameters characterizing the cloud. 
  In this letter we introduce a generalization of the Knizhnik- Zamolodchikov equations from affine Lie algebras to a wide class of conformal field theories (not necessarily rational). The new equations describe correlations functions of primary fields and of a finite number of their descendents. Our proposal is based on Nahm's concept of small spaces which provide adequate substitutes for the lowest energy subspaces in modules of affine Lie algebras. We explain how to construct the first order differential equations and investigate properties of the associated connections, thereby preparing the grounds for an analysis of quantum symmetries. The general considerations are illustrated in examples of Virasoro minimal models. 
  The equivalence between the Dirac method and Faddeev-Jackiw analysis for gauge theories is proved. In particular we trace out, in a stage by stage procedure, the standard classification of first and second class constraints of Dirac's method in the F-J approach. We also find that the Darboux transformation implied in the F-J reduction process can be viewed as a canonical transformation in Dirac approach. Unlike Dirac's method the F-J analysis is a classical reduction procedure, then the quantization can be achieved only in the framework of reduce and then quantize approach with all the know problems that this type of procedures presents. Finally we illustrate the equivalence by means of a particular example. 
  We show that extreme and nonextreme Reissner--Nordstrom black holes in five dimensions can be described by closed fundamental strings with two charges in a magnetic five brane. Due to the five brane background the string oscillator number and tension are rescaled whereas the mass remains fixed. The black hole mass is given by the sum of the five brane and string masses. Its entropy however is given by only that of the string with the tension rescaling taken into account. We also show that the emission of a low energy scalar from the slightly nonextreme string matches the Hawking radiation expected from the black hole. 
  The Ashtekar-Mason-Newman equations are used to construct the hyperk\"ahler metrics on four dimensional manifolds. These equations are closely related to anti self-dual Yang-Mills equations of the infinite dimensional gauge Lie algebras of all volume preserving vector fields. Several examples of hyperk\"ahler metrics are presented through the reductions of anti self-dual connections. For any gauge group anti self-dual connections on hyperk\"ahler manifolds are constructed using the solutions of both Nahm and Laplace equations. 
  The paper is withdrawn because we realized triviality of the main considered example. Less trivial examples are provided in other our papers on the subject. 
  We present a comparative study of the dynamical behaviour of topological systems of recent interest, namely the non-Abelian Chern-Simons Higgs system and the Yang-Mills Chern-Simons Higgs system. By reducing the full field theories to temporal differential systems using the assumption of spatially homogeneous fields , we study the Lyapunov exponents for two types of initial conditions. We also examine in minute detail the behaviour of the Lyapunov spectra as a function of the various coupling parameters in the system. We compare and contrast our results with those for Abelian Higgs, Yang-Mills Higgs and Yang-Mills Chern-Simons systems which have been discussed by other authors recently. The role of the various terms in the Hamiltonians for such systems in determining the order-disorder transitions is emphasized and shown to be counter-intuitive in the Yang-Mills Chern-Simons Higgs systems. 
  It is shown that the three-dimensional isotropic oscillator with coordinates belonging to the two-dimensional half-up cone is dual to the cyon , i.e. the planar particle-vortex bound system provided by fractional statistics. 
  The problem of separation of variables in a dyon--dyon system is discussed. A linear transformation is obtained between fundamental bases of this system. Comparison of the dyon--dyon system with a 4D isotropic oscillator is carried out. 
  The Beckenstein-Hawking black hole entropy in string theory and its extensions, as expressed in terms of charges that correspond to central extensions of the supersymmetry algebra, has more symmetries than U-duality. It is invariant under transformations of the charges, involving a 12th (or 13th) ``dimension''. This is an indication that the secret theory behind string theory has a superalgebra involving Lorentz non-scalar extensions (that are not strictly central), as suggested in S-theory, and which could be hidden in M- or F- theories. It is suggested that the idea of spacetime is broader than usual, and that a larger ``spacetime" is partially present in black holes. 
  The axial gauge is applied to the analysis of Euclidean quantum gravity on manifolds with boundary. A set of boundary conditions which are completely invariant under infinitesimal diffeomorphisms require that spatial components of metric perturbations should vanish at the boundary, jointly with all components of the ghost one-form and of the gauge-averaging functional. If the latter is taken to be of the axial type, all components of metric perturbations obey Dirichlet conditions, and all ghost modes are forced to vanish identically. The one-loop divergence coincides with the contribution resulting from three-dimensional transverse-traceless perturbations. 
  We obtain the elliptic curve and the Seiberg-Witten differential for an $N=2$ superconformal field theory which has an $E_8$ global symmetry at the strong coupling point $\tau=e^{\pi i/3}$. The differential has 120 poles corresponding to half the charged states in the fundamental representation of $E_8$, with the other half living on the other sheet. Using this theory, we flow down to $E_7$, $E_6$ and $D_4$. A new feature is a $\lambda_{SW}$ for these theories based on their adjoint representations. We argue that these theories have different physics than those with $\lambda_{SW}$ built from the fundamental representations. 
  The supersymmetric p-branes of Type II string theory can be interpreted after compactification as extremal black holes with zero entropy and infinite temperature. We show how the p-branes avoid this apparent, catastrophic instability by developing an infinite mass gap. Equivalently, these black holes behave like elementary particles: they are dressed by effective potentials that prevent absorption of impinging particles. In contrast, configurations with 2, 3, and 4 intersecting branes and their nonextremal extensions, behave increasingly like conventional black holes. These results extend and clarify earlier work by Holzhey and Wilczek in the context of four dimensional dilaton gravity. 
  It is shown that the phase space of a dynamical system subject to second class constraints can be extended by ghost variables in such a way that some formal analogies of the $\Omega$-charge and the unitarizing Hamiltonian can be constructed. Then BFV-type path integral representation for the generating functional of Green's functions is written and shown to coincide with the standard one. 
  We consider an algebraic structure of the $XXZ$ model in the gapless regime. We argue that a certain degeneration limit of the elliptic algebra $A_{q,p}(\widehat{sl_2})$ is a relevant object. We give a free boson realization of this limiting algebra and derive an integral formula for the correlation function. The result agrees with the one obtained by solving a system of difference equations. We also discuss the relation of our algebra to the deformed Virasoro algebra and Lukyanov's bosonization of the sine-Gordon theory. 
  In this note the Polyakov equation [Phys. Rev. E {\bf 52} (1995) 6183] for the velocity-difference PDF, with the exciting force correlation function $\kappa (y)\sim1-y^{\alpha}$ is analyzed. Several solvable cases are considered, which are in a good agreement with available numerical results. Then it is shown how the method developed by A. Polyakov can be applied to turbulence with short-scale-correlated forces, a situation considered in models of self-organized criticality. 
  We supplement the discussion of Moore and Reshetikhin and others by finding new semiclassical nonabelian vertex operators for the chiral, antichiral and nonchiral primary fields of WZW theory. These new nonabelian vertex operators are the natural generalization of the familiar abelian vertex operators: They involve only the representation matrices of Lie $g$, the currents of affine $(g \times g)$ and certain chiral and antichiral zero modes, and they reduce to the abelian vertex operators in the limit of abelian algebras. Using the new constructions, we also discuss semiclassical operator product expansions, braid relations and relations to the known form of the semiclassical affine-Sugawara conformal blocks. 
  We study the dynamics of D-particles (D0-branes) in type I' string theory and of the corresponding states in the dual heterotic description. We account for the presence of the two 8-orientifolds (8 dimensional orientifold planes) and sixteen D8-branes by deriving the appropriate quantum mechanical system. We recover the familiar condition of eight D8-branes for each 8-orientifold. We investigate bound states and compute the phase shifts for the scattering of such states and find that they agree with the expectations from the supergravity action. In the type I' regime we study the motion transverse to the 8-orientifold and find an interesting cancellation effect. 
  We show that constraints on the generating functional have direct BRST-extensions in terms of nilpotent operators $\Delta$ that annihilate this generating functional, and which may be of arbitrarily high order. The free energy $F$ in the presence of external sources thus satisfies a ``Master Equation'' which is described in terms of a tower of higher antibrackets. 
  We formulate the general construction for singular vectors in Verma modules of the affine sl(2|1) superalgebra. We then construct sl(2|1) representations out of the fields of the non-critical N=2 string. This allows us to extend naturally to sl(2|1) several crucial properties of the N=2 superconformal algebra, first of all the construction of extremal states (an analogue of different pictures for non-free fermions) and the spectral flow transform (which then affects the Liouville sector). We further evaluate the affine sl(2|1) singular vectors in the realization of sl(2|1) provided by the N=2 string. We establish that, with a notable exception, the respective singular vectors are in a 2:1 correspondence, namely two different sl(2|1) singular vectors evaluate as an N=2 superconformal singular vector (however, those singular vectors that are labelled by a pair of positive integers get these integers transposed under the reduction). We also analyse the `exceptional' cases, which amount to a series of sl(2|1) singular vectors, labelled by r>=1, which do not have an N=2 counterpart, and discuss the mechanism by which the multiplicity of singular vectors becomes equal to two at certain points in the weight spaces of both algebras. 
  We report on recent advances in the understanding of non-perturbative phenomena in the quantum theory of fields and strings. 
  We call a state "vacuum-bounded" if every measurement performed outside a specified interior region gives the same result as in the vacuum. We compute the maximum entropy of a vacuum-bounded state with a given energy for a one-dimensional model, with the aid of numerical calculations on a lattice. The maximum entropy is larger than it would be for rigid wall boundary conditions by an amount $\delta S$ which for large energies is less than or approximately $(1/6) ln (L_in T)$, where L_in is the length of the interior region. Assuming that the state resulting from the evaporation of a black hole is similar to a vacuum-bounded state, and that the similarity between vacuum-bounded and rigid-wall-bounded problems extends from 1 to 3 dimensions, we apply these results to the black hole information paradox. We conclude that large amounts of information cannot be emitted in the final explosion of a black hole. 
  A deformed $q$-calculus is developed on the basis of an algebraic structure involving graded brackets. A number operator and left and right shift operators are constructed for this algebra, and the whole structure is related to the algebra of a $q$-deformed boson. The limit of this algebra when $q$ is a $n$-th root of unity is also studied in detail. By means of a chain rule expansion, the left and right derivatives are identified with the charge $Q$ and covariant derivative $D$ encountered in ordinary/fractional supersymmetry and this leads to new results for these operators. A generalized Berezin integral and fractional superspace measure arise as a natural part of our formalism. When $q$ is a root of unity the algebra is found to have a non-trivial Hopf structure, extending that associated with the anyonic line. One-dimensional ordinary/fractional superspace is identified with the braided line when $q$ is a root of unity, so that one-dimensional ordinary/fractional supersymmetry can be viewed as invariance under translation along this line. In our construction of fractional supersymmetry the $q$-deformed bosons play a role exactly analogous to that of the fermions in the familiar supersymmetric case. 
  It is shown that the recently obtained quantum wave functionals in terms of the CJZ variables for generic 2d dilaton gravity are equivalent to the previously reported exact quantum wave functionals in geometrical variables. A third representation of these exact quantum states is also presented. 
  We study dynamical gaugino condensation in superstring effective theories using the linear multiplet representation for the dilaton superfield. An interesting necessary condition for the dilaton to be stabilized, which was first derived in generic models of static gaugino condensation, is shown to hold for generic models of dynamical gaugino condensation. We also point out that it is stringy non-perturbative effects that stabilize the dilaton and allow dynamical supersymmetry breaking via the field-theoretical non-perturbative effect of gaugino condensation. As a typical example, a toy S-dual model of a dynamical E_8 condensate is constructed and the dilaton is explicitly shown to be stabilized with broken supersymmetry and (fine-tuned) vanishing cosmological constant. 
  The deformation of a topological field theory, namely the pure BF theory, gives the first order formulation of Yang-Mills theory; Feynman rules are given and the standard uv-behaviour is recovered. In this formulation new non local observables can be introduced following the topological theory and giving an explicit realization of `t Hooft algebra. 
  We present a mechanism to construct four-dimensional charged massless Ramond states using the discrete states of a fivebrane Liouville internal conformal field theory. This conformal field theory has background charge, and admits an inner product which allows positive norm states. A connection among supergravity soliton solutions, Liouville conformal field theory, non-critical string theory and their gauge symmetry properties is given. A generalized construction of the SU(2) super Kac-Moody algebra mixing with the N=1 super Virasoro algebra is analyzed. How these Ramond states evade the DKV no-go theorem is explained. 
  We derive a first-order formalism for solving $(2+1)$ gravity on Riemann surfaces, analogously to the recently discovered classical solutions for $N$ moving particles. We choose the York time gauge and the conformal gauge for the spatial metric. We show that Moncrief's equations of motion can be generally solved by the solution f of a $O(2,1$) sigma-model. We build out of f a mapping from a regular coordinate system to a Minkowskian multivalued coordinate system. The polydromy is in correspondence with the branch cuts on the complex plane representing the Riemann surface. The Poincar\'e holonomies, which define the coupling of Riemann surfaces to gravity, describe simply the Minkowskian free motion of the branch points. By solving f we can find the dynamics of the branch points in the physical coordinate system. We check this formalism in some cases, i.e. for the torus and for every Riemann surface with $SO(2,1)$ holonomies. 
  We discuss the problem of determining the spacetime structure. We show that when we are using only topological methods the spacetime can be modelled as an R- or Q-compact space although the R-compact spaces seem to be more appropriate. Demanding the existence of a differential structure substantially narrows the choice of possible models. The determination of the differential structure may be difficult if it is not unique. By using the noncommutative geometry construction of the standard model we show that fundamental interactions determine the spacetime in the class of R-compact spaces. Fermions are essential for the process of determining the spacetime structure. 
  We diagonalise the transfer matrix of boundary ABF models using bosonized vertex operators. We compute the boundary S-matrix and check the scaling limit against known results for perturbed boundary conformal field theories. 
  A method of MC simulations including quantum interference, proposed recently by A.Krzywicki and the present author, is explained. 
  We consider a class of abstract nonlinear evolution equations in supermanifolds (smf's) modelled over Z_2-graded locally convex spaces. We show uniqueness, local existence, smoothness, and an abstract version of causal propagation of the solutions. If an a-priori estimate prevents the solutions from blowing-up then an infinite-dimensional smf of "all" solutions can be constructed.   We apply our results to a class of systems of nonlinear field equations with anticommuting fields which arise in classical field models used for realistic quantum field theoretic models. In particular, we show that under suitable conditions, the smf of smooth Cauchy data with compact support is isomorphic with an smf of corresponding classical solutions of the model. 
  The generalized Killing equations for the configuration space of spinning particles (spinning space) are analysed. Solutions of these equations are expressed in terms of Killing-Yano tensors. In general the constants of motion can be seen as extensions of those from the scalar case or new ones depending on the Grassmann-valued spin variables. 
  It is proposed the generalized action functional for N=1 superparticle in D=3,4,6 and 10 space-time dimensions. The superfield geometric approach equations describing superparticle motion in terms of extrinsic geometry of the worldline superspace are obtained on the base of the generalized action. The off-shell superdiffeomorphism invariance (in the rheonomic sense) of the superparticle generalized action is proved. It was demonstrated that the half of the fermionic and one bosonic (super)fields disappear from the generalized action in the analytical basis. Superparticle interaction with Abelian gauge theory is considered in the framework of this formulation. The geometric approach equations describing superparticle motion in Abelian background are obtained. 
  We find the solitons of massive (1,1)-supersymmetric sigma models with target space the groups $SO(2)$ and $SU(2)$ for a class of scalar potentials and compute their charge, mass and moduli space metric. We also investigate the massive sigma models with target space any semisimple Lie group and show that some of their solitons can be obtained from embedding the $SO(2)$ and $SU(2)$ solitons. 
  It is shown that the critical (or deconfinement) temperature for the Nambu-Goto string connecting the point-like masses (quarks) does not depend on the value of these masses. This temperature turns out to be the same as that in the case of string with fixed ends (infinitely heavy immobile quarks). 
  We investigate 't Hooft-Mandelstam monopoles in QCD in the presence of a single classical instanton configuration. The solution to the Maximal Abelian projection is found to be a circular monopole trajectory with radius $R$ centered on the instanton. At zero loop radius, there is a marginally stable (or flat) direction for loop formation to $O(R^4 logR)$. We argue that loops will form, in the semi-classical limit, due to small perturbations such as the dipole interaction between instanton anti-instanton pairs. As the instanton gas becomes a liquid, the percolation of the monopole loops may therefore provide a semi-classical precursor to the confinement mechanism. 
  We study d=2, N=(2,2) non-linear sigma-models in (2,2) superspace. By analyzing the most general constraints on a superfield, we show that through an appropriate choice of coordinates, there are no other superfields than chiral, twisted chiral and semi-chiral ones. We study the resulting sigma-models and we speculate on the possibility that all (2,2) non-linear sigma-models can be described using these fields. We apply the results to two examples: the SU(2) x U(1) and the SU(2) x SU(2) WZW model. Pending upon the choice of complex structures, the former can be described in terms of either one semi-chiral multiplet or a chiral and a twisted chiral multiplet. The latter is formulated in terms of one semi-chiral and one twisted chiral multiplet. For both cases we obtain the potential explicitely. 
  We review non-linear sigma-models with (2,1) and (2,2) supersymmetry. We focus on off-shell closure of the supersymmetry algebra and give a complete list of (2,2) superfields. We provide evidence to support the conjecture that all N=(2,2) non-linear sigma-models can be described by these fields. This in its turn leads to interesting consequences about the geometry of the target manifolds. One immediate corollary of this conjecture is the existence of a potential for hyper-Kahler manifolds, different from the Kahler potential, which does not only allow for the computation of the metric, but of the three fundamental two-forms as well. Several examples are provided: WZW models on SU(2) x U(1) and SU(2) x SU(2) and four-dimensional special hyper-Kahler manifolds. 
  We implement the inverse scattering method in the case of the $A_n$ affine Toda field theories, by studying the space-time evolution of simple poles in the underlying loop group. We find the known single soliton solutions, as well as additional solutions with non-linear modes of oscillation around the standard solution, by studying the particularly simple case where the residue at the pole is a rank one projection. We show that these solutions with extra modes have the same mass and topological charges as the standard solutions, so we do not shed any light on the missing topological charge problem in these models. We also show that the integrated energy-momentum density can be calculated from the central extension of the loop group. 
  We use heterotic/type-II prepotentials to study quantum/classical black holes with half the $N=2, D=4$ supersymmetries unbroken. We show that, in the case of heterotic string compactifications, the perturbatively corrected entropy formula is given by the tree-level entropy formula with the tree-level coupling constant replaced by the perturbative coupling constant. In the case of type-II compactifications, we display a new entropy/area formula associated with axion-free black-hole solutions, which depends on the electric and magnetic charges as well as on certain topological data of Calabi--Yau three-folds, namely the intersection numbers, the second Chern class and the Euler number of the three-fold. We show that, for both heterotic and type-II theories, there is the possibility to relax the usual requirement of the non-vanishing of some of the charges and still have a finite entropy. 
  We give a classification of 3-family SO(10) and E_6 grand unification in string theory within the framework of conformal field theory and asymmetric orbifolds. We argue that the construction of such models in the heterotic string theory requires certain Z_6 asymmetric orbifolds that include a Z_3 outer-automorphism, the latter yielding a level-3 current algebra for the grand unification gauge group SO(10) or E_6. We then classify all such Z_6 asymmetric orbifolds that result in models with a non-abelian hidden sector. All models classified in this paper have only one adjoint (but no other higher representation) Higgs field in the grand unified gauge group. In addition, all of them are completely anomaly free. There are two types of such 3-family models. The first type consists of the unique SO(10) model with SU(2) X SU(2) X SU(2) as its hidden sector (which is not asymptotically-free at the string scale). This SO(10) model has 4 left-handed and 1 right-handed 16s. The second type is described by a moduli space containing 17 models (distinguished by their massless spectra). All these models have an SU(2) hidden sector, and 5 left-handed and 2 right-handed families in the grand unified gauge group. One of these models is the unique E_6 model with an asymptotically-free SU(2) hidden sector. The others are SO(10) models, 8 of them with an asymptotically free hidden sector at the string scale. 
  We obtain a large class of cosmological solutions in the toroidally-compactified low energy limits of string theories in $D$ dimensions. We consider solutions where a $p$-dimensional subset of the spatial coordinates, parameterising a flat space, a sphere, or an hyperboloid, describes the spatial sections of the physically-observed universe. The equations of motion reduce to Liouville or $SL(N+1,R)$ Toda equations, which are exactly solvable. We study some of the cases in detail, and find that under suitable conditions they can describe four-dimensional expanding universes. We discuss also how the solutions in $D$ dimensions behave upon oxidation back to the $D=10$ string theory or $D=11$ M-theory. 
  These two lectures give a pedagogical introduction to the ``string-inspired'' worldline technique for perturbative calculations in quantum field theory. This includes an overview over the present range of its applications. Several examples are calculated in detail, up to the three-loop level. The emphasis is on photon scattering in quantum electrodynamics. 
  We show that it is possible, in opposition to a previous conjecture, to derive a Nielsen identity for the effective action in the case of the generalized $R_\xi $-gauge, where the gauge function explicitly depends on the gauge parameter $\xi $. Also the Nielsen identity for the effective potential is verified to one-loop in the Abelian Higgs model and the corresponding identity for the physical Higgs mass is derived. 
  The quantum consistency of sigma-models describing the dynamics of extended objects in a curved background requires the cancellation of their world-volume anomalies, which are conformal anomalies for the heterotic string and $SO(1,5)$ Lorentz-anomalies for the heterotic five-brane, and of their ten dimensional target space anomalies. In determining these anomalies in a $D=10$ Lorentz-covariant back-ground gauge we find that for the heterotic string the worldvolume anomalies cancel for 32 heterotic fermions while for the conjectured heterotic five-brane they cancel for only 16 heterotic fermions, this result being in contrast with the string/five-brane duality conjecture. For what concerns the target space anomalies we find that the five-brane eight-form Lorentz-anomaly polynomial differs by a factor of $1/2$ from what is expected on the basis of duality. Possible implications of these results are discussed. 
  We study F-theory on elliptic threefold Calabi-Yau near colliding singularities. We demonstrate that resolutions of those singularities generically correspond to transitions to phases characterized by new tensor multiplets and enhanced gauge symmetry. These are governed by the dynamics of tensionless strings. We also find new transition points which are associated with several small instantons simultaneously shrinking to zero size. 
  We present a class of Kaluza-Klein electrically charged black p-brane solutions of ten-dimensional, type IIA superstring theory. Uplifting to eleven dimensions these solutions are studied in the context of M-theory. They can be interpreted either as a p+1 extended object trapped around the eleventh dimension along which momentum is flowing or as a boost of the following backgrounds: the Schwarzschild black (p+1)-brane or the product of the (10-p)-dimensional Euclidean Schwarzschild manifold with the (p+1)-dimensional Minkowski spacetime. 
  We exhibit the dynamical scattering of multi-solitons in the Skyrme model for configurations with charge two, three and four. First, we construct maximally attractive configurations from a simple profile function and the product ansatz. Then using a sophisticated numerical algorithm, initially well-separated skyrmions in approximately symmetric configurations are shown to scatter through the known minimum energy configurations. These scattering events illustrate a number of similarities to BPS monopole configurations of the same charge. A simple modification of the dynamics to a dissipative regime, allows us to compute the minimal energy skyrmions for baryon numbers one to four to within a few percent. 
  We describe a few properties of the non semi-simple associative algebra H = M_3 + (M_{2|1}(Lambda2))_0, where Lambda2 is the Grassmann algebra with two generators. We show that H is not only a finite dimensional algebra but also a (non co-commutative) Hopf algebra, hence a finite dimensional quantum group. By selecting a system of explicit generators, we show how it is related with the quantum enveloping algebra of SLq(2) when the parameter q is a cubic root of unity. We describe its indecomposable projective representations as well as the irreducible ones. We also comment about the relation between this object and the theory of modular representations of the group SL(2,F3), i.e. the binary tetrahedral group. Finally, we briefly discuss its relation with the Lorentz group and, as already suggested by A.Connes, make a few comments about the possible use of this algebra in a modification of the Standard Model of particle physics (the unitary group of the semi-simple algebra associated with H is U(3) x U(2) x U(1)). 
  Scalar radiation, represented by a massless scalar field in a Robertson-Walker metric, is taken into account. By using a weak non minimum vacuum definition, the radiation temperature as a time dependent function is obtained. When the universe evolution is nearly but non equal to $t^{1/2}$, it is possible to fit the temperature of the microwave background. A particular massive case is compared with the massless one. When the mass of the matter field is next to the Planck one and the time is going to infinite, a similar result to the Hawking radiation of the blackhole is obtained. 
  A generalization of ``Termo Field Dynamics'' to a curved geometry is proposed. In particular a neutral scalar field minimally coupled to gravity is considered as matter content in a Robertson-Walker metric. A non linear amplification in the particle creation is obtained, due to the altogether action of thermal and geometric effects. As a consequence the frequencies in the system look like red shifted with respect to the case where the thermal creation is not taken into account. 
  Taking into account a neutral massive scalar field minimally coupled to gravity, in a Robertson-Walker metric, it is shown that when the final state is connected with the initial one by means of a Bogoliubov transformation, which does not include the single-mode rotation operator, the mean value of created particles is conserved. When the rotation operator is considered, it is still possible to use the approach of single-mode squeezed operators and get the entropy as the logarithm of the created particles. 
  Quantum tunneling between degenerate ground states through the central barrier of a potential is extended to excited states with the instanton method. This extension is achieved with the help of an LSZ reduction technique as in field theory and may be of importance in the study of macroscopic quantum phenomena in magnetic systems. 
  We review here some of the checks of string-string duality between N=2, D=4 Heterotic and Type II models. The heterotic low energy field theory is reproduced on the type II side. It is also shown to be a generalization in the string context of the rigid $N=2$ Yang-Mills theory of Seiberg and Witten which is exactly known. The non perturbative information of this rigid theory is then recovered on the type II side. This talk is based on a work done in collaboration with I. Antoniadis. 
  We analyze terms subleading to Rutherford in the $S$-matrix between black hole and probes of successively high energies. We show that by an appropriate choice of the probe one can read off the quantum state of the black hole from the S-matrix, staying asymptotically far from the BH all the time. We interpret the scattering experiment as scattering off classical stringy backgrounds which explicitly depend on the internal quantum numbers of the black hole. 
  We investigate the effect of a constant threshold correction to a general non-extreme, static, spherically symmetric, electrically charged black hole solution of the dilatonic Einstein-Maxwell Lagrangian, with an arbitrary coupling $\beta$ between the electromagnetic tensor and the dilaton field. For a small $\beta$, an exact analytical solution is obtained. For an arbitrary $\beta$, a close form solution, up to first order in the threshold correction, of the metric and the dilaton are presented. In the extremal limit, the close form solution is reduced to an exact analytical form. 
  Infinite-dimensional Grassmannian manifold contains moduli spaces of Riemann surfaces of all genera. This well known fact leads to a conjecture that non-perturbative string theory can be formulated in terms of Grassmannian. We present new facts supporting this hypothesis. In particular, it is shown that Grassmannians can be considered as generalized moduli spaces; this statement permits us to define corresponding "string amplitudes" (at least formally). One can conjecture, that it is possible to explain the relation between non-perturbative and perturbative string theory by means of localization theorems for equivariant cohomology; this conjecture is based on the characterization of moduli spaces, relevant to string theory, as sets consisting of points with large stabilizers in certain groups acting on Grassmannian. We describe an involution on the Grassmannian that could be related to S-duality in string theory. 
  We consider a recently proposed approach to bosonization in which the original fermionic partition function is expressed as a product of a $G/G$-coset model and a bosonic piece that contains the dynamics. In particular we show how the method works when topological backgrounds are taken into account. We also discuss the application of this technique to the case of massive fermions. 
  We first introduce and discuss the formalism of $SU_q(N)$-bosons and fermions and consider the simplest Hamiltonian involving these operators. We then calculate the grand partition function for these models and study the high temperature (low density) case of the corresponding gases for $N=2$. We show that quantum group gases exhibit anyonic behavior in $D=2$ and $D=3$ spatial dimensions. In particular, for a $SU_q(2)$ boson gas at $D=2$ the parameter $q$ interpolates within a wider range of attractive and repulsive systems than the anyon statistical parameter. 
  A zero-brane is used to probe non-threshold BPS bound states of (p, p+2,p+4)-branes. At long distances the stringy calculation agrees with the supergravity calculations. The supergravity description is given, using the interpretation of the D=8 dyonic membrane as the bound state of a two-brane inside a four-brane. We investigate the short distance structure of these bound states, compute the phase shift of the scattered zero-brane and find the bound states characteristic size. It is found that there should be a supersymmetric solution of type IIa supergravity, describing a bound state of a zero-brane and two orthogonal two-brane, all inside a four-brane , with an additional unbound zero-brane. We comment on the relationship between p-branes and (p-2)-branes. 
  The local SUSY symmetry of the loop dynamics of QCD is found. The remarkable thing is, there is no einbein-gravitino on this theory, which makes it a 1D topological supergravity, or locally SUSY quantum mechanics. Using this symmetry, we derive the large $N_c$ loop equation in momentum superloop space. Introducing as before the position operator $\X{\mu}$ we argue that the superloop equation is equivalent to invariance of correlation functions of products of these operators with respect to certain quadrilinear transformation. The applications to meson and glueball sectors as well as the chiral symmetry breaking are discussed. The 1D field theory with Quark propagating around the loop in superspace is constructed. 
  We study the large N limit in the presence of magnetic monopoles in the Yang-Mills/Higgs model in three dimensions. The physics in the limit depends strongly on the distribution of eigenvalues of the Higgs field in the vacuum, and we propose a particular, nondegenerate configuration. It minimizes the free energy at the moment of symmetry breaking. Given this, the magnetic monopoles show a wide hierarchy of masses, and some are vanishing as 1/N. The dilute gas picture, then, provides an interesting structure in the large N limit. 
  We investigate Feynman diagrams which are calculable in terms of generalized one-loop functions, and explore how the presence or absence of transcendentals in their counterterms reflects the entanglement of link diagram constructed from them and explains unexpected relations between them. 
  The theory described by the sum of the Einstein-Hilbert action and the action of conformal scalar field possesses the duality symmetry which includes some special conformal transformation of the metric, and also inversion of scalar field and of the gravitational constant. In the present paper the conformal duality is generalized for arbitrary space-time dimension $n \neq 2$ and for the general sigma-model type conformal scalar theory. We also consider to apply the conformal duality for the investigation of quantum gravity in the strong curvature regime. The trace of the first coefficient of the Schwinger-DeWitt expansion is derived and it's dependence on the gauge fixing condition is considered. After that we discuss the way to extract the gauge-fixing independent result and also it's possible physical applications. 
  Analytical harmonic superfields are the basic variables of a standard harmonic formalism of SYM^2_4-theory. We consider superfield actions for alternative formulations of this theory using the unconstrained harmonic prepotentials. The corresponding equations of motion are equivalent to the component field equations of SYM^2_4-theory. The analyticity condition appears only on-shell as the zero-curvature equation in the alternative formulations. 
  In this paper we explicitly prove the invariance of the time-dependent string gravity Lagrangian with up to four derivatives under the global $O(d,d)$ symmetry. 
  These lectures provide an introduction to lattice methods for nonperturbative studies of quantum field theories, with an emphasis on Quantum Chromodynamics.   Lecture 1 (Ch. 2): gauge field basics   lecture 2 (Ch. 3): Abelian duality with a lattice regulator             (Ch. 4): simple lattice intuition   lecture 3 (Ch. 5): standard methods (and results) for hadron spectroscopy   lecture 4 (Ch. 6): bare actions and physics   lecture 5 (Ch. 7): two case studies, mass of the glueball and $\alpha_s(M_Z)$ 
  Recent work on the quantization of Maxwell theory has used a non-covariant class of gauge-averaging functionals which include explicitly the effects of the extrinsic-curvature tensor of the boundary, or covariant gauges which, unlike the Lorentz case, are invariant under conformal rescalings of the background four-metric. This paper studies in detail the admissibility of such gauges at the classical level. It is proved that Euclidean Green functions of a second- or fourth-order operator exist which ensure the fulfillment of such gauges at the classical level, i.e. on a portion of flat Euclidean four-space bounded by three-dimensional surfaces. The admissibility of the axial and Coulomb gauges is also proved. 
  Superstring field theory was recently used to derive a covariant action for a self-dual five-form field strength. This action is shown to be a ten-dimensional version of the McClain-Wu-Yu action. By coupling to D-branes, it can be generalized in the presence of sources. In four dimensions, this gives a local Maxwell action with electric and magnetic sources. 
  An approach based on considerations of the non-classical energy momentum tensor outside the event horizon of a black hole provides additional physical insight into the nature of discrete quantum hair on black holes and its effect on black hole temperature. Our analysis both extends previous work based on the Euclidean action techniques, and corrects an omission in that work. We also raise several issues related to the effects of instantons on black hole thermodynamics and the relation between these effects and results in two dimensional quantum field theory. 
  We consider a gauge symmetry in a quantum Hilbert space. The symmetry leads to that of the heat-kernel and of the anomaly formulae which were previously obtained by the authors. This greatly simplifies and clarifies the structure of the formulae. We explicitly obtain the anomaly formulae in two and four dimensions, which ``unify'' all kinds of anomaly. The symmetry corresponds to that of the counterterm formulae in the background field method. As an example, the non-abelian anomaly is considered. 
  In this paper we solve the following problems: (i) find two differential operators P and Q satisfying [P,Q]=P, where P flows according to the KP hierarchy \partial P/\partial t_n = [(P^{n/p})_+,P], with p := \ord P\ge 2; (ii) find a matrix integral representation for the associated $\t au$-function. First we construct an infinite dimensional space {\cal W}=\Span_\BC \{\psi_0(z),\psi_1(z),... \} of functions of z\in\BC invariant under the action of two operators, multiplication by z^p and A_c:= z \partial/\partial z - z + c. This requirement is satisfied, for arbitrary p, if \psi_0 is a certain function generalizing the classical H\"ankel function (for p=2); our representation of the generalized H\"ankel function as a double Laplace transform of a simple function, which was unknown even for the p=2 case, enables us to represent the \tau-function associated with the KP time evolution of the space \cal W as a ``double matrix Laplace transform'' in two different ways. One representation involves an integration over the space of matrices whose spectrum belongs to a wedge-shaped contour \gamma := \gamma^+ + \gamma^- \subset\BC defined by \gamma^\pm=\BR_+\E^{\pm\pi\I/p}. The new integrals above relate to the matrix Laplace transforms, in contrast with the matrix Fourier transforms, which generalize the Kontsevich integrals and solve the operator equation [P,Q]=1. 
  We generalise all the known supersymmetric composite M-branes to the corresponding black configurations. Thermodynamical formulae is written by using the simple rules to construct these black branes. 
  We give necessary criteria for N=1 supersymmetric theories to be in a smoothly confining phase without chiral symmetry breaking and with a dynamically generated superpotential. Using our general arguments we find all such confining SU and Sp theories with a single gauge group and no tree level superpotential. 
  D-brane technology and strong/weak coupling duality supplement traditional orbifold techniques by making certain background geometries more accessible. In this spirit, we consider some of the geometric properties of the type IIB theory on R^6 \times M where M is an `Asymptotically Locally Euclidean (ALE)' gravitational instanton. Given the self-duality of the theory, we can extract the geometry (both singular and resolved) seen by the weakly coupled IIB string by studying the physics of a D1-brane probe. The construction is both amusing and instructive, as the physics of the probe completely captures the mathematics of the construction of ALE instantons via `HyperKahler Quotients', as presented by Kronheimer. This relation has been noted by Douglas and Moore for the A-series. We extend the explicit construction to the case of the D- and E-series -- uncovering a quite beautiful structure -- and highlight how all of the elements of the mathematical construction find their counterparts in the physics of the type IIB D-string. We discuss the explicit ALE metrics which may be obtained using these techniques, and comment on the role duality plays in relating gauged linear sigma models to conformal field theories. 
  We calculate explicitly in terms of complete elliptic integrals the metric on the moduli space of tetrahedrally-symmetric, charge four, SU(2) monopoles. Using this we verify that in the asymptotic regime the metric of Gibbons and Manton is exact up to exponentially suppressed corrections. 
  The weak coupling spectrum of BPS saturated states of pure $N=2$ supersymmetric SU$(n)$ gauge theory is investigated. The method uses known results on the dyon spectrum of the analogous theory with $N=4$ supersymmetry, along with the action on these states of the semi-classical monodromy transformations. For dyons whose magnetic charge is not a simple root of the Lie algebra, it is found that the weak coupling region is divided into a series of domains, for which the dyons have different electric charge, separated by walls on which the dyons decay. The proposed spectrum is shown to be consistent with the exact solution of the theory at strong coupling in the sense that the states at weak coupling can account for the singularities at strong coupling. 
  The phase structure of the scalar field theory with arbitrary powers of the gradient operator and a local non-analytic potential is investigated by the help of the RG in Euclidean space. The RG equation for the generating function of the derivative part of the action is derived. Infinitely many non-trivial fixed points of the RG transformations are found. The corresponding effective actions are unbounded from below and do probably not exhibit any particle content. Therefore they do not provide physically sensible theories. 
  Our aim is to give a self-contained review of recent advances in the analytic description of the deconfinement transition and determination of the deconfinement temperature in lattice QCD at large N. We also include some new results, as for instance in the comparison of the analytic results with Montecarlo simulations. We first review the general set-up of finite temperature lattice gauge theories, using asymmetric lattices, and develop a consistent perturbative expansion in the coupling $\beta_s$ of the space-like plaquettes. We study in detail the effective models for the Polyakov loop obtained, in the zeroth order approximation in $\beta_s$, both from the Wilson action (symmetric lattice) and from the heat kernel action (completely asymmetric lattice). The distinctive feature of the heat kernel model is its relation with two-dimensional QCD on a cylinder; the Wilson model, on the other hand, can be exactly reduced to a twisted one-plaquette model via a procedure of the Eguchi-Kawai type. In the weak coupling regime both models can be related to exactly solvable Kazakov-Migdal matrix models. The instability of the weak coupling solution is due in both cases to a condensation of instantons; in the heat kernel case, it is directly related to the Douglas-Kazakov transition of QCD2. A detailed analysis of these results provides rather accurate predictions of the deconfinement temperature. In spite of the zeroth order approximation they are in good agreement with the Montecarlo simulations in 2+1 dimensions, while in 3+1 dimensions they only agree with the Montecarlo results away from the continuum limit. 
  We do a critical review of the Faraday-Maxwell concept of classical field and of its quantization process. With the hindsight knowledge of the essentially quantum character of the interactions, we use a naive classical model of field, based on exchange of classical massless particles, for a comparative and qualitative analysis of the physical content of the Coulomb's and Gauss's laws. It enlightens the physical meaning of a field singularity and of a static field. One can understand the problems on quantizing a classical field but not the hope of quantizing the gravitational field right from General Relativity. 
  We discuss topologically massive QED --- the Abelian gauge theory in which (2+1)-dimensional QED with a Chern-Simons term is minimally coupled to a spinor field. We quantize the theory in covariant gauges, and construct a class of unitary transformations that enable us to embed the theory in a Fock space of states that implement Gauss's law. We show that when electron (and positron) creation and annihilation operators represent gauge-invariant charged particles that are surrounded by the electric and magnetic fields required by Gauss's law, the unitarity of the theory is manifest, and charged particles interact with photons and with each other through nonlocal potentials. These potentials include a Hopf-like interaction, and a planar analog of the Coulomb interaction. The gauge-invariant charged particle excitations that implement Gauss's law obey the identical anticommutation rules as do the original gauge-dependent ones. Rotational phases, commonly identified as planar `spin', are arbitrary, however. 
  Exact loop-variables formulation of pure gauge lattice QCD_3 is derived from the Wilson version of the model. The observation is made that the resulting model is two-dimensional. This significant feature is shown to be a unique property of the gauge field. The model is defined on the infinite genus surface which covers regularly the original three-dimensional lattice. Similar transformation applied to the principal chiral field model in two and three dimensions for comparison with QCD. 
  We give the full supersymmetric and kappa-symmetric action for the Dirichlet three-brane, including its coupling to background superfields of ten-dimensional type IIB supergravity. 
  The light-front (LF) quantization of the bosonized Schwinger model is discussed in the "continuum formulation". The proposal, successfully used earlier for describing the spontaneous symmetry breaking (SSB) on the LF, of separating first the scalar field into the dynamical condensate and the fluctuation fields before employing the "standard" Dirac method works here as well. The condensate variable, however, is now shown to be a q-number operator in contrast to the case of SSB where it was shown to be a c-number or a background field. The "condensate or Theta-vacua" emerge straightforwardly together with their continuum normalization which avoids the violation of the cluster decomposition property in the theory. Some topics on the "front form" theory are summarized in the Appendices and attention is drawn to the fact that "the theory quantized, say, at equal $x^{+}$ seems already to carry information on equal $x^{-}$ commutators as well". 
  We compute the $1/N$ correction to the location of the previously found first-order phase transition in the Gross-Neveu model at a chemical potential $\mu = \mu_c = {1 \over \sqrt{2}} m$, where $m$ is the fermion mass. We employ an expression for the free energy $f(\mu)$ given by the thermodynamic Bethe ansatz under the approximation that the fundamental fermions dominate the ground state, and combine it with the effective potential evaluated at zero chemical potential. Our result is $\mu_c = {m \over \sqrt{2}} [1 - {0.47 \over N}]$. 
  Type IIB strings are compactified on a Calabi-Yau three-fold. When Calabi-Yau-valued expectation values are given to the NS-NS and RR three-form field strengths, the dilaton hypermultiplet becomes both electrically and magnetically charged. The resultant classical potential is calculated, and minima are found. At singular points in the moduli space, such as Argyres-Douglas points, supersymmetric minima are found. A formula for the classical potential in $N=2$ supergravity is given which holds in the presence of both electric and magnetic charges. 
  A method of constructing a canonical gauge invariant quantum formulation for a non-gauge classical theory depending on a set of parameters is advanced and then applied to the theory of closed bosonic string interacting with massive background fields. Choosing an ordering prescription and developing a suitable regularization technique we calculate quantum guage algebra up to linear order in background fields. Requirement of closure for the algebra leads to equations of motion for massive background fields which appear to be consistent with the structure of string spectrum. 
  We review here the microcanonical and canonical ensembles constructed on an underlying generalized quantum dynamics and the algebraic properties of the conserved quantities. We discuss the structure imposed on the microcanonical entropy by the equilibrium conditions. 
  We present two methods for studying fibrations of Calabi-Yau manifolds embedded in toric varieties described by single weight systems. We analyse 184,026 such spaces and identify among them 124,701 which are K3 fibrations. As some of the weights give rise to two or three distinct types of fibrations, the total number we find is 167,406. With our methods one can also study elliptic fibrations of 3-folds and K3 surfaces. We also calculate the Hodge numbers of the 3-folds obtaining more than three times as many as were previously known. 
  Geometric Killing spinors which exist on AdS_{p+2} X S^{d-p-2} sometimes may be identified with supersymmetric Killing spinors. This explains the enhancement of unbroken supersymmetry near the p-brane horizon in d dimensions. The corresponding p-brane interpolates between two maximally supersymmetric vacua, at infinity and at the horizon. New case is studied here: p=0, d=5. The details of supersymmetric version of the very special geometry are presented. We find the area-entropy formula of the supersymmetric 5d black holes via the volume of S^3 which depends on charges and intersection matrix. 
  We clarify the mass dependence of the effective prepotential in N=2 supersymmetric SU(N_c) gauge theories with an arbitrary number N_f<2N_c of flavors. The resulting differential equation for the prepotential extends the equations obtained previously for SU(2) and for zero masses. It can be viewed as an exact renormalization group equation for the prepotential, with the beta function given by a modular form. We derive an explicit formula for this modular form when N_f=0, and verify the equation to 2-instanton order in the weak-coupling regime for arbitrary N_f and N_c. 
  We study classical and quantum aspects of D=4, N=2 BPS black holes for T_2 compactification of D=6, N=1 heterotic string vacua. We extend dynamical relaxation phenomena of moduli fields to background consisting of a BPS soliton or a black hole and provide a simpler but more general derivation of the Ferrara-Kallosh's extremized black hole mass and entropy. We study quantum effects to the BPS black hole mass spectra and to their dynamical relaxation. We show that, despite non-renormalizability of string effective supergravity, quantum effect modifies BPS mass spectra only through coupling constant and moduli field renormalizations. Based on target-space duality, we establish a perturbative non-renormalization theorem and obatin exact BPS black hole mass and entropy in terms of renormalized string loop-counting parameter and renormalized moduli fields. We show that similar conclusion holds, in the large T_2 limit, for leading non- perturbative correction. We finally discuss implications to type-I and type-IIA Calabi -Yau black holes. 
  We investigate the consistency of the background-field formalism when applying various regularizations and renormalization schemes. By an example of a two-dimensional $\sigma$ model it is demonstrated that the background-field method gives incorrect results when the regularization (and/or renormalization) is noninvariant. In particular, it is found that the cut-off regularization and the differential renormalization belong to this class and are incompatible with the background-field method in theories with nonlinear symmetries. 
  In this talk, some aspects of duality symmetries are presented. 
  In this work we raise the question whether nonsymmetric gravity and string theory are related. We start making the observation, that the gravitational field $ g_{\mu\nu}$ and the nonsymmetric gauge field $ A_{\mu\nu}$ arising in the low energy limit in the string theory are exactly the same two basic fields used in four dimensions in nonsymmetric gravity. We argue, that this connection between nonsymmetric gravity and string theory at the level of the gauge fields $ g_{\mu\nu}$ and $ A_{\mu\nu}$ is not, however, reflected at the level of the corresponding associated actions. In an effort to find a connection between such an actions we discover a new gravitational action, which suggests an alternative version of the bosonic string in which the target and the world-volume metrics are unified. 
  We extend a previously developed technique for computing spin-spin critical correlators in the 2d Ising model, to the case of multiple correlations. This enables us to derive Kadanoff-Ceva's formula in a simple and elegant way. We also exploit a doubling procedure in order to evaluate the critical exponent of the polarization operator in the Baxter model. Thus we provide a rigorous proof of the relation between different exponents, in the path-integral framework. 
  We study aspects of Calabi-Yau four-folds as compactification manifolds of F-theory, using mirror symmetry of toric hypersurfaces. Correlation functions of the topological field theory are determined directly in terms of a natural ring structure of divisors and the period integrals, and subsequently used to extract invariants of moduli spaces of rational curves subject to certain conditions. We then turn to the discussion of physical properties of the space-time theories, for a number of examples which are dual to $E_8\times E_8$ heterotic N=1 theories. Non-critical strings of various kinds, with low tension for special values of the moduli, lead to interesting physical effects. We give a complete classification of those divisors in toric manifolds that contribute to the non-perturbative four-dimensional superpotential; the physical singularities associated to it are related to the apppearance of tensionless strings. In some cases non-perturbative effects generate an everywhere non-zero quantum tension leading to a combination of a conventional field theory with light strings hiding at a low energy scale related to supersymmetry breaking. 
  We study the cosmological constant problem in a three-dimensional N=2 supergravity theory with gauge group SU[2]_{global}xU[1]_{local}. The model we consider is known to admit string-like configurations, the so-called semi-local cosmic strings. We show that the stability of these solutions is provided by supersymmetry through the existence of a lower bound for the energy, even though the manifold of the Higgs vacuum does not contain non-contractible loops. Charged Killing spinors do exist over configurations that saturate the Bogomolnyi bound, as a consequence of an Aharonov-Bohm-like effect. Nevertheless, there are no physical fermionic zero modes on these backgrounds. The exact vanishing of the cosmological constant does not imply, then, Bose-Fermi degeneracy. This provides a non-trivial example of the recent claim made by Witten on the vanishing of the cosmological constant in three dimensions without unphysical degeneracies. 
  An extra term generally appears in the q-deformed $su(2)$ algebra for the deformation parameter $q = \exp{ 2 \pi i\theta}$, if one combines the Biedenharn-Macfarlane construction of q-deformed $su(2)$, which is a generalization of Schwinger's construction of conventional $su(2)$, with the representation of the q-deformed oscillator algebra which is manifestly free of negative norm. This extra term introduced by the requirement of positive norm is analogous to the Schwinger term in current algebra. Implications of this extra term on the Bloch electron problem analyzed by Wiegmann and Zabrodin are briefly discussed. 
  A split dimensional regularization, which was introduced for the Coulomb gauge by Leibbrandt and Williams, is used to regularize the spurious singularities of Yang-Mills theory in the temporal gauge. Typical one-loop split dimensionally regularized temporal gauge integrals, and hence the renormalization structure of the theory are shown to be the same as those calculated with some nonprincipal-value prescriptions. 
  We construct 4-dimensional cosmological FRW--models by compactifying a black 5-brane solution of type IIB supergravity, which carries both magnetic NS-NS-charge and RR-charge. The influence of nontrivial RR-fields on the dynamics of the cosmological models is investigated. 
  In this thesis steps are taken in the direction of formulating non-critical strings in the framework of the $G/G$ approach. A major part of the thesis is concerned with conformal field theory based on affine $SL(2)$ current algebra, in particular for admissible representations which are relevant in the $SL(2)/SL(2)$ description of coupling conformal minimal matter to 2D gravity. By virtue of fractional calculus, free field realizations are made applicable of producing integral representations of chiral blocks, even in the case of admissible representations where ghost fields raised to fractional powers are inherent. The famous work on minimal models by Dotsenko and Fateev is then generalised, working out monodromy invariant 4-point Greens functions and the operator algebra coefficients. Furthermore, the fusion rules are re-derived within the context of free field realizations by introducing the notion of over-screening. For higher groups an explicit Wakimoto free field realization is presented of affine current algebras based on simple Lie groups. Furthermore, a study is undertaken of primary fields in the framework of extra $x$ variables, and screening currents of both kinds are discussed. Several new and definite results are presented. A quantum group structure relying on both kinds of screening currents is discussed along the lines of Gomez and Sierra. 
  We study the correlation functions of logarithmic conformal field theories. First, assuming conformal invariance, we explicitly calculate two-- and three-- point functions. This calculation is done for the general case of more than one logarithmic field in a block, and more than one set of logarithmic fields. Then we show that one can regard the logarithmic field as a formal derivative of the ordinary field with respect to its conformal weight. This enables one to calculate any $n$-- point function containing the logarithmic field in terms of ordinary $n$--point functions. At last, we calculate the operator product expansion (OPE) coefficients of a logarithmic conformal field theory, and show that these can be obtained from the corresponding coefficients of ordinary conformal theory by a simple derivation. 
  We present the technique of derivation of a theory to obtain an $(n+1)f$-degrees-of-freedom theory from an $f$-degrees-of-freedom theory and show that one can calculate all of the quantities of the derived theory from those of the original one. Specifically, we show that one can use this technique to construct, from an integrable system, other integrable systems with more degrees of freedom. 
  The properties of composite black holes in the background of electric or magnetic flux tubes are analyzed, both when the black holes remain in static equilibrium and when they accelerate under a net external force. To this effect, we present a number of exact solutions (generalizing the Melvin, C and Ernst solutions) describing these configurations in a theory that admits composite black holes with an arbitrary number of constituents. The compositeness property is argued to be independent of supersymmetry. Even if, in general, the shape of the horizon is distorted by the fields, the dependence of the extreme black hole area on the charges is shown to remain unchanged by either the external fields or the acceleration. We also discuss pair creation of composite black holes. In particular, we extend a previous analysis of pair creation of massless holes. Finally, we give the generalization of our solutions to include non-extreme black holes. 
  It is shown that many modes of the gravitational field exist only inside the horizon of an extreme black hole in string theory. At least in certain cases, the number of such modes is sufficient to account for the Bekenstein-Hawking entropy. These modes are associated with sources which carry Ramond-Ramond charge, and so may be viewed as the strong coupling limit of D-branes. Although these sources naturally live at the singularity, they are well defined and generate modes which extend out to the horizon. This suggests that the information in an extreme black hole is not localized near the singularity or the horizon, but extends between them. 
  We calculate the emission and absorption rates of fixed scalars by the near-extremal five-dimensional black holes that have recently been modeled using intersecting D-branes. We find agreement between the semi-classical and D-brane computations. At low energies the fixed scalar absorption cross-section is smaller than for ordinary scalars and depends on other properties of the black hole than just the horizon area. In the D-brane description, fixed scalar absorption is suppressed because these scalars must split into at least four, rather than two, open strings running along the D-brane. Consequently, this comparison provides a more sensitive test of the effective string picture of the D-brane bound state than does the cross-section for ordinary scalars. In particular, it allows us to read off the value of the effective string tension. That value is precisely what is needed to reproduce the near-extremal 5-brane entropy. 
  The axial anomaly and fermion condensate in the light cone Schwinger model are studied following path integral methods. This formalism allows for a simple and direct calculation for these and other vacuum dependent phenomena. 
  This is a summary of the analysis of the one-loop effective action in Z_2-orbifold compactifications of type-I theory presented in references [1,2]. We show how, for non-abelian group factors, the threshold effects are ultraviolet finite though given entirely by a six-dimensional field theory expression. We also discuss implications for the equivalence between Type-I and Heterotic four dimensional N=2 superstring theories. 
  Real-time perturbation theory is formulated for complex scalar fields away from thermal equilibrium in such a way that dissipative effects arising from the absorptive parts of loop diagrams are approximately resummed into the unperturbed propagators. Low order calculations of physical quantities then involve quasiparticle occupation numbers which evolve with the changing state of the field system, in contrast to standard perturbation theory, where these occupation numbers are frozen at their initial values. The evolution equation of the occupation numbers can be cast approximately in the form of a Boltzmann equation. Particular attention is given to the effects of a non-zero chemical potential, and it is found that the thermal masses and decay widths of quasiparticle modes are different for particles and antiparticles. 
  We find the comlete solution to ten-dimensional supergravity coupled to a three-form field strength, given the ``standard ansatz" for the fields, and show that in addition to the well-known elementary and solitonic (heterotic) string solutions, one of the possibilities is an (unstable) elementary type I string solution. 
  We discuss $w_\infty$ and $sl_q(2)$ symmetries in Chern-Simons theory and Landau problem on a torus. It is shown that when the coefficient of the Chern-Simons term, or when the total flux passing through the torus is a rational number, there exist in general two $w_\infty$ and two $sl_q(2)$ algebras, instead of one set each discussed in the literature. The general wavefunctions for the Landau problem with rational total flux is also presented. 
  We discuss $w_\infty$ and $sl_q(2)$ symmetries in multiple Chern-Simons theory on a torus. It is shown that these algebraic structures arise from the dynamics of the non-integrable phases of the Chern-Simons fields. The generators of these algebras are constructed from the Wilson line operators corresponding to these phases. The vacuum states form the basis of cyclic representation of $sl_q (2)$. 
  We discuss some perturbative techniques suitable for the gauge-invariant treatment of the scalar and tensor inhomogeneities of an anisotropic and homogeneous background geometry whose spatial section naturally decomposes into the direct product of two maximally symmetric Eucledian manifolds, describing a general situation of dimensional decoupling in which $d$ external dimensions evolve (in conformal time) with scale factor $a(\eta)$ and $n$ internal dimensions evolve with scale factor $b(\eta)$. We analyze the growing mode problem which typically arises in contracting backgrounds and we focus our attention on the situation where the amplitude of the fluctuations not only depends on the external space-time but also on the internal spatial coordinates. In order to illustrate the possible relevance of this analysis we compute the gravity waves spectrum produced in some highly simplified model of cosmological evolution and we find that the spectral amplitude, whose magnitude can be constrained by the usual bounds applied to the stochastic gravity waves backgrounds, depends on the curvature scale at which the compactification occurs and also on the typical frequency of the internal excitations. 
  The optical manifold method to compute the one-loop effective action in a static space-time is extended from the massless scalar field to the Maxwell field in any Feynman-like covariant gauge. The method is applied to the case of the Rindler space obtaining the same results as the point-splitting procedure. The result is free from Kabat's surface terms which instead affect the $\zeta$-function or heat-kernel approaches working directly in the static manifold containing conical singularities. The relation between the optical method and the direct $\zeta$-function approach on the Euclidean Rindler manifold is discussed both in the scalar and the photon cases. Problems with the thermodynamic self-consistency of the results obtained from the stress tensor in the case of the Rindler space are pointed out. 
  Dynamics of a free point particle on a multi world-line is presented and shown to reduce to that of a bosonic string theory at the appropriate limit. Other higher dimensional extended objects are argued to appear at other regions of the space of configurations of the theory. 
  The 727-dimensional root space associated with the level-2 root $\bLambda_1$ of the hyperbolic Kac--Moody algebra $E_{10}$ is determined using a recently developed string theoretic approach to hyperbolic algebras. The explicit form of the basis reveals a complicated structure with transversal as well as longitudinal string states present. 
  Einstein's $E = mc^{2}$ unifies the momentum-energy relations for massive and massless particles. According to Wigner, the internal space-time symmetries of massive and massless particles are isomorphic to $O(3)$ and $E(2)$ respectively. According to Inonu and Wigner, $O(3)$ can be contracted to $E(2)$ in the large-radius limit. It is noted that the $O(3)$-like little group for massive particles can be contracted to the $E(2)$-like little group for massless particles in the limit of large momentum and/or small mass. It is thus shown that transverse rotational degrees of freedom for massive particles become contracted to gauge degrees of freedom for massless particles. 
  The geometries of spaces having as groups the real orthogonal groups and some of their contractions are described from a common point of view. Their central extensions and Casimirs are explicitly given. An approach to the trigonometry of their spaces is also advanced. 
  The heat kernel $M_{xy} = <x\mid exp [ 1/\sqrt{g} \partial_\mu g^{\mu\nu} \sqrt{g} \partial_\nu ]t \mid y>$ is of central importance when studying the propagation of a scalar particle in curved space. It is quite convenient to analyze this quantity in terms of classical variables by use of the quantum mechanical path integral; regrettably it is not entirely clear how this path integral can be mathematically well defined in curved space. An alternate approach to studying the heat kernel in terms of classical variables was introduced by Onofri. This technique is shown to be applicable to problems in curved space; an unambiguous expression for $M_{xy}$ is obtained which involves functional derivatives of a classical quantity. We illustrate how this can be used by computing $M_{xx}$ to lowest order in the curvature scalar R. 
  A generalized action for strings which is a sum of the Nambu-Goto and the extrinsic curvature (the energy integral of the surface) terms, is used to couple strings to gravity. It is shown that the conical singularity has deficit angle that has contributions from both the above terms. It is found that the effect of extrinsic curvature is to oppose that of the N-G action for the temperature of the black-hole and to modify the entropy-area relation. 
  We investigate the structure of an infinite-dimensional symmetry of the four-dimensional K\"ahler WZW model, which is a possible extension of the two-dimensional WZW model. We consider the SL(2,R) group and, using the Gauss decomposition method, we derive a current algebra identified with a two-toroidal Lie algebra, a generalization of the affine Kac-Moody algebra. We also give an expression of the energy-momentum tensor in terms of currents and extra terms. 
  We extend the notion of space shifts introduced by L. D. Faddeev and A. Yu. Volkov (Phys. Lett. B 315 (1993)) for certain quantum light cone lattice equations of sine-Gordon type at root of unity. As a result we obtain a compatibility equation for the roots of central elements within the algebra of observables (also called current algebra). The equation which is obtained by exponentiating these roots is exactly the evolution equation for the "classical background" as described in V. Bazhanov, A. Bobenko, N. Reshetikhin (Comm. Math. Phys. 175 (1996)).   As an application for the introduced constructions, a one to one correspondence between a special case of the quantum light cone lattice equations of sine-Gordon type and free massive fermions on a lattice as constructed in Destri and de Vega (Nucl. Phys. B 290 (1987)) is derived. 
  The pressure of a system in thermal equilibrium is expressed as a mass integral over a sum of thermal propagators. This allows a Dyson resummation and is used to demonstrate that potential infrared divergences are rendered harmless. 
  We give a model for composite quarks and leptons based on the semisimple gauge group SU(4), with the preons in the 10 representation; this choice of gauge gluon and preon multiplets is motivated by the possibility of embedding them in an N=6 supergravity multiplet. Hypercolor singlets are forbidden in the fermionic sector of this theory; we propose that SU(4) symmetry spontaneously breaks to $SU(3) \times U(1)$, with the binding of triality nonzero preons and gluons into composites, and with the formation of a color singlet condensate that breaks the initial $Z_{12}$ vacuum symmetry to $Z_{6}$. The spin 1/2 fermionic composites have the triality structure of a quark lepton family, and the initial $Z_{12}$ symmetry implies that there are six massless families, which mix to give three distinct families, two massless with massive partners and one with both states massive, at the scale of the condensate. The spin 1 triality zero composites of the color triplet SU(4) gluons, when coupled to the condensate and with the color singlet representation of the 10 acting as a doorway state, lead to weak interactions of the fermionic composites through an exact SU(2) gauge algebra. The initial $Z_{12}$ symmetry implies that this SU(2) gauge algebra structure is doubled, which in turn requires that the corresponding independent gauge bosons must couple to chiral components of the composite fermions. Since the U(1) couples to the 10 representation as $B-L$, an effective $SU(2)_L \times SU(2)_R \times U(1)_{B-L}$ electroweak theory arises at the condensate scale, with all composites having the correct electric charge structure. A renormalization group analysis shows that the conversion by binding of one 10 of SU(4) to 12 triplets of SU(3) can give a very large, calculable hierarchy ratio between 
  We investigate the usefulness of the "string-inspired technique" for gauge theory calculations in a constant external field background. Our approach is based on Strassler's worldline path integral approach to the Bern-Kosower formalism, and on the construction of worldline (super--) Green's functions incorporating external fields as well as internal propagators. The worldline path integral representation of the gluon loop is reexamined in detail. We calculate the two-loop effective actions induced for a constant external field by a scalar and spinor loop, and the corresponding one-loop effective action in the gluon loop case. 
  An analogue of the classical Frobenius-Schur indicator is introduced in order to distinguish between real and pseudo-real self-conjugate primary fields, and an explicit expression for this quantity is derived from the trace of the braiding operator. 
  Using methods of high performance computing, we have found indications that knotlike structures appear as stable finite energy solitons in a realistic 3+1 dimensional model. We have explicitly simulated the unknot and trefoil configurations, and our results suggest that all torus knots appear as solitons. Our observations open new theoretical possibilities in scenarios where stringlike structures appear, including physics of fundamental interactions and early universe cosmology. In nematic liquid crystals and 3He superfluids such knotted solitons might actually be observed. 
  A reduction of the Dirac-Maxwell equations in the case of static cylindrical symmetry is performed. The behaviour of the resulting system of o.d.e.'s is examined analytically and numerical solutions presented. There are two classes of solutions.  The first type of solution is a Dirac field surrounding a charged "wire". The Dirac field is highly localised, concentrated in cylindrical shells about the wire. A comparison with the usual linearized theory demonstrates that this localization is entirely due to the non-linearities in the equations which result from the inclusion of the "self-field".  The second class of solutions have the electrostatic potential finite along the axis of symmetry but unbounded at large distances from the axis. 
  We consider splitting type phase transitions between Calabi-Yau fourfolds. These transitions generalize previously known types of conifold transitions between threefolds. Similar to conifold configurations the singular varieties mediating the transitions between fourfolds connect moduli spaces of different dimensions, describing ground states in M- and F-theory with different numbers of massless modes as well as different numbers of cycles to wrap various p-branes around. The web of Calabi-Yau fourfolds obtained in this way contains the class of all complete intersection manifolds embedded in products of ordinary projective spaces, but extends also to weighted configurations. It follows from this that for some of the fourfold transitions vacua with vanishing superpotential are connected to ground states with nonzero superpotential. 
  We give an elementary derivation of the de Rham cohomology of SO(n) in terms of supersymmetric quantum mechanics. Our analysis is based on Witten's Morse theory. We show reflection symmetries of the theory are useful to select true vacuums. The number of the selected vacuums will agree with the de Rham cohomology of SO(n). 
  These lecture notes want to illustrate the close connection between statistical mechanics and field theory not only on the formal level, i.e. that many concepts of one area can easily be taken over to the other one, but also on the level of actual calculations. To this purpose, the last section will demonstrate that a special statistical system, the binary fluid system, can be described by field theory in its critical behaviour. 
  Aspects of Poisson-Lie T-duality are reviewed in more algebraic way than in our, rather geometric, previous papers. As a new result, a moment map is constructed for the Poisson-Lie symmetry of the system consisting of open strings propagating in a Poisson-Lie group manifold. 
  We study the combinatorics of solitons in $D<2$ (or $c<1$) string theory. The weights in the summation over multi-solitons are shown to be automatically determined if we further require that the partition function with soliton background be a $\tau$ function of the KP hierarchy, in addition to the $W_{1+\infty}$ constraint. 
  We consider a nonlinear field equation which can be derived from a binomial lattice as a continuous limit. This equation, containing a perturbative friction-like term and a free parameter $\gamma$, reproduces the Toda case (in absence of the friction-like term) and other equations of physical interest, by choosing particular values of $\gamma$. We apply the symmetry and the approximate symmetry approach, and the prolongation technique. Our main purpose is to check the limits of validity of different analytical methods in the study of nonlinear field equations. We show that the equation under investigation with the friction-like term is characterized by a finite-dimensional Lie algebra admitting a realization in terms of boson annhilation and creation operators. In absence of the friction-like term, the equation is linearized and connected with equations of the Bessel type. Examples of exact solutions are displayed, and the algebraic structure of the equation is discussed. 
  We investigate the behavior of generic, matter-coupled, 2D dilaton gravity theories under dilaton-dependent Weyl rescalings of the metric. We show that physical observables associated with 2D black holes, such as the mass, the temperature and the flux of Hawking radiation are invariant under the action of both Weyl transformations and dilaton reparametrizations. The field theoretical and geometrical meaning of these invariances is discussed. 
  The canonical quantization of the WZNW model provides a complete set of exchange relations in the enlarged chiral state spaces that include the Gauss components of the monodromy matrices. Regarded as new dynamical variables, the elements of the latter cannot be identified -- they satisfy different exchange relations. Accordingly, the two dimensional theory expressed in terms of the left and right movers' fields does not automatically respect monodromy invariance. Continuing our recent analysis of the problem by gauge theory methods we conclude that physical states (on which the two dimensional field acts as a single valued operator) are invariant under the (permuted) coproduct of the left and right $U_q(sl(n))$. They satisfy additional constraints fully described for n=2. 
  We investigate the effective potential for Abelian Maxwell--Chern--Simons systems. The calculations follow an alternate approach, recently proposed as a gauge invariant formulation of the effective potential, constructed in terms of a gauge invariant order parameter. We compare the results with another investigation, obtained within a standard route of calculating the effective potential. 
  We construct modular invariants on the moduli space of quantum vacua of N=2 SYM with gauge group SU(2). We also introduce a nonchiral function K which is expressed in terms of the Seiberg-Witten and Poincare' metrics. It turns out that K has all the expected properties of the next to leading term in the Wilsonian effective action whose modular properties are considered in the framework of the dimensional regularization. 
  We present several new examples of nontrivial 4d N=1 superconformal field theories. Some of these theories exhibit exotic global symmetries, including non-simply laced groups (such as $F_4$). They are obtained by studying threebrane probes in F-theory compactifications on elliptic Calabi-Yau threefolds. The geometry of the compactification encodes in a simple way the behavior of the gauge coupling and the K\"ahler potential on the Coulomb branch of these theories. 
  Matrix model approach to multicolor induced QCD based on the quenched momentum prescription is presented. It is shown that this model exhibits the reduction of spatial degrees of freedom: the partition function is determined by the solution of one dimensional quantum mechanical problem while the D-dimensional scalar field correlators coinside with the same type correlators in the two-dimensional induced QCD. 
  The KZB equations for conformal blocks of the WZNW theory are written on the moduli space of holomorphic principal bundles on the surface. They become the multi-time Schroedinger equation for the nonstationary Hitchin system. From the known form of the equations we learn about the covariance of quantization with respect to changes of the coordinate frame. 
  A new method is applied to solve the Baxter equation for three coupled, noncompact spins. Due to the equivalence with the system of three reggeized gluons, the intercept of the odderon trajectory is predicted for the first time, as the analytic function of the two relevant parameters. 
  The quantum mechanical transition amplitudes are calculated perturbatively on the basis of the stochastic quantization method of Parisi and Wu. It is shown that the stochastic scheme reproduces the ordinary result for the amplitude and systematically incorporates higher-order effects, even at the lowest order. 
  Taking the complex nature of quantum mechanics which we observe today as a low energy effect of a broken quaternionic theory we explore the possibility that dark matter arises as a consequence of this underlying quaternionic structure to our universe. We introduce a low energy, effective, Lagrangian which incorporates the remnants of a local quaternionic algebra, investigate the stellar production of the resultant exotic bosons and explore the possible low energy consequences of our remnant extended Hilbert space. 
  Instanton calculations are demonstrated from a viewpoint of twisted topological field theory. Various properties become manifest such that perturbative corrections are terminated at one-loop, and norm cancellations occur between bosonic and fermionic excitations in any instanton background. We can easily observe that for a suitable choice of Green functions the infinite dimensional path integration reduces to a finite dimensional integration over a supersymmetric instanton moduli space. 
  The parameter space of the Feigin-Fuks representations of the N=4 SU(2)$_k$ superconformal algebras is studied from the viewpoint of the specral flow. The $\eta$ phase of the spectral flow is nicely incorporated through twisted fermions and the spectral flow resulting from the inner automorphism of the N=4 superconformal algebras is explicitly shown to be operating as identiy relations among the generators. Conditions for the unitary representations are also investigated in our Feigin-Fuks parameter space. 
  Yang-Mills theory is studied in a variant of 't Hooft's maximal Abelian gauge. In this gauge magnetic monopoles arise in the Abelian magnetic field. We show, however, that the full (non-Abelian) magnetic field does not possess any monopoles, but rather strings of magnetic fluxes. We argue that these strings are the relevant infrared degrees of freedom. The properties of the magnetic strings which arise from a dilute instanton gas are investigated for the gauge group SU(2). 
  Two-step mechanisms in the $N\bar{N}$ annihilation and their role in the OZI rule violating reactions are discussed. In particular the two meson rescattering mechanism for $\pi\phi$ channel including all off-shell effects is typically two orders of magnitude bigger than the OZI tree level expectation and explains the observed ratio $\phi \pi/\omega \pi$ in the annihilation at rest. The rates for the final states including photons, $\gamma\omega$ and $\gamma\phi$, can be explained in the vector dominance model. The observed rate for $p\bar{p}\to\gamma\omega$ is suppressed due to destructive interference between the intermediate $\rho$ and $\omega$ states while the interference in $p\bar{p}\to\gamma\phi$ is required to be constructive leading to a large ratio $\gamma\phi/\gamma\omega$. 
  We study the production and the equilibration of a non-Abelian $q\bar{q}$ plasma in an external chromoelectric field, by solving the Boltzmann equation with the non-Abelian features explicitly incorporated. We consider the gauge group $SU(2)$ and show that the colour degree of freedom has a major and dominant role in the dynamics of the system. It is seen that the assumption of the so called Abelian dominance is not justified. Finally, it is also shown that many of the features of microscopic studies of the system appear naturally in our studies as well. 
  Further formulas are presented involving quantum mechanics, thermodynamics, and integrable systems. Modifications of dispersionless theory are developed. 
  At $c=3$, two of the three integrable quantum $N=2$ supersymmetric Korteweg-de Vries equations become identical (SKdV$_1$ and SKdV$_4$). Quite remarkably, all their conservation laws can be written in closed form, which provides thus a simple constructive integrability proof. 
  We compute the dependence on the classical action "gauge" parameters of the beta functions of the standard topological sigma model in flat space. We thus show that their value is a "gauge" artifact indeed. We also show that previously computed values of these beta functions can be continuously connected to one another by smoothly varying those gauge parameters. 
  We discuss the unitarity relation of the Aharonov-Bohm scattering amplitude with the hope that it distinguishes between the differing treatments which employ different incident waves. We find that the original Aharonov-Bohm scattering amplitude satisfies the unitarity relation under the regularization prescription whose theoretical foundation does not appear to be understood. On the other hand, the amplitude obtained by Ruijsenaars who uses plane wave as incident wave also satisfies the unitarity relation but in an unusual way. 
  The supersymmetric descent equations in superspace are discussed by means of the introduction of two operators which allow to decompose the supersymmetric covariant derivatives as BRS commutators. 
  All Lie symmetries of the Burgers equation driven by an external random force are found. Besides the generalized Galilean transformations, this equation is also invariant under the time reparametrizations. It is shown that the Gaussian distribution of a pumping force is not invariant under the symmetries and breaks them down leading to the nontrivial vacuum (instanton). Integration over the volume of the symmetry groups provides the description of fluctuations around the instanton and leads to an exactly solvable quantum mechanical problem. 
  A class of the $D=4$ gravity models describing a coupled system of $n$ Abelian vector fields and the symmetric $n \times n$ matrix generalizations of the dilaton and Kalb-Ramond fields is considered. It is shown that the Pecci-Quinn axion matrix can be entered and the resulting equations of motion possess the $Sp(2n, R)$ symmetry in four dimensions. The stationary case is studied. It is established that the theory allows a $\sigma$-model representation with a target space which is invariant under the $Sp[2(n+1), R]$ group of isometry transformations. The chiral matrix of the coset $Sp[2(n+1), R]/U(n+1)$ is constructed. A K\"ahler formalism based on the use of the Ernst $(n+1) \times (n+1)$ complex symmetric matrix is developed. The stationary axisymmetric case is considered. The Belinsky-Zakharov chiral matrix depending on the original field variables is obtained. The Kramer-Neugebauer transformation, which algebraically maps the original variables into the target space ones, is presented. 
  The equation of state of the universality class of the 3D Ising model is determined numerically in the critical domain from quantum field theory and renormalization group techniques. The starting point is the five loop perturbative expansion of the effective potential (or free energy) in the framework of renormalized $\phi^4_3$ field theory. The 3D perturbative expansion is summed, using a Borel transformation and a mapping based on large order behaviour results. It is known that the equation of state has parametric representations which incorporate in a simple way its scaling and regularity properties. We show that such a representation can be used to accurately determine it from the knowledge of the few first coefficients of the expansion for small magnetization. Revised values of amplitude ratios are deduced. Finally we compare the 3D values with the results obtained by the same method from the $\epsilon=4-d$ expansion. 
  We study the free field realization of the two-dimensional osp(1|2) current algebra. We consider the case in which the level of the affine osp(1|2) symmetry is a positive integer. Using the Coulomb gas technique we obtain integral representations for the conformal blocks of the model. In particular, from the behaviour of the four-point function, we extract the structure constants for the product of two arbitrary primary operators of the theory. From this result we derive the fusion rules of the osp(1|2) conformal field theory and we explore the connections between the osp(1|2) affine symmetry and the N=1 superconformal field theories. 
  We construct a model of a chiral transition using the well known large N transition in two dimensional U(N) lattice gauge theory. Restricting the model to a single plaquette, we introduce Grassmann variables on the corners of the plaquette with the natural phase factors of staggered fermions and couple them to the U(N) link variables. The classical theory has a continuous chiral symmetry which is broken at strong couplings, but is restored for weak couplings in the $N \to \infty$ limit. 
  Superstring field theory was recently used to derive a four-dimensional Maxwell action with manifest duality. This action is related to the McClain-Wu-Yu Hamiltonian and can be locally coupled to electric and magnetic sources.   In this letter, the manifestly dual Maxwell action is supersymmetrized using N=1 and N=2 superspace. The N=2 version may be useful for studying Seiberg-Witten duality. 
  Flux-string models can be used to study the deconfining phase transition. In this note, we study the models proposed by Patel. We also study the large N_c limits of Patel's model. To discuss the validity of the mean field theory results, the one-loop Coleman-Weinberg effective potential is calculated for N_c=3. We argue that the quantum corrections vanish at large N_c when the energy of the so-called baryonic vertices scale with N_c. 
  We examine some properties of supermultiplet consisting of the U(1)_{J} current, extended supercurrents, energy-momentum tensor and the central charge in N=2 supersymmetric Yang-Mills theory. The superconformal improvement requires adding another supermultiplet beginning with the U(1)_{R} current. We determine the anomalous (quantum mechanical) supersymmetry transformation associated with the central charge and the energy-momentum tensor to one-loop order. 
  The analytic properties of the zeta-function for a Laplace operator on a generalised cone are studied in some detail using the Cheeger's approach and explicit expressions are given. In the compact case, the zeta-function of the Laplace operator turns out to be singular at the origin. As a result, strictly speaking, the zeta-function regularisation does not ``regularise'' and a further subtraction is required for the related one-loop effective potential. 
  The difficulties with the measurability of classical space-time distances are considered. We outline the framework of quantum deformations of D=4 space-time symmetries with dimensionfull deformation parameter, and present some recent results. 
  We give a simple derivation of a new formal expression for the number of particles produced from a conformal scalar field vacuum due to the creation of a gauge cosmic string. We find that the number of particles released in string formation may be substantially lower than what previous estimates have indicated. Our derivation also indicates that there always exists at least one critical angle deficit less than $2\pi$, at which the particle production attains a maximum value. At the end we argue that additional (quantum) effects will occur in string formation. In particular, a new mechanism to produce small scale structure on strings is proposed. 
  In recent times quantum corrections for N=2 black holes in 4 dimensions have been addressed in the framework of double extreme black hole solutions, which are characterised by constant scalar fields. In this paper we generalize these solutions to non-constant scalar fields. This enables us to discuss quantum corrections for massless black holes and for configurations that are classically singular. We also discuss the relation to the 5-dimensional magnetic string solution. 
  We discuss the intimate connection between the chaotic dynamics of a classical field theory and the instability of the one-loop effective action of the associated quantum field theory. Using the example of massless scalar electrodynamics, we show how the radiatively induced spontaneous symmetry breaking stabilizes the vacuum state against chaos, and we speculate that monopole condensation can have the same effect in non-Abelian gauge theories. 
  On the world-volume of an $M$-theory five-brane propagates a two-form with self-dual field strength. As this field is non-Lagrangian, there is no obvious framework for determining its partition function. An analogous problem exists in Type IIB superstring theory for the self-dual five-form. The resolution of these problems and definition of the partition function is explained. A more complete analysis of perturbative anomaly cancellation for $M$-theory five-branes is also presented, uncovering some surprising details. 
  In the presence of membranes, M-theory becomes in the low energy limit 11 dimensional supergravity action coupled to a supermembrane action. The fields of the first action are the same fields which couple to the membrane. It is shown that the axionic moduli of the membrane obtained by wrapping the three form potential about three-cycles of a Calabi-Yau manifold can take nonzero integer values. This novel property allows M-theory to have smooth transition from the Kahler cone of a geometrical phase to a Kahler cone of another geometrical phase. Nongeometrical phases which define the boundary of the extended Kahler cone of the geometrical phases have discrete spectrum, and are continuously connected to the geometric phases. Using this new property, we relate the M-theory model dependent axion to the type IIA model dependent axion and show that a potential develops for the type IIA axion in the strong coupling regime which does not seem to be generated by instantons. Evidence is presented, using these moduli, which supports the Strominger conjecture on the winding p-branes. 
  We propose a construction of five-branes which fill both light-cone dimensions in Banks, Fischler, Shenker and Susskind's matrix model of M theory. We argue that they have the correct long-range fields and spectrum of excitations. We prove Dirac charge quantization with the membrane by showing that the five-brane induces a Berry phase in the membrane world-volume theory, with a familiar magnetic monopole form. 
  We compute the gravitational coupling $F_1$ for IIA string theory on $K3 \times T^2$ and use string-string duality to deduce the corresponding term for heterotic string on $T^6$. The latter is an infinite sum of gravitational instanton effects which we associate with the effects of Euclidean fivebranes wrapped on $T^6$. These fivebranes are the neutral fivebranes or zero size instantons of heterotic string theory. 
  A general framework for studying a large class of cosmological solutions of the low-energy limit of type II string theory and of M-theory, with non-trivial Ramond form fields excited, is presented. The framework is applicable to spacetimes decomposable into a set of flat or, more generally, maximally symmetric spatial subspaces, with multiple non-trivial form fields spanning one or more of the subspaces. It is shown that the corresponding low-energy equations of motion are equivalent to those describing a particle moving in a moduli space consisting of the scale factors of the subspaces together with the dilaton. The choice of which form fields are excited controls the potential term in the particle equations. Two classes of exact solutions are given, those corresponding to exciting only a single form and those with multiple forms excited which correspond to Toda theories. Although typically these solutions begin or end in a curvature singularity, there is a subclass with positive spatial curvature which appears to be singularity free. Elements of this class are directly related to certain black p-brane solutions. 
  A surface functional theory for p-dimensional extended objects, the p-branes, was proposed in previous papers. The field equations for toroidal p-branes was exactly solved in $d=p+2$ dimensions, yielding equally spaced mass-squared spectrum with massless states. In this paper, we obtain the asymptotic distribution of mass spectrum in the point-particle limit of the theory with sphere-like membranes ($p=2$) in $d=4$ dimensions. Similarity between this spectrum and that obtained in the Dirac's membrane model of electron is discussed. 
  We present a gauge independent Lagrangian method of abstracting the reduced space of a solvable model with Gribov-like ambiguity, recently proposed by Friedberg, Lee, Pang and Ren. The reduced space is found to agree with the explicit solutions obtained by these authors. Complications related to gauge fixing are analysed. The Gribov ambiguity manifests by a nonuniqueness in the canonical transformations mapping the hamiltonian in the afflicted gauge with that obtained gauge independently. The operator ordering problem in this gauge is investigated and a prescription is suggested so that the results coincide with the usual hamiltonian formalism using the Schr\"odinger representation. Finally, a Dirac analysis of the model is elaborated. In this treatment it is shown how the existence of a nontrivial canonical set in the ambiguity-ridden gauge yields the connection with the previous hamiltonian formalism. 
  The $N_f$-flavour Schwinger Model on a finite space $0\leq x^1\leq L$ and subject to bag-type boundary-conditions at $x^1=0$ und $x^1=L$ is solved at finite temperature $T=1/\beta$. The boundary conditions depend on a real parameter $\theta$ and break the axial flavour symmetry. We argue that this approach is more appropriate to study the broken phases than introducing small quark masses, since all calculations can be performed analytically. In the imaginary time formalism we determine the thermal correlators for the fermion-fields and the determinant of the Dirac-operator in arbitrary background gauge-fields. We show that the boundary conditions induce a CP-odd $\theta$-term in the effective action. The chiral condensate, and in particular its T- and L- dependence, is calculated for $N_f$ fermions. It is seen to depend on the order in which the two lengths $\beta=1/T$ and $L$ are sent to infinity. 
  I review the appearance of classical integrable systems as an effective tool for the description of non-perturbative exact results in quantum string and gauge theories. Various aspects of this relation: spectral curves, action-angle variables, Whitham deformations and associativity equations are considered separately demonstrating hidden parallels between topological 2d string theories and naively non-topological 4d theories. The proofs are supplemented by explicit illustrative examples. 
  The null string equations of motion and constraints in the Schwarzschild spacetime are given. The solutions are those of the null geodesics of General Relativity appended by a null string constraint in which the "constants of motion" depend on the world-sheet spatial coordinate. Because of the extended nature of a string, the physical interpretation of the solutions is completely different from the point particle case. In particular, a null string is generally not propagating in a plane through the origin, although each of its individual points is. Some special solutions are obtained and their physical interpretation is given. Especially, the solution for a null string with a constant radial coordinate $r$ moving vertically from the south pole to the north pole around the photon sphere, is presented. A general discussion of classical null/tensile strings as compared to massless/massive particles is given. For instance, tensile circular solutions with a constant radial coordinate $r$ do not exist at all. The results are discussed in relation to the previous literature on the subject. 
  It is proposed that asymptotically nonfree gauge theories are consistently interpreted as theories of composite gauge bosons. It is argued that when hidden local symmetry is introduced, masslessness and coupling universality of dynamically generated gauge boson are ensured. To illustrate these ideas we take a four dimensional Grassmannian sigma model as an example and show that the model should be regarded as a cut-off theory and there is a critical coupling at which the hidden local symmetry is restored. Propagator and vertex functions of the gauge field are calculated explicitly and existence of the massless pole is shown. The beta function determined from the $ Z $ factor of the dynamically generated gauge boson coincides with that of an asymptotic nonfree elementary gauge theory. Using these theoretical machinery we construct a model in which asymptotic free and nonfree gauge bosons coexist and their running couplings are related by the reciprocally proportional relation. 
  In this letter we present a topological observation related to the CP problem in non-Abelian gauge theories, based on significant differences between space-times and their Wick-rotation. We point out that models containing stationary black holes the problem of CP breaking does not appear. 
  It has been argued that a certain large $N$ matrix model may provide a non-perturbative definition of $M$-theory. This model is the truncation to $0+1$ dimensions of ten-dimensional supersymmetric Yang-Mills theory. It is crucial to this identification that terms with four derivatives in the effective action for the quantum mechanics should not be renormalized. We offer a perturbative proof of this result. 
  We study the nonrelativistic limit of the quantum theory of a real scalar field with quartic self-interaction. The two body scattering amplitude is written in such way as to separate the contributions of high and low energy intermediary states. From this result and the two loop computation of the self energy correction, we determine an effective nonrelativistic action. 
  Weyl-Wigner-Moyal formalism is used to describe the large-$N$ limit of reduced SU$(N)$ quenching gauge theory. Moyal deformation of Schild-Eguchi action is obtained. 
  We formulate world-volume actions that describe the dynamics of Dirichlet p-branes in a flat 10d background. The fields in these theories consist of the 10d superspace coordinates and an abelian world-volume gauge field. The global symmetries are given by the N=2A or N=2B super-Poincare group, according to whether p is even or odd. The local symmetries in the (p + 1)-dimensional world volume are general coordinate invariance and a fermionic kappa symmetry. 
  Fractional strings in the spectrum of states of open strings attached to a multiply wound D-brane is explained. We first describe the fractional string states in the low-energy effective theory where the topology of multiple winding is encoded in the gauge holonomy. The holonomy induces twisted boundary conditions responsible for the fractional moding of these states. We also describe fractional strings in world sheet formulation and compute simple scattering amplitudes for Hawking emission/absorption. Generalization to fractional DN-strings in a 1-brane 5-brane bound state is described. When a 1-brane and a 5-brane wraps $Q_1$ and $Q_5$ times respectively around a circle, the momentum of DN-strings is quantized in units of $2 \pi/L Q_1 Q_5$. These fractional states appear naturally in the perturbative spectrum of the theory. 
  We consider compactifications of the N=1, d=6, E_8 theory on tori to five, four, and three dimensions and learn about some properties of this theory. As a by-product we derive the SL(2,\IZ) duality of the N=2, d=4, SU(2) theory with N_f=4. Using this theory on a D-brane probe we shed new light on the singularities of F-theory compactifications to eight dimensions. As another application we consider compactifications of F-theory, M-theory and the IIA string on (singular) Calabi-Yau spaces where our theory appears in spacetime. Our viewpoint leads to a new perspective on the nature of the singularities in the moduli space and their spacetime interpretations. In particular, we have a universal understanding of how the singularities in the classical moduli space of Calabi--Yau spaces are modified by worldsheet instantons to singularities in the moduli space of the corresponding conformal field theories. 
  We show that a class of 3+1 dimensional Friedmann-Robertson-Walker cosmologies can be embedded within a variety of solutions of string theory. In some realizations the apparent singularities associated with the big bang or big crunch are resolved at non-singular horizons of higher-dimensional quasi-black hole solutions (with compactified real time); in others plausibly they are resolved at D-brane bound states having no conventional space-time interpretation. 
  In this letter the algebraic renormalization method, which is independent of any kind of regularization scheme, is presented for the parity-preserving QED_3 coupled to scalar matter in the symmetric regime, where the scalar assumes vanishing vacuum expectation value, $<\phi>=0$. The model shows to be stable under radiative corrections and anomaly free. 
  In this letter the algebraic renormalization method, which is independent of any kind of regularization scheme, is presented for the parity-preserving QED_3 coupled to scalar matter in the broken regime, where the scalar assumes a finite vacuum expectation value, $<\vf> = v$. The model shows to be stable under radiative corrections and anomaly free. 
  Contents: Generalities, Chiral supermultiplets, Super Yang-Mills theory, Superspace Feynman graphs, Renormalization, Supercurrent, Finite theories. 
  Euler sums (also called Zagier sums) occur within the context of knot theory and quantum field theory. There are various conjectures related to these sums whose incompletion is a sign that both the mathematics and physics communities do not yet completely understand the field. Here, we assemble results for Euler/Zagier sums (also known as multidimensional zeta/harmonic sums) of arbitrary depth, including sign alternations. Many of our results were obtained empirically and are apparently new. By carefully compiling and examining a huge data base of high precision numerical evaluations, we can claim with some confidence that certain classes of results are exhaustive. While many proofs are lacking, we have sketched derivations of all results that have so far been proved. 
  A system of SU(N)-matrix difference equations is solved by means of a nested version of a generalized Bethe Ansatz, also called "off shell" Bethe Ansatz. The highest weight property of the solutions is proved. (Part I of a series of articles on the generalized nested Bethe Ansatz and difference equations.) 
  A system of U(N)-matrix difference equations is solved by means of a nested version of a generalized Bethe Ansatz. The highest weight property of the solutions is proved and some examples of solutions are calculated explicitly. (Part II of a series of articles on the generalized nested Bethe Ansatz and difference equations.) 
  We use non-perturbative U-duality symmetries of type II strings to construct new vacuum solutions. In some ways this generalizes the F-theory vacuum constructions. We find the possibilities of new vacuum constructions are very limited. Among them we construct new theories with N=2 supersymmetry in 3-dimensions and (1, 1) supersymmetry in 2-dimensions. 
  The equations of motion of the super five-brane in D=11 dimensions are derived using the formalism of superembeddings. The equations describe highly nonlinear self-interactions of a tensor multiplet in the six dimensional worlsurface, and they have manifest worldsurface local supersymmetry. The geometry of the target space corresponds to D=11 supergravity. 
  Generalizing the concept of primary fields, we find a new representation of the Virasoro algebra, which we call it a pseudo-conformal representation. In special cases, this representation reduces to ordinary- or logarithmic-conformal field theory. There are, however, other cases in which the Green functions differ from those of ordinary- or logarithmic-conformal field theories. This representation is parametrized by two matrices. We classify these two matrices, and calculate some of the correlators for a simple example. 
  Higher order renormalons beyond the chain of one-loop bubbles are discussed. A perturbation method for the infrared renormalon residue is found. The large order behavior of the current-current correlation function due to the first infrared renormalon is determined in both QED and QCD to the first three orders. 
  We formulate and solve a class of two-dimensional matrix gauge models describing ensembles of non-folding surfaces covering an oriented, discretized, two-dimensional manifold. We interpret the models as string theories characterized by a set of coupling constants associated to worldsheet ramification points of various orders. Our approach is closely related to, but simpler than, the string theory describing two-dimensional Yang-Mills theory. Using recently developed character expansion methods we exactly solve the models for target space lattices of arbitrary internal connectivity and topology. 
  I review, with some pedagogy, two different approaches to the computation of BPS spectra in N=2 supersymmetric QCD with gauge group SU(2). The first one is semiclassical and has been widely used in the literature. The second one makes use of constraints coming from the non perturbative, global structure of the Coulomb branch of these theories. The second method allows for a description of discontinuities in the BPS spectra at strong coupling, and should lead to accurate test of duality conjectures in N=2 theories. 
  We discuss the role of the CPT transformation in first quantized string theories, both on the world-sheet and in the space-time. We explicitly show that the space-time CPT theorem holds for all first-quantized (perturbative) string theories in a Minkowski background of even dimension $D>2$. 
  We consider the group theoretical properties of R--R scalars of string theories in the low-energy supergravity limit and relate them to the solvable Lie subalgebra $\IG_s\subset U$ of the U--duality algebra that generates the scalar manifold of the theory: $\exp[\IG_s]= U/H$. Peccei-Quinn symmetries are naturally related with the maximal abelian ideal ${\cal A} \subset \IG_s $ of the solvable Lie algebra. The solvable algebras of maximal rank occurring in maximal supergravities in diverse dimensions are described in some detail. A particular example of a solvable Lie algebra is a rank one, $2(h_{2,1}+2)$--dimensional algebra displayed by the classical quaternionic spaces that are obtained via c-map from the special K\"ahlerian moduli spaces of Calabi-Yau threefolds. 
  The vev's of the magnetic order-disorder operators in QCD are found in an explicit calculation using the first order formulation of Yang-Mills theory. 
  Seiberg and Witten's proposed solution of N=2 SQCD with N_c=2 and N_F=4 is known to conflict with instanton calculations in three distinct ways. Here we show how to resolve all three discrepancies, simply by reparametrizing the elliptic curve in terms of quantities $\tau^0_{eff}$ and $\tilde{u}$ rather than $\tau$ and $u = < Tr A^2 > $. SL(2,Z) invariance of the curve is preserved. However, there is now an infinite ambiguity in the relation between $\tau^0_{eff}$ and $\tau$ and between $\tilde{u}$ and $u$, corresponding to an infinite number of unknown coefficients in the instanton expansion. Thus the reinterpreted curve (unlike the cases N_F<4) no longer determines the quantum modulus u as a function of the classical VEV a. 
  We test the conjectured Type I-Heterotic Duality in four dimensions by analyzing a given class of higher derivative F-terms of the form $F_gW^{2g}$, with W the N=2 gravitational superfield. We study a particular dual pair of theories, the O(2,2) heterotic model and a type I model based on the K3 $Z_2$ orbifold theory constructed by Gimon and Polchinski, further compactified on a torus. The $F_g$ couplings appear at 1-loop on both theories; because of the weak-weak nature of this duality in four dimensions, it is meaningful to compare the heterotic $F_g$'s with the corresponding type I couplings perturbatively. We compute the $F_g$'s in type I, showing that they receive contributions only from N=2 BPS states and that in the appropriate limit they coincide with the heterotic couplings, in agreement with the given duality. 
  A method is presented by which a hidden N=2 superconformal symmetry can be exhibited in a string theory or indeed in a topological conformal field theory. More precisely, we present strong evidence, based on calculations with string theories, in favour of the conjecture that any topological conformal field theory can be obtained by twisting an N=2 superconformal field theory. (Talk given at the Workshop on Gauge Theories, Applied Supersymmetry and Quantum Gravity held at Imperial College, London, 5-10 July 1996.) 
  We evaluate the exact $QED_{2+1}$ effective action for fermions in the presence of a family of static but spatially inhomogeneous magnetic field profiles. This exact result yields an all-orders derivative expansion of the effective action, and indicates that the derivative expansion is an asymptotic, rather than a convergent, expansion. 
  We note that the accidental symmetries which are present in some examples of duality imply the existence of continuously infinite sets of theories with the same infrared behavior. These sets interpolate between theories of different flavors and colors; the change in color and flavor is compensated by interactions (often non-perturbative) induced by operators in the superpotential. As an example we study the behavior of SU(2) gauge theories with 2\nf doublets; these are dual to SU(\nf-2) gauge theories whose ultraviolet flavor symmetry is SU(\nf)xSU(\nf)xU(1) but whose flavor symmetry is SU(2\nf) in the infrared. The infrared SU(2\nf) flavor symmetry is implemented in the ultraviolet as a non-trivial transformation on the Lagrangian and matter content of the magnetic theory, involving (generally non-renormalizable) baryon operators and non-perturbative dynamics. We discuss various implications of this fact, including possible new chiral fixed points and interesting examples of dangerously irrelevant operators. 
  Covariant forms are given to a gauge theory of massive tensor field. This is accomplished by introducing another auxiliary field of scalar type to the system composed of a symmetric tensor field and an auxiliary field of vector type. The situation is compared to the case of the theory in which a tensor field describes a scalar ghost as well as an ordinary massive tensor. In this case only an auxiliary vector field is needed to give covariant expressions for the gauge theory. 
  The free field theory for Regge trajectory is described in the framework of the BRST - quantization method. The physical spectrum includes daugther trajectories along with parent one. The applicability of the BRST approach to the description of a single Regge trajectory without its daughter trajectories is discussed. The simple example illustrates the appropriately modified BRST construction for the needed second class constraints. 
  We report on recent progress in the use of string techniques for the computation of field theory amplitudes. We show how one-loop renormalization constants in Yang-Mills theory can be computed using the open spinning string, we review the calculation of two-loop scalar amplitudes with the bosonic string, and we briefly indicate how the technique can be applied to the two-loop vacuum bubbles of Yang-Mills theory. 
  We review the theory of higher-spin gauge fields in four and three space-time dimensions and present some new results on higher-spin gauge interactions of matter fields in two dimensions. 
  A new method is applied to solve the Baxter equation for the one dimensional system of noncompact spins. Dynamics of such an ensemble is equivalent to that of a set of reggeized gluons exchanged in the high energy limit of QCD amplitudes. The technique offers more insight into the old calculation of the intercept of hard Pomeron, and provides new results in the odderon channel. 
  We use a simple algebraic method to find a special class of composite p-brane solutions of higher dimensional gravity coupled with matter. These solutions are composed of n constituent p-branes corresponding n independent harmonic functions. A simple algebraic criteria of existence of such solutions is presented. Relations with D=11,10 known solutions are discussed. 
  We determine the |p|/m expansion of the two body scattering amplitude of the quantum theory of a Chern-Simons field minimally coupled to a scalar field with quartic self-interaction. It is shown that the existence of a critical value of the self-interaction parameter for which the 2-particle amplitude reduces to the Aharonov-Bohm one is restricted to the leading, nonrelativistic, order. The subdominant terms correspond to relativistic corrections to the Aharonov-Bohm scattering. 
  String-inspired 1+1-dimensional gravity is coupled to Yang-Mills fields in the Cangemi-Jackiw gauge-theoretical formulation, based on the extended Poincar\'e group. A family of couplings, which involves metrics obtainable from the physical metric with a conformal rescaling, is considered, and the resulting family of models is investigated both at the classical and the quantum level. In particular, also using a series of Kirillov-Kostant phases, the wave functionals that solve the constraints are identified. 
  We review the evidence for the various dualities amongst the five D=10 superstring theories and for the existence of M-theory using the associated effective supergravity theories. We also summarise the combinatorial technics developed for constructing BPS solutions in D=11 supergravity theory and conjecture that all the BPS solutions of D<11 supergravity theories can be derived from the BPS solutions of D=11 supergravity that preserve 1/2 the supersymmetry. To demonstrate this, we derive the dyonic p-brane solutions from eleven dimensions. 
  We analyse the one-loop fermionic contribution for the scalar effective potential in the temperature dependent Yukawa model. In order to regularize the model a mix between dimensional and analytic regularization procedures is used. We find a general expression for the fermionic contribution in arbitrary spacetime dimension. It is found that in D=3 this contribution is finite. 
  It is shown that the Callan-Giddings-Harvey-Strominger theory on the cylinder can be consistently quantized (using Dirac's approach) without imposing any constraints on the sign of the gravitational coupling constant or the sign (or value) of the cosmological constant. The quantum constraints in terms of the original geometrical variables are also derived. 
  We present results obtained by a consideration of the non-classical energy momentum tensor associated with Euclidean Instantons outside the event horizon of black holes. We demonstrate here how this allows an analytic estimate to be made of the effect of discrete quantum hair on the temperature of the black hole, in which the role of violations of the weak energy condition associated with instantons is made explicit, and in which the previous results of Coleman, Preskill, and Wilczek are extended. Last, we demonstrate how the existence of a non-classical electric field outside the event horizon of black holes, uncovered by these authors, can be identified with a well-known effect in the Abelian-Higgs model in two dimensions. In this case, there is a one-to-one connection between the discrete charge of a black hole and a topological phase in two dimensions. 
  We present a new integrable extension of the a=-2, N=2 SKdV hierarchy, with the "small" N=4 superconformal algebra (SCA) as the second hamiltonian structure. As distinct from the previously known N=4 supersymmetric KdV hierarchy associated with the same N=4 SCA, the new system respects only N=2 rigid supersymmetry. We give for it both matrix and scalar Lax formulations and consider its various integrable reductions which complete the list of known SKdV systems with the N=2 SCA as the second hamiltonian structure. We construct a generalized Miura transformation which relates our system to the $\alpha = -2$, N=2 super Boussinesq hierarchy and, respectively, the ``small'' N=4 SCA to the N=2 W_3 superalgebra. 
  The work proposes a geometric background of the theory of field interactions and strings in spaces with higher order anisotropy. Our approach proceeds by developing the concept of higher order anisotropic superspace which unifies the logical and mathematical aspects of modern Kaluza-Klein theories and generalized Lagrange and Finsler geometry and leads to modelling of physical processes on higher order fiber bundles provided with nonlinear and distingushed connections and metric structures. The view adopted is that a general field theory should incorporate all possible anisotropic and stochastic manifestations of classical and quantum interactions and, in consequence, a corresponding modification of basic principles and mathematical methods in formulation of physical theories. The presentation is divided into two parts. The first five sections cover the higher order anisotropic superspaces. We focus on the geometry of distinguished by nonlinear connection vector superbundles, consider different supersymmetric extensions of Finsler and Lagrange spaces and analyze the structure of basic geometric objects on such superspaces. The remaining five sections are devoted to the theory of higher order anisotropic superstrings. In the framework of supersymmetric nonlinear sigma models in Finser extended backgrounds we prove that the low-energy dynamics of such strings contains motion equations for locally anisotropic field interactions. 
  The Higgs mixing term coefficient $\mu_{eff}$ is calculated in the scalar potential in supergravity theories with string origin, in a model independent approach. A general low energy effective expression is derived, where new contributions are included which depend on the modular weights $q_{1,2}$ of the Higgs superfields, the moduli and derivative terms. We find that in a class of models obtained in the case of compactifications of the heterotic superstring, the derivative terms are identically zero. Further, the total $\mu_{eff}$-term vanishes identically if the sum of the two modular weights $q_1+q_2$ is equal to two. Subleading $\mu$- corrections, in the presence of intermediate gauge symmetries predicted in viable string scenarios, are also discussed. 
  Using U-duality transformations we map perturbative Type IIA string theory compactified on a class of Joyce 7-manifolds to a D-strings on D-manifold description in Type IIB theory. For perturbative Type IIB theory on the same class of Joyce manifolds we use duality transformations to map to an M-theory, M-manifold description, which is an orientifold with fivebrane twisted sectors. D and M-manifold descriptions of discrete torsion are found. For the same class of compactifications we show that Type IIA/IIB theory on a Joyce orbifold without (with) discrete torsion is T-dual to Type IIB/IIA theory on the same orbifold with (without) discrete torsion. For this class of Type II compactifications this proves an extension of the Papadopoulos-Townsend conjecture, which states that the Type IIA and IIB theories compactified on the same Joyce 7-manifold are equivalent. Finally we note that the Papadopoulos-Townsend conjecture is a special case of the Generalised Mirror Conjecture. 
  The static color-Coulomb potential is calculated as the solution of a non-linear integral equation. This equation has been derived recently as a self-consistency condition which arises in the Coulomb Hamiltonian formulation of lattice gauge theory when the restriction to the interior of the Gribov horizon is implemented. The potential obtained is in qualitative agreement with expectations, being Coulombic with logarithmic corrections at short range and confining at long range. The values obtained for the string tension and $\Lambda_{\overline{MS}}$ are in semi-quantitative agreement with lattice Monte Carlo and phenomenological determinations. 
  We propose a gauge-invariant version of Wilson Renormalization Group for thermal field theories in real time. The application to the computation of the thermal masses of the gauge bosons in an SU(N) Yang-Mills theory is discussed. 
  We discuss general features of the $\beta$-function equations for spatially flat, $(d+1)$-dimensional cosmological backgrounds at lowest order in the string-loop expansion, but to all orders in $\alpha'$. In the special case of constant curvature and a linear dilaton these equations reduce to $(d+1)$ algebraic equations in $(d+1)$ unknowns, whose solutions can act as late-time regularizing attractors for the singular lowest-order pre-big bang solutions. We illustrate the phenomenon in a first order example, thus providing an explicit realization of the previously conjectured transition from the dilaton to the string phase in the weak coupling regime of string cosmology. The complementary role of $\alpha'$ corrections and string loops for completing the transition to the standard cosmological scenario is also briefly discussed. 
  In this paper we view the sigma-model couplings of appropriate vertex operators describing the interaction of string matter with a certain type of string solitons (0-branes) as the quantum phase space of a point particle. The sigma-model is slightly non critical, and therefore one should dress it with a Liouville mode. Quantization is achieved by summing over world-sheet genera (in the pinched approximation). To leading order in the coupling constant expansion, the quantization reproduces the usual quantum mechanical commutator. We attempt to go beyond leading order and we reproduce the generalized string uncertainty principle. 
  We investigate the string theory on three dimensional black holes discovered by Ba\~{n}ados, Teitelboim and Zanelli in the framework of conformal field theory. The model is described by an orbifold of the $\widetilde{SL}(2,R)$ WZW model. The spectrum is analyzed by solving the level matching condition and we obtain winding modes. We then study the ghost problem and show explicit examples of physical states with negative norms. We discuss the tachyon propagation and the target space geometry, which are irrelevant to the details of the spectrum. We find a self-dual T-duality transformation reversing the black hole mass. We also discuss difficulties in string theory on curved spacetime and possibilities to obtain a sensible string theory on three dimensional black holes. This work is the first attempt to quantize a string theory in a black hole background with an infinite number of propagating modes. 
  We consider Dirichlet p-branes in type II string theory on a space which has been toroidally compactified in d dimensions. We give an explicit construction of the field theory description of this system by putting a countably infinite number of copies of each brane on the noncompact covering space, and modding out the resulting gauge theory by Z^d. The resulting theory is a gauge theory with graded fields corresponding to strings winding around the torus an arbitrary number of times. In accordance with T-duality, this theory is equivalent to the gauge theory for the dual system of (d + p)-branes wrapped around the compact directions, where the winding number is exchanged for momentum in the compact direction. 
  We obtain a continuous Wick rotation for Dirac, Majorana and Weyl spinors $\psi \to \exp ({1\over 2} \theta \gamma^4 \gamma^5)\psi$ which interpolates between Minkowski and Euclidean field theories. 
  In the framework of the worldline path integral approach to QFTH we discuss spin and relativistic couplings, in particular Yukawa and axial couplings to spin 1/2, and the case of spin 1 in the loop. 
  We explain the concept of worldline Green functions on classes of multiloop graphs. The QED beta function and the 2-loop Euler-Heisenberg Lagrangian are discussed for illustration. 
  An algebraic restriction of the nonabelian self-dual Chern-Simons-Higgs systems leads to coupled-abelian self-dual models with intricate mass spectra. The vacua are characterized by embeddings of SU(2) into the gauge algebra; and in the broken phases, the gauge and real scalar masses are related to the exponents of the gauge algebra. In this paper we compute the gauge-gauge-Higgs couplings in the broken phases and use this to compute the finite renormalizations of the Chern-Simons coefficient in the various vacua. 
  Certain type II string non-threshold BPS bound states are shown to be related to non-static backgrounds in 11-dimensional theory. The 11-d counterpart of the bound state of NS-NS and R-R type IIB strings wound around a circle is a pure gravitational wave propagating along a generic cycle of 2-torus. The extremal (q_1,q_2) string with non-vanishing momentum along the circle (or infinitely boosted black string) corresponds in D=11 to a 2-brane wrapped around 2-torus with momentum flow along the (q_1,q_2) cycle. Applying duality transformations to the string-string solution we find type IIA background representing a bound state of 2-brane and 0-brane. Its lift to 11 dimensions is simply a 2-brane finitely boosted in transverse direction. This 11-d solution interpolates between a static 2-brane (zero boost) and a gravitational wave in 11-th dimension (infinite boost). Similar interpretations are given for various bound states involving 5-branes. Relations between transversely boosted M-branes and 1/2 supersymmetric non-threshold bound states 2+0 and 5+0 complement relations between M-branes with momentum in longitudinal direction and 1/4 supersymmetric threshold bound states 1+0 and 4+0. In the second part of the paper we establish the correspondence between the BPS states of type IIB strings on a circle and oscillating states of a fundamental supermembrane wrapped around a 2-torus. We show that the (q_1,q_2) string spectrum is reproduced by the membrane BPS spectrum, determined using a certain limit. This supports the picture suggested by Schwarz. 
  It is shown how to construct many-particle quantum-mechanical spectra of particles obeying multispecies exclusion statistics, both in one and in two dimensions. These spectra are derived from the generalized exclusion principle and yield the same thermodynamic quantities as deduced from Haldane's multiplicity formula. 
  We consider N=1 SU(N_c) gauge theory with an adjoint matter field $\Phi$, N_f flavors of fundamentals Q and antifundamentals \tQ, and tree-level superpotential of the form $\tQ\Phi^l Q$. This superpotential is relevant or marginal for $lN_f\leq 2N_c$. The theory has a Coulomb branch which is not lifted by quantum corrections. We find the exact effective gauge coupling on the Coulomb branch in terms of a family of hyperelliptic curves, thus providing a generalization of known results about N=2 SUSY QCD to N=1 context. The Coulomb branch has singular points at which mutually nonlocal dyons become massless. These singularities presumably correspond to new N=1 superconformal fixed points. We discuss them in some detail for N_c=2, N_f=1. 
  This is an introduction to the properties of D-branes, topological defects in string theory on which string endpoints can live. D-branes provide a simple description of various nonperturbative objects required by string duality, and give new insight into the quantum mechanics of black holes and the nature of spacetime at the shortest distances. The first two thirds of these lectures closely follow the earlier ITP lectures hep-th/9602052, written with S. Chaudhuri and C. Johnson. The final third includes more extensive applications to string duality. 
  In the lectures given at '96 Kashikojima Summer Institute, Amagi-Highland Seminars, and Yukawa Institute Workshop, the old-fashioned dualities and the theory of extended objects are reviewed, starting from 't Hooft-Mandelstam duality and following the author's old works. Application of the old-fashioned dualities to new issues are also given on the D-branes, on the phase transition of B-F theory to Einstein gravity and on the swimming of microorganisms. 
  We consider a configuration of strings and solitons in the type IIB superstring theory on $M^5\times T^5$, which is composed of a set of arbitrarily-wound D-fivebranes on $T^5$ and a set of arbitrarily-wound D-strings on $S^1$ of the torus. For the configuration, it is shown that number of microscopic states is bounded from above by the exponential of the Hawking-Bekenstein entropy of the corresponding black hole and the temperature of closed string radiation from the D-branes is bounded from below by the Hawking temperature of the black hole. After discussing the necessary and sufficient condition to saturate these bounds, we give some speculations about black hole thermodynamics. 
  We give an explicit procedure which computes for degree $d \leq 3$ the correlation functions of topological sigma model (A-model) on a projective Fano hypersurface $X$ as homogeneous polynomials of degree $d$ in the correlation functions of degree 1 (number of lines). We extend this formalism to the case of Calabi-Yau hypersurfaces and explain how the polynomial property is preserved. Our key tool is the construction of universal recursive formulas which express the structural constants of the quantum cohomology ring of $X$ as weighted homogeneous polynomial functions in the constants of the Fano hypersurface with the same degree and dimension one more. We propose some conjectures about the existence and the form of the recursive formulas for the structural constants of rational curves of arbitrary degree. Our recursive formulas should yield the coefficients of the hypergeometric series used in the mirror calculation. Assuming the validity of the conjectures we find the recursive laws for rational curves of degree 4 and 5. 
  The supporting worldsheet of a string, membrane, or other higher dimensional brane, is analysed in terms of its first, second, and third fundamental tensors, and its inner and outer curvature tensors. The dynamical equations governing the models appropriate for phenomena such as (superconducting) cosmic strings and cosmic domain walls are developed in a general framework (allowing for both electromagnetic and Kalb Ramond background coupling). It is shown how the surface stress momentum energy density tensor determines the propagation characteristics of small ``wiggle" perturbations of the worldsheet. Attention is then focussed on special features of strings (using the transonic model with tension T inversely proportional to the energy density U as a particularly important example). A quadratic Hamilton-Jacobi formulation is shown to govern equilibium states and other conservative string configurations sharing a symmetry of the (gravitational, electromagnetic, and Kalb-Ramond) background, including stable ring states that may be cosmologically important. 
  The interplay of paramagnetism, zero modes of the Dirac operator and fermionic mass singularities on the fermionic determinants in quantum electrodynamics in two, three and four dimensions is discussed. 
  Higher order conservation laws, associated with conserved antisymmetric tensors $j^{\mu_1 ... \mu_k}$ fulfilling $\partial_{\mu_1} j^{\mu_1 ... \mu_k} \approx 0$, are shown to define rigid symmetries of the master equation. They thus lead to independent Ward identities which are explicitly derived. 
  The well known domain wall type solutions are nowadays of great physical interest in classical field theory. These solutions can mostly be found only approximately. Recently the Hilbert-Chapman-Enskog method was succesfully applied to obtain this type solutions in phi**4 theory. The goal of the present paper is to verify these perturbative results by numerical computations. 
  We describe the classical Schwinger model as a study of the projective modules over the algebra of complex-valued functions on the sphere. On these modules, classified by $\pi_2(S^2)$, we construct hermitian connections with values in the universal differential envelope which leads us to the Schwinger model on the sphere. The Connes-Lott program is then applied using the Hilbert space of complexified inhomogeneous forms with its Atiyah-Kaehler structure. It splits in two minimal left ideals of the Clifford algebra preserved by the Dirac-Kaehler operator D=i(d-delta). The induced representation of the universal differential envelope, in order to recover its differential structure, is divided by the unwanted differential ideal and the obtained quotient is the usual complexified de Rham exterior algebra over the sphere with Clifford action on the "spinors" of the Hilbert space. The subsequent steps of the Connes-Lott program allow to define a matter action, and the field action is obtained using the Dixmier trace which reduces to the integral of the curvature squared. 
  We examine the quantum corrections to the static energy for Higgs winding configurations in order to ascertain whether such corrections may stabilize solitons in the standard model. We evaluate the effective action for winding configurations in Weinberg-Salam theory without U(1)-gauge fields or fermions. For a configuration whose size, $a \ll m^{-1}$ where $m = \max{m_W,m_H}$, m_W is the W-mass, and m_H is the Higgs mass, the static energy goes like $g^{-2}m_W^2a [1+b_0g^2\ln(1/ma)]^{c_0/b_0}$ in the semiclassical limit. Here g is the SU(2)-gauge coupling constant and b_0, c_0 are positive numbers determined by renormalization-group techniques. We discuss the limitations of this result for extremely small configurations and conclude that quantum fluctuations do not stabilize winding configurations where we have confidence in SU(2)-Higgs as a renormalizable field theory. 
  We gauge the non-abelian isometries of a sigma model with boundaries. Forcing the field strength of the gauge fields to vanish renders the gauged model equivalent to the ungauged one provided that boundary conditions are taken into account properly. Integrating out the gauge fields gives the T-dual model. We observe that T-duality interchanges Neumann (or mixed) boundary conditions with Dirichlet boundary conditions. 
  We study gaugino condensation in the presence of an intermediate mass scale in the hidden sector. S-duality is imposed as an approximate symmetry of the effective supergravity theory. Furthermore, we include in the K\"ahler potential the renormalization of the gauge coupling and the one-loop threshold corrections at the intermediate scale. It is shown that confinement is indeed achieved. Furthermore, a new running behaviour of the dilaton arises which we attribute to S-duality. We also discuss the effects of the intermediate scale, and possible phenomenological implications of this model. 
  The advantages of using more than one renormalization group (RG) in problems with more than one important length scale are discussed. It is shown that: i) using different RG's can lead to complementary information, i.e. what is very difficult to calculate with an RG based on one flow parameter may be much more accessible using another; ii) using more than one RG requires less physical input in order to describe via RG methods the theory as a function of its parameters; iii) using more than one RG allows one to solve problems with more than one diverging length scale. The above points are illustrated concretely in the context of both particle physics and statistical physics using the techniques of environmentally friendly renormalization. Specifically, finite temperature $\lambda\phi^4$ theory, an Ising-type system in a film geometry, an Ising-type system in a transverse magnetic field, the QCD coupling constant at finite temperature and the crossover between bulk and surface critical behaviour in a semi-infinite geometry are considered. 
  We construct and analyze dual N=4 supersymmetric gauge theories in three dimensions with unitary and symplectic gauge groups. The gauge groups and the field content of the theories are encoded in quiver diagrams. The duality exchanges the Coulomb and Higgs branches and the Fayet-Iliopoulos and mass parameters. We analyze the classical and the quantum moduli spaces of the theories and construct an explicit mirror map between the mass parameters and the the Fayet-Iliopoulos parameters of the dual. The results generalize the relation between ALE spaces and moduli spaces of SU(n) and SO(2n) instantons. We interpret some of these results from the string theory viewpoint, for SU(n) by analyzing T-duality and extremal transitions in type II string compactifications, for SO(2n) by using D-branes as probes. Finally, we make a proposal for the moduli space of vacua of these theories in the absence of matter. 
  Boundary condition changing operators in conformal field theory describe various types of "sudden switching" problems in condensed matter physics such as the X-ray edge singularity. We review this subject and give two extensions of previous work. A general derivation of a connection between the X-ray edge singularity, the Anderson orthogonality catastrophe and finite-size scaling of energies is given. The formalism is also extended to include boundstates. 
  Dimensional reduction of a self-dual tensor gauge field in 6d gives an Abelian vector gauge field in 5d. We derive the conditions under which an interacting 5d theory of an Abelian vector gauge field is the dimensional reduction of a 6d Lorentz invariant interacting theory of a self-dual tensor. Then we specialize to the particular 6d theory that gives 5d Born-Infeld theory. The field equation and Lagrangian of this 6d theory are formulated with manifest 5d Lorentz invariance, while the remaining Lorentz symmetries are realized nontrivially. A string soliton with finite tension and self-dual charge is constructed. 
  We briefly review the covariant formulation of the Green-Schwarz superstring by Berkovits, and describe how a detailed tree-level and one-loop analysis of this model leads, for the first time, to a derivation of the low-energy effective action of the heterotic superstring while keeping target-space supersymmetry manifest. The resulting low-energy theory is old-minimal supergravity coupled to tensor multiplet. The dilaton is part of the compensator multiplet. 
  We introduce the notion of a fused quantum superplane by allowing for terms $\theta\theta\sim x$ in the defining relations. We develop the differential calculus for a large class of fused quantum superplanes related to particular solutions of the Yang-Baxter equation. 
  We find a mapping between antisymmetric tensor matter fields and the Weinberg's 2(2j+1)- component "bispinor" fields. Equations which describe the j=1 antisymmetric tensor field coincide with the Hammer-Tucker equations entirely and with the Weinberg ones within a subsidiary condition, the Klein-Gordon equation. The new Lagrangian for the Weinberg theory is proposed which is scalar and Hermitian. It is built on the basis of the concept of the `Weinberg doubles'. Origins of a contradiction between the classical theory, the Weinberg theorem B-A=\lambda for quantum relativistic fields and the claimed `longitudity' of the antisymmetric tensor field (transformed on the (1,0)\oplus (0,1) Lorentz group representation) after quantization are clarified. Analogs of the j=1/2 Feynman-Dyson propagator are presented in the framework of the j=1 Weinberg theory. It is then shown that under the definite choice of field functions and initial and boundary conditions the massless j=1 Weinberg-Tucker-Hammer equations contain all information that the Maxwell equations for electromagnetic field have. Thus, the former appear to be of use in describing some physical processes for which that could be necessitated or be convenient. 
  Triple systems are closely related to Yang-Baxter symmetries. Utilizing a non-parameter-dependent triple product, we derive the BCS interaction. The enlargement of the notion of symmetry leads in some sense to a regular vertex function. The connection to the effect of running coupling constants is outlined, which leads to the recently discussed anisotropic effective local interactions. Furthermore, a discussion of the physical nature of q-symmetries is given. 
  A renormalizable rigid supersymmetry for the four dimensional antisymmetric tensor field model in a curved space-time background is constructed. A closed algebra between the BRS and the supersymmetry operators is only realizable if the vector parameter of the supersymmetry is a covariantly constant vector field. This also guarantees that the corresponding transformations lead to a genuine symmetry of the model. The proof of the ultraviolet finiteness to all orders of perturbation theory is performed in a pure algebraic manner by using the rigid supersymmetry. 
  We present analytic expressions for the single particle excitation energies of the 8 quasi-particles in the lattice $E_8$ Ising model and demonstrate that all excitations have an extended Brillouin zone which, depending on the excitation, ranges from 0<P < 4\pi to 0< P< 12 \pi. These are compared with exact diagonalizations for systems through size 10 and with the E_8 fermionic representations of the characters of the critical system in order to study the counting statistics. 
  We argue that the space-time uncertainty relation of the form $\Delta X \Delta T \gtrsim \alpha'$ for the observability of the distances with respect to time, $\Delta T$, and space, $\Delta X$, is universally valid in string theory including D-branes. This relation has been previously proposed by one (T.Y.) of the present authors as a simple qualitative representation of the perturbative short distance structure of fundamental string theory. We show that the relation, combined with the usual quantum mechanical uncertainty principle, explains the key qualitative features of D-particle dynamics. 
  Differential calculus on the space of asymptotically linear curves is developed. The calculus is applied to the vortex filament equation in its Hamiltonian description. The recursion operator generating the infinite sequence of commuting flows is shown to be hereditary. The system is shown to have a description with a Hamiltonian pair. Master symmetries are found and are applied to deriving an expression of the constants of motion in involution. The expression agrees with the inspection of Langer and Perline. 
  The superconformal properties of N=4 Yang-Mills theory are most naturally studied using the formalism of harmonic superspace. Superconformal invariance is shown to imply that the Green's functions of analytic operators are invariant holomorphic sections of a line bundle on a product of certain harmonic superspaces and it is argued that the theory is soluble for a class of such operators. 
  The superconformal invariants in analytic superspace are found. Superconformal invariance is shown to imply that the Green's functions of analytic operators are invariant holomorphic sections of a line bundle on a product of certain harmonic superspaces. It is argued that the correlation functions for a class of sufficiently low dimension gauge invariant operators in N=2 and N=4 supersymmetric Yang-Mills theory can be evaluated up to constants. 
  In this lecture moduli dependent charges for p-extended objects are analyzed for generic N-extended supergravities in dimensions 4 \leq D <10. Differential relations and sum rules among the charges are derived. 
  Following systematically the generalized Hamiltonian approach of Batalin and Fradkin, we demonstrate the equivalence of a self-dual model with the Maxwell-Chern-Simons theory by embedding the former second-class theory into a first-class theory. 
  The Casimir energy or stress due to modes in a D-dimensional volume subject to TM (mixed) boundary conditions on a bounding spherical surface is calculated. Both interior and exterior modes are included. Together with earlier results found for scalar modes (TE modes), this gives the Casimir effect for fluctuating ``electromagnetic'' (vector) fields inside and outside a spherical shell. Known results for three dimensions, first found by Boyer, are reproduced. Qualitatively, the results for TM modes are similar to those for scalar modes: Poles occur in the stress at positive even dimensions, and cusps (logarithmic singularities) occur for integer dimensions $D\le1$. Particular attention is given the interesting case of D=2. 
  We discuss the vertical dimensional reduction of M-branes to domain walls in D=7 and D=4, by dimensional reduction on Ricci-flat 4-manifolds and 7-manifolds. In order to interpret the vertically-reduced 5-brane as a domain wall solution of a dimensionally-reduced theory in D=7, it is necessary to generalise the usual Kaluza-Klein ansatz, so that the 3-form potential in D=11 has an additional term that can generate the necessary cosmological term in D=7. We show how this can be done for general 4-manifolds, extending previous results for toroidal compactifications. By contrast, no generalisation of the Kaluza-Klein ansatz is necessary for the compactification of M-theory to a D=4 theory that admits the domain wall solution coming from the membrane in D=11. 
  Based on the generalized gauge theory on $M^4\times Z_2\times Z_3$, we reconstructed the realistic SU(5) Grand Unified model by a suitable assignment of fermion fields. The action of group elements $Z_2$ on fermion fields is the charge conjugation while the action of $Z_3$ elements represent generation translation. We find that to fit the spontaneous symmetry breaking and gauge hierarchy of SU(5) model a linear term of curvature has to be introduced. A new mass relation is obtained in our reconstructed model. 
  By relating the two-dimensional U(N) Principal Chiral Model to a simple linear system we obtain a free-field parametrisation of solutions. Obvious symmetry transformations on the free-field data give symmetries of the model. In this way all known `hidden symmetries' and B\"acklund transformations, as well as a host of new symmetries, arise. 
  The macroscopic dynamics of a rotating superfluid deviates from that of a simple perfect fluid due to the effect of vorticity quantisation, which gives rise to a substructure of cosmic string type line defects that results in a local anisotropy whereby the effective average pressure in the direction of the vortex lines is reduced below its value in lateral directions. Whereas previous descriptions of this effect have been restricted to a non-relativistic framework that is adequate for the treatment of liquid helium in a laboratory context, the present work provides a fully relativistic description of the kind required for application to rotating neutron star models. To start with, the general category of vortex fibration models needed for this purpose is set up on the basis of a Kalb-Ramond type variational principle. The appropriate specification of the particular model to be chosen within this category will ultimately be governed by the conclusions of microscopic investigations that have not yet been completed, but the results available so far suggest that a uniquely simple kind of model with an elegant dilatonic formulation should be tentatively adopted as a provisional choice so long as there is no indication that a more complicated alternative is needed. 
  A generalized Wakimoto realization of $\widehat{\cal G}_K$ can be associated with each parabolic subalgebra ${\cal P}=({\cal G}_0 +{\cal G}_+)$ of a simple Lie algebra ${\cal G}$ according to an earlier proposal by Feigin and Frenkel. In this paper the proposal is made explicit by developing the construction of Wakimoto realizations from a simple but unconventional viewpoint. An explicit formula is derived for the Wakimoto current first at the Poisson bracket level by Hamiltonian symmetry reduction of the WZNW model. The quantization is then performed by normal ordering the classical formula and determining the required quantum correction for it to generate $\widehat{\cal G}_K$ by means of commutators. The affine-Sugawara stress-energy tensor is verified to have the expected quadratic form in the constituents, which are symplectic bosons belonging to ${\cal G}_+$ and a current belonging to ${\cal G}_0$. The quantization requires a choice of special polynomial coordinates on the big cell of the flag manifold $P\backslash G$. The effect of this choice is investigated in detail by constructing quantum coordinate transformations. Finally, the explicit form of the screening charges for each generalized Wakimoto realization is determined, and some applications are briefly discussed. 
  The Wen-Wu c=-2 model describes the d-wave paired singlet factor of the Haldane-Rezayi (quantum Hall) wave function in terms of an SU(2) doublet of dimension 1 fermions. The resulting CFT involves an infinite sequence of Virasoro primary fields. The superalgebra generated by these fields, on the other hand, admits a set of ground state representations giving rise to a rational c=-2 CFT. We compute the corresponding characters as well as the characters of the bosonic subalgebra in the Z_2 twisted sector and write down a modular invariant partition function for the ensuing 2-dimensional conformal theory. 
  A boson mapping of pair field operators is presented. The mapping preserves all hermiticity properties and the Poisson bracket relations between fields and momenta. The most practical application of the boson mapping is to field theories which exhibit bound states of pairs of fields. As a concrete application we consider, in the low energy limit, the Wick-Cutkosky model with equal mass for the charged fields. 
  Motivated by the concept of shape invariance in supersymmetric quantum mechanics, we obtain potentials whose spectrum consists of two shifted sets of equally spaced energy levels. These potentials are similar to the Calogero-Sutherland model except the singular term $\alpha x^{-2}$ always falls in the transition region $-1/4 < \alpha < 3/4$ and there is a delta-function singularity at x=0. 
  It is shown that the operator methods of supersymmetric quantum mechanics and the concept of shape invariance can profitably be used to derive properties of spherical harmonics in a simple way. The same operator techniques can also be applied to several problems with non-central vector and scalar potentials. As examples, we analyze the bound state spectra of an electron in a Coulomb plus an Aharonov-Bohm field and/or in the magnetic field of a Dirac monopole. 
  We discuss our attempts to generalize the known examples of dualities in N=1 supersymmetric gauge theories to exceptional gauge groups. We derive some dual pairs from known examples connected to exceptional groups and find an interesting phenomenon: sometimes the full global symmetry is ``hidden'' on the magnetic side. It is not realized as a symmetry on the fundamental fields in the Lagrangian. Rather, it emerges as a symmetry of the quantum theory. We then focus on an approach based on self-dual models. We construct duals for some very special matter content of $E_ 7$, $E_6$ and $F_4$. Again we find that the full global symmetry is not realized on the fundamental fields. 
  We show that locally Lorentz invariant, third order, topological massive gravity can not be broken down neither to the local diffeomorphism subgroup nor to the rigid Poincar\'e group. On the other hand, the recently formulated, locally diffeomorphism invariant, second order massive triadic (translational) Chern-Simons gravity breaks down on rigid Minkowski space to a double massive spin-two system. This flat double massive action is the uniform spin-two generalization of the Maxwell-Chern-Simons-Proca system which one is left with after U(1) abelian gauge invariance breaks down in the presence of a sextic Higgs potential. 
  We construct local geometric model in terms of F- and M-theory compactification on Calabi-Yau fourfolds which lead to N=1 Yang-Mills theory in d=4 and its reduction on a circle to d=3. We compute the superpotential in d=3, as a function of radius, which is generated by the Euclidean 5-brane instantons. The superpotential turns out to be the same as the potential for affine Toda theories. In the limit of vanishing radius the affine Toda potential reduces to the Toda potential. 
  We present a geometric approach to the field theory with higher order anisotropic interactions. The concepts of higher order space, or locally anisotropic, space (in brief, h-space, or la-space) are introduced as general ones for various types of higher order extensions of Lagrange and Finsler geometry and higher dimension (Kaluza-Klein type) spaces. The spinors on h-spaces are defined in the framework of the geometry of Clifford bundles provided with compatible nonlinear and distinguished connections and metric structures (d-connection and d-metric). There are discussed some related issues connected with the geometric aspects of higher order anisotropc interactions for gravitational, gauge, spinor, Dirac spinor and Proca fields. The nearly autoparallel maps are introduced as maps with deformation of connections extending the class of geodesic and conformal transforms. We propose two variants of solution of the problem of definition of conservation laws on h-spaces. A general background of the theory of field interactions and strings in spaces with higher order anisotropy is fromulated. The conditions for consistent propagation of closed strings in higher order anisotropic background spaces are analyzed. The connection between the conformal invariance, the vanishing of the renormalization group beta-function of the generalized sigma-model and field equations of higher order anisotropic gravity are studied in detail. 
  We consider the moduli space of flat G-bundles over the twodimensional torus, where G is a real, compact, simple Lie group which is not simply connected. We show that the connected components that describe topologically non-trivial bundles are isomorphic as symplectic spaces to moduli spaces of topologically trivial bundles with a different structure group. Some physical applications of this isomorphism which allows to trade topological non-triviality for a change of the gauge group are sketched. 
  This talk is a brief review of gaugino condensation in superstring effective field theories and some related issues (such as renormalization of the gauge coupling in the effective supergravity theories and modular anomaly cancellation). As a specific example, we discuss a model containing perturbative (1-loop) corrections to the K\"ahler potential and approximate S-duality symmetry. 
  Five-dimensional stringy rotating black holes are embedded into N=2 supergravity interacting with one vector multiplet. The existence of an unbroken supersymmetry of the rotating solution is proved directly by solving the Killing spinor equations. The asymptotic enhancement of supersymmetry near the horizon in the presence of rotation is established via the calculation of the super-curvature. The area of the horizon of the rotating supersymmetric black holes is found to be $\sqrt {Z_{fix}^{3 }- J^2}$, where $Z_{fix}$ is the extremal value of the central charge in moduli space. 
  We construct a four-dimensional BPS saturated heterotic string solution from the Taub-NUT solution. It is a non-extremal black hole solution since its Euler number is non-zero. We evaluate its black hole entropy semiclassically. We discuss the relation between the black hole entropy and the degeneracy of string states. The entropy of our string solution can be understood as the microscopic entropy which counts the elementary string states without any complications. 
  The spherically and cylindrically symmetric solutions of the $SU(3)$ Yang - Mills theory are obtained. The corresponding gauge potential has the confining properties. It is supposed that: a) the spherically symmetric solution is a field distribution of the classical ``quark'' and in this sense it is similar to the Coulomb potential; b) the cylindrically symmetric solution describes a classical field ``string'' (flux tube) between two ``quarks''. It is noticed that these solutions are typically for the classical $SU(3)$ Yang - Mills theory in contradiction to monopole that is an exceptional solution. This allows to conclude that the confining properties of the classical $SU(3)$ Yang - Mills theory are general properties of this theory. 
  We derive the singularity conditions of the N=1 generalized (general yukawa couplings and quark masses) form of hyperelliptic curves of SU(N_c) with N_f flavors. The results reproduce the known form of N=2 curves when the yukawa couplings and the quark masses reduce to those of N=2. We obtained these curves by determining the dependence of the unbroken SU(2) gaugino condensation on the couplings in the moduli source terms which break N=2 SQCD to N=1 SU(N_c) gauge theory with the quarks and the adjoint matter, \Phi. The degenerate component of the diagonalized classical vacuum expectation value of \Phi is shown to be explicitly written in terms of these couplings, which enables us to determine the form of the gaugino condensation. 
  We study the relationship between the continuum overlap and its corresponding chiral determinant, showing that the former amounts to an unregularised version of the latter. We then construct a regularised continuum overlap, and consider the chiral anomalies that follow therefrom. The relation between these anomalies and the ones derived from the formal (i.e., unregularised) overlap is elucidated. 
  We consider solutions to the string effective action corresponding to p-Branes, D-Branes and M-Branes and discuss some of their properties. 
  We demonstrate how a Lorentz covariant formulation of the chiral p-form model in D=2(p+1) containing infinitely many auxiliary fields is related to a Lorentz covariant formulation with only one auxiliary scalar field entering a chiral p-form action in a nonpolynomial way. The latter can be regarded as a consistent Lorentz-covariant truncation of the former. We make the Hamiltonian analysis of the model based on the nonpolynomial action and show that the Dirac constraints have a simple form and are all of the first class. In contrast to the Siegel model the constraints are not the square of second-class constraints. The canonical Hamiltonian is quadratic and determines energy of a single chiral p-form. In the case of d=2 chiral scalars the constraint can be improved by use of `twisting' procedure (without the loss of the property to be of the first class) in such a way that the central charge of the quantum constraint algebra is zero. This points to possible absence of anomaly in an appropriate quantum version of the model. 
  Self-dual perturbiner in the Yang-Mills theory is constructed by the twistor methods both in topologically trivial and topologically nontrivial cases. Maximally helicity violating amplitudes and their instanton induced analogies are briefly discussed. 
  We derive the Holstein-Primakoff oscillator realization on the coadjoint orbits of the $SU(N+1)$ and $SU(1,N)$ group by treating the coadjoint orbits as a constrained system and performing the symplectic reduction. By using the action-angle variables transformations, we transform the original variables into Darboux variables. The Holstein-Primakoff expressions emerge after quantization in a canonical manner with a suitable normal ordering. The corresponding Dyson realizations are also obtained and some related issues are discussed. 
  It is shown that the world-line can be eliminated in the matrix quantum mechanics conjectured by Banks, Fischler, Shenker and Susskind to describe the light-cone physics of M theory. The resulting matrix model has a form that suggests origins in the reduction to a point of a Yang-Mills theory. The reduction of the Nishino-Sezgin 10+2 dimensional supersymmetric Yang-Mills theory to a point gives a matrix model with the appropriate features: Lorentz invariance in 9+1 dimensions, supersymmetry, and the correct number of physical degrees of freedom. 
  Type IIB strings compactified on K3 have a rich structure of solitonic strings, transforming under SO(21,5,Z). We derive the BPS tension formula for these strings, and discuss their properties, in particular, the points in the moduli space where certain strings become tensionless. By examining these tensionless string limits, we shed some further light on the conjectured dual M-Theory description of this compactification. 
  A simple derivation of the free energy and expectation values of Polyakov-loops in $QCD_2$ via path integral methods is given. In the chosen gauge (which can be generalized to 4 dimensions) without Gribov-copies the Fadeev-Popov determinant and the integration over the space component of the gauge field cancel exactly and we are left only with an integration over the zero components of the gauge field in the Cartan sub-algebra. This way the Polyakov-loop operators become Vertex-operators in a simple quantum mechanical model. The number of fermionic zero modes is related to the winding-numbers of $A_0$ in this gauge. 
  It is shown how monopoles and dyons decay on curves of marginal stability in the moduli space of vacua at weak coupling in pure N=2 gauge theory with arbitrary gauge group. The analysis involves a semi-classical treatment of the monopole and rests on the fact that the monopole moduli space spaces for a magnetic charge vector equal to a non-simple root enlarge discontinuously at the curves of marginal stability. This enlargement of the moduli space describes the freedom for the monopole to be separated into stable constituent monopoles. Such decays do not occur in the associated theory with N=4 supersymmetry because in this case there exist bound-states at threshold. 
  In this letter we show how string winding modes can be constructed using topological membranes. We use the fact that monopole-instantons in compact topologically massive gauge theory lead to charge non-conservation inside the membrane which, in turn, enables us to construct vertex operators with different left and right momenta. The amount of charge non-conservation inside the membrane is interpreted as giving the momentum associated with the string winding mode and is shown to match precisely the full mass spectrum of compactified string theory. 
  A new integrable N=2 supersymmetric f-Toda mapping in (1|2) superspace, acting like the symmetry transformation of N=2 supersymmetric NLS hierarchy, is proposed. The first two Hamiltonian structures and the recursion operator connecting all evolution systems and Hamiltonian structures of the N=2 super-NLS hierarchy are constructed in explicit form using only invariance conditions with respect to the f-Toda mapping. A new representation for its Hamiltonians is observed. 
  The leading short-time behaviour of the Yang-Mills Schroedinger functional is obtained within a local expansion in the fields. 
  We construct solitonic string solutions of N=2 four-dimensional heterotic models of rank three, four and five. These finite energy configurations have constant dilaton while the moduli fields vary over space-time with jumps at the location of the string cores consistent with the T-duality groups SL(2,Z), SL(2,Z) X SL(2,Z) and Sp(4,Z). The solutions are expressed in terms of modular forms of the T-duality group. They break half of the supersymmetries and the vacuum contain a certain number of solitonic strings in order the singularities to be resolved in a Ricci flat way. 
  We comment on the structure of intersecting black p-brane solutions in string theory explaining how known solutions can be obtained from Schwarzschild solution simply by sequences of boosts and dualities. This implies, in particular, that dimensional reduction in all internal world-volume directions including time leads to a metric (related by analytic continuation to a cosmological metric) which does not depend on p-brane charges, i.e. is the same as the metric following by reduction from a higher-dimensional `neutral' Schwarzschild black hole. 
  The essential elements in the construction of the couplings of vector multiplets to supergravity using the conformal approach are repeated. This approach leads automatically to the basic quantities on which the symplectic transformations, the basic tools for duality transformations, are defined. A recent theorem about the existence of a basis allowing for a prepotential is discussed. 
  We parametrize the gauge-fixing freedom in choosing the Lagrangian of a topological gauge theory. We compute the gauge-fixing dependence of correlators of equivariant operators when the compactified moduli space has a non-empty boundary and verify that only a subset of these has a gauge independent meaning. We analyze in detail a simple example of such anomalous topological theories, 4D topological Yang-Mills on the four-sphere and instanton number k=1. 
  Equivariant cohomology is a mathematical framework particularly well adapted to a kinematical understanding of topological gauge theories of the cohomological type. It also sheds some light on gauge fixing, a necessary field theory operation connected with the non compactness of the gauge group. The respective roles of fields and observables are emphasized throughout. 
  La fixation de jauge est d\'efinie comme l'op\'eration permettant d'exprimer une int\'egrale sur un espace d'orbite comme int\'egrale sur le fibr\'e principal correspondant. Quand la fibre est non compacte cette op\'eration met en jeu une classe de cohomologie \`a support compact -ou \`a d\'ecroissance rapide- de celle-ci. La sym\'etrie de Slavnov est l'expression alg\'ebrique de l'ambiguit\'e de cette construction. 
  Equivariant cohomology is suggested as an alternative algebraic framework for the definition of topological field theories constructed by E. Witten circa 1988. It also enlightens the classical Faddeev Popov gauge fixing procedure. 
  Exact solutions to the quantum S-matrices for solitons in simply-laced affine Toda field theories are obtained, except for certain factors of simple type which remain undetermined in some cases. These are found by postulating solutions which are consistent with the semi-classical limit, $\hbar\rightarrow 0$, and the known time delays for a classical two soliton interaction. This is done by a `$q$-deformation' procedure, to move from the classical time delay to the exact S-matrix, by inserting a special function called the `regularised' quantum dilogarithm, which only holds when $|q|=1$. It is then checked that the solutions satisfy the crossing, unitarity and bootstrap constraints of S-matrix theory. These properties essentially follow from analogous properties satisfied by the classical time delay. Furthermore, the lowest mass breather S-matrices are computed by the bootstrap, and it is shown that these agree with the particle S-matrices known already in the affine Toda field theories, in all simply-laced cases. 
  Talk presented by the second author at the Inaugural Coference of the Asia Pacific Center for Theoretical Physics, Seoul, June 1996. The purpose of this note is to give a resume of the Seiberg-Witten theory in the simplest possible mathematical terms. 
  We present orientifold and F-theory duals of the heterotic string compactification constructed by Chaudhuri, Hockney and Lykken (CHL) which has the maximal supersymmetry but gauge group of reduced rank. The 8-dimensional dual is given by the Type IIA orientifold on the M\"{o}bius band. We show the non-trivial monodromy on the base induces non-simply laced gauge groups on the F-theory orbifolds dual to the CHL strings. The F-theory models dual to the CHL strings in six dimensions are examples of N=2 F-theory vacua. We discuss the other N=2 F-theory vacua in six dimensions. 
  Three dimensional topological field theories associated with the three dimensional version of Abelian and non-Abelian Seiberg-Witten monopoles are presented. These three dimensional monopole equations are obtained by a dimensional reduction of the four dimensional ones. The starting actions to be considered are Gaussian types with random auxiliary fields. As the local gauge symmetries with topological shifts are found to be first stage reducible, Batalin-Vilkovisky algorithm is suitable for quantization. Then BRST transformation rules are automatically obtained. Non-trivial observables associated with Chern classes are obtained from geometric sector and are found to correspond to those of the topological field theory of Bogomol'nyi monopoles. 
  Two-dimensional gravity in the light-cone gauge was shown to exhibit an underlying sl(2,R) current algebra. It is the purpose of this note to offer a possible explanation about the origin of this important algebra. The essential point is that two-dimensional gravity is governed by a topological field theory. The gauge group is sl(2,R) and it is this enhanced gauge group that yields Polyakov's current algebra. 
  We demonstrate, by a simple analysis, that cosmological line elements related by scale factor duality also exhibit a duality with respect to the conservation/violation of the Weak Energy Condition (WEC) by the matter that acts as the source in the one-loop beta function equations for the metric coupling written explicitly in the form of the Einstein equations. Furthermore, a study of specific pairs of line elements (obtained via O(d,d) transformations) hints at a possible generalisation of the above duality w.r.t. WEC for the case of O(d,d) related spacetimes. Consequences and extensions thereof are also pointed out. 
  The low energy effective Lagrangian for $N\es 2$ supersymmetric Yang-Mills theory, proposed by Seiberg and Witten is shown to be the unique solution, assuming only that supersymmetry is unbroken and that the number of strong-coupling singularities is finite. Duality is then a consequence rather than an input. 
  The conjecture of Fuchs, Schellekens and Schweigert on the relation of mapping class group representations and fixed point resolution in simple current extensions is investigated, and a cohomological interpretation of the untwisted stabilizer is given. 
  It has been observed recently that many properties of some near extremal black holes can be described in terms of bound states of D-branes. Using a non-renormalization theorem we argue that the D-brane description is the correct quantum gravity description of the black hole at low energies. The low energy theory includes the black hole degrees of freedom that account for the entropy and describes also Hawking radiation. The description is unitary and there seems to be no information loss at low energies. 
  We investigate the convergence properties of the cluster expansion of equal-time Green functions in scalar theories with quartic self-coupling in (0+1), (1+1), and (2+1) space-time dimensions. The computations are carried out within the equal-time correlation dynamics approach. We find that the cluster expansion shows good convergence as long as the system is in a single phase configuration and that it breaks down in a two phase configuration, as one would naively expect. In the case of dynamical calculations with a time dependent Hamiltonian we find two timescales determining the adiabaticity of the propagation; these are the time required for adiabaticity in the single phase region and the time required for tunneling into the non-localized lowest energy state in the two phase region. 
  We conjecture a simple relationship between the one-loop maximally helicity violating gluon amplitudes of ordinary QCD (all helicities identical) and those of N=4 supersymmetric Yang-Mills (all but two helicities identical). Because the amplitudes in self-dual Yang Mills have been shown to be the same as the maximally helicity violating ones in QCD, this conjecture implies that they are also related to the maximally helicity violating ones of N=4 supersymmetric Yang-Mills. We have an explicit proof of the relation up to the six-point amplitude; for amplitudes with more external legs, it remains a conjecture. A similar conjecture relates amplitudes in self-dual gravity to maximally helicity violating N=8 supergravity amplitudes. 
  We discuss space-time chaos and scaling properties for classical non-Abelian gauge fields discretized on a spatial lattice. We emphasize that there is a ``no go'' for simulating the original continuum classical gauge fields over a long time span since there is a never ending dynamical cascading towards the ultraviolet. We note that the temporal chaotic properties of the original continuum gauge fields and the lattice gauge system have entirely different scaling properties thereby emphasizing that they are entirely different dynamical systems which have only very little in common. Considered as a statistical system in its own right the lattice gauge system in a situation where it has reached equilibrium comes closest to what could be termed a ``continuum limit'' in the limit of very small energies (weak non-linearities). We discuss the lattice system both in the limit for small energies and in the limit of high energies where we show that there is a saturation of the temporal chaos as a pure lattice artifact. Our discussion focuses not only on the temporal correlations but to a large extent also on the spatial correlations in the lattice system. We argue that various conclusions of physics have been based on monitoring the non-Abelian lattice system in regimes where the fields are correlated over few lattice units only. This is further evidenced by comparison with results for Abelian lattice gauge theory. How the real time simulations of the classical lattice gauge theory may reach contact with the real time evolution of (semi-classical aspects of) the quantum gauge theory (e.g. Q.C.D.) is left as an important question to be further examined. 
  Using the method of Green's functions within the framework of a Bethe-Salpeter formalism characterized by a pairwise $qq$ interaction with a 3D support to its kernel (expressed in a Lorentz-covariant manner), the 4D BS wave function for a system of three identical relativistic spinless quarks is reconstructed from the corresponding 3D form which satisfies a {\it fully connected} 3D BSE. This result is a 3-body generalization of a similar interconnection between the 3D and 4D 2-body wave functions that had been found earlier under identical conditions of a 3D support to the corresponding BS kernel, using the ansatz of Covariant Instaneity for the pairwise $q\bar q$ interaction. The generalization from spinless to fermion quarks is straightforward. 
  Using the simple current method we study a class of $(0,2)$ SCFTs which we conjecture to be equivalent to (0,2) sigma models constructed in the framework of gauged linear sigma models. 
  We demonstrate the role of Drinfeld's quantum double D(G) as a spontaneously broken hidden symmetry in a large class of massive quantum field theories in 1+1 dimensions with compact symmetry group G. Our considerations are independent of exact integrability. The main technical ingredient is an assumption concerning the statistical independence of fields localized in spacelike separated regions which should hold in all reasonable massive models. The present note is an abridged version of hep-th/9606175. 
  A new form of the Wilson renormalization group equation is derived, in which the flow equations are, up to linear terms, proportional to a gradient flow. A set of co\"ordinates is found in which the flow of marginal, low-energy, couplings takes a gradient form, if relevant couplings are tuned to vanish. 
  Two-loop contributions to the anomalous correlation function <J_mu(x)J_nu(y)J_rho(z)> of three chiral currents are calculated by a method based on the conformal properties of massless field theories. The method was previously applied to virtual photon diagrams in quantum electrodynamics, and it is extended here to diagrams with scalars and chiral spinors in the abelian Higgs model and in the SU(3)xSU(2)xU(1) standard model. In each case there are nonvanishing contributions to the gauge current correlator from self-energy insertions, vertex insertions and nonplanar diagrams, but their sum exactly vanishes. The two-loop contribution to the anomaly therefore also vanishes, in agreement with the Adler-Bardeen theorem. An application of the method to the correlator <R_mu(x)R_nu(y)K_rho(z)> of the R and Konishi axial currents in supersymmetric gauge theories which was reported in hep-th/9608125 is discussed here. The net two-loop contribution to this correlator also vanishes. 
  In the usual procedure for toroidal Kaluza-Klein reduction, all the higher-dimensional fields are taken to be independent of the coordinates on the internal space. It has recently been observed that a generalisation of this procedure is possible, which gives rise to lower-dimensional ``massive'' supergravities. The generalised reduction involves allowing gauge potentials in the higher dimension to have an additional linear dependence on the toroidal coordinates. In this paper, we show that a much wider class of generalised reductions is possible, in which higher-dimensional potentials have additional terms involving differential forms on the internal manifold whose exterior derivatives yield representatives of certain of its cohomology classes. We consider various examples, including the generalised reduction of M-theory and type II strings on K3, Calabi-Yau and 7-dimensional Joyce manifolds. The resulting massive supergravities support domain-wall solutions that arise by the vertical dimensional reduction of higher-dimensional solitonic p-branes and intersecting p-branes. 
  Boundary states for D-branes are constructed using the light cone gauge. The D-brane breaks half the spacetime supersymmetry giving rise to fermionic zero modes living on the brane. The nonlinear realization of the broken supersymmetry on the open string degrees of freedom is analysed and the influence of boundary terms coming from closed string vertex operators is discussed. 
  String theories with (N,N') local world-sheet supersymmetries are related to each other by marginal deformations. This connects N=1 and N=0 theories in which the target-spaces are interpreted as space-times, N=2 theories in which the target spaces can be interpreted as world-volumes, and theories with $N\ge 3$, in which the central charge vanishes -- theories with zero target-space dimensions. 
  The primary purpose of these lecture notes is to explore the moduli space of type IIA, type IIB, and heterotic string compactified on a K3 surface. The main tool which is invoked is that of string duality. K3 surfaces provide a fascinating arena for string compactification as they are not trivial spaces but are sufficiently simple for one to be able to analyze most of their properties in detail. They also make an almost ubiquitous appearance in the common statements concerning string duality. We review the necessary facts concerning the classical geometry of K3 surfaces that will be needed and then we review "old string theory" on K3 surfaces in terms of conformal field theory. The type IIA string, the type IIB string, the E8 x E8 heterotic string, and Spin(32)/Z2 heterotic string on a K3 surface are then each analyzed in turn. The discussion is biased in favour of purely geometric notions concerning the K3 surface itself. These are an extended form of the notes from lectures given at TASI 96. 
  The formalism, which permits to study the phase structure of gauged NJL-model for arbitrary external fields, is developed. The effective potential in the gauged NJL model in the weak magnetic field is found. It is shown that in fixed gauge coupling case the weak magnetic field doesn't influence chiral symmetry breaking condition. The analogy with the situation near black hole is briefly mentioned. 
  A three dimensional supergravity theory which generalizes the super IG theory of Witten and resembles the model discussed recently by Mann and Papadopoulos is displayed. The partition function is computed, and is shown to be a three-manifold invariant generalizing the Casson invariant. 
  In this paper we discuss the M-brane description for a N=2 black hole. This solution is a result of the compactification of M-5-brane configurations over a Calabi-Yau threefold with arbitrary intersection numbers $C_{ABC}$. In analogy to the D-brane description where one counts open string states we count here open 2-branes which end on the M-5-brane. 
  Izawa's gauge-fixing procedure based on BRS symmetry is applied twice to the massive tensor field theory of Fierz-Pauli type. It is shown the second application can remove massless singularities which remain after the first application. Massless limit of the theory is discussed. 
  The conception of the conformal phase transiton (CPT), which is relevant for the description of non-perturbative dynamics in gauge theories, is introduced and elaborated. The main features of such a phase transition are established. In particular, it is shown that in the CPT there is an abrupt change of the spectrum of light excitations at the critical point, though the phase transition is continuous. The structure of the effective action describing the CPT is elaborated and its connection with the dynamics of the partially conserved dilatation current is pointed out. The applications of these results to QCD, models of dynamical electroweak symmetry breaking, and to the description of the phase diagram in (3+1)-dimensional $ SU(N_c)$ gauge theories are considered. 
  A simple backreaction problem in quantum mechanics, the full quantum anharmonic oscillator, and quantum parametric resonance are studied using Renormalization Group techniques for global asymptotic analysis. In this short note this technique is adapted for the first time to operator problems. 
  Can one represent quantum group covariant q-commuting ``creators, annihilators'' $A^+_i,A^j$ as operators acting on standard bosonic/fermionic Fock spaces? We briefly address this general problem and show that the answer is positive (at least) in some simplest cases. 
  We present a simple argument which determines the critical value of the anomaly coefficient in four dimensional conformal factor quantum gravity, at which a phase transition between a smooth and elongated phase should occur. The argument is based on the contribution of singular configurations ("spikes") which dominate the partition function in the infrared. The critical value is the analog of c=1 in the theory of random surfaces, and the phase transition is similar to the Berezenskii-Kosterlitz-Thouless transition. The critical value we obtain is in agreement with the previous canonical analysis of physical states of the conformal factor and may explain why a smooth phase of quantum gravity has not yet been observed in simplicial simulations. We also rederive the scaling relations in the smooth phase in light of this determination of the critical coupling. 
  An M-brane and anti-M-brane scheme is proposed to study nonextremal 4D and 5D black holes. The improved nonextremal intersecting M-brane solutions proposed here, involve two sets of harmonic functions. The constraints among the pressures are found, and new features in the M-brane and anti-M-brane picture are demonstrated, which resolve the discrepancy in the number of free parameters in the D-brane picture. In terms of the ``numbers'' of M-branes and anti-M-branes, the prefactors of the entropies are found to be model independent, and the Bekenstein-Hawking entropy assumes the duality invariant form which is consistent with the microscopic explanation of the black hole entropy. 
  Asymptotic states in field theories containing non-local kinetic terms are analyzed using the canonical method, naturally defined in Minkowski space. We apply our results to study the asymptotic states of a non-local Maxwell-Chern-Simons theory coming from bosonization in 2+1 dimensions. We show that in this case the only asymptotic state of the theory, in the trivial (non-topological) sector, is the vacuum. 
  A few new N=2 superintegrable mappings in the (1|2) superspace are proposed and their origin is analyzed. Using one of them, acting like the discrete symmetry transformation of the N=2 supersymmetric modified NLS hierarchy, the recursion operator and hamiltonian structures of the hierarchy are constructed. 
  Using the linear multiplet formulation for the dilaton superfield, we construct an effective lagrangian for hidden-sector gaugino condensation in string effective field theories with arbitrary gauge groups and matter. Nonperturbative string corrections to the K\"ahler potential are invoked to stabilize the dilaton at a supersymmetry breaking minimum of the potential. When the cosmological constant is tuned to zero the moduli are stabilized at their self-dual points, and the vev's of their F-component superpartners vanish. Numerical analyses of one- and two-condensate examples with massless chiral matter show considerable enhancement of the gauge hierarchy with respect to the E_8 case. The nonperturbative string effects required for dilaton stabilization may have implications for gauge coupling unification. As a comparison, we also consider a parallel approach based on the commonly used chiral formulation. 
  The sum of all ladder and rainbow diagrams in $\phi^3$ theory near 6 dimensions leads to self-consistent higher order differential equations in coordinate space which are not particularly simple for arbitrary dimension D. We have now succeeded in solving these equations, expressing the results in terms of generalized hypergeometric functions; the expansion and representation of these functions can then be used to prove the absence of renormalization factors which are transcendental for this theory and this topology to all orders in perturbation theory. The correct anomalous scaling dimensions of the Green functions are also obtained in the six-dimensional limit. 
  The problem of asymptotic density of quantum states of fundamental extended objects is revised in detail. We argue that in the near-extremal regime the fundamental $p$-brane approach can yield a microscopic interpretation of the black hole entropy. The asymptotic behavior of partition functions, associated with the $p$-branes, and the near-extremal entropy of five-dimensional black holes are explicitly calculated. 
  Some notions in non-perturbative dynamics of supersymmetric gauge theories are being reviewed. This is done by touring through a few examples. 
  We consider quantum holonomy of some three-dimensional general covariant non-Abelian field theory in Landau gauge and confirm a previous result partially proven. We show that quantum holonomy retains metric independence after explicit gauge fixing and hence possesses the topological property of a link invariant. We examine the generalized quantum holonomy defined on a multi-component link and discuss its relation to a polynomial for the link. 
  Problems in lattice gauge models with fermions are discussed. A new bosonic Hermitean effective action for lattice QCD with dynamical quarks is presented. In distinction of the previous version, it does not include constraints and is better suited for Monte-Carlo simulations. 
  We give an exposition of a technique, based on the Zwanzig projection formalism, to construct the evolution equation for the reduced density matrix corresponding to the n-particle sector of a field theory. We consider the case of a scalar field with a $g \phi^3$ interaction as an example and construct the master equation at the lowest non-zero order in perturbation theory. 
  In this paper we are interested in the studying coarse-graining in field theories using the language of quantum open systems. Motivated by the ideas of Calzetta and Hu on correlation histories we employ the Zwanzig projection technique to obtain evolution equations for relevant observables in self-interacting scalar field theories. Our coarse-graining operation consists in concentrating solely on the evolution of the correlation functions of degree less than $n$, a treatment which corresponds to the familiar from statistical mechanics truncation of the BBKGY hierarchy at the n-th level. We derive the equations governing the evolution of mean field and two-point functions thus identifying the terms corresponding to dissipation and noise. We discuss possible applications of our formalism, the emergence of classical behaviour and the connection to the decoherent histories framework. 
  A class of quantum field theories invariant with respect to the action of an odd vector field Q on a source supermanifold $\Sigma$ is considered. We suppose that Q satisfies the conditions under which an integral of any Q-invariant function over $\Sigma$ localizes to the zero locus of Q. The Q-invariant sector of a field theory from the class above is shown then to be equivalent to the quantum field theory defined on zero locus of the vector field Q. 
  Algebraic-geometrical n-orthogonal curvilinear coordinate systems in a flat space are constructed. They are expressed in terms of the Riemann theta function of auxiliary algebraic curves. The exact formulae for the potentials of algebraic geometrical Egoroff metrics and the partition functions of the corresponding topological field theories are obtained. 
  We give the full supersymmetric and kappa-symmetric actions for the Dirichlet p-branes, including their coupling to background superfields of ten-dimensional type IIA and IIB supergravity. 
  Motivated by the necessity to include so-called logarithmic operators in conformal field theories (Gurarie, 1993) at values of the central charge belonging to the logarithmic series c_{1,p}=1-6(p-1)^2/p, reducible but indecomposable representations of the Virasoro algebra are investigated, where L_0 possesses a nontrivial Jordan decomposition. After studying `Jordan lowest weight modules', where L_0 acts as a Jordan block on the lowest weight space (we focus on the rank two case), we turn to the more general case of extensions of a lowest weight module by another one, where again L_0 cannot be diagonalized. The moduli space of such `staggered' modules is determined. Using the structure of the moduli space, very restrictive conditions on submodules of `Jordan Verma modules' (the generalization of the usual Verma modules) are derived. Furthermore, for any given lowest weight of a Jordan Verma module its `maximal preserving submodule' (the maximal submodule, such that the quotient module still is a Jordan lowest weight module) is determined. Finally, the representations of the W-algebra W(2,3^3) at central charge c=-2 are investigated yielding a rational logarithmic model. 
  Some bosonic solutions of supergravities admit Killing spinors of unbroken supersymmetry. The anti-Killing spinors of broken supersymmetry can be used to generate the superpartners of stringy black holes. This has a consequent feedback on the metric and the graviphoton. We have found however that the fixed scalars for the black hole superpartners remain the same as for the original black holes. Possible phenomenological implications of this result are discussed. 
  It is pointed out that the energy of the bound states of D-branes and strings is determined by the central charge of the space-time supersymmetry. The universality which is seen at the black hole horizon appears also on the D-brane side: the total energy of the bound states of a given number of branes has a minimum when considered as a function of the independent parameters (moduli). This provides a new evidence that the near-horizon space-time geometry of the dilaton black holes can be represented by the bound states of branes. The axion-dilaton dyonic black holes have the mass formula of the non-threshold type bound state. Upon uplifting to higher dimensions they may give information about such states. 
  Using a simple hypothesis about the degrees of freedom of intersecting branes we find a microscopic counting argument that reproduces the entropy of a class of BPS black holes of type IIA string theory on general Calabi Yau three folds. 
  The matrix model formulation of M theory can be generalized to compact transverse backgrounds such as tori. If the number of compact directions is K then the matrix model must be generalized to K+1 dimensional super Yang Mills theory on a compact space. If K is greater than or equal to 3, there are T dualities which which require highly nontrivial identifications between different SYM theories. In the simplest case we will see that the requirement reduces to the well known electric- magnetic duality of N=4 SYM theory in 3+1 dimensions. 
  We study toroidal compactifications of Type II string theory with D-branes and nontrivial antisymmetric tensor moduli and show that turning on these fields modifies the supersymmetry projections imposed by D-branes. These modifications are seen to be necessary for the consistency of T-duality. We also show the existence of unusual BPS configurations of branes at angles that are supersymmetric because of conspiracies between moduli fields. Analysis of the problem from the point of view of the effective field theory of massless modes shows that the presence of a 2-form background must modify the realization of supersymmetry on the brane. In particular, the appropriate supersymmetry variation of the physical gaugino vanishes in any constant field strength background. These considerations are relevant for the $E_7$-symmetric counting of states of 4-dimensional black holes in Type II string theory compactified on $T^6$. 
  A path integration formulation for the finite density and temperature problems is shown to be consistent with the thermodynamics using an 8 component ``real'' representation for the fermion fields by applying it to a free fermion system. A relativistic quantum field theory is shown to be smoothly approached at zero temperature by a real-time thermal field theory so derived even at a finite density. The analysis leads to a new representation for the fermion fields which is shown to be inequivalent to the conventional 4 component theory at the quantum level by having a mirror universe with observable effects and to be better behaved at short distances. 
  A short review is given of how to apply the algebraic Heisenberg quantization scheme to a system of identical particles. For two particles in one dimension the approach leads to a generalization of the Bose and Fermi description which can be expressed in the form of a 1/x^2 statistics interaction between the particles. For an N-particle system it is shown how a particular infinite-dimensional algebra arises as a generalization of the su(1,1) algebra which is present for the two-particle system. 
  We discuss the algebraic way of solving the descent equations corresponding to the BRST consistency condition for the gauge anomalies and the Chern--Simons terms on a nontrivial bundle. The method of decomposing the exterior derivative as a BRST commutator is extended to the present case. 
  We consider a generalized three-dimensional theory of gravity which is specified by two fields, the graviton and the dilaton, and one parameter. This theory contains, as particular cases, three-dimensional General Relativity and three-dimensional String Theory. Stationary black hole solutions are generated from the static ones using a simple coordinate transformation. The stationary black holes solutions thus obtained are locally equivalent to the corresponding static ones, but globally distinct. The mass and angular momentum of the stationary black hole solutions are computed using an extension of the Regge and Teitelboim formalism. The causal structure of the black holes is described. 
  The large N limit of a one-dimensional infinite chain of random matrices is investigated. It is found that in addition to the expected Kosterlitz--Thouless phase transition this model exhibits an infinite series of phase transitions at special values of the lattice spacing \epsilon_{pq}=\sin(\pi p/2q). An unusual property of these transitions is that they are totally invisible in the double scaling limit. A method which allows us to explore the transition regions analytically and to determine certain critical exponents is developed. It is argued that phase transitions of this kind can be induced by the interaction of two-dimensional vortices with curvature defects of a fluctuating random lattice. 
  We consider branching of baby universes off parent one in (1+1)-dimensional dilaton gravity with 24 types of conformal matter fields. This theory is equivalent to string theory in a certain background in D=26-dimensional target space, so this process may be also viewed as the emission of a light string state by a heavy string. We find that bare energy is not conserved in (1+1) dimensions due to the emission of baby universes, and that the probability of this process is finite even for local distribution of matter in the parent universe. We present a scenario suggesting that the non-conservation of bare energy may be consistent with the locality of the baby universe emission process in (1+1) dimensions and the existence of the long ranged dilaton field whose source is bare energy. This scenario involves the generation of longitudinal gravitational waves in the parent universe. 
  We study orbifolds of (0,2) models, including some cases with discrete torsion. Our emphasis is on models which have a Landau-Ginzburg realization, where we describe part of the massless spectrum by computing the elliptic genus for the orbifolded theory. Somewhat surprisingly, we find simple examples of (0,2) mirror pairs that are related by a quotient action. We present a detailed description of a family of such pairs. 
  We present a manifestly Lorentz invariant, spacetime supersymmetric, and `$\kappa$-invariant' worldvolume action for all type II Dirichlet p-branes, $p\le9$, in a general type II supergravity background, including massive backgrounds in the IIA case. The $p=0,2$ cases are rederived from D=11. The $p=9$ case provides a supersymmetrization of the D=10 Born-Infeld action. 
  The low-energy background field solutions corresponding to D-brane bound states which possess a difference in dimension of two are presented. These solutions are constructed using the T-duality map between the type IIA and IIB superstring theories. Since supersymmetry is preserved by T-duality, the bound state solutions retain the supersymmetric properties of the initial (single) D-brane states from which they are produced, i.e., they preserve one half of the supersymmetries. 
  We give a microscopic description of extreme and near-extreme Reissner-Nordstrom black holes in four dimensions in terms of fundamental strings in a background of magnetic five branes and monopoles. The string oscillator numbers and tension are rescaled due to the Rindler space background with the string mass fixed. The entropy of the black holes is reproduced correctly by that of the string with the tension rescaling taken into account. 
  We consider the automorphic forms which govern the gravitational threshold correction $F_1$ in models of heterotic/IIA duality with N=2 supersymmetry in four dimensions. In particular we derive the full nonperturbative formula for $F_1$ for the dual pair originally considered by Ferrara, Harvey, Strominger and Vafa (FHSV). The answer involves an interesting automorphic product constructed by Borcherds which is associated to the ``fake Monster Lie superalgebra.'' As an application of this result we rederive a result of Jorgenson & Todorov on determinants of $\bar \partial$ operators on $K3$ surfaces. 
  We study the full unitary matrix models. Introducing a new term $l log U$, l plays the role of the discrete time. On the other hand, the full unitary matrix model contains a topological term. In the continuous limit it gives rise to a phase transition at $\theta=\pi$. The ground state is characterize by the discrete time l. The discrete time l plays like the instanton number. 
  We get the general static, spherically symmetric solutions of the d-dimensional Einstein-Maxwell-Dilaton theories by dimensionally reducing them to a class of 2-dimensional dilaton gravity theories. By studying the symmetries of the actions for the static equations of motion, we find field redefinitions that nearly reduce these theories to the d-dimensional Einstein-Maxwell-Scalar theories, and therefore enable us to get the exact solutions. We do not make any assumption about the asymptotic space-time structure. As a result, our 4-dimensional solutions contain the asymptotically flat Garfinkle-Horowitz-Strominger (GHS) solutions and the non-asymptotically flat Chan-Horne-Mann (CHM) solutions. Besides, we find some new solutions with a finite range of allowed radius of the transversal sphere. These results generalize to an arbitrary space-time dimension d (d>3). 
  We obtain the general static solutions of the axially symmetric (2+1)-dimensional Einstein-Maxwell-Dilaton theory by dimensionally reducing it to a 2-dimensional dilaton gravity theory. The solutions consist of the magnetically charged sector and the electrically charged sector. We illuminate the relationship between the two sectors by pointing out the transformations between them. 
  The spontaneous magnetization in Chern-Simons QED_3 is discussed in a finite temperature system. The thermodynamical potential is analyzed within the weak field approximation and in the fermion massless limit. We find that there is a linear term with respect to the magnetic field with a negative coefficient at any finite temperature. This implies that the spontaneous magnetic field does not vanish even at high temperature. In addition, we examine the photon spectrum in the system. We find that the bare Chern-Simons coefficient is cancelled by the radiative effects. The photons then become topologically massless according to the magnetization, though they are massive by finite temperature effects. Thus the magnetic field is a long-range force without the screening even at high temperature. 
  Quantum Liouville theory is analyzed in terms of the infinite dimensional representations of $U_Qsl(2,C)$ with q a root of unity. Making full use of characteristic features of the representations, we show that vertex operators in this Liouville theory are factorized into `classical' vertex operators and those which are constructed from the finite dimensional representations of $U_qsl(2,C)$. We further show explicitly that fusion rules in this model also enjoys such a factorization. Upon the conjecture that the Liouville action effectively decouples into the classical Liouville action and that of a quantum theory, correlation functions and transition amplitudes are discussed, especially an intimate relation between our model and geometric quantization of the moduli space of Riemann surfaces is suggested. The most important result is that our Liouville theory is in the strong coupling region, i.e., the central charge c_L satisfies $1<c_L<25$. An interpretation of quantum space-time is also given within this formulation. 
  Relying on the geometrical set up of Special K\"ahler Geometry and Quaternionic Geometry, which I discussed at length in my Lectures at the 1995 edition of this Spring School, I present here the recently obtained fully general form of N=2 supergravity with completely arbitrary couplings. This lagrangian has already been used in the literature to obtain various results: notably the partial breaking of supersymmetry and various extremal black--hole solutions. My emphasis, however, is only on providing the reader with a completely explicit and ready to use component expression of the supergravity action. All the details of the derivation are omitted but all the definitions of the items entering the lagrangian and the supersymmetry transformation rules are given. 
  We demonstrate the existence of stable time dependent solutions of the Landau-Lifshitz model with a constant external magnetic field. We find such solutions in all topological sectors, including N=0. We discuss some of their properties. 
  In these lectures we review the properties of holomorphic couplings in the effective action of four-dimensional N=1 and N=2 closed string vacua. We briefly outline their role in establishing a duality among (classes of) different string vacua. (Lectures presented by J. Louis at the Trieste Spring School 1996.) 
  I analyze the one-dimensional, cubic Schr\"odinger equation, with nonlinearity constructed from the current density, rather than, as is usual, from the charge density. A soliton solution is found, where the soliton moves only in one direction. Relation to higher-dimensional Chern--Simons theory is indicated. The theory is quantized and results for the two-body quantum problem agree at weak coupling with those coming from a semiclassical quantization of the soliton. 
  A T-dual version of the Gimon-Polchinski orientifold can be described by a configuration of intersecting Dirichlet seven branes and orientifold seven planes in the classical limit. We study modification of this background due to quantum corrections. It is shown that non-perturbative effects split each orientifold plane into a pair of nearly parallel seven branes. Furthermore, a pair of intersecting orientifold planes, instead of giving rise to two pairs of intersecting seven branes, gives just one pair of seven branes, each representing a pair of nearly orthogonal seven branes smoothly joined to each other near the would be intersection point. Interpretation of these results from the point of view of the dynamics on a three brane probe is also discussed. 
  We prove the uniqueness of the ground state for a supersymmetric quantum mechanical system of two fermions and two bosons, which is closely related to the N=1 WZ-model. The proof is constructive and gives detailed information on what the ground state looks like. 
  Generic partial supersymmetry breaking of N=2 supergravity with zero vacuum energy and with surviving unbroken arbitrary gauge groups is exhibited. Specific examples are given. 
  The emission of a scalar with low energy $\omega$, from a $D (4\le D\le 8 )$ dimensional black hole with n charges is studied in both string and semiclassical calculations. In the lowest order in $\omega$, the weak coupling string and semiclassical calculations agree provided that the Bekenstein--Hawking formula is valid and the effective central charge $c_{eff}=6$ for any D. When the next order in $\omega$ is considered however, there is no agreement between the two schemes unless D=5, n=3 or D=4, n=4. 
  We give an elementary introduction to the recent solution of $N=2$ supersymmetric Yang-Mills theory. In addition, we review how it can be re-derived from string duality. 
  In this paper the Maxwell field theory is considered on a closed and orientable Riemann surface of genus $h>1$. The solutions of the Maxwell equations corresponding to nontrivial values of the first Chern class are explicitly constructed for any metric in terms of the prime form. 
  Contour gauges are discussed in the framework of canonical formalism. We find flux operator algebras with the structure constants of underlying Yang-Mills theory. 
  We describe the ``universal'' action for massless superfields of all superspins in N = 1, D = 4 anti-de Sitter superspace as a gauge theory of unconstrained superfields taking their values in the commutative algebra of analytic functions over a one-sheeted hyperboloid in $R^{3,1}$. The action is invariant under N = 2 supersymmetry transformations which form a closed algebra off the mass-shell. 
  A method is presented to compute the order of the untwisted stabilizer of a simple current orbit, as well as some results about the properties of the resolved fields in a simple current extension. 
  Two classes of stringy instanton effects, stronger than standard field theory instantons, are identified in the heterotic string theory. These contributions are established using type IIA/heterotic and type I/heterotic dualities. They provide examples for the heterotic case of the effects predicted by Shenker based on the large-order behavior of perturbation theory. The corrections vanish as the radius of the compactification goes to infinity. For appropriate amplitudes, they are computable worldsheet or worldline instanton effects on the dual side. Some potential applications are discussed. 
  In this paper we work out explicit lagrangians describing superpotential coupling to the boundary of a 5D orientifold, as relevant to a number of quasi-realistic models of nature. We also make a number of general comments on orientifold compactifications of M theory. 
  We study the patterns in the duality of a wide class of N=1 supersymmetric gauge theories in four dimensions. We present many new generalizations of the classic duality models of Kutasov and Schwimmer, which have themselves been generalized numerous times in works of Intriligator, Leigh and the present authors. All of these models contain one or two fields in a two-index tensor representation, along with fields in the defining representation. The superpotential for the two-index tensor(s) resembles A_k or D_k singularity forms, generalized from numbers to matrices. Looking at the ensemble of these models, classifying them by superpotential, gauge group, and ``level'' -- for terminology we appeal to the architecture of a typical European-style theatre -- we identify emerging patterns and note numerous interesting puzzles. 
  Based on the Schwinger-Dyson (SD) equation, the fermion mass generation is further studied in the D(2<D<4)-dimensional Thirring model as a gauge theory previously proposed. By using a certain approximation to the kernel, we analytically obtained explicit form of the dynamical mass of fermion and the critical line in (N,1/g) space, where N is the number of fermions and g is the dimensionless vector-type four-fermion coupling constant. This analytical result is confirmed by the numerical solution for the SD equation with exact form of the kernel in (2+1) dimensions. 
  We review aspects of Poisson-Lie T-duality which we explicitly formulate as a canonical transformation on the world-sheet. Extensions of previous work on T-duality in relation to supersymmetry are also discussed. (Contribution to the proceedings of the 30th International Symposium Ahrenshoop on the Theory of Elementary Particles, Buckow, Germany, 26-31 August 1996) 
  We define certain extensions of affine Weyl groups (distinct from these considered by K. Saito [S1] in the theory of extended affine root systems), prove an analogue of Chevalley theorem for their invariants, and construct a Frobenius structure on their orbit spaces. This produces solutions $F(t_1, ..., t_n)$ of WDVV equations of associativity polynomial in $t_1, ..., t_{n-1}, \exp t_n$. 
  We present M-theory compactifications on $K_3 \times K_3$ with membranes near the $A_n$ or $D_n$ singularities of the $K_3$ spaces. By realizing each of these compactifications in two different ways as type I' models with 2- and 6-branes, we explain the three-dimensional duality between gauge theories recently found by Intriligator and Seiberg. We also find new pairs of dual gauge theories, which we briefly describe. 
  We use the T-duality transformation which relates M-theory on T^3 to M-theory on a second T^3 with inverse volume to test the Banks-Fischler-Shenker-Susskind suggestion for the matrix model description of M-theory. We find evidence that T-duality is realized as S-duality for $U(\infty)$ N=4 Super-Yang-Mills in 3+1D. We argue that Kaluza Klein states of gravitons correspond to electric fluxes, wrapped membranes become magnetic fluxes and instantonic membranes are related to Yang-Mills instantons. The T-duality transformation of gravitons into wrapped membranes is interpreted as the duality between electric and magnetic fluxes. The identification of M-theory T-duality as SYM S-duality provides a natural framework for studying the M-theory 5-brane as the S-dual object to the unwrapped membrane. Using the equivalence between compactified M(atrix) theory and SYM, we find a natural candidate for a description of the light-cone 5-brane of M-theory directly in terms of matrix variables, analogous to the known description of the M(atrix) theory membrane. 
  We give an elementary introduction to the theory of supermembranes. 
  We show how non-trivial form fields can induce an effective potential for the dilaton and metric moduli in compactifications of type II string theory and M-theory. For particular configurations, the potential can have a stable minimum. In cosmological compactifications of type II theories, we demonstrate that, if the metric moduli become fixed, this mechanism can then lead to the stabilization of the dilaton vacuum. Furthermore, we show that for certain cosmological M-theory solutions, non-trivial forms lead to the stabilization of moduli. We present a number of examples, including cosmological solutions with two solitonic forms and examples corresponding to the infinite throat of certain p-branes. 
  We discuss $F_{\mu\nu}^4$ terms in torroidal compactifications of type-I and heterotic SO(32) string theory. We give a simple argument why only short BPS multiplets contribute to these terms at one loop, and verify heterotic-type-I duality to this order. 
  Quantum mechanics of a test particle interacting with a spinning string (torsion vortex) is the quantum mechanics of the celebrated Aharonov-Bohm effect. The angular momentum per unit length $J$ characterizing the spinning string corresponds to the magnetic flux $\Phi$ of the Aharonov-Bom fluxon and the gravitational mass-energy $E$ of an incoming particle corresponds to the electric charge $q$. The characteristic periodicity of the gravitational Aharonov-Bohm scattering cross-section in the product of $GEJ$ has suggested a new quantization of mass-energy relation. We encounter a new physical situation where the gravitational mass-energy is quantized and the quantization condition contains both the Newton constant $G$ and the Planck constant $h$. 
  We present a new class of matrix models which are manifestly symmetric under the T-duality transformation of the target space. The models may serve as a nonperturbative regularization for the T-duality symmetry in continuum string theory. In particular, it now becomes possible to extract winding modes explicitly in terms of extended matrix variables. 
  Correlation functions of gauged WZNW models are shown to satisfy a differential equation, which is a gauge generalization of the Knizhnik-Zamolodchikov equation. 
  An elementary introduction into the Seiberg-Witten theory is given. Many efforts are made to get it as pedagogical as possible, within a reasonable size. The selection of the relevant material is heavily oriented towards graduate students. The basic ideas about solitons, monopoles, supersymmetry and duality are reviewed from first principles, and they are illustrated on the simplest examples. The exact Seiberg-Witten solution to the low-energy effective action of the four-dimensional N=2 supersymmetric pure Yang-Mills theory with the gauge group SU(2) is the main subject of the review. Other gauge groups are also considered. Some related issues (like adding matter, confinement, string dualities) are outlined. 
  We discuss the entropy and the transformation properties of classical extremal N=2 black hole solutions in supergravity theories associated with the minimal coupling models CP(n-1,1). The entropy is given by a manifestly invariant quantity under the embedding of the duality group SU(1,n) into Sp(2n+2) which is a symmetry of the classical BPS mass formula. 
  In this letter the Chern-Simons field theories are studied in the Coulomb gauge using the Dirac's canonical formalism for constrained systems. As a strategy, we first work out the constraints and then quantize, replacing the Dirac brackets with quantum commutators. We find that the Chern-Simons field theories become two dimensional models with no propagation along the time direction. Moreover, we prove that, despite of the presence of non-trivial self-interactions in the gauge fixed functional, the commutation relations between the fields are trivial at any order in perturbation theory in the absence of couplings with matter fields. If these couplings are present, instead, the commutation relations become rather involved, but it is still possible to study their main properties and to show that they vanish at the tree level. 
  It is shown that the Regge-Teitelboim criterion for fixing the unique boundary contribution to the Hamiltonian compatible with free boundary conditions should be modified if the Poisson structure is noncanonical. The new criterion requires cancellation of boundary contributions to the Hamiltonian equations of motion. In the same time, boundary contributions to the variation of Hamiltonian are allowed. The Ashtekar formalism for gravity and hydrodynamics of the ideal fluid with a free surface in the Clebsch variables are treated as examples. 
  Supersymmetry breaking and compactification of extra space-time dimensions may have a common dynamical origin if our universe is spontaneously generated in the form of a four-dimensional topological or non-topological defect in higher dimensional space-time. Within such an approach the conventional particles are zero modes trapped in the core of the defect. In many cases solutions of this type spontaneously break all supersymmetries of the original theory, so that the low-energy observer from ``our'' universe inside the core would not detect supersymmetry. Since the extra dimensions are not compact but, rather, inaccessible to low-energy observers, the usual infinite tower of the Kaluza-Klein excitations does not exist. Production of superpartners at the energy scale of SUSY restoration will be accompanied by four-momentum non-conservation. (Depending on the nature of the solution at hand, the non-conservation may either happen above some threshold energy or be continuous). In either case, the door to extra dimensions may be not very far from the energies accessible at present colliders. 
  We review a number of perturbative calculations describing the interactions of D-branes with massless elementary string states. The form factors for the scattering of closed strings off D-branes are closely related to the Veneziano amplitude. They show that, in interactions with strings, D-branes acquire many of their physical features: the effective size of D-branes is of order the string scale and expands with the energy of the probe, while the fixed angle scattering amplitudes fall off exponentially. We also calculate the leading process responsible for the absorption of closed strings: the amplitude for a closed string to turn into a pair of open strings attached to the D-brane. The inverse of this process describes the Hawking radiation by an excited D-brane. 
  We analyze some of the kinematical and dynamical properties of flat infinite membrane solutions in the conjectured M theory proposed by Banks, Fischler, Shenker and Susskind. In particular, we compute the long range potential between membranes and anti-membranes, and between membranes and gravitons, and compare it with the supergravity results. We also discuss membranes with finite relative longitudinal velocities, providing some evidence for the eleven dimensional Lorentz invariance of the theory. 
  The foundation for the theory of correlation functions of exactly solvable models is determinant representation. Determinant representation permit to describe correlation functions by classical completely integrable differential equations [Barough, McCoy, Wu]. In this paper we show that determinant represents works not only for free fermionic models. We obtained determinant representation for the correlation function $<\psi(0,0)\psi^\dagger(x,t)>$ of the quantum nonlinear Schr\"odinger equation, out of free fermionic point. In the forthcoming publications we shall derive completely integrable equation and asymptotic for the quantum correlation function of this model of interacting fermions. 
  We study various classical solutions of the baby-Skyrmion model in $(2+1)$ dimensions. We point out the existence of higher energy states interpret them as resonances of Skyrmions and anti-Skyrmions and study their decays. Most of the discussion involves a highly exited Skyrmion-like state with winding number one which decays into an ordinary Skyrmion and a Skyrmion-anti-Skyrmion pair. We also study wave-like solutions of the model and show that some of such solutions can be constructed from the solutions of the sine-Gordon equation. We also show that the baby-Skyrmion has non-topological stationary solutions. We study their interactions with Skyrmions. 
  An explanation of the origin of the hidden eleventh dimension in string theory is given. It is shown that any two sigma models describing the propagation of string backgrounds are related to each other by a Weyl transformation of the world-sheet metric. To avoid this ambiguity in defining two-dimensional sigma models, extra fields are needed. An interesting connection is established with Abelian T-duality. 
  We construct a one-loop effective metric describing the evaporation phase of a Schwarzschild black hole in a spherically symmetric null-dust model. This is achieved by quantising the Vaidya solution and by chosing a time dependent quantum state. This state describes a black hole which is initially in thermal equilibrium and then the equilibrium is switched off, so that the black hole starts to evaporate, shrinking to a zero radius in a finite proper time. The naked singularity appears, and the Hawking flux diverges at the end-point. However, a static metric can be imposed in the future of the end-point. Although this end-state metric cannot be determined within our construction, we show that it cannot be a flat metric. 
  We introduce higher-order (or multibracket) simple Lie algebras that generalize the ordinary Lie algebras. Their `structure constants' are given by Lie algebra cohomology cocycles which, by virtue of being such, satisfy a suitable generalization of the Jacobi identity. Finally, we introduce a nilpotent, complete BRST operator associated with the l multibracket algebras which are based on a given simple Lie algebra of rank l. 
  New generalized Poisson structures are introduced by using skew-symmetric contravariant tensors of even order. The corresponding `Jacobi identities' are given by the vanishing of the Schouten-Nijenhuis bracket. As an example, we provide the linear generalized Poisson structures which can be constructed on the dual spaces of simple Lie algebras. 
  We discuss the field theory of 3-brane probes in F-theory compactifications in two configurations, generalizing the work of Sen and of Banks, Douglas and Seiberg. One configuration involves several parallel 3-brane probes in F-theory compactified on $T^4/Z_2$, while the other involves a compactification of F-theory on $T^6/Z_2 x Z_2$ (which includes intersecting $D_4$ singularities). In both cases string theory provides simple pictures of the spacetime theory, whose implications for the three-brane world-volume theories are discussed. In the second case the field theory on the probe is an unusual N=1 superconformal theory, with exact electric-magnetic duality. Several open questions remain concerning the description of this theory. 
  We present a class of graviton-dilaton models which leads to a singularity free evolution of the universe. We study the evolution of a homogeneous isotropic universe. We follow an approach which enables us to analyse the evolution and obtain its generic features even in the absence of explicit solutions, which are not possible in general. We describe the generic evolution of the universe and show, in particular, that it is singularity free in the present class of models. Such models may stand on their own as interesting models for singularity free cosmology, and may be studied accordingly. They may also arise from string theory. We discuss critically a few such possibilities. 
  In this paper we analyse the perturbative aspects of Chern-Simons field theories in the Coulomb gauge. We show that in the perturbative expansion of the Green functions there are neither ultraviolet not infrared divergences. Moreover, all the radiative corrections are zero at any loop order. Some problems connected with the Coulomb gauge fixing, like the appearance of spurious singularities in the computation of the Feynman diagrams, are discussed and solved. The regularization used here for the spurious singularities can be easily applied also to the Yang-Mills case, which is affected by similar divergences. 
  The integral representations of the $\hat {sl}(2,C)$ Spin 1/2 - Spin 1/2 Kac-Moody Blocks on the torus, arising from the free field representation of the $\hat {sl}(2,C)$ Kac-Moody algebra of Wakimoto and Bernard and Felder, are used to derive an infinite class of representations of the mapping class group of the two punctured torus. 
  By taking a product of two sl(2) representations, we obtain the differential operators preserving some space of polynomials in two variables. This allows us to construct the representations of osp(2,2) in terms of matrix differential operators in two variables. The corresponding operators provide the building blocks for the construction of quasi exactly solvable systems of two and four equations in two variables. Some generalisations are also sketched. The peculiar labelling used for the generators allows us to elaborate a nice deformation of osp(2,2). This gives an appropriate basis for analyzing the quasi exactly solvable systems of finite difference equations. 
  Assuming the monopole dominance, that has been proved in the lattice gluodynamics, to hold in the continuum limit, we develop an effective scalar field theory for QCD at large distances to describe confinement. The approach is based on a gauge (or projection) independent formulation of the monopole dominance and manifestly Lorentz invariant. 
  A projection (gauge) independent formulation of the monopole dominance, discovered in lattice QCD for the maximal abelian projection, is given. A new dynamical abelian projection of continuum QCD, which does not rely on any explicit gauge condition imposed on gauge fields, is proposed. Under the assumption that the results of numerical simulations holds in the continuum limit, the monopole dominance is proved for the dynamical abelian projection. The latter enables us to develop an effective scalar field theory for dominant (monopole) configurations of gauge fields. The approach is manifestly gauge and Lorentz invariant. 
  A classification scheme of hadrons is proposed on the basis of the division algebra H of quaternions and an appropriate geometry. This scheme suggests strongly to understand flavour symmetry in another manner than from standard symmetry schemes. In our approach, we do not start from `exact' symmetry groups like SU(2) \times SU(2) chiral symmetry and impose various symmetry breaking mechanisms which collide with theorems wellknown from quantum field theory. On the contrary, the approximate symmetry properties of the hadron spectrum at low energies, usually classified by `appropriately' broken compact flavour groups, emerge very naturally as a low energy reduction of the noncompact (dynamical) symmetry group Sl(2,H). This quaternionic approach not only avoids most of the wellknown conceptual problems of Chiral Dynamics but it also allows for a general treatment of relativistic flavour symmetries as well as it yields a direct connection towards classical relativistic symmetry. 
  We propose an explanation via string theory of the correspondence between the Coulomb branch of certain three-dimensional supersymmetric gauge theories and certain moduli spaces of magnetic monopoles. The same construction also gives an explanation, via $SL(2,\Z)$ duality of Type IIB superstrings, of the recently discovered ``mirror symmetry'' in three dimensions. New phase transitions in three dimensions as well as new infrared fixed points and even new coupling constants not present in the known Lagrangians are predicted from the string theory construction. An important role in the construction is played by a novel aspect of brane dynamics in which a third brane is created when two branes cross. 
  Following on from earlier work relating modules of meromorphic bosonic conformal field theories to states representing solutions of certain simple equations inside the theories, we show, in the context of orbifold theories, that the intertwiners between twisted sectors are unique and described explicitly in terms of the states corresponding to the relevant modules. No explicit knowledge of the structure of the twisted sectors is required. Further, we propose a general set of sufficiency conditions, illustrated in the context of a third order no-fixed-point twist of a lattice theory, for verifying consistency of arbitrary orbifold models in terms of the states representing the twisted sectors. 
  We show that a U(1) gauge theory defined in the configuration space for closed p-branes yields the gauge theory of a massless rank-(p+1) antisymmetric tensor field and the Stueckelberg formalism for a massive vector field. 
  We apply noncommutative geometry to a system of N parallel D-branes, which is interpreted as a quantum space. The Dirac operator defining the quantum differential calculus is identified to be the supercharge for strings connecting D-branes. As a result of the calculus, Connes' Yang-Mills action functional on the quantum space reproduces the dimensionally reduced U(N) super Yang-Mills action as the low energy effective action for D-brane dynamics. Several features that may look ad hoc in a noncommutative geometric construction are shown to have very natural physical or geometric origin in the D-brane picture in superstring theory. 
  Based on the technique of derivation of a theory, presented in our recent paper, we investigate the properties of the derived quantum system. We show that the derived quantum system possesses the (nonanomalous) symmetries of the original one, and prove that the exact Green functions of the derived theory are expressed in terms of the semiclassically approximated Green functions of the original theory. 
  Supersymmetric Yang-Mills theories are considered in 1+1 dimensions. Firstly physical mass spectra of supersymmetric Yang-Mills theories in 1+1 dimensions are evaluated in the light-cone gauge with a compact spatial dimension. The supercharges are constructed in order to provide a manifestly supersymmetric infrared regularization for the discretized light-cone approach. By exactly diagonalizing the supercharge matrix between up to several hundred color singlet bound states, we find a rapidly increasing density of states as mass increases. Interpreting this limiting density of states as the stringbehavior, we obtain the Hagedron temperature $\beta_H=0.676 \sqrt{\pi \over g^2 N}$. Secondly we have examined the vacuum structure of supersymmetric Yang-Mills theories in 1+1 dimensions. SUSY allows only periodic boundary conditions for both fermions and bosons. By using the Born-Oppenheimer approximation for the weak coupling limit, we find that the vacuum energy vanishes, and hence the SUSY is unbroken. Other boundary conditions are also studied. The first part is based on a work in collaboration with Y. Matsumura and T. Sakai. The second part is based on a work in collaboration with H. Oda and T. Sakai. 
  The character problems of SU(2) and SU(1,1) are reexamined from the standpoint of a physicist by employing the Hilbert space method which is shown to yield a completely unified treatment for SU(2) and the discrete series of representations of SU(1,1). For both the groups the problem is reduced to the evaluation of an integral which is invariant under rotation for SU(2) and Lorentz transformation for SU(1,1). The integrals are accordingly evaluated by applying a rotation to a unit position vector in SU(2) and a Lorentz transformation to a unit SO(2,1) vector which is time-like for the elliptic elements and space-like for the hyperbolic elements in SU(1,1). The details of the procedure for the principal series of representations of SU(1,1) differ substantially from those of the discrete series.  
  Intersecting D-brane configurations are related to black holes in D=4. Using the standard way of compactification only the Reissner-Nordstr{\o}m black hole is non-singular. In this paper we argue, that also the other black holes are non-singular if i) we compactify over a periodic array and ii) we allow the string metric after reaching a critical curvature to choose the dual geometry. Effectively this means that near the horizon the solution completely decompactifies and chooses a non-singular D-brane configuration. 
  We propose an explicit expression for vacuum expectation values of the exponential fields in the sine-Gordon model. Our expression agrees both with semi-classical results in the sine-Gordon theory and with perturbative calculations in the Massive Thirring model. We use this expression to make new predictions about the large-distance asymptotic form of the two-point correlation function in the XXZ spin chain. 
  The asymptotic conformal invariance of some SU(2) model and Standard Model in curved space-time are investigated. We have examined the conditions for asymptotic conformal invariance for these models numerically. 
  We comment on the regularization of the expectation values of Wilson surfaces for the bosonic string in the (3+1) dimensions. We analyze the singular behaviors of propagator for the Chern-Simons action with the additional higher order terms. 
  First steps in incorporating Nottale's scale-relativity principle to string theory and extended objects are taken. Scale Relativity is to scales what motion Relativity is to velocities. The universal, absolute, impassible, invariant scale under dilatations, in Nature, is taken to be the Planck scale which is not the same as the string scale. Starting with Nambu-Goto actions for strings and other extended objects, we show that the principle of scale-relativity invariance of the world-volume measure associated with the extended objects ( Lorentzian-scalings transformations with respect to the resolutions of the world-volume coordinates) is compatible with the vanishing of the scale-relativity version of the $\beta$ functions : $\beta^G_{\mu\nu}=\beta^X=0$, of the target spacetime metric and coordinates, respectively. Preliminary steps are taken to merge motion relativity with scale relativity and, in this fashion, analogs of Weyl-Finsler geometries make their appearance. The quantum case remains to be studied. 
  The Hamiltonian of the wrapping and KK modes of the supermembrane is identified with the SL(2, Z) symmetric axion-dilaton black hole mass formula. It means that the supermembrane with KK modes wrapped $m$ times around the space-time torus of compactified dimensions induces the superpotential breaking spontaneously N=2 down to N=1 SUSY. The supersymmetry breaking parameter Lambda is inversely proportional to the area A of the space-time torus around which the supermembrane wraps. The restoration of supersymmetry as well as decompactification of higher dimensions of space-time are forbidden by the requirement of stability of the wrapped supermembrane. This may be an ultimate reason why supersymmetry is broken in the real world. 
  Under certain conditions, imposed on the viscosity of the fluid, initial data and the class of contours under consideration, the Cauchy problem with finite values of time for the loop equation in turbulence with Gaussian random forces is solved by making use of the smearing procedure for the loop space functional Laplacian. The solution obtained depends on the initial data and its functional derivatives and on the potential of the random forces. 
  We investigate how a uniformly rotating frame is defined as the rest frame of an observer rotating with constant angular velocity $\Omega$ around the $z$ axis of an inertial frame. Assuming that this frame is a Lorentz one, we second quantize a free massless scalar field in this rotating frame and obtain that creation-anihilation operators of the field are not the same as those of an inertial frame. This leads to a new vacuum state --- a rotating vacuum --- which is a superposition of positive and negative frequency Minkowski particles. After this, introducing an apparatus device coupled linearly with the field we obtain that there is a strong correlation between number of rotating particles (in a given state) obtained via canonical quantization and via response function of the rotating detector. Finally, we analyse polarization effects in circular accelerators in the proper frame of the electron making a connection with the inertial frame point of view. 
  We construct dyonic states in 2+1-dimensional lattice Z_N-Higgs models, i.e., states which are both, electrically and magnetically charged. The associated Hilbert spaces carry charged representations of the observable algebra, the global transfer matrix and a unitary implementation of the group of spatial lattice translations. We prove that for coinciding total charges these representations are dynamically equivalent and we construct a local intertwiner connection depending on a path in the space of charge distributions. The holonomy of this connection is given by Z_N-valued phases. This will be the starting point for a construction of scattering states with anyon statistics in a subsequent paper. 
  We discuss supersymmetry in twelve dimensions and present a covariant supersymmetric action for a brane with worldsheet signature (2,2), called a super (2+2)-brane, propagating in the osp(64,12) superspace. This superspace is explicitly constructed, and is trivial in the sense that the spinorial part is a trivial bundle over spacetime, unlike the twisted superspace of usual Poincare supersymmetry. For consistency, it is necessary to take a projection of the superspace. This is the same as the projection required for worldvolume supersymmetry. Upon compactification of this superspace, a torsion is naturally introduced and we produce the membrane and type IIB string actions in 11 and 10 dimensional Minkowski spacetimes. In addition, the compactification of the twelve dimensional supersymmetry algebra produces the correct algebras for these theories, including central charges. These considerations thus give the type IIB string and M-theory a single twelve dimensional origin. 
  Simplicial versions of topological abelian gauge theories are constructed which reproduce the continuum expressions for the partition function and Wilson expectation value of linked loops, expressible in terms of R-torsion and linking numbers respectively. The new feature which makes this possible is the introduction of simplicial fields (cochains) associated with the dual triangulation of the background manifold, as well as with the triangulation itself. This doubling of fields, reminiscent of lattice fermion doubling, is required because the natural simplicial analogue of the Hodge star operator maps between cochains of a triangulation and cochains of the dual triangulation. The simplicial analogue of Hodge-de Rham theory is developed, along with a natural simplicial framework for considering linking numbers of framed loops. When the loops represent torsion elements of the homology of the manifold then Q/Z-valued torsion pairings appear in place of linking numbers for certain discrete values of the coupling parameter of the theory. 
  We find that the overall UV divergences of a renormalizable field theory with trivalent vertices fulfil a four-term relation. They thus come close to establish a weight system. This provides a first explanation of the recent successful association of renormalization theory with knot theory. 
  At four loops there first occurs a test of the four-term relation derived by the second author in the course of investigating whether counterterms from subdivergence-free diagrams form a weight system. This test relates counterterms in a four-dimensional field theory with Yukawa and $\phi^4$ interactions, where no such relation was previously suspected. Using integration by parts, we reduce each counterterm to massless two-loop two-point integrals. The four-term relation is verified, with $<G_1-G_2+G_3-G_4> = 0 - 3\zeta_3 + 6\zeta_3 - 3\zeta_3 = 0$, demonstrating non-trivial cancellation of the trefoil knot and thus supporting the emerging connection between knots and counterterms, via transcendental numbers assigned by four-dimensional field theories to chord diagrams. Restrictions to scalar couplings and renormalizable interactions are found to be necessary for the existence of a pure four-term relation. Strong indications of richer structure are given at five loops. 
  Multiple zeta values (MZVs) are under intense investigation in three arenas -- knot theory, number theory, and quantum field theory -- which unite in Kreimer's proposal that field theory assigns MZVs to positive knots, via Feynman diagrams whose momentum flow is encoded by link diagrams. Two challenging problems are posed by this nexus of knot/number/field theory: enumeration of positive knots, and enumeration of irreducible MZVs. Both were recently tackled by Broadhurst and Kreimer (BK). Here we report large-scale analytical and numerical computations that test, with considerable severity, the BK conjecture that the number, $D_{n,k}$, of irreducible MZVs of weight $n$ and depth $k$, is generated by $\prod_{n\ge3}\prod_{k\ge1}(1-x^n y^k) ^{D_{n,k}}=1-\frac{x^3y}{1-x^2}+\frac{x^{12}y^2(1-y^2)}{(1-x^4)(1-x^6)}$, which is here shown to be consistent with all shuffle identities for the corresponding iterated integrals, up to weights $n=44, 37, 42, 27$, at depths $k=2, 3, 4, 5$, respectively, entailing computation at the petashuffle level. We recount the field-theoretic discoveries of MZVs, in counterterms, and of Euler sums, from more general Feynman diagrams, that led to this success. 
  We simplify, to a single integral of dilogarithms, the least tractable O(1/N^3) contribution to the large-N critical exponent $\eta$ of the non-linear sigma-model, and hence $\phi^4$-theory, for any spacetime dimensionality, D. It is the sole generator of irreducible multiple zeta values in epsilon-expansions with $D=2-2\epsilon$, for the sigma-model, and $D=4-2\epsilon$, for $\phi^4$-theory. In both cases we confirm results of Broadhurst, Gracey and Kreimer (BGK) that relate knots to counterterms. The new compact form is much simpler than that of BGK. It enables us to develop 8 new terms in the epsilon-expansion with $D=3-2\epsilon$. These involve alternating Euler sums, for which the basis of irreducibles is larger. We conclude that massless Feynman diagrams in odd spacetime dimensions share the greater transcendental complexity of massive diagrams in even dimensions, such as those contributing to the electron's magnetic moment and the electroweak $\rho$-parameter. Consequences for the perturbative sector of Chern-Simons theory are discussed. 
  We examine the pair creation of black holes in the presence of supergravity domain walls with broken and unbroken supersymmetry. We show that black holes will be nucleated in the presence of non- extreme, repulsive walls which break the supersymmetry, but that as one allows the parameter measuring deviation from extremality to approach zero the rate of creation will be suppressed. In particular, we show that the probability for creation of black holes in the presence of an extreme domain wall is identically zero, even though an extreme vacuum domain wall still has repulsive gravitational energy. This is consistent with the fact that the supersymmetric, extreme domain wall configurations are BPS states and should be stable against quantum corrections. We discuss how these walls arise in string theory, and speculate about what string theory might tell us about such objects. 
  As a further test of the conjectured equivalence of string states and extremal black holes, we compute the dipole moments of black holes with arbitrary spin and superspin in D=4,N=4 supergravity coupled to 22 vector multiplets and compare them with the dipole moments of states in the heterotic string on $T^6$ or the Type IIA string on $K3 \times T^2$. Starting from a purely bosonic black hole with Kerr angular momentum L, the superpartners are generated by acting with fermion zero modes, thus filling out the complete supermultiplet. $L$ is then identified with the superspin. On the heterotic side, elementary states belong only to short to long multiplets, but Type IIA elementary states can belong to intermediate multiplets as well. We find that the black hole gyromagnetic ratios are in perfect agreement with the string states not only for the BPS states belonging to short multiplets but also for those belonging to intermediate multiplets. In fact, these intermediate multiplets provide a stronger test of the black-hole/string-state equivalence because the gyromagnetic ratios are not determined by supersymmetry alone, in contrast to those of the short multiplets. We even find agreement between the non-supersymmetric (but still extremal) black holes and non-BPS string states belonging to long supermultiplets. In addition to magnetic dipole moments we also find electric dipole moments even for purely electrically charged black holes. The electric dipole moments of the corresponding string states have not yet been calculated directly but are consistent with heterotic/Type IIA duality. 
  On the basis of graded RTT formalism,the defining relation of the super-Yangian Y(gl(1|1)) is derived and its oscillator realization is constructed. 
  We consider a new D=2 nonrelativistic classical mechanics model providing via the Noether theorem the (2+1)-Galilean symmetry algebra with two central charges: mass m and the coupling constant k of a Chern-Simons-like term. In this way we provide the dynamical interpretation of the second central charge of the (2+1)-dimensional Galilean algebra. We discuss also the interpretation of k as describing the noncommutativity of D=2 space coordinates. The model is quantized in two ways: using the Ostrogradski-Dirac formalism for higher order Lagrangians with constraints and the Faddeev-Jackiw method which describes constrained systems and produces nonstandard symplectic structures. We show that our model describes the superposition of a free motion in noncommutative D=2 space as well as the "internal" oscillator modes. We add a suitably chosen class of velocity-dependent two-particle interactions, which is descrobed by local potentials in D=2 noncommutative space. We treat, in detail, the particular case of a harmonic oscillator and describe its quantization. It appears that the indefinite metric due to the third order time derivative term in the field equations, even in the presence of interactions, can be eliminated by the imposition of a subsidiary condition. 
  Using the path integral method, we calculate the partition function and the generating functional (of the field strengths) of the generalized 2D Yang-Mills theories in the Schwinger--Fock gauge. Our calculation is done for arbitrary 2D orientable, and also nonorientable surfaces. 
  We study the gauge dependence of the effective average action Gamma_k and Newtonian gravitational constant using the RG equation for Gamma_k. Then we truncate the space of action functionals to get a solution of this equation. We solve the truncated evolution equation for the Einstein gravity in the De Sitter background for a general gauge parameter alpha and obtain a system of equations for the cosmological and the Newtonian constants. Analyzing the running of the gravitational constant we find that the Newtonian constant depends strongly on the gauge parameter. This leads to the appearance of antiscreening and screening behavior of the quantum gravity. The resolution of the gauge dependence problem is suggested. For physical gauges like the Landau-De Witt gauge the Newtonian constant shows an antiscreening. 
  A numerical study of static, spherically symmetric sphaleron solutions in the standard model coupled to the dilaton field is presented. We show that sphaleron is surrounded by strong dilaton cloud which vanishes inside the sphaleron. 
  We consider a dimensional reduction of 3+1 dimensional SU(N) Yang-Mills theory coupled to adjoint fermions to obtain a class of 1+1 dimensional matrix field theories. We derive the quantized light-cone Hamiltonian in the light-cone gauge A_- = 0 and large-N limit, and then solve for the masses, wavefunctions and structure functions of the color singlet ``meson-like'' and ``baryon-like'' boundstates. Among the states we study are many massless string-like states that can be solved for exactly. 
  A novel probe of D-brane dynamics is via scattering of a high energy ripple traveling along an attached string. The inelastic processes in which the D-brane is excited through emission of an additional attached string is considered. Corresponding amplitudes can be found by factorizing a one-loop amplitude derived in this paper. This one-loop amplitude is shown to have the correct structure, but extraction of explicit expressions for the scattering amplitudes is difficult. It is conjectured that the exponential growth of available string states with energy leads to an inclusive scattering rate that becomes large at the string scale, due to excitation of the ``string halo,'' and meaning that such probes do not easily see structure at shorter scales. 
  A recently found new free field realization of the affine Sugawara operators at arbitrary level is reviewed, which involves exponentials of the well-known DDF operators in string theory. 
  It is shown that time-harmonic motions of spherical and toroidal surfaces can be deformed non-locally without loosing the existence of infinitely many constants of the motion. 
  We investigate the quantum effects of the non-minimal matter-gravity couplings derived by Cangemi and Jackiw in the realm of a specific fermionic theory, namely the abelian Thirring model on a Riemann surface of genus zero and one. The structure and the strength of the new interactions are seen to be highly constrained, when the topology of the underlying manifold is taken into account. As a matter of fact, by requiring to have a well-defined action, we are led both to quantization rules for the coupling constants and to selection rules for the correlation functions. Explicit quantum computations are carried out in genus one (torus). In particular the two-point function and the chiral condensate are carefully derived for this case. Finally the effective gravitational action, coming from integrating out the fermionic degrees of freedoom, is presented. It is different from the standard Liouville one: a new non-local functional of the conformal factor arises and the central charge is improved, depending also on the Thirring coupling constant. This last feature opens the possibility of giving a new explicit representation of the minimal series in terms of a fermionic interacting model. 
  We prove a determinantal formula for quantities related to the problem of enumeration of (semi-) meanders, namely the topologically inequivalent planar configurations of non-self-intersecting loops crossing a given (half-) line through a given number of points. This is done by the explicit Gram-Schmidt orthogonalization of certain bases of subspaces of the Temperley-Lieb algebra. 
  We study a class of noncommutative geometries that give rise to dimensionally reduced Yang-Mills theories. The emerging geometries describe sets of copies of an even dimensional manifold. Similarities to the D-branes in string theory are discussed. 
  We obtain a self-dual formulation of the conventional nonlinear Schr\"odinger equation (NLSE) in the 1+1 dimension by studying the dimensional reduction of the self-dual Chern-Simons nonlinear Schr\"odinger model (NLSM) in the 2+1 dimension. It is found that this self-dual formulation allows us to find not only the well-known soliton solutions from the Bogomol'nyi bound and the Galilean boost, but also other soliton solutions in the presence of the background sources. 
  We present an elementary derivation of the soliton-like solutions in the $A_n^{(1)}$ Toda models which is alternative to the previously used Hirota method. The solutions of the underlying linear problem corresponding to the N-solitons are calculated. This enables us to obtain explicit expression for the element which by dressing group action, produces a generic soliton solution. In the particular example of monosolitons we suggest a relation to the vertex operator formalism, previously used by Olive, Turok and Underwood. Our results can also be considered as generalization of the approach to the sine-Gordon solitons, proposed by Babelon and Bernard. 
  We examine the statistical mechanics of a 1-dimensional gas of both adjoint and fundamental representation quarks which interact with each other through 1+1-dimensional U(N) gauge fields. Using large-N expansion we show that, when the density of fundamental quarks is small, there is a first order phase transition at a critical temperature and adjoint quark density which can be interpreted as deconfinement. When the fundamental quark density is comparable to the adjoint quark density, the phase transition becomes a third order one. We formulate a way to distinguish the phases by considering the expectation values of high winding number Polyakov loop operators. 
  We demonstrate the close relationship between Chern-Simons gauge theory with gauge group Osp(1|2) and N=1 induced supergravity in two dimensions. More precisely, the inner product of the physical states in the former yields the partition function of the latter evaluated in the Wess-Zumino supergauge. It is also shown that the moduli space of flat Osp(1|2) connections naturally includes super Teichm\"uller space of super Riemann surfaces. 
  We study a vertex operator algebra containing a tensor product of Ising models. It is a direct sum of code vertex operator algebra and its irreducible modules. Therefore, we classify all irreducible modules of code vertex operator algebras and show a new construction of vertex operator algebras. 
  We pick up a method originally developed by Cheng and Tsai for vacuum perturbation theory which allows to test the consistency of different sets of Feynman rules on a purely diagrammatic level, making explicit loop calculations superfluous. We generalize it to perturbative calculations in thermal field theory and we show that it can be adapted to check the gauge invariance of physical quantities calculated in improved perturbation schemes. Specifically, we extend this diagrammatic technique to a simple resummation scheme in imaginary time perturbation theory. As an application, we check up to O(g^4) in general covariant gauge the gauge invariance of the result for the QCD partition function which was recently obtained in Feynman gauge. 
  The supersymmetric generalization of Poisson-Lie T-duality in superconformal WZNW models is considered. It is shown that the classical N=2 superconformal WZNW models posses a natural Poisson-Lie symmetry which allows to construct dual $\sigma$- models. 
  We study the properties of ultraviolet renormalons in the vectorial $(\vec{\phi}^{2})^{2}$ model. This is achieved by studying the effective potential of the theory at next to leading order of the $1/N$ expansion, the appearence ofthe renormalons in the perturbative series and their relation to the imaginary part of the potential. We also consider the mechanism of renormalon cancellation by `irrelevant" higher dimensional operators. 
  For a variety of diffeomorphism-invariant field theories describing hypersurface motions (such as relativistic M-branes in space-time dimension M+2) we perform a Hamiltonian reduction ``at level 0'', showing that a simple algebraic function of the normal velocity is canonically conjugate to the shape \Sigma of the hypersurface. The Hamiltonian dependence on \Sigma is solely via the domain of integration, raising hope for a consistent, reparametrisation-invariant quantization. 
  We present a systematic description of the mathematical techniques for studying multiloop Feynman diagrams which constitutes a full-fledged and inherently more powerful alternative to the BPHZ theory. The new techniques emerged as a formalization of the reasoning behind a recent series of record multiloop calculations in perturbative quantum field theory. It is based on a systematic use of the ideas and notions of the distribution theory. We identify the problem of asymptotic expansion of products of singular functions in the sense of distributions as a key problem of the theory of asymptotic expansions of multiloop Feynman diagrams. Its complete solution for the case of Euclidean Feynman diagrams (the so-called Euclidean asymptotic operation for products of singular functions) is explicitly constructed and studied. 
  The results of the mathematical theory of asymptotic operation developed in hep-th/9612037 are applied to problems of immediate physical interest. First, the problem of UV renormalizationis analyzed from the viewpoint of asymptotic behaviour of integrands in momentum representation. A new prescription for UV renormalization in momentum space representation is presented (generalized minimal subtraction scheme); it ensures UV convergence of renormalized diagrams by construction, makes no use of special (e.g. dimensional) regularizations, and comprizes massless renormalization schemes (including the MS scheme). Then we present formal regularization-independent proofs of general formulae for Euclidean asymptotic expansions of renormalized Feynman diagrams (inlcuding short-distance OPE, heavy mass expansions and mixed asymptotic regimes etc.) derived earlier in the context of dimensional regularization. This result, together with the new variant of UV renormalization, demonstrates the power of the new techniques based on a systematic use of the theory of distributions and establishes the method of As-operation as a comprehensive full-fledged---and inherently more powerful---alternative to the BPHZ approach. 
  The off-shell dynamics of the O(3) nonlinear sigma-model is probed in terms of spectral densities and two-point functions by means of the form factor approach. The exact form factors of the Spin field, Noether-current, EM-tensor and the topological charge density are computed up to 6-particles. The corresponding $n\leq 6$ particle spectral densities are used to compute the two-point functions, and are argued to deviate at most a few per mille from the exact answer in the entire energy range below 10^3 in units of the mass gap. To cover yet higher energies we propose an extrapolation scheme to arbitrary particle numbers based on a novel scaling hypothesis for the spectral densities. It yields candidate results for the exact two-point functions at all energy scales and allows us to exactly determine the values of two, previously unknown, non-perturbative constants. 
  We study $su(2)_k$ WZW model perturbed by a multiplet of primary fields. The theory has a rich variety of particles. Presence of nontrivial decay processes is a peculiarity of the model. We prove integrability by explicit construction of quantum conserved currents. The scattering theory is briefly discussed. 
  A new kind of fundamental superfield is proposed, with an Ising-like Euclidean action. Near the Planck energy it undergoes its first stage of symmetry-breaking, and the ordered phase is assumed to support specific kinds of topological defects. This picture leads to a low-energy Lagrangian which is similar to that of standard physics, but there are interesting and observable differences. For example, the cosmological constant vanishes, fermions have an extra coupling to gravity, the gravitational interaction of W-bosons is modified, and Higgs bosons have an unconventional equation of motion. 
  We construct the Wess-Zumino terms from anomalies in case of quasigroups for the following situations. One is effective gauge field theories of Nambu-Goldstone fields associated with spontaneously broken global symmetries and the other is anomalous gauge theories. The formalism that we will develop can be seen as a generalization of the non-linear realization method of Lie groups. As an example we consider 2d gravity with a Weyl invariant regularization 
  By generalizing a fermionic construction, a natural relation is found between SL(2) degenerate conformal field theories and some N=2 discrete superconformal series. These non-unitary models contain, as a subclass, N=2 minimal models. The construction permits one to investigate the properties of chiral operators in the N=2 models. A chiral ring reveals a close connection with underlying quantum group structures. 
  In analogy to the KP theory, the second Poisson structure for the dispersionless KP hierarchy can be defined on the space of commutative pseudodifferential operators $L=p^n+\sum_{j=-\infty}^{n-1}u_j p^j$. The reduction of the Poisson structure to the symplectic submanifold $u_{n -1}=0$ gives rise to the w-algebras. In this paper, we discuss properties of this Poisson structure, its Miura transformation and reductions. We are particularly interested in the following two cases: a) L is pure polynomial in p with multiple roots and b) L has multiple poles at finite distance. The w-algebra corresponding to the case a) is defined as $w_ {[m_1,m_2, ... ,m_r]}$, where m_i means the multiplicity of roots and to the case b) is defined by $w(n,[m_1,m_2, ... ,m_r])$ where m_i is the multiplicity of poles. We prove that w(n,[m_1, m_2, ... , m_r])$-algebra is isomorphic via a transformation to $w_{[m_1,m_2, ... ,m_r]} \bigoplus w_{n+m} \bigoplus U(1) with $m=\sum m_i$. We also give the explicit free fields representations for these w-algebras. 
  Fundamental electroweak properties arise from a spacetime theory of matter, permitting evaluation of low energy coupling constants from first principles. 
  Various aspects of the theory of quantum integrable systems are reviewed. Basic ideas behind the construction of integrable ultralocal and nonultralocal quantum models are explored by exploiting the underlying algebraic structures related to the Yang-Baxter equations. Physical meaning of abstract mathematical notions like universal R-matrix, quantized algebras, Sklyanin algebra, braided algebra, Hopf algebra etc. and the role played by them in integrable systems are highlighted. Systematic construction of quantum integrable lattice as well as field models and their exact excitation spectra are presented through examples. The coordinate and algebraic formulations of the Bethe ansatz are illustrated with comparison, along with the description of nested and functional Bethe ansatzes. The techniques for deriving quantum Hamiltonians from the Lax operators are demonstrated on concrete models. The exposition of this review is kept in a fairly elementary level with emphasis on the physical contents. 
  We derive an expression for the relation between two scattering transition amplitudes which reflect the same dynamics, but which differ in the description of their initial and final state vectors. In one version, the incident and scattered states are elements of a perturbative Fock space, and solve the eigenvalue problem for the `free' part of the Hamiltonian --- the part that remains after the interactions between particle excitations have been `switched off'. Alternatively, the incident and scattered states may be coherent states that are transforms of these Fock states. In earlier work, we reported on the scattering amplitudes for QED, in which a unitary transformation relates perturbative and non-perturbative sets of incident and scattered states. In this work, we generalize this earlier result to the case of transformations that are not necessarily unitary and that may not have unique inverses. We discuss the implication of this relationship for Abelian and non-Abelian gauge theories in which the `transformed', non-perturbative states implement constraints, such as Gauss's law. 
  Using a simple ansatz for the solutions of the three dimensional generalization of the sine--Gordon and Toda model introduced by Konopelchenko and Rogers, a class of solutions is found by elementary methods. It is also shown that these equations are not evolution equations in the sense that solution to the initial value problem is not unique. 
  The conjecture that the states of the fermionic quasi-particles in minimal conformal field theories are eigenstates of the integrals of motion to certain eigenvalues is checked and shown to be correct only for the Ising model. 
  We demonstrate in two minisuperspace models that a perturbation expansion of quasiclassical Euclidean gravity has a factorial dependence on the order of the term at large orders. This behavior indicates that the expansion is an asymptotic series which is suggestive of an effective field theory. The series may or may not be Borel summable depending on the classical solution expanded around. We assume that only the positive action classical solution contributes to path integrals. We close with some speculative discussion on possible implications of the asymptotic nature of the expansion. 
  We perform the classical gravity calculations of the fixed scalar absorption cross-sections by D=5 black holes with three charges and by D=4 black holes with four charges. We obtain analytic results for the cases where the energy and the left and right moving temperatures are sufficiently low but have arbitrary ratios. In D=5 the greybody factor is in perfect agreement with the recent calculation performed in the context of the effective string model for black holes. In D=4 the formula for the greybody factor in terms of the energy and the temperatures differs from that in D=5 only by the overall normalization. This suggests that the fixed scalar coupling to the effective string in D=4 is identical to that in D=5. 
  We consider geometric engineering of N=1 supersymmetric QFTs with matter in terms of a local model for compactification of F-theory on Calabi-Yau fourfold. By bringing 3-branes near 7-branes we engineer N=1 supersymmetric $SU(N_c)$ gauge theory with $N_f$ flavors in the fundamental. We identify the Higgs branch of this system with the instanton moduli space on the compact four dimensional space of the 7-brane worldvolume. Moreover we show that the Euclidean 3-branes wrapped around the compact part of the 7-brane worldvolume can generate superpotential for $N_f=N_c-1$ as well as lead to quantum corrections to the moduli space for $N_f=N_c$. Finally we argue that Seiberg's duality for N=1 supersymmetric QCD may be mapped to T-duality exchanging 7-branes with 3-branes. 
  There are numerous examples of approximately degenerate states of opposite parity in molecular physics. Theory indicates that these doubles can occur in molecules that are reflection-asymmetric. Such parity doubles occur in nuclear physics as well, among nuclei with odd A $\sim$ 219-229. We have also suggested elsewhere that such doubles occur in particle physics for baryons made up of `cbu' and `cbd' quarks. In this article, we discuss the theoretical foundations of these doubles in detail, demonstrating their emergence as a surprisingly subtle consequence of the Born-Oppenheimer approximation, and emphasizing their bundle-theoretic and topological underpinnings. Starting with certain ``low energy'' effective theories in which classical symmetries like parity and time reversal are anomalously broken on quantization, we show how these symmetries can be restored by judicious inclusion of ``high-energy'' degrees of freedom. This mechanism of restoring the symmetry naturally leads to the aforementioned doublet structure. A novel by-product of this mechanism is the emergence of an approximate symmetry (corresponding to the approximate degeneracy of the doubles) at low energies which is not evident in the full Hamiltonian. We also discuss the implications of this mechanism for Skyrmion physics, monopoles, anomalies and quantum gravity. 
  A multidimensional gravitational model on the manifold $M = M_0 \times \prod_{i=1}^{n} M_i$, where M_i are Einstein spaces ($i \geq 1$), is studied. For $N_0 = dim M_0 > 2$ the $\sigma$ model representation is considered and it is shown that the corresponding Euclidean Toda-like system does not satisfy the Adler-van-Moerbeke criterion. For $M_0 = R^{N_0}$, $N_0 = 3, 4, 6$ (and the total dimension $D = dim M = 11, 10, 11$, respectively) nonsingular spherically symmetric solutions to vacuum Einstein equations are obtained and their generalizations to arbitrary signatures are considered. It is proved that for a non-Euclidean signature the Riemann tensor squared of the solutions diverges on certain hypersurfaces in $R^{N_0}$. 
  We analyse in a systematic way the four dimensionnal Einstein-Weyl spaces equipped with a diagonal K\"ahler Bianchi IX metric. In particular, we show that the subclass of Einstein-Weyl structures with a constant conformal scalar curvature is the one with a conformally scalar flat - but not necessarily scalar flat - metric ; we exhibit its 3-parameter distance and Weyl one-form. This extends previous analysis of Pedersen, Swann and Madsen , limited to the scalar flat, antiself-dual case. We also check that, in agreement with a theorem of Derdzinski, the most general conformally Einstein metric in the family of biaxial K\"ahler Bianchi IX metrics is an extremal metric of Calabi, conformal to Carter's metric, thanks to Chave and Valent's results. 
  Using world-sheet parity we show that mixed D and N components of D-strings are dual to the anti-symmetric $B_{\mu\nu}$ field. The contribution of the latter is responsible for the reduction and even removal of all the interactions between two dissimilar D-branes. 
  A supersymmetry anomaly is found in the presence of non-perturbative fields. When the action is expressed in terms of the correct quantum variables, anomalous surface terms appear in its supersymmetric variation - one per each collective coordinate. The anomalous surface terms do not vanish in general when inserted in two- or higher-loop bubble diagrams, and generate a violation of the SUSY Ward identities. 
  The formulation of Berry for the Aharonov-Bohm effect is generalized to the relativistic regime. Then, the problem of finding the self-adjoint extensions of the (2+1)-dimensional Dirac Hamiltonian, in an Aharonov-Bohm background potential, is solved in a novel way. The same treatment also solves the problem of finding the self-adjoint extensions of the Dirac Hamiltonian in a background Aharonov-Casher. 
  The noncovariant duality symmetric action put forward by Schwarz-Sen is quantized by means of the Dirac bracket quantization procedure. The resulting quantum theory is shown to be, nevertheless, relativistically invariant. 
  We associate to outer automorphisms of generalized Kac-Moody algebras generalized character-valued indices, the twining characters. A character formula for twining characters is derived which shows that they coincide with the ordinary characters of some other generalized Kac-Moody algebra, the so-called orbit Lie algebra. Some applications to problems in conformal field theory, algebraic geometry and the theory of sporadic simple groups are sketched. 
  Formulating the statistical mechanics for a scalar field with non-minimal $\xi R \phi^2$ coupling in a black hole background we propose modification of the original 't Hooft ``brick wall'' prescription. Instead of the Dirichlet condition we suggest some scattering ansatz for the field functions at the horizon. This modifies the energy spectrum of the system and allows one to obtain the statistical entropy dependent on the non-minimal coupling. For $\xi<0$ the entropy renormalizes the classical Bekenstein-Hawking entropy in the correct way and agrees with the result previously obtained within the conical singularity approach. For a positive $\xi$, however, the results differ. 
  We study multiple 3-branes on an F theory orientifold. The world-volume theory of the 3-branes is d=4, N=2 Sp(2k) gauge theory with an antisymmetric tensor and four flavors of matter in the fundamental. The solution of this gauge theory is found for vanishing bare mass of the antisymmetric tensor matter, and massive fundamental matter. The integrable system underlying this theory is constructed. 
  We consider the (2+1)-dimensional gauged Heisenberg ferromagnet model coupled with the Chern-Simons gauge fields. Self-dual Chern-Simons solitons, the static zero energy solution saturating Bogomol'nyi bounds, are shown to exist when the generalized spin variable is valued in the Hermitian symmetric spaces G/H. By gauging the maximal torus subgroup of H, we obtain self-dual solitons which satisfy vortex-type nonlinear equations thereby extending the two dimensional instantons in a nontrivial way. An explicit example for the CP(N) case is given. 
  By developing an appropriate path-integral formalism, we compute, in bosonic string theory, the disk amplitude for the scattering of closed string states from a D-particle, in which the collective coordinate of the D-particle is fully quantized. As a consequence, the recoil of the D-particle is naturally taken into account. Our result can be readily factorized in the closed string channel to yield the boundary state describing the recoiling D-particle. This turned out to agree with the BRST invariant vertex recently proposed by Ishibashi to the leading order in the derivative expansion, but it will receive corrections in subsequent orders. The advantage of our formalism is that it is extendable to deal with more general processes involving multiple D-particles. A viewpoint regarding our work as describing a dynamical transition of CFT's is also discussed. 
  In these lectures we report recent work on the exact quantization of dimensionally reduced gravity, i.e. 2d non-linear (G/H)-coset space sigma-models coupled to gravity and a dilaton. Using methods developed in the context of flat space integrable systems, the Wheeler-DeWitt equations for these models can be reduced to a modified version of the Knizhnik-Zamolodchikov equations from conformal field theory, the insertions given by singularities in the spectral parameter plane. This basic result in principle permits the explicit construction of solutions, i.e. physical states of the quantized theory. In this way, we arrive at integrable models of quantum gravity with infinitely many self-interacting propagating degrees of freedom. 
  ``Anomalous'' U(1) gauge symmetry with Green-Schwarz anomaly cancellation mechanism is discussed in the orbifold construction of four-dimensional heterotic string models. Some conditions are given as criteria to have ``anomalous'' U(1) in orbifold string models. In particular, ``anomalous'' U(1) is absent if the massless twisted matter has no mixing between visible and hidden sectors or if a certain type of discrete symmetries are found. We then give a general procedure for classifying orbifold models with ``anomalous'' U(1) and for identifying the ``anomalous'' U(1) basis. We illustrate our procedure in Z_3 and Z_4 orbifold models. According to our procedure, the classification of ``anomalous'' U(1) can be reduced to the classification in the absence of a Wilson line. We also discuss discrete symmetries left unbroken after the ``anomalous'' U(1) breaking. This includes a possible relation between ``anomalous'' U(1) and discrete R-symmetries. 
  We study the Calabi-Yau phase of a certain class of (0,2) models. These are conjectured to be equivalent to exact (0,2) superconformal field theories which have been constructed recently. Using the methods of toric geometry we discuss in a few examples the problem of resolving the singularities of such models and calculate the Euler characteristic of the corresponding gauge bundles. 
  A new, configuration-space picture of a formalism of group quantization, the GAQ formalism, is presented in the context of a previous, algebraic generalization. This presentation serves to make a comprehensive discussion in which other extensions of the formalism, particularly to incorporate gauge symmetries, are developed as well. Both images are combined in order to analyse, in a systematic manner and with complete generality, the case of linear fields (abelian current groups). To ilustrate these developments we particularize them for several fields and, in particular, we carry out the quantization of the abelian Chern-Simons models over an arbitrary closed surface in detail. 
  We discuss the geometry of dynamical triangulations associated with 3-dimensional and 4-dimensional simplicial quantum gravity. We provide analytical expressions for the canonical partition function in both cases, and study its large volume behavior. In the space of the coupling constants of the theory, we characterize the infinite volume line and the associated critical points. The results of this analysis are found to be in excellent agreement with the MonteCarlo simulations of simplicial quantum gravity. In particular, we provide an analytical proof that simply-connected dynamically triangulated 4-manifolds undergo a higher order phase transition at a value of the inverse gravitational coupling given by 1.387, and that the nature of this transition can be concealed by a bystable behavior. A similar analysis in the 3-dimensional case characterizes a value of the critical coupling (3.845) at which hysteresis effects are present. 
  The property of some finite W algebras to be the commutant of a particular subalgebra of a simple Lie algebra G is used to construct realizations of G. When G=so(4,2), unitary representations of the conformal and Poincare algebras are recognized in this approach, which can be compared to the usual induced representation technique. When G=sp(2,R) or sp(4,R), the anyonic parameter can be seen as the eigenvalue of a W generator in such W representations of G. The generalization of such properties to the affine case is also discussed in the conclusion, where an alternative of the Wakimoto construction for sl(2) level k is briefly presented. This mini review is based on invited talks presented by P. Sorba at the ``Vth International Colloquium on Quantum Groups and Integrable Systems'', Prague (Czech Republic), June 1996; ``Extended and Quantum Algebras and their Applications to Physics'', Tianjin (China), August 1996; ``Selected Topics of Theoretical and Modern Mathematical Physics'', Tbilisi (Georgia), September 1996; to be published in the Proceedings. 
  We present an algebraic approach to string theory, using a Hamiltonian reduction of N=2 WZW models. An embedding of sl(1|2) in a Lie superalgebra determines a niltopent subalgebra. Chirally gauging this subalgebra in the corresponding WZW action leads to an extension of the N=2 superconformal algebra. We classify all the embeddings of sl(1|2) into Lie superalgebras: this provides an exhaustive classification and characterization of all extended N=2 superconformal algebras. Then, twisting these algebras, we obtain the BRST structure of a string theory. We characterize and classify all the string theories which can be obtained in this way.   Based on a common work of E.  Ragoucy, A. Sevrin and P. Sorba, presented by E. Ragoucy at ``Extended and Quantum Algebras and their Applications to Physics'', Nankai Institute in Tianjin (China) August 19-24 1996 
  In this paper we describe a covariant canonical formalism for a free time-like (massive) as well as space-like (tachyonic) particle in the framework of nonstandard synchronization scheme. In this scheme one is able to introduce absolute causality without breaking the Poincar\'e invariance. 
  We consider SU(2) Yang-Mills theory in 1+1 dimensions coupled to massless adjoint fermions. With all fields in the adjoint representation the gauge group is actually SU(2)/Z_2, which possesses nontrivial topology. In particular, there are two distinct topological sectors and the physical vacuum state has a structure analogous to a \theta vacuum. We show how this feature is realized in light-front quantization, with periodicity conditions used to regulate the infrared and treating the gauge field zero mode as a dynamical quantity. We find expressions for the degenerate vacuum states and construct the analog of the \theta vacuum. We then calculate the bilinear condensate in the model. We argue that the condensate does not affect the spectrum of the theory, although it is related to the string tension that characterizes the potential between fundamental test charges when the dynamical fermions are given a mass. We also argue that this result is fundamentally different from calculations that use periodicity conditions in x^1 as an infrared regulator. 
  We build up a class of N=2 supersymmetric non-linear $\sigma$-models in an N=1 superspace based on the Atiyah-Ward space-time of (2+2)-signature metric. We also discuss the gauging of isometries of the associated hyper-K\"ahlerian target spaces and present the resulting gauge-covariant supersymmetric action functional. 
  String theory has already motivated, suggested, and sometimes well-nigh proved a number of interesting and sometimes unexpected mathematical results, such as mirror symmetry. A careful examination of the behavior of string propagation on (mildly) singular varieties similarly suggests a new type of (co)homology theory. It has the `good behavior' of the well established intersection (co)homology and $L^2$-cohomology, but is markedly different in some aspects. For one, unlike the intersection (co)homology and the $L^2$-cohomology (or any other known thus far), this new cohomology is symmetric with respect to the mirror map. Among the available choices, this makes it into a prime candidate for describing the string theory zero modes in geometrical terms. 
  We have found the entropy of N=2 extreme black holes associated with general Calabi-Yau moduli space. We show that for arbitrary d_{ABC} and black hole charges the entropy-area formula depends on combinations of these charges and parameters d_{ABC}. These combinations are the solutions of the simple system of algebraic equations. We gave a few examples of particular Calabi-Yau moduli space for which this system has an explicit solution. For special case when one of black hole charges is equal to zero (p^0=0) the solution always exists. 
  We discuss the dependence of superpotential terms in 4D F-theory on moduli parameters. Two cases are studied: the dependence on world-filling 3-brane positions and the dependence on 2-form VEVs. In the first case there is a zero when the 3-brane hits the divisor responsible for the superpotential. In the second case, which has been extensively discussed by Witten in 3D M-theory, there is a zero for special values of 2-form VEVs when the M-theory divisor contains non-trivial 3-cycles. We give an alternative derivation of this fact for the special case of F-theory. 
  Chiral orbifold models are defined as gauge field theories with a finite gauge group $\Gamma$. We start with a conformal current algebra A associated with a connected compact Lie group G and a negative definite integral invariant bilinear form on its Lie algebra. Any finite group $\Gamma$ of inner automorphisms or A (in particular, any finite subgroup of G) gives rise to a gauge theory with a chiral subalgebra $A^{\Gamma}\subset A$ of local observables invariant under $\Gamma$. A set of positive energy $A^{\Gamma}$ modules is constructed whose characters span, under some assumptions on $\Gamma$, a finite dimensional unitary representation of $SL(2,Z)$. We compute their asymptotic dimensions (thus singling out the nontrivial orbifold modules) and find explicit formulae for the modular transformations and hence, for the fusion rules.    As an application we construct a family of rational conformal field theory (RCFT) extensions of $W_{1+\infty}$ that appear to provide a bridge between two approaches to the quantum Hall effect. 
  Quantum field theories and Matrix models have a far richer solution set than is normally considered, due to the many boundary conditions which must be set to specify a solution of the Schwinger-Dyson equations. The complete set of solutions of these equations is obtained by generalizing the path integral to include sums over various inequivalent contours of integration in the complex plane. We discuss the importance of these exotic solutions. While naively the complex contours seem perverse, they are relevant to the study of theta vacua and critical phenomena. Furthermore, it can be shown that within certain phases of many theories, the physical vacuum does not correspond to an integration over a real contour. We discuss the solution set for the special case of one component zero dimensional scalar field theories, and make remarks about matrix models and higher dimensional field theories that will be developed in more detail elsewhere. Even the zero dimensional examples have much structure, and show some analogues of phenomena which are usually attributed to the effects of taking a thermodynamic limit. 
  The first part of this paper presents actions for Dirichlet p-branes embedded in a flat 10-dimensional space-time. The fields of the (p+1)-dimensional world-volume theories are the 10d space-time coordinates $ X^m$, a pair of Majorana-Weyl spinors $\theta_1$ and $\theta_2$, and a U(1) gauge field $A_{\mu}$. The N = 2A or 2B super-Poincare group in ten dimensions is realized as a global symmetry. In addition, the theories have local symmetries consisting of general coordinate invariance of the world volume, a local fermionic symmetry (called ``kappa''), and U(1) gauge invariance. A detailed proof of the kappa symmetry is given that applies to all cases (p = 0,1, . . ., 9). The second part of the paper presents gauge-fixed versions of these theories. The fields of the 10d (p = 9) gauge-fixed theory are a single Majorana-Weyl spinor $\lambda$ and the U(1) gauge field $A_{\mu}$. This theory, whose action turns out to be surprisingly simple, is a supersymmetric extension of 10d Born-Infeld theory. It has two global supersymmetries: one represents an unbroken symmetry, and the second corresponds to a broken symmetry for which $\lambda$ is the Goldstone fermion. The gauge-fixed supersymmetric D-brane theories with $p<9$ can be obtained from the 10d one by dimensional reduction. 
  A detailed consideration of the maximally nonabelian Toda systems based on the classical semisimple Lie groups is given. The explicit expressions for the general solution of the corresponding equations are obtained. 
  The possibility of the existence of small correction terms to the canonical commutation relations and the uncertainty relations has recently found renewed interest. In particular, such correction terms could induce finite lower bounds $\Delta x_0, \Delta p_0$ to the resolution of distances and/or momenta. I review a general framework for the path integral formulation of quantum field theories on such generalised geometries, and focus then on the mechanisms by which $\Delta p_0>0$, and/or $\Delta x_0>0$ lead to IR and/or UV regularisation. 
  We discuss the problem of vacuum structure in light-front field theory in the context of (1+1)-dimensional gauge theories. We begin by reviewing the known light-front solution of the Schwinger model, highlighting the issues that are relevant for reproducing the $\theta$-structure of the vacuum. The most important of these are the need to introduce degrees of freedom initialized on two different null planes, the proper incorporation of gauge field zero modes when periodicity conditions are used to regulate the infrared, and the importance of carefully regulating singular operator products in a gauge-invariant way. We then consider SU(2) Yang-Mills theory in 1+1 dimensions coupled to massless adjoint fermions. With all fields in the adjoint representation the gauge group is actually SU(2)$/Z_2$, which possesses nontrivial topology. In particular, there are two topological sectors and the physical vacuum state has a structure analogous to a $\theta$ vacuum. We formulate the model using periodicity conditions in $x^\pm$ for infrared regulation, and consider a solution in which the gauge field zero mode is treated as a constrained operator. We obtain the expected $Z_2$ vacuum structure, and verify that the discrete vacuum angle which enters has no effect on the spectrum of the theory. We then calculate the chiral condensate, which is sensitive to the vacuum structure. The result is nonzero, but inversely proportional to the periodicity length, a situation which is familiar from the Schwinger model. The origin of this behavior is discussed. 
  Studies in string theory and quantum gravity suggest the existence of a finite lower limit $\Delta x_0$ to the possible resolution of distances, at the latest on the scale of the Planck length of $10^{-35}m$. Within the framework of the euclidean path integral we explicitly show ultraviolet regularisation in field theory through this short distance structure. Both rotation and translation invariance can be preserved. An example geometry is studied in detail. 
  We study limits of four-dimensional type II Calabi-Yau compactifications with vanishing four-cycle singularities, which are dual to $\IT^2$ compactifications of the six-dimensional non-critical string with $E_8$ symmetry. We define proper subsectors of the full string theory, which can be consistently decoupled. In this way we obtain rigid effective theories that have an intrinsically stringy BPS spectrum. Geometrically the moduli spaces correspond to special geometry of certain non-compact Calabi-Yau spaces of an intriguing form. An equivalent description can be given in terms of Seiberg-Witten curves, given by the elliptic simple singularities together with a peculiar choice of meromorphic differentials. We speculate that the moduli spaces describe non-perturbative non-critical string theories. 
  We discuss a new type of Landau-Ginzburg potential for the E_6 singularity of the form $W=const+(Q_1(x)+P_1(x)\sqrt{P_2(x)})/x^3$ which featured in a recent study of heterotic/typeII string duality. Here $Q_1,P_1$ and $P_2$ are polynomials of degree 15,10 and 10, respectively. We study the properties of the potential in detail and show that it gives a new and consistent description of the E_6 singularity. 
  We calculate the potential between various configurations of membranes and gravitons in M(atrix) theory. The computed potentials agree with the short distance potentials between corresponding 2-branes and 0-brane configurations in Type IIA string theory, bound to a large number of 0-branes to account for the boost to the infinite momentum frame. We show that, due to the large boost, these type IIA configurations are almost supersymmetric, so that the short and long distance potentials actually agree. Thus the M(atrix) theory is able to reproduce correct long distance behavior in these cases. 
  The validity of dilute gas approximation is explored by making use of the large-sized instanton in quantum mechanical model. It is shown that the Euclidean probability amplitude derived through a dilute gas approximation not only cannot explain the result of the linear combination of atomic orbitals approximation, but also does not exhibit a proper limiting case when the size of instanton is very large. 
  Multidimensional gravitational model on the manifold $M = M_0 \times \prod_{i=1}^{n} M_i$, where $M_i$ are Einstein spaces ($i \geq 1$), is considered. The action contains $m = 2^n -1$ dilatonic scalar fields $\phi^I$ and $m$ (antisymmetric) forms $A^I$. When all fields and scale factors of the metric depend (essentially) on the point of $M_0$ and any $A^I$ is "proportional" to the volume form of submanifold $M_{i_1} \times ... \times M_{i_k}$, $1 \leq i_1 < ... < i_k \leq n$, the sigma-model representation is obtained. A family of "Majumdar-Papapetrou type" solutions are obtained, when all $M_{\nu}$ are Ricci-flat. A special class of solutions (related to the solution of some Diophantus equation on dimensions of $M_{\nu}$) is singled out. Some examples of intersecting p-branes (e.g. solution with seven Euclidean 2-branes for D = 11 supergravity) are considered.} 
  We consider the 3-manifold invariant I(M) which is defined by means of the Chern-Simons quantum field theory and which coincides with the Reshetikhin-Turaev invariant. We present some arguments and numerical results supporting the conjecture that, for nonvanishing I(M), the absolute value  | I(M) | only depends on the fundamental group \pi_1 (M) of the manifold M. For lens spaces, the conjecture is proved when the gauge group is SU(2). In the case in which the gauge group is SU(3), we present numerical computations confirming the conjecture. 
  The concept of perturbative gauge invariance formulated exclusively by means of asymptotic fields is generalized to massive gauge fields. Applying it to the electroweak theory leads to a complete fixing of couplings of scalar and ghost fields and of the coupling to leptons, in agreement with the standard theory. The W/Z mass ratio is also determined, as well as the chiral character of the fermions. We start directly with massive gauge fields and leptons and, nevertheless, obtain a theory which satisfies perturbative gauge invariance. 
  General permutation invariant statistics in the second quantized approach are considered. Simple interpolations between dual statistics are constructed. Particularly, we present a new minimal interpolation between parabosons and parafermions of any order. The connection with a simple mixing between bosons and fermions is established. The construction is extended to anyonic-like statistics. 
  We summarize recent progress in the understanding of fixed point resolution for conformal field theories. Fixed points in both coset conformal field theories and non-diagonal modular invariants which describe simple current extensions of chiral algebras are investigated. A crucial role is played by the mathematical structures of twining characters and orbit Lie algebras. 
  We demonstrate that the generalization of the relativistic Toda chain (RTC) is a special reduction of two-dimensional Toda Lattice hierarchy (2DTL). We also show that the RTC is gauge equivalent to the discrete AKNS hierarchy and the unitary matrix model. Relativistic Toda molecule hierarchy is also considered, along with the forced RTC. The simple approach to the discrete RTC hierarchy based on Darboux-B\"acklund transformation is proposed. 
  We give a classification of all multiple intersections of D-branes in ten dimensions and M-branes in eleven dimensions that corresponds to threshold BPS bound states. The residual supersymmetry of these composite branes is determined. By dimensional reduction composite p-branes in lower dimensions can be constructed. We emphasize in dimensions D greater or equal than two, those solutions which involve a single scalar and depend on a single harmonic function. For these extremal branes we obtain the strength of the coupling between the scalar and the gauge field. In particular we give a D-brane and M-brane interpretation of extreme p-branes in two, three and four dimensions. 
  Motivated by twisted N=4 supersymmetric Yang-Mills theory in 4 dimensions, a natural extension of the monad (ADHM) construction relevant to D-instantons is considered. We show that a family of Yang-Mills instantons can be constructed from D-instantons. We discuss some possible roles of reciprocity in D-brane physics. We conjecture the existence of universal instantons together with a generalized Fourier-Nahm transformation as an unifying framework of D-brane physics. 
  The notion of the optimized perturbation, which has been successfully applied to energy eigenvalues, is generalized to treat wave functions of quantum systems. The key ingredient is to construct an envelope of a set of perturbative wave functions. This leads to a condition similar to that obtained from the principle of minimal sensitivity. Applications of the method to quantum anharmonic oscillator and the double well potential show that uniformly valid wave functions with correct asymptotic behavior are obtained in the first-order optimized perturbation even for strong couplings. 
  In these notes I briefly outline SL(2) degenerate conformal field theories and their application to some related models, namely 2d gravity and N=2 discrete superconformal series. 
  We review and summarize recent works on the relation between form factors in integrable quantum field theory and deformation of geometrical data associated to hyper-elliptic curves. This relation, which is based on a deformation of the Riemann bilinear identity, in particular leads to the notion of null vectors in integrable field theory and to a new description of the KdV hierarchy. 
  A generalization of the classical one-dimensional Darboux transformation to arbitrary n-dimensional oriented Riemannian manifolds is constructed using an intrinsic formulation based on the properties of twisted Hodge Laplacians. The classical two-dimensional Moutard transformation is also generalized to non-compact oriented Riemannian manifolds of dimension n greater than one. New examples of quasi-exactly solvable multidimensional matrix Schr\"odinger operators on curved manifolds are obtained by applying the above results. 
  We construct static axially symmetric solutions of SU(2) Einstein-Yang-Mills-dilaton theory. Like their spherically symmetric counterparts, these solutions are nonsingular and asymptotically flat. The solutions are characterized by the winding number n and the node number k of the gauge field functions. For fixed n with increasing k the solutions tend to ``extremal'' Einstein-Maxwell-dilaton black holes with n units of magnetic charge. 
  The string field theory of N=(2,1) heterotic strings describes a set of self-dual Yang-Mills fields coupled to self-dual gravity in 2+2 dimensions. We show that the exact classical action for this field theory is a certain complexification of the Green-Schwarz/Dirac-Born-Infeld string action, closely related to the four dimensional Wess-Zumino action describing self-dual gauge fields. This action describes the world-volume of a 2+2d ``M-brane'', which gives rise upon different null reductions to critical strings and membranes. We discuss a number of further properties of N=2 heterotic strings, such as the geometry of null reduction, general features of a covariant formulation, and possible relations to BPS and GKM algebras. 
  We consider several aspects of `confining strings', recently proposed to describe the confining phase of gauge field theories.  We perform the exact duality transformation that leads to the confining string action and show that it reduces to the Polyakov action in the semiclassical approximation. In 4D we introduce a `$\theta$-term' and compute the low-energy effective action for the confining string in a derivative expansion. We find that the coefficient of the extrinsic curvature (stiffness) is negative, confirming previous proposals. In the absence of a $\theta$-term, the effective string action is only a cut-off theory for finite values of the coupling e, whereas for generic values of $\theta$, the action can be renormalized and to leading order we obtain the Nambu-Goto action plus a topological `spin' term that could stabilize the system. 
  We study the effect of mirror symmetry for K3 surfaces on D-brane probe physics. The case of elliptically fibered K3 surfaces is considered in detail. In many cases, mirror can transform a singular fiber of Kodaira's type ADE into sets of singular fibers of type I_1 (II) with equal total Euler number, but vanishing contribution to the Picard number of the mirror surface. This provides a geometric model of quantum splitting phenomena. Mirror for three dimensional gauge theories, interchanging Fayet-Iliopoulos and mass terms, is also briefly discussed. 
  A geometric formulation which describes extended supergravities in any dimension in presence of electric and magnetic sources is presented. In this framework the underlying duality symmetries of the theories are manifest. Particular emphasis is given to the construction of central and matter charges and to the symplectic structure of all D=4, N-extended theories. The latter may be traced back to the existence, for N>2, of a flat symplectic bundle which is the N>2 generalization of N=2 Special Geometry. 
  The tunneling effect of a periodic potential with an asymmetric twin barrier per period is calculated using the instanton method. The model is derived from the Hamiltonian of a small ferromagnetic particle in an external magnetic field using the spin-coherent-state path integral. The instantons in two neighbouring barriers differ and lead to different level shifts $\triangle\epsilon_1, \triangle\epsilon_2$. We derive with Bloch theory the energy spectrum which has formally the structure of an energy band. The spectrum depends on both level shifts. The removal of Kramer's degeneracy by an external magnetic field is discussed. In addition we find a new kind of quenching of macroscopic quantum coherence which is irrelevant to Kramer's degeneracy. 
  The explicit formulae for m-soliton solutions of (1+2)-dimensional matrix Davey-Stewartson equation are represented. They are found by means of known general solution of the matrix Toda chain with the fixed ends [1]. These solutions are expressed trough m+m independent solutions of a pair of linear Shrodinger equations with Hermitian potentials. 
  We consider heterotic string theories compactified on a K3 surface which lead to an unbroken perturbative gauge group of Spin(32)/Z2. All solutions obtained are combinations of two types of point-like instanton --- one ``simple type'' as discovered by Witten and a new type associated to the ``generalized second Stiefel-Whitney class'' as introduced by Berkooz et al. The new type of instanton is associated to an enhancement of the gauge symmetry by Sp(4) and the addition of a massless tensor supermultiplet. It is shown that if four simple instantons coalesce at an orbifold point in the K3 surface then a massless tensor field appears which may be used to interpolate between the two types of instanton. By allowing various combinations of point-like instantons to coalesce, large gauge groups (e.g., rank 128) with many massless tensor supermultiplets result. The analysis is done in terms of F-theory. 
  A correction to the Hamiltonian of the quark-antiquark system, arising due to the rigidity term in the gluodynamics string effective action, is obtained. This correction contains additional contributions to the orbital momentum of the system and several higher derivative operators. With the help of the derived Hamiltonian a rigid string-induced term in the Hamiltonian of the relativistic quark model is evaluated for the case of large masses of a quark and antiquark. 
  Certain higher dimensional operators of the lagrangian may render the vacuum inhomogeneous. A rather rich phase structure of the phi4 scalar model in four dimensions is presented by means of the mean-field approximation. One finds para- ferro- ferri- and antiferromagnetic phases and commensurate-incommensurate transitions. There are several particles described by the same quantum field in a manner similar to the species doubling of the lattice fermions. It is pointed out that chiral bosons can be introduced in the lattice regularized theory. 
  It is shown that the four dimensional antiferromagnetic lattice phi4 model has the usual non-asymptotically free scaling law in the UV regime around the chiral symmetrical critical point. The theory describes a scalar and a pseudoscalar particle. A continuum effective theory is derived for low energies. A possibility of constructing a model with a single chiral boson is mentioned. 
  We show that the assumption of quasiperiodic boundary conditions (those that interpolate continuously periodic and antiperiodic conditions) in order to compute partition functions of relativistic particles in 2+1 space-time can be related with anyonic physics. In particular, in the low temperature limit, our result leads to the well known second virial coefficient for anyons. Besides, we also obtain the high temperature limit as well as the full temperature dependence of this coefficient. 
  A new formalism is given for the renormalization of quantum field theories to all orders of perturbation theory, in which there are manifestly no overlapping divergences. We prove the BPH theorem in this formalism, and show how the local subtractions add up to counterterms in the action. Applications include the renormalization of lattice perturbation theory, the decoupling theorem, Zimmermann oversubtraction, the renormalization of operator insertions, and the operator product expansion. 
  These lectures give a self-contained introduction to supersymmetry from a modern perspective. Emphasis is placed on material essential to understanding duality. Topics include: central charges and BPS-saturated states, supersymmetric nonlinear sigma models, N=2 Yang-Mills theory, holomorphy and the N=2 Yang-Mills beta function, supersymmetry in 2, 6, 10, and 11 spacetime dimensions. 
  A matrix model which has the manifest ten-dimensional N=2 super Poincare invariance is proposed. Interactions between BPS-saturated states are analyzed to show that massless spectrum is the same as that of type IIB string theory. It is conjectured that the large-N reduced model of ten-dimensional super Yang-Mills theory can be regarded as a constructive definition of this model and therefore is equivalent to superstring theory. 
  We study the simultaneous influence of boundary conditions and external fields on quantum fluctuations by considering vacuum zero-point energies for quantum fields in the presence of a magnetic fluxon confined by a bag, circular and spherical for bosons and circular for fermions. The Casimir effect is calculated in a generalized cut-off regularization after applying zeta-function techniques to eigenmode sums and using recent techniques about Bessel zeta functions at negative arguments. 
  We show that non-perturbative fixed points of the exact renormalization group, their perturbations and corresponding massive field theories can all be determined directly in the continuum -- without using bare actions or any tuning procedure. As an example, we estimate the universal couplings of the non-perturbative three-dimensional one-component massive scalar field theory in the Ising model universality class, by using a derivative expansion (and no other approximation). These are compared to the recent results from other methods. At order derivative-squared approximation, the four-point coupling at zero momentum is better determined by other methods, but factoring this out appropriately, all our other results are in very close agreement with the most powerful of these methods. In addition we provide for the first time, estimates of the n-point couplings at zero momentum, with n=12,14, and the order momentum-squared parts with n=2 ... 10. 
  The Batalin Vilkovisky (BV) quantization provides a general procedure for calculating anomalies associated to gauge symmetries. Recent results show that even higher loop order contributions can be calculated by introducing an appropriate regularization-renormalization scheme. However, in its standard form, the BV quantization is not sensible to quantum violations of the classical conservation of Noether currents, the so called global anomalies. We show here that the BV field antifield method can be extended in such a way that the Ward identities involving divergencies of global Abelian currents can be calculated from the generating functional, a result that would not be obtained by just associating constant ghosts to global symmetries. This extension, consisting of trivially gauging the global Abelian symmetries, poses no extra obstruction to the solution of the master equation, as it happens in the case of gauge anomalies. We illustrate the procedure with the axial model and also calculating the Adler Bell Jackiw anomaly. 
  We propose a new supersymmetry in field theory that generalizes standard supersymmetry and we construct field theoretic models that provide some of its representations. This symmetry combines a finite number of standard 4D supersymmetry multiplets into a single multiplet with a new type of Kaluza-Klein embedding in higher dimensions. We suggest that this mechanism may have phenomenological applications in understanding family unification. The algebraic structure, which has a flavor of W-algebras, is directly motivated by S-theory and its application in black holes. We show connections to previous proposals in the literature for 12 dimensional supergravity, Yang-Mills, (2,1) heterotic superstrings and Matrix models that attempt to capture part of the secret theory behind string theory. 
  We construct a calculational scheme for handling the matrix ordering problems connected with the appearance of D-brane positions taking values in the same Lie algebra as the nonabelian gauge field living on the D-brane. The formalism is based on the use of an one-dimensional auxiliary field living on the boundary of the string world sheet and taking care of the order of the matrix valued fields. The resulting system of equations of motion for both the gauge field and the D-brane position is derived in lowest order of the $\alpha'$ -expansion. 
  Synopsis: (i) how superstring theories are unified by M-theory; (ii) how superstring and supermembrane properties follow from the D=10 and D=11 supersymmetry algebras; (iii) how D=10 and D=11 supergravity theories determine the strong coupling limit of superstring theories; (iv) how properties of Type II p-branes follow from those of M-branes. 
  A new mechanism for symmetry breaking is proposed which naturally avoids the constraints following from the usual theorems of symmetry breaking. In the context of super-symmetry, for example, the breaking may be consistent with a vanishing vacuum energy. A 2+1 dimensional super-symmetric gauge field theory is explicitly shown to break super-symmetry through this mechanism while maintaining a zero vacuum energy. This mechanism may provide a solution to two long standing problems, namely, dynamical super-symmetry breaking and the cosmological constant problem. 
  We consider perturbations in the fields of the 't Hooft-Polyakov monopole, dyon and point electric solutions. We find a series of bound modes where the fields are confined to the core of the respective solutions, and these are interpreted as bound states of a gauge boson with the respective solutions. We discuss the spectra of these bound states of the various two particle systems of the Yang-Mills-Higgs model in connection with the duality of Sen. 
  Directly interacting particles are considered in the multitime formalism of predictive relativistic mechanics. When the equations of motion leave a phase-space volume invariant, it turns out that the phase average of any first integral, covariantly defined as a flux across a $7n$-dimensional surface, is conserved. The Hamiltonian case is discussed, a class of simple models is exhibited, and a tentative definition of equilibrium is proposed. 
  Higher dimensional supersymmetric quantum mechanics is studied. General properties of the two dimensional case are presented. For three spatial dimesions or higher, a spin structure is shown to arise naturally from the nonrelativistic supersymmetry algebra. 
  We discuss the origin of enhanced gauge symmetry in ALE (and K3) compactification of M theory, either defined as the strong coupling limit of the type IIa superstring, or as defined by Banks et al. In the D-brane formalism, wrapped membranes are D0 branes with twisted string boundary conditions, and appear on the same footing with the Kaluza-Klein excitations of the gauge bosons. In M(atrix) theory, the construction appears to work for arbitrary ALE metric. 
  The potential between two separated D-instantons at fixed (super) space-time points is obtained by a simple explicit integration over the `massive' variables of the zero-dimensional reduction of ten-dimensional U(2) super Yang-Mills theory. This potential vanishes for asymptotically large separations, becoming significant at separations of around the ten-dimensional Planck scale with a singularity at the origin, which is resolved by the extra `massless' internal Yang-Mills super-coordinates. 
  Domain walls in strongly coupled gauge theories are discussed. A general mechanism is suggested automatically leading to massless gauge bosons localized on the wall. In one of the models considered, outside the wall the theory is in the non-Abelian confining phase, while inside the wall it is in the Abelian Coulomb phase. Confining property of the non-Abelian theories is a key ingredient of the mechanism which may be of practical use in the context of the dynamic compactification scenarios.   In supersymmetric (N=1) Yang-Mills theories the energy density of the wall can be exactly calculated in the strong coupling regime. This calculation presents a further example of non-trivial physical quantities that can be found exactly by exploiting specific properties of supersymmetry. A key observation is the fact that the wall in this theory is a BPS-saturated state. 
  We obtain the internal degrees of freedom of the skyrmion (spin and isospin) within a manifestly Lorentz covariant quantization framework based on defining Green functions for skyrmions and then, the S-matrix via LSZ reduction. Our method follows Frohlich and Marchetti's definition of Euclidean soliton Green functions, supplemented with a careful treatment of the boundary conditions around the singularities. The covariant two-point function obtained propagates a tower of spin equal to isospin particles. Our treatment contains the usual method of collective coordinates, as a non-relativistic limit and, because of the new topology introduced, it leads, in a natural way, to the inequivalent (boson/fermion) quantizations of the SU(2) skyrmion. 
  The D-brane counting of black hole entropy is commonly understood in terms of excitations carrying fractional charges living on long, multiply-wound branes (e.g. open strings with fractional Kaluza-Klein momentum). This paper addresses why the branes become multiply wound. Since multiply wound branes are T-dual to branes evenly spaced around the compact dimension, this tendency for branes to become multiply wound can be seen as an effective repulsion between branes in the T-dual picture. We also discuss how the fractional charges on multiply wound branes conspire to always form configurations with integer charge. 
  We construct intersecting D-brane configurations that encode the gauge groups and field content of dual N=4 supersymmetric gauge theories in three dimensions. The duality which exchanges the Coulomb and Higgs branches and the Fayet-Iliopoulos and mass parameters is derived from the SL(2,Z) symmetry of the type IIB string. Using the D-brane configurations we construct explicitly this mirror map between the dual theories and study the instanton corrections in the D-brane worldvolume theory via open string instantons. A general procedure to obtain mirror pairs is presented and illustrated. We encounter transitions among different field theories that correspond to smooth movements in the D-brane moduli space. We discuss the relation between the duality of the gauge theories and the level-rank duality of affine Lie algebras. Examples of other dual theories are presented and explained via T-duality and extremal transitions in type II string compactifications. Finally we discuss a second way to study instanton corrections in the gauge theory, by wrapping five-branes around six-cycles in M-theory compactified on a Calabi-Yau 4-fold. 
  We discuss the recent developments in the generalized continuous Heisenberg ferromagnet model formulated as a nonrelativistic field theory defined on the target space of the coadjoint orbits. Hermitian symmetric spaces are special because they provide completely integrable field theories in 1+1 dimension and self-dual Chern-Simons solitons and vortices in 2+1 dimension. 
  We derive closed recursion equations for the symmetric polynomials occuring in the form factors of $D_n^{(1)}$ affine Toda field theories. These equations follow from kinematical- and bound state residue equations for the full form factor. We also discuss the equations arising from second and third order forward channel poles of the S-matrix. The highly symmetric case of $D_4^{(1)}$ form factors is treated in detail. We calculate explicitly cases with a few particles involved. 
  We consider the close relation between duality in N=2 SUSY gauge theories and integrable models. Various integrable models ranging from Toda lattices, Calogero models, spinning tops, and spin chains are related to the quantum moduli space of vacua of N=2 SUSY gauge theories. In particular, SU(3) gauge theories with two flavors of massless quarks in the fundamental representation can be related to the spectral curve of the Goryachev-Chaplygin top, which is a Nahm's equation in disguise. This can be generalized to the cases with massive quarks, and N_f = 0,1,2, where a system with seven dimensional phase space has the relevant hyperelliptic curve appear in the Painlev\'e test. To understand the stringy origin of the integrability of these theories we obtain exact nonperturbative point particle limit of type II string compactified on a Calabi-Yau manifold, which gives the hyperelliptic curve of SU(2) QCD with N_f=1 hypermultiplet. 
  We propose the theory of quantum gravity with interactions introduced by topological principle. The fundamental property of such a theory is that its energy-momentum tensor is an BRST anticommutator. Physical states are elements of BRST cohomology group. The model with only topological excitations, introduced recently by Witten is discussed from the point of view of induced gravity program. We find that the mass scale is induced dynamically by gravitational instantons. The low energy effective theory has gravitons, which occur as the collective excitations of geometry, when the metric becomes dynamical. Applications of cobordism theory to QG are discussed. 
  Topology change in quantum gravity is considered. An exact wave function of the Universe is calculated for topological Chern-Simons 2+1 dimensional gravity. This wave function occurs as the effect of a quantum anomaly which leads to the induced gravity. We find that the wave function depends universally on the topology of the two-dimensional space. Indeed, the property of the ground state wave function of Chern-Simons gravity which has an attractive physical interpretation is that it becomes large in the infrared (large distances) if the Universe has ``classical'' topology $S^2\times R$. On the other hand, nonclassical topologies ${\Sigma}_g\times R$, where ${\Sigma}_g$ is the Riemann surface of genus g, are driven by quantum effects into the Planckian regime (``space-time foam''). The similar behavior of the quantum gravitational measure on four-manifolds constructed recently is discussed as the next example. We discuss the new phenomenon of the nonperturbative instability of black holes discovered recently. One finds that the Planck- sized black holes are unstable due to topology change. The decay rate is estimated using the instanton approximation. A possible solution to the primordial black hole problem in quantum cosmology is suggested. 
  In Thermal Field Dynamics, thermal states are obtained from restrictions of vacuum states on a doubled field algebra. It is shown that the suitably doubled Fock representations of the Heisenberg algebra do not need to be introduced by hand but can be canonically handed down from deformations of the extended Heisenberg bialgebra. No artificial redefinitions of fields are necessary to obtain the thermal representations and the case of arbitrary dimension is considered from the beginning. Our results support a possibly fundamental role of bialgebra structures in defining a general framework for Thermal Field Dynamics. 
  Talks given at the workshop ``Frontiers in Quantum Field Theory'', held in Urumqi, Xinjiang Uygur Autonomous Region, People's Republic of China, 11-19 August 1996. 
  The Ernst-like matrix representation of the multidimensional Einstein-Kalb-Ramond theory is developed and the O(d,d)-symmetry is presented in the matrix-valued SL(2,R)-form. The analogy with the Einstein and Einstein-Maxwell-Dilaton-Axion theories is discussed. 
  The Chern-Simons topological term coefficient is derived at arbitrary finite density. As it occures that $\mu^2 = m^2$ is the crucial point for Chern-Simons. So when $\mu^2 < m^2 \mu$--influence disappears and we get the usual Chern-Simons term. On the other hand when $\mu^2 > m^2$ the Chern-Simons term vanishes because of non-zero density of background fermions. In particular for massless case parity anomaly is absent at any finite density. This result holds in any odd dimension as in abelian so as in nonabelian cases. 
  We consider several orbifold compactifications of M-theory and their corresponding type II duals in two space-time dimensions. In particular, we show that while the orbifold compactification of M-theory on $T^9/J_9$ is dual to the orbifold compactification of type IIB string theory on $T^8/I_8$, the same orbifold $T^8/I_8$ of type IIA string theory is dual to M-theory compactified on a smooth product manifold $K3 \times T^5$. Similarly, while the orbifold compactification of M-theory on $(K3 \times T^5)/\sigma ... J_5$ is dual to the orbifold compactification of type IIB string theory on $(K3 \times T^4)/\sigma ... I_4$, the same orbifold of type IIA string theory is dual to the orbifold $T^4 \times (K3 \times S^1)/\sigma ... J_1$ of M-theory. The spectrum of various orbifold compactifications of M-theory and type II string theories on both sides are compared giving evidence in favor of these duality conjectures. We also comment on a connection between Dasgupta-Mukhi-Witten conjecture and Dabholkar-Park-Sen conjecture for the six-dimensional orbifold models of type IIB string theory and M-theory. 
  We study general black-hole solutions of the low-energy string effective action in arbitrary dimensions using a general metric that can describe them all in a unified way both in the extreme and non-extreme cases. We calculate the mass, temperature and entropy and study which relations amongst the charges and the mass lead to extremality. We find that the temperature always vanishes in the extreme limit and we find that, for a set of n charges (no further reducible by duality) there are 2^{(n-1)} combinations of the charges that imply extremality. Not all of these combinations can be central charge eigenvalues and, thus, there are in general extreme black holes which are not supersymmetric (or ``BPS-saturated''). In the N=8 supergravity case we argue that the existence of roughly as many supersymmetric and non-supersymmetric extreme black holes suggests the existence of an underlying twelve-dimensional structure. 
  We comment on various aspects of topological gauge theories possessing N_{T}\geq 2 topological symmetry: (1) We show that the construction of Vafa-Witten and Dijkgraaf-Moore of `balanced' topological field theories is equivalent to an earlier construction in terms of N_{T}=2 superfields inspired by Susy QM. (2) We explain the relation between topological field theories calculating signed and unsigned sums of Euler numbers of moduli spaces. (3) We show that the topological twist of N=4 d=4 Yang-Mills theory recently constructed by Marcus is formally a deformation of four-dimensional super-BF theory. (4) We construct a novel N_{T}=2 topological twist of N=4 d=3 Yang-Mills theory, a `mirror' of the Casson invariant model, with some unusual features. (5) We give a complete classification of the topological twists of N=8 d=3 Yang-Mills theory and show that they are realised as world-volume theories of Dirichlet two-brane instantons wrapping supersymmetric three-cycles of Calabi-Yau three-folds and G_{2}-holonomy Joyce manifolds. (6) We describe the topological gauge theories associated to D-string instantons on holomorphic curves in K3s and Calabi-Yau 3-folds. 
  We discuss how to construct open membranes in the recently proposed matrix model of M theory. In order to sustain an open membrane, two boundary terms are needed in the construction. These boundary terms are available in the system of the longitudinal five-branes and D0-branes. 
  We apply the SL(2,C) lattice Kac-Moody algebra of Alekseev, Faddeev and Semenov-Tian-Shansky to obtain a new lattice description of the SU(2) chiral model in two dimensions. The system has a global quantum group symmetry and it can be regarded as a deformation of two different theories. One is the nonabelian Toda lattice which is obtained in the limit of infinite central charge, while the other is a nonstandard Hamiltonian description of the chiral model obtained in the continuum limit. 
  For most black holes in string theory, the Schwarzschild radius in string units decreases as the string coupling is reduced. We formulate a correspondence principle, which states that (i) when the size of the horizon drops below the size of a string, the typical black hole state becomes a typical state of strings and D-branes with the same charges, and (ii) the mass does not change abruptly during the transition. This provides a statistical interpretation of black hole entropy. This approach does not yield the numerical coefficient, but gives the correct dependence on mass and charge in a wide range of cases, including neutral black holes. 
  A new framework for a high energy limit of quantum gauge field theories is introduced. Its potency is illustrated on a new derivation of the reggeization of the gluon. 
  Possible Dirichlet boundary states for WZW models with untwisted affine super Kac-Moody symmetry are classified for all compact simple Lie groups. They are obtained by inner- and outer-automorphism of the group. D-brane world-volume turns out to be a group manifold of a symmetric subgroup, so that the moduli space of D-brane is a irreducible Riemannian symmetric space. It is also clarified how these D-branes are transformed to each other under abelian T-duality of WZW model. Our result implies, for example, there is no D-particle on the compact simple group manifold. When the D-brane world-volume contains $S^1$ factor, the D-brane moduli space becomes hermitian symmetric space and the open string world-sheet instantons are allowed. 
  In this article we show in some detail how the full action functional of the standard model of elementary particle physics can be described within the geometrical setting of generalized Dirac operators. We thereby introduce a new model building kit for (a certain class of) gauge invariant theories which provides a unified geometrical description of Einstein's theory of gravity and Yang-Mills gauge theories on the "classical" level. Moreover, when the gauge symmetry is spontaneously broken, the Higgs sector as well has a natural geometrical interpretation. It turns out that the Higgs field is related to the gravitational potential.   Since the full action functional of the standard model is derived in one stroke, the appropriate parameters of the model have to satisfy certain relations similar to those in the Connes-Lott approach. Likewise, this may yield some phenomenological consequences, which is illustrated by using the gauge group of the standard model in the case of $N-$generations of leptons and quarks. 
  The classical integrability the O(N) nonlinear sigma model on a half-line is examined, and the existence of an infinity of conserved charges in involution is established for the free boundary condition. For the case N=3 other possible boundary conditions are considered briefly. 
  The theory of the usual, constrained p-branes is embedded into a larger theory in which there is no constraints. In the latter theory the Fock-Schwinger proper time formalism is extended from point-particles to p-branes which can be considered as a points in an infinite dimensional space M. The quantization appears to be straightforward and elegant. The conventional p-brane states are particular stationary solutions to the functional Schr\"odinger equation which describes the evolution of a membrane's state with respect to the invariant evolution parameter $\tau$. It is also shown that states of a lower dimensional p-brane can be considered as particular states of a higher dimensional p-brane. 
  In this paper the Maxwell field theory is considered on the $Z_n$ symmetric algebraic curves. As a first result, a large family of nondegenerate metrics is derived for general curves. This allows to treat many differential equations arising in quantum mechanics and field theory on Riemann surfaces as differential equations on the complex sphere. The examples of the scalar fields and of an electron immersed in a constant magnetic field will be briefly investigated. Finally, the case of the Maxwell equations on curves with $Z_n$ group of automorphisms is studied in details. These curves are particularly important because they cover the entire moduli space spanned by the Riemann surfaces of genus $g\le 2$. The solutions of these equations corresponding to nontrivial values of the first Chern class are explicitly constructed. 
  A couple of issues concerning the effective dynamics of D-branes in string theory are discussed. Primarily, I am concerned with linearization of the actions by introduction of non-propagating fields and a full super- and kappa-symmetric description of D-branes. 
  The most relevant thermal perturbation of the continuous d=2 minimal conformal theory with c=7/10 (Tricritical Ising Model) is treated here. This model describes the scaling region of the phi^6 universality class near the tricritical point. The problematic IR divergences of the naive perturbative expansion around conformal theories are dealt within the OPE approach developed at all orders by the authors. The main result is a description of the short distance behaviour of correlators that is compared with existing long distance expansion (form factors approach) related to the integrability of the model. 
  Functional determinants for Dirac operators on manifolds with boundary are considered. Ellipticity of boundary value problems is discussed in terms of the Calderon projector.   The functional determinant for a Dirac operator on a bidimensional disk, in the presence of an Abelian gauge field, subject to global boundary conditions of the type introduced by Atiyah-Patodi-Singer, is evaluated. The relationship with the index theorem is also commented. 
  We study the renormalizable abelian vector-field models in the presence of the Wess-Zumino interaction with the pseudoscalar matter. The renormalizability is achieved by supplementing the standard kinetic term of vector fields with higher derivatives. The appearance of fourth power of momentum in the vector-field propagator leads to the super-renormalizable theory in which the beta-function, the vector-field renormalization constant and the anomalous mass dimension are calculated exactly. It is shown that this model has the infrared stable fixed point and its low-energy limit is non-trivial. The modified effective potential for the pseudoscalar matter leads to the occurrence of the quantum dynamical breaking of Lorentz symmetry. 
  Various aspects of branes in the recently proposed matrix model for M theory are discussed. A careful analysis of the supersymmetry algebra of the matrix model uncovers some central charges which can be activated only in the large $N$ limit. We identify the states with non-zero charges as branes of different dimensions. 
  Liouville field theory is quantized by means of a Wilsonian effective action and its associated exact renormalization group equation. For $c<1$, an approximate solution of this equation is obtained by truncating the space of all action functionals. The Ward identities resulting from the Weyl invariance of the theory are used in order to select a specific universality class for the renormalization group trajectory. It is found to connect two conformal field theories with central charges $25-c$ and $26-c$, respectively. 
  The Demazure character formula is applied to the Verlinde formula for affine fusion rules. We follow Littelmann's derivation of a generalized Littlewood-Richardson rule from Demazure characters. A combinatorial rule for affine fusions does not result, however. Only a modified version of the Littlewood-Richardson rule is obtained that computes an (old) upper bound on the fusion coefficients of affine $A_r$ algebras. We argue that this is because the characters of simple Lie algebras appear in this treatment, instead of the corresponding affine characters. The Bruhat order on the affine Weyl group must be implicated in any combinatorial rule for affine fusions; the Bruhat order on subgroups of this group (such as the finite Weyl group) does not suffice. 
  The exact quantum integrability aspects of a sector of the membrane is investigated. It is found that spherical membranes ( in the lightcone gauge) moving in flat target spacetime backgrounds admit a class of integrable solutions linked to $SU(\infty)$ SDYM equations ( dimensionally reduced to one temporal dimension) which, in turn, are related to Plebanski 4D SD Gravitational equations. A further rotational Killing-symmetry reduction yields the 3D continuous Toda theory. It is precisely the latter which bears a direct relationship to non critical $W_\infty$ string theory. The expected critical dimensions for the ( super) membrane , (D=11) and D=27, are easily obtained. This suggests that this particular sector of the membrane's spectrum (connected to the $SU(\infty)$ SDYM equations ) bears a direct connection to a critical $W_\infty$ string spectrum adjoined to a q=N+1 unitary minimal model of the W_N algebra in the $N\rightarrow \infty$ limit. Final comments are made about the connection to Jevicki's observation that the 4D quantum membrane is linked to dilatonic-self dual gravity plus matter . 2D dilatonic ( super) gravity was studied by Ikeda and its relation to nonlinear $W_\infty$ algebras from nonlinear integrable deformations of 4D self dual gravity was studied by the author.The full $SU(\infty)$ YM theory remains to be explored as well as the incipient role that noncritical nonlinear $W_\infty$ strings might have in the full quantization program. 
  We investigate the exact results of the Navier-Stokes equations using the methods developed by Polyakov. It is shown that when the velocity field and the density are not independent, the Burgers equation is obtained leading to exact N-point generating functions of velocity field. Our results show that, the operator product expansion has to be generalized both in the absence and the presence of pressure. We find a method to determine the extra terms in the operator product expansion and derive its coefficients and find the first correction to probablity distribuation function. In the general case and for small pressure, we solve the problem perturbatively and find the probablity distribuation function for the Navier-Stokes equation in the mean field approximation. 
  We discuss the appearance of $E_8\times E_8$ gauge bosons in Banks, Fischler, Shenker, and Susskind's zero brane quantum mechanics approach to M theory, compactified on the interval $S^1/Z_2$. The necessary bound states of zero branes are proven to exist by a straightforward application of T-duality and heterotic $Spin(32)/Z_2$-Type I duality. We then study directly the zero brane Hamiltonian in Type I' theory. This Hamiltonian includes couplings between the zero branes and background Dirichlet 8 branes localized at the orientifold planes. We identify states, localized at the orientifold planes, with the requisite gauge boson quantum numbers. An interesting feature is that $E_8$ gauge symmetry relates bound states of different numbers of zero branes. 
  Light-front Hamiltonian methods are being developed to attack bound-state problems in QCD. In this paper we advance the state of the art for these methods by computing the well-known Lamb shift in hydrogen starting from first principles of QED. There are obvious but significant qualitative differences between QED and QCD. In this paper, we discuss the similarities that may survive in a non-perturbative QCD calculation in the context of a precision non-perturbative QED calculation. Central to the discussion are how a constituent picture arises in a gauge field theory, how bound-state energy scales emerge to guide the renormalization procedure, and how rotational invariance emerges in a light-front calculation. 
  0-brane of type IIA string theory can be interpreted as a dimensional reduction of a gravitational wave in 11 dimensions. We observe that a similar interpretation applies also to the D-instanton background of type IIB theory: it can be viewed as a reduction (along one spatial and one time-like direction) of a wave in a 12-dimensional theory. The instanton charge is thus related to a linear momentum in 12 dimensions. This suggests that the instanton should play as important role in type IIB theory as the 0-brane is supposed to play in type IIA theory. 
  We apply Borel resummation method to the conventional perturbation series of ground state energy in a metastable potential, $V(x)=x^2/2-gx^4/4$. We observe numerically that the discontinuity of Borel transform reproduces the imaginary part of energy eigenvalue, i.e., total decay width due to the quantum tunneling. The agreement with the exact numerical value is remarkable in the whole tunneling regime $0<g\lsim0.7$. 
  The exact operator solutions of two-dimensional anomaly-free chiral abelian gauge theories are obtained. We show that anomaly-cancellation conditions arise as consistency requirements of these solutions. For a certain class of flavour symmetries, fermion condensates are constructed. These are shown to violate the fermion-number conservation rule. The models are extended to include massive fermions. We propose a bosonised lagrangian for the massive theory and verify that it complies with the Gupta-Bleuler condition. 
  We discuss a previously discovered [hep-th/9401027] extension of the infinite-dimensional Lie algebras Map(M,g) which generalizes the Kac-Moody algebras in 1+1 dimensions and the Mickelsson-Faddeev algebras in 3+1 dimensions to manifolds M of general dimensions. Furthermore, we review the method of regularizing current algebras in higher dimensions using pseudodifferential operator (PSDO) symbol calculus. In particular, we discuss the issue of Lie algebra cohomology of PSDOs and its relation to the Schwinger terms arising in the quantization process. Finally, we apply this regularization method to the algebra of the above reference with partial success, and discuss the remaining obstacles to the construction of a Fock space representation. 
  The large N limit of the 3-d Gross-Neveu model is here studied on manifolds with positive and negative constant curvature. Using the $\zeta$-function regularization we analyze the critical properties of this model on the spaces $S^2 \times S^1$ and $H^2\times S^1$. We evaluate the free energy density, the spontaneous magnetization and the correlation length at the ultraviolet fixed point. The limit $S^1\to R$, which is interpreted as the zero temperature limit, is also studied. 
  The dymamical chiral symmetry breaking in higher- derivative quantum gravity has been investigated on the flat background. The Schwinger- Dyson equations numerical solutions have been found in the ladder approximation. Both two- and four- dimensional cases have been considered. The dymamical fermion mass generation accompanied by the second- order phase transition has been shown to take place in a different gauges. 
  The exact effective field equations of motion, corresponding to the perturbative mixed theory of open and closed (2,2) world-sheet supersymmetric strings, are investigated. It is shown that they are only integrable in the case of an abelian gauge group. The gravitational equations are then stationary with respect to the Born-Infeld-type effective action. 
  The (2,2) world-sheet supersymmetric string theory is discussed from the viewpoint of string/membrane unification. The effective field theory in the closed string target space is known to be the 2+2 dimensional (integrable) theory of self-dual gravity (SDG). A world-volume supersymmetrization of the Pleba'nski action for SDG naturally implies the maximal N=8 world-volume supersymmetry, while the maximal supersymmetrization of the dual covariant K"ahler-Lorentz-Chern-Simons action for SDG implies gauging a self-dual part of the super-Lorentz symmetry in 2+10 dimensions. The proposed OSp(32|1) supersymmetric action for the M-brane may be useful for a fundamental formulation of uncompactified F theory, with the self-duality being playing the central role both in the world-volume and in the target space of the M-brane. 
  A `canonical mapping' is established between the c=-1 system of bosonic ghosts and the c=2 complex scalar theory and, a similar mapping between the c=-2 system of fermionic ghosts and the c=1 Dirac theory. The existence of this mapping is suggested by the identity of the characters of the respective theories. The respective c<0 and c>0 theories share the same space of states, whereas the spaces of conformal fields are different. Upon this mapping from their c<0 counterparts, the (c>0) complex scalar and the Dirac theories inherit hidden nonlocal sl(2) symmetries. 
  The energy spectrum of the three-particle Hamiltonian obtained by replacing the two-body trigonometric potential of the Sutherland problem by a three-body one of a similar form is derived. When expressed in appropriate variables, the corresponding wave functions are shown to be expressible in terms of Jack polynomials. The exact solvability of the problem with three-body interaction is explained by a hidden sl(3,\R) symmetry. 
  Born-Infeld theory is formulated using an infinite set of gauge fields, along the lines of McClain, Wu and Yu. In this formulation electromagnetic duality is generated by a fully local functional. The resulting consistency problems are analyzed and the formulation is shown to be consistent. 
  The Ohnuki-Kitakado (O-K) scheme of quantum mechanics on $S^D$ embedded in $R^{D+1}$ is investigated. Generators satisfying the O-K algebra are written down explicitly in term of the induced gauge potential. A direct method is developed to obtain the generators in covariant form. It is seen that there exists an induced gauge configuration which is trivial on $S^D$ but might cause a nontrivial physical effect in $R^{D+1}$. The relation of the O-K scheme to extended objects such as the 't Hooft-Polyakov monopole is discussed. 
  We extend the notion of self-duality to spaces built from a set of representations of the Lorentz group with bosonic or fermionic behaviour, not having the traditional spin-one upper-bound of super Minkowski space. The generalized derivative vector fields on such superspaces are assumed to form a superalgebra. Introducing corresponding gauge potentials and hence covariant derivatives and curvatures, we define generalized self-duality as the Lorentz covariant vanishing of certain irreducible parts of the curvatures. 
  As an application of the renormalization method introduced by the second author we give a causal definition of the phase of the quantum scattering matrix for fermions in external Yang-Mills potentials. The phase is defined using parallel transport along the path of renormalized time evolution operators. The time evolution operators are elements of the restricted unitary group $U_{res}$ of Pressley and Segal. The central extension of $U_{res}$ plays a central role. 
  We construct global observable algebras and global DHR morphisms for the Virasoro minimal models with central charge c(2,q), q odd. To this end, we pass {}from the irreducible highest weight modules to path representations, which involve fusion graphs of the c(2,q) models. The paths have an interpretation in terms of quasi-particles which capture some structure of non-conformal perturbations of the c(2,q) models. The path algebras associated to the path spaces serve as algebras of bounded observables. Global morphisms which implement the superselection sectors are constructed using quantum symmetries: We argue that there is a canonical semi-simple quantum symmetry algebra for each quasi-rational CFT, in particular for the c(2,q) models. These symmetry algebras act naturally on the path spaces, which allows to define a global field algebra and covariant multiplets therein. 
  This contribution reviews recent progress in constructing affine Lie algebras at arbitrary level in terms of vertex operators. The string model describes a completely compactified subcritical chiral bosonic string whose momentum lattice is taken to be the (Lorentzian) affine weight lattice. The main feature of the new realization is the replacement of the ordinary string oscillators by physical DDF operators, whereas the unphysical position operators are substituted by certain linear combinations of the Lorentz generators. As a side result we obtain simple expressions for the affine Weyl translations as Lorentz boosts. Various applications of the construction are discussed. 
  Dynamical systems, described by Lagrangians with first- and second-class constraints, are investigated. In the Dirac approach to the generalized Hamiltonian formalism, the classification and separation of the first- and second-class constraints are presented with the help of passing to an equivalent canonical set of constraints. The general structure of second-class constraints is clarified. The method of constructing the generator of symmetry transformations in the second Noether theorem is given. It is proved that second-class constraints do not contribute to the transformation law of the local symmetry which entirely is entirely stipulated by all the first-class constraints. 
  We discuss some of the classical and quantum geometry associated to the degeneration of cycles within a Calabi-Yau compactification. In particular, we focus on the definition and properties of quantum volume, especially as it applies to identifying the physics associated to loci in moduli space where nonperturbative effects become manifest. We discuss some unusual features of quantum volume relative to its classical counterpart. 
  In the context of D-dimensional Euclidean gravity, we define the natural generalisation to D-dimensions of the self-dual Yang-Mills equations, as duality conditions on the curvature 2-form of a Riemannian manifold. Solutions to these self-duality equations are provided by manifolds of SU(2), SU(3), G_2 and Spin(7) holonomy. The equations in eight dimensions are a master set for those in lower dimensions. By considering gauge fields propagating on these self-dual manifolds and embedding the spin connection in the gauge connection, solutions to the D-dimensional equations for self-dual Yang-Mills fields are found. We show that the Yang-Mills action on such manifolds is topologically bounded from below, with the bound saturated precisely when the Yang-Mills field is self-dual. These results have a natural interpretation in supersymmetric string theory. 
  We study domain wall solitons in the relativistic self-dual Chern-Simons Higgs systems by the dimensional reduction method to two dimensional spacetime. The Bogomolny bound on the energy is given by two conserved quantities in a similar way that the energy bound for BPS dyons is set in some Yang-Mills-Higgs systems in four dimensions. We find the explicit soliton configurations which saturate the energy bound and their nonrelativistic counter parts. We also discuss the underlying N=2 supersymmetry. 
  In this paper, we construct a new topological quantum field theory of cohomological type and show that its partition function is a crossing number. 
  The following is a short report about recent work on discrete physics/mathematics on the Planckscale and the use of the concept of ''random graphs'' in this business, appearing in the group21-proceedings (Gosslar 1996) 
  The super or Z_2-graded Schouten-Nijenhuis bracket is introduced. Using it, new generalized super-Poisson structures are found which are given in terms of certain graded-skew-symmetric contravariant tensors \Lambda of even order. The corresponding super `Jacobi identities' are expressed by stating that these tensors have zero super Schouten-Nijenhuis bracket with themselves [\Lambda,\Lambda]=0. As a particular case, we provide the linear generalized super-Poisson structures which can be constructed on the dual spaces of simple superalgebras with a non-degenerate Killing metric. The su(3,1) superalgebra is given as a generic example. 
  This paper is withdrawn since there is a problem in the previous paper on which it is based. 
  We study supersymmetric compactifications of type II strings on eightfolds to two dimensions. It is demonstrated that the type IIB string on an eightfold is free of gravitational anomalies. T-duality requires that this theory when further compactified on a circle must have a vacuum momentum; this is explicitly shown to be present and to have the right value. A subtlety in the relation of IIB compactifications and M-Theory orientifolds to two dimensions is pointed out. 
  The theory of free relativistic fields is shown to arise in a unified manner from higher-order, configuration-space, irreducible representations of the Poincar\'e group. A de Sitter subalgebra, in the massive case, and a Poincar\'e subalgebra, in the massless case, of the enveloping algebra of the Poincar\'e group are the suitable higher-order polarizations. In particular, a simple group-theoretic derivation of the Dirac equation is given. 
  The effective potential of composite fermion fields in three-dimensional Thirring model in curved spacetime is calculated in linear curvature approximation. The phase transition accompanied by the creation of non-zero chiral invariant bifermionic vector-like condensate is shown to exist. The type of this phase transition is discussed. 
  By dimensional reduction of a self dual p-form theory on some compact space, we determine the duality generators of the gauge theory in 4 dimensions. In this picture duality is seen as a consequence of the geometry of the compact space. We describe the dimensional reduction of 10-dimensional self dual 4-form Maxwell theory to give a theory in 4-dimensions with scalar, one form and two form fields that all transform non trivially under duality. 
  In this lecture we review some of the recent developments in string theory on an introductory and qualitative level. In particular we focus on S-T-U dualities of toroidally compactified ten-dimensional string theories and outline the connection to M-theory. Dualities among string vacua with less supersymmetries in six and four space-time dimensions is discussed and the concept of F-theory is briefly presented. (Lecture given by J. Louis at the Workshop on Gauge Theories, Applied Supersymmetry and Quantum Gravity, Imperial College, London, UK, July 5--10, 1996.) 
  The Standard MS renormalization prescription is inadequate for dealing with multi-scale problems. To illustrate this we consider the computation of the effective potential in the Higgs-Yukawa model. It is argued that it is natural to employ a two-scale renormalization group. We give a modified version of a two-scale scheme introduced by Einhorn and Jones. In such schemes the beta functions necessarily contain potentially large logarithms of the RG scale ratios. For credible perturbation theory one must implement a large logarithms resumation on the beta functions themselves. We show how the integrability condition for the two RG equations allows one to perform this resummation. 
  We compute the exact finite temperature effective action in a 0+1-dimensional field theory containing a topological Chern-Simons term, which has many features in common with 2+1-dimensional Chern-Simons theories. This exact result explains the origin and meaning of puzzling temperature dependent coefficients found in various naive perturbative computations in the higher dimensional models. 
  We study the low-energy effective Hamiltonian of N=2 super Yang-Mills theories. We find that the BPS equations are unchanged outside a quantum core where higher dimension contributions are expected to be important. We find two quantum generalizations of the BPS soliton. The leading higher-derivative correction to the effective action is shown not to contribute to the BPS mass formula. 
  We present a new 4D, N = 1 supersymmetric nonlinear sigma-model using complex linear and chiral superfields that generalizes the massless limit of the QCD effective action of Gasser and Leutwyler. 
  We point out that the celebrated Hawking effect of quantum instability of black holes seems to be related to a nonperturbative effect in string theory. Studying quantum dynamics of strings in the gravitational background of black holes we find classical instability due to emission of massless string excitations. The topology of a black hole seems to play a fundamental role in developing the string theory classical instability due to the effect of sigma model instantons. We argue that string theory allows for a qualitative description of black holes with very small masses and it predicts topological solitons with quantized spectrum of masses. These solitons would not decay into string massless excitations but could be pair created and may annihilate also. Semiclassical mass quantization of topological solitons in string theory is based on the argument showing existence of nontrivial zeros of beta function of the renormalization group. 
  A proposal for the matrix model formulation of the M-theory on a space with a boundary is given. A general machinery for modding out a symmetry in M(atrix) theory is used for a Z_2 symmetry changing the sign of the X_1 coordinate. The construction causes the elements of matrices to be equivalent to real numbers or quaternions and the symmetry U(2N) of the original model is reduced to O(2N) or USp(2N)=U(N,H). We also show that membranes end on the boundary of the spacetime correctly in this construction. 
  In this paper we study the motion of a rigidly rotating Nambu-Goto test string in a stationary axisymmetric background spacetime. As special examples we consider the rigid rotation of strings in flat spacetime, where explicit analytic solutions can be obtained, and in the Kerr spacetime where we find an interesting new family of test string solutions. We present a detailed classification of these solutions in the Kerr background. 
  We consider four-parameter $D=4,N=2$ string models with Hodge numbers $(4,214- 12n)$ and $(4,148)$, and we express their perturbative Wilsonian gravitational coupling $F_1$ in terms of Siegel modular forms. 
  Quantum theory of 2d gravity for $c>1$ is examined as a non-critical string theory by taking account of the loop-correction of open strings whose end points are on the 2d world surface of the closed string. This loop-correction leads to a conformal anomaly, and we obtain a modified target-space action which implies a new phase of the non-critical closed-string. In this phase, the dual field of the gauge field, which lives on the boundary, condenses and the theory can be extended to $c>1$ without any instability. 
  We study some applications of solvable Lie algebras in type IIA, type IIB and M theories. RR and NS generators find a natural geometric interpretation in this framework. Special emphasis is given to the counting of the abelian nilpotent ideals (translational symmetries of the scalar manifolds) in arbitrary D dimensions. These are seen to be related, using Dynkin diagram techniques, to one-form counting in D+1 dimensions. A recipy for gauging isometries in this framework is also presented. In particular, we list the gauge groups both for compact and translational isometries. The former agree with some results already existing in gauged supergravity. The latter should be possibly related to the study of partial supersymmetry breaking, as suggested by a similar role played by solvable Lie algebras in N=2 gauged supergravity. 
  The off-shell vector-tensor multiplet is considered in an arbitrary background of N=2 vector supermultiplets. We establish the existence of two inequivalent versions, characterized by different Chern-Simons couplings. In one version the vector field of the vector-tensor multiplet is contained quadratically in the Chern-Simons term, which implies nonlinear terms in the supersymmetry transformations and equations of motion. In the second version, which requires a background of at least two abelian vector supermultiplets, the supersymmetry transformations remain at most linear in the vector-tensor components. This version is of the type known to arise from reduction of tensor supermultiplets in six dimensions. Our work applies to any number of vector-tensor multiplets. 
  We show that the M-theory/IIA and IIA/IIB superstring dualities together with the diffeomorphism invariance of the underlying theories require the presence of certain p-brane bound states in IIA and IIB superstring theories preserving 1/2 of the spacetime supersymmetry. We then confirm the existence of IIA and IIB supergravity solutions having the appropriate p-brane bound states interpretation. 
  We reconsider differential geometry from the point of view of the quantum theory of non-relativistic spinning particles, which provides examples of supersymmetric quantum mechanics. This enables us to encode geometrical structure in algebraic data consisting of an algebra of functions on a manifold and a family of supersymmetry generators represented on a Hilbert space. We show that known types of differential geometry can be classified in terms of the supersymmetries they exhibit. Replacing commutative algebras of functions by non-commutative *-algebras of operators, while retaining supersymmetry, we arrive at a formulation of non-commutative geometry encompassing and extending Connes' original approach. We explore different types of non-commutative geometry and introduce notions of non-commutative manifolds and non-commutative phase spaces. One of the main motivations underlying our work is to construct mathematical tools for novel formulations of quantum gravity, in particular for the investigation of superstring vacua. 
  The time evolution of correlation functions in statistical systems is described by an exact functional differential equation for the corresponding generating functionals. This allows for a systematic discussion of non-equilibrium physics and the approach to equilibrium without the need of solving the nonlinear microscopic equations of motion or computing the time dependence of the probability distribution explicitly. 
  We develop a systematic approach to confinement in N=1 supersymmetric theories. We identify simple necessary conditions for theories to confine without chiral symmetry breaking and to generate a superpotential non-perturbatively (s-confine). Applying these conditions we identify all N=1 theories with a single gauge group and no tree-level superpotential which s-confine. We give a complete list of the confined spectra and superpotentials. Some of these theories are of great interest for model building. We give several new examples of models which break supersymmetry dynamically. 
  The geometry of (2,1) supersymmetric sigma-models is reviewed and the conditions under which they have isometry symmetries are analysed. Certain potentials are constructed that play an important role in the gauging of such symmetries. The gauged action is found for a special class of models. 
  We give a new interpretation for the super loop space that has been used to formulate supersymmetry. The fermionic coordinates in the super loop space are identified as the odd generators of the Weil algebra. Their bosonic superpartners are the auxiliary fields. The general N=1 supermultiplet is interpreted in terms of Weil algebras. As specific examples we consider supersymmetric quantum mechanics, Wess-Zumino model and supersymmetric Yang-Mills theory in four dimensions. Some comments on the formulation of constrained systems and integrable models and non-Abelian localization are given. 
  Using the coset construction, we compute the root multiplicities at level three for some hyperbolic Kac-Moody algebras including the basic hyperbolic extension of $A_1^{(1)}$ and $E_{10}$. 
  The consideration of the bound skyrmions with large strangeness content is continued. The connection between B=2 SO(3)-hedgehog and SU(2)-torus is investigated and the quantization of the dipole- type configuration with large strangeness content is described. 
  We consider a particular 4-dimensional generalization of the transition from the Heisenberg to the Schr\"odinger picture. The space-time independent expansion with respect to the unitary irreducible representations of the Lorentz group is applied, as Fourier transformation in the Heisenberg picture, to the states of a massive relativistic particle. A new Hamiltonian operator has been found for such a particle with spin one. 
  Using an ``action at a distance'' formulation we probe the possible classical interactions for tensionless strings, (the $T\rightarrow 0$ limit of the ordinary bosonic string.) We find $G_{\mu\nu}$ and $B_{\mu\nu}$ type interactions but no dilaton interactions. 
  We formulate a renormalized running coupling expansion for the $\beta$--function and the potential of the renormalized $\phi^4$--trajectory on four dimensional Euclidean space-time. Renormalization invariance is used as a first principle. No reference is made to bare quantities. The expansion is proved to be finite to all orders of perturbation theory. The proof includes a large momentum bound on the connected free propagator amputated vertices. 
  Crossing symmetry appears in Dbrane-anti-Dbrane dynamics in the form of an analytic continuation from U(N) for N brane amplitudes to U(N-p,p) for the interactions of N-p branes with p anti-branes. I consider the consequences for supersymmetry and brane-anti-brane forces. 
  We study a 3-dimensional topological sigma-model, whose target space is a hyper-Kahler manifold X. A Feynman diagram calculation of its partition function demonstrates that it is a finite type invariant of 3-manifolds which is similar in structure to those appearing in the perturbative calculation of the Chern-Simons partition function.   The sigma-model suggests a new system of weights for finite type invariants of 3-manifolds, described by trivalent graphs. The Riemann curvature of X plays the role of Lie algebra structure constants in Chern-Simons theory, and the Bianchi identity plays the role of the Jacobi identity in guaranteeing the so-called IHX relation among the weights.   We argue that, for special choices of X, the partition function of the sigma-model yields the Casson-Walker invariant and its generalizations. We also derive Walker's surgery formula from the SL(2,Z) action on the finite-dimensional Hilbert space obtained by quantizing the sigma-model on a two-dimensional torus. 
  A new action of the Yangians in the WZW models is displayed. Its structure is generic and level independent. This Yangian is the natural extension at the conformal point of the one unravelled in massive theories with current algebras. Expectingly, this new symmetry of WZW models will lead to a deeper understanding of the integrable structure of conformal field theories and their deformations. 
  A complete treatment of the (2,2) NSR string in flat (2+2) dimensional space-time is given, from the formal path integral over N=2 super Riemann surfaces to the computational recipe for amplitudes at any loop or gauge instanton number. We perform in detail the superconformal gauge fixing, discuss the spectral flow, and analyze the supermoduli space with emphasis on the gauge moduli. Background gauge field configurations in all instanton sectors are constructed. We develop chiral bosonization on punctured higher-genus surfaces in the presence of gauge moduli and instantons. The BRST cohomology is recapitulated, with a new space-time interpretation for picture-changing. We point out two ways of combining left- and right-movers, which lead to different three-point functions. 
  I argue that the first-order formalism recently found to describe classical 2+1-Gravity with matter, is also able to include higher topologies. The present gauge, which is conformal with vanishing York time, is characterized by an analytic mapping from single-valued coordinates to Minkowskian ones. In the torus case, this mapping is based on four square-root branch points, whose location is related to the modulus, which has a well defined time dependence. In the general case, it is connected with the hyperelliptic representation of Riemann surfaces. 
  We present the chiral truncation of the eleven dimensional M-algebra down to ten and six dimensions. In ten dimensions, we obtain a topological extension of the $(1,0)$ Poincar\'e superalgebra that includes super one-form and super five-form charges. Closed super three- and seven-forms associated with this algebra are constructed. In six dimensions, we obtain a topological extension of the $(2,0)$ and $(1,0)$ Poincar\'e superalgebras. The former includes a quintet of super one-form charges, and a decuplet of super three-form charges, while the latter includes a triplet of super three-form charges. 
  We show how the reduced Self-dual Yang-Mills theory described by the Nahm equations can be carried over to the Weyl-Wigner-Moyal formalism employed recently in Self-dual gravity. Evidence of the existence of correspondence between BPS magnetic monopoles and space-time hyper-K\"ahler metrics is given. 
  D-string action is constructed from IIB matrices, a spacetime commutator is essential in this construction. This hints at the central role of the spacetime uncertainty relation in a unified formulation of strings. Vertex operators of fundamental strings are also discussed. 
  We discuss the proposed description of configurations with four-branes and six-branes in m(atrix) theory. Computing the velocity dependent potential between these configurations and gravitons and membranes, we show that they agree with the short distance string results computed in type IIa string theory. Due to the ``closeness'' of these configuration to a supersymmetric configuration the m(atrix) theory reproduces the correct long distance behavior. 
  We study the relationship between static p-brane solitons and cosmological solutions of string theory or M-theory. We discuss two different ways in which extremal p-branes can be generalised to non-extremal ones, and show how wide classes of recently discussed cosmological models can be mapped into non-extremal p-brane solutions of one of these two kinds. We also extend previous discussions of cosmological solutions to include some that make use of cosmological-type terms in the effective action that can arise from the generalised dimensional reduction of string theory or M-theory. 
  We comment on the calculation of the Chern-Simons coefficient in (2+1)-dimensional gauge theories at finite chemical potential made by A.N.Sissakian, O.Yu.Shevchenko and S.B.Solganik (hep-th/9608159 and hep-th/9612140). 
  The renormalized trajectory of massless $\phi^4$-theory on four dimensional Euclidean space-time is investigated as a renormalization group invariant curve in the center manifold of the trivial fixed point, tangent to the $\phi^4$-interaction. We use an exact functional differential equation for its dependence on the running $\phi^4$-coupling. It is solved by means of perturbation theory. The expansion is proved to be finite to all orders. The proof includes a large momentum bound on amputated connected momentum space Green's functions. 
  The N component scalar tricritical theory is considered in a non-perturbative setting. We derive non-perturbative beta functions for the relevant couplings in $d\leq 3$. The beta functions are obtained through the use of an exact evolution equation for the so called effective average action. In d=3 it is established the existence of an ultraviolet stable fixed point for N>4. This confirms earlier results obtained using the 1/N expansion where such a fixed point is believed to exist at least for $N\gtrsim 1000$. 
  Connes has extended Einstein's principle of general relativity to noncommutative geometry. The new principle implies that the Dirac operator is covariant with respect to Lorentz and internal gauge transformations and the Dirac operator must include Yukawa couplings. It further implies that the action for the metric, the gauge potentials and the Higgs scalar is coded in the spectrum of the covariant Dirac operator. This ``universal'' action has been computed by Chamseddine & Connes, it is the coupled Einstein-Hilbert and Yang-Mills-Higgs action. This result is rederived and we discuss the physical consequences. 
  We present intersecting p-brane solutions of eleven-dimensional supergravity (M-branes) which upon toroidal compactification reduce to non-extreme ``rotating'' black holes. We identify harmonic functions, associated with each M-brane, and non-extremality functions, specifying a deviation from the BPS limit. These functions are modified due to the angular momentum parameters, which specify the rotation along the transverse directions of the M-branes. We spell out the intersection rules for the eleven-dimensional space-time metric for intersecting (up to three) rotating M-brane configurations (and a boost along the common intersecting direction). 
  Self-dual 2-forms in D=2n dimensions are characterised by an eigenvalue criterion. The equivalence of various definitions of self-duality is proven. We show that the self-dual 2-forms determine a n^2-n+1 dimensional manifold S_{2n} and the dimension of the maximal linear subspaces of S_{2n}$ is equal to the Radon-Hurwitz number of linearly independent vector fields on the sphere S^{2n-1}. The relation between the maximal linear subspaces and the representations of Clifford algebras is noted. A general procedure based on this relation for the explicit construction of linearly self-dual 2-forms is given. The construction of the octonionic instanton solution in D=8 dimensions is discussed. 
  We construct the n-instanton action for the above model with gauge group SU(2), as a function of the collective coordinates of the general self-dual configurations of Atiyah, Drinfeld, Hitchin and Manin (ADHM). We calculate the quantum modulus u = <Tr(A^2)> at the 1-instanton level, and find a discrepancy with Seiberg and Witten's proposed exact solution. As in related models (N=2, N_F=3 or 4), this discrepancy may be resolved by modifying their proposed relation between $\tilde u$ (the parameter in the elliptic curve) and u. 
  The (3+d)-dimensional Einstein-Kalb-Ramond theory reduced to two dimensions is considered. It is shown that the theory allows two different Ernst-like $d \times d$ matrix formulations: the real non-dualized target space and the Hermitian dualized non-target space ones. The O(d, d) symmetry is written in a SL(2,R) matrix-valued form in both cases. The Kramer-Neugebauer transformation, which algebraically maps the non-dualized Ernst potential into the dualized one, is presented. 
  Virasoro-type symmetries and their roles in solvable models are reviewed. These symmetries are described by the two-parameter Virasoro-type algebra $Vir_{p,q}$ by choosing the parameters p and q suitably. 
  We study vertex operators for super Yang-Mills and multi D-branes in covariant form using Green-Schwarz formalism. We introduce the contact terms naturally and prove space-time supersymmetry and gauge invariance. The nonlinear realization of broken supersymmetry in the presence of D-branes is also discussed. The shift of fermionic coordinate \delta^{(-)}\Psi =\eta becomes exact symmetry of D-brane in the static gauge, where $\eta$ is a constant spinor in U(1) direction. 
  As the Yangian double with center,which is deformed from affine algebra by the additive loop parameter $\hbar$ ,we get the commuting relation and the bosonization of quantum $\hbar$-deformed Virasoro algebra. The corresponding Miura transformation, associated screening operators and the BRST charge have been studied. Moreover, we also constructe the bosonization for type I and type II intertwiner vertex operators. Finally, we show that the commuting relation of these vertex operators in the case of $p'=r p=r-1$ and $\hbar =\pi$ actually gives the exact scattering matrix of the Restricted sine-Gordon model. 
  An algebraic method for a general construction of intersecting p-brane solutions in diverse spacetime dimensions is discussed. An incidence matrix describing configurations of electric and magnetic fields is introduced. Intersecting p-branes are specified by solutions of a system of characteristic algebraic equations for the incidence matrix. This set of characteristic equations generalizes a single characteristic equation found before for a special "flower" ansatz. The characteristic equations admit solutions only for quantized values of scalar coupling parameters. A wide list of examples including solutions with regular horizons and non-zero entropy in D=11 and D=10 theories is presented. 
  We study critical and universal behaviors of unitary invariant non-gaussian random matrix ensembles within the framework of the large-N renormalization group. For a simple double-well model we find an unstable fixed point and a stable inverse-gaussian fixed point. The former is identified as the critical point of single/double-arc phase transition with a discontinuity of the third derivative of the free energy. The latter signifies a novel universality of large-N correlators other than the usual single arc type. This phase structure is consistent with the universality classification of two-level correlators for multiple-arc models by Ambjorn and Akemann. We also establish the stability of the gaussian fixed point in the multi-coupling model. 
  We suggest the brane interpretation of the integrable dynamics behind the exact solution to the N=2 SUSY YM theory. Degrees of freedom of the Calogero type integrable system responsible for the appearance of the spectral Riemann surfaces originate from the collective coordinates of the dynamical branes. The second Whitham type integrable system corresponds to the low energy scattering of branes similar to the scattering of the magnetic monopoles. 
  We present some mathematical aspects of Landau-Ginzburg string vacua in terms of toric geometry. The one-to-one correspondence between toric divisors and some of (-1,1) states in Landau-Ginzburg model is presented for superpotentials of typical types. The Landau-Ginzburg interpretation of non-toric divisors is also presented. Using this interpretation, we propose a method to solve the so-called "twisted sector problem" by orbifold construction. Moreover,this construction shows that the moduli spaces of the original Landau-Ginzburg string vacua and their orbifolds are connected. By considering the mirror map of Landau-Ginzburg models, we obtain the relation between Mori vectors and the twist operators of our orbifoldization. This consideration enables us to argue the embedding of the Seiberg-Witten curve in the defining equation of the Calabi-Yau manifoulds on which the type II string gets compactified. Related topics concerning the Calabi-Yau fourfolds and the extremal transition are discussed. 
  We show how to give the expression for periods, Higgs field and its dual of N=2 supersymmetric Yang-Mills theory around the conformal point. This is achieved by evaluating the integral representation in the weak coupling region, and by using analytic continuation to the conformal point. The explicit representation is shown for the SU(2) theory with matter fields and also for pure SU(N) and pure SO(2N) theory around the conformal point where the relation to the beta function of the theory is clarified. We also discuss a relation between the fixed points in the SU(2) theories with matter fields and the Landau-Ginzburg point of 2-D N=2 SCFT. 
  The exact quantum integrability aspects of a sector of the membrane is investigated. It is found that spherical membranes moving in flat target spacetime backgrounds admit a class of integrable solutions linked to SU(infty) SDYM equations (dimensionally reduced to one temporal dimension). After a suitable ansatz, the SDYM equations can be recast in the form of the continuous Toda molecule equations whose symmetry algebra is the dimensional reduction of the W (infty} plus {\bar W}(infty} algebra. The latter algebra is explicitly constructed. Highest weight representations are built directly from the infinite number of defining relations among the highest weight states of W(\infty) algebras and the quantum states of the Toda molecule. Discrete states are also constructed. The full (dimensionaly reduced) quantum SU(infty) YM theory remains to be explored. 
  Random matrix theory (RMT) provides a common mathematical formulation of distinct physical questions in three different areas: quantum chaos, the 1-d integrable model with the $1/r^2$ interaction (the Calogero-Sutherland-Moser system), and 2-d quantum gravity. We review the connection of RMT with these areas. We also discuss the method of loop equations for determining correlation functions in RMT, and smoothed global eigenvalue correlators in the 2-matrix model for gaussian orthogonal, unitary and symplectic ensembles. 
  In the framework of the four dimensional heterotic superstring with free fermions we discuss the rank eight and/or sixteen Grand Unified String Theories (GUST) which contain the SU(3)_H - gauge family symmetry. We explicitly investigate the paths of the unification in the GUST with gauge symmetry G x G = [SU(5) x U(1) x (SU(3) x U(1))_H]^2. We show that the GUSTs with the G x G gauge group allow to make the scale of unification to be consistent with the string scale M_SU = g_{string} * 5 * 10^17 GeV. 
  We examine the detail of the analytic structure of an exact analytic solution of three anyons, which interpolates to the fermion ground state in a harmonic potential well. The analysis is done on the fundamental domain with appropriate boundary conditions. Some remarks on the hard-core conditions and self-adjointness are made. 
  It is shown how to obtain superconformal Toda models as reductions of WZNW theories based on any Lie or super--Lie algebra. 
  We discuss the existence of $\theta$-vacua in pure Yang-Mills theory in two space-time dimensions. More precisely, a procedure is given which allows one to classify the distinct quantum theories possessing the same classical limit for an arbitrary connected gauge group G and compact space-time manifold M (possibly with boundary) possessing a special basepoint. For any such G and M it is shown that the above quantizations are in one-to-one correspondence with the irreducible unitary representations (IUR's) of $\pi_1(G)$ if M is orientable, and with the IUR's of $\pi_1(G)/2\pi_1(G)$ if M is nonorientable. 
  Using a description of defects in solids in terms of three-dimensional gravity, we study the propagation of electrons in the background of disclinations and screw dislocations. We study the situations where there are bound states that are effectively localized on the defect and hence can be described in terms of an effective 1+1 dimensional field theory for the low energy excitations. In the case of screw dislocations, we find that these excitations are chiral and can be described by an effective field theory of chiral fermions. Fermions of both chirality occur even for a given direction of the magnetic field. The ``net'' chirality of the system however is not always the same for a given direction of the magnetic field, but changes from one sign of the chirality through zero to the other sign as the Fermi momentum or the magnitude of the magnetic flux is varied. On coupling to an external electromagnetic field, the latter becomes anomalous, and predicts novel conduction properties for these materials. 
  Recent developments in string theory have brought forth a considerable interest in time-dependent hair on extended objects. This novel new hair is typically characterized by a wave profile along the horizon and angular momentum quantum numbers $l,m$ in the transverse space. In this work, we present an extensive treatment of such oscillating black objects, focusing on their geometric properties. We first give a theorem of purely geometric nature, stating that such wavy hair cannot be detected by any scalar invariant built out of the curvature and/or matter fields. However, we show that the tidal forces detected by an infalling observer diverge at the `horizon' of a black string superposed with a vibration in any mode with $l \ge 1$. The same argument applied to longitudinal ($l=0$) waves detects only finite tidal forces. We also provide an example with a manifestly smooth metric, proving that at least a certain class of these longitudinal waves have regular horizons. 
  We study $p=-1$ D-brane in type IIB superstring theory. In addition to RR instanton, we obtain the RR charged wormhole solution in the Einstein frame. This corresponds to the ten-dimensional singular wormhole solution with infinite euclidean action. 
  Among various approaches in proving gauge independence, models containing an explicit gauge dependence are convenient. The well-known example is the gauge parameter in the covariant gauge fixing which is of course most suitable for the perturbation theory but a negative metric prevents us from imaging a dynamical picture. Noncovariant gauge such as the Coulomb gauge is on the contrary used for many physical situations. Therefore it is desirable to include both cases. More than ten years ago, Steinmann introduced a function (distribution) which can play this role in his attempt on discussing quantum electrodynamics (QED) in terms of the gauge invariant fields solely. The method is, however, broken down in the covariant case: the invariant operators are ill-defined because of 1/p^2 singularity in the Minkowski space. In this paper, we apply his function to the path integral: utilizing the arbitrariness of the function we first restrict it to be able to have a well- defined operator, and then a Hamiltonian with which we can build up the (Euclidean) path integral formula. Although the formula is far from covariant, a full covariant expression is recovered by reviving the components which have been discarded under the construction of the Hamiltonian. There is no pathological defects contrary to the operator formalism. With the aid of the path integral formula, the gauge independence of the free energy as well as the S-matrix is proved. Moreover the reason is clarified why it is so simple and straightforward to argue gauge transformations in the path integral. Discussions on the quark confinement is also presented. 
  We discuss some properties of a supersymmetric matrix model that is the dimensional reduction of supersymmetric Yang-Mills theory in ten dimensions and which has been recently argued to represent the short-distance structure of M theory in the infinite momentum frame. We describe a reduced version of the matrix quantum mechanics and derive the Nicolai map of the simplified supersymmetric matrix model. We use this to argue that there are no phase transitions in the large-N limit, and hence that S-duality is preserved in the full eleven dimensional theory. 
  Correlation functions of exactly solvable models can be described by differential equation [Barough, McCoy, Wu]. In this paper we show that for non free fermionic case differential equations should be replaced by integro-differential equations.   We derive an integro-differential equation, which describes time and temperature dependent correlation function $<\psi(0,0)\psi^\dagger(x,t)>_T$ of penetrable Bose gas. The integro-differential equation turns out be the continuum generalization of classical nonlinear Schr\"odinger equation. 
  A complete set of supertraces on the algebras of observables of the rational Calogero models with harmonic interaction based on the classical root systems of B_N, C_N and D_N types is found. These results extend the results known for the case A_N. It is shown that there exist Q independent supertraces where Q(B_N)=Q(C_N) is a number of partitions of N into a sum of positive integers and Q(D_N) is a number of partitions of N into a sum of positive integers with even number of even integers. 
  These lecture notes are based on a course on string theories given by Hirosi Ooguri in the first week of TASI 96 Summer School at Boulder, Colorado. It is an introductory course designed to provide students with minimum knowledge before they attend more advanced courses on non-perturbative aspects of string theories in the School. The course consists of five lectures: 1. Bosonic String, 2. Toroidal Compactifications, 3. Superstrings, 4. Heterotic Strings, and 5. Orbifold Compactifications. 
  The polysymplectic $(n+1)$-form is introduced as an analogue of the symplectic form for the De Donder-Weyl polymomentum Hamiltonian formulation of field theory. The corresponding Poisson brackets on differential forms are constructed. The analogues of the Poisson algebra are shown to be generalized (non-commutative and higher-order) Gerstenhaber algebras defined in the text. 
  We consider d-dimensional Riemanian manifolds which admit d-2 commuting space-like Killing vector fields, orthogonal to a surface, containing two one-parametric families of light-like curves. The condition of the Ricci tensor to be zero gives Ernst equations for the metric. We write explicitly a family of local solutions of this equations corresponding to arbitrary initial data on two characteristics in terms of a series. These metrics describe scattering of 2 gravitational waves, and thus we expect they are very interesting. Ernst equations can be written as equations of motion for some 2D Lagrangian, which governs fluctuations of the metric, constant in the Killing directions. This Lagrangian looks essentially as a 2D chiral field model, and thus is possibly treatable in the quantum case by standart methods. It is conceivable that it may describe physics of some specially arranged scattering experiment, thus giving an insight for 4D gravity, not treatable by standart quantum field theory methods. The renormalization flow for our Lagrangian is different from the flow for the unitary chiral field model, the difference is essentially due to the fact that here the field is taking values in a non-compact space of symmetric matrices. We investigate the model and derive the renormalized action in one loop. 
  The claim put forward in [hep-th/9512051, hep-th/9612244] that the energies of the ``missing'' states of three anyons in a harmonic potential depend linearly on the statistics parameter, is incorrect because the wave functions proposed do not satisfy the anyonic interchange conditions. 
  Supersymmetric and parasupersymmetric quantum mechanics are now recognized as two further parts of quantum mechanics containing a lot of new informations enlightening (solvable) physical applications. Both contents are here analysed in connection with generalized quantum deformations. In fact, the parasupersymmetric context is visited when the order of paraquantization p is limited to the first nontrivial value p = 2. 
  We study the factorial divergences of Euclidean $\phi^3_5$, a problem with connections both to high-energy multiparticle scattering in d=4 and to d=3 (or high-temperature) gauge theory, which like $\phi^3_5$ is infrared-unstable and superrenormalizable. At large external momentum p (or small mass M) and large order N one might expect perturbative bare skeleton graphs to behave roughly like $N!(ag^2/p)^N$ with a>0, so that no matter how large p is there is an $N\sim g^2/p$ giving rise to strong perturbative amplitudes. The semi- classical Lipatov technique (which works only in the presence of a mass) is blind to this momentum dependence, so we proceed by direct summation of bare skeleton graphs. We find that the various limits of large N, large p, and small M do not commute, and that when $N\gg p^2/M^2$ there is a Borel singularity associated with $g^2/M$, not $g^2/p$. This is described by the zero-momentum Lipatov technique, and we find the necessary soliton for $\phi^3_5$; the corresponding sphaleron-like solution for unbroken Yang-Mills theory has long been known. We also show that the massless theories have no classical solitons. We discuss non-perturbative effects based partly on known physical arguments concerning the cancellation by solitons of imaginary parts due to the pert- urbative Borel singularity, and partly on the dressing of bare skeleton graphs by dressed propagators showing non-perturbative mass generation, as happens in d=3 gauge theory. 
  It is shown that, if generators of supersymmetry transformations (supercharges) can be defined in a spatially homogeneous physical state, then this state describes the vacuum. Thus, supersymmetry is broken in any thermal state and it is impossible to proceed from it by ``symmetrization'' to states on which an action of supercharges can be defined. So, unlike the familiar spontaneous breakdown of bosonic symmetries, there is a complete collapse of supersymmetry in thermal states. It is also shown that spatially homogeneous superthermal ensembles are never supersymmetric. 
  Gluodynamics in 3D spacetime with one spatial direction compactified into a circle of length $L$ is studied. The confinement order parameters, such as the Polyakov loops, are analyzed in both the limits $L \to 0$ and $L \to \infty$. In the latter limit the behavior of the confinement order parameters is shown to be described by a 2D non-linear sigma-model on the compact coset space $G/ad G$, where $G$ is the gauge group and $ad G$ its adjoint action on $G$. Topological vortex-like excitations of the compact field variable cause a Kosterlitz-Thouless phase transition which is argued to be associated with the confinement phase transition in the 3D gluodynamics. 
  An argument is presented for a certain universality of finite size corrections in two dimensional gauge theories. In the abelian case a direct calculation is carried out for a particular chiral model. The analytical result confirms the above universality and that the 't Hooft vertex previously measured using the overlap smoothly approaches the correct continuum limit within statistical errors. 
  A recent result concerning interacting theories of self-dual tensor gauge fields in six dimensions is generalized to include coupling to gravity. The formalism makes five of the six general coordinate invariances manifest, whereas the sixth one requires a non-trivial analysis. The result should be helpful in formulating the world-volume action of the M theory five-brane. 
  We study low-energy effective superpotentials for the phase with a confined photon in N=1 supersymmetric gauge theories with an adjoint matter $\Phi$ and fundamental flavors $Q, \tilde Q$. Arbitrary classical gauge groups are considered. The results are used to derive the hyperelliptic curves which describe the Coulomb phase of N=2 supersymmetric QCD with classical gauge groups. These curves are in agreement with those proposed earlier by several authors. Our results also produce the curves relevant to describe the Coulomb phase of N=1 theories with a superpotential of the form $\tilde{Q} \Phi^{l} Q$. 
  A new global approach in the study of duality transformations is introduced. The geometrical structure of complex line bundles is generalized to higher order U(1) bundles which are classified by quantized charges and duality maps are formulated over these structures. Quantum equivalence is shown between dual theories. A global constraint is proven to be needed to achieve well defined bundles. These global structures are used to refine the proof of the duality equivalence between d=11 supermembrane and d=10 IIA Dirichlet supermembrane, giving a complete topological interpretation to their quantized charges. 
  We investigate the most general N=1 graded extension of the Poincare algebra, and find the corresponding supersymmetry transformations and the associated superspaces. We find that the supersymmetry for which {Q,Q} = P is not special, and in fact must be treated democratically with a whole class of supersymmetries. We show that there are two distinct types of grading, and a new class of general spinors is defined. The associated superspaces are shown to be either of the usual type, or flat with no torsion. p-branes are discussed in these general superspaces and twelve dimensions emerges as maximal. New types of brane are discovered which could explain many features of the standard p-brane theories. 
  Results about the phase structure of certain N=1 supersymmetric gauge theories, which have been obtained as a consequence of holomorphy and `electric-magnetic' duality, are shown to be in quantitative agreement with corresponding consequences of analyticity and superconvergence of the gauge field propagator. This connection is of interest, because the superconvergence arguments for confinement are not restricted to theories with supersymmetry. The method of reduction in the space of coupling parameters is used in order to define, beyond the matching conditions, an asymptotically free, dual magnetic theory involving Yukawa couplings. 
  We derive the (matrix-valued) Feynman rules of the mass perturbation theory and use it for the resummation of the $n$-point functions with the help of the Dyson-Schwinger equations. We use these results for a short review of the complete spectrum of the model and for a discussion of scattering processes. We find that in scattering cross sections all the resonances and higher particle production thresholds of the model are properly taken into account by our resummed mass perturbation theory, without the need of further approximations. 
  A class of solutions to the WDVV equations is provided by period matrices of hyperelliptic Riemann surfaces, with or without punctures. The equations themselves reflect associativity of explicitly described multiplicative algebra of (possibly meromorphic) 1-differentials, which holds at least in the hyperelliptic case. This construction is direct generalization of the old one, involving the ring of polynomials factorized over an ideal, and is inspired by the study of the Seiberg-Witten theory. It has potential to be further extended to reveal algebraic structures underlying the theory of quantum cohomologies and the prepotentials in string models with N=2 supersymmetry. 
  Extending recent work on SU gauge theory, we engineer local string models for N=1 four-dimensional SO and USp gauge theories coupled to matter in the fundamental. The local models are type IIB orientifolds with D7 branes on a curved orientifold 7-plane, and matter realized by adding D3 branes on the orientifold plane. The Higgs branches of the SO and USp theories can be matched with the moduli spaces of SO and USp instantons on the compact four-dimensional part of the D7 branes worldvolume. The R-charge of the gauge theories is identified with a U(1) symmetry on the worldvolume of an Euclidean D3 brane instanton. We argue that the quantum field theory dualities of these gauge theories arise from T-dualities of type IIB strings exchanging D7 and D3 charges. A crucial role is played by the induced D3 charge of D7 branes and an orientifold 7-plane, both partially compactified on a $Z_2$ orbifold of K3. 
  We examine harmonic oscillator defects coupled to a photon field in the environs of an optical fiber. Using techniques borrowed or extended from the theory of two dimensional quantum fields with boundaries and defects, we are able to compute exactly a number of interesting quantities. We calculate the scattering S-matrices (i.e. the reflection and transmission amplitudes) of the photons off a single defect. We determine using techniques derived from thermodynamic Bethe ansatz (TBA) the thermodynamic potentials of the interacting photon-defect system. And we compute several correlators of physical interest. We find the photon occupancy at finite temperature, the spontaneous emission spectrum from the decay of an excited state, and the correlation functions of the defect degrees of freedom. In an extension of the single defect theory, we find the photonic band structure that arises from a periodic array of harmonic oscillators. In another extension, we examine a continuous array of defects and exactly derive its dispersion relation. With some differences, the spectrum is similar to that found for EM wave propagation in covalent crystals. We then add to this continuum theory isolated defects, so as to obtain a more realistic model of defects embedded in a frequency dependent dielectric medium. We do this both with a single isolated defect and with an array of isolated defects, and so compute how the S-matrices and the band structure change in a dynamic medium. 
  We obtain a formal solution of an integral equation for $q\bar q$ bound states, depending on a parameter \eta which interpolates between 't Hooft's (\eta=0) and Wu's (\eta=1) equations. We also get an explicit approximate expression for its spectrum for a particular value of the ratio of the coupling constant to the quark mass. The spectrum turns out to be in qualitative agreement with 't Hooft's as long as \eta \neq 1. In the limit \eta=1 (Wu's case) the entire spectrum collapses to zero, in particular no rising Regge trajectories are found. 
  The quantization of gauge fields and gravitation on manifolds with boundary makes it necessary to study boundary conditions which involve both normal and tangential derivatives of the quantized field. The resulting one-loop divergences can be studied by means of the asymptotic expansion of the heat kernel, and a particular case of their general structure is here analyzed in detail. The interior and boundary contributions to heat-kernel coefficients are written as linear combinations of all geometric invariants of the problem. The behaviour of the differential operator and of the heat kernel under conformal rescalings of the background metric leads to recurrence relations which, jointly with the boundary conditions, may determine these linear combinations. Remarkably, they are expressed in terms of universal functions, independent of the dimension of the background and invariant under conformal rescalings, and new geometric invariants contribute to heat-kernel asymptotics. Such technique is applied to the evaluation of the A(1) coefficient when the matrices occurring in the boundary operator commute with each other. Under these assumptions, the form of the A(3/2) and A(2) coefficients is obtained for the first time, and new equations among universal functions are derived. A generalized formula, relating asymptotic heat kernels with different boundary conditions, is also obtained. 
  This is a concise foreword to, rather than a review of D-brane physics. 
  We investigate the confining phase vacuum structure of supersymmetric SO(11) gauge theories with one spinor matter field and Nf \le 6 vectors. We describe several useful tricks and tools that facilitate the analysis of these chiral models and many other theories of similar type. The forms of the Nf=5 and Nf=6 quantum moduli spaces are deduced by requiring that they reproduce known results for SU(5) SUSY QCD along the spinor flat direction. After adding mass terms for vector fields and integrating out heavy degrees of freedom, we also determine the dynamically generated superpotentials in the Nf \le 4 quantum theories. We close with some remarks regarding magnetic duals to the Nf \ge 7 electric SO(11) theories. 
  We calculate, to one-loop order, the ln (T) contributions of 3-point functions in the \phi^3 and Yang-Mills theory at high temperature. We find that these terms are Lorentz invariant and have the same structure as the ultraviolet divergent contributions which occur at zero temperature. A simple argument, valid for all N-point Green functions, is given for this behavior. 
  In this letter, the problem of radiation in a fiber geometry interacting with a two level atom is mapped onto the anisotropic Kondo model. Thermodynamical and dynamical properties are then computed exploiting the integrability of this latter system. We compute some correlation functions, decay rates and Lamb shifts. In turn this leads to an analysis of the classical limit of the anisotropic Kondo model. 
  We investigate topological properties of Calabi-Yau fourfolds and consider a wide class of explicit constructions in weighted projective spaces and, more generally, toric varieties. Divisors which lead to a non-perturbative superpotential in the effective theory have a very simple description in the toric construction. Relevant properties of them follow just by counting lattice points and can be also used to construct examples with negative Euler number. We study nets of transitions between cases with generically smooth elliptic fibres and cases with ADE gauge symmetries in the N=1 theory due to degenerations of the fibre over codimension one loci in the base. Finally we investigate the quantum cohomology ring of this fourfolds using Frobenius algebras. 
  By an explicit construction, it is shown that the geometry of the SU(3) pion multiplet with respect to the group manifold SU_L(3) x SU_R(3) maybe deformed to admit a second pseudoscalar multiplet that is analogous to the Z_0 in unified theories of the electroweak interactions. This observation is found to play a key role in the construction of the N = 1 supersymmetric models with pions and Dirac-like spin-1/2 superpartners ("pionini"). 
  We show a possibility that the matrix models recently proposed to explain (almost) all the physics of M-theory may include the superstring theories that we know perturbatively. The ``1st quantized'' physical system of one IIA string seems to be an exact consequence of M(atrix) theory with a proper mechanism to mod out a symmetry. The central point of the paper is the representation of strings with P^+/epsilon greater than one. I call the mechanism ``screwing strings to matrices''. I also give the first versions of the proof of 2/3 power law between the compactification radius and the coupling constant in this formulation. Multistring states are involved in a M(atrix) theory fashion, replacing the 2nd quantization that I briefly review. We shortly discuss the T-dualities, type I string theory and involving of FP ghosts to all the systems including the original one of Banks et al. 
  Exact relations between gauge-invariant vacuum correlators in QCD are derived. Derivatives of the correlators are expressed in terms of higher orders correlators. The behaviour of the correlators at large and small distances due to these relations is discussed. 
  A non-dissipative model for vortex motion in thin superconductors is considered. The Lagrangian is a Galilean invariant version of the Ginzburg--Landau model for time-dependent fields, with kinetic terms linear in the first time derivatives of the fields. It is shown how, for certain values of the coupling constants, the field dynamics can be reduced to first order differential equations for the vortex positions. Two vortices circle around one another at constant speed and separation in this model. 
  Using Wegner-Houghton equation, within the Local Potential Approximation, we study critical properties of O(N) vector models. Fixed Points, together with their critical exponents and eigenoperators, are obtained for a large set of values of N, including N=0 and N\to\infty. Polchinski equation is also treated. The peculiarities of the large N limit, where a line of Fixed Points at d=2+2/n is present, are studied in detail. A derivation of the equation is presented together with its projection to zero modes. 
  We calculate within an algebraic Bjorken-Johnson-Low (BJL) method anomalous Schwinger terms of fermionic currents and the Gauss law operator in chiral gauge theories. The current algebra is known to violate the Jacobi identity in an iterative computation. Our method takes the subtleties of the equal-time limit into account and leads to an algebra that fulfills the Jacobi identity. The non-iterative terms appearing in the double commutators can be traced back directly to the projective representation of the gauge group. 
  In this paper we use a non-covariant gauge which makes some of the YM features manifest which otherwise do not appear explicitly in the Landau gauge used by Nielsen and Olesen.We show, in fact, that the unstable mode which makes a large contribution to the vacuum energy is non-perturbative even classically. 
  We investigate the possibility of computing energy by means of operators associated to the Wheeler-DeWitt equation. By choosing three dimensional wormholes as a framework, we apply such calculation scheme to the black hole pair creation. We compare our results with the recent ones appeared in the literature. 
  We study the effective action in 2D dilaton-Maxwell quantum gravity. Working with the one-loop renormalizable subset of such theories, we construct the improved effective Lagrangian which contains curvature under logarithm. This effective Lagrangian leads to new classical dilatonic gravity inspired by quantum effects. The static black holes (BH) solutions which may play the role of a remnant after the Hawking radiation for such theory are carefully investigated. The effective Lagrangian for Gross-Neveu-dilaton gravity is also constructed (in 1/N expansion). 
  The gravitational measure on an arbitrary topological three-manifold is constructed. The nontrivial dependence of the measure on the conformal factor is discussed. We show that only in the case of a compact manifold with boundary the measure acquires a nontrivial dependence on the conformal factor which is given by the Liouville action. A nontrivial Jacobian (the divergent part of it) generates the Einstein-Hilbert action. The Hartle-Hawking wave function of Universe is given in terms of the Liouville action. In the gaussian approximation to the Wheeler-DeWitt equation this result was earlier derived by Banks et al. Possible connection with the Chern-Simons gravity is also discussed. 
  Following the work with Jimbo and Miwa, we introduce a certain degeneration of the elliptic algebra $A_{q,p}(\widehat{sl_2})$ and its boson realization. We investigate its rational limit. The limit is the central extension of the Yangian double DY(sl_2) at level one. We give a new boson realization of it. Based on these algebras, we reformulate the Smirnov's form factor bootstrap approach to the sine-Gordon theory and the SU(2) invariant Thirring model. A conjectural integral formula for form factor in the sine-Gordon theory is derived. 
  We discuss in detail how to consistently impose boundary conditions in quantum string cosmology. Since a classical time parameter is absent in quantum gravity, such conditions must be imposed with respect to intrinsic variables. Constructing wave packets for minisuperspace models from different tree-level string effective actions, we explain in particular the meaning of a transition between ``pre-big-bang" and ``post-big-bang" branches. This leads to a scenario different from previous considerations. 
  We formulate field theories in fractal space and show the phase diagrams of the coupling versus the fractal dimension for the dynamical symmetry breaking. We first consider the 4-dimensional Gross-Neveu (GN) model in the (4-d)-dimensional randomized Cantor space where the fermions are restricted to a fractal space by the high potential barrier of Cantor fractal shape. By the statistical treatment of this potential, we obtain an effective action depending on the fractal dimension. Solving the 1/N leading Schwinger-Dyson (SD) equation, we get the phase diagram of dynamical symmetry breaking with a critical line similar to that of the d-dimensional (2<d<4) GN model except for the system-size dependence. We also consider QED4 with only the fermions formally compactified to d dimensions. Solving the ladder SD equation, we obtain the phase diagram of dynamical chiral symmetry breaking with a linear critical line, which is consistent with the known results for d=4 (the Maskawa-Nakajima case) and d=2 (the case with the external magnetic field). 
  We propose a complete Born-Infeld-like action for a bosonic 5-brane with the worldvolume chiral field in a background of gravitational and antisymmetric gauge fields of D=11 supergravity. When the five-brane couples to a three-rank antisymmetric gauge field local worldvolume symmetries of the five-brane require the addition to the action of an appropriate Wess-Zumino term. To preserve general coordinate and Lorentz invariance of the model we introduce a single auxiliary scalar field. The auxiliary field can be eliminated by gauge fixing a corresponding local symmetry at the price of the loss of manifest d=6 worldvolume covariance. The double dimensional reduction of the five-brane model results in the Born-Infeld action with the Wess-Zumino term for a D=10 four-D-brane. 
  We show that the topological charge of the n-soliton solution of the sine-Gordon equation n is related to the genus g > 1 of a constant negative curvature compact surface described by this configuration. The relation is n=2(g-1), where n is even. The moduli space of complex dimension B(g)=3(g-1) corresponds precisely to the freedom to choosing the configuration with n solitons of arbitrary positions and velocities. We speculate also that the odd soliton states will describe the unoriented surfaces. 
  It is explained how techniques from microlocal analysis can be used to settle some long-standing questions that arise in the study of the interaction of quantum matter fields with a classical gravitational background field. 
  By treating magnetic charge as a gauge symmetry through the introduction of a ``magnetic'' pseudo four-vector potential, it is shown that it is possible, using the 't Hooft-Polyakov construction, to obtain a topological electric charge. The mass of this electrically charged particle is found to be on the order of {1 / 137} M_W as opposed to the much larger mass (on the order of 137 M_W) of the magnetic soliton. Some model building possibilities are discussed. 
  We analyze several issues concerning the singular vectors of the Topological N=2 Superconformal algebra. First we investigate which types of singular vectors exist, regarding the relative U(1) charge and the BRST-invariance properties, finding four different types in chiral Verma modules and twenty-nine different types in complete Verma modules. Then we study the family structure of the singular vectors, every member of a family being mapped to any other member by a chain of simple transformations involving the spectral flows. The families of singular vectors in chiral Verma modules follow a unique pattern (four vectors) and contain subsingular vectors. We write down these families until level 3, identifying the subsingular vectors. The families of singular vectors in complete Verma modules follow infinitely many different patterns, grouped roughly in five main kinds. We present a particularly interesting thirty-eight-member family at levels 3, 4, 5, and 6, as well as the complete set of singular vectors at level 1 (twenty-eight different types). Finally we analyze the D\"orrzapf conditions leading to two linearly independent singular vectors of the same type, at the same level in the same Verma module, and we write down four examples of those pairs of singular vectors, which belong to the same thirty-eight-member family. 
  We present a general rule determining how extremal branes can interesect in a configuration with zero binding energy. The rule is derived in a model independent way and in arbitrary spacetime dimensions $D$ by solving the equations of motion of gravity coupled to a dilaton and several different $n$-form field strengths. The intersection rules are all compatible with supersymmetry, although derived without using it. We then specialize to the branes occurring in type II string theories and in M-theory. We show that the intersection rules are consistent with the picture that open branes can have boundaries on some other branes. In particular, all the D-branes of dimension $q$, with $1\leq q \leq6$, can have boundaries on the solitonic 5-brane. 
  Highest-weight type representation theories of the affine sl(2) and N=2 superconformal algebras are shown to be equivalent modulo the respective spectral flows. 
  A theory containing both electric and magnetic charges is formulated using two vector potentials, $A_{\mu}$ and $C_{\mu}$. This has the aesthetic advantage of treating electric and magnetic charges both as gauge charges, but it has the experimental disadvantage of introducing a second massless gauge boson (the ``magnetic'' photon) which is not observed. This problem is dealt with by using the Higgs mechanism to give a mass to one of the gauge bosons while the other remains massless. This effectively ``hides'' the magnetic charge, and the symmetry associated with it, when one is at an energy scale far enough removed from the scale of the symmetry breaking. 
  We consider N=1 supersymmetric gauge theories with a simple classical gauge group, one adjoint $\Phi, N_f$ pairs ($Q_i,\tilde{Q_i}$) of (fundamental, anti-fundamental) and a tree-level superpotential with terms of the Landau-Ginzburg form $\tilde{Q}_i\Phi^lQ_j$. The quantum moduli space of these models includes a Coulomb branch. We find hyperelliptic curves that encode the low energy effective gauge coupling for the groups SO(N_c) and USp(N_c) (the corresponding curve for SU(N_c) is already known). As a consistency check, we derive the sub-space of some vacua with massless dyons via confining phase superpotentials. We also discuss the existence and nature of the non-trivial superconformal points appearing when singularities merge in the Coulomb branch. 
  We review the fundamentals of Jahn-Teller interactions and their field theoretical modelings and show that a 2+1 dimensional gauge theory where the gauge field couples to "flavored fermions" arises in a natural way from a two-band model describing the dynamical Jahn-Teller effect. The theory exhibits a second order phase transition to novel finite-temperature superconductivity. 
  I calculate the potential of a pointlike particle carrying SU$(N_c)$ charge in a gauge theory with a dilaton. The solution depends on boundary conditions imposed on the dilaton: For a dilaton that vanishes at infinity the resulting potential is of the form $(r+r_\phi)^{-1}$, with $r_\phi$ inverse proportional to the decay constant of the dilaton. Another natural constraint for the dilaton $\phi$ is independence of $\frac{1}{g^2}\exp(\frac{\phi}{f_\phi})$ from the gauge coupling $g$. This requirement yields a potential proportional to $r$ and makes it impossible to create an isolated SU$(N_c)$ charge. 
  We analyze in detail the anomaly cancellation conditions for the strongly coupled $E_8 \times E_8$ heterotic string introduced by Horava and Witten and find new features compared to the ten-dimensional Green-Schwarz mechanism. We project onto ten dimensions the corresponding Lagrangian of the zero-mode fields. We find that it has a simple interpretation provided by the conjectured heterotic string/fivebrane duality. The part which originates from eleven-dimensions is naturally described in fivebrane language. We discuss physical couplings and scales in four dimensions. 
  Recent results of Trudinger on Isoperimetric Inequalities for non-convex bodies are applied to the gravitational collapse of a lightlike shell of matter to form a black hole. Using some integral identities for co-dimension two surfaces in Minkowski spacetime, the area $A$ of the apparent horizon is shown to be bounded above in terms of the mass $M$ by the $16 \pi G^2 M^2$, which is consistent with the Cosmic Censorship Hypothesis. The results hold in four spacetime dimensions and above. 
  We show that the abelian Proca model, which is gauge non-invariant with second class constraints can be converted into gauge theories with first class constraints. The method used, which we call Gauge Unfixing employs a projection operator defined in the original phase space. This operator can be constructed in more than one way, and so we get more than one gauge theory. Two such gauge theories are the Stuckelberg theory, and the theory of Maxwell field interacting with an antisymmetric tensor field. We also show that the application of the projection operator does not affect the Lorentz invariance of this model. 
  We investigate the way in which the Gribov problem is manifested in the BRST quantization of simple quantum mechanical models by comparing models with and without a Gribov problem. We show that the hermiticity and nilpotency of the BRST charge together with the Batalin-Vilkovisky theorem yield non-trivial supplementary conditions on gauge fixing fermions. If the gauge fixing fermion satisfies the supplementary conditions, the BRST physical states form a space isomorphic to the Dirac space, and the BRST formal path integral does not suffer from the Gribov problem. The conventional gauge fixing fermion, that gives rise to the Faddeev-Popov integral, fails to satisfy the supplementary conditions due to the Gribov problem. Alternatively, enforcing the conventional gauge fixing fermion, these supplementary conditions imply restrictions on the BRST physical states for which the Batalin-Vilkovisky theorem holds. We find that these BRST physical states are not isomorphic the Dirac states. This can be interpreted as a violation of the Batalin-Vilkovisky theorem on the space of Dirac states and implies a breakdown of unitarity and a general dependence of physical quantities on the gauge condition. 
  We prove that three-dimensional N=1 supersymmetric Yang-Mills-Chern-Simons theory is finite to all loops. This leaves open the possibility that different regularization methods give different finite effective actions. We show that for this model dimensional regularization and regularization by dimensional reduction yield the same effective action. 
  In this talk I address some aspects in the recent developments for N=2 black holes in 4 dimensions. I restrict myself on axion-free solutions that can classically be related to intersections of isotropic $D$- or $M$-branes. After reviewing of some classical properties I include corrections coming from a general cubic prepotential. On the heterotic side these are quantum corrections for these black hole solutions. Finally, I discuss a microscopic interpretation of the entropy for the extremal as well as near-extremal black hole. 
  In this lecture I review recent results on the first order equations describing BPS extremal states, in particular N=2 extremal black-holes. The role of special geometry is emphasized also in the rigid theory and a comparison is drawn with the supersymmetric derivation of instantons and hyperinstantons in topological field theories. Work in progress on the application of solvable Lie algebras to the discussion of BPS states in maximally extended supergravities is outlined. 
  We set up a systematic expansion of the prepotential for ${\cal N}=2$ supersymmetric Yang-Mills theories with SU(N) gauge group in the region of strong coupling where a maximal number of mutually local fields become massless. In particular, we derive the first non-trivial non-perturbative correction, which is the first term in the strong coupling expansion providing a coupling between the different U(1) factors, and which governs for example the strength of gaugino Yukawa couplings. 
  We discuss the physics of topological vortices moving on an arbitrary surface M in a Yang-Mills-Higgs theory in which the gauge group G breaks to a finite subgroup H. We concentrate on the case where M is compact and/or nonorientable. Interesting new features arise which have no analog on the plane. The consequences for the quantum statistics of vortices are discussed, particularly when H is nonabelian. 
  Requiring that supersymmetric SU(5) and SU(6) grand unifications in the heterotic string theory must have 3 chiral families, adjoint (or higher representation) Higgs fields in the grand unifiedgauge group, and a non-abelian hidden sector, we construct such string models within the framework of conformal field theory and asymmetric orbifolds. Within this framework, we construct all such string models via Z_6 asymmetric orbifolds that include a Z_3 outer-automorphism, the latter yielding a level-3 current algebra for the grand unification gauge group SU(5) or SU(6). We then classify all such Z_6 asymmetric orbifolds that result in models with a non-abelian hidden sector. All models classified in this paper have only one adjoint (but no otherhigher representation) Higgs field in the grand unified gauge group. This Higgs field is neutral under all other gauge symmetries. The list of hidden sectors for 3-family SU(6) string models are SU(2), SU(3) and $SU(2) \otimes SU(2)$. In addition to these, 3-family SU(5) string models can also have an SU(4) hidden sector. Some of the models have an anomalous U(1). 
  We use the method due to Batalin, Fradkin, Fradkina, and Tyutin (BFFT) in order to convert second-class into first-class constraints for some quantum mechanics supersymmetric theories. The main point to be considered is that the extended theory, where new auxiliary variables are introduced, has to be supersymmetric too. This leads to some additional restrictions with respect the conventional use of the BFFT formalism. 
  We examine the precise structure of the loop algebra of `dressing' symmetries of the Principal Chiral Model, and discuss a new infinite set of abelian symmetries of the field equations which preserve a symplectic form on the space of solutions. 
  The multiplicative anomaly associated with the zeta-function regularized determinant is computed for the Laplace-type operators $L_1=-\lap+V_1$ and $L_2=-\lap+V_2$, with $V_1$, $V_2$ constant, in a D-dimensional compact smooth manifold $ M_D$, making use of several results due to Wodzicki and by direct calculations in some explicit examples. It is found that the multiplicative anomaly is vanishing for $D$ odd and for D=2. An application to the one-loop effective potential of the O(2) self-interacting scalar model is outlined. 
  We introduce an orbifold induction procedure which provides a systematic construction of cyclic orbifolds, including their twisted sectors. The procedure gives counterparts in the orbifold theory of all the current-algebraic constructions of conformal field theory and enables us to find the orbifold characters and their modular transformation properties. 
  We compute the form factors of exponential operators $e^{kg\varphi(x)}$ in the two-dimensional integrable Bullough-Dodd model ($a_2^{(2)}$ Affine Toda Field Theory). These form factors are selected among the solutions of general nonderivative scalar operators by their asymptotic cluster property. Through analitical continuation to complex values of the coupling constant these solutions permit to compute the form factors of scaling relevant primary fields in the lightest-breather sector of integrable $\phi_{1,2}$ and $\phi_{1,5}$ deformations of conformal minimal models. We also obtain the exact wave-function renormalization constant Z(g) of the model and the properly normalized form factors of the operators $\varphi(x)$ and $:\varphi^2(x):$. 
  We have found that supersymmetry (SUSY) in curved space is broken softly. It is also found that Pauli-Villars regularization preserves the remaining symmetry, softly broken SUSY. Using it we computed the one-loop effective potential along a (classical) flat direction in a Wess-Zumino model in de Sitter space. The analysis is relevant to the Affleck-Dine mechanism for baryogenesis. The effective potential is unbounded from below: $V_{eff}(\phi)\to -3g^2H^2\phi ^2 ln \phi ^2 /16\pi ^2$, where $\phi$ is the scalar field along the flat direction, g is a typical coupling constant, and H is the Hubble parameter. This is identical with the effective potential which is obtained by using proper-time cutoff regularization. Since proper-time cutoff regularization is exact even at the large curvature region, the effective potential possesses softly broken SUSY and reliability in the large curvature region. 
  The role of the conformal group in electrodynamics in four space-time dimensions is re-examined. As a pedagogic example we use the application of conformal transformations to find the electromagnetic field for a charged particle moving with a constant relativistic acceleration from the Coulomb electric field for the particle at rest. We also re-consider the reformulation of Maxwell's equations on the projective cone, which is isomorphic to a conformal compactification on Minkowski space, so that conformal transformations, belonging to the group O(4,2), are realised linearly. The resulting equations are different from those postulated previously and respect additional gauge invariances which play an essential role in ensuring consistency with conventional electrodynamics on Minkowski space. The solution on the projective cone corresponding to a constantly accelerating charged particle is discussed. 
  It is shown that the deformed Heisenberg algebra involving the reflection operator R (R-deformed Heisenberg algebra) has finite-dimensional representations which are equivalent to representations of paragrassmann algebra with a special differentiation operator. Guon-like form of the algebra, related to the generalized statistics, is found. Some applications of revealed representations of the R-deformed Heisenberg algebra are discussed in the context of OSp(2|2) supersymmetry. It is shown that these representations can be employed for realizing (2+1)-dimensional supersymmetry. They give also a possibility to construct a universal spinor set of linear differential equations describing either fractional spin fields (anyons) or ordinary integer and half-integer spin fields in 2+1 dimensions. 
  We consider gapless models of statistical mechanics. At zero temperatures correlation functions decay asymptotically as powers of distance in these models. Temperature correlations decay exponentially. We used an example of solvable model to find the formula, which describes long distance and large time asymptotic of correlation function of local fields. The formula describes correlation at any temperature and arbitrary coupling constant. 
  We find that the local character of field theory requires the parity degree of freedom of the fields to be considered as an additional dicrete fifth dimension which is an artifact emerging due to the local description of space-time. Higgs field arises as the gauge field corresponding to this discrete dimension. Hence the noncommutative geometric derivation of the standard model follows as a manifestation of the local description of the usual space-time. 
  A nonperturbative regularization of UV-divergencies, caused by finite discontinuities in the field configuration, is discussed in the context of 1+1-dimensional kink models. The relationship between this procedure and the appearance of "quantum copies" of classical kink solutions is studied in detail and confirmed by conventional methods of soliton quantization. 
  In these notes we attempt to give a pedagogical introduction to the work of Seiberg and Witten on S-duality and the exact results of N=2 supersymmetric gauge theories with and without matter. The first half is devoted to a review of monopoles in gauge theories and the construction of supersymmetric gauge theories. In the second half, we describe the work of Seiberg and Witten. 
  Conditions which must be satisfied by the gauge-fixing fermion $\chi$ used in the BRST quantisation of constrained systems are established. These ensure that the extension of the Hamiltonian by the gauge-fixing term $[\Omega, \chi]$ (where $\Omega$ is the BRST charge) gives the correct path integral. (Lecture given at the conference Constrained Dynamics and Quantum Gravity II, Santa Margherita, Italy, September 1996) 
  We present the results for the local state probabilities (LSP) of the solvable lattice models, constructed around rational conformal field theory given by WZW model on $SO(3)_{4 R}=SU(2)_{4 R} / Z_{2}$ together with primary field $\phi_{1}$(symmetric tensor of degree 2). Some conjectures for the LSP for some higher rank relatives of $A_{n}^{(1)}$ face models are also presented. 
  We consider the method due to Batalin, Fradkin, Fradkina, and Tyutin (BFFT) that makes the conversion of second-class constraints into first-class ones for the case of nonlinear theories. We first present a general analysis of an attempt to simplify the method, showing the conditions that must be fulfilled in order to have first-class constraints for nonlinear theories but that are linear in the auxiliary variables. There are cases where this simplification cannot be done and the full BFFT method has to be used. However, in the way the method is formulated, we show with details that it is not practicable to be done. Finally, we speculate on a solution for these problems. 
  Algebraic Yang-Mills-Higgs theories based on noncommutative geometry have brought forth novel extensions of gauge theories with interesting applications to phenomenology. We sketch the model of Connes and Lott, as well as variants of it, and the model developed by a Mainz-Marseille group, by comparing them in a schematic way. The role of fermion masses and mixings is discussed, and the question of possible parameter relations is briefly touched. 
  For slowly varying fields on the scale of the lightest mass the logarithm of the vacuum functional of a massive quantum field theory can be expanded in terms of local functionals satisfying a form of the Schrodinger equation, the principal ingredient of which is a regulated functional Laplacian. We extend a previous work to construct the next to leading order terms of the Laplacian for the Schrodinger equation that acts on such local functionals. Like the leading order the next order is completely determined by imposing rotational invariance in the internal space together with closure of the Poincare algebra. 
  We define fermionic collective coordinates for type-IIB Dirichlet instantons and discuss some effects of the associated fermionic zero modes within the dilute gas framework. We show that the standard rules for clustering of zero modes in the dilute limit, and the fermion-exchange interactions follow from world-sheet Ward identities. Fermion exchange is strongly attractive at string-scale distances, which makes the short-distance Hagedorn singularity between instantons and anti-instantons even stronger. 
  An earlier proposed theory with linear-gonihedhic action for quantum gravity is reviewed. One can consider this theory as a "square root" of classical gravity with a new fundamental constant of dimension one. We demonstrate also, that the partition function for the discretized version of the Einstein-Hilbert action found by Regge in 1961 can be represented as a superposition of random surfaces with Euler character as an action and in the case of linear gravity as a superposition of three-dimensional manifolds with an action which is proportional to the total solid angle deficit of these manifolds. This representation allows to construct the transfer matrix which describes the propagation of space manifold. We discuss the so called gonihedric principle which allows to defind a discrete version of high derivative terms in quantum gravity and to introduce intrinsic rigidity of spacetime. This note is based on a talk delivered at the II meeting on constrained dynamics and quantum gravity at Santa Margherita Ligure. 
  We examine the geometry near the event horizon of a family of black string solutions with traveling waves. It has previously been shown that the metric is continuous there. Contrary to expectations, we find that the geometry is not smooth, and the horizon becomes singular whenever a wave is present. Both five dimensional and six dimensional black strings are considered with similar results. 
  These lectures notes are an intoduction for physicists to several ideas and applications of noncommutative geometry. The necessary mathematical tools are presented in a way which we feel should be accessible to physicists. We illustrate applications to Yang-Mills, fermionic and gravity models, notably we describe the spectral action recently introduced by Chamseddine and Connes. We also present an introduction to recent work on noncommutative lattices. The latter have been used to construct topologically nontrivial quantum mechanical and field theory models, in particular alternative models of lattice gauge theory.   Here is the list of sections:    1. Introduction. 2. Noncommutative Spaces and Algebras of Functions. 3. Noncommutative Lattices. 4. Modules as Bundles. 5. The Spectral Calculus. 6. Noncommutative Differential Forms. 7. Connections on Modules. 8. Field Theories on Modules. 9. Gravity Models. 10. Quantum Mechanical Models on Noncommutative Lattices.    Appendices: Basic Notions of Topology. The Gel'fand-Naimark-Segal Construction. Hilbert Modules. Strong Morita Equivalence. Partially Ordered Sets. Pseudodifferential Operators 
  A definition is given and the physical meaning of quantum transformations of a non-commutative configuration space (quantum group coactions) is discussed. It is shown that non-commutative coordinates which are transformed by quantum groups are the natural generalization of the notion of a tensor operator for usual groups and that the quantum group coactions induce semigroups of transformations of states of a system. Two examples of non-commutative transformations and the corresponding semigroups are considered. 
  Using the general notions of Batalin, Fradkin, Fradkina and Tyutin to convert second class systems into first class ones, we present a gauge invariant formulation of the massive Yang-Mills theory by embedding it in an extended phase space. The infinite set of correction terms necessary for obtaining the involutive constraints and Hamiltonian is explicitly computed and expressed in a closed form. It is also shown that the extra fields introduced in the correction terms are exactly identified with the auxiliary scalars used in the generalized St\"uckelberg formalism for converting a gauge noninvariant Lagrangian into a gauge invariant form. 
  It is known that the spin structure on a Riemannian manifold can be extended to noncommutative geometry using the notion of a spectral triple. For finite geometries, the corresponding finite spectral triples are completely described in terms of matrices and classified using diagrams. When tensorized with the ordinary space-time geometry, finite spectral triples give rise to Yang-Mills theories with spontaneous symmetry breaking, whose characteristic features are given within the diagrammatic approach: vertices of the diagram correspond to gauge multiplets of chiral fermions and links to Yukawa couplings. 
  We study axion-dilaton cosmologies derived from the low-energy string effective action. We present the classical homogeneous Friedmann-Robertson-Walker solutions and derive the semi-classical perturbation spectra in the dilaton, axion and moduli fields in the pre-Big Bang scenario. By constructing the unique S-duality invariant field perturbations for the axion and dilaton fields we derive S-duality invariant solutions, valid when the axion field is time-dependent as well as in a dilaton-vacuum cosmology. Whereas the dilaton and moduli fields have steep blue perturbation spectra (with spectral index n=4) we find that the axion spectrum depends upon the expansion rate of the internal dimensions (0.54<n<4) which allows scale-invariant (n=1) spectra. We note that for n<1 the metric is non-singular in the conformal frame in which the axion is minimally coupled. 
  We investigate Georgi-Glashow model in terms of a set of explicitly SO(3) gauge invariant dynamical variables. In the new description a novel compact abelian gauge invariance emerges naturally. As a consequence magnetic monopoles occur as point like "defects" in space time. Their non-perturbative contribution to the partition function is explicitly included. This procedure corresponds to dynamical "abelian projection" without gauge fixing. In the Higgs phase the above abelian invariance is to be identified with electromagnetism. We also study the effect of $\theta$ term in the above abelian theory. 
  We show that for every algebra of creation and annihilation operators with a Fock-like representation,one can define extended Haldane statistical parameters in a unique way. Specially for parastatistics, we calculate extended Haldane parameters and discuss the corresponding partition functions. 
  We present exact inhomogeneous and anisotropic cosmological solutions of low-energy string theory containing dilaton and axion fields. The spacetime metric possesses cylindrical symmetry. The solutions describe ever-expanding universes with an initial curvature singularity and contain known homogeneous solutions as subcases. The asymptotic form of the solution near the initial singularity has a spatially-varying Kasner-like form. The inhomogeneous axion and dilaton fields are found to evolve quasi-homogeneously on scales larger than the particle horizon. When the inhomogeneities enter the horizon they oscillate as non-linear waves and the inhomogeneities attentuate. When the inhomogeneities are small they behave like small perturbations of homogeneous universes. The manifestation of duality and the asymptotic behaviour of the solutions are investigated. 
  The adiabatic evolution of two doubly-degenerate (Kramers) levels is considered. The general five-parameter Hamiltonian describing the system is obtained and shown to be equivalent to one used in the $\Gamma_8 \otimes(\tau_2\oplus\epsilon)$ Jahn-Teller system. It is shown explicitly that the resulting SU(2) non-Abelian geometric vector potential is that of the (SO(5) symmetric) SU(2) instanton. Various forms of the potentials are discussed. 
  By applying an inverse Landau-Khalatnikov transformation, connecting (resummed) Schwinger-Dyson treatments in non-local and Landau gauges of $QED_3$, we derive the infrared behaviour of the wave-function renormalization in the Landau gauge, and the associated critical exponents in the normal phase of the theory (no mass generation). The result agrees with the one conjectured in earlier treatments. The analysis involves an approximation, namely an expansion of the non-local gauge in powers of momenta in the infrared. This approximation is tested by reproducing the critical number of flavours necessary for dynamical mass generation in the chiral-symmetry-broken phase of $QED_3$. 
  We review the properties of classical p-brane solutions to supergravity theories, i.e. solutions that may be interpreted as Poincare-invariant hyperplanes in spacetime. Topics covered include the distinction between elementary/electric and solitonic/magnetic solutions, examples of singularity and global structure, relations between mass densities, charge densities and the preservation of unbroken supersymmetry, diagonal and vertical Kaluza-Klein reduction families, Scherk-Schwarz reduction and domain walls, and the classification of multiplicities using duality symmetries. 
  Bell's inequality has been traditionally used to explore the relationship between hidden variables and the Copenhagen interpretation of quantum mechanics. In this paper, another use is found. Bell's inequality is used to derive a coupling principle for elementary particles and to give a deeper understanding of baryonic structure. We also give a derivation of the Pauli exclusion principle from the coupling principle. Pacs: 14.20-c, 12.90+b, 3.65.Bz, 12.40.Ee 
  Using the result that an electric charge - magnetic charge system carries an internal field angular momentum of $e g / 4 \pi$ we arrive at two restrictions on magnetic monopoles via the requirement of angular momentum quantization and/or conservation. First we show that magnetic charge should scale in the opposite way from electric charge. Second we show that free, unconfined monopoles seem to be inconsistent when one considers a magnetic charge in the vicinity of more than one electric charge. 
  A universality of deformed Heisenberg algebra involving the reflection operator is revealed. It is shown that in addition to the well-known infinite-dimensional representations related to parabosons, the algebra has also finite-dimensional representations of the parafermionic nature. We demonstrate that finite-dimensional representations are representations of deformed parafermionic algebra with internal Z_2-grading structure. On the other hand, any finite- or infinite-dimensional representation of the algebra supply us with irreducible representation of osp(1|2) superalgebra. We show that the normalized form of deformed Heisenberg algebra with reflection has the structure of guon algebra related to the generalized statistics. 
  We review an algebraic method of finding the composite p-brane solutions for a generic Lagrangian, in arbitrary spacetime dimension, describing an interaction of a graviton, a dilaton and one or two antisymmetric tensors. We set the Fock--De Donder harmonic gauge for the metric and the "no-force" condition for the matter fields. Then equations for the antisymmetric field are reduced to the Laplace equation and the equation of motion for the dilaton and the Einstein equations for the metric are reduced to an algebraic equation. Solutions composed of n constituent p-branes with n independent harmonic functions are given. The form of the solutions demonstrates the harmonic functions superposition rule in diverse dimensions. Relations with known solutions in D=10 and D=11 dimensions are discussed. 
  Scattering of fundamental states of type IIB supergravity and superstring theory is discussed at low orders in perturbation theory in the background of a D-instanton. The integration over fermionic zero modes in both the low energy supergravity and in the string theory leads to explicit nonperturbative terms in the effective action. These include a single instanton correction to the known tree-level and one-loop $R^4$ interactions. The `spectrum' of multiply-charged D-instantons is deduced by T-duality in nine dimensions from multiply-wound world-lines of marginally-bound D-particles. This, and other clues, lead to a conjectured SL(2,Z) completion of the $R^4$ terms which suggests that they are not renormalized by perturbative corrections in the zero-instanton sector beyond one loop. The string theory unit-charged D-instanton gives rise to point-like effects in fixed-angle scattering, raising unresolved issues concerning distance scales in superstring theory. 
  The conifold singularities in the type-II string are considered as the points of phase transition. In some cases, these singularities can be understood in the framework of the conventional fields theores as the points of enhanced gauge symmetry. We consider a class of three moduli Type-II strings. It is shown the periods can be written in the form of hypergeometric series around the singular points in these models. The leading expansion around the conifold locus turns out to be described by Appell functions. In one singular point, we observe the enhanced gauge symmetry of $SU(2)\times SU(2)$ independent of the models. Around another conifold locus, however, the resulting expression of the Appell functions depends on the models. We examine the result by considering a relation between these Appell functions and underlying Riemann surfaces. 
  We investigate the possibility of extending non-extreme black hole solutions made of intersecting M-branes to those with two non-extreme deformation parameters, similar to Reissner-Nordstr{\o}m solutions. General analysis of possible solutions is carried out to reduce the problem of solving field equations to a simple algebraic one for static spherically-symmetric case in $D$ dimensions. The results are used to show that the extension to two-parameter solutions is possible for $D=4,5$ dimensions but not for higher dimensions, and that the area of horizon always vanishes in the extreme limit for black hole solutions for $D \geq 6$ except for two very special cases which are identified. Various solutions are also summarized. 
  A supersymmetric theory in two-dimensions has enough data to define a noncommutative space thus making it possible to use all tools of noncommutative geometry. In particular, we apply this to the N=1 supersymmetric non-linear sigma model and derive an expression for the generalized loop space Dirac operator, in presence of a general background, using canonical quantization. The spectral action principle is then used to determine a spectral action valid for the fluctuations of the string modes. 
  It is well known through a recent work of Bernard, Gaudin, Haldane and Pasquier (BGHP) that the usual spin Calogero-Sutherland (CS) model, containing particles with $M$ internal degrees of freedom, respects the $Y(gl_M)$ Yangian symmetry. By following and suitably modifying the approach of BGHP, in this article we construct a novel class of spin CS models which exhibit multi-parameter deformed or `nonstandard' variants of $Y(gl_M)$ Yangian symmetry. An interesting feature of such CS Hamiltonians is that they contain many-body spin dependent interactions, which can be calculated directly from the associated rational solutions of Yang-Baxter equation. Moreover, these spin dependent interactions often lead to `anyon like' representations of permutation algebra on the combined internal space of all particles. We also find out the general forms of conserved quantities as well as Lax pairs for the above mentioned class of spin CS models, and describe the method of constructing their exact wave functions. 
  I outline the basics of numerical solution of quantum field theory using the Source Galerkin method. This method is based on an analysis of the functional differential equations of a field theory in the presence of external sources. This approach is particularly powerful because it naturally allows the inclusion of symmetry information and avoids the fermion determinant problem associated with Monte Carlo calculations. The technique is also suitable for continuum calculations. 
  The local $\zeta-$function approach is implemented to regularize the natural path integral definition of the geometric entropy in the large mass black hole Euclidean manifold. The case of a massless field coupled with the (off-shell) singular curvature is considered. It is proved that the geometric entropy is independent of the curvature coupling parameter $\xi$ avoiding negative values obtained in other approaches. 
  We are studying finite fermion density states in Maxwell QED$_{2+1}$ with external magnetic field. It is shown that at any fermion density the energy of some magnetized states may be less than that of the state with the same density, but no magnetic field. Magnetized states are described by the effective Maxwell-Chern-Simons QED$_{2+1}$ Lagrangian with gauge field mass proportional to the number of filled Landau levels. 
  In this paper,we derive a $\hbar$-deformation of the $W_{N}$ algebra and its quantum Miura tranformation. The vertex operators for this $\hbar$-deformed $W_{N}$ algebra and its commutation relations are also obtained. 
  We apply newly improved Batalin-Fradkin-Tyutin Hamiltonian method to the chiral Schwinger Model in the case of the regularization ambiguity $a>1$. We show that one can systematically construct the first class constraints by the BFT Hamiltonian method, and also show that the well-known Dirac brackets of the original phase space variables are exactly the Poisson brackets of the corresponding modified fields in the extended phase space. Furthermore, we show that the first class Hamiltonian is simply obtained by replacing the original fields in the canonical Hamiltonian by these modified fields. Performing the momentum integrations, we obtain the corresponding first class Lagrangian in the configuration space. 
  We review a method of construction of exceptional graphs generalising the ADE Dynkin diagrams which encode the spectrum of conformal field theories described by conformal embeddings of $\widehat{sl}(n)_k$. 
  The quantum evolution equations for the field expectation value are analytically solved to cubic order in the field amplitude and to one-loop level in the lambda phi-fourth model. We adapt and use the renormalization group (RG) method for such non-linear and non-local equations. The time dependence of the field expectation value is explicitly derived integrating the RG equations. It is shown that the field amplitude for late times approaches a finite limit as the time to the power -3/2. This limiting value is expressed as a function of the initial field amplitude. 
  The equation of state of a one-dimensional classical nonrelativistic Coulomb gas of particles in the adjoint representation of SU(2) is given. The problem is solved both with and without sources in the fundamental representation at either end of the system. The gas exhibits confining properties at low densities and temperatures and deconfinement in the limit of high densities and temperatures. However, there is no phase transition to a regime where the string tension vanishes identically; true deconfinement only happens for infinite densities and temperatures. In the low density, low temperature limit, a new type of collective behavior is observed. 
  The string-black hole correspondence is considered in the context of the correspondence principle proposed recently by Horowitz and Polchinski. We demonstrate that the entropy of string states and the entropy of a Schwarzschild black hole can be matched including the subleading terms which depend on mass logarithmically. We argue the necessity to include the string interaction (with coupling $g$) in the consideration and propose the $g^2$-dependent modification of the string entropy. The matching of it with the entropy of Schwarzschild black hole is analyzed. 
  The NLIE (the non-linear integral equation equivalent to the Bethe Ansatz equations for finite size) is generalized to excited states, that is states with holes and complex roots over the antiferromagnetic ground state. We consider the sine-Gordon/massive Thirring model (sG/mT) in a periodic box of length $L$ using the light-cone approach, in which the sG/mT model is obtained as the continuum limit of an inhomogeneous six vertex model. This NLIE is an useful starting point to compute the spectrum of excited states both analytically in the large $L$ (perturbative) and small $L$ (conformal) regimes as well as numerically. 
  The study of the convergence of power series expansions of energy eigenvalues for anharmonic oscillators in quantum mechanics differs from general understanding, in the case of quasi-exactly solvable potentials. They provide examples of expansions with finite radius and suggest techniques useful to analyze more generic potentials. 
  The quantum integrability of a class of massive perturbations of the parafermionic conformal field theories associated to compact Lie groups is established by showing that they have quantum conserved densities of scale dimension 2 and 3. These theories are integrable for any value of a continuous vector coupling constant, and they generalize the perturbation of the minimal parafermionic models by their first thermal operator. The classical equations-of-motion of these perturbed theories are the non-abelian affine Toda equations which admit (charged) soliton solutions whose semi-classical quantization is expected to permit the identification of the exact S-matrix of the theory. 
  Using a new technique for the extraction of leading-log energy dependence from the non-Abelian portion of quark-line interactions, hadronic scattering amplitudes may be represented in terms of functional integrals over gluonic material exchanged between scattering quarks. If that material is further represented by a condensed tube of gluonic flux (obtained by dimensional transmutation), these functional integrals will produce effective propagators corresponding to plasma oscillations, or ``condensons'', in this condensed gluonic material. These condensons yield a non-zero, non-tachyonic, finite, gauge-invariant contribution only when the gluonic flux tube corresponds to a ``rigid string'' of negligible thickness, so that this formalism provides a natural mechanism for a ``dynamical string'' in QCD4. Summation over a subset of such relevant condensons (at least in SU(2)) generates a form of the Donnachie-Landshoff Pomeron for the scattering amplitude. Other, non-condensate approaches are also available. 
  It is known that in the WKB approximation of multicomponent systems like Dirac equation or Born-Oppenheimer approximation, an additional phase appears apart from the Berry phase. So far, this phase was only examined in special cases, or under certain restrictive assumptions, namely that the eigenspaces of the matrix or endomorphism valued symbol of the Hamiltonian form trivial bundles.   We give a completely global derivation of this phase which does not depend on any choice of local trivializing sections. This is achieved using a star product approach to quantization. Furthermore, we give a systematic and global approach to a reduction of the problem to a problem defined completely on the different ``polarizations''. Finally, we discuss to what extent it is actually possible to reduce the problem to a really scalar one, and make some comments on obstructions to the existence of global quasiclassical states. 
  The pseudo--rigid body represents an example of a constrained system with a nonunimodular gauge group. This system is treated along the guidelines of an algebraic constraint quantization scheme which focusses on observable quantities, translating the vanishing of the constraints into representation conditions on the algebra of observables. The constraint which is responsible for the nonunimodularity of the gauge group is shown not to contribute to the observable content of the constraints, i.e., not to impose any restrictions on the construction of the quantum theory of the system. The application of the algebraic constraint quantization scheme yields a unique quantization of the physical degrees of freedom, which are shown to form a realization of the so-called CM(N) model of collective motions. 
  This is a colloquium-style review lecture for physicists and non-physicists, as part of the requirements for ``Habilitation'' at the university of Bern: At a pressure of 220 atm. and a temperature of 374 Celsius there is a second-order phase transition between water and steam. Understanding it requires the concept of the renormalization group. Images from computer simulations of the lattice gas model (included) are used to explain its basic ideas. It is briefly reviewed how the renormalization group is used to compute critical coefficients for the water-steam phase transition, in good agreement with experiment. Applications in particle physics and string theory are mentioned. The appendix contains a sample of the author's results on renormalization group flows in theories with dynamical gravity and their relation to perturbative string theory: gravity modifies critical coefficients and phase diagrams, in agreement with numerical calculations, and leads to curious phenomena such as oscillating flows and quantum mechanical flows. 
  We have studied the kinetics of $q$-deformed bosons and fermions, within a semiclassical approach. This investigation is realized by introducing a generalized exclusion-inclusion principle, intrinsically connected with the quantum $q$-algebra by means of the creation and annihilation operators matrix elements. In this framework, we have derived a non-linear Fokker-Planck equation for $q$-deformed bosons and fermions which can be seen as a time evolution equation, appropriate to consider non-equilibrium or near-equilibrium systems in a semiclassical approximation. The steady state of this equation reproduces in a simple mode the $q$-oscillators equilibrium statistics. 
  We study the classical equations of motion and the corresponding Wheeler-De Witt equation for tree level string effective action with the dilaton and axion. The graceful exit problem in certain cases is then analysed. 
  We study the Giddings-Strominger wormholes in string theories. We found negative modes among O(4)-symmetric fluctuations about the non-singular wormhole background. Hence the stringy wormhole contribution to the euclidean functional integral is purely imaginary. This means that the stringy wormhole is a bounce (not an instanton) and describes the nucleation and growth of wormholes in the Minkowski spacetime. 
  We study the D-brane solutions to type IIB superstring in ten dimensions and find interpretation in terms of compactification of a twelve dimensional three-brane of (a specific) F-theory on a torus $T^2$. In this frame-work, there also exist a two-brane which may be argued to be equivalent to the three-brane by utilizing the electric-magnetic duality in eleven dimensions. In this context, we propose for the existence of an isometry in one of the transverse directions to the three-brane in F-theory. As a consequence the two-brane may be identified with the three-brane in twelve dimensions itself. The twelve dimensional picture of D-branes in type IIB theory suggests for the reformulation of type IIB superstring in terms of three-brane of F-theory. 
  An exact S-matrix is conjectured for the imaginary coupled d_4(3) affine Toda field theory, using the U_q(g_2(1)) symmetry. It is shown that this S-matrix is consistent with the results for the case of real coupling using the breather-particle correspondence. For q a root of unity it is argued that the theory can be restricted to yield Phi(11|14) perturbations of WA_2 minimal models and the restriction is performed for the (3,p') minimal models. 
  We recall a formulation of super-membrane theory in terms of certain matrix models. These models are known to have a mass spectrum given by the positive half-axis. We show that, for the simplest such matrix model, a normalizable zero-mass ground state does not exist. 
  Possible physical consequences of a recently discovered nonabelian dual symmetry are explored in the standard model. It is found that both Higgs fields and fermion generations can be assigned a natural place in the dual framework, with Higgs fields appearing as frames (or `N-beins') in internal symmetry space, and generations appearing as spontaneously broken dual colour. Fermions then occur in exactly 3 generations and have a factorizable mass matrix which gives automatically one generation much heavier than the other two. The CKM matrix is the identity at zeroth order, but acquires mixing through higher loop corrections. Preliminary considerations are given to calculating the CKM matrix and lower generation masses. New vector and Higgs bosons are predicted. 
  Black holes in several dimensions and in several theories are studied and discussed. The theories are, general relativity, Kaluza-Klein, Brans-Dicke, Lovelock gravity and string theory. 
  We study the microcanonical density of states and the thermal properties of a bosonic string gas starting from a calculation of the Helmholtz free energy in the S-representation. By adding more and more strings to the single string system, we induce that, for infinite volume, there is no negative specific heat region but a transition at a finite value of the energy per string from the low energy regime to a region of infinite specific heat at the Hagedorn temperature. Forcing the description of this phase in terms of strings gives a picture in which there is a very fat string in a sea of low energetic ones. We argue that the necessary changing of this description should not change the fact that perturbatively $T_H$ is a maximum temperature of the system. 
  We consider 4d and 5d N=2 supersymmetric theories and demonstrate that in general their Seiberg-Witten prepotentials satisfy the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations. General proof for the Yang-Mills models (with matter in the first fundamental representation) makes use of the hyperelliptic curves and underlying integrable systems. A wide class of examples is discussed, it contains few understandable exceptions. In particular, in perturbative regime of 5d theories in addition to naive field theory expectations some extra terms appear, like it happens in heterotic string models. We consider also the example of the Yang-Mills theory with matter hypermultiplet in the adjoint representation (related to the elliptic Calogero-Moser system) when the standard WDVV equations do not hold. 
  A basis of Lorentz and gauge-invariant monomials in non--Abelian gauge theories with matter is described, applicable for the inverse mass expansion of effective actions. An algorithm to convert an arbitrarily given invariant expression into a linear combination of the basis elements is presented. The linear independence of the basis invariants is proven. 
  We show that the part of the tree-level open string effective action for the non-abelian vector field which depends on the field strength but not on its covariant derivatives, is given by the symmetrised trace of the direct non-abelian generalisation of the Born-Infeld invariant. We discuss applications to D-brane dynamics. 
  Explicit open single and multi-membrane solutions of the low energy limit of M-theory on the orbifold $R^{10}\times S^1/Z_2$ are presented. This low energy action is described by an 11-dimensional supergravity action coupled to two $E_8$ super Yang-Mills fields, which propagate only on the 10-dimensional boundaries of the target space. The membrane solutions we construct preserve half the supersymmetries. They carry electric charge and current with respect to the gauge fields, whose generators are in the Cartan subalgebra of the two $E_8$ gauge groups present at the boundaries. 
  The action for the D=10 type II Dirichlet super-p-branes, which has been obtained recently, is reconstructed in a more geometrical form involving Lorentz harmonic variables. This new (Lorentz-harmonic) formulation possesses kappa-symmetry in an irreducible form and is used as a basis for applying a generalized action principle that provides the superfield equations of motion and clarifies the geometrical nature of the kappa-symmetry of these models. The case of a Dirichlet super-3-brane is considered in detail. 
  We review the relations between (twisted) supersymmetric gauge theories in four dimensions and moduli problems in four-dimensional topology, and we study in detail the non-abelian monopole equations from this point of view. The relevance of exact results in N=1 and N=2 supersymmetric gauge theories to the computation of topological invariants is emphasized. Some background material is provided, including an introduction to Donaldson theory, the twisting procedure and the Mathai-Quillen formalism. 
  Generalizing the work of Sen, we analyze special points in the moduli space of the compactification of the F-theory on elliptically fibered Calabi-Yau threefolds where the coupling remains constant. These contain points where they can be realized as orbifolds of six torus $T^6$ by $Z_m \times Z_n (m, n=2, 3, 4, 6)$. At various types of intersection points of singularities, we find that the enhancement of gauge symmetries arises from the intersection of two kinds of singularities. We also argue that when we take the Hirzebruch surface as a base for the Calabi- Yau threefold, the condition for constant coupling corresponds to the case where the point like instantons coalesce, giving rise to enhanced gauge group of $Sp(k)$. 
  In order to gain insight into the possible Ground State of Quantized Einstein's Gravity, we have devised a variational calculation of the energy of the quantum gravitational field in an open space, as measured by an asymptotic observer living in an asymptotically flat space-time. We find that for Quantum Gravity (QG) it is energetically favourable to perform its quantum fluctuations not upon flat space-time but around a ``gas'' of wormholes, whose size is the Planck length $a_p$ ($a_p\simeq 10^{-33}$cm). As a result, assuming such configuration to be a good approximation to the true Ground State of Quantum Gravity, space-time, the arena of physical reality, turns out to be well described by Wheeler's Quantum Foam and adequately modeled by a space-time lattice with lattice constant $a_p$, the Planck lattice. 
  We herein set forth intrinsically four-dimensional string solutions and analyze some of its properties. The solutions are constructed as gauged WZW models of the coset $SO(2,2)/SO(2)\times SO(1,1)$. We recover backgrounds having metric and antisymmetric tensors, dilaton fields and two electromagnetic fields. The theories describe anisotropically expanding and static universes for some time values. 
  We study soliton solutions in 1+1 dimensional gauged sigma models, obtained by dimensional reduction from its 2+1 dimensional counterparts. We show that the Bogomol'nyi bound of these models can be expressed in terms of two conserved charges in a similar way to that of the BPS dyons in 3+1 dimensions. Purely magnetic vortices of the 2+1 dimensional completely gauged sigma model appear as charged solitons in the corresponding 1+1 dimensional theory. The scale invariance of these solitons is also broken because of the dimensional reduction. We obtain exact static soliton solutions of these models saturating the Bogomol'nyi bound. 
  In this talk the main features of the operator formalism for the $b-c$ systems on general algebraic curves developed in refs. [1-2] are reviewed. The first part of the talk is an introduction to the language of algebraic curves. Some explicit techniques for the construction of meromorphic tensors are explained. The second part is dedicated to the discussion of the $b-c$ systems. Some new results concerning the concrete representation of the basic operator algebra of the $b-c$ systems and the calculation of divisors on algebraic curves have also been included. 
  Renormalization group methods are used to determine the evolution of the low energy Wilson effective action for supersymmetric nonlinear sigma models in four dimensions. For the case of supersymmetric $CP^{(N-1)}$ models, the K\"ahler potential is determined exactly and is shown to exhibit a nontrivial ultraviolet fixed point in addition to a trivial infrared fixed point. The strong coupling behavior of the theory suggests the possible existence of additional relevant operators or nonperturbative degrees of freedom. 
  Manifestly covariant formulation of discrete-spin, real-mass unitary representations of the Poincar\'e group is given. We begin with a field of spin-frames associated with 4-mometa p and use them to simplify the eigenvalue problem for the Pauli-Lubanski vector projection in a direction given by a world-vector t. As opposed to the standard treatments where t is a constant time direction, our t is in general p-dependent and timelike, spacelike or null. The corresponding eigenstates play a role of a basis used to define Bargmann-Wigner spinors which form a carrier space of the unitary representation. The construction does not use the Wigner-Mackey induction procedure, is manifestly covariant and works simultaneously in both massive and massless cases (in on- and off-shell versions). Of particular interest are special Bargmann-Wigner spinors ($\omega$-spinors) associated with flag pole directions of the spin-frame field $\omega_A(p)$. 
  It is shown that there is a precise sense in which the Heisenberg uncertainty between fluxes of electric and magnetic fields through finite surfaces is given by (one-half $\hbar$ times) the Gauss linking number of the loops that bound these surfaces. To regularize the relevant operators, one is naturally led to assign a framing to each loop. The uncertainty between the fluxes of electric and magnetic fields through a single surface is then given by the self-linking number of the framed loop which bounds the surface. 
  A non-supersymmetric ten-dimensional open string theory is constructed as an orbifold of type I string theory, and as an orientifold of the bosonic type B theory. It is purely bosonic, and cancellation of massless tadpoles requires the gauge group to be SO(32)xSO(32). The spectrum of the theory contains a closed string tachyon, and open string tachyons in the (32,32) multiplet. The D-branes of this theory are analyzed, and it is found that the massless excitations of one of the 1-branes coincide with the world-sheet degrees of freedom of the D=26 bosonic string theory compactified on the SO(32) lattice. This suggests that the two theories are related by S-duality. 
  In this letter, classical chiral $QCD_{2}$ is studied in the lightcone gauge $A_{-}=0$. The once integrated equation of motion for the current is shown to be of the Lax form, which demonstrates an infinite number of conserved quantities. Specializing to gauge group SU(2), we show that solutions to the classical equations of motion can be identified with a very large class of curves. We demonstrate this correspondence explicitly for two solutions. The classical fermionic fields associated with these currents are then obtained. 
  M(atrix) theory on an orbifold and classical two-branes therein are studied with particular emphasis to heterotic M(atrix) theory on $S_1/Z_2$ relevant to strongly coupled heterotic and dual Type IA string theories. By analyzing orbifold condition on Chan-Paton factors, we show that three choice of gauge group are possible for heterotic M(atrix) theory: SO(2N), SO(2N+1) or USp(2N). By examining area-preserving diffeomorphism that underlies the M(atrix) theory, we find that each choices of gauge group restricts possible topologies of two-branes. The result suggests that only the choice of SO(2N) or SO(2N+1) groups allows open two-branes, hence, relevant to heterotic M(atrix) theory. We show that requirement of both local vacuum energy cancellation and of worldsheet anomaly cancellation of resulting heterotic string identifies supersymmetric twisted sector spectra with sixteen fundamental representation spinors from each of the two fixed points. Twisted open and closed two-brane configurations are obtained in the large N limit. 
  We analyze the question of screening versus confinement in bosonized massless QCD in two dimensions. We deduce the screening behavior of massless $SU(N_c)$ QCD with flavored fundamental fermions and fermions in the adjoint representation. This is done by computing the potential between external quarks as well as by bosonizing also the external sources and analyzing the states of the combined system. We write down novel "non-abelian Schwinger like" solutions of the equations of motion, compute their masses and argue that an exchange of massive modes of this type is associated with the screening mechanism. 
  Conformal isometry algebras of plane wave geometry are studied. Then, based on the requirement of conformal invariance, a definition of masslessness is introduced and gauge invariant equations of motion, subsidiary conditions, and corresponding gauge transformations for all plane wave geometry massless spin fields are constructed. Light cone representation for elements of conformal algebra acting as differential operators on wavefunctions of massless higher spin fields is also evaluated. Interrelation of plane wave geometry massless higher spin fields with ladder representation of $u(2,2)$ algebra is investigated. 
  We discuss Dirichlet instanton effects on type-IIB string Thermodynamics. We review some general properties of dilute D-instanton gases and use the low-energy supergravity solutions to define the normalization of the instanton measure, as well as the effects of long-range interactions. Thermal singularities in the single-instanton sector are due to tachyonic winding modes of Dirichlet open strings. Purely bosonic D-instantons induce in this way hard infrared singularities that ruin the weak-coupling expansion in the microcanonical ensemble. However, type-IIB D-instantons, give smooth contributions at the Hagedorn temperature, and the induced mass and coupling of the axion field are insufficient to change the first-order character of the phase transition in the mean field approximation. 
  We study axion-graviton scattering from a system of two D$0$-branes in a Type II superstring theory, a process which does not occur on a single brane. The two D$0$-branes interact via the exchange of closed string states which form a cylinder joining them. By compactifying on the $Z_3$ orbifold we find a non vanishing amplitude coming from the odd spin structure sector, thus from the exchanged RR states. We compute, in particular, the leading term of the amplitude at large distance from the branes, which corresponds to taking a field theory limit. This seems to suggest that the process takes place through the coupling of an axion to the RR states exchanged between the 0-branes. 
  Within a Liouville approach to non-critical string theory, we argue for a non-trivial commutation relation between space and time observables, leading to a non-zero space-time uncertainty relation $\delta x \delta t > 0$, which vanishes in the limit of weak string coupling. 
  In a 2+1-dimensional pure LGT at finite temperature the critical coupling for the deconfinement transition scales as $\beta_c(n_t) = J_c n_t + a_1$, where $n_t$ is the number of links in the ``time-like'' direction of the symmetric lattice. We study the effective action for the Polyakov loop obtained by neglecting the space-like plaquettes, and we are able to compute analytically in this context the coefficient $a_1$ for any SU(N) gauge group; the value of $J_c$ is instead obtained from the effective action by means of (improved) mean field techniques. Both coefficients have already been calculated in the large N limit in a previous paper. The results are in very good agreement with the existing Monte Carlo simulations. This fact supports the conjecture that, in the 2+1-dimensional theory, space-like plaquettes have little influence on the dynamics of the Polyakov loops in the deconfined phase. 
  We discuss the possibility of quantum transitions from the string perturbative vacuum to cosmological configurations characterized by isotropic contraction and decreasing dilaton. When the dilaton potential preserves the sign of the Hubble factor throughout the evolution, such transitions can be represented as an anti-tunnelling of the Wheeler--De Witt wave function in minisuperspace or, in a third-quantization language, as the production of pairs of universes out of the vacuum. 
  We investigate D=4, N=1 F theory models realized by type IIB string compactification on toric threefolds. Massless spectra, gauge symmetries, phase transitions associated with divisor contractions and flops, and non-perturbative superpotentials are analyzed using elementary toric methods. 
  The large spacing phase of the infinite random matrix chain, which represents the strongly coupled two-dimensional O(2) model on a random planar lattice, is explored. A class of solutions valid for large lattice spacings is constructed. It is proved that these solutions exhibit the critical exponents characteristic of pure two-dimensional gravity. The character expansion for the chain model is developed and an order parameter governing the Kosterlitz--Thouless phase transition is identified. 
  We propose a complete, d=6 covariant and kappa-symmetric, action for an M-theory five-brane propagating in D=11 supergravity background. 
  We analize instanton generated superpotentials for three dimensional N=2 supersymmetric gauge theories obtained by compactifying on S^1 N=1 four dimensional theories. In the N_f=1 case, we find that the vacua in the decompactification limit is given by the singular points of the Coulomb branch of the N=2 four dimensional theory (we also consider the massive case). The decompactification limit of the superpotential for pure gauge theories without chiral matter is interpreted in terms of `t Hooft's fractional instanton amplitudes. 
  We discuss properties of D-brane configurations in the matrix model of type IIB superstring recently proposed by Ishibashi, Kawai, Kitazawa and Tsuchiya. We calculate central charges in supersymmetry algebra at infinite N and associate them with one- and five-branes present in IIB superstring theory. We consider classical solutions associated with static three- and five-branes and calculate their interactions at one loop in the matrix model. We discuss some aspects of the matrix-model formulation of IIB superstring. 
  We review the macroscopic and microscopic properties of black holes of toroidally compactified heterotic and Type II string theory in dimensions 4<=D<=9. General charged rotating black hole solutions are obtained by acting on a generating solution with classical duality symmetries. In D=4, D=5 and 6<=D<=9, the generating solution for both toroidally compactified Type II and heterotic strings is specified by the ADM mass, [(D-1)/2]-angular momentum components and five, three and two charges, respectively. We give the Bekenstein-Hawking entropy for these solutions, address the BPS-saturated limit and compare the results to calculations of the microscopic entropy both in the NS-NS sector and the R-R sector of the theory. We also interpret such black hole solutions as dimensionally reduced intersecting p-branes of M-theory. 
  We construct the Euclidean on-shell action for the five-dimensional non-extremal black hole with multiple electric charges. We show that this thermal action agrees with one half of the entropy. This agreement is argued to be related to the generalized Smarr formula of the five-dimensional black hole mass. Through the calculation of the specific heat far off extremality we observe that a phase transition occurs. 
  We consider the effective action for strings and describe in detail the evolution of a four dimensional homogeneous isotropic universe with matter included. We find that the evolution, which is singular in general, becomes singularity free if during certain Phase of the evolution, when the scale factor increases and the effective string coupling becomes strong, the universe is dominated by solitonic p-branes, p = 0 and/or - 1, or by `matter' for which $(pressure) \le - \frac{1}{\sqrt{3}} (density)$. The mechanism in the case of branes is reminiscent of the recently discovered field theory mechanism where heavy states become light and resolve the moduli space singularities. 
  Duality in supersymmetric SU(N) gauge theory with a symmetric tensor is studied using the technique of deconfining and Seiberg's duality. By construction the gauge group of the dual theory necessarily becomes a product group. In order to check the duality, several nontrivial consistency conditions are examined. In particular we find that by deforming along a flat direction, the duality flows to the Seiberg's duality of SO(N) gauge theory. 
  A scaling hypothesis for the n-particle spectral densities of the O(3) nonlinear sigma-model is described. It states that for large particle numbers the n-particle spectral densities are ``self-similar'' in being basically rescaled copies of a universal shape function. This can be viewed as a 2-dimensional, but non-perturbative analogue of the KNO scaling in QCD. Promoted to a working hypothesis, it allows one to compute the two point functions at ``all'' energy or length scales. In addition, the values of two non-perturbative constants (needed for a parameter-free matching of the perturbative and the non-perturbative regime) are determined exactly. 
  After defining the concept of duality in the context of general $n$-form abelian gauge fields in 2$n$ dimensions, we show by explicit example the difference between apparent but unrealizable duality transformations, namely those in $D=4k+2$, and those, in $D=4k$, that can be implemented by explicit dynamical generators. We then consider duality transformations in Maxwell theory in the presence of gravitation, particularly electrically and magnetically charged black hole geometries. By comparing actions in which both the dynamical variables and the charge parameters are "rotated," we show their equality for equally charged electric and magnetic black holes, thus establishing their equivalence for semiclassical processes which depend on the value of the action itself. 
  The effective low-energy hyper-K"ahler potential for a massive N=2 matter in the N=2 super-QCD is investigated. The N=2 extended supersymmetry severely restricts that N=2 matter self-couplings so that their exact form can be fixed by a few parameters, which is apparent in the N=2 harmonic superspace. In the N=2 QED with a single matter hypermultiplet, the one-loop perturbative calculations lead to the Taub-NUT hyper-K"ahler metric in the massive case, and a free metric in the massless case. It is remarkable that the naive non-renormalization `theorem' does not apply. There exists a manifestly N=2 supersymmetric duality transformation converting the low-energy effective action for the N=2 QED hypermultiplet into a sum of the quadratic and the improved (non-polynomial) actions for an N=2 tensor multiplet. The duality transformation also gives a simple connection between the low-energy effective action in the N=2 harmonic superspace and the component results. 
  Despite the many successes of the relativistic quantum theory developed by Horwitz, et. al., certain difficulties persist in the associated covariant classical mechanics. In this paper, we explore these difficulties through an examination of the classical Coulomb problem in the framework of off-shell electrodynamics. As the local gauge theory of a covariant quantum mechanics with evolution parameter $\tau$, off-shell electrodynamics constitutes a dynamical theory of spacetime events, interacting through five $\tau$-dependent pre-Maxwell potentials. We present a straightforward solution of the classical equations of motion, which is seen to be unsatisfactory, and reveals the essential difficulties in the formalism at the classical level. We then offer a new model of the particle current -- as a certain distribution of the event currents on the worldline -- which eliminates these difficulties and permits comparison of classical off-shell electrodynamics with the standard Maxwell theory. In this model, the ``fixed'' event induces a Yukawa-type potential, permitting a semi-classical identification of the pre-Maxwell time scale $\lambda$ with the inverse mass of the intervening photon. Numerical solutions to the equations of motion are compared with the standard Maxwell solutions, and are seen to coincide when $\lambda \gtrsim 10^{-6}$ seconds, providing an initial estimate of this parameter. It is also demonstrated that the proposed model provides a natural interpretation for the photon mass cut-off required for the renormalizability of the off-shell quantum electrodynamics. 
  We show how to extend the standard functional approach to bosonisation, based on a decoupling change of path-integral variables, to the case in which a finite temperature is considered. As examples, in order to both illustrate and check the procedure, we derive the thermodynamical partition functions for the Thirring and Schwinger models. 
  I discuss the applicability in Weyl-invariant induced gravity and topologically massive gravity of certain formulas originally derived by Knizhnik, Polyakov, and Zamolodchikov in the context of diffeomorphism-invariant induced gravity. 
  To understand in detail duality between heterotic string and F theory compactifications, it is important to understand the construction of holomorphic G bundles over elliptic Calabi-Yau manifolds, for various groups G. In this paper, we develop techniques to describe the bundles, and make several detailed comparisons between the heterotic string and F theory. 
  The role of Bertotti-Robinson geometry in the attractor mechanism of extremal black holes is described for the case of N = 2 supersymmetry. Its implication for a model-independent derivation of the Bekenstein-Hawking entropy formula is discussed. 
  We started from computer experiments with simple one-dimensional ergodic dynamical systems called interval exchange transformations. Correlators in these systems decay as a power of time. In the simplest non-trivial case the exponent is equal to 1/3. We found a formula connecting characteristic exponents with explicit integrals over moduli spaces of algebraic curves with additional structures. Moreover, these integrals can be interpreted as correlators in a topological string theory. Also a new analogy arose between ergodic theory and complex algebraic geometry. 
  Branches of moduli space of F-theory in four dimensions are investigated. The transition between two branches is described as a 3-brane-instanton transition on a 7-brane. A dual heterotic picture of the transition is presented and the F-theory - heterotic theory map is given. The F-theory data - complex structure of the Calabi-Yau fourfold and the instanton bundle on the 7-brane is mapped to the heterotic bundle on the elliptic Calabi-Yau threefold CY_3. The full moduli space has a web structure which is also found in the moduli space of semi-stable bundles on $CY_3$. Matter content of the four-dimensional theory is discussed in both F-theory and heterotic theory descriptions. 
  This paper presents a 6d world-volume action that describes the dynamics of the M theory five-brane in a flat 11d space-time background. The world-volume action has global 11d super-Poincare invariance, as well as 6d general coordinate invariance and kappa symmetry, which are realized as local symmetries. The paper mostly considers a formulation in which general coordinate invariance is not manifest in one direction. However, it also describes briefly an alternative formulation, due to Pasti, Sorokin, and Tonin, in which general coordinate invariance is manifest. The latter approach requires auxiliary fields and new gauge invariances. 
  Euclidean field theory on 4-dimensional sphere is suggested for the study of high energy multiparticle production. The singular classical field configurations are found in scalar and SU(2)-gauge theories and the cross section of 2->n processes is calculated. It is shown,that the cross section has a maximum at the energy compared to the sphaleron mass. 
  We find p-brane solutions to the recently proposed IKKT IIB matrix theory for all odd p. We also propose central charges for the p-branes. 
  We review models of supersymmetric quantum mechanics that are important in the description of supermembranes and of Dirichlet particles, which play a role in the context of M-theory. 
  We use the microscopic instanton calculus to determine the one-instanton contribution to the quantum modulus u_3=<Tr(\phi^3)> in N=2 SU(N_c) supersymmetric QCD with N_f<2N_c fundamental flavors. This is compared with the corresponding prediction of the hyperelliptic curves which are expected to give exact solutions in this theory. The results agree up to certain regular terms which appear when N_f\geq 2N_c-3. The curve prediction for these terms depends upon the curve parameterization which is generically ambiguous when N_f\geq N_c. In SU(3) theory our instanton calculation of the regular terms is found to disagree with the predictions of all of the suggested curves. For this theory we employ our results as input to improve the curve parameterization for N_f=3,4,5. 
  We propose a new approach for deriving the string field equations from a general sigma model on the world sheet. This approach leads to an equation which combines some of the attractive features of both the renormalization group method and the covariant beta function treatment of the massless excitations. It has the advantage of being covariant under a very general set of both local and non-local transformations in the field space. We apply it to the tachyon, massless and first massive level, and show that the resulting field equations reproduce the correct spectrum of a left-right symmetric bosonic string. 
  Using the nonperturbative Schwinger-Dyson equation, we show that chiral symmetry is dynamically broken in QED at weak couplings when an external magnetic field is present, and that chiral symmetry is restored at temperatures above $T_c \simeq \alpha\pi^2/\sqrt{2 \pi |eH|}$, where $\alpha$ is the fine structure constant and $H$ is the magnetic field strength. 
  A novel ansatz for solving the string equations of motion and constraints in generic curved backgrounds, namely the planetoid ansatz, was proposed recently by some authors. We construct several specific examples of planetoid strings in curved backgrounds which include Lorentzian wormholes, spherical Rindler spacetime and the 2+1 dimensional black hole. A semiclassical quantisation is performed and the Regge relations for the planetoids are obtained. The general equations for the study of small perturbations about these solutions are written down using the standard, manifestly covariant formalism. Applications to special cases such as those of planetoid strings in Minkowski and spherical Rindler spacetimes are also presented. 
  We discuss the algebraic structure of the spin chains related to high energy scattering in QCD. We study the sl(2) Yangian symmetry and possible generalizations to nonzero spin and anisotropy parameter. 
  We present a general scheme for identifying fibrations in the framework of toric geometry and provide a large list of weights for Calabi--Yau 4-folds. We find 914,164 weights with degree $d\le150$ whose maximal Newton polyhedra are reflexive and 525,572 weights with degree $d\le4000$ that give rise to weighted projective spaces such that the polynomial defining a hypersurface of trivial canonical class is transversal. We compute all Hodge numbers, using Batyrev's formulas (derived by toric methods) for the first and Vafa's fomulas (obtained by counting of Ramond ground states in N=2 LG models) for the latter class, checking their consistency for the 109,308 weights in the overlap. Fibrations of k-folds, including the elliptic case, manifest themselves in the N lattice in the following simple way: The polyhedron corresponding to the fiber is a subpolyhedron of that corresponding to the k-fold, whereas the fan determining the base is a linear projection of the fan corresponding to the k-fold. 
  A gauge-averaging functional of the axial type is studied for simple supergravity at one loop about flat Euclidean four-space bounded by a three-sphere, or two concentric three-spheres. This is a generalization of recent work on the axial gauge in quantum supergravity on manifolds with boundary. Ghost modes obey nonlocal boundary conditions of the spectral type, in that half of them obey Dirichlet or Neumann conditions at the boundary. In both cases, they give a vanishing contribution to the one-loop divergence. The admissibility of noncovariant gauges at the classical level is also proved. 
  We address the issue of T-duality and U-duality symmetries in the toroidally-compactified type IIA string. It is customary to take as a starting point the dimensionally-reduced maximal supergravity theories, with certain field strengths dualised such that the classical theory exhibits a global $E_{n(n)}$ symmetry, where n=11-D in D dimensions. A discrete subgroup then becomes the conjectured U-duality group. In dimensions D\le 6, these necessary dualisations include NS-NS fields, whose potentials, rather than merely their field strengths, appear explicitly in the couplings to the string worldsheet. Thus the usually-stated U-duality symmetries act non-locally on the fundamental fields of perturbative string theory. At least at the perturbative level, it seems to be more appropriate to consider the symmetries of the versions of the lower-dimensional supergravities in which no dualisations of NS-NS fields are required, although dualisations of the R-R fields are permissible since these couple to the string through their field strengths. Taking this viewpoint, the usual T-duality groups survive unscathed, as one would hope since T-duality is a perturbative symmetry, but the U-duality groups are modified in D\le 6. 
  It is a well known fact that the classical (``Buscher'') transformations of T-duality do receive, in general, quantum corrections. It is interesting to check whether classical T-duality can be exact as a quantum symmetry. The natural starting point is a $\sigma$-model with N=4 world sheet supersymmetry. Remarkably, we find that (owing to the fact that N=4 models with torsion are not off-shell finite as quantum theories),the T-duality transformations for these models get in general quantum corrections, with the only known exception of warped products of flat submanifolds or orbifolds thereof with other geometries. 
  Using the formalism of superconnections, we show the existence of a bosonic action functional for the standard K-cycle in noncommutative geometry, giving rise, through the spectral action principle, only to the Einstein gravity and Standard Model Yang-Mills-Higgs terms. It provides an effective nonminimal coupling in the bosonic sector of the Lagrangian. 
  In this article we propose a `second quantization' scheme especially suitable to deal with non-trivial, highly symmetric phase spaces, implemented within a more general Group Approach to Quantization, which recovers the standard Quantum Field Theory (QFT) for ordinary relativistic linear fields. We emphasize, among its main virtues, greater suitability in characterizing vacuum states in a QFT on a highly symmetric curved space-time and the absence of the usual requirement of global hyperbolicity. This can be achieved in the special case of the Anti-de Sitter universe, on which we explicitly construct a QFT. 
  We extend recent work by Elizalde et al. to incorporate curvatures which are not small and backgrounds which are not just $S^2\times R^2, S^1\times S^1 \times R^2$. Some possible problems in their paper is also pointed out. 
  A new non-perturbative approach to quantum theory in curved spacetime and to quantum gravity, based on a generalisation of the Wigner equation, is proposed. Our definition for a Wigner equation differs from what have otherwise been proposed, and does not imply any approximations. It is a completely exact equation, fully equivalent to the Heisenberg equations of motion. The approach makes different approximation schemes possible, e.g. it is possible to perform a systematic calculation of the quantum effects order by order. An iterative scheme for this is also proposed. The method is illustrated with some simple examples and applications. A calculation of the trace of the renormalised energy-momentum tensor is done, and the conformal anomaly is thereby related to non-conservation of a current in d=2 dimensions and a relationship between a vector and an axial-vector current in d=4 dimensions.   The corresponding ``hydrodynamic equations'' governing the evolution of macroscopic quantities are derived by taking appropriate moments. The emphasis is put on the spin-1/2 case, but it is shown how to extend to arbitrary spins. Gravity is treated first in the Palatini formalism, which is not very tractable, and then more successfully in the Ashtekar formalism, where the constraints lead to infinite order differential equations for the Wigner functions. 
  We study the critical behavior of the D (2<D<4) dimensional Gross-Neveu model with a Thirring interaction, where a vector-vector type four-fermi interaction is on equal terms with a scalar-scalar type one. By using inversion method up to the next-to-leading order of 1/N expansion, we construct a gauge invariant effective potential. We show the existence of the chiral order phase transition, and determine explicitly the critical surface. It is observed that the critical behavior is mainly controlled by the Gross-Neveu coupling g. The critical surface can be divided into two parts by the surface g=1 which is the critical coupling in the Gross-Neveu model at the 1/N next-to-leading order, and the form of the critical surface is drastically change at g=1. Comparison with the Schwinger-Dyson(SD) equation is also discussed. Our result is almost the same as that derived in the SD equation. Especially, in the case of pure Gross-Neveu model, we succeed in deriving exactly the same critical line as the one derived in the SD equation. 
  In this work I present a numerical study of the Finite Size Scaling (FSS) of a correlation length in the framework of the $CP ^{N-1}$ model by means of the 1/N expansion. This study has been thought as propedeutical to the application of FSS to the measure on the lattice of a new coupling constant $f_{x}(1/R)$, defined in terms or rectangular Wilson Loops. I give also a perturbative expansion of $f_{x}(1/R)$ in powers of the corresponding coupling constant in the $\overline{MS}$ scheme together with some preliminary numerical results obtained from the Polyakov ratio and I point out the conceptual problems that limit this approach. 
  Employing Sen's picture of BPS states on a 3-brane probe world volume field theory in a F-theory background. we determine some selection rules for the allowed spectrum in massless $N_{f}\leq 4$ SU(2) Seiberg-Witten theory. The spectrum for any $N_f \leq 4$ is consistent with previous conjectures and analysis. 
  We show that the SL(2,R) duality symmetry of type IIB superstring theory can be formulated as the canonical transformation interchanging momenta and magnetic degrees of freedom associated to the abelian world-volume gauge field of the D-3-brane. D-strings are shown to be connected under the corresponding transformation in the world-sheet to the (m,n) family of string solutions of type IIB supergravity constructed by Schwarz. For the type IIA superstring the D-2-brane is mapped under the three dimensional world-volume electric-magnetic duality to the dimensional reduction of the membrane of M-theory. 
  We compute the greybody factors for classical black holes in a domain where two kinds of charges and their anticharges are excited by the extra energy over extremality. We compare the result to the greybody factors expected from an effective string model which was earlier shown to give the correct entropy. In the regime where the left and right moving temperatures are much smaller than the square root of the effective string tension, we find a non-trivial greybody factor which agrees with the effective string model. However, if the temperatures are comparable with the square root of the effective string tension, the greybody factors agree only at the leading order in energy. Nevertheless, there are several interesting relations between the two results, suggesting that a modification of the effective string model might lead to better agreement. 
  The (q_1,q_2) SL(2,Z) string bound states of type IIB superstring theory admit two inequivalent (T-dual) representations in eleven dimensions in terms of a fundamental 2-brane. In both cases, the spectrum of membrane oscillations can be determined exactly in the limit $g^2\to \infty $, where $g^2$ is the type IIA string coupling. We find that the BPS mass formulas agree, and reproduce the BPS mass spectrum of the $(q_1,q_2)$ string bound state. In the non-BPS sector, the respective mass formulas apply in different corners of the moduli space. The axiomatic requirement of T-duality in M-theory permits to derive a discrete mass spectrum in a (thin torus) region where standard supermembrane theory undergoes instabilities. 
  We prove that the Calogero-Sutherland Model with reflections (the BC_N model) possesses a property of duality relating the eigenfunctions of two Hamiltonians with different coupling constants. We obtain a generating function for their polynomial eigenfunctions, the generalized Jacobi polynomials. The symmetry of the wave-functions for certain particular cases (associated to the root systems of the classical Lie groups B_N, C_N and D_N) is also discussed. 
  We determine the boundary state for both the NS-NS and R-R sectors of superstring theory. We show how they are modified under a boost. The boosted boundary state is then used for computing the interaction of two D-branes moving with constant velocity reproducing with a completely different method a recent calculation by Bachas. 
  We present a large set of new self-dual N=1 SUSY gauge theories. Examples include SU(N) theories with tensors and SO(N) theories with spinors. Using these dualities as starting points, new non-trivial duals can be derived by higgsing the gauge group or by integrating out matter. General lessons that can be learned from these duals are: ``accidental" infrared symmetries play an important role in duality, many theories have more than one ``dual", and there seems to be no simple organizing pattern which relates duals of theories with different number of flavors. 
  Framework for constructing Fock spaces associated either with certain solutions of the quantum Yang-Baxter equation or with infinite dimensional Hecke algebra is presented. For the former case, the quantum deformed oscillator algebra associated with the solution of the quantum Yang-Baxter equation is found. 
  The canonical description is presented for the string with pointlike masses at the ends in 1+1 dimensions in two different gauges: in the proper time gauge and in the light cone one. The classical canonical transformation is written out explicitly, which relates physical variables in both gauges, and equivalence of two classical theories is demonstrated in such a way. Both theories are quantized, and it is shown that quantum theories are not unitary equivalent. It happens due to the fact that the canonical transformation depends on interaction. The quantum Poincare algebra proves to be closed in both cases, so that the requirement of Poincare covariance is not able to distinguish between two versions of the theory. 
  We obtain a Ha type formula for n-point correlations in the log-gas at rational temperature (or, equivalently, n-point one-time ground state correlations in the quantum Calogero-Sutherland model for rational coupling constant). 
  The BRST quantization of the Abelian Proca model is performed using the Batalin-Fradkin-Tyutin and the Batalin-Fradkin-Vilkovisky formalism. First, the BFT Hamiltonian method is applied in order to systematically convert a second class constraint system of the model into an effectively first class one by introducing new fields. In finding the involutive Hamiltonian we adopt a new approach which is more simpler than the usual one. We also show that in our model the Dirac brackets of the phase space variables in the original second class constraint system are exactly the same as the Poisson brackets of the corresponding modified fields in the extended phase space due to the linear character of the constraints comparing the Dirac or Faddeev-Jackiw formalisms. Then, according to the BFV formalism we obtain that the desired resulting Lagrangian preserving BRST symmetry in the standard local gauge fixing procedure naturally includes the St\"uckelberg scalar related to the explicit gauge symmetry breaking effect due to the presence of the mass term. We also analyze the nonstandard nonlocal gauge fixing procedure. 
  This paper treats some basic points in general relativity and in its perturbative analysis. Firstly a systematic classification of global SO(n) invariants, which appear in the weak-field expansion of n-dimensional gravitational theories, is presented. Through the analysis, we explain the following points: a) a graphical representation is introduced to express invariants clearly; b) every graph of invariants is specified by a set of indices; c) a number, called weight, is assigned to each invariant. It expresses the symmetry with respect to the suffix-permutation within an invariant. Interesting relations among the weights of invariants are given. Those relations show the consistency and the completeness of the present classification; d) some reduction procedures are introduced in graphs for the purpose of classifying them. Secondly the above result is applied to the proof of the independence of general invariants with the mass-dimension $M^6$ for the general geometry in a general space dimension. We take a graphical representation for general invariants too. Finally all relations depending on each space-dimension are systematically obtained for 2, 4 and 6 dimensions. 
  New features of systems with non-trivial topology such as fractional quantum numbers, inequivalent quantizations, good operators, topological anomalies, etc. are described in the framework of an algebraic quantization procedure on a group. Modular invariance naturally appears as a subgroup of good operators in the particular case of the torus. 
  The correlation functions of the multi-arc complex matrix model are shown to be universal for any finite number of arcs. The universality classes are characterized by the support of the eigenvalue density and are conjectured to fall into the same classes as the ones recently found for the hermitian model. This is explicitly shown to be true for the case of two arcs, apart from the known result for one arc. The basic tool is the iterative solution of the loop equation for the complex matrix model with multiple arcs, which provides all multi-loop correlators up to an arbitrary genus. Explicit results for genus one are given for any number of arcs. The two-arc solution is investigated in detail, including the double-scaling limit. In addition universal expressions for the string susceptibility are given for both the complex and hermitian model. 
  Duality between the E_8 x E_8 heterotic string and Type I' theory predicts a tower of D(irichlet)-particle bound states corresponding to perturbative heterotic string states. In the limit of infinite Type I' coupling, some of these bound states become massless, giving rise to enhanced E_8 x E_8 gauge symmetry. By taking a different infinite coupling limit, one can recover the E_8 x E_8 gauge bosons of M-theory, compactified on S^1/Z_2. In this paper we use the matrix model description of the D-particle dynamics to study these bound states. We find results consistent with the chain of dualities and clarify a number of issues that arise in the application of the matrix mechanics to this system. 
  In a recent series of papers, Schwinger discussed a process that he called the Dynamical Casimir Effect. The key essence of this effect is the change in zero-point energy associated with any change in a dielectric medium. (In particular, if the change in the dielectric medium is taken to be the growth or collapse of a bubble, this effect may have relevance to sonoluminescence.) The kernel of Schwinger's result is that the change in Casimir energy is proportional to the change in volume of the dielectric, plus finite-volume corrections. Other papers have called into question this result, claiming that the volume term should actually be discarded, and that the dominant term remaining is proportional to the surface area of the dielectric. In this communication, which is an expansion of an earlier letter on the same topic, we present a careful and critical review of the relevant analyses. We find that the Casimir energy, defined as the change in zero-point energy due to a change in the medium, has at leading order a bulk volume dependence. This is in full agreement with Schwinger's result, once the correct physical question is asked. We have nothing new to say about sonoluminescence itself. 
  The component form of the equations of motion for the 5-brane in eleven-dimensions is derived from the superspace equations. These equations are fully covariant in six-dimensions. It is shown that double-dimensional reduction of the bosonic equations gives the equations of motion for a 4-brane in ten dimensions governed by the Born-Infeld action. 
  We calculate the effective potentials for scalar, Dirac and Yang-Mills fields in curved backgrounds using a new method for the determination of the heat kernel involving a re-summation of the Schwinger-DeWitt series. Self-interactions are treated both to one loop order as usual and slightly beyond one-loop order by means of a mean-field approximation. The new approach gives the familiar result for scalar fields, the Coleman-Weinberg potential plus corrections such as the leading-log terms, but the actual calculation is much faster. We furthermore show how to go systematically to higher loop order. The Schwarzschild space-time is used to exemplify the procedure. Next we consider phase transitions and we show that for a classical critical point to be a critical point of the effective potential too, certain restrictions must be imposed on as well its value as on the form of the classical potential and the background geometry. We derive this extra condition for scalar fields with arbitrary self couplings and comment on the case of fermions and gauge bosons too. Critical points of the effective action which are not there classically are also discussed. This has implications for inflation. The renormalised energy-momentum tensor for a scalar field with arbitrary self-interaction and non-minimal coupling to the gravitational background is also calculated to this improved one-loop order as is the resulting conformal anomaly. Conditions for the violation of energy conditions are described. Finally we discuss metric fluctuations and a self-consistency condition for such fluctuations is written down for spin 0,1/2,1 quantum fields. This is of importance for the study of cosmic density fluctuations. All calculations are performed in the physically relevant case of d=4 dimensions. 
  We employ the principle of minimal central charge and study the entropy of N=2 black holes corresponding to the most general quadratic prepotential. We also give a static black hole solution for these models in which the scalar moduli are non constants. Finally, we speculate on the microscopic origin for our solution. 
  We consider theories with a spontaneously broken gauged R-symmetry, which can only occur in supergravity models. These give rise to cosmic R-strings upon which gravitino zero modes can exist. We construct solutions to the Rarita-Schwinger spin-3/2 equation describing the gravitino in the field of these cosmic strings and show that under some conditions these solutions may give rise to gravitino currents on the string. We discuss further mathematical and physical questions associated with these solutions. 
  We introduce some techniques for making a more global analysis of the existence of geodesics on a Seiberg-Witten Riemann surface with metric $ds^2 = |\lambda_{SW}|^2$. Because the existence of such geodesics implies the existence of BPS states in N=2 supersymmetric Yang-Mills theory, one can use these methods to study the BPS spectrum in various phases of the Yang-Mills theory. By way of illustration, we show how, using our new methods, one can easily recover the known results for the N=2 supersymmetric SU(2) pure gauge theory, and we show in detail how it also works for the N=2, SU(2) theory coupled to a massive adjoint matter multiplet. 
  Reflection equation for the scattering of lines moving in half-plane is obtained. The corresponding geometric picture is related with configurations of half-planes touching the boundary plane in 2+1 dimensions. This equation can be obtained as an additional to the tetrahedron equation consistency condition for a modified Zamolodchikov algebra. 
  We propose a construction of dual pairs in four dimensional N=1 supersymmetric Yang-Mills theory using branes in type IIA string theory. 
  Fundamental string theory has been used to show that low energy excitations of certain black holes are described by a two dimensional conformal field theory. This picture has been found to be extremely robust. In this paper it is argued that many essential features of the low energy effective theory can be inferred directly from a semiclassical analysis of the general Kerr-Newman solution of supersymmetric four-dimensional Einstein-Maxwell gravity, without using string theory. We consider the absorption and emission of scalars with orbital angular momentum, which provide a sensitive probe of the black hole. We find that the semiclassical emission rates -including superradiant emission and greybody factors - for such scalars agree in striking detail with those computed in the effective conformal field theory, in both four and five dimensions. Also the value of the quantum mass gap to the lowest-lying excitation of a charge-$Q$ black hole, $E_{gap}=1/8Q^3$ in Planck units, can be derived without knowledge of fundamental string theory. 
  We study the algebra of SU(n)-currents in the many-flavor chiral Gross-Neveu model. The general structure of the current-current OPE leading to non-local quantum conserved charges is reviewed. We calculate the OPE in the one-flavor and the many-flavor models perturbatively and use renormalization group invariance to prove that our results are not altered by higher-order corrections. We conclude that in these models the non-local quantum charge exists which is the first step towards the proof of the absence of particle production and factorization. 
  We introduce N=1 supersymmetric generalization of the mechanical system describing a particle with fractional spin in D=1+2 dimensions and being classically equivalent to the formulation based on the Dirac monopole two-form. The model introduced possesses hidden invariance under N=2 Poincar\'e supergroup with a central charge saturating the BPS bound. At the classical level the model admits a Hamiltonian formulation with two first class constraints on the phase space $T^*(R^{1,2})\times {\cal L}^{1|1}$, where the K\"ahler supermanifold ${\cal L}^{1|1}\cong OSp(2|2)/U(1|1)$ is a minimal superextension of the Lobachevsky plane. The model is quantized by combining the geometric quantization on ${\cal L}^{1|1}$ and the Dirac quantization with respect to the first class constraints. The constructed quantum theory describes a supersymmetric doublet of fractional spin particles. The space of quantum superparticle states with a fixed momentum is embedded into the Fock space of a deformed harmonic oscillator. 
  This contribution gives in sigma-model language a short review of recent work on T-duality for open strings in the presence of abelian or non-abelian gauge fields. Furthermore, it adds a critical discussion of the relation between RG beta-functions and the Born-Infeld action in the case of a string coupled to a D-brane. 
  We studied the lowest order quantum corrections to the macroscopic wave functions $\Gamma (A,\ell)$ of non-critical string theory using the semi-classical expansion of Liouville theory. By carefully taking the perimeter constraint into account we obtained a new type of boundary condition for the Liouville field which is compatible with the reparametrization invariance of the boundary and which is not only a mixture of Dirichlet and Neumann types but also involves an integral of an exponential of the Liouville field along the boundary. This condition contains an unknown function of $A/\ell^2$. We determined this function by computing part of the one-loop corrections to $\Gamma (A,\ell)$. 
  We consider the gl(N)-invariant Calogero-Sutherland Models with N=1,2,3,... in a unified framework, which is the framework of Symmetric Polynomials. By the framework we mean an isomorphism between the space of states of the gl(N)-invariant Calogero-Sutherland Model and the space of Symmetric Laurent Polynomials. In this framework it becomes apparent that all gl(N)-invariant Calogero-Sutherland Models are manifestations of the same entity, which is the commuting family of Macdonald Operators. Macdonald Operators depend on two parameters $q$ and $t$. The Hamiltonian of gl(N)-invariant Calogero-Sutherland Model belongs to a degeneration of this family in the limit when both $q$ and $t$ approach the N-th elementary root of unity. This is a generalization of the well-known situation in the case of Scalar Calogero-Sutherland Model (N=1). In the limit the commuting family of Macdonald Operators is identified with the maximal commutative sub-algebra in the Yangian action on the space of states of the gl(N)-invariant Calogero-Sutherland Model. The limits of Macdonald Polynomials which we call gl(N)-Jack Polynomials are eigenvectors of this sub-algebra and form Yangian Gelfand-Zetlin bases in irreducible components of the Yangian action. The gl(N)-Jack Polynomials describe the orthogonal eigenbasis of gl(N)-invariant Calogero-Sutherland Model in exactly the same way as Jack Polynomials describe the orthogonal eigenbasis of the Scalar Model (N=1). For each known property of Macdonald Polynomials there is a corresponding property of gl(N)-Jack Polynomials. As a simplest application of these properties we compute two-point Dynamical Spin-Density and Density Correlation Functions in the gl(2)-invariant Calogero-Sutherland Model at integer values of the coupling constant. 
  We present a systematic and unifying treatment of the problem of spontaneous nucleation of particle-antiparticle pairs in a (2+1)-dimensional system due to a static and uniform electromagnetic-like field, in the presence of quantum dissipation. We first describe a direct derivation of the Caldeira-Leggett type of mechanism for quantum dissipation within the context of string theory and of the ensuing Born-Infeld action, pointing out the difference with the physical context in which vacuum decay can occur. We then evaluate the particle-antiparticle pair production rate, working out all the details of the calculation and including also the effects of a possible periodic background potential and of the Coulomb-like particle-antiparticle attraction. The former induces a dissipation-driven localization which interferes with the effect of the driving electric-like field. We also hint at a possible application to the problem of the decay of a supercurrent in a superconducting thin film due to vortex-antivortex nucleation in the presence of a pinning lattice. 
  Using the Schwinger-Keldysh technique we discuss how to derive the transport equations for the system of massless quantum fields. We analyse the scalar field models with quartic and cubic interaction terms. In the $\phi^4$ model the massive quasiparticles appear due to the self-interaction of massless bare fields. Therefore, the derivation of the transport equations strongly resembles that one of the massive fields, but the subset of diagrams which provide the quasiparticle mass has to be resummed. The kinetic equation for the finite width quasiparticles is found, where, except the mean-field and collision terms, there are terms which are absent in the standard Boltzmann equation. The structure of these terms is discussed. In the massless $\phi^3$ model the massive quasiparticles do not emerge and presumably there is no transport theory corresponding to this model. It is not surprising since the $\phi^3$ model is anyhow ill defined. 
  The response of isodoublet fermions to classical backgrounds of essentially 2-dimensional boson fields in SU(2) Yang-Mills-Higgs theory is investigated. In particular, the spectral flow of Dirac eigenvalues is calculated for a non-contractible sphere of configurations passing through the vacuum and the Z-string (the embedded vortex solution). Also, a non-vanishing Berry phase is established for adiabatic transport "around" the Z-string. These results imply the existence of a new type of global (non-perturbative) gauge anomaly in SU(2) Yang-Mills-Higgs quantum field theory with a single doublet of left-handed fermions. Possible extensions to other chiral gauge field theories are also discussed. 
  We discuss symplectic structures for the chiral boson in (1+1) dimensions and the self-dual field in (4k+2) dimensions. Dimensional reduction of the six-dimensional field on a torus is also considered. 
  In a spacetime with nonvanishing torsion there can occur topologically stable configurations associated with the frame bundle which are independent of the curvature. The relevant topological invariants are integrals of local scalar densities first discussed by Nieh and Yan (N-Y). In four dimensions, the N-Y form $N= (T^a \wedge T_a - R_{ab} \wedge e^a \wedge e^b)$ is the only closed 4-form invariant under local Lorentz rotations associated with the torsion of the manifold. The integral of $N$ over a compact D-dimensional (Euclidean) manifold is shown to be a topological invariant related to the Pontryagin classes of SO(D+1) and SO(D). An explicit example of a topologically nontrivial configuration carrying nonvanishing instanton number proportional to $\int N$ is costructed. The chiral anomaly in a four-dimensional spacetime with torsion is also shown to contain a contribution proportional to $N$, besides the usual Pontryagin density related to the spacetime curvature. The violation of chiral symmetry can thus depend on the instanton number of the tangent frame bundle of the manifold. Similar invariants can be constructed in D>4 dimensions and the existence of the corresponding nontrivial excitations is also discussed. 
  We use the recently established higher-level Bailey lemma and Bose-Fermi polynomial identities for the minimal models $M(p,p')$ to demonstrate the existence of a Bailey flow from $M(p,p')$ to the coset models $(A^{(1)}_1)_N\times (A^{(1)}_1)_{N'}/(A^{(1)}_1)_{N+N'}$ where $N$ is a positive integer and $N'$ is fractional, and to obtain Bose-Fermi identities for these models. The fermionic side of these identities is expressed in terms of the fractional-level Cartan matrix introduced in the study of $M(p,p')$. Relations between Bailey and renormalization group flow are discussed. 
  This is a talk presented at the conference ``Historical and Philosophical Reflections on the Foundations of Quantum Field Theory,'' at Boston University, March 1996. It will be published in the proceedings of this conference. 
  In this paper we derive the exact expression of the four-spinon contribution to the dynamical correlation function of the spin S= 1/2 anisotropic (XXZ) Heisenberg model in the antiferromagnetic regime. We extensively study its isotropic (XXX) limit and derive perturbatively the Ising one. Our method relies on the quantum affine symmetry of the model, which allows for a systematic diagonalization of the Hamiltonian in the thermodynamic limit and for an exact calculation of matrix elements of local spin operators. In fact, we argue that the familiar criticism of this method related to the complication of these matrix elements is not justified. First, we give, in the form of contour integrals, an exact expression for the n-spinon contribution. After we compile recently found results concerning the two-spinon contribution, we specialize the n-spinon formula to the new case n=4. Then we give an explicit series representation of this contribution in the isotropic limit. Finally, after we show that this representation is free of divergences, we discuss the Ising limit in which a simple expression is found up to first order in the anisotropy parameter. 
  We find the static vortex solutions of the model of Maxwell-Chern-Simons gauge field coupled to a (2+1)-dimensional four-fermion theory. Especially, we introduce two matter currents coupled to the gauge field minimally: the electromagnetic current and a topological current associated with the electromagnetic current. Unlike other Chern-Simons solitons the N-soliton solution of this theory has binding energy and the stability of the solutions is maintained by the charge conservation laws. 
  We discuss some aspects of perturbative $(0,2)$ Calabi-Yau moduli space. In particular, we show how models with different $(0,2)$ data can meet along various sub-loci in their moduli space. In the simplest examples, the models differ by the choice of desingularization of a holomorphic V-bundle over the same resolved Calabi-Yau base while in more complicated examples, even the smooth Calabi-Yau base manifolds can be topologically distinct. These latter examples extend and clarify a previous observation which was limited to singular Calabi-Yau spaces and seem to indicate a multicritical structure in moduli space. This should have a natural F-theory counterpart in terms of the moduli space of Calabi-Yau four-folds. 
  We study the enhanced gauge symmetries at constant couplings in the moduli space of the Gimon-Polchinski Orientifold. We show that whenever there is a singularity of E_6, E_7 and E_8 type we get enhanced gauge symmetry U(5), U(6) and U(8) respectively. This is due to the additional twisting involved in the definition of the Gimon-Polchiniski orientifold. 
  We discuss the comment by V.Zeitlin on our recent papers concerning Chern-Simons term generation at finite density. 
  The interaction of a nontrivial topological field configuration with the external fields is considered. The approach is based on the calculation of the zero modes excitation probability. We consider the interaction of the t'Hooft-Polyakov monopole with an external weak uniform magnetic field and the field of another monopole. 
  We extend work done previously concerning the formation of vortices in a U(1)->1 gauge theory at a first order phase transition to the symmetry breaking SU(2)xU(1)->U(1). It is shown that the collision of bubbles, appearing at the phase transition, allow the possibility for forming non-topological strings associated with the gauge group. The method used also shows clearly how these vortices are related to the Nielsen-Olesen vortices. 
  Yang-Mills theory in the first order formalism appears as the deformation of a topological field theory, the pure BF theory. We discuss this formulation at the quantum level, giving the Feynman rules of the BF-YM theory, the structure of the renormalization and checking its uv-behaviour in the computation of the beta-function which agrees with the expected result. 
  In this letter we quantize a previously proposed non-local lagrangean for the classical dual electrodynamics (Phys.Lett.B 384(1996)197), showing how it can be used to construct probability amplitudes. Our results are shown to agree with those obtained in the context of Schwinger and Zwanzinger formulations, but without necessity of introducing strings. 
  We develop an operator formalism to compute scattering amplitudes of arbitrary bosonic string states in the background of many D-branes. Specifically, we construct a suitable boundary state which we use to saturate the multi-Reggeon vertex in order to obtain the generator of multi-membrane scattering amplitudes. We explicitly show that the amplitudes with $h$ parallel D-branes are similar to amplitudes with $h-1$ open string loops. 
  We discuss a set of generalized, necessary conditions for non-trivial, interacting fixed points in six dimensional supersymmetric field theories. We use string theory to argue for the existence of infinite families of interacting RG fixed point theories. The theories are based on certain gauge groups and matter content, which we identify, along with additional tensor multiplets. They are conjectured to arise in the world-volume of Type I D5 branes at orbifold singularities. 
  We show that the Einstein-Maxwell-Dilaton-Axion system with multiple vector fields (bosonic sector of the D=4, N=4 supergravity) restricted to spacetimes possessing a non-null Killing vector field admits a concise representation in terms of the Ernst-type matrix valued potentials. A constructive derivation of the SWIP solutions is given and a colliding waves counterpart of the DARN-NUT solution is obtained. SU(m,m) chiral representation of the two-dimensionally reduced system is derived and the corresponding Kramer-Neugebauer-type map is presented. 
  The Faddeev-Jackiw Hamiltonian Reduction approach to constrained dynamics is applied to the collective coordinates analysis of non-linear waves, and compared with the alternative procedure known as symplectic formalism. 
  We consider vacuum tunneling of a new kind where the false vacua are not translationally invariant, but have topological defects that break some of their translational symmetries. In the particular case where the topological defects are cosmic strings, we show the existence of an O(2) \times O(2) symmetric bounce configuration that allows a semi-classical estimate of the rate of cosmic string induced tunneling. A method of reduction is then suggested for simplifying the computation of the bounce action. Some phenomenological applications are described. 
  We completely compute the local BRST cohomology $H(s|d)$ of the combined Yang-Mills-2-form system coupled through the Yang-Mills Chern-Simons term ("Chapline-Manton model"). We consider the case of a simple gauge group and explicitely include in the analysis the sources for the BRST variations of the fields ("antifields"). We show that there is an antifield independent representative in each cohomological class of $H(s|d)$ at ghost number 0 or 1. Accordingly, any counterterm may be assumed to preserve the gauge symmetries. Similarly, there is no new candidate anomaly beside those already considered in the literature, even when one takes the antifields into account. We then characterize explicitly all the non-trivial solutions of the Wess-Zumino consistency conditions. In particular, we provide a cohomological interpretation of the Green-Schwarz anomaly cancellation mechanism. 
  String cosmology solutions are examined in a generalized phase-space including sources representing arbitrary corrections to lowest order string-dilaton-gravity effective action. We find a set of necessary conditions for a graceful exit transition from a dilaton-driven inflationary phase to a radiation dominated era. We show that sources allowing such a transition have to violate energy conditions similar to those appearing in singularity theorems of general relativity. Since familiar classical sources, excepting spatial curvature, obey these energy conditions we conclude that a generic graceful exit in string cosmology requires a new effective phase of matter. Our results clarify and generalize previous analyses and enable us to critically reexamine proposed non-singular cosmologies. 
  The question of the stability of the four dimensional Gross-Perry-Sorkin Kaluza-Klein magnetic monopole solution is investigated within the framework of a N=2, D=5 supergravity theory. We show that this solution does not support a spin structure of the Killing type and is therefore, contrary to previous expectations, not necessarily stable. 
  Near extreme black holes can lose their charge and decay by the emission of massive BPS charged particles. We calculate the greybody factors for low energy charged and neutral scalar emission from four and five dimensional near extremal Reissner-Nordstrom black holes. We use the corresponding emission rates to obtain ratios of the rates of loss of excess energy by charged and neutral emission, which are moduli independent, depending only on the integral charges and the horizon potentials. We consider scattering experiments, finding that evolution towards a state in which the integral charges are equal is favoured, but neutral emission will dominate the decay back to extremality except when one charge is much greater than the others. The implications of our results for the agreement between black hole and D-brane emission rates and for the information loss puzzle are then discussed. 
  A geometric formulation of the Moyal deformation for the Self-dual Yang-Mills theory and the Chiral Model approach to Self-dual gravity is given. We find in Fedosov's geometrical construction of deformation quantization the natural geometrical framework associated to the Moyal deformation of the six-dimensional version of the second heavenly equation and the Park-Husain heavenly equation. The Wess-Zumino-Witten-like Lagrangian of Self-dual gravity is re-examined within this context. 
  A guiding principle to determine the Kaehler potential in the low energy effective theory of the supersymmetric chiral gauge theory with no flat direction is proposed. The guiding principle is applied to the SU(5) gauge theory with chiral superfields in the 5* and 10 representation, and the low energy effective theory is consistently constructed. The spontaneous supersymmetry breaking takes place in the low energy effective theory as expected. The particle mass spectrum in the low energy is explicitly calculated. 
  We propose and canonically quantize a generalization of the two-dimensional massive fermion theory described by a Lagrangian containing third-order derivatives. In our approach the mass term contains a derivative coupling. The quantum solution is expressed in terms of three usual fermions. Employing the standard bosonization scheme, the equivalent boson theory is derived. The results obtained are used to solve a theory including a current-current interaction. 
  An efficient way of resolving Gauss' law in Yang-Mills theory is presented by starting from the projected gauge invariant partition function and integrating out one spatial field variable. In this way one obtains immediately the description in terms of unconstrained gauge invariant variables which was previously obtained by explicitly resolving Gauss' law in a modified axial gauge. In this gauge, which is a variant of 't Hooft's Abelian gauges, magnetic monopoles occur. It is shown how the Pontryagin index of the gauge field is related to the magnetic charges. It turns out that the magnetic monopoles are sufficient to account for the non-trivial topological structure of the theory. 
  A five-dimensional dyonic black hole in Type-I theory is considered that is extremal but non-supersymmetric. It is shown that the Bekenstein-Hawking entropy of this black hole counts precisely the microstates of a D-brane configuration with the same charges and mass, even though there is no apparent supersymmetric nonrenormalization theorem for the mass. A similar result is known for the entropy at the stretched horizon of electrically charged, extremal, but non-supersymmetric black holes in heterotic string theory. It is argued that classical nonrenormalization of the mass may partially explain this result. 
  We study the beta-function of the N=2 sigma-model coupled to N=2 induced supergravity. We compute corrections to first order in the semiclassical limit, $c \to -\infty$, beyond one-loop in the matter fields. As compared to the corresponding bosonic, metric sigma-model calculation, we find new types of contributions arising from the dilaton coupling automatically accounted for, once the K\"ahler potential is coupled to N=2 supergravity. 
  We consider a new MacDowell-Mansouri R^2-type of supergravity theory, an extension of conformal supergravity, based on the superalgebra Osp(1|8). Invariance under local symmetries with negative Weyl weight is achieved by imposing chirality-duality and double-duality constraints on curvatures, along with the usual constraint of vanishing supertorsion. An analysis of the remaining gauge symmetries shows that those with vanishing Weyl weight are invariances of the action at the linearized level. For the symmetries with positive Weyl weight we find that invariance of the action would require further modifications of the transformation rules. This conclusion is supported by a kinematical analysis of the closure of the gauge algebra. 
  A recent construction of the electroweak theory, based on perturbative quantum gauge invariance alone, is extended to the case of more generations of fermions with arbitrary mixing. The conditions implied by second order gauge invariance lead to an isolated solution for the fermionic couplings in agreement with the standard model. Third order gauge invariance determines the Higgs potential. The resulting massive gauge theory is manifestly gauge invariant, after construction. 
  We show that the baryon number of N=2 supersymmetric QCD can be twisted in order to couple the topological field theory of non-abelian monopoles to $Spin^c$-structures. To motivate the construction, we also consider some aspects of the twisting procedure as a gauging of global currents in two and four dimensions, in particular the role played by anomalies. 
  The Lorentz group is the fundamental language for space-time symmetries of relativistic particles. This group can these days be derived from the symmetries observed in other branches of physics. It is shown that this group can be derived from optical filters. The group O(2,1) is appropriate for attenuation filters, while the O(3) group describes phase-shift filters. The combined operation leads to a two-by-two representation of the six-parameter Lorentz group. It is shown also that the bilinear representation of this group is the natural language for the polarization optics. 
  I define central functions c(g) and c'(g) in quantum field theory, useful to study the flow of the numbers of vector, spinor and scalar degrees of freedom from the UV limit to the IR limit and basic ingredients for a description of quantum field theory as an interpolating theory between pairs of 4D conformal field theories. The key importance of the correlator of four stress-energy tensors is outlined in this respect. Then I focus the analysis on the behaviours of the central functions in QCD, computing their slopes in the UV critical point. To two-loops, c(g) and c'(g) point towards the expected IR directions. As a possible physical application, I argue that a closer study of the central functions might allow us to lower the upper bound on the number of generations to the observed value. Candidate all-order expressions for the central functions are compared with the predictions of electric-magnetic duality. 
  Recently we proposed a new Wick rotation for Dirac spinors which resulted in a hermitean action in Euclidean space. Our work was in a path integral context, however, in this note, we provide the canonical formulation of the new Wick rotation along the lines of the proposal of Osterwalder and Schrader. 
  The plot of allowed p and D values for p-brane solitons in D-dimensional supergravity is the same whether the solitons are extremal or non-extremal. One of the useful tools for relating different points on the plot is vertical dimensional reduction, which is possible if periodic arrays of p-brane solitons can be constructed. This is straightforward for extremal p-branes, since the no-force condition allows arbitrary multi-centre solutions to be constructed in terms of a general harmonic function on the transverse space. This has also been shown to be possible in the special case of non-extremal black holes in D=4 arrayed along an axis. In this paper, we extend previous results to include multi-scalar black holes, and dyonic black holes. We also consider their oxidation to higher dimensions, and we discuss general procedures for constructing the solutions, and studying their symmetries. 
  Expanding upon earlier work of Pouliot and Strassler, we construct chiral magnetic duals to nonchiral supersymmetric electric theories based upon SO(7), SO(8) and SO(9) gauge groups with various numbers of vector and spinor matter superfields. Anomalies are matched and gauge invariant operators are mapped within each dual pair. Renormalization group flows along flat directions are also examined. We find that confining phase quantum constraints in the electric theories are recovered from semiclassical equations of motion in their magnetic counterparts when the dual gauge groups are completely Higgsed. 
  Using the Non-Abelian Batalin-Vilkovisky formalism introduced recently, we present a generalization of the Yang-Mills gauge transformations , to include antisymmetric tensor fields as gauge bosons. The Freedman-Townsend transformation for the two-form gauge field is automatically recovered. New characteristic classes involving this two-form field and the Yang-Mills one-form field are derived. We also show how to include, in an unified way, a gauge invariant coupling of the new gauge bosons to fermionic and bosonic matter. 
  We establish the equivalence of the Gimon-Polchinski orientifold and F-theory on an elliptically fibered Calabi-Yau three fold on base $CP^1 \times CP^1$ by comparing the gauge symmetry breaking pattern, local deformations in the moduli space, as well as the axion-dilaton background in the weak coupling limit in the two theories. We also provide an explanation for an apparent discrepancy between the F-theory and the orientifold results for constant coupling configuration. 
  Hamiltonian BRST formalism (FV formalism) includes many auxiliary fields without explanation. Its path-integration has a simple form by using BRST charge, but its construction is quite mechanically and hard to understand physical meaning. In this paper we perform the phase space path-integral with requiring BRST invariance for action and measure, and show that the resultant form is equivalent to the Hamiltonian BRST (FV) formalism in gravitational theory. This explains why so many auxiliary fields are necessary to be introduced. We also find the gauge fixing is automatically done by requiring the BRST invariance of the path-integral measure. This is a pedagogical introduction to Hamiltonian BRST formalism. 
  The evidently supersymmetric four-dimensional Wess-Zumino model with quenched disorder is considered at the one-loop level. The infrared fixed points of a beta-function form the moduli space $M = RP^2$ where two types of phases were found: with and without replica symmetry. While the former phase possesses only a trivial fixed point, this point become unstable in the latter phase which may be interpreted as a spin glass phase. 
  This paper introduces the way of the embedding of spinning particle quantum mechanically. Schr\"odinger equation on its submanifold obtains the gauge field as spin connection, and it reduces to the ones obtained by Ohnuki and Kitakado when we consider $S^2$ in $R^3$. PACS numbers: 03.65 
  The continuous extension of the discrete duality symmetry of the Schwarz-Sen model is studied. The corresponding infinitesimal generator $Q$ turns out to be local, gauge invariant and metric independent. Furthermore, $Q$ commutes with all the conformal group generators. We also show that $Q$ is equivalent to the non---local duality transformation generator found in the Hamiltonian formulation of Maxwell theory. We next consider the Batalin--Fradkin-Vilkovisky formalism for the Maxwell theory and demonstrate that requiring a local duality transformation lead us to the Schwarz--Sen formulation. The partition functions are shown to be the same which implies the quantum equivalence of the two approaches. 
  The algebra of observables for identical particles on a line is formulated starting from postulated basic commutation relations. A realization of this algebra in the Calogero model was previously known. New realizations are presented here in terms of differentiation operators and in terms of SU(N)-invariant observables of the Hermitian matrix models. Some particular structure properties of the algebra are briefly discussed. 
  Effective actions are derived for (2,0) and (2,1) superstrings by studying the corresponding sigma-models. The geometry is a generalisation of Kahler geometry involving torsion and the field equations imply that the curvature with torsion is self-dual in four dimensions, or has SU(n,m) holonomy in other dimensions. The Yang-Mills fields are self-dual in four dimensions and satisfy a form of the Uhlenbeck-Yau equation in higher dimensions. In four dimensions with Euclidean signature, there is a hyperkahler structure and the sigma-model has (4,1) supersymmetry, while for signature (2,2) there is a hypersymplectic structure consisting of a complex structure squaring to -1 and two real structures squaring to 1. The theory is invariant under a twisted form of the (4,1) superconformal algebra which includes an SL(2,R) Kac-Moody algebra instead of an SU(2) Kac-Moody algebra. Kahler and related geometries are generalised to ones involving real structures. 
  In string-bit models, string is described as a polymer of point-like constituents. We attempt to use string-bit ideas to investigate how the size of string is affected by string interactions in a non-perturbative context. Lacking adequate methods to deal with the full complications of bit rearrangement interactions, we study instead a simplified analog model with only ``direct'' potential interactions among the bits. We use the variational principle in an approximate calculation of the mean-square size of a polymer as a function of the number of constituents/bits for various interaction strengths g in three specific models. 
  Within the frame of a Group Approach to Quantization anomalies arise in a quite natural way. We present in this talk an analysis of the basic obstructions that can be found when we try to translate symmetries of the Newton equations to the Quantum Theory. They fall into two classes: algebraic and topologic according to the local or global character of the obstruction. We present here one explicit example of each. 
  Some general remarks are made about the quantum theory of scalar fields and the definition of momentum in curved space. Special emphasis is given to field theory in anti-de Sitter space, as it represents a maximally symmetric space-time of constant curvature which could arise in the local description of matter interactions in small regions of space-time. Transform space rules for evaluating Feynman diagrams in Euclidean anti-de Sitter space are initially defined using eigenfunctions based on generalized plane waves. It is shown that, for a general curved space, the rules associated with the vertex are dependent on the type of interaction being considered. A condition for eliminating this dependence is given. It is demonstrated that the vacuum and propagator in conformally flat coordinates in anti-de Sitter space are equivalent to those analytically continued from $H^4$ and that transform space rules based on these coordinates can be used more readily. A proof of the analogue of Goldstone's theorem in anti-de Sitter space is given, using a generalized plane wave representation of the commutator of the current and the scalar field. It is shown that the introduction of curvature in the space-time shifts the momentum by an amount which is determined by the Riemann tensor to first order, and it follows that there is a shift in both the momentum and mass scale in anti-de Sitter space. 
  We provide the theoretic value 1/2 for the quantity $\sin**2\theta_W=e**2/g**2$, as opposite to the experimental value of $\sin**2\theta_W=1-M_W**2/M_Z**2 \sim 0.23$, in the electro-weak gauge group $SU(2)\otimes U(1)_Y$. The arguments we use are of topological nature and have to do with natural requirements of globality in the transformations of the gauge algebra. This results supports the idea that the two definitions of $\sin**2\theta_W$ cannot be simultaneously identified except, maybe, in an asymptotic limit corresponding to an earlier stage of the Universe, before the spontaneous symmetry breaking had taken place. 
  For slowly varying fields the vacuum functional of a quantum field theory may be expanded in terms of local functionals. This expansion satisfies its own form of the Schr\"odinger equation from which the expansion coefficents can be found. For scalar field theory in 1+1 dimensions we show that this approach correctly reproduces the short-distance properties as contained in the counter-terms. We also describe an approximate simplification that occurs for the Sine-Gordon and Sinh-Gordon vacuum functionals. 
  We completely describe presentations of Lie superalgebras with Cartan matrix if they are simple Z-graded of polynomial growth. Such matrices can be neither integer nor symmetrizable. There are non-Serre relations encountered. In certain cases there are infinitely many relations. Our results are applicable to the Lie algebras with the same Cartan matrices as the Lie superalgebras considered. 
  There exist two different languages, the ^sl(2) and N=2 ones, to describe similar structures; a dictionary is given translating the key representation-theoretic terms related to the two algebras. The main tool to describe the structure of ^sl(2) and N=2 modules is provided by diagrams of extremal vectors. The ^sl(2) and N=2 representation theories of certain highest-weight types turn out to be equivalent modulo the respective spectral flows. 
  We study the emission of scalar particles from a class of near-extremal five dimensional black holes and the corresponding D-brane configuration at high energies. We show that the distribution functions and the black hole greybody factors are modified in the high energy tail of the Hawking spectrum in such way that the emission rates exactly match. We extend the results to charged scalar emission and to four dimensions. 
  We calculate classical cross sections for absorption of massless scalars by the extremal 3-branes of type IIB theory, and by the extremal 2- and 5-branes of M-theory. The results are compared with corresponding calculations in the world volume effective theories. For all three cases we find agreement in the scaling with the energy and the number of coincident branes. For 3-branes, whose stringy description is known in detail in terms of multiple D-branes, the string theoretic absorption cross section for low energy dilatons is in exact agreement with the classical gravity. This suggests that scattering from extremal 3-branes is a unitary process well described by perturbative string theory. 
  General properties of intersecting extremal p-brane solutions of gravity coupled with dilatons and several different d-form fields in arbitrary space-time dimensions are considered. It is show that heuristically expected properties of the intersecting p-branes follow from the explicit formulae for solutions. In particular, harmonic superposition and S-duality hold for all p-brane solutions. Generalized T-duality takes place under additional restrictions on the initial theory parameters . 
  Within the framework of "anomalously gauged" Wess-Zumino-Witten (WZW) models, we construct solutions which include nonabelian fields. Both compact and noncompact groups are discussed. In the case of compact groups, as an example of background containing nonabelian fields, we discuss conformal theory on the $SO(4)/SO(3)$ coset, which is the natural generalization of the 2D monopole theory corresponding to the $SO(3)/SO(2)$ coset. In noncompact case, we consider examples with $SO(2,1)/SO(1,1)$ and $SO(3,2)/SO(3,1)$ cosets. 
  Non-extremal intersecting p-brane solutions of gravity coupled with several antisymmetric fields and dilatons in various space-time dimensions are constructed. The construction uses the same algebraic method of finding solutions as in the extremal case and a modified "no-force" conditions. We justify the "deformation" prescription. It is shown that the non-extremal intersecting p-brane solutions satisfy harmonic superposition rule and the intersections of non-extremal p-branes are specified by the same characteristic equations for the incidence matrices as for the extremal p-branes. We show that S-duality holds for non-extremal p-brane solutions. Generalized T-duality takes place under additional restrictions to the parameters of the theory which are the same as in the extremal case. 
  It is shown that the measurement algebra of Schwinger, a characterization of the properties of Pauli measurements of the first and second kinds, forming the foundation of his formulation of quantum mechanics over the complex field, has a quaternionic generalization. In this quaternionic measurement algebra some of the notions of quaternionic quantum mechanics are clarified. The conditions imposed on the form of the corresponding quantum field theory are studied, and the quantum fields are constructed. It is shown that the resulting quantum fields coincide with the fermion or boson annihilation-creation operators obtained by Razon and Horwitz in the limit in which the number of particles in physical states $N \to \infty$. 
  The quantization of the tensionless p-brane is discussed. Inspection of the constraint algebra reveals that the central extensions for the p-branes have a simple form. Using a Hamiltonian BRST scheme we find that the quantization is consistent in any space-time dimension while the quantization of the conformal tensionless p-brane gives a critical dimension $d=2$. 
  The Polchinski version of the exact renormalization group equation is discussed and its applications in scalar and fermionic theories are reviewed. Relation between this approach and the standard renormalization group is studied, in particular the relation between the derivative expansion and the perturbation theory expansion is worked out in some detail. 
  The properties of the N=2 SUSY gauge theories underlying the Seiberg-Witten hypothesis are discussed. The main ingredients of the formulation of the finite-gap solutions to integrable equations in terms of complex curves and generating 1-differential are presented, the invariant sense of these definitions is illustrated. Recently found exact nonperturbative solutions to N=2 SUSY gauge theories are formulated using the methods of the theory of integrable systems and where possible the parallels between standard quantum field theory results and solutions to integrable systems are discussed. 
  We present a relatively simple argument showing that the H-dyon states required by S-duality of the heterotic string on $T^6$ are present provided that the BPS dyons required by S-duality of N=4 supersymmetric Yang-Mills theory are present. We also conjecture and provide evidence that H-dyons at singularities where the nonperturbative gauge symmetry is completely broken are actually BPS dyons. 
  Anomalous contributions to the Jacobi identity of chromo-electric fields and non-Abelian vector currents are calculated using a non-perturbative approach that combines operator product expansion and a generalization of Bjorken-Johnson-Low limit. The failure of the Jacobi identity and the associated 3-cocycles are discussed. 
  We review an approach to the construction and classification of p-brane solitons in arbitrary dimensions, with an emphasis on those that arise in toroidally-compactified M-theory. Procedures for constructing the low-energy supergravity limits in arbitrary dimensions, and for studying the supersymmetry properties of the solitons are presented. Wide classes of p-brane solutions are obtained, and their properties and classification in terms of bound states and intersections of M-branes are described. (Based on lectures presented at the Summer School in High-Energy Physics and Cosmology, Trieste, Italy, 10 Jun - 26 Jul 1996.) 
  The Zeroth and First Laws of Black Hole Mechanics are derived in the context of non-linear electrodynamics coupled to gravity. The Zeroth Law is shown to hold quite generally even if the Dominant Energy Condition is violated. The derivation of the First Law is discussed in detail for general matter fields coupled to gravity. The general mass variation formula obtained includes a term previously omitted in some of the literature. This is then applied to the case of non-linear electrodynamics and the usual First Law is found to hold true. As an example, Born-Infeld theory is discussed. The results are extended to include scalar fields in a very general way, including additional terms arising from the variation of the asymptotic values of the scalars. 
  We show how the spectral flow between the Neveu-Schwarz and Ramond sectors of N=2 superconformal field theories can be described in three dimensions in terms of the propagation of charged particles coupled to a a Chern-Simons gauge theory. Quantum mechanical mixing between the degenerate Chern-Simons vacua interpolates between the different boundary conditions of the two sectors and so provides a dynamical picture for the GSO-projection. 
  We present candidates for the global minimum energy solitons of charge one to nine in the Skyrme model, generated using sophisticated numerical algorithms. Assuming the Skyrme model accurately represents the low energy limit of QCD, these configurations correspond to the classical nuclear ground states of the light elements. The solitons found are particularly symmetric, for example, the charge seven skyrmion has icosahedral symmetry, and the shapes are shown to fit a remarkable sequence defined by a geometric energy minimization (GEM) rule. We also calculate the energies and sizes to within at least a few percent accuracy. These calculations provide the basis for a future investigation of the low energy vibrational modes of skyrmions and hence the possibility of testing the Skyrme model against experiment. 
  We describe how it is possible to introduce the interaction between superconformal fields of the same conformal dimensions. In the classical case such construction can be used to the construction of the Hirota - Satsuma equation. We construct supersymmetric Poisson tensor for such fields, which generates a new class of Hamiltonin systems. We found Lax representation for one of equation in this class by supersymmetrization Lax operator responsible for Hirota - Satsuma equation. Interestingly our supersymmetric equation is not reducible to classical Hirota - Satsuma equation. We show that our generalized system is reduced to the one of the supersymmetric KDV equation (a=4) but in this limit integrals of motion are not reduced to integrals of motion of the supersymmetric KdV equation. 
  QCD without matter and quantized on a light-cone spatial cylinder is considered. For the gauge group SU(N) the theory has N-1 quantum mechanical degrees of freedom, which describe the color fux that circulates around the the spatial cylinder. In 1+1 dimensions this problem can be solved analytically. I use the solution for SU(2) to compute the Wilson loop phase on the surface of the cylinder and find that it is equal to g^2 area/4. This result is different from the well known result for flat space. I argue that for SU(N) the Wilson loop phase for a contour on a light-cone spatial cylinder is g^2(area) (N-1)/4. The underlying reason for this result is that only the N-1 dimensional Cartan subgroup of SU(N) is dynamical in this problem. 
  Higher order coefficients of the inverse mass expansion of one-loop effective actions are obtained from a one-dimensional path integral representation. For the case of a massive scalar loop in the background of both a scalar potential and a (non Abelian) gauge field explicit results to $O(T^5)$ in the proper time parameter are presented. 
  The perturbative analysis of models of open and closed superstrings presents a number of surprises. For instance, variable numbers of antisymmetric tensors ensure their consistency via generalized Green-Schwarz cancellations and a novel type of singularity occurs in their moduli spaces. All these features are related, in one way or another, to the presence of boundaries on the world sheet or, equivalently, of extended objects (branes) interacting with the bulk theory in space time. String dualities have largely widened the interest in these models, that exhibit a wealth of generic non-perturbative features of String Theory. 
  These lectures provide an introduction to the behavior of strongly-coupled supersymmetric gauge theories. After a discussion of the effective Lagrangian in nonsupersymmetric and supersymmetric field theories, I analyze the qualitative behavior of the simplest illustrative models. These include supersymmetric QCD for $N_f < N_c$, in which the superpotential is generated nonperturbatively, N=2 SU(2) Yang-Mills theory (the Seiberg-Witten model), in which the nonperturbative behavior of the effective coupling is described geometrically, and supersymmetric QCD for N_f large, in which the theory illustrates a non-Abelian generalization of electric-magnetic duality. [Lectures presented at the 1996 TASI Summer School, to appear in the proceedings.] 
  The gauged SL(2,R)/U(1) Wess-Zumino-Novikov-Witten (WZNW) model is classically an integrable conformal field theory. We have found a Lax pair representation for the non-linear equations of motion, and a B"acklund transformation. A second-order differential equation of the Gelfand-Dikii type defines the Poisson bracket structure of the theory, and its fundamental solutions describe the general solution of the WZNW model as well. The physical and free fields are related by non-local transformations. The (anti-)chiral component of the energy-momentum tensor which factorizes into conserved quantities satisfying a non-linear algebra characteristic for parafermions assumes the expected canonical free field form. So the black hole model seems to be prepared for an exact canonical quantization. 
  A gap in the mathematical logic in derivations in quantum field theory arises as consequence of variation before quantization. To close this gap the present paper introduces a mathematically rigorous variational calculus for operator fields. Using quantization before variation it is demonstrated that the so-called naive results are correct; in particular both Noether's theorem and the Ward-Takahashi identities retain full validity in quantum field theory. 
  We investigate the effect of an external gravitational fields to the chiral symmetry breaking in the SUSY (supersymmetric) NJL (Nambu-Jona-Lasinio) model non-minimally interacting with external supergravity. Evaluating the effective potential in the leading order of the $1/N_{c}$-expansion and in the linear curvature approximation it is found the possibility of the chiral symmetry breaking in the SUSY NJL model in an external gravitational fields. In the broken phase the dynamically generated mass is analytically and numerically calculated. 
  Notwithstanding the central role of D-branes in many recently proposed string dualities, several interesting type I vacua have been found without resorting directly to D-brane technology. In this talk, we analyze a three-generation SO(8)xU(12) chiral type I model with N=1 supersymmetry in D=4. It descends from the type IIB compactification on the Z orbifold and requires only Neumann boundary conditions, i.e. only the ubiquitous D9-branes (pan-branes). We also discuss a large class of 6D type I vacua that display rich patterns of Chan-Paton symmetry breaking/enhancement and various numbers of tensor multiplets. Finally, we briefly address issues raised by the conjectured heterotic - type I duality and by the relation between type I vacua and compactifications of the putative F-theory. 
  We interpret instantons on a torus with twisted boundary conditions, in terms of bound states of branes. The interplay between the SU(N) and U(1) parts of the U(N) theory of N 4-branes allows the construction of a variety of bound states. The SU(N) and U(1) parts can contribute fractional amounts to the total instanton number which is integral. The geometry of non-self intersecting two-cycles in $T^4$ sheds some light on a number of properties of these solutions. 
  The basic AB problem is to determine how an unshielded tube of magnetic flux $\Phi$ affects arbitrarily long-wavelength charged particles impinging on it. For spin-1 at almost all $\Phi$ the particles do not penetrate the tube, so the interaction essentially is periodic in $\Phi$ (AB effect). Below-threshold bound states move freely only along the tube axis, and consequent induced vacuum currents supplement rather than screen $\Phi$. For a pure magnetic interaction the tube must be broader than the particle Compton wavelength, i.e., only the nonrelativistic spin-1 AB problem exists. 
  The matrix model formulation of M-theory can be generalized by compactification to ten-dimensional type II string theory, formulated in the infinite momentum frame. Both the type IIA and IIB string theories can be formulated in this way. In the M-theory and type IIA cases, the transverse rotational invariance is manifest, but in the IIB case, one of the transverse dimensions materializes in a completely different way from the other seven. The full O(8) rotational symmetry then follows in a surprising way from the electric-magnetic duality of supersymmetric Yang-Mills field theory. 
  The conversion of second-class constraints into first-class constraints is used to extend the coordinate-free path integral quantization, achieved by a flat-space Brownian motion regularization of the coherent-state path integral measure, to systems with second-class constraints. 
  We study the stabilization of scalars near a supersymmetric black hole horizon using the equation of motion of a particle moving in a potential and background metric. When the relevant 4-dimensional theory is described by special geometry, the generic properties of the critical points of this potential can be studied. We find that the extremal value of the central charge provides the minimal value of the BPS mass and of the potential under the condition that the moduli space metric is positive at the critical point. We relate these ideas to the Weinhold and Ruppeiner metrics introduced in the geometric approach to thermodynamics and used for study of critical phenomena. 
  This paper introduces the modified version of Schwinger's quantization method, in which the information on constraints and the choice of gauge conditions are included implicitly in the choice of variations used in quantization scheme. A proof of equivalence between Schwinger- and Dirac-methods for constraint systems is given. 
  In the context of the field theory limit of superstrings, we consider an almost realistic model of supersymmetry breaking by gaugino condensation which includes, through nonperturbative corrections to the K\"ahler potential, dilaton stabilization at a value compatible with a weak coupling regime. Invariance under modular transformations is ensured through a Green-Schwarz term and string threshold corrections, which lead to moduli stabilization at the self-dual point. We are thus in a position to discuss several issues of physical relevance: gravitino, dilaton and moduli masses, axion, soft supersymmetry breaking parameters and gauge coupling unification. 
  We present a detailed description of the three inequivalent twists of N=4 supersymmetric gauge theories. The resulting topological quantum field theories are reobtained in the framework of the Mathai-Quillen formalism and the corresponding moduli spaces are analyzed. We study their geometric features in each case. In one of the twists we make contact with the theory of non-abelian monopoles in the adjoint representation of the gauge group. In another twist we obtain a topological quantum field theory which is orientation reversal invariant. For this theory we show how the functional integral contributions to the vacuum expectation values leading to topological invariants notably simplify. 
  Motivated by the recent D-brane constructions of world-volume monopoles and instantons, we study the supersymmetric SU(N) Yang-Mills theory on $S^1 \times R^{3+1}$, spontaneously broken by a Wilson loop. In addition to the usual N-1 fundamental monopoles, the N-th BPS monopole appears from the Kaluza-Klein sector. When all N monopoles are present, net magnetic charge vanishes and the solution can be reinterpreted as a Wilson-loop instanton of unit Pontryagin number. The instanton/multi-monopole moduli space is explicitly constructed, and seen to be identical to a Coulomb phase moduli space of a U(1)^N gauge theory in 2+1 dimensions related to Kronheimer's gauge theory of SU(N) type. This extends the results by Intriligator and Seiberg to the finite couplings that, in the infrared limit of Kronheimer's theory, the Coulomb phase parameterizes a centered SU(N) instanton. We also elaborate on the case of restored SU(N) symmetry. 
  We demonstrate the precise numerical correspondence between long range scattering of supergravitons and membranes in supergravity in the infinite momentum frame and in M(atrix)-Theory, both in 11 dimensions and for toroidal compactifications. We also identify wrapped membranes in terms of topological invariants of the vector bundles associated to the field theory description of compactified M(atrix)-Theory. We use these results to check the realization of T-duality in M(atrix)-Theory. 
  An expression for the spacetime absorption coefficient of a scalar field in a five dimensional, near extremal black hole background is derived, which has the same form as that presented by Maldacena and Strominger, but is valid over a larger, U-duality invariant region of parameter space and in general disagrees with the corresponding D-brane result. We develop an argument, based on D-brane thermodynamics, which specifies the range of parameters over which agreement should be expected. For neutral emission, the spacetime and D-brane results agree over this range. However, for charged emission, we find disagreement in the `Fat Black Hole' regime, in which charge is quantized in smaller units on the brane, than in the bulk of spacetime. We indicate a possible problem with the D-brane model in this regime. We also use the Born approximation to study the high frequency limit of the absorption coefficient and find that it approaches unity, for large black hole backgrounds, at frequencies still below the string scale, again in disagreement with D-brane results. 
  We derive the universal threshold corrections in heterotic string theory including a continuous Wilson line. Unification of gauge and gravitational couplings is shown to be possible even within perturbative string theory. The relative importance of gauge group dependent and independent thresholds on unification is clarified. Equipped with these results we can then attempt an extrapolation to the strongly coupled heterotic string -- M-theory. We argue that such an extrapolation might be meaningful because of the holomorphic structure of the gauge coupling function and the close connection of the threshold corrections to the anomaly cancelation mechanism. 
  The equations of motion for a self-interacting self-dual tensor in six dimensions are extracted from the equations describing the M-theory five-brane. These equations are presented in a self-contained, six-dimensional Lorentz-covariant form. In particular, it is shown that the field-strength tensor satisfies a non-linear generalised self-duality constraint. The self-duality equation is rewritten in five-dimensional notation and shown to be identical to the corresponding equation in the non-covariant formalism. 
  In this paper we obtain the orthogonality relations for the supergroup U(m|n), which are remarkably different from the ones for the U(N) case. We extend our results for ordinary representations, obtained some time ago, to the case of complex conjugated and mixed representations. Our results are expressed in terms of the Young tableaux notation for irreducible representations. We use the supersymmetric Harish-Chandra-Itzykson-Zuber integral and the character expansion technique as mathematical tools for deriving these relations. As a byproduct we also obtain closed expressions for the supercharacters and dimensions of some particular irreducible U(m|n) representations. A new way of labeling the U(m|n) irreducible representations in terms of m + n numbers is proposed. Finally, as a corollary of our results, new identities among the dimensions of the irreducible representations of the unitary group U(N) are presented. 
  A simple, anomaly-free chiral gauge theory can be perturbatively quantised and renormalised in such a way as to generate fermion and gauge boson masses. This development exploits certain freedoms inherent in choosing the unperturbed Lagrangian and in the renormalisation procedure. Apart from its intrinsic interest, such a mechanism might be employed in electroweak gauge theory to generate fermion and gauge boson masses without a Higgs sector. 
  A review is given of the status of the program of classical reduction to Dirac's observables of the four interactions (standard SU(3)xSU(2)xU(1) particle model and tetrad gravity) with the matter described either by Grassmann-valued fermion fields or by particles with Grassmann charges. 
  We review in detail the recently discovered phenomenon of partial spontaneous breaking of supersymmetry in the case of a N=2 pure gauge U(1) theory, and recall how the standard lore no-go theorem is evaded. We discuss the extension of this mechanism to theories with charged matter, and surprisingly find that the gauging forbids the existence of a magnetic Fayet-Iliopoulos term. 
  The role of instantons in three dimensional N=2 supersymmetric SU(2) Yang-Mills theory is studied, especially in relation to the issue of confinement. The instanton-induced low energy effective action is derived by extending the dilute gas approximation to the super-moduli space of instantons. Following Polyakov's description of confinement in compact U(1) gauge theory, it is argued that there is no confinement in N=2 supersymmetric Yang-Mills theory. 
  It is shown that the operator $B(C) = Tr [P \exp i \tilde{g} \oint \tilde{A}_i(x) dx^i]$ constructed with the recently derived dual potential $\tilde{A}(x)$ and a coupling $\tilde{g}$ related to $g$ by the Dirac quantization condition satisfies the correct commutation relation with the Wilson operator $Tr [P \exp ig \oint A_i(x) dx^i]$ as required by 't Hooft for his order-disorder parameters. 
  We study the structure and properties of vortices in a recently proposed Abelian Maxwell-Chern-Simons model in $2 +1 $ dimensions. The model which is described by gauge field interacting with a complex scalar field, includes two parity and time violating terms: the Chern-Simons and the anomalous magnetic terms. Self-dual relativistic vortices are discussed in detail. We also find one dimensional soliton solutions of the domain wall type. The vortices are correctly described by the domain wall solutions in the large flux limit. 
  The existence of normalizable zero modes of the twisted Dirac operator is proven for a class of static Einstein-Yang-Mills background fields with a half-integer Chern-Simons number. The proof holds for any gauge group and applies to Dirac spinors in an arbitrary representation of the gauge group. The class of background fields contains all regular, asymptotically flat, CP-symmetric configurations with a connection that is globally described by a time-independent spatial one-form which vanishes sufficiently fast at infinity. A subset is provided by all neutral, spherically symmetric configurations which satisfy a certain genericity condition, and for which the gauge potential is purely magnetic with real magnetic amplitudes. 
  We define and describe simple complex Lie superalgbras of vector fields on "supercircles" - simple stringy superalgebras. There are four series of such algebras and four exceptional stringy superalgebras. The 13 of the simple stringy Lie superalgebras are distinguished: only they have nontrivial central extensions; since two of the distinguish algebras have 3 nontrivial central extensions each, there are exactly 16 superizations of the Liouville action, Schroedinger equation, KdV hierarchy, etc. We also present the three nontrivial cocycles on the N=4 extended Neveu-Schwarz and Ramond superalgebras in terms of primary fields and describe the "classical" stringy superalgebras close to the simple ones. One of these stringy superalgebras is a Kac-Moody superalgebra G(A) with a nonsymmetrizable Cartan matrix A. Unlike the Kac-Moody superalgebras of polynomial growth with symmetrizable Cartan matrix, it can not be interpreted as a central extension of a twisted loop algebra.The stringy superalgebras are often referred to as superconformal ones. We discuss how superconformal stringy superalgebras really are. 
  The five simple exceptional complex Lie superalgbras of vector fields are described. One of them is new; the other four are explicitely described for the first time. All of the exceptional Lie superalgebras are obtained with the help of the Cartan prolongation or a generalized prolongation.   The description of several of the exceptional Lie superalgebras is associated with the Lie superalgebra AS - the nontrivial central extension of the supertraceless subalgebra SPE(4) of the periplectic Lie superalgebra PE(4) that preserves the nondegenerate odd bilinear form on the (4|4)-dimensional superspace. (A nontrivial central extension of SPE(n) only exists for n=4.) 
  Dynkin's classification of maximal subalgebras of simple finite dimensional complex Lie algebras is generalized to Lie subsuperalgebras of the general linear Lie superalgebras. 
  We analyze a new MacDowell-Mansouri $R^2$-type supergravity action based on the superalgebra Osp(1|8). This contribution summarizes the work of hep-th/9702052. 
  We construct parity and time reversal invariant Maxwell-Chern-Simons gauge theory coupled to fermions with adding the parity partner to the matter and the gauge fields, which can give nontopological vortex solutions depending on the sign of the Chern-Simons coupling constant. 
  The time-dependence of correlation functions under the influence of classical equations of motion is described by an exact evolution equation. For conservative systems thermodynamic equilibrium is a fixed point of these equations. We show that this fixed point is not universally stable, since infinitely many conserved correlation functions obstruct the approach to equilibrium. Equilibrium can therefore be reached at most for suitably averaged quantities or for subsystems, similar to quantum statistics. The classical time evolution of correlation functions shows many dynamical features of quantum mechanics. 
  We investigate the correlators in unitary minimal conformal models coupled to two-dimensional gravity from the two-matrix model. We show that simple fusion rules for all of the scaling operators exist. We demonstrate the role played by the boundary operators and discuss its connection to how loops touch each other. 
  We find in our quaternionic version of the electroweak theory an apparently hopeless problem: In going from complex to quaternions, the calculation of the real-valued parameters of the CKM matrix drastically changes. We aim to explain this quaternionic puzzle. 
  A detailed analysis is performed for the Nambu--Jona-Lasinio model coupled with constant (external) magnetic and/or electric fields in two, three, and four dimensions. The infrared cut-off is essential for a well-defined functional determinant by means of the proper time method. Contrary to the previous observation, the critical coupling remains nonzero even in three dimensions. It is also found that the asymptotic expansion has an excellent matching with the exact value. 
  A nonperturbative approach to the vacuum polarization for quantized fermions in external vector potentials is discussed. It is shown that by a suitable choice of counterterms the vacuum polarization phase is both gauge and renormalization independent, within a large class of nonperturbative renormalizations. 
  The maximal acceleration corrections to the Lamb shift of one--electron atoms are calculated starting from the Dirac equation and splitting the spinor into large and small components. The results depend on $Z^8$ and a cut-off $\Lambda$. Sizeable values are obtained even at Z=1 for $\Lambda\sim a_0/2$, where $a_0$ is the Bohr radius. These values are compatible with theoretical and experimental results. 
  We calculate the first order maximal acceleration corrections to the classical electrodynamics of a particle in external electromagnetic fields. These include additional dissipation terms, the presence of a critical electric field, a correction to the cyclotron frequency of an electron in a constant magnetic field and the power radiated by the particle. The electric effects are sizeble at the fields that are considered attainable with ultrashort $TW$ laser pulses on plasmas. 
  We compute the exact effective string vacuum backgrounds of the level $k = 81/19 SU(2,1)/U(1)$ coset model. A compact SU(2) isometry present in this seven dimensional solution allows to interpreting it after compactification as a four dimensional non-abelian SU(2) charged instanton with a singular submanifold and an $SO(3) \times U(1)$ isometry. The semiclassical backgrounds, solutions of the type II strings, present similar characteristics. 
  Dual super Dp-brane actions are constructed by carrying out a duality transformation of the world-volume U(1) gauge field. The resulting world-volume actions, which contain a (p - 2)-form gauge field, are shown to have the expected properties. Specifically, the D1-brane and D3-brane transform in ways that can be understood on the basis of the SL(2, Z) duality of type IIB superstring theory. Also, the D2-brane and the D4-brane transform in ways that are expected on the basis of the relationship between type IIA superstring theory and 11d M theory. For example, the dual D4-brane action is shown to coincide with the double-dimensional reduction of the recently constructed M5-brane action. The implications for gauge-fixed D-brane actions are discussed briefly. 
  We present the newly improved Batalin-Fradkin-Tyutin (BFT) Hamiltonian formalism and the generalization to the Lagrangian formulation, which provide the much more simple and transparent insight to the usual BFT method, with application to the non-Abelian Proca model which has been an difficult problem in the usual BFT method. The infinite terms of the effectively first class constraints can be made to be the regular power series forms by ingenious choice of $X_{\alpha \beta}$ and $\omega^{\alpha \beta}$-matrices. In this new method, the first class Hamiltonian, which also needs infinite correction terms is obtained simply by replacing the original variables in the original Hamiltonian with the BFT physical variables. Remarkably all the infinite correction terms can be expressed in the compact exponential form. We also show that in our model the Poisson brackets of the BFT physical variables in the extended phase space are the same structure as the Dirac brackets of the original phase space variables. With the help of both our newly developed Lagrangian formulation and Hamilton's equations of motion, we obtain the desired classical Lagrangian corresponding to the first class Hamiltonian which can be reduced to the generalized St\"uckelberg Lagrangian which is non-trivial conjecture in our infinitely many terms involved in Hamiltonian and Lagrangian. 
  We propose a generalization of the stochastic gauge fixing procedure for the stochastic quantization of gauge theories where not only the drift term of the stochastic process is changed but also the Wiener process itself. All gauge invariant expectation values remain unchanged. As an explicit example we study the case of an abelian gauge field coupled with three bosonic matter fields in 0+1 dimensions. We nonperturbatively prove quivalence with the path integral formalism. 
  We demonstrate the emergence of the U-duality group in compactification of Matrix theory on a 4-torus. The discussion involves non-trivial effects in strongly coupled 4+1 dimensional gauge theory, and highlights some interesting phenomena in the Matrix theory description of compactified M-theory. 
  Finite quantum field theories may be constructed from the most general renormalizable quantum field theory by forbidding, order by order in the perturbative loop expansion, all ultraviolet-divergent renormalizations of the physical parameters of the theory. The relevant finiteness conditions resulting from this requirement relate all dimensionless couplings in the theory. At first sight, Yukawa couplings which are equivalent to the generators of some Clifford algebra with identity element represent a very promising type of solutions of the condition for one-loop finiteness of the Yukawa couplings. However, under few reasonable and simplifying assumptions about their particular structure, these Clifford-like Yukawa couplings prove to be in conflict with the requirements of one- and two-loop finiteness of the gauge coupling and of the absence of gauge anomalies, at least for all simple gauge groups up to and including rank 8. 
  In a finite zone KdV context we show relations between the duality variables of Faraggi-Matone and those involved in Seiberg-Witten type duality. 
  I present the ``Chern--Simons'' formulation of generalized 2d dilaton gravity, summarize its Hamiltonian quantization (reduced phase space and Dirac quantization) and briefly discuss the statistical mechanical entropy of 2d black holes. Focus is put on the close relation to finite dimensional point particle systems. 
  To produce an isomorphism between the light-cone and equal-time representations some additional formalism beyond that originally proposed for the light-cone representation may sometimes be required. The additional formalism usually involves zero modes and is most likely to affect delicate, high energy aspects of the solution such as condensates. In this talk I will review some of the information which has been obtained in the past few years on these issues with particular emphasis on the Schwinger model as an example. 
  A method of constructing a canonical gauge invariant quantum formulation for a non-gauge classical theory depending on a set of parameters is advanced and then applied to the theory of closed bosonic string interacting with massive background fields. It is shown that within the proposed formulation the correct linear equations of motion for background fields arise. 
  By investigating the critical behavior appearing at the extremal limit of the non-dilatonic, black p-branes in (d+p) dimensions, we find that some critical exponents related to the critical point obey the scaling laws. From the scaling laws we obtain that the effective spatial dimension of the non-dilatonic black holes and black strings is one, and is p for the non-dilatonic black p-branes. For the dilatonic black holes and black p-branes, the effective dimension will depend on the parameters in theories. Thus, we give an interpretation why the Bekenstein-Hawking entropy may be given a simple world volume interpretation only for the non-dilatonic black p-branes. 
  W_k structure underlying the tensverse realization of SU(2) at level k is analyzed. Extension of the equivalence existing between covariant and light-cone gauge realization of affine Kac-Moody algebra to W_k algebras is given. Higher spin generators related to parafermions are extracted from the operator product algebra of the generators and are showed to be written in terms of only one free boson compactified on a circle. 
  Infinite-dimensional algebras of hidden symmetries of the self-dual Yang-Mills equations are considered. A current-type algebra of symmetries and an affine extension of conformal symmetries introduced recently are discussed using the twistor picture. It is shown that the extended conformal symmetries of the self-dual Yang-Mills equations have a simple description in terms of Ward's twistor construction. 
  We extend the ''modular localization'' principle from free to interacting theories and test its power for the special class of d=1+1 factorizing models. 
  We investigate 4-dim gauge theories and gravitational theories with nonpolynomial actions containing an infinite series in covariant derivatives of the fields representing the expansion of a transcendental entire function. A class of entire functions is explicitly constructed such that: (i) the theory is perturbatively superrenormalizable; (ii) no (gauge-invariant) unphysical poles are introduced in the propagators. The nonpolynomial nature is essential; it is not possible to simultaneously satisfy (i) and (ii) with any polynomial series in derivatives. Cutting equations are derived verifying the absence of unphysical cuts and the Bogoliubov causality condition within the loop expansion. A generalized KL representation for the 2-point function is obtained exhibiting the consistency of physical positivity with the improved convergence of the propagators. Some physical effects, such as extended bound excitations in the spectrum, are briefly discussed. 
  A systematic consideration of the problem of the reduction and extension of the structure group of a principal bundle is made and a variety of techniques in each case are explored and related to one another. We apply these to the study of the Dixmier-Douady class in various contexts including string structures, U-res bundles and other examples motivated by considerations from quantum field theory. 
  We consider an analogue of the Aharonov-Bohm effect in quantum field theory: the fermionic vacuum attains nontrivial quantum numbers in the background of a magnetic vortex even in the case when the spatial region of nonvanishing external field strength is excluded. The dependence of the vacuum quantum numbers on the value of the vortex flux and the choice of the condition on the boundary of the excluded region is determined. 
  We shortly illustrate how the field-theoretic approach to critical phenomena takes place in the more complete Wilson theory of renormalization and qualitatively discuss its domain of validity. By the way, we suggest that the differential renormalization functions (like the beta-function) of the perturbative scalar theory in four dimensions should be Borel summable provided they are calculated within a minimal subtraction scheme. 
  A key problem in the attempt to quantize the gravitational field is the choice of boundary conditions. These are mixed, in that spatial and normal components of metric perturbations obey different sets of boundary conditions. In the covariant quantization scheme this leads to a boundary operator involving both normal and tangential derivatives of metric perturbations. On studying the corresponding heat-kernel asymptotics, one finds that universal, tensorial, nonpolynomial structures contribute through the integrals over the boundary of linear combinations of all geometric invariants of the problem. These universal functions are independent of conformal rescalings of the background metric, and they might lead to a deep revolution in the current understanding of quantum gravity. 
  The existence of run-away solutions in classical and non-relativistic quantum electrodynamics is reviewed. It is shown that the less singular high energy behavior of relativistic spin 1/2 quantum electrodynamics precludes an analogous behavior in that theory. However, a Landau-like anomalous pole in the photon propagation function or in the electron-massive photon foward scattering amplitude would generate a new run-away, characterized by an energy scale omega ~ m_e exp (1/alpha). This contrasts with the energy scale omega ~ (m_e/alpha) associated with the classical and non-relativistic quantum run-aways. 
  The K\"ahler formulation of 5-dimensional Einstein-Kalb-Ramond (EKR) theory admitting two commuting Killing vectors is presented. Three different Kramer-Neugebauer-like maps are established for the 2-dimensional case. A class of solutions constructed on the double Ernst one is obtained. It is shown that the double Kerr solution corresponds to a EKR dipole configuration with horizon. 
  The statistical mechanics of black holes arbitrarily far from extremality is modeled by a gas of weakly interacting strings. As an effective low energy description of black holes the model provides a number of highly non-trivial consistency checks and predictions. Speculations on a fundamental origin of the model suggest surprising simplifications in non-perturbative string theory, even in the absence of supersymmetry. 
  We use brane configurations and SL(2,Z) symmetry of the type IIB string to construct mirror N=2 supersymmetric gauge theories in three dimensions. The mirror map exchanges Higgs and Coulomb branches, Fayet-Iliopoulos and mass parameters and U(1)R symmetries. Some quantities that are determined at the quantum level in one theory are determined at the classical level of the mirror. One such example is the complex structure of the Coulomb branch of one theory which is determined quantum mechanically. It is mapped to the complex structure of the Higgs branch of the mirror theory, which is determined classically. We study the generation of N=2 superpotentials by open D-string instantons in the brane configurations. 
  These lectures are devoted to introducing some of the basic features of quantum geometry that have been emerging from compactified string theory over the last couple of years. The developments discussed include new geometric features of string theory which occur even at the classical level as well as those which require non-perturbative effects. These lecture notes are based on an evolving set of lectures presented at a number of schools but most closely follow a series of seven lectures given at the TASI-96 summer school on Strings, Fields and Duality. 
  A metric is introduced on the space of parameters (couplings) describing the large N limit of the O(N) model in Euclidean space. The geometry associated with this metric is analysed in the particular case of the infinite volume limit in 3 dimensions and it is shown that the Ricci curvature diverges at the ultra-violet (Gaussian) fixed point but is finite and tends to constant negative curvature at the infra-red (Wilson-Fisher) fixed point. The renormalisation group flow is examined in terms of geodesics of the metric. The critical line of cross-over from the Wilson-Fisher fixed point to the Gaussian fixed point is shown to be a geodesic but all other renormalisation group trajectories, which are repulsed from the Gaussian fixed point in the ultra-violet, are not geodesics. The geodesic flow is interpreted in terms of a maximisation principle for the relative entropy. 
  We construct integrable modifications of 2d lattice gauge theories with finite gauge groups. 
  Pair creation of electrically charged black holes and its dual process, pair creation of magnetically charged black holes, are considered. It is shown that the creation rates are equal provided the boundary conditions for the two processes are dual to one another. This conclusion follows from a careful analysis of boundary terms and boundary conditions for the Maxwell action. 
  We study the critical behavior in the black p-branes and four dimensional charged dilaton black holes. We calculate the thermodynamic fluctuations in the various (microcanonical, canonical, and grandcanonical) ensembles. It is found that the extremal limit of some black configurations has a critical point and a phase transition takes place from the extremal to nonextremal black configurations. Some critical exponents are obtained, which satisfy the scaling laws. This is related to the fact that the entropy of these black configurations is a homogeneous function. 
  In order to prevent ``unavoidable'' break-down of the ``peaceful coexistence'' between foundations of quantum theory and relativity I propose a new type of a quantum gauge theory (superrelativity). This differs from ordinary gauge theories in the sense that the affine connection of this theory is constructed from first derivatives of the Fubini-Study metric tensor in the projective Hilbert space of the pure quantum states CP(N). That is we have not merely analogy with general relativity but this construction should presumably provide a unification of general relativity and quantum theory. 
  The construction of new hyper-Kaehler manifolds by taking the infinite monopole mass limit of certain Bogomol'nyi-Prasad-Sommerfield monopole moduli spaces is considered. The one-parameter family of hyperkaehler manifolds due to Dancer is shown to be an example of such manifolds. A new family of fixed monopole spaces is constructed. They are the moduli spaces of four SU(4) monopoles, in the infinite mass limit of two of the monopoles. These manifolds are shown to be nonsingular when the fixed monopole positions are distinct. 
  The infinitesimal unitary transformation, introduced recently by F.Wegner, to bring the Hamiltonian to diagonal (or band diagonal) form, is applied to the Hamiltonian theory as an exact renormalization scheme. We consider QED on the light front to illustrate the method. The low-energy generated interaction, induced in the renormalized Hamiltonian to the order \alpha, is shown to be negative to insure together with instantaneous term and perturbative photon exchange the bound states for positronium. It is possible to perform the complete complete elimination of the ee\gamma-vertex in the instant form frame; this gives rise to the cutoff independent e\bar{e}-interaction governed by generated and instantaneous terms. The well known result for the singlet-triplet splitting $7/6 \alpha^2 Ryd$ is recovered in the nonrelativistic limit as long as $\la << m$.   We examine the mass and wave function renormalization. The ultraviolet divergencies, associated with a large transverse momentum, are regularized by the regulator arising from the unitary transformation. The severe infrared divergencies are removed if all diagrams to the second order, arising from flow equations method and normal-ordering Hamiltonian, are taken into account. The electron (photon) mass in the renormalized Hamiltonian vary with UV cutoff in accordance with 1-loop renormalization group equations.This indicates to an intimete connection between Wilson's renormalization and the flow equation method.   The advantages of the method in comparison with the naive renormalisation group approach are discussed. 
  We give an overview of various composite BPS configurations of string theory and M-theory p-branes represented as classical supergravity solutions. Type II string backgrounds can be obtained by S- and T- dualities from NS-NS string - 5-brane configurations corresponding to exact conformal sigma models. The single-center solutions can be also generated from the Schwarzschild solution by applying a sequence of boosts, dualities and taking the extremal limit. Basic `marginal' backgrounds representing threshold BPS bound states of branes are parametrised by a number of independent harmonic functions. `Non-marginal' BPS configurations in D=10 can be constructed from marginal ones by using U-duality and thus depend also on a number of O(d,d) and SL(2,R) parameters. We present a new more general class of configurations in which some of the branes or their intersections are localised on other branes. In particular, we find the type IIB supergravity background describing the BPS configuration of a 3-brane, RR 5-brane and NS-NS 5-brane, and related `localised' 2-5-5 D=11 solution. We also consider the classical action for a 3-brane probe moving in such backgrounds and determine the structure of the corresponding moduli space metrics. 
  We give a model-independent derivation of general intersecting rules for non-extreme $p$-branes in arbitrary dimensions $D$. This is achieved by directly solving bosonic field equations for supergravity coupled to a dilaton and antisymmetric tensor fields with minimal ans\"{a}tze. We compare the results with those in eleven-dimensional supergravity. Supersymmetry is recovered in the extreme limit if the backgrounds are taken to be independent. Consistency with non-supersymmetric solutions is also discussed. Finally the general formulae for the ADM mass, entropy and Hawking temperature are given. 
  We show how an F-theory compactified on a Calabi-Yau (n+1)-fold in appropriate weak coupling limit reduces formally to an orientifold of type IIB theory compactified on an auxiliary complex n-fold. In some cases (but not always) if the original (n+1)-fold is singular, then the auxiliary n-fold is also singular. We illustrate this by analysing F-theory on elliptically fibered Calabi-Yau 3-folds on base $F_n$. 
  It is shown that all the (p,q) dyon bound states exist and are unique in N=4 and N=2 with four massless flavours supersymmetric SU(2) Yang-Mills theories, where p and q are any relatively prime integers. The proof can be understood in the context of field theory alone, and does not rely on any duality assumption. We also give a general physical argument showing that these theories should have at least an exact Gamma(2) duality symmetry, and then deduce in particular the existence of the (2p,2q) vector multiplets in the Nf=4 theory. The corresponding massive theories are studied in parallel, and it is shown that though in these cases the spectrum is no longer self-dual at a given point on the moduli space, it is still in perfect agreement with an exact S duality. We also discuss the interplay between our results and both the semiclassical quantization and the heterotic-type II string-string duality conjecture. 
  In this lecture I review recent results on the use of Solvable Lie Algebras as an efficient description of the scalar field sector of supergravities in relation with their non perturbative structure encoded in the U-duality group. I also review recent results on the construction of BPS saturated states as solution of the first differential equations following from imposing preservation of a fraction of the original supersymmetries. In particular I discuss N=2 extremal black holes that are approximated by a Bertotti Robinson metric near their horizon. The extension of this construction to maximally extended supergravities in all dimensions from 4 to 11 is work in progress where the use of the Solvable Lie algebra approach promises to be of decisive usefulness. 
  We present subleading corrections to the N=2 supersymmetric black hole entropy. These subleading contributions correspond to instanton corrections of the Type II string theory. In particular we consider an axion free black hole solution of low-energy effective Type II string theory. We present a procedure to include successively all instanton corrections. Expanding these corrections at particular points in moduli space yields polynomial and logarithmic instanton corrections to the classical black hole entropy. We comment on a microscopic interpretation of these instanton corrections and find that the logarithmic corrections correspond to subleading terms in the degeneracy of the spectrum of an underlying quantum theory. 
  Some exact solutions to the classical matrix model equations that arise in the context of M(embrane) theory are given, and their topological nature is identified. 
  The exact factorisable quantum S-matrices are known for simply laced as well as non-simply laced affine Toda field theories. Non-simply laced theories are obtained from the affine Toda theories based on simply laced algebras by folding the corresponding Dynkin diagrams. The same process, called classical `reduction', provides solutions of a non-simply laced theory from the classical solutions with special symmetries of the parent simply laced theory. In the present note we shall elevate the idea of folding and classical reduction to the quantum level. To support our views we have made some interesting observations for S-matrices of non-simply laced theories and give prescription for obtaining them through the folding of simply laced ones. 
  We compare the conventional description of the interaction of matter with the four known forces in the standard model with an alternative Weyl description in which the chiral coupling is extended to include gravity. The two are indistinguishable at the low energy classical level of equations of motion, but there are subtle differences at the quantum level when nonvanishing torsion and the Adler-Bell-Jackiw anomaly is taken into account. The spin current and energy-momentum of the chiral theory then contain non-Hermitian terms which are not present in the conventional theory. In the chiral alternative, CPT invariance is not automatic because chirality supersedes Hermiticity but full Lorentz invariance holds. New fermion loop processes associated with the theory are discussed together with a perturbative regularization which explicitly maintains the chiral nature and local symmetries of the theory. 
  We investigate the quantization of two-dimensional version of the generalized Chern-Simons actions which were proposed previously. The models turn out to be infinitely reducible and thus we need infinite number of ghosts, antighosts and the corresponding antifields. The quantized minimal actions which satisfy the master equation of Batalin and Vilkovisky have the same Chern-Simons form. The infinite fields and antifields are successfully controlled by the unified treatment of generalized fields with quaternion algebra. This is a universal feature of generalized Chern-Simons theory and thus the quantization procedure can be naturally extended to arbitrary even dimensions. 
  We construct T-duality on K3 surfaces. The T-duality exchanges a 4-brane R-R charge and a 0-brane R-R charge. We study the action of the T-duality on the moduli space of 0-branes located at points of K3 and 4-branes wrapping it. We apply the construction to F-theory compactified on a Calabi-Yau 4-fold and study the duality of N=2 SU(N_c) gauge theories in four dimensions. We discuss the generalization to the N=1 duality scenario. 
  We argue that supersymmetric gluodynamics (theory of gluons and gluinos) has a condensate-free phase. Unlike the standard phase, the discrete axial symmetry of the Lagrangian is unbroken in this phase, and the gluino condensate does not develop. Extra unconventional vacua are supersymmetric and are characterized by the presence of (bosonic and fermionic) massless bound states. A set of arguments in favor of the conjecture includes: (i) analysis of the effective Lagrangian of the Veneziano-Yankielowicz type which we amend to properly incorporate all symmetries of the model; (ii) consideration of an unsolved problem with the Witten index; (iii) interpretation of a mismatch between the strong-coupling and weak coupling instanton calculations of the gluino condensate detected previously. Impact on Seiberg's results is briefly discussed. 
  It was found in the two-dimensional quantum gravity both in the de Donder gauge and in the lightcone gauge that one of field equations breaks down at the level of the representation, though the breakdown is very little. It is shown that this anomalous behavior occurs also in a very simple model analogous to the lightcone-gauge two-dimensional quantum gravity. This model, however, can be transformed into a free field theory by a nonsingular transformation. Of course, the latter has no anomaly. The cause of the discrepancy is analyzed. 
  A general construction of an sh Lie algebra from a homological resolution of a Lie algebra is given. It is applied to the space of local functionals equipped with a Poisson bracket, induced by a bracket for local functions along the lines suggested by Gel'fand, Dickey and Dorfman. In this way, higher order maps are constructed which combine to form an sh Lie algebra on the graded differential algebra of horizontal forms. The same construction applies for graded brackets in field theory such as the Batalin-Fradkin-Vilkovisky bracket of the Hamiltonian BRST theory or the Batalin-Vilkovisky antibracket. 
  One more paradox of classical Lorentz-Dirac preaccelerative solution is found: the formation of the event horizon. 
  Let M be a compact Riemannian manifold with smooth boundary. We study the vacuum expectation value of an operator Q by studying Tr Qe^{-tD}, where D is an operator of Laplace type on M, and where Q is a second order operator with scalar leading symbol; we impose Dirichlet or modified Neumann boundary conditions. 
  Starting from a generalization of a recent result on self-duality we systematically analyze self-dual models. We find a criterion to judge whether a given model is self-dual or not. With this tool we construct some new self-dual pairs, focussing on examples with exceptional gauge groups. 
  We discuss how N=1 dualities in four dimensions are geometrically realized by wrapping D-branes about 3-cycles of Calabi-Yau threefolds. In this setup the N=1 dualities for SU, SO and USp gauge groups with fundamental fields get mapped to statements about the monodromy and relations among 3-cycles of Calabi-Yau threefolds. The connection between the theory and its dual requires passing through configurations which are T-dual to the well-known phenomenon of decay of BPS states in N=2 field theories in four dimensions. We compare our approach to recent works based on configurations of D-branes in the presence of NS 5-branes and give simple classical geometric derivations of various exotic dynamics involving D-branes ending on NS-branes. 
  We propose a generalization of meanders, i.e., configurations of non-selfintersecting loops crossing a line through a given number of points, to SU(N). This uses the reformulation of meanders as pairs of reduced elements of the Temperley-Lieb algebra, a SU(2)-related quotient of the Hecke algebra, with a natural generalization to SU(N). We also derive explicit formulas for SU(N) meander determinants, defined as the Gram determinants of the corresponding bases of the Hecke algebra. 
  We seek to clarify some of the physical aspects of the Ruijsenaars-Schneider models. This important class of models was presented as a relativistic generalisation of the Calogero-Moser models but, as we shall argue, this description is misleading. It is far better to simply view the models as a one-parameter generalisation of Calogero-Moser models. By viewing the models as describing certain eigenvalue motions we can appreciate the generic nature of the models. 
  Defining the $WBC_n$ algebras as the commutant of certain screening charges a special form for the classical generators is obtained which does not change under quantisation. This enables us to give explicitly the first few generators in a compact form for arbitrary $WBC_n$ algebras. 
  We investigate duality properties of N-form fields, provide a symmetric way of coupling them to electric/magnetic sources, and check that these charges obey the appropriate quantization requirements. First, we contrast the D=4k case, in which duality is a well-defined SO(2) rotation generated by a Chern-Simons form leaving the action invariant, and D=4k+2 where the corresponding ostensibly SO(1,1) rotation is not only not an invariance but does not even have a generator. When charged sources are included we show explicitly in the Maxwell case how the usual Dirac quantization arises in a fully symmetric approach attaching strings to both types of charges. Finally, for D=4k+2 systems, we show how charges can be introduced for self-dual (2k)-forms, and obtain the D=4k models with sources by dimensional reduction, tracing their duality invariance to a partial invariance in the higher dimensions. 
  The origin of the string conformal anomaly is studied in detail. We use a reformulated string Lagrangian which allows to consider the string tension $T_{0}$ as a small perturbation. The expansion parameter is the worldsheet speed of light c, which is proportional to $T_{0}$ . We examine carefully the interplay between a null (tensionless) string and a tensionful string which includes orders $ c^{2} $ and higher. The conformal algebra generated by the constraints is considered. At the quantum level the normal ordering provides a central charge proportional to $ c^{2} $. Thus it is clear that quantum null strings respect conformal invariance and it is the string tension which generates the conformal anomaly. 
  It is known that the high-energy quark-quark scattering amplitude can be described by the expectation value of two lightlike Wilson lines, running along the classical trajectories of the two colliding particles. Generalizing the results of a previous paper, we give here the general proof that the expectation value of two infinite Wilson lines, forming a certain hyperbolic angle in Minkowski space-time, and the expectation value of two infinite Euclidean Wilson lines, forming a certain angle in Euclidean four-space, are connected by an analytic continuation in the angular variables. This result could be used to evaluate the high-energy scattering amplitude directly on the lattice. 
  We identify Type IIA and IIB strings, as excitations in the matrix model of M theory. 
  We discuss how concepts such as geodesic length and the volume of space-time can appear in 2d topological gravity. We then construct a detailed mapping between the reduced Hermitian matrix model and 2d topological gravity at genus zero. This leads to a complete solution of the counting problem for planar graphs with vertices of even coordination number. The connection between multi-critical matrix models and multi-critical topological gravity at genus zero is studied in some detail. 
  The parameter-space orbifold construction of open and unoriented toroidal and (target-space) orbifold compactifications is briefly reviewed, with emphasis on the underlying geometrical framework. A class of chiral four-dimensional type-I vacua with three generations is also discussed. 
  We propose an explicit expression for vacuum expectation values of the boundary field e^{ia\phi_{B}} in the boundary sine-Gordon model with zero bulk mass. This expression agrees with known exact results for the boundary free energy and with perturbative calculations. 
  We explain in a context different from that of Maraner the formalism for describing motion of a particle, under the influence of a confining potential, in a neighbourhood of an n-dimensional curved manifold M^n embedded in a p-dimensional Euclidean space R^p with p >= n+2. The effective Hamiltonian on M^n has a (generally non-Abelian) gauge structure determined by geometry of M^n. Such a gauge term is defined in terms of the vectors normal to M^n, and its connection is called the N-connection. In order to see the global effect of this type of connections, the case of M^1 embedded in R^3 is examined, where the relation of an integral of the gauge potential of the N-connection (i.e., the torsion) along a path in M^1 to the Berry's phase is given through Gauss mapping of the vector tangent to M^1. Through the same mapping in the case of M^1 embedded in R^p, where the normal and the tangent quantities are exchanged, the relation of the N-connection to the induced gauge potential on the (p-1)-dimensional sphere S^{p-1} (p >= 3) found by Ohnuki and Kitakado is concretely established. Further, this latter which has the monopole-like structure is also proved to be gauge-equivalent to the spin-connection of S^{p-1}. Finally, by extending formally the fundamental equations for M^n to infinite dimensional case, the present formalism is applied to the field theory that admits a soliton solution. The resultant expression is in some respects different from that of Gervais and Jevicki. 
  Talk presented at the 9th Max Born Symposion, Karpacz, September 1996 
  We consider a 1+1 dimensional field theory which contains both a complex fermion field and a real scalar field. We then construct a unitary operator that, by a similarity transformation, gives a continuum of equivalent theories which smoothly interpolate between the massive Thirring model and the sine-Gordon model. This provides an implementation of smooth bosonization proposed by Damgaard et al. as well as an example of a quantum canonical transformation for a quantum field theory. 
  This is an introduction to the basic ideas and to a few further selected topics in conformal quantum field theory and in the theory of Kac-Moody algebras. 
  We construct a vertex operator realization for the simple current primary fields of WZW theories which are based on simply laced affine Lie algebras g. This is achieved by employing an embedding of the integrable highest weight modules of g into the Fock space for a bosonic string compactified on the weight lattice of g. Our vertex operators are universal in the sense that a single expression for the vertex operator holds simultaneously for all positive integral values of the level of g. 
  The R-matrix of the U_q(d_4(3)) algebra is constructed in the 8-dimensional fundamental representation. Using this result an exact S-matrix is conjectured for the imaginary coupled g_2(1) affine Toda field theory, the structure of which is found to be very similar to the previously investigated S-matrix of d_4(3) Toda theory. It is shown that this S-matrix is consistent with the results for the case of real coupling using the breather-particle correspondence. For q a root of unity it is argued that the theory can be restricted to yield Phi(11|12) perturbations of WA_2 minimal models. 
  This paper investigates the algebraic structure that exists on perturbative BPS-states in the superstring, compactified on the product of a circle and a Calabi-Yau fourfold. This structure was defined in a recent article by Harvey and Moore. It shown that for a toroidal compactification this algebra is related to a generalized Kac-Moody algebra. The BPS-algebra itself is not a Lie-algebra. However, it turns out to be possible to construct a Lie-algebra with the same graded dimensions, in terms of a half-twisted model. The dimensions of these algebras are related to the elliptic genus of the transverse part of the string algebra. Finally, the construction is applied to an orbifold compactification of the superstring. 
  We discuss five-dimensional supersymmetric gauge theories. An anomaly renders some theories inconsistent and others consistent only upon including a Wess-Zumino type Chern-Simons term. We discuss some necessary conditions for existence of nontrivial renormalization group fixed points and find all possible gauge groups and matter content which satisfy them. In some cases, the existence of these fixed points can be inferred from string duality considerations. In other cases, they arise from M-theory on Calabi-Yau threefolds. We explore connections between aspects of the gauge theory and Calabi-Yau geometry. A consequence of our classification of field theories with nontrivial fixed points is a fairly complete classification of a class of singularities of Calabi-Yau threefolds which generalize the ``del Pezzo contractions'' and occur at higher codimension walls of the K\"{a}hler cone. 
  We study orbifolds of (0,2) models and their relation to (0,2) mirror symmetry. In the Landau-Ginzburg phase of a (0,2) model the superpotential features a whole bunch of discrete symmetries, which by quotient action lead to a variety of consistent (0,2) vacua. We study a few examples in very much detail. Furthermore, we comment on the application of (0,2) mirror symmetry to the calculation of Yukawa couplings in the space-time superpotential. 
  We review our recent work on the BPS magnetic monopoles and its relation to the electromagnetic duality in the N=4 supersymmetric Yang-Mills systems with an arbitrary gauge group. The gauge group can be maximally or partially broken. The low energy dynamics of the massive and massless magnetic monopoles are approximated by the moduli space metric. We emphasize the possible connection between the nature of the monopole moduli space with unbroken gauge group and the physics of mesons and baryons in QCD. 
  In this set of lectures I review recent developments in string theory emphasizing their non-perturbative aspects and their recently discovered duality symmetries. The goal of the lectures is to make the recent exciting developments in string theory accessible to those with no previous background in string theory who wish to join the research effort in this area. Topics covered include a brief review of string theory, its compactifications, solitons and D-branes, black hole entropy and web of string dualities. (Lectures presented at ICTP summer school, June 1996) 
  Generalized membrane solutions of D=11 supergravity, for which the transverse space is a toric hyper-K{\" a}hler manifold, are shown to have IIB duals representing the intersection of parallel 3-branes with 5-branes whose orientations are determined by their $Sl(2;\bZ)$ charge vectors. These IIB solutions, which generically preserve 3/16 of the supersymmetry, can be further mapped to solutions of D=11 supergravity representing the intersection of parallel membranes with any number of fivebranes at arbitrary angles. Alternatively, a subclass (corresponding to non-singular D=11 solutions) can be mapped to solutions representing the intersection on a string of any number of D-5-branes at arbitrary angles, again preserving 3/16 supersymmetry, as we verify in a special case by a quaternionic extension of the analysis of Berkooz, Douglas and Leigh. We also use similar methods to find new 1/8 supersymmetric solutions of orthogonally intersecting branes. 
  We discuss issues concerning M(atrix) theory compactifications on curved spaces. We argue from the form of the graviton propagator on curved space that excited string states do not decouple from the annulus D0-brane $v^4$ amplitude, unlike the flat space case. This argument shows that a large class of quantum mechanical systems with a finite number of degrees of freedom cannot reproduce supergravity answers. We discuss the specific example of an ALE space and suggest sources of possible higher derivative terms that might help reproduce supergravity results. 
  In an abelian gauge theory, the Coulombic potential between two static charges is obtained most directly when a correct separation between gauge-invariant and gauge degrees of freedom is made. This motivates a similar separation in the nonabelian theory. When a careful identification of the Hilbert space is made, along with the proper analyticity requirements, it is then possible to find the appropriate wavefunctionals describing heavy color sources in the theory. This treatment is consistent with, and realizes in a simple way the center symmetry Z_N of SU(N) gauge theories. 
  We found BPS-saturated solutions of M-theory and Type II string theory which correspond to (non-marginally) bound states of p-branes intersecting at angles different from pi/2. These solutions are obtained by starting with a BPS marginally bound (orthogonally) intersecting configurations of two p-branes (e.g, two four-branes of Type II string theory), performing a boost transformation at an angle with respect to the world-volume of the configuration, performing T-duality transformation along the boost-direction, S-duality transformation, and T- transformations along the direction perpendicular to the boost transformation. The resulting configuration is non-marginally bound BPS-saturated solution whose static metric possesses the off-diagonal term which cannot be removed by a coordinate transformation, and thus signifies an angle (different from pi/2) between the resulting intersecting p-branes. Additional new p-branes are bound to this configuration, in order to ensure the stability of such a static, tilted configuration. 
  The instability of "physical" preaccelerative solution of the Lorentz-Dirac equation is explicitly shown 
  In this paper we present the principal construction of the vertex operator representation for toroidal Lie algebras. 
  Recent results on duality between string theories and connectedness of their moduli spaces seem to go a long way toward establishing the uniqueness of an underlying theory. For the large class of Calabi-Yau 3-folds that can be embedded as hypersurfaces in toric varieties the proof of mathematical connectedness via singular limits is greatly simplified by using polytopes that are maximal with respect to certain single or multiple weight systems. We identify the multiple weight systems occurring in this approach. We show that all of the corresponding Calabi-Yau manifolds are connected among themselves and to the web of CICY's. This almost completes the proof of connectedness for toric Calabi-Yau hypersurfaces. 
  We discuss two types of deformations of a 2D black hole carrying an electric charge. Type I gives rise to a space-time configuration similar to the one described by McGuigan, Nappi and Yost. Whereas type II results in a space-time configuration which has a rather peculiar geometry. 
  The structure of diagonal singularities of Green functions of partial differential operators of even order acting on smooth sections of a vector bundle over a Riemannian man ifold is studied. A special class of operators formed by the products of second-order operators of Laplace type defined with the help of a unique Riemannian metric and a unique bundle connection but with different potential terms is investigated. Explicit simple formulas for singularities of Green functions of such operators in terms of the usual heat kernel coefficients are obtained. 
  The time development of equal-time correlation functions in quantum mechanics and quantum field theory is described by an exact evolution equation for generating functionals. This permits a comparison between classical and quantum evolution in non-equilibrium systems. 
  We compare the N=1 F-theory compactification of Donagi, Grassi and Witten with modular superpotential - and some closely related models - to dual heterotic models. We read of the F-theory spectrum from the cohomology of the fourfold and discuss on the heterotic side the gauge bundle moduli sector (including the spectral surface) and the necessary fivebranes. Then we consider the N=1 superpotential and show how a heterotic superpotential matching the F-theory computation is built up by worldsheet instantons. Finally we discuss how the original modular superpotential should be corrected by an additional modular correction factor, which on the F-theory side matches nicely with a `curve counting function' for the del Pezzo surface. On the heterotic side we derive the same factor demanding correct T-duality transformation properties of the superpotential. 
  The paper discusses the connection between S-duality and string-theoretic picture-changing formalism. The singular limit of the picture-changing transformation at zero momentum leads to the presence of the topological 5-form charge in the type IIB superalgebra which is attributed to the M5-brane. The topological charge defines the boundary D-brane state, which also is the analogue of the monopole part of the Olive-Witten's result in field theory.The correlation functions involving this non-perturbative state are computed.The non-perturbative D-brane states are associated with the ghost number cohomologies, introduced in the paper. Connections with M(atrix) theory are discussed. 
  The SL(2,R) invaraint ten dimensional type IIB superstring effective action is compactified on a torus to D spacetime dimensions. The transformation properties of scalar, vector and tensor fields, appearing after the dimensional reduction, are obtained in order to maintain the SL(2,R)} invariance of the reduced effective action. The symmetry of the action enables one to generate new string vacua from known configurations. As illustrative examples, new black hole solutions are obtained in five and four dimensions from a given set of solutions of the equations of motion. 
  The local free field theory for Regge trajectory is described in the framework of the BRST - quantization method. The corresponding BRST - charge is constructed with the help of the method of dimensional reduction. 
  I review recent work on the infrared structure of (2+1)-dimensional Abelian gauge theories and their application to condensed matter physics. In particular, within a large-N Schwinger-Dyson treatment, and including an `infrared momentum cut-off', I demonstrate the existence of a non-trivial infrared fixed point of the renormalization group. I connect this property to non-fermi liquid low-energy behaviour, and I attempt to draw some conclusions about the possible application of this approach to an understanding of the normal and superconducting phases of planar high-temperature superconducting cuprates. 
  We make a proposal for a bosonic field theory in twelve dimensions that admits the bosonic sector of eleven-dimensional supergravity as a consistent truncation. It can also be consistently truncated to a ten-dimensional Lagrangian that contains all the BPS p-brane solitons of the type IIB theory. The mechanism allowing the consistent truncation in the latter case is unusual, in that additional fields with an off-diagonal kinetic term are non-vanishing and yet do not contribute to the dynamics of the ten-dimensional theory. They do, however, influence the oxidation of solutions back to twelve dimensions. We present a discussion of the oxidations of all the basic BPS solitons of M-theory and the type IIB string to D=12. In particular, the NS-NS and R-R strings of the type IIB theory arise as the wrappings of membranes in D=12 around one or other circle of the compactifying 2-torus. 
  For spontaneous breaking of global or gauge symmetry, it is superfluous to assume that the vacuum expectation value of the scalar field manifesting the symmetry is nonvanishing. The vacuum with spontaneous symmetry breaking simply corresponds to nonzero number of particles of one or more components of the real scalar field. 
  We study a class of lattice field theories in two dimensions that includes gauge theories. Given a two dimensional orientable surface of genus $g$, the partition function $Z$ is defined for a triangulation consisting of $n$ triangles of area $\epsilon$. The reason these models are called quasi-topological is that $Z$ depends on $g$, $n$ and $\epsilon$ but not on the details of the triangulation. They are also soluble in the sense that the computation of their partition functions can be reduced to a soluble one dimensional problem. We show that the continuum limit is well defined if the model approaches a topological field theory in the zero area limit, i.e., $\epsilon \to 0$ with finite $n$. We also show that the universality classes of such quasi-topological lattice field theories can be easily classified. Yang-Mills and generalized Yang-Mills theories appear as particular examples of such continuum limits. 
  Different finite difference replacements for the derivative are analyzed in the context of the Heisenberg commutation relation. The type of the finite difference operator is shown to be tied to whether one can naturally consider $P$ and $X$ to be self-adjoint and skew self-adjoint or whether they have to be viewed as creation and annihilation operators. The first class, generalizing the central difference scheme, is shown to give unitary equivalent representations. For the second case we construct a large class of examples, generalizing previously known difference operator realizations of $[P,X]=Id$. 
  In this paper we study duality properties of the M(atrix) theory compactified on a circle. We establish the equivalence of this theory to the strong coupling limit of type IIB string theory compactified on a circle. In the M(atrix) theory context, our major evidence for this duality consists of identifying the BPS states of IIB strings in the spectrum and finding the remnant symmetry of SL(2,Z) and the associated tau moduli. By this IIB/M duality, a number of insights are gained into the physics of longitudinal membranes in the infinite momentum frame. We also point out an accidental affine Lie symmetry in the theory. 
  The Stueckelberg model for massive vector fields is cast into a BRS invariant, polynomial form. Its symmetry algebra simplifies to an abelian gauge symmetry which is sufficient to decouple the negative norm states. The propagators fall off like $1/k^2$ and the Lagrangean is polynomial but it is not powercounting renormalizable due to derivative couplings. 
  Starting from the von Neumann equation, we construct the quantum evolution equation for the effective action for systems in mixed states. This allows us to find the hierarchy of equations which describe the time evolution of equal time correlators. The method is applied explicitly to a scalar theory with quartic self-interaction. 
  The characterization of the Nambu-Poisson n-tensors as a subfamily of the Generalized-Poisson ones recently introduced (and here extended to the odd order case) is discussed. The homology and cohomology complexes of both structures are compared, and some physical considerations are made. 
  We propose a systematic method of dealing with the canonical constrained structure of reducible systems in the Dirac and symplectic approaches which involves an enlargement of phase and configuration spaces, respectively. It is not necessary, as in the Dirac approach, to isolate the independent subset of constraints or to introduce, as in the symplectic analysis, a series of lagrange multipliers-for-lagrange multipiers. This analysis illuminates the close connection between the Dirac and symplectic approaches of treating reducible theories, which is otherwise lacking. The example of p-form gauge fields (p=2,3) is analyzed in details. 
  The realizations of the exceptional non-linear (quadratically generated, or W-type) N=8 and N=7 superconformal algebras with Spin(7) and G_2 affine symmetry currents are reviewed. Both the N=8 and N=7 algebras admit unitary realizations in terms of a single boson and free fermions in 8 of Spin(7) and 7 of G_2, with the central charges c=26/5 and c=5, respectively. They also have realizations over the coset spaces SO(8)XU(1)/SO(7) and SO(7)X U(1)/G_2 for some fixed values of their central charges. The coset space SO(8)/SO(7) is the seven-sphere , whereas the space SO(7)/G_2 represents the seven-sphere with torsion. We conclude with a discussion of a novel 'hybrid' method developed recently that yields unitary realizations of the exceptional N=8 and N=7 algebras for all allowed values of their central charges. 
  The exact operator solution for quantum Liouville theory constructed for the generic quantum deformation parameter $q$ is extended to the case with $q$ being a root of unity. The screening charge operator becomes nilpotent in such cases and arbitrary Liouville exponentials can be obtained in finite polynomials of the screening charge. 
  In this paper we study isotropic integrable systems based on the braid-monoid algebra. These systems constitute a large family of rational multistate vertex models and are realized in terms of the B_n, C_n and D_n Lie algebra and by the superalgebra Osp(n|2m). We present a unified formulation of the quantum inverse scattering method for many of these lattice models. The appropriate fundamental commutation rules are found, allowing us to construct the eigenvectors and the eigenvalues of the transfer matrix associated to the B_n, C_n, D_n, Osp(2n-1|2), Osp(2|2n-2), Osp(2n-2|2) and Osp(1|2n) models. The corresponding Bethe Ansatz equations can be formulated in terms of the root structure of the underlying algebra. 
  This paper addresses an issue essential to the study of hidden supersymmetries (meaning here ones that do not close on the Hamiltonian) for one-dimensional non-linear supersymmetric sigma models. The issue relates to ambiguities, due to partial integrations in superspace, both in the actual definition of these supersymmetries and in the Noether definition of the associated supercharges. The unique consistent forms of both these definitions have to be determined simultaneously by a process that adjusts the former definitions so that the associated supercharges do indeed correctly generate them with the aid of the canonical formalism. The paper explains and illustrates these matters and gives some new results. 
  The new class of integrable mappings and chains is introduced. Corresponding (1+2) integrable systems invariant with respect to such discrete transformations are represented in explicit form. Soliton like solutions of them are represented in terms of matrix elements of fundamental representations of semisimple A_n algebras for a given group element. 
  We give further support to Smirnov's conjecture on the exact kink S-matrix for the massive Quantum Field Theory describing the integrable perturbation of the c=0.7 minimal Conformal Field theory (known to describe the tri-critical Ising model) by the operator $\phi_{2,1}$. This operator has conformal dimensions $(7/16,7/16)$ and is identified with the subleading magnetic operator of the tri-critical Ising model. In this paper we apply the Thermodynamic Bethe Ansatz (TBA) approach to the kink scattering theory by explicitly utilising its relationship with the solvable lattice hard hexagon model. Analytically examining the ultraviolet scaling limit we recover the expected central charge c=0.7 of the tri-critical Ising model. We also compare numerical values for the ground state energy of the finite size system obtained from the TBA equations with the results obtained by the Truncated Conformal Space Approach and Conformal Perturbation Theory. 
  We show that if for fixed negative (physical) square of the momentum transfer t, the differential cross-section ${d\sigma\over dt}$ tends to zero and if the total cross-section tends to infinity, when the energy goes to infinity, the real part of the even signature amplitude cannot have a constant sign near t = 0. 
  We study correlation functions of coset constructions by utilizing the method of gauge dressing. As an example we apply this method to the minimal models and to the Witten 2D black hole. We exhibit a striking similarity between the latter and the gravitational dressing. In particular, we look for logarithmic operators in the 2D black hole. 
  We give some simple examples of mirror Calabi-Yau fourfolds in Type II string theory. These are realised as toroidal orbifolds. Motivated by the Strominger, Yau, Zaslow argument we give explicitly the mirror transformation which maps Type IIA/IIB on such a fourfold to Type IIA/IIB on the mirror. The mirrors are related by the inclusion/exclusion of discrete torsion. Implicit in the result is a confirmation of mirror symmetry to genus $g$ in the string path integral. Finally, by considering the relationship between M-theory and Type IIA theory, we show how in M-theory on mirror Calabi-Yau fourfolds, mirror symmetry exchanges the Coulomb branch with (one of the) Higgs branches of the theory. This result is relevant to duality in N=2 supersymmetric gauge theories in three dimensions. 
  Via compactification on a circle, the matrix model of M-theory proposed by Banks et al suggests a concrete identification between the large N limit of two-dimensional N=8 supersymmetric Yang-Mills theory and type IIA string theory. In this paper we collect evidence that supports this identification. We explicitly identify the perturbative string states and their interactions, and describe the appearance of D-particle and D-membrane states. 
  We consider a one-brane probe in the presence of a five-dimensional black hole in the classical limit. The velocity-dependent force on a slowly-moving probe is characterized by a metric on the probe moduli space. This metric is computed for large black holes using low-energy supergravity, and for small black holes using D-brane gauge theory. The results are compared. 
  We define quantum field theory by taking the Lagrangian action to be given as a sequence of mathematically well-defined functionals written in terms of operator fields fulfilling given \hbox{local} commutation relations. The renormalized solution fields have a fully defined Fock space expansion and are \hbox{multi-local}; thus Haag's theorem does not apply, i.e., the interaction picture exists. Also, the formalism allows immediately the definition of a wave function and the description of many-body bound-state systems. 
  It is found that the 2-index potential in nonabelian theories does not behave geometrically as a connection but that, considered as an element of the second de Rham cohomology group twisted by a flat connection, it fits well with all the properties assigned to it in various physical contexts. We also prove some results on the Euler characteristic of the twisted de Rham complex. 
  Vacuum polarization and particle production effects in classical electromagnetic and gravitational backgrounds can be studied by the effective lagrangian method. Background field configurations for which the effective lagrangian is zero are special in the sense that the lowest order quantum corrections vanishes for such configurations. We propose here the conjecture that there will be neither particle production nor vacuum polarization in classical field configurations for which all the scalar invariants are zero. We verify this conjecture, by explicitly evaluating the effective lagrangian, for non-trivial electromagnetic and gravitational backgrounds with vanishing scalar invariants. The implications of this result are discussed. 
  We consider a parabolic-like systems of differential equations involving geometrical quantities to examine uniformization theorems for two- and three-dimensional closed orientable manifolds. We find that in the two-dimensional case there is a simple gauge theoretic flow for a connection built from a Riemannian structure, and that the convergence of the flow to the fixed points is consistent with the Poincare Uniformization Theorem. We construct a similar system for the three-dimensional case. Here the connection is built from a Riemannian geometry, an SO(3) connection and two other 1-form fields which take their values in the SO(3) algebra. The flat connections include the eight homogeneous geometries relevant to the three-dimensional uniformization theorem conjectured by W. Thurston. The fixed points of the flow include, besides the flat connections (and their local deformations), non-flat solutions of the Yang-Mills equations. These latter "instanton" configurations may be relevant to the fact that generic 3-manifolds do not admit one of the homogeneous geometries, but may be decomposed into "simple 3-manifolds" which do. 
  We study the decompactification limit of M-theory superpotentials for N=1 four dimensional supersymmetric gauge theories. These superpotentials can be interpreted as generated by toron configurations. The connection with the confinement picture in the maximal abelian gauge is discussed. 
  Duality relations for the 2D nonhomogeneous Ising model on the finite square lattice wrapped on the torus are obtained. The partition function of the model on the dual lattice with arbitrary combinations of the periodical and antiperiodical boundary conditions along the cycles of the torus is expressed through some specific combination of the partition functions of the model on the original lattice with corresponding boundary conditions. It is shown that the structure of the duality relations is connected with the topological peculiarities of the dual transformation of the model on the torus. 
  We address the problem of a non-perturbative formulation of superstring theory by means of the recently proposed matrix models. For the model by Ishibashi, Kawai, Kitazawa and Tsuchiya (IKKT), we perform one-loop calculation of the interaction between operator-like solutions identified with D-brane configurations of type IIB superstring (in particular, for parallel moving and rotated static p-branes). Comparing to the superstring calculations, we show that the matrix model reproduces the superstring results only at large distances or small velocities, corresponding to keeping only the lowest mass closed string modes. We propose a modification of the IKKT matrix model introducing an integration over an additional Hermitian matrix required to have positive definite eigenvalues, which is similar to the square root of the metric in the continuum Schild formulation of IIB superstrings. We show that for this new matrix action the Nambu-Goto version of the Green-Schwarz action is reproduced even at quantum level. 
  We calculate relativistic phase-shifts resulting from the large impact parameter scattering of 0-branes off p-branes within supergravity. Their full functional dependence on velocity agrees with that obtained by identifying the p-branes with D-branes in string theory. These processes are also described by 0-brane quantum mechanics, but only in the non-relativistic limit. We show that an improved 0-brane quantum mechanics based on a Born-Infeld type Lagrangian also does not yield the relativistic results. Scattering of 0-branes off bound states of arbitrary numbers of 0-branes and 2-branes is analyzed in detail, and we find agreement between supergravity and string theory at large distances to all orders in velocity. Our careful treatment of this system, which embodies the 11 dimensional kinematics of 2-branes in M(atrix) theory, makes it evident that control of 1/n corrections will be necessary in order to understand our relativistic results within M(atrix) theory. 
  Low energy absorption cross-sections for various particles falling into extreme non-dilatonic branes are calculated using string theory and world-volume field theory methods. The results are compared with classical absorption by the corresponding gravitational backgrounds. For the self-dual threebrane, earlier work by one of us demonstrated precise agreement of the absorption cross-sections for the dilaton, and here we extend the result to Ramond-Ramond scalars and to gravitons polarized parallel to the brane. In string theory, the only absorption channel available to dilatons and Ramond-Ramond scalars at leading order is conversion into a pair of gauge bosons on the threebrane. For gravitons polarized parallel to the brane, scalars, fermions and gauge bosons all make leading order contributions to the cross-section, which remarkably add up to the value predicted by classical gravity. For the twobrane and fivebrane of M-theory, numerical coefficients fail to agree, signalling our lack of a precise understanding of the world-volume theory for large numbers of coincident branes. In many cases, we note a remarkable isotropy in the final state particle flux within the brane. We also consider the generalization to higher partial waves of minimally coupled scalars. We demonstrate agreement for the threebrane at l=1 and indicate that further work is necessary to understand l>1. 
  A low-energy background field solution is presented which describes several D-membranes oriented at angles with respect to one another. The mass and charge densities for this configuration are computed and found to saturate the BPS bound, implying the preservation of one-quarter of the supersymmetries. T-duality is exploited to construct new solutions with nontrivial angles from the basic one. 
  We investigate quantum dynamics of self-gravitating spherical dust shell. The wave functions of discrete spectrum are not localized inside the Schwarzschild radius. We argue that such shells can transform into white holes (in another space). It is plausible that shells with bare masses larger than the Planck mass loose their mass emitting lighter shells. 
  We generalize the notion of the 'noncommutative coupling constant' given by Kastler and Sch"ucker by dropping the constraint that it commute with the Dirac-operator. This leads essentially to the vanishing of the lower bound for the Higgsmass and of the upper bound for the W-mass. 
  We find exactly solvable dilaton gravity theories containing a U(1) gauge field in two dimensional space-time. The classical general solutions for the gravity sector (the metric plus the dilaton field) of the theories coupled to a massless complex scalar field are obtained in terms of the stress-energy tensor and the U(1) current of the scalar field. We discuss issues that arise when we attempt to use these models for the study of the gravitational back-reaction. 
  The classification of the coadjoint orbits of the Virasoro algebra is reviewed and is then applied to analyze the so-called global Liouville equation. The review is self-contained, elementary and is tailor-made for the application. It is well-known that the Liouville equation for a smooth, real field $\phi$ under periodic boundary condition is a reduction of the SL(2,R) WZNW model on the cylinder, where the WZNW field g in SL(2,R) is restricted to be Gauss decomposable. If one drops this restriction, the Hamiltonian reduction yields, for the field $Q=\kappa g_{22}$ where $\kappa\neq 0$ is a constant, what we call the global Liouville equation. Corresponding to the winding number of the SL(2,R) WZNW model there is a topological invariant in the reduced theory, given by the number of zeros of Q over a period. By the substitution $Q=\pm\exp(- \phi/2)$, the Liouville theory for a smooth $\phi$ is recovered in the trivial topological sector. The nontrivial topological sectors can be viewed as singular sectors of the Liouville theory that contain blowing-up solutions in terms of $\phi$. Since the global Liouville equation is conformally invariant, its solutions can be described by explicitly listing those solutions for which the stress-energy tensor belongs to a set of representatives of the Virasoro coadjoint orbits chosen by convention. This direct method permits to study the `coadjoint orbit content' of the topological sectors as well as the behaviour of the energy in the sectors. The analysis confirms that the trivial topological sector contains special orbits with hyperbolic monodromy and shows that the energy is bounded from below in this sector only. 
  We propose a formulation of the massive spinning particle in terms of physical bosonic and fermionic fields only. We make use neither of auxiliary objects of the type of $\gamma_5$ nor of gauge fields. The model is to be used as a suitable starting point for a forthcoming study of the rigid superparticle. 
  We examine if there exists a zero-energy supersymmetric ground state for the fundamental five-brane. Looking for an $SO(6)\times SO(2)$- invariant ground state, we construct, in the light-cone gauge, perturbatively a Nicolai map up to third order in the inverse five-brane tension. We show that the Nicolai map equilibrates and the five-brane has a zero-energy normalizable supersymmetric vacuum state. For the other p-branes, we argue that only the three-brane has a zero-energy ground state. 
  We discuss Abelian and non-Abelian three dimensional bosonization within the path-integral framework. We present a systematic approach leading to the construction of the bosonic action which, together with the bosonization recipe for fermion currents, describes the original fermion system in terms of vector bosons. 
  We study the moduli dependence of a class of couplings in $K3\times T^2$ compactifications of type I string theory, for which one-loop amplitudes can be written in terms of an N=2 supersymmetric index. This index is determined for generic models as a function of the BPS spectrum. As an application we compute the one-loop moduli dependence of the $F_g W^{2g}$ couplings, where W is the N=2 gravitational superfield, for type I compactifications based on the Gimon-Johnson K3 orientifolds, showing explicitly their dependence on the aforementioned index. 
  Starting with the representation of the Wilson average in the Euclidean 4D compact QED as a partition function of the Universal Confining String Theory, we derive for it the corresponding loop equation, alternative to the familiar one.  In the functional momentum representation the obtained equation decouples into two independent ones, which describe the dynamics of the transverse and longitudinal components of the area derivative of the Wilson loop. At some critical value of the momentum discontinuity, which can be determined from a certain equation, the transverse component does not propagate.  Next, we derive the equation for the momentum Wilson loop, where on the left-hand side stands the sum of the squares of the momentum discontinuities, multiplied by the loop, which describes its free propagation, while the right-hand side describes the interaction of the loop with the functional vorticity tensor current.  Finally, using the method of inversion of the functional Laplacian, we obtain for the Wilson loop in the coordinate representation a simple Volterra type-II linear integral equation, which can be treated perturbatively. 
  We consider configurations of rotated NS-branes leading to a family of four-dimensional N=1 super-QCD theories, interpolating between four-dimensional analogues of the Hanany-Witten vacua, and the Elitzur-Giveon-Kutasov configuration for N=1 duality. The rotation angle is the N=2 breaking parameter, the mass of the adjoint scalar in the N=2 vector multiplet. We add some comments on the relevance of these configurations as possible stringy proofs of N=1 duality. 
  We study the symmetry structure of N=8 quantum mechanics, and apply it to the physics of D0-brane probes in type I' string theory. We focus on the theory with a global $Spin(8)$ R symmetry which arises upon dimensional reduction from $2d$ field theory with $(0,8)$ supersymmetry. There are several puzzles involving supersymmetry which we resolve. In particular, by taking into account the gauge constraint and central charge we explain how the system preserves supersymmetry despite having different numbers of bosonic and fermionic variables. The resulting zero-point energy leads to a linear potential consistent with supersymmetry, and the metric is largely unconstrained. We discuss implications for type I' string theory and the matrix model proposal for M theory. 
  We show that the recently proposed matrix model for M theory obeys the cyclic trace assumptions underlying generalized quantum or trace dynamics. This permits a verification of supersymmetry as an operator calculation, and a calculation of the supercharge density algebra by using the generalized Poisson bracket, in a basis-independent manner that makes no reference to individual matrix elements. Implications for quantization of the model are discussed. Our results are a special case of a general result presented elsewhere, that all rigid supersymmetry theories can be extended to give supersymmetric trace dynamics theories, in which the supersymmetry algebra is represented by the generalized Poisson bracket of trace supercharges, constructed from fields that form a noncommutative trace class graded operator algebra. 
  These lectures on the Batalin-Vilkovisky method of quantization were delivered at "VII Mexican School of Particles and Fields", M\'erida, M\'exico, October 30-November 6, 1996.   In section II, we study the derivation of BV from Schwinger-Dyson-BRST symmetry; in section III, we consider some generalizations of the BV structure suggested by our approach. In section IV, we present a connection between the Poisson bracket and the antibracket and use it to relate the Nambu bracketc with the generalized $n$-brackets of section III. 
  We present some important corrections to our work which appeared in Nucl. Phys. B426 (1994) 534 (hep-th/9402144). Our previous results for the correlation functions $\langle e^{i\alpha \Phi(x)} e^{i\alpha' \Phi (0) } \rangle$ were only valid for $\alpha = \alpha'$, due to the fact that we didn't find the most general solution to the differential equations we derived. Here we present the solution corresponding to $\alpha \neq \alpha'$. 
  We obtain actions for N D-branes occupying points in a manifold with arbitrary Kahler metric. In one complex dimension, the action is uniquely determined (up to second order in commutators) by the requirement that it reproduce the masses of stretched strings and by imposing supersymmetry. These conditions are very restrictive in higher dimensions as well. The results provide a noncommutative extension of Kahler geometry. 
  The twice-dimensionally reduced Seiberg-Witten monopole equations admit solutions depending on two real parameters (b,c) and an arbitrary analytic function f(z) determining a solution of Liouville's equation. The U(1) and manifold curvature 2-forms F and R^1_2 are invariant under fractional SL(2,R) transformations of f(z). When b=1/2 and c=0 and f(z) is the Fuchsian function uniformizing an algebraic function whose Riemann surface has genus p \geq 2 , the solutions, now SL(2,R) invariant, are the same surfaces accompanied by a U(1) bundle of c_1=\pm (p-1) and a 1-component constant spinor. 
  It is pointed out that the strong CP problem may have a natural solution in the context of a recently proposed dualized version of the Standard Model where Higgs fields and generations emerge naturally. Although fermions have finite pole-masses, the fermionic mass matrix itself is factorizable (having only one nonzero eigenvalue) to all orders in perturbation theory thus allowing one to perform a chiral transformation $\psi\to\psi'=e^{-i\gamma_{5}\alpha}\psi$ and to rotate the theta-angle to zero. 
  We show the existence of a supersymmetry breaking mechanism in string theory, where N=4 supersymmetry is broken spontaneously to N=2 and N=1 with moduli dependent gravitino masses. The spectrum of the spontaneously broken theory with lower supersymmetry is in one-to-one correspondence with the spectrum of the heterotic N=4 string. The mass splitting of the N=4 spectrum depends on the compactification moduli as well as the three R-symmetry charges. We also show that, in string theory, chiral theories can be obtained after spontaneous breaking of extended supersymmetry. This was impossible at the level of field theory. In the large moduli limit a restoration of the N=4 supersymmetry is obtained. As expected the graviphotons and some of the gauge bosons become massive in N=1 vacua. At some special points of the moduli space some of the N=4 states with non-zero winding numbers and with spin 0 and {1/2} become massless chiral superfields of the unbroken N=1 supersymmetry. Such vaccua have a dual type II description, in which there are magnetically charged states with spin 0 and {1/2} that become massless. The heterotic-type II duality suggests some novel non-perturbative transitions on the type II side. Such transitions do not seem to have a geometric interpretation, since they relate type II vaccua with symmetric worlsheet structure to assymetric ones. The heteroric interpretation of such a transition is an ordinary Higgsing of an SU(2) factor. In the case of N=4 --> N=2, the perturbative N=2 prepotential is determined by the perturbative N=4 BPS states. This observation let us to suggest a method which determines the exact non-perturbative prepotential of the effective N=2 supergravity using the shifted spectrum of the non-perturbative BPS states of the underlying N=4 theory. 
  General considerations on the unification of A-type and B-type supersymmetries in the context of interacting p-branes strongly suggest that the signature of spacetime includes two timelike dimensions. This leads to the puzzle of how ordinary physics with a single timelike dimension emerges. In this letter we suggest that the two timelike dimensions could be real, and belong to two physical sectors of a single theory each containing its own timelike dimension. Effectively there is a single time evolution parameter. We substantiate this idea by constructing certain actions for interacting p-branes with signature (n,2) that have gauge symmetries and constraints appropriate for a physical interpretation with no ghosts. In combination with related ideas and general constraints in S-theory, we are led to a cosmological scenario in which, after a phase transition, the extra timelike dimension becomes part of the compactified universe residing inside microscopic matter. The internal space, whose geometry is expected to determine the flavor quantum numbers of low energy matter, thus acquires a Minkowski signature. The formalism meshes naturally with a new supersymmetry in the context of field theory that we suggested in an earlier paper. The structure of this supersymmetry gives rise to a new Kaluza-Klein type mechanism for determining the quantum numbers of low energy families, thus suggesting that the extra timelike dimension would be taken into account in understanding the Standard Model of particle physics. 
  The Frobenius-Schwinger-Dyson equations are a rather high-brow abstract nonsense type of equations describing n-point functions of arbitrarily high composite insertions. It is not clear how to solve or even find approximate solutions of these equations in general, but they are worth investigating because (a certain preferred type of) renormalization of composite insertions has been performed in advance: it just remains to find solutions given an action and renormalization conditions. Earlier work in this field involved only Gaussian actions or variable transformations thereof. In this work we illustrate the use of Frobenius-Schwinger-Dyson at a less obviously trivial level, that of the Thirring model. 
  Using a recently constructed M5-brane world-volume action, we demonstrate that wrapping the M5-brane on K3 gives the heterotic string in seven dimensions. To facilitate the comparison, a new version of the world-sheet action for the Narain-compactified heterotic string, with manifest T duality invariance, is formulated. 
  We construct the N=2, D=9 supergravity theory up to the quartic fermionic terms and derive the supersymmetry transformation rules for the fields modulo cubic fermions. We consider a class of p-brane solutions of this theory, the stainless p-branes, which cannot be isotropically oxidized into higher dimensions. The new stainless elementary membrane and elementary particle solutions are found. It is explicitly verified that these solutions preserve half of the supersymmetry. 
  The time development of a model of (2+1)-dimensional torus universe is studied based on background field equations which follow from a string theory. The metrics in various cases are characterized by a real parameter which specifies a ratio of the lengths of two independent cycles. When the parameter is a rational number, the space is asymptotically stretched along a cycle while the other cycle kept finite. When the parameter is an irrational number, the lengths of two cycles, as well as the space volume (area), grow in proportion to the proper time $t$ for an observer sitting at rest in this universe in the asymptotic region. 
  An analysis is given of the structure of a general two-dimensional Toda field theory involving bosons and fermions which is defined in terms of a set of simple roots for a Lie superalgebra. It is shown that a simple root system for a superalgebra has two natural bosonic root systems associated with it which can be found very simply using Dynkin diagrams; the construction is closely related to the question of how to recover the signs of the entries of a Cartan matrix for a superalgebra from its Dynkin diagram. The significance for Toda theories is that the bosonic root systems correspond to the purely bosonic sector of the integrable model, knowledge of which can determine the bosonic part of the extended conformal symmetry in the theory, or its classical mass spectrum, as appropriate. These results are applied to some special kinds of models and their implications are investigated for features such as supersymmetry, positive kinetic energy and generalized reality conditions for the Toda fields. As a result, some new families of integrable theories with positive kinetic energy are constructed, some containing a mixture of massless and massive degrees of freedom, others being purely massive and supersymmetric, involving a number of coupled sine/sinh-Gordon theories. 
  I discuss the scattering of a graviton (or a dilaton, or an anti-symmetric tensor) on two parallel static Dp-branes. The graviton belongs to a type II string in 10D. 
  We present a starting point for the search for a Lagrangian density for M-Theory using characteristic classes for flat foliations of bundles. 
  For minimally coupled scalars at low frequencies, the D-brane model has the same spectrum of radiation as the Hawking radiation from a black hole. We perform a similar comparison for another type of scalar which we call an intermediate scalar. In this case, we find that there is a discrepancy between the D-brane model and the black hole even for very low frequency scalars. This suggests that the model is only valid within the moduli space approximation. 
  We use the homological algebra context to give a more rigorous proof of Polyakov's basic variational formula for loop spaces. 
  We construct stringy cosmic string solutions corresponding to compactifications of F-theory on several elliptic Calabi-Yau manifolds by solving the equations of motion of low energy effective action of ten dimensional type IIB superstring theory. Existence of such solutions supports the compactifications of F-theory. 
  We examine the relations between observables in two- and three-dimensional quantum gravity by studying the coupling of topologically massive gravity to matter fields in non-trivial representations of the three-dimensional Lorentz group. We show that the gravitational renormalization of spin up to one-loop order in these theories reproduces the leading orders of the KPZ scaling relations for quantum Liouville theory. We demonstrate that the two-dimensional scaling dimensions can be computed from tree-level Aharonov-Bohm scattering amplitudes between the charged particles in the limit where the three-dimensional theory possesses local conformal invariance. We show how the three-dimensional description defines scale-dependent weights by computing the one-loop order anomalous magnetic moment of fermions in a background electromagnetic field due to the renormalization by topologically massive gravity. We also discuss some aspects concerning the different phases of three-dimensional quantum gravity and argue that the topological ones may be related to the branched polymer phase of two-dimensional quantum gravity. 
  We present a general analysis of all the possible soft breakings of N=2 supersymmetric QCD, preserving the analytic properties of the Seiberg-Witten solutions for the SU(2) group with Nf=1, 2, 3 hypermultiplets. We obtain all the couplings of the spurion fields in terms of properties of the Seiberg-Witten periods, which we express in terms of elementary elliptic functions by uniformizing the elliptic curves associated to each number of flavors. We analyze in detail the monodromy properties of the softly broken theory, and obtain them by a particular embedding into a pure gauge theory with higher rank group. This allows to write explicit expressions of the effective potential, which are close to the exact answer for moderate values of the supersymmetry breaking parameters. The vacuum structures and phases of the broken theories will be analyzed in the forthcoming second part of this paper. 
  A unitary matrix model is proposed as the large-N matrix formulation of M theory on flat space with toroidal topology. The model reproduces the motion of elementary D-particles on the compact space, and admits membrane states with nonzero wrapping around nontrivial 2-tori even at finite N. 
  Two dimensional QCD is quantized on the light front coordinate. We solve the Einstein-Schr\"odinger equation by the use of Tamm-Dancoff truncation and find that the simplest wavefunction produces the $M/g$ versus $m/g$ relation in agreement with other calculations, where $M$ and $m$ are the masses of the ground state and quarks, respectively. 
  We describe the couplings of six-dimensional supergravity, which contain a self-dual tensor multiplet, to $n_T$ anti-self-dual tensor matter multiplets, $n_V$ vector multiplets and $n_H$ hypermultiplets. The scalar fields of the tensor multiplets form a coset $SO(n_T,1)/SO(n_T)$, while the scalars in the hypermultiplets form quaternionic K\"ahler symmetric spaces, the generic example being $Sp(n_H,1)/Sp(n_H)\otimes Sp(1)$. The gauging of the compact subgroup $Sp(n_H) \times Sp(1)$ is also described. These results generalize previous ones in the literature on matter couplings of $N=1$ supergravity in six dimensions. 
  We discuss type I -- heterotic duality in four-dimensional models obtained as a Coulomb phase of the six-dimensional U(16) orientifold model compactified on T^2 with arbitrary SU(16) Wilson lines. We show that Kahler potentials, gauge threshold corrections and the infinite tower of higher derivative F-terms agree in the limit that corresponds to weak coupling, large T^2 heterotic compactifications. On the type I side, all these quantities are completely determined by the spectrum of N=2 BPS states that originate from D=6 massless superstring modes. 
  We present a solution of M(atrix) theory describing type IIA fundamental string. Our construction is based on the central charge of the longitudinal membrane (= fundamental string), the BPS saturation condition and the relation between M(atrix) theory and supersymmetric Yang-Mills theory. The fundamental string corresponds to a photon in supersymmetric Yang-Mills theory. 
  We show that the path-integral quantization of relativistic strings with the Schild action is essentially equivalent to the usual Polyakov quantization at critical space-time dimensions. We then present an interpretation of the Schild action which points towards a derivation of superstring theory as a theory of quantized space-time where the squared string scale plays the role of the minimum quantum for space-time areas. A tentative approach towards such a goal is proposed, based on a microcanonical formulation of large N supersymmetric matrix model. 
  We discuss the principles to be used in the construction of discrete time classical and quantum mechanics as applied to point particle systems. In the classical theory this includes the concept of virtual path and the construction of system functions from classical Lagrangians, Cadzow's variational principle applied to the action sum, Maeda-Noether and Logan invariants of the motion, elliptic and hyperbolic harmonic oscillator behaviour, gauge invariant electrodynamics and charge conservation, and the Grassmannian oscillator. First quantised discrete time mechanics is discussed via the concept of system amplitude, which permits the construction of all quantities of interest such as commutators and scattering amplitudes. We discuss stroboscopic quantum mechanics, or the construction of discrete time quantum theory from continuous time quantum theory and show how this works in detail for the free Newtonian particle. We conclude with an application of the Schwinger action principle to the important case of the quantised discrete time inhomogeneous oscillator. 
  We apply the principles discussed in an earlier paper to the construction of discrete time field theories. We derive the discrete time field equations of motion and Noether's theorem and apply them to the Schrodinger equation to illustrate the methodology. Stationary solutions to the discrete time Schrodinger wave equation are found to be identical to standard energy eigenvalue solutions except for a fundamental limit on the energy. Then we apply the formalism to the free neutral Klein Gordon system, deriving the equations of motion and conserved quantities such as the linear momentum and angular momentum. We show that there is an upper bound on the magnitude of linear momentum for physical particle-like solutions. We extend the formalism to the charged scalar field coupled to Maxwell's electrodynamics in a gauge invariant way. We apply the formalism to include the Maxwell and Dirac fields, setting the scene for second quantisation of discrete time mechanics and discrete time Quantum Electrodynamics. 
  The quantization of the complex linear superfield requires an infinite tower of ghosts. We use the Batalin-Vilkovisky method to obtain a gauge-fixed action. In superspace, the method brings in some novel features. 
  The scalars in vector multiplets of N=2 supersymmetric theories in 4 dimensions exhibit `special Kaehler geometry', related to duality symmetries, due to their coupling to the vectors. In the literature there is some confusion on the definition of special geometry. We show equivalences of some definitions and give examples which show that earlier definitions are not equivalent, and are not sufficient to restrict the Kaehler metric to one that occurs in N=2 supersymmetry. We treat the rigid as well as the local supersymmetry case. The connection is made to moduli spaces of Riemann surfaces and Calabi-Yau 3-folds. The conditions for the existence of a prepotential translate to a condition on the choice of canonical basis of cycles. 
  Local gauge freedom in relativistic quantum mechanics is derived from a measurement principle for space and time. For the Dirac equation, one obtains local U(2,2) gauge transformations acting on the spinor index of the wave functions. This local U(2,2) symmetry allows a unified description of electrodynamics and general relativity as a classical gauge theory. 
  We study the embedding of Kac-Moody algebras into Borcherds (or generalized Kac-Moody) algebras which can be explicitly realized as Lie algebras of physical states of some completely compactified bosonic string. The extra ``missing states'' can be decomposed into irreducible highest or lowest weight ``missing modules'' w.r.t. the relevant Kac-Moody subalgebra; the corresponding lowest weights are associated with imaginary simple roots whose multiplicities can be simply understood in terms of certain polarization states of the associated string. We analyse in detail two examples where the momentum lattice of the string is given by the unique even unimodular Lorentzian lattice $II_{1,1}$ or $II_{9,1}$, respectively. The former leads to the Borcherds algebra $g_{1,1}$, which we call ``gnome Lie algebra", with maximal Kac-Moody subalgebra $A_1$. By the use of the denominator formula a complete set of imaginary simple roots can be exhibited, whereas the DDF construction provides an explicit Lie algebra basis in terms of purely longitudinal states of the compactified string in two dimensions. The second example is the Borcherds algebra $g_{9,1}$, whose maximal Kac-Moody subalgebra is the hyperbolic algebra $E_{10}$. The imaginary simple roots at level 1, which give rise to irreducible lowest weight modules for $E_{10}$, can be completely characterized; furthermore, our explicit analysis of two non-trivial level-2 root spaces leads us to conjecture that these are in fact the only imaginary simple roots for $g_{9,1}$. 
  We define an infinite class of integrable theories with a defect which are formulated as chiral defect perturbations of a conformal field theory. Such theories can be interacting in the bulk, and are purely transmitting through the defect. The examples of the sine-Gordon theory and Ising model are worked out in some detail. 
  We propose that the Virasoro algebra controls quantum cohomologies of general Fano manifolds $M$ ($c_1(M)>0$) and determines their partition functions at all genera. We construct Virasoro operators in the case of complex projective spaces and show that they reproduce the results of Kontsevich-Manin, Getzler etc. on the genus-0,1 instanton numbers. We also construct Virasoro operators for a wider class of Fano varieties. The central charge of the algebra is equal to $\chi(M)$, the Euler characteristic of the manifold $M$. 
  A direct method for obtaining the differential conservation laws in the field theory from the principle of stationary action is proposed. The method is based on a variation of field functions through small local transformation of a special kind. The action of general theory of relativity are considered. 
  The thermal partition functions of photons in any covariant gauge and gravitons in the harmonic gauge, propagating in a Rindler wedge, are computed using a local zeta-function approach. The relation with the surface terms previously obtained by D. Kabat is studied. The results are discussed in relation to the quantum corrections to the black hole entropy. 
  The thermodynamic free energy F is calculated for a gas whose particles are the quantum excitations of a piecewise uniform bosonic string. The string consists of two parts of length L_I and L_II, endowed with different tensions and mass densities, adjusted in such a way that the velocity of sound always equals the velocity of light. The explicit calculation is done under the restrictive condition that the tension ratio x = T_I/T_II approaches zero. Also, the length ratio s = L_II/L_I is assumed to be an integer. The expression for F is given on an integral form, in which s is present as a parameter. For large values of s, the Hagedorn temperature becomes proportional to the square root of s. 
  A Galilean Chern-Simons field theory is formulated for the case of two interacting spin-1/2 fields of distinct masses M and M'. A method for the construction of states containing N particles of mass M and N' particles of mass M' is given which is subsequently used to display equivalence to the spin-1/2 Aharonov-Bohm effect in the N = N' =1 sector of the model. The latter is then studied in perturbation theory to determine whether there are divergences in the fourth order (one loop) diagram. It is found that the contribution of that order is finite (and vanishing) for the case of parallel spin projections while the antiparallel case displays divergences which are known to characterize the spin zero case in field theory as well as in quantum mechanics. 
  These are notes of a seminar given at the 30th International Symposium on the Theory of Elementary Particles, Berlin-Buckow, August 1996. The material is derived from collaborations with E. Cremmer and J.-L. Gervais, and C. Klimcik, and is partially new. Within the general framework of Poisson-Lie symmetry, we discuss two approaches to the problem of constructing moment maps, or q-deformed Noether charges, that generate the quantum group symmetry which appears in many conformal field theories. Concretely, we consider the case of $U_q(sl(2))$ and the operator algebra that describes Liouville theory and other models built from integer powers of screenings in the Coulomb gas picture. 
  In this letter we show how the covariant anomaly emerges in the overlap scheme. We also prove that the overlap scheme correctly reproduces the anomaly in the flavour currents such as $j^5_\mu$ in vector like theories like QCD. 
  We use the Wilson-Fisher $\epsilon$ expansion to study quantum critical behavior in gauged Yukawa matrix field theories with weak quenched disorder. We find that the resulting quantum critical behavior is in the universality class of the pure system. As in the pure system, the phase transition is typically first order, except for a limited range of parameters where it can be second order with computable critical exponents. Our results apply to the study of two-dimensional quantum antiferromagnets with weak quenched disorder and provide an example for fluctuation-induced first order phase transitions in circumstances where naively none is expected. 
  Since the lightcone self dual spherical membrane, moving in flat target backgrounds, has a direct correspondence with the $SU(\infty)$ Nahm equations and the continuous Toda theory, we construct the Moyal deformations of the self dual membrane in terms of the Moyal deformations of the continuous Toda theory. This is performed by using the Weyl-Wigner-Moyal quantization technique of the 3D continuous Toda field equation, and its associated 2D continuous Toda molecule, based on Moyal deformations of rotational Killing symmetry reductions of Plebanski first heavenly equation associated with 4D Self Dual Gravity. 
  We investigate the multi-loop correlators and the multi-point functions for all of the scaling operators in unitary minimal conformal models coupled to two-dimensional gravity from the two-matrix model. We show that simple fusion rules for these scaling operators exist, and these are summarized in a compact form as fusion rules for loops. We clarify the role of the boundary operators and discuss its connection to how loops touch each other. We derive a general formula for the n-resolvent correlators, and point out the structure similar to the crossing symmetry of underlying conformal field theory. We discuss the connection of the boundary conditions of the loop correlators to the touching of loops for the case of the four-loop correlators. 
  We show the quantum equivalence between certain symmetric space sine-Gordon models and the massive free fermions. In the massless limit, these fermions reduce to the free fermions introduced by Goddard, Nahm and Olive (GNO) in association with symmetric spaces $K/G$. A path integral formulation is given in terms of the Wess-Zumino-Witten action where the field variable $g$ takes value in the orthogonal, unitary, and symplectic representations of the group $G$ in the basis of the symmetric space. We show that, for example, such a path integral bosonization is possible when the symmetric spaces $K/G$ are $SU(N) \times SU(N)/SU(N); N \le 3, ~ Sp(2)/U(2)$ or $SO(8)/U(4)$. We also address the relation between massive GNO fermions and the nonabelian solitons, and explain the restriction imposed on the fermion mass matrix due to the integrability of the bosonic model. 
  A consistent procedure for regularization of divergences and for the subsequent renormalization of the string tension is proposed in the framework of the one-loop calculation of the interquark potential generated by the Polyakov-Kleinert string. In this way, a justification of the formal treatment of divergences by analytic continuation of the Riemann and Epstein-Hurwitz zeta functions is given. A spectral representation for the renormalized string energy at zero temperature is derived, which enables one to find the Casimir energy in this string model at nonzero temperature very easy. 
  We investigate consistency conditions for supersymmetric gauge theories in higher dimensions. First, we give a survey of Seiberg's necessary conditions for the existence of such theories with simple groups in five and six dimensions. We then make some comments on how theories in different dimensions are related. In particular, we discuss how the Landau pole can be avoided in theories that are not asymptotically free in four dimensions, and the mixing of tensor and vector multiplets in dimensional reduction from six dimensions. 
  A generalization of the embedding approach for d-dimensional gravity based upon p-brane theories is considered. We show that the D-dimensional p-brane coupled to an antisymmetric tensor field of rank (p+1) provides the dynamical basis for the description of d=(p+1) dimensional gravity in the isometric embedding formalism. ''Physical'' matter appears in such an approach as a manifestation of a D-dimensional antisymmetric tensor (generalized Kalb- Ramond) background. For the simplest case, the Lorentz harmonic formulation of the bosonic string in a Kalb-Ramond background and its relation to a first order Einstein-Cartan approach for d=2 dimensional gravity is analysed in some detail. A general Poisson-sigma-model structure emerges. For the minimal choice of D=3 an interesting ``dual'' formulation is found which has the structure of a Jackiw-Teitelboim theory, coupled minimally to a massive scalar field. Our approach is intended to serve as a preparation for the study of d- dimensional supergravity theory, either starting from the generalized action of free supersymmetric (d-1)-branes or $D_{(d-1)}$-branes, or from the corresponding geometric equations ('rheotropic' conditions). 
  We study the structure of the moduli spaces of vacua and superpotentials of N=2 supersymmetric gauge theories in three dimensions. By analyzing the instanton corrections, we compute the exact superpotentials and determine the quantum Coulomb and Higgs branches of the theories in the weak coupling regions. We find candidates for non-trivial N=2 superconformal field theories at the singularities of the moduli spaces. The analysis is carried out explicitly for gauge groups U(N_c) and SU(N_c) with N_f flavors. We show that the field theory results are in complete agreement with the intersecting branes picture. We also compute the exact superpotentials for arbitrary gauge groups and arbitrary matter content. 
  We find general static BPS black hole solutions for general N=2, d=4 supergravity theories with an arbitrary number of vector multiplets. These solutions are completely specified by the K\"ahler potential of the underlying special K\"ahler geometry and a set of constrained harmonic functions. 
  Using M(atrix) Theory, the dualities of toroidally compactified M-theory can be formulated as properties of super Yang Mills theories in various dimensions. We consider the cases of compactification on one, two, three, four and five dimensional tori. The dualities required by string theory lead to conjectures of remarkable symmetries and relations between field theories as well as extremely unusual dynamical properties. By studying the theories in the limit of vanishingly small tori, a wealth of information is obtained about strongly coupled fixed points of super Yang-Mills theories in various dimensions. Perhaps the most striking behavior, as noted by Rozali in this context, is the emergence of an additional dimension of space in the case of a four torus. 
  Supersymmetry is studied in 2+1 dimensions. In addition to the multiplets corresponding to those in 3+1 dimensions the Clifford algebra allows an extra set. When the extra chiral multiplet is included, formulating supersymmetric QED3 in the manner of Wess and Zumino yields a lagrangian with a richer 'chiral' symmetry than that derived from supersymmetric QED by dimensional reduction. 
  The new extensions of the Poincar\'e superalgebra recently found in ten and eleven dimensions are shown to admit a linear realization. The generators of the nonlinear and linear group transformations are shown to fall into equivalent representations of the superalgebra. The parametrization of the coset space $G/H$, with $G$ a given extended supergroup and $H$ the Lorentz subgroup, that corresponds to the linear transformations is presented. 
  We reformulate time evolution of systems in mixed states in terms of the classical observables of correlators using the Weyl correspondence rule. The resulting equation of motion for the Wigner functional of the density matrix is found to be of the Liouville type. To illustrate the methods developed, we explicitly consider a scalar theory with quartic self-interaction and derive the short time behaviour with the non-interacting thermal density matrix as initial condition. In the scalar case, the complete correlator hierarchy is studied and restrictions are derived for spatially homogeneous initial conditions and systems with unbroken symmetry. 
  We study the QED bound-state problem in a light-front hamiltonian approach. It is important to establish the equivalence (or not) of equal-time and light-front approaches in the well-understood arena of Quantum Electrodynamics. Along these lines, the singlet-triplet ground state spin splitting in positronium is calculated. The well-known result, $(7/6) \alpha^2 Ryd$, is obtained analytically, which establishes the equivalence between the equal-time and light-front approaches (at least to this order). The true equivalence of the two approaches can only be established after higher-order calculations. It was previously shown that this light-front result could be obtained analytically, but a simpler method is presented in this paper. 
  Spontaneous symmetry breaking of the light-front Gross-Neveu model is studied in the framework of the discretized light-cone quantization. Introducing a scalar auxiliary field and adding its kinetic term, we obtain a constraint on the longitudinal zero mode of the scalar field. This zero-mode constraint is solved by using the $1/N$ expansion. In the leading order, we find a nontrivial solution which gives the fermion nonzero mass and thus breaks the discrete symmetry of the model. It is essential for obtaining the nontrivial solution to treat adequately an infrared divergence which appears in the continuum limit. We also discuss the constituent picture of the model. The Fock vacuum is trivial and an eigenstate of the light-cone Hamiltonian. In the large $N$ limit, the Hamiltonian consists of the kinetic term of the fermion with dressed mass and the interaction term of these fermions. 
  We consider a dimensional reduction of 3+1 dimensional SU(N) Yang-Mills theory coupled to adjoint fermions to obtain a class of 1+1 dimensional gauge theories. We derive the quantized light-cone Hamiltonian in the light-cone gauge A^+ = 0$ and large-N limit, then solve for the masses, wavefunctions and of the color singlet boson and fermion boundstates. We find that the theory has many exact massless state that are similar to the t'Hooft pion. 
  In three dimensions, the effective action for the gauge field induced by integrating out a massless Dirac fermion is known to give either a parity-invariant or a parity-violating result, depending on the regularization scheme. We construct a lattice formulation of the massless Dirac fermion using the overlap formalism. We show that the result is parity invariant in contrast to the formulation using Wilson fermions in the massless limit. This facilitates a non-perturbative study of three-dimensional massless Dirac fermions interacting with a gauge field in a parity invariant setting with no need for fine-tuning. 
  We consider general aspects of N=2 gauge theories in three dimensions, including their multiplet structure, anomalies and non-renormalization theorems. For U(1) gauge theories, we discuss the quantum corrections to the moduli space, and their relation to ``mirror symmetries'' of 3d N=4 theories. Mirror symmetry is given an interpretation in terms of vortices. For SU(N_c) gauge groups with N_f fundamental flavors, we show that, depending on the number of flavors, there are quantum moduli spaces of vacua with various phenomena near the origin. 
  A many variable $q$-calculus is introduced using the formalism of braided covector algebras. Its properties when certain of its deformation parameters are roots of unity are discussed in detail, and related to fractional supersymmetry. The special cases of two dimensional supersymmetry and fractional supersymmetry are developed in detail. 
  We extend ideas developed for the loop representation of quantum gravity to diffeomorphism-invariant gauge theories coupled to fermions. Let P -> Sigma be a principal G-bundle over space and let F be a vector bundle associated to P whose fiber is a sum of continuous unitary irreducible representations of the compact connected gauge group G, each representation appearing together with its dual. We consider theories whose classical configuration space is A x F, where A is the space of connections on P and F is the space of sections of F, regarded as a collection of Grassmann-valued fermionic fields. We construct the `quantum configuration space a x f as a completion of A x F. Using this we construct a Hilbert space L^2(a x f) for the quantum theory on which all automorphisms of P act as unitary operators, and determine an explicit `spin network basis' of the subspace L^2((a x f)/G) consisting of gauge-invariant states. We represent observables constructed from holonomies of the connection along paths together with fermionic fields and their conjugate momenta as operators on L^2((a x f)/G). We also construct a Hilbert space H_diff of diffeomorphism-invariant states using the group averaging procedure of Ashtekar, Lewandowski, Marolf, Mourao and Thiemann. 
  In this article we will examine a "generalized topological sigma model." This so-called "generalized topological sigma model" is the M-Theoretic analog of the standard topological sigma model of string theory. We find that the observables of the theory are elements in the cohomology ring of the moduli space of supersymmetric maps; in addition, we find that the correlation functions of such observables allow us to compute non-perturbative corrections to the four-fermion terms present in M-Theory on a six-dimensional Calabi-Yau. 
  The screening nature of the potential between external quarks in massless $SU(N_c)$ $QCD_2$ is derived using an expansion in $N_f$- the number of flavors. Applying the same method to the massive model, we find a confining potential. We consider the N=1 super Yang Mills theory, reveal certain problematic aspects of its bosonized version and show the associated screening behavior by applying a point splitting method to the scalar current. 
  We study our Schwinger-Dyson equation as well as the large $N_{c}$ loop equation for supersymmetric Yang-Mills theory in four dimensions by the N=1 superspace Wilson-loop variable. We are successful in deriving a new manifestly supersymmetric form in which a loop splitting and joining are represented by a manifestly supersymmetric as well as supergauge invariant operation in superspace. This is found to be a natural extension from the abelian case. We solve the equation to leading order in perturbation theory or equivalently in the linearized approximation, obtaining a desirable nontrivial answer. The super Wilson-loop variable can be represented as the system of one-dimensional fermion along the loop coupled minimally to the original theory. One-loop renormalization of the one-point Wilson-loop average is explicitly carried out, exploiting this property. The picture of string dynamics obtained is briefly discussed. 
  The mass generation in the (3+1)-dimensional supersymmetric Nambu-Jona-Lasinio model in a constant magnetic field is studied. It is shown that the external magnetic field catalyzes chiral symmetry breaking. 
  I consider a three-dimensional string theory whose action, besides the standard area term, contains one of the form $\int_{\Sigma} \epsilon_{\mu\nu\sigma} X^{\mu} d X^{\nu} \wedge d X^{\sigma}$. In the case of closed strings this extra term has a simple geometrical interpretation as the volume enclosed by the surface. The associated variational problem yields as solutions constant mean curvature surfaces. One may then show the equivalence of this equation of motion to that of an SU(2) principal chiral model coupled to gravity. It is also possible by means of the Kemmotsu representation theorem, restricted to constant curvature surfaces, to map the solution space of the string model into the one of the $CP^1$ nonlinear sigma model. I also show how a description of the Gauss map of the surface in terms of SU(2) spinors allows for yet a different description of this result by means of a Gross-Neveu spinorial model coupled to 2-D gravity. The standard three-dimensional string equations can also be recovered by setting the current-current coupling to zero. 
  This is an introductory review on the eleven-dimensional description of the BPS bound states of type II superstring theories, and on the role of supermembranes in M-theory. The first part describes classical solutions of 11d supergravity which upon dimensional reduction and T-dualities give bound states of NS-NS and R-R p-branes of type IIA and IIB string theories. In some cases (e.g. (q_1,q_2) string bound states of type IIB string theory), these non-perturbative objects admit a simple eleven-dimensional description in terms of a fundamental 2-brane. The BPS excitations of such 2-brane are calculated and shown to exactly match the mass spectrum for the BPS (q_1,q_2) string bound states. Different 11d representations of the same bound state can be used to provide inequivalent (T-dual) descriptions of the oscillating BPS states. This permits to test T-duality beyond perturbation theory and, in certain cases, to evade membrane instabilities by going to a stable T-dual representation. We finally summarize the results indicating in what regions of the modular parameter space a supermembrane description for M-theory on R^9 x T^2 seems to be adequate. 
  The Hamiltonian structure of spin generalization of the rational Ruijsenaars-Schneider model is found by using the Hamiltonian reduction technique. It is shown that the model possesses the current algebra symmetry. The possibility of generalizing the found Poisson structure to the trigonometric case is discussed and degeneration to the Euler-Calogero-Moser system is examined. 
  We revisit the treatment of the multiflavor massive Schwinger model by non-Abelian Bosonization. We compare three different approximations to the low-lying spectrum: i) reading it off from the bosonized Lagrangian (neglecting interactions), ii) semi-classical quantization of the static soliton, iii) approximate semi-classical quantization of the ``breather'' solitons. A number of new points are made in this process. We also suggest a different ``effective low-energy Lagrangian'' for the theory which permits easy calculation of the low-energy scattering amplitudes. It correlates an exact mass formula of the system with the requirement of the Mermin-Wagner theorem. 
  The Maxwell-Chern-Simons gauge theory with charged scalar fields is analyzed at two loop level. The effective potential for the scalar fields is derived in the closed form, and studied both analytically and numerically. It is shown that the U(1) symmetry is spontaneously broken in the massless scalar theory. Dimensional transmutation takes place in the Coleman-Weinberg limit in which the Maxwell term vanishes. We point out the subtlety in defining the pure Chern-Simons scalar electrodynamics and show that the Coleman-Weinberg limit must be taken after renormalization. Renormalization group analysis of the effective potential is also given at two loop. 
  A global analysis of duality transformations is presented. It is shown that duality between quantum field theories exists only when the geometrical structure of the quantum configuration spaces of the theories comply with certain precise conditions. Applications to S-dual actions and to T duality of string theories and D-branes are briefly discussed.It is shown that a new topological term in the dual open string actions is required. We also study an extension of the procedure to construct duality maps among abelian gauge theories to the non abelian case. 
  A supersymmetric Yang-Mills system in (11,3) dimensions is constructed with the aid of two mutually orthogonal null vectors which naturally arise in a generalized spacetime superalgebra. An obstacle encountered in an attempt to extend this result to beyond 14 dimensions is described. A null reduction of the (11,3) model is shown to yield the known super Yang-Mills model in (10,2) dimensions. An (8,8) supersymmetric super Yang-Mills system in (3,3) dimensions is obtained by an ordinary dimensional reduction of the (11,3) model, and it is suggested there may exist a superbrane with (3,3) dimensional worldvolume propagating in (11,3) dimensions. 
  We consider intersecting M-brane solutions of supergravity in eleven dimensions. Supersymmetry turns out to be a powerful tool in obtaining such solutions and their generalizations. 
  The one-loop effective potential for gauge models in static de Sitter space at finite temperatures is computed by means of the $\zeta$--function method. We found a simple relation which links the effective potentials of gauge and scalar fields at all temperatures.   In the de Sitter invariant and zero-temperature states the potential for the scalar electrodynamics is explicitly obtained, and its properties in these two vacua are compared. In this theory the two states are shown to behave similarly in the regimes of very large and very small radii a of the background space. For the gauge symmetry broken in the flat limit ($a \to \infty$) there is a critical value of a for which the symmetry is restored in both quantum states. Moreover, the phase transitions which occur at large or at small a are of the first or of the second order, respectively, regardless the vacuum considered. The analytical and numerical analysis of the critical parameters of the above theory is performed. We also established a class of models for which the kind of phase transition occurring depends on the choice of the vacuum. 
  Based on the bosonization of vertex operators for $A_{n-1}^{(1)}$ face model by Asai,Jimbo, Miwa and Pugai, using vertex-face correspondence we obtain vertex operators for Zn symmetric Belavin model,which are constructed by deformed boson oscilllators. The correlation functions are also obtained. 
  We show that the field equations for the supercoordinates and the self--dual antisymmetric tensor field derived from the recently constructed kappa-invariant action for the M theory five-brane are equivalent to the equations of motion obtained in the doubly supersymmetric geometrical approach at the worldvolume component level. 
  We discuss three dimensional compact QED with a theta term due to an axionic field. The variational gauge invariant functional is considered and it is shown that the ground state energy is independent of theta in a leading approximation. The mass gap of the axionic field is found to be dependent upon theta, the mass gap of the photon field and the scalar potential. The vacuum expectation of the Wilson loop is shown to be independent of theta in a leading approximation, to obey the area law and to lead to confinement. We also briefly discuss the properties of axionic confining strings. 
  Starting with a conformal Quantum Field Theory on the real line, we show that the dual net is still conformal with respect to a new representation of the Moebius group. We infer from this that every conformal net is normal and conormal, namely the local von Neumann algebra associated with an interval coincides with its double relative commutant inside the local von Neumann algebra associated with any larger interval. The net and the dual net give together rise to an infinite dimensional symmetry group, of which we study a class of positive energy irreducible representations. We mention how superselsection sectors extend to the dual net and we illustrate by examples how, in general, this process generates solitonic sectors. We describe the free theories associated with the lowest weight n representations of PSL(2,R), showing that they violate 3-regularity for n>2. When n>1, we obtain examples of non Moebius-covariant sectors of a 3-regular (non 4-regular) net. 
  Although axial QED suffers from a gauge anomaly, gauge invariance may be maintained by the addition of a nonlocal counterterm. Such nonlocal conterterms, however, are expected to ruin unitarity of the theory. We explicitly investigate some relevant Feynman diagrams and show that, indeed, unitarity is violated, contrary to recent claims. 
  We identify a class of 2+1 dimensional models, involving multiple Chern-Simons gauge fields, in which a form of classical confinement occurs. This confinement is not cumulative, but allows finite mass combinations of individually confined objects, as in baryons. The occurrence and nature of the phenomena depends on number theoretic properties of the couplings and charges. 
  We show that rigid supersymmetry theories in four dimensions can be extended to give supersymmetric trace (or generalized quantum) dynamics theories, in which the supersymmetry algebra is represented by the generalized Poisson bracket of trace supercharges, constructed from fields that form a trace class noncommutative graded operator algebra. In particular, supersymmetry theories can be turned into supersymmetric matrix models this way. We demonstrate our results by detailed component field calculations for the Wess-Zumino and the supersymmetric Yang-Mills models (the latter with axial gauge fixing), and then show that they are also implied by a simple and general superspace argument. 
  The matrix model for IIB Superstring proposed by Ishibashi, Kawai, Kitazawa and Tsuchiya is investigated. Consideration of planar and non-planar diagrams suggests that the large N perturbative expansion is consistent with the double scaling limit proposed by the above authors. We write down a Wilson loop that can be interpreted as a fundamental string vertex operator. The one point tadpole in the presence of a D-string has the right form and this can be viewed as a matrix model derivation of the boundary conditions that define a D-string. We also argue that if world sheet coordinates $\sigma$ and $\tau$ are introduced for the fundamental string, then the conjugate variable ${d}/{d\sigma}$ and ${d}/{d\tau}$ can be interpreted as the D-string world sheet coordinates. In this way the $SL(2Z)$ duality group of the IIB superstring becomes identified with the symplectic group acting on ($p,q$). 
  We present duality invariant structure of the thermodynamic quantities of non-extreme black hole solutions of torodially compactified Type II (M-theory) and heterotic string in five and four dimensions. These quantities are parameterized by duality invariant combinations of charges and the non-extremality parameter, which measures a deviation from the BPS-saturated limit. In particular, in D=5 we find explicit S- and T-duality [U-duality] invariant expressions for solutions of toroidally compactified heterotic string [Type II string]. In D=4, we consider general S-duality invariant expressions for non-extreme solutions of pure N=4 supergravity and find to the leading order in non-extremality parameter the T- and S-duality invariant expressions of toroidally compactified heterotic string. General non-extreme solutions of toroidally compactified string in D=4 are awaiting further investigation. 
  Bialek, Callan and Strong have recently given a solution of the problem of determining a continuous probability distribution from a finite set of experimental measurements by formulating it as a one-dimensional quantum field theory. This letter gives a reparametrization-invariant solution of the problem, obtained by coupling to gravity. The case of a large number of dimensions may involve quantum gravity restricted to metrics of vanishing Weyl curvature. 
  Notes of my 14 `lectures on everything' given at the 1995 Les Houches school. An introductory course in topological and conformal field theory, strings, gauge fields, supersymmetry and more. The presentation is more mathematical then usual and takes a modern point of view stressing moduli spaces, duality and the interconnectedness of the subject. An apocryphal lecture on BPS states and D-branes is added. 
  The properties of the deformed bosonic oscillator, and the quantum groups ${\cal U}_q(SL(2))$ and $GL_q(2)$ in the limit as their deformation parameter $q$ goes to a root of unity are investigated and interpreted physically. These properties are seen to be related to fractional supersymmetry and intrinsic anyonic spin. A simple deformation of the Klein-Gordon equation is introduced, based on $GL_q(2)$. When $q$ is a root of unity this equation is a root of the undeformed Klein-Gordon equation. 
  We discuss different criteria for `classical size' of extremal Dirichlet p-branes in type-II supergravity. Using strong-weak coupling duality, we find that the size of the strong-coupling region at the core of the (p<3)-branes, is always given by the asymptotic string scale, if measured in the weakly coupled dual string metric. We also point out how the eleven-dimensional Planck scale arises in the classical 0-brane solution, as well as the ten-dimensional Planck scale in the D-instanton solution. 
  We use a unitary operator constructed in earlier work to transform the Hamiltonian for QCD in the temporal ($A_0=0$) gauge into a representation in which the quark field is gauge-invariant, and its elementary excitations -- quark and antiquark creation and annihilation operators -- implement Gauss's law. In that representation, the interactions between gauge-dependent parts of the gauge field and the spinor (quark) field have been transformed away and replaced by long-range non-local interactions of quark color charge densities. These long-range interactions connect SU(3) color charge densities through an infinite chain of gauge-invariant gauge fields either to other SU(3) color charge densities, or to a gluon "anchor". We discuss possible implications of this formalism for low-energy processes, including confinement of quarks that are not in color singlet configurations. 
  The V-algebras are the non-local matrix generalization of the well-known W-algebras. Their classical realizations are given by the second Poisson brackets associated with the matrix pseudodifferential operators. In this paper, by using the general Miura transformation, we give the decomposition theorems for the second Poisson brackets, from which we are able to construct the free field realizations for a class of V-algebras including V_{(2k,2)}-algebras that corresponds to the Lie algebra of C_k-type as the particular examples. The reduction of our discussion to the scalar case provides the similar result for the W_{BKP}-algebra. 
  We consider new objects in bosonic open string theory -- ND tadpoles, which have N(euman) boundary conditions at one end of the world-sheet and D(irichlet) at the other, must exist due to s-t duality in a string theory with both NN strings and D-branes. We demonstrate how to interpolate between N and D boundary conditions. In the case of mixed boundary conditions the action for a quantum particle is induced on the boundary. Quantum-mechanical particle-wave duality, a dual description of a quantum particle in either the coordinate or the momentum representation, is induced by world-sheet T-duality. The famous relation between compactification radii is equivalent to the quantization of the phase space area of a Planck cell. We also introduce a boundary operator - a ``Zipper'' which changes the boundary condition from N into D and vice versa. 
  An exactly soluble non-perturbative model of the pure gauge QCD is derived as a weak coupling limit of the lattice theory in plaquette formulation. The model represents QCD as a theory of the weakly interacting field strength fluxes. The area law behavior of the Wilson loop average is a direct result of this representation: the total flux through macroscopic loop is the additive (due to the weakness of the interaction) function of the elementary fluxes. The compactness of the gauge group is shown to be the factor which prevents the elementary fluxes contributions from cancellation. There is no area law in the non-compact theory. 
  We discuss the application of T-duality to massive supersymmetric sigma models. In particular (1,1) supersymmetric models with off-shell central charges reveal an interesting structure. The T-duality transformations of the BPS states of these theories are also discussed and an explicit example of Q-kinks is given. 
  We discuss massive scalar field with conformal coupling in Friedmann-Robertson-Walker (FRW) Universe of special type with constant electromagnetic field. Treating an external gravitational-electromagnetic background exactly, at first time the proper-time representations for out-in, in-in, and out-out scalar Green functions are explicitly constructed as proper-time integrals over the corresponding (complex) contours. The vacuum-to-vacuum transition amplitudes and number of created particles are found and vacuum instability is discussed. The mean values of the current and energy-momentum tensor are evaluated, and different approximations for them are investigated. The back reaction of the particles created to the electromagnetic field is estimated in different regimes. The connection between proper-time method and effective action is outlined. The effective action in scalar QED in weakly-curved FRW Universe (De Sitter space) with weak constant electromagnetic field is found as derivative expansion over curvature and electromagnetic field strength. Possible further applications of the results are briefly mentioned. 
  Conformal mapping techniques are used to determine analytically the geodesics on the Seiberg-Witten Riemann surface which correspond to the BPS dyon states in N=2 SUSY QCD with gauge group SU(2) and up to three flavors of massless fundamental matter. The results are exact for zero and two flavors, and approximate in the weak-coupling limit for one and three flavors. The presence of states of magnetic charge 2, in the three-flavor case only, is confirmed. 
  It is known that weak coupling calculations of absorption or emission by slightly non-extremal D-brane configurations are in exact agreement with semiclassical results for the black holes they describe at strong couplings. We investigate one open string loop corrections to processes involving single and parallel D-branes and show that a class of relevant terms vanish, indicating that these processes are not renormalized. Our results have implications for five dimensional black holes and extremal 3-branes. 
  We present, in the N=2, D=4 harmonic superspace formalism, a general method for constructing the off-shell effective action of an N=2 abelian gauge superfield coupled to matter hypermultiplets. Using manifestly N=2 supersymmetric harmonic supergraph techniques, we calculate the low-energy corrections to the renormalized one-loop effective action in terms of N=2 (anti)chiral superfield strengths. For a harmonic gauge prepotential with vanishing vacuum expectation value, corresponding to massless hypermultiplets, the only non-trivial radiative corrections to appear are non-holomorphic. For a prepotential with non-zero vacuum value, which breaks the U(1)-factor in the N=2 supersymmetry automorphism group and corresponds to massive hypermultiplets, only non-trivial holomorphic corrections arise at leading order. These holomorphic contribution are consistent with Seiberg's quantum correction to the effective action, while the first non-holomorphic contribution in the massless case is the N=2 supersymmetrization of the Heisenberg-Euler effective Lagrangian. 
  We construct, as hypersurfaces in toric varieties, Calabi-Yau manifolds corresponding to F-theory vacua dual to E8*E8 heterotic strings compactified to six dimensions on K3 surfaces with non-semisimple gauge backgrounds. These vacua were studied in the recent work of Aldazabal, Font, Ibanez and Uranga. We extend their results by constructing many more examples, corresponding to enhanced gauge symmetries, by noting that they can be obtained from previously known Calabi-Yau manifolds corresponding to K3 compactification of heterotic strings with simple gauge backgrounds by means of extremal transitions of the conifold type. 
  We construct the Lagrangeans of N=3 and N=4 two-form supergravities. The two-form gravity theories are classically equivalent to the Einstein gravity theories and can be formulated as gauge theories. The gauge algebras used here can be identified with the subalgebra of N=3 superconformal algebra and SU(2)\times SU(2)\times U(1)-extended N=4 superconformal algebra. 
  An inhomogeneous version of pre--Big Bang cosmology emerges, within string theory, from quite generic initial conditions, provided they lie deeply inside the weak-coupling, low-curvature regime. Large-scale homogeneity, flatness, and isotropy appear naturally as late-time outcomes of such an evolution. 
  The general, linear equations with constant coefficients on quantum Minkowski spaces are considered and the explicit formulae for their conserved currents are given. The proposed procedure can be simplified for *-invariant equations. The derived method is then applied to Klein-Gordon, Dirac and wave equations on different classes of Minkowski spaces. In the examples also symmetry operators for these equations are obtained. They include quantum deformations of classical symmetry operators as well as an additional operator connected with deformation of the Leibnitz rule in non-commutative differential calculus. 
  The first order formalism for 3D Yang-Mills theory is considered and two different formulations are introduced, in which the gauge theory appears to be a deformation of the topological BF theory. We perform the quantization and the algebraic analysis of cohomology and renormalization for both the models, which are found to be anomaly free. We discuss also their stability against radiative corrections, giving the full structure of possible counterterms, requiring an involved matricial renormalization of fields and sources. 
  N flavor QED in two dimensions is reduced to a quantum mechanics problem with N degrees of freedom for which the potential is determined by the ground state of the problem itself. The chiral condensate is determined at all values of temperature, fermion masses, and the $\theta$ parameter. In the single flavor case, the anomalous mass (m) dependence of the chiral condensate at $\theta=\pi$ at low temperature is found. The critical value is given by $m_c \sim .437 e/\sqrt{\pi}$. 
  We give a gauge and manifestly SO(2,2) covariant formulation of the field theory of the self-dual string. The string fields are gauge connections that turn the super-Virasoro generators into covariant derivatives. 
  Starting with free massless scalar and spinor fields described by a globally N=1 supersymmetric action, infalling on a Schwarzschild black hole, the outgoing Hawking radiation is shown to break supersymmetry spontaneously, exactly as induced by a heat bath in Minkowski space, with no generation of Nambu-Goldstone fermions. 
  The absolute (moduli-independent) U-invariants of all N>2 extended supergravities at D=4 are derived in terms of (moduli-dependent) central and matter charges. These invariants give a general definition of the ``topological'' Bekenstein-Hawking entropy formula for extremal black-holes and reduce to the square of the black-hole ADM mass for ``fixed scalars'' which extremize the black-hole ``potential'' energy. The Hessian matrix of the black-hole potential at ``fixed scalars'', in contrast to N=2 theories, is shown to be degenerate, with rank (N-2)(N-3) + 2 n (N being the number of supersymmetries and n the number of matter multiplets) and semipositive definite. 
  We study certain compactifications of the type I string on K3. The three topologically distinct choices of gauge bundle for the type I theory are shown to be equivalent to type IIB orientifolds with different choices of background anti-symmetric tensor field flux. Using a mirror transformation, we relate these models to orientifolds with fixed seven planes, and without any antisymmetric tensor field flux. This map allows us to relate these type I vacua to particular six-dimensional F theory and heterotic string compactifications. 
  Using U_q(a_n^(1))- and U_q(a_2n^(2))-invariant R-matrices we construct exact S-matrices in two-dimensional space-time. These are conjectured to describe the scattering of solitons in affine Toda field theories. In order to find the spectrum of soliton bound states we examine the pole structure of these S-matrices in detail. We also construct the S-matrices for all scattering processes involving scalar bound states. In the last part of this paper we discuss the connection of these S-matrices with minimal N=1 and N=2 supersymmetric S-matrices. In particular we comment on the folding from N=2 to N=1 theories. 
  We review mechanisms of dynamical supersymmetry breaking. Several observations that narrow the search for possible models of dynamical supersymmetry breaking are summarized. These observations include the necessary and sufficient conditions for supersymmetry breaking. The two conditions are based on non-rigorous arguments, and we show examples where they are too restrictive. Dynamical effects present in models with product gauge groups are given special attention. 
  We consider dimensional reduction of the eleven-dimensional supergravity to less than four dimensions. The three-dimensional $E_{8(+8)}/SO(16)$ nonlinear sigma model is derived by direct dimensional reduction from eleven dimensions. In two dimensions we explicitly check that the Matzner-Misner-type $SL(2,R)$ symmetry, together with the $E_8$, satisfies the generating relations of $E_9$ under the generalized Geroch compatibility (hypersurface-orthogonality) condition. We further show that an extra $SL(2,R)$ symmetry, which is newly present upon reduction to one dimension, extends the symmetry algebra to a real form of $E_{10}$. The new $SL(2,R)$ acts on certain plane wave solutions propagating at the speed of light. To show that this $SL(2,R)$ cannot be expressed in terms of the old $E_9$ but truly enlarges the symmetry, we compactify the final two dimensions on a two-torus and confirm that it changes the conformal structure of this two-torus. 
  The critical points of the continuous series are characterized by two complex numbers l_1,l_2 (Re(l_1,l_2)< 0), and a natural number n (n>=3) which enters the string susceptibility constant through gamma = -2/(n-1). The critical potentials are analytic functions with a convergence radius depending on l_1 or l_2. We use the orthogonal polynomial method and solve the Schwinger-Dyson equations with a technique borrowed from conformal field theory. 
  We discuss how to represent the non-associative octonionic structure in terms of the associative matrix algebra using the left and right octonionic operators. As an example we construct explicitly some Lie and Super Lie algebra. Then we discuss the notion of octonionic Grassmann numbers and explain its possible application for giving a superspace formulation of the minimal supersymmetric Yang-Mills models. 
  We address the derivation of the effective conformal field theory description of the 5-dimensional black hole, modelled by a collection of D1- and D5- branes, from the corresponding low energy U(Q_1)xU(Q_5) gauge theory. Finite horizon size at weak coupling requires both Q_1 and Q_5 to be large. We derive the result in the moduli space approximation (say for Q_1>Q_5) and appeal to supersymmetry to argue its validity beyond weak coupling. As a result of a combination of quenched Z_{Q_1} Wilson lines and a residual Weyl symmetry, the low-lying excitations of the U(Q_1)xU(Q_5) gauge theory are described by an effective N=4 superconformal field theory with c=6 in 1+1 dimensions, where the space is a circle of radius RQ_1Q_5. We also discuss the appearance of a marginal perturbation of the effective conformal field theory for large but finite values of Q_5. 
  We generalize a family of Lagrangians with values in the Poincar\'e group ISO(d-1,1), which contain the description of spinning strings in flat (d-1)+1 dimensions, by including symmetric terms in the world-sheet coordinates. Then, by promoting a subgroup H=R^n, n less than or equal to d, which acts invariantly from the left on the element of ISO(d-1,1), to a gauge symmetry of the action, we obtain a family of sigma-models. They describe bosonic strings moving in (generally) curved, and in some cases degenerate, space-times with an axion field. Further, the space-times of the effective theory admit in general T-dual geometries. We give explicit results for two non degenerate cases. 
  We extend to a general class of covariant gauges an approach which relates the thermal Green functions to forward scattering amplitudes of thermal particles. A brief discussion of the non-transversality of the thermal gluon polarization tensor is given in this context. This method is then applied to the calculation of the ln(T) contributions associated with general configurations of 2 and 3-point gluon functions. The results are Lorentz covariant and have the same structure as the ultraviolet divergent contributions which occur at zero temperature. 
  N=2 supersymmetric gauge theories in four dimensions are studied by formulating them as the quantum field theories derived from configurations of fourbranes, fivebranes, and sixbranes in Type IIA superstrings, and then reinterpreting those configurations in M theory. This approach leads to explicit solutions for the Coulomb branch of a large family of four-dimensional N=2 field theories with zero or negative beta function. 
  We show the existence of global gauge anomalies in six dimensions for gauge groups SU(2),SU(3) and G_2 coupled to matter, characterized by an element of Z_{12},Z_6 and Z_3 respectively. Consideration of this anomaly rules out some of the recently proposed 6 dimensional N=1 QFT's which were conjectured to possess IR fixed point at infinite coupling. We geometrically engineer essentially all the other models with one tensor multiplet using F-theory. In addition we construct 3 infinite series using F-theory geometry which do not have field theory analogs. All these models in the maximally Higgsed phase correspond to the strong coupling behaviour of E_8 x E_8 heterotic string compactification on K3 with instanton numbers (12+n,12-n). 
  We show that collective dynamics of a curved domain wall in a (3+1)-dimensional relativistic scalar field model is represented by Nambu-Goto membrane and (2+1)-dimensional scalar fields defined on the worldsheet of the membrane. Our argument is based on a recently proposed by us version of the expansion in the width. Derivation of the expansion is significantly reformulated for the present purpose. Third and fourth order corrections to the domain wall solution are considered. We also derive an equation of motion for the core of the domain wall. Without the (2+1)-dimensional scalar fields this equation would be nonlocal. 
  It is studied time dependence of the evolution operator kernel for the Schr\"odinger equation with a help of the Schwinger -- DeWitt expansion. For many of potentials this expansion is divergent. But there were established nontrivial potentials for which the Schwinger -- DeWitt expansion is convergent. These are, e.g., V=g/x^2, V=-g/cosh^2 x, V=g/sinh^2 x, V=g/sin^2 x. For all of them the expansion is convergent when $g=\lambda (\lambda -1)/2$ and $\lambda$ is integer. The theories with these potentials have no divergences and in this meaning they are "good" potentials contrary to other ones. So, it seems natural to pay special attention namely to these "good" potentials. Besides convergence they have other interesting feature: convergence takes place only for discrete values of the charge $g$. Hence, in the theories of this class the charge is quantized. 
  Using the differential equation approach to W-algebras, we discuss the inclusion of punctures in W-string theory. The key result is the existence of different kinds of punctures in W-strings. This is similar to the NS and R punctures occuring in superstring theories. We obtain the moduli associated with these punctures and present evidence in existing W-string theories for these punctures. The $W_3$ case is worked out in detail. It is conjectured that the $(1,3)$ minimal model coupled to two dimensional gravity corresponds to topological $W_3$-gravity. 
  The instanton-induced effective vertex is derived for N=2 supersymmetric QCD (SQCD) with arbitrary mass matter hypermultiplets for the case of SU(2). The leading term of the low energy effective lagrangian obtained from this vertex agrees with one-instanton effective term of the Seiberg-Witten result. 
  We analyze three dimensional gauge theories with $Sp$ gauge group. We find that in some regime the theory should be described in terms of a dual theory, very much in the spirit of Seiberg duality in four dimensions. This duality does not coincide with mirror symmetry. 
  We give here a covariant definition of the path integral formalism for the Lagrangian, which leaves a freedom to choose anyone of many possible quantum systems that correspond to the same classical limit without adding new potential terms nor searching for a strange measure, but using as a framework the geometry of the spaces considered. We focus our attention on the set of paths used to join succesive points in the discretization if the time-slicing definition is used to calculate the integral.If this set of paths is not preserved when performing a point transformation, the integral may change. The reasons for this are geometrically explained. Explicit calculation of the Kernel in polar coordinates is made, yielding the same system as in Cartesian coordinates. 
  A Chern-Simons theory in 11 dimensions, which is a piece of the 11 dimensional supergravity action, is considered as a quantum field theory in its own right. We conjecture that it defines a non-perturbative phase of M theory in which the metric and gravitino vanish. The theory is diffeomorphism invariant but not topological in that there are local degrees of freedom. Nevertheless, there are a countable number of momentum variables associated with relative cobordism classes of embeddings of seven dimensional manifolds in ten dimensional space. The canonical theory is developed in terms of an algebra of gauge invariant observables. We find a sector of the theory corresponding to a topological compactification in which the geometry of the compactified directions is coded in an algebra of functions on the base manifold. The diffeomorphism invariant quantum theory associated to this sector is constructed, and is found to describe diffeomorphism classes of excitations of three surfaces wrapping homology classes of the compactified dimensions. 
  I now agree with conclusion of author that there is a problem with unitarity in discussed models. 
  A concise presentation of the PF equations for N=2 Seiberg-Witten theories for the classical groups of rank r with N_f massless hypermultiplets in the fundamental representation is provided. For N_f=0, all r PF equations can be given in a generic form. For certain cases with N_f\neq zero, not all equations are generic. However, in all cases there are at least r-2 generic PF equations. For these cases the classical part of the equations is generic, while the quantum part can be formulated using a method described in a previous paper by the authors, which is well suited to symbolic computer calculations. 
  We consider the loop quantization of Maxwell theory. A quantization of this type leads to a quantum theory in which the fundamental excitations are loop-like rather than particle-like. Each such loop plays the role of a quantized Faraday's line of electric flux. We find that the quantization depends on an arbitrary choice of a parameter e that carries the dimension of electric charge. For each value of e an electric charge that can be contained inside a bounded spatial region is automatically quantized in units of hbar/4*pi*e. The requirement of consistency with the quantization of electric charge observed in our Universe fixes a value of the, so far arbitrary, parameter e of the theory. Finally, we compare the ambiguity in the choice of parameter e with the beta-ambiguity that, as pointed by Immirzi, arises in the loop quantization of general relativity, and comment on a possible way this ambiguity can be fixed. 
  The mechanism of generation of the Bekenstein-Hawking entropy $S^{BH}$ of a black hole in the Sakharov's induced gravity is proposed. It is suggested that the "physical" degrees of freedom, which explain the entropy $S^{BH}$, form only a finite subset of the standard Rindler-like modes defined outside the black hole horizon. The entropy $S_R$ of the Rindler modes, or entanglement entropy, is always ultraviolet divergent, while the entropy of the "physical" modes is finite and it coincides in the induced gravity with $S^{BH}$. The two entropies $S^{BH}$ and $S_R$ differ by a surface integral Q interpreted as a Noether charge of non-minimally coupled scalar constituents of the model. We demonstrate that energy E and Hamiltonian H of the fields localized in a part of space-time, restricted by the Killing horizon $\Sigma$, differ by the quantity $T_H Q$, where $T_H$ is the temperature of a black hole. The first law of the black hole thermodynamics enables one to relate the probability distribution of fluctuations of the black hole mass, caused by the quantum fluctuations of the fields, to the probability distribution of "physical" modes over energy E. The latter turns out to be different from the distribution of the Rindler modes. We show that the probability distribution of the "physical" degrees of freedom has a sharp peak at E=0 with the width proportional to the Planck mass. The logarithm of number of "physical" states at the peak coincides exactly with the black hole entropy $S^{BH}$. It enables us to argue that the energy distribution of the "physical" modes and distribution of the black hole mass are equivalent in the induced gravity. Finally it is shown that the Noether charge Q is related to the entropy of the low frequency modes propagating in the vicinity of the bifurcation surface $\Sigma$ of the horizon. 
  This paper illustrates the derivation of the low-energy background field solutions of D2-branes and D4-branes intersecting at non-trivial angles by solving directly the bosonic equations of motion of II supergravity coupled to a dilaton and antisymmetric fields. We also argue for how a similar analysis can be performed for any similar Dp-branes oriented at angles. Finally, the calculation presented here serves as a basis in the search for a systematic derivation of the background fields of the more general configuration of a p-brane `angled' with a q-brane ($p \neq q$). 
  We study the low-energy effective theory of N=2 supersymmetric Yang-Mills theory with the exceptional gauge group $G_{2}$. We obtain the Picard-Fuchs equations for the $G_{2}$ spectral curve and compute multi-instanton contribution to the prepotential. We find that the spectral curve is consistent with the microscopic supersymmetric instanton calculus. It is also shown that $G_{2}$ hyperelliptic curve does not reproduce the microscopic result. 
  Embedding of Klein-Gordon and Dirac particle onto Riemannian submanifold in higher dimensional Minkowski space is given by using Hamiltonian BRST formalism. Up to the ordering and quantum potential term induced by embedding, obtained K-G equation is the usual one in Riemannian space, instead, the obtained Dirac equation is essentially different from the usual well known form using vierbein. The requirement of equivalence between two Dirac equations gives the property of natural-frame for spinor. 
  We review several aspects of heterotic, type II, F-theory, and M-theory compactifications on Calabi-Yau threefolds and fourfolds. In the context of dualities we focus on the heterotic gauge structure determined by the various types of fibration relevant in the framework of heterotic/type II duality in D=4 as well as 4D F-theory. We also consider transitions between Calabi-Yau manifolds in both three and four dimensions and review some of the consequences for the behavior of the superpotential. 
  We present a new geometrical approach to superstrings based on the geometrical theory of integration on supermanifolds. This approach provides an effective way to calculate multi-loop superstring amplitudes for arbitrary backgrounds. It makes possible to calculate amplitudes for the physical states defined as BRST cohomology classes using arbitrary representatives. Since the new formalism does not rely on the presence of primary representatives for the physical states it is particulary valuable for analyzing the discrete states for which no primary representatives are available. We show that the discrete states provide information about symmetries of the background including odd symmetries which mix Bose and Fermi states. The dilaton is an example of a non-discrete state which cannot be covariantly represented by a primary vertex. The new formalism allows to prove the dilaton theorem by a direct calculation. 
  The interaction of a spin 1/2 particle (described by the non-relativistic "Dirac" equation of L\'evy-Leblond) with Chern-Simons gauge fields is studied. It is shown, that similarly to the four dimensional spinor models, there is a consistent possibility of coupling them also by axial or chiral type currents. Static self dual vortex solutions together with a vortex-lattice are found with the new couplings. 
  We discuss the N=2 supersymmetric extension of the gauged O(3) sigma model in (2+1) dimensions with an abelian Chern-Simons term. It is shown that the self-dual potential and the Bogomolny relations naturally appear as consequences of extended supersymmetry. 
  A closed form of the Picard-Fuchs equations for N=2 supersymmetric Yang-Mills theories with massless hypermultiplet are obtained for classical Lie gauge groups. We consider any number of massless matter in fundamental representation so as to keep the theory asymptotically free. 
  The Casimir energy for the transverse oscillations of a piecewise uniform closed string is calculated. The string consists of 2N pieces of equal length, of alternating type I and type II material, and is taken to be relativistic in the sense that the velocity of sound always equals the velocity of light. By means of a new recursion formula we manage to calculate the Casimir energy for arbitrary integers N. Agreement with results obtained in earlier works on the string is found in all special cases. As basic regularization method we use the contour integration method. As a check, agreement is found with results obtained from the \zeta function method (the Hurwitz function) in the case of low N (N = 1-4). The Casimir energy is generally negative, and the more so the larger is the value of N. We illustrate the results graphically in some cases. The generalization to finite temperature theory is also given. 
  We give a complete classification of the real forms of simple nonlinear superconformal algebras (SCA) and quasi-superconformal algebras (QSCA) and present a unified realization of these algebras with simple symmetry groups. This classification is achieved by establishing a correspondence between simple nonlinear QSCA's and SCA's and quaternionic and super-quaternionic symmetric spaces of simple Lie groups and Lie supergroups, respectively. The unified realization involves a dimension zero boson (dilaton), dimension one symmetry currents and dimension 1/2 free bosons for QSCA'a and dimension 1/2 free fermions for SCA's. The dimension 1/2 free bosons and fermions are associated with the quaternionic and super-quaternionic symmetric spaces of corresponding Lie groups and Lie supergroups, respectively. We conclude with a discussion of possible applications of our results. 
  We present a complex matrix gauge model defined on an arbitrary two-dimensional orientable lattice. We rewrite the model's partition function in terms of a sum over representations of the group U(N). The model solves the general combinatorial problem of counting branched covers of orientable Riemann surfaces with any given, fixed branch point structure. We then define an appropriate continuum limit allowing the branch points to freely float over the surface. The simplest such limit reproduces two-dimensional chiral U(N) Yang-Mills theory and its string description due to Gross and Taylor. 
  An integral representation for form-factors of exponential fields in the sine-Gordon model is proposed. 
  N=1, d=4 superconformal group is studied and its representations are discussed. Under superconformal transformations, left invariant derivatives and some class of superfields, including supercurrents, are shown to follow these representations. In other words, these superfields are quasi-primary by analogy with two dimensional conformal field theory. Based on these results, we find the general forms of the two-point and the three-point correlation functions of the quasi-primary superfields in a group theoretical way. In particular, we show that the two-point function of the supercurrent is unique up to a constant and the general form of the three-point function of the supercurrent has two free parameters. 
  We demonstrate linear stability of the dilatonic Black Holes appearing in a string-inspired higher-derivative gravity theory with a Gauss- Bonnet curvature-squared term. The proof is accomplished by mapping the system to a one-dimensional Schrodinger problem which admits no bound states. This result is important in that it constitutes a linearly stable example of a black hole that bypasses the `no-hair conjecture'. However, the dilaton hair is `secondary'in the sense that it is not accompanied by any new quantum number for the black hole solution. 
  We consider a uniaxial planar ferromagnet coupled minimally to an Abelian Chern-Simons gauge field and study self-dual solitons which saturate the Bogomol'nyi bound. We find a rich structure of rotationally symmetric static soliton solutions for various uniform background charge densities. For a given ferromagnet material, the properties of these solitons are controlled only by the external magnetic field and the background charge. 
  We discuss N=2 supersymmetric compactifications to four dimensions from the point of view of F-theory and heterotic theory. In a relatively simple setup, we illustrate the spectral theory for vector bundles on K3xT^2 and discuss the heterotic -- F-theory map. The moduli space of instantons on the 7-brane wrapped around K3 is discussed from the point of view of Higgs mechanism in the effective four-dimensional N=2 theory. This allows us to elaborate on the transitions between various branches of the moduli space of the heterotic/F-theory. We also describe the F-theory compactification on smooth K3xK3 without the 3-branes in which the anomaly is cancelled by the gauge fields. 
  We consider special geometry of the vector multiplet moduli space in compactifications of the heterotic string on $K3 \times T^2$ or the type IIA string on $K3$-fibered Calabi-Yau threefolds. In particular, we construct a modified dilaton that is invariant under $SO(2, n; Z)$ T-duality transformations at the non-perturbative level and regular everywhere on the moduli space. The invariant dilaton, together with a set of other coordinates that transform covariantly under $SO(2, n; Z)$, parameterize the moduli space. The construction involves a meromorphic automorphic function of $SO(2, n; Z)$, that also depends on the invariant dilaton. In the weak coupling limit, the divisor of this automorphic form is an integer linear combination of the rational quadratic divisors where the gauge symmetry is enhanced classically. We also show how the non-perturbative prepotential can be expressed in terms of meromorphic automorphic forms, by expanding a T-duality invariant quantity both in terms of the standard special coordinates and in terms of the invariant dilaton and the covariant coordinates. 
  The weak energy condition is known to fail in general when applied to expectation values of the the energy momentum tensor in flat space quantum field theory. It is shown how the usual counter arguments against its validity are no longer applicable if the states $|\psi \r$ for which the expectation value is considered are restricted to a suitably defined subspace. A possible natural restriction on $|\psi \r$ is suggested and illustrated by two quantum mechanical examples based on a simple perturbed harmonic oscillator Hamiltonian. The proposed alternative quantum weak energy condition is applied to states formed by the action of scalar, vector and the energy momentum tensor operators on the vacuum. We assume conformal invariance in order to determine almost uniquely three-point functions involving the energy momentum tensor in terms of a few parameters. The positivity conditions lead to non trivial inequalities for these parameters. They are satisfied in free field theories, except in one case for dimensions close to two. Further restrictions on $|\psi \r$ are suggested which remove this problem. The inequalities which follow from considering the state formed by applying the energy momentum tensor to the vacuum are shown to imply that the coefficient of the topological term in the expectation value of the trace of the energy momentum tensor in an arbitrary curved space background is positive, in accord with calculations in free field theories. 
  We discuss the introduction of soft breaking terms into the exact solutions of N=1 SQCD using a spurion analysis. The spurion symmetries are not sufficient to determine the behavior of models in which squark or gaugino masses alone are introduced. However, a controlled approximation is obtained in some cases if a supersymmetric mass is first introduced for the matter fields. We present low-energy solutions for two models with perturbing soft breaking terms, one with a gaugino mass and one with squark mixing. These models have non-trivial theta angle dependence and exhibit phase transitions at non-zero theta angle analogous to those found in the chiral Lagrangian description of QCD. 
  We show the existence of a supersymmetry-breaking mechanism in string theory, where N=4 supersymmetry is broken spontaneously to N=2 and N=1 with moduli-dependent gravitino masses. The BPS spectrum of the theory with lower supersymmetry is in one-to-one correspondence with the spectrum of the heterotic N=4 string. The mass splitting of the N=4 spectrum depends on the moduli as well as the three R-symmetry charges. In the case of N=4 \to N=2, the perturbative N=2 prepotential is determined by the perturbative N=4 BPS states. This observation led us to suggest a method that determines the exact non-perturbative prepotential of the effective N=2 supergravity using the shifted spectrum of the non-perturbative BPS states of the underlying N=4 theory. 
  We consider the description of second-class constraints in a Lagrangian path integral associated with a higher-order $\Delta$-operator. Based on two conjugate higher-order $\Delta$-operators, we also propose a Lagrangian path integral with $Sp(2)$ symmetry, and describe the corresponding system in the presence of second-class constraints. 
  We study the nonrelativistic limit of the theory of a quantum Chern--Simons field minimally coupled to Dirac fermions. To get the nonrelativistic effective Lagrangian one has to incorporate vacuum polarization and anomalous magnetic moment effects. Besides that, an unsuspected quartic fermionic interaction may also be induced. As a by product, the method we use to calculate loop diagrams, separating low and high loop momenta contributions, allows to identify how a quantum nonrelativistic theory nests in a relativistic one. 
  We describe a way to compute scattering amplitudes in M(atrix) quantum mechanics, that involve the transverse five-brane. We then compute certain scattering processes and show that they have the expected SO(5) invariance, give the correct transverse-five-brane mass, and agree with the supergravity result. 
  In first-quantized string theory, spacetime symmetries are described by inner automorphisms of the underlying conformal field theory. In this paper we use this approach to illustrate the Higgs effect in string theory. We consider string propagation on M^{24,1} \times S^1, where the circle has radius R, and study SU(2) symmetry breaking as R moves away from its critical value. We find a gauge-covariant equation of motion for the broken-symmetry gauge bosons and the would-be Goldstone bosons. We show that the Goldstone bosons can be eliminated by an appropriate gauge transformation. In this unitary gauge, the Goldstone bosons become the longitudinal components of massive gauge bosons. 
  We review here a path-integral approach to classical mechanics and explore the geometrical meaning of this construction. In particular we bring to light a universal hidden BRS invariance and its geometrical relevance for the Cartan calculus on symplectic manifolds. Together with this BRS invariance we also show the presence of a universal hidden genuine non-relativistic supersymmetry. In an attempt to understand its geometry we make this susy local following the analogous construction done for the supersymmetric quantum mechanics of Witten. 
  We start with BPS-saturated configurations of two (orthogonally) intersecting M-branes and use the electro-magnetic duality or dimensional reduction along a boost, in order to obtain new p-brane bound states. In the first case the resulting configurations are interpreted as BPS-saturated non-threshold bound states of intersecting p-branes, and in the second case as p-branes intersecting at angles and their duals. As a by-product we deduce the enhancement of supersymmetry as the angle approaches zero. We also comment on the D-brane theory describing these new bound states, and a connection between the angle and the world-volume gauge fields of the D-brane system. We use these configurations to find new embeddings of the four and five dimensional black holes with non-zero entropy, whose entropy now also depends on the angle and world-volume gauge fields. The corresponding D-brane configuration sheds light on the microscopic entropy of such black holes. 
  A straightforward derivation of the effective quark Lagrangian is presented for the topologically neutral chiral broken phase of the dilute instanton gas. The resulting quark Lagrangian is a nonlocal NJL-type and contains 4q, 6q,... vertices for any number of flavours. Correspondence of this result with previously known in literature is discussed in detail. 
  String theory one-loop threshold corrections are studied in a background field approach due to Kiritsis and Kounnas which uses space-time curvature as an infrared regulator. We review the conformal field theory aspects using the semiwormhole space-time solution. The comparison of string and effective field theories vacuum functionals is made for the low derivative order, as well as for certain higher-derivative, gauge and gravitational interactions. We study the dependence on the infrared cut-off. Numerical applications are considered for a sample of four-dimensional abelian orbifold models. The implications on the perturbative string theory unification are examined. We present numerical results for the gauge interactions coupling constants as well as for the quadratic order gravitational ($R^2$) and the quartic order gauge ($F^4$) interactions. 
  We study the bosonic super Liouville system which is a statistical transmutation of super Liouville system. Lax pair for the bosonic super Liouville system is constructed using prolongation method, ensuring the Lax integrability, and the solution to the equations of motion is also considered via Leznov-Saveliev analysis. 
  We study the T^5/Z_2 orbifold compactification of the M-theory matrix model. This model was originally studied by Dasgupta, Mukhi, and Witten. It was found that one had to add 16 5-branes to the system to make the compactification consistent. We demonstrate how this is mimicked in the matrix model. 
  We describe the structure of string vacuum states in the supersymmetric matrix model for M theory compactified on a circle in the large-N limit. We show that the theory admits topological instanton field configurations which at short-distance scales reduce to ordinary Yang-Mills instantons that interpolate between degenerate vacua of the theory. We show that there exists further classes of hadronic strings associated with the D-string super-fields. We discuss the relationships between these non-perturbative string states and rigid QCD strings, critical strings, and membrane states. 
  Recently, a D-brane construction in type IIA string theory was shown to yield the electric/magnetic duality of four dimensional N=1 supersymmetric U(N_c) gauge theories with N_f flavours of quark. We present here an extension of that construction which yields the electric/magnetic duality for the SO(N_c) and USp(N_c) gauge theories with N_f quarks, by adding an orientifold plane which is consistent with the supersymmetry. Due to the intersection of the orientifold plane with the NS-NS fivebranes already present, new features arise which are crucial in determining the correct final structure of the dualities. 
  Four and five dimensional extremal black holes with nonzero entropy have simple presentations in M-theory as gravitational waves bound to configurations of intersecting M-branes. We discuss realizations of these objects in matrix models of M-theory, investigate the properties of zero-brane probes, and propose a measure of their internal density. A scenario for black hole dynamics is presented. 
  We construct a model for n-level atoms coupled to quantized electromagnetic fields in a fibrillar geometry. In the slowly varying envelope and rotating wave approximations, the equations of motion are shown to satisfy a zero curvature representation, implying integrability of the quantum system. 
  The left-right symmetric gauge model with the symmetry of $SU(3)_c\times SU(2)_L\times SU(2)_R\times U(1)$ is reconstructed in a new scheme of the noncommutative differential geometry (NCG) on the discrete space $M_4\times Z_4$ which is a product space of Minkowski space and four points space. The characteristic point of this new scheme is to take the fermion field to be a vector in a 24-dimensional space which contains all leptons and quarks. Corresponding to this specification, all gauge and Higgs boson fields are represented in $24\times 24$ matrix forms. We incorporate two Higgs doublet bosons $h$ and $SU(2)_R$ adjoint Higgs $\xi_R$ which are as usual transformed as $(2,2^\ast,0)$ and $(1,3,-2)$ under $SU(2)_L\times SU(2)_R\times U(1)$, respectively. Owing to the revise of the algebraic rules in a new NCG, we can obtain the necessary potential and interacting terms between these Higgs bosons which are responsible for giving masses to the particles included. Among the Higgs doublet bosons, one CP-even scalar boson survives in the weak energy scale and other four scalar bosons acquire the mass of the $SU(2)_R\times U(1)$ breaking scale, which is similar to the situation in the standard model. $\xi_R$ is responsible to spontaneously break $SU(2)\ma{R} \times U(1)$ down to $U(1)\ma{Y}$ and so well explains the seesaw mechanism. Up and down quarks have the different masses through the vacuum expectation value of $h$. 
  We present a consistent theory of N=1 supergravity in twelve-dimensions with the signature (10,2). Even though the formulation uses two null vectors violating the manifest Lorentz covariance, all the superspace Bianchi identities are satisfied. After a simple dimensional reduction to ten-dimensions, this theory reproduces the N=1 supergravity in ten-dimensions, supporting the consistency of the system. We also show that our supergravity can be the consistent backgrounds for heterotic or type-I superstring in Green-Schwarz formulation, by confirming the kappa-invariance of the total action. This theory is supposed to be the purely N=1 supergravity sector for the field theory limit of the recently predicted F-theory in twelve-dimensions. 
  We suggest IR-dual descriptions for d=3 N=2 supersymmetric gauge theories with gauge groups USp(2N_c) and U(N_c) and matter in the fundamental representation. We relate this duality to the IR duality of d=4 N=1 SQCD theories, and in one case also to mirror symmetry. 
  We solve the problem of mixing between the fixed scalar and metric fluctuations. First, we derive the decoupled fixed scalar equation for the four-dimensional black hole with two different charges. We proceed to the five-dimensional black hole with different electric (1-brane) and magnetic (5-brane) charges, and derive two decoupled equations satisfied by appropriate mixtures of the original fixed scalar fields. The resulting greybody factors are proportional to those that follow from coupling to dimension (2,2) operators on the effective string. In general, however, the string action also contains couplings to chiral operators of dimension (1,3) and (3,1), which cause disagreements with the semiclassical absorption cross-sections. Implications of this for the effective string models are discussed. 
  We consider the spectra of excitations around diagonal and intersecting D-brane configurations on tori. These configurations are described by constant curvature connections in a dual gauge theory description. The low-energy string fluctuation spectrum is reproduced exactly by the gauge theory in the case of vanishing field strength; however, this correspondence breaks down for fixed nonzero field strength. We show that in many cases the full Born-Infeld action correctly captures the low-energy spectrum in the case of non-vanishing field strength. This gives a field theory description of the low-energy physics of systems of diagonally wound branes and branes at angles as considered by Berkooz, Douglas and Leigh. This description extends naturally to non-supersymmetric configurations, where the tachyonic instability associated with brane-anti-brane systems appears as an instability around a saddle point solution of the corresponding Yang-Mills/Born-Infeld theory. In some cases, the field theory description requires a non-abelian generalization of the Born-Infeld action. We follow Tseytlin's recent proposal for formulating such an action. In the case of intersecting branes, the non-abelian Born-Infeld theory produces a transcendental relation which comes tantalizingly close to reproducing the correct spectrum; however, a discrepancy remains which indicates that a further clarification of the non-abelian Born-Infeld action may be necessary. 
  We propose a nonperturbative definition of heterotic string theory on arbitrary multidimensional tori. 
  Differential regularization is used to investigate the one-loop quantum corrections to Chern-Simons-Maxwell spinor and scalar electrodynamics. We illustrate the techniques to write the loop amplitudes in coordinate space. The short-distance expansion method is developed to perform the Fourier transformation of the amplitudes into momentum space and the possible renormalization ambiguity in Chern-Simons type gauge theories in terms of differential regularization is discussed. We also stress that the surface terms appearing in the differential regularization should be kept along for finite theories and they will result in the finite renormalization ambiguity. 
  Callan-Symanzik and renormalization group equation are discussed for the $U(1)$-axial model and it is shown that the symmetric model is not the asymptotic version of the spontaneously broken one due to mass logarithms in the $\beta$-functions. The Callan-Symanzik equation of the standard model is seen to have the same form as the one of the simple model. 
  Rigid gauge invariance comprises the symmetry content for physical quantities in a local gauge theory. Its derivation from BRS invariance is thus crucial for determining the physical consequences of the symmetry. 
  Using a unified and systematic scheme, the free field realization of irreducible representations of osp(2|2) is constructed. By using these realization, the correlation functions of N=2 super-conformal model based on osp(2|2) symmetry and free field representation of ${\hat{osp(2|2)}}$ generators are calculated. Free field representation of currents are used to determine the stress-energy tensor and the central charge of the model. 
  A method for calculation of the DWSG coefficients for operators in spaces with metric incompatible with connection is suggested based on a generalization of the pseudodifferential operators technique. By using the proposed method, the lowest DWSG coefficients are calculated for minimal operators of the second and fourth order and for nonminimal operators of the type $H^{\mu\nu} = - g^{\mu\nu} g_{\alpha\beta}\nabla^{\alpha}\nabla^{\beta} + a\nabla^{\mu}\nabla^{\nu} + X^{\mu\nu}$ in spaces with metric incompatible with connection. 
  N=2 extension of affine algebra $\hat{sl(2)\oplus u(1)}$ possesses a hidden global N=4 supersymmetry and provides a second hamiltonian structure for a new N=4 supersymmetric integrable hierarchy defined on N=2 affine supercurrents. This system is an N=4 extension of at once two hierarchies, N=2 NLS and N=2 mKdV ones. It is related to N=4 KdV hierarchy via a generalized Sugawara-Feigin-Fuks construction which relates N=2 $\hat{sl(2)\oplus u(1)}$ algebra to ``small'' N=4 SCA. We also find the underlying affine hierarchy for another integrable system with the N=4 SCA second hamiltonian structure, ``quasi'' N=4 KdV hierarchy. It respects only N=2 supersymmetry. For both new hierarchies we construct scalar Lax formulations. We speculate that any N=2 affine algebra admitting a quaternionic structure possesses N=4 supersymmetry and so can be used to produce N=4 supersymmetric hierarchies. This suggests a way of classifying all such hierarchies. 
  We discuss the role that quantum group symmetries, in particular $SU_q(2)$, play in a thermodynamic system at high temperatures. We show that the interactions introduced by the quantum group symmetries, are such that a quantum group gas describe repulsive and attractive behavior in two and three spatial dimensions. 
  Mass-gap calculations in three-dimensional gauge theories are discussed. Also we present a Chern--Simons-like mass-generating mechanism which preserves parity and is realized non-perturbatively. 
  Using a simple observation based on holomorphy, we argue that any model which spontaneously breaks supersymmetry for some range of a parameter will do so generically for all values of that parameter, modulo some isolated exceptional points. Conversely, a model which preserves supersymmetry for some range of a parameter will also do so everywhere except at isolated exceptional points. We discuss how these observations can be useful in the construction of new models which break supersymmetry and discuss some simple examples. We also comment on the relation of these results to the Witten index. 
  We investigate quantum effects on the Coulomb branch of three-dimensional N=4 supersymmetric gauge theory with gauge group SU(2). We calculate perturbative and one-instanton contributions to the Wilsonian effective action using standard weak-coupling methods. Unlike the four-dimensional case, and despite supersymmetry, the contribution of non-zero modes to the instanton measure does not cancel. Our results allow us to fix the weak-coupling boundary conditions for the differential equations which determine the hyper-Kahler metric on the quantum moduli space. We confirm the proposal of Seiberg and Witten that the Coulomb branch is equivalent, as a hyper-Kahler manifold, to the centered moduli space of two BPS monopoles constructed by Atiyah and Hitchin. 
  Modified Laplace transformation method is applied to N component $\phi^4$ theory and the finite temperature problem in the massless limit is re-examined in the large N limit. We perform perturbation expansion of the dressed thermal mass in the massive case to several orders and try the massless approximation with the help of modified Laplace transformation. The contribution with fractional power of the coupling constant is recovered from the truncated massive series. The use of inverse Laplace transformation with respect to the mass square is crucial in evaluating the coefficients of fractional power terms. 
  On the base of the notion of N=2 abelian Yang-Mills supergeometry with constant curvature, we propose an off-shell formulation for the massive q-hypermultiplet and complex $\omega$-hypermultiplet in the standard harmonic superspase without central charge variables. The corresponding superpropagators are derived. 
  We introduce a new type of deformation of the chiral symmetry based on the deformation of the Laurent expansion of the conformal energy momentum tensor. Two kinds of solutions of the deformed equations of continuity are worked out. Known results are recovered, others features are also discussed. 
  We study the infrared perturbative properties of a class of non supersymmetric gauge theories with the same field content of N=4 Super Yang-Mills and we show that the N=4 supersymmetric model represents an IR unstable fixed point for the renormalization group equations. 
  We show that the D-brane configurations for the five and four-dimensional black holes give the geometry of two and three-dimensional ones as well. The emergence of these lower dimensional black holes from the D-brane configurations for those of higher dimensions comes from the choice of the integration constant of harmonic functions, which decides the asymptotic behavior of the metric and other fields. We show that they are equivalent, which are connected by U-dual transformations. This means that stringy black holes in various dimensions are effectively in the same universality class and many properties of black holes in the same class can be infered from the study of those of the three-dimensional black holes. 
  We study bound states of D-p-branes and D-(p+2)-branes. By switching on a large magnetic field F on the (p+2) brane, the problem is shown to admit a perturbative analysis in an expansion in inverse powers of F. It is found that, to the leading order in 1/F, the quartic potential of the tachyonic state from the open string stretched between the p- and (p+2)-brane gives a vacuum energy which agrees with the prediction of the BPS mass formula for the bound state. We generalize the discussion to the case of m p-branes plus 1 (p+2)-brane with magnetic field. The T dual picture of this, namely several (p+2)-branes carrying some p-brane charges through magnetic flux is also discussed, where the perturbative treatment is available in the small F limit. We show that once again, in the same approximation, the tachyon condensates give rise to the correct BPS mass formula. The role of 't Hooft's toron configurations in the extension of the above results beyond the quartic approximation as well as the issue of the unbroken gauge symmetries are discussed. We comment on the connection between the present bound state problem and Kondo-like problems in the context of relevant boundary perturbations of boundary conformal field theories. 
  The scale in the D-brane theory is discussed from different points of view. It is shown that scattering of a $D_4$-brane by a $D_0$-brane gives the condition to penetrate beyond the string scale. 
  In this paper the general form of scattering amplitudes for massless particles with equal spins s ($s s \to s s$) or unequal spins ($s_a s_b \to s_a s_b$) are derived. The imposed conditions are that the amplitudes should have the lowest possible dimension, have propagators of dimension $m^{-2}$, and obey gauge invariance. It is shown that the number of momenta required for amplitudes involving particles with s > 2 is higher than the number implied by 3-vertices for higher spin particles derived in the literature. Therefore, the dimension of the coupling constants following from the latter 3-vertices has a smaller power of an inverse mass than our results imply. Consequently, the 3-vertices in the literature cannot be the first interaction terms of a gauge-invariant theory. When no spins s > 2 are present in the process the known QCD, QED or (super) gravity amplitudes are obtained from the above general amplitudes. 
  We apply the techniques of the ``universal string theory'' to the ``manifold'' paradigm for superstring/M-theory and come up with a candidate manifold: the manifold of F-theory vacua, defined in conformal field theoretical terms. It contains the five known superstring theories as particular vacua; although the natural vacua are (10+2)-dimensional. As a byproduct, a natural explanation emerges for the compactness of the extra two coordinates in F-theory. 
  We construct a generalization of the two-dimensional Wess-Zumino-Witten model on a $2n$-dimensional K\"ahler manifold as a group-valued non-linear sigma model with an anomaly term containing the K\"ahler form. The model is shown to have an infinite-dimensional symmetry which generates an $n$-toroidal Lie algebra. The classical equation of motion turns out to be the Donaldson-Uhlenbeck-Yau equation, which is a $2n$-dimensional generalization of the self-dual Yang-Mills equation. 
  At a generic point in the moduli space of vacua of an N=4 supersymmetric gauge theory with arbitrary gauge group the Higgs force does not cancel the magneto-static force between magnetic monopoles of distinct charge. As a consequence the moduli space of magnetically charged solutions is related in a simple way to those of the SU(2) theory. This leads to a rather simple test of S-duality. On certain subspaces of the moduli space of vacua the forces between distinct monopoles cancel and the test of S-duality becomes more complicated. 
  The Skyrme model of nuclear physics requires quantisation if it is to match observed nuclear properties. A simple technique is used to find the normal mode spectrum of the baryon number B=4 Skyrme soliton, representing the $\alpha$ particle. We find sixteen vibrational modes and classify them under the cubic symmetry group $O_h$ of the static solution. The spectrum possesses a remarkable structure, with the lowest energy modes lying in those representations expected from an approximate correspondence between Skyrmions and BPS monopoles. The next mode up is the `breather', and above that are higher multipole breathing modes. 
  We consider the gauge dyonic string solution of the K3 compactified heterotic string theory in a four dimensional cosmological context. Since for this solution Green-Schwarz as well as Chern-Simons corrections have been taken into account it contains both world sheet and string loop corrections. The cosmological picture is obtained by rotating the world volume of the gauge dyonic string into two space like dimensions and compactifying those dimensions on a two torus. We compare the result with gauge neutral extreme and non-extreme cosmologies and find that the non-trivial Yang Mills background leads to a solution without any singularities whereas for trivial Yang-Mills backgrounds some of the fields become always singular at the big bang. 
  We study the one-loop effective action for a generic two-dimensional dilaton gravity theory conformally coupled to $N$ matter fields. We obtain an explicit expression for the effective action in the weak-coupling limit under a suitable restriction of the dilaton potential asymptotics. Our result applies to the CGHS model as well as to the spherically symmetric general relativity. The effective action is obtained by using the background-field method, and we take into account the loop contributions from all the fields in the classical action and from the ghosts. In the large-$N$ limit, and after an appropriate field redefinition, the one-loop correction takes the form of the Polyakov-Liouville action. 
  We introduce a set of gauge invariant fermion fields in fermionic coset models and show that they play a very central role in the description of several Conformal Field Theories (CFT's). In particular we discuss the explicit realization of primaries and their OPE in unitary minimal models, parafermion fields in $Z_k$ CFT's and that of spinon fields in $SU(N)_k, k=1$ Wess-Zumino-Witten models (WZW) theories. The higher level case ($k>1$) will be briefly discussed. Possible applications to QHE systems and spin-ladder systems are addressed. 
  This paper is devoted to mathematical and physical properties of the Dirac operator and spectral geometry. Spin-structures in Lorentzian and Riemannian manifolds, and the global theory of the Dirac operator, are first analyzed. Elliptic boundary-value problems, index problems for closed manifolds and for manifolds with boundary, Bott periodicity and K-theory are then presented. This makes it clear why the Dirac operator is the most fundamental, in the theory of elliptic operators on manifolds. The topic of spectral geometry is developed by studying non-local boundary conditions of the Atiyah-Patodi-Singer type, and heat-kernel asymptotics for operators of Laplace type on manifolds with boundary. The emphasis is put on the functorial method, which studies the behaviour of differential operators, boundary operators and heat-kernel coefficients under conformal rescalings of the background metric. In the second part, a number of relevant physical applications are studied: non-local boundary conditions for massless spin-1/2 fields, massless spin-3/2 potentials on manifolds with boundary, geometric theory of massive spin-3/2 potentials, local boundary conditions in quantum supergravity, quark boundary conditions, one-loop quantum cosmology, conformally covariant operators and Euclidean quantum gravity. 
  We show that a recently proposed Yang-Mills matrix model with an auxiliary field, which is a candidate for a non-perturbative description of type IIB superstrings, captures the Euler characteristic of moduli space of Riemann surfaces. This happens at the saddle point for the Yang-Mills field. It turns out that the large-n limit in this matrix model corresponds to a double scaling limit in the Penner model. 
  We derive the world-volume theory, the (non)-extremal entropy and background geometry of black holes and black strings constructed out of the NS IIA fivebrane within the framework of matrix theory. The CFT description of strings propagating in the black hole geometry arises as an effective field theory. 
  In the Dirac approach to the generalized Hamiltonian formalism, dynamical systems with first- and second-class constraints are investigated. The classification and separation of constraints into the first- and second-class ones are presented with the help of passing to an equivalent canonical set of constraints. The general structure of second-class constraints is clarified. 
  We analyze the chiral symmetries of flavored quantum chromodynamics in two dimensions and show the existence of chiral condensates within the path-integral approach. The massless and massive cases are discussed as well, for arbitrary finite and infinite number of colors. Our results put forward the question of topological issues when matter is in the fundamental representation of the gauge group. 
  We obtain forms of Born-Infeld and D-brane actions that are quadratic in derivatives of $X$ and linear in $F_{\mu \nu}$ by introducing an auxiliary `metric' which has both symmetric and anti-symmetric parts, generalising the simplification of the Nambu-Goto action for $p$-branes using a symmetric metric. The abelian gauge field appears as a Lagrange multiplier, and solving the constraint gives the dual form of the $n$ dimensional action with an $n-3$ form gauge field instead of a vector gauge field. We construct the dual action explicitly, including cases which could not be covered previously. The generalisation to supersymmetric D-brane actions with local fermionic symmetry is also discussed. 
  We speculate that above or just below the electroweak phase transition magnetic fields are generated which have a net helicity (otherwise said, a Chern-Simons term) of order of magnitude $N_B + N_L$, where $N_{B,L}$ is the baryon or lepton number today. (To be more precise requires much more knowledge of B,L-generating mechanisms than we currently have.) Electromagnetic helicity generation is associated (indirectly) with the generation of electroweak Chern-Simons number through B+L anomalies. This helicity, which in the early universe is some 30 orders of magnitude greater than what would be expected from fluctuations alone in the absence of B+L violation, should be reasonably well-conserved through the evolution of the universe to around the times of matter dominance and decoupling, because the early universe is an excellent conductor. Possible consequences include early structure formation; macroscopic manifestations of CP violation in the cosmic magnetic field (measurable at least in principle, if not in practice); and an inverse-cascade dynamo mechanism in which magnetic fields and helicity are unstable to transfer to larger and larger spatial scales. We give a quasi-linear treatment of the general-relativistic MHD inverse cascade instability, finding substantial growth for helicity of the assumed magnitude out to scales $\sim l_M\epsilon^{-1}$, where $\epsilon$ is roughly the B+L to photon ratio and $l_M$ is the magnetic correlation length. We also elaborate further on an earlier proposal of the author for generation of magnetic fields above the EW phase transition. 
  The (perturbative) renormalization properties of the BF formulation of Yang-Mills gauge models are shown to be identical to those of the usual, second order formulation. This result holds in any number of spacetime dimensions and is a direct consequence of cohomological theorems established by G. Barnich, F. Brandt and the author (Commun.Math.Phys., 174 (1995) 57). 
  We extend the work of Foda et al and propose an elliptic quantum algebra $A_{q,p}(\hat {sl_n})$. Similar to the case of $A_{q,p}(\hat {sl_2})$, our presentation of the algebra is based on the relation $RLL=LLR^*$, where $R$ and $R^*$ are $Z_n$ symmetric R-matrices with the elliptic moduli chosen differently and a factor is also involved. With the help of the results obtained by Asai et al, we realize type I and type II vertex operators in terms of bosonic free fields for $Z_n$ symmetric Belavin model. We also give a bosonization for the elliptic quantum algebra $A_{q,p}(\hat {sl_n})$ at level one. 
  Path-integral approach in imaginary and complex time has been proven successful in treating the tunneling phenomena in quantum mechanics and quantum field theories. Latest developments in this field, the proper valley method in imaginary time, its application to various quantum systems, complex time formalism, asympton theory for the large order analysis of the perturbation theory, are reviewed in a self-contained manner. 
  It is shown that the well-known non-Abelian static SU(2) black hole solutions have rotating generalizations, provided that the hypothesis of linearization stability is accepted. Surprisingly, this rotating branch has an asymptotically Abelian gauge field with an electric charge that cannot vanish, although the non-rotating limit is uncharged. We argue that this may be related to our second finding, namely that there are no globally regular slowly rotating excitations of the particle-like Bartnik-McKinnon solutions. 
  The low energy effective Lagrangian for N= 2 SU(2) supersymmetric Yang-Mills theory coupled to N_F<4 massless matter fields is derived from the BPS mass formula using asymptotic freedom and assuming that the number of strong coupling singularities is finite. 
  The normal mode spectrum of the deuteron in the Skyrme model is computed. We find a bound doublet mode below the pion mass, which can be related to the well-known $90^{\circ}$ scattering of two skyrmions. We also find a singlet `breather' mode and another doublet above the pion mass. The qualitative pattern of the spectrum is similar to that recently found for the B=4 multiskyrmion. The symmetries of all the vibrational modes are presented. 
  Membrane scattering in m(atrix) theory is related to dynamics in three-dimensional $SU(2)$ gauge theory, with transfer of $p^{11}$ being an instanton process. We calculate the instanton amplitude and find precise agreement with the amplitude in eleven dimensional supergravity. 
  Matrix string theory (or more generally U-Duality) requires Super Yang-Mills theory to reflect a stringy degeneracy of BPS short multiplets. These are found as supersymmetric states in the Yang-Mills carrying (fractionated) momentum, or in some cases, instanton number. Their energies also agree with those expected from M(atrix) theory. A nice parallel also emerges in the relevant cases, between momentum and instanton number, (both integral as well as fractional) providing evidence for a recent conjecture relating the two. 
  Since the lightcone self dual spherical membrane, moving in flat target backgrounds, has a direct correspondence with the SU(\infty) Nahm equations and the continuous Toda theory, we construct the quantum/Moyal deformations of the self dual membrane in terms of the q-Moyal star product . The q deformations of the SU(\infty) Nahm equations are studied and explicit solutions are given. The continuum limit of the q Toda chain equations are obtained furnishing q deformations of the self dual membrane. Finally, the continuum Moyal-Toda chain equation is embedded into the SU(\infty) Moyal-Nahm equations, rendering the relation with the Moyal deformations of the self dual membrane. W_{\infty} and q-W_{\infty} algebras arise as the symmetry algebras and the role of ( the recently developed ) quantum Lie algebras associated with quantized universal enveloping algebras is pointed out pertaining the formulation of a q Toda theory. We review as well the Weyl-Wigner-Moyal quantization of the 3D continuous Toda field equation, and its associated 2D continuous Toda molecule, based on Moyal deformations of rotational Killing symmetry reductions of Plebanski first heavenly equation. 
  We present a summary of the applications of duality to Donaldson-Witten theory and its generalizations. Special emphasis is made on the computation of Donaldson invariants in terms of Seiberg-Witten invariants using recent results in N=2 supersymmetric gauge theory. A brief account on the invariants obtained in the theory of non-abelian monopoles is also presented. 
  Physical aspects of the thermofield dynamics of the D=10 heterotic thermal string theory are exemplified through the infrared behaviour of the one-loop dual symmetric cosmological constant in association with the global phase structure of the thermal string ensemble. 
  We discuss the quantum equivalence, to all orders of perturbation theory, between the Yang-Mills theory and its first order formulation through a second rank antisymmetric tensor field. Moreover, the introduction of an additional nonphysical vector field allows us to interpret the Yang-Mills theory as a kind of perturbation of the topological BF model. 
  The Ray-Singer torsion for a compact smooth hyperbolic 3-dimensional manifold ${\cal H}^3$ is expressed in terms of Selberg zeta-functions, making use of the associated Selberg trace formulae. Applications to the evaluation of the semiclassical asymptotics of the Witten's invariant for the Chern-Simons theory with gauge group SU(2) as well as to the sum over topologies in 3-dimensional quantum gravity are presented. 
  We generalise the notions of supersymmetry and superspace by allowing generators and coordinates transforming according to more general Lorentz representations than the spinorial and vectorial ones of standard lore. This yields novel SO(3,1)-covariant superspaces, which we call hyperspaces, having dimensionality greater than (4|4) of traditional super-Minkowski space. As an application, we consider gauge fields on complexifications of these superspaces; and extending the concept of self-duality, we obtain classes of completely solvable equations analogous to the four-dimensional self-duality equations. 
  Cubic interactions are considered in 3 and 7 space dimensions, respectively, for bosonic membranes in Poisson Bracket form. Their symmetries and vacuum configurations are discussed. Their associated first order equations are transformed to Nahm's equations, and are hence seen to be integrable, for the 3-dimensional case, by virtue of the explicit Lax pair provided. The constructions introduced also apply to commutator or Moyal Bracket analogues. 
  The method of the effective action for the composite operators $\Phi^2(x)$ and $\Phi^4(x)$ is applied to the termodynamics of the scalar quantum field with $\lambda\Phi^4$ interaction. An expansion of the finite temperature effective potential in powers of $\hbar$ provides successive approximations to the free energy with an effective mass and an effective coupling determined by the gap equations. The numerical results are studied in the space-time of one dimension, when the theory is equivalent to the quantum mechanics of an anharmonic oscillator. The approximations to the free energy show quick convergence to the exact result. 
  We re-examine the OSp(N,4) invariant interacting model of massless chiral and gauge superfields, whose superconformal invariance was instrumental, both in proving the all-order no-renormalization of the mass and chiral self-interaction lagrangians, and in determining the linear superfield renormalization needed. We show that the renormalization of the gravitational action modifies only the cosmological term, without affecting higher-order tensors. This could explain why the effect of the cosmological constant is shadowed by the effects of newtonian gravity. 
  A new heterotic N=2 string with manifest target space supersymmetry is constructed by combining a conventional N=2 string in the right-moving sector and a Green-Schwarz-Berkovits type string in the left-moving sector. The corresponding sigma model is then obtained by turning on background fields for the massless excitations. We compute the beta functions and we partially check the OPE's of the superconformal algebra perturbatively in $\alpha'$, all in superspace. The resulting field equations describe N=1 self-dual supergravity. 
  M-theory suggests the large N limit of the matrix description of a collection of N Type IA D-particles should provide a nonperturbative formulation of heterotic string theory. In this paper states in the matrix theory corresponding to fundamental heterotic strings are identified, and their interactions are studied. Comments are made about analogous states in Type IIA string theory, which correspond to bound states of D-particles and D-eightbranes. 
  It is proven that if more than a single magnetic charge exists it is impossible to define a proper quantum mechanical angular momentum operator for an electrically charged particle in the field of the magnetic charges. Assuming that quantum mechanics is correct we conclude that free magnetic charges (i.e. magnetic charges with a Coulomb-like magnetic field) can not exist. The only apparent way to avoid this conclusion is if magnetic charges do exist, they must be permanently confined in monopole anti-monopole pairs, much in the same way quarks are thought to be confined. 
  We study N=1 four dimensional gauge theories as the world volume theory of D4-branes between NS 5-branes. We find constructions for a number of known field theory dualities involving $SU(N_c)\times SU(N_c')$ groups, coupled by matter fields F in the $(N_c, \bar N_c)$ representation, in terms of branes of type IIA string theory. The dual gauge group follows from simply reversing the ordering of the NS 5-branes and the D6-branes while conserving magnetic charge on the world volume of the branes. We interpret many field theory phenomena such as deformation of the superpotential $W = \Tr (F\tilde F)^{k+1}$ in terms of the position of branes. By looking to D-branes for guidance, we find new N=1 dualities involving arbitrary numbers of gauge groups. We propose a mechanism for enhanced chiral symmetry in the brane construction which, we conjecture, is associated with tensionless threebranes in six dimensions. 
  We study brane configurations which correspond to field theories in four dimension with N=2 and N=1 supersymmetry. In particular we discuss brane motions that translate to Seiberg's duality in N=1 models recently studied by Elitzur, Giveon and Kutasov. We investigate, using the brane picture, the moduli spaces of the dual theories. Deformations of these models like mass terms and vacuum expectation values of scalar fields can be identified with positions of branes. The map of these deformations between the electric and dual magnetic theories is clarified. The models we study reproduce known field theory results and we provide an example of new dual pairs with N=1 supersymmetry. Possible relations between brane configurations and non-supersymmetric field theories are discussed. 
  We discuss membranes in four-dimensional N=1 superspace. The kappa-invariance of the Green-Schwarz action implies that there is a dual version of N=1 supergravity with a three-form potential. We formulate this new supergravity in terms of a three-form superfield in curved superspace, giving the relevant constraints on the field strength. We find the corresponding membrane soliton in the new supergravity and discuss how the extended supersymmetry algebra emerges from the symmetries of the flat superspace background. 
  Using deformation theory based on BRST cohomology, a supergravity model is constructed which interpolates through a continuous deformation parameter between new minimal supergravity with an extra U(1) gauge multiplet and standard supergravity with local R-symmetry in a formulation with a nonstandard set of auxiliary fields. The deformation implements an electromagnetic duality relating the extra U(1) to the R-symmetry. A consistent representative of the R-anomaly in the model is proposed too. 
  We show that the evolution of the universe is singularity free in a class of graviton-dilaton models. 
  The behavior of the two-particle Green's function in QED is analyzed in the limit when one of the particles becomes infinitely massive. It is found that the dependences of the Green's function on the relative times of the ingoing and outgoing particles factorize and that the bound state spectrum is the same as that of the Dirac equation with the static potential created by the heavy particle. The Bethe-Salpeter wave function is also determined in terms of the Dirac wave function. The present result excludes the existence, in the above limit, of abnormal solutions due to relative time excitations as predicted by the Bethe-Salpeter equation in the ladder approximation. 
  We give a one dimensional octonionic representation of the different Clifford algebra Cliff(5,5)\sim Cliff(9,1), Cliff(6,6)\sim Cliff(10,2) and lastly Cliff(7,6)\sim Cliff(10,3) which can be given by (8x8) real matrices taking into account some suitable manipulation rules. 
  The BRST structure of a current satisfying a non abelian affine algebra in two dimensions was studied by Isidro and Ramallo and an N=2 Superconformal Algebra was obtained. In this paper, we study the total BRST and anti BRST structure of the topological algebra. We end up with an N=4 Superconformal algebra in which the central charge drops out of most of the OPE's. The price one has to pay is that the no. of operators proliferates tremendously and the algebra becomes infinite dimensional. 
  We use a phase space description to (re)derive a first order form of the Born-Infeld action for D-branes. This derivation also makes it possible to consider the limit where the tension of the D-brane goes to zero. We find that in this limit, which can be considered to be the strong coupling limit of the fundamental string theory, the world-volume of the D-brane generically splits into a collection of tensile strings. 
  In this paper we consider Wakimoto free field realizations of simple affine Lie algebras, a subject already much studied. We present three new sets of results. (i) Based on quantizing differential operator realizations of the corresponding Lie algebras we provide general universal very simple expressions for all currents, more compact than has been established so far. (ii) We supplement the treatment of screening currents of the first kind, known in the literature, by providing a direct proof of the properties for screening currents of the second kind. Finally (iii) we work out explicit free field realizations of primary fields with general non-integer weights. We use a formalism where the (generally infinite) multiplet is replaced by a generating function primary operator. These results taken together allow setting up integral representations for correlators of primary fields corresponding to non-integrable degenerate (in particular admissible) representations. 
  Neutron stars are supposed to be mainly formed by a neutron superfluid. The angular momentum is given by the vortex array within the fluid, and a good account of the observable effects is determined by its coupling with the crust. In this article we show that the gravitational field introduces important modifications in the vortex distribution and shape. The inertial frame dragging on the quantum fluid produces a decrease in the vortex density, which for realistic models is in the order of 15%. This effect is relevant for neutron star rotation models and can provide a good framework for checking the quantum effect of the frame dragging. 
  Extended superalgebras of types A,B,C, heterotic and type-I are all derived as solutions to a BPS equation in 14 dimensions with signature ( 11,3). The BPS equation as well as the solutions are covariant under SO( 11,3). This shows how supersymmetries with N<=8 in four dimensions have their origin in the same superalgebra in 14D. The solutions provide different bases for the same superalgebra in 4D, and the transformations among bases correspond to various dualities. 
  Using the graded eigenvalue method and a recently computed extension of the Itzykson-Zuber integral to complex matrices, we compute the $k$-point spectral correlation functions of the Dirac operator in a chiral random matrix model with a deterministic diagonal matrix added. We obtain results both on the scale of the mean level spacing and on the microscopic scale. We find that the microscopic spectral correlations have the same functional form as at zero temperature, provided that the microscopic variables are rescaled by the temperature-dependent chiral condensate. 
  We demonstrate the universality of the spectral correlation functions of a QCD inspired random matrix model that consists of a random part having the chiral structure of the QCD Dirac operator and a deterministic part which describes a schematic temperature dependence. We calculate the correlation functions analytically using the technique of Itzykson-Zuber integrals for arbitrary complex super-matrices. An alternative exact calculation for arbitrary matrix size is given for the special case of zero temperature, and we reproduce the well-known Laguerre kernel. At finite temperature, the microscopic limit of the correlation functions are calculated in the saddle point approximation. The main result of this paper is that the microscopic universality of correlation functions is maintained even though unitary invariance is broken by the addition of a deterministic matrix to the ensemble. 
  Actions for two-superparticle system in (10,2) dimensions and three-superparticle systems in (11,3) dimensions are constructed. These actions have worldline bosonic and fermionic local symmetries, and target space global supersymmety generalizing the reparametrization, kappa-symmetry and Poincare supersymmetry of the usual superparticle. With the second particle, or the second and third particles on-shell, they describe a superparticle propagating in the background of a second superparticle in (10,2) dimensions, or two other superparticles in (11,3) dimensions. Symmetries of the action are shown to exist in presence of super Yang-Mills background as well. 
  It is shown that there are large static black holes for which all curvature invariants are small near the event horizon, yet any object which falls in experiences enormous tidal forces outside the horizon. These black holes are charged and near extremality, and exist in a wide class of theories including string theory. The implications for cosmic censorship and the black hole information puzzle are discussed. 
  In this paper we discuss some aspects of N=1 type I-heterotic string duality in four dimensions. We consider a particular example of a (weak-weak) dual pair where on the type I side there are only D9-branes corresponding to perturbative heterotic description in a certain region of the moduli space. We match the perturbative type I and heterotic tree-level massless spectra via giving certain scalars appropriate vevs, and point out the crucial role of the perturbative superpotential (on the heterotic side) for this matching. We also discuss the role of anomalous U(1) gauge symmetry present in both type I and heterotic models. In the perturbative regime we match the (tree-level) moduli spaces of these models. Since both type I and heterotic models can be treated perturbatively, we are able to discuss a dictionary that in generic models maps type I description onto heterotic one, and vice-versa. Finally, we discuss possible directions to study perturbative quantum corrections to the moduli space, as well as outline ways to learn about the non-perturbative effects in both descriptions. 
  We classify all the Heisenberg and conformal vectors and determine the full automorphism group of the free bosonic vertex algebra without gradation. To describe it we introduce a notion of inner automorphisms of a vertex algebra. 
  We consider N=1 dualities in four dimensional supersymmetric gauge theories as a geometrical realization of wrapping D 6-branes around 3-cycles of Calabi-Yau threefolds in type IIA string theory. By extending the recent work of Ooguri and Vafa to the case of $SU, SO and Sp$ gauge groups with  additional fields together with defining fields, we give simple geometrical descriptions of the interrelation between the electric theory and its magnetic dual in terms of the configuration of D 6-branes wrapped 3-cycles. 
  A field dependent $su(2)$ gauge transformation connects between the adiabatic and diabatic pictures in the (Landau-Zener-Stueckelberg) potential curve crossing problem. It is pointed out that weak and strong potential curve crossing interactions are interchanged under this transformation, and thus realizing a naive strong and weak duality. A reliable perturbation theory should thus be formulated in the both limits of weak and strong interactions. In fact, main characteristics of the potential crossing phenomena such as the Landau-Zener formula including its numerical coefficient are well-described by simple (time-independent) perturbation theory without referring to Stokes phenomena. We also show that quantum coherence in a double well potential is generally suppressed by the effect of potential curve crossing, which is analogous to the effect of Ohmic dissipation on quantum coherence. 
  Planar Chern-Simons (CS) theories in which a compact abelian gauge group U(1) x U(1) is spontaneously broken to U(1) x Z_N are investigated. Among other things, it is noted that the theories just featuring the mixed CS term coupling the broken to the unbroken U(1) gauge fields in general exhibits an interesting form of confinement: only particles carrying certain multiples of the fundamental vortex flux unit and certain multiples of the fundamental charge of the unbroken U(1) gauge field can appear as free particles. Adding the usual CS term for the broken U(1) gauge fields does not change much. It merely leads to additional Aharonov-Bohm interactions among these particles. Upon introducing the CS term for the unbroken U(1) gauge fields, in contrast, the confinement phenomenon completely disappears. 
  In this article we give a detailed discussion of the mass perturbation theory of the massive Schwinger model. After discussing some general features and briefly reviewing the exact solution of the massless case, we compute the vacuum energy density of the massive model and some related quantities. We derive the Feynman rules of mass perturbation theory and discuss the exact $n$-point functions with the help of the Dyson-Schwinger equations. Further we identify the stable and unstable bound states of the theory and compute some bound-state masses and decay widths. Finally we discuss scattering processes, where the resonances and particle production thresholds of the model are properly taken into account by our methods. 
  The fusion of Verma modules of the osp(1|2) current algebra is studied. In the framework of an isotopic formalism, the singular vector decoupling conditions are analyzed. The fusion rules corresponding to the admissible representations of the osp(1|2) algebra are determined. A relation between the characters of these last representations and those corresponding to the minimal superconformal models is found. A series of equations that relate the descendants of the highest weight vectors resulting from a fusion of Verma modules are obtained. Solving these equations the singular vectors of the theory can be determined. 
  In this paper we discuss the possible existing correlation functions in the N=4 topological model. Due to the distinguished feature that no anomaly exists in N=4 supersymmetric theories, the positive-negative ghost number balance has to be taken into account while considering the correlation functions. On restriction to Kahler manifolds we may find a perturbative mass term which breaks the N=4 supersymmetry down to N=1. In all of these, a nonelectromagnetic duality plays an important role. Moreover, to get a computable generating functional the existence of a proper vanishing theorem is required. 
  We study the low-energy behavior of N=1 supersymmetric gauge theories with product gauge groups SU(N)^M and M chiral superfields transforming in the fundamental representation of two of the SU(N) factors. These theories are in the Coulomb phase with an unbroken U(1)^(N-1) gauge group. For N >= 3, M >= 3 the theories are chiral. The low-energy gauge kinetic functions can be obtained from hyperelliptic curves which we derive by considering various limits of the theories. We present several consistency checks of the curves including confinement through the addition of mass perturbations and other limits. 
  The new principle of constrained twistor-like variables is proposed for construction of the Cartan 1-forms on the worldsheet of the D=3,4,6 bosonic strings. The corresponding equations of motion are derived. Among them there are two well-known Liouville equations for real and complex worldsheet functions W(s,\bar s). The third one in which W(s,\bar s) is replaced by the quaternionic worldsheet function is unknown and can be thought of as that of the SU(2) nonlinear sigma-model governing the classical dynamics of the bosonic string in D=6. 
  The zero modes of the monodromy extended SU(2) WZNW model give rise to a gauge theory with a finite dimensional state space. A generalized BRS operator $A$ such that $A^h=0 (h=k+2=3,4,...$ being the height of the current algebra representation) acts in a (2h-1)-dimensional indefinite metric space $H_I$ of quantum group invariant vectors. The generalized cohomologies $Ker A^n/ Im A^{h-n} (n=1,..., h-1)$ are 1-dimensional. Their direct sum spans the physical subquotient of $H_I$. 
  A recursive formula for an infinity of integrals of motion for the super-Liouville theory is derived. The integrable boundary interactions for this theory and the super-Toda theory based on the affine superalgebra $B^{(1)} (0,1)$ are computed. In the first case the boundary interactions are unambiguously determined by supersymmetry, whilst in the latter case there are free parameters. 
  Gotay showed that a representation of the whole Poisson algebra of the torus given by geometric quantization is irreducible with respect to the most natural overcomplete set of observables. We study this representation and argue that it cannot be considered as physically acceptable. In particular, classically bounded observables are quantized by operators with unbounded spectrum. Effectively, the latter amounts to lifting the constraints that compactify both directions in the torus. 
  There are known problems with the standard Lorentz-Dirac description of radiation reaction in classical electrodynamics. The model of extended in one dimension particle is proposed and is shown that for this model there is no total change in particle momentum due to radiation reaction 
  We develop an adequate description of non-topological solitons with a small charge, for which the thin-wall approximation is not valid. There is no classical lower limit on the charge of a stable Q-ball. We examine the parameters of these small-charge solitons and discuss the limits of applicability of the semiclassical approximation. 
  We elaborate the trigonometric version of intertwining vectors and factorized L-operators. The starting point is the corresponding elliptic construction with Belavin's R-matrix. The naive trigonometric limit is singular and a careful analysis is needed. It is shown that the construction admits several different trigonometric degenerations. As a by-product, a quantum Lax operator for the trigonometric Ruijsenaars model intertwined by a non-dynamical R-matrix is obtained. The latter differs from the standard trigonometric R-matrix of $A_n$ type. A connection with the dynamical R-matrix approach is discussed. 
  These are short notes of three introductory lectures on recently proposed matrix models of Superstrings and M theory given at 5th Nordic Meeting on Supersymmetric Field and String Theories in Helsinki (March 10-12, 1997). Contents: M(atrix) theory of BFSS, From IIA to IIB with IKKT, The NBI matrix model. 
  When the gauge instantons on the N=2 string worldsheet are properly included in the sum over topologies, the breaking of SO(2,2) Lorentz symmetry in R^{2,2} is parametrized by a spacetime twistor containing the string coupling and theta angle. The resulting (tree-level) effective action for the open string is not Yang's but Leznov's cubic action for self-dual Yang-Mills in a light-cone gauge. In the closed case, Plebanski's action for self-dual gravity gets modified analogously. In contrast to the N=1 NSR string, picture-changing is not locally invertible, but produces a semi-infinite tower of massless physical states with ever-increasing spin, perhaps related to harmonic superspace. A truncation yields the two-field action of Chalmers and Siegel. 
  We give the general solution for the elementary and solitonic D-brane configurations as a result of a reinterpretation of the already known p-branes. These solutions are found by means of a relevant conformal transformation on the string inspired action and its dual form. From this point of view, the nature of the electric and magnetic charge is clearer and the elementary and solitonic behaviour dependence on the initial lagrangian set. We give a complete characterisation of the spacetime defined by these solutions. The dual pair of instanton and 7-brane solution is presented as an example. 
  We present simple diagrammatic rules to write down Euclidean n-point functions at finite temperature directly in terms of 3-dimensional momentum integrals, without ever performing a single Matsubara sum. The rules can be understood as describing the interaction of the external particles with those of the thermal bath. 
  We study the asymptotic solutions of the Schr\"odinger equation for the color-singlet reggeon compound states in multi-color QCD. We show that in the leading order of asymptotic expansion, quasiclassical reggeon trajectories have a form of the soliton waves propagating on the 2-dimensional plane of transverse coordinates. Applying methods of the finite-gap theory we construct their explicit form in terms of Riemann theta-functions and examine their properties. 
  The current understanding of M(atrix) theory is that in the large N limit certain supersymmetric Yang Mills theories become equivalent to M-theory in the infinite momentum frame. In this paper the conjecture is put forward that the equivalence between M and M(atrix) theory is not limited to the large N limit but is valid for finite N. It is argued that a light cone description of M-theory exists in which one of the light like coordinates is periodically identified. In the light cone literature this is called Discrete Light Cone Quantization (DLCQ). In this framework an exact light cone description exists for each quantized value N of longitudinal momentum. The new conjecture states that the sector of the DLCQ of M-theory is exactly described by a U(N) matrix theory. Evidence is presented for the conjecture. 
  Quantum Gravity admits topological excitations of microscopic scale which can manifest themselves as particles --- topological geons. Non-trivial spatial topology also brings into the theory free parameters analogous to the $\theta$-angle of QCD. We show that these parameters can be interpreted in terms of geon properties. We also find that, for certain values of the parameters, the geons exhibit new patterns of particle identity together with new types of statistics. Geon indistinguishability in such a case is expressed by a proper subgroup of the permutation group and geon statistics by a (possibly projective) representation of the subgroup. 
  We define transformation of multiplets of fields (Jordan cells) under the D-dimensional conformal group, and calculate two and three point functions of fields, which show logarithmic behaviour. We also show how by a formal differentiation procedure, one can obtain n-point function of logarithmic field theory from those of ordinary conformal field theory. 
  A new matrix representation for low-energy limit of heterotic string theory reduced to three dimensions is considered. The pair of matrix Ernst Potentials uniquely connected with the coset matrix is derived. The action of the symmetry group on the Ernst potentials is established. 
  We solve the Cauchy problem for the relativistic closed string in Minkowski space $M^{3+1}$, including the cases where the initial data has a knot like topology. We give the general conditions for the world sheet of a closed knotted string to be a time periodic surface. In the particular case of zero initial string velocity the period of the world sheet is proportional to half the length ($\ell$) of the initial string and a knotted string always collapses to a link for $t=\ell/4$. Relativistic closed strings are dynamically evolving or pulsating structures in spacetime, and knotted or unknotted like structures remain stable over time. The generation of arbitrary $n$-fold knots, starting with an initial simple link configuration with non zero velocity is possible. 
  In the framework of the generalized Hamiltonian formalism by Dirac, the local symmetries of dynamical systems with first- and second-class constraints are investigated. For theories with an algebra of constraints of special form (to which a majority of the physically interesting theories belongs) the method of constructing the generator of local-symmetry transformations is obtained from the requirement of the quasi-invariance of an action. It is proved that second-class constraints do not contribute to the transformation law of the local symmetry which entirely is stipulated by all the first-class constraints. It is thereby shown that degeneracy of special form theories with the first- and second-class constraints is due to their quasi-invariance under local-symmetry transformations. 
  We propose a state in 5+1 super Yang-Mills theory which corresponds to the wrapped transverse five brane of M(atrix) theory on $T^5$. This state is a magnetic flux quantized in units of $1/g_6^2$ through a plane defined by one side of the box and a new direction which is not manifest. 
  By identifying the moduli space of coupling constants in the SYM description of toroidal compactifications of M(atrix)-Theory, we construct the M(atrix) description of the moduli spaces of Type IIA string theory compactified on T^n. Addition of theta terms to the M(atrix) SYM produces the shift symmetries necessary to recover the correct global structure of the moduli spaces. Up to n=3, the corresponding BPS charges transform under the proper representations of the U-duality groups. For n=4,5, if we make the ansatz of including the BPS charges corresponding to the wrapped M-theory 5-brane, the correspondence with Type IIA continues to hold. However, for n=6, we find additional charges for which there are no obvious candidates in M(atrix)-Theory. 
  We generalize the Hamilton-Jacobi formulation for higher order singular systems and obtain the equations of motion as total differential equations. To do this we first study the constraint structure present in such systems. 
  We study the Matrix theory description of M-theory compactified on $T^4$ and $T^5$. M-theory on $T^4$ is described by the six dimensional (2,0) fixed point field theory compactified on a five torus, $\widetilde T^5$. For M-theory on $T^5$ we suggest the existence of a new theory which is compactified on a $\widetilde T^5$. The IR description of this theory is given by the (2,0) theory with a compactified moduli space. This new theory appears to be a new kind of a non-critical string theory. Clearly, these two descriptions differ from the ``standard'' Super-Yang-Mills on the dual torus prescription. 
  We study algebro-geometric (finite-gap) and elliptic solutions of fully discretized KP or 2D Toda equations. In bilinear form they are Hirota's difference equation for $\tau$-functions. Starting from a given algebraic curve, we express the $\tau$-function and the Baker-Akhiezer function in terms of the Riemann theta function. We show that the elliptic solutions, when the $\tau$-function is an elliptic polynomial, form a subclass of the general algebro-geometric solutions. We construct the algebraic curves of the elliptic solutions. The evolution of zeros of the elliptic solutions is governed by the discrete time generalization of the Ruijsenaars-Schneider many body system. The zeros obey equations which have the form of nested Bethe-Ansatz equations, known from integrable quantum field theories. We discuss the Lax representation and the action-angle-type variables for the many body system. We also discuss elliptic solutions to discrete analogues of KdV, sine-Gordon and 1D Toda equations and describe the loci of the zeros. 
  We study a relativistic two-dimensional free electron gas in the presence of a uniform external magnetic field and a random static scalar potential. We compute the expectation values of the charge density and the conductivity tensor, averaged over the random potential in first order perturbation theory. We discuss the effect of a zero vs. finite range correlation length of the random potential. 
  The five-dimensional supersymmetric SU(N) gauge theory is studied in the framework of the relativistic Toda chain. This equation can be embedded in two-dimensional Toda lattice hierarchy. This system has the conjugate structure. This conjugate structure corresponds to the charge conjugation. 
  Pasti, Sorokin and Tonin have recently constructed manifestly Lorentz-invariant actions for self-dual field strengths and for Maxwell fields with manifest electromagnetic duality. Using the method of Deser, Gomberoff, Henneaux and Teitelboim, we generalize these actions in the presence of sources. 
  In the large N limit, we show that the Local Potential Approximation to the flow equation for the Legendre effective action, is in effect no longer an approximation, but exact - in a sense, and under conditions, that we determine precisely. We explain why the same is not true for the Polchinski or Wilson flow equations and, by deriving an exact relation between the Polchinski and Legendre effective potentials (that holds for all N), we find the correct large N limit of these flow equations. We also show that all forms (and all parts) of the renormalization group are exactly soluble in the large N limit, choosing as an example, D dimensional O(N) invariant N-component scalar field theory. 
  We consider axion-free quantum corrected black hole solutions in the context of the heterotic S-T model with half the N=2, D=4 supersymmetries unbroken. We express the perturbatively corrected entropy in terms of the electric and magnetic charges in such a way, that target-space duality invariance is manifest. We also discuss the microscopic origin of particular quantum black hole configurations. We propose a microscopic interpretation in terms of a gas of closed membranes for the instanton corrections to the entropy. 
  We consider string theory corrections to 4D black holes which solve the 5D vacuum Einstein equations. We find that the corrections vanish only for the extremal electric solution. We also show that for the non-extremal electric black hole the mass corrections are related to the charge corrections. The implications to string states counting and the correspondence principle for black holes and strings are discussed. 
  We study F-theory compactified on elliptic Calabi-Yau threefolds that are realised as hypersurfaces in toric varieties. The enhanced gauge group as well as the number of massless tensor multiplets has a very simple description in terms of toric geometry. We find a large number of examples where the gauge group is not a subgroup of E8xE8, but rather, is much bigger (with rank as high as 296). The largest of these groups is the group recently found by Aspinwall and Gross. Our algorithm can also be applied to elliptic fourfolds, for which the groups can become extremely large indeed (with rank as high as 121328). We present the gauge content for two of the fourfolds recently studied by Klemm et al. 
  We consider the Witten index ${\cal I}= Tr (-1)^F$ of SU(2) Super-Yang-Mills quantum mechanics (SYMQ) with N=16, 8, 4 supersymmetries. The theory governs the interactions between a pair of D-branes under various circumstances, and our goal is to count the number of the threshold bound states directly from the low-energy effective theory. The string theory and M theory have predicted that ${\cal I}=1$ for N=16, which in fact forms an underlying hypothesis of the M(atrix) theory formulation. Also the consistency of conifold transitions in type II theories is known to require ${\cal I}=0$ for N=8 and 4. Here, the bulk contribution to ${\cal I}$ is computed explicitly, and for N=16, 8, 4, found to be 5/4, 1/4, 1/4 respectively, suggesting a common defect contribution of -1/4. We illustrate how the defect term of -1/4 may arise in the SU(2) SYMQ by considering the effective dynamics along the asymptotic region. 
  We classify elementary particles according to their behaviour under the action of the full inhomogeneous Lorentz group. For fundamental fermions, this approach leads us to delineate fermions into eight basic families or `types', corresponding to the eight simply connected double covering groups of the inhomogeneous Lorentz group (the `pin' groups). Given this classification, it is natural to ask whether or not fermion type determines a superselection rule. It is also important to determine what observable effects fermion type might have; for example, can the type of a given fermion be determined by laboratory experiments? We address these questions by arguing that if multiple fermion types really did occur in nature, then it would be mathematically equivalent and also much simpler to think of the different types as being different states of a {\it single} particle, which would be a particle which lived in the direct sum of Hilbert spaces associated with the different particle types. In the language of group theory, these are pinor supermultiplets. We discuss the possible experimental ramifications of this proposal. In particular, following work of J. Giesen, we show that the symmetries of the electric dipole moment of a particle would be definitely affected by this proposal. In fact, we show that it would be possible to use the electric dipole moment of a particle to determine the type. We also present an argument that M-theory may provide the mechanism which selects a {\it unique} pin bundle. 
  A difference operator realization of quantum deformed oscillator algebra $H_q(1)$ with a Casimir operator freedom is introduced. We show that this $H_q(1)$ have a nonlinear mapping to the deformed quantum su(2) which was introduced by Fujikawa et al. We also examine the cyclic representation obtained by this difference operator realization and the possibility to analyze a Bloch electron problem by $H_q(1)$. 
  We provide world sheet non-supersymmetrical actions that describe the coupling of a bosonic string to the tachyon and massless states of both the type-B and type-O theories. The type-B theory is derived as a truncation and chiral doubling of the Ramond-Ramond sector in our previous model that connected the (1,0) heterotic string to a 10 D type IIB supergravity background. The type-O theory then follows from a ``fermionization'' of the type-B theory. 
  A combined model of the Kerr spinning particle and superparticle is considered. The structure of the Kerr geometry is presented in a complex form as being created by a complex source. A natural supergeneralization of this construction is obtained corresponding to a complex "supersource". Peforming a supershift to the Kerr and Kerr-Sen solutions we obtain metrics of supergravity black holes with a nonlinear realization of broken supersymmetry. 
  We propose a method for the numerical computation of microcanonical expectation values--i.e. averages over energy eigenstates with the same eigenvalue-without any prior knowledge about the spectrum of the Hamiltonian. This is accomplished by first defining a new Gaussian ensemble to approximate the microcanonical one. We then develop a ``lattice theory'' for this Gaussian ensemble and propose a Monte Carlo integration of the expectation values of the lattice theory. 
  We discuss some aspects of the relation between space-time properties of branes in string theory, and the gauge theory on their worldvolume, for models invariant under four supercharges in three and four dimensions. We show that a simple set of rules governing brane dynamics reproduces many features of gauge theory. We study theories with $U(N_c)$, $SO(N_c)$ and $Sp(N_c)$ gauge groups and matter in the fundamental and two index tensor representations, and use the brane description to establish Seiberg's electric-magnetic duality for these models. 
  On the basis of Borel resummation, we propose a systematical improvement of bounce calculus of quantum bubble nucleation rate. We study a metastable super-renormalizable field theory, $D$ dimensional O(N) symmetric $\phi^4$ model ($D<4$) with an attractive interaction. The validity of our proposal is tested in D=1 (quantum mechanics) by using the perturbation series of ground state energy to high orders. We also present a result in D=2, based on an explicit calculation of vacuum bubble diagrams to five loop orders. 
  QCD is constructed as a lattice gauge theory in which the elements of the link matrices are represented by non-commuting operators acting in a Hilbert space. The resulting quantum link model for QCD is formulated with a fifth Euclidean dimension, whose extent resembles the inverse gauge coupling of the resulting four-dimensional theory after dimensional reduction. The inclusion of quarks is natural in Shamir's variant of Kaplan's fermion method, which does not require fine-tuning to approach the chiral limit. A rishon representation in terms of fermionic constituents of the gluons is derived and the quantum link Hamiltonian for QCD with a U(N) gauge symmetry is expressed in terms of glueball, meson and constituent quark operators. The new formulation of QCD is promising both from an analytic and from a computational point of view. 
  We briefly review the general structure of integrable particle theories in 1+1 dimensions having N=1 supersymmetry. Examples are specific perturbed superconformal field theories (of Yang-Lee type) and the N=1 supersymmetric sine-Gordon theory. We comment on the modifications that are required when the N=1 supersymmetry algebra contains non-trivial topological charges. 
  We discuss conformally covariant differential operators, which under local rescalings of the metric, \delta_\sigma g^{\mu\nu} = 2 \sigma g^{\mu\nu}, transform according to \delta_\sigma \Delta = r \Delta \sigma + (s-r) \sigma \Delta for some r if \Delta is s-th order. It is shown that the flat space restrictions of their associated Green functions have forms which are strongly constrained by flat space conformal invariance. The same applies to the variation of the Green functions with respect to the metric. The general results are illustrated by finding the flat space Green function and also its first variation for previously found second order conformal differential operators acting on $k$-forms in general dimensions. Furthermore we construct a new second order conformally covariant operator acting on rank four tensors with the symmetries of the Weyl tensor whose Green function is similarly discussed. We also consider fourth order operators, in particular a fourth order operator acting on scalars in arbitrary dimension, which has a Green function with the expected properties.   The results obtained here for conformally covariant differential operators are generalisations of standard results for the two dimensional Laplacian on curved space and its associated Green function which is used in the Polyakov effective gravitational action. It is hoped that they may have similar applications in higher dimensions. 
  It was earlier shown that an SO(9,1) $\theta^\a$ spinor variable can be constructed from RNS matter and ghost fields. $\theta^\a$ has a bosonic worldsheet super-partner $\lambda^\a$ which plays the role of a twistor variable, satisfying $\lambda\Gamma^\mu\lambda = \partial x^\mu +i\theta\Gamma^\mu \partial\theta$. For Type IIA superstrings, the left-moving $[\theta_L^\a,\lambda_L^\a]$ and right-moving $[\theta_{R\a},\lambda_{R\a}]$ can be combined into 32-component SO(10,1) spinors $[\theta^A,\lambda^A]$.     This suggests that $\lambda^A \Gamma^{11}_{AB}\lambda^B= 2\lambda_L^\a \lambda_{R\a}$ can be interpreted as momentum in the eleventh direction. Evidence for this interpretation comes from the zero-momentum vertex operators of the Type IIA superstring and from consideration of $D_0$-branes. As in the work of Bars, one finds an SO(10,2) structure for the Type IIA superstring and an SO(9,1) x SO(2,1) structure for the Type IIB superstring. 
  These lectures are intended as an introduction to some of the basic aspects of string solitons, duality and black holes. We begin with a discussion of the role of classical solutions in duality, then focus on string/string duality and fundamental membranes. Finally, we examine the feature of compositeness of string solitons, and its implications for bound states and black hole thermodynamics. As these lectures are aimed primarily at those less familiar with this field, technical details are minimized. 
  We classify degeneration patterns of Verma modules over the N=2 superconformal algebra in two dimensions. Explicit formulae are given for singular vectors that generate maximal submodules in each of the degenerate cases. The mappings between Verma modules defined by these singular vectors are embeddings; in particular, their compositions never vanish. As a by-product, we also obtain general formulae for N=2 subsingular vectors. 
  We consider five-dimensional black holes modeled by D-strings bound to D5-branes, with momentum along the D-strings. We study the greybody factors for the non-minimally coupled scalars which originate from the gravitons and R-R antisymmetric tensor particles polarized along the 5-brane, with one index along the string and the other transverse to the string. These scalars, which we call intermediate, couple to the black holes differently from the minimal and the fixed scalars which were studied previously. Analysis of their fluctuations around the black hole reveals a surprising mixing between these NS-NS and R-R scalars. We disentangle this mixing and obtain two decoupled scalar equations. These equations have some new features, and we are able to calculate the greybody factors only in certain limits. The results agree with corresponding calculations in the effective string model provided one of the intermediate scalars couples to an operator of dimension (1,2), while the other to an operator of dimension (2,1). Thus, the intermediate scalars are sensitive probes of the chiral operators in the effective string action. 
  Using the bosonic supercurrent (or covariant lattice) formalism, we review how to compute scattering amplitudes in asymmetric orbifold string models. This method is particularly useful for calculating scattering of multiple asymmetrically twisted string states, where the twisted states are rewritten as ordinary momentum states. We show how to reconstruct some of the 3-family grand unified string models in this formalism, and identify the quantum numbers of the massless states in their spectra. The discrete symmetries of these models are rather intricate. The superpotentials for the 3-family E_6 model and a closely related SO(10) model are discussed in some detail. The forms of the superpotentials of the two 3-family SU(6) models (with asymptotically-free hidden sectors SU(3) and SU(2) \otimes SU(2)) are also presented. 
  I give an introductory review of recent, fascinating developments in supersymmetric gauge theories. I explain pedagogically the miraculous properties of supersymmetric gauge dynamics allowing one to obtain exact solutions in many instances. Various dynamical regimes emerging in supersymmetric Quantum Chromodynamics and its generalizations are discussed. I emphasize those features that have a chance of survival in QCD and those which are drastically different in supersymmetric and non-supersymmetric gauge theories.   Unlike most of the recent reviews focusing almost entirely on the progress in extended supersymmetries (the Seiberg-Witten solution of N=2 models), these lectures are mainly devoted to N=1 theories. The primary task is extracting lessons for non-supersymmetric theories. 
  The generalized kinetic term of a dilaton gives the classical superinflation without recourse to any potential, and the quantum version of the dilaton gravity exhibits the finite curvature and graceful exit. For $p=2$ case, the model corresponds to the RST quantization of the s-wave sector of the four-dimensional Einstein cosmology. Further, the de Sitter universe is realized for $p=8$ and the smooth transition to the Minkowski space-time is possible. Even in the accelerating contraction case of the universe for $-4<p<0$, the curvature singularity does not appear in a certain branch. 
  We describe a method of writing down the exact interacting gauge invariant equations for all the modes of the bosonic open string. It is a generalization of the loop variable approach that was used earlier for the free, and lowest order interacting cases. The generalization involves, as before, the introduction of a parameter to label the different strings involved in an interaction. The interacting string has thus becomes a ``band'' of finite width. As in the free case, the fields appear to be massless in one higher dimension. Although a proof of the consistency and gauge invariance to all orders (and thus of equivalence with string theory) is not yet available, plausibility arguments are given. We also give some simple illustrations of the procedure. 
  A framework of second-quantization of D5-branes is proposed. It is based on the study of topology of the moduli space of their low energy effective worldvolume theory. Among the topological cycles which resolve singularities caused by overlapping D5-branes, it is introduced those cycles which duals, constituting a subspace of cohomology group of the moduli space, turn out to define the Fock space of the second-quantized D5-branes. The second-quantized operators are given by creation and annihilation operators of those cycles or their duals. 
  We construct a Chern-Simons action for q-deformed gauge theory which is a simple and straightforward generalization of the usual one. Space-time continues to be an ordinary (commuting) manifold, while the gauge potentials and the field strengths become q-commuting fields. Our approach, which is explicitly carried out for the case of `minimal' deformations, has the advantage of leading naturally to a consistent Hamiltonian structure that has essentially all of the features of the undeformed case. For example, using the new Poisson brackets, the constraints form a closed algebra and generate q-deformed gauge transformations. 
  We generalize the Drinfeld-Sokolov formalism of bosonic integrable hierarchies to superspace, in a way which systematically leads to the zero curvature formulation for the supersymmetric integrable systems starting from the Lax equation in superspace. We use the method of symmetric space as well as the non-Abelian gauge technique to obtain the supersymmetric integrable hierarchies of the AKNS type from the zero curvature condition in superspace with the graded algebras, sl(n+1,n), providing the Hermitian symmetric space structure. 
  We consider intersections in eleven dimensions involving Kaluza-Klein monopoles and Brinkmann waves. Besides these purely gravitational configurations we also construct solutions to the equations of motion that involve additional M2- and M5-branes. The maximal number of independent objects in these intersections is nine, and such maximal configurations, when reduced to two dimensions, give rise to a 0-brane solution with dilaton coupling a=-4/9. 
  In a series of papers published in this Journal (J. Math. Phys.), a discussion was started on the significance of a new definition of projective representations in quaternionic Hilbert spaces. The present paper gives what we believe is a resolution of the semantic differences that had apparently tended to obscure the issues. 
  Halpern's field strength's formulation of gauge theories is applied to effective QED_3, namely, a gauge invariant theory for an Abelian gauge field $A_\mu$ with non-localities and self-interactions. The resulting description in terms of the pseudovector field $\tilde{F}_\mu = \epsilon_{\mu\nu\lambda}\partial_\nu A_\lambda$ is applied to different examples. 
  We conjecture that M-theory compactified on an ALE space (or K3) is described by 0-branes moving on the ALE space. We give evidence for this by showing that if we compactify another circle, we recover string theory on the ALE space. This guarantees that in the large N limit, the matrix model correctly describes the force law between gravitons moving in an ALE background. We also show the appearance in M(atrix) theory of the duality of M-theory on K3 with the heterotic string on a three-torus. 
  We propose a model in which a spliced vector bundle (with an arbitrary number of gauge structures in the splice) possesses a geometry which do not split. The model employs connection 1-forms with values in a space-product of Lie algebras, and therefore interlaces the various gauge structures in a non-trivial manner. Special attention is given to the structure of the geometric ghost sector and the super-algebra it possesses: The ghosts emerge as $x$-dependent deformations at the gauge sector, and the associated BRST super algebra is realized as constraints that follow from the invariance of the curvature. 
  We construct a system of bosonic closed string field theory coupled to a D-brane. The interaction between the D-brane and closed string field is introduced using the boundary state which is a function of constant field strength $F_{\mu\nu}$ and the tilt $\theta_\mu^i$ of the D-brane. We find that the gauge invariance requirement on the system determines the $(F_{\mu\nu},\theta_\mu^i)$-dependence of the normalization factor of the boundary state as well as the form of the purely $(F_{\mu\nu},\theta_\mu^i)$ term of the action. Correspondence between the action in the present formalism and the low energy effective action (bulk + D-brane actions) in the $\sigma$-model approach is studied. 
  We construct a new supersymmetric two boson (sTB-B) hierarchy and study its properties. We derive the conserved quantities and the Hamiltonian structures (proving the Jacobi identity) for the system. We show how this system gives the sKdV-B equation and its Hamiltonian structures upon appropriate reduction. We also describe the zero curvature formulation of this hierarchy both in the superspace as well as in components. 
  We discuss long-distance, low-velocity interaction potentials for processes involving longitudinally boosted M5-brane (corresponding in type IIA theory language to the 1/4 supersymmetric bound state of 4-brane and 0-brane). We consider the following scattering configurations: (a) D=11 graviton off longitudinal M5-brane, or, equivalently, 0-branes off marginal 4+0 bound state; (b) M2-brane off longitudinal M5-brane, or a non-marginal 2+0 bound state off marginal 4+0 bound state; (c) two parallel longitudinal M5-branes, or two 4+0 marginal bound states. We demonstrate the equivalence between the classical closed string theory (supergravity) and M(atrix) model (one-loop super Yang-Mills) results for the leading terms in the interaction potentials. The supergravity results are obtained using a generalisation of the classical probe method which allows one to treat bound states of D-branes as probes by introducing non-trivial world-volume gauge field backgrounds. 
  By studying classical realizations of the sl(2,R) algebra in a two dimensional phase space $(q,\pi)$, we have derived a continuous family of new actions for free anyons in 2+1 dimensions. For the case of light-like spin vector $(S_\mu S^\mu =0)$, the action is remarkably simple. We show the appearence of the Zitterbewegung in the solutions of the equations of motion, and relate the actions to others in the literature at classical level. 
  We explain the observation by Candelas and Font that the Dynkin diagrams of nonabelian gauge groups occurring in type IIA and F-theory can be read off from the polyhedron $\Delta^*$ that provides the toric description of the Calabi-Yau manifold used for compacification. We show how the intersection pattern of toric divisors corresponding to the degeneration of elliptic fibers follows the ADE classification of singularities and the Kodaira classification of degenerations. We treat in detail the cases of elliptic K3 surfaces and K3 fibered threefolds where the fiber is again elliptic. We also explain how even the occurrence of monodromy and non-simply laced groups in the latter case is visible in the toric picture. These methods also work in the fourfold case. 
  The first two Hamiltonian structures and the recursion operator connecting all evolution systems and Hamiltonian structures of the N=2 supersymmetric (n,m)-GNLS hierarchy are constructed in terms of N=2 superfields in two different superfield bases with local evolution equations. Their bosonic limits are studied in detail. New local and nonlocal bosonic and fermionic integrals both for the N=2 supersymmetric (n,m)-GNLS hierarchy and its bosonic counterparts are derived. As an example, in the n=1, m=1 case, the algebra and the symmetry transformations for some of them are worked out and a rich N=4 supersymmetry structure is uncovered. 
  We show, in two complementary ways, that D=11 supergravity---in contrast to all its lower dimensional versions---forbids a cosmological extension. First, we linearize the putative model about an Anti de Sitter background and show that it cannot even support a "global" supersymmetry invariance; hence there is no Noether construction that can lead to a local supersymmetry. This is true with the usual 4-form field as well as for a "dual", 7-form, starting point. Second, a cohomology argument, starting from the original full nonlinear theory, establishes the absence of deformations involving spin 3/2 mass and cosmological terms. In both approaches, it is the form field that is responsible for the obstruction. ``Dualizing'' the cosmological constant to an 11-form field also fails. 
  Lagrangian formulation of free massive fields corresponding to irreducible representations of the Poincare group of arbitrary integer and half-integer spins in three-dimensional space-time is presented. A relationship of the theory under consideration with the self-duality equations in four dimensions is discussed. 
  I give an overview of recent progress in constructing the KdV, mKdV and NLS type hierarchies with extended N=4 supersymmetry. 
  We compute the entropy of 5d black holes carrying up to three charges using matrix theory. 
  Starting from the QCD Lagrangian, we present the QCD Hamiltonian for near light cone coordinates. We study the dynamics of the gluonic zero modes of this Hamiltonian. The strong coupling solutions serve as a basis for the complete problem. We discuss the importance of zero modes for the confinement mechanism. 
  We study the most general renormalization transformations for the first-order formulation of the Yang-Mills theory. We analyze, in particular, the trivial sector of the BRST cohomology of two possible formulations of the model: the standard one and the extended one. The latter is a promising starting point for the interpretation of the Yang-Mills theory as a deformation of the topological BF theory. This work is a necessary preliminary step towards any perturbative calculation, and completes some recently obtained results. 
  Coupling constant renormalization is investigated in 2 dimensional sigma models related by non Abelian duality transformations. In this respect it is shown that in the one loop order of perturbation theory the duals of a one parameter family of models, interpolating between the SU(2) principal model and the O(3) sigma model, exhibit the same behaviour as the original models. For the O(3) model also the two loop equivalence is investigated, and is found to be broken just like in the already known example of the principal model. 
  The potential that generates the cohomology ring of the Grassmannian is given in terms of the elementary symmetric functions using the Waring formula that computes the power sum of roots of an algebraic equation in terms of its coefficients. As a consequence, the fusion potential for $su(N)_K$ is obtained. This potential is the explicit Chebyshev polynomial in several variables of the first kind. We also derive the fusion potential for $sp(N)_K$ from a reciprocal algebraic equation. This potential is identified with another Chebyshev polynomial in several variables. We display a connection between these fusion potentials and generalized Fibonacci and Lucas numbers. 
  Talk presented at the conference ``Historical and Philosophical Reflections on the Foundations of Quantum Field Theory,'' at Boston University, March 1996. It will be published in the proceedings of this conference. 
  In the framework of the generalized Hamiltonian formalism by Dirac, the local symmetries of dynamical systems with first- and second-class constraints are investigated in the general case without restrictions on the algebra of constraints. The method of constructing the generator of local-symmetry transformations is obtained from the requirement for them to map the solutions of the Hamiltonian equations of motion into the solutions of the same equations. It is proved that second-class constraints do not contribute to the transformation law of the local symmetry entirely stipulated by all the first-class constraints (and only by them) of an equivalent set passing to which from the initial constraint set is always possible and is presented. A mechanism of occurrence of higher derivatives of coordinates and group parameters in the symmetry transformation law in the Noether second theorem is elucidated. In the latter case it is shown that the obtained transformations of symmetry are canonical in the extended (by Ostrogradsky) phase space. It is thereby shown in the general case that the degeneracy of theories with the first- and second-class constraints is due to their invariance under local-symmetry transformations. It is also shown in the general case that the action functional and the corresponding Hamiltonian equations of motion are invariant under the same quasigroup of local-symmetry transformations. 
  We analyse higher order background independence conditions arising from multiple commutators of background deformations in quantum closed string field theory. The conditions are shown to amount to a vanishing theorem for $\Delta_S$ cohomology classes. This holds by virtue of the existence of moduli spaces of higher genus surfaces with two kinds of punctures. Our result is a generalisation of a previous genus zero analysis relevant to the classical theory. 
  We study critical points of the BPS mass $Z$, the BPS string tension $Z_m$, the black hole potential $V$ and the gauged central charge potential $P$ for M-theory compactified on Calabi-Yau three-folds. We first show that the stabilization equations for $Z$ (determining the black hole entropy) take an extremely simple form in five dimensions as opposed to four dimensions. The stabilization equations for $Z_m$ are also very simple and determine the size of the infinite $adS_3$-throat of the string. The black hole potential in general exhibits two classes of critical points: supersymmetric critical points which coincide with those of the central charge and non-supersymmetric critical points. We then generalize the discussion to the entire extended K\"ahler cone encompassing topologically different but birationally equivalent Calabi-Yau three-folds that are connected via flop transitions. We examine behavior of the four potentials to probe the nature of these phase transitions. We find that $V$ and $P$ are continuous but not smooth across the flop transition, while $Z$ and its first two derivatives, as well as $Z_m$ and its first derivative, are continuous. This in turn implies that supersymmetric stabilization of $Z$ and $Z_m$ for a given configuration takes place in at most one point throughout the entire extended K\"ahler cone. The corresponding black holes (or string states) interpolate between different Calabi-Yau three-folds. At the boundaries of the extended K\"ahler cone we observe that electric states become massless and/or magnetic strings become tensionless. 
  We construct the most general supersymmetric configuration of D2-branes and D6-branes on a 6-torus. It contains arbitrary numbers of branes at relative U(3) angles. The corresponding supergravity solutions are constructed and expressed in a remarkably simple form, using the complex geometry of the compact space. The spacetime supersymmetry of the configuration is verified explicitly, by solution of the Killing spinor equations. Our configurations can be interpreted as a 16-parameter family of regular extremal black holes in four dimensions. Their entropy is interpreted microscopically by counting the degeneracy of bound states of D-branes. Our result agrees in detail with the prediction for the degeneracy of BPS states in terms of the quartic invariant of the E(7,7) duality group. 
  We construct an explicit covariant Majorana formulation of Maxwell electromagnetism which does not make use of vector 4-potential. This allows to write a ``Dirac'' equation for the photon containing all the known properties of it. In particular, the spin and (intrinsic) boost matrices are derived and the helicity properties of the photon are studied. 
  The R^4 terms in the effective action for M-theory compactified on a two-torus are motivated by combining one-loop results in type II superstring theories with the Sl(2,Z) duality symmetry. The conjectured expression reproduces precisely the tree-level and one-loop R^4 terms in the effective action of the type II string theories compactified on a circle, together with the expected infinite sum of instanton corrections. This conjecture implies that the R^4 terms in ten-dimensional string type II theories receive no perturbative corrections beyond one loop and there are also no non-perturbative corrections in the ten-dimensional IIA theory. Furthermore, the eleven-dimensional M-theory limit exists, in which there is an R^4 term that originates entirely from the one-loop contribution in the type IIA theory and is related by supersymmetry to the eleven-form C^{(3)}R^4. The generalization to compactification on T^3 as well as implications for non-renormalization theorems in D-string and D-particle interactions are briefly discussed. 
  We show how different modifications of Kantowski-Sachs cosmologies emerge in four dimensions as dimensional reduction of the gauged Wess-Zumino-Witten model based on $SO(2,2)/SO(2)$. 
  We present dyonic BPS static black hole solutions for general d=4, N=2 supergravity theories coupled to vector and hypermultiplets. These solutions are generalisations of the spherically symmetric Majumdar-Papapetrou black hole solutions of Einstein-Maxwell gravity and are completely characterised by a set of constrained harmonic functions. In terms of the underlying special geometry, these harmonic functions are identified with the imaginary part of the holomorphic sections defining the special K\"ahler manifold and the metric is expressed in terms of the symplectic invariant K\"ahler potential. The relations of the holomorphic sections to the harmonic functions constitute the generalised stabilisation equations for the moduli fields. In addition to asymptotic flatness, the harmonic functions are also constrained by the requirement that the K\"ahler connection of the underlying Hodge-K\"ahler manifold has to vanish in order to obtain static solutions. The behaviour of these solutions near the horizon is also explained. 
  The scaling limit of the spectrum, $S$ matrix, and of the form factors of the polarization operator in the six vertex model has been found. The result for the form factors is consistent with the form factors of the sine-Gordon model found recently by Lukyanov. We discuss the origin of the structure of the free field representation for the sine-Gordon model at the critical coupling from the point of view of the lattice model. 
  We resolve an ambiguity in the sign of the gaugino determinant in supersymmetric models. The result, that the gaugino determinant can be taken positive for all background gauge configurations, is necessary for application of QCD inequalities and lattice Monte Carlo methods to supersymmetric Yang-Mills models. 
  A new discretisation of a doubled, i.e. BF, version of the pure abelian Chern-Simons theory is presented. It reproduces the continuum expressions for the topological quantities of interest in the theory, namely the partition function and correlation function of Wilson loops. Similarities with free spinor field theory are discussed which are of interest in connection with lattice fermion doubling. 
  We study topological properties of the D-brane resolution of three-dimensional orbifold singularities, C^3/Gamma, for finite abelian groups Gamma. The D-brane vacuum moduli space is shown to fill out the background spacetime with Fayet--Iliopoulos parameters controlling the size of the blow-ups. This D-brane vacuum moduli space can be classically described by a gauged linear sigma model, which is shown to be non-generic in a manner that projects out non-geometric regions in its phase diagram, as anticipated from a number of perspectives. 
  We consider the exact solution of a model of correlated electrons based on the superalgebra $Osp(2|2)$. The corresponding Bethe ansatz equations have an interesting form. We derive an expression for the ground state energy at half filling. We also present the eigenvalue of the transfer matrix commuting with the Hamiltonian. 
  Some aspects of the fields of charge two SU(3) monopoles with minimal symmetry breaking are discussed. A certain class of solutions look like SU(2) monopoles embedded in SU(3) with a transition region or ``cloud'' surrounding the monopoles. For large cloud size the relative moduli space metric splits as a direct product AH\times R^4 where AH is the Atiyah-Hitchin metric for SU(2) monopoles and R^4 has the flat metric. Thus the cloud is parametrised by R^4 which corresponds to its radius and SO(3) orientation. We solve for the long-range fields in this region, and examine the energy density and rotational moments of inertia. The moduli space metric for these monopoles, given by Dancer, is also expressed in a more explicit form. 
  The theory of the usual, constrained p-branes is embedded into a larger theory in which there is no constraints. In the latter theory the Fock-Schwinger proper time formalism is extended from point-particles to membranes of arbitrary dimension. For this purpose the tensor calculus in the infinite dimensional membrane space M is developed and an action which is covariant under reparametrizations in M is proposed. The canonical and Hamiltonian formalism is elaborated in detail. The quantization appears to be straightforward and elegant. No problem with unitarity arises. The conventional p-brane states are particular stationary solutions to the functional Schroedinger equation which describes the evolution of a membrane's state with respect to the invariant evolution parameter tau. A tau-dependent solution which corresponds to the wave packet of a null p-brane is found. It is also shown that states of a lower dimensional membrane can be considered as particular states of a higher dimensional membrane. 
  The general properties of the abelian projection are reviewed. We derive the explicit expression for the abelian functional integral for U(1) abelian theory which corresponds to the abelian projection of the SU(2) gluodynamics. The numerical results for the temperature dependence of the monopole condensate are presented. 
  We consider the absorption of higher angular momentum modes of scalars into black holes, at low energies, and ask if the resulting cross sections are reproduced by a D-brane model. To get the correct dependence on the volume of the compactified dimensions, we must let the absorbing element in the brane model have a tension that is the geometric mean of the tensions of the D-string and an effective stringlike tension obtained from the D-5-brane; this choice is also motivated by T-duality. In a dual model we note that the correct dependence on the volume of the compact dimensions and the coupling arise if the absorbing string is allowed to split into many strings in the process of absorbing a higher angular momentum wave. We obtain the required energy dependence of the cross section by carrying out the integrals resulting from partitioning the energy of the incoming quantum into vibrations of the string. 
  Manifest T-duality covariance of the one-loop renormalization group flows is shown for a generic bosonic sigma model with an abelian isometry, by referring a set of previously derived consistency conditions to the tangent space of the target. For a restricted background, T-duality transformations are then studied at the next order, and the ensuing consistency conditions are found to be satisfied by the two-loop Weyl anomaly coefficients of the model. This represents an extremely non-trivial test of the covariance of renormalization group flows under T-duality, and a stronger condition than T-duality invariance of the string background effective action. 
  Following recent proposal of Dijkgraaf, Verlinde and Verlinde, we show that the M(atrix) theory compactified on $S_1/Z_2$ provides with a non-perturbative description of second-quantized light-cone heterotic string. This so-called heterotic M(atrix) string theory is defined by two-dimensional (8,0) supersymmetric chiral gauge theory with gauge group SO(2N) in the large N limit. We argue that at strong coupling fixed point the chiral gauge theory flows to a (8,0) superconformal field theory defined via $S_N$ symmetric product space orbifold. We show that the leading order correction to the strong coupling expansion corresponds to a unique irrelevant operator of scaling dimension three and describes joining and splitting cubic interactions of light-cone heterotic string. We also speculate on M(atrix) description of bosonic strings via dimensional reduction of d=26 Yang-Mills theory. 
  A refined expression for the Faddeev-Popov determinant is derived for gauge theories quantised around a reducible classical solution. We apply this result to Chern-Simons perturbation theory on compact spacetime 3-manifolds with quantisation around an arbitrary flat gauge field isolated up to gauge transformations, pointing out that previous results on the finiteness and formal metric-independence of perturbative expansions of the partition function continue to hold. 
  It was shown that the non-perturbative properties of the vacuum are described by the quantum fluctuations around the classical background with zero canonical momentum. The vacuum state has been built and checked in the framework of the sigma models in two dimentions. 
  The action for an 11-dimensional supermembrane contains a chiral Wess-Zumino-Witten model coupling to the E8 super-Yang-Mills theory on the end-of-the-world 9-brane. It is demonstrated that this boundary string theory is dictated both by gauge invariance and by kappa-symmetry. 
  In the context of simple models, it is shown that demanding finiteness for physical masses with respect to a longitudinal cutoff, can be used to fix the ambiguity in the renormalization of fermions masses in the Hamiltonian light-front formulation. Difficulties that arise in applications of finiteness conditions to discrete light-cone quantization are discussed. 
  We present a spontaneously broken N=2 supergravity model that reduces in the flat limit to a globally supersymmetric N=2 system with explicit soft supersymmetry breaking terms. These soft terms generate a mass O(100 GeV) for mirror quarks and leptons, while leaving the physical fermions light, thereby overcoming one of the major obstacles towards the construction of a realistic N=2 model of elementary interactions. 
  In classical two-dimensional pure dilaton gravity, and in particular in spherically symmetric pure gravity in d dimensions, the generalized Birkhoff theorem states that, for a suitable choice of coordinates, the metric coefficients are only functions of a single coordinate. It is interesting to see how this result is recovered in quantum theory by the explicit construction of the Hilbert space. We examine the CGHS model, enforce the set of auxiliary conditions that select physical states a` la Gupta-Bleuler, and prove that the matrix elements of the metric and of the dilaton field obey the classical requirement. We introduce the mass operator and show that its eigenvalue is the only gauge invariant label of states. Thus the Hilbert space is equivalent to that obtained by quantum mechanical treatment of the static case. This is the quantum form of the Birkhoff theorem for this model. 
  We consider open supermembranes in an eleven dimensional background. We show that, in a flat space-time, the world-volume action is kappa-symmetric and has global space-time supersymmetry if space-time has even dimensional topological defects where the membrane can end. An example of such topological defects is provided by the space-time with boundaries considered by Horava and Witten. In that case the world-volume action has reparametrisation anomalies whose cancellation requires the inclusion of a current algebra on the boundaries of the membrane. The role of kappa-anomalies in a general background is discussed. The tension of the membrane is related to the eleven dimensional gravitational constant with the aid of the Green-Schwarz mechanism allowing a consistency check of M-theory. 
  The heat kernel associated with an elliptic second-order partial differential operator of Laplace type acting on smooth sections of a vector bundle over a Riemannian manifold, is studied. A general manifestly covariant method for computation of the coefficients of the heat kernel asymptotic expansion is developed. The technique enables one to compute explicitly the diagonal values of the heat kernel coefficients, so called Hadamard-Minackshisundaram-De Witt-Seeley coefficients, as well as their derivatives. The elaborated technique is applicable for a manifold of arbitrary dimension and for a generic Riemannian metric of arbitrary signature. It is very algorithmic, and well suited to automated computation. The fourth heat kernel coefficient is computed explicitly for the first time. The general structure of the heat kernel coefficients is investigated in detail. On the one hand, the leading derivative terms in all heat kernel coefficients are computed. On the other hand, the generating functions in closed covariant form for the covariantly constant terms and some low-derivative terms in the heat kernel coefficients are constructed by means of purely algebraic methods. This gives, in particular, the whole sequence of heat kernel coefficients for an arbitrary locally symmetric space. 
  We build nearly topological quantum field theories in various dimensions. We give special attention to the case of 8 dimensions for which we first consider theories depending only on Yang-Mills fields. Two classes of gauge functions exist which correspond to the choices of two different holonomy groups in SO(8), namely SU(4) and Spin(7). The choice of SU(4) gives a quantum field theory for a Calabi-Yau fourfold. The expectation values for the observables are formally holomorphic Donaldson invariants. The choice of Spin(7) defines another eight dimensional theory for a Joyce manifold which could be of relevance in M- and F-theories. Relations to the eight dimensional supersymmetric Yang-Mills theory are presented. Then, by dimensional reduction, we obtain other theories, in particular a four dimensional one whose gauge conditions are identical to the non-abelian Seiberg-Witten equations. The latter are thus related to pure Yang-Mills self-duality equations in 8 dimensions as well as to the N=1, D=10 super Yang-Mills theory. We also exhibit a theory that couples 3-form gauge fields to the second Chern class in eight dimensions, and interesting theories in other dimensions. 
  It is shown that the 1+1-dimensional matter-coupled Jackiw-Teitelboim model and the model with an exponential potential can be converted by means of appropriate canonical transformations into a bosonic string theory propagating on a flat target space with an indefinite signature. This makes it possible to consistently quantize these models in the functional Schroedinger representation thus generalizing recent results on CGHS theory. 
  Within the general framework of Liouville string theory, we construct a model for quantum D-brane fluctuations in the space-time background through which light closed-string states propagate. The model is based on monopole and vortex defects on the world sheet, which have been discussed previously in a treatment of 1+1-dimensional black-hole fluctuations in the space-time background, and makes use of a T-duality transformation to relate formulations with Neumann and Dirichlet boundary conditions. In accordance with previous general arguments, we derive an open quantum-mechanical description of this D-brane foam which embodies momentum and energy conservation and small mean energy fluctuations. Quantum decoherence effects appear at a rate consistent with previous estimates. 
  We analyze various brane configurations corresponding to field theories in three, four and five dimensions. We find brane configurations which correspond to three dimensional N=2 and four dimensional N=1 supersymmetric QCD theories with quartic superpotentials, in which what appear to be ``hidden parameters'' play an important role. We discuss the construction of five dimensional N=1 supersymmetric gauge theories and superconformal fixed points using branes, which leads to new five dimensional N=1 superconformal field theories. The same five dimensional theories are also used, in a surprising way, to describe new superconformal fixed points of three dimensional N=2 supersymmetric theories, which have both ``electric'' and ``magnetic'' Coulomb branches. 
  We study the zero mode and the spontaneous symmetry breaking on the light front (LF). We use the discretized light-cone quantization (DLCQ) of Maskawa-Yamawaki to treat the zero mode in a clean separation from all other modes. It is then shown that the Nambu-Goldstone (NG) phase can be realized on the trivial LF vacuum only when an explicit symmetry-breaking mass of the NG boson $m_{\pi}$ is introduced. The NG-boson zero mode integrated over the LF must exhibit singular behavior $ \sim 1/m_{\pi}^2$ in the symmetric limit $m_{\pi}\to 0$, which implies that current conservation is violated at zero mode, or equivalently the LF charge is not conserved even in the symmetric limit. We demonstrate this peculiarity in a concrete model, the linear sigma model, where the role of zero-mode constraint is clarified. We further compare our result with the continuum theory. It is shown that in the continuum theory it is difficult to remove the zero mode which is not a single mode with measure zero but the accumulating point causing uncontrollable infrared singularity. A possible way out within the continuum theory is also suggested based on the ``$\nu$ theory''. We finally discuss another problem of the zero mode in the continuum theory, i.e., no-go theorem of Nakanishi-Yamawaki on the non-existence of LF quantum field theory within the framework of Wightman axioms, which remains to be a challenge for DLCQ, ``$\nu$ theory'' or any other framework of LF theory. 
  We calculate the absorption rates of fixed scalars by near extremal five dimensional black holes carrying general one-brane and five-brane charges by semi-classical and D-brane methods. We find that the absorption cross-sections do not in general agree for either fixed scalar and we discuss possible explanations of the discrepancy. 
  We present a general algorithm which permits to construct solutions in string cosmology for heterotic and type-IIB superstrings in four dimensions. Using a chain of transformations applied in sequence: conformal, T-duality and SL(2,R) rotations, along with the usual generating techniques associated to Geroch transformations in Einstein frame, we obtain solutions with all relevant low-energy remnants of the string theory. To exemplify our algorithm we present an inhomogeneous string cosmology with S^3 topology of spatial sections, discuss some properties of the solution and point out some subtleties involved in the concept of homogeneity and isotropy in string cosmology. 
  A new method to obtain the Picard-Fuchs equations of effective, N=2 supersymmetric gauge theories with massive matter hypermultiplets in the fundamental representation is presented. It generalises a previously described method to derive the Picard-Fuchs equations of both pure super Yang-Mills and supersymmetric gauge theories with massless matter hypermultiplets. The techniques developed are well suited to symbolic computer calculations. 
  We briefly review the status of the ``graceful exit'' problem in superstring cosmology and present a possible resolution. It is shown that there exists a solution to this problem in two-dimensional dilaton gravity provided quantum corrections are incorporated. This is similar to the recently proposed solution of Rey. However, unlike in his case, in our one-loop corrected model the graceful exit problem is solved for any finite number of massless scalar matter fields present in the theory. 
  We compute the quark-antiquark potential in three dimensional massive Quantum Electrodynamics. The result indicates that screening prevails for large quark masses, contrary to the classical expectations. The classical result is reproduced for small separation of the quarks. 
  We investigate compactifications of M-theory from $11\to 5\to 4$ dimensions and discuss geometrical properties of 4-d moduli fields related to the structure of 5-d theory. We study supersymmetry breaking by compactification of the fifth dimension and find that an universal superpotential is generated for the axion-dilaton superfield $S$. The resulting theory has a vacuum with $<S>=1$, zero cosmological constant and a gravitino mass depending on the fifth radius as $m_{3/2} \sim R_5^{-2}/M_{Pl}$. We discuss phenomenological aspects of this scenario, mainly the string unification and the decompactification problem. 
  We study U-duality symmetries of toroidally compactified M theory from the membrane worldvolume point of view. This is done taking the most general set of bosonic background fields into account. Upon restriction to pure moduli backgrounds, we are able to find the correct U-duality groups and moduli coset parameterizations for dimensions D>6 as symmetries of the membrane worldvolume theory. In particular, we derive the D=8 U-duality group SL(2)xSL(3). For general background fields, we concentrate on the case D=8. Though the SL(2) part of the symmetry appears to be obstructed by certain terms in the equations of motion, we are able to read off the transformation properties for the background fields. These transformations are verified by comparison with 11-dimensional supergravity dimensionally reduced to D=8. 
  We suggest in this Letter that the Bekenstein-Hawking black hole entropy accounts for the degrees of freedom which are excited at low temperatures only and hence it leads to the negative specific heat. Taking into account the physical degrees of freedom which are excited at high temperatures, the existence of which we postulate, we compute the total specific heat of the quantum black hole that appears to be positive. This is done in analogy to the Planck's treatment of the black body radiation problem. Other thermodynamic functions are computed as well. Our results and the success of the thermodynamic description of the quantum black hole suggest an underlying atomic (discrete) structure of gravitation. The basic properties of these gravitational atoms are found. 
  The thermofield dynamics of the D = 10 heterotic thermal string theory is described in proper reference to the thermal duality symmetry as well as the thermal stability of modular invariance in association with the global phase structure of the D = 10 heterotic thermal string ensemble. 
  In the paper we begin a description of functional methods of quantum field theory for systems of interacting q-particles. These particles obey exotic statistics and are the q-generalization of the colored particles which appear in many problems of condensed matter physics, magnetism and quantum optics. Motivated by the general ideas of standard field theory we prove the q-functional analogues of Hori's formulation of Wick's theorems for the different ordered q-particle creation and annihilation operators. The formulae have the same formal expressions as fermionic and bosonic ones but differ by a nature of fields. This allows us to derive the perturbation series for the theory and develop analogues of standard quantum field theory constructions in q-functional form. 
  A twistor model is proposed for the free relativistic anyon.  The Hamiltonian reduction of this model by the action of the spin generator leads to the minimal covariant model; whereas that by the action of spin and mass generators, to the anyon model with free phase space that is a cotangent bundle of the Lobachevsky plane with twisted symplectic structure. Quantum mechanics of that model is described by irreducible representations of the (2+1)-dimensional Poincare' group. 
  Gauge-invariant systems of a general form with higher order derivatives of gauge parameters are investigated within the framework of the BFV formalism. Higher order terms of the BRST charge and BRST-invariant Hamiltonian are obtained. It is shown that the identification rules for Lagrangian and Hamiltonian ghost variables depend on the choice of the extension of constraints from the primary constraint surface. 
  In a recent paper we pointed out the presence of extra fermionic degrees of freedom in a chiral gauge theory based on Connes Noncommutative Geometry. Here we propose a mechanism which provides a high mass to these mirror states, so that they decouple from low energy physics. 
  We provide evidence that some four-dimensional N=1 string vacua with different numbers of generations are connected through phase transitions. The transitions involve going through a point in moduli space where there is a nontrivial fixed point governing the low energy field theory. In an M-theory description, the examples involve wrapped 5-branes leaving one of the ends of the world. 
  We present four-dimensional M-theory vacua with N>0 supersymmetry which, from the perspective of perturbative Type IIA string theory, have N=0. Such vacua can appear when the compactifying 7-manifold is a U(1) fibration. The missing superpartners are Dirichlet 0-branes. Someone unable to detect Ramond-Ramond charge would thus conclude that these worlds have no unbroken supersymmetry. In particular, the gravitinos (and also some of the gauge bosons) are 0-branes not seen in perturbation theory but which curiously remain massless however weak the string coupling. 
  It is pointed out that if we allow for the possibility of a multilayered universe, it is possible to maintain exact supersymmetry and arrange, in principle, for the vanishing of the cosmological constant. Superpartner(s) of a known particle will then be associated with the other layers of such a universe. A concrete model realizing this scenario is exhibited in \2 dimensions, and it is suggested that it may be realizable in 3+1 dimensions. The connection between this nonclassical geometry and noncommutative geometries is discussed. 
  We study the small oscillations regime (RPA approximation) of the time-dependent mean-field equations, obtained in a previous work, which describe the time evolution of one-body dynamical variables of a uniform Chiral Gross-Neveu system. In this approximation we obtain an analytical solution for the time evolution of the one-body dynamical variables. The two-fermion physics can be explored through this solution. The condition for the existence of bound states is examined. 
  We examine the supersymmetry of classical D-brane and M-brane configurations and explain the dependence of Killing spinors on coordinates. We find that one half supersymmetry is broken in the bulk and that supersymmetry near the D-brane horizon is restored for $p\leq 3$, for solutions in the stringy frame, but only for $p=3$ in the10d canonical frame. We study the enhancement for the case of four intersecting D-3-branes in 10 dimensions and the implication of this for the size of the infinite throat of the near horizon geometry in non-compactified theory. We found some indications of universality of near horizon geometries of various intersecting brane configurations. 
  We relate, in 10 and 11 dimensional supergravities, configurations of intersecting closed branes with vanishing binding energy to configurations where one of the branes opens and has its boundaries attached to the other. These boundaries are charged with respect to fields living on the closed brane. The latter hosts electric and magnetic charges stemming from dual pairs of open branes terminating on it. We show that charge conservation, gauge invariance and supersymmetry entirely determine these charges and these fields, which can be seen as Goldstone fields of broken supersymmetry. Open brane boundary charges can annihilate, restoring the zero binding energy configuration. This suggests emission of closed branes by branes, a generalization of closed string emission by D-branes. We comment on the relation of the Goldstone fields to matrix models approaches to M-theory. 
  We review our recent discussion of fivebrane central terms that appear in the space-time superalgebra in D=10 provided that the space-time supercharges are taken in non-canonical pictures. We correct the mistake contained in the original version of the earlier paper hep-th/9703008 which suggested that the naive picture-changing of the superalgebra could give rise to the non-perturbative five-form term. The relation between picture-changing and p-brane central terms appears to be much more subtle, as is pointed out in the revised version of the mentioned paper. 
  The main principles of two-dimensional quantum field theories, in particular two-dimensional QCD and gravity are reviewed. We study non-perturbative aspects of these theories which make them particularly valuable for testing ideas of four-dimensional quantum field theory. The dynamics of confinement and theta vacuum are explained by using the non-perturbative methods developed in two dimensions. We describe in detail how the effective action of string theory in non-critical dimensions can be represented by Liouville gravity. By comparing the helicity amplitudes in four-dimensional QCD to those of integrable self-dual Yang-Mills theory, we extract a four dimensional version of two dimensional integrability. 
  After a review of theoretical motivations to consider theories with direct couplings of scalar fields to Ricci and gauge curvature terms, we consider the dynamics and non-perturbative stabilization of a dilaton in three and in four dimensions. In particular, we derive generalized Coulomb potentials in the presence of a dilaton and discuss a low energy effective dilaton potential induced by instanton effects and the S--dual coupling to axions. We conclude with a discussion of cosmological implications of a light dilaton. 
  We clarify a discrepancy between two previous calculations of the two-loop QED Euler-Heisenberg Lagrangian, both performed in proper-time regularization, by calculating this quantity in dimensional regularization. 
  The semi-classical cross-sections for arbitrary partial waves of ordinary scalars to fall into certain five-dimensional black holes have a form that seems capable of explanation in terms of the effective string model. The kinematics of these processes is analyzed in detail on the effective string and is shown to reproduce the correct functional form of the semi-classical cross-sections. But it is necessary to choose a peculiar value of the effective string tension to obtain the correct scaling properties. Furthermore, the assumptions of locality and statistics combine to forbid the effective string from absorbing more than a finite number of partial waves. The relation of this limitation to cosmic censorship is discussed. 
  The supersymmetric generalization of dilatations in the presence of the dilaton is defined. This is done by defining the supersymmetric dilaton geometry which is motivated by the supersymmetric volume preserving diffeomorphisms. The resulting model is classical superconformal field theory with an additional dilaton-axion supermultiplet coupled to the supersymmetric gauge theory, where the dilaton-axion couplings are nonrenormalizable. The possibility of spontaneous scale symmetry breaking is investigated in this context. There are three different types of vacua with broken scale symmetry depending on the details of the dilaton sector: unbroken supersymmetry, spontaneously broken supersymmetry and softly broken supersymmetry. If the scale symmetry is broken in the bosonic vacuum, then the Poincar\'e supersymmetry must be broken at the same time. If the scale symmetry is broken in the fermionic vacuum but the bosonic vacuum remains invariant, then the Poincar\'e supersymmetry can be preserved as long as the R-symmetry breaking is specifically related to the scale symmetry breaking. 
  We calculate multi-instanton effects in a three-dimensional gauge theory with N=8 supersymmetry and gauge group SU(2). The k-instanton contribution to an eight-fermion correlator is found to be proportional to the Gauss-Bonnet-Chern integral of the Gaussian curvature over the centered moduli-space of charge-k BPS monopoles, \tilde{M}_{k}. For k=2 the integral can be evaluated using the explicit metric on \tilde{M}_{2} found by Atiyah and Hitchin. In this case the integral is equal to the Euler character of the manifold. More generally the integral is the volume contribution to the index of the Euler operator on \tilde{M}_{k}, which may differ from the Euler character by a boundary term. We conjecture that the boundary terms vanish and evaluate the multi-instanton contributions using recent results for the cohomology of \tilde{M}_{k}. We comment briefly on the implications of our result for a recently proposed test of M(atrix) theory. 
  We study brane configurations which correspond to N=1 field theories in four dimensions. By inverting the order of the NS 5-branes and D6-branes, a check on dualities in four dimensional theories can be made.  We consider a brane configuration which yields electric/magnetic duality for gauge theories with $SO(N_{c1})\times Sp(N_{c2})$ product gauge group.  We also discuss the possible extension to any alternating product of SO and Sp groups. The new features arising from the intersection of the NS 5-branes on the orientifold play a crucial role in our construction. 
  We perform the perturbation analysis of the black holes in the 4D, N=4 supergravity. Analysis around the black holes reveals a complicated mixing between the dilaton and other fields (metric and two U(1) Maxwell fields). It turns out that considering both s-wave (l=0) and higher momentum modes (l \neq 0), the dilaton as a fixed scalar is the only propagating mode with $P=Q, h_1=h_2=0$ and $F = -G = 2\phi$. We calculate the absorption cross-section for scattering of low frequency waves of fixed scalar and U(1) Maxwell fields off the extremal black hole. 
  Measuring distances on a lattice in noncommutative geometry involves square, symmetric and real ``three-diagonal'' matrices, with the sum of their elements obeying a supremum condition, together with a constraint forcing the absolute value of the maximal eigenvalue to be equal to 1. In even dimensions, these matrices are unipotent of order two, while in odd dimensions only their squares are Markovian. We suggest that these bi-graded Markovian matrices (i.e. consisting in the square roots of Markovian matrices) can be thought of as non-local Dirac operators. The eigenvectors of these matrices are spinors. Treating these matrices as determining the stochastic time evolution of states might explain why one observes only left handed neutrinos. Some other physical interpretations are suggested. We end by presenting a mathematical conjecture applying to q-graded Markovian matrices. 
  After summarizing the development of black hole thermodynamics in the seventies, we describe a recent microscopic model. This model suggests that the Bekenstein-Hawking area formula holds for extremal black holes as well as for ordinary (non-extremal) ones. On the other hand, semiclassical studies have suggested a discontinuity between non-extremal and extremal cases. We indicate how a reconciliation has been brought about by summing over topologies. 
  We apply a derivative expansion to the Legendre effective action flow equations of O(N) symmetric scalar field theory, making no other approximation. We calculate the critical exponents eta, nu, and omega at the both the leading and second order of the expansion, associated to the three dimensional Wilson-Fisher fixed points, at various values of N. In addition, we show how the derivative expansion reproduces exactly known results, at special values N=infinity,-2,-4, ... . 
  It is shown that vielbeins and connections of any (super)space are naturally described in terms of nonlinear realizations of infinite - dimensional diffeomorphism groups of the corresponding (super)space. The method of construction of integral invariants from the invariant Cartan's differential $\Omega$ - forms is generalized to the case of superspace. 
  The recently proposed ``correspondence principle'' of Horowitz and Polchinski provides a concrete means to relate (among others) black holes with electric NS-NS charges to fundamental strings and correctly match their entropies. We test further this correspondence by examining the greybody factors in the absorption rates of neutral, minimally coupled scalars by a near extremal black hole. Perhaps surprisingly, the results disagree in general with the absorption by weakly coupled strings. Though this does not disprove the correspondence, it indicates that it might not be simple in this region of the black hole parameter space. 
  This is a short review of the results on the associativity algebras and WDVV equations found recently for the Seiberg-Witten solutions of N=2 4d SUSY gauge theories. The presentation is mostly based on the integrable treatment of these solutions. 
  We introduce new local gauge invariant variables for N=1 supersymmetric Yang-Mills theory, explicitly parameterizing the physical Hilbert space of the theory. We show that these gauge invariant variables have a geometrical interpretation, and can be constructed such that the emergent geometry is that of N=1 supergravity: a Riemannian geometry with vector-spinor generated torsion. Full geometrization of supersymmetric Yang-Mills theory is carried out, and geometry independent divergences associated to the inversion of a differential operator with zero modes -- that were encountered in the non-supersymmetric case -- do not arise in this situation. 
  We develop here a simple formalism that converts the second-class constraints into first-class ones for a particle moving on the $n$-dimensional sphere. The Poisson algebra generated by the Hamiltonian and the constraints closes and by quantization transforms into a Lie algebra. The observable of the theory is given by the Casimir operator of this algebra and coincides with the square of the angular momentum. 
  We present a numerical simulation of the scattering of a topological soliton off finite size attractive impurities, repulsive impurities and a combination of both. The attractive and attractive-repulsive cases show similar features to those found for $\delta$ function type of impurities. For the repulsive case, corresponding to a finite width barrier, the soliton behaves completely classically. No tunneling occurs for sub-barrier kinetic energies despite the extended nature of the soliton. 
  A general field-antifield BV formalism for antisymplectic first class constraints is proposed. It is as general as the corresponding symplectic BFV-BRST formulation and it is demonstrated to be consistent with a previously proposed formalism for antisymplectic second class constraints through a generalized conversion to corresponding first class constraints. Thereby the basic concept of gauge symmetry is extended to apply to quite a new class of gauge theories potentially possible to exist. 
  To facilitate the treatment of electromagnetic effects in applications such as dynamically perturbed vortons, this work employs a covariantly formulated string-source Green measure to obtain a coherent relativistic scheme for describing the self interaction of electromagnetic currents in string models of a very general kind, at leading order in the relevant field gradients, using a regularised gradient operator given by $\hat\nabla_\nu=$ $\bar\nabla_\nu +{1\over 2} K_\nu$ where $\bar\nabla_\nu$ is the usual tangential gradient operator and $K_\nu$ is the extrinsic curvature vector. 
  The small momentum fraction x behaviour of quarks in mesons is analysed in the 1+1-dimensional reduced model of large-N QCD by light-cone quantisation. 
  In the DHR theory of superselection sectors, one usually considers states which are local excitations of some vacuum state. Here, we extend this analysis to local excitations of a class of "infravacuum" states appearing in models with massless particles. We show that the corresponding superselection structure, the statistics of superselection sectors and the energy-momentum spectrum are the same as with respect to the vacuum state. (The latter result is obtained with a novel method of expressing the shape of the spectrum in terms of properties of local charge transfer cocycles.) These findings provide evidence to the effect that infravacua are a natural starting point for the analysis of the superselection structure in theories with long-range forces. 
  A free field representation for form-factors of exponential operators in the affine A^{(1)}_{N-1} Toda model is proposed. The one and two particle form-factors are calculated explicitly. 
  The background field method for N=2 super Yang-Mills theories in harmonic superspace is developed. The ghost structure of the theory is investigated. It is shown that the ghosts include two fermionic real omega-hypermultiplets (Faddeev-Popov ghosts) and one bosonic real omega-hypermultiplet (Nielsen-Kallosh ghost), all in the adjoint representation of the gauge group. The one-loop effective action is analysed in detail and it is found that its structure is determined only by the ghost corrections in the pure super Yang-Mills theory. As applied to the case of N=4 super Yang-Mills theory, realized in terms of N=2 superfields, the latter result leads to the remarkable conclusion that the one-loop effective action of the theory does not contain quantum corrections depending on the N=2 gauge superfield only. We show that the leading low-energy contribution to the one-loop effective action in the N=2 SU(2) super Yang-Mills theory coincides with Seiberg's perturbative holomorphic effective action. 
  Front form dynamics is not a manifestly rotational invariant formalism. In particular, the requirement of an invariance under rotations around the transverse axes is difficult to fulfill.In the present work it is investigated, to which extent rotational invariance is restored in the solution of a light-cone quantized field theory. The positronium spectrum in full (3+1) dimensions is calculated at an unphysically large coupling of alpha=0.3 to be sensitive for terms breaking rotational invariance and to accustomize the theory to future QCD applications. The numerical accuracy of the formalism is improved to allow for the calculation of mass eigenvalues for arbitrary components J_z of the total angular momentum. We find numerically degenerate eigenvalues as expected from rotationally invariant formalisms and the right multiplet structures up to large principal quantum numbers. The results indicate that rotational invariance is unproblematic even in front form dynamics. Another focus of the work relies on the inclusion of the annihilation channel. This enlargement of the model is non-trivial and a consistency check of the underlying theory of effective interactions. The correct numerical eigenvalue shifts, and especially the right hyperfine splitting are obtained. Moreover, the cutoff dependence of the eigenvalues is improved drastically by the annihilation channel for the triplet states. The implications of the applied effective Hamiltonian approach are discussed in detail. 
  It has been pointed out that Nielsen-Olesen vortices may be able to decay by pair production of black holes. We show that when the abelian Higgs model is embedded in a larger theory, the additional fields may lead to selection rules for this process - even in the absence of fermions - due to the failure of a charge quantization condition. We show that, when there is topology change, the criterion based on the charge quantization condition supplements the usual criterion based on $\pi_0(H)$. In particular, we find that, unless $2\sin^2\theta_W$ is a rational number, the thermal splitting of electroweak Z-strings by magnetically neutral black holes is impossible, even though $\pi_0(H)$ is trivial. 
  We discuss the relation among some disk amplitudes with non-trivial boundary conditions in two-dimensional quantum gravity. They are obtained by the two-matrix model as well as the three-matirx model for the case of the tricritical Ising model. We examine them for simple spin configurations, and find that a finite number of insertions of the different spin states cannot be observed in the continumm limit. We also find a closed set of eight Schwinger-Dyson equations which determines the disk amplitudes in the three-matrix model. 
  We examine the low energy structure of N=1 supersymmetric SO(10) gauge theory with matter chiral superfields in N_Q spinor and N_f vector representations. We construct a dual to this model based upon an SU(N_f+2N_Q-7) x Sp(2N_Q-2) gauge group without utilizing deconfinement methods. This product theory generalizes all previously known Pouliot-type duals to SO(N_c) models with spinor and vector matter. It also yields large numbers of new dual pairs along various flat directions. The dual description of the SO(10) theory satisfies multiple consistency checks including an intricate renormalization group flow analysis which links it with Seiberg's duality transformations. We discuss its implications for building grand unified theories that contain all Standard Model fields as composite degrees of freedom. 
  We study N=1 dualities in four dimensional supersymmetric gauge theories in terms of wrapping D 6-branes around 3-cycles of Calabi-Yau threefolds in type IIA string theory. We generalize the recent work of geometrical realization for the models which have the superpotential corresponding to an $A_k$ type singularity, to various models presented by Brodie and Strassler, consisting of $D_{k+2}$ superpotential of the form $W=Tr X^{k+1} + Tr XY^2$. We discuss a large number of representations for the field $Y$, but with $X$ always in the adjoint (symmetric) [antisymmetric] representation for $SU (SO) [Sp]$ gauge groups. 
  We prove the Lorentz symmetry of supermembrane theory in the light cone gauge to complete the program initiated by de Wit, Marquard and Nicolai. We give some comments on extending the formulation to the M(atrix) theory. 
  It is shown that the Dirac sea can be uniquely defined for the Dirac equation with general interaction, if we impose a causality condition on the Dirac sea. We derive an explicit formula for the Dirac sea in terms of a power series in the bosonic potentials.   The construction is extended to systems of Dirac seas. If the system contains chiral fermions, the causality condition yields a restriction for the bosonic potentials. 
  The antifield formalism is extended so as to incorporate the rigid symmetries of a given theory. To that end, it is necessary to introduce global ghosts not only for the given rigid symmetries, but also for all the higher order conservation laws, associated with conserved antisymmetric tensors. Otherwise, one may encounter obstructions of the type discussed in [13]. These higher order conservation laws are shown to define additional rigid symmetries of the master equation and to form -- together with the standard symmetries -- an interesting algebraic structure. They lead furthermore to independent Ward identities which are derived in the standard manner, because the resulting master ("Zinn-Justin") equation capturing both the gauge symmetries and the rigid symmetries of all orders takes a known form. Issues such as anomalies or consistent deformations of the action preserving some set of rigid symmetries can be also systematically analysed in this framework. 
  We discuss exact renormalization group (RG) in $R^2$-gravity using effective average action formalism. The truncated evolution equation for such a theory on De Sitter background leads to the system of nonperturbative RG equations for cosmological and gravitational coupling constants. Approximate solution of these RG equations shows the appearence of antiscreening and screening behaviour of Newtonian coupling what depends on higher derivative coupling constants. 
  The helix model describes the minimal coupling of an abelian gauge field with three bosonic matter fields in 0+1 dimensions; it is a model without a global Gribov obstruction. We perform the stochastic quantization in configuration space and prove nonperturbatively equivalence with the path integral formalism. Major points of our approach are the geometrical understanding of separations into gauge independent and gauge dependent degrees of freedom as well as a generalization of the stochastic gauge fixing procedure which allows to extract the equilibrium Fokker-Planck probability distribution of the model. 
  In the path integral formulation of quantum mechanics, Feynman and Hibbs noted that the trajectory of a particle is continuous but nowhere differentiable. We extend this result to the quantum mechanical path of a relativistic string and find that the ``trajectory'', in this case, is a fractal surface with Hausdorff dimension three. Depending on the resolution of the detecting apparatus, the extra dimension is perceived as ``fuzziness'' of the string world-surface. We give an interpretation of this phenomenon in terms of a new form of the uncertainty principle for strings, and study the transition from the smooth to the fractal phase. 
  BPS configurations of intersecting branes have many applications in string theory. We attempt to provide an introductory and pedagogical review of supergravity solutions corresponding to orthogonal BPS intersections of branes with an emphasis on eleven and ten space-time dimensions. Recent work on BPS solutions corresponding to non-orthogonally intersecting branes is also discussed. These notes are based on lectures given at the APCTP Winter School "Dualities of Gauge and String Theories", Korea, February 1997. 
  We study the Landau-Ginzburg models which correspond to Calabi-Yau four-folds. We construct the index of the typical states which correspond to toric divisors. This index shows that whether a corresponding divisor can generate a non-perturbative superpotential. For an application, we consider the phase transition in terms of the orbifold construction. We obtain the simple method by which the divisor, which can not generate a superpotential in the original theory, can generate a superpotential after orbifoldization. 
  We argue that black p-branes will occur in the collision of D0-branes at Planckian energies. This extents the Amati, Ciafaloni and Veneziano and 't Hooft conjecture that black holes occur in the collision of two light particles at Planckian energies. We discuss a possible scenario for such a process by using colliding plane gravitational waves. D-branes in the presence of black holes are discussed. M(atrix) theory and matrix string in curved space are considered. A violation of quantum coherence in M(atrix) theory is noticed. 
  We calculate an effective action and measure induced by the integration over the auxiliary field in the matrix model recently proposed to describe IIB superstrings. It is shown that the measure of integration over the auxiliary matrix is uniquely determined by locality and reparametrization invariance of the resulting effective action. The large--$N$ limit of the induced measure for string coordinates is discussed in detail. It is found to be ultralocal and, thus, possibly is irrelevant in the continuum limit. The model of the GKM type is considered in relation to the effective action problem. 
  We prove a C-theorem within the framework of two dimensional quantum field theories at finite temperature. There exists a function C(g) of coupling constants which is non-increasing along renormalization group trajectories and non-decreasing along temperature trajectory and stationary only at the fixed points. The connection between the C-theorem at zero temperature and the C-theorem at finite temperature is discussed. We also consider the thermodynamical aspects of the C-theorem. If we define the C-function in an arbitrary number of dimensions in anology to the two dimensional case, we can show that its behavior is not universal. The phase transitions destroy the monotonic properties of the C-function. The proof of the C-theorem is also presented within the framework of the Kallen-Lehmann spectral representation at finite temperature. 
  It has been shown recently that the toroidally compactified type IIB string effective action possesses an SL(2, R) invariance as a consequence of the corresponding symmetry in ten dimensions when the self-dual five-form field strength is set to zero. By working in the string frame we clarify how a Z$_2$ subgroup of this SL(2, R) group responsible for producing the strong-weak coupling duality in the ten dimensional theory produces the same symmetry for the reduced theory. In the absence of the full covariant action of type IIB supergravity theory, we show that the T-dual version of type IIA string effective action (including the R-R terms) in D=9 also possesses the SL(2, R) invariance indicating that this symmetry is present for the full type IIB string effective action compactified on torus. 
  Decay of a near-extremal black hole down to the extremal state is studied in the background field approximation to determine the fate of injected matter and Hawking pairs. By examining the behavior of light rays and solutions to the wave equation it is concluded that the singularity at the origin is irrelevant. Furthermore, there is most likely an instability of the event horizon arising from the accumulation of injected matter and Hawking partners there. The possible role of this instability in reconciling the D-brane and black hole pictures of the decay process is discussed. 
  We consider the complete normal field net with compact symmetry group constructed by Doplicher and Roberts starting from a net of local observables in >=2+1 spacetime dimensions and its set of localized (DHR) representations. We prove that the field net does not possess nontrivial DHR sectors, provided the observables have only finitely many sectors. Whereas the superselection structure in 1+1 dimensions typically does not arise from a group, the DR construction is applicable to `degenerate sectors', the existence of which (in the rational case) is equivalent to non-invertibility of Verlinde's S-matrix. We prove Rehren's conjecture that the enlarged theory is non-degenerate, which implies that every degenerate theory is an `orbifold' theory. Thus, the symmetry of a generic model `factorizes' into a group part and a pure quantum part which still must be clarified. 
  We show that a large class of massive quantum field theories in 1+1 dimensions, characterized by Haag duality and the split property for wedges, does not admit locally generated superselection sectors in the sense of Doplicher, Haag and Roberts. Thereby the extension of DHR theory to 1+1 dimensions due to Fredenhagen, Rehren and Schroer is vacuous for such theories. Even charged representations which are localizable only in wedge regions are ruled out. Furthermore, Haag duality holds in all locally normal representations. These results are applied to the theory of soliton sectors. Furthermore, the extension of localized representations of a non-Haag dual net to the dual net is reconsidered. It must be emphasized that these statements do not apply to massless theories since they do not satisfy the above split property. In particular, it is known that positive energy representations of conformally invariant theories are DHR representations. 
  The stability problem for the O(N) nonlinear sigma model in the 2+\epsilon dimensions is considered. We present the results of the 1/N^{2} order calculations of the critical exponents (in the 2<d<4 dimensions) of the composite operators relevant for this problem. The arguments in the favor of the scenario with the conventional fixed point are given. 
  Quantum field theories in front-form dynamics are not manifestly rotationally invariant. We study a model bound-state equation in 3+1 dimensional front-form dynamics, which was shown earlier to reproduce the Bohr and hyperfine structure of positronium. We test this model with regard to its rotational symmetry and find that rotational invariance is preserved to a high degree. Also, we find and quantify the expected dependence on the cut-off. 
  We analyze brane configurations corresponding to non-trivial six dimensional fixed points. Several results previously obtained from a pure field theoretical analysis are rederived in the brane language. 
  The scattering of NS-NS antisymmetric tensor states in the presence of D-instantons in type IIB superstring theory is studied. It is shown that in order to preserve gauge invariance, spacetime supersymmetry and picture changing symmetry the inclusion of boundary contact terms for closed string antisymmetric tensor vertex operators is necessary. 
  We find the topological entropy formula for all $N>2$ extended supergravities in five dimensions and the fixed scalar condition for the black-hole ``potential energy'' which extremizes the BPS mass. We comment upon the interpretation of these results in a string and M-theory setting. 
  Exact duality in supersymmetric gauge theories leads to highly non-trivial predictions about the moduli spaces of BPS monopole solutions. These notes attempt to be a pedagogical review of the current status of these investigations and are based on lectures given at the 33rd Karpacz Winter School: Duality - Strings and Fields, February 1997. 
  We use dimensional reduction techniques to relate real time finite T correlation functions in (2+1) dimensional QCD to bound state parameters in a generalized 't Hooft model with an infinite number of heavy quark and adjoint scalar fields. While static susceptibilities and correlation functions of the DeTar type can be calculated using only the light (static) gluonic modes, the dynamical correlators require the inclusion of the heavy modes. In particular we demonstrate that the leading T perturbative result can be understood in terms of the bound states of the 2d model and that consistency requires bound state trajectories composed of both quarks and adjoint scalars. We also propose a non-perturbative expression for the dynamical DeTar correlators at small spatial momenta. 
  We describe the Higgs mechanism for general N=2 super Yang-Mills theories in a manifestly supersymmetric form based on the harmonic superspace. 
  Various gauge invariant but non-Yang-Mills dynamical models are discussed: Pr\'ecis of Chern-Simons theory in (2+1)-dimensions and reduction to (1+1)-dimensional B-F theories; gauge theories for (1+1)-dimensional gravity-matter interactions; parity and gauge invariant mass term in (2+1)-dimensions. 
  The existence two S-dual descriptions of (N,1) string bound states suggests that the strong coupling behaviour of electric flux lines in large N 2D SYM theory has a dual description in terms of weakly coupled IIB string theory. In support of this identification, we propose a dual interpretation of the SYM loop equation as a perturbative string Ward identity, expressing the conformal invariance of the corresponding boundary interaction. This correspondence can be viewed as a weak coupling check of the matrix string conjecture. 
  We discuss consistency conditions for branes at orbifold singularities. The conditions have a world-sheet interpretation in terms of tadpole cancellation and a space-time interpretation in terms of anomalies. As examples, we consider type II and type I branes on $C^2/Z_M$ orbifolds. We give orientifold constructions of phases of type I or heterotic string theory, involving branches with extra tensor multiplets, which arise when small SO(32) instantons sit on orbifold singularities. 
  5-brane configurations describing 5d field theories are promoted to an M theory description a la Witten in terms of polynomials in two complex variables. The coefficients of the polynomials are the Coulomb branch. This picture resolves apparent singularities at vertices and reveals exponentially small corrections. These corrections ask to be compared to world line instanton corrections. From a different perspective this procedure may be used to define a diagrammatic representation of polynomials in two variables. 
  This manuscript was published as a JINR Preprint E17-10550 in 1977. In view of the recent interest in various new kinds of statistics it seems the results it contains may be still of some interest. It is shown that the second quantization axioms can, in principle, be satisfied with creation and annihilation operators generating (in the case of $n$ pairs of such operators) the Lie algebra $A_n$ of the group $SL(n+1)$. A concept of a Fock space is introduced. The matrix elements of these operators are found. 
  Kadanoff's "correlations along a line" in the critical two-dimensional Ising model (1969) are reconsidered. They are the analytical aspect of a representation of abelian chiral vertex operators as quadratic polynomials, in the sense of operator valued distributions, in non-abelian exchange fields. This basic result has interesting applications to conformal coset models. It also gives a new explanation for the remarkable relation between the "doubled" critical Ising model and the free massless Dirac theory. As a consequence, analogous properties as for the Ising model order/disorder fields with respect both to doubling and to restriction along a line are established for the two-dimensional local fields with chiral level 2 SU(2) symmetry. 
  A universal minimal spinor set of linear differential equations describing anyons and ordinary integer and half-integer spin fields is constructed with the help of deformed Heisenberg algebra with reflection. The construction is generalized to some d=2+1 supersymmetric field systems. Quadratic and linear forms of action functionals are found for the universal minimal as well as for supersymmetric spinor sets of equations. A possibility of constructing a universal classical mechanical model for d=2+1 spin systems is discussed. 
  The pre-big-bang cosmology inspired by superstring theories has been suggested as an alternative to slow-roll inflation. We analyze, in both the Jordan and Einstein frames, the effect of spatial curvature on this scenario and show that too much curvature --- of either sign --- reduces the duration of the inflationary era to such an extent that the flatness and horizon problems are not solved. Hence, a fine-tuning of initial conditions is required to obtain enough inflation to solve the cosmological problems. 
  A multidimensional gravitational model containing several dilatonic scalar fields and antisymmetric forms is considered. The manifold is chosen in the form M = M_0 x M_1 x ... x M_n, where M_i are Einstein spaces (i > 0). The block-diagonal metric is chosen and all fields and scale factors of the metric are functions on M_0. For the forms composite (electro-magnetic) p-brane ansatz is adopted. The model is reduced to gravitating self-interacting sigma-model with certain constraints. In pure electric and magnetic cases the number of these constraints is m(m - 1)/2 where m is number of 1-dimensional manifolds among M_i. In the "electro-magnetic" case for dim M_0 = 1, 3 additional m constraints appear. A family of "Majumdar-Papapetrou type" solutions governed by a set of harmonic functions is obtained, when all factor-spaces M_k are Ricci-flat. These solutions are generalized to the case of non-Ricci-flat M_0 when also some additional "internal" Einstein spaces of non-zero curvature are added to M. As an example exact solutions for D = 11 supergravity and related 12-dimensional theory are presented. 
  We argue that supersymmetry breaking by gaugino condensation in the strongly coupled heterotic string can be described by an analogue of Scherk-Schwarz compactification on the eleventh dimension in M-theory. The M-theory scale is identified with the gauge coupling unification mass, whereas the radius of the eleventh dimension $\rho$ is at an intermediate scale $\rho^{-1}\sim 10^{12}$ GeV. At the lowest order, supersymmetry is broken only in the gravitational and moduli sector at a scale $m_{3/2}\sim\rho^{-1}$, while it is mediated by gravitational interactions to the observable world. Computation of the mass splittings yields in general a hierarchy of soft masses at the TeV scale $(\sim\rho^{-2}/M_p)$ with matter scalars much heavier than gauginos. 
  The possible tensor constructions of open string theories are analyzed from first principles. To this end the algebraic framework of open string field theory is clarified, including the role of the homotopy associative A_\infty algebra, the odd symplectic structure, cyclicity, star conjugation, and twist. It is also shown that two string theories are off-shell equivalent if the corresponding homotopy associative algebras are homotopy equivalent in a strict sense.   It is demonstrated that a homotopy associative star algebra with a compatible even bilinear form can be attached to an open string theory. If this algebra does not have a spacetime interpretation, positivity and the existence of a conserved ghost number require that its cohomology is at degree zero, and that it has the structure of a direct sum of full matrix algebras. The resulting string theory is shown to be physically equivalent to a string theory with a familiar open string gauge group. 
  It is demonstrated that several series of conformal field theories, while satisfying braid group statistics, can still be described in the conventional setting of the DHR theory, i.e. their superselection structure can be understood in terms of a compact DHR gauge group. Besides theories with only simple sectors, these include (the untwisted part of) c=1 orbifold theories and level two so(N) WZW theories. We also analyze the relation between these models and theories of complex free fermions. 
  We present a new form of kappa-symmetry transformations for D-branes in which the dependence on the Born-Infeld field strength is expressed as a relative rotation on the left- and right-moving fields with opposite parameters. Then, we apply this result to investigate the supersymmetry preserved by certain intersecting brane configurations at arbitrary angles and with non-vanishing constant Born-Infeld fields. We also comment on the covariant quantization of the D-brane actions. 
  A first principles calculation of the quantum corrections to the electric charge of a dyon in an N=2 gauge theory with arbitrary gauge group is presented. These corrections arise from the fermion fields via the mechanism of fermion fractionalization. For a dyon whose magnetic charge is a non-simple co-root, the correction is a discontinuous function on the moduli space of vacua and the discontinuities occur precisely on co-dimension one curves on which the dyon decays. In this way, the complete spectrum of dyons at weak coupling is found for a theory with any gauge group. It is shown how this spectrum is consistent with the semi-classical monodromies. 
  The neutral Kaon system is used to test the quantum theory of resonance scattering and decay phenomena. The two dimensional Lee-Oehme-Yang theory with complex Hamiltonian is obtained by truncating the complex basis vector expansion of the exact theory in Rigged Hilbert space. This can be done for K_1 and K_2 as well as for K_S and K_L, depending upon whether one chooses the (self-adjoint, semi-bounded) Hamiltonian as commuting or non-commuting with CP. As an unexpected curiosity one can show that the exact theory (without truncation) predicts long-time 2 pion decays of the neutral Kaon system even if the Hamiltonian conserves CP. 
  We show that deformed Heisenberg algebra with reflection emerging in parabosonic constructions is also related to parafermions. This universality is discussed in different algebraic aspects and is employed for the description of spin-j fields, anyons and supersymmetry in 2+1 dimensions. 
  We discuss type II and type I branes at general ADE type orbifold singularities. We show that there are new phases of type I or heterotic string theory in six dimensions, involving extra tensor multiplets, which arise when small instantons sit on orbifold singularities. The theories with extra tensor multiplets are explicitly constructed via orientifolds. The world-volume theories in type IIB or type I five-branes at orbifold singularities lead to the existence of several infinite classes of six dimensional, interacting, renormalization group fixed point theories. 
  We discuss the issue of renormalization and the derivation of effective interactions for light-cone Hamiltonians in the context of large-N scalar matrix models with $\Phi^3$ interactions. For various space-time dimensions $D \geq 3$, we deduce appropriate mass, coupling constant, and wavefunction renormalizations which are necessary for finiteness of the Hamiltonian at leading order. We also outline how higher order corrections may be derived within this framework, and discuss the relevance of this approach in the light-cone quantization of gauge theories. 
  We study the existence of D-brane bound states at threshold in Type II string theories. In a number of situations, we can reduce the question of existence to quadrature, and the study of a particular limit of the propagator for the system of D-branes. This involves a derivation of an index theorem for a family of non-Fredholm operators. In support of the conjectured relation between compactified eleven-dimensional supergravity and Type IIA string theory, we show that a bound state exists for two coincident zero-branes. This result also provides support for the conjectured description of M-theory as a matrix model. In addition, we provide further evidence that there are no BPS bound states for two and three-branes twice wrapped on Calabi-Yau vanishing cycles. 
  We show that a particular many-matrix model gives rise, upon hamiltonian reduction, to a multidimensional version of the Calogero-Sutherland model and its spin generalizations. Some simple solutions of these models are demonstrated by solving the corresponding matrix equations. A connection of this model to the dimensional reduction of Yang-Mills theories to (0+1)-dimensions is pointed out. In particular, it is shown that the low-energy dynamics of D0-branes in sectors with nontrivial fermion content is that of spin-Calogero particles. 
  We present results of investigation into a problem of cubic interaction vertices of massless higher spin fields which transform in arbitrary irreps of the Poincare algebra 
  The underlying reasons for the difficulty of unitarily implementing the whole conformal group $SO(4,2)$ in a massless Quantum Field Theory (QFT) are investigated in this paper. Firstly, we demonstrate that the singular action of the subgroup of special conformal transformations (SCT), on the standard Minkowski space $M$, cannot be primarily associated with the vacuum radiation problems, the reason being more profound and related to the dynamical breakdown of part of the conformal symmetry (the SCT subgroup, to be more precise) when representations of null mass are selected inside the representations of the whole conformal group. Then we show how the vacuum of the massless QFT radiates under the action of SCT (usually interpreted as transitions to a uniformly accelerated frame) and we calculate exactly the spectrum of the outgoing particles, which proves to be a generalization of the Planckian one, this recovered as a given limit. 
  We investigate the generation of curvature and isocurvature (dilaton, moduli and axion) perturbations in a general class of axion-dilaton-moduli models,including the pre-big bang scenario. Allowing for an arbitrary coupling constant between the dilaton field and the axion field, we exploit the SL(2,R) symmetry of the theory to obtain the spectral indices of the field perturbations in a pre-big bang type scenario. Axion field fluctuations about a homogeneous background field can yield a scale-invariant (Harrison-Zel'dovich) spectrum. As an example we present a string-motivated case with SL(2,R)xSL(2,R) symmetry, where a second axion field arises from the compactification of the ten-dimensional theory to four dimensions. 
  It is well known that anomaly cancellations for D_16 Lie algebra are at the root of the first string revolution. For E_8 Lie algebra, cancellation of anomalies is the principal fact leading to the existence of heterotic string. They are in fact nothing but the 6th order cohomologies of corresponding Lie algebras. Beyond 6th order, the calculations seem to require special care and it could be that their study will be worthwhile in the light of developments of the second string revolution.   As we have shown in a recent article, for A_N Lie algebras, there is a method which are based on the calculations of Casimir eigenvalues. This is extended to E_8 Lie algebra in the present article. In the generality of any irreducible representation of E_8 Lie algebra, we consider 8th and 12th order cohomologies while emphasizing the diversities between the two. It is seen that one can respectively define 2 and 8 basic invariant polinomials in terms of which 8th and 12th order Casimir eigenvalues are always expressed as linear superpositions. All these can be easily investigated because each one of these invariant polinimials gives us a linear equation to calculate E_8 weight multiplicities. Our results beyond order 12 are not included here because they get more complicated though share the same characteristic properties with 12th order calculations. 
  The effective gauge field actions generated by charged fermions in $QED_3$ and $QCD_3$ can be made invariant under both small and large gauge transformations at any temperature by suitable regularization of the Dirac operator determinant, at the price of parity anomalies. We resolve the paradox that the perturbative expansion is not invariant, as manifested by the temperature dependence of the induced Chern-Simons term, by showing that large (unlike small) transformations and hence their Ward identities, are not perturbative order-preserving. Our results are illustrated through concrete examples of field configurations. 
  We calculate the one loop effective action for D-brane probes moving in the presence of near BPS D-branes. The $v^2$ term agrees with supergravity in all cases and the static force agrees for a five dimensional black hole with two large charges. It also agrees qualitatively in all the other cases. We make some comments on the M(atrix) theory interpretation of these results. 
  A black hole solution to low energy type IIA string theory which is extremal, non-supersymmetric, and carries 0- and 6-brane charge is presented. For large values of the charges it is metastable and a corresponding D-brane picture can be found. The mass and statistical entropy of the two descriptions agree at a correspondence point up to factors of order one, providing more evidence that the correspondence principle for black holes and strings of Horowitz and Polchinski may be extended to include black holes with more than one Ramond-Ramond charge. 
  We consider compactifications of the matrix model of M-theory on $S^1/Z_2\times T^d$ for $d>0$, and interpret them as orbifolds of the supersymmetric U(N) Yang-Mills theory on $R\times T^{d+1}$. The orbifold group acts both on the gauge group and on the $T^{d+1}$, reduces the gauge group to O(N) over 1+1 dimensional fixed-point submanifolds, and breaks half of the supersymmetry. We clarify some puzzling aspects of the gauge anomaly cancellation in the presence of space-time Wilson lines; in general, the Yang-Mills theory requires certain Chern-Simons couplings to supergravity background fields. We discuss the possibility that D8-branes are present as certain matrix configurations in the Yang-Mills theory, and the fundamental fermions emerge as zero modes. Finally, we point out that the correspondence between matrix theory and string theory suggests the existence of a multitude of non-trivial RG fixed points and dualities in orbifold Yang-Mills theories with eight supercharges in various dimensions. 
  We have found that kappa-symmetry allows a covariant quantization provided the ground state of the theory is strictly massive. For D-p-branes a Hamiltonian analysis is performed to explain the existence of a manifestly supersymmetric and Lorentz covariant description of the BPS states of the theory. The covariant quantization of the D-0-brane is presented as an example. 
  We study the leading irrelevant operators along the flat directions of certain supersymmetric theories. In particular, we focus on finite N=2 (including N=4) supersymmetric field theories in four dimensions and show that these operators are completely determined by the symmetries of the problem. This shows that they are generated only at one loop and are not renormalized beyond this order. An instanton computation in similar three dimensional theories shows that these terms are renormalized. Hence, the four dimensional non-renormalization theorem of these terms is not valid in three dimensions. 
  The ten-dimensional type IIA string effective action with cosmological constant term is dimensionally reduced on a d-dimensional torus to derive lower dimensional effective action. The symmetries of the reduced effective action are examined. It is shown that the resulting six-dimensional theory does not remain invariant under $SO(4,4)$ symmetry whereas the reduced action, in the absence of the cosmological constant respects the symmetry as was shown by Sen and Vafa. New class of black hole solutions are obtained in five and four dimensions in the presence of cosmological constant. For the six-dimensional theory, a four-brane solution is also presented. 
  The radiation reaction of a BPS monopole in the presence of incident electromagnetic waves as well as massless Higgs waves is analyzed classically. The reactive forces are compared to those of $W$ boson that is interpreted as a dual partner of the BPS monopole. It is shown that the damping of acceleration is dual to each other, while in the case of finite size effects the duality is broken explicitly. Their implications on the duality are discussed. 
  A method which uses a generalized tensorial $\zeta$-function to compute the renormalized stress tensor of a quantum field propagating in a (static) curved background is presented. The starting point of the method is the direct computation of the functional derivatives of the Euclidean one-loop effective action with respect to the background metric. This method, when available, gives rise to a conserved stress tensor and produces the conformal anomaly formula directly. It is proven that the obtained stress tensor agrees with statistical mechanics in the case of a finite temperature theory. The renormalization procedure is controlled by the structure of the poles of the stress-tensor $\zeta$ function. The infinite renormalization is automatic and is due to a ``magic'' cancellation of two poles. The remaining finite renormalization involves conserved geometrical terms arising by a certain residue. Such terms renormalize coupling constants of the geometric part of Einstein's equations (customary generalized through high-order curvature terms). The method is checked on particular cases (closed and open Einstein`s universe) finding agreement with other approaches. The method is also checked considering a massless scalar field in the presence of a conical singularity in the Euclidean manifold (i.e. Rindler spacetimes/large mass black hole manifold/cosmic string manifold). There, the method gives rise to the stress tensor already got by the point-splitting approach for every coupling with the curvature regardless of the presence of the singular curvature. Comments on the measure employed in the path integral, the use of the optical manifold and different approaches to renormalize the Hamiltonian are made. 
  We present analytic cosmological solutions in a model of two-dimensional dilaton gravity with back reaction. One of these solutions exhibits a graceful exit from the inflationary to the FRW phase and is nonsingular everywhere. A duality related second solution is found to exist only in the ``pre-big-bang'' epoch and is singular at $\tau = 0$. In either case back reaction is shown to play a crucial role in determining the specific nature of these geometries. 
  We show, using purely classical considerations and logical extrapolation of results belonging to point particle theories, that the metric background field in which a string propagates must satisfy an Einstein or an Einstein-like equation. Additionally, there emerge restrictions on the worldsheet curvature, which seems to act as a source for spacetime gravity, even in the absence of other matter fields. 
  The horizon area and curvature of three-charge BPS black strings are studied in the D-brane ensemble for the stationary black string. The charge distributions along the string are used to translate the classical expressions for the horizon area and curvature of BPS black strings with waves into operators on the D-brane Hilbert space. Despite the fact that any `wavy' black string has smaller horizon area and divergent curvature, the typical values of the horizon area and effects of the horizon curvature in the D-brane ensemble deviate negligibly from those of the original stationary black string in the limit of large integer charges. Whether this holds in general will depend on certain properties of the quantum bound states. 
  We review a generic structure of conventional (Nambu-Goto and Dirac-Born-Infeld-like) worldvolume actions for the superbranes and show how it is connected through a generalized action construction with a doubly supersymmetric geometrical approach to the description of super-p-brane dynamics as embedding world supersurfaces into target superspaces. 
  We consider the type IIB supergravity in ten dimensions compactified on S^1 \times T^4, with intersecting one and five D-branes in the compact dimensions. By imposing the spherical symmetry in the resulting five dimensional theory, we further reduce the s-wave sector of the theory to a two dimensional dilaton gravity. Via this construction, the techniques developed for the general two dimensional dilaton gravities are applicable in this context. Specifically, we obtain the bosonic sector general static solutions. In addition to the well-known asymptotically flat black hole solutions, they include solutions with naked singularities and non-asymptotically flat black holes. 
  The second quantization of M(atrix) theory in the free (Boltzmannian) Fock space is considered. It provides a possible framework to the recent Susskind proposal that U(N) supersymmetic Yang-Mills theories for all N might be embedded in a single dynamical system. The second quantization of M(atrix) theory can also be useful for the study of the Lorentz symmetry and for the consideration of processes with creation and annihilation of D-branes. 
  We discuss classical composite p-brane solutions and their quantization using the conjecture that their fluctuations may be described via degrees of freedom of Dirichlet strings ended on these p-branes. We work with Dirichlet (super)strings in framework of string field theory for open (super)strings. To elaborate in this scheme the eleventh dimension modes we take just a collection of Dirichlet strings which in their middle points have jumps in eleventh dimension. This theory can be seen as string field theory in infinite momentum frame of an eleven dimensional object. 
  We derive four dimensional gauge theories with exceptional groups $F_4$, $E_8$, $E_7$, and $E_7$ with matter, by starting from the duality between the heterotic string on $K3$ and F-theory on a elliptically fibered Calabi-Yau 3-fold. This configuration is compactified to four dimensions on a torus, and by employing toric geometry, we compute the type IIB mirrors of the Calabi-Yaus of the type IIA string theory. We identify the Seiberg-Witten curves describing the gauge theories as ALE spaces fibered over a $P^1$ base. 
  We study deformations of dualities in finite N=2 supersymmetric QCD. Adding mass terms for some quarks and the adjoint matter to the finite N=2 theory, which is known to have dual descriptions, the correspondence of gauge invariant operators between the original and dual theory is deformed. As a result, we naturally obtain N.Seiberg's N=1 duality. Furthermore, we discuss the origin of the meson and superpotential in the dual theory. This approach can be applied to SU(N), SO(N), and USp(2N) gauge theories, and we analyze all these cases. 
  We perform the dimensional reduction of the linear $\sigma$ model at one-loop level. The effective potential of the reduced theory obtained from the integration over the nonzero Matsubara frequencies is exhibited. Thermal mass and coupling constant renormalization constants are given, as well as the thermal renormalization group equation which controls the dependence of the counterterms on the temperature. We also recover, for the reduced theory, the vacuum unstability of the model for large N. 
  This is a general introduction to electric-magnetic duality in non-abelian gauge theories. In chapter I, I review the general ideas which led in the late 70s to the idea of electric/magnetic duality in quantum field theory. In chapters II and III, I focus mainly on N=2 supersymmetric theories. I present the lagrangians and explain in more or less detail the non-renormalization theorems, rigid special geometry, supersymmetric instanton calculus, charge fractionization, the semiclassical theory of monopoles, duality in Maxwell theory and the famous Seiberg-Witten solution. I discuss various physical applications, as electric charge confinement, chiral symmetry breaking or non-trivial superconformal theories in four dimensions. In Section II.3 new material is presented, related to the computation of the eta invariant of certain Dirac operators coupled minimally to non-trivial monopole field configurations. I explain how these invariants can be obtained exactly by a one-loop calculation in a suitable N=2 supersymmetric gauge theory. This is an unexpected application of the holomorphy properties of N=2 supersymmetry, and constitutes a tremendous simplification of the usual computation. An expanded version of these new results will be published soon. 
  The annihilation channel is implemented into the front form calculations of the positronium spectrum presented in a previous publication. The effective Hamiltonian is calculated analytically. Its eigensolutions are obtained numerically. A complete separation of the dynamical and instantaneous part of the annihilation interaction is observed. We find the remarkable effect that the annihilation channel stabilizes the cutoff behavior of the spectrum. 
  Perturbation theory in the nonperturbative QCD vacuum and the non-Abelian Stokes theorem, representing a Wilson loop in the SU(2) gluodynamics as an integral over all the orientations in colour space, are applied to derivation of the correction to the background-induced string effective action. This correction is due to accounting in the lowest order of perturbation theory for the interaction of perturbative gluons with the string world sheet. It occurs that this interaction affects only the coupling constant of the rigidity term, while its contribution to the string tension of the Nambu-Goto term vanishes. The obtained correction to the rigid string coupling constant multiplicatively depends on the spin of the representation of the Wilson loop under consideration, the QCD coupling constant and a certain path integral, which includes the background Wilson average. 
  In certain circumstances when two branes pass through each other a third brane is produced stretching between them. We explain this phenomenon by the use of chains of dualities and the inflow of charge that is required for the absence of chiral gauge anomalies when pairs of D-branes intersect. 
  Born-Infeld nonlinear electrodynamics are considered. Main attention is given to existence of singular point at static field configuration that M.Born and L.Infeld are considered as a model of electron. It is shown that such singularities are forbidden within the framework of the Born-Infeld model. It is proposed a modernized action that make possible an existence of the singularities. It is obtained main relations in view of the singularities. In initial approximation this model gives the usual linear electrodynamics with point charged particles. 
  Lectures presented at the 33rd Karpacz Winter School ``Duality: Strings and Fields'' briefly introducing dualities in four-dimensional quantum field theory, and summarizing results found in supersymmetric field theories. The first lecture describes physical aspects of electric-magnetic (EM), strong-weak coupling (S), and infrared (IR) dualities. The second lecture focuses on results and conjectures concerning S-duality in N=2 supersymmetric gauge theories. The third lecture discusses IR-dualities and their relation to S-duality in N=1 supersymmetric field theories. 
  We use path-integral methods to analyze the vacuum properties of a recently proposed extension of the Thirring model in which the interaction between fermionic currents is non-local. We calculate the exact ground state wave functional of the model for any bilocal potential, and also study its long-distance behavior. We show that the ground state wave functional has a general factored Jastrow form. We also find that it posess an interesting symmetry involving the interchange of density-density and current-current interactions. 
  D-branes have been used to describe many properties of extremal and near extremal black holes. These lecture notes provide a short review of these developments. 
  A low-energy background field solution describing D-membrane configurations is constructed which is distinguished by the appearance of a Hermitian metric on the internal space. This metric is composed of a number of independent harmonic functions on the transverse space. Thus this construction generalizes the usual harmonic superposition rule. The BPS bound of these solutions is shown to be saturated indicating that they are supersymmetric. By means of T-duality, we construct more solutions of the IIA and IIB theories. 
  The paper has been withdrawn by the author. 
  A massive relativistic spinning point particle in any number of dimensions has in a previous article been shown to be described by first class constraints, which define a gauge theory. In the present paper we find the corresponding finite gauge transformations. By comparing the integrated gauge transformations to transformation equations found by Pryce, we conclude that the selection of gauge corresponds to selection of the relativistic center of mass frame in the model of Pryce, where a spinning particle is considered a composite object. The Lorentz group is identified as the gauge group, and as gauge field we identify the relativistic angular velocity. We also show that an analogous physical interpretation is possible for the relativistic spherical top of Hanson and Regge. 
  The coherent states associated to the discrete serie representations $D(E_o,s)$ of $SO(3,2)$ are constructed in terms of (spin-)tensor fields on $D=4$ anti-de Sitter space. For $E_o>s+5$ the linear space ${\cal H}_{E_o,s}$ spanned by these states is proved to carry the unitary irreducible representation $D(E_o,s)$. The $SO(3,2) $-covariant generalized Fourier transform in this space is exhibited. The quasiclassical properties of the coherent states are analyzed. In particular, these states are shown to be localized on the time-like geodesics of anti-de Sitter space. 
  It is shown how a pure background tensor formalism provides a concise but explicit and highly flexible machinery for the generalised curvature analysis of individual embedded surfaces and foliations such as arise in the theory of topological defects in cosmological and other physical contexts. The unified treatment provided here shows how the relevant extension of the Raychaudhuri identity is related to the correspondingly extended Codazzi identity. 
  We study the force balance between orthogonally positioned $p$-brane and $(8-p)$-brane. The force due to graviton and dilaton exchange is repulsive in this case. We identify the attractive force that balances this repulsion as due to one-half of a fundamental string stretched between the branes. As the $p$-brane passes through the $(8-p)$-brane, the connecting string changes direction, which may be interpreted as creation of one fundamental string. We show this directly from the structure of the Chern-Simons terms in the D-brane effective actions. We also discuss the effect of string creation on the 0-brane quantum mechanics in the type I' theory. The creation of a fundamental string is related by U-duality to the creation of a 3-brane discussed by Hanany and Witten. Both processes have a common origin in M-theory: as two M5-branes with one common direction cross, a M2-brane stretched between them is created. 
  We subject the methodology used to derive the effective dynamics of topological defects to a critical reappraisal, using the two-dimensional kink as an illustrative example. Special care is taken on how the zero modes should be handled in order to avoid overcounting of degrees of freedom. This is an issue that has been overlooked in many recent contributions on the derivation of domain wall effective actions. We show that, unless such redundancy is completely removed by means of a sort of gauge-fixing, the expression obtained for the effective action will not be consistent. We readdress some earlier calculations over the existence of curvature corrections in the light of the previous discussion and briefly comment on the application of this method to higher dimensional topological defects. 
  In the paper we give consecutive description of functional methods of quantum field theory for systems of interacting q-particles. These particles obey exotic statistics and appear in many problems of condensed matter physics, magnetism and quantum optics. Motivated by the general ideas of standard field theory we derive formulae in q-functional derivatives for the partition function and Green's functions generating functional for systems of exotic particles. This leads to a corresponding perturbation series and a diagram technique. Results are illustrated by a consideration of an one-dimensional q-particle system and compared with some exact expressions obtained earlier. 
  We carry out the Hamiltonian analysis of non-Abelian gauge theories in (2+1) dimensions in a gauge-invariant matrix parametrization of the fields. A detailed discussion of regularization issues and the construction of the renormalized Laplace operator on the configuration space, which is proportional to the kinetic energy, are given. The origin of the mass gap is analyzed and the lowest eigenstates of the kinetic energy are explicitly obtained; these have zero charge and exhibit a mass gap . The nature of the corrections due to the potential energy, the possibility of an improved perturbation theory and a Schrodinger-like equation for the states are also discussed. 
  Within the Local Potential Approximation to Wilson's, or Polchinski's, exact renormalization group, and for general spacetime dimension, we construct a function, c, of the coupling constants; it has the property that (for unitary theories) it decreases monotonically along flows, and is stationary only at fixed points ---where it `counts degrees of freedom', i.e. is extensive, counting one for each Gaussian scalar. Furthermore, by choosing restrictions to some sub-manifold of coupling constant space, we arrive at a very promising variational approximation method. 
  We discuss a simple but instructive model in which Gauss' law holds for a class of charged states. In spite of the non-localizability of these charges, the corresponding superselection sectors can be labelled by the spectrum of some internal symmetry group and have well defined statistics. More interestingly, the properties of these charged states seem to point to a general argument allowing one to establish these features for any theory with charges of electric or magnetic type. 
  Non-extreme black hole solutions of four dimensional, N=2 supergravity theories with Calabi-Yau prepotentials are presented, which generalize certain known double-extreme and extreme solutions. The boost parameters characterizing the nonextreme solutions must satisfy certain constraints, which effectively limit the functional independence of the moduli scalars. A necessary condition for being able to take certain boost parameters independent is found to be block diagonality of the gauge coupling matrix. We present a number of examples aimed at developing an understanding of this situation and speculate about the existence of more general solutions. 
  We consider the scattering of two Dirichlet zero-branes in M(atrix) theory. Using the formulation of M(atrix) theory in terms of ten-dimensional super Yang-Mills theory dimensionally reduced to $(0+1)$-dimensions, we obtain the effective (velocity dependent) potential describing these particles. At one-loop we obtain the well known result for the leading order of the effective potential $V_{eff}\sim v^4/r^7$, where $v$ and $r$ are the relative velocity and distance between the two zero-branes respectively. A calculation of the effective potential at two-loops shows that no renormalizations of the $v^4$-term of the effective potential occur at this order. 
  Brane actions with chiral bosons present special challenges. Recent progress in the description of the two main examples -- the M theory five-brane and the heterotic string -- is described. Also, double dimensional reduction of the M theory five-brane on K3 is shown to give the heterotic string. 
  Some aspects of the geometry of superembeddings and its application to supersymmetric extended objects are discussed. In particular, the embeddings of (3|16) and (6|16) dimensional superspaces into (11|32) dimensional superspace, corresponding to supermembranes and superfivebranes in eleven dimensions, are treated in some detail. 
  We study the transformation under the String Theory duality group of the observable charges (mass, angular momentum, NUT charge, electric, magnetic and different scalar charges) of four dimensional point-like objects whose asymptotic behavior constitutes a subclass closed under duality. The charges fall into two complex four-dimensional representations of the duality group. T duality (including Buscher's) has an O(1,2) action on them and S duality a U(1) action. The generalized Bogomol'nyi bound is an U(2,2)-invariant built out of one representations while the other representation (which includes the angular momentum) never appears in it. The bound is manifestly duality-invariant. Consistency between T duality and supersymmetry requires that primary scalar hair is included in the Bogomol'nyi bound. Four-dimensional supersymmetric massless black holes are the T duals in time of massive supersymmetric black holes. Non-extreme massless ``black holes'' are the T duals of the non-extreme black holes and have primary scalar hair and naked singularities. 
  We give a brief overview of black-hole solutions in four-dimensional supergravity theories and their extremal and supersymmetric limits. We also address problems like cosmic censorship and no-hair theorems in supergravity theories. While supergravity by itself seems not to be enough to enforce cosmic censorship and absence of primary scalar hair, superstring theory may be. 
  We study the invariant unstable manifold of the trivial renormalization group fixed point tangent to the $\phi^{4}$-vertex in three dimensions. We parametrize it by a running $\phi^{4}$-coupling with linear step $\beta$-function. It is shown to have a renormalized double expansion in the running coupling and its logarithm. 
  We show that the low-lying excitations of the one-dimensional Bose gas are described, at all orders in a 1/N expansion and at the first order in the inverse of the coupling constant, by an effective hamiltonian written in terms of an extended conformal algebra, namely the Cartan subalgebra of the $W_{1+\infty}\times \bar{W}_{1+\infty}$ algebra. This enables us to construct the first interaction term which corrects the hamiltonian of free fermions equivalent to a hard-core boson system. 
  A simple algebraic model for charged particle moving in two dimensional space under influence of singular magnetic field is given. The fundamental assumption for the model is that every charged particle coupled to the magnetic field is transformed into a system of quasiparticles. It is also assumed for simplicity that there is in average N fluxes per particle. If the number N of fluxes is even, then the system can be identified as composite fermions. If N is odd, then we obtain composite bosons. Quantization is described as gradation by certain abelian group G. Statistics is determined by a commutation factor on the grading group G. It is schown that for N = 2 there is a state for which the magnetic field is completely compensatad by the system of quasiparticles and for N = 3 the Landau levells are fractionally filled like in the fractional quantum Hall effect. 
  Brandenberger and Vafa have proposed the string cosmological model based on T-duality. In this model, they took the toroidal target space and introduced the new position \tilde{x} conjugate to the winding mode, in addition to the position x conjugate to the momentum mode. In this way they can describe a universe larger than the string scale with the coordinate x and one smaller with the coordinate \tilde{x}. Resultingly, they never encounter the singularity seen in the standard Big Bang scenario. The most interesting phenomenon in this model is the transition from \tilde{x}-space to x-space when the size of universe is nearly the string scale. Here, we define the dispersion of the momentum number m times the winding number w as the `correlation' of momentum modes and winding modes. Then using the statistical mechanics of strings on a torus, we calculate the correlation in low and high temperature limits, and we consider the possibility that we can observe this effect today, but we will see that this is unlikely. 
  In recent years, Susskind, Thorlacius and Uglum have proposed a model for strings near a black hole horizon in order to represent the quantum mechanical entropy of the black hole and to resolve the information loss problem. However, this model is insufficient because they did not consider the metric modification due to massive strings and did not explain how to carry information from inside of the horizon to the outside world. In this paper, we present a possible, intuitive model for the time development of a black hole in order to solve the information loss problem. In this model, we assume that a first order phase transition occurs near the Hagedorn temperature and the string gas changes to hypothetical matter with vanishing entropy and energy which we call `the Planck solid'. We also study the background geometry of black holes in this picture and find out that there is no singularity within the model. 
  We calculate the constraints on the constants of hypothetical long-range interactions which follow from the recent measurement of the Casimir force. A comparison with previous constraints is given. The new constraints are up to a factor of 3000 stronger in some parameter regions . 
  Explicit computations of the partition function and correlation functions of Wilson and Polyakov loop operators in theta-sectors of two dimensional Yang-Mills theory on the line cylinder and torus are presented. Several observations about the correspondence of two dimensional Yang-Mills theory with unitary matrix quantum mechanics are presented. The incorporation of the theta-angle which characterizes the states of two dimensional adjoint QCD is discussed. 
  In this article, we will discuss geometric quantization of 2d QCD with fermionic and bosonic matter fields. We identify the respective large-N_c phase spaces as the infinite dimensional Grassmannian and the infinite dimensional Disc. The Hamiltonians are quadratic functions, and the resulting equations of motion for these classical systems are nonlinear. In a previous publication, the first author has shown that the linearization of the equations of motion for the Grassmannian gave the `t Hooft equation. We will see that the linearization in the bosonic case leads to the scalar analog of the `t Hooft equation found by Tomaras. 
  The map between the moduli space of F-theory (or type II string) compactifications and heterotic string compactifications can be considerably simplified by using "stable degenerations". We discuss how this method applies to both the E8 x E8 and the Spin(32)/Z2 heterotic string. As a simple application of this method we derive some basic properties of the nonperturbative physics of collections of E8 or Spin(32)/Z2 point-like instantons sitting at A-D-E singularities on a K3 surface. 
  Starting from a consistency requirement between T-duality symmetry and renormalization group flows, the two-loop metric beta function is found for a d=2 bosonic sigma model on a generic, torsionless background. The result is obtained without Feynman diagram calculations, and represents further evidence that duality symmetry severely constrains renormalization flows. 
  We study N=1 dualities in four dimensional supersymmetric gauge theories by D 6-branes wrapping around 3-cycles of Calabi-Yau threefolds in the type IIA string theory. We consider the models involving $SU(N_{c1})\times SU(N_{c2})$ product gauge group without or with adjoint matter in terms of geometrical realisation of the configuration of D 6-branes wrapped 3-cycles. We also find simple geometric description for the triple product gauge group $SU(N_{c1})\times SU(N_{c2})\times SU(N_{c3})$ interpreted recently by Brodie and Hanany in the context of D brane configurations together with NS 5-branes. Their introduction of semi-infinite D 4-branes appears naturally by looking at the flavor group in the dual theory. We generalize to product of arbitrary number of gauge groups. 
  We examine five dimensional extreme black holes with three charges in the matrix model. We build configurations of the 5+1 super Yang-Mills theory which correspond to black holes with transverse momentum charge. We calculate their mass and entropy from the super Yang-Mills theory and find that they match the semi-classical black hole results. We extend our results to nonextreme black holes in the dilute gas approximation. 
  We postulate that physical states are equivalent under coordinate transformations. We then implement this equivalence principle first in the case of one-dimensional stationary systems showing that it leads to the quantum analogue of the Hamilton-Jacobi equation which in turn implies the Schroedinger equation. In this context the Planck constant plays the role of covariantizing parameter. The construction is deeply related to the GL(2,C)-symmetry of the second-order differential equation associated to the Legendre transformation which selects, in the case of the quantum analogue of the Hamiltonian characteristic function, self-dual states which guarantee its existence for any physical system. The universal nature of the self-dual states implies the Schroedinger equation in any dimension. 
  It is pointed out that affine Lie algebras appear to be the natural mathematical structure underlying the notion of integrability for two-dimensional systems. Their role in the construction and classification of 2D integrable systems is discussed. The super- symmetric case will be particularly enphasized. The fundamental examples will be outlined. 
  We consider eight-brane configurations in M(atrix) theory and compute their interaction potentials with gravitons, membranes, and four-branes. We compare these results with the interactions of D8-branes with D0-branes, D2-branes, and D4-branes in IIA string theory. We find agreement between the two approaches for eight-brane interactions with two-branes and four-branes. A discrepancy is noted in the case with zero-branes. 
  As a step toward satisfactory understanding of the quantum dynamics of Dirichlet \break (D-) particles, the amplitude for the basic process describing the scattering of two quantized D-particles is computed in bosonic string theory. The calucluation is performed and cross-checked using three different methods, namely, (i) path integral, (ii) boundary state, and (iii) open-channel operator formalism. The analysis is exact in $\al'$ and includes the first order correction in the expansion with respect to the acceleration of the D-particles. The resultant Lorentz-invariant amplitude is capable of describing general non-forward scattering with recoil effects fully taken into account and it reproduces the known result for the special case of forward scattering in the limit of infinitely large D-particle mass. The expected form of the amplitude for the supersymmetric case is also briefly discussed. 
  We review recent results for the low-lying glueball spectrum on the three-sphere in intermediate volumes that incorporate instanton effects. The latter are implemented through boundary conditions on the fundamental domain obtained by minimising the norm of the gauge field along the gauge orbit. Non-perturbative corrections due to the boundary conditions in field space are seen to be crucial. 
  We consider M(atrix) theory compactifications to seven dimensions with eight unbroken supersymmetries. We conjecture that both M(atrix) theory on K3 and Heterotic M(atrix) theory on T^3 are described by the same 5+1 dimensional theory with N=2 supersymmetry which is broken to N=1 by the base space. The emergence of the extra dimension follows from a recent result of Rozali[hep-th/9702136]. We show that the seven dimensional duality between M-theory on K3 and Heterotic string theory on T^3 is realised in M(atrix) theory as the exchange of one of the dimensions with this new dimension. 
  A supereigenvalue model with purely positive bosonic eigenvalues is presented and solved by considering its superloop equations. This model represents the supersymmetric generalization of the complex one matrix model, in analogy to the relation between the supereigenvalue and the hermitian one matrix model. Closed expressions for all planar multi-superloop correlation functions are found. Moreover an iterative scheme allows the calculation of higher genus contributions to the free energy and to the correlators. Explicit results for genus one are given. 
  I discuss features required for preserving unitarity in black hole decay and concepts underlying such a perspective. Unitarity requires that correlations extend on the scale of the horizon. I show, in a toy model inspired by string theories, that such correlations can indeed arise. The model suggests that, after a time of order 4M ln M following the onset of Hawking radiation, quantum effects could maintain throughout the decay a collapsing star within a Planck distance of its Schwarzschild radius. In this way information loss would be avoided. The concept of black hole ``complementarity'', which could reconcile these macroscopic departures from classical physics with the equivalence principle, is interpreted in terms of weak values of quantum operators. 
  A Yang-Mills solution is constructed on T^6 which corresponds to a brane configuration composed purely of 0-branes and 6-branes. This configuration breaks all supersymmetries and has an energy greater than the sum of the energies of its components; nonetheless, the configuration is stable classically, at least to quadratic order. An analogous construction is also given for a system of 0-branes and 8-branes on T^8. These constructions may prove to be useful for describing 6-branes and 8-branes in M(atrix) theory. 
  We survey the various field theories with 16 real supercharges. The most widely known theory in this class is the N=4 theory in four dimensions. The moduli space of vacua of these theories are described and the physics at the singularities of the moduli spaces are studied. 
  The action of supersymmetric Born-Infeld theory (D-9-brane in a Lorentz covariant static gauge) has a geometric form of the Volkov-Akulov-type. The first non-linearly realized supersymmetry can be made manifest, the second world-volume supersymmetry is not manifest. We also study the analogous 2 supersymmetries of the quadratic action of the covariantly quantized D-0-brane. We show that the Hamiltonian and the BRST operator are build from these two supersymmetry generators. 
  Within the framework of a local expansion of the logarithm of the O(N) sigma-model vacuum functional, valid for slowly varying fields, the modified Virasoro algebra is studied. The operator-like central charge term is given, up to second order, for a general functional. 
  We compute long-distance interaction potentials between certain 1/2 and 1/4 supersymmetric D-brane configurations of type IIB theory, demonstrating detailed agreement between classical supergravity and one-loop instanton matrix model results. This confirms the interpretation of D-branes as described by classical matrix model backgrounds as being `populated' by large number of D-instantons, i.e. as corresponding to non-marginal bound states of branes of lower dimensions. In the process, we establish precise relation between matrix model expressions and non-abelian F^4 terms in the super Yang-Mills effective action. 
  We study the unitary matrix model with a topological term. We call the topological term the theta term. In the symmetric model there is the phase transition between the strong and weak coupling regime at $\lambda_{c}=2$. If the Wilson term is bigger than the theta term, there is the strong-weak coupling phase transition at the same $\lambda_{c}$. On the other hand, if the theta term is bigger than the Wilson term, there is only the strong coupling regime. So the topological phase transition disappears in this case. 
  This is the written version of a series of lectures reviewing the basics of duality as applied to p-forms and sigma-models. The ideas are introduced by way of worked examples, often quite detailed. Our approach is very pedestrian and the presentation is aimed at non-specialists, such as e.g. graduate students. 
  The classical action for pure Yang--Mills gauge theory can be formulated as a deformation of the topological $BF$ theory where, beside the two-form field $B$, one has to add one extra-field $\eta$ given by a one-form which transforms as the difference of two connections. The ensuing action functional gives a theory that is both classically and quantistically equivalent to the original Yang--Mills theory. In order to prove such an equivalence, it is shown that the dependency on the field $\eta$ can be gauged away completely. This gives rise to a field theory that, for this reason, can be considered as semi-topological or topological in some but not all the fields of the theory. The symmetry group involved in this theory is an affine extension of the tangent gauge group acting on the tangent bundle of the space of connections. A mathematical analysis of this group action and of the relevant BRST complex is discussed in details. 
  We compute the exact induced parity-breaking part of the effective action for 2+1 massive fermions in $QED_3$ at finite temperature by calculating the fermion determinant in a particular background. The result confirms that gauge invariance of the effective action is respected even when large gauge transformations are considered. 
  The planar Yang-Mills theory in three spatial dimensions is examined in a particular representation which explicitly embodies factorization. The effective planar Yang-Mills theory Hamiltonian is constructed in this representation. 
  Matrix String Theory of Banks, Fischler, Shenker and Susskind can be understood as a generalized quantum theory (provisionally named "quansical" theory) which differs from Adler's generalized trace quantum dynamics. The effective Matrix String Theory Hamiltonian is constructed in a particular fermionic realization of Matrix String Theory treated as an example of "quansical" theory. 
  We construct nearly topological Yang-Mills theories on eight dimensional manifolds with a special holonomy group. These manifolds are the Joyce manifold with $Spin(7)$ holonomy and the Calabi-Yau manifold with SU(4) holonomy. An invariant closed four form $T_{\mu\nu\rho\sigma}$ on the manifold allows us to define an analogue of the instanton equation, which serves as a topological gauge fixing condition in BRST formalism. The model on the Joyce manifold is related to the eight dimensional supersymmetric Yang-Mills theory. Topological dimensional reduction to four dimensions gives non-abelian Seiberg-Witten equation. 
  We derive Schwinger-Dyson equations for the Wilson loops of a type IIB matrix model. Superstring coordinates are introduced through the construction of the loop space. We show that the continuum limit of the loop equation reproduces the light-cone superstring field theory of type IIB superstring in the large-N limit. We find that the interacting string theory can be obtained in the double scaling limit as it is expected. 
  The connection between the anomalous dimension and some invariance properties of the fixed point actions within exact RG is explored. As an application, Polchinski equation at next-to-leading order in the derivative expansion is studied. For the Wilson fixed point of the one-component scalar theory in three dimensions we obtain the critical exponents \eta=0.042, \nu=0.622 and \omega=0.754. 
  The potential of a configuration of two Dirichlet branes for which the number of ND-directions is eight is determined. Depending on whether one of the branes is an anti-brane or a brane, the potential vanishes or is twice as large as the dilaton-gravitational potential. This is shown to be related to the fact that a fundamental string is created when two such branes cross. Special emphasis is given to the D0-D8 system, for which an interpretation of these results in terms of the massive IIA supergravity is presented. It is also shown that the branes cannot move non-adiabatically in the transverse direction. The configuration of a zero brane and an orientifold 8-plane is analyzed in a similar way, and some implications for the type IA-heterotic duality and the heterotic matrix theory are discussed. 
  An introduction to Seiberg-Witten theory and its relation to theories which include gravity. 
  We study M(atrix) theory description of M theory compactified on T5/Z2 orbifold. In the large volume limit we show that M theory dynamics is described by N=8 supersymmetric USp(2N) M(atrix) quantum mechanics. Via zero-brane parton scattering, we show that each orbifold fixed point carries anomalous G-flux $\oint [G/2 \pi]= - 1/2$. To cancel the anomalous G-flux, we introduce twisted sector consisting of sixteen five-branes represented by fundamental representation hypermultiplets. In the small volume limit we show that M theory dynamics is described by by (5+1)-dimensional (8,0) supersymmetric USp(2N) chiral gauge theory. We point out that both perturbative and global gauge anomalies are cancelled by the sixteen fundamental representation hyper- multiplets in the twisted sector. We show that M(atrix) theory is capable of turning on spacetime background with the required sixteen five-branes out of zero-brane partons as bound-states. We determine six-dimensional spacetime spectrum from the M(atrix) theory for both untwisted and twisted sectors and find a complete agreement with the spectrum of (2,0) supergravity. We discuss M(atrix) theory description of compactification moduli space, symmetry enhance- ment thereof as well as further toroidal compactifications. 
  We construct a non-perturbative method to investigate the phase structure of the scalar theory at finite temperature. The derivative of the effective potential with respect to the mass square is expressed in terms of the full propagator. Under a certain approximation this expression reduces to the partial differential equation for the effective potential. We numerically solve the partial differential equation and obtain the effective potential non-perturbatively. It is found that the phase transition is of the second order. The critical exponents calculated in this method are consistent with the results obtained in Landau approximation. 
  We discuss the canonical quantization of non-unitary time evolution in inflating Universe. We consider gravitational wave modes in the FRW metrics in a de Sitter phase and show that the vacuum is a two-mode SU(1,1) squeezed state of thermo field dynamics, thus exhibiting the link between inflationary evolution and thermal properties. In particular we discuss the entropy and the free energy of the system. The state space splits into many unitarily inequivalent representations of the canonical commutation relations parametrized by time $t$ and non-unitary time evolution is described as a trajectory in the space of the representations: the system evolves in time by running over unitarily inequivalent representations. The generator of time evolution is related to the entropy operator. A central ingredient in our discussion is the doubling of the degrees of freedom which turns out to be the bridge to the unified picture of non-unitary time evolution, squeezing and thermal properties in inflating metrics. 
  An imploding shell of radiation is shown to create a 2-D black hole within the framework of the ``R=T'' theory. The radius of the horizon is given by 1/(2M), where M is the mass of the black hole. The topology of the central singularity is that of a corner. The radiation emitted very far from the black hole is thermal with temperature M/(2\pi). The back-reaction problem is solved to one-loop order. 
  The beta-function is calculated for an SU(N) Yang-Mills theory from an ansatz for the vacuum wavefunctional. Direct comparison is made with the results of calculations of the beta-function of QCD. In both cases the theories are asymptotically free. The only difference being in the numerical coefficient of the beta-function, which is found to be -4 from the ansatz and -4+1/3 from other QCD calculations. This is because, due to the constraint of Gauss' law applied to the wavefunctional, transverse gluons (which contribute the 1/3) are omitted. The renormalisation procedure is understood in terms of `tadpole' and `horse-shoe' Feynman diagrams which must be interpreted with a non-local propagator. 
  It has been proposed recently that, in the framework of M(atrix) theory, N=8 supersymmetric U(N) Yang-Mills theory in 1+1 dimensions gives rise to type IIA long string configurations. We point out that the quantum moduli space of $SYM_{1+1}$ gives rise to two quantum numbers, which fit very well into the M(atrix) theory. The two quantum numbers become familiar if one switches to a IIB picture, where they represent configurations of D-strings and fundamental strings. We argue that, due to the SL(2,Z) symmetry, of the IIB theory, such quantum numbers must represent configurations that are present also in the IIA framework. 
  We present a Donaldson-Witten type field theory in eight dimensions on manifolds with $Spin(7)$ holonomy. We prove that the stress tensor is BRST exact for metric variations preserving the holonomy and we give the invariants for this class of variations. In six and seven dimensions we propose similar theories on Calabi-Yau threefolds and manifolds of $G_2$ holonomy respectively. We point out that these theories arise by considering supersymmetric Yang-Mills theory defined on such manifolds. The theories are invariant under metric variations preserving the holonomy structure without the need for twisting. This statement is a higher dimensional analogue of the fact that Donaldson-Witten field theory on hyper-K\"ahler 4-manifolds is topological without twisting. Higher dimensional analogues of Floer cohomology are briefly outlined. All of these theories arise naturally within the context of string theory. 
  We discuss issues involving p(D)- brane charge quantization and the normalization of effective actions, in string/M-theory. We also construct the action of (the bosonic sector of) eleven dimensional supergravity in the presence of two- and five- branes and discuss (perturbative) anomaly cancellation. 
  We present a counter example which shows the violation of the S-matrix factorization in the massive Thirring model. This is done by solving the PBC equations of the massive Thirring model exactly but numerically.   The violation of the S-matrix factorization is related to the fact that the crossing symmetry and the factorization do not commute with each other. This confirms that the soliton antisoliton S-matrix factorization picture of the sine-Gordon model is semiclassical and does not lead to a full quantization procedure of the massive Thirring model. 
  We construct a new class of quasi-exactly solvable many-body Hamiltonians in arbitrary dimensions, whose ground states can have any correlations we choose. Some of the known correlations in one dimension and some recent novel correlations in two and higher dimensions are reproduced as special cases. As specific interesting examples, we also write down some new models in two and higher dimensions with novel correlations. 
  It is shown that the problem of calculating form factors in ADE affine Toda field theories can be reduced to the nonperturbative recursive calculation of polynomials symmetric in each sort of variables. We determine these recursion equations explicitly for the ADE series and characterize the polynomial solutions by an interplay between the weight space of the underlying Lie algebra and representations of the symmetric group. 
  Talk presented in the 97 Karpatcz winter school. We describe Sen's work on F theory on K3 and its reflection on the world-volume field theory on a D3-brane probe. Field theories on a multiple of probes are analyzed. We construct a 4d N=1 superconformal probe theory which is invariant under electric-magnetic duality via a compactification to six dimensions on T^6/Z_2 x Z_2. 
  In a recent paper (hep-th/9703084) it was conjectured that the imaginary simple roots of the Borcherds algebra $g_{II_{9,1}}$ at level 1 are its only ones. We here propose an independent test of this conjecture, establishing its validity for all roots of norm $\geq -8$. However, the conjecture fails for roots of norm -10 and beyond, as we show by computing the simple multiplicities down to norm -24, which turn out to be remakably small in comparison with the corresponding $E_{10}$ multiplicities. Our derivation is based on a modified denominator formula combining the denominator formulas for $E_{10}$ and $g_{II_{9,1}}$, and provides an efficient method for determining the imaginary simple roots. In addition, we compute the $E_{10}$ multiplicities of all roots up to height 231, including levels up to $\ell =6$ and norms -42. 
  We show that a canonical tranformation converts, up to a boundary term, a generic 2d dilaton gravity model into a bosonic string theory with a Minkowskian target space. 
  We discuss a resummed perturbation theory based on the Wilson renormalization group. In this formulation the Wilsonian flowing couplings, which generalize the running coupling, enter directly into the loop expansion. In the case of an asymptotically free theory the flowing coupling is well defined since the infrared Landau pole is absent. We show this property in the case of the $\phi^3_6$ theory. We also extend this formulation to the QED theory and we prove that it is consistent with gauge invariance. 
  In quantum gravity, fields may lose quantum coherence by scattering off vacuum fluctuations in which virtual black hole pairs appear and disappear. Although it is not possible to properly compute the scattering off such fluctuations, we argue that one can get useful qualitative results, which provide a guide to the possible effects of such scattering, by considering a quantum field on the $C$ metric, which has the same topology as a virtual black hole pair. We study a scalar field on the Lorentzian $C$ metric background, with the scalar field in the analytically-continued Euclidean vacuum state. We find that there are a finite number of particles at infinity in this state, contrary to recent claims made by Yi. Thus, this state is not determined by data at infinity, and there is loss of quantum coherence in this semi-classical calculation. 
  Dualities between certain supersymmetric gauge field theories in three and four dimensions have been studied in considerable detail recently, by realizing them as geometric manipulations of configurations of extended objects in type II string theory. These extended objects include `D-branes' and `NS-(five)branes'. In constructing the brane configurations which realize dualities for orthogonal and symplectic gauge groups, an `orientifold' was introduced, which results in non-orientable string sectors. Certain features of orientifolded NS-branes -such as their existence- were assumed in the original construction, which have not been verified directly. However, those features fit very well together with the properties of the relevant field theories, and subsequently yielded the known dualities. This letter describes how orientifolded NS-branes can exist in type II string theory by displaying explicitly that the assumed combinations of world-sheet and space-time symmetries do indeed leave the NS-brane invariant and therefore can be gauged. The resulting orientifolded NS-brane can be described in terms of background fields, and furthermore as an exact conformal field theory, to exactly the same extent as the standard NS-brane. 
  We construct a one-to-one map between the primary fields of the N=2 superconformal Kazama-Suzuki models G(m,n,k) and G(k,n,m) based on complex Grassmannian cosets, using level-rank duality of Wess-Zumino-Witten models. We then show that conformal weights, superconformal U(1) charges, modular transformation matrices, and fusion rules are preserved under this map, providing strong evidence for the equivalence of these coset models. 
  N=2 supersymmetric quantum black holes in the heterotic S-T-U model are presented. In particular three classes of axion-free quantum black holes with half the N=2, D=4 supersymmetries unbroken are considered. First, these quantum black holes are investigated at generic points in moduli space. Then linearized non-abelian black holes are investigated representing a subset of non-abelian black hole solutions at critical points of perturbative gauge symmetry enhancement in moduli space. It is shown that the entropy of linearized non-abelian black holes can be obtained, starting at non-critical points in moduli space, by continuous variation of the moduli and a proper identification of the non-abelian charges. 
  We discuss the similarities between BPS monopoles and Skyrmions, and point to an underlying connection in terms of rational maps between Riemann spheres. This involves the introduction of a new ansatz for Skyrme fields. We use this to construct good approximations to several known Skyrmions, including all the minimal energy configurations up to baryon number nine, and some new solutions such as a baryon number seventeen Skyrme field with the truncated icosahedron structure of a buckyball.   The new approach is also used to understand the low-lying vibrational modes of Skyrmions, which are required for quantization. Along the way we discover an interesting Morse function on the space of rational maps which may be of use in understanding the Sen forms on the monopole moduli spaces. 
  We present a lattice formulation which gives super Yang-Mills theories in any dimensions with simple supersymmetry as well as extended supersymmetry in the continuum limit without fine-tuning. We first formulate super Yang-Mills theories with simple supersymmetry in 3,4,6,10 dimensions, incorporating the gluino on the lattice using the overlap formalism. In 4D, exact chiral symmetry forbids gluino mass, which ensures that the continuum limit is supersymmetric without fine-tuning. In 3D, exact parity invariance plays the same role. 6D and 10D thories being anomalous, we formulate them as anomalous chiral gauge theories as they are. Dimensional reduction within lattice formulation is then applied to the theories in 3,4,6 and 10D to obtain super Yang-Mills theories in arbitrary dimensions with either simple or extended supersymmetry. 
  A supersymmetric theory in two-dimensions has enough data to define a noncommutative space thus making it possible to use all the tools of noncommutative geometry. In particular, we apply this to the N=1 supersymmetric non-linear sigma model and derive an expression for the generalized loop space Dirac operator, in presence of a general background, using canonical quantization. The spectral action principle is used to show that the superstring partition function is also a spectral action valid for the fluctuations of the string modes. 
  Talk at the International Workshop ``New Non Perturbative Methods and Quantization on the Light Cone", Les Houches February 24 - March 7, 1997. 
  After a summary of a recently proposed new type of instant form of dynamics (the Wigner-covariant rest-frame instant form), the reduced Hamilton equations in the covariant rest-frame Coulomb gauge for the isolated system of N scalar particles with pseudoclassical Grassmann-valued electric charges plus the electromagnetic field are studied. The Lienard-Wiechert potentials of the particles are evaluated and it is shown how the causality problems of the Abraham-Lorentz-Dirac equation are solved at the pseudoclassical level. Then, the covariant rest-frame description of scalar electrodynamics is given. Applying to it the Feshbach-Villars formalism, the connection with the particle plus electromagnetic field system is found. 
  The system of N scalar particles with Grassmann-valued color charges plus the color SU(3) Yang-Mills field is reformulated on spacelike hypersurfaces. The Dirac observables are found and the physical invariant mass of the system in the Wigner-covariant rest-frame instant form of dynamics (covariant Coulomb gauge) is given. From the reduced Hamilton equations we extract the second order equations of motion both for the reduced transverse color field and the particles. Then, we study this relativistic scalar quark model, deduced from the classical QCD Lagrangian and with the color field present, in the N=2 (meson) case. A special form of the requirement of having only color singlets, suited for a field-independent quark model, produces a ``pseudoclassical asymptotic freedom" and a regularization of the quark self-energy. 
  We use an extension of the method due to Batalin, Fradkin, Fradkina, and Tyutin (BFFT) for transforming the nonlinear $\sigma$ model in a non-Abelian gauge theory. We deal with both supersymmetric and nonsupersymmetric cases. The bosonic case was already considered in literature but just gauged with an Abelian algebra. We show that the supersymmetric version is only compatible with a non-Abelian gauge theory. The usual BFFT method for this case leads to a nonlocal algebra. 
  We calculate the effective potential of the scalar theory at finite temperature under the super-daisy approximation, after expressing its derivative with respect to mass square in terms of the full propagator. This expression becomes the self-consistent equation for the derivative of the effective potential. We find the phase transition is first order with this approximation. We compare our result with others. 
  We investigate the mesonic light-front bound-state equations of the 't Hooft and Schwinger model in the two-particle, i.e. valence sector, for small fermion mass. We perform a high precision determination of the mass and light-cone wave function of the lowest lying meson by combining fermion mass perturbation theory with a variational approach. All calculations are done entirely in the fermionic representation without using any bosonization scheme. In a step-by-step procedure we enlarge the space of variational parameters. For the first two steps, the results are obtained analytically. Beyond that we use computer algebraic and numerical methods. We achieve good convergence so that the calculation of the meson mass squared can be extended to third order in the fermion mass. Within the numerical treatment we include higher Fock states up to six particles. Our results are consistent with all previous numerical investigations, in particular lattice calculations. For the massive Schwinger model, we find a small discrepancy (less than 2 percent) in comparison with known bosonization results. Possible resolutions of this discrepancy are discussed. 
  A manifestly S-dual, and `12 dimensional', IIB superstring action with an $Sl(2;\bR)$ doublet of `Born-Infeld' fields is presented. The M-theory origin of the 12th dimension is the M-2-brane tension, which can be regarded as the flux of a 3-form worldvolume field strength. The latter is required by the fact that the M-2-brane can have a boundary on an M-5-brane. 
  The cohomology of a compact Kaehler (resp. hyperKaehler) manifold admits the action of the Lie algebra so(2,1) (resp. so(4,1)). In this paper we show, following an idea of Witten, how this action follows from supersymmetry, in particular from the symmetries of certain supersymmetric sigma models. In addition, many of the fundamental identities in Hodge-Lefschetz theory are also naturally derived from supersymmetry. 
  It is argued that D=10 type II strings and M-theory in D=11 have D-5 branes and 9-branes that are not standard p-branes coupled to anti-symmetric tensors. The global charges in a D-dimensional theory of gravity consist of a momentum $P_M$ and a dual D-5 form charge $K_{M_1...M_{D-5}}$, which is related to the NUT charge. On dimensional reduction, P gives the electric charge and K the magnetic charge of the graviphoton. The charge K is constructed and shown to occur in the superalgebra and BPS bounds in $D\ge 5$, and leads to a NUT-charge modification of the BPS bound in D=4. $K$ is carried by Kaluza-Klein monopoles, which can be regarded as D-5 branes. Supersymmetry and U-duality imply that the type IIB theory has (p,q) 9-branes. Orientifolding with 32 (0,1) 9-branes gives the type I string, while modding out by a related discrete symmetry with 32 (1,0) 9-branes gives the SO(32) heterotic string. Symmetry enhancement, the effective world-volume theories and the possibility of a twelve dimensional origin are discussed. 
  In this paper we consider Type I string theory compactified on a Z_7 orbifold. The model has N=1 supersymmetry, a U(4) \otimes U(4) \otimes U(4) \otimes SO(8) gauge group, and chiral matter. There are only D9-branes (for which we discuss tadpole cancellation conditions) in this model corresponding to a perturbative heterotic description in a certain region of the moduli space. We construct the heterotic dual, match the perturbative type I and heterotic tree-level massless spectra via giving certain scalars appropriate vevs, and point out the crucial role of the perturbative superpotential (on the heterotic side) for this matching. The relevant couplings in this superpotential turn out to be non-renormalizable (unlike the Z-orbifold case discussed in Ref [1], where Yukawa couplings sufficed for duality matching). We also discuss the role of the anomalous U(1) gauge symmetry present in both type I and heterotic models. In the perturbative regime we match the (tree-level) moduli spaces of these models. We point out possible generalizations of the Z_3 and Z_7 cases to include D5-branes which would help in understanding non-perturbative five-brane dynamics on the heterotic side. 
  The N=1 supersymmetric gauge SU(5) theory with one antisymmetric tensor, n+3 fundamentals and n+4 antifundamentals has dual magnetic descriptions in the infrared. By introducing extra singlet fields and tree level superpotential terms to the electric SU(5) theories, we are able to make the dual theories flow to the known SU(n)xSU(2) gauge theories which break supersymmetry dynamically. In the n=2 case, the lifting of the pseudo-flat direction is estimated by using dual operator mappings. 
  The entropy of a near-extremal black hole made of parallel D-branes has been shown to agree, upto a numerical factor, with that of the gas of massless open string states on the brane worldvolume when the string coupling is chosen suitably. We investigate the process of emission or absorption of massless S-wave neutral scalars by these black holes. We show that with rather mild assumptions about the nature of the interactions between the scalar and open string states, the D-brane cross-section generally fails to reproduce the universal low energy black hole cross-section except for 1-branes and 3-branes. 
  In SU(2) Seiberg-Witten theory, it is known that the dual pair of fields are expressed by hypergeometric functions. As for the theory with SU(3) gauge symmetry without matters, it was shown that the dual pairs of fields can be expressed by means of the Appell function of type F_4. These expressions are convenient for analyzing analytic properties of fields. We investigate the relation between Seiberg-Witten theory of rank two gauge group without matters and hypergeometric series of two variables. It is shown that the relation between gauge theories and Appell functions can be observed for other classical gauge groups of rank two. For B_2 and C_2, the fields are written in terms of Appell functions of type H_5. For D_2, we can express fields by Appell functions of type F_4 which can be decomposed to two hypergeometric functions, corresponding to the fact SO(4)\sim SU(2)\times SU(2). We also consider the integrable curve of type C_2 and show how the fields are expressed by Appell functions. However in the case of exceptional group G_2, our examination shows that they can be represented by hypergeometric series which does not correspond to the Appell functions. 
  Energy eigenstates for N=2 supersymmetric gauged quantum mechanics are found for the gauges groups SU(n) and U(n). The analysis is aided by the existence of an infinite number of conserved operators. The spectum is continuous. Perturbative eigenstates for $N>2$ are also presented, a case which is relevant for the conjectured description of M theory in the infinite momentum frame. 
  A method for solving Schwinger-Dyson equations for the Green function generating functional of non-Abelian gauge theory is proposed. The method is based on an approximation of Schwinger-Dyson equations by exactly soluble equations. For the SU(2) model the first step equations of the iteration scheme are solved which define a gauge field propagator. Apart from the usual perturbative solution, a non-perturbative solution is found which corresponds to the spontaneous symmetry breaking and obeys infrared finite behaviour of the propagator. 
  We discuss general bosonic stationary configurations of N=2, D=4 supergravity coupled to vector multiplets. The requirement of unbroken supersymmetries imposes constraints on the holomorphic symplectic section of the underlying special K\"ahler manifold. The corresponding solutions of the field equations are completely determined by a set of harmonic functions. As examples we discuss rotating black holes, Taub-NUT and Eguchi-Hanson like instantons for the STU model. In addition, we discuss, in the static limit, worldsheet instanton corrections to the STU black hole solution, in the neighbourhood of a vanishing 4-cycle of the Calabi-Yau manifold. Our procedure is quite general and includes all known black hole solutions that can be embedded into N=2 supergravity. 
  A vacuum solution of anomaly-free N=1, D=10 Dual Supergravity is constructed. This vacuum corresponds to the presence of two 5-branes, intersecting along M_4, and possesses N=1, D=4 supersymmetry. 
  An efficient procedure for constructing quasi-exactly solvable matrix models is suggested. It is based on the fact that the representation spaces of representations of the algebra sl(2,R) within the class of first-order matrix differential operators contain finite dimensional invariant subspaces. 
  This course provides a self contained introduction to the general theory of relativistic brane models, of the category that includes point particle, string, and membrane representations for phenomena that can be considered as being confined to a worldsheet of the corresponding dimension (respectively one, two, and three) in a thin limit approximation. The first part of the course is concerned with purely kinematic aspects: it is shown how, to second differential order, the geometry (and in particular the inner and outer curvature) of a brane worldsheet of arbitrary dimension is describable in terms of the first, second, and third fundamental tensor; the extension to a foliation by a congruence of such worldsheets is also briefly discussed. In the next part, it is shown how -- to lowest order in the thin limit -- the evolution of such a brane worldsheet will always be governed by a simple tensorial equation of motion whose left hand side is the contraction of the second fundamental tensor with the relevant surface stress tensor, while the right hand side will simply vanish in the case of free motion and will otherwise be just the orthogonal projection of any external force density that may happen to act on the brane. (Allowance for first order deviations from such a thin limit treatment would require evolution equations of a more complicated kind of which a prototype example is presented.) The last part of the course concentrates on the case of a string, and particularly on the stationary (centrifugally supported) configurations known as vortons, which, if they are sufficiently stable, may be of considerable cosmological significance. 
  We review several topics of interest in the study of 4d N=1 supersymmetric compactifications of the heterotic string. After a brief introduction to the construction of such models, our focus is on the novel physics which occurs at singularities in the moduli space of vacua. Among the phenomena we discuss are nonperturbative superpotentials, dynamical generation of poles in various low-energy couplings, and phase transitions which change the net number of chiral generations. 
  We construct two chains of fourdimensional F-theory/heterotic dual string pairs with N=1 supersymmetry. On the F-theory side as well as on the heterotic side the geometry of the involved manifolds relies on del Pezzo surfaces. We match the massless spectra by using, for one chain of models, an index formula to count the heterotic bundle moduli and determine the dual F-theory spectra from the Hodge numbers of the fourfolds $X^4$ and of the type IIB base spaces. 
  We suggest that the (2,0) six dimensional field theory compactified on $S^1\times K3$ is the Matrix model description of both M-theory on $K3$ and the Heterotic string on $T^3$. This proposal is different from existing proposals for the Heterotic theory. Different limits of the base space geometry give the different space-time interpretations, making M-theory/Heterotic duality manifest. We also present partial results on Heterotic/F-theory duality. 
  Previously we have proposed that in certain relativistic quantum field theories knotlike configurations may appear as stable solitons. Here we present a detailed investigation of the simplest knotted soliton, the torus-shaped unknot. 
  We investigate the two-loop gap equation for the thermal mass of hot massless $g^2\phi^4$ theory and find that the gap equation itself has a non-zero finite imaginary part. This indicates that it is not possible to find the real thermal mass as a solution of the gap equation beyond $g^2$ order in perturbation theory. We have solved the gap equation and obtain the real and the imaginary part of the thermal mass which are correct up to $g^4$ order in perturbation theory. 
  We study the influence of the anomaly on the physical quantum picture of the generalized chiral Schwinger model defined on the circle. We show that the anomaly i) results in the background linearly rising electric field and ii) makes the spectrum of the physical Hamiltonian nonrelativistic without a massive boson. The physical matter fields acquire exotic statistics . We construct explicitly the algebra of the Poincare generators and show that it differs from the Poincare one. We exhibit the role of the vacuum Berry phase in the failure of the Poincare algebra to close. We prove that, in spite of the background electric field, such phenomenon as the total screening of external charges characteristic for the standard Schwinger model takes place in the generalized chiral Schwinger model, too. 
  We consider asymptotically flat static spherically symmetric black hole solutions in SU(N) Einstein-Yang-Mills theory. Embedding the N-dimensional representation of $su(2)$ in $su(N)$, the purely magnetic gauge field ansatz contains $N-1$ functions. When one or more of these gauge field functions are identically zero, magnetically charged EYM black hole solutions emerge, consisting of a neutral and a charged gauge field part, based on non-abelian subalgebras and the Cartan subalgebra of $su(N)$, respectively. We classify these charged black hole solutions in general and present numerical solutions for SU(5) EYM theory. 
  We study the N -> 0 limit of the O(N) Gross-Neveu model in the framework of the massless form-factor approach. This model is related to the continuum limit of the Ising model with random bonds via the replica method. We discuss how this method may be useful in calculating correlation functions of physical operators. The identification of non-perturbative fixed points of the O(N) Gross-Neveu model is pursued by its mapping to a WZW model. 
  We consider N=2 SUSY QCD with gauge group SU(2) and N_f flavours of matter with nonzero mass. Using the method of the instanton-induced effective vertex we calculate higher derivative corrections to the Seiberg-Witten result in the momentum expansion of the low energy effective Lagrangian in various regions of the modular space. Then we focus on a certain higher derivative operator on the Higgs branch. We show that the singular behavior of this operator comes from values of mass of matter at which charge singularity on the Coulomb branch collides with the monopole or dyon one. Given the behavior of this operator at weak coupling coming from instantons as well as its behavior near points of colliding singularities we find the exact solution for this operator. 
  We consider the role of supersymmetry breaking soft terms that are present in generalized Narain compactifications of heterotic string theory, in which local supersymmetry is spontaneously broken, with gravitino masses being inversely proportional to the radii of compact dimensions. Such compactifications and their variants, are thought to be the natural application of the Scherk-Schwarz mechanism to string theory. We show that in the case where this mechanism leads to spontaneous breaking of N=4, d=4 local supersymmetry, the limit \kappa goes to zero, yields a 2-parameter class of soft terms whose precise form are shown to preserve the ultraviolet properties of N=4 super Yang-Mills theory. This result is in broad agreement with that of the field theory Scherk-Schwarz mechanism, as applied to N=1, d=10 supergravity coupled to super Yang-Mills, although the detailed structure of the soft terms are different in general. 
  The Hopf term in the $2 + 1$ dimensional O(3) nonlinear sigma model, which is known to be responsible for fractional spin and statistics, is re-examined from the viewpoint of quantization ambiguity. It is confirmed that the Hopf term can be understood as a term induced quantum mechanically, in precisely the same manner as the $\theta$-term in QCD. We present a detailed analysis of the topological aspect of the model based on the adjoint orbit parametrization of the spin vectors, which is not only very useful in handling topological (soliton and/or Hopf) numbers, but also plays a crucial role in defining the Hopf term for configurations of nonvanishing soliton numbers. The Hopf term is seen to arise explicitly as a quantum effect which emerges when quantizing an $S^1$ degree of freedom hidden in the configuration space. 
  The O(3) sigma model in two spatial dimensions admits topological (Bogomol'nyi) lower bound on its energy. This paper proposes a lattice version of this system which maintains the Bogomol'nyi bound and allows the explicit construction of static solitons on the lattice. Numerical simulations show that these lattice solitons are unstable undersmall perturbations; in fact, their size changes linearly with time. 
  The computation of the microcanonical density of states for a string gas in a finite volume needs a one by one count because of the discrete nature of the spectrum. We present a way to do it using geometrical arguments in phase space. We take advantage of this result in order to obtain the thermodynamical magnitudes of the system. We show that the results for an open universe exactly coincide with the infinite volume limit of the expression obtained for the gas in a box. For any finite volume the Hagedorn temperature is a maximum one, and the specific heat is always positive. We also present a definition of pressure compatible with R-duality seen as an exact symmetry, which allows us to make a study on the physical phase space of the system. Besides a maximum temperature the gas presents an asymptotic pressure. 
  I discuss in this talk a bosonization approach recently developed. It leads to the (exact) bosonization rule for fermion currents in d > 2 dimensions and also provides a systematic way of constructing the bosonic action in different regimes. 
  The antiferromagnetic Heisenberg spin chain with N spins has a sector with N=odd, in which the number of excitations is odd. In particular, there is a state with a single one-particle excitation. We exploit this fact to give a simplified derivation of the boundary S matrix for the open antiferromagnetic spin-1/2 Heisenberg spin chain with diagonal boundary magnetic fields. 
  Scattering of zero branes off the fixed point in $R^8/Z_2$, as described by a super-quantum mechanics with eight supercharges, displays some novel effects relevant to Matrix theory in non-compact backgrounds. The leading long distance behaviour of the moduli space metric receives no correction at one loop in Matrix theory, but does receive a correction at two loops. There are no contributions at higher loops. We explicitly calculate the two-loop term, finding a non-zero result. We find a discrepancy with M(atrix)-theory. Although the result has the right dependence on $v$ and $b$ for the scattering of zero branes off the fixed point the factors of $N$ do not match. We also discuss scattering in the orbifolds, $R^5/Z_2$ and $R^9/Z_2$ where we find the predicted fractional charges. 
  We clarify the notion of Wilsonian renormalization group (RG) invariance in supersymmetric gauge theories, which states that the low-energy physics can be kept fixed when one changes the ultraviolet cutoff, provided appropriate changes are made to the bare coupling constants in the Lagrangian. We first pose a puzzle on how a quantum modified constraint (such as Pf(Q^i Q^j) = \Lambda^{2(N+1)} in SP(N) theories with N+1 flavors) can be RG invariant, since the bare fields Q^i receive wave function renormalization when one changes the ultraviolet cutoff, while we naively regard the scale \Lambda as RG invariant. The resolution is that \Lambda is not RG invariant if one sticks to canonical normalization for the bare fields as is conventionally done in field theory. We derive a formula for how \Lambda must be changed when one changes the ultraviolet cutoff. We then compare our formula to known exact results and show that their consistency requires the change in \Lambda we have found. Finally, we apply our result to models of supersymmetry breaking due to quantum modified constraints. The RG invariance helps us to determine the effective potential along the classical flat directions found in these theories. In particular, the inverted hierarchy mechanism does not occur in the original version of these models. 
  In this paper we consider features of graviton scattering in Matrix theory compactified on a 2-torus. The features which interest us can only be determined by nonperturbative effects in the corresponding 2+1 dimensional super Yang Mills theory. We show that the superconformal symmetry of strongly coupled Super Yang Mills Theory in 2+1 dimensions almost determines low energy, large impact parameter ten dimensional graviton scattering at zero longitudinal momentum in the Matrix model of IIB string theory. We then show that amplitudes involving arbitrary transverse momentum transfer are governed by instanton processes similar to the Polchinski Pouliot process. Finally we consider the influence of instantons on a conjectured nonrenormalization theorem. This theorem is violated by instanton processes. Far from being a problem, this fact is seen to be crucial to the consistency of the IIB interpretation. We suggest that the SO(8) invariance of strongly coupled SYM theory may lead to a proof of eleven dimensional Lorentz invariance. 
  A new way how to calculate the off-shell renormalization functions within the $R^2$-gravity has been proposed. The one-loop renormalization group equations in the approach suggested have been constructed. The behaviour of effective potential for an massless scalar field interacting with the quantum gravitational field has been analyzed in this approach. 
  We derive the wave equation for a minimally coupled scalar field in the background of a general rotating five-dimensional black hole. It is written in a form that involves two types of thermodynamic variables, defined at the inner and outer event horizon, respectively. We model the microscopic structure as an effective string theory, with the thermodynamic properties of the left and right moving excitations related to those of the horizons. Previously known solutions to the wave equation are generalized to the rotating case, and their regime of validity is sharpened. We calculate the greybody factors and interpret the resulting Hawking emission spectrum microscopically in several limits. We find a U-duality invariant expression for the effective string length that does not assume a hierarchy between the charges. It accounts for the universal low-energy absorption cross-section in the general non-extremal case. 
  In this article we give a calculation of the two-loop $\sigma$-model corrections to the T-duality map in string theory. We use the effective action approach, and analyze two-loop corrections in a specific subtraction scheme. Focusing on backgrounds which have a single Abelian isometry, we find the explicit form for the $O(\alpha')$ modifications of the lowest order duality transformations. Rather surprisingly, the manifest two-loop duality depends crucially on the torsion field. In contrast to the dilaton and metric fields, which are merely passive spectators, the torsion plays a more active role, because of the anomalous couplings to the gauge fields that arise via dimensional reduction. Our results support the interpretation of T-duality as an expansion in the inverse string tension $\alpha'$, and its order-by-order realization as a manifest symmetry of the full string theory. 
  We review and develop the formalism of ghost number cohomologies, outlined in our previous work, to classify the quantum states of M-theory. We apply this formalism to the matrix formulation of M-theory to obtain NSR superstring action from dimensionally reduced matrix model. The BPS condition of the matrix theory is related to the worldsheet reparametrizational invariance in superstring theory, underlining the connection between unbroken supersymmetries in M-theory and superstring gauge symmetries. 
  In the Brans-Dicke(BD) theory on $M_{4}\times Z_{2}$ geometry the geometrical meaning of the torsion is clarified. The BD theory on $M_{4}\times Z_{2}$ is rederived by taking into account of a new isometry condition. 
  We consider a (2+1) dimensional nonlinear O(3) sigma model with its U(1) subgroup gauged along with the inclusion of a self-interaction having symmetry breaking minima.The gauge field dynamics is governed by the Maxwell term.The model is shown to support topologically stable purely magnetic self-dual vortices. 
  We compare the spectrum of chiral multiplets in the N=1 vacua of the heterotic string on a Calabi-Yau together with an $E_8\times E_8$ vector bundle and F-theory on a smooth Calabi-Yau fourfold. Under suitable restrictions we show agreement using an index-computation. 
  We consider several renormalizable, scale free models in three space-time dimensions which involve scalar and spinor fields. The Yukawa couplings are bilinear in both the spinor and scalar fields and the potential is of sixth order in the scalar field. In a model with a single scalar field and a complex Fermion field in three Euclidean dimensions, the couplings in the theory are both asymptotically free. This property is not retained in 2+1 dimensional Minkowski space, as we illustrate by considering a renormalizable scale-free supersymmetric model. This is on account of the different properties of the Dirac matrices in Euclidean and Minkowski space. We also examine a model in 2+1 dimensional Minkowski space in which two species of Fermions, associated with the two unitarily inequivalent representations of the $2 \times 2$ Dirac matrices, couple in two different ways to two distinct scalar fields. There are two types of Yukawa couplings in this model, and either one or the other of them can be asymptotically free (but not both simultaneously). 
  Starting with configurations of fourbranes, fivebranes, sixbranes and orientifolds in Type IIA string theory we derive via M-theory the curves solving N=2 supersymmetric gauge theories with gauge groups SO(N) and Sp(2N). We also obtain new curves describing theories with product gauge groups. A crucial role in the discussion is played by the interaction of the orientifolds with the NS-fivebranes. 
  For nonsupersymmetric theories, the one-loop effective action can be computed via zeta function regularization in terms of the functional trace of the heat kernel associated with the operator which appears in the quadratic part of the action. A method is developed for computing this functional trace by exploiting its similarity to a Gaussian integral. The procedure is extended to superspace, where it is used to compute the low energy effective action obtained by integrating out massive scalar supermultiplets in the presence of a supersymmetric Yang-Mills background. 
  We discuss the topological properties of the manifold of coupling constants for multi-coupling deformations of conformal field theories. We calculate the Euler and Betti numbers and briefly discuss physical applications of these results. 
  We construct a reparametrization invariant two-point function for c=-2 conformal matter coupled to two-dimensional quantum gravity. From the two-point function we extract the critical indices \nu and \eta. The results support the quantum gravity version of Fisher's scaling relation. Our approach is based on the transfer matrix formalism and exploits the formulation of the c=-2 string as an O(n) model on a random lattice. 
  The procedure for Abelian conversion of second class constraints due to Batalin, Fradkin, Fradkina and Tyutin is considered at quantum level, by using the field-antifield formalism. It is argued that quantum effects can obstruct the process. In this case, Wess-Zumino fields may be introduced in order to restore the lost symmetries. 
  The 27-dimensional Hopf algebra A(F), defined by the exact sequence of quantum groups A(SL(2,C))->A(SL_q(2))->A(F), q^3=1, is studied as a finite quantum group symmetry of the matrix algebra M(3,C), describing the color sector of Alain Connes' formulation of the Standard Model. The duality with the Hopf algebra H,investigated in a recent work by Robert Coquereaux, is established and used to define a representation of H on M(3,C) and two commuting representations of H on A(F). 
  An action for a string and a particle with two timelike dimensions is proposed and analyzed. Due to new gauge symmetries and associated constraints, the motion of each system in the background of the other is equivalent to effective motion with a single timelike dimension. The quantum constraints are consistent only in critical dimensions. For the bosonic system in flat spacetime the critical dimension is 27 or 28, with signature (25,2) or (26,2), depending on whether the particle is massive or massless respectively. For the supersymmetric case the critical dimensions are 11 or 12, with signature (9,2) or (10,2), under the same circumstances. Generalizations to multiparticles, strings and p-branes are outlined. 
  A global uniqueness theorem for stationary black holes is proved as a direct consequence of the Topological Censorship Theorem and the topological classification of compact, simply connected four-manifolds. 
  We demonstrate how Sakharov's idea of induced gravity allows one to explain the statistical-mechanical origin of the entropy of a black hole. According to this idea, gravity becomes dynamical as the result of quantum effects in the system of heavy constituents of the underlying theory. The black hole entropy is related to the properties of the vacuum in the induced gravity in the presence of the horizon. We obtain the Bekenstein-Hawking entropy by direct counting the states of the constituents. 
  Spectra of a string in $ SL(2,R) $ and three dimensional (BTZ) black hole geometry are discussed. We consider a free field realization of ^sl (2,R) different from the standard ones in treatment of zero-modes. Applying this to the string model in SL(2,R), we show that the spectrum is ghost-free. The essence of the argument is the same as Bars' resolution to the ghost problem, but there are differences; for example, the currents do not contain logarithmic cuts. Moreover, we obtain a modular invariant partition function. This realization is also applicable to the analysis of the string in the three dimensional black hole geometry, the model of which is described by an orbifold of the SL(2,R) WZW model. We obtain ghost-free and modular invariant spectra for the black hole theory as well. These spectra provide examples of few sensible spectra of a string in non-trivial backgrounds with curved time and, in particular, in a black hole background with an infinite number of propagating modes. 
  We investigate quantum aspects of the three dimensional (BTZ) black holes. The discussions are devoted to two subjects: the thermodynamics of quantum scalar fields and the string theory in the three dimensional black hole backgrounds. We take two approaches to the thermodynamics. In one approach we use mode expansion, and in the other we use Hartle-Hawking Green functions. We obtain exact expressions of mode functions, Hartle-Hawking Green functions, Green functions on a cone geometry, and thermodynamic quantities. The entropies depend largely upon methods of calculation including regularization schemes and boundary conditions. This indicates the importance of precise discussions on the definition of the thermodynamic quantities for understanding black hole entropy. Then we investigate the string theory in the framework of conformal field theory. The model is described by an orbifold of the SL(2,R) WZW model. We discuss the spectrum by solving the level matching condition and obtain winding modes. We analyze the physical states and investigate the ghost problem. Explicit examples of negative-norm physical states (ghosts) are found. Thus we discuss possibilities for obtaining a sensible theory. The tachyon propagation and the target-space geometry are also discussed. This is the first attempt to quantize a string in a black hole background with an infinite number of propagating modes. Although we cannot overcome all the problems, our results may provide a useful basis for both subjects. 
  In this paper we study extremal transitions between heterotic string compactifications, i.e., transitions between pairs (M,V) where M is a Calabi-Yau manifold and V a gauge bundle. Bundle transitions are described using language recently espoused by Friedman, Morgan, Witten. In addition, partly as a check on our methods, we also study how small instantons are described in the same language, and also describe the sheaves corresponding to small instantons. 
  Quantum groups in general and the quantum Anti-de Sitter group $U_q(so(2,3))$ in particular are studied from the point of view of quantum field theory. We show that if $q$ is a suitable root of unity, there exist finite-dimensional, unitary representations corresponding to essentially all the classical one-particle representations with (half)integer spin, with the same structure at low energies as in the classical case. In the massless case for spin $\geq 1$, the "naive" representations are unitarizable only after factoring out a subspace of "pure gauges", as classically. Unitary many-particle representations are defined, with the correct classical limit. Furthermore, we identify a remarkable element $Q$ in the center of $U_q(g)$, which plays the role of a BRST operator in the case of $U_q(so(2,3))$ at roots of unity, for any spin $\geq 1$. The associated ghosts are an intrinsic part of the indecomposable representations. We show how to define an involution on algebras of creation and anihilation operators at roots of unity, in an example corresponding to non-identical particles. It is shown how nonabelian gauge fields appear naturally in this framework, without having to define connections on fiber bundles. Integration on Quantum Euclidean space and sphere and on Anti-de Sitter space is studied as well. We give a conjecture how $Q$ can be used in general to analyze the structure of indecomposable representations, and to define a new, completely reducible associative (tensor) product of representations at roots of unity, which generalizes the standard "truncated" tensor product as well as our many-particle representations. 
  S-duality of hetertotic / type II string theory compactified on a six dimensional torus requires the existence of Kaluza-Klein dyons, carrying winding charge. We identify the zero modes of the Kaluza-Klein monopole solution which are responsible for these dyonic excitations, and show that we get the correct degeneracy of dyons as predicted by S-duality. The self-dual harmonic two form on the Euclidean Taub-NUT space plays a crucial role in this construction. 
  We present finite energy analytic monopole and dyon solutions whose size is fixed by the electroweak scale. Our result shows that genuine electroweak monopole and dyon could exist whose mass scale is much smaller than the grand unification scale. 
  This paper has been withdrawn. A more satisfactory account of the results it was supposed to present will appear in a series of papers starting with hep-th/9712256 and hep-th/9712258 
  We discuss further a recent space-time interpretation of the $c=1$ matrix model which retains both sides of the inverted harmonic oscillator potential in the underlying free fermion theory and reproduces the physics of the discrete state moduli of two-dimensional string theory. We show that within this framework the linear tachyon background in flat space arises from the fermi vacuum. We argue that this framework does not suffer from any obvious nonperturbative inconsistency. We also identify and discuss a class of nearly static configurations in the free fermion theory which are interpreted as static metric backgrounds in space-time. These backgrounds are classically absorbing --- a beam of tachyons thrown at such a background is only partly reflected back --- and are tentatively identified with the eternal back-hole of 2-dimensional string theory. 
  We show how the widely used concept of spontaneous symmetry breaking can be explained in causal perturbation theory by introducing a perturbative version of quantum gauge invariance. Perturbative gauge invariance, formulated exclusively by means of asymptotic fields, is discussed for the simple example of Abelian U(1) gauge theory (Abelian Higgs model). Our findings are relevant for the electroweak theory, as pointed out elsewhere. 
  In this work we consider an Abelian O(3) sigma model coupled nonminimally with a gauge field governed by a Maxwell and Chern-Simons terms. Bogomol'nyi equations are constructed for a specific form of the potential and generic nonminimal coupling constant. Furthermore, topological and nontopological self-dual soliton solutions are obtained for a critical value of the nonminimal coupling constant. Some particular static vortex solutions (topological and nontopological) satisfying the Bogomol'nyi bound are numerically solved and presented. 
  Dynamical chiral symmetry breaking (D$\chi$SB) is studied in the Nambu-Jona-Lasinio model for an arbitrary combination of external constant electric and magnetic fields. In 3+1 dimensions it is shown that the critical coupling constant increases with increasing of the value of the second invariant of electromagentic field $\vec{E}\cdot\vec{B}$, i.e. the second invariant inhibits $D\chi SB$. The case of 2+1 dimensions is simpler because there is only one Lorentz invariant of electromagnetic field and any combination of constant fields can be reduced to cases either purely magnetic or purely electric field. 
  We propose to parametrize the configuration space of one-dimensional quantum systems of N identical particles by the elementary symmetric polynomials of bosonic and fermionic coordinates. It is shown that in this parametrization the Hamiltonians of the A_N, BC_N, B_N, C_N and D_N Calogero and Sutherland models, as well as their supersymmetric generalizations, can be expressed -- for arbitrary values of the coupling constants -- as quadratic polynomials in the generators of a Borel subalgebra of the Lie algebra gl(N+1) or the Lie superalgebra gl(N+1|N) for the supersymmetric case. These algebras are realized by first order differential operators. This fact establishes the exact solvability of the models according to the general definition given by one of the authors in 1994, and implies that the Calogero and Jack-Sutherland polynomials, as well as their supersymmetric generalizations, are related to finite-dimensional irreducible representations of the Lie algebra gl(N+1) and the Lie superalgebra gl(N+1|N). 
  We show how Higgs mechanism for non-abelian N=2 gauge theories in four dimensions is geometrically realized in the context of type II strings as transitions among compactifications of Calabi-Yau threefolds. We use this result and T-duality of a further compacitification on a circle to derive N=4, d=3 dual field theories. This reduces dualities for N=4 gauge systems in three dimensions to perturbative symmetries of string theory. Moreover we find that the dual of a gauge system always exists but may or may not correspond to a lagrangian system. In particular we verify a conjecture of Intriligator and Seiberg that an ordinary gauge system is dual to compactification of Exceptional tensionless string theory down to three dimensions. 
  We present four infinite series of new quantum theories with super-Poincare symmetry in six dimensions, which are not local quantum field theories. They have string like excitations but the string coupling is of order one. Compactifying these theories on $T^5$ we find a Matrix theory description of M theory on $T^5$ and on $T^5/\IZ_2$, which is well defined and is manifestly U-duality invariant. 
  Backreaction of excitations on a planar domain wall in a real scalar field model is investigated in the cases of homogeneous, plane wave and wave packet type excitations. It is found that the excited domain wall radiates. The method of calculating backreaction for the general forms of excitations is also presented. 
  I examine the pair creation of black holes in spacetimes with a cosmological constant of either sign. I consider cosmological C-metrics and show that the conical singularities in this metric vanish only for three distinct classes of black hole metric, two of which have compact event horizons on each spatial slice. One class is a generalization of the Reissner-Nordstrom (anti) de Sitter black holes in which the event horizons are the direct product of a null line with a 2-surface with topology of genus $g$. The other class consists of neutral black holes whose event horizons are the direct product of a null conoid with a circle. In the presence of a domain wall, black hole pairs of all possible types will be pair created for a wide range of mass and charge, including even negative mass black holes. I determine the relevant instantons and Euclidean actions for each case. Only for spherical are non-static solutions possible. 
  In this paper we consider a class of systems of two coupled real scalar fields in bidimensional spacetime, with the main motivation of studying classical or linear stability of soliton solutions. Firstly, we present the class of systems and comment on the topological profile of soliton solutions one can find from the first-order equations that solve the equations of motion. After doing that, we follow the standard approach to classical stability to introduce the main steps one needs to obtain the spectra of Schr\"odinger operators that appear in this class of systems. We consider a specific system, from which we illustrate the general calculations and present some analytical results. We also consider another system, more general, and we present another investigation, that introduces new results and offers a comparison with the former investigations. 
  We study supermembranes in the light-cone gauge in eleven flat dimensions with compact directions. The membrane states are subject to only the central charges associated with closed two-branes, which, in this case, are generated by the winding itself. We present arguments why this winding leaves the mass spectrum continuous. The lower bound on the mass spectrum is set by the winding number and corresponds to a BPS state. 
  We discuss duality invariant interactions between electromagnetic fields and matter. The case of scalar fields is treated in some detail. 
  We review the construction of the multiparametric inhomogeneous orthogonal quantum group ISO_qr(N) as a projection from SO_qr(N+2), and recall the conjugation that for N=4 leads to the quantum Poincare group. We study the properties of the universal enveloping algebra U_qr(iso(N)), and give an R-matrix formulation. A quantum Lie algebra and a bicovariant differential calculus on twisted ISO(N) are found. 
  Talk at the International Workshop "New Non Perturbative Methods and Quantization on the Light Cone", Les Houches, France, Feb.24-March 7, 1997 
  For certain situations relations are indicated between the space-wave function duality of Faraggi-Matone, enhanced dispersionless KdV, and Whitham dynamics for appropriate hyperelliptic Riemann surfaces related to Seiberg-Witten theory. This paper gives refinements of hep-th/9702138 and some new ideas. 
  Variational (Rayleigh-Ritz) methods are applied to local quantum field theory. For scalar theories the wave functional is parametrized in the form of a superposition of Gaussians and the expectation value of the Hamiltonian is expressed in a form that can be minimized numerically. A scheme of successive refinements of the superposition is proposed that may converge to the exact functional. As an illustration, a simple numerical approximation for the effective potential is worked out based on minimization with respect to five variational parameters. A variational principle is formulated for the fermion vacuum energy as a functional of the scalar fields to which the fermions are coupled. The discussion in this paper is given for scalar and fermion interactions in 1+1 dimensions. The extension to higher dimensions encounters a more involved structure of ultraviolet divergences and is deferred to future work. 
  We calculate the number spectrum of particles radiated during a scattering into a heat bath using the thermal largest-time equation and the Dyson-Schwinger equation. We show how one can systematically calculate {d<N(\omega)>}/{d\omega} to any order using modified real time finite-temperature diagrams. Our approach is demonstrated on a simple model where two scalar particles scatter, within a photon-electron heat bath, into a pair of charged particles and it is shown how to calculate the resulting changes in the number spectra of the photons and electrons. 
  We discuss N=2 supersymmetric Type IIA brane configurations within M theory. This is a generalization of the work of Witten to all classical groups. 
  The supersymmetric generalization of Pisson-Lie T-duality in N=2 superconformal WZNW models on the compact groups is considered. It is shown that the role of Drinfeld's doubles play the complexifications of the corresponding compact groups. These complex doubles are used to define the natural actions of the isotropic subgroups forming the doubles on the group manifolds of the N=2 superconformal WZNW models. The Poisson- Lie T-duality in N=2 superconformal U(2)-WZNW model considered in details. It is shown that this model admits Poisson-Lie symmetries with respect to the isotropic subgroups forming Drinfeld's double Gl(2,C). Poisson-Lie T-duality transformation maps this model into itself but acts nontrivialy on the space of classical solutions. Supersymmetric generalization of Poisson-Lie T-duality in N=2 superconformal WZNW models on the compact groups of higher dimensions is proposed. 
  We calculate characters and supercharacters for irreducible, admissible representations of the affine superalgebra sl(2|1) in both the Ramond and Neveu-Schwarz sectors and discuss their modular properties in the special case of level k=-1/2. We also show that the non-degenerate integrable characters coincide with some N=4 superconformal characters. 
  The dimensional properties of fields in classical general relativity lead to a tangent tower structure which gives rise directly to quantum mechanical and quantum field theory structures without quantization. We derive all of the fundamental elements of quantum mechanics from the tangent tower structure, including fundamental commutation relations, a Hilbert space of pure and mixed states, measurable expectation values, Schroedinger time evolution, collapse of a state and the probability interpretation. The most central elements of string theory also follow, including an operator valued mode expansion like that in string theory as well as the Virasoro algebra with central charges. 
  Conformal scalar fields coupled to the dilaton appear naturally in two-dimensional models of black hole evaporation. We calculate their trace anomaly. It follows that an RST-type counterterm appears naturally in the one-loop effective action. 
  We use the effective action of the $E_n$ non-critical strings to study its BPS spectrum for $0 \le n \le 8$. We show how to introduce mass parameters, or Wilson lines, into the effective action, and then perform the appropriate asymptotic expansions that yield the BPS spectrum. The result is the $E_n$ character expansion of the spectrum, and is equivalent to performing the mirror map on a Calabi-Yau with up to nine K\"ahler moduli. This enables a much more detailed examination of the $E_n$ structure of the theory, and provides extensive checks on the effective action description of the non-critical string. We extract some universal ($E_n$ independent) information concerning the degeneracies of BPS excitations. 
  We construct Calabi-Yau manifolds and their mirrors from K3 surfaces. This method was first developed by Borcea and Voisin. We examined their properties torically and checked mirror symmetry for Calabi-Yau 4-fold case. From Borcea-Voisin 3-fold or 4-fold examples, it may be possible to probe the S-duality of Seiberg -Witten. 
  It is argued that the quantum correction to the mass of some very massive, nonsupersymmetric states vanishes in inverse proportion to their tree-level mass to all orders in string loops. This approximate nonrenormalization can explain the agreement between the perturbative degeneracy of these states and the Sen entropy of the associated black holes. 
  We study logarithmic operators in Coulomb gas models, and show that they occur when the ``puncture'' operator of the Liouville theory is included in the model. We also consider WZNW models for $SL(2,R)$, and for SU(2) at level 0, in which we find logarithmic operators which form Jordan blocks for the current as well as the Virasoro algebra. 
  String theory on D-brane backgrounds is open-closed string theory. Given the relevance of this fact, we give details and elaborate upon our earlier construction of oriented open-closed string field theory. In order to incorporate explicitly closed strings, the classical sector of this theory is open strings with a homotopy associative A_\infty algebraic structure. We build a suitable Batalin-Vilkovisky algebra on moduli spaces of bordered Riemann surfaces, the construction of which involves a few subtleties arising from the open string punctures and cyclicity conditions. All vertices coupling open and closed strings through disks are described explicitly. Subalgebras of the algebra of surfaces with boundaries are used to discuss symmetries of classical open string theory induced by the closed string sector, and to write classical open string field theory on general closed string backgrounds. We give a preliminary analysis of the ghost-dilaton theorem. 
  In this paper we discuss a class of 1+1 dimensional matrix field theories in the light-cone Hamiltonian approach that are obtained from a dimensional reduction of 3+1 dimensional Yang-Mills theory. We present exact massless solutions of the boundstate integral equations in the large N limit. These massless states have constant wavefunctions in momentum space and are therefore fundamental excitations of the theory. 
  The semi-classical phase structure of two-dimensional QED and QCD are briefly reviewed. The non-abelian theory is reformulated to closely resemble the Schwinger model. It is shown that, contrary to the abelian theory, the phase structure of two-dimensional QCD is unaffected by the structure of the theta vacuum. We make parallel calculations in the two theories and conclude that massless Schwinger model is in the screening and the massive theory is in the confining phase, whereas both massless and massive QCD are in the screening phase. 
  The possibility of spontaneous breaking of CP symmetry by the expectation values of orbifold moduli is investigated with particular reference to $CP$ violating phases in soft supersymmetry breaking terms. The effect of different mechanisms for stabilizing the dilaton and the form of the non-perturbative superpotential on the existence and size of these phases is studied. Non-perturbative superpotentials involving the absolute modular invariant $j(T)$, such as may arise from F-theory compactifications, are considered. 
  In this paper we develop a technique of computation of correlation functions in theories with action being cubic or higher degree form in terms of discriminants of corresponding tensors. These are analogues of formula $\int \exp (iT(x))dx\sim \det T^{-1/2}$ for symmetric tensors of rank two. 
  We study de Wit-Hoppe-Nicolai supermembrane with emphasis on the winding in M-direction. We propose a SUSY algebra of the supermembrane in the Lorentz invariant form. We analyze the BPS conditions and argue that the area preserving diffeomorphism constraints associated with the harmonic vector fields play an essential role. We derive the first order partial differential equation that describes the BPS state with one quarter SUSY. 
  We define a $(k\oplus l|q)$-dimensional supermanifold as a manifold having q odd coordinates and k + l even coordinates with l of them taking only nilpotent values. We show that this notion can be used to formulate superconformal field theories with different number of supersymmetries in holomorphic and antiholomorphic sectors. 
  The conformal structure of Brans-Dicke gravity action is carefully studied. It is discussed that Brans-Dicke gravity action has definitely no conformal invariance. It is shown, however, that this lack of conformal invariance enables us to demonstrate that Brans-Dicke theory appears to have a better short-distance behavior than Einstein gravity as far as Euclidean path integral formulation for quantum gravity is concerned. 
  We consider F-theory compactifications on a mirror pair of elliptic Calabi-Yau threefolds. This yields two different six-dimensional theories, each of them being nonperturbatively equivalent to some compactification of heterotic strings on a K3 surface S with certain bundle data E --> S. We find evidence for a transformation of S together with the bundle that takes one heterotic model to the other. 
  Based on a class of exactly solvable models of interacting bose and fermi liquids, we compute the single-particle propagators of these systems exactly for all wavelengths and energies and in any number of spatial dimensions. The field operators are expressed in terms of bose fields that correspond to displacements of the condensate in the bose case and displacements of the fermi sea in the fermi case.  Unlike some of the previous attempts, the present attempt reduces the answer for the spectral function in any dimension in both fermi and bose systems to quadratures.  It is shown that when only the lowest order sea-displacement terms are included, the random phase approximation in its many guises is recovered in the fermi case, and Bogoliubov's theory in the bose case. The momentum distribution is evaluated using two different approaches, exact diagonalisation and the equation of motion approach.  The novelty being of course, the exact computation of single-particle properties including short wavelength behaviour. 
  We discuss the contribution of ADHM multi-instantons to the higher-derivative terms in the gradient expansion along the Coulomb branch of N=2 and N=4 supersymmetric SU(2) gauge theories. In particular, using simple scaling arguments, we confirm the Dine-Seiberg nonperturbative nonrenormalization theorems for the 4-derivative/8-fermion term in the two finite theories (N=4, and N=2 with N_F=4). 
  By applying Newman's method, the AdS_3 rotating black hole solution is "derived" from the nonrotating black hole solution of Banados, Teitelboim and Zanelli (BTZ). The rotating BTZ solution derived in this fashion is given in "Boyer-Lindquist-type" coordinates whereas the form of the solution originally given by BTZ is given in a kind of an "unfamiliar" coordinates which are related to each other by a transformation of time coordinate alone. The relative physical meaning between these two coordinates is carefully studied by evaluating angular momentum per unit mass, angular velocity, surface gravity and area of the event horizon in two alternative coordinates respectively. The result of this study leads us to the conclusion that the BTZ time coordinate must be the time coordinate of an observer who rotates around the axis of the spinning hole in opposite direction to that of the hole outside its static limit. 
  Using the results of the calculation of the one-loop effective action (E. Elizalde et al , Phys.Rev. D49 (1994) 2852), we find the trace anomaly for most general conformally invariant 2D dilaton coupled scalar-dilaton system (the contribution of dilaton itself is included). The non-local effective action induced by conformal anomaly for such system is found. That opens new possibilities in generalizing of CGHS-like model for the study of back reaction of matter to 2D black holes. 
  Superradiant scattering, which can be thought of as the wave analogue of the Penrose process is revisited. As is well-known, boson fields display superradiance provided they have frequency in a certain range whereas fermion fields do not. A succinct superradiance-checking algorithm employing particle number or energy current is formally reviewed and then applied to the case of fermion field. The demonstrations of the absence of fermionic superradiance in terms of the particle number current exist in the literature but they are in the context of two-component SL(2,C) spinor formalism for massive spinor and SO(3,1) Dirac spinor formalism for massless spinor. Here we present an alternative demonstration in terms of both particle number and energy current but in a different context of local SO(3,1) Dirac spinor formalism for both massless and massive spinors. It appears that our presentation confirms the absence of fermionic superradiance in a more simple and systematic manner. 
  In this paper the large $N$ limit of one hermitian matrix models coupled to an external matrix is considered. It is shown that in the large N limit the number of degrees of freedom are reduced to be order N even though it is order $N^{2}$ for finite N. It is claimed that this result is the origin of the factorization of observables in the path integral formalism. 
  We discuss supersymmetric quantum mechanical models with periodic potentials. The important new feature is that it is possible for both isospectral potentials to support zero modes, in contrast to the standard nonperiodic case where either one or neither (but not both) of the isospectral pair has a zero mode. Thus it is possible to have supersymmetry unbroken and yet also have a vanishing Witten index. We present some explicit exactly soluble examples for which the isospectral potentials have identical band spectra, and which are ``self-isospectral'' in the sense that the potentials have identical shape, but are translated by one half period relative to one another. 
  We have carried out a two loop computation of the low-energy effective action for the four-dimensional N=2 supersymmetric Yang-Mills system coupled to hypermultiplets, with the chiral superfields of the vector multiplet lying in an abelian subalgebra. We have found a complete cancellation at the level of the integrands of Feynman amplitudes, and therefore the two loop contribution to the action, effective or Wilson, is identically zero. 
  The long-range fields associated with non-abelian vortices generally obstruct full realization, in the spectrum, of the symmetry of the ground state. In the context of 2+1 dimensional field theories, we show how this effect manifests itself concretely in altered conditions for the angular momentum and in the energy spectrum. A particularly interesting case is supersymmetry, which is obstructed by the gravitational effect of any mass. 
  As a system which is known to admit classical wormhole instanton solutions, Einstein-Kalb-Ramond (KR) antisymmetric tensor theory is revisited. As an untouched issue, the existence of fermionic zero modes in the background of classical axionic wormhole spacetime and its physical implications is addressed. In particular, in the context of a minisuperspace quantum cosmology model based on this Einstein-KR antisymmetric tensor theory, ``quantum wormhole'', defined as a state represented by a solution to the Wheeler-DeWitt equation satisfying an appropriate wormhole boundary condition, is discussed. An exact, analytic wave function for quantum wormholes is actually found. Finally, it is proposed that the minisuperspace model based on this theory in the presence of the cosmological constant may serve as an interesting simple system displaying an overall picture of entire universe's history from the deep quantum domain all the way to the classical domain. 
  We study how the decay properties of particles are changed by acceleration. It is shown that under the influence of acceleration (1) the lifetime of particles is modified and (2) new processes (like the decay of the proton) become possible. This is illustrated by considering scalar models for the decay of muons, pions, and protons. We discuss the close conceptual relation between these processes and the Unruh effect. 
  In this letter we propose to use an extension of the variational approach known as Truncated Conformal Space to compute numerically the Vacuum Expectation Values of the operators of a conformal field theory perturbed by a relevant operator. As an example we estimate the VEV's of all (UV regular) primary operators of the Ising model and of some of the Tricritical Ising Model conformal field theories when perturbed by any choice of the relevant primary operators. We compare our results with some other independent predictions. 
  Following systematically the generalized Hamiltonian approach of Batalin, Fradkin and Tyutin, we embed the second-class non-abelian self-dual model of P. K. Townsend et al into a gauge theory. The strongly involutive Hamiltonian and constraints are obtained as an infinite power series in the auxiliary fields. By formally summing the series we obtain a simple interpretation for the first-class Hamiltonian, constraints and observables. 
  A rectangular Wilson loop with sides parallel to space and time directions is perturbatively evaluated in two light-cone gauge formulations of Yang-Mills theory in 1+1 dimensions, with ``instantaneous'' and ``causal'' interactions between static quarks. In the instantaneous formulation we get Abelian-like exponentiation of the area in terms of $C_F$. In the ``causal'' formulation the loop depends not only on the area, but also on the dimensionless ratio $\beta = {L \over T}$, $2L$ and $2T$ being the lengths of the rectangular sides. Besides it also exhibits dependence on $C_A$. In the limit $T \to \infty$ the area law is recovered, but dependence on $C_A$ survives. Consequences of these results are pointed out. 
  The string equations of motion for some homogeneous (Kantowski-Sachs, Bianchi I and Bianchi IX) background spacetimes are given, and solved explicitly in some simple cases. This is motivated by the recent developments in string cosmology, where it has been shown that, under certain circumstances, such spacetimes appear as string-vacua.   Both tensile and null strings are considered. Generally, it is much simpler to solve for the null strings since then we deal with the null geodesic equations of General Relativity plus some additional constraints.   We consider in detail an ansatz corresponding to circular strings, and we discuss the possibility of using an elliptic-shape string ansatz in the case of homogeneous (but anisotropic) backgrounds. 
  In this paper we consider space-times containing matter expanding or contracting according to a time-dependent scale factor. Cosmologies with vanishing, positive or negative cosmological constant are considered. In the case of vanishing or negative cosmological constant open and closed spatial surfaces are solutions while in the case of positive cosmological constant only closed surfaces exist. The gravitational field is solved explicitly in the case of 1 or 2 particles, 1 black-hole, and 1 black-hole vacuum state. 
  Since the spin of real particles is of order of $\hbar$, it is difficult to distinguish in a quantum mechanical experiment involving spinning particles what part of the outcome is related to the spin contribution and what part is a pure quantum mechanical effect. We analyze in detail a classical model of a nonrelativistic spinning particle under the action of a potential barrier and compute numerically the crossing for different potentials. In this way it is shown that because of the spin structure there is a nonvanishing contribution to crossing for energies above a certain minimum value, even below the top of the potential barrier. Results are compared with the quantum tunnel effect. 
  We study an infinite family of N=2 $Sp(2n)$ gauge theories that naturally arise from the D3-brane probe dynamics in F-theory. The matter sector consists of four fundamental and one antisymmetric tensor hyper multiplets. We propose that, in the limit of vanishing bare masses, the theory has exact $SO(8)\sdtimes SL(2,Z)$ duality. We examine the semiclassical BPS spectrum in the Coulomb phase by quantizing various monopole moduli space dynamics, and show that it is indeed consistent with the exact S-duality. 
  The free action for the massless sector of the Type II superstring was recently constructed using closed RNS superstring field theory. The supersymmetry transformations of this action are shown to satisfy an N=2 D=10 SUSY algebra with Ramond-Ramond central charges. 
  We analyze the vector multiplet prepotential of d=4, N=2 type IIA compactifications. We find that the worldsheet instanton corrections have a natural interpretation as one-loop corrections in five dimensions, with the extra dimension being compactified on a circle of radius $g_{s}\ell_{s}$. We argue that the relation between spacetime and worldsheet instantons is natural from this point of view. We also discuss the map between the type IIA worldsheet instantons and the spacetime instantons in the heterotic dual. 
  In this note we give a new construction of the N=2 superconformal algebra using currents of the affine superalgebra $\hat{sl}(2 | 1)$ and free bosonic fields, and also the N=4 superconformal algebra without central charge in terms of currents of $\hat{sl}(2 | 2)$ and free bosonic fields. Arguments on commutants and invariant subspaces are added. 
  A Lorentz covariant matrix regularization of membrane thories is studied.It is shown that the action for a bosonic membrane can be defined by matrix regularization in a Lorentz covariant manner. The generator of area preserving diffeomorphism can also be consistently defined by matrix regularization, and we can make the area preserving gauge symmetry manifest. However, the reparametrization BRST charge explicitly depends on a specific basis set introduced to define the matrix regularization. We also briefly comment on an extension of the present formulation to a supermembrane. 
  We investigate the natural occurence of exponentially small couplings in effective field theories deduced from higher dimensional models. We calculate the coupling between twisted fields of the Z_3 Abelian orbifold compactification of the heterotic string. Due to the propagation of massive Kaluza-Klein modes between the fixed points of the orbifold, the massless twisted fields located at these singular points become weakly coupled. The resulting small couplings have an exponential dependence on the mass of the intermediate states and the distance between the fixed points. 
  A systematical description of possible symmetry breakings in the SO(3) gauge theory and an algorithmical method to construct SU(2) or SO(3) invariant Higgs potentials in an arbitrary irreducible representation is given. We close our paper with the explicit construction of the Lagrangian of the simplest SO(3) theory violated to its subgroup A_4. 
  We compute the transition amplitudes between charged particles of mass $M$ and $m$ accelerated by a constant electric field and interacting by the exchange of quanta of a third field. We work in second quantization in order to take into account both recoil effects induced by transitions and the vacuum instability of the charged fields. In spite of both effects, when the exchanged particle is neutral, the equilibrium ratio of the populations is simply $\exp(\pi (M^2 - m^2)/eE)$. Thus, in the limit $(M-m)/M \to 0$, one recovers Unruh's result characterized by the temperature $a/2\pi$ where $a$ is the acceleration. When the exchanged particle is charged, its vacuum instability prevents a simple description of the equilibrium state. However, in the limit wherein the charge of the exchanged particle tends to zero, the equilibrium distribution is once more Boltzmanian, but characterized not only by a temperature but also by the electric potential felt by the exchanged particle. This work therefore confirms that thermodynamics in the presence of horizons does not rely on a semi-classical treatment. The relationship with thermodymanics of charged black holes is stressed. 
  We describe the deformed covariant phase space corresponding to the so-called kappa-deformation of D=4 relativistic symmetries, with quantum ``time'' coordinate and Heisenberg algebra obtained according to the Heisenberg double construction. The associated modified uncertainty relations are analyzed, and in particular it is shown that these relations are consistent with independent estimates of quantum-gravity limitations on the measurability of space-time distances. 
  We consider the low-energy effective field theory of heterotic string theory compactified on a seven-torus, and we construct electrically charged as well as more general solitonic solutions. These solutions preserve 1/2, 1/4 and 1/8 of N=8, D=3 supersymmetry and have Killing spinors which exist due to cancellation of holonomies. The associated space-time line elements do not exhibit the conical structure that often arises in 2+1 dimensional gravity theories. 
  Geometrical meaning of superstring pictures is discussed in details. An off-shell generalization of the picture changing operation and its inverse are constructed. It is demonstrated that the generalised operations are inverse to each other on-shell while off-shell their product is a projection operator. 
  A new approach to probability theory based on quantum mechanical and Lie algebraic ideas is proposed and developed. The underlying fact is the observation that the coherent states of the Heisenberg-Weyl, $su(2)$, $su(r+1)$, $su(1,1)$ and $su(r,1)$ algebras in certain symmetric (bosonic) representations give the ``probability amplitudes'' (or the ``square roots'') of the well-known Poisson, binomial, multinomial, negative binomial and negative multinomial distributions in probability theory. New probability distributions are derived based on coherent states of the classical algebras $B_r$, $C_r$ and $D_r$ in symmetric representations. These new probability distributions are simple generalisation of the multinomial distributions with some added new features reflecting the quantum and Lie algebraic construction. As byproducts, simple proofs and interpretation of addition theorems of Hermite polynomials are obtained from the `coordinate' representation of the (negative) multinomial states. In other words, these addition theorems are higher rank counterparts of the well-known generating function of Hermite polynomials, which is essentially the `coordinate' representation of the ordinary (Heisenberg-Weyl) coherent state. 
  We propose a stringy mechanism whereby a large hierarchy between symmetry breaking scales is generated. This mechanism is based upon the existence of a fifth dimension compactified on a segment. We focus on a simple supersymmetric model with one massless Higgs field in the 3 of SU(3) and another one in the $\bar 3$ on each four dimensional end-point of the fifth dimension. Along supersymmetric flat directions the gauge symmetry is broken down to SU(2) due to the vacuum expectation value of the Higgs fields on one of the end points. The remaining massless mode on the other end point acquires a potential due to a massive five dimensional state propagating between the end points. This potential breaks the SU(2) symmetry at an exponentially suppressed scale compared to the SU(3) breaking scale. The suppression factor depends exponentially on the mass M of the massive state and the length $\pi R$ of the fifth dimension. For reasonably large values of the length scale R one can achieve a factor of order $M_{W}/M_GUT}$. 
  The canonical front form Hamiltonian for non-Abelian SU(N) gauge theory in 3+1 dimensions is mapped non-perturbatively on an effective Hamiltonian which acts only in the Fock space of a quark and an antiquark. The approach is based on the novel method of iterated resolvents and on discretized light-cone quantization, driven to the continuum limit. It is free of the usual Tamm-Dancoff truncations of the Fock space, rather the perturbative series are consistently resumed to all orders in the coupling constant. Emphasis is put on dealing with the many-body aspects of gauge field theory. The effective interaction turns out to be the kernel of an integral equation in the momentum space of a single quark, which is frame-independent and solvable on comparatively small computers. Important is that the higher Fock-space amplitudes can be retrieved self-consistently from these solutions. 
  We study universality properties of the Weingarten hyper-cubic random surfaces. Since a long time ago the model with a local restriction forbidding surface self-bendings has been thought to be in a different universality class from the unrestricted model defined on the full set of surfaces. We show that both models in fact belong to the same universality class with the entropy exponent gamma = 1/2 and differ by finite size effects which are much more pronounced in the restricted model. 
  We study disk amplitudes whose boundary conditions on matter configurations are not restricted to homogeneous ones. They are examined in the two-matrix model as well as in the three-matrix model for the case of the tricritical Ising model. Comparing these amplitudes, we demonstrate relations between degrees of freedom of matter states in the two models. We also show that they have a simple geometrical interpretation in terms of interactions of the boundaries. It plays an important role that two parts of a boundary with different matter states stick each other. We also find two closed sets of Schwinger-Dyson equations which determine disk amplitudes in the three-matrix model. 
  The long-range, spin-dependent forces between D0-branes are related to long-range fundamental string interactions using duality. These interactions can then be computed by taking the long distance non-relativistic expansion of string four-point amplitudes. The results are in accord with the general constraints of Matrix Theory. 
  The weakly-coupled heterotic string is known to have problems of dilaton/moduli stabilization, supersymmetry breaking (by hidden-sector gaugino condensation), gauge coupling unification (or the Newton's constant), QCD axion, as well as cosmological problems. We study these problems by adopting the viewpoint that they arise mostly due to our limited calculational power, little knowledge of the vacuum structure, and an inappropriate treatment of gaugino condensation. It turns out that these problems can be solved or are much less severe after a more consistent and complete treatment. There are two kinds of non-perturbative effects in the construction of effective field theory: the field-theoretical non-perturbative effects of gaugino condensation (with a constraint ignored in the past) and the stringy non-perturbative effects conjectured by Shenker, which are best described using the linear multiplet formalism. Stringy non-perturbative corrections to the K\"ahler potential are invoked to stabilize the dilaton at a value compatible with a weak coupling regime. Modular invariance is ensured through the Green-Schwarz counterterm and string threshold corrections which, together with hidden matter condensation, lead to moduli stabilization at the self-dual point where the vev's of moduli's F-components vanish. In the vacuum, supersymmetry is broken at a realistic scale with vanishing cosmological constant. As for soft supersymmetry breaking, our model always leads to a dilaton-dominated scenario. For the strong CP problem, the model-independent axion has the right properties to be the QCD axion. Furthermore, there is a natural mass hierarchy between the dilaton/moduli and the gravitino, which could solve the cosmological moduli problem and the cosmological problem of the model-independent axion. 
  We derive conjectures for the N=2 "chiral" determinant formulae of the Topological algebra, the Antiperiodic NS algebra, and the Periodic R algebra, corresponding to incomplete Verma modules built on chiral topological primaries, chiral and antichiral NS primaries, and Ramond ground states, respectively. Our method is based on the analysis of the singular vectors in chiral Verma modules and their spectral flow symmetries, together with some computer exploration and some consistency checks. In addition, and as a consequence, we uncover the existence of subsingular vectors in these algebras, giving examples (subsingular vectors are non-highest-weight null vectors which are not descendants of any highest-weight singular vectors). 
  Various effective field theories in four dimensions are shown to have exact non-trivial solutions in the limit as the number $N$ of fields of some type becomes large. These include extended versions of the U(N) Gross-Neveu model, the non-linear O(N) $\sigma$-model, and the $CP^{N-1}$ model. Although these models are not renormalizable in the usual sense, the infinite number of coupling types allows a complete cancellation of infinities. These models provide qualitative predictions of the form of scattering amplitudes for arbitrary momenta, but because of the infinite number of free parameters, it is possible to derive quantitative predictions only in the limit of small momenta. For small momenta the large-$N$ limit provides only a modest simplification, removing at most a finite number of diagrams to each order in momenta, except near phase transitions, where it reduces the infinite number of diagrams that contribute for low momenta to a finite number. 
  We comment on a recent puzzle regarding renormalization group invariance of exact results in SUSY theories, and argue that a purported renormalization of the dynamical scale \Lambda does not in fact occur. 
  A previously proposed generalized BRST quantization on inner product spaces for second class constraints is further developed through applications. This BRST method involves a conserved generalized BRST charge Q which is not nilpotent but which satisfies Q=\delta+\delta^{\dagger}, \delta^2=0, and by means of which physical states are obtained from the projection \delta|ph>=\delta^{\dagger}|ph>=0. A simple model is analyzed in detail from which some basic properties and necessary ingredients are extracted. The method is then applied to a massive vector field. An effective theory is derived which is close to the one of the Stueckelberg model. However, since the scalar field here is introduced in order to have inner product solutions, a massive Yang-Mills theory with polynomial interaction terms might be possible to construct. 
  Method of derivation of the duality relations for two-dimensional Z(N)-symmetric spin models on finite square lattice wrapped on the torus is proposed. As example, exact duality relations for the nonhomogeneous Ising model (N=2) and the Z(N)-Berezinsky-Villain model are obtained. 
  We formulate the Rindler space description of rotating black holes in string theories. We argue that the comoving frame is the natural frame for studying thermodynamics of rotating black holes and statistical analysis of rotating black holes gets simplified in this frame. We also calculate statistical entropies of general class of rotating black holes in heterotic strings on tori by applying D-brane description and the correspondence principle. We find at least qualitative agreement between the Bekenstein-Hawking entropies and the statistical entropies of these black hole solutions. 
  We propose a mechanism in which, using an eightbrane, a sixbrane ends on a NS brane in type IIA superstring theory. We use this mechanism to construct N=1 supersymmetric gauge theories in four dimensions with chiral matter localized in different points in space. Anomaly cancellation for the gauge theories is satisfied by requiring RR charge conservation for the various type IIA fields. The construction allows us to study a curious phase transition in which the number of flavors in supersymmetric QCD depends on the value of the ten dimensional cosmological constant. These phenomena are related to the fact that every time a D8-brane crosses a NS brane, a D6-brane is created in between them. 
  We study the one-loop corrections to the effective on-shell action of N=2 supergravity in the background of the Reissner-Nordstrom black hole. In the extreme case the contributions from graviton, gravitino and photon to the one-loop corrections to the entropy are shown to cancel. This gives the first explicit example of the supersymmetric non-renormalization theorem for the on-shell action (entropy) for BPS configurations which admit Killing spinors. We display the residual supersymmetry of the perturbations of a general supersymmetric theory in a bosonic BPS background. 
  We introduce a Wheeler-De Witt approach to quantum cosmology based on the low-energy string effective action, with an effective dilaton potential included to account for non-perturbative effects and, possibly, higher-order corrections. We classify, in particular, four different classes of scattering processes in minisuperspace, and discuss their relevance for the solution of the graceful exit problem. 
  5-branes of nontrivial topology are associated in the Diaconescu-Hanany-Witten-Witten (DHWW) approach with the Seiberg-Witten (SW) theory of low-energy effective actions. There are two different "pictures", related to the IIA and IIB phases of M-theory. They differ by the choice of $6d$ theory on the 5-brane world volume. In the IIB picture it is just the 6d SUSY Yang-Mills, while in the IIA picture it is a theory of SUSY self-dual 2-form. These two pictures appear capable to describe the (non-abelian) Lax operator and (abelian) low-energy effective action respectively. Thus IIB-IIA duality is related to the duality between Hitchin and Whitham integrable structures. 
  In this paper we consider compactifications of type I strings on Abelian orbifolds. We discuss the tadpole cancellation conditions for the general case with D9-branes only. Such compactifications have (perturbative) heterotic duals which are also realized as orbifolds (with non-standard embedding of the gauge connection). The latter have extra twisted states that become massive once orbifold singularities are blown-up. This is due to the presence of perturbative heterotic superpotential with couplings between the extra twisted states, the orbifold blow-up modes, and (sometimes) untwisted matter fields. Anomalous U(1) (generically present in such models) also plays an important role in type I-heterotic (tree-level) duality matching. We illustrate these issues on a particular example of Z_3 \otimes Z_3 orbifold type I model (and its heterotic dual). The model has N=1 supersymmetry, U(4)^3 \otimes SO(8) gauge group, and chiral matter. We also consider compactifications of type I strings on Abelian orbifolds with both D9- and D5-branes. We discuss tadpole cancellation conditions for a certain class of such models. We illustrate the model building by considering a particular example of type I theory compactified on Z_6 orbifold. The model has N=1 supersymmetry, [U(6)\otimes U(6)\otimes U(4)]^2 gauge group, and chiral matter. This would correspond to a non-perturbative chiral vacuum from the heterotic point of view. 
  We present a nonperturbative field theoretic method based on the Liouville-Neumann (LN) equation. The LN approach provides a unified formulation of nonperturbative quantum fields and also nonequilibrium quantum fields, which makes use of mean-field type equations and whose results at the lowest level are identically the same as those of the Gaussian effective potential approach and the mean-field approach. The great advantageous point of this formulation is its readiness of applicability to time-dependent quantum systems and to finite temperature field theory, and its possibility to go beyond the Gaussian approximation. 
  The canonical front form Hamiltonian for non-Abelian SU(N) gauge theory in 3+1 dimensions is mapped non-perturbatively on an effective Hamiltonian which acts only in the Fock space of a quark and an antiquark. The approach is based on the novel method of iterated resolvents and on discretized light-cone quantization, driven to the continuum limit. It is free of the usual Tamm-Dancoff truncations of the Fock space, rather the perturbative series are consistently resumed to all orders in the coupling constant. Emphasis is put on dealing with the many-body aspects of gauge field theory. Important is that the higher Fock-space amplitudes can be retrieved self-consistently from these solutions. (Proceedings Les Houches 1997) 
  The Lie algebraic structures of the S-matrices for the affine Toda field theories based on the dual pairs (X_N^{(1)}, Y_M^{(l)}) are discussed. For the non-simply-laced horizontal subalgebra X_N and the simply-laced horizontal subalgebra Y_M, we introduce a ``q-deformation'' of a Coxeter element and a ``p-deformation'' of a twisted Coxeter element respectively. Using these deformed objects, expressions for the generating function of the multiplicities of the building block of the S-matrices are obtained. The relation between the deformed version of Dorey's rule and generalized Dorey's rule due to Chari and Pressley are discussed. 
  Numerical results on the positronium spectrum in the front form of QED at large coupling constant are presented. Emphasis is put on the question whether one can derive an effective interaction at all, and whether this effective interaction is rotationally invariant. (Proceedings of Les Houches 1997) 
  An original regular approach to constructing special type symmetries for boundary value problems, namely renormgroup symmetries, is presented. Different methods of calculating these symmetries, based on modern group analysis are described. Application of the approach to boundary value problems is demonstrated with the help of a simple mathematical model. 
  We study an orientifold of the solitonic fivebrane of type II string theory. The consideration is restricted to a space-time domain which can be described by an exact conformal field theory. There are no IR divergent contributions to tadpole diagrams and thus no consistency conditions arise. However, extrapolating the results to spatial infinity leads to consistency conditions implying that there are four (physical) D-6-branes sitting at each of the two orientifold fixed planes. 
  The complete structure of curvature squared terms is analyzed in the context of chirally extended supergravity, with special emphasis on the gravitationally induced Fayet-Iliopoulos D--term. Coupling of (chiral) matter is discussed in relation with a possible extension to U(1) supergravity of the equivalence mechanism between $\cR+\al\cR^2$ and General Relativity coupled to a scalar. 
  Sigma models describing low energy effective actions on D0-brane probes with N=8 supercharges are studied in detail using a manifestly d=1, N=4 super-space formalism. Two 0+1 dimensional N=4 multiplets together with their general actions are constructed. We derive the condition for these actions to be N=8 supersymmetric and apply these techniques to various D-brane configurations. We find that if in addition to N=8 supersymmetry the action must also have Spin(5) invariance, the form of the sigma model metric is uniquely determined by the one-loop result and is not renormalized perturbatively or non-perturbatively. 
  A Higgs mechanism for Abelian theories over non-trivial background flat connections is proposed. It is found that the mass generated for the spin 1 excitation is the same as the one obtained from the standard Higgs mechanism over trivial backgrounds, however, the dynamical structure of the action for the Higgs scalar is completely different from the usual approach. There is a topological contribution to the mass term of the Higgs field. After functional integration over all backgrounds, it is shown that the action for the massive spin 1 excitation is dual to the Topologically Massive Models in any dimension. 
  We calculate for the first time the finite size corrections in the massive Thirring model. This is done by numerically solving the equations of periodic boundary conditions of the Bethe ansatz solution. It is found that the corresponding central charge extracted from the $1/L$ term is around 0.4 for the coupling constant of ${g_0}=-{\pi\over 4} $ and decreases down to zero when ${g_0}=-{\pi\over{3}}$. This is quite different from the predicted central charge of the sine-Gordon model. 
  We study exact renormalization group (RG) in O(4) gauged supergravity using the effective average action formalism. The nonperturbative RG equations for cosmological and newtonian coupling constants are found. It is shown the existence of (nonstable) fixed point of these equations. The solution of RG equation for newtonian coupling constant is qualitatively the same as in Einstein gravity(i.e. it is growing at large distances). 
  Two dimensional QCD is bosonized to be an integrably deformed Wess-Zumino-Witten model under proper limit. Fermions are identified having indices of the Grassmann manifold. Conditions for integrability are analyzed and their physical meanings are discussed. We also address the nature of the exactly solvable part of the theory and find the infinitely many conserved quantities. 
  We study the composite Skyrme model, proposed by Cheung and G\"{u}rsey, introducing vector mesons in a chiral Lagrangian. We calculate the static properties of baryons and compare with results obtained from models without vector mesons. 
  Using standard coordinates, the Maxwell equations in the Reissner-Nordstr\"om geometry are written in terms of a couple of scalar fields satisfying Klein-Gordon like equations. The density of states is derived in the semi-classical approximation and the first quantum corrections to the black hole entropy is computed by using the brick-wall model. 
  We calculate the complete one-loop effective action for a spherical scalar field collapse in the large radius approximation. This action gives the complete trace anomaly, which beside the matter loop contributions, receives a contribution from the graviton loops. Our result opens a possibility for a systematic study of the back-reaction effects for a real black hole. 
  The three-particle Hamiltonian obtained by replacing the two-body trigonometric potential of the Sutherland problem by a three-body one of a similar form is shown to be exactly solvable. When written in appropriate variables, its eigenfunctions can be expressed in terms of Jack symmetric polynomials. The exact solvability of the problem is explained by a hidden $sl(3,R)$ symmetry. A generalized Sutherland three-particle problem including both two- and three-body trigonometric potentials and internal degrees of freedom is then considered. It is analyzed in terms of three first-order noncommuting differential-difference operators, which are constructed by combining SUSYQM supercharges with the elements of the dihedral group~$D_6$. Three alternative commuting operators are also introduced. 
  We make a detailed investigation on the quantum corrections to Chern-Simons spinor electrodynamics. Starting from Chern-Simons spinor quantum electrodynamics with the Maxwell term $-1/(4\gamma){\int}d^3x F_{\mu\nu}F^{\mu\nu}$ and by calculating the vacuum polarization tensor, electron self-energy and on-shell vertex, we explicitly show that the Ward identity is satisfied and hence verify that the physical quantities are independent of the procedure of taking ${\gamma}{\to}{\infty}$ at one-loop and tree levels. In particular, we find the three-dimensional analogue of the Schwinger anomalous magnetic moment term of the electron produced from the quantum corrections. 
  We show that the strength of the leading non-perturbative effects in non-critical string theory is of the order $e^{-O(1/{\beta_{st}})}$. We show how this restricts the space of consistent theories. We also identify non-critical one dimensional D-instantons as dynamical objects which exchange closed string states and calculate the order of their size. 
  We examine several zero-range potentials in non-relativistic quantum mechanics. The study of such potentials requires regularization and renormalization. We contrast physical results obtained using dimensional regularization and cutoff schemes and show explicitly that in certain cases dimensional regularization fails to reproduce the results obtained using cutoff regularization. First we consider a delta-function potential in arbitrary space dimensions. Using cutoff regularization we show that for $d \ge 4$ the renormalized scattering amplitude is trivial. In contrast, dimensional regularization can yield a nontrivial scattering amplitude for odd dimensions greater than or equal to five. We also consider a potential consisting of a delta function plus the derivative-squared of a delta function in three dimensions. We show that the renormalized scattering amplitudes obtained using the two regularization schemes are different. Moreover we find that in the cutoff-regulated calculation the effective range is necessarily negative in the limit that the cutoff is taken to infinity. In contrast, in dimensional regularization the effective range is unconstrained. We discuss how these discrepancies arise from the dimensional regularization prescription that all power-law divergences vanish. We argue that these results demonstrate that dimensional regularization can fail in a non-perturbative setting. 
  We present the wave equation for a minimally coupled scalar field in the background of a rotating four-dimensional black hole that is parametrized by its mass, angular momentum, and four independent U(1) charges. The near horizon structure is identical to the five-dimensional case, and suggestive of an underlying description in string theory that is valid in the general non-extremal case. We calculate the greybody factors for the Hawking radiation. For sufficiently large partial wave number the emission spectrum can be calculated for general non-extremal black holes and any particle energy. We interpret this spectrum in terms of a multi-body process in an effective string theory. 
  In matrix theory the effective action for graviton-graviton scattering is a double expansion in the relative velocity and inverse separation. We discuss the systematics of this expansion and subject matrix theory to a new test. Low energy supergravity predicts the coefficient of the $v^6/r^{14}$ term, a two-loop effect, in agreement with explicit matrix model calculation. 
  We study the off-diagonal blocks in the M(atrix) model that are supposed to correspond to open strings stretched between a Dp-brane and a Dp'-brane. It is shown that the spectrum, including the quantum numbers, of the zero modes in the off-diagonal blocks can be determined from the index theorem and unbroken supersymmetry, and indeed reproduces string theory predictions for p-p' strings. Previously the matrix description of a longitudinal fivebrane needed to introduce extra degrees of freedom corresponding to 0-4 strings by hand. We show that they are naturally associated with the off-diagonal zero modes, and the supersymmetry transformation laws and low energy effective action postulated for them are now derivable from the M(atrix) theory. 
  I present a set of theories which display non-trivial `t Hooft anomaly matching for QCD with $F$ flavors. The matching theories are non-Abelian gauge theories with "dual" quarks and baryons, rather than the purely confining theories of baryons that `t Hooft originally searched for. The matching gauge groups are required to have an $F\pm 6$ dimensional representation. Such a correspondence is reminiscent of Seiberg's duality for supersymmetric (SUSY) QCD, and these theories are candidates for non-SUSY duality. However anomaly matching by itself is not sufficiently restrictive, and duality for QCD cannot be established at present. At the very least, the existence of multiple anomaly matching solutions should provide a note of caution regarding conjectured non-SUSY dualities. 
  Sufficient conditions are proven for 't Hooft's consistency conditions to hold at points in the moduli space of supersymmetric gauge theories. Known results for anomaly matching in supersymmetric QCD are rederived as a sample application of the results. The results can be used to show that the anomaly matching conditions hold for s-confining theories. 
  We obtain a low-energy effective superpotential for a phase with a single confined photon in N=1 gauge theory with an adjoint matter with ADE gauge groups. The expectation values of gauge invariants built out of the adjoint field parametrize the singularities of moduli space of the Coulomb phase. The result can be used to derive the N=2 curve in the form of a foliation over $CP^1$. Our N=1 theory exhibits non-trivial fixed points which naturally inherit the properties of the ADE classification of N=2 superconformal field theories in four dimensions. We also discuss how to include matter hypermultiplets toward deriving the Riemann surface which describes N=2 QCD with exceptional gauge groups. 
  We investigate how the gap equations are obtained in the light-front formalism within the four-Fermi theories. Instead of the zero-mode constraint, we find that the "fermionic constraints" on nondynamical spinor components are responsible for symmetry breaking. Careful treatment of the infrared divergence is indispensable for obtaining the gap equation. 
  We calculate the charged hypermultiplet low-energy effective action in the Coulomb branch of the 4D, N=2 gauge theory, by using the harmonic superspace approach. We find that the unique leading contribution is given by the harmonic-analytic Lagrangian of the fourth order in the hypermultiplet superfields, with the induced coupling constant being proportional to central charges of N=2 SUSY algebra. The central charges are identified with Cartan generators of the internal symmetry, and they give BPS masses to the hypermultiplets. The induced hyper-K"ahler metrics appear to be the Taub-NUT metric or its higher-dimensional generalizations. Simultaneously, non-trivial scalar potentials are produced. Within the harmonic superspace method, we show an equivalence between the two known approaches to the 4D central charges. In the first one, the central charges are obtained via the Scherk-Schwarz dimensional reduction from six dimensions, whereas in the second one they are generated by a covariantly-constant background gauge superfield. Our analysis is extended to a more general situation with the non-vanishing Fayet-Iliopoulos (FI) terms. The perturbatively-induced Taub-NUT self-coupling in the Coulomb branch is found to be stable against adding the FI term, whereas the non-perturbatively generated (Eguchi-Hanson type) hypermultiplet self-interaction is proposed for the Higgs branch. 
  I attempt to analyse the next-to-leading-order non-holomorphic contribution to the Wilsonian low-energy effective action in the four-dimensional N=2 gauge theories with matter, from the manifestly N=2 supersymmeric point of view, by using the harmonic superspace. The perturbative one-loop correction is found to be in agreement with the N=1 superfield calculations of de Wit, Grisaru and Rocek. The previously unknown coefficient in front of this non-holomorphic correction is calculated. A special attention is devoted to the N=2 superconformal gauge theories, whose one-loop non-holomorphic contribution is likely to be exact, even non-perturbatively. This leading (one-loop) non-holomorphic contribution to the LEEA of the N=2 superconformally invariant gauge field theories is calculated, and it does not vanish, similarly to the case of the N=4 super-Yang-Mills theory. 
  We show that the one-dimensional projection of Chern-Simons gauged Nonlinear Schrodinger model is equivalent to an Abelian gauge field theory of continuum Heisenberg spin chain. In such a theory, the matter field has geometrical meaning of coordinates in tangent plane to the spin phase space, while the U(1) gauge symmetry relates to rotation in the plane. This allows us to construct the infinite hierarchy of integrable gauge theories and corresponding magnetic models. To each of them a U(1) invariant gauge fixing constraint of non-Abelian BF theory is derived. The corresponding moving frames hierarchy is obtained and the spectral parameter is interpreted as a constant-valued statistical gauge potential constrained by the 1-cocycle condition. 
  We perform a microcanonical study of classical lattice phi^4 field models in 3 dimensions with O(n) symmetries. The Hamiltonian flows associated to these systems that undergo a second order phase transition in the thermodynamic limit are here investigated. The microscopic Hamiltonian dynamics neatly reveals the presence of a phase transition through the time averages of conventional thermodynamical observables. Moreover, peculiar behaviors of the largest Lyapunov exponents at the transition point are observed. A Riemannian geometrization of Hamiltonian dynamics is then used to introduce other relevant observables, that are measured as functions of both energy density and temperature. On the basis of a simple and abstract geometric model, we suggest that the apparently singular behaviour of these geometric observables might probe a major topological change of the manifolds whose geodesics are the natural motions. 
  It has been known that the fivebrane of type IIA theory can be used to give an exact low energy description of N=2 supersymmetric gauge theories in four dimensions. We follow the recent M theory description by Witten and show that it can be used to study theories with N=1 supersymmetry. The N=2 supersymmetry can be broken to N=1 by turning on a mass for the adjoint chiral superfield in the N=2 vector multiplet. We construct the configuration of the fivebrane for both finite and infinite values of the adjoint mass. The fivebrane describes strong coupling dynamics of N=1 theory with SU(N_c) gauge group and N_f quarks. For N_c > N_f, we show how the brane configuration encodes the information of the Affleck-Dine-Seiberg superpotential. For N_c = and < N_f, we study the deformation space of the brane configuration and compare it with the moduli space of the N=1 theory. We find agreement with field theory results, including the quantum deformation of the moduli space at N_c = N_f. We also prove the type II s-rule in M theory and find new non-renormalization theorems for N=1 superpotentials. 
  We show that the growth of the size with the number of partons holds in a Thomas-Fermi analysis of the threshold bound state of D0--branes. Our results sharpen the evidence that for a fixed value of the eleven dimensional radius the partonic velocities can be made arbitrarily small as one approaches the large N limit. 
  The asymptotic behaviour of cubic field theories is investigated in the Regge limit using the techniques of environmentally friendly renormalization, environmentally friendly in the present context meaning asymmetric in its momentum dependence. In particular we consider the crossover between large and small energies at fixed momentum transfer for a model scalar theory of the type phi^2 psi. The asymptotic forms of the crossover scaling functions are exhibited for all two particle scattering processes in this channel to one loop in a renormalization group improved perturbation theory. 
  Wormhole spacetimes may be responsible for the possible loss of quantum coherence and the introduction of additional fundamental quantum indeterminancy of the values of constants of nature. As a system which is known to admit such classical wormhole solutions, Einstein-Yang-Mills (EYM) theory is revisited. Since the classical wormhole instanton solution in this theory has been studied extensively thus far, in the present work, ``quantum wormholes'' are explored. Namely in the context of a minisuperspace quantum cosmology model based on this EYM theory, ``quantum wormhole'', defined as a state represented by a solution to the Wheeler-DeWitt equation satisfying an appropriate wormhole boundary condition, is discussed. Finally, it is proposed that the minisuperspace model based on this theory in the presence of the cosmological constant may serve as a simple yet interesting system displaying an overall picture of entire universe's history from the deep quantum domain all the way to the classical domain. 
  We consider the XXZ model in the infinite volume limit with spin half quantum space and higher spin auxiliary space. Using perturbation theory arguments, we relate the half infinite transfer matrices of this class of models to certain $U_q(\hat{sl_2})$ intertwiners introduced by Nakayashiki. We construct the monodromy matrices, and show that the one with spin one auxiliary space gives rise to the L operator. 
  The form of the Seiberg-Witten differential is derived from the M-theory approach to N=2 supersymmetric Yang-Mills theories by directly imposing the BPS condition for twobranes ending on fivebranes. The BPS condition also implies that the pullback of the Kahler form onto the space part of the twobrane world-volume vanishes. 
  We study the universal critical behaviour near weakly first-order phase transitions for a three-dimensional model of two coupled scalar fields -- the cubic anisotropy model. Renormalization-group techniques are employed within the formalism of the effective average action. We calculate the universal form of the coarse-grained free energy and deduce the ratio of susceptibilities on either side of the phase transition. We compare our results with those obtained through Monte Carlo simulations and the epsilon-expansion. 
  We present a detailed analysis of the domain walls in supersymmetric gluodynamics and SQCD. We use the (corrected) Veneziano-Yankielowicz effective Lagrangians to explicitely obtain the wall profiles and check recent results of Dvali and Shifman (Phys. Lett. B396, (1997) 64: (i) the BPS-saturated nature of the walls; (ii) the exact expressions for the wall energy density which depend only on global features of dynamics (the existence of a non-trivial central extension of N=1 superalgebra in the theories which admit wall-like solutions). If supersymmetry is softly broken by the gluino mass, the degeneracy of the distinct vacua is gone, and one can consider the decay rate of the "false" vacuum into the genuine one. We do this calculation in the limit of the small gluino mass. Finally, we comment on the controversy regarding the existence of $N$ distinct chirally asymmetric vacua in SU(N) SUSY gluodynamics. 
  We prove D=11 supermembrane theories wrapping around in an irreducible way over $S^{1} \times S^{1}\times M^{9}$ on the target manifold, have a hamiltonian with strict minima and without infinite dimensional valleys at the minima for the bosonic sector. The minima occur at monopole connections of an associated U(1) bundle over topologically non trivial Riemann surfaces of arbitrary genus. Explicit expressions for the minimal connections in terms of membrane maps are presented. The minimal maps and corresponding connections satisfy the BPS condition with half SUSY. 
  In this paper free field realizations of affine current superalgebras are considered. Based on quantizing differential operator realizations of the corresponding basic Lie superalgebras, general and simple expressions for both the bosonic and the fermionic currents are provided. Screening currents of the first kind are also presented. Finally, explicit free field realizations of primary fields with general, possibly non-integer, weights are worked out. A formalism is used where the (generally infinite) multiplet is replaced by a generating function primary operator. The results allow setting up integral representations for correlators of primary fields corresponding to integrable representations. The results are generalizations to superalgebras of a recent work on free field realizations of affine current algebras by Petersen, Yu and the present author. 
  This is an introductory survey of the theory of $p$-form conservation laws in field theory. It is based upon a series of lectures given at the Second Mexican School on Gravitation and Mathematical Physics held in Tlaxcala, Mexico from December 1--7, 1996. Proceedings available online at http://kaluza.physik.uni-konstanz.de/2MS/ProcMain.html. 
  In this note we show that in dual N=1 string vacua provided by the heterotic string on an elliptic Calabi-Yau together with a vector bundle respectively F-theory on Calabi-Yau fourfold the number of heterotic fivebranes necessary for anomaly cancellation matches the number of F-theory threebranes necessary for tadpole cancellation. This extends to the general case the work of Friedman, Morgan and Witten, who treated the case of embedding a heterotic $E_8\times E_8$ bundle, leaving no unbroken gauge group, where one has a smooth Weierstrass model on the F-theory side. 
  The superfield equations of massive IIA supergravity, in the form of constraints on the superspace geometry, are shown to be implied by $\kappa$-symmetry of the topologically massive D-2-brane. 
  S-dualities in scale invariant N=2 supersymmetric field theories are derived by embedding those theories in asymptotically free N=2 theories with higher rank gauge groups. S-duality transformations on the coupling of the scale invariant theory follow from global symmetries acting on the Coulomb branch of the higher rank theory. Since these global symmetries are exact in the asymptotically free theory, this shows that S-duality is an exact equivalence of N=2 theories and not just a property of their supersymmetric states. 
  We discuss the T-duality between the solutions of type IIA versus IIB superstrings compactified on Calabi-Yau threefolds. Within the context of the N=2, D=4 supergravity effective Lagrangian, the T-duality transformation is equivalently described by the c-map, which relates the special Kahler moduli space of the IIA N=2 vector multiplets to the quaternionic moduli space of the N=2 hyper multiplets on the type IIB side (and vice versa). Hence the T-duality, or c-map respectively, transforms the IIA black hole solutions, originating from even dimensional IIA branes, of the special Kahler effective action, into IIB D-instanton solutions of the IIB quaternionic sigma-model action, where the D-instantons can be obtained by compactifying odd IIB D-branes on the internal Calabi-Yau space. We construct via this mapping a broad class of D-instanton solutions in four dimensions which are determinded by a set of harmonic functions plus the underlying topological Calabi-Yau data. 
  In extended supergravity theories there are $p$-brane solutions preserving different numbers of supersymmetries, depending on the charges, the spacetime dimension and the number of original supersymmetries (8, 16 or 32). We find U-duality invariant conditions on the quantized charges which specify the number of supersymmetries preserved with a particular charge configuration. These conditions relate U-duality invariants to the picture of intersecting branes. The analysis is carried out for all extended supergravities with 16 or 32 supersymmetries in various dimensions. 
  Recent work has shown that complex quantum field theory emerges as a statistical mechanical approximation to an underlying noncommutative operator dynamics based on a total trace action. In this dynamics, scale invariance of the trace action becomes the statement $0=Re Tr T_{\mu}^{\mu}$, with $T_{\mu \nu}$ the operator stress energy tensor, and with $Tr$ the trace over the underlying Hilbert space. We show that this condition implies the vanishing of the cosmological constant and vacuum energy in the emergent quantum field theory. However, since the scale invariance condition does not require the operator $T_{\mu}^{\mu}$ to vanish, the spontaneous breakdown of scale invariance is still permitted. 
  In this paper we first investigate the Ansatz of one of the present authors for K(\Psi,\bar\Psi), the adimensional modular invariant non-holomorphic correction to the Wilsonian effective Lagrangian of an N=2 globally supersymmetric gauge theory. The renormalisation group beta-function of the theory crucially allows us to express K(\Psi,\bar\Psi) in a form that easily generalises to the case in which the theory is coupled to N_F hypermultiplets in the fundamental representation of the gauge group. This function satisfies an equation which should be viewed as a fully non-perturbative ``non-chiral superconformal Ward identity". We also determine its renormalisation group equation. Furthermore, as a first step towards checking the validity of this Ansatz, we compute the contribution to K(\Psi,\bar\Psi) from instantons of winding number k=1 and k=2. As a by-product of our analysis we check a non-renormalisation theorem for N_F=4. 
  The absorption of photons and fermions into four-dimensional black holes is described by equations which in certain cases can be analyzed using dyadic index techniques. The resulting absorption cross-sections for near-extremal black holes have a form at low energies suggestive of the effective string model. A coupling to the effective string is proposed for spin-0 and spin-1/2 fields of pure N=4 supergravity which respects the unbroken supersymmetry of extreme black holes and correctly predicts dilaton, axion, and fermion cross-sections up to an overall normalization. 
  Non-commutative differential geometry allows a scalar field to be regarded as a gauge connection, albeit on a discrete space. We explain how the underlying gauge principle corresponds to the independence of physics on the choice of vacuum state, should it be non-unique. A consequence is that Yang-Mills-Higgs theory can be reformulated as a generalised Yang-Mills gauge theory on Euclidean space with a $Z_2$ internal structure. By extending the Hodge star operation to this non-commutative space, we are able to define the notion of self-duality of the gauge curvature form in arbitrary dimensions. It turns out that BPS monopoles, critically coupled vortices, and kinks are all self-dual solutions in their respective dimensions. We then prove, within this unified formalism, that static soliton solutions to the Yang-Mills-Higgs system exist only in one, two and three spatial dimensions. 
  The moduli space dynamics of vortices in the Jackiw-Pi model where a non-relativistic Schrodinger field couples minimally to Chern-Simons gauge field, is considered. It is shown that the difficulties in direct application of Manton's method to obtain a moduli-space metric in the first order system can be circumvented by turning the Lagrangian into a second order system. We obtain exact metrics for some simple cases and describe how the vortices respond to an external U(1) field. We then construct an effective Lagrangian describing dynamics of the vortices. In addition, we clarify strong-weak coupling duality between fundamental particles and vortices. 
  We analyse the BRST constraints and corresponding Hilbert-space structure of chiral QCD$_2$ in the decoupled formulation for the case of the Jackiw-Rajaraman parameter $a=2$. We show that despite formal similarities this theory is not equivalent to QCD$_2$, and that its extension to U(N) does not lead to an infinite vacuum degeneracy. 
  We have explored the possibility that the universe at very early stage was dominated by (macroscopic) heterotic strings. We have found that the dimensionless parameter $G\mu$ for the heterotic strings varies from $10^{-2}$ to $10^{-4}$ as the universe evolve from the matter dominance to radiation dominance. This led to the interesting consequence of epoch dependent gauge coupling constant. The gauge coupling constant at early times was found to be much stronger than the present strong interaction. 
  Connes' view at Yang-Mills theories is reviewed with special emphasis on the gauge invariant scalar product. This landscape is shown to contain Chamseddine and Connes' noncommutative extension of general relativity restricted to flat space-time, if the top mass is between 172 and 204 GeV. Then the Higgs mass is between 188 and 201 GeV. 
  It is shown that in some cases higher (covariant) derivative regularization for spinor field is equivalent to the gauge invariant Pauli-Villars one. 
  The structure of spacetime duality and discrete worldsheet symmetries of compactified string theory is examined within the framework of noncommutative geometry. The full noncommutative string spacetime is constructed using the Frohlich-Gawedzki spectral triple which incorporates the vertex operator algebra of the string theory. The duality group appears naturally as a subgroup of the automorphism group of the vertex operator algebra and spacetime duality is shown to arise as the possibility of associating two independent Dirac operators, arising from the chiral structure of the worldsheet theory, to the noncommutative geometry. 
  We present a new, alternative interpretation of the vector-tensor multiplet as a 2-form in central charge superspace. This approach provides a geometric description of the (non-trivial) central charge transformations ab initio and is naturally generalized to include couplings of Chern-Simons forms to the antisymmetric tensor gauge field, giving rise to a N=2 supersymmetric version of the Green-Schwarz anomaly cancellation mechanism. 
  A brane configuration is described that is relevant to understanding the dynamics of N=1 supersymmetric Yang-Mills theory. Confinement and spontaneous breaking of a discrete chiral symmetry can be understood as consequences of the topology of the brane. Because of the symmetry breaking, there can be domain walls separating different vacua; the QCD string can end on such a domain wall. The model in which these properties can be understood semiclassically does not coincide with supersymmetric Yang-Mills theory but is evidently in the same universality class. 
  Using geometric engineering in the context of type II strings, we obtain exact solutions for the moduli space of the Coulomb branch of all N=2 gauge theories in four dimensions involving products of SU gauge groups with arbitrary number of bi-fundamental matter for chosen pairs, as well as an arbitrary number of fundamental matter for each factor. Asymptotic freedom restricts the possibilities to SU groups with bi-fundamental matter chosen according to ADE or affine ADE Dynkin diagrams. Many of the results can be derived in an elementary way using the self-mirror property of K3. We find that in certain cases the solution of the Coulomb branch for N=2 gauge theories is given in terms of a three dimensional complex manifold rather than a Riemann surface. We also study new stringy strong coupling fixed points arising from the compactification of higher dimensional theories with tensionless strings and consider applications to three dimensional N=4 theories. 
  Families of solutions to the field equations of the covariant BRST invariant effective action of the membrane theory are constructed. The equations are discussed in a double dimensional reduction, they lead to a nonlinear equation for a one dimensional extended object. One family of solutions of these equations are solitary waves with several properties of solitonic solutions in integrable systems, giving evidence that in this double dimensional reduction the nonlinear equations are an integrable system. The other family of solutions found, exploits the property that the non linear system under some assumptions is equivalent to a non linear Schr$\ddot {o}$dinger equation. 
  Classical solutions of the self-interacting, non-abelian antisymmetric tensor gauge theory of Freedman and Townsend coupled to Einstein gravity is discussed. Particularly, it is demonstrated that the theory admits a classical metric solution which, depending on the value of the gauge coupling parameter of the theory, exhibits a black hole with an exotic non-abelian hair or a spacetime showing the ``violation of the cosmic censorship hypothesis'' which should be distinguished from white holes. 
  A new class of completely integrable models is constructed. These models are deformations of the famous integrable and exactly solvable Gaudin models. In contrast with the latter, they are quasi-exactly solvable, i.e. admit the algebraic Bethe ansatz solution only for some limited parts of the spectrum. An underlying algebra responsible for both the phenomena of complete integrability and quasi-exact solvability is constructed. We call it "quasi-Gaudin algebra" and demonstrate that it is a special non-Lie-algebraic deformation of the ordinary Gaudin algebra. 
  A possibility of an extension of the minimality principle for electromagnetic interactions of charged and neutral particles having spin 1/2 is investigated. 
  Inclusive absorption cross section of fundamental IIB string to D-string is calculated perturbatively. The leading order result agrees with estimate based on stringy Higgs mechanism via Cremmer-Scherk coupling. It is argued that the subleading order correction is dominated by purely planar diagrams in the large mass limit. The correction represents conversion of binding energy into local recoil process of the fundamental string and D-string bound state. We show their presence explicitly in the next leading order. 
  The Liouville field theory on Z_N-Riemann surfaces is studied and it is shown that it decomposes into a Liouville field theory on the sphere and N-1 free boson theories. Also, the partition function of the Liouville field theory on the Z_N-Riemann surfaces is expressed as a product of the correlation function for the Liouville vertex operators on the sphere and a number of twisted fields. 
  We propose an effective action for the eleven-dimensional (bosonic) Kaluza-Klein monopole solution. The construction of the action requires that the background fields admit an Abelian isometry group. The corresponding sigma-model is gauged with respect to this isometry. The gauged sigma-model is the source for the monopole solution. A direct (double) dimensional reduction of the action leads to the effective action of a 10-dimensional D-6-brane (IIA Kaluza-Klein monopole). We also show that the effective action of the 10-dimensional heterotic Kaluza-Klein monopole (which is a truncation of the IIA monopole action) is T-dual to the effective action of the solitonic 5-brane. We briefly discuss the kappa-symmetric extension of our proposal and the possible role of gauged sigma-models in connection with the conjectured M-theory 9-brane. 
  The identities satisfied by two-dimensional chiral quantum fields are studied from the point of view of vertex algebras. The Cauchy-Jacobi identity (or the Borcherds identity) for three mutually local fields is proved and consequently a direct proof of Li's theorem on a local system of vertex operators is provided. Several characterizations of vertex algebras are also discussed. 
  We give the complete list of all first-order consistent interaction vertices for a set of exterior form gauge fields of form degree >1, described in the free limit by the standard Maxwell-like action. A special attention is paid to the interactions that deform the gauge transformations. These are shown to be necessarily of the Noether form "conserved antisymmetric tensor" times "p-form potential" and exist only in particular spacetime dimensions. Conditions for consistency to all orders in the coupling constant are given. For illustrative purposes, the analysis is carried out explicitly for a system of forms with two different degrees p and q (1<p<q<n). 
  The global additive and multiplicative properties of Laplace type operators acting on irreducible rank 1 symmetric spaces are considered. The explicit form of the zeta function on product spaces and of the multiplicative anomaly is derived. 
  With the help of M-theory configuration of N=1 supersymmetric QCD, we analyze the strong coupling moduli of N=1 theory and its relation with S-duality transformation in N=2 supersymmetric QCD. As a result we confirm that two type-IIA descriptions for N=1 supersymmetric QCD, one of them being strong coupling description and another being weak, correspond to a single M-theory description for N=1 supersymmetric QCD. The existence of singlet fields is also discussed. 
  We classify all untwisted (2,1) heterotic strings. The only solutions are the three already known cases, having massless spectra consisting either of 24 chiral fermions, or of 24 bosons, or of 8 scalars and 8 fermions of each chirality. 
  The second alternative conformal limit of the recently proposed general higher derivative dilaton quantum theory in curved spacetime is explored. In this version of the theory the dilaton is transformed, along with the metric, to provide the conformal invariance of the classical action. We find the corresponding quantum theory to be renormalizable at one loop, and the renormalization constants for the dimensionless parameters are explicitly shown to be universal for an arbitrary parametrization of the quantum field. The renormalization group equations indicate an asymptotic freedom in the IR limit. In this respect the theory is similar to the well-known model based on the anomaly-induced effective action. 
  We generalize the supermembrane solution of D=11 supergravity by permitting the 4-form $G$ to be either self-dual or anti-self-dual in the eight dimensions transverse to the membrane. After analyzing the supergravity field equations directly, and also discussing necessary conditions for unbroken supersymmetry, we focus on two specific, related solutions. The self-dual solution is not asymptotically flat. The anti-self-dual solution is asymptotically flat, has finite mass per unit area and saturates the same mass=charge Bogomolnyi bound as the usual supermembrane. Nevertheless, neither solution preserves any supersymmetry. Both solutions involve the octonionic structure constants but, perhaps surprisingly, they are unrelated to the octonionic instanton 2-form $F$, for which $TrF \wedge F$ is neither self-dual nor anti-self-dual. 
  We have argued previously that black holes may be represented in a D-brane approach by monopole and vortex defects in a sine-Gordon field theory model of Liouville dynamics on the world sheet. Supersymmetrizing this sine-Gordon system, we find critical behaviour in 11 dimensions, due to defect condensation that is the world-sheet analogue of D-brane condensation around an extra space-time dimension in M theory. This supersymmetric description of Liouville dynamics has a natural embedding within a 12-dimensional framework suggestive of F theory. 
  Baker, Ball and Zachariasen have developed an elegant formulation of the dual superconducting model of quantum chromodynamics (QCD), which allows one to use the field equations to eliminate the gluon and Higgs degrees of freedom and thus to express the interaction between quarks as an effective potential. Carrying out an expansion in inverse powers of the constituent quark masses, these authors succeeded in identifying the central part, the spin-dependent part and the leading relativistic corrections to the central potential. The potential offers a good account of the energies and splittings of charmonium and the upsilon system. Since all of the flavor dependence of the interaction is presumed to enter through the constituent masses, it is possible to test the potential in other systems. Logical candidates are the heavy B-flavor charmed system and the heavy-light systems, which should be more sensitive to the relativistic corrections. Lattice gauge calculations furnish an additional point of contact for the components of the BBZ potential. Some preliminary calculations of the energies of B and D mesons are presented and the challenge of agreement with experiment is discussed. The spinless Salpeter equation is used to account for the effects of relativistic kinematics. 
  We use the M theory approach of Witten to investigate N=1 SU($N_c$) SQCD with $N_f$ flavors. We reproduce the field theoretical results and identify in M theory the gluino condensate and the eigenvalues of the meson matrix. This approach allows us to identify the constant piece of the effective field theory coupling from which the coefficient of the one-loop $\beta$-function can be identified. By studying the area of the M-theory five-brane we investigate the stability of type IIA brane configurations. We prove that in a supersymmetric setup there is no force between static D4-branes that end on NS five-branes. The force in the case that there is a relative velocity between the branes is computed. We show that at the regions of intersecting IIA branes the curvature of the M theory five-brane is singular in the type IIA limit. 
  We show that it is algebraically consistent to express some string field theory operators as inner derivations of the B-V algebra of string vertices. In this approach, the recursion relations for the string vertices are found to take the form of a `geometrical' quantum master equation. We also show that the B-V delta operator cannot be an inner derivation on the algebra. 
  We calculate the heat-kernel coefficients, up to $a_2$, for a U(1) bundle on the 4-Ball for boundary conditions which are such that the normal derivative of the field at the boundary is related to a first-order operator in boundary derivatives acting on the field. The results are used to place restrictions on the general forms of the coefficients. In the specific case considered, there can be a breakdown of ellipticity. 
  The recent developments in string theory suggest that the space-time coordinates should be generalized to non-commuting matrices. Postulating this suggestion as the fundamental geometrical principle, we formulate a candidate for covariant second quantized RNS superstrings as a topological field theory in two dimensions. Our construction is a natural non-Abelian extension of the RNS string. It also naturally leads to a model with manifest 11-dimensional covariance, which we conjecture to be a formulation of M theory. The non-commuting space-time coordinates of the strings are further generalized to non-commuting anti-symmetric tensors. The usual space-time picture and the free superstrings appear only in certain special phases of the model. We derive a simple set of algebraic equations, which determine the moduli space of our model. We test some aspects of our conjectual M theory for the case of compactification on $T^2$. 
  We describe, in the context of M-theory on elliptically fibered Calabi-yau fourfolds, the change of variables that allow us to pass from the U(1) invariant elliptic fibration to the one describing the uncompactified four dimensional limit, where the U(1) symmetry is broken. These changes of variables are the analog to the ones used to derive from the Atiyah-Hitchin space the complex structure of the Seiberg-Witten solution for N=2 supersymmetric Yang-Mills. The connection between these changes of variables and the recently introduced rotation of branes is discussed. 
  This is an expanded version of the notes to a course taught by the first author at the 1995 Les Houches Summer School. Constraints on a tentative reconciliation of quantum theory and general relativity are reviewed. It is explained what supersymmetric quantum theory teaches us about differential topology and geometry. Non-commutative differential topology and geometry are developed in some detail. As an example, the non-commutative torus is studied. An introduction to string theory and $M$(atrix) models is provided, and it is outlined how tools of non-commutative geometry can be used to explore the geometry of string theory and conformal field theory. 
  An algebraic cohomological characterization of a class of linearly broken Ward identities is provided. The examples of the topological vector supersymmetry and of the Landau ghost equation are discussed in detail. The existence of such a linearly broken Ward identities turns out to be related to BRST exact antifield dependent cocycles with negative ghost number. 
  In this note we describe the physics of equivalence of the Seiberg-Witten invariants of 4-manifolds and certain Gromov-Witten invariants defined by pseudo-holomorphic curves. We show that physics of the pseudo-holomorphic curves should be governed by the N=2 Green-Schwarz string. 
  A new variant of the exact Fradkin representation of the Green's function $G_c(x,y|gU)$, defined for arbitrary external potential $U$, is presented. Although this new approach is very similar in spirit to that previously derived by Fried and Gabellini, for certain calculations this specific variant, with its prescribed approximations, is more readily utilizable. Application of the simplest of these forms is made to the $\lambda\Phi^4$ theory in four dimensions.   As an independent check of these approximate forms, an improved version of the Schwinger-DeWitt asymptotic expansion of parametrix function is derived. 
  The two-dimensional simple supergravity is reexamined from the point of view of super-Weyl group cohomologies. The non-local form of the effective action of 2d supergravity which generalise the famous $R 1/\Box R$ is obtained. 
  A specific Friedmann-Roberstson-Walker (FRW) model derived from the theory of N=1 supergravity with gauged supermatter is considered in this paper. The supermatter content is restricted to a vector supermultiplet. The corresponding Lorentz and supersymmetry quantum constraints are then derived. Non-trivial solutions are subsquently found. A no-boundary solution is identified while another state may be interpreted as a wormhole solution. 
  Bianchi class-A models and Reissner-Nordst\"rom (RN) black hole scenarios are considered from the point of view of quantum N=2 Supergravity. It is shown that the presence of Maxwell fields in the supersymmetry constraints implies a non-conservation of the fermion number present in Bianchi class-A models. This effect corresponds to a mixing between different (Lorentz invariant) fermionic sectors in the wave function of the Universe. Quantum states are constituted by exponentials of N=2 supersymmetric Chern-Simons functionals. With respect to the RN case, we analyse some problems and features present in a reduced model with supersymmetry.    Lines of subsequent research work are then provided. 
  In this paper we deal with defects inside defects in systems of two scalar fields in 3+1 dimensions. The systems we consider are defined by potentials containing two real scalar fields, and so we are going to investigate domain ribbons inside domain walls. After introducing some general comments on the possibility of finding defects that support internal structure in two specific systems, we introduce thermal effects to show how the picture for domain walls hosting domain ribbons appears at high temperature. 
  The method of analytic continuation is used to find exact integral equations for a selection of finite-volume energy levels for the non-unitary minimal models $M_{2,2N+3}$ perturbed by their $\varphi_{13}$ operators. The N=2 case is studied in particular detail. Along the way, we find a number of general results which should be relevant to the study of excited states in other models. 
  Domain-wall solutions in four-dimensional supersymmetric field theories with distinct discrete vacuum states lead to the spontaneous breaking of supersymmetry, either completely or partially. We consider in detail the case when the domain walls are the BPS-saturated states, and 1/2 of supersymmetry is preserved. Several useful criteria that relate the preservation of 1/2 of supersymmetry on the domain walls to the central extension appearing in the N=1 superalgebras are established. We explain how the central extension can appear in N=1 supersymmetry and explicitly obtain the central charge in various models: the generalized Wess-Zumino models, and supersymmetric Yang-Mills theories with or without matter. The BPS-saturated domain walls satisfy the first-order differential equations which we call the creek equations, since they formally coincide with the (complexified) equations of motion of an analog high-viscosity fluid on a profile which is given by the superpotential of the original problem. Some possible applications are considered. 
  The M-theory origin of brane creation processes is discussed. 
  The spherically symmetric reduction of higher dimensional Einstein-scalar theory leads to lower dimensional dilatonic gravity with dilaton coupled scalar (for example, from 4D to 2D system). We calculate trace anomaly and anomaly induced effective action for 2D and 4D dilaton coupled scalars. The large-N effective action for 2D quantum dilaton-scalar gravity is also found. These 2D results maybe applied for analysis of 4D spherical collapse. The role of new, dilaton dependent terms in trace anomaly for 2D black holes and Hawking radiation is investigated in some specific models of dilatonic gravity which represent modification of CGHS model. The conformal sector for 4D dilatonic gravity is constructed. Quantum back-reaction of dilaton coupled matter is briefly discussed (it may lead to the inflationary Universe with non-trivial dilaton). 
  We investigate the zero modes of the Kaluza-Klein monopole in M-theory and show that the Born-Infeld action and the Chern-Simons term of a D6-brane are reproduced to quadratic order in the field strength of the U(1) field on the brane. 
  We present a detailed study of the analytic structure, BPS spectra and superconformal points of the $ N=2 $ susy $ SU(2) $ gauge theories with $ N_f=1,2,3 $ massive quark hypermultiplets. We compute the curves of marginal stability with the help of the explicit solutions for the low energy effective actions in terms of standard elliptic functions. We show that only a few of these curves are relevant. As a generic example, the case of $ N_f=2 $ with two equal bare masses is studied in depth. We determine the precise existence domains for each BPS state, and show how they are compatible with the RG flows. At the superconformal point, where two singularities coincide, we prove that (for $ N_f=2 $) the massless spectrum consists of four distinct BPS states and is S-invariant. This is due to the monodromy around the superconformal point being S, providing strong evidence for exact S-duality of the SCFT. For all $ N_f $, we compute the slopes $ \omega $ of the $ \beta $-functions at the fixed point couplings and show that they are related to the anomalous dimensions $ \alpha $ of $ u= < tr \phi^2 > $ by $ \omega= 2 (\alpha -1) $. 
  We consider a matrix model with d matrices NxN and show that in the limit of large N and d=0 the model describes the knot diagrams. The same limit in matrix string theory is also discussed. We speculate that a prototypical M(atrix) without matrix theory exists in void. 
  We consider the dynamical correlation functions of the quantum Nonlinear Schrodinger equation. In a previous paper we found that the dynamical correlation functions can be described by the vacuum expectation value of an operator-valued Fredholm determinant. In this paper we show that a Riemann-Hilbert problem can be associated with this Fredholm determinant. This Riemann-Hilbert problem formulation permits us to write down completely integrable equations for the Fredholm determinant and to perform an asymptotic analysis for the correlation function. 
  We present a theory of N=2 chiral supergravity in (10+2)-dimensions. This formulation is similar to N=1 supergravity presented recently using null-vectors in 12D. In order to see the consistency of this theory, we perform a simple dimensional reduction to ten-dimensions, reproducing the type IIB chiral supergravity. We also show that our supergravity can be consistent background for super (2+2)-brane theory, satisfying fermionic invariance of the total action. Such supergravity theory without manifest Lorentz invariance had been predicted by the recent F-theory in twelve-dimensions. 
  A matrix model of an asymptotically free theory with a bound state is solved using a perturbative similarity renormalization group for hamiltonians. An effective hamiltonian with a small width, calculated including the first three terms in the perturbative expansion, is projected on a small set of effective basis states. The resulting small hamiltonian matrix is diagonalized and the exact bound state energy is obtained with accuracy of order 10%. Then, a brief description and an elementary illustration are given for a related light-front Fock space operator method which aims at carrying out analogous steps for hamiltonians of QCD and other theories. 
  We solve for the exact atom-field eigenstates of a single atom in a three dimensional spherical cavity, by mapping the problem onto the anisotropic Kondo model. The spectrum has a rich bound state structure in comparison with models where the rotating wave approximation is made. It is shown how to obtain the Jaynes-Cummings model states in the limit of weak coupling. Non-perturbative Lamb shifts and decay rates are computed. The massive Kondo model is introduced to model light localization in the form of photon-atom bound states in photonic crystals. 
  When we vary the moduli of a compactification it may become entropically favourable at some point for a state of branes and strings to rearrange itself into a new configuration. We observe that for the elementary string with two large charges such a rearrangement happens at the `correspondence point' where the string becomes a black hole. For smaller couplings it is entropically favourable for the excitations to be vibrations of the string, while for larger couplings the favoured excitations are pairs of solitonic 5-branes attached to the string; this helps resolve some recently noted difficulties with matching emission properties of the string to emission properties of the black hole. We also examine the change of state when a black hole is placed in a spacetime with an additional compact direction, and the size of this direction is varied. These studies suggest a mechanism that might help resolve the information paradox. 
  We suggest a D=11 super Poincar\'e invariant action for the superstring which has free dynamics in the physical variables sector. Instead of the standard approach based on the searching for an action with local $\kappa$-symmetry (or, equivalently, with corresponding first class constraints), we propose a theory with fermionic constraints of second class only. Then the $\kappa$-symmetry and the well known $\Gamma$-matrix identities are not necessary for the construction. Thus, at the classical level, the superstring action of the type described can exist in any spacetime dimensions and the known brane-scan can be revisited. 
  We present a systematic analysis of possible bound states of M-brane solutions (including waves and monopoles) by using the solution generating technique of reduction of M-brane to 10 dimensions, use of T-duality and then lifting back to 11 dimensions. We summarize a list of bound states for one- and two-charge cases including tilted brane solutions. Construction rules for these non-marginal solutions are also discussed. 
  We study the graceful exit problem and the role of the stress-energy-momentum tensors in the two-dimensional string cosmology. The one-loop quantum correction of conformal fields is incorporated in the arbitrary large $N$ limit to ensure exact quantum solvability. The only solution which gives the bounded curvature with the asymptotic flatness is restricted to the first branch under some conditions. However, even in this case, the accelerating expansion is forever. We show that the only nonvanishing quantum stress-momentum tensor is the pressure part($T_{xx}$) which is of relevance to the dynamical evolution of the universe in the comoving coordinate. The quantum energy part is zero since the negative contribution of the induced conformal matters always cancels the positive quantity of the induced dilaton part in terms of the constraint equation. 
  A duality relationship between certain brane configurations in type IIA and type IIB string theory is explored by exploiting the geometrical origins of each theory in M-theory. The configurations are dual ways of realising the non-perturbative dynamics of a four dimensional N=2 supersymmetric SU(2) gauge theory with four or fewer favours of fermions in the fundamental, and the spectral curve which organizes these dynamics plays a prominent role in each case. This is an illustration of how non-trivial F-theory backgrounds follow from M-theory ones, hopefully demystifying somewhat the origins of the former. 
  In this letter, we provide evidence for a classical sector of states in the Hilbert space of Finite Quantum Mechanics (FQM). We construct a subset of states whose the minimum bound of position -momentum uncertainty (equivalent to an effective $\hbar$) vanishes. The classical regime, contrary to standard Quantum Mechanical Systems of particles and fields, but also of strings and branes appears in short distances of the order of the lattice spacing. {}For linear quantum maps of long periods, we observe that time evolution leads to fast decorrelation of the wave packets, phenomenon similar to the behavior of wave packets in t' Hooft and Susskind holographic picture. Moreoever, we construct explicitly a non - dispersive basis of states in accordance with t' Hooft's arguments about the deterministic behavior of FQM. 
  On the basis of recently discovered connections between D-branes and black holes, I show how the information puzzle is solved by superstring theory as the fundamental theory of quantum gravity. The picture that emerges is that a well-defined quantum state does not give rise to a black hole even if the apparent distribution of energy, momenta, charges, etc. would predict one on classical grounds. Indeed, geometry - general relativistic space time description - is unwarranted at the quantum microstate level. It is the decoherence leading to macrostates (average over degenerate microstates) that provides - on the same token - the loss of quantum coherence, the emergence of a space time description with causal properties and, thus, the formation of a black hole and its Hawking evaporation 
  We consider D=6, N=1, Z_M orbifold compactifications of heterotic strings in which the usual modular invariance constraints are violated. It is argued that in the presence of non-perturbative effects many of these vacua are nevertheless consistent. The perturbative massless sector can be computed explicitly from the perturbative mass formula subject to an extra shift in the vacuum energy. This shift is associated to a non-trivial antisymmetric B-field flux at the orbifold fixed points. The non-perturbative piece is given by five-branes either moving in the bulk or stuck at the fixed points, giving rise to Coulomb phases with tensor multiplets. The heterotic duals of some Type IIB orientifolds belong to this class of orbifold models. We also discuss how to carry out this type of construction to the D=4, N=1 case and specific $Z_M\times Z_M$ examples are presented in which non-perturbative transitions changing the number of chiral generations do occur. 
  We study, through path-integral methods, an extension of the massive Thirring model in which the interaction between currents is non-local. By examining the mass-expansion of the partition function we show that this non-local massive Thirring model is equivalent to a certain non-local extension of the sine-Gordon theory. Thus, we establish a non-local generalization of the famous Coleman's equivalence. We also discuss some possible applications of this result in the context of one-dimensional strongly correlated systems and finite-size Quantum Field Theories. 
  For any principal bundle $P$, one can consider the subspace of the space of connections on its tangent bundle $TP$ given by the tangent bundle $T{\cal A}$ of the space of connections ${\cal A}$ on $P$. The tangent gauge group acts freely on $T{\cal A}$. Appropriate BRST operators are introduced for quantum field theories that include as fields elements of $T{\cal A}$, as well as tangent vectors to the space of curvatures. As the simplest application, the BRST symmetry of the so-called $BF$-Yang-Mills theory is described and the relevant gauge fixing conditions are analyzed. A brief account on the topological $BF$ theories is also included and the relevant Batalin-Vilkovisky operator is described. 
  We study multi-soliton states in two-dimensional N=2 supersymmetric theories. We calculate their energy exactly as a function of mass and volume in the simplest integrable N=2 theory, the sine-Gordon model at a particular coupling. These energies are related to the expectation value $I= tr [\exp(in\pi F) \exp(-H/T)]$, where F is the fermion number. For n=1, this is Witten's index; for n an odd integer, we argue that $I$ is an index in the sense that it is independent of all D-term variations. 
  The variational perturbation theory for wave functions, which has been shown to work well for bound states of the anharmonic oscillator, is applied to resonance states of the anharmonic oscillator with negative coupling constant. We obtain uniformly accurate wave functions starting from the bound states. 
  We propose a new formulation of chiral fermions on a lattice, on the basis of a lattice extension of the covariant regularization scheme in continuum field theory. The species doublers do not emerge. The real part of the effective action is just one half of that of Dirac-Wilson fermion and is always gauge invariant even with a finite lattice spacing. The gauge invariance of the imaginary part, on the other hand, sets a severe constraint which is a lattice analogue of the gauge anomaly free condition. For real gauge representations, the imaginary part identically vanishes and the gauge invariance becomes exact. 
  The exactly and quasi-exactly solvable problems for spin one-half in one dimension on the basis of a hidden dynamical symmetry algebra of Hamiltonian are discussed. We take the supergroup, $OSP(2|1)$, as such a symmetry. A number of exactly solvable examples are considered and their spectrum are evaluated explicitly. Also, a class of quasi-exactly solvable problems on the basis of such a symmetry has been obtained. 
  It has been shown recently that the toroidally compactified type IIB string effective action possesses an SL(2, R) invariance. Using this symmetry we construct an infinite family of macroscopic string-like solutions permuted by SL(2, Z) group for type II superstrings in $4 \leq D < 10$. These solutions, which formally look very similar to the corresponding solutions in $D = 10$, are characterized by two relatively prime integers corresponding to the `electric' charges associated with the two antisymmetric tensor fields of the strings. Stability of these solutions is discussed briefly in the light of charge conservation and the tension gap equation. 
  Using numerical simulations, the stability and scattering properties of the O(3) model on a two-dimensional torus are studied. Its solitons are found to be unstable but can be stabilized by the addition of a Skyrme term to the Lagrangian. Scattering at right angles with respect to the initial direction of motion is observed in all cases considered. The model has no solutions of degree one, so when a field configuration that resembles a soliton is considered, it shrinks to become infinitely thin. A comparison of these results with those of the model defined on the sphere is made. 
  It is argued that, in the two dimensional principal chiral model, the Wess-Zumino term can be induced quantum mechanically, allowing the model with the critical value of the coupling constant $\lambda^2 = 8\pi/|k|$ to turn into the Wess-Zumino-Novikov-Witten model at the quantum level. The Wess-Zumino term emerges from the inequivalent quantizations possible on a sphere hidden in the configuration space of the original model. It is shown that the Dirac monopole potential, which is induced on the sphere in the inequivalent quantizations, turns out to be the Wess-Zumino term in the entire configuration space. 
  This is a brief description of what has been accomplished and what remains to be done in the construction of a nonperturbative formulation of "The Theory Formerly Known as String". It is culled from two short talks given by the author at SUSY 97 and Strings 97. 
  We describe the vector-tensor multiplet and derive its Chern-Simons couplings to the N=2 Yang-Mills gauge superfield in harmonic superspace. 
  We show a general method to solve 2+1 dimensional dilatonic Maxwell-Einstein equation with a positive or negative cosmological constant. All the physical solutions are listed with assumptions that they are static, rotationally symmetric, and has a nonzero magnetic field and a nonzero dilaton field. On the contrary to the magnetic solution without a dilaton field, some of the present solutions with a dilaton field possess a horizon. 
  We examine the possibility that the strong CP problem is solved by string-theoretic axions in strong-coupling limit of the E_8 x E_8 heterotic string theory (M-theory). We first discuss some generic features of gauge kinetic functions in compactified M-theory, and examine in detail the axion potential induced by the explicit breakings other than the QCD anomaly of the non-linear U(1)_{PQ} symmetries of string-theoretic axions. It is argued based on supersymmetry and discrete gauge symmetries that if the compactification radius is large enough, there can be a U(1)_{PQ}-symmetry whose breaking other than the QCD anomaly, whatever its microscopic origin is, is suppressed enough for the axion mechanism to work. Phenomenological viability of such a large radius crucially depends upon the quantized coefficients in gauge kinetic functions. We note that the large radius required for the axion mechanism is viable only in a limited class of models. For instance, for compactifications on a smooth Calabi-Yau manifold with a vanishing second E_8 field strength, it is viable only when the quantized flux of the antisymmetric tensor field in M-theory has a minimal nonzero value. It is also stressed that this large compactification radius allows the QCD axion in M-theory to be cosmologically viable in the presence of a late time entropy production. 
  A derivation of the cyclic form factor equation from quantum field theoretical principles is given; form factors being the matrix elements of a field operator between scattering states. The scattering states are constructed from Haag-Ruelle type interpolating fields with support in a `comoving' Rindler spacetime. The cyclic form factor equation then arises from the KMS property of the modular operators Delta associated with the field algebras of these Rindler wedges. The derivation in particular shows that the equation holds in any massive 1+1 dim. relativistic QFT, regardless of its integrability. 
  We perform Monte Carlo simulations of a gauge invariant spin system which describes random surfaces with gonihedric action in four dimensions. The Hamiltonian is a mixture of one-plaquette and additional two- and three-plaquette interaction terms with specially adjusted coupling constants. For the system with the large self-intersection coupling constant $k$ we observe the second-order phase transition at temperature $\beta_{c}\simeq 1.75$. The string tension is generated by quantum fluctuations as it was expected theoretically. This result suggests the existence of a noncritical string in four dimensions. For smaller values of $k$ the system undergoes the first order phase transition and for $k$ close to zero exhibits a smooth crossover. 
  In the Chern-Simons gauge theory formulation of the spinning (2+1) dimensional black hole, we may treat the horizon and the spatial infinity as boundaries. We obtain the actions induced on both boundaries, applying the Faddeev and Shatashvili procedure. The action induced on the boundary of the horizon is precisely the gauged $SL(2,R)/U(1)$ Wess-Zumino-Witten (WZW) model, which has been studied previously in connection with a Lorentz signature black hole in (1+1) dimensions. The action induced on the boundary of spatial infinity is also found to be a gauged $SL(2,R)$ WZW model, which is equivalent to the Liouville model, the covariant action for the (1+1) dimensional quantum gravity. Thus, the (2+1) dimensional black hole is intimately related to the quantum gravity in (1+1) dimensions. 
  Four-graviton scattering in eleven-dimensional supergravity is considered at one loop compactified on one, two and three-dimensional tori. The dependence on the toroidal geometry determines the known perturbative and non-perturbative terms in the corresponding processes in type II superstring theories in ten, nine and eight dimensions. The ultra-violet divergence must be regularized so that it has a precisely determined finite value that is consistent both with T-duality in nine dimensions and with eleven-dimensional supersymmetry. 
  Based on the quantum superspace construction of $q$-deformed algebra, we discuss a supersymmetric extension of the deformed Virasoro algebra, which is a subset of the $q$-$W_{\infty}$ algebra recently appeared in the context of two-dimensional string theory. We analyze two types of deformed super-Virasoro algebra as well as their $osp(1,2)$ subalgebras. Applying our quantum superspace structure to conformal field theory, we find the same type of deformation of affine $sl(2)$ algebra. 
  This is a summary of old work on connections between discrete area preserving diffeomorphisms, reduced SU(N) Yang-Mills, strings, and the quantum Hall effect on a Riemann surface of genus g. It is submitted to the archives due to the interest expressed by colleagues who are currently working on matrix models, and who could not have access to the proceedings in which the article was published. The text that follows is the version published in 1991. 
  We apply a new and mathematically rigorous method for the quantization of constrained systems to two-dimensional gauge theories. In this method, which quantizes Marsden-Weinstein symplectic reduction, the inner product on the physical state space is expressed through a certain integral over the gauge group. The present paper, the first of a series, specializes to the Minkowski theory defined on a cylinder. The integral in question is then constructed in terms of the Wiener measure on a loop group. It is shown how $\th$-angles emerge in the new method, and the abstract theory is illustrated in detail in an example. 
  We discuss the existence of de Sitter inflationary solutions for the string-inspired fourth-derivative gravity theories with dilaton field. We consider a space-time of arbitrary dimension D and an arbitrary parametrization of the target space metric. The specific features of the theory in dimension D=4 and those of the special ghost-free parametrization of the metric are found. We also consider similar string-inspired theories with torsion and construct an inflationary solution with torsion and dilaton for D=4. The stability of the inflationary solutions is also investigated. 
  The free non-critical string quantum model is constructed directly in the light-cone variables in the range of dimensions $1<D<25$. The longitudinal degrees of freedom are described by an abstract Verma module. The central charge of this module is restricted by the requirement of the closure of the nonlinear realization of the Poincare algebra. The spin content of the model is analysed. In particular for D=4 the explicit formulae for the character generating functions of the open and closed massive strings are given and the spin spectrum of first 12 excited levels is calculated. It is shown that for the space-time dimension in the range $1<D<25$ the non-critical light-cone string is equivalent to the critical massive string and to the non-critical Nambu-Goto string. 
  Cosmological models that are locally consistent with general relativity and the standard model in which an object transported around the universe undergoes P, C and CP transformations, are constructed. This leads to generalization of the gauge fields that describe electro-weak and strong interactions by enlarging the gauge groups to include anti-unitary transformations. Gedanken experiments show that if all interactions obey Einstein causality then P, C and CP cannot be violated in these models. But another model, which would violate charge superselection rule even for an isolated system, is allowed. It is suggested that the fundamental physical laws must have these discrete symmetries which are broken spontaneously, or they must be non causal. 
  1. The 2-Toda lattice and its generic symmetries   2. A Larger class of symmetries for special initial conditions   3. Borel decomposition of Moment matrices, tau-functions and string-orthogonal polynomials   4. From string-orthogonal Polynomials to the 2-Toda lattice and the string equation   5. Virasoro constraints on two-matrix integrals 
  The statistical mechanics of a mixed gas of adjoint and fundamental representation charges interacting via 1+1-dimensional U(N) gauge fields is investigated. In the limit of large N we show that there is a first order deconfining phase transition for low densities of fundamental charges. As the density of fundamental charges becomes comparable to the adjoint charge density the phase transition becomes a third order one. 
  We calculate the second virial coefficient of anyons whose wave function does not vanish at coincidence points. This kind of anyons appear naturally when one generalizes the hard-core boundary condition according to self-adjoint extension method in quantum mechanics, and also when anyons are treated field theoretically by applying renormalization procedure to nonrelativistic Chern-Simons field theory. For the anyons which do not satisfy hard-core boundary condition, it is argued that the other scale-invariant limit is more relevant in high-temperature limit where virial expansion is useful. Furthermore, the cusp existing at the bosonic point for hard-core anyons disappears in all the other cases; instead it is shown that a new cusp is generated at the fermionic point. A physical explanation is given. 
  Type-IIB supersymmetric theories have an SL(2,Z) invariance, known as U-duality, which controls the non-perturbative behavior of the theory. Under SL(2,Z) the supercharges are doublets, implying that the bosonic charges would be singlets or triplets. However, among the bosonic charges there are doublet strings and doublet fivebranes which are in conflict with the doublet property of the supercharges. It is shown that the conflict is resolved by structure constants that depend on moduli, such as the tau parameter, which transform under the same SL(2,Z). The resulting superalgebra encodes the non-perturbative duality properties of the theory and is valid for any value of the string coupling constant. The usefulness of the formalism is illustrated by applying it to purely algebraic computations of the tension of (p,q) strings, and the mass and entropy of extremal blackholes constructed from D-1-branes and D-5-branes. In the latter case the non-perturbative coupling dependence of the BPS mass and metric is computed for the first time in this paper. It is further argued that the moduli dependence of the superalgebra provides hints for four more dimensions beyond ten, such that the superalgebra is embedded in a fundamental theory which would be covariant under SO(11,3). An outline is given for a matrix theory in 14 dimensions that would be consistent with M(atrix) theory as well as with the above observations. 
  We use the boundary state formalism to study the interaction of two moving identical D-branes in the Type II superstring theory compactified on orbifolds. By computing the velocity dependence of the amplitude in the limit of large separation we can identify the nature of the different forces between the branes. In particular, in the Z_3 orbifold case we find a boundary state which is coupled only to the N=2 graviton multiplet containing just a graviton and a vector like in the extremal Reissner-Nordstr\"{o}m configuration. We also discuss other cases including $T_4/Z_2$. 
  We show that the supersymmetric Wilson loops in IIB matrix model give a transition operator from reduced supersymmetric Yang-Mills theory to supersymmetric space-time theory. In comparison with Green-Schwarz superstring we identify the supersymmetric Wilson loops with the asymptotic states of IIB superstring. It is pointed out that the supersymmetry transformation law of the Wilson loops is the inverse of that for the vertex operators of massless modes in the U(N) open superstring with Dirichlet boundary condition. 
  The non-relativistic Maxwell-Chern-Simons model recently introduced by Manton is shown to admit self-dual vortex solutions with non-zero electric field. The interrelated ``geometric'' and ``hidden'' symmetries are explained. The theory is also extended to (non-relativistic) spinors. A relativistic, self-dual model, whose non-relativistic limit is the Manton model is also presented. The relation to previous work is discussed. 
  In this paper we show how to construct a Dirac operator on a lattice in complete analogy with the continuum. In fact we consider a more general problem, that is, the Dirac operator over an abelian finite group (for which a lattice is a particular example). Our results appear to be in direct connexion with the so called fermion doubling problem. In order to find this Dirac operator we need to introduce an algebraic structure (that generalizes the Clifford algebras) where we have quantities that work as square-root of the translation operator. Quantities like these square-roots have been used recently in order to provide an approach to fermions on the lattice that is free from doubling and has chiral invariance in the massless limit, and our studies seem to give a mathematical basis to it. 
  The usual description of 2+1 dimensional Einstein gravity as a Chern-Simons (CS) theory is extended to a one parameter family of descriptions of 2+1 Einstein gravity. This is done by replacing the Poincare' gauge group symmetry by a q-deformed Poincare' gauge group symmetry, with the former recovered when q-> 1. As a result, we obtain a one parameter family of Hamiltonian formulations for 2+1 gravity. Although formulated in terms of noncommuting dreibeins and spin-connection fields, our expression for the action and our field equations, appropriately ordered, are identical in form to the ordinary ones. Moreover, starting with a properly defined metric tensor, the usual metric theory can be built; the Christoffel symbols and space-time curvature having the usual expressions in terms of the metric tensor, and being represented by c-numbers. In this article, we also couple the theory to particle sources, and find that these sources carry exotic angular momentum. Finally, problems related to the introduction of a cosmological constant are discussed. 
  A previously introduced method to renormalize the one-loop stress tensor and the one-loop vacuum fluctuations in a curved background by a direct local $\zeta$-function approach is checked in some thermal and nonthermal cases. First the method is checked in the case of a conformally coupled massless field in the static Einstein universe where all hypotheses initially requested by the method hold true. Secondly, dropping the hypothesis of a closed manifold, the method is checked in the open static Einstein universe. Finally, the method is checked for a massless scalar field in the presence of a conical singularity in the Euclidean manifold (i.e. Rindler spacetimes/ large mass black hole manifold/cosmic string manifold). In all cases, a complete agreement with other approaches is found. Concerning the last case in particular, the method is proved to give rise to the stress tensor already got by the point-splitting approach for every coupling with the curvature regardless of the presence of the singular curvature. In the last case, comments on the measure employed in the path integral, the use of the optical manifold and the different approaches to renormalize the Hamiltonian are made. 
  We consider a series of duality transformations that leads to a constant shift in the harmonic functions appearing in the description of a configuration of branes. This way, for several intersections of branes, we can relate the original brane configuration which is asymptotically flat to a geometry of the type $adS_k \xx E^l \xx S^m$. The implications of our results for supersymmetry enhancement, M(atrix) theory at finite N, and for supergravity theories in diverse dimensions are discussed. 
  The importance and usefulness of renormalization are emphasized in nonrelativistic quantum mechanics. The momentum space treatment of both two-body bound state and scattering problems involving some potentials singular at the origin exhibits ultraviolet divergence. The use of renormalization techniques in these problems leads to finite converged results for both the exact and perturbative solutions. The renormalization procedure is carried out for the quantum two-body problem in different partial waves for a minimal potential possessing only the threshold behavior and no form factors. The renormalized perturbative and exact solutions for this problem are found to be consistent with each other. The useful role of the renormalization group equations for this problem is also pointed out. 
  We reinterpret N=(2,1) strings as describing the continuum limit of matrix theory with all spatial dimensions compactified. Thus they may characterize the full set of degrees of freedom needed to formulate the theory. 
  The universal hypermultiplet arises as a subsector of every Calabi-Yau compactification of M-theory or Type II string theory. Classically its moduli space is the quaternionic space $SU(2,1)/U(2)$. We show that this moduli space receives a one-loop correction proportional to the Euler character of the Calabi-Yau, which can locally be absorbed by a certain constant shift of the fields. The correction vanishes in the limit that the Planck mass is taken to infinity, and hence is essentially gravitational in nature. 
  We study the effective actions of various brane configurations in Matrix theory. Starting from the 0+1 dimensional quantum mechanics, we replace coordinate matrices by covariant derivatives in the large N limit, thereby obtaining effective field theories on the brane world volumes. Even for noncompact branes, these effective theories are of Yang-Mills type, with constant background magnetic fields. In the case of a D2-brane, we show explicitly how the effective action equals the large magnetic field limit of the Born-Infeld action, and thus derive from Matrix theory the action used by Polchinski and Pouliot to compute M-momentum transfer between membranes. We also consider the effect of compactifying transverse directions. Finally, we analyze a scattering process involving a recently proposed background representing a classically stable D6+D0 brane configuration. We compute the potential between this configuration and a D0-brane, and show that the result agrees with supergravity. 
  BPS saturated p-branes play an important role in recent progress in understanding superstring theory and M theory. One approach to understanding the dynamics of p-branes is to formulate an effective (p+1)-dimensional world-volume theory. The construction of such brane actions involves a number of interesting issues. One such issue is how to formulate the action for theories that contain chiral bosons. The two main examples, which are the M theory five-brane and the heterotic string, are described in this lecture. Also, double dimensional reduction of the M theory five-brane on K3 is shown to give the heterotic string. 
  It is shown that the scattering amplitude for contact-interacting anyons does exhibit a genuine non-perturbative sector. This means that the corresponding perturbative field theoretical formulation, based on the {\it 2+1} non-relativistic Chern-Simons gauge model coupled to self-interacting complex scalar field, is not generally able to reproduce, order by order in perturbation theory, the exact result. It is proven that the full agreement between the exact scattering amplitude and the resummation of the perturbative expansion of the renormalized 1PI amplitude actually occurs only for some continuous sub-family of self-adjoint extensions of the quantum Hamiltonians, which entail the absence of bound states. A comparison with previously obtained results is carefully worked out. 
  Poisson-Lie T-duality in N=2 superconformal WZNW models on the real Lie groups is considered. It is shown that Poisson-Lie T-duality is governed by the complexifications of the corresponding real groups endowed with Semenov-Tian-Shansky symplectic forms, i.e. Heisenberg doubles. Complex Heisenberg doubles are used to define on the group manifolds of the N=2 superconformal WZNW models the natural actions of the isotropic complex subgroups forming the doubles. It is proved that with respect to these actions N=2 superconformal WZNW models admit Poisson-Lie symmetries. Poisson-Lie T-duality transformation maps each model into itself but acts nontrivialy on the space of classical solutions. 
  The mathematical apparatus of non commutative geometry and operator algebras which Connes has brought to bear to construct a rational scheme for the internal symmetries of the standard model is presented from the physicist's point of view. Gauge symmetry, anomaly freedom, conservation of electric charge, parity violation and charge conjugation all play a vital role. When put together with a relatively simple set of algebraic algorithms they deliver many of the features of the standard model which otherwise seem rather ad hoc. 
  A magnetic field, coherent over the horizon size at the decoupling and strong enough to rotate the polarization plane of the CMBR, can be generated from the electromagnetic vacuum fluctuations amplified by the space-time evolution of the dilaton coupling. The possible relevance of this result for superstring inspired cosmological models is discussed. Particular attention will be paid to the connection between Faraday rotation signals and stochastic gravity-wave backgrounds. 
  In the generalized Hamiltonian formalism by Dirac, the method of constructing the generator of local-symmetry transformations for systems with first- and second-class constraints (without restrictions on the algebra of constraints) is obtained from the requirement for them to map the solutions of the Hamiltonian equations of motion into the solutions of the same equations. It is proved that second-class constraints do not contribute to the transformation law of the local symmetry entirely stipulated by all the first-class constraints (and only by them). A mechanism of occurrence of higher derivatives of coordinates and group parameters in the symmetry transformation law in the Noether second theorem is elucidated. It is shown that the obtained transformations of symmetry are canonical in the extended (by Ostrogradsky) phase space. An application of the method in theories with higher derivatives is demonstrated with an example of the spinor Christ -- Lee model. 
  The semi-classical spectrum of the Homogeneous sine-Gordon theories associated with an arbitrary compact simple Lie group G is obtained and shown to be entirely given by solitons. These theories describe quantum integrable massive perturbations of Gepner's G-parafermions whose classical equations-of-motion are non-abelian affine Toda equations. One-soliton solutions are constructed by embeddings of the SU(2) complex sine-Gordon soliton in the regular SU(2) subgroups of G. The resulting spectrum exhibits both stable and unstable particles, which is a peculiar feature shared with the spectrum of monopoles and dyons in N=2 and N=4 supersymmetric gauge theories. 
  We derive the low energy absorption cross section of minimal scalars for a very generic class of geometries, that includes, among others, black holes, strings, $p$-branes, as well as intersecting brane configurations. The scalar field can be absorbed across a regular surface (a horizon) of finite area, or across a zero area throat, which may be singular or regular. Then we focus on some particular cases and compare with the results obtained using microscopic models at weak coupling. Exact agreement is found for the absorption by the non-degenerate ground state of effective strings. For geometries where the throat is singular, we show that absorption through the stretched horizon yields correct results for degenerate states. For non-degenerate (ground) states we introduce a stretched throat, and show that it accounts for the correct dependence on frequency, charges, and moduli for the absorption by ground states of BPS fundamental strings. It is also shown to lead to the correct dependence on frequency for extremal D-branes. 
  Continuing the thrust of our recent work, but with an important new idea, we find a cut-off regularization of the determinant of a scalar particle in a classical Euclidean gravitational field. The field is assumed asymptotically flat, and the regularization is diffeomorphism-invariant under coordinate changes that are the identity at infinity. The scalar field is expanded in term of variables that depend on the gravitational field, with wavelet localization of each variable. A renormalization group structure is thus automatically present. A similar construction is carried out for the determinant of a scalar field in a background Yang-Mills field. 
  By imposing self-duality conditions, we obtain the explicit form in which gauge theories spontaneously breakdown in the Bogomol'nyi limit. In this context, we reconsider the Abelian Higgs, Chern-Simons Higgs and Maxwell-Chern-Simons Higgs models. On the same footing, we find a topological Higgs potential for a Maxwell-Chern-Simons extended theory presenting both minimal and nonminimal coupling. Finally, we perform a numerical calculation in the asymmetric phase and show the solutions to the self-dual equations of motion of the topological theory. We argue about certain relations among the parameters of the model in order to obtain such vortex configurations. 
  We consider Shiraishi's metrics on the moduli space of extreme black holes. We interpret the simplification in the pattern of N-body interactions that he observed in terms of the recent picture of black holes in four and five dimensions as composites, made up of intersecting branes. We then show that the geometry of the moduli space of a class of black holes in five and nine dimensions is hyper-K\"ahler with torsion, and octonionic-K\"ahler with torsion, respectively. For this, we examine the geometry of point particle models with extended world-line supersymmetry and show that both of the above geometries arise naturally in this context. In addition, we construct a large class of hyper-K\"ahler with torsion and octonionic-K\"ahler with torsion geometries in various dimensions. We also present a brane interpretation of our results. 
  We reformulate the Born-Infeld action, coupled to an axion and a dilaton in a duality manifest way. This action is the generalization of the Schwarz-Sen action for non-linear electrodynamics. We show that this action may be obtained by dimensional reduction on a torus of a self-dual theory in 6 dimensions. The dilaton-axion being identified with the complex structure of the torus. Applications to M-theory and the self-dual IIB three brane are investigated. 
  Scale invariance provides a principled reason for the physical importance of Hilbert space, the Virasoro algebra, the string mode expansion, canonical commutators and Schroedinger evolution of states, independent of the assumptions of string theory and quantum theory. The usual properties of dimensionful fields imply an infinite, projective tower of conformal weights associated with the tangent space to scale-invariant spacetimes. Convergence and measurability on this tangent tower are guaranteed using a scale-invariant norm, restricted to conformally self-dual vectors. Maps on the resulting Hilbert space are correspondingly restricted to semi-definite conformal weight. We find the maximally- and minimally-commuting, complete Lie algebras of definite-weight operators. The projective symmetry of the tower gives these algebras central charges, giving the canonical commutator and quantum Virasoro algebras, respectively. Using a continuous, m-parameter representation for rank-m tower tensors, we show that the parallel transport equation for the momentum vector of a particle is the Schroedinger equation, while the associated definite-weight operators obey canonical commutation relations. Generalizing to the set of integral curves of general timelike, self-dual vector-valued weight maps gives a lifting such that the action of the curves parallel transports arbitrary tower vectors. We prove that the full set of Schroedinger-lifted integral curves of a general self-dual map gives an immersion of its 2-dim parameter space into spacetime, inducing a Lorentzian metric on the parameter space. The immersion is shown to satisfy the variational equations of open string. 
  This is a general introduction to duality in field theories. The existence and breaking of global symmetries is used as a guideline to systematically prove duality between different field theories. Systems discussed include abelian and non-abelian T-duality in string theory, abelian and nonabelian bosonization, and duality for massless and massive antisymmetric tensor field theories in arbitrary number of dimensions. Open questions regarding these techniques are also discussed. (Lectures given at 33rd Karpacz Winter School `Duality: Strings and Fields' .) 
  We present a review of heterotic-type I string duality. In particular, we discuss the effective field theory of six- and four-dimensional compactifications with N>1 supersymmetries. We then describe various duality tests by comparing gauge couplings, N=2 prepotentials, as well as higher-derivative F-terms. Based on invited lectures delivered at: 33rd Karpacz Winter School of Theoretical Physics ``Duality, Strings and Fields,'' Przesieka, Poland, 13 - 22 February 1997; Trieste Conference on Duality Symmetries in String Theory, Trieste, Italy, 1 - 4 April 1997; Cargese Summer School ``Strings, Branes and Dualities,'' Cargese, France, 26 May - 14 June 1997. 
  We outline an ultraviolet renormalization procedure for hamiltonians acting in the light-front Fock space. The hamiltonians are defined and calculated using creation and annihilation operators with no limitation of the space of states. Both, the regularization of the initial hamiltonian and the definition of the renormalized effective hamiltonians, preserve the light-front frame kinematical symmetries. The general equations for the effective hamiltonians are illustrated by second order calculations of the self-energy and two-particle interaction terms in Yukawa theory, QED and QCD. Infrared singularities are regulated but not renormalized. 
  We study functional analytic aspects of two types of correction terms to the Heisenberg algebra. One type is known to induce a finite lower bound $\Delta x_0$ to the resolution of distances, a short distance cutoff which is motivated from string theory and quantum gravity. It implies the existence of families of self-adjoint extensions of the position operators with lattices of eigenvalues. These lattices, which form representations of certain unitary groups cannot be resolved on the given geometry. This leads us to conjecture that, within this framework, degrees of freedom that correspond to structure smaller than the resolvable (Planck) scale turn into internal degrees of freedom with these unitary groups as symmetries. The second type of correction terms is related to the previous essentially by "Wick rotation", and its basics are here considered for the first time. In particular, we investigate unitarily inequivalent representations. 
  A new 8-dim conformal gauging solves the auxiliary field problem and eliminates unphysical size change from Weyl's electromagnetic theory. We derive the Maurer-Cartan structure equations and find the zero curvature solutions for the conformal connection. By showing that every one-particle Hamiltonian generates the structure equations we establish a correspondence between phase space and the 8-dim base space, and between the action and the integral of the Weyl vector. Applying the correspondence to generic flat solutions yields the Lorentz force law, the form and gauge dependence of the electromagnetic vector potential and minimal coupling. The dynamics found for these flat solutions applies locally in generic spaces. We then provide necessary and sufficient curvature constraints for general curved 8-dimensional geometries to be in 1-1 correspondence with 4-dimensional Einstein-Maxwell spacetimes, based on a vector space isomorphism between the extra four dimensions and the Riemannian tangent space. Despite part of the Weyl vector serving as the electromagnetic vector potential, the entire class of geometries has vanishing dilation, thereby providing a consistent unified geometric theory of gravitation and electromagnetism. In concluding, we discuss the observability of the extra dimensions. 
  A new 8-dimensional conformal gauging avoids the unphysical size change, third order gravitational field equations, and auxiliary fields that prevent taking the conformal group as a fundamental symmetry. We give the structure equations, gauge transformations and intrinsic metric structure for the new biconformal spaces. We prove that a torsion-free biconformal space with exact Weyl form, closed dilational curvature and trace-free spacetime curvature admits a sub-bundle of vanishing Weyl form homeomorphic to the Whitney sum bundle of the tangent bundle and the bundle of orthonormal Lorentz frames over 4-dimensional spacetime. Conversely, any 4-dimensional spacetime extends uniquely to such a normal biconformal space. The Einstein equation holds if and only if the biconformal basis is orthonormal. Unconstrained antisymmetric trace of the spacetime curvature provides a closed 2-form, independent of the Weyl vector, consistently interpretable as the electromagnetic field. The trace of the spacetime co-torsion decouples from gravitational sources and serves as electromagnetic source. 
  A two-form formulation for the N=2 vector-tensor multiplet is constructed using superfield methods in central charge superspace. The N=2 non-Abelian standard supergauge multiplet in central charge superspace is also discussed, as is with the associated Chern-Simons form. We give the constraints, solve the Bianchi identities and present the action for a theory of the vector-tensor multiplet coupled to the non-Abelian supergauge multiplet via the Chern-Simons form. 
  We study the domain walls connecting different chirally asymmetric vacua in supersymmetric QCD. We show that BPS - saturated solutions exist only in the limited range of mass. When m exceeds some critical value, the domain wall either ceases to be BPS - saturated or disappears altogether. In any case, the properties of the system are qualitatively changed. 
  We consider a general N=(2,2) non-linear sigma-model in (2,2) superspace. Depending on the details of the complex structures involved, an off-shell description can be given in terms of chiral, twisted chiral and semi-chiral superfields. Using superspace techniques, we derive the conditions the potential has to satisfy in order to be ultra-violet finite at one loop. We pay particular attention to the effects due to the presence of semi-chiral superfields. A complete description of N=(2,2) strings follows from this. 
  In N=1 supersymmetric theories, quasi Nambu-Goldstone (QNG) bosons appear in addition to ordinary Nambu-Goldstone (NG) bosons when the global symmetry G breaks down spontaneously. We investigate two-body scattering amplitudes of these bosons in the low-energy effective Lagrangian formalism. They are expressed by the curvature of Kahler manifold. The scattering amplitudes of QNG bosons are shown to coincide with those of NG bosons though the effective Lagrangian contains an arbitrary function, and those with odd number of QNG bosons all vanish. 
  We quantize massive vector theory in such a way that it has a well-defined massless limit. In contrast to the approach by St\"uckelberg where ghost fields are introduced to maintain manifest Lorentz covariance, we use reduced phase space quantization with nonlocal dynamical variables which in the massless limit smoothly turn into the photons, and check explicitly that the Poincare algebra is fullfilled. In contrast to conventional covariant quantization our approach leads to a propagator which has no singularity in the massless limit and is well behaved for large momenta. For massive QED, where the vector field is coupled to a conserved fermion current, the quantum theory of the nonlocal vector fields is shown to be equivalent to that of the standard local vector fields. An inequivalent theory, however, is obtained when the reduced nonlocal massive vector field is coupled to a nonconserved classical current. 
  We generalize previous work on inhomogeneous pre-big bang cosmology by including the effect of non-trivial moduli and antisymmetric-tensor/axion fields. The general quasi-homogeneous asymptotic solution---as one approaches the big bang singularity from perturbative initial data---is given and its range of validity is discussed, allowing us to give a general quantitative estimate of the amount of inflation obtained during the perturbative pre-big bang era. The question of determining early-time ``attractors'' for generic pre-big bang cosmologies is also addressed, and a motivated conjecture is advanced. We also discuss S-duality-related features of the solutions, and speculate on the way an asymptotic T-duality symmetry may act on moduli space as one approaches the big bang. 
  We examine the weak coupling limit of Euclidean SU(n) gauge theory in covariant gauges. Following an earlier suggestion, an equivariant BRST-construction is used to define the continuum theory on a finite torus. The equivariant gauge fixing introduces constant ghost fields as moduli of the model. We study the parameter- and moduli- space perturbatively. For $n_f \leq n$ quark flavors, the moduli flow to a non-trivial fixed point in certain critical covariant gauges and the one-loop effective potential indicates that the global SU(n) color symmetry of the gauge fixed model is spontaneously broken to $U(1)^{n-1}$. Ward identities and renormalization group arguments imply that the longitudinal gauge boson propagator at long range is dominated by $n(n-1)$ Goldstone bosons in these critical covariant gauges. In the large $n$ limit, we derive a nonlinear integral equation for the expectation value of large Wilson loops assuming that the exchange of Goldstone bosons dominates the interaction at long range in critical covariant gauges. We find numerically that the expectation value of large circular Wilson loops decreases exponentially with the enclosed area in the absence of dynamical fermions. The gauge invariance of this mechanism for confinement in critical covariant gauges is discussed. 
  We explore the quantum properties of self-dual gravity formulated as a large $N$ two-dimensional WZW sigma model. Using a non-trivial classical background, we show that a $(2,2)$ space-time is generated. The theory contains an infinite series of higher point vertices. At tree level we show that, in spite of the presence of higher than cubic vertices, the on-shell 4 and higher point functions vanish, indicating that this model is related with the field theory of closed N=2 strings. We examine the one-loop on-shell 3-point amplitude and show that it is ultra-violet finite. 
  We derive, in path integral approach, the (anomalous) master Ward identity associated with an infinite set of nonlocal conservation laws in two-dimensional principal chiral models 
  Euclidean supersymmetric theories are obtained from Minkowskian theories by performing a reduction in the time direction. This procedure elucidates certain mysterious features of Zumino's N=2 model in four dimensions, provides manifestly hermitian Euclidean counterparts of all non-mimimal SYM theories, and is also applicable to supergravity theories. We reanalyse the twists of the 4d N=2 and N=4 models from this point of view. Other applications include SYM theories on special holonomy manifolds. In particular, we construct a twisted SYM theory on Kaehler 3-folds and clarify the structure of SYM theory on hyper-Kaehler 4-folds. 
  We show that the F-theory dual of the heterotic string with unbroken Spin(32)/Z_2 symmetry in eight dimensions can be described in terms of the same polyhedron that can also encode unbroken E_8\times E_8 symmetry. By considering particular compactifications with this K3 surface as a fiber, we can reproduce the recently found `record gauge group' in six dimensions and obtain a new `record gauge group' in four dimensions. Our observations relate to the toric diagram for the intersection of components of degenerate fibers and our definition of these objects, which we call `tops', is more general than an earlier definition by Candelas and Font. 
  We study the effect of the solitonic degree of freedom in string cosmology following the line of Rama. The gas of solitonic p-brane is treated as a perfect fluid in a Brans-Dicke type theory.  In this paper, we find analytic solutions for a few solvable cases of equation of states with general Brans-Dicke parameter $\omega$ and study the cosmology of the solutions. In some cases the solutions are free from initial singularity. 
  The purpose of this talk is to review some considerations by the present author on the possible role of a simple space-time uncertainty relation toward nonperturbative string theory. We first motivate the space-time uncertainty relation as a simple space-time characterization of the fundamental string theory. We then argue that the relation captures some of the important aspects of the short-distance dynamics of D-particles described by the effective super Yang-Mills matrix quantum mechanics, and also that the recently proposed type IIB matrix model can be regarded as a possible realization of the space-time uncertainty principle. 
  We study the effect of the solitonic degrees of freedom in string cosmology following the line of Rama. The gas of solitonic p-brane is treated as a perfect fluid in a Brans-Dicke type theory. In this paper, we find exact cosmological solutions for any Brans-Dicke parameter $\omega$ and for general parameter $\gamma$ of equation of state and classify the cosmology of the solutions on a parameter space of $\gamma$ and $\omega$. 
  Dynamics has been generalized to a noncommutative phase space. The noncommuting phase space is taken to be invariant under the quantum group $GL_{q,p}(2)$. The $q$-deformed differential calculus on the phase space is formulated and using this, both the Hamiltonian and Lagrangian forms of dynamics have been constructed. In contrast to earlier forms of $q$-dynamics, our formalism has the advantage of preserving the conventional symmetries such as rotational or Lorentz invariance. 
  Several dynamical aspects of the SU(2) Seiberg-Witten models with N_f quark hypermultiplets are explored. We first clarify the meaning of the number of the singularities of the space of vacua. CP invariance of the theories are then studied and periodicities of theories in \theta with and without bare quark masses are obtained ((4-N_f)\pi and \pi, respectively). CP noninvariance at a generic point of QMS manifests itself as the electric and quark-number charge fractionalizations for the dyons; we show that the exact Seiberg-Witten solution contains such effects correctly, in agreement with the semiclassical analysis recently made by F.Ferrari. Upon N=1 perturbation the low energy effective theories at the singularities display confinement, and in most cases chiral symmetry breaking as a consequence. In one of the vacua for N_f=3 confinement is not accompanied by chiral symmetry breaking: we interpret it as an example of oblique confinement of 't Hooft. We discuss further the consistency of the physical picture found here by studying the effects of soft supersymmetry breaking as well as the behavior of the theory in the N=1 limit. 
  There are known problems of Lorentz-Dirac equation for moving with acceleration charged particle in classical electrodynamics. The model of extended in one dimension particle is proposed and shown that electromagnetic self-interaction can lead (with appropriate choice of retarded and advanced interactions) to zero change in particle momentum. The hypothesis is formulated: all relativistic internal forces of various nature can give zero change in particle momentum 
  A "reduced" differential geometry adapted to the presence of abelian isometries is constructed.Classical T-duality diagonalizes in this setting, allowing us to get conveniently the transformation of the relevant geometrical objects such as connections, pullbacks and generalized curvatures.Moreover we can induce privileged maps from the viewpoint of the covariant derivatives in the target-space and in the world-sheet generalizing previous results, at the same time that we can correct connections and curvatures covariantly in order to have a proper transformation under T-duality. 
  We explicitly calculate one-loop divergences for an arbitrary field theory model using the higher derivative regularization and nonsingular gauge condition. They are shown to agree with the results found in the dimensional regularization and do not depend on the form of regularizing term. So, the consistency of the higher derivative regularization is proven at the one-loop level. The result for the Yang-Mills theory is reproduced. 
  We review classical BPS monopoles, their moduli spaces, twistor descriptions and dynamics. Particular emphasis is placed upon symmetric monopoles, where recent progress has been made. Some remarks on the role of monopoles in S-duality and Seiberg-Witten theory are also made. 
  We comment on the recent paper by Balog and Niedermaier [hep-th/9701156]. 
  We find an eleven dimensional description of the D-2-brane of the massive type IIA theory as a first step towards an understanding of this theory in eleven dimensions. By means of a world-volume IIA/M theory duality transformation we show that the massive D-2-brane is equivalent to the dimensional reduction of the eleven dimensional membrane coupled to an auxiliary vector field. The role of this vector field is to preserve the invariance under massive gauge transformations in the world-volume and has non-trivial dynamics, governed by a Chern-Simons term proportional to $1/m$. 
  We investigate a (0,2) gauge theory realized on the world volume of the type IIB D1-brane at the singular point of a Calabi-Yau fourfold. It is argued that the gauge anomaly can be canceled via coupling to the R-R chiral bosons in bulk IIB string. We find that for a generic choice of the Fayet-Iliopoulos parameters on the world volume, the Higgs moduli space is a smooth fourfold birational to the original Calabi-Yau fourfold, but is not necessarily a Calabi-Yau manifold. 
  We discuss several implications of R^4 couplings in M theory when compactified on Calabi-Yau (CY) manifolds. In particular, these couplings can be predicted by supersymmetry from the mixed gauge-gravitational Chern-Simons couplings in five dimensions and are related to the one-loop holomorphic anomaly in four-dimensional N=2 theories. We find a new contribution to the Einstein term in five dimensions proportional to the Euler number of the internal CY threefold, which corresponds to a one-loop correction of the hypermultiplet geometry. This correction is reproduced by a direct computation in type II string theories. Finally, we discuss a universal non-perturbative correction to the type IIB hyper-metric. 
  If one begins with the assertion that the type IIA string compactified on a K3 surface is equivalent to the heterotic string on a four-torus one may try to find a statement about duality in ten dimensions by decompactifying the four-torus. Such a decompactification renders the K3 surface highly singular. The resultant K3 surface may be analyzed in two quite different ways - one of which is natural from the point of view of differential geometry and the other from the point of view of algebraic geometry. We see how the former leads to a "squashed K3 surface" and reproduces the Horava-Witten picture of the heterotic string in M-theory. The latter produces a "stable degeneration" and is tied more closely to F-theory. We use the relationship between these degenerations to obtain the M-theory picture of a point-like E8-instanton directly from the F-theory picture of the same object. 
  We suggest that the $\phi^4$ model is only a polynomial approximation to a more fundamental theory. As a consequence the high temperature regime might not be correctly described by this model. If this turns out to be true then several results concerning e.g., critical temperatures, symmetry restoration at high temperature and high temperature expansions should be reconsidered. We illustrate our conjecture by using the Nambu-Goto string model. We compare a two-loop calculation of the free energy or quark-antiquark static potential at finite temperature with a previous exact calculation in the large-d limit and show how the perturbative expansion fails to reproduce important features in the neighborhood of the critical temperature. It becomes clear why this happens in the Nambu-Goto model and we suggest that perhaps something similar occurs with the $\phi^4$ model. 
  We study a variety of supersymmetric systems describing sixth-order interactions between two coupled real superfields in 2+1 dimensions. We search for BPS domain ribbon solutions describing minimum energy static field configurations that break one half of the supersymmetries. We then use the supersymmetric system to investigate the behavior of mesons and fermions in the background of the defects. In particular, we show that certain BPS domain ribbons admit internal structure in the form of bosonic kinks and fermionic condensate, for a given range of the two parameters that completely identify the class of systems. 
  We analyze the vacuum structure of N=2, SU(2) QCD with massive quark hypermultiplets, once supersymmetry is softly broken down to N=0 with dilaton and mass spurions. We give general expressions for the low energy couplings of the effective potential in terms of elliptic functions to have a complete numerical control of the model. We study in detail the possible phases of the theories with Nf = 1, 2 flavors for different values of the bare quark masses and the supersymmetry breaking parameters and we find a rich structure of first order phase transitions. The chiral symmetry breaking pattern of the Nf = 2 theory is considered, and we obtain the pion Lagrangian for this model up to two derivatives. Exact expressions are given for the pion masses and the pion decay constant in terms of the magnetic monopole description of chiral symmetry breaking. 
  We obtain the exact non-perturbative thresholds of $R^4$ terms in IIB string theory compactified to eight and seven dimensions. These thresholds are given by the perturbative tree-level and one-loop results together with the contribution of the D-instantons and of the $(p,q)$-string instantons. The invariance under U-duality is made manifest by rewriting the sum as a non-holomorphic modular function of the corresponding discrete U-duality group. In the eight-dimensional case, the threshold is the sum of a order-1 Eisenstein series for $SL(2,Z)$ and a order-3/2 Eisenstein series for $SL(3,Z)$. The seven-dimensional result is given by the order-3/2 Eisenstein series for $SL(5,Z)$. We also conjecture formulae for the non-perturbative thresholds in lower dimensional compactifications and discuss the relation with M-theory. 
  Certain limits of the duality between M-theory on ${T^5/Z_2}$ and IIB on K3 are analyzed in Matrix theory. The correspondence between M-theory five-branes and ALE backgrounds is realized as three dimensional mirror symmetry. Non-critical strings dual to open membranes are explicitly described as gauge theory excitations. We also comment on Type IIA on K3 and the appearance of gauge symmetry enhancement at special points in the moduli space. 
  This paper starts with a personal memoir of how some significant ideas arose and events took place during the period from 1972, when I first encountered Ted Bastin, to 1979, when I proposed the foundation of ANPA. I then discuss program universe, the fine structure paper and its rejection, the quantitative results up to ANPA 17 and take a new look at the handy-dandy formula. Following this historical material is a first pass at establishing new foundations for bit-string physics. An abstract model for a laboratory notebook and an historical record are developed, culminating in the bit-string representation. I set up a tic-toc laboratory with two synchronized clocks and show how this can be used to analyze arbitrary incoming data. This allows me to discuss (briefly) finite and discrete Lorentz transformations, commutation relations, and scattering theory. Earlier work on conservation laws in 3- and 4-events and the free space Dirac and Maxwell equations is cited. The paper concludes with a discussion of the quantum gravity problem from our point of view and speculations about how a bit-string theory of strong, electromagnetic, weak and gravitational unification could take shape. 
  The moduli space metric for an arbitrary number of extremal D=5 black holes with arbitrary relatively supersymmetric charges is found. 
  We show how to evaluate the periods in Seiberg-Witten theories and in K3-fibered Calabi-Yau manifolds by using fibrations of the theories. In the Seiberg-Witten theories, it is shown that the dual pair of fields can be constructed from the classical fields in a simple form. As for Calabi-Yau manifolds which are fibrations of K3 surface, we obtain the solutions of the Picard-Fuchs equations from the periods of K3 surface. By utilizing the expression of periods for two-parameter models of type-II string, we derive the solutions of the Picard-Fuchs equations around the points of enhanced gauge symmetry and show a simple connection to the SU(2) Seiberg-Witten theory. 
  A local world volume Q-supersymmetric Weyl invariant Lagrangian for the membrane is presented. An analysis is provided which solves the problems raised by some authors in the past concerning the algebraic elimination of the auxiliary fields belonging to the coupling function supermultiplet. The starting bosonic action is the one given by Dolan and Tchrakian with vanishing cosmological constant and with quadratic, quartic derivative terms. Our Lagrangian differs from the one of Lindstrom and Rocek in the fact that is polynomial in the fields facilitating the quantization process. It is argued, rigorously, that if one wishes to construct polynomial actions without curvature terms and where supersymmetry is linearly realized, after the elimination of auxiliary fields, one must relinquish S supersymmetry and concentrate solely on the Q-supersymmetry associated with the superconformal algebra in three dimensions. The role that this spinning membrane action may have in the theory of D-branes, Skyrmions and BPS monopoles is also pointed out. 
  The propagator of the spinless Aharonov-Bohm-Coulomb system is derived by following the Duru-Kleinert method. We use this propagator to explore the spin-1/2 Aharonov-Bohm-Coulomb system which contains a point interaction as a Zeeman term. Incorporation of the self-adjoint extension method into the Green's function formalism properly allows us to derive the finite propagator of the spin-1/2 Aharonov-Bohm-Coulomb system. As a by-product, the relation between the self-adjoint extension parameter and the bare coupling constant is obtained. Bound-state energy spectra of both spinless and spin-1/2 Aharonov-Bohm-Coulomb systems are examined. 
  Weyl Anomaly in the dilaton-scalar system in 2 dimensional gravity is examined. We take the heat-kernel regularization for the ultraviolet divergences. Generally the Weyl anomaly is determined by the 2nd order differential (elliptic) operator of the system and the definition of the measure. We have the freedom of the operator choice caused by the arbitrariness of total divergences (surface terms) in the action. We examine the Weyl anomaly in connection with such points and the hermiticity of the operator. 
  It is shown that only the infinite angular momentum quantum states contribute to the incident wave in Aharonov-Bohm (AB) scattering. This result is clearly shown by recalculating the AB calculation with arbitrary decomposition of summation over the angular momentum quantum numbers in wave function. It is motivated from the fact that the pole contribution in the integral representation used by Jackiw is given by only the infinite angular momentum states, in which the closed contour integration involving this pole gives just the incident wave. 
  We study N=1 dualities in four dimensional supersymmetric gauge theories as the worldvolume theory of D4 branes with one compact direction in type IIA string theory. We generalize the previous work for SO(N_{c1}) x Sp(N_{c2}) with the superpotential W=Tr X^4 to the case of W= Tr X^4(k+1) in terms of brane configuration. We conjecture that the new dualities for the product gauge groups of SO(N_{c1}) x Sp(N_{c2}) x SO(N_{c3}), SO(N_{c1}) x Sp(N_{c2}) x SO(N_{c3}) x Sp(N_{c4}) and higher multiple product gauge groups can be obtained by reversing the ordering of NS5 branes and D6 branes while preserving the linking numbers. We also describe the above dualities in terms of wrapping D6 branes around 3 cycles of Calabi-Yau threefolds in type IIA string theory. The theory with adjoint matter can be regarded as taking multiple copies of NS5 brane in the configuration of brane or geometric approaches. 
  We study a model of asymptotically free theories with bound states using the similarity renormalization group for hamiltonians. We find that the renormalized effective hamiltonians can be approximated in a large range of widths by introducing similarity factors and the running coupling constant. This approximation loses accuracy for the small widths on the order of the bound state energy and it is improved by using the expansion in powers of the running coupling constant. The coupling constant for small widths is order 1. The small width effective hamiltonian is projected on a small subset of the effective basis states. The resulting small matrix is diagonalized and the exact bound state energy is obtained with accuracy of the order of 10% using the first three terms in the expansion. We briefly describe options for improving the accuracy. 
  We apply the principles discussed in earlier papers to the construction of discrete time quantum field theories. We use the Schwinger action principle to find the discrete time free field commutators for scalar fields, which allows us to set up the reduction formalism for discrete time scattering processes. Then we derive the discrete time analogue of the Feynman rules for a scalar field with a cubic self interaction and give examples of discrete time scattering amplitude calculations. We find overall conservation of total linear momentum and overall conservation of total theta parameters, which is the discrete time analogue of energy conservation and corresponds to the existence of a Logan invariant for the system. We find that temporal discretisation leads to softened vertex factors, modifies propagators and gives a natural cutoff for physical particle momenta. 
  We apply the principles of discrete time mechanics discussed in earlier papers to the first and second quantised Dirac equation. We use the Schwinger action principle to find the anticommutation relations of the Dirac field and of the particle creation operators in the theory. We find new solutions to the discrete time Dirac equation, referred to as oscillons on account of their extraordinary behaviour. Their principal characteristic is that they oscillate with a period twice that of the fundamental time interval T of our theory. Although these solutions can be associated with definite charge, linear momentum and spin, such objects should not be observable as particles in the continuous time limit. We find that for non-zero T they correspond to states with negative squared norm in Hilbert space. However they are an integral part of the discrete time Dirac field and should play a role in particle interactions analogous to the role of longitudinal photons in conventional quantum electrodynamics. 
  We provide further evidence for the screening behavior of massless SU(N_c) bosonized QCD by (i) computing the potential between external quarks, (ii) bosonizing also the external sources and analyzing the states of the combined system and (iii) using an expansion in N_f- the number of flavors. We write down novel "non-abelian Schwinger like" solutions of the equations of motion, compute their masses and argue that an exchange of massive modes of this type is associated with the screening mechanism. Confinement for massive dynamical fermions is shown using (ii) and (iii). We show the screening behavior of the N=1 super Yang Mills theory, by applying a point splitting method to the scalar current. 
  Jacobi brackets (a generalization of standard Poisson brackets in which Leibniz's rule is replaced by a weaker condition) are extended to brackets involving an arbitrary (even) number of functions. This new structure includes, as a particular case, the recently introduced generalized Poisson structures. The linear case on simple group manifolds is also studied and non-trivial examples (different from those coming from generalized Poisson structures) of this new construction are found by using the cohomology ring of the given group. 
  We develop a new approach for bosonization based on the direct comparison of current correlation functions and apply it to the case of the Massive Thirring Model in three dimensions in the weak coupling regime, but with an arbitrary mass. Explicit bosonized forms for the lagrangian and the current are obtained in terms of a vector gauge field. Exact results for the corresponding expressions are also obtained in the case of a free massive fermion. Finally, a comment on the derivation of the current algebra directly from the bosonized expressions is included. 
  The aim of this lecture is to present the concept of C-algebra and to illustrate its applications in two contexts: the study of reflection groups and their folding on the one hand, the structure of rational conformal field theories on the other. For simplicity the discussion is restricted to finite Coxeter groups and conformal theories with a $\hat{sl}(2)$ current algebra, but it may be extended to a larger class of groups and theories associated with $\hat{sl}(N)$. (Proceedings of the RIMS Symposium, Kyoto, 16-19 December 1996.) 
  We develop a systematic method of the perturbative expansion around the Gaussian effective action based on the background field method. We show, by applying the method to the quantum mechanical anharmonic oscillator problem, that even the first non-trivial correction terms greatly improve the Gaussian approximation. 
  The 't Hooft anomaly matching conditions are a standard tool to study and test non-perturbative issues in quantum field theory. We give a new, simple proof of the anomaly matching conditions in 2D Poincare` invariant theories. We consider the case of invariance under a large class of generalized symmetries, which include abelian and non-abelian internal symmetries, space-time symmetries generated by the stress tensor, and W-type of symmetries generated by higher spin currents. 
  Following recent work by Lambiase and Nesterenko we study in detail the interquark potential for a Nambu-Goto string with point masses attached to its ends. We obtain exact solutions to the gap equations for the Lagrange multipliers and metric components and determine the potential without simplifying assumptions. We also discuss L\"{u}scher term and argue that it remains universal. 
  We present finite energy analytic monopole and dyon solutions whose size is fixed by the electroweak scale. The new solutions are obtained by regularizing the recent Cho-Maison solutions of the Weinberg-Salam model. Our result shows that genuine electroweak monopole and dyon could exist whose mass scale is much smaller than the grand unification scale. 
  The spherically symmetric solution in classical SU(3) Yang - Mills theory is found. It is supposed that such solution describes a classical quark. It is regular in origin and hence the interaction between two quarks is small on the small distance. The obtained solution has the singularity on infinity. It is possible that is the reason why the free quark cannot exist. Evidently, nonlocality of this object leads to the fact that in quantum chromodynamic the difficulties arise connected with investigation of quarks interaction on large distance. 
  The Green functions of the partial differential operators of even order acting on smooth sections of a vector bundle over a Riemannian manifold are investigated via the heat kernel methods. We study the resolvent of a special class of higher-order operators formed by the products of second-order operators of Laplace type defined with the help of a unique Riemannian metric but with different bundle connections and potential terms. The asymptotic expansion of the Green functions near the diagonal is studied in detail in any dimension. As a by-product a simple criterion for the validity of the Huygens principle is obtained. It is shown that all the singularities as well as the non-analytic regular parts of the Green functions of such high-order operators are expressed in terms of the usual heat kernel coefficients $a_k$ for a special Laplace type second-order operator. 
  It is well known that spherically symmetric reduction of General Relativity (SSG) leads to non-minimally coupled scalar matter. We generalize (and correct) recent results to Hawking radiation for a class of dilaton models which share with the Schwarzschild black hole non-minimal coupling of scalar fields and the basic global structure. An inherent ambiguity of such models (if they differ from SSG) is discussed. However, for SSG we obtain the rather disquieting result of a negative Hawking flux at infinity, if the usual recipe for such calculations is applied. 
  We analyse the world-volume theory of multiple Kaluza-Klein monopoles in string and M-theory by identifying the appropriate zero modes of various fields. The results are consistent with supersymmetry, and all conjectured duality symmetries. In particular for M-theory and type IIA string theory, the low energy dynamics of N Kaluza-Klein monopoles is described by supersymmetric U(N) gauge theory, and for type IIB string theory, the low energy dynamics is described by a (2,0) supersymmetric field theory in (5+1) dimensions with N tensor multiplets and tensionless self-dual strings. It is also argued that for the Kaluza-Klein monopoles in heterotic string theory, the apparently flat moduli space gets converted to the moduli space of BPS monopoles in SU(2) gauge theory when higher derivative corrections to the supergravity equations of motion are taken into account. 
  I comment on a curious relation between Siegel's model of random lattice strings and type IIB matrix model. The comparison of the two theories suggests that there may exist extra terms in the latter which are overlooked in the weak string coupling limit. 
  Covariant actions for the bosonic fields of D=10 IIB supergravity are constructed with the help of a single auxiliary scalar field and in a formulation with an infinite series of auxiliary (anti)-self-dual 5-form fields. 
  We obtain the exact operator solution of two-dimensional quantum Born-Infeld theory. This theory has a Lagrangian density non-polynomial in the fundamental fields. So this analysis might shed some light on the analysis of non-perturbative effects of field theories. We find the new exact soluble class of quantum field theories. 
  Graphs which generalize the simple or affine Dynkin diagrams are introduced. Each diagram defines a bilinear form on a root system and thus a reflection group. We present some properties of these groups and of their natural "Coxeter element". The folding of these graphs and groups is also discussed, using the theory of C-algebras. (Proceedings of the Taniguchi Symposium {Topological Field Theory, Primitive Forms and Related Topics}, Kyoto Dec 1996) 
  Singularities in the Yukawa and gauge couplings of N=1 compactifications of the SO(32) heterotic string are discussed. 
  We develop possible versions of supersymmetric single particle quantum mechanics, with application to superstring-bit models in view. We focus principally on space dimensions $d=1,2,4,8$, the transverse dimensionalities of superstring in $3,4,6,10$ space-time dimensions. These are the cases for which ``classical'' superstring makes sense, and also the values of $d$ for which Hooke's force law is compatible with the simplest superparticle dynamics. The basic question we address is: When is it possible to replace such harmonic force laws with more general ones, including forces which vanish at large distances? This is an important question because forces between string-bits that do not fall off with distance will almost certainly destroy cluster decomposition. We show that the answer is affirmative for $d=1,2$, negative for $d=8$, and so far inconclusive for $d=4$. 
  We present an algorithm for obtaining the matter content of effective six-dimensional theories resulting from compactification of F-theory on elliptic Calabi-Yau threefolds which are hypersurfaces in toric varieties. The algorithm allows us to read off the matter content of the theory from the polyhedron describing the Calabi-Yau manifold. This is based on the generalized Green-Schwarz anomaly cancellation condition. 
  In this work we pursue the singular-vector analysis of the integrable perturbations of conformal theories that was initiated in hep-th/9603088. Here we consider the detailed study of the N=1 superconformal theory and show that all integrable perturbations can be identified from a simple singular-vector argument. We identify these perturbations as theories based on affine Lie superalgebras and show that the results we obtain relating two perturbations can be understood by the extension of affine Toda duality to these theories with fermions. We also discuss how this duality is broken in specific cases. 
  A generalized Chan-Paton construction is presented which is analogous to the tensor product of vector bundles. To this end open string theories are considered where the space of states decomposes into sectors whose product is described by a semigroup. The cyclicity properties of the open string theory are used to prove that the relevant semigroups are direct unions of Brandt semigroups. The known classification of Brandt semigroups then implies that all such theories have the structure of a theory with Dirichlet-branes. We also describe the structure of an arbitrary orientifold group, and show that the truncation to the invariant subspace defines a consistent open string theory. Finally, we analyze the possible orientifold projections of a theory with several kinds of branes. 
  With asymptotic method developed from weak turbulent theory, the kinetic equations for QGP are expanded in fluctuation field potential $A^T_\mu $. Considering the second-order and third-order currents, we derive the nonlinear permeability tensor function from Yang-Mills field equation, and find that the third-order current is more important in turbulent theory. The nonlinear permeability formulae for longitudinal color oscillations show that the non-Abelian effects are more important than the Abelian-like effects. To compare with other works, we give the numerical result of the damping rate for the modes with zero wave vector. 
  We present a class of static, spherically symmetric, non-singular solutions of the tree-level string effective action, truncated to first order in $\alpha'$. In the string frame the solutions approach asymptotically (at $r\to 0$ and $r\to \infty$) two different anti-de Sitter configurations, thus interpolating between two maximally symmetric states of different constant curvature. The radial-dependent dilaton defines a string coupling which is everywhere finite, with a peak value that can be chosen arbitrarily small so as to neglect quantum-loop corrections. This example stresses the possible importance of finite-size $\alpha'$ corrections, typical of string theory, in avoiding space-time singularities. 
  We investigate the dynamics of classical and quantum N-component phi^4 oscillators in the presence of an external field. In the large N limit the effective dynamics is described by two-degree-of-freedom classical Hamiltonian systems. In the classical model we observe chaotic orbits for any value of the external field, while in the quantum case chaos is strongly suppressed. A simple explanation of this behaviour is found in the change in the structure of the orbits induced by quantum corrections. Consistently with Heisenberg's principle, quantum fluctuations are forced away from zero, removing in the effective quantum dynamics a hyperbolic fixed point that is a major source of chaos in the classical model. 
  The nonadiabatic geometric phase in a time dependent quantum evolution is shown to provide an intrinsic concept of time having dual properties relative to the external time. A nontrivial extension of the ordinary quantum mechanics is thus obtained with interesting scaling laws. A fractal like structure in time is thus revealed. 
  We investigate the nonlinear realization of spontaneously broken N=2 superconformal symmetry in 4 dimensions. We particularly study Nambu-Goldstone degrees of freedom for the partial breaking of N=2 superconformal symmetry down to N=1 super-Poincar{\'e} symmetry, where we get the chiral NG multiplet of dilaton and the vector NG multiplet of NG fermion of broken Q-supersymmetry. Evaluating the covariant differentials and supervielbeins for the chiral as well as the full superspace, we obtain the nonlinear effective lagrangians. 
  We study the similarity renormalization scheme for hamiltonians to the fourth order in perturbation theory using a model hamiltonian for fermions coupled to bosons. We demonstrate that the free finite parts of counterterms can be chosen in such a way that the T-matrix is covariant up to the fourth order and the eigenvalue equation for the physical fermion reduces to the Dirac equation. Through this choice, the systematic renormalization scheme reproduces the model solution originally proposed by G{\l}azek and Perry. 
  We have constructed the N=1/2 supersymmetric general Abelian model with asymmetric chiral couplings. This leads to a N=1/2 supersymmetrization of the Schwinger model. We show that the supersymmetric general model is plagued with problems of infrared divergence. Only the supersymmetric chiral Schwinger model is free from such problems and is dynamically equivalent to the chiral Schwinger model because of the peculiar structure of the N=1/2 multiplets. 
  We propose a new model of inflation based on the soft-breaking of N=2 supersymmetric SU(2) Yang-Mills theory. The advantage of such a model is the fact that we can write an exact expression for the effective scalar potential, including non-perturbative effects, which preserves the analyticity and duality properties of the Seiberg-Witten solution. We find that the scalar condensate that plays the role of the inflaton can drive a long period of cosmological expansion, produce the right amount of temperature anisotropies in the microwave background, and end inflation when the monopole acquires a vacuum expectation value. Duality properties relate the weak coupling Higgs region where inflation takes place with the strong coupling monopole region, where reheating occurs, creating particles corresponding to the light degrees of freedom in the true vacuum. 
  We consider perturbation of a conformal field theory by a pair of relevant logarithmic operators and calculate the beta function up to two loops. We observe that the beta function can not be derived from a potential. Thus the renormalization group trajectories are not always along decreasing values of the central charge. However there exists a domain of structure constants in which the c-theorem still holds. 
  We show that the tensor gauge multiplet of N=1 supersymmetry can serve as the Goldstone multiplet for partially broken rigid N=2 supersymmetry. We exploit a remarkable analogy with the Goldstone-Maxwell multiplet of hep-th/9608177 to find its nonlinear transformation law and its invariant Goldstone action. We demonstrate that the tensor multiplet has two dualities. The first transforms it into the chiral Goldstone multiplet; the other leaves it invariant. 
  We propose a general mechanism for stabilizing the dilaton against runaway to weak coupling. The method is based on features of the effective superpotential which arise for supersymmetric gauge theories which are not asymptotically free. Consideration of the 2PI effective action for bilinear operators of matter and gauge superfields allows one to overcome the obstacles to constructing a nonvanishing superpotential. 
  We analyse the classical symmetries of bosonic D-string actions and generalizations thereof. Among others, we show that the simplest actions of this type have infinitely many nontrivial rigid symmetries which act nontrivially and nonlinearly both on the target space coordinates and on the U(1) gauge field, and form a Kac-Moody version of the Weyl algebra (= Poincare algebra + dilatations). 
  We discuss the possibility of generalizing some aspects of the C-theorem within two different approaches, the conventional RG and the Wilson RG flows. We show that the original Zamolodchikov's theorem is related to the existence of the phase transitions in finite temperature QFT. We present some arguments related to the holomorphic property of the low energy Wilson effective action. 
  We present two different Lax operators for a manifestly N=2 supersymmetric extension of "a=-2" Boussinesq hierarchy . The first is the supersymmetric generalization of the Lax operator of the Modified KdV equation. The second is the generalization of the supersymmetric Lax operator of the N=2 supersymmetric a=-2 KdV system. The gauge transformation of the first Lax operator provide the Miura link between the "small" N=4 supersymmetric conformal algebra and the supersymmetric $W_{3}$ algebra . 
  Free massless fermionic fields of arbitrary spins $s>0$ corresponding to totally (anti)symmetric tensor-spinor representations of the $SO(d-1)$ compact subgroup and in $d$-dimensional anti-de Sitter space are investigated. We propose the free equations of motion, subsidiary conditions and corresponding gauge transformations for such fields. The equations obtained are used to derive the lowest energy values for the above-mentioned representations. A new representation for equations of motion and gauge transformations in terms of generators of anti-de Sitter group $SO(d-1,2)$ is found. It is demonstrated that in contrast to the symmetric case the gauge parameter of the antisymmetric massless field is also a massless field. 
  Semiclassical quantization of the SU(3)-skyrmions is performed by means of the collective coordinate method. The quantization condition known for the SU(2)-solitons quantized with SU(3) collective coordinates is generalized for the SU(3) skyrmions with strangeness content different from zero. Quantization of the dipole-type configuration with large strangeness content found recently is considered as an example, the spectrum and the mass splitting of the quantized states are estimated. The energy and baryon number density of SU(3) skyrmions are presented in the form emphasizing their symmetry in different SU(2) subgroups of SU(3), and the lower boundary for the static energy of SU(3) skyrmions is derived. 
  We show that the boundary state description of a Dp-brane is strictly related to the corresponding classical solution of the low-energy string effective action. By projecting the boundary state on the massless states of the closed string we obtain the tension, the R-R charge and the large distance behavior of the classical solution. We discuss both the case of a single D-brane and that of bound states of two D-branes. We also show that in the R-R sector the boundary state, written in a picture which treats asymmetrically the left and right components, directly yields the R-R gauge potentials. 
  By exploring the description of chiral blocks in terms of co-invariants, a derivation of the Verlinde formula for WZW models is obtained which is entirely based on the representation theory of affine Lie algebras. In contrast to existing proofs of the Verlinde formula, this approach works universally for all untwisted affine Lie algebras. As a by-product we obtain a homological interpretation of the Verlinde multiplicities as Euler characteristics of complexes built from invariant tensors of finite-dimensional simple Lie algebras. Our results can also be used to compute certain traces of automorphisms on the spaces of chiral blocks. Our argument is not rigorous; in its present form this paper will therefore not be submitted for publication. 
  A phenomenon of classical quantization is discussed. This is revealed in the class of pseudoclassical gauge systems with nonlinear nilpotent constraints containing some free parameters. Variation of parameters does not change local (gauge) and discrete symmetries of the corresponding systems, but there are some special discrete values of them which give rise to the maximal global symmetries at the classical level. Exactly the same values of the parameters are separated at the quantum level, where, in particular, they are singled out by the requirement of conservation of the discrete symmetries. The phenomenon is observed for the familiar pseudoclassical model of 3D P,T-invariant massive fermion system and for a new pseudoclassical model of 3D P,T-invariant system of topologically massive U(1) gauge fields. 
  We analyze the low energy behavior of N=1 supersymmetric gauge theories with SU(N_c) x SU(N_c) gauge group and a Landau-Ginzburg type superpotential. These theories contain fundamentals transforming under one of the gauge groups as well as bifundamental matter which transforms as a fundamentals under each. We obtain the parametrization of the gauge coupling on the Coulomb branch in terms of a hyperelliptic curve. The derivation of this curve involves making use of Seiberg's duality for SQCD as well as the classical constraints for N_f=N_c+1 and the quantum modified constraints for N_f=N_c. 
  The complete, missing, Hamiltonian treatment of the standard SU(3)xSU(2)xU(1) model with Grassmann-valued fermion fields in the Higgs phase is given. We bypass the complications of the Hamiltonian theory in the Higgs phase, resulting from the spontaneous symmetry breaking with the Higgs mechanism, by studying the Hamiltonian formulation of the Higgs phase for the gauge equivalent Lagrangian in the unitary gauge. A canonical basis of Dirac's observables is found and the reduced physical Hamiltonian is evaluated. Its self-energy part is nonlocal for the electromagnetic and strong interactions, but local for the weak ones. Therefore, the Fermi 4-fermion interaction reappears at the nonperturbative level. 
  In this paper we take a deeper look at the technically elementary but physically robust viewpoint in which the Casimir energy in dielectric media is interpreted as the change in the total zero point energy of the electromagnetic vacuum summed over all states. Extending results presented in previous papers [hep-th/9609195; hep-th/9702007] we approximate the sum over states by an integral over the density of states including finite volume corrections. For an arbitrarily-shaped finite dielectric, the first finite-volume correction to the density of states is shown to be proportional to the surface area of the dielectric interface and is explicitly evaluated as a function of the permeability and permitivity. Since these calculations are founded in an elementary and straightforward way on the underlying physics of the Casimir effect they serve as an important consistency check on field-theoretic calculations. As a concrete example we discuss Schwinger's suggestion that the Casimir effect might be the underlying physical basis behind sonoluminescence}. The recent controversy concerning the relative importance of volume and surface contributions is discussed. For sufficiently large bubbles the volume effect is always dominant. Furthermore we can explicitly calculate the surface area contribution as a function of refractive index. 
  Functional relations among the fusion hierarchy of quantum transfer matrices give a novel derivation of the TBA equations, namely without string hypothesis. This is demonstrated for two important models of 1D highly correlated electron systems, the supersymmetric $t-J$ model and the supersymmetric extended Hubbard model. As a consequence, "the excited state TBA" equations, which characterize correlation lengths, are explicitly derived for the $t-J$ model. To the authors' knowledge, this is the first explicit derivation of excited state TBA equations for 1D lattice electron systems. 
  We review the structure $D=6, N=1$ string vacua with emphasis on the different connections due to T-dualities and S-dualities. The topics discussed include: Anomaly cancellation; K3 and orbifold $D=6, N=1$ heterotic compactifications; T-dualities between $E_8\times E_8$ and $Spin(32)/Z_2$ heterotic vacua; non-perturbative heterotic vacua and small instantons; N=2 Type-II/Heterotic duality in D=4 ; F-theory/heterotic duality in D=6; and heterotic/heterotic duality in six and four dimensions. 
  We suggest an alternative approach to deconfine N =1 SU(N) supersymmetric gauge theory with a symmetric tensor, fundamentals, anti-fundamentals, and no superpotential. It is found that although the dual prescription derived by this new method of deconfinement is different from that by the original method, both dual prescriptions are connected by duality transformations. By deforming the theory, it is shown that both dual theories flow properly so that the Seiberg's duality is preserved. 
  Quantum mechanical operators and quantum fields are interpreted as realizations of timespace manifolds. Such causal manifolds are parametrized by the classes of the positive unitary operations in all complex operations, i.e. by the homogenous spaces $\D(n)=\GL(\C^n_\R)/\U(n)$ with $n=1$ for mechanics and $n=2$ for relativistic fields. The rank $n$ gives the number of both the discrete and continuous invariants used in the harmonic analysis, i.e. two characteristic masses in the relativistic case. 'Canonical' field theories with the familiar divergencies are inappropriate realizations of the real 4-dimensional causal manifold $\D(2)$. Faithful timespace realizations do not lead to divergencies. In general they are reducible, but nondecomposable - in addition to representations with eigenvectors (states, particle) they incorporate principal vectors without a particle (eigenvector) basis as exemplified by the Coulomb field. 
  We illustrate the phase structure of a deformed two-dimensional Gross-Neveu model which is defined by undeformed field contents plus deformed Pauli matrices. This deformation is based on two motives to find a more general polymer model and to estimate how $q$-deformed field theory affects on its effective potential. There found some regions where chiral symmetry breaking and restoration take place repeatedly as temperature increasing. 
  We propose descriptions of interacting (2,0) supersymmetric theories without gravity in six dimensions in the infinite momentum frame. They are based on the large $N$ limit of quantum mechanics or 1+1 dimensional field theories on the moduli space of $N$ instantons in $\IR^4$. 
  There has been some controversies at the large $N$ behaviour of the 2D Yang-Mills and chiral 2D Yang-Mills theories. To be more specific, is there a one parameter family of minima of the free energy in the strong region, or the minimum is unique. We show that there is a missed equation which, added to the known equations, makes the minimum unique. 
  Using the standard saddle-point method, we find an explicit relation for the large-N limit of the free energy of an arbitrary generalized 2D Yang-Mills theory in the weak ($A<A_c$) region. In the strong ($A>A_c$) region, we investigate carefully the specific fourth Casimir theory, and show that the ordinary integral equation of the density function is not adequate to find the solution. There exist, however, another equation which restricts the parameters. So one can find the free energy in strong region and show that the theory has a third order phase transition. 
  Starting from the working hypothesis that both physics and the corresponding mathematics have to be described by means of discrete concepts on the Planck-scale, one of the many problems one has to face in this enterprise is to find the discrete protoforms of the building blocks of continuum physics and mathematics. A core concept is the notion of dimension. In the following we develop such a notion for irregular structures like (large) graphs and networks and derive a number of its properties. Among other things we show its stability under a wide class of perturbations which is important if one has 'dimensional phase transitions' in mind. Furthermore we systematically construct graphs with almost arbitrary 'fractal dimension' which may be of some use in the context of 'dimensional renormalization' or statistical mechanics on irregular sets. 
  The vacuum energies corresponding to massive Dirac fields with the boundary conditions of the MIT bag model are obtained. The calculations are done with the fields occupying the regions inside and outside the bag, separately. The renormalization procedure for each of the situations is studied in detail, in particular the differences occurring with respect to the case when the field extends over the whole space. The final result contains several constants undergoing renormalization, which can be determined only experimentally. The non-trivial finite parts which appear in the massive case are found exactly, providing a precise determination of the complete, renormalized zero-point energy for the first time, in the fermionic case. The vacuum energy behaves like inverse powers of the mass for large masses. 
  We show that the method of S. Wu to study topological 4d-gravity can be understood within a now standard method designed to produce equivariant cohomology classes. Next, this general framework is applied to produce some observables of the topological 4d-gravity. 
  The one-loop effective potential of a thermodynamic fermion loop under constant magnetic field is studied. As expected, it can be interpreted literally as a discretized sum of $(D-2)$-dimensional energy density above the Dirac sea. Large/small mass expansions of the potential are also examined. 
  Modified similarity renormalization of Hamiltonians is proposed, that performes by means of flow equations the similarity transformation of Hamiltonian in the particle number space. This enables to renormalize in the energy space the field theoretical Hamiltonian and makes possible to work in a severe trancated Fock space for the renormalized Hamiltonian. 
  We study the general equations determining BPS Black Holes by using a Solvable Lie Algebra representation for the homogenous scalar manifold U/H of extended supergravity. In particular we focus on the N=8 case and we perform a general group theoretical analysis of the Killing spinor equation enforcing the BPS condition. Its solutions parametrize the U-duality orbits of BPS solutions that are characterized by having 40 of the 70 scalars fixed to constant values. These scalars belong to hypermultiplets in the N=2 decomposition of the N=8 theory. Indeed it is shown that those decompositions of the Solvable Lie algebra into appropriate subalgebras which are enforced by the existence of BPS black holes are the same that single out consistent truncations of the N=8 theory to intereacting theories with lower supersymmetry. As an exemplification of the method we consider the simplified case where the only non-zero fields are in the Cartan subalgebra H of Solv(U/H) and correspond to the radii of string toroidal compactification. Here we derive and solve the mixed system of first and second order non linear differential equations obeyed by the metric and by the scalar fields. So doing we retrieve the generating solutions of heterotic black holes with two charges. Finally, we show that the general N=8 generating solution is based on the 6 dimensional solvable subalgebra Solv [(SL(2,\IR) /U(1))^3]. 
  We study the initial value problem for actions which contain non-trivial functions of integrals of local functions of the dynamical variable. In contrast to many other non-local actions, the classical solution set of these systems is at most discretely enlarged, and may even be restricted, with respect to that of a local theory. We show that the solutions are those of a local theory whose (spacetime constant) parameters vary with the initial value data according to algebraic equations. The various roots of these algebraic equations can be plausibly interpreted in quantum mechanics as different components of a multi-component wave function. It is also possible that the consistency of these algebraic equations imposes constraints upon the initial value data which appear miraculous from the context of a local theory. 
  We review some recent results concerning gauge theories in various dimensions. In particular, we discuss RG fixed points and ``mirror'' symmetry duality in 3d N=4 supersymmetric gauge theories and a classification of non-trivial RG fixed points in 5d N=1 supersymmetric theories. 
  Null vectors are generalized to the case of indecomposable representations which are one of the main features of logarithmic conformal field theories. This is done by developing a compact formalism with the particular advantage that the stress energy tensor acting on Jordan cells of primary fields and their logarithmic partners can still be represented in form of linear differential operators. Since the existence of singular vectors is subject to much stronger constraints than in regular conformal field theory, they also provide a powerful tool for the classification of logarithmic conformal field theories. 
  We discuss an application of the method of the angular quantization to reconstruction of form-factors of local fields in massive integrable models. The general formalism is illustrated with examples of the Klein-Gordon, sinh-Gordon and Bullough-Dodd models. For the latter two models the angular quantization approach makes it possible to obtain free field representations for form-factors of exponential operators. We discuss an intriguing relation between the free field representations and deformations of the Virasoro algebra. The deformation associated with the Bullough-Dodd models appears to be different from the known deformed Virasoro algebra. 
  In a solvable model of two dimensional SU(N) (N \to \infty) gauge fields interacting with matter in both adjoint and fundamental representations we investigate the nature of the phase transition separating the strong and weak coupling regions of the phase diagram. By interpreting the large N solution of the model in terms of SU(N) representations it is shown that the strong coupling phase corresponds to a region where a gap occurs in the spectrum of irreducible representations. We identify a gauge invariant order parameter for the generalized confinement-deconfinement transition and give a physical meaning to each phase in terms of the interaction of a pair of test charges. 
  We study 1+1-dimensional theories of vector and hypermultiplets with (4,4) supersymmetry. Despite strong infrared fluctuations, these theories flow in general to distinct conformal field theories on the Coulomb and Higgs branches. In some cases there may be a quantum Higgs theory even when there is no classical Higgs branch. The Higgs branches of certain such theories provide a framework for a matrix model of Type IIA fivebranes and the associated exotic six-dimensional string theories. Proposals concerning the interactions of these string theories are evaluated. 
  We extend the Wilson renormalization group (RG) formulation to chiral gauge theories and show that local gauge symmetry can be implemented by a suitable choice of the RG flow boundary conditions. Since the space-time dimension is four, there is no ambiguity in handling the matrix $\g_5$ and left and right fermions are not coupled. As a result the ultraviolet action contains all possible globally chiral invariant interactions. Nevertheless, the correct chiral anomaly is reproduced. 
  We discuss the physics of four-dimensional compact U(1) lattice gauge theory from the point of view of softly broken N=2 supersymmetric SU(2) Yang-Mills theory. We provide arguments in favor of (pseudo-)critical mass exponents 1/3, 5/11 and 1/2, in agreement with the values observed in the computer simulations. We also show that the J^{CP} assignment of some of the lowest lying states can be naturally explained. 
  Instantons and their quantisation in pure Yang-Mills theory formulated in the background of de Sitter spacetime represented by spatially-closed ($k = 1$) Friedmann-Robertson-Walker metric are discussed. As for the classical treatment of the instanton physics, first, explicit instanton solutions are found and next, quantities like Pontryagin index and the semiclassical approximation to the inter-vacua tunnelling amplitude are evaluated. The Atiyah-Patodi-Singer index theorem is checked as well by constructing explicitly the normalizable fermion zero modes in this de Sitter spacetime instanton background. Finally, following the kink quantisation scheme originally proposed by Dashen, Hasslacher and Neveu, the quantisation of our instanton is performed. Of particular interest is the estimate of the lowest quantum correction to the inter-vacua tunnelling amplitude arising from the quantisation of the instanton. It turns out that the inter-vacua tunnelling amplitude gets enhanced upon quantising the instanton. 
  Perturbation theory in the nonperturbative QCD vacuum and the non-Abelian Stokes theorem, representing a Wilson loop in the SU(2) gluodynamics as an integral over all the orientations in colour space, are applied to a derivation of the correction to the string effective action in the lowest order in the coupling constant $g$. This correction is due to the interaction of perturbative gluons with the string world sheet and affects only the coupling constant of the rigidity term, while its contribution to the string tension of the Nambu-Goto term vanishes. The obtained correction to the rigidity coupling constant multiplicatively depends on the colour "spin" of the representation of the Wilson loop under consideration and a certain path integral, which includes the background Wilson loop average. 
  We compute the finite size spectrum for the spin 1/2 XXZ chain with twisted boundary conditions, for anisotropy in the regime $0< \gamma <\pi/2$, and arbitrary twist $\theta$. The string hypothesis is employed for treating complex excitations. The Bethe Ansatz equtions are solved within a coupled non-linear integral equation approach, with one equation for each type of string. The root-of-unity quantum group invariant periodic chain reduces to the XXZ_1/2 chain with a set of twist boundary conditions ($\pi/\gamma\in Z$, $\theta$ an integer multiple of $\gamma$). For this model, the restricted Hilbert space corresponds to an unitary conformal field theory, and we recover all primary states in the Kac table in terms of states with specific twist and strings. 
  We study the M(atrix) theory which describes the $E_8 \times E_8$ heterotic string compactified on $S^1$, or equivalently M-theory compactified on an orbifold $(S^1/\integer_2) \times S^1$, in the presence of a Wilson line. We formulate the corresponding M(atrix) gauge theory, which lives on a dual orbifold $S^1 \times (S^1 / \integer_2)$. Thirty-two real chiral fermions must be introduced to cancel gauge anomalies. In the absence of an $E_8 \times E_8$ Wilson line, these fermions are symmetrically localized on the orbifold boundaries. Turning on the Wilson line moves these fermions into the interior of the orbifold. The M(atrix) theory action is uniquely determined by gauge and supersymmetry anomaly cancellation in 2+1 dimensions. The action consistently incorporates the massive IIA supergravity background into M(atrix) theory by explicitly breaking (2+1)-dimensional Poincare invariance. The BPS excitations of M(atrix) theory are identified and compared to the heterotic string. We find that heterotic T-duality is realized as electric-magnetic S-duality in M(atrix) theory. 
  We describe two distinct approaches for bosonization in higher dimensions; one is based on a direct comparison of current correlation functions while the other relies on a Master lagrangean formalism. These are used to bosonise the Massive Thirring Model in three and four dimensions in the weak coupling regime but with an arbitrary fermion mass. In both approaches the explicit bosonised lagrangean and current are derived in terms of gauge fields. The complete equivalence of the two bosonization methods is established. Exact results for the free massive fermion theory are also obtained. Finally, the two-dimensional theory is revisited and the possibility of extending this analysis for arbitrary dimensions is indicated. 
  The topology and geometry of the moduli space, M_2, of degree 2 static solutions of the CP^1 model on a torus (spacetime T^2 x R) are studied. It is proved that M_2 is homeomorphic to the left coset space G/G_0 where G is a certain eight-dimensional noncompact Lie group and G_0 is a discrete subgroup of order 4. Low energy two-lump dynamics is approximated by geodesic motion on M_2 with respect to a metric g defined by the restriction to M_2 of the kinetic energy functional of the model. This lump dynamics decouples into a trivial ``centre of mass'' motion and nontrivial relative motion on a reduced moduli space. It is proved that (M_2,g) is geodesically incomplete and has only finite diameter. A low dimensional geodesic submanifold is identified and a full description of its geodesics obtained. 
  We consider configurations of D6-branes with D0-brane charge given by recent work of Taylor and compute interaction potentials with various D-brane probes using a 1-loop open string calculation. These results are compared to a supergravity calculation using the solution given by Sheinblatt of an extremal black hole carrying 0-brane and 6-brane charge. 
  Generalized quantum mechanics is used to examine a simple two-particle scattering experiment in which there is a bounded region of closed timelike curves (CTCs) in the experiment's future. The transitional probability is shown to depend on the existence and distribution of the CTCs. The effect is therefore acausal, since the CTCs are in the experiment's causal future. The effect is due to the non-unitary evolution of the pre- and post-scattering particles as they pass through the region of CTCs. We use the time-machine spacetime developed by Politzer [1], in which CTCs are formed due to the identification of a single spatial region at one time with the same region at another time. For certain initial data, the total cross-section of a scattering experiment is shown to deviate from the standard value (the value predicted if no CTCs existed). It is shown that if the time machines are small, sparsely distributed, or far away, then the deviation in the total cross-section may be negligible as compared to the experimental error of even the most accurate measurements of cross-sections. For a spacetime with CTCs at all points, or one where microscopic time machines pervade the spacetime in the final moments before the big crunch, the total cross-section is shown to agree with the standard result (no CTCs) due to a cancellation effect. 
  Using a supergeometric interpretation of field functionals, we show that for a class of classical field models used for realistic quantum field theoretic models, an infinite-dimensional supermanifold (smf) of classical solutions in Minkowski space can be constructed. That is, we show that the smf of smooth Cauchy data with compact support is isomorphic with an smf of corresponding classical solutions of the model. 
  The possibility of spontaneous breaking of CP symmetry by the expectation values of orbifold moduli is investigated with particular reference to $CP$ violating phases in soft supersymmetry breaking terms. The effect of different mechanisms for stabilizing the dilaton and the form of the non-perturbative superpotential on the existence and size of these phases is studied. Models with modular symmetries which are subgroups of $PSL(2,Z)$, as well as the single overall modulus $T$ case with the full $PSL(2,Z)$ modular symmetry, are discussed. Non-perturbative superpotentials involving the absolute modular invariant $j(T)$, such as may arise from F-theory compactifications, are considered. 
  We develop a geometrical structure of the manifolds $\Gamma$ and $\hat\Gamma$ associated respectively to the gauge symmetry and to the BRST symmetry. Then, we show that ($\hat\Gamma,\hat\zeta,\Gamma$), where $\hat\zeta$ is the group of BRST transformations, is endowed with the structure of a principle fiber bundle over the base manifold $\Gamma$. Furthermore, in this geometrical set up due to the nilpotency of the BRST operator, we prove that the effective action of a gauge theory is a BRST-exact term up to the classical action. Then, we conclude that the effective action where only the gauge symmetry is fixed, is cohomologically equivalent to the action where the gauge and the BRST symmetries are fixed. 
  The duality properties of perturbative string characters associated with the transverse space-time rotations are studied. T duality is achieved by suitably integrating over the total momentum, contrary to earlier discussions. The O(8) triality properties of the characters for critical superstrings, are derived. This shows the existence of a third formulation (unnoticed so far to our knowledge) equivalent to, but different from the ones of Neveu-Schwarz-Ramond, and Green-Schwarz. Projectors in the NS and R sectors are defined in the GS formalism. The consequences of supersymmetry are neatly derived at once for all massive states by factorising the character of the long SUSY multiplet. 
  We discuss the renormalizability of quantum gravity near two dimensions based on the results obtained by a computation of the BRST-antibracket cohomology in the space of local functionals of the fields and antifields. We justify the assumption on the general structure of the counterterms which have been used in the original proof of renormalizability of the quantum gravity near two dimensions. 
  Anyon gas with interparticle (retarded) Coulomb interaction has been studied. The resulting system is shown to be a collection of dressed anyons, with a screening factor introduced in their spin. Close structural similarity with the Chern-Simons construction of anyons has helped considerably in computing the screening effect. Finally the present model is compared with the conventional Chern-Simons construction. 
  A new first order action for type IIB Dirichlet 3-brane is proposed. Its form is inspired by the superfield equations of motion obtained recently from the generalized action principle. The action involves auxiliary symmetric spin tensor fields. It seems promising for a reformulation of the generalized action in a structure most adequate for investigating the extrinsic geometry of the super-3- brane, but also for further studies of string dualities. 
  We point out a map between the dynamics of a non-relativistic system of $N$ particles in one dimension interacting via the pair-wise potentials $U_I(q) = (\nu^2/4R^2)\sin^2(q/2R)$ and the one of the particles with the pair potential $U_{II}(q) = \nu^2/q^2$ and the external potential $U_{ext} = \omega^2 q^2/2$. The natural relation between the frequency $\omega$ and the radius $R$ is: $\omega R = 1$. 
  We analyze the generalized point-splitting method and Jo's result for the commutator anomaly. We find that certain classes of general regularization kernels satisfying integral conditions provide a unique result, which, however differs from Faddeev's cohomological result. 
  Gauging the M-2-brane effective action with respect to an Abelian isometry in such a way that the invariance under gauge transformations of the 3-form potential is maintained (slightly modified) we obtain a fully covariant action with 11-dimensional target space that gives the massive D-2-brane effective action upon dimensional reduction in the direction of the gauged isometry. 
  We describe an effective method for calculating certain infinite sums, generalizations of the classical Bernoulli polynomials. As shown by Edward Witten in his papers on two-dimensional gauge theories, the correlation functions of two-dimensional topological Yang-Mills theory (or intersection numbers on moduli spaces of flat connections) can be given in the form of such infinite sums. Thus, in particular, our results give finite expressions for these correlation functions in the case of arbitrary compact structure groups G. 
  Performing an nonperturbative path integral for the geometric part of a large class of 2d theories without kinetic term for the dilaton field, the quantum effects from scalar matter fields are treated as a perturbation. When integrated out to two-loops they yield a correction to the Polyakov term which is still exact in the geometric part. Interestingly enough the effective action only experiences a renormalization of the dilaton potential. 
  We present an arbitrary model based on the trefoil knot to construct objects of the same spectrum as that of elementary particles. It includes `waves' and three identical sets of sources. Due to Lorentz invariance, `waves' group into 3 types of 1, 3 and 8 objects and `sources' consists of 3 identical sets of 30+2 elements, which separate into: 1 * 1 * 2 + 1 * 2 * 2 + 3 * 2 * 2 + 3 * 1 * 2 + 3 * 1 * 2 and another 1 * 1 * 2 group (which does not match classification of the Standard Model fields). On the other hand, there is no room in this construction for objects directly corresponding to Higgs-like degrees of freedom. 
  It is shown that physical states of a non-abelian Yang-Mills-Higgs dyon are invariant under large gauge transformations that do not commute with its magnetic field. This result is established within an enlarged Hamiltonian formalism where surface terms are kept as dynamical variables. These additional variables are parameters of large gauge transformations, and are potential collective coordinates for the quantization of the monopole. Our result implies that there are no physical effects associated to some large gauge transformations and therefore their parameters should not be counted as collective coordinates. 
  We consider euclidean D-branes wrapping around manifolds of exceptional holonomy in dimensions seven and eight. The resulting theory on the D-brane---that is, the dimensional reduction of 10-dimensional supersymmetric Yang-Mills theory---is a cohomological field theory which describes the topology of the moduli space of instantons. The 7-dimensional theory is an N_T = 2 (or balanced) cohomological theory given by an action potential of Chern-Simons type. As a by-product of this method, we construct a related cohomological field theory which describes the monopole moduli space on a 7-manifold of G_2 holonomy. 
  The fivebrane of M theory -- the M5-brane -- is an especially interesting object. It plays a central role in a geometric understanding of the Seiberg-Witten solution of N=2 D=4 gauge theories as well as in certain new 6d quantum theories. The low energy effective action is an interacting theory of a (2,0) tensor multiplet. The fact that this multiplet contains a two-form gauge field with a self-dual field strength poses special challenges. Recent progress in addressing those challenges is reviewed. 
  We discuss supersymmetric Yang-Mills theories with the multiple scales in the brane language. The issue concerns N=2 SUSY gauge theories with massive fundamental matter including the UV finite case of $n_{f}=2n_c$, theories involving products of SU(n) gauge groups with bifundamental matter, and the systems with several parameters similar to $\Lambda_{QCD}$. We argue that the proper integrable systems are, accordingly, twisted XXX SL(2) spin chain, $SL(p)$ magnets and degenerations of the spin Calogero system. The issue of symmetries underlying integrable systems is addressed. Relations with the monopole systems are specially discussed. Brane pictures behind all these integrable structures in the IIB and M theory are suggested. We argue that degrees of freedom in integrable systems are related to KK excitations in M theory or D-particles in the IIA string theory, which substitute the infinite number of instantons in the field theory. This implies the presence of more BPS states in the low-energy sector. 
  We discuss the geometrical connection between 2D conformal field theories, random walks on hyperbolic Riemann surfaces and knot theory. For the wide class of CFTs with monodromies being the discrete subgroups of SL(2,R), the determination of four-point correlation functions are related to construction of topological invariants for random walks on multipunctured Riemann surfaces 
  The experimental observation of intense light emission by acoustically driven, periodically collapsing bubbles of air in water (sonoluminescence) has yet to receive an adequate explanation. One of the most intriguing ideas is that the conversion of acoustic energy into photons occurs quantum mechanically, through a dynamical version of the Casimir effect. We have argued elsewhere that in the adiabatic approximation, which should be reliable here, Casimir or zero-point energies cannot possibly be large enough to be relevant. (About 10 MeV of energy is released per collapse.) However, there are sufficient subtleties involved that others have come to opposite conclusions. In particular, it has been suggested that bulk energy, that is, simply the naive sum of ${1\over2}\hbar\omega$, which is proportional to the volume, could be relevant. We show that this cannot be the case, based on general principles as well as specific calculations. In the process we further illuminate some of the divergence difficulties that plague Casimir calculations, with an example relevant to the bag model of hadrons. 
  Two different mechanisms exist in non-perturbative String / M- theory for enhanced SU(N) (SO(2N)) gauge symmetries. It can appear in type IIA string theory or M-theory near an $A_{N-1}$ (D_N) type singularity where membrnes wrapped around two cycles become massless, or it can appear due to coincident D-branes (and orientifold planes) where open strings stretched between D-branes become massless. In this paper we exhibit the relationship between these two mechanisms by displaying a configuration in M-theory, which, in one limit, can be regarded as membranes wrapped around two cycles with $A_{N-1}$ (D_N) type intersection matrix, and in another limit, can be regarded as open strings stretched between N Dirichlet 6-branes (in the presence of an orientifold plane). 
  The semiclassical grey-body factor for massless fermion emission from the four dimensional black hole described by an ensemble of intersecting triplets of D- five-branes is shown to be consistent with the (statistical) decay rate of the branes (in the `long' D-string approximation) into massless fermionic closed string states, subject to assumptions regarding the energy distribution of colliding open string states. 
  We describe eight-dimensional vacuum configurations with varying moduli consistent with the U-duality group $SL(2,Z) \times SL(3,Z)$. Focusing on the latter less-well understood SL(3,Z) properties, we construct a class of fivebrane solutions living on lines on a three-dimensional base space. The resulting U-manifolds, with five scalars transforming under SL(3), admit a Ricci-flat Kahler metric. Based on the connection with special lagrangian $T^3$ fibered Calabi-Yau 3-folds, this construction provides a simple framework for the investigation of Calabi-Yau mirrors. 
  We study heterotic/type I duality in d=8,9 uncompactified dimensions. We consider the special (``BPS saturated'') F^4 and R^4 terms in the effective one-loop heterotic action, which are expected to be non-perturbatively exact. Under the standard duality map these translate to tree-level, perturbative and non-perturbative contributions on the type I side. We check agreement with the one-loop open string calculation, and discuss the higher-order perturbative contributions, which arise because of the mild non-holomorphicities of the heterotic elliptic genus. We put the heterotic world-sheet instanton corrections in a form that can be motivated as arising from a D-brane instanton calculation on the type-I side. 
  We calculate the coefficients of the operator product expansion (OPE), in Polyakov's approach for Burgers turbulence. We show that the OPE has to be generalized and it is shown that the extra term gives us the instanton solution (shock solution) of Burgers equation. We consider the effect of the new-term in the OPE, on the right and left-tail of probability distribution function (PDF). It is shown that the left-tail of PDF, where is dominated by the well-separated shocks behaves as $ W(u)\sim u^{-7/2}$. Finally we calculate the assymptotic behaviour of the N-point generating function of the velocity field, using the new OPE. 
  The perturbation of the Dirac sea to first order in the external potential is calculated in an expansion around the light cone. It is shown that the perturbation consists of a causal contribution, which describes the singular behavior of the Dirac sea on the light cone and contains bounded line integrals over the potential and its partial derivatives, and a non-causal contribution, which is a smooth function. As a preparatory step, we construct a formal solution of the inhomogeneous Klein-Gordon equation in terms of an infinite series of line integrals.   More generally, the method presented can be used for an explicit analysis of Feynman diagrams of the Dirac, Klein-Gordon, and wave equations in position space. 
  We prove that there is no power-counting renormalizable nonabelian generalization of the abelian topological mass mechanism in four dimensions. The argument is based on the technique of consistent deformations of the master equation developed by G. Barnich and one of the authors. Recent attempts involving extra fields are also commented upon. 
  We perform a generalised Scherk-Schwarz reduction of the effective action of the heterotic string on T^6 to obtain a massive N=4 supergravity theory in four dimensions. The local symmetry-group of the resulting d=4 theory includes a Heisenberg group, which is a subgroup of the global O(6,6+n) obtained in the standard reduction. We show explicitly that the same theory can be obtained by gauging this Heisenberg group in d=4, N=4 supergravity. 
  We show how $N=4, D=4$ duality of Montonen and Olive can be derived for all gauge groups using geometric engineering in the context of type II strings, where it reduces to T-duality. The derivation for the non-simply laced cases involves the use of some well known facts about orbifold conformal theories. 
  We analyse the velocity-dependent potentials seen by D0 and D4-brane probes moving in Type I' background for head-on scattering off the fixed planes. We find that at short distances (compared to string length) the D0-brane probe has a nontrivial moduli space metric, in agreement with the prediction of Type I' matrix model; however, at large distances it is modified by massive open strings to a flat metric, which is consistent with the spacetime equations of motion of Type I' theory. We discuss the implication of this result for the matrix model proposal for M-theory. We also find that the nontrivial metric at short distances in the moduli space action of the D0-brane probe is reflected in the coefficient of the higher dimensional v^4 term in the D4-brane probe action. 
  There have been known "exact" beta functions for the gauge coupling in N=1 supersymmetric gauge theories, the so-called NSVZ beta functions. Shifman and Vainshtein (SV) further related these beta functions to the exact 1-loop running of the "Wilsonian" gauge coupling. All these results, however, remain somewhat mysterious. We attempt to clarify these issues by presenting new perspectives on the NSVZ beta function. Our interpretation of the results is somewhat different than the one given by SV, having nothing to do with the distinction between "Wilsonian" and "1PI" effective actions. Throughout we work in the context of the Wilsonian Renormalization Group; namely, as the cutoff of the theory is changed from M to M', we determine the appropriate changes in the bare couplings needed to keep the low energy physics fixed. The entire analysis is therefore free of infrared subtleties. When the bare Lagrangian given at the cutoff is manifestly holomorphic in the gauge coupling, we show that the required change in the holomorphic gauge coupling is exhausted at 1-loop to all orders of perturbation theory, and even non-perturbatively in some cases. On the other hand, when the bare Lagrangian has canonically normalized kinetic terms, we find that the required change in the gauge coupling is given by the NSVZ beta function. The higher order contributions in the NSVZ beta function are due to anomalous Jacobians under the rescaling of the fields done in passing from holomorphic to canonical normalization. We also give prescriptions for regularizing certain N=1 theories with an ultraviolet cutoff M preserving manifest holomorphy, starting from finite N=4 and N=2 theories. It is then in principle possible to check the validity of the exact beta function by higher order calculations in these theories. 
  We find explicit expression for the one-loop four-graviton amplitude in eleven-dimensional supergravity compactified on a circle. Represented in terms of the string coupling (related to the compactification radius) it takes the form of an infinite sum of perturbative string loop corrections. We also compute the amplitude in the case of compactification on a 2-torus which is given by an SL(2,Z) invariant expansion in powers of the torus area. We discuss the structure of quantum corrections in eleven-dimensional theory and their relation to string theory. 
  We find the potential per unit length between two non-intersecting D1-branes as a function of their relative angle. 
  The SU(2) gauge invariant Dirac-Yang-Mills mechanics of spatially homogeneous isospinor and gauge fields is considered in the framework of the generalized Hamiltonian approach. The unconstrained Hamiltonian system equivalent to the model is obtained using the gaugeless method of Hamiltonian reduction. The latter includes the Abelianization of the first class constraints, putting the second class constraints into the canonical form and performing a canonical transformation to a set of adapted coordinates such that a subset of the new canonical pairs coincides with the second class constraints and part of the new momenta is equal to the Abelian constraints. In the adapted basis the pure gauge degrees of freedom automatically drop out from the consideration after projection of the model onto the constraint shell. Apart from the elimination of these ignorable degrees of freedom a further Hamiltonian reduction is achieved due to the three dimensional group of rigid symmetry possessed by the system. 
  The Schwinger model, defined in the space interval $-L \le x \le L$, with (anti)periodic boundary conditions, is canonically quantized in the light-cone gauge $A_-=0$ by means of equal-time (anti)commutation relations. The transformation diagonalizing the complete Hamiltonian is explicitly constructed, thereby giving spectrum, chiral anomaly and condensate. The structures of Hilbert spaces related both to free and to interacting Hamiltonians are completely exhibited. Besides the usual massive field, two chiral massless fields are present, which can be consistently expunged from the physical space by means of a subsidiary condition of a Gupta-Bleuler type. The chiral condensate does provide the correct non-vanishing value in the decompactification limit $L \to \infty$. 
  A crucial problem in quantum cosmology is a careful analysis of the one-loop semiclassical approximation for the wave function of the universe, after an appropriate choice of mixed boundary conditions. The results for Euclidean quantum gravity in four dimensions are here presented, when linear covariant gauges are implemented by means of the Faddeev-Popov formalism. On using zeta-function regularization and a mode-by-mode analysis, one finds a result for the one-loop divergence which agrees with the Schwinger-DeWitt method only after taking into account the non-trivial effect of gauge and ghost modes. For the gravitational field, however, the geometric form of heat-kernel asymptotics with boundary conditions involving tangential derivatives of metric perturbations is still unknown. Moreover, boundary effects are found to be responsible for the lack of one-loop finiteness of simple supergravity, when only one bounding three-surface occurs. This work raises deep interpretative issues about the admissible backgrounds and about quantization techniques in quantum cosmology. 
  We consider the most general form for eleven dimensional supersymmetry compatible with on-shell superfields. This allows for the introduction of a conformal Spin(1,10) connection. In eleven dimensional Minkowski space this modification is trivial and can be removed by a field redefinition, however, upon compactification on S^1 it is possible to introduce a non-trivial `Wilson line'. The resulting ten dimensional supergravity has massive 1-form and 3-form potentials and a cosmological constant. This theory does not possess a supersymmetric eightbrane soliton but it does admit a supersymmetric non-static cosmological solution. 
  We develop an electromagnetic symplectic structure on the space-time manifold by defining a Poisson bracket in terms of an invertible electromagnetic tensor F_{\mu\nu}. Moreover, we define electromagnetic symplectic diffeomorphisms by imposing a special condition on the electromagnetic tensor. This condition implies a special constraint on the Levi-Civita tensor. Then, we express geometrically the electromagnetic duality by defing a dual local coordinates system. 
  We discuss the spontaneous symmetry breaking (SSB) on the light front (LF) in view of the zero mode. We first demonstrate impossibility to remove the zero mode in the continuum LF theory by two examples: The Lorentz invariance forbids even a free theory on the LF and the trivial LF vacuum is lost in the SSB phase, both due to the zero mode as the accumulating point causing uncontrollable infrared singularity. We then adopt the Discretized Light-Cone Quantization (DLCQ) which was first introduced by Maskawa and Yamawaki to establish the trivial LF vacuum and was re-discovered by Pauli and Brodsky in a different context. It is shown in DLCQ that the SSB phase can be realized on the trivial LF vacuum only when an explicit symmetry-breaking mass of the Nambu-Goldstone (NG) boson m_pi is introduced as an infrared regulator. The NG-boson zero mode integrated over the LF must exhibit singular behavior sim 1/m_{pi}^2 in the symmetric limit m_{pi} rightarrow 0 in such a way that the LF charge is not conserved even in the symmetric limit; dot{Q} ne 0. There exists no NG theorem on the LF. Instead, this singular behavior establishes existence of the massless NG boson coupled to the current whose charge satisfies Q |0 >=0 and dot{Q} ne 0, in much the same as the NG theorem in the equal-time quantization which ensures existence of the massless NG boson coupled to the current whose charge satisfies Q|0 > ne 0 and dot{Q} = 0. We demonstrate such a peculiarity in a concrete model, the linear sigma model, where the role of zero-mode constraint is clarified. 
  We discuss the one-loop effective potential and static (large $d$) potential for toroidal D-brane described by DBI-action in constant magnetic and in constant electric fields. Explicit calculation is done for membrane case ($p=2$) for both types of external fields and in case of static potential for an arbitrary $p$. In the case of one-loop potential it is found that the presence of magnetic background may stabilize D-brane giving the possibility for non-pointlike ground state configuration. On the same time, constant electrical field acts against stabilization and the correspondent one-loop potential is unbounded from below. The properties of static potential which also has stable minimum are described in detail. The back-reaction of quantum gauge fields to one-loop potential is also evaluated. 
  We investigate gaugino condensation in the framework of the strongly coupled heterotic $E_8 \times E_8$ string (M--theory). Supersymmetry is broken in a hidden sector and gravitational interactions induce soft breaking parameters in the observable sector. The resulting soft masses are of order of the gravitino mass. The situation is similar to that in the weakly coupled $E_8 \times E_8$ theory with one important difference: we avoid the problem of small gaugino masses which are now comparable to the gravitino mass. 
  We consider a model of quantum field theory with higher derivatives for a spinor field with quartic selfinteraction. With the help of the Bethe-Salpeter equation we study the problem of the two particle bound states in the "chain" approximation. The existence of a scalar bound state is established. 
  Using the harmonic superspace techniques in D=2 N=4, we present an explicit derivation of a new hyper-Kahler metric associated to the Toda like self interaction $H ^{4+}(\omega, u)= (\frac{\xi^{++}}{\lambda})^{2}\exp(2\lambda \omega)$. Some important features are also discussed. 
  We consider the 1+1 dimensional supersymmetric matrix field theory obtained from a dimensional reduction of ten dimensional ${\cal N} = 1$ super Yang-Mills, which is a matrix model candidate for non-perturbative Type IIA string theory. The gauge group here is U($N$), where $N$ is sent to infinity. We adopt light-cone coordinates to parametrize the string world sheet, and choose to work in the light-cone gauge. Quantizing this theory via Discretized Light-Cone Quantization (DLCQ) introduces an integer, K, which restricts the light-cone momentum-fraction of constituent quanta to be integer multiples of 1/K. We show how a double scaling limit involving the integers $K$ and $N$ implies the existence of an extra (free) parameter in the Yang-Mills theory, which plays the role of an effective string coupling constant. The formulation here provides a natural framework for studying quantitatively string dynamics and conventional Yang-Mills in a unified setting. 
  Using the theory of supersymmetric anyons, I extend the definition of the Witten index to 2+1 dimensions so as to accommodate the existence of anyon spin and statistics. I then demonstrate that, although in general the index receives irrational and complex contributions from anyonic states, the overall index is always integral, and I consider some of the implications and interpretations of this result. 
  We extend strong/weak coupling duality to string theories without spacetime supersymmetry, and focus on the case of the unique ten-dimensional, nonsupersymmetric, tachyon-free $SO(16)\times SO(16)$ heterotic string. We construct a tachyon-free heterotic string model that interpolates smoothly between this string and the ten-dimensional supersymmetric $SO(32)$ heterotic string, and we construct a dual for this interpolating model. We find that the perturbative massless states of our dual theories precisely match within a certain range of the interpolation. Further evidence for this proposed duality comes from a calculation of the one-loop cosmological constant in both theories, as well as the presence of a soliton in the dual theory. This is therefore the first known duality relation between nonsupersymmetric tachyon-free string theories. Using this duality, we then investigate the perturbative and nonperturbative stability of the $SO(16)\times SO(16)$ string, and present a conjecture concerning its ultimate fate. 
  We consider the BPS states of the $E_8$ non-critical string wound around one of the circles of a toroidal compactification to four dimensions. These states are indexed by their momenta and winding numbers. We find explicit expressions, $G_n$, for the momentum partition functions for the states with winding number $n$. The $G_n$ are given in terms of modular forms. We give a simple algorithm for generating the $G_n$, and we show that they satisfy a recurrence relation that is reminiscent of the holomorphic anomaly equations of Kodaira-Spencer theory. 
  We apply the coadjoint orbit method to construct relativistic nonlinear sigma models (NLSM) on the target space of coadjoint orbits coupled with the Chern-Simons (CS) gauge field and study self-dual solitons. When the target space is given by Hermitian symmetric space (HSS), we find that the system admits self-dual solitons whose energy is Bogomol'nyi bounded from below by a topological charge. The Bogomol'nyi potential on the Hermitian symmetric space is obtained in the case when the maximal torus subgroup is gauged, and the self-dual equation in the $CP(N-1)$ case is explored. We also discuss the self-dual solitons in the non-compact $SU(1,1)$ case and present a detailed analysis for the rotationally symmetric solutions. 
  We introduce a new concept of quasi-Yang-Baxter algebras. The quantum quasi-Yang-Baxter algebras being simple but non-trivial deformations of ordinary algebras of monodromy matrices realize a new type of quantum dynamical symmetries and find an unexpected and remarkable applications in quantum inverse scattering method (QISM). We show that applying to quasi-Yang-Baxter algebras the standard procedure of QISM one obtains new wide classes of quantum models which, being integrable (i.e. having enough number of commuting integrals of motion) are only quasi-exactly solvable (i.e. admit an algebraic Bethe ansatz solution for arbitrarily large but limited parts of the spectrum). These quasi-exactly solvable models naturally arise as deformations of known exactly solvable ones. A general theory of such deformations is proposed. The correspondence ``Yangian --- quasi-Yangian'' and ``$XXX$ spin models --- quasi-$XXX$ spin models'' is discussed in detail. We also construct the classical conterparts of quasi-Yang-Baxter algebras and show that they naturally lead to new classes of classical integrable models. We conjecture that these models are quasi-exactly solvable in the sense of classical inverse scattering method, i.e. admit only partial construction of action-angle variables. 
  We investigate a 3+1 dimensional toy model that exhibits spontaneous breakdown of chiral symmetry, both in a light-front (LF) Hamiltonian and in a Euclidean Schwinger-Dyson (SD) formulation. We show that both formulations are completely equivalent --- provided the renormalization is properly done. For the model considered, this means that if one uses the same transverse momentum cutoff on the SD and LF formulations then the vertex mass in the LF calculation must be taken to be the same as the current quark mass in the SD calculation. The kinetic mass term in the LF calculation is renormalized non-trivially, which is eventually responsible for the mass generation of the physical fermion of the model. 
  A new formal scheme is presented in which Einstein's classical theory of General Relativity appears as the common, invariant sector of a one-parameter family of different theories. This is achieved by replacing the Poincare` group of the ordinary tetrad formalism with a q-deformed Poincare` group, the usual theory being recovered at q=1. Although written in terms of noncommuting vierbein and spin-connection fields, each theory has the same metric sector leading to the ordinary Einstein-Hilbert action and to the corresponding equations of motion. The Christoffel symbols and the components of the Riemann tensor are ordinary commuting numbers and have the usual form in terms of a metric tensor built as an appropriate bilinear in the vierbeins. Furthermore we exhibit a one-parameter family of Hamiltonian formalisms for general relativity, by showing that a canonical formalism a` la Ashtekar can be built for any value of q. The constraints are still polynomial, but the Poisson brackets are not skewsymmetric for q different from 1. 
  A representation for the phase of a chiral determinant in terms of a path integral of a local action is constructed. This representation is used to modify the action of chiral SU(2) fermions removing the global anomaly. 
  We give an explicit proof within the framework of the Bethe Ansatz/string hypothesis of the factorization of multiparticle scattering in the antiferromagnetic spin-1/2 Heisenberg spin chain, for the case of 3 particles. 
  We discuss the sigma model description of a D-string bound to k D-fivebranes in type I string theory. The effective theory is an (0,4) supersymmetric hyper-Kahler with torsion sigma model on the moduli space of Sp(k) instantons on R^4. Upon toroidal compactification to five dimensions the model is related to the type II picture where the target space is a symmetric product of K3's. 
  Additional evidence is presented for a recently proposed effective string model, conjectured to hold throughout the parameter space of the basic 5 dimensional, triply charged black holes, which includes the effects of brane excitations, as well as momentum modes. We compute the low energy spacetime absorption coefficient $\sigma$ for the scattering of a triply-charged scalar field in the near extremal case, and conjecture an exact form for $\sigma$. It is shown that this form of $\sigma$ arises simply from the effective string model. This agreement encompasses both statistical factors coming from the Bose distributions of string excitations and a prefactor which depends on the effective string radius. An interesting feature of the effective string model is that the change in mass of the effective string system in an emission process is not equal to the change in the energies of the effective string excitations. If the model is valid, this may hold clues towards understanding back reaction due to Hawking radiation. A number of weak spots and open questions regarding the model are also noted. 
  We study (4,4) supersymmetric field theories in two dimensions with a one dimensional Coulomb branch. These theories have applications in string theory. Our analysis explains the known relation between $A-D-E$ groups and modular invariants of affine SU(2). 
  The massive phase of two-layer integrable systems is studied by means of RSOS restrictions of affine Toda theories. A general classification of all possible integrable perturbations of coupled minimal models is pursued by an analysis of the (extended) Dynkin diagrams. The models considered in most detail are coupled minimal models which interpolate between magnetically coupled Ising models and Heisenberg spin-ladders along the $c<1$ discrete series. 
  Both the supersymmetric $SO(32)$ and $E_8\times E_8$ heterotic strings in ten dimensions have known strong-coupling duals. However, it has not been known whether there also exist strong-coupling duals for the non-supersymmetric heterotic strings in ten dimensions. In this paper, we construct explicit open-string duals for the circle-compactifications of several of these non-supersymmetric theories, among them the tachyon-free $SO(16)\times SO(16)$ string. Our method involves the construction of heterotic and open-string interpolating models that continuously connect non-supersymmetric strings to supersymmetric strings. We find that our non-supersymmetric dual theories have exactly the same massless spectra as their heterotic counterparts within a certain range of our interpolations. We also develop a novel method for analyzing the solitons of non-supersymmetric open-string theories, and find that the solitons of our dual theories also agree with their heterotic counterparts. These are therefore the first known examples of strong/weak coupling duality relations between non-supersymmetric, tachyon-free string theories. Finally, the existence of these strong-coupling duals allows us to examine the non-perturbative stability of these strings, and we propose a phase diagram for the behavior of these strings as a function of coupling and radius. 
  An imaginary part of the false-vacuum energy density in a metastable system, i.e., the decay width due to quantum tunneling, might be reproduced by Borel resummation of vacuum bubble diagrams. We examine the convergence of this prescription in the Gaussian propagator model, in which the analytical expression of vacuum bubbles to the ninth order of loop expansion is available. 
  We propose a unitary matrix model as a regularization of the IIB matrix model of Ishibashi-Kawai-Kitazawa-Tsuchiya (IKKT). The fermionic part is incorporated using the overlap formalism in order to avoid unwanted ``doublers'' while preserving the global gauge invariance. This regularization, unlike the one adopted by IKKT, has manifest U(1)^10 symmetry, which corresponds to the ten-dimensional translational invariance of the space time. We calculate one-loop effective action around some typical BPS-saturated configurations in the weak coupling limit. We also discuss a possible scenario for the dynamical generation of the four-dimensional space time through spontaneous breakdown of the U(1)^10 symmetry in the double scaling limit. 
  We reanalyse the question whether the quantum Bogomolnyi bound is saturated in the two-dimensional supersymmetric kink and sine-Gordon models. Our starting point is the usual expression for the one-loop correction to the mass of a soliton in terms of sums over zero-point energies. To regulate these sums, most authors put the system in a box with suitable boundary conditions, and impose an ultraviolet cut-off. We distinguish between an energy cut-off and a mode number cut-off, and show that they lead to different results. We claim that only the mode cut-off yields correct results, and only if one considers exactly the same number of bosonic and fermionic modes in the total sum over bound-state and zero-point energies. To substantiate this claim, we show that in the sine-Gordon model only the mode cut-off yields a result for the quantum soliton mass that is consistent with the exact result for the spectrum as obtained by Dashen et al. from quantising the so-called breather solution. In the supersymmetric case, our conclusion is that contrary to previous claims the quantum Bogomolnyi bound is not saturated in any of the two-dimensional models considered. 
  Type II compactifications with varying string coupling can be described elegantly in F-theory/M-theory as compactifications on U - manifolds. Using a similar approach to describe Super Yang-Mills with a varying coupling constant, we argue that at generic points in Narain moduli space, the $E_8 \times E_8$ Heterotic string compactified on $T^2$ is described in M(atrix) theory by N=4 SYM in 3+1 dimensions with base $S^1 \times CP^1$ and a holomorphically varying coupling constant. The $CP^1$ is best described as the base of an elliptic K3 whose fibre is the complexified coupling constant of the Super Yang-Mills theory leading to manifest U-duality. We also consider the cases of the Heterotic string on $S^1$ and $T^3$. The twisted sector seems to (almost) naturally appear at precisely those points where enhancement of gauge symmetry is expected and need not be postulated. A unifying picture emerges in which the U-manifolds which describe type II orientifolds (dual to the Heterotic string) as M- or F- theory compactifications play a crucial role in Heterotic M(atrix) theory compactifications. 
  We examine the dependence on all gauge parameters in the example of the Abelian Higgs model by applying a general algebraic method which roots in an extension of the usual Slavnov-Taylor identity. This method automatically yields all information about the gauge parameter dependence of Green functions and therefore especially allows to control the range of ``good'' normalization conditions. In this context we show that the physical on-shell normalization conditions are in complete agreement with the restrictions dictated by the enlarged Slavnov-Taylor identity and that the coupling can be fixed in an easily handleable way on the Ward identity of local gauge invariance. As an application of the general method we also study the Callan-Symanzik equation and the renormalization group equation of the Abelian Higgs model. 
  This extended write-up of a talk gives an introductory survey of mathematical problems of the quantization of gauge systems. Using the Schwinger model as an exactly tractable but nontrivial example which exhibits general features of gauge quantum field theory, I cover the following subjects: The axiomatics of quantum field theory, formulation of quantum field theory in terms of Wightman functions, reconstruction of the state space, the local formulation of gauge theories, indefiniteness of the Wightman functions in general and in the special case of the Schwinger model, the state space of the Schwinger model, special features of the model. New results are contained in the Mathematical Appendix, where I consider in an abstract setting the Pontrjagin space structure of a special class of indefinite inner product spaces - the so called quasi-positive ones. This is motivated by the indefinite inner product space structure appearing in the above context and generalizes results of Morchio and Strocchi [J. Math. Phys. 31 (1990) 1467], and Dubin and Tarski [J. Math. Phys. 7 (1966) 574]. 
  The thermodynamics of a gas of strings and D-branes near the Hagedorn transition is described by a coupled set of Boltzmann equations for weakly interacting open and closed long strings. The resulting distributions are dominated by the open string sector, indicating that D-branes grow to fill space at high temperature. 
  The zero-point energy of a conducting spherical shell is evaluated by imposing boundary conditions on the potential, and on the ghost fields. The scheme requires that temporal and tangential components of perturbations of the potential should vanish at the boundary, jointly with the gauge-averaging functional, first chosen of the Lorenz type. Gauge invariance of such boundary conditions is then obtained provided that the ghost fields vanish at the boundary. Normal and longitudinal modes of the potential obey an entangled system of eigenvalue equations, whose solution is a linear combination of Bessel functions under the above assumptions, and with the help of the Feynman choice for a dimensionless gauge parameter. Interestingly, ghost modes cancel exactly the contribution to the Casimir energy resulting from transverse and temporal modes of the potential, jointly with the decoupled normal mode of the potential. Moreover, normal and longitudinal components of the potential for the interior and the exterior problem give a result in complete agreement with the one first found by Boyer, who studied instead boundary conditions involving TE and TM modes of the electromagnetic field. The coupled eigenvalue equations for perturbative modes of the potential are also analyzed in the axial gauge, and for arbitrary values of the gauge parameter. The set of modes which contribute to the Casimir energy is then drastically changed, and comparison with the case of a flat boundary sheds some light on the key features of the Casimir energy in non-covariant gauges. 
  It is proved that the moduli space of static solutions of the CP^1 model on spacetime Sigma x R, where Sigma is any compact Riemann surface, is geodesically incomplete with respect to the metric induced by the kinetic energy functional. The geodesic approximation predicts, therefore, that lumps can collapse and form singularities in finite time in these models. 
  We study the configuration of a typical highly excited string as one slowly increases the string coupling. The dominant interactions are the long range dilaton and gravitational attraction. In four spacetime dimensions, the string slowly contracts from its initial (large) size until it approaches the string scale where it forms a black hole. In higher dimensions, the string stays large until the coupling reaches a critical value, and then it rapidly collapses to a black hole. The implications for the recently proposed correspondence principle are discussed. 
  Nottale's special scale-relativity principle was proposed earlier by the author as a plausible geometrical origin to string theory and extended objects. Scale Relativity is to scales what motion Relativity is to velocities. The universal, absolute, impassible, invariant scale under dilatations in Nature is taken to be the Planck scale, which is not the same as the string scale. Starting with ordinary actions for strings and other extended objects, we show that gauge theories of volume-resolutions scale-relativistic symmetries, of the world volume measure associated with the extended ``fuzzy'' objects, are a natural and viable way to formulate the geometrical principle underlying the theory of all extended objects. Gauge invariance can only be implemented if the extendon actions in $D$ target dimensions are embedded in $D+1$ dimensions with an extra temporal variable corresponding to the scaling dimension of the original string coordinates. This is achieved upon viewing the extendon coordinates, from the fuzzy worldvolume point of view, as noncommuting matrices valued in the Lie algebra of Lorentz-scale relativistic transformations. Preliminary steps are taken to merge motion relativity with scale relativity by introducing the gauge field that gauges the Lorentz-scale symmetries in the same vain that the spin connection gauges ordinary Lorentz transformations and, in this fashion, one may go beyond string theory to construct the sought-after General Theory of Scale-Motion Relativity. Such theory requires the introduction of the scale-graviton (in addition to the ordinary graviton) which is the field that gauges the symmetry which converts motion dynamics into scaling-resolutions dynamics and vice versa (the analog of the gravitino that gauges supersymmetry). To go beyond the quantum string geometry most probably 
  It is shown that an extended q-deformed $su(2)$ algebra with an extra (``Schwinger '') term can describe Bloch electrons in a uniform magnetic field with an additional periodic potential. This is a generalization of the analysis of Bloch electrons by Wiegmann and Zabrodin. By using a representation theory of this q-deformed algebra, we obtain functional Bethe ansatz equations whose solutions should be functions of finite degree. It is also shown that the zero energy solution is expressed in terms of an Askey Wilson polynomial. 
  We investigate exact results of isotropic turbulence in three-dimensions when the pressure gradient is negligible. We derive exact two-point correlation functions of density in three-dimensions and show that the density-density correlator behaves as $ |{x_1 - x_2}|^{-\alpha_3}$, where $\alpha_3 = 2 + \frac{\sqrt{33}}{6}$. It is shown that, in three-dimensions, the energy spectrum $E(k)$ in the inertial range scales with exponent $ 2 - \frac {\sqrt{33}}{12} \simeq 1.5212$. We also discuss the time scale for which our exact results are valid for strong 3D--turbulence in the presence of the pressure. We confirm our predictions by using the recent results of numerical calculations and experiment. 
  We describe special supersymmetric gauge theories in three, five, seven and nine dimensions, whose compactification on two-, four-, six- and eight-folds produces a supersymmetric quantum mechanics on moduli spaces of holomorphic bundles and/or solutions to the analogues of instanton equations in higher dimensions. The theories may occur on the worldvolumes of D-branes wrapping manifolds of special holonomy. We also discuss the theories with matter. 
  The geodesic motion of pseudo-classical spinning particles in Euclidean Taub-NUT space is analysed. The constants of motion are expressed in terms of Killing-Yano tensors. Some previous results from the literature are corrected. 
  We study static, spherically symmetric, and purely magnetic solutions of SU(2) $\times$ SU(2) gauge supergravity in four dimensions. A systematic analysis of the supersymmetry conditions reveals solutions which preserve 1/8 of the supersymmetries and are characterized by a BPS-monopole-type gauge field and a globally hyperbolic, everywhere regular geometry. These present the first known example of non-Abelian backgrounds in gauge supergravity and in leading order effective string theory. 
  A quantum mechanical path integral derivation is given of a thermal propagator in non-static Gui spacetime. The thermal nature of the propagator is understood in terms of homotopically non-trivial paths in the configuration space appropriate to tortoise coordinates. The connection to thermal emission from collapsing black holes is discussed. 
  We show that there are no nontrivial solutions of the Seiberg-Witten equations on R^8 with constant standard spin^c structure. 
  We study the spectrum of BPS states in N=4 supersymmetric U(N) Yang-Mills theory. This theory has been proposed to describe M-theory on T^3 in the discrete light-cone formalism. We find that the degeneracy of irreducible BPS bound states in this model exhibits a (partially hidden) SL(5,Z) duality symmetry. Besides the electro-magnetic symmetry, this duality group also contains Nahm-like transformations that interchange the rank N of the gauge group with some of the magnetic or electric fluxes. In the M-theory interpretation, this mapping amounts to a reflection that interchanges the longitudinal direction with one of the transverse directions. 
  We investigate consequences of the effective colour-dielectric formulation of lattice gauge theory using the light-cone Hamiltonian formalism with a transverse lattice (hep-ph/9704408). As a quantitative test of this approach, we have performed extensive analytic and numerical calculations for 2+1-dimensional pure gauge theory in the large $N$ limit. We study the structure of coupling constant space for our effective potential by comparing with results available from conventional Euclidean lattice Monte Carlo simulations of this system. In particular, we calculate and measure the scaling behaviour of the entire low-lying glueball spectrum, glueball wavefunctions, string tension, asymptotic density of states, and deconfining temperature. 
  We explore the relationship between black holes in Jackiw-Teitelboim(JT) dilaton gravity and solitons in sine-Gordon field theory. Our analysis expands on the well known connection between solutions of the sine-Gordon equation and constant curvature metrics. In particular, we show that solutions to the dilaton field equations for a given metric in JT theory also solve the sine-Gordon equation linearized about the corresponding soliton. Since the dilaton generates Killing vectors of the constant curvature metric, it is interesting that it has an analoguous interpretation in terms of symmetries of the soliton solution. We also show that from the B${\ddot a}$cklund transformations relating different soliton solutions, it is possible to construct a flat SL(2,R) connection which forms the basis for the gauge theory formulation of JT dilaton gravity. 
  The low energy effective actions which arise from string theory or M-theory are considered in the cosmological context, where the graviton, dilaton and antisymmetric tensor field strengths depend only on time. We show that previous results can be extended to include cosmological solutions that are related to the E_N Toda equations. The solutions of the Wheeler-DeWitt equation in minisuperspace are obtained for some of the simpler cosmological models by introducing intertwining operators that generate canonical transformations which map the theories into free theories. We study the cosmological properties of these solutions, and also briefly discuss generalised Brans-Dicke models in our framework. The cosmological models are closely related to p-brane solitons, which we discuss in the context of the E_N Toda equations. We give the explicit solutions for extremal multi-charge (D-3)-branes in the truncated system described by the D_4 =O(4,4) Toda equations. 
  The null string's equations of motion and constraints in the Kerr spacetime are given. We assume a generic ansatz for the null strings in the Kerr spacetime and we present the resulting solutions in quadratures. Some specific string configurations, that follow from the generic one, are considered separately. In each case we also extract the corresponding solutions in the Schwarzschild spacetime. 
  It is shown that the equations of motion of eleven-dimensional supergravity follow from setting the dimension zero components of the superspace torsion tensor equal to the Dirac matrices. The proof of this assertion is facilitated by the introduction of a connection taking its values in the Lie algebra of $Spin(1,10)\times R^+$. 
  We present a reformulation of SU(2) Yang-Mills theory in the maximal Abelian gauge, where the non-Abelian gauge field components are exactly integrated out at the expense of a new Abelian tensor field. The latter can be treated in a semiclassical approximation and the corresponding saddle point equation is derived. Besides the non-trivial solutions, which are presumably related to non-perturbative interactions for the Abelian gauge field, the equation of motion for the tensor fields allows for a trivial solution as well. We show that the semiclassical expansion around this trivial solution is equivalent to the standard perturbation theory. In particular, we calculate the one-loop $\beta$-function for the running coupling constant in this approach and reproduce the standard result. 
  We consider Type II string theories on ${\bf T^n}/{{\bf Z_2}^m}$ Joyce orbifolds. This class contains orbifolds which can be desingularised to give manifolds of $G_2$ $({\bf n}$$=$$7)$ and $Spin(7)$ holonomy $({\bf n}$$=$$8)$. In the $G_2$ holonomy case we present two types of $T$-duality transformation which are clearly generalisations of the $T$-duality/mirror transformation in Calabi-Yau spaces. The first maps Type IIA theory on one such space from this class to Type IIB theory on another such space. The second maps Type IIA (IIB) to Type IIA (IIB). In the case of $Spin(7)$ holonomy we present a $T$-duality transformation which maps Type IIA (IIB) theory on one such space to Type IIA (IIB) on another such space. As orbifold conformal field theories these $T$-dual target spaces are related via the inclusion/exclusion of discrete torsion and the $T$-duality is proven to genus $g$ in string perturbation theory. We then apply a Strominger, Yau, Zaslow type argument which suggests that manifolds of $G_2$ holonomy which have a ``mirror'' of the first (second) type admit supersymmetric ${\bf T^3}$ (${\bf T^4}$) fibrations and that manifolds of $Spin(7)$ holonomy for which a mirror exists admit fibration by supersymmetric $4$-tori. Further evidence for this suggestion is given by examining the moduli space structure of wrapped D-branes. 
  Feynman diagrams are the best tool we have to study perturbative quantum field theory. For this very reason the development of any new technique which allows us to compute Feynman integrals is welcome. By the middle of the 80's, Halliday and Ricotta suggested the possibility of using negative dimensional integrals to tackle the problem. The aim of this work is to revisit the technique as such and check up on its possibilities. For this purpose, we take a box diagram integral contributing to the photon-photon scattering amplitude in quantum electrodynamics using the negative dimensional integration method. The reason for this choice of ours is twofold: Firstly, it is a well-studied integral with well-known results, and secondly because it bears in its integrand the complexities associated with four massive propagators of the intermediate states. 
  The exact decay rate for emission of massless minimally coupled scalar fields by a non-extremal black hole in 2+1 dimensions is obtained. At low energy, the decay rate into scalars with zero angular momentum is correctly reproduced within conformal field theory. The conformal field theory has both left- and right-moving sectors and their contribution to the decay rate is associated naturally with left and right temperatures of the black hole. 
  We apply the string inspired worldline formalism to the calculation of the higher derivative expansion of one-loop effective actions in non-Abelian gauge theory. For this purpose, we have completely computerized the method, using the symbolic manipulation programs FORM, PERL and M. Explicit results are given to sixth order in the inverse mass expansion, reduced to a minimal basis of invariants specifically adapted to the method. Detailed comparisons are made with other gauge-invariant algorithms for calculating the same expansion. This includes an explicit check of all coefficients up to fifth order. 
  The infinite limit of Matrix Theory in 4 and 10 dimensions is described in terms of Moyal Brackets. In those dimensions there exists a Bogomol'nyi bound to the Euclideanized version of these equations, which guarantees that solutions of the first order equations also solve the second order Matrix Theory equations. A general construction of such solutions in terms of a representation of the target space co-ordinates as non-local spinor bilinears, which are generalisations of the standard Wigner functions on phase space, is given. 
  We present a candidate of anomaly and Wess Zumino action of the two dimensional supergravity coupling with matters in a super-Weyl invariant regularization. It is a generalization of the Weyl and the area preserving \Diff invariant formulation of two dimensional gravity theory. 
  Type IIA brane configurations are used to construct N=2 supersymmetric gauge theories in two dimensions. Using localization of chiral multiplets in ten-dimensional spacetime, supersymmetric non-linear sigma models with target space such as $\CP^{n-1}$ and the Grassmann manifolds are studied in detail. The quantum properties of these models are realized in $M$ theory by taking the strong Type IIA coupling limit. The brane picture implies an equivalence between the parameter space of N=2 supersymmetric theories in two dimensions and the moduli space of vacua of N=2 supersymmetric gauge theories in four dimensions. Effects like level-rank duality are interpreted in the brane picture as continuation past infinite coupling. The BPS solitons of the $\CP^{n-1}$ model are identified as topological excitations of a membrane and their masses are computed. This provides the brane realization of higher rank tensor representations of the flavor group. 
  The lagrangian of N=2, D=6 supergravity coupled to E_7 X SU(2) vector- and hyper-multiplets is derived. For this purpose the coset manifold E_8/E_7 X SU(2), parametrized by the scalars of the hypermultiplet, is constructed. A difference from the case of Sp(n)-matter is pointed out. This model can be considered as an intermediate step in the compactification of D=10 supergravity coupled to E_8 X E_8 matter to four-dimensional model of E_6 unification. 
  A duality invariant action for (1,1) supersymmetric extension of Poisson-Lie dualizable $\sigma$-models is constructed. 
  A particle spectrum below the string scale in accordance with predictions from heterotic string theory yields a Planck mass $m_{Pl}=(8\pi G_N)^{-1/2}$ which exceeds the string scale by a factor $\simeq 61.9$. A Planck mass $m_{Pl}=2.43\times 10^{18}$ GeV then corresponds to a string scale $m_s=3.9\times 10^{16}$ GeV. Such a low value for the string scale in turn implies that the relative strength of graviton and vector exchange in the string/M-theory phase exceeds the corresponding ratio in the low energy field theory. 
  The geometric aspect of Wick rotation in quantum field theory and its localization on manifolds are explored. After the explanation of the notion and its related geometric objects, we study the topology of the set of landing $W$ for Wick rotations and its natural stratification. These structures in two, three, and four dimensions are computed explicitly. We then focus on more details in two dimensions. In particular, we study the embedding of $W$ in the ambient space of Wick rotations, the resolution of the generic metric singularities of a Lorentzian surface $\Sigma$ by local Wick rotations, and some related $S^1$-bundles over $\Sigma$. 
  We review progress in studying the solutions of SQCD in the presence of explicit, soft SUSY breaking terms. Massive N=1 SQCD in the presence of a small gaugino mass leads to a controlled solution which has interesting phase structure with changing theta angle reminiscent of the QCD chiral Lagrangian. Current attempts to test the solutions of pure glue SQCD on the lattice require a theoretical understanding of the theory with small gaugino mass in order to understand the approach to the SUSY point. We provide such a description making predictions for the gaugino condensate and lightest bound state masses. Finally we briefly review recent D-brane constructions of 4D gauge theories in string theory including a non-supersymmetric configuration. We identify this configuration with softly broken N=2 SQCD. 
  The moduli spaces of two (0,2) compactifications of the heterotic string can share the same Landau-Ginzburg model even though at large radius they look completely different. It was argued that such a pair of (0,2) models might be connected via a perturbative transition at the Landau-Ginzburg point. Situations of this kind are studied for some explicit models. By calculating the exact dimensions of the generic moduli spaces at large radius, strong indications are found in favor of a different scenario. The two moduli spaces are isomorphic and complex, K\"ahler and bundle moduli get exchanged. 
  We compute the exact canonically induced parity breaking part of the effective action for 2+1 massive fermions in particular Abelian and non Abelian gauge field backgrounds. The method of computation resorts to the chiral anomaly of the dimensionally reduced theory. 
  We study open supermembranes in 11 dimensional rigid superspace with 6 dimensional topological defects (M-theory five-branes). After rederiving in the Green-Schwarz formalism the boundary conditions for open superstrings in the type IIA theory, we determine the boundary conditions for open supermembranes by imposing kappa symmetry and invariance under a fraction of 11 dimensional supersymmetry. The result seems to imply the self-duality of the three-form field strength on the five-brane world volume. We show that the light-cone gauge formulation is regularized by a dimensional reduction of a 6 dimensional N=1 super Yang-Mills theory with the gauge group SO(N\to\infty). We also analyze the SUSY algebra and BPS states in the light-cone gauge. 
  Using the tools of q--differential calculus and quantum Lie algebras associated to quantum groups, we find a one--parameter family of q-gauge theories associated to the quantum group $ISO_q(3,1)$. Although the gauge fields, that is the spin--connection and the vierbeins are non--commuting objects depending on a deformation parameter, $q$, it is possible to construct out of them a metric theory which is insensitive to the deformation. The Christoffel symbols and the Riemann tensor are ordinary commuting objects. Hence it is argued that torsionless Einstein's General Relativity is the common invariant sector of a one--parameter family of deformed gauge theories. 
  We examine the structure of spacetime symmetries of toroidally compactified string theory within the framework of noncommutative geometry. Following a proposal of Frohlich and Gawedzki, we describe the noncommutative string spacetime using a detailed algebraic construction of the vertex operator algebra. We show that the spacetime duality and discrete worldsheet symmetries of the string theory are a consequence of the existence of two independent Dirac operators, arising from the chiral structure of the conformal field theory. We demonstrate that these Dirac operators are also responsible for the emergence of ordinary classical spacetime as a low-energy limit of the string spacetime, and from this we establish a relationship between T-duality and changes of spin structure of the target space manifold. We study the automorphism group of the vertex operator algebra and show that spacetime duality is naturally a gauge symmetry in this formalism. We show that classical general covariance also becomes a gauge symmetry of the string spacetime. We explore some larger symmetries of the algebra in the context of a universal gauge group for string theory, and connect these symmetry groups with some of the algebraic structures which arise in the mathematical theory of vertex operator algebras, such as the Monster group. We also briefly describe how the classical topology of spacetime is modified by the string theory, and calculate the cohomology groups of the noncommutative spacetime. A self-contained, pedagogical introduction to the techniques of noncommmutative geometry is also included. 
  In these lectures we give a geometrical formulation of N-extended supergravities which generalizes N=2 special geometry of N=2 theories. In all these theories duality symmetries are related to the notion of "flat symplectic bundles" and central charges may be defined as "sections" over these bundles. Attractor points giving rise to "fixed scalars" of the horizon geometry and Bekenstein-Hawking entropy formula for extremal black-holes are discussed in some details. 
  We show in a precise way, either in the fermionic or its bosonized version, that Bose symmetry provides a systematic way to carry out the chiral decomposition of the two dimensional fermionic determinant. Interpreted properly, we show that there is no obstruction of this decomposition to gauge invariance, as is usually claimed. Finally, a new way of interpreting the Polyakov-Wiegman identity is proposed. 
  We give a confirmation of U-duality of type II superstring by discussing mass spectrum of the BPS states. We first evaluate the mass spectrum of BPS solitons with one kind of R-R charges. Our analysis is based on the 1-loop effective action of D-brane, which is known as ``Dirac-Born-Infeld (DBI) action'', and the fact that BPS states correspond to the SUSY cycles with minimal volumes. We show the mass formula derived in this manner is completely fitted with that given by U-duality. We further discuss the cases of BPS solitons possessing several kinds of R-R charges. These are cases of ``intersecting D-branes'', which cannot be described by simple DBI actions. We claim that, in these cases, higher loop corrections should be incorporated as binding energies between the branes. It is remarkable that the summation of the contributions from all loops reproduces the correct mass formula predicted by U-duality. 
  We are used to thinking of an operator acting once, twice, and so on. However, an operator acting integer times can be consistently analytic continued to an operator acting complex times. Applications: (s,r) diagrams and an extension of Fractional Calculus where commutativity of fractional derivatives is preserved, generating integrals and non-standard derivations of theorems in Number Theory, non-integer power series and breaking of Leibniz and Chain rules, pseudo-groups and symmetry deforming models in particle physics and cosmology, non-local effect in analytic continued matrix representations, particle-physics-like scatterings of zeros of analytic continued Bernoulli polynomials (physics/9705021), analytic continuation of operators in QM, QFT and Strings. 
  We show that there exist nonlinearly realised duality symmetries that are independent of the standard supergravity global symmetries, and which provide active spectrum-generating symmetries for the fundamental BPS solitons. The additional ingredient, in any spacetime dimension, is a single scaling transformation that allows one to map between BPS solitons with different masses. Without the inclusion of this additional transformation, which is a symmetry of the classical equations of motion, but not the action, it is not possible to find a spectrum-generating symmetry. The necessity of including this scaling transformation highlights the vulnerability of duality multiplets to quantum anomalies. We argue that fundamental BPS solitons may be immune to this threat. 
  We present an additional test of the recent proposal for describing supersymmetry breaking due to gaugino condensation in the strong coupling regime, by a Scherk-Schwarz mechanism on the eleventh dimension of M-theory. An analysis of supersymmetric transformations in the infinite-radius limit reveals the presence of a discontinuity in the spinorial parameter, which coincides with the result found in the presence of gaugino condensation. The condensate is then identified with the quantized parameter entering the modification of the Scherk-Schwarz boundary conditions. This mechanism provides an alternative perturbative explanation of the gauge hierarchy that determines the scale of low-energy supersymmetry breaking in terms of the unification gauge coupling 
  Lectures given at the First School on Field Theory and Gravitation, Vit\'{o}ria, Esp\'{\i}rito Santo, Brazil, 15-19 April, 1997. 
  We show that, with the help of a general BRST symmetry, different theories in 3 dimensions can be connected through a fundamental topological field theory related to the classical limit of the Chern-Simons model. 
  There are two different spectral flows on the N=2 superconformal algebras (four in the case of the Topological algebra). The usual spectral flow, first considered by Schwimmer and Seiberg, is an even transformation, whereas the spectral flow previously considered by the author and Rosado is an odd transformation. We show that the even spectral flow is generated by the odd spectral flow, and therefore only the latter is fundamental. We also analyze thoroughly the four ``topological'' spectral flows, writing two of them here for the first time. Whereas the even and the odd spectral flows have quasi-mirrored properties acting on the Antiperiodic or the Periodic algebras, the topological even and odd spectral flows have drastically different properties acting on the Topological algebra. The other two topological spectral flows have mixed even and odd properties. We show that the even and the even-odd topological spectral flows are generated by the odd and the odd-even topological spectral flows, and therefore only the latter are fundamental. 
  In this paper we discuss the singularities in the Yukawa and gauge couplings of N=1 compactifications of the SO(32) heterotic string in four space-time dimensions. Such singularities can arise from the strong coupling dynamics of a confined non-perturbative gauge group. 
  An algorithm for the isolation of any singularity of f-matrix models in the double scaling limit is presented. In particular it is proved by construction that only those universality classes exist that are known from 2-matrix models. 
  We calculate the metric on the D-brane vacuum moduli space for backgrounds of the form C^3/Gamma for cyclic groups Gamma. In the simplest procedure --- starting with a flat ``seed'' metric on the covering space --- we find that the resulting D-brane metric is not Ricci-flat. We argue that this is likely to be true of the true 0-brane metric at weak string coupling. 
  We construct superstring theories that obey the new supersymmetry algebra {Q_a , Q_b}=\gamma_{ab}^{mn} P_{1m} P_{2n}, in a Green-Schwarz formalism, with kappa supersymmetry also of the new type. The superstring is in a system with a superparticle so that their total momenta are $P_{2n},P_{1m}$ respectively. The system is covariant and critical in (10,2) dimensions if the particle is massless and in (9,2) dimensions if the particle is massive. Both the superstring and superparticle have coordinates with two timelike dimensions but each behaves effectively as if they have a single timelike dimension. This is due to gauge symmetries and associated constraints. We show how to generalize the gauge principle to more intricate systems containing two parts, 1 and 2. Each part contains interacting constituents, such as p-branes, and each part behaves effectively as if they have one timelike coordinate, although the full system has two timelike coordinates. The examples of two superparticles, and of a superparticle and a superstring, discussed in more detail are a special cases of such a generalized interacting system. 
  A possible model for quantum kinematics of a test particle in a curved space-time is proposed. Every reasonable neighbourhood V_e of a curved space-time can be equipped with a nonassociative binary operation called the geodesic multiplication of space-time points. In the case of the Minkowski space-time, left and right translations of the geodesic multiplication coincide and amount to a rigid shift of the space-time x->x+a. In a curved space-time infinitesimal geodesic right translations can be used to define the (geodesic) momentum operators. The commutation relations of position and momentum operators are taken as the quantum kinematic algebra. As an example, detailed calculations are performed for the space-time of a weak plane gravitational wave. The uncertainty relations following from the commutation rules are derived and their physical meaning is discussed. 
  We study the exceptional U duality group $E_d$ of M-theory compactified on a d-torus and its representations using Matrix theory. We exhibit the $E_d$ structure and show that p-branes wrapped or unwrapped around the longitudinal direction form representations of the U duality group together with other, more mysterious, states. 
  It is shown, by using Grassmann space to describe the internal degrees of freedom of fermions and bosons, that the Weyl like equation exists not only for massless fermions but also for massless gauge bosons. The corresponding states have well defined helicity and handedness. It is shown that spinors and gauge bosons of the same handedness only interact. 
  We construct the Euclidean Green functions for the soliton (magnetic monopole) field in the U(1)_4 Lattice Gauge Theory with Wilson action. We show that in the strong coupling regime there is monopole condensation while in the QED phase the physical Hilbert space splits into orthogonal soliton sectors labelled by integer magnetic charge. 
  I show that the spin-statistics theorem has been confused with another theorem that I call the spin-locality theorem. I also argue that the spin-statistics theorem properly depends on the properties of asymptotic fields which are free fields. In addition, I discuss how ghosts evade both theorems, give the basis of the spin-statistics theorem for fields without asymptotic limits such as quark and gluon fields, and emphasise the weakness of the requirements for the $TCP$ theorem. 
  The canonical quantum theory of a free field using arbitrary foliations of a flat two-dimensional spacetime is investigated. It is shown that dynamical evolution along arbitrary spacelike foliations is unitarily implemented on the same Fock space as that associated with inertial foliations. It follows that the Schrodinger picture exists for arbitrary foliations as a unitary image of the Heisenberg picture for the theory. An explicit construction of the Schrodinger picture image of the Heisenberg Fock space states is provided. The results presented here can be interpreted in terms of a Dirac constraint quantization of parametrized field theory. In particular, it is shown that the Schrodinger picture physical states satisfy a functional Schrodinger equation which includes a slice-dependent c-number quantum correction, in accord with a proposal of Kuchar. The spatial diffeomorphism invariance of the Schrodinger picture physical states is established. Fundamental difficulties arise when trying to generalize these results to higher-dimensional spacetimes. 
  We identify and evaluate a class of physical amplitudes in four-dimensional N=4 superstring theory, which receive, in the weak coupling limit, contributions of order e^{-1/\lambda}, where \lambda is the type II superstring coupling constant. They correspond to four-derivative \Ftilde_1 interaction terms involving the universal type II dilaton supermultiplet. The exact result, obtained by means of a one-loop computation in the dual heterotic theory compactified on T^6, is compared with the perturbation theory on the type II side, and the e^{-1/\lambda} contributions are associated to non-perturbative effects of Euclidean solitons (D-branes) wrapped on K3 x T^2. The ten-dimensional decompactification limit on the type IIB side validates the recent conjecture for the D-instanton-induced R^4 couplings. 
  We derive the explicit form, and discuss some properties of the moduli dependent effective potential arising from M-theory compactified on $M_4 \times X\times S^1 / Z_2 $, when one of the boundaries supports a strongly interacting gauge sector and induces gaugino condensation. We discuss the relation between the explicit gaugino condensate and effective superpotential formulations and find interesting differences with respect to the situation known from the weakly coupled heterotic string case. The moduli dependence of the effective potential turns out to be more complicated than expected, and perhaps offers new clues to the stabilization problem. 
  The interplay between gravitational couplings on branes and the occurrence of fractional flux in low dimensional orientifolds is examined. It is argued that gravitational couplings need to be assigned not only to D-branes but also to orientifold planes. The fractional charges of the orientifold $d$-planes can be understood in terms of flux quantization of the $d-3$ form potential and modified Bianchi identities. Detailed results are presented for the case of the type IIB orientifold on $T^6/Z_2$, which is dual to F-theory on a complex 4-fold with terminal singularities. 
  The three-dimensional topologies of the membrane of M-theory can be constructed by performing Dehn surgery along knot lines. We investigate membranes wrapped around a circle and the correponding subset of topologies (Seifert manifolds). The knot lines are interpreted as magnetic flux tubes in an XY model coupled to Maxwell theory. In this model the eleventh dimension of M-theory gets ``eaten'' by the world-brane metric. There is argued to be a second-order phase transition at a critical value of the string coupling constant. The topology fluctuations that correspond to the knot lines are irrelevant in one phase while they condense in the other phase. 
  We investigate the geometry of the moduli spaces of dual electric and magnetic N=1 supersymmetric field theories. Using the SU(N_c) gauge group as a guideline we show that the electric and magnetic moduli spaces coincide for a suitable choice of the Kahler potential of the magnetic theory. We analyse the Kahler structure of the dual moduli spaces. 
  The algebra of linear and quadratic functions of basic observables on the phase space of either the free particle or the harmonic oscillator possesses a finite-dimensional anomaly. The quantization of these systems outside the critical values of the anomaly leads to a new degree of freedom which shares its internal character with spin, but nevertheless features an infinite number of different states. Both are associated with the transformation properties of wave functions under the Weyl-symplectic group $WSp(6,\Re)$. The physical meaning of this new degree of freedom can be established, with a major scope, only by analysing the quantization of an infinite-dimensional algebra of diffeomorphisms generalizing string symmetry and leading to more general extended objects. 
  We discuss the relation between supersymmetric gauge theory of branes and supergravity; as it was discovered in D-brane physics, and as it appears in Matrix theory, with emphasis on motion in curved backgrounds. We argue that gauged sigma model Lagrangians can be used as definitions of Matrix theory in curved space. Lecture given at Strings '97; June 20, 1997. 
  Correlation functions of Toda field vertices are investigated by applying the method of integrating zero-mode developed for Liouville theory. We generalize the relations among the zero-, two- and three-point couplings known in Liouville case to arbitrary Toda theories. Two- and three-point functions of Toda vertices associated with the simple roots are obtained. 
  The content of this paper is incorporated into hep-th/9805093 
  The utility of lattice discretization technique is demonstrated for solving nonrelativistic quantum scattering problems and specially for the treatment of ultraviolet divergences in these problems with some potentials singular at the origin in two and three space dimensions. This shows that lattice discretization technique could be a useful tool for the numerical solution of scattering problems in general. The approach is illustrated in the case of the Dirac delta function potential. 
  This is a summary of the work in the author's recent paper with this title written with Jerome Gauntlett, George Papadopoulos and Paul Townsend, hep-th/9702202. We showed how to construct hyper-K\"ahler 8-metrics in terms of arrangements of three-dimensional hyperplanes in six-dimensional euclidean space. The slopes of the planes define two relatively prime integers $(p,q)$. Under reduction to ten dimensions and T-duality we get a geometric picture of the action of $SL(2,Z)$ on the $(NS \otimes NS,q R \otimes R)$ 5-branes of type IIB string theory. Configurations are exhibited with $3 \over 16$'th SUSY. 
  The colliding plane wave metric discovered by Ferrari and Iba\~{n}ez to be locally isometric to the interior of a Schwarzschild black hole is extended to the case of general axion-dilaton black holes. Because the transformation maps either black hole horizon to the focal plane of the colliding waves, this entire class of colliding plane wave spacetimes only suffers from the formation of spacetime singularities in the limits where the inner horizon itself is singular, which occur in the Schwarzschild and dilaton black hole limits. The supersymmetric limit corresponding to the extreme axion-dilaton black hole yields the Bertotti-Robinson metric with the axion and dilaton fields flowing to fixed constant values. The maximal analytic extension of this metric across the Cauchy horizon yields a spacetime in which two sandwich waves in a cylindrical universe collide to produce a semi-infinite chain of Reissner-Nordstrom-like wormholes. The focussing of particle and string geodesics in this spacetime is explored. 
  The functional Schrodinger picture formulation of quantum field theory and the variational Gaussian approximation method based on the formulation are briefly reviewed. After presenting recent attempts to improve the variational approximation, we introduce a new systematic method based on the background field method, which enables one to compute the order-by-order correction terms to the Gaussian approximation of the effective action. 
  The sinh-Gordon model with integrable boundary conditions is considered in low order perturbation theory. It is pointed out that results obtained by Ghoshal for the sine-Gordon breather reflection factors suggest an interesting dual relationship between models with different boundary conditions. Ghoshal's formula for the lightest breather is checked perturbatively to $O(\beta^2)$ in the special set of cases in which the $\phi\to -\phi$ symmetry is maintained. It is noted that the parametrisation of the boundary potential which is natural for the semi-classical approximation also provides a good parametrisation at the `free-fermion' point. 
  In this paper we present a superspace formulation of $N = 1, D = 6$ supergravity with one tensor-multiplet and an arbitrary number of vector- and hypermultiplets, in which the bosonic abelian superforms of the theory, the dilaton, the abelian gauge fields and the two-form are replaced by their S-duals i.e. four, three and two-superforms respectively, in compatibility with supersymmetry. As usual this replacement interchanges Bianchi identities with equations of motion. This formulation holds in the presence of one tensor multiplet and arbitrary numbers of hypermultiplets and abelian super-Maxwell multiplets if all couplings are minimal. We determine the consistency conditions for non-minimal couplings in $N = 1$, $D = 6$ supergravity, for which we present a particularly significant solution, namely the one associated with the Chern-Simons-Lorentz three-form which entails the Green-Schwarz anomaly cancellation mechanism. In the case of non minimal couplings it is found that the gauge fields and the two-form can still be dualized while the dilaton has to remain a zero-form. 
  The implementation of modular invariance on the torus as a phase space at the quantum level is discussed in a group-theoretical framework. Unlike the classical case, at the quantum level some restrictions on the parameters of the theory should be imposed to ensure modular invariance. Two cases must be considered, depending on the cohomology class of the symplectic form on the torus. If it is of integer cohomology class $n$, then full modular invariance is achieved at the quantum level only for those wave functions on the torus which are periodic if $n$ is even, or antiperiodic if $n$ is odd. If the symplectic form is of rational cohomology class $\frac{n}{r}$, a similar result holds --the wave functions must be either periodic or antiperiodic on a torus $r$ times larger in both direccions, depending on the parity of $nr$. Application of these results to the Abelian Chern-Simons is discussed. 
  We show that there is no off-shell Palatini formulation of minimal supergravity. Nonetheless, we have been able to generalize the multiplet and Lagrangians of this theory to the case of vanishing determinant of the vierbein. Unfortunately, the requirement of regularity does not single out a unique action. 
  We compute the running of the cosmological constant and Newton's constant taking into account the effect of quantum fields with any spin between 0 and 2. We find that Newton's constant does not vary appreciably but the cosmological constant can change by many orders of magnitude when one goes from cosmological scales to typical elementary particle scales. In the extreme infrared, zero modes drive the cosmological constant to zero. 
  We propose a simple method for constructing representations of (super)conformal and nonlinear W-type algebras in terms of their subalgebras and corresponding Nambu-Goldstone fields. We apply it to N=2 and N=1 superconformal algebras and describe in this way various embeddings of strings and superstrings for which these algebras and their subalgebras define world-sheet symmetries. Besides reproducing the known examples, we present some new ones, in particular an embedding of the bosonic string with additional U(1) affine symmetry into N=2 superstring. We also apply our method to the nonlinear $W_3^{(2)}$ algebra and demonstrate that the linearization procedure worked out for it some time ago gets a natural interpretation as a kind of string embedding. All these embeddings include the critical ones as particular cases. 
  A natural SL(2,Z) invariant generalization of the Veneziano amplitude in type IIB superstring theory is investigated. It includes certain perturbative and non-perturbative (D-instanton) contributions, and it reduces to the correct expressions in different limits. The singularities are poles in the $s$-$t$-$u$ channels, corresponding to the exchange of particles with a mass spectrum coinciding with that of $(p,q)$ string states. We describe the general structure of the associated perturbative corrections to the effective action. 
  In these proceedings for the First School on Field Theory and Gravitation (Vit\'oria, Brasil), a brief introduction is given to superstring theory and its duality symmetries. This introduction is intended for beginning graduate students with no prior knowledge of string theory. 
  We study a class of four dimensional, anisotropic string backgrounds and analyse their expansion and singularity structure. In particular we will see how O(3,3) duality acts on this class. 
  We study aspects of confinement in the M theory fivebrane version of QCD (MQCD). We show heavy quarks are confined in hadrons (which take the form of membrane-fivebrane bound states) for N=1 and softly broken N=2 SU(Nc) MQCD. We explore and clarify the transition from the exotic physics of the latter to the standard physics of the former. In particular, the many strings and quark-antiquark mesons found in N=2 field theory by Douglas and Shenker are reproduced. It is seen that in the N=1 limit all but one such meson disappears while all of the strings survive. The strings of softly broken N=2, N=1, and even non-supersymmetric SU(Nc) MQCD have a common ratio for their tensions as a function of the amount of flux they carry. We also comment on the almost BPS properties of the Douglas-Shenker strings and discuss the brane picture for monopole confinement on N=2 QCD Higgs branches. 
  The effective string theory emerging from the bilocal approximation to the Method of Vacuum Correlators in gluodynamics is shown to be well described by the 4D theory of the massive Abelian Kalb-Ramond field interacting with the string, which is known to be the low-energy limit of the Universal Confining String Theory. This correspondence follows from the agreement of the behaviour of the coefficient functions, which parametrize the gauge-invariant correlator of two gluonic field strength tensors, known from the lattice data, with their values obtained from the propagator of the Kalb-Ramond field. We discuss this correspondence in several aspects and demonstrate that the mass of the Kalb-Ramond field in this approach plays the role of the inverse correlation length of the vacuum, so that in the massless limit string picture disappears. Next, we apply the background field method, known in the theory of nonlinear sigma models, to obtain the action, which is quadratic in quantum fluctuations around a given (e.g. minimal) string world-sheet. Several nontrivial types of couplings of these fluctuations with the background world-sheet are obtained and discussed. 
  We consider open supermembranes in eleven dimensions in the presence of closed M-Theory five-branes. It has been shown that, in a flat space-time, the world-volume action is kappa invariant and preserves a fraction of the eleven dimensional supersymmetries if the boundaries of the membranes lie on the five-branes. We calculate the reparametrisation anomalies due to the chiral fermions on the boundaries of the membrane and examine their cancellation mechanism. We show that these anomalies cancel with the aid of a classical term in the world-volume action, provided that the tensions of the five-brane and the membrane are related to the eleven dimensional gravitational constant in a way already noticed in M-Theory. 
  We consider brane configurations in M-theory describing N=1 supersymmetric gauge theories and using the parametric representation of the brane configurations, we calculate the superpotentials for various cases including multiple gauge groups or fermions. For SU(n) N=1 SQCD with $N_f$ fermion case ($N_f < N_c)$, we find that the superpotential from M-theory and the gauge theory agree precisely. This gives a direct evidence of the validity of Witten's M-theory method for calculating the super potential. 
  Critical dynamics of the Nambu-Jona-Lasinio model, coupled to a constant electromagnetic field in D=2, 3, and 4, is reconsidered from a viewpoint of infrared behavior and vacuum instability. The latter is associated with constant electric fields and cannot be avoidable in the nonperturbative framework obtained through the proper time method. As for magnetic fields, an infrared cut-off is essential to investigate the critical phenomena. The result reconfirms the fact that the critical coupling in D=3 and 4 goes to zero even under an infinitesimal magnetic field. There also shows that a non-vanishing $F_{\mu\nu}\widetilde F^{\mu\nu}$ causes instability. A perturbation with respect to external fields is adopted to investigate critical quantities, but the resultant asymptotic expansion excellently matches with the exact value. 
  We briefly discuss the status of three-family grand unified string models. 
  We consider M theory 5-branes with compact transverse dimensions. In certain limits the theory on the 5-brane decouples and defines ``little string theories'' in 5+1 dimensions. We show that the familiar structure of IIA/IIB,M,F- theory in 10,11,12 dimensions respectively has a perfect parallel in a theory of strings and membranes in 6,7,8 dimensions. We call these theories $a/b,m,f$ theories. They have a coupling constant but no gravity. This construction clarifies some mysteries in F-theory and leads to several speculations about the phase structure of M theory. 
  We consider the realization of four-dimensional theories with N = 2 supersymmetry as M-theory configurations including a five-brane. Our emphasis is on the spectrum of massive states, that are realized as two-branes ending on the five-brane. We start with a determination of the supersymmetries that are left unbroken by the background metric and five-brane. We then show how the central charge of the N = 2 algebra arises from the central charge associated with the M-theory two-brane. This determines the condition for a two-brane configuration to be BPS-saturated in the four-dimensional sense. By imposing certain conditions on the moduli, we can give concrete examples of such two-branes. This leads us to conjecture that vectormultiplet and hypermultiplet BPS-saturated states correspond to two-branes with the topology of a cylinder and a disc respectively. We also discuss the phenomenon of marginal stability of BPS-saturated states. 
  We perform the dimensional reduction of the nonrelativistic CP(1) model coupled to an Abelian Chern-Simons gauge field in the self-dual limit, and investigate the soliton and domain wall solutions of the emerging 1+1 dimensional self-dual spin system. This system is described by inhomogeneous Landau-Lifshitz system with an extra non-local term. The Hamiltonian is Bogomol'nyi bounded from below and has four adjusting parameters. The Bogomol'nyi equation is described in detail in analogy with the Newtonian equation of motion and its numerical solution is presented. 
  A simple method for calculating the Casimir energy for a sphere is developed which is based on a direct mode summation and counter integration in a complex plane of eigenfrequencies. The method uses only classical equations determining the eigenfrequencies of the quantum field under consideration. Efficiency of this approach is demonstrated by calculation of the Casimir energy for a perfectly conducting spherical shell and for a massless scalar field obeying the Dirichlet and Neumann boundary conditions on sphere. The possibility of rationalizing the removal of divergences in this problem as a renormalization of both the energy and the radius of the sphere is discussed. 
  We introduce a new concept of infinite quasi-exactly solvable models which are constructable through multi-parameter deformations of known exactly solvable ones. The spectral problem for these models admits exact solutions for infinitely many eigenstates but not for the whole spectrum. The hermiticity of their hamiltonians is guaranteed by construction. The proposed models have quasi-exactly solvable classical conterparts. 
  The randomly driven Burgers equation with pressure is considered as a 1D model of strong turbulence of compressible fluid. It is shown that infinitely small pressure provides a finite effect on the velocity and density statistics and this case therefore is qualitatively different from turbulence without pressure. We establish the corresponding operator product expansion and predict the intermittent velocity- difference and mass-difference PDFs. We then apply the developed methods to the statistics of a passive scalar advected by the Burgers field. 
  Within a BRST formulation, we determine the expressions of the consistent anomaly for superstrings with extended worldsheet supersymmetries of rank N. We consider the O(N) superconformal algebras up to N=4, as well as the `small N=4' superalgebra. This is done using a superfield formalism, allowing to recover previous results that were expressed in components. Moreover, we identify the `small N=4' algebra as the constrained `large N=4' via a self-duality like condition in superspace. 
  After recalling the precise existence conditions of the zeta function of a pseudodifferential operator, and the concept of reflection formula, an exponentially convergent expression for the analytic continuation of a multidimensional inhomogeneous Epstein-type zeta function of the general form \zeta_{A,\vec{b},q} (s) = \sum_{\vec{n}\in Z^p (\vec{n}^T A \vec{n} +\vec{b}^T \vec{n}+q)^{-s}, with $A$ the $p\times p$ matrix of a quadratic form, $\vec{b}$ a $p$ vector and $q$ a constant, is obtained. It is valid on the whole complex $s$-plane, is exponentially convergent and provides the residua at the poles explicitly. It reduces to the famous formula of Chowla and Selberg in the particular case $p=2$, $\vec{b}= \vec{0}$, $q=0$. Some variations of the formula and physical applications are considered. 
  Higgs branch of N=2 SQCD is studied from the M theory viewpoint. With a differential geometrical proof of the s-rule besides an investigation on the global symmetry of M theory brane configurations, an exact description of the baryonic and non-baryonic branches in terms of M theory is presented. The baryonic branch root is also studied. The ``electric'' and ``magnetic'' descriptions of the root are shown to be related with each other by the brane exchange in M theory. 
  We give some evidence that the worldvolume theory of the M-theory KK 6-brane is governed by a non-critical membrane theory. We use this theory to give a matrix description of M-theory on $T^6$. 
  Supersymmetric Yang Mills theory is directly accessible to lattice simulations using current methodology, and can provide a non-trivial check of recent exact results in SQCD. In order to tune the lattice simulation to the supersymmetric point it is neccessary to understand the behavior of the theory with a small supersymmetry breaking gaugino mass. We introduce a soft breaking gaugino mass in a controlled fashion using a spurion analysis. We compute the gluino condensate, vacuum energy and bound-state masses as a function of the gaugino mass, providing more readily accessible predictions which still test the supersymmetric results. Finally we discuss diagnostics for obtaining the bare lattice parameters that correspond to the supersymmetric continuum limit. 
  The supersymmetry algebra for supermembranes, quantized in the light-cone gauge, exhibits central charges induced by wrapping the membrane around compact dimensions. These central charges are manifestly consistent with Lorentz symmetry. While the central charges raise the mass of the membrane states, they still leave the mass spectrum continuous, at least generically. The lower bound on the mass spectrum is set by the winding number and corresponds to a BPS state. 
  We give a detailed analysis of pairs of vector and hypermultiplet theories with N=2 supersymmetry in four spacetime dimensions that are related by the (classical) mirror map. The symplectic reparametrizations of the special K\"ahler space associated with the vector multiplets induce corresponding transformations on the hypermultiplets. We construct the Sp(1)$\times$Sp($n$) one-forms in terms of which the hypermultiplet couplings are encoded and exhibit their behaviour under symplectic reparametrizations. Both vector and hypermultiplet theories allow vectorial central charges in the supersymmetry algebra associated with integrals over the K\"ahler and hyper-K\"ahler forms, respectively. We show how these charges and the holomorphic BPS mass are related by the mirror map. 
  We show that a system of bosons in a T=0 quantum field theory can present metastable ground states with spontaneous symmetry breaking, even in the absence of an imaginary mass term. This gives a natural explanation to the Davis-Shellard background field \exp(-i \omega_0 t) and adds a new degree of freedom in boson systems, with possible applications in cosmology, condensed matter and high energy physics. 
  We present the covariant BRST quantization of the super-D-string. The non-vanishing supersymmetric U(1) field strength ${\cal F}$ is essential for the covariant quantization of the super-D-string as well as for its static picture. A SO(2) parameter parametrizes a family of local supersymmetric (kappa symmetric) systems including the super-D-string with ${\cal F}\ne 0$ and the Green-Schwarz superstring with ${\cal F}= 0$. We suggest that $E^1$ (canonical conjugate of U(1) gauge field) plays a role of the order parameter in the Green-Schwarz formalism: the super-D-string exists for $E^1 \ne 0$ while the fundamental Green-Schwarz superstring exists only for $E^1 =0$. 
  We study some aspects of the strong coupling dynamics of Dirichlet six branes, anti- six branes, and orientifold planes by using the equivalence of type IIA string theory and M-theory on S^1. In the strong coupling limit there exists static configuration of brane and anti-brane at arbitrary separation, suspended in an external magnetic field. The mass of the open string stretched between the brane and the anti-brane approaches a finite positive value even when the branes coincide. Similar result is obtained for a Dirichlet six brane on top of an orientifold six plane. We also derive the anomalous gravitational interaction on the brane and the orientifold plane from M-theory. 
  The algebra of Noether supercharges of the M-5-brane effective action is shown to include both 2-form and 5-form central charges. Surprisingly, only the 5-form charge is entirely due to the Wess-Zumino term because the `naive' algebra is the M-2-brane supertranslation algebra. The full structure of central charges is shown to be directly related to the projector arising in the proof of $\kappa$-symmetry of the M-5-brane action. It is also shown to allow `mixed' M-brane configurations preserving 1/2 supersymmetry that include the (non-marginal) M-2-brane/M-5-brane `bound state' as a special case. 
  Magnetic monopoles in Yang-Mills-Higgs theory with a non-abelian unbroken gauge group are classified by holomorphic charges in addition to the topological charges familiar from the abelian case. As a result the moduli spaces of monopoles of given topological charge are stratified according to the holomorphic charges. Here the physical consequences of the stratification are explored in the case where the gauge group SU(3) is broken to U(2). The description due to A. Dancer of the moduli space of charge two monopoles is reviewed and interpreted physically in terms of non-abelian magnetic dipole moments. Semi-classical quantisation leads to dyonic states which are labelled by a magnetic charge and a representation of the subgroup of U(2) which leaves the magnetic charge invariant (centraliser subgroup). A key result of this paper is that these states fall into representations of the semi-direct product $U(2) \semidir R^4$. The combination rules (Clebsch-Gordan coefficients) of dyonic states can thus be deduced. Electric-magnetic duality properties of the theory are discussed in the light of our results, and supersymmetric dyonic BPS states which fill the SL(2,Z)-orbit of the basic massive W-bosons are found. 
  We study how coincident Dirichlet 3-branes absorb incident gravitons polarized along their world volume. We show that the absorption cross-section is determined by the central term in the correlator of two stress-energy tensors. The existence of a non-renormalization theorem for this central charge in four-dimensional N=4 supersymmetric Yang-Mills theories shows that the leading term at low energies in the absorption cross-section is not renormalized. This guarantees that the agreement of the cross-section with semiclassical supergravity, found in earlier work, survives all loop corrections. The connection between absorption of gravitons polarized along the brane and Schwinger terms in the stress-energy correlators of the world volume theory holds in general. We explore this connection to deduce some properties of the stress-energy tensor OPE's for 2-branes and 5-branes in 11 dimensions, as well as for 5-branes in 10 dimensions. 
  We discuss the ADHMN construction for SU(N) monopoles and show that a particular simplification arises in studying charge N-1 monopoles with minimal symmetry breaking. Using this we construct families of tetrahedrally symmetric SU(4) and SU(5) monopoles. In the moduli space approximation, the SU(4) one-parameter family describes a novel dynamics where the monopoles never separate, but rather, a tetrahedron deforms to its dual. We find a two-parameter family of SU(5) tetrahedral monopoles and compute some geodesics in this submanifold numerically. The dynamics is rich, with the monopoles scattering either once or twice through octahedrally symmetric configurations. 
  A global supersymmetry (SUSY) in supersymmetric gauge theory is generally broken by gauge fixing. A prescription to extract physical information from such SUSY algebra broken by gauge fixing is analyzed in path integral framework. If $\delta_{SUSY}\delta_{BRST}\Psi = \delta_{BRST}\delta_{SUSY}\Psi$ for a gauge fixing ``fermion'' $\Psi$, the SUSY charge density is written as a sum of the piece which is naively expected without gauge fixing and a BRST exact piece. If $\delta_{SUSY}\delta_{SUSY}\delta_{BRST}\Psi = \delta_{BRST}\delta_{SUSY}\delta_{SUSY}\Psi$, the equal-time anti-commutator of SUSY charge is written as a sum of a physical piece and a BRST exact piece. We illustrate these properties for N=1 and N=2 supersymmetric Yang-Mills theories and for a D=10 massive superparticle (or ``D-particle'') where the $\kappa$-symmetry provides extra complications. 
  In this article, we examine the possibility that there exist special scalar-tensor theories of gravity with completely nonsingular FRW solutions. Our investigation in fact shows that while most probes living in such a Universe never see the singularity, gravity waves always do. This is because they couple to both the metric and the scalar field, in a way which effectively forces them to move along null geodesics of the Einstein conformal frame. Since the metric of the Einstein conformal frame is always singular for configurations where matter satisfies the energy conditions, the gravity wave world lines are past inextendable beyond the Einstein frame singularity, and hence the geometry is still incomplete, and thus singular. We conclude that the singularity cannot be entirely removed, but only be made invisible to most, but not all, probes in the theory. 
  We review aspects of N=2 duality between the heterotic and the type IIA string. After a description of string duality intended for the non-specialist the computation of the heterotic prepotential and the $F_1$ function for the ST, STU and STUV model (V a Wilson line) and the matching with the Calabi-Yau instanton expansions are given in detail. Relations with BPS spectral sums in various connections are pointed out. 
  Continuing previous work we develop a certain piece of functional analysis on general graphs and use it to create what Connes calls a 'spectral triple', i.e. a Hilbert space structure, a representation of a certain (function) algebra and a socalled 'Dirac operator', encoding part of the geometric/algebraic properties of the graph. We derive in particular an explicit expression for the 'Connes-distance function' and show that it is in general bounded from above by the ordinary distance on graphs (being, typically, strictly smaller(!) than the latter). We exhibit, among other things, the underlying reason for this phenomenon. 
  The $SL(2,R)$ duality symmetric action for the Born-Infeld theory in terms of two potentials, coupled with non-trivial backgroud fields in four dimensions is established. This construction is carried out in detail by analysing the hamiltonian structure of the Born-Infeld theory. The equivalence with the usual Born-Infeld theory is shown. 
  We consider actions for N D-branes at points in a general Kahler manifold, which satisfy the axioms of D-geometry, and could be used as starting points for defining Matrix theory in curved space.   We show that the axioms cannot be satisfied unless the metric is Ricci flat, and argue that such actions do exist when the metric is Ricci flat. This may provide an argument for Ricci flatness in Matrix theory. 
  The 2-dimensional version of the Schwarz and Sen duality model (Tseytlin model) is analyzed at the classical and quantum levels. The solutions are obtained after removing the gauge dependent sector using the Dirac method. The Poincar\`e invariance is verified at both levels. An extension with global supersymmetry is also proposed. 
  In this work a relation between topology and thermodynamical features of gravitational instantons is shown. The expression for the Euler characteristic, through the Gauss-Bonnet integral, and the one for the entropy of gravitational instantons are proposed in a form that makes the relation between them self-evident. A new formulation of the Bekenstein-Hawking formula, where the entropy and the Euler characteristic are related by $S=\chi A/8$, is obtained. This formula provides the correct results for a wide class of gravitational instantons described by both spherically and axially symmetric metrics. 
  We investigate Seiberg's N=1 field theory duality for four-dimensional supersymmetric QCD with the M-theory 5-brane. We find that the M-theory configuration for the magnetic dual theory arises via a smooth deformation of the M-theory configuration for the electric theory. The creation of Dirichlet 4-branes as Neveu-Schwarz 5-branes are passed through each other in Type IIA string theory is given a nice derivation from M-theory. 
  It is well-known that the SO(32) and $E_8\times E_8$ heterotic strings can be continuously connected to each other in nine dimensions. Since the strong-coupling duals of these theories are respectively the SO(32) Type I theory and M-theory compactified on a line segment, there should be a corresponding continuous connection between the Type I string and M-theory. In this paper, we give an explicit construction of this dual connecting theory. Our construction also enables us to realize the $E_8\times E_8$ heterotic string as a D-string soliton of the Type I theory. This provides a useful alternative description of the D-brane bound states previously discussed from a Type I' point of view. 
  We study the Picard-Fuchs differential equations for the Seiberg-Witten period integrals in N=2 supersymmetric Yang-Mills theory. For A-D-E gauge groups we derive the Picard-Fuchs equations by using the flat coordinates in the A-D-E singularity theory. We then find that these are equivalent to the Gauss-Manin system for two-dimensional A-D-E topological Landau-Ginzburg models and the scaling relation for the Seiberg-Witten differential. This suggests an interesting relationship between four-dimensional N=2 gauge theories in the Coulomb branch and two-dimensional topological field theories. 
  In this paper we consider the class of exact solutions of the Schrodinger equation with the Razavi potential. By means of this we obtain some wavefunctions and mass spectra of the relativistic scalar field model with spontaneously broken symmetry near the static kink solution. Appearance of the bosons, which have two different spins, will be shown in the theory, thereby the additional breaking of discrete symmetry between the quantum mechanical kink particles with the opposite spins (i.e. the T-violation) takes place. 
  Functional measures for lattice quantum gravity should agree with their continuum counterparts in the weak field, low momentum limit. After showing that the standard simplicial measure satisfies the above requirement, we prove that a class of recently proposed non-local measures for lattice gravity do not satisfy such a criterion, already to lowest order in the weak field expansion. We argue therefore that the latter cannot represent acceptable discrete functional measures for simplicial geometries. 
  We argue that the account of Reference Frames quantum properties must change the standard space-time picture accepted in Quantum Mechanics. If RF is connected with some macroscopic solid object then its free quantum motion - wave packet smearing results in additional uncertainty into the measurement of test particle coordinate. It makes incorrect the use of Galilean or Lorentz space-time transformations between two RF and the special quantum space-time transformations are formulated. It results in generalized Klein- Gordon equation which depends on RF mass. Both space and time coordinates become the operators. In particular RF proper time becomes the operator depending of momentums spectra of this RF wave packet, from the point of view of other observer. 
  It is shown that the integration measure over the matrix $Y$ in the matrix representation of the Schild action can be fixed by comparing the Schild matrix model with the random lattice string model for D=0. It is further checked that the given measure is consistent with the case D=1 as well. 
  We consider the scattering of zero branes off an elementary string in Matrix theory or equivalently gravitons off a longitudinally wrapped membrane. The leading supergravity result is recovered by a one-loop calculation in zero brane quantum mechanics.  Simple scaling arguments are used to show that there are no further corrections at higher loops, to the leading term in the large impact parameter, low velocity expansion. The mechanism for this agreement is identified in terms of properties of a recently discovered boundary conformal field theory. 
  I review findings of various research groups regarding perturbative heterotic string model building in the last 12 months. Attention is given to recent studies of extra U(1)'s and local discrete symmetries (LDS's) in generic string models. Issues covered include the role of U(1)'s and LDS's in limiting proton decay, developments in classification of models containing anomalous U(1), and possible complications resulting from kinetic mixing between observable and hidden sector U(1)'s. Additionally, recent string-derived and string-inspired models are briefly reviewed.     Talk Presented at SUSY '97. 
  An application of the Newman-Penrose dyad formalism for description of tensionless (null) string dynamics in 4D Minkowski spacetime is studied in a background of antisymmetric fields. We present a special class of background antisymmetric fields which admits exact solutions of the null string motion equations and formulate a sufficient condition of the absence of interaction of a null string with such fields. 
  We give an invariant classification of orbits of the fundamental representations of exceptional groups $E_{7(7)}$ and $E_{6(6)}$ which classify BPS states in string and M theories toroidally compactified to d=4 and d=5. The exceptional Jordan algebra and the exceptional Freudenthal triple system and their cubic and quartic invariants play a major role in this classification. The cubic and quartic invariants correspond to the black hole entropy in d=5 and d=4, respectively. The classification of BPS states preserving different numbers of supersymmetries is in close parallel to the classification of the little groups and the orbits of timelike, lightlike and space-like vectors in Minkowski space. The orbits of BPS black holes in N=2 Maxwell-Einstein supergravity theories in d=4 and d=5 with symmetric space geometries are also classified including the exceptional N=2 theory that has $E_{7(-25)}$ and $E_{6(-26)}$ as its symmety in the respective dimensions. 
  We examine the behavior of the vortex defects, which are represented by the sine-Gordon potential, on the world-sheet when a solitonic d-5 brane is existing in the d-dimensional target space. The d-5 brane solution is obtained as an extension of 5-brane in 10d supergravity. It is shown that the vortices of any charge should be dissociated to the free gas when they approach to the brane. The meanings of this fact are addressed through the analysis of the renormalization group equations on the world sheet. 
  We investigate the zero temperature chiral phase transition in (2+1)-dimensional QED in the presence of a Chern-Simons term, changing the number of fermion flavors. In the symmetric phase, there are no light degrees of freedom even at the critical point. Unlike the case without a Chern-Simons term, the phase transition is first-order. 
  We investigate the behavior of small perturbations around the Kaluza-Klein monopole in the five dimensional space-time. We find that the even parity gravitational wave does not propagate in the five dimensional space-time with Kaluza-Klein monopole provided that the gravitational wave is constant in the fifth direction. We conclude that a gravitational wave and a U(1) magnetic monopole do not coexist in five dimensional Kaluza-Klein spacetime. 
  F theory and M theory are formulated as gauge theories of area preserving diffeomorphism algebra. Our M theory is shown to be 1-brane formulation rather than 0-brane formulation of M theory of Banks, Fischler, Shenker and Susskind and the F theory is shown to be 1-brane formulation rather than -1-brane formulation of type IIB matrix theory of Ishibashi, Kawai, Kitazawa and Tsuchiya. 
  The thermodynamics of 3d adjoint Higgs model is considered. We study the properties of the Polyakov loop correlators and the critical behavior at the deconfinement phase transition. Our main tool is a reduction to the 2d sine-Gordon model. The Polyakov loops appear to be connected with the soliton operators in it. The known exact results in the sine-Gordon theory allow us to study in detail the temperature dependence of the string tension, as well as to get some information about a nonperturbative dynamics in the confinement phase. We also consider the symmetry restoration at high temperature which makes it possible to construct the phase diagram of the model completely. 
  Reviewing the cancellation of local anomalies of M-theory on R^10 x S^1/Z_2 the Yang-Mills coupling constant on the boundaries is rederived. The result is lambda^2 = 2^(1/3) (2 pi) (4 pi kappa^2)^(2/3) corresponding to eta = lambda^6/kappa^4 = 256 pi^5 in the `upstairs' units used by Horava and Witten and differs from their calculation. It is shown that these values are compatible with the standard membrane and fivebrane tensions derived from the M-theory bulk action. In view of these results it is argued that the natural units for M-theory on R^10 x S^1/Z_2 are the `downstairs' units where the brane tensions take their standard form and the Yang-Mills coupling constant is lambda^2 = 4 pi (4 pi kappa^2)^(2/3). 
  A recently proposed renormalization scheme can be used to deal with nonrelativistic potential scattering exhibiting ultraviolet divergence in momentum space. A numerical application of this scheme is made in the case of potential scattering with $r^{-2}$ divergence for small r, common in molecular and nuclear physics, by the use of cut-offs in momentum and configuration spaces. The cut-off is finally removed in terms of a physical observable and model-independent result is obtained at low energies. The expected variation of the off-shell behavior of the t matrix arising from the renormalization scheme is also discussed. 
  We apply stochastic quantization method to real symmetric matrix-vector models for the second quantization of non-orientable strings, including both open and closed strings. The Fokker-Planck hamiltonian deduces a well-defined non-orientable open-closed string field theory at the double scaling limit of the matrix model. There appears a new algebraic structure in the continuum F-P hamiltonian including a Virasoro algebra and a $SO(r)$ current algebra. 
  An `algebraic' approach to M-theory is briefly reviewed, and a proposal is made for a similar algebraic structure underlying the $T^9$ compactification of `M(embrane) theory', i.e. the M(atrix) model with area-preserving diffeomorphism gauge group. 
  A BPHZ renormalized form for the master equation of the field antifiled (or BV) quantization has recently been proposed by De Jonghe, Paris and Troost. This framework was shown to be very powerful in calculating gauge anomalies. We show here that this equation can also be applied in order to calculate a global anomaly (anomalous divergence of a classically conserved Noether current), considering the case of QED. This way, the fundamental result about the anomalous contribution to the Axial Ward identity in standard QED (where there is no gauge anomaly) is reproduced in this BPHZ regularized BV framework. 
  We propose explicit formulae for the integration measure on the moduli space of charge-n ADHM multi-instantons in N=1 and N=2 supersymmetric gauge theories. The form of this measure is fixed by its (super)symmetries as well as the physical requirement of clustering in the limit of large spacetime separation between instantons. We test our proposals against known expressions for n < 3. Knowledge of the measure for all n allows us to revisit, and strengthen, earlier N=2 results, chiefly: (1) For any number of flavors N_F, we provide a closed formula for F_n, the n-instanton contribution to the Seiberg-Witten prepotential, as a finite-dimensional collective coordinate integral. This amounts to a solution, in quadratures, of the Seiberg-Witten models, without appeal to electric-magnetic duality. (2) In the conformal case N_F=4, this means reducing to quadratures the previously unknown finite renormalization that relates the microscopic and effective coupling constants, \tau_{micro} and \tau_{eff}. (3) Similar expressions are given for the 4-derivative/8-fermion term in the gradient expansion of N=2 supersymmetric QCD. 
  We discuss M(atrix) theory compactification on T^6. This theory is described by the large N limit of the world volume theory, of N Kaluza-Klein monopoles in eleven dimensions. We discuss the BPS states, and their arrangement in E_6 multiplets. We then propose the formulation of the world volume theory of KK monopoles in eleven dimensions that decouples from the bulk. This is given by a large N_1 m(atrix) theory with eight supercharges, corresponding to the quantum mechanics theory of N_1 zero-branes inside the Type IIA Kaluza-Klein monopole. Various limits of the construction are considered. 
  A reformulation of the Thirring model as a gauge theory on both continuum spacetime and discretized lattice is reviewed. In (1+1) dimensions, our result reproduces consistently the bosonization of the massless Thirring model. In (2+1) dimensions, the analysis by use of Schwinger-Dyson equation is shown to exhibit dynamical fermion mass generation when the number $N$ of four-component fermions is less than the critical value $N_{cr}= 128/3\pi^{2}$. 
  Starting from the primal principle based on the noncommutative nature of (9+1)-dimensional spacetime, we construct a topologically twisted version of the supersymmetric reduced model with a certain modification. Our formulation automatically provides extra 1+1 dimensions, thereby the dimensions of spacetime are promoted to 10+2. With a suitable gauge choice, we can reduce the model with (10+2)-dimensional spacetime to the one with (9+1)-dimensions and thus we regard this gauge as the light-cone gauge. It is suggested that the model so obtained would describe the light-cone F-theory. From this viewpoint we argue the relation of the reduced model to the matrix model of M-theory and the SL(2,Z) symmetry of type IIB string theory. We also discuss the general covariance of the matrix model in a broken phase, and make some comments on the background independence. 
  We calculate the tadpole equations and their solutions for a class of four-dimensional orientifolds with orbifold group Z_N X Z_M, and we present the massless bosonic spectra of these models. Surprisingly we find no consistent solutions for the models with Z_2 X Z_4 and Z_4 X Z_4 orbifold groups. 
  We consider in more detail the role of D0-branes as instantons in the construction of SU(N) Super Yang-Mills and Super QCD theories in four space-time dimensions with D4, D6 and NS-branes. In particular, we show how the D0-branes describe both the exact and constrained instantons and reproduce the correct pattern of lifting of zero modes on the various branches of these models. 
  For quantum field theories that flow between ultraviolet and infrared fixed points, central functions, defined from two-point correlators of the stress tensor and conserved currents, interpolate between central charges of the UV and IR critical theories. We develop techniques that allow one to calculate the flows of the central charges and that of the Euler trace anomaly coefficient in a general N=1 supersymmetric gauge theory. Exact, explicit formulas for $SU(N_c)$ gauge theories in the conformal window are given and analysed. The Euler anomaly coefficient always satisfies the inequality $% a_{UV}-a_{IR}>0$. This is new evidence in strongly coupled theories that this quantity satisfies a four-dimensional analogue of the $c$-theorem, supporting the idea of irreversibility of the RG flow. Various other implications are discussed. 
  The physics of the Wilson line leads to new developments in high temperature particle physics. The main tool is the effective action for a given fixed value of the phase of the Wilson line. It furnishes a gauge invariant infrared cut off, and yields for small values of the phases a systematic procedure for obtaining a power series in the coupling g and glog(1/g). It breaks the centergroup symmetry of the gauge group only at high temperature so leads to domain walls disappearing at low temperatures. It shows long lived metastable states in the standard model, SU(5), SO(10) and its SUSY partners, with possibilities for CP violation and thermal inflation. 
  The fivebrane of M theory is used in order to study the moduli space of vacua of confining phase N=1 supersymmetric gauge theories in four dimensions. The supersymmetric vacua correspond to the condensation of massless monopoles and confinement of photons. The monopole and meson vacuum expectation values are computed using the fivebrane configuration. The comparison of the fivebrane computation and the field theory analysis shows that at vacua with a classically enhanced gauge group SU(r) the effective superpotential obtained by the "integrating in" method is exact for r=2 but is not exact for r > 2. The fivebrane configuration corresponding to N=1 gauge theories with Landau-Ginzburg type superpotentials is studied. N=1 non-trivial fixed points are analyzed using the brane geometry. 
  The classical and quantum aspects of planar Coulomb interactions have been studied in detail. In the classical scenario, Action Angle Variables are introduced to handle relativistic corrections, in the scheme of time-independent perturbation theory. Complications arising due to the logarithmic nature of the potential are pointed out. In the quantum case, harmonic oscillator approximations are considered and effects of the perturbations on the excited (oscillator) states have been analysed. In both the above cases, the known 3+1-dimensional analysis is carried through side by side, for a comparison with the 2+1-dimensional (planar) results. 
  We construct, for the first time, a model of graceful exit transition from a dilaton-driven inflationary phase to a decelerated Friedman-Robertson-Walker era. Exploiting a demonstration that classical corrections can stabilize a high curvature string phase while the evolution is still in the weakly coupled regime, we show that if additional terms of the type that may result from quantum corrections to the string effective action exist, and induce violation of the null energy condition, then evolution towards a decelerated Friedman-Robertson-Walker phase is possible. We also observe that stabilizing the dilaton at a fixed value, either by capture in a potential minimum or by radiation production, may require that these quantum corrections are turned off, perhaps by non-perturbative effects or higher order contributions which overturn the null energy condition violation. 
  The geometry of (2,1) supersymmetric sigma-models with isometry symmetries is discussed. The gauging of such symmetries in superspace is then studied. We find that the coupling to the (2,1) Yang-Mills supermultiplet can be achieved provided certain geometric conditions are satisfied. We construct the general gauged action, using an auxiliary vector to generate the full non-polynomial structure. 
  A non-abelian generalisation of a theory of gravity coupled to a 2-form gauge field and a dilaton is found, in which the metric and 3-form field strength are Lie algebra-valued. In the abelian limit, the curvature with torsion is self-dual in four dimensions, or has SU(n) holonomy in $2n$ dimensions. The coupling to self-dual Yang-Mills fields in 4 dimensions, or their higher dimensional generalisation, is discussed. The abelian theory is the effective action for (2,1) strings, and the non-abelian generalisation is relevant to the study of coincident branes in the (2,1) string approach to M-theory. The theory is local when expressed in terms of a vector pre-potential. 
  Stabilizing a heterotic string vacuum with a large expectation value of the dilaton and simultaneously breaking low-energy supersymmetry is a long-standing problem of string phenomenology. We reconsider these issues in light of the recent developments in F-theory. 
  The Virasoro master equation describes a large set of conformal field theories known as the affine-Virasoro constructions, in the operator algebra (affine Lie algebra) of the WZW model, while the Einstein equations of the general non-linear sigma model describe another large set of conformal field theories. This talk summarizes recent work which unifies these two sets of conformal field theories, together with a presumable large class of new conformal field theories. The basic idea is to consider spin-two operators of the form $L_{ij} \partial x^i \partial x^j$ in the background of a general sigma model. The requirement that these operators satisfy the Virasoro algebra leads to a set of equations called the unified Einstein-Virasoro master equation, in which the spin-two spacetime field $L_{ij}$ couples to the usual spacetime fields of the sigma model. The one-loop form of this unified system is presented, and some of its algebraic and geometric properties are discussed. 
  We employ nonperturbative flow equations for an investigation of the effective action in Yang-Mills theories. We compute the effective action $\Gamma[B]$ for constant color magnetic fields $B$ and examine Savvidy's conjecture of an unstable perturbative vacuum. Our results indicate that the absolute minimum of $\Gamma[B]$ occurs for B=0. Gluon condensation is described by a nonvanishing expectation value of the regularized composite operator $F_{\mu\nu}F^{\mu\nu}$ which agrees with phenomenological estimates. 
  A superconnection is a supermatrix whose even part contains the gauge-potential one-forms of a local gauge group, while the odd parts contain the (0-form) Higgs fields; the combined grading is thus odd everywhere. We demonstrate that the simple supergroup ${\bar P}(4,R)$ (rank=3) in Kac' classification (even subgroup $\bar {SL}(4,R)$) prverline {SL}(4,R)$) provides for the most economical spontaneous breaking of $\bar{SL}(4,R)$ as gauge group, leaving just local $\bar{SO}(1,3)$ unbroken. As a result, post-Riemannian SKY gravity yields Einstein's theory as a low-energy (longer range) effective theory. The theory is renormalizable and may be unitary. 
  We construct a manifestly dual formulation of Dirichlet three-brane in the framework of Pasti-Sorokin-Tonin approach. 
  Modified similarity renormalization (MSR) of Hamiltonians is proposed, that performes by means of flow equations the similarity transformation of Hamiltonian in the particle number space. This enables to renormalize in the energy space the field theoretical Hamiltonian and makes possible to work in a severe trancated Fock space for the renormalized Hamiltonian.   Original works of Wegner in solid state physics has served us as the guiding principle in constructing the MSR scheme.   The renormalized to the second order effective $QED_{3+1}$ Hamiltonian on the light front is obtained. This Hamiltonian reproduces in $|e\bar{e}>$ sector the standard singlet-triplet splitting of positronium and recorves rotational symmetry of the canonical theory. The lowest and next-to-lowest energy states of positronium are almost independent on the cutoff when in both cases the 'same' (state and sector independent) counterterms are included.   The electron (photon) mass counterterms are IR (collinear) finite if all diagrams to the second order, arising from the flow equations and normal-ordering Hamiltonian, are taken into account, and vary with UV cutoff in accordance with $1$-loop renormalization group equations.   Both approximations of perturbation theory and Fock space trancation are under the control and can be improved systematically within the proposed renormalization scheme. 
  We give a full classification of the multi-charge supersymmetric $p$-brane solutions in the massless and massive maximal supergravities in dimensions $D\ge2$ obtained from the toroidal reduction of eleven-dimensional supergravity. We derive simple universal rules for determining the fractions of supersymmetry that they preserve. By reversing the steps of dimensional reduction, the $p$-brane solutions become intersections of $p$-branes, NUTs and waves in D=10 or D=11. Having classified the lower-dimensional $p$-branes, this provides a classification of all the intersections in D=10 and D=11 where the harmonic functions depend on the space transverse to all the individual objects. We also discuss the structure of U-duality multiplets of $p$-brane solutions, and show how these translate into multiplets of harmonic and non-harmonic intersections. 
  The propagation of a massless field in attractive and repulsive potentials is considered. It is shown that though the group velocity in such potentials can be larger than one, the wave front propagates with the speed of light. A larger-than-one group velocity leads only to a strong deformation of the wave packet. The results obtained are applied to the light propagation in a gravitational field. An erroneous assertion concerning the last problem, recently made in the literature, is refuted. 
  We prove the Polyakov conjecture on the supertorus $(ST_2)$: we dermine an iterative solution at any order of the superconformal Ward identity and we show that this solution is resumed by the Wess-Zumino-Polyakov (WZP) action that describes the $(1,0)$ 2D-supergravity. The resolution of the superBeltrami equation for the Wess-Zumino (WZ) field is done by using on the one hand the Cauchy kernel defined on $ST_2$ and on the other hand, the formalism developed to get the general solution on the supercomplex plane. Hence, we determine the n-points Green functions from the (WZP) action expressed in terms of the (WZ) field. 
  In this paper we formulate a general method for building completely integrable quantum systems. The method is based on the use of the so-called multi-parameter spectral equations, i.e. equations with several spectral parameters. We show that any such equation, after eliminating some spectral parameters by means of the so-called inverse procedure of separation of variables can be reduced to a certain completely integrable model. Starting with exactly or quasi-exactly solvable multi-parameter spectral equations we, respectively, obtain exactly or quasi-exactly solvable integrable models. 
  In this paper we propose a simple method for building exactly solvable multi-parameter spectral equations which in turn can be used for constructing completely integrable and exactly solvable quantum systems. The method is based on the use of a special functional relation which we call the scalar triangle equation because of its similarity to the classical Yang-Baxter equation. 
  In a wide class of supersymmetric theories degenerate families of the BPS-saturated domain walls exist. The internal structure of these walls can continuously vary, without changing the wall tension. This is described by hidden parameters (collective coordinates). Differentiating with respect to the collective coordinates one gets a set of the bosonic zero modes localized on the wall. Neither of them is related to the spontaneous breaking of any symmetry. Through the residual 1/2 of supersymmetry each bosonic zero mode generates a fermionic partner. 
  We show that in supersymmetric pure Yang Mills theories with arbitrary simple gauge group, the spontaneous breaking of chiral fermionic and bosonic charge by the associated gaugino and gauge boson condensates implies the spontaneous breaking of supersymmetry by the condensate of the underlying Lagrangian density. The explicit breaking of the restricted fermionic charge through the chiral anomaly is deferred to a secondary stage in the elimination of infrared singularities or long range forces. 
  We compute and analyse a variety of four-derivative gravitational terms in the effective action of six- and four-dimensional type II string ground states with N=4 supersymmetry. In six dimensions, we compute the relevant perturbative corrections for the type II string compactified on K3. In four dimensions we do analogous computations for several models with (4,0) and (2,2) supersymmetry. Such ground states are related by heterotic-type II duality or type II-type II U-duality. Perturbative computations in one member of a dual pair give a non-perturbative result in the other member. In particular, the exact CP-even R^2 coupling on the (2,2) side reproduces the tree-level term plus NS 5-brane instanton contributions on the (4,0) side. On the other hand, the exact CP-odd coupling yields the one-loop axionic interaction a.R\wedge R together with a similar instanton sum. In a subset of models, the expected breaking of the SL(2,Z)_S S-duality symmetry to a \Gamma(2)_S subgroup is observed on the non-perturbative thresholds. Moreover, we present a duality chain that provides evidence for the existence of heterotic N=4 models in which N=8 supersymmetry appears at strong coupling. 
  We study the relation between the effective Lagrangian in Matrix Theory and eleven dimensional supergravity. In particular, we provide a relationship between supergravity operators and the corresponding terms in the post-Newtonian approximation of Matrix Theory. 
  We present supersymmetric Yang-Mills theories in arbitrary even dimensions with the signature (9+m,1+m) where $m=0,1,2,...$ beyond ten-dimensions up to infinity. This formulation utilizes null-vectors and is a generalization of our previous work in 10+2 dimensions to arbitrary even dimensions with the above signature. We have overcome the previously-observed obstruction beyond 11+3 dimensions, by the aid of projection operators. Both component and superspace formulations are presented. This also suggests the possibility of consistent supergravity theories in any even dimensions beyond 10+1 dimensions. 
  In this paper we discuss compactifications of type II superstrings where the moduli of the internal Calabi-Yau space vary over four-dimensional space time. The corresponding solutions of four-dimensional N=2 supergravity are given by charged, extremal BPS black hole configurations with non-constant scalar field values. In particular we investigate the behaviour of our solutions near those points in the Calabi-Yau moduli space where some internal cycles collapse and topology change (flop transitions, conifold transitions) can take place. The singular loci in the internal space are related to special points in the uncompactified space. The phase transition can happen either at spatial infinity (for positive charges) or on spheres (with at least one negative charge). The corresponding BPS configuration has zero ADM mass and can be regarded as a domain wall that separates topologically different vacua of the theory. 
  We introduce a classical field theory based on a concept of extended causality that mimics the causality of a point-particle Classical Mechanics by imposing constraints that are equivalent to a particle initial position and velocity. It results on a description of discrete (pointwise) interactions in terms of localized particle-like fields. We find the propagators of these particle-like fields and discuss their physical meaning, properties and consequences. They are conformally invariant, singularity-free, and describing a manifestly covariant $(1+1)$-dimensional dynamics in a $(3+1)$ spacetime. Remarkably this conformal symmetry remains even for the propagation of a massive field in four spacetime dimensions. The standard formalism with its distributed fields is retrieved in terms of spacetime average of the discrete fields. Singularities are the by-products of the averaging proccess. This new formalism enlightens the meaning and the problems of field theory, and may allow a softer transition to a quantum theory. 
  A supersymmetric theory with several scalar superfields generically has several domain wall type classical configurations which interpolate between various supersymmetric vacua of the scalar fields. Depending on the couplings, some of these configurations develop instability and decay into multiple domain walls, others can form intersections in space. These phenomena are considered here in a simplest, yet non-trivial, model with two scalar superfields. 
  It was observed recently, that the low energy effective action of the four-dimensional supersymmetric theories may be obtained as a certain limit of M Theory. From this point of view, the BPS states correspond to the minimal area membranes ending on the M Theory fivebrane. We prove that for the configuration, corresponding to the SU(2) Super Yang-Mills theory, the BPS spectrum is correctly reproduced, and develop techniques for analyzing the BPS spectrum in more general cases. We show that the type of the supermultiplet is related to the topology of the membrane: disks correspond to hypermultiplets, and cylinders to vector multiplets. We explain the relation between minimal surfaces and geodesic lines, which shows that our description of BPS states is closely related to one arising in Type II string compactification on Calabi-Yau threefolds. 
  The Fock space of bosons and fermions and its underlying superalgebra are represented by algebras of functions on a superspace. We define Gaussian integration on infinite dimensional superspaces, and construct superanalogs of the classical function spaces with a reproducing kernel -- including the Bargmann-Fock representation -- and of the Wiener-Segal representation. The latter representation requires the investigation of Wick ordering on Z2-graded algebras. As application we derive a Mehler formula for the Ornstein-Uhlenbeck semigroup on the Fock space. 
  I review the properties of a matrix action of relevance for IIB superstrings. This model generalizes the action proposed by Ishibashi, Kawai, Kitazawa, and Tsuchiya by introducing an auxillary field Y, which is the matrix version of the auxillary field g in the Schild action. 
  We present a mathematical framework of gauge theories that is based upon a skew-adjoint Lie algebra and a generalized Dirac operator, both acting on a Hilbert space. 
  Utilizing the conformal-flatness nature of 3-dim. Anti-de Sitter (AdS_3) black hole solution of Banados, Teitelboim and Zanelli, the quantisation of conformally-coupled scalar and spinor fields in this background spacetime is explicitly carried out. In particular, mode expansion forms and propagators of the fields are obtained in closed forms. The vacuum in this conformally-coupled field theories in AdS_3 black hole spacetime, which is conformally-flat, is the conformal vacuum which is unique and has global meaning. This point particularly suggests that now the particle production by AdS_3 black hole spacetime should be absent. General argument establishing the absence of real particle creation by AdS_3 black hole spacetime for this case of conformal triviality is provided. Then next, using the explicit mode expansion forms for conformally-coupled scalar and spinor fields, the bosonic and fermionic superradiances are examined and found to be absent confirming the expectation. 
  We obtain the Picard-Fuchs equations of N=2 supersymmetric Yang-Mills theory with the exceptional gauge group E_6. Such equations are based on E_6 spectral curve. 
  Supersymmetric configurations of non-orthogonally intersecting M-5-branes can be obtained by rotation of one of a pair of parallel M-5-branes. Examples preserving 1/4, 3/16 and 1/8 supersymmetry are reviewed. 
  Motivated by the work of Polchinski and Strominger on type IIA theory, where the effect of non-trivial field strengths for p-form potentials on a Calabi-Yau space was discussed, we study four-dimensional heterotic string theory in the presence of magnetic field on a 2-cycle in the internal manifold, for both N=4 and N=2 cases. We show that at special points in the moduli space, certain perturbative charged states become tachyonic and stabilize the vacuum by acquiring vacuum expectation values, thereby restoring supersymmetry. We discuss both the cases where the tachyons appear with a tower of Landau levels, which become light in the limit of large volume of the 2-cycle, and the case where such Landau levels are not present. In the latter case it is sufficient to restrict the analysis to the quartic potential for the tachyon. On the other hand, in the former case it is necessary to include the Landau levels in the analysis of the potential; for toroidal and orbifold examples, we give an explicit CFT description of the new supersymmetric vacuum. The resulting new vacuum turns out to be in the same class as the original supersymmetric one. Finally, using duality, we discuss the role of the Landau levels on the type IIA side. 
  We apply the light-cone hamiltonian approach to D2-brane, and derive the equivalent gauge-invariant Lagrangian. The later appears to be that of three-dimensional Yang-Mills theory, interacting with matter fields, in the special external induced metric, depending on matter fields. The duality between this theory and 11d membrane is shown. 
  In the triplectic quantization of general gauge theories, we prove a `triplectic' analogue of the Darboux theorem: we show that the doublet of compatible antibrackets can be brought to a weakly-canonical form provided the general triplectic axioms of [BMS] are imposed together with some additional requirements that can be formulated in terms of marked functions of the antibrackets. The weakly-canonical antibrackets involve an obstruction to bringing them to the canonical form. We also classify the `triplectic' odd vectors fields compatible with the weakly-canonical antibrackets and construct the Poisson bracket associated with the antibrackets and the odd vector fields. We formulate the Sp(2)-covariance requirement for the antibrackets and the vector fields; whenever the obstruction to the canonical form of the antibrackets vanishes, the Sp(2)-covariance condition implies the canonical form of the triplectic vector fields. 
  D-branes are string theory solitons defined by boundary conditions on open strings that end on them. They play an important role in the non-perturbative dynamics of string theory and have found a wide range of applications. These lectures give a basic introduction to D-branes and their dynamics. 
  The hierarchy of equations of motion for equal-time Green functions in temporal gauge SU(N) Yang-Mills theory is truncated using an expansion in terms of connected Green functions. A second hierarchy of constraint equations arises from Gauss law and can be truncated in a similar way. Within this approximation scheme we investigate SU(2) Yang-Mills theory on a torus in 2+1 spacetime dimensions in a finite basis of plane wave states and focus on infrared and ultraviolet properties of the approach. We study the consequences of restoring the hierarchy of Gauss law constraints and of different momentum cutoffs for the 2- and the 3-point functions. In all truncation schemes considered up to the 4-point level the connected Green function approach is found to be UV divergent and either violating gauge invariance and/or energy conservation. The problems associated with adiabatically generating a perturbed ground state are discussed as well. 
  In this paper the dynamics of the classical chiral $QCD_{2}$ currents is studied. We describe how the dynamics of the theory can be summarized in an equation of the Lax form, thereby demonstrating the existence of an infinite set of conserved quantities. Next, the $r$ matrix of a fundamental Poisson relation is obtained and used to demonstrate that the conserved charges Poisson commute. An underlying diffeomorphism symmetry of the equations of motion which is not a symmetry of the action is used to provide a geometric interpretation for the case of gauge group SU(2). This enables us to show that the solutions to the classical equations of motion can be identified with a large class of curves, to demonstrate an auto-B\"acklund transformation and to demonstrate a non linear superposition principle. A link between the spectral problem for $QCD_{2}$ and the solution to the closed curve problem is also demonstrated. We then go on to provide a systematic inverse scattering treatment. This formalism is used to obtain the reflectionless single boundstate eigenvalue soliton solution. 
  In this letter we present the calculation of the $a_{5}$ heat kernel coefficient of the heat operator trace for a partial differential operator of Laplace type on a compact Riemannian manifold with Dirichlet and Robin boundary conditions. 
  We describe fivebrane configurations in M theory whose 4-d spacetime contains N=1 supersymmetric Sp or SO gauge fields and fundamentals of these groups. We show how field-theory dualities for Sp and SO groups can be derived using these fivebrane configurations in M theory. 
  The finite N version of Matrix theory describes M-theory and superstrings in so-called discretized light cone quantization (DLCQ). Its role has been explained for M-theory in 11 dimensions and for type IIA theory. We show novelties which arise by generalizing the ideas to heterotic strings. The states which arise in O(N) theories with odd N we interpret as fields which are antiperiodic in the longitudinal direction. Consistency of these ideas provides a new evidence for the conjecture that finite N models describe sectors with a given longitudinal momentum. 
  We use minimal area metrics to generate all nonorientable string diagrams. The surfaces in unoriented string theory have nontrivial open curves and nontrivial closed curves whose neighborhoods are either annuli or Mobius strips. We define a minimal area problem by imposing length conditions on open curves and on annular closed curves only. We verify that the minimal area conditions are respected by the sewing operations. The natural objects that satisfy recursion relations involving the propagator, which performs both orientable and nonorientable sewing, are classes of moduli spaces grouped by Euler characteristic. 
  We study the theory of NS five-branes in string theory with a smooth non-trivial transverse space. We show that in the limit that the bulk physics decouples, these theories become equivalent to theories with a flat and non-compact transverse space. We present a matrix model description of the type IIA theory on $\IR^9\times S^1$ with NS five-branes located at points on the circle. Consequently, we obtain a description of the dual configuration of Kaluza-Klein monopoles in the type IIB theory. 
  A fundamental string with non-zero winding number can unwind in the presence of a Kaluza-Klein monopole. We use this fact to deduce the presence of a zero mode for the Kaluza-Klein monopole corresponding to excitations carrying H electric charge and we study the coupling of this zero mode to fundamental strings. We also a describe a T-dual process in which the momentum of a fundamental string ``unwinds'' in the presence of a H monopole. We use the coupling of string winding modes to the dyon collective coordinate of the Kaluza-Klein monopole to argue that there are stringy corrections to the Kaluza-Klein monopole which are in accordance with T-duality. 
  Solutions of the superspace Schwinger-Dyson equations, describing mass generation and chiral symmetry breaking in supersymmetric gauge theory, need not be constrained to vanish by no-renormalization theorems, nor by special choices of gauge parameter. Thus symmetry breaking vacuum structures remain possible (as in non-supersymmetric gauge theory), inviting comparison with predictions of an alternative approach based on holomorphy and the Wilsonian effective action. 
  The zero-signature Killing metric of a new, real-valued, 8-dimensional gauging of the conformal group accounts for the complex character of quantum mechanics. The new gauge theory gives manifolds which generalize curved, relativistic phase space. The difference in signature between the usual momentum space metric and the Killing metric of the new geometry gives rise to an imaginary proportionality constant connecting the momentumlike variables of the two spaces. Path integral quantization becomes an average over dilation factors, with the integral of the Weyl vector taking the role of the action. Minimal U(1) electromagnetic coupling is predicted. 
  The moduli space of 0-branes on $T^4$ gives a prediction for the moduli space of torons in U(n) Super Yang Mills theory which preserve 16 supersymmetries. The zero brane number corresponds to the greatest common denominator of the rank $n$, magnetic fluxes and the instanton number. This prediction is derived using U-duality. We explicitly check this prediction by analyzing U(n) bundles with non-zero first as well as second Chern classes. The argument is extended to deduce the moduli space of torons which preserve 8 supersymmetries. Parts of the discussion extend naturally to $T^2$ and $T^3$. Some of the U-dualities involved are related to Lorentz boosts along the eleventh direction in M theory. 
  We review the classical thermodynamics and the greybody factors of general (rotating) non-extreme black holes and discuss universal features of their near-horizon geometry. We motivate a microscopic interpretation of general black holes that relates the thermodynamics of an effective string theory to the geometry of the black hole in the vicinity of both the outer and the inner event horizons. In this framework we interpret several near-extreme examples, the universal low-energy absorption cross-section, and the emission of higher partial waves from general black holes. 
  The moduli space metric for an arbitrary number of extremal black holes in four dimensions with arbitrary relatively supersymmetric charges is found. 
  We suggest a generalization of the Lie algebraic approach for constructing quasi-exactly solvable one-dimensional Schroedinger equations which is due to Shifman and Turbiner in order to include into consideration matrix models. This generalization is based on representations of Lie algebras by first-order matrix differential operators. We have classified inequivalent representations of the Lie algebras of the dimension up to three by first-order matrix differential operators in one variable. Next we describe invariant finite-dimensional subspaces of the representation spaces of the one-, two-dimensional Lie algebras and of the algebra sl(2,R). These results enable constructing multi-parameter families of first- and second-order quasi-exactly solvable models. In particular, we have obtained two classes of quasi-exactly solvable matrix Schroedinger equations. 
  We present the Hamiltonian formulation of the bosonic Dirichlet p-brane action. We rewrite the recently proposed quadratic D-brane action in terms of generalized shift vector and lapse function. The first class and the second class constraints are explicitly separated for the bosonic case. We then impose the gauge conditions in such a way that only time-independent gauge transformations are left. In this gauge we obtain the light-cone Hamiltonian which is quadratic in the field momenta of scalar and vector fields. The constraints are explicitly solved to eliminate part of the canonical variables. The Dirac brackets between the remaining variables are computed and shown to be equal to simple Poisson brackets. 
  We review some results recently obtained for the conformal field theories based on the affine Lie superalgebra osp(1|2). In particular, we study the representation theory of the osp(1|2) current algebras and their character formulas. By means of a free field representation of the conformal blocks, we obtain the structure constants and the fusion rules of the model. (Lecture delivered at the CERN-Santiago de Compostela-La Plata Meeting, "Trends in Theoretical Physics", La Plata, Argentina, April-May 1997). 
  The concept of scaling algebra provides a novel framework for the general structural analysis and classification of the short distance properties of algebras of local observables in relativistic quantum field theory. In the present article this method is applied to the simple example of massive free field theory in s = 1,2 and 3 spatial dimensions. Not quite unexpectedly, one obtains for s = 2,3 in the scaling (short distance) limit the algebra of local observables in massless free field theory. The case s =1 offers, however, some surprises. There the algebra of observables acquires in the scaling limit a non-trivial center and describes charged physical states satisfying Gauss' law. The latter result is of relevance for the interpretation of the Schwinger model at short distances and illustrates the conceptual and computational virtues of the method. 
  We discuss the physical meaning and the geometric interpretation of causality implementation in classical field theories. Causality is normally implemented through kinematical constraints on fields but we show that in a zero-distance limit they also carry a dynamical information, which calls for a revision of our standard concepts of interacting fields. The origin of infinities and other inconsistencies in field theories is traced to fields defined with support on the lightcone; a finite and consistent field theory requires a lightcone generator as the field support. 
  Charged matter spin-1 fields enjoy a nonelectromagnetic gauge symmetry when interacting with vacuum electromagnetism, provided their gyromagnetic ratio is 2. 
  Classical three dimensional Yang-Mills is seen to be related to the topological Chern-Simons term through a nonlinear but fully local and covariant gauge field redefinition. A classical recursive cohomological argument is provided. 
  We discuss the role of fixed scalars($\nu,\lambda$) in scattering off a five-dimensional balck hole. The issue is to explain the disagreement of the greybody factor for $\lambda$ between the semiclassical and effective string calculations. In the effective string approach, this is related to the operators with dimension (3,1) and (1,3). On the semiclassical calculation, this originates from a complicated mixing between $\lambda$ and other fields. Hence it may depend on the decoupling procedure. It is shown that $\lambda$ depends on the gauge choices such as the harmonic, dilaton gauges, and the Krasnitz-Klebanov setting for $h_{\mu\nu}$. It turns out that $\nu$ plays a role of test field well, while the role of $\lambda$ is obscure. 
  We study in detail the semi-infinite or BRST cohomology of general affine Lie algebras. This cohomology is relevant in the BRST approach to gauged WZNW models. We prove the existence of an infinite sequence of elements in the cohomology for non-zero ghost numbers. This will imply that the BRST approach to topological WZNW model admits many more states than a conventional coset construction. This conclusion also applies to some non-topological models. Our work will also contain results on the structure of Verma modules over affine Lie algebras. In particular, we generalize the results of Verma and Bernstein-Gel'fand-Gel'fand,for finite dimensional Lie algebras, on the structure and multiplicities of Verma modules. 
  We generalize the scale invariant gravity by allowing a negative kinetic energy term for the classical scalar field. This gives birth to a new scalar-tensor theory of gravity, in which the scalar field is in fact an auxiliary field. For a pure gravity theory without matter, the scale symmetric phase represents an equivalent class of gravity theories, which the Einstein gravity plus a cosmological constant belongs to under a special gauge choice. The one-loop quantum correction of the theory is calculated by using the Vilkovisky-DeWitt's method. We find that the scale symmetry is broken dynamically, and that the Einstein gravity is the ground state of the broken phase. We also briefly discuss the consequent cosmological implications. It is shown that the time-delay experiment restricts the present universe to be very close to the ground state. 
  When graviton loops are taken into account, the background metric obtained as a solution to the one-loop corrected Einstein equations turns out to be gauge fixing dependent. Therefore it is of no physical relevance. Instead we consider a physical observable, namely the trajectory of a test particle in the presence of gravitons. We derive a quantum corrected geodesic equation that includes backreaction effects and is explicitly independent of any gauge fixing parameter. 
  We construct general static black hole configuration for the theory of N=2, d=5 supergravity coupled to an arbitrary number of Abelian vector multiplets. The underlying very special geometry structure plays a major role in this construction. From the viewpoint of M-theory compactified on a Calabi-Yau threefold, these black holes are identified with BPS winding states of the membrane around 2-cycles of the Calabi-Yau threefold, and thus are of importance in the probing of the phase transitions in the moduli space of M-theory compactified on a Calabi-Yau threefold. 
  A new approach is demonstrated that QFTs can be UV finite if they are viewed as the low energy effective theories of a fundamental underlying theory (that is complete and well-defined in all respects) according to the nowaday's standard point of view. No subtraction procedure, counter terms and hence bare parameters are needed. It can also be viewed as a new formulation of the wilson's renormalization program. In contrast to the old ones, this new approach works for any interaction model and spacetime dimensions. Its important implications are sketched. 
  In discussing the construction of a consistent theory of quantum gravity unified with the gauge interactions we are naturally led to a string theory. We review its properties and the five consistent supersymmetric string theories in ten dimensions. We finally discuss the evidence that these theories are actually special limits of a unique 11-dimensional theory, called M-theory, and a recent conjecture for its explicit formulation as a supersymmetric Matrix theory. 
  On the basis of the principle that topological quantum phases arise from the scattering around space-time defects in higher dimensional unification, a geometric model is presented that associates with each quantum phase an element of a transformation group. 
  We investigate the anisotropic integrable spin chain consisting of spins $s={1/2}$ and $s=1$ by means of thermodynamic Bethe ansatz for the anisotropy $\gamma>\pi/3$, where the analysis of the Takahashi conditions leads to a more complicated string picture. We give the phase diagram with respect to the two real coupling constants $\bar{c}$ and $\tilde{c}$, which contains a new region where the ground state is formed by strings with infinite Fermi zones. In this region the velocities of sound for the two physical excitations have been calculated from the dressed energies. This leads to an additional line of conformal invariance not known before. 
  We use analyticity arguments to conjecture a one-loop gravity scattering amplitude with an arbitrary number of external legs possessing the same helicity. This result also gives the complete perturbative S-matrix of self-dual gravity. 
  There exist simple single-charge and multi-charge BPS p-brane solutions in the D-dimensional maximal supergravities. From these, one can fill out orbits in the charge vector space by acting with the global symmetry groups. We give a classification of these orbits, and the associated cosets that parameterise them. 
  The requirement that duality and renormalization group transformations commute as motions in the space of a theory has recently been explored to extract information about the renormalization flows in different statistical and field theoretical systems. After a review of what has been accomplished in the context of 2d sigma models, new results are presented which set up the stage for a fully generic calculation at two-loop order, with particular emphasis on the question of scheme dependence. 
  The Renormalization Group Flow Equations of the Scalar-QED model near Planck's scale are computed within the framework of the average effective action. Exact Flow Equations, corrected by Einstein Gravity, for the running self-interacting scalar coupling parameter and for the running v.e.v. of $\phi^* \phi$, are computed taking into account threshold effects. Analytic solutions are given in the infrared and ultraviolet limits. 
  The realistic free fermionic models have had remarkable success in providing plausible explanations for various properties of the Standard Model which include the natural appearance of three generations, the explanation of the heavy top quark mass and the qualitative structure of the fermion mass spectrum in general, the stability of the proton and more. These intriguing achievements makes evident the need to understand the general space of these models. While the number of possibilities is large, general patterns can be extracted. In this paper I present a detailed discussion on the construction of the realistic free fermionic models with the aim of providing some insight into the basic structures and building blocks that enter the construction. The role of free phases in the determination of the phenomenology of the models is discussed in detail. I discuss the connection between the free phases and mirror symmetry in (2,2) models and the corresponding symmetries in the case of the (2,0) models. The importance of the free phases in determining the effective low energy phenomenology is illustrated in several examples. The classification of the models in terms of boundary condition selection rules, real world-sheet fermion pairings, exotic matter states and the hidden sector is discussed. 
  We propose a toy model to illustrate how the effective Lagrangian for super QCD might go over to the one for ordinary QCD by a mechanism whereby the gluinos and squarks in the fundamental theory decouple below a given supersymmetry breaking scale $m$. The implementation of this approach involves a suitable choice of possible supersymmetry breaking terms. An amusing feature of the model is the emergence of the ordinary QCD degrees of freedom which were hidden in the auxiliary fields of the supersymmetric effective Lagrangian. 
  An overview of the quantum integrable systems (QIS) is presented. Basic concepts of the theory are highlighted stressing on the unifying algebraic properties, which not only helps to generate systematically the representative Lax operators of different models, but also solves the related eigenvalue problem in an almost model independent way. Difference between the approaches in the integrable ultralocal and nonultralocal quantum models are explained and the interrelation between the QIS and other subjects are focussed on. 
  Massless Majorana fermions in the adjoint representation of SU(N_c) are expected to screen gauge interactions in 1+1 dimensions, analogous to a similar Higgs phenomena known for 1+1-dimensional U(1) gauge theory with massless fundamental fermions (Schwinger model). Using the light-cone formalism and large-N_c limit, a non-abelian analogue of the Schwinger boson is shown to be responsible for the screening between heavy test charges. This adjoint boson does not exist simply as a physical state, but boundstates are built entirely from this particle. 
  We interpret a class of 4k-dimensional instanton solutions found by Ward, Corrigan, Goddard and Kent as four-dimensional instantons at angles. The superposition of each pair of four-dimensional instantons is associated with four angles which depend on some of the ADHM parameters. All these solutions are associated with the group $Sp(k)$ and are examples of Hermitian-Einstein connections on $\bE^{4k}$. We show that the eight-dimensional solutions preserve 3/16 of the ten-dimensional N=1 supersymmetry. We argue that under the correspondence between the BPS states of Yang-Mills theory and those of M-theory that arises in the context of Matrix models, the instantons at angles configuration corresponds to the longitudinal intersecting 5-branes on a string at angles configuration of M-theory. 
  We present several classes of new 6d string theories which arise via branes at orbifold singularities. They have compact moduli spaces, associated with tensor multiplets, given by Weyl alcoves of non-Abelian groups. We discuss T-duality and Matrix model applications upon compactification. 
  We consider configurations of Neveu-Schwarz fivebranes, Dirichlet fourbranes and an orientifold sixplane in type IIA string theory. Upon lifting the configuration to M-theory and proposing a description of how to include the effects of the orientifold sixplane we derive the curves describing the Coulomb branch of N=2 gauge theories with orthogonal and symplectic gauge groups, product gauge groups of the form SU(k_1)...SU(k_i) x SO(N) and SU(k_1)...SU(k_i) x Sp(N). We also propose new curves describing theories with unitary gauge groups and matter in the symmetric or antisymmetric representation. 
  We propose a general method that allows to detect the existence of normalizable ground states in supersymmetric quantum mechanical systems with non-Fredholm spectrum. We apply our method to show the existence of bound states at threshold in two important cases: 1) the quantum mechanical system describing H-monopoles; 2) the quantum mechanics of D0 branes. 
  Explicit formulae describing the genus one characters and modular transformation properties of permutation orbifolds of arbitrary Rational Conformal Field Theories are presented, and their relation to the theory of covering surfaces is investigated. 
  A non-Grassmanian path integral representation is given for the solution of the Klein-Gordon and the Dirac equations. The trajectories of the path integral are rendered differentiable by the relativistic corrections. The nonrelativistic limit is briefly discussed from the point of view of the renormalization group. 
  We identify the 4D field theories living on the world volume of D4 branes in non-supersymmetric type IIA string theory constructions. They are softly broken N=2 SQCD with the breakings introduced through vevs of the auxilliary fields in the spurion coupling field. Exact solutions of these theories for perturbing soft breakings exist in the literature. We calculate the ratios of string tensions in softly broken N=2 SU(N) gauge theory testing the recently proposed M-theory prediction. The semi-classical result of M-theory is renormalized in the non-supersymmetric models. 
  We present a zero dimensional matrix model based on $USp(2k)$ with supermultiplets in symmetric, antisymmetric and fundamental representations. The four dimensional compactification of this model naturally captures the exact results of Sen \cite{Sen} in $F$ theory. Eight dynamical and eight kinematical supercharges are found, which is required for critical string interpretation. Classical vacuum has ten coordinates and is equipped with orbifold structure. We clarify the issue of spacetime dimensions which $F$ theory represented by this matrix model produces. 
  In an effort to understand the physical implications of the newly discovered non-trivial directions in scalar field theory, we compute lowest order scattering amplitudes, cross sections, and the 1-loop effective potential. To lowest order, the primary effect of the nonpolynomial nature of the theories is a renormalization of the field. The high energy scaling of the 2-->2 cross section is studied and found to differ significantly from that of pure phi^4 theory. From the 1-loop effective potential we determine that in some cases radiative corrections destroy classical symmetry breaking, resulting in a phase boundary between symmetry broken and unbroken theories. No radiatively induced symmetry breaking is observed. 
  The path integral for the relativistic spinless Aharonov-Bohm-Coulomb system is solved, and the energy spectra are extracted from the resulting amplitude. 
  We describe in superspace a theory of a massive superparticle coupled to a version of two dimensional N=1 dilaton supergravity. The (1+1) dimensional supergravity is generated by the stress-energy of the superparticle, and the evolution of the superparticle is reciprocally influenced by the supergravity. We obtain exact superspace solutions for both the superparticle worldline and the supergravity fields. We use the resultant non-trivial compensator superfield solution to construct a model of a two-dimensional supersymmetric black hole. 
  We analyze M theory fivebrane in order to study the moduli space of vacua of N=1 supersymmetric $Sp(N_c)$ gauge theories with $N_f$ flavors in four dimensions. We show how the N=2 Higgs branch can be encoded in M theory by studying the orientifold which plays a crucial role in our work. When all the quark masses are the same, the surface of the M theory spacetime representing a nontrivial ${\bf S^1}$ bundle over ${\bf R^3}$ develops $A_{N_f-1}$ type singularities at two points where D6 branes are located. Furthermore, by turning off the masses, two singular points on the surface collide and produce $A_{2N_f-1}$ type singularity. The sum of the multiplicities of rational curves on the resolved surface gives the dimension of N=2 Higgs branch which agrees with the counting from the brane configuration picture of type IIA string theory. By rotating M theory fivebranes we get the strongly coupled dynamics of N=1 theory and describe the vacuum expectation values of the meson field parameterizing Higgs branch which are in complete agreement with the field theory results. Finally, we take the limit where the mass of adjoint chiral multiplet goes to infinity and compare with field theory results. For massive case, we comment on some relations with recent work which deals with N=1 duality in the context of M theory. 
  Recent progress in quantum field theory and quantum gravity relies on mixed boundary conditions involving both normal and tangential derivatives of the quantized field. In particular, the occurrence of tangential derivatives in the boundary operator makes it possible to build a large number of new local invariants. The integration of linear combinations of such invariants of the orthogonal group yields the boundary contribution to the asymptotic expansion of the integrated heat-kernel. This can be used, in turn, to study the one-loop semiclassical approximation. The coefficients of linear combination are now being computed for the first time. They are universal functions, in that are functions of position on the boundary not affected by conformal rescalings of the background metric, invariant in form and independent of the dimension of the background Riemannian manifold. In Euclidean quantum gravity, the problem arises of studying infinitely many universal functions. 
  Four tachyon scattering amplitude is derived from the $S^N\R^{24}$ orbifold sigma model in the large $N$ limit. The closed string interaction is described by a vertex which is a bosonic analog of the supersymmetric one, recently proposed by Dijkgraaf, Verlinde and Verlinde. 
  These are a set of lectures presented at the CERN-La Plata-Santiago de Compostella School of Physics, La Plata, May 1997. A brief introduction to the non-perturbative structure of string theory is presented. Various non-perturbative dualities in ten and six dimensions as well as D-branes are discussed. 
  We explicitly compute the Compton amplitude for the scattering of a photon and a (massless) ``electron/positron'' at tree level and one loop, in a four-dimensional fermionic heterotic string model. We comment on the relationship between the amplitudes we compute in string theory and the corresponding ones in field theory. 
  The existence of non-local charges, generating a Yangian symmetry is discussed in generalized chiral Gross-Neveu models. Their conservation can be proven by a finite-loop perturbative computation, the order of which is determined from group theoretic constants and is independent of the number of flavors. Examples, where the 1-loop calculation is sufficient, include the SO(n)-models and other more exotic groups and representations. 
  The idea of viewing gravitation as a many body phenomenon is put forward here. Physical arguments supporting this idea are briefly reviewed. The basic mathematical object of the new gravitational mechanics is a matrix of operators. Striking similarity of the method of R-matrix (QISM) to the mathematical formulation of the new gravitational mechanics is pointed out. The s-wave difference Schrodinger equation describing a process of emission of radiation by a gravitating particle is shown to be analogous to the Baxter equation of the QISM. 
  We construct brane configurations leading to chiral four dimensional N=1 supersymmetric gauge theories. The brane realizations consist of intersecting Neveu-Schwarz five-branes and Dirichlet four-branes in non-flat spacetime backgrounds. We discuss in some detail the construction in a C^2/Z_M orbifold background. The infrared theory on the four-brane worldvolume is a four dimensional N=1 SU(N)^M gauge theory with chiral matter representations. We discuss various consistency checks and show that the spectral curves describing the Coulomb phase of the theory can be obtained once the orbifold brane construction is embedded in M-theory. We also discuss the addition of extra vectorlike matter and other interesting generalizations. 
  We describe a general way of constructing integrable defect theories as perturbations of conformal field theory by local defect operators. The method relies on folding the system onto a boundary field theory of twice the central charge. The classification of integrable defect theories obtained in this way parallels that of integrable bulk theories which are a perturbation of the tensor product of two conformal field theories. These include local defect perturbations of all $c<1$ minimal models, as well as of the coset theories based on SO(2n), obtained in this way. We discuss in detail the former case of all the Virasoro minimal models. In the Ising case our construction corresponds to having a spin field as a defect operator; in the folded formulation this is mapped onto an orbifolding of the boundary sine-Gordon theory at $\beta^2/8\pi = 1/8$, or a version of the anisotropic Kondo model. 
  We discuss the asymptotic properties of quantum states density for fundamental (super) membrane in the semiclassical approach. The matching of BPS part of spectrum for superstring and supermembrane gives the possibility to get stringy results via membrane calculations and vice versa. The brane-black hole correspondence (on the level of black hole states and brane microstates) is also studied. 
  We discuss, in the context of M(atrix) theory, the creation of a membrane suspendend between two longitudinal five-branes when they cross each other. It is shown that the membrane creation is closely related to the degrees of freedom in the off-diagonal blocks which are related via dualities to the chiral fermionic zero mode on a 0-8 string. In the dual system of a D0-brane and a D8-brane in type \IIA theory the half-integral charges associated with the ``half''-strings are found to be connected to the well-known fermion-number fractionalization in the presence of a fermionic zero mode. At sufficiently short distances, the effective potential between the two five-branes is dominated by the zero mode contribution to the vacuum energy. 
  The existence of topological invariants analogous to Chern/Pontryagin classes for a standard $SO(D)$ or $SU(N)$ connection, but constructed out of the torsion tensor, is discussed. These invariants exhibit many of the features of the Chern/Pontryagin invariants: they can be expressed as integrals over the manifold of local densities and take integer values on compact spaces without boundary; their spectrum is determined by the homotopy groups $\pi_{D-1}(SO(D))$ and $\pi_{D-1}(SO(D+1))$.   These invariants are not solely determined by the connection bundle but depend also on the bundle of local orthonormal frames on the tangent space of the manifold. It is shown that in spacetimes with nonvanishing torsion there can occur topologically stable configurations associated with the frame bundle which are independent of the curvature.   Explicit examples of topologically stable configurations carrying nonvanishing instanton number in four and eight dimensions are given, and they can be conjectured to exist in dimension $4k$. It is also shown that the chiral anomaly in a spacetime with torsion receives a contribution proportional to this instanton number and hence, chiral theories in $4k$-dimensional spacetimes with torsion are potentially anomalous. 
  We construct the theory of 2d dilatonic supergravity(SG) with matter and dilaton supermultiplets coupled to dilaton functions. Trace anomaly and induced effective action for matter supermultiplet are calculated (what gives also large-N effective action for dilatonic SG). Study of black holes and Hawking radiation which turns out to be zero in supersymmetric CGHS model with dilaton coupled matter is presented. In the same way one can study spherically symmetric collapse for other 4d SG using simplified 2d approach. 
  We present a superfield formulation of the quantization program for theories with first class constraints. An exact operator formulation is given, and we show how to set up a phase-space path integral entirely in terms of superfields. BRST transformations and canonical transformations enter on equal footing, and they allow us to establish a superspace analog of the BFV theorem. We also present a formal derivation of the Lagrangian superfield analogue of the field-antifield formalism, by an integration over half of the phase-space variables. 
  We introduce a finite-dimensional algebra that controls the possible boundary conditions of a conformal field theory. For theories that are obtained by modding out a Z_2 symmetry (corresponding to a so-called D_odd-type, or half-integer spin simple current, modular invariant), this classifying algebra contains the fusion algebra of the untwisted sector as a subalgebra. Proper treatment of fields in the twisted sector, so-called fixed points, leads to structures that are intriguingly close to the ones implied by modular invariance for conformal field theories on closed orientable surfaces. 
  The black hole solutions to Einstein's vacuum field equations are also solutions to the equations of motion of the low energy limit of superstring theory. At the same time, string theory boasts a much broader and richer collection of black hole solutions. Fortunately, string theories also possess a remarkable set of duality symmetries relating states within and between different string theories. These duality symmetries can be exploited to construct new black hole solutions from known solutions, giving us powerful tools with which to explore the black hole solutions of string theory. Here we introduce and demonstrate these techniques of solution generating. 
  We show how the usual chiral perturbation theory description of phenomenological pion physics admits an interpretation as a low-energy string-like model associated with QCD. By naive and straightforward generalization within the context of a new class of supersymmetrical models, it is shown that this string-like structure admits a 4D, N = 1 supersymmetrical extension. The presence of a WZNW term in the model implies modifications of certain higher order processes involving the ordinary SU(3) pion octet. 
  The path integral for relativistic three-dimensional spinless Aharonov-Bohm-Coulomb system is solved, and the energy spectra are extracted from the resulting amplitude. 
  We compute by means of the Bethe Ansatz the boundary S matrix for the open anisotropic spin-1/2 chain with diagonal boundary magnetic fields in the noncritical regime (Delta > 1). Our result, which is formulated in terms of q-gamma functions, agrees with the vertex-operator result of Jimbo et al. 
  We study a model for two-dimensional skyrmions on a sphere of radius L. Such model simulates a skyrmion lattice of density W/(2 \pi L^2), where W is the skyrmion winding number. We show that, to a very good approximation, physical results depend only on the product \alpha L^4, where \alpha is the strength of potential term. In the range \alpha L^4 approx. or less than 3 the order parameter vanishes, there is a uniform distribution of the density over the whole surface and the energy of the W=2 sector lies above twice the energy of the W=1 sector. If \alpha L^4 approx. or greater than 6 the order parameter approaches unity and the density concentrates near one of the poles. Moreover the disoliton is always bound. We also present a variational solution to the field equations for which the pure \alpha L^4-dependence is exact. Finally, some consequences of our results for the Quantum Hall Effect are discussed. 
  We use the abelian Born-Infeld action for the worldvolume gauge field and transverse displacement scalars to explore new aspects of D-brane structure and dynamics. We study several classic gauge field configurations, including point charges in any worldvolume dimension and vortices in two worldvolume dimensions, and show that, with an appropriate excitation of the transverse coordinate field, they are BPS-saturated solutions. The Coulomb point charge solutions turn out to represent, with considerable fidelity, fundamental strings attached to the brane (their magnetic counterparts describe D1-branes attached to D3-branes). We also show that S-matrix for small excitations propagating on the point charge solution is consistent with (and gives further illuminating information about) Polchinski's effective open string boundary condition. 
  We present new expressions of form factors of the XXZ model which satisfy Smirnov's three axioms. These new form factors are obtained by acting the affine quantum group $U_q (\hat{\frak s \frak l_2})$ to the known ones obtained in our previous works. We also find the relations among all the new and known form factors, i.e., all other form factors can be expressed as kind of descendents of a special one. 
  Based on a detailed analysis of nonlinear field equations of the SU(2) Yang-Mills-Higgs system, we obtain the effective field theory describing low-energy interaction of BPS dyons and massless particles (i.e., photons and Higgs particles). Our effective theory manifests electromagnetic duality and spontaneously broken scale symmetry, and reproduces the multimonopole moduli space dynamics of Manton in a suitable limit. Also given is a generalization of our approach to the case of BPS dyons in a gauge theory with an arbitrary gauge group that is maximally broken. 
  In this work we construct a gauge invariant description of free massive particle with an arbitrary integer spin. Such description allows one to investigate the problem of consistent interactions for massive high spin particles using the requirement that the whole interaction Lagrangian must be gauge invariant. As a first example of such approach, we consider the case of electromagnetic interaction of massive spin-2 particle: a linear approximation in a case of the arbitrary field and a full theory for the homogeneous electromagnetic field in the space-time of any dimensionality. 
  In this paper, for massive fields of spins 2 and 3 with non-canonical Lagrangians, we build Hamiltonians and full systems of constraints and show that the use of derivatives in a redefinition of fields can give rise to a change of number of physical degrees of freedom. 
  Using index-free notation, we present the diagonal values of the first five heat kernel coefficients associated with a general Laplace-type operator on a compact Riemannian space without boundary. The fifth coefficient appears here for the first time. For a flat space with a gauge connection, the sixth coefficient is given too. Also provided are the leading terms for any coefficient, both in ascending and descending powers of the Yang-Mills and Riemann curvatures, to the same order as required for the fourth coefficient. These results are obtained by directly solving the relevant recursion relations, working in Fock-Schwinger gauge and Riemann normal coordinates. Our procedure is thus noncovariant, but we show that for any coefficient the `gauged' respectively `curved' version is found from the corresponding `non-gauged' respectively `flat' coefficient by making some simple covariant substitutions. These substitutions being understood, the coefficients retain their `flat' form and size. In this sense the fifth and sixth coefficient have only 26 and 75 terms respectively, allowing us to write them down. Using index-free notation also clarifies the general structure of the heat kernel coefficients. In particular, in flat space we find that from the fifth coefficient onward, certain scalars are absent. This may be relevant for the anomalies of quantum field theories in ten or more dimensions. 
  We discuss the symmetry properties of the low-energy effective action of the type IIB superstring that may be employed to derive four-dimensional solutions. A truncated effective action, compactified on a six-torus, but including both Neveu/Schwarz-Neveu/Schwarz and Ramond-Ramond field strengths, can be expressed as a non-linear sigma model which is invariant under global SL(3,R) transformations. This group contains as a sub-group the SL(2,R) symmetry of the ten-dimensional theory and a discrete Z2 reflection symmetry which leads to a further SL(2,R) sub-group. The symmetries are employed to analyse a general class of spatially homogeneous cosmological solutions with non-trivial Ramond-Ramond fields. 
  The continuous and discrete symmetries of a dimensionally reduced type IIB superstring action are employed to generate four-dimensional cosmological solutions with non-trivial Neveu-Schwarz/Neveu-Schwarz and Ramond-Ramond form-fields from the dilaton-moduli-vacuum solutions. 
  The effective potential of $\lambda\phi^4_{1+3}$ model with both sign of parameter $m^2$ is evaluated at T=0 by means of a simple but effective method for regularization and renormalization. Then at $T\ne 0$, the effective potential is evaluated in imaginary time Green Function approach, using the Plana formula. A critical temperature for restoration of symmetry breaking in the standard model of particle physics is estimated to be $T_c\simeq 510$ GeV. 
  Basing on the fundamental symmetry that the space-time inversion is equivalent to particle-antiparticle transformation, a relativistic modification on the stationary Schrodinger equation for many-particle system is made. The eigenvalue in the center of mass system is no longer equal to the negative of binding energy simply. The possible applications in various fields (e.g. the model of quarkonium) are discussed. 
  Multidimensional cosmological model describing the evolution of n+1 Einstein spaces in the theory with several scalar fields and forms is considered. When a (electro-magnetic composite) p-brane Ansatz is adopted the field equations are reduced to the equations for Toda-like system. The Wheeler-De Witt equation is obtained. In the case when n "internal" spaces are Ricci-flat, one space M_0 has a non-zero curvature, and all p-branes do not "live" in M_0, the classical and quantum solutions are obtained if certain orthogonality relations on parameters are imposed. Spherically-symmetric solutions with intersecting non-extremal p-branes are singled out. A non-orthogonal generalization of intersection rules corresponding to (open, closed) Toda lattices is obtained. A chain of bosonic D > 11 models (that may be related to hypothetical higher dimensional supergravities and F-theories) is suggested. 
  The aim of this article is to calculate (to first order in $\hbar$) the renormalized effective action of a self interacting massive scalar field propagating in the space-time due to a cylindrically symmetric, rotating body. The vacuum (exterior space-time) contribution is model independent, we also consider the simplest case of a core (interior space-time) model, namely a cylindrical shell. The heat kernel of the system is calculated, and used to obtain an expression for the determinant of the Klein-Gordon operator on the space-time manifold. New ultra-violet poles are discovered, and regularization techniques are then employed to render finite the Klein-Gordon determinant and consequently extract the regularized one loop effective action for a self interacting scalar field theory. The coupling constants of the theory are then renormalized. As a test case a conical singularity with non-zero flux is also considered. 
  The anti-BRST transformation, in its Sp(2)-symmetric version, for the general case of any stage-reducible gauge theories is implemented in the usual BV approach. This task is accomplished not by duplicating the gauge symmetries but rather by duplicating all fields and antifields of the theory and by imposing the acyclicity of the Koszul-Tate differential. In this way the Sp(2)-covariant quantization can be realised in the standard BV approach and its equivalence with BLT quantization can be proven by a special gauge fixing procedure. 
  Fermionic and bosonic ghost systems are defined each in terms of a single vertex algebra which admits a one-parameter family of conformal structures. The observation that these structures are related to each other provides a simple way to obtain character formulae for a general twisted module of a ghost system. The U(1) symmetry and its subgroups that underly the twisted modules also define an infinite set of invariant vertex subalgebras. Their structure is studied in detail from a W-algebra point of view with particular emphasis on Z_N-invariant subalgebras of the fermionic ghost system. 
  The thermofield dynamics of the $D = 10$ heterotic thermal string theory is exemplified at any finite temperature through the infrared behaviour of the one-loop cosmological constant in proper reference to the thermal duality symmetry in association with the global phase structure of the thermal string ensemble. 
  We study ten-dimensional N=2 maximal chiral supergravity in the context of Lie superalgebra SU(8/1). The possible successive superalgebraic truncations from ten dimensional N=2 chiral theory to the lower dimensional supergravity theories are systematically realized as sub-superalgebraic chains of SU(8/1) by using the Kac-Dynkin weight techniques. 
  Recent work in Euclidean quantum gravity has studied boundary conditions which are completely invariant under infinitesimal diffeomorphisms on metric perturbations. On using the de Donder gauge-averaging functional, this scheme leads to both normal and tangential derivatives in the boundary conditions. In the present paper, it is proved that the corresponding boundary value problem fails to be strongly elliptic. The result raises deep interpretative issues for Euclidean quantum gravity on manifolds with boundary. 
  We will report on recent advances in the understanding of non-perturbative interconnections between different string dualities. Weak-strong coupling duality (S-duality) and T-duality (symmetry under compactification on dual tori) allows one to compare and explore the strong coupling regime of seemingly unrelated theories. These theories naturally merge in a quantum version of supergravity called M-theory. The dynamical role of `branes' of different nature and the new dynamical tool of (M)atrix formulation of M-theory will be briefly mentioned. 
  The reduced O(3)-sigma model with an O(3)->O(2) symmetry breaking potential is considered with an additional Skyrmionic term, i.e. a totally antisymmetric quartic term in the field derivatives. This Skyrme term does not affect the classical static equations of motionwhich, however, allow an unstable sphaleron solution. Quantum fluctuations around the static classical solution are considered for the determination of the rate of thermally induced transitions between topologically distinct vacua mediated by the sphaleron. The main technical effect of the Skyrme term is to produce an extra measure factor in one of the fluctuation path integrals which is therefore evaluated using a measure-modified Fourier-Matsubara decomposition (this being one of the few cases permitting this explicit calculation). The resulting transition rate is valid in a temperature region different from that of the original Skyrme-less model, and the crossover from transitions dominated by thermal fluctuations to those dominated by tunneling at the lower limit of this range depends on the strength of the Skyrme coupling. 
  We investigate boundary states of D-branes wrapped around supersymmetric cycles in Kazama--Suzuki models. We show that the geometry of the D-branes corresponds to a generalisation of calibrated geometry. We comment on the link with the geometry of the coset space and discuss how T-duality maps between these boundary states. 
  We consider the perturbations of the 3-state Potts conformal field theory introduced by Cardy as a description of the chiral 3-state Potts model. By generalising Zamolodchikov's counting argument and by explicit calculation we find new inhomogeneous conserved currents for this theory. We conjecture the existence of an infinite set of conserved currents of this form and discuss their relevance to the description of the chiral Potts models. 
  SU(N_L) X SU(N_R) gauge theories are investigated as effective field theories on D_4 branes in type IIA string theory. The classical gauge configuration is shown to match quantitatively with a corresponding classical U(N_L) X U(N_R) gauge theory. Quantum effects freeze the U(1) gauge factors and turn some parameters into moduli. The SU(N_L) X SU(N_R) quantum model is realized in M theory. Starting with an N=2 configuration (parallel NS fivebranes), the rotation of a single NS fivebrane is considered. Generically this leads to a complete lifting of the Coulomb moduli space. The implications of this result to field theory and the dynamics of branes are discussed. When the initial M fivebrane is reducible, part of the Coulomb branch may survive. Some such situations are considered, leading to curves describing the effective gauge couplings for N=1 models. The generalization to models with more gauge group factors is also discussed. 
  In this article we review a recent calculation of the two-loop $\sigma$-model corrections to the T-duality map in string theory. Using the effective action approach, and focusing on backgrounds with a single Abelian isometry, we give the $O(\alpha')$ modifications of the lowest-order duality transformations. The torsion plays an important role in the theory to $O(\alpha')$, because of the Chern-Simons couplings to the gauge fields that arise via dimensional reduction. 
  We develop a Hamiltonian theory for 2D soliton equations. In particular, we identify the spaces of doubly periodic operators on which a full hierarchy of commuting flows can be introduced, and show that these flows are Hamiltonian with respect to a universal symplectic form $\omega={1\over 2}\r_{\infty} <\Psi_0^*\delta L\wedge\delta\Psi_0>\d k$. We also construct other higher order symplectic forms and compare our formalism with the case of 1D solitons. Restricted to spaces of finite-gap solitons, the universal symplectic form agrees with the symplectic forms which have recently appeared in non-linear WKB theory, topological field theory, and Seiberg-Witten theories. We take the opportunity to survey some developments in these areas where symplectic forms have played a major role. 
  In this note it is demonstrated how the Seiberg-Witten solutions and related integrable systems may arise from certain brane configurations in M-theory. Some subtleties of the formulation of the Seiberg-Witten theory via integrable systems are discussed and interpreted along the lines of general picture of string/M-theory dualities. 
  We present a manifestly Sl(2;Z)-covariant action for the type IIB superstring, and prove kappa-symmetry for on-shell IIB supergravity backgrounds. 
  We describe gauge-fixing at the level of virtual paths in the path integral as a non-symplectic BRST-type of flow on the path phase space. As a consequence a gauge-fixed, non-local symplectic structure arises. Restoring of locality is discussed. A pertinent anti-Lie-bracket and an infinite dimensional group of gauge fermions are introduced. Generalizations to Sp(2)-symmetric BLT-theories are made. 
  For the gauge couplings, which arise after toroidal compactification of six-dimensional heterotic N=1 string theories from the T^2 torus, we calculate their one-loop corrections. This is performed by considering string amplitudes involving two gauge fields and moduli fields. We compare our results with the equations following from N=2 special geometry and the underlying prepotential of the theory. Moreover we find relations between derivatives of the N=2, d=4 prepotential and world-sheet tau-integrals which appear in various string amplitudes of any T^2-compactification. 
  Starting from topological quantum field theory, we derive space-time uncertainty relation with respect to the time interval and the spatial length proposed by Yoneya through breakdown of topological symmetry in the large N matrix model. This work suggests that the topological symmetry might be an underlying higher symmetry behind the space-time uncertainty principle of string theory. 
  In the M(atrix) theory by making the expansions of the matrices around the membrane and four-brane solutions we derive the three- and five-dimensional gauge theories on the dual tori. The explicit forms of solutions yield the dual coordinates and each expansion is related to a toroidal compactification of the M(atrix) theory. From the derived Lorentz and gauge invariant actions the gauge coupling constants are shown to be characterized by the volume of the dual tori. 
  We show here that matrix Darboux-Toda transformation can be written as a product of a number of mappings. Each of these mappings is a symmetry of the matrix nonlinear Shrodinger system of integro-differential equations. We thus introduce a completely new type of discrete transformations for this system. The discrete symmetry of the vector nonlinear Shrodinger system is a particular realization of these mappings. 
  In the framework of three dimensional extended supergravity theories, we demonstrate that there exist non-trivial Killing spinors over BPS soliton configurations, even when the space-time is asymptotically conical. We also show that there are no physical fermionic zero modes on these backgrounds. We further generalize these results to the case of semilocal systems. 
  We investigate the geometric interpretation of the Standard Model based on noncommutative geometry. Neglecting the $S_0$-reality symmetry one may introduce leptoquarks into the model. We give a detailed discussion of the consequences (both for the Connes-Lott and the spectral action) and compare the results with physical bounds. Our result is that in either case one contradicts the experimental results. 
  We construct M-theory curves associated with brane configurations of SU(N), SO(N) and $Sp(2N)$ 5d supersymmetric gauge theories compactified on a circle. From the curves we can account for all the existing different SU(N) field theories with $N_f \leq 2 N$. This is the correct bound for $N \geq 3$. We remark on the exceptional case SU(2). The bounds obtained for SO(N) and $Sp(2N)$ are $N_f\leq N-4$ and $N_f\leq 2N+4$, respectively. 
  A renormalized perturbative expansion of interacting quantum fields on a globally hyperbolic spacetime is performed by adapting the Bogoliubov Epstein Glaser method to a curved background. The results heavily rely on techniques from microlocal analysis, in particular on Radzikowski's characterization of Hadamard states by wave front sets of Wightman functions. 
  The anomaly due to the chiral fermions on the world-volume of the SO(32) five-brane is calculated. It is shown that this contribution has the correct structure for it to be cancelled by the variation of the classical world-volume action. The cancellation mechanism requires a Green-Schwarz-like term in the classical action. The result confirms the field content of the SO(32) five-brane proposed by Witten. 
  We consider type IIB theory compactified on a two-sphere in the presence of mutually nonlocal 7-branes. The BPS states associated with the gauge vectors of exceptional groups are seen to arise from open strings connecting the 7-branes, and multi-pronged open strings capable of ending on more than two 7-branes. These multi-pronged strings are built from open string junctions that arise naturally when strings cross 7-branes. The different string configurations can be multiplied as traditional open strings, and are shown to generate the structure of exceptional groups. 
  We obtain a BPS soliton of the M theory fivebrane's equations of motion representing a supersymmetric self-dual string. The resulting solution is then dimensionally reduced and used to obtain 0-brane and (p-2)-brane solitons on D-p-branes. 
  A formulation of new six-dimensional theories with (1,0) supersymmetry and E_8 global symmetry is proposed. The model is based on the large n theory describing n D-strings interacting with parallel D-fivebranes in Type I string theory. 
  We construct a realization of the Yangian double DY_\hbar(gl_N) and DY_\hbar(sl_N) of an arbitrary level k in terms of free boson fields with a continuous parameter. In the \hbar \to 0 limit this realization becomes the Wakimoto realization of kac-Moody algebra gl_N and sl_N, respectively. The vertex operators and the screening currents are also constructed with the same spirits; the screening currents commute with DY_\hbar(sl_N) modulo total difference. 
  It is well-known that the conjectured SL(2, Z) invariance of type IIB string theory in ten dimensions also persists in lower dimensions when the theory is compactified on tori. By making use of this recent observation, we construct an infinite family of magnetically charged black hole solutions of type II superstring theory in four space-time dimensions. These solutions are characterized by two relatively prime integers corresponding to the magnetic charges associated with the two gauge fields (from NS-NS and R-R sectors) of the theory and form an SL(2, Z) multiplet. In the extremal limit these solutions are stable as they are prevented from decaying into black holes of lower masses by a `mass gap' equation. 
  It is well-known that the quantum tunneling makes conventional perturbation series non-Borel summable. We use this fact reversely and attempt to extract the decay width of the false-vacuum from the actual perturbation series of the vacuum energy density (vacuum bubble diagrams). It is confirmed that, at least in quantum mechanical examples, our proposal provides a complementary approach to the conventional instanton calculus in the strong coupling region. 
  We show how the multiloop amplitudes of $\Phi^3$ theory (in the worldline formulation of Schmidt and Schubert) are recovered from the N-tachyon $(h+1)$-loop amplitude in bosonic string theory in the $\alpha' \to 0$ limit, if one keeps the tachyon coupling constant fixed. In this limit the integral over string moduli space receives contributions only from those corners where the world-sheet resembles a $\Phi^3$ particle diagram. Technical issues involved are a proper choice of local world-sheet coordinates, needed to take the string amplitudes off-shell, and a formal procedure for introducing a free mass parameter $M^2$ instead of the tachyonic value $-4/\alpha'$. 
  We analyze the gauge symmetry of a topological mass generating action in four dimensions which contains both a vector and a second rank antisymmetric tensor fields. In the Abelian case, this system induces an effective mass for the vector gauge field via a topological coupling $B \wedge F$ in the presence of a kinetic term for the antisymmetric tensor field $B$, while maintaining a gauge symmetry. On the other hand, for the non-Abelian case the $B$ field does not have a gauge symmetry unless an auxiliary vector field is introduced to the system. We analyze this change of symmetry in the Faddeev-Jackiw formalism, and show how the auxiliary vector field enhances the symmetry. At the same time this enhanced gauge symmetry becomes reducible. We also show this phenomenon in this analysis. 
  A four dimensional gauge theory with nonpolynomial but local interactions of 1-form and 2-form gauge potentials is constructed. The model is a nontrivial deformation of a free gauge theory with nonpolynomial dependence on the deformation parameter (= gauge coupling constant). 
  We compute spectra of particles produced during a dilaton-driven kinetic inflation phase within string cosmology models. The resulting spectra depend on the parameters of the model and on the type of particle and are quite varied, some increasing and some decreasing with frequency. We use an approximation scheme in which all spectra can be expressed in a nice symmetric form, perhaps hinting at a deeper symmetry of the underlying physics. Our results may serve as a starting point for detailed studies of relic abundances, dark matter candidates, and possible sources of large scale anisotropy. 
  We describe the breaking of supersymmetry in M-theory by coordinate dependent (Scherk-Schwarz) compactification of the eleventh dimension. Supersymmetry is spontaneously broken in the gravitational and moduli sector and communicated to the observable sector, living at the end-point of the semicircle, by radiative gravitational interactions. This mechanism shares the generic features of non-perturbative supersymmetry breaking by gaugino condensation, in the presence of a constant antisymmetric field strength, in the weakly coupled regime of the heterotic string, which suggests that both mechanisms could be related by duality. In particular an analysis of supersymmetric transformations in the infinite-radius limit reveals the presence of a discontinuity in the spinorial parameter, which coincides with the result found in the presence of gaugino condensation, while the condensate is identified with the quantized parameter entering the boundary conditions 
  Negative dimensional integration method (NDIM) is a technique to deal with D-dimensional Feynman loop integrals. Since most of the physical quantities in perturbative Quantum Field Theory (pQFT) require the ability of solving them, the quicker and easier the method to evaluate them the better. The NDIM is a novel and promising technique, ipso facto requiring that we put it to test in different contexts and situations and compare the results it yields with those that we already know by other well-established methods. It is in this perspective that we consider here the calculation of an on-shell two-loop three point function in a massless theory. Surprisingly this approach provides twelve non-trivial results in terms of double power series. More astonishing than this is the fact that we can show these twelve solutions to be different representations for the same well-known single result obtained via other methods. It really comes to us as a surprise that the solution for the particular integral we are dealing with is twelvefold degenerate. 
  A covariant action for closed D=11 superstring with local $\kappa$-symmetry and global supersymmetry transformations obeying the algebra $\{Q_\alpha,Q_\beta\}=C\Gamma^{\mu\nu}P_\mu n_\nu$ is suggested. Physical sector variables of the model and their dynamics exactly coincide with those of the D=10 type IIA Green-Schwarz superstring. It is shown that action of the D=10 type IIA Green-Schwarz superstring can be considered as a partially gauge fixed version of the D=11 superstring action. 
  Using physical arguments (Higgs mechanism, superconductivity, infrared regime, duality) and a geometric-topological construction (scalar curvature distribution compatible with surgery), we propose a topological interpretation of 3D SW theory in terms of the abelian Casson invariant. Further algebraic reasoning shows equivalence of that invariant to the Alexander ``polynomial''. Our starting point is a 3D version of the original SW theory. Observing that the scalar curvature $R$ plays the role of a mass-squared parameter for the monopole field we can use that observation to control the theory in low-energy limit. 
  Born-Infeld theory admits finite energy point particle solutions with $\delta$-function sources, BIons. I discuss their role in the theory of Dirichlet $p$-branes as the ends of strings intersecting the brane when the effects of gravity are ignored. There are also topologically non-trivial electrically neutral catenoidal solutions looking like two $p$-branes joined by a throat. The general solution is a non-singular deformation of the catenoid if the charge is not too large and a singular deformation of the BIon solution for charges above that limit. The intermediate solution is BPS and Coulomb-like. Performing a duality rotation we obtain monopole solutions, the BPS limit being a solution of the abelian Bogolmol'nyi equations. The situation closely resembles that of sub and super extreme black-brane solutions of the supergravity theories. I also show that certain special Lagrangian submanifolds of ${\Bbb C}^p$, $p=3,4,5$, may be regarded as supersymmetric configurations consisting of $p$-branes at angles joined by throats which are the sources of global monopoles. Vortex solutions are also exhibited. 
  We consider the two-dimensional quantum field theory of a scalar field self-interacting via two periodic terms of frequencies $\alpha$ and $\beta$. Looking at the theory as a perturbed Sine-Gordon model, we use Form Factor Perturbation Theory to analyse the evolution of the spectrum of particle excitations. We show how, within this formalism, the non-locality of the perturbation with respect to the solitons is responsible for their confinement in the perturbed theory. The effects of the frequency ratio $\alpha/\beta$ being a rational or irrational number and the occurrence of massless flows from the gaussian to the Ising fixed point are also discussed. A generalisation of the Ashkin-Teller model and the massive Schwinger model are presented as examples of application of the formalism. 
  It is shown that all possible N sheeted coverings of the cylinder are contained in type IIA matrix string theory as non-trivial gauge field configurations. Using these gauge field configurations as backgrounds the large $N$ limit is shown to lead to the type IIA conformal field theory defined on the corresponding Riemann surfaces. The sum over string diagrams is identified as the sum over non-trivial gauge backgrounds of the SYM theory. 
  A manifestly U-duality covariant approach to M-theory cosmology is developed and applied to cosmologies in dimensions D=4,5. Cosmological properties such as expansion powers and Hubble parameters turn out to be U-duality invariant in certain asymptotic regions. U-duality transformations acting on cosmological solutions, on the other hand, shift the transition time between two asymptotic regions and determine the details of the transition. Moreover, in D=5, we show that U-duality can map expanding negative and positive branch solutions into each other. 
  We describe an approach to understanding exponential decay of correlation functions in asymptotically free theories. This approach is systematic; it does not start from any conjectured mechanism or picture. We begin by studying the metric on the space of configurations and the behavior of the potential-energy function on this space.   We describe how these ideas fit in the framework of QCD, as discussed earlier by one of us (P.O.). We then consider the 1+1-dimensional O(2) and O(3) nonlinear sigma models and show that no gap exists in the former at weak coupling. In the O(3) model a new kind of strong/weak-coupling duality is realized. We briefly outline our proposals for understanding the spectrum. 
  In these two lectures given at the 1997 Zakopane workshop on "New Developments in Quantum Field Theory" we review recent results on universal fluctuations in QCD Dirac spectra. We start the first lecture with a review of some general properties of Dirac spectra. It will be argued that there is an intimate relation between chiral symmetry breaking and correlations of Dirac eigenvalues. In particular, we will focus on the microscopic spectral density density, i.e. the spectral density near zero virtuality on the scale of a typical level spacing. The relation with Leutwyler-Smilga sum-rules will be discussed. Standard methods for the statistical analysis of quantum spectra will be reviewed. Recent results on the application of Random Matrix Theory to spectra of 'complex' systems will be summarized. This leads to the introduction of a chiral Random Matrix Theory (chRMT) with the global symmetries of the QCD partition function. In the second lecture the chiral random matrix model will be compared to QCD and some of its properties will be discussed. Our central conjecture is that correlations of QCD Dirac spectra are described by chRMT. We will review recent results showing that the microscopic spectral density and eigenvalue correlations near zero virtuality are strongly universal. Lattice QCD results for the microscopic spectral density and for correlations in the bulk of the spectrum will be presented. In all cases that have been considered, the correlations are in perfect agreement with chRMT. We will end the second lecture with a review of chiral Random Matrix Theory at nonzero chemical potential. New features of spectral universality in nonhermitean matrices will be discussed. 
  Configurations of two or more branes wrapping different homology cycles of space-time are considered and the amount of supersymmetry preserved is analysed, generalising the analysis of multiple branes in flat space. For K3 compactifications, these give the Type II or M theory origin of certain supersymmetric four-dimensional heterotic string solutions that fit into spin-3/2 multiplets and which become massless at certain points in moduli space. The interpretation of these BPS states and the possibility of supersymmetry enhancement are discussed. 
  Exact expectation values of the fields e^{a\phi} in the Bullough-Dodd model are derived by adopting the ``reflection relations'' which involve the reflection S-matrix of the Liouville theory, as well as special analyticity assumption. Using this result we propose explicit expressions for expectation values of all primary operators in the c<1 minimal CFT perturbed by the operator \Phi_{1,2} or Phi_{2,1}. Some results concerning the $\Phi_{1,5}$ perturbed minimal models are also presented. 
  For background gauge field configurations reducible to the form Amu = (A3, A(x)) where A3 is a constant, we provide an elementary derivation of the recently obtained result for the exact induced Chern-Simons (CS) effective action in QED3 at finite temperature. The method allows us to extend the result in several useful ways: to obtain the analogous result for the `mixed' CS term in the Dorey-Mavromatos model of parity-conserving planar superconductivity, thereby justifying their argument for flux quantization in the model; to the induced CS term for a tau-dependent flux; and to the term of second order in A(x) (and all orders in A3) in the effective action. 
  We discuss the quantum Lax-Phillips theory of scattering and unstable systems. In this framework, the decay of an unstable system is described by a semigroup. The spectrum of the generator of the semigroup corresponds to the singularities of the Lax-Phillips $S$-matrix. In the case of discrete (complex) spectrum of the generator of the semigroup, associated with resonances, the decay law is exactly exponential. The states corresponding to these resonances (eigenfunctions of the generator of the semigroup) lie in the Lax-Phillips Hilbert space, and therefore all physical properties of the resonant states can be computed.   We show that the Lax-Phillips $S$-matrix is unitarily related to the $S$-matrix of standard scattering theory by a unitary transformation parametrized by the spectral variable $\sigma$ of the Lax-Phillips theory. Analytic continuation in $\sigma$ has some of the properties of a method developed some time ago for application to dilation analytic potentials.   We work out an illustrative example using a Lee-Friedrichs model for the underlying dynamical system. 
  We suggest that the static configurations of M-theory may be described by the matrix regularisation of the supermembrane theory in static regime. We compute the long range interaction between a M2-brane and an anti-M2-brane in agreement with the 11 dimensional supergravity result. 
  I discuss the relation between M-theory and M(atrix)-theory in flat space by considering the effective potential for the scattering of two groups of D0-branes in both theories. An explicit calculation of this potential up to two loop order in M(atrix)-theory reveals a fascinating agreement. Lecture given at Strings '97; June 17, 1997. 
  Using the path-integral approach, the quantum massive Thirring and sine-Gordon models are proven to be equivalent at finite temperature. This result is an extension of Coleman's proof of the equivalence between both theories at zero temperature. The usual identifications among the parameters of these models also remain valid at $T \neq 0$. 
  Using the formalism of noncommutative geometric gauge theory based on the superconnection concept, we construct a new type of vector gauge theory possessing a shift-like symmetry and the usual gauge symmetry. The new shift-like symmetry is due to the matrix derivative of the noncommutative geometric gauge theory, and this gives rise to a mass term for the vector field without introducing the Higgs field. This construction becomes possible by using a constant one form even matrix for the matrix derivative, for which only constant zero form odd matrices have been used so far. The fermionic action in this formalism is also constructed and discussed. 
  We expand on previous work involving "vacuum-bounded" states, i.e., states such that every measurement performed outside a specified interior region gives the same result as in the vacuum. We improve our previous techniques by removing the need for a finite outside region in numerical calculations. We apply these techniques to the limit of very low energies and show that the entropy of a vacuum-bounded state can be much higher than that of a rigid box state with the same energy. For a fixed $E$ we let $L_in'$ be the length of a rigid box which gives the same entropy as a vacuum-bounded state of length $L_in$. In the $E\to 0$ limit we conjecture that the ratio $L_in'/L_in$ grows without bound and support this conjecture with numerical computations. 
  This is a pedagogical digest of results reported in Phys Lett B405 (1997) 37, and an explicit implementation of Euler's construction for the solution of the Poisson Bracket dual Nahm equation. But it does not cover 9 and 10-dimensional systems, and subsequent progress on them [hep-th/9707190]. Cubic interactions are considered in 3 and 7 space dimensions, respectively, for bosonic membranes in Poisson Bracket form. Their symmetries and vacuum configurations are explored. Their associated first order equations are transformed to Nahm's equations, and are hence seen to be integrable, for the 3-dimensional case, by virtue of the explicit Lax pair provided. Most constructions introduced also apply to matrix commutator or Moyal Bracket analogs. 
  We analyze in detail supersymmetry breaking by compactification of the fifth dimension in M-theory in the compactification pattern $11d \to 5d \to 4d$ and find that a superpotential is generated for the complex fields coming from $5d$ hypermultiplets, namely the dilaton $S$ and the complex structure moduli. Using general arguments it is shown that these fields are always stabilized such that they don't contribute to supersymmetry breaking, which is completely saturated by the K\"ahler moduli coming from vector multiplets.   It is shown that this mechanism is the strong-coupling analog of the Rohm-Witten quantization of the antisymmetric tensor field strength of string theories. The effect of a gaugino condensate on one of the boundaries is also considered. 
  An $SO(9,1)$ invariant formulation of an 11-dimensional supermembrane is presented by combining an $SO(10,1)$ invariant treatment of reparametrization symmetry with an $SO(9,1)$ invariant $\theta_{R} = 0$ gauge of $\kappa$-symmetry. The Lagrangian thus defined consists of polynomials in dynamical variables (up to quartic terms in $X^{\mu}$ and up to the eighth power in $\theta$), and reparametrization BRST symmetry is manifest. The area preserving diffeomorphism is consistently incorporated and the area preserving gauge symmetry is made explicit. The $SO(9,1)$ invariant theory contains terms which cannot be induced by a naive dimensional reduction of higher dimensional supersymmetric Yang-Mills theory. The $SO(9,1)$ invariant Hamiltonian and the generator of area preserving diffeomorphism together with the supercharge are matrix regularized by applying the standard procedure. As an application of the present formulation, we evaluate the possible central charges in superalgebra both in path integral and in canonical (Dirac) formalism, and we find only the two-form charge $[X^\mu, X^\nu]$. 
  The high-energy quark-quark scattering amplitude is calculated first in the case of scalar QCD, using Fradkin's approach to derive the scalar quark propagator in an external gluon field and computing it in the eikonal approximation. The results are then extended to the case of ``real'' (i.e., fermion) QCD. The high-energy quark-quark scattering amplitude turns out to be described by the expectation value of two lightlike Wilson lines, running along the classical trajectories of the two colliding particles. Interesting analytic properties of the high-energy quark-quark scattering amplitude can be derived, going from Minkowskian to Euclidean theory: they could open the possibility of evaluating the high-energy scattering amplitude directly on the lattice. 
  A Schwinger-Dyson equation for the quark propagator is derived in the context of a Bethe-Salpeter second order formalism developped in preceding papers and of the Minimal Area Law model for the evaluation of the Wilson loop. We discuss how the equal time straight line approximation has to be modified to include correctly trajectories going backwards in time. We also show, by an appropriate selection of the solution of the SD equation, that in the limit of zero quark mass chiral symmetry breaking and a zero mass pseudoscalar meson actually occur. The inclusion of backward quark trajectories proves to be essential to make the model consistent with Goldstone's theorem. 
  We present a twisted commutator deformation for $N=1,2$ super Virasoro algebras based on $GL_q(1,1)$ covariant noncommutative superspace. 
  We examine conformal rescaling and T-duality in the context of four-dimensional HKT geometries. The closure of the torsion forces the conformal factor to satisfy a modified harmonic equation. Because of this equation the conformal factors form non-commutative groups acting on the HKT geometries. Using conformal rescalings and T-duality transformations we generate from flat space new families of HKT geometries with tri-holomorphic Killing vectors. We also find ultraviolet-finite (4,0) supersymmetric sigma models which are not conformally invariant. 
  We study the problem of baryon stability in M theory, starting from realistic four-dimensional string models constructed using the free-fermion formulation of the weakly-coupled heterotic string. Suitable variants of these models manifest an enhanced custodial gauge symmetry that forbids to all orders the appearance of dangerous dimension-five baryon-decay operators. We exhibit the underlying geometric (bosonic) interpretation of these models, which have a $Z_2 \times Z_2$ orbifold structure similar, but not identical, to the class of Calabi-Yau threefold compactifications of M and F theory investigated by Voisin and Borcea. A related generalization of their work may provide a solution to the problem of proton stability in M theory. 
  We investigate the properties of a class of near-extreme static black hole solutions called naked black holes. These black holes, which occur in string theory, have small curvature invariants but large tidal forces outside their event horizons. We show that these large tidal forces are due to a concentration of dilaton stress-energy near the horizon. We study infalling test strings and find that they are highly excited by the large tidal forces, but remain small. Perhaps most importantly, it turns out that a small amount of infalling matter will cause the curvature invariants to become large outside the horizon. Nevertheless, an exact calculation shows that both the matter trajectories and the classical black hole background are not significantly altered. 
  The physical fields (electromagnetic and electron fields) considered in the framework of Clifford algebras $\C_2$ and $\C_4$. The electron field described by the algebra $\C_4$ which in spinor representation is realized by well-known Dirac $\gamma$-matrices, and by force of isomorphism $\C_{4}\cong\C_{2}\otimes \C_{2}$ is represented as a tensor product of two photon fields. By means of this introduced a system of electron field equations, which in particular cases is coincide with Dirac's and Maxwell's equations. 
  The path integral for relativistic spinless dionium atom is solved, and the energy spectra are extracted from the resulting amplitude. 
  The Seiberg-Witten curve and differential for ${\cal N}=2$ supersymmetric SU(N) gauge theory, with a massive hypermultiplet in the adjoint representation of the gauge group, are analyzed in terms of the elliptic Calogero-Moser integrable system. A new parametrization for the Calogero-Moser spectral curves is found, which exhibits the classical vacuum expectation values of the scalar field of the gauge multiplet. The one-loop perturbative correction to the effective prepotential is evaluated explicitly, and found to agree with quantum field theory predictions. A renormalization group equation for the variation with respect to the coupling is derived for the effective prepotential, and may be evaluated in a weak coupling series using residue methods only. This gives a simple and efficient algorithm for the instanton corrections to the effective prepotential to any order. The 1- and 2- instanton corrections are derived explicitly. Finally, it is shown that certain decoupling limits yield ${\cal N}=2$ supersymmetric theories for simple gauge groups $SU(N_1)$ with hypermultiplets in the fundamental representation, while others yield theories for product gauge groups $SU(N_1) \times ...\times SU(N_p)$, with hypermultiplets in fundamental and bi-fundamental representations. The spectral curves obtained this way for these models agree with the ones proposed by Witten using D-branes and M-theory. 
  In this article we consider open strings with mixed boundary conditions (a combination of Neumann and Dirichlet conditions at each end). We discuss how their end points show a $D_p$-brane with NS-NS charge, i.e. a bound state of a D-brane with a fundamental strings. We show that these branes are BPS saturated. In the case of one-branes, we show that their mass densities are in agreement with IIb SUGRA which is Sl(2,Z) invariant. Using Chan-Paton factors, we extend our results to the case of bound states of $n$ D-strings and $m$ F-strings. These string theoretic results are also checked in the effective field theory limit. 
  We derive a two-dimensional effective dilaton - gravity - matter action that describes the dynamics of an uncharged black p-brane in N dimensions. We show that this effective theory is completely integrable in the static sector and establish its general static solution. The solution includes, as a particular case, the boost symmetric p-brane solution investigated in hep-th/9510202 . 
  The spectra of BPS states in F-theory on elliptic fibred fourfolds are investigated. 
  We calculate the second virial coefficient of spin-1/2 anyon gas in the various values of the self-adjoint extension parameter by incorporating the self-adjoint extension method into the Green's function formalism. Especially, the completely different cusp- and discontinuity-structures from the result of previous literature are obtained when the self-adjoint extension parameter goes to infinity. This is originated from the different condition for the occurrence of irregular states. 
  We continue our study of heterotic/type-I duality in D<10 dimensions. We consider the heterotic and type-I theories compactified on tori to lower dimensions. We calculate the special (``BPS saturated'') F^4 and R^4 terms in the effective one-loop heterotic action. These terms are expected to be non-perturbatively exact for D>4.   The heterotic result is compared with the associated type-I result. In D<9 dimensions, the type-I theory has instanton corrections due to D1 instantons. In D=8 we use heterotic-type I duality to give a simple prescription of the D-instanton calculation on the type I side. We allow arbitrary Wilson lines and show that the D1-instanton determinant is the affine character-valued elliptic genus evaluated at the induced complex structure of the D1-brane world-volume. The instanton result has an expansion in terms of Hecke operators that suggest an interpretation in terms of an SO(N) matrix model of the D1-brane. The total result can be written in terms of generalized prepotentials revealing an underlying holomorphic structure.   In D<8 we calculate again the heterotic perturbative thresholds and show that they agree with the D1-instanton calculation using the rules derived in D=8. 
  A model of a relativistic particle moving in the Liouville field is investigated. Symmetry group of the system is $SL(2,R)/Z_2$. The corresponding dynamical integrals describe full set of classical trajectories. Dynamical integrals are used for the gauge-invariant Hamiltonian reduction. The new scheme is proposed for quantization of the reduced system. Obtained quantum system reproduces classical symmetry. Physical aspects of the model are discussed. 
  The renormalization group is applied to the phi4 model in the symmetry broken phase in order to identify different scaling regimes. The new scaling laws reflect nonuniversal behavior at the phase transition. The extension of the analysis to finite temperature is briefly outlined. It is mentioned that the coupling constants can be found in the mixed phase by taking into account the saddle points of the blocking procedure. 
  The N = 3 string models are special solutions of the type II perturbative string theories. We present explicit expressions for the helicity supertraces,which count the number of the perturbative BPS multiplets. Assuming the non-perturbative duality $(S\leftrightarrow T)$ of the heterotic string on T^6 and type II on K_3 x T^2 valid in N = 4 theories, we derive the N = 3 non-perturbative BPS mass formula by "switching off" some of the N = 4 charges and "fixing" to special values some of the N = 4 moduli. This operation corresponds to a well-defined Z_2 projection acting freely on the compactification manifold. The consistency of this projection and the precise connection of the N = 4 and N = 3 BPS spectrum is shown explicitly in several type II string constructions. The heterotic N = 3 and some asymmetric type II constructions turn out to be non-perturbative with the S moduli fixed at the self-dual point S=i. Some of the non-perturbative N = 3 type II are defined in the context of F-theory. The bosonic sector of the N=3 string effective action is also presented. This part can be useful for the study of 4d black holes in connection with the asymptotic density of BPS states in N = 3 string theory. 
  In these lecture notes, an introduction to superstring theory is presented. Classical strings, covariant and light-cone quantization, supersymmetric strings, anomaly cancelation, compactification, T-duality, supersymmetry breaking, and threshold corrections to low-energy couplings are discussed. A brief introduction to non-perturbative duality symmetries is also included. 
  We study the spin dependence of D-brane dynamics in the Green-Schwarz formalism of boundary states. In particular we show how to interpret insertion of supercharges on the boundary state as sources of non-universal spin effects in D-brane potentials. In this way we find for a generic (D)p-brane, potentials going like $v^{4-n}/r^{7-p+n}$ corresponding to interactions between the different components of the D-brane supermultiplet. From the eleven dimensional point of view, these potentials arise from the exchange of field strengths corresponding to the graviton and the three form, coupled non-minimally to the branes. We show how an annulus computation truncated to its massless contribution is enough to reproduce these next-to-leading effects, meaning in particular that the one-loop (M)atrix theory effective action should encode all the spin dependence of low-energy supergravity interactions. 
  We consider the extremal limit of a black hole geometry of the Reissner-Nordstrom type and compute the quantum corrections to its entropy. Universally, the limiting geometry is the direct product of two 2-dimensional spaces and is characterized by just a few parameters. We argue that the quantum corrections to the entropy of such extremal black holes due to a massless scalar field have a universal behavior. We obtain explicitly the form of the quantum entropy in this extremal limit as function of the parameters of the limiting geometry. We generalize these results to black holes with toroidal or higher genus horizon topologies. In general, the extreme quantum entropy is completely determined by the spectral geometry of the horizon and in the ultra-extreme case it is just a determinant of the 2-dimensional Laplacian. As a byproduct of our considerations we obtain expressions for the quantum entropy of black holes which are not of the Reissner-Nordstrom type: the extreme dilaton and extreme Kerr-Newman black holes. In both cases the classical Bekenstein-Hawking entropy is modified by logarithmic corrections. 
  An explicit solution in super Yang-Mills theory, which after T-duality describes two sets of D4-branes at angles, is constructed. The gauge configuration possesses 3/16 unbroken supersymmetry for the equal magnitudes of field strengths, and can be considered as the counterpart of the solution of D=11 supergravity with the same amount of supersymmetry in the solutions given by Gauntlett et al. The energy of the Born-Infeld action of the gauge configuration gives further evidence for the geometrical interpretation as two sets of D4-branes at angles. The energy of super Yang-Mills theory is shown to coincide with that of M(atrix) theory. This fact shows that the configuration with 3/16 supersymmetry can be realized in M(atrix) theory, which describes two sets of longitudinal M5-branes (with common string direction) at angles. 
  In this paper, we derive an expression for the grand canonical partition function for a fluid of hot, rotating massless scalar field particles in the Einstein universe. We consider the number of states with a given energy as one increases the angular momentum so that the fluid rotates with an increasing angular velocity. We find that at the critical value when the velocity of the particles furthest from the origin reaches the speed of light, the number of states tends to zero. We illustrate how one can also interpret this partition function as the effective action for a boosted scalar field configuration in the product of three dimensional de Sitter space and $S^1$. In this case, we consider the number of states with a fixed linear momentum around the $S^1$ as the particles are given more and more boost momentum. At the critical point when the spacetime is about to develop closed timelike curves, the number of states again tends to zero. Thus it seems that quantum mechanics naturally enforces the chronology protection conjecture by superselecting the causality violating field configurations from the quantum mechanical phase space. 
  We present a procedure of differential renormalization at the one loop level which avoids introducing unnecessary renormalization constants and automatically preserves abelian gauge invariance. The amplitudes are expressed in terms of a basis of singular functions. The local terms appearing in the renormalization of these functions are determined by requiring consistency with the propagator equation. Previous results in abelian theories, with and without supersymmetry, are discussed in this context. 
  We study the crossing symmetry of the conformal blocks of the conformal field theory based on the affine Lie superalgebra osp(1|2). Within the framework of a free field realization of the osp(1|2) current algebra, the fusion and braiding matrices of the model are determined. These results are related in a simple way to those corresponding to the su(2) algebra by means of a suitable identification of parameters. In order to obtain the link invariants corresponding to the osp(1|2) conformal field theory, we analyze the corresponding topological Chern-Simons theory. In a first approach we quantize the Chern-Simons theory on the torus and, as a result, we get the action of the Wilson line operators on the supercharacters of the affine osp(1|2). From this result we get a simple expression relating the osp(1|2) polynomials for torus knots and links to those corresponding to the su(2) algebra. Further, this relation is verified for arbitrary knots and links by quantizing the Chern-Simons theory on the punctured two-sphere. 
  A general set of rules is given how to convert a local kappa-symmetry of a brane action and space-time supersymmetry into the global supersymmetry of the worldvolume. A Killing spinor adapted gauge for quantization of kappa-symmetry is defined for this purpose. As an application of these rules we perform the gauge-fixing of the M-5-brane to get the theory of the (0, 2) tensor supermultiplet in d=6. 
  A method of deriving solutions to Nahm's equations based on root structure of simple Lie algebras is given. As an illustration of this method the recently found solutions to Nahm's equations with tetrahedral and octahedral symmetries are shown to correspond to $A_2$ and $A_3$ root systems. 
  Non-Abelian duality transformations built on non-semi-simple isometry groups are analysed. We first give the conditions under which the original non-linear sigma model and its non-Abelian dual are equivalent. The existence of an invariant and non-degenerate bilinear form for the isometry Lie algebra is crucial for this equivalence. The non-Abelian dual of a conformally invariant sigma model, with non-semi-simple isometries, is then constructed and its beta functions are shown to vanish. This study resolves an apparent obstruction to the conformal invariance of sigma models obtained via non-Abelian duality based on non-semi-simple groups. 
  Extending recent N=1 and N=2 results, we propose an explicit formula for the integration measure on the moduli space of charge-n ADHM multi-instantons in N=4 supersymmetric SU(2) gauge theory. As a consistency check, we derive a renormalization group relation between the N=4, N=2, and N=1 measures. We then use this relation to construct the purely bosonic (``N=0'') measure as well, in the classical approximation in which the one-loop small-fluctuations determinants is not included. 
  We establish the isomorphism between a nonlinear $\sigma$-model and the abelian gauge theory on an arbitrary curved background, which allows us to derive integrable models and the corresponding Lax representations from gauge theoretical point of view. In our approach the spectral parameter is related to the global degree of freedom associated with the conformal or Galileo transformations of the spacetime. The B$\ddot{\rm a}$cklund transformations are derived from Chern-Simons theory where the spectral parameter is defined in terms of the extract compactified space dimension coordinate. 
  Scalar fields at finite temperature are considered in four dimensional ultrastatic curved spacetime. One loop nonlocal effective action at finite temperature is found up to the second order in curvature expansion. This action is explicitly infrared finite. In the high temperature expansion of free energy, essentially nonlocal terms linear in temperature are derived. 
  The soliton structure of a gauge theory recently proposed to describe chiral excitations in the Fractional Quantum Hall Effect is investigated. A new type of non-linear derivative Schr\"odinger equation emerges as an effective description of the system that supports novel chiral solitons. We discuss the classical properties of solutions with vanishing and non-vanishing boundary conditions (dark solitons) and we explain their relation to integrable systems. The quantum analysis is also addressed in the framework of a semiclassical approximation improved by Renormalization Group arguments. 
  By considering constraints on the dimensions of the Lie algebra corresponding to the weight one states of Z_2 and Z_3 orbifold models arising from imposing the appropriate modular properties on the graded characters of the automorphisms on the underlying conformal field theory, we propose a set of constructions of all but one of the 71 self-dual meromorphic bosonic conformal field theories at central charge 24. In the Z_2 case, this leads to an extension of the neighborhood graph of the even self-dual lattices in 24 dimensions to conformal field theories, and we demonstrate that the graph becomes disconnected. 
  We investigate the effects of the external gravitational and constant magnetic fields to the dynamical symmetrybreaking. As simple models of the dynamical symmetry breaking we consider the Nambu-Jona-Lasinio (NJL) model and the supersymmetric Nambu-Jona-Lasinio (SUSY NJL) model non-minimally interacting with the external gravitational field and minimally interacting with constant magnetic field. The explicit expressions for the scalar and spinor Green functions are found up to the linear terms on the spacetime curvature and exactly for a constant magnetic field. We obtain the effective potential of the above models from the Green functions in the magnetic field in curved spacetime. Calculating the effective potential numerically with the varying curvature and/or magnetic fields we show the effects of the external gravitational and magnetic fields to the phase structure of the theories. In particular, increase of the curvature in the spontaneously broken chiral symmetry phase due to the fixed magnetic field makes this phase to be less broken. On the same time strong magnetic field quickly induces chiral symmetry breaking even at the presence of fixed gravitational field within nonbroken phase. 
  We construct integrals of motion (IM) for the sine-Gordon model with boundary at the free Fermion point which correctly determine the boundary S matrix. The algebra of these IM (``boundary quantum group'' at q=1) is a one-parameter family of infinite-dimensional subalgebras of twisted affine sl(2). We also propose the structure of the fractional-spin IM away from the free Fermion point. 
  The parity-violating topological term in the effective action for 2+1 massive fermions is computed at finite temperature in the presence of a constant background field strength tensor. Gauge invariance of the finite-temperature effective action is also discussed. 
  Starting from Nahm's equations, we explore BPS magnetic monopoles in the Yang-Mills Higgs theory of gauge group $Sp(4)$ which is broken to $SU(2)\times U(1)$. A family of BPS field configurations with purely Abelian magnetic charge describe two identical massive monopoles and one massless monopole. We construct the field configurations with axial symmetry by employing the ADHMN construction, and find the explicit expression of the metrics for the 12-dimensional moduli space of Nahm data and its submanifolds. 
  Four-dimensional N=1 supersymmetric Spin(N) gauge theories with matter in the vector and spinor representations are considered. Dual descriptions are known for some of these theories. It is noted that when masses are given to all fields in the spinor representation, the dual gauge group G breaks to a group H such that \pi_2(G/H)=Z_2. The quantum numbers of the associated Z_2 monopole and those of the massive spinors are shown to agree, suggesting that the monopole is the image of the massive spinors under duality. It follows that electric sources in the spinor representation, needed as test charges to determine the phase of an SO(N) gauge theory, can be introduced as Z_2-valued magnetic sources in the dual nonabelian gauge theory. This fact is used to study the phases of SO(N) gauge theories with matter in the vector representation. 
  Using twistor methods we derive a generating function which leads to the hyperk\" ahler metric on a deformation of the Atiyah-Hitchin monopole moduli space. This deformation was first considered by Dancer through the quotient construction and is related to a charge two monopole configuration in a completely broken SU(3) gauge theory. The manifold and metric are the first members of a family of hyperk\" ahler manifolds which are deformations of the $D_k$ rational singularities of $C^2$. 
  For a calculation of divergent fermion string amplitudes a regularization procedure invariant under the supermodular group is constructed. By this procedure superstring amplitudes of an arbitrary genus are calculated using both partition functions and superfield vacuum correlators computed early. A finiteness of superstring amplitudes and related topics are discussed. 
  In the present paper we study the Faddeev-Popov path integral quantization of electrodynamics in an inhomogenious dielectric medium. We quantize all polarizations of the photons and introduce the corresponding ghost fields. Using the heat kernel technique, we express the heat kernel coefficients in terms of the dielectricity $\epsilon (x)$ and calculate the ultra violet divergent terms in the effective action. No cancellation between ghosts and "non-physical" degrees of freedom of the photon is observed. 
  In this talk I describe a recently introduced field-theoretical approach that can be used as an alternative framework to study one-dimensional systems of highly correlated particles. 
  We consider the Dirac equation in cylindrically symmetric magnetic fields and find its normal modes as eigenfunctions of a complete set of commuting operators. This set consists of the Dirac operator itself, the $z$-components of the linear and the total angular momenta, and of one of the possible spin polarization operators. The spin structure of the solution is completely fixed independently of the radial distribution of the magnetic field which influences only the radial modes. We solve explicitly the radial equations for the uniform magnetic field inside a solenoid of a finite radius and consider in detail the scattering of scalar and Dirac particles in this field. For particles with low energy the scattering cross section coincides with the Aharonov-Bohm scattering cross section. We work out the first order corrections to this result caused by the fact that the solenoid radius is finite. At high energies we obtain the classical result for the scattering cross section. 
  We address the issue of correspondence between classical supergravity and quantum super Yang-Mills (or Matrix theory) expressions for the long-distance, low-velocity interaction potentials between 0-branes and various bound states of branes. The leading-order potentials are known to be reproduced by the F^4 terms in the 1-loop SYM effective action. Using self-consistency considerations, we determine a universal combination of F^6 terms in the 2-loop SYM effective action that corresponds to the subleading terms in the supergravity potentials in many cases, including 0-brane scattering off 1/8 supersymmetric 410 and 4440 bound states representing extremal D=5 and D=4 extremal black holes. We give explicit descriptions of the these configurations in terms of 1/4 supersymmetric SYM backgrounds on dual tori. Under a proper choice of the gauge field backgrounds, the 2-loop F^6 SYM action reproduces the full expression for the subleading term in the supergravity potentials, including its subtle v^2 part. 
  We construct a consistent self-coupling for the vector-tensor multiplet in N = 2 harmonic central charge superspace. 
  The matrix theory description of the discrete light cone quantization of $M$ theory on a $T^{2}$ is studied. In terms of its super Yang- Mills description, we identify symmetries of the equations of motion corresponding to independent rescalings of one of the world sheet light cone coordinates, which show how the $S$ duality of Type IIB string theory is realized as a Nahm-type transformation. In the $M$ theory description this corresponds to a simple $9-11$ flip. 
  The story of (Ward-)Takahashi relations and their impact on physical theory is reviewed. 
  We consider Matrix theory compactified on T^3 and show that it correctly describes the properties of Schwarzschild black holes in 7+1 dimensions, including the energy-entropy relation, the Hawking temperature and the physical size, up to numerical factors of order unity. The most economical description involves setting the cut-off N in the discretized light-cone quantization to be of order the black hole entropy. A crucial ingredient necessary for our work is the recently proposed equation of state for 3+1 dimensional SYM theory with 16 supercharges. We give detailed arguments for the range of validity of this equation following the methods of Horowitz and Polchinski. 
  We present a functional Schr\"{o}dinger picture formalism of the (1+1)-dimensional $O(N) $ nonlinear sigma model. The energy density has been calculated to two-loop order using the wave functional of a gaussian form, and from which the nonperturbative mass gap of the boson fields has been obtained. The functional Schr\"{o}dinger picture approach combined with the variational technique is shownto describe the characteristics of the ground state of the nonlinear sigma model in a transparent way. 
  Dilaton stabilization may occur in a theory based on a single asymptotically free gauge group with matter due to an interplay between quantum modification of the moduli space and tree-level superpotential. We present a toy model where such a mechanism is realized. Dilaton stabilization in this mechanism tends to occur at strong coupling values unless some unnatural adjustment of parameters is involved. 
  We use the time-dependent invariant method in a geometric approach (Jacobi fields) to quantize the motion of a free falling point particle in the Schwarzschild black hole. Assuming that the particle comes from infinity, we obtain the relativistic Schr\"{o}dinger wave function for this system. 
  We consider QED - processes in the presence of an infinitely thin and infinitely long straight string with a magnetic flux inside it. The bremsstrahlung from an electron passing by the magnetic string and the electron-positron pair production by a single photon are reviewed. Based on the exact electron and positron solutions of the Dirac equation in the external Aharonov-Bohm potential we present matrix elements for these processes. The dependence of the resulting cross sections on energies, directions and polarizations of the involved particles is discussed for low energies. 
  We study M theory fivebranes to understand the moduli space of vacua of N=1 supersymmetric SO(N_c) gauge theory with N_f flavors in four dimensions. We discuss how the various branches of this theory arise in the string/M theory brane configurations and compare our results with the ones obtained earlier by Intriligator and Seiberg in the context of field theory.   In the M theory approach, we explain the various branches from the asymptotic position of semi-infinite D4 branes in the $w=x^8+i x^9$ direction, which is closely related to the eigenvalues of the meson matrix $M^{ij} = Q^{i}_a Q^{j}_a$ where $Q^{i}_a$ is a squark multiplet ( $i=1, \cdots, 2N_f$ and $ a=1, \cdots, N_c$).In M theory, these branches are explained by observing a new phenomena which did not occur for the gauge groups SU(N_c) or Sp(N_c). 
  We compute the string tension in massive $QCD_2$. It is shown that the string tension vanishes when the mass of the dynamical quark is zero, with no dependence on the representations of the dynamical or of the external charges. When a small mass ($m\ll e$) is added, a tension appears and we calculate its value as a function of the representations. 
  We calculate the potential between bound states of D-branes of different dimension in IIB matrix model upto one loop order and find nice agreement with the open string calculations in short and large distance limit. We also consider the scattering of bound states of D-branes, calculate the scattering phase shift and analyze the effective potential in different limits. 
  We consider a brane moving close to a large number of coincident branes. We compare the calculation of the effective action using the gauge theory living on the brane and the calculation using the supergravity approximation. We discuss some general features about the correspondence between large N gauge theories and black holes. Then we do a one loop calculation which applies for extremal and near extremal black holes. We comment on the expected results for higher loop calculations. We make some comments on the Matrix theory interpretation of these results. 
  Over the last several years, there has been a resurgence of interest in using non-perturbative approximation methods based on Wilson's continuous renormalization group. In this lecture, I review progress particularly in the past year, concentrating on theoretical issues in the structure of the exact renormalization group and its approximations. 
  We find the exact non-perturbative expression for a simple Wilson loop of arbitrary shape for U(N) and SU(N) Euclidean or Minkowskian two-dimensional Yang-Mills theory regulated by the Wu-Mandelstam-Leibbrandt gauge prescription. The result differs from the standard pure exponential area-law of YM_2, but still exhibits confinement as well as invariance under area-preserving diffeomorphisms and generalized axial gauge transformations. We show that the large N limit is NOT a good approximation to the model at finite N and conclude that Wu's N=infinity Bethe-Salpeter equation for QCD_2 should have no bound state solutions. The main significance of our results derives from the importance of the Wu-Mandelstam-Leibbrandt prescription in higher-dimensional perturbative gauge theory. 
  It is well known that charges coupled to a pure Chern-Simons gauge field in (2+1) dimensions undergo an effective change of statistics, i.e., become anyons. We will consider several generalizations thereof, arising when the gauge field is more general. The first one is ``multispecies anyons''---charged particles of several species coupled to one, or possibly several, Chern-Simons fields. The second one is finite-size anyons, which are charged particles coupled to a gauge field described by the Chern-Simons term plus some other term. In fact, rigorously speaking, quasielectrons and quasiholes in the fractional quantum Hall effect are multispecies finite-size anyons. The third one is an analog of finite-size anyons which arises in a model with a mixed Chern-Simons term; notably, this model is P,T-invariant, which opens the way for practical applications even when there is no parity-breaking magnetic field. 
  We derive the fusion rules for a basic series of admissible representations of $\hat{sl}(3)$ at fractional level $3/p-3$. The formulae admit an interpretation in terms of the affine Weyl group introduced by Kac and Wakimoto. It replaces the ordinary affine Weyl group in the analogous formula for the fusion rules multiplicities of integrable representations. Elements of the representation theory of a hidden finite dimensional graded algebra behind the admissible representations are briefly discussed. 
  Preliminary evidence is presented that a long overlooked and critical element in the fundamental definition of a general theory of integration over curved Wess-Zumino superspace lies with the imposition of ``the Ethereal Conjecture'' which states the necessity of the superspace to be topologically ``close'' to its purely bosonic sub-manifold. As a step in proving this, a new theory of integration of closed super p-forms is proposed. 
  We develop a technique that solders the dual aspects of some symmetry following from the bosonisation of two distinct fermionic models, thereby leading to new results which cannot be otherwise obtained. Exploiting this technique, the two dimensional chiral determinants with opposite chirality are soldered to reproduce either the usual gauge invariant expression leading to the Schwinger model or, alternatively, the Thirring model. Likewise, two apparently independent three dimensional massive Thirring models with same coupling but opposite mass signatures, in the long wavelegth limit, combine by the process of bosonisation and soldering to yield an effective massive Maxwell theory. The current bosonisation formulas are given, both in the original independent formulation as well as the effective theory, and shown to yield consistent results for the correlation functions. Similar features also hold for quantum electrodynamics in three dimensions. 
  We study the leading corrections to the emergent canonical commutation relations arising in the statistical mechanics of matrix models, by deriving several related Ward identities, and give conditions for these corrections to be small. We show that emergent canonical commutators are possible only in matrix models in complex Hilbert space for which the numbers of fermionic and bosonic fundamental degrees of freedom are equal, suggesting that supersymmetry will play a crucial role. Our results simplify, and sharpen, those obtained earlier by Adler and Millard. 
  We review some recent developments in the study of M-theory compactifications via Matrix theory. In particular we highlight the appearance of IIA strings and their interactions, and explain the unifying role of the M-theory five-brane for describing the spectrum of the T^5 compactification and its duality symmetries. The 5+1-dimensional micro-string theory that lives on the fivebrane world-volume takes a central place in this presentation. 
  In a recent paper it was shown that the properties of Schwarzschild black holes in 8 dimensions are correctly described up to factors of order unity by Matrix theory compactified on T^3. Here we consider compactifications on tori of general dimension d. Although in general little is known about the relevant d+1 dimensional theories on the dual tori, there are hints from their application to near-extreme parallel Dirichlet d-branes. Using these hints we get the correct mass-entropy scaling for Schwarzschild black holes in (11-d) dimensions. 
  Starting from SU(2) Yang-Mills theory in 3+1 dimensions, we prove that the abelian-projected effective gauge theories are written in terms of the maximal abelian gauge field and the dual abelian gauge field interacting with monopole current. This is performed by integrating out all the remaining non-Abelian gauge field belonging to SU(2)/U(1). We show that the resulting abelian gauge theory recovers exactly the same one-loop beta function as the original Yang-Mills theory. Moreover, the dual abelian gauge field becomes massive if the monopole condensation occurs. This result supports the dual superconductor scenario for quark confinement in QCD. We give a criterion of dual superconductivity and point out that the monopole condensation can be estimated from the classical instanton configuration. Therefore there can exist the effective abelian gauge theory which shows both asymptotic freedom and quark confinement based on the dual Meissner mechanism. Inclusion of arbitrary number of fermion flavors is straightforward in this approach. Some implications to lower dimensional case will also be discussed. 
  We show how the state of an unstable particle can be defined in terms of stable asymptotic states. This general definition is used to discuss and to solve some old problems connected with the short-time and large-time behaviour of the non-decay amplitude. 
  If the gravitino is light and all the other supersymmetric particles are heavy, we can consider the effective theory describing the interactions of its goldstino components with ordinary matter. To discuss the model-dependence of these interactions, we take the simple case of spontaneously broken supersymmetry and only two chiral superfields, associated with the goldstino and a massless matter fermion. We derive the four-point effective coupling involving two matter fermions and two goldstinos, by explicit integration of the heavy spin-0 degrees of freedom in the low-energy limit. Surprisingly, our result is not equivalent to the usual non-linear realization of supersymmetry, where a pair of goldstinos couples to the energy-momentum tensor of the matter fields. We solve the puzzle by enlarging the non-linear realization to include a second independent invariant coupling, and we show that there are no other independent couplings of this type up to this order in the low-energy expansion. We conclude by commenting on the interpretation of our results and on their possible phenomenological implications. 
  Electromagnetic field is considered in the framework of Clifford algebra $\C_2$ over a field of complex numbers. It is shown here that a modulo 2 periodicity of complex Clifford algebras may be connected with electromagnetic field. 
  We report on some general results on the physics of extremal BPS black holes in four and five dimensions. The duality-invariant entropy-formula for all $N>2$ extended supergravities is derived. Its relation with the fixed-scalar condition for the black-hole ``potential energy'' wich extremizes the BPS mass is obtained. BPS black holes preserving different fractions of supersymmetry are classified in a U-duality invariant set up. The latter deals with different orbits of the fundamental representations of the exceptional groups $E_{7(7)}$ and $E_{6(6)}$. We comment upon the interpretation of these results in a string and M-theory framework. 
  We review recent progress in understanding black hole structure and dynamics via matrix theory. 
  The Polyakov loop variable serves as an order parameter to characterize the confined and deconfined phases of Yang-Mills theory. By integrating out the vector fields in the SU(2) Yang-Mills partition function in one-loop approximation, an effective action is obtained for the Polyakov loop to second order in a derivative expansion. The resulting effective potential for the Polyakov loop is capable of describing a second-order deconfinement transition as a function of temperature. 
  Using linearized superfields, $R^4$ terms in the Type II superstring effective action compactified on $T^2$ are constructed as integrals in N=2 D=8 superspace. The structure of these superspace integrals allows a simple proof of the $R^4$ non-renormalization theorems which were first conjectured by Green and Gutperle. 
  Much use has been made of the techniques of supersymmetric quantum mechanics (SUSY QM) for studying bound-state problems characterized by a superpotential $\phi(x)$. Under the analytic continuation $\phi(x) \to i\phi(x)$, a pair of superpartner bound-state problems is transformed into a two-state level-crossing problem in the continuum. The description of matter-enhanced neutrino flavor oscillations involves a level-crossing problem. We treat this with the techniques of supersymmetric quantum mechanics. For the benefit of those not familiar with neutrino oscillations and their description, enough details are given to make the rest of the paper understandable. Many other level-crossing problems in physics are of exactly the same form. Particular attention is given to the fact that different semiclassical techniques yield different results. The best result is obtained with a uniform approximation that explicitly recognizes the supersymmetric nature of the system. 
  We propose descriptions of interacting (1,0) supersymmetric theories without gravity in six dimensions in the infinite momentum frame. They are based on the large N limit of quantum mechanics or 1+1 dimensional field theories with SO(N) gauge group and four supercharges. We argue that this formulation allows for a concrete description of the chirality-changing phase transitions which connect (1,0) theories with different numbers of tensor multiplets. 
  We compute the one-loop effective action and the conformal anomaly associated with the product $\bigotimes_p{\cal L}_p$ of the Laplace type operators ${\cal L}_p, p=1,2$, acting in irreducible rank 1 symmetric spaces of non-compact type. The explicit form of the zeta functions and the conformal anomaly of the stress-energy momentum tensor is derived. 
  Gauge transformations of type-II spinors are considered in the Majorana-Ahluwalia construct for self/anti-self charge conjugate states. Some speculations on the relations of this model with the earlier ones are given. 
  On the basis of the first principles we argue that self/anti-self charge conjugate states of the (1/2,0)+(0,1/2) representation can possess the axial charge. Finally, we briefly discuss recent claims of the \sim \vec \sigma \cdot [ \vec A \times \vec A^\ast ] interaction term for the particles of this representation. 
  We propose a formula for the effective action of Matrix Theory which succesfully reproduces a large class of Born-Infeld type D-brane probe actions. The formula is motivated by demanding consistency with known results, and is tested by comparing with a wide range of source-probe calculations in supergravity. In the case of D0-brane sources and Dp-brane probes, we study the effect of boosts, rotations, and worldvolume electric fields on the probe, and find agreement with supergravity to all orders in the gravitational coupling. We also consider D4-brane sources at the one loop level and recover the correct probe actions for a D0-brane, and for a D4-brane rotated at an angle with respect to the source. 
  We review some tests of the 0-brane and instanton matrix models based on comparing long-distance interaction potentials between branes and their bound states (with 1/2,1/4 or 1/8 of supersymmetry) in supergravity and in super Yang-Mills descriptions. We first consider the supergravity-SYM correspondence at the level of the leading term in the interaction potential, and then describe some recent results concerning the subleading term and their implications for the structure of the 2-loop F^6 term in the SYM effective action. 
  We discuss the role of string solitons in duality and examine the feature of compositeness, which allows for the interpretation of general solutions as bound states of supersymmetric fundamental constituents. This feature lies at the heart of the recent success of string theory in reproducing the Bekenstein-Hawking black hole entropy formula. Talk given at 19th annual MRST meeting, Syracuse, NY, May 12-13, 1997. 
  We find that QCD in covariant gauge yields zero for the topological susceptibility, even at the nonperturbative level. The result is derived in two ways, one using translational invariance, and the other using the BRST Hamiltonian. Comparison with the canonical formalism suggests that QCD is not uniquely defined at the nonperturbative level. Supporting evidence is also provided in 1+1 dimensions. Our results imply that the strong CP problem admits a trivial resolution in covariant gauge, but obstacles remain for the U(1) problem. 
  We complete the set of string vertices of non-negative dimension by introducing in a consistent manner those moduli spaces which had previously been excluded. As a consequence we obtain a `geometrised' string action taking the simple form $S=f(\B)$ where `$\B$' is the sum of the string vertices. That the action satisfies the B-V master equation follows from the recursion relations for the string vertices which take the form of a `geometrical' quantum master equation. 
  We present evidence that an interplay of the laws of microphysics and cosmology renders the Planck momentum unattainable by an elementary particle. Several categories of accelerators are analyzed and shown to fail. 
  In this paper we study energy radiation from a moving mirror in 1+1 dimensional space-time. The mirror is assumed to have finite mass and accordingly to receive back reaction from scalar photon field. The mode expansion of the scalar field becomes different from that without back reaction though the trajectory of the mirror is not changed. Then energy density of the vacuum becomes to have finite value proportional to square of the mass of the mirror. Moreover we compute the energy momentum tensor of the radiation in the case that acceleration of the mirror is small. As a result we show that the mirror creates energy radiation whose quantity does not depend on its mass but on its acceleration even if the acceleration is uniform. 
  We consider d=4 N=2 supergravity theories which serve as low-energy effective actions for heterotic strings on K_3 \times T^2. At the perturbative level we construct a new version of the heterotic effective action in which the axion has been traded for an antisymmetric tensor field. In the string frame the antisymmetric tensor doesn't transform under Poincar\'e supersymmetry into the dilaton-dilatini system. This indicates that in this frame the antisymmetric tensor field and the dilaton are not contained in an N=2 vector-tensor multiplet. Instead, we find that the heterotic dilaton is part of a compensating hypermultiplet, whereas the antisymmetric tensor is part of the gravitational multiplet. In order to obtain our results we use superconformal techniques. This enables us to comment on the range of applicability of this particular framework. 
  The 3-loop effective action and effective potential in Nambu - Jona-Lasinio model are calculated. The problem of vanishing contributions in the higher orders is discussed. The general form of such contributions is obtained. 
  The reduced covariant phase space associated with the three-dimensional Euclidean Nambu-Goto action can be identified, via the Enneper-Weierstrass representation of minimal surfaces, with the space of complex analytic functions plus three translational zero modes. The symplectic structure induced trough the Enneper-Weierstrass map can be explicitly computed. Quantization is then straightforward, yielding as a result a target-space Euclidean-invariant, positive-definite, two-dimensional quantum field theory. The physical states are shown to correspond with particles states of integer spin and arbitrary mass. 
  For the SU(N) invariant supersymmetric matrix model related to membranes in 4 space-time dimensions, the general solution to the equation(s) $Q^{\dagger}\Psi=0$ $(Q\chi =0)$ is determined for N odd. For any such (bosonic) solution of $Q^{\dagger}\Psi=0$, a (fermionic) $\Phi$ is found that (formally) satisfies $Q^{\dagger}\Phi=\Psi$.   For the analogous model in 11 dimensions the solution of $Q^{\dagger}\Psi=0 (Q\Psi=0)$ is outlined. 
  We investigate the scattering of an electron by an infinitely thin and infinitely long straight magnetic flux tube in the framework of QED. We discuss the solutions of the Dirac and Maxwell fields in the related external pure AB potential and evaluate matrix elements and differential probabilities for the bremsstrahlung process. The dependence of the resulting cross section on the energy, direction and polarization of the involved particles is analyzed. In the low energy regime a surprising angular asymmetry is found which results from the interaction of the electron's magnetic moment with the magnetic field. 
  In the framework of QED we evaluate the cross section for electron-positron pair production by a single photon in the presence of the external Aharonov-Bohm potential in first order of perturbation theory. We analyse energy, angular and polarization distributions at different energy regimes: near the threshold and at high photon energies. 
  This is an article on the interaction between topology and physics which will appear in 1998 in a book called: A History of Topology, edited by Ioan James and published by Elsevier-North Holland. 
  Averaged spin-spin correlation function squared $\overline{<\sigma(0)\sigma(R)>^{2}}$ is calculated for the ferromagnetic random bond Potts model. The technique being used is the renormalization group plus conformal field theory. The results are of the $\epsilon$ - expansion type fixed point calculation, $\epsilon$ being the deviation of the central charge (or the number of components) of the Potts model from the Ising model value. Calculations are done both for the replica symmetric and the replica symmetry broken fixed points. The results obtained allow for the numerical simulation tests to decide between the two different criticalities of the random bond Potts model. 
  A family of degenerate domain wall configurations, partially preserving supersymmetry, is discussed in a generalized Wess-Zumino model with two scalar superfields. We establish some general features inherent to the models with continuously degenerate domain walls. For instance, for purely real trajectories additional "integrals of motion" exist. The solution for the profile of the scalar fields for any wall belonging to the family is found in quadratures for arbitrary ratio of the coupling constants. For a special value of this ratio the solution family is obtained explicitly in terms of elementary functions. We also discuss the threshold amplitudes for multiparticle production generated by these solutions. New unexpected nullifications of the threshold amplitudes are found. 
  As another evidence for the matrix Discrete Light Cone formulation of M theory, we show how general integrable Hamiltonian systems emerge from BPS bound states of k longitudinal fivebranes. Such configurations preserve eight supercharges and by chain of dualities can be related to the solution of N=2 four-dimensional gauge theories. Underlying Hitchin systems on the bare spectral curve with k singular points arise from the Matrix theory compactification on the dual curve. 
  We study consistency conditions on a M(atrix)-model which would describe M-theory on $T^6$. We argue that there is a limit in moduli space for which it becomes a 6+1D theory and study the low-energy description of extended objects in the decompactified limit. We discuss the requirements from a M(atrix)-model which would describe such an $E_{6(6)}$ theory. We suggest that it could be a 5+1D theory and that a 1+1D theory with $(0,4)$ supersymmetry might be the M(atrix)-model for the M(atrix)-model of the $E_{6(6)}$ theory. 
  Through introducing a notion of renormalization of particle-number density, a simple perturbation scheme of nonequilibrium quantum-field theory is proposed. In terms of the renormalized particle-distribution functions, which characterize the system, the structure of the scheme (and then also the structure of amplitudes and reaction rates) are the same as in the equilibrium thermal field theory. Then, as an obvious consequence, the amplitudes and reaction rates computed in this scheme are free from pinch singularities due to multiple products of $\delta$-functions, which inevitably present in traditional perturbation scheme. 
  In this paper we consider classical point particles in full interaction with an arbitrary number of dynamical scalar and (abelian) vector fields. It is shown that the requirement of stability ---vanishing self-force--- is sufficient to remove the well-known inconsistencies of the classical theory: the divergent self-energy, as well as the failure of Lorentz-covariance of the energy-momentum when including the contributions of the fields. As a result, in these models the mass of a point particle becomes finitely computable. It is shown how these models are connected to quantum field theory via the path-integral representation of the propagator. 
  We describe solutions of type IIA (N=2, D=10) supergravity built under the assumption of the existence of at least one residual chiral supersymmetry. Their geometry is of pp-wave type. Explicit parametrization of the metric and matter field components, in terms of Killing spinors and arbitrary functions, is provided. 
  The path integral description of the Wess-Zumino-Witten $\to$ Liouville reduction is formulated in a manner that exhibits the conformal invariance explicitly at each stage of the reduction process. The description requires a conformally invariant generalization of the phase space path integral methods of Batalin, Fradkin, and Vilkovisky for systems with first class constraints. The conformal anomaly is incorporated in a natural way and a generalization of the Fradkin-Vilkovisky theorem regarding gauge independence is proved. This generalised formalism should apply to all conformally invariant reductions in all dimensions. A previous problem concerning the gauge dependence of the centre of the Virasoro algebra of the reduced theory is solved. 
  In this work we calculate two two-loop massless Feynman integrals pertaining to self-energy diagrams using NDIM (Negative Dimensional Integration Method). We show that the answer we get is 36-fold degenerate. We then consider special cases of exponents for propagators and the outcoming results compared with known ones obtained via traditional methods. 
  The link between chirality in the fermion sector and (anti-)self-duality in the boson sector is reexamined in the light of Connes' noncommutative geometry approach to the Standard Model. We find it to impose that the noncommutative Yang-Mills action be symmetrized in an analogous way to the Dirac-Yukawa operator itself. 
  We show how certain non-perturbative superpotentials W, which are the two-dimensional analogs of the Seiberg-Witten prepotential in 4d, can be computed via geometric engineering from 4-folds. We analyze an explicit example for which the relevant compact geometry of the 4-fold is given by $P^1$ fibered over $P^2$. In the field theory limit, this gives an effective U(1) gauge theory with N=(2,2) supersymmetry in two dimensions. We find that the analog of the SW curve is a K3 surface, and that the complex FI coupling is given by the modular parameter of this surface. The FI potential itself coincides with the middle period of a meromorphic differential. However, it only shows up in the effective action if a certain 4-flux is switched on, and then supersymmetry appears to be non-perturbatively broken. This can be avoided by tuning the bare FI coupling by hand, in which case the supersymmetric minimum naturally corresponds to a singular K3. 
  The semiclassical approximation for the partition function in Chern-Simons gauge theory is derived using the invariant integration method. Volume and scale factors which were undetermined and had to be fixed by hand in previous derivations are automatically taken account of in this framework. Agreement with Witten's exact expressions for the partition function in the weak coupling (large k) limit is verified for gauge group SU(2) and spacetimes S^3, S^2 x S^1, S^1 x S^1 x S^1 and L(p,q). 
  Using the manifestly spacetime supersymmetric description of the four-dimensional open superstring, we construct the vertex operator in superspace for the first massive state. This construction provides an N=1 D=4 superspace representation of the massive spin-2 multiplet. 
  It is shown that purely bosonic field theories can have configurations with half-integral angular momentum even when the topological magnetic charge of the configuration vanishes. This result is applicable whenever there is a non-Abelian gauge theory with particles that transform in the fundamental representation of the non-Abelian symmetry group. 
  It is shown that, in QED$_{3}$, the Pauli-Villars regularization involving a pair of auxiliary fermion fields with masses of opposite sign leads to results that are consistent with those obtained using all other parity-preserving schemes of regularization. At the same time, ambiguity problems remain unsolved in non-Abelian models (QCD$_{3}$). 
  Semiclassical approximation based on extracting a c-number classical component from quantum field is widely used in the quantum field theory. Semiclassical states are considered then as Gaussian wave packets in the functional Schrodinger representation and as Gaussian vectors in the Fock representation. We consider the problem of divergences and renormalization in the semiclassical field theory in the Hamiltonian formulation. Although divergences in quantum field theory are usually associated with loop Feynman graphs, divergences in the Hamiltonian approach may arise even at the tree level. For example, formally calculated probability of pair creation in the leading order of the semiclassical expansion may be divergent. This observation was interpretted as an argumentation for considering non-unitary evolution transformations, as well as non-equivalent representations of canonical commutation relations at different time moments. However, we show that this difficulty can be overcomed without the assumption about non-unitary evolution. We consider first the Schrodinger equation for the regularized field theory with ultraviolet and infrared cutoffs. We study the problem of making a limit to the local theory. To consider such a limit, one should impose not only the requirement on the counterterms entering to the quantum Hamiltonian but also the requirement on the initial state in the theory with cutoffs. We find such a requirement in the leading order of the semiclassical expansion and show that it is invariant under time evolution. This requirement is also presented as a condition on the quadratic form entering to the Gaussian state. 
  In our previous article we have proposed that the Virasoro algebra controls the quantum cohomology of Fano varieties at all genera. In this paper we construct a free field description of Virasoro operators and quantum cohomology. We shall show that to each even (odd) homology class of a K\"{a}hler manifold we have a free bosonic (fermionic) field and Virasoro operators are given by a simple bilinear form of these fields. We shall show that the Virasoro condition correctly reproduces the Gromov-Witten invariants also in the case of manifolds with non-vanishing non-analytic classes ($h^{p,q}\not=0,p\not=q$) and suggest that the Virasoro condition holds universally for all compact smooth K\"{a}hler manifolds. 
  On basis of an algebraic analysis of symmetry breaking in general and the Higgs mechanism in the standard model of elementary particles we generalize the concept of symmetry breaking to systems with non-compact groups but not necessarily caused by a potential. Thereto we give some simple, but unfamiliar examples of symmetry breaking with and without potentials. The analysis of the concept of mass in space-time and in the Higgs mechanism will lead to a model unifying both structures in terms of symmetry breaking of GL(2,C). 
  We give the renormalization of the standard model of electroweak interactions to all orders of perturbation theory by using the method of algebraic renormalization, which is based on general properties of renormalized perturbation theory and not on a specific regularization scheme. The Green functions of the standard model are uniquely constructed to all orders, if one defines the model by the Slavnov-Taylor identity,  Ward-identities of rigid symmetry and a specific form of the abelian local gauge Ward-identity, which continues the Gell-Mann Nishijima relation to higher orders. Special attention is directed to the mass diagonalization of massless and massive neutral vectors and ghosts. For obtaining off-shell infrared finite expressions it is required to take into account higher order corrections into the functional symmetry operators. It is shown, that the normalization conditions of the on-shell schemes are in agreement with the most general symmetry transformations allowed by the algebraic constraints. 
  A momentum dependent projection of the Wegner-Hougton equation is derived for a scalar theory coupled to an external field. This formalism is useful to discuss the phase diagram of the theory. In particular we study some properties of the Gaussian fixed point. 
  General aspects of fundamental physics are considered. We comment the Wigner's logical scheme and modify it to adjust to modern theoretical physics. Then, we discuss the role and indicate the place of renormalization group in the logic of fundamental physics. 
  A dynamical theory is studied in which a scalar field $\phi$ in Einstein- Minkowski space is coupled to the four-velocity $N_{\mu}$ of a preferred inertial observer in that space. As a consistent requirement on this coupling we study a principle of duality invariance of the dynamical mass- term of $\phi$ at some universal length in the small-distance regime. In the large-distance regime duality breaking can be introduced by giving a back- ground value to $\phi$ and a back-ground direction to $N_{\mu}$. It is shown that, in an appropriate approximation, duality breaking can be related to the emergence of a characteristic phase in which the condensation of the ground state allows massive excitations with a characteristic scale of squared mass which agrees with present observational bound for the cosmological constant. 
  The Hadamard renormalization prescription is used to derive a two dimensional analog of the renormalized stress tensor for a minimally coupled scalar field in Schwarzschild-de Sitter space time. In the two dimensional analog the minimal coupling reduces to the conformal coupling and the stress tensor is found to be determined by the (nonlocal) contribution of the anomalous trace and some additional parameters in close relation to the work [1]. To properly relate the stress tensor to the state of outwards signals coming from the direction of the black hole at late times we propose a cut-off hypothesis which excludes the contribution of the anomalous trace close to the black hole horizon. The corresponding cut-off scale is found to be related to the equilibrium-temperature of the cosmological horizon in a leading order estimate. Finally, we establish a relation between the radiation-temperature of the black hole horizon at large distance from the hole and the the anomalous trace and determine the correction term to the Hawking temperature due to the presence of the cosmological horizon. 
  We show how F-theory on a Calabi-Yau (n+1)-fold, in appropriate limit, can be identified as an orientifold of type IIB string theory compactified on a Calabi-Yau n-fold. 
  We review two types of D-branes processes where open strings are created. In the first type, a closed string incident on a collection of D-branes is converted into a number of open strings running along them. For the case of threebranes we compare the leading absorption rate with that in semiclassical gravity, and find exact agreement. A supersymmetric non-renormalization theorem guarantees that this agreement survives all corrections in powers of the string coupling times the number of branes. The second type of process is creation of stretched open strings by crossing D-branes. We show that this is possible whenever a p-brane passes through an (8-p)-brane positioned orthogonally to it. The extra attractive force exerted by the stretched open string is crucial for finding that the net force cancels in this BPS system. 
  The matching of global anomalies of a supersymmetric gauge theory and its dual is seen to follow from similarities in their classical chiral rings. These similarities provide a formula for the dimension of the dual gauge group. As examples we derive 't Hooft consistency conditions for the duals of supersymmetric QCD and SU(N) theories with matter in the adjoint, and obtain the dimension of the dual groups. 
  A new approach to analyze the properties of the energy-momentum tensor $T(z)$ of conformal field theories on generic Riemann surfaces (RS) is proposed. $T(z)$ is decomposed into $N$ components with different monodromy properties, where $N$ is the number of branches in the realization of RS as branch covering over the complex sphere. This decomposition gives rise to new infinite dimensional Lie algebra which can be viewed as a generalization of Virasoro algebra containing information about the global properties of the underlying RS. In the simplest case of hyperelliptic curves the structure of the algebra is calculated in two ways and its central extension is explicitly given. The algebra possess an interesting symmetry with a clear interpretation in the framework of the radial quantization of CFT's with multivalued fields on the complex sphere. 
  We show how to bosonize two-dimensional non-abelian models using finite chiral determinants calculated from a Gauss decomposition. The calculation is quite straightforward and hardly more involved than for the abelian case. In particular, the counterterm $A\bar A$, which is normally motivated from gauge invariance and then added by hand, appears naturally in this approach. 
  In the first part the sh Lie structure of brackets in field theory, described in the jet bundle context along the lines suggested by Gel'fand, Dickey and Dorfman, is analyzed. In the second part, we discuss how this description allows us to find a natural relation between the Batalin-Vilkovisky antibracket and the Poisson bracket. 
  A genuine continuum treatment of the massive \phi^4_{1+1}-theory in light-cone quantization is proposed. Fields are treated as operator valued distributions thereby leading to a mathematically well defined handling of ultraviolet and light cone induced infrared divergences and of their renormalization. Although non-perturbative the continuum light cone approach is no more complex than usual perturbation theory in lowest order. Relative to discretized light cone quantization, the critical coupling increases by 30% to a value r = 1.5. Conventional perturbation theory at the corresponding order yields r_1=1, whereas the RG improved fourth order result is r_4 = 1.8 +-0.05. 
  We study the Hawking process on lattices falling into static black holes. The motivation is to understand how the outgoing modes and Hawking radiation can arise in a setting with a strict short distance cutoff in the free-fall frame. We employ two-dimensional free scalar field theory. For a falling lattice with a discrete time-translation symmetry we use analytical methods to establish that, for Killing frequency $\omega$ and surface gravity $\kappa$ satisfying $\kappa\ll\omega^{1/3}\ll 1$ in lattice units, the continuum Hawking spectrum is recovered. The low frequency outgoing modes arise from exotic ingoing modes with large proper wavevectors that "refract" off the horizon. In this model with time translation symmetry the proper lattice spacing goes to zero at spatial infinity. We also consider instead falling lattices whose proper lattice spacing is constant at infinity and therefore grows with time at any finite radius. This violation of time translation symmetry is visible only at wavelengths comparable to the lattice spacing, and it is responsible for transmuting ingoing high Killing frequency modes into low frequency outgoing modes. 
  In this sequel calculation of the one-loop Feynman integral pertaining to a massive box diagram contributing to the photon-photon scattering amplitude in quantum electrodynamics, we present the six solutions as yet unknown in the literature. These six new solutions arise quite naturally in the context of negative dimensional integration approach, revealing a promising technique to handle Feynman integrals. 
  Local M-operators for the classical sine-Gordon model in discrete space-time are constructed by convolution of the quantum trigonometric 4$\times$4 R-matrix with certain vectors in its "quantum" space. Components of the vectors are identified with $\tau$-functions of the model. This construction generalizes the known representation of M-operators in continuous time models in terms of Lax operators and classical $r$-matrix. 
  We call a state ``vacuum bounded'' if every measurement performed outside a specified interior region gives the same result as in the vacuum. We compute the maximum entropy of a vacuum-bounded state with a given energy for a one-dimensional model, with the aid of numerical calculations on a lattice. For large energies we show that a vacuum-bounded system with length $L_in$ and a given energy has entropy no more than $S^rb + (1/6) \ln S^rb$, where $S^rb$ is the entropy in a rigid box with the same size and energy. Assuming that the state resulting from the evaporation of a black hole is similar to a vacuum-bounded state, and that the similarity between vacuum-bounded and rigid box problems extends from 1 to 3 dimensions, we apply these results to the black hole information paradox. Under these assumptions we conclude that large amounts of information cannot be emitted in the final explosion of a black hole.   We also consider vacuum-bounded states at very low energies and come to the surprising conclusion that the entropy of such a state can be much higher than that of a rigid box state with the same energy. For a fixed $E$ we let $L_in'$ be the length of a rigid box which gives the same entropy as a vacuum-bounded state of length $L_in$. In the $E\to 0$ limit we conjecture that the ratio $L_in'/L_in$ grows without bound and support this conjecture with numerical computations. 
  In this note we study the structure of diffeomorphism anomalies in 3+1 canonical gravity coupled to a chiral massless fermion. We find that when the spatial manifold is S^3 or a Lens space L(p,q), the first homotopy group of the related diffeomorphism group can be nontrivial and hence the question of global anomalies becomes relevant. Here we show that for gravity coupled to SU(2) chiral fermions, assuming the strong form of the Hatcher conjecture, SU(2)-induced diffeomorphism anomalies do not occur whenever the spatial manifold is S^3 or a Lens space. 
  D-particle quantum mechanics in a type I' background is reviewed. It is also discussed how a string is created when a D-particle is taken through a D8-brane. The process is found to be dual to the creation of a D3-brane when a NS5 and D5 brane are passed through each other. 
  This is a revised and shortened version of a MSc thesis submitted to the University of Sussex, UK. An introduction into the pre-string physics of black holes and related thermodynamics is given. Then, starting with an introduction of how superstring theory is approaching the problem of black hole entropy, work on that and closely related topics like Hawking radiation and the information paradox is reviewed. 
  A general method of constructing canonical gauge invariant actions is used to establish the equivalence between 2D induced gravity and a WZNW system, defined by a difference of two simple WZNW actions fo the SL(2,R) group. The diffeomorphism invariance of the induced gravity is generated by the SL(2,R) Kac-Moody structure of the WZNW system. 
  We show that a large class of supersymmetric solutions to the low-energy effective field theory of heterotic string theory compactified on a seven torus can have finite energy, which we compute. The mechanism by which these solutions are turned into finite energy solutions is similar to the one occurring in the context of four-dimensional stringy cosmic string solutions. We also describe the solutions in terms of intersecting eleven-dimensional M-branes, M-waves and M-monopoles. 
  A natural geometry, arising from the embedding into a Hilbert space of the parametrised probability measure for a given lattice model, is used to study the symmetry properties of real-space renormalisation group (RG) flow. In the projective state space this flow is shown to have two contributions: a gradient term, which generates a projective automorphism of the state space for each given length scale; and an explicit correction. We then argue that this structure implies the absence of any symmetry of a geodesic type for the RG flow when restricted to the parameter space submanifold of the state space. This is demonstrated explicitly via a study of the one dimensional Ising model in an external field. In this example we construct exact expressions for the beta functions associated with the flow induced by infinitesimal rescaling. These constitute a generating vector field for RG diffeomorphisms on the parameter space manifold, and we analyse the symmetry properties of this transformation. The results indicate an approximate conformal Killing symmetry near the critical point, but no generic symmetry of the RG flow globally on the parameter space. 
  We discuss the problem of gauge fixing for the partition function in generalized quantum (or trace) dynamics, deriving analogs of the De Witt-Faddeev-Popov procedure and of the BRST invariance familiar in the functional integral context. 
  We compare gap equation predictions for the spontaneous breaking of global symmetries in supersymmetric Yang-Mills theory to nonperturbative results from holomorphic effective action techniques. In the theory without matter fields, both approaches describe the formation of a gluino condensate. With $N_f$ flavors of quark and squark fields, and with $N_f$ below a certain critical value, the coupled gap equations have a solution for quark and gluino condensate formation, corresponding to breaking of global symmetries and of supersymmetry. This appears to disagree with the newer nonperturbative techniques, but the reliability of gap equations in this context and whether the solution represents the ground state remain unclear. 
  We consider the dimensional reduction of supersymmetric Yang-Mills on a Calabi-Yau 3-fold. We show by construction how the resulting cohomological theory is related to the balanced field theory of the Kaehler Yang-Mills equations introduced by Donaldson and Uhlenbeck-Yau. 
  The Gross-Neveu model with chemical potential is investigated as a low-energy effective theory of polyacetylene. In particular, we focus on the abrupt change in the features of electric conductivity such as sharp rise in the Pauli paramagnetism at dopant concentration of about 6%. We will try to explain it by the finite density phase transition in the Gross-Neveu model. The thermodynamic Bethe ansatz is combined with the large-N expansion to construct thermodynamics of the Gross-Neveu model. A first-order phase transition is found in leading order in the 1/N expansion and it appears to be stable against the 1/N correction. The next to leading order correction to the critical dopant concentration is explicitly calculated. 
  These lectures give an introduction to duality in Quantum Field Theory. We discuss the phases of gauge theories and the implications of the electric-magnetic duality transformation to describe the mechanism of confinement. We review the exact results of N=1 supersymmetric QCD and the Seiberg-Witten solution of N=2 super Yang-Mills. Some of its extensions to String Theory are also briefly discussed. 
  Recent results concerning the internal structure of static spherically-symmetric non-Abelian black holes in the Einstein-Yang-Mills (EYM) theory and its generalizations including scalar fields are reviewed and discussed with an emphasis on the problem of a generic singularity in black holes. It is argued that in the theories admitting a violation of the naive no-hair conjecture the structure of singularity is essentially affected by the "hair roots". This invalidates an image of a non-Abelian black hole as a Schwarzschild black hole sitting inside the soliton. We give an analytic description of the generic oscillatory approach to the singularity in the pure EYM theory in terms of a divergent discrete sequence and show that the mass function is exponentially growing "in average". The second type of a generic approach to the singularity in hairy black holes is a "power-law mass inflation" which is realized in the theories including scalar fields. Both singularities are spacelike and no Cauchy horizons are met in the full interior region in conformity with the Strong Cosmic Censorship conjecture. Black holes violating this conjecture exist only for certain discrete values of the event horizon radius thus forming a subset of zero measure. 
  We formulate criteria of applicability of the Faddeev-Popov trick to gauge theories on manifolds with boundaries. With the example of Euclidean Maxwell theory we demonstrate that the path integral is indeed gauge independent when these criteria are satisfied, and depends on a gauge choice whenever these criteria are violated. In the latter case gauge dependent boundary conditions are required for a self-consistent formulation of the path intgral. 
  In the talk, the relationship between black holes in Jackiw-Teitelboim(JT) dilaton gravity and solitons in sine-Gordon field theory is described. The well-known connection between solutions of the sine-Gordon equation and constant curvature metrics is reviewed and expanded. In particular, it is shown that solutions to the dilaton field equations for a given metric in JT theory also solve the sine-Gordon equation linearized about the corresponding soliton. Next, it is shown that from the B${\ddot a}$cklund transformations relating different soliton solutions, it is possible to construct a flat SL(2,R) connection which forms the basis for the gauge theory formulation of JT dilaton gravity. Finally, the supersymmetric generalization of sine-Gordon theory is reviewed, leading to a new approach to the problem of the origin of low dimensional black hole entropy. 
  The general solution to the quantum master equation (and its $Sp(2)$ symmetric counterpart) is constructed explicitly in case of finite number of variables. It is shown that the finite-dimensional solution is physically trivial and thus can not be extended directly to cover the case of a local field theory. In this way we conclude that the locality condition plays an important role by making it possible to obtain nontrivial physical results when quantizing gauge field theories on the basis of field-antifield formalism. 
  We apply the method of the Renormalization Group (GR), following the Polchinski point of view, to a model of well developed and isotropic fluid turbulence. The Galilei-invariance is preserved and a universal behavior, related to the change of the stochastic stirring force, is evident by the numerical results in the inertial region, where a scale-invariant behavior also appear. The expected power law of the energy spectrum ($q^{-{5\over 3}}$) is obtained and the Kolmogorov constant $C_K$ agrees with experimental data. 
  We investigate the phase structures of various N=1 supersymmetric gauge theories including even the exceptional gauge group from the viewpoint of superconvergence of the gauge field propagator. Especially we analyze in detail whether a new type of duality recently discovered by Oehme in $SU(N_c)$ gauge theory coupled to fundamental matter fields can be found in more general gauge theories with more general matter representations or not. The result is that in the cases of theories including matter fields in only the fundamental representation, Oehme's duality holds but otherwise it does not. In the former case, superconvergence relation might give good criterion to describe the interacting non-Abelian Coulomb phase without using some information from dual magnetic theory. 
  We study classical solutions of the vector O(3) sigma model in (2+1) dimensions, spontaneously broken to O(2)xZ2. The model possesses Skyrmion-type solutions as well as stable domain walls which connect different vacua. We show that different types of waves can propagate on the wall, including waves carrying a topological charge. The domain wall can also absorb Skyrmions and, under appropriate initial conditions, it is possible to emit a Skyrmion from the wall. 
  We analyze the problem of graceful exit from superinflationary pre-big bang phase of string cosmology within the context of lowest-order string effective action. The previous no go theorems are generalized for the case when higher genus terms of general form and additional matter fields are included. It is shown that the choice of the E-frame essentially simplifies the consideration. For the example of pure gravi-dilaton case the comparison of E-frame and string frame approaches, based on phase space analysis, is carried out. 
  We use a prescription to gauge the su(2) Skyrme model with a U(1) field, characterised by a conserved Baryonic current. This model reverts to the usual Skyrme model in the limit of the gauge coupling constant vanishing. We show that there exist axially symmetric static solutions with zero magnetic charge, which can be electrically either charged or uncharged. The energies of the (uncharged) gauged Skyrmions are less than the energy of the (usual) ungauged Skyrmion. For physical values of the parameters the impact of the U(1) field is very small, so that it can be treated as a perturbation to the (ungauged) spherically symmetric Hedgehog. This allows the perturbative calculation of the magnetic moment. 
  We briefly review the structure and properties of self-dual field actions. 
  Under the assumption of axial symmetry we introduce a map from dilatonic gravity to a string-like action. This map allows one to introduce, in a rather simple way, the equivalent of string theory T-duality in dilatonic gravity. Here we choose the duality group to be an $SO(2,1)$ group and, for a particular rotation, we recover a symmetry of dilatonic gravity discussed previously in the literature. 
  In these lectures we present a general introduction to topological quantum field theories. These theories are discussed in the framework of the Mathai-Quillen formalism and in the context of twisted N=2 supersymmetric theories. We discuss in detail the recent developments in Donaldson-Witten theory obtained from the application of results based on duality for N=2 supersymmetric Yang-Mills theories. This involves a description of the computation of Donaldson invariants in terms of Seiberg-Witten invariants. Generalizations of Donaldson-Witten theory are reviewed, and the structure of the vacuum expectation values of their observables is analyzed in the context of duality for the simplest case. 
  We analyze the u-plane contribution to Donaldson invariants of a four-manifold X. For $b_2^+(X)>1$, this contribution vanishes, but for $b_2^+=1$, the Donaldson invariants must be written as the sum of a u-plane integral and an SW contribution. The u-plane integrals are quite intricate, but can be analyzed in great detail and even calculated. By analyzing the u-plane integrals, the relation of Donaldson theory to N=2 supersymmetric Yang-Mills theory can be described much more fully, the relation of Donaldson invariants to SW theory can be generalized to four-manifolds not of simple type, and interesting formulas can be obtained for the class numbers of imaginary quadratic fields. We also show how the results generalize to extensions of Donaldson theory obtained by including hypermultiplet matter fields. 
  Self-consistent Ansaetze are presented for the left- and right-handed isodoublet fermion zero-modes of the constrained instanton Istar in the vacuum sector of euclidean SU(2) Yang-Mills-Higgs theory. These left- and right-handed fermion wave functions do not coincide and, most likely, have maxima at different positions. This may be important for the fermion zero-mode contribution to the euclidean 4-point Green's function in chiral Yang-Mills-Higgs theory and the high-energy behaviour of fermion-fermion scattering processes. 
  An approximation is used that permits one to explicitly solve the two-point Schwinger-Dyson equations of the U(N) lattice chiral models. The approximate solution correctly predicts a phase transition for dimensions $d$ greater than two. For $d \le 2 $, the system is in a single disordered phase with a mass gap. The method reproduces known $N=\infty$ results well for $d=1$. For $d=2$, there is a moderate difference with $N=\infty$ results only in the intermediate coupling constant region. 
  Gravitational analogues of the nonlinear electrodynamics of Born and of Born and Infeld are introduced and applied to the black hole problem. This work is mainly devoted to the 2-dimensional case in which the relevant lagrangians are nonpolynomial in the scalar curvature. 
  We summarize recent work showing how the Thermodynamic Bethe Ansatz may be used to study the finite-density first-order phase transition in the Gross-Neveu model. The application to trans-polyacetylene is discussed, and the significance of the results is addressed. 
  The structure of S-duality in U(1) gauge theory on a 4-manifold M is examined using the formalism of noncommutative geometry. A noncommutative space is constructed from the algebra of Wilson-'t Hooft line operators which encodes both the ordinary geometry of M and its infinite-dimensional loop space geometry. S-duality is shown to act as an inner automorphism of the algebra and arises as a consequence of the existence of two independent Dirac operators associated with the spaces of self-dual and anti-selfdual 2-forms on M. The relations with the noncommutative geometry of string theory and T-duality are also discussed. 
  We study the leading order spin dependence of graviton scattering in eleven dimensions, and show that the results obtained from supergravity and from Matrix Theory precisely agree. 
  The nontrivial transformation of the phase space path integral measure under certain discretized analogues of canonical transformations is computed. This Jacobian is used to derive a quantum analogue of the Hamilton-Jacobi equation for the generating function of a canonical transformation that maps any quantum system to a system with a vanishing Hamiltonian. A formal perturbative solution of the quantum Hamilton-Jacobi equation is given. 
  We exhibit the one-loop multi-gluon effective Lagrangian in any dimension for a field theory with a quasilocal background, using the background-field formalism. Specific results, including counter terms (up to 12 spacetime dimensions), have been derived, applied to the Yang-Mills theory and found to be in agreement with other string-inspired approaches. 
  Killing spinors of space-time BPS configurations play an important role in quantization of theories with the fermionic worldvolume local symmetry. We show here how it works for the GS superstring, BST supermembrane and M-5-brane. We show that the non-linear generalization of the (2,0) d=6 tensor supermultiplet action is the M-5-brane action in a Killing gauge. For D-p-branes the novel feature of quantization is that they can be quantized Lorentz covariantly, in particular, for D-0-brane a gauge exists where the action is covariant and free. We present a general condition on possible choice of gauges for the kappa-symmetric branes. 
  A role of reducible connections in Non-Abelian Seiberg-Witten invariants is analyzed with massless Topological QCD where monopole is extended to non-Abelian groups version. By giving small external fields, we found that vacuum expectation value can be separated into a part from Donaldson theory, a part from Abelian Monopole theory and a part from non-Abelian monopole theory. As a by-product, we find identities of U(1) topological invariants. In our proof, the duality relation and Higgs mechanism are not necessary. 
  We study models of quantum statistical mechanics which can be solved by the algebraic Bethe ansatz. The general method of calculation of correlation functions is based on the method of determinant representations. The auxiliary Fock space and auxiliary Bose fields are introduced in order to remove the two body scattering and represent correlation functions as a mean value of a determinant of a Fredholm integral operator; the representation has a simple form for large space and time separations. In this paper we explain how to calculate the mean value in the auxiliary Fock space of asymptotic expression of the Fredholm determinant. It is necessary for the evaluation of the asymptotic form of the physical correlation functions. 
  Motion of a non-relativistic particle on a cone with a magnetic flux running through the cone axis (a ``flux cone'') is studied. It is expressed as the motion of a particle moving on the Euclidean plane under the action of a velocity-dependent force. Probability fluid (``quantum flow'') associated with a particular stationary state is studied close to the singularity, demonstrating non trivial Aharonov-Bohm effects. For example, it is shown that near the singularity quantum flow departs from classical flow. In the context of the hydrodynamical approach to quantum mechanics, quantum potential due to the conical singularity is determined and the way it affects quantum flow is analysed. It is shown that the winding number of classical orbits plays a role in the description of the quantum flow. Connectivity of the configuration space is also discussed. 
  Tree level decay amplitudes of near-BPS D-brane configurations are known to exactly reproduce Hawking radiation rates from corresponding black holes at low energies even though the brane configurations describe semiclassical black holes only when the open string couplings are large. We show that a large class of one (open string) loop corrections to emission processes from D-branes vanish at low energies and nonvanishing loop contributions have an energy dependence consistent with black hole answers, thus providing a justification for the agreement of the tree level results with semiclassical answers. 
  We investigate the classical limit of the Knizhnik-Zamolodchikov-Bernard equations, considered as a system of non-stationar Schr\"{o}odinger equations on singular curves, where times are the moduli of curves. It has a form of reduced non-autonomous hamiltonian systems which include as particular examples the Schlesinger equations, Painlev\'{e} VI equation and their generalizations. In general case, they are defined as hierarchies of isomonodromic deformations (HID) with respect to changing the moduli of underling curves. HID are accompanying with the Whitham hierarchies. The phase space of HID is the space of flat connections of $G$ bundles with some additional data in the marked points. HID can be derived from some free field theory by the hamiltonian reduction under the action of the gauge symmetries and subsequent factorization with respect to diffeomorphisms of curve. This approach allows to define the Lax equations associated with HID and the linear system whose isomonodromic deformations are provided by HID. In addition, it leads to description of solutions of HID by the projection method.   In some special limit HID convert into the Hitchin systems. In particular, for $\SL$ bundles over elliptic curves with a marked point we obtain in this limit the elliptic Calogero $N$-body system 
  In this report we describe quantum Reissner-Nordstr\"om (RN) black-holes interacting with a complex scalar field. Our analysis is characterized by solving a Wheeler-DeWitt equation in the proximity of an apparent horizon of the RN space-time. Subsequently, we obtain a wave-function $\Psi_{RN}[M, Q]$ representing the RN black-hole. A special emphasis is given to the evolution of the mass-charge rate affected by Hawking radiation. More details can be found in gr-qc/9709080. 
  I describe recent examples of phase transitions in four-dimensional M theory vacua in which the net generation number changes. There are naive obstructions to transitions lifting chiral matter, but loopholes exist which enable us to avoid them. I first review how chirality arises in the heterotic limit of M theory, previously known forms of topology change in string theory, and chirality-changing phase transitions in six dimensions. This leads to the construction of the four-dimensional examples, which involve wrapped M-theory fivebranes at an $E_8$ wall. 
  The degeneracy structure of the eigenspace of the N-particle Calogero-Sutherland model is studied from an algebraic point of view. Suitable operators satisfying SU(2) algebras and acting on the degenerate eigenspace are explicitly constructed for the two particle case and then appropriately generalized to the N-particle model. The raising and lowering operators of these algebras connect the states, in a subset of the degenerate eigenspace, with each other. 
  Talk presented at Strings `97 in Amsterdam (June 16 - 21, 1997) 
  The present-day crisis in quantum field theory is described. 
  The relation between the covariant Euclidean free-energy $F^E$ and the canonical statistical-mechanical free energy $F^C$ in the presence of the Killing horizons is studied. $F^E$ is determined by the covariant Euclidean effective action. The definition of $F^C$ is related to the Hamiltonian which is the generator of the evolution along the Killing time. At arbitrary temperatures $F^E$ acquires additional ultraviolet divergences because of conical singularities. The divergences of $F^C$ are different and occur since the density ${dn \over d\omega}$ of the energy levels of the system blows up near the horizon in an infrared way. We show that there are regularizations that make it possible to remove the infrared cutoff in ${dn \over d\omega}$. After that the divergences of $F^C$ become identical to the divergences of $F^E$. The latter property turns out to be crucial to reconcile the covariant Euclidean and the canonical formulations of the theory. The method we use is new and is based on a relation between ${dn \over d\omega}$ and heat kernels on hyperbolic-like spaces. Our analysis includes spin 0 and spin 1/2 fields on arbitrary backgrounds. For these fields the divergences of ${dn \over d\omega}$, $F^C$ and $F^E$ are presented in the most complete form. 
  We explicitly construct soliton solutions in the low energy description of M-theory on S^1/Z_2. It is shown that the 11-dimensional membrane is a BPS solution of this theory if stretched between the Z_2 hyperplanes. A similar statement holds for the 11-dimensional 5--brane oriented parallel to the hyperplanes. The parallel membrane and the orthogonal 5-brane, though solutions, break all supersymmetries. Furthermore, we construct the analog of the gauge 5-brane with gauge instantons on the hyperplanes. This solution varies nontrivially along the orbifold direction due to the gauge anomalies located on the orbifold hyperplanes. Its zero mode part is identical to the weakly coupled 10-dimensional gauge 5-brane. 
  The set of string vertices is extended to include moduli spaces with genus and numbers of ordinary and special punctures ranging over all integral values $g,n,\bar n\geq0$. It is argued that both the string background and the B-V delta operator should be associated with the vertex $\B^0_{0,1}$ corresponding to the once-punctured sphere. This leads naturally to the proposal that the manifestly background independent formulation of quantum closed string field theory is given by the sum $\B$ of the completed set of string vertices satisfying the classical master equation $\{\B,\B\}=0$. 
  A direct link between a one-loop N-point Feynman diagram and a geometrical representation based on the N-dimensional simplex is established by relating the Feynman parametric representations to the integrals over contents of (N-1)-dimensional simplices in non-Euclidean geometry of constant curvature. In particular, the four-point function in four dimensions is proportional to the volume of a three-dimensional spherical (or hyperbolic) tetrahedron which can be calculated by splitting into birectangular ones. It is also shown that the known formula of reduction of the N-point function in (N-1) dimensions corresponds to splitting the related N-dimensional simplex into N rectangular ones. 
  For the SU(N) invariant supersymmetric matrix model related to membranes in 11 space-time dimensions, the general (bosonic) solution to the equations $Q_\beta^\dagger \Psi =0$ ($Q_\beta \Psi=0$) is determined. 
  It is pointed out that there exists an interesting strong and weak duality in the Landau-Zener-Stueckelberg potential curve crossing. A reliable perturbation theory can thus be formulated in the both limits of weak and strong interactions. It is shown that main characteristics of the potential crossing phenomena such as the Landau-Zener formula including its numerical coefficient are well-described by simple (time-independent) perturbation theory without referring to Stokes phenomena. A kink-like topological object appears in the ``magnetic'' picture, which is responsible for the absence of the coupling constant in the prefactor of the Landau-Zener formula. It is also shown that quantum coherence in a double well potential is generally suppressed by the effect of potential curve crossing, which is analogous to the effect of Ohmic dissipation on quantum coherence. 
  We examine some six-dimensional orientifold models with $N = 1$ supersymmetry, which can be realised as intersecting 7-branes and 7-planes. These models are studied in the light of recent work showing that orientifold planes carry anomalous gravitational couplings on their world-volume. We show that gravitational anomalies can be locally cancelled by these new couplings at every point in the internal space, under the assumption that the anomaly residing on orientifold planes is distributed in a particular way among brane-plane and plane-plane intersections. 
  The Hamiltonian describing Matrix theory on T^n is identified with the Hamiltonian describing the dynamics of D0-branes on T^n in an appropriate weak coupling limit for all n up to 5. New subtleties arise in taking this weak coupling limit for n=6, since the transverse size of the D0 brane system blows up in this limit. This can be attributed to the appearance of extra light states in the theory from wrapped D6 branes. This subtlety is related to the difficulty in finding a Matrix formulation of M-theory on T^6. 
  Recently it has been found that the structure of Skyrmions has a close analogy to that of fullerene shells in carbon chemistry. In this letter we show that this analogy continues further, by presenting a Skyrme field that describes a lattice of Skyrmions with hexagonal symmetry. This configuration, a novel `domain wall' in the Skyrme model, has low energy per baryon (about 6% above the Faddeev-Bogomolny bound) and in many ways is analogous to graphite. By comparison to the energy per baryon of other known Skyrmions and also the Skyrme crystal, we discuss the possibility of finding Skyrmion shells of higher charge. 
  We review the duality between heterotic and F--theory string vacua with N=1 space-time supersymmetry in eight, six and four dimensions. In particular, we discuss two chains of four-dimensional F--theory/heterotic dual string pairs, where F--theory is compactified on certain elliptic Calabi-Yau fourfolds, and the dual heterotic vacua are given by compactifications on elliptic Calabi-Yau threefolds plus the specification of the $E_8\times E_8$ gauge bundles. We show that the massless spectra of the dual pairs agree by using, for one chain of models, an index formula to count the heterotic bundle moduli and determine the dual F--theory spectra from the Hodge numbers of the fourfolds and of the type IIB base spaces. Moreover as a further check, we demonstrate that for one particular heterotic/F--theory dual pair the N=1 superpotentials are the same. 
  We employ the influence functional technique to trace out the photonic contribution from full quantum electrodynamics. The reduced density matrix propagator for the spinor field is then constructed. We discuss the role of time-dependent renormalization in the propagator and focus on the possibility of obtaining dynamically induced superselection rules. Finally, we derive the master equation for the case of the field being in an one-particle state in a non-relativistic regime and discuss whether EM vacuumm fluctuations are sufficient to produce decoherence in the position basis. 
  We study the quantum evolution of black holes immersed in a de Sitter background space. For black holes whose size is comparable to that of the cosmological horizon, this process differs significantly from the evaporation of asymptotically flat black holes. Our model includes the one-loop effective action in the s-wave and large N approximation. Black holes of the maximal mass are in equilibrium. Unexpectedly, we find that nearly maximal quantum Schwarzschild-de Sitter black holes anti-evaporate. However, there is a different perturbative mode that leads to evaporation. We show that this mode will always be excited when a pair of cosmological holes nucleates. 
  We calculate the entropy of six dimensional Schwarzschild black holes in matrix theory. We use the description of the matrix model on $T^5$ as the world-volume theory of NS five-branes and show that the black hole entropy is reproduced by noncritical closed strings with fractional tension living on the five-brane. 
  We investigate multidimensional gravity with the Gauss-Bonnet term and with torsion on the space of extra dimensions chosen to be the group manifold of a simple Lie group. We take the Robertson-Walker ansatz for the 4-dimensional space-time and study the potential of a dilaton and torsion fields. It is shown that for certain values of the parameters of the initial theory the potential has classically stable minima, corresponding to the spontaneous compactification of the extra dimensions. However, these minima have zero torsion. 
  A set of new canonical variables for $d=11$ supergravity is proposed which renders the supersymmetry variations and the supersymmetry constraint polynomial. The construction is based on the $SO(1,2)\times SO(16)$ invariant reformulation of $d=11$ supergravity given in previous work, and has some similarities with Ashtekar's reformulation of Einstein's theory. The new bosonic variables fuse the gravitational degrees of freedom with those of the three-index photon $A_{MNP}$ in accordance with the hidden symmetries of the dimensionally reduced theory. Although $E_8$ is not a symmetry of the theory, the bosonic sector exhibits a remarkable $E_8$ structure, hinting at the existence of a novel type of ``exceptional geometry''. 
  We construct two dimensional gauge theories with $N= (4,4)$ supersymmetry from branes of type IIA string theory. Quantum effects in the two dimensional gauge theory are analyzed by embedding the IIA brane construction into M-theory. We find that the Coulomb branch of one theory and the Higgs branch of a mirror theory become equivalent at strong coupling. A relationship to the decoupling limit of the type IIA and IIB 5-branes in Matrix theory is shown. T-duality between the ALE metric and the wormhole metric of Callan, Harvey, and Strominger is discussed from a brane perspective and some puzzles regarding string duality resolved. We comment on the existence of a quantum Higgs branch in two dimensional theories. Branes prove to be useful tools in analyzing singular conformal field theories. 
  Canonical structure of the space-time symmetric analogue of the Hamiltonian formalism in field theory based on the De Donder-Weyl (DW) theory is studied. In $n$ space-time dimensions the set of $n$ polymomenta is associated to the space-time derivatives of field variables. The polysymplectic $(n+1)$-form generalizes the simplectic form and gives rise to a map between horizontal forms playing the role of dynamical variables and vertical multivectors generalizing Hamiltonian vector fields. Graded Poisson bracket is defined on forms and leads to the structure of a Z-graded Lie algebra on the subspace of the so-called Hamiltonian forms for which the map above exists. A generalized Poisson structure arises in the form of what we call a ``higher-order'' and a right Gerstenhaber algebra. Field euations and the equations of motion of forms are formulated in terms of the graded Poisson bracket with the DW Hamiltonian $n$-form $H\vol$ ($\vol$ is the space-time volume form and $H$ is the DW Hamiltonian function). A few applications to scalar fields, electrodynamics and the Nambu-Goto string, and a relation to the standard Hamiltonian formalism in field theory are briefly discussed. This is a detailed and improved account of our earlier concise communications (hep-th/9312162, hep-th/9410238, and hep-th/9511039). 
  We begin the process of classifying all supersymmetric theories with quantum modified moduli. We determine all theories based on a single SU or Sp gauge group with quantum modified moduli. By flowing among theories we have calculated the precise modifications to the algebraic constraints that determine the moduli at the quantum level. We find a class of theories, those with a classical constraint that is covariant but not invariant under global symmetries, that have a singular modification to the moduli, which consists of a new branch. 
  We find examples of duality among quantum theories that are related to arithmetic functions by identifying distinct Hamiltonians that have identical partition functions at suitably related coupling constants or temperatures. We are led to this after first developing the notion of partial supersymmetry-in which some, but not all, of the operators of a theory have superpartners-and using it to construct fermionic and parafermionic thermal partition functions, and to derive some number theoretic identities. In the process, we also find a bosonic analogue of the Witten index, and use this, too, to obtain some number theoretic results related to the Riemann zeta function. 
  The functional integral of the massless Schwinger model in $(1+1)$ dimensions is reduced to an integral in terms of local gauge invariant quantities. It turns out that this approach leads to a natural bosonisation scheme, yielding, in particular the famous `bosonisation rule'' and giving some deeper insight into the nature of the bosonisation phenomenon. As an application, the chiral anomaly is calculated within this formulation. 
  I review some recent developments in our understanding of the dynamics of Z_2 orientifolds of type IIA and IIB strings and M-theory. Interesting physical phenomena that occur in each spacetime dimension from 10 to 1 are summarized, along with some relationships to nonperturbative effects and to M- and F-theory. The conceptual aspects that are highlighted are: (i) orientifolds of M-theory, (ii) exceptional gauge symmetry from open strings, (iii) certain similarities between branes and orientifold planes. Some comments are also made on disconnected components of orientifold moduli space. 
  We present a matrix model which interpolates between type IIA and type IIB NS five-branes. The matrix description involves a three-dimensional bulk quantum field theory interacting with impurities localized in one spatial direction. We obtain a dual matrix formulation for the exotic six-dimensional theory on coincident type IIB NS five-branes by studying the T-dual description in terms of Kaluza-Klein monopoles in type IIA string theory. After decoupling the bulk physics, the matrix description reduces to the conformal field theory of the Coulomb branch for the type IIA matrix string propagating on certain singular spaces. In many ways, this dual realization of superconformal theories is the two-dimensional analogue of three-dimensional mirror symmetry. 
  A new nonperturbative approach is used to investigate the Gross-Neveu model of four fermion interaction in the space-time dimensions 2, 3 and 4, the number $N$ of inner degrees of freedom being a fixed integer. The spontaneous symmetry breaking is shown to exist in $D=2,3$ and the running coupling constant is calculated. The four dimensional theory seems to be trivial. 
  We propose a method for constructing super-brane actions where every background tensor potential corresponds to a world-volume field strength. The procedure provides a natural coupling to the background and automatically displays the SL(2;Z) symmetry of the IIB string theory. The Dirichlet 3-brane is used as a test ground for these ideas. A polynomial action consistent with non-linear self-duality is presented. Invariance of the action under kappa-symmetry is demonstrated for arbitrary on-shell type IIB supergravity backgrounds and is shown to require self-duality. 
  A set of generalized superalgebras containing arbitrary tensor p-form operators is considered in dimensions $D=2n+1$ for $n=1,4 mod 4$ and the general conditions for its existence expressed in the form of generalized Jacobi identities is established. These are then solved in a univoque way and some lowest dimensional cases $ D= 3, 9, 11 $ of possible interest are made explicit. 
  We consider the compactification of M theory on a light-like circle as a limit of a compactification on a small spatial circle boosted by a large amount. Assuming that the compactification on a small spatial circle is weakly coupled type IIA theory, we derive Susskind's conjecture that M theory compactified on a light-like circle is given by the finite $N$ version of the Matrix model of Banks, Fischler, Shenker and Susskind. This point of view provides a uniform derivation of the Matrix model for M theory compactified on a transverse torus $T^p$ for $p=0,...,5$ and clarifies the difficulties for larger values of $p$. 
  We study the vacuum polarization of supersymmetric toroidal D-brane placed in the constant electromagnetic field. Explicit calculation of the one-loop effective potential is performed for membrane with constant magnetic or electric background. We find the one-loop potentials vanish as the effect of supersymmetry, 
  We study the 3+1 dimensional Skyrme model with a mass term different from the usual one. We show that this new model possesses domain walls solutions. We describe how, in the equivalent 2+1 dimensional model, the Skyrmion is absorbed by the wall. 
  We report our observations that the baby-Skyrme model in (2+1) dimensions possesses non-topological stationary solutions which we call pseudo-breathers. We discuss their properties and present our results on their interaction with the topological skyrmions. 
  When an asymptotically non-free theory possesses a mass parameter, the ultraviolet (UV) renormalon gives rise to non-perturbative contributions to dimension-four operators and dimensionless couplings, thus has a similar effect as the instanton. We illustrate this phenomenon in O(N) symmetric massive $\lambda\phi^4$ model in the $1/N$ expansion. This effect of UV renormalon is briefly compared with non-perturbative corrections in the magnetic picture of the Seiberg-Witten theory. 
  We argue that a near-extremal charge-$k$ type II NS fivebrane can be reliably described in semiclassical string perturbation theory as long as both $k$ and $\mu \over k$ are large, where $\mu$ is the energy density in string units. For a small value of the asymptotic string coupling $g$, the dynamics in the throat surrounding the fivebrane reduces to the CGHS model with massive fields. We find that the energy density leaks off the brane in the form of Hawking radiation at a rate of order $1 \over k^{7/2}$ in string units independently of $g$ to leading order. In the $g\to 0$ limit the radiation persists but never reaches asymptotic infinity because the throat becomes infinitely long. 
  The simple method for the calculating of the anomalous dimensions of the composite operators up to 1/N^2 order is developed. We demonstrate the effectiveness of this approach by computing the critical exponents of the $(\otimes\vec\Phi)^{s}$ and $\vec\Phi\otimes(\otimes\vec\partial)^{n}\vec\Phi$ operators in the 1/N^2 order in the nonlinear sigma model. The special simplifications due to the conformal invariance of the model are discussed. 
  It has recently become apparent that the elliptic genera of K3 surfaces (and their symmetric products) are intimately related to the Igusa cusp form of weight ten. In this contribution, I survey this connection with an emphasis on string theoretic viewpoints. 
  The present paper deals with N=1 2D supersymmetric integrable quantum field theory. The S-matrix proposed to describe the interactions between supersymmetric particles is applied to theories involving topological excitations of zero central charge. Bound states can fit consistently within this type of theories, since the bootstrap can be shown to close. The topological character of the excitations and the similarity with the scattering of particles is fully understood when a kink sector is introduced in the theory. 
  We study the evolution of non-linear spherically symmetric inhomogeneities in string cosmology. Friedmann solutions of different spatial curvature are matched to produce solutions which describe the evolution of non-linear density and curvature inhomogeneities. The evolution of bound and unbound inhomogeneities are studied. The problem of primordial black hole formation is discussed in the string cosmological context and the pattern of evolution is determined in the pre- and post-big-bang phases of evolution. 
  We use the results of integrable field theory to determine the universal amplitude ratios in the two-dimensional Ising model. In particular, the exact values of the ratios involving amplitudes computed at nonzero magnetic field are provided. 
  We construct type II A brane configuration of N=(4,4) supersymmetric two dimensional gauge theory with gauge group U(1) and N_f hypermultiplets in the fundamental representation. By lifting to M-theory (strong coupling), we can see the origin of the R-symmetry enhancement of the Coulomb branch. One can also find two theories which become equivalent at strong coupling. 
  It is shown that the recently proposed target space duality for (0,2) models is not limited to models admitting a Landau-Ginzburg description. By studying some generic examples it is established for the broader class of vector bundles over complete intersections in toric varieties. Instead of sharing a common Landau-Ginzburg locus, a pair of dual models agrees in more general non-geometric phases. The mathematical tools for treating reflexive sheaves are provided, as well. 
  Two closely related topological phenomena are studied at finite density and temperature. These are chiral anomaly and Chern-Simons term. By using different methods it is shown that $\mu^2 = m^2$ is the crucial point for Chern-Simons at zero temperature. So when $\mu^2 < m^2$ $\mu$--influence disappears and we get the usual Chern-Simons term. On the other hand when $\mu^2 > m^2$ the Chern-Simons term vanishes because of non-zero density of background fermions. It is occurs that the chiral anomaly doesn't depend on density and temperature. The connection between parity anomalous Chern-Simons and chiral anomaly is generalized on finite density. These results hold in any dimension as in abelian, so as in nonabelian cases. 
  We propose a four-point effective action for the graviton, antisymmetric two-forms, dilaton and axion of type IIB superstring in ten dimensions. It is explicitly SL(2,Z)-invariant and reproduces the known tree-level results. Perturbatively, it has only one-loop corrections for the NS-NS sector, generalizing the non-renormalization theorem of the R^4 term. Finally, the non-perturbative corrections are of the expected form, namely, they can be interpreted as arising from single D-instantons of multiple charge. 
  The structure of the moduli space of N=1 supersymmetric gauge theories is analyzed from an algebraic geometric viewpoint. The connection between the fundamental fields of the ultraviolet theory, and the gauge invariant composite fields of the infrared theory is explained in detail. The results are then used to prove an anomaly matching theorem. The theorem is used to study anomaly matching for supersymmetric QCD, and can explain all the known anomaly matching results for this case. 
  Starting from a relativistic quantum field theory, we study the low energy scattering of two fermions of opposite spins interacting through a Chern-Simons field. Using the Coulomb gauge we implement the one loop renormalization program and discuss vacuum polarization and magnetic moment effects. We prove that the induced magnetic moments for spin up and spin down fermions are the same. Next, using an intermediary auxiliary cutoff the scattering amplitude is computed up to one loop. Similarly to Aharonov-Bohm effect for spin zero particles, the low energy part of the amplitude contains a logarithmic divergence in the limit of very high intermediary cutoff. In our approach however the needed counterterm is automatically provided without any additional hypothesis. 
  We study the propagations of gravitational wave and D-particle on D6-brane and orientifold 6-plane backgrounds in the M-theory framework. In the case of orientifold plane, D-particle number is not conserved and gravitational wave can convert into D-particle. For the simplest case, we calculate its amplitude numerically. 
  We discuss theories with 16 and 8 supercharges in 6 and 7 dimensions. These theories are defined as world-volume theories of 5- and 6-branes of type II and M theories, in the limit in which bulk modes decouple. We analyze in detail the spectrum of BPS extended objects of these theories, and show that the 6 dimensional ones can be interpreted as little (non-critical) string theories. The little 5-branes of the 6 dimensional theories with 16 supercharges are used to find new string theories with 8 supercharges, which have additional group structure. We describe the web of dualities relating all these theories. We show that the theories with 16 supercharges can be used for a Matrix description of M-theory on T^6 in the general case, and that they also reproduce Matrix theory on T^5 and T^4 in some particular limit. 
  We couple the Hopf term to the relativistic $CP^1$ model and carry out the Hamiltonian analysis at the classical level. The symplectic structure of the model given by the set of Dirac Brackets among the phase space variables is found to be the same as that of the pure $CP^1$ model. This symplectic structure is shown to be inherited from the global SU(2) invariant $S^3$ model, and undergoes no modification upon gauging the U(1) subgroup, except the appearance of an additional first class constraint generating U(1) gauge transformation. We then address the question of fractional spin as imparted by the Hopf term at the classical level. For that we construct the expression of angular momentum through both symmetric energy-momentum tensor as well as through Noether's prescription. Both the expressions agree for the model indicating no fractional spin is imparted by this term at the classical level-a result which is at variance with what has been claimed in the literature. We provide an argument to explain the discrepancy and corroborate our argument by considering a radiation gauge fixed Hopf term coupled to $CP^1$ model, where the desired fractional spin is reproduced and is given in terms of the soliton number. Finally, by making the gauge field of the $CP^1$ model dynamical by adding the Chern-Simons term, the model ceases to be a $CP^1$ model, as is the case with its nonrelativistic counterpart. This model is also shown to reveals the existence of `anomalous' spin. This is however given in terms of the total charge of the system, rather than any soliton number. 
  Configurations of fivebranes, twobranes and fourbranes in type IIA string theory, which give (1+1) dimensional supersymmetric gauge theories in the low energy limit, are constructed. It is shown that these brane configurations are equivalent to a certain class of matrix string theories. This opens up an avenue for the investigation of matrix string dynamics via the geometry of the brane configurations in type IIA string theory. 
  Starting with topological field theory, we derive space-time uncertainty relation proposed by Yoneya through breakdown of topological symmetry in the large $N$ matrix model. Next, on the basis of only two basic principles, those are, generalized space-time uncertainty principle containing spinor field and topological symmetry, we construct a new matrix model. If we furthermore impose a requirement of N=2 supersymmetry, this new matrix model exactly reduces to the IKKT model or the Yoneya model for IIB superstring depending on an appropriate choice for a scalar function. A key feature of these formulations is an appearance of the nontrivial "dynamical" theory through breakdown of topological symmetry in the matrix model. It is closely examined why the nontrivial "dynamical" theory appears from the trivial topological field theory. 
  In this technical note we describe a pair of results on heterotic compactifications. First, we give an example demonstrating that the usual statement of the anomaly-freedom constraint for perturbative heterotic compactifications (meaning, matching second Chern characters) is incorrect for compactifications involving torsion-free sheaves. Secondly, we correct errors in the literature regarding the counting of massless particles in heterotic compactifications. 
  We consider the classical and quantum dynamics in M(atrix) theory. Using a simple ansatz we show that a classical trajectory exhibits a chaotic motion. We argue that the holographic feature of M(atrix) theory is related with the repulsive feature of energy eigenvalues in quantum chaotic system. Chaotic dynamics in N=2 supersymmetric Yang-Mills theory is also discussed. We demonstrate that after the separation of "slow" and "fast" modes there is a singular contribution from the "slow" modes to the Hamiltonian of the "fast" modes. 
  We discuss the supersymmetry algebra of the M theory fivebrane and obtain a new threebrane soliton preserving half of the six-dimensional supersymmetry. This solution is dimensionally reduced to various D-p-branes. 
  We obtain the complete quantum Seiberg-Witten effective action for N=2 supersymmetric SU(N) Yang-Mills theory from the classical M-fivebrane equations of motion with N threebranes moving in its worldvolume. 
  We imagine the strings on the stretched horizon of any $d$ space-time dimensional black hole to be bits of polymer. Then, proposing an interaction between these bits we obtain the size of the configuration, and thus of the black hole, using the scaling laws. The transition from a typical black hole state to a typical string state has a simple explanation, which also holds for the extremal black holes. 
  Operator quantization of the WZNW theory invariant with respect to an affine Kac-Moody algebra $\hat g$ with constrained $\hat {u}(1)^d$ currents is performed using Dirac's procedure. Upon quantization the initial energy-momentum tensor is replaced by the $g/u(1)^d$ coset construction. The $\hat {su}(2)$ WZNW  theory with a constrained $\hat u(1)$ current is equivalent to the $su(2)/u(1)$ conformal field theory. 
  The transformation properties of the N=2 Virasoro superalgebra generators under Poisson-Lie T-duality in (2,2)-superconformal WZNW and Kazama-Suzuki models is considered. It is shown that Poisson-Lie T-duality acts on the N=2 super-Virasoro algebra generators as a mirror symmetry does: it unchanges the generators from one of the chirality sectors while in another chirality sector it changes the sign of U(1) current and interchanges spin-3/2 currents. We discuss Kazama-Suzuki models generalization of this transformation and show that Poisson-Lie T-duality acts as a mirror symmetry also. 
  Continuous dual symmetry in electrodynamics, Yang-Mills theory and gravitation is investigated. Dual invariant which leads to badly nonlinear motion equations is chosen as a Lagrangian of the pure classical dual nonlinear electrodynamics. In a natural manner some dual angle which is determined by the electromagnetic strengths at the point of the time-space appears in the model. Motion equations may well be interpreted as the equations of the standard Maxwell theory with source. Alternative interpretation is the quasi-Maxwell linear theory with magnetic charge. Analogous approach is possible in the Yang-Mills theory. In this case the dual-invariant non-Abelian theory motion equations possess the same instanton solutions as the conventional Yang--Mills equations have. An Abelian two-parameter dual group is found to exist in Gravitation. Irreducible representations have been obtained: the curvature tensor was expanded into the sum of twice anti-self-dual and self-dual parts. Gravitational instantons are defined as (real) solutions to the usual duality equations. Central-symmetry solutions to these equations are obtained. The twice anti-self-dual part of the curvature tensor may be used for introduction of new gravitational equations generalizing Einstein's equations. However, the theory obtained reduces to the conformal-flat Nordstrom theory. 
  Just like decent classical difference-difference systems define symplectic maps on suitable phase spaces, their counterparts with properly ordered noncommutative entries come as Heisenberg equations of motion for corresponding quantum discrete-discrete models. We observe how this idea applies to a difference-difference counterpart of the Liouville equation. We produce explicit forms of of its evolution operator for the two natural space-time coordinate systems. We discover that discrete-discrete models inherit crucial features of their continuous-time parents like locality and integrability while the new-found algebraic transparency promises a useful progress in some branches of Quantum Inverse Scattering Method. 
  An operator formalism for bosonic $\beta-\gamma$ systems on arbitrary algebraic curves is introduced. The classical degrees of freedom are identified and their commutation relations are postulated. The explicit realization of the algebra formed by the fields is given in a Hilbert space equipped with a bilinear form. The construction is based on the "gaussian" representation for $\beta-\gamma$ systems on the complex sphere [Alvarez-Gaum\' e et al, Nucl. Phys. B 311 (1988) 333]. Detailed computations are provided for the two and four points correlation functions. 
  We present in detail a class of solutions to the $4D SU(\infty)$ Moyal Anti Self Dual Yang Mills equations that are related to $reductions$ of the generalized Moyal Nahm quations using the Ivanova-Popov ansatz. The former yields solutions to the ASDYM/SDYM equations for arbitary gauge groups. A further dimensional reduction yields solutions to the Moyal Anti Self Dual Gravitational equations. The Self Dual Yang Mills /Self Dual Gravity case requires a separate study. SU(2) and $SU(\infty)$ (continuous) Moyal Toda equations are derived and solutions to the latter equations in $implicit$ form are proposed via the Lax-Brockett double commutator formalism . An explicit map taking the Moyal heavenly form (after a rotational Killing symmetry reduction) into the SU(2) Moyal Toda field is found. Finally, the generalized Moyal Nahm equations are conjectured that contain the continuous $SU(\infty)$ Moyal Toda equation after a suitable reduction. Three different embeddings of the three different types of Moyal Toda equations into the Moyal Nahm equations are discussed. 
  We derive all covariant and consistent divergence and commutator anomalies of chiral QED$_2$ within the framework of canonical quantization of the fermions. Further, we compute the time evolution of all occurring operators and find that all commutators evolve canonically. We comment on the relation of our results to the finding of a nontrivial U(1)-curvature in gauge-field space. 
  Using mass perturbation theory, we infer the bound-state spectrum of massive QED$_2$ and compute some decay widths of unstable bound states. Further, we discuss scattering processes, where all the resonances and particle production thresholds are properly taken into account by our methods. 
  The effect of topology on the thermodynamics of a gas of adjoint representation charges interacting via 1+1 dimensional SU(N) gauge fields is investigated. We demonstrate explicitly the existence of multiple vacua parameterized by the discrete superselection variable k=1,...,N. In the low pressure limit, the k dependence of the adjoint gas equation of state is calculated and shown to be non-trivial. Conversely, in the limit of high system pressure, screening by the adjoint charges results in an equation of state independent of k. Additionally, the relation of this model to adjoint QCD at finite temperature in two dimensions and the limit of large N are discussed. 
  In 70's A.A. Kirillov interpreted the stationary Schroedinger (Sturm-Liouville) operator as an element of the dual space to the Virasoro algebra, i.e., the nontrivial central extension of the Witt algebra. He interpreted the KdV operator in terms of the stabilizer of the Schroedinger operator. By studying the coadjoint representation of the simplest nontrivial central extension of a simplest stringy superalgebra, the Neveu--Schwarz superalgebra, Kirillov connected solutions of the KdV and Schroedinger equations. We extend Kirillov's results and find all supersymmetric extension of the Schroedinger and Korteweg-de Vries operators associated with the 12 distinguished stringy superalgebras. We also take into account the odd parameters and the possibility for Time to be a $(1|1)$-dimensional supermanifold. The superization of construction due to Khesin e.a. (Drinfeld--Sokolov's reduction for the pseudodifferential operators) relates the complex powers of the Schroedinger operators we describe to the superized KdV-type hierarchies labeled by complex parameter. Our construction brings the KdV-type equations directly in the Lax form guaranteeing their complete integrability. 
  We review various aspects of classical solutions in string theories. Emphasis is placed on their supersymmetry properties, their special roles in string dualities and microscopic interpretations. Topics include black hole solutions in string theories on tori and N=2 supergravity theories; p-branes; microscopic interpretation of black hole entropy. We also review aspects of dualities and BPS states. 
  We propose a geometric method to parameterize inequivalent vacua by dynamical data. Introducing quantum Clifford algebras with arbitrary bilinear forms we distinguish isomorphic algebras --as Clifford algebras-- by different filtrations resp. induced gradings. The idea of a vacuum is introduced as the unique algebraic projection on the base field embedded in the Clifford algebra, which is however equivalent to the term vacuum in axiomatic quantum field theory and the GNS construction in C^*-algebras. This approach is shown to be equivalent to the usual picture which fixes one product but employs a variety of GNS states. The most striking novelty of the geometric approach is the fact that dynamical data fix uniquely the vacuum and that positivity is not required. The usual concept of a statistical quantum state can be generalized to geometric meaningful but non-statistical, non-definite, situations. Furthermore, an algebraization of states takes place. An application to physics is provided by an U(2)-symmetry producing a gap-equation which governs a phase transition. The parameterization of all vacua is explicitly calculated from propagator matrix elements. A discussion of the relation to BCS theory and Bogoliubov-Valatin transformations is given. 
  The boundary-value problem for Laplace-type operators acting on smooth sections of a vector bundle over a compact Riemannian manifold with generalized local boundary conditions including both normal and tangential derivatives is studied. The condition of strong ellipticity of this boundary-value problem is formulated. The resolvent kernel and the heat kernel in the leading approximation are explicitly constructed. As a result, the previous work in the literature on heat-kernel asymptotics is shown to be a particular case of a more general structure. For a bosonic gauge theory on a compact Riemannian manifold with smooth boundary, the problem is studied of obtaining a gauge-field operator of Laplace type, jointly with local and gauge-invariant boundary conditions, which should lead to a strongly elliptic boundary-value problem. The scheme is extended to fermionic gauge theories by means of local and gauge-invariant projectors. After deriving a general condition for the validity of strong ellipticity for gauge theories, it is proved that for Euclidean Yang--Mills theory and Rarita--Schwinger fields all the above conditions can be satisfied. For Euclidean quantum gravity, however, this property no longer holds, i.e. the corresponding boundary-value problem is not strongly elliptic. Some non-standard local formulae for the leading asymptotics of the heat-kernel diagonal are also obtained. It is shown that, due to the absence of strong ellipticity, the heat-kernel diagonal is non-integrable near the boundary. 
  We compute the amplitude for the radiation of massless NS-NS closed string states from the interaction of two moving D-branes. We consider particle-like D-branes with reference to 4-dimensional spacetime, in toroidal and orbifold compactifications, and we work out the relevant world sheet propagators within the moving boundary state formalism. We find no on-shell axion emission. For large inter-brane separation, we compute the spacetime graviton emission amplitude and estimate the average energy radiated, whereas the spacetime dilaton amplitude is found to vanish in this limit. The possibility of emission of other massless states depends on the nature of the branes and of the compactification scheme. 
  We consider generating functionals of Green's functions with external fields in the framework of BV and BLT quantization schemes for general gauge theories. The corresponding Ward identities are obtained, and the gauge dependence is studied. 
  Among the usual constraints of (1,1) supergravity in d=2 the condition of vanishing bosonic torsion is dropped. Using the inverse supervierbein and the superconnection considerably simplifies the formidable computational problems. It allows to solve the constraints for those fields before taking into account the (identically fulfilled) Bianchi identities. The relation of arbitrary functions in the seminal paper of Howe to supergravity multiplets is clarified. The local supersymmetry transformations remain the same, but, somewhat surprisingly, the transformations of zweibein and Rarita-Schwinger field decouple from those of the superconnection multiplet. A method emerges naturally, how to construct `non-Einsteinian' supergravity theories with nontrivial curvature and torsion in d=2 which, apart from their intrinsic interest, may be relevant for models of super black holes and for novel generalizations in superstring theories. Several explicit examples of such models are presented, some of which immediately allow a dilatonic formulation for the bosonic part of the action. 
  Parity and flavor symmetry is not broken in QCD. Using this fact, we propose a new method to measure the chiral condensate in lattice QCD using Wilson fermions with an operator that breaks parity and flavor in addition to the chiral symmetry. 
  We study the two-point correlators of the currents of the $E_8$ global symmetry in the $N=(1,0)$ superconformal six-dimensional theory as well as in the 4D superconformal theories upon toroidal compactification. From the high-energy behavior of the correlator we deduce that in 4D 10 copies of the superconformal theory with $E_8$ global symmetry can be coupled to an N=2 $E_8$ gauge theory. We present three alternative derivations for the expression for the correlators. One from field-theory, one from M-theory and one from F-theory. 
  We analyze the Poisson structure of the time-dependent mean-field equations for bosons and construct the Lie-Poisson bracket associated to these equations. The latter follow from the time-dependent variational principle of Balian and Veneroni when a gaussian Ansatz is chosen for the density operator. We perform a stability analysis of both the full and the linearized equations. We also search for the canonically conjugate variables. In certain cases, the evolution equations can indeed be cast in a Hamiltonian form. 
  The time-dependent variational principle proposed by Balian and Veneroni is used to provide the best approximation to the generating functional for multi-time Green's functions of a set of (bosonic) observables $Q_{\mu)$. By suitably restricting the trial spaces, the computation of the two-time Green's function, obtained by a second order expansion in the sources, is considerably simplified. This leads to a tractable formalism suited to quantum fields out of equilibrium. We propose an illustration on the finite temperature ${\bf\Phi^4}$ theory in curved space and coupled to gravity. 
  Evaluating the propagator by the usual time-sliced manner, we use it to compute the second virial coefficient of an anyon gas interacting through the repulsive potential of the form $g/r^2 (g > 0)$. All the cusps for the unpolarized spin-1/2 as well as spinless cases disappear in the $\omega \to 0$ limit, where $\omega$ is a frequency of harmonic oscillator which is introduced as a regularization method. As $g$ approaches to zero, the result reduces to the noninteracting hard-core limit. 
  We construct the Matrix theory descriptions of M-theory on the Mobius strip and the Klein bottle. In a limit, these provide the matrix string theories for the CHL string and an orbifold of type IIA string theory. 
  The effects of boundary conditions of the fields for the compactified space directions on the supersymmetric theories are discussed. The boundary conditions can be taken to be periodic up to the degrees of freedom of localized $U(1)_{R}$ transformations. The boundary condition breaks the supersymmery to yield universal soft supersymmetry breaking terms. The 4-dimensional supersymmetric QED with one flavour and the pure supersymmetric QCD are studied as toy models when one of the space coordinates is compactified on $S^1$. 
  Keeping in mind the several models of M(atrix) theory we attempt to understand the possible structure of the topological M(atrix) theory ``underlying'' these approaches. In particular we raise the issue about the nature of the structure of the vacuum of the topological M(atrix) theory and how this could be related to the vacuum of the electroweak theory. In doing so we are led to a simple Topological Matrix Model. Moreover it is expected from the current understanding that the noncommutative nature of ``spacetime'' and background independence should lead to Topological Model. The main purpose of this note is to propose a simple Topological Matrix Model which bears relation to F and M theories. Suggestions on the origin of the chemical potential term appearing in the matrix models are given. 
  We develop a systematic method of obtaining duality symmetric actions in different dimensions. This technique is applied for the quantum mechanical harmonic oscillator, the scalar field theory in two dimensions and the Maxwell theory in four dimensions. In all cases there are two such distinct actions. Furthermore, by soldering these distinct actions in any dimension a master action is obtained which is duality invariant under a much bigger set of symmetries than is usually envisaged. The concept of swapping duality is introduced and its implications are discussed. The effects of coupling to gravity are also elaborated. Finally, the extension of the analysis for arbitrary dimensions is indicated. 
  We obtain for the attractive Dirac delta-function potential in two-dimensional quantum mechanics a renormalized formulation that avoids reference to a cutoff and running coupling constant. Dimensional transmutation is carried out before attempting to solve the system, and leads to an interesting eigenvalue problem in N-2 degrees of freedom (in the center of momentum frame) when there are N particles. The effective Hamiltonian for N-2 particles has a nonlocal attractive interaction, and the Schrodinger equation becomes an eigenvalue problem for the logarithm of this Hamiltonian. The 3-body case is examined in detail, and in this case a variational estimate of the ground-state energy is given. 
  We compute the exact QED_{3+1} effective action for fermions in the presence of a family of static but spatially inhomogeneous magnetic field profiles. An asymptotic expansion of this exact effective action yields an all-orders derivative expansion, the first terms of which agree with independent derivative expansion computations. These results generalize analogous earlier results by Cangemi et al in QED_{2+1}. 
  We discuss several mechanisms to cancel the anomalies of a 5-brane embedded in M-theory. Two of them work, provided we impose suitable conditions either on the 11-dimensional manifold of M-theory or on the 4-form field strength of M-theory. 
  In this work we study the recently introduced octonionic duality for membranes. Restricting the self - duality equations to seven space dimensions, we provide various forms for them which exhibit the symmetries of the octonionic and quaternionic structure. These forms may turn to be useful for the question of the integrability of this system. Introducing a consistent quadratic Poisson algebra of functions on the membrane we are able to factorize the time dependence of the self - duality equations. We further give the general linear embeddings of the three dimensional system into the seven dimensional one using the invariance of the self-duality equations under the exceptional group G_2. 
  More general constructions are given of six-dimensional theories that look at low energy like six-dimensional super Yang-Mills theory. The constructions start with either parallel fivebranes in Type IIB, or M-theory on $(\C^2\times\S^1)/\Gamma$ for $\Gamma$ a suitable finite group. Via these constructions, one can obtain six-dimensional theories with any simple gauge group, and $SU(r)$ theories with any rational theta angle. A matrix construction of these theories is also possible. 
  We formulate new gauge principles for n supersymmetric particles in a worldline formalism with N supersymmetries. The models provide realizations of the more general supersymmetries that are encountered in sectors of S-theory or Matrix theory, with a superalgebra that involves products of momenta. Due to local gauge and kappa symmetries the n superparticle momenta and N supercharges are constrained. The constraints have solutions only in a space with n timelike dimensions and SO(d+n-2,n) spacetime symmetry. The cases SO(9,1), SO(10,2) and SO(10,3), with one, two and three timelike dimensions respectively, are of special interest. In each case, due to the constraints, the classical motion and quantum theory of each superparticle are equivalent to the physics with a single time-like dimension in an effective 10D superspace with SO(9,1) Lorentz symmetry. 
  An ambiguity inherent in the partial integration procedure leading to the Bern-Kosower rules is fixed in a way which preserves the complete permutation symmetry in the scattering states. This leads to a canonical version of the Bern-Kosower representation for the one-loop N - photon/gluon amplitudes, and to a natural decomposition of those amplitudes into permutation symmetric gauge invariant partial amplitudes. This decomposition exhibits a simple recursive structure. 
  We obtain the periods and one-instanton coefficient of the N=2 superrsymmetric Yang-Mills theory with the exceptional gauge group E_6. These calculations are based on the E_6 spectral curve and the obtained one-instanton coefficient is in agreement with the microscopic results. 
  A few generalizations of a Poisson algebra to field theory canonically formulated in terms of the polymomentum variables are discussed. A graded Poisson bracket on differential forms and an $(n+1)$-ary bracket on functions are considered. The Poisson bracket on differential forms gives rise to various generalizations of a Gerstenhaber algebra: the noncommutative (in the sense of Loday) and the higher-order (in the sense of the higher order graded Leibniz rule). The $(n+1)$-ary bracket fulfills the properties of the Nambu bracket including the ``fundamental identity'', thus leading to the Nambu-Poisson algebra. We point out that in the field theory context the Nambu bracket with a properly defined covariant analogue of Hamilton's function determines a joint evolution of several dynamical variables. 
  We apply the long-wavelength approximation to the low energy effective string action in the context of Hamilton-Jacobi theory. The Hamilton-Jacobi equation for the effective string action is explicitly invariant under scale factor duality. We present the leading order, general solution of the Hamilton-Jacobi equation. The Hamilton-Jacobi approach yields a solution consistent with the Lagrange formalism. The momentum constraints take an elegant, simple form. Furthermore this general solution reduces to the quasi-isotropic one, if the evolution of the gravitational field is neglected. Duality transformation for the general solution is written as a coordinate transformation in an abstract field space. 
  We analyse the Spherical Model with frustration induced by an external gauge field. In infinite dimensions, this has been recently mapped onto a problem of q-deformed oscillators, whose real parameter q measures the frustration. We find the analytic solution of this model by suitably representing the q-oscillator algebra with q-Hermite polynomials. We also present a related Matrix Model which possesses the same diagrammatic expansion in the planar approximation. Its interaction potential is oscillating at infinity with period log(q), and may lead to interesting metastability phenomena beyond the planar approximation. The Spherical Model is similarly q-periodic, but does not exhibit such phenomena: actually its low-temperature phase is not glassy and depends smoothly on q. 
  We find that the target space of two-dimensional (4,0) supersymmetric sigma models with torsion coupled to (4,0) supergravity is a QKT manifold, that is, a quaternionic K\"ahler manifold with torsion. We give four examples of geodesically complete QKT manifolds one of which is a generalisation of the LeBrun geometry. We then construct the twistor space associated with a QKT manifold and show that under certain conditions it is a K\"ahler manifold with a complex contact structure. We also show that, for every 4k-dimensional QKT manifold, there is an associated 4(k+1)-dimensional hyper-K\"ahler one. 
  A complete definition of the cycles, on the auxiliary Riemann surface defined by Martinec and Warner for describing pure N=2 gauge theories with arbitrary group, is provided. The strong coupling monodromies around the vanishing cycles are shown to arise from a set of dyons which becomes massless at the singularities. It is shown how the correct weak coupling monodromies are reproduced and how the dyons have charges which are consistent with the spectrum that can be calculated at weak coupling using conventional semi-classical methods. In particular, the magnetic charges are co-root vectors as required by the Dirac-Schwinger-Zwanziger quantization condition. 
  Within the Schroedinger Electric Representation we analytically calculate the complete wave functional obeying Gauss' law with static SU(2) sources in (1+1)-dimensions. The effective potential is found to be linear in the distance between the sources as expected. 
  We present a method for computing the spectrum of black hole radiation of a scalar field satisfying a wave equation with high frequency dispersion. The method involves a combination of Laplace transform and WKB techniques for finding approximate solutions to ordinary differential equations. The modified wave equation is obtained by adding a higher order derivative term suppressed by powers of a fundamental momentum scale $k_0$ to the ordinary wave equation. Depending on the sign of this new term, high frequency modes propagate either superluminally or subluminally. We show that the resulting spectrum of created particles is thermal at the Hawking temperature, and further that the out-state is a thermal state at the Hawking temperature, to leading order in $k_0$, for either modification. 
  Recently, it was observed that self-interacting scalar quantum field theories having a non-Hermitian interaction term of the form $g(i\phi)^{2+\delta}$, where $\delta$ is a real positive parameter, are physically acceptable in the sense that the energy spectrum is real and bounded below. Such theories possess PT invariance, but they are not symmetric under parity reflection or time reversal separately. This broken parity symmetry is manifested in a nonzero value for $<\phi>$, even if $\delta$ is an even integer. This paper extends this idea to a two-dimensional supersymmetric quantum field theory whose superpotential is ${\cal S}(\phi)=-ig(i\phi)^{1+\delta}$. The resulting quantum field theory exhibits a broken parity symmetry for all $\delta>0$. However, supersymmetry remains unbroken, which is verified by showing that the ground-state energy density vanishes and that the fermion-boson mass ratio is unity. 
  Starting from a relativistic s-wave scattering length model for the two particle input we construct an unambiguous, unitary solution of the relativistic three body problem given only the masses $m_a,m_b,m_c$ and the masses of the two body bound states $\mu_{bc},\mu_{ca},\mu_{ab}$. 
  We investigate D-brane instanton contributions to R^4 couplings in any toroidal compactification of type II theories. Starting from the 11D supergravity one-loop four-graviton amplitude computed by Green, Gutperle and Vanhove, we derive the non-perturbative O(e^{-1/\lambda}) corrections to R^4 couplings by a sequence of T-dualities, and interpret them as precise configurations of bound states of D-branes wrapping cycles of the compactification torus. Dp-branes explicitely appear as fluxes on D(p+2)-branes, and as gauge instantons on D(p+4)-branes. Specific rules for weighting these contributions are obtained, which should carry over to more general situations. Furthermore, it is shown that U-duality in D<=6 relates these D-brane configurations to O(e^{-1/\lambda^2}) instantons for which a geometric interpretation is still lacking. 
  Generalisations of the familiar Euler top equations in three dimensions are proposed which admit a sufficiently large number of conservation laws to permit integrability by quadratures. The usual top is a classical analogue of the Nahm equations. One of the examples discussed here is a seven-dimensional Euler top, which arises as a classical counterpart to the eight-dimensional self-dual equations which are currently believed to play a role in new developments in string theory. 
  Evidence is discussed, from lattice simulations, that QCD vacuum is a dual superconductor in the confining phase, and undergoes a phase transition to normal at the deconfining temperature. 
  We present soliton and soliton-antisoliton solutions for the integrable chiral model in 2+1 dimensions with nontrivial (elastic) scattering. These solutions can be obtained either as the limiting cases of the ones already constructed by Ward or by adapting Uhlenbeck's method. 
  In four-dimensional gauge theory there exists a well-known correspondence between instantons and holomorphic curves, and a similar correspondence exists between certain octonionic instantons and triholomorphic curves. We prove that this latter correspondence stems from the dynamics of various dimensional reductions of ten-dimensional supersymmetric Yang-Mills theory. More precisely we show that the dimensional reduction of the (5+1)-dimensional supersymmetric sigma model with hyperkaehler (but otherwise arbitrary) target X to a four-dimensional hyperkaehler manifold M is a topological sigma model localising on the space of triholomorphic maps M -> X (or hyperinstantons). When X is the moduli space M_K of instantons on a four-dimensional hyperkaehler manifold K, this theory has an interpretation in terms of supersymmetric gauge theory. In this case, the topological sigma model can be understood as an adiabatic limit of the dimensional reduction of ten-dimensional supersymmetric Yang-Mills on the eight-dimensional manifold M x K of holonomy Sp(1) x Sp(1) in Spin(7), which is a cohomological theory localising on the moduli space of octonionic instantons. 
  We study Abelian Chern-Simons field theories with matter fields and global SU(N) symmetry in the presence of random weak quenched disorder. In the absence of disorder these theories possess N=2 supersymmetric fixed points and N=1 supersymmetric fixed lines in the infra-red limit. We show that although the presence of disorder forbids any supersymmetry of the bare action, infra-red stable supersymmetric fixed points and fixed lines are realized in the disorder-averaged effective theories. 
  The collective dynamics of solitons with a coset space G/H as moduli space is studied. It is shown that the collective band for a vibrational state is given by the inequivalent coset space quantization corresponding to the representation of H carried by the vibration. 
  Exact solutions to the low-energy effective action of the four-dimensional, N=2 supersymmetric gauge theories with matter (including N=2 super-QCD) are discussed from the three different viewpoints: (i) instanton calculus, (ii) N=2 harmonic superspace, and (iii) M theory. The emphasis is made on the foundations of all three approaches and their relationship. 
  The effective field equations of motion for a mixed theory of open and closed (2,2) world-sheet supersymmetric critical strings are shown to be integrable in the case of an abelian gauge group. The Born-Infeld-type effective action in 2+2 dimensions is intrinsically non-covariant, and it can be interpreted as (a part of) the F-brane world-volume action. The covariant F-brane action is unambiguously restored by its maximal (N=8) world-volume supersymmetry. The 32 supercharges, the local SO(2,1) x SO(8) and rigid SL(2,R) symmetries of the F-brane action naturally suggest its interpretation as the hypothetical (non-covariant) self-dual `heterotic' (1,0) supergravity in 2+10 dimensions. 
  The renormalization group method is applied for obtaining the asymptotic form of the wave function of the quantum anharmonic oscillator by resumming the perturbation series. It is shown that the resumed series is the cumulant of the naive perturbation series. Working out up to the sixth order and performing a further resummation proposed by Bender and Bettencourt, we find that the agreement with the WKB result becomes worse in the higher orders than the fourth at which the agreement is the best. 
  We examine the claim that the effective action of four-dimensional SU(2)_L gauge theory at high and low temperature contains a three-dimensional Chern-Simons term with coefficient being the chemical potential for baryon number. We perform calculations in a two-dimensional toy model and find that the existence of the term is rather subtle. 
  The question that guides our discussion is how did the geometry and particles come into being. The present theory reveals primordial deeper structures underlying fundamental concepts of contemporary physics. We begin with a drastic revision of a role of local internal symmetries in physical concept of curved geometry. A standard gauge principle of local internal symmetries is generalized. The gravitation gauge group is proposed, which is generated by hidden local internal symmetries. Last two parts address to the question of physical origin of geometry and basic concepts of particle physics such as the fields of quarks with the spins and various quantum numbers, internal symmetries and so forth; also four basic principles of Relativity, Quantum, Gauge and Color Confinement, which are, as it was proven, all derivative and come into being simultaneously. The most promising aspect of our approach so far is the fact that many of the important anticipated properties, basic concepts and principles of particle physics are appeared quite naturally in the framework of suggested theory. 
  We derive the spectral representations of QED 3-point functions and then explicitly calculate the 3-point spectral densities in hard thermal loop approximation within the real time formalism. The Ward identities obeyed by the retarded and advanced 2- and 3-point functions are discussed. We compare our results with those for hot QCD . 
  We study the functional integrals that appear in a path integral bosonization procedure for more than two spacetime dimensions. Since they are not in general exactly solvable, their evaluation by a suitable loop expansion would be a natural procedure, even if the exact fermionic determinant were known. The outcome of our study is that we can consistently ignore loop corrections in the functional integral defining the bosonized action, if the same is done for the functional integral corresponding to the bosonic representation of the generating functional. If contributions up to some order $l$ in the number of loops are included in both integrals, all but the lowest terms cancel out in the final result for the generating functional. 
  We develop a bosonization procedure on the half line. Different boundary conditions, formulated in terms of the vector and axial fermion currents, are implemented by using in general the mixed boundary condition on the bosonic field. The interplay between symmetries and boundary conditions is investigated in this context, with a particular emphasis on duality. As an application, we explicitly construct operator solutions of the massless Thirring model on the half line, respecting different boundary conditions. 
  The gauge dependence of the time-ordered products for Yang-Mills theories is analysed in perturbation theory by means of the causal method of Epstein and Glaser together with perturbative gauge invariance. This approach allows a simple inductive proof of the gauge independence of the physical S-matrix. 
  Within the framework of algebraic quantum field theory a general method is presented which allows one to compute and classify the short distance (scaling) limit of any algebra of local observables. The results can be used to determine the particle and symmetry content of a theory at very small scales and thereby give an intrinsic meaning to notions such as ``parton'' and ``confinement''. The method has been tested in models. (Invited talk given at the International Congress of Mathematical Physics, July 1997, Brisbane) 
  In these lectures I present a basic introduction to supersymmetry, especially to N=1 supersymmetric gauge theories and their renormalization, in the Wess-Zumino gauge. I also discuss the various ways supersymmetry may be broken in order to account for the lack of exact supersymmetry in the actual world of elementary particles. 
  A system of gravity coupled to a 2-form gauge field, a dilaton and Yang-Mills fields in $2n$ dimensions arises from the (2,1) sigma model or string. The field equations imply that the curvature with torsion and Yang-Mills field strength are self-dual in four dimensions, or satisfy generalised self-duality equations in $2n$ dimensions. The Born-Infeld-type action describing this system is simplified using an auxiliary metric and shown to be classically Weyl invariant only in four dimensions. A dual form of the action is found (no isometries are required). In four dimensions, the dual geometry is self-dual gravity without torsion coupled to a scalar field. In $D>4$ dimensions, the dual geometry is hermitian and determined by a $D-4$ form potential $K$, generalising the K\"{a}hler potential of the four dimensional case, with the fundamental 2-form given by $\tilde J= i*\partial \bar \partial K$. The coupling to Yang-Mills is through a term $K\wedge tr (F\wedge F)$ and leads to a Uhlenbeck-Yau field equation $\tilde J^{ij}F_{ij}=0$. 
  Recently, solutions of the Born-Infeld theory representing strings emanating from a Dirichlet p-brane have been constructed. We discuss the embedding of these Born-Infeld solutions into the non-abelian theory appropriate to multiple overlapping p-branes. We also prove supersymmetry of the solutions explicitly in the full nonlinear theory. We then study transverse fluctuations, both from the worldbrane point of view and by analyzing a test-string in the supergravity background of a Dp-brane. We find agreement between the two approaches for the cases p=3,4. 
  We calculate instanton corrections to three dimensional gauge theories with N=4 and N=8 supersymmetry and SU(n) gauge groups. The N=4 results give new information about the moduli space of n BPS SU(2) monopoles, including the leading order non-pairwise interaction terms. A few comments are made on the relationship of the N=8 results to membrane scattering in matrix theory. 
  A free field representation for the type $I$ vertex operators and the corner transfer matrices of the eight-vertex model is proposed. The construction uses the vertex-face correspondence, which makes it possible to express correlation functions of the eight-vertex model in terms of correlation functions of the SOS model with a nonlocal insertion. This new nonlocal insertion admits of a free field representation in terms of Lukyanov's screening operator. The spectrum of the corner transfer matrix and the Baxter--Kelland formula for the average staggered polarization have been reproduced. 
  We consider two-dimensional gravity with dynamical torsion in the Batalin - Vilkovisky and Batalin - Lavrov - Tyutin formalisms of gauge theories quantization as well as in the background field method. 
  The Casimir energy of a solid ball placed in an infinite medium is calculated by a direct frequency summation using the contour integration. It is assumed that the permittivity and permeability of the ball and medium satisfy the condition $\epsilon_1 \mu_1=\epsilon_2\mu_2$. Upon deriving the general expression for the Casimir energy, a dilute compact ball is considered $(\epsilon_1 -\epsilon_2)^2/(\epsilon_1+\epsilon_2)^2\ll 1$. In this case the calculations are carried out which are of the first order in $\xi ^2$ and take account of the five terms in the Debye expansion of the Bessel functions involved. The implication of the obtained results to the attempts of explaining the sonoluminescence via the Casimir effect is shortly discussed. 
  I compare the calculations of special $F^4$ and $R^4$ terms in the (toroidally-compactified) heterotic and type-I effective actions. Besides checking duality, this elucidates the quantitative rules of D-brane calculus. I explain in particular (a) why D-branes do not run in loops, and (b) how their instanton contributions arise from orbifold fixed points of their moduli space. The instanton sum has a simple representation as a sum of the elliptic genera of matrix models. 
  We discuss random matrix models in terms of elementary operations on Blue's functions (functional inverse of Green's functions). We show that such operations embody the essence of a number of physical phenomena whether at/or away from the critical points. We illustrate these assertions by borrowing on a number of recent results in effective QCD in vacuum and matter. We provide simple physical arguments in favor of the universality of the continuum QCD spectral oscillations, whether at zero virtuality, in the bulk of the spectrum or at the chiral critical points. We also discuss effective quantum systems of disorder with strong or weak dissipation (Hatano-Nelson localization). 
  We present a study of M(atrix) theory from a purely canonical viewpoint. In particular, we identify free particle asymptotic states of the model corresponding to the supergraviton multiplet of eleven dimensional supergravity. These states have a natural interpretation as excitations in the flat directions of the matrix model potential. Furthermore, we provide the split of the matrix model Hamiltonian into a free part describing the free propagation of these particle states along with the interaction Hamiltonian describing their interactions. Elementary quantum mechanical perturbation theory then yields an effective potential for these particles as an expansion in their inverse separation. Remarkably we find that the leading velocity independent terms of the effective potential cancel in agreement with the fact that there is no force between stationary D0 branes. The scheme we present provides a framework in which one can perturbatively compute the M(atrix) theory result for the eleven dimensional supergraviton S matrix. 
  We extend the well-known 't Hooft anomaly matching conditions for continuous global symmetries to discrete groups. We state the matching conditions for all possible anomalies which involve discrete symmetries explicitly. There are two types of discrete anomalies. For Type I anomalies, the matching conditions have to be always satisfied regardless of the details of the massive bound state spectrum. The Type II anomalies have to be also matched except if there are fractionally charged massive bound states in the theory. We check discrete anomaly matching in recent solutions of certain N=1 supersymmetric gauge theories, most of which satisfy these constraints. The excluded examples include the chirally symmetric phase of N=1 pure supersymmetric Yang-Mills theories described by the Veneziano-Yankielowicz Lagrangian and certain non-supersymmetric confining theories. The conjectured self-dual theories based on exceptional gauge groups do not satisfy discrete anomaly matching nor mapping of operators, and are viable only if the discrete symmetry in the electric theory appears as an accidental symmetry in the magnetic theory and vice versa. 
  A realization of the Yangian double with center $DY_\hbar(sl_2)_k$ of level $k(\not=0,-2)$ in terms of free boson fields is constructed. The screening currents are also presented, which commute with $DY_\hbar(sl_2)$ modulo total difference. In the $\hbar\to 0$ limit, the currents of Yangian double $DY_\hbar(sl_2)_k$ becomes the Feigin-Fuchs realization of affine Lie $sl(2)_k$, while the screening currents of Yangian double $DY_\hbar(sl_2)_k$ becomes the screening currents of the affine Lie algebra $sl(2)_k$. 
  A set of physical operators which are responsible for touching interactions in the framework of c<1 unitary conformal matter coupled to 2D quantum gravity is found. As a special case the non-critical bosonic strings are considered. Some analogies with four dimensional quantum gravity are also discussed, e.g. creation-annihilation operators for baby universes, Coleman mechanism for the cosmological constant. 
  On a nonrelativistic contact four-fermion model we have shown that the simple Lambda-cut-off prescription together with definite fine-tuning of the Lambda dependency of "bare"quantities lead to self-adjoint semi-bounded Hamiltonian in one-, two- and three-particle sectors. The fixed self-adjoint extension and exact solutions in two-particle sector completely define three-particle problem. The renormalized Faddeev equations for the bound states with Fredholm properties are obtained and analyzed. 
  Various exact solutions of two-particle eigenvalue problems for nonrelativistic contact four-fermion current-current interaction are obtained. Specifics of Goldstone mode is investigated. The connection between a renormalization procedure and construction of self-adjoint extensions is revealed. 
  Generalization of harmonic superposition rule for the case of dependent choice of harmonic functions is given. Dependence of harmonic functions from all (relative and overall) transverse coordinates is considered using the Beltrami-Laplace operator. Supersymmetry of IIB 10D supergravity solutions with only non-vanished 5-form field and 11D supergravity solutions is discussed. 
  We reconsider the quasi exactly solvable matrix models constructed recently by R. Zhdanov. The 2$\times$2 matrix operators representing the algebra sl(2) are generalized to matrices of arbitrary dimension and a similar construction is achieved for the algebra sl(n). 
  The constraints of N=2 supersymmetry, in combination with several other quite general assumptions, have recently been used to show that N=2 supersymmetric Yang-Mills theory has a low energy quantum parameter space symmetry characterised by the discrete group $\gu$. We show that if one also assumes the commutativity of renormalization group flow with the action of this group on the complexified coupling constant $\ta$, then this is sufficient to determine the non-perturbative $\beta$-function, given knowledge of its weak coupling behaviour. The result coincides with the outcome of direct calculations from the Seiberg-Witten solution. 
  We study an Abelian Maxwell-Chern-Simons model in $2 +1 $ dimensions which includes a magnetic moment interaction. We show that this model possesses domain wall as well as vortex solutions. 
  In these lectures we review recent results on universal fluctuations of QCD Dirac spectra and applications of Random Matrix Theory (RMT) to QCD. We review general properties of Dirac spectra and discuss the relation between chiral symmetry breaking and correlations of Dirac eigenvalues. In particular, we will focus on the microscopic spectral density density, i.e. the spectral density near zero virtuality on the scale of a typical level spacing. The relation with Leutwyler-Smilga sum-rules will be discussed. The success of applications of RMT to spectra of 'complex' systems leads us to the introduction of a chiral Random Matrix Theory (chRMT) with the global symmetries of the QCD partition function. Our central conjecture is that it decribes correlations of QCD Dirac spectra. We will review recent universality proofs supporting this conjecture. Lattice QCD results for the microscopic spectral density and for correlations in the bulk of the spectrum are shown to be in perfect agreement with chRMT. We close with a review of chRMT at nonzero chemical potential. Novel features of spectral universality in nonhermitean matrices will be discussed. As an illustration of mathematical methods used in RMT several important recent results will be derived in all details. We mention the derivation of the microscopic spectral density, the universality proof by Akemann, Damgaard, Magnea and Nishigaki, the spectral density of a chRMT at nonzero temperature and the Stephanov solution for chRMT at nonzero chemical potential. 
  We discuss supersymmetry breaking in some supersymmetric quantum mechanical models with periodic potentials. The sensitivity to the parameters appearing in the superpotential is more acute than in conventional nonperiodic models. We present some simple elliptic models to illustrate these points. 
  We continue to study 5d N=1 supersymmetric field theories and their compactifications on a circle through brane configurations. We develop a model, which we call (p,q) Webs, which enables simple geometrical computations to reproduce the known results, and facilitates further study. The physical concepts of field theory are transparent in this picture, offering an interpretation for global symmetries, local symmetries, the effective (running) coupling, the Coulomb and Higgs branches, the monopole tensions, and the mass of BPS particles. A rule for the dimension of the Coulomb branch is found by introducing Grid Diagrams. Some known classifications of field theories are reproduced. In addition to the study of the vacuum manifold we develop methods to determine the BPS spectrum. Some states, such as quarks, correspond to instantons inside the 5-brane which we call strips. In general, these may not be identified with (p,q) strings. We describe how a strip can bend out of a 5-brane, becoming a string. A general BPS state corresponds to a Web of strings and strips. For special values of the string coupling a few strips can combine and leave the 5-brane as a string. 
  We prove that the simple condition on the potential V, \int exp(-t V) < \infty for all t>0, is sufficient for the lattice approximation of the trace Tr[A exp(-b H)] with (Re b)>0 to work for all bounded functions A and a large class of potentials. As a by-product we obtain an explicit bound for the real-temperature lattice kernels. 
  We propose a new integrable N=2 supersymmetric Toda lattice hierarchy which may be relevant for constructing a supersymmetric one-matrix model. We define its first two Hamiltonian structures, the recursion operator and Lax--pair representation. We provide partial evidence for the existence of an infinite-dimensional N=2 superalgebra of its flows. We study its bosonic limit and introduce new Lax-pair representations for the bosonic Toda lattice hierarchy. Finally we discuss the relevance this approach for constructing N=2 supersymmetric generalized Toda lattice hierarchies. 
  We analyse the global (rigid) symmetries that are realised on the bosonic fields of the various supergravity actions obtained from eleven-dimensional supergravity by toroidal compactification followed by the dualisation of some subset of fields. In particular, we show how the global symmetries of the action can be affected by the choice of this subset. This phenomenon occurs even with the global symmetries of the equations of motion. A striking regularity is exhibited by the series of theories obtained respectively without any dualisation, with the dualisation of only the Ramond-Ramond fields of the type IIA theory, with full dualisation to lowest degree forms, and finally for certain inverse dualisations (increasing the degrees of some forms) to give the type IIB series. These theories may be called the GL_A, D, E and GL_B series respectively. It turns out that the scalar Lagrangians of the E series are sigma models on the symmetric spaces K(E_{11-D})\backslash E_{11-D} (where K(G) is the maximal compact subgroup of G) and the other three series lead to models on homogeneous spaces K(G) \backslash G\semi \R^s. These can be understood from the E series in terms of the deletion of positive roots associated with the dualised scalars, which implies a group contraction. We also propose a constrained Lagrangian version of the even dimensional theories exhibiting the full duality symmetry and begin a systematic analysis of abelian duality subalgebras. 
  It was conjectured by Witten that a BPS-saturated domain wall exists in the M-theory fivebrane version of QCD (MQCD) and can be represented as a supersymmetric three-cycle in the sense of Becker et al with an appropriate asymptotic behavior. We derive the differential equation which defines an associative cycle in $G_2$ holonomy seven-manifold corresponding to the supersymmetric three-cycle and show that it contains a sum of the Poisson brackets. We study solutions of the differential equation with prescribed asymptotic behavior. 
  In this paper, we consider two D-branes rotated with respect to each other, and argue that in this way one can find brane configurations preserving ${1 \f 16}$ of SUSY. Also we classify different brane configurations preserving ${1 \f 2}$, ${1 \f 4}$, ${3 \f 16}$,${1 \f 8}$, ${1 \f 16}$ of SUSY. 
  N=2 heterotic strings may provide a window into the physics of M-theory radically different than that found via the other supersymmetric string theories. In addition to their supersymmetric structure, these strings carry a four-dimensional self-dual structure, and appear to be completely integrable systems with a stringy density of states. These lectures give an overview of N=2 heterotic strings, as well as a brief discussion of possible applications of both ordinary and heterotic N=2 strings to D-branes and matrix theory. 
  Solving numerically the equations of motion for the effective lagrangian describing supersymmetric QCD with the SU(2) gauge group, we find a menagerie of complex domain wall solutions connecting different chirally asymmetric vacua. Some of these solutions are BPS saturated walls; they exist when the mass of the matter fields does not exceed some critical value m < m* < 4.67059... There are also sphaleron branches (saddle points of the ebergy functional). In the range m* < m < m** \approx 4.83, one of these branches becomes a local minimum (which is not a BPS saturated one). At m > m*, the complex walls disappear altogether and only the walls connecting a chirally asymmetric vacuum with the chirally symmetric one survive. 
  The Matrix theory description of Type IIA string theory on a compact K3 surface as the theory of Neveu-Schwarz five-branes on $\tilde{K3}\times S^1$ is analyzed. The full multiplet of space-time BPS states is identified in the five-brane world-volume as fluxes. 
  We study the sypersymmetric pure Yang-Mills theory with semisimple Lie groups. We show that the general form of the gluino condensate is determined solely by the symmetries of the theory and it is in disagreement with the recently proposed existence of a conformal phase in SYM theory. We discuss the peculiarities of the Veneziano-Yankielowicz effective Lagrangian approach and explain how it is related to the calculation of the gluino condensate 
  We first consider M-theory formulated on an open eleven-dimensional spin-manifold. There is then a potential anomaly under gauge transformations on the E_8 bundle that is defined over the boundary and also under diffeomorphisms of the boundary. We then consider M-theory configurations that include a five-brane. In this case, diffeomorphisms of the eleven-manifold induce diffeomorphisms of the five-brane world-volume and gauge transformations on its normal bundle. These transformations are also potentially anomalous. In both of these cases, it has previously been shown that the perturbative anomalies, i.e. the anomalies under transformations that can be continuously connected to the identity, cancel. We extend this analysis to global anomalies, i.e. anomalies under transformations in other components of the group of gauge transformations and diffeomorphisms. These anomalies are given by certain topological invariants, that we explicitly construct. 
  We show that the recently found covariant formulation for chiral $p$--forms in $2(p+1)$ dimensions with $p$ even, can be naturally extended to supersymmetric theories. We present the general method for writing covariant actions for chiral bosons and construct, in particular, in six dimensions covariant actions for one tensor supermultiplet, for pure supergravity and for supergravity coupled to an arbitrary number of tensor supermultiplets. 
  Recently several workers have attempted determinations of the so-called magnetic mass of d=3 non-Abelian gauge theories through a one-loop gap equation, using a free massive propagator as input. Self-consistency is attained only on-shell, because the usual Feynman-graph construction is gauge-dependent off-shell. We examine two previous studies of the pinch technique proper self-energy, which is gauge-invariant at all momenta, using a free propagator as input, and show that it leads to inconsistent and unphysical result. In one case the residue of the pole has the wrong sign (necessarily implying the presence of a tachyonic pole); in the second case the residue is positive, but two orders of magnitude larger than the input residue, which shows that the residue is on the verge of becoming ghostlike. This happens because of the infrared instability of d=3 gauge theory. A possible alternative one-loop determination via the effective action also fails. The lesson is that gap equations must be considered at least at two-loop level. 
  We determine the possible fractions of supersymmetry preserved by two intersecting M-5-branes. These include the fractions 3/32 and 5/32 which have not occurred previously in intersecting brane configurations. Both occur in non-orthogonal pointlike intersections of M-5-branes but 5/32 supersymmetry is possible only for specific fixed angles. 
  Singular configuration of an external static magnetic field in the form of a string polarizes vacuum in the secondly quantized theory on a plane which is orthogonal to the string axis. We consider the most general boundary conditions at the punctured singular point, which are compatible with the self-adjointness of the two-dimensional Dirac Hamiltonian. The dependence of the induced vacuum quantum numbers on the self-adjoint extension parameter and the flux of the string is determined. 
  Kraemmer and Rebhan claimed the gauge independence of the conformal anomaly of bosonic string for various gauge fixings in the framework of the perturbation theory of two-dimensional quantum gravity. It is pointed out that their proof is wrong. The gauge independence is proved for the gauge-fixings which reduce to the linearized de Donder gauge in the flat limit of the background metric. Similar remarks are made also for the Rebhan-Kraemmer current anomaly. 
  In the covariant-gauge two-dimensional quantum gravity, various derivations of the critical dimension D=26 of the bosonic string are critically reviewed, and their interrelations are clarified. It is shown that the string theory is not identical with the proper framework of the two-dimensional quantum gravity, but the former should be regarded as a particular aspect of the latter. The appearance of various anomalies is shown to be explainable in terms of a new type of anomaly in a unified way. 
  The technique of functional integration over velocities is applied to the calculation of the propagator of a spinning particle with and without anomalous magnetic moment. A representation for the spin factor is obtained in this context for the particle in a constant electromagnetic field. As a by-product, we also obtain a Schwinger representation for the first case. 
  The rational conformal field theory (RCFT) extensions of W_{1+infinity} at c=1 are in one-to-one correspondence with 1-dimensional integral lattices L(m). Each extension is associated with a pair of oppositely charged ``vertex operators" of charge square m in N. Their product defines a bilocal field V_m(z_1,z_2) whose expansion in powers of z_{12}=z_1-z_2 gives rise to a series of (neutral) local quasiprimary fields V^l(z,m) (of dimension l+1). The associated bilocal exponential of a normalized current generates the W_{1+infinity} algebra spanned by the V^l(z,1) (and the unit operator). The extension of this construction to higher (integer) values of the central charge c is also considered.   Applications to a quantum Hall system require computing characters (i.e., chiral partition functions) depending not just on the modular parameter tau, but also on a chemical potential zeta. We compute such a zeta dependence of orbifold characters, thus extending the range of applications of a recent study of affine orbifolds. 
  I briefly discuss three phenomena arising in recent work where the realization of symmetries in quantum mechanics is unusual. The first of these is, I believe, the very simplest realization of non-trivial confinement by a mechanism of charge-flux frustration. It arises in models having several coupled abelian Chern-Simons gauge symmetries, which closely resemble effective theories used for the quantum Hall effect. The second is symmetry obstruction by non-abelian flux, which has implications for 2+1 dimensional supergravity. Third is the possibility of new varieties of quantum statistics: non-abelian and projective. 
  I attempt to give a pedagogical introduction to the matrix model of M-theory as developed by Banks, Fischler, Shenker and Susskind (BFSS). In the first lecture, I introduce and review the relevant aspects of D-branes with the emergence of the matrix model action. The second lecture deals with the appearance of eleven-dimensional supergravity and M-theory in strongly coupled type IIA superstring theory. The third lecture combines the material of the two previous ones to arrive at the BFSS conjecture and explains the evidence presented by these authors. The emphasis is not on most recent developments but on a hopefully pedagogical presentation. 
  We calculate the one-loop effective potential for the toroidal non-abelian D-brane in the constant magnetic SU(2) gauge field. The study of its properties shows that the potential is unbounded below. This fact indicates the instability of the non-abelian D-brane in the background under consideration like the instability of chromomagnetic vacuum in SU(2) gauge theory. 
  We construct new maximally symmetric solutions for the metric. We then show that for a string moving in a background consisting of maximally symmetric gravity, dilaton field and second rank antisymmetric tensor field, the O(d) $\otimes$ O(d) transformation on the vacuum solutions, in general, gives inequivalent solutions that are not maximally symmetric. 
  We show that the usual physical meaning of maximal symmetry can be made to remain unaltered even if torsion is present. All that is required is that the torsion fields satisfy some mutually consistent constraints. We also give an explicit realisation of such a scenario by determining the torsion fields, the metric and the associated Killing vectors. 
  The Jackiw-Teitelboim gauge formulation of the 1+1 dimensional gravity allows us to relate different gauge fixing conditions with integrable hierarchies of evolution equations. We show that the equations for the Zweibein fields can be written as a pair of time reversed evolution equations of the reaction-diffusion type, admitting dissipative solutions. The spectral parameter for the related Lax pair appears as the constant valued spin connection associated with the $SO(1,1)$ gauge symmetry. Spontaneous breaking of the non-compact symmetry and the irreversible evolution are discussed. 
  We present a lagrangian formulation for recently-proposed supersymmetric Yang-Mills theory in twelve dimensions. The field content of our multiplet has an additional auxiliary vector field in the adjoint representation. The usual Yang-Mills field strength is modified by a Chern-Simons form containing this auxiliary vector field. This formulation needs no constraint imposed on the component field from outside, and a constraint on the Yang-Mills field is generated as the field equation of the auxiliary vector field. The invariance check of the action is also performed without any reference to constraints by hand. Even though the total lagrangian takes a simple form, it has several highly non-trivial extra symmetries. We couple this twelve-dimensional supersymmetric Yang-Mills background to Green-Schwarz superstring, and confirm fermionic kappa-invariance. As another improvement of this theory, we present a set of fully Lorentz-covariant and supercovariant field equations with no use of null-vectors. This system has an additional scalar field, whose gradient plays a role of the null-vector. This system exhibits spontaneous breaking of the original Lorentz symmetry SO(10,2) for twelve-dimensions down to SO(9,1) for ten-dimensions. 
  Using the harmonic superspace background field formulation for general D=4, N=2 super Yang-Mills theories, with matter hypermultiplets in arbitrary representations of the gauge group, we present the first rigorous proof of the N=2 non-renormalization theorem; specifically, the absence of ultraviolet divergences beyond the one-loop level. Another simple consequence of the background field formulation is the absence of the leading non-holomorphic correction to the low-energy effective action at two loops. 
  The Schwinger - DeWitt expansion for the evolution operator kernel of the Schrodinger equation is studied for convergence. It is established that divergence of this expansion which is usually implied for all continuous potentials, excluding ones of the form V(q)=aq^2+bq+c, really takes place only if the coupling constant g is treated as independent variable. But the expansion may be convergent for some kinds of the potentials and for some discrete values of the charge, if the latter is considered as fixed parameter. Class of such potentials is interesting because inside of it the property of discreteness of the charge in the nature is reproduced in the theory in natural way. 
  Making use of the duality transformation, we derive in the Londons' limit of the Abelian Higgs Model string representation for the 't Hooft loop average defined on the string world-sheet, which yields the values of two coefficient functions parametrizing the bilocal correlator of the dual field strength tensors. The asymptotic behaviours of these functions agree with the ones obtained within the Method of Vacuum Correlators in QCD in the lowest order of perturbation theory. We demonstrate that the bilocal approximation to the Method of Vacuum Correlators is an exact result in the Londons' limit, i.e. all the higher cumulants in this limit vanish. We also show that at large distances, apart from the integration over metrics, the obtained string effective theory (which in this case reduces to the nonlinear massive axionic sigma model) coincides with the low-energy limit of the dual version of 4D compact QED, the so-called Universal Confining String Theory. We derive string tension of the Nambu-Goto term and the coupling constant of the rigidity term for the obtained string effective theory and demonstrate that the latter one is always negative, which means the stability of strings, while the positiveness of the former is confirmed by the present lattice data. These data enable us to find the Higgs boson charge and the vacuum expectation value of the Higgs field, which model QCD best of all. We also study dynamics of the weight factor of the obtained string representation for the 't Hooft average in the loop space. In conclusion, we obtain string representation for the partition function of the correlators of an arbitrary number of Higgs currents, by virtue of which we rederive the structure of the bilocal correlator of the dual field strength tensors, which yields the surface term in the string effective action. 
  The Wilsonian exact renormalization group gives a natural framework in which ultraviolet and infrared divergences can be treated separately. In massless QED we introduce, as the only mass parameter, a renormalization scale $\L_R > 0$. We prove, using the flow equation technique, that infrared convergence is a necessary consequence of any zero-momentum renormalization condition at $\L_R$ compatible with the effective Ward identities and axial symmetry. The same formalism is applied to renormalize gauge-invariant composite operators and to prove their infrared finiteness; in particular we consider the case of the axial current operator and its anomaly. 
  We derive modular anomaly equations from the Seiberg-Witten-Donagi curves for softly broken N=4 SU(n) gauge theories. From these equations we can derive recursion relations for the pre-potential in powers of m^2, where m is the mass of the adjoint hypermultiplet. Given the perturbative contribution of the pre-potential and the presence of ``gaps'' we can easily generate the m^2 expansion in terms of polynomials of Eisenstein series, at least for relatively low rank groups. This enables us to determine efficiently the instanton expansion up to fairly high order for these gauge groups, e. g. eighth order for SU(3). We find that after taking a derivative, the instanton expansion of the pre-potential has integer coefficients. We also postulate the form of the modular anomaly equations, the recursion relations and the form of the instanton expansions for the SO(2n) and E_n gauge groups, even though the corresponding Seiberg-Witten-Donagi curves are unknown at this time. 
  The zero curvature representation for two dimensional integrable models is generalized to spacetimes of dimension d+1 by the introduction of a d-form connection. The new generalized zero curvature conditions can be used to represent the equations of motion of some relativistic invariant field theories of physical interest in 2+1 dimensions (BF theories, Chern-Simons, 2+1 gravity and the CP^1 model) and 3+1 dimensions (self-dual Yang-Mills theory and the Bogomolny equations). Our approach leads to new methods of constructing conserved currents and solutions. In a submodel of the 2+1 dimensional CP^1 model, we explicitly construct an infinite number of previously unknown nontrivial conserved currents. For each positive integer spin representation of sl(2) we construct 2j+1 conserved currents leading to 2j+1 Lorentz scalar charges. 
  We give a brief introduction to the Gauge Theory of Arbitrage. Treating a calculation of Net Present Values (NPV) and currencies exchanges as a parallel transport in some fibre bundle, we give geometrical interpretation of the interest rate, exchange rates and prices of securities as a proper connection components. This allows us to map the theory of capital market onto the theory of quantized gauge field interacted with a money flow field. The gauge transformations of the matter field correspond to a dilatation of security units which effect is eliminated by a gauge transformation of the connection. The curvature tensor for the connection consists of the excess returns to the risk-free interest rate for the local arbitrage operation. Free quantum gauge theory is equivalent to the assumption about the log-normal walks of assets prices. In general case the consideration maps the capital market onto lattice QED. 
  We review the construction and classification of three-family grand unified models within the framework of asymmetric orbifolds in perturbative heterotic superstring. We give a detailed survey of all such models which is organized to aid analysis of their phenomenological properties. We compute tree-level superpotentials for these models. These superpotentials are used to analyze the issues of proton stability (doublet-triplet splitting and R-parity violating terms) and Yukawa mass matrices. To have agreement with phenomenological data all these models seem to require certain degree of fine-tuning. We also analyze the possible patterns of supersymmetry breaking in these models.   We find that the supersymmetry breaking scale comes out either too high to explain the electroweak hierarchy problem, or below the electroweak scale unless some degree of fine-tuning is involved. Thus, none of the models at hand seem to be phenomenologically flawless. 
  We discuss new bounce-like (but non-time-reversal-invariant-) solutions to Euclidean equations of motion, which we dub boomerons. In the Euclidean path integral approach to quantum theories, boomerons make an imaginary contribution to the vacuum energy. The fake vacuum instabilty can be removed by cancelling boomeron contributions against contributions from time reversed boomerons (anti-boomerons). The cancellation rests on a sign choice whose significance is not completely understood in the path integral method. 
  Certain one-loop processes in eleven-dimensional supergravity compactified on T**2 determine exact, non-perturbative, terms in the effective action of type II string theories compactified on a circle. One example is the modular invariant U(1)-violating interaction of sixteen complex spin-1/2 fermions of ten-dimensional type IIB theory. This term, together with the (curvature)**4 term, and many other terms of the same dimension are all explicitly related by supersymmetry. 
  The lattice SU(2) gluodynamics in the maximal abelian projection is reduced to the abelian theory, in which the natural small parameter exists. We show that in the zeroth order of the expansion in this parameter the theory is equivalent to the compact abelian gauge theory coupled to ghosts of charge 2.   The theory is a renormalizable and asymptotically free. This theory is represented in the form of the theory of open string with the boundary consisting of the worldline of the quark. The ghosts live on the worldsheet of the string. The naive continuum limit of such string representation gives a simple expression for the chromoelectric string action. 
  Gauge theories are formulated on the noncommutative two-sphere. These theories have only finite number of degrees of freedom, nevertheless they exhibit both the gauge symmetry and the SU(2) "Poincar\'e" symmetry of the sphere. In particular, the coupling of gauge fields to chiral fermions is naturally achieved. 
  A geometric formulation of the Moyal deformation for the Self-dual Yang-Mills theory and the Chiral Model approach to Self-dual gravity is given. We find in Fedosov's geometrical construction of deformation quantization the natural geometrical framework associated to the Moyal deformation of Self-dual gravity. 
  Intersecting p-branes can be viewed as higher-dimensional interpretations of multi-charge extremal p-branes, where some of the individual p-branes undergo diagonal dimensional oxidation, while the others oxidise vertically. Although the naive vertical oxidation of a single p-brane gives a continuum of p-branes, a more natural description arises if one considers a periodic array of p-branes in the higher dimension, implying a dependence on the compactification coordinates. This still reduces to the single lower-dimensional p-brane when viewed at distances large compared with the period. Applying the same logic to the multi-charge solutions, we are led to consider more general classes of intersecting p-brane solutions, again depending on the compactification coordinates, which turn out to be described by interacting functions rather than independent harmonic functions. These new solutions also provide a more satisfactory interpretation for the lower-dimensional multi-charge p-branes, which otherwise appear to be nothing more than the improbable coincidence of charge-centres of individual constituents with zero binding energy. 
  We consider the canonical symplectic form for sine-Gordon evaluated explicitly on the solitons of the model. The integral over space in the form, which arises because the canonical argument uses the Lagrangian density, is done explicitly in terms of functions arising in the group doublecrossproduct formulation of the inverse scattering procedure, and we are left with a simple expression given by two boundary terms. The expression is then evaluated explicitly in terms of the changes in the positions and momenta of the solitons, and we find agreement with a result of Babelon and Bernard who have evaluated the form using a different argument, where it is diagonal in terms of `in' or `out' co-ordinates. Using the result, we also investigate the higher conserved charges within the inverse scattering framework, check that they Poisson commute and evaluate them on the soliton solutions. 
  We review the canonical quantization of continuum Yang-Mills theory, and derive the continuum Coulomb-gauge Hamiltonian by a simplification of the Christ-Lee method. We then analogously derive, by a simple and elementary method, the lattice Coulomb-gauge Hamiltonian in the minimal Coulomb gauge (and in other Coulomb gauges) from the known Kogut-Susskind Hamiltonian. 
  Extremal six-dimensional black string solutions with some non-trivial momentum distribution along the wave are considered. These solutions were recently shown to contain a singularity at the would-be position of the event horizon. In the black string geometry, all curvature invariants are finite at the horizon. It is shown that if the effects of infalling matter are included, there are curvature invariants which diverge there. This implies that quantum corrections will be important at the would-be horizon. The effect of this singularity on test strings is also considered, and it is shown that it leads to a divergent excitation of the string. The quantum corrections will therefore be important for test objects. 
  On the basis of the first principles we derive the Barut-Wilson-Fushchich second-order equation in the (1/2,0)+(0,1/2) representation. Then we discuss the possibility of the description of various mass and spin states in such a framework. 
  The aim of this talk is to give a brief introduction to the problem of confinement in QCD and to N=2 globally supersymmetric Yang-Mills gauge theories (SYM). While avoiding technicalities as much as possible I will try to emphasize the physical ideas which lie behind the picture of confinement as a consequence of the vacua of QCD to be a dual superconductor. Finally I review the implementation of this picture in the framework of N=2 SYM. 
  It is shown that the beta functions for four dimensional N=2 supersymmetric Yang-Mills theory without matter give integral curves on the moduli space some of which are geodesics of the natural metric on the moduli space. In particular the flow lines which cross-over from from the weak coupling limit (asymptotically free theory) to the singular points, representing the strong coupling limit, are geodesics. A possible connection with irreversibility is discussed. 
  Using a regularization with the properties of dimensional regularization, higher order local consistency conditions on one loop anomalies and divergent counterterms are given. They are derived without any a priori assumption on the form of the BRST cohomology and can be summarized by the statements that (i) the antibracket involving the first order divergent counterterms, respectively the first order anomaly, with any BRST cocycle is BRST exact, (ii) the first order divergent counterterms can be completed into a local deformation of the solution of the master equation and (iii) the first order anomaly can be deformed into a local cocycle of the deformed solution. 
  We prove that a transformation, conjectured in our previous work, between phase-space variables in $\s$-models related by Poisson-Lie T-duality is indeed a canonical one. We do so by explicitly demonstrating the invariance of the classical Poisson brackets. This is the first example of a class of $\s$-models with no isometries related by canonical transformations. In addition we discuss generating functionals of canonical transformations in generally non-isometric, bosonic and supersymmetric $\s$-models and derive the complete set of conditions that determine them. We apply this general formalism to find the generating functional for Poisson-Lie T-duality. We also comment on the relevance of this work to D-brane physics and to quantum aspects of T-duality. 
  We find the general hyper-elliptic solutions to the two-component reduced Nahm equations proposed by Hitchin et al. Elliptic solutions are a special case and can appear only for specific values of the monopole charges. 
  Higher dimensional generalisations of self-duality conditions and of theta angle terms are analysed in Yang-Mills theories. For the theory on a torus, the torus metric and various antisymmetric tensors are viewed as coupling constants related by U-duality, arising from background expectation values of supergravity fields for D-brane or matrix theories. At certain special points in the moduli space of coupling constants certain branes or instantons are found to dominate the functional integral. The possibility of lifting chiral or supersymmetric theories to higher dimensions is discussed. 
  We compute in superspace the one-loop beta-function for the nonlinear sigma-model defined in terms of the nonminimal scalar multiplet. The recently proposed quantization of this complex linear superfield, viewed as the field strength of an unconstrained gauge spinor superfield, allows to handle efficiently the infinite tower of ghosts via the Batalin-Vilkovisky formalism. We find that the classical duality of the nonminimal scalar and chiral multiplets is maintained at the quantum one-loop level. 
  The S-matrix in the static limit of a dispersion relation is a matrix of a finite order N of meromorphic functions of energy $\omega$ in the plane with cuts $(-\infty,-1],[+1,+\infty)$. In the elastic case it reduces to N functions $S_{i}(\omega)$ connected by the crossing symmetry matrix A. The scattering of a neutral pseodoscalar meson with an arbitrary angular momentum l at a source with spin 1/2 is considered (N=2). The Regge trajectories of this model are explicitly found. 
  We present a novel formulation of the instanton equations in 8-dimensional Yang-Mills theory. This formulation reveals these equations as the last member of a series of gauge-theoretical equations associated with the real division algebras, including flatness in dimension 2 and (anti-)self-duality in 4. Using this formulation we prove that (in flat space) these equations can be understood in terms of moment maps on the space of connections and the moduli space of solutions is obtained via a generalised symplectic quotient: a Kaehler quotient in dimension 2, a hyperkaehler quotient in dimension 4 and an octonionic Kaehler quotient in dimension 8. One can extend these equations to curved space: whereas the 2-dimensional equations make sense on any surface, and the 4-dimensional equations make sense on an arbitrary oriented manifold, the 8-dimensional equations only make sense for manifolds whose holonomy is contained in Spin(7). The interpretation of the equations in terms of moment maps further constraints the manifolds: the surface must be orientable, the 4-manifold must be hyperkaehler and the 8-manifold must be flat. 
  In this paper we derive the generalisations of Gauss-Codazzi, Raychaudhuri and area change equations for classical relativistic branes and multidimensional fluids in arbitrary background manifolds with metricity and torsion. The kinematical description we develop is fully covariant and based on the use of projection tensors tilted with respect to the brane worldsheets. 
  We review the resolvent technique for computing the effective action in planar QED. For static magnetic backgrounds the effective action yields (minus) the effective energy of the fermions, while for electric backgrounds the imaginary part of the effective action gives (half) the probability of fermion-antifermion pair creation. For some special `solitonic' background profiles, these effective actions can be computed exactly. 
  The one-loop partition function for a charged self-interacting Bose gas at finite temperature in D-dimensional spacetime is evaluated within a path integral approach making use of zeta-function regularization. For D even, a new additional vacuum term ---overlooked in all previous treatments and coming from the multiplicative anomaly related to functional determinants-- is found and its dependence on the mass and chemical potential is obtained. The presence of the new term is shown to be crucial for having the factorization invariance of the regularized partition function. In the non interacting case, the relativistic Bose-Einstein condensation is revisited. By means of a suitable charge renormalization, for D=4 the symmetry breaking phase is shown to be unaffected by the new term, which, however, gives actually rise to a non vanishing new contribution in the unbroken phase. 
  Recently derived results for the exact induced parity-breaking term in 2+1 dimensions at finite temperature are shown to be relevant to the determination of the free energy for fixed-charge ensembles. The partition functions for fixed total charge corresponding to massive fermions in the presence of Abelian and non-Abelian magnetic fields are discussed. We show that the presence of the induced Chern-Simons term manifests itself in that the free energy depends strongly on the relation between the external magnetic flux and the value of the fixed charge. 
  The linear $\delta$ expansion (LDE) is applied to the Hamiltonian $H=(p^2 +m^2 x^2)/2 + igx^3$, which arises in the study of Lee-Yang zeros in statistical mechanics. Despite being non-Hermitian, this Hamiltonian appears to possess a real, positive spectrum. In the LDE, as in perturbation theory, the eigenvalues are naturally real, so a proof of this property devolves on the convergence of the expansion. A proof of convergence of a modified version of the LDE is provided for the $ix^3$ potential in zero dimensions. The methods developed in zero dimensions are then extended to quantum mechanics, where we provide numerical evidence for convergence. 
  We consider scattering processes in the matrix model with three incoming and three outgoing gravitons. We find a discrepancy between the amplitude calculated from the matrix model and the supergravity prediction. Possible sources for this discrepancy are discussed. 
  Light-front coordinates offer a scenario in which a constituent approximation of hadron structure can emerge from QCD. This requires cutoffs that violate Lorentz covariance and gauge invariance, and a new renormalization group formalism based on a similarity transformation is used with coupling coherence to fix counterterms that restore these symmetries. The counterterms contain functions of longitudinal momentum fractions which severely complicate renormalization, but they also offer possible resolutions of apparent contradictions between the constituent picture and QCD. The similarity transformation and coupling coherence are applied to QED; and it is shown that the resultant Hamiltonian leads to the correct bound state results. The same techniques are applied to QCD and it is shown that a simple confinement mechanism and a reasonable description of heavy quark bound states emerge naturally. 
  We analyse the structure of the perturbative series expansion of Chern-Simons gauge theory in the light-cone gauge. After introducing a regularization prescription that entails the consideration of framed knots, we present the general form of the vacuum expectation value of a Wilson loop. The resulting expression turns out to give the same framing dependence as the one obtained using non-perturbative methods and perturbative methods in covariant gauges. It also contains the Kontsevich integral for Vassiliev invariants of framed knots. 
  We present a simple construction of the instantonic type equation over octonions where its similarities and differences with the quaternionic case are very clear. We use the unified language of Clifford Algebra. We argue that our approach is the pure algebraic formulation of the geometric based soft Lie algebra. The topological criteria for the stability of our solution is given explicitly to establish its solitonic property. Many beautiful features of the parallelizable ring division spheres and Absolute Parallelism (AP) reveal their presence in our formulation. 
  Recently Sen and Seiberg gave a prescription for constructing the matrix theory in any superstring background. We use their prescription to test the finite N matrix theory conjecture on an ALE space. Based on our earlier work with Shenker, we find a sharper discrepancy between matrix theory computation and supergravity prediction. We discuss subtleties in the light-front quantization which may lead to a resolution to the discrepancy. 
  We describe mirror symmetry in N=2 superconformal field theories in terms of a dynamical topology changing process of the principal fiber bundle associated with a topological membrane. We show that the topological symmetries of Calabi-Yau sigma-models can be obtained from discrete geometric transformations of compact Chern-Simons gauge theory coupled to charged matter fields. We demonstrate that the appearence of magnetic monopole-instantons, which interpolate between topologically inequivalent vacua of the gauge theory, implies that the discrete symmetry group of the worldsheet theory is realized kinematically in three dimensions as the magnetic flux symmetry group. From this we construct the mirror map and show that it corresponds to the interchange of topologically non-trivial matter field and gauge degrees of freedom. We also apply the mirror transformation to the mean field theory of the quantum Hall effect. We show that it maps the Jain hierarchy into a new hierarchy of states in which the lowest composite fermions have the same filling fractions. 
  Locally supersymmetric systems in odd dimensions whose Lagrangians are Chern-Simons forms for supersymmetric extensions of anti-de Sitter gravity are discussed. The construction is illustrated for D=7 and 11. In seven dimensions the theory is an N=2 supergravity whose fields are the vielbein ($e_{\mu}^{a}$), the spin connection ($\omega_{\mu}^{ab}$), two gravitini ($\psi_{\mu}^{i}$) and an $sp(2)$ gauge connection ($a_{\mu j}^{i}$). These fields form a connection for $osp(2|8)$. In eleven dimensions the theory is an N=1 supergravity containing, apart from $e_{\mu}^{a}$ and $\omega_{\mu}^{ab}$, one gravitino $\psi_{\mu}$, and a totally antisymmetric fifth rank Lorentz tensor one-form, $b_{\mu}^{abcde}$. These fields form a connection for $osp(32|1)$. The actions are by construction invariant under local supersymmetry and the algebra closes off shell without requiring auxiliary fields. The $N=2^{[D/2]}$-theory can be shown to have nonnegative energy around an AdS background, which is a classical solution that saturates the Bogomolnyi bound obtained from the superalgebra. 
  Solutions of Born-Infeld theory, representing strings extending from a Dirichlet p-brane, are also solutions of the higher derivative generalization of the Born-Infeld equations defining an exact open string vacuum configuration. 
  The dyonic quantum states of magnetic monopoles in Yang-Mills-Higgs theory with a non-abelian unbroken gauge group display a subtle interplay between magnetic and electric properties. This is described in detail in the theory with the gauge group SU(3) broken to U(2), and shown to be captured by the representation theory of the semi-direct product of U(2) with R^4. The implications of this observation for the fusion rules and electric-magnetic duality properties of dyonic states are pointed out. 
  The combined effect of the magnetic field background in the form of a singular vortex and the Dirichlet boundary condition at the location of the vortex on the vacuum of quantized scalar field is studied. We find the induced vacuum energy density and current to be periodic functions of the vortex flux and holomorphic functions of the space dimension. 
  We give an extension of Casimir of Casimir $\cal{WA_N}$ algebras including a vertex operator which depends on non-simple roots of $A_{N-1}$. 
  Charged BPS hypermultiplets can develop a non-trivial self-interaction in the Coulomb branch of an N=2 supersymmetric gauge theory, whereas neutral BPS hypermultiplets in the Higgs branch may also have a non-trivial self-interaction in the presence of Fayet-Iliopoulos terms. The exact hypermultiplet low-energy effective action (LEEA) takes the form of the non-linear sigma-model (NLSM) with a hyper-K"ahler metric. A non-trivial scalar potential is also quantum-mechanically generated at non-vanishing central charges, either perturbatively (Coulomb branch), or non-perturbatively (Higgs branch). We calculate the effective scalar potentials for (i) a single charged hypermultiplet in the Coulomb branch and (ii) a single neutral hypermultiplet in the Higgs branch. The first case corresponds to the NLSM with the Taub-NUT (or KK-monopole) metric for the kinetic LEEA, whereas the second one is attached to the NLSM having the Eguchi-Hanson instanton metric. 
  Clifford geometric algebras of multivectors are treated in detail. These algebras are build over a graded space and exhibit a grading or multivector structure. The careful study of the endomorphisms of this space makes it clear, that opposite Clifford algebras have to be used also. Based on this mathematics, we give a fully Clifford algebraic account on generating functionals, which is thereby geometric. The field operators are shown to be Clifford and opposite Clifford maps. This picture relying on geometry does not need positivity in principle. Furthermore, we propose a transition from operator dynamics to corresponding generating functionals, which is based on the algebraic techniques. As a calculational benefit, this transition is considerable short compared to standard ones. The transition is not injective (unique) and depends additionally on the choice of an ordering. We obtain a direct and constructive connection between orderings and the explicit form of the functional Hamiltonian. These orderings depend on the propagator of the theory and thus on the ground state. This is invisible in path integral formulations. The method is demonstrated within two examples, a non-linear spinor field theory and spinor QED. Antisymmetrized and normal-ordered functional equations are derived in both cases. 
  The largely open problem of scaling laws in fully developed turbulence is discussed, with the stress put on similarities and differences with scaling in field theory. A soluble model of the passive advection is examined in more detail in order to illustrate the principal ideas. 
  Within two specific string cosmology scenarios --differing in the way the pre- and post-big bang phases are joined-- we compute the size and spectral slope of various types of cosmologically amplified quantum fluctuations that arise in generic compactifications of heterotic string theory. By further imposing that these perturbations become the dominant source of energy at the onset of the radiation era, we obtain physical bounds on the background's moduli, and discuss the conditions under which both a (quasi-) scale-invariant spectrum of axionic perturbations and sufficiently large seeds for the galactic magnetic fields are generated. We also point out a potential problem with achieving the exit to the radiation era when the string coupling is near its present value. 
  In this report we obtain explicit string-like solutions of equations of motion of massive heterotic supergravity recently obtained by Bergshoeff, Roo and Eyras. We also find consistent string source which can be embedded in these backgrounds when space-time dimension is greater than or equal to six. 
  The mechanism of a confining medium is investigated within the Nambu-Jona-Lasinio (NJL) approach. It is shown that a confining medium can be realized in the bosonized phase of the NJL model due to vacuum fluctuations of both fermion and Higgs (scalar fermion-antifermion collective excitation) fields. In such an approach there is no need to introduce Dirac strings. 
  A Lorentz covariant quantization of membrane dynamics is defined, which also leaves unbroken the full three dimensional diffeomorphism invariance of the membrane. Among the applications studied are the reduction to string theory, which may be understood in terms of the phase space and constraints, and the interpretation of physical,zero-energy states. A matrix regularization is defined as in the light cone gauged fixed theory but there are difficulties implementing all the gauge symmetries. The problem involves the non-area-preserving diffeomorphisms which are realized non-linearly in the classical theory. In the quantum theory they do not seem to have a consistent implementation for finite N. Finally, an approach to a genuinely background independent formulation of matrix dynamics is briefly described. 
  M(atrix) theory description is investigated for M-theory compactified on non-orientable manifolds. Relevant M(atrix) theory is obtained by Fourier transformation in a way consistent with T-duality. For nine-dimensional compactification on Klein bottle and M\"obius strip, we show that M(atrix) theory is (2+1)-dimensional ${\cal N} = 8$ supersymmetric U(N) gauge theory defined on dual Klein bottle and dual Moebius strip parameter space respectively. The latter requires a twisted sector consisting of sixteen chiral fermions localized parallel to the boundary of dual Moebius strip and defines Narain moduli space of Chaudhuri-Hockney-Lykken heterotic string. For six-dimensional CHL compactification on S1/Z2*T4 we show that low-energy dynamics of M(atrix) theory is described by (5+1)-dimensional ${\cal N} = 8$ supersymmetric U(N)$\times$U(N) gauge theory defined on dual orbifold parameter space of S1*K3/Z2. Spacetime spectrum is deduced from BPS gauge field configurations consistent with respective involutions and is shown to agree with results from M-theory analysis. 
  The three-dimensional Gross-Neveu model in $R^{1} \times S^{2}$ spacetime is considered at finite particles number density. We evaluate an effective potential of the composite scalar field $\sigma(x)$, which is expressed in terms of a scalar curvature $R$ and nonzero chemical potential $\mu$. We then derive the critical values of $(R,\mu)$ at which the system undergoes the first order phase transition from the phase with broken chiral invariance to the symmetric phase. 
  The next-to-leading term in the weight factor of the string representation of the 't Hooft loop average defined on the string world-sheet is found in the Abelian Higgs Model near the Londons' limit. This term emerges due to the finiteness of the coupling constant and, in contrast to the Londons' limit, where only the bilocal cumulant in the expansion of the 't Hooft average survived, leads to the appearance of the quartic cumulant. Apart from the Londons' penetration depth of the vacuum, which was a typical fall-off scale of the bilocal cumulant, the quartic cumulant depends also on the other characteristic length of the Abelian Higgs Model, the correlation radius of the Higgs field. 
  The perturbiner approach to the multi-gluonic amplitudes in Yang-Mills theory is reviewed. 
  We present here an explicit self-dual classical solution of the type of perturbiner in gravity. This solution is a generating function for tree gravitational form-factors with all on-shell gravitons in the same helicity state. 
  An explicit self-dual classical solution of the type of perturbiner in Yang-Mills theory interacting with gravity is obtained. This allows one to describe the tree self-dual gluonic form-factors including any number of positive-helicity gravitons in addition to the positive helicity gluons. 
  This is a summary of a review talk at the Trieste conference on duality symmetries in string theory, April 1997. We review some work that has been done on the relation of BPS states, automorphic forms and the geometry of the quantum ground states of string compactifications with extended supersymmetry, emphasizing the (hypothetically) central role of BPS algebras. 
  The soldering mechanism has been shown to represent the quantum interference effect between self and anti-self dual aspects of a given symmetry. This mechanism was used to show that the massive mode of the 2D Schwinger model results from the constructive interference between the right and the left massless modes of chiral Schwinger models. Similarly, the topologically massive modes resulting from the bosonization of 3D massive Thirring models of opposite mass signatures, are fused into the two massive modes of the 3D Proca model, thanks to the interference of dualities characteristic of the soldering mechanism. In this work, we show that the field theoretical analog of destructive quantum mechanical interference may also be represented by the soldering mechanism. This phenomenon is illustrated by the fusion of two (diffeomorphism) invariant self-dual scalars described by right and left chiral-WZW actions, producing a Hull non-mover field. After fusion, right and left moving modes disappear from the spectrum, displaying the claimed (destructive) interference of dualities. 
  A new model for anyon is proposed, which exhibits the classical analogue of the quantum phenomenon - Zitterbewegung. The model is derived from existing spinning particle model and retains the essential features of anyon in the non-relativistic limit. 
  In this letter we study $D$ particle quantum mechanics on a torus in the limit that one or more cycles of the torus have a zero length. 
  By imposing on the most general renormalizable quantum field theory the requirement of the absence of ultraviolet-divergent renormalizations of the physical parameters (masses and coupling constants) of the theory, finite quantum field theories in four space-time dimensions may be constructed. Famous prototypes of these form certain well-known classes of supersymmetric finite quantum field theories. Within a perturbative evaluation of the quantum field theories under consideration, the starting point of all such investigations is represented by the conditions for one- and two-loop finiteness of the gauge couplings as well as for one-loop finiteness of the Yukawa couplings. Particularly attractive solutions of the one-loop Yukawa finiteness condition involve Yukawa couplings which are equivalent to generators of Clifford algebras with identity element. However, a closer inspection shows, at least for all simple gauge groups up to and including rank 8, that these Clifford-like solutions prove to be inconsistent with the requirements of one- and two-loop finiteness of the gauge coupling and of absence of gauge anomalies. 
  An investigation of the Nambu-Jona-Lasino model with external constant electric and weak gravitational fields is carried out in three- and four- dimensional spacetimes. The effective potential of the composite bifermionic fields is calculated keeping terms linear in the curvature, while the electric field effect is treated exactly by means of the proper- time formalism.   A rich dynamical symmetry breaking pattern, accompanied by phase transitions which are ruled, independently, by both the curvature and the electric field strength is found. Numerical simulations of the transitions are presented. 
  The proposal of Dijkgraaf, Verlinde and Verlinde for the emergence of smooth strings from a supersymmetric U(N) Yang-Mills theory assumes that conjugacy classes in the symmetric group with a few long cycles dominate the dynamics about the infrared fixed point. It is shown that the average number of cycles in a conjugacy class of $S_{N}$ is bounded below by $const.\sqrt N\ln\sqrt N,$ implying that some physical mechanism is needed to ensure the assumed dominance. It is shown that if individual cycles have positive energies that depend very weakly on their lengths, then long cycles dominate the dynamics at low temperatures. 
  The monopole confinement mechanism in the abelian projection of lattice gluodynamics is reviewed. The main topics are: the abelian projection on the lattice and in the continuum, a numerical study of the abelian monopoles in the lattice gauge theory. Additionally, we briefly review the notation of differential forms, duality, and the BKT transformation in the lattice gauge theories. 
  When a D-brane wraps around a cycle of a curved manifold, the twisting of its normal bundle can induce chiral asymmetry in its worldvolume theory. We obtain the general form of the resulting anomalies for D-branes and their intersections. They are not cancelled among themselves, and the standard inflow mechanism does not apply at first sight because of their apparent lack of factorizability and the apparent vanishing of the corresponding inflow. We show however after taking into consideration the effects of the nontrivial topology of the normal bundles, the anomalies can be transformed into factorized forms and precisely cancelled by finite inflow from the Chern-Simons actions for the D-branes as long as the latter are well defined. We then consider examples in type II compactifications where the twisting of the normal bundles occurs and calculate the changes in the induced Ramond-Ramond charges on the D-branes. 
  Multidimensional cosmological, static spherically symmetric and Euclidean configurations are described in a unified way for gravity interacting with several dilatonic fields and antisymmetric forms, associated with electric and magnetic p-branes. Exact solutions are obtained when certain vectors, built from the input parameters of the model, are either orthogonal in the minisuperspace, or form mutually orthogonal subsystems. Some properties of black-hole solutions are indicated, in particular, a no-hair-type theorem and restrictions emerging in models with multiple times. From the non-existence of Lorentzian wormholes, a universal restriction is obtained, applicable to orthogonal or block-orthogonal subsystems of any p-brane systems. Euclidean wormhole solutions are found, their actions and radii are explicitly calculated. 
  The low-energy D=4, N=1 effective action of the strongly coupled heterotic string is explicitly computed by compactifying Horava-Witten theory on the deformed Calabi-Yau three-fold solution due to Witten. It is shown that, to order kappa^{2/3}, the Kahler potential is identical to that of the weakly coupled theory. Furthermore, the gauge kinetic functions are directly computed to order kappa^{4/3} and shown to receive a non-vanishing correction. Also, we compute gauge matter terms in the Kahler potential to the order kappa^{4/3} and find a nontrivial correction to the dilaton term. Part of those corrections arise from background fields that depend on the orbifold coordinate and are excited by four-dimensional gauge field source terms. 
  A particle in quantum mechanics on manifolds couples to the induced topological gauge field that characterises the possible inequivalent quantizations. For instance, the gauge potential induced on $S^2$ is that of a magnetic monopole located at the center of $S^2$. We find that the gauge potential induced on $S^3$ ($S^{2n+1}$) is that of a meron (generalized meron) also sitting at the center of $S^3$ ($S^{2n+1}$). 
  Dimensional reduction of various gravity and supergravity models leads to effectively two-dimensional field theories described by gravity coupled G/H coset space sigma-models. The transition matrices of the associated linear system provide a complete set of conserved charges. Their Poisson algebra is a semi-classical Yangian double modified by a twist which is a remnant of the underlying coset structure. The classical Geroch group is generated by the Lie-Poisson action of these charges. Canonical quantization of the structure leads to a twisted Yangian double with fixed central extension at a critical level. 
  Recently it has been shown by Cho and Kimm that the gauged $CP^1$ model, obtained by gauging the global SU(2) group of $CP^1$ model and adding a corresponding Chern-Simons term, has got its own soliton. These solitons are somewhat distinct from those of pure $CP^1$ model, as they cannot always be characterised by $\pi_2(CP^1)=Z$. In this paper, we first carry out the Hamiltonian analysis of this gauged $CP^1$ model. Then we couple the Hopf term, associated to these solitons and again carry out its Hamiltonian analysis. The symplectic structures, along with the structures of the constraints, of these two models (with or without Hopf term) are found to be essentially the same. The model with Hopf term, is then shown to have fractional spin, which however depends not only on the soliton number $N$ but also on the nonabelian charge. 
  We discuss the coupling of vector-tensor multiplets to N=2 supergravity. 
  An "expanded" description is introduced to examine the spinor-monopole identification proposed by Strassler for four-dimensional $\cal N$ = 1 supersymmetric Spin(10) gauge theories with matter in F vector and N spinor representations. It is shown that a Z_2 monopole in the "expanded" theory is associated with massive spinors of the Spin(10) theory. For N=2, two spinor case, we confirm this identification by matching the transformation properties of the two theories under SU(2) flavor symmetry. However, for N $\ge$ 3, the transformation properties are not matched between the spinors and the monopole. This disagreement might be due to the fact that the SU(N) flavor symmetry of the Spin(10) theory is partially realized as an SU(2) symmetry in the "expanded" theory. 
  We show that the D=6 SU(2) gauged supergravity of van Nieuwenhuizen et al, obtained by dimensional reduction of the D=7 topologically massive gauged supergravity and previously thought not to be dimensionally reducible, can be further reduced to five and four dimensions. On reduction to D=4 one recovers the special case of the SU(2)XSU(2) gauged supergravity of Freedman and Schwarz for which one of the SU(2) coupling constants vanishes. Previously known supersymmetric electrovacs of this model then imply new ground states in 7-D. We construct a supersymmetric electrovac solution of N=2 SU(2) gauged supergravity in 7-D. We also investigate the domain wall solutions of these theories and show they preserve a half of the supersymmetry. 
  Motivated by manifest Lorentz symmetry and a well-defined large-N limit prescription, we study the supersymmetric quantum mechanics proposed as a model for the collective dynamics of D0-branes from the point of view of the 11-dimensional supermembrane. We argue that the continuity of the spectrum persists irrespective of the presence of winding around compact target-space directions and discuss the central charges in the superalgebra arising from winding membrane configurations. Along the way we comment on the structure of open supermembranes. 
  The classical electromagnetic field of a spinless point electron is described in a formalism with extended causality by discrete finite transverse point-vector fields with discrete and localized point interactions. These fields are taken as a classical representation of photons, ``classical photons". They are all transversal photons; there are no scalar nor longitudinal photons as these are definitely eliminated by the gauge condition. The angular distribution of emitted photons coincides with the directions of maximum emission in the standard formalism. The Maxwell formalism and its standard field are retrieved by the replacement of these discrete fields by their space-time averages, and in this process scalar and longitudinal photons are necessarily created and added. Divergences and singularities are by-products of this averaging process. This formalism enlighten the meaning and the origin of the non-physical photons, the ones that violate the Lorentz condition in manifestly covariant quantization methods. 
  The recent suggestion that the entropy of Schwarzschild black holes can be computed in matrix theory using near-extremal D-brane thermodynamics is examined. It is found that the regime in which this approach is valid actually describes black strings stretched across the longitudinal direction, near the transition where black strings become unstable to the formation of black holes. It is argued that the appropriate dynamics on the other (black hole) side of the transition is that of the zero modes of the corresponding super Yang-Mills theory. A suggestive mean field theory argument is given for the entropy of black holes in all dimensions. Consequences of the analysis for matrix theory and the holographic principle are discussed. 
  The potential between two D4-branes at angles with partially unbroken supersymmetry is computed, and is used to discuss the creation of a fundamental string when two such D4-branes cross each other in M(atrix) theory. The effective Lagrangian is shown to be anomalous at 1-loop approximation, but it can be modified by bare Chern-Simons terms to preserve the invariance under the large gauge transformation. The resulting effective potential agrees with that obtained from the string calculations. The result shows that a fundamental string is created in order to cancel the repulsive force between two D4-branes at proper angles. 
  The scaling limit used recently to derive matrix models, and a certain analyticity assumption, are invoked to argue that the agreement between some matrix model calculations and supergravity is a consequence of string world sheet duality. 
  We propose a new, transport-theoretic (tt) class of relativistic extensions of quantum field theories of fundamental interactions. Its concepts are inspired by Feynman's atomistic idea about the physical world and by the extension of fluid dynamics to shorter distances through the Boltzmann transport equation. The extending tt Lagrangians imply the original Lagrangians as path-integralwise approximations. By constructing a tt Lagrangian that extends a general gauge-invariant Lagrangian, we show that a tt extension of the standard model is feasible. We define a tt Lagrangian in terms of tt fields of the spacetime variable and an additional, four-vector variable. We explain the fields of quantum field theories as certain covariant, local averages of tt fields. Only two tt fields may be needed for modeling fundamental interactions: (i) a four-vector one unifying all fundamental forces, and (ii) a two-component-spinor one unifying all fundamental matter particles. We comment on the new physics expected within the tt framework put forward, and point out some open questions. 
  We calculate the one loop corrections to the Chern-Simons coefficient $\kappa$ in the Higgs phase of Yang-Mills Chern-Simons Higgs theories. When the gauge group is SU(N), we show, by taking into account the effect of the would be Chern-Simons term, that the corrections are always integer multiples of ${1\over 4\pi}$, as they should for the theories to be quantum-mechanically consistent. In particular, the correction is vanishing for SU(2). The same method can also be applied to the case that the gauge group is SO(N). The result for SO(2) agrees with that found in the abelian Chern-Simons theories. Therefore, the calculation provides with us a unified understanding of the quantum correction to the Chern-Simons coefficient. 
  We study N=2 supersymmetric Abelian gauge model with the Fayet-Iliopoulos term and an arbitrary number of chiral matter multiplets in two dimensions. By analyzing the instanton contribution we compute the non-perturbative corrections to the mass spectrum of the theory and the quantum deformation of the classical vacua space. In contrast to known examples the non-perturbative bosonic potential is saturated by the one-instanton contribution and can be directly found within the semiclassical expansion around the one-instanton saddle point. 
  Covariant field equations of M-fivebrane in eleven dimensional curved superspace are obtained from the requirement of kappa-symmetry of an open supermembrane ending on a fivebrane. The worldvolume of the latter is a (6|16) dimensional supermanifold embedded in the (11|32) dimensional target superspace. The kappa-symmetry of the system imposes a constraint on this embedding, and a constraint on a modified super 3-form field strength on the fivebrane worldvolume. These constraints govern the dynamics of the M-fivebrane. 
  We study some reparametrization invariant theories in context of the BRST-co-BRST quantization method. The method imposes restrictions on the possible gauge fixing conditions and leads to well defined inner product states through a gauge regularisation procedure. Two explicit examples are also treated in detail. 
  In Causal Perturbation Theory the process of renormalization is precisely equivalent to the extension of time ordered distributions to coincident points. This is achieved by a modified Taylor subtraction on the corresponding test functions. I show that the pullback of this operation to the distributions yields expressions known from Differential Renormalization. The subtraction is equivalent to BPHZ subtraction in momentum space. Some examples from Euclidean scalar field theory in flat and curved spacetime will be presented. 
  Based on a gas picture of D0-brane partons, it is shown that the entropy, as well as the geometric size of an infinitely boosted Schwarzschild black hole, can be accounted for in matrix theory by interactions involving spins, or interactions involving more than two bodies simultaneously. 
  We study the generating functional for the massless Schwinger model in the torus, at nonzero chemical potential and temperature. The lack of hermiticity of the Dirac operator yields a non-trivial phase in the effective action, which is a topological contribution induced by the chemical potential. In the sector with no zero modes, we evaluate exactly the generating functional, the partition function, the boson propagator and the thermally averaged Polyakov loop. The system bosonizes at finite T and \mu, with the same mass as in vacuum. From the solution obtained for the Schwinger model we derive also exactly the generating functional and the partition function for the massless Thirring model at nonzero T and \mu. 
  The Bekenstein-Hawking entropy of BPS black holes can be obtained as the minimum of the mass (= largest central charge). In this letter we investigate the analog procedure for the matrix model of M-theory. Especially we discuss the configurations: (i) $2 \times 2 \times 2$ corresponding to the 5-d black hole and (ii) the $5 \times 5 \times 5$ configuration yielding the 5-d string. After getting their matrix-entropy, we discuss a way of counting of microstates in matrix theory. As Yang Mills field theory we propose the gauged world volume theory of the 11-d KK monopole. 
  We obtain the parity invariant generating functional for the canonical transformation mapping the Liouville theory into a free scalar field and explain how it is related to the pseudoscalar transformation 
  We discuss the relation between the Ramond-Ramond charges of D-branes and the topology of Chan-Paton vector bundles. We show that a topologically nontrivial normal bundle induces RR charge and that the result fits in perfectly with the proposal that D-brane charge is the topology of the Chan-Paton bundle, regarded as an element of K-theory. 
  This is an expanded version of talks given by the author at the Trieste Spring School on Supergravity and Superstrings in April of 1997 and at the accompanying workshop. The manuscript is intended to be a mini-review of Matrix Theory. The motivations and some of the evidence for the theory are presented, as well as a clear statement of the current puzzles about compactification to low dimensions. 
  The quantum mechanics of an N=1 supersymmetric dynamical system constrained to a hypersurface embedded in the higher dimensional Euclidean space is investigated by using the projection-operator method (POM) of constrained systems. It is shown that the Hamiltonian obtained by the successive operations of projection operators contains the additional terms, which are completely missed when imposing constraints before the quantization. We derive the conditions the additional terms should satisfy when the N=1 supersymmetry holds in the resulting system, and present the geometrical interpretations of these additional terms. 
  The method of flow equations is applied to QED on the light front. Requiring that the partical number conserving terms in the Hamiltonian are considered to be diagonal and the other terms off-diagonal an effective Hamiltonian is obtained which reduces the positronium problem to a two-particle problem, since the particle number violating contributions are eliminated. No infrared divergencies appear. The ultraviolet renormalization can be performed simultaneously. 
  The recently proposed constructive approach to nonperturbative QFT, based on modular localization, is reviewed and extended. It allows to unify black holes physics and H-temperatures (H standing for Hawking or Horizon) with the bootstrap-formfactor program for nonperturbative construction of low dimensional QFT. In case of on-shell particle number conservation, the equations characterizing the modular localization spaces for wedges are Bethe-Ansatz equation in the form as recently obtained in the treatment of factorizable models. 
  The properties of the ground state wavefunctional in gluodynamics responsible for confinement are considered. It is shown that confinement arises due to the generation of a mass gap in the averaging over the gauge group which is necessary to ensure the gauge invariance of the ground state. The string tension is calculated in assumption of a particular infrared behavior of the vacuum wavefunctional. 
  We study peculiarities of realization of N=2 supersymmetry in N=2 abelian gauge theory with two sorts of FI terms, electric and magnetic ones, within manifestly supersymmetric formulations via the Mezincescu and harmonic-analytic prepotentials. We obtain a `magnetic', duality- transformed superfield form of the N=2 Maxwell effective holomorphic action with standard electric $FI$ term and demonstrate that in such a system off-shell N=2 supersymmetry is inevitably realized in an unusual Goldstone mode corresponding to the {\it partial} spontaneous breaking down to N=1. On shell, the standard total breaking occurs. In a system with the two sorts of FI terms, off-shell N=2 supersymmetry is realized in the partial breaking mode both in the electric and magnetic representations. This regime is retained on shell due to the Antoniadis- Partouche-Taylor mechanism. We show that the off-shell algebra of N=2 supersymmetry in the partial breaking realization is modified on gauge-variant objects like potentials and prepotentials. The closure of spinor charges involves some special gauge transformations before any gauge-fixing. 
  We consider the Compton amplitude for the scattering of a photon and a (massless) ``electron/positron'' at one loop (i.e. genus one) in a four-dimensional fermionic heterotic string model. Starting from the bosonization of the world-sheet fermions needed to explicitly construct the spin-fields representing the space-time fermions, we present all the steps of the computation which leads to the explicit form of the amplitude as an integral of modular forms over the moduli space. 
  We compute the Schwinger terms in the energy-momentum tensor commutator algebra from the anomalies present in Weyl-invariant and diffeomorphism-invariant effective actions for two dimensional massless scalar fields in a gravitational background. We find that the Schwinger terms are not sensitive to the regularization procedure and that they are independent of the background metric. 
  Five and six dimensional SUSY gauge theories, with one or two compactified directions, are discussed. The 5d theories with the matter hypermultiplets in the fundamental representation are associated with the twisted $XXZ$ spin chain, while the group product case with the bi-fundamental matter corresponds to the higher rank spin chains. The Riemann surfaces for $6d$ theories with fundamental matter and two compact directions are proposed to correspond to the $XYZ$ spin chain based on the Sklyanin algebra. We also discuss the obtained results within the brane and geometrical engeneering frameworks and explain the relation to the toric diagrams. 
  We study the spectrum of 1+1 dimensional large $N$ QCD coupled to an adjoint Majorana fermion of mass $m$. As $m\to 0$ this model makes a transition from confinement to screening. We argue that in this limit the spectrum becomes continuous for mass greater than twice the mass of the lightest bound state. This critical mass is nothing but the threshold for a decay into two lightest states. We present numerical results based on DLCQ that appear to support our claim. 
  The relative entropy in two-dimensional Field Theory is studied for its application as an irreversible quantity under the Renormalization Group, relying on a general monotonicity theorem for that quantity previously established. In the cylinder geometry, interpreted as finite-temperature field theory, one can define from the relative entropy a monotonic quantity similar to Zamolodchikov's c function. On the other hand, the one-dimensional quantum thermodynamic entropy also leads to a monotonic quantity, with different properties. The relation of thermodynamic quantities with the complex components of the stress tensor is also established and hence the entropic c theorems are proposed as analogues of Zamolodchikov's c theorem for the cylinder geometry. 
  We investigate excitations in Matrix Theory on T^2 corresponding to bound states of strings. We demonstrate the Dirichlet aspect of R-R charged vacua through a non-trivial connection between the U(1) and SU(n) sectors of the matrix SYM. 
  We study some geometrical and topological aspects of the generalised dimensional reduction of supergravities in D=11 and D=10 dimensions, which give rise to massive theories in lower dimensions. In these reductions, a global symmetry is used in order to allow some of the fields to have a non-trivial dependence on the compactifying coordinates. Global consistency in the internal space imposes topological restrictions on the parameters of the compactification as well as the structure of the space itself. Examples that we consider include the generalised reduction of the type IIA and type IIB theories on a circle, and also the massive ten-dimensional theory obtained by the generalised reduction of D=11 supergravity. 
  We present the form of the Dirac quantisation condition for the p-form charges carried by p-brane solutions of supergravity theories. This condition agrees precisely with the conditions obtained in lower dimensions, as is necessary for consistency with Kaluza-klein dimensional reduction. These considerations also determine the charge lattice of BPS soliton states, which proves to be a universal modulus-independent lattice when the charges are defined to be the canonical charges corresponding to the quantum supergravity symmetry groups. 
  M(atrix) theory compactified on an orbifold ${\bf T}_9/{\bf Z}_2$ is studied. Via zero-brane parton scattering we find that each of the $2^9 = 512$ orbifold fixed points carry $-1/32$ units of zero-brane charge. The anomalous flux is cancelled by introducing a twisted sector consisting of 32 zero-branes that are spacetime supersymmetry singlets. These twisted sector zero-branes are nothing but gravitational waves propagating along the M-theory direction. There is no D0-partons in the untwisted sector, a fact consistent with holographic principle. For low-energy excitations, the orbifold compactification is described by ten-dimensional supersymmetric Yang-Mills theory with gauge group $SO(32)$. 
  After briefly reviewing basic concepts of perturbative string theory, we explain in simple terms some of the new findings that created excitement among the string physicists. These developments include non-perturbative dualities and a unified picture that embraces the so-far known theories. (An overview for non-experts) 
  We give a detailed discussion of the matching of the BPS states of heterotic, type I and type I' theories in d=9 for general backgrounds. This allows us to explicitly identify these (composite) brane states in the type I' theory that lead to gauge symmetry enhancement at critical points in moduli space. An example is the enhancement of $SO(16)\times SO(16)$ to $E_8\times E_8$. 
  The Seiberg-Witten solution of N=2 supersymmetric SU(2) gauge theories with matter is analysed as an isomonodromy problem. We show that the holomorphic section describing the effective action can be deformed by moving its singularities on the moduli space while keeping their monodromies invariant. Well-known examples of isomonodromic sections are given by the correlators of two-dimensional rational conformal field theories -- the conformal blocks. The Seiberg-Witten section similarly admits the operations of braiding and fusing of its singularities, which obey the Yang-Baxter and Pentagonal identities, respectively. Using them, we easily find the complete expressions of the monodromies with affine term, and the full quantum numbers of the BPS spectrum. While the braiding describes the quark-monopole transmutation, the fusing implies the superconformal points in the moduli space. In the simplest case of three singularities, the supersymmetric sections are directly related to the conformal blocks of the logarithmic minimal models. 
  We improve and extend a method introduced in an earlier paper for deriving string field equations. The idea is to impose conformal invariance on a generalized sigma model, using a background field method that ensures covariance under very general non-local coordinate transformations. The method is used to derive the free string equations, as well as the interacting equations for the graviton-dilaton system. The full interacting string equations derived by this method should be manifestly background independent. 
  Manifestly N=2 supersymmetric Feynman rules are found for different off-shell realizations of the massless hypermultiplet in projective superspace. When we reduce the Feynman rules to an N=1 superspace we obtain the correct component propagators. The Feynman rules are shown to be compatible with a ``duality'' that acts only on the auxiliary fields, as well as with the usual duality relating the hypermultiplet to the tensor multiplet. 
  We compare the $(0,2)$ theory of the single M5 brane decoupled from gravity in the lightcone with transverse $R^4$, and a matrix model description in terms of quantum mechanics on instanton moduli space. We give some tests of the Matrix model in the case of multi fivebranes on $R^4$. We extract constraints on the operator content of the field theory of the multi-fivebrane system by analyzing the Matrix model. We also begin a study of compactifications of the $(0,2)$ theory in this framework, arguing that for large compactification scale the $(0,2)$ theory is described by super-quantum mechanics on appropriate instanton moduli spaces. 
  This article is based on a talk given at the ``Strings'97'' conference. It discusses the search for the universality class of confining strings. The key ingredients include the loop equations, the zigzag symmetry, the non-linear renormalization group. Some new tests for the equivalence between gauge fields and strings are proposed. 
  $D$ dimensional neutral black strings wrapped on a circle are related to $(D-1)$ dimensional charged black holes by boosts. We show that the boost has to be performed in the covering space and the boosted coordinate has to be compactified on a circle with a Lorentz contracted radius. Using this fact we show that the transition between Schwarzschild black holes to black p-branes observed recently in M theory is the well-known black hole- black string transition viewed in a boosted frame. In a similar way the correspondence point where an excited string state goes over to a neutral black hole is mapped exactly to the correspondence point for black p-branes. In terms of the $p$ brane quantities the equation of state for an excited string state becomes identical to that of a 3+1 dimensional massless gas for all $p$. Finally, we show how boosts can be used to relate Hawking radiation rates. Using the known microscopic derivation of absorption by extremal 3-branes and near-extremal 5D holes with three large charges we provide a microscopic derivation of absorption of 0-branes by seven and five dimensional Schwarzschild black holes in a certain regime. 
  New CP1-soliton behaviour on a flat torus is reported. Defined by the Weierstrass elliptic function and numerically-evolved from rest, each soliton splits up in two lumps which eventually reunite, divide and get back together again, etc.. This result invites speculation on the question of fractional topological charge. 
  We present a crude Matrix Theory model for Schwarzchild black holes in uncompactified dimension greater than $5$. The model accounts for the size, entropy, and long range state interactions of black holes. The key feature of the model is a Boltzmann gas of D0 branes, a concept which depends on certain qualitative features of Matrix Theory which have not previously been utilized in studies of black holes. 
  Recent work on the action of T duality on Dirichlet-branes is generalized to the case in which the open string satisfies boundary conditions that are neither Neumann nor Dirichlet. This is achieved by implementing T duality as a canonical transformation of the $\sigma$-model path integral. A class of boundary interactions that violate conformal symmetry is found to be T-dual of a correspondingly non-conformal class of boundary conditions. The analogy with some problems in boundary-non-critical quantum mechanics of interest for condensed matter is pointed out. 
  We propose doubly supersymmetric actions in terms of n=2(D-2) worldline superfields for N=2 superparticles in D=3,4 and Type IIA D=6 superspaces. These actions are obtained by dimensional reduction of superfield actions for N=1 superparticles in D=4,6 and 10, respectively. We show that in all these models geometrodynamical constraints on target superspace coordinates do not put the theory on the mass shell, so the actions constructed consistently describe the dynamics of the corresponding N=2 superparticles. We also find that in contrast to the IIA D=6 superparticle a chiral IIB D=6 superparticle, which is not obtainable by dimensional reduction from N=1, D=10, is described by superfield constraints which produce dynamical equations. This implies that for the IIB D=6 superparticle the doubly supersymmetric action does not exist in the conventional form. 
  We analyze the Ising model on a random surface with a boundary magnetic field using matrix model techniques. We are able to exactly calculate the disk amplitude, boundary magnetization and bulk magnetization in the presence of a boundary field. The results of these calculations can be interpreted in terms of renormalization group flow induced by the boundary operator. In the continuum limit this RG flow corresponds to the flow from non-conformal to conformal boundary conditions which has recently been studied in flat space theories. 
  We derive the fully extended supersymmetry algebra carried by D-branes in a massless type IIA superspace vacuum. We find that the extended algebra contains not only topological charges that probe the presence of compact spacetime dimensions but also pieces that measure non-trivial configurations of the gauge field on the worldvolume of the brane. Furthermore there are terms that measure the coupling of the non-triviality of the worldvolume regarded as a U(1)-bundle of the gauge field to possible compact spacetime dimensions. In particular, the extended algebra carried by the D-2-brane can contain the charge of a Dirac monopole of the gauge field. In the course of this work we derive a set of generalized Gamma-matrix identities that include the ones presently known for the IIA case.   In the first part of the paper we give an introduction to the basic notions of Noether current algebras and charge algebras; furthermore we find a Theorem that describes in a general context how the presence of a gauge field on the worldvolume of an embedded object transforming under the symmetry group on the target space alters the algebra of the Noether charges, which otherwise would be the same as the algebra of the symmetry group. This is a phenomenon recently found by Sorokin and Townsend in the case of the M-5-brane, but here we show that it holds quite generally, and in particular also in the case of D-branes. 
  We generalize the Gibbons-Wiltshire solution of four dimensional Kaluza-Klein black holes in order to describe Type IIA solutions of bound states of D6 and D0-branes. We probe the solutions with a D6-brane and a D0-brane. We also probe a system of D2+D0-branes and of a D2-brane bound to a F1-string with a D2-brane. A precise agreement between the SYM and the SUGRA calculations is found for the static force as well as for the $v^2$ force in all cases. 
  We investigate the effects of thermal fluctuations on the spontaneous magnetic condensate in three dimensional QED coupled with P-odd Dirac fermions. Our results show that the phenomenon of the spontaneous generation of the constant background magnetic field survives to the thermal corrections even at infinite temperature. We also study the thermal corrections to the fermionic condensate in presence of the magnetic field. 
  We describe a dynamical worldsheet origin for the Lagrangian describing the low-energy dynamics of a system of parallel D-branes. We show how matrix-valued collective coordinate fields for the D-branes naturally arise as couplings of a worldsheet sigma-model, and that the quantum dynamics require that these couplings be mutually noncommutative. We show that the low-energy effective action for the sigma-model couplings describes the propagation of an open string in the background of the multiple D-brane configuration, in which all string interactions between the constituent branes are integrated out and the genus expansion is taken into account, with a matrix-valued coupling. The effective field theory is governed by the non-abelian Born-Infeld target space action which leads to the standard one for D-brane field theory. 
  We show that toric geometry can be used rather effectively to translate a brane configuration to geometry. Roughly speaking the skeletons of toric space are identified with the brane configurations. The cases where the local geometry involves hypersurfaces in toric varieties (such as P^2 blown up at more than 3 points) presents a challenge for the brane picture. We also find a simple physical explanation of Batyrev's construction of mirror pairs of Calabi-Yau manifolds using T-duality. 
  In the absence of a complete M-theory, we gather certain quantum aspects of this theory, namely, M-2 and M-5 brane duality and their tension quantization rule $2\kappa^2 T_2 T_5 = 2 \pi n$, the M-2 brane tension quantization $T_2 =\bigg({(2\pi)^2/2\kappa^2 m}\bigg)^{1/3}$, supersymmetry, perturbative gauge and gravitational anomaly cancellations, and the half-integral quantization of [$G^W/2\pi$], and study the consistency among these quantum effects. We find: (1) The complete determination of Ho\u{r}ava-Witten's  $\eta = \lambda^6 /\kappa^4$ for M-theory on $R^{10} \times S^1/ Z_2$ requires not only the cancellation of M-theory gauge anomaly but also that of the gravitational anomaly, the quantization of M-2 brane tension, and the recently recognized half-integral quantization of $[G^W / 2 \pi]$. (2) A well-defined quantum M-theory necessarily requires the presence of both M-2 and M-5 branes and allows only $n = 1$ and $m =1$ for the respectively quantized M-2 and M-5 brane tensions. Implications of the above along with other related issues are discussed. 
  We discuss the basic properties of the gonihedric string and the problem of its formulation in continuum. We propose a generalization of the Dirac equation and of the corresponding gamma matrices in order to describe the gonihedric string. The wave function and the Dirac matrices are infinite-dimensional. The spectrum of the theory consists of particles and antiparticles of increasing half-integer spin lying on quasilinear trajectories of different slope. Explicit formulas for the mass spectrum allow to compute the string tension and thus demonstrate the string character of the theory. 
  We investigate the effect of a constant external non-Abelian field on chiral symmetry breaking in a (2+1)-dimensional Nambu-Jona-Lasinio model and in 3D QCD by solving the gap equation and the Bethe-Salpeter equation, and also by RG analysis. In the (2+1)-dimensional NJL model chiral symmetry breaking occurs for any weak coupling constant but in 3D QCD catalysis of chiral symmetry breaking does not occur. 
  A recent proposal for 6d "gauge" theories with rational theta angle is discussed. These theories were constructed from n_C coinciding (p,q) 5-branes of IIB in the limit of vanishing string constant. They have (1,1) supersymmetry and the low energy theory is 6d Yang-Mills with gauge group SU(n_C) and a rational theta angle theta_6/(2 pi)=f/q, where fp=1 (mod q). By changing the point of view, considering the (p,q) 5-branes to be D5-branes, the 6d theta angle is identified with a 10d theta angle. This point of view together with some assumptions suggests a generalization of the previous limit to arbitrary theta. This limit seems to define a decoupled 6d theory even though the 10d theory does not become free in general. 
  A volume form $H$ on the $n$--dimensional sphere $S^n$ is closed $(dH=0)$, so that it is locally written as $H=dB$, where B is a $(n-1)$--form. In the first half we give an explicit form to B and, moreover, a speculation concerning higher order U(1) bundles. In the second half we apply our B in the case $n=3$ to a string soliton solution discussed by Perry and Schwarz. 
  We present a hermitian matrix chain representation of the general solution of the Hirota bilinear difference equation of three variables. In the large N limit this matrix model provides some explicit particular solutions of continuous differential Hirota equation of three variables. A relation of this representation to the eigenvalues of transfer matrices of 2D quantum integrable models is discussed. 
  We discuss D-brane dynamics in orbifold compactifications of type II superstring theory. We compute the interaction potential between two D-branes moving with constant velocities and give a field theory interpretation of it in the large distance limit. 
  We discuss various D-brane configurations in 4-dimensional orbifold compactifications of type II superstring theory which are point-like 0-branes from the 4-dimensional space-time point of view. We analyze their interactions and compute the amplitude for the emission of a massless NSNS boson from them, in the case where the branes have a non vanishing relative velocity. In the large distance limit, we compare our computation to the expected field theory results, finding complete agreement. 
  Light-front Hamiltonian for Yukawa type models is determined without the framework of canonical light-front formalism. Special attention is given to the contribution of zero modes. 
  We derive the large-N spectral correlators of complex matrix ensembles with weights that in the context of Dirac spectra correspond to N_f massive fermions, and prove that the results are universal in the appropriate scaling limits. The resulting microscopic spectral densities satisfy exact spectral sum rules of massive Dirac operators in QCD. 
  We discuss various bosonization schemes from a path integral perspective. Our analysis shows that the existence of different bosonization schemes, such as abelian bosonization of non-abelian models and non-abelian bosonization of fermions with colour and flavour indices, can be understood as different ways of factoring out a dynamically trivial coset which contains the fermions. From this perspective follows the importance of the coset model in ensuring the correct superselection rules on the bosonic level. 
  We gauge the central charge of the N=2 supersymmetry algebra in rigid superspace. The Fayet Sohnius hypermultiplet with gauged central charge coincides on-shell with N=2 supersymmetric electrodynamics. The gauge couplings of the vector-tensor multiplet turn out to be nonpolynomial. 
  The $CP^2$ model, with and without a generalized Hopf term, is studied using the collective coordinate approximation. In the spirit of this approximation, an ansatz is given which in previous numerical studies was seen to give a good parameterization of the numerical solution. The equations of motion for the collective coordinates are then solved analytically, for solitons close together and for solitons far apart. The solutions show how the generalized Hopf term changes the scattering angle which in its absence is $90^{\circ}$. 
  We consider solutions to the cosmological equations of motion in 11 dimensions with and without 4-form charges. We show explicitly the correspondence between some of these solutions and known solutions in 10 dimensional string gravity. New solutions involving combinations of 4-form charges are explored. We also speculate on the possibility of removing curvature singularities present in 10D theories by oxidizing to 11D. 
  Recently we showed that the postulated diffeomorphic equivalence of states implies quantum mechanics. This approach takes the canonical variables to be dependent by the relation p=\partial_q S_0 and exploits a basic GL(2,C)-symmetry which underlies the canonical formalism. In particular, we looked for the special transformations leading to the free system with vanishing energy. Furthermore, we saw that while on the one hand the equivalence principle cannot be consistently implemented in classical mechanics, on the other it naturally led to the quantum analogue of the Hamilton-Jacobi equation, thus implying the Schroedinger equation. In this letter we show that actually the principle uniquely leads to this solution. Furthermore, we find the map reducing any system to the free one with vanishing energy and derive the transformations on S_0 leaving the wave function invariant. We also express the canonical and Schroedinger equations by means of the brackets recently introduced in the framework of N=2 SYM. These brackets are the analogue of the Poisson brackets with the canonical variables taken as dependent. 
  There have been many remarkable developments in our understanding of superstring theory in the past few years, a period that has been described as ``the second superstring revolution.'' Several of them are discussed here. The presentation is intended primarily for the benefit of nonexperts. 
  We consider a family of non-supersymmetric MQCD five-brane configurations introduced by Witten, and discuss the dependence of the curves on the microscopic theta angle and its relation with CP. We find evidence for a non-trivial spectral flow of the curves (vacua) and for the level-crossing of adjacent curves at a particular value of the theta angle, with spontaneous breaking of CP symmetry, providing an MQCD analogue of the phase transitions in theta proposed by 't Hooft. 
  By scaling arguments we show that the presence of a $R^4$-term in the eleven dimensional supergravity effective lagrangian, if it is visible in (M)atrix theory, should produce a correction to the five-loops effective lagrangian of two moving D0-branes. 
  We consider the N=1 supersymmetric QCD with SU(N_c) gauge group involving  N_f = N_c - 1 pairs of chiral matter multiplets in fundamental and antifundamental color representations. For this theory in the framework of the effective lagrangian approach, we solve the BPS equations for the domain walls interpolating between different vacua. The equations always have a unique solution for the walls interpolating between the chirally symmetric and a chirally asymmetric vacua. For the walls interpolating between different chirally asymmetric vacua, the equations admit two different solutions which exist when the mass of the matter field is below some critical value m*. At m = m*, two branches join together and, at m > m*, no BPS - saturated complex domain walls exist. 
  For a supersymmetric Hamiltonian appearing in the matrix model related to 11 dimensional supermembranes, zero energy states are constructed. A useful symmetry, and an energy-equipartition property is pointed out. 
  A possible avenue towards the covariant formulation of the bosonic Matrix Theory is explored. The approach is guided by the known covariant description of the bosonic membrane. We point out various problems with this particular covariantization scheme, stemming from the central question of how to enlarge the original U(N) symmetry of Matrix Theory while preserving all of its essential features in the infinite momentum frame. 
  The relation between the trace and R-current anomalies in supersymmetric theories implies that the U$(1)_RF^2$, U$(1)_R$ and U$(1)_R^3$ anomalies which are matched in studies of N=1 Seiberg duality satisfy positivity constraints. Some constraints are rigorous and others conjectured as four-dimensional generalizations of the Zamolodchikov $c$-theorem. These constraints are tested in a large number of N=1 supersymmetric gauge theories in the non-Abelian Coulomb phase, and they are satisfied in all renormalizable models with unique anomaly-free R-current, including those with accidental symmetry. Most striking is the fact that the flow of the Euler anomaly coefficient, $a_{UV}-a_{IR}$, is always positive, as conjectured by Cardy. 
  In this paper we give an inherently toric description of a special class of sheaves (known as equivariant sheaves) over toric varieties, due in part to A. A. Klyachko. We apply this technology to heterotic compactifications, in particular to the (0,2) models of Distler, Kachru, and also discuss how knowledge of equivariant sheaves can be used to reconstruct information about an entire moduli space of sheaves. Many results relevant to heterotic compactifications previously known only to mathematicians are collected here -- for example, results concerning whether the restriction of a stable sheaf to a Calabi-Yau hypersurface remains stable are stated. We also describe substructure in the Kahler cone, in which moduli spaces of sheaves are independent of Kahler class only within any one subcone. We study F theory compactifications in light of this fact, and also discuss how it can be seen in the context of equivariant sheaves on toric varieties. Finally we briefly speculate on the application of these results to (0,2) mirror symmetry. 
  We study field theories in the limit that a compactified dimension becomes lightlike. In almost all cases the amplitudes at each order of perturbation theory diverge in the limit, due to strong interactions among the longitudinal zero modes. The lightlike limit generally exists nonperturbatively, but is more complicated than might have been assumed. Some implications for the matrix theory conjecture are discussed. 
  We propose a harmonic superspace description of the non-linear vector-tensor N=2 multiplet. We show that there exist two inequivalent version: the old one in which one of the vectors is the field-strength of a gauge two-form, and a new one in which this vector satisfies a non-linear constraint and cannot be expressed in terms of a potential. In this the new version resembles the non-linear N=2 multiplet. We perform the dualization of both non-linear versions and discuss the resulting K\"ahler potentials. Finally, we couple the non-linear vector-tensor multiplet to an abelian background gauge multiplet. 
  We construct models in 1+1 dimensions with chiral (0,N) supersymmetry by taking orientifolds of type IIB on an eight-torus identified by different numbers of reflections. The resulting models have Dirichlet strings, fivebranes and ninebranes stretched along different directions. The cases we study in detail have residual chiral supersymmetry (0,8), (0,4) and (0,2). The gravitational anomaly in all cases is shown to cancel. 
  We present a class of BPS solutions of the IIB Matrix Theory which preserve 1/4 supersymmetry. The solutions desrcibe D-string configurations with left-moving oscillations. We demonstrate that the one-loop quantum effective action of Matrix Theory vanishes for this solution, confirming its BPS nature. We also study the world-volume gauge theory of oscillating strings and show its connection with static D-strings. 
  A symmetry reduction of the Dirac equation is shown to yield the system of ordinary differential equations whose integrability by quadratures is closely connected to the stationary mKdV hierarchy. 
  Eleven dimensional supergravity has electric type currents arising from the Chern-Simon and anomaly terms in the action. However the bulk charge integrates to zero for asymptotically flat solutions with topological trivial spatial sections. We show that by relaxing the boundary conditions to generalisations of the ALE and ALF boundary conditions in four dimensions one can obtain static solutions with a bulk charge preserving between 1/16 and 1/4 of the supersymmetries. One can introduce membranes with the same sign of charge into these backgrounds. This raises the possibility that these generalized membranes might decay quantum mechanically to leave just a bulk distribution of charge. Alternatively and more probably, a bulk distribution of charge can decay into a collection of singlely charged membranes. Dimensional reductions of these solutions lead to novel representations of extreme black holes in four dimensions with up to four charges. We discuss how the eleven-dimensional Kaluza-Klein monopole wrapped around a space with non-zero first Pontryagin class picks up an electric charge proportional to the Pontryagin number. 
  It is known that the M-branes of M-theory correspond to p-form charges in the D=11 spacetime supersymmetry algebra. Here we show that their intersections are encoded in the p-form charges of their worldvolume supersymmetry algebras. Triple intersections are encoded in double intersection worldvolume algebras with eight supercharges. 
  We present some methods of determining explicit solutions for self-dual supermembranes in 4+1 and 8+1 dimensions with spherical or toroidal topology. For configurations of axial symmetry, the continuous SU(\infty) Toda equation turns out to play a central role, and a specific method of determining all the periodic solutions are suggested. A number of examples are studied in detail. 
  Barton Zwiebach constructed the `string products' on the Hilbert space of combined conformal field theory of matter and ghosts. It is well-known that the `tree level' specialization of these products forms a strongly homotopy Lie algebra. A strongly homotopy Lie algebra is given by a square zero coderivation on the cofree cocommutative connected coalgebra, on the other hand, strongly homotopy Lie algebras are algebras over the cobar construction on the commutative algebras operad. The aim of our paper is to give two similar characterizations of the structure formed by the `string products' of arbitrary genera. Our first characterization will be based on the notion of a higher order coderivation, the second characterization will be based on the machinery of modular operads. We will also discuss possible generalizations to open string field theory. 
  We evaluate one-loop finite-time amplitudes for graviton scattering in Matrix theory and compare to the corresponding amplitudes in supergravity. We find agreement for arbitrary time intervals at leading order in distance, providing a functional agreement between supergravity and Matrix theory. At subleading order, we find corrections to the effective potential found from previous phase shift calculations in Matrix theory. 
  Massive spectral sum rules are derived for Dirac operators of $SU(N_c)$ gauge theories with $N_f$ flavors. The universal microscopic massive spectral densities of random matrix theory, where known, are all consistent with these sum rules. 
  Decomposition of the solvable Lie algebras of maximal supergravities in D=4, 5 and 6 indicates, at least at the geometrical level, the existence of an N=(4,2) chiral supergravity theory in D=6 dimensions. This theory, with 24 supercharges, reduces to the known N=6 supergravity after a toroidal compactification to D=5 and D=4. Evidence for this theory was given long ago by B. Julia. We show that this theory suffers from a gravitational anomaly equal to 4/7 of the pure N=(4,0) supergravity anomaly. However, unlike the latter, the absence of N=(4,2) matter to cancel the anomaly presumably makes this theory inconsistent. We discuss the obstruction in defining this theory in D=6, starting from an N=6 five-dimensional string model in the decompactification limit. The set of massless states necessary for the anomaly cancellation appears in this limit; as a result the N=(4,2) supergravity in D=6 is extended to N=(4,4) maximal supergravity theory. 
  Bianchi type IX, 'Mixmaster' universes are investigated in low-energy-effective-action string cosmology. We show that, unlike in general relativity, there is no chaos in these string cosmologies for the case of the tree-level action. The characteristic Mixmaster evolution through a series of Kasner epochs is studied in detail. In the Einstein frame an infinite sequence of chaotic oscillations of the scale factors on approach to the initial singularity is impossible, as it was in general relativistic Mixmaster universes in the presence of a massless scalar field.  A finite sequence of oscillations of the scale factors described by approximate Kasner metrics is possible, but it always ceases when all expansion rates become positive. In the string frame the evolution through Kasner epochs changes to a new form which reflects the duality symmetry of the theory. Again, we show that chaotic oscillations must end after a finite time. The need for duality symmetry appears to be incompatible with the presence of chaotic behaviour as $t \to 0$. We also obtain our results using the Hamiltonian qualitative cosmological picture for Mixmaster models. We also prove that a time-independent pseudoscalar axion field $h$ is not admitted by the Bianchi IX geometry. 
  Nonabelian gauge theories with a generic background field A_mu in nonlinear gauges due to Delbourgo and Jarvis are investigated. The A_mu-dependence is completely determined by the help of a linear differential equation which obtaines from the Kluberg-Stern-Zuber and the Lee identity. Its integration leads to a relation between the one-particle irreducible vertex functional in the background field A_mu and the corresponding functional for A_mu = 0. An analogous relation holds for the generating functional of the complete Green functions which, after restriction to physical Green functions, is used to confirm a result obtained by Rouet in the case of linear background gauge. 
  We compute the vacuum one loop amplitude for two D-strings at an angle compactified on $T^2$ as a function of the transverse separation and the winding numbers. We show that in certain cases the amplitude is independent on whether the D-strings are compactified or not. 
  We obtain the symmetry algebra of multi-matrix models in the planar large N limit. We use this algebra to associate these matrix models with quantum spin chains. In particular, certain multi-matrix models are exactly solved by using known results of solvable spin chain systems. 
  Extremal black holes in M-theory compactification on $M\times S^1$ are microscopically represented by fivebranes wrapping $P\times S^1$, where $M$ is a Calabi-Yau threefold and $P$ is a four-cycle in $M$. Additional spacetime charges arise from momentum around the $S^1$ and expectation values for the self-dual three-form field strength in the fivebrane. The microscopic entropy of the fivebrane as a function of all the charges is determined from a two-dimensional $(0,4)$ sigma model whose target space includes the fivebrane moduli space. This entropy is compared to the macroscopic formula. Precise agreement is found for both the tree-level and one-loop expressions. 
  We show that the holomorphic Wilsonian beta-function of a renormalizable asymptotically free supersymmetric gauge theory with an arbitrary semi-simple gauge group, matter content, and renormalizable superpotential is exhausted at 1-loop with no higher loops and no non-perturbative contributions. This is a non-perturbative extension of the well known result of Shifman and Vainshtein. 
  The standard eleven-dimensional supergravity action depends on a three-form gauge field and does not allow direct coupling to five-branes. Using previously developed methods, we construct a covariant eleven-dimensional supergravity action depending on a three-form and six-form gauge field in a duality-symmetric manner. This action is coupled to both the M-theory two-brane and five-brane, and corresponding equations of motion are obtained. Consistent coupling relates D=11 duality properties with self-duality properties of the M-5-brane. From this duality-symmetric formulation, one can derive an action describing coupling of the M-branes to standard D=11 supergravity. 
  We show the existence of a co(dual)-BRST symmetry for the usual BRST invariant Lagrangian density of an Abelian gauge theory in two dimensions of space-time where a U(1) gauge field is coupled to the Noether conserved current (constructed by the Dirac fields). Under this new symmetry, it is the gauge-fixing term that remains invariant and the symmetry transformations on the Dirac fields are analogous to the chiral transformations. This interacting theory is shown to provide a tractable field theoretical model for the Hodge theory. The Hodge dual operation is shown to correspond to a discrete symmetry in the theory and the extended BRST algebra for the generators of the underlying symmetries turns out to be reminiscent of the algebra obeyed by the de Rham cohomology operators of differential geometry. 
  We compute the microscopic spectrum of the QCD Dirac operator in the presence of dynamical fermions in the framework of random-matrix theory for the chiral Gaussian unitary ensemble. We obtain results for the microscopic spectral correlators, the microscopic spectral density, and the distribution of the smallest eigenvalue for an arbitrary number of flavors, arbitrary quark masses, and arbitrary topological charge. 
  This paper presents a construction of isospectral problems on the torus. The construction starts from an SU(n) version of the XYZ Gaudin model recently studied by Kuroki and Takebe in the context of a twisted WZW model. In the classical limit, the quantum Hamiltonians of the generalized Gaudin model turn into classical Hamiltonians with a natural $r$-matrix structure. These Hamiltonians are used to build a non-autonomous multi-time Hamiltonian system, which is eventually shown to be an isomonodromic problem on the torus. This isomonodromic problem can also be reproduced from an elliptic analogue of the KZ equation for the twisted WZW model. Finally, a geometric interpretation of this isomonodromic problem is discussed in the language of a moduli space of meromorphic connections. 
  We construct the complete coupling of $(1,0)$ supergravity in six dimensions to $n$ tensor multiplets, extending previous results to all orders in the fermi fields. We then add couplings to vector multiplets, as dictated by the generalized Green-Schwarz mechanism. The resulting theory embodies factorized gauge and supersymmetry anomalies, to be disposed of by fermion loops, and is determined by corresponding Wess-Zumino consistency conditions, aside from a quartic coupling for the gaugini. The supersymmetry algebra contains a corresponding extension that plays a crucial role for the consistency of the construction. We leave aside gravitational and mixed anomalies, that would only contribute to higher-derivative couplings. 
  A specific SU(2) gauge configuration yielding a magnetic field concentrated at a point is investigated. Its relation to the Aharonov-Bohm gauge potential and its cohomological meaning in a three dimensional space are clarified. Quantum mechanics of a spinning particle in such a gauge configuration is briefly discussed. 
  The problem considered here is the determination of the hamiltonian of a first quantized nonrelativistic particle by the help of some measurements of the location with a finite resolution. The resulting hamiltonian depends on the resolution of the measuring device. This dependence is reproduced by the help of a blocking transformation on the wave function. The systems with quadratic hamiltonian are studied in details. The representation of the renormalization group in the space of observables is identified. 
  It is shown that canonical quantization of the $4d$ Siegel superparticle yields massless Wess-Zumino model as an effective field theory. Quantum states of the superparticle are realized in terms of the real scalar superfields which prove to be decomposed into the sum of on-shell chiral and antichiral superfields. 
  In this brief note we would like to discuss, in a simple model system, the conditions under which the discrete light cone quantization framework should be trusted as an approximation scheme, with regard, in particular, to the size and mass of the system. Specifically, we are going to discuss ``quark-antiquark'' bound states in 1+1 dim., for which a natural size is provided by analogy with a ``two points and a spring'' system, and show that the condition for obtaining a reliable estimate is the same as the one derived in a recent paper for black holes in matrix theory. 
  The explicit expression of all the WZW effective actions for a simple group G broken down to a subgroup H is established in a simple and direct way, and the formal similarity of these actions to the Chern-Simons forms is explained. Applications are also discussed. 
  We study the finite-temperature phase transition of the generalized Gross-Neveu model with continous chiral symmetry in $2 < d \leq 4$ euclidean dimensions. The critical exponents are computed to the leading order in the $1/N$ expansion at both zero and finite temperatures. A dimensionally reduced theory is obtained after the introduction of thermal counterterms necessary to cancel thermal divergences that arise in the limit of high temperature. Although at zero temperature we have an infinitely and continously degenerate vacuum state, we show that at finite temperature this degeneracy is discrete and, depending on the values of the bare parameters, we may have either total or partial restoration of symmetry. Finally we determine the universality class of the reduced theory by a simple analysis of the infrared structure of thermodynamic quantities computed using the reduced action as starting point. 
  A C*-algebra containing the CCR and CAR algebras as its subalgebras and naturally described as the semidirect product of these algebras is discussed. A particular example of this structure is considered as a model for the algebra of asymptotic fields in quantum electrodynamics, in which Gauss' law is respected. The appearence in this algebra of a phase variable related to electromagnetic potential leads to the universal charge quantization. Translationally covariant representations of this algebra with energy-momentum spectrum in the future lightcone are investigated. It is shown that vacuum representations are necessarily nonregular with respect to total electromagnetic field. However, a class of translationally covariant, irreducible representations is constructed excplicitly, which remain as close as possible to the vacuum, but are regular at the same time. The spectrum of energy-momentum fills the whole future lightcone, but there are no vectors with energy-momentum lying on a mass hyperboloid or in the origin. 
  We compute the microscopic entropy of certain 4 and 5 dimensional extermal black holes which arise for compactification of M-theory and type IIA on Calabi-Yau 3-folds. The results agree with macroscopic predictions, including some subleading terms. The macroscopic entropy in the 5 dimensional case predicts a surprising growth in the cohomology of moduli space of holomorphic curves in Calabi-Yau threefolds which we verify in the case of elliptic threefolds. 
  Accurate and powerful computational methods developed by the author for the wave scattering by black holes allow to obtain the highly non trivial total absorption spectrum of the Black Hole. As well as partial wave phase shifts and cross sections (elastic and inelastic), the angular distribution of absorbed and scattered waves, and the Hawking emission rates. The exact total absorption spectrum of waves by the Black Hole has as a function of frequency a remarkable oscillatory behaviour characteristic of a diffraction pattern. This is an unique distinctive feature of the black hole absorption, and due to its r = 0 singularity. Ordinary absorptive bodies and optical models do not present these features. The unitarity optical theorem is generalized to the Black Hole case explicitly showing that absorption ocurrs only at the origin. All these results allow to understand and reproduce the Black Hole absorption spectrum in terms of Fresnel-Kirchoff diffraction theory: by the interference of the absorbed rays arriving at the origin through different optical paths. These fundamental features of the Black Hole Absorption will be present for generic higher dimensional Black Hole backgrounds, and whatever the low energy effective theory they arise from. In recent and increasing litterature devoted to compute absorption cross sections (grey body factors) of black holes (whatever ordinary, stringy, D-braned), the fundamental remarkable features of the Black Hole Absorption spectrum are overlooked. 
  We analyze the propagation of open and unoriented strings on the Neveu-Schwarz pentabrane (N5-brane) along the lines of a similar analysis for the SU(2) WZNW models. We discuss the two classes of open descendants of the diagonal models and a series of Z_2 projected models which exist only for even values of the level k and correspond to branes at D-type orbifold singularities. The resulting configurations of branes and planes are T-dual to those relevant to the study of dualities in super Yang-Mills theories. The association of Chan-Paton factors to D-brane multiplicities is possible in the semi-classical limit k -> infinity, but due to strong curvature effects is unclear for finite k. We show that the introduction of a magnetic field implies a twist of the SU(2) current algebra in the open-string sector leading to spacetime supersymmetry breaking. 
  This lecture is dedicated to the memory of Vladimir Gribov.   We discuss the procedure of complete gauge fixing to avoid Gribov copies, as well as results for the low-lying glueball spectrum in intermediate volumes for the torus and for the sphere that became available recently. 
  We present a numerical scheme for calculating the first quantum corrections to the properties of static solitons. The technique is applicable to solitons of arbitrary shape, and may be used in 3+1 dimensions for multiskyrmions or other complicated solitons. We report on a test computation in 1+1 dimensions, where we accurately reproduce the analytical result with minimal numerical effort. 
  We calculate semiclassically the emission rate of spin 1/2 particles from charged, nonrotating black holes in D=5,N=8 supergravity. The relevant Dirac equation is solved by the same approximation as in the bosonic case. The resulting expression for the emission rate has a form which is predicted from D-brane effective field theory. 
  We investigate N_c=2 case of IIB matrix model, which is exactly soluble. We calculate the partition function exactly and obtain a finite result without introducing any cut-off. We also evaluate some correlation functions consisting of Wilson loops. 
  We study N=1 four dimensional gluodynamics in the context of M-theory compactifications on elliptically fibered Calabi-Yau fourfolds. Gaugino condensates, \theta-dependence, Witten index and domain walls are considered for singularities of type $\hat{A}_{n-1}$ and $\hat{D}_{n+4}$. It is shown how the topology of intersections among the irreducible components defining the singular elliptic fiber, determine the entanglement of vacua and the appareance of domain walls. 
  In two recent preprints (hep-th/9710131 and hep-th/9710132), Abe and Nakanishi have claimed that the proof of the gauge independence of the conformal anomaly of the bosonic string as given by us in 1988 was wrong. A similar allegation has been made concerning our proof of the gauge independence of the sum of the ghost number and Lagrange multiplier anomalies in non-conformal gauges. In this short note we refute their criticism by explaining the simple logic of our proofs and emphasizing the points that have been missed by Abe and Nakanishi. 
  I review my work together with Piljin Yi on the spectrum of BPS-saturated states in N = 2 supersymmetric Yang-Mills theories. In an M-theory description, such states are realized as certain two-brane configurations. We first show how the central charge of the N = 2 algebra arises from the two-form central charge of the eleven-dimensional supersymmetry algebra, and derive the condition for a two-brane configuration to be BPS-saturated. We then discuss how the topology of the two-brane determines the type of BPS-multiplet. Finally, we discuss the phenomenon of marginal stability and show how it is related to the mutual non-locality of states. 
  We review some results on the complete coupling between tensor and vector multiplets in six-dimensional $(1,0)$ supergravity. 
  We consider membranes of spherical topology in uncompactified Matrix theory. In general for large membranes Matrix theory reproduces the classical membrane dynamics up to 1/N corrections; for certain simple membrane configurations, the equations of motion agree exactly at finite N. We derive a general formula for the one-loop Matrix potential between two finite-sized objects at large separations. Applied to a graviton interacting with a round spherical membrane, we show that the Matrix potential agrees with the naive supergravity potential for large N, but differs at subleading orders in N. The result is quite general: we prove a pair of theorems showing that for large N, after removing the effects of gravitational radiation, the one-loop potential between classical Matrix configurations agrees with the long-distance potential expected from supergravity. As a spherical membrane shrinks, it eventually becomes a black hole. This provides a natural framework to study Schwarzschild black holes in Matrix theory. 
  A 3-dimensional non-abelian gauge theory was proposed by Jackiw and Pi to create mass for the gauge fields. However, the quadratic action obtained by switching off the non-abelian interactions possesses more gauge symmetries than the original one, causing some difficulties in quantization. Jackiw and Pi proposed another action by introducing new fields, whose gauge symmetries are consistent with the quadratic part. It is shown that all of these theories have the same number of physical degrees of freedom in the hamiltonian framework. Hence, as far as the physical states are considered there is no inconsistency. Nevertheless, perturbation expansion is still problematic. To cure this we propose to modify one of the constraints of the non-abelian theory without altering neither its canonical hamiltonian nor the number of physical states. 
  Recent developments involving strongly coupled superstrings are discussed from a phenomenological point of view. In particular, strongly coupled $E_8\times E'_8$ is described as an appropriate long-wavelength limit of M-theory, and some generic phenomenological implications are analyzed, including a long sought downward shift of the string unification scale and a novel way to break supersymmetry. A specific scenario is presented that leads to a rather light, and thus presently experimentally testable, sparticle spectrum. 
  Q-ball configuration that represents oscillating or spinning closed membrane is constructed via M(atrix) theory. Upon gravitational collapse Q-balls are expected to form Schwarzschild black holes. For quasi-static spherical membrane, we probe spacetime geometry induced by monopole moment via D0-parton scattering off the Q-ball. We find a complete agreement with long distance potential calculated using eleven-dimensional supergravity. Generalizing to heterotic M(atrix) theory, we also construct Q-ball configurations of real projective and disk membranes. The latter Q-ball configuration arises as twisted sector of heterotic M(atrix) theory, hence, are expected to form a charged black hole after gravitational collapse. 
  The nature of M-theory on K3 X I, where I is a line interval, is considered, with a view towards formulating a `matrix theory' representation of that situation. Various limits of this compactification of M-theory yield a number of well known N=1 six dimensional compactifications of the heterotic and type I string theories. Geometrical relations between these limits give rise to string/string dualities between some of these compactifications. At a special point in the moduli space of compactifications, this motivates a partial definition of the matrix theory representation of the M-theory on K3 X I as the large N limit of a certain type IA orientifold model probed by a conglomerate of N D-branes. Such a definition in terms of D-branes and orientifold planes is suggestive, but necessarily incomplete, due to the low amount of superymmetry. It is proposed - following hints from the orientifold model - that the complete matrix theory representation of the K3 X I compactified M-theory is given by the large N limit of compactification - on a suitable `dual' surface - of the `little heterotic string' N = 1 six dimensional quantum theories. 
  N=1 supersymmetric QCD is considered using the recently proposed picture of it as the world volume theory of a single M-theory fivebrane with two of its dimensions wrapped on a Riemann surface. The conditions under which a second M-theory brane can be introduced preserving some supersymmetry are analysed. Such configurations represent BPS saturated extended objects in the quantum field theory on the brane worldvolume. Formulae for the tension of these extended objects are derived. An explicit intersecting fivebrane configuration is found which is interpreted as a BPS domain wall in 4-dimensional MQCD. 
  Four-fermion interaction models are considered to be prototype models for dynamical symmetry breaking.The present review deals with recent developments in the studies of dynamical symmetry breaking in the four-fermion interaction models and their extension in curved spacetime. Starting with the Minkowski spacetime in dimension $D$ ($2 \leq D \leq 4$) the effective potential in the leading order of $1/N$-expansion is calculated and the phase structure of the theory is investigated. In curved spacetime curvature-induced phase transitions are discussed in the circumstances where fermion masses are dynamically generated. Subsequently a possibility of curvature- and temperature-induced or curvature- and topology-induced phase transitions is discussed. It is also argued that the chiral symmetry broken by a weak magnetic field may be restored due to the presence of gravitational field. Finally some applications of four-fermion models in quantum gravity are briefly described. 
  The statistics of soliton sectors of massive 2D field theories is analysed. In the soliton field algebra, the non-local commutation relations are determined and Weak Locality, Spin-Statistics and CPT theorems are proven. These theorems depart from their usual appearance due to the broken symmetry connecting the inequivalent vacua. An interpretation in terms of modular theory is given. For the neutral subalgebra, the theorems hold in the familiar form, and Twisted Locality is derived. 
  We discuss charged black-hole solutions to the equations of motion of the string-loop-corrected effective action. At the string-tree level, these solutions provide backgrounds for the "chiral null model". The effective action contains gravity, dilaton and moduli fields. Analytic solutions of the one-loop-corrected equations of motion are presented for the extremal magnetic and dyonic black holes. Using the fact that in magnetic solution the loop-corrected dilaton is non-singular at the origin, we apply the correspondence principle to show that the entropy of the loop-corrected magnetic black hole can be interpreted as the microscopic entropy of the D-brane system. 
  Closed string physical states are BRST cohomology classes computed on the space of states annihilated by $b_0^-$. Since $b_0^-$ does not commute with the operations of picture changing, BRST cohomologies at different pictures need not agree. We show explicitly that Ramond-Ramond (RR) zero-momentum physical states are inequivalent at different pictures, and prove that non-zero momentum physical states are equivalent in all pictures. We find that D-brane states represent BRST classes that are nonpolynomial on the superghost zero modes, while RR gauge fields appear as polynomial BRST classes. We also prove that in $x$-cohomology, the cohomology where the zero mode of the spatial coordinates is included, there is a unique ghost-number one BRST class responsible for the Green-Schwarz anomaly, and a unique ghost number minus one BRST class associated with RR charge. 
  We write general one-loop anomalies of string field theory as path integrals on a torus for the corresponding nonlinear sigma model. This extends the work of Alvarez-Gaum\'e and Witten from quantum mechanics to two dimensions. Higher world-volume loops contribute in general to nontopological anomalies and a formalism to compute these is developed. We claim that (i) for general anomalies one should not use the propagator widely used in string theory but rather the one obtained by generalization from quantum mechanics, but (ii) for chiral anomalies both propagators give the same result. As a check of this claim in a simpler model we compute trace anomalies in quantum mechanics. The propagator with a center-of-mass zero mode indeed does not give the correct result for the trace anomaly while the propagator for fluctuations $q^i (\tau)$ satisfying $q^i (\tau = -1) = q^i (\tau = 0) = 0$ yields in $d=2$ and $d=4$ dimensions the correct results from two- and three-loop graphs.   We then return to heterotic string theory and calculate the contributions to the anomaly from the different spin structures for $d=2$. We obtain agreement with the work of Pilch, Schellekens and Warner and that of Li in the sector with spacetime fermions. In the other sectors, where no explicit computations have been performed in the past and for which one needs higher loops, we find a genuine divergence, whose interpretation is unclear to us. We discuss whether or not this leads to a new anomaly. 
  We show that six-dimensional supergravity coupled to tensor and Yang-Mills multiplets admits not one but two different theories as global limits, one of which was previously thought not to arise as a global limit and the other of which is new. The new theory has the virtue that it admits a global anti-self-dual string solution obtained as the limit of the curved-space gauge dyonic string, and can, in particular, describe tensionless strings. We speculate that this global model can also represent the worldvolume theory of coincident branes. We also discuss the Bogomol'nyi bounds of the gauge dyonic string and show that, contrary to expectations, zero eigenvalues of the Bogomol'nyi matrix do not lead to enhanced supersymmetry and that negative tension does not necessarily imply a naked singularity. 
  We consider D-branes on an orbifold $C^3/Z_n$ and investigate the moduli space of the D-brane world-volume gauge theory by using toric geometry and gauged linear sigma models. For $n=11$, we find that there are five phases, which are topologically distinct and connected by flops to each other. We also verify that non-geometric phases are projected out for $n=7,9,11$ cases as expected. Resolutions of non-isolated singularities are also investigated. 
  We study the structure of the normal matrix model (NMM). We show that all correlation functions of the model with axially symmetric potentials can be expressed in terms of holomorphic functions of one variable. This observation is used to demonstrate the exact solvability of the model. The two-point correlation function is calculated in the scaling limit by solving the BBGKY chain of equations. The answer is shown to be universal (i.e. potential independent up to a change of the scale). We then develop a two-dimensional free fermion formalism and construct a family of completely integrable hierarchies (which we call the extended-KP(N) hierarchies) of non-linear differential equations. The well-known KP hierarchy is a lower-dimensional reduction of this family. The extended-KP(1) hierarchy contains the (2+1)-dimensional Burgers equations. The partition function of the N*N NMM is the tau function of the extended-KP(N) hierarchy invariant with respect to a subalgebra of an algebra of all infinitesimal diffeomorphisms of the plane. 
  Following systematically the generalized Hamiltonian approach of Batalin, Fradkin and Tyutin (BFT), we embed the second-class non-abelian SU(2) Higgs model in the unitary gauge into a gauge invariant theory. The strongly involutive Hamiltonian and constraints are obtained as an infinite power series in the auxiliary fields. Furthermore, comparing these results with those obtained from the gauged second class Lagrangian, we arrive at a simple interpretation for the first class Hamiltonian, constraints and observables. 
  We study the BPS black hole solutions of the (truncated) action for heterotic string theory compactified on a six-torus. The O(3,Z) duality symmetry of the theory, together with the bound state interpretation of extreme black holes, is used to generate the whole spectrum of the solutions. The corresponding spacetime structures, written in terms of the string metric, are analyzed in detail. In particular, we show that only the elementary solutions present naked singularities. The bound states have either null singularities (electric solutions) or are regular (magnetic or dyonic solutions) with near-horizon geometries given by the product of two 2d spaces of constant curvature. The behavior of some of these solutions as supersymmetric attractors is discussed. We also show that our approach is very useful to understand some of the puzzling features of charged black hole solutions in string theory. 
  We study BPS-saturated classical solutions for the world-sheet theory of a D-string in the presence of a point charge. These solutions are interpreted as describing planar 3-string junctions, which arise because the original D-string is deformed by the presence of the inserted charge. We compute the angles of the junctions and show that the vector sum of string tensions is zero, confirming a conjecture of Schwarz that such configurations are BPS states. 
  We show that the vacuum functional of 3+1 dimensional non-abelian gauge theories vanishes for some classical field configurations $\psi_0(A)=0$ when the coefficient of the CP violating $\theta$--term becomes $\theta=\pi$ (mod. $2\pi$). Some of these classical configurations are explicitly identified and include sphalerons. The results shed new light into the non-perturbative behavior of non-abelian gauge theories and suggest a relevant role for these classical configurations in the confinement mechanism at $\theta=0$. 
  Based on random matrix theory in the unitary ensemble, we derive the double-microscopic massive spectral correlators corresponding to the Dirac operator of QCD_3 with an even number of fermions N_f. We prove that these spectral correlators are universal, and demonstrate that they satisfy exact massive spectral sum rules of QCD_3 in a phase where flavor symmetries are spontaneously broken according to U(N_f) -> U(N_f/2) x U(N_f/2). 
  We examine the system where a string stretches between pair of D-branes, and study the bending of the D-brane caused by the tension of the string. If the distance between the pair of D-branes is sent to infinity, the tension of the string stretching between them is strong enough to pull the spike all the way to infinity. We study the shape of these spikes when the branes are finite distance apart using two different methods. First, we consider a string stretched between a pair of D2-branes in type IIA theory by going to the M-theory limit in which all of these branes are M-theory 2-branes embedded along a holomorphic curve. Second, we consider a D-string stretched between a pair of D3-branes in type IIB theory and infer the geometry of the D3-brane embeddings from the configuration of the adjoint scalar field in the magnetic monopole solution of Prasad and Sommerfield. The case of fundamental string stretching between a pair of D3-branes follows from S-duality. The energy of these configurations matches the expected value based on fundamental string and D-string tensions. 
  The BPS spectrum of type I' string theory in a generic background is derived using the duality with the nine-dimensional heterotic string theory with Wilson lines. It is shown that the corresponding mass formula has a natural interpretation in terms of type I', and it is demonstrated that the relevant states in type I' preserve supersymmetry. By considering certain BPS states for different Wilson lines an independent confirmation of the string creation phenomenon in the D0-D8 system is found. We also comment on the non-perturbative realization of gauge enhancement in type I', and on the predictions for the quantum mechanics of type I' D0-branes. 
  We propose a class of field theories featuring solitonic solutions in which topological defects can end when they intersect other defects of equal or higher dimensionality. Such configurations may be termed ``Dirichlet topological defects'', in analogy with the D-branes of string theory. Our discussion focuses on defects in scalar field theories with either gauge or global symmetries, in (3+1) dimensions; the types of defects considered include walls ending on walls, strings on walls, and strings on strings. 
  The string field theory for unoriented open-closed string mixed system is constructed up to quadratic order based on the joining-splitting type vertices. The gauge invariance with closed string transformation parameter is proved. The infinity cancellation mechanism between disk and projective plane amplitudes plays an essential role for the gauge invariance of the theory. 
  We calculate, using zeta function regularization method, semiclassical energy of the Nambu-Goto string supplemented with the boundary, Gauss-Bonnet term in the action and discuss the tachyonic ground state problem. 
  Lecture notes reviewing most recent developments in string/M/brane theory given by C. G. at the CIME Summer International Center of Mathematics at Cetraro. July 1997. 
  Continuing our work hep-th/9609135 where a explicit formula for the two-point functions of the two dimensional Z-invariant Ising model were found. I obtain here different results for the higher correlation functions and several consistency checks are done. 
  We present a few remarks on disconnected components of the moduli space of heterotic string compactifications on $T_2$. We show in particular how the eight dimensional CHL heterotic string can be understood in terms of topologically non-trivial $E_8\times E_8$ and $\Spin(32)/Z_2$ vector bundles over the torus, and that the respective moduli spaces coincide. 
  The Matrix description of certain $E_8\times E_8$ heterotic theories on $T^4/Z_2$ is shown to correspond to the world-volume theory of $Spin(32)/Z_2$ heterotic five-branes on the dual $T^4/Z_2\times S^1$. 
  Recent work on solutions to the Born-Infeld theory used to describe D-branes suggests that fundamental strings can be viewed, in a certain limit, as D-branes whose worldvolumes have collapsed to string-like configurations. Here we address the possibility of undoing this collapse, by inducing the string to tunnel to an extended brane configuration. Using semiclassical methods we argue that, by putting one of these Born-Infeld strings in the background of a uniform 4-form RR field strength, the string can nucleate a spheroidal bulge of D2-brane, or, if the string is wrapped around a small enough circle, it can tunnel to a toroidal D2-brane. This process can also be interpreted in terms of the M2-brane. We also address the extension to other D$p$-branes, and discuss the range of validity of the approximations involved. 
  A connection is made between the Witten index of relevance to threshold bound states of D-particles in the type IIA superstring theory and the measure that enters D-instanton sums for processes dominated by single multiply-charged D-instantons in the type IIB theory. 
  We discuss topological theories, arising from the general $\CN=2$ twisted gauge theories. We initiate a program of their study in the Gromov-Witten paradigm. We re-examine the low-energy effective abelian theory in the presence of sources and study the mixing between the various $p$-observables. We present the twisted superfield formalism which makes duality transformations transparent. We propose a scheme which uniquely fixes all the contact terms. We derive a formula for the correlation functions of $p$-observables on the manifolds of generalized simple type for $0 \leq p \leq 4$ and on some manifolds with $b_{2}^{+} =1$. We study the theories with matter and explore the properties of universal instanton. We also discuss the compactifications of higher dimensional theories. Some relations to sigma models of type $A$ and $B$ are pointed out and exploited. 
  Different BPS M-brane configurations including single and two parallel M$p$-branes ($p$= even) and M5-branes are introduced as the classical solutions of the recently proposed Static Matrix Model. Also the long range interactions of two relatively rotated M$p$-branes (one and two angles) and M$p$-brane--anti-M$p$-brane are calculated. The results are in agreement with 11 dimensional supergravity results. 
  Based on the relation to random matrix theory, exact expressions for all microscopic spectral correlators of the Dirac operator can be computed from finite-volume partition functions. This is illustrated for the case of $SU(N_c)$ gauge theories with $N_c\geq 3$ and $N_f$ fermions in the fundamental representation. 
  A generating function for the Parke-Taylor amplitudes with any number of positive helicity gravitons in addition to the positive helicity gluons is obtained using the recently constructed self-dual classical solution of the type of perturbiner in Yang-Mills theory interacting with gravity. 
  An $(n, 1)$ string is a bound state of a D-string and $n$ fundamental strings. It may be described by a D-string with a world volume electric field turned on. As the electric field approaches its critical value, $n$ becomes large. We calculate the 4-point function for transverse oscillations of an $(n, 1)$ string, and the two-point function for massless closed strings scattering off an $(n, 1)$ string. In both cases we find a set of poles that becomes dense in the large $n$ limit. The effective tension that governs the spacing of these poles is the fundamental string tension divided by $1+(n\lambda)^2$, where $\lambda$ is the closed string coupling. We associate this effective tension with the open strings attached to the $(n, 1)$ string, thereby governing its dynamics. We also argue that the effective coupling strenth of these open strings is reduced by the electric field and approaches zero in the large $n$ limit. 
  Employing factorized versions of characters as products of quantum dilogarithms corresponding to irreducible representations of the Virasoro algebra, we obtain character formulae which admit an anyonic quasi-particle interpretation in the context of minimal models in conformal field theories. We propose anyonic thermodynamic Bethe ansatz equations, together with their corresponding equation for the Virasoro central charge, on the base of an analysis of the classical limit for the characters and the requirement that the scattering matrices are asymptotically phaseless. 
  The partition function of type IIA and B strings on R^6xK3, in the T^4/Z_2 orbifold limit, is explicitly computed as a modular invariant sum over spin strutures required by perturbative unitarity in order to extend the analysis to include type II strings on R^6 x W4, where W4 is associated with the tube metric conformal field theory, given by the degrees of freedom transverse to the Neveu-Schwarz fivebrane solution. This generates partition functions and perturbative spectra of string theories in six space-time dimensions, associated with the modular invariants of the level k affine SU(2) Kac-Moody algebra. These theories provide a conformal field theory (i.e. perturbative) probe of non-perturbative (fivebrane) vacua. We contrast them with theories whose N=(4,4) sigma-model action contains n_H=k+2 hypermultiplets as well as vector supermultiplets, and where k is the level just mentioned. In Appendix B we also give a D=6, N=(1,1) `free fermion' string model which has a different moduli space of vacua from the 81 parameter space relevant to the above examples. 
  For non-abelian non-supersymmetric gauge theories, generic dual theories have been constructed. In these theories the couplings appear inverted. However, they do not possess a Yang-Mills structure but rather are a kind of non-linear sigma model. It is shown that for a topological gravitational model an analog to this duality exists. 
  We consider the maximal central extension of the supertranslation algebra in d=4 and 3, which includes tensor central charges associated to topological defects such as domain walls (membranes) and strings. We show that for all N-extended superalgebras these charges are related to nontrivial configurations on the scalar moduli space. For N=2 theories obtained from compactification on Calabi-Yau threefolds, we give an explicit realization of the moduli-dependent charges in terms of wrapped branes. 
  We study configurations of a single $M$ fivebrane in the geometry ${\bf R}^7 \times Q$ where $Q$ is the Taub-NUT space. Taking the IIA limit at each value of their modulus, two possibilities of the IIA configuration are revealed. One consists of a NS fivebrane and a D sixbrane while the other consists of a NS fivebrane, a D sixbrane and a D fourbrane. In the latter case the fourbrane is shown to be suspended by the fivebrane and the sixbrane. This appearance of fourbrane can be interpreted in Type IIA picture as the result of the crossing of the fivebrane and the sixbrane 
  In this contribution I summarize the achievements of separation of variables in integrable quantum systems from the point of view of path integrals. This includes the free motion on homogeneous spaces, and motion subject to a potential force, and I would like to propose systematic investigations of parametric coordinate systems on homogeneous spaces. 
  In this Letter I present an alternative solution of the path integral for the radial Coulomb problem which is based on a two-dimensional singular version of the Levi-Civita transformation. 
  Using normal coordinate expansions we derive by purely superspace methods the density formula giving the component action corresponding to a superspace supergravity-matter action. 
  A quantum theory is developed for the scattering of a nonrelativistic particle in the field of a cosmic string regarded as a combination of a magnetic and gravitational strings. Allowance is made for the effects due to the finite transverse dimensions of the string under fairly general assumptions about the distribution of the magnetic field and spatial curvature in the string. It is shown that in a definite range of angles the differential cross section at all absolute values of the wave vector of the incident particle depends strongly on the magnetic flux of the string. 
  The geometry of antisymmetric fields with nontrivial transitions over a base manifold is described in terms of exact sequences of cohomology groups. This formulation leads naturally to the appearance of nontrivial topological charges associated to the periods of the curvature of the antisymmetric fields. The relation between the partition functions of dual theories is carefully studied under the most general assumptions, and new topological factors related to zero modes and the Ray Singer torsion are found. 
  A non-relativistic version of the 2+1 dimensional gauged Chern-Simons O(3) sigma model, augmented by a Maxwell term, is presented and shown to support topologically stable static self-dual vortices. Exactly like their counterparts of the ungauged model, these vortices are shown to exhibit Hall behaviour in their dynamics. 
  We study D-branes on three-dimensional orbifold backgrounds that admit topologically distinct resolutions differing by flop transitions. We show that these distinct phases are part of the vacuum moduli space of the super Yang-Mills gauge theory describing the D-brane dynamics. In this way we establish that D-branes --- like fundamental strings --- allow for physically smooth topology changing transitions. 
  In this letter we study a coupled system of six-dimensional N=1 tensor and super Yang-Mills multiplets. We identify some of the solitonic states of this system which exhibit stringy behaviour in six dimensions. A discussion of the supercharges and energy for the tensor multiplet as well as zero modes is also given. We speculate about the possible relationship between our solution and what is known as tensionless strings. 
  Chiral symmetry is dynamically broken in quenched, ladder QED at weak gauge couplings when an external magnetic field is present. In this paper, we show that chiral symmetry is restored above a critical chemical potential and the corresponding phase transition is of first order. In contrast, the chiral symmetry restoration at high temperatures (and at zero chemical potential) is a second order phase transition. 
  Spacetime superalgebras with 64 or less number of real supercharges, containing the type IIB Poincare superalgebra in (9,1) dimensions and the N=1 Poincare superalgebra in (10,1) are considered. The restriction D<14, and two distinct possibilities arise: The N=(1,0) superalgebra in (11,3) dimensions, and the N=(2,0) superalgebra in (10,2) dimensions. Emphasizing the former, we describe superparticle and super Yang-Mills systems in (11,3) dimensions. We also propose an N=(2,1) superstring theory in (n,n) dimensions as a possible origin of super Yang-Mills in (8+n,n) dimensions. 
  A model for $n$ superparticles in $(d-n,n)$ dimensions is studied. The target space supersymmetry involves a product of $n$ momentum generators, and the action has $n(n+1)/2$ local bosonic symmetries and $n$ local fermionic symmetries. The precise relation between the symmetries presented here and those existing in the literature is explained. A new model is proposed for superparticles in arbitrary dimensions where coordinates are associated with all the $p$-form charges occuring in the superalgebra. The model naturally gives rise to the BPS condition for the charges. 
  An indication of spontaneous symmetry breaking is found in the two-dimensional $\lambda\phi^4$ model, where attention is paid to the functional form of an effective action. An effective energy, which is an effective action for a static field, is obtained as a functional of the classical field from the ground state of the hamiltonian $H[J]$ interacting with a constant external field. The energy and wavefunction of the ground state are calculated in terms of DLCQ (Discretized Light-Cone Quantization) under antiperiodic boundary conditions. A field configuration that is physically meaningful is found as a solution of the quantum mechanical Euler-Lagrange equation in the $J\to 0$ limit. It is shown that there exists a nonzero field configuration in the broken phase of $Z_2$ symmetry because of a boundary effect. 
  Type IIB string theory admits a BPS configuration in which three strings (of different type) meet at a point. Using this three string configuration we construct a string network and study its properties. In particular we prove supersymmetry of this configuration. We also consider string lattices, which can be used to construct BPS states in toroidally compactified string theory. 
  In this note we study the resolution of conifold singularity by D-branes by considering compactification of D-branes on $\C^3/(\Z_2\times\Z_2)$. The resulting vacuum moduli space of D-branes is a toric variety which turns out to be a resolved conifold, that is a nodal variety in $\C^4$. This has the implication that all the corresponding phases of Type--II string theory are geometrical and are accessible to the D-branes, since they are related by flops. 
  Mass perturbations of the twisted N=4 supersymmetric gauge theory considered by Vafa and Witten to test S-duality are studied for the case of Kahler four-manifolds. It is shown that the resulting mass-perturbed theory can be regarded as an equivariant extension associated to a U(1) symmetry of the twisted theory, which is only present for Kahler manifolds. In addition, it is shown that the partition function, the only topological invariant of the theory, remains invariant under the perturbation. 
  This paper reviews recent work on a new geometric object called a bundle gerbe and discusses some new examples arising in quantum field theory. One application is to an Atiyah-Patodi-Singer index theory construction of the bundle of fermionic Fock spaces parametrized by vector potentials in odd space dimensions and a proof that this leads in a simple manner to the known Schwinger terms (Mickelsson-Faddeev cocycle) for the gauge group action. This gives an explicit computation of the Dixmier-Douady class of the associated bundle gerbe. The method works also in other cases of fermions in external fields (external gravitational field, for example) provided that the APS theorem can be applied; however, we have worked out the details only in the case of vector potentials. Another example, in which the bundle gerbe curvature plays a role, arises from the WZW model on Riemann surfaces. A further example is the `existence of string structures' question. We conclude by showing how global Hamiltonian anomalies fit within this framework. 
  The graceful exit transition from a dilaton-driven inflationary phase to a decelerated Friedmann-Robertson-Walker era requires certain classical and quantum corrections to the string effective action. Classical corrections can stabilize a high curvature string phase while the evolution is still in the weakly coupled regime, and quantum corrections can induce violation of the null energy condition, allowing evolution towards a decelerated phase. 
  Manifest N=2 supersymmetric hypermultiplet mass terms can be introduced in the projective N=2 superspace formalism. In the case of complex hypermultiplets, where the gauge covariantized spinor derivatives have an explicit representation in terms of gauge prepotentials, it is possible to interpret such masses as vacuum expectation values of an Abelian vector multiplet. The duality transformation that relates the N=2 off-shell projective description of the hypermultiplet to the on-shell description involving two N=1 chiral superfields allows us to obtain the massive propagators of the N=1 complex linear fields in the projective hypermultiplet. The N=1 massive propagators of the component superfields in the projective hypermultiplet suggest a possible ansatz for the N=2 massive propagator, which agrees with an explicit calculation in N=2 superspace. 
  BRST quantization is carried out for a model of p-branes with second class constraints. After extension of the phase space the constraint algebra coincides with the one of null string when p=1. It is shown that in this case one can or can not obtain critical dimension for the null string, depending on the choice of the operator ordering and corresponding vacuum states. When p>1, operator orderings leading to critical dimension in the p=1 case are not allowed. Admissable orderings give no restrictions on the dimension of the embedding space-time. Finally, a generalization to supersymmetric null branes is proposed. 
  The problem of gauge invariance of the physical sector of (2+1)-dimensional Maxwell-Chern-Simons quantum electrodynamics (QED$_{2+1}$) is studied. It is shown that using Proca mass term for the infrared regularization one obtains gauge-invariant fermion mass and the physical mass shell of QED$_{2+1}$ is well-defined in all orders of the perturbation theory. We are demonstrating also a class of gauges in the framework of QED$_{2+1}$, including transversal and Feynman-like ones, where the physical sector is well defined and independent of the gauge parameter. 
  We derive the Bekenstein-Hawking entropy formula for four and five dimensional non-supersymmetric black holes (which include the Schwarzchild ones) by counting microscopic states. This is achieved by first showing that these black holes are U-dual to the three-dimensional black hole of Banados-Teitelboim-Zanelli and then counting microscopic states of the latter following Carlip's approach. Higher than five dimensional black holes are also considered. We discuss the connection of our approach to the D-brane picture. 
  Well known methods of measure theory on infinite dimensional spaces are used to study physical properties of measures relevant to quantum field theory. The difference of typical configurations of free massive scalar field theories with different masses is studied. We apply the same methods to study the Ashtekar-Lewandowski (AL) measure on spaces of connections. We prove that the diffeomorphism group acts ergodically, with respect to the AL measure, on the Ashtekar-Isham space of quantum connections modulo gauge transformations. We also prove that a typical, with respect to the AL measure, quantum connection restricted to a (piecewise analytic) curve leads to a parallel transport discontinuous at every point of the curve. 
  We report on investigations of local (and non-local) charges in bosonic and supersymmetric principal chiral models in 1+1 dimensions. In the bosonic PCM there is a classically conserved local charge for each symmetric invariant tensor of the underlying group. These all commute with the non-local Yangian charges. The algebra of the local charges amongst themselves is rather more subtle. We give a universal formula for infinite sets of mutually commuting local charges with spins equal to the exponents of the underlying classical algebra modulo its Coxeter number. Many of these results extend to the supersymmetric PCM, but with local conserved charges associated with antisymmetric invariants in the Lie algebra. We comment briefly on the quantum conservation of local charges in both the bosonic and super PCMs. 
  The modified Seiberg-Witten monopole equations are presented in this letter. These equations have analytic solutions in the whole 1+3 Minkowski space with finite energy. The physical meaning of the equations and solutions are discussed here. 
  We reconsider the Abelian pure Chern-Simons theory in three dimensions by using our improved Batalin-Fradkin-Tyutin Hamiltonian formalism. As a result, we show several novel features, including the connection of the Dirac brackets. In particular, through the path integral quantization, we obtain the desired new type of the Wess-Zumino action. 
  The construction of four dimensional supersymmetric gauge theories via the fivebrane of M theory wrapped around a Riemann surface has been successfully applied to the computation of holomorphic quantities of field theory. In this paper we compute non-holomorphic quantities in the eleven dimensional supergravity limit of M theory. While the Kahler potential on the Coulomb of N=2 theories is correctly reproduced, higher derivative terms in the N=2 effective action differ from what is expected for the four dimensional gauge theory. For the Kahler potential of N=1 theories at an abelian Coulomb phase, the result again differs from what is expected for the four-dimensional gauge theory. Using a gravitational back reaction method for the fivebrane we compute the metric on the Higgs branch of N=2 gauge theories. Here we find an agreement with the results expected for the gauge theories. A similar computation of the metric on N=1 Higgs branches yields information on the complex structure associated with the flavor rotation in one case and the classical metric in another. We discuss what four-dimensional field theory quantities can be computed via the fivebrane in the supergravity limit of M theory. 
  Imposing initial conditions to nonequilibrium systems at some time t_0 leads, in renormalized quantum field theory, to the appearance of singularities in the variable t-t_0 in relevant physical quantities, such as energy density and pressure. The origin of these `initial singularities' can be traced back to the choice of initial state. We construct here, by a Bogoliubov transformation, initial states such that these singularities are eliminated. While the construction is not unique it can be considered a minimal way of taking into account the nonequilibrium evolution of the system prior to t_0. 
  According to the proposal of Hanany and Witten, Coulomb branches of N=4 SU(n) gauge theories in three dimensions are isometric to moduli spaces of BPS monopoles. We generalize this proposal to gauge theories with matter, which allows us to describe the metrics on their spaces of vacua by means of the hyperk\"ahler quotient construction. To check the identification of moduli spaces a comparison is made with field theory predictions. For SU(2) theory with k fundamental hypermultiplets the Coulomb branch is expected to be the D_k ALF gravitational instanton, so our results lead to a construction of such spaces. In the special case of SU(2) theory with four or fewer fundamental hypermultiplets we calculate the complex structures on the moduli spaces and compare them with field-theoretical results. We also discuss some puzzles with brane realizations of three-dimensional N=4 theories. 
  Form factor sequences of an integrable QFT can be defined axiomatically as solutions of a system of recursive functional equations, known as ``form factor equations''. We show that their solution can be replaced with the study of the representation theory of a novel algebra F(S). It is associated with a given two-particle S-matrix and has the following features: (i) It contains a double TTS algebra as a subalgebra. (ii) Form factors arise as thermal vector states over F(S) of temperature 1/2\pi. The thermal ground states are in correspondence to the local operators of the QFT. (iii) The underlying `finite temperature structure' is indirectly related to the ``Unruh effect'' in Rindler spacetime. In F(S) it is manifest through modular structures (j,\delta) in the sense of algebraic QFT, which can be implemented explicitly in terms of the TTS generators. 
  Basing on the field theory of gravity and observable parameters of the expanding Universe the upper limit of $m_g \leq 4.5 . 10^{-66} g$. on the value of possible graviton mass is derived. 
  We review some of the progress in affine Toda field theories in recent years, explain why known dualities cannot easily be extended, and make some suggestions for what should be sought instead. 
  We study the compatiblity between the higher dimension dualities and the Yang-Mills equation of motion. Taking a 't Hooft solution as a starting point, we come to the conclusion that for only 4 dimensions the self duality implies the equation of motion for generic instanton size. Whereas in higher dimensions, the self duality is compatable with the equation of motion, approximately, for small instanton size i.e. the zero curvature condition. At the mathematical level, the self duality is still useful since it transforms a second order into a first order differential equation. 
  We construct an RG potential for N=2 supersymmetric SU(2) Yang-Mills theory, and extract a positive definite metric by comparing its gradient with the recently discovered beta-function for this system, thus proving that the RG flow is gradient in this four-dimensional field theory. We also discuss how this flow might change after supersymmetry breaking, provided the quantum symmetry group does not, emphasizing the non-trivial problem of asymptotic matching of automorphic functions to perturbation theory. 
  We give efficient superspace methods for deriving component actions for supergravity coupled to matter. One method uses normal coordinates to covariantly expand the superfield action, and can be applied straightforwardly to any superspace. The other interprets the component lagrangian as a differential form on a bosonic hypersurface in superspace, and gives a simple derivation for pertinent cases such as chiral superspace. 
  This paper considers the Schroedinger propagator on a cone with the conical singularity carrying magnetic flux (``flux cone''). Starting from the operator formalism and then combining techniques of path integration in polar coordinates and in spaces with constraints, the propagator and its path integral representation are derived. "Quantum correction" in the Lagrangian appears naturally and no a priori assumption is made about connectivity of the configuration space. 
  In this review we try to give a pedagogical introduction to the recent progress in the resolution of old problems of black hole thermodynamics within superstring theory. We start with a brief description of classical black hole dynamics. Then, follow with the consideration of general properties of supersymmetric black holes. We conclude with the review of the statistical explanation of the black hole entropy and string theory description of the black hole evaporation. 
  We show in a model-independent way that, in the background of a topological soliton or instanton that saturates a Bogomol'nyi bound, the fermion and boson excitation spectra of non-zero modes cancel at the one-loop level. This generalizes D'Adda and DiVecchia's result for some specific instanton models. Our method also establishes, again in a model-independent way, the generality of the connection between zero modes in topologically non-trivial backgrounds and index theorems. 
  We study the logarithmic conformal field theories in which conformal weights are continuous subset of real numbers. A general relation between the correlators consisting of logarithmic fields and those consisting of ordinary conformal fields is investigated. As an example the correlators of the Coulomb-gas model are explicitly studied. 
  We discuss $5d$ and $6d$ supersymmetric gauge theories in the target-space with compactified directions and with the matter hypermultiplets in fundamental representations in the framework of integrable systems. In particular, we consider the prepotentials of these theories and derive explicit formulas for their perturbative parts. 
  We search for dual gauge theories of all-loop finite, N = 1 supersymmetric gauge theories. It is shown how to find explicitly the dual gauge theories of almost all chiral, N = 1, all-loop finite gauge theories, while several models have been discussed in detail, including a realistic finite SU(5) unified theory. Out of our search only one all-loop, N = 1 finite SO(10) theory emerges, so far, as a candidate for exhibiting also S-duality. 
  We provide a simple macroscopic analysis of the four-dimensional effective supergravity of the Ho\v{r}ava-Witten M-theory which is expanded in powers of $\kappa^{2/3}/\rho V^{1/3}$ and $\kappa^{2/3}\rho/V^{2/3}$ where $\kappa^2$, $V$ and $\rho$ denote the eleven-dimensional gravitational coupling, the Calabi-Yau volume and the eleventh length respectively. Possible higher order terms in the K\"ahler potential are identified and matched with the heterotic string corrections. In the context of this M-theory expansion, we analyze the soft supersymmetry-breaking terms under the assumption that supersymmetry is spontaneously broken by the auxiliary components of the bulk moduli superfields. It is examined how the pattern of soft terms changes when one moves from the weakly coupled heterotic string limit to the M-theory limit. 
  We calculate a temperature dependent part of the one-loop thermodynamic potential (and the free energy) for charged massive fields in a general class of irreducible rank 1 symmetric spaces. Both low- and high-temperature expansions are derived and the role of non-trivial topology influence on asymptotic properties of the potential is discussed. 
  We construct BPS states in the matrix description of M-theory. Starting from a set of basic M-theory branes, we study pair intersections which preserve supersymmetry. The fractions of the maximal supersymmetry obtained in this way are 1/2, 1/4, 1/8, 3/16 and 1/16. In explicit examples we establish that the matrix BPS states correspond to (intersecting) brane configurations that are obtained from the d=11 supersymmetry algebra. This correspondence for the 1/2 supersymmetric branes includes the precise relations between the charges. 
  We present a gauge-fixed M 5-brane action: a 6-dimensional field theory of a self-interacting (0,2) tensor multiplet with 32 worldvolume supersymmetries. The quadratic part of this action is shown to be invariant under rigid OSp(6,2|4) superconformal symmetry, with 16 supersymmetries and 16 special supersymmetries. We explore a deep relation between the superconformal symmetry on the worldvolume of the brane and symmetry of the near horizon anti-de Sitter infinite throat geometry of the M 5-brane in space-time. 
  We study toroidal compactification of Matrix theory, using ideas and results of non-commutative geometry. We generalize this to compactification on the noncommutative torus, explain the classification of these backgrounds, and argue that they correspond in supergravity to tori with constant background three-form tensor field. The paper includes an introduction for mathematicians to the IKKT formulation of Matrix theory and its relation to the BFSS Matrix theory. 
  We discuss the scattering of two light particles in a D-brane background. It is known that, if one light particle strikes the D brane at small impact parameter, quantum recoil effects induce entanglement entropy in both the excited D brane and the scattered particle. In this paper we compute the asymptotic `out' state of a second light particle scattering off the D brane at large impact parameter, showing that it also becomes mixed as a consequence of quantum D-brane recoil effects. We interpret this as a non-factorizing contribution to the superscattering operator S-dollar for the two light particles in a Liouville D-brane background, that appears when quantum D-brane excitations are taken into account. 
  The semiclassical solution of quantum Dirac constraints in generic constrained system is obtained by directly calculating in the one-loop approximation the gauge field path integral with relativistic gauge fixing procedure. The gauge independence property of this path integral is analyzed by the method of Ward identities with a special emphasis on boundary conditions for gauge fields. The calculations are based on the known reduction algorithms for functional determinants extended to gauge theories. The mechanism of transition from relativistic gauge conditions to unitary gauges, participating in the construction of this solution, is explicitly revealed. Implications of this result in problems with spacetime boundaries, quantum gravity and cosmology are briefly discussed. 
  We show that in certain superstring compactifications, gauge theories on noncommutative tori will naturally appear as D-brane world-volume theories. This gives strong evidence that they are well-defined quantum theories. It also gives a physical derivation of the identification proposed by Connes, Douglas and Schwarz of Matrix theory compactification on the noncommutative torus with M theory compactification with constant background three-form tensor field. 
  We propose a generalization of the Veneziano-Yankielowicz effective low-energy action for N=1 SUSY Yang-Mills theory which includes composite operators interpolating pure gluonic bound states. The chiral supermultiplet of anomalies is embedded in a larger three-form multiplet and an extra term in the effective action is introduced. The mass spectrum and mixing of the lowest-spin bound states are studied within the effective Lagrangian approach. The physical mass eigenstates form two multiplets, each containing a scalar, pseudoscalar and Weyl fermion. The multiplet containing the states which are most closely related to glueballs is the lighter one. 
  A consistent local approach to the study of interacting relativistic fermion systems with a condensation of bare particles in its ground or vacuum state, which may has a finite matter density, is developed. The attention is payed to some of the not so well explored quantum aspects that survive the thermodynamic limit. A 4-vector local field, called the primary statistical gauge field, and a statistical blocking parameter are introduced for a consistent treatment of the problem. The effects of random fluctuations of the fields on local observables are discussed. It is found that quasiparticle contributions are not sufficient to saturate local observables. The property of the primary statistical gauge field are discussed in some detail. Two models for the strong interaction are then introduced and studied using the general framework developed. Four possible phases for these models are found. The possibility of spontaneous CP violation and local fermion creation in two of the four phases is revealed. The implications of the finding on our understanding of some of the strong interaction processes are discussed. 
  Using the mode-by-mode summation technique the zero point energy of the electromagnetic field is calculated for the boundary conditions given on the surface of an infinite solid cylinder. It is assumed that the dielectric and magnetic characteristics of the material which makes up the cylinder $(\epsilon_1, \mu_1)$ and of that which makes up the surroundings $(\epsilon_2, \mu_2)$ obey the relation $\epsilon_1\mu_1= \epsilon_2\mu_2$. With this assumption all the divergences cancel. The divergences are regulated by making use of zeta function techniques. Numerical calculations are carried out for a dilute dielectric cylinder and for a perfectly conducting cylindrical shell. The Casimir energy in the first case vanishes, and in the second is in complete agreement with that obtained by DeRaad and Milton who employed a Green's function technique with an ultraviolet regulator. 
  For the SU(N) invariant supersymmetric matrix model related to membranes in 4 space-time dimensions we argue that <Psi,chi> = 0 for the previously obtained solution of Q chi = 0, Q^{dagger} Psi = 0. 
  We discuss the phenomenon of classical anomaly. It is observed for 3D Berezin-Marinov (BM), Barducci-Casalbuoni-Lusanna (BCL) and Cortes-Plyushchay-Velazquez (CPV) pseudoclassical spin particle models. We show that quantum mechanically these different models correspond to the same P,T-invariant system of planar fermions, but the quantum system has global symmetries being not reproducible classically in full in any of the models. We demonstrate that the specific U(1) gauge symmetry characterized by the opposite coupling constants of spin s=+1/2 and s=-1/2 states has a natural classical analog in CPV model but can be reproduced in BM and BCL models in an obscure and rather artificial form. We also show that BM and BCL models quantum mechanically are equivalent in any odd-dimensional space-time, but describe different quantum systems in even space-time dimensions. 
  The formulation of the local BRST cohomology on infinite jet bundles and its relation and reduction to gauge covariant algebras are reviewed. As an illustration, we compute the local BRST cohomology for geodesic motion in (pseudo-) Riemannian manifolds and discuss briefly the result (symmetries, constants of the motion, consistent deformations). 
  We propose an extension of a recent non-perturbative method suited for solving the N-body problem in (2+1)-gravity to the case of Chern-Simons supergravity. Coupling with supersymmetric point particles is obtained implicitly by extending the DJH matching conditions of gravity. The consistent solution of the interacting case is obtained by building a general non-trivial mapping, extending the superanalytic mapping, between a flat polydromic $X^M$ supercoordinate system and a physical one $x^N$, representing the DJH matching conditions around the superparticles. We show how to construct such a mapping in terms of analytic functions, and we give their exact expressions for the two body case. The extension to the N body case is also discussed. In the Minkoskian coordinates the superparticles move freely, and in particular the fermionic coordinates $\Theta(\xi^N_{(i)})$ are constants, whose values can be fixed by using the monodromy properties. While the bosonic part of the supergeodesic equations are obtained, as in gravity, by measuring the bosonic distance in Minkowskian space-time, we find that the fermionic geodesic equations can be defined only by requiring that a non-perturbative divergence of the $X^M = X^M(x^N)$ mapping cancels out on the world-lines of the superparticles.  
  We derive sum rules for the magnetic and electric dipole moments of all particle states of an N=2 supermultiplet. For short representations, we find agreement with previously determined N=1 sum rules, while there is added freedom for long representations (expressed as certain scalar expectation values). With mild assumptions we find the simple result that the supersymmetry generated spin adds to the magnetic (electric) dipole moment with strength corresponding to $g=2$ ($g_e=0$). This result is equally valid for N=1, this time without any further assumptions. 
  We show how the gauge invariant formulation of QCD in terms of loops is free from a hidden $\theta$ parameter and offers an alternative way to solve the $U(1)_A$ puzzle. 
  We discuss on the possible existence of a supersymmetric invariance in purely fermionic planar systems and its relation to the fermion-boson mapping in three-dimensional quantum field theory. We consider, as a very simple example, the bosonization of free massive fermions and show that, under certain conditions on the masses, this model displays a supersymmetric-like invariance in the low energy regime. We construct the purely fermionic expression for the supercurrent and the non-linear supersymmetry transformation laws. We argue that the supersymmetry is absent in the limit of massless fermions where the bosonized theory is non-local. 
  An arbitrary renormalizable quantum field theory is considered as finite if its dimensionless couplings conspire to yield, at every order of its perturbative expansion, no ultraviolet-divergent renormalizations of the physical parameters of the theory. The "finiteness conditions" resulting from these requirements form highly complicated, non-linear systems of relations. A promising type of solution to the condition for one-loop finiteness of the Yukawa couplings involves Yukawa couplings which are equivalent to the generators of Clifford algebras with identity element. However, our attempt to construct even one finite model based on such Clifford-like Yukawa couplings fails: a Clifford structure of the Yukawa couplings spoils the finiteness of the gauge couplings, at least for every simple gauge group of rank less than or equal to 8. 
  A geometrical analysis is given of Dirichlet fourbrane creation, when sixbrane crosses fivebrane in M-theory. A special property of the Taub-NUT space leads to the consequence. When brane configurations are considered for four dimensional N=2 field theories, sixbrane contributes to the beta-function through the Dirac string of the Taub-NUT space. 
  We develop techniques to classify D- and F-flat directions for N=1 supersymmetric string vacua of the perturbative heterotic string theory, which possess an anomalous U(1) gauge group at the tree level. Genus-one corrections generate a Fayet-Iliopoulos term for the D-term of U(1)_A, which is canceled by non-zero vacuum expectation values (VEVs) of certain massless multiplets in such a way that the anomalous U(1) is broken, while maintaining the D- and F-flatness of the effective field theory. A systematic analysis of flat directions is given for non-zero VEVs of non-Abelian singlets, and the techniques are illustrated for a specific model. The approach sets the stage to classify the D- and F-flat directions for a large class of perturbative string vacua. This classification is a prerequisite to address systematically the phenomenological consequences of these models. 
  Using U-duality, the properties of the matrix theories corresponding to the compactification of M-theory on $T^d$ are investigated. The couplings of the $d+1$ dimensional effective Super-Yang-Mills theory to all the M-theory moduli is deduced and the spectrum of BPS branes in the SYM gives the corresponding spectrum of the matrix theory.Known results are recovered for $d\le 5$ and predictions for $d>5$ are proposed. For $d>3$, the spectrum includes $d-4$ branes arising from YM instantons, and U-duality interchanges momentum modes with brane wrapping modes.For $d=6$, there is a generalised $\th $-angle which couples to instantonic 3-branes and which combines with the SYM coupling constant to take values in $SL(2,\R)/U(1)$, acted on by an $SL(2,\Z)$ subgroup of the U-duality group $E_6(\Z)$. For $d=4,7,8$, there is an $SL(d+1)$ symmetry, suggesting that the matrix theory could be a scale-invariant $d+2$ dimensional theory on $T^{d+1} \times \R$ in these cases, as is already known to be the case for $d=4$; evidence is found suggesting this happens for $d=8$ but not $d=7$. 
  The metric-affine gauge theory of gravity provides a broad framework in which gauge theories of gravity can be formulated. In this article we fit metric-affine gravity into the covariant BRST--antifield formalism in order to obtain gauge fixed quantum actions. As an example the gauge fixing of a general two-dimensional model of metric-affine gravity is worked out explicitly. The result is shown to contain the gauge fixed action of the bosonic string in conformal gauge as a special case. 
  We study static, spherically symmetric, and purely magnetic solutions of the N=4 gauged supergravity in four dimensions. A systematic analysis of the supersymmetry conditions reveals solutions which preserve 1/4 of the supersymmetries and are characterized by a BPS-monopole-type gauge field and a globally hyperbolic, everywhere regular geometry. We show that the theory in which these solutions arise can be obtained via compactification of ten-dimensional supergravity on the group manifold. This result is then used to lift the solutions to ten dimensions. 
  The non-local regularization is a powerfull method to regularize theories with an action that can be decomposed in a kinetic and an interacting part. Recently it was shown how to regularize the Batalin-Vilkovisky field-antifield formalism of quantization of gauge theories with the non-local regularization. We compute precisely the anomaly of the Chiral Schwinger model with this extended non-local regularization. 
  The Wigner phase-space distribution function provides the basis for Moyal's deformation quantization alternative to the more conventional Hilbert space and path integral quantizations. General features of time-independent Wigner functions are explored here, including the functional ("star") eigenvalue equations they satisfy; their projective orthogonality spectral properties; their Darboux ("supersymmetric") isospectral potential recursions; and their canonical transformations. These features are illustrated explicitly through simple solvable potentials: the harmonic oscillator, the linear potential, the Poeschl-Teller potential, and the Liouville potential. 
  We study the exact equivalence between the self-dual model minimally coupled with a Dirac field and the Maxwell-Chern-Simons model with non-minimal magnetic coupling to fermions. We show that the fermion sectors of the models are equivalent only if a Thirring like interaction is included. Using functional methods we verify that, up to renormalizations, the equivalence persists at the quantum level. 
  We study the scaling limits of the L-state Restricted Solid-on-Solid (RSOS) lattice models and their fusion hierarchies in the off-critical regimes. Starting with the elliptic functional equations of Klumper and Pearce, we derive the Thermodynamic Bethe Ansatz (TBA) equations of Zamolodchikov. Although this systematic approach, in principle, allows TBA equations to be derived for all the excited states we restrict our attention here to the largest eigenvalue or groundstate in Regimes III and IV. In Regime III the TBA equations are massive while in Regime IV there is massless scattering describing the renormalization group flow between distinct A_1^{(1)} coset conformal field theories. Regimes I and II, pertaining to Z_{L-1} parafermions, will be treated in a subsequent paper. 
  We quantize the (2+1)-dimensional self-dual and Maxwell-Chern-Simons theories by using the Faddeev-Jackiw formulation and compare the results with those of the Dirac formalism. 
  Classical vacuum - pure gauge - solutions of Euclidean two-dimensional SU(2) Yang-Mills theories are studied. Topologically non-trivial vacua are found in a class of gauge group elements isomorphic to $S_2$. These solutions are unexpectedly related to the solution of the non-linear O(3) model and to the motion of a particle in a periodic potential. 
  It is by now well established that, by means of the integration by part identities, all the integrals occurring in the evaluation of a Feynman graph of given topology can be expressed in terms of a few independent master integrals. It is shown in this paper that the integration by part identities can be further used for obtaining a linear system of first order differential equations for the master integrals themselves. The equations can then be used for the numerical evaluation of the amplitudes as well as for investigating their analytic properties, such as the asymptotic and threshold behaviours and the corresponding expansions (and for analytic integration purposes, when possible). The new method is illustrated through its somewhat detailed application to the case of the one loop self-mass amplitude, by explicitly working out expansions and quadrature formulas, both in arbitrary continuous dimension n and in the n \to 4 limit. It is then shortly discussed which features of the new method are expected to work in the more general case of multi-point, multi-loop amplitudes. 
  The variational ansatz for the ground state wavefunctional of QCD is found to capture the anti-screening behaviour that contributes the dominant `-4' to the beta-function and leads to asymptotic freedom. By considering an SU(N) purely gauge theory in the Hamiltonian formalism and choosing the Coulomb gauge, the origins of all screening and anti-screening contributions in gluon processes are found in terms of the physical degrees of freedom. The overwhelming anti- screening contribution of `-4' is seen to originate in the renormalisation of a Coulomb interaction by a transverse gluon. The lesser screening contribution of `1/3' is seen to originate in processes involving transverse gluon interactions. It is thus apparent how the variational ansatz must be developed to capture the full running of the QCD coupling constant. 
  We obtain using Schwinger's proper time approach the Casimir-Euler-Heisenberg effective action of fermion fluctuations for the case of an applied magnetic field. We implement here the compactification of one space dimension into a circle through anti-periodic boundary condition. Aside of higher order non-linear field effects we identify a novel contribution to the vacuum permeability. These contributions are exceedingly small for normal electromagnetism due to the smallness of the electron Compton wavelength compared to the size of the compactified dimension, if we take the latter as the typical size of laboratory cavities, but their presence is thought provoking, also considering the context of strong interactions. 
  Considering three-dimensional Chern-Simons theory, either coupled to matter or with a Yang-Mills term, we show the validity of a trace identity, playing the role of a local form of the Callan-Symanzik equation, in all orders of perturbation theory. From this we deduce the vanishing of the $\beta$-function associated to the Chern-Simons coupling constant and the full finiteness in the case of the Yang-Mills Chern-Simons theory. The main ingredient in the proof of the latter property is the noninvariance of the Chern-Simons form under the gauge transformations. Our results hold for the three-dimensional Chern-Simons model in a general Riemannian manifold. 
  In the framework of the four dimensional heterotic superstring with free world-sheet fermions we study the GUSTs which contain non-abelian horizontal symmetry. To solve the problem of breaking the G_GUT gauge symmetry of the level one current algebra we extend the gauge symmetry to G x G. Using the autoduality of the fermionic charge lattice we study the possibilities of constructing of some GUST gauge groups. We also study the string amplitudes and for one realistic model explicitly calculate the renormalizable and non-renormalizable (N=4,5,6) contributions to the superpotential. We observe that the existence of the mass matrix ansatz depends on the way of the breaking of the horizontal gauge symmetry. 
  An investigation of the Nambu-Jona-Lasinio model with external constant electromagnetic and weak gravitational fields is carried out in three- and four-dimensional spacetimes. The effective potential of the composite bifermionic fields is calculated keeping terms linear in the curvature, while the electromagnetic field effect is treated exactly by means of the proper-time formalism. Numerical simulations of the dynamical symmetry breaking phenomenon accompanied by some phase transitions are presented. 
  In the theory of quantum cohomologies the WDVV equations imply integrability of the system $(I\partial_\mu - zC_\mu)\psi = 0$. However, in generic situation -- of which an example is provided by the Seiberg-Witten theory -- there is no distinguished direction (like $t^0$) in the moduli space, and such equations for $\psi$ appear inconsistent. Instead they are substituted by $(C_\mu\partial_\nu - C_\nu\partial_\mu)\psi^{(\mu)} \sim (F_\mu\partial_\nu - F_\nu\partial_\mu)\psi^{(\mu)} = 0$, where matrices $(F_\mu)_{\alpha\beta} = \partial_\alpha \partial_\beta \partial_\mu F$. 
  We derive an exact equation of motion for a non-relativistic vortex in two- and three-dimensional models with a complex field. The velocity is given in terms of gradients of the complex field at the vortex position. We discuss the problem of reducing the field dynamics to a closed dynamical system with non-locally interacting strings as the fundamental degrees of freedom. 
  We suggest a super Poincar\'e invariant action for closed eleven dimensional superstring. The sector of physical variables $x^i$, $\theta_a$, $\bar\theta_{\dot a}$, with $a,\dot a=1...8$ and $x^i$ the transverse part of the D=11 $x^\mu$ coordinate is shown to possess free dynamics. 
  In the low energy limit of for M-theory on S^1/Z_2, we calculate the gaugino condensate potential in four dimensions using the background solutions due to Horava. We show that this potential is free of delta-function singularities and has the same form as the potential in the weakly coupled heterotic string. A general flux quantization rule for the three-form field of M-theory on S^1/Z_2 is given and checked in certain limiting cases. This rule is used to fix the free parameter in the potential originating from a zero mode of the form field. Finally, we calculate soft supersymmetry breaking terms. We find that corrections to the Kahler potential and the gauge kinetic function, which can be large in the strongly coupled region, contribute significantly to certain soft terms. In particular, for supersymmetry breaking in the T-modulus direction, the small values of gaugino masses and trilinear couplings that occur in the weakly coupled, large radius regime are enhanced to order m_3/2 in M-theory. The scalar soft masses remain small even, in the strong coupling M-theory limit. 
  A variational calculation is presented of the ADM-energy of the quantized gravitational field around a wormhole solution of the classical Einstein's equations. One finds the energy of such state to be in general lower than the perturbative ground state, in which the quantized gravity field fluctuates around flat (Euclidean) space-time. As a result the strong indication emerges that a gas (or a lattice) of wormholes of Planck mass and average distance $l_p$, the Planck length, may be a good approximation of the Ground State of Quantum Gravity, some implications of which are reviewed. 
  In this paper the complete geometrical setting of (lowest order) abelian T-duality is explored with the help of some new geometrical tools (the reduced formalism). In particular, all invariant polynomials (the integrands of the characteristic classes) can be explicitly computed for the dual model in terms of quantities pertaining to the original one and with the help of the canonical connection whose intrinsic characterization is given. Using our formalism the physically, and T-duality invariant, relevant result that top forms are zero when there is an isometry without fixed points is easily proved. 
  We show that the large $N$ limit of certain conformal field theories in various dimensions include in their Hilbert space a sector describing supergravity on the product of Anti-deSitter spacetimes, spheres and other compact manifolds. This is shown by taking some branes in the full M/string theory and then taking a low energy limit where the field theory on the brane decouples from the bulk. We observe that, in this limit, we can still trust the near horizon geometry for large $N$. The enhanced supersymmetries of the near horizon geometry correspond to the extra supersymmetry generators present in the superconformal group (as opposed to just the super-Poincare group). The 't Hooft limit of 4-d ${\cal N} =4$ super-Yang-Mills at the conformal point is shown to contain strings: they are IIB strings. We conjecture that compactifications of M/string theory on various Anti-deSitter spacetimes are dual to various conformal field theories. This leads to a new proposal for a definition of M-theory which could be extended to include five non-compact dimensions. 
  We show that various disconnected components of the moduli space of superstring vacua with 16 supercharges admit a rationale in terms of BPS un-orientifolds, i.e. type I toroidal compactifications with constant non-vanishing but quantized vacuum expectation values of the NS-NS antisymmetric tensor. These include various heterotic vacua with reduced rank, known as CHL strings, and their dual type II (2,2) superstrings below D=6. Type I vacua without open strings allow for an interpretation of several disconnected components with N_V=10-D. An adiabatic argument relates these unconventional type I superstrings to type II (4,0) superstrings without D-branes. The latter are connected by U-duality below D=6 to type II (2,2) superstrings. We also comment on the relation between some of these vacua and compactifications of the putative M-theory on unorientable manifolds as well as F-theory vacua. 
  We study dynamics of triple junction of (p,q) strings in Type IIB string theory. We probe tension and mass density of (p,q) strings by studying harmonic fluctuations of the triple junction. We show that they agree perfectly with BPS formula provided suitable geometric interpretation of the junction is given. We provide a precise statement of BPS limit and force-balance property. At weak coupling and sufficiently dense limit, we argue that (p,q)-string embedded in string network is a `wiggly string', whose low-energy dynamics can be described via renormalization group evolved, smooth effective non-relativistic string. We also suggest the possibility that, upon Type IIB strings are promoted to M-theory membrane, there can exist `evanescent' bound-states at triple junction in the continuum. 
  We examine general properties of superembeddings, i.e., embeddings of supermanifolds into supermanifolds. The connection between an embedding procedure and the method of non-linearly realised supersymmetry is clarified, and we demonstrate how the latter arises as a special case of the former. As an illustration, the super-5-brane in 7 dimensions, containing a self-dual 3-form world-volume field strength, is formulated in both languages, and provides an example of a model where the embedding condition does not suffice to put the theory on-shell. 
  Assume that in a fundamental theory of quantum gravity spatial information is encoded through elements x_i of an associative, complex and possibly noncommutative algebra in which the involution acts as x^*_i = x_i. Without further assumptions it can be shown that such x_i can describe only three different types of short distance structures: I. a lattice, II. a continuum or III. a finite lower bound on the uncertainty in positions, as e.g. described in a stringy uncertainty relation. All other cases are mixtures of the three. We briefly review recent results on the case III short distance structure, in particular its ultraviolet regularity and a possible new mechanism that turns the external degrees of freedom lost through the UV-cutoff into internal degrees of freedom. 
  The worldvolume field equations of M-branes and D-branes are known to admit p-brane soliton solutions. These solitons are shown to saturate a BPS-type bound on their p-volume tensions, which are expressed in terms of central charges that are expected to appear in the worldvolume supertranslation algebra. The cases we consider include vortices, `BIons', instantons and dyons (both abelian and non-abelian), and the string boundaries of M-2-branes in the M-5-brane. 
  Quantum Dirac constraints in generic constrained system are solved by directly calculating in the one-loop approximation the path integral with relativistic gauge fixing procedure. The calculations are based on the reduction algorithms for functional determinants extended to gauge theories. Explicit mechanism of transition from relativistic gauge conditions to unitary gauges, participating in the construction of this solution, is revealed by the method of Ward identities. 
  In this paper we introduce the $(n+2)$-dimensional Born-Infeld action with a dual field strength $\tilde{H}$. We compute the field equation by using Schur polynomials and give a soliton solution. 
  String representations of the Wilson loop and of the non-Abelian analog of the 't Hooft loop, defined on the string world-sheet, are studied in the Londons' limit of the 4D Georgi-Glashow model. In the first case, massive gauge bosons yield only perimeter type terms, whereas in the second case, they also interact with the string, which enables one to derive the coefficient functions parametrizing the bilocal correlator of the dual field strength tensors. The asymptotic behaviours of these functions at small and large distances are argued to be in a good agreement with the observable lattice data. 
  These lectures explore what can be learnt about M-theory from its superalgebra. The first three lectures introduce the 'basic' branes of M-theory, and type II superstring theories, and show how the duality relations between them are encoded in the respective spacetime superalgebras. The fourth lecture introduces brane intersections and explains how they are encoded in the worldvolume superalgebras. 
  The moduli space of vacua for the confining phase of N=1 $SO(N_c)$ supersymmetric gauge theories in four dimensions is analyzed by studying the M theory fivebrane. The type IIA brane configuration consists of a single NS5 brane, multiple copies of NS'5 branes, D4 branes between them, and D6 branes intersecting D4 branes. We construct M theory fivebrane configuration corresponding to the superpotential perturbation where the power of adjoint field is connected to the number of NS'5 branes. At a singular point in the moduli space where mutually local dyons become massless, the quadratic degeneracy of the N=2 $SO(N_c)$ hyperelliptic curve determines whether this singular point gives a N=1 vacua or not. The comparison of the meson vevs in M theory fivebrane configuration with field theory results turns out that the effective superpotential by the integrating in method with enhanced gauge group SU(2) is exact. 
  We find the microscopic spectral densities and the spectral correlators associated with multicritical behavior for both hermitian and complex matrix ensembles, and show their universality. We conjecture that microscopic spectral densities of Dirac operators in certain theories without spontaneous chiral symmetry breaking may belong to these new universality classes. 
  Using an infinite number of fields, we construct actions for D=4 self-dual Yang-Mills with manifest Lorentz invariance and for D=10 super-Yang-Mills with manifest super-Poincar\'e invariance. These actions are generalizations of the covariant action for the D=2 chiral boson which was first studied by McClain, Wu, Yu and Wotzasek. 
  We show that the recently proposed confining string theory describes smooth surfaces with long-range correlations for the normal components of tangent vectors. These long-range correlations arise as a consequence of a "frustrated antiferromagnetic" interaction whose two main features are non-locality and a negative stiffness. 
  The Inonu-Wigner contraction is applied to special relativity and the little groups of the Lorentz group. If the O(3) symmetry group for massive particle is boosted to an infinite-momentum frame, it becomes contracted to a combination of the cylindrical group and the two-dimensional Euclidean group. The Euclidean component becomes the Lorentz condition applicable to the electromagnetic four-potential, and the cylindrical component leads to the helicity and gauge degrees of freedom. The rotation around the cylindrical axis corresponds to the helicity, while the translation parallel to the axis on the cylindrical surface leads to a gauge transformation. 
  All supersymmetric gauge theories based on simple groups which have an affine quantum moduli space, i.e. one generated by gauge invariants with no relations, W=0, and anomaly matching at the origin, are classified. It is shown that the only theories with no gauge invariants (and moduli space equal to a single point) are the two known examples, SU(5) with 5-bar + 10 and SO(10) with a spinor. The index of the matter representation must be at least as big as the index of the adjoint in theories which have a non-trivial relation among the gauge invariants. 
  We find exact solutions for N=2 supersymmetric SO(N), N=7,9,10,11,12 gauge theories with matter in the fundamental and spinor representation. These theories, with specific numbers of vectors and spinors, arise naturally in the compactification of type IIA string theory on suitably chosen Calabi-Yau threefolds. Exact solutions are obtained by using mirror symmetry to find the corresponding type IIB compactification. We propose generalizations of these results to cases with arbitrary numbers of massive vectors and spinors. 
  I review the appearance, within Matrix theory, of the $SL(5,Z)$ U-duality group of M-theory on $T^4$, and the duality between M-theory on K3 and the Heterotic string on $T^3$. In both cases the duality is geometrical and manifest. 
  We study graviton scattering in the presence of higher dimensional operators - particularly, R^4 - arising from loop effects. We find that the results do not correspond to any known terms in the effective action of Matrix Theory, thus lending support to the idea that the finite N Matrix Theory has no simple relation to supergravity with large compactification radii. 
  In the generalized Schwinger model the vector and axial vector currents are linearly coupled, with arbitrary coefficients, to the gauge connection. Therefore it represents an interesting example of a theory where both gauge anomalies and anomalous diver gences of global currents show up in general. We derive results for these two kinds of quantum corrections inside the field antifield framework. 
  The reduction of 4D Einstein gravity with $N$ minimal scalars leads to specific 2D dilaton gravity with dilaton coupled scalars. Applying s-wave and large $N$ approximation (where large $N$ quantum contribution due to dilaton itself is taken into account) we study 2D cosmology for CGHS model and for reduced Einstein gravity (in both cases with dilaton coupled scalars). Numerical study shows that in most cases we get 2D singular Universe which may correspond to big bang. Nevertheless, big crunch or non-singular Universes are also possible. For reduced 4D Einstein gravity one can regard the obtained 2D cosmology with time-dependent dilaton as 4D Kantowski-Sacks Universe of singular,non-singular or big crunch type. 
  A possibility is discussed that new nontrivial Chern-Simons like terms may be generated in (3+1)--dimensional QED Lagrangian, when induced by quark loops interacting with a non-abelian background field in an SU(2) model of QCD. 
  We consider BPS configurations in theories with two timelike directions from the perspective of the supersymmetry algebra. We show that whereas a BPS state in a theory with one timelike variable must have positive energy, in a theory with two times any BPS state must have positive angular momentum in the timelike plane, in that $Z_{0\tilde{0}}>0$, where $0$ and $\tilde{0}$ are the two timelike directions. We consider some generic BPS solutions of theories with two timelike directions, and then specialise to the study of the (10,2) dimensional superalgebra for which the spinor operators generate 2-forms and 6-forms. We argue that the BPS configurations of this algebra relate to F-theory in the same way that the BPS configurations of the eleven dimensional supersymmetry algebra relate to M-theory. We show that the twelve dimensional theory is one of fundamental 3-branes and 7-branes, along with their dual partners. We then formulate the new intersection rules for these objects. Upon reduction of this system we find the algebraic description of the IIB-branes and the M-branes. Given these correspondences we may begin an algebraic study of F-theory. 
  We calculate the instanton corrections in the effective prepotential for N=2 supersymmetric Yang-Mills theory with all A-D-E gauge groups from the Seiberg-Witten geometry constructed out of the spectral curves of the affine Toda lattice. The one-instanton contribution is determined explicitly by solving the Gauss-Manin system associated with the A-D-E singularity. Our results are in complete agreement with the ones obtained from the microscopic instanton calculations. 
  We consider the heat-kernel on a manifold whose boundary is piecewise smooth. The set of independent geometrical quantities required to construct an expression for the contribution of the boundary discontinuities to the C_{2} heat-kernel coefficient is derived in the case of a scalar field with Dirichlet and Robin boundary conditions. The coefficient is then determined using conformal symmetry and evaluation on some specific manifolds. For the Robin case a perturbation technique is also developed and employed. The contributions to the smeared heat-kernel coefficient and cocycle function are calculated. Some incomplete results for spinor fields with mixed conditions are also presented. 
  In the fruitful interplay between gauge fields and strings and in many conjectured M-theory dualities, open strings play a prominent role. We review the construction of open-string descendants (un-orientifolds) of closed-string theories admitting a generalized orientation reversal involution. We then specialize the construction to some classes of non-supersymmetric models in D=10 that have been recently considered in the context of duality without supersymmetry. We also discuss the propagation of open and unoriented strings on the NS pentabrane (N5-brane). This background is a prototype of the configurations of branes and orientifold planes that represent a powerful alternative to the geometric engineering of Supersymmetric Yang-Mills Theories. The resulting description of D-branes in non-trivial backgrounds looks very different from the one naively expected. In particular the very distinction between different Dp-branes becomes ambiguous in the presence of strong curvature effects. 
  Certain non-asymptotically flat but supersymmetric classical solution of the type IIA supergravity can be interpreted as the infinitely-boosted version of the D-particle solution along the M-theory circle. By a chain of T-dual transformations, this analysis also applies to yield non-asymptotically flat solutions from the asymptotically flat and (non)-extremal solutions with intersecting D-strings and D five-branes of the type IIB supergravity compactified on a five-torus. Under S-duality, the non-asymptotically flat solutions in this context can in particular be used to describe the (1+1)-dimensional CGHS type black holes via spontaneous compactifications. 
  We study classical BPS five-brane solutions in the Horava-Witten supergravity. The presence of the eleventh dimension add a new feature, namely the dependence of the solution on this new coordinate. For gauge five-branes with an instanton size less than the eleventh radius and in the neighborhood of the center of the neutral five-brane, important corrections to the ten-dimensional solution appear for all values of the string coupling constant. We compute the mass and magnetic charge of the five-brane solitons and the result is shown to agree with the membrane and five-brane quantization conditions. Compactified to four dimensions, our solutions are interpreted as axionic strings. 
  We present an explicit method for translating between the linear sigma model and the spectral cover description of SU(r) stable bundles over an elliptically fibered Calabi-Yau manifold. We use this to investigate the 4-dimensional duality between (0,2) heterotic and F-theory compactifications. We indirectly find that much interesting heterotic information must be contained in the `spectral bundle' and in its dual description as a gauge theory on multiple F-theory 7-branes.  A by-product of these efforts is a method for analyzing semistability and the splitting type of vector bundles over an elliptic curve given as the sheaf cohomology of a monad. 
  The osp(1,2)-covariant Lagrangian quantization of general gauge theories is formulated which applies also to massive fields. The formalism generalizes the Sp(2)-covariant BLT approach and guarantees symplectic invariance of the quantized action. The dependence of the generating functional of Green's functions on the choice of gauge in the massive case disappears in the limit m = 0. Ward identities related to osp(1,2) symmetry are derived. Massive gauge theories with closed algebra are studied as an example. 
  We determine the exact beta function and a RG flow Lyapunov function for N=2 SYM with gauge group SU(n). It turns out that the classical discriminants of the Seiberg-Witten curves determine the RG potential. The radial irreversibility of the RG flow in the SU(2) case and the non-perturbative identity relating the $u$-modulus and the superconformal anomaly, indicate the existence of a four dimensional analogue of the c-theorem for N=2 SYM which we formulate for the full SU(n) theory. Our investigation provides further evidence of the essentially topological nature of the theory. 
  We consider the supergravity solution describing a configuration of intersecting D-4-branes with non-vanishing worldvolume gauge fields. The entropy of such a black hole is calculated in terms of the D-branes quantised charges. The non-extreme solution is also considered and the corresponding thermodynamical quantities are calculated in terms of a D-brane/anti-D-brane system. To perform the quantum mechanical D-brane analysis we study open-strings with their ends on branes with a magnetic condensate. Applying the results to our D-brane system we managed to have a perfect agreement between the D-brane entropy counting and the corresponding semi-classical result. The Landau degeneracy of the open string states describing the excitations of the D-brane system enters in a crucial way. We also derive the near-extreme results which agree with the semi-classical calculations. 
  We re-examine the threshold bound state problem on the wrong sign Taub-Nut space; the metric on which describes the relative moduli space of well separated BPS monopoles. The quantum mechanics gives rise to a continuous family of threshold bound states, in distinction to the unique one found on the Atiyah-Hitchin metric. 
  Many important ideas about string duality that appear in conventional $\T^2$ compactification have analogs for $\T^2$ compactification without vector structure. We analyze some of these issues and show, in particular, how orientifold planes associated with $Sp(n)$ gauge groups can arise from T-duality and how they can be interpreted in F-theory. We also, in an appendix, resolve a longstanding puzzle concerning the computation of $\Tr (-1)^F$ in four-dimensional supersymmetric Yang-Mills theory with gauge group SO(n). 
  We investigate the brane exchange in the framework of N=2 MQCD by using a specific family of M fivebrane configurations relevant to describe the baryonic branch root. An exchange of M fivebranes is realized in the Taub-NUT geometry and controlled by the moduli parameter of the configurations. This family also provides two different descriptions of the root. These descriptions are examined carefully using the Taub-NUT geometry. It is shown that they have the same baryonic branch and are shifted each other by the brane exchange. 
  Boundary equations for the relativistic string with masses at ends are formulated in terms of geometrical invariants of world trajectories of masses at the string ends. In the three-dimensional Minkowski space $E^1_2$, there are two invariants of that sort, the curvature $K$ and torsion $\kappa$. For these equations of motion with periodic $\kappa_i(\tau+n l)=\kappa(\tau)$, constants of motion are obtained. 
  I discuss a realization of stress-tensor for parafermion theories following the generalized Frenkel-Kac construction for higher level Kac-Moody algebras. All the fields are obtained from $d$=rank free bosons compactified on torus. This gives an alternative realization of Virasoro algebra in terms of a non-local correction of a free field construction which does not fit the usual background charge of Feigin-Fuchs approach. 
  We present a continuum formulation for $\theta$-vacua in the massive Schwinger model on the light-front, where $\theta$ enters as a background electric field. The effective coupling of the external field is partially screened due to vacuum polarization processes. For small fermion masses and small $\theta$ we calculate the mass of the meson and find agreement with results from bosonization. 
  The most general solution to the form factor problem in the sinh-Gordon model is presented in an explicit way. The linearly independent classes of solutions correspond to powers of the elementary field. We show how the form factors of exponential operators can be obtained from the general solution by adjusting free parameters. The general formula obtained in the sinh-Gordon case reproduces the form factors of the scaling Lee-Yang model in a certain limit of the coupling constant. 
  We apply Gauge Theory of Arbitrage (GTA) {hep-th/9710148} to derivative pricing. We show how the standard results of Black-Scholes analysis appear from GTA and derive correction to the Black-Scholes equation due to a virtual arbitrage and speculators reaction on it. The model accounts for both violation of the no-arbitrage constraint and non-Brownian price walks which resemble real financial data. The correction is nonlocal and transform the differential Black-Scholes equation to an integro-differential one. 
  We find M-theory (Type IIA) duals for compactifications of the 9d CHL string to 5d (4d) on K3 (K3 x S^1). The IIA duals are Calabi-Yau orbifolds with nontrivial RR U(1) backgrounds turned on. 
  We repeat the known procedure of the derivation of the set of Proca equations. It is shown that it can be written in various forms. The importance of the normalization is point out for the problem of the correct description of spin-1 quantized fields. Finally, the discussion of the so-called Kalb-Ramond field is presented. 
  In this small note we ask several questions which are relevant to the construction of the self-consistent neutrino theory of light. The previous confusions in such attempts are explained in the more detailed publication. 
  A derivative expansion of the Wegner-Houghton equation is derived for a scalar theory. The corresponding full non-perturbative renormalization group equations for the potential and the wave-function renormalization function are presented. We also show that the two loop perturbative anomalous dimension for the O(N) theory can be obtained by means of a polynomial truncation in the field dependence in our equations. 
  We show that the exact $beta$--function of 4D N=2 SYM plays the role of the metric whose inverse satisfies the WDVV--like equations $\F_{ikl}\beta^{lm} \F_{mnj}=\F_{jkl}\beta^{lm}\F_{mni}$. The conjecture that the WDVV--like equations are equivalent to the identity involving the $u$--modulus and the prepotential $\F$, seen as a superconformal anomaly, sheds light on the recently considered c-theorem for the N=2 SYM field theories. 
  In this paper we obtain both the vector and scalar equations of motion of an M-fivebrane in the presence of threebrane solitons. The resulting equations of motion are precisely those obtained from the Seiberg-Witten low energy effective action for N=2 Yang-Mills, including all quantum corrections. This analysis extends the work of a previous paper which derived the scalar equations of motion but not in detail the vector equations. We also discuss some features of an infinite number of higher derivative terms predicted by M theory. 
  Perturbation series for the electron propagator in the Schwinger Model is summed up in a direct way by adding contributions coming from individual Feynman diagrams. The calculation shows the complete agreement between nonperturbative and perturbative approaches. 
  We give an intrinsic definition of the special geometry which arises in global N=2 supersymmetry in four dimensions. The base of an algebraic integrable system exhibits this geometry, and with an integrality hypothesis any special Kahler manifold is so related to an integrable system. The cotangent bundle of a special Kahler manifold carries a hyperkahler metric. We also define special geometry in supergravity in terms of the special geometry in global supersymmetry. 
  We show that the physical degrees of freedom of the critical open string with N=2 superconformal symmetry on the worldsheet are described by a self-dual Yang-Mills field on a hyperspace parametrised by the coordinates of the target space R^{2,2} together with a commuting chiral spinor. A prepotential for the self-dual connection in the hyperspace generates the infinite tower of physical fields corresponding to the inequivalent pictures or spinor ghost vacua of this string. An action is presented for this tower, which describes consistent interactions amongst fields of arbitrarily high spin. An interesting truncation to a theory of five fields is seen to have no graphs of two or more loops. 
  In this paper we first consider the null-plane bound-state equation for a $q \bar q$ pair in 1+3 dimensions and in the lowest-order Tamm-Dancoff approximation. Light-cone gauge is chosen with a causal prescription for the gauge pole in the propagator. Then we show that this equation, when dimensionally reduced to 1+1 dimensions, becomes 't Hooft's bound-state equation, which is characterized by an $x^+$-instantaneous interaction. The deep reasons for this coincidence are carefully discussed. 
  We study the conformally invariant quantum field theory in spaces of even dimension D >= 4. The conformal transformations of current j_\mu and energy-momentum tensor T_{\mu\nu} are examined. It is shown that the set of conformal transformations of particular kind corresponds to the canonical (unlike anomalous) dimensions l_j=D-1 and l_T=D of those fields. These transformations cannot be derived by a smooth transiton from anomalous dimensions. The structure of representations of the conformal group, which correspond to these canonical dimensions, is analyzed, and new expressions for the propagators < j_\mu j_\nu > and < T_{\mu\nu} T_{\rho\sigma}> are derived. The latter expressions have integrable singularities. It is shown that both propagators satisfy non-trivial Ward identities. The higher Green functions of the fields j_\mu and T_{\mu\nu} are considered. The conformal QED and linear conformal gravity are discussed. We obtain the expressions for invariant propagators of electromagnetic and gravitational fields. The integrations over internal photon and graviton lines are performed. The integrals are shown to be conformally invariant and convergent, provided that the new expressions for the propagators are used. 
  Various features of domain walls in supersymmetric gluodynamics are discussed. We give a simple field-theoretic interpretation of the phenomenon of strings ending on the walls recently conjectured by Witten. An explanation of this phenomenon in the framework of gauge field theory is outlined. The phenomenon is argued to be particularly natural in supersymmetric theories which support degenerate vacuum states with distinct physical properties. The issue of existence (or non-existence) of the BPS saturated walls in the theories with glued (super)potentials is addressed. The amended Veneziano-Yankielowicz effective Lagrangian belongs to this class. The physical origin of the cusp structure of the effective Lagrangian is revealed, and the limitation it imposes on the calculability of the wall tension is explained. Related problems are considered. In particular, it is shown that the so called discrete anomaly matching, when properly implemented, does not rule out the chirally symmetric phase of supersymmetric gluodynamics, contrary to recent claims. 
  We explore various aspects of implementing the full M-theory U-duality group E_{d+1}, and thus Lorentz invariance, in the finite N matrix theory (DLCQ of M-theory) on d-tori: (1) We generalize the analysis of U-duality orbits of BPS states by Elitzur et al. (hep-th/9707217) from E_{d} to E_{d+1}. (2) We identify the new E_{d+1}-symmetries with Nahm-duality-like symmetries (N-duality) exchanging the rank N of the matrix theory gauge group with other quantum numbers. (3) We describe the action of N-duality on BPS bound states, thus making testable predictions for the Lorentz invariance of matrix theory. (4) We discuss the problems that arise in the matrix theory limit for BPS states with no top-dimensional branes, i.e. configurations with N=0. (5) We show that N-duality maps the matrix theory SYM picture to the matrix string picture and argue that, for d even, the latter should be thought of as an M-theory membrane description (which appears to be well defined even for d>5). (6) We find a compact and unified expression for a U-duality invariant of E_{d+1} for all d and show that in d=5,6 it reduces to the black hole entropy cubic E_{6}- and quartic E_{7}-invariants respectively. (7) We describe some of the solitonic states in d=6,7 and give an example (a `rolled-up' Taub-NUT 6-brane) of a configuration exhibiting the unusual 1/g_{s}^{3}-behaviour. 
  We consider bound states of D-branes wrapped around cycles with non-trivial fundamental groups of finite order. We find a new mechanism for binding D-branes by turning on flat discrete abelian and non-abelian gauge fields on their worldvolume. As a concrete application we study type IIB in the background where an S^3/G shrinks, where G is a discrete subgroup of SU(2) acting freely on S^3. 
  We examine several aspects of the formulation of M(atrix)-Theory on ALE spaces. We argue for the existence of massless vector multiplets in the resolved $A_{n-1}$ spaces, as required by enhanced gauge symmetry in M-Theory, and that these states might have the correct gravitational interactions. We propose a matrix model which describes M-Theory on an ALE space in the presence of wrapped membranes. We also consider orbifold descriptions of matrix string theories, as well as more exotic orbifolds of these models, and present a classification of twisted matrix string theories according to Reid's exact sequences of surface quotient singularities. 
  We study the three string junctions and string networks in Type IIB string theory by explicity constructing the holomorphic embeddings of the M-theory membrane that describe such configurations. The main feature of them such as supersymmetry, charge conservation and balance of tensions are derived in a more unified manner. We calculate the energy of the string junction and show that there is no binding energy associated with the junction. 
  I shall present a proof of universality of the microscopic spectral correlations in Verbaarschot's random matrix models of QCD, to corroborate the beautiful agreement between the predictions from the gaussian model and the numerical data. Rather than following closely the original proof, I shall disguise it with the conventional approach of Q, P operators and with the method employed in an alternative proof by Kanzieper-Freilikher, in a hope that borrowing notions from quantum mechanics may add the proof some pedagogical flavor. I shall also discuss a problem associated with the multi-criticality. 
  We analyze constrained quantum systems where the dynamics do not preserve the constraints. This is done in particular for the restriction of a quantum particle in Euclidean n-space to a curved submanifold, and we propose a method of constraining and dynamics adjustment which produces the right Hamiltonian on the submanifold when tested on known examples. This method we hope will become the germ of a full Dirac algorithm for quantum constraints. We take a first step in generalising it to the situation where the constraint is a general selfadjoint operator with some additional structures. 
  We quantize the spontaneously broken abelian U(1) Higgs model by using the improved BFT and BFV formalisms. We have constructed the BFT physical fields, and obtain the first class observables including the Hamiltonian in terms of these fields. We have also explicitly shown that there are exact form invariances between the second class and first class quantities. Then, according to the BFV formalism, we have derived the corresponding Lagrangian having U(1) gauge symmetry. We also discuss at the classical level how one easily gets the first class Lagrangian from the symmetry-broken second class Lagrangian. 
  The behavior of the beta-function of the low-energy effective coupling in the N=2 supersymmetric SU(2) QCD with several massive matter hypermultiplets and in the SU(3) Yang-Mills theory is determined near the superconformal points in the moduli space. The renormalization group flow is unambiguously fixed by looking at limited types of deformation near the superconformal points. It is pointed out that the scaling dimension of the beta-function is controlled by the scaling behavior of moduli parameters and the relation between them is explicitly worked out. Our scaling dimensions of the beta-functions are consistent in part with the results obtained recently by Bilal and Ferrari in a different method for the SU(2) QCD. 
  M(atrix) theory defines light-front description of M-theory boosted along positive direction of eleventh, M-coordinate. Rank of M(atrix) gauge group is directly related to M-momentum $P_{11} = N / R_{11}$ or, equivalently, to total number of D0-partons. Alternatively, M-theory may be boosted along opposite direction of M-coordinate, for which the theory consists only of anti-D0 partons. In M(atrix) theory description, we interpret this as analytic continuation of dimension of the gauge group: $U(-N) \sim U(N)$, $SO(-2N) \sim USp(2N)$ and $USp(-2N) \sim SO(2N)$. We check these reciprocity relations explicitly for uncompactified, heterotic, and CHL M(atrix) theories as well as effective M(atrix) gauge theories of $T_5/Z_2$ and $T_9/Z_2$ compactifications. In all cases, we show that absence of parity, gauge and supersymmetry anomalies require introduction of a twisted sector with negative numbers of matter multiplets. They are interpreted as massless open string excitations connected to anti-D-brane background. 
  Deriglazov and Galajinsky have recently proposed a new covariant action for the Green-Schwarz superstring which can be constructed in any spacetime dimension. In this short note, I show that their action contains extra on-shell degrees of freedom as compared with the standard action, and is therefore inequivalent. 
  A survey is given of the formulation of a $\sigma$-model describing an open string moving in general target space background fields and coupling to both a matrix-valued D-brane position and a matrix-valued gauge field on the D-brane. The equations of motion for the D-brane and the gauge field are derived from the conformal invariance condition on the string world sheet in lowest order of $\ap$. The ordering problem of the involved matrices is solved. In addition to our previous work we discuss a conflict between the classical T-duality rules and renormalization. The calculation of the RG $\beta$-functions does not yield the mass term obtained by formal application of these rules in the case of target space separated D-brane copies. 
  We show that the spectral dimension on non-generic branched polymer models with susceptibility exponent $\gamma$ is given by $2/(1+\gamma)$. For those models with negative $\gamma$ we find that the spectral dimension is 2. 
  An alternative path is taken for deriving an action for the supersymmetric 5-brane in 11 dimensions. Selfduality does not follow from the action, but is consistent with the equations of motion for arbitrary supergravity backgrounds. The action involves a 2-form as well as a 5-form world-volume potential; inclusion of the latter makes the action, as well as the non-linear selfduality relation for the 3-form field strength, polynomial. The requirement of invariance under kappa-transformations determines the form of the selfduality relation, as well as the action. The formulation is shown to be equivalent to earlier formulations of 5-brane dynamics. 
  We discuss epsilon-expansion in curved spacetime for asymptotically free and asymptotically non-free theories. The esistence of stable and unstable fixed points is investigated for $f \phi^4$ and SU(2) gauge theory. It is shown that epsilon-expansion maybe compatible with asymptotic freedom on special solutions of the RG equations in a special case (supersymmetric theory). Using epsilon-expansion RG technique the effective Lagrangian for covariantly constant gauge SU(2) field and effective potential for gauged NJL-model are found in 4-epsilon- dimensional curved space (in linear curvature approximation). The curvature- induced phase transitions from symmetric phase to asymmetric phase (chromomagnetic vacuum and chiral symmetry broken phase, respectively) are discussed for the above two models. 
  In the IR limit the Matrix string theory is expected to be described by the $S^N\R^{8}$ supersymmetric orbifold sigma model. Recently Dijkgraaf, Verlinde and Verlinde proposed a vertex that may describe the type IIA string interaction. In this paper using this interaction vertex we derive the four graviton scattering amplitude from the orbifold model in the large $N$ limit. 
  We review the relation of certain integrals over the Coulomb phase of $d=4$, N=2 SO(3) supersymmetric Yang-Mills theory with Donaldson-Witten theory. We describe a new way to write an important contact term in the theory and show how the integrals generalize to higher rank gauge groups. 
  We consider a system of D0-branes in toroidally compactified space with interactions described by a Born-Infeld-type generalisation of the leading v^2 + v^4/r^{D-4} terms (D is the number of non-compact directions in M-theory, including the longitudinal one). This non-linear action can be interpreted as an all-loop large N super Yang-Mills effective action and has a remarkable scaling property. We first study the classical dynamics of a brane probe in the field of a central brane source and observe the interesting difference between the D=5 and D > 5 cases: for D >5 the center acts as a completely absorbing black hole of effective size proportional to a power of the probe energy, while for D=5 there is no absorption for any impact parameter. A similar dependence on D is found in the behaviour of the Boltzmann partition function Z of an ensemble of D0-branes. For D=5 (i.e. for compactification on 6-torus) Z is convergent at short distances and is analogous to the ideal gas one. For D > 5 the system has short-distance instability. For sufficiently low temperature Z is shown to describe the thermodynamics of a Schwarzschild black hole in D > 5 dimensions, supporting recent discussions of black holes in Matrix theory. 
  We study the extent to which D=11 supergravity can be deformed and show in two very different ways that, unlike lower D versions, it forbids an extension with cosmological constant. Some speculations about other invariants are made, in connection with the possible counterterms of the theory. 
  We re-examine the quantization of a class of non-polynomial scalar field theories which interpolates continuously from a free one to $\phi^4$ theory. The quantization of such theories is problematic because the Feynman rules may not be directly obtained. We give a means for calculating the correlation functions in this theory. The Feynman rules developed here shall enable further progress in the understanding of the triviality of $\phi^4$ theory in four dimensions. 
  We analyze and give explicit representations for the effective abelian vector gauge field actions generated by charged fermions with particular attention to the thermal regime in odd dimensions, where spectral asymmetry can be present. We show, through $\zeta-$function regularization, that both small and large gauge invariances are preserved at any temperature and for any number of fermions at the usual price of anomalies: helicity/parity invariance will be lost in even/odd dimensions, and in the latter even at zero mass. Gauge invariance dictates a very general ``Fourier'' representation of the action in terms of the holonomies that carry the novel, large gauge invariant, information. We show that large (unlike small) transformations and hence their Ward identities, are not perturbative order-preserving, and clarify the role of (properly redefined) Chern-Simons terms in this context. From a powerful representation of the action in terms of massless heat kernels, we are able to obtain rigorous gauge invariant expansions, for both small and large fermion masses, of its separate parity even and odd parts in arbitrary dimension. The representation also displays both the nonperturbative origin of a finite renormalization ambiguity, and its physical resolution by requiring decoupling at infinite mass. Finally, we illustrate these general results by explicit computation of the effective action for some physical examples of field configurations in the three dimensional case, where our conclusions on finite temperature effects may have physical relevance. Nonabelian results will be presented separately. 
  In (3+1) Hamiltonian form, the conditions for the electric/magnetic invariance of generic self-interacting gauge vector actions and the definition of the duality generator are obvious. Instead, (3+1) actions are not intrinsically Lorentz invariant. Imposing the Dirac-Schwinger stress tensor commutator requirement to enforce the latter yields a differential constraint on the Hamiltonian which translates into the usual Lagrangian form of the duality invariance condition obeyed by Maxwell and Born-Infeld theories. We also discuss covariance properties of some analogous scalar models. 
  We find Wess-Zumino actions for kappa invariant IIB D-branes in the explicit form. A simple and compact expression is obtained by the grace of spinor variables which are defined as power series of differential forms. Once explicit form of the Wess-Zumino actions is determined, global supersymmetry (SUSY) charges and constraint equations including local supersymmetry (kappa) generators can be determined. The SUSY algebra and the central charges are also calculated explicitly. 
  The classical electromagnetic field of a spinless point electron is described in a formalism with extended causality by discrete finite point-vector fields with discrete and localized point interactions. These fields are taken as a classical representation of photons, "classical photons". They are all transversal photons; there are no scalar nor longitudinal photons and the Lorentz gauge condition is automatically satisfied. The angular distribution of emitted photons reproduces the directions of maximum emission of the standard formalism. The Maxwell formalism in terms of continuous and distributed fields is retrieved by the smearing of these discrete fields over the light-cone, and in this process scalar and longitudinal photons are necessarily created and added. Divergences and singularities are by-products of this averaging process. The discrete and the continuous formalisms are not equivalent. The discrete one is superior for not having divergencies, singularities, unphysical degrees of freedom, for describing processes of creation and annihilation of particles instead of advanced solutions, for having a natural explanation for the photon, and for generating the continuous formalism as an effective one. It enlightens the meaning and the origin of the non-physical photons in the standard formalism. The standard theory based on average continuous fields is more convenient and appropriate for dealing with a large number of charges and for relatively large distances, but for few charges or for the field configuration in a charge close neighborhood the discrete field description is mandatorily required for avoiding inconsistencies. 
  We give a simple derivation of BPS condition of string junction from M theory 
  We describe various approaches that give matrix descriptions of compactified NS five-branes. As a result, we obtain matrix models for Yang-Mills theories with sixteen supersymmetries in dimensions $2,3,4$ and 5. The equivalence of the various approaches relates the Coulomb branch of certain gauge theories to the moduli space of instantons on $T^4$. We also obtain an equivalence between certain six-dimensional string theories. Further, we discuss how various perturbative and non-perturbative features of these Yang-Mills theories appear in their matrix formulations. The matrix model for four-dimensional Yang-Mills is manifestly S-dual. In this case, we describe how electric fluxes, magnetic fluxes and the interaction between vector particles are realized in the matrix model. 
  In this article we present a self contained review of the principles of Matrix Theory including the basics of light cone quantization, the formulation of 11 dimensional M-Theory in terms of supersymmetric quantum mechanics, the origin of membranes and the rules of compactification on 1,2 and 3 tori. We emphasize the unusual origins of space time and gravitation which are very different than in conventional approaches to quantum gravity. Finally we discuss application of Matrix Theory to the quantum mechanics of Schwarzschild black holes. This work is based on lectures given by the second author at the Cargese ASI 1997 and at the Institute for Advanced Study in Princeton. 
  Kappa-symmetric worldvolume actions of the D3-, M5- and M2-branes can be coupled consistently to their near horizon bosonic geometry background. We study the gauge-fixed action in the approximation in which only the transverse radial direction of the brane is allowed to fluctuate. The generalized special conformal symmetry of these self-interacting actions is established. This opens up a possibility to find out if the full superconformal symmetry of the free actions of these branes survives in the presence of coupling defined by the size of the anti-deSitter throat. 
  We derive unitarity restrictions on the scaling dimensions of primary operators in a superconformal quantum field theory, in d=3,4,5,6. 
  It is shown that the BPS spectrum of Super-Yang-Mills theory on $T^d\times \R$, which fits into representations of the U-duality group for M-theory compactified on $T^{d}$, in accordance with the matrix-theory conjecture, in fact fits into representations of the U-duality group for M-theory on $T^{d+1}$, the extra dualities realised as generalised Nahm transformations. The spectrum of BPS M-branes is analysed, new branes are discussed and matrix theory applications described. 
  In this talk some recent results in the quantization of Chern-Simons field theories in the Coulomb gauge will be presented. In the first part, the consistency of the Chern-Simons field theories in this gauge is proven using the Dirac's canonical formalism for constrained systems. Despite the presence of non-trivial self-interactions in the gauge fixed functional, it will be shown that the commutation relations between the fields are trivial at any perturbative order in the absence of couplings with matter fields. If these couplings are present, instead, the commutation relations become rather involved, but it is still possible to study their main properties and to show that they vanish at the tree level. In the second part of the talk the perturbative aspects of Chern-Simons field theories in the Coulomb gauge will be analysed. In particular, it will be shown by explicit computations and in a regularization independent way that there are no radiative contributions to the $n-$point correlation functions. Finally the Feynman rules in the Coulomb gauge will be derived on a three dimensional manifold with a spatial section given by a closed and orientable Riemann surface. 
  We study the relationship between M theory on a nearly lightlike circle and U(N) gauge theory in p+1 dimensions. We define large N limits of these theories in which low energy supergravity is valid. The regularity of these limits implies an infinite series of nonrenormalization theorems for the gauge theory effective action, and the leading large N terms sum to a Born-Infeld form. Compatibility of two different large N limits that describe the same decompactified M theory leads to a conjecture for a relation between two limits of string theories. 
  We show that any $d$-dimensional strictly stationary, asymptotically Minkowskian solution $(d\ge 4)$ of a null reduction of $d+1$-dimensional pure gravity must saturate the BPS bound provided that the KK vector field can be identified appropriately. We also argue that it is consistent with the field equations. 
  The study of the special F^4 and R^4 in the effective action for the Spin(32)/Z_2 and type II strings sheds some light on D-brane calculus and on instanton contribution counting. The D-instanton case is discussed separately. 
  Infra-red divergences obscure the underlying soft dynamics in gauge theories. They remove the pole structures associated with particle propagation in the various Green's functions of gauge theories. Here we present a solution to this problem. We give two equations which describe how charged particles must be dressed by gauge degrees of freedom. One follows from gauge invariance, the other, which is new, from velocity superselection rules familiar from the heavy quark effective theory. The solution to these equations in the abelian theory is proven to lead to on-shell Green's functions that are free of soft divergences at all orders in perturbation theory. 
  We study the Born-Infeld equation from a Lagrangian point of view emphasizing the duality symmetry present in such systems. We obtain the Hamiltonian formulation directly from the Lagrangian. We also show that this system admits a dispersionless nonstandard Lax representation which directly leads to all the conserved charges (including the ones not previously obtained). We also present the generating function for these charges and point out various other properties of this system. 
  We briefly review recent developments in the theory of supermembranes and supermatrix models. In a second part we discuss their interaction with background fields. In particular, we present the full background field coupling for the bosonic case. This is a short summary of the talk at the workshop. A more extended version will appear elsewhere. 
  The motion of a one-dimensional kink and its energy losses are considered as a model of interaction of nontrivial topological field configurations with external fields. 
  The full U-duality symmetry of toroidally compactified M-theory can only be displayed by allowing non-rectangular tori with expectation values of the gauge fields. We construct an E_d(Z) U-duality invariant mass formula incorporating non-vanishing gauge backgrounds of the M-theory three-form C. We interpret this mass formula from the point of view of the Matrix gauge theory, and identify the coupling of the three-form to the gauge theory as a topological theta term, in agreement with earlier conjectures. We give a derivation of this fact from D-brane analysis, and obtain the Matrix gauge theory description of other gauge backgrounds allowed by the Discrete Light-Cone Quantization. We further show that the conjectured extended U-duality symmetry of Matrix theory on T^d in the Discrete Light-Cone Quantization has an implementation as an action of E_{d+1}(Z) on the BPS spectrum. Some implications for the proper interpretation of the rank N of the Matrix gauge theory are discussed. 
  We present subsingular vectors of the N=2 superconformal algebras other than the ones which become singular in chiral Verma modules, reported recently by Gato-Rivera and Rosado. We show that two large classes of singular vectors of the Topological algebra become subsingular vectors of the Antiperiodic NS algebra under the topological untwistings. These classes consist of BRST- invariant singular vectors with relative charges $q=-2,-1$ and zero conformal weight, and no-label singular vectors with $q=0,-1$. In turn the resulting NS subsingular vectors are transformed by the spectral flows into subsingular and singular vectors of the Periodic R algebra. We write down these singular and subsingular vectors starting from the topological singular vectors at levels 1 and 2. 
  The low-energy supersymmetric quantum mechanics describing D-particles in the background of D8-branes and orientifold planes is analyzed in detail, including a careful discussion of Gauss' law and normal ordering of operators. This elucidates the mechanism that binds D-particles to an orientifold plane, in accordance with the predictions of heterotic/type I duality. The ocurrence of enhanced symmetries associated with massless bound states of a D-particle with one orientifold plane is illustrated by the enhancement of $SO(14) \times U(1)$ to $E_8$ and $SO(12)\times U(1)$ to $E_7$ at strong type I' coupling. Enhancement to higher-rank groups involves both orientifold planes. For example, the enhanced $E_8 \times E_8 \times SU(2)$ symmetry at the self-dual radius of the heterotic string is seen as the result of two D8-branes coinciding midway between the orientifold planes, while the enhanced $SU(18)$ symmetry results from the coincidence of all sixteen D8-branes and $SO(34)$ when they also coincide with an orientifold plane. As a separate by-product, the s-rule of brane-engineered gauge theories is derived by relating it through a chain of dualities to the Pauli exclusion principle. 
  Reducing a 3-dimensional Chern-Simons term by a symmetry yields other topologically interesting structures. Specifically, reducing by radial symmetry results in a 1-dimensional quantum mechanical model, which has recently been used in an analysis of finite-temperature Chern-Simons theory. The radially symmetric expression may be inserted into 3-dimensional monopole or (2+1)-dimensional instanton equations, where it eliminates the monopole or instanton solutions. 
  We present a simple discussion of the appearance of light-front partons in local field theory.The description in terms of partons provides a dimensional reduction which relates a 2+1 with a 3+1 dimensional theory for example. The possibility for existence of Lorentz symmetry and a connection to the relativistic membrane is described. It is shown how the reconstruction of the full relativistic field theory is possible with a proper treatment of the parton configuration space. Compared with the case of identical particles this involves keeping configurations where the partons are at the same points. 
  Low-energy dynamics in the unit-charge sector of the CP^1 model on spherical space (space-time S^2 x R) is treated in the approximation of geodesic motion on the moduli space of static solutions, a six-dimensional manifold with non-trivial topology and metric. The structure of the induced metric is restricted by consideration of the isometry group inherited from global symmetries of the full field theory. Evaluation of the metric is then reduced to finding five functions of one coordinate, which may be done explicitly. Some totally geodesic submanifolds are found and the qualitative features of motion on these described. 
  We have discovered that the gauge invariant observables of matrix models invariant under U($N$) form a Lie algebra, in the planar large-N limit. These models include Quantum Chromodynamics and the M(atrix)-Theory of strings. We study here the gauge invariant states corresponding to open strings (`mesons'). We find that the algebra is an extension of a remarkable new Lie algebra ${\cal V}_{\Lambda}$ by a product of more well-known algebras such as $gl_{\infty}$ and the Cuntz algebra. ${\cal V}_{\Lambda}$ appears to be a generalization of the Lie algebra of vector fields on the circle to non-commutative geometry. We also use a representation of our Lie algebra to establish an isomorphism between certain matrix models (those that preserve `gluon number') and open quantum spin chains. Using known results on quantum spin chains, we are able to identify some exactly solvable matrix models. Finally, the Hamiltonian of a dimensionally reduced QCD model is expressed explicitly as an element of our Lie algebra. 
  The classical evolution of a homogeneous cosmological model is investigated within the framework of low - energy effective string gravity with higher genus corrections. Various conformal frames are considered. For the general case of correction functions in the Lagrangian we give the cosmological solutions with arbitrary curvature and dilaton, modulus and Kalb-Ramond fields. They generalize previously known tree level solutions. The features of the solutions are discussed. 
  A symplectically invariant definition of special K\"ahler geometry is discussed. Certain aspects hereof are illustrated by means of Calabi-Yau moduli spaces. 
  We study real-time correlation functions in scalar quantum field theories at temperature $T=1/\beta$. We show that the behaviour of soft, long wavelength modes is determined by classical statistical field theory. The loss of quantum coherence is due to interactions with the soft modes of the thermal bath. The soft modes are separated from the hard modes by an infrared cutoff $\L \ll 1/(\hbar\beta)$. Integrating out the hard modes yields an effective theory for the soft modes. The infrared cutoff $\L$ controls corrections to the classical limit which are $\cO{\hbar\beta\L}$. As an application, the plasmon damping rate is calculated. 
  The Casimir energy of quantum fluctuations about the classical kink configuration is computed numerically for a recently proposed lattice sine-Gordon model. This energy depends periodically on the kink position and is found to be approximately sinusoidal. 
  The present status of Connes' noncommutative view at the four forces is reviewed. 
  Variables parametrized by closed and open curves are defined to reformulate compact U(1) Quantum Electrodynamics in the circle with a massless fermion field. It is found that the gauge invariant nature of these variables accommodates into a regularization scheme for the Hamiltonian and current operators that is specially well suited for the study of the compact case. The zero mode energy spectrum, the value of the axial anomaly and the anomalous commutators this model presents are hence determined in a manifestly gauge invariant manner. Contrary to the non compact case, the zero mode spectrum is not equally spaced and consequently the theory does not lead to the spectrum of a free scalar boson. All the states are invariant under large gauge transformations. In particular, that is the case for the vacuum, and consequently the $\theta$-dependence does not appear. 
  A pseudoclassical model for P,T-invariant system of topologically massive U(1) gauge fields is analyzed. The model demonstrates a nontrivial relationship between continuous and discrete symmetries and reveals a phenomenon of ``classical quantization''. It allows one to identify SU(1,1) symmetry and S(2,1) supersymmetry as hidden symmetries of the corresponding quantum system. We show this P,T-invariant quantum system realizes an irreducible representation of a non-standard super-extension of the (2+1)-dimensional Poincare group. 
  We show an application of the Wilson Renormalization Group (RG) method to a SU(2 ) gauge field theory in interaction with a massive fermionic doublet. By choosing suitable boundary conditions to the RG equation, i.e. by requiring the relevant monomials not present in the classical action to satisfy the Slavnov-Taylor identities once the cutoffs are removed, we succeed in implementing the local gauge symmetry. In this way the so called fine-tuning problem, due to the assignation of boundary conditions in terms of the bare parameters, is avoided. In this framework, loop expansion is equivalent to the iterative solution of the RG equation; we perform one loop calculations in order to determine how much the fermionic matter modifies the asymptotic form of the couplings. Then we compute the beta-function and we check gluon transversality. Finally, a proof of perturbative renormalizability is shown. 
  Taking the St\"uckelberg Lagrangian associated with the abelian self-dual model of P.K. Townsend et al as a starting point, we embed this mixed first- and second-class system into a pure first-class system by following systematically the generalized Hamiltonian approach of Batalin, Fradkin and Tyutin. The resulting Lagrangian possesses an extended gauge invariance and provides a non-trivial example for a general Lagrangian approach to unravelling the full set of local symmetries of a Lagrangian. 
  We analyse the renormalizability of an Abelian N=1 super-Chern-Simons model coupled to parity-preserving matter on the light of the regularization independent algebraic method. The model shows to be stable under radiative corrections and to be gauge anomaly free. 
  The mechanism for the generation of a dipole moment due to an external field is presented for the Born-Infeld charged particle. The 'polarizability coefficient' is calculated: it is proportional to the third power of the characteristic length in the Born-Infeld theory. Some physical implications are briefly discussed. 
  We classify and explicitly construct the embedding diagrams of Verma modules over the N=2 supersymmetric extension of the Virasoro algebra. The essential ingredient of the solution consists in drawing the distinction between two different types of submodules appearing in N=2 Verma modules. The problem is simplified by associating to every N=2 Verma module a relaxed Verma module over the affine algebra ^sl(2) with an isomorphic embedding diagram. We then make use of the mechanism according to which the structure of the N=2/relaxed-sl(2) embedding diagrams can be found knowing the standard embedding diagrams of ^sl(2) Verma modules. The resulting classification of the N=2/relaxed-^sl(2) embedding diagrams follows the I-II-III pattern extended by an additional indication of the number (0, 1, or 2) and the twists of the standard ^sl(2) embedding diagrams contained in a given N=2/relaxed-^sl(2) embedding diagram. 
  We discuss continuous duality transformations and the properties of classical theories with invariant interactions between electromagnetic fields and matter. The case of scalar fields is treated in some detail. Special discrete elements of the continuous group are shown to be related to the Legendre transformation with respect to the field strengths. 
  Negative dimensional integration method (NDIM) seems to be a very promising technique for evaluating massless and/or massive Feynman diagrams. It is unique in the sense that the method simultaneously gives solutions in different regions of external momenta. Moreover, it is a technique whereby the difficulties associated with performing parametric integrals --- the standard approach --- are transferred to a simpler solving of a system of linear algebraic equations. Employing this method, we calculate a massless two-loop three point vertex with all the external legs off-shell. Then NDIM approach allows us to obtain twenty-one distinct new power series representations for the integral in question. In order to verify the correctness of our results, we consider five particular cases where either two of the external legs are put on-shell, or one of them amputated or one exponent of the propagators is set to zero, and compare our results thus obtained with the ones calculated with standard methods in positive dimension. 
  We present a construction in Matrix theory of longitudinal 5-branes whose geometry in transverse space corresponds to a 4-sphere. We describe these branes through an explicit construction in terms of N*N matrices for a particular infinite series of values of N. The matrices used in the construction have a number of properties which can be interpreted in terms of the 4-sphere geometry, in analogy with similar properties of the SU(2) generators used in the construction of a spherical membrane. The physical properties of these systems correspond with those expected from M-theory; in particular, these objects have an energy and a leading long-distance interaction with gravitons which agrees with 11D supergravity at leading order in N. 
  The mechanism for the generation of multipole moments due to an external field is presented for the Born-Infeld charged particle. The 'polarizability coefficient' for arbitrary l-pole moment is calculated. The l-th coefficient is proportional to the (2l+1)-th power of the characteristic length in the Born-Infeld theory. Physical implications are discussed. 
  The construction of Neveu-Schwarz superconformal field theories for any N is given via a superfield formalism. We also review some results and definitions of superconformal manifolds and we generalise contour integration and Taylor expansion to superconformal spaces. For arbitrary N we define (uncharged) primary fields and give their infinitesimal change under superconformal transformations. This leads us to the operator product expansion of the stress-energy tensor with itself and with primary fields. In this way we derive the well-known commutation relations of the Neveu-Schwarz superconformal algebras K_N. In this context we observe that the central extension term disappears for N>=4 for the Neveu-Schwarz theories. Finally, we give the global transformation rules of primary fields under the action of the algebra generators. 
  Negative dimensional integration is a step further dimensional regularization ideas. In this approach, based on the principle of analytic continuation, Feynman integrals are polynomial ones and for this reason very simple to handle, contrary to the usual parametric ones. The result of the integral worked out in $D<0$ must be analytically continued again --- of course --- to real physical world, $D>0$, and this step presents no difficulties. We consider four two-loop three-point vertex diagrams with arbitrary exponents of propagators and dimension. These original results give the correct well-known particular cases where the exponents of propagators are equal to unity. 
  We derive a new set of WDVV equations for N=2 SYM in which the renormalization scale $\Lambda$ is identified with the distinguished modulus which naturally arises in topological field theories. 
  We collect a number of facts and conjectures concerning Whitham theory and the renormalization group. Some explicit relations and problems are indicated in the context of N=2 susy Yang-Mills. 
  We consider the scaling limit of the two-dimensional $q$-state Potts model for $q\leq 4$. We use the exact scattering theory proposed by Chim and Zamolodchikov to determine the one and two-kink form factors of the energy, order and disorder operators in the model. Correlation functions and universal combinations of critical amplitudes are then computed within the two-kink approximation in the form factor approach. Very good agreement is found whenever comparison with exact results is possible. We finally consider the limit $q\to 1$ which is related to the isotropic percolation problem. Although this case presents a serious technical difficulty, we predict a value close to 74 for the ratio of the mean cluster size amplitudes above and below the percolation threshold. Previous estimates for this quantity range from 14 to 220. 
  In this paper we perform a detailed investigation of the Dirichlet eight-brane of the Type IIA string theory, when the effects of gravity are included. In particular, consider what happens when one allows the ten-form field strength $F_{10}$ to vary discontinuously across the worldvolume of the brane. Since the ten-form is constant on each side of the brane ($d*F_{10} = 0$), a variation in the bulk term $\int F_{10}*F_{10}$ gives rise to a net pressure acting on the surface of the brane. This means that the infinite `planar' eight-brane is no longer a static configuration with these boundary conditions. Instead, a static configuration is found only when the brane `compactifies' to the topology of an eight-sphere, $S^8$. These spherical eight-branes are thus bubbles which form boundaries between different phases of the massive Type IIA supergravity theory. While these bubbles are generically unstable and will want to expand (or contract), we show that in certain cases there is a critical radius, $r_c$, at which the (inward) tension of the brane is exactly counterbalanced by the (outward) force exerted by the pressure terms. Intuitively, these `compactified' branes are just spherical bubbles where the effective cosmological constant jumps by a discrete amount as you cross a brane worldsheet. We argue that these branes will be unstable to various semi-classical decay processes. We discuss the implications of such processes for the open strings which have endpoints on the eight-brane. 
  4x4 Dirac (gamma) matrices (irreducible matrix representations of the Clifford algebras C(3,1), C(1,3), C(4,0)) are an essential part of many calculations in quantum physics. Although the final physical results do not depend on the applied representation of the Dirac matrices (e.g. due to the invariance of traces of products of Dirac matrices), the appropriate choice of the representation used may facilitate the analysis. The present paper introduces a particularly symmetric real representation of 4x4 Dirac matrices (Majorana representation) which may prove useful in the future. As a byproduct, a compact formula for (transformed) Pauli matrices is found. The consideration is based on the role played by isoclinic 2-planes in the geometry of the real Clifford algebra C(3,0) which provide an invariant geometric frame for it. It can be generalized to larger Clifford algebras. 
  The one-loop free energy of the four-dimensional compact QED, which is known to be equivalent to the vector Sine-Gordon model, is calculated in the strong coupling regime. In the case, when the norm of the strength tensor of the saddle-point value of the corresponding Sine-Gordon model is much larger than the typical inverse area of a loop in the gas of the monopole rings, the obtained free energy decays exponentially versus this norm. In the opposite case, when the dominant configuration of the Sine-Gordon model is identically zero, the resulting free energy decays with the growth of loops as an exponent of the inverse square of their typical area. 
  We investigate the effective worldvolume theories of branes in a background given by (the bosonic sector of) 10-dimensional massive IIA supergravity (``massive branes'') and their M-theoretic origin. In the case of the solitonic 5-brane of type IIA superstring theory the construction of the Wess-Zumino term in the worldvolume action requires a dualization of the massive Neveu-Schwarz/Neveu-Schwarz target space 2-form field. We find that, in general, the effective worldvolume theory of massive branes contains new worldvolume fields that are absent in the massless case, i.e. when the mass parameter m of massive IIA supergravity is set to zero. We show how these new worldvolume fields can be introduced in a systematic way. In particular, we find new couplings between the massive solitonic 5-brane and the target space background, involving an additional worldvolume 1-form and 6-form. These new couplings have implications for the anomalous creation of branes. In particular, when a massive solitonic 5-brane passes through a D8-brane a stretched D6-brane is created. Similarly, in M-theory we find that when an M5-brane passes through an M9-brane a stretched Kaluza-Klein monopole is created. Pairs of massive branes of type IIA string theory can be viewed as the direct and double dimensional reduction of a single ``massive M-brane'' whose worldvolume theory is described by a gauged sigma model. 
  The low energy structures of N=1 supersymmetric models with E_6, F_4 and E_7 gauge groups and fundamental irrep matter contents are studied herein. We identify sets of gauge invariant composites which label all flat directions in the confining/Higgs phases of these theories. The impossibility of mapping several of these primary operators rules out previously conjectured exceptional self duals reported in the literature. 
  We study the (2,0) superconformal theories in six dimensions, which arise from the low-energy limit of k coincident 5-branes, using their discrete light-cone formulation as a superconformal quantum mechanical sigma model. We analyze the realization of the superconformal symmetry in the quantum mechanics, and the realization of primary operators. As an example we compute the spectrum of chiral primary states in symmetric Spin(5)_R representations. To facilitate the analysis we introduce and briefly discuss a new class of Lorentz non-invariant theories, which flow in the IR to the (2,0) superconformal field theories but differ from them in the UV. 
  We compute the emission spectrum of minimally coupled particles with spin that are Hawking radiated from four dimensional black holes in string theory. For a range of the black hole parameters the result has a product structure that may be interpreted in terms of the respective right- and left-moving thermal correlation functions of an effective string model. For spin-one and spin-two particles a novel cancellation between contributions to the wave function is needed to ensure this outcome. The form of the spectra suggests that the four-dimensional effective string description is ``heterotic'': particles with spin are emitted from the right-moving sector, only. 
  We show that a quantum field theoretic model of anyons cannot be ``free'' in the (restrictive) sense that the basic fields create only one-particle states out of the vacuum. 
  A simple analytic proof of the formula known as the non-Abelian Stokes theorem is given. It is explicitly shown that the consistency of the formula is guaranteed by the Bianchi identity for the gauge field. An attempt is made to construct the Lagrangian for the gauge field in terms of loop variables. 
  The exact perturbation approach is used to derive the elementary correlation lengths $\xi_i$ and related mass gaps $m_i$ of the two-dimensional dilute A_L lattice model in regimes 1 and 2 for L odd from the Bethe Ansatz solution. In regime 2 the A_3 model is the E_8 lattice realisation of the two-dimensional Ising model in a magnetic field at T=T_c. The calculations for the A_3 model in regime 2 start from the eight thermodynamically significant string types found in previous numerical studies. These string types are seen to be consistent in the ordered high field limit. The eight masses obtained reduce with the approach to criticality to the E_8 masses predicted by Zamolodchikov, thus providing a further direct lattice determination of the E_8 mass spectrum. 
  We construct some examples of analytic solutions of the low energy (i.e. tree-level) string cosmological effective action. We work with the ``minimal'' field content (i.e. graviton and dilaton) in the absence of any dilaton potential. Provided the metric is sufficiently inhomogeneous we find solutions whose curvature invariants are bounded and everywhere defined in time and space. The dilaton coupling and its associated energy density are regular and homogeneous. A phase of growing (and non-singular) dilaton coupling compatible with the regularity of the curvature invariants without the addition of higher curvature (or higher genus) corrections to the tree-level effective action. We discuss the symmetries of the obtained solutions. 
  The gauged Nambu-Jona-Lasinio model in the quenched-ladder approximation has non-trivial dynamics near a critical scaling region (critical curve) separating a chiral symmetric and a dynamically chiral symmetry broken phase. Scalar and pseudoscalar composites corresponding to the four-fermion interaction become relevant degrees of freedom at short distances, which is reflected in the appearance of a large anomalous dimension of the four-fermion operators. A method is introduced for solving the Schwinger-Dyson equation for the Yukawa vertex in specific kinematic regimes. This allows one to derive an analytic expression for the scalar propagator, which is valid along the entire critical curve. The mass and width of the scalar composites in the critical scaling region are reexamined and the conformal phase transition at the critical gauge coupling is discussed. 
  In this paper I continue the study of the new framework of modular localization and its constructive use in the nonperturbative d=1+1 Karowski-Weisz-Smirnov formfactor program. Particular attention is focussed on the existence of semilocal generators of the wedge-localized algebra without vauum polarization (FWG-operators) which are closely related to objects fulfilling the Zamolodchikov-Faddeev algebraic structure. They generate a ``thermal Hilbert space'' and allow to understand the equivalence of the KMS conditions with the so-called cyclicity equation for formfactors which was known to be closely related to crossing symmetry properties. The modular setting gives rise to interesting new ideas on ``free'' d=2+1 anyons and plektons. 
  We analyse completely the BRST cohomology on local functionals for two dimensional sigma models coupled to abelian world sheet gauge fields, including effective bosonic D-string models described by Born-Infeld actions. In particular we prove that the rigid symmetries of such models are exhausted by the solutions to generalized Killing vector equations which we have presented recently, and provide all the consistent first order deformations and candidate gauge anomalies of the models under study. For appropriate target space geometries we find nontrivial deformations both of the abelian gauge transformations and of the world sheet diffeomorphisms, and antifield dependent candidate anomalies for both types of symmetries separately, as well as mixed ones. 
  We show that parity symmetry is not spontaneously broken in the CP^N sigma model for any value of N when the coefficient of the $\theta$--term becomes $\theta=\pi$ (mod $2\pi$). The result follows from a non-perturbative analysis of the nodal structure of the vacuum functional $\psi_0(z)$. The dynamical role of sphalerons turns out to be very important for the argument. The result introduces severe constraints on the possible critical behavior of the models at $\theta=\pi$ (mod $2\pi$). 
  We study the singular Landau surfaces of planar diagrams contributing to scattering of a massless quark and antiquark in 3+1 dimensions. In particular, we look at singularities which remain after integration with respect to the various angular degrees of freedom. We derive a general relation between these singularities and the singularities of quark- antiquark scattering in 1+1 dimensions. We then classify all Landau surfaces of the 1+1 dimensional system. Combining these results, we deduce that the singular surfaces of the angle- integrated 3+1 dimensional amplitude must satisfy at least one of three conditions, which we call the planar light-cone conditions. We discuss the extension of our results to non-perturbative processes by means of the non-perturbative operator product expansion. Our findings offer new insights into the connection between the 't Hooft model and large-N_c mesons in 3+1 dimensions and may prove useful in studies of confinement in relativistic meson systems. 
  The kinetic action of the N=2 Yang-Mills vector multiplet can be written in projective N=2 superspace using projective multiplets. It is possible to perform a simple N=2 gauge fixing, which translated to N=1 component language makes the kinetic terms of gauge potentials invertible. After coupling the Yang-Mills multiplet to unconstrained sources it is very simple to integrate out the gauge fixed vector multiplet from the path integral of the free theory and obtain the N=2 propagator. Its reduction to N=1 components agrees with the propagators of the gauge fixed N=1 component superfields. The coupling of Yang-Mills multiplets and hypermultiplets in N=2 projective superspace allows us to define Feynman rules in N=2 superspace for these two fields. 
  We generalize to the case of compactified superstrings a construction given previously for critical superstrings of finite one loop amplitudes that are well-defined for all external momenta. The novel issues that arise for compactified strings are the appearance of infrared divergences from the propagation of massless strings in four dimensions and, in the case of orbifold schemes, the contribution of tachyons in partial amplitudes with given spin structure and twist sectors. Methods are presented for the resolution of these problems and expressions for finite amplitudes are given in terms of double and single dispersion relations, with explicit spectral densities. 
  We suggest that M-theory could be non-perturbatively equivalent to a local quantum field theory. More precisely, we present a ``renormalizable'' gauge theory in eleven dimensions, and show that it exhibits various properties expected of quantum M-theory, most notably the holographic principle of 't~Hooft and Susskind. The theory also satisfies Mach's principle: A macroscopically large space-time (and the inertia of low-energy excitations) is generated by a large number of ``partons'' in the microscopic theory. We argue that at low energies in large eleven dimensions, the theory should be effectively described by eleven-dimensional supergravity. This effective description breaks down at much lower energies than naively expected, precisely when the system saturates the Bekenstein bound on energy density. We show that the number of partons scales like the area of the surface surrounding the system, and discuss how this holographic reduction of degrees of freedom affects the cosmological constant problem. We propose the holographic field theory as a candidate for a covariant, non-perturbative formulation of quantum M-theory. 
  It is shown that some analog of the ``second quantization'' exists in the framework of CP(N) theory. I analyse conditions under that ``geometrical bosons'' may be identified with real physical fields. The compact character of a state manifold should preserve the quantities of dynamical variables from divergences. 
  We present two compact derivations of the correct definition of the Chern-Simons term in the topologically non trivial context of thermal $QED_3$. One is based on a transgression descent from a D=4 background connection, the other on embedding the abelian model in SU(2). The results agree with earlier cohomology conclusions and can be also used to justify a recent simple heuristic approach. The correction to the naive Chern-Simons term, and its behavior under large gauge transformations are displayed. 
  We introduce a complete set of gauge-invariant variables and a generalized Born-Oppenheimer formulation to search for normalizable zero-energy asymptotic solutions of the Schrodinger equation of SU(2) matrix theory. The asymptotic method gives only ground state candidates, which must be further tested for global stability. Our results include a set of such ground state candidates, including one state which is a singlet under spin(9). 
  Energy-dependent Green's functions for the two and three dimensional $\delta$-function plus harmonic oscillator potential systems are derived by incorporating the renormalization and the self-adjoint extension into the Green's function formalism, respectively. It is shown that both methods yield an identical Green's function if a certain relation between the self-adjoint extension parameter and the renormalized coupling constant is imposed. 
  We discuss the role of D0-branes as instantons in the construction of SU(N) Super Yang-Mills and Super QCD theories in four space-time dimensions with D4-, D6- and NS-branes.   [To appear in the proceedings of the conference "Quantum Aspects of Gauge Theories, Supersymmetry and Unification", held at Neuchatel University, Neuchatel, Switzerland, 18--23 September 1997.] 
  A Newtonian matrix cosmology, correspoding to the BFSS model of eleven-dimensional M-theory in the IMF as a (0+1) M(atrix) model is constructed. Interesting new results are obtained, such as the existence of (much sought for in the past) static solutions. The possible interpretation of the off-diagonal entries as a background geometry is also briefly discussed. 
  We consider the coupling of nonminimal scalar multiplets to supersymmetric Yang-Mills in four dimensions and compute the one-loop contribution to the low-energy effective action in the abelian sector. We show that the resulting theory realizes the dual version of the corresponding one from N=2 supersymmetric Yang-Mills. 
  We extend the critical point self-consistency method used to solve field theories at their d-dimensional fixed point in the large N expansion to include superfields. As an application we compute the beta-function of the Wess-Zumino model with an O(N) symmetry to O(1/N^2). This result is then used to study the effect the higher order corrections have on the radius of convergence of the 4-dimensional beta-function at this order in 1/N. The critical exponent relating to the wave function renormalization of the basic field is also computed to O(1/N^2) and is shown to be the same as that for the corresponding field in the supersymmetric O(N) sigma model in d-dimensions. We discuss how the non-renormalization theorem prevents the full critical point equivalence between both models. 
  We construct gauge invariant operators for singular knots in the context of Chern-Simons gauge theory. These new operators provide polynomial invariants and Vassiliev invariants for singular knots. As an application we present the form of the Kontsevich integral for the case of singular knots. 
  We solve the Schwinger Dyson equations of the O(N) symmetric Wess-Zumino model at O(1/N^3) at the non-trivial fixed point of the d-dimensional beta-function and deduce a critical exponent for the wave function renormalization at this order. By developing the epsilon-expansion of the result, which agrees with known perturbation theory, we examine the distribution of transcendental coefficients and show that only the Riemann zeta series arises at this order in 1/N. Unlike the analogous calculation at the same order in the bosonic O(N) phi^4-theory non-zeta transcendentals, associated with for example the (3,4)-torus knot, cancel. 
  We review the construction of open descendants of the type IIB superstring on the Z-orbifold. It results in a chiral four-dimensional model with gauge group $SO(8) \otimes U(12)$ and three generations of matter in the $(8,12^*)\oplus (1,66)$ representations. As a test of type I - heterotic duality, that reduces to a weak/weak duality in D=4, a heterotic model on the same orbifold is also presented. The massless spectrum reproduces exactly the one found in the type I case apart from additional twisted matter charged with respect to the SO(8) gauge group. The puzzle is solved by noting that at generic points in the moduli space these states get masses. 
  Like the M-theory itself also the worldvolume theory of the M5-brane contains brane excitations, which can be extracted from the supersymmetry algebra. Bound states (intersecting branes) of the worldvolume can be translated into bound states of {11-d} SUGRA. In this paper, we discuss the matrix description for these bound states and their entropy (=degeneracy). In order to decouple the worldvolume field theory from the bulk gravity we have especially to assume that all charges are large, which gives a nice agreement with the Bekenstein-Hawking entropy of black holes. 
  We reconstruct non-trivial 6d theories obtained by Blum and Intriligator by considering IIB or SO(32) 5 branes at ALE spaces in the language of Hanany Witten setups. Using ST duality we make the equivalence of the two approaches manifest, thereby uncovering several new T-duality relations between the group theoretic data describing the embedding of the instantonic 5 brane in the ALE and brane positions in the Hanany Witten language. We construct several new 6d theories, which can be understood as arising on 5 branes in IIB orientifolds with oppositely charged orientifold planes recently introduced by Witten. 
  We examine finite temperature perturbation theory for Chern-Simons theories, in the context of an analogue 0+1-dimensional model. In particular, we show how nonextensive terms arise in the perturbative finite temperature effective action, using both the real-time and imaginary-time formalisms. We illustrate how large gauge invariance is restored at all orders, despite being broken at any given order in perturbation theory. We discuss which aspects generalize to a perturbative analysis of finite temperature Chern-Simons terms in higher dimensions. 
  We consider configurations of six-branes, five-branes and eight-branes in various superstring backgrounds. These configurations give rise to $(0,1)$ supersymmetric theories in six dimensions. The condition for RR charge conservation of a brane configuration translates to the condition that the corresponding field theory is anomaly-free. Sets of infinitely many models with non-trivial RG fixed points at strong coupling are demonstrated. Some of them reproduce and generalise the world-volume theories of SO(32) and $E_8\times E_8$ small instantons. All the models are shown to be connected by smooth transitions. In particular, the small instanton transition for which a tensor multiplet is traded for 29 hypermultiplets is explicitly demonstrated. The particular limit in which these theories can be considered as six-dimensional string theories without gravity are discussed. New fixed points (string theories) associated with $E_n$ global symmetries are discovered by taking the strong string coupling limit. 
  A compactification of the heterotic string is considered which describes a particle propagating in the symmetric space M = SO(9,1;Z)\SO(9,1;R)/SO(9;R) in interaction with excited states whose dynamics is (1+1)-dimensional. The model appears to describe D-particle dynamics in type IIA string theory on M. 
  The motivation and the challenge in applying the renormalization group for systems with several scaling regimes is briefly outlined. The four dimensional $\phi^4$ model serves as an example where a nontrivial low energy scaling regime is identified in the vicinity of the spinodal instability region. It is pointed out that the effective theory defined in the vicinity of the spinodal instability offers an amplification mechanism, a precursor of the condensation that can be used to explore nonuniversal forces at high energies. 
  We describe abstract (p,q) string networks which are the string networks of Sen without the information about their embedding in a background spacetime. The non-perturbative dynamical formulation invented for spin networks, in terms of causal evolution of dual triangulations, is applied to them. The formal transition amplitudes are sums over discrete causal histories that evolve (p,q) string networks. The dynamics depend on two free SL(2,Z) invariant functions which describe the amplitudes for the local evolution moves. 
  The moduli space of vacua for the confining phase of N=1 $Sp(N_c)$ supersymmetric gauge theories in four dimensions is studied by M theory fivebrane. We construct M theory fivebrane configuration corresponding to the perturbation of superpotential in which the power of adjoint field is related to the number of NS'5 branes in type IIA brane configuration. We interpret the dyon vacuum expectation values in field theory results as the brane geometry and the comparison with meson vevs will turn out that the low energy effective superpotential with enhanced gauge group SU(2) is exact. 
  We compare the logarithm of the degeneracy of BPS saturated elementary string states and the string modified Bekenstein-Hawking entropy of the corresponding black holes in N=2 supersymmetric heterotic string compactification to four dimensions. As in the case of N=4 supersymmetric theory, the two results match up to an overall undetermined numerical factor. We also show that this undetermined numerical constant is identical in the N=2 and N=4 supersymmetric theories, therby showing that the agreement between the Bekenstein-Hawking entropy and the microscopic entropy for N=2 theories does not require any new identity, other than the one already required for the N=4 theory. A similar result holds for type II string compactification as well. 
  The N-extended self-dual supergravity in the ultra-hyperbolic four-dimensional spacetime of kleinian signature (2+2) is given in the N-extended harmonic superspace. We reformulate the on-shell N-extended self-dual supergravity constraints of Siegel to a `zero-curvature' representation, and solve all of them but one in terms of a single superfield prepotential, by using a covariant Frobenius gauge in the Devchand-Ogievetsky approach. An off-shell superspace action, whose equation of motion yields the remaining constraint, is found. Our manifestly Lorentz-covariant action in harmonic superspace is very similar to the non-covariant Chern-Simons-type action, which was proposed earlier by Siegel in the light-cone N=8 superspace. Our action is also invariant under the residual superdiffeomorphisms and the residual local OSp(8|2) super-Lorentz rotations, which are left after imposing the Frobenius gauge. The infinitesimal superfield parameters of the residual symmetries are expressed in terms of independent analytic superfields. 
  Some exact solutions to the hypermultiplet low-energy effective action in N=2 supersymmetric four-dimensional gauge field theories with massive `quark' hypermultiplets are discussed. The need for a spontaneous N=2 supersymmetry breaking is emphasized, because of its possible relevance in the search for an ultimate theoretical solution to the confinement problem. 
  An iterative map of the unit disc in the complex plane (Appendix) is used to explore certain aspects of selfdual, four dimensional gauge fields (quasi)periodic in the Euclidean time. These fields are characterized by two topological numbers and contain standard instantons and monopoles as different limits. The iterations do not correspond directly to a discretized time evolution of the gauge fields. They are implemented in an indirect fashion. First, (t,r,\theta,\phi) being the standard coordinates, the (r,t) half plane is mapped on the unit disc in an appropriate way. This provides an (r,t) parametrization (Sec.1) of Z_0, the starting point of the iterations and makes the iterates increasingly complex functions of r and t. These are then incorporated as building blocks in the generating function of the fields (Sec.2). We explain (starting in Sec.1 and at different stages) in what sense and to what extent some remarkable features of our map (indicated in the title) are thus carried over into the continuous time development of the fields. Special features for quasiperiodicity are studied (Sec.3). Spinor solutions (Sec.4) and propagators (Sec.5) are discussed from the point of view of the mapping. Several possible generalizations are indicated (Sec.6). Some broader topics are discussd in conclusion (Sec.7). 
  The goal of this introduction to symmetries is to present some general ideas, to outline the fundamental concepts and results of the subject and to situate a bit the following lectures of this school. [These notes represent the write-up of a lecture presented at the fifth ``Seminaire Rhodanien de Physique: Sur les Symetries en Physique" held at Dolomieu (France), 17-21 March 1997. Up to the appendix and the graphics, it is to be published in "Symmetries in Physics", F.Gieres, M.Kibler,C.Lucchesi and O.Piguet, eds. (Editions Frontieres, 1998).] 
  We review exact results obtained for R^4 couplings in maximally supersymmetric type II string theories. These couplings offer a privileged scene to understand the rules of semiclassical calculus in string theory. Upon expansion in weak string coupling, they reveal an infinite sum of non-perturbative e^{-1/g} effects that can be imputed to euclidean D-branes wrapped on cycles of the compactification manifolds. They also shed light on the relation between Dp-branes and D-(p-2)branes, D-strings and (p,q) strings, instanton sums and soliton loops. The latter interpretation takes over in D<=6 in order to account for the e^{-1/g^2} effects, still mysterious from the point of view of instanton calculus.  [To appear in the proceedings of the conference "Quantum Aspects of Gauge Theories, Supersymmetry and Unification" held at Neuchatel University, Switzerland, 18-23 September 1997.] 
  An overview over effects of D-instantons in ten dimensional IIB superstring theory is given, including the supergravity instanton solution, instanton induced effective interaction vertices, the conjectured $SL(2,Z)$ invariant completion of such terms and the connection of such terms to a one loop calculation in eleven dimensional supergravity. 
  This is a survey of `Cohomological Physics', a phrase that first appeared in the context of anomalies in gauge theory. Differential forms were implicit in physics at least as far back as Gauss (1833) (cf. his electro-magnetic definition of the linking number), and more visibly in Dirac's magnetic monopole (1931). The magnetic charge was given by the first Chern number. Thus were characteristic classes (and by implication the cohomology of Lie algebras and of Lie groups) introduced into physics. The `ghosts' introduced by Fade'ev and Popov were incorporated into what came to be known as BRST cohomology. Later the ghosts were reinterpreted as generators of the Chevalley-Eilenberg complex for Lie algebra cohomology. Cohomological physics also makes use of group theoretic cohomology, algebraic deformation theory and especially a novel extension of homological algebra, combining Lie algebra cohomology with the Koszul-Tate resolution, the major emphasis of the talk. This synergistic combination of both kinds of cohomology appeared in the Batalin-Fradkin-Vilkovisky approach to the cohomological reduction of constrained Poisson algebras. An analogous `odd' version was developed in the Batalin-Vilkovisky approach to quantizing particle Lagrangians and Lagrangians of string field theory. A revisionist view of the Batalin- Vilkovisky machinery recognizes parts of it as a reconstruction of homological algebra with some powerful new ideas undreamt of in that discipline. 
  We show that the spontaneous compactification of the Abelian and non-Abelian two-form gauge field theories from $D=4+1$ to $D=3+1$ leads to the same theories plus the Maxwell and Yang-Mills ones, respectively. The vector potential comes from the zero mode of the fifth component of the tensor gauge field in D=5. Concerning to the non-Abelian case, it is necessary to make a more refined definition of the three-form stress tensor in order to be compatible, after the compactification, with the two-form stress tensor of the Yang-Mills theory. 
  We consider subleading terms in the one-loop Matrix theory potential between a classical membrane state and a supergraviton. Nontrivial terms arise at order v/r^8 and v^3/r^8 which are proportional to the angular momentum of the membrane state. The effective potential for a graviton moving in a boosted Kerr-type metric is computed and shown to agree precisely with the Matrix theory calculation at leading order in the long-distance expansion for each power of the graviton velocity. This result generalizes to arbitrary order; we show that terms in the membrane-graviton potential corresponding to nth moments of the membrane stress-energy tensor are reproduced correctly to all orders in the long-distance expansion by terms of the form F^4 X^n in the one-loop Matrix theory calculation. 
  We consider a supersymmetric system of D-5-branes compactified on a 5-torus with a self-dual background field strength on a 4-torus and carrying left-moving momentum along a circle. The corresponding supergravity solution describes a 5-dimensional black hole with a regular horizon. The entropy of this black hole may be explained in terms of the Landau degeneracy for open strings stretching between different branes. In the gauge theory approximation this D-5-brane system is described by a super Yang-Mills theory with a t'Hooft twist. By choosing a supersymmetric branch of the theory we obtain perfect agreement with the entropy formula. The result relies on the number of massless torons associated with the gauge field components that obey twisted boundary conditions. 
  We show that the nonlinear Born-Infeld field equations supplemented by the "dynamical condition" (certain boundary condition for the field along the particle's trajectory) define perfectly deterministic theory, i.e. particle's trajectory is determined without any equations of motion. It is a first step towards constructing the consistent theory of point particles interacting with nonlinear electromagnetism. 
  We show that the fluxes of the various six-dimensional "gauge" theories are associated to below threshold bound states of D-branes with the NS-5-branes and KK-monopoles which preserve half of bulk supersymmetry. We then present the supergravity solutions that correspond to these bound states. In addition using the worldvolume solitons of IIA and IIB NS-5-branes and KK-monopoles, we investigate the sectors of the "gauge" theories that preserve one quarter of bulk supersymmetry. This leads to a generalization a supergravity solution which has the interpretation of two intersecting NS-5-branes at a 3-brane and to the construction of some of the worldvolume solitons of IIA and M-theory KK-monopoles. Furthermore, using the IIA/IIB T-duality of the bulk theories, we give the T-duality transformations of the worldvolume solitons of NS-5-branes and KK-monopoles. We find that the worldvolume 0-brane, self-dual string and 2-brane solitons of NS-5-branes appear in the same T-duality chain. 
  We discuss an alternative form of the supersymmetric D-$p$-brane action which is quadratic in derivatives of $X$ and linear in $F_{\mu \nu}$. This action involves an auxiliary worldvolume tensor and generalises the simplification of the Nambu-Goto action for $p$-branes using a symmetric metric. When the worldvolume gauge field is abelian, it appears as a Lagrange multiplier, and solving the constraint gives the dual form of the ($p+1$)-dimensional action with a $p-2$ form gauge field instead of a vector gauge field. This is illustrated by the example of the dual D-2-brane action, for which the known result is recovered. 
  We discuss higher derivative interactions in the type IIB superstring in ten dimensions. From the fundamental string point of view, the non-perturbative corrections are due to D-instantons. We argue that they can alternatively be understood as arising from $(p,q)$-strings. We derive a non-renormalization theorem for eight-derivative bosonic interactions, which states that terms involving either NS-NS or R-R fields occur at tree-level and one-loop only. By using the $SL(2, Z)$ symmetry of M-theory on $T^2$, we show that in order for the possible $R^{3m+1} (m=1,2,...)$ interactions in M-theory to have a consistent perturbative expansion in nine dimensions, $m$ must be odd. Thus, only $R^{6N+4} (N=0,1,...)$ terms can be present in M-theory and their string theory counterparts arise at $N$ and $2N+1$ loops. Finally, we treat an example of fermionic term. 
  We derive the embedding structure of unitary N=2 minimal models and show as a result that these representations have a degeneration of uncharged singular states. This corrects some earlier mistakes made in the literature. We discuss the connexion to the N=2 character formulae and finally give a proof for the embedding diagrams. 
  We propose examples, which involve orbifolds by elements of the U-duality group, with M-theory moduli fixed at the eleven-dimensional Planck scale. We begin by reviewing asymmetric orbifold constructions in perturbative string theory, which fix radial moduli at the string scale. Then we consider non-perturbative aspects of those backgrounds (brane probes and the orbifold action from the eleven-dimensional point of view). This leads us to consider mutually non-perturbative group actions. Using a combination of dualities, matrix theory, and ideas for the generalization of the perturbative orbifold prescription, we present evidence that the examples we construct are consistent M-theory backgrounds. In particular we argue that there should be consistent non-supersymmetric compactifications of M-theory. 
  I consider N=1 supersymmetric SU(N_c) gauge theories with matter fields consisting of one antisymmetric representation, five flavors, and enough anti-fundamental representations to cancel the gauge anomaly. Previous analyses are extended to the case of even N_c with no superpotential. Using holomorphy I show that the theory has an interacting infrared fixed point for sufficiently large N_c. These theories are interesting due to the fact that in going from five to four flavors the theory goes from a non-trivial infrared fixed point to confinement, in contradistinction to SUSY QCD, but in analogy to the behavior expected in non-SUSY QCD. 
  We study the new ``gauge'' theories in 5+1 dimensions, and their non-commutative generalizations. We argue that the $\theta$-term and the non-commutative torus parameters appear on an equal footing in the non-critical string theories which define the gauge theories. The use of these theories as a Matrix description of M-theory on $T^5$, as well as a closely related realization as 5-branes in type IIB string theory, proves useful in studying some of their properties. 
  We introduce a new localization principle which is a generalized canonical transformation. It unifies BRST localization, the non-abelian localization principle and a special case of the conformal Duistermaat-Heckman integration formula of Paniak, Semenoff and Szabo. The heat kernel on compact Lie groups is localized in two ways. First using a non-abelian generalization of the derivative expansion localization of Palo and Niemi and secondly using the BRST localization principle and a configuration space path integral. In addition we present some new formulas on homogeneous spaces which might be useful in a possible localization of Selberg's trace formula on locally homogeneous spaces. 
  A general rule determining how extremal branes can intersect in a configuration with zero binding energy is presented. It is derived in a model independent way and without explicit use of supersymmetry, solving a set of classical equations of motion. When specializing to M and type II theories, it is shown that some intersection rules can be consistently interpreted as boundary rules for open branes ending on other branes. 
  The Dualized Standard Model which has a number of very interesting physical consequences is itself based on the concept of a nonabelian generalization to electric-magnetic duality. This paper explains first the reasons why the ordinary (Hodge) * does not give duality for the nonabelian theory and then reviews the steps by which these difficulties are surmounted, leading to a generalized duality transform formulated in loop space. The significance of this in relation to the Dualized Standard Model is explained, and possibly also to some other areas. 
  We study the moduli space of vacua of four dimensional N=1 and N=2 supersymmetric gauge theories with the gauge groups $Sp(2 N_c)$, $SO(2 N_c)$ and $SO(2 N_c +1)$ using the M theory fivebrane. Higgs branches of the N=2 supersymmetric gauge theories are interpreted in terms of the M theory fivebrane and the type IIA $s$-rule is realized in it. In particular we construct the fivebrane configuration which corresponds to a special Higgs branch root. This root is analogous to the baryonic branch root in the $SU(N_c)$ theory which remains as a vacuum after the adjoint mass perturbation to break N=2 to N=1. Furthermore we obtain the monopole condensations and the meson vacuum expectation values in the confining phase of N=1 supersymmetric gauge theories using the fivebrane technique. These are in complete agreement with the field theory results for the vacua in the phase with a single confined photon. 
  We show how to obtain the dual of any lattice model with inhomogeneous local interactions based on an arbitrary Abelian group in any dimension and on lattices with arbitrary topology. It is shown that in general the dual theory contains disorder loops on the generators of the cohomology group of a particular dimension. An explicit construction for altering the statistical sum to obtain a self-dual theory, when these obstructions exist, is also given. We discuss some applications of these results, particularly the existence of non-trivial self-dual 2-dimensional Z_N theories on the torus. In addition we explicitly construct the n-point functions of plaquette variables for the U(1) gauge theory on the 2-dimensional g-tori. 
  We prove unboundedness and boundedness of the unsmeared and smeared chiral vertex operators, respectively. We use elementary methods in bosonic Fock space, only. Possible applications to conformal two - dimensional quantum field theory, perturbation thereof, and to the perturbative construction of the sine-Gordon model by the Epstein-Glaser method are discussed. From another point of view the results of this paper can be looked at as a first step towards a Hilbert space interpretation of vertex operator algebras. 
  A massive gauge invariant formulation for scalar ($\phi$) and antisymmetric ($C_{mnp}$) fields with a topological coupling, which provides a mass for the axion field, is considered. The dual and local equivalence with the non-gauge invariant proposal is established, but on manifolds with non-trivial topological structure both formulations are not globally equivalent. 
  We construct the complete coupling of $(2,0)$ supergravity in six dimensions to $n$ tensor multiplets, extending previous results to all orders in the fermi fields. The truncation to $(1,0)$ supergravity coupled to tensor multiplets exactly reproduces the complete couplings recently obtained. 
  The (generalized) WDVV equations for the prepotentials in $2d$ topological and $4,5d$ Seiberg-Witten models are covariant with respect to non-linear transformations, described in terms of solutions of associated linear problem. Both time-variables and the prepotential change non-trivially, but period matrix (prepotential's second derivatives) remains intact. 
  The structure of the models of section 3.2 has been found to be richer than previously thought; a revised version has appeared as hep-th/9808087. 
  We study a configuration of a parallel F- (fundamental) and D- string in IIB string theory by considering its T-dual configuration in the matrix model description of M-theory. We show that certain non-perturbative features of string theory such as $O(e^{-\frac{1}{g_{s}}})$ effects due to soliton loops, the existence of bound state (1,1) strings and manifest S-duality, can be seen in matrix models. We discuss certain subtleties that arise in the large-N limit when membranes are wrapped around compact dimensions. 
  We discuss the BPS configurations of IIB strings in the 7-brane background from M-theory viewpoint. We first obtain the hyperkahler geometry background of M-theory expected from the 7-brane solutions of the type IIB supergravity. We choose the appropriate complex structures of the background geometries and embed a membrane of M-theory holomorphically to obtain a BPS string configuration. The recently discussed BPS string configurations such as 3-string junctions and string networks in the flat background are generalized to the cases with the 7-brane backgrounds. The property of the BPS string configurations in the 7-brane backgrounds is in agreement with the previously known results from the IIB string viewpoint. 
  We consider Yang-Mills theory in a general class of Abelian gauges. Exploiting the residual Abelian symmetry on a quantum level, we derive a set of Ward identities in functional form, valid to all orders in perturbation theory. As a consequence, the coupling constant is only renormalised through the Abelian two-point function. This implies that asymptotic freedom in all Abelian gauges can be understood from an effective Abelian theory alone, which can be interpreted as Abelian dominance in the high energy regime. 
  We compute the BRST cohomology of the holomorphic part of the N=2 string at arbitrary ghost and picture number. We confirm the expectation that the relative cohomology at non-zero momentum consists of a single massless state in each picture. The absolute cohomology is obtained by an independent method based on homological algebra. For vanishing momentum, the relative and absolute cohomologies both display a picture dependence -- a phenomenon discovered recently also in the relative Ramond sector of N=1 strings by Berkovits and Zwiebach. 
  We investigate the (noncommutative) geometry defined by the standard model, which turns out to be of Kaluza-Klein type. We find that spacetime points are replaced by extended two-dimensional objects which resemble the surface of a gyro. Their size is of the order of the inverse top quark mass. 
  We review brane engineering of mirror pairs of 3d N=4 theories. It reveals aspects of 3d physics not known from previous field theoretic studies: novel QFT's without Lagrangian description and transitions to them from conventional QFT's. 
  We show that the linearized supergravity potential between two objects arising from the exchange of quanta with zero longitudinal momentum is reproduced to all orders in 1/r by terms in the one-loop Matrix theory potential. The essential ingredient in the proof is the identification of the Matrix theory quantities corresponding to moments of the stress tensor and membrane current. We also point out that finite-N Matrix theory violates the equivalence principle. 
  We discuss D-branes from a conformal field theory point of view. In this approach, branes are described by boundary states providing sources for closed string modes, independently of classical notions. The boundary states must satisfy constraints which fall into two classes: The first consists of gluing conditions between left- and right-moving Virasoro or further symmetry generators, whereas the second encompasses non-linear consistency conditions from world sheet duality, which severely restrict the allowed boundary states. We exploit these conditions to give explicit formulas for boundary states in Gepner models, thereby computing excitation spectra of brane configurations. From the boundary states, brane tensions and RR charges can also be read off directly. 
  The occupied and unoccupied fermionic BPS quantum states of a type-IIA string stretched between a D6-brane and an orthogonal D2-brane are described in M-theory by two particular holomorphic curves embedded in a Kaluza-Klein monopole. The absence of multiply-occupied fermionic states --- the Pauli exclusion principle --- is manifested in M-theory by the absence of any other holomorphic curves satisfying the necessary boundary conditions. Stable, non-BPS states with multiple strings joining the D6-brane and D2-brane are described M-theoretically by non-holomorphic curves. 
  We outline a method of deriving boost invariant hamiltonians for effective particles in quantum field theory. The hamiltonians are defined and calculated using creation and annihilation operators in light-front dynamics. The renormalization group equations are written for a sequence of unitary transformations which gradually transform the bare canonical creation and annihilation operators of a local theory to the creation and annihilation operators of effective particles in an effective theory with the same dynamical content but a finite range of energy transfers due to form factors in the interaction vertices. The boost invariant effective hamiltonians can be used to describe the constituent dynamics in relativistically moving systems including the rest and the infinite momentum frame. The general equations are illustrated in perturbation theory by second-order calculations of self-energy and two-particle interaction terms in Yukawa theory, QED and QCD. In Yukawa theory, one obtains the generalized Yukawa potential including its full off-energy-shell extension and form factors in the vertices. In QED, the effective hamiltonian eigenvalue problem converges for small coupling constants to the Schr\"odinger equation but the typical relativistic ultraviolet singularities at short distances between constituents are regularized by the similarity form factors. In the second-order QCD effective hamiltonian one obtains a boost invariant logarithmically confining quark-anti-quark interaction term which may remain uncanceled in the non-abelian dynamics of effective quarks and gluons. 
  We discuss dyons, charge quantization and electric-magnetic duality for self-interacting, abelian, p-form theories in the spacetime dimensions D=2(p+1) where dyons can be present. The corresponding quantization conditions and duality properties are strikingly different depending on whether p is odd or even. If p is odd one has the familiar eg'-ge'= 2nh, whereas for even p one finds the opposite relative sign, eg'+ge'= 2nh. These conditions are obtained by introducing Dirac strings and taking due account of the multiple connectedness of the configuration space of the strings and the dyons. A two-potential formulation of the theory that treats the electric and magnetic sources on the same footing is also given.  Our results hold for arbitrary gauge invariant self-interaction of the fields and are valid irrespective of their duality properties. 
  Intrinsically stable or `fundamental' solitons may be decorated with conserved charges which are pieces of those carried by elementary particles in the same medium. These `hairs' are always significant in principle, and in the strong-coupling regime (where solitons and particles exchange roles) may become major factors in dynamics. 
  We introduce a self-dual field strength which replaces the gauge field in spontaneously broken Yang-Mills theory, reformulating it as a Lorentz covariant non-linear sigma model. This dualized theory is in both a unitary and renormalizable gauge: The self-dual field strength has exactly the three components necessary to describe a massive vector field. In future work we shall utilize this new formulation as a calculational tool in spontaneously broken gauge theories. 
  Jackiw and Rebbi found two types of intrinsically stable or `fundamental' soliton (kinks in 1+1 D and magnetic monopoles in 3+1 D) which can carry pieces of elementary-particle charges. After two decades there are no more, and it is argued here why that is inevitable. 
  We consider a brane configuration consisting of intersecting Neveu-Schwarz five-branes, Dirichlet four-branes, and an orientifold four-plane in a C^2/Z_3 orbifold background. We show that the low-energy dynamics is described by a four dimensional gauge theory with N=1 supersymmetry and SO(N+4) X SU(N) or SP(2M) X SU(2M+4) gauge symmetry. The matter content of this theory is chiral. In particular, the SU group has one matter field in the antisymmetric tensor or symmetric tensor representation and several fields in the fundamental and antifundamental representations. We discuss various consistency checks on these theories. By considering the brane configuration in M theory we deduce the spectral curves for these theories. Finally, we consider the effects of replacing the orbifold background with a non-singular ALE space (both with and without an orientifold plane) and show that it leaves the spectral curves unchanged. 
  We investigate the canonical equivalence of a matter-coupled 2D dilaton gravity theory defined by the action functional $S = \int d^2x \sqrt{-g} (R\phi + V(\phi) - 1/2 H(\phi ) (\nablaf)^2)$, and a free field theory. When the scalar field $f$ is minimally coupled to the metric field$(H(\phi)=1)$ the theory is equivalent, up to a boundary contribution,to a theory of three free scalar fields with indefinite kinetic terms, irrespective of the particular form of the potential $V(\phi)$. If the potential is an exponential function of the dilaton one recovers a generalized form of the classical canonical transformation of Liouville theory. When $f$ is a dilaton coupled scalar $(H(\phi)=\phi)$ and the potential is an arbitrary power of the dilaton the theory is also canonically equivalent to a theory of three free fields with a Minkowskian target space. In the simplest case $(V(\phi)=0)$ we provide an explicit free field realization of the Einstein-Rosen midisuperspace. The Virasoro anomaly and the consistence of the Dirac operator quantization play a central role in our approach. 
  This article reviews the non-perturbative structure of certain higher derivative terms in the type II string theory effective action and their connection to one-loop effects in eleven-dimensional supergravity compactified on a torus. New material is also included that was not presented in the talks. 
  In order to study the discrepancy between the supersymmetry bound and the extremality bound for rotating black holes, the effect of duality transformations on the class of stationary and axially symmetric string backgrounds, called the TNbh, is considerd. It is shown that the Bogomolnyi bound is invariant under those duality transformations that transform the TNbh into itself, meaning that duality does not constrain the angular momentum in such a way as to reconcile the aforementioned bounds. A physical reason for the existence of the discrepancy is given in terms of superradiance. 
  We study the spectrum of the scaling Lee-Yang model on a finite interval from two points of view: via a generalisation of the truncated conformal space approach to systems with boundaries, and via the boundary thermodynamic Bethe ansatz. This allows reflection factors to be matched with specific boundary conditions, and leads us to propose a new (and non-minimal) family of reflection factors to describe the one relevant boundary perturbation in the model. The equations proposed previously for the ground state on an interval must be revised in certain regimes, and we find the necessary modifications by analytic continuation. We also propose new equations to describe excited states, and check all equations against boundary truncated conformal space data. Access to the finite-size spectrum enables us to observe boundary flows when the bulk remains massless, and the formation of boundary bound states when the bulk is massive. 
  In this work we calculate the low-energy effective action for gravity with torsion, obtained after the integration of scalar and fermionic matter fields, using the local momentum representation based on the Riemann normal coordinates expansion. By considering this expansion around different space-time points, we also compute the nonlocal terms together with the more usual divergent ones. Finally we discuss the applicability of our results to the calculation of particle production probabilities. 
  In this article, applying different types of boundary conditions; Dirichlet, Neumann, or Mixed, on open strings we realize various new brane bound states in string theory. Calculating their interactions with other D-branes, we find their charge densities and their tension. A novel feature of $(p-2,p)$ brane bound state is its "non-commutative" nature which is manifestly seen both in the open strings mode expansions and in their scattering off a $D_p$-brane. Moreover we study three or more object bound states in string theory language. Finally we give a M-theoretic picture of these bound states. 
  We show that the D=5 dilaton-axion gravity compactified on a 2-torus possesses the SL(4,R)/SO(4) matrix formulation. It is used for construction of the SO(2,2)-invariant BPS solution depended on the one harmonic function. 
  In this paper we study generic M(atrix) theory compactifications that are specified by a set of quotient conditions. A procedure is proposed, which both associates an algebra to each compactification and leads deductively to general solutions for the matrix variables. The notion of noncommutative geometry on the dual space is central to this construction. As examples we apply this procedure to various orbifolds and orientifolds, including ALE spaces and quotients of tori. While the old solutions are derived in a uniform way, new solutions are obtained in several cases. Our study also leads to a new formulation of gauge theory on quantum spaces. 
  The concept of infinite statistics is applied to the analysis of black hole thermodynamics in Matrix Theory. It is argued that Matrix Theory partons, D0-branes, satisfy quantum infinite statistics, and that this observation justifies the recently proposed Boltzmann gas model of Schwarzschild black holes in Matrix Theory. 
  We derive and solve the Hamiltonian flow equations for a Dirac particle in an external static potential. The method shows a general procedure for the set up of continuous unitary transformations to reduce the Hamiltonian to a quasidiagonal form. 
  The generating functionals of Green's functions with composite and external fields are considered in the framework of BV and BLT quantization methods for general gauge theories. The corresponding Ward identities are derived and the gauge dependence is investigated 
  We review some basic notions on anomalies in field theories and superstring theories, with particular emphasis on the concept of locality. The aim is to prepare the ground for a discussion on anomalies in theories with branes. In this light we review the problem of chiral anomaly cancellation in M-theory with a 5-brane. 
  We illustrate the various ways in which the algebraic framework of noncommutative geometry naturally captures the short-distance spacetime properties of string theory. We describe the noncommutative spacetime constructed from a vertex operator algebra and show that its algebraic properties bear a striking resemblence to some structures appearing in M Theory, such as the noncommutative torus. We classify the inner automorphisms of the space and show how they naturally imply the conventional duality symmetries of the quantum geometry of spacetime. We examine the problem of constructing a universal gauge group which overlies all of the dynamical symmetries of the string spacetime. We also describe some aspects of toroidal compactifications with a light-like coordinate and show how certain generalized Kac-Moody symmetries, such as the Monster sporadic group, arise as gauge symmetries of the resulting spacetime and of superstring theories. 
  The paper containes a classification of consistent free string models in physical dimensions and a brief discussion of recent results concerning relations between various models. 
  A slightly extended version, with a footnote added on December 19, 1997, of a contributed Abstract to the Eight Marcel Grossmann Meeting, Jerusalem, June 1997. 
  We propose a string-theoretic ansatz describing the dynamics of SU(N) Yang-Mills theories in the limit of large N in D=4. The construction uses in a crucial way open-string vertex operators that describe non-perturbative brane dynamics. According to our proposal, various gauge theories are described by string theories with the same action, but with different measures in the functional integral. The choice of measure defines the gauge group, as well as the effective space-time dimension of the resulting gauge theory. 
  We extend a recently proposed non-local and non-covariant version of the Thirring model to the finite-temperature case. We obtain a completely bosonized expression for the partition function, describing the thermodynamics of the collective modes which are the underlying excitations of this system. From this result we derive closed formulae for the free-energy, specific-heat, two-point correlation functions and momentum distribution, as functionals of electron-electron coupling potentials. 
  We provide an explicit construction of 1/4 BPS states in four-dimensional N=4 Super-Yang-Mills theory with a gauge group SU(3). These states correspond to three-pronged strings connecting three D3-branes. We also find curves of marginal stability in the moduli space of the theory, at which the above states can decay into two 1/2 BPS states. 
  We show that the abelian topological mass mechanism in four dimensions, described by the Cremmer-Sherk action, can be obtained from dimensional reduction in five dimensions. Starting from a gauge invariant action in five dimensions, where the dual equivalence between a massless vector field and a massless second-rank antisymmetric field in five dimensions is established, the dimensional reduction is performed keeping only one massive mode. Furthermore, the Kalb-Ramond action and the Stuckelberger formulation for massive spin-1 are recovered. 
  We study the hypermultiplet moduli space of an N=4, U(Q_1)xU(Q_5) gauge theory in 1+1 dimensions to extract the effective SCFT description of near extremal 5-dimensional black holes modelled by a collection of D1- and D5-branes. On the moduli space, excitations with fractional momenta arise due to a residual discrete gauge invariance. It is argued that, in the infra-red, the lowest energy excitations are described by an effective c=6, N=4 SCFT on T^4, also valid in the large black hole regime. The ``effective string tension'' is obtained using T-duality covariance. While at the microscopic level, minimal scalars do not couple to (1,5) strings, in the effective theory a coupling is induced by (1,1) and (5,5) strings, leading to Hawking radiation. These considerations imply that, at least for such black holes, the calculation of the Hawking decay rate for minimal scalars has a sound foundation in string theory and statistical mechanics and, hence, there is no information loss. 
  In the strong-coupling limit of the heterotic string theory constructed by Horava and Witten, an 11-dimensional supergravity theory is coupled to matter multiplets confined to 10-dimensional mirror planes. This structure suggests that realistic unification models are obtained, after compactification of 6 dimensions, as theories of 5-dimensional supergravity in an interval, coupling to matter fields on 4-dimensional walls. Supersymmetry breaking may be communicated from one boundary to another by the 5-dimensional fields. In this paper, we study a toy model of this communication in which 5-dimensional super-Yang-Mills theory in the bulk couples to chiral multiplets on the walls. Using the auxiliary fields of the Yang-Mills multiplet, we find a simple algorithm for coupling the bulk and boundary fields. We demonstrate two different mechanisms for generating soft supersymmetry breaking terms in the boundary theory. We also compute the Casimir energy generated by supersymmetry breaking. 
  N=8 supergravity has a rich spectrum of black holes charged under the 56 U(1) gauge fields of the theory. Duality predicts that the entropy of these black holes is related to the quartic invariant of the E(7,7) group. We verify this prediction in detail by constructing black holes that correspond to supersymmetric bound states of 2-branes at angles and 6-branes. The general bound state contains an arbitrary number of branes rotated relative to each other, and we derive the condition for these rotations to preserve supersymmetry. The microscopic bound state degeneracy matches the black hole entropy in detail. The entire 56 charge spectrum of extremal black holes in N=8 supergravity can be displayed as the orbit under duality of a five parameter generating solution. We exhibit a new generating configuration consisting of D3-branes at angles and discuss its entropy. 
  The two-component formulation of quantum electrodynamics is studied. The relation with the usual Dirac formulation is exhibited, and the Feynman rules for the two-component form of the theory are presented in terms of familiar objects. The transformation from the Dirac theory to the two-component theory is quite amusing, involving Faddeev-Popov ghost loops of a fermion type with bose statistics. The introduction of an anomalous magnetic moment in the two-component formalism is simple; it is not equivalent to a Pauli term in the Dirac formulation. Such an anomalous magnetic moment appears not to destroy the renormalizability of the theory but violates unitarity. 
  The neutrino field is considered in the framework of a complex Clifford algebra $\C_3\cong\C_2\oplus\stackrel{\ast}{\C}_2$. The factor-algebras ${}^{\epsilon}\C_2$ and ${}^{\epsilon}\stackrel{\ast}{\C}_2$, which are obtained by means of homomorphic mappings $\C_3\to\C_2$ and $\C_3\to\stackrel{\ast}{\C}_2$, are identified with the neutrino and antineutrino fields, respectively. In this framework we have natural explanation for absence of right-handed neutrino and left-handed antineutrino. 
  We study instanton effects along the Coulomb branch of an N=2 supersymmetric Yang-Mills theory with gauge group SU(2) on Asymptotically Locally Euclidean (ALE) spaces. We focus our attention on an Eguchi-Hanson gravitational background and on gauge field configurations of lowest Chern class. 
  The dual Schrodinger equation for x in terms of psi derived by Faraggi and Matone [hep-th/9606063] with techniques inspired by field theory duality can be derived almost trivially from differential calculus. 
  We derive the fermionic polynomial generalizations of the characters of the integrable perturbations $\phi_{2,1}$ and $\phi_{1,5}$ of the general minimal $M(p,p')$ conformal field theory by use of the recently discovered trinomial analogue of Bailey's lemma. For $\phi_{2,1}$ perturbations results are given for all models with $2p>p'$ and for $\phi_{1,5}$ perturbations results for all models with ${p'\over 3}<p< {p'\over 2}$ are obtained. For the $\phi_{2,1}$ perturbation of the unitary case $M(p,p+1)$ we use the incidence matrix obtained from these character polynomials to conjecture a set of TBA equations. We also find that for $\phi_{1,5}$ with $2<p'/p < 5/2$ and for $\phi_{2,1}$ satisfying $3p<2p'$ there are usually several different fermionic polynomials which lead to the identical bosonic polynomial. We interpret this to mean that in these cases the specification of the perturbing field is not sufficient to define the theory and that an independent statement of the choice of the proper vacuum must be made. 
  We give an explicit form of the classical entropy for four-dimensional static near-BPS-saturated black holes of generic N=2 superstring vacua. The expression is obtained by determining the leading corrections in the non-extremality parameter to the corresponding BPS-saturated black hole solutions. These classical results are quantitatively compared with the microscopic leading order corrections to the microscopic result of Maldacena Strominger and Witten for N=2 BPS-saturated black holes. 
  A set of coupled non-linear integral equations is derived for a class of models connected with the quantum group $U_q(\hat g)$ ($g$ simply laced Lie algebra), which are solvable using the Bethe Ansatz; these equations describe arbitrary excited states of a system with finite spatial length $L$. They generalize the Destri-De Vega equation for the Sine-Gordon/massive Thirring model to affine Toda field theory with imaginary coupling constant. As an application, the central charge and all the conformal weights of the UV conformal field theory are extracted in a straightforward manner. The quantum group truncation for $q$ at a root of unity is discussed in detail; in the UV limit we recover through this procedure the RCFTs with extended $W(g)$ conformal symmetry. 
  We point out that the matrix description of M-theory compactified on Calabi-Yau threefolds is in many respects simpler than the matrix description of a $T^6$ compactification. This is largely because of the differences between D6 branes wrapped on Calabi-Yau threefolds and D6 branes wrapped on six-tori. In particular, if we define the matrix theory following the prescription of Sen and Seiberg, we find that the remaining degrees of freedom are decoupled from gravity. 
  We study the Euler numbers and the entropies of the non-extremal intersecting D-branes in ten-dimensions. We use the surface gravity to constrain the compactification radii. We correctly obtain the integer valued Euler numbers for these radii. Moreover, the entropies are found to be invariant under the T-duality transformation. In the extremal limit, we obtain the finite entropies only for two intersecting D-branes. We observe that these entropies are proportional to the product of the charges of each D-brane. We further study the entropies of the boosted metrics. We find that their entropies can be interpreted in term of the microscopic states of D-branes. 
  A large class of equivalence relations between the moduli spaces of instantons on ALE spaces and the Higgs branches of supersymmetric Yang-Mills theories, are found by means of a certain kind of duality transformation between brane configurations in superstring theories. 4d, N=2 and 5d, N=1 supersymmetric gauge theories with product gauge groups turn out to correspond to the ALE-instanton moduli of type II B and type II A superstring theories, respectively. 
  The cohomological approach to the problem of consistent interactions between fields with a gauge freedom is reviewed. The role played by the BRST symmetry is explained. Applications to massless vector fields and 2-form gauge fields are surveyed. 
  We discuss certain integrable quantum field theories in (1+1)-dimensions consisting of coupled sine/sinh-Gordon theories with N=1 supersymmetry, positive kinetic energy, and bosonic potentials which are bounded from below. We show that theories of this type can be constructed as Toda models based on the exceptional affine Lie superalgebra $D(2,1;\A)^{(1)}$ (or on related algebras which can be obtained as various limits) provided one adopts appropriate reality conditions for the fields. In particular, there is a continuous family of such models in which the couplings and mass ratios all depend on the parameter $\A$. The structure of these models is analyzed in some detail at the classical level, including the construction of conserved currents with spins up to 4. We then show that these currents generalize to the quantum theory, thus demonstrating quantum-integrability of the models. 
  We derive the classical transport equation, in scalar field theory with a V(phi) interaction, from the equation of motion for the quantum field. We obtain a very simple, but iterative, expression for the effective action which generates all the n-point Green functions in the high-temperature limit. An explicit closed form is given in the static case. 
  Basic features of the conservation laws in the Hamiltonian approach to the Poincar\'e gauge theory are presented. It is shown that the Hamiltonian is given as a linear combination of ten first class constraints. The Poisson bracket algebra of these constraints is used to construct the gauge generators. By assuming that the asymptotic symmetry is the global Poincar\'e symmetry, we derived the improved form of the asymptotic generators, and discussed the related conservation laws of energy, momentum, etc. 
  We discuss the ``fractional D-branes'' which arise in orbifold resolution. We argue that they arise as subsectors of the Coulomb branch of the quiver gauge theory used to describe both string theory D-brane and Matrix theory on an orbifold, and thus must form part of the full physical Hilbert space. We make further observations confirming their interpretation as wrapped membranes. 
  We review a recently developed covariant Lagrangian formulation for $p$--forms with (anti)self-dual field-strengths and present its extension to the supersymmetric case. As explicit examples we construct covariant Lagrangians for six-dimensional models with N=1 rigid and curved supersymmetry. 
  We compute the genus one correction to the integrable hierarchy describing coupling to gravity of a 2D topological field theory. The bihamiltonian structure of the hierarchy is given by a classical W-algebra; we compute the central charge of this algebra. We also express the generating function of elliptic Gromov - Witten invariants via tau-function of the isomonodromy deformation problem arising in the theory of WDVV equations of associativity. 
  We consider novel nonperturbative effects of type I theories compactified on singular ALE spaces obtained by adding NS branes. Such effects include a description of small $E_8$ instantons at singularities. 
  In the conventional $p$-brane theory, a gauge ($p+1$)-form field $\phi^{(p+1)}$ mediates the interaction between $p$-branes, which is gauge invariant in the sense $\phi^{(p+1)}\to \phi^{(p+1)}+d\Lambda^{(p)}$ with $\Lambda^{(p)}$, an arbitrary $p$-form and $d$, a boundary operator. We, on the contrary, propose to introduce a new gauge field $\phi^{(p+2)}$ mediating the interaction, and we have a new type of gauge transformation: $\phi^{(p+2)}\to \phi^{(p+2)} +\delta\Lambda^{(p+3)}$, with $\delta$, a coboundary operator. 
  In this report, in the framework of an analytical approach and with help of the generalized version of the Hurwitz transformation the five-dimensional SU(2)--monopole model is constructed from the eight-dimensional quantum oscillator. The SU(2)--monopole fields, the Clebsh-Gordan expansion stimulated by the space-gauge coupling, the hyperangle and the radial parts of the total wave function, the energy spectrum of the charge-monopole bound system and the corresponding degeneracy are calculated. 
  Recently, in collaboration with Susskind, we proposed a model of Schwarzschild black holes in Matrix theory. A large Schwarzschild black hole is described by a metastable bound state of a large number of D0-branes which are held together by a background, whose structure has so far been understood only in 8 and 11 dimensions. The Hawking radiation proceeds by emission of small clusters of D0-branes. We estimate the Hawking rate in the Matrix theory model of Schwarzschild black holes and find agreement with the semiclassical rate up to an undetermined numerical coefficient of order 1. 
  It has been a puzzle that rotating detector may respond even in the appropriate vacuum defined via canonical quantization. We solve this puzzle by taking back reaction of the detector into account. The influence of the back reaction, even in the detector's mass infinite limit, appears in the response function. It makes the detector possible to respond in the vacuum if the detector is rotating, though the detector in linear uniform motion never respond in the vacuum as expected from Poincare invariance. 
  We compute graviton scattering amplitudes in M-theory using Feynman rules for a scalar particle coupled to gravity in eleven dimensions. The processes that we consider describe the single graviton exchange and the double graviton exchange, that in M(atrix) theory correspond to the $v^4/r^7$- and $v^6/r^{14}$-term respectively. We further show that the $ v^6/r^{14}$-term appearing in M(atrix) theory at two-loops can be obtained from the covariant eleven-dimensional four-graviton amplitude. Finally, we calculate the $v^8/r^{18}$-term appearing at two-loops in M(atrix) theory. It has been previously conjectured that this term is related to a four graviton scattering amplitude involving the $R^4$-vertex of M-theory. 
  We examine the boundary conditions associated with extended supersymmetric Maxwell theory in 5-dimensional anti-De Sitter space. Excitations on the boundary are identical to those of ordinary 4-dimensional conformal invariant super electrodynammics. Extrapolations of these excitations give rise to a 5-dimensional topological gauge theory of the singleton type. The possibility of a connection of this phenomenon to the world volume theory of 3-branes in IIB string theory is discussed. 
  We study a model of free massless scalar fields on a two dimensional cylinder with metric that admits a change of signature between Lorentzian and Euclidean type (ET), across the two timelike hypersurfaces (with respect to Lorentzian region). Considering a long strip-shaped region of the cylinder, denoted by an angle \theta, as the signature changed region it is shown that the energy spectrum depends on the angle \theta and in a sense differs from ordinary one for low energies. Morever diffeomorphism algebra of corresponding infinite conserved charges is different from '' Virasoro'' algebra and approaches to it at higher energies. The central term is also modified but does not approach to the ordinary one at higher energies. 
  We develop some useful techinques for integrating over Higgs branches in supersymmetric theories with 4 and 8 supercharges. In particular, we define a regularized volume for hyperkahler quotients. We evaluate this volume for certain ALE and ALF spaces in terms of the hyperkahler periods. We also reduce these volumes for a large class of hyperkahler quotients to simpler integrals. These quotients include complex coadjoint orbits, instanton moduli spaces on R^4 and ALE manifolds, Hitchin spaces, and moduli spaces of parabolic Higgs bundles on Riemann surfaces. In the case of Hitchin spaces the evaluation of the volume reduces to a summation over solutions of Bethe Ansatz equations for the non-linear Schroedinger system. We discuss some applications of our results. 
  The cosmological compactification of D=10, N=1 supergravity-super-Yang-Mills theory obtained from superstring theory is studied. The constraint of unbroken N=1 supersymmetry is imposed. A duality transformation is performed on the resulting consistency conditions. The original equations as well as the transformed equations are solved numerically to obtain new configurations with a nontrivial scale factor and a dynamical dilaton. It is shown that various classes of solutions are possible, which include cosmological solutions with no initial singularity. 
  The conformal field theory based on the $g/u(1)^d$ coset construction is treated as the WZNW theory for the affine Lie algebra $\hat g$ with the constrained $\hat u(1)^d$ subalgebra.Using a modification of the generalized canonical quantization method generators and primary fields of an extended symmetry algebra are found for arbitrary d. 
  We investigate a lattice version of QED by numerical simulations. For the renormalized charge and mass we find results which are consistent with the renormalized charge vanishing in the continuum limit. A detailed study of the relation between bare and renormalized quantities reveals that the Landau pole lies in a region of parameter space which is made inaccessible by spontaneous chiral symmetry breaking. 
  The content of the comment [hep-th/9712219] is the derivation of Eq.(13) in Phys. Rev. Lett. 78 (1997) 163 by direct differential calculus: which is precisely the same method we used to derive it (it is in fact difficult to imagine any other possible derivation). 
  We formulate a non-linear system of equations which describe higher-spin gauge interactions of massive matter fields in 2+1 dimensional space-time and explain some properties of the deformed oscillator algebra which underlies this formulation. In particular we show that the parameter of mass $M$ of matter fields is related to the deformation parameter in this algebra. 
  We examine the general conditions for the existence of the complex structure intrinsic in the Gupta-Bleuler quantization method for the specific case of mixed first and second class fermionic constraints in an arbitrary space-time dimension. The cases d=3 and 10 are shown to be of prime importance. The explicit solution for d=10 is presented. 
  We study adjoint and fundamental Wilson loops in the center-vortex picture of confinement, for gauge group SU(N) with general N. There are N-1 distinct vortices, whose properties, including collective coordinates and actions, we study. In d=2 we construct a center-vortex model by hand so that it has a smooth large-N limit of fundamental-representation Wilson loops and find, as expected, confinement. Extending an earlier work by the author, we construct the adjoint Wilson-loop potential in this d=2 model for all N, as an expansion in powers of $\rho/M^2$, where $\rho$ is the vortex density per unit area and M is the vortex inverse size, and find, as expected, screening. The leading term of the adjoint potential shows a roughly linear regime followed by string breaking when the potential energy is about 2M. This leading potential is a universal (N-independent at fixed fundamental string tension $K_F$) of the form $(K_F/M)U(MR)$, where R is the spacelike dimension of a rectangular Wilson loop. The linear-regime slope is not necessarily related to $K_F$ by Casimir scaling. We show that in d=2 the dilute vortex model is essentially equivalent to true d=2 QCD, but that this is not so for adjoint representations; arguments to the contrary are based on illegal cumulant expansions which fail to represent the necessary periodicity of the Wilson loop in the vortex flux. Most of our arguments are expected to hold in d=3,4 also. 
  We study the relation between the group-algebraic approach and the dressing symmetry one to the soliton solutions of the $A_n^{(1)}$ Toda field theory in 1+1 dimensions. Originally solitons in the affine Toda models has been found by Olive, Turok and Underwood. Single solitons are created by exponentials of elements which ad-diagonalize the principal Heisenberg subalgebra. Alternatively Babelon and Bernard exploited the dressing symmetry to reproduce the known expressions for the fundamental tau functions in the sine-Gordon model. In this paper we show the equivalence between these two methods to construct solitons in the $A_n^{(n)}$ Toda models. 
  We propose that spacetime is fundamentally a property of matter, inseparable from it. This leads us to suggest that all properties of matter must be elevated to the same status as that of spacetime in quantum field theories of matter. We suggest a specific method for extending field theories to accomodate this, and point out how this leads to the evolution of fields through channels other than the spacetime channel. 
  Black holes whose near-horizon geometries are locally, but not necessarily globally, AdS$_3$ (three-dimensional anti-de Sitter space) are considered. Using the fact that quantum gravity on AdS$_3$ is a conformal field theory, we microscopically compute the black hole entropy from the asymptotic growth of states. Precise numerical agreement with the Bekenstein-Hawking area formula for the entropy is found. The result pertains to any consistent quantum theory of gravity, and does not use string theory or supersymmetry. 
  In this note it is shown that near-extremal four dimensional dyonic black holes, where the dilaton is not constant, can be described by a microscopic model consisting of a one-dimensional gas of massless particles. 
  A review of recent progress in string theory concerning the Bekenstein formula for black hole entropy is given. Topics discussed include p-branes, D-branes and supersymmetry; the correspondence principle; the D- and M-brane approach to black hole entropy; the D-brane analogue of Hawking radiation, and information loss; D-branes as probes of black holes; and the Matrix theory approach to charged and neutral black holes. Some introductory material is included. 
  The Geroch group, isomorphic to the SL(2,R) affine Kac-Moody group, is an infinite dimensional solution generating group of Einstein's equations with two surface orthogonal commuting Killing vectors. We introduce another solution generating group for these equations, the dressing group, and discuss its connection with the Geroch group. We show that it acts transitively on a dense subset of moduli space. We use a new Lax pair expressing a twisted self-duality of this system and we study the dressing problem associated to it. We also describe how to use vertex operators to solve the reduced Einstein's equations. In particular this allows to find solutions by purely algebraic computations. 
  The soliton structure of a gauge theory proposed to describe chiral excitations in the multi-Layer Fractional Quantum Hall Effect is investigated. A new type of derivative multi-component nonlinear Schr\"{o}dinger equation emerges as effective description of the system that supports novel chiral solitons. We discuss the classical properties of the solutions and study relations to integrable systems. 
  A closed formula for the structure constants in the SL(2,C)/SU(2) WZNW model is derived by a method previously used in Liouville theory. With the help of a reflection amplitude that follows from the structure constants one obtains a proposal for the fusion rules from canonical quantization. Taken together these pieces of information allow an unambigous definition of any genus zero n-point function. 
  Motivated by recent developments in string theory, we study the structure of boundary conditions in arbitrary conformal field theories. A boundary condition is specified by two types of data: first, a consistent collection of reflection coefficients for bulk fields on the disk; and second, a choice of an automorphism $\omega$ of the fusion rules that preserves conformal weights. Non-trivial automorphisms $\omega$ correspond to D-brane configurations for arbitrary conformal field theories. The choice of the fusion rule automorphism $\omega$ amounts to fixing the dimension and certain global topological features of the D-brane world volume and the background gauge field on it. We present evidence that for fixed choice of $\omega$ the boundary conditions are classified as the irreducible representations of some commutative associative algebra, a generalization of the fusion rule algebra. Each of these irreducible representations corresponds to a choice of the moduli for the world volume of the D-brane and the moduli of the flat connection on it. 
  Many qualitatively new features of WZNW models associated to noncompact cosets are due to zero modes with continuous spectrum. Insight may be gained by reducing the theory to its zero-mode sector, the mini-superspace limit. This will be discussed in some detail for the example of SL(2,C)/SU(2)-WZNW model. The mini-superspace limit of this model can be formulated as baby-CFT. Spectrum, structure constants and fusion rules as well as factorization of four point functions are obtained from the harmonic analysis on SL(2,C)/SU(2). The issues of operator-state correspondence or the appearance of non-normalizable intermediate states in correlation functions can be discussed transparently in this context. 
  The motion of a one-dimensional kink and its energy losses are considered as a model of interaction of nontrivial topological field configurations with external fields. The approach is based on the calculation of the zero modes excitation probability in the external field. We study in the same way the interaction of the t'Hooft-Polyakov monopole with weak external fields. The basic idea is to treat the excitation of a monopole zero mode as the monopole displacement. The excitation is found perturbatively. As an example we consider the interaction of the t'Hooft-Polyakov monopole with an external uniform magnetic field. 
  Using a six-orientifold on top of a NS-fivebrane we construct a chiral N=1 supersymmetric gauge theory in four dimensions with gauge group SU(N_c) and matter in the symmetric, antisymmetric and (anti)fundamental representations. Anomaly cancellation is fulfilled by the requirement of a smooth RR 7-form charge distribution and leads us to the introduction of 8 half D-sixbranes ending on the NS-fivebrane. We obtain the dual model from branes by a linking number argument. We check explicitly the 't Hooft anomaly matching conditions and the map between deformations in the original and the dual model. 
  We review some recent results on the effective interactions of a light gravitino with ordinary particles. In particular, we discuss on a simple example a novel aspect of the low-energy theorems for broken supersymmetry: in the effective lagrangian describing the goldstino couplings to matter, there are terms bilinear in the goldstino that, already at the lowest non-trivial order, are not entirely controlled by the supersymmetry-breaking scale, and introduce additional free parameters. We conclude by mentioning some phenomenological implications, including a lower bound on the gravitino mass from collider data. 
  Through more detailed calculations on QED$_{1+1}$ and QED$_{3+1}$ emplying a new treatment of Feynman Amplitudes, we attribute the regularization independent and hence definite origin of chiral anomaly in perturbation theory to an unambiguous term which is a rational function in momentum space. Some relevant remarks are presented. 
  This is a further explanation of a new and simple renormalization approach recently proposed by the author (hep-th/9708104, Ref. [1], that is somewhat sketchy) for any ordinary QFT (whether renormalizable or not) in any spacetime dimension. We discussed the physical motivations of the new approach and its efficiency when compared to the existent renormalization approaches. Some other important issues related are briefly touched. 
  The interconnection between quantum mechanics and probabilistic classical mechanics for a free relativistic particle is derived in terms of Wigner functions (WF) for both Dirac and Klein-Gordon (K-G) equations. Construction of WF is achieved by first defining a bilocal 4-current and then taking its Fourier transform w.r.t. the relative 4-coordinate. The K-G and Proca cases also lend themselves to a closely parallel treatment provided the Kemmer- Duffin beta-matrix formalism is employed for the former. Calculation of WF is carried out in a Lorentz-covariant fashion by standard `trace' techniques. The results are compared with a recent derivation due to Bosanac. 
  Some formal aspects of supersymmetry breaking are reviewed. The classic "requirements" for supersymmetry breaking include chiral matter, a dynamical superpotential, and a classical superpotential which completely lifts the moduli space. These "requirements" may be evaded in theories with large matter representations. The mechanisms of supersymmetry breaking by confinement and quantum deformation of the moduli space are explained, with emphasis on the importance of identifying the relevant degrees of freedom in the ground state. Supersymmetry breaking and the behavior of the Witten index in non-chiral theories are discussed. The quantum removal of directions which are classically unlifted is also illustrated. Examples of product gauge group theories that admit dual descriptions of the non-supersymmetric ground state are also presented. 
  A string cosmology scenario ("pre-big-bang") postulates that the evolution of the Universe starts from a state of very small curvature and coupling, undergoes a long phase of dilaton-driven kinetic inflation and at some later time joins smoothly standard radiation dominated cosmological evolution, thus giving rise to a singularity free inflationary cosmology. I report on recent progress in understanding some outstanding issues such as initial conditions, graceful exit transition and generation of inhomogeneity perturbations. 
  Using the conformal compensator superfields of N=2 D=4 supergravity, the Type IIB S-duality transformations are expressed as a linear rotation which mixes the compensator and matter superfields. The classical superspace action for D=4 compactifications of Type IIB supergravity is manifestly invariant under this transformation. Furthermore, the introduction of conformal compensators allows a Fradkin-Tseytlin term to be added to the manifestly SL(2,Z)-covariant sigma model action of Townsend and Cederwall. 
  We construct topological recursion relations (TRR's) at higher genera $g\ge2$ for general 2-dimensional topological field theories coupled to gravity. These TRR's when combined with Virasoro conditions enable one to determine the number of higher genus holomorphic curves in any Fano varieties. In the case of $CP^2$ we reproduce the known results at genus $g=2$. 
  We determine all Lie groups compatible with the gauge structure of the Standard Elementary Particle Model (SM) and their representations. The groups are specified by congruence equations of quantum numbers. By comparison with the experimental results, we single out one Lie group and show that this choice implies certain old and new correlations between the quantum numbers of the SM quantum fields as well as some hitherto unknown group theoretical properties of the Higgs mechanism. 
  Exact solutions of Dirac equation in two spatial dimensions in the Coulomb field are obtained. Equation which determines the so-called critical charge of the Coulomb field is derived and solved for a simple model. 
  Superstring amplitudes of an arbitrary genus are calculated through super-Schottky parameters by a summation over the fermion strings. For a calculation of divergent multi-loop fermion string amplitudes a supermodular invariant regularization procedure is used. A cancellation of divergences in the superstring amplitudes is established. Grassmann variables are integrated, the superstring amplitudes are obtained to be explicitly finite and modular invariant. 
  We classify possible `self-duality' equations for p-form gauge fields in space-time dimension up to D=16, generalizing the pioneering work of Corrigan et al. (1982) on Yang-Mills fields (p=1) for D from 5 to 8. We impose two crucial requirements. First, there should exist a 2(p+1)-form T invariant under a sub-group H of SO(D). Second, the representation for the SO(D) curvature of the gauge field must decompose under H in a relevant way. When these criteria are fulfilled, the `self-duality' equations can be candidates as gauge functions for SO(D)-covariant and H-invariant topological quantum field theories. Intriguing possibilities occur for dimensions greater than 9, for various p-form gauge fields. 
  The gap between a microscopic theory for quantum spacetime and the semiclassical physics of blackholes is bridged by treating the blackhole spacetimes as highly excited states of a class of nonlocal field theories. All the blackhole thermodynamics is shown to arise from asymptotic form of the dispersion relation satisfied by the elementary excitations of these field theories. These models involve, quite generically, fields which are: (i) smeared over regions of the order of Planck length and (ii) possess correlation functions which have universal short distance behaviour. 
  We study realization of N=2 SUSY in N=2 abelian gauge theory with electric and magnetic $FI$ terms within a manifestly supersymmetric formulation. We find that after dualization of even one $FI$ term N=2 SUSY is realized in a partial breaking mode off shell. In the case of two $FI$ terms, this regime is preserved on shell. The N=2 SUSY algebra is shown to be modified on gauge-variant objects. 
  We study brane realizations of chiral matter in N=1 supersymmetric gauge theories in four dimensions. A "cross" configuration which leads to "flavor doubling" is found to have a superpotential. The main example is realized using a special "fork" configuration. Many of the results are found by studying a SU times SU product gauge group first. The chiral theory is then an orientifold projection of the product gauge group. An interesting observation in the brane picture is that there are transitions between chiral and non chiral models. These transitions are closely related to small instanton transitions in six dimensions. 
  The method of flow equations is applied to QED in the light-front dynamics. To second order in the coupling the particle number conserving part of the effective QED Hamiltonian has two terms of different structure. The first term gives the Coulomb interaction and the correct spin splittings of positronium; the contribution of the second term to mass spectrum depends on the explicit form of unitary transformation and may influence the spin-orbit. 
  It is observed that the three-dimensional BTZ black hole is a supersymmetric solution of the low-energy field equations of heterotic string theory compactified on an Einstein space. The solution involves a non-zero dilaton and NS-NS H-field. The entropy of the extreme black hole can then be computed using string theory and the asymptotic properties of anti-de Sitter space, without recourse to a D-brane analysis. This provides an explicit example of a black hole whose entropy can be computed using fundamental string theory, as advocated by Susskind. 
  We discuss some aspects of the physics of branes in the presence of orientifolds and the corresponding worldvolume gauge dynamics. We show that at strong coupling orientifolds sometimes turn into bound states of orientifolds and branes, and give a worldsheet argument for the flip of the sign of an orientifold plane split into two disconnected parts by an NS fivebrane. We also describe the moduli space of vacua of N=2 supersymmetric gauge theories with symplectic and orthogonal gauge groups, and analyze a set of four dimensional N=1 supersymmetric gauge theories with chiral matter content using branes. 
  Three dimensional Yang-Mills gauge theories in the presence of the Chern-Simons action are seen as being generated by the pure topological Chern-Simons term through nonlinear covariant redefinitions of the gauge field 
  A connection between non-perturbative formulations of quantum gravity and perturbative string theory is exhibited, based on a formulation of the non-perturbative dynamics due to Markopoulou. In this formulation the dynamics of spin network states and their generalizations is described in terms of histories which have discrete analogues of the causal structure and many fingered time of Lorentzian spacetimes. Perturbations of these histories turn out to be described in terms of spin systems defined on 2-dimensional timelike surfaces embedded in the discrete spacetime. When the history has a classical limit which is Minkowski spacetime, the action of the perturbation theory is given to leading order by the spacetime area of the surface, as in bosonic string theory. This map between a non-perturbative formulation of quantum gravity and a 1+1 dimensional theory generalizes to a large class of theories in which the group SU(2) is extended to any quantum group or supergroup. It is argued that a necessary condition for the non-perturbative theory to have a good classical limit is that the resulting 1+1 dimensional theory defines a consistent and stable perturbative string theory. 
  The partition function in Matrix theory is constructed by Euclidean path integral method. The D0-branes, which move around in the finite region with a typical size of Schwarzschild radius, are chosen as the background. The mass and entropy of the system obtained from the partition function contain the parameters of the background. After averaging the mass and entropy over the parameters, we find that they match the properties of 11D Schwarzschild black holes. The period $\b$ of Euclidean time can be identified with the reciprocal of the boosted Hawking temperature. The entropy $S$ is shown to be proportional to the number $N$ of Matrix theory partons, which is a consequence of the D0-brane background. 
  We propose a reformulation of Yang-Mills theory as a perturbative deformation of a novel topological (quantum) field theory. We prove that this reformulation of the four-dimensional QCD leads to quark confinement in the sense of area law of the Wilson loop. First, Yang-Mills theory with a non-Abelian gauge group G is reformulated as a deformation of a novel topological field theory. Next, a special class of topological field theories is defined by both BRST and anti-BRST exact action corresponding to the maximal Abelian gauge leaving the maximal torus group H of G invariant. Then we find the topological field theory ($D>2$) has a hidden supersymmetry for a choice of maximal Abelian gauge. As a result, the D-dimensional topological field theory is equivalent to the (D-2)-dimensional coset G/H non-linear sigma model in the sense of Parisi and Sourlas dimensional reduction. After maximal Abelian gauge fixing, the topological property of magnetic monopole and anti-monopole of four-dimensional Yang-Mills theory is translated into that of instanton and anti-instanton in two-dimensional equivalent model. It is shown that the linear static potential in four-dimensions follows from the instanton--anti-instanton gas in the equivalent two-dimensional non-linear sigma model obtained from the four-dimensional topological field theory by dimensional reduction, while the remaining Coulomb potential comes from the perturbative part in four-dimensional Yang-Mills theory. The dimensional reduction opens a path for applying various exact methods developed in two-dimensional quantum field theory to study the non-perturbative problem in low-energy physics of four-dimensional quantum field theories. 
  This review gives an introduction into problems, concepts and techniques when quantizing matter fields near black holes. The first part focusses on quantum fields in general curved space-times. The second part is devoted to a detailed treatment of the Unruh effect in uniformly accelerated frames and the Hawking radiation of black holes. Paricular emphasis is put on the induced energy momentum tensor near black holes 
  Using a group-invariant version of the Faddeev-Popov method we explicitly obtain the partition functions of the Self-Dual Model and Maxwell-Chern-Simons theory. We show that their ratio coincides with the partition function of abelian Chern-Simons theory to within a phase factor depending on the geometrical properties of the manifold. 
  Physical quark-number charges of dyons are determined, via a formula which generalizes that of Witten for the electric charge, in N=2 supersymmetric theories with $SU(2) \times U(1)^{N_f} $ gauge group. The quark numbers of the massless monopole at a nondegenerate singularity of QMS turn out to vanish in all cases. A puzzle related to CP invariant cases is solved. Generalization of our results to $SU(N_c)\times U(1)^{N_f}$ gauge theories is straightforward. 
  A modified algorithm for the construction of nonlinear realizations (sigma models) is applied to the general covariance group. Our method features finite dimensionality of the coset manifold by letting the vacuum stability group be infinite. No decomposition of the symmetry group to its finite-dimensional subgroups is required.   The expected result, i.e. Einstein-Cartan gravity, is finally obtained. 
  In this note we discuss the theory of super-threebranes in a spacetime of signature (10,2). Upon reduction, the threebrane provides us with the classical representations of the M-2-brane and the type IIB superstring. Many features of the original super (2+2)-brane theory are clarified. In particular, the (10,2) superspace and the spinors required to construct the brane action are discussed. 
  We consider hidden sector supersymmetry breakdown in the strongly coupled heterotic $E_8\times E_8$ theory of Ho\v{r}ava and Witten. Using effective field theory methods in four dimensions, we can show that gravitational interactions induce soft breaking terms in the observable sector that are of order of the gravitiono mass. We apply these methods to the mechanism of gaugino condensation at the hidden wall. Although the situation is very similar to the weakly coupled case, there is a decisive difference concerning the observable sector gaugino mass; with desirable phenomenological as well as cosmological consequences. 
  The algebraic structure of Thermo Field Dynamics lies in the $q$-deformation of the algebra of creation and annihilation operators. Doubling of the degrees of freedom, tilde-conjugation rules, and Bogoliubov transformation for bosons and fermions are recognized as algebraic properties of $h_{q}(1)$ and of $h_{q}(1|1)$, respectively. 
  We study the holomorphic properties of the MQCD by comparing the super-potentials in MQCD and the gauge theory. First we show that the super-potential defined as an integral of three form is NOT appropriate for generic situation with quarks. We report a resolution of the problem which works for the brane configurations of 90 degree rotation, including the true SQCD. The new definition does not need auxiliary surface and can be reduced to a contour integral for some cases. We find relation beetween the new and old definitions, which is verified by explicit calculation for SU(N), SO(N), Sp(N) simple groups with $F$ of massive quarks. 
  We show that the stationary quantum Hamilton-Jacobi equation of non-relativistic 1D systems, underlying Bohmian mechanics, takes the classical form with $\partial_q$ replaced by $\partial_{\hat q}$ where $d\hat q={dq\over \sqrt{1-\beta^2}}$. The $\beta^2$ term essentially coincides with the quantum potential that, like $V-E$, turns out to be proportional to a curvature arising in projective geometry. In agreement with the recently formulated equivalence principle, these ``quantum transformations'' indicate that the classical and quantum potentials deform space geometry. 
  We consider Heterotic string theories in the DLCQ. We derive that the matrix model of the Spin(32)/$Z_2$ Heterotic theory is the theory living on N D-strings in type I wound on a circle with no Spin(32)/$Z_2$ Wilson line on the circle. This is an O(N) gauge theory. We rederive the matrix model for the $\e8$ Heterotic string theory, explicitly taking care of the Wilson line around the lightlike circle. The result is the same theory as for Spin(32)/$Z_2$ except that now there is a Wilson line on the circle. We also see that the integer N labeling the sector of the O(N) matrix model is not just the momentum around the lightlike circle, but a shifted momentum depending on the Wilson line. We discuss the aspect of level matching, GSO projections and why, from the point of view of matrix theory the $\e8$ theory, and not the Spin(32)/$Z_2$, develops an 11'th dimension for strong coupling. Furthermore a matrix theory for type I is derived. This is again the O(N) theory living on the D-strings of type I. For small type I coupling the system is 0+1 dimensional quantum mechanics. 
  The generalization to N=1 superconformal minimal models of the relation between the modular transformation matrix and the fusion rules in rational conformal field theories, the Verlinde theorem, is shown to provide complete information about the fusion rules, including their fermionic parity. The results for the superconformal Tricritical Ising and Ashkin-Teller models agree with the known rational conformal formulation. The Coulomb gas description of correlation functions in the Ramond sector of N=1 minimal models is also discussed and a previous formulation is completed. 
  Five dimensional supersymmetric gauge theory compactified on a circle defines an effective N=2 supersymmetric theory for massless fields in four dimensions. Based on the relativistic Toda chain Hamiltonian proposed by Nekrasov, we derive the Picard-Fuchs equation on the moduli space of the Coulomb branch of SU(2) gauge theory. Our Picard-Fuchs equation agrees with those from other approaches; the spectral curve of XXZ spin chain and supersymmetric cycle in compactified M theory. By making use of a relation to the Picard-Fuchs equation of SU(2) Seiberg-Witten theory, we obtain the prepotential and the effective coupling constant that incorporate both a perturbative effect of Kaluza-Klein modes and a non-perturbative one of four dimensional instantons. In the weak coupling regime we check that the prepotential exhibits a consistent behavior in large and small radius limits of the circle. 
  A rule of thumb derivation of the Dirac-Born-Infeld action for D-branes is studied \`a la Fradkin and Tseytlin, by simply integrating out of the superstring coordinates in a narrow strip attached to the D-branes. In case of superstrings, the coupling of Ramond-Ramond fields as well as the Dirac-Born-Infeld type coupling of the Neveu Schwarz-Neveu Schwarz fields come out in this way. 
  In this paper we obtain supersymmetric brane-like configurations in the vacuum of N=4 gauged $SU(2)\times SU(2)$ supergravity theory in four spacetime dimensions. Almost all of these vacuum solutions preserve either half or one quarter of the supersymmetry in the theory. We also study the solutions in presence of nontrivial axionic and gauge field backgrounds. In the case of pure gravity with axionic charge the geometry of the spacetime is $AdS_3\times R^1$ with N=1 supersymmetry. An interesting observation is that the domain walls of this theory cannot be given an interpretation of a 2-brane in four dimensions. But it still exists as a stable vacuum. This feature is quite distinct from the domain-wall configuration in massive type IIA supergravity in ten dimensions. 
  We discuss some consequences of our previous work on rigid special geometry in hypermultiplets in 4-dimensional Minkowski spacetime for supersymmetric gauge dynamics when one of the spatial dimensions is compactified on a circle. 
  Within the mathematical framework of Quillen, one interprets the Higgs field as part of the superconnection on a superbundle. We propose to take as superbundle the exterior algebra obtained from a Hermitian bundle with structure group U(n). Spontaneous symmetry breaking appears as a consequence of a non-vanishing scalar curvature. The U(1) Higgs model reformulates the Ginzburg-Landau theory, while the U(2) model relates to the electroweak theory with the relation $g^2=3g4^2$ for the gauge coupling constants, the formula $\sin^2\theta=1/4$ for the Weinberg angle, and the ratio                $ m_W^2 : m_Z^2 : m_H^2 = 3 : 4 : 12 $   for the masses (squared) of the W, Z, and Higgs boson (at tree level). 
  We formulate a set of simple sufficient conditions for the existence of Q-balls in gauge theories. 
  In this paper the kernel for the spectral correlation functions of the invariant chiral random matrix ensembles with real ($\beta =1$) and quaternion real ($\beta = 4$) matrix elements is expressed in terms of the kernel of the corresponding complex Hermitean random matrix ensembles ($\beta=2$). Such identities are exact in case of a Gaussian probability distribution and, under certain smoothness assumptions, they are shown to be valid asymptotically for an arbitrary finite polynomial potential. They are proved by means of a construction proposed by Br\'ezin and Neuberger. Universal behavior at the hard edge of the spectrum for all three chiral ensembles then follows from microscopic universality for $\beta =2$ as shown by Akemann, Damgaard, Magnea and Nishigaki. 
  To appear in proceedings of Solitons, A CRM-Fields-CAP Summer Workshop in Theoretical Physics, July 20-26, Kingston, Ontario, Canada 1997. (Talk delivered by M. Rocek.) 
  We present here an explicit classical solution of the type of perturbiner in Sin(h)-Gordon model. This solution is a generating function for form-factors in the tree approximation. 
  In a previous paper (hep-th/9801040), the superconnection formalism was used to fit the Higgs field into a U(n) gauge theory with particular emphasis on the n=2 case, aiming at the reconstruction of certain parts of the Standard Model. The approach provides an alternative to the one bases on non-commutative geometry. This work is continued by including matter fields (leptons only). We extend the Standard Model by including the right-handed neutrino field. The possibility of a finite neutrino mass is thus accounted for. 
  We consider the quantum sinh-Gordon model in this paper. Using known formulae for form factors we sum up all their contributions and obtain a closed expression for a correlation function. This expression is a determinant of an integral operator. Similar determinant representations were proven to be useful not only in the theory of correlation functions, but also in the matrix models. 
  In perturbative quantum field theory the limit of compactification on an almost light-like circle has recently been shown to be plagued by divergences. We argue that the light-like limit for M-theory probably is free of such divergences due to, among others, the existence of the wrapping modes of the membranes. To illustrate this, we consider superstring theory compactified on an almost light-like circle. Specifically, we compute a one-loop four-point amplitude in type II theory. As is well known, if the external states have vanishing momenta in the compact dimension, the divergence in the light-like limit is even stronger than in field theory. However, in the case of present interest, where these external momenta are non-vanishing, there is a subtle compensation and the resulting amplitude has a well-defined and finite light-like limit. The net effect of taking the light-like limit is to replace the integration over one of the moduli of the 4-punctured torus by a sum over a discrete modulus taking values in a finite lattice on the torus. The same result can also be obtained from a suitably "Wick rotated" amplitude computed directly with a compact light-like circle. 
  The entropy of a seven dimensional Schwarzschild black hole of arbitrary large radius is obtained by a mapping onto a near extremal self-dual three-brane whose partition function can be evaluated. The three-brane arises from duality after submitting a neutral blackbrane, from which the Schwarzschild black hole can be obtained by compactification, to an infinite boost in non compact eleven dimensional space-time and then to a Kaluza-Klein compactification. This limit can be defined in precise terms and yields the Bekenstein-Hawking value up to a factor of order one which can be set to be exactly one with the extra assumption of keeping only transverse brane excitations. The method can be generalized to five and four dimensional black holes. 
  We determine an intersection rule for extremal p-branes which are localized in their relative transverse coordinates by solving, in a purely bosonic context, the equations of motion of gravity coupled to a dilaton and n-form field strengths. The unique algebraic rule we obtained does not lead to new solutions while it manages to collect, in a systematic way, most of the solutions (all those compatible with our ansatz) that have appeared in the literature. We then consider bound states of zero binding energy where a third brane is accomodated in the common and overall transverse directions. They are given in terms of non-harmonic functions. A different algebraic rule emerges for these last intersections, being identical to the intersection rule for p-branes which only depend on the overall transverse coordinates. We clarify the origin of this coincidence. The whole set of solutions in ten and eleven dimensional theories is connected by dualities and dimensional reductions. They are related to brane configurations recently used to study non-perturbative phenomena in supersymmetric gauge theories. 
  The theory of the strong interactions, Quantum Chromodynamics (QCD), has been addressed by a variety of non-perturbative techniques over the decades since its introduction. We have investigated Hamiltonian formulations with different quantization methods and approximation schemes. In one method, we utilize light-front coordinates to investigate the role of bosonic zero modes in leading to confinement. In another method we are able to obtain spectra for the mesons and baryons using constituent quark masses but no phenomenological confinement. We survey our principal accomplishments to date and indicate our future directions. 
  A class of background independent matrix models is made for which the structure of both local gauge symmetries and classical solutions is clarified. These matrix models do not involve a space-time metric and provide the matrix analogs of topological Chern-Simons and BF theories. It is explicitly shown that the BF type of matrix model can be formulated in any space-time dimension and include 3+1 dimensional gravity as a special case. Moreover, we discuss some generalization of the model to include a fermionic BRST-like symmetry whose partition function is related to the Casson invariant. 
  The derivation of the conformal anomaly for dilaton coupled electromagnetic field in curved space is presented. The models of this sort naturally appear in stringy gravity or after spherical reduction of multidimensional Einstein-Maxwell theory. It is shown that unlike the case of minimal vector in curved space or dilaton coupled scalar the anomaly induced effective action cannot be derived. The reason is the same why anomaly induced effective action cannot be constructed for interacting theories (like QED) in curved space. 
  The entropy of the four dimensional Schwarzschild black hole is derived by mapping it onto a configuration of intersecting branes with four charges. This configuration is obtained by performing several boosts and dualities on a neutral black brane of M-theory to which the Schwarzschild black hole is related by trivial compactification. The infinite boost limit is well-defined and corresponds to extremality where the intersecting brane configuration is a marginal one on which a standard microscopic counting of the entropy can be safely performed. The result reproduces exactly the Bekenstein-Hawking entropy of the four dimensional black hole. 
  A quantum field described by the field operator $\Delta_{a}=\Delta+ a\delta_\Sigma$ involving a $\delta$-like potential is considered. Mathematically, the treatment of the $\delta$-potential is based on the theory of self-adjoint extension of the unperturbed operator $\Delta$. We give the general expressions for the resolvent and the heat kernel of the perturbed operator $\Delta_{a}$. The main attention is payed to $d=2$ $\delta$-potential though $d=1$ and $d=3$ cases are considered in some detail. We calculate exactly the heat kernel, Green's functions and the effective action for the operator $\Delta_{a}$ in diverse dimensions and for various spaces $\Sigma$. The renormalization phenomenon for the coupling constant $a$ of $d=2$ and $d=3$ $\delta$-potentials is observed. We find the non-perturbative behavior of the effective action with respect to the renormalized coupling $a_{ren}$. 
  We extend the general solution to the Cauchy problem for the relativistic closed string (Phys. Lett. B404 (1997) 57-65, hep-th/9704084) to the case of open strings attached to Dp-branes, including the cases where the initial data has a knotlike topology. We use this extended solution to derive intrinsic dynamical properties of open and closed relativistic strings attached to Dp-branes. We also study the singularity structure and the oscillating periods of this extended solution. 
  The general structure of the conformal anomaly and the dilaton's effect to it are analysed. First we give a new formal proof of the statement that the conformal anomaly, in the theory which is conformal invariant at the classical level, is conformal invariant. The heat-kernel regularization and Fujikawa's method are taken for the analysis. We present a new explicit result of the conformal anomaly in 4 dimensional photon-dilaton gravity. This result is examined from the point of the general structure. 
  The moduli in a 4D N=1 heterotic compactification on an elliptic CY, as well as in the dual F-theoretic compactification, break into "base" parameters which are even (under the natural involution of the elliptic curves), and "fiber" or twisting parameters; the latter include a continuous part which is odd, as well as a discrete part. We interpret all the heterotic moduli in terms of cohomology groups of the spectral covers, and identify them with the corresponding F-theoretic moduli in a certain stable degeneration. The argument is based on the comparison of three geometric objects: the spectral and cameral covers and the ADE del Pezzo fibrations. For the continuous part of the twisting moduli, this amounts to an isomorphism between certain abelian varieties: the connected component of the heterotic Prym variety (a modified Jacobian) and the F-theoretic intermediate Jacobian. The comparison of the discrete part generalizes the matching of heterotic 5brane / F-theoretic 3brane impurities. 
  In this article we summarise the brane solutions (instantons, black holes and strings) in 4 dimensions when embedded in N=2 supergravity. Like in 10 dimensions these solutions are related by duality transformations (T-duality, c-map). For the case that the graviphoton has no magnetic charge, we discuss the conformal field theory description of the near-horizon geometry. The decoupling of the massless modes in the matrix limit of the M-theory compactified on a Calabi-Yau threefold is discussed. 
  Following to the lines drawn in my previous paper about the S=0 relativistic oscillator I build up an oscillatorlike system which can be named as the S=1 Proca oscillator. The Proca field function is obtained in the framework of the Bargmann-Wigner prescription and the interaction is introduced similarly to the S=1/2 Dirac oscillator case regarded by Moshinsky and Szczepaniak. We obtained the intriguing rule of quantization: E = \hbar \omega /2 for the parity states (-1)^j and E = \pm \hbar \omega (j+1/2) for the parity states -(-1)^j. There are no radial excitations. Finally, I apply the above-mentioned procedure to the case of the two-body relativistic oscillator. 
  We study dynamical supersymmetry breaking in four dimensions using the fivebrane of M theory, in particular for the Izawa-Yanagida-Intriligator-Thomas (IYIT) model, which we realize as the worldvolume theory of a certain M-theory fivebrane configuration. From the brane point of view, supersymmetry is broken when a holomorphic configuration with the proper boundary conditions does not exist. We discuss the difference between explicit and spontaneous supersymmetry breaking and between runaway behavior and having a stable vacuum. As a preparation for the study of the IYIT model, we examine a realization of the orientifold four-plane in M theory. We derive known as well as new results on the moduli spaces of N=2 and N=1 theories with symplectic gauge groups. These results are based on a hypothesis that a certain intersection of the fivebrane and the Z_2 fixed plane breaks supersymmetry. In the IYIT model, we show that the brane exhibits runaway behavior when the flavor group is gauged. On the other hand, if the flavor group is not gauged, we find that the brane does not run away. We suggest that a stable supersymmetry-breaking vacuum is realized in the region beyond the reach of the supergravity approximation. 
  We propose a few tests of Seiberg-Witten solutions of $\CN=2$ supersymmetric gauge theories by the instanton calculus in twisted gauge theories. We re-examine the low-energy effective abelian theory in the presence of sources and present the formalism which makes duality transformations transparent and easily fixes all the contact terms in a broad class of theories. We also discuss ADHM integration and its relevance to the stated problems. 
  We formulate a manifestly supersymmetric gauge covariant regularization of supersymmetric chiral gauge theories. In our scheme, the effective action in the superfield background field method above one-loop is always supersymmetric and gauge invariant. The gauge anomaly has a covariant form and can emerge only in one-loop diagrams with all the external lines being the background gauge superfield. We also present several illustrative applications in the one-loop approximation: the self-energy part of the chiral multiplet and of the gauge multiplet; the super-chiral anomaly and the superconformal anomaly; as the corresponding anomalous commutators, the Konishi anomaly and an anomalous supersymmetric transformation law of the supercurrent (the ``central extension'' of N=1 supersymmetry algebra) and of the R-current. 
  Bogomolnyi-type bound is constructed for the topological solitons in O(3) nonlinear $\sigma$ model coupled to gravity with a negative cosmological constant. Spacetimes made by self-dual solutions form a class of G\"{o}del-type universe. In the limit of a spinless massive point particle, the obtained stationary metric does not violate the causality and it is a new point particle solution different from the known static hyperboloid and black hole. We also showed that static Nielsen-Olesen vortices saturate Bogomolnyi-type bound only when the cosmological constant vanishes. 
  A one-dimensional gas model has been constructed and shown to provide correct expressions for entropies for extremal and near-extremal BTZ black holes. Recently suggested boosting of black strings is used to compute the entropy for the Schwarzschild black hole also from this gas model. 
  The covariant propagator of the notivarg is found. It has the Feynmann - like form. 
  We study N=4 supersymmetric Yang-Mills theory on a Kaehler manifold with $b_2^+ \geq 3$. Adding suitable perturbations we show that the partition function of the N=4 theory is the sum of contributions from two branches: (i) instantons, (ii) a special class of Seiberg-Witten monopoles. We determine the partition function for the theories with gauge group SU(2) and SO(3), using S-duality. This leads us to a formula for the Euler characteristic of the moduli space of instantons. 
  We study the BPS spectrum of supersymmetric 5 dimensional field theories and their representations as string webs. It is found that a state of given charges exists when it has a representation as an irreducible string web. Its spin is determined by the string web. The number of fermionic zero modes is 8g+4b, where g is the number of internal faces and b is the number of boundaries. In the lift to M theory of 4d field theories such states are described by membranes ending on the 5-brane, breaking SUSY from 8 to 4, and g becomes the genus of the membrane. Mathematically, we obtain a diagrammatic method to find the spectrum of curves on a toric complex surface, and the number of their moduli. 
  This is a short account of our work on the statistical mechanics of `cold' quantum black holes in the constituent model of a black hole$^{3,4,6}$. A quantum Schwarzschild black hole consists of gravitational atoms of Planckian mass scale$^{3,4,6}$. The gapless collective excitations of a bound state of N gravitational atoms dominate the thermodynamics of a cold quantum black hole. It turns out that it is only in the limit of large $N$, with the observable mean values of the gravitational mass-energy kept fixed, that we recover the Bekenstein thermodynamics$^1$ of spin-zero black holes. This is also the limit of the Boltzmann statistics. 
  Duality is considered for the perturbation theory by deriving, given a series solution in a small parameter, its dual series with the development parameter being the inverse of the other. A dual symmetry in perturbation theory is identified. It is then shown that the dual to the Dyson series in quantum mechanics is given by a recent devised series having the adiabatic approximation as leading order. A simple application of this result is given by rederiving a theorem for strongly perturbed quantum systems. 
  Black holes in matrix theory may consist of interacting clusters (correlated domains) which saturate the uncertainty principle. We show that this assumption qualitatively accounts for the thermodynamic properties of both charged and neutral black holes, and reproduces the asymptotic geometry seen by probes. 
  We calculate vacuum energy for twisted SUSY D-brane on toroidal background with constant magnetic or constant electric field. Its behaviour for toroidal D-brane (p=2) in constant electric field shows the presence of stable minimum for twisted versions of the theory. That indicates such a background maybe reasonable groundstate. 
  Dirichlet 0-branes, considered as extreme Type IIA black holes with spin carried by fermionic hair, are shown to have the anomalous gyromagnetic ratio g=1, consistent with their interpretation as Kaluza-Klein modes. 
  We show that in a generic case of the pre-big-bang scenario, inflation will solve cosmological problems only if the universe at the onset of inflation is extremely large and homogeneous from the very beginning. The size of a homogeneous part of the universe at the beginning of the stage of pre-big-bang (PBB) inflation must be greater than $10^{19}$ $l_s$, where $l_s$ is the stringy length. The total mass of an inflationary domain must be greater than $10^{72} M_{s}$, where $M_{s} \sim l_s^{-1}$. If the universe is initially radiation dominated, then its total entropy at that time must be greater than $10^{68}$. If the universe is closed, then at the moment of its formation it must be uniform over $10^{24}$ causally disconnected domains. The natural duration of the PBB stage in this scenario is $M_p^{-1}$. We argue that the initial state of the open PBB universe could not be homogeneous because of quantum fluctuations. Independently of the issue of homogeneity, one must introduce two large dimensionless parameters, $g_0^{-2} > 10^{53}$, and $B > 10^{91}$, in order to solve the flatness problem in the PBB cosmology. A regime of eternal inflation does not occur in the PBB scenario. This should be compared with the simplest versions of the chaotic inflation scenario, where the regime of eternal inflation may begin in a universe of size $O(M_{p}^{-1})$ with vanishing initial radiation entropy, mass $O(M_p)$, and geometric entropy O(1). We conclude that the current version of the PBB scenario cannot replace usual inflation even if one solves the graceful exit problem in this scenario. 
  We consider the entropy of near extremal black holes with multiple charges in the context of the recently proposed correspondence principle of Horowitz and Polchinski, including black holes with two, three and four Ramond-Ramond charges. We find that at the matching point the black hole entropy can be accounted for by massless open strings ending on the D-branes for all cases except a black hole with four Ramond-Ramond charges, in which case a possible resolution in terms of brane-antibrane excitations is considered. 
  Effects of boundary conditions of fields for compactified space directions on the supersymmetric gauge theories are discussed. For general and possible boundary conditions the supersymmetry is explicitly broken to yield universal soft supersymmetry breaking terms, and the gauge symmetry of the theory can also be broken through the dynamics of non-integrable phases, depending on number and the representation under the gauge group of matters. The 4-dimensional supersymmetric QCD is studied as a toy model when one of the space coordinates is compactified on $S^1$. 
  We consider a series of duality transformations that leads to a constant shift in the harmonic functions appearing in the description of a configuration of branes. This way, for several intersections of branes, we can relate the original brane configuration which is asymptotically flat to a geometry which is locally isometric to adS_k x E^l x S^m. These results imply that certain branes are dual to supersingleton field theories. We also discuss the implications of our results for supersymmetry enhancement and for supergravity theories in diverse dimensions. 
  Phenomenological evidence and analytic approximations to the QCD ground state suggest a complex gluon condensate structure. Exclusion of elementary fermion excitations by the generation of infinite mass corrections is a consequence. In addition the existence of vacuum condensates in unbroken non-abelian gauge theories, endows SU(3) and higher order groups with a non-trivial structure in the manifold of possible vacuum solutions, which is not present in SU(2). This may be related to the existence of particle generations. 
  We consider N=1 supersymmetric QCD based on the G_2 gauge group and including 3 chiral matter 7-plets. In that case, the gauge symmetry is broken completely and the instanton-generated superpotential on the classical moduli space is present. If the theory involves the Yukawa term, there are six chirally asymmetric vacua. In the limit when the Yukawa coupling vanishes, two of the vacua run away to infinity and only 4 asymmetric vacuum states are left. Besides, a chirally symmetric state is always present. We consider also an O(7) model with 4 chiral multiplets in spinor representation. In that case, there are 4 extra "virtual vacua" dwelling at infinity of the moduli space. In a non-renormalizable theory involving a quartic term in the superpotential, they show up at finite moduli values. 
  The general solution of a 7D analogue of the 3D Euler top equation is shown to be given by an integration over a Riemann surface with genus 9. The 7D model is derived from the 8D $Spin(7)$ invariant self-dual Yang-Mills equation depending only upon one variable and is regarded as a model describing self-dual membrane instantons. Several integrable reductions of the 7D top to lower target space dimensions are discussed and one of them gives 6, 5, 4D descendants and the 3D Euler top associated with Riemann surfaces with genus 6, 5, 2 and 1, respectively. 
  The off-shell description of N=(2,2) supersymmetric non-linear sigma-models is reviewed. The conditions for ultra-violet finiteness are derived and T-duality is discussed in detail. 
  Based on special geometry, we consider corrections to N=2 extremal black-hole solutions and their entropies originating from higher-order derivative terms in N=2 supergravity. These corrections are described by a holomorphic function, and the higher-order black-hole solutions can be expressed in terms of symplectic Sp(2$n$+2) vectors. We apply the formalism to N=2 type-IIA Calabi-Yau string compactifications and compare our results to recent related results in the literature. 
  Recently, a geometric model for the confinement of magnetic charges in the context of type II string compactifications was constructed by Greene, Morrison and Vafa. This model assumes the existence of stable magnetic vortices with quantized flux in the low energy theory. However, quantization of flux alone does not imply that the vortex is stable, since the flux may not be confined to a tube of definite size. We show that in the field theoretical model which underlies the geometric model of confinement, static, cylindrically symmetric magnetic vortices do not exist. While our results do not preclude the existence of confinement in a different low-energy regime of string theory, they show that confinement is not a universal outcome of the string picture, and its origin in the low energy theory remains to be understood. 
  We calculate some one loop corrections to the effective action of theories in $d$ dimensions that arise on the dimensional reduction of a Weyl fermion in $D$ dimensions. The terms that we are interested in are of a topological nature. Special attention is given to the effective actions of the super Yang Mills theories that arise on dimensional reduction of the N=1 theory in six dimensions or on the dimensional reduction of the N=1 theory in ten dimensions. In the latter case we suggest an interpretation of the quantum effect as a coupling of the gauge field on the brane to a relative background gauge field. 
  We discuss theoretical implications of the large k USp(2k) matrix model in zero dimension. The model appears as the matrix model of type IIB superstrings on a large $T^{6}/Z^{2}$ orientifold via the matrix twist operation. In the small volume limit, the model behaves four dimensional and its T dual is six-dimensional worldvolume theory of type I superstrings in ten spacetime dimensions. Several theoretical considerations including the analysis on planar diagrams, the commutativity of the projectors with supersymmetries and the cancellation of gauge anomalies are given, providing us with the rationales for the choice of the Lie algebra and the field content. A few classical solutions are constructed which correspond to Dirichlet p-branes and some fluctuations are evaluated. The particular scaling limit with matrix T duality transformation is discussed which derives the F theory compactification on an elliptic fibered K3. 
  Starting from the Schild action for membrane, we present an alternative formulation of Matrix Theory. First of all, we construct the Schild action for general bosonic p-brane which is classically equivalent to the Nambu-Goto action for p-brane. Next, based on the constraint obtained from the variational equation for the auxiliary field in the case of $p = 2$ (membrane), we construct a new matrix model which is closely related to the matrix model of M-theory as developed by Banks, Fischler, Shenker and Susskind (BFSS). Our present formulation is a natural extension of the construction of type IIB matrix model by Yoneya to the case of M-theory. 
  The effective potential in the model introduced by Buchbinder-Inagaki-Odintsov (BIO) which represents SUSY NJL model non-minimally coupled with the external gravitational field is found. The topology of the space is considered to be non-trivial. Chiral symmetry breaking under the action of external curvature and non-trivial topology is investigated. 
  We derive the ten-dimensional effective action of the strongly coupled heterotic string as the low energy limit of M-theory on S^1/Z_2. In contrast to a conventional dimensional reduction, it is necessary to integrate out nontrivial heavy modes which arise from the sources located on the orbifold fixed hyperplanes. This procedure, characteristic of theories with dynamical boundaries, is illustrated by a simple example. Using this method, we determine a complete set of R^4, F^2R^2, and F^4 terms and the corresponding Chern-Simons and Green-Schwarz terms in ten dimensions. As required by anomaly cancelation and supersymmetry, these terms are found to exactly coincide with their weakly coupled one-loop counterparts. 
  A generalization of the Yang-Mills covariant derivative, that uses both vector and scalar fields and transforms as a 4-vector contracted with Dirac matrices, is used to simplify and unify the Glashow-Weinberg-Salam model. Since SU(3) assigns the wrong hypercharge to the Higgs boson, it is necessary to use a special representation of U(3) to obtain all the correct quantum numbers. A surplus gauge scalar boson emerges in the process, but it uncouples from all other particles. 
  Boundary conditions changing operators have played an important role in conformal field theory. Here, we study their equivalent in the case where a mass scale is introduced, in an integrable way, either in the bulk or at the boundary. More precisely, we propose an axiomatic approach to determine the general scalar products ${}_b<\theta_1, ... ,\theta_m||\theta'_1, ... ,\theta'_{n}>_a$ between asymptotic states in the Hilbert spaces with $a$ and $b$ boundary conditions respectively, and compute these scalar products explicitely in the case of the Ising and sinh-Gordon models with a mass and a boundary interaction. These quantities can be used to study statistical systems with inhomogeneous boundary conditions, and, more interestingly maybe, dynamical problems in quantum impurity problems. As an example, we obtain a series of new exact results for the transition probability in the double well problem of dissipative quantum mechanics. 
  This contribution gives a personal view on recent attempts to find a unified framework for non-perturbative string theories, with special emphasis on the hidden symmetries of supergravity and their possible role in this endeavor. A reformulation of $d=11$ supergravity with enlarged tangent space symmetry SO(1,2) $\times$ SO(16) is discussed from this perspective, as well as an ansatz to construct yet further versions with SO(1,1) $\times$ SO(16)$^\infty$ and possibly even SO(1,1)$_+$ $\times$ ISO(16)$^\infty$ tangent space symmetry. It is suggested that upon ``third quantization'', dimensionally reduced maximal supergravity may have an equally important role to play in this unification as the dimensionally reduced maximally supersymmetric $SU(\infty)$ Yang Mills theory. 
  We show that, in the region where monopoles are well separated, the L^2-metric on the moduli space of n-monopoles is exponentially close to the T^n-invariant hyperk\"ahler metric proposed by Gibbons and Manton. The proof is based on a description of the Gibbons-Manton metric as a metric on a certain moduli space of solutions to Nahm's equations, and on twistor methods. In particular, we show how the twistor description of monopole metrics determines the asymptotic metric. 
  We compute the asymptotic metrics for moduli spaces of SU(N) monopoles with maximal symmetry breaking. These metrics are exponentially close to the exact monopole metric as soon as, for each simple root, the individual monopoles corresponding to that root are well separated. We also show that the estimates can be differentiated term by term in natural coordinates, which is a new result even for SU(2) monopoles. 
  We study the Matrix theory from a purely canonical viewpoint. In particular, we identify free particle asymptotic states of the model corresponding to the 11D supergraviton multiplet along with the split of the matrix model Hamiltonian into a free and an interacting part. Elementary quantum mechanical perturbation theory then yields an effective potential for these particles as an expansion in their inverse separation. We discuss how our scheme can be used to compute the Matrix theory result for the 11D supergraviton S matrix and briefly comment on non-eikonal and longitudinal momentum exchange processes. 
  We establish the second quantized solution of the nonlinear Schrodinger equation on the half line with a mixed boundary condition. The solution is based on a new algebraic structure, which we call boundary exchange algebra and which substitutes, in the presence of boundaries, the familiar Zamolodchikov-Faddeev algebra. 
  We report on duality invariant constraints, which allow a classification of BPS black holes preserving different fractions of supersymmetry. We then relate this analysis to the orbits of the exceptional groups $E_{6(6)}, E_{7(7)}$, relevant for black holes in five and four dimensions. 
  We re-examine three issues, the Hopf term, fractional spin and the soliton operators, in the 2+1 dimensional O(3) nonlinear sigma model based on the adjoint orbit parameterization (AOP) introduced earlier. It is shown that the Hopf Term is well-defined for configurations of any soliton charge $Q$ if we adopt a time independent boundary condition at spatial infinity. We then develop the Hamiltonian formulation of the model in the AOP and thereby argue that the well-known $Q^2$-formula for fractional spin holds only for a restricted class of configurations. Operators which create states of given classical configurations of any soliton number in the (physical) Hilbert space are constructed. Our results clarify some of the points which are crucial for the above three topological issues and yet have remained obscure in the literature. 
  We explore the anomaly induced effective QCD meson potential in the framework of the effective Lagrangian approach. We suggest a decoupling procedure, when a flavored quark becomes massive, which mimics the one employed by Seiberg for supersymmetric gauge theories. It is seen that, after decoupling, the QCD potential naturally converts to the one with one less flavor. We study the $N_c$ and $N_f$ dependence of the $\eta^{\prime}$ mass. 
  Starting from the known representation of the partition function of the 2- and 3-D Ising models as an integral over Grassmann variables, we perform a hopping expansion of the corresponding Pfaffian. We show that this expansion is an exact, algebraic representation of the loop- and surface expansions (with intrinsic geometry) of the 2- and 3-D Ising models. Such an algebraic calculus is much simpler to deal with than working with the geometrical objects. For the 2-D case we show that the algebra of hopping generators allows a simple algebraic treatment of the geometry factors and counting problems, and as a result we obtain the corrected loop expansion of the free energy. We compute the radius of convergence of this expansion and show that it is determined by the critical temperature. In 3-D the hopping expansion leads to the surface representation of the Ising model in terms of surfaces with intrinsic geometry. Based on a representation of the 3-D model as a product of 2-D models coupled to an auxiliary field, we give a simple derivation of the geometry factor which prevents overcounting of surfaces and provide a classification of possible sets of surfaces to be summed over. For 2- and 3-D we derive a compact formula for 2n-point functions in loop (surface) representation. 
  We show that the spectral dimension d_s of two-dimensional quantum gravity coupled to Gaussian fields is two for all values of the central charge c <= 1. The same arguments provide a simple proof of the known result d_s= 4/3 for branched polymers. 
  We show that the BRS operator of the topological string B model is not holomorphic in the complex structure of the target space. This implies that the so-called holomorphic anomaly of topological strings should not be interpreted as a BRS anomaly. 
  Using the correlation function of chiral vertex operators of the Coulomb gas model, we find the Laughlin wavefunctions of quantum Hall effect, with filling factor $\nu =1/m$, on Riemann sufaces with Poincare metric. The same is done for quasihole wavefunctions. We also discuss their plasma analogy. 
  We consider N=2 supersymmetric nonlinear sigma-models in two dimensions defined in terms of the nonminimal scalar multiplet. We compute in superspace the one-loop beta function and show that the classical duality between these models and the standard ones defined in terms of chiral superfields is maintained at the quantum one-loop level. Our result provides an explicit application of the recently proposed quantization of the nonminimal scalar multiplet via the Batalin-Vilkovisky procedure. 
  We verify the QED Ward identity for the two- and three -point functions at non-equilibrium in the HTL limit. We use the Keldysh formalism of real time finite temperature field theory. We obtain an identity of the same form as the Ward identity for a set of one loop self-energy and one loop three-point vertex diagrams which are constructed from HTL effective propagators and vertices. 
  In this paper we provide a manifestly N=2 supersymmetric formulation of the M-fivebrane in the presence of threebrane solitons. The superspace form of four-dimensional effective equations for the threebranes are readily obtained and yield the complete Seiberg-Witten equations of motion for N=2 super-Yang-Mills. A particularly simple derivation is given by introducing an N=2 superfield generalisation of the Seiberg-Witten differential. 
  A simple, often invoked, regularization scheme of quantum mechanical path integrals in curved space is mode regularization: one expands fields into a Fourier series, performs calculations with only the first $M$ modes, and at the end takes the limit $M \to \infty$. This simple scheme does not manifestly preserve reparametrization invariance of the target manifold: particular noncovariant terms of order $\hbar^2$ must be added to the action in order to maintain general coordinate invariance. Regularization by time slicing requires a different set of terms of order $\hbar^2$ which can be derived from Weyl ordering of the Hamiltonian. With these counterterms both schemes give the same answers to all orders of loops. As a check we perform the three-loop calculation of the trace anomaly in four dimensions in both schemes. We also present a diagrammatic proof of Matthews' theorem that phase space and configuration space path integrals are equal. 
  I describe some recent work in which classical solutions of Dirac-Born-Infeld theory may be used to throw light on some properties of M-theory. The sources of Born-Infeld theory are the ends of strings ending on the world volume. Equivalently the fundamental string may be regarded as merely a thin and extended piece of the world volume. 
  We show that the generators of canonical transformations in the triplectic manifold must satisfy constraints that have no parallel in the usual field antifield quantization. A general form for these transformations is presented. Then we consider gauge fixing by means of canonical transformations in this Sp(2) covariant scheme, finding a relation between generators and gauge fixing functions. The existence of a wide class of solutions to this relation nicely reflects the large freedom of the gauge fixing process in the triplectic quantization. Some solutions for the generators are discussed. Our results are then illustrated by the example of Yang Mills theory. 
  We determine the low energy description of N=2 supersymmetric SU(k) product group theories with bifundamental and fundamental matter based on M-theory fivebrane configurations. The dependence on moduli and scales of the coefficients in the non-hyperelliptic Seiberg-Witten curves for these theories is determined by considering various field theory and brane limits. A peculiarity in the interpretation of these curves for the vanishing beta-function case is noted. 
  We describe some of the novel 6d quantum field theories which have been discovered in studies of string duality. The role these theories (and their 4d descendants) may play in alleviating the vacuum degeneracy problem in string theory is reviewed. The DLCQ of these field theories is presented as one concrete way of formulating them, independent of string theory. 
  We present a general description of electromagnetic RR interactions between pairs of magnetically dual D-branes, focusing on the interaction of a magnetically charged brane with an electrically charged one. In the boundary state formalism, it turns out that while the electric-electric and/or magnetic-magnetic interaction corresponds to the usual RR even spin structure, the magnetic-electric interaction is described by the RR odd spin structure. As representative of the generic case of a dual pair of p and 6-p branes, we discuss in detail the case of the self-dual 3-brane wrapped on a T_6/Z_3, which looks like an extremal dyonic black hole in 4 dimensions. 
  This is a short introduction to the study of compactifications of F-theory on elliptic Calabi-Yau threefolds near colliding singularities. In particular we consider the case of non-transversal intersections of the singular fibers. 
  This is a r\'esum\'e of an extensive investigation of some examples in which one obtains the rigid limit of N=2 supergravity by means of an expansion around singular points in the moduli space of a Calabi-Yau 3-fold. We make extensive use of the K3 fibration of the Calabi-Yau manifolds which are considered. At the end the fibration parameter becomes the coordinate of the Riemann surface whose moduli space realises rigid N=2 supersymmetry. 
  We review various three-dimensional solutions in the low-energy description of M-theory on $S^1/Z_2 \times T_7$. These solutions have an eleven dimensional interpretation in terms of intersecting M-branes. 
  We construct a one loop amplitude for any Abelian orientifold point group for arbitary complex dimensions. From this we show several results for orientifolds in this general class of models as well as for low dimensional compactifications. We also discuss the importance and structure of the contribution of orientifold planes to the dynamics of D-branes, and give a physical explaination for the inconsistency of certain Z_4 models as discovered by Zwart. 
  The U(1) gauge field is usually induced from the gauge principle, that is, the extension of global U(1) phase transformation for matter field. However the phase itself is realized only for quantum theory. In this paper we introduce the U(1) gauge field and gauge coupling from the gauge principle classically. The gauge symmetry is spontaneously broken from the out set. The Higgs mechanism occurs and we obtain the London equation. The Hydrodynamical interpretation of classical field we utilized is given, and the relation to super conductivity is discussed. 
  The derivation of absolute (moduli-independent) U-invariants for all N>2 extended supergravities at D=4 in terms of (moduli-dependent) central and matter charges is reported. These invariants give a general definition of the ``topological'' Bekenstein-Hawking entropy formula for extremal black-holes and reduce to the square of the black-hole ADM mass for ``fixed scalars'' which extremize the black-hole ``potential'' energy. 
  We discuss the precise relation of the open N=2 string to a self-dual Yang-Mills (SDYM) system in 2+2 dimensions. In particular, we review the description of the string target space action in terms of SDYM in a ``picture hyperspace'' parametrised by the standard vectorial R^{2,2} coordinate together with a commuting spinor of SO(2,2). The component form contains an infinite tower of prepotentials coupled to the one representing the SDYM degree of freedom. The truncation to five fields yields a novel one-loop exact lagrangean field theory. 
  We show that the super D-string action is canonically equivalent to the type IIB superstring action with a world-sheet gauge field. Canonical transformation to the type IIB theory with dynamical tension is also constructed to establish the SL(2,Z) covariance beyond the semi-classical approximations. 
  In super-symmetric quantum theory, or in string theory, (including generalizations of these theories to underlying quantum spaces) we study a certain partition function Z(Q,A,g). Here Q denotes a supercharge, A denotes an observable with the property A^2 = I, and g denotes an element of a symmetry group of Q. The supercharge may depend on a parameter lambda, namely Q = Q(lambda). We give an elementary argument to show that Z, as defined, does not actually depend on lambda. 
  We compute the interaction potential between two parallel transversely boosted wrapped membranes (with fixed momentum $p_-$) in D=11 supergravity with compact light-like direction. We show that the supergravity result is in exact agreement with the potential following from the all-order Born-Infeld-type action conjectured to be the leading planar infra-red part of the quantum super Yang-Mills effective action. This provides a non-trivial test of consistency of the arguments relating Matrix theory to a special limit of type II string theory. We also find the potential between two (2+0) D-brane bound states in D=10 supergravity (corresponding to the case of boosted membrane configuration in 11-dimensional theory compactified on a space-like direction). We demonstrate that the result reduces to the SYM expression for the potential in the special low-energy ($\a'\to 0$) limit, in agreement with previous suggestions. In appendix we derive the action obtained from the D=11 membrane action by the world-volume duality transformation of the light-like coordinate $x^-$ into a 3-vector. 
  Higher derivative terms are computed in the one-loop effective action governing the interactions of D3-branes, in two ways: (1) in a formalism with N=2 supersymmetry preserved off-shell, and (2) in the standard background field formalism, with only on-shell supersymmetry. It is shown that these calculations only agree using tree-level equations of motion. The off-shell supersymmetric calculation exhibits acceleration terms that appear in terms with four derivatives. These may imply disagreement at two-loop order between supergravity and Yang-Mills descriptions of D-brane dynamics. 
  The trace anomaly for nonminimally coupled scalars in spherically reduced gravity obtained by Bousso and Hawking (hep-th/9705236) is incorrect. We explain the reasons for the deviations from our correct (published) result which is supported by several other recent papers. 
  Keeping N=1 supersymmetry in 4-dimension and in the leading order, we disuss the various orbifold compactifications of M-theory suggested by Horava and Witten on $T^6/Z_3$, $T^6/Z_6$, $T^6/Z_{12}$, and the compactification by keeping singlets under $SU(2)\times U(1)$ symmetry, then the compactification on $S^1/Z_2$. We also discuss the next to leading order K\"ahler potential, superpotential, and gauge kinetic function in the $Z_{12}$ case. In addition, we calculate the SUSY breaking soft terms and find out that the universality of the scalar masses will be violated, but the violation might be very small. 
  We project the Wilson/Polchinski renormalization group equation onto its uniform external field dependent effective free energy and connected Green's functions. The result is a hierarchy of equations which admits a choice of "natural" truncation and closure schemes for nonperturbative approximate solution. In this way approximation schemes can be generated which avoid power series expansions in either fields or momenta. When following one closure scheme the lowest order equation is the mean field approximation, while another closure scheme gives the "local potential approximation." Extension of these closure schemes to higher orders leads to interesting new questions regarding truncation schemes and the convergence of nonperturbative approximations. One scheme, based on a novel "momentum cluster decomposition" of the connected Green's functions, seems to offer new possibilities for accurate nonperturbative successive approximation. 
  We discuss the propagation of scalars in a large class of non-extremal black hole and black p-brane geometries in generic dimensions. We show that the radial wave equation near the horizon possesses the SL(2,R) structure in every case; approximately it takes the form of the wave equation in the SL(2,R) (AdS_3) background and has a symmetry related to the T-duality of the string model in that geometry. We see a close connection to two and three dimensional black holes. We also find that, in some parameter region, the absorption cross-sections by the black objects take the form expected from a conformal field theory. Our results indicate that some of the properties known about a certain class of four and five dimensional black holes hold more generally. 
  This paper analyses the Noether symmetries and the corresponding conservation laws for Chern-Simons Lagrangians in dimension $d=3$. In particular, we find an expression for the superpotential of Chern-Simons gravity. As a by-product the general discussion of superpotentials for 3rd order natural and quasi-natural theories is also given. 
  We attempt to settle the issue as to what is the correct non-abelian generalisation of the Born-Infeld action, via a consideration of the two-loop $\beta$--function for the non-abelian background gauge field in open string theory. An analysis of the bosonic theory alone shows the recent proposal of Tseytlin's to be somewhat lacking. For the superstring, however, this proposal would seem to be correct, and not just within the approximation used in \cite{tseytlin}. Since it is this latter case that is relevant to the description of D-branes we, in effect, obtain an independent verification of Tseytlin's result. Some issues involved in the concept of non-abelian T--duality are discussed; and it is shown how the interaction between separated and parallel branes, in the form of massive string states, emerges. 
  We study the infra-red limit of the O(N) gauge theory that describes the low energy modes of a system of $N$ type I D-strings and provide some support to the conjecture that, in this limit, the theory flows to an orbifold conformal theory. We compute the elliptic genus of the orbifold theory and argue that its longest string sector describes the bound states of D-strings. We show that, as a result, the masses and multiplicities of the bound states are in agreement with the predictions of heterotic-type I duality in 9 dimensions, for all the BPS charges in the lattice $\Gamma_{(1,17)}$. 
  The first radiative correction to the Casimir energy of a perfectly conducting spherical shell is calculated. The calculation is performed in the framework of covariant perturbation theory with the boundary conditions implemented as constraints. The formalism is briefly reviewed and its use is explained by deriving the known results for two parallel planes.   The ultraviolet divergencies are shown to have the same structure as those for a massive field in zeroth order of $\alpha$. In the zeta-functional regularization employed by us no divergencies appear.   If the radius of the sphere is large compared to the Compton wavelength of the electron the radiative correction is of order $\alpha/(R^2m_e)$ and contains a logarithmic dependence on $m_eR$. It has the opposite sign but the same order of magnitude as in the case of two parallel planes. 
  Using the observation that configurations of N polymers with hard core interactions on a closed random surface correspond to random surfaces with N boundary components we calculate the free energy of a gas of polymers interacting with fully quantized two-dimensional gravity. We derive the equation of state for the polymer gas and find that all the virial coefficients beyond the second one vanish identically. 
  The problem of the scattering of a charged test particle in the gravitational background of axially symmetrical wormhole in the presence of the Aharonov-Bohm type magnetic field is considered. It is shown that the natural mathematical framework appropriate for the problem is the scattering theory in the pair of Hilbert spaces. Both relevant modified wave operators and $S$-matrices are found. 
  In these lectures we review how some of the recent developments in string theory are closely related to generic properties of supersymmetry. Lectures given at the Nato Advanced Study Institute on "Strings, Branes and Dualities", Cargese, May 26 - June 14, 1997. 
  Exact results from random matrix theory are used to systematically analyse the relationship between microscopic Dirac spectra and finite-volume partition functions. Results are presented for the unitary ensemble, and the chiral analogs of the three classical matrix ensembles: unitary, orthogonal and symplectic, all of which describe universality classes of $SU(N_c)$ gauge theories with $N_f$ fermions in different representations. Random matrix theory universality is reconsidered in this new light. 
  We consider a general brane construction for realizing chiral four-dimensional gauge theories. The advantage of the construction is the simplicity and the possibility of realizing a large class of models existing in the literature. We start the study of these models by determining the matter content and the superpotential which naturally arise in the brane construction. 
  We study the creation of a fundamental string between D4-branes at angles in string theory. It is shown that $R(-1)^{F}$ part of the one-loop potential of open string changes its sign due to the change of fermionic zero-mode vacua when the branes cross each other. As a result the effective potential is independent of the angles when supersymmetry is partially unbroken, and leads to a consistent picture that a fundamental string is created in the process. We also discuss the s-rule in the configuration. The same result is obtained from the one-loop potential for the orthogonal D4-branes with non-zero field strength. The result is also confirmed from the tension obtained by deforming the Chern-Simons term on one D4-brane, which is induced by another tilted D4-brane. 
  By studying ordinary chiral fermions in background gauge fields we show that in the case of gauge group SU(3) and space-time dimension 5+1 localized solitons obey $q-$ commutation relations with $q$ not equal to $\pm 1$ but a third root of unity. 
  It is shown that the notoph propagator in the noncovariant longitudinal gauge is equivalent to the covariant Feynmann - like propagator. 
  An attempt is made to describe the `thermodynamics' of semiclassical spacetime without specifying the detailed `molecular structure' of the quantum spacetime, using the known properties of blackholes. I give detailed arguments, essentially based on the behaviour of quantum systems near the event horizon, which suggest that event horizon acts as a magnifying glass to probe Planck length physics even in those contexts in which the spacetime curvature is arbitrarily low. The quantum state describing a blackhole, in any microscopic description of spacetime, has to possess certain universal form of density of states which can be ascertained from general considerations. Since a blackhole can be formed from the collapse of any physical system with a low energy Hamiltonian H, it is suggested that when such a system collapses to form a blackhole, it should be described by a modified Hamiltonian of the form $H^2_{\rm mod} =A^2 \ln (1+ H^2/A^2)$ where $A^2 \propto E_P^2$.I also show that it is possible to construct several physical systems which have the blackhole density of states and hence will be indistinguishable from a blackhole as far as thermodynamic interactions are concerned. In particular, blackholes can be thought of as one-particle excitations of a class of {\it nonlocal} field theories with the thermodynamics of blackholes arising essentially from the asymptotic form of the dispersion relation satisfied by these excitations. These field theoretic models have correlation functions with a universal short distance behaviour, which translates into the generic behaviour of semiclassical blackholes. Several implications of this paradigm are discussed. 
  Extending previous work on geometric engineering of N=1 Yang-Mills in four dimensions for simply laced ($A_n,D_n,E_{6,7,8}$) gauge groups, we construct local models for all other gauge groups ($B_n,C_n,F_4,G_2$) in terms of F-theory. We compute the radius dependent superpotential upon further compactification on a circle to $d=3$ in the dual M-theory and use it to show that the number of vacua in four dimensions for each group is given by its dual coxeter number, in accordance with expectations based on gaugino condensates. 
  We discuss, in the framework of special Kahler geometry, some aspects of the "rigid limit" of type IIB string theory compactified on a Calabi-Yau threefold. We outline the general idea and demonstrate by direct analysis of a specific example how this limit is obtained. The decoupling of gravity and the reduction of special Kahler geometry from local to rigid is demonstrated explicitly, without first going to a noncompact approximation of the Calabi-Yau. In doing so, we obtain the Seiberg-Witten Riemann surfaces corresponding to different rigid limits as degenerating branches of a higher genus Riemann surface, defined for all values of the moduli. Apart from giving a nice geometrical picture, this allows one to calculate easily some gravitational corrections to e.g. the Seiberg-Witten central charge formula. We make some connections to the 2/5-brane picture, also away from the rigid limit, though only at the formal level. 
  We discuss the generalization of recently discovered BPS configurations, corresponding to the planar string networks, to non-planar ones by considering the U-duality symmetry of type II string theory in various dimensions. As an explicit example, we analyze the string solutions in 8-dimensional space-time, carrying SL(3) charges, and show that by aligning the strings along various directions appropriately, one can obtain a string network which preserves 1/8 supersymmetry. 
  We study the spectrum of the domain walls interpolating between different chirally asymmetric vacua in supersymmetric QCD with the SU(3) gauge group and including 2 pairs of chiral matter multiplets in fundamental and anti-fundamental representations. For small enough masses m < m* = .286... (in the units of \Lambda), there are two different domain wall solutions which are BPS-saturated and two types of ``wallsome sphalerons''. At m = m*, two BPS branches join together and, in the interval m* < m < m** = 3.704..., BPS equations have no solutions but there are solutions to the equations of motion describing a non-BPS domain wall and a sphaleron. For m > m**, there are no solutions whatsoever. 
  The unimodularity condition in Connes' formulation of the standard model is rewritten in terms of group representations. 
  The purpose of the present thesis is to give a self-contained review of the solvable Lie algebra approach to supergravity problems related with S, T and U dualities. After recalling the general features of dualities in both Superstring theory and Supergravity, we introduce the solvable Lie algebra formalism as an alternative description of the scalar manifold in a broad class of supergravity theories. It is emphasized how this mathematical technique on one hand allows to achieve a geometrical intrinsic characterization of Ramond-Ramond, Neveu-Schwarz and Peccei-Quinn scalars, once the supergravity theory is interpreted as the low energy limit of a suitably compactified superstring theory, on the other hand provides a convenient framework in which to deal with several non-perturbative problems. Using solvable Lie algebras for instance we find a general mechanism for spontaneous N=2 to N=1 local supersymmetry breaking. Moreover solvable Lie algebras are used to define a general method for studying systematically BPS saturated Black Hole solutions in supergravity. 
  It is shown that the five-dimensional anti-de Sitter black hole is a supersymmetric solution of the low-energy field equations of type IIB string theory compactified on an Einstein space. A statistical interpretation of the mass dependence of the entropy can be obtained from considerations of the three-dimensional BTZ black hole. 
  We study Seiberg duality for N=1 supersymmetric QCD with soft supersymmetry-breaking terms. We generate the soft terms through gauge mediation by coupling two theories related by Seiberg duality to the same supersymmetry-breaking sector. In this way, we know what a supersymmetry-breaking perturbation in one theory maps into in its ``dual''. Assuming a canonical Kahler potential we calculate the soft terms induced in the magnetic theory and find that some of the scalars acquire negative masses squared. If duality is still good for small supersymmetry breaking, this may imply some specific symmetry breaking patterns for supersymmetric QCD with small soft supersymmetry-breaking masses, in the case that its dual theory is weakly coupled in the infrared. In the limit of large supersymmetry breaking, the electric theory becomes ordinary QCD. However, the resulting symmetry breaking in the magnetic theory is incompatible with that expected for QCD. 
  We present a general framework for Matrix theory compactified on a quotient space R^n/G, with G a discrete group of Euclidean motions in R^n. The general solution to the quotient conditions gives a gauge theory on a noncommutative space. We characterize the resulting noncommutative gauge theory in terms of the twisted group algebra of G associated with a projective regular representation. Also we show how to extend our treatments to incorporate orientifolds. 
  A classical model of N=2, D=3 fractional spin superparticle (superanyon) is presented, whose first-quantization procedure combines the Berezin quantization for the superspin degrees of freedom and the canonical quantization for the space-time ones. To provide the supersymmetry for the quantised theory, certain quantum corrections are required to the N=2 supersymmetry generators as compared to the Berezin procedure. The renormalized generators are found and the first quantised theory of N=2 superanyon is constructed. 
  This note gives a brief review of the integrable structures presented in the Seiberg-Witten approach to the N=2 SUSY gauge theories with emphasize on the case of the gauge theories with matter hypermultiplets included (described by spin chains). The web of different N=2 SUSY theories is discussed. 
  We discuss the average length l of the shortest non-contractible loop on surfaces in the two-dimensional pure quantum gravity ensemble. The value of $\gamma_{str}$ and the explicit form of the loop functions indicate that l diverges at the critical point. Scaling arguments suggest that the critical exponent of l is 1/2. We show that this value of the critical exponent is also obtained for branched polymers where the calculation is straightforward. 
  The three dimensional abelian fermionic determinant of a two component massive spinor in flat euclidean space-time is resetted to a pure Chern-Simons action through a nonlinear redefinition of the gauge field. 
  It is shown that all string fields except dilaton are non-propagating in the (2+1)-dimensional black string. One finds that the perturbation around the black string reveals a mixing between the dilaton and other fields. Under the new gauge(dilaton gauge), we disentangle this mixing and obtain one decoupled dilaton equation. It turns out that this black string is stable. From the scattering of dilaton off the neutral black string(N=0), we find the absorption cross-section. Further the absorption cross-section for minimally coupled scalar is obtained and we compared it with that of dilaton. 
  A unified description of spacetime and matter at the Planck scale is proposed by using the irreducible representation of N=10 extended Super-Poincar\'e algebra, where all matters and all forces except the graviton are the supersymmetric composites made of the fundamental objects with spin 1/2,  superon quintet. All the local gauge interactions in GUTs are investigated systematically by using the superon diagrams. The proton is stable and the flavor changing neutral current process is suppressed in the superon pictures of GUTs. The fundamental action of the superon model is proposed. The characteristic predictions which can be tested in the (coming) high energy experiments are discussed briefly. 
  We review the problems associated with Matrix compactifications on T^6. 
  We apply a global and geometrically well-defined formalism for spinor-dilaton-gravity to two-dimensional manifolds. We discuss the general formalism and focus attention on some particular choices of the dilatonic potential. For constant dilatonic potential the model turns out to be completely solvable and the general solution is found. For linear and exponential dilatonic potentials we present the class of exact solutions with a Killing vector. 
  We review the construction of six dimensional N=1 fixed points in a brane picture involving D6 branes stretching between NS 5 branes. 
  We study supersymmetric QCD with N_f<N_c in the limit of small supersymmetry-breaking masses and smaller quark masses using the weak-coupling Kahler potential. We calculate the full spectrum of this theory, which manifests a chiral symmetry breaking pattern similar to that caused by the strong interactions of the standard model. We derive the chiral effective lagrangian for the pion degrees of freedom, and discuss the behavior in the formal limit of large squark and gluino masses and for large N_c. We show that the resulting scalings of the pion decay constant and pion masses in these limits differ from those expected in ordinary nonsupersymmetric QCD. Although there is no weak coupling expansion with N_f=N_c, we extend our results to this case by constructing a superfield quantum modified constraint in the presence of supersymmetry breaking. 
  We show that all branes admit worldvolume domain wall solutions. We find one class of solutions for which the tension of the brane changes discontinuously along the domain wall. These solutions are not supersymmetric. We argue that there is another class of domain wall solutions which is supersymmetric. A particular case concerns supersymmetric domain wall solutions on IIB D-5- and NS-5-branes. 
  A set of non-supersymmetric minimal area embeddings of an M-theory 5-brane are considered. The field theories on the surface of the 5-brane have the field content of N=2 SQCD with fundamental representation matter fields. By suitable choice of curve parameters the N=2 and N=1 superpartners may be decoupled leaving a semi-classical approximation to QCD with massive quarks. As supersymmetry breaking is introduced a quark condensate grows breaking the low energy $Z_{F}$ flavour symmetry. At $\theta =$ (odd) $\pi$ spontaneous CP violation is observed consistent with that of the QCD chiral lagrangian. 
  Starting with the D-dimensional Einstein-dilaton-antisymmetric form equations and assuming a block-diagonal form of a metric we derive a $(D-d)$-dimensional $\sigma$-model with the target space $SL(d,R)/SO(d) \times SL(2,R)/SO(2) \times R$ or its non-compact form. Various solution-generating techniques are developed and applied to construct some known and some new $p$-brane solutions. It is shown that the Harrison transformation belonging to the $SL(2,R)$ subgroup generates black $p$-branes from the seed Schwarzschild solution. A fluxbrane generalizing the Bonnor-Melvin-Gibbons-Maeda solution is constructed as well as a non-linear superposition of the fluxbrane and a spherical black hole. A new simple way to endow branes with additional internal structure such as plane waves is suggested. Applying the harmonic maps technique we generate new solutions with a non-trivial shell structure in the transverse space (`matrioshka' $p$-branes). It is shown that the $p$-brane intersection rules have a simple geometric interpretation as conditions ensuring the symmetric space property of the target space. Finally, a Bonnor-type symmetry is used to construct a new magnetic 6-brane with a dipole moment in the ten-dimensional IIA theory. 
  BPS black hole configurations which break half of supersymmetry in the theory of N=2, d=5 supergravity coupled to an arbitrary number of abelian vector multiplets are discussed. A general class of solutions comprising all known BPS rotating black hole solutions is obtained. 
  We show that the recent observation that near extremal NS fivebranes decay due to Hawking radiation, can be understood in terms of their coupling to RR states which does not vanish even as the string coupling goes to zero. 
  The Coulomb branch of the potential between two static colored sources is calculated for the Yang-Mills theory using the electric Schroedinger representation. 
  We study constraints among coupling constants of the standard model obtained in the noncommutative geometry (NCG) method. First, we analyze the evolution of the Higgs boson mass under the renormalization group by adopting the idea of \'Alvarez et al. For this analysis we derive two certain constraints by modifying Connes's way of constructing the standard model. Next, we find renormalization group invariant (RGI) constraints in the NCG method. We also consider the relation between the condition that a constraint among coupling constants of a model becomes RGI and the condition that the model becomes multiplicative renormalizable by using a simple example. 
  Domain wall and electrovac solutions of gauged N=4 D=4 supergravity, with gauge group SU(2) or SU(2)xSU(2), are interpreted as supersymmetric Kaluza-Klein vacua of N=1 D=10 supergravity. These vacua are shown to be the near-horizon geometries of certain intersecting brane solutions. 
  We study Witten's proposal that a domain wall exists in M-theory fivebrane version of QCD (MQCD) and that it can be represented as a supersymmetric three-cycle in G_2 holonomy manifold. It is shown that equations defining the U(1) invariant domain wall for SU(2) group can be reduced to the Monge-Ampere equation. A proof of an algebraic formula of Kaplunovsky, Sonnenschein and Yankielowicz is presented. The formal solution of equations for domain wall is constructed. 
  We consider the one-dimensional Hubbard model with the infinitely strong repulsion. The two-point dynamical temperature correlation functions are calculated. They are represented as Fredholm determinants of linear integrable integral operators. 
  We present results for some infinite series appearing in Feynman diagram calculations, many of which are similar to the Euler series. These include both one-dimensional and two-dimensional series. Most of these series can be expressed in terms of zeta(2), zeta(3), the Catalan constant G and Cl{2}(pi/3) where Cl{2}(theta) is Clausen's function. 
  We discuss the construction of the physical configuration space for Yang-Mills quantum mechanics and Yang-Mills theory on a cylinder. We explicitly eliminate the redundant degrees of freedom by either fixing a gauge or introducing gauge invariant variables. Both methods are shown to be equivalent if the Gribov problem is treated properly and the necessary boundary identifications on the Gribov horizon are performed. In addition, we analyze the significance of non-generic configurations and clarify the relation between the Gribov problem and coordinate singularities. 
  In this paper we study the compactification conditions of the M theory on D-dimensional noncommutative tori. The main tool used for this analysis is the algebra A(Z^D) of the projective representations of the abelian group Z^D. We exhibit the explicit solutions in the space of the multiplication algebra of A(Z^D), that is the algebra generated by right and left multiplications. 
  We analyze the induction and restriction of sectors for nets of subfactors defined by Longo and Rehren. Picking a local subfactor we derive a formula which specifies the structure of the induced sectors in terms of the original DHR sectors of the smaller net and canonical endomorphisms. We also obtain a reciprocity formula for induction and restriction of sectors, and we prove a certain homomorphism property of the induction mapping.   Developing further some ideas of F. Xu we will apply this theory in a forthcoming paper to nets of subfactors arising from conformal field theory, in particular those coming from conformal embeddings or orbifold inclusions of SU(n) WZW models. This will provide a better understanding of the labeling of modular invariants by certain graphs, in particular of the A-D-E classification of SU(2) modular invariants. 
  Here is presented a detailed work on the (1+1) dimensional SU(N) Yang-Mills theory with static sources. By studying the structure of the SU(N) group and of the Gauss' law we construct in the electric representation the appropriate wave functionals, which are simultaneously eigenstates of the Gauss' operator and of the Hamiltonian. The Fourier transformation between the A- and the E-representations connecting the Wilson line and a superposition of our solutions is given. 
  We present a systematic study of N=1 supersymmetric gauge theories which are in the Coulomb phase. We show how to find all such theories based on a simple gauge group and no tree-level superpotential. We find the low-energy solution for the new theories in terms of a hyperelliptic Seiberg-Witten curve. This work completes the study of all N=1 supersymmetric gauge theories where the Dynkin index of the matter fields equals the index of the adjoint (mu=G), and consequently all theories for which mu<G. 
  We show that for a string moving in a background consisting of maximally symmetric gravity, dilaton field and second rank antisymmetric tensor field, the $O(d) \otimes O(d)$ transformation on the vacuum solutions gives inequivalent solutions that are not maximally symmetric. We then show that the usual physical meaning of maximal symmetry can be made to remain unaltered even if torsion is present and illustrate this through two toy models by determining the torsion fields, the metric and Killing vectors. Finally we show that under the $O(d) \otimes O(d)$ transformation this generalised maximal symmetry can be preserved under certain conditions. This is interesting in the context of string related cosmological backgrounds. 
  M-theory compactification leads one to consider 7-manifolds obtained by rolling Calabi-Yau threefolds in the web of Calabi-Yau moduli spaces. The resulting 7-space in general has singularities governed by the extremal transition undergone. After providing some background in Sec. 1, the simplest case of conifold transitions is studied in Sec. 2. In Sec. 3 we employ topological methods, Smale's classification theorem of smooth simply-connected spin closed 5-manifolds, and a computer code in the Appendix to understand the 5-manifolds that appear as the link of the singularity of a singuler Calabi-Yau threefolds from a Type II primitive contraction of a smooth one.   From this we obtain many locally admissible extremal transition pairs of Calabi-Yau threefolds, listed in Sec. 4. Their global realization will require further study. As a mathematical byproduct in the pursuit of the subject, we obtain a formula to compute the topology of the boundary of the tubular neighborhood of a Gorenstein rational singular del Pezzo surface embedded in a smooth Calabi-Yau threefold as a divisor. 
  It is shown that recently pointed out by Berkovits on-shell degrees of freedom of the D=11 superstring do not make contributions into the quantum states spectrum of the theory. As a consequence, the spectrum coincides with that of the D=10 type IIA superstring. 
  Behaviour of singularities of the rotating black holes under an arbitrary boost is considered on the basis of a complex representation of the Kerr theorem. We give a simple algorithm allowing to get explicit expressions for the metric and the position of the singularities for arbitrary direction and magnitude of the boost, including the ultrarelativistic case. The non-smoothness of the ultrarelativistic limit is discussed. The Kerr-Sen BH-solution to low energy string theory is also analyzed. 
  The spinorial local world-line $\kappa$-symmetry of the covariant Brink-Schwarz formulation of the 4-$D$ superparticle is abelian in an off-shell phase-space formulation. The result is shown to generalize to the extended spinorial transformations of the spinning superparticle. 
  The "Hodge strings" construction of solutions to associativity equations is proposed. From the topological string theory point of view this construction formalizes the "integration over the position of the marked point" procedure for computation of amplitudes. From the mathematical point of view the "Hodge strings" construction is just a composition of elements of harmonic theory (known among physicists as a $t$-part of $t-t^*$ equations) and the K.Saito construction of flat coordinates (starting from flat connection with a spectral parameter).    We also show how elements of K.Saito theory of primitive form appear naturally in the "Landau-Ginzburg" version of harmonic theory if we consider the holomorphic pieces of germs of harmonic forms at the singularity. 
  The action for 2d dilatonic supergravity with dilaton coupled matter and dilaton multiplets is constructed. Trace anomaly and anomaly induced effective action (in components as well as in supersymmetric form) for matter supermultiplet on bosonic background are found. The one-loop effective action and large-$N$ effective action for quantum dilatonic supergravity are also calculated. Using induced effective action one can estimate the back-reaction of dilaton coupled matter to the classical black hole solutions of dilatonic supergravity. That is done on the example of supersymmetric CGHS model with dilaton coupled quantum matter where Hawking radiation which turns out to be zero is calculated. Similar 2d analysis maybe used to study spherically symmetric collapse for other models of 4d supergravity. 
  A worldvolume action for membrane is considered to study the target space local symmetries. We introduce a set of generators of canonical transformations to exhibit the target space symmetries such as the general coordinate transformation and the gauge transformation of antisymmetric tensor field. Similar results are derived for type IIB string with manifestly S-duality-invariant worldsheet action. 
  These notes give a pedagogical introduction to D-branes and Matrix theory. The development of the material is based on super Yang-Mills theory, which is the low-energy field theory describing multiple D-branes. The main goal of these notes is to describe physical properties of D-branes in the language of Yang-Mills theory, without recourse to string theory methods. This approach is motivated by the philosophy of Matrix theory, which asserts that all the physics of light-front M-theory can be described by an appropriate super Yang-Mills theory. 
  We study spin interactions between two moving D-branes using the Green-Schwarz formalism of boundary states. We focus our attention on the leading terms for small velocities v, of the form v^{4-n}/r^{7-p+n} (v^{2-n}/r^{3-p+n}) for p-p (p-p+4) systems, with 16 (8) supercharges. In analogy with standard G-S computations of massless four-point one-loop amplitudes in Type I theory, the above terms are governed purely by zero modes, massive states contributions cancelling as expected by the residual supersymmetry. This implies the scale invariance of these leading spin-effects, supporting the relevant matrix model descriptions of supergravity interactions; in this context, we also discuss similar results for more general brane configurations. We then give a field theory interpretation of our results, that allows in particular to deduce the gyromagnetic ratio g=1 and the presence of a quadrupole moment for D0-branes. 
  It is shown how a metric structure can be induced in a simple way starting with a gauge structure and a preferred volume, by spontaneous symmetry breaking. A polynomial action, including coupling to matter, is constructed for the symmetric phase. It is argued that assuming a preferred volume, in the context of a metric theory, induces only a limited modification of the theory. 
  The complete UV-divergent contribution to the one-loop 1PI four-point function of Yang-Mills theory in the light-cone gauge is computed in this paper. The formidable UV-divergent contributions arising from each four-point Feynman diagram yield a succinct final result which contains nonlocal terms as expected. These nonlocal contributions are consistent with gauge symmetry, and correspond to a nonlocal renormalization of the wave function. Renormalization of Yang-Mills theory in the light-cone gauge is thus shown explicitly at the one-loop level. 
  The conformal-gauge two-dimensional quantum gravity is formulated in the framework of the BRS quantization and solved completely in the Heisenberg picture: All n-point Wightman functions are explicitly obtained. The field-equation anomaly is shown to exist as in other gauges, but there is no other subtlety. At the critical dimension D=26 of the bosonic string, the field-equation anomaly is shown to be absent. However, this result is not equivalent to the statement that the conformal anomaly is proportional to D-26. The existence of the FP-ghost number current anomaly is seen to be an illusion. 
  We study the confining phase structure on {\cal N}=1 supersymmetric SO(12) gauge theory with N_f \le 7 vectors and one spinor. The explicit form of low energy superpotentials for N_f \le 7 are derived after gauge invariant operators relevant in the effective theory are identified via gauge symmetry breaking pattern. The resulting confining phase structure is analogous to N_f \le N_c+1 SUSY QCD. Finally, we conclude with some comments on the search for duals to N_f \ge 8 SO(12) theory. 
  The unitary transformation of path-integral differential measure is described. The main properties of perturbation theory in the phase space of action-angle, energy-time variables are investigated. The measure in cylindrical coordinates is derived also. The dependence of perturbation theory contributions from global (topological) properties of corresponding phase spaces is shown. 
  We calculate the spin dependent static force between two D0-branes in Matrix theory. Supersymmetry relates velocity dependent potentials to spin dependent potentials. The well known v^4/r^7 term is related to a theta^8/r^11 term, where theta is the relative spin of the D0-branes. We calculate this term, confirming that it is the lowest order contribution to the static potential, and find its structure consistent with supergravity. 
  We outline the structure of boundary conditions in conformal field theory. A boundary condition is specified by a consistent collection of reflection coefficients for bulk fields on the disk together with a choice of an automorphism \omega of the fusion rules that preserves conformal weights. Non-trivial automorphisms \omega correspond to D-brane configurations for arbitrary conformal field theories. 
  The boundary conditions of a non-trivial string background are classified. To this end we need traces on various spaces of conformal blocks, for which generalizations of the Verlinde formula are presented. 
  We study the nonrelativistic limit of the quantum theory of a Chern-Simons field minimally coupled to a scalar field with quartic self-interaction. The renormalization of the relativistic model, in the Coulomb gauge, is discussed. We employ a procedure to calculate scattering amplitudes for low momenta that generates their $|p|/m$ expansion and separates the contributions coming from high and low energy intermediary states. The two body scattering amplitude is calculated up to order $p^2/m^2$. It is shown that the existence of a critical value of the self-interaction parameter for which the 2-particle scattering amplitude reduces to the Aharonov-Bohm one is a strictly nonrelativistic feature. The subdominant terms correspond to relativistic corrections to the Aharonov-Bohm scattering. A nonrelativistic reduction scheme and an effective nonrelativistic Lagrangian to account for the relativistic corrections are proposed. 
  We study the domain wall soliton solutions in the relativistic self-dual Maxwell Chern-Simons model in 1+1 dimensions obtained by the dimensional reduction of the 2+1 model. Both topological and nontopological self-dual solutions are found in this case. A la BPS dyons here the Bogomol'ny bound on the energy is expressed in terms of two conserved quantities. We discuss the underlying supersymmetry. Nonrelativistic limit of this model is also considered and static, nonrelativistic self-dual soliton solutions are obtained. 
  We consider the gravitational scattering of point particles in four dimensions, at Planckian centre of mass energy and low momentum transfer, or the eikonal approximation. The scattering amplitude can be exactly computed by modelling point particles by very generic metrics. A class of such metrics are black hole solutions obtained from dimensional reduction of p-brane solutions with one or more Ramond-Ramond charges in string theory. At weak string coupling, such black holes are replaced by a collection of wrapped D-branes. Thus, we investigate eikonal scattering at weak coupling by modelling the point particles by wrapped D-branes and show that the amplitudes exactly match the corresponding amplitude found at strong coupling. We extend the calculation for scattering of charged particles. 
  The differential algebra on the fuzzy sphere is constructed by applying Connes' scheme. The U(1) gauge theory on the fuzzy sphere based on this differential algebra is defined. The local U(1) gauge transformation on the fuzzy sphere is identified with the left $U(N+1)$ transformation of the field, where a field is a bimodule over the quantized algebra $\CA_N$. The interaction with a complex scalar field is also given. 
  The perturbative renormalization group(RG) equation is applied to resum divergent series of perturbative wave functions of quantum anharmonic oscillator. It is found that the resummed series gives the cumulant of the naive perturbation series. It is shown that a reorganization of the resummed series reproduce the correct asymptotic form of the wave function at $x\to \infty$ when the perturbation expansion is stopped at the fourth order. A brief comment is given on the relation between the present method and the delta-expansion method, which is based on a kind of a nonperturbative RG equation. 
  With non-equilibrium applications in mind we present in this paper a self-contained calculation of the hydrostatic pressure of the O(N)\lambda \phi^4 theory at finite temperature. By combining the Keldysh-Schwinger closed-time path formalism with thermal Dyson-Schwinger equations we compute in the large N limit the hydrostatic pressure in a fully resumed form. We also calculate the high-temperature expansion for the pressure (in D=4) using the Mellin transform technique. The result obtained extends the results found by Drummond et al. [hep-ph/9708426] and Amelino-Camelia and Pi [hep-ph/9211211]. The latter are reproduced in the limits m_r(0)\to 0, T \to \infty and T \to \infty, respectively. Important issues of renormalizibility of composite operators at finite temperature are addressed and the improved energy-momentum tensor is constructed. The utility of the hydrostatic pressure in the non-equilibrium quantum systems is discussed. 
  This article deals with a nonrelativistic quantum mechanical study of a charge-dyon system with the SU(2)--monopole in five dimensions. The Schr\"odinger equation for this system is separable in the hyperspherical and parabolic coordinates. The problem of interbasis expansion of the wave functions is completely solved. The coefficients for the expansion of the parabolic basis in terms of the hyperspherical basis can be expressed through the Clebsch-Gordan coefficients of the group SU(2). 
  Starting from the working hypothesis that both physics and the corresponding mathematics have to be described by means of discrete concepts on the Planck-scale, one of the many problems one has to face is to find the discrete protoforms of the building blocks of continuum physics and mathematics. In the following we embark on developing such concepts for irregular structures like (large) graphs or networks which are intended to emulate (some of) the generic properties of the presumed combinatorial substratum from which continuum physics is assumed to emerge as a coarse grained and secondary model theory. We briefly indicate how various concepts of discrete (functional) analysis and geometry can be naturally constructed within this framework, leaving a larger portion of the paper to the systematic developement of dimensional concepts and their properties, which may have a possible bearing on various branches of modern physics beyond quantum gravity. 
  We review the coupling of N=2 supergravity to vector-tensor multiplets, based on the method of superconformal multiplet calculus. 
  The superspace geometry of Chern-Simons forms is shown to be closely related to that of the 3-form multiplet. This observation allows to simplify considerably the geometric structure of supersymmetric Chern-Simons forms and their coupling to linear multiplets. The analysis is carried through in U_K(1) superspace, relevant at the same time for supergravity-matter couplings and for chirally extended supergravity. 
  The covariant field equations of ten-dimensional super D-branes are obtained by considering fundamental strings whose ends lie in the superworldsurface of the D-brane. By considering in a similar fashion Dp-branes ending on D(p+2)-branes we derive equations describing D-branes with dual potentials, as well as the vector potentials. 
  We describe in superspace a classical theory of two dimensional $(1,1)$ cosmological dilaton supergravity coupled to a massive superparticle. We give an exact non-trivial superspace solution for the compensator superfield that describes the supergravity, and then use this solution to construct a model of a two-dimensional supersymmetric black hole. 
  The construction of effective field theories describing M-theory compactified on $S^1/{\bf Z}_2$ is revisited, and new insights into the parameters of the theory are explained. Particularly, the web of constraints which follow from supersymmetry and anomaly cancelation is argued to be more rich than previously understood. In contradistinction to the lore on the subject, a consistent classical theory describing the coupling of eleven dimensional supergravity to super Yang-Mills theory constrained to the orbifold fixed points is suggested to exist. 
  It is confirmed that geodesic string junctions are necessary to describe the gauge vectors of symmetry groups that arise in the context of IIB superstrings compactified in the presence of nonlocal 7-branes. By examining the moduli space of 7-brane backgrounds for which the dilaton and axion fields are constant, we are able to describe explicitly and geometrically how open string geodesics can fail to be smooth, and how geodesic string junctions then become the relevant BPS representatives of the gauge bosons. The mechanisms that guarantee the existence and uniqueness of the BPS representative of any gauge vector are also shown to generalize to the case where the dilaton and axion fields are not constant. 
  The bosonic actions for M2, D3 and M5 branes in their own d-dimensional near-horizon background are given in a manifestly SO(p+1,2) x SO(d-p-1) invariant form (p=2,3,5). These symmetries result from a breakdown of ISO(d,2) (with d=10 for D3 and d=11 for M2 and M5) symmetry by the Wess-Zumino term and constraints. The new brane actions, reduce after gauge-fixing and solving constraints to (p+1) dimensional interacting field theories with a non-linearly realized SO(p+1,2) conformal invariance. We also present an interacting two-dimensional conformal field theory on a D-string in the near-horizon geometry of a D1+D5 configuration. 
  We present brane constructions in Type IIA string theory for N=1 supersymmetric SO and Sp gauge theories with tensor representations using an orientifold 6-plane. One limit of these set-ups corresponds to N=2 theories previously constructed by Landsteiner and Lopez, while a different limit yields N=1 SO or Sp theories with a massless tensor and no superpotential. For the Sp-type orientifold projection comparison with the field theory moduli space leads us to postulate two new rules governing the stability of configurations of D-branes intersecting the orientifold. Lifting one of our configurations to M-theory by finding the corresponding curves, we re-derive the N=1 dualities for SO and Sp groups using semi-infinite D4 branes. 
  This presentation is intended to give a flavour of the physics to be studied and discovered by exploiting the interface between string theory, M-theory and field theory provided by the dynamics of extended objects called `branes', and their relationships under string duality. It is quite non-technical, with very few equations but many pictorial illustrations to aid the reader. There is only a moderate amount of text, as the content is based upon transparencies which were intended to be accompanied by extra words. (Hyperlinked transparencies of a talk given at the annual UK Particle Theory Meeting,Rutherford Appleton Laboratories, Didcot, Oxfordshire, England, 15th December 1997.) 
  A dual description of 3-dimensional topological Seiberg-Witten theory in terms of the Alexander invariant on manifolds obtained via surgery on a knot is proposed. The description directly follows from a low-energy analysis of the corresponding SUSY theory, in full analogy to the 4-dimensional case. 
  We consider the scaling Lee-Yang model. It corresponds to the unique perturbation of the minimal CFT model M(2,5). This is not a unitary model. We used known expression for form factors in order to obtain a closed expression for a correlation function of a trace of energy-momentum tensor. This expression is a determinant of an integral operator. Similar determinant representation were proven to be useful not only for quantum correlation functions but also in matrix models. 
  Recently, a non-hyperelliptic curve describing the Coulomb branch of N=2 SUSY $SU(N_c)$ Yang-Mills theory with an antisymmetric tensor matter was proposed using a configuration of a single M theory five-brane. We study the singular surface in the moduli space of the curve to compare it with results from the ``integrating in'' method in field theory. In order to achieve the consistency, we find it necessary to take account of an additional superpotential $W_{\Delta}$ which has been neglected so far. The explicit form of $W_{\Delta}$ is worked out. 
  We study membrane scattering in a curved space with non-zero M-momentum p_{11} transfer. In the low-energy short-distance region, the membrane dynamics is described by a three-dimensional N=4 supersymmetric gauge theory. We study an n-instanton process of the gauge theory, corresponding to the exchange of n units of p_{11}, and compare the result with the scattering amplitude computed in the low-energy long-distance region using supergravity. We find that they behave differently. We show that this result is not in contradiction with the large-N Matrix Theory conjecture, by pointing out that cutoff scales of the supergravity and the gauge theory are complementary to each other. 
  Some points concerning the relation of strings to interfaces in statistical systems are discussed. 
  The second derivatives of prepotential with respect to Whitham time-variables in the Seiberg-Witten theory are expressed in terms of Riemann theta-functions. These formulas give a direct transcendental generalization of algebraic ones for the Kontsevich matrix model. In particular case they provide an explicit derivation of the renormalization group (RG) equation proposed recently in papers on the Donaldson theory. 
  The problem of understanding the role of large gauge transformations in thermal field theories has recently inspired a number of studies of a one dimensional field theory. Such work has led to the conclusion that gauge invariance is restored only when the entire perturbation expansion can be summed. A careful reexamination of that model is shown, h0owever, to lead to vastly different conclusions when the constraint implied by the field equations is explicitly taken into account. In particular it is found that none of the relevant propagators has any temperature dependence and that the effective action is essentially trivial. A generalization of the model to include bosons as well as fermions is also solved with qualitatively identical results being obtained. 
  We present an alternative formulation of duality-symmetric eleven-dimensional supergravity with both three-form and six-form gauge fields. Instead of the recently-proposed scalar auxiliary field, we use a simpler lagrangian with a non-propagating auxiliary multiplier tensor field with eight-indices. We also complete the superspace formulation in a duality-symmetric manner. An alternative super M-5-brane action coupled to this eleven-dimensional background is also presented. This formulation bypasses the usual obstruction for an invariant lagrangian for a self-dual three-form field strength, by allowing the self-duality only as a solution for field equations, but not as a necessary condition. 
  In this review we describe statistical mechanics of quantum systems in the presence of a Killing horizon and compare statistical-mechanical and one-loop contributions to black hole entropy. Studying these questions was motivated by attempts to explain the entropy of black holes as a statistical-mechanical entropy of quantum fields propagating near the black hole horizon. We provide an introduction to this field of research and review its results. In particular, we discuss the relation between the statistical-mechanical entropy of quantum fields and the Bekenstein-Hawking entropy in the standard scheme with renormalization of gravitational coupling constants and in the theories of induced gravity. 
  If Einstein's photon is $E = cp = \hbar\omega$, Wigner's photon is its helicity which is a Lorentz-invariant concept coming from the E(2)-like little group for massless particles. In addition, the E(2)-like little group has two translation-like degrees of freedom. What happens to them? They are associated with the gauge degree of freedom. Since the physics of polarized light waves can be formulated within the framework of the Lorentz group, it is now possible to use polarization experiments to study the E(2)-like little group in terms of quantities that can be measured in laboratories. 
  We investigate Yang-Mills theories with arbitrary gauge group on $R^3\times S^1$, whose symmetry is spontaneously broken by the Wilson loop. We show that instantons are made of fundamental magnetic monopoles, each of which has a corresponding root in the extended Dynkin diagram. The number of constituent magnetic monopoles for a single instanton is the dual Coxeter number of the gauge group, which also accounts for the number of instanton zero modes. In addition, we show that there exists a novel type of the $S^1$ coordinate dependent magnetic monopole solutions in $G_2,F_4,E_8$. 
  We present a solvable model of two-dimensional dilaton-gravity coupled to a massless scalar field. We locally integrate the field equations and briefly discuss the properties of the solutions. For a particular choice of the coupling between the dilaton and the scalar field the model can be interpreted as the two-dimensional effective theory of 2+1 cylindrical gravity minimally coupled to a massless scalar field. 
  We study the long wavelength limit of a spin S Heisenberg antiferromagnetic chain. The fermionic Lagrangian obtained corresponds to a perturbed level 2S SU(2) Wess-Zumino-Witten model. This effective theory is then mapped into a compact U(1) boson interacting with Z_{2S} parafermions. The analysis of this effective theory allows us to show that when S is an integer there is a mass gap to all excitations, whereas this gap vanishes in the half-odd-integer spin case. This gives a field theory treatment of the so-called Haldane's conjecture for arbitrary values of the spin S. 
  We develop an unambiguous and practical method to calculate one-loop quantum corrections to the energies of classical time-independent field configurations in renormalizable field theories. We show that the standard perturbative renormalization procedure suffices here as well. We apply our method to a simplified model where a charged scalar couples to a neutral "Higgs" field, and compare our results to the derivative expansion. 
  We consider topological closed string theories on Calabi-Yau manifolds which compute superpotential terms in the corresponding compactified type II effective action. In particular, near certain singularities we compare the partition function of this topological theory (the Kodaira-Spencer theory) to $SU(\infty)$ Chern-Simons theory on the vanishing 3-cycle. We find agreement between these theories, which we check explicitly for the case of shrinking $S^3$ and Lens spaces, at the perturbative level. Moreover, the gauge theory has non-perturbative contributions which have a natural interpretation in the Type IIB picture. We provide a heuristic explanation for this agreement as well as suggest further equivalences in other topological gravity/gauge systems. 
  We explore a novel way of deriving the effective Higgs Lagrangian from strongly interacting vector-like gauge theories. We consider the N=1 supersymmetric extension of gauge theories and interpret the auxiliary field associated with the low energy effective "meson" superfield as the Higgs field. By introducing an explicit supersymmetry breaking term and computing the one-loop effective action at the effective theory level we show that the kinetic term for the Higgs field is generated, while the negative mass squared term is already present at the tree level. We further propose a scenario by which the complete Higgs potential can be generated and the fermion in the low energy effective theory acquires a mass. 
  In this letter we discuss the supersymmetry issue of the self dual supermembranes in (8+1) and (4+1)-dimensions. We find that all genuine solutions of the (8+1)-dimensional supermembrane, based on the exceptional group G_2, preserve one of the sixteen supersymmetries while all solutions in (4+1)-dimensions preserve eight of them. 
  A general method of regularisation of classical self interaction in strings is extended from the electromagnetic case (for which it was originally developed) to the gravitation case, for which the result can also be represented as a renormalisation. 
  We show that the ``time'' t_s defined via spin clusters in the Ising model coupled to 2d gravity leads to a fractal dimension d_h(s) = 6 of space-time at the critical point, as advocated by Ishibashi and Kawai. In the unmagnetized phase, however, this definition of Hausdorff dimension breaks down. Numerical measurements are consistent with these results. The same definition leads to d_h(s)=16 at the critical point when applied to flat space. The fractal dimension d_h(s) is in disagreement with both analytical prediction and numerical determination of the fractal dimension d_h(g), which is based on the use of the geodesic distance t_g as ``proper time''. There seems to be no simple relation of the kind t_s = t_g^{d_h(g)/d_h(s)}, as expected by dimensional reasons. 
  A systematic analysis of a continuous version of a binomial lattice, containing a real parameter $\gamma$ and covering the Toda field equation as $\gamma\to\infty$, is carried out in the framework of group theory. The symmetry algebra of the equation is derived. Reductions by one-dimensional and two-dimensional subalgebras of the symmetry algebra and their corresponding subgroups, yield notable field equations in lower dimensions whose solutions allow to find exact solutions to the original equation. Some reduced equations turn out to be related to potentials of physical interest, such as the Fermi-Pasta-Ulam and the Killingbeck potentials, and others. An instanton-like approximate solution is also obtained which reproduces the Eguchi-Hanson instanton configuration for $\gamma\to\infty$. Furthermore, the equation under consideration is extended to $(n+1)$--dimensions. A spherically symmetric form of this equation, studied by means of the symmetry approach, provides conformally invariant classes of field equations comprising remarkable special cases. One of these $(n=4)$ enables us to establish a connection with the Euclidean Yang-Mills equations, another appears in the context of Differential Geometry in relation to the socalled Yamabe problem. All the properties of the reduced equations are shared by the spherically symmetric generalized field equation. 
  Recently, we have studied the general Virasoro construction at one loop in the background of the general non-linear sigma model. Here, we find the action formulation of these new conformal field theories when the background sigma model is itself conformal. In this case, the new conformal field theories are described by a large class of new spin-two gauged sigma models. As examples of the new actions, we discuss the spin-two gauged WZW actions, which describe the conformal field theories of the generic affine-Virasoro construction, and the spin-two gauged g/h coset constructions. We are able to identify the latter as the actions of the local Lie h-invariant conformal field theories, a large class of generically irrational conformal field theories with a local gauge symmetry. 
  A geometric derivation of $W_\infty$ Gravity based on Fedosov's deformation quantization of symplectic manifolds is presented. To lowest order in Planck's constant it agrees with Hull's geometric formulation of classical nonchiral $W_\infty$ Gravity. The fundamental object is a ${\cal W}$-valued connection one form belonging to the exterior algebra of the Weyl algebra bundle associated with the symplectic manifold. The ${\cal W} $-valued analogs of the Self Dual Yang Mills equations, obtained from a zero curvature condition, naturally lead to the Moyal Plebanski equations, furnishing Moyal deformations of self dual gravitational backgrounds associated with the complexified cotangent space of a two dimensional Riemann surface. Deformation quantization of $W_\infty$ Gravity is retrieved upon the inclusion of all the $\hbar$ terms appearing in the Moyal bracket. Brief comments on Non Commutative Geometry and M(atrix)theory are made. 
  We propose an integrable extension of nonlinear sigma model on the target space of Hermitian symmetric space (HSS). Starting from a discussion of soliton solutions of O(3) model and an integrally extended version of it, we construct general theory defined on arbitrary HSS by using the coadjoint orbit method. It is based on the exploitation of a covariantized canonical structure on HSS. This term results in an additional first-order derivative term in the equation of motion, which accommodates the zero curvature representation. Infinite conservation laws of nonlocal charges in this model are derived. 
  We suggest to take the fluctuation-dissipation theorem of Callen and Welton as a basis to study quantum dissipative phenomena (such as macroscopic quantum tunneling) in a manner analogous to the Nambu-Goldstone theorem for spontaneous symmetry breakdown. It is shown that the essential physical contents of the Caldeira-Leggett model such as the suppression of quantum coherence by Ohmic dissipation are derived from general principles only, namely, the fluctuation-dissipation theorem and unitarity and causality (i.e., dispersion relations), without referring to an explicit form of the Lagrangian. An interesting connection between quantum tunneling with Ohmic dissipation and the Anderson's orthogonality theorem is also noted. 
  By following Seiberg's prescriptions on DLCQ of $M$ theory, we give the background geometries of DLCQ supergravity associated with $N$ sector of DLCQ of $M$ theory on $T^p$ with vanishingly small radii. Most of these are the product of anti-de Sitter spacetimes and spheres, which have been found as the spontaneous compactifications of eleven dimensional supergravity long time ago and also are revisited recently by Maldacena by considering the near horizon geometry of various D-branes in appropriate limit. Those geometries are maximally symmetric and have full 32 supersymmetries of eleven dimensional supergravity, which agrees with the number of supersymmetries of DLCQ of $M$ theory. This suggests that DLCQ of $M$ theory is the $M$/string theory on these nontrivial background. 
  The supercurrent components of the N=1, D=4 Super-Yang-Mills theory in the Wess-Zumino gauge are coupled to the components of a background supergravitation field in the ``new minimal'' representation, in order to describe the various conservation laws in a functional way through the Ward identities for the diffeomorphisms and for the local supersymmetry, Lorentz and R-transformations. We also incorporate in the same functional formalism the supertrace identities, which leads however to a slight modification of the new minimal representation for supergravity, thus leading to a conformal version of it. The most general classical action obeying all the symmetry constraints is constructed. 
  In a quantum theory of gravity, fluctuations about the vacuum may be considered as Planck scale virtual black holes appearing and annihilating in pairs. Incident fields scattering from such fluctuations would lose quantum coherence.   In a recent paper (hep-th/9705147), Hawking and Ross obtained an estimate for the magnitude of this loss in the case of a scalar field. Their calculation exploited the separability of the conformally invariant scalar wave equation in the electrovac C metric background, which is justified as a sufficiently good description of a virtual black hole pair in the limit considered.   In anticipation of extending this result, the Teukolsky equations for incident fields of higher spin are separated on the vacuum C metric background and solved in the same limit. With the exception of spin 2 fields, these equations are shown in addition to be valid on the electrovac C metric background. The angular solutions are found to reduce to the spin- weighted spherical harmonics, and the radial solutions are found to approach hypergeometrics close to the horizons.   By defining appropriate scattering boundary conditions, these solutions are then used to estimate the transmission and reflection coefficients for an incident field of spin s. The transmission coefficient is required in order to estimate the loss of quantum coherence of an incident field through scattering off virtual black holes. 
  The discrepancy between the anomaly found by Bousso and Hawking (hep-th/9705236) and that of other workers is explained by the omission of a zero mode contribution to the effective action. 
  We show that within the framework of a definite proposal for the initial conditions for the universe, the Hartle-Hawking `no boundary' proposal, open inflation is generic and does not require any special properties of the inflaton potential. In the simplest inflationary models, the semiclassical approximation to the Euclidean path integral and a minimal anthropic condition lead to $\Omega_0\approx 0.01$. This number may be increased in models with more fields or extra dimensions. 
  The asymptotic high momentum behaviour of quantum field theories with cubic interactions is investigated using renormalization group techniques in the asymmetric limit x << 1. Particular emphasis is paid to theories with interactions involving more than one field where it is found that a matrix renormalization is necessary. Asymptotic scaling forms, in agreement with Regge theory, are derived for the elastic two-particle scattering amplitude and verified in one-loop renormalization group improved perturbation theory, corresponding to the summation of leading logs to all orders. We give explicit forms for the Regge trajectories of different scalar theories in this approximation and determine the signatures. 
  Superfield equations of motion for D=10 type IIB Dirichlet super-9-brane are obtained from the generalized action principle. The geometric equations containing fermionic superembedding equations and constraints on the generalized field strength of Abelian gauge field are separated from the proper dynamical equations and are found to contain these dynamical equations among their consequences. The set of superfield equations thus obtained involves a Spin(1,9) group valued superfield $h_\a^{~\b}$ whose leading component appears in the recently obtained simplified expression for the kappa-symmetry projector of the D9-brane. The Cayley image of this superfield coincides (on the mass shell) with the field strength tensor of the world volume gauge field characteristic for the Dirichlet brane. The superfield description of the super-9-brane obtained in this manner is known to be, on the one hand, the nonlinear (Born-Infeld) generalization of supersymmetric Yang-Mills theory and, on the other hand, the theory of partial spontaneous breaking of D=10, N=IIB supersymmetry down to D=10, N=1. 
  We argue that certain BPS states in the D3-brane probe realization of N=2 SU(2) Super-Yang-Mills theory correspond to multi-pronged strings connecting the D3-brane to the background 7-branes. This provides a physical realization of the decay of these states on the curve of marginal stability, and explains their absence in the strong coupling regime. 
  String configurations have been identified in compactified Matrix theory at vanishing string coupling. We show how the interactions of these strings are determined by the Yang-Mills gauge field on the worldsheet. At finite string coupling, this suggests the underlying dynamics is not well-approximated as a theory of strings. This may explain why string perturbation theory diverges badly, while Matrix string perturbation theory presumably has a perturbative expansion with properties similar to the strong coupling expansion of 2d Yang-Mills theory. 
  In order to construct examples for interacting quantum field theory models, the methods of euclidean field theory turned out to be powerful tools since they make use of the techniques of classical statistical mechanics.   Starting from an appropriate set of euclidean n-point functions (Schwinger distributions), a Wightman theory can be reconstructed by an application of the famous Osterwalder-Schrader reconstruction theorem. This procedure (Wick rotation), which relates classical statistical mechanics and quantum field theory, is, however, somewhat subtle. It relies on the analytic properties of the euclidean n-point functions.   We shall present here a C*-algebraic version of the Osterwalder-Scharader reconstruction theorem. We shall see that, via our reconstruction scheme, a Haag-Kastler net of bounded operators can directly be reconstructed.   Our considerations also include objects, like Wilson loop variables, which are not point-like localized objects like distributions. This point of view may also be helpful for constructing gauge theories. 
  Variational calculations using Gaussian wave functionals combined with an approximate projection on gauge invariant states are presented. We find that the energy exhibits a minimum for a wave functional centered around a non vanishing background magnetic field. We show that divergences can be removed by a renormalization of the coupling constant. The resulting expectation value of the gluon condensate is found to be in qualitative agreement with phenomenological estimates. 
  A series of lectures are given to discuss the zero-mode problem on the light-front (LF) quantization with special emphasis on the peculiar realization of the trivial vacuum, the spontaneous symmetry breaking (SSB) and the Lorentz invariance. We discuss Discrete Light-Cone Quantization (DLCQ) which was first introduced by Maskawa and Yamawaki (MY). Following MY, we present canonical formalism of DLCQ and the zero-mode constraint through which the zero mode can actually be solved away in terms of other modes,thus establishing the trivial vacuum. Due to this trivial vacuum, existence of the massless Nambu-Goldstone (NG) boson coupled to the current is guaranteed by the non-conserved charge such that Q |0> = 0 and dot{Q} ne 0. The SSB (NG phase) in DLCQ can be realized on the trivial vacuum only when an explicit symmetry-breaking mass of the NG boson m_{pi} is introduced so that the NG-boson zero mode integrated over the LF exhibits singular behavior sim 1/m_{pi}^2 in such a way that dot{Q} ne 0 in the symmetric limit m_{pi} -> 0. We also demonstrate this realization more explicitly in the linear sigma model where the role of zero-mode constraint is clarified. We fur ther point out, in disagreement with Wilson et al., that for SSB in the continuum LF theory, the trivial vacuum collapses due to the special nature of the zero mode as the accumulating point P^+ -> 0, in sharp contrast to DLCQ. Finally, we discuss the no-go theorem of Nakanishi and Yamawaki, which forbids exact LF r estriction of the field theory. Thus DLCQ as well as any other regularization on the exact LF has no Lorentz-invariant limit as the theory itself, although the Lorentz-invariant limit can be realized on the c-number quantity like S matrix which has no reference to the fixed LF. 
  The Barut--Zanghi (BZ) theory can be regarded as the most satisfactory picture of a classical spinning electron and constitutes a natural "classical limit" of the Dirac equation. The BZ model has been analytically studied in some previous papers of ours in the case of free particles. By contrast, in this letter we consider the case of external fields, and a previously found equation of the motion is generalized for a non-free spin-1/2 particle. In the important case of a spinning charge in a uniform magnetic field, we find that its angular velocity (along its circular orbit around the magnetic field direction) is slightly different from the classical "cyclotron frequency" eH/m which is expected to hold for spinless charges. As a matter of fact, the angular velocity results to depend on the spin orientation. As a consequence, the electrons with magnetic moment mu parallel to the magnetic field do rotate with a frequency greater than that of electrons endowed with mu antiparallel to H. 
  These two lectures cover some of the advances that underpin recent progress in deriving continuum solutions from the exact renormalization group. We concentrate on concepts and on exact non-perturbative statements, but in the process will describe how real non-perturbative calculations can be done, particularly within derivative expansion approximations. An effort has been made to keep the lectures pedagogical and self-contained. Topics covered are the derivation of the flow equations, their equivalence, continuum limits, perturbation theory, truncations, derivative expansions, identification of fixed points and eigenoperators, and the role of reparametrization invariance. Some new material is included, in particular a demonstration of non-perturbative renormalizability, and a discussion of ultraviolet renormalons. 
  We analyze the finite temperature chiral restoration transition of the $(D=d+1)$-dimensional Gross-Neveu model for the case of a large number of flavors and fixed total fermion number. This leads to the study of the model with a nonzero imaginary chemical potential. In this formulation of the theory, we have obtained that, in the transition region, the model is described by a chiral conformal field theory where the concepts of dimensional reduction and universality do apply due to a transmutation of statistics which makes fermions act as if they were bosons, having zero energy. This result should be generic for theories with dynamical symmetry breaking, such as Quantum Chromodynamics. 
  We review cosmological solutions of type II superstrings and M-theory, emphasizing the role of non-vanishing Ramond form backgrounds. Compactifications on flat and, more generally, maximally symmetric spatial subspaces are presented. We give a physical discussion of both inflating and subluminally expanding cosmological solutions of such theories and explore their singularity structure. An explicit example, in the context of the type IIA superstring, is constructed. We then analyze compactifications of M-theory on Ricci flat manifolds. The external part of U--duality and its relation to cosmological solutions is studied in the low energy theory. In particular, we investigate the behaviour of important cosmological properties, such as the Hubble parameters and the transition time between two asymptotic regions, under U-duality transformations. Motivated by Horava-Witten theory, we present an explicit example of manifestly U-duality covariant M--theory cosmology in a five-dimensional model resulting from compactification on a Calabi-Yau three-fold. 
  We consider field theories with sixteen supersymmetries, which includes U(N) Yang-Mills theories in various dimensions, and argue that their large N limit is related to certain supergravity solutions. We study this by considering a system of D-branes in string theory and then taking a limit where the brane worldvolume theory decouples from gravity. At the same time we study the corresponding D-brane supergravity solution and argue that we can trust it in certain regions where the curvature (and the effective string coupling, where appropriate) are small. The supergravity solutions typically have several weakly coupled regions and interpolate between different limits of string-M-theory. 
  The conformal anomaly for 4D gravity-matter theories, which are non-minimally coupled with the dilaton, is systematically studied. Special care is taken for: rescaling of fields, treatment of total derivatives, hermiticity of the system operator and choice of measure. Scalar, spinor and vector fields are taken as the matter quantum fields and their explicit conformal anomalies in the gravity-dilaton background are found. The cohomology analysis is done and some new conformal invariants and trivial terms, involving the dilaton, are obtained. The symmetry of the constant shift of the dilaton field plays an important role. The general structure of the conformal anomaly is examined. It is shown that the dilaton affects the conformal anomaly characteristically for each case: 1)[Scalar] The dilaton changes the conformal anomaly only by a new conformal invariant, $I_4$; 2)[Spinor] The dilaton does {\it not} change the conformal anomaly; 3)[Vector] The dilaton changes the conformal anomaly by three new (generalized) conformal invariants, $I_4,I_2,I_{1}$. We present some new anomaly formulae which are useful for practical calculations. Finally, the anomaly induced action is calculated for the dilatonic Wess-Zumino model. We point out that the coefficient of the total derivative term in the conformal anomaly for the 2D scalar coupled to a dilaton is ambiguous. This resolves the disagreement between calculations in refs.\cite{ENO,NO,SI97,KLV} and the result of Hawking-Bousso\cite{BH}. 
  It is found that, in addition to the conventional ones, a local approach to the relativistic quantum field theories at both zero and finite density consistent with the violation of Bell like inequalities should contain, and provide solutions to at least three additional problems, namely, 1) the statistical gauge invariance 2) the dark components of the local observables and 3) the fermion statistical block effects, base upon an asymptotic non-thermo ensemble. An application to models are presented to show the relevance of the discussions. 
  In this talk I review both accomplished results and work in progress on the use of solvable Lie algebras as an intrinsic algebraic characterization of the scalar field sector of M--theory low energy effective lagrangians. In particular I review the application of these techniques in obtaining the most general form of BPS black hole solutions. 
  The content of this paper is incorporated into hep-th/9805093 
  We identify the two dimensional AdS subsupergroup $OSp(16/2,R)$ of the M-theory supergroup $OSp(1/32,R)$ which captures the dynamics of $n$ $D0$-branes in the large $n$ limit of Matrix theory. The $Sp(2,R)$ factor in the even subgroup $SO(16) \times Sp(2,R)$ of $OSp(16/2,R)$ corresponds to the AdS extension of the Poincare symmetry of the longitudinal directions. The infinite number of $D0$-branes with ever increasing and quantized values of longitudinal momenta are identified with the Fourier modes of the singleton supermultiplets of $OSp(16/2,R)$,which consist of 128 bosons and 128 fermions. The large $n$ limit of N=16 U(n) Yang-Mills quantum mechanics which describes Matrix theory is a conformally invariant N=16 singleton quantum mechanics living on the boundary of $AdS_{2}$. We also review some of the earlier results on the spectra of Kaluza-Klein supergravity theories in relation to the recent conjecture of Maldacena relating the dynamics of $n$ $Dp$-branes to certain AdS supergravity theories. We point out the remarkable parallel between the conjecture of Maldacena and the construction of the spectra of $11-d$ and type $IIB$ supergravity theories compactified over various spheres in terms of singleton or doubleton supermultiplets of corresponding AdS supergroups. 
  The derivation of the explicit formula for the vacuum expectation value of the Wilson loop functional for an arbitrary gauge group on an arbitrary orientable two-dimensional manifold is considered both in the continuum case and on the lattice. A contribution to this quantity, coming from the space of invariant connections, is also analyzed and is shown to be similar to the contribution of monopoles. 
  We determine all SU(2) caloron solutions with topological charge one and arbitrary Polyakov loop at spatial infinity (with trace 2.cos(2.pi.omega)), using the Nahm duality transformation and ADHM. By explicit computations we show that the moduli space is given by a product of the base manifold R^3 X S^1 and a Taub-NUT space with mass M=1/sqrt{8.omega(1-2.omega)}, for omega in [0, 1/2], in units where S^1=R/Z. Implications for finite temperature field theory and string duality between Kaluza-Klein and H-monopoles are briefly discussed 
  It is shown that SU(N) gauge theory coupled to adjoint Higgs can be explicitly re-written in terms of SU(N) gauge invariant dynamical variables with local physical interactions. The resultant theory has a novel compact abelian $U(1)^{(N - 1)}$ gauge invariance. The above abelian gauge invariance is related to the adjoint Higgs field and not to the gauge group SU(N). In this abelianized version the magnetic monopoles carrying the magnetic charges of $(N-1)$ types have a natural origin and therefore appear explicitly in the partition function as Dirac monopoles along with their strings. The gauge invariant electric and magnetic charges with respect to $U(1)^{(N-1)}$ gauge groups are shown to be vectors in root and co-root lattices of SU(N) respectively. Therefore, the Dirac quantization condition corresponds to SU(N) Cartan matrix elements being integers. We also study the effect of the $\theta$ term in the abelian version of the theory. 
  In this review I discuss some basic aspects of non-perturbative string theory. The topics include test of duality symmetries based on the analysis of the low energy effective action and the spectrum of BPS states, relationship between different duality symmetries, an introduction to M- and F-theories, black hole entropy in string theory, and Matrix theory. 
  D0-brane theory on a torus with a nonvanishing B field is embedded into a string theory in the weak coupling limit. It is shown that the usual supersymmetric Yang-Mills theory on a noncommutative torus can not be the whole story. The Born-Infeld action survives the noncommutative torus limit. 
  We study supersymmetric quantum mechanics on RP_{n},SO(n),G_{2} and U(2) to examine Witten's Morse theory concretely. We confirm the simple instanton picture of the de Rham cohomology that has been given in a previous paper. We use a reflection symmetry of each theory to select the true vacuums. The number of selected vacuums agrees with the de Rham cohomology for each of the above manifolds. 
  A singular configuration of an external static vector field in the form of a magnetic string polarizes the vacuum of a second-quantized theory on the plane orthogonal to the string axis. The most general boundary conditions at the punctured singular point that are compatible with the self-adjointness of two-dimensional Dirac Hamiltonian are considered. The dependences of the induced vacuum quantum numbers on the parameter of the self-adjoint extension, on the string flux, and on the choice of irreducible representation of the matrices in (2+1)-dimensional spacetime are disscussed. 
  Effects due to fermion-vacuum polarization by an external static magnetic field are considered in a two-dimensional noncompact curved space with a nontrivial topology. An expression for the vacuun angular momentum is obtained. Like the vacuum fermion number, it proves to be dependent on the global characteristics of the field and space. 
  The effects bringing about by the finiteness of the photon mass due to the Debye screening in the monopole gas in three-dimensional compact QED are studied. In this respect, a representation of the partition function of this theory as an integral over monopole densities is derived. Dual formulation of the Wilson loop yields a new theory of confining strings, which in the low-energy limit almost coincides with the one corresponding to the case when the photon is considered to be massless, whereas in the high-energy limit these two theories are quite different from each other. The confining string mass operator in the low-energy limit is also found, and its dependence on the volume of observation is studied. 
  A simple, non-technical introduction to the pre-big bang scenario is given, emphasizing physical motivations, considerations, and consequences over formalism. 
  Certain vertex algebras and Lie algebras arising in superstring theory are investigated.   We show that the Fock space of a compactified Neveu-Schwarz superstring, i.e. a Neveu-Schwarz superstring moving on a torus, carries the structure of a vertex superalgebra with a Neveu-Schwarz element. This implies that the physical states of such a string form a Lie algebra. The same is true for the GSO-projected states. The structure of these Lie algebras is investigated in detail. In particular there is a natural invariant form on them. In case that the torus has Lorentzian signature the quotient of these Lie algebras by the kernel of this form is a generalized Kac-Moody algebra. The roots can be easily described. If the dimension of space-time is smaller than or equal to 10 we can even determine their multiplicities. 
  Since the Maxwell theory of electromagnetic phenomena is a gauge theory, it is quite important to evaluate the zero-point energy of the quantized electromagnetic field by a careful assignment of boundary conditions on the potential and on the ghost fields. Recent work by the authors has shown that, for a perfectly conducting spherical shell, it is precisely the contribution of longitudinal and normal modes of the potential which enables one to reproduce the result first due to Boyer. This is obtained provided that one works with the Lorenz gauge-averaging functional, and with the help of the Feynman choice for a dimensionless gauge parameter. For arbitrary values of the gauge parameter, however, covariant and non-covariant gauges lead to an entangled system of three eigenvalue equations. Such a problem is crucial both for the foundations and for the applications of quantum field theory. 
  We reexamine the solvable model problem of two static, fundamental quarks interacting with a SU(2) Yang-Mills field on a spatial circle, introduced by Engelhardt and Schreiber. If the quarks are at the same point, the model exhibits a quantum mechanical supersymmetry. At finite separation, the supersymmetry is explicitly broken in a way which naturally explains the geometrical nature of spectrum and state vectors of this system. 
  We discuss in this paper the behaviour of minimal models of conformal theory perturbed by the operator $\Phi_{13}$ at the boundary. Using the RSOS restriction of the sine-Gordon model, adapted to the boundary problem, a series of boundary flows between different set of conformally invariant boundary conditions are described. Generalizing the "staircase" phenomenon discovered by Al. Zamolodchikov, we find that an analytic continuation of the boundary sinh-Gordon model provides a flow interpolation not only between all minimal models in the bulk, but also between their possible conformal boundary conditions. In the particular case where the bulk sinh-Gordon coupling is turned to zero, we obtain a boundary roaming trajectory in the $c=1$ theory that interpolates between all the possible spin $S$ Kondo models. 
  We have constructed a four-fermion theory coupled to a Yang-Mills-Chern-Simons gauge field which admits static multi-vortex solutions. This is achieved through the introduction of an anomalous magnetic interation term in addition to the usual minimal coupling, and the appropriate choice of the fermion quartic coupling constant. 
  Strong-weak duality invariance can only be defined for particular sectors of supersymmetric Yang-Mills theories. Nevertheless, for full non-Abelian non-supersymmetric theories, dual theories with inverted couplings, have been found. We show that an analogous procedure allows to find the dual action to the gauge theory of gravity constructed by the MacDowell-Mansouri model plus the superposition of a $\Theta$ term. 
  We present a formulation of non-Abelian gauge theories in general axial gauges using a Wilsonian (or 'Exact') Renormalisation Group. No 'spurious' propagator divergencies are encountered in contrast to standard perturbation theory. Modified Ward identities, compatible with the flow equation, ensure gauge invariance of physical Green functions. The axial gauge $n A=0$ is shown to be a fixed point under the flow equation. Possible non-perturbative approximation schemes and further applications are outlined. 
  The role of $SL(2,IR)$ symmetry in two-dimensional gravity is investigated in the context of the extended hamiltonian formalism.   Using our results we clarify previous works on the subject. 
  A study of the center symmetric phase of SU(2) Yang Mills theory is presented. Realization of the center symmetry is shown to result from non-perturbative gauge fixing. Dictated by the center symmetry, this phase exhibits already at the perturbative level confinement like properties. The analysis is performed by investigating the dynamics of the Polyakov loops. The ultralocality of these degrees of freedom implies significant changes in the vacuum structure of the theory. General properties of the confined phase and of the transition to the deconfined phase are discussed. Perturbation theory built upon the vacuum of ultralocal Polyakov loops is presented and used to calculate, via the Polyakov loop correlator, the static quark-antiquark potential. 
  We review some aspects of the interplay between the dynamics of branes in string theory and the classical and quantum physics of gauge theories with different numbers of supersymmetries in various dimensions. 
  We show that the resolution of moduli space of ideal instantons parameterizes the instantons on non-commutative $\IR^{4}$. This moduli space appears as a Higgs branch of the theory of $k$ $D0$-branes bound to $N$ $D4$-branes by the expectation value of the $B$ field. It also appears as a regularized version of the target space of supersymmetric quantum mechanics arising in the light cone description of $(2,0)$ superconformal theories in six dimensions. 
  We argue that by taking a limit of SYM on a non-commutative torus one can obtain a theory on non-compact space with a finite non-locality scale. We also suggest that one can also obtain a similar generalization of the (2,0) field theory in 5+1 dimensions, and that the DLCQ of this theory is known. 
  With the third order Monge-Amp\`ere equation as an example, we show that there exists an infinite number of nonlocal conserved charges associated with the Witten-Dijkgraaf-Verlinde-Verlinde equations. A general prescription for the construction of these charges is given and the charge algebra is calculated bringing out various other interesting features associated with such systems. 
  An expansion in the number of spatial covariant derivatives is carried out to compute the $\zeta$-function regularized effective action of 2+1-dimensional fermions at finite temperature in an arbitrary non-Abelian background. The real and imaginary parts of the Euclidean effective action are computed up to terms which are ultraviolet finite. The expansion used preserves gauge and parity symmetries and the correct multivaluation under large gauge transformations as well as the correct parity anomaly are reproduced. The result is shown to correctly reproduce known limiting cases, such as massless fermions, zero temperature, and weak fields as well as exact results for some Abelian configurations. Its connection with chiral symmetry is discussed. 
  We show that the vacuum expectation value of the stress-energy tensor of a scalar particle on the background of a spherical gravitational shock wave does not give a finite expression in second order perturbation theory, contrary to the case seen for the impulsive wave. No infrared divergences appear at this order. This result shows that there is a qualitative difference between the shock and impulsive wave solutions which is not exhibited in first order. 
  We review recent developments in the theory of supermembranes and their relation to matrix models. 
  We fix the long-standing ambiguity in the 1-loop contribution to the mass of a 1+1-dimensional supersymmetric soliton by adopting a set of boundary conditions which follow from the symmetries of the action and which depend only on the topology of the sector considered, and by invoking a physical principle that ought to hold generally in quantum field theories with a topological sector: for vanishing mass and other dimensionful constants, the vacuum energies in the trivial and topological sectors have to become equal. In the two-dimensional N=1 supersymmetric case we find a result which for the supersymmetric sine-Gordon model agrees with the known exact solution of the S-matrix but seems to violate the BPS bound. We analyze the nontrivial relation between the quantum soliton mass and the quantum BPS bound and find a resolution. For N=2 supersymmetric theories, there are no one-loop corrections to the soliton mass and to the central charge (and also no ambiguities) so that the BPS bound is always saturated. Beyond 1-loop there are no ambiguities in any theory, which we explicitly check by a 2-loop calculation in the sine-Gordon model. 
  We point out that the claims made in the paper ``Non-thermalizability of a Quantum Field Theory'' (hep-th/9802008) by C. R. Hagen are irrelevant to our recent results concerning large gauge invariance of the effective action in thermal QED. 
  We calculate the density of states of the 2+1 dimensional BTZ black hole in the micro- and grand-canonical ensembles. Our starting point is the relation between 2+1 dimensional quantum gravity and quantised Chern-Simons theory. In the micro-canonical ensemble, we find the Bekenstein--Hawking entropy by relating a Kac-Moody algebra of global gauge charges to a Virasoro algebra with a classical central charge via a twisted Sugawara construction. This construction is valid at all values of the black hole radius. At infinity it gives the asymptotic isometries of the black hole, and at the horizon it gives an explicit form for a set of deformations of the horizon whose algebra is the same Virasoro algebra. In the grand-canonical ensemble we define the partition function by using a surface term at infinity that is compatible with fixing the temperature and angular velocity of the black hole. We then compute the partition function directly in a boundary Wess-Zumino-Witten theory, and find that we obtain the correct result only after we include a source term at the horizon that induces a non-trivial spin-structure on the WZW partition function. 
  The quantum Wess-Zumino-Witten $\to$ Liouville reduction is formulated using the phase space path integral method of Batalin, Fradkin, and Vilkovisky, adapted to theories on compact two dimensional manifolds. The importance of the zero modes of the Lagrange multipliers in producing the Liouville potential and the WZW anomaly, and in proving gauge invariance, is emphasised. A previous problem concerning the gauge dependence of the Virasoro centre is solved. 
  The phase space path integral Wess-Zumino-Witten $\to$ Toda reductions are formulated in a manifestly conformally invariant way. For this purpose, the method of Batalin, Fradkin, and Vilkovisky, adapted to conformal field theories, with chiral constraints, on compact two dimensional Riemannian manifolds, is used. It is shown that the zero modes of the Lagrange multipliers produce the Toda potential and the gradients produce the WZW anomaly. This anomaly is crucial for proving the Fradkin-Vilkovisky theorem concerning gauge invariance. 
  In a short review of recent work, we discuss the general problem of constructing the actions of new conformal field theories from old conformal field theories. Such a construction follows when the old conformal field theory admits new conformal stress tensors in its chiral algebra, and it turns out that the new conformal field theory is generically a new spin-two gauge theory. As an example we discuss the new spin-two gauged sigma models which arise in this fashion from the general conformal non-linear sigma model. 
  It is well-known that the low energy string theory admits a non-singular solitonic super five-brane solution which is the magnetic dual to the fundamental string solution. By using the symmetry of the type IIB string theory, we construct an SL(2, Z) multiplet of magnetically charged super five-branes starting from this solitonic solution. These solutions are characterized by two integral three-form charges $(q_1, q_2)$ and are stable when the integers are coprime. We obtain an expression for the tension of these $(q_1, q_2)$ five-branes as envisaged by Witten. The SL(2, Z) multiplets of black strings and black fivebranes and the existence of similar magnetic dual solutions of strings in type II string theory in $D < 10$ have also been discussed. 
  We derive Ward-Takahashi identities including composite fields in Abelian gauge theories and the matching condition between the elementary field description and the composite field description. With these we develop an approach to dynamical symmetry breaking in Abelian gauge theories including the study of the dynamically generated masses of the gauge boson, the fermions and the composite Higgs field. The Cornwall-Norton, Jackiw-Johnson and Schwinger models are taken as examples of the application. The obtained gauge boson masses are in agreement with the existing results. In this appraoch, we are able to further obtain new results for the mass of the composite Higgs boson and the goldstone boson decay constant. 
  We study the two-dimensional Eguchi-Kawai model as a toy model of the IIB matrix model, which has been recently proposed as a nonperturbative definition of the type IIB superstring theory. While the planar limit of the model is known to reproduce the two-dimensional Yang-Mills theory, we find through Monte Carlo simulation that the model allows a different large N limit, which can be considered as the double scaling limit in matrix models. 
  Since the appearance of the paper by Bilal & al. in 1991, it has been widely assumed that W-algebras originating from the Hamiltonian reduction of an SL(n,C)-bundle over a Riemann surface give rise to a flat connection, in which the Beltrami differential may be identified. In this Letter, it is shown that the use of the Beltrami parametrisation of complex structures on a compact Riemann surface over which flat complex vector bundles are considered, allows to construct the above mentioned flat connection. It is stressed that the modulus of the Beltrami differential is of necessity less than one, and that solutions of the so-called Beltrami equation give rise to an orientation preserving smooth change of local complex coordinates. In particular, the latter yields a smooth equivalence between flat complex vector bundles. The role of smooth diffeomorphisms which induce equivalent complex structures is specially emphasized. Furthermore, it is shown that, while the construction given here applies to the special case of the Virasoro algebra, the extension to flat complex vector bundles of arbitrary rank does not provide "generalizations" of the Beltrami differential usually considered as central objects for such non-linear symmetries. 
  We derive the string representation of the Abelian Higgs theory in which dyons are condensed. It occurs that in such representation the topological interaction exists in the expectation value of the Wilson loop. Due to this interaction the dynamics of the string spanned on the Wilson loop is non-trivial. 
  We derive a long distance effective action for space-time coordinates from a IIB matrix model. It provides us an effective tool to study the structures of space-time. We prove the finiteness of the theory for finite $N$ to all orders of the perturbation theory. Space-time is shown to be inseparable and its dimensionality is dynamically determined. The IIB matrix model contains a mechanism to ensure the vanishing cosmological constant which does not rely on the manifest supersymmetry. We discuss possible mechanisms to obtain realistic dimensionality and gauge groups from the IIB matrix model. 
  In complete analogy with the Beltrami parametrization of complex structures on a (compact) Riemann surface, we use in this paper the Kodaira-Spencer deformation theory of complex structures on a (compact) complex manifold of higher dimension. According to the Newlander-Nirenberg theorem, a smooth change of local complex coordinates can be implemented with respect to an integrable complex structure parametrized by a Beltrami differential. The question of constructing a local field theory on a complex compact manifold is addressed and the action of smooth diffeomorphisms is studied in the BRS algebraic approach. The BRS cohomology for the diffeomorphisms gives rise to generalized Gel'fand-Fuchs cocycles provided that the Kodaira-Spencer integrability condition is satisfied. The diffeomorphism anomaly is computed and turns out to be holomorphically split as in the bidimensional Lagrangian conformal models. Moreover, its algebraic structure is much more complicated than the one proposed in a quite recent paper hep-th/9606082 (Nucl. Phys. B484 (1997) 196). 
  In this article we review the quantization of the Dirac-field on a curved spacetime. For that purpose we describe the construction of the local observable algebras in the algebraic approach to quantum field theory. Among the possible states we single out the so called Hadamard-states, which are the ones relevant for physics. Finally, as an example, we give a definition for an adiabatic vacuum state of the Dirac-field on a Robertson-Walker spacetime. We believe that these states are physical in the sense that they have the singularity structure of Hadamard-form, although we cannot give a formal proof of this conjecture. 
  Using the boundary state formalism, we perform a microscopic string analysis of the interaction between two D-branes and provide a local interpretation for the R-R force in the D0-D8 brane system. To do so, we construct BRST invariant vertex operators for the massless R-R states in the asymmetric picture that are proportional to potentials rather than field strengths. The Hilbert space of such R-R states contains combinations of two vectors that decouple from all physical amplitudes, even in the presence of boundaries. Identifying these vectors, we remove the null states and recover duality relations among R-R potentials. If we specify to the D0-D8 brane system, this mechanism implies that the R-R 1-form state has a non-zero overlap with both the D0-brane and the D8-brane, thus explaining from a local point of view the non-vanishing R-R contribution in the interaction for the D0-D8 brane system and those related to it by duality. 
  Wilsonian effective actions are interpreted as free energies in ensembles with prescribed field expectation values and prescribed connected two-point functions. Since such free energies are directly obtained from two-particle-irreducible functionals, it follows that Wilsonian effective actions satisfy elementary perturbative consistency conditions, and non-perturbative convexity conditions. In particular, the exact determination of a Wilsonian action by other means (e.g. supersymmetry) allows one to extract restrictions on the particular cutoff scheme and field reparametrization that would lead to such a Wilsonian action from an underlying microscopic action. 
  The construction of $SL(2,Z)$ invariant amplitudes that generalizethe Virasoro amplitude is investigated in detail. We describe a number of mathematical properties that characterize the simplest example, and present pieces of evidence that it represents the tree-level four-graviton scattering amplitude in membrane theory on ${\bf R}^9\times T^2$ in the limit that the torus area goes to zero. In particular, we show that the poles of the $S$-dual amplitude are in precise correspondence with the states of membrane theory that survive in the type IIB limit. These are shown to be the states that span the Cartan subspaces of area preserving diffeomorphisms of the 2-torus; all other states become infinitely massive, and membrane world-volume theory acquires the structure of a free theory. 
  According to the Matrix theory proposal of Banks, Fischler, Shenker and Susskind M-theory in the infinite momentum frame is the large N limit of super Yang-Mills theory in a flat background. To address some physical issues of classical gravity such as gravitational collapse and cosmological expansion we consider an extension of the BFSS proposal by defining M-theory in curved space as the large N limit of super Yang-Mills theory in a curved background. Motivations and possible implications of this extension are discussed. 
  We investigate the low energy dynamics of N=1 supersymmetric SO(N) gauge theories with a single symmetric tensor matter field. These theories exhibit non-trivial matching of global 't Hooft anomalies at the origin of moduli space. We argue that their quantum moduli spaces possess distinct Higgs and confining branches which touch at the origin in an interacting non-Abelian Coulomb phase. The matching of anomalies between microscopic degrees of freedom and colorless moduli therefore appears to be coincidental. We discuss a formal mathematical relation between the SO(N) model and an analogous Sp(2N) theory with a single antisymmetric matter field which provides an explanation for the anomaly matching coincidence. 
  The heterotic string compactified on an (n-1)-dimensional elliptically fibered Calabi-Yau Z-->B is conjectured to be dual to F-theory compactified on an n-dimensional Calabi-Yau X-->B, fibered over the same base with elliptic K3 fibers. In particular, the moduli of the two theories should be isomorphic. The cases most relevant to the physics are n=2, 3, 4, i.e. the compactification is to dimensions d=8, 6 or 4 respectively. Mathematically, the richest picture seems to emerge for n=3, where the moduli space involves an analytically integrable system whose fibers admit rather different descriptions in the two theories. The purpose of this talk is to review some of what is known and what is not yet known about this conjectural isomorphism. Some of the underlying mathematics of principal bundles on elliptic fibrations is reviewed in the accompanying Taniguchi talk (hep-th/9802094). 
  In this talk we discuss the description of the moduli space of principal G-bundles on an elliptic fibration X-->S in terms of cameral covers and their distinguished Prym varieties. We emphasize the close relationship between this problem and the integrability of Hitchin's system and its generalizations. The discussion roughly parallels that of [D2], but additional examples are included and some important steps of the argument are illustrated. Some of the applications to heterotic/F-theory duality were described in the accompanying ICMP talk (hep-th/9802093). 
  Recently a non-supersymmetric analog of Veneziano-Yankielowicz (VY) effective Lagrangian has been proposed and applied for the analysis of the theta dependence in pure Yang-Mills theory. This effective Lagrangian is similar in many respects to the VY construction and, in particular, exhibits a kind of low energy holomorphy which is absent in the full YM theory. Here we incorporate a heavy fermion into this effective theory by using the "integrating in" technique. We find that, in terms of this extended theory, holomorphy of the effective Lagrangian for pure YM theory naturally implies a holomorphic dependence on the heavy fermion mass.   It is shown that this analysis fixes, under certain assumptions, a dimensionless parameter which enters the effective Lagrangian and determines the number of nondegenerate vacuum sectors in pure YM theory. We also compare our results for the vacuum structure and theta dependence to those obtained recently by Witten on the basis of AdS/CFT correspondence. 
  An N=1--supersymmetric version of the Cremmer-Scherk-Kalb-Ramond model with non-minimal coupling to matter is built up both in terms of superfields and in a component-field formalism. By adopting a dimensional reduction procedure, the N=2--D=3 counterpart of the model comes out, with two main features: a genuine (diagonal) Chern-Simons term and an anomalous magnetic moment coupling between matter and the gauge potential. 
  Arbitrary spin free massless fermionic fields corresponding to mixed symmetry representations of the \hbox{$SO(d-1)$} compact group and propagating in even $d$-dimensional anti-de Sitter spacetime are investigated. Free wave equations of motion, subsidiary conditions and the corresponding gauge transformations for such fields are proposed. The lowest eigenvalues of the energy operator for the massless fields and the gauge parameter fields are derived. The results are formulated in $SO(d-1,2)$ covariant form as well as in terms of intrinsic coordinates. 
  To solve the quantum-mechanical problem the procedure of mapping onto linear space $W$ of generators of the (sub)group violated by given classical trajectory is formulated. The formalism is illustrated by the plane H-atom model. The problem is solved noting conservation of the Runge-Lentz vector $n$ and reducing the 4-dimensional incident phase space $T$ to the 3-dimensional linear subspace $W=T^* V\times R^1$, where $T^* V$ is the (angular momentum ($l$) - angle ($\vp$)) phase space and $R^1 =n$. It is shown explicitly that (i) the motion in $R^1$ is pure classical as the consequence of the reduction, (ii) motion in the $\vp$ direction is classical since the Kepler orbits are closed independently from initial conditions and (iii) motion in the $l$ direction is classical since all corresponding quantum corrections are defined on the bifurcation line ($l=\infty$) of the problem. In our terms the H-atom problem is exactly quasiclassical and is completely integrable by this reasons. 
  The perturbative prepotential and the K\"ahler metric of the vector multiplets of the N=2 effective low-energy heterotic strings is calculated directly in N=1 six-dimensional toroidal compactifications of the heterotic string vacua. This method provides the solution for the one loop correction to the N=2 vector multiplet prepotential for compactifications of the heterotic string for any rank three and four models, as well for compactifications on $K_3 \times T^2$. In addition, we complete previous calculations, derived from string amplitudes, by deriving the differential equation for the third derivative of the prepotential with respect of the usual complex structure U moduli of the $T^2$ torus. Moreover, we calculate the one loop prepotential, using its modular properties, for N=2 compactifications of the heterotic string exhibiting modular groups similar with those appearing in N=2 sectors of N=1 orbifolds based on non-decomposable torus lattices and on N=2 supersymmetric Yang-Mills. 
  The intercept of the odderon trajectory is derived, by finding the spectrum of the second integral of motion of the three reggeon system in high energy QCD. When combined with earlier solution of the appropriate Baxter equation, this leads to the determination of the low lying states of that system. In particular, the energy of the lowest state gives the intercept of the odderon alpha_O(0)=1-0.2472 alpha_s N_c/pi. 
  We study dynamical symmetry breaking in three-dimensional QED with a Chern-Simons (CS) term, considering the screening effect of $N$ flavor fermions. We find a new phase of the vacuum, in which both the fermion mass and a magnetic field are dynamically generated, when the coefficient of the CS term $\kappa$ equals $N e^2/4 \pi$. The resultant vacuum becomes the finite-density state half-filled by fermions. For $\kappa=N e^2/2 \pi$, we find the fermion remains massless and only the magnetic field is induced. For $\kappa=0$, spontaneous magnetization does not occur and should be regarded as an external field. 
  A new relation between two-dimensional conformal field theories and three-dimensional topologically massive gauge theories is found, where the dynamical nature of the 3d theory is ultimately important.  It is shown that the those primary states in CFT which have non-unitary descendants correspond in the 3d theory to supercritical charges and cause vacuum instability. It is also shown that logarithmic operators separating the unitary sector from a non-unitary one correspond to an exact zero energy ground state in which case the 3d Hamiltonian naturally has a Jordan structure. 
  Supersymmetric N=1, D=4 string vacua are known to be finite in perturbation theory. However, the effective low energy D=4, N=1 field theory lagrangian does not yield in general finite theories. In this note we present the first (to our knowledge) such an example. It may be constructed in three dual ways: i) as a $Z_3$, SO(32) heterotic orbifold; ii) as a Type -IIB, $Z_3$ orientifold with only ninebranes and a Wilson line or iii) as a Type-IIB, $Z_6$ orientifold with only fivebranes. The gauge group is $SU(4)^3$ with three chiral generations. Although chiral, a subsector of the model is continuosly connected to a model with global N=4 supersymmetry. From the $Z_6$, Type IIB orientifold point of view the above connection may be understood as a transition of four dynamical fivebranes from a fixed point to the bulk. The N=1 model is thus also expected to be S-dual. We also remark that, using the untwisted dilaton and moduli fields of these constructions as spurion fields, yields soft SUSY-breaking terms which preserve finiteness even for N=0. 
  One discusses here the connection between \sigma-model gauge anomalies and the existence of a connection with torsion that does not flatten the Ricci tensor of the target manifold, by considering a number of non-symmetric coset spaces. The influence of an eventual anisotropy on a certain direction of the target manifold is also contemplated. 
  We treat in this paper non-linear sigma models such as $CP^1$-model, $QP^1$-model and etc, in 1+2 dimensions. For submodels of such ones we definitely construct an infinite number of nontrivial conserved currents. Our result is a generalization of that of authors (Alvarez, Ferreira and Guillen). 
  In this note, we first obtain the decomposition of the non-relativistic field velocity into the classical part (i.e., the velocity w=p/m OF the center-of-mass (CM), and the so-called quantum part (i.e., the velocity V of the motion IN the CM frame (namely, the internal spin-motion or Zitterbewegung), these two parts being orthogonal. Our starting point is the Pauli current. Then, by inserting such a composite expression of the velocity into the kinetic energy term of the non-relativistic newtonian lagrangian, we get the appearance of the so-called "quantum potential" (which makes the difference between classical and quantum behaviour) as a pure consequence of the internal motion. Such a result carries further evidence about the possibility that the quantum behaviour of micro-systems be a direct consequence of the fundamental existence of spin. 
  We discuss higher-dimensional gravitational instantons by studying appropriate self-duality equations for the spin connection. In seven and in eight dimensions, the corresponding spaces admit a covariantly constant spinor and have holonomies in G_2 and Spin(7), respectively. We find a non-compact solution to the self-duality equations in eight dimensions in which the self-dual space has an elliptically-fibered structure. 
  We investigate the self-dual Yang-Mills gauge configurations on $R^3\times S^1$ when the gauge symmetry SU(2) is broken to U(1) by the Wilson loop. We construct the explicit field configuration for a single instanton by the Nahm method and show that an instanton is composed of two self-dual monopoles of opposite magnetic charge. We normalize the moduli space metric of an instanton and study various limits of the field configuration and its moduli space metric. 
  We suggest a means of obtaining certain Green's functions in 3+1-dimensional ${\cal N} = 4$ supersymmetric Yang-Mills theory with a large number of colors via non-critical string theory. The non-critical string theory is related to critical string theory in anti-deSitter background. We introduce a boundary of the anti-deSitter space analogous to a cut-off on the Liouville coordinate of the two-dimensional string theory. Correlation functions of operators in the gauge theory are related to the dependence of the supergravity action on the boundary conditions. From the quadratic terms in supergravity we read off the anomalous dimensions. For operators that couple to massless string states it has been established through absorption calculations that the anomalous dimensions vanish, and we rederive this result. The operators that couple to massive string states at level $n$ acquire anomalous dimensions that grow as $2\left (n g_{YM} \sqrt {2 N} )^{1/2}$ for large `t Hooft coupling. This is a new prediction about the strong coupling behavior of large $N$ SYM theory. 
  A natural combined model of the Kerr spinning particle and superparticle is obtained leading to a non-trivial super black hole solution. By analogue with complex structure of the Kerr solution we perform a supershift on the Kerr geometry, and then select a "body"-submanifold of superspace that yields a non-trivial supergeneralization of the Kerr metric with a nonlinear realization of (2,0)-supersymmetry. For the known parameters of spinning particles this "black hole" is to be in a specific state without horizons and very far from extreme. The naked Kerr singular ring has to be hidden inside a rotating superconducting disk, built of a supermultiplet of matter fields. Stringy wave excitations of the Kerr singular ring (traveling waves) yield an extra axial singular line modulated by de Broglie periodicity. 
  The generic observable sector gaugino mass in the weakly-coupled heterotic string compactified to four dimensions by the Scherk-Schwarz scheme (together with hidden sector gaugino condensation inducing the super-Higgs effect with a vanishing cosmological constant) is shown to be non-zero at tree level, being of the order of the gravitino mass, modulo reasonable assumptions regarding the magnitude of the condensate and the Scherk-Schwarz mass parameters 
  The correlation of the fractionally represented hypercharge group with the isospin and colour group in the standard model determines as faithfully represented internal group the quotient group ${\U(1)\x\SU(2)\x\SU(3)\over\Z_2\x\Z_3}$. The discrete cyclic central abelian-nonabelian internal correlation involved is considered with respect to its consequences for the representations by the standard model fields, the electroweak mixing angle and the symmetry breakdown. There exists a further discrete $\Z_2$-correlation between chirality and Lorentz properties and also a continuous $\U(1)$-external-internal one between hyperisospin and chirality. 
  For N=1 Super-Yang-Mills theory we generalize the effective average action Gamma_k in a manifest supersymmetric way using the superspace formalism. The exact evolution equation for Gamma_k is derived and, introducing as an application a simple truncation, the standard one-loop beta-function of N=1 SYM theory is obtained. 
  We show how the topological susceptibility in the Minkowskian theory of QCD is related to the corresponding quantity in the Euclidean theory, which is measured on the lattice. We discuss both the zero-temperature case (T = 0) and the finite-temperature case (T > 0). It is shown that the two quantities are equal when T = 0, while the relation between them is much less trivial when T > 0. The possible existence of ``Kogut-Susskind poles'' in the matrix elements of the topological charge density between states with equal four-momenta turns out to invalidate the equality of these two quantities in a strict sense. However, an equality relation is recovered after one re-defines the Minkowskian topological susceptibility by using a proper infrared regularization. 
  We carry out the extension of the Ostrogradski method to relativistic field theories. Higher-derivative Lagrangians reduce to second differential-order with one explicit independent field for each degree of freedom. We consider a higher-derivative relativistic theory of a scalar field and validate a powerful order-reducing covariant procedure by a rigorous phase-space analysis. The physical and ghost fields appear explicitly. Our results strongly support the formal covariant methods used in higher-derivative gravity. 
  Recently, Maldacena proposed that the large N limit of the N=4 supersymmetric gauge theory in four dimensions with U(N) gauge group is dual to the type IIB superstring theory on AdS_5 x S^5. We use this proposal to study the spectrum of the large N gauge theory on R x S^3 in a low energy regime. We find that the spectrum is discrete and evenly spaced, and the number of states at each energy level is smaller than the one predicted by the naive extrapolation of the Bekenstein-Hawking formula to the low energy regime. We also show that the gauge theory describes a region of spacetime behind the horizon as well as the region in front. 
  It is argued that the existence of constant dilaton field solutions is a generic feature of string-inspired dilaton gravity. Such solutins arise in the extreme limit of black hole metrics. It is shown that in a strong coupling region quantum effects give rise to two horizons in thermal equilibrium that has no classical counterpart. 
  The local BRST cohomology of the gauged non-linear sigma model on a group manifold is worked out for any Lie group G. We consider both, the case where the gauge field is dynamical and the case where it has no kinetic term (G/G topological theory). Our results shed a novel light on the problem of gauging the WZW term as well as on the nature of the topological terms introduced a few years ago by De Wit, Hull and Rocek. We also consider the BRST cohomology of the rigid symmetries of the ungauged model and recover the results of D'Hoker and Weinberg on the most general effective actions compatible with the symmetries. 
  We study the appearance of induced parity-violating magnetic moment, in the presence of external magnetic fields, for even-number of fermion species coupled to dynamical fields in three dimensions. Specifically, we use a SU(2)xU(1) gauge model for dynamical gauge symmetry breaking, which is also proposed recently as a field theoretical model for high-temperature superconductors. By decomposing the fermionic degrees of freedom in terms of Landau levels, we show that, in the effective theory with the lowest Landau levels, a parity-violating magnetic moment interaction is induced by the higher Landau levels when the fermions are massive. The possible relevance of this result for a recently observed phenomenon in high-temperature superconductors is also discussed. 
  We propose a solution to the problem of renormalizing light-cone Hamiltonian theories while maintaining Lorentz invariance and other symmetries. The method uses generalized Pauli--Villars regulators to render the theory finite. We discuss the method in the context of Yukawa theory at one loop and for a soluble model in 3+1 dimensions. The model is studied nonperturbatively. Numerical results obtained with use of discretized light-cone quantization, special integration weighting factors, and the complex symmetric Lanczos diagonalization algorithm compare well with the analytic answers. 
  The block-orthogonal generalization of the Majumdar-Papapetrou type solutions for the sigma-model studied earlier are obtained and corresponding solutions with p-branes are considered. The existence of solutions and the number of independent harmonic functions is defined by the matrix of scalar products of vectors $U^s$, governing the sigma-model target space metric. For orthogonal $U^s$, when target space is a symmetric homogeneous space, the solutions reduce to the previous ones. Two special classes of solutions with $U^s$ related to finite dimensional Lie algebras and hyperbolic (Kac-Moody) algebras are singled out and investigated. The affine Cartan matrices do not arise in the scheme under consideration. Some examples of obtained solutions and intersection rules for D=11 supergravity, related D=12 theory and extending them $B_D$-models are considered. For special multicenter solutions criterions for the existence of horizon and curvature singularity are found. 
  This paper deals with classical solutions of the modified chiral model on $R^{2+1}$. Such solutions are shown to correspond to products of various factor which we call time-dependent unitons. Then the problem of solving the system of second-order partial differential equations for the chiral field is reduced to solving a sequence of systems of first-order partial differential equations for the unitons. 
  We obtain the distribution functions for anyonic excitations classified into equivalence classes labeled by Hausdorff dimension, $h$ and as an example of such anyonic systems, we consider the collective excitations of the Fractional Quantum Hall Effect (FQHE). 
  The normal mode spectra of multiskyrmions play a key role in their quantisation. We present a general method capable of predicting all the low-lying vibrational modes of known minimal energy multiskyrmions. In particular, we explain the origin of the higher multipole breathing modes, previously observed but not understood. We show how these modes may be classified according to the symmetry group of the static solution. Our results provide strong hints that the N-skyrmion moduli space, for N>3, may have a richer structure than previously thought, incorporating 8N-4 degrees of freedom. 
  We study supersymmetric SU(5) chiral gauge theories with 2 fields in the 10 representation, $2+N_F$ fields in the $\bar{5}$ representation and $N_F$ fields in the 5 representation, for $N_F=0,1,2$. With a suitable superpotential, supersymmetry is shown to be broken dynamically for each of these values of $N_F$. We analyze the calculable limit for the model with $N_F=0$ in detail, and determine the low energy effective sigma model in this case. For $N_F=1$ we find the quantum moduli space, and for $N_F=2$ we construct the s--confining potential. 
  We investigate representations of the conformal group that describe "massless" particles in the interior and at the boundary of anti-de Sitter space. It turns out that massless gauge excitations in anti-de Sitter are gauge "current" operators at the boundary. Conversely, massless excitations at the boundary are topological singletons in the interior. These representations lie at the threshold of two "unitary bounds" that apply to any conformally invariant field theory. Gravity and Yang-Mills gauge symmetry in anti-De Sitter is translated to global translational symmetry and continuous $R$-symmetry of the boundary superconformal field theory. 
  Counting the contribution rate of a world-line formula to Feynman diagrams in $\phi^3$ theory, we explain the idea how to determine precise combinatorics of Bern-Kosower-like amplitudes derived from a bosonic string theory for $N$-point two-loop Feynman amplitudes. In this connection we also present a method to derive simple and compact world-line forms for the effective action. 
  An open string in four dimensions is supplemented by forty four Majorana fermions. The fermions are grouped in such a way that the resulting action is supersymmetric. The super-Virasoro algebra is constructed and closed by the use of Jacobi identity. The tachyonic ground state decouples from the physical states. After a GSO projection, the resulting physical mass spectrum is shown to be $\alpha' M_n^2 =n$ where $n=0, 1, 2, ...$. There are fermions and bosons in each mass level. The internal symmetry group of the string breaks to $SU(3) \times SU(2)\times U(1) \times U(1)$. 
  Since the subject of noncommutative geometry is now entering maturity, we felt there is need for presentation of the material at an undergraduate course level. Our review is a zero order approximation to this project. Thus, the present paper attempts to offer some motivations and mathematical prerequisites for a deeper study or at least to serve as support in glancing at recent results in theoretical physics. 
  We organize and review some material from various sources about prepotentials, Riemann surfaces and kernels, WDVV, and the renormalization group, provide some further connections and information, and indicate some directions and problems. 
  A one parameter generalization of Ward's chiral model in 2+1 dimensions is given. Like the original model the present one is integrable and possesses a positive-definite and conserved energy and $y$-momentum. The details of the scattering depend on the value of the parameter of the generalisation. 
  We construct the quasi-classical approximation of the form factors in finite volume using the separation of variables. The latter is closely related to the Baxter equation. 
  We find the general expression for the open superstring partition function on the annulus in a constant abelian gauge field background and at finite temperature. We use the approach based on Green-Schwarz string path integral in the light-cone gauge and compare it with NSR approach. We discuss the super Yang-Mills theory limit and mention some D-brane applications. 
  A contour gauge of general type is analysed where 1-form (vector potential) is expressed as a contour integral of the 2-form (field strength) along an arbitrary contour $C$. For a special class of contours the gauge condition reduces to $k_{\mu}(x) A_{\mu}(x) = 0 $ where $k_{\mu}(x)$ is a tangent vector to the contour $C$. A simple proof of the nonabelian Stokes theorem is given demonstrating the advantage of the gauge. 
  We obtain for any spin, $s$, the Hausdorff dimension, $h_{i}$, for fractional spin particles and we discuss the connection between this number, $h_{i}$, and the Chern-Simons potential. We also define the topological invariants, $W_s$, in terms of the statistics of these particles. 
  The effective action for local composite operators in $QED_3$ is considered. The effective potential is calculated in leading order in $1/N_f$ ($N_f$ is the number of fermion flavors) and used to describe the features of the phase transition at $N_f=N_{\rm cr}$, $3<N_{\rm cr}<5$. It is shown that this continuous phase transition satisfies the criteria of the conformal phase transition, considered recently in the literature. In particular, there is an abrupt change of the spectrum of light excitations at the critical point, although the phase transition is continuous, and the structure of the equation for the divergence of the dilatation current is essentially different in the symmetric and nonsymmetric phases. The connection of this dynamics with the dynamics in $QCD_4$ is briefly discussed. 
  Existence of breather (spatially localized, time periodic, oscillatory) solutions of the topological discrete sine-Gordon (TDSG) system, in the regime of weak coupling, is proved. The novelty of this result is that, unlike the systems previously considered in studies of discrete breathers, the TDSG system does not decouple into independent oscillator units in the weak coupling limit. The results of a systematic numerical study of these breathers are presented, including breather initial profiles and a portrait of their domain of existence in the frequency-coupling parameter space. It is found that the breathers are uniformly qualitatively different from those found in conventional spatially discrete systems. 
  Supergravities in four and higher dimensions are reviewed. We discuss the action and its local symmetries of N=1 supergravity in four dimensions, possible types of spinors in various dimensions, field contents of supergravity multiplets, non-compact bosonic symmetries, non-linear sigma models, duality symmetries of antisymmetric tensor fields and super p-branes. (An expanded version of a review talk at YITP workshop on Supersymmetry, 27 - 30 March, 1996) 
  We consider N=1 two-dimensional (2d) dilatonic supergravity (SG), 2d dilatonic SG obtained by dimensional reduction from N=1 four-dimensional (4d) SG, N=2 2d dilatonic SG and string-inspired 4d dilatonic SG. For all the theories, the corresponding action on a bosonic background is constructed and the interaction with $N$ (dilatonic) Wess-Zumino (WZ) multiplets is presented. Working in the large-N approximation, it is enough to consider the trace anomaly induced effective action due to dilaton-coupled conformal matter as a quantum correction (for 2d models s-waves approximation is additionally used). The equations of motion for all such models with quantum corrections are written in a form convenient for numerical analysis. Their solutions are numerically investigated for 2d and 4d Friedmann-Robertson-Walker (FRW) or 4d Kantowski-Sacks Universes with a time-dependent dilaton via exponential dilaton coupling. The evolution of the corresponding quantum cosmological models is given for different choices of initial conditions and theory parameters. In most cases we find quantum singular Universes. Nevertheless, there are examples of Universe non-singular at early times. Hence, it looks unlikely that quantum matter back reaction on dilatonic background (at least in large $N$ approximation) may really help to solve the singularity problem. 
  Exact black hole solutions of the five dimensional heterotic $S$-$T$-$U$ model including all perturbative quantum corrections and preserving $1/2$ of $N=2$ supersymmetry are studied. It is shown that the quantum corrections yield a bound on electric charges and harmonic functions of the solutions. 
  After a short introduction on Clifford algebras of polynomials, we give a general method of constructing a matrix representation. This process of linearization leads naturally to two fundamental structures: the generalized Clifford algebra (GCA) and the generalized Grassmann algebra (GGA) which are studied. Then, it is proved that if we equip the GGA with a differential structure, we obtain the $q-$deformed Heisenberg algebra or the $q-$oscillators. Finally, it is shown that the $q-$deformed Heisenberg algebra is the basic tool to define an adapted superspace leading to the extension of supersymmetry called fractional supersymmetry of order $F$ (FSUSY), $F=2$ corresponding to the usual supersymmetry. Local FSUSY in one dimension is then contructed in the world-line formalism, and an extension of the Dirac equation is obtained. In two dimensions, it turns out that FSUSY is a conformal field theory and in addition to the stress energy tensor, a supercurrent of conformal weight $1+1/F$, which generates a symmetry between the primary fields of conformal weight $(0,1, \cdots, 1-1/F$), is obtained. The algebra is explicitly constructed. We also show that in $1+2$ dimensions FSUSY is a non-trivial extension of the Poincar\'e algebra which generates a symmetry among fractional spin states or anyons. Unitarity of the representation is checked. Finally, we prove that, independently of the dimension, a natural classification emerges according to the decomposition of $F$ as a product of prime numbers and that FSUSY is a symmetry which closes non-linearly, and is sustained by mathematical structures that go beyond Lie or super-Lie algebras. 
  The personal and scientific history of the discovery of spontaneous symmetry breaking in gauge theories is outlined and its scientific content is reviewed 
  SU(4) Einstein-Yang-Mills theory possesses sequences of static spherically symmetric globally regular and black hole solutions. Considering solutions with a purely magnetic gauge field, based on the 4-dimensional embedding of $su(2)$ in $su(4)$, these solutions are labelled by the node numbers $(n_1,n_2,n_3)$ of the three gauge field functions $u_1$, $u_2$ and $u_3$. We classify the various types of solutions in sequences and determine their limiting solutions. The limiting solutions of the sequences of neutral solutions carry charge, and the limiting solutions of the sequences of charged solutions carry higher charge. For sequences of black hole solutions with node structure $(n,j,n)$ and $(n,n,n)$, several distinct branches of solutions exist up to critical values of the horizon radius. We determine the critical behaviour for these sequences of solutions. We also consider SU(4) Einstein-Yang-Mills-dilaton theory and show that these sequences of solutions are analogous in most respects to the corresponding SU(4) Einstein-Yang-Mills sequences of solutions. 
  An overview of supersymmetry and its different applications is presented. We motivate supersymmetry in particle physics. We then explain how supersymmetry helps us analyze field theories exactly, and what dynamical lessons these solutions teach us. Finally, we describe how supersymmetry is used to derive exact results in string theory. These results have led to a revolution in our understanding of the theory. 
  We build the two dimensional Gross-Neveu model by a new method which requires neither cluster expansion nor discretization of phase-space. It simply reorganizes the perturbative series in terms of trees. With this method we can for the first time define non perturbatively the renormalization group differential equations of the model and at the same time construct explicitly their solution. 
  The composite operator effective potential is compared with the conventional Dyson-Schwinger method as a calculational tool for (2+1)-dimensional quantum electrodynamics. It is found that when the fermion propagator ansatz is put directly into the effective potential, it reproduces exactly the usual gap equations derived in the Dyson-Schwinger approach. 
  We consider the problem of extremizing the tension for BPS strings in D=6 supergravities with different number of supersymmetries. General formulae for fixed scalars and a discussion of degenerate directions is given. Quantized moduli, according to recent analysis, are supposed to be related to conformal field theories which are the boundary of three dimensional anti-de Sitter space time. 
  We compactify M-theory in the Horava-Witten formulation on S^1/Z_2 \times K3 \times T^2. Focusing on the moduli-space of vector multiplets of the resulting four-dimensional N=2 theory, we determine the prepotential as an expansion in two dimensionless parameters which both scale as \kappa^{2/3}. We determine the prepotential completely to relative order \kappa^{4/3} and compare the expression with the results obtained for the perturbative string theories. We find complete agreement to relative order \kappa^{4/3} between the strong and weak coupling regimes. The sources of higher order perturbative and non-perturbative corrections to the prepotential are also briefly discussed from the M-theory perspective. 
  A new version of the Casimir effect where the two plates conduct in specific, different, directions is considered. By direct functional integration the evaluation of the Casimir energy as a function of the angle between the conduction directions is reduced to quadratures. Other applications of the method and a magnetic Casimir variant are mentioned. 
  Recently, it has been proposed by Maldacena that large $N$ limits of certain conformal field theories in $d$ dimensions can be described in terms of supergravity (and string theory) on the product of $d+1$-dimensional $AdS$ space with a compact manifold. Here we elaborate on this idea and propose a precise correspondence between conformal field theory observables and those of supergravity: correlation functions in conformal field theory are given by the dependence of the supergravity action on the asymptotic behavior at infinity. In particular, dimensions of operators in conformal field theory are given by masses of particles in supergravity. As quantitative confirmation of this correspondence, we note that the Kaluza-Klein modes of Type IIB supergravity on $AdS_5\times {\bf S}^5$ match with the chiral operators of $\N=4$ super Yang-Mills theory in four dimensions. With some further assumptions, one can deduce a Hamiltonian version of the correspondence and show that the $\N=4$ theory has a large $N$ phase transition related to the thermodynamics of $AdS$ black holes. 
  We study two dimensional $N=(4,4)$ supersymmetric gauge theories with various gauge groups and various hypermultiplets in the fundamental as well as bi-fundamental and adjoint representations. They have " mirror theories " which become equivalent to them at the strong coupling. The theory with one fundamental and one adjoint has a Higgs branch which is parametrized by the adjoint matter. We also consider theories which involve an orientifold plane. The brane realization of the Matrix theory formulation of NS 5-branes in Type II string theories is also considered. 
  We show that the super D-string action is exactly equivalent to the IIB Green-Schwarz superstring action with some "theta term" in terms of the path integral. Since the "theta term" imposes the Gauss law constraint on the physical state but contributes to neither the mass operator nor the constraints associated with the kappa symmetry and the reparametrization, this exact equivalence implies that the impossibility to disentangle the first and second class fermionic constraints covariantly in the super D-string action is generally inherited from the IIB Green-Schwarz superstring action except specific gauge choices which make the ground state massive, such as the static gauge. Moreover, it is shown that if the electric field is quantized to be integers, the super D-string action can be transformed to the IIB Green-Schwarz superstring action with $SL(2, Z)$ covariant tension. 
  We consider test strings and test branes ending on D$p$-branes ($p\le 6$) and NS5-branes in the background, for a heuristic understanding of the dynamics. Whenever some supersymmetry is preserved, a simple BPS bound appears, but the central charge in question is measured by certain isotropic coordinate system, rather than by the actual spacetime geometry. This way, the ground state energy is independent of the gravitational radii of the solitonic background. Furthermore, a perturbation around the supersymmetric ground states reveals that the appropriate Dirichlet boundary condition is dynamically induced. We close with comments. 
  An equivariant BRST-construction is used to define the continuum SU(3) gauge theory on a finite torus. I corroborate previous results using renormalization group techniques by explicitly computing the measure on the moduli-space of the model with 3 quark flavors to two loops. I find that the correction to the maximum of the one-loop effective action is indeed of order $g^2$ in the critical covariant gauge. The leading logarithmic corrections from higher loops are also shown to be suppressed by at least one order of $g^2$. I therefore am able to relate the expectation value of the moduli to the asymptotic scale parameter of the modified minimal subtraction scheme. An immediate consequence is the determination of the non-perturbative proportionality constant in the relation between the vacuum expectation value of the trace of the energy momentum tensor and $\Lambda_{QCD}$ for the modified minimal subtraction scheme with three quark flavors. The result compares favorably with phenomenological estimates of the gluon condensate from QCD sumrules for the charmonium system and $\Lambda_{QCD}$ from $\tau$-decay. 
  The paper is a brief review of the works devoted to problem of effective action in N=2 super Yang-Mills theories in harmonic superspace approach. The formulation of N=2 superfield models in harmonic superspace is discussed, background field method for N=2 super Yang-Mills theory is constructed and general structure of effective action in harmonic superspace is investigated. It is shown how the holomorphic and non-holomorphic contributions to effective action can be calculated within the harmonic superspace approach. 
  This paper is a brief review of background field method and some of its applications in N=2 super Yang-Mills theories with a matter within harmonic superspace approach. A general structure of effective action is discussed, an absence of two-loop quantum corrections to first non-leading term in effective action is proved and N=2 non-renormalization theorem in this approach is considered. 
  In the possible scaling region for lattice chiral fermions advocated in  hep-lat/9609037, no hard spontaneous symmetry breaking occurs and doublers are gauge-invariantly decoupled via mixing with composite three-fermion-states. However the strong coupling expansion breaks down due to no ``static limit'' for the low-energy limit ($p\sim 0$). We further analyze relevant Green functions of three-fermion-operators. It is shown that in the low-energy limit, the propagators of three-fermion-states with the ``wrong'' chiralities positively vanish due to the generalized form factors (the wave-function renormalizations) of these composite states vanishing as $O(p^4)$. This strongly implies that three-fermion-states with ``wrong'' chirality are ``decoupled'' in this limit. 
  We develop a simple scheme of quantization for the dilaton CGHS model without scalar fields, that uses the Gupta-Bleuler approach for the string fields. This is possible because the constraints can be linearized classically, due to positivity conditions that are present in the model (and not in the general string case). There is no ambiguity nor anomalies in the quantization. The expectation values of the metric and dilaton fields obey the classical requirements, thus exhibiting at the quantum level the Birkhoff theorem. 
  The large distance behavior of the Maxwell- Chern-Simons (MCS) equations is analyzed, and it is found that the pure Chern-Simons limit, (when the Maxwell term is dropped from the equations), does not describe the large distance limit of the MCS model. This necessitates the solution of the original problem. The MCS gauge theory coupled to a nonrelativistic matter field, (governed by the gauged non-linear Schr\"odinger equation), is studied. It turns out, that there are no regular self-dual solutions as in the pure Chern-Simons case, but the model admits interesting, though singular self-dual solutions. The properties of these solutions, and their large distance limits are analyzed. 
  We study the one-loop effective action for $N$ 4D conformally invariant scalars on the spherically symmetric background. The main part of effective action is given by integration of 4D conformal anomaly. This effective action (in large $N$ approximation and partial curvature expansion) is applied to investigate the quantum evolution of Schwarzschild-de Sitter (SdS) black holes of maximal mass. We find that the effect (recently discovered by Bousso and Hawking for $N$ minimal scalars and another approximate effective action) of anti-evaporation of nearly maximal SdS (Nariai) black holes takes also place in the model under consideration. Careful treatment of quantum corrections and perturbations modes of Nariai black hole is given being quite complicated. It is shown that exists also perturbation where black hole radius shrinks, i.e. black hole evaporates. We point out that our result holds for wide class of models including conformal scalars, spinors and vectors. Hence,anti-evaporation of SdS black holes is rather general effect which should be taken into account in quantum gravity considerations. 
  We analyse dependence of the partition function on the boundary condition for the longitudinal component of the electric field strength in gauge field theories. In a physical gauge the Gauss law constraint may be resolved explicitly expressing this component via an integral of the physical transversal variables. In particular, we study quantum electrodynamics with an external charge and SU(2) gluodynamics. We find that only a charge distribution slowly decreasing at spatial infinity can produce a nontrivial dependence in the Abelian theory. However, in gluodynamics for temperatures below some critical value the partition function acquires a delta-function like dependence on the boundary condition, which leads to colour confinement. 
  String theory implies that field theories containing gravity are in a certain sense `products' of gauge theories. We make this product structure explicit up to two loops for the relatively simple case of N=8 supergravity four-point amplitudes, demonstrating that they are `squares' of N=4 super-Yang-Mills amplitudes. This is accomplished by obtaining an explicit expression for the $D$-dimensional two-loop contribution to the four-particle S-matrix for N=8 supergravity, which we compare to the corresponding N=4 Yang-Mills result. From these expressions we also obtain the two-loop ultraviolet divergences in dimensions D=7 through D=11. The analysis relies on the unitarity cuts of the two theories, many of which can be recycled from a one-loop computation. The two-particle cuts, which may be iterated to all loop orders, suggest that squaring relations between the two theories exist at any loop order. The loop-momentum power-counting implied by our two-particle cut analysis indicates that in four dimensions the first four-point divergence in N=8 supergravity should appear at five loops, contrary to the earlier expectation, based on superspace arguments, of a three-loop counterterm. 
  Non-extremal overlapping p-brane supergravity solutions localised in their relative transverse coordinates are constructed. The construction uses an algebraic method of solving the bosonic equations of motion. It is shown that these non-extremal solutions can be obtained from the extremal solutions by means of the superposition of two deformation functions defined by vacuum solutions of M-theory. Vacuum solutions of M-theory including irrational powers of harmonic functions are discussed. 
  We discuss the open string ending on D $p$-branes in IKKT framework. First we determine the boundary conditions of Green-Schwarz superstring which are consistent with supersymmetry and $\kappa$-symmetry. We point out some subtleties arising from taking the Schild gauge and show that in this gauge the system incorporates the limited dimensional D $p$-branes ($p=3,7$). The matrix regularization for the Dirichlet open string is given by gauge group SO(N). When $p=3$, the matrix model becomes the dimensional reduction of a 6 dimensional ${\cal N}=1$ super Yang-Mills theory. 
  The author has previously suggested that the ground state for 4-dimensional quantum gravity can be represented as a condensation of non-linear gravitons connected by Dirac strings. In this note we suggest that the low-lying excitations of this state can be described by a quasi-topological action corresponding to a trilinear coupling of solitonic 8-branes and 7-branes. It is shown that when the 7-brane excitations are neglected, the effective action can be interpreted as a theory of conformal gravity in four dimensions. This suggests that ordinary gravity as well as supersymmetric matter and phenomenological gauge symmetries arise from the spontaneous breaking of topological invariance. 
  We present a topological mechanism of discretization, which gives for the fundamental electric charge a value equal to the square root of the Planck constant times the velocity of light, which is about 3.3 times the electron charge. Its basis is the following recently proved property of the standard linear classical Maxwell equations: they can be obtained by change of variables from an underlying topological theory, using two complex scalar fields, the level curves of which coincide with the magnetic and the electric lines, respectively. 
  Some new expressions are found, concerning the one-loop effective action of four dimensional massive and massless Dirac fermions in the presence of general uniform electric and magnetic fields, with $\vec E\cdot \vec H\neq 0$ and $\vec E ^2\neq \vec H ^2$. The rate of pair-production is computed and briefly discussed. 
  We study certain properties of six-dimensional tensionless E-strings (arising from zero size $E_8$ instantons). In particular we show that $n$ E-strings form a bound string which carries an $E_8$ level $n$ current algebra as well as a left-over conformal system with $c=12n-4-{248n \over n+30}$, whose characters can be computed. Moreover we show that the characters of the $n$-string bound state are captured by N=4 U(n) topological Yang-Mills theory on $\half K3$. This relation not only illuminates certain aspects of E-strings but can also be used to shed light on the properties of N=4 topological Yang-Mills theories on manifolds with $b_2^+=1$. In particular the E-string partition functions, which can be computed using local mirror symmetry on a Calabi-Yau three-fold, give the Euler characteristics of the Yang-Mills instanton moduli space on $\half K3$. Moreover, the partition functions are determined by a gap condition combined with a simple recurrence relation which has its origins in a holomorphic anomaly that has been conjectured to exist for N=4 topological Yang-Mills on manifolds with $b_2^+=1$ and is also related to the holomorphic anomaly for higher genus topological strings on Calabi-Yau threefolds. 
  Linearization of homogeneous polynomials of degree n and k variables leads to generalized Clifford algebras. Multicomplex numbers are then introduced in analogy to complex numbers with respect to usual Clifford algebra. In turn multicomplex extensions of trigonometric functions are constructed in terms of `compact' and `non-compact' variables. It gives rise to the natural extension of the d-dimensional sine-Gordon field theory in the n-dimensional multicomplex space. In dimension 2, the cases n=1,2,3,4 are identified as the quantum integrable Liouville, sine-Gordon and known deformed Toda models. The general case is discussed. 
  In the class of (0,2) heterotic compactifications which has been constructed in the framework of gauged linear sigma models the Calabi-Yau varieties X are realized as complete intersections of hypersurfaces in toric varieties IP and the corresponding gauge bundles (or more generally gauge sheaves) E are defined by some short exact sequences. We show that there is yet another degree of freedom in resolving singularities in such models which is related to the possible choices of nef partitions of the anticanonical divisors in Gorenstein Fano toric varieties IP. 
  We consider renormalization groups of transformations composed of a Gaussian convolution and a field dilatation. As an example, we consider perturbations of a single component real Euclidean free field $\phi$ with covariance $(-\bigtriangleup)^{-1+\frac{\epsilon}{2}}$. We show that the renormalization group admits two equivalent formulations called massless picture and massive picture respectively. We then show in the massive picture that the renormalization group has a symmetry. The symmetry consists of global scale transformations composed with certain Gaussian convolutions. We translate the symmetry back to the massless picture. The relation between the symmetry and the notion of an anomalous dimension is briefly discussed. 
  Statistical mechanics explains thermodynamics in terms of (quantum) mechanics by equating the entropy of a microstate of a closed system with the logarithm of the number of microstates in the macrostate to which it belongs, but the question `what is a macrostate?' has never been answered except in a vague, subjective, way. However Hawking's discovery of black hole evaporation led to a formula for black hole entropy with no subjective element. In this letter, we argue from this result, together with the assumption that `black hole thermodynamics is just ordinary thermodynamics applied to black holes', that a macrostate for a general (quantum gravitational) closed system is an equivalence class of matter-gravity microstates with the same expectation values for the matter degrees of freedom alone. Not only does this finally answer the question `what is entropy?', but it also predicts the equality of the thermodynamic entropy of a black hole with the matter and the gravity entropy-like quantities derived from the Euclidean path integral. Furthermore it gives us a clear glimpse of an ultimate synthesis of quantum theory and gravity in which we see that (a) gravity acts as a universal environment, thus predicting that, if the initial state of the universe is unentangled, its entropy must go on increasing forever, (b) the gravitational field has degrees of freedom, but no observables, thus enabling gravity to perform the trick of providing an objective continual process of decoherence. All the above rests on the validity of unitarity. The `information-loss puzzle' had raised doubts about that. But we suggest a resolution for this puzzle. 
  The statistical entropy of black holes in M-theory is considered. Assuming Matrix theory is the discretized light-cone quantization of a theory with eleven-dimensional Lorentz invariance, we map the counting problem onto the original Gibbons-Hawking calculation of the thermodynamic entropy. 
  Using relations from random matrix theory, we derive exact expressions for all $n$-point spectral correlation functions of Dirac operator eigenvalues in terms of finite-volume partition functions. This is done for both chiral symplectic and chiral unitary random matrix ensembles, which correspond to $SU(N_c \geq 3)$ gauge theories with $N_f$ fermions in the adjoint and fundamental representations, respectively. In the latter case we infer from this an infinite sequence of consistency conditions that must be satisfied by the corresponding finite-volume partition functions. 
  We obtain electrically charged vortex solutions for the Born-Infeld Higgs system with a Chern Simons term. We analyse numerically these solutions, comparing their properties with those of ``normal'' Nielsen-Olesen vortices and also show that no charged vortex solutions exist in Born-Infeld theory when the Chern Simons term is absent. 
  We construct a locally supersymmetric worldsheet formulation of a non-Abelian Ramond-Neveu-Schwarz (NARNS) string theory where the string coordinates are noncommuting matrices in a group U(N). This is described by the two dimensional supergravity coupled to supersymmetric Yang-Mills fields and adjoint matters in the gauge group U(N). We show that our NARNS string theory has a free string limit where it becomes N-copies of usual RNS string which can be described by the orbifold conformal field theory corresponding to the covariant worldsheet version of the Matrix string theory of Dijkgraaf, Verlinde and Verlinde. In the weak coupling limit, i.e. $g_s \to 0$ where $g_s$ is the coupling constant of our theory related with the Yang-Mills coupling as $g_{YM}^{-2}=\alpha' g_s^2$, a new additional dimension appears in the string spectrum and it can be speculatively interpreted as the compactified eleven dimensional coordinate whose dynamics is given by an orbifold O(N) sigma model. 
  We present a Lagrangian formulation for the general modified chiral model. We use it to discuss the Hamiltonian formalism for this model and to derive the commutation relations for the chiral field. We look at some explicit examples and show that the Hamiltonian, containing a contribution involving a Wess-Zumino term, is conserved, as required. 
  We analyse in detail the behaviour of supersymmetric QCD with a number of flavours M smaller than the number of colours N, for quark masses smaller than the dynamically generated scale. In this regime, we find it useful to move from meson superfields to Nambu-Goldstone-like variables. In particular we work out the mass spectrum and the set of decay constants that specify the interactions of the low-energy theory. We explicitly check that masses and decay constants have a consistent behaviour under decoupling and that they satisfy current algebra requirements. Finally we speculate about the massless limit. For vanishing quark masses, and only in this case, the relation between mesons and Nambu-Goldstone variables becomes singular. When analysed in terms of the Nambu-Goldstone superfields, the massless limit exhibits a spontaneous breaking of the flavour symmetry, with massless Goldstone modes embedded in an M^2-dimensional complex moduli space. The symmetry-breaking order parameter is formally infinite, but this has the only effect of turning off the interactions between the chiral superfields. 
  New forms of Born-Infeld, D-brane and M theory five-brane actions are found which are quadratic in the abelian field strength. The gauge fields couple both to a background or induced metric and a new auxiliary metric, whose elimination reproduces the non-polynomial Born-Infeld action. This is similar to the introduction of an auxiliary metric to simplify the Nambu-Goto string action. This simplifies the quantisation and dualisation of the gauge fields. 
  We review the Coulomb gauge in QCD and report some recent results. The minimal Coulomb gauge is defined and the fundamental modular region, a region without Gribov copies, is described. The Coulomb gauge action is expressed in terms of the dynamical degrees of freedom with an instantaneous Coulomb interaction, and its physical meaning is discussed. The local Coulomb gauge action with phase-space and ghost variables is derived, and its BRS-invariance and renormalizability are reviewed. It is shown that the instantanteous part $V(R)$ of $g^2 D_{00}(R, t)$, the time-time component of the gluon propagator, is a renormalization-group invariant $V(R) = f(R\Lambda_{QCD})/R$, and that the contribution of $V(R)$ to the Wilson loop exponentiates. It is conjectured that $V(R) \sim \kappa_{coul}R$ at large $R$, and that $\kappa_{coul}$ provides an order parameter for confinement of color even in the presence of dynamical quarks. 
  The usual proof of renormalizability using the Callan-Symanzik equation makes explicit use of normalization conditions. It is shown that demanding that the renormalization group functions take the form required for minimal subtraction allows one to prove renormalizability using the Callan-Symanzik equation, without imposing normalization conditions. Scalar field theory and quantum electrodynamics are treated. 
  We introduce a $N_c\times N_c$ matrix model with $\cN=2$ supersymmetries and show its relation to the topological rigid string and the topological YM$_2$. This allows to connect the latter two theories directly. Moreover the construction leads to a new insight in the \ninfty limit. Finally a quantum mechanical matrix theory is proposed which may describe light-cone (light-front) dynamics of gauge fields. 
  We propose correspondences between 4d quantum field theories with N=2,1,0 (super)conformal invariance and Type IIB string theory on various orbifolds. We argue using the spacetime string theory, and check using the beta functions (exactly for N=2,1 and so far at 1-loop for the gauge couplings in the N=0 case), that these theories have conformal fixed lines. The latter case potentially gives well-defined non-supersymmetric vacua of string theory, with a mechanism for making the curvature and cosmological constant small at nontrivial string coupling. We suggest a correspondence between nonsupersymmetric conformal fixed lines and nonsupersymmetric string vacua with vanishing vacuum energy. 
  Recently, it has been observed that a quantum field theory need not be Hermitian to have a real, positive spectrum. What seems to be required is symmetry under combined parity and time-reversal transformations. This idea is extended to massless electrodynamics, in which the photon couples to the axial-vector current with an imaginary coupling constant. The eigenvalue condition necessary for the finiteness of the theory can now be solved; the value for the charge appears to be stable order-by-order. Similarly, the semiclassical Casimir model for the fine-structure constant yields a positive value. 
  We study correlation functions in topologically twisted $\CN=2, d=4$ supersymmetric Yang-Mills theory for gauge groups of rank larger than one on compact four-manifolds $X$. We find that the topological invariance of the generator of correlation functions of BRST invariant observables is not spoiled by noncompactness of field space. We show how to express the correlators on simply connected manifolds of $b_{2,+}(X)>0$ in terms of Seiberg-Witten invariants and the classical cohomology ring of $X$. For manifolds $X$ of simple type and gauge group SU(N) we give explicit expressions of the correlators as a sum over $\CN=1$ vacua. We describe two applications of our expressions, one to superconformal field theory and one to large $N$ expansions of SU(N) $\CN=2, d=4$ supersymmetric Yang-Mills theory. 
  We compute the moduli metrics of worldvolume 0-brane solitons of D-branes and the worldvolume self-dual string solitons of the M-5-brane and examine their geometry. We find that the moduli spaces of 0-brane solitons of D-4-branes and D-8-branes are hyper-K\"ahler manifolds with torsion and octonionic K\"ahler manifolds with torsion, respectively. The moduli space of the self-dual string soliton of the M-5-brane is also a hyper-K\"ahler manifold with torsion. 
  In this paper we consider brane solutions of the form $G/H$ in M(atrix) theory, showing the emergence of world volume coordinates for the cases where $G=SU(N)$. We examine a particular solution with a world volume geometry of the form ${\bf CP}^2 \times S^1$ in some detail and show how a smooth manifold structure emerges in the large $N$ limit. In this limit the solution becomes static; it is not supersymmetric but is part of a supersymmetric set of configurations. Supersymmetry in small locally flat regions can be obtained, but this is not globally defined. A general group theoretic analysis of the previously known spherical membrane solutions is also given. 
  Quantum $A_2$-Toda field theory in two dimensions is investigated based on the method of quantizing canonical free field. Toda exponential operators associated with the fundamental weights are constructed to the fourth order in the cosmological constant. This leads us to a conjecture for the exact operator solution. 
  We analyze gauge symmetry enhancements $SO(16)\to E_8$ on eight D7-branes and $SO(14)\times U(1)\to E_8$ on seven D7-branes from open strings. String configurations which we present in this paper are closely related to the ones given by Gaberdiel and Zwiebach. Our construction is based on $SO(8)\times SO(8)$ decomposition and its relation to the D8-brane case via T-duality is clearer. Then we study supersymmetric Yang-Mills theory on D3-brane near the D7-branes. This theory has flavour symmetry group which is equal to the gauge group on D7-branes. We suggest that when this symmetry is enhanced, two dyons make bound states which, together with elementary quarks, constitute an $E_8$ multiplet. 
  The theory of eleven dimensional supergravity on R^10 x S^1/Z_2 with super Yang-Mills theory on the boundaries is reconsidered. We analyse the general solution of the modified Bianchi identity for the four-form field strength using the equations of motion for the three-form and find that the four-form field strength has a unique value on the boundaries of R^10 x S^1/Z_2. Considering the local supersymmetry in the "downstairs" approach this leads to a relation between the eleven dimensional supergravity coupling constants in the "upstairs" and "downstairs" approaches. Moreover, it is shown using flux quantization that the brane tensions only have their standard form in the "downstairs" units. We consider the gauge variation of the classical theory and find that it cannot be gauge invariant, contrary to a recent claim. Finally we consider anomaly cancellation in the "downstairs" and "upstairs" approaches and obtain the values of lambda^6/kappa^4 and the two- and five-brane tensions. 
  We consider pure Yang Mills theory on the four torus. A set of non-Abelian transition functions is presented which encompass all instanton sectors. It is argued that these transition functions are a convenient starting point for gauge fixing. In particular, we give an extended Abelian projection with respect to the Polyakov loop, where $A_0$ is independent of time and in the Cartan subalgebra. In the non-perturbative sectors such gauge fixings are necessarily singular. These singularities can be restricted to Dirac strings joining monopole and anti-monopole like ``defects''. 
  The problem of gauging a closed form is considered. When the target manifold is a simple Lie group G, it is seen that there is no obstruction to the gauging of a subgroup H\subset G if we may construct from the form a cocycle for the relative Lie algebra cohomology (or for the equivariant cohomology), and an explicit general expression for these cocycles is given. The common geometrical structure of the gauged closed forms and the D'Hoker and Weinberg effective actions of WZW type, as well as the obstructions for their existence, is also exhibited and explained. 
  Making use of non-perturbative U-duality symmetries of type II strings we construct new `superstring' vacua in three dimensions with N=1 supersymmetry. This has an interpretation as compactifying formally from 13 dimensions (S-theory) on Calabi-Yau 5-folds possessing a $T^3 \x T^2$ fibration. We describe some part of the massless multiplets, given by the Hodge spectrum, and point to a corresponding 5-brane configuration. 
  A type IIA string (or F-theory) compactified on a Calabi-Yau threefold is believed to be dual to a heterotic string on a K3 surface times a 2-torus (or on a K3 surface). We consider how the resulting moduli space of hypermultiplets is identified between these two pictures in the case of the E8xE8 heterotic string. As examples we discuss SU(2)-bundles and G2-bundles on the K3 surface and the case of point-like instantons. We are lead to a rather beautiful identification between the integral cohomology of the Calabi-Yau threefold and some integral structures on the heterotic side somewhat reminiscent of mirror symmetry. We discuss the consequences for probing nonperturbative effects in the both the type IIA string and the heterotic string. 
  We suggest a supergravity dual for the $(1,0)$ superconformal field theory in six dimensions which has $E_8$ global symmetry. Compared to the description of the (2,0) field theory, the 4-sphere is replaced by a 4-hemisphere, or by orbifolding the 4-sphere. 
  We extend the Wilson renormalization group (RG) to supersymmetric theories. As this regularization scheme preserves supersymmetry, we exploit the superspace technique. To set up the formalism we first derive the RG flow for the massless Wess-Zumino model and deduce its perturbative expansion. We then consider N=1 supersymmetric Yang-Mills and show that the local gauge symmetry -broken by the regularization- can be recovered by a suitable choice of the RG flow boundary conditions. We restrict our analysis to the first loop, the generalization to higher loops presenting no difficulty due to the iterative nature of the procedure. Furthermore, adding matter fields, we reproduce the one-loop supersymmetric chiral anomaly to the second order in the vector field. 
  Cosmological solutions are obtained by continuation of black D-brane solutions into the region between the horizons. It is investigated whether one can find exponential expansion when probing the cosmology with D-branes. A unique configuration exhibiting exponential expansion is discussed. 
  A large class of extremal and near-extremal four dimensional black holes in M-theory feature near horizon geometries that contain three dimensional asymptotically anti-de Sitter spaces. Globally, these geometries are derived from AdS_3 by discrete identifications. The microstates of such black holes can be counted by exploiting the conformal symmetry induced on the anti-de Sitter boundary, and the result agrees with the Bekenstein-Hawking area law. This approach, pioneered by Strominger, clarifies the physical nature of the black hole microstates. It also suggests that recent analyses of the relationship between boundary conformal field theory and supergravity can be extended to orbifolds of AdS spaces. 
  We construct the (bosonic) effective worldvolume action of an M-theory Kaluza-Klein monopole in a background given by the bosonic sector of eleven-dimensional massive supergravity, i.e. a "massive Kaluza-Klein monopole". As a consistency check we show that the direct dimensional reduction along the isometry direction of the Taub-NUT space leads to the massive D-6-brane. We furthermore perform a double dimensional reduction in the massless case and obtain the effective worldvolume action of a type IIA Kaluza-Klein monopole. 
  We derive the Dirac brackets for the O(N) nonlinear sigma model in the lightfront description with and without the constraint. We bring out various subtleties that arise including the fact that anti-periodic boundary condition seems to be preferred. 
  We consider conformal non-Abelian Toda theories obtained by hamiltonian reduction from Wess-Zumino-Witten models based on general real Lie groups. We study in detail the possible choices of reality conditions which can be imposed on the WZW or Toda fields and prove correspondences with sl(2,R) embeddings into real Lie algebras and with the possible real forms of the associated W-algebras. We devise a a method for finding all real embeddings which can be obtained from a given embedding of sl(2,C) into a complex Lie algebra. We then apply this to give a complete classification of real embeddings which are principal in some simple regular subalgebra of a classical Lie algebra. 
  In this note we derive the net number of generations of chiral fermions in heterotic string compactifications on Calabi-Yau threefolds with certain SU(n) vector bundles, for n odd, using the parabolic approach for bundles. We compare our results with the spectral cover construction for bundles and make a comment on the net number interpretation in F-theory. 
  We discuss the spectrum of states of IIB supergravity on $AdS_5\times S^5$ in a manifest $SU(2,2/4)$ invariant setting. The boundary fields are described in terms of N=4 superconformal Yang-Mills theory and the proposed correspondence between supergravity in $AdS_5$ and superconformal invariant singleton theory at the boundary is formulated in an N=4 superfield covariant language. 
  The Topological N=2 Superconformal algebra has 29 different types of singular vectors (in complete Verma modules) distinguished by the relative U(1) charge and the BRST-invariance properties of the vector and of the primary on which it is built. Whereas one of these types only exists at level zero, the remaining 28 types exist for general levels and can be constructed already at level 1. In this paper we write down one-to-one mappings between 16 of these types of topological singular vectors and the singular vectors of the Antiperiodic NS algebra. As a result one obtains construction formulae for these 16 types of topological singular vectors using the construction formulae for the NS singular vectors due to Doerrzapf. 
  We study some aspects of Maldacena's large $N$ correspondence between N=4 superconformal gauge theory on D3-brane and maximal supergravity on AdS_5xS_5 by introducing macroscopic strings as heavy (anti)-quark probes. The macroscopic strings are semi-infinite Type IIB strings ending on D3-brane world-volume. We first study deformation and fluctuation of D3-brane when a macroscopic BPS string is attached. We find that both dynamics and boundary conditions agree with those for macroscopic string in anti-de Sitter supergravity. As by-product we clarify how Polchinski's Dirichlet / Neumann open string boundary conditions arise dynamically. We then study non-BPS macroscopic string anti-string pair configuration as physical realization of heavy quark Wilson loop. We obtain quark-antiquark static potential from the supergravity side and find that the potential exhibits nonanalyticity of square-root branch cut in `t Hooft coupling parameter. We put forward the nonanalyticity as prediction for large-N gauge theory at strong `t Hooft coupling limit. By turning on Ramond-Ramond zero-form potential, we also study theta-vacuum angle dependence of the static potential. We finally discuss possible dynamical realization of heavy N-prong string junction and of large-N loop equation via local electric field and string recoil thereof. Throughout comparisons of the AdS-CFT correspondence, we find crucial role played by `geometric duality' between UV and IR scales on directions perpendicular to D3-brane and parallel ones, explaining how AdS5 spacetime geometry emerges out of four-dimensional gauge theory at strong coupling. 
  We propose a method to calculate the expectation values of an operator similar to the Wilson loop in the large N limit of field theories. We consider N=4 3+1 dimensional super-Yang-Mills. The prescription involves calculating the area of a fundamental string worldsheet in certain supergravity backgrounds. We also consider the case of coincident M-theory fivebranes where one is lead to calculating the area of M-theory two-branes. We briefly discuss the computation for 2+1 dimensional super-Yang-Mills with sixteen supercharges which is non-conformal. In all these cases we calculate the energy of quark-antiquark pair. 
  In recent work by Kabat and Taylor, certain Matrix theory quantities have been identified with the spatial moments of the supergravity stress-energy tensor, membrane current, and fivebrane current. In this note, we determine the relations between these moments required by current conservation, and prove that these relations hold as exact Matrix Theory identities at finite N. This establishes conservation of the effective supergravity currents (averaged over the compact circle). In addition, the constraints of current conservation allow us to deduce Matrix theory quantities corresponding to moments of the spatial current of the longitudinal fivebrane charge, not previously identified. 
  We discuss a 4-dimensional nonsupersymmetric black hole solution to low energy type IIA string theory which carries D0- and D6-brane charges. For equal charges this solution reduces to the one discussed recently by Sheinblatt. We present a new parametrization of the solution in terms of four numbers which reveals the underlying brane and antibrane structure of the black hole arbitrarily far from extremality. In this parametrization, the entropy of the general nonextremal black hole takes on a simple U-duality invariant form. A Yang-Mills solution for the brane configuration corresponding to the extremal case is constructed and a computation of the 1-loop matrix theory potential for the scattering of a 0-brane probe off this brane configuration done. We find that this agrees with the 1-loop potential obtained from a supergravity calculation in the limit in which the ratio of the 0-brane to 6-brane charges is large. 
  We derive the thermodynamic Bethe ansatz equation for the situation inwhich the statistical interaction of a multi-particle system is governed by Haldane statistics. We formulate a macroscopical equivalence principle for such systems. Particular CDD-ambiguities play a distinguished role in compensating the ambiguity in the exclusion statistics. We derive Y-systems related to generalized statistics. We discuss several fermionic, bosonic and anyonic versions of affine Toda field theories and Calogero-Sutherland type models in the context of generalized statistics. 
  A subclass of recently discovered class of solutions in multidimensional gravity with intersecting p-branes related to Lie algebras and governed by a set of harmonic functions is considered. This subclass in case of three Euclidean p-branes (one electric and two magnetic) contains a cosmological-type solution (in 11-dimensional model with two 4-forms) related to hyperbolic Kac-Moody algebra ${\cal F}_3$ (of rank 3). This solution describes the non-Kasner power-law inflation. 
  We derive the hole probability and the distribution of the smallest eigenvalue of chiral hermitian random matrices corresponding to Dirac operators coupled to massive quarks in QCD. They are expressed in terms of the QCD partition function in the mesoscopic regime. Their universality is explicitly related to that of the microscopic massive Bessel kernel. 
  In recent times some interesting field theoretical descriptions of the statistical mechanics of entangling polymers have been proposed by various authors. In these approaches, a single test polymer fluctuating in a background of static polymers or in a lattice of obstacles is considered. The extension to the case in which the configurations of two or more polymers become non-static is not straightforward unless their trajectories are severely constrained. In this paper we present another approach, based on Chern--Simons field theory, which is able to describe the topological entanglements of two fluctuating polymers in terms of gauge fields and second quantized replica fields. 
  Performing functional integration of the free Lagrangian, we find the vacuum energy of a field. The functional integration is performed in a way which easily generalizes to systems at non-zero temperature. We use this technique to obtain the Casimir energy density and pressure at arbitrary temperatures. 
  The bound system composed of the Yang monopole coupled to a particle of the isospin by the SU(2) and Coulomb interaction is considered. The generalized Runge--Lenz vector and the SO(6) group of hidden symmetry are established. It is also shown that the group of hidden symmetry make it possible to calculate the spectrum of the system by a pure algebraic method. 
  We show for the case of interacting massless vector bosons, how the structure of Yang-Mills theories emerges automatically from a more fundamental concept, namely perturbative quantum gauge invariance. It turns out that the coupling in a non-abelian gauge theory is necessarily of Yang-Mills type plus divergence- and coboundary couplings. The extension of the method to massive gauge theories is briefly discussed. 
  The classical linearised gravitational self interaction of a Goto-Nambu string is examined in four spacetime dimensions. Using a conveniently gauge independent tensorial treatment, the divergent part of the self-force is shown to be exactly zero. This is due to cancelation by a contribution that was neglected in the previous treatments. This result has implications for many applications. 
  There is a chance that singleton fields, that in the context of strings and membranes have been regarded as topological gauge fields that can interact only at the boundary of anti-De Sitter space, at spatial infinity, may have a more physical manifestation as costituents of massless fields in space time. The composite character of massless fields is expressed by field - current identities that relate ordinary massless field operators to singleton currents and stress-energy tensors. Naive versions of such identities do not make sense, but when the singletons are described in terms of dipole structures, then such constructions are at least formally possible. The new proposal includes and generalizes an early composite version of QED, and includes quantum gravity, super gravity and models of QCD. Unitarity of such theories is conjectural. 
  We propose Seiberg-Witten geometry for N=2 gauge theory with gauge group $E_6$ with massive $N_f$ fundamental hypermultiplets. The relevant manifold is described as a fibration of the ALE space of $E_6$ type. It is observed that the fibering data over the base ${\bf CP}^1$ has an intricate dependence on hypermultiplet bare masses. 
  Extending recent work of Kachru and Silverstein, we consider ``orbifolds'' of 4-dimensional $\CN=4$ SU(n) super-Yang-Mills theories with respect to discrete subgroups of the SU(4) $R$-symmetry which act nontrivially on the gauge group. We show that for every discrete subgroup of SU(4) there is a canonical choice of imbedding of the discrete group in the gauge group which leads to theories with a vanishing one-loop beta-function. We conjecture that these give rise to (generically non-supersymmetric) conformal theories. The gauge group is $\otimes_i SU(Nn_i)$ where $n_i$ denote the dimension of the irreducible representations of the corresponding discrete group; there is also bifundamental matter, dictated by associated quiver diagrams. The interactions can also be read off from the quiver diagram. For SU(3) and SU(2) subgroups this leads to superconformal theories with $\CN=1$ and $\CN=2$ respectively. In the $\CN=1$ case we prove the vanishing of the beta functions to two loops. 
  We compute the amount of inflation required to solve the horizon problem of cosmology in the pre-big-bang scenario. First we give a quick overview of string cosmology as developed by Veneziano and collaborators. Then we show that the amount of inflation in this background solves the horizon problem. We discuss fine-tuning. 
  $BF$ theories defined over non trivial line bundles are studied. It is shown that such theories describe a realization of a non trivial higher order bundle. The partition function differs from the usual one -in terms of the Ray Singer Torsion- by a factor that arises from the non triviality of the line bundles. 
  Cosmological perturbation equations derived from low-energy effective actions are shown to be invariant under a duality transformation reminiscent of electric-magnetic, strong-weak coupling, S-duality. A manifestly duality-invariant approximation for perturbations far outside the horizon is introduced, and it is argued to be useful even during a high curvature epoch. Duality manifests itself through a remnant symmetry acting on the classical moduli of cosmological models, and implying lower bounds on the number and energy density of produced particles. 
  In order to study quantum aspects of $\s$-models related by Poisson--Lie T-duality, we construct three- and two-dimensional models that correspond, in one of the dual faces, to deformations of $S^3$ and $S^2$. Their classical canonical equivalence is demonstrated by means of a generating functional, which we explicitly compute. We examine how they behave under the renormalization group and show that dually related models have the same 1-loop beta functions for the coupling and deformation parameters. We find non-trivial fixed points in the ultraviolet, where the theories do not become asymptotically free. This suggests that the limit of Poisson--Lie T-duality to the usual Abelian and non-Abelian T-dualities does not exist quantum mechanically, although it does so classically. 
  We consider a simple action for a fractional spin particle and a path integral representation for the propagator is obtained in a gauge such that the constraint embodied in the Lagrangian is not an obstacle. We obtain a propagator for the particle in a constant electromagnetic field via the path integral representation over velocities, which is characterized by arbitrary boundary conditions and the absence of time derivatives following integration over bosonic variables. 
  We discuss a scenario of ``the path to physics at the Planck scale'' where todays theory of the interactions of elementary particles, the so called Standard Model (SM), emerges as a low energy effective theory describing the long distance properties of a sub--observable medium existing at the Planck scale, which we call ``ether''. Properties of the ether can only be observable to the extent that they are relevant to characterize the universality class of the totality of systems which exhibit identical low energy behavior. In such a picture the SM must be embedded into a ``Gaussian extended SM'' (GESM), a quantum field theory (QFT) which includes the SM but is extended in such a way that it exhibits a quasi infrared (IR) stable fixed point in all its couplings. Some phenomenological consequences of such a scenario are discussed. 
  The branching functions of the affine superalgebra $sl(2/1)$ characters into characters of the affine subalgebra $sl(2)$ are calculated for fractional levels $k=1/u-1$, u positive integer. They involve rational torus $A_{u(u-1)}$ and $Z_{u-1}$ parafermion characters. 
  We investigate a version of fixed scalars for non-dilatonic branes which correspond to dilatations of the brane world-volume. We obtain a cross-section whose world-volume interpretation falls out naturally from an investigation of the breaking of conformal invariance by the irrelevant Born-Infeld corrections to Yang-Mills theory. From the same irrelevant world-volume operator we obtain the leading correction to the cross-sections of minimal scalars. This correction can be obtained in supergravity via an improved matching of inner and outer solutions to the minimal wave equation. 
  The four fermionic currents of the affine superalgebra $sl(2/1)$ at fractional level $k=1/u-1$, u positive integer, are shown to be realised in terms of a free scalar field, an $sl(2)$ doublet field and a primary field of the parafermionic algebra $Z_{u-1}$. 
  We derive the effective action for classical strings coupled to dilatonic, gravitational, and axionic fields. We show how to use this effective action for: (i) renormalizing the string tension, (ii) linking ultraviolet divergences to the infrared (long-range) interaction between strings, (iii) bringing additional light on the special cancellations that occur for fundamental strings, and (iv) pointing out the limitations of Dirac's celebrated field-energy approach to renormalization. 
  In these lectures we present a detailed description of various aspects of gauge theories with extended N=2 and N=4 supersymmetry that are at the basis of recently found exact results. These results include the exact calculation of the low energy effective action for the light degrees of freedom in the N=2 super Yang-Mills theory and the conjecture, supported by some checks, that the N=4 super Yang-Mills theory is dual in the sense of Montonen-Olive. 
  We consider the dimensional reduction of N=1 {SYM}_{2+1} to 1+1 dimensions. The gauge groups we consider are U(N) and SU(N), where N is finite. We formulate the continuum bound state problem in the light-cone formalism, and show that any normalizable SU(N) bound state must be a superposition of an infinite number of Fock states. We also discuss how massless states arise in the DLCQ formulation for certain discretizations. 
  It was proposed by Maldacena that the large $N$ limit of certain conformal field theories can be described in terms of supergravity on anti-De Sitter spaces (AdS). Recently, Gubser, Klebanov and Polyakov and Witten have conjectured that the generating functional for certain correlation functions in conformal field theory is given by the classical supergravity action on AdS. It was shown that the spectra of states of the two theories are matched and the two-point correlation function was studied. We discuss the interacting case and compare the three- and four-point correlation functions computed from a classical action on AdS with the large N limit of conformal theory. We discuss also the large N limit for the Wilson loop and suggest that singletons which according to Flato and Fronsdal are constituents of composite fields in spacetime should obey the quantum Boltzmann statistics. 
  Using the first order formalism (BFYM) of the Yang-Mills theory we show that it displays an embedded topological sector corresponding to the field content of the Topological Yang-Mills theory (TYM). This picture arises after a proper redefinition of the fields of BFYM and gives a clear representation of the non perturbative part of the theory in terms of the topological sector. In this setting the calculation of the $vev$ of a YM observable is translated into the calculation of a corresponding (non topological) quantity in TYM. We then compare the topological observables of TYM with a similar set of observables for BFYM and discuss the possibility of describing topological observables in YM theory. 
  We present a general method for constructing consistent quantum field theories with global symmetries. We start from a free non-interacting quantum field theory with given global symmetries and we determine all consistent perturbative quantum deformations assuming the construction is not obstructed by anomalies. The method is established within the causal Bogoliubov-Shirkov-Epstein-Glaser approach to perturbative quantum field theory (which leads directly to a finite perturbative series and does not rely on an intermediary regularization). Our construction can be regarded as a direct implementation of Noether's method at the quantum level. We illustrate the method by constructing the pure Yang-Mills theory (where the relevant global symmetry is BRST symmetry), and the N=1 supersymmetric model of Wess and Zumino. The whole construction is done before the so-called adiabatic limit is taken. Thus, all considerations regarding symmetry, unitarity and anomalies are well-defined even for massless theories. 
  We study D0-branes in type IIA on $T^2$ with a background B-field turned on. We calculate explicitly how the background B-field modifies the D0-brane action. The effect of the B-field is to replace ordinary multiplication with a noncommutative product. This enables us to find the matrix model for M-theory on $T^2$ with a background 3-form potential along the torus and the lightlike circle. This matrix model is exactly the non-local 2+1 dim SYM theory on a dual $T^2$ proposed by Connes, Douglas and Schwarz. We calculate the radii and the gauge coupling for the SYM on the dual $T^2$ for all choices of longitudinal momentum and membrane wrapping number on the $T^2$. 
  We consider mirrors of the spherical shape, that can expand or contract. Due to the excitation of the vacuum around, some spherical waves radiated from vibrating mirrors are encountered. Using experience from well-known literature on studies of two-dimensional conformal models, we adopt a similar framework to investigate such quantum phenomena in four dimensions. We calculate quantum averages of the energy-momentum tensor for s-wave approximation. 
  We consider N=1 supersymmetric SU(N_c) gauge theories, using the type IIB brane construction recently proposed by Hanany and Zaffaroni. At non-zero string coupling, we find that the bending of branes imposes consistency conditions that allow only non-anomalous gauge theories with stable vacua, i.e., N_f >= N_c, to be constructed. We find qualitative differences between the brane configurations for N_f <= 3N_c and N_f > 3N_c, corresponding to asymptotically free and infrared free theories respectively. We also discuss some properties of the brane configurations that may be relevant to constructing Seiberg's duality in this framework. 
  The supersymmetric Lagrangian compatible with the presence of torsion in the background spacetime requires, in addition to the minimal coupling, an interaction between the spin and the torsion of the form ${1/2} N_{\mu\nu\lambda\rho}\psi^\mu\psi^\nu\psi^\lambda\psi^\rho$, where $N$ is the Nieh-Yan 4-form. This gives rise to a coupling between helicity ($h$) and the Nieh-Yan density of the form $hN$. The classical Lagrangian allows computing the index for the Dirac operator on a four-dimensional compact manifold with curvature and torsion using the path integral representation for the index. This calculation provides an independent check for the recent result of the chiral anomaly in spaces with torsion. 
  A lattice gauge theory with an action polynomial in independent field variables is considered. The link variables are described by unconstrained complex matrices instead of unitary ones. A mechanism which permits to switch off in the continuous limit the arising extra fields is discussed. The Euclidean version of the theory with an 4-dimensional lattice is described. The canonical form of this theory in Lorentz coordinates with continuous time is given. The canonical formulation in the light-front coordinates is investigated for the lattice in 2-dimensional transverse space and for the 3-dimensional lattice including one of the light-like coordinates. The light-front zero mode problem is considered in the framework of this canonical formulation. 
  Recent work in the literature has studied fourth-order elliptic operators on manifolds with boundary. This paper proves that, in the case of the squared Laplace operator, the boundary conditions which require that the eigenfunctions and their normal derivative should vanish at the boundary lead to self-adjointness of the boundary-value problem. On studying, for simplicity, the squared Laplace operator in one dimension, on a closed interval of the real line, alternative conditions which also ensure self-adjointness set to zero at the boundary the eigenfunctions and their second derivatives, or their first and third derivatives, or their second and third derivatives, or require periodicity, i.e. a linear relation among the values of the eigenfunctions at the ends of the interval. For the first four choices of boundary conditions, the resulting one-loop divergence is evaluated for a real scalar field on the portion of flat Euclidean four-space bounded by a three-sphere, or by two concentric three-spheres. 
  We make a first step to extend to the supersymmetric arena the effective action method, which is used to covariantly deduce the low energy dynamics of topological defects directly from their parent field theory. By focussing on two-dimensional supersymmetric theories we are able to derive the appropriate change of variables that singles out the low energy degrees of freedom. These correspond to super-worldline embeddings in superspace which are subject to a geometrical constraint. We obtain a supersymmetric and $\kappa$--invariant low energy expansion, with the standard superparticle action as the leading term, which can be used for the determination of higher-order corrections. Our formulation fits quite naturally with the present geometrical description of $\kappa$--symmetry in terms of the so-called geometrodynamical constraints. It also provides a basis for the exploration of these issues in higher-dimensional supersymmetric theories. 
  We consider classical and quantum strings in the conformally invariant background corresponding to the SL(2,R) WZWN model. This background is locally anti-de Sitter spacetime with non-vanishing torsion. Conformal invariance is expressed as the torsion being parallelized. The precise effect of the conformal invariance on the dynamics of both circular and generic classical strings is extracted. In particular, the conformal invariance gives rise to a repulsive interaction of the string with the background which precisely cancels the dominant attractive term arising from gravity. We perform both semi-classical and canonical string-quantization, in order to see the effect of the conformal invariance of the background on the string mass spectrum. Both approaches yield that the high-mass states are governed by m sim HN (N,`large integer'), where m is the string mass and H is the Hubble constant. It follows that the level spacing grows proportionally to N: d(m^2 alpha')/dN sim N, while the entropy goes like: S sim sqrt{m}. Moreover, it follows that there is no Hagedorn temperature,so that the partition function is well defined at any positive temperature. All results are compared with the analogue results in Anti- de Sitter spacetime, which is a non conformal invariant background. Conformal invariance simplifies the mathematics of the problem but the physics remains mainly unchanged. Differences between conformal and non-conformal backgrounds only appear in the intermediate region of the string spectrum, but these differences are minor. For low and high masses, the string mass spectra in conformal and non-conformal backgrounds are identical. Interestingly enough, conformal invariance fixes the value of the spacetime curvature to be -69/(26 alpha'). 
  We prove the existence of a new class of BPS saturated M-branes. They are in one-to-one correspondence with the Freund--Rubin compactifications of M-theory on either (AdS_4) x (G/H) or (AdS_7) x (G/H), where G/H is the seven (or four) dimensional Einstein coset manifolds classified long ago in the context of Kaluza Klein supergravity. The G/H M-branes are solitons that interpolate between flat space at infinity and the old Kaluza-Klein compactifications at the horizon. They preserve N/2 supersymmetries where N is the number of Killing spinors of the (AdS) x (G/H) vacuum. A crucial ingredient in our discussion is the identification of a solvable Lie algebra parametrization of the Lorentzian non compact coset SO(2,p+1)/SO(1,p+1) corresponding to anti de Sitter space AdS_{p+2} . The solvable coordinates are those naturally emerging from the near horizon limit of the G/H p-brane and correspond to the Bertotti Robinson form of the anti-de-Sitter metric. The pull-back of anti-de-Sitter isometries on the p-brane world-volume contain, in particular, the broken conformal transformations recently found in the literature. 
  We derive the dynamics of M-brane intersections from the worldvolume action of one brane in the background supergravity solution of another one. In this way we obtain an effective action for the self-dual string boundary of an M2-brane in an M5-brane, and show that the dynamics of the 3-brane intersection of two M5-branes is described by a Dirac-Born-Infeld action. 
  Open branes ending on other branes, which may be referred to as the host branes, are studied in the superembedding formalism. The open brane, host brane and the target space in which they are both embedded are all taken to be supermanifolds. It is shown that the superspace constraints satisfied by the open brane are sufficient to determine the corresponding superspace constraints for the host branes, whose dynamics are determined by these constraints. As a byproduct, one also obtains information about the boundary of the open brane propagating in the host brane. 
  We consider the twisted N = 4 SYM on \Sigma \times S^2. In the limit that S^2 shrinks to zero size the four dimensional theory reduces to a two dimensional SYM theory. We compute the correlation functions of a set of BRST cohomology classes in the reduced theory perturbed by mass. 
  We obtain the Hausdorff dimension, $h=2-2s$, for particles with fractional spins in the interval, $0\leq s \leq 0.5$, such that the manifold is characterized by a topological invariant given by, ${\cal W}=h+2s-2p$. This object is related to fractal properties of the path swept out by fractional spin particles, the spin of these particles, and the genus (number of anyons) of the manifold. We prove that the anyonic propagator can be put into a path integral representation which gives us a continuous family of Lagrangians in a convenient gauge. The formulas for, $h$ and ${\cal W}$, were obtained taking into account the anyon model as a particle-flux system and by a qualitative inference of the topology. 
  We consider the compactification of Matrix theory on tori with background antisymmetric tensor field. Douglas and Hull have recently discussed how noncommutative geometry appears on the tori. In this paper, we demonstrate the concrete construction of this compactification of Matrix theory in a similar way to that previously given by Taylor. 
  We find a representation for the determinant of a Dirac operator in an even number $D= 2 n$ of Euclidean dimensions as an overlap between two different vacua, each one corresponding to a bosonic theory with a quadratic action in $2 n + 1$ dimensions, with identical kinetic terms, but differing in their mass terms. This resembles the overlap representation of a fermionic determinant (although bosonic fields are used here). This representation may find applications to lattice field theory, as an alternative to other bosonized representations for Dirac determinants already proposed. 
  Confining strings in 4D are effective, thick strings describing the confinement phase of compact U(1) and, possibly, also non-Abelian gauge fields. We show that these strings are dual to the gauge fields, inasmuch as their perturbative regime corresponds to the strong coupling (large e) regime of the gauge theory. In this regime they describe smooth surfaces with long-range correlations and Hausdorff dimension two. For lower couplings e and monopole fugacities z, a phase transition takes place, beyond which the smooth string picture is lost. On the critical line intrinsic distances on the surface diverge and correlators vanish, indicating that world-sheets become fractal. 
  We study the gauge invariant fermions in the fermion coset representation of $SU(N)_k$ Wess-Zumino-Witten models which create, by construction, the physical excitations (quasiparticles) of the theory. We show that they provide an explicit holomorphic factorization of $SU(N)_k$ Wess-Zumino-Witten primaries and satisfy non-Abelian braiding relations. 
  We study the properties of wrapped membranes in matrix theory on ALE spaces. We show that the only BPS bound states of wrapped membranes that can form are roots of the $A$-$D$-$E$ group. We determine a bound on the energy of a bound state and find the correct dependence on the blow-up parameters and longitudinal momentum expected from M-Theory. For the $A_{n-1}$ series, we construct explicit classical solutions for the wrapped membrane bound states. These states have a very rich structure and have a natural interpretation in terms of noncommutative geometry. In the $A_1$ case, we examine the spectrum of excitations around the wrapped membrane solution and provide an explicit calculation of their energies. The results agree exactly with supergravity calculations. 
  We present some obvious physical requirements on gravitational avatars of non-linear electrodynamics and illustrate them with explicit determinantal Born-Infeld-Einstein models. A related procedure, using compensating Weyl scalars, permits us to formulate conformally invariant versions of these systems as well. 
  Novel 3+1 dimensional N=2 superconformal field theories (with tensionless BPS string solitons) are believed to arise when two sets of M5 branes intersect over a 3+1 dimensional hyperplane. We derive a DLCQ description of these theories as supersymmetric quantum mechanics on the Higgs branch of suitable 4d N=1 supersymmetric gauge theories. Our formulation allows us to determine the scaling dimensions of certain chiral primary operators in the conformal field theories. We also discuss general criteria for quantum mechanical DLCQ descriptions of supersymmetric field theories (and the resulting multiplicities and scaling dimensions of chiral primary operators). 
  We study the large N limit of the interacting superconformal field theories associated with N M5 branes or M2 branes using the recently proposed relation between these theories and M theory on AdS spaces. We first analyze the spectrum of chiral operators of the 6d (0,2) theory associated with M5 branes in flat space, and find full agreement with earlier results obtained using its DLCQ description as quantum mechanics on a moduli space of instantons. We then perform a similar analysis for the D_N type 6d (0,2) theories associated with M5 branes at an R^5/Z_2 singularity, and for the 3d N=8 superconformal field theories associated with M2 branes in flat space and at an R^8/Z_2 singularity respectively. Little is known about these three theories, and our study yields for the first time their spectrum of chiral operators (in the large N limit). 
  We obtain effective equations of inflationary dynamics for the mean inflaton and metric fields in the no-boundary and tunneling quantum states of the Universe. In the slow roll approximation (taking the form of the local Schwiger-DeWitt expansion) effective equations follow from the Euclidean effective action on the DeSitter gravitational instanton. Effective equations are applied in the model of the inflaton scalar field coupled to the GUT sector of matter fields and also having a strong nonminimal coupling to the curvature. The inverse of its big negative nonminimal coupling constant, serves as a small parameter of the slow roll expansion and semiclassical expansion of quantum gravitational effects. As a source of initial conditions we use a sharp probability peak recently obtained in the one-loop approximation for the no-boundary and tunneling quantum states and belonging (in virtue of a strong nonminimal coupling) to the GUT energy scale much below the Planck scale. The obtained equations in the tunneling quantum state predict a finite duration of inflationary stage compatible with the observational status of inflation theory, whereas for the no-boundary state they lead to the infinite inflationary epoch with a constant inflaton field. 
  1 identify a correspondence between the various spherical harmonic modes of massless 11 dimensional fields propagating on the $AdS_{4/7}$ in an $AdS_{4/7} \times S^{7\4}$ compactification of M theory, and the corresponding operators, primary under the conformal group, on the world volume of $M_2,M_5$ branes. This is achieved by matching representations of the superconformal algebra on the two sides of the correspondence. 
  We study the introduction of orientifold six-planes in the type IIA brane configurations known as elliptic models. The N=4 SO(n) and $Sp(k)$ theories softly broken to N=2 through a mass for the adjoint hypermultiplet can be realized in this framework in the presence of two orientifold planes with opposite RR charge. A large class of $\b=0$ models is solved for vanishing sum of hypermultiplet masses by embedding the type IIA configuration into M-theory. We also find a geometric interpretation of Montonen-Olive duality based on the properties of the curves. We make a proposal for the introduction of non-vanishing sum of hypermultiplet masses in a sub-class of models.   In the presence of two negatively charged orientifold planes and four D6-branes other interesting $\beta=0$ theories are constructed, e.g. $Sp(k)$ with four flavours and a massive antisymmetric hypermultiplet. We comment on the difficulties in obtaining the curves within our framework due to the arbitrary positions of the D6-branes. 
  A comment is given to the reply of Kraemmer and Rebhan (hep-th/9711075) to our paper (hep-th/9710131). 
  The truncation scheme dependence of the exact renormalization group equations is investigated for scalar field theories in three dimensions. The exponents are numerically estimated to the next-to-leading order of the derivative expansion. It is found that the convergence property in various truncations in the number of powers of the fields is remarkably improved if the expansion is made around the minimum of the effective potential. It is also shown that this truncation scheme is suitable for evaluation of infrared effective potentials. The physical interpretation of this improvement is discussed by considering O(N) symmetric scalar theories in the large-N limit. 
  For massive and conformal quantum field theories in 1+1 dimensions with a global gauge group we consider soliton automorphisms, viz. automorphisms of the quasilocal algebra which act like two different global symmetry transformations on the left and right spacelike complements of a bounded region. We give a unified treatment by providing a necessary and sufficient condition for the existence and Poincare' covariance of soliton automorphisms which is applicable to a large class of theories. In particular, our construction applies to the QFT models with the local Fock property -- in which case the latter property is the only input from constructive QFT we need -- and to holomorphic conformal field theories. In conformal QFT soliton representations appear as twisted sectors, and in a subsequent paper our results will be used to give a rigorous analysis of the superselection structure of orbifolds of holomorphic theories. 
  We present the way the Lorentz invariant canonical partition function for Matrix Theory as a light-cone formulation of M-theory can be computed. We explicitly show how when the eleventh dimension is decompactified, the N = 1 eleven dimensional SUGRA partition function appears. We also provide a high temperature expansion which captures some structure of the canonical partition function when interactions amongst D-particles are on. The connection with the semi-classical computations thermalizing the open superstrings attached to a D-particle is also clarified through a Born-Oppenheimer approximation. Some ideas about how Matrix Theory would describe the complementary degrees of freedom of the massless content of eleven dimensional SUGRA are discussed. Comments about possible connections to black hole physics are also made. 
  Integrable models of 1+1 dimensional gravity coupled to scalar and vector fields are briefly reviewed. A new class of integrable models with nonminimal coupling to scalar fields is constructed and discussed. 
  We consider D3 branes world-volume theories substaining $N=1,2$ superconformal field theories. Under the assumption that these theories are dual to $N=2,4$ supergravities in $AdS_5$, we explore the general structure of the latter and discuss some issues when comparing the bulk theory to the boundary singleton theory. 
  Noting that T-duality untwists S^5 to CP^2 x S^1, we construct the duality chain: n=4 super Yang-Mills --> Type IIB superstring on AdS_5 x S^5 --> Type IIA superstring on AdS_5 x CP^2 x S^1 --> M-theory on AdS_5 x CP^2 x T^2. This provides another example of supersymmetry without supersymmetry: on AdS_5 x CP^2 x S^1, Type IIA supergravity has SU(3) x U(1) x U(1) x U(1) and N=0 supersymmetry but Type IIA string theory has SO(6) and N=8. The missing superpartners are provided by stringy winding modes. We also discuss IIB compactifications to AdS_5 with N=4, N=2 and N=0. 
  Using the method of Green's functions within a Bethe-Salpeter framework characterized by a pairwise qq interaction with a Lorentz-covariant 3D support to its kernel, the 4D BS wave function for a system of 3 identical relativistic spinless quarks is reconstructed from the corresponding 3D form which satisfies a fully connected 3D BSE. This result is a 3-body generalization of a similar 2-body result found earlier under identical conditions of a 3D support to the corresponding qq-bar BS kernel under Covariant Instaneity (CIA for short). (The generalization from spinless to fermion quarks is straightforward).         To set the CIA with 3D BS kernel support ansatz in the context of contemporary approaches to the qqq baryon problem, a model scalar 4D qqq BSE with pairwise contact interactions to simulate the NJL-Faddeev equations is worked out fully, and a comparison of both vertex functions shows that the CIA vertex reduces exactly to the NJL form in the limit of zero spatial range. This consistency check on the CIA vertex function is part of a fuller accounting for its mathematical structure whose physical motivation is traceable to the role of `spectroscopy' as an integral part of the dynamics. 
  We give an outline of a recent proof that the low-energy effective gauge theory exhibiting quark confinement due to magnetic monopole condensation can be derived from QCD without any specific assumption. We emphasize that the low-energy effective abelian gauge theories obtained here give the dual description of the same physics in the low-energy region. They show that the QCD vacuum is nothing but the dual (type II) superconductor. 
  The statistical entropy of a five-dimensional black hole in Type II string theory was recently derived by showing that it is U-dual to the three-dimensional Banados-Teitelboim-Zanelli black hole, and using Carlip's method to count the microstates of the latter. This is valid even for the non-extremal case, unlike the derivation which relies on D-brane techniques. In this letter, I shall exploit the U-duality that exists between the five-dimensional black hole and the two-dimensional charged black hole of McGuigan, Nappi and Yost, to microscopically compute the entropy of the latter. It is shown that this result agrees with previous calculations using thermodynamic arguments. 
  Using standard field theory techniques we compute perturbative and instanton contributions to the Coulomb branch of three-dimensional supersymmetric QCD with N=2 and N=4 supersymmetry and gauge group SU(2). For the N=4 theory with one massless flavor, we confirm the proposal of Seiberg and Witten that the Coulomb branch is the double-cover of the centered moduli space of two BPS monopoles constructed by Atiyah and Hitchin. Introducing a hypermultiplet mass term, we show that the asymptotic metric on the Coulomb branch coincides with the metric on Dancer's deformation of the monopole moduli space. For the N=2 theory with $N_f$ flavors, we compute the one-loop corrections to the metric and complex structure on the Coulomb branch. We then determine the superpotential including one-loop effects around the instanton background. These calculations provide an explicit check of several results previously obtained by symmetry and holomorphy arguments. 
  Recently interest in using generalized reductions to construct massive supergravity theories has been revived in the context of M-theory and superstring theory. These compactifications produce mass parameters by introducing a linear dependence on internal coordinates in various axionic fields. Here we point out that by extending the form of this simple ansatz, it is always possible to introduce the various mass parameters simultaneously. This suggests that the various ``distinct'' massive supergravities in the literature should all be a part of a single massive theory. 
  We derive the noncommutative torus compactification of M(atrix) theory directly from the string theory by imposing mixed boundary conditions on the membranes. The relation of various dualities in string theory and M(atrix) theory compactification on the noncommutative torus are studied. 
  We discuss the large N limit of the (2,0) field theory in six dimensions. We do this by assuming the validity of Maldacena's conjecture of the correspondence between large N gauge theories and supergravity backgrounds, here $AdS_7\times S^4$. We review the spectrum of the supergravity theory and compute the spectrum of primary operators of the conformal algebra of arbitrary spin. 
  We study BPS monopoles in 4 dimensional N=4 SO(N) and $Sp(N)$ super Yang-Mills theories realized as the low energy effective theory of $N$ (physical and its mirror) parallel D3 branes and an {\it Orientifold 3 plane} with D1 branes stretched between them in type IIB string theory. Monopoles on D3 branes give the natural understanding by embedding in SU(N) through the constraints on both the asymptotic Higgs field (corresponding to the horizontal positions of D3 branes) and the magnetic charges (corresponding to the number of D1 branes) imposed by the O3 plane. The compatibility conditions of Nahm data for monopoles for these groups can be interpreted very naturally through the D1 branes in the presence of O3 plane. 
  We study the time development of correlation functions at both zero and finite temperature with Shibata-Hashitsume's projection operator method and carry out the renormalization of ultraviolet divergence that appears in a time-dependent frequency shift, using a mass counter term. A harmless divergence of log t-type remains at an initial time t=0 after the lowest order renormalization. 
  We study the logarithmic superconformal field theories. Explicitly, the two-point functions of N=1 logarithmic superconformal field theories (LSCFT) when the Jordan blocks are two (or more) dimensional, and when there are one (or more) Jordan block(s) have been obtained. Using the well known three-point fuctions of N=1 superconformal field theory (SCFT), three-point functions of N=1 LSCFT are obtained. The general form of N=1 SCFT's four-point functions is also obtained, from which one can easily calculate four-point functions in N=1 LSCFT. 
  In this article we review four-dimensional string vacua with N=2 space-time supersymmetry. In particular, we will discuss several aspects of the string-string duality between the heterotic string, compactified on $K3\times T^2$, and the type II superstring compactified on a Calabi-Yau three-fold. We investigate the massless supersymmetric spectra, showing agreement for a large class of dual heterotic/type II string pairs. Some emphasis is given to non-perturbative heterotic phenomena, such as non-perturbative transitions among different vacua and strong coupling singularities, and to their geometric Calabi-Yau description on the type II side. We compare the effective N=2 supergravity actions of dual heterotic/type II string compactifications, and show that the N=2 prepotentials and also higher order gravitational couplings nicely agree in the weak heterotic coupling limit. Finally we consider extremal black hole solutions of N=2 supergravity which arise in the context of heterotic or type II N=2 compactifications. For the type II backgrounds we show how the entropies of these black holes depend on the topological data of the underlying Calabi-Yau spaces; we also construct massless black holes which are relevant for the conifold transition among different Calabi-Yau vacua. 
  We derive model independent, non-perturbative supersymmetric sum rules for the magnetic and electric multipole moments of any theory with N=1 supersymmetry. We find that in any irreducible N=1 supermultiplet the diagonal matrix elements of the l-multipole moments are completely fixed in terms of their off-diagonal matrix elements and the diagonal (l-1)-multipole moments. 
  The non-perturbative Schwinger-Dyson equation is used to show that chiral symmetry is dynamically broken in QED at weak gauge couplings when an external uniform magnetic field is present. A complete analysis of this phenomenon may shed light on the Pauli problem, namely, why $\alpha$ = 1/137. 
  I discuss the general principles underlying quantum field theory, and attempt to identify its most profound consequences. The deepest of these consequences result from the infinite number of degrees of freedom invoked to implement locality. I mention a few of its most striking successes, both achieved and prospective. Possible limitations of quantum field theory are viewed in the light of its history. 
  We consider string perturbative expansion in the presence of D-branes imbedded in orbifolded space-time. In the regime where the string coupling is weak and $\alpha'\to 0$, the string perturbative expansion coincides with `t Hooft's large N expansion. We specifically concentrate on theories with d=4 and ${\cal N}=0,1,2,4$, and use world-sheet orbifold techniques to prove vanishing theorems for the field theory beta functions to all orders in perturbation theory in the large N limit. This is in accord with recent predictions. 
  We calculate the dimensions of operators in three and six dimensional superconformal field theories by using the duality between these theories at large N and D=11 supergravity on $AdS_{4/7} \times S^{7/4}$. We find that for the duality relations to work the Kaluza--Klein masses given in the supergravity literature must be rescaled and/or shifted. 
  Recently a method has been developed for relating four dimensional Schwarzschild black holes in M-theory to near-extremal black holes in string theory with four charges, using suitably defined ``boosts'' and T-dualities. We show that this method can be extended to obtain the emission rate of low energy massless scalars for the four dimensional Schwarzschild hole from the microscopic picture of radiation from the near extremal hole. 
  We consider the (2+1)-dimensional massive Thirring model as a gauge theory, with one fermion flavor, in the framework of the causal perturbation theory and address the problem of dynamical mass generation for the gauge boson. In this context we get an unambiguous expression for the coefficient of the induced Chern-Simons term. 
  We study the dynamical behavior of the BTZ (Banados-Teitelboim-Zanelli) black hole with the low-energy string effective action. The perturbation analysis around the BTZ black hole reveals a mixing between the dilaton and other fields. Introducing the new gauge (dilaton gauge), we disentangle this mixing completely and obtain one decoupled dilaton equation. We obtain the decay rate $\Gamma$ of BTZ black hole. 
  We consider two-dimensional Brans-Dicke theory to study the initial singularity problem. It turns out that the initial curvature singularity can be finite for a certain Brans-Dicke constant $\omega$ by considering the quantum back reaction of the geometry. For $\omega=1$, the universe starts with the finite curvature scalar and evolves into the flat spacetime. Furthermore the divergent gravitational coupling at the initial time can be finite effectively with the help of quantum correction. The other type of universe is studied for the case of $0<\omega<1$. 
  We calculate the absorption cross section on a black threebrane of two-form perturbations polarized along the brane. The equations are coupled and we decouple them for s-wave perturbations. The Hawking rate is suppressed at low energies, and this is shown to be reflected in the gauge theory by a coupling to a higher dimension operator. 
  Four-dimensional magnetic black holes including dilaton and abelian gauge fields which are solutions of supergravity can also be obtained by dimensional reduction of the Einstein-Maxwell gravity in five dimensions. In the extremal case the five-dimensional solutions have horizon and their near-horizon geometry is $AdS_3\times S^2 $. In the non-extremal case the near-horizon geometry is shown to be the product of the three-dimensional Ba\~nados-Teitelboim-Zanelli black hole and $S^2$. This allows to perform microscopic counting of statistical entropy of magnetic black holes. Exact agreement with the geometric entropy is found.   The microstates responsible for statistical entropy are located in the near-horizon region. 
  Singular instantons of the type introduced by Hawking and Turok (hep-th/9802030) lead to unacceptable physical consequences and cannot, therefore, be used to describe the creation of open universes. 
  We point out that the recent conjecture relating large N gauge theories to string theory in anti-de Sitter spaces offers a resolution in principle of many problems in black hole physics. This is because the gauge theory also describes spacetimes which are not anti-de Sitter, and include black hole horizons and curvature singularities. 
  We show how to use D and NS fivebranes in Type IIB superstring theory to construct large classes of finite N=1 supersymmetric four dimensional field theories. In this construction, the beta functions of the theories are directly related to the bending of branes; in finite theories the branes are not bent, and vice versa. Many of these theories have multiple dimensionless couplings. A group of duality transformations acts on the space of dimensionless couplings; for a large subclass of models, this group always includes an overall $SL(2,\ZZ)$ invariance. In addition, we find even larger classes of theories which, although not finite, also have one or more marginal operators. 
  We study the low-energy dynamics of all N=1 supersymmetric gauge theories whose basic gauge invariant fields are unconstrained. This set includes all theories whose matter Dynkin index is less than the index of the adjoint representation. We study the dynamically generated superpotential in these theories, and show that there is a W=0 branch if and only if anomaly matching is satisfied at the origin. An interesting example studied in detail is SO(13) with a spinor, a theory with a dynamically generated W and no anomaly matching at the origin. It flows via the Higgs mechanism to SU(6) with a three-index antisymmetric tensor, a theory with a W=0 branch and anomaly matching at the origin. 
  We develop a matrix model for the SO(32) Heterotic string with certain Wilson lines on the lightlike circle. This is done by using appropriate T-dualities. The method works for an infinite number of Wilson lines, but not for all. The matrix model is the theory on the D-string of type I wrapped on a circle with a SO(32) Wilson line on the circle. The radius of the circle depends on the Wilson line. This is a 1+1 dimensional O(N) theory on a circle. N depends not only on the momentum around the lightlike circle but also on the winding and SO(32) quantum numbers of the state. Perspectives for obtaining a matrix model for all Wilson lines are discussed. 
  Scalar field theory with an asymmetric potential is studied at zero temperature and high-temperature for phi^4 theory with both phi and phi^3 symmetry breaking. The equations of motion are solved numerically to obtain O(4) symmetric and O(3) cylindrical symmetric bounce solutions. These solutions control the rates for tunneling from the false vacuum to the true vacuum by bubble formation. The range of validity of the thin-wall approximation (TWA) is investigated. An analytical solution for the bounce is presented, which reproduces the action in the thin-wall as well as the thick-wall limits. 
  The large N limit of D3-branes is expected to correspond to a superconformal field theory living on the boundary of the anti-de Sitter space appearing in the near-horizon geometry. Dualizing the D3-brane to a D-instanton, we show that this limit is equivalent to a type IIB S-duality. In both cases one effectively reaches the near-horizon geometry. This provides an alternative approach to an earlier derivation of the same result that makes use of the properties of a gravitational wave instead of the D-instanton. 
  In each of the 10 cases with propagators of unit or zero mass, the finite part of the scalar 3-loop tetrahedral vacuum diagram is reduced to 4-letter words in the 7-letter alphabet of the 1-forms $\Omega:=dz/z$ and $\omega_p:=dz/ (\lambda^{-p}-z)$, where $\lambda$ is the sixth root of unity. Three diagrams yield only $\zeta(\Omega^3\omega_0)=1/90\pi^4$. In two cases $\pi^4$ combines with the Euler-Zagier sum $\zeta(\Omega^2\omega_3\omega_0)=\sum_{m> n>0}(-1)^{m+n}/m^3n$; in three cases it combines with the square of Clausen's $Cl_2(\pi/3)=\Im \zeta(\Omega\omega_1)=\sum_{n>0}\sin(\pi n/3)/n^2$. The case with 6 masses involves no further constant; with 5 masses a Deligne-Euler-Zagier sum appears: $\Re \zeta(\Omega^2\omega_3\omega_1)= \sum_{m>n>0}(-1)^m\cos(2\pi n/3)/m^3n$. The previously unidentified term in the 3-loop rho-parameter of the standard model is merely $D_3=6\zeta(3)-6 Cl_2^2(\pi/3)-{1/24}\pi^4$. The remarkable simplicity of these results stems from two shuffle algebras: one for nested sums; the other for iterated integrals. Each diagram evaluates to 10 000 digits in seconds, because the primitive words are transformable to exponentially convergent single sums, as recently shown for $\zeta(3)$ and $\zeta(5)$, familiar in QCD. Those are SC$^*(2)$ constants, whose base of super-fast computation is 2. Mass involves the novel base-3 set SC$^*(3)$. All 10 diagrams reduce to SC$^*(3)\cup$SC$^* (2)$ constants and their products. Only the 6-mass case entails both bases. 
  We establish conditions under which the worldsheet beta-functions of logarithmic conformal field theories can be derived as the gradient of some scalar function on the moduli space of running coupling constants. We derive a renormalization group invariant version of this function and relate it to the usual Zamolodchikov C-function expressed in terms of correlation functions of the worldsheet energy-momentum tensor. The results are applied to the example of D-brane recoil in string theory. 
  Several arguments are given for the summability of the superstring perturbation series. Whereas the Schottky group coordinatization of moduli space may be used to provide refined estimates of large-order bosonic string amplitudes, the super-Schottky group variables define a measure for the supermoduli space integral which leads to upper bounds on superstring scattering amplitudes. 
  We consider smeared zeta functions and heat-kernel coefficients on the bounded, generalized cone in arbitrary dimensions. The specific case of a ball is analysed in detail and used to restrict the form of the heat-kernel coefficients $A_n$ on smooth manifolds with boundary. Supplemented by conformal transformation techniques, it is used to provide an effective scheme for the calculation of the $A_n$. As an application, the complete $A_{5/2}$ coefficient is given. 
  N = 1, all-loop Finite Unified Theories (FUTs) are very interesting not only since they realize an old theoretical dream, but also due the remarkable predictive power of particular models as well as for providing candidates that might shed light in non-perturbative Physics.  Here we discuss (a) the recent developments concerning the soft supersymmetry breaking (SSB) sector of these theories and the resulting predictions in very interesting realistic models, and (b) the results of a recent search for duals of N=1, all-loop FUTs. 
  We show how an extremal Reissner-Nordstrom black hole can be obtained by wrapping a dyonic D3-brane on a Calabi-Yau manifold. In the orbifold limit T^6/Z_3, we explicitly show the correspondence between the solution of the supergravity equations of motion and the D-brane boundary state description of such a black hole. 
  We analyze scattering of string modes at string junctions of type IIB string theory. In the infrared limit, certain orthogonal linear combinations of the fields on the different strings satisfy either Dirichlet or Neumann boundary conditions. We confirm that the worldsheet theory of a general string network has eight conserved supercharges in agreement with target space BPS considerations. As an application, we obtain the band spectrum of some simple string lattices. 
  Some textbooks and reports claim that the Jacobian which arises in the discussion of the Faddeev-Popov method to quantize non-Abelian gauge theories and which is given by the derivative of the gauge fixing conditions over the gauge group parameters, is gauge invariant. Other references however prove the opposite. In this brief report we present a discussion about this matter. 
  A perturbative non-renormalization theorem is presented that applies to general supersymmetric theories, including non-renormalizable theories in which the $\int d^2\theta$ integrand is an arbitrary gauge-invariant function $F(\Phi,W)$ of the chiral superfields $\Phi$ and gauge field-strength superfields $W$, and the $\int d^4\theta$-integrand is restricted only by gauge invariance. In the Wilsonian Lagrangian, $F(\Phi,W)$ is unrenormalized except for the one-loop renormalization of the gauge coupling parameter, and Fayet-Iliopoulos terms can be renormalized only by one-loop graphs, which cancel if the sum of the U(1) charges of the chiral superfields vanishes. One consequence of this theorem is that in non-renormalizable as well as renormalizable theories, in the absence of Fayet-Iliopoulos terms supersymmetry will be unbroken to all orders if the bare superpotential has a stationary point. 
  In the previous paper (hep-th/9712161) it was shown that the nonlinear Born-Infeld field equations supplemented by the "dynamical condition" (certain boundary condition for the field along the particle's trajectory) define perfectly deterministic theory, i.e. particle's trajectory is determined without any equations of motion. In the present paper we show that this theory possesses mathematically consistent Lagrangian and Hamiltonian formulations. Moreover, it turns out that the "dynamical condition" is already present in the definition of the physical phase space and, therefore, it is a basic element of the theory. 
  The paper gives a quick account of the simplest cases of the Hitchin integrable systems and of the Knizhnik-Zamolodchikov-Bernard connection at genus 0, 1 and 2. In particular, we construct the action-angle variables of the genus 2 Hitchin system with group SL(2) by exploiting its relation to the classical Neumann integrable systems. 
  Local effective action is derived to describe Regge asymptotic of Yang-Mills theories. Local symmetries of the effective action originating from the gauge symmetry of the underlying Yang-Mills theory are studied. Multicomponent effective action is introduced to express the symmetry transformations as field transformations. The algebra of these symmetries is decomposed onto a semi-direct sum of commutative algebras and four copies of the gauge algebra of the underlying Yang-Mills theory. Possibility of existence of solitons corresponding to the commutative subalgebra of the symmetry algebra is mentioned. 
  We construct supergravity solutions describing branes (D2-branes or NS5-branes or waves) localized within D6-branes in the region close to the core of the D6-branes. Other similar string-theory and M-theory `near-core' localized solutions can be found by applying U-duality and/or lifting D=10 solutions to D=11. In particular, the D2-branes localized on D6-branes is T-dual to a special case of the background describing (D)strings localized on (D)5-branes and thus is also related to a localized intersection of M2-branes and M5-branes. D6+wave configuration is U-dual to a D0-brane localized on a Kaluza-Klein 5-brane or to the fundamental string intersecting a D5-brane with the point of intersection localized on the D5-brane. 
  We develop Truncated Conformal Space (TCS) technique for perturbations of c=1 Conformal Field Theories. We use it to give the first numerical evidence of the validity of the non-linear integral equation (NLIE) derived from light-cone lattice regularization at intermediate scales. A controversy on the quantization of Bethe states is solved by this numerical comparison and by using the locality principle at the ultra- violet fixed point. It turns out that the correct quantization for pure hole states is the one with half-integer quantum numbers originally proposed by Mariottini et al. Once the correct rule is imposed, the agreement between TCS and NLIE for pure hole states turns out to be impressive. 
  The Casimir energy of a solid ball (or cavity in an infinite medium) is calculated by a direct frequency summation using the contour integration. The dispersion is taken into account, and the divergences are removed by making use of the zeta function technique. The Casimir energy of a dielectric ball (or cavity) turns out to be positive, it being increased when the radius of the ball decreases. The latter eliminates completely the possibility of explaining, via the Casimir effect, the sonoluminescence for bubbles in a liquid. Besides, the Casimir energy of the air bubbles in water proves to be immensely smaller than the amount of the energy emitted in a sonoluminescent flash. The dispersive effect is shown to be inessential for the final result. 
  We present two novel derivations of the recently established $(-)^p$ factor in the charge quantization condition for $p$-brane dyon sources in spacetime dimension $D$=$2p$+2. The first requires consistency of the condition under the charge shifts produced by (generalized) $\theta$-terms. The second traces the sign difference between adjoining dimensions to compactification effects. 
  Dynamical supersymmetry breaking is a fascinating theoretical problem. It is also of phenomenological significance. A better understanding of this phenomenon can help in model building, which in turn is useful in guiding the search for supersymmetry. In this article, we review the recent developments in the field. We discuss a few examples, which allow us to illustrate the main ideas in the subject. In the process, we also show how the techniques of holomorphy and duality come into play. Towards the end we indicate how these developments have helped in the study of gauge mediated supersymmetry breaking. The review is intended for someone with a prior knowledge of supersymmetry who wants to find out about the recent progress in this field. 
  We study non-linear corrections to the low-energy description of the (2,0) theory. We argue for the existence of a topological correction term similar to the C3 wedge X8(R) in M-theory. This term can be traced to a classical effect in supergravity and to a one-loop diagram of the effective 4+1D Super Yang-Mills. We study other terms which are related to it by supersymmetry and discuss the requirements on the subleading correction terms from M(atrix)-theory. We also speculate on a possible fundamental formulation of the theory. 
  We consider superconformal field theories in three and six dimensions with eight supercharges which can be realized on the world-volume of M-theory branes sitting at orbifold singularities. We find that they should admit a N=4 and N=2 supergravity dual in AdS_4 and AdS_7, respectively. We discuss the characteristics of the corresponding gauged supergravities. 
  We have studied a problem of the tachyon mediated D-brane - D-brane annihilation from the underlying world-volume gauge field theory point of view. The initial state was chosen in the form of two D-branes crossing at a non-zero angle, which is non-supersymmetric configuration generically. This state was not a groundstate of the theory and the problem at hand was a model, where we had rather precise control over the behavior of the theory.Some applications of this model to the D-brane physics and conclusions about stability of several configurations were made. By taking a T-dual picture of this process on a T^2 torus, we derived known conclusion about the instability of a system consisting of D-0 and D-2 branes, and found its decay modes. 
  We compute the quark-monopole potential for ${\cal N}=4$ super Yang-Mills in the large $N$ limit. We find an attractive potential that falls off as 1/L and is manifestly invariant under $g\to 1/g$. The strength of the potential is less than the quark-antiquark and monopole-antimonopole potentials. 
  We construct D_k asymptotically locally flat gravitational instantons as moduli spaces of solutions of Nahm equations. This allows us to find their twistor spaces and Kahler potentials. 
  Heat kernel coefficients encode the short distance behavior of propagators in the presence of background fields, and are thus useful in quantum field theory. We present a Mathematica program for computing these coefficients and their derivatives, based on an algorithm by Avramidi\cite{Avramidi:npb91}. See http://fermi.pha.jhu.edu/~booth/HeatK/ for source and examples. 
  In this letter the explicit form of general two-point functions in affine SL(N) current algebra is provided for all representations, integrable or non-integrable. The weight of the conjugate field to a primary field of arbitrary weight is immediately read off. 
  The effect of retardation on dynamical mass generation in is studied, in the imaginary time formalism. The photon porarization tensor is evaluated to leading order in 1/N (N is the number of flavours), and simple closed form expressions are found for the fully retarded longitudinal and transverse propagators, which have the correct limit when T goes to zero. The resulting S-D equation for the fermion mass (at order 1/N) has an infrared divergence associated with the contribution of the transverse photon propagator; only the longitudinal contribution is retained, as in earlier treatments. For solutions of constant mass, it is found that the retardation reduces the value of the parameter r (the ratio of twice the mass to the critical temperature) from about 10 to about 6. The gap equation is then solved allowing for the mass to depend on frequency. It was found that the r value remained close to 6. Possibilities for including the transverse propagator are discussed. 
  This review considers the properties of classical solutions to supergravity theories with partially unbroken supersymmetry. These solutions saturate Bogomol'ny-Prasad-Sommerfield bounds on their energy densities and are the carriers of the $p$-form charges that appear in the supersymmetry algebra. The simplest such solutions have the character of $(p+1)$-dimensional Poincar\'e-invariant hyperplanes in spacetime, i.e. $p$-branes. Topics covered include the relations between mass densities, charge densities and the preservation of unbroken supersymmetry; interpolating-soliton structure; diagonal and vertical Kaluza-Klein reduction families; multiple-charge solutions and the four D=11 elements; duality-symmetry multiplets; charge quantisation; low-velocity scattering and the geometry of worldvolume supersymmetric $\sigma$-models; and the target-space geometry of BPS instanton solutions obtained by the dimensional reduction of static $p$-branes. 
  We discuss supersymmetric Yang-Mills theory dimensionally reduced to zero dimensions and evaluate the SU(2) and SU(3) partition functions by Monte Carlo methods. The exactly known SU(2) results are reproduced to very high precision. Our calculations for SU(3) agree closely with an extension of a conjecture due to Green and Gutperle concerning the exact value of the SU(N) partition functions. 
  We consider both closed and open integrable antiferromagnetic chains constructed with the SU(N)-invariant R matrix. For the closed chain, we extend the analyses of Sutherland and Kulish-Reshetikhin by considering also complex ``string'' solutions of the Bethe Ansatz equations. Such solutions are essential to describe general multiparticle excited states. We also explicitly determine the SU(N) quantum numbers of the states. In particular, the model has particle-like excitations in the fundamental representations [k] of SU(N), with k = 1, ..., N-1. We directly compute the complete two-particle S matrices for the cases [1] X [1] and [1] X [N-1]. For the open chain with diagonal boundary fields, we show that the transfer matrix has the symmetry SU(l) X SU(N-l) X U(1), as well as a new ``duality'' symmetry which interchanges l and N - l. With the help of these symmetries, we compute by means of the Bethe Ansatz for particles of types [1] and [N-1] the corresponding boundary S matrices. 
  We propose supergravity duals to non-trivial fixed points of the renormalization group in three dimensions with ADE global symmetries. All the fixed point symmetries are identified with space-time symmetries. 
  Banks, Fischler, Klebanov and Susskind have proposed a model for black hole thermodynamics based on the principle that the entropy is of order the number of particles at the phase transition point in a Boltzmann gas of D0-branes. We show that the deviations from Boltzmann scaling found in $d<6$ noncompact spatial dimensions have a simple explanation in the analysis of self-gravitating random walks due to Horowitz and Polchinski. In the special case of $d=4$ we find evidence for the onset of a phase transition in the Boltzmann gas analogous to the well-known Hagedorn transition in a gas of free strings. Our result relies on an estimate of the asymptotic density of states in a dilute gas of D0-branes. 
  In the no-boundary univers e the universe is created from an instanton. However, there does not exist any instanton for the ``realistic'' $FRW$ universe with a scalar field. The ``instanton'' leading to its quantum creation may be modified and reinterpreted as a constrained gravitational instanton. 
  The classical and quantum algebras of a class of conformal NA-Toda models are studied. It is shown that the $SL(2,R)_q$ Poisson brackets algebra generated by certain chiral and antichiral charges of the nonlocal currents and the global U(1) charge appears as an algebra of the symmetries of these models. 
  We consider brane configurations in elliptic models which represent softly broken N=4 USp(2 N_c) and SO(N_c) theory. We generalize the notion of the O4 plane, so that it is compatible with the symmetry in the covering space of the elliptic models. By using this notion of the O4 plane, we find the curve for softly broken N=4 USp(2 N_c) and that for SO(N_c) theory as infinite series expansions. For the USp case, we can present the expansion as a polynomial. 
  Limits of a system of N Dn-branes in which the bulk and string degrees of freedom decouple to leave a `matter' theory are investigated and, for n>4, either give a free theory or require taking $N \to \infty$. The decoupled matter theory is described at low energies by the $N \to \infty$ limit of n+1 dimensional \sym, and at high energies by a free type II string theory in a curved space-time. Metastable bound states of D6-branes with mass $M$ and D0-branes with mass $m$ are shown to have an energy proportional to $M^{1/3}m^{2/3}$ and decouple, whereas in matrix theory they only decouple in the large N limit. 
  It is well-known that the winding number of the Skyrmion can be identified as the baryon number. We show in this paper that this result can also be established using the Atiyah-Singer index theorem and spectral flow arguments. We argue that this proof suggests that there are light quarks moving in the field of the Skyrmion. We then show that if these light degrees of freedom are averaged out, the low energy excitations of the Skyrmion are in fact spinorial. A natural consequence of our approach is the prediction of a $(1/2)^{-}$ state and its excitations in addition to the nucleon and delta. Using the recent numerical evidence for the existence of Skyrmions with discrete spatial symmetries, we further suggest that the the low energy spectrum of many light nuclei may possess a parity doublet structure arising from a subtle topological interaction between the slow Skyrmion and the fast quarks. We also present tentative experimental evidence supporting our arguments. 
  We present a simple proof of the WDVV equations for the prepotential of four-dimensional N=2 supersymmetric Yang-Mills theory with all ADE gauge groups. According to our proof it is clearly seen that the WDVV equations in four dimensions have their origin in the associativity of the chiral ring in two-dimensional topological Landau-Ginzburg models. The WDVV equations for the BC gauge groups are also studied in the Landau-Ginzburg framework. We speculate about the topological field theoretic interpretation of the Seiberg-Witten solution of N=2 Yang-Mills theory. 
  We construct BPS saturated regular configurations of N=4 SU(3) supersymmetric Yang-Mills theory carrying non-parallel electric and magnetic charges. These field theory BPS states correspond to the string theory BPS states of 3-string junctions connecting three different D3-branes by regarding the N=4 supersymmetric Yang-Mills theory as an effective field theory on parallel D3-branes. 
  A canonical realization of the BMS (Bondi-Metzner-Sachs) algebra is given on the phase space of the classical real Klein-Gordon field .   By assuming the finiteness of the generators of the Poincar\'e group, it is shown that a countable set of conserved quantities exists (supertranslations); this set transforms under a particular Lorentz representation, which is uniquely determined by the requirement of having an invariant four-dimensional subspace, which corresponds to the Poincar\'e translations. This Lorentz representation is infinite-dimensional, non unitary, reducible and indecomposable. Its representation space is studied in some detail. It determines the structure constants of the infinite-dimensional canonical algebra of the Poincar\'e generators together with the infinite set of the new conserved quantities.   It is shown that this algebra is isomorphic with that of the BMS group. 
  We classify the possible discrete (finite) symmetries of two--dimensional critical models described by unitary minimal conformally invariant theories. We find that all but six models have the group Z_2 as maximal symmetry. Among the six exceptional theories, four have no symmetry at all, while the other two are the familiar critical and tricritical 3--Potts models, which both have an S_3 symmetry. These symmetries are the expected ones, and coincide with the automorphism groups of the Dynkin diagrams of simply--laced simple Lie algebras ADE. We note that extended chiral algebras, when present, are almost never preserved in the frustrated sectors. 
  We calculate finite temperature effects on a correlation function in the two dimensional supersymmetric nonlinear O(3) sigma model. The correlation function violates chiral symmetry and at zero temperature it has been shown to be a constant, which gives rise to a double-valued condensate. Within the bilinear approximation we find an exact result in a one-instanton background at finite temperature. In contrast to the result at zero temperature we find that the correlation function decays exponentially at large distances. 
  The correspondence between supergravity (and string theory) on $AdS$ space and boundary conformal field theory relates the thermodynamics of ${\cal N}=4$ super Yang-Mills theory in four dimensions to the thermodynamics of Schwarzschild black holes in Anti-de Sitter space. In this description, quantum phenomena such as the spontaneous breaking of the center of the gauge group, magnetic confinement, and the mass gap are coded in classical geometry. The correspondence makes it manifest that the entropy of a very large $AdS$ Schwarzschild black hole must scale ``holographically'' with the volume of its horizon. By similar methods, one can also make a speculative proposal for the description of large $N$ gauge theories in four dimensions without supersymmetry. 
  Bekenstein has proposed the bound S < pi M_P^2 L^2 on the total entropy S in a volume L^3. This non-extensive scaling suggests that quantum field theory breaks down in large volume. To reconcile this breakdown with the success of local quantum field theory in describing observed particle phenomenology, we propose a relationship between UV and IR cutoffs such that an effective field theory should be a good description of Nature. We discuss implications for the cosmological constant problem. We find a limitation on the accuracy which can be achieved by conventional effective field theory: for example, the minimal correction to (g-2) for the electron from the constrained IR and UV cutoffs is larger than the contribution from the top quark. 
  We show that the four-dimensional U(1) gauge theory in the continuum formulation has a confining phase (exhibiting area law of the Wilson loop) in the strong coupling region above a critical coupling $g_c$. This result is obtained by taking into account topological non-trivial sectors in U(1) gauge theory. The derivation is based on the reformulation of gauge theory as a deformation of a topological quantum field theory and subsequent dimensional reduction of the D-dimensional topological quantum field theory to the (D-2)-dimensional nonlinear sigma model. The topological quantum field theory part of the four-dimensional U(1) gauge theory is exactly equivalent to the two-dimensional O(2) nonlinear sigma model. The confining (resp. Coulomb) phase of U(1) gauge theory corresponds to the high (resp. low) temperature phase of O(2) nonlinear sigma model and the critical point $g_c$ is determined by the Berezinskii-Kosterlitz-Thouless phase transition temperature. The quark (charge) confinement in the strong coupling phase is caused by the vortex condensation. Thus the continuum gauge theory has the direct correspondence to the compact formulation of lattice gauge theory. 
  We study gauge theories in the context of a gravitational theory without the cosmological constant problem (CCP). The theory is based on the requirement that the measure of integration in the action is not necessarily $\sqrt{-g}$ but it is determined dynamically through additional degrees of freedom. Realization of these ideas in the framework of the first order formalism solves the CCP. Incorporation of a condensate of a four index field strength allows, after a conformal transformation to the Einstein frame, to represent the system of gravity and matter in the standard GR form. Now, however, the effective potential vanishes at a vacuum state due to the exact balance to zero of the gauge fields condensate and the original scalar fields potential. As a result it is possible to combine the solution of the CCP with: a) inflation and transition to a $\Lambda =0$ phase without fine tuning after a reheating period; b) spontaneously broken gauge unified theories (including fermions). The model opens new possibilities for a solution of the hierarchy problem. 
  Aspects of d=4, N=4 superconformal U(N) gauge theory are studied at finite temperature. Utilizing dual description of large $N$ and strong coupling limit via Type IIB string theory compactification on Schwarzschild anti-de Sitter spacetime, we study correlations of Wilson-Polyakov loops and heavy quark potential thereof. We find that the heavy quark potential is Coulomb-like and has a finite range, as expected for gauge theory in high temperature, deconfinement phase. The potential exhibits finite temperature scaling consistent with underlying conformal invariance. We also study isolated heavy quark on probe D3-brane world-volume and find supporting evidence that near extremal D3-branes are located at Schwarzschild horizon. 
  We study the integrable and supersymmetric massive $\hat\phi_{(1,3)}$ deformation of the tricritical Ising model in the presence of a boundary. We use constraints from supersymmetry in order to compute the exact boundary $S$-matrices, which turn out to depend explicitly on the topological charge of the supersymmetry algebra. We also solve the general boundary Yang-Baxter equation and show that in appropriate limits the general reflection matrices go over the supersymmetry preserving solutions. Finally, we briefly discuss the possible connection between our reflection matrices and boundary perturbations within the framework of perturbed boundary conformal field theory. 
  Using a proposal of Maldacena we compute in the framework of the supergravity description of N coincident D3 branes the energy of a quark anti-quark pair in the large N limit of U(N) N=4 SYM in four dimensions at finite temperature. 
  We review the oscillator construction of the unitary representations of noncompact groups and supergroups and study the unitary supermultiplets of OSp(1/32,R) in relation to M-theory. OSp(1/32,R) has a singleton supermultiplet consisting of a scalar and a spinor field. Parity invariance leads us to consider OSp(1/32,R)_L X OSp(1/32,R)_R as the "minimal" generalized AdS supersymmetry algebra of M-theory corresponding to the embedding of two spinor representations of SO(10,2) in the fundamental representation of Sp(32,R). The contraction to the Poincare superalgebra with central charges proceeds via a diagonal subsupergroup OSp(1/32,R)_{L-R} which contains the common subgroup SO(10,1) of the two SO(10,2) factors. The parity invariant singleton supermultiplet of OSp(1/32,R)_L \times OSp(1/32,R)_R decomposes into an infinite set of "doubleton" supermultiplets of the diagonal  OSp(1/32,R)_{L-R}. There is a unique "CPT self-conjugate" doubleton supermultiplet whose tensor product with itself yields the "massless" generalized AdS_{11} supermultiplets. The massless graviton supermultiplet contains fields corresponding to those of 11-dimensional supergravity plus additional ones. Assuming that an AdS phase of M-theory exists we argue that the doubleton field theory must be the holographic superconformal field theory in ten dimensions that is dual to M-theory in the same sense as the duality between the N=4 super Yang-Mills in d=4 and the IIB superstring over AdS_5 X S^5. 
  We complete the process of classifying all supersymmetric theories with quantum modified moduli. We present all the supersymmetric gauge theories based on a simple orthogonal or exceptional group that exhibit a quantum modified moduli space. The quantum modified constraints of theories derived from s-confining theories are invariant under all symmetries. However, theories that cannot be obtained by a deformation of an s-confining theory may have constraints that are covariant, rather than invariant. 
  We examine the effects of instantons in four dimensional N=1 supersymmetric gauge theory by including D0-branes in type IIA brane constructions. We examine instanton generated superpotentials in supersymmetric QCD and find that they are due to a repulsive force between D4-branes bound to D0-branes ending on NS 5-branes. We study situations where instanton effects break supersymmetry such as the Intriligator-Thomas-Izawa-Yangagida model and relate this to a IIA brane construction. We also argue how confinement due to a condensate of fractional instantons manifests itself in Super Yang-Mills theory using fractional D0 branes, D4 branes, and NS strings. 
  We consider six dimensional N=1 space-time supersymmetric Type IIB orientifolds with non-zero untwisted NS-NS sector B-field. The B-field is quantized due to the requirement that the Type IIB spectrum be left-right symmetric. The presence of the B-field results in rank reduction of both 99 and 55 open string sector gauge groups. We point out that in some of the models with non-zero B-field there are extra tensor multiplets in the Z_2 twisted closed string sector, and we explain their origin in a simple example. Also, the 59 open string sector states come with a multiplicity that depends on the B-field. These two facts are in accord with anomaly cancellation requirements. We point out relations between various orientifolds with and without the B-field, and also discuss the F-theory duals of these models. 
  We study the BPS spectrum of the theory on a D3-brane probe in F theory. The BPS states are realized by multi-string configurations in spacetime. Only certain configurations obeying a selection rule give rise to BPS states in the four-dimensional probe theory. Using these string configurations, we determine the spectrum of N=2 SU(2) Yang-Mills. We also explore the relation between multi-string configurations, M theory membranes and self-dual strings. 
  In this paper we discuss quantum modified moduli spaces in supergravity. We examine a model suggested by Izawa and Yanagida and by Intriligator and Thomas that breaks global supersymmetry by a quantum deformation of the classical moduli space. We determine the minimum of the supergravity potential when the gauge coupling is taken to depend on a dynamical field, typically a modulus of string theory. We find that the only minimum is at the trivial configuration of vanishing coupling constant and unbroken supersymmetry. We also discuss models involving more complicated superpotentials and find that the gauge coupling is only stabilized in a supersymmetric ground state. 
  Using the Worldline formalism of QED we compute the two-loop effective action induced by a charged scalar, respectively spinor particle in a general constant electromagnetic field. 
  We propose $SL(2,Z)$ (and $SL(3,Z)$) invariant conjectures for all $R^4 H^{4g-4}$ couplings of Type IIB strings on $R^{10}$ (and $R^{8}\times T^2$), generalizing conjectures of Green and Gutperle (and Kiritsis and Pioline) for the $R^4$ coupling. A strong check for our conjectures is that on $T^2$ at weak coupling, they reproduce the multiloop scattering amplitudes which had been previously computed using N=2 strings in the N=4 topological formalism. Applications to $(p,q)$ string production in a background $H$ field, generalizing Schwinger's computation for pair production in constant $F$ field, are suggested. 
  We study softly broken N=1 supersymmetric QCD with the gauge group $SU(N_c)$ and $N_f$ flavours of quarks for $N_f > N_c+1$. We investigate the phase structure of its dual theory adding generic soft supersymmetry breking terms, i.e. soft scalar masses, trilinear coupling terms of scalar fields and gaugino masses. It is found that the trilinear coupling terms play an improtant role in determining the potential minima. Also we compare softly broken original and dual theories in the broken phase. 
  We present a general analysis of the field theoretical properties which guarantee the recovery, at the renormalized level, of symmetries broken by regularization. We also discuss the anomalous case. 
  We study three-dimensional N=4 gauge theories with product gauge groups constructed from ADE Dynkin diagrams. One-loop corrections to the metric on the Coulomb branch are shown to coincide with the metric on the moduli space of well-seperated ADE monopoles. We propose that this correspondence is exact. 
  The large momentum expansion for the inverse propagator of the auxiliary field $\lambda(x)$ in the conformally invariant O(N) vector model is calculated to leading order in 1/N, in a strip-like geometry with one finite dimension of length $L$ for $2<d<4$. Its leading terms are identified as contributions from $\lambda(x)$ itself and the energy momentum tensor, in agreement with a previous calculation based on conformal operator product expansions. It is found that a non-trivial cancellation takes place by virtue of the gap equation. The leading coefficient of the energy momentum tensor contribution is shown to be related to the free energy density. 
  A matrix model is constructed to compute characteristic numbers of the space of subsets of $R^d $ with $N$ elements. This matrix model is found to be a constrained null dimensional reduction to a point of a Yang-Mills theory with anticommuting matter in $d+2$ dimensions. The constraints and equations of motion are similar to those found by Nishino and Sezgin in their 10+2 dimensional supersymmetric Yang-Mills equations. It is conjectured that the $d=10$ topological model is a twisted form of the Nishino-Sezgin model, dimensionally reduced to a point, due to SO(8) triality. Topological matrix models are constructed for non-commutative spaces, entirely in terms of algebras associated with such spaces. These models exhibit degrees of freedom analogous to massive modes in string theory. 
  The relation between the Dirac quantization condition of magnetic charge and the quantization of the Chern-Simons coefficient is obtained. It implies that in a (2+1)-dimensional QED with the Chern-Simons topological mass term and the existence of a magnetic monopole with magnetic charge $g$, the Chern-Simons coefficient must be also quantized, just as in the non-Abelian case. 
  We present string-like soliton solutions of three-dimensional gravity, coupled to a compact scalar field $x^{11}$ and Kaluza-Klein reduced on a circle. These solitons carry fractional magnetic flux with respect to the Kaluza-Klein gauge field. Summing over such ``Kaluza-Klein flux tubes'' is shown to imply summing over a subclass of three-dimensional topologies (Seifert manifolds). It is also shown to imply an area law for the Wilson loop of the Kaluza-Klein gauge field; the confined charge is nothing but Kaluza-Klein momentum. Applied to the membrane of M-theory, this is interpreted as ``dynamical wrapping'' of the M-brane around its eleventh embedding dimension $x^{11}$. 
  This paper is dedicated to formulate the interaction picture dynamics of the self-dual field minimally coupled to fermions. To make this possible, we start by quantizing the free self-dual model by means of the Dirac bracket quantization procedure. We obtain, as result, that the free self-dual model is a relativistically invariant quantum field theory whose excitations are identical to the physical (gauge invariant) excitations of the free Maxwell-Chern-Simons theory. The model describing the interaction of the self-dual field minimally coupled to fermions is also quantized through the Dirac bracket quantization procedure. One of the self-dual field components is found not to commute, at equal times, with the fermionic fields. Hence, the formulation of the interaction picture dynamics is only possible after the elimination of the just mentioned component. This procedure brings, in turns, two new interaction terms, which are local in space and time while non-renormalizable by power counting. Relativistic invariance is tested in connection with the elastic fermion-fermion scattering amplitude. We prove that all the non-covariant pieces in the interaction Hamiltonian are equivalent to the covariant minimal interaction of the self-dual field with the fermions. The high energy behavior of the self-dual field propagator corroborates that the coupled theory is non-renormalizable. Certainly, the self-dual field minimally coupled to fermions bears no resemblance with the renormalizable model defined by the Maxwell-Chern-Simons field minimally coupled to fermions. 
  We investigate the perturbative part of Seiberg's low-energy effective action of N=2 supersymmetric Yang-Mills theory in Wess-Zumino gauge in the conventional effective field theory technique. Using the method of constant field approximation and restricting the effective action with at most two derivatives and not more than four-fermion couplings, we show some features of the low-energy effective action given by Seiberg based on $U(1)_R$ anomaly and non-perturbative $\beta$-function arguments. 
  We study the graviton self-energy function in a general gauge, using a hard thermal loop expansion which includes terms proportional to T^4, T^2 and log(T). We verify explicitly the gauge independence of the leading T^4 term and obtain a compact expression for the sub-leading T^2 contribution. It is shown that the logarithmic term has the same structure as the ultraviolet pole part of the T=0 self-energy function. We argue that the gauge-dependent part of the T^2 contribution is effectively canceled in the dispersion relations of the graviton plasma, and present the solutions of these equations. 
  Fundamental theories of quantum gravity such as supergravity include a four form field strength which contributes to the cosmological constant. The inclusion of such a field into our theory of open inflation (hep-th/9802030) allows an anthropic solution to the cosmological constant problem in which the cosmological constant gives a small but non-negligible contribution to the density of today's universe. We include a discussion of the role of the singularity in our solution and a reply to Vilenkin's recent criticism (hep-th/9803084). 
  We present the exact formula for neutrino oscillations. By resorting to recent results of Quantum Field Theory of fermion mixing, we work out the Green's function formalism for mixed neutrinos. The usual quantum mechanical Pontecorvo formula is recovered in the relativistic limit. 
  We discuss the generalized Plateau problem in the 3+1 dimensional Schwarzschild background. This represents the physical situation, which could for instance have appeared in the early universe, where a cosmic membrane (thin domain wall) is located near a black hole. Considering stationary axially symmetric membranes, three different membrane-topologies are possible depending on the boundary conditions at infinity: 2+1 Minkowski topology, 2+1 wormhole topology and 2+1 black hole topology.    Interestingly, we find that the different membrane-topologies are connected via phase transitions of the form first discussed by Choptuik in investigations of scalar field collapse. More precisely, we find a first order phase transition (finite mass gap) between wormhole topology and black hole topology; the intermediate membrane being an unstable wormhole collapsing to a black hole. Moreover, we find a second order phase transition (no mass gap) between Minkowski topology and black hole topology; the intermediate membrane being a naked singularity.    For the membranes of black hole topology, we find a mass scaling relation analogous to that originally found by Choptuik. However, in our case the parameter $p$ is replaced by a 2-vector $\vec{p}$ parametrizing the solutions. We find that $Mass\propto|\vec{p}-\vec{p}_*|^\gamma$ where $\gamma\approx 0.66$. We also find a periodic wiggle in the scaling relation.    Our results show that black hole formation as a critical phenomenon is far more general than expected. 
  Symmetry, in particular gauge symmetry, is a fundamental principle in theoretical physics. It is intimately connected to the geometry of fibre bundles. A refinement to the gauge principle, known as ``spontaneous symmetry breaking'', leads to one of the most successful theories in modern particle physics. In this short talk, I shall try to give a taste of this beautiful and exciting concept. 
  We model A_k and D_k asymptotically locally flat gravitational instantons on the moduli spaces of solutions of U(2) Bogomolny equations with prescribed singularities. We study these moduli spaces using Ward correspondence and find their twistor description. This enables us to write down the K\"ahler potential for A_k and D_k gravitational instantons in a relatively explicit form. 
  We consider a gas of Newtonian self-gravitating particles in two-dimensional space, finding a phase transition, with a high temperature homogeneous phase and a low temperature clumped one. We argue that the system is described in terms of a gas with fractal behaviour. 
  A `reduced' action formulation for a general class of the supergravity solutions, corresponding to the `marginally' bound `distributed' systems of various types of branes at arbitrary angles, is developed. It turns out that all the information regarding the classical features of such solutions is encoded in a first order Lagrangian (the `reduced' Lagrangian) corresponding to the desired geometry of branes. The marginal solution for a system of $N$ such distributions (for various distribution functions) span an $N$ dimensional submanifold of the fields' configuration (target) space, parametrised by a set of $N$ independent harmonic functions on the transverse space. This submanifold, which we call it as the `$H$-surface', is a null surface with respect to a metric on the configuration space, which is defined by the reduced Lagrangian. The equations of motion then transform to a set of equations describing the embedding of a null geodesic surface in this space, which is identified as the $H$-surface. Using these facts, we present a very simple derivation of the conventional orthogonal solutions together with their intersection rules. Then a new solution for a (distributed) pair of $p$-branes at SU(2) angles in $D$ dimensions is derived. 
  We investigate the solutions of Nambu-Goto-type actions associated with calibrations. We determine the supersymmetry preserved by these solutions using the contact set of the calibration and examine their bulk interpretation as intersecting branes. We show that the supersymmetry preserved by such solutions is closely related to the spinor singlets of the subgroup $G$ of $Spin (9,1)$ or $Spin (10,1)$ that rotates the tangent spaces of the brane. We find that the supersymmetry projections of the worldvolume solutions are precisely those of the associated bulk configurations. We also investigate the supersymmetric solutions of a Born-Infeld action. We show that in some cases this problem also reduces to counting spinor singlets of a subgroup of $Spin (9,1)$ acting on the associated spinor representations. We also find new worldvolume solutions which preserve 1/8 of the supersymmetry of the bulk and give their bulk interpretation. 
  We construct multimonopole solutions containing N-1 distinct fundamental monopoles in SU(N) gauge theory. When the gauge symmetry is spontaneously broken to U(1)^{N-1}, the monopoles are all massive, and we show that the fields can be written in terms of elementary function for all values of the monopole positions and phases. In the limit of unbroken U(1) X SU(N-2) X U(1) symmetry, the configuration can be viewed as containing a pair of massive monopoles, each carrying both U(1) and SU(N-2) magnetic charges, together with N-3 massless monopoles that condense into a cloud of non-Abelian fields. We obtain explicit expressions for the fields in the latter case and use these to analyze the properties of the non-Abelian cloud. 
  We consider a description of membranes by (2,1)-dimensional field theory, or alternatively a description of irrotational, isentropic fluid motion by a field theory in any dimension. We show that these Galileo-invariant systems, as well as others related to them, admit a peculiar diffeomorphism symmetry, where the transformation rule for coordinates involves the fields. The symmetry algebra coincides with that of the Poincare group in one higher dimension. Therefore, these models provide a nonlinear representation for a dynamical Poincare group. 
  Following the recent work of Connes, Douglas and Schwarz, we study the M(atrix) model compactified on a torus with a background of the three-form field. This model is given by a super Yang-Mills theory on a quantum torus. To consider twisted gauge field configurations, we construct twisted U(n) bundles on the quantum torus as a deformation of its classical counterpart. By properly taking into account membranes winding around the light-cone direction, we derive from the M(atrix) model the BPS spectrum which respects the full SL(2,Z)*SL(2,Z) U-duality in M theory. 
  We study the relation between the large Nc limit of four dimensional N=2,1,0 conformal field theories and supergravity on orbifolds of AdS_5xS^5. We analyze the Kaluza-Klein states of the supergravity theory and relate them to the spectrum of (chiral) primary operators of the (super) conformal field theories. 
  We study F-theory duals of six dimensional heterotic vacua in extreme regions of moduli space where the heterotic string is very strongly coupled. We demonstrate how to use orientifold limits of these F-theory duals to regain a perturbative string description. As an example, we reproduce the spectrum of a $T^4/\ZZ_{4}$ orientifold as an F-theory vacuum with a singular $K3$ fibration. We relate this vacuum to previously studied heterotic $E_8\times E_8$ compactifications on $K3$. 
  The Hamiltonian of the $N$-particle Calogero model can be expressed in terms of generators of a Lie algebra for a definite class of representations. Maintaining this Lie algebra, its representations, and the flatness of the Riemannian metric belonging to the second order differential operator, the set of all possible quadratic Lie algebra forms is investigated. For N=3 and N=4 such forms are constructed explicitly and shown to correspond to exactly solvable Sutherland models. The results can be carried over easily to all $N$. 
  We consider the dimensional reduction of N = 1 SYM_{2+1} to 1+1 dimensions, which has (1,1) supersymmetry. The gauge groups we consider are U(N) and SU(N), where N is a finite variable. We implement Discrete Light-Cone Quantization to determine non-perturbatively the bound states in this theory. A careful analysis of the spectrum is performed at various values of N, including the case where N is large (but finite), allowing a precise measurement of the 1/N effects in the quantum theory. The low energy sector of the theory is shown to be dominated by string-like states. The techniques developed here may be applied to any two dimensional field theory with or without supersymmetry. 
  We show that the K-K spectrum of IIB string on AdS_5 x S_5 is described by ``twisted chiral'' N=4 superfields, naturally described in ``harmonic superspace'', obtained by taking suitable gauge singlets polynomials of the D3-brane boundary SU(n) superconformal field theory. To each p-order polynomial is associated a massive K-K short representation with 256 x 1/12 p^2(p^2 -1) states. The p=2 quadratic polynomial corresponds to the ``supercurrent multiplet'' describing the ``massless'' bulk graviton multiplet. 
  We study the problem of axial and gauge anomalies in a reducible theory involving vector and tensor gauge fields coupled in a topological way. We consider that vector and axial fermionic currents couple with the tensor field in the same topological manner as the vector gauge one. This kind of coupling leads to an anomalous axial current, contrarily to the results found in literature involving other tensor couplings, where no anomaly is obtained. 
  The finite-temperature renormalization group is formulated via the Wilson-Kadanoff blocking transformation. Momentum modes and the Matsubara frequencies are coupled by constraints from a smearing function which plays the role of an infrared cutoff regulator. Using the scalar lambda phi^4 theory as an example, we consider four general types of smearing functions and show that, to zeroth-order in the derivative expansion, they yield qualitatively the same temperature dependence of the running constants and the same critical exponents within numerical accuracy. 
  Using a proposal of Maldacena one describes the large N limit of gauge theories in terms of supergravity solutions on anti-de Sitter space. From this point of view we discuss a possible scenario for quark confinement in gauge theory by describing hadrons as strongly curved universes. In particular an interpretation of black hole as a bag model in SQCD is discussed. One relates the mystery of curvature singularities in classical general relativity with the mystery of quark confinement. The AdS bag model is defined by computing the probe membrane action in supergravity background. It naturally implies the "Cheshire Cat bag" principle. The confining pressure in the MIT bag model is related with the cosmological constant in the AdS bag model. The Skyrme model is interpreted as an effective theory describing black holes. 
  Poisson-Lie target space duality is a framework where duality transformations are properly defined. In this letter we investigate the pair of sigma models defined by the double SO(3,1) in the Iwasawa decomposition. 
  We explicitly demonstrate that the perturbative holomorphic contribution to the off-shell effective action of N=2 U(1) gauge supermultiplet is an entire effect of the minimal coupling to a hypermultiplet with the mass generated by a central charge in N=2 superalgebra. The central charge is induced by a constant vacuum N=2 gauge superfield strength spontaneously breaking the automorphism U(1)_R symmetry of N=2 superalgebra. We use the manifestly off-shell supersymmetric harmonic superspace techniques of quantum calculations with the central charge-massive hypermultiplet propagator. 
  An investigation of singular fields emerging in the process of transforming QCD to the axial gauge is presented. The structure of the singularities is analyzed. It is shown that apart from well known neutral magnetic monopole singularities, the field configurations also exhibit singularities in their charged and transverse components. This complex singularity structure guarantees finite non-Abelian field strength and thus finite action if expressed in terms of gauge fixed fields. A relation between the monopole charges of singular field configurations and their topological charge is derived. Qualitative dynamical aspects of the role of the monopoles are discussed. It is argued that the entropy associated with monopoles increases with decreasing temperature and that the coupling to quantum fluctuations favors monopole-antimonopole binding. 
  We discuss the coordinate complexification necessary to obtain a spatially homogeneous open universe in the singular instantonic solution recently suggested by Hawking and Turok (hep-th/9802030), concluding that only the time coordinate $\sigma$ of the spatially inhomogeneous De Sitter-like Lorentzian solution and its scale factor should be rotated in order to continue into a Lorentzian open universe. 
  String theory and supersymmetry are theoretical ideas that go beyond the standard model of particle physics and show promise for unifying all forces. After a brief introduction to supersymmetry, we discuss the prospects for its experimental discovery in the near future. We then show how the magic of supersymmetry allows us to solve certain quantum field theories exactly, thus leading to new insights about field theory dynamics related to electric-magnetic duality. The discussion of superstring theory starts with its perturbation expansion, which exhibits new features including ``stringy geometry.'' We then turn to more recent non-perturbative developments. Using new dualities, all known superstring theories are unified, and their strong coupling behavior is clarified. A central ingredient is the existence of extended objects called branes. 
  Generalizing previous quantum gravity results for Schwarzschild black holes from 4 to D>4 spacetime dimensions yields an energy spectrum E_n = n^{1-1/(D-2)} sigma E_P, n=1,2,..., sigma = O(1). Assuming the degeneracies of these levels to be given by g^n, g>1, leads to a partition function which is the same as that of the primitive droplet nucleation model for 1st-order phase transitions in D-2 spatial dimensions. Exploiting the well-known properties of the so-called critical droplets of this model immediately leads to the Hawking temperature and the Bekenstein-Hawking entropy of Schwarzschild black holes. Thus, the "holographic principle" of 't Hooft and Susskind is naturally realised. The values of temperature and entropy appear closely related to the imaginary part of the partition function which describes metastable states. Finally some striking conceptual similarities ("correspondence point" etc.) between the droplet nucleation picture and the very recent approach to the quantum statistics of Schwarzschild black holes in the framework of the DLCQ Matrix theory are pointed out. 
  We systematically embed the SU(2)$\times$U(1) Higgs model in the unitary gauge into a fully gauge-invariant theory by following the generalized BFT formalism. We also suggest a novel path to get a first-class Lagrangian directly from the original second-class one using the BFT fields. 
  The two-point functions in affine current algebras based on simple Lie algebras are constructed for all representations, integrable or non-integrable. The weight of the conjugate field to a primary field of arbitrary weight is immediately read off. 
  Geometry of the solution space of the self-dual Yang-Mills (SDYM) equations in Euclidean four-dimensional space is studied. Combining the twistor and group-theoretic approaches, we describe the full infinite-dimensional symmetry group of the SDYM equations and its action on the space of local solutions to the field equations. It is argued that owing to the relation to a holomorphic analogue of the Chern-Simons theory, the SDYM theory may be as solvable as 2D rational conformal field theories, and successful nonperturbative quantization may be developed. An algebra acting on the space of self-dual conformal structures on a 4-space (an analogue of the Virasoro algebra) and an algebra acting on the space of self-dual connections (an analogue of affine Lie algebras) are described. Relations to problems of topological and N=2 strings are briefly discussed. 
  Elizalde, Vanzo, and Zerbini have shown that the effective action of two free Euclidean scalar fields in flat space contains a `multiplicative anomaly' when zeta-function regularization is used. This is related to the Wodzicki residue. I show that there is no anomaly when using a wide range of other regularization schemes and further that this anomaly can be removed by an unusual choice of renormalisation scales. I define new types of anomalies and show that they have similar properties. Thus multiplicative anomalies encode no novel physics. They merely illustrate some dangerous aspects of zeta-function and Schwinger proper time regularization schemes. 
  We investigate nonperturbative effects in N=1 and N=2 supersymmetric theories using a relation between perturbative and exact anomalies as a starting point. For N=2 supersymmetric SU(n) Yang-Mills theory we derive the general structure of the Picard-Fuchs equations; for N=1 supersymmetric Yang-Mills theories we find holomorphic part of the superpotential (with gluino condensate) exactly. 
  We use type IIB brane configurations which were recently suggested by Hanany and Zaffaroni to study four dimensional N=1 supersymmetric gauge theories. We calculate the one loop beta function and realize Seiberg's duality using a particular configuration. We also comment on the anomaly cancelation condition in the case of chiral theories and the beta function in the case of chiral and SO/Sp theories. 
  The coupling to a 2+1 background geometry of a quantized charged test particle in a strong magnetic field is analyzed. Canonical operators adapting to the fast and slow freedoms produce a natural expansion in the inverse square root of the magnetic field strength. The fast freedom is solved to the second order.  At any given time, space is parameterized by a couple of conjugate operators and effectively behaves as the `phase space' of the slow freedom. The slow Hamiltonian depends on the magnetic field norm, its covariant derivatives, the scalar curvature and presents a peculiar coupling with the spin-connection. 
  We construct a duality between several simple physical systems by showing that they are different aspects of the same quantum theory. Examples include the free relativistic massless particle and the hydrogen atom in any number of dimensions. The key is the gauging of the Sp(2) duality symmetry that treats position and momentum (x,p) as a doublet in phase space. As a consequence of the gauging, the Minkowski space-time vectors (x^\mu, p^\mu) get enlarged by one additional space-like and one additional time-like dimensions to (x^M,p^M). A manifest global symmetry SO(d,2) rotates (x^M,p^M) like d+2 dimensional vectors. The SO(d,2) symmetry of the parent theory may be interpreted as the familiar conformal symmetry of quantum field theory in Minkowski spacetime in one gauge, or as the dynamical symmetry of a totally different physical system in another gauge. Thanks to the gauge symmetry, the theory permits various choices of ``time'' which correspond to different looking Hamiltonians, while avoiding ghosts. Thus we demonstrate that there is a physical role for a spacetime with two times when taken together with a gauged duality symmetry that produces appropriate constraints. 
  In this talk I give a preliminary account of original results, obtained in collaboration with John Ellis. Details and further elaboration will be presented in a forthcoming publication. We present a proposal for a non-critical (Liouville) string approach to confinement of four-dimensional (non-abelian) gauge theories, based on recent developments on the subject by Witten and Maldacena. We discuss the effects of vortices and monopoles on the open world-sheets whose boundaries are Wilson loops of the target-space (non Abelian) Gauge theory. By appropriately employing `D-particles', associated with the target-space embedding of such defects, we argue that the apprearance of five-dimensional Anti-De-Sitter (AdS) space times is quite natural, as a result of Liouville dressing.We isolate the world-sheet defect contributions to the Wilson loop by constructing an appropriate observable, which is the same as the second observable in the supersymmetric U(1) theory of Awada and Mansouri, but in our approach supersymmetry is not necessary.When vortex condensation occurs, we argue in favour of a (low-temperature) confining phase, in the sense of an area law, for a large-$N_c$ (conformal) gauge theory at finite temperatures. A connection of the Berezinski-Kosterlitz-Thouless (BKT) transitions on the world-sheet with the critical temperatures in the thermodynamics of Black Holes in the five-dimensional AdS space is made. 
  This paper proposes a variational principle for the solutions of quantum field theories in which the ``trial functions'' are chosen from the algebra of asymptotic fields, and illustrates this variational principle in simple cases. 
  We discuss the connection between Matrix string theory and the DLCQ of string theory. Using this connection we describe the sense in which perturbative string amplitudes are reproduced in the Matrix string theory. Using recent realization of the connection between SYM and Supergravity, we suggest how to describe Matrix theory with non-flat backgrounds. 
  We study issues pertaining to the Ricci-flatness of metrics on orbifolds resolved by D-branes. We find a K\"ahler metric on the three-dimensional orbifold $\C^3/\Z_3$, resolved by D-branes, following an approach due to Guillemin. This metric is not Ricci-flat for any finite value of the blow-up parameter. Conditions for the envisaged Ricci-flat metric for finite values of the blow-up parameter are formulated in terms of a correction to the K\"ahler potential. This leads to an explicit construction of a Ricci-flat K\"ahler metric on the resolved orbifold. The correction can be interpreted as a part of the superspace-interaction in the corresponding gauged linear sigma-model. 
  We obtain three and four dimensional conformal field theories with less than maximal supersymmetry by using their supergravity duals. These supergravity theories are type II on $AdS_5 \times CP^2$, IIA on $AdS_4 \times CP^3$, IIB on $AdS_5 \times S^5/Z_k$ and D=11 supergravity on $AdS_4 \times S^7/Z_k$. They are obtained from the spherically compactified ten and eleven dimensional theories by either Hopf reduction or by winding the U(1) fiber over the base. 
  Many string theories contain states which are not BPS, but are stable due to charge conservation. In many cases the description of these states in the strong coupling limit remains unknown despite the existence of a weakly coupled dual theory. However, we show that in some cases duality symmetries in string theory do enable us to identify these states in the strong coupling limit and calculate their masses. We also speculate that in some of the other cases the missing states might arise from non-supersymmetric analog of D0-branes. 
  D-brane backgrounds are specified in closed string theories by holes with appropriate mixed Dirichlet and Neumann boundary conditions on the string worldsheet. As presently stated, the prescription defining D-brane backgrounds is such that the Einstein equation is not equivalent to the condition for scale invariance on the string worldsheet. A modified D-brane prescription is found, that leads to the desired equivalence, while preserving all known D-brane lore. A possible interpretation is that the worldsheet cutoff is finite. Possible connections to recent work of Maldacena and Strominger, and Gopakumar and Vafa are suggested. 
  We show that the symmetries of effective D-string actions in constant dilaton backgrounds are directly related to homothetic motions of the background metric. In presence of such motions, there are infinitely many nonlinearly realized rigid symmetries forming a loop (or loop like) algebra. Near horizon (AdS) D3 and D1+D5 backgrounds are discussed in detail and shown to provide 2d interacting field theories with infinite conformal symmetry. 
  We use the M theory approach in the presence of an orientifold O6 plane to understand some aspects of the moduli space of vacua for N=1 supersymmetric $SO(N_c)/Sp(N_c)$ gauge theories in four dimensions. By exploiting some general properties of the O6 orientifold, we reproduce some results obtained previously with an orientifold O4 plane when the flavor group arises from the worldvolume dynamics of D6 branes. By using semi-infinite D4 branes instead of D6 branes, we derive the most general form of the rotated curve describing the moduli space of vacua for N=1 supersymmetric gauge theory with massive matter. 
  In the Matrix theories compactified on the $d$ dimensional torus with $d = 5, 6, 7$ which are described by the theories of D$d$-branes in the type IIA or IIB theory on the $d$ dimensional dual torus we analyze the BPS bound states of the background D$d$-brane with other objects by taking the matrix theory limit. Through the Nahm duality transformation the flux and the momentum multiplets for these BPS states are shown to be associated with the black holes and the black strings respectively from the viewpoint of the weakly coupled type IIA string theory compactified on the $d-1$ dimensional torus. 
  Using noncommutative geometry, the standard tools of differential geometry can be extended to a broad class of spaces whose coordinates are noncommuting operators acting on a Hilbert space. In the simplest case of coordinates being matrix valued functions on space-time, the standard model of particle physics can be reconstructed out of a few basic principles. Following these ideas, we investigate the general case of models arising from matrices and give the resulting constraints on the scalar potential and gauge couplings constants, as well as some relations between fermionic and bosonic masses. 
  We shed doubt on a commonly used manipulation in computing the partition function for a matrix valued operator together with its attendant invocation of the multiplicative anomaly. 
  We show for any oriented surface, possibly with a boundary, how to generalize Kramers-Wannier duality to the world of quantum groups. The generalization is motivated by quantization of Poisson-Lie T-duality from the string theory. Cohomologies with quantum coefficients are defined for surfaces and their meaning is revealed. They are functorial with respect to some glueing operations and connected with q-invariants of 3-folds. 
  A modification of the harmonic superfield formalism in $D=4, N=2$ supergravity using a subsidiary condition of covariance under the background supersymmetry with a central charge ($B$-covariance) is considered. Conservation of analyticity together with the $B$-covariance leads to the appearance of linear gravitational superfields. Analytic prepotentials arise in a decomposition of the background linear superfields in terms of spinor coordinates and transform in a nonstandard way under the background supersymmetry. The linear gravitational superfields can be written via spinor derivatives of nonanalytic spinor prepotentials. The perturbative expansion of the extended supergravity action in terms of the $B$-covariant superfields and the corresponding version of the differential-geometric formalism are considered. We discuss the dual harmonic representation of the linearized extended supergravity, which corresponds to the dynamical condition of Grassmann analyticity. 
  A BIon may be defined as a finite energy solution of a non-linear field theory with distributional sources. By contrast a soliton is usually defined to have no sources. I show how harmonic coordinates map the exteriors of the topologically and causally non-trivial spacetimes of extreme p-branes to BIonic solutions of the Einstein equations in a topologically trivial spacetime in which the combined gravitational and matter energy momentum is located on distributional sources. As a consequence the tension of BPS p-branes is classically unrenormalized. The result holds equally for spacetimes with singularities and for those, like the M-5-brane, which are everywhere singularity free. 
  We study the role of bosonic zero modes in light-cone quantisation on the invariant mass spectrum for the simplified setting of two-dimensional SU(2) Yang-Mills theory coupled to massive scalar adjoint matter. Specifically, we use discretised light-cone quantisation where the momentum modes become discrete. Two types of zero momentum mode appear -- constrained and dynamical zero modes. In fact only the latter type of modes turn out to mix with the Fock vacuum. Omission of the constrained modes leads to the dynamical zero modes being controlled by an infinite square-well potential. We find that taking into account the wavefunctions for these modes in the computation of the full bound state spectrum of the two dimensional theory leads to 21% shifts in the masses of the lowest lying states. 
  We study gravitational anomalies for fivebranes in M theory. We show that an apparent anomaly in diffeomorphisms acting on the normal bundle is cancelled by a careful treatment of the M theory Chern-Simons coupling in the presence of fivebranes. One interesting aspect of our treatment is the way in which a magnetic object (the fivebrane) is smoothed out through coupling to gravity and the resulting relation between antisymmetric tensor gauge transformations and diffeomorphisms in the presence of a fivebrane. 
  Branes may be approximated semi-classically by solutions of supergravity theories with event and Cauchy horizons. I suggest that if one wishes to avoid singularities and to capture accurately some of the properties of branes then these classical spacetimes must be identified so as to render them periodic in time. 
  The BRST approach is applied to the description of irreducible massless higher spins representations of the Poincare group in arbitrary dimensions. The total system of constraints in such theory includes both the first and the second class constraints. The corresponding nilpotent BRST charge contains terms up to the seventh degree in ghosts. 
  A new gauge invariant formulation of the relativistic scalar field interacting with Chern-Simons gauge fields is considered. This formulation is consistent with the gauge fixed formulation. Furthermore we find that canonical (Noether) Poincar\'e generators are not gauge invariant even on the constraints surface and do not satisfy the (classical) Poincar\'e algebra. It is the improved generators, constructed from the symmetric energy-momentum tensor, which are (manifestly) gauge invariant and obey the classical Poincar\'e algebra. 
  We construct part of the superspace vielbein and tensor gauge field in terms of the component fields of 11-dimensional on-shell supergravity. The result can be utilized to describe supermembranes and corresponding matrix models for Dirichlet particles in nontrivial supergravity backgrounds to second order in anticommuting coordinates. We exhibit the kappa-invariance of the corresponding supermembrane action, which at this order holds for unrestricted supergravity backgrounds, the supersymmetry covariance and the resulting surface terms in the action. 
  The singularity in Hawking and Turok's model (hep-th/9802030) of open inflation has some appealing properties. We suggest that this singularity should be regularized with matter. The singular instanton can then be obtained as the limit of a family of ``no-boundary'' solutions where both the geometry and the scalar field are regular. Using this procedure, the contribution of the singularity to the Euclidean action is just 1/3 of the Gibbons-Hawking boundary term. Unrelated to this question, we also point out that gravitational backreaction improves the behaviour of scalar perturbations near the singularity. As a result, the problem of quantizing scalar perturbations and gravity waves seems to be very well posed. 
  We consider various ways of treating the infrared divergence which appears in the dynamically generated fermion mass, when the transverse part of the photon propagator in N flavour $QED_{3}$ at finite temperature is included in the Matsubara formalism. This divergence is likely to be an artefact of taking into account only the leading order term in the $1 \over N$ expansion when we calculate the photon propagator and is handled here phenomenologically by means of an infrared cutoff. Inserting both the longitudinal and the transverse part of the photon propagator in the Schwinger-Dyson equation we find the dependence of the dynamically generated fermion mass on the temperature and the cutoff parameters. It turns out that consistency with certain statistical physics arguments imposes conditions on the cutoff parameters. For parameters in the allowed range of values we find that the ratio $r=2*Mass(T=0)/critical temperature$ is approximately 6, consistently with previous calculations which neglected the transverse photon contribution. 
  We derive the Gell-Mann and Low renormalization group equation in the Wilsonian approach to renormalization of massless $g\phi^4$ in four dimensions, as a particular case of a non-linear equation satisfied at any scale by the Wilsonian effective action.   We give an exact expression for the $\beta$ and $\gamma_{\phi}$ functions in terms of the Wilsonian effective action at the Wilsonian renormalization scale $\L_R$; at the first two loops they are simply related to the gradient of the flow of the relevant couplings and have the standard values; beyond two loops this relation is spoilt by corrections due to irrelevant couplings. We generalize this analysis to the case of massive $g\phi^4$, introducing a mass-independent Wilsonian renormalization scheme; using the flow equation technique we prove renormalizability and we show that the limit of vanishing mass parameter exists. We derive the corresponding renormalization group equation, in which $\beta$ and $\gamma_{\phi}$ are the same as in the massless case; $\gamma_m$ is also mass-independent; at one loop it is the gradient of a relevant coupling and it has the expected value. 
  The associative superalgebra of observables of 3-particle Calogero model giving all wavefunctions of the model via standard Fock procedure has 2 independent supertraces. It is shown here that when the coupling constant \nu is equal to n+1/3, n-1/3 or n+1/2 for any integer n the existence of 2 independent supertraces leads to existence of nontrivial two-sided ideal in the superalgebra of observables. 
  Extending the recent work in hep-th/9803076, we consider string perturbative expansion in the presence of D-branes and orientifold planes imbedded in orbifolded space-time. In the $\alpha'\to 0$ limit the weak coupling string perturbative expansion maps to `t Hooft's large N expansion. We focus on four dimensional ${\cal N}=1,2,4$ supersymmetric theories, and also discuss possible extensions to ${\cal N}=0$ cases. Utilizing the string theory perturbation techniques we show that computation of any M-point correlation function in these theories reduces to the corresponding computation in the parent ${\cal N}=4$ theory. In particular, we discuss theories (which are rather constrained) with vanishing beta-functions to all orders in perturbation theory in the large N limit. We also point out that in theories with non-vanishing beta-functions the gauge coupling running is suppressed in the large N limit. Introduction of orientifold planes allows to construct certain gauge theories with SO, Sp and SU gauge groups and various matter (only unitary gauge groups with bi-fundamental/adjoint matter arise in theories without orientifold planes). 
  Repulsive singularities (repulsons) in extended supergravity theories are investigated. These repulsive singularities are related to attractive singularities (black holes) in moduli space of extended supergravity vacua. In order to study these repulsive singularities a scalar test-particle in the background of a repulson is investigated. It is shown, using a partial wave expansion, that the wave function of the scalar particle vanishes at the curvature singularity at the origin. In addition the connection to higher dimensional p-brane solutions including anti-branes is discussed. 
  The fivebrane worldvolume theory in eleven dimensions is known to contain BPS threebrane solitons which can also be interpreted as a fivebrane whose worldvolume is wrapped around a Riemann surface. By considering configurations of intersecting fivebranes and hence intersecting threebrane solitons, we determine the Bogomol'nyi equations for more general BPS configurations. We obtain differential equations, generalising Cauchy-Riemann equations, which imply that the worldvolume of the fivebrane is wrapped around a calibrated geometry. 
  The $u$-plane integrals of topologically twisted $N = 2$ supersymmetric gauge theories generally contain contact terms of nonlocal topological observables. This paper proposes an interpretation of these contact terms from the point of view of integrable hierarchies and their Whitham deformations. This is inspired by Mari\~no and Moore's remark that the blowup formula of the $u$-plane integral contains a piece that can be interpreted as a single-time tau function of an integrable hierarchy. This single-time tau function can be extended to a multi-time version without spoiling the modular invariance of the blowup formula. The multi-time tau function is comprised of a Gaussian factor $e^{Q(t_1,t_2,...)}$ and a theta function. The time variables $t_n$ play the role of physical coupling constants of 2-observables $I_n(B)$ carried by the exceptional divisor $B$. The coefficients $q_{mn}$ of the Gaussian part are identified to be the contact terms of these 2-observables. This identification is further examined in the language of Whitham equations. All relevant quantities are written in the form of derivatives of the prepotential. 
  We make a systematic development of the non-Abelian formulation of two-form gauge fields with topological coupling with the Yang-Mills one-form connection. An analysis of the gauge structure, reducibility conditions and physical degrees of freedom is presented. We employ the Batalin-Vilkovisky formalism to quantize the resulting theory. 
  We revisit two-point function approaches used to study vacuum fluctuation in wedge-shaped regions and conical backgrounds. Appearance of divergent integrals is discussed and circumvented. The issue is considered in the context of a massless scalar field in cosmic string spacetime. 
  Within recent Maldacena's proposal to relate gauge theories in the large N limit to the supergravity in the AdS background and recipe for calculation the Wilson loop, we compute corrections to the energy of quark/anti-quark pair in the large N limit. 
  We investigate hidden symmetries of P,T-invariant system of topologically massive U(1) gauge fields. For this purpose, we propose a pseudoclassical model giving rise to this field system at the quantum level. The model contains a parameter, which displays a quantization property at the classical and the quantum levels and demonstrates a nontrivial relationship between continuous and discrete symmetries. Analyzing the integrals of motion of the pseudoclassical model, we identify U(1,1) symmetry and S(2,1) supersymmetry as hidden symmetries of the corresponding quantum system. Representing the hidden symmetries in a covariant form, we show that one-particle states realize an irreducible representation of a non-standard super-extension of the (2+1)-dimensional Poincare group labelled by the zero eigenvalue of the superspin. 
  The general conditions for the applicability of the Faddeev-Jackiw approach to gauge theories are studied. When the constraints are effective a new proof in the Lagrangian framework of the equivalence between this method and the Dirac approach is given. We find, however, that the two methods may give different descriptions for the reduced phase space when ineffective constraints are present. In some cases the Faddeev-Jackiw approach may lose some constraints or some equations of motion. We believe that this inequivalence can be related to the failure of the Dirac conjecture (that says that the Dirac Hamiltonian can be enlarged to an Extended Hamiltonian including all first class constraints, without changes in the dynamics) and we suggest that when the Dirac conjecture fails the Faddeev-Jackiw approach fails to give the correct dynamics. Finally we present some examples that illustrate this inequivalence. 
  Some questions were recently raised about the equivalence of two methods commonly used to compute the Casimir energy: the mode summation approach and the one-loop effective potential. In this respect, we argue that the scale dependence induced by renormalization effects, displayed by the effective potential approach, also appears in the MS method. 
  In the framework of N=1 supersymmetric string models given by the heterotic string on an elliptic Calabi-Yau $\pi :Z\ra B$ together with a SU(n) bundle we compute the chiral matter content of the massless spectrum. For this purpose the net generation number, i.e. half the third Chern class, is computed from data related to the heterotic vector bundle in the spectral cover description; a non-technical introduction to that method is supplied. This invariant is, in the class of bundles considered, shown to be related to a discrete modulus which is the heterotic analogue of the $F$-theory four-flux. We consider also the relevant matter which is supported along certain curves in the base $B$ and derive the net generation number again from the independent matter-related computation. We then illustrate these considerations with two applications. First we show that the construction leads to numerous 3 generation models of unbroken gauge group $SU(5), SO(10)$ or $E_6$. Secondly we discuss the closely related issue of the heterotic 5-brane/instanton transition resp. the F-theoretic 3-brane/instanton transition. The extra chiral matter in these transitions is related to the Hecke transform of the direct sum of the original bundle and the dissolved 5-brane along the intersection of their spectral covers. Finally we point to the corresponding $F$-theory interpretation of chiral matter from the intersection of 7-branes where the influence of four-flux on the twisting along the intersection curve plays a crucial role. 
  A new method of deriving the Higgs Lagrangian from vector-like gauge theories is explored. After performing a supersymmetric extension of gauge theories we identify the auxiliary field associated with the "meson" superfield, in the low energy effective theory, as the composite Higgs field. The auxiliary field, at tree level, has a "negative squared mass". By computing the one-loop effective action in the low energy effective theory, we show that a kinetic term for the auxiliary field emerges when an explicit non-perturbative mechanism for supersymmetry breaking is introduced. We find that, due to the naive choice of the Kaehler potential, the Higgs potential remains unbounded from the below. A possible scenario for solving this problem is presented. It is also shown that once chiral symmetry is spontaneously broken via a non-zero vacuum expectation value of the Higgs field, the low energy composite fermion field acquires a mass and decouples, while in the supersymmetric limit it was kept massless by the 't Hooft anomaly matching conditions. 
  We derive the Ward-Takahashi identity obeyed by the fermion-antifermion-gauge boson vertex in ladder QED in the presence of a constant magnetic field. The general structure in momentum space of the fermion mass operator with external electromagnetic field is discussed. Using it we find the solutions of the ladder WT identity with magnetic field. The consistency of our results with the solutions of the corresponding Schwinger-Dyson equation ensures the gauge invariance of the magnetic field induced chiral symmetry breaking recently found in ladder QED. 
  We introduce a set of scalar fields as test fields to study the dynamical behaviors of the BTZ (Banados-Teitelboim-Zanelli) black hole. These include minimally coupled scalar, conformally coupled scalar, dilaton, and tachyon. To calculate the decay rate of the BTZ black hole, we consider both the Dirichlet boundary condition at spatial infinity and the stability condition. It turns out that the tachyon may be a relevant field to get information of the BTZ black hole. 
  The limiting procedure of special Kahler manifolds to their rigid limit is studied for moduli spaces of Calabi-Yau manifolds in the neighbourhood of certain singularities. In two examples we consider all the periods in and around the rigid limit, identifying the nontrivial ones in the limit as periods of a meromorphic form on the relevant Riemann surfaces. We show how the Kahler potential of the special Kahler manifold reduces to that of a rigid special Kahler manifold. We extensively make use of the structure of these Calabi-Yau manifolds as K3 fibrations, which is useful to obtain the periods even before the K3 degenerates to an ALE manifold in the limit. We study various methods to calculate the periods and their properties. The development of these methods is an important step to obtain exact results from supergravity on Calabi-Yau manifolds. 
  We discuss the relation between string quantization based on the Schild path integral and the Nambu-Goto path integral. The equivalence between the two approaches at the classical level is extended to the quantum level by a saddle--point evaluation of the corresponding path integrals. A possible relationship between M-Theory and the quantum mechanics of string loops is pointed out. Then, within the framework of ``loop quantum mechanics'', we confront the difficult question as to what exactly gives rise to the structure of spacetime. We argue that the large scale properties of the string condensate are responsible for the effective Riemannian geometry of classical spacetime. On the other hand, near the Planck scale the condensate ``evaporates'', and what is left behind is a ``vacuum'' characterized by an effective fractal geometry. 
  We study renormalizable nonlinear sigma-models in two dimensions with N=2 supersymmetry described in superspace in terms of chiral and complex linear superfields. The geometrical structure of the underlying manifold is investigated and the one-loop divergent contribution to the effective action is computed. The condition of vanishing beta-function allows to identify a class of models which satisfy this requirement and possess N=4 supersymmetry. 
  We construct a class of intersecting brane solutions with horizon geometries of the form adS_k x S^l x S^m x E^n. We describe how all these solutions are connected through the addition of a wave and/or monopoles. All solutions exhibit supersymmetry enhancement near the horizon. Furthermore we argue that string theory on these spaces is dual to specific superconformal field theories in two dimensions whose symmetry algebra in all cases contains the large N=4 algebra A_{gamma}. Implications for gauged supergravities are also discussed. 
  We present an explicit relation between the Hanany-Witten and Geometric Engineering approaches of realising gauge theories in string theory. The last piece in the puzzle is a T-duality relating arbitrary Hanany-Witten setups and fractional branes. 
  We study the cluster properties of thermal equilibrium states in theories with a maximal propagation velocity (such as relativistic QFT). Our analysis, carried out in the setting of algebraic quantum field theory, shows that there is a tight relation between spectral properties of the generator of time translations and the decay of spatial correlations in thermal equilibrium states, in complete analogy to the well understood case of the vacuum state. 
  We investigate a class of conformal Non-Abelian-Toda models representing a noncompact $SL(2,R)/U(1)$ parafermionions (PF) interacting with a specific abelian Toda theories and having a global U(1) symmetry. A systematic derivation of the conserved currents, their algebras and the exact solution of these models is presented. An important property of this class of models is the affine $SL(2,R)_q$ algebra spanned by charges of the chiral and antichiral nonlocal currents and the U(1) charge. The classical (Poisson Brackets) algebras of symmetries $V{G_n}$ of these models appears to be of mixed PF-$W{G_n}$ type. They contain together with the local quadratic terms specific for the $W_n$-algebras the nonlocal terms similar to the ones of the classical PF-algebra. The renormalization of the spins of the nonlocal currents is the main new feature of the quantum $V{A_n}$-algebras. The quantum $V{A_2}$-algebra and its degenerate representations are studied in detail. 
  It is shown that the effective five-dimensional theory of the strongly coupled heterotic string is a gauged version of N=1 five-dimensional supergravity with four-dimensional boundaries. For the universal supermultiplets, this theory is explicitly constructed by a generalized dimensional reduction procedure on a Calabi-Yau manifold. A crucial ingredient in the reduction is the retention of a "non-zero mode" of the four-form field strength, leading to the gauging of the universal hypermultiplet by the graviphoton. We show that this theory has an exact three-brane domain wall solution which reduces to Witten's "deformed" Calabi-Yau background upon linearization. This solution consists of two parallel three-branes with sources provided by the four-dimensional boundary theories and constitutes the appropriate background for a reduction to four dimensions. Four-dimensional space-time is then identified with the three-brane worldvolume. 
  We explain how static multi-vortex solutions arise in non-linear field theories, by taking the non-linear Schr\"odinger equation coupled to Chern-Simons field (Jackiw-Pi model) and a fermion Chern-Simons theory as simple examples. We then construct a fermion Maxwell-Chern-Simons theory which has consistent static field equations, and show that it has the same vortex solutions as the Jackiw-Pi model, but gives rise to quite different vortex dynamics. 
  Renormalization group transformations for Schr\"odinger equation are performed in $\phi^4$ and in Yang-Mills theories. The dependence of the ground state wave functional on rapidly oscillating fields is found. For Yang-Mills theory, this dependence restricts a possible form of variational ansatz compatible with asymptotic freedom. 
  We present some approaches to the perturbative analysis of the classical and quantum gravity. First we introduce a graphical representation for a global SO(n) tensor $(\pl)^d h_\ab$, which generally appears in the weak field expansion around the flat space: $g_\mn=\del_\mn+h_\mn$. Making use of this representation, we explain 1) Generating function of graphs (Feynman diagram approach), 2) Adjacency matrix (Matrix approach), 3) Graphical classification in terms of "topology indices" (Topology approach), 4) The Young tableau (Symmetric group approach). We systematically construct the global SO(n) invariants. How to show the independence and completeness of those invariants is the main theme. We explain it taking simple examples of $\pl\pl h-, {and} (\pl\pl h)^2-$ invariants in the text. The results are applied to the analysis of the independence of general invariants and (the leading order of) the Weyl anomalies of scalar-gravity theories in "diverse" dimensions (2,4,6,8,10 dimensions). 
  We consider the dimensional reduction/compactification of supergravity, string and M-theories on tori with one time-like circle. We find the coset spaces in which the massless scalars take their values, and identify the discrete duality groups. 
  Gauge theories coupled to fermions in generation are reformulated in a modified version of extended differential geometry with the symbol $\chi$. After discussing several toy models, we will reformulate in our framework the standard model based on Connes' real structure. It is shown that for the most general bosonic lagrangin which is required to also reconstruct N=2 super Yang-Mills theory Higgs mechanism operates only for more than one generation as first pointed out by Connes and Lott. 
  The enhanced gauge groups in F-theory compactified on elliptic Calabi-Yau fourfolds are investigated in terms of toric geometry. 
  For a large class of finite W algebras, the defining relations of a Yangian are proved to be satisfied. Therefore such finite W algebras appear as realisations of Yangians. This result is useful to determine properties of such W algebra representations. 
  We construct an algebra homomorphism between the Yangian Y(sl(n)) and the finite W-algebras W(sl(np),n.sl(p)) for any p. We show how this result can be applied to determine properties of the finite dimensional representations of such W-algebras. 
  The point-splitting regularization technique for composite operators is discussed in connection with anomaly calculation. We present a pedagogical and self-contained review of the topic with an emphasis on the technical details. We also develop simple algebraic tools to handle the path ordered exponential insertions used within the covariant and non-covariant version of the point-splitting method. The method is then applied to the calculation of the chiral, vector, trace, translation and Lorentz anomalies within diverse versions of the point-splitting regularization and a connection between the results is described. As an alternative to the standard approach we use the idea of deformed point-split transformation and corresponding Ward-Takahashi identities rather than an application of the equation of motion, which seems to save the complexity of the calculations. 
  Given a thermal field theory for some temperature $\beta^{-1}$, we construct the theory at an arbitrary temperature $ 1 / \beta'$. Our work is based on a construction invented by Buchholz and Junglas, which we adapt to thermal field theories. In a first step we construct states which closely resemble KMS states for the new temperature in a local region $\O_\circ \subset \rr^4$, but coincide with the given KMS state in the space-like complement of a slightly larger region $\hat{\O}$. By a weak*-compactness argument there always exists a convergent subnet of states as the size of $ \O_\circ$ and $ \hat{\O}$ tends towards $ \rr^4$. Whether or not such a limit state is a global KMS state for the new temperature, depends on the surface energy contained in the layer in between the boundaries of $ \O_\circ$ and $ \hat{\O}$. We show that this surface energy can be controlled by a generalized cluster condition. 
  We consider the gravitationally induced particle production from the quantum vacuum which is defined by a free, massless and minimally coupled scalar field during the formation of a gauge cosmic string. Previous discussions of this topic estimate the power output per unit length along the string to be of the order of $10^{68}$ ergs/sec/cm in the s-channel. We find that this production may be completely suppressed. A similar result is also expected to hold for the number of produced photons. 
  The ``extended'' BF-Yang-Mills theory in 3 dimensions, which contains a minimally coupled scalar field, is shown to be ultraviolet finite. It obeys a trivial Callan-Symanzik equation, with all beta-functions and anomalous dimensions vanishing. The proof is based on an anomaly-free trace identity valid to all orders of perturbation theory. 
  New single soliton solutions to the affine Toda field theories are constructed, exhibiting previously unobserved topological charges. This goes some of the way in filling the weights of the fundamental representations, but nevertheless holes in the representations remain. We use the group doublecross product form of the inverse scattering method, and restrict ourselves to the rank one solutions. 
  We consider certain orbifoldization of the ${\cal N}=4$ field theories that leads to ${\cal N}=2,1,0$ field theories in 4 dimensions. These theories were recently analyzed using the string theory perturbation technique. It was found that in the large $N$ limit all correlation functions of the orbifold theories coincide with those of ${\cal N}=4$, modulo the rescaling of the gauge coupling constant. In this paper we repeat the same analysis using the field theoretical language. 
  We examine a nonlocal interaction that results from expressing the QCD Hamiltonian entirely in terms of gauge-invariant quark and gluon fields. The interaction couples one quark color-charge density to another, much as electric charge densities are coupled to each other by the Coulomb interaction in QED. In QCD, this nonlocal interaction also couples quark color-charge densities to gluonic color. We show how the leading part of the interaction between quark color-charge densities vanishes when the participating quarks are in a color singlet configuration, and that, for singlet configurations, the residual interaction weakens as the size of a packet of quarks shrinks. Because of this effect, color-singlet packets of quarks should experience final state interactions that increase in strength as these packets expand in size. For the case of an SU(2) model of QCD based on the {\em ansatz} that the gauge-invariant gauge field is a hedgehog configuration, we show how the infinite series that represents the nonlocal interaction between quark color-charge densities can be evaluated nonperturbatively, without expanding it term-by-term. We discuss the implications of this model for QCD with SU(3) color and a gauge-invariant gauge field determined by QCD dynamics. 
  We consider a free massive spinor field in Euclidean Anti-de Sitter space. The usual Dirac action in bulk is supplemented by a certain boundary term. The boundary conditions of the field are parametrized by a spinor on the boundary, subject to a projection. We calculate the dependence of the partition function on this boundary spinor. The result agrees with the generating functional of the correlation functions of a quasi-primary spinor operator, of a certain scaling dimension, in a free conformal field theory on the boundary. 
  The point where a D2-brane intersecting a stack of D2-branes is proposed as a candidate for the 't Hooft vortex in the world-volume theory of N D2-branes. This straightforwardly generalizes to D3-branes, where a vortex line is generated by the intersection. Similarly, there are such objects on M-branes. We use Maldacena's conjecture to compute the static potential between a vortex and an anti-vortex in each case, in the large N limit. 
  We study realizations of polynomial deformations of the sl(2,R)- Lie algebra in terms of differential operators strongly related to bosonic operators. We also distinguish their finite- and infinite-dimensional representations. The linear, quadratic and cubic cases are explicitly visited but the method works for arbitrary degrees in the polynomial functions. Multi-boson Hamiltonians are studied in the context of these ``nonlinear'' Lie algebras and some examples dealing with quantum optics are pointed out. 
  A topological fact about eleven dimensions is used to motivate a potential new duality in M-theory. We complete the discussion of consistent limits of M-theory raised in a previous paper to include gravitational anomaly cancelation and four-form flux quantization in the context of the $M^{10}\times S^1/Z_2$ compactifications. A suprise is found: If one includes a $Z_2$ anomaly which exists only in 8k+3 dimensions then there are two distinct quantum limits, one related to each of two equivalency classes of the orbifolds $M^{10}\times S^1/Z_2$. 
  The well-known formal analogy between time and absolute temperature, existing on the quantum level, is considered as a profound duality relationship requiring some modifications in the conventional quantum dynamics. They consist of tiny deviations from uniform time run in the physical spacetime, as well as of tiny deviations from unitary time evolution characteristic for the conventional quantum theory. The first deviations are conjectured to be produced by local changes of total average particle number. Then, they imply the second deviations exerting in turn influence upon this particle number. Two examples of the second deviations are described: hypothetic tiny violation of optical theorem for particle scattering, and hypothetic slow variation of the average number of probe particles contained in a sample situated in proximity of a big accelerator (producing abundantly particles on a target). 
  Using an analogy between Bogomol'nyi bound and the harmonic oscillator in quantum mechanics we propose a possible explanation of the coupling constant numerical value at the grand unification scale. It is found to be $1/8\pi$. 
  We discuss the relationship between two-dimensional (2D) dilaton gravity models and sine-Gordon-like field theories. We show that there is a one-to-one correspondence between the solutions of 2D dilaton gravity and the solutions of a (two fields) generalization of the sine-Gordon model. In particular, we find a connection between the soliton solutions of the generalized sine-Gordon model and extremal black hole solutions of 2D dilaton gravity. As a by-product of our calculations we find a easy way to generate cosmological solutions of 2D dilaton gravity. 
  The collective coordinates expansion of the Skyrme soliton particle model gives rise to the second class constraints. We use the non-abelian BFFT formalism to convert this system into the one with only first class constraints. Choosing two different structure functions of the non-abelian algebra, we obtain simplified algebraic expressions for the first class non-abelian Hamiltonians. This result shows that the non-abelian BFFT method is, in many aspects, richer than the abelian BFFT formalism. For both of the first class Hamiltonians, we derive the Lagrangians which lead to the new theory. When one puts the extended phase space variables equal to zero, the original Skyrmion Lagrangian is reproduced. The method of the Dirac first class constraints is employed to quantize these two systems. We achieve the same spectrum, a result which confirms the consistency of the non-abelian BFFT formalism. 
  We study the Euclidean-signature supergravities that arise by compactifying D=11 supergravity or type IIB supergravity on a torus that includes the time direction. We show that the usual T-duality relation between type IIA and type IIB supergravities compactified on a spatial circle no longer holds if the reduction is performed on the time direction. Thus there are two inequivalent Euclidean-signature nine-dimensional maximal supergravities. They become equivalent upon further spatial compactification to D=8. We also show that duality symmetries of Euclidean-signature supergravities allow the harmonic functions of any single-charge or multi-charge instanton to be rescaled and shifted by constant factors. Combined with the usual diagonal dimensional reduction and oxidation procedures, this allows us to use the duality symmetries to map any single-charge or multi-charge p-brane soliton, or any intersection, into its near-horizon regime. Similar transformations can also be made on non-extremal p-branes. We also study the structures of duality multiplets of instanton and (D-3)-brane solutions. 
  In a recent paper, Ohta and Townsend studied the conditions which must be satisfied for a configuration of two intersecting M5-branes at angles to be supersymmetric. In this paper we extend this result to any number of M5-branes or any number of M2-branes. This is accomplished by interpreting their results in terms of calibrated geometry, which is of independent interest. 
  Some thermodynamic properties of the Ba\~nados-Teitelboim-Zanelli (BTZ) black hole are studied to get the effective dimension of its corresponding statistical model. For this purpose, we make use of the geometrical approach to the thermodynamics: Considering the black hole as a thermodynamic system with two thermodynamic variables (the mass $M$ and the angular momemtum $J$), we obtain two-dimensional Riemannian thermodynamic geometry described by positive definite Ruppeiner metric. From the thermodynamic curvature we find that the extremal limit is the critical point. The effective spatial dimension of the statistical system corresponding to the near-extremal BTZ black holes is one. Far from the extremal point, the effective dimension becomes less than one, which leads to one possible speculation on the underlying structure for the corresponding statistical model. 
  We construct a Landau-Ginzburg model with the same data and symmetries as a $Z_2\times Z_2$ orbifold that corresponds to a class of realistic free-fermion models. Within the class of interest, we show that this orbifolding connects between different $Z_2\times Z_2$ orbifold models and commutes with the mirror symmetry. Our work suggests that duality symmetries previously discussed in the context of specific $M$ and $F$ theory compactifications may be extended to the special $Z_2\times Z_2$ orbifold that characterizes realistic free-fermion models. 
  We use the recently proposed supergravity approach to large $N$ gauge theories to calculate ordinary and spatial Wilson loops of gauge theories in various dimensions. In this framework we observe an area law for spatial Wilson loops in four and five dimensional supersymmetric Yang-Mills at finite temperature. This can be interpreted as the area law of ordinary Wilson loops in three and four dimensional non-supersymmetric gauge theories at zero temperature which indicates confinement in these theories. Furthermore, we show that super Yang Mills theories with 16 supersymmetries at finite temperature do not admit phase transitions between the weakly coupled super Yang Mills and supergravity regimes. This result is derived by analyzing the entropy and specific heat of those systems as well as by computing ordinary Wilson loops at finite temperature. The calculation of the entropy was carried out in all different regimes and indicates that there is no first order phase transition in these systems. For the same theories at zero temperature we also compute the dependence of the quark anti-quark potential on the separating distance. 
  Critical exponents for the 3D O(n)-symmetric model with n > 3 are estimated on the base of six-loop renormalization-group (RG) expansions. A simple Pade-Borel technique is used for the resummation of the RG series and the Pade approximants [L/1] are shown to give rather good numerical results for all calculated quantities. For large n, the fixed point location g_c and the critical exponents are also determined directly from six-loop expansions without addressing the resummation procedure. An analysis of the numbers obtained shows that resummation becomes unnecessary when n exceeds 28 provided an accuracy of about 0.01 is adopted as satisfactory for g_c and critical exponents. Further, results of the calculations performed are used to estimate the numerical accuracy of the 1/n-expansion. The same value n = 28 is shown to play the role of the lower boundary of the domain where this approximation provides high-precision estimates for the critical exponents. 
  We compute the principal contribution to the index in the supersymmetric quantum mechanical systems which are obtained by reduction to 0+1 dimensions of $\CN=1$, $D=4,6,10$ super-Yang-Mills theories with gauge group SU(N). The results are: ${1\over{N^{2}}}$ for $D=4,6$, $\sum_{d | N} {1\over{d^{2}}}$ for D=10. We also discuss the D=3 case. 
  By exploiting the relation between Fredholm modules and the Segal-Shale-Stinespring version of canonical quantization, and taking as starting point the first-quantized fields described by Connes' axioms for noncommutative spin geometries, a Hamiltonian framework for fermion quantum fields over noncommutative manifolds is introduced. We analyze the ultraviolet behaviour of second-quantized fields over noncommutative 3-tori, and discuss what behaviour should be expected on other noncommutative spin manifolds. 
  We consider the problem of designing an Ansatz for the fermion-photon vertex function, using three-dimensional quantum electrodynamics as a test case. In many existing studies, restrictions have been placed on the form of the vertex Ansatz by making the unsubstantiated assumption that in the quenched, massless limit the Landau gauge Dyson-Schwinger equations admit a trivial solution. We demonstrate, without recourse to this assumption, the existence of a non-local gauge in which the fermion propagator is the bare propagator. This result is used to provide a viable Ansatz for part of the vertex function. 
  We propose a new general BRST approach to string and string-like theories which have a wider range of applicability than e g the conventional conformal field theory method. The method involves a simple general regularization of all basic commutators which makes all divergent sums to be expressible in terms of zeta functions from which finite values then may be extracted in a rigorous manner. The method is particular useful in order to investigate possible state space representations to a given model. The method is applied to three string models: The ordinary bosonic string, the tensionless string and the conformal tensionless string. We also investigate different state spaces for these models. The tensionless string models are treated in details. Although we mostly rederive known results they appear in a new fashion which deepens our understanding of these models. Furthermore, we believe that our treatment is more rigorous than most of the previous ones. In the case of the conformal tensionless string we find a new solution for d=4. 
  We give in this paper some formulas which are useful in the construction of nontrivial conserved currents for submodels of CP^1-model or QP^1-model in (1+2) dimensions.   These are full generalization of our results in the previous paper  (hep-th/9802105). 
  We analyse the Hilbert space structure of the isomorphic gauge non-invariant and gauge invariant bosonized formulations of chiral $QCD_2$ for the particular case of the Jackiw-Rajaraman parameter $ a = 2$. The BRST subsidiary conditions are found not to provide a sufficient criterium for defining physical states in the Hilbert space and additional superselection rules must to be taken into account. We examine the effect of the use of a redundant field algebra in deriving basic properties of the model. We also discuss the constraint structure of the gauge invariant formulation and show that the only primary constraints are of first class. 
  We explore the connection of anti-de-Sitter supergravity in six dimensions, based on the exceptional F(4) superalgebra, and its boundary superconformal singleton theory. The interpretation of these results in terms of a D4-D8 system and its near horizon geometry is suggested. 
  S-dualities in scale invariant N=2 supersymmetric field theories with product gauge groups are derived by embedding those theories in asymptotically free theories with higher rank gauge groups. S-duality transformations on the couplings of the scale invariant theory follow from the geometry of the embedding of the scale invariant theory in the Coulomb branch of the asymptotically free theory. 
  The renormalization group flows of the coupling constants for the gauged U(N) vector model, with N_f massless fermions in the defining representation, are studied in the large N limit, to all orders in the scalar coupling lambda, leading order in 1/N, and lowest two orders in the gauge coupling g^2. It is shown that the restrictions of asymptotic freedom, and the reality of the coupling constants throughout the flows, places important restrictions on N_f/N. For the case with massless mesons, these conditions are sufficiently restrictive to imply the existence of an infrared fixed-point (g_*,lambda_*) in both couplings. Thus, the consistent massless theory is scale invariant, and in a non-abelian Coulomb phase. The case of massive mesons, and of spontaneously broken symmetry is also discussed, with similar, but not identical, conclusions. Speculations related to the possibility that there is a non-perturbative (in g^2) breakdown of chiral symmetry are presented. 
  We present a compendium of results for ADHM multi-instantons in SU(N) SUSY gauge theories, followed by applications to N=2 supersymmetric models. Extending recent SU(2) work, and treating the N=1 and N=2 cases in parallel, we construct: (i) the ADHM supermultiplet, (ii) the multi-instanton action, and (iii) the collective coordinate integration measure. Specializing to N=2, we then give a closed formula for F_k, the k-instanton contribution to the prepotential, as a finite-dimensional collective coordinate integral. This amounts to a weak-coupling solution, in quadratures, of the low-energy dynamics of N=2 SQCD, without appeal to duality. As an application, we calculate F_1 for all SU(N) and any number of flavors N_F; for N_F<2N-2 and N_F=2N-1 we confirm previous instanton calculations and agree with the proposed hyper-elliptic curve solutions. For N_F=2N-2 and N_F=2N with N>3 we obtain new results, which in the latter case we do not understand how to reconcile with the curves. 
  The effective action of N=2 gauge multiplets in general includes higher-dimension UV finite nonholomorphic corrections integrated with the full N=2 superspace measure. By adding a hypermultiplet in the adjoint representation we study the effective action of N=4 SYM. The nonanomalous SU(4) R-symmetry of the classical N=4 theory must be also present in the on-shell effective action, and therefore we expect to find similar nonholomorphic terms for each of the scalars in the hypermultiplet. The N=2 path integral quantization formalism developed in projective superspace allows us to compute these hypermultiplet nonholomorphic terms directly in N=2 superspace. The corresponding gauge multiplet expression can be successfully compared with the result inferred from a N=1 calculation in the abelian subsector. 
  A version of the kappa-symmetric super D-p-brane action is presented in which the tension is a dynamical variable, equal to the flux of a p-form worldvolume gauge field. The Lagrangian is shown to be invariant under all (super)isometries of the background for appropriate transformations of the worldvolume gauge fields, which determine the central charges in the symmetry algebra. We also present the Hamiltonian form of the action in a general supergravity background. 
  The first massive level of closed bosonic string theory is studied. Free-field equations are derived by imposing Weyl invariance on the world sheet. A two-parameter solution to the equation of motion and constraints is found in two dimensions with a flat linear-dilaton background. One-to-one tachyon scattering is studied in this background. The results support Dhar, Mandal and Wadia's proposal that 2D critical string theory corresponds to the c=1 matrix model in which both sides of the Fermi sea are excited. 
  In this paper we study the Abelian Anti-ghost equation for the Standard Model quantized in the 't Hooft-Background gauge. We show that this equation assures the non-renormalization of the abelian ghost fields and prevents possible abelian anomalies. 
  For a Calabi-Yau threefold admitting both a $K3$ fibration and an elliptic fibration (with some extra conditions) we discuss candidate asymptotic expressions of the genus 0 and 1 Gromov-Witten potentials in the limit (possibly corresponding to the perturbative regime of a heterotic string) where the area of the base of the $K3$ fibration is very large. The expressions are constructed by lifting procedures using nearly holomorphic Weyl-invariant Jacobi forms. The method we use is similar to the one introduced by Borcherds for the constructions of automorphic forms on type IV domains as infinite products and employs in an essential way the elliptic polylogarithms of Beilinson and Levin. In particular, if we take a further limit where the base of the elliptic fibration decompactifies, the Gromov-Witten potentials are expressed simply by these elliptic polylogarithms. The theta correspondence considered by Harvey and Moore which they used to extract the expression for the perturbative prepotential is closely related to the Eisenstein-Kronecker double series and hence the real versions of elliptic polylogarithms introduced by Zagier. 
  We use (nonconservative) dynamical semigroups to investigate the decay law of a quantum unstable system weakly coupled with a large environment. We find that the deviations from the classical exponential law are small and can be safely ignored in any actual experiment. 
  An effective string theory emerging from the bilocal approximation to the Method of Vacuum Correlators in gluodynamics is shown to be well described by the 4D theory of the massive Abelian Kalb-Ramond field interacting with the string, which is known to be the low-energy limit of the Universal Confining String Theory. The mass of the Kalb-Ramond field in this approach plays the role of the inverse correlation length of the vacuum, and it is shown that in the massless limit string picture disappears. The background field method, known in the theory of nonlinear sigma models, is applied to derivation of the effective action, quadratic in quantum fluctuations around a given (e.g. minimal) string world-sheet. 
  The basic ingredients of Tomita-Takesaki modular theory are used to establish cluster estimates. Applications to thermal quantum field theory are discussed. 
  We show how to reduce the non abelian Born-Infeld action describing the interaction of two D-particles to the sum of elliptic integrals depending on simple kinematic invariants. This representation gives explicitly all alpha' corrections to D-particle dynamics. The alpha' corrections induce a stabilization of the classical trajectories such as the ``eikonal'' which are unstable within the Yang-Mills approximation. 
  The assumptions behind the recently conjectured relation between gauge theory and supergravity are elaborated on. It is pointed out that the scaling limit that preserves supergravity solutions, gives the entire DBI action on the gauge theory side, but in the low energy limit the relation between the conformal field theory and Anti-de Sitter supergravity emerges. We also argue that recent work on these issues may help in understanding the physics of five (four) dimensional black hole with three (four) charges in the so-called dilute gas region. 
  We discuss the meaning of a Casher-Banks relation for the Dirac operator eigenvalues in MQCD. It suggests the interpretaion of the eigenvalue as a coordinate involved in the brane configuration. 
  Classical dynamics in SU(2) Matrix theory is investigated. A classical chaos-order transition is found. For the angular momentum small enough (even for small coupling constant) the system exhibits a chaotic behavior, for angular momentum large enough the system is regular. 
  We compute the anomalous divergence of currents associated with global transformations in the antifield formalism, by introducing compensating fields that gauge these transformations. We consider the explicit case of the global axial current in QCD but the method applies to any global transformation of the fields. 
  Exact non-perturbative results have been conjectured for R^4 couplings in type II maximally supersymmetric string theory. Strong evidence has already been obtained, but contributions of cusp forms, invisible in perturbation theory, have remained an open possibility. In this note, we use the D=8 N=2 superfield formalism of Berkovits to prove that supersymmetry requires the exact R^4 threshold to be an eigenmode of the Laplacian on the scalar manifold with a definite eigenvalue. Supersymmetry and U-duality invariance then identify the exact result with the order-3/2 Eisenstein series, and rule out cusp form contributions. 
  In the light-cone Fock state expansion of gauge theories, the influence of non-valence states may be significant in precision non-perturbative calculations. In two-dimensional gauge theories, it is shown how these states modify the behaviour of the light-cone wavefunction in significant ways relative to endemic choices of variational ansatz. Similar effects in four-dimensional gauge theories are briefly discussed. 
  We present the thermodynamic Bethe ansatz as a way to factorize the partition function of a 2d field theory, in particular, a conformal field theory and we compare it with another approach to factorization due to K. Schoutens which consists of diagonalizing matrix recursion relations between the partition functions at consecutive levels. We prove that both are equivalent, taking as examples the SU(2) spinons and the 3-state Potts model. In the latter case we see that there are two different thermodynamic Bethe ansatz equation systems with the same physical content, of which the second is new and corresponds to a one-quasiparticle representation, as opposed to the usual two-quasiparticle representation. This new thermodynamic Bethe ansatz system leads to a new dilogarithmic formula for the central charge of that model. 
  We study different aspects of the construction of D=4, N=1 type IIB orientifolds based on toroidal Z_N and Z_M x Z_N, D=4 orbifolds. We find that tadpole cancellation conditions are in general more constraining than in six dimensions and that the standard Gimon-Polchinski orientifold projection leads to the impossibility of tadpole cancellations in a number of Z_N orientifolds with even N including Z_4, Z_8, Z_8' and Z_{12}'. We construct D=4, Z_N and Z_N x Z_M orientifolds with different configurations of 9-branes, 5-branes and 7-branes, most of them chiral. Models including the analogue of discrete torsion are constructed and shown to have features previously conjectured on the basis of F-theory compactified on four-folds. Different properties of the D=4, N=1 models obtained are discussed including their possible heterotic duals and effective low-energy action. These models have in general more than one anomalous U(1) and the anomalies are cancelled by a D=4 generalized Green-Schwarz mechanism involving dilaton and moduli fields. 
  We consider supersymmetric gauge theories with impurities in various dimensions. These systems arise in the study of intersecting branes. Unlike conventional gauge theories, the Higgs branch of an impurity theory can have compact directions. For models with eight supercharges, the Higgs branch is a hyperKahler manifold given by the moduli space of solutions of certain differential equations. These equations are the dimensional reductions of self-duality equations with boundary conditions determined by the impurities. They can also be interpreted as Nahm transforms of self-duality equations on toroidally compactified spaces. We discuss the application of our results to the light-cone formulation of Yang-Mills theories and to the solution of certain N=2 d=4 gauge theories. 
  We establish a duality between the free massless relativistic particle in d dimensions, the non-relativistic hydrogen atom (1/r potential) in (d-1) space dimensions, and the harmonic oscillator in (d-2) space dimensions with its mass given as the lightcone momentum of an additional dimension. The duality is in the sense that the classical action of these systems are gauge fixed forms of the same worldline gauge theory action at the classical level, and they are all described by the same unitary representation of the conformal group SO(d,2) at the quantum level. The worldline action has a gauge symmetry Sp(2) which treats canonical variables (x,p) as doublets and exists only with a target spacetime that has d spacelike dimensions and two timelike dimensions. This spacetime is constrained due to the gauge symmetry, and the various dual solutions correspond to solutions of the constraints with different topologies. For example, for the H-atom the two timelike dimensions X^{0'},X^{0} live on a circle. The model provides an example of how realistic physics can be viewed as existing in a larger covariant space that includes two timelike coordinates, and how the covariance in the larger space unifies different looking physics into a single system. 
  We consider generic MQCD configurations with matter described by semi-infinte D4-branes and softly broken supersymmetry. We show that the matter sector does not introduce supersymmetry breaking parameters so that the most relevant supersymmetry breaking operator at low energies is the gaugino mass term. By studying the run-away properties of these models in the decoupling limit of the adjoint matter, we argue that these softly broken MQCD configurations fail to capture the infrared physics of QCD at scales below the gaugino mass scale. 
  By making use of the duality transformation, gauge field correlators of the Abelian Higgs Model are studied in the London limit. The obtained results are in a good agreement with the dual Meissner scenario of confinement and with the Stochastic Model of QCD vacuum. The nontrivial contribution to the quartic correlator arising due to accounting for the finiteness of the coupling constant is discussed. 
  We present and analyze solutions of D=11 supergravity describing the ``near-horizon'' (i.e., asymptotically AdS_4 x S^7) geometry of M2-branes wrapped on surfaces of arbitrary genus. We study the forces experienced by test M2-branes in such backgrounds, and find evidence that extremal branes on surfaces of genera higher than the torus are unstable. Using the holographic connection between AdS spaces and superconformal field theories in the large N limit, we discuss the phases of the associated 2+1 dimensional theories. Finally, we also study the extension of these solutions to other branes, in particular to D2-branes. 
  We give a simple prescription for computing, in the framework of the bosonic string theory, off-shell one-loop amplitudes with any number of external massless particles, both for the open and for the closed string. We discuss their properties and, in particular, for the two-string one-loop amplitudes we show their being transverse. 
  We calculate the one-loop effective action for conformal matter (scalars, spinors and vectors) on spherically symmetric background. Such effective action (in large $N$ approximation and expansion on curvature) is used to study quantum aspects of Schwarzschild-de Sitter black holes (SdS BHs) in nearly degenerated limit (Nariai BH). We show that for all types of above matter SdS BHs may evaporate or anti-evaporate in accordance with recent observation by Bousso and Hawking for minimal scalars. Some remarks about energy flow for SdS BHs in regime of evaporation or anti-evaporation are also done. Study of no boundary condition shows that this condition supports anti-evaporation for nucleated BHs (at least in frames of our approximation). That indicates to the possibility that some pair created cosmological BHs may not only evaporate but also anti-evaporate. Hence, cosmological primordial BHs may survive much longer than it is expected. 
  With the aim of clarifying the eleven dimensional content of Matrix theory, we examine the dependence of a theory in the infinite momentum frame (IMF) on the (purely spatial) longitudinal compactification radius R. It is shown that in a point particle theory the generic scattering amplitude becomes independent of R in the IMF. Processes with zero longitudinal momentum transfer are found to be exceptional. The same question is addressed in a theory with extended objects. A one-loop type II string amplitude is shown to be R-independent in the IMF, and to coincide with that of the uncompactified theory. No exceptional processes exist in this case. The possible implications of these results for M-theory are discussed. In particular, if amplitudes in M-theory are independent of R in the IMF, Matrix theory can be rightfully expected (in the N -> infty limit) to describe uncompactified M-theory. 
  We use the correspondence between scalar field theory on $AdS_{d+1}$ and a conformal field theory on $R^d$ to calculate the 3- and 4-point functions of the latter. The classical scalar field theory action is evaluated at tree level. 
  A hidden gauge theory structure of quantum mechanics which is invisible in its conventional formulation is uncovered. Quantum mechanics is shown to be equivalent to a certain Yang-Mills theory with an infinite-dimensional gauge group and a nondynamical connection. It is defined over an arbitrary symplectic manifold which constitutes the phase-space of the system under consideration. The ''matter fields'' are local generalizations of states and observables; they assume values in a family of local Hilbert spaces (and their tensor products) which are attached to the points of phase-space. Under local frame rotations they transform in the spinor representation of the metaplectic group Mp(2N), the double covering of Sp(2N). The rules of canonical quantization are replaced by two independent postulates with a simple group theoretical and differential geometrical interpretation. A novel background-quantum split symmetry plays a central role. 
  We analyze the strong coupling behavior of the large N gauge theories in 4-dimensions with N=4 supersymmetry by making use of S-duality. We show that at large values of the coupling constant $\lambda=g_{YM}^2N$ the j-th non-planar amplitude $f_j(\lambda) (j=0,1,2 ...)$ behaves as $f_j(\lambda)\approx \lambda^j$. Implication of this behavior is discussed in connection with the supergravity theory on $AdS_5\times S^5$ suggested by the CFT/AdS correspondence. S-duality of the gauge theory corresponds to the duality between the closed and open string loop expansions in the gravity/string theory. 
  We study the soft breaking of N=2 self-dual gauge theories down to N=0 by promoting the coupling constant and hypermultiplet masses to spurions, and we analyze the microscopic duality symmetry in the resulting models. Explicit formulae are given for the Seiberg-Witten periods and couplings in the case of SU(2), and we perform a numerical study of the non-supersymmetric vacuum structure in the case of the mass-deformed N=4 SU(2) gauge theory. Although the softly broken model has a well-defined behavior under the duality symmetry, the stable vacua are in a confining phase. We also extend some of the results to the self-dual theories with classical gauge groups, and we obtain the RG equation for these models. 
  Solutions to the equations of motion of the low energy, effective field theory emerging out of compactified heterotic string theory are constructed by making use of the well-known duality symmetries. Beginning with four-dimensional solutions of the Einstein-massless scalar field theory in the canonical frame we first rewrite the corresponding solutions in the string frame. Thereafter, using the T and S duality symmetries of the low energy string effective action we arrive at the corresponding uncharged, electrically charged and magnetically charged solutions. Brief comments on the construction of dual versions of the Kerr-Sen type using the dilatonic Kerr solution as the seed are also included. Thereafter, we verify the status of the energy conditions for the solutions in the string frame. Several of the metrics found here are shown to possess naked singularities although the energy conditions are obeyed. Dual solutions exhibit a duality in the conservation/violation of the Null and Averaged Null Energy Conditions (NEC/ANEC), a fact demonstrated earlier in the context of black holes (hep-th/9604047) and cosmologies (hep-th/9611122). Additionally, those backgrounds which conserve the NEC/ANEC in spite of possesing naked singularities serve as counterexamples to cosmic censorship in the context of low energy, effective string theory. 
  New heterotic modular invariants are found using the level-rank duality of affine Kac-Moody algebras. They provide strong evidence for the consistency of an infinite list of heterotic Wess-Zumino-Witten (WZW) conformal field theories. We call the basic construction the dual-flip, since it flips chirality (exchanges left and right movers) and takes the level-rank dual. We compare the dual-flip to the method of conformal subalgebras, another way of constructing heterotic invariants. To do so, new level-one heterotic invariants are first found; the complete list of a specified subclass of these is obtained. We also prove (under a mild hypothesis) an old conjecture concerning exceptional $A_{r,k}$ invariants and level-rank duality. 
  The O(N) symmetric vector model is considered on both ordinary and fuzzy sphere. It is shown that in both cases master fields exist and their explicit forms are presented. They are found to mix the internal symmetry and the (fuzzy) space-time symmetry. It is also argued that the cutoff brought by the fuzzy sphere plays an essential role in constructing the master field. 
  Making use of the duality transformation, we construct string representation for the partition function of the London limit of Abelian Higgs Model with an additional axionic term. In the lowest order of perturbation theory, this term leads to the appearance in the resulting string effective action of a new threelocal interaction between the elements of the string world-sheet. Consequently, there emerges a threelocal correlator of the dual field strength tensors, which does not contain the average over world-sheets, and is therefore nontrivial even in the sector of the theory with a single small vortex. The relation between the obtained correlator and the bilocal one is established. Finally, it is argued that the vacuum structure of the London limit of the Abelian Higgs Model with an additional axionic interaction is much more similar to that of gluodynamics rather than without this interaction. 
  We present a detailed algebraic study of the N=2 cohomological set--up describing the balanced topological field theory of Dijkgraaf and Moore. We emphasize the role of N=2 topological supersymmetry and $sl(2,R)$ internal symmetry by a systematic use of superfield techniques and of an $sl(2,R)$ covariant formalism. We provide a definition of N=2 basic and equivariant cohomology, generalizing Dijkgraaf's and Moore's, and of N=2 connection. For a general manifold with a group action, we show that: $i$) the N=2 basic cohomology is isomorphic to the tensor product of the ordinary N=1 basic cohomology and a universal $sl(2,R)$ group theoretic factor: $ii$) the affine spaces of N=2 and N=1 connections are isomorphic. 
  We discuss hidden symmetries of three-dimensional field configurations revealed at the one-particle level by the use of pseudoclassical particle models. We argue that at the quantum field theory level, these can be naturally explained in terms of manifest symmetries of the reduced phase space Hamiltonian of the corresponding field systems. 
  Arguing that the equation for the gonihedric string should have a generalized Dirac form, we found a new equation which corresponds to a symmetric solution of the Majorana commutation relations and has non-Jacobian form. The corresponding generalized gamma-matrices are anticommuting. Explicit formulas for the mass spectrum lead to nonzero string tension $M^{2}_{j} \geq M^{2}(j+1)^{2}$. We discuss also new dual transformation of the Dirac equation and of the proposed generalizations. 
  We investigate the SO(10)-unification model in a Lie algebraic formulation of noncommutative geometry. The SO(10)-symmetry is broken by a 45-Higgs and the Majorana mass term for the right neutrinos (126-Higgs) to the standard model structure group. We study the case that the fermion masses are as general as possible, which leads to two 10-multiplets, four 120-multiplets and two additional 126-multiplets of Higgs fields. This Higgs structure differs considerably from the two Higgs multiplets 16 \otimes 16^* and 16^c \otimes 16^* used by Chamseddine and Fr\"ohlich. We find the usual tree-level predictions of noncommutative geometry m_W=(1/2)m_t, \sin^2\theta_W=(3/8) and g_2=g_3 as well as m_H \leq m_t. 
  The ground, one- and two-particle states of the (1+1)-dimensional massive sine-Gordon field theory are investigated within the framework of the Gaussian wave-functional approach. We demonstrate that for a certain region of the model-parameter space, the vacuum of the field system is asymmetrical. Furthermore, it is shown that two-particle bound state can exist upon the asymmetric vacuum for a part of the aforementioned region. Besides, for the bosonic equivalent to the massive Schwinger model, the masses of the one boson and two-boson bound states agree with the recent second-order results of a fermion-mass perturbation calculation when the fermion mass is small. 
  We evaluate the one-loop fermion self-energy for the gauged Thirring model in (2+1) dimensions, with one massive fermion flavor, in the framework of the causal perturbation theory. In contrast to QED$_3$, the corresponding two-point function turns out to be infrared finite on the mass shell. Then, by means of a Ward identity, we derive the on-shell vertex correction and discuss the role played by causality for nonrenormalizable theories. 
  It was conjectured by Jackson et.al. that the finite volume effective partition function of QCD with the topological charge $M-N$ coincides with the Itzyskon-Zuber type integral for $M\times N$ rectangular matrices. In the present article we give a proof of this conjecture, in which the original Itzykson-Zuber integral is utilized. 
  Two point correlation functions of the off-critical primary fields \phi_{1, 1+s} are considered in the perturbed minimal models M_{2, 2N+3} + \phi_{1,3}. They are given as infinite series of form factor contributions. The form factors of \phi_{1, 1+s} are conjectured from the known results for those of \phi_{1,2} and \phi_{1,3}. The conjectured form factors are rewritten in the form which is convenient for summing up. The final expression of the two point functions is written as a determinant of an integral operator. 
  Actions for branes, with or without worldsurface gauge fields, are discussed in a unified framework. A simple algorithm is given for constructing the component Green-Schwarz actions. Superspace actions are also discussed. Three examples are given to illustrate the general procedure: the membrane in D=11 and the D2-brane, which both have on-shell worldsurface supermultiplets, and the membrane in D=4, which has an off-shell multiplet. 
  Superalgebras including generators having spins up to two and realisable as tangent vector fields on Lorentz covariant generalised superspaces are considered. The latter have a representation content reminiscent of configuration spaces of (super)gravity theories. The most general canonical supercommutation relations for the corresponding phase space coordinates allowed by Lorentz covariance are discussed. By including generators transforming according to every Lorentz representation having spin up to two, we obtain, from the super Jacobi identities, the complete set of quadratic equations for the Lorentz covariant structure constants. These defining equations for spin two Heisenberg superalgebras are highly overdetermined. Nevertheless, non-trivial solutions can indeed be found. By making some simplifying assumptions, we explicitly construct several classes of these superalgebras. 
  We investigate the superconformal transformation properties of Green functions with one or more insertions of the supercurrent in N=1 supersymmetric quantum field theories. These Green functions are conveniently obtained by coupling the supercurrent and its trace to a classical supergravity background. We derive flat space superconformal Ward identities from diffeomorphisms and Weyl transformations on curved superspace. For the classification of potential quantum superconformal anomalies in the massless Wess-Zumino model on curved superspace a perturbative approach is pursued, using the BPHZ scheme for renormalisation. By deriving a local Callan-Symanzik equation the usual dilatational anomalies are identified and it is shown that no further superconformal anomalies involving the dynamical fields are present. 
  The new method based on the operator formalism proposed by Abe and Nakanishi is applied to the quantum nonlinear abelian gauge theory in two dimension. The soluble models in this method are extended to wider class of quantum field theories. We obtain the exact solution in the canonical-quantization operator formalism in the Heisenberg picture. So this analysis might shed some light on the analysis of gravitational theory and non-polynomial field theories. 
  A manifestly Poincare invariant formulations for $SO(1,10)$ and SO(2,9) superstring actions are proposed. The actions are invariant under a local fermionic $\kappa$-symmetry as well as under a number of global symmetries, which turn out to be on-shell realization of the known ``new supersymmetry`` S-algebra. Canonical quantization of the theory is performed and relation of the quantum state spectrum with that of type IIA Green-Schwarz superstring is discussed. Besides, a mechanical model is constructed, which is a zero tension limit of the D=11 superstring and which incorporates all essential features of the latter. A corresponding action invariant under off-shell closed realization of the S-algebra is obtained. 
  In this paper we reconsider, for N=8 supergravity, the problem of gauging the most general electric subgroup.   We show that admissible theories are fully characterized by a single algebraic equation to be satisfied by the embedding of the gauge group G within the electric subalgebra SL(8,\IR) of E_{7(7)}. The complete set of solutions to this equation contains 36 parameters. Modding by the action of SL(8,\IR) conjugations that yield equivalent theories all continuous parameters are eliminated except for an overall coupling constant and we obtain a discrete set of orbits. This set is in one--to--one correspondence with 36 Lie subalgebras of SL(8,\IR), corresponding to all possible real forms of the SO(8) Lie algebra plus a set of contractions thereof. By means of our analysis we establish the theorem that the N=8 gaugings constructed by Hull in the middle eighties constitute the exhaustive set of models. As a corollary we show that there exists a unique 7--dimensional abelian gauging. The corresponding abelian algebra is not contained in the maximal abelian ideal of the solvable Lie algebra generating the scalar manifold E_{7(7)}/SU(8). 
  We find aspects of electrically confining large $N$ Yang-Mills theories on $T^2 \times R^{d-2}$ which are consistent with a $GL(2,Z)$ duality. The modular parameter associated with this $GL(2,Z)$ is given by ${m\over N} + i\Lambda^2 A$, where $A$ is the area of the torus, $m$ is the t'Hooft twist on the torus, and $\Lambda^2$ is the string tension. $N$ is taken to infinity keeping $m\over N$ and $g^2N$ fixed. This duality may be interpreted as T-duality of the QCD string if one identifies the magnetic flux with a two-form background in the string theory. Our arguments make no use of supersymmetry. While we are not able to show that this is an exact self duality of conventional QCD, we conjecture that it may be applicable within the universality class of QCD. We discuss the status of the conjecture for the soluble case of pure two dimensional Euclidean QCD on $T^2$, which is almost but not exactly self dual. For higher dimensional theories, we discuss qualitative features consistent with duality. For $m=0$, such a duality would lead to an equivalence between pure QCD on $R^4$ and QCD on $R^2$ with two adjoint scalars. When $\Lambda^2 A << m^2/N^2$, the proposed duality includes exchanges of rank with twist. This exchange bears some resemblance, but is not equivalent, to Nahm duality. A proposal for an explicit perturbative map which implements duality in this limit is discussed. 
  Conformal techniques are applied to the calculation of integrals on AdS(d+1) space which define correlators of composite operators in the superconformal field theory on the d-dimensional boundary. The 3-point amplitudes for scalar fields of arbitrary mass and gauge fields in the AdS supergravity are calculated explicitly. For 3 gauge fields we compare in detail with the known conformal structure of the SU(4) flavor current correlator <J_i^a J_j^b J_k^c> of the N=4, d=4 SU(N) SYM theory. Results agree with the free field approximation as would be expected from superconformal non-renormalization theorems. In studying the Ward identity relating <J_i^a O^I O^J> to <O^I O^J> for (non-marginal) scalar composite operators O^I, we find that there is a subtlety in obtaining the normalization of <O^I O^J> from the supergravity action integral. 
  Three dimensional N=2 gauge theories with arbitrary gauge group and fundamental flavors are engineered from degenerations of Calabi-Yau four-folds. We show how Coulomb and Higgs branches emerge in the geometric picture. The analysis of instanton generated superpotentials unravels interesting aspects of the five-brane effective action in M theory. 
  We analyze the large $N$ spectrum of chiral primary operators of three dimensional fixed points of the renormalization group. Using the space-time picture of the fixed points and the correspondence between anti-de Sitter compactifications and conformal field theories we are able to extract the dimensions of operators in short superconformal multiplets. We write down some of these operators in terms of short distance theories flowing to these non-trivial fixed points in the infrared. 
  We discuss the effects of instantons in partially broken gauge groups on the low-energy effective gauge theory. Such effects arise when some of the instantons of the original gauge group G are no longer contained in (or can not be gauge rotated into) the unbroken group H. In cases of simple G and H, a good indicator for the existence of such instantons is the ``index of embedding.'' However, in the general case one has to examine \pi_3(G/H) to decide whether there are any instantons in the broken part of the gauge group. We give several examples of supersymmetric theories where such instantons exist and leave their effects on the low-energy effective theory. 
  We compute the entropy of extremal black strings in three dimensions, using Strominger's approach to relate the Anti-de-Sitter near-horizon geometry and the conformal field theory at the asymptotic infinity of this geometry. The result is identical to the geometric Bekenstein-Hawking entropy. We further discuss an embedding of three-dimensional black strings in $N=1 D=10$ supergravity and demonstrate that the extremal strings preserve 1/4 of supersymmetries. 
  We consider the analogue of the 6-vertex model constructed from alternating spin n/2 and spin m/2 lines, where $1\leq n<m$. We identify the transfer matrix and the space on which it acts in terms of the representation theory of $U_q(sl_2)$. We diagonalise the transfer matrix and compute the S-matrix. We give a trace formula for local correlation functions. When n=1, the 1-point function of a spin m/2 local variable for the alternating lattice with a particular ground state is given as a linear combination of the 1-point functions of the pure spin m/2 model with different ground states. The mixing ratios are calculated exactly and are expressed in terms of irreducible characters of $U_q(sl_2)$ and the deformed Virasoro algebra. 
  We consider four-dimensional charged black-holes occuring in toroidally compactified heterotic string theory, whose ten-dimensional interpretation involves a Kaluza-Klein monopole and a five-brane. We show that these four-dimensional black-holes can be connected to two-dimensional charged heterotic black-holes upon removal of the constants appearing in the harmonic functions associated with the Kaluza-Klein monopole and the five-brane. 
  The formal extension of the T-duality rules for open strings from Abelian to non-Abelian gauge field background leads in a well known manner to the notion of matrix valued D-brane position. The application of this concept to the non-Abelian gauge field RG $\beta $-function of the corresponding $\sigma $-model yields a mass term in the gauge field dynamics on the matrix D-brane. The direct calculation in a corresponding D-brane model does $not$ yield such a mass term, if the Dirichlet boundary condition is implemented as a constraint on the integrand in the defining functional integral. However, the mass term arises in the direct calculation for a D-brane model with dynamically realized boundary condition. 
  In this paper we conjecture a reformulation of the monomial-divisor mirror map for (2,2) mirror symmetry, valid at a boundary of the moduli space, that is easily extended to also include tangent bundle deformations -- an important step towards understanding (0,2) mirror symmetry. We check our conjecture in a few simple cases, and thereby illustrate how to perform calculations using a description of sheaves recently published by Knutson, Sharpe. 
  Polchinski and Pouliot have shown that M-momentum transfer between membranes in supergravity can be understood as a non-perturbative instanton effect in gauge theory. Here we consider a dual process: electric flux transmission between D-branes. We show that this process can be described in perturbation theory as virtual string pair creation, and is closely related to Schwinger's treatment of the pair creation of charged particles in a uniform electric field. Through the application of dualities, our perturbative calculation gives results for various non-perturbative amplitudes, including M-momentum transfer between gravitons, membranes, and longitudinal fivebranes. Thus perturbation theory plus dualities are sufficient to demonstrate agreement between supergravity and gauge theory for a number of M-momentum transferring processes. A variety of other processes where branes are transmitted between branes, e.g. (p,q)-string transmission in IIB-theory, can also be studied. We discuss the implications of our results for proving the 11 dimensional Lorentz invariance of Matrix theory. 
  We consider strongly coupled supersymmetric gauge theories softly broken by the addition of gaugino masses $m_\lambda$ and (non-holomorphic) scalar masses $m^2$, taken to be small relative to the dynamical scale $\Lambda$. For theories with a weakly coupled dual description in the infrared, we compute exactly the leading soft masses for the "magnetic" degrees of freedom, with uncalculable corrections suppressed by powers of $(m_{\lambda}/\Lambda), (m/\Lambda)$. The exact relations hold between the infrared fixed point "magnetic" soft masses and the ultraviolet fixed point "electric" soft masses, and correspond to a duality mapping for soft terms. We briefly discuss implications of these results for the vacuum structure of these theories. 
  A number of N=2 gauge theories can be realized by brane configurations in Type IIA string theory. One way of solving them involves lifting the brane configuration to M-theory. In this paper we present an alternative way of analyzing a subclass of these theories (elliptic models). We observe that upon compactification on a circle one can use a version of mirror symmetry to map the original brane configuration into one containing only D-branes. Simultaneously the Coulomb branch of the four-dimensional theory is mapped to the Higgs branch of a five-dimensional theory with three-dimensional impurities. The latter does not receive quantum corrections and can be analyzed exactly. The solution is naturally formulated in terms of an integrable system, which is a version of a Hitchin system on a punctured torus. 
  We derive a dual theory of the four-dimensional anomalous U(1) gauge theory with a Wess--Zumino (WZ) term and with a St\"uckelberg type mass term by means of a duality transformation at each of the classical and quantum levels. It is shown that in the dual anomalous U(1) gauge theory, the $BF$ term with a rank-two antisymmetric tensor field plays the roles of the WZ term as well as the mass term of the U(1) gauge field. Similar anomalous U(1) gauge theory with $BF$ term is considered in six-dimensions by introducing a rank-four antisymmetric tensor field. In addition to this theory, we propose a six-dimensional anomalous U(1) gauge theory including an extended $BF$ term with a rank-two antisymmetric tensor field, discussing a difference between the two theories. We also consider a four-dimensional anomalous SU(2)$\times$U(1) gauge theory with $BF$ term and recognize a crucial role of the $BF$ term in cancelling the non-abelian chiral anomaly. 
  In a recent work, T.S. Evans has claimed that the multiplicative anomaly associated with the zeta-function regularization of functional determinants is regularization dependent. We show that, if one makes use of consistent definitions, this is not the case and clarify some points in Evans' argument. 
  In a recent work, S. Dowker has shed doubt on a recipe used in computing the partition function for a matrix valued operator. This recipe, advocated by Benson, Bernstein and Dodelson, leads naturally to the so called multiplicative anomaly for the zeta-function regularized functional determinants. In this letter we present arguments in favour of the mentioned prescription, showing that it is the valid one in calculations involving the relativistic charged bosonic ideal gas in the framework of functional analysis. 
  In D=10 N=1 super Yang-Mills theory we give the background breaking a half of supersymmetry. In the background there is a six-dimensional object so called D=10 5-brane. 
  In this paper we discuss some aspects of the behavior of superconformal N=1 models under Seiberg's duality. Our claim is that if an electric gauge theory is superconformal on some marginal subspace of all coupling constants then its magnetic dual must be also superconformal on a corresponding moduli space of dual couplings. However this does not imply that the magnetic dual of a completely finite N=1 gauge theory is again finite. Moreover we generalize this statement conjecturing that also for non-superconformal N=1 models the determinant of the beta-function equations is invariant under Seiberg duality. During the course of this investigation we construct some superconformal N=1 gauge theories which were not yet discussed before. 
  In this article we review some recent developments in heterotic compactifications. In particular we review an ``inherently toric'' description of certain sheaves, called equivariant sheaves, that has recently been discussed in the physics literature. We outline calculations that can be performed with these objects, and also outline more general phenomena in moduli spaces of sheaves. 
  We derive the complete covariant action for the type IIA superstring in a simple D=10 background which represents a 7-brane with a magnetic Ramond-Ramond vector field (and is U-dual to the Kaluza-Klein Melvin solution). This curved background can be obtained by dimensional reduction from a flat (but topologically non-trivial) D=11 space-time. The action of a supermembrane propagating in this flat D=11 space is straightforward to write down. The explicit form of the superstring action is then obtained by double dimensional reduction of the supermembrane action. In the light-cone gauge the action contains only quadratic and quartic terms in fermions. 
  Conventional non-Abelian SO(4) gauge theory is able to describe gravity provided the gauge field possesses a specific polarized vacuum state. In this vacuum the instantons and anti-instantons have a preferred direction of orientation. Their orientation plays the role of the order parameter for the polarized phase of the gauge field. The interaction of a weak and smooth gauge field with the polarized vacuum is described by an effective long-range action which is identical to the Hilbert action of general relativity. In the classical limit this action results in the Einstein equations of general relativity. Gravitational waves appear as the mode describing propagation of the gauge field which strongly interacts with the oriented instantons. The Newton gravitational constant describes the density of the considered phase of the gauge field. The radius of the instantons under consideration is comparable with the Planck radius. 
  We demonstrate how some problems arising in simplicial quantum gravity can be successfully addressed within the framework of combinatorial group theory. In particular, we argue that the number of simplicial 3-manifolds having a fixed homology type grows exponentially with the number of tetrahedra they are made of. We propose a model of 3D gravity interacting with scalar fermions, some restriction of which gives the 2-dimensional self-avoiding-loop-gas matrix model. We propose a qualitative picture of the phase structure of 3D simplicial gravity compatible with the numerical experiments and available analytical results. 
  We point out that there is a missing portion in the two-loop effective potential of the massless O(N) phi^4 theory obtained by Jackiw in his classic paper, Phys. Rev. D 9, 1686 (1974). 
  We introduce a new family of integrable theories with $N$ bosons and $N$ freely adjustable mass parameters. These theories restrict in particular limits to the ``generalized supersymmetric'' sine-Gordon models, as well as to the flavor anisotropic chiral Gross Neveu models (studied recently by N. Andrei and collaborators). The scattering theory involves scalar particles that are no bound states, and bears an intriguing resemblance wih the results of a sharp cut-off analysis of the Thirring model carried out by Korepin in (1980). Various physical applications are discussed. In particular, we demonstrate that our theories are the appropriate continuum limit of integrable quantum spin chains with mixtures of spins. 
  We derive certain boundary conditions in Nahm's equations by considering a system of N parallel D1-branes perpendicular to a D3-brane in type IIB string theory. 
  The beautiful scenario of pre-big-bang cosmology is appealling not only because it is more or less derived from string theory, but also because it separates clearly the problem of the initial conditions for the universe from that of high curvatures. Recently, the pre-big-bang program was subject to attack from on the grounds that pre-big-bang cosmology does not solve the horizon and flatness problems in a ``natural'' way, as customary exponential ``new'' inflation does. In particular, it appears that an arbitrarily small deviation from perfect flatness in the initial state can not be accommodated. For this analysis, matter in the universe before the big bang was assumed to be radiation. We perform a similar analysis to theirs, but using the equation of state for ``string matter'' $\rho=-3p$ which seems more appropriate to the physical situation and, also, is motivated by the scale factor duality (in the flat case) with respect to our expanding, radiation dominated, universe. For an open universe we find, exactly, the same time-dependence of the scale factor as in the Milne universe, recently found to represent the universal attractor at $t=-\infty $ of all pre big bang cosmologies. We conclude that our radiation dominated universe comes from a flat rather than a curved region. 
  We consider the role of N=4 conformal supergravity in the relation between N=4 SYM theory and D=5 gauged supergravity expanded near the Anti de Sitter background. We discuss the structure of the SYM effective action in conformal supergravity background, in particular, terms related to conformal anomaly. Solving the leading-order Dirichlet problem for the metric perturbation in AdS background we explicitly compute the bilinear graviton term in the D=5 Einstein action, demonstrating its equivalence to the linearized Weyl tensor squared part of the gravitational effective action induced by SYM theory. We also compute the graviton-dilaton-dilaton 3-point function which is found to have the form consistent with conformal invariance of the boundary theory. 
  We first review the interpretation of world-sheet defects as $D$ branes described by a critical theory in 11 dimensions, that we interpret as $M$ theory. We then show that $D$-brane recoil induces dynamically an anti-de-Sitter (AdS) space-time background, with criticality restored by a twelfth time-like dimension described by a Liouville field. Local physics in the bulk of this AdS$_{11}$ may be described by an $Osp(1|32,R) \otimes Osp(1|32,R)$ topological gauge theory (TGT), with non-local boundary states in doubleton representations. We draw analogies with structures previously exhibited in two-dimensional black-hole models. Wilson loops of `matter' in the TGT may be described by an effective string action, and defect condensation may yield string tension and cause a space-time metric to appear. 
  The duality relating near-horizon microstates of black holes obtained as orbifolds of a subset of AdS3 to the states of a conformal field theory is analyzed in detail. The SL(2,R) invariant vacuum on AdS3 corresponds to the NS-NS vacuum of the conformal field theory. The effect of the orbifolding is to produce a density matrix, the temperature and entropy of which coincide with the black hole. For string theory examples the spectrum of chiral primaries agrees with the spectrum of multi-particle BPS states for particle numbers less than of order the central charge. An upper bound on the BPS particle number follows from the upper bound on the U(1) charge of chiral primaries. This is a stringy exclusion principle which cannot be seen in perturbation theory about AdS3. 
  A number of formulas are displayed concerning Whitham theory for a simple example of pure N=2 susy YM with gauge group SU(2). In particular this serves to illuminate the role of Lambda and T derivatives and the interaction with prepotentials F based on Seiberg-Witten and Whitham theory. 
  We present Wess-Zumino actions for general IIA D-p-branes in explicit forms. We perform the covariant and irreducible separation of the fermionic constraints of IIA D-p-branes into the first class and the second class. A necessary condition which guarantees this separation is discussed. The generators of the local supersymmetry (kappa symmetry) and the kappa algebra are obtained. We also explicitly calculate the conserved charge of the global supersymmetry (SUSY) and the SUSY algebra which contains topological charges. 
  The existence of a local solution to the Sp(2) master equation for gauge field theory is proven in the framework of perturbation theory and under standard assumptions on regularity of the action. The arbitrariness of solutions to the Sp(2) master equation is described, provided that they are proper. It is also shown that the effective action can be chosen to be Sp(2) and Lorentz invariant (under the additional assumption that the gauge transformation generators are Lorentz tensors). 
  New approach to exact solvability of dilaton gravity theories is suggested which appeals directly to structure of field equations. It is shown that black holes regular at the horizon are static and their metric is found explicitly. If a metric possesses singularities the whole spacetime can be divided into different sheets with one horizon on each sheet between neighboring singularities with a finite value of dilaton field (addition horizons may arise at infinite value of it), neighboring sheets being glued along the singularity. The position of singularities coincide with the values of dilaton in solutions with a constant dilaton field. Quantum corrections to the Hawking temperature vanish. For a wide subset of these models the relationship between the total energy and the total entropy of the quantum finite size system is the same as in the classical limit. For another subset the metric itself does not acquire quantum corrections. The present paper generalizes Solodukhin's results on the RST model in that instead of a particular model we deal with whole classes of them. Apart from this, the found models exhibit some qualitatively new properties which are absent in the RST model. The most important one is that there exist quantum black holes with geometry regular everywhere including infinity. 
  A. Ghosh and P. Mitra made the proposal how to explain the area law for the entropy of extreme black holes in some model calculations. I argue that their approach implicitly operates with strongly singular geometries and says nothing about the contribution of regular metrics of extreme black holes into the partition function. 
  The problem of Wu-Yang ambiguities in 3 dimensions is related to the problem of existence of torsion free driebeins for an arbitrary potential. The ambiguity is only at the level of boundary conditions. We also find that in 3 dimensions any smooth Yang-Mills field tensor can be uniquely written as the non-Abelian magnetic field of a smooth Yang-Mills potential. 
  We consider six and four dimensional ${\cal N}=1$ supersymmetric orientifolds of Type IIB compactified on orbifolds. We give the conditions under which the perturbative world-sheet orientifold approach is adequate, and list the four dimensional ${\cal N}=1$ orientifolds (which are rather constrained) that satisfy these conditions. We argue that in most cases orientifolds contain non-perturbative sectors that are missing in the world-sheet approach. These non-perturbative sectors can be thought of as arising from D-branes wrapping various collapsed 2-cycles in the orbifold. Using these observations, we explain certain ``puzzles'' in the literature on four dimensional orientifolds. In particular, in some four dimensional orientifolds the ``naive'' tadpole cancellation conditions have no solution. However, these tadpole cancellation conditions are derived using the world-sheet approach which we argue to be inadequate in these cases due to appearance of additional non-perturbative sectors. The main tools in our analyses are the map between F-theory and orientifold vacua and Type I-heterotic duality. Utilizing the consistency conditions we have found in this paper, we discuss consistent four dimensional chiral ${\cal N}=1$ Type I vacua which are non-perturbative from the heterotic viewpoint. 
  We analyze the relation between the large N limit of 6 dimensional superconformal field theories with eight supercharges and M theory on orbifolds of AdS_7xS^4. We use the known spectrum of Kaluza-Klein harmonics of supergravity on AdS_7xS^4 and we take their orbifold projection to get information about the chiral primary operators of 6 dimensional SCFT which is realized on the worldvolume of M5 brane sitting at the orbifold singularities. 
  We have extended the perturbative expansion method around the Gaussian effective action to the fermionic field theory, by taking the 2-dimensional Gross-Neveu model as an example. We have computed both the zero temperature and the finite temperature effective potentials of the Gross-Neveu model up to the first perturbative correction terms, and have found that the critical temperature, at which dynamically broken symmetry is restored, is significantly improved for small value of the flavour number. 
  We study the 5D black holes in the type IIB superstring theory compactified on $S^1 \times T^4$. Far from horizon, we have flat space-time. Near horizon, we have $AdS_3(BTZ black hole) \times S^3 \times T^4$. We calculate the greybody factor of a minimally coupled scalar by replacing the original geometry($M_5 \times S^1 \times T^4$) by $AdS_3 \times S^3 \times T^4$. In the low-energy scattering, it turns out that the result agrees with the greybody factor of the 5D black hole (or D1 + D5 branes)in the dilute gas approximation. This confirms that the $AdS$-theory($AdS_3 \times S^3 \times T^4$) contains the essential information about the bulk 5D black holes. 
  We provide the first example of a cosmological solution of the Horava-Witten supergravity. This solution is obtained by exchanging the role of time with the radial coordinate of the transverse space to the five-brane soliton. On the boundary this corresponds to rotating an instanton solution into a tunneling process in a space with Lorentzian signature, leading to an expanding universe. Due to the freedom to choose different non-trivial Yang-Mills backgrounds on the boundaries, the two walls of the universe ( visible and hidden worlds) expand differently. However at late times the anisotropy is washed away by gravitational interactions. 
  The curves that describe the M-theoretic extension of type IIA string configurations with non-supersymmetric field theories on their surface exhibit a duality map. The map suggests a continued link between a SU(N) gauge theory with F flavours and an SU(F-N) gauge theory with F flavours (the duality of supersymmetric QCD) even when the gaugino mass is taken to infinity. Within the context of the field theory such a duality only continues to make sense if the scalar fields remain light. We discuss the difficulties of decoupling the scalars within this framework. 
  A foundational investigation of the basic structural properties of two-dimensional anomalous gauge theories is performed. The Hilbert space is constructed as the representation of the intrinsic local field algebra generated by the fundamental set of field operators whose Wightman functions define the model. We examine the effect of the use of a redundant field algebra in deriving basic properties of the models and show that different results may arise, as regards the physical properties of the generalized chiral model, in restricting or not the Hilbert space as representation of the intrinsic local field algebra. The question referring to considering the vector Schwinger model as a limit of the generalized anomalous model is also discussed. We show that this limit can only be consistently defined for a field subalgebra of the generalized model. 
  We show that the background field method applied to supergravity in adS space-time provides the path integral for the theory in the bulk with conformal symmetry associated with the isometry of the adS space. This in turn allows to establish the rigid conformal invariance of the generating functional for the supergravity correlators on the boundary. 
  We consider two-body problem in the self-field (1+1)-dimensional quantum electrodynamics on the circle. We present two formulations of the problem which correspond to two different types of variational principles and prove that both formulations lead to the same spectrum of the two-body Hamiltonian with massless matter fields. We give the exact and complete solution of the relativistic two-body equation in the massless case. 
  Recently a non-perturbative formula for the RG flow between UV and IR fixed points of the coefficient in the trace of the energy momentum tensor of the Euler density has been obtained for N=1 SUSY gauge theories by relating the trace and R-current anomalies. This result is compared here with an earlier perturbation theory analysis based on a naturally defined metric on the space of couplings for general renormalisable quantum field theories. This approach is specialised to N=1 supersymmetric theories and extended, using consistency arguments, to obtain the Euler coefficient at fixed points to 4-loops. The result agrees completely, to this order, with the exact formula. 
  The target space M for the sigma-model appearing in theories with p-branes is considered. It is proved that M is a homogeneous space G/H. It is symmetric if and only if the U-vectors governing the sigma-model metric are either coinciding or mutually orthogonal. For nonzero noncoinciding U-vectors the Killing equations are solved. Using a block-orthogonal decomposition of the set of the U-vectors it is shown that under rather general assumptions the algebra of Killing vectors is a direct sum of several copies of sl(2,R) algebras (corresponding to 1-vector blocks), several solvable Lie algebras (corresponding to multivector blocks) and the Killing algebra of a flat space. The target space manifold is decomposed in a product of a flat space, several 2-dimensional spaces of constant curvature (e.g. Lobachevsky space, part of anti-de Sitter space) and several solvable Lie group manifolds. 
  We discuss the renormalisation of the ground state energy of massive fields obeying boundary conditions, i.e., of the Casimir effect, and emphasise the role of the mass for its understanding. This is an extended abstract of a talk given at the topical group meeting on Casimir Forces at the Harvard-Smithsonian Center for Astrophysics on March 15-29, 1998. 
  We study Coulomb branch (``u-plane'') integrals for $\CN=2$ supersymmetric $SU(2),SO(3)$ Yang-Mills theory on 4-manifolds $X$ of $b_1(X)>0, b_2^+(X)=1$. Using wall-crossing arguments we derive expressions for the Donaldson invariants for manifolds with $b_1(X)>0, b_2^+(X)>0$. Explicit expressions for $X=\IC P^1 \times F_g$, where $F_g$ is a Riemann surface of genus $g$ are obtained using Kronecker's double series identity. The result might be useful in future studies of quantum cohomology. 
  One-instanton predictions are obtained from the Seiberg-Witten curve derived from M-theory by Landsteiner and Lopez for the Coulomb branch of N=2 supersymmetric SU(N) gauge theory with a matter hypermultiplet in the antisymmetric representation. Since this cubic curve describes a Riemann surface that is non-hyperelliptic, a systematic perturbation expansion about a hyperelliptic curve is developed, with a comparable expansion for the Seiberg-Witten differential. Calculation of the period integrals of the SW differential by the method of residues of D'Hoker, Krichever, and Phong enables us to compute the prepotential explicitly to one-instanton order. It is shown that the one-instanton predictions for SU(2), SU(3), and SU(4) agree with previously available results. For SU(N), N > 4, our analysis provides explicit predictions of a curve derived from M-theory at the one-instanton level in field theory. 
  We present a non-singular instanton describing the creation of an open universe with a compactified extra dimension. The four dimensional section of this solution is a singular instanton of the type introduced by Hawking and Turok. The ``singularity'' is viewed in five dimensions as a smooth bubble of ``nothing'' which eats up a portion of spacetime as it expands. Flat space with a compact extra dimension is shown to be gravitationally metastable, but sufficiently long lived if the size of the extra dimension is large compared with the Planck length. 
  We consider the compactification of N=1, D=10 supergravity with E_8 x E_8 Yang-Mills matter to N=1, D=4 model with 3 generations. With help of embedding SU(5)->SO(10)->E_6->E_8 we find the value of the top Yukawa coupling $\lambda_t=\sqrt{16\pi\alpha_{GUT}/3}$ at the GUT scale. 
  I show that physical quantities in several two-dimensional condensed-matter models are related to the Seiberg-Witten calculation of exact quantities in supersymmetric gauge theory. In particular, the magnetization in the Kondo problem and the current in the boundary sine-Gordon model can each be expressed in the form $\int dx/y$, where for example in the latter $y^2 = x + x^g - u^2$ with u related to the boundary mass scale (the analog of \Lambda_{QCD}) and g proportional to the radius of the boson squared. Thus for irrational g, the curve y(x) is of infinite genus, while for rational g it is of finite genus. The models are integrable and possess a quantum-group symmetry for any g, but are supersymmetric only at g=2/3. Both models also possess unique forms of g to 1/g duality. 
  We use a gauge-invariant regularization procedure, called ``split dimensional regularization'', to evaluate the quark self-energy $\Sigma (p)$ and quark-quark-gluon vertex function $\Lambda_\mu (p^\prime,p)$ in the Coulomb gauge, $\vec{\bigtriangledown}\cdot\vec{A}^a = 0$. The technique of split dimensional regularization was designed to regulate Coulomb-gauge Feynman integrals in non-Abelian theories. The technique which is based on two complex regulating parameters, $\omega$ and $\sigma$, is shown to generate a well-defined set of Coulomb-gauge integrals. A major component of this project deals with the evaluation of four-propagator and five-propagator Coulomb integrals, some of which are nonlocal. It is further argued that the standard one-loop BRST identity relating $\Sigma$ and $\Lambda_\mu$, should by rights be replaced by a more general BRST identity which contains two additional contributions from ghost vertex diagrams. Despite the appearance of nonlocal Coulomb integrals, both $\Sigma$ and $\Lambda_\mu$ are local functions which satisfy the appropriate BRST identity. Application of split dimensional regularization to two-loop energy integrals is briefly discussed. 
  We construct a four dimensional chiral N=1 space-time supersymmetric Type I vacuum corresponding to a compactification on a toroidal Z_2 X Z_2 X Z_3 orbifold. Using recent results in four dimensional orientifolds, we argue that this model has a well defined world-sheet description. An interesting feature of this model is that the gauge group contains an SU(6) subgroup with three chiral generations. Moreover, this model contains D5-branes and therefore corresponds to a non-perturbative heterotic vacuum. This is the first example of a consistent chiral N=1 supersymmetric string vacuum which is non-perturbative from the heterotic viewpoint, has a perturbative description in a dual theory, and possesses some phenomenologically interesting characteristics. We also compute the tree-level superpotential in this theory 
  We explore the connection between anti-deSitter supergravity and gauge theory, in the context of bound states of many D1 and D5 branes. The near-horizon $AdS_3$ supergravity describes the identity sector of the conformal field theory produced by the brane dynamics. A variant of anomaly inflow (for the 2d conformal anomaly) is involved. Dynamical matter fields on $AdS_3$ couple to the chiral ring and its descendant fields on the branes. We propose a map between boundary conformal field theory and bulk supergravity/matter dynamics, which is strongly reminiscent of matrix models of 2d gravity. 
  Super-matrix KdV and super-generalized non-linear Schrodinger equations are shown to arise from a symmetry reduction of ordinary self-dual Yang-Mills equations with supergauge groups 
  A multidimensional gravitational model with several scalar fields, fields of forms and cosmological constant is considered. When scalar fields are constant and composite p-brane monopole-like ansatz for the fields of forms is adopted, a wide class of solutions on product of n+1 Einstein spaces is obtained. These solutions are the composite p-brane generalizations of the Freund-Rubin solution. Some examples including the AdS_m x S^k x... solutions are considered. 
  We clarify some ambiguous points in a derivation of duality via brane exchange using M-theory language, and propose a ``proof'' of duality in MQCD. Actually, duality in MQCD is rather trivial and does not need a complicated proof.  The problem is how to interpret it in field theory language. We examine BPS states in N=2 theory and find the particle correspondence under duality. In the process, we also find some exotic particles in N=2 MQCD, and we observe an interesting phenomenon in type IIA string theory, namely, that fundamental strings are converted into D2-branes via the exchange of two NS5-branes. We also discuss how we should understand Seiberg's N=1 duality from exact duality in MQCD. 
  We examine the D-3 brane from the point of view of the double dimensionally reduced M theory 5 brane on a torus. M-theory, IIB identifications are explicitly constructed and a possible reformulation of the D-3 brane is discussed. The duality transformation of the reduced 3-brane necessary to make the identification is discussed in detail. 
  We show the existence of quasi-supersymmetry as formulated by Nambu in the top-sector of the standard model. We present the explicit form of the quasi-supersymmetric charge. We also deduce,like Nambu, a quasi supersymmetric mass relation which is the same as suggested by Veltman. 
  The quadratic form of the Dirac equation in a Riemann spacetime yields a gravitational gyromagnetic ratio \kappa_S = 2 for the interaction of a Dirac spinor with curvature. A gravitational gyromagnetic ratio \kappa_S = 1 is also found for the interaction of a vector field with curvature. It is shown that the Dirac equation in a curved background can be obtained as the square--root of the corresponding vector field equation only if the gravitational gyromagnetic ratios are properly taken into account. 
  In the framework leading to the multiplicative anomaly formula ---which is here proven to be valid even in cases of known spectrum but non-compact manifold (very important in Physics)--- zeta-function regularisation techniques are shown to be extremely efficient. Dirac like operators and harmonic oscillators are investigated in detail, in any number of space dimensions. They yield a non-zero anomaly which, on the other hand, can always be expressed by means of a simple analytical formula. These results are used in several physical examples, where the determinant of a product of differential operators is not equal to the product of the corresponding functional determinants. The simplicity of the Hamiltonian operators chosen is aimed at showing that such situation may be quite widespread in mathematical physics. However, the consequences of the existence of the determinant anomaly have often been overlooked. 
  We consider the GUT-like model with two scalar fields which has infinitesimal deviation from the conformal invariant fixed point at high energy region. In this case the dominating quantum effect is the conformal trace anomaly and the interaction between the anomaly-generated propagating conformal factor of the metric and the usual dimensional scalar field. This interaction leads to the renormalization group flow from the conformal point. In the supersymmetric conformal invariant model such an effect produces a very weak violation of sypersymmetry at lower energies. 
  We aim to connect the non commutative geometry ``quotient space'' viewpoint with the standard super Yang Mills theory approach in the spirit of Connes-Douglas-Schwartz and Douglas-Hull description of application of noncommutative geometry to matrix theory. This will result in a relation between the parameters of a rational foliation of the torus and the dimension of the group U(N). Namely, we will be provided with a prescription which allows to study a noncommutative geometry with rational parameter p/N by means of a U(N) gauge theory on a torus of size \Sigma / N with the boundary conditions given by a system with p units of magnetic flux. The transition to irrational parameter can be obtained by letting N and p tend to infinity with fixed ratio. The precise meaning of the limiting process will presumably allow better clarification. 
  In this paper we use the matrix string approach to begin a study of high energy scattering processes in M-theory. In particular we exhibit an instanton-type configuration in 1+1 super-Yang-Mills theory that can be interpreted as a non-perturbative description of a string interaction. This solution is used to describe high energy processes with non-zero longitudinal momentum exchange, in which an arbitrary number of eigenvalues get transferred between the two scattering states. We describe a direct correspondence between these semi-classical SYM configurations and the Gross-Mende saddle points. We also study in detail the pair production of D-particles via a one-loop calculation which in the 1+1D gauge theory language is described by the (perturbative) transition between states with different electric flux. Finally, we discuss a possible connection between these calculations in which D-particle production gives important corrections to the Gross-Mende process. 
  Chern-Simons theories, which are topological quantum field theories, provide a field theoretic framework for the study of knots and links in three dimensions. These are rare examples of quantum field theories which can be exactly and explicitly solved. Expectation values of Wilson link operators yield a class of link invariants, the simplest of them is the famous Jones polynomial. Other invariants are more powerful than that of Jones. These new invariants are sensitive to the chirality of all knots at least upto ten crossing number unlike those of Jones which are blind to the chirality of some of them. However, all these invariants are still not good enough to distinguish a class of knots called mutants. These link invariants can be alternately obtained from two dimensional vertex models. The $R$-matrix of such a model in a particular limit of the spectral parameter provides a representation of the braid group. This in turn is used to construct the link invariants. Exploiting theorems of Lickorish and Wallace and also those of Kirby, Fenn and Rourke which relate three-manifolds to surgeries on framed links, these link invariants in $S^3$ can also be used to construct three-manifold invariants. 
  Duality arguments are used to determine D-instanton contributions to certain effective interaction terms of type II supergravity theories in various dimensions. This leads to exact expressions for the partition functions of the finite N D-instanton matrix model in d=4 and 6 dimensions that generalize our previous expression for the case d=10. These results are consistent with the fact that the Witten index of the T-dual D-particle process should only be non-vanishing for d=10. 
  We construct a Lax pair with spectral parameter for the elliptic Calogero-Moser Hamiltonian systems associated with each of the finite dimensional Lie algebras, of the classical and of the exceptional type. When the spectral parameter equals one of the three half periods of the elliptic curve, our result for the classical Lie algebras reduces to one of the Lax pairs without spectral parameter that were known previously. These Calogero-Moser systems are invariant under the Weyl group of the associated untwisted affine Lie algebra. For non-simply laced Lie algebras, we introduce new integrable systems, naturally associated with twisted affine Lie algebras, and construct their Lax operators with spectral parameter (except in the case of $G_2$). 
  The elliptic Calogero-Moser Hamiltonian and Lax pair associated with a general simple Lie algebra $\G$ are shown to scale to the (affine) Toda Hamiltonian and Lax pair. The limit consists in taking the elliptic modulus $\tau$ and the Calogero-Moser couplings $m$ to infinity, while keeping fixed the combination $M = m e^{i \pi \delta \tau}$ for some exponent $\delta$. Critical scaling limits arise when $1/\delta$ equals the Coxeter number or the dual Coxeter number for the untwisted and twisted Calogero-Moser systems respectively; the limit consists then of the Toda system for the affine Lie algebras $\G^{(1)}$ and $(\G ^{(1)})^\vee$. The limits of the untwisted or twisted Calogero-Moser system, for $\delta$ less than these critical values, but non-zero, consists of the ordinary Toda system, while for $\delta =0$, it consists of the trigonometric Calogero-Moser systems for the algebras $\G$ and $\G^\vee$ respectively. 
  The Seiberg-Witten curves and differentials for $\N=2$ supersymmetric Yang-Mills theories with one hypermultiplet of mass $m$ in the adjoint representation of the gauge algebra $\G$, are constructed for arbitrary classical or exceptional $\G$ (except $G_2$). The curves are obtained from the recently established Lax pairs with spectral parameter for the (twisted) elliptic Calogero-Moser integrable systems associated with the algebra $\G$. Curves and differentials are shown to have the proper group theoretic and complex analytic structure, and to behave as expected when $m$ tends either to 0 or to $\infty$. By way of example, the prepotential for $\G = D_n$, evaluated with these techniques, is shown to agree with standard perturbative results. A renormalization group type equation relating the prepotential to the Calogero-Moser Hamiltonian is obtained for arbitrary $\G$, generalizing a previous result for $\G = SU(N)$. Duality properties and decoupling to theories with other representations are briefly discussed. 
  It is shown that Zaslavskii's misunderstanding of our published proof of the irrelevance of all extremal black hole configurations (whether with equal charge and mass or not) rests on his refusal to see the essential difference between the correct inequality governing extremal and non-extremal actions and his incorrect version. 
  General solution of the non-abelian Gauss law in terms of covariant curls and gradients is presented. Also two non-abelian analogs of the Hodge decomposition in three dimensions are addressed. i) Decomposition of an isotriplet vector field $V_{i}^{a}(x)$ as sum of covariant curl and gradient with respect to an arbitrary background Yang-Mills potential is obtained. ii) A decomposition of the form $V_{i}^{a}=B_{i}^{a}(C)+D_{i}(C) \phi^{a} $ which involves non-abelian magnetic field of a new Yang-Mills potential C is also presented. These results are relevant for duality transformation for non-abelian gauge fields. 
  We computed the statistical entropy of nonextremal 4D and extremal 5D Calabi-Yau black holes and found exact agreement with the Bekenstein-Hawking entropy. The computation is based on the fact that the near-horizon geometry of equivalent representations contains as a factor the Ba\~nados-Teitelboim-Zanelli black hole and on subsequent use of Strominger's proposal generalizing the statistical count of microstates of the BTZ black hole due to Carlip. 
  We study the quantum mechanics of a system of topologically interacting particles in 2+1 dimensions, which is described by coupling the particles to a Chern-Simons gauge field of an inhomogeneous group. Analysis of the phase space shows that for the particular case of ISO(3) Chern-Simons theory the underlying symmetry is that of the quantum double D(SU(2)), based on the homogeneous part of the gauge group. This in contrast to the usual q-deformed gauge group itself, which occurs in the case of a homogeneous gauge group. Subsequently, we describe the structure of the quantum double of a continuous group and the classification of its unitary irreducible representations. The comultiplication and the R-element of the quantum double allow for a natural description of the fusion properties and the nonabelian braid statistics of the particles. These typically manifest themselves in generalised Aharonov-Bohm scattering processes, for which we compute the differential cross sections. Finally, we briefly describe the structure of D(SO(2,1)), the underlying quantum double symmetry of (2+1)-dimensional quantum gravity. 
  We define the notion of mirror of a Calabi-Yau manifold with a stable bundle in the context of type II strings in terms of supersymmetric cycles on the mirror. This allows us to relate the variation of Hodge structure for cohomologies arising from the bundle to the counting of holomorphic maps of Riemann surfaces with boundary on the mirror side. Moreover it opens up the possibility of studying bundles on Calabi-Yau manifolds in terms of supersymmetric cycles on the mirror. 
  We present an analytical continuum calculation, starting from first principles, of the vacuum wavefunction and string tension for pure Yang-Mills theories in $(2+1)$ dimensions, extending our previous analysis using gauge-invariant matrix variables. The vacuum wavefunction is consistent with what is expected at both high and low momentum regimes. The value of the string tension is in very good agreement with recent lattice Monte Carlo evaluations. 
  We perform the stochastic quantization of Yang-Mills theory in configuration space and derive the Faddeev-Popov path integral density. Based on a generalization of the stochastic gauge fixing scheme and its geometrical interpretation this result is obtained as the exact equilibrium solution of the associated Fokker--Planck equation. Included in our discussion is the precise range of validity of our approach. 
  We show that contrary to the claim made by Hallin and Liljenberg in Phys. Rev. D52 1150,(1995), (hep-th/9412188) the thermal correction to the thermal decay or pair production rate for a system placed in a heat bath in the presence of an external electric field, is always nonzero in the finite as well as infinite time limit. Using the formalism outlined there, we reestimate the decay rate for different values of temperature, mass and time.We also try to identify the parameter ranges where the quantity of interest agrees with that computed previously, at high temperature (in the infinite time limit), from the imaginary part of the effective action using imaginary time and real time formalism of thermal field theory. We also point out that in the strictly infinite time limit, the correct decay rate as obtained from the work of Hallin et. al. tends to diverge. 
  The CP-violating phases in the soft supersymmetry-breaking sector in orbifold compactifications with a continuous Wilson line are investigated. In this case the modular symmetry is the Siegel modular group $Sp(4,Z)$ of genus two. In particular, we study the case that the hidden sector non-perturbative superpotential is determined by the Igusa cusp form ${\cal C}_{12}$ of modular weight 12. The effect of large non-perturbative corrections to the dilaton K\"ahler potential on the resulting CP-violating phases is also investigated. 
  We develop a systematic method for computing a renormalized light-front field theory Hamiltonian that can lead to bound states that rapidly converge in an expansion in free-particle Fock-space sectors. To accomplish this without dropping any Fock sectors from the theory, and to regulate the Hamiltonian, we suppress the matrix elements of the Hamiltonian between free-particle Fock-space states that differ in free mass by more than a cutoff. The cutoff violates a number of physical principles of the theory, and thus the Hamiltonian is not just the canonical Hamiltonian with masses and couplings redefined by renormalization. Instead, the Hamiltonian must be allowed to contain all operators that are consistent with the unviolated physical principles of the theory. We show that if we require the Hamiltonian to produce cutoff-independent physical quantities and we require it to respect the unviolated physical principles of the theory, then its matrix elements are uniquely determined in terms of the fundamental parameters of the theory. This method is designed to be applied to QCD, but for simplicity, we illustrate our method by computing and analyzing second- and third-order matrix elements of the Hamiltonian in massless phi-cubed theory in six dimensions. 
  The two-dimensional anyon system, when reduced to one dimension, yields models related to the Calogero-Sutherland model. One such reduction leads to a new model with a class of exact solutions. This model is one of a family of models obtained upon dimensional reduction of spherically symmetric models in arbitrary dimensions. 
  We investigate the quantization of even-dimensional topological actions of Chern-Simons form which were proposed previously. We quantize the actions by Lagrangian and Hamiltonian formulations {\`a} la Batalin, Fradkin and Vilkovisky. The models turn out to be infinitely reducible and thus we need infinite number of ghosts and antighosts. The minimal actions of Lagrangian formulation which satisfy the master equation of Batalin and Vilkovisky have the same Chern-Simons form as the starting classical actions. In the Hamiltonian formulation we have used the formulation of cohomological perturbation and explicitly shown that the gauge-fixed actions of both formulations coincide even though the classical action breaks Dirac's regularity condition. We find an interesting relation that the BRST charge of Hamiltonian formulation is the odd-dimensional fermionic counterpart of the topological action of Chern-Simons form. Although the quantization of two dimensional models which include both bosonic and fermionic gauge fields are investigated in detail, it is straightforward to extend the quantization into arbitrary even dimensions. This completes the quantization of previously proposed topological gravities in two and four dimensions. 
  We construct the classical configurations of BPS states with 1/4 unbroken supersymmetries in four-dimensional N=4 SU(n+1) supersymmetric Yang-Mills theory, and discuss that these configurations correspond to string networks connecting (n+1) D3-branes in Type IIB string theory. 
  Chamorro and Virbhadra studied, using the energy-momentum complex of Einstein, the energy distribution associated with static spherically symmetric charged dilaton black holes for an arbitrary value of the coupling parameter $\gamma$ which controls the strength of the dilaton to the Maxwell field. We study the same in Tolman's prescription and get the same result as obtained by Chamorro and Virbhadra. The energy distribution of charged dilaton black holes depends on the value of $\gamma$ and the total energy is independent of this parameter. 
  We study a system of external particles of various charges in N =4 super Yang-Mills in the large N limit at finite temperature. We demonstrate that at high enough temperature partial or complete screening of the particles can occur. At zero temperature the total electric or magnetic charge cannot be screened, while higher multipole moments of these charges can be screened. The specific case of a quark, a monopole and a dyon is worked out and the above properties are verified. We also discuss the free energy of isolated particles and show that their entropy is independent of the temperature. 
  A zero mode quantization of the minimal energy SU(2) Skyrmions for nucleon numbers four to nine and seventeen is described. This involves quantizing the rotational and isorotational modes of the configurations. For nucleon numbers four, six and eight the ground states obtained are in agreement with the observed nuclear states of Helium, Lithium and Beryllium. However, for nucleon numbers five, seven, nine and seventeen the spins obtained conflict with the observed isodoublet nuclear states. 
  The derivative expansion of the one-loop effective action in QED$_3$ and QED$_4$ is considered. The first term in such an expansion is the effective action for a constant electromagnetic field. An explicit expression for the next term containing two derivatives of the field strength $F_{\mu\nu}$, but exact in the magnitude of the field strength, is obtained. The general results for fermion and scalar electrodynamics are presented. The cases of pure electric and pure magnetic external fields are considered in details. The Feynman rules for the perturbative expansion of the one-loop effective action in the number of derivatives is developed. 
  We confirm the non-integrability of the multi-deformed Ising Model, an already expected result. After deforming with the energy operator $\phi_{1,3}$ we use the Majorana free fermionic representation for the massive theory to show that, besides the trivial one, no local integrals of motion can be built in the theory arising from perturbing with both energy and spin operators. 
  In this paper, we discuss the behavior of conformal field theories interacting at a single point. The edge states of the quantum Hall effect (QHE) system give rise to a particular representation of a chiral Kac-Moody current algebra. We show that in the case of QHE systems interacting at one point we obtain a ``twisted'' representation of the current algebra. The condition for stationarity of currents is the same as the classical Kirchoff's law applied to the currents at the interaction point. We find that in the case of two discs touching at one point, since the currents are chiral, they are not stationary and one obtains current oscillations between the two discs. We determine the frequency of these oscillations in terms of an effective parameter characterizing the interaction. The chiral conformal field theories can be represented in terms of bosonic Lagrangians with a boundary interaction. We discuss how these one point interactions can be represented as boundary conditions on fields, and how the requirement of chirality leads to restrictions on the interactions described by these Lagrangians. By gauging these models we find that the theory is naturally coupled to a Chern-Simons gauge theory in 2+1 dimensions, and this coupling is completely determined by the requirement of anomaly cancellation. 
  One loop effects due to virtual gauge field propagation in 2+1 dimensional Born-Infeld theory are investigated. Although this field theory model is not power counting renormalizable, it can be consistently interpreted as an effective field theory. We derive the one-loop effective action in this framework. Halpern's field strength's formulation is then applied to derive an effective description for the interaction between magnetically charged particles, when the gauge field dynamics is determined by a Born-Infeld action. We compare the results with those of the Maxwell theory. 
  We study anti-de Sitter black holes in 2+1 dimensions in terms of Chern Simons gauge theory of anti-de Sitter group coupled to a source. Taking the source to be an anti-de Sitter state specified by its Casimir invariants, we show how all the relevant features of the black hole are accounted for. The requirement that the source be a unitary representation leads to a discrete tower of states which provide a microscopic model for the black hole. 
  We consider the canonical quantization of a generalized two-dimensional massive fermion theory containing higher odd-order derivatives. The requirements of Lorentz invariance, hermiticity of the Hamiltonian and absence of tachyon excitations suffice to fix the mass term, which contains a derivative coupling. We show that the basic quantum excitations of a higher-derivative theory of order 2N+1 consist of a physical usual massive fermion, quantized with positive metric, plus 2N unphysical massless fermions, quantized with opposite metrics. The positive metric Hilbert subspace, which is isomorphic to the space of states of a massive free fermion theory, is selected by a subsidiary-like condition. Employing the standard bosonization scheme, the equivalent boson theory is derived. The results obtained are used as a guideline to discuss the solution of a theory including a current-current interaction. 
  Recently, a relation between N=4 Super Yang Mills in 3+1 dimensions and supergravity in an $AdS_5$ background has been proposed. In this paper we explore the idea that the correspondence between operators in the Yang Mills theory and modes of the supergravity theory can be obtained by using the D3 brane action. Specifically, we consider two form gauge fields for this purpose. The supergravity analysis predicts that the operator which corresponds to this mode has dimension six. We show that this is indeed the leading operator in the three brane Dirac-Born-Infeld and Wess-Zumino action which couples to this mode. It is important in the analysis that the brane action is expanded around the anti de-Sitter background. Also, the Wess-Zumino term plays a crucial role in cancelling a lower dimension operator which appears in the the Dirac-Born-Infeld action. 
  The most general gauge-invariant marginal deformation of four-dimensional abelian BF-type topological field theory is studied. It is shown that the deformed quantum field theory is topological and that its observables compute, in addition to the usual linking numbers, smooth intersection indices of immersed surfaces which are related to the Euler and Chern characteristic classes of their normal bundles in the underlying spacetime manifold. Canonical quantization of the theory coupled to non-dynamical particle and string sources is carried out in the Hamiltonian formalism and explicit solutions of the Schroedinger equation are obtained. The wavefunctions carry a one-dimensional unitary representation of the particle-string exchange holonomies and of non-topological string-string intersection holonomies given by adiabatic limits of the worldsheet Euler numbers. They also carry a multi-dimensional projective representation of the deRham complex of the underlying spatial manifold and define a generalization of the presentation of its motion group from Euclidean space to an arbitrary 3-manifold. Some potential physical applications of the topological field theory as a dual model for effective vortex strings are discussed. 
  We consider N=2 supersymmetric Yang-Mills theories in four dimensions with gauge group SU(N) for N larger than two. Using the cubic curve for a matter hypermultiplet transforming in the symmetric representation, obtained from M-theory by Landsteiner and Lopez, we calculate the prepotential up to the one instanton correction. We treat the curve to be approximately hyperelliptic and perform a perturbation expansion for the Seiberg-Witten differential to get the one instanton contribution. We find that it reproduces the correct result for one-loop, and we obtain the prediction for that curve for the one instanton correction term. 
  The canonical formulation of d=2, N=16 supergravity is presented. We work out the supersymmetry generators (including all higher order spinor terms) and the N=16 superconformal constraint algebra. We then describe the construction of the conserved non-local charges associated with the affine E_9 symmetry of the classical equations of motion. These charges are shown to commute weakly with the supersymmetry constraints, and hence with all other constraints. Under commutation, they close into a quadratic algebra of Yangian type, which is formally the same as that of the bosonic theory. The Lie-Poisson action of E_9 on the classical solutions is exhibited explicitly. Further implications of our results are discussed. 
  It is outlined how deformations of field theoretical rigid symmetries can be constructed and classified by cohomological means in the extended antifield formalism. Special attention is devoted to deformations referring only to a subset of the rigid symmetries of a given model and leading to a nontrivial extension of the graded Lie algebra associated with that subset. The method is illustrated for a D=4, N=2 supersymmetric model where the central extension of the supersymmetry algebra emerges via a deformation. Deformations of gauge fixed actions with a BRST symmetry are discussed too and illustrated by the Curci-Ferrari model. 
  We discuss some of the experimental motivation for the need for semigroup decay laws, and the quantum Lax-Phillips theory of scattering and unstable systems. In this framework, the decay of an unstable system is described by a semigroup.  The spectrum of the generator of the semigroup corresponds to the singularities of the Lax-Phillips $S$-matrix. In the case of discrete (complex) spectrum of the generator of the semigroup, associated with resonances, the decay law is exactly exponential. The states corresponding to these resonances (eigenfunctions of the generator of the semigroup) lie in the Lax-Phillips Hilbert space, and therefore all physical properties of the resonant states can be computed. We show that the parametrized relativistic quantum theory is a natural setting for the realization of the Lax-Phillips theory. 
  The gauge compensation fields induced by the differential operators of the Stueckelberg-Schr\"odinger equation are discussed, as well as the relation between these fields and the standard Maxwell fields. An action is constructed and the second quantization of the fields carried out using a constraint procedure. Some remarks are made on the properties of the second quantized matter fields. 
  We study the geometry of the Lagrangian Batalin--Vilkovisky theory on an antisymplectic manifold. We show that gauge symmetries of the BV-theory are essentially the symmetries of an even symplectic structure on the stationary surface of the master action. 
  Starting from the bosonic sector of the M-theory super-five-brane we obtain the action for duality-symmetric three-brane and construct the consistent coupling of the proposed action with the bosonic sector of type IIB supergravity. 
  This paper presents an analysis of the radiation seen by an observer in circular acceleration, for a magnetic spin. This is applied to an electron in a storage ring, and the subtilty of the interaction of the spin with the spatial motion of the electron is explicated. This interaction is shown to be time dependent (in the radiating frame), which explains the strange results found for the electron's residual polarisation in the literature. Finally, some brief comments about the radiation emitted by an accelerating detector are made where it is shown that the spectrum is correlated in that particles are emitted in pairs. 
  We present a derivation of the Bekenstein-Hawking entropy from conformal field theories. In particular we consider a six-dimensional string configuration with background metric AdS_3 * S^3 near the horizon. In addition we introduce momentum modes along the string, corresponding to a Banados-Teitelboim-Zanelli (BTZ) black hole in the anti-de Sitter (AdS) space-time, and a Taub-NUT soliton in the transverse Euclidean space, projecting out a discrete subgroub of S^3. This spherical part is described by a SU(2)/Z(m) Wess-Zumino-Witten (WZW) model. The AdS_3 space-time, on the other hand, is determined by two conformal field theories living at the two boundaries: One SL(2,R)/U(1) WZW model on the horizon of the BTZ black hole and one Liouville model at infinity. The extremal BTZ black hole interpolates between these two conformal field theories. Moreover, we argue that the sum of the three conformal field theories yields the correct microscopic state-counting including all alpha' corrections. 
  We argue for the existence of many new 1/4 BPS states in N=4 SU(N_c) Super-Yang-Mills theory with N_c>=3, by constructing them from supersymmetric string webs whose external strings terminate on parallel D3-branes. The masses of the string webs are shown to agree with the BPS bound for the corresponding states in SYM. We identify the curves of marginal stability, at which these states decay into other BPS states. We find the bosonic and fermionic zero modes of the string webs, and thereby the degeneracy and spin content of some of the BPS states. States of arbitrarily high spin are predicted in this manner, all of which become massless at the conformal point. For N_c>=4 we find BPS states which transform in long multiplets, and are therefore not protected against becoming stable non-BPS states as moduli are varied. The mass of these extremal non-BPS states is constrained as they are connected to BPS states. Analogous geometric phenomena are anticipated. 
  The SU(5) grand unified theory (GUT) is derived from the geometrical point of view of gauge theory on three-sheeted space-time, i.e., $M_4 \times Z_3$ manifold without recourse to noncommutative geometry. A derivation of SO(10) GUT is also discussed in the same point of view. 
  We show that parity is conserved in vector-like supersymmetric theories, such as supersymmetric QCD with massive quarks with no cubic couplings among chiral multiplets, based on fermionic path-integrals, originally developed by Vafa and Witten. We also look into the effect of supersymmetric breaking through gluino masses, and see that the parity-conservation is intact also in this case. Our conclusion is valid, when only bosonic parity-breaking observable terms are considered in path-integrals like the original Vafa-Witten formulation. 
  We show that Dirichlet p-brane can be expressed as a configuration of infinitely many Dirichlet (p-2)-branes in the bosonic string theory. Using this fact, we interpret the massless fields on the p-brane worldvolume as deformations of the configuration of the (p-2)-branes. Especially we find that the worldvolume gauge field parametrizes part of the group of diffeomorphisms on the worldvolume. 
  Extending our previous work on SU(3), we construct spherically symmetric BPS saturated regular configurations of N=4 SU(N) supersymmetric Yang-Mills theory preserving 1/4 supersymmetry, and investigate their features. We also give exact solutions in the case some of the free parameters of the general solutions take certain values. These field theory BPS states correspond to the string theory BPS states of multi-pronged strings connecting N different D3-branes by regarding the N=4 supersymmetric Yang-Mills theory as an effective field theory on parallel D3-branes. We compare our solutions with multi-pronged strings in string picture. 
  Principles of discrete time mechanics are applied to the quantisation of Maxwell's equations. Following an analysis of temporal node and link variables, we review the classical discrete time equations in the Coulomb and Lorentz gauges and conclude that electro-magneto duality does not occur in pure discrete time electromagnetism. We discuss the role of boundary conditions in our mechanics and how temporal discretisation should influence very early universe dynamics. Quantisation of the Maxwell potentials is approached via the discrete time Schwinger action principle and the Faddeev-Popov path integral. We demonstrate complete agreement in the case of the Coulomb gauge, obtaining the vacuum functional and the discrete time field commutators in that gauge. Finally, we use the Faddeev-Popov method to construct the discrete time analogues of the photon propagator in the Landau and Feynman gauges, which casts light on the break with relativity and possible discrete time analogues of the metric tensor. 
  The complete spectrum of D=6, N=4b supergravity with n tensor multiplets compactified on AdS_3 x S^3 is determined. The D=6 theory obtained from the K_3 compactification of Type IIB string requires that n=21, but we let n be arbitrary. The superalgebra that underlies the symmetry of the resulting supergravity theory in AdS_3 coupled to matter is SU(1,1|2)_L x SU(1,1|2)_R. The theory also has an unbroken global SO(4)_R x SO(n) symmetry inherited from D=6. The spectrum of states arranges itself into a tower of spin-2 supermultiplets, a tower of spin-1, SO(n) singlet supermultiplets, a tower of spin-1 supermultiplets in the vector representation of SO(n) and a special spin-1/2 supermultiplet also in the vector representation of SO(n). The SU(2)_L x SU(2)_R Yang-Mills states reside in the second level of the spin-2 tower and the lowest level of the spin-1, SO(n) singlet tower and the associated field theory exhibits interesting properties. 
  Three-dimensional field theories with N=3 and N=4 supersymmetries are considered in the framework of the harmonic-superspace approach. Analytic superspaces of these supersymmetries are similar; however, the geometry of gauge theories with the manifest N=3 is richer and admits construction of the topological mass term. 
  We analyse the one-loop effective action of N=4 SYM theory in the framework of the background field formalism in N=2 harmonic superspace. For the case of on-shell background N=2 vector multiplet we prove that the effective action is free of harmonic singularities. When the lowest N=1 superspace component of the N=2 vector multiplet is switched off, the effective action of N=4 SYM theory is shown to coincide with that obtained by Grisaru et al on the base of the N=1 background field method. We compute the leading non-holomorphic corrections to the N=4 SU(2) SYM effective action. 
  It is pointed out that ambiguities in the regularization of actions with second derivatives seem to happen with the same multiplicity that the standard model of elementary particles 
  We show that five-dimensional anti de-Sitter space remains a solution to low-energy type IIB supergravity when the leading higher-derivative corrections to the classical supergravity (which are non-perturbative in the string coupling) are included. Furthermore, at this order in the low energy expansion of the IIB theory the graviton two-point and three-point functions in $AdS_5 \times S^5$ are shown not to be renormalized and a precise expression is obtained for the four-graviton and related S-matrix elements. By invoking Maldacena's conjectured connection between IIB superstring theory and supersymmetric Yang-Mills theory corresponding statements are obtained concerning correlation functions of the energy-momentum tensor and related operators in the large-N Yang-Mills theory. This leads to interesting non-perturbative statements and insights into the r\^ole of instantons in the gauge theory. 
  Exact gauge structures arise in the evolution of spin-1/2 particles in conformally flat space-times. The corresponding Berry potentials can be Abelian or non-Abelian depending on the mass degeneracy of the system considered. Examples include de Sitter universes and maximal acceleration. 
  We consider the AdS/CFT correspondence in the context of 2d CFT and find that essentially all ``single particle'' primary fields with higher spin, predicted from the CFT side are missing from the Kaluza-Klein excitations of the AdS supergravity. The high mass extension of these missing states gives rise to the macroscopic entropy of extremal 5d black holes. 
  The low energy decay rates of four- and five dimensional dyonic black holes in string theory are equivalently described in terms of an effective near horizon AdS_3 (BTZ) black hole. It is then argued that AdS_3 gravity provides an universal microscopic description of the low energy dynamics these black holes. 
  We construct and explore BPS states that preserve 1/4 of supersymmetry in N=4 Yang-Mills theories. Such states are also realized as three-pronged strings ending on D3-branes. We correct the electric part of the BPS equation and relate its solutions to the unbroken abelian gauge group generators. Generic 1/4-BPS solitons are not spherically symmetric, but consist of two or more dyonic components held apart by a delicate balance between static electromagnetic force and scalar Higgs force. The instability previously found in three-pronged string configurations is due to excessive repulsion by one of these static forces. We also present an alternate construction of these 1/4-BPS states from quantum excitations around a magnetic monopole, and build up the supermultiplet for arbitrary (quantized) electric charge. The degeneracy and the highest spin of the supermultiplet increase linearly with a relative electric charge. We conclude with comments. 
  It is shown that a pair of vortex and anti-vortex is completely screened in 2+1 dimensional Yang-Mills theory and 3+1 dimensional Yang-Mills theory in the strong coupling limit, based on the recent conjecture of Maldacena. This is consistent with the fact that these theories exhibit confinement. 
  We compute certain one-loop corrections to F^4 couplings of the heterotic string compactified on T^2, and show that they can be characterized by holomorphic prepotentials. We then discuss how some of these couplings can be obtained in F-theory, or more precisely from 7-brane geometry in type IIB language. We in particular study theories with E_8 x E_8 and SO(8)^4 gauge symmetry, on certain one-dimensional sub-spaces of the moduli space that correspond to constant IIB coupling. For these theories, the relevant geometry can be mapped to Riemann surfaces. Physically, the computations amount to non-trivial tests of the basic F-theory -- heterotic duality in eight dimensions. Mathematically, they mean to associate holomorphic 5-point couplings of the form (del_t)^5 G = sum[ g_l l^5 q^l/(1-q^l) ] to K3 surfaces. This can be seen as a novel manifestation of the mirror map, acting here between open and closed string sectors. 
  The dynamics of a (super)particle near the horizon of an extreme Reissner-Nordstrom black hole is shown to be governed by an action that reduces to a (super)conformal mechanics model in the limit of large black hole mass. 
  It is shown that the matter content of F-theory compactifications on elliptic Calabi-Yau threefolds is encoded in the Gromov-Witten invariants. 
  Quantum effects for electrons in a storage ring are studied in a co-moving, accelerated frame. The polarization effect due to spin flip synchrotron radiation is examined by treating the electron as a simple quantum mechanical two-level system coupled to the orbital motion and to the radiation field. The excitations of the spin system are related to the Unruh effect, i.e. the effect that an accelerated radiation detector is thermally excited by vacuum fluctuations. The importance of orbital fluctuations is pointed out and the vertical fluctuations are examined. 
  We argue that the trace structure of the non-abelian Born-Infeld action can be fixed by demanding that the action be linearised by certain energy-minimising BPS-like configurations. It is shown how instantons in D4-branes, SU(2) monopoles and dyons in D3-branes, and vortices in D2-branes are all BPS states of the action recently proposed by Tseytlin. The T-dual worldvolume theories of D-strings and D0-branes are also considered. All such configurations can be dealt with exactly within the context of non-abelian Born-Infeld theory since, given the relevant BPS-like condition, the action reduces to that of Yang-Mills theory. The worldvolume energy of such configurations is an absolute minimum. It would seem, moreover, that such an analysis holds for the symmetrised trace structure of Tseytlin's proposal only. 
  We present a decomposition formula for $U_n$, an integral of time-ordered products of operators, in terms of sums of products of the more primitive quantities $C_m$, which are the integrals of time-ordered commutators of the same operators. The resulting factorization enables a summation over $n$ to be carried out to yield an explicit expression for the time-ordered exponential, an expression which turns out to be an exponential function of $C_m$. The Campbell-Baker-Hausdorff formula and the nonabelian eikonal formula obtained previously are both special cases of this result. 
  The renormalization of the boundary action in the AdS/CFT correspondence is considered and the breaking of conformal symmetry is discussed. 
  A non standard super extensions of the Poincare algebra (S-algebra [1,2]), which seems to be relevant for construction of various D=11 models, are studied. We present two examples of actions for point-like dynamical systems, which are invariant under off-shell closed realization of the S-algebra as well as under local fermionic $\kappa$-symmetry. On this ground, an explicit form of the S-algebra is advocated. 
  We consider four dimensional ${\cal N}=1$ supersymmetric gauge theories obtained via orientifolds of Type IIB on Abelian C^3/G orbifolds. We construct all such theories that have well defined world-sheet expansion. The number of such orientifolds is rather limited. We explain this fact in the context of recent developments in four dimensional Type IIB orientifolds. In particular, we elaborate these issues in some examples of theories where world-sheet description is inadequate due to non-perturbative (from the orientifold viewpoint) states arising from D-branes wrapping (collapsed) 2-cycles in the orbifold. We find complete agreement with the corresponding statements recently discussed in the context of Type I compactifications on toroidal orbifolds. This provides a non-trivial check for correctness of the corresponding conclusions. We also find non-trivial agreement with various field theory expectations, and point out their origin in string language. The orientifold gauge theories that do possess well defined world-sheet description have the property that in the large N limit computation of any M-point correlation function in these theories reduces to the corresponding computation in the parent ${\cal N}=4$ oriented theory. 
  The Turaev-Viro state sum invariant is known to give the transition amplitude for the three dimensional BF theory with cosmological term, and its deformation parameter hbar is related with the cosmological constant via hbar=sqrt{Lambda}. This suggests a way to find the expectation value of the spacetime volume by differentiating the Turaev-Viro amplitude with respect to the cosmological constant. Using this idea, we find an explicit expression for the spacetime volume in BF theory. According to our results, each labelled triangulation carries a volume that depends on the labelling spins. This volume is explicitly discrete. We also show how the Turaev-Viro model can be used to obtain the spacetime volume for (2+1) dimensional quantum gravity. 
  The large-mass behaviour of loop variables in Maxwell-Chern-Simons theory is analysed by means of a gauge-field transformation which allows to reset the Maxwell-Chern-Simons action to pure Chern-Simons. 
  An approach to calculating approximate solutions to the continuum Schwinger-Dyson equations is outlined, with examples for \phi^4 in D=1. This approach is based on the source Galerkin methods developed by Garcia, Guralnik and Lawson. Numerical issues and opportunities for future calculations are also discussed briefly. 
  Matrix theory compactifications on tori have associated Yang-Mills theories on the dual tori with sixteen supercharges. A noncommutative description of these Yang-Mills theories based in deformation quantization theory is provided. We show that this framework allows a natural generalization of the `Moyal B-deformation' of the Yang-Mills theories to non-constant background B-fields on curved spaces. This generalization is described through Fedosov's geometry of deformation quantization. 
  On the perturbatively non-renormalizable and non-perturbatively finite examples (delta-function type potential in non-relativistic quantum mechanics and the mathematical model of the propagator by Redmond and Uretsky in quantum field theory) we illustrate that one can develop a perturbative approach for non-renormalizable theory. The key idea is the introduction of finite number of additional expansion parameters which allows us to eliminate all infinities from the perturbative expressions. The generated perturbative series reproduce the expansions of the exact analytical solutions. 
  Herein we propose a new numerical technique for solving field theories: the large momentum frame (LMF). This technique combines several advantages of lattice gauge theory with the simplicity of front form quantisation. We apply the LMF on QED(1+1) and on the $\phi^4(3+1)$ theory. We demonstrate both analytically and in practical examples (1) that the LMF does neither correspond to the infinite momentum frame (IMF) nor to the front-form (FF) (2) that the LMF is not equivalent to the IMF (3) that the IMF is unphysical since it violates the lattice scaling window and (4) that the FF is even more unphysical because FF propagators violate micro-causality, causality and the finiteness of the speed of light. We argue that distribution functions measured in deep inelastic scattering should be interpreted in the LMF (preferably in the Breit frame) rather than in the FF formalism. In particular, we argue that deep inelastic scattering probes space-like distribution functions. 
  Using the underlying algebraic structures of Natanzon potentials, we discuss conditions that generate shape invariant potentials. In fact, these conditions give all the known shape invariant potentials corresponding to a translational change of parameters. We also find that while the algebra for the general Natanzon potential is $SO(2,2)$, a subgroup $SO(2,1)$ suffices for all the shape invariant problems of Natanzon type. 
  The dependence of the energies of axially symmetric monopoles of magnetic charges 2 and 3, on the Higgs self-interaction coupling constant, is studied numerically. Comparing the energy per unit topological charge of the charge-2 monopole with the energy of the spherically symmetric charge-1 monopole, we confirm that there is only a repulsive phase in the interaction energy between like monopoles 
  U(1) gauge theory with the Villain action on a cubic lattice approximation of three- and four-dimensional torus is considered. The naturally chosen correlation functions converge to the correlation functions of the R-gauge electrodynamics on three- and four-dimensional torus as the lattice spacing approaches zero only for the special scaling. This special scaling depends on a choice of a correlation function system. Another scalings give the degenerate continuum limits. The Wilson criterion for the confinement is ambiguous. The asymptotics of the smeared Wilson loop integral for the large loop perimeters is defined by the density of the loop smearing over a torus which is transversal to the loop plane. When the initial torus radius tends to infinity the correlation functions converge to the correlation functions of the R-gauge Euclidean electrodynamics. 
  In the framework of the four-dimensional effective theory of heterotic superstrings at low energies, which is considered as a generalised theory of gravity, we search for new black hole solutions. The analytical expressions of all the scalar fields of the effective theory is determined in a Kerr-Newman black hole background and the evasion of the existing "no-hair" theorems in the presence of higher curvature gravitational terms is demonstrated. New black hole solutions, such as the Dilatonic and the Coloured black holes, are determined by use of numerical integration and their properties are analysed. Finally, the linear stability of Dilatonic black holes under small time-dependent perturbations is exhibited through a semi-analytic method. 
  Starting from renormalised Effective Lagrangian, in the presence of an external Chromo-Electric field at finite temperature, the expression for thermal coupling constant ($\alpha = (g^2)/(4 \pi)$) as a function of temperature and external field is derived, using finite temperature two parameter renormalisation group equation of Matsumoto, Nakano and Umezawa. For some values of the parameters, the coupling constant is seen to be approaching a value $\sim unity$. 
  We construct new bosonic boundary states in the light-cone gauge which describe D(p)-branes moving at the speed of light. 
  We introduced a dynamical system given by a difference of two simple SL(2,R) WZNW actions in 2D, and defined the related gauge theory in a consistent way. It is shown that gauge symmetry can be fixed in such a way that, after integrating out some dynamical variables in the functional integral, one obtains the induced gravity action. 
  In this work we present a proof of the discreteness of the spectrum for bosonic membrane, in a flat minkowski space. This may be useful to show the quantum mechanical consistence of others bosonics extended models. This proof includes the BRST residual symmetry and was directly performed over the discretized membrane model. The BRST residual invariant effective action is explicity constructed. 
  We use Monte Carlo methods to directly evaluate D-dimensional SU(N) Yang-Mills partition functions reduced to zero Euclidean dimensions, with and without supersymmetry. In the non-supersymmetric case, we find that the integrals exist for D=3, N>3 and D=4, N>2 and, lastly, D >= 5, N >= 2. We conclude that the D=3 and D=4 integrals exist in the large N limit, and therefore lead to a well-defined, new type of Eguchi-Kawai reduced gauge theory. For the supersymmetric case, we check, up to SU(5), recently proposed exact formulas for the D=4 and D=6 D-instanton integrals, including the explicit form of the normalization factor needed to interpret the integrals as the bulk contribution to the Witten index. 
  We study the origins of the five ten-dimensional ``matrix superstring'' theories, supplementing old results with new ones, and find that they all fit into a unified framework. In all cases the matrix definition of the string in the limit of vanishingly small coupling is a trivial 1+1 dimensional infra-red fixed point (an orbifold conformal field theory) characterized uniquely by matrix versions of the appropriate Green-Schwarz action. The Fock space of the matrix string is built out of winding T-dual strings. There is an associated dual supergravity description in terms of the near horizon geometry of the fundamental string solution of those T-dual strings. The singularity at their core is related to the orbifold target space in the matrix theory. At intermediate coupling, for the IIB and SO(32) systems, the matrix string description is in terms of non-trivial 2+1 dimensional fixed points. Their supergravity duals involve Anti de-Sitter space (or an orbifold thereof) and are well-defined everywhere, providing a complete description of the fixed point theory. In the case of the type IIB system, the two extra organizational dimensions normally found in F-theory appear here as well. The fact that they are non-dynamical has a natural interpretation in terms of holography. 
  In SO(32) heterotic string theory, the space-time at the core of N coincident NS-fivebranes is an infinite throat, R x S^3. As shown by Witten, the throat signals a singularity in the usual heterotic string conformal field theory and a non--perturbative USp(2N) gauge group appears, due to the N small instantons at the fivebranes' core. Nevertheless, we look for some trace of the non-perturbative physics in a description of the heterotic string infinitely far down the throat. Our guide is a D1-brane probing N D5-branes in type I, which yields a 1+1 dimensional (0,4) supersymmetric model with ADHM data in its couplings, as shown by Douglas. The neighbourhood of the classical boundary of the hypermultiplet moduli space of the theory flows to an exact conformal field theory description of the throat theory. Ironically, the remnant of the non-perturbative symmetry is indeed found in the conformal field theory, lurking in the structure of the partition function, and encoded in a family of deformations of the theory along flat directions. The deformations have an explicit description using the flow from the type I theory, and have a hyperKahler structure. Similar results hold true for the analogous (4,4) supersymmetric situation in the type IIB theory, as is evident in the work of Diaconescu and Seiberg. 
  We derive the spacetime superalgebras explicitly from ``test'' M-brane actions in M-brane backgrounds to the lowest order in $\theta$ via canonical formalism, and discuss various BPS saturated configurations on the basis of their central charges which depend on the harmonic functions determined by the backgrounds. All the 1/4 supersymmetric intersections of two M-branes obtained previously are deduced from the requirement of the test branes to be so ``gauge fixed'' in the brane backgrounds as to preserve 1/4 supersymmetry. Furthermore, some of 1/2-supersymmetric bound states of two M-branes are deduced from the behavior of the harmonic functions in the limits of vanishing distances of the two branes. The possibilities of some triple intersections preserving 1/4 supersymmetry are also discussed. 
  In this talk, we recall the most important features of the Dilatonic Black Holes which arise in the framework of the Einstein-Dilaton-Gauss-Bonnet theory and which are dressed with a classical long range dilaton field in contradiction with the existing "no-hair" theorems of the Theory of General Relativity. We demonstrate linear stability of these black hole solutions under small spacetime-dependent perturbations by making use of a semi-analytic method based on the Fubini-Sturm's theorem. As a result, the Dilatonic Black Holes constitute one of the very few examples of stable black hole solutions with non-trivial "hair" that arise in the framework of a more generalised theory of gravity. 
  We derive Bogomolny equations for an Einstein-Yang-Mills-dilaton-$\sigma$ model (EYMD-$\sigma$) on a static spacetime, showing that the Einstein equations are satisfied if and only if the associated (conformally scaled) three-metric is flat. These are precisely the static metrics for which super-covariantly constant spinors exist. We study some general properties of these equations and then consider the problem of obtaining axially symmetric solutions for the gauge group SU(2). 
  The massive N-flavor Schwinger model is analyzed by the bosonization method. The problem is reduced to the quantum mechanics of N degrees of freedom in which the potential needs to be self-consistently determined by its ground-state wave function and spectrum with given values of the $\theta$ parameter, fermion masses, and temperature. Boson masses and fermion chiral condensates are evaluated. In the N=1 model the anomalous behavior is found at $\theta \sim \pi$ and $m/\mu \sim 0.4$. In the N=3 model an asymmetry in fermion masses $(m_1 < m_2 \ll m_3)$ removes the singularity at $\theta=\pi$ and T=0. The chiral condensates at $\theta=0$ are insensitive to the asymmetry in fermion masses, but are significantly sensitive at $\theta=\pi$. The resultant picture is similar to that obtained in QCD by the chiral Lagrangian method. 
  We review some aspects of the construction of self-dual gravity and the associated field theory of ${\cal N}=2$ strings in terms of two-dimensional sigma models at large $N$. The theory is defined through a large $N$ Wess-Zumino-Witten model in a nontrivial background and in a particular double scaling limit. We examine the canonical structure of the theory and describe an infinite-dimensional Poisson algebra of currents. 
  We present a new approximation technique for quantum field theory. The standard one-loop result is used as a seed for a recursive formula that gives a sequence of improved Gaussian approximations for the generating functional. In a different setting, the basic idea of this recursive scheme is used in the second part of the paper to substantialy speed up the standard Monte Carlo algorithm. 
  This is an introduction to orientifolds with emphasis on applications to duality. Based on lectures given at the 1997 Trieste Summer School on Particle Physics and Cosmology, Italy. 
  A one-parameter family of new solutions representing Einstein spaces in $d=5,7$ is presented, and used to construct non-supersymmetric backgrounds in type IIB and M-theory that asymptotically approach $AdS_5\times S^5$ and $AdS_7\times S^4$ . Upon dimensional reduction, the latter gives a type IIA solution representing a 4-brane with Ramond-Ramond charge, which interpolates between the "near-horizon" non-extremal D4 brane and a geometry connected by T-duality to a new constant dilaton solution in type IIB. We discuss the possibility that M-theory on this space may be related to a (0,2) six-dimensional field theory on $S^1\times S^1$, with fermions obeying antiperiodic boundary conditions in both circles. 
  We consider string junctions with endpoints on a set of branes of IIB string theory defining an ADE-type gauge Lie algebra. We show how to characterize uniquely equivalence classes of junctions related by string/brane crossing through invariant charges that count the effective number of prongs ending on each brane. Each equivalence class defines a point on a lattice of junctions. We define a metric on this lattice arising from the intersection pairing of junctions, and use self-intersection to identify junctions in the adjoint and fundamental representations of all ADE algebras. This information suffices to determine the relation between junction lattices and the Lie-algebra weight lattices. Arbitrary representations are built by allowing junctions with asymptotic (p,q) charges, on which the group of conjugacy classes of representations is represented additively. One can view the (p,q) asymptotic charges as Dynkin labels associated to two new fundamental weight vectors. 
  We compute the super Liouville action for a two dimensional Regge surface by exploiting the invariance of the theory under the superconformal group for sphere topology and under the supermodular group for torus topology. For sphere topology and torus topology with even spin structures, the action is completely fixed up to a term which in the continuum limit goes over to a topological invariant, while the overall normalization of the action can be taken from perturbation theory. For the odd spin structure on the torus, due to the presence of the fermionic supermodulus, the action is fixed up to a modular invariant quadratic polynomial in the fermionic zero modes. 
  The instability of Liouville theory coupled to $c>1$ matter fields is shown to persist even when the ``spikes'' which represent highly singular geometries are allowed to interact in a natural way. 
  Keeping the two fundamental postulates of the special theory of relativity, the principle of relativity and the constancy of the one-way velocity of light in all inertial frames of reference, and assuming two generalized Finslerian structures of gravity-free space and time in the usual inertial coordinate system, we can modify the special theory of relativity. The modified theory is still characterized by the localized Lorentz transformation between any two usual inertial coordinate systems. It together with the quantum mechanics theory features a convergent and invariant quantum field theory. The modified theory also involves a new velocity distribution for free particles that is different from the Maxwell distribution. It is claimed that the deviation of the new distribution from its previous formula will provide experimental means of judging the modified special relativity theory. 
  Besides the two fundamental postulates, (i) the principle of relativity and (ii) the constancy of the one-way velocity of light in all inertial frames of reference, the special theory of relativity employs another assumption. This assumption concerns the flat structures of gravity-free space and time in the usual inertial coordinate system. We introduce the primed inertial coordinate system, in addition to the usual inertial coordinate system, for each inertial frame of reference, and assume the flat structures of gravity-free space and time in the primed inertial coordinate system and their generalized Finslerian structures in the usual inertial coordinate system. Combining this alternative assumption with (i) and (ii), we modify the special theory of relativity. The modified theory involves two versions of the light speed, infinite speed c' in the primed inertial coordinate system and finite speed c in the usual inertial coordinate system. It involves the c'-type Galilean transformation between two primed inertial coordinate systems and the localized Lorentz transformation between two corresponding usual inertial coordinate systems. It also involves a new physical principle. This principle is applied to reform of mechanics, field theory and quantum field theory. The validity of relativistic mechanics in the usual inertial coordinate system remains, while field theory is freshened. Based on the establishment of a transformation law for the quantized field systems, we construct a convergent and invariant quantum field theory, in full agreement with experimental facts, founded on the modified special relativity theory and the quantum mechanics theory. 
  We address the important issue of stabilizing the dilaton in the context of superstring cosmology. Scalar potentials which arise out of gaugino condensates in string models are generally exponential in nature. In a cosmological setting this allows for the existence of quasi scaling solutions, in which the energy density of the scalar field can, for a period, become a fixed fraction of the background density, due to the friction of the background expansion. Eventually the field can be trapped in the minimum of its potential as it leaves the scaling regime. We investigate this possibility in various gaugino condensation models and show that stable solutions for the dilaton are far more common than one would have naively thought. 
  We find exact D-brane configurations in the Nappi-Witten background using the boundary state approach and describe how they are related by T-duality transformations. We also show that the classical boundary conditions of the associated sigma model correspond to a field dependent automorphism relating the chiral currents and discuss the correspondence between the boundary state approach and the sigma model approach. 
  $M$-theory is believed to be described in various dimensions by large $N$ field theories. It has been further conjectured that at finite $N$, these theories describe the discrete light cone quantization (DLCQ) of $M$ theory. Even at low energies, this is not necessarily the same thing as the DLCQ of supergravity. It is believed that this is only the case for quantities which are protected by non-renormalization theorems. In 0+1 and 1+1 dimensions, we provide further evidence of a non-renormalization theorem for the $v^4$ terms, but also give evidence that there are not such theorems at order $v^8$ and higher. 
  In this paper we fill some gaps in the arguments of our previous papers [hep-th/9412229,hep-th/9604044]. In particular, we give a proof that the L operators of Conformal Field Theory indeed satisfy the defining relations of the Yang-Baxter algebra. Among other results we present a derivation of the functional relations satisfied by ${\bf T}$ and ${\bf Q}$ operators and a proof of the basic analyticity assumptions for these operators used in [hep-th/9412229,hep-th/9604044]. 
  The Hamiltonian formulation for the mechanical systems with reparametrization-invariant Lagrangians, depending on the worldline external curvatures is given, which is based on the use of moving frame.   A complete sets of constraints are found for the Lagrangians with quadratic dependence on curvatures, for the lagrangians, proportional to an arbitrary curvature, and for the Lagrangians, linear on the first and second curvatures. 
  We examine classical and quantum aspects of the planar non-compact spin system coupled with Chern-Simons gauge field in the presence of background charge. We first define our classical spin system as non- relativistic non-linear sigma model in which the order parameter spin takes value in the non-compact manifold ${\cal M}=SU(1,1)/U(1)$. Although the naive model does not allow any finite energy self dual solitons, it is shown that the gauged system admits static Bogomol'nyi solitons with finite energy whose rotationally symmetric soliton solutions are analyzed in detail. We also discuss the large spin limit in which the self-dual equation reduces to the well-known gauged non- linear Schr\"odinger model or Abelian Higgs model, depending on the choice of the background charge term. Then, we perform quantization of the model. We find that the spin algebra satisfies anomalous commutation relations, and the system is a field theoretic realization of the anyons. 
  I discuss a formula decomposing the integral of time-ordered products of operators into sums of products of integrals of time-ordered commutators. The resulting factorization enables summation of an infinite series to be carried out to yield an explicit formula for the time-ordered exponentials. The Campbell-Baker-Hausdorff formula and the nonabelian eikonal formula obtained previously are both special cases of this result. 
  We show that the instability of the singular Vilenkin instanton describing the creation of an open universe can be avoided using, instead of a minimally coupled scalar field, an axionic massless scalar field which gives rise to the Giddings-Strominger instanton. However, if we replace the singularity of the Hawking Turok instanton for an axionic wormhole some interpretational and technical difficulties would appear which can be overcome by introducing a positive cosmological constant in the action. This would make the instanton finite and free constant in the action. This would make the instanton finite and free from any instabilities. 
  Asymptotic freedom arises from the dominance of anti-screening over screening in non-abelian gauge theories. In this paper we will present a simple and physically appealing derivation of the anti-screening contribution to the interquark potential. Our method allows us to identify the dominant gluonic distribution around static quarks. Extensions are discussed. 
  Recent work has uncovered a correspondence between theories in anti-de Sitter space, and those on its boundary. This has important implications for black holes in string theory which have near-horizon AdS geometries. Using the effective coupling to the boundary conformal field theory, I compute the low-energy, s-wave absorption cross-sections for a minimally coupled scalar in the near-extremal four- and five-dimensional black holes. The results agree precisely with semi-classical gravity calculations. Agreement for fixed scalars, and for the BTZ black hole, is also found. 
  We present a proof of the irreversibility of renormalization group flows, i.e. the c-theorem for unitary, renormalizable theories in four (or generally even) dimensions. Using Ward identities for scale transformations and spectral representation arguments, we show that the c-function based on the trace of the energy-momentum tensor (originally suggested by Cardy) decreases monotonically along renormalization group trajectories. At fixed points this c-function is stationary and coincides with the coefficient of the Euler density in the trace anomaly, while away from fixed points its decrease is due to the decoupling of positive--norm massive modes. 
  We construct an irreducible representation for the extended affine algebra of type $sl_2$ with coordinates in a quantum torus. We explicitly give formulas using vertex operators similar to those found in the theory of the infinite rank affine algebra $A_{\infty}$. 
  It is shown how some field theories in the target-space induce the splitting of the space-time into a continuous of branes, which can be p-branes or D-branes depending on what the field theory it is. The basic symmetry underlying the construction is used to build an invariant action, which is proved to be off-shell identical to the p-brane (D-brane) action. The coupling with the abelian ($p+1$)-form in this formulation it is also found. While the classical brane's embedding couple to the field strenght, the classical fields couple with its dual (in the Hodge sense), therefore providing an explicit electric-magnetic duality. Finally, the generic role of the underlying symmetry in the connection between the target-space theory and the world-volume one, is completely elucidated. 
  We consider quantum mechanical gauge theories with sixteen supersymmetries. The Hamiltonians or Lagrangians characterizing these theories can contain higher derivative terms. In the operator approach, we show that the free theory is essentially the unique abelian theory with up to four derivatives in the following sense: any small deformation of the free theory, which preserves the supersymmetries, can be gauged away by a unitary conjugation. We also present a method for deriving constraints on terms appearing in an effective Lagrangian. We apply this method to the effective Lagrangian describing the dynamics of two well-separated clusters of D0-branes. As a result, we prove a non-renormalization theorem for the $v^4$ interaction. 
  S-duality symmetry of type IIB string theory predicts the existence of a stable non-BPS state on an orbifold five plane of the type IIB theory if the orbifold group is generated by the simultaneous action of (-1)^{F_L} and the reversal of sign of the four coordinates transverse to the orbifold plane. We calculate the mass of this state by starting from a pair of D-strings carrying the same charge as this state, and then identifying the point in the moduli space where this pair develops a tachyonic mode, signalling the appearance of a bound state of this configuration into the non-BPS state. 
  The Kirillov-Souriau-Kostant construction is applied to derive the classical and quantum mechanics for the massive spinning particle in arbitrary dimension. 
  There is a conceptual error in the main argument of this paper (essentially a regularization scheme is changed in the middle of a calculation), and therefore it is withdrawn. Interested readers are instead referred to hep-th/9811137. 
  The importance of a rigorous definition of the singular degree of a distribution is demonstrated on the case of two-dimensional QED (Schwinger model). Correct mathematical treatment of second order vacuum polarization in the perturbative approach is crucial in order to obtain the Schwinger mass of the photon by resummation. 
  We apply the theory of $\alpha$-induction of sectors which we elaborated in our previous paper to several nets of subfactors arising from conformal field theory. The main application are conformal embeddings and orbifold inclusions of SU(n) WZW models. For the latter, we construct the extended net of factors by hand. Developing further some ideas of F. Xu, our treatment leads canonically to certain fusion graphs, and in all our examples we rediscover the graphs Di Francesco, Petkova and Zuber associated empirically to the corresponding SU(n) modular invariants. We establish a connection between exponents of these graphs and the appearance of characters in the block-diagonal modular invariants, provided that the extended modular S-matrices diagonalize the endomorphism fusion rules of the extended theories. This is proven for many cases, and our results cover all the block-diagonal SU(2) modular invariants, thus provide some explanation of the A-D-E classification. 
  We develop a Lagrangian approach for constructing a symplectic structure for singular systems. It gives a simple and unified framework for understanding the origin of the pathologies that appear in the Dirac-Bergmann formalism, and offers a more general approach for a symplectic formalism, even when there is no Hamiltonian in a canonical sense. We can thus overcome the usual limitations of the canonical quantization, and perform an algebraically consistent quantization for a more general set of Lagrangian systems. 
  In 3 dimensions, the Ising model is in the same universality class as $\phi^4$-theory, whose massive 3-loop tetrahedral diagram, $C^{Tet}$, was of an unknown analytical nature. In contrast, all single-scale 4-dimensional tetrahedra were reduced, in hep-th/9803091, to special values of exponentially convergent polylogarithms. Combining dispersion relations with the integer-relation finder PSLQ, we find that $C^{Tet}/2^{5/2} = Cl_2(4\alpha) - Cl_2(2\alpha)$, with $Cl_2(\theta):=\sum_{n>0}\sin(n\theta)/n^2$ and $\alpha:=\arcsin\frac13$. This empirical relation has been checked at 1,000-digit precision and readily yields 50,000 digits of $C^{Tet}$, after transformation to an exponentially convergent sum, akin to those studied in math.CA/9803067. It appears that this 3-dimensional result entails a polylogarithmic ladder beginning with the classical formula for $\pi/\sqrt2$, in the manner that 4-dimensional results build on that for $\pi/\sqrt3$. 
  On the bundles of WZW chiral blocks over the moduli space of a punctured rational curve we construct isomorphisms that implement the action of outer automorphisms of the underlying affine Lie algebra. These bundle-isomorphisms respect the Knizhnik-Zamolodchikov connection and have finite order. When all primary fields are fixed points, the isomorphisms are endomorphisms; in this case, the bundle of chiral blocks is typically a reducible vector bundle. A conjecture for the trace of such endomorphisms is presented; the proposed relation generalizes the Verlinde formula. Our results have applications to conformal field theories based on non-simply connected groups and to the classification of boundary conditions in such theories. 
  The quantum corrections to the counting of statistical entropy for the 5+1-dimensional extremal black string in type-IIB supergravity with two observers are studied using anomalous Wess-Zumino actions for the corresponding intersecting D-brane description. The electric-magnetic duality symmetry of the anomalous theory implies a new symmetry between D-string and D-fivebrane sources and renders opposite sign for the RR charge of one of the intersecting D-branes relative to that of the black string. The electric-magnetic symmetric Hilbert space decomposes into subspaces associated with interior and exterior regions and it is shown that, for an outside observer, the expectation value of a horizon area operator agrees with the deviation of the classical horizon area in going from extremal to near-extremal black strings. 
  We construct the covariant kappa-symmetric superstring action for a type IIB superstring on AdS_5 x S^5 background. The action is defined as a 2d sigma-model on the coset superspace SU(2,2|4) / SO(4,1) x SO(5) and is shown to be the unique one that has the correct bosonic and flat space limits. 
  We study M5-brane configuration of the chiral gauge theory whose Type IIA brane configuration with orientifold 6-plane(O6) is studied by various authors. Much of the paper is devoted to studying M-theory picture of SO/Sp gauge theory with fundamental flavors realized in Type IIA setup with O6-plane. The Higgs branch of N=2 SO/Sp gauge theory is studied and the curve corresponding to rotated brane configuration is presented. In the chiral gauge theory, the middle NS$^{\prime}$5-brane on top of the O6 plane is realized as a detached rational curve. Depending on a location of the rational curve in $x^7$ direction, the same curve plus the rational curve can be interpreted as describing the Coulomb branch of $SU(2N_c)$ chiral gauge theory, $SO(2N_c)/Sp(N_c)$ gauge theory with $N_f/(N_f+4)$ fundamental hypermultiplets. Various consistency checks for this proposal are made. By introducing two more rational curves corresponding to NS$^{\prime}$5-branes, one can produce a non-trivial fixed point which mediates chiral non-chiral transition. 
  In the context of algebraic renormalization, the extended antifield formalism is used to derive the general forms of the anomaly consistency condition and of the Callan-Symanzik equation for generic gauge theories. A local version of the latter is used to derive sufficient conditions for the vanishing of beta functions associated to terms whose integrands are invariant only up to a divergence for an arbitrary non trivial non anomalous symmetry of the Lagrangian. These conditions are independent of power counting restrictions and of the form of the gauge fixation. 
  We analyse various world-sheet properties of the $SL(2,Z)$ covariant type IIB string action by coupling it with $SL(2,Z)$ covariant source. 
  We look for instanton solutions in a class of two scalar field gravity models, which includes the low energy string action in four dimensions. In models where the matter field has a potential with a false vacuum, we find that non-singular instantons exist as long as the Dilaton field found in string theory has a potential with a minimum, and provide an example of such an instanton. The class of singular instanton solutions are also examined, and we find that depending on the parameter values, the volume factor of the Euclidean region does not always vanish fast enough at the singularity to make the action finite. 
  It is shown that (2+1)-dimensional QED reveals several unusual effects due to the surface-term contributions. It is also shown that this system provides a new pairing mechanism for the high-$T_c$ superconductivity on the plane. 
  It was shown by Connes, Douglas, Schwarz[1] that one can compactify M(atrix) theory on noncommutative torus. We prove that compactifications on Morita equivalent tori are physically equivalent. This statement can be considered as a generalization of non-classical duality conjectured in [1] for two-dimensional tori. 
  A new formula for the composite Higgs boson mass is given, based on the Ward-Takahashi identity and the Schwinger-Dyson(SD) equation. In this formula the dominant asymptotic solution of the SD equation yields a correct answer, in sharp contrast to the Partially Conserved Dilatation Current(PCDC) approach where the sub- and sub-sub-dominant solutions should be taken into account carefully. In the gauged Nambu-Jona-Lasinio model we find M_H \simeq \sqrt{2}M for the composite Higgs boson mass M_H and the dynamical mass of the fermion M in the case of the constant gauge coupling(with large cut off), which is consistent with the PCDC approach and the renormalization-group approach. As to the case of the running gauge coupling, we find M_H \simeq 2 \sqrt{(A-1)/(2A-1)}M, where A \equiv 18 C_2 /(11N_c - 2N_f) with C_2 being the quadratic Casimir of the fermion representation. We also discuss a straightforward application of our formula to QCD(without 4-Fermi coupling), which yields M_{\sigma} \sim \sqrt{2}M_{dyn}, with M_{\sigma} and M_{dyn} being the light scalar(``\sigma-meson'') mass and mass of the constituent quark, respectively. 
  We prove that, in (2+1) dimensions, the S-wave phase shift, $ \delta_0(k)$,  k being the c.m. momentum, vanishes as either $\delta_0 \to {c\over \ln (k/m)} or \delta_0 \to O(k^2)$ as $k\to 0$. The constant $c$ is universal and $c=\pi/2$. This result is established first in the framework of the Schr\"odinger equation for a large class of potentials, second for a massive field theory from proved analyticity and unitarity, and, finally, we look at perturbation theory in $\phi_3^4$ and study its relation to our non-perturbative result. The remarkable fact here is that in n-th order the perturbative amplitude diverges like $(\ln k)^n$ as $k\to 0$, while the full amplitude vanishes as $(\ln k)^{-1}$. We show how these two facts can be reconciled. 
  Using the well-known Chern-Weil formula and its generalization, we systematically construct the Chern-Simons forms and their generalization induced by torsion as well as the Nieh-Yan (N-Y) forms. We also give an argument on the vanishing of integration of N-Y form on any compact manifold without boundary. A systematic construction of N-Y forms in D=4n dimension is also given. 
  We derive the moduli space for the global symmetry in N=1 supersymmetric theories. We show, at the generic points, it coincides with the space of quasi-Nambu-Goldstone (QNG) bosons, which appear besides the ordinary Nambu-Goldstone (NG) bosons when global symmetry G breaks down spontaneously to its subgroup H with preserving N=1 supersymmetry. At the singular points, most of the NG bosons change to the QNG bosons and the unbroken global symmetry is enhanced. The G-orbits parametrized by the NG bosons are the fibre at the moduli space and the singular points correspond to the point where H-orbit (in G-orbit) shrinks. We also show the low-energy effective Lagrangian is the arbitrary function of the orbit map. 
  The magnetic response of the charged anyon fluid at temperatures larger than the fermion energy gap is investigated in the self-consistent field approximation. In this temperature region a new phase, characterized by an inhomogeneous magnetic penetration, is found. The inhomogeneity is linked to the existence of an imaginary magnetic mass which increases with the temperature. The system stability in the new phase is proved by investigating the electromagnetic field rest-energy spectrum. 
  Reparametrization invariance being treated as a gauge symmetry shows some specific peculiarities. We study these peculiarities both from a general point of view and on concrete examples. We consider the canonical treatment of reparametrization invariant systems in which one fixes the gauge on the classical level by means of time-dependent gauge conditions. In such an approach one can interpret different gauges as different reference frames. We discuss the relations between different gauges and the problem of gauge invariance in this case. Finally, we establish a general structure of reparametrizations and its connection with the zero-Hamiltonian phenomenon. 
  We argue that supersymmetric higher-dimension operators in the effective actions of M-theory and IIB string theory do not affect the maximally supersymmetric vacua: $adS_4\times S^7$ and $adS_7\times S^4$ in M-theory and $adS_5\times S^5$ in IIB string theory. All these vacua are described in superspace by a fixed point with all components of supertorsion and supercurvature being supercovariantly constant. This follows from 32 unbroken supersymmetries and allows us to prove that such vacua are exact. 
  The approach to the dynamics of a charged particle in the Born-Infeld nonlinear electrodynamics developed in [Phys. Lett. A 240 (1998) 8] is generalized to include a Born-Infeld dyon. Both Hamiltonian and Lagrangian structures of many dyons interacting with nonlinear electromagnetism are constructed. All results are manifestly duality invariant. 
  We study, in the framework put forward by Siegel\cite{WS} and by Floreanini and Jackiw\cite{FJ} (FJ), the relationship between different chiral bosonization schemes (CBS). This is done in the context of the soldering formalism\cite{MS}, that considers the phenomenon of interference in the quantum field theory\cite{ABW}. We propose a field redefinition that discloses the presence of a noton, a nonmover field, in Siegel's formulation for chiral bosons. The presence of a noton in the Siegel CBS is a new and surprising result, that separates dynamics from symmetry. While the FJ component describes the dynamics, it is the noton that carries the symmetry contents, acquiring dynamics upon quantization and is fully responsible for the Siegel anomaly. The diagonal representation proposed here is used to study the effect of quantum interference between gauged rightons and leftons. 
  We study Siegel superparticle moving in $R^{4|4}$ flat superspace. Canonical quantization is accomplished yielding the massless Wess-Zumino model as an effective field theory. Path integral representation for the corresponding superpropagator is constructed and proven to involve the Siegel action in a gauge fixed form. It is shown that higher order fermionic constraints intrinsic in the theory, though being a consequence of others in $d=4$, make a crucial contribution into the path integral. 
  We study the compactification of the $(2,0)$ and type-II little-string theories on $S^1$, $T^2$ and $T^3$ with an R-symmetry twist that preserves half the supersymmetry. We argue that it produces the same moduli spaces of vacua as compactification of the $(1,0)$ theory with $E_8$ Wilson lines given by a maximal embedding of SU(2). In certain limits, this reproduces the moduli space of SU(2) with a massive adjoint hyper-multiplet. In the type-II little-string theory case, we observe a peculiar phase transition where the strings condense. We conjecture a generalization to more than two 5-branes which involves instantons on non-commutative $T^4$. We conclude with open questions. 
  We compute the leading contribution to the effective Hamiltonian of SU(N) matrix theory in the limit of large separation. We work with a gauge fixed Hamiltonian and use generalized Born-Oppenheimer approximation, extending the recent work of Halpern and Schwartz for SU(2). The answer turns out to be a free Hamiltonian for the coordinates along the flat directions of the potential. Applications to finding ground state candidates and calculation of the correction (surface) term to Witten index are discussed. 
  The four six-dimensional ``little string'' theories are all described in the infinite momentum frame (IMF) as matrix theories by non-trivial 1+1 dimensional infra-red fixed points. We characterize these fixed points using supergravity. Starting from the matrix theory definition of M5-branes, we derive an associated dual supergravity description of the fixed point theories, arising as the near horizon geometry of certain brane configurations. These supergravity solutions are all smooth, and involve three dimensional Anti-de Sitter space AdS_3. They therefore provide a complete description of the fixed point theories, and hence the IMF little string theories, if the AdS/CFT correspondence holds. 
  The field theories on the surface of non-supersymmetric D-brane constructions are identified. By moving to M-theory a semi-classical, strong coupling expansion to the IR non-supersymmetric gauge dynamics is obtained. The solution is consistent with the formation of a quark condensate but there is evidence that in moving to strong coupling scalar degrees of freedom have not decoupled. 
  The spontaneous symmetry breaking in (1+1)-dimensional $\phi^{4}$ theory is studied with discretized light-front quantization, that is, by solving the zero-mode constraint equation. The symmetric ordering is assumed for the operator-valued constraint equation. The commutation relation between the zero mode and each oscillator mode is calculated with $\hbar$ expansion. A critical coupling evaluated from the first some terms in the expansion is $28.8\mu ^{2}/\hbar \le \lambda_{cr} \le 31.1\mu ^{2}/\hbar$ consistent with the equal-time one $22\mu ^{2}/\hbar \le \lambda_{cr} \le 55.5\mu ^{2}/\hbar$. The same analysis is also made under another operator ordering. 
  A 5D black hole(M$_5$) is investigated in the type IIB superstring theory compactified on S$^1 \times $T$^4$. This corresponds to AdS$_3 \times $S$^3 \times $T$^4$ in the near horizon with asymptotically flat space. Here the harmonic gauge is introduced to decouple the mixing between the dilaton and others. On the other hand we obtain the BTZ balck hole(AdS$_3\times$S$^3\times$T$^4$) as the non-dilatonic solution. We calculate the greybody factor of the dilaton as a test scalar both for a 5D black hole(M$_5 \times $S$^1 \times $T$^4$) and the BTZ black hole(AdS$_3 \times $S$^3 \times $T$^4$). The result of the BTZ black hole agrees with the greybody factor of the dilaton in the dilute gas approximation of a 5D black hole. 
  A wide class of Seiberg-Witten models constructed by M-theory techniques and described by non-hyperelliptic Riemann surfaces are shown to possess an associative algebra of holomorphic differentials. This is a first step towards proving that also these models satisfy the Witten-Dijkgraaf-Verlinde-Verlinde equation. In this way, similar results known for simpler Seiberg-Witten models (described by hyperelliptic Riemann surfaces and constructed without recourse to M-theory) are extended to certain non-hyperelliptic cases constructed in M-theory. Our analysis reveals a connection between the algebra of holomorphic differentials on the Riemann surface and the configuration of M-theory branes of the corresponding Seiberg-Witten model. 
  In analogy to the class structure $\GL(\R^4)/\O(1,3)$ for general relativity with a local Lorentz group as stabilizer and a basic tetrad field for the parametrization, a corresponding class structure $\GL(\C^2)/\U(2)$ is investigated for the standard model with a local hyperisospin group $\U(2)$. The lepton, quark, Higgs and gauge fields, used in the standard model, cannot be basic in a coset interpretation, they may to be taken as first order terms in a flat spacetime, particle oriented expansion of a basic field (as the analogue to the tetrad) and its products. 
  We show how to extend the 't Hooft anomaly matching conditions to discrete symmetries. We check these discrete anomaly matching conditions on several proposed low-energy spectra of certain strongly interacting gauge theories. The excluded examples include the proposed chirally symmetric vacuum of pure N=1 supersymmetric Yang-Mills theories, certain non-supersymmetric confining theories and some self-dual N=1 supersymmetric theories based on exceptional groups. 
  We study N=2 supersymmetric Born-Infeld-Higgs theory in 3 dimensions and derive Bogomol'nyi relations in its bosonic sector. A peculiar coupling between the Higgs and the gauge field (with dynamics determined by the Born-Infeld action) is forced by supersymmetry. The resulting equations coincide with those arising in the Maxwell-Higgs model. Concerning Bogomol'nyi bounds for the vortex energy, they are derived from the N=2 supersymmetry algebra. 
  In these lectures we discuss the supersymmetry algebra and its irreducible representations. We construct the theories of rigid supersymmetry and gave their superspace formulations. The perturbative quantum properties of the extended supersymmetric theories are derived, including the superconformal invariance of a large class of these theories. We also discuss flat directions, non-holomorphicity and the perturbative chiral effective action for N=2 Yang-Mills theory. The superconformal transformations in four dimensional superspace are derived and encoded into one superconformal Killing superfield. It is shown that the anomalous dimensions of chiral operators in a superconformal quantum field are related to their $R$ weight. 
  We present evidence that the CHL string in eight dimensions is dual to F-theory compactified on an elliptic K3 with a $\Gamma_{0}(2)$ monodromy group. The monodromy group $\Gamma_{0}(2)$ allows one to turn on the flux of an antisymmetric two form along the base. The $B_{\mu \nu}$ flux is quantized and therefore the moduli space of the CHL string is disconnected from the moduli space of F-theory/Heterotic strings (as expected). The non-zero $B_{\mu \nu}$ flux obstructs certain deformations restricting the moduli of elliptic K3 to a 10 dimensional moduli space. We also discuss how one can reconstruct the gauge groups from the elliptic fibration structure. 
  A relevant part of the quantum algebra of observables for the closed bosonic strings moving in 1+3-dimensional Minkowski space is presented in the form of generating relations involving still one, as yet undetermined, real free parameter. 
  We study topological A-model disk amplitudes with Calabi-Yau target spaces by mirror symmetries. This allows us to calculate holomorphic instantons of Riemann surfaces with boundaries that are mapped into susy cycles in Calabi-Yau d-folds. Also we analyse disk amplitudes in Fano manifold cases by considering fusion relations between A-model operators. 
  We make a numerical study of the classical solutions of the combined system consisting of the Georgi-Glashow model and the SO(3) gauged Skyrme model. Both monopole-Skyrmion and dyon-Skyrmion solutions are found. A new bifurcation is shown to occur in the gauged Skyrmion solution sector. 
  Using many-body techniques we obtain the time-dependent Gaussian approximation for interacting fermion-scalar field models. This method is applied to an uniform system of relativistic spin-1/2 fermion field coupled, through a Yukawa term, to a scalar field in 3+1 dimensions, the so-called quantum scalar plasma model. The renormalization for the resulting Gaussian mean-field equations, both static and dynamical, are examined and initial conditions discussed. We also investigate solutions for the gap equation and show that the energy density has a single minimum. 
  The Gaussian-time-dependent variational equations are used to explored the physics of $(\phi^4)_{3+1}$ field theory. We have investigated the static solutions and discussed the conditions of renormalization. Using these results and stability analysis we show that there are two viable non-trivial versions of $(\phi^4)_{3+1}$. In the continuum limit the bare coupling constant can assume $b\to 0^{+}$ and $b\to 0^{-}$, which yield well defined asymmetric and symmetric solutions respectively. We have also considered small oscillations in the broken phase and shown that they give one and two meson modes of the theory. The resulting equation has a closed solution leading to a ``zero mode'' and vanished scattering amplitude in the limit of infinite cutoff. 
  We propose a systematic way of introducing the (nongravitational) low energy dilaton and a scheme for spontaneous breaking of scale symmetry (SBSS) is explained. 
  This paper has been withdrawn by the author as the conjecture proposed in it is wrong. 
  We study the absorption of a class of fields in the geometry produced by extremal three-branes. We consider fields that do not mix with the ten-dimensional graviton. For these fields we solve the wave equations and find the absorption probabilities for all partial waves at leading order in the energy. We note that in some of these cases one needs an `intermediate' region which interpolates between flat Minkowski space at infinity and the AdS geometry near the branes. 
  We obtain the Hamiltonian form of the worldvolume action for the M5-brane in a general D=11 supergravity background. We use this result to obtain a new version of the covariant M5-brane Lagrangian in which the tension appears as a dynamical variable, although this Lagrangian has some unsatisfactory features which we trace to peculiarities of the null limit. We also show that the M5-brane action is invariant under all (super)isometries of the background. 
  I investigate a class of dynamical systems in which finite pieces of spacetime contain finite amounts of information. Most of the guiding principles for designing these systems are drawn from general relativity: the systems are deterministic; spacetime may be foliated into Cauchy surfaces; the law of evolution is local (there is a light-cone structure); and the geometry evolves locally (curvature may be present; big bangs are possible). However, the systems differ from general relativity in that spacetime is a combinatorial object, constructed by piecing together copies of finitely many types of allowed neighborhoods in a prescribed manner. Hence at least initially there is no metric. The role of diffeomorphism is played by a combinatorial equivalence map which is local and preserves information content. Most of my results come in the 1+1-dimensional oriented case. There sets of spaces may be described equivalently by matrices of nonnegative integers, directed graphs, or symmetric tensors; local equivalences between space sets are generated by simple matrix transformations. These equivalence maps turn out to be closely related to the flow equivalence maps between subshifts of finite type studied in symbolic dynamics. Also, the symmetric tensor algebra generated by equivalence transformations turns out to be isomorphic to the abstract tensor algebra generated by commutative cocommutative bialgebras. In higher dimensions I study the case where space is a special type of colored graph (discovered by Pezzana) which may be interpreted as an n-dimensional pseudomanifold. Finally, I show how one may study the behavior of combinatorial spacetimes by searching for constants of motion, which typically are associated with local flows and often may be interpreted in terms of particles. 
  We report on an attempt to solve the gauge hierarchy problem in the framework of higher dimensional gauge theories. Both classical Higgs mass and quadratically divergent quantum correction to the mass are argued to vanish. Hence the hierarchy problem in its original sense is solved. The remaining finite mass correction is shown to depend crucially on the choice of boundary condition for matter fields, and a way to fix it dynamically is presented. We also point out that on the simply-connected space $S^2$ even the finite mass correction vanishes. 
  By undertaking additional analyses postponed in a previous paper, we complete our construction of a manifestly supersymmetric gauge-covariant regularization of supersymmetric chiral gauge theories. We present the following: An evaluation of the covariant gauge anomaly; a proof of the integrability of the covariant gauge current in anomaly-free cases; a calculation of a one-loop superconformal anomaly in the gauge supermultiplet sector. On the last point, we find that the ghost-anti-ghost supermultiplet and the Nakanishi-Lautrup supermultiplet give rise to BRST exact contributions which, due to ``tree-level'' Slavnov-Taylor identities in our regularization scheme, can safely be neglected, at least at the one-loop level. 
  Motivated by the space-time uncertainty principle, we establish a conformal symmetry in the dynamics of D-particles. The conformal symmetry, combined with the supersymmetric non-renormalization theorem, uniquely determines the classical form of the effective action for a probe D-particle in the background of a heavy D-particle source, previously constructed by Becker-Becker-Polchinski-Tseytlin. Our results strengthen the conjecture proposed by Maldacena on the correspondence, in the case of D-particles, between the supergravity and the supersymmetric Yang-Mills matrix models in the large $N$-limit, the latter being the boundary conformal field theory of the former in the classical D-particle background in the near horizon limit. 
  We investigate whether DLCQ of M-theory can be defined as a limit of M-theory compactified on an almost light-like circle. This is of particular interest since the proofs of the matrix description of M-theory by Seiberg and Sen rely on this assumption. By the standard relation between M-theory on $S^1$ and IIA string theory, we translate this question into the corresponding one about the existence of the light-like limit of IIA superstring theory for any string coupling $g_s$. We argue that perturbative string loop amplitudes should have a finite and well-defined light-like limit provided the external momenta are chosen to correspond to a well-defined DLCQ set-up. On the non-perturbative side we consider states and amplitudes. We show that an appropriate class of non-perturbative states (wrapped D-branes) precisely have the right light-like limit. We give some indications that non-perturbative corrections to string amplitudes, too, may behave as required in the light-like limit. Having perturbative and non-perturbative evidence, this suggests that type IIA superstring theory as a whole has a well-defined light-like limit (for any string coupling $g_s$) and hence that the same is true for M-theory. 
  Extending a recent result of S.B. Giddings, F. Hacquebord and H. Verlinde, we show that in the U(N) SYM Matrix theory there exist classical BPS instantons which interpolate between different closed string configurations via joining/splitting interactions similar to those of string field theory. We construct them starting from branched coverings of Riemann surfaces. For the class of them which we analyze in detail the construction can be made explicit in terms U(N) affine Toda field theories. 
  Starting from certain 3D non-abelian dual systems, we discuss a number of related dual systems in 2D, some of which are obtained by dimensional reduction. The dualities relate massive scalar and vector fields, and may be relevant for string theory in the context of massive type IIA supergravity. Supersymmetric extensions of the models are also presented. 
  This is the first of a series of papers devoted to the group-theoretical analysis of the conditions which must be satisfied for a configuration of intersecting M5-branes to be supersymmetric. In this paper we treat the case of static branes. We start by associating (a maximal torus of) a different subgroup of Spin(10) with each of the equivalence classes of supersymmetric configurations of two M5-branes at angles found by Ohta & Townsend. We then consider configurations of more than two intersecting branes. Such a configuration will be supersymmetric if and only if the branes are G-related, where G is a subgroup of Spin(10) contained in the isotropy of a spinor. For each such group we determine (a lower bound for) the fraction of the supersymmetry which is preserved. We give examples of configurations consisting of an arbitrary number of non-coincident intersecting fivebranes with fractions: 1/32, 1/16, 3/32, 1/8, 5/32, 3/16 and 1/4, and we determine the resulting (calibrated) geometry. 
  We consider target space duality transformations for heterotic sigma models and strings away from renormalization group fixed points. By imposing certain consistency requirements between the T-duality symmetry and renormalization group flows, the one loop gauge beta function is uniquely determined, without any diagram calculations. Classical T-duality symmetry is a valid quantum symmetry of the heterotic sigma model, severely constraining its renormalization flows at this one loop order. The issue of heterotic anomalies and their cancelation is addressed from this duality constraining viewpoint. 
  The generating functional of two dimensional $BF$ field theories coupled to fermionic fields and conserved currents is computed in the general case when the base manifold is a genus g compact Riemann surface. The lagrangian density $L=dB{\wedge}A$ is written in terms of a globally defined 1-form $A$ and a multi-valued scalar field $B$. Consistency conditions on the periods of $dB$ have to be imposed. It is shown that there exist a non-trivial dependence of the generating functional on the topological restrictions imposed to $B$. In particular if the periods of the $B$ field are constrained to take values $4\pi n$, with $n$ any integer, then the partition function is independent of the chosen spin structure and may be written as a sum over all the spin structures associated to the fermions even when one started with a fixed spin structure. These results are then applied to the functional bosonization of fermionic fields on higher genus surfaces. A bosonized form of the partition function which takes care of the chosen spin structure is obtained 
  We analyze string networks in 7-brane configurations in IIB string theory. We introduce a complex parameter M characterizing equivalence classes of networks on a fixed 7-brane background and specifying the BPS mass of the network as M_{BPS} = | M |. We show that M can be calculated without knowing the particular representative of the BPS state. Based on detailed examination of backgrounds with three and four 7-branes we argue that equivalent networks may not be simultaneously BPS, an essential requirement of consistency. 
  We are unable to formulate lattice gauge theories in the framework of Connes' spectral triples. 
  We derive fully covariant expressions for disk scattering amplitudes of any two massless closed strings in which mixed Neumann and Dirichlet world-sheet boundary conditions are included. From the two-point amplitudes, we derive the long range background fields and verify that they correspond to D$p$-brane bound state. Also, from the scattering amplitudes, we calculate the linear coupling of closed string fields to D-brane world-volume and show that they are consistent with Born-Infeld and Chern-Simons actions in the presence of a background field. 
  We reconsider the issue of the existence of a complex structure in the Gupta-Bleuler quantization scheme. We prove an existence theorem for the complex structure associated with the $d=10$ Casalbuoni-Brink-Schwarz superparticle, based on an explicitly constructed Lagrangian that allows a holomorphic-antiholomorphic splitting of the fermionic constraints consistent with the vanishing of all first class constraints on the physical states. 
  We calculate the power law decay, and asymptotic form of a (unique) SO(9) and SU(2) invariant wave function satisfying, to leading and sub-leading order, $Q_{\hat{\beta}} \psi = 0$ for all 16 supercharges of the matrix model corresponding to supermembranes in 11 space-time dimensions. 
  I show that de Sitter space disintegrates into an infinite number of copies of itself. This occurs iteratively through a quantum process involving two types of topology change. First a handle is created semiclassically, on which multiple black hole horizons form. Then the black holes evaporate and disappear, splitting the spatial hypersurfaces into large parts. Applied to cosmology, this process leads to the production of a large or infinite number of universes in most models of inflation and yields a new picture of global structure. 
  We consider certain supersymmetric Born-Infeld couplings to the D3 brane action and show that they give rise to massless and massive KK excitations of type IIB on $AdS_5\times S_5$, in terms of singleton Yang-Mills superfields. 
  Duality symmetries of supergravity theories are powerful tools to restrict the number of possible actions, to link different dimensions and number of supersymmetries and might help to control quantisation. (Hodge-Dirac-)Dualisation of gauge potentials exchanges Noether and topological charges, equations of motion and Bianchi identities, internal rigid symmetries and gauge symmetries, local transformations with nonlocal ones and most exciting particles and waves. We compare the actions of maximally dualised supergravities (ie with gauge potential forms of lowest possible degree) to the non-dualised actions coming from 11 (or 10) dimensions by plain dimensional reduction as well as to other theories with partial dualisations. The effect on the rigid duality group is a kind of contraction resulting from the elimination of the unfaithful generators associated to the (inversely) dualised scalar fields. New gauge symmetries are introduced by these (un)dualisations and it is clear that a complete picture of duality (F(ull)-duality) should include all gauge symmetries at the same time as the rigid symmetries and the spacetime symmetries. We may read off some properties of F-duality on the internal rigid Dynkin diagram: field content, possible dualisations, increase of the rank according to the decrease of space dimension... Some recent results are included to suggest the way towards unification via a universal twisted self-duality (TS) structure. The analysis of this structure had revealed several profound differences according to the parity mod 4 of the dimension of spacetime (to be contrasted with the (Bott) period 8 of spinor properties). 
  A binary expression in terms of operators is given which satisfies all the quantum counterparts of the algebraic properties of the classical antibracket. This quantum antibracket has therefore the same relation to the classical antibracket as commutators to Poisson brackets. It is explained how this quantum antibracket is related to the classical antibracket and the \Delta-operator in the BV-quantization. Higher quantum antibrackets are introduced in terms of generating operators, which automatically yield all their subsequent Jacobi identities as well as the consistent Leibniz' rules. 
  A free spinor field on a noncommutative sphere is described starting from a canonical realization of the enveloping algebra U(u(2|1)). The gauge extension of the model - the Schwinger model on a noncommutative sphere is defined and the model is quantized. The model contains only finite number degrees of freedom and is nonperturbatively UV-regular. The chiral anomaly and the effective actions are calculated. In the nomcommutative limit standard formulas are recovered. 
  As argued by Witten on the basis of M-theory, QCD strings can end on domain walls. We present an explanation of this effect in the framework of effective field theories for the Polyakov loop and the gluino condensate in N=1 supersymmetric QCD. The domain walls separating confined phases with different values of the gluino condensate are completely wet by a layer of deconfined phase at the high-temperature phase transition. As a consequence, even at low temperatures, the Polyakov loop has a non-vanishing expectation value on the domain wall. Thus, close to the wall, the free energy of a static quark is finite and the string emanating from it can end on the wall. 
  Anti-de Sitter space with identified points give rise to black-hole structures. This was first pointed out in three dimensions, and generalized to higher dimensions by Aminneborg et al. In this paper, we analyse several aspects of the five dimensional anti-de Sitter black hole including, its relation to thermal anti-de Sitter space, its embedding in a Chern-Simons supergravity theory, its global charges and holonomies, and the existence of Killing spinors. 
  Naively, helicity flip amplitudes for fermions seem to vanish in the chiral limit of light-front QCD, which would make it nearly impossible to generate a small pion mass in this framework. Using a simple model, it is illustrated how a large helicity flip amplitude is generated dynamically by summing over an infinite number of Fock space components. While the kinetic mass is basically generated by a zero-mode induced counter-term, the vertex mass is generated dynamically by infinitesimally small, but nonzero, momenta. Implications for the renormalization of light-front Hamiltonians for fermions are discussed. 
  The semiclassical theory for the large-N field models is developed from an unusual point of view. Analogously to the procedure of the second quantization in quantum mechanics, the functional Schrodinger large-N equation is presented in a third-quantized form. The third-quantized creation and annihilation operators depend on the field $\phi({\bf x})$. If the coefficient of the $\phi^4$-term is of order 1/N (this is a usual condition of applicability of the 1/N-expansion), one can rescale the third-quantized operators in such a way that their commutator will be small, while the Heisenberg equations will not contain large or small parameters. This means that classical equation of motion is an equation on the functional $\Phi[\phi(\cdot)]$. This equation being a nonlinear analog of the functional Schrodinger equation for the one-field theory is investigated. The exact solutions are constructed and the renormalization problem is analysed. We also perform a quantization procedure about found classical solutions. The corresponding semiclassical theory is a theory of a variable number of fields. The developed third-quantized semiclassical approach is applied to the problem of finding the large-N spectrum. The results are compared with obtained by known methods. We show that not only known but also new energy levels can be found. 
  We show from first principles, using explicitly invariant Pauli-Villars regularization of chiral fermions, that the Nieh-Yan form does contribute to the Adler-Bell-Jackiw (ABJ) anomaly for spacetimes with generic torsion, and comment on some of the implications. There are a number of interesting and important differences with the usual ABJ contributions in the absence of torsion. For dimensional reasons, the Nieh-Yan contribution is proportional to the square of the regulator mass. In spacetimes with flat vierbein but non-trivial torsion, the associated diagrams are actually vacuum polarization rather than triangle diagrams, and the Nieh-Yan contribution to the ABJ anomaly arises from the fact that the axial torsion "photon" is not transverse. 
  We use different techniques to analyze the system formed by a D0 brane and a D6 brane (with background gauge fields) in relative motion. In particular, using the closed string formalism of boosted boundary states, we show the presence of a term linear in the velocity, corresponding to the Lorentz force experienced by the D0 brane moving in the magnetic background produced by the D6 brane. This term, that was missed in previous analyses of this system, comes entirely from the R-R odd spin structure and is also reproduced by a M(atrix) theory calculation. 
  We show that in a Wilsonian renormalization scheme with zero-momentum subtraction point the massless Wess-Zumino model satisfies the non-renormalization theorem; the finite renormalization of the superpotential appearing in the usual non-zero momentum subtraction schemes is thus avoided. We give an exact expression of the beta and gamma functions in terms of the Wilsonian effective action; we prove the expected relation $\beta = 3g\gamma$. We compute the beta function at the first two loops, finding agreement with previous results. 
  The main content of this treatise is a new concept in nonperturbative non-Lagrangian QFT which explains and extends the ad hoc constructions in low-dimensional models and incorporates them together with the higher dimensional theories into a new construction method. Thermal and entropical properties, which were hitherto restricted to situations with classical horizons (Killing vectors), are now generic (nonperturbative) aspects of "modular localization". The underlying more algebraic (and less geometric) mode of thinking also gives rise to interesting questions in renormalizable deformatioms of higher spin fields, in particular gauge theories. 
  Following a recent proposal for integrable theories in higher dimensions based on zero curvature, new Lorentz invariant submodels of the principal chiral model in 2+1 dimensions are found. They have infinite local conserved currents, which are explicitly given for the su(2) case. The construction works for any Lie algebra and in any dimension, and it is given explicitly also for su(3). We comment on the application to supersymmetric chiral models. 
  Certain (3+1)-dimensional chiral non-Abelian gauge theories have been shown to exhibit a new type of global gauge anomaly, which in the Hamiltonian formulation is due to the fermion zero-modes of a Z-string-like configuration of the gauge potential and the corresponding spectral flow. Here, we clarify the relation between this Z-string global gauge anomaly and other anomalies in both 3+1 and 2+1 dimensions. We then point out a possible trade-off between the (3+1)-dimensional Z-string global gauge anomaly and the violation of CPT and Lorentz invariance. 
  Maldacena's duality between conformal field theories and supergravity is applied to some conformal invariant models with 8 supercharges appearing in the F-theory moduli space on a locus of constant coupling. This includes Sp(2N) gauge theories describing the worldvolume dynamics of D3-branes in the presence of D7-branes and an orientifold plane. Other examples of this kind are models with exceptional global symmetries which have no perturbative field theory description. In all these cases the duality is used to describe perturbations by primary marginal and relevant operators. 
  We interpret the general rotating black holes in five dimensions as rotating black strings in six dimensions. In the near horizon limit the geometry is locally AdS_3 x S_3, as in the nonrotating case. However, the global structure couples the AdS_3 and the S_3, giving angular velocity to the S_3. The asymptotic geometry is exploited to count the microstates and recover the precise value of the Bekenstein- Hawking entropy, with rotation taken properly into account. We discuss the perturbation spectrum of the rotating black hole, and its relation to the underlying conformal field theory. 
  We reinvestigate Kreimer's Hopf algebra structure of perturbative quantum field theories with a special emphasis on overlapping divergences. Kreimer first disentangles overlapping divergences into a linear combination of disjoint and nested ones and then tackles that linear combination by the Hopf algebra operations. We present a formulation where the Hopf algebra operations are directly defined on any type of divergence. We explain the precise relation to Kreimer's Hopf algebra and obtain thereby a characterization of their primitive elements. 
  We suggest that compactifications on Anti-de-Sitter (AdS) spaces of type IIA, IIB, heterotic strings and eleven dimensional vacuua of M-theory are related by a combination of $T$ and strong/weak dualities. Maldacena conjecture relates then all these vacuua to a conformal theory on the boundaries. Furthermore acting with discrete groups on part of the internal spaces of these theories will lead to further dual theories with less or no supersymmetry. 
  The randomly driven Navier-Stokes equation without pressure in d-dimensional space is considered as a model of strong turbulence in a compressible fluid. We derive a closed equation for the velocity-gradient probability density function. We find the asymptotics of this function for the case of the gradient velocity field (Burgers turbulence), and provide a numerical solution for the two-dimensional case. Application of these results to the velocity-difference probability density function is discussed. 
  In this paper we study the boundary effects for off-critical integrable field theories which have close analogs with integrable lattice models. Our models are the $SU(2)_{k}\otimes SU(2)_{l}/SU(2)_{k+l}$ coset conformal field theories perturbed by integrable boundary and bulk operators. The boundary interactions are encoded into the boundary reflection matrix. Using the TBA method, we verify the flows of the conformal BCs by computing the boundary entropies. These flows of the BCs have direct interpretations for the fusion RSOS lattice models. For super CFTs ($k=2$) we show that these flows are possible only for the Neveu-Schwarz sector and are consistent with the lattice results. The models we considered cover a wide class of integrable models. In particular, we show how the the impurity spin is screened by electrons for the $k$-channel Kondo model by taking $l\to\infty$ limit. We also study the problem using an independent method based on the boundary roaming TBA. Our numerical results are consistent with the boundary CFTs and RSOS TBA analysis. 
  Dual form of 3+1 dimensional Yang-Mills theory is obtained as another SO(3) gauge theory. Duality transformation is realized as a canonical transformation. The The non-abelian Gauss law implies the corresponding Gauss law for the dual theory. The dual theory is non-local. There is a non-local version of Yang-Mills theory which is self dual. 
  In this work we study the dynamical generation of a fermion mass induced by a constant and uniform external magnetic field in an Abelian gauge model with a Yukawa term. We show that the Yukawa coupling not only enhances the dynamical generation of the mass, but it substantially decreases the magnetic field required for the mass to be generated at temperatures comparable to the electroweak critical temperature. These results indicate that if large enough primordial magnetic fields were present during the early universe evolution, the field-induced generation of fermion masses, which in turn corresponds to the generation of fermion bound states, may play an important role in the electroweak phase transition. 
  We compute, in the large N limit, the quark potential for ${\cal N}=4$ supersymmetric SU(N) Yang-Mills theory broken to $SU(N_1) \times SU(N_2)$. At short distances the quarks see only the unbroken gauge symmetry and have an attractive potential that falls off as 1/L. At longer distances the interquark interaction is sensitive to the symmetry breaking, and other QCD states appear. These states correspond to combinations of the quark-antiquark pair with some number of W-particles. If there is one or more W-particles then this state is unstable because of the coulomb interaction between the W-particles and between the W's and the quarks. As L is decreased the W-particles delocalize and these coulomb branches merge onto a branch with a linear potential. The quarks on this branch see the unbroken gauge group, but the flux tube is unstable to the production of W-particles. 
  We test the conjectured relationship between N=4 super Yang-Mills theory in four dimensions and IIB supergravity compactified on $AdS_5\times S_5$ by computing the two- and three-point functions of R-symmetry currents. We observe that the integral expressions describing the general three-point correlator on the supergravity side have a structure similar to one-loop triangle diagrams in N=4 super Yang-Mills theory. This allows us to compare the expressions on both sides of the AdS/CFT correspondence without the technical complications of the loop integrations. We confirm that the two- and three-point correspondence arises at only one-loop in the N=4 super Yang-Mills theory. Higher-point functions as well as further three-point functions may be analyzed similarly. 
  A new formulation of Calogero-Moser models based on root systems and their Weyl group is presented. The general construction of the Lax pairs applicable to all models based on the simply-laced algebras (ADE) are given for two types which we call `root' and `minimal'. The root type Lax pair is new; the matrices used in its construction bear a resemblance to the adjoint representation of the associated Lie algebra, and exist for all models, but they do not contain elements associated with the zero weights corresponding to the Cartan subalgebra. The root type provides a simple method of constructing sufficiently many number of conserved quantities for all models, including the one based on $E_{8}$, whose integrability had been an unsolved problem for more than twenty years. The minimal types provide a unified description of all known examples of Calogero-Moser Lax pairs and add some more. In both cases, the root type and the minimal type, the formulation works for all of the four choices of potentials: the rational, trigonometric, hyperbolic and elliptic. 
  We review M-theory description of 4d N=2 SQCD. Configurations of M-theory fivebranes relevant to describe the moduli spaces of the Coulomb and Higgs branches are studied using the Taub-NUT geometry. Minimal area membranes related with the BPS states of N=2 SQCD are given explicitly. They almost saturate the BPS bounds. The deviation from the bounds is due to their boundary condition constrained by the fivebrane. The electric-magnetic duality at the baryonic branch root is also examined from the M-theory viewpoint. In this course, novel concepts such as creation of brane and exchange of branes in Type II theory are explained in the framework of M-theory. 
  We formulate a non-perturbative lattice model of two-dimensional Lorentzian quantum gravity by performing the path integral over geometries with a causal structure. The model can be solved exactly at the discretized level. Its continuum limit coincides with the theory obtained by quantizing 2d continuum gravity in proper-time gauge, but it disagrees with 2d gravity defined via matrix models or Liouville theory. By allowing topology change of the compact spatial slices (i.e. baby universe creation), one obtains agreement with the matrix models and Liouville theory. 
  We develop a technique that solders the dual aspects of some symmetry. Using this technique it is possible to combine two theories with such symmetries to yield a new effective theory. Some applications in two and three dimensional bosonisation are discussed. In particular, it is shown that two apparently independent three dimensional massive Thirring models with same coupling but opposite mass signatures, in the long wavelegth limit, combine by the process of bosonisation and soldering to yield an effective massive Maxwell theory. Similar features also hold for quantum electrodynamics in three dimensions. We also provide a systematic derivation of duality symmetric actions and show that the soldering mechanism leads to a master action which is duality invariant under a bigger set of symmetries than is usually envisaged. The concept of duality swapping is introduced and its implications are analysed. The example of electromagnetic duality is discussed in details. 
  Using examples of a D=2 chiral scalar and a duality-symmetric formulation of D=4 Maxwell theory we study duality properties of actions for describing chiral bosons. In particular, in the D=4 case, upon performing a duality transform of an auxiliary scalar field, which ensures Lorentz covariance of the action, we arrive at a new covariant duality-symmetric Maxwell action, which contains a two-form potential as an auxiliary field. When the two-form field is gauge fixed this action reduces to a duality-symmetric action for Maxwell theory constructed by Zwanziger. We consider properties of this new covariant action and discuss its coupling to external dyonic sources. We also demonstrate that the formulations considered are self-dual with respect to a dualization of the field-strengths of the chiral fields. 
  The picture of CP-violation in orbifold compactifications in which the $T$-modulus is at a complex fixed point of the modular group is studied. CP-violation in the neutral kaon system and in the neutron electric dipole moment are both discussed. The situation where the $T$-modulus takes complex values on the unit circle which are not at a fixed point is also discussed. 
  In the mapping from four-dimensional gauge theories to string theory in $AdS$ space, many features of gauge theory can be described by branes wrapped in different ways on $\S^5$, $\RP^5$, or subspaces thereof. These include a baryon vertex coupling N external charges in the fundamental representation of SU(N), a bound state of $k$ gluons in SO(2k) gauge theory, strings coupled to external charges in the spinor representation of the gauge group, and domain walls across which the low energy gauge group changes. 
  A family of Majumdar-Papapetrou type solutions in sigma-model of p-brane origin is obtained for all direct sums of finite-dimensional simple Lie algebras. Several examples of p-brane dyonic configurations in D=10 (IIA) and D=11 supergravities corresponding to the Lie algebra sl(3,C) are considered. 
  The correspondence between string theory in Anti-de Sitter space and super Yang Mills theory is an example of the Holographic principle according to which a quantum theory with gravity must be describable by a boundary theory. However, arguments given so far are incomplete because, while the bulk theory has been related to a boundary theory, the holographic bound saying that the boundary theory has only one bit of information per Planck area has not been justified. We show here that this bound is the physical interpretation of one of the unusual aspects of the correspondence between Anti-de Sitter space and the boundary conformal field theory, which is that infrared effects in the bulk theory are reflected as ultraviolet effects in the boundary theory. 
  Large N gauge theories have so called Gross-Witten phase transitions which typically can occur in finite volume systems. In this paper we relate these transitions in supersymmetric gauge theories to transitions that take place between black hole solutions in general relativity. The correspondence between gauge theory and gravitation is through matrix theory which represents the gravitational system in terms of super Yang Mills theories on finite tori. We also discuss a related transition that was found by Banks, Fischler, Klebanov and Susskind. 
  We review the geodesic motion of pseudo-classical spinning particles in curved spaces. Investigating the generalized Killing equations for spinning spaces, we express the constants of motion in terms of Killing-Yano tensors. The general results are applied to the case of the four-dimensional Euclidean Taub-NUT spinning space. A simple exact solution, corresponding to trajectories lying on a cone, is given. 
  We examine the connection between the nonlinear integral equation (NLIE) derived from light-cone lattice and sine-Gordon quantum field theory, considered as a perturbed c=1 conformal field theory. After clarifying some delicate points of the NLIE deduction from the lattice, we compare both analytic and numerical predictions of the NLIE to previously known results in sine-Gordon theory. To provide the basis for the numerical comparison we use data from Truncated Conformal Space method. Together with results from analysis of infrared and ultraviolet asymptotics, we find evidence that it is necessary to change the rule of quantization proposed by Destri and de Vega to a new one which includes as a special case that of Fioravanti et al. This way we find strong evidence for the validity of the NLIE as a description of the finite size effects of sine-Gordon theory. 
  The master differential equations in the external square momentum p^2 for the master integrals of the two-loop sunrise graph, in n-continuous dimensions and for arbitrary values of the internal masses, are derived. The equations are then used for working out the values at p^2 = 0 and the expansions in p^2 at p^2 =0, in (n-4) at n to 4 limit and in 1/p^2 for large values of p^2 . 
  The generalized gluing and resmoothing theorem originally proved by LeClair, Peskin and Preitschopf, gives a powerful formula for the fused vertex obtained by contracting any two vertices in string field theories. Although the theorem is naturally expected to hold for the vertices at any loop level, the original proof was restricted to the vertices at tree level. Here we present a simplified proof for the tree level theorem and then prove explicitly the extended version at one-loop level. We also find that a non-trivial sign factor, which depends on the string states to be contracted, appears in the theorem. This sign factor turns out to be essential for reproducing correctly the conformal field theory correlation function on the torus. 
  We discuss the treatment of quantum-gravitational fluctuations in the space-time background as an `environment', using the formalism for open quantum-mechanical systems, which leads to a microscopic arrow of time. After reviewing briefly the open-system formalism, and the motivations for treating quantum gravity as an `environment', we present an example from general relativity and a general framework based on non-critical strings, with a Liouville field that we identify with time. We illustrate this approach with calculations in the contexts of two-dimensional models and $D$ branes. Finally, some prospects for observational tests of these ideas are mentioned. 
  We address the delocalization of low dimensional D-branes and NS-branes when they are a part of a higher dimensional BPS black brane, and the homogeneity of the resulting horizon. We show that the effective delocalization of such branes is a classical effect that occurs when localized branes are brought together. Thus, the fact that the few known solutions with inhomogeneous horizons are highly singular need not indicate a singularity of generic D- and NS-brane states. Rather, these singular solutions are likely to be unphysical as they cannot be constructed from localized branes which are brought together from a finite separation. 
  We discuss theories containing higher-order forms in various dimensions. We explain how Chern--Simons-type theories of forms can be defined from TQFTs in one less dimension. We also exhibit new TQFTs with interacting Yang--Mills fields and higher--order forms. They are obtained by the dimensional reduction of TQFTs whose gauge functions are free self-duality equations. Interactions are due to the gauging of global internal symmetries after dimensional reduction. We list possible symmetries and give a brief discussion on the possible relation of such systems to interacting field theories. 
  Confinement and Screening are investigated in SUSY gauge theories, realized by an M5 brane configuration, extending an approach applied previously to N=1 SYM theory, to other models. The electric flux tubes are identified as M2 branes ending on the M5 branes and the conserved charge they carry is identified as a topological property. The group of charges carried by the flux tubes is calculated and the results agree in all cases considered with the field theoretical expectations. In particular, whenever the dynamical matter is expected to screen the confining force, this is reproduced correctly in the M theory realization. 
  We consider the N_f-flavour Schwinger Model on a thermal cylinder of circumference $\beta=1/T$ and of finite spatial length $L$. On the boundaries $x^1=0$ and $x^1=L$ the fields are subject to an element of a one-dimensional class of bag-inspired boundary conditions which depend on a real parameter $\theta$ and break the axial flavour symmetry. For the cases $N_f=1$ and $N_f=2$ all integrals can be performed analytically. While general theorems do not allow for a nonzero critical temperature, the model is found to exhibit a quasi-phase-structure: For finite $L$ the condensate - seen as a function of $\log(T)$ - stays almost constant up to a certain temperature (which depends on $L$), where it shows a sharp crossover to a value which is exponentially close to zero. In the limit $L \to \infty$ the known behaviour for the one-flavour Schwinger model is reproduced. In case of two flavours direct pictorial evidence is given that the theory undergoes a phase-transition at $T_c=0$. The latter is confirmed - as predicted by Smilga and Verbaarschot - to be of second order but for the critical exponent $\delta$ the numerical value is found to be 2 which is at variance with their bosonization-rule based prediction $\delta=3$. 
  The product of two N=8 supersingletons yields an infinite tower of massless states of higher spin in four dimensional anti de Sitter space. All the states with spin s > 1/2 correspond to generators of Vasiliev's super higher spin algebra shs^E (8|4) which contains the D=4, N=8 anti de Sitter superalgebra OSp(8|4). Gauging the higher spin algebra and introducing a matter multiplet in a quasi-adjoint representation leads to a consistent and fully nonlinear equations of motion as shown sometime ago by Vasiliev. We show the embedding of the N=8 AdS supergravity equations of motion in the full system at the linearized level and discuss the implications for the embedding of the interacting theory. We furthermore speculate that the boundary N=8 singleton field theory yields the dynamics of the N=8 AdS supergravity in the bulk, including all higher spin massless fields, in an unbroken phase of M-theory. 
  We use the Matrix formalism to investigate what happens to strings above the Hagedorn temperaure. We show that it is not a limiting temperature but a temperature at which the continuum string picture breaks down. We study a collection of $N$ D-0-branes arranged to form a string having $N$ units of light cone momentum. We find that at high temperatures the favoured phase is one where the string world sheet has disappeared and the low energy degrees of freedom consists of $N^2$ massless particles (``gluons''). The nature of the transition is very similar to the deconfinement transition in large-N Yang Mills theories. 
  We formulate classical actions for N=1 supergravity in D=(1,3) as a gauge theory of OSp(1|4). One may choose the action such that it does not include a cosmological term. 
  We find self-dual vortex solutions in a Maxwell-Chern-Simons model with anomalous magnetic moment. From a recently developed N=2-supersymmetric extension, we obtain the proper Bogomol'nyi equations together with a Higgs potential allowing both topological and non-topological phases in the theory. 
  In this paper we explore some of the features of large N supersymmetric and nonsupersymmetric gauge theories using Maldacena's duality conjectures. We shall show that the resulting strong coupling behavior of the gauge theories is consistent with our qualitative expectations of these theories. Some of these consistency checks are highly nontrivial and give additional evidence for the validity of the proposed dualities. 
  We construct transformations that decouple fermionic fields in interaction with a gauge field, in the path integral representation of the generating functional. Those transformations express the original fermionic fields in terms of non-interacting ones, through non-local functionals depending on the gauge field. This procedure, holding true in any number of spacetime dimensions both in the Abelian and non-Abelian cases, is then applied to the path integral bosonization of the Thirring model in 3 dimensions. Knowledge of the decoupling transformations allows us, contrarily to previous bosonizations, to obtain the bosonization with an explicit expression of the fermion fields in terms of bosonic ones and free fermionic fields. We also explain the relation between our technique, in the two dimensional case, and the usual decoupling in 2 dimensions. 
  We discuss a D3-D7 system in type IIB string theory. The near-horizon geometry is described by AdS^5 x X^5 where X^5 is a U(1) bundle over a Kahler-Einstein complex surface S with positive first Chern class c_1>0. The surface S can either be P^1 x P^1, P^2 or P_{n_1,...,n_k}, a blow up of P^2 at k points with 3\leq k\leq 8. The P^2 corresponds to the maximally supersymmetric AdS^5 x S^5 vacuum while the other cases lead to vacua with less supersymmetries. In the F-theory context they can be viewed as compactifications on elliptically fibered almost Fano 3-folds. 
  We generalize the recently proposed noncommutative ADHM construction to the case of $\Gamma$-equivariant instantons over $\R^4$, with $\Gamma$ a Kleinian group. We show that a certain form of the inhomogeneous ADHM equations describes instantons over a noncommutative deformation of the Kleinian orbifold $\C^2/\Gamma$ and we discuss the relation of this with Nakajima's description of instantons over ALE spaces. In particular, we obtain a noncommutative interpretation of the minimal resolution of Kleinian singularities. 
  In quasi-realistic string models that contain an anomalous U(1) the non-zero Fayet-Iliopoulos term triggers the shifting of the original vacuum to a new one along some flat direction, so that SUSY is preserved but the gauge group is partially broken. The phenomenological study of these models thus requires as a first step the mapping of the space of flat directions. We investigate F- and D-flat directions in several three-generation SU(3)_C x SU(2)_L x U(1)_Y free-fermionic string models and discuss the typical scenarios that generically arise. When they exist, we systematically construct the flat directions that preserve hypercharge, only break Abelian group factors, and can be proven to remain F-flat to all orders in the non-renormalizable superpotential. 
  We examine the decoupling properties of the N=2 SQCD vacua when the adjoint mass term is turned on and then the N=1 limit is taken. The BPS domain wall tension in N=1 MQCD and SQCD is also examined. The correspondence of the MQCD integrals with the superpotential and the gaugino condensate is shown. 
  We present our recent results on the odderon intercept in perturbative QCD, obtained through the solution of the Baxter equation and investigation of the spectrum of the relevant constant of motion. 
  Via supergravity, we argue that the infinite Lorentz boost along the M theory circle a la Seiberg toward the DLCQ M theory compactified on a p-torus (p<5) implies the holographic description of the microscopic theory. This argument lets us identify the background geometries of DLCQ $M$ theory on a p-torus; for p=0 (p=1), the background geometry turns out to be eleven-dimensional (ten-dimensional) flat Minkowski space-time, respectively. Holography for these cases results from the localization of the light-cone momentum. For p = 2,3,4, the background geometries are the tensor products of an Anti de Sitter space and a sphere, which, according to the AdS/CFT correspondence, have the holographic conformal field theory description. These holographic descriptions are compatible to the microscopic theory of Seiberg based on $\tilde{M}$ theory on a spatial circle with the rescaled Planck length, giving an understanding of the validity of the AdS/CFT correspondence. 
  We study the gauge-invariant gaussian ansatz for the vacuum wave functional and show that it potentially possesses many desirable features of the Yang--Mills theory, like asymptotic freedom, mass generation through the transmutation of dimensions and a linear potential between static quarks. We point out that these (and other) features can be studied in a systematic way by combining perturbative and 1/n expansions. Contrary to the euclidean approach, confinement can be easily formulated and easily built in, if not derived, in the variational Schroedinger approach. 
  We explore the Dirac equation in external electromagnetic and torsion fields. Motivated by the previous study of quantum field theory in an external torsion field, we include a nonminimal interaction of the spinor field with torsion. As a consequence, the torsion axial vector and the electromagnetic potential enter the action in a similar form. The existence of an extra local symmetry is emphasized and the Foldy-Wouthuysen transformation is performed to an accuracy of next to the leading order. We also discuss the motion of a classical test particle in a constant torsion field. 
  Brane Box Models of intersecting NS and D5 branes are mapped to D3 branes at C^3/Gamma orbifold singularities and vise versa, in a setup which gives rise to N=1 supersymmetric gauge theories in four dimensions. The Brane Box Models are constructed on a two-torus. The map is interpreted as T-duality along the two directions of the torus. Some Brane Box Models contain NS fivebranes winding around (p,q) cycles in the torus, and our method provides the geometric T-dual to such objects. An amusing aspect of the mapping is that T-dual configurations are calculated using D=4 N=1 field theory data. The mapping to the singularity picture allows the geometrical interpretation of all the marginal couplings in finite field theories. This identification is further confirmed using the AdS/CFT correspondence for orbifold theories. The AdS massless fields coupling to the marginal operators in the boundary appear as stringy twisted sectors of S^5/Gamma. The mapping for theories which are non-finite requires the introduction of fractional D3 branes in the singularity picture. 
  We consider a minimal scalar in the presence of a three-brane in ten dimensions. The linearized equation of motion, which is just the wave equation in the three-brane metric, can be solved in terms of associated Mathieu functions. An exact expression for the reflection and absorption probabilities can be obtained in terms of the characteristic exponent of Mathieu's equation. We describe an algorithm for obtaining the low-energy behavior as a series expansion, and discuss the implications for the world-volume theory of D3-branes. 
  We derive some consistency conditions for fivebrane in M theory on R^5/Z_2 orbifold from the quantization law for the antisymmetric tensor field. We construct consistent fivebrane configurations in R^5/Z_2 type orbifold that exhibit the correct low energy dynamics of N=2 SQCD in four dimensions with symplectic and orthogonal gauge groups. This leads us to propose the M theory realization of orientifold four-planes of various types, and we study their properties by applying the consistency conditions. 
  We study the Type IIA limit of the M theory fivebrane configuration corresponding to N=1 supersymmetric QCD with massless quarks. We identify the effective gauge coupling constant that fits with Novikov-Shifman-Veinshtein-Zakharov exact beta function. We find two different Type IIA limits that correspond to the electric and magnetic descriptions of SQCD, as observed in the massive case by Schmaltz and Sundrum. The analysis is extended to the case of symplectic and orthogonal gauge groups. In any of the cases considered in this paper, the electric and magnetic configurations are smoothly interpolated via $M$ theory. This is in sharp contrast with the proposed derivation of N=1 duality within the weakly coupled Type IIA string theory where a singularity is inevitable unless one turns on a parameter that takes the theory away from an interesting point. 
  We study the dimensionality manifested in the AdS/CFT correspondence. We show that the dimensionality as expressed by the high temperature behavior of a system has a holographic nature also at the quantum level. The emergence of the AdS black hole as a master field at high temperature leads to the screening of the extra dimensions in its excluded volume. 
  The Wess-Zumino term is constructed for supersymmetric QCD with two colors and flavors, and is shown to correctly reproduce the anomalous Ward identities. Supersymmetric QCD is also shown not to have topologically stable skyrmion solutions because of baryon flat directions, which allow them to unwind. The generalization of these results to other supersymmetric theories with quantum modified constraints is discussed. 
  We use the AdS/CFT correspondence to calculate CFT correlation functions of vector and spinor fields. The connection between the AdS and boundary fields is properly treated via a Dirichlet boundary value problem. 
  We show that a class of four-dimensional rotating black holes allow five-dimensional embeddings as black rotating strings. Their near-horizon geometry factorizes locally as a product of the three-dimensional anti-deSitter space-time and a two-dimensional sphere (AdS_3 x S^2), with angular momentum encoded in the global space-time structure. Following the observation that the isometries on the AdS_3 space induce a two-dimensional (super)conformal field theory on the boundary, we reproduce the microscopic entropy with the correct dependence on the black hole angular momentum. 
  Two-dimensional N=1,2 supersymmetric chiral models and their dual extensions are introduced and canonically quantized. Working within a superspace formalism, the non-manifest invariance under 2D-superPoincare' transformations is proven. The N=1,2 superVirasoro algebras are recovered as current algebras. The non-anomalous quantum invariances under 1D-superdiffeomorphisms (for chiral models) and N=1,2 superconformal transformations (for dual models) are shown to be a consequence of an N=1,2 super-Coulomb gas representation. 
  The 12d supersymmetry algebra is considered, and classification of BPS states for some canonical form of second-rank central charge is given. It is shown, that possible fractions of survived supersymmetry can be 1/16, 1/8, 3/16, 1/4, 5/16 and 1/2, the values 3/8, 7/16 cannot be achieved in this way. The consideration of a special case of non-zero sixth-rank tensor charge also is included. 
  Quantum integrable systems generalizing Calogero-Sutherland systems were introduced by Olshanetsky and Perelomov (1977). Recently, it was proved that for systems with trigonometric potential, the series in the product of two wave functions is a deformation of the Clebsch-Gordan series. This yields recursion relations for the wave functions of those systems. In this note, this approach is used to compute the explicit expressions for the three-body Calogero-Sutherland wave functions, which are the Jack polynomials. We conjecture that similar results are also valid for the more general two-parameters deformation introduced by Macdonald. 
  We calculate one-loop quantum energies in a renormalizable self-interacting theory in one spatial dimension by summing the zero-point energies of small oscillations around a classical field configuration, which need not be a solution of the classical field equations. We unambiguously implement standard perturbative renormalization using phase shifts and the Born approximation. We illustrate our method by calculating the quantum energy of a soliton/antisoliton pair as a function of their separation. This energy includes an imaginary part that gives a quantum decay rate and is associated with a level crossing in the solutions to the classical field equation in the presence of the source that maintains the soliton/antisoliton pair. 
  We derive simple general expressions for the explicit Killing spinors on the n-sphere, for arbitrary n. Using these results we also construct the Killing spinors on various AdS x Sphere supergravity backgrounds, including AdS_5 x S^5$, AdS_4 x S^7 and AdS_7 x S^4. In addition, we extend previous results to obtain the Killing spinors on the hyperbolic spaces H^n. 
  We develop a general setting for N=2 rigid supersymmetric field theories with gauged central charge in harmonic superspace. We consider those N=2 multiplets which have a finite number of off-shell components and exist off shell owing to a non-trivial central charge. This class includes, in particular, the hypermultiplet with central charge and various versions of the vector-tensor multiplet. For such theories we present a manifestly supersymmetric universal action. Chern-Simons couplings to an external N=2 super Yang-Mills multiplet are given, in harmonic superspace, for both the linear and nonlinear vector-tensor multiplets with gauged central charge. We show how to deduce the linear version of the vector-tensor multiplet from six dimensions. 
  We prove Abelian magnetic monopole dominance in the string tension of QCD. Abelian and monopole dominance in low energy physics of QCD has been confirmed for various quantities by recent Monte Carlo simulations of lattice gauge theory. In order to prove this dominance, we use the reformulation of continuum Yang-Mills theory in the maximal Abelian gauge as a deformation of a topological field theory of magnetic monopoles, which was proposed in the previous article by the author. This reformulation provides an efficient way for incorporating the magnetic monopole configuration as a topological non-trivial configuration in the functional integral. We derive a version of the non-Abelian Stokes theorem and use it to estimate the expectation value of the Wilson loop. This clearly exhibits the role played by the magnetic monopole as an origin of the Berry phase in the calculation of the Wilson loop in the manifestly gauge invariant manner. We show that the string tension derived from the diagonal (abelian) Wilson loop in the topological field theory (studied in the previous article) converges to that of the full non-Abelian Wilson loop in the limit of large Wilson loop. Therefore, within the above reformulation of QCD, this result (together with the previous result) completes the proof of quark confinement in QCD based on the criterion of the area law of the full non-Abelian Wilson loop. 
  The large N limit of extremal non-supersymmetric Type-I five-dimensional string black holes is studied from the point of view of D-branes.  We find that the agreement between the D-brane and the black-hole picture is due to an asymptotic restoration of supersymmetry in the large $N$ limit in which both pictures are compared.  In that limit Type-I string perturbation theory is effectively embedded into a Type-IIB perturbation theory with unbroken supersymmetric charges whose presence guarantees the non-renormalization of mass and entropy as the effective couplings are increased. In this vein, we also study the near-horizon geometry of the Type-I black hole using D5-brane probes to find that the low energy effective action for the probe is identical to the corresponding one in the auxiliary Type-IIB theory in the large N limit. 
  Massless $QCD_2$ is dominated by classical configurations in the large $N_f$ limit. We use this observation to study the theory by finding solutions to equations of motion, which are the non-Abelian generalization of the Schwinger equation. We find that the spectrum consists of massive mesons with $M^2={e^2 N_f\over 2\pi}$ which correspond to Abelian solutions. We generalize previously discovered non-Abelian solutions and discuss their interpretation. We prove a no-go theorem ruling out the existence of soliton solutions. Thus the semi-classical approximation shows no baryons in the case of massless quarks, a result derived before in the strong-coupling limit only. 
  The free energy of the maximally supersymmetric SU(N) gauge theory at temperature T is expected to scale, in the large N limit, as N^2 T^4 times a function of the 't Hooft coupling, f(g_{YM}^2 N). In the strong coupling limit the free energy has been deduced from the near-extremal 3-brane geometry, and its normalization has turned out to be 3/4 times that found in the weak coupling limit. In this paper we calculate the leading correction to this result in inverse powers of the coupling, which originates from the R^4 terms in the tree level effective action of type IIB string theory. The correction to 3/4 is positive and of order (g_{YM}^2 N)^{-3/2}. Thus, f(g_{YM}^2 N) increases as the 't Hooft coupling is decreased, in accordance with the expectation that it should be approaching 1 in the weak coupling limit. We also discuss similar corrections for other conformal theories describing coincident branes. In particular, we suggest that the coupling-independence of the near extremal entropy for D1-branes bound to D5-branes is related to the vanishing of the Weyl tensor of AdS_3\times S^3. 
  Utilizing the idea of extra large dimensions, it has been suggested that the gauge and gravity couplings unification can happen at a scale as low as 1 TeV. In this paper, we explore this phenomenological possibility within string theory. In particular, we discuss how the proton decay bound can be satisfied in Type I string theory. The string picture also suggests different scenarios of gauge and gravitational couplings unification. The various scenarios are explicitly illustrated with a specific 4-dimensional N=1 supersymmetric chiral Type I string model with Pati-Salam-like gauge symmetry. We point out certain features that should be generic in other Type I strings. 
  We study brane configurations in the presence of orientifold six-planes. After deriving the curves for N=2 supersymmetric SU(Nc) gauge theories with one flavor in the symmetric or antisymmetric representation and Nf fundamental flavors, we rotate the brane configuration, reducing the supersymmetry to N=1. For the case of an antisymmetric flavor and less than two fundamental flavors, nonperturbative effects lead to a brane configuration that is topologically a torus. Using the description of the orientifold six-planes as Dn singularities we discuss the Higgs branches for N=2 brane configurations with Sp/SO gauge groups and the related N=1 theories with tensor representations. 
  The phenomenon of the magnetic catalysis of dynamical symmetry breaking is based on the dimensional reduction $D\to D-2$ in the dynamics of fermion pairing in a magnetic field. We discuss similarities between this phenomenon and the Aharonov-Bohm effect. This leads to the interpretation of the dynamics of the (1+1)-dimensional Gross-Neveu model with a non-integer number of fermion colors as a quantum field theoretical analogue of the Aharonov-Bohm dynamics. 
  A graded generalization of the Z_k parafermionic current osp(1|2)/U(1) coset conformal field theory. The structure of the parafermionic highest-weight modules is analyzed and the dimensions of the fields of the theory are determined. A free field realization of the graded parafermionic system is obtained and the structure constants of the current algebra are found. Although the theory is not unitary, it presents good reducibility properties. 
  The most general vortex solution of the Liouville equation (which arises in non-relativistic Chern-Simons theory) is associated with rational functions, $f(z)=P(z)/Q(z)$ where $P(z)$ and $Q(z)$ are both polynomials, $\deg P<\deg Q\equiv N$. This allows us to prove that the solution depends on $4N$ parameters without the use of an index theorem, as well as the flux quantization~: $\Phi=-4N\pi(sign \kappa)$. 
  By taking the interacting spinor-scalar theory on the $AdS_{d+1}$ space we calculate the boundary CFT correlation functions using AdS/CFT correspondence. 
  It is observed that the magnetic charges of classical monopole solutions in Yang-Mills-Higgs theory with non-abelian unbroken gauge group $H$ are in one-to-one correspondence with coherent states of a dual or magnetic group $\tilde H$. In the spirit of the Goddard-Nuyts-Olive conjecture this observation is interpreted as evidence for a hidden magnetic symmetry of Yang-Mills theory. SU(3) Yang-Mills-Higgs theory with unbroken gauge group U(2) is studied in detail. The action of the magnetic group on semi-classical states is given explicitly. Investigations of dyonic excitations show that electric and magnetic symmetry are never manifest at the same time: Non-abelian magnetic charge obstructs the realisation of electric symmetry and vice-versa. On the basis of this fact the charge sectors in the theory are classified and their fusion rules are discussed. Non-abelian electric-magnetic duality is formulated as a map between charge sectors. Coherent states obey particularly simple fusion rules, and in the set of coherent states S-duality can be formulated as an SL(2,Z)-mapping between sectors which leaves the fusion rules invariant. 
  The open inflation model recently proposed by Hawking and Turok is investigated in scalar-tensor gravity context. If the dilaton-like field has no potential, the instanton of our model is singular but has a finite action. The Gibbons-Hawking surface term vanishes and hence, can not be used to make $\Omega_0$ nonzero. To obtain a successful open inflation one should introduce other matter fields or a potential for the dilaton-like fields. 
  Anti-de Sitter supergravity models are considered in three dimensions. Precise asymptotic conditions involving a chiral projection are given on the Rarita-Schwinger fields. Together with the known boundary conditions on the bosonic fields, these ensure that the asymptotic symmetry algebra is the superconformal algebra. The classical central charge is computed and found to be equal to the one of pure gravity. It is also indicated that the asymptotic degrees of freedom are described by 2D "induced supergravity" and that the boundary conditions "transmute" the non-vanishing components of the WZW supercurrent into the supercharges. 
  The magnetic monopole condensate is calculated in the dual Monopole Nambu-Jona-Lasinio model with dual Dirac strings suggested in Refs.[1,2] as a functional of the dual Dirac string shape. The calculation is carried out in the tree approximation in the scalar monopole-antimonopole collective excitation field. The integration over quantum fluctuations of the dual-vector monopole-antimonopole collective excitation field around the Abrikosov flux line and string shape fluctuations are performed explicitly. We claim that there are important contributions of quantum and string shape fluctuations to the magnetic monopole condensate. 
  The four-form field recently considered by Hawking and Turok couples naturally to a charged membrane, across which the effective cosmological constant has a discontinuity. We present instantons for the creation of an open inflationary universe surrounded by a membrane. They can also be used to describe the nucleation of a membrane on a pre-existing inflationary background. This process typically decreases the value of the effective cosmological constant and may lead to a novel scenario of eternal inflation. Moreover, by coupling the inflaton field to the membrane, the troublesome singularities which arise in the Hawking-Turok model can be eliminated. 
  We present the detailed derivation of the charge one periodic instantons - or calorons - with non-trivial holonomy for SU(2). We use a suitable combination of the Nahm transformation and ADHM techniques. Our results rely on our ability to compute explicitly the relevant Green's function in terms of which the solution can be conveniently expressed. We also discuss the properties of the moduli space, R^3 X S^1 X Taub-NUT/Z_2 and its metric, relating the holonomy to the Taub-NUT mass parameter. We comment on the monopole constituent description of these calorons, how to retrieve topological charge in the context of abelian projection and possible applications to QCD. 
  The nature of the interaction of a soliton with an attractive well is elucidated using a model of two interacting point particles. The model explains the existence of trapped states at positive kinetic energy, as well as reflection by an attractive impurity. The transition from a trapped soliton state to a bound state is studied. Bound states of the soliton in a well are also found. 
  A coincident D-brane - anti-D-brane pair has a tachyonic mode. We present an argument showing that at the classical minimum of the tachyonic potential the negative energy density associated with the potential exactly cancels the sum of the tension of the brane and the anti-brane, thereby giving a configuration of zero energy density and restoring space-time supersymmetry. 
  We investigate the details of the bulk-boundary correspondence in Lorentzian signature anti-de Sitter space. Operators in the boundary theory couple to sources identified with the boundary values of non-normalizable bulk modes. Such modes do not fluctuate and provide classical backgrounds on which bulk excitations propagate. Normalizable modes in the bulk arise as a set of saddlepoints of the action for a fixed boundary condition. They fluctuate and describe the Hilbert space of physical states. We provide an explicit, complete set of both types of modes for free scalar fields in global and Poincar\'e coordinates. For $\ads{3}$, the normalizable and non-normalizable modes originate in the possible representations of the isometry group $\SL_L\times\SL_R$ for a field of given mass. We discuss the group properties of mode solutions in both global and Poincar\'e coordinates and their relation to different expansions of operators on the cylinder and on the plane. Finally, we discuss the extent to which the boundary theory is a useful description of the bulk spacetime. 
  We study N=2 super Yang-Mills theory with gauge group SU(N) from the point of view of the Whitham hierarchy. We develop a new recursive method to compute the whole instanton expansion of the prepotential using the theta function associated to the root lattice of the group. Explicit results for the one and two-instanton corrections in SU(N) are presented. Interpreting the slow times of the hierarchy as additional couplings, we show how they can be promoted to spurion superfields that softly break N=2 supersymmetry down to N=0. This provides a family of nonsupersymmetric deformations of the theory, associated to higher Casimir operators of the gauge group. The SU(3) theory is analyzed in some detail. 
  It has been known for some time that the SL(2,R) WZWN model reduces to Liouville theory. Here we give a direct and physical derivation of this result based on the classical string equations of motion and the proper string size. This allows us to extract precisely the physical effects of the metric and antisymmetric tensor, respectively, on the {\it exact} string dynamics in the SL(2,R) background. The general solution to the proper string size is also found. We show that the antisymmetric tensor (corresponding to conformal invariance) generally gives rise to repulsion, and it precisely cancels the dominant attractive term arising from the metric.     Both the sinh-Gordon and the cosh-Gordon sectors of the string dynamics in non-conformally invariant AdS spacetime reduce here to the Liouville equation (with different signs of the potential), while the original Liouville sector reduces to the free wave equation. Only the very large classical string size is affected by the torsion. Medium and small size string behaviours are unchanged.     We also find illustrative classes of string solutions in the SL(2,R) background: dynamical closed as well as stationary open spiralling strings, for which the effect of torsion is somewhat like the effect of rotation in the metric. Similarly, the string solutions in the 2+1 BH-AdS background with torsion and angular momentum are fully analyzed. 
  In the given work we study an interaction of second massive state of an open boson string with the constant electromagnetic field. This state contains massive fields with spins 3 and 1. Using the method of an open string BRST quantization, we receive gauge-invariant lagrangian, describing the electromagnetic interaction of these fields. From the explicit view of transformations and lagrangian it follows that the presence of external constant e/m field leads to the mixing of the given level states. Most likely that the presence of the external field will lead to the mixing of the states at other mass string levels as well. 
  We consider interaction of a probe monopole-antimonopole pair in the vacuum of the Abelian Higgs model. For simplicity, the mass of the Higgs particle is assumed to be much larger than the mass of the photon (London limit). In case of a massive photon the straightforward application of the Zwanziger formalism to accommodate both magnetic and electric charges is known to result in gauge dependence and infrared instabilities. We argue that the use of the string representation of the Abelian Higgs model allows to ameliorate both difficulties. In particular, we arrive at a well defined expression for the potential energy of the static monopole sources. We argue that the monopole-antimonopole interaction cannot be described by a massive photon exchange with a definite propagator having simple analytical properties. 
  We complete the classification of supersymmetric configurations of two M5-branes, started by Ohta and Townsend. The novel configurations not considered before are those in which the two branes are moving relative to one another. These configurations are obtained by starting with two coincident branes and Lorentz-transforming one of them while preserving some supersymmetry. We completely classify the supersymmetric configurations involving two M5-branes, and interpret them group-theoretically. We also present some partial results on supersymmetric configurations involving an arbitrary number of M5-branes. We show that these configurations correspond to Cayley planes in eight-dimensions which are null-rotated relative to each other in the remaining (2+1) dimensions. The generic configuration preserves 1/32 of the supersymmetry, but other fractions (up to 1/4) are possible by restricting the planes to certain subsets of the Cayley grassmannian. We discuss some examples with fractions 1/32, 1/16, 3/32, 1/8, 1/4 involving an arbitrary number of branes, as well as their associated geometries. 
  The best candidate for a fundamental unified theory of all physical phenomena is no longer ten-dimensional superstring theory but rather eleven-dimensional {\it M-theory}. In the words of Fields medalist Edward Witten, ``M stands for `Magical', `Mystery' or `Membrane', according to taste''. New evidence in favor of this theory is appearing daily on the internet and represents the most exciting development in the subject since 1984 when the superstring revolution first burst on the scene. (Talk delivered at the Abdus Salam Memorial Meeting, ICTP, Trieste, November 1997.) 
  We derive the Kac-Moody algebra and Virasoro algebra in Chern-Simons theory with boundary by using the symplectic reduction method and the Noether procedures. 
  We evaluate characters of irreducible representations of the N=2 supersymmetric extension of the Virasoro algebra. We do so by deriving the BGG-resolution of the admissible N=2 representations and also a new 3,5,7...-resolution in terms of twisted massive Verma modules. We analyse how the characters behave under the automorphisms of the algebra, whose most significant part is the spectral flow transformations. The possibility to express the characters in terms of theta functions is determined by their behaviour under the spectral flow. We also derive the identity expressing every $\hat{sl}(2)$ character as a linear combination of spectral-flow transformed N=2 characters; this identity involves a finite number of N=2 characters in the case of unitary representations. Conversely, we find an integral representation for the admissible N=2 characters as contour integrals of admissible $\hat{sl}(2)$ characters. 
  U-duality p-branes in toroidally compactified type II superstring theories in space-time dimensions $10 > D \ge 4$ can be constructed explicitly based on the conjectured U-duality symmetries and the corresponding known single-charge super p-brane configurations. As concrete examples, we first construct explicitly the $SL(3, Z)$ superstrings and $SL(3, Z) \times SL(2, Z)$ 0-branes as well as their corresponding magnetic duals in $D = 8$. For the $SL(3, Z)$ superstrings (3-branes), each of them is characterized by a triplet of integers corresponding to the electric-like (magnetic-like) charges associated with the three 2-form gauge potentials present in the theory. For the $SL(3, Z)\times SL(2,Z)$ 0-branes (4-branes), each of them is labelled by a pair of triplets of integers corresponding to the electric-like (magnetic-like) charges associated with the two sets of three 1-form gauge potentials. The string (3-brane) tension and central charge are shown to be given by $SL(3, Z)$ invariant expressions. It is argued that when any two of the three integers in the integral triplet are relatively prime to each other, the corresponding string (3-brane) is stable and does not decay into multiple strings (3-branes) by a `tension gap' or `charge gap' equation. Similar results hold also for the 0-branes (4-branes). Alongwith the $SL(2, Z)$ dyonic membranes of Izquierdo et. al., these examples provide a further support for the conjectured $SL(3, Z) \times SL(2, Z)$ U-duality symmetry in this theory. Moreover, the study of these examples along with the previous ones provides us a recipe for constructing the U-duality p-branes of various supergravity theories in diverse dimensions. Constructions for these U-duality p-branes are also given. 
  I review recent results in three topics of the M-world: (i) Scales. (ii) New dark matter candidates. (iii) Cosmological solutions from p-branes. The three topics are discussed in the framework of Ho\v{r}ava-Witten compactifications. Part (iii) includes comments on cosmological solutions in M-theory describing nucleation of universes through instanton effects and expansions toward asymptotically flat or anti-de-Sitter spaces. 
  The dual relationship between the supergravity in the anti-de Sitter(AdS) space and the superconformal field theory is discussed in the simplest form. We show that a topological Ward identity holds in the three dimensional Chern-Simons gravity. In this simple case the proposed dual relationship can be understood as the topological Ward identity. Extensions to the supersymmetric theories and higher dimensional ones are also briefly discussed. 
  An explicit proof of the existence of nontrivial vacua in the pure supersymmetric Yang-Mills theories with higher orthogonal SO(N), N>=7 or the G_2 gauge group defined on a 3-torus with periodic boundary conditions is given. Extra vacuum states are separated by an energy barrier from the perturbative vacuum A_i=0 and its gauge copies. 
  We discuss the role of the multiplicative anomaly for a complex scalar field at finite temperature and density. It is argued that physical considerations must be applied to determine which of the many possible expressions for the effective action obtained by the functional integral method is correct. This is done by first studying the non-relativistic field where the thermodynamic potential is well-known. The relativistic case is also considered. We emphasize that the role of the multiplicative anomaly is not to lead to new physics, but rather to preserve the equality among the various expressions for the effective action. 
  We provide detailed arguments on how to derive properties of generalized form factors, originally proposed by one of the authors (M.K.) and Weisz twenty years ago, solely based on the assumption of "minimal analyticity" and the validity of the LSZ reduction formalism. These properties constitute consistency equations which allow the explicit evaluation of the n-particle form factors once the scattering matrix is known. The equations give rise to a matrix Riemann-Hilbert problem. Exploiting the "off-shell" Bethe ansatz we propose a general formula for form factors for an odd number of particles. For the Sine-Gordon model alias the massive Thirring model we exemplify the general solution for several operators. We carry out a consistency check for the solution of the three particle form factor against the Thirring model perturbation theory and thus confirm the general formalism. 
  In this paper we generalize the fermionic approach to the KP hierarchy sudgested in the papers of Kyoto school 1981-1984 (Sato,Jimbo, Miwa...). The main idea is that the components of the intertwiningoperators are in some sense a generalization of free fermions for $gl_{\infty}$. We formulate in terms of intertwining operators the integrable hierarchies related to Kac-Moody Lie algebra symmetries. We write down explicitly the bosonization of these operators for different choices of Heisenberg subalgebras. These different realizations lead to different hierarchies of soliton equations. For example, for $sl_N$-symmetries we get hierarchies obtained as $(n_1,..., n_s)$-reduction from $s$-component KP $(n_1+...+n_s = N)$ introduced by V.Kac and J.Van de Leur. 
  The invariant integration method for Chern-Simons theory for gauge group SU(2) and manifold \Gamma\H^3 is verified in the semiclassical approximation. The semiclassical limit for the partition function associated with a connected sum of hyperbolic 3-manifolds is presented. We discuss briefly L^2 - analytical and topological torsions of a manifold with boundary. 
  We consider SU(N) QCD_{1+1} coupled to massless adjoint Majorana fermions, where N is finite but arbitrary. We examine the spectrum for various values of N, paying particular attention to the formation of multi-particle states, which were recently identified by Gross, Hashimoto and Klebanov in the N -> infinity limit of the theory. It is believed that in the limit of vanishing fermion mass, there is a transition from confinement to screening in which string-like states made out of adjoint fermion bits dissociate into stable constituent ``single particles''. In this work, we provide numerical evidence that such a transition into stable constituent particles occurs not only at large N, but for any finite value of N. In addition, we discuss certain issues concerning the ``topological'' properties exhibited by the DLCQ spectrum. 
  We describe type IIB compactifications with varying coupling constant in d=6,7,8,9 dimensions, where part of the ten-dimensional SL(2,Z) symmetry is broken by a background with Gamma_0(n) or Gamma(n) monodromy for n=2,3,4. This extends the known class of F-theory vacua to theories which are dual to heterotic compactifications with reduced rank. On compactifying on a further torus, we obtain a description of the heterotic moduli space of G bundles over elliptically fibered manifolds without vector structure in terms of complex geometries. 
  We present the last missing details of our algorithm for the classification of reflexive polyhedra in arbitrary dimensions. We also present the results of an application of this algorithm to the case of three dimensional reflexive polyhedra. We get 4319 such polyhedra that give rise to K3 surfaces embedded in toric varieties. 16 of these contain all others as subpolyhedra. The 4319 polyhedra form a single connected web if we define two polyhedra to be connected if one of them contains the other. 
  The non-commutative algebra which defines the theory of zero-branes on $T^4/Z_2$ allows a unified description of moduli spaces associated with zero-branes, two-branes and four-branes on the orbifold space. Bundles on a dual space $\hat T^4/Z_2$ play an important role in this description. We discuss these moduli spaces in the context of dualities of K3 compactifications, and in terms of properties of instantons on $T^4$. Zero-branes on the degenerate limits of the compact orbifold lead to fixed points with six-dimensional scale but not conformal invariance. We identify some of these in terms of the ADS dual of the $(0,2)$ theory at large $N$, giving evidence for an interesting picture of "where the branes live" in ADS. 
  In our previously published papers, it was argued that a massive non-Abelian gauge field theory in which all gauge fields have the same mass can well be set up on the gauge-invariance principle. The quantization of the fields was performed by different methods. In this paper, It is proved that the quantum theory is invariant with respect to a kind of BRST-transformations. From the BRST-invariance of the theory, the Ward-Takahashi identities satisfied by the generating functionals of full Green's functions, connected Green's functions and proper vertex functions are successively derived. As an application of the above Ward-Takahashi identity, the Ward-Takahashi identity obeyed by the massive gauge boson propagator is derived and the renormalization of the propagator is discussed. Furthermore, based on the Ward-Takahashi identity, it is exactly proved that the S-matrix elements given by the quantum theory are gauge-independent and hence unitary. 
  ~It is shown that the quantum massive non-Abelian field theory established in the former papers is renormalizable. This conclusion is achieved with the aid of the Ward-Takahashi identities satisfied by the generating functionals which were derived in the preceding paper based on the BRST-symmetry of the theory. By the use of the Ward-Takahashi identity, it is proved that the divergences occurring in the perturbative calculations for the massive gauge field theory can be eliminated by introducing a finite number of counterterms in the effective action. As a result of the proof, it is found that the renormalization constants for the massive gauge field theory comply with the same Slavnov-Taylor identity as that for the massless gauge field theory. The latter identity is re-derived from the Ward-Takahashi identities satisfied by the gluon proper vertices and their renormalization. 
  To illustrate the unitarity of the massive gauge field theory described in the foregoing papers, we calculate the scattering amplitudes up to the fourth order of perturbation by the optical theorem and the Landau-Cutkosky rule. In the calculations, it is shown that for a given process, if all the diagrams are taken into account, the contributions arising from the unphysical intermediate states included in the longitudinal part of the gauge boson propagator and in the ghost particle propagator are completely cancelled out with each other in the S-matrix elements. Therefore, the unitarity of the S-matrix is perfectly ensured. 
  According to the conventional concept of the gauge field theory, the local gauge invariance excludes the possibility of giving a mass to the gauge boson without resorting to the Higgs mechanism because the Lagrangian constructed by adding a mass term to the Yang-Mills Lagrangian is not only gauge-non-invariant, but also unrenormalizable. On the contrary, we argue that the principle of gauge invariance actually allows a mass term to enter the Lagrangian if the Lorentz constraint condition is taken into account at the same time. The Lorentz condition, which implies vanishing of the unphysical longitudinal field, defines a gauge-invariant physical space for the massive gauge field. The quantum massive gauge field theory without Higgs mechanism may well be established by using a BRST-invariant action which is constructed by the Lagrange undetermined multiplier procedure of incorporating the Lorentz condition and another condition constraining the gauge group into the original massive Yang-Mills action. The quantum theory established in this way shows good renormalizability. 
  It is argued that the massive non-Abelian gauge field theory without involving Higgs bosons may be well established on the basis of gauge-invariance principle because the dynamics of the field is gauge-invariant in the physical space defined by the Lorentz constraint condition. The quantization of the field can readily be performed in the Hamiltonian path-integral formalism and leads to a quantum theory which shows good renormalizability and unitarity. 
  The exact quantum $S$-matrices and conserved charges are known for affine Toda field theories(ATFTs). In this note we report on a new type of bi-linear sum rules of conserved quantities derived from these exact $S$ matrices. They exist when there is a multiplicative identity among $S$-matrices of a particular ATFT. Our results are valid for simply laced as well as non-simply laced ATFTs. We also present a few explicit examples. 
  A quantum dynamical $\check{R}$-matrix with spectral parameter is constructed by fusion procedure. This spin-1 $\check{R}$-matrix is connected with Lie algebra $so(3)$ and does not satisfy the condition of translation invariance. 
  We derive consistent superfield constraints for the linear vector-tensor multiplet with gauged central charge. The central charge transformations and the action turn out to be nonpolynomial in the gauge field. 
  We describe a generalization of Yang--Mills topological field theory for Abelian two-forms in six dimensions. The connection of this theory by a twist to Poincar\'e supersymmetric theories is given. We also briefly consider interactions and the case of self-dual three-forms in eight dimensions. 
  We discuss the newly found exact instanton solutions at finite temperature with a non-trivial Polyakov loop at infinity. They can be described in terms of monopole constituents and we discuss in this context an old result due to Taubes how to make out of monopoles non-trivial topological charge configurations, with possible applications to abelian projection. 
  We study scattering of $D0$ branes in eleven dimensional supergravity (SUGRA) using the tree level four point amplitude. The range of validity of SUGRA allows us to make reliable calculations of relativistic brane scattering, inaccessible in string theory because of halo effects. We also compare the rate of annihilation of $D0-\bar{D0}$ branes to the rate of elastic $D0-\bar{D0}$ scattering and find that the former is always smaller than the latter. We also find a pole in $\sigma_{D0-\bar{D0}}$. We exploit the analogy with the positronium to argue that the brane anti-brane pairs form branium atoms in 3 spatial dimensions. We also derive a long range effective potential for interacting branes which explicitly depends on their polarizations. We compare two approaches to large impact parameter brane scattering: in GR, polarizations of the branes are kept constant during the interaction, while in QFT, one sums over all possible polarizations. We show that the GR and QFT approaches give the same answer. 
  Lagrangians for gauge fields and matter fields can be constructed from the infinite dimensional Kac-Moody algebra and group. A continuum regularization is used to obtain such generic lagrangians, which contain new nonlinear and asymmetric interactions not present in gauge theories based on compact Lie groups. This technique is applied to deriving the Yang-Mills and Chern-Simons lagrangians for the Kac-Moody case. The extension of this method to D=4, N=(1/2,0) supersymmetric Kac-Moody gauge fields is also made. 
  We study the 0+1 dimensional Chern-Simons theory at finite temperature within the framework of derivative expansion. We obtain various interesting relations, solve the theory within this framework and argue that the derivative expansion is not a suitable formalism for a study of the question of large gauge invariance. 
  I discuss two novel results in D=11 Supergravity. The first establishes, in two complementary ways, a no-go theorem that, in contrast to all D<11, a cosmological extension of the theory does not exist. The second deals with the structure of (on-shell) four-point invariants. These are important both for establishing existence of the lowest (2-loop) order candidate counter-terms in the theory proper, as well as for comparison with the form of eventual "zero-slope" QFT limit of M-theory. 
  We consider the F-theory description of non-simply-connected gauge groups appearing in the E8 x E8 heterotic string. The analysis is closely tied to the arithmetic of torsion points on an elliptic curve. The general form of the corresponding elliptic fibration is given for all finite subgroups of E8 which are applicable in this context. We also study the closely-related question of point-like instantons on a K3 surface whose holonomy is a finite group. As an example we consider the case of the heterotic string on a K3 surface having the E8 gauge symmetry broken to (E6 x SU(3))/Z3 or SU(9)/Z3 by point-like instantons with Z3 holonomy. 
  We discuss quotients of Anti-de Sitter (AdS) spacetime by a discrete group in light of the AdS-CFT correspondence. Some quotients describe closed universes which expand from zero volume to a maximum size and then contract. Maldacena's conjecture suggests that they should be represented in string theory by suitable quotients of the boundary conformal field theory. We discuss the required identifications, and construct the states associated with the linearized supergravity modes in the cosmological background. 
  The near horizon geometry of four-dimensional black holes in the dilute gas regime is AdS_3 x S^2, and the global symmetry group is SU(2) x USp(6). This is exploited to calculate their perturbation spectrum using group theoretical methods. The result is interpreted in terms of three extreme M5-branes, orthogonally intersecting over a common string. We also consider N=8 black holes in five dimensions, and compute the spectrum by explicit decoupling of the equations of motion, extending recent work on N=4 black holes. This result is interpreted in terms of D1- and D5-branes that are wrapped on a small four-torus. The spectra are compared with string theory. 
  We derive the spacetime superalgebras explicitly from ``test'' p-brane actions in a D-8-brane (i.e. a massive IIA) background to the lowest order in $\theta$ via canonical formalism, and show that the forms of the superalgebras are the same as those in all the other D-brane (i.e. massless IIA) backgrounds, that is, they are indifferent to the presence of the Chern-Simons terms which are proportional to the mass and added to the D-brane actions in the case of massive IIA backgrounds. Thus, we can say that all the D-brane background solutions including a D-8-brane are on equal footing from the viewpoint of the superalgebras via brane probes. We also deduce from the algebra all the previously known 1/4 supersymmetric intersections of a p-brane or a fundamental string with a D-8-brane, as the supersymmetric ``gauge fixings'' of the test branes in the D-8-brane background. 
  We study the Principal Chiral Ginzburg-Landau-Wilson model around two dimensions within the Local Potential Approximation of an Exact Renormalization Group equation. This model, relevant for the long distance physics of classical frustrated spin systems, exhibits a fixed point of the same universality class that the corresponding Non-Linear Sigma model. This allows to shed light on the long-standing discrepancy between the different perturbative approaches of frustrated spin systems. 
  It is shown that when the underlying sigma model of bosonic string theory is written in terms of single-valued fields, which live in the covering space of the target space, Abelian T-duality survives lattice regularization of the world-sheet. The projection onto the target-space is implemented through a sum over cohomology, which bears resemblance to summing over topological sectors in Yang-Mills theories. In particular, the case of string theory on a circle is shown to be explicitly self-dual in the lattice regulated model and automatically forbids vortex excitations which would otherwise destroy the duality. For other target spaces a generalized notion of T-duality is observed in which the target space and the cohomology coefficient group are interchanged under duality. Specific examples show that the fundamental group of the target space may not be preserved in the T-dual theory. Generalized models which exhibit T-duality behaviour, with dynamical variables that live on the k-dimensional cells of (p+1)-dimensional world-volumes, are also constructed. These models correspond to gauge theories, and higher-dimensional analogues, in which one sums over various topological sectors of the theory. 
  We investigate the NBI matrix model with the potential $X\Lambda+X^{-1}+(2\eta+1)\log X$ recently proposed to describe IIB superstrings. With the proper normalization, using Virasoro constraints, we prove the equivalence of this model and the Kontsevich matrix model for $\eta\ne0$ and find the explicit transformation between the two models. 
  We review some aspects of minimal cycles in string compactifications and their role in constructing new critical theories in six and lower dimensions as well as in accounting for black hole entropy. (Based on a talk presented at the Salam Memorial Meeting, the Abdus Salam International Center for Theoretical Physics, Fall 1997) 
  The structure of the state-vector space for the two-mode parabose system is investigated and a complete set of state-vectors is constructed. The basis vectors are orthonormal in order $p=2$. In order $p=2$, conserved-charge parabose coherent states are constructed and an explicit completeness relation is obtained. 
  We prove that any gauged WZNW model has a Lax pair representation, and give explicitly the general solution of the classical equations of motion of the SL(2,R)/U(1) theory. We calculate the symplectic structure of this solution by solving a differential equation of the Gelfand-Dikii type with initial state conditions at infinity, and transform the canonical physical fields non-locally onto canonical free fields. The results will, finally, be collected in a local B\"acklund transformation. These calculations prepare the theory for an exact canonical quantization. 
  The nearest neighbor two-point correlation function of the $Z$-invariant inhomogeneous eight-vertex model in the thermodynamic limit is computed using the free field representation. 
  The adS_{p+2} x S^{d-p-2} geometry of the near horizon branes is promoted to a supergeometry: the solution of the supergravity constraints for the vielbein, connection and form superfields are found. This supergeometry can be used for the construction of new superconformal theories. We also discuss the Green-Schwarz action for a type IIB string on adS_5 x S_5. 
  Starting from Seiberg's electric-magnetic duality for supersymmetric QCD, we construct dual pairs of non-supersymmetric gauge theories. This is accomplished by first taking the large N limit of supersymmetric QCD and its dual partner and then performing a special ``orbifold projection'' recently introduced by Kachru and Silverstein. We argue that in the large N limit the two projected theories remain dual. The non-supersymmetric gauge theories which can be studied in this fashion have non-supersymmetric field content, chiral fermions and exactly massless scalar matter. 
  I argue that for the case of fermions with nonzero bare mass there is a term in the matter density operator in the light-cone representation which has been omitted from previous calculations. The new term provides agreement with previous results in the equal-time representation for mass perturbation theory in the massive Schwinger model. For the DLCQ case the physics of the new term can be represented by an effective operator which acts in the DLCQ subspace, but the form of the term might be hard to guess and I do not know how to determine its coefficient from symmetry considerations. 
  BPS states of N=2, D=4 Super Yang-Mills theories with ADE flavor symmetry arise as junctions joining a D3-brane to a set of 7-branes defining the enhanced flavor algebra. We show that the familiar BPS spectrum of SU(2) theories with N_f <= 4 is simply given by the set of junctions whose self-intersection is bounded below as required by supersymmetry. This constraint, together with the relations between junction and weight lattices, is used to establish the appearance of arbitrarily large flavor representations for the case of D_{n>=5} and E_n symmetries. Such representations are required by consistency with decoupling down to smaller flavor symmetries. 
  We study the (p,q)5-brane dynamics from the viewpoint of Matrix string theory in the T-dualized ALE background. The most remarkable feature in the (p,q)5-brane is the existence of ``fractional string'', which appears as the instanton of 5-brane gauge theory. We approach to the physical aspects of fractional string by means of the two types of Matrix string probes: One of which is that given in hep-th/9710065. As the second probe we present the Matrix string theory describing the fractional string itself. We calculate the moduli space metrics in the respective cases and argue on the specific behaviors of fractional string. Especially, we show that the ``joining'' process of fractional strings can be realized as the transition from the Coulomb branch to the Higgs branch of the fractional string probe. In this argument, we emphasize the importance of some monodromies related with the theta-angle of the 5-brane gauge theory. 
  We consider quantum dynamical systems whose degrees of freedom are described by $N \times N$ matrices, in the planar limit $N \to \infty$. Examples are gauge theoires and the M(atrix)-theory of strings. States invariant under U(N) are `closed strings', modelled by traces of products of matrices. We have discovered that the U(N)-invariant opertors acting on both open and closed string states form a remarkable new Lie algebra which we will call the heterix algebra. (The simplest special case, with one degree of freedom, is an extension of the Virasoro algebra by the infinite-dimensional general linear algebra.) Furthermore, these operators acting on closed string states only form a quotient algebra of the heterix algebra. We will call this quotient algebra the cyclix algebra. We express the Hamiltonian of some gauge field theories (like those with adjoint matter fields and dimensionally reduced pure QCD models) as elements of this Lie algebra. Finally, we apply this cyclix algebra to establish an isomorphism between certain planar matrix models and quantum spin chain systems. Thus we obtain some matrix models solvable in the planar limit; e.g., matrix models associated with the Ising model, the XYZ model, models satisfying the Dolan-Grady condition and the chiral Potts model. Thus our cyclix Lie algebra described the dynamical symmetries of quantum spin chain systems, large-N gauge field theories, and the M(atrix)-theory of strings. 
  We construct new non-diagonal solutions to the boundary Yang-Baxter-Equation corresponding to a two-dimensional field theory with U_q(a_2^(1)) quantum affine symmetry on a half-line. The requirements of boundary unitarity and boundary crossing symmetry are then used to find overall scalar factors which lead to consistent reflection matrices. Using the boundary bootstrap equations we also compute the reflection factors for scalar bound states (breathers). These breathers are expected to be identified with the fundamental quantum particles in a_2^(1) affine Toda field theory and we therefore obtain a conjecture for the affine Toda reflection factors. We compare these factors with known classical results and discuss their duality properties and their connections with particular boundary conditions. 
  In the context of the bulk-boundary correspondence we study the correlation functions arising on a boundary for different types of boundary conditions. The most general condition is the mixed one interpolating between the Neumann and Dirichlet conditions. We obtain the general expressions for the correlators on a boundary in terms of Green's function in the bulk for the Dirichlet, Neumann and mixed boundary conditions and establish the relations between the correlation functions. As an instructive example we explicitly obtain the boundary correlators corresponding to the mixed condition on a plane boundary $R^d$ of a domain in flat space $R^{d+1}$. The phases of the boundary theory with correlators of the Neumann and Dirichlet types are determined. The boundary correlation functions on sphere $S^d$ are calculated for the Dirichlet and Neumann conditions in two important cases: when sphere is a boundary of a domain in flat space $R^{d+1}$ and when it is a boundary at infinity of Anti-De Sitter space $AdS_{d+1}$. For massless in the bulk theory the Neumann correlator on the boundary of AdS space is shown to have universal logarithmic behavior in all AdS spaces. In the massive case it is found to be finite at the coinciding points. We argue that the Neumann correlator may have a dual two-dimensional description. The structure of the correlators obtained, their conformal nature and some recurrent relations are analyzed. We identify the Dirichlet and Neumann phases living on the boundary of AdS space and discuss their evolution when the location of the boundary changes from infinity to the center of the AdS space. 
  The soldering mechanism is a new technique to work with distinct manifestations of dualities that incorporates interference effects, leading to new physical results that includes quantum contributions. This approach was used to investigate the cases of electromagnetic dualities, and $D\geq 2$ bosonization. In the former context this technique is applied for the quantum mechanical harmonic oscillator, the scalar field theory in two dimensions and the Maxwell theory in four dimensions. The soldered actions in any dimension leads to a master action which is duality invariant under a much bigger set of symmetries. The effects of coupling to gravity are also elaborated. In the later context, a technique is developed that solders the dual aspects of some symmetry following from the bosonisation of two distinct fermionic models, leading to new results which cannot be otherwise obtained. Exploiting this technique, the two dimensional chiral determinants with opposite chirality are soldered to reproduce either the usual gauge invariant expression leading to the Schwinger model or, alternatively, the Thirring model. Likewise, two apparently independent three dimensional massive Thirring models with same coupling but opposite mass signatures, in the long wavelegth limit, combine by the process of bosonisation and soldering to yield an effective massive Maxwell theory. The current bosonisation formulas are given, both in the original independent formulation as well as the effective theory, and shown to yield consistent results for the correlation functions. 
  We construct a class of one-dimensional Lie-algebraic problems based on sl(2) where the spectrum in the algebraic sector has a dynamical symmetry E -> - E. All 2j+1 eigenfunctions in the algebraic sector are paired, and inside each pair are related to each other by simple analytic continuation x -> ix, except the zero mode appearing if j is integer. At j-> infinity the energy of the highest level in the algebraic sector can be calculated by virtue of the quasiclassical expansion, while the energy of the ground state can be calculated as a weak coupling expansion. The both series coincide identically. 
  It is very plausible that confinement in QCD comes from linking of Wilson loops to finite-thickness vortices with magnetic fluxes corresponding to the center of the gauge group. The vortices are solitons of a gauge-invariant QCD action representing the generation of gluon mass. There are a number of other solitonic states of this action. We discuss here what we call nexus solitons, in which for gauge group SU(N), up to N vortices meet a a center, or nexus, provided that the total flux of the vortices adds to zero (mod N). There are fundamentally two kinds of nexuses: Quasi-Abelian, which can be described as composites of Abelian imbedded monopoles, whose Dirac strings are cancelled by the flux condition; and fully non-Abelian, resembling a deformed sphaleron. Analytic solutions are available for the quasi-Abelian case, and we discuss variational estimates of the action of the fully non-Abelian nexus solitons in SU(2). The non-Abelian nexuses carry Chern-Simons number (or topological charge in four dimensions). Their presence does not change the fundamentals of confinement in the center-vortex picture, but they may lead to a modified picture of the QCD vacuum. 
  We consider four dimensional N=1 supersymmetric Type I compactifications on toroidal orbifolds T^6/G. In particular, we focus on the Type I vacua which are perturbative from the orientifold viewpoint, that is, on the compactifications with well defined world-sheet expansion. The number of such models is rather constrained. This allows us to study all such vacua. This, in particular, involves considering compactifications with non-trivial NS-NS antisymmetric tensor backgrounds. We derive massless spectra for these models, and also compute superpotentials. We review the reasons responsible for such a limited number of perturbative Type I compactifications on toroidal orbifolds (which include Abelian as well as non-Abelian cases). As an aside, we generalized the recent work on large N gauge theories from orientifolds to include a non-Abelian orbifold. This also provides an important independent check for perturbative consistency of the corresponding Type I compactification. 
  Exact solutions to the low-energy effective action (LEEA) of the four-dimensional N=2 supersymmetric gauge field theories are known to be obtained either by quantum field theory methods from S-duality in the Seiberg-Witten approach, or by the Type-IIA superstring/M-Theory methods of brane technology. After a brief review of the standard field-theoretical results for the N=2 gauge (Seiberg-Witten) LEEA, we consider a field-theoretical derivation of the exact hypermultiplet LEEA by using the N=2 harmonic superspace methods. We illustrate our techniques on a number of explicit examples. Our main purpose, however, is to discuss the existing analytical (calculational) support for the alternative methods of brane technology. We summarize known exact solutions to the eleven-dimensional and ten-dimensional type-IIA supergravities, which describe classical configurations of intersecting BPS branes with eight supercharges relevant to the non-perturbative N=2 gauge field theory with fundamental hypermultiplet matter. The crucial role of M-Theory in providing a classical resolution of singularities in the ten-dimensional (Type-IIA superstring) brane picture, as well as the N=2 extended supersymmetry in four dimensions, are made manifest. The two approaches to a derivation of the exact N=2 gauge theory LEEA are thus seen to be complementary to each other and mutually dependent. 
  The absorption cross section of dilatinos by D3-branes is calculated by means of both classical type IIB supergravity and the effective gauge field theory on their worldvolume. The two methods give the same results, supporting the microscopic description of black holes in terms of D-branes and giving another evidence of AdS/superconformal reciprocity. 
  The string theory description of BPS states in D-brane world-volume field theories may undergo transitions from open strings to string webs, as well as between different string webs, as one moves in the field theory moduli space. These transitions are driven by the string creation phenomenon. We demonstrate such transitions in the D3-brane realization of N=2 SU(2) Super-Yang-Mills theory. 
  Many noncompact Type I orbifolds satisfy tadpole constraints yet are anomalous. We present a generalization of the anomaly inflow mechanism for some of these cases in six and four dimensions. 
  We point out that if spatial information is encoded through linear operators $X_i$, or `infinite-dimensional matrices' with an involution $X_i^*=X_i$ then these $X_i$ can only describe either continuous, discrete or certain "fuzzy" space-time structures. We argue that the fuzzy space structure may be relevant at the Planck scale. The possibility of this fuzzy space-time structure is related to subtle features of infinite dimensional matrices which do not have an analogue in finite dimensions. For example, there is a slightly weaker version of self-adjointness: symmetry, and there is a slightly weaker version of unitarity: isometry. Related to this, we also speculate that the presence of horizons may lead to merely isometric rather than unitary time evolution. 
  In this work, a general definition of Convolution between two arbitrary Tempered Ultradistributions is given. When one of the Tempered Ultradistributions is rapidly decreasing this definition coincides with the definition of J. Sebastiao e Silva.   The product of two arbitrary distributions of exponential type is defined via the Convolution of its corresponding Fourier Transforms.   Several examples of Convolution of two Tempered Ultradistributions and singular products are given. In particular, we reproduce the results obtained by A. Gonzales Dominguez and A. Bredimas. 
  We study the Donaldson-Witten function in four-dimensional topological gauge theory which is constructed from N=2 supersymmetric SU(2) gauge theory with $N_f < 4$ massless fundamental hypermultiplets. When $N_f = 2,3$, the strong-coupling singularities with multiple massless monopoles appear in the moduli space (the u-plane) of the Coulomb branch. We show that the invariants made out of such singularities exhibit a property which is similar to the one expected for four-manifolds of generalized simple type. 
  We compute an SL(6,Z) invariant partition function for the chiral two-form of the M theory fivebrane compactified on the six-torus. From a manifestly SL(5,Z) invariant formalism, we prove that the partition function has an additional SL(2,Z) symmetry. The combination of these two symmetries ensures SL(6,Z) invariance. Thus, whether or not a fully covariant Lagrangian is available, the fivebrane on the six-torus has a consistent quantum theory. 
  In Einstein-Maxwell theory, magnetic flux lines are `expelled' from a black hole as extremality is approached, in the sense that the component of the field strength normal to the horizon goes to zero. Thus, extremal black holes are found to exhibit the sort of `Meissner effect' which is characteristic of superconducting media. We review some of the evidence for this effect, and do present new evidence for it using recently found black hole solutions in string theory and Kaluza-Klein theory. We also present some new solutions, which arise naturally in string theory, which are non-superconducting extremal black holes. We present a nice geometrical interpretation of these effects derived by looking carefully at the higher dimensional configurations from which the lower dimensional black hole solutions are obtained. We show that other extremal solitonic objects in string theory (such as p-branes) can also display superconducting properties. In particular, we argue that the relativistic London equation will hold on the worldvolume of `light' superconducting p-branes (which are embedded in flat space), and that minimally coupled zero modes will propagate in the adS factor of the near-horizon geometries of `heavy', or gravitating, superconducting p-branes. 
  We compare the amplitudes for the long-distance scattering of three gravitons in eleven dimensional supergravity and matrix theory at finite N. We show that the leading supergravity term arises from loop contributions to the matrix theory effective action that are not required to vanish by supersymmetry. We evaluate in detail one type of diagram---the setting sun with only massive propagators---reproducing the supergravity behavior. 
  Many physical systems like supersymmetric Yang-Mills theories are formulated as quantum matrix models. We discuss how to apply the Beth ansatz to exactly solve some supersymmetric quantum matrix models in the large-N limit. Toy models are constructed out of the one-dimensional Hubbard and t-J models as illustrations. 
  In this technical note we describe a new (to the physics literature) construction of bundles on Calabi-Yaus. We primarily study this construction in the special case of K3 surfaces, for which interesting results can be obtained. For example, we use this construction to give plausibility arguments for a relationship between spaces of solutions of Hitchin's equations and moduli spaces of bundles on K3s. Also, in a recent paper it was proposed by C. Vafa that the mirror to a bundle on a Calabi-Yau n-fold is, in a particular sense, a supersymmetric n-cycle on the mirror Calabi-Yau. We use this new construction to observe that for the special case of K3s, Vafa's mirror data also specifies a bundle directly on the mirror K3, and so we potentially have a duality between bundles on any one K3 and other bundles on the mirror K3. 
  We calculate the spectrum of glueball masses in non-supersymmetric Yang-Mills theory in three and four dimensions, based on a conjectured duality between supergravity and large N gauge theories. The glueball masses are obtained by solving supergravity wave equations in a black hole geometry. We find that the mass ratios are in good numerical agreement with the available lattice data. We also compute the leading (g_{YM}^2 N)^{-1} corrections to the glueball masses, by taking into account stringy corrections to the supergravity action and to the black hole metric. We find that the corrections to the masses are negative and of order (g_{YM}^2N)^{-3/2}. Thus for a fixed ultraviolet cutoff the masses decrease as we decrease the 't Hooft coupling, in accordance with our expectation about the continuum limit of the gauge theories. 
  We discuss simple cosmological solutions of Horava-Witten theory describing the strongly coupled heterotic string. At energies below the grand-unified scale, the effective theory is five- not four-dimensional, where the additional coordinate parameterizes a S^1/Z_2 orbifold. Furthermore, it admits no homogeneous solutions. Rather, the vacuum state, appropriate for a reduction to four-dimensional supersymmetric models, is a BPS domain wall. Relevant cosmological solutions are those associated with this BPS state. In particular, such solutions must be inhomogeneous, depending on the orbifold coordinate as well as on time. We present two examples of this new type of cosmological solution, obtained by separation of variables rather that by exchange of time and radius coordinate applied to a brane solution, as in previous work. The first example represents the analog of a rolling radii solution with the radii specifying the geometry of the domain wall. This is generalized in the second example to include a nontrivial ``Ramond-Ramond'' scalar. 
  The structure of counterterms in higher derivative quantum gravity is reexamined. Nontrivial dependence of charges on the gauge and parametrization is established. Explicit calculations of two-loop contributions are carried out with the help of the generalized renormgroup method demonstrating consistency of the results obtained. 
  A complete proof of the No-ghost Theorem for bosonic and fermionic string theories on AdS_3, or the group manifold of SU(1,1), is given. It is then shown that the restriction on the spin (in terms of the level) that is necessary to obtain a ghost-free spectrum corresponds to the stringy exclusion principle of Maldacena and Strominger. 
  A serious problem with the Schwinger-Dyson approach to dynamical mass generation in QED3 at finite temperature is that the contribution from the transverse part of the photon propagator, in the Landau gauge, leads to infrared divergences in both the mass function and the wavefunction renormalisation. We show how, by using a simple choice of vertex anatz and a choice of non-local gauge (the `D-gauge') both quantities can be made finite. We formulate an equation for the physical mass M. and show that it reduces to the coresponding equation obtained in the constant physical mass approximation M=M(0,pi T) (which is finite). There for at finite temperature, we are able to justify a `constant' mass approximation for M, and show that the value of r (the ratio of twice the physical mass at zero temperature to the critical temperature) remains close to the value obtained in previous calculations with retardation. 
  With the recent discovery that many aspects of black hole thermodynamics can be effectively reduced to problems in three spacetime dimensions, it has become increasingly important to understand the ``statistical mechanics'' of the (2+1)-dimensional black hole of Banados, Teitelboim, and Zanelli (BTZ). Several conformal field theoretic derivations of the BTZ entropy exist, but none is completely satisfactory, and many questions remain open: there is no consensus as to what fields provide the relevant degrees of freedom or where these excitations live. In this paper, I review some of the unresolved problems and suggest avenues for their solution. 
  By the concurrent use of two different resummation methods, the composite operator formalism and the Dyson-Schwinger equation, we re-examinate the behavior at finite temperature of the O(N)-symmetric $\lambda\phi^{4}$ model in a generic D-dimensional Euclidean space. In the cases D=3 and D=4, an analysis of the thermal behavior of the renormalized squared mass and coupling constant are done for all temperatures. It results that the thermal renormalized squared mass is positive and increases monotonically with the temperature. The behavior of the thermal coupling constant is quite different in odd or even dimensional space. In D=3, the thermal coupling constant decreases up to a minimum value diferent from zero and then grows up monotonically as the temperature increases. In the case D=4, it is found that the thermal renormalized coupling constant tends in the high temperature limit to a constant asymptotic value. Also for general D-dimensional Euclidean space, we are able to obtain a formula for the critical temperature of the second order phase transition. This formula agrees with previous known values at D=3 and D=4. 
  Higher derivative terms in the effective action of certain Yang-Mills theories can be severely constrained by supersymmetry. We show that requiring sixteen supersymmetries in quantum mechanical gauge theory determines the $v^6$ term in the effective action. Even the numerical coefficient of the $v^6$ term is fixed in terms of lower derivative terms in the effective action. 
  Two different families of abelian chiral gauge theories on the torus are investigated: the aim is to test the consistency of two-dimensional anomalous gauge theories in the presence of global degrees of freedom for the gauge field. An explicit computation of the partition functions shows that unitarity is recovered in particular regions of parameter space and that the effective dynamics is described in terms of fermionic interacting models. For the first family, this connection with fermionic models uncovers an exact duality which is conjectured to hold in the nonabelian case as well. 
  We propose some analogue of the Narain lattice for CHL string. The symmetries of this lattice are the symmetries of the perturbative spectrum. We explain in this language the known results about the possible gauge groups in compactified theory. For the four-dimensional theory, we explicitly describe the action of S-duality on the background fields. We show that the moduli spaces of the six, seven and eight-dimensional compactifications coincide with the moduli spaces of the conjectured Type IIA, M Theory and F Theory duals. We classify the rational components of the boundary of the moduli space in seven, eight and nine dimensions. 
  Some exact solutions of the SU(2) Seiberg-Witten equations in Minkowski spacetime are given. 
  We consider a twisted version of the four-dimensional N=4 supersymmetric Yang-Mills theory with gauge groups SU(2) and SO(3), and bare masses for two of its chiral multiplets, thereby breaking N=4 down to N=2. Using the wall-crossing technique introduced by Moore and Witten within the u-plane approach to twisted topological field theories, we compute the partition function and all the topological correlation functions for the case of simply-connected spin four-manifolds of simple type. By including 't Hooft fluxes, we analyse the properties of the resulting formulae under duality transformations. The partition function transforms in the same way as the one first presented by Vafa and Witten for another twist of the N=4 supersymmetric theory in their strong coupling test of S-duality. Both partition functions coincide on K3. The topological correlation functions turn out to transform covariantly under duality, following a simple pattern which seems to be inherent in a general type of topological quantum field theories. 
  The problem of duality symmetry in free field models is examined in details by performing a mode expansion of these fields which provides a mapping with the purely quantum mechanical example of a harmonic oscillator. By analysing the duality symmetry in the harmonic oscillator, we show that the massless scalar theory in two dimensions display, along with the expected discrete $Z_2$ symmetry, the continuous SO(2) symmetry as well. The same holds for the free Maxwell theory in four dimensions, which is usually regarded to manifest only the SO(2) symmetry. This leads to the new result that, following a proper interpretation, the duality groups in two and four dimensions become identical. Incidentally, duality in quantum mechanics is generally not covered in the literature that considers only $D=4k$ or $D=4k+2$ spacetime dimensions, for integral $k$. 
  We present a simple result for the action density of the SU(n) charge one periodic instantons - or calorons - with arbitrary non-trivial Polyakov loop P_oo at spatial infinity. It is shown explicitly that there are n lumps inside the caloron, each of which represents a BPS monopole, their masses being related to the eigenvalues of P_oo. A suitable combination of the ADHM construction and the Nahm transformation is used to obtain this result. 
  We solve the Cauchy problem of the Ward model in light-cone coordinates using the inverse spectral (scattering) method. In particular we show that the solution can be constructed by solving a $2\times 2$ local matrix Riemann-Hilbert problem which is uniquely defined in terms of the initial data. These results are also directly applicable to the 2+1 Chiral model. 
  We consider a class of extremal and non-extremal four-dimensional black-hole solutions occuring in toroidally compactified heterotic string theory, whose ten-dimensional interpretation involves a Kaluza-Klein monopole and a five-brane. We show that these four-dimensional solutions can be connected to extremal and non-extremal two-dimensional heterotic black-hole solutions through a change in the asymptotic behaviour of the harmonic functions associated with the Kaluza-Klein monopole and with the five-brane. This change in the asymptotic behaviour can be achieved by a sequence of S and T-S-T duality transformations in four dimensions. These transformations are implemented by performing a reduction on a two-torus with Lorentzian signature. We argue that the same mechanism can be applied to extremal and non-extremal black-hole solutions in the FHSV model. 
  The exact Wilson loop expression for the pure Yang-Mills U(N) theory on a sphere $S^2$ of radius $R$ exhibits, in the decompactification limit $R\to \infty$, the expected pure area exponentiation. This behaviour can be understood as due to the sum over all instanton sectors. If only the zero instanton sector is considered, in the decompactification limit one exactly recovers the sum of the perturbative series in which the light-cone gauge Yang-Mills propagator is prescribed according to Wu-Mandelstam-Leibbrandt. When instantons are disregarded, no pure area exponentiation occurs, the string tension is different and, in the large-N limit, confinement is lost. 
  Utilizing sets of super-vector fields (derivations), explicit expressions are obtained for; (a.) the 1D, N-extended superconformal algebra, (b.) the 1D, N-extended super Virasoro algebra for N = 1, 2 and 4 and (c.) a geometrical realization (GR) covering algebra that contains the super Virasoro algebra for arbitrary values of N. By use of such vector fileds, the super Virasoro algebra is embedded as a geometrical and model-independent structure in 1D and 2D Aleph-null-extended superspace. 
  A cosmological version of the holographic principle is proposed. Various consequences are discussed including bounds on equation of state and the requirement that the universe be infinite. 
  We present a survey of the calibrated geometries arising in the study of the local singularity structure of supersymmetric fivebranes in M-theory. We pay particular attention to the geometries of 4-planes in eight dimensions, for which we present some new results as well as many details of the computations. We also analyse the possible generalised self-dualities which these geometries can afford. 
  We study the correspondence between the large $N$ limit of ${\cal N} = 2$ three dimensional superconformal field theories and M theory on orbifolds of $AdS_4 \times {\bf S^7}$. We identify the brane configuration which gives $C^3/Z_3$ as a background for the M theory in terms of a Brane Box Model or a (p, q) web model.   By taking the orbifold projection on the known spectrum of Kaluza-Klein harmonics of supergravity, we obtain information about the chiral primary operators at the orbifold singularities. 
  We study the unitary supermultiplets of the N=8, d=5 anti-de Sitter (AdS) superalgebra SU(2,2|4) which is the symmetry algebra of the IIB string theory on AdS_5 X S^5. We give a complete classification of the doubleton supermultiplets of SU(2,2|4) which do not have a Poincare limit and correspond to d=4 conformal field theories (CFT) living on the boundary of AdS_5. The CPT self-conjugate irreducible doubleton supermultiplet corresponds to d=4, N = 4 super Yang-Mills theory. The other irreducible doubleton supermultiplets come in CPT conjugate pairs. The maximum spin range of the general doubleton supermultiplets is 2. In particular, there exists a CPT conjugate pair of doubleton supermultiplets corresponding to the fields of N=4 conformal supergravity in d=4 which can be coupled to N=4 super Yang-Mills theory in d=4. We also study the "massless" supermultiplets of SU(2,2|4) which can be obtained by tensoring two doubleton supermultiplets. The CPT self-conjugate "massless" supermultiplet is the N=8 graviton supermultiplet in AdS_5. The other "massless" supermultiplets generally come in conjugate pairs and can have maximum spin range of 4. We discuss the implications of our results for the conjectured CFT/AdS dualities. 
  The exact prepotential for $N = 2$ supersymmetric Yang-Mills theory is derived from the superconformal anomalous Ward identity for the gauge group SU(2) and SU(3) which can be generalized to any other rank two gauge group. 
  We construct a four dimensional chiral N=1 space-time supersymmetric perturbative Type I vacuum corresponding to a compactification on a toroidal Z_2 X Z_2 X Z_3 orbifold with a discrete Wilson line. This model is non-perturbative from the heterotic viewpoint. It has three chiral families in the SU(4)_c X SU(2)_w X U(1) subgroup of the total gauge group. We compute the tree-level superpotential in this model. There appears to be no obvious obstruction to Higgs the gauge group down to SU(3)_c X SU(2)_w X U(1)_Y and obtain the Standard Model gauge group with three chiral families. 
  The self-field approach to quantum electrodynamics (QED) is used to study the bound state problem in light-front two-dimensional QED with massive matter fields. A composite matter field describing bound states is introduced and the relativistic bound state equation for the composite field including a self-potential is obtained. The Hamiltonian form of the bound state equation in terms of the invariant mass squared operator is given. The eigenvalue problem of this operator is solved for a fixed value of the self-potential, the corresponding eigenfunctions and the mass spectrum are found. In the case of massless matter fields, there are no self-field terms in the bound state equation, and the invariant mass spectrum can be evaluated explicitly. Possible ways of deriving more complete information about the bound state spectrum are briefly discussed. 
  The presence of chiral fermions in the physical Hilbert space implies consistency conditions on the spectral action. These conditions are equivalent to the absence of gauge and gravitational anomalies. Suggestions for the fermionic part of the spectral action are made based on the supersymmetrisation of the bosonic part. 
  Two closely related topological phenomena are studied at finite density and temperature. These are chiral anomaly and Chern--Simons term. It occurs that the chiral anomaly doesn't depend on density and temperature. Chern-Simons term appearance in even dimensions is studied under two types of constraints: chiral and usual charges conservation. In odd dimensions, by using different methods it is shown that $\mu^2 = m^2$ is the crucial point for Chern--Simons at zero temperature. So when $\mu^2 < m^2$ $\mu$--influence disappears and we get the usual Chern-Simons term. On the other hand, when $\mu^2 > m^2$ the Chern-Simons term vanishes because of non--zero density of background fermions. The connection between parity anomalous Chern-Simons in odd dimension and chiral anomaly in even dimension is established at arbitrary density and temperature. These results hold in any dimension as in abelian, so as in nonabelian cases. 
  We study the pair production of open strings in constant electric fields, using a general framework which encodes both relativistic string theory and generic linearly extended systems as well. In the relativistically invariant case we recover previous results, both for pair production and for the effective Born-Infeld action. We then derive a non-relativistic limit - where the propagation velocity along the string is much smaller than the velocity of light - obtaining quantum dissipation. We calculate the pair nucleation rate for this case, which could be relevant for applications. 
  If G is a simple non-compact Lie group, with K its maximal compact subgroup, such that K contains a one-dimensional center C, then the coset space G/K is an Hermitian symmetric non-compact space. SL(2,R)/U(1) is the simplest example of such a space. It is only when G/K is an Hermitian symmetric space that there exists unitary discrete representations of G. We will here study string theories defined as G/K', K'=K/C, WZNW models. We will establish unitarity for such string theories for certain discrete representations. This proof generalizes earlier results on SL(2,R), which is the simplest example of this class of theories. We will also prove unitarity of G/K conformal field theories generalizing results for SL(2,R)/U(1). We will show that the physical space of states lie in the subspace of the G/K state space. 
  We consider the (1,0) supersymmetric Yang-Mills multiplet coupled to a self-dual tensor multiplet in six dimensions. It is shown that the counterterm required to cancel the one-loop gauge anomaly modifies the classical equations of motion previously obtained by Bergshoeff, Sezgin and Sokatchev (BSS). We discuss the supermultiplet structure of the anomalies exhibited in the resulting equations of motion. The anomaly corrected field equations agree with the global limit, recently obtained by Duff, Liu, Lu and Pope, of a matter coupled supergravity theory in six dimensions. We also obtain the dual formulation of the BSS model in which the tensor multiplet is free while the field equations of the Yang-Mills multiplet contain the fields of the tensor multiplet at the classical level. 
  We derive the five-dimensional effective action of strongly coupled heterotic string theory for the complete (1,1) sector of the theory by performing a reduction, on a Calabi-Yau three-fold, of M-theory on S^1/Z_2. A crucial ingredient for a consistent truncation is a non-zero mode of the antisymmetric tensor field strength which arises due to magnetic sources on the orbifold planes. The correct effective theory is a gauged version of five-dimensional N=1 supergravity coupled to Abelian vector multiplets, the universal hypermultiplet and four-dimensional boundary theories with gauge and gauge matter fields. The gauging is such that the dual of the four-form field strength in the universal multiplet is charged under a particular linear combination of the Abelian vector fields. In addition, the theory has potential terms for the moduli in the bulk as well as on the boundary. Because of these potential terms, the supersymmetric ground state of the theory is a multi-charged BPS three-brane domain wall, which we construct in general. We show that the five-dimensional theory together with this solution provides the correct starting point for particle phenomenology as well as early universe cosmology. As an application, we compute the four-dimensional N=1 supergravity theory for the complete (1,1) sector to leading nontrivial order by a reduction on the domain wall background. We find a correction to the matter field Kahler potential and threshold corrections to the gauge kinetic functions. 
  We develop a method to analyze systematically the configuration space of a D-brane localized at the orbifold singular point of a Calabi--Yau $d$-fold of the form ${\Bbb C}^d/\Gamma$ using the theory of toric quotients. This approach elucidates the structure of the K\"ahler moduli space associated with the problem. As an application, we compute the toric data of the $\Gamma$-Hilbert scheme. 
  Fermionic model of Superconformal field theory with boundary is considered. There were written the ''boundary'' Ward Identity for this theory and also constructed boundary states for fermionic and spin models. For this model were derived ''bootstrap'' equations for boundary structure constants. 
  String dynamics in a curved space-time is studied on the basis of an action functional including a small parameter of rescaled tension $\epsilon=\gamma/\alpha^{\prime}$, where $\gamma$ is a metric parametrizing constant. A rescaled slow worldsheet time $T=\epsilon\tau$ is introduced, and general covariant non-linear string equation are derived.   It is shown that in the first order of an $\epsilon $-expansion these equations are reduced to the known equation for geodesic derivation but complemented by a string oscillatory term. These equations are solved for the de Sitter and Friedmann -Robertson-Walker spaces. The primary string constraints are found to be split into a chain of perturbative constraints and their conservation and consistency are proved. It is established that in the proposed realization of the perturbative approach the string dynamics in the de Sitter space is stable for a large Hubble constant $H  (\alpha^{\prime}H^{2}\gg1)$. 
  The quantum properties of charged black holes (BHs) in 2D dilaton-Maxwell gravity (spontaneously compactified from heterotic string) with $N$ dilaton coupled scalars are studied. We first investigate 2D BHs found by McGuigan, Nappi and Yost. Kaluza-Klein reduction of 3D gravity with minimal scalars leads also to 2D dilaton-Maxwell gravity with dilaton coupled scalars and the rotating BH solution found by Ba\~nados, Teitelboim and Zanelli (BTZ) which can be also described by 2D charged dilatonic BH. Evaluating the one-loop effective action for dilaton coupled scalars in large $N$ (and s-wave approximation for BTZ case), we show that quantum-corrected BHs may evaporate or else anti-evaporate similarly to 4D Nariai BH as is observed by Bousso and Hawking. Higher modes may cause the disintegration of BH in accordance with recent observation by Bousso. 
  The vacuum structure and spectra of two-dimensional gauge theories with N=(2,2) supersymmetry are investigated. These theories admit a twisted mass term for charged chiral matter multiplets. In the case of a U(1) gauge theory with N chiral multiplets of equal charge, an exact description of the BPS spectrum is obtained for all values of the twisted masses. The BPS spectrum has two dual descriptions which apply in the Higgs and Coulomb phases of the theory respectively. The two descriptions are related by a massive analog of mirror symmetry: the exact mass formula which is given by a one-loop calculation in the Coulomb phase gives predictions for an infinite series of instanton corrections in the Higgs phase. The theory is shown to exhibit many phenomena which are usually associated with N=2 theories in four dimensions. These include BPS-saturated dyons which carry both topological and Noether charges, non-trivial monodromies of the spectrum in the complex parameter space, curves of marginal stability on which BPS states can decay and strongly coupled vacua with massless solitons and dyons. 
  We perform a one-dimensional complexified quaternionic version of the Dirac equation based on $i$-complex geometry. The problem of the missing complex parameters in Quaternionic Quantum Mechanics with $i$-complex geometry is overcome by a nice ``trick'' which allows to avoid the Dirac algebra constraints in formulating our relativistic equation. A brief comparison with other quaternionic formulations is also presented. 
  The use of complexified quaternions and $i$-complex geometry in formulating the Dirac equation allows us to give interesting geometric interpretations hidden in the conventional matrix-based approach. 
  This note is supposed to be an introduction to those concepts of toric geometry that are necessary to understand applications in the context of string and F-theory dualities. The presentation is based on the definition of a toric variety in terms of homogeneous coordinates, stressing the analogy with weighted projective spaces. We try to give both intuitive pictures and precise rules that should enable the reader to work with the concepts presented here. 
  The technique of generating new solutions to 4D gravity/matter systems by dimensional reduction to a sigma-model is extended to supersymmetric configurations of supergravity. The conditions required for the preservation of supersymmetry under isometry transformations in the sigma-model target space are found. Some examples illustrating the technique are given. 
  Ideas from physics are used to show that the prime pairs have the density conjectured by Hardy and Littlewood. The proof involves dealing with infinities like in quantum field theory. 
  Two issues regarding chiral $p$-forms are addressed. First, we investigate the topological conditions on spacetime under which the action for a non-chiral $p$-form can be split as the sum of the actions for two chiral $p$-forms, one of each chirality. When these conditions are not met, we exhibit explicitly the extra topological degrees of freedom and their couplings to the chiral modes. Second, we study the problem of constructing Lorentz-invariant self-couplings of a chiral $p$-form in the light of the Dirac-Schwinger condition on the energy-momentum tensor commutation relations. We show how the Perry-Schwarz condition follows from the Dirac-Schwinger criterion and point out that consistency of the gravitational coupling is automatic. 
  We report here the status of different gauge conditions in the canonical formulation of quantum electrodynamics on light-front surfaces. We start with the massive vector fields as pedagogical models where all basic concepts and possible problems manifestly appear. Several gauge choices are considered for both the infinite and the finite volume formulation of massless gauge field electrodynamics. We obtain the perturbative Feynman rules in the first approach and the quantum Hamiltonian for all sectors in the second approach. Different space-time dimensions are discussed in all models where they crucially change the physical meaning. Generally, fermions are considered as the charged matter fields but also one simple 1+1 dimensional model is discussed for scalar fields. Finally the perspectives for further research projects are discussed. 
  The Chern-Simons (CS) theory in three dimensions with a compact gauge group G is studied. Starting from the BRST quantization of the theory defined in R^3, the values of gauge invariants observables are computed in any closed and orientable three manifold M by constructing a surgery operator associated with a Dehn's surgery presentation of M. We have given the general rules to find the reduced tensor algebra whose elements give the physical inequivalent quantum numbers that characterize uniquely the CS observables. Some general properties of the reduced tensor algebra and of the surgery operator are studied. Many examples of the general construction are given in the case of G=SU(2) and G=SU(3), in particular the expectation value of a Wilson line associated with the unknot and the Hopf link in R^3(S^3) is computed for any irreducible representation of the gauge group. The relation of the topological invariant I(M), defined as a suitable modified partition function of the CS theory in M, and the fundamental group of M is studied. Finally, the relationship between CS theory and WZWN model in two dimensions is exploited to derive from a full three-dimensional point of view some classical results in conformal field theory. 
  We compute -in the saddle point approximation- the partition function for the 2+1 black hole using the Gibbons-Hawking approach. Some issues concerning the definition of thermodynamical ensembles are clarified. It is pointed out that the right action in covariant form is exactly equal to the Chern-Simons action with no added boundary terms. This action is finite, yields the right canonical free energy and has an extremum when the temperature and angular velocity are fixed. The correspondence with a 1+1 $CFT$ is indicated. 
  We consider a subset of the terms in the effective potential describing three-graviton interactions in Matrix theory and in classical eleven-dimensional supergravity. In agreement with the results of Dine and Rajaraman, we find that these terms vanish in Matrix theory. We show that the absence of these terms is compatible with the classical supergravity theory when the theory is compactified in a lightlike direction, resolving an apparent discrepancy between the two theories. A brief discussion is given of how this calculation might be generalized to compare the Matrix theory and supergravity descriptions of an arbitrary 3-body system. 
  In this letter we describe an approach to the current algebra based in the Path Integral formalism. We use this method for abelian and non-abelian quantum field theories in 1+1 and 2+1 dimensions and the correct expressions are obtained. Our results show the independence of the regularization of the current algebras. 
  The problem of quantum equivalence between non-linear sigma models related by Abelian or non-Abelian T-duality is studied in perturbation theory. Using the anomalous Ward identity for Weyl symmetry we derive a relation between the Weyl anomaly coefficients of the original and dual theories. The analysis is not restricted to conformally invariant backgrounds. The formalism is applied to the study of two examples. The first is a model based on SU(2) non-Abelian T duality. The second represents a simple realization of Poisson-Lie T duality involving the Drinfeld double based on SU(2). In both cases quantum T duality is established at the 1-loop level. 
  We discuss some properties of the conjectured M-9-brane. We investigate both the worldvolume action as well as the target space solution. The worldvolume action is given by a gauged sigma model which, via dimensional reduction and duality, is shown to be related to the worldvolume actions of the branes of ten- dimensional superstring theory. The effective tension of the M-9-brane scales as (R_{11})^3, where R_{11} is the radius of an S^1-isometry direction. This isometry enables us to add a cosmological constant to eleven-dimensional supergravity. The target space solution corresponding to the M-9-brane is a (wrapped) domain wall of massive eleven-dimensional supergravity. This solution breaks half of the bulk supersymmetry. We consider both single M-9-branes as well as a system of two M-9-branes. In both cases one can define regions in spacetime, separated by the domain walls, with zero cosmological constant. In these regions the limit R_{11}-> \infty can be taken in which case the M-9-brane is unwrapped and a massless eleven-dimensional supergravity is obtained. 
  The non-relativistic formalism introduced by Berry and Robbins that naturally incorporates the spin-statistics connection is generalized relativistically. It is then extended to an arbitrary Kaluza-Klein space-time by a suitable generalization of the Schwinger treatment of angular momenta. This leads, in this approach, to the inclusion of the `internal' quantum numbers in the spin-statistics connection on an equal footing with spin. 
  At low energy the near horizon geometry of nonextreme black holes in four dimensions exhibits an effective SL(2,R)_L x SL(2,R)_R symmetry. The parameters of the corresponding induced conformal field theory gives the correct expression for the black hole entropy. The resulting spectrum of the Schwarzchild black hole is compared with another proposal. 
  We show that $p$-forms on $AdS_{2p+1}$ describe both singletons and massless particles. On the $2p$-dimensional boundary the singleton $p$-form Lagrangian reduces to the conformally invariant functional $\int F^2$. All the representations, singletons as well as massless, are zero center modules and involve a vacuum mode. Two- and three-form singleton fields are required by supersymmetry in AdS$_5$ and AdS$_7$ supergravity respectively. 
  While perturbative techniques work extremely well for weakly interacting field theories (e.g. QED), they are not useful when studying strongly interacting field theories (e.g. QCD at low energies). In this paper we review Heisenberg's idea about quantizing strongly interacting, non-linear fields, and an approximate method of solving the infinite set of Tamm-Dankoff equations is suggested. We then apply this procedure to an infinite energy, classical flux tube-like solution of SU(2) Yang-Mills theory and show that this quantization procedure ameliorates some of the bad behaviour of the classical solution. We also discuss the possible application of this quantization procedure to a recently proposed strongly interacting phonon model of High-$T_c$ superconductors. 
  We study all three-point functions of normalized chiral operators in D=4, $\CN=4$, U(N) supersymmetric Yang-Mills theory in the large $N$ limit. We compute them for small 't Hooft coupling $\lambda=g_{YM}^2N<<1$ using free field theory and at strong coupling $\lambda=g_{YM}^2>>1$ using the $AdS$/CFT correspondence. Surprisingly, we find the same answers in the two limits. We conjecture that at least for large $N$ the exact answers are independent of $\lambda $ . 
  We explore the consequences of imposing Polyakov's zig/zag-invariance in the search for a confining string. We first find that the requirement of zig/zag-invariance seems to be incompatible with spacetime supersymmetry. We then try to find zig/zag-invariant string backgrounds on which to implement the minimal-area prescription for the calculation of Wilson loops considering different possibilities. 
  Starting with two supersymmetric dual theories, we imagine adding a chiral perturbation that breaks supersymmetry dynamically. At low energy we then get two theories with soft supersymmetry-breaking terms that are generated dynamically. With a canonical Kahler potential, some of the scalars of the "magnetic" theory typically have negative mass-squared, and the vector-like symmetry is broken. Since for large supersymmetry breaking the "electric" theory becomes ordinary QCD, the two theories are then incompatible. For small supersymmetry breaking, if duality still holds, the magnetic theory analysis implies specific patterns of chiral symmetry breaking in supersymmetric QCD with small soft masses. 
  Two supersymmetric classical mechanical systems are discussed. Concrete realizations are obtained by supposing that the dynamical variables take values in a Grassmann algebra with two generators. The equations of motion are explicitly solved. 
  It is argued that degrees of freedom responsible for the Bekenstein-Hawking entropy of a black hole in induced gravity are described by two dimensional quantum field theory defined on the bifurcation surface of the horizon. This result is proved for a class of induced gravity models with scalar, spinor and vector heavy constituents. 
  We describe the low-energy dynamics of N=1 supersymmetric gauge theories with the Dynkin index of matter fields less than or equal to the Dynkin index of the adjoint plus two. We explain what kinds of nonperturbative phenomena take place in this class of supersymmetric gauge theories. 
  We study Type I string theory compactified on a T^6/Z_3 orientifold. The low-energy dynamics is most conveniently analyzed in terms of D3-branes. We show that a sector of the theory, which corresponds to placing an odd number of D3-branes at orientifold fixed points, can give rise to an SU(5) gauge theory with three generations of chiral matter fields. The resulting model is not fully realistic, but the relative ease with which an adequate gauge group and matter content can be obtained is promising. The model is also of interest from the point of view of supersymmetry breaking. We show that, for fixed values of the closed string modes, the model breaks supersymmetry due to a conflict between a non-perturbatively generated superpotential and an anomalous U(1) D-term potential. 
  The technology required for eikonal scattering amplitude calculations in Matrix theory is developed. Using the entire supersymmetric completion of the v^4/r^7 Matrix theory potential we compute the graviton-graviton scattering amplitude and find agreement with eleven dimensional supergravity at tree level. 
  The potential describing long-range interactions between D0-branes contains spin-dependent terms. In the matrix model, these should be reproduced by the one-loop effective action computed in the presence of a nontrivial fermionic background $\psi.$ The $\frac{v^3 \psi^2}{r^8}$ term in the effective action has been computed by Kraus and shown to correspond to a spin-orbit interaction between D0-branes, and the $\frac{\psi^8}{r^{11}}$ term in the static potential has been obtained by Barrio et al. In this paper, the $\frac{v^2 \psi^4}{r^9}$ term is computing in the matrix model and compared with the corresponding results of Morales et al obtained using string theoretic methods. The technique employed is adapted to the underlying supersymmetry of the matrix model, and should be useful in the calculation of spin-dependent effects in more general Dp-brane scatterings. 
  The integrability of the one-dimensional (1D) fermion chain model is investigated in the framework of the Quantum Inverse Scattering Method (QISM). We introduce a new R-operator for the fermion chain model, which is expressed in terms of the fermion operators. The R-operator satisfies a new type of the Yang-Baxter relation with fermionic L-operator. We derive the fermionic Sutherland equation from the relation, which is equivalent to the fermionic Lax equation. It also provides a mathematical foundation of the boost operator approach for the fermion model. In fact, we obtain some higher conserved quantities of the fermion model using the boost operator. 
  We first consider nonlinear Grassmann sigma models in any dimension and next construct their submodels. For these models we construct an infinite number of nontrivial conserved currents. Our result is independent of time-space dimensions and, therfore, is a full generalization of that of authors (Alvarez, Ferreira and Guillen). Our result also suggests that our method may be applied to other nonlinear sigma models such as chiral models, $G/H$ sigma models in any dimension. 
  We show that a simple OSp(1/2) worldline gauge theory in 0-brane phase space (X,P), with spin degrees of freedom, formulated for a d+2 dimensional spacetime with two times X^0,, X^0', unifies many physical systems which ordinarily are described by a 1-time formulation. Different systems of 1-time physics emerge by choosing gauges that embed ordinary time in d+2 dimensions in different ways. The embeddings have different topology and geometry for the choice of time among the d+2 dimensions. Thus, 2-time physics unifies an infinite number of 1-time physical interacting systems, and establishes a kind of duality among them. One manifestation of the two times is that all of these physical systems have the same quantum Hilbert space in the form of a unique representation of SO(d,2) with the same Casimir eigenvalues. By changing the number n of spinning degrees of freedom the gauge group changes to OSp(n/2). Then the eigenvalue of the Casimirs of SO(d,2) depend on n and then the content of the 1-time physical systems that are unified in the same representation depend on n. The models we study raise new questions about the nature of spacetime. 
  We discuss a magnetic black-hole solution to the equations of motion of the string-loop-corrected effective action. At the string-tree level, this solution is the extremal magnetic black hole described by the "chiral null model." In the extremal case, the string-loop correction is constant, and this fact is used to analytically solve the loop-corrected equations of motion. In distinction to the tree-level solution, the resulting loop-corrected solution has the horizon at a finite distance from the origin; its location is a function of the loop correction. The loop-corrected configuration is compared with a string-tree-level non-extremal magnetic black hole solution which also has the horizon at a finite distance from the origin. We find that for an appropriate choice of the free parameters of solutions, the loop-corrected magnetic black hole can be approximated by a tree-level non-extremal solution. We compare the thermodynamic properties of the loop-corrrected and non-extremal solutions. 
  We calculate the Weyl anomaly for conformal field theories that can be described via the adS/CFT correspondence. This entails regularizing the gravitational part of the corresponding supergravity action in a manner consistent with general covariance. Up to a constant, the anomaly only depends on the dimension d of the manifold on which the conformal field theory is defined. We present concrete expressions for the anomaly in the physically relevant cases d = 2, 4 and 6. In d = 2 we find for the central charge c = 3 l/ 2 G_N in agreement with considerations based on the asymptotic symmetry algebra of adS_3. In d = 4 the anomaly agrees precisely with that of the corresponding N = 4 superconformal SU(N) gauge theory. The result in d = 6 provides new information for the (0, 2) theory, since its Weyl anomaly has not been computed previously. The anomaly in this case grows as N^3, where N is the number of coincident M5 branes, and it vanishes for a Ricci-flat background. 
  Generalisations of the 't Hooft-Polyakov monopole which can exhibit repulsion only, attraction only, and both attraction and repulsion, between like monopoles, are studied numerically. The models supporting these solitons are SO(3) gauged Higgs models featuring Skyrme-like terms. 
  This paper collects the various ways of computing the central charge $c=3l/2G$ arising in 3d asymptotically anti-de Sitter spaces, in the Chern-Simons formulation. Their similarities and differences are displayed. 
  We propose an alternative description of the spectrum of local fields in the classical limit of the integrable quantum field theories. It is close to similar constructions used in the geometrical treatment of W-gravities. Our approach provides a systematic way of deriving the null-vectors that appear in this construction. We present explicit results for the case of the A_1^{1}-(m)KdV and the A_2^{2}-(m)KdV hierarchies, different classical limits of 2D CFT's. In the former case our results coincide with the classical limit of the construction of Babelon, Bernard and Smirnov.Some hints about quantization and off-critical treatment are also given. 
  We construct anomaly free non-supersymmetric large N gauge theories from orientifolds of Type IIB on C^3/G orbifolds. In particular, massless as well as tachyonic one-loop tadpoles are cancelled in these models. This is achieved by starting with ${\cal N}=1,2$ supersymmetric orientifolds with well defined world-sheet description and including discrete torsion (which breaks supersymmetry) in the orbifold action. In this way we obtain non-trivial non-chiral as well as anomaly free chiral large N gauge theories. We point out certain subtleties arising in the chiral cases. Subject to certain assumptions, these theories are shown to have the property that computation of any M-point correlation function in these theories reduces to the corresponding computation in the parent ${\cal N}=4$ oriented theory. This generalizes the analogous results recently obtained in supersymmetric large N gauge theories from orientifolds, as well as in (non)supersymmetric large N gauge theories without orientifold planes. 
  We study the type IIB brane box configurations recently introduced by Hanany and Zaffaroni. We show that even at finite string coupling, one can construct smooth configurations of branes with fairly arbitrary gauge and flavor structure. Limiting our attention to the better understood case where NS-branes do not intersect over a four dimensional surface gives some restrictions on the theories, but still permits many examples, both anomalous and non-anomalous. We give several explicit examples of such configurations and discuss what constraints can be imposed on brane-box theories from bending considerations. We also discuss the relation between brane bending and beta-functions for brane-box configurations. 
  We propose the mechanism of quantum creation of the open Universe in the observable range of values of $\Omega$. This mechanism is based on the no-boundary quantum state with the Hawking-Turok instanton in the model with nonminimally coupled inflaton field and does not use any anthropic considerations. Rather, the probability distribution peak with necessary parameters of the inflation stage is generated on this instanton due to quantum loop effects. In contrast with a similar mechanism for closed models, existing only for the tunneling quantum state of the Universe, open inflation originates from the no-boundary cosmological wavefunction. 
  We consider the compactification of the E8xE8 heterotic string on a K3 surface with "the spin connection embedded in the gauge group" and the dual picture in the type IIA string (or F-theory) on a Calabi-Yau threefold X. It turns out that the same X arises also as dual to a heterotic compactification on 24 point-like instantons. X is necessarily singular, and we see that this singularity allows the Ramond-Ramond moduli on X to split into distinct components, one containing the (dual of the heterotic) tangent bundle, while another component contains the point-like instantons. As a practical application we derive the result that a heterotic string compactified on the tangent bundle of a K3 with ADE singularities acquires nonperturbatively enhanced gauge symmetry in just the same fashion as a type IIA string on a singular K3 surface. On a more philosophical level we discuss how it appears to be natural to say that the heterotic string is compactified using an object in the derived category of coherent sheaves. This is necessary to properly extend the notion of T-duality to the heterotic string on a K3 surface. 
  We find the space-time supersymmetric and kappa-invariant action for a D3-brane propagating in the AdS_5 x S^5 background. As in the previous construction of the fundamental string action in this maximally supersymmetric string vacuum the starting point is the corresponding superalgebra su(2,2|4). We comment on the super Yang-Mills interpretation of the gauge-fixed form of the action. 
  We illustrate a physical situation in which topological symmetry, its breakdown, space-time uncertainty principle, and background independence may play an important role in constructing and understanding matrix models. First, we show that the space-time uncertainty principle of string may be understood as a manifestation of the breakdown of the topological symmetry in the large $N$ matrix model. Next, we construct a new type of matrix models which is a matrix model analog of the topological Chern-Simons and BF theories. It is of interest that these topological matrix models are not only completely independent of the background metric but also have nontrivial "p-brane" solutions as well as commuting classical space-time as the classical solutions. In this paper, we would like to point out some elementary and unsolved problems associated to the matrix models, whose resolution would lead to the more satisfying matrix model in future. 
  A general framework is presented for the renormalization of Hamiltonians via a similarity transformation. Divergences in the similarity flow equations may be handled with dimensional regularization in this approach, and the resulting effective Hamiltonian is finite since states well-separated in energy are uncoupled. Specific schemes developed several years ago by Glazek and Wilson and contemporaneously by Wegner correspond to particular choices within this framework, and the relative merits of such choices are discussed from this vantage point. It is shown that a scheme for the transformation of Hamiltonians introduced by Dyson in the early 1950's also corresponds to a particular choice within the similarity renormalization framework, and it is argued that Dyson's scheme is preferable to the others for ease of computation. As an example, it is shown how a logarithmically confining potential arises simply at second order in light-front QCD within Dyson's scheme, a result found previously for other similarity renormalization schemes. Steps toward higher order and nonperturbative calculations are outlined. In particular, a set of equations analogous to Dyson-Schwinger equations is developed. 
  The non-Abelian Stokes theorem for loop variables associated with nontrivial loops (knots and links) is derived. It is shown that a loop variable is in general different from unity even if the field strength vanishes everywhere on the surface surrounded by the loop. 
  We construct a new gauge theory on a pair of d-dimensional noncommutative tori. The latter comes from an intimate relationship between the noncommutative geometry associated with a lattice vertex operator algebra A and the noncommutative torus. We show that the (truncated) tachyon subalgebra of A is naturally isomorphic to a class of twisted modules representing quantum deformations of the algebra of functions on the torus. We construct the corresponding even real spectral triples and determine their Morita equivalence classes using string duality arguments. These constructions yield simple proofs of the O(d,d;Z) Morita equivalences between $d$-dimensional noncommutative tori and give a natural physical interpretation of them in terms of the target space duality group of toroidally compactified string theory. We classify the automorphisms of the twisted modules and construct the most general gauge theory which is invariant under the automorphism group. We compute bosonic and fermionic actions associated with these gauge theories and show that they are explicitly duality-symmetric. The duality-invariant gauge theory is manifestly covariant but contains highly non-local interactions. We show that it also admits a new sort of particle-antiparticle duality which enables the construction of instanton field configurations in any dimension. The duality non-symmetric on-shell projection of the field theory is shown to coincide with the standard non-abelian Yang-Mills gauge theory minimally coupled to massive Dirac fermion fields. 
  We determine the representations of the ``conformal'' group ${\bar{SO}}_0(2, n)$, the restriction of which on the ``Poincar\'e'' subgroup ${\bar{SO}}_0(1, n-1).T_n$ are unitary irreducible. We study their restrictions to the ``De Sitter'' subgroups ${\bar{SO}}_0(1, n)$ and ${\bar{SO}}_0(2, n-1)$ (they remain irreducible or decompose into a sum of two) and the contraction of the latter to ``Poincar\'e''. Then we discuss the notion of masslessness in $n$ dimensions and compare the situation for general $n$ with the well-known case of 4-dimensional space-time, showing the specificity of the latter. 
  Various properties of two kinds of massless representations of the n-conformal (or (n+1)-De Sitter) group $\tilde{G}_n=\widetilde{SO}_0(2,n)$ are investigated for $n\ge2$. It is found that, for space-time dimensions $n\ge3$, the situation is quite similar to the one of the n=4 case for $S_n$-massless representations of the n-De Sitter group $\widetilde{SO}_0(2,n-1)$. These representations are the restrictions of the singletons of $\tilde{G}_n$. The main difference is that they are not contained in the tensor product of two UIRs with the same sign of energy when n>4, whereas it is the case for another kind of massless representation. Finally some examples of Gupta-Bleuler triplets are given for arbitrary spin and $n\ge3$. 
  Using the method of correlation dynamics we investigate the properties of a field-theory for fermions and scalar bosons coupled via a Yukawa interaction. Within this approach, which consists in an expansion of full equal-time Green functions into connected equal-time Green functions and a corresponding truncation of the hierarchy of equations of motion we carry out calculations up to 4th order in the connected Green functions and evaluate the effective potential of the theory in 1+1 dimensions on a torus. Comparing the different approximations we find a strong influence of the connected 4-point functions on the properties of the system. 
  Following the observation of Banks and Green that the D-instantons in AdS_5 correspond to the instantons in 4-dimensional supersymmetric Yang-Mills theory, we study in more detail this correspondence for individual instantons. The supergravity solution for a D-instanton in AdS_5 is found using the ansatz used previously for D-instantons in flat space. We check that the actions and supersymmetries match between the D-instanton solution and the Yang-Mills instanton. Generalizing this result, we propose that any supergravity solution satisfying the ansatz corresponds to a (anti-)self-dual Yang-Mills configuration. Using this ansatz a family of identities for correlation functions in the supersymmetric Yang-Mills theory are derived. 
  In this paper we study the relation between six-dimensional supergravity compactified on S^3 X AdS_3 and certain two-dimensional conformal field theories. We compute the Kaluza-Klein spectrum of supergravity using representation theory; these methods are quite general and can also be applied to other compactifications involving anti-de Sitter spaces. A detailed comparison between the spectrum of the two-dimensional conformal field theory and supergravity is made, and we find complete agreement. This applies even at the level of certain non-chiral primaries, and we propose a resolution to the puzzle of the missing states recently raised by Vafa. As a further illustration of the method the Kaluza-Klein spectra of F-theory on M^6 X S^3 X AdS_3 and of M-theory on M^6 X S^2 X AdS_3 are computed, with M^6 some Calabi-Yau manifold. 
  We study the self-duality of Born-Infeld-Chern-Simons theory which can be interpreted as a massive D2 brane in IIA string theory and exhibit the self-dual formulation in terms of the gauge invariant master Lagrangian. The proposed master Lagrangian contains the nonlocal auxiliary field and approaches self-dual formulation of Maxwell-Chern-Simons theory in a point-particle limit with the weak string-coupling limit. The consistent canonical brackets of dual system are derived. 
  We introduce a doubled formalism for the bosonic sector of the maximal supergravities, in which a Hodge dual potential is introduced for each bosonic field (except for the metric). The equations of motion can then be formulated as a twisted self-duality condition on the total field strength \G, which takes its values in a Lie superalgebra. This doubling is invariant under dualisations; it allows a unification of the gauge symmetries of all degrees, including the usual U-dualities that have degree zero. These ``superdualities'' encompass the dualities for all choices of polarisation (i.e. the choices between fields and their duals). All gauge symmetries appear as subgroups of finite-dimensional supergroups, with Grassmann coefficients in the differential algebra of the spacetime manifold. 
  GSO projected Superconformal field theory (Spin Model) with boundary is considered. There were written the boundary states. For this model were derived one-point structure constants and "bootstrap" equations for boundary-bulk structure constants. 
  We present detailed analyses of the 3-body interactions of D-particles from both sides of 11 dimensional supergravity and Matrix theory. In supergravity, we derive a complete expression for the classical bosonic effective action for D-particles including 2-and 3-body interaction terms. In Matrix theory, we compute 1-particle irreducible contributions to the eikonal phase shift in the two-loop approximation. The results precisely agree with the predictions from supergravity and thus provide a strong support to the discrete light-cone interpretation of the Matrix-theory conjecture as a possible nonperturbative definition of M-theory. 
  We show by explicit calculation that the matrix model effective action does not contain the term $v_{12}^2 v_{23}^2 v_{13}^2/{R^7 r^7}$, in the limit $R \gg r$, contradicting a result reported recently. 
  We study the spectrum of the QCD Dirac operator by means of the valence quark mass dependence of the chiral condensate in partially quenched Chiral Perturbation Theory (pqChPT) in the supersymmetric formulation of Bernard and Golterman. We consider valence quark masses both in the ergodic domain ($m_v \ll E_c$) and the diffusive domain ($m_v \gg E_c$). These domains are separated by a mass scale $E_c \sim F^2/\Sigma_0 L^2$ (with $F$ the pion decay constant, $\Sigma_0$ the chiral condensate and $L$ the size of the box). In the ergodic domain the effective super-Lagrangian reproduces the microscopic spectral density of chiral Random Matrix Theory (chRMT). We obtain a natural explanation of Damgaard's relation between the spectral density and the finite volume partition function with two additional flavors. We argue that in the ergodic domain the natural measure for the superunitary integration in the pqChPT partition function is noncompact. We find that the tail of the two-point spectral correlation function derived from pqChPT agrees with the chRMT result in the ergodic domain. In the diffusive domain we extend the results for the slope of the Dirac spectrum first obtained by Smilga and Stern. We find that the spectral density diverges logarithmically for nonzero topological susceptibility. We study the transition between the ergodic and the diffusive domain and identify a range where chRMT and pqChPT coincide. 
  We discuss the problem of time in spherically symmetric pure Einstein gravity with the cosmological term by using an exact solution to the Wheeler-DeWitt equation. A positive definite inner product is defined, based on the momentum constraint rather than the hamiltonian constraint. A natural notion of time is introduced via the Heisenberg equation. This notion enables one to reproduce the time-time component of the classical metric. Non-Hermiticity of the hamiltonian is essential in the definition of time. 
  We present an efficient method to understand the p-brane dynamics in a unified framework. For this purpose, we reformulate the action for super p-branes in the form appropriate to incorporate the pointlike (parton) structure of higher dimensional p-branes and intend to interpret the p-brane dynamics as the collective dynamics of superparticles. In order to examine such a parton picture of super p-branes, we consider various superparticle configurations that can be reduced from super p-branes, especially, a supermembrane, and study the partonic structure of classical p-brane solutions. 
  We construct an action, which governs the dynamics of the Ba\~nados-Teitelboim-Zanelli (BTZ) black hole and perform the canonical quantization. The quantum action is given by a $SL(2,R)$ Wess-Zumino-Witten model on the boundary coupled to the classical anti-de Sitter background, representing a massless BTZ black hole. The coupling, determined by a one-cocyle condition, is found to give dominant contribution to the central charge of Virasoro algebra. The entropy of the BTZ black hole is discussed from the point view of the AdS/CFT correspondence and an explanation is given to the puzzle of black hole entropy in the BTZ case. The BTZ black hole is a quantum object and the BTZ black hole with finite mass should be considered as a quantum excitation of the massless one. 
  A new type of quantum master equation is presented which is expressed in terms of a recently introduced quantum antibracket. The equation involves only two operators: an extended nilpotent BFV-BRST charge and an extended ghost charge. It is proposed to determine the generalized quantum Maurer-Cartan equations for arbitrary open groups. These groups are the integration of constraints in arbitrary involutions. The only condition for this is that the constraint operators may be embedded in an odd nilpotent operator, the BFV-BRST charge. The proposal is verified at the quasigroup level. The integration formulas are also used to construct a generating operator for quantum antibrackets of operators in arbitrary involutions. 
  The SO(32) theory, in the limit where it is an open superstring theory, is completely specified in the light-cone gauge as a second-quantized string theory in terms of a ``matrix string'' model. The theory is defined by the neighbourhood of a 1+1 dimensional fixed point theory, characterized by an Abelian gauge theory with type IB Green-Schwarz form. Non-orientability and SO(32) gauge symmetry arise naturally, and the theory effectively constructs an orientifold projection of the (weakly coupled) matrix type IIB theory (also discussed herein). The fixed point theory is a conformal field theory with boundary, defining the free string theory. Interactions involving the interior of open and closed strings are governed by a twist operator in the bulk, while string end-points are created and destroyed by a boundary twist operator. 
  We calculate asymptotic expansions of elliptic genera for a supersymmetric sigma model on the N - fold symmetric product S^NM of a Kahler manifold M and for N = 2 superconformal field theory. Asymptotic expansions for the degeneracy of dyonic black hole spectrum are also derived. 
  The osp(1,2)-covariant Lagrangian quantization of irreducible gauge theories [hep-th/9712204] is generalized to L-stage reducible theories. The dependence of the generating functional of Green's functions on the choice of gauge in the massive case is dicussed and Ward identities related to osp(1,2) symmetry are given. Massive first stage theories with closed gauge algebra are studied in detail. The generalization of the Chapline-Manton model and topological Yang-Mills theory to the case of massive fields is consedered as examples. 
  We will establish the connection between the Lorentz covariant and so-called single-time formulation for the quark Wigner operator. To this end we will discuss the initial value problem for the Wigner operator of a field theory and give a discussion of the gauge-covariant formulation for the Wigner operator including some new results concerning the chiral limit. We discuss the gradient or semi-classical expansion and the color and spinor decomposition of the equations of motion for the Wigner operator. The single-time formulation will be derived from the covariant formulation by taking energy moments of the equations for the Wigner operator. For external fields we prove that only the lowest energy moments of the quark Wigner operator contain dynamical information. 
  I consider the (2+1)-dimensional Kerr-De Sitter space and its statistical entropy computation. It is shown that this space has only one (cosmological) event horizon and there is a phase transition between the stable horizon and the evaporating horizon at a point $M^2={1/3}J^2/l^2$ together with a lower bound of the horizon temperature. Then, I compute the statistical entropy of the space by using a recently developed formulation of Chern-Simons theory with boundaries, and extended Cardy's formula. This is in agreement with the thermodynamics formula. 
  We show that only by performing generalized dimensional reductions all possible brane configurations are taken into account and one gets the complete lower-dimensional theory. We apply this idea to the reduction of type IIB supergravity in an SL(2,R)-covariant way and establish T duality for the type II superstring effective action in the context of generalized dimensional reduction giving the corresponding generalized Buscher's T duality rules. The full (generalized) dimensional reduction involves all the S duals of D-7-branes: Q-7-branes and a sort of composite 7-branes. The three species constitute an SL(2,Z) triplet. Their presence induces the appearance of the triplet of masses of the 9-dimensional theory. The T duals, including a ``KK-8A-brane'', which must have a compact transverse dimension have to be considered in the type IIA side. Compactification of 11-dimensional KK-9M-branes (a.k.a. M-9-branes) on the compact transverse dimension give D-8-branes while compactification on a worldvolume dimension gives KK-8A-branes. The presence of these KK-monopole-type objects breaks translation invariance and two of them given rise to an SL(2,R)-covariant ``massive 11-dimensional supergravity'' whose reduction gives the massive 9-dimensional type II theories. 
  We show that the conformally invariant boundary conditions for the three-state Potts model are exhausted by the eight known solutions. Their structure is seen to be similar to the one in a free field theory that leads to the existence of D-branes in string theory. Specifically, the fixed and mixed boundary conditions correspond to Neumann conditions, while the free boundary condition and the new one recently found by Affleck et al [1] have a natural interpretation as Dirichlet conditions for a higher-spin current. The latter two conditions are governed by the Lee\hy Yang fusion rules. These results can be generalized to an infinite series of non-diagonal minimal models, and beyond. 
  2+1-dimensional Anti-deSitter gravity is quantized in the presence of an external scalar field. We find that the coupling between the scalar field and gravity is equivalently described by a perturbed conformal field theory at the boundary of AdS_3. We derive the explicit form of this coupling, which allows us to perform a microscopic computation of the transition rates between black hole states due to absorption and induced emission of the scalar field. Detailed thermodynamic balance then yields Hawking radiation as spontaneous emission, and we find agreement with the semiclassical result, including greybody factors. This result also has application to four and five dimensional black holes in supergravity. However, since we only deal with gravitational degrees of freedom, the approach is not based on string theory, and does not depend, either, on the validity of Maldacena's AdS/CFT conjecture. 
  This work is devoted to a study of some of the non-perturbative aspects of superstring theory that have come within reach thanks to dualities. The first chapter gives a non-technical zero-equation introduction to these developments intended for readers with a knowledge of quantum field theory (and french). The second chapter introduces the non-perturbative dualities that have been observed in gauge and supergravity theories as well as the non-perturbative BPS spectrum, and presents a novel duality relating the Higgs branches of certain supersymmetric N=2 gauge theories. In the third chapter, we briefly introduce the perturbative string theories and give some checks on the duality between four-dimensional N=4 string theories. From duality arguments, we obtain exact non-perturbative results for various couplings in the low energy effective action, and interpret the corresponding non-perturbative effects in terms of instantonic configurations of p-branes wrapped on supersymmetric cycles of the compactification manifold. Finally, we discuss the M(atrix) theory proposal of definition of M-theory in terms of large N supersymmetric U(N) gauge theories; we extend it to toroidal compactifications in constant background fields, and interpret the resulting spectrum of BPS states in terms of excitations in the gauge theory. 
  The ground state of string theory may lie at a point of ``maximally enhanced symmetry", at which all of the moduli transform under continuous or discrete symmetries. This hypothesis, along with the hypotheses that the theory at high energies has N=1 supersymmetry and that the gauge couplings are weak and unified, has definite consequences for low energy physics. We describe these, and offer some suggestions as to how these assumptions might be compatible. 
  In the framework of the conjectured duality relation between large $N$ gauge theory and supergravity the spectra of masses in large $N$ gauge theory can be determined by solving certain eigenvalue problems in supergravity. In this paper we study the eigenmass problem given by Witten as a possible approximation for masses in QCD without supersymmetry. We place a particular emphasis on the treatment of the horizon and related boundary conditions. We construct exact expressions for the analytic expansions of the wave functions both at the horizon and at infinity and show that requiring smoothness at the horizon and normalizability gives a well defined eigenvalue problem. We show for example that there are no smooth solutions with vanishing derivative at the horizon. The mass eigenvalues up to $m^{2}=1000$ corresponding to smooth normalizable wave functions are presented. We comment on the relation of our work with the results found in a recent paper by Cs\'aki et al., hep-th/9806021, which addresses the same problem. 
  We explicitly write down the invariant supersymmetry conditions for branes with generic values of moduli and U-duality charges in various space-time dimensions $D \leq 10$. We then use these results to obtain new BPS states, corresponding to network type structure of such branes. 
  We find the inconsistency of dimensional reduction and naive dimensional regularization in their applications to Chern-Simons type gauge theories. Further we adopt a consistent dimensional regularization to investigate the quantum correction to non-Abelian Chern-Simons term coupled with fermionic matter. Contrary to previous results, we find that not only the Chern-Simons coefficient receives quantum correction from spinor fields, but the spinor field also gets a finite quantum correction. 
  A conjectured duality between supergravity and $N=\infty$ gauge theories gives predictions for the glueball masses as eigenvalues for a supergravity wave equations in a black hole geometry, and describes a physics, most relevant to a high-temeperature expansion of a lattice QCD. We present an analytical solution for eigenvalues and eigenfunctions, with eigenvalues given by zeroes of a certain well-computable function $r(p)$, which signify that the two solutions with desired behaviour at two singular points become linearly dependent. Our computation shows corrections to the WKB formula $m^2= 6n(n+1)$ for eigenvalues corresponding to glueball masses QCD-3, and gives the first states with masses $m^2=$ 11.58766; 34.52698; 68.974962; 114.91044; 172.33171; 241.236607; 321.626549, ... . In $QCD_4$, our computation gives squares of masses 37.169908; 81.354363; 138.473573; 208.859215; 292.583628; 389.671368; 500.132850; 623.97315 ... for $O++$. In both cases, we have a powerful method which allows to compute eigenvalues with an arbitrary precision, if needed so, which may provide quantative tests for the duality conjecture. Our results matches with the numerical computation of [5] well withing precision reported there in both $QCD_3$ and $QCD_4$ cases. As an additional curiosity, we report that for eigenvalues of about 7000, the power series, although convergent, has coefficients of orders ${10}^{34}$; tricks we used to get reliably the function $r(p)$, as also the final answer gets small, of order ${10}^{-6}$ in $QCD_4$. In principle we can go to infinitely high eigenavalues, but such computations maybe impractical due to corrections. 
  In this note we clarify some issues in six-dimensional (1,0) supergravity coupled to vector and tensor multiplets. In particular, we show that, while the low-energy equations embody tensor-vector couplings that contribute only to gauge anomalies, the divergence of the energy-momentum tensor is properly non-vanishing. In addition, we show how to revert to a supersymmetric formulation in terms of covariant non-integrable field equations that embody corresponding covariant anomalies. 
  We consider g coincident M-5-branes on top of each other, in the KK monopole background Q of multiplicity N. The worldvolume of each M-5-brane is supposed to be given by the local product of the four-dimensional spacetime and an elliptic curve. In the coincidence limit, all these curves yield a single (Seiberg-Witten) hyperelliptic curve, while the gauge symmetry is enhanced to U(N). We make this gauge symmetry enhancement manifest by considering the hypermultiplet LEEA which is given by the spacetime N=2 non-linear sigma-model (NLSM) having Q as the target space. The hyper-K"ahler manifold Q is given by the multicentre Taub-NUT space, which in the coincidence limit amounts to the multiple Eguchi-Hanson (ALE) space Q. The NLSM is most naturally described in terms of the hyper-K"ahler coset construction on SU(N,N)/U(N) in harmonic superspace, by using the auxiliary (in classical theory) N=2 vector superfields as Lagrange multipliers, with FI terms resolving the singularity. The Maldacena limit, in which the hypermultiplet LEEA becomes extended to the N=4 SYM with the gauge group U(N), arises in quantum field theory due to a dynamical generation of the N=2 vector and hypermultipet superfields, when sending the FI terms to zero. 
  We compute the massless spectra of some four dimensional, N=1 supersymmetric compactifications of the type I string. The backgrounds are non-toroidal Calabi-Yau manifolds described at special points in moduli space by Gepner models. Surprisingly, the abstract conformal field theory computation reveals Chan-Paton gauge groups as big as SO(12) x SO(20) or SO(8)^4 x SO(4)^3. 
  We consider scattering of minimally coupled scalars from a six-dimensional black string carrying one and five brane charges but no Kaluza-Klein momentum. The leading correction to the absorption cross section is found by improved matching of inner and outer solutions to the wave equation. The world sheet interpretation of this correction follows from the breaking of conformal invariance by irrelevant Born-Infeld corrections. We note that discrepancies in normalisation are caused by there being two effective length scales in the black string geometry but only one in the effective string model and comment on the implications of our results for the effective string model. 
  We consider the 1+1 dimensional N = (8,8) supersymmetric matrix field theory obtained from a dimensional reduction of ten dimensional N = 1 super Yang-Mills. The gauge groups we consider are U(N) and SU(N), where N is finite but arbitrary. We adopt light-cone coordinates, and choose to work in the light-cone gauge. Quantizing this theory via Discretized Light-Cone Quantization (DLCQ) introduces an integer, K, which restricts the light-cone momentum-fraction of constituent quanta to be integer multiples of 1/K. Solutions to the DLCQ bound state equations are obtained for K=2,3 and 4 by discretizing the light-cone super charges, which preserves supersymmetry manifestly. We discuss degeneracies in the massive spectrum that appear to be independent of the light-cone compactification, and are therefore expected to be present in the decompactified limit K -> infinity. Our numerical results also support the claim that the SU(N) theory has a mass gap. 
  We consider a model for tensionless (null) super p-branes in the Hamiltonian approach and in the framework of a harmonic superspace. The obtained algebra of Lorentz-covariant, irreducible, first class constraints is such that the BRST charge corresponds to a first rank system. 
  Starting from the working hypothesis that both physics and the corresponding mathematics have to be described by means of discrete concepts on the Planck scale, one of the many problems one has to face in this enterprise is to find the discrete protoforms of the building blocks of our ordinary continuum physics and mathematics. We base our own approach on what we call `cellular networks', consisting of cells (nodes) interacting with each other via bonds (figuring as elementary interactions) according to a certain `local law'. Geometrically our dynamical networks are living on graphs. Hence a substantial amount of the investigation is devoted to the developement of various versions of discrete (functional) analysis and geometry on such (almost random) webs. Another important topic we address is a suitable concept of intrinsic (fractal) dimension on erratic structures of this kind. In the course of the investigation we make comments concerning both different and related approaches to quantum gravity as, say, the spin network framework. It may perhaps be said that certain parts of our programme seem to be a realisation of ideas sketched by Smolin some time ago (see the introduction). 
  We compute the effective action for scattering of three well-separated extremal brane solutions, in 11d supergravity, with zero p_ transfer and small transverse velocities. Using an interpretation of the conjecture of Maldacena, following Hyun, this can be viewed as the large N limit of the Matrix theory description of three supergraviton scattering at leading order. The result is consistent with the perturbative supergravity calculation. 
  We reinvestigate the twisted N=4 supersymmetry present in Schwarz type topological field models. We show that Chern-Simons theory in three dimensions can be untwisted to a kind of sigma-model with reversed statistics only in the free case. By dimensional reduction we define then the two-dimensional BF-model. We establish an analog result concerning the untwisting. As a consequence of the definition through dimensional reduction we find new fermionic scalar symmetries that have been overlooked so far in the literature. 
  We investigate the radiative decay of the axion into two photons in an external electromagnetic field to one loop order. Our approach is based on the world-line formalism, which is very suitable to take into account the external field to all orders. Afterwards we discuss how the calculation could be generalized to finite temperature. 
  Berry's connection is computed in the USp(2k) matrix model. In T dualized quantum mechanics, the Berry phase exhibits a residual interaction taking place at a distance m_(f) from the orientifold surface via the integration of the fermions in the fundamental representation. This is interpreted as a coupling of the magnetic D2 with the electric D4 branes. We make a comment on the Berry phase associated with the 6D nonabelian gauge anomaly whose cancellation selects the number of flavours n_f=16. 
  We present a manifestly Lorentz invariant and supersymmetric component field action for $D = 10$, type $IIB$ supergravity, using a newly developed method for the construction of actions with chiral bosons, which implies only a single scalar non propagating auxiliary field. With the same method we construct also an action in which the complex two-form gauge potential and its Hodge-dual, a complex six-form gauge potential, appear in a symmetric way in compatibility with supersymmetry and Lorentz invariance. The duals of the two physical scalars of the theory turn out to be described by a $SL(2,R)$ triplet of eight-forms whose curvatures are constrained by a single linear relation.  We present also a supersymmetric action in which the basic fields and their duals, six-form and eight-form potentials, appear in a symmetric way. All these actions are manifestly invariant under the global $SL(2,R)$-duality group of $D = 10$, $IIB$ supergravity and are equivalent to each other in that their dynamics corresponds to the well known equations of motion of $ D=10$, $IIB$ supergravity. 
  Five-dimensional non-extreme rotating black holes with large NS-NS five-brane and fundamental string charge are shown to be described by a conformal sigma model, which is a marginal integrable deformation of six-dimensional SL(2,R) x SU(2) WZW model. The two WZW levels are equal to the five-brane charge, while the parameters of the two marginal deformations generated by the left and right chiral SU(2) currents are proportional to the two angular momentum components of the black hole. The near-horizon description is effectively in terms of a free fundamental string whose tension is rescaled by the five-brane charge. The microstates are identified with those of left and right moving superconformal string oscillations in the four directions transverse to the five-brane. Their statistical entropy reproduces precisely the Bekenstein-Hawking entropy of the rotating black hole. 
  We investigate the low energy structure of the Kahler potential in SUSY QCD with Nf=Nc+1 quark flavors. Since this theory's moduli space is everywhere smooth, a systematic power series expansion of its Kahler potential can be developed in terms of confined meson and baryon fields. Perturbation theory in the supersymmetric sigma model based upon a momentum expansion consistent with naive dimensional analysis and 1/Nf power counting exhibits some similarities with ordinary QCD chiral perturbation theory along with several key differences. We compute meson and baryon wavefunction renormalization as well as Kahler potential operator mixing to leading nontrivial order. We also deduce the asymptotic dependence of the lowest dimension operators' coefficients upon moduli space location along flat directions where the theory is Higgsed down to Nf-1=(Nc-1)+1 SUSY QCD. Although an exact form for the confining phase Kahler potential remains unknown, we find that some detailed Kahler sector information can nevertheless be derived from first principles. 
  Generalizing the recent work on three-family Type I compactifications, we classify perturbative Type I vacua obtained via compactifying on the T^6/Z_2 X Z_2 X Z_3 orbifold with all possible Wilson lines. In particular, we concentrate on models with gauge groups containing the Standard Model gauge group SU(3)_c X SU(2)_w X U(1)_Y as a subgroup. All of the vacua we obtain contain D5-branes and are non-perturbative from the heterotic viewpoint. The models we discuss have three-chiral families. We study some of their phenomenological properties, and point out non-trivial problems arising in these models in the phenomenological context. 
  One-instanton predictions are obtained from certain non-hyperelliptic Seiberg-Witten curves derived from M-theory for N=2 supersymmetric gauge theories. We consider SU(N_1) x SU(N_2) gauge theory with a hypermultiplet in the bifundamental representation together with hypermultiplets in the defining representations of SU(N_1) and SU(N_2). We also consider SU(N) gauge theory with a hypermultiplet in the symmetric or antisymmetric representation, together with hypermultiplets in the defining representation. The systematic perturbation expansion about a hyperelliptic curve together with the judicious use of an involution map for the curve of the product groups provide the principal tools of the calculations. 
  We show that a massive scalar field on a 2-dimensional Anti-De Sitter space can be mapped by means of a time-dependent canonical transformation into a massless free field theory when the mass square of the field is equal to minus the curvature of the background metric. We also provide the (hidden) conformal symmetry of the massive scalar field. 
  T-Duality on a timelike circle does not interchange IIA and IIB string theories, but takes the IIA theory to a type $IIB^*$ theory and the IIB theory to a type $IIA^*$ theory. The type $II^*$ theories admit E-branes, which are the images of the type II D-branes under timelike T-duality and correspond to imposing Dirichlet boundary conditions in time as well as some of the spatial directions. The effective action describing an E$n$-brane is the $n$-dimensional Euclidean super-Yang-Mills theory obtained by dimensionally reducing 9+1 dimensional super-Yang-Mills on $9-n$ spatial dimensions and one time dimension. The $IIB^*$ theory has a solution which is the product of 5-dimensional de Sitter space and a 5-hyperboloid, and the E4-brane corresponds to anon-singular complete solution which interpolates between this solution and flat space. This leads to a duality between the large $N$ limit of the Euclidean 4-dimensional U(N) super-Yang-Mills theory and the $IIB^*$ string theory in de Sitter space, and both are invariant under the same de Sitter supergroup. This theory can be twisted to obtain a large $N$ topological gauge theory and its topological string theory dual. Flat space-time may be an unstable vacuum for the type $II^*$ theories, but they have supersymmetric cosmological solutions. 
  We analyse the relationship between the N=2 harmonic and projective superspaces which are the only approaches developed to describe general N=2 super Yang-Mills theories in terms of off-shell supermultiplets with conventional supersymmetry. The structure of low-energy hypermultiplet effective action is briefly discussed. 
  2+1-dimensional Yang-Mills theory is reinterpreted in terms of metrics on 3-manifolds. The dual gluons are related to diffeomorphisms of the 3-manifold. Monopoles are identified with points where the Ricci tensor has triply degenerate eigenvalues. The dual gluons have the desired interaction with these monopoles. This would give a mass for the dual gluons resulting in confinement. 
  The Dirac-Born-Infeld action with transverse scalar fields is considered to study the dynamics of various BPS states. We first describe the characteristic properties of the so-called 1/2 and 1/4 BPS states on the D3 brane, which can be interpreted as F/D-strings ending on a D3-brane in Type IIB string theory picture. We then study the response of the BPS states to low energy excitations of massless fields on the brane, the scalar fields representing the shape fluctuation of the brane and U(1) gauge fields describing the open string excitations on the D-brane. This leads to an identification of interactions between BPS states including the static potentials and the kinetic interactions. 
  Static charges are introduced in Yang-Mills theory via coupling to heavy fermions. The states containing static color charges are constructed using integration over gauge transformations. A functional representation for interquark potential is obtained. This representation provides a simple criterion for confinement. 
  This is the written version of an invited talk delivered at the workshop ``Quantum gravity in the Southern Cone'' held in San Carlos de Bariloche, Argentina, January 7-10, 1998. After giving a brief introduction to the concept of branes and their role in string theory, this talk describes a method for formulating the dynamics of branes, especially those containing non-scalar moduli. Emphasis is put on the coupling of branes to fields in the low-energy background supergravity theories, and on preservation of maximal amount of manifest symmetry. Due to the nature of the workshop, the presentation is aimed at physicists who are not experts in string theory. 
  The method introduced in [hep-th/9805020] is simplified, and used to calculate the asymptotic form of all SU(2) \times SO(d=3, resp. 5) invariant wave functions satisfying $Q_{\hat{\beta}} \Psi = 0, \hat{\beta} = 1 ... 4$ resp. 8, where $Q_{\hat{\beta}}$ are the supercharges of the SU(2) matrix model related to supermembranes in d+2=5 (resp. 7) space-time dimensions. For d=3, there exist 2 asymptotic solutions, both of which are constant (hence non-normalizable) in the flat directions, confirming previous arguments that gauge-invariant zero energy states should not exist for d<9. For d=5, however, out of 4 asymptotic singlet solutions (3 with orbital angular momentum $l=0$, one having $l=1$) the one with $l=1$ does fall off fast enough to be asymptotically normalizable, hence requiring further analysis to be excluded as being extendable to a global solution. 
  By making use of the path integral duality transformation, we derive the string representation for the partition function of an extended Dual Abelian Higgs Model containing gauge fields of external currents of electrically charged particles. By the same method, we obtain the corresponding representations for the generating functionals of gauge field and monopole current correlators. In the case of bilocal correlators, the obtained results are found to be in agreement with the dual Meissner scenario of confinement and with the Stochastic Model of the QCD vacuum. 
  We present a simple algorithm to obtain solutions that generalize the Israel--Wilson--Perj\'es class for the low-energy limit of heterotic string theory toroidally compactified from D=d+3 to three dimensions. A remarkable map existing between the Einstein--Maxwell (EM) theory and the theory under consideration allows us to solve directly the equations of motion making use of the matrix Ernst potentials connected with the coset matrix of heterotic string theory. For the particular case d=1 (if we put n=6, the resulting theory can be considered as the bosonic part of the action of D=4, N=4 supergravity) we obtain explicitly a dyonic solution in terms of one real 2\times 2--matrix harmonic function and 2n real constants (n being the number of Abelian vector fields). By studying the asymptotic behaviour of the field configurations we define the charges of the system. They satisfy the Bogomol'nyi--Prasad--Sommmerfeld (BPS) bound. 
  It is shown that the orbifold of type IIB string theory by (-1)^{F_L} I_4 admits a stable non-BPS Dirichlet particle that is stuck on the orbifold fixed plane. It is charged under the SO(2) gauge group coming from the twisted sector, and transforms as a long multiplet of the D=6 supersymmetry algebra. This suggests that it is the strong coupling dual of the perturbative stable non-BPS state that appears in the orientifold of type IIB by \Omega I_4. 
  The one-loop finite temperature effective potential of QED in an external electromagnetic field is obtained using the worldline method. The general structure of the temperature dependent part of the effective action in an arbitrary external inhomogeneous magnetic field is established. The two-derivative effective action of spinor and scalar QED in a static magnetic background at $T\neq 0$ is derived. 
  The potential during inflation must be very flat in, at least, the direction of the inflaton. In renormalizable global supersymmetry, flat directions are ubiquitous, but they are not preserved in a generic supergravity theory. It is known that at least some of them are preserved in no-scale supergravity, and simple generalizations of it. We here study a more realistic generalization, based on string-derived supergravity, using the linear supermultiplet formalism for the dilaton. We consider a general class of hybrid inflation models, where a Fayet-Illiopoulos $D$ term drives some fields to large values. The potential is dominated by the $F$ term, but flatness is preserved in some directions. This allows inflation, with the dilaton stabilized in its domain of attraction, and some moduli stabilized at their vacuum values. Another modulus may be the inflaton. 
  We study the construction of baryons via supergravity along the line suggested recently by Witten and by Gross and Ooguri. We calculate the energy of the baryon as a function of its size. As expected the energy is linear with N. For the non-supersymmetric theories (in three and four dimensions) we find a linear relation which is an indication of confinement. For the {\cal N} = 4 theory we obtain the result (E L= - {const.}) which is compatible with conformal invariance. Surprisingly, our calculation suggests that there is a bound state of k quarks if N\geq k\geq 5N/8. We study the {\cal N} = 4 theory also at finite temperature and find the zero temperature behavior for small size of the baryon, and screening behaviour for baryon, whose size is large compared to the thermal wavelength. 
  We consider field theories arising from a large number of D3-branes near singularities in F-theory. We study the theories at various conformal points, and compute, using their conjectured string theory duals, their large $N$ spectrum of chiral primary operators. This includes, as expected, operators of fractional conformal dimensions for the theory at Argyres-Douglas points. Additional operators, which are charged under the (sometimes exceptional) global symmetries of these theories, come from the 7-branes. In the case of a $D_4$ singularity we compare our results with field theory and find agreement for large $N$. Finally, we consider deformations away from the conformal points, which involve finding new supergravity solutions for the geometry produced by the 3-branes in the 7-brane background. We also discuss 3-branes in a general background. 
  For D=4 theories of a single U(1) gauge field strength coupled to gravity and matters, we show that the electric-magnetic duality can be formulated as an invariance of the actions. The symmetry is associated with duality rotation acting directly on the gauge field. The rotation is constructed in flat space, and an extension to curved spaces is also given. It is non-local and non-covariant, yet generates off-shell extended transformation of the field strength. The algebraic condition of Gaillard and Zumino turns out to be a necessary and sufficient condition for the invariance of actions. It may be used as a guiding principle in constructing self-dual actions in string and field theories. 
  We establish self-duality of super D3-brane theory as an exact symmetry of the action both in the Lagrangian and Hamiltonian formalism. In the Lagrangian formalism, the action is shown to satisfy the Gaillard-Zumino condition. This algebraic relation is recognized in our previous paper to be a necessary and sufficient condition for generic action of U(1) gauge field strength coupled with gravity and matters to be self-dual. For the super D3-brane action, SO(2) duality transformation of a world-volume gauge field should be associated with SO(2) rotation of fermionic brane coordinates in N=2 SUSY multiplet. This SO(2) duality symmetry is lifted to SL(2,R) symmetry in the presence of a dilaton and an axion background fields. In the canonical formalism, we show that the duality rotation is described by a canonical transformation, and the Hamiltonian of the D3-brane action is invariant under the transformation. 
  We calculate the baryon mass in N=4 large $N$ gauge theory by means of AdS/CFT correspondence and show that it is a truly bound state, at least in some situations. We find that a phase transition occurs at a critical temperature. Furthermore, we find there are bound states of W-bosons in the Higgs phase, where the gauge group is broken to SU(N_1)xSU(N_2). 
  A new light-front formulation of Q.E.D. is developed, within the framework of standard perturbation theory, in which $x^+$ plays the role of the evolution parameter and the gauge choice is $A_+=0$ (light-front "temporal" gauge). It is shown that this formulation leads to the Mandelstam-Leibbrandt causal prescription for the non-covariant singularities in the photon propagator. Furthermore, it is proved that the dimensionally regularized one loop off-shell amplitudes exactly coincide with the correct ones, as computed within the standard approach using ordinary space-time coordinates. 
  The recently proposed probability representation of quantum mechanics is generalized to quantum field theory. We introduce a probability distribution functional for field configurations and find an evolution equation for such a distribution. The connection to the time-dependent generating functional of Green's functions is elucidated and the classical limit is discussed. 
  We examine in non-Abelian gauge theory the heavy quark limit in the presence of the (anti-)self-dual homogeneous background field and see that a confining potential emerges, consistent with the Wilson criterion, although the potential is quadratic and not linear in the quark separation. This builds upon the well-known feature that propagators in such a background field are entire functions. The way in which deconfinement can occur at finite temperature is then studied in the static temporal gauge by calculation of the effective potential at high temperature. Finally we discuss the problems to be surmounted in setting up the calculation of the effective potential nonperturbatively on the lattice. 
  It is briefly explained why recent claims about the vanishing of the one-loop effective potential in Matrix theory, thus invalidating the possible agreement with supergravity, do not hold. 
  We look at the possibility of superinflationary behavior in a class of anisotropic Type-IIB superstring cosmologies in the context of Pre-big Bang scenario and find that there exists a rather narrow range of parameters for which these models inflate. We then show that, although in general this behavior is left untouched by the introduction of a Ramond-Ramond axion field through a SL(2,R) rotation, there exists a particular class of axions for which inflation disappears completely. Asymptotic past initial conditions are briefly discussed, and some speculations on the possible extension of Pre-big Bang ideas to gravitational collapse are presented. 
  We present a method for the calculation of the $a_{3/2}$ heat kernel coefficient of the heat operator trace for a partial differential operator of Laplace type on a compact Riemannian manifold with oblique boundary conditions. Using special case evaluations, restrictions are put on the general form of the coefficients, which, supplemented by conformal transformation techniques, allows the entire smeared coefficient to be determined. 
  We construct the explicit worldvolume effective actions of the type IIB NS-5-brane and KK-monopole. These objects are obtained through a T-duality transformation from the IIA KK-monopole and the IIA NS-5-brane respectively. We show that the worldvolume field content of these actions is precisely that necessary to describe their worldvolume solitons. The IIB NS-5-brane effective action is shown to be related to the D-5-brane's by an S-duality transformation, suggesting the way to construct (p,q) 5-brane multiplets. The IIB KK-monopole is described by a gauged sigma model, in agreement with the general picture for KK-monopoles, and behaves as a singlet under S-duality. We derive the explicit T-duality rules NS-5-brane -- KK, which we use for the construction of the previous actions, as well as NS-5 -- NS-5, and KK -- KK. 
  We generalize the M-theoretic duality of Schmaltz and Sundrum to the product group SU(N) x SU(N') case. We show that the type IIA brane configurations for dual gauge theories are in fact two special limits of the same M-theory 5-brane, just as in the case of the simple SU(N) group. 
  Spectra of glueball masses in non-supersymmetric Yang-Mills theory in three and four dimensions have recently been computed using the conjectured duality between superstring theory and large N gauge theory. The Kaluza-Klein states of supergravity do not correspond to any states in the Yang-Mills theory and therefore should decouple in the continuum limit. On the other hand, in the supergravity limit g_{YM}^2 N -> \infty, we find that the masses of the Kaluza-Klein states are comparable to those of the glueballs. We also show that the leading (g_{YM}^2N)^{-1} corrections do not make these states heavier than the glueballs. Therefore, the decoupling of the Kaluza-Klein states is not evident to this order. 
  A thought experiment is formulated to unify quantum mechanics and general relativity in a topological manner. An analysis of the interactions in Nature is then presented. The universal ground state of the constructed theory derives from the cyclic properties ($S^1$ homotopy) of the topological manifold $Q=2T^3\oplus 3S^1\times S^2$ which has 23 intrinsic degrees of freedom, discrete $Z_3$ and $Z_2\times Z_3$ internal groups, an SU(5) or SO(10) gauge group, and leads to an anomalous U(1) symmetry on a lattice. These properties can in principle reproduce the standard model with a stable proton. The general equation of motion for the unified theory is derived up to the Planck energy and leads to a Higgs field with possible inflation. The thermodynamic properties of $Q$ are discussed and yield a consistent amplitude for the cosmic microwave background fluctuations. The manifold $Q$ possesses internal energy scales which are independent of the field theory defined on it, but which constrain the predicted mass hierarchy of such theories. In particular the electron and its neutrino are identified as particle ground states and their masses are predicted. The mass of the electron agrees very well with observations. A heuristic argument for the occurrence and magnitude of CP violation is given. Future extensions of the presented framework are discussed. 
  We study the loop expansion for the low energy effective action for matrix string theory. For long string configurations we find the result depends on the ordering of limits. Taking $g_s\to 0$ before $N\to\infty$ we find free strings. Reversing the order of limits however we find anomalous contributions coming from the large $N$ limit that invalidate the loop expansion. We then embed the classical instanton solution into a long string configuration. We find the instanton has a loop expansion weighted by fractional powers of $N$. Finally we identify the scaling regime for which interacting long string configurations have a well defined large $N$ limit. The limit corresponds to large "classical" strings and can be identified with the "dual of the 't Hooft limit, $g_{SYM}^2\sim N$. 
  In hep-th/9805025, a result for the symmetric 3-loop massive tetrahedron in 3 dimensions was found, using the lattice algorithm PSLQ. Here we give a more general formula, involving 3 distinct masses. A proof is devised, though it cannot be accounted as a derivation; rather it certifies that an Ansatz found by PSLQ satisfies a more easily derived pair of partial differential equations. The result is similar to Schl\"afli's formula for the volume of a bi-rectangular hyperbolic tetrahedron, revealing a novel connection between 3-loop diagrams and 1-loop boxes. We show that each reduces to a common basis: volumes of ideal tetrahedra, corresponding to 1-loop massless triangle diagrams. Ideal tetrahedra are also obtained when evaluating the volume complementary to a hyperbolic knot. In the case that the knot is positive, and hence implicated in field theory, ease of ideal reduction correlates with likely appearance in counterterms. Volumes of knots relevant to the number content of multi-loop diagrams are evaluated; as the loop number goes to infinity, we obtain the hyperbolic volume of a simple 1-loop box. 
  Coupling of a membrane and a five-brane to the bosonic sector of D=11 supergravity is considered. The five--brane is a dyonic object which carries both an electric and a magnetic charge of the D=11 three-form gauge field $A^3$, and it couples not only to $A^3$ but also minimally couples to a six-form field $A^6$ dual to $A^3$. This implies that the 5-brane should more naturally couple to a version of D=11 supergravity where both gauge fields are present in a duality-symmetric fashion. We demonstrate how an action of duality-symmetric D=11 supergravity looks like, couple it to the five-brane and then reduce the resulting system to an action, which describes an interaction of the 5-brane with the standard D=11 supergavity. 
  We study curved space versions of matrix string theory taking as a definition of the theory a gauged matrix sigma model. By analyzing the divergent terms in the loop expansion for the effective action we reduce the problem to a simple matrix generalization of the standard string theory beta function calculation. It is then demonstrated that the model can only be consistent for Ricci flat manifolds with vanishing six-dimensional Euler density. 
  We study type IIA configurations of D4 branes and three kinds of NS fivebranes. The D4 brane world-volume has finite extent in three directions, giving rise to a two-dimensional low-energy field theory. The models have generically $(0,2)$ supersymmetry. We determine the rules to read off the spectrum and interactions of the field theory from the brane box configuration data. We discuss the construction of theories with enhanced $(0,4)$, $(0,6)$ and $(0,8)$ supersymmetry. Using T-duality along the directions in which the D4 branes are finite, the configuration can be mapped to D1 branes at $\IC^4/\Gamma$ singularities, with $\Gamma$ an abelian subgroup of SU(4). This provides a rederivation of the rules in the brane box model. The enhancement of supersymmetry has a nice geometrical interpretation in the singularity picture in terms of the holonomy group of the four-fold singularity. 
  We review the assumptions and the logic underlying the derivation of DLCQ Matrix models. In particular we try to clarify what remains valid at finite $N$, the role of the non-renormalization theorems and higher order terms in the supergravity expansion. The relation to Maldacena's conjecture is also discussed. In particular the compactification of the Matrix model on $T_3$ is compared to the $AdS_5\times S_5$ ${\cal N}=4$ super Yang-Mills duality, and the different role of the branes in the two cases is pointed out. 
  This thesis discusses the topological aspects of quantum gravity, focusing on the connection between 2D quantum gravity and 2D topological gravity. The mathematical background for the discussion is presented in the first two chapters. The possible gauge formulations of 2D topological gravity as a BF or a Super BF theory are presented and compared against 2D quantum gravity in the dynamical triangulation scheme. A new identification between topological gravity in the Super BF formulation and the reduced hermitian matrix model at genus zero is explained in depth. 
  We discuss twisted states of AdS orbifolds which couple to N=2 chiral primary operators not invariant under exchange of the gauge factors. Kaluza-Klein reduction on the fixed circle gives the correct conformal dimensions of operators in the superconformal theory and involves some aspects of monopole dynamics in the non-trivial background. As a byproduct we found evidence for decoupling of U(1) factors in the four-dimensional gauge theory. 
  It is shown that N=4 gauged supergravity in four dimensions is obtained by compactifying N=1 supergravity in ten dimensions on the group manifold $S^3\times S^3$. This could be further related to supergravity in eleven dimensions. Analysis of supersymmetry conditions of N=4 gauged supergravity in four dimensions reveals solutions which preserve 1/4 of the supersymmetries and are characterized by a BPS-monopole-type gauge field. These solutions are lifted to solutions of the ten and eleven dimensional theories. 
  We study the doubly charged Callan-Giddings-Harvey-Strominger (CGHS) model, which has black hole solutions that were found to be U-dual to the D1-D5-KK black holes of the IIB supergravity. We derive the action of the model via a spontaneous compactification on S^3 of the IIB supergravity on S^1*T^4 and obtain the general static solutions including black holes corresponding to certain non-asymptotically flat black holes in the IIB supergravity. Thermodynamics of them is established by computing the entropy, temperature, chemical potentials, and mass in the two-dimensional setup, and the first law of thermodynamics is explicitly verified. The entropy is in precise agreement with that of the D1-D5-KK black holes, and the mass turns out to be consistent with the infinite Lorentz boost along the M theory circle that is a part of the aforementioned U-dual chain. 
  We prove that the bosons and massless fermions of one generation of the standard model are supersymmetric partners of each other. Except for one additional auxilliary vector boson, there are no other SUSY particles. 
  The lowest representatives of the Form Factors relative to the trace operators of N=1 Super Sinh-Gordon Model are exactly calculated. The novelty of their determination consists in solving a coupled set of unitarity and crossing equations. Analytic continuations of the Form Factors as functions of the coupling constant allows the study of interesting models in a uniform way, among these the latest model of the Roaming Series and the minimal supersymmetric models as investigated by Schoutens. A fermionic version of the $c$-theorem is also proved and the corresponding sum-rule derived. 
  ``Constants of Nature'' and cosmological parameters may in fact be variables related to some slowly-varying fields. In models of eternal inflation, such fields will take different values in different parts of the universe. Here I show how one can assign probabilities to values of the ``constants'' measured by a typical observer. This method does not suffer from ambiguities previously discussed in the literature. 
  We probe a non-supersymmetric D6 + D0 state with D6-branes and find agreement at subleading order between the supergravity and super Yang-Mills description of the long-distance, low-velocity interaction. 
  We propose a new method to calculate the 4-dimensional divergent integrals. By calculating the one loop integral as an example, the regularization of the integrals in 3-dimension momentum space are given in details. We find that the new method gives the same results as the traditional dimensional regularization method gives, but the new method has the advantage that it gives the real and the imaginary part separately. 
  We report a sign that M(atrix) theory conjecture and the Maldacena conjecture for the case of D0-branes are compatible. Furthermore Maldacena point of view implies a restriction of range of validity in the DLCQ version of M(atrix) theory. The analysis is based on the uplift of type IIA supersymetric solution in the Maldacena approach to eleven dimensions, using a boost as a main tool. The relation is explored on both, IMF and DLCF versions of M(atrix) theory 
  We construct generalizations of the Calogero-Sutherland-Moser system by appropriately reducing a model involving many unitary matrices. The resulting systems consist of particles on the circle with internal degrees of freedom, coupled through modifications of the inverse-square potential. The coupling involves SU(M) non-invariant (anti)ferromagnetic interactions of the internal degrees of freedom. The systems are shown to be integrable and the spectrum and wavefunctions of the quantum version are derived. 
  A rectangular Wilson loop centered at the origin, with sides parallel to space and time directions and length $2L$ and $2T$ respectively, is perturbatively evaluated ${\cal O}(g^4)$ in Feynman gauge for Yang--Mills theory in $1+(D-1)$ dimensions. When $D>2$, there is a dependence on the dimensionless ratio $L/T$, besides the area. In the limit $T \to \infty$, keeping $D>2$, the leading expression of the loop involves only the Casimir constant $C_F$ of the fundamental representation and is thereby in agreement with the expected Abelian-like time exponentiation (ALTE). At $D= 2$ the result depends also on $C_A$, the Casimir constant of the adjoint representation and a pure area law behavior is recovered, but no agreement with ALTE in the limit $T\to\infty$. Consequences of these results concerning two and higher-dimensional gauge theories are pointed out. 
  We find new solutions of IIA supergravity which have the interpretation of intersecting NS-5-branes at $Sp(2)$-angles on a string preserving at least 3/32 of supersymmetry. We show that the relative position of every pair of NS-5-branes involved in the superposition is determined by four angles. In addition we explore the related configurations in IIB strings and M-theory. 
  In this paper we demonstrate that there exists a close relationship between quasi-exactly solvable quantum models and two special classes of classical dynamical systems. One of these systems can be considered a natural generalization of the multi-particle Calogero-Moser model and the second one is a classical matrix model. 
  The Casimir energy for a conducting spherical shell of radius $a$ is computed using a direct mode summation approach. An essential ingredient is the implementation of a recently proposed method based on Cauchy's theorem for an evaluation of the eigenfrequencies of the system. It is shown, however, that this earlier calculation uses an improper set of modes to describe the waves exterior to the sphere. Upon making the necessary corrections and taking care to ensure that no mathematically ill-defined expressions occur, the technique is shown to leave numerical results unaltered while avoiding a longstanding criticism raised against earlier calculations of the Casimir energy. 
  We study string propagation on $AdS_3$ times a compact space from an ``old fashioned'' worldsheet point of view of perturbative string theory. We derive the spacetime CFT and its Virasoro and current algebras, thus establishing the conjectured $AdS$/CFT correspondence for this case in the full string theory. Our results have implications for the extreme IR limit of the $D1-D5$ system, as well as to 2+1 dimensional BTZ black holes and their Bekenstein-Hawking entropy. 
  We present a detailed analysis of $AdS_3$ gravity, the BTZ black hole and the associated conformal field theories (CFTs). In particular we focus on the non-extreme six-dimensional string solution with background metric $AdS_3 \times S^3$ near the horizon. In addition we introduce momentum modes along the string, corresponding to a BTZ black hole, and a Taub-NUT soliton in the transverse Euclidean space. We show that the $AdS_3$ space-time of this configuration has the spatial geometry of an annulus with a Liouville model at the outer boundary and a two-dimensional black hole at the inner boundary. These CFTs provide the dynamical degrees of freedom of the three-dimensional effective model and, together with the CFT corresponding to $S^3$, provide a statistical interpretation of the corresponding Bekenstein-Hawking entropy. We test the proposed exact black hole entropy, which should hold to all orders in $\alpha^\prime$, by an independent field theoretical analysis including higher-order curvature corrections. We find consistent results that yield a renormalization of the classical parameters, only. In addition we find a logarithmic subleading black hole entropy coming from gravitational fivebrane instantons in a special limit in moduli space. 
  We review the light-front Hamiltonian approach for the Abelian gauge theory in 3+1 dimensions, and then study electromagnetic duality in this framework. 
  The Maldacena's proposal has established an intriguing connection between string theory in AdS spaces and gauge theory. In this paper we study the effects of adding D(-1)-branes to the system of D3- or (D1-D5)-branes and we give arguments indicating that D(-1)-branes are necessary to describe four and two dimensional instantons. 
  The Wigner-Weyl- Moyal approach to Quantum Mechanics is recalled, and similarities to classical probability theory emphasised. The Wigner distribution function is generalised and viewed as a construction of a bosonic object, a target space co-ordinate, for example, in terms of a bilinear convolution of two fermionic objects, e.g. a quark antiquark pair. This construction is essentially non-local, generalising the idea of a local current. Such Wigner functions are shown to solve a BPS generalised Moyal-Nahm equation. 
  This is an introduction to the physics of D-branes. Topics covered include Polchinski's original calculation, a critical assessment of some duality checks, D-brane scattering, and effective worldvolume actions. 
  This talk is in two parts both entitled mass production requires precision engineering. The first is about the dynamical generation of mass for matter particles in gauge theories. I will explain how the details of this depend on a precision knowledge of the interactions. The second is about tests of the mechanism of chiral symmetry breaking in QCD that the precision engineering of high luminosity colliders and particle detectors will shortly make possible.   Since the latter topic has been described in Ref. 1, here I will just discuss the first: the production of mass from nothing. 
  It is argued that quantum propagation of D-particles in the limit \alpha'-> 0 can represent the "joining-splitting" processes of Feynman graphs of a certain field theory in the light-cone frame. So basically it provides the possibility to define a field theory by its Feynman graphs. The application of this observation to define M-theory by an energetic expansion approach is discussed. 
  Recently proposed by Hwang, Marnelius and Saltsidis zeta regularization of basic commutators in string theories is generalized to the string models with non-trivial vacuums. It is shown that implementation of this regularization implies the cancellation of dangerous terms in the commutators between Virasoro generators, which break Jacobi identity. 
  High frequency dispersion does not alter the low frequency spectrum of Hawking radiation from a single black hole horizon, whether the dispersion entails subluminal or superluminal group velocities. We show here that in the presence of an inner horizon as well as an outer horizon the superluminal case differs dramatically however. The negative energy partners of Hawking quanta return to the outer horizon and stimulate more Hawking radiation if the field is bosonic or suppress it if the field is fermionic. This process leads to exponential growth or damping of the radiated flux and correlations among the quanta emitted at different times, unlike in the usual Hawking effect. These phenomena may be observable in condensed matter black hole analogs that exhibit "superluminal" dispersion. 
  We study the spectrum of the softly broken generalized Veneziano-Yankielowicz effective action for N=1 SUSY Yang-Mills theory. Two dual formulations of the effective action are given. The spurion method is used for the soft SUSY breaking. Masses of the bound states are calculated and mixing patterns are discussed. Mass splittings of pure gluonic states are consistent with predictions of conventional Yang-Mills theory. The results can be tested in lattice simulations of the SUSY Yang-Mills model. 
  A nonlinear vector supersymmetry for three-dimensional topological massive Yang-Mills is obtained by making use of a nonlinear but local and covariant redefinition of the gauge field. 
  It is shown that for any elastic string model with energy density $U$ and tension $T$, the divergent contribution from gravitational self interaction can be allowed for by an action renormalisation proportional to $(U-T)^2$. This formula is applied to the important special case of a bare model of the transonic type (characterised by a constant value of the product $UT$) that represents the macroscopically averaged effect of shortwavelength wiggles on an underlying microscopic model of the Nambu-Goto type (characterised by $U=T$). 
  Existence theorem is proven for the generating equations of the split involution constraint algebra. The structure of the general solution is established, and the characteristic arbitrariness in generating functions is described. 
  The effects of an external field on the dynamics of chiral symmetry breaking are studied using quenched, ladder QED as our model gauge field theory. It is found that a uniform external magnetic field enables the chiral symmetry to be spontaneously broken at weak gauge couplings, in contrast with the situation when no external field is present. The broken chiral symmetry is restored at high temperatures as well as at high chemical potentials. The nature of the two chiral phase transitions is different: the transition at high temperatures is a continuous one whereas the phase transition at high chemical potentials is discontinuous. 
  The effect of the Gaussian curvature in the rigid string action on the interquark potential is investigated. The linearized equations of motion and boundary conditions, following from the modified string action, are obtained. The equation, defining the eigenfrequency spectrum of the string oscillations is derived. On this basis, the interquark potential, generated by the string is calculated in one-loop approximation. A substantial influence of the topological term in the string action on the interquark potential at the distances of hadronic size order or less is revealed. 
  In some recent papers it is claimed that the physical significance of the vacuum angle theta for QCD-like theories depends on the chosen gauge condition. We criticise the arguments that were given in support of this claim, and show by explicit construction for the case of QED$_2$ that and why they fail, confirming thereby the commonly accepted point of view. 
  Within mass perturbation theory, already the first order contribution to the chiral condensate of the massive Schwinger model is UV divergent. We discuss the problem of choosing a proper normalization and, by making use of some bosonization results, we are able to choose a normalization so that the resulting chiral condensate may be compared, e.g., with lattice data. 
  The massive Schwinger model may be analysed by a perturbation expansion in the fermion mass. However, the results of this mass perturbation theory are sensible only for sufficiently small fermion mass. By performing a renormal-ordering, we arrive at a chiral perturbation expansion where the expansion parameter remains small even for large fermion mass. We use this renormal-ordered chiral perturbation theory for a computation of the Schwinger mass and compare our results with lattice computations. 
  We analyze the potentials which arise from the D1--D5 brane (5D black hole). In the sufficiently low energy ($\omega \ll 1$), we can derive the Schrodinger-type equation with potential $V_N$ from the linearized equations. In this case one can understand the difference between absorption cross section for a free and two fixed scalars intuitively in terms of their potentials. In the low temperature limit ($\omega \gg T_H$), one expects the logarithmic correction to the cross section of a free scalar. However, we cannot obtain the Schrodinger equation with potential for this case. Finally we comment on the stability of 5D black hole. 
  Any probe which crosses the horizon of a black hole should be absorbed. In M(atrix) theory, for 0-brane probes of Schwarzschild black holes, we argue that the relevant absorption mechanism is a tachyon instability which sets in at the horizon. We give qualitative arguments, and some quantitative large-N calculations, in support of this claim. The tachyon instability provides an attractive mechanism for infalling matter to be captured and thermalized by a Schwarzschild black hole. 
  The theory on M_4 x Z_2 geometry is applied to the Einstein gravity to yield the Brans-Dicke theory on M_4 geometry. The geometrical meaning and the relation between the curvatures and the torsions are clarified. The cosmological constant is also introduced into the pure Einstein action on M_4 x Z_2 in order to determine the explicit form of the cosmological term in the Brans-Dicke theory on M_4 geometry. 
  The standard formulation of the AdS/CFT correspondence is incomplete since it requires adding to a supergravity action some a priori unknown boundary terms. We suggest a modification of the correspondence principle based on the Hamiltonian formulation of the supergravity action, which does not require any boundary terms. Then all the boundary terms of the standard formulation naturally appear by passing from the Hamiltonian version to the Lagrangian one. As examples the graviton part of the supergravity action on the product of $AdS_{d+1}$ with a compact Einstein manifold $\cal E$ and fermions on $AdS_{d+1}$ are considered. We also discuss conformal transformations of gravity fields on the boundary of $AdS$ and show that they are induced by the isometries of $AdS$. 
  The correspondence between the four-dimensional SU(N), $\cN = 4$ SYM taken at large $N$ and the type II B SUGRA on the $AdS_5\times S_5$ background is considered. We argue that the classical equations of motion in the SUGRA picture can be interpreted as that of the renormalization group on the SYM side. In fact, when the D3-brane is slightly excited higher derivative terms in the field theory on its world-volume deform it form the conformal $\cN = 4$ SYM limit. We give arguments in favor of that the deformation goes in the way set by the SUGRA equations of motion. Concrete example of the s-wave dilaton is considered. 
  Universality of multicritical unitary matrix models is shown and a new scaling behavior is found in the microscopic region of the spectrum, which may be relevant for the low energy spectrum of the Dirac operator at the chiral phase transition. 
  We analyze the formation of fermionic condensates in two dimensional quantum chromodynamics for matter in the fundamental representation of the gauge group. We show that a topological regular instanton background is crucial in order to obtain nontrivial correlators. We discuss both massless and massive cases. 
  We study the renormalizable abelian vector-field models in the presence of the Wess-Zumino interaction with the pseudoscalar matter. The renormalizability is achieved by supplementing the standard kinetic term of vector fields with higher derivatives. The appearance of fourth power of momentum in the vector-field propagator leads to the super-renormalizable theory in which the $\beta$-function, the vector-field renormalization constant and the anomalous mass dimension are calculated exactly. It is shown that this model has the infrared stable fixed point and its low-energy limit is non-trivial. The modified effective potential for the pseudoscalar matter leads to the possible occurrence of dynamical breaking of the Lorentz symmetry. This phenomenon is related to the modification of Electrodynamics by means of the Chern-Simons (CS) interaction polarized along a constant CS vector. Its presence makes the vacuum optically active that has been recently estimated from astrophysical data. We examine two possibilities for the CS vector to be time-like or space-like, under the assumption that it originates from v.e.v. of some pseudoscalar matter and show that only the latter one is consistent in the framework of the AWZ model, because a time-like CS vector makes the vacuum unstable under pairs creation of tachyonic photon modes with the finite vacuum decay rate. 
  We investigate canonical structure of the Abelian Higgs model within the framework of DLCQ. Careful boundary analysis of differential equations, such as the Euler-Lagrange equations, leads us to a novel situation where the canonical structure changes in a drastic manner depending on whether the (light-front) spatial Wilson line is periodic or not. In the former case, the gauge-field ZM takes discrete values and we obtain the so-called ``Zero-Mode Constraints'' (ZMCs), whose semiclassical solutions give a nonzero vev to the scalar fields. Contrary, in the latter case, we have no ZMC and the scalar ZMs remain dynamical as well as the gauge-field ZM. In order to give classically nonzero vev to the scalar field, we work in a background field which minimizes the light-front energy. 
  We consider the SU(2) Principal Chiral Model (at level $k=1$) on the half-line with scale invariant boundary conditions. By looking at the IR limiting conformal field theory and comparing with the Kondo problem, we propose the set of permissible boundary conditions and the corresponding reflection factors. 
  We present a unified description of temperature and entropy in spaces with either "true" or "accelerated observer" horizons: In their (higher dimensional) global embedding Minkowski geometries, the relevant detectors have constant accelerations a_{G}; associated with their Rindler horizons are temperature a_{G}/2\pi and entropy equal to 1/4 the horizon area. Both quantities agree with those calculated in the original curved spaces.   As one example of this equivalence, we obtain the temperature and entropy of Schwarzschild geometry from its flat D=6 embedding. 
  We study the M-theory five-brane wrapped around the Seiberg-Witten curves for pure classical and exceptional groups given by an integrable system. Generically, the D4-branes arise as cuts that collapse to points after compactifying the eleventh dimension and going to the semiclassical limit, producing brane configurations of NS5- and D4-branes with N=2 gauge theories on the world volume of the four-branes. We study the symmetries of the different curves to see how orientifold planes are related to the involutions needed to obtain the distinguished Prym variety of the curve. This explains the subtleties encountered for the Sp(2n) and SO(2n +1). Using this approach we investigate the curves for exceptional groups, especially G_2 and E_6, and show that unlike for classical groups taking the semiclassical ten dimensional limit does not reduce the cuts to D4-branes. For G_2 we find a genus two quotient curve that contains the Prym and has the right properties to describe the G_2 field theory, but the involutions are far more complicated than the ones for classical groups. To realize them in M-theory instead of an orientifold plane we would need another object, a kind of curved orientifold surface. 
  We study the description of Schwarzschild black holes, of entropy S, within matrix theory in the regime $N \ge S \gg 1$. We obtain the most general matrix theory equation of state by requiring that black holes admit a description within this theory. It has a recognisable form in various cases. In some cases a D dimensional black hole can plausibly be thought of as a $\tilde{D} = D + 1$ dimensional black hole, described by another auxiliary matrix theory, but in its $\tilde{N} \sim S$ regime. We find what appears to be a matrix theory generalisation to higher dynamical branes of the normalisation of dynamical string tension, seen in other contexts. We discuss a further possible generalisation of the matrix theory equation of state. In a special case, it is governed by $N^3$ dynamical degrees of freedom. 
  We obtain an M--theory interpretation of different IIA orientifold planes by compactifying them on a circle and use a chain of dualities to get a new IIA limit of these objects using this circle as the eleventh dimension. Using the combination of the two IIA description, we give an interpretation for all orientifold four-planes in M-theory, including a mechanism for freezing M5-branes at singularities. 
  It is shown that the one-loop coefficients of on-shell operators of standard supergravity with canonical gauge kinetic energy can be regulated by the introduction of Pauli-Villars chiral and abelian gauge multiplets, subject to a condition on the matter representations of the gauge group. Aspects of the anomaly structure of these theories under global nonlinear symmetries and an anomalous gauge symmetry are discussed. 
  In quantum mechanics, symmetry groups can be realized by projective, as well as by ordinary unitary, representations. For the permutation symmetry relevant to quantum statistics of N indistinguishable particles, the simplest properly projective representation is highly non-trivial, of dimension $2^{(N-1)/2}$, and is most easily realized starting with spinor geometry. Quasiparticles in the Pfaffian quantum Hall state realize this representation. Projective statistics is a consistent theoretical possibility in any dimension. 
  We present a ``toy'' model for breaking supersymmetric gauge theories at the effective Lagrangian level. We show that it is possible to achieve the decoupling of gluinos and squarks, below a given supersymmetry breaking scale m, in the fundamental theory for super QCD once a suitable choice of supersymmetry breaking terms is made. A key feature of the model is the description of the ordinary QCD degrees of freedom via the auxiliary fields of the supersymmetric effective Lagrangian. Once the anomaly induced effective QCD meson potential is deduced we also suggest a decoupling procedure, when a flavored quark becomes massive, which mimics the one employed by Seiberg for supersymmetric theories. It is seen that, after quark decoupling, the QCD potential naturally converts to the one with one less flavor. Finally we investigate the N_c and N_f dependence of the \eta^{\prime} mass. 
  We formulate the basic postulate of pre-big bang cosmology as one of ``asymptotic past triviality'', by which we mean that the initial state is a generic perturbative solution of the tree-level low-energy effective action. Such a past-trivial ``string vacuum'' is made of an arbitrary ensemble of incoming gravitational and dilatonic waves, and is generically prone to gravitational instability, leading to the possible formation of many black holes hiding singular space-like hypersurfaces. Each such singular space-like hypersurface of gravitational collapse becomes, in the string-frame metric, the usual big-bang t=0 hypersurface, i.e. the place of birth of a baby Friedmann universe after a period of dilaton-driven inflation. Specializing to the spherically-symmetric case, we review and reinterpret previous work on the subject, and propose a simple, scale-invariant criterion for collapse/inflation in terms of asymptotic data at past null infinity. Those data should determine whether, when, and where collapse/inflation occurs, and, when it does, fix its characteristics, including anisotropies on the big bang hypersurface whose imprint could have survived till now. Using Bayesian probability concepts, we finally attempt to answer some fine-tuning objections recently moved to the pre-big bang scenario. 
  We describe a non-perturbative procedure for solving from first principles the light-front Hamiltonian problem of SU(N) pure gauge theory in D spacetime dimensions (D>2), based on enforcing Lorentz covariance of observables. A transverse lattice regulator and colour-dielectric link fields are employed, together with an associated effective potential. We argue that the light-front vacuum is necessarily trivial for large enough lattice spacing, and clarify why this leads to an Eguchi-Kawai dimensional reduction of observables to 1+1-dimensions in the infinite N limit. The procedure is then tested by explicit calculations for 2+1-dimensional SU(infinity) gauge theory, within a first approximation to the lattice effective potential. We identify a scaling trajectory which produces Lorentz covariant behaviour for the lightest glueballs. The predicted masses, in units of the measured string tension, are in agreement with recent results from conventional Euclidean lattice simulations. In addition, we obtain the potential between heavy sources and the structure of the glueballs from their light-front wavefunctions. Finally, we briefly discuss the extension of these calculations to 3+1-dimensions. 
  We construct a relativistically covariant stochastic model for systems of non-interacting spinless particles whose number undergoes random fluctuations. The model is compared with the canonical quantization of the free scalar field in the limit of infinite volume. 
  We solve exactly the (linear order) equations for tensor and scalar perturbations over the homogeneous, isotropic, open pre-big bang model recently discussed by several authors. We find that the parametric amplification of vacuum fluctuations (i.e. particle production) remains negligible throughout the perturbative pre-big bang phase. 
  We argue that the topology of the quantum coupling space and the low energy effective action on the Coulomb branch of scale invariant N=2 SU(n) gauge theories pick out a preferred nonperturbative definition of the gauge coupling up to non-singular holomorphic reparametrizations. 
  We point out certain unexpected features of the planar QCD3 confining potential, as computed from a classical worldsheet action in an AdS metric via the Maldacena conjecture. We show that there is no Luscher c/R term in the static-quark potential, which is contrary to both the prediction of various effective string models, and the results of some recent lattice Monte Carlo studies. It is also noted that the glueball masses extracted from classical supergravity tend to finite, coupling-independent constants in the strong coupling limit, even as the string tension tends to infinity in the same limit; this is a counter-intuitive result. 
  A remarkable feature of the models with interactions exhibiting higher-spin (HS) gauge symmetries in $d>2$ is that their most symmetric vacua require (anti)-de Sitter (AdS) geometry rather than the flat one. In striking parallelism to what might be expected of M theory HS gauge theories describe infinite towers of fields of all spins and possess naturally space-time SUSY and Chan-Paton type inner symmetries. In this paper, we analyze at the level of the equations of motion the simplest non-trivial HS model which describes HS gauge interactions (on the top of the usual supergravitational and (Chern-Simons) Yang-Mills interactions) of massive spin-0 and spin-1/2 matter fields in d=2+1 AdS space-time. The parameter of mass of the matter fields is identified with the vev of a certain auxiliary field in the model. The matter fields are shown to be arranged into d3 N=2 massive hypermultiplets in certain representations of $U(n)\times U(m)$ Yang-Mills gauge groups. Discrete symmetries of the full system are studied, and the related N=1 supersymmetric truncations with O(n) and Sp(n) Yang-Mills symmetries are constructed. The simplicity of the model allows us to elucidate some general properties of the HS models. In particular, a new result, which can have interesting implications to the higher-dimensional models, is that our model is shown to admit an "integrating" flow that proves existence of a non-local B\"acklund-Nicolai-type mapping to the free system. 
  We study the energy density of two distinct fundamental monopoles in SU(3) and Sp(4) theories with an arbitrary mass ratio. Several special limits of the general result are checked and verified. Based on the analytic expression of energy density the coefficient of the internal part of the moduli space metric is also computed, which gives it a nice "mechanical" interpretation. We then investigate the interaction energy density for both cases. By analyzing the contour of the zero interaction energy density we propose a detailed picture of what happens when one gets close to the massless limit. The study of the interaction energy density also sheds light on the formation of the non-Abelian cloud. 
  D-branes can end on orbifold planes if the action of the orbifold group includes (-1)^{F_L}. We consider configurations of D-branes ending on such orbifolds and study the low-energy theory on their worldvolume. We apply our results to gauge theories with eight supercharges in three and four dimensions. We explain how mirror symmetry for N=4 d=3 gauge theories with gauge group Sp(k) and matter in the antisymmetric tensor and fundamental representations follows from S-duality of IIB string theory. We argue that some of these theories have hidden Fayet-Iliopoulos deformations, not visible classically. We also study a class of finite N=2 d=4 theories (so-called D_n quiver theories) and find their exact solution. The integrable model corresponding to the exact solution is a Hitchin system on an orbifold Riemann surface. We also give a simple derivation of the S-duality group of these theories based on their relationship to SO(2n) instantons on R^2\times T^2. 
  It is well known that rational 2D conformal field theories are connected with Chern-Simons theories defined on 3D real manifolds. We consider holomorphic analogues of Chern-Simons theories defined on 3D complex manifolds (six real dimensions) and describe 4D conformal field theories connected with them. All these models are integrable. We describe analogues of the Virasoro and affine Lie algebras, the local action of which on fields of holomorphic analogues of Chern-Simons theories becomes nonlocal after pushing down to the action on fields of integrable 4D conformal field theories. Quantization of integrable 4D conformal field theories and relations to string theories are briefly discussed. 
  We briefly review the present status of string theory from the viewpoint of its implications on the short-distance space-time structure and black hole physics. Special emphases are given on two closely related issues in recent developments towards nonperturbative string theory, namely, the role of the space-time uncertainty relation as a qualitative but universal characterization of the short-distance structure of string theory and the microscopic formulation of black-hole entropies. We will also suggest that the space-time uncertainty relation can be an underlying principle for the holographic property of M theory, by showing that the space-time uncertainty relation naturally explains the UV/IR relation used in a recent derivation of the holographic bound for D3 brane by Susskind and Witten. 
  No theory of four-dimensional quantum gravity exists as yet. In this situation the two-dimensional theory, which can be analyzed by conventional field-theoretical methods, can serve as a toy model for studying some aspects of quantum gravity. It represents one of the rare settings in a quantum-gravitational context where one can calculate quantities truly independent of any background geometry. We review recent progress in our understanding of 2d quantum gravity, and in particular the relation between the Euclidean and Lorentzian sectors of the quantum theory. We show that conventional 2d Euclidean quantum gravity can be obtained from Lorentzian quantum gravity by an analytic continuation only if we allow for spatial topology changes in the latter. Once this is done, one obtains a theory of quantum gravity where space-time is fractal: the intrinsic Hausdorff dimension of usual 2d Euclidean quantum gravity is four, and not two. However, certain aspects of quantum space-time remain two-dimensional, exemplified by the fact that its so-called spectral dimension is equal to two. 
  D0-brane gas picture of Schwarzschild black hole (SBH) is considered in the large N regime of Matrix theory. An entropy formula, which reproduces the thermodynamics of SBH in the large N limit for all dimensions ($D\geq 6$), is proposed. The equations of states for low temperature are obtained. We also give a proof of the Newton gravitation law between two SBHs, whose masses are not equal. Our result in some extent supports that the physics of Matrix theory is Lorentz invariant. 
  Lorentz covariant generalisations of the notions of supersymmetry, superspace and self-duality are discussed. The essential idea is to extend standard constructions by allowing tangent vectors and coordinates which transform according to more general Lorentz representations than solely the spinorial and vectorial ones of standard lore. Such superspaces provide model configuration spaces for theories of arbitrary spin fields. Our framework is an elegant one for handling higher-dimensional theories in a manifestly SO(3,1) covariant fashion. A further application is the construction of a hierarchy of solvable Lorentz covariant systems generalising four-dimensional self-duality. 
  We generalise a result of Aste and Scharf saying that, under some reasonable assumptions, consistent with renormalisation theory, the non-Abelian gauge theories describes the only possibility of coupling the gluons. The proof is done using Epstein-Glaser approach to renormalisation theory. 
  By using the D0-brane cluster picture, we consider the Hawking radiation of Schwarszchild black hole (SBH) in large N Matrix model. We get the correct formula for the the Hawing evaporation rate. Our results give some evidence on the Lorentz invariance of the physics of Matrix model. 
  We construct the duality groups for N=2 Supersymmetric QCD with gauge group SU(2n+1) and N_f=4n+2 hypermultiplets in the fundamental representation. The groups are generated by two elements $S$ and $T$ that satisfy a relation $(STS^{-1}T)^{2n+1}=1$. Thus, the groups are not subgroups of $SL(2,Z)$. We also construct automorphic functions that map the fundamental region of the group generated by $T$ and $STS$ to the Riemann sphere. These automorphic functions faithfully represent the duality symmetry in the Seiberg-Witten curve. 
  We derive a set of complex potentials which linearize the action of charging symmetries of the stationary Einstein-Maxwell dilaton-axion theory. 
  A connection between one-loop $N$-point Feynman diagrams and certain geometrical quantities in non-Euclidean geometry is discussed. A geometrical way to calculate the corresponding Feynman integrals is considered. (This paper contains a brief review of the results presented in hep-th/9709216). 
  We consider a general formalism for treating a Hamiltonian (canonical) field theory with a spatial boundary. In this formalism essentially all functionals are differentiable from the very beginning and hence no improvement terms are needed. We introduce a new Poisson bracket which differs from the usual ``bulk'' Poisson bracket with a boundary term and show that the Jacobi identity is satisfied. The result is geometrized on an abstract world volume manifold. The method is suitable for studying systems with a spatial edge like the ones often considered in Chern-Simons theory and General Relativity. Finally, we discuss how the boundary terms may be related to the time ordering when quantizing. 
  We consider the hungry Volterra hierarchy from the view point of the multi boson KP hierarchy. We construct the hungry Volterra equation as the B\"{a}cklund transformations (BT) which are not the ordinary ones. We call them ``fractional '' BT. We also study the relations between the (discrete time) hungry Volterra equation and two matrix models. From this point of view we study the reduction from (discrete time) 2d Toda lattice to the (discrete time) hungry Volterra equation. 
  We embed the semilocal Chern-Simons-Higgs theory into an N=2 supersymmetric system. We construct the corresponding conserved supercharges and derive the Bogomol'nyi equations of the model from supersymmetry considerations. We show that these equations hold provided certain conditions on the coupling constants as well as on the Higgs potential of the system, which are a consequence of the huge symmetry of the theory, are satisfied. They admit string-like solutions which break one half of the supersymmetries --BPS Chern-Simons semilocal cosmic strings-- whose magnetic flux is concentrated at the center of the vortex. We study such solutions and show that their stability is provided by supersymmetry through the existence of a lower bound for the energy, even though the manifold of the Higgs vacuum does not contain non-contractible loops. 
  We give a B\"acklund transformation connecting a generic 2D dilaton gravity theory to a generally covariant free field theory. This transformation provides an explicit canonical transformation relating both theories. 
  We find a general expression for the free energy of $G(\phi)=\phi^{2k}$ generalized 2D Yang-Mills theories in the strong ($A>A_c$) region at large $N$. We also show that in this region, the density function of Young tableau of these models is a three-cut problem. In the specific $\phi^6$ model, we show that the theory has a third order phase transition, like $\phi^2$ (YM_2) and $\phi^4$ models. We have some comments for $k \geq 4$ cases. At the end, we study the phase structure of $\phi^2 + g \phi^4$ model for $g \leq A/4$ region. 
  The problem of the classical non-relativistic electromagnetically kicked oscillator can be cast into the form of an iterative map on phase space. The original work of Zaslovskii {\it et al} showed that the resulting evolution contains a stochastic flow in phase space to unbounded energy. Subsequent studies have formulated the problem in terms of a relativistically charged particle in interaction with the electromagnetic field. We review the standard derivation of the covariant Lorentz force, and review the structure of the relativistic equations used to study this problem.   We show that the Lorentz force equation can be derived as well from the manifestly covariant mechanics of Stueckelberg in the presence of a standard Maxwell field. We show how this agreement is achieved, and criticize some of the fundamental assumptions underlying these derivations. We argue that a more complete theory, involving ``off-shell'' electromagnetic fields should be utilized. We then discuss the formulation of the off-shell electromagnetism implied by the full gauge invariance of the Stueckelberg mechanics (based on its quantized form), and show that a more general class of physical phenomena can occur. 
  We show that the full global symmetry groups of all the D-dimensional maximal supergravities can be described in terms of the closure of the internal general coordinate transformations of the toroidal compactifications of D=11 supergravity and of type IIB supergravity, with type IIA/IIB T-duality providing an intertwining between the two pictures. At the quantum level, the part of the U-duality group that corresponds to the surviving discretised internal general coordinate transformations in a given picture leaves the internal torus invariant, while the part that is not described by internal general coordinate transformations can have the effect of altering the size or shape of the internal torus. For example, M-theory compactified on a large torus T^n can be related by duality to a compactification on a small torus, if and only if n\ge 3. We also discuss related issues in the toroidal compactification of the self-dual string to D=4. An appendix includes the complete results for the toroidal reduction of the bosonic sector of type IIB supergravity to arbitrary dimensions D\ge3. 
  The field equations for both generic bosonic and generic locally supersymmetric 2D dilatonic gravity theories in the absence of matter are written as free differential algebras. This constitutes a generalization of the gauge theoretic formulation. Moreover, it is shown that the condition of free differential algebra can be used to obtain the equations in the locally supersymmetric case. Using this formulation, the general solution of the field equations is found in the language of differential forms. The relation with the ordinary formulation and the coupling to supersymmetric conformal matter are also studied. 
  In this paper we discuss two aspects of duality transformations on the Neveu-Schwarz (NS) 5-brane solutions in type II and heterotic string theories. First we demonstrate that the non-extremal NS 5-brane background is U-dual to its CGHS limit, a two-dimensional black hole times $S^3\times T^5$; an intermediate step is provided by the near horizon geometry which is given by the three-dimensional $BTZ_3$ black hole (being closely related to $AdS_3$) times $S^3\times T^4$. In the second part of the paper we discuss the T-duality between $k$ NS 5-branes and the Taub-NUT spaces respectively ALE spaces, which are related to the resolution of the $A_{k-1}$ singularities of the non-compact orbifold ${\bf C}^2/{\bf Z}_{k}$. In particular in the framework of N=1 supersymmetric gauge theories related to brane box constructions we give the metric dual to two sets of intersecting NS 5-branes. In this way we get a picture of a dual orbifold background ${\bf C}^3/ \Gamma$ which is fibered together out of two N=2 models ($\Gamma={\bf Z}_k\times {\bf Z}_{k'}$). Finally we also discuss the intersection of NS 5-branes with D branes, which can serve as probes of the dual background spaces. 
  We obtain extremal stationary solutions that generalize the Israel-Wilson-Perj\'es class for the low-energy limit of heterotic string theory with n>=3 U(1) gauge fields toroidally compactified from five to three dimensions. A dyonic solution is obtained using the matrix Ernst potential (MEP) formulation and expressed in terms of a single real 3X3-matrix harmonic function. By studying the asymptotic behaviour of the field configurations we define the physical charges of the field system. The extremality condition makes the charges to saturate the Bogomol'nyi-Prasad-Sommerfield (BPS) bound. 
  We study the half advanced and half retarded Wheeler Green function and its relation to Feynman propagators. First for massless equation. Then, for Klein-Gordon equations with arbitrary mass parameters; real, imaginary or complex. In all cases the Wheeler propagator lacks an on-shell free propagation. The Wheeler function has support inside the light-cone (whatever the mass). The associated vacuum is symmetric with respect to annihilation and creation operators.   We show with some examples that perturbative unitarity holds, whatever the mass (real or complex). Some possible applications are discussed. 
  We study supersymmetry breaking by Scherk-Schwarz compactifications in type I string theory. While in the gravitational sector all mass splittings are proportional to a (large) compactification radius, supersymmetry remains unbroken for the massless excitations of D-branes orthogonal to the large dimension. In this sector, supersymmetry breaking can then be mediated by gravitational interactions alone, that are expected to be suppressed by powers of the Planck mass. The mechanism is non perturbative from the heterotic viewpoint and requires a compactification radius at intermediate energies of order 10^{12}-10^{14} GeV. This can also explain the value of Newton's constant if the string scale is close to the unification scale, of order 10^{16} GeV. 
  Using the ADM formulation of the Einstein-Maxwell axion-dilaton gravity we derived the formulas for the variation of mass and other asymptotic conserved quantities in the theory under consideration. Generalizing this kind of reasoning to the initial dota for the manifold with an interior boundary we got the generalized first law of black hole mechanics. We consider an asymptotically flat solution to the Einstein-Maxwell axion-dilaton gravity describing a black hole with a Killing vector field timelike at infinity, the horizon of which comprises a bifurcate Killing horizon with a bifurcate surface. Supposing that the Killing vector field is asymptotically orthogonal to the static hypersurface with boundary S and compact interior, we find that the solution is static in the exterior world, when the timelike vector field is normal to the horizon and has vanishing electric and axion- electric fields on static slices. 
  This paper has been withdrawn by the author due to a misunderstanding. 
  We present fivedimensional extreme black hole solutions of M-theory compactified on Calabi-Yau threefolds and study these solutions in the context of flop transitions in the extended Kahler cone. In particular we consider a specific model and present black hole solutions, breaking half of N=2 supersymmetry, in two regions of the extended Kahler cone, which are connected by a flop transition. The conditions necessary to match both solutions at the flop transition are analysed. Finally we also discuss the conditions to obtain massless black holes at the flop transition. 
  The problem of the gauge dependence of the fermion mass in the Maxwell-Chern-Simons QED$_{2+1}$ is revisited. Using Proca mass term as an intermediate infrared regulator we are demonstrating gauge-invariance of the fermion mass shell in QED$_{2+1}$ in all orders of the perturbation theory. 
  We analyze the BRST field-antifield construction for generalized gauge fields consisting of massless mixed representations of the Lorentz Group and we calculate all the strictly gauge invariant interactions between them. All these interactions are higher derivative terms constructed out from the derivatives of the curl of field strength. 
  A gas of $N$ Bogomol'nyi vortices in the Abelian Higgs model is studied on a compact Riemann surface of genus $g$ and area $A$. The volume of the moduli space is computed and found to depend on $N, g$ and $A$, but not on other details of the shape of the surface. The volume is then used to find the thermodynamic partition function and it is shown that the thermodynamical properties of such a gas do not depend on the genus of the Riemann surface. 
  In non-relativistic quantum mechanics, singular potentials in problems with spherical symmetry lead to a Schrodinger equation for stationary states with non-Fuchsian singularities both as r tends to zero and as r tends to infinity. In the sixties, an analytic approach was developed for the investigation of scattering from such potentials, with emphasis on the polydromy of the wave function in the r variable. The present paper extends those early results to an arbitrary number of spatial dimensions. The Hill-type equation which leads, in principle, to the evaluation of the polydromy parameter, is obtained from the Hill equation for a two-dimensional problem by means of a simple change of variables. The asymptotic forms of the wave function as r tends to zero and as r tends to infinity are also derived. The Darboux technique of intertwining operators is then applied to obtain an algorithm that makes it possible to solve the Schrodinger equation with a singular potential containing many negative powers of r, if the exact solution with even just one term is already known. 
  We find a simple relation between the first subleading terms in the asymptotic expansion of the metric field in AdS$_3$, obeying the Brown-Henneaux boundary conditions, and the stress tensor of the underlying Liouville theory on the boundary. We can also provide an more explicit relation between the bulk metric and the boundary conformal field theory when it is described in terms of a free field with a background charge. 
  At Bradlow's limit, the moduli space of Bogomol'nyi vortices on a compact Riemann surface of genus $g$ is determined. The K\"{a}hler form, and the volume of the moduli space is then computed. These results are compared with the corresponding results previously obtained for a general vortex moduli space. 
  An $\alpha$-parameter representation is derived for gauge field theories.It involves, relative to a scalar field theory, only constants and derivatives with respect to the $\alpha$-parameters. Simple rules are given to obtain the $\alpha$-representation for a Feynman graph with an arbitrary number of loops in gauge theories in the Feynman gauge. 
  We show that the string tension in N=1 two-dimensional super Yang-Mills theory vanishes independently of the representation of the quark anti-quark external source. We argue that this result persists in SQCD_2 and in two-dimensional gauge theories with extended supersymmetry or in chiral invariant models with at least one massless dynamical fermion. We also compute the string tension for the massive Schwinger model, as a demonstration of the method of the calculation. 
  We study the geometry of the triplectic quantization of gauge theories. We show that underlying the triplectic geometry is a Kaehler manifold N endowed with a pair of transversal polarizations. The antibrackets can be brought to the canonical form if and only if N admits a flat symmetric connection that is compatible with the complex structure and the polarizations. 
  To study ``physical'' gauges such as the Coulomb, light-cone, axial or temporal gauge, we consider ``interpolating'' gauges which interpolate linearly between a covariant gauge, such as the Feynman or Landau gauge, and a physical gauge. Lorentz breaking by the gauge-fixing term of interpolating gauges is controlled by extending the BRST method to include not only the local gauge group, but also the global Lorentz group. We enumerate the possible divergences of interpolating gauges, and show that they are renormalizable, and we show that the expectation value of physical observables is the same as in a covariant gauge. In the second part of the article we study the Coulomb-gauge as the singular limit of the Landau-Coulomb interpolating gauge. We find that unrenormalized and renormalized correlation functions are finite in this limit. We also find that there are finite two-loop diagrams of ``unphysical'' particles that are not present in formal canonical quantization in the Coulomb gauge. We verify that in the same limit, the Gauss-BRST Ward identity holds, which is the functional analog of the operator statement that a BRST transformation is generated by the Gauss-BRST charge. As a consequence, $gA_0$ is invariant under renormalization, whereas in a covariant gauge, no component of the gluon field has this property. 
  The Gaussian wavefunctional approach is developed in thermofield dynamics. We manufacture thermal vacuum wavefunctional, its creation as well as annihilation operators,and accordingly thermo-particle excited states. For a (D+1)-dimensional scalar field system with an arbitrary potential whose Fourier representation exists in a sense of tempered distributions, we calculate the finite temperature Gaussian effective potential (FTGEP), one- and two-thermo-particle-state energies. The zero-temperature limit of each of them is just the corresponding result in quantum field theory, and the FTGEP can lead to the same one of each of some concrete models as calculated by the imaginary time Green function. 
  Recent work on the spectrum of the Euclidean Dirac operator spectrum show that the exact microscopic spectral density can be computed in both random matrix theory, and directly from field theory. Exact relations to effective Lagrangians with additional quark species form the bridge between the two formulations. Taken together with explicit computations in the chGUE random matrix ensemble, a series of universality theorems are used to prove that the finite-volume QCD partition function coincides exactly with the universal double-microscopic limit of chUE random matrix partition functions. In the limit where N_f and N_c both go to infinity with the ratio N_f/N_c fixed, the relevant effective Lagrangian undergoes a third order phase transition of Gross-Witten type. 
  We study higher--derivative supergravity with curvature squared terms in different bases. Performing a Weyl rescaling only on the metric or on all the superfield components does not allow to obtain a normalized kinetic Einstein term from a $\cR+\cR^2$ theory. It is necessary to combine a Legendre transformation and a Weyl rescaling on a $\cR+\cR^2$ theory to arrive at a theory of supergravity coupled to matter. This mechanism is applied to supergravity coupled to a general function $k(R,\BR,\Phi,\bar\Phi)$, where $R$ is one of the supergravity chiral superfields and $\Phi$ a chiral matter superfield. 
  Dirac's approach to the canonical quantization of constrained systems is applied to $N = 1$ supergravity, with or without gauged supermatter. Two alternative types of boundary condition applicable to quantum field theory or quantum gravity are contrasted. The first is the `coordinate' boundary condition as used in quantum cosmology; the second type is scattering boundary conditions, as used in Feynman diagrams, applicable to asymptotically flat space-time. The first yields a differential-equation form of the theory, dual to the integral version appropriate to the second. Here, the first (Dirac) approach is found to be extremely streamlined for the calculation of loop amplitudes in these locally supersymmetric theories. By contrast, Feynman-diagram methods have led to calculations which are typically so large as to be unmanageable. Remarkably, the Riemannian quantum amplitude for coordinate boundary conditions in $N = 1$ supergravity (without matter) is exactly semi-classical, being of the form $exp(-I/\hbar)$, where $I$ is the classical action, allowing for the presence of fermions as well as gravity on the boundaries. Even when supermatter is included, typical one-loop amplitudes are often very simple, sometimes not even involving an infinite sum or integral. Specifically, the boundary conditions considered for a number of concrete one-loop examples are set on a pair of concentric 3-spheres in Euclidean 4-space. In the non-trivial cases the amplitudes appear to be exponentially convergent. 
  Recently proposed supergravity theories in odd dimensions whose fields are connection one-forms for the minimal supersymmetric extensions of anti-de Sitter gravity are discussed. Two essential ingredients are required for this construction: (1) The superalgebras, which extend the adS algebra for different dimensions, and (2) the lagrangians, which are Chern-Simons $(2n-1)$-forms. The first item completes the analysis of van Holten and Van Proeyen, which was valid for N=1 only. The second ensures that the actions are invariant by construction under the gauge supergroup and, in particular, under local supersymmetry. Thus, unlike standard supergravity, the local supersymmetry algebra closes off-shell and without requiring auxiliary fields. \\   The superalgebras are constructed for all dimensions and they fall into three families: $osp(m|N)$ for $D=2,3,4$, mod 8, $osp(N|m)$ for $D=6,7,8$, mod 8, and $su(m-2,2|N)$ for D=5 mod 4, with $m=2^{[D/2]}$. The lagrangian is constructed for $D=5, 7$ and 11. In all cases the field content includes the vielbein ($e_{\mu}^{a}$), the spin connection ($\omega_{\mu}^{ab}$), $N$ gravitini ($\psi_{\mu}^{i}$), and some extra bosonic "matter" fields which vary from one dimension to another. 
  This paper investigates a new formalism to describe real time evolution of quantum systems at finite temperature. A time correlation function among subsystems will be derived which allows for a probabilistic interpretation. Our derivation is non-perturbative and fully quantized. Various numerical methods used to compute the needed path integrals in complex time were tested and their effectiveness was compared. For checking the formalism we used the harmonic oscillator where the numerical results could be compared with exact solutions. Interesting results were also obtained for a system that presents tunneling. A ring of coupled oscillators was treated in order to try to check selfconsistency in the thermodynamic limit. The short time distribution seems to propagate causally in the relativistic case. Our formalism can be extended easily to field theories where it remains to be seen if relevant models will be computable. 
  We apply the resolvent technique to the computation of the QED effective action in time dependent electric field backgrounds. The effective action has both real and imaginary parts, and the imaginary part is related to the pair production probability in such a background. The resolvent technique has been applied previously to spatially inhomogeneous magnetic backgrounds, for which the effective action is real. We explain how dispersion relations connect these two cases, the magnetic case which is essentially perturbative in nature, and the electric case where the imaginary part is nonperturbative. Finally, we use a uniform semiclassical approximation to find an expression for very general time dependence for the background field. This expression is remarkably similar in form to Schwinger's classic result for the constant electric background. 
  We propose a method for the calculation of vacuum expectation values (VEVs) given a non-trivial, long-distance vacuum wave functional (VWF) of the kind that arises, for example, in variational calculations. The VEV is written in terms of a Schr\"odinger-picture path integral, then a local expansion for (the logarithm of the) VWF is used. The integral is regulated with an explicit momentum cut-off, $\Lambda$. The resulting series is not expected to converge for $\Lambda$ larger than the mass-gap but studying the domain of analyticity of the VEVs allows us to use analytic continuation to estimate the large-$\Lambda$ limit. Scalar theory in 1+1 dimensions is analyzed, where (as in the case of Yang-Mills) we do not expect boundary divergences. 
  The one-instanton contributions to various correlation functions of supercurrents in four-dimensional N=4 supersymmetric SU(2) Yang-Mills theory are evaluated to the lowest order in perturbation theory.Expressions of the same form are obtained from the leading effects of a single D-instanton extracted from the IIB superstring effective action around the AdS5*S5 background. This is in line with the suggested AdS/Yang-Mills correspondence. The relation between Yang--Mills instantons and D-instantons is further confirmed by the explicit form of the classical D-instanton solution in the AdS5*S5 background and its associated supermultiplet of zero modes. Speculations are made concerning instanton effects in the large-N_c limit of the SU(N_c) Yang-Mills theory. 
  We generalize the Maldacena correspondence to the logarithmic conformal field theories. We study the correspondence between field theories in (d+1)-dimensional AdS space and the d-dimensional logarithmic conformal field theories in the boundary of $AdS_{d+1}$. Using this correspondence, we get the n-point functions of the corresponding logarithmic conformal field theory in d-dimensions. 
  We show under suitable assumptions that zero-modes decouple from the dynamics of non-zero modes in the light-front formulation of some supersymmetric field theories. The implications for Lorentz invariance are discussed. 
  We study the gauge covariance of the fermion-photon vertex in quenched, massless 3-dimensional quantum electrodynamics. A class of vertex Ans\"{a}tze obtained by generalizing that proposed by Dong et al. is tested using the invariance of the photon polarization scalar under the Landau-Khalatnikov transformation. 
  A new attempt is demonstrated that QFTs can be UV finite if they are viewed as the low energy effective theories of a fundamental underlying theory (that is complete and well-defined in all respects) according to the modern standard point of view. This approach works for any interaction model and space-time dimension. It is much simpler in principle and in technology comparing to any known renormalization program.Unlike the known renormalization methods, the importance of the procedure for defining the ambiguities (corresponding to the choice of the renormalization conditions in the conventional program) is fully appreciated in the new approach. It is shown that the high energy theory(s) or the underlying theory(s) in fact 'stipulates (stipulate)' the low energy and effective ones through these definitions within our approach while all the conventional methods miss this important point. Some simple but important nonperturbative examples are discussed to show the power and plausibility of the new approach. Other related issues (especially the IR problem and the implication of our new approach for the canonical quantization procedure) are briefly touched. 
  Massless and massive scalar fields and massless spinor fields are considered at arbitrary temperatures in four dimensional ultrastatic curved spacetime. Scalar models under consideration can be either conformal or nonconformal and include selfinteraction. The one-loop nonlocal effective action at finite temperature and free energy for these quantum fields are found up to the second order in background field strengths using the covariant perturbation theory. The resulting expressions are free of infrared divergences. Spectral representations for nonlocal terms of high temperature expansions are obtained. 
  The problem of renormalization of the semiclassical one-loop equations used in the non-equilibrium field theory is considered. Recently, the renormalizability of such equations has been justified for some special cases of classical field configurations. In this paper the general case of arbitrary spatially inhomogeneous field configuration is investigated. It is shown that for certain quantum states the divergences arising in one-loop equations can be eliminated by usual perturbation-theory counterterms. 
  We formulate the effective field theory of a D-particle on orbifolds of $T^4$ by a cyclic group as a gauge theory in a $V$-bundle over the dual orbifold. We argue that this theory admits Fayet-Iliopoulos terms analogous to those present in the case of noncompact orbifolds. In the $n=2$ case, we present some evidence that turning on such terms resolves the orbifold singularities and may lead to a $K3$ surface realized as a blow up of the fixed points of the cyclic group action. 
  A computational scheme is developed to determine the response of a quantum field theory (QFT) with a factorized scattering operator under a variation of the Unruh temperature. To this end a new family of integrable systems is introduced, obtained by deforming such QFTs in a way that preserves the bootstrap S-matrix. The deformation parameter \beta plays the role of an inverse temperature for the thermal equilibrium states associated with the Rindler wedge, \beta = 2\pi being the QFT value. The form factor approach provides an explicit computational scheme for the \beta \neq 2\pi systems, enforcing in particular a modification of the underlying kinematical arena. As examples deformed counterparts of the Ising model and the Sinh-Gordon model are considered. 
  The relationship between N-soliton solutions to the Euclidean sine-Gordon equation and Lorentzian black holes in Jackiw-Teitelboim dilaton gravity is investigated, with emphasis on the important role played by the dilaton in determining the black hole geometry. We show how an N-soliton solution can be used to construct ``sine-Gordon'' coordinates for a black hole of mass M, and construct the transformation to more standard ``Schwarzchild-like'' coordinates. For N=1 and 2, we find explicit closed form solutions to the dilaton equations of motion in soliton coordinates, and find the relationship between the soliton parameters and the black hole mass. Remarkably, the black hole mass is non-negative for arbitrary soliton parameters. In the one-soliton case the coordinates are shown to cover smoothly a region containing the whole interior of the black hole as well as a finite neighbourhood outside the horizon. A Hamiltonian analysis is performed for slicings that approach the soliton coordinates on the interior, and it is shown that there is no boundary contribution from the interior. Finally we speculate on the sine-Gordon solitonic origin of black hole statistical mechanics. 
  The renormalized energy density of a massless scalar field defined in a D-dimensional flat spacetime is computed in the presence of "soft" and "semihard" boundaries, modeled by some smoothly increasing potential functions. The sign of the renormalized energy densities for these different confining situations is investigated. The dependence of this energy on $D$ for the cases of "hard" and "soft/semihard" boundaries are compared. 
  Using an octonionic formalism, we introduce a new mechanism for reducing 10 spacetime dimensions to 4 without compactification. Applying this mechanism to the free, 10-dimensional, massless (momentum space) Dirac equation results in a particle spectrum consisting of exactly 3 generations. Each generation contains 1 massive spin-1/2 particle with 2 spin states, 1 massless spin-1/2 particle with only 1 helicity state, and their antiparticles --- precisely one generation of leptons. There is also a single massless spin-1/2 particle/antiparticle pair with the opposite helicity and no generation structure.   We conclude with a discussion of some further consequences of this approach, including those which could arise when using the formalism on a curved spacetime background, as well as the implications for the nature of spacetime itself. 
  We construct explicit multivortex solutions for the first and second complex sine-Gordon equations. The constructed solutions are expressible in terms of the modified Bessel and rational functions, respectively. The vorticity-raising and lowering Backlund transformations are interpreted as the Schlesinger transformations of the fifth Painleve equation. 
  By using the Hamiltonian version of the AdS/CFT correspondence, we compute the two-point Green function of a local operator in D=4 N=4 super Yang-Mills theory, which corresponds to a massive antisymmetric tensor field of the second rank on the $AdS_{5}$ background. We discuss the conformal transformations induced on the boundary by isometries of $AdS_{5}$. 
  Dirac's contour representation is extended to parabose and parafermi systems by use of deformed algebra techniques. In this analytic representation the action of the paraparticle annihilation operator is equivalent to a deformed differentiation which encodes the statistics of the paraparticle. In the parafermi case, the derivative's ket-domain is degree $p$ polynomials. 
  Eigenstates of the parabose and parafermi creation operators are constructed. In the Dirac contour representation, the parabose eigenstates correspond to the dual vectors of the parabose coherent states. In order $p=2$, conserved-charge parabose creation operator eigenstates are also constructed. The contour forms of the associated resolutions of unity are obtained. 
  There are no reasons why the singularity in the growth of the dilaton coupling should not be regularised, in a string cosmological context, by the presence of classical inhomogeneities. We discuss a class of inhomogeneous dilaton-driven models whose curvature invariants are all bounded and regular in time and space. We prove that the non-space-like geodesics of these models are all complete in the sense that none of them reaches infinity for a finite value of the affine parameter. We conclude that our examples represent truly singularity-free solutions of the low energy beta functions. We discuss some symmetries of the obtained solutions and we clarify their physical interpretation. We also give examples of solutions with spherical symmetry. In our scenario each physical quantity is everywhere defined in time and space, the big-bang singularity is replaced by a maximal curvature phase where the dilaton kinetic energy reaches its maximum. The maximal curvature is always smaller than one (in string units) and the coupling constant is also smaller than one and it grows between two regimes of constant dilaton, implying, together with the symmetries of the solutions, that higher genus and higher curvature corrections are negligible. We argue that our examples describe, in a string cosmological context, the occurrence of ``little bangs''(i.e. high curvature phases which never develop physical singularities). They also suggest the possibility of an unexplored ``pre-little-bang'' phase. 
  We review the structure and symmetry properties of the worldvolume action for the M-theory 5-brane and of its equations of motion. 
  The recent proposal by Hawking and Turok for obtaining an open inflationary universe from singular instantons makes use of low-energy effective Lagrangians describing gravity coupled to scalars and non-propagating antisymmetric tensors. In this paper we derive some exact results for Lagrangians of this type, obtained from spherical compactifications of M-theory and string theory. In the case of the S^7 compactification of M-theory, we give a detailed discussion of the cosmological solutions. We also show that the lower-dimensional Lagrangians admit domain-wall solutions, which preserve one half of the supersymmetry, and which approach AdS spacetimes near their horizons. 
  A new spinning particle with a definite sign of the energy is defined on spacelike hypersurfaces after a critical discussion of the standard spinning particles. They are the pseudoclassical basis of the positive energy $({1\over 2},0)$ [or negative energy $(0,{1\over 2})$] parts of the $({1\over 2},{1\over 2})$ solutions of the Dirac equation. The study of the isolated system of N such spinning charged particles plus the electromagnetic field leads to their description in the rest-frame Wigner-covariant instant form of dynamics on the Wigner hyperplanes orthogonal to the total 4-momentum of the isolated system (when it is timelike). We find that on such hyperplanes these spinning particles have a nonminimal coupling only of the type "spin-magnetic field" like the nonrelativistic Pauli particles to which they tend in the nonrelativistic limit. The Lienard-Wiechert potentials associated with these charged spinning particles are found. Then, a comment on how to quantize the spinning particles respecting their fibered structure describing the spin structure is done. 
  The BRST-anti-BRST covariant extension is suggested for the split involution quantization scheme for the second class constrained theories. The constraint algebra generating equations involve on equal footing a pair of BRST charges for second class constraints and a pair of the respective anti-BRST charges. Formalism displays explicit Sp(2) \times Sp(2) symmetry property. Surprisingly, the the BRST-anti-BRST algebra must involve a central element, related to the nonvanishing part of the constraint commutator and having no direct analogue in a first class theory. The unitarizing Hamiltonian is fixed by the requirement of the explicit BRST-anti-BRST symmetry with a much more restricted ambiguity if compare to a first class theory or split involution second class case in the nonsymmetric formulation. The general method construction is supplemented by the explicit derivation of the extended BRST symmetry generators for several examples of the second class theories, including self--dual nonabelian model and massive Yang Mills theory. 
  Grassmann-valued Dirac fields together with the electromagnetic field (the pseudoclassical basis of QED) are reformulated on spacelike hypersurfaces in Minkowski spacetime and then restricted to Wigner hyperplanes to get their description in the rest-frame Wigner-covariant instant form of dynamics. The canonical reduction to the Wigner-covariant Coulomb gauge is done in the rest frame. It is shown, on the basis of a geometric incosistency, that the description of fermions is incomplete, because there is no bosonic carrier of the spin structure describing the trajectory of the electric current in Minkowski spacetime, as it was already emphasized in connection with the first quantization of spinning particles in a previous paper. 
  A geometrical construction of superconformal transformations in six dimensional (2,0) superspace is proposed. Superconformal Killing vectors are determined. It is shown that the transformation of the tensor multiplet involves a zero curvature non-trivial cochain. 
  We consider attractor varieties arising in the construction of dyonic black holes in Calabi-Yau compactifications of IIB string theory. We show that the attractor varieties are constructed from products of elliptic curves with complex multiplication for $\CN=4,8$ compactifications. The heterotic dual theories are related to rational conformal field theories. The emergence of curves with complex multiplication suggests many interesting connections between arithmetic and string theory. This paper is a brief overview of a longer companion paper entitled ``Arithmetic and Attractors,'' hep-th/9807087. 
  We show that the geometry of K3 surfaces with singularities of type A-D-E contains enough information to reconstruct a copy of the Lie algebra associated to the given Dynkin diagram. We apply this construction to explain the enhancement of symmetry in F and IIA theories compactified on singular K3's. 
  In this work we have constructed the most general action for a set of complex homogeneous scalar supermultiplets interacting with the scale factor in the supersymmetric FRW model. It is shown, that local conformal time supersymmetry leads to the scalar fields potential, which is defined in the same combination: K\"ahler potential and superpotential as in supergravity (or effective superstring) theories. This scalar fields potential depends on arbitrary parameter $\alpha$, which is not fixed by conformal time supersymmetry. 
  The vacuum structure of N=2 (and N=4) SUSY Yang-Mills theory is analyzed in detail by considering the effective potential for constant background scalar- magnetic fields within different approximations. We compare the one-loop approximation with- or without instanton improved effective coupling with the one-loop result in the dual desription. For N=2 we find that non-perturbative monopole degrees of freedom remove the non-trivial minima present in the (improved) one-loop potential in the strong-coupling regime. The combination of Yang-Mills and dual desription leads to a self-consistent effective potential over the full range of background fields. 
  Membranes holomorphically embedded in flat noncompact space are constructed in terms of the degrees of freedom of an infinite collection of 0-branes. To each holomorphic curve we associate infinite-dimensional matrices which are static solutions to the matrix theory equations of motion, and which can be interpreted as the matrix theory representation of the holomorphically embedded membrane. The problem of finding such matrix representations can be phrased as a problem in geometric quantization, where epsilon -> l_P^3/R plays the role of the Planck constant and parametrizes families of solutions. The concept of Bergman projection is used as a basic tool, and a local expansion for the action of the projection in inverse powers of curvature is derived. This expansion is then used to compute the required matrices perturbatively in epsilon. The first two terms in the expansion correspond to the standard geometric quantization result and to the result obtained using the metaplectic correction to geometric quantization. 
  The Hamiltonian approach to the quantum field theory is considered. Since there are additional difficulties such as the Haag theorem and Stueckelberg divergences, renormalization of the time-dependent dynamical quantum field theory is much more complicated than renormalization of the S-matrix. It is necessary to consider the regularized theory with ultraviolet and infrared cutoffs and impose the conditions not only on the dependence of the Hamiltonian on the cutoffs (as usual) but also on the dependence of the initial states. It happens that one should consider the initial states to be singulary dependent on the cutoffs in order to avoid the Stueckelberg divergences. Different types of semiclassical approximations to quantum theory are discussed. It happens that the method of quantizing classical solutions to field equations corresponds not to the WKB-approach but to the complex-WKB theory. The problem of imposing conditions on the semiclassical initial states is discussed. Different prescriptions for choice of initial conditions are analysed. 
  We construct a bi-linear form on the periods of Calabi-Yau spaces. These are used to obtain the prepotentials around conifold singularities in type-II strings compactified on Calabi-Yau space. The explicit construction of the bi-linear forms is achieved for the one-moduli models as well as two moduli models with K3-fibrations where the enhanced gauge symmetry is known to be observed at conifold locus. We also show how these bi-linear forms are related with the existence of flat coordinates. We list the resulting prepotentials in two moduli models around the conifold locus, which contains alpha' corrections of 4-D N=2 SUSY SU(2) Yang-Mills theory as the stringy effect. 
  The minimal length uncertainty principle of Kempf, Mangano and Mann (KMM), as derived from a mutilated quantum commutator between coordinate and momentum, is applied to describe the modes and wave packets of Hawking particles evaporated from a black hole. The transplanckian problem is successfully confronted in that the Hawking particle no longer hugs the horizon at arbitrarily close distances. Rather the mode of Schwarzschild frequency $\omega$ deviates from the conventional trajectory when the coordinate $r$ is given by $| r - 2M|\simeq \beta_H \omega / 2 \pi$ in units of the non local distance legislated into the uncertainty relation. Wave packets straddle the horizon and spread out to fill the whole non local region. The charge carried by the packet (in the sense of the amount of "stuff" carried by the Klein--Gordon field) is not conserved in the non--local region and rapidly decreases to zero as time decreases. Read in the forward temporal direction, the non--local region thus is the seat of production of the Hawking particle and its partner. The KMM model was inspired by string theory for which the mutilated commutator has been proposed to describe an effective theory of high momentum scattering of zero mass modes. It is here interpreted in terms of dissipation which gives rise to the Hawking particle into a reservoir of other modes (of as yet unknown origin). On this basis it is conjectured that the Bekenstein--Hawking entropy finds its origin in the fluctuations of fields extending over the non local region. 
  We study the Born-Infeld D-brane action in the limit when the string coupling goes to infinity. The resulting actions is presented in an arbitrary background and shown to describe a foliation of the world-volume by strings. Using a recently developed ``degenerate'' supergravity the parton picture is shown to be applicable also to supersymmetric D-branes. 
  We show that the transfer matrix of the A_{N-1}^(1) open spin chain with diagonal boundary fields has the symmetry U_q (SU(L)) x U_q (SU(N-L)) x U(1), as well as a ``duality'' symmetry which interchanges L and N - L. We exploit these symmetries to compute exact boundary S matrices in the regime with q real. 
  We present the full action for the unoriented open-closed string field theory which is based on the \alpha=p^+ HIKKO type vertices. The BRS invariance of the action is proved up to the terms which are expected to cancel the anomalous one-loop contributions. This implies that the system is invariant under the gauge transformations with open and closed string field parameters up to the anomalies. 
  We analyse a class of four-dimensional heterotic ground states with N=2 space-time supersymmetry. From the ten-dimensional perspective, such models can be viewed as compactifications on a six-dimensional manifold with SU(2) holonomy, which is locally but not globally K3 x T^2. The maximal N=4 supersymmetry is spontaneously broken to N=2. The masses of the two massive gravitinos depend on the (T,U) moduli of T^2. We evaluate the one-loop threshold corrections of gauge and R^2 couplings and we show that they fall in several universality classes, in contrast to what happens in usual K3 x T^2 compactifications, where the N=4 supersymmetry is explicitly broken to N=2, and where a single universality class appears. These universality properties follow from the structure of the elliptic genus. The behaviour of the threshold corrections as functions of the moduli is analysed in detail: it is singular across several rational lines of the T^2 moduli because of the appearance of extra massless states, and suffers only from logarithmic singularities at large radii. These features differ substantially from the ordinary K3 x T^2 compactifications, thereby reflecting the existence of spontaneously-broken N=4 supersymmetry. Although our results are valid in the general framework defined above, we also point out several properties, specific to orbifold constructions, which might be of phenomenological relevance. 
  In this paper we describe a new type of topological defect, called a homilia string, which is stabilized via interactions with the string network. Using analytical and numerical techniques, we investigate the stability and dynamics of homilia strings, and show that they can form stable electroweak strings. In SU(2)xU(1) models of symmetry breaking the intersection of two homilia strings is identified with a sphaleron. Due to repulsive forces, the homilia strings seperate, resulting in sphaleron annihilation. It is shown that electroweak homilia string loops cannot stabilize as vortons, which circumvents the adverse cosmological problems associated with stable loops. The consequences for GUT scale homilia strings are also discussed. 
  We propose a reformulation of SU(2) Yang-Mills theory in terms of new variables. These variables are appropriate for describing the theory in its infrared limit, and indicate that it admits knotlike configurations as stable solitons. As a consequence we arrive at a dual picture of the Yang-Mills theory where the short distance limit describes asymptotically free, massless point gluons and the large distance limit describes extended, massive knotlike solitons. 
  In this lecture we discuss correlations of the QCD Dirac eigenvalues. We find that below a scale of $E_c\sim \Lambda/L^2$ they are given by chiral Random Matrix Theory. This follows from analytical arguments based on partially quenched Chiral Perturbation Theory and is substantiated by lattice QCD and instanton liquid simulations. 
  In this paper we study the scattering of gravitons off a five orientifold in eleven dimensions. We compare the supergravity result with a two loop M(atrix) model calculation and find exact agreement. The supergravity calculation involves nonlinear three graviton effects. 
  We consider the one-loop effective action due to a spinor loop coupled to an abelian vector and axial vector field background. After rewriting this effective action in terms of an auxiliary non-abelian gauge connection, we use the De Witt expansion to analyze both its anomalous and non-anomalous content. The same transformation allows us to obtain a novel worldline path integral representation for this effective action which avoids the usual separation into the real and imaginary parts of the Euclidean effective action, as well as the introduction of auxiliary dimensions. 
  We study a massive Thirring-like model in 2-dimensional space-time, which contains two different species of fermions. This model is a field theoretical version of the quantum mechanical model originally proposed by Gl\"{o}ckle, Nogami and Fukui, where different fermions interact with each other through $\delta$-function potentials. We derive a corresponding boson model by the bosonization technique in the path integral formulation. This is a simple but non-trivial extension of the freedom of the bosonization technique. Operator correspondences between fermion and boson fields are given. One of these could not be realistically expected from the naive correspondence of the original single-species models. It is essential for this point that in our model fermions of the same kind do not interact with each other directly. We find that for a specific value of the coupling constant, one boson field becomes free while the other is a Sine-Gordon field. For this case, therefore, our two-species model is equivalent to the ordinary Sine-Gordon model of a single boson field. 
  We construct new trigonometric solutions of the Yang-Baxter equation, using the Fuss-Catalan algebras, a set of multi-colored versions of the Temperley-Lieb algebra, recently introduced by Bisch and Jones. These lead to new two-dimensional integrable lattice models, describing dense gases of colored loops. 
  In the light of the recent Super-Kamiokande experiments which demonstrate neutrino oscillation, and therefore a non zero mass, it is pointed out how such a mass has also been deduced theoretically. 
  We present a nonsupersymmetric orbifold of type II string theory and show that it has vanishing cosmological constant at the one and two loop level. We argue heuristically that the cancellation persists at higher loops. 
  A well-defined chirally split functional integrating the 2D chirally split diffeomorphism anomaly is exhibited on an arbitrary compact Riemann surface without boundary. The construction requires both the use of the Beltrami parametrisation of complex structures and the introduction of a background metric possibly subject to a Liouville equation. This formula reproduces in the flat case the so-called Polyakov action. Although it works on the torus (genus 1), the proposed functional still remains to be related to a Wess-Zumino action for diffeomorphisms. 
  Interacting fields can be constructed as formal power series in the framework of causal perturbation theory. The local field algebra $\tilde {\cal F}({\cal O})$ is obtained without performing the adiabatic limit; the (usually bad) infrared behavior plays no role. To construct the observables in gauge theories we use the Kugo-Ojima formalism; we define the BRST-transformation $\tilde s$ as a graded derivation on the algebra of interacting fields and use the implementation of $\tilde s$ by the Kugo-Ojima operator $Q_{\rm int}$. Since our treatment is local, the operator $Q_{\rm int}$ differs from the corresponding operator $Q$ of the free theory. We prove that the Hilbert space structure present in the free case is stable under perturbations. All assumptions are shown to be satisfied in QED. 
  We discuss, in the context of the strongly coupled E_8 \times E_8 heterotic string proposed by Horava and Witten, the appearance of anomalous U(1)_X symmetries of a nonperturbative origin, related to the presence, after compactification, of five-branes in the five-dimensional bulk of the theory. We compute the gauge anomalies and the induced Fayet-Iliopoulos terms on each boundary, which we find to be lower than the universal one induced in the weakly coupled case. 
  Just as parallel threebranes on a smooth manifold are related to string theory on $AdS_5\times {\bf S}^5$, parallel threebranes near a conical singularity are related to string theory on $AdS_5\times X_5$, for a suitable $X_5$. For the example of the conifold singularity, for which $X_5=(SU(2)\times SU(2))/U(1)$, we argue that string theory on $AdS_5\times X_5$ can be described by a certain ${\cal N}=1$ supersymmetric gauge theory which we describe in detail. 
  Motivated by the shape invariance condition in supersymmetric quantum mechanics, we develop an algebraic framework for shape invariant Hamiltonians with a general change of parameters. This approach involves nonlinear generalizations of Lie algebras. Our work extends previous results showing the equivalence of shape invariant potentials involving translational change of parameters with standard $SO(2,1)$ potential algebra for Natanzon type potentials. 
  Certain classes of chiral four-dimensional gauge theories may be obtained as the worldvolume theories of D5-branes suspended between networks of NS5-branes, the so-called brane box models. In this paper, we derive the stringy consistency conditions placed on these models, and show that they are equivalent to anomaly cancellation of the gauge theories. We derive these conditions in the orbifold theories which are T-dual to the elliptic brane box models. Specifically, we show that the expression for tadpoles for unphysical twisted Ramond-Ramond 4-form fields in the orbifold theory are proportional to the gauge anomalies of the brane box theory. Thus string consistency is equivalent to worldvolume gauge anomaly cancellation. Furthermore, we find additional cylinder amplitudes which give the $\beta$-functions of the gauge theory. We show how these correspond to bending of the NS-branes in the brane box theory. 
  In this work we construct the vacuum configuration of supergravity interacting with homogeneous complex scalar matter fields. The corresponding configuration is of the FRW model invariant under the $n=2$ local conformal time supersymmetry, which is a subgroup of the four dimensional space-time supersymmetry. We show, that the potential of the scalar matter fields is a function of the K\"ahler potential and of the arbitrary parameter $\alpha$. This parameter enumerates the vacuum states. The scalar matter potential induces the spontaneous breaking of supersymmetry in supergravity. On the quantum level our model is a specific supersymmetric quantum mechanics, which admits quantum states in supergravity, and the states with zero energy are described by the wave function of the FRW universe. 
  If cosmological constant is positive, a black hole is naturally described by the Schwarzschild-de Sitter solution with two horizons. We use the global method to extract the topological information and the selection rule for the Gibbons-Hawking temperature for the thermal vacua. These are related to the Euler number of the Euclidean section whose topology is more complicated than expected. We also point out the failure of the usual local method of conical singularity approach in dealing with multi-horizon scenarios. 
  We give a reinterpretation of the matrix theory discussed by Moore, Nekrasov and Shatashivili (MNS) in terms of the second quantized operators which describes the homology class of the Hilbert scheme of points on surfaces. It naturally relate the contribution from each pole to the inner product of orthogonal basis of free boson Fock space. These basis can be related to the eigenfunctions of Calogero-Sutherland (CS) equation and the deformation parameter of MNS is identified with coupling of CS system. We discuss the structure of Virasoro symmetry in this model. 
  Usually we consider the symmetry of action as the symmetry of the theory, however, in the Keplar problem the scaling symmetry existing in equa tion of motion is not the ones for action. It changes the multiplicative c onstant of action and the time boundary. In such a case that the scale tran sformation does not leave the action invariant but keeping the equation inva riant, the following statement is proved. The time integration of Lagrangian is explicitly performed and the action ca n be expressed by the difference of formal (non-conserved) Noether charges a t time boundaries. In field theory the action can be expressed by the bound ary integration of the formal Noether current. 
  We study relations between some topics in number theory and supersymmetric black holes. These relations are based on the ``attractor mechanism'' of N=2 supergravity. In IIB string compactification this mechanism singles out certain ``attractor varieties.'' We show that these attractor varieties are constructed from products of elliptic curves with complex multiplication for N=4 and N=8 compactifications. The heterotic dual theories are related to rational conformal field theories. In the case of N=4 theories U-duality inequivalent backgrounds with the same horizon area are counted by the class number of a quadratic imaginary field. The attractor varieties are defined over fields closely related to class fields of the quadratic imaginary field. We discuss some extensions to more general Calabi-Yau compactifications and explore further connections to arithmetic including connections to Kronecker's Jugendtraum and the theory of modular heights. The paper also includes a short review of the attractor mechanism. A much shorter version of the paper summarizing the main points is the companion note entitled ``Attractors and Arithmetic'' (hep-th/9807056). 
  By using the enlarged BRS transformations we control the gauge parameter dependence of Green functions in the background field gauge. We show that it is unavoidable -- also if we consider the local Ward identity -- to introduce the normalization gauge parameter $\xi_o$, which enters the Green functions of higher orders similarly to the normalization point $\kappa$. The dependence of Green functions on $\xi_o$ is governed by a further partial differential equation. By modifying the Ward identity we are able to construct in 1-loop order a gauge parameter independent combination of 2-point vector and background vector functions. By explicit construction of the next orders we show that this combination can be used to construct a gauge parameter independent RG-invariant charge. However, it is seen that this RG-invariant charge does not satisfy the differential equation of the normalization gauge parameter $\xi_o$, and, hence, is not $\xi_o$-independent as required. 
  A massive vector gauge theory constructed from the matrix derivative approach of noncommutative geometry is compared with the Higgs-Connes-Lott theory. In the massive vector gauge theory, a new extra shift-like symmetry which is due to the one form constant matrix derivative allows the theory to have a mass term while keeping the gauge symmetry intact. In the Higgs-Connes-Lott theory, the transformation of scalar field makes up the deficiency of symmetry due to the mass term. Thus, when the scalar field is absent there remains no gauge symmetry just like the Proca model. In the massive vector gauge theory, the shift-like symmetry makes up the deficiency of symmetry due to the mass term even in the absence of the scalar field. 
  We describe conformal operators living at the boundary of $AdS_{d+1}$ in a general setting. Primary conformal operators at the threshold of the unitarity bounds of UIR's of O(d,2) correspond to singletons and massless fields in $AdS_{d+1}$, respectively. For maximal supersymmetric theories in $AdS_{d+1}$ we describe ``chiral'' primary short supermultiplets and non-chiral primary long supermultiplets. Examples are exhibited which correspond to KK and string states. We give the general contribution of a primary conformal operator to the OPE and Green's functions of primary fields, which may be relevant to compute string corrections to the four-point supergraviton amplitude in Anti-de-Sitter space. The material in this paper was presented by one of the authors (S.F) at String 98, Santa Barbara. 
  The local logarithmic conformal field theory corresponding to the triplet algebra at c=-2 is constructed. The constraints of locality and crossing symmetry are explored in detail, and a consistent set of amplitudes is found. The spectrum of the corresponding theory is determined, and it is found to be modular invariant. This provides the first construction of a non-chiral rational logarithmic conformal field theory, establishing that such models can indeed define bona fide conformal field theories. 
  We propose a new systematic approach that allows one to derive the spin foam (state sum) model of a theory starting from the corresponding classical action functional. It can be applied to any theory whose action can be written as that of the BF theory plus a functional of the B field. Examples of such theories include BF theories with or without cosmological term, Yang-Mills theories and gravity in various spacetime dimensions. Our main idea is two-fold. First, we propose to take into account in the path integral certain distributional configurations of the B field in which it is concentrated along lower dimensional hypersurfaces in spacetime. Second, using the notion of generating functional we develop perturbation expansion techniques, with the role of the free theory played by the BF theory. We test our approach on various theories for which the corresponding spin foam (state sum) models are known. We find that it exactly reproduces the known models for BF and 2D Yang-Mills theories. For the BF theory with cosmological term in 3 and 4 dimensions we calculate the terms of the transition amplitude that are of the first order in the cosmological constant, and find an agreement with the corresponding first order terms of the known state sum models. We discuss implications of our results for existing quantum gravity models. 
  Within the classical approximation we calculate the static $Q\bar Q$ potential via the AdS/CFT relation for nonzero temperature and arbitrary internal orientation of the quarks. We use a higher order curvature corrected target space background. For timelike Wilson loops there arises a critical line in the orientation-distance plane which is shifted to larger distances relative to the calculation with uncorrected background. Beyond that line there is no $Q\bar Q$-force. The overall vanishing of the force for antipodal orientation known from zero tempera ture remains valid. The spacelike Wilson loops yield a string tension for a (2+1)-dimensional gauge theory, independent of the relative internal orientation, but sensitive to the background correction. 
  Unlike the extremal Reissner - Nordstr\"{o}m black hole in ordinary spacetime, the one in anti-de Sitter spacetime is a minimum of action and has zero entropy if quantization is carried out after extremalization. However, if extremalization is carried out after quantization, then the entropy is a quarter of the area as in the usual case. 
  We study the rigid limit of type IIB string theory, compactified on a K3 fibration, which, near its conifold limit, contains the Seiberg-Witten curve for N=2 SU(2) Super-Yang-Mills with a massive hypermultiplet in the fundamental representation. Instead of working with an ALE approximation, we treat the K3 fibration globally. The periods we get in this way, allow for an embedding of the field theory into a supergravity model. 
  In this lecture we give an elementary introduction to the natural realization of non-perturbative N=2 quantum field theories as a low energy limit of  classical string theory. We review a systematic construction of six, five, and four dimensional gauge theories using geometrical data, which provides the exact, non-perturbative solution via mirror symmetry. This construction has lead to the exact solution of a large class of gravity free gauge theories, including Super Yang Mills (SYM) theories as well as non-conventional quantum field theories without a known Lagrangian description. 
  We discuss the properties of four-point functions in the context of the correspondence between a classical supergravity theory in the bulk of the Anti de Sitter space and quantum conformal field theory at the boundary. The contribution to a four-point function from the exchange of a scalar field of arbitrary mass in AdS space is explicitly identified with that of the corresponding operator in the conformal partial wave expansion of a four-point function on the CFT side. Integral representations are found for the massless vector and graviton exchanges. We also discuss some aspects of the four-point functions of Tr F^2 and Tr F F^* (`dilaton' and `axion') operators in the N =4 supersymmetric Yang-Mills. 
  The order g^2 radiative corrections to all 2- and 3-point correlators of the composite primary operators Tr X^k are computed in {\cal N} = 4 supersymmetric Yang-Mills theory with gauge group SU(N). Corrections are found to vanish for all N. For k=2 this is a consequence of known superconformal nonrenormalization theorems, and for general k the result confirms an N-to-infinity, fixed large g^2N supergravity calculation and further conjectures in hep-th/9806074. A 3-point correlator involving {\cal N} = 4 descendents of Tr X^2 is calculated, and its order g^2 contribution also vanishes, giving evidence for the absence of radiative corrections in correlators of descendent operators. 
  It is shown that, under quite general conditions, thermal correlation functions in relativistic quantum field theory have stronger analyticity properties in configuration space than those imposed by the KMS-condition. These analyticity properties may be understood as a remnant of the relativistic spectrum condition in the vacuum sector and lead to a Lorentz-covariant formulation of the KMS-condition involving all space-time variables. 
  We investigate cosmological solutions of eleven dimensional supergravity compactified on a squashed seven manifold. The effective action for the four dimensional theory contains scalar fields describing the size and squashing of the compactifying space. The potential for these fields consists of a sum of exponential terms. At early times only one such term is expected to dominate. The condition for an exponential potential to admit inflationary solutions is derived and it is shown that inflation is not possible in our model. The criterion for an exponential potential to admit a Hawking-Turok instanton is also derived. It is shown that the instanton remains singular in eleven dimensions. 
  Models of black holes in (1+1)-dimensions provide a theoretical laboratory for the study of semi-classical effects of realistic black holes in Einstein's theory. Important examples of two-dimensional models are given by string theory motivated dilaton gravity, by ordinary general relativity in the case of spherical symmetry, and by {\em Poincar\'e gauge gravity} in two spacetime dimensions. In this paper, we present an introductory overview of the exact solutions of two-dimensional classical Poincar\'e gauge gravity (PGG). A general method is described with the help of which the gravitational field equations are solved for an arbitrary Lagrangian. The specific choice of a torsion-related coframe plays a central role in this approach. Complete integrability of the general PGG model is demonstrated in vacuum, and the structure of the black hole type solutions of the quadratic models with and without matter is analyzed in detail. Finally, the integrability of the general dilaton gravity model is established by recasting it into an effective PGG model. 
  The algebraic method of renormalization is applied to the standard model of electroweak interactions. We present the most important modifications compared to theories with simple groups. 
  A systematic investigation is given of finite size effects in $d=2$ quantum gravity or equivalently the theory of dynamically triangulated random surfaces. For Ising models coupled to random surfaces, finite size effects are studied on the one hand by numerical generation of the partition function to arbitrary accuracy by a deterministic calculus, and on the other hand by an analytic theory based on the singularity analysis of the explicit parametric form of the free energy of the corresponding matrix model. Both these reveal that the form of the finite size corrections, not surprisingly, depend on the string susceptibility. For the general case where the parametric form of the matrix model free energy is not explicitly known, it is shown how to perform the singularity analysis. All these considerations also apply to other observables like susceptibility etc. In the case of the Ising model it is shown that the standard Fisher-scaling laws are reproduced. 
  By using BPS closed string, the entropy is calculated of the extremal five dimensional black hole consisting of Dirichlet onebranes, Dirichlet fivebranes and Kaluza-Klein momentum in the flat background approximation. In our formulations we consider two kinds of BPS closed strings with or without a winding number. In the former case heavy excitation modes of closed strings are used to derive the entropy. In the latter case we have no oscillator modes and consider collective motion of such massless closed strings. The entropy is given by the number of the ways how we divide the Kaluza-Klein momentum among the massless closed strings. In both cases the black hole entropy is the same as the Bekenstein-Hawking entropy. We argue that the collective modes of closed strings without winding is equivalent to a single closed string with winding. We propose that the two closed string pictures are connected with the open string pictures by the modular transformation. 
  We invistigate exact solutions for the two-dimensional quantum field theories called Wess-Zumino-Novikov-Witten (WZNW) models. A WZNW model is a sigma model whose classical fields are applications from a bidimensional space-time (a Riemann surface in the euclidian case) to a Lie group, the target space. We construct (and we compute in genus zero and one) the metric connection, called the Knizhnik-Zamolodchikov-Bernard (KZB) connection, on the bundle of conformal blocks of the WZNW model. The KZB connection may be viewed as a quantization of Hitchin integrable systems whose configuration space is the moduli space of principal holomorphic bundles over a Riemann surface and whose phase space is the (holomorphic) cotangent bundle to the configuration space. For these systems, we construct explicitly a complete familly of Hamiltonians in involution in genus zero, one and two, with (complex) group SL(2) for the last case. The main result is the self-duality property of the Hitchin system at genus two, that is the invariance of the Hamiltonians with respect to the interchange of positions and momenta in the phase space. We finally realize the (geometric) quantization of the Hitchin systems. 
  We examine the 1+1 dimensional CP(N) model in the large N limit by using the Schroedinger representation. Starting from the Hamiltonian analysis of the model, we present the variational gap equation resulting from the Gaussian trial wave functional. The renormalization of the theory is performed with insertion of mass and energy counter-terms, and the dynamical generation of mass and the energy eigenvalue are derived. 
  In this work, we consider matroid theory. After presenting three different (but equivalent) definitions of matroids, we mention some of the most important theorems of such theory. In particular, we note that every matroid has a dual matroid and that a matroid is regular if and only if it is binary and includes no Fano matroid or its dual. We show a connection between this last theorem and octonions which at the same time, as it is known, are related to the Englert's solution of D = 11 supergravity. Specifically, we find a relation between the dual of Fano matroid and D = 11 supergravity. Possible applications to M-theory are speculated upon. 
  We review some of the most important results obtained over the years on the study of Yang-Mills fields on the four dimensional torus at the classical level. 
  The theta dependent of pure gauge theories in four dimensions can be studied using a duality of large N gauge theories with string theory on a certain spacetime. Via this duality, one can argue that for every theta, there are infinitely many vacua that are stable in the large N limit. The true vacuum, found by minimizing the energy in this family, is a smooth function of theta except at theta equal to pi, where it jumps. This jump is associated with spontaneous breaking of CP symmetry. Domain walls separating adjacent vacua are described in terms of wrapped sixbranes. 
  We examine a possibility to introduce a non-trivial classical background metric into the 2-d Liouville gravity theory. The classical background appears as a part of the Weyl factor of the physical metric of 2-d surfaces with some conformal dimension. On the other hand, the rest part of the factor corresponds to the quantum fluctuating sector, having another conformal dimension such that these two conformal dimensions sum up to just (1,1). Consequently we conclude that, in the 2-d Liouville gravity, the target space dimensions D can be beyond 1. 
  It has been known that D=5 simple supergravity resembles D=11 supergravity in many respects. We present their further resemblances in (1) the duality groups upon dimensional reduction, and (2) the worldsheet structure of the solitonic string of the D=5 supergravity. We show that the D=3, G_{2(+2)}/SO(4) (bosonic) nonlinear sigma model is obtained by using Freudenthal's construction in parallel to the derivation of the D=3, E_{8(+8)}/SO(16) sigma model from D=11 supergravity. The zero modes of the string solution with unbroken (4,0) supersymmetry consist of three (non-chiral) scalars, four Majorana-Weyl spinors of the same chirality and one chiral scalar, which suggests a duality to a certain six-dimensional chiral string theory. The worldsheet gravitational anomaly indicates a quantum correction to the Bianchi identity for the dualized two-form gauge field in the bulk just like the M5-brane case. 
  In this Thesis we study quantum corrections to the classical dynamics for mean values in field theory. To that end we make use of the formalism of the closed time path effective action to get real and causal equations of motion.   We introduce a coarse grained effective action, which is useful in the study of phase transitions in field theory. We derive an exact renormalization group equation that describes how this action varies with the coarse graining scale. We develop different approximation methods to solve that equation, and we obtain non perturbative improvements to the effective potential for a self interacting scalar field theory. We also discuss the stochastic aspects contained in this action.   On the other hand, using the effective action, we find low energy and large distance quantum corrections for the gravitational potential, treating relativity as an effective low energy theory. We include the effect of scalar fields, fermions and gravitons. The inclusion of metric fluctuations causes Einstein semiclassical equations to depend on the gauge fixing parameters, and they are therefore non physical. We solve this problem identifying as a physical observable the trayectory of a test particle. We explicitly show that the geodesic equation for such particle is independent of the arbitrary parameters of the gauge fixing. 
  We show that each rigid symmetry of a D-string action is contained in a family of infinitely many symmetries. In particular, kappa-invariant D-string actions have infinitely many supersymmetries. The result is not restricted to standard D-string actions, but holds for any two-dimensional action depending on an abelian world-sheet gauge field only via the field strength. It applies thus also to manifestly $SL(2,Z)$ covariant D-string actions. Furthermore, it extends analogously to $d$-dimensional actions with $(d-1)$-form gauge potentials, such as brane actions with dynamical tension. 
  Previous work involving Born-regulated gravity theories in two dimensions is extended to four dimensions. The action we consider has two dimensionful parameters. Black hole solutions are studied for typical values of these parameters. For masses above a critical value determined in terms of these parameters, the event horizon persists. For masses below this critical value, the event horizon disappears, leaving a ``bare mass'', though of course no singularity. 
  Our goal is to study the supermembrane on an $AdS_4 \times \cal M_7$ background, where $\cal M_7$ is a 7--dimensional Einstein manifold with $N$ Killing spinors. This is a direct way to derive the $Osp(N|4)$ singleton field theory with all the additional properties inherited from the geometry of the internal manifold. As a first example we consider the maximally supersymmetric $Osp(8|4)$ singleton corresponding to the choice $\cal M_7 = S^7$. We find the explicit form of the action of the membrane coupled to this background geometry and show its invariance under non--linearly realised superconformal transformations. To do this we introduce a supergroup generalisation of the solvable Lie algebra parametrisation of non--compact coset spaces. We also derive the action of quantum fluctuations around the classical configuration, showing that this is precisely the singleton action. We find that the singleton is simply realised as a free field theory living on flat Minkowski space. 
  Sigma models in which the integer coefficient of the Wess-Zumino term vanishes are easy to construct. This is the case if all flavor symmetries are vectorlike. We show that there is a subset of SU(N)XSU(N) vectorlike sigma models in which the Wess-Zumino term vanishes for reasons of symmetry as well. However, there is no chiral sigma model in which the Wess-Zumino term vanishes for reasons of symmetry. This can be understood in the sigma model basis as a consequence of an index theorem for the axialvector coupling matrix. We prove this index theorem directly from the SU(N)XSU(N) algebra. 
  In a recent paper DeWolfe et al. have shown how to use the self-intersection number of junctions to constrain the BPS spectrum of N=2, D=4 theories with ADE flavor symmetry arising on a single D3-brane probe in a 7-brane background. Motivated by the existence of more general N=2, D=4 theories arising on the worldvolume of multiple D3-brane probes we show how to compute the self-intersection number of junctions in the presence of 7-branes and multiple D3-branes. 
  We study $n$-point functions at finite temperature in the closed time path formalism. With the help of two basic column vectors and their dual partners we derive a compact decomposition of the time-ordered $n$-point functions with $2^n$ components in terms of $2^{n-1} -1$ independent retarded/advanced $n$-point functions. This representation greatly simplifies calculations in the real-time formalism. 
  The general lines of the derivation and the main properties of the master equations for the master amplitudes associated to a given Feynman graph are recalled. Some results for the 2-loop self-mass graph with 4 propagators are then presented. 
  We consider evidence for the existence of gauge configurations with fractional charge in pure N=1 supersymmetric Yang-Mills theory . We argue that these field configurations are singular and have to be treated as distributions. It is shown that the path integral representation of constant Green's functions can be reduced to a finite dimensional integral. The fractional configurations are essentially the zero size limit of the usual instantons and they have a reduced number of fermionic zero modes. At the end we comment on the status of the D-instanton/YM-instanton correspondence within AdS/CFT correspondence. 
  The leading radiative correction to the Casimir energy for two parallel penetrable mirrors is calculated within QED perturbation theory. It is found to be of the order $\alpha$ like the known radiative correction for ideally reflecting mirrors from which it differs only by a monotonic, powerlike function of the frequency at which the mirrors become transparent. This shows that the $O(\alpha^2)$ radiative correction calculated recently by Kong and Ravndal for ideally reflecting mirrors on the basis of effective field theory methods remains subleading even for the physical case of penetrable mirrors. 
  It is well-known that all 2d models of gravity---including theories with nonvanishing torsion and dilaton theories---can be solved exactly, if matter interactions are absent. An absolutely (in space and time) conserved quantity determines the global classification of all (classical) solutions. For the special case of spherically reduced Einstein gravity it coincides with the mass in the Schwarzschild solution. The corresponding Noether symmetry has been derived previously by P. Widerin and one of the authors (W.K.) for a specific 2d model with nonvanishing torsion. In the present paper this is generalized to all covariant 2d theories, including interactions with matter. The related Noether-like symmetry differs from the usual one. The parameters for the symmetry transformation of the geometric part and those of the matterfields are distinct. The total conservation law (a zero-form current) results from a two stage argument which also involves a consistency condition expressed by the conservation of a one-form matter ``current''. The black hole is treated as a special case. 
  Recently proposed 2D anomaly induced effective actions for the matter-gravity system are critically reviewed. Their failure to correctly reproduce Hawking's black hole radiation or the stability of Minkowski space-time led us to a modification of the relevant ``quantum'' matter stress energy tensor that allows physically meaningful results to be extracted. 
  We work out the relation between automorphic forms on SO(s+2,2) and gauge one-loop corrections of heterotic K3 x T^2 string compactifications for the cases s=0,1. We find that one-loop gauge corrections of any orbifold limit of K3 can always be expressed by their instanton numbers and generic automorphic forms.These functions classify also one-loop gauge thresholds of N=1 (0,2) heterotic compactifications based on toroidal orbifolds T^6/Z_N. We compare these results with the gauge couplings of M-theory compactified on S^1/Z_2 x T^6/Z_N using Witten's Calabi-Yau strong coupling expansion. 
  We summarize recent results connecting multiloop Feynman diagram calculations to different parts of mathematics, with special attention given to the Hopf algebra structure of renormalization. 
  A review is presented of some recent progress in spectral geometry on manifolds with boundary: local boundary-value problems where the boundary operator includes the effect of tangential derivatives; application of conformal variations and other functorial methods to the evaluation of heat-kernel coefficients; conditions for strong ellipticity of the boundary-value problem; fourth-order operators on manifolds with boundary; non-local boundary conditions in Euclidean quantum gravity. Many deep developments in physics and mathematics are therefore in sight. 
  Versions of M-theory are found in spacetime signatures (9,2) and (6,5), in addition to the usual M-theory in 10+1 dimensions, and these give rise to type IIA string theories in 10-dimensional spacetime signatures (10,0),(9,1),(8,2),(6,4) and (5,5), and to type IIB string theories in signatures (9,1),(7,3) and (5,5). The field theory limits are  10 and 11 dimensional supergravities in these signatures. These theories are all linked by duality transformations which can change the number of time dimensions as well as the number of space dimensions, so that each should be a different limit of the same underlying theory. 
  This work presents a first study of a radiative calculation for the gravitational axial anomaly in the massless Abelian Higgs model. The two loop contribution to the anomalous correlation function of one axial current and two energy-momentum tensors, <A_alpha(z) T_\mu\nu(y) T_\rho\sigma(x)>, is computed at an order that involves only internal matter fields. Conformal properties of massless field theories are used in order to perform the Feynman diagram calculations in the coordinate space representation. The two loop contribution is found not to vanish, due to the presence of two independent tensor structures in the anomalous correlator. 
  In this work we present a new hidden symmetry in gravity for the scale factor in the FRW model, for $k=0$. This exact symmetry vanishes the cosmological constant. We interpret this hidden symmetry as a dual symmetry in the sense that appears in the string theory. 
  The hyperplane and proper time formalisms are discussed mainly for the spin-half particles in the quantum case. A connection between these covariant Hamiltonian formalisms is established. It is showed that choosing the space-like hyperplanes instantaneously orthogonal to the direction of motion of the particle the proper time formalism is retrieved on the mass shell. As a consequence, the relation between the St\"uckelberg-Feynman picture and the standard canonical picture of quantum field theory is clarified. 
  A great effort has been devoted to formulate a classical relativistic theory of spin compatible with quantum relativistic wave equations. The main difficulty in order to connect classical and quantum theories rests in finding a parameter which plays the role of proper time at a purely quantum level. We present a partial review of several proposals of classical and quantum spin theories from the pioneer works of Thomas and Frenkel, revisited in the classical BMT work, to the semiclassical model of Barut and Zanghi [Phys. Rev. Lett. 52, 2009 (1984)]. We show that the last model can be obtained from a semiclassical limit of the Feynman proper time parametrization of the Dirac equation. At the quantum level we derive spin precession equations in the Heisenberg picture. Analogies and differences with respect to classical theories are discussed in detail. 
  The genesis of Feynman's original approach to QED is reviewed. The main ideas of his original presentation at the Pocono Conference are discussed and compared with the ones involved in his action-at-distance formulation of classical electrodynamics. The role of the de Sitter group in Feynman's visualization of space-time processes is pointed out. 
  A way of covariantizing duality symmetric actions is described. 
  Large-N field systems are considered from an unusual point of view. The Hamiltonian is presented in a third-quantized form analogously to the second-quantized formulation of the quantum theory of many particles. The semiclassical approximation is applied to the third-quantized Hamiltonian. The advantages of this approach in comparison with 1/N-expansion are discussed. 
  In the strong coupling limit type IIA superstring theory develops an eleventh dimension that is not apparent in perturbation theory. This suggests the existence of a consistent 11d quantum theory, called M theory, which is approximated by 11d supergravity at low energies. In this review we describe some of the evidence for this picture and some of its implications. 
  In the context of N=8 supergravity we construct the general form of BPS 0--branes that preserve either 1/2 or 1/4 of the original supersymmetry. We show how such solutions are related to suitable decompositions of the 70 dimensional solvable Lie algebra that describes the scalar field sector. We compare our new results to those obtained in a previous paper for the case of 1/8 supersymmetry preserving black holes. Each of the three cases is based on a different solvable Lie algebra decomposition and leads to a different structure of the scalar field evolution and of their fixed values at the horizon of the black hole. 
  We extend the correspondence between adS-supergravities and superconformal field theories on the adS boundary to a correspondence between gauged supergravities (typically with non-compact gauge groups) and quantum field theories on domain walls. 
  We consider Type IIB superstring theory with the addition of n 9-branes and n anti-9-branes (and no orientifolds). The result is a ten-dimensional chiral theory of open and closed oriented strings with gauge group U(n) \times U(n). There is, however, a tachyonic instability which can be understood as the consequence of brane-antibrane annihilation. We therefore expect to recover the usual IIB theory as the tachyon rolls to infinity. 
  We study a BPS configuration in which four strings (of different type) meet at a point in $N = 2, D = 8$ supergravity, i.e., the low energy effective theory of $T^2$-compactified type II string theory. We demonstrate that the charge conservation of the four strings implies the vanishing of the net force (due to the tensions of various strings) at the junction and vice versa, using the tension formula for $SL(3, Z)$ strings obtained recently by the present authors. We then show that a general 4-string junction preserves 1/8 of the spacetime supersymmetries. Using 4-string junctions as building blocks, we construct a string network which also preserves 1/8 of the spacetime supersymmetries. 
  We consider a model which in a certain limit reduces to the large N ${\cal N}=1$ supersymmetric SU(N) gauge theory without matter. The gaugino condensate in this model is controlled by the dynamics of an additional singlet superfield. Using this model we explicitly construct BPS domain walls arising due to the chiral symmetry breaking. In particular, in the large N limit we obtain the exact shapes of the domain walls corresponding to solitons, and also of the domain walls interpreted as D-branes on which the SQCD string can end, whose existence was previously argued by Witten in the context of the large N SQCD. We also discuss various points which appear to support the consistency of the D-brane interpretation for these domain walls within the SQCD string context. 
  A new cosmological model leads to testable predictions that are different from those of both standard cosmology and models with a cosmological constant. The prediction that q_0=0 is the same as in other ``coasting universe'' models, but arises without the need for any exotic form of matter or other ad hoc assumptions. 
  The $sl(2)$ minimal theories are labelled by a Lie algebra pair $(A,G)$ where $G$ is of $A$-$D$-$E$ type. For these theories on a cylinder we conjecture a complete set of conformal boundary conditions labelled by the nodes of the tensor product graph $A\otimes G$. The cylinder partition functions are given by fusion rules arising from the graph fusion algebra of $A\otimes G$. We further conjecture that, for each conformal boundary condition, an integrable boundary condition exists as a solution of the boundary Yang-Baxter equation for the associated lattice model. The theory is illustrated using the $(A_4,D_4)$ or 3-state Potts model. 
  We study the simplest geometrical particle model associated with null paths in four-dimensional Minkowski space-time. The action is given by the pseudo-arclength of the particle worldline. We show that the reduced classical phase space of this system coincides with that of a massive spinning particle of spin $s=\alpha^2/M$, where $M$ is the particle mass, and $\alpha$ is the coupling constant in front of the action. Consistency of the associated quantum theory requires the spin $s$ to be an integer or half integer number, thus implying a quantization condition on the physical mass $M$ of the particle. Then, standard quantization techniques show that the corresponding Hilbert spaces are solution spaces of the standard relativistic massive wave equations. Therefore this geometrical particle model provides us with an unified description of Dirac fermions ($s=1/2$) and massive higher spin fields. 
  In this note we calculate the fusion coefficients for minimal series representations of the N=2 superconformal algebra by using a modified Verlinde's formula, and obtain associative and commutative fusion algebras with non-negative integral fusion coefficients at each level. Some references are added. 
  We discuss differential-- versus integral--equation based methods describing out--of thermal equilibrium systems and emphasize the importance of a well defined reduction to statistical observables. Applying the projection operator approach, we investigate on the time evolution of expectation values of linear and quadratic polynomials in position and momentum for a statistical anharmonic oscillator with quartic potential. Based on the exact integro-differential equations of motion, we study the first and naive second order approximation which breaks down at secular time-scales. A method is proposed to improve the expansion by a non--perturbative resummation of all quadratic operator correlators consistent with energy conservation for all times. Motion cannot be described by an effective Hamiltonian local in time reflecting non-unitarity of the dissipative entropy generating evolution. We numerically integrate the consistently improved equations of motion for large times. We relate entropy to the uncertainty product, both being expressible in terms of the observables under consideration. 
  The $O(\beta^2)$ quantum correction to the classical reflection factor is calculated for one of the integrable boundary conditions of $a_2^{(1)}$ affine Toda field theory. This is found to agree with the conjectured exact reflection factor of the quantum theory. We consider the existence of other exact reflection factors consistent with our perturbative answer and examine the question of how duality transformations might relate theories with different boundary conditions. 
  In this work we propose an action which unifies self-dual gravity and self-dual Yang-Mills in the context of the Macdowell-Mansouri formalism. We claim that such an action may be used to find the S-dual action for both self-dual gravity and self-dual Yang-Mills. 
  The wave function of the universe is usually taken to be a functional of the three-metric on a spacelike section, Sigma, which is measured. It is sometimes better, however, to work in the conjugate representation, where the wave function depends on a quantity related to the second fundamental form of Sigma. This makes it possible to ensure that Sigma is part of a Lorentzian universe by requiring that the argument of the wave function be purely imaginary. We demonstrate the advantages of this formalism first in the well-known examples of the nucleation of a de Sitter or a Nariai universe. We then use it to calculate the pair creation rate for sub-maximal black holes in de Sitter space, which had been thought to vanish semi-classically. 
  This is the written version of my talk at SUSY '98. It presents a geometric characterisation of the allowed near-horizon geometries of supersymmetric branes. We focus primarily on the M2-brane, but results for other branes (e.g., the D3-brane) are also presented. Some new examples are discussed. 
  We consider some long multiplets describing bulk massive excitations of M-theory two-branes and IIB string three-branes which correspond to ``non chiral'' primary operators of the boundary OSp(8/4) and SU(2,2/4) superconformal field theories. Examples of such multiplets are the ``radial'' modes on the branes, including up to spin 4 excitations, which may be then considered as prototypes of states which are not described by the K-K spectrum of the corresponding supergravity theories on AdS_4 x S_7 and AdS_5 x S_5 respectively. 
  We recall the special features of quantum dynamics on a light-front (in an infinite momentum frame) in string and field theory. The reason this approach is more effective for string than for fields is stressed: the light-front dynamics for string is that of a true Newtonian many particle system, since a string bit has a fixed Newtonian mass. In contrast, each particle of a field theory has a variable Newtonian mass P^+, so the Newtonian analogy actually requires an infinite number of species of elementary Newtonian particles. This complication substantially weakens the value of the Newtonian analogy in applying light-front dynamics to nonperturbative problems. Motivated by the fact that conventional field theories can be obtained as infinite tension limits of string theories, we propose a way to recast field theory as a standard Newtonian system. We devise and analyze some simple quantum mechanical systems that display the essence of the proposal, and we discuss prospects for applying these ideas to large N_c QCD. 
  We show how Schwinger's proper time method can be used to evaluate directly the determinant of first order operators associated with fermionic theories. Several examples are worked out in detail. 
  In this paper we develop a general method for constructing 3-point functions in conformal field theory with affine Lie group symmetry, continuing our recent work on 2-point functions. The results are provided in terms of triangular coordinates used in a wave function description of vectors in highest weight modules. In this framework, complicated couplings translate into ordinary products of certain elementary polynomials. The discussions pertain to all simple Lie groups and arbitrary integrable representation. An interesting by-product is a general procedure for computing tensor product coefficients, essentially by counting integer solutions to certain inequalities. As an illustration of the construction, we consider in great detail the three cases SL(3), SL(4) and SO(5). 
  The space of solutions to the Hitchin equations on the dual torus with punctures determines the Higgs branch of certain impurity theories. An alternative description of this Higgs branch is provided, in terms of the proper deformation of Hitchin system with deformation parameter given by a $B$-field. For the dual torus minus the singular points we construct explicit solutions to the B-deformed Hitchin equations by reducing them to principal chiral model equations and then using deformation quantization methods. 
  We analyse the perturbative series expansion of the vacuum expectation value of a Wilson loop in Chern-Simons gauge theory in the temporal gauge. From the analysis emerges the notion of the kernel of a Vassiliev invariant. The kernel of a Vassiliev invariant of order n is not a knot invariant, since it depends on the regular knot projection chosen, but it differs from a Vassiliev invariant by terms that vanish on knots with n singular crossings. We conjecture that Vassiliev invariants can be reconstructed from their kernels. We present the general form of the kernel of a Vassiliev invariant and we describe the reconstruction of the full primitive Vassiliev invariants at orders two, three and four. At orders two and three we recover known combinatorial expressions for these invariants. At order four we present new combinatorial expressions for the two primitive Vassiliev invariants present at this order. 
  We show that the D-instanton in the $AdS_5 \times S^5$ background is a wormhole connecting the background $AdS_5 \times S^5$ to the flat space $\bf R^{10}$ located at the position of the D-instanton. By a $SL(2,{\bf R})$ rotation of type IIB theory, we can make the {\it global} geometry flat in string frame. We also find that, due to the tight relation between the dilaton and the axion, there is no $SL(2,{\bf R})$ element that takes strong string coupling to weak one without making the axion ill defined. We also discuss the case of $AdS_3$ as well as the instanton gases. A subtlety on the D-instanton at the boundary or at the horizon is discussed. 
  This paper is an extended version of hep-th/9802134. Dual QCD Lagrangian is derived by making use of the generalized coordinate gauge, where 1-form (vector potential) is expressed as an integral of the 2-form (field strength) along an (arbitrary) contour. As another application a simple proof of the nonabelian Stokes theorem is given. 
  Negative dimensional integration method (NDIM) seems to be a very promising technique for evaluating massless and/or massive Feynman diagrams. It is unique in the sense that the method gives solutions in different regions of external momenta simultaneously. Moreover, it is a technique whereby the difficulties associated with performing parametric integrals in the standard approach are transferred to a simpler solving of a system of linear algebraic equations, thanks to the polynomial character of the relevant integrands. We employ this method to evaluate a scalar integral for a massless two-loop three-point vertex with all the external legs off-shell, and consider several special cases for it, yielding results, even for distinct simpler diagrams. We also consider the possibility of NDIM in non-covariant gauges such as the light-cone gauge and do some illustrative calculations, showing that for one-degree violation of covariance (i.e., one external, gauge-breaking, light-like vector $n_{\mu}$) the ensuing results are concordant with the ones obtained via either the usual dimensional regularization technique, or the use of principal value prescription for the gauge dependent pole, while for two-degree violation of covariance --- i.e., two external, light-like vectors $n_{\mu}$, the gauge-breaking one, and (its dual) $n^{\ast}_{\mu}$ --- the ensuing results are concordant with the ones obtained via causal constraints or the use of the so-called generalized Mandelstam-Leibbrandt prescription. 
  The local symmetry transformations of the quantum effective action for general gauge theory are found. Additional symmetries arise under consideration of background gauges. Together with "trivial" gauge transformations, vanishing on mass shell, they can be used for construction simple gauge generators. For example, for the Yang-Mills theory the classically invariant effective action is obtained, reproducing DeWitt's result. For rank one theories a natural generalization is proposed. 
  A non-linear integral equation (NLIE) governing the finite size effects of excited states of even topological charge in the sine-Gordon (sG) / massive Thirring (mTh) field theory, deducible from a light-cone lattice formulation of the model, has been known for some time. In this letter we conjecture an extension of this NLIE to states with odd topological charge, thus completing the spectrum of the theory. The scaling functions obtained as solutions to our conjectured NLIE are compared successfully with Truncated Conformal Space data and the construction is shown to be compatible with all other facts known about the local Hilbert spaces of sG and mTh models. With the present results we have achieved a full control over the finite size behaviour of energy levels of sG/mTh theory. 
  We discuss the effect of relevant boundary terms on the nature of branes. This is done for toroidal and orbifold compactifications of the bosonic string. Using the relative minimalization of the boundary entropy as a guiding principle, we uncover the more stable boundary conditions at different regions of moduli space. In some cases, Neumann boundary conditions dominate for small radii while Dirichlet boundary conditions dominate for large radii. The c=1 and c=2 moduli spaces are studied in some detail. The antisymmetric background field B is found to have a more limited role in the case of Dirichlet boundary conditions. This is due to some topological considerations. The results are subjected to T-duality tests and the special role of the points in moduli space fixed under T-duality is explained from least-action considerations. 
  We compute the Casimir energy between an unusual pair of parallel plates at finite temperature, namely, a perfectely conducting plate ($\epsilon\to\infty$) and an infinitely permeable one ($\mu\to\infty$) by applying the generalized zeta function method. We also compute the Casimir pressure and discuss the high and the low temperature limits. 
  For slowly varying fields the Yang-Mills Schroedinger functional can be expanded in terms of local functionals. We show how analyticity in a complex scale parameter enables the Schroedinger functional for arbitrarily varying fields to be reconstructed from this expansion. We also construct the form of the Schroedinger equation that determines the coefficients. Solving this in powers of the coupling reproduces the results of the `standard' perturbative solution of the functional Schroedinger equation which we also describe. In particular the usual result for the beta-function is obtained illustrating how analyticity enables the effects of rapidly varying fields to be computed from the behaviour of slowly varying ones. 
  In light of the AdS/CFT correspondence, it is natural to try to define a conformal field theory in a large N, strong coupling limit via a supergravity compactification on the product of an Einstein manifold and anti-de Sitter space. We consider the five-dimensional manifolds T^{pq} which are coset spaces (SU(2) x SU(2))/U(1). The central charge and a part of the chiral spectrum are calculated, respectively, from the volume of T^{pq} and the spectrum of the scalar laplacian. Of the manifolds considered, only T^{11} admits any supersymmetry: it is this manifold which characterizes the supergravity solution corresponding to a large number of D3-branes at a conifold singularity, discussed recently in hep-th/9807080. Through a field theory analysis of anomalous three point functions we are able to reproduce the central charge predicted for the T^{11} theory by supergravity: it is 27/32 of the central charge of the N=2 Z_2 orbifold theory from which it descends via an RG flow. 
  A new synthesis of the principles of relativity and quantum mechanics is developed by replacing the Poincar\'e group for the de Sitter one. The new relativistic quantum mechanics is an indefinite mass theory which is reduced to the standard theory on the mass shell. The charge conjugation acquires a geometrical meaning and the Stueckelberg interpretation for antiparticles naturally arises in the formalism. So the idea of the Dirac sea in the second quantized formalism proves to be superfluous. The off-shell theory is free from ultraviolet divergences, which only appear in the process of mass shell reduction. 
  Supersymmetric domain walls are reviewed. The main emphasis is made on saturated walls in strongly coupled gauge theories. The central charge of the N=1 superalgebra appears in supersymmetric gluodynamics as a quantum anomaly. I also consider some rather unusual general features of supersymmetric domain walls. 
  Recently Faddeev and Niemi proposed a low energy effective action for pure SU(2) Yang Mills theory in 4 dimension to describe its long distance physics. The effective action is O(3) $\sigma$ model with a mass parameter, a dimensionless coupling constant e and a topological term. In this work, choosing a new set of variables, we relate this $\sigma$ model to a U(1) gauge theory with electric and magnetic charges of charge e and $4 \pi e^{-1}$ respectively. In the new formulation the connection of the mass parameter with the monopole condensate is discussed. This theory after lattice regularisation is the compact U(1) gauge theory coupled to electric charges. 
  We have proposed the entropy formula of the black hole which is constructed by the intersecting D1-brane and D5-brane with no momentum, whose compactification radii are constrained by the surface gravities in ten-dimensions. We interpret the entropy of the black hole as the statistical entropy of the effective string living on the D5-brane. We further study the behavior of the absorption cross-section of the black hole using our entropy formula. 
  We study static spherically symmetric gravitating dyon solutions and dyonic black holes in Einstein-Yang-Mills-Higgs theory. The gravitating dyon solutions share many features with the gravitating monopole solutions. In particular, gravitating dyon solutions and dyonic black holes exist up to a maximal coupling constant, and beside the fundamental dyon solutions there are excited dyon solutions. 
  Five dimensional field theories with exceptional gauge groups are engineered from degenerations of Calabi-Yau threefolds. The structure of the Coulomb branch is analyzed in terms of relative K\"ahler cones. For low number of flavors, the geometric construction leads to new five dimensional fixed points. 
  The thesis begins with an introduction to M-theory (at a graduate student's level), starting from perturbative string theory and proceeding to dualities, D-branes and finally Matrix theory. The following chapter treats, in a self-contained way, of general classical p-brane solutions. Black and extremal branes are reviewed, along with their semi-classical thermodynamics. We then focus on intersecting extremal branes, the intersection rules being derived both with and without the explicit use of supersymmetry. The last three chapters comprise more advanced aspects of brane physics, such as the dynamics of open branes, the little theories on the world-volume of branes and how the four dimensional Schwarzschild black hole can be mapped to an extremal configuration of branes, thus allowing for a statistical interpretation of its entropy. The original results were already reported in hep-th/9701042, hep-th/9704190, hep-th/9710027 and hep-th/9801053. 
  By studying zero modes of the Dirac equation on the lattice, we explicitly construct the Nahm transform of some topologically non-trivial gauge field configurations. 
  We consider type IIB configurations carrying both NS-NS and R-R electric and magnetic 3-form charges, and whose near horizon geometry contains AdS_3 x S^3. Noting that S^3 is a U(1) bundle over CP^1 \sim S^2, we construct the dual type IIA configurations by a Hopf T-duality along the U(1) fibre. In the case where there are only R-R charges, the S^3 is untwisted to S^2 x S^1 (in analogy with a previous treatment of AdS_5 x S^5.) However, in the case where there are only NS-NS charges, the S^3 becomes the cyclic lens space S^3/Z_p with its round metric (and is hence invariant when p=1), where p is the magnetic NS-NS charge. In the generic case with NS-NS and R-R charges, the S^3 not only becomes S^3/Z_p but is also squashed, with a squashing parameter that is related to the values of the charges. Similar results apply if we regard AdS_3 as a bundle over AdS_2 and T-dualise along the fibre. We show that Hopf T-dualities relate different black holes, and that they preserve the entropy. The AdS_3 x S^3 solutions arise as the near-horizon limits of dyonic strings. We construct an O(2,2;Z) multiplet of such dyonic strings, where O(2,2;Z) is a subgroup of the O(5,5) or O(5,21) six-dimensional duality groups, which captures the essence of the NS-NS/R-R and electric/magnetic dualities. 
  We discuss non-relativistic scattering by a Newtonian potential. We show that the gray-body factors associated with scattering by a black hole exhibit the same functional dependence as scattering amplitudes in the Newtonian limit, which should be the weak-field limit of any quantum theory of gravity. This behavior arises independently of the presence of supersymmetry. The connection to two-dimensional conformal field theory is also discussed. 
  Recent hints from observations of distant supernovae of a positive cosmological constant with magnitude comparable to the average density of matter seem to point in the direction of a two fluid model for spacetime; where the "normal" component consists of ordinary matter, while the "superfluid" component is a zero entropy condensate. Such a two fluid model for spacetime provides an immediate and simple explanation for why information seems to be lost when objects fall into a classical black hole. 
  The theory of a complex scalar interacting with a pure Chern-Simons gauge field is quantized canonically. Dynamical and nondynamical variables are separated in a gauge-independent way. In the physical subspace of the full Hilbert space, this theory reduces to a pure scalar theory with nonlocal interaction. Several scattering processes are studied and the cross sections are calculated. 
  The theory of a spinor field interacting with a pure Chern-Simons gauge field in 2+1 dimensions is quantized. Dynamical and nondynamical variables are separated in a gauge-independent way. After the nondynamical variables are dropped, this theory reduces to a pure spinor field theory with nonlocal interaction. Several two-body scattering processes are studied and the cross sections are obtained in explicitly Lorentz invariant forms. 
  In previous work, the conformal-gauge two-dimensional quantum gravity in the BRS formalism has been solved completely in terms of Wightman functions. In the present paper, this result is extended to the closed and open bosonic strings of finite length; the open-string case is nothing but the Kato-Ogawa string theory. The field-equation anomaly found previously, which means a slight violation of a field equation at the level of Wightman functions, remains existent also in the finite-string cases. By using this fact, a BRS charge nilpotent even for $D\not=26$ is explicitly constructed in the framework of the Kato-Ogawa string theory. The FP-ghost vacuum structure of the Kato-Ogawa theory is made more transparent; the appearance of half-integral ghost numbers and the artificial introduction of indefinite metric are avoided. 
  We study supersymmetry breaking due to the presence of branes on anti-de Sitter space and obtain conditions for brane orientations not to break too many supersymmetries. Using the conditions, we construct a brane configuration corresponding to a baryon in large N gauge theory, and it is shown that the baryon is a marginal bound state of quarks as is expected from supersymmetry. 
  The second class constraints algebra of the abelian Chern-Simons theory is rigorously studied in terms of the Hamiltonian embedding in order to obtain the first class constraint system. The symplectic structure of fields due to the second class constraints disappears in the resulting system. Then we obtain a new type of Chern-Simons action which has an infinite set of the irreducible first class constraints and exhibits new extended local gauge symmetries implemented by these first class constraints. 
  We propose a way of protecting a Dirac fermion interacting with a scalar field from acquiring a mass from the vacuum. It is obtained through an implementation of translational symmetry when the theory is formulated with a momentum cutoff, which forbids the usual Yukawa term. We consider that this mechanism can help to understand the smallness of neutrino masses without a tuning of the Yukawa coupling. The prohibition of the Yukawa term for the neutrino forbids at the same time a gauge coupling between the right-handed electron and neutrino. We prove that this mechanism can be implemented on the lattice. 
  The complete result for the effective potential for two graviton exchange at two loops in M(atrix) theory can be expressed in terms of a generalized hypergeometric function. 
  In view of two-dimensional topological gravity coupled to matter, we study the Seiberg-Witten theory for the low-energy behavior of N=2 supersymmetric Yang-Mills theory with ADE gauge groups. We construct a new solution of the Picard-Fuchs equations obeyed by the Seiberg-Witten periods. Our solution is expressed as the linear sum over the infinite set of one-point functions of gravitational descendants in $d<1$ topological strings. It turns out that our solution provides the power series expansion around the origin of the quantum moduli space of the Coulomb branch. For SU(N) gauge group we show how the Seiberg-Witten periods are reconstructed from the present solution. 
  An ambiguity in the computation of the one-loop effective action for fields living on a cone is illustrated. It is shown that the ambiguity arises due to the non-commutativity of the regularization of ultraviolet and (conical) boundary divergencies. 
  A D-5-brane bound state with a self-dual field strength on a 4-torus is considered. In a particular case this model reproduces the D5-D1 brane bound state usually used in the string theory description of 5-dimensional black holes. In the limit where the brane dynamics decouples from the bulk the Higgs and Coulomb branches of the theory on the brane decouple. Contrasting with the usual instanton moduli space approximation to the problem the Higgs branch describes fundamental excitations of the gauge field on the brane. Upon reduction to 2-dimensions it is associated with the so-called instanton strings. Using the Born-Infeld action for the D-5-brane we determine the coupling of these strings to a minimally coupled scalar in the black hole background. The supergravity calculation of the cross section is found to agree with the D-brane absorption probability rate calculation. We consider the near horizon geometry of our black hole and elaborate on the corresponding duality with the Higgs branch of the gauge theory in the large N limit. A heuristic argument for the scaling of the effective string tension is given. 
  Superconformal symmetry in six-dimensions is analyzed in terms of coordinate transformations on superspace. A superconformal Killing equation is derived and its solutions are identified in terms of supertranslations, dilations, Lorentz transformations, R-symmetry transformations and special superconformal transformations. The full superconformal symmetry, which is shown to form the group OSp(2,6|N), is possible only if the supersymmetry algebra has N spinorial generators of the same chirality, corresponding to (N,0) supersymmetry. The R-symmetry group is then Sp(N) and the corresponding superspace is R^{6|8N}. We define superinversion as a map to the associated superspace of opposite chirality. General formulae for two-point and three-point correlation functions of quasi-primary superfields are exhibited. The superconformal group in six-dimensions is reduced to a corresponding extended superconformal group in four-dimensions. Superconformally covariant differential operators are also discussed. 
  BPS black hole solutions of U(1) gauged five-dimensional supergravity are obtained by solving the Killing spinor equations. These extremal static black holes live in an asymptotic AdS_5 space time. Unlike black holes in asymptotic flat space time none of them possess a regular horizon. We also calculate the influence, of a particular class of these solutions, on the Wilson loops calculation. 
  Recently a strong-weak coupling duality in non-abelian non-supersymmetric theories in four dimensions has been found. An analogous procedure is reviewed, which allows to find the `dual action' to the gauge theory of dynamical gravity constructed by the MacDowell-Mansouri model plus the superposition of a $\Theta$ term. 
  We restrict affine Toda field theory to the half-line by imposing certain boundary conditions at $x=0$. The resulting theory possesses the same spectrum of solitons and breathers as affine Toda theory on the whole line. The classical solutions describing the reflection of these particles off the boundary are obtained from those on the whole line by a kind of method of mirror images. Depending on the boundary condition chosen, the mirror must be placed either at, in front, or behind the boundary. We observe that incoming solitons are converted into outgoing antisolitons during reflection. Neumann boundary conditions allow additional solutions which are interpreted as boundary excitations (boundary breathers). For $a_n^{(1)}$ and $c_n^{(1)}$ Toda theories, on which we concentrate mostly, the boundary conditions which we study are among the integrable boundary conditions classified by Corrigan et.al. As applications of our work we study the vacuum solutions of real coupling Toda theory on the half-line and we perform semiclassical calculations which support recent conjectures for the $a_2^{(1)}$ soliton reflection matrices by Gandenberger. 
  We examine how the consequences which follow from a recent model, both in cosmology and at the elementary particle level have since been observationally and experimentally confirmed. Some of the considerations of the model are also justified from alternative viewpoints. It is also shown how the standard Big Bang and quark models can be recovered from the above theory. 
  We propose an algorithm, based on Algebraic Renormalization, that allows the restoration of Slavnov-Taylor invariance at every order of perturbation expansion for an anomaly-free BRS invariant gauge theory. The counterterms are explicitly constructed in terms of a set of one-particle-irreducible Feynman amplitudes evaluated at zero momentum (and derivatives of them). The approach is here discussed in the case of the abelian Higgs-Kibble model, where the zero momentum limit can be safely performed. The normalization conditions are imposed by means of the Slavnov-Taylor invariants and are chosen in order to simplify the calculation of the counterterms. In particular within this model all counterterms involving BRS external sources (anti-fields) can be put to zero with the exception of the fermion sector. 
  The transition amplitude is obtained for a free massive particle of arbitrary spin by calculating the path integral in the index-spinor formulation within the BFV-BRST approach. None renormalizations of the path integral measure were applied. The calculation has given the Weinberg propagator written in the index-free form with the use of index spinor. The choice of boundary conditions on the index spinor determines holomorphic or antiholomorphic representation for the canonical description of particle/antiparticle spin. 
  The meander problem is a combinatorial problem which provides a toy model of the compact folding of polymer chains. In this paper we study various questions relating to the enumeration of meander diagrams, using diagrammatical methods. By studying the problem of folding tree graphs, we derive a lower bound on the exponential behaviour of the number of connected meander diagrams. A different diagrammatical method, based on a non-commutative algebra, provides an approximate calculation of the behaviour of the generating functions for both meander and semi-meander diagrams. 
  We construct exactly solvable models for four particles moving on a real line or on a circle with translation invariant two- and four-particle interactions. 
  Superstring theory, and a recent extension called M theory, are leading candidates for a quantum theory that unifies gravity with the other forces. As such, they are certainly not ordinary quantum field theories. However, recent duality conjectures suggest that a more complete definition of these theories can be provided by the large N limits of suitably chosen U(N) gauge theories associated to the asymptotic boundary of spacetime. 
  The AdS/CFT correspondence provides valuable constraints on the possible exact form of various physical quantities in the ${\cal N}=4$ super Yang-Mills theory in the large N limit. We examine the free energy as the expansions in a small as well as in a large 't Hooft parameter $\lambda$. We argue that it is impossible to smoothly extrapolate from the weak coupling regime to the strong coupling regime, thus there must exist a large N phase transition in $\lambda$ at a finite temperature. We also argue that there is no world-sheet instanton in the background of the Euclidean anti-de Sitter black hole. 
  We discuss Bogomol'nyi equations for general gauge theories (depending on the two Maxwell invariants $F^{\mu \nu} F_{\mu \nu}$ and $\tilde F^{\mu \nu} F_{\mu \nu}$) coupled to Higgs scalars. By analysing their supersymmetric extension, we explicitly show why the resulting BPS structure is insensitive to the particular form of the gauge Lagrangian: Maxwell, Born-Infeld or more complicated non-polynomial Lagrangians all satisfy the same Bogomol'nyi equations and bounds which are dictated by the underlying supersymmetry algebra. 
  After a brief review of matrix theory compactification leading to noncommutative supersymmetric Yang-Mills gauge theory, we present solutions for the fundamental and adjoint sections on a two-dimensional twisted quantum torus in two different gauges. We also give explicit transformations connecting different representations which have appeared in the literature. Finally we discuss the more mathematical concept of Morita equivalence of C*-algebras as it applies to our specific case. 
  We construct supergravity theories in twelve and thirteen dimensions with the respective signatures (10,2) and (11,2) with some technical details. Starting with N=1 supergravity in 10+2 dimensions coupled to Green-Schwarz superstring, we give N=2 chiral supergravity in 10+2 dimensions with its couplings to super (2+2)-brane. We also build an N=1 supergravity in 11+2 dimensions, coupled to supermembrane. All of these formulations utilize scalar (super)fields intact under supersymmetry, replacing the null-vectors introduced in their original formulations. This method makes all the equations SO(10,2) or SO(11,2) Lorentz covariant, up to modified Lorentz generators. We inspect the internal consistency of these formulations, in particular with the usage of the modified Lorentz generators for the extra coordinates. 
  We discuss stability of non-supersymmetric compactification of M-theory and string theory of the form AdS * X, and their dual non-supersymmetric interacting conformal field theories. We argue some of the difficulties in controlling 1/N-corrections disappear in the cases that the large-N dual conformal field theory has no invariant marginal operators (and in some cases with no exactly marginal operators only). We provide several examples of such compactifications of M-theory down to AdS4. 
  The finite XXZ spin chain with boundaries is studied. We derive the transfer matrix from the q-difference equation discovered by Cherednik and construct its eigenstates by the vertex operator approach. We point out that the eigenstates with no magnetic fields have a symmetry called the turning symmetry. Making use of this symmetry we calculate the spontaneous magnetization in the thermodynamic limit, which is roughly twice as large as that in the half-infinite XXZ spin chain. 
  A brief introduction to superembedding approach (SEA) in its variant based on the generalized action principle (GAP) for super-p-branes is given. A role of harmonic variables for Lorentz group is stressed. A relation of the GAP with complete superfield actions is noted. Recent applications in studying of Dirichlet branes (super-Dp-branes) and M-branes are discussed. 
  We study the spectrum of the domain walls interpolating between different chirally asymmetric vacua in supersymmetric QCD with the SU(N) gauge group and including N-1 pairs of chiral matter multiplets in fundamental and anti-fundamental representations. There are always "real walls" interpolating between the chirally symmetric and a chirally asymmetric vacua which are BPS saturated. For small enough masses, there are two different "complex" BPS wall solutions interpolating between different chirally asymmetric vacua and two types of "wallsome sphalerons". At some m = m_*, two BPS branches join together and, in some interval m_* < m < m_{**}, BPS equations have no solutions, but there are solutions to the equations of motion describing a non--BPS domain wall and a sphaleron. For m > m_{**}, there are no complex wall solutions whatsoever. 
  The gauge symmetry inherent in the concept of manifold has been discussed. Within the scope of this symmetry the linear connection or displacement field can be considered as a natural gauge field on the manifold. The gauge invariant equations for the displacement field have been derived. It has been shown that the energy-momentum tensor of this field conserves and hence the displacement field can be treated as one that transports energy and gravitates. To show the existence of the solutions of the field equations we have derived the general form of the displacement field in Minkowski space-time which is invariant under rotation and space and time inversion. With this anzats we found spherically-symmetric solutions of the equations in question. 
  Upon compactification on a circle, SU(N) gauge theory with all fields in the adjoint representation acquires a $Z_N$ global symmetry because the center of the gauge group is $Z_N$. For N=4 super Yang-Mills theory, we show how this $Z_N$ "topological symmetry" arises in the context of the AdS/CFT correspondence, and why the symmetry group is $Z_N$ rather than U(1). This provides a test of the AdS/CFT correspondence for finite N. If the theory is formulated on $R^3 \times S^1$ with anti-periodic boundary conditions for fermions around the $S^1$, the topological symmetry is spontaneously broken; we show that the domain walls are D-strings, and hence that flux tubes associated with magnetic confinement can end on the domain walls associated with the topological symmetry. For the (0,2) $A_{N-1}$ superconformal field theory in six dimensions, we demonstrate an analogous phenomenon: a $Z_N$ global symmetry group arises if this theory is compactified on a Riemann surface. In this case, the domain walls are M-theory membranes. 
  The classical superconformal actions of branes in adS superspaces have a closed form depending on a matrix $M^2$ quadratic in fermions, as found in hep-th/9805217. One can gauge-fix the local $\kappa$-symmetry using the Killing spinors of the brane in the bulk. We show that in such gauges the superconformal actions are simplified dramatically since $M^2=0$ in all cases. The relation between classical and gauge-fixed actions for these theories reflects the relation between the full superconformal algebra and its supersolvable subalgebra. 
  Complete wetting is a universal phenomenon associated with interfaces separating coexisting phases. For example, in the pure gluon theory, at $T_c$ an interface separating two distinct high-temperature deconfined phases splits into two confined-deconfined interfaces with a complete wetting layer of confined phase between them. In supersymmetric Yang-Mills theory, distinct confined phases may coexist with a Coulomb phase at zero temperature. In that case, the Coulomb phase may completely wet a confined-confined interface. Finally, at the high-temperature phase transition of gluons and gluinos, confined-confined interfaces are completely wet by the deconfined phase, and similarly, deconfined-deconfined interfaces are completely wet by the confined phase. For these various cases, we determine the interface profiles and the corresponding complete wetting critical exponents. The exponents depend on the range of the interface interactions and agree with those of corresponding condensed matter systems. 
  We study a target space geometry of the form $AdS_3 \times {\bf R}$ for the $(2,1)$ heterotic string. This target space arises as the near horizon limit of a solitonic configuration in 2+2 dimensions. We investigate the null isometries of this space and discuss the reduction to 1+1 dimensions of the target space geometry arising from the consistent gauging of one of these isometries. 
  We explore the connection between D-branes and black holes in one particular case: a $D3$-brane compactified to four dimensions on $T^6/Z_3$. Using the $D$-brane boundary state description we show the equivalence with a double extremal N=2 black hole solution of four dimensional supergravity. 
  The most convenient tool to study the renormalization of a Lagrangian field theory invariant under non linear local or global symmetries is the proper solution to the master equation of the extended antifield formalism. It is shown that, from the knowledge of the BRST cohomology, it is possible to explicitly construct a further extension of the formalism containing all the observables of the theory and satisfying an extended master equation, with some of the features of the quantum Batalin-Vilkovisky master equation already present at the classical level. This solution has the remarkable property that all its infinitesimal deformations can be extended to complete deformations. The deformed solutions differs from the original one through the addition of terms related to coupling constant and anticanonical field-antifield redefinitions. As a consequence, all theories admitting an invariant regularization scheme are shown to be renormalizable while preserving the symmetries, in the sense that both the subtracted and the effective action satisfy the extended master equation, and this independently of power counting restrictions. The anomalous case is also studied and a suitable definition of the Batalin-Vilkovisky ``Delta'' operator in the context of dimensional renormalization is proposed. 
  This work is a comment on Ryder's derivation of the Dirac equation, with emphasis on the physical contents of this equation: the notion of particles and antiparticles according to the Stueckelberg-Feynman interpretation, the opposite intrinsic parity between particles and antiparticles, and the spin. 
  Several quantum proper time derivatives are obtained from the Beck one in the usual framework of relativistic quantum mechanics (spin 1/2 case). The ``scalar Hamiltonians'' of these derivatives should be thought of as the conjugate variables of the proper time. Then, the Hamiltonians would play the role of mass operators, suggesting the formulation of an adequate extended indefinite mass framework. We propose and briefly develop the framework corresponding to the Feynman parametrization of the Dirac equation. In such a case we derive the other parametrizations known in the literature, linking the extension of the different proposals of quantum proper time derivatives again. 
  In recent work Kachru, Kumar and Silverstein introduced a special class of non-supersymmetric type II string theories in which the cosmological constant vanishes at the first two orders of perturbation theory. Heuristic arguments suggest the cosmological constant may vanish in these theories to all orders in perturbation theory leading to a flat potential for the dilaton. A slight variant of their model can be described in terms of a dual heterotic theory. The dual theory has a non-zero cosmological constant which is non-perturbative in the coupling of the original type II theory. The dual theory also predicts a mismatch between Bose and Fermi degrees of freedom in the non-perturbative D-brane spectrum of the type II theory. 
  We formulate the theory of field interactions with higher order anisotropy. The concepts of higher order anisotropic space and locally anisotropic space (in brief, ha-space and la-space) are introduced as general ones for various types of higher order extensions of Lagrange and Finsler geometry and higher dimension (Kaluza-Klein type) spaces. The spinors on ha-spaces are defined in the framework of the geometry of Clifford bundles provided with compatible nonlinear and distinguished connections and metric structures (d-connection and d-metric). The spinor differential geometry of ha-spaces is constructed. There are discussed some related issues connected with the physical aspects of higher order anisotropic interactions for gravitational, gauge, spinor, Dirac spinor and Proca fields. Motion equations in higher order generalizations of Finsler spaces, of the mentioned type of fields, are defined by using bundles of linear and affine frames locally adapted to the nonlinear connection structure. 
  To avoid the problems which are connected with the long distance behavior of perturbative gauge theories we present a local construction of the observables which does not involve the adiabatic limit. First we construct the interacting fields as formal power series by means of causal perturbation theory. The observables are defined by BRST invariance where the BRST-transformation $\tilde s$ acts as a graded derivation on the algebra of interacting fields. Positivity, i.e. the existence of Hilbert space representations of the local algebras of observables is shown with the help of a local Kugo-Ojima operator $Q_{\rm int}$ which implements $\tilde s$ on a local algebra and differs from the corresponding operator $Q$ of the free theory. We prove that the Hilbert space structure present in the free case is stable under perturbations. All assumptions are shown to be satisfied in QED in a finite spatial volume with suitable boundary conditions. As a by-product we find that the BRST-quantization is not compatible with periodic boundary conditions for massless free gauge fields. 
  We compute the canonical partition function of 2+1 dimensional de Sitter space using the Euclidean $SU(2)\times SU(2)$ Chern-Simons formulation of 3d gravity with a positive cosmological constant. Firstly, we point out that one can work with a Chern-Simons theory with level $k=l/4G$, and its representations are therefore unitary for integer values of $k$. We then compute explicitly the partition function using the standard character formulae for SU(2) WZW theory and find agreement, in the large $k$ limit, with the semiclassical result. Finally, we note that the de Sitter entropy can also be obtained as the degeneracy of states of representations of a Virasoro algebra with $c=3l/2G$. 
  We investigate some properties of a system of Dirac fermions in 2+1 dimensions, with a space dependent mass having domain wall like defects.These defects are defined by the loci of the points where the mass changes sign. In general, they will be curves lying on the spatial plane. We show how to treat the dynamics of the fermions in such a way that the existence of localized fermionic zero modes on the defects is transparent. Moreover, effects due to the higher, non zero modes, can be quantitatively studied. We also consider the relevance of the profile of the mass near the region where it changes sign. Finally, we apply our general results to the calculation of the induced fermionic current, in the linear response approximation, in the presence of an external electric field and defects. 
  Electric self-dual vortices arising as BPS states in the strong coupling limit of N=2 supersymmetric Yang-Mills theory, softly broken to N=1, are reported. 
  The nineteen-vertex models of Zamolodchikov-Fateev, Izergin-Korepin and the supersymmetric osp(1|2) with periodic boundary conditions are studied. We find the spectrum of these quantum spin chains using the Coordinate Bethe Ansatz. The approche is a suitable parametrization of their wavefunctions. We also applied the Algebraic Bethe Ansatz in order to obtain the eigenvalues and eigenvectors of the corresponding transfer matrices. 
  Orthosympletic Hamiltonians derived from representations of the Temperley-Lieb algebra are presented and solved via the coordinate Bethe Ansatz. The spectra of these Hamiltonians are obtained using open and closed boundary conditions. 
  The temperature dependence of the anomalous sector of the effective action of fermions coupled to external gauge and pseudo-scalar fields is computed at leading order in an expansion in the number of Lorentz indices in two and four dimensions. The calculation preserves chiral symmetry and confirms that a temperature dependence is compatible with axial anomaly saturation. The result checks soft-pions theorems at zero temperature as well as recent results in the literature for the pionic decay amplitude into static photons in the chirally symmetric phase. The case of chiral fermions is also considered. 
  We give a classification and overview of the confining N=1 supersymmetric gauge theories. For simplicity we consider only theories based on simple gauge groups and no tree-level superpotential. Classification of these theories can be done according to whether or not there is a superpotential generated for the confined degrees of freedom. The theories with the superpotential include s-confining theories and also theories where the gauge fields participate in the confining spectrum, while theories with no superpotential include theories with a quantum deformed moduli space and theories with an affine moduli space. 
  We propose a world-sheet realization of the zigzag-invariant bosonic and fermionic strings as a perturbed Wess-Zumino-Novikov-Witten model at large negative level $k$ on a group manifold $G$ coupled to 2D gravity. In the large $k$ limit the zigzag symmetry can be obtained as a result of a self-consistent solution of the gravitationally dressed RG equation. The only solution found for simple group is $G=SL(2)$. More general target-space geometries can be obtained via tensoring of various cosets based on SL(2). In the supersymmetric case the zigzag symmetry fixes the maximal target-space dimension of the confining fermionic string to be seven. 
  The relation between the trace and R-current anomalies in 4D supersymmetric theories implies that the U(1)$_R$F$^2$, U(1)$_R$ and U(1)$^3_R$ anomalies which matched in studies of N=1 Seiberg duality satisfy positivity constraints. These constraints are tested in a large number of N=1 supersymmetric gauge theories in the non-Abelian Coulomb phase, and they are satisfied in all renormalizable models with unique anomaly-free R-current, including those with accidental symmetry. Most striking is the fact that the flow of the Euler anomaly coefficient, $a_{UV}-a_{IR}$, is always positive, as conjectured by Cardy. 
  We consider (3 + 1)-dimensional N=4 super Yang-Mills theory with a nonvanishing scalar Higgs vacuum expectation value, and compare this theory to AdS supergravity with branes in the bulk. We show that the one-loop effective potential for excitations of the Yang-Mills field agrees with the classical linearized potential for brane waves in the AdS picture in the limit of long wavelengths. This supports the idea that the AdS/CFT correspondance fits into string theory as expected from previous work. 
  In this paper the whole geometrical set-up giving a conformally invariant holographic projection of a diffeomorphism invariant bulk theory is clarified. By studying the renormalization group flow along null geodesic congruences a holographic version of Zamolodchikov's c-theorem is proven. 
  It is shown that all affine Toda theories admit (1,0) supersymmetric extensions. The construction is based on classical Lie algebras and supersymmetric massive sigma models. The supersymmetrized affine Toda theories have a unique, supersymmetric vacuum, their mass matrix is well defined and their energy functional is positive semi-definite. 
  In gauge theories with an extended Higgs sector the classical equations of motion can have solutions that describe stable, closed finite energy vortices. Such vortices separate two disjoint Higgs vacua, with one of the vacua embedded in the other in a manner that forms a topologically nontrivial knot. The knottedness stabilizes the vortex against shrinkage in 3+1 dimensional space-time. But in a world with extra large dimensions we expect the configuration to decay by unknotting. As an example we consider the semilocal $\theta_W \to \frac{\pi}{2}$ limit of the Weinberg-Salam model. We present numerical evidence for the existence of a stable closed vortex, twisted into a toroidal configuration around a circular Higgs vacuum at its core. 
  The supermembrane in light-cone gauge gives rise to a supersymmetric quantum mechanics system with SU(N) gauge symmetry when the group of area preserving diffeomorphisms is suitably regulated. de Wit, Marquard and Nicolai showed how eleven-dimensional Lorentz generators can be constructed from these degrees of freedom at the classical level. In this paper, these considerations are extended to the quantum level and it is shown the algebra closes to leading nontrivial order at large N. A proposal is made for extending these results to Matrix theory by realizing longitudinal boosts as large N renormalization group transformations. 
  We suggest a model of the large N limit ${\cal N}=4$ D=4 SU(N) SYM as a gas of 3-branes in a 10 dimensional space. Field theory analysis suggests that this 10 dimensional space does not carry the usual gravity dynamics but rather a contraction of it. Using a non-local transformation some aspects of the dynamics of this system are mapped to the dynamics of standard gravitons on $AdS_5\times S^5$. In particular some of the correspondence between operator in the CFT and states on $AdS$ is more transparent. 
  The quantum algebra of observables postulated in hep-th/9805057 is constructed up to degree five. All independent relations of degree four are given; they involve three as yet undetermined parameters. Definitions and symbols are used as introduced in the above-mentioned article. 
  The Matrix String Theory, i.e. the two dimensional U(N) SYM with N=(8,8) supersymmetry, has classical BPS solutions that interpolate between an initial and a final string configuration via a bordered Riemann surface. The Matrix String Theory amplitudes around such a classical BPS background, in the strong Yang--Mills coupling, are therefore candidates to be interpreted in a stringy way as the transition amplitude between given initial and final string configurations. In this paper we calculate these amplitudes and show that the leading contribution is proportional to the factor g_s^{-\chi}, where \chi is the Euler characteristic of the interpolating Riemann surface and g_s is the string coupling. This is the factor one expects from perturbative string interaction theory. 
  Within the Electric Schroedinger Representation of the Yang-Mills theory the Hamiltonian eigenstate and eigenvalue, as well as the Coulomb and confining potentials are presented for a special regularization-approximation scheme, which focuses on the ultra-local behavior of the propagator. 
  Verma modules of superconfomal algebras can have singular vector spaces with dimensions greater than 1. Following a method developed for the Virasoro algebra by Kent, we introduce the concept of adapted orderings on superconformal algebras. We prove several general results on the ordering kernels associated to the adapted orderings and show that the size of an ordering kernel implies an upper limit for the dimension of a singular vector space. We apply this method to the topological N=2 algebra and obtain the maximal dimensions of the singular vector spaces in the topological Verma modules: 0, 1, 2 or 3 depending on the type of Verma module and the type of singular vector. As a consequence we prove the conjecture of Gato-Rivera and Rosado on the possible existing types of topological singular vectors (4 in chiral Verma modules and 29 in complete Verma modules). Interestingly, we have found two-dimensional spaces of singular vectors at level 1. Finally, by using the topological twists and the spectral flows, we also obtain the maximal dimensions of the singular vector spaces for the Neveu-Schwarz N=2 algebra (0, 1 or 2) and for the Ramond N=2 algebra (0, 1, 2 or 3). 
  We show that discrete torsion is implemented in a D-brane world-volume theory by using a projective representation of the orbifold point group. We study the example of C^3/Z_2 x Z_2 and show that the resolution of singularities agrees with that proposed by Vafa and Witten. A new type of fractional brane appears. 
  We obtain exactly the vacuum expectation values $<(\partial\phi)^2 ({\bar\partial}\phi) e^{i\alpha\phi}>$ in the sine-Gordon model and  $<L_{-2}{\bar L}_{-2} \Phi_{l,k}>$ in $\Phi_{1,3}$ perturbed minimal CFT. We discuss applications of these results to short-distance expansions of two-point correlation functions. 
  Various string theory realizations of three-dimensional gauge theories relate them to gravitational instantons, Nahm equations and monopoles. We use this correspondence to model self-dual gravitational instantons of $D_k$-type as moduli spaces of singular monopoles, find their twistor spaces and metrics. 
  The charged anyon fluid in the presence of an externally applied constant and homogeneous magnetic field is investigated at temperatures larger than the energy gap ($T\gg \omega_{c}$). It is shown that the applied magnetic field inhomogeneously penetrates the sample with a spatial periodicity depending on a wavelength that decreases with temperature. The distribution of charges in the ($T\gg \omega_{c}$)-phase acquires a periodic spatial arrangement. 
  A path integral formalism is developed to study the interaction of an arbitrary curved Dirichlet (D-) string with elementary excitations of the fundumental (F-) string in bosonic string theory. Up to the next to leading order in the derivative expansion, we construct the properly renormalized vertex operator, which generalizes the one previously obtained for a D-particle moving along a curved trajectory. Using this vertex, an attempt is further made to quantize the D-string coordinates and to compute the quantum amplitude for scattering between elementary excitations of the D- and F-strings. By studying the dependence on the Liouville mode for the D-string, it is found that the vertex in our approximation consists of an infinite tower of local vertex operators which are conformally invariant on their respective mass-shell. This analysis indicates that, unlike the D-particle case, an off-shell extension of the interaction vertex would be necessary to compute the full amplitude and that the realization of symmetry can be quite non-trivial when the dual extended objects are simultaneously present. Possible future directions are suggested. 
  We briefly review the status of three-family grand unified string models. 
  D-instantons of Type IIB string theory are Ramond-Ramond counterpart of Giddings-Strominger wormholes connecting two asymptotic regions of spacetime. Such wormholes, according to Coleman, might lead to spacetime topology change, third-quantized baby universes and probabilistic determination of fundamental coupling parameters. Utilizing correspondence between AdS5 x M5 Type IIB supergravity and d=4 super Yang-Mills theory, we point out that topology change and sum over topologies not only take place in string theory but also are required for consistency with holography. Nevertheless, the effects of D-instanton wormholes remain completely deterministic, in sharp contrast to Coleman's scenario. 
  We have developed a variational perturbation theory based on the Liouville-Neumann equation, which enables one to systematically compute the perturbative correction terms to the variationally determined wave functions of the time-dependent systems. We then apply the method to the time-independent anharmonic oscillator, and show that the results agree with those of other variational perturbation theories. We also show that the system has an interesting algebraic structure at the first order correction level. 
  We propose a new method to calculate the greybody factor in the $AdS_3$. This is based on both the non-normalizable modes of a test field($\Phi_i$) and $AdS_3$/CFT correspondence. Such non-normalizable modes serve as classical, non-fluctuating background and encode the choice of operator insertion(${\cal O}^i$) in the boundary. Actually specifying the boundary condition at infinity of $AdS_3$ corresponds to turning on these non-normalizable modes. Hence we can calculate the greybody factor of test fields without the Dirichlet(or Neumann) boundary condition. The result is consistent with those of the boundary CFT and effective string calculations. 
  We compute the quark--antiquark potential in three dimensional massive Quantum Electrodynamics for arbitrary fermion mass. The result indicates that screening prevails for any quark masses, contrary to the classical expectations, generalizing our previous result obtained for large masses. We also test the validity of several approximation schemes using a detailed numerical analysis. The classical result is still reproduced for small separation of the quarks. 
  We study higher spin tensor currents in quantum field theory. Scalar, spinor and vector fields admit unique "improved" currents of arbitrary spin, traceless and conserved. Off-criticality as well as at interacting fixed points conservation is violated and the dimension of the current is anomalous. In particular, currents J^(s,I) with spin s between 0 and 5 (and a second label I) appear in the operator product expansion of the stress tensor. The TT OPE is worked out in detail for free fields; projectors and invariants encoding the space-time structure are classified. The result is used to write and discuss the most general OPE for interacting conformal field theories and off-criticality. Higher spin central charges c_(s,I) with arbitrary s are defined by higher spin channels of the many-point T-correlators and central functions interpolating between the UV and IR limits are constructed. We compute the one-loop values of all c_(s,I) and investigate the RG trajectories of quantum field theories in the conformal window following our approach. In particular, we discuss certain phenomena (perturbative and nonperturbative) that appear to be of interest, like the dynamical removal of the I-degeneracy. Finally, we address the problem of formulating an action principle for the RG trajectory connecting pairs of CFT's as a way to go beyond perturbation theory. 
  We find an expression for the effective superpotential describing the $N_c$ vacua of $SU(N_c)$ SUSY gluodynamics. The superpotential reduces in some approximation to the Veneziano-Yankielowicz expression amended by the term restoring the discrete $Z_{2N_c}$ symmetry. Moreover, the superpotential, being restricted to one particular vacuum state, yields the expression which was derived recently to describe all the lowest-spin physical states of the theory. The corresponding scalar potential has no cusp singularities and can be used to study the domain walls interpolating between the chirally asymmetric vacua of the model. 
  We study the four--point function of chiral primaries corresponding to the dilaton--axion sector in supergravity in the $AdS_5$/CFT$_4$ correspondence. We find relations between some of the supergravity graphs and compute their leading singularities. We discuss the issue of logarithmic singularities and their significance for the OPE structure of the CFT. 
  Dynamical Higgs mechanism on the light-front (LF) is studied using a (1+1) dimensional model, with emphasis on the infrared divergence problem. The consideration of the zero mode $ k^+ = 0 $ is not sufficient for investigating dynamical symmetry breaking on the LF. It also needs to treat properly an infrared divergence caused by internal momentum $ p^+ \to 0 ~ (p^+ \neq 0) $ in the continuum limit. In order to avoid the divergence, we introduce an infrared cutoff function $ F_{\rm IR} (p, \Lambda) $ which is not Lorentz invariant. It is then shown that the gauge boson obtains mass dynamically on the LF. 
  Lattice calculations performed in Abelian gauges give strong evidence that confinement is realized as a dual Meissner effect, implying that the Yang--Mills vacuum consists of a condensate of magnetic monopoles. We show in Polyakov gauge how the Pontrjagin index of the gauge field is related to the magnetic monopole charges. 
  We propose a method for the approximate computation of the Green function of a scalar massless field Phi subjected to potential barriers of given size and shape in spacetime. This technique is applied to the case of a 3D gaussian ellipsoid-like barrier, placed on the axis between two pointlike sources of the field. Instead of the Green function we compute its temporal integral, that gives the static potential energy of the interaction of the two sources. Such interaction takes place in part by tunneling of the quanta of Phi across the barrier. We evaluate numerically the correction to the potential in dependence on the size of the barrier and on the barrier-sources distance. 
  Static, spherically symmetric monopole solutions of a spontaneously broken SU(2) gauge theory coupled to a massive dilaton field are studied in detail in function of the dilaton coupling strength and of the dilaton mass. 
  The RG expansions for renormalized coupling constants g_6 and g_8 of the 3D n-vector model are calculated in the 4-loop and 3-loop approximations respectively. Resummation of the RG series for g_6 by the Pade-Borel-Leroy technique results in the estimates for its universal critical values g_6^*(n) which are argued to deviate from the exact numbers by less than 0.3%. The RG expansion for g_8 demonstrates stronger divergence being much less suitable for getting reliable numerical estimates. 
  We formulate the notion of parity for the periodic XXZ spin chain within the Quantum Inverse Scattering Method. We also propose an expression for the eigenvalues of the charge conjugation operator. We use these discrete symmetries to help classify low-lying S^z=0 states in the critical regime, and we give a direct computation of the S matrix. 
  We demonstrate the equality between the universal chiral partition function, which was first found in the context of conformal field theory and Rogers-Ramanujan identities, and the exclusion statistics introduced by Haldane in the study of the fractional quantum Hall effect. The phenomena of multiple representations of the same conformal field theory by different sets of exclusion statistics is discussed in the context of the ${\hat u}(1)$ theory of a compactified boson of radius $R.$ 
  For supergavrity solutions which are the product of an anti-de Sitter space with an Einstein space X, we study the relation between the amount of supersymmetry preserved and the geometry of X. Depending on the dimension and the amount of supersymmetry, the following geometries for X are possible, in addition to the maximally supersymmetric spherical geometry: Einstein-Sasaki in dimension 2k+1, 3-Sasaki in dimension 4k+3, 7-dimensional manifolds of weak G_2 holonomy and 6-dimensional nearly Kaehler manifolds. Many new examples of such manifolds are presented which are not homogeneous and have escaped earlier classification efforts. String or M theory in these vacua are conjectured to be dual to superconformal field theories. The brane solutions interpolating between these anti-de Sitter near-horizon geometries and the product of Minkowski space with a cone over X lead to an interpretation of the dual superconformal field theory as the world-volume theory for branes at a conical singularity (cone branes). We propose a description of those field theories whose associated cones are obtained by (hyper-)Kaehler quotients. 
  We investigate the behaviour of a particle moving on the quotient manifold $M=C^2/Z_$ which is derived from the EH metric as the two centers approach each other. In the classical region of the configuration space we specify the physically acceptable solutions and observe a tendency of the radial wave function to concentrate around the conical singularity. Fot the quantum case, using Schr\"{o}dinger's equation, we determine the energy spectra and the radial eigenfunctions for a class of potentials. 
  We explore the extent to which a local string theory dynamics in anti-de Sitter space can be determined from its proposed Conformal Field Theory (CFT) description. Free fields in the bulk are constructed from the CFT operators, but difficulties are encountered when one attempts to incorporate interactions. We also discuss general features of black hole dynamics as seen from the CFT perspective. In particular, we argue that the singularity of AdS_3 black holes is resolved in the CFT description. 
  We describe probes of anti-de Sitter spacetimes in terms of conformal field theories on the AdS boundary. Our basic tool is a formula that relates bulk and boundary states -- classical bulk field configurations are dual to expectation values of operators on the boundary. At the quantum level we relate the operator expansions of bulk and boundary fields. Using our methods, we discuss the CFT description of local bulk probes including normalizable wavepackets, fundamental and D-strings, and D-instantons. Radial motions of probes in the bulk spacetime are related to motions in scale on the boundary, demonstrating a scale-radius duality. We discuss the implications of these results for the holographic description of black hole horizons in the boundary field theory. 
  We introduce new modified Abelian lattice models, with inhomogeneous local interactions, in which a sum over topological sectors are included in the defining partition function. The dual models, on lattices with arbitrary topology, are constructed and they are found to contain sums over topological sectors, with modified groups, as in the original model. The role of the sum over sectors is illuminated by deriving the field-strength formulation of the models in an explicitly gauge-invariant manner. The field-strengths are found to satisfy, in addition to the usual local Bianchi constraints, global constraints. We demonstrate that the sum over sectors removes these global constraints and consequently softens the quantization condition on the global charges in the system. Duality is also used to construct mappings between the order and disorder variables in the theory and its dual. A consequence of the duality transformation is that correlators which wrap around non-trivial cycles of the lattice vanish identically. For particular dimensions this mapping allows an explicit expression for arbitrary correlators to be obtained. 
  We briefly review some recent developments in large N gauge theories which utilize the power of string perturbation techniques. 
  We describe the type IIB supergravity background on AdS_{5} X S_{5} using the potentials of AdS_{5|4} X S_{5} and we use the supersolvable algebra associated to AdS_{5|4} to compute the k gauge fixed type IIB string action. 
  The Weyl-Weinberg-Salam model is presented. It is based on the local conformal gauge symmetry. The model identifies the Higgs scalar field in SM with the Penrose-Chernikov-Tagirov scalar field of the conformal theory of gravity. Higgs mechanism for generation of particle masses is replaced by the originated in Weyl's ideas conformal gauge scale fixing. Scalar field is no longer a dynamical field of the model and does not lead to quantum particle-like excitations that could be observed in HE experiments. Cosmological constant is naturally generated by the scalar quadric term. The model admits Weyl vector bosons that can mix with photon and weak bosons. 
  We obtain the Seiberg-Witten geometry for four-dimensional N=2 gauge theory with gauge group SO(2N_c) (N_c \leq 5) with massive spinor and vector hypermultiplets by considering the gauge symmetry breaking in the N=2 $E_6$ theory with massive fundamental hypermultiplets. In a similar way the Seiberg-Witten geometry is determined for N=2 SU(N_c) (N_c \leq 6) gauge theory with massive antisymmetric and fundamental hypermultiplets. Whenever possible we compare our results expressed in the form of ALE fibrations with those obtained by geometric engineering and brane dynamics, and find a remarkable agreement. We also show that these results are reproduced by using N=1 confining phase superpotentials. 
  The connection between the theory of permutation orbifolds, covering surfaces and uniformization is investigated, and the higher genus partition functions of an arbitrary permutation orbifold are expressed in terms of those of the original theory. 
  We investigate the non-perturbative equivalence of some heterotic/type II dual pairs with N=2 supersymmetry. The perturbative heterotic scalar manifolds are respectively SU(1, 1)/U(1) x SO(2, 2+NV)/ SO(2) x SO(2+NV) and SO(4, 4+NH)/ SO(4) x SO(4+NH) for moduli in the vector multiplets and hypermultiplets. The models under consideration correspond, on the type II side, to self-mirror Calabi-Yau threefolds with Hodge numbers h(1,1)= NV +3= h(2,1)= NH +3, which are K3 fibrations. We consider three classes of dual pairs, with NV=NH=8, 4 and 2. The models with h(1,1)=7 and 5 provide new constructions, while the h(1,1)=11, already studied in the literature, is reconsidered here. Perturbative R2-like corrections are computed on the heterotic side by using a universal operator whose amplitude has no singularities in the (T,U) space, and can therefore be compared with the type II side result. We point out several properties connecting K3 fibrations and spontaneous breaking of the N=4 supersymmetry to N=2. As a consequence of the reduced S- and T- duality symmetries, the instanton numbers in these three classes are restricted to integers, which are multiples of 2, 2 and 4, for NV=8, 4 and 2, respectively. 
  Starting from a manifestly Lorentz- and diffeomorphism-invariant classical action we perform a perturbative derivation of the gravitational anomalies for chiral bosons in 4n+2 dimensions. The manifest classical invariance is achieved using a newly developed method based on a scalar auxiliary field and two new bosonic local symmetries. The resulting anomalies coincide with the ones predicted by the index theorem. In the two-dimensional case, moreover, we perform an exact covariant computation of the effective action for a chiral boson (a scalar) which is seen to coincide with the effective action for a two-dimensional complex Weyl-fermion. All these results support the quantum reliability of the new, at the classical level manifestly invariant, method. 
  It has been shown that a procedure analogous to orbifolding in string theory, when applied to certain large N field theories, leaves correlators invariant perturbatively. We test nonperturbative agreement of some aspects of the orbifolded and non-orbifolded theories. More specifically, we find that the period matrices of parent and orbifolded Seiberg-Witten theories are related, even away from the 't Hooft limit. We also check that any large N theory which has an infrared conformal fixed point and satisfies certain anomaly positivity constraints required by theories with fixed points will continue to satisfy those constraints after orbifolding. We discuss extensions of these results to finite N. 
  In an effort to further understand the structure of effective actions for fermions in an external gauge background at finite temperature, we study the example of 1+1 dimensional fermions interacting with an arbitrary Abelian gauge field. We evaluate the effective action exactly at finite temperature. This effective action is non-analytic as is expected at finite temperature. However, contrary to the structure at zero temperature and contrary to naive expectations, the effective action at finite temperature has interactions to all (even) orders (which, however, do not lead to any quantum corrections). The covariant structure thus obtained may prove useful in studying 2+1 dimensional models in arbitrary backgrounds. We also comment briefly on the solubility of various 1+1 dimensional models at finite temperature. 
  We discuss the transformation of the QCD temporal-gauge Hamiltonian to a representation in which it can be expressed as a functional of gauge-invariant quark and gluon fields. We show how this objective can be realized by implementing the non-Abelian Gauss's law, and by using the mathematical apparatus developed for that purpose to also construct gauge-invariant quark and gluon fields. We demonstrate that, in the transformed QCD Hamiltonian, the interactions of pure-gauge components of the gauge field with color-current densities are replaced by nonlocal interactions connecting quark color-charge densities to each other and to `glue'-color. We discuss the nonperturbative evaluation of these nonlocal interactions, which are non-Abelian analogs of the Coulomb interaction in QED, and we explore their implications for QCD in the low-energy regime. 
  Duality symmetries are discussed for non-linear gauge theories of (n-1)-th rank antisymmetric tensor fields in general even dimensions d=2n. When there are M field strengths and no scalar fields, the duality symmetry groups should be compact. We find conditions on the Lagrangian required by compact duality symmetries and show an example of duality invariant non-linear theories. We also discuss how to enlarge the duality symmetries to non-compact groups by coupling scalar fields described by non-linear sigma models. 
  We review the application of the Wigner-Weisskopf model for the neutral K meson system in the resolvent formalism. The Wigner-Weisskopf model is not equivalent to the Lee-Oehme-Yang-Wu formulation (which provides an accurate representation of the data).   The residues in the pole approximation in the Wigner-Weisskopf model are not orthogonal, leading to additional interference terms in the $K_S-K_L 2\pi$ channel. We show that these terms would be detectable experimentally in the decay pattern of the beam emitted from a regenerator if the Wigner-Weisskopf theory were correct. The consistency of the data with the Lee-Oehme-Yang-Wu formulation appears to rule out the applicability of the Wigner-Weisskopf theory to the problem of neutral K meson decay. 
  We consider properties of confining strings in 2+1 dimensional SU(2) nonabelian gauge theory with the Higgs field in adjoint representation. The analysis is carried out in the context of effective dual Lagrangian which describes the dynamics of t'Hooft's $Z_{N}$ vorices. We point out that the same Lagrangian should be interpreted as an effective Lagrangian for the lightest glueballs. It is shown how the string tension for a fundamental string arises in this description. We discuss the properties of the adjoint string and explain how its breaking occurs when the distance between the charges exceeds a critical value. The interaction between the fundamental strings is studied. It is shown that they repel each other in the weak coupling regime. We argue that in the confining regime (pure Yang-Mills theory, or a theory with a heavy Higgs field) the strings actually attract each other and the crossover between the two regimes corresponds to the crossover between the dual superconductors of first and second kind. 
  We consider a class of black hole solutions to Einstein's equations in d dimensions with a negative cosmological constant. These solutions have the property that the horizon is a (d-2)-dimensional Einstein manifold of positive, zero, or negative curvature. The mass, temperature, and entropy are calculated. Using the correspondence with conformal field theory, the phase structure of the solutions is examined, and used to determine the correct mass dependence of the Bekenstein-Hawking entropy. 
  An elementary introduction is provided to the phase space quantization method of Moyal and Wigner. We generalize the method so that it applies to 2-dimensional surfaces, where it has an interesting connection with quantum holography. In the case of Riemann surfaces the connection between Moyal quantization and holography provides new insights into the Torelli theorem and the quantization of non-linear integrable models. Quantum holography may also serve as a model for a quantum theory of membranes. 
  The elucidation of the properties of the instantons in the topologically trivial sector has been a long-standing puzzle. Here we claim that the properties can be summarized in terms of the geometrical structure in the configuration space, the valley. The evidence for this claim is presented in various ways. The conventional perturbation theory and the non-perturbative calculation are unified, and the ambiguity of the Borel transform of the perturbation series is removed. A `proof' of Bogomolny's ``trick'' is presented, which enables us to go beyond the dilute-gas approximation. The prediction of the large order behavior of the perturbation theory is confirmed by explicit calculations, in some cases to the 478-th order. A new type of supersymmetry is found as a by-product, and our result is shown to be consistent with the non-renormalization theorem. The prediction of the energy levels is confirmed with numerical solutions of the Schr\"{o}dinger equation. 
  In this note further evidence is collected in support of the claim that the space-time uncertainty principle implies holography, both within the context of Matrix Theory and the framework of the proposed duality between certain conformal field theories and M-theory/string theory on AdS backgrounds. 
  We consider models in which nonrelativistic matter fields interact with gauge fields whose dynamics are governed by the Chern-Simons term. The relevant equations of motion are derived and reduced dimensionally in time or in space. Interesting solitonic equations emerge and their solutions are described. Finally, we consider a Chern-Simons term in three-dimensional Euclidean space, reduced by spherical symmetry, and we discuss its effect on monopole and instanton solutions. 
  Recently, the AdS/CFT conjecture of Maldacena has been investigated in Lorentzian signature by Balasubramanian et. al. We extend this investigation to Lorentzian BTZ black hole spacetimes, and study the bulk and boundary behaviour of massive scalar fields both in the non-extremal and extremal case. Using the bulk-boundary correspondence, we also evaluate the two-point correlator of operators coupling to the scalar field at the boundary of the spacetime, and find that it satisfies thermal periodic boundary conditions relevant to the Hawking temperature of the BTZ black hole. 
  We present a simple form of the Type IIB string action on $AdS_5\times S^5$. The result is achieved by fixing $\kappa$-symmetry in the Killing spinor gauge defined by the projector of the Killing spinor of the D3 brane. We show explicitly that in this gauge the superspace is greatly simplified which is the crucial ingredient for the simple string action. 
  We show how the linear special conformal transformation in four-dimensional N=4 super Yang-Mills theory is metamorphosed into the nonlinear and field-dependent transformation for the collective coordinates of Dirichlet 3-branes, which agrees with the transformation law for the space-time coordinates in the anti-de Sitter (AdS) space-time. Our result provides a new and strong support for the conjectured relation between AdS supergravity and super conformal Yang-Mills theory (SYM). Furthermore, our work sheds elucidating light on the nature of the AdS/SYM correspondence. 
  We discuss the Becchi-Rouet-Stora-Tyutin (BRST) cohomology and Hodge decomposition theorem for the two dimensional free U(1) gauge theory. In addition to the usual BRST charge, we derive a local, conserved and nilpotent co(dual)-BRST charge under which the gauge-fixing term remains invariant. We express the Hodge decomposition theorem in terms of these charges and the Laplacian operator. We take a single photon state in the quantum Hilbert space and demonstrate the notion of gauge invariance, no-(anti)ghost theorem, transversality of photon and establish the topological nature of this theory by exploiting the concepts of BRST cohomology and Hodge decomposition theorem. In fact, the topological nature of this theory is encoded in the vanishing of the Laplacian operator when equations of motion are exploited. On the two dimensional compact manifold, we derive two sets of topological invariants with respect to the conserved and nilpotent BRST- and co-BRST charges and express the Lagrangian density of the theory as the sum of terms that are BRST- and co-BRST invariants. Mathematically, this theory captures together some of the key features of both Witten- and Schwarz type of topological field theories. 
  The implications of N=1 superconformal symmetry for four dimensional quantum field theories are studied. Superconformal covariant expressions for two and three point functions of quasi-primary superfields of arbitrary spin are found and connected with the operator product expansion. The general formulae are specialised to cases involving a scalar superfield L, which contains global symmetry currents, and the supercurrent, which contains the energy momentum tensor, and the consequences of superconformal Ward identities analysed. The three point function of L is shown to have unique completely antisymmetric or symmetric forms. In the latter case the superspace version of the axial anomaly equation is obtained. The three point function for the supercurrent is shown to have two linearly independent forms. A linear combination of the associated coefficients for the general expression is shown to be related to the scale of the supercurrent two point function through Ward identities. The coefficients are given for the two free field superconformal theories and are also connected with the parameters present in the supercurrent anomaly for supergravity backgrounds. Superconformal invariants, which are possible even in three point functions, are discussed. 
  We explore the relation between the Hopf algebra associated to the renormalization of QFT and the Hopf algebra associated to the NCG computations of transverse index theory for foliations. 
  Using recently developed methods of character expansions we solve exactly in the large N limit a new two-matrix model of hermitean matrices A and B with the action S={1\over 2}(\tr A^2+\tr B^2)-{\alpha\over 4}(\tr A^4+\tr B^4) -{\beta\over 2} \tr(AB)^2. This model can be mapped onto a special case of the 8-vertex model on dynamical planar graphs. The solution is parametrized in terms of elliptic functions. A phase transition is found: the critical point is a conformal field theory with central charge c=1 coupled to 2D quantum gravity. 
  We construct a set of states that implement the non-Abelian Gauss's law for QCD. We also construct a set of gauge-invariant operator-valued quark and gluon fields by establishing an explicit unitary equivalence between the Gauss's law operator and the `pure glue' part of the Gauss's law operator. This unitary equivalence enables us to use the `pure glue' Gauss's law operator to represent the entire Gauss's law operator in a new representation. Since the quark field commutes with the `pure glue' Gauss's law operator, it is a gauge-invariant field in this new representation. We use the unitary equivalence of the new and the conventional representations to construct gauge-invariant quark and gluon fields in both representations, and to transform the QCD Hamiltonian in the temporal gauge so that it is expressed entirely in terms of gauge-invariant quantities. In that form, all interactions between quark fields mediated by `pure gauge' components of the gluon field have been transformed away, and replaced by a nonlocal interaction between gauge-invariant color-charge densities. This feature --- that, in gauge-invariant formulations, interactions mediated by pure gauge components of gauge fields have been replaced by nonlocal interactions --- is shared by many gauge theories. In QED, the resulting nonlocal interaction is the Coulomb interaction, which is the Abelian analog of the QCD interaction we have identified and are describing in this work. The leading term, in a multipole expansion, of this nonlocal QCD interaction vanishes for quarks in color-singlet configurations, suggesting a dynamical origin for color confinement; higher order terms of this multipole expansion suggest a QCD mechanism for color transparency. We also show how, in an SU(2) model, this nonlocal interaction can be evaluated nonperturbatively. 
  We investigate the three dimensional Georgi-Glashow model with a Chern-Simons term. We find that there exist complex monopole solutions of finite action. They dominate the path integral and disorder the Higgs vacuum, but electric charges are not confined. Subtleties in the gauge fixing procedure in the path integral and issues related to Gribov copies are noted. 
  Interactions of relatively rotated Dp-branes in 1, 2, 3 and 4 angles in M(atrix) model are calculated and it is found to be in agreement with string theory calculations. In 4 angles case the agreement is achieved after subtracting the contribution of the single chiral fermionic zero mode. 
  Some explanations and implications of the underlying theory approach for quantum theories (QM or QFT) are discussed and suggested. This simple idea seems to have significantly nontrivial effects for our understanding of the quantum theories. 
  We consider D-branes and orientifold planes embedded in non-compact (orbifolded) space-time. We point out that even in the non-compact cases we can turn on non-zero (quantized) NS-NS antisymmetric B-field. In particular, we study the effect of the B-field on four dimensional large N gauge theories from orientifolds. Thus, in theories with both D3- and D7-branes, the effect of the B-field is non-trivial: the number of D7-branes (of each species) is reduced from 8 (which is the required number if the B-field is trivial) to 4. This results in a different orientifold string theory, and, subsequently, the corresponding large N gauge theory is also different. We explicitly construct large N gauge theories from orientifolds with non-zero B-field backgrounds with ${\cal N}=2,1,0$ supersymmetries. These theories, just as their counterparts without the B-field, have the property that in the large N limit computation of any M-point correlation function reduces to the corresponding computation in the parent ${\cal N}=4$ supersymmetric theory with a unitary gauge group. 
  We study duality properties of actions for chiral boson fields in various space-time dimensions using D=2 and D=6 cases as examples. As a result we get dual covariant formulations of chiral bosons. 
  The paper has been withdrawn for further elaboration. 
  General properties of coordinate-space holographic projections of fields in AdS/CFT correspondence, which respect the Ward identity, are investigated. To show the usefulness of this methodology it is applied to the computation of correlators of massive gauge fields. 
  We construct the supermembrane action in an AdS_4 x S^7 and AdS_7 x S^4 background to all orders in anticommuting coordinates. The result is compared to and agrees completely with results obtained earlier for generic supergravity backgrounds through gauge completion at low orders in \theta. 
  By considering N=2 string amplitudes we determine the (2+2)-dimensional target space action for the physical degrees of freedom: self-dual gravity and self-dual Yang-Mills, together with their respective infinite towers of higher-spin inequivalent picture states. Novel `stringy' couplings amongst these fields are essential ingredients of an action principle for the effective target space field theory. We discuss the covariant description of this theory in terms of self-dual fields on a hyperspace parametrised by the target space coordinate and a commuting chiral spinor. 
  In view of the presence of a superpotential, the dual of a gauge theory like SQCD contains two coupling parameters. The method of the Reduction of Couplings is used in order to express the parameter of the superpotential in terms of the dual gauge coupling. In the conformal window and above it, a unique, isolated solution is obtained. The coupling parameter of the superpotential is given simply by f times the square of the gauge coupling. Here f is a function of the the number of colors and the number of flavors, and it is known explicitly. The solution is valid to all orders in the asymptotic expansion, and it is the appropriate choice for the dual theory. The same solution exists in the free magnetic interval. A `general' solution with non-integer powers is discussed, as are some exceptional cases. 
  We describe forms with non-Abelian charges. We avoid the use of theories with flat curvatures by working in the context of topological field theory. We obtain TQFTs for a form and its dual. We leave open the question of getting gauges in which the form, or its dual, can be gauged away, in such way that the model has two dual formulations. We give the example of charged two-forms in six dimensions. 
  It has recently been proposed that certain nonsupersymmetric type II orbifolds have vanishing perturbative contributions to the cosmological constant. We show that techniques of Sen and Vafa allow one to construct dual type II descriptions of these models (some of which have no weakly coupled heterotic dual). The dual type II models are given by the same orbifolds with the string coupling $S$ and a $T^2$ volume $T$ exchanged. This allows us to argue that in various strongly coupled limits of the original type II models, there are weakly coupled duals which exhibit the same perturbative cancellations as the original models. 
  In this talk, we give a brief discussion of complex monopole solutions in the three dimensional Georgi-Glashow model with a Chern-Simons term. We find that there exist complex monopole solutions of finite action. They dominate the path integral and disorder the Higgs vacuum, but electric charges are not confined. Subtleties in the model and issues related to Gribov copies are also noted. 
  Bosonisation of the massive Thirring model, with a non-minimal and non-abelian gauging is studied in 2+1-dimensions. The static abelian model is solved completely in the large fermion mass limit and the spectrum is obtained. The non-abelian model is solved for a restricted class of gauge fields. In both cases explicit expressions for bosonic currents corresponding to the fermion currents are given. 
  The Cremmer-Scherk mechanism is generalised in a non-Abelian context. In the presence of the Higgs scalars of the standard model it is argued that fields arising from the low energy effective string action may contribute to the mass generation of the observed vector bosons that mediate the electroweak interactions and that future analyses of experimental data should consider the possibility of string induced radiative corrections to the Weinberg angle coming from physics beyond the standard model. 
  We use the anomaly cancellation of the M-theory fivebrane to derive the R-symmetry anomalies of the $A_{N}$ $(0,2)$ tensor-multiplet theories. This result leads to a simple derivation of black hole entropy in $d=4, \CN=2$ compactifications of $M$-theory. We also show how the formalism of normal bundle anomaly cancellation clarifies the Kaluza-Klein origin of Chern-Simons terms in gauged supergravity theories. The results imply the existence of interesting 1/N corrections in the AdS/CFT correspondence. 
  Supersymmetry is used to derive conditions on higher derivative terms in the effective action of type IIB supergravity. Using these conditions, we are able to prove earlier conjectures that certain modular invariant interactions of order alpha' **3 relative to the Einstein-Hilbert term are proportional to eigenfunctions of the Laplace operator on the fundamental domain of SL(2,Z). We also discuss how these arguments generalize to terms of higher order in alpha', as well as to compactifications of supergravity. 
  Recently it was observed by one of the authors that supersymmetric quantum mechanics (SUSYQM) admits a formulation in terms of only one bosonic degree of freedom. Such a construction, called the minimally bosonized SUSYQM, appeared in the context of integrable systems and dynamical symmetries. We show that the minimally bosonized SUSYQM can be obtained from Witten's SUSYQM by applying to it a nonlocal unitary transformation with a subsequent reduction to one of the eigenspaces of the total reflection operator. The transformation depends on the parity operator, and the deformed Heisenberg algebra with reflection, intimately related to parabosons and parafermions, emerges here in a natural way. It is shown that the minimally bosonized SUSYQM can also be understood as supersymmetric two-fermion system. With this interpretation, the bosonization construction is generalized to the case of N=1 supersymmetry in 2 dimensions. The same special unitary transformation diagonalises the Hamiltonian operator of the 2D massive free Dirac theory. The resulting Hamiltonian is not a square root like in the Foldy-Wouthuysen case, but is linear in spatial derivative. Subsequent reduction to `up' or `down' field component gives rise to a linear differential equation with reflection whose `square' is the massive Klein-Gordon equation. In the massless limit this becomes the self-dual Weyl equation. The linear differential equation with reflection admits generalizations to higher dimensions and can be consistently coupled to gauge fields. The bosonized SUSYQM can also be generated applying the nonlocal unitary transformation to the Dirac field in the background of a nonlinear scalar field in a kink configuration. 
  This note is a summary of the work reported in hep-th/9801073. We give a brief discussion of the fine tuning problem in pre-big bang cosmology. We use the flatness problem as our test case, and in addition to the exact numerical limits on initial conditions, we highlight the differences between pre-big bang and standard inflation. The main difference is that in pre-big bang the universe must be smooth and flat in an exponentially large domain already at the beginning of the dilaton-driven inflation. 
  In order to get the general framework describing a nonlocalizable object beyond the bilocal field theory early proposed by Markov and Yukawa, the quantization of space-time is reconsidered and further developed. Space-time quantities are there not only noncommutative with U-field describing the nonlocalizable object, as in the bilocal field theory, but also become noncommutative among themselves. Under the U-field representation, where the basis vectors of representation are chosen to be eigenvectors of operator U, space-time quantities get a matrix representation of infinite dimension in general. Field equation is considered, which determines the relation between space-time quantities and U-field. The possible inner relation between the recent topics of matrix model in superstring theory and the present approach is discussed. 
  Due to a computational mistake this paper has been withdrawn. 
  Due to a computational mistake this paper has been withdrawn. 
  We show how to compute vacuum expectation values from derivative expansions of the vacuum wave functional. Such expansions appear to be valid only for slowly varying fields, but by exploiting analyticity in a complex scale parameter we can reconstruct the contribution from rapidly varying fields. 
  We find the critical charge for a topologically massive gauge theory for any gauge group, generalising our earlier result for SU(2). The relation between critical charges in TMGT, singular vectors in the WZNW model and logarithmic CFT is investigated. 
  Dualities link M-theory, the 10+1 dimensional strong coupling limit of the IIA string, to other 11-dimensional theories in signatures 9+2 and 6+5, and to type II string theories in all 10-dimensional signatures. We study the Freund-Rubin-type compactifications and brane-type solutions of these theories, and find that branes with various world-volume signatures are possible. For example, the 9+2 dimensional M* theory has membrane-type solutions with world-volumes of signature (3,0) and (1,2), and a solitonic solution with world-volume signature (5,1). 
  We compute the intrinsic Hausdorff dimension of spacetime at the infrared fixed point of the quantum conformal factor in 4D gravity. The fractal dimension is defined by the appropriate covariant diffusion equation in four dimensions and is determined by the coefficient of the Gauss-Bonnet term in the trace anomaly to be generally greater than 4. In addition to being testable in simplicial simulations, this scaling behavior suggests a physical mechanism for the screening of the effective cosmological `constant' and inverse Newtonian coupling at very large distance scales, which has implications for the dark matter content and large scale structure of the universe. 
  We renormalize QCD to one loop in coordinate space using constrained differential renormalization, and show explicitly that the Slavnov-Taylor identities are preserved by this method. 
  We describe new $N$-extended 2D supergravities on a $(p+1)$-dimensional (bosonic) space. The fundamental objects are moving frame densities that equip each $(p+1)$-dimensional point with a 2D ``tangent space''. The theory is presented in a $[p+1, 2]$ superspace. For the special case of $p=1$ we recover the 2D supergravities in an unusual form. The formalism has been developed with applications to the string-parton picture of $D$-branes at strong coupling in mind. 
  As shown in hep-th/9709081, non-BPS saturated solitons play an important role in the duality transformations of N=1 supersymmetric gauge theories. In particular, a massive spinor in an SO(N) gauge theory with massless matter in the vector representation appears in the dual description as a magnetic monopole with a Z_2 charge. This claim is supported by numerous tests, including detailed matching of flavor quantum numbers. This fact makes it possible to test the phase of an SO(N) gauge theory using massive spinors as a probe. It is thereby shown explicitly that the free magnetic phase which appears in supersymmetric theories is a non-confining phase. A fully non-abelian version of the Dual Meissner effect is also exhibited, in which the monopoles are confined by non-BPS string solitons with Z_2 charges. 
  We compute scattering amplitudes involving both R-R and NS-NS fields in the presence of a D-brane or orientifold plane. These provide direct evidence for the anomalous couplings in the D-brane and orientifold actions. The D9-brane and O9-plane are found to couple to the first Pontrjagin class with the expected relative strength. 
  Coincident D3-branes placed at a conical singularity are related to string theory on $AdS_5\times X_5$, for a suitable five-dimensional Einstein manifold $X_5$. For the example of the conifold, which leads to $X_5=T^{1,1}=(SU(2)\times SU(2))/U(1)$, the infrared limit of the theory on $N$ D3-branes was constructed recently. This is ${\cal N}=1$ supersymmetric $SU(N)\times SU(N)$ gauge theory coupled to four bifundamental chiral superfields and supplemented by a quartic superpotential which becomes marginal in the infrared. In this paper we consider D3-branes wrapped over the 3-cycles of $T^{1,1}$ and identify them with baryon-like chiral operators built out of products of $N$ chiral superfields. The supergravity calculation of the dimensions of such operators agrees with field theory. We also study the D5-brane wrapped over a 2-cycle of $T^{1,1}$, which acts as a domain wall in $AdS_5$. We argue that upon crossing it the gauge group changes to $SU(N)\times SU(N+1)$. This suggests a construction of supergravity duals of ${\cal N}=1$ supersymmetric $SU(N_1)\times SU(N_2)$ gauge theories. 
  We study a three-dimensional symmetric Chern-Simons field theory with a general covariance and it turns out that the original Chern-Simons theory is just a gauge fixed action of the symmetric Chern-Simons theory whose constraint algebra belongs to fully first class constraint system. The Abelian Chern-Simons theory with matter coupling is studied for the construction of anyon operators without any ordering ambiguity with the help of this symmetric Chern-Simons action. Finally we shall discuss some connections between the present symmetric formulation of Chern-Simons theory and the St\"ukelberg mechanism. 
  In this note we address the problem of solving for the positronium mass spectrum. We use front-form dynamics together with the method of flow equations. For a special choice of the similarity function, the calculations can be simplified by analytically integrating over the azimuthal angle. One obtains an effective Hamiltonian and we solve numerically for its spectrum. Comparing our results with different approaches we find encouraging properties concerning the cutoff dependence of the results. 
  Using a novel approach to renormalization in the Hamiltonian formalism, we study the connection between asymptotic freedom and the renormalization group flow of the configuration space metric. It is argued that in asymptotically free theories the effective distance between configuration decreases as high momentum modes are integrated out. 
  We examine the AdS/CFT correspondence when the gauge theory is considered on a compactified space with supersymmetry breaking boundary conditions. We find that the corresponding supergravity solution has a negative energy, in agreement with the expected negative Casimir energy in the field theory. Stability of the gauge theory would imply that this supergravity solution has minimum energy among all solutions with the same boundary conditions. Hence we are lead to conjecture a new positive energy theorem for asymptotically locally Anti-de Sitter spacetimes. We show that the candidate minimum energy solution is stable against all quadratic fluctuations of the metric. 
  Boundary states corresponding to wrapped D-branes in Calabi-Yau compactifications of type II strings are discussed using Gepner models. In particular boundary conditions corresponding to D-0 branes and D-instantons in four dimensions are investigated. The boundary states constructed by Recknagel and Schomerus are analyzed in the light-cone gauge and the broken and conserved space-time supersymmetry charges are found. The geometrical interpretation of these algebraically constructed boundary states is clarified in some simple cases. Moreover, the action of mirror symmetry and other discrete symmetries of the Gepner model on the boundary states are discussed. As an application the boundary states are used to calculate instanton induced corrections to metric on the hypermultiplets in the N=2 effective action. 
  In the context of the conjectured AdS-CFT correspondence of string theory, we consider a class of asymptotically Anti-de Sitter black holes whose conformal boundary consists of a single connected component, identical to the conformal boundary of Anti-de Sitter space. In a simplified model of the boundary theory, we find that the boundary state to which the black hole corresponds is pure, but this state involves correlations that produce thermal expectation values at the usual Hawking temperature for suitably restricted classes of operators. The energy of the state is finite and agrees in the semiclassical limit with the black hole mass. We discuss the relationship between the black hole topology and the correlations in the boundary state, and speculate on generalizations of the results beyond the simplified model theory. 
  The unconstrained system equivalent to SU(2) Yang-Mills field theory is obtained in the framework of the generalized Hamiltonian formalism using the method of Hamiltonian reduction. The reduced system is expressed in terms of fields which transform as spin zero and spin two under spatial rotations. 
  The symmetries, especially those related to the $R$-transformation, of the reflection equation(RE) for two-component systems are analyzed. The classification of solutions to the RE for eight-, six- and seven-vertex type $R$-matrices is given. All solutions can be obtained from those corresponding to the standard $R$-matrices by $K$-transformation. For the free-Fermion models, the boundary matrices have property $tr K_+(0)=0$, and the free-Fermion type $R$-matrix with the same symmetry as that of Baxter type corresponds to the same form of $K_-$-matrix for the Baxter type. We present the Hamiltonians for the open spin systems connected with our solutions. In particular, the boundary Hamiltonian of seven-vertex models was obtained with a generalization to the Sklyanin's formalism. 
  The space-time disclination is studied by making use of the decomposition theory of gauge potential in terms of antisymmetric tensor field and $\phi$-mapping method. It is shown that the self-dual and anti-self-dual parts of the curvature compose the space-time disclinations which are classified in terms of topological invariants--winding number. The projection of space-time disclination density along an antisymmetric tensor field is quantized topologically and characterized by Brouwer degree and Hopf index. 
  The underlying reason for the existence of gravitational entropy is traced to the impossibility of foliating topologically non-trivial Euclidean spacetimes with a time function to give a unitary Hamiltonian evolution. In $d$ dimensions the entropy can be expressed in terms of the $d-2$ obstructions to foliation, bolts and Misner strings, by a universal formula. We illustrate with a number of examples including spaces with nut charge. In these cases, the entropy is not just a quarter the area of the bolt, as it is for black holes. 
  The emission rate of fermions from 2+1 dimensional BTZ black holes is shown to have a form which can be reproduced from a conformal field theory at finite temperature. The rate obtained for fermions is identical to the rate of non-minimally coupled fermions emitted from a five dimensional black hole, whose near horizon geometry is BTZ X M, where M is a compact manifold. 
  We study the dilaton-dependence of the effective action for N=1, SU(N1) x SU(N2) models with one generation of vectorlike matter transforming in the fundamental of both groups. We treat in detail the confining and Coulomb phases of these models writing explicit expressions in many cases for the effective superpotential. We can do so for the Wilson superpotentials of the Coulomb phase when N2=2, N1=2,4.   In these cases the Coulomb phase involves a single U(1) gauge multiplet, for which we exhibit the gauge coupling in terms of the modulus of an elliptic curve. The SU(4) x SU(2) model reproduces the weak-coupling limits in a nontrivial way. In the confining phase of all of these models, the dilaton superpotential has a runaway form, but in the Coulomb phase the dilaton enjoys flat directions. Had we used the standard moduli-space variables: Tr M^k, k=1,..., N2, with M the quark condensate matrix, to parameterize the flat directions instead of the eigenvalues of M, we would find physically unacceptable behaviour, illustrating the importance to correctly identify the moduli. 
  Type IIB string action on AdS_5 x S^5 constructed in hep-th/9805028 is put into a form where it becomes quadratic in fermions. This is achieved by performing 2-d duality (T-duality) on the action in which kappa-symmetry was fixed in the Killing gauge in hep-th/9808038. We discuss some properties and possible applications of the resulting action. 
  The variables appropriate for the infrared limit of unconstrained SU(2) Yang-Mills field theory are obtained in the Hamiltonian formalism. It is shown how in the infrared limit an effective nonlinear sigma model type Lagrangian can be derived which out of the six physical fields involves only one of three scalar fields and two rotational fields summarized in a unit vector. Its possible relation to the effective Lagrangian proposed recently by Faddeev and Niemi is discussed. 
  Topological defects can arise in symmetry breaking models where the scalar field potential $V(\phi)$ has no minima and is a monotonically decreasing function of $|\phi|$. The properties of such vacuumless defects are quite different from those of the ``usual'' strings and monopoles. In some models such defects can serve as seeds for structure formation, or produce an appreciable density of mini-black holes. 
  The basic features of a quantum field theory which is Poincar\'e invariant, gauge invariant, finite and unitary to all orders of perturbation theory are reviewed. Quantum gravity is finite and unitary to all orders of perturbation theory. The Bekenstein-Hawking entropy formula for a black hole is investigated in a conical Rindler space approximation to a black hole event horizon. A renormalization of the gravitational coupling constant is performed leading to a finite Bekenstein-Hawking entropy at the horizon. 
  The asymptotic behaviour of the vacuum energy density, ${\bar E}(\theta)$, at $\theta \ra \pm i \infty$ is found out. A new interpretation and a qualitative discussion of the ${\bar E}(\theta)$ behaviour are presented. It is emphasized that the vacuum is doubly degenerate at $\theta=\pi$, and the quark electric string can terminate on the domain wall interpolating between these two vacua. The potential of the monopole field condensing in the Yang-Mills vacuum is obtained. 
  It is shown that there is no chirally symmetric vacuum state in the {cal N}=1 supersymmetric Yang-Mills theory. The values of the gluino condensate and the vacuum energy density are found out through a direct instanton calculation. A qualitative picture of domain wall properties is presented, and a new explanation of the phenomenon of strings ending on the wall is proposed. 
  3d Chern-Simons gauge theory has a strong connection with 2d CFT and link invariants in knot theory. We impose some constraints on the $D(2|1;\alpha)$ CS theory in the similar context of the hamiltonian reduction of 2d superconformal algebras. There Hilbert states in $D(2|1;\alpha)$ CS theory are partly identified with characters of the large N=4 SCFT by their transformation properties. 
  Following Kachru, Kumar and Silverstein, we construct a set of non-supersymmetric Type II string models which have equal numbers of bosons and fermions at each mass level. The models are asymmetric {\bf Z}_2 \otimes {\bf Z}_2^{\prime} orbifolds. We demonstrate that this bose-fermi degeneracy feature implies that both the one-loop and the two-loop contributions to the cosmological constant vanish. We conjecture that the cosmological constant actually vanishes to all loops. We construct a strong-weak dual pair of models, both of which have bose-fermi degeneracy. This implies that at least some of the non-perturbative corrections to the cosmological constant are absent. 
  Working at the level of Poisson brackets, we describe the extension of the generalized Wakimoto realization of a simple Lie algebra valued current, J, to a corresponding realization of a group valued chiral primary field, b, that has diagonal monodromy and satisfies $Kb'=Jb$. The chiral WZNW field b is subject to a monodromy dependent exchange algebra, whose derivation is reviewed, too. 
  We examine supersymmetry of four-dimensional asymptotically anti-de Sitter (AdS) dyonic black holes in the context of gauged N=2 supergravity. Our calculations concentrate on black holes with unusual topology and their rotating generalizations, but we also reconsider the spherical rotating dyonic Kerr-Newman-AdS black hole, whose supersymmetry properties have previously been investigated by Kosteleck\'{y} and Perry within another approach. We find that in the case of spherical, toroidal or cylindrical event horizon topology, the black holes must rotate in order to preserve some supersymmetry; the non-rotating supersymmetric configurations representing naked singularities. However, we show that this is no more true for black holes whose event horizons are Riemann surfaces of genus $g>1$, where we find a nonrotating extremal solitonic black hole carrying magnetic charge and permitting one Killing spinor. For the nonrotating supersymmetric configurations of various topologies, all Killing spinors are explicitly constructed. 
  It is observed that some structures recently uncovered in the study of Calogero-Sutherland models and anyons are close analogs of well-known structures of boundary conformal field theory. These examples of ``boundary conformal quantum mechanics'', in spite of their apparent simplicity, have a rather reach structure, including some sort of T-duality, and could provide useful frameworks for testing general properties of boundary conformal theories. Of particular interest are the duality properties of anyons and Calogero-Sutherland particles in presence of boundary-violations of conformal invariance; these are here briefly analyzed leading to the conjecture of a general interconnection between (deformed) boundary conformal quantum mechanics, T-type duality, and (``exchange'' or ``exclusion'') exotic statistics. These results on the point-particle quantum-mechanics side are compared with recent results on the action of T-duality on open strings that satisfy conformal-invariance-violating boundary conditions. Moreover, it is observed that some of the special properties of anyon and Calogero-Sutherland quantum mechanics are also enjoyed by the M(atrix) quantum mechanics which has recently attracted considerable attention. 
  The considered model of baryon consists of three pointlike masses (quarks) bounded pairwise by relativistic strings forming a curvilinear triangle. Classic analytic solutions for this model corresponding to a planar uniform rotation about the system center of mass are found and investigated. These solutions describe a rotating curve composed of segments of a hypocycloid. The curve is a curvilinear triangle or --- a more complicated configuration with a set of internal massless points moving at the speed of light. Different topological types of these motions are classified in connection with different forms of hypocycloids in zero quark mass limit. An application of these solutions to description of baryon states on the Regge trajectories is considered. 
  There has recently been a revival of interest in anti de-Sitter space (AdS) brought about by the conjectured duality beteeen physics in the bulk of AdS and a conformal field theory on the boundary. Since the whole subject of branes, singletons and superconformal field theories on the AdS boundary was an active area of research about ten years ago, I begin with a historical review, including the ``Membrane at the end of the universe'' idea. Next I discuss two recent papers with Lu and Pope on on $AdS_{5} \times S^{5}$ and on $AdS_{3} \times S^{3}$, respectively. In each case we note that odd-dimensional spheres $S^{{2n+1}}$ may be regarded as U(1) bundles over $CP^{n}$ and that this permits an unconventional ``Hopf''duality along the U(1) fibre. This leads in particular to the phenomenon of BPS without BPS whereby states which appear to be non-BPS in one picture are seen to be BPS in the dual picture. 
  We construct vacua of M-theory on S^1/Z_2 associated with Calabi-Yau three-folds. These vacua are appropriate for compactification to N=1 supersymmetry theories in both four and five dimensions. We allow for general E_8 x E_8 gauge bundles and for the presence of five-branes. The five-branes span the four-dimensional uncompactified space and are wrapped on holomorphic curves in the Calabi-Yau space. Properties of these vacua, as well as of the resulting low-energy theories, are discussed. We find that the low-energy gauge group is enlarged by gauge fields that originate on the five-brane world-volumes. In addition, the five-branes increase the types of new E_8 x E_8 breaking patterns allowed by the non-standard embedding. Characteristic features of the low-energy theory, such as the threshold corrections to the gauge kinetic functions, are significantly modified due to the presence of the five-branes, as compared to the case of standard or non-standard embeddings without five-branes. 
  SU(2) Yang-Mills field theory is considered in the framework of the generalized Hamiltonian approach and the equivalent unconstrained system is obtained using the method of Hamiltonian reduction. A canonical transformation to a set of adapted coordinates is performed in terms of which the Abelianization of the Gauss law constraints reduces to an algebraic operation and the pure gauge degrees of freedom drop out from the Hamiltonian after projection onto the constraint shell. For the remaining gauge invariant fields two representations are introduced where the three fields which transform as scalars under spatial rotations are separated from the three rotational fields. An effective low energy nonlinear sigma model type Lagrangian is derived which out of the six physical fields involves only one of the three scalar fields and two rotational fields summarized in a unit vector. Its possible relation to the effective Lagrangian proposed recently by Faddeev and Niemi is discussed. Finally the unconstrained analog of the well-known nonnormalizable groundstate wave functional which solves the Schr\"odinger equation with zero energy is given and analysed in the strong coupling limit. 
  We construct solutions to the chiral Thirring model in the framework of algebraic quantum field theory. We find that for all positive temperatures there are fermionic solutions only if the coupling constant is $\lambda = \sqrt{2(2n + 1)\pi}, n \in \bf N$. 
  The constrained structure of the duality invariant form of Maxwell theory is considered in the Hamiltonian formulation of Dirac as well as from the symplectic viewpoint. Compared to the former the latter approach is found to be more economical and elegant. Distinctions from the constrained analysis of the usual Maxwell theory are pointed out and their implications are also discussed. 
  Complex monopole solutions exist in the three dimensional Georgi-Glashow model with the Chern-Simons term. They dominate the path integral and disorder the Higgs vacuum. Gribov copies of the vacuum and monopole configurations are studied in detail. 
  Quons are particles characterized by the parameter $q$, which permits smooth interpolation between Bose and Fermi statistics; $q=1$ gives bosons, $q=-1$ gives fermions.   In this paper we give a heuristic argument for an extension of conservation of statistics to quons with trilinear couplings of the form $\bar{f}fb$, where $f$ is fermion-like and $b$ is boson-like. We show that $q_f^2=q_b$. In particular, we relate the bound on $q_{\gamma}$ for photons to the bound on $q_e$ for electrons, allowing the very precise bound for electrons to be carried over to photons. An extension of this argument suggests that all particles are fermions or bosons to high precision. 
  Closed forms are derived for the effective actions for free, massive spinless fields in anti-de Sitter spacetimes in arbitrary dimensions. The results have simple expressions in terms of elementary functions (for odd dimensions) or multiple Gamma functions (for even dimensions). We use these to argue against the quantum validity of a recently-proposed duality relating such theories with differing masses and cosmological constants. 
  It is proposed to make formulation of second quantizing a bosonic theory by generalizing the method of filling the Dirac negative energy sea for fermions. We interpret that the correct vacuum for the bosonic theory is obtained by adding minus one boson to each single particle negative energy states while the positive energy states are empty. The boson states are divided into two sectors ; the usual positive sector with positive and zero numbers of bosons and the negative sector with negative numbers of bosons. Once it comes into the negative sector it cannot return to the usual positive sector by ordinary interaction due to a barrier. It is suggested to use as a playround models in which the filling of empty fermion Dirac sea and the removal of boson from the negative energy states are not yet performed. We put forward such a naive vacuum world and propose a CPT-like theorem for it.We study it in detail and give a proof for $\lambda/4 (\phi^+\phi)^2$ theory. The CPT-like theorem is a strong reflection, but does not include inversion of operator order which is involved in the ordinary CPT theorem. Instead it needs certain analytic continuation of the multiple wave function when the state is formulated as a finite number of particles present. 
  Dilaton coupled electromagnetic field is essential element of low-energy string effective action or it may be considered as result of spherical compactification of Maxwell theory in higher dimensions. The large $N$ and large curvature effective action for $N$ dilaton coupled vectors is calculated. Adding such quantum correction to classical dilaton gravity action we show that effective dilaton-Maxwell gravity under consideration may generate Schwarzschild-de Sitter black holes (SdS BHs) with constant dilaton as solutions of the theory. That suggests a mechanism (alternative to BHs production) for quantum generation of SdS BHs in early universe (actually, for quantum creation of inflationary Universe) due to back-reaction of dilaton coupled matter. The possibility of proliferation of anti-de Sitter space is briefly discussed. 
  It is argued that an electronically charged dilaton black hole can support a long range field of a Nielsen-Olesen string. Combining both numerical and perturbative techniques we examine the properties of an Abelian-Higgs vortex in the presence of the black hole under consideration. Allowing the black hole to approach extremality we found that all fields of the vortex are expelled from the extreme black hole. In the thin string limit we obtained the metric of a conical electrically charged dilaton black hole. The effect of the vortex can be measured from infinity justifying its characterization as black hole hair. 
  We study three-dimensional gauge dynamics by using type IIB superstring brane configurations, which can be obtained from the M-theory configuration of M2-branes stretched between two M5-branes with relative angles. Our construction of brane configurations includes (p,q)5-brane and gives a systematic classification of possible three-dimensional gauge theories. The explicit identification of gauge theories are made and their mirror symmetry is discussed. As a new feature, our theories include interesting Maxwell-Chern-Simons system whose vacuum structure is also examined in detail, obtaining results consistent with the brane picture. 
  We construct a class of representations of the Heisenberg algebra in terms of the complex shift operators subject to the proper continuous limit imposed by the correspondence principle. We find a suitable Hilbert space formulation of our construction for two types of shifts: (1) real shifts, (2) purely imaginary shifts. The representations involving imaginary shifts are free of spectrum doubling. We determine the corresponding coordinate and momentum operators satisfying the canonical commutation relations. The eigenvalues of the coordinate operator are in both cases discrete. 
  Explicit expressions are considered for the generating functions concerning the number of planar diagrams with given numbers of 3- and 4-point vertices. It is observed that planar renormalization theory requires diagrams with restrictions, in the sense that one wishes to omit `tadpole' inserions and `seagull' insertions; at a later stage also self-energy insertions are to be removed, and finally also the dressed 3-point inserions and the dressed 4-point insertions. Diagrams with such restrictions can all be counted exactly. This results in various critical lines in the $\lambda$-$g$ plane, where $\lambda$ and $g$ are effective zero-dimensional coupling constants. These lines can be localized exactly. 
  We study softly broken N=1 supersymmetric QCD with the gauge group $SU(N_c)$ and $N_f$ flavors of quark pairs. We investigate vacuum structure of the theory with generic soft supersymmetry breaking terms. Trilinear soft breaking terms play an essential role in determining vacua. For $N_f=N_c+1$, chiral symmetry is broken for a sufficiently large magnitude of trilinear couplings, while it is unbroken in the case with only soft masses. In the case where appearance of trilinear coupling terms is allowed, i.e. for $N_f \geq N_c+1$, we have two possible vacua, the trivial and non-trivial ones. Otherwise, we only have the non-trivial vacuum, which corresponds to the non-trivial vacuum in the $N_f \geq N_c+1$ theory. 
  A general theory of the Berezinsky-Kosterlitz-Thouless (BKT) type phase transitions in low-dimensional systems is proposed. It is shown that in d-dimensional case the necessary conditions for it can take place are 1) conformal invariance of kinetic part of model action and 2) vacuum homotopy group $\pi_{d-1}$ must be nontrivial and discrete. It means a discrete vacuum degeneracy for $1d$ systems and continuous vacuum degeneracy for higher $d$ systems. For such systems topological exitations have logariphmically divergent energy and they can be described by corresponding effective field theories. In general case the sufficient conditions for existence of the BKT type phase transition are 1) constraint $d \le 2$ and 2) $\pi_{d-1}$ must have some crystallographic symmetries. Critical properties of possible low-dimensional effective theories are determined and it is shown that in two-dimensional case they are characterized by the Coxeter numbers $h_G$ of lattices from the series ${A,D,E,Z}$ and can be interpreted as those of conformal field theories with integer central charge $c=r,$ where $r$ is a rank of groups $\pi_1$ and $G.$ In one-dimensional case analogous critical properties have ferromagnetic Dyson chains with discrete Cartan-Ising spins. In contrast, critical properties of one-dimensional models with periodic potentials have a weak dependence on group $G.$ 
  Non equilibrium effective field theory is presented as an inhomogeneous field theory, using a formulation which is analogous to that of a gauge theory. This formulation underlines the importance of structural aspects of non-equilibrium, effective field theories. It is shown that, unless proper attention is paid to such structural features, hugely different answers can be obtained for a given model. The exactly soluble two-level atom is used as an example of both the covariant methodology and of the conclusions. 
  We construct supergravity backgrounds representing non-homogeneous compactifications of d=10,11 supergravities to four dimensions, which cannot be written as a direct product. The geometries are regular and approach $AdS_7\times S^4$ or $AdS_5\times S^5$ at infinity; they are generically non-supersymmetric, except in a certain "extremal" limit, where a Bogomol'nyi bound is saturated and a naked singularity appears. By using these spaces, one can construct a model of QCD that generalizes by one (or two) extra parameters a recently proposed model of QCD based on the non-extremal D4 brane. This allows for some extra freedom to circumvent some (but not all) limitations of the simplest version. 
  Solutions to the Thirring model are constructed in the framework of algebraic QFT. It is shown that for all positive temperatures there are fermionic solutions only if the coupling constant is $\lambda = \sqrt{2(2n+1)\pi}, n\in \bf N$, otherwise solutions are anyons. Different anyons (which are uncountably many) live in orthogonal spaces, so the whole Hilbert space becomes non-separable and in each of its sectors a different Urgleichung holds. This feature certainly cannot be seen by any power expansion in $\lambda$. Moreover, if the statistic parameter is tied to the coupling constant it is clear that such an expansion is doomed to failure and will never reveal the true structure of the theory.   On the basis of the model in question, it is not possible to decide whether fermions or bosons are more fundamental since dressed fermions can be constructed either from bare fermions or directly from the current algebra. 
  We show that the four derivative terms in the effective action of three-dimensional N=8 Yang-Mills theory are determined by supersymmetry. These terms receive both perturbative and non-perturbative corrections. Using our technique for constraining the effective action, we are able to determine the exact form of the eight fermion terms in the supersymmetric completion of the $F^4$ term, including all instanton corrections. As a consequence, we argue that the integral of the Euler density over $k$ monopole moduli space in SU(2) Yang-Mills is determined by our non-renormalization theorem for all values of $k$. 
  We consider the 1+1 dimensional N = (2,2) supersymmetric matrix model which is obtained by dimensionally reducing N = 1 super Yang-Mills from four to two dimensions. The gauge groups we consider are U(Nc) and SU(Nc), where Nc is finite but arbitrary. We adopt light-cone coordinates, and choose to work in the light-cone gauge. Quantizing this theory via Discretized Light-Cone Quantization (DLCQ) introduces an integer, K, which restricts the light-cone momentum-fraction of constituent quanta to be integer multiples of 1/K. Solutions to the DLCQ bound state equations are obtained for K=2,3,...,6 by discretizing the light-cone supercharges, which results in a supersymmetric spectrum. Our numerical results imply the existence of normalizable massless states in the continuum limit K -> infinity, and therefore the absence of a mass gap. The low energy spectrum is dominated by string-like (or many parton) states. Our results are consistent with the claim that the theory is in a screening phase. 
  We consider the detailed renormalization of two (1+1)-dimensional gauge theories which are quantized without preserving gauge invariance: the chiral and the "anomalous" Schwinger models. By regularizing the non-perturbative divergences that appear in fermionic Green's functions of both models, we show that the "tree level" photon propagator is ill-defined, thus forcing one to use the complete photon propagator in the loop expansion of these functions. We perform the renormalization of these divergences in both models to one loop level, defining it in a consistent and semi-perturbative sense that we propose in this paper. 
  We examine the stabilization of the two typical moduli, the length $\rho$ of the eleventh segment and the volume $V$ of the internal six manifold, in compactified heterotic $M$-theory. It is shown that, under certain conditions, the phenomenologically favored vacuum expectation values of $\rho$ and $V$ can be obtained by the combined effects of multi-gaugino condensations on the hidden wall and the membrane instantons wrapping the three cycle of the internal six manifold. 
  For the generalized chiral Schwinger model defined on the circle, a direct calculation of the zero curvature part of the vacuum Berry phase connection is given. Although this part does not contribute to the curvature, it is shown to attach several features to the total connection and to produce a physical background of linearly rising electric fields. 
  We describe the structure of the moduli space of $\sigma$-model couplings for the worldsheet description of a system of $N$ D-particles, in the case that the couplings are represented by a pair of logarithmic recoil operators. We derive expressions for the canonical momenta conjugate to the D-particle couplings and the Zamolodchikov metric to the first few orders in $\sigma$-model perturbation theory. We show, using only very general properties of the operator product expansion in logarithmic conformal field theories, that the canonical dynamics on moduli space agree with the predictions of the non-abelian generalization of the Born-Infeld effective action for D-particles with a symmetrized trace structure. We demonstrate that the Zamolodchikov metric naturally encodes the short-distance structure of spacetime, and from this we derive uncertainty relations for the D-particle coordinates directly from the quantum string theory. We show that the moduli space geometry naturally leads to new forms of spacetime indeterminancies involving only spatial coordinates of target space and illustrate the manner in which the open string interactions between D-particles lead to a spacetime quantization. We also derive appropriate non-abelian generalizations of the string-modified Heisenberg uncertainty relations and the space--time uncertainty principle. The non-abelian uncertainties exhibit decoherence effects suggesting the interplay of quantum gravity in multiple D-particle dynamics. 
  Using a six-orientifold, fourbranes and four fivebranes in type IIA string theory we construct $\mathcal{N}$=1 supersymmetric gauge theories in four dimensions with product group $SU(M)\times SO(N)$ or $SU(M)\times Sp(2N)$, a bifundamental flavor and quarks. We obtain the Seiberg dual for these theories and rederive it via branes. To obtain the complete dual group via branes we have to add semi-infinite fourbranes. We propose that the theory derived from branes has a meson deformation switched on. This deformation implies higgsing in the dual theory. The addition of the semi-infinite fourbranes compensates this effect. 
  We compute $O(\alpha'^3)$ corrections to the $AdS_5\times S^5$ black hole metric. We find that the radius of the $S^5$ depends on the radial $AdS_5$ coordinate. This completes the computation of Gubser, Klebanov and Tseytlin (hep-th/9805156). The fact that the metric no longer factorizes should modify the value of the Wilson line at finite temperature and the glueball mass spectrum. 
  Faddeev and Niemi have proposed a reformulation of SU(2) Yang-Mills theory in terms of new variables, appropriate for describing the theory in its infrared limit based on the intuitive picture of colour confinement due to monopole condensation. I generalize their proposal (with some differences) to SU(N) Yang-Mills theory. The natural variables are $N-1$ mutually commuting traceless $N\times N$ Hermitian matrices, an element of the maximal torus defined by these commuting matrices, $N-1$ Abelian gauge fields for the maximal torus gauge group, and an invariant symmetric two-index tensor on the tangent space of the maximal torus, adding up to the requisite 2$(N^{2}-1)$ physical degrees of freedom. 
  We calculate the Hamiltonian of a compactified D4-brane, with general fluxes and moduli, and find the BPS-mass. The results are invariant under the complete U-duality SO(5,5,Z). 
  It has been suggested recently that knots might exist as stable soliton solutions in a simple three-dimensional classical field theory, opening up a wide range of possible applications in physics and beyond. We have re-examined and extended this work in some detail using a combination of analytic approximations and sophisticated numerical algorithms. For charges between one and eight, we find solutions which exhibit a rich and spectacular variety of phenomena, including stable toroidal solitons with twists, linked loops and also knots. The physical process which allows for this variety is the reconnection of string-like segments. 
  The class of relativistic spin particle models reveals the `quantization' of parameters already at the classical level. The special parameter values emerge if one requires the maximality of classical global continuous symmetries. The same requirement applied to a non-relativistic particle with odd degrees of freedom gives rise to supersymmetric quantum mechanics. Coupling classical non-relativistic superparticle to a `U(1) gauge field', one can arrive at the quantum dynamical supersymmetry. This consists in supersymmetry appearing at special values of the coupling constant characterizing interaction of a system of boson and fermion but disappearing in a free case. Possible relevance of this phenomenon to high-temperature superconductivity is speculated. 
  We explicitly evaluate the low energy coupling $F_g$ in a $d=4,\CN=2$ compactification of the heterotic string. The holomorphic piece of this expression provides the information not encoded in the holomorphic anomaly equations, and we find that it is given by an elementary polylogarithm with index $3-2g$, thus generalizing in a natural way the known results for $g=0,1$. The heterotic model has a dual Calabi-Yau compactification of the type II string. We compare the answer with the general form expected from curve-counting formulae and find good agreement. As a corollary of this comparison we predict some numbers of higher genus curves in a specific Calabi-Yau, and extract some intersection numbers on the moduli space of genus $g$ Riemann surfaces. 
  The product space configuration $AdS_2\times S_2$ (with $l$ and $r$ being radiuses of the components) carrying the electric charge $Q$ is demonstrated to be an exact solution of the semiclassical Einstein equations in presence of the Maxwell field. If the logarithmic UV divergences are absent in the four-dimensional theory the solution we find is identical to the classical Bertotti-Robinson space ($r=l=Q$) with no quantum corrections added. In general, the analysis involves the quadratic curvature coupling $\lambda$ appearing in the effective action. The solutions we find are of the following types: i) (for arbitrary $\lambda$) charged configuration which is quantum deformation of the Bertotti-Robinson space; ii) ($\lambda >\lambda_{cr}$) Q=0 configuration with $l$ and $r$ being of the Planck order; iii) ($\lambda<\lambda_{cr}$) $Q\neq 0$ configuration ($l$ and $r$ are of the Planck order) not connected analytically to the Bertotti-Robinson space. The interpretation of the solutions obtained and an indication on the internal structure of the Schwarzschild black hole are discussed. 
  Adiabatic vacua are known to be Hadamard states. We show, however that the energy-momentum tensor of a linear Klein-Gordon field on Robertson-Walker spaces developes a generic singularity on the initial hypersurface if the adiabatic vacuum is of order less than four. Therefore, adiabatic vacua are physically reasonable only if their order is at least four.   A certain non-local large momentum expansion of the mode functions has recently been suggested to yield the subtraction terms needed to remove the ultraviolet divergences in the energy-momentum tensor. We find that this scheme fails to reproduce the trace anomaly and therefore is not equivalent to adiabatic regularisation. 
  We discuss the brane interpretation of the integrable dynamics behind the exact solution to the N=2 SUSY YM theory. Degrees of freedom in the first integrable system responsible for the spectral Riemann surfaces comes from the hidden Higgs branch of the moduli space. The second integrable system of the Whitham type yields the dynamics on the Coulomb branch and can be considered as the scattering of branes. 
  We briefly present two-dimensional dilaton gravity from the point of view of integrable systems. 
  I briefly summarize recent results on classical and quantum dilaton gravity in 1+1 dimensions. 
  We derive a general WKB energy splitting formula in a double-well potential by incorporating both phase loss and anharmonicity effect in the usual WKB approximation. A bare application of the phase loss approach to the usual WKB method gives better results only for large separation between two potential minima. In the range of substantial tunneling, however, the phase loss approach with anharmonicity effect considered leads to a great improvement on the accuracy of the WKB approximation. 
  Recently, a new framework for solving the hierarchy problem was proposed which does not rely on low energy supersymmetry or technicolor. The fundamental Planck mass is at a TeV and the observed weakness of gravity at long distances is due the existence of new sub-millimeter spatial dimensions. In this letter, we study how the properties of black holes are altered in these theories. Small black holes---with Schwarzschild radii smaller than the size of the new spatial dimensions---are quite different. They are bigger, colder, and longer-lived than a usual $(3+1)$-dimensional black hole of the same mass. Furthermore, they primarily decay into harmless bulk graviton modes rather than standard-model degrees of freedom. We discuss the interplay of our scenario with the holographic principle. Our results also have implications for the bounds on the spectrum of primordial black holes (PBHs) derived from the photo-dissociation of primordial nucleosynthesis products, distortion of the diffuse gamma-ray spectrum, overclosure of the universe, gravitational lensing, as well as the phenomenology of black hole production. For example, the bound on the spectral index of the primordial spectrum of density perturbations is relaxed from 1.25 to 1.45-1.60 depending on the epoch of the PBH formation. In these scenarios PBHs provide interesting dark matter candidates; for 6 extra dimensions MACHO candidates with mass $\sim 0.1M_\odot$ can arise. For 2 or 3 extra dimensions PBHs with mass $\sim 2000 M_\odot$ can occur and may act as both dark matter and seeds for early galaxy and QSO formation. 
  We study the cancellation of U(1) anomalies in Type I and Type IIB D=4, N=1 string vacua. We first consider the case of compact toroidal $Z_N$ Type IIB orientifolds and then proceed to the non-compact case of Type IIB D3 branes at orbifold and orientifold singularities. Unlike the case of the heterotic string we find that for each given vacuum one has generically more than one U(1) with non-vanishing triangle anomalies. There is a generalized Green-Schwarz mechanism by which these anomalies are cancelled. This involves only the Ramond-Ramond scalars coming from the twisted closed string spectrum but not those coming from the untwisted sector. Associated to the anomalous U(1)'s there are field-dependent Fayet-Illiopoulos terms whose mass scale is fixed by undetermined vev's of the NS-NS partners of the relevant twisted RR fields. Thus, unlike what happens in heterotic vacua, the masses of the anomalous U(1)'s gauge bosons may be arbitrarily light. In the case of D3 branes at singularities, appropriate factorization of the U(1)'s constrains the Chan-Paton matrices beyond the restrictions from cancellation of non-abelian anomalies. These conditions can be translated to constraints on the T-dual Type IIB brane box configurations. We also construct a new large family of N=1 chiral gauge field theories from D3 branes at orientifold singularities, and check its non-abelian and U(1) anomalies cancel. 
  We consider one-loop quantum corrections to soliton energies and central charges in the supersymmetric $\phi^4$ and sine-Gordon models in 1+1 dimensions. In both models, we unambiguously calculate the correction to the energy in a simple renormalization scheme and obtain $\Delta H = - m/(2\pi)$, in agreement with previous results. Furthermore, we show that there is an identical correction to the central charge, so that the BPS bound remains saturated in the one-loop approximation. We extend these results to arbitrary 1+1 dimensional supersymmetric theories. 
  We construct the SO(32) spinor state in weakly coupled type I string theory as a kink solution of the tachyon field on the D-string - anti-D-string pair and calculate its mass. We also give a description of this system in terms of an exact boundary conformal field theory and show that in this description this state can be regarded as a non-supersymmetric D0-brane in type I string theory. This construction can be generalised to represent the D0-brane in type IIA string theory as a vortex solution of the tachyon field on the membrane anti-membrane pair, and the D-string of type I string theory as a topological soliton of the tachyon field on the D5-brane anti- D5-brane pair. 
  Cosmological inflation is studied in the case where the inflaton is the overall modulus $T$ for an orbifold. General forms of the (non-perturbative) superpotential are considered to ensure that $G=K+{\rm ln}|W|^2$ is modular invariant. We find generically that these models do not produce a potential flat enough for slow roll to a supersymmetric minimum, although we do find a model which produces up to 20 e-folds of inflation to a non-supersymmetric minimum. 
  We study AdS/CFT correspondence in four dimensional N=1 field theories realized on the worldvolume of D3 branes near the intersections between D7 and D7' branes in F theory studied by Aharony et al. We consider the compactification of F theory on elliptically fibered Calabi-Yau threefolds corresponding to two sets of parallel D7 branes sharing six spacetime directions. This can be viewed as orbifolds of six torus T^6 by Z_p x Z_q (p, q=2, 3, 4, 6). We find the large N spectrum of chiral primary operators by exploiting the property of AdS/CFT correspondence. Moreover, we discuss supergravity solutions for D3 branes in D7 and D7' branes background. 
  We have studied the underlying algebraic structure of the anharmonic oscillator by using the variational perturbation theory. To the first order of the variational perturbation, the Hamiltonian is found to be factorized into a supersymmetric form in terms of the annihilation and creation operators, which satisfy a q-deformed algebra. This algebraic structure is used to construct all the eigenstates of the Hamiltonian. 
  We develop a method for extracting accurate critical exponents from perturbation expansions of the O(n)-symmetric nonlinear sigma-model in D=2+ epsilon dimensions. This is possible by considering the epsilon-expansions in this model as strong-coupling expansions of functions of the variable tildevarepsilon = 2(4-D)/(D-2), whose first five weak-coupling expansion coefficients of powers of tildevarepsilon are known from varepsilon-expansions of critical exponents in O(n)-symmetric phi^4-theory in D=4-epsilon dimensions. 
  We show explicitly that the two recently proposed actions for the type IIB superstring propagating on AdS_{5} X S_{5} agree completely. 
  Generalized matter couplings to four-dimensional supersymmetric sigma models on general Kaehler manifolds are presented, preserving all holomorphic symmetries. Our generalization allows assignment of arbitrary U(1) charges to additional matter fermions, in all representations of (the holomorphic part of) the isometry group. This can be used to eliminate unwanted gamma_5 anomalies, in particular for the U(1) symmetry arising from the complex structure of the target space. A consistent gauging of this isometry group, or any of its subgroups, then becomes possible. When gauged in the presence of a chiral scalar multiplet, the U(1) symmetry is broken spontaneously, generating a mass for the U(1) vector multiplet via the supersymmetric Higgs effect. As an example we discuss the case of the homogeneous coset space E6/SO(10) x U(1). 
  We argue that the self/anti-self charge conjugate states of the (1/2,0)\oplus (0,1/2) representation possess the axial charge. Furthermore, we analyze the recent claims of the \sim \sigma \cdot [A \times A^*] interaction terms for `fermions'. Finally, we briefly discuss the problem in the (1,0)\oplus (0,1) representation. 
  We argue that vacua of string theory which asymptote at weak coupling to linear dilaton backgrounds are holographic. The full string theory in such vacua is ``dual'' to a theory without gravity in fewer dimensions. The dual theory is generically not a local quantum field theory. Excitations of the string vacuum, which can be studied in the weak coupling region using worldsheet methods, give rise to observables in the dual theory. An interesting example is string theory in the near-horizon background of parallel NS5-branes, the CHS model, which is dual to the decoupled NS5-brane theory (``little string theory''). This duality can be used to study some of the observables in this theory and some of their correlation functions. Another interesting example is the ``old'' matrix model, which gives a holographic description of two dimensional string theory. 
  We study two-dimensional classically integrable field theory with independent boundary condition on each end, and obtain three possible generating functions for integrals of motion when this model is an ultralocal one. Classically integrable boundary condition can be found in solving boundary $K_{\pm}$ equations. In quantum case, we also find that unitarity condition of quantum $R$- matrix is sufficient to construct commutative quantities with boundary, and its reflection equations are obtained. 
  A global extension of the Batalin-Marnelius proposal for a BRST inner product to gauge theories with topologically nontrivial gauge orbits is discussed. It is shown that their (appropriately adapted) method is applicable to a large class of mechanical models with a semisimple gauge group in the adjoint and fundamental representation. This includes cases where the Faddeev-Popov method fails. Simple models are found also, however, which do not allow for a well-defined global extension of the Batalin-Marnelius inner product due to a Gribov obstruction. Reasons for the partial success and failure are worked out and possible ways to circumvent the problem are briefly discussed. 
  In the framework of a generalized iterative scheme introduced previously to account for the non-analytic coupling dependence associated with the renormalization-group invariant mass scale Lambda, we establish the self-consistency equations of the extended Feynman rules (Lambda-modified vertices of zeroth perturbative order) for the three-gluon vertex, the two ghost vertices, and the two vertices of massless quarks. Calculations are performed to one-loop-order, in Landau gauge, and at the lowest approximation level (r=1) of interest for QCD. We discuss the phenomenon of compensating poles inherent in these equations, by which the formalism automatically cancels unphysical poles on internal lines, and the role of composite-operator information in the form of equation-of-motion condensate conditions. The observed near decoupling of the four-gluon conditions permits a solution to the 2-and-3-point conditions within an effective one-parameter freedom. There exists a parameter range in which one solution has all vertex coefficients real, as required for a physical solution, and a narrower range in which the transverse-gluon and massless-quark propagators both exhibit complex-conjugate pole pairs. 
  We analyse the famous Baxter's $T-Q$ equations for $XXX$ ($XXZ$) spin chain and show that apart from its usual polynomial (trigonometric) solution, which provides the solution of Bethe-Ansatz equations, there exists also the second solution which should corresponds to Bethe-Ansatz beyond $N/2$. This second solution of Baxter's equation plays essential role and together with the first one gives rise to all fusion relations. 
  We glance back at the short period of the great discoveries between 1970 and 1974 that led to the restablishment of Quantum Field Theory and the discovery of the Standard Model of Elementary Particles, in particular Quantum Chromodynamics, and ask ourselves where we stand now. 
  We study the self-consistency problem of the generalized Feynman rule (nonperturbatively modified vertex of zeroth perturbative order) for the 4-gluon vertex function in the framework of an extended perturbation scheme accounting for non-analytic coupling dependence through the Lambda scale. Tensorial structure is restricted to a minimal dynamically closed basis set. The self-consistency conditions are obtained at one loop, in Landau gauge, and at the lowest approximation level (r=1) of interest for QCD. At this level, they are found to be linear in the nonperturbative 4-gluon coefficients, but strongly overdetermined due to the lack of manifest Bose symmetry in the relevant Dyson-Schwinger equation. The observed near decoupling from the 2-and-3-point conditions permits least-squares quasisolutions for given 2-and-3-point input within an effective one-parameter freedom. We present such solutions for N_F=2 massless quarks and for the pure gluon theory, adapted to the 2-and-3-point coefficients determined previously. 
  The classical order parameters for the $\N=2$ supersymmetric SU(N) gauge theory with matter in the adjoint representation are exhibited explicitly as conservation laws for the elliptic Calogero-Moser system. Central to the construction are certain elliptic function identities, which arise from considering Feynman diagrams in a theory of free fermions with twisted boundary conditions. 
  We examine a certain 16-fermion correlator in {\cal N}=4 supersymmetric SU(N) gauge theory in 4 dimensions. Generalizing recent SU(2) results of Bianchi, Green, Kovacs and Rossi, we calculate the exact N-dependence of the effective 16-fermion vertex at the 1-instanton level, and find precise agreement in the large-N limit with the prediction of the type IIB superstring on AdS_5 x S^5. This suggests that the string theory prediction for the 1-instanton amplitude considered here is not corrected by higher-order terms in the alpha' expansion. 
  We examine general aspects of parity functions arising in rational conformal field theories, as a result of Galois theoretic properties of modular transformations. We focus more specifically on parity functions associated with affine Lie algebras, for which we give two efficient formulas. We investigate the consequences of these for the modular invariance problem. 
  We review aspects of N=1 duality between the heterotic string and F-theory. After a description of string duality intended for the non-specialist the framework and the constraints for heterotic/F-theory compactifications are presented. The computations of the necessary Calabi-Yau manifold and vector bundle data, involving characteristic classes and bundle moduli, are given in detail. The matching of the spectrum of chiral multiplets and of the number of heterotic five-branes respectively F-theory three-branes, needed for anomaly cancellation in four-dimensional vacua, is pointed out. Several examples of four-dimensional dual pairs are constructed where on both sides the geometry of the involved manifolds relies on del Pezzo surfaces. 
  We discuss superconformally invariant systems of hypermultiplets coupled to gauge fields associated with target-space isometries. 
  We construct the T duals of certain type IIA brane configurations with one compact dimension (elliptic models) which contain orientifold planes. These configurations realize four-dimensional $\NN=2$ finite field theories. For elliptic models with two negatively charged orientifold six-planes, the T duals are given by D3 branes at singularities in the presence of O7-planes and D7-branes. For elliptic models with two oppositely charged orientifold planes, the T duals are D3 branes at a different kind of orientifold singularities, which do not require D7 branes. We construct the adequate orientifold groups, and show that the cancellation of twisted tadpoles is equivalent to the finiteness of the corresponding field theory. One family of models contains orthogonal and symplectic gauge factors at the same time. These new orientifolds can also be used to define some six-dimensional RG fixed points which have been discussed from the type IIA brane configuration perspective. 
  Three-point functions of analytic (chiral primary) operators in N=4 Yang-Mills theory in four dimensions are calculated using the harmonic superspace formulation of this theory. In the case of the energy-momentum tensor multiplet anomaly considerations determine the coefficient. Analyticity in N=2 harmonic superspace is explicitly checked in a two-loop calculation. 
  An outline is given of a recently discovered technique for building a quantum effective action that is completely independent of gauge-fixing choices and ghost determinants. One makes maximum use of the geometry and fibre-bundle structure of the space of field histories and introduces a set of nonlocal composite fields: the geodesic normal fields based on Vilkovisky's connection on the space of histories. The closed-time-path formalism of Schwinger, Bakshi, Mahantappa {\it et al} can be adapted for these fields, and a set of gauge-fixing-independent dynamical equations for their expectation values (starting from given initial conditions) can be computed. An obvious application for such equations is to the study of the formation and radiative decay of black holes, and to other back-reaction problems. 
  In the context of the bosonic closed string theory, by using the operatorial formalism, we give a simple expression of the off-shell amplitude with an arbitrary number of external massless states inserted on the Klein bottle. 
  The pure-Skyrme limit of a scale-breaking Skyrmed O(3) sigma model in 1+1 dimensions is employed to study the effect of the Skyrme term on the semiclassical analysis of a field theory with instantons. The instantons of this model are self-dual and can be evaluated explicitly. They are also localised to an absolute scale, and their fluctuation action can be reduced to a scalar subsystem. This permits the explicit calculation of the fluctuation determinant and the shift in vacuum energy due to instantons. The model also illustrates the semiclassical quantisation of a Skyrmed field theory. 
  We study noncompact and static membrane solutions in Matrix theory. Demanding axial symmetry on a membrane embedded in three spatial dimensions, we obtain a wormhole solution whose shape is the same with the catenoidal solution of Born-Infeld theory. We also discuss another interesting class of solutions, membranes embedded holomorphically in four spatial dimensions, which are 1/4 BPS. 
  An investigation of the center symmetric phase of SU(2) QCD is presented. The role of the center-symmetry, the dynamics of Polyakov loops and the structure of Abelian monopoles are studied within the axial gauge representation of QCD. Realization of the center symmetry is shown to result from non-perturbative gauge fixing and concomitant confinement like properties emerging even at the perturbative level are displayed. In an analysis of the Polyakov loop dynamics, non-perturbative gauge fixing is also shown to inevitably lead to singular gauge field configurations whose dynamics are briefly discussed. 
  We present a complete derivation of absorption cross-section and Hawking radiation of minimal and fixed scalars from the Strominger-Vafa model of five-dimensional black hole, starting right from the moduli space of the D1-D5 brane system. We determine the precise coupling of this moduli space to bulk modes by using the AdS/CFT correspondence. Our methods resolve a long-standing problem regarding emission of fixed scalars. We calculate three-point correlators of operators coupling to the minimal scalars from supergravity and from SCFT, and show that both vanish. We make some observations about how the AdS/SYM correspondence implies a close relation between large $N$ equations of motion of $d$-dimensional gauge theory and supergravity equations on $AdS_{d+1}$-type backgrounds. We compare with the explicit nonlocal transform relating 1 and 2 dimensions in the context of $c=1$ matrix model. 
  The periods of arbitrary abelian forms on hyperelliptic Riemann surfaces, in particular the periods of the meromorphic Seiberg-Witten differential, are shown to be in one-to-one correspondence with the conformal blocks of correlation functions of the rational logarithmic conformal field theory with central charge c=c(2,1)=-2. The fields of this theory precisely simulate the branched double covering picture of a hyperelliptic curve, such that generic periods can be expressed in terms of certain generalised hypergeometric functions, namely the Lauricella functions of type F_D. 
  We derive some explicit expressions for correlators on Grassmannian G_r(C^n) as well as on the moduli space of holomorphic maps, of a fixed degree d, from sphere into the Grassmannian. Correlators obtained on the Grassmannain are a first step generalization of the Schubert formula for the self-intersection. The intersection numbers on the moduli space for r=2,3 are given explicitly by two closed formulas, when r=2 the intersection numbers, are found to generate the alternate Fibonacci numbers, the Pell numbers and in general a random walk of a particle on a line with absorbing barriers. For r=3 the intersection numbers form a well organized pattern. 
  I comment on the paper hep-th/9808013 by A. Berkovich and B.M. McCoy. 
  We present a Liouville-string approach to confinement in four-dimensional gauge theories, which extends previous approaches to include non-conformal theories. We consider Liouville field theory on world sheets whose boundaries are the Wilson loops of gauge theory, which exhibit vortex and spike defects. We show that world-sheet vortex condensation occurs when the Wilson loop is embedded in four target space-time dimensions, and show that this corresponds to the condensation of gauge magnetic monopoles in target space. We also show that vortex condensation generates a effective string tension corresponding to the confinement of electric degrees of freedom. The tension is independent of the string length in a gauge theory whose electric coupling varies logarithmically with the length scale. The Liouville field is naturally interpreted as an extra target dimension, with an anti-de-Sitter (AdS) structure induced by recoil effects on the gauge monopoles, interpreted as D branes of the effective string theory. Black holes in the bulk AdS space correspond to world-sheet defects, so that phases of the bulk gravitational system correspond to the different world-sheet phases, and hence to different phases of the four-dimensional gauge theory. Deconfinement is associated with a Berezinskii-Kosterlitz-Thouless transition of vortices on the Wilson-loop world sheet, corresponding in turn to a phase transition of the black holes in the bulk AdS space. 
  Starting from the Kac--Moody structure of the WZNW model for SL(2,R) and using the general canonical formalism, we formulate a gauge theory invariant under local SL(2,R) x SL(2,R) and diffeomorphisms. This theory represents a gauge extension of the WZNW system, defined by a difference of two simple WZNW actions. By performing a partial gauge fixing and integrating out some dynamical variables, we prove that the resulting effective theory coincides with the induced gravity in 2D. The geometric properties of the induced gravity are obtained out of the gauge properties of the WZNW system with the help of the Dirac bracket formalism. 
  The equivalence between rank-2 anti-symmetric tensor fields, considered as gauge potentials, and torsionless non-linear $\sigma$-models suggests us to study the possibility of coupling tensorial matter with Yang-Mills fields, through the gauging of the isometries of the target space. We show that this coupling is actually possible; however, the matter appears no longer as an elementary field, but rather as a composite one, expressed in terms of the bosonic degrees of freedom of the $\sigma$-model. A possible phenomenological application is presented that describes the interactions among vector mesons in terms of the geometrical properties of the target manifold. Also, spin-2 meson resonances may naturally be accommodated whenever the $\sigma$-model's target manifold is non-symmetric. 
  We consider M-theory on (T^2\times R^2)/Z_n with M5 branes wrapped on R^2. One can probe this background with M5 branes wrapped on T^2. The theories on the probes provide many new examples of N=2 field theories without Lagrangian description. All these theories have Coulomb branches, and we find the corresponding Seiberg-Witten curves. The exact solution is encoded in a Hitchin system on an orbifolded torus with punctures. The theories we consider also arise from D3 probes in F-theory on K3\times K3 orbifolds. Interestingly, the relevant F-theory background has frozen Z_n singularities which are analogous to frozen Z_2 singularities in Type I string theory. We use the F-theory description to find supergravity duals of the probe SCFT's in the large N limit and compute the spectrum of relevant and marginal operators. We also explain how the decoupling of U(1) factors is manifested in the supergravity description. 
These notes review the effective lagrangian treatment of Goldstone and pseudo-Goldstone bosons, taking examples from high-energy/nuclear and condensed-matter physics. The contents are:   1. Goldstone Bosons   2. Pions: A Relativistic Application   3. Magnons: Nonrelativistic Applications   4. SO(5) Invariance and Superconductors   5. Bibliography 
  Recent results in the literature concerning holography indicate that the thermodynamics of quantum gravity (at least with a negative cosmological constant) can be modeled by the large N thermodynamics of quantum field theory. We emphasize that this suggests a completely unitary evolution of processes in quantum gravity, including black hole formation and decay; and even more extreme examples involving topology change. As concrete examples which show that this correspondence holds even when the space-time is only locally asymptotically AdS, we compute the thermodynamical phase structure of the AdS-Taub-NUT and AdS-Taub-Bolt spacetimes, and compare them to a 2+1 dimensional conformal field theory (at large N) compactified on a squashed three sphere, and on the twisted plane. 
  Applying the recently obtained results on the renormalization of soft supersymmetry-breaking parameters, we investigate the infrared behaviour of the softly broken supersymmetric QCD as well as its dual theory in the conformal window. Under general assumptions on $\beta$-functions, it is shown that the soft supersymmetry-breaking parameters asymptotically vanish in the infrared limit so that superconformal symmetry in softly broken supersymmetric QCD and in its dual theory revives at the infrared fixed point, provided the soft scalar masses satisfy certain renormalization group invariant relations. If these relations are not satisfied, there exist marginal operators in both theories that lead to the breaking of supersymmetry and also colour symmetry. 
  The W-infinity minimal models are conformal field theories which can describe the edge excitations of the hierarchical plateaus in the quantum Hall effect. In this paper, these models are described in very explicit terms by using a bosonic Fock space with constraints, or, equivalently, with a non-trivial Hamiltonian. The Fock space is that of the multi-component Abelian conformal theories, which provide another possible description of the hierarchical plateaus; in this space, the minimal models are shown to correspond to the sub-set of states which satisfy the constraints. This reduction of degrees of freedom can also be implemented by adding a relevant interaction to the Hamiltonian, leading to a renormalization-group flow between the two theories. Next, a physical interpretation of the constraints is obtained by representing the quantum incompressible Hall fluids as generalized Fermi seas. Finally, the non-Abelian statistics of the quasi-particles in the W-infinity minimal models is described by computing their correlation functions in the Coulomb Gas approach. 
  We perform a foliation of a four dimensional Riemannian space-time with respect to a discrete time which is an integer multiple of the Planck time. We find that the quantum fluctuations of the metric have a discrete energy spectrum. The metric field is expanded in stationary eigenstates, and this leads to the description of a de Sitter-like universe. At the Planck scale the model describes a Planckian Euclidean black hole. 
  In this note we extend our work in a previous paper hep-th/9801038. We show here that various intersecting brane-like configurations can be found in the vacuum of $D=4, N=4$ supergravity with gauged R-symmetry group $SU(2)\times SU(2)$. These include intersections of domain-walls, strings and point-like objects. Some of these intersecting configurations preserve 1/2 and 1/4 of supersymmetry. We observe that the previously obtained $AdS_3\times R^1$ pure axionic vacuum or `axio-vac' is an intersection of domain-wall with extended string with 1/2 supersymmetries. Also the solutions known as `electro-vac' with geometry $AdS_2\times R^2$ can be simply interpreted as the intersection of domain-wall with point-like object. 
  We study M theory on $AdS_7 \times \RP^4$ corresponding to 6 dimensional SO(2N) $(0, 2)$ superconformal field theory on a circle which becomes 5 dimensional super Yang-Mills theory at low energies. For SU(N) (0,2) theory, a wrapped D4 brane on $\S^4$ which is connected to a D4 brane on the boundary of $AdS_7$ by N fundamental strings can be interpreted as baryon vertex. For SO(2N) (0,2) theory, by using the property of homology of $\RP^4$, we classify various wrapping branes. Then we consider particles, strings, twobranes, domain walls and the baryon vertex in Type IIA string theory. 
  The effective action for the membrane dynamics on the background geometry of the $N$-sector DLCQ $M$ theory compactified on a two-torus is computed via supergravity. We compare it to the effective action obtained from the matrix theory, i.e., the (2+1)-dimensional supersymmetric Yang-Mills (SYM) theory, including the one-loop perturbative and full non-perturbative instanton effects. Consistent with the DLCQ prescription of $M$ theory {\em a la} Susskind, we find the precise agreement for the finite $N$-sector (off-conformal regime), as well as for the large $N$ limit (conformal regime), providing us with a concrete example of the correspondence between the matrix theory and the DLCQ $M$ theory. Non-perturbative instanton effects in the SYM theory conspire to yield the eleven-dimensionally covariant effective action. 
  We solve the Dirichlet boundary value problem for the massless gravitino on $AdS_{d+1}$ space and compute the two-point function of the dual CFT supersymmetry currents using the $AdS$/CFT correspondence principle. We find analogously to the spinor case that the boundary data for the massless $(d+1)$ dimensional bulk gravitino field consists of only a $(d-1)$ dimensional gravitino. 
  We study the three-pronged strings (three string junctions) from the point of view of D-string worldsheet gauge theory. We justify interpreting kink solutions in the 2-dimensional gauge theory as pronged strings by examining BPS energy bounds obtained from this theory. When the ends of the pronged string are on D3-branes, the configuration preserves 1/4 supersymmetry with BPS conditions including the Nahm equation. Using solutions of the Nahm equation explicitly, we treat string junction configuration ending on the D3-branes from the viewpoint of the D-string. In particular, when two separated pronged strings end on the same D3-branes, they interact with each other through the D3-branes and the resultant trajectories of three-pronged strings are found to be curved non-trivially. 
  I propose to reformulate the gauge field theory as the perturbative deformation of a novel topological quantum field theory. It is shown that this reformulation leads to quark confinement in QCD$_4$. Similarly, the fractional charge confinement is also derived in the strong coupling phase of QED$_4$. As a confinement criterion, we use the area decay of the expectation value of the Wilson loop. 
  A four-dimensional N=2 supersymmetric non-linear sigma-model with the Eguchi-Hanson (ALE) target space and a non-vanishing central charge is rewritten to a classically equivalent and formally renormalizable gauged `linear' sigma-model over a non-compact coset space in N=2 harmonic superspace by making use of an N=2 vector gauge superfield as the Lagrange multiplier. It is then demonstrated that the N=2 vector gauge multiplet becomes dynamical after taking into account one-loop corrections due to quantized hypermultiplets. This implies the appearance of a composite gauge boson, a composite chiral spinor doublet and a composite complex Higgs particle, all defined as the physical states associated with the propagating N=2 vector gauge superfield. The composite N=2 vector multiplet is further identified with the zero modes of a superstring ending on a D-6-brane. Some non-perturbative phenomena, such as the gauge symmetry enhancement for coincident D-6-branes and the Maldacena conjecture, turn out to be closely related to our NLSM via M-theory. Our results support a conjecture about the composite nature of superstrings ending on D-branes. 
  As a continuation of our previous work on the multi-body forces of D-particles in supergravity and Matrix theory, we investigate the problem of motion. We show that the scattering of D-particles including recoil derived in Matrix theory is precisely reproduced by supergravity with the discrete light-cone prescription up to the second order in 11 dimensional Newton constant. An intimate connection of recoil and Galilei invariance in supergravity is pointed out and elucidated. 
  Static color charges in Yang-Mills theory are considered in the Schr\"odinger picture. Stationary states containing color sources, interquark potential and confinement criterion are discussed within the framework based on integration over gauge transformations which projects the vacuum wave functional on the space of physical, gauge-invariant states. 
  The light-cone little group, SO(9), classifies the massless degrees of freedom of eleven-dimensional supergravity, with a triplet of representations. We observe that this triplet generalizes to four-fold infinite families with the quantum numbers of massless higher spin states. Their mathematical structure stems from the three equivalent ways of embedding SO(9) into the exceptional group $F_4$. 
  We show that there is a sector of quantum general relativity which may be expressed in a completely holographic formulation in terms of states and operators defined on a finite boundary. The space of boundary states is built out of the conformal blocks of SU(2)_L + SU(2)_R, WZW field theory on the n-punctured sphere, where n is related to the area of the boundary. The Bekenstein bound is explicitly satisfied. These results are based on a new lagrangian and hamiltonian formulation of general relativity based on a constrained Sp(4) topological field theory. The hamiltonian formalism is polynomial, and also left-right symmetric. The quantization uses balanced SU(2)_L + SU(2)_R spin networks and so justifies the state sum model of Barrett and Crane. By extending the formalism to Osp(4/N) a holographic formulation of extended supergravity is obtained, as will be described in detail in a subsequent paper. 
  Work in progress is described which aims to construct a background independent formulation of M theory by extending results about background independent states and observables from quantum general relativity and supergravity to string theory. A list of principles for such a theory is proposed which is drawn from results of both string theory and background independent approaches to quantum gravity. Progress is reported on a background independent membrane field theory and on a realization of the holographic principle based on finite surfaces. 
  We use a variant of the classical Segal-Bargmann transform to understand the canonical quantization of Yang-Mills theory on a space-time cylinder. This transform gives a rigorous way to make sense of the Hamiltonian on the gauge-invariant subspace. Our results are a rigorous version of the widely accepted notion that on the gauge-invariant subspace the Hamiltonian should reduce to the Laplacian on the compact structure group. We show that the infinite-dimensional classical Segal-Bargmann transform for the space of connections, when restricted to the gauge-invariant subspace, becomes the generalized Segal-Bargmann transform for the the structure group. 
  We investigate linear combinations of characters for minimal Virasoro models which are representable as a products of several basic blocks. Our analysis is based on consideration of asymptotic behaviour of the characters in the quasi-classical limit. In particular, we introduce a notion of the secondary effective central charge. We find all possible cases for which factorization occurs on the base of the Gauss-Jacobi or the Watson identities. Exploiting these results, we establish various types of identities between different characters. In particular, we present several identities generalizing the Rogers-Ramanujan identities. Applications to quasi-particle representations, modular invariant partition functions, super-conformal theories and conformal models with boundaries are briefly discussed. 
  Cosmic string solutions of the abelian Higgs model with conformal coupling to gravity are shown to exist. The main characteristics of the solutions are presented and the differences with respect to the minimally coupled case are studied. An important difference is the absence of Bogomolnyi cosmic string solutions for conformal coupling. Several new features of the abelian Higgs cosmic strings of both types are discussed. The most interesting is perhaps a relation between the angular deficit and the central magnetic field which is bounded by a critical value. 
  The link between (super)-affine Lie algebras as Poisson brackets structures and integrable hierarchies provides both a classification and a tool for obtaining superintegrable hierarchies. The lack of a fully systematic procedure for constructing matrix-type Lax operators, which makes the supersymmetric case essentially different from the bosonic counterpart, is overcome via the notion of Poisson embeddings (P.E.), i.e. Poisson mappings relating affine structures to conformal structures (in their simplest version P.E. coincide with the Sugawara construction). A full class of hierarchies can be recovered by using uniquely Lie-algebraic notions. The group-algebraic properties implicit in the super-affine picture allow a systematic derivation of reduced hierarchies by imposing either coset conditions or hamiltonian constraints (or possibly both). 
  A version of the Wilson Renormalization Group Equation consistent with gauge symmetry is presented. A perturbative renormalizability proof is established. A wilsonian derivation of the Callan-Symanzik equation is given. 
  The canonical front form Hamiltonian for non-Abelian SU(N) gauge theory in 3+1 dimensions and in the light-cone gauge is mapped non-perturbatively on an effective Hamiltonian which acts only in the Fock space of a quark and an antiquark. Emphasis is put on the many-body aspects of gauge field theory, and it is shown explicitly how the higher Fock-space amplitudes can be retrieved self-consistently from solutions in the $q\bar q$-space. The approach is based on the novel method of iterated resolvents and on discretized light-cone quantization driven to the continuum limit. It is free of the usual perturbative Tamm-Dancoff truncations in particle number and coupling constant and respects all symmetries of the Lagrangian including covariance and gauge invariance. Approximations are done to the non-truncated formalism. Together with vertex as opposed to Fock-space regularization, the method allows to apply the renormalization programme non-perturbatively to a Hamiltonian. The conventional QCD scale is found arising from regulating the transversal momenta. It conspires with additional mass scales to produce possibly confinement. 
  We study the strong coupling limit of the 2-flavor lattice Schwinger model in the Hamiltonian formalism using staggered fermions. We show that the problem of finding the low-lying states is equivalent to solving the Heisenberg antiferromagnetic spin chain. We find good agreement with the continuum theory. 
  The study of fibrations of the target manifolds of string/M/F-theories has provided many insights to the dualities among these theories or even as a tool to build up dualities since the work of Strominger, Yau, and Zaslow on the Calabi-Yau case. For M-theory compactified on a Joyce manifold $M^7$, the fact that $M^7$ is constructed via a generalized Kummer construction on a 7-torus ${\smallBbb T}^7$ with a torsion-free $G_2$-structure $\phi$ suggests that there are natural fibrations of $M^7$ by ${\smallBbb T}^3$, ${\smallBbb T}^4$, and K3 surfaces in a way governed by $\phi$. The local picture of some of these fibrations and their roles in dualities between string/M-theory have been studied intensively in the work of Acharya. In this present work, we explain how one can understand their global and topological details in terms of bundles over orbifolds. After the essential background is provided in Sec. 1, we give general discussions in Sec. 2 about these fibrations, their generic and exceptional fibers, their monodromy, and the base orbifolds. Based on these, one obtains a 5-step-routine to understand the fibrations, which we illustrate by examples in Sec. 3. In Sec. 4, we turn to another kind of fibrations for Joyce manifolds, namely the fibrations by the Calabi-Yau threefolds constructed by Borcea and Voisin. All these fibrations arise freely and naturally from the work of Joyce. Understanding how the global structure of these fibrations may play roles in string/M-theory duality is one of the major issues for further pursuit. 
  A TQFT in terms of general gauge fixing functions is discussed. In a covariant gauge it yields the Donaldson-Witten TQFT. The theory is formulated on a generalized phase space where a simplectic structure is introduced. The Hamiltonian is expressed as the anticommutator of off-shell nilpotent BRST and anti-BRST charges. Following original ideas of Witten a time reversal operation is introduced and an inner product is defined in terms of it. A non-covariant gauge fixing is presented giving rise to a manifestly time reversal invariant Lagrangean and a positive definite Hamiltonian, with the inner product previously introduced. As a consequence, the indefiniteness problem of some of the kinetic terms of the Witten's action is resolved. The construction allows then a consistent interpretation of Floer groups in terms of the cohomology of the BRST charge which is explicitly independent of the background metric.  The relation between the BRST cohomology and the ground states of the Hamiltonian is then completely stablished. The topological theories arising from the covariant, Donaldson-Witten, and non-covariant gauge fixing are shown to be quantum equivalent by using the operatorial approach. 
  A free Rarita-Schwinger field in the Anti-de Sitter space is considered. We show that the usual action can be supplemented by a boundary term that can be interpreted as the generating functional of the correlation functions in a conformal field theory on the boundary of the Anti-de Sitter space. 
  The nature of the transition from quantum tunneling at low temperatures to thermal hopping at high temperatures is investigated in a scalar field theory with cubic symmetry breaking. The bounce solution which interpolates between the zero-temperature and high-temperature solutions is obtained numerically, using a multigrid method. It is found that, for a small value of the symmetry-breaking coupling f, the transition is first-order. For higher values of f, the transition continues to be first-order, though weakly so. 
  In the light of $\phi $--mapping method and topological current theory, the topological structure and the topological quantization of arbitrary dimensional topological defects are investigated. It is pointed out that the topological quantum numbers of the defects are described by the Winding numbers of $\phi $--mapping which are determined in terms of the Hopf indices and the Brouwer degrees of $\phi$--mapping. Furthermore, it is shown that all the topological defects are generated from where $\vec \phi =0$, i.e. from the zero points of the $\phi $--mapping. 
  The asymptotic expansion of the product of an operator raised to an arbitrary power and an exponential function of this operator is obtained. With the aid of this expansion, the density of vacuum energy induced by a static external magnetic field of an Abelian or non-Abelian nature is expressed in terms of the DeWitt-Seeley-Gilkey coefficients. 
  We use the background field method along with a special gauge condition, to derive the hard thermal loop effective action in a simple manner. The new point in the paper is to relate the effective action explicitly to the S-matrix from the onset. 
  In the light of $\phi $--mapping method and topological current theory, the topological structure of the vortex state in TDGL model and the topological quantization of the vortex topological charges are investigated. It is pointed out that the topological charges of the vortices in TDGL model are described by the Winding numbers of $\phi $--mapping which are determined in terms of the Hopf indices and the Brouwer degrees of $\phi $--mapping. 
  The renormalization group flow of the worldvolume theory depends very much from the number of unbroken supersymmetries. In the dual $AdS$ picture we break supersymmetry by adding different types of BPS black holes. We argue, that this BPS black hole causes a non-trivial renormalization group flow in the worldvolume field theory and especially a regular horizon translates into a non-trivial IR fixpoint. For this interpretation we have to rewrite the $AdS$ models into a flat space description with a linear dilaton vacuum. The dual models (linear dilaton and the $AdS$ vacuum) can be seen as the different sides of a domain wall. We discuss the cases of $AdS_3$ and $AdS_5$. 
  A nonlinear generalization of the Fluctuation-Dissipation Theorem (FDT) for the n-point Green functions and the amputated 1PI vertex functions at finite temperature is derived in the framework of the Closed Time Path formalism. We verify that this generalized FDT coincides with known results for n=2 and 3. New explicit relations among the 4-point nonlinear response and correlation (fluctuation) functions are presented. 
  In this article an attempt is made to present very recent conceptual and computational developments in QFT as new manifestations of old and well establihed physical principles. The vehicle for converting the quantum-algebraic aspects of local quantum physics into more classical geometric structures is the modular theory of Tomita. As the above named laureate to whom I have dedicated has shown together with his collaborator for the first time in sufficient generality, its use in physics goes through Einstein causality. This line of research recently gained momentum when it was realized that it is not only of structural and conceptual innovative power (see section 4), but also promises to be a new computational road into nonperturbative QFT (section 5) which, picturesquely speaking, enters the subject on the extreme opposite (noncommutative) side. 
  The string representation of the Abelian projected SU(3)-gluodynamics partition function is derived by using the path-integral duality transformation. On this basis, we also derive analogous representations for the generating functionals of correlators of gluonic field strength tensors and monopole currents, which are finally applied to the evaluation of the corresponding bilocal correlators. The large distance asymptotic behaviours of the latter turn out to be in a good agreement with existing lattice data and the Stochastic Model of the QCD vacuum. 
  We study the Dirac sea in the presence of external chiral and scalar/pseudoscalar potentials. In preparation, a method is developed for calculating the advanced and retarded Green's functions in an expansion around the light cone. For this, we first expand all Feynman diagrams and then explicitly sum up the perturbation series. The light-cone expansion expresses the Green's functions as an infinite sum of line integrals over the external potential and its partial derivatives.   The Dirac sea is decomposed into a causal and a non-causal contribution. The causal contribution has a light-cone expansion which is closely related to the light-cone expansion of the Green's functions; it describes the singular behavior of the Dirac sea in terms of nested line integrals along the light cone. The non-causal contribution, on the other hand, is, to every order in perturbation theory, a smooth function in position space. 
  We investigate non-Abelian gauge theories within a Wilsonian Renormalisation Group approach. The cut-off term inherent in this approach leads to a modified Ward identity (mWI). It is shown that this mWI is compatible with the flow and that the full effective action satisfies the usual Ward identity (WI). The universal 1-loop beta-function is derived within this approach and the extension to the 2-loop level is briefly outlined. 
  In the context of the AdS/CFT correspondence, an explicit relation between the physical degrees of freedom of 2+1d gravity and the stress tensor of 1+1d conformal field theory is exhibited. Gravity encodes thermodynamic state variables of conformal field theory, but does not distinguish among different CFT states with the same expectation value for the stress tensor. Simply put, gravity is thermodynamics; gauge theory is statistical mechanics. 
  We point out that two distinct distance--energy relations have been discussed in the AdS/CFT correspondence. In conformal backgrounds they differ only in normalization, but in nonconformal backgrounds they differ in functional form. We discuss the relation to probe processes, the holographic principle, and black hole entropies. 
  General Axial Gauges within a perturbative approach to QCD are plagued by 'spurious' propagator singularities. Their regularisation has to face major conceptual and technical problems. We show that this obstacle is naturally absent within a Wilsonian or 'Exact' Renormalisation Group approach and explain why this is so. The axial gauge turns out to be a fixed point under the flow, and the universal 1-loop running of the gauge coupling is computed. 
  In this paper a set of canonical collective variables is defined for a classical Klein-Gordon field and the problem of the definition of a set of canonical relative variables is discussed. This last point is approached by means of a harmonic analysis is momentum space. This analysis shows that the relative variables can be defined if certain conditions are fulfilled by the field configurations. These conditions are expressed by the vanishing of a set of conserved quantities, referred to as supertranslations since as canonical observables they generate a set of canonical transformations whose algebra is the same as that which arises in the study of the asymptotic behaviour of the metric of an isolated system in General Relativity. 
  The quarks of quark models cannot be identified with the quarks of the QCD Lagrangian. We review the restrictions that gauge field theories place on any description of physical (colour) charges. A method to construct charged particles is presented. The solutions are applied to a variety of applications. Their Green's functions are shown to be free of infra-red divergences to all orders in perturbation theory. The interquark potential is analysed and it is shown that the interaction responsible for anti-screening results from the force between two separately gauge invariant constituent quarks. A fundamental limit on the applicability of quark models is identified. 
  We consider the realization of affine ADE Lie algebras as string junctions on mutually non-local 7-branes in Type IIB string theory. The existence of the affine algebra is signaled by the presence of the imaginary root junction ``delta'', which is realized as a string encircling the 7-brane configuration. The level k of an affine representation partially constrains the asymptotic (p,q) charges of string junctions departing the configuration. The junction intersection form reproduces the full affine inner product, plus terms in the asymptotic charges. 
  AdS2 has an SL(2,R) isometry group. It is argued that in the context of quantum gravity on AdS2 this group is enlarged to the full infinite-dimensional 1+1 conformal group. Massive scalar fields are coupled to AdS2 gravity and shown to have associated conformal weights h(m) shifted by their mass. For integral values of h primary boundary operators are obtained as h normal derivatives of the scalar field. AdS2 string theories arise in the `very-near-horizon' limit of S1-compactified AdS3 string theories. This limit corresponds to energies far below the compactification scale. The dual conformal field theory has one null dimension and can in certain cases be described as the discrete light cone quantization of a two-dimensional deformed symmetric-product conformal field theory. Evidence is given that the AdS2 Virasoro algebra is related to the right-moving AdS3 Virasoro algebra by a topological twist which shifts the central charge to zero. 
  We show that the continuum limit of the integrable XYZ spin-1/2 chain on a half-line gives rise to the boundary sine-Gordon theory using the perturbation method. 
  We examine the claim of Chamblin et. al. that extreme black holes cannot support abelian Higgs hair. We provide evidence that contradicts this claim and discuss reasons for this discrepancy. 
  We study at finite temperature the energy-momentum tensor $T_{\mu\nu}(x)$ of (i) a scalar field in arbitrary dimension, and (ii) a spinor field in 1+1 dimensions, interacting with a static background electromagnetic field. $T_{\mu\nu}$ separates into an UV divergent part $T_{\mu\nu}^{sea}$ representing the virtual sea, and an UV finite part $T_{\mu\nu}^{plasma}$ describing the thermal plasma of the matter field. $T_{\mu\nu}^{sea}$ remains uniform in the presence of a \underbar{uniform} electric field $\vec E$, while $T_{\mu\nu}^{plasma}$ becomes a periodic function with period $\Delta x=2\pi T/E$ in the direction parallel to $\vec E$. A related periodicity is found for a uniform static magnetic field if one spatial direction perpendicular to the magnetic field is compactified to a circle. 
  The squared Laplace operator acting on symmetric rank-two tensor fields is studied on a (flat) Riemannian manifold with smooth boundary. Symmetry of this fourth-order elliptic operator is obtained provided that such tensor fields and their first (or second) normal derivatives are set to zero at the boundary. Strong ellipticity of the resulting boundary-value problems is also proved. Mixed boundary conditions are eventually studied which involve complementary projectors and tangential differential operators. In such a case, strong ellipticity is guaranteed if a pair of matrices are non-degenerate. These results find application to the analysis of quantum field theories on manifolds with boundary. 
  We study the free energy of \N=4 super Yang-Mills theory in the Higgs phase with a mass scale M corresponding to non-zero v.e.v. of the scalar fields. At zero temperature this theory describes a system of parallel separated extremal D3-branes. Non-zero temperature corresponds to non-extremality in supergravity description. We interpret the supergravity interaction potential between a non-extremal D3-brane and a D3-brane probe as contribution of massive states to the free energy of N --> infinity SYM theory at strong 't Hooft coupling (N g^2_{YM} >> 1). Both low (M >> T) and high (T >> M) temperature regimes are considered. For low temperature we find that the structure of terms that appear in the free energy at strong and weak coupling is the same. The analysis of the high-temperature regime depends on a careful identification of the scalar field v.e.v. in terms of the distance between branes in the supergravity description and again predicts strong coupling terms similar to those found in the weak-coupling \N=4 SYM theory. We consider also the corrections to the strong coupling results by taking into account the leading \a'^3 R^4 string contribution to the supergravity effective action. This gives rise to the (N g^2_{YM})^{-3/2} corrections in the coefficient functions of 't Hooft coupling which multiply different terms in free energy. 
  We consider aspects of the role of stringy scales and Hagedorn temperatures in the correspondence between various field theories and AdS-type spaces. The boundary theory is set on a toroidal world-volume to enable small scales to appear in the supergravity backgrounds also for low field-theory temperatures. We find that thermodynamical considerations tend to favour background manifolds with no string-size characteristic scales. The gravitational dynamics censors the reliable exposure of Hagedorn physics on the supergravity side, and the system does not allow the study of the Hagedorn scale by low-temperature field theories. These results are obtained following some heuristic assumptions on the character of stringy modifications to the gravitational backgrounds. A rich phenomenology appears on the supergravity side, with different string backgrounds dominating in different regions, which should have field-theoretic consequences. Six-dimensional world volumes turn out to be borderline cases from several points of view. For lower dimensional world-volumes, a fully holographic behaviour is exhibited to order 1/N^2, and open strings in their presence are found to have a thermodynamical Hagedorn behaviour similar to that of closed strings in flat space. 
  We give an overview of the correspondance between one-time-physics and two-time-physics. This is characterized by the presence of an SO(d,2) symmetry and an Sp(2) duality among diverse one-time-physics systems all of which can be lifted to the same more symmetric two-time-physics system by the addition of gauge degrees of freedom. We provide several explicit examples of physical systems that support this correspondance. The example of a particle moving in (AdS_D) X (S^n), with SO(D+n,2) symmetry which is larger than the popularly known symmetry SO(D-1,2) X SO(n+1) for this case, should be of special current interest in view of the proposed AdS-CFT duality. 
  It has been proposed that spacetimes with a U(1) isometry group have contributions to the entropy from Misner strings as well as from the area of $d-2$ dimensional fixed point sets. In this paper we test this proposal by constructing Taub-Nut-AdS and Taub-Bolt-AdS solutions which are examples of a new class of asymptotically locally anti-de Sitter spaces. We find that with the additional contribution from the Misner strings, we exactly reproduce the entropy calculated from the action by the usual thermodynamic relations. This entropy has the right parameter dependence to agree with the entropy of a conformal field theory on the boundary, which is a squashed three-sphere, at least in the limit of large squashing. However the conformal field theory and the normalisation of the entropy remain to be determined. 
  We consider the Chern-Simons parameter shift with the hybrid regularization consisting of the higher covariant derivative (HCD) and the Pauli-Villars (PV) regulators. We show that the shift is closely related to the parity of the regulators and get the shift and no-shift results by a suitable choice of the PV regulators. A naive treatment of the HCD term leads incorrect value of the shift. 
  We show that there are solutions of the SU(2) Yang-Mills classical equations of motion in R^4, which are self-dual and vortex-like(fluxons). The action density is concentrated along a thick two-dimensional wall (the world sheet of a straight infinite vortex line). The configurations are constructed from self-dual R^2 x T^2 configurations. 
  A general functional definition of the infinite dimensional quantum R-matrix satisfying the Yang-Baxter equation is given. A procedure for extracting a finite dimensional R-matrix from the general definition is demonstrated for the particular cases of the group O(2) and of the group of translations. 
  This work is intended as a pedagogical introduction to M-theory and to its maximally supersymmetric toroidal compactifications, in the frameworks of 11D supergravity, type II string theory and M(atrix) theory. U-duality is used as the main tool and guideline in uncovering the spectrum of BPS states. We review the 11D supergravity algebra and elementary 1/2-BPS solutions, discuss T-duality in the perturbative and non-perturbative sectors from an algebraic point of view, and apply the same tools to the analysis of U-duality at the level of the effective action and the BPS spectrum, with a particular emphasis on Weyl and Borel generators. We derive the U-duality multiplets of BPS particles and strings, U-duality invariant mass formulae for 1/2- and 1/4-BPS states for general toroidal compactifications on skew tori with gauge backgrounds, and U-duality multiplets of constraints for states to preserve a given fraction of supersymmetry. A number of mysterious states are encountered in D<=3, whose existence is implied by T-duality and 11D Lorentz invariance. We then move to the M(atrix) theory point of view, give an introduction to Discrete Light Cone Quantization (DLCQ) in general and DLCQ of M-theory in particular. We discuss the realization of U-duality as electric-magnetic dualities of the Matrix gauge theory, display the Matrix gauge theory BPS spectrum in detail, and discuss the conjectured extended U-duality group in this scheme. 
  Many Euclidean Einstein manifolds possess continuous symmetry groups of at least one parameter and we consider here a classification scheme of $d$ dimensional compact manifolds based on the existence of such a one parameter group in terms of the fixed point sets of the isometries. We discuss applications of such a classification scheme, including the geometric interpretation of the entropy; there are intrinsic contributions to the entropy from the volumes of $(d-2)$ dimensional fixed point sets and contributions related to the cohomology structure of the orbit space of the isometry. We consider the relevance of such a decomposition of the entropy in the context of the no boundary proposal and cosmological processes, and generalise the discussion to compact solutions of gravity coupled to scalar and gauge fields. 
  We discuss the action of a circle isometry group on non compact Euclidean Einstein manifolds. We discuss approaches to a decomposition of the action and entropy for non compact manifolds in terms of the characteristics of the orbit space of a suitable isometry. There is entropy associated with non trivial cohomology of the orbit space of the isometry, and we consider a class of non compact solutions for which such contributions do not vanish. To obtain suitable solutions we generalise the Bais-Batenburg construction of higher dimensional Taub-Nut type solutions to obtain the corresponding bolt solutions. We consider the generalisations to non compact solutions of gravity coupled to scalar and gauge fields. 
  we present a new topological invariant to describe the space-time defect which is closely related to torsion tensor in Riemann-Cartan manifold. By virtue of the topological current theory and $\phi$-mapping method, we show that there must exist many strings objects generated from the zero points of $\phi$-mapping, and these strings are topological quantized and the topological quantum numbers is the Winding numbers described by the Hopf indices and the Brouwer degrees of the $\phi$-mapping. 
  We look at the vertical dimensional reduction of the supermembrane of M-theory to the D2-brane of Type IIA string theory. Our approach considers the soliton solutions of the two low energy field limits, D=11 and D=10 Type IIA supergravities, rather than the worldvolume actions. It is thus necessary to create a periodic array. The standard Kaluza-Klein procedure requires that the brane is smeared over a transverse direction, but we will keep the dependence on the compactification coordinate, seeing how the eleventh dimension comes into play when we close up on the D2-brane. 
  In this contribution some aspects of supergravity and super Yang-Mills systems in D=6 are briefly reviewed and, in some cases, are contrasted with the analogous features in D=4. Particular emphasis is laid on the stringy solutions of the D=6 super Yang-Mills systems. 
  The world-volume action of the M2 brane and the M5-brane in an adS_4 x S^7 and a adS_7 x S^4 background is derived to all orders in anticommuting superspace coordinates. Contrary to recent constructions of super p-brane actions relying on supercoset methods, we only use 11 dimensional supergravity torsion and curvature constraints. Complete agreement between the two methods is found. The simplification of the actions by choosing a suitable kappa-gauge is discussed. 
  {\it Perturbiner}, that is, the solution of field equations which is a generating function for tree form-factors in N=3 $(N=4)$ supersymmetric Yang-Mills theory, is studied in the framework of twistor formulation of the N=3 superfield equations. In the case, when all one-particle asymptotic states belong to the same type of N=3 supermultiplets (without any restriction on kinematics), the solution is described very explicitly. It happens to be a natural supersymmetrization of the self-dual perturbiner in non-supersymmetric Yang-Mills theory, designed to describe the Parke-Taylor amplitudes. In the general case, we reduce the problem to a neatly formulated algebraic geometry problem (see Eqs(\ref{5.15i}),(\ref{5.15ii}),(\ref{5.15iii})) and propose an iterative algorithm for solving it, however we have not been able to find a closed-form solution. Solution of this problem would, of course, produce a description of all tree form-factors in non-supersymmetric Yang-Mills theory as well. In this context, the N=3 superfield formalism may be considered as a convenient way to describe a solution of the non-supersymmetric Yang-Mills theory, very much in the spirit of works by E.Witten \cite{Witten} and by J.Isenberg, P.B.Yasskin and P.S.Green \cite{2}. 
  A discussion is made of the strategy to check dual superconductivity of the vacuum as a mechanism of colour confinement. Recent evidence from Lattice is reviewed. 
  A reformulation of the fluctuation-dissipation theorem of Callen and Welton is presented in such a manner that the basic idea of Feynman-Vernon and Caldeira -Leggett of using an infinite number of oscillators to simulate the dissipative medium is realized manifestly without actually introducing oscillators. If one assumes the existence of a well defined dissipative coefficient $R(\omega)$ which little depends on the temperature in the energy region we are interested in, the spontanous and induced emissions as well as induced absorption of these effective oscillators with correct Bose distribution automatically appears.   Combined with a dispersion relation, we reproduce the tunneling formula in the presence of dissipation at finite temperature without referring to an explicit model Lagrangian. The fluctuation-dissipation theorem of Callen-Welton is also generalized to the fermionic dissipation (or fluctuation) which allows a transparent physical interpretation in terms of second quantized fermionic oscillators. This fermionic version of fluctuation-dissipation theorem may become relevant in the analyses of, for example, fermion radiation from a black hole and also supersymmetry at the early universe. 
  It is shown that in general the energy ${\cal E}$ and the Hamiltonian ${\cal H}$ of matter fields on the black hole exterior play different roles. ${\cal H}$ is a generator of the time evolution along the Killing time while ${\cal E}$ enters the first law of black hole thermodynamics. For non-minimally coupled fields the difference ${\cal H}-{\cal E}$ is not zero and is a Noether charge $Q$ analogous to that introduced by Wald to define the black hole entropy. If fields vanish at the spatial boundary, $Q$ is reduced to an integral over the horizon. The analysis is carried out and an explicit expression for $Q$ is found for general diffeomorphism invariant theories. As an extension of the results by Wald et al, the first law of black hole thermodynamics is derived for arbitrary weak matter fields. 
  When dealing with zeta-function regularized functional determinants of matrix valued differential operators, an additional term, overlooked until now and due to the multiplicative anomaly, may arise. The presence and physical relevance of this term is discussed in the case of a charged bosonic field at finite charge density and other possible applications are mentioned. 
  The quantization of the massless Thirring model in the light-cone using functional methods is considered. The need to compactify the coordinate $x^-$ in the light-cone spacetime implies that the quantum effective action for left-handed fermions contains excitations similar to abelian instantons produced by composite of left-handed fermions. Right-handed fermions don't have a similar effective action. Thus, quantum mechanically, chiral symmetry must be broken as a result of the topological excitations. The conserved charge associated to the topological states is quantized. Different cases with only fermionic excitations or bosonic excitations or both can occur depending on the boundary conditions and the value of the coupling. 
  For a class of generalized integrable hierarchies associated with affine (twisted or untwisted) Kac-Moody algebras, an explicit representation of their local conserved densities by means of a single scalar tau-function is deduced. This tau-function acts as a partition function for the conserved densities, which fits its potential interpretation as the effective action of some quantum system. The class consists of multi-component generalizations of the Drinfel'd-Sokolov and the two-dimensional affine Toda lattice hierarchies. The relationship between the former and the approach of Feigin, Frenkel and Enriquez to soliton equations of KdV and mKdV type is also discussed. These results considerably simplify the calculation of the conserved charges carried by the soliton solutions to the equations of the hierarchy, which is important to establish their interpretation as particles. By way of illustration, we calculate the charges carried by a set of constrained KP solitons recently constructed. 
  The issue discussed is a thermodynamic version of the Bern-Kosower master amplitude formula, which contains all necessary one-loop Feynman diagrams. It is demonstrated how the master amplitude at finite values of temperature and chemical potential can be formulated within the framework of the world-line formalism. In particular we present an elegant method how to introduce a chemical potential for a loop in the master formula. Various useful integral formulae for the master amplitude are then obtained. The non-analytic property of the master formula is also derived in the zero temperature limit with the value of chemical potential kept finite. 
  We show that the attempt to introduce all of the discrete space-time transformations into the spinor representation of the Lorentz group as wholly independent transformations (as in the vectorial representation) leads to an 8-component spinor representation in general. The first indications seem to imply that CPT can be violated in this formulation without going outside of field theory. However one needs further study to reach a final conclusion. 
  We study Maldacena's conjecture and the AdS/SYM correspondence on the Coulomb branch. Several interesting aspects of this conjectured AdS/SYM correspondence on the Coulomb branch are pointed out and clarified. 
  In a new approach to the theory of integration over Wess-Zumino supermanifolds, we suggest that a fundamental principle is their consistency with an ``Ethereal Conjecture'' that asserts the topology of the supermanifold must be generated essentially from its bosonic submanifold. This naturally leads to a theory of ``ectoplasmic'' integration based on super p-forms. One consequence of this approach is that the derivation of ``density projection operators'' becomes trivial in a number of supergravity theories. 
  In this article old and new relations between gauge fields and strings are discussed. We add new arguments that the Yang Mills theories must be described by the non-critical strings in the five dimensional curved space. The physical meaning of the fifth dimension is that of the renormalization scale represented by the Liouville field. We analyze the meaning of the zigzag symmetry and show that it is likely to be present if there is a minimal supersymmetry on the world sheet. We also present the new string backgrounds which may be relevant for the description of the ordinary bosonic Yang-Mills theories. The article is written on the occasion of the 40-th anniversary of the IHES. 
  The XYZ spin chain with boundaries is studied. We construct the vacuum state by the vertex operators in the level one modules of the elliptic algebra, and compact it through a geometric symmetry of the model called the turning symmetry. From this simplified expression the ``finite size formula'' for magnetizations in the bounded and in the half-infinite chains is deduced. Applying this formula we calculate the spontaneous magnetization in the bounded XYZ model. 
  We point out an interesting analogy between the BTZ black hole and QHE (Quantum Hall effect) in the (2+1)-dimensional bulk/boundary theories. It is shown that the Chern-Simons/Liouville(Chern-Simons/chiral boson) is an effective description for the BTZ black hole (QHE). Also the IR(bulk)-UV(boundary) connection for a black hole information bound is realized as the UV(low-lying excitations on bulk)-IR(long-range excitations on boundary) connection in the QHE. An inflow of conformal anomaly($c=1$ central charge) onto the timelike boundary of AdS$_3$ by the Noether current corresponds to an inflow of chiral anomaly onto the edge of disk by the Hall current. 
  On the occasion of the 25th anniversary of Asymptotic Freedom, celebrated at the QCD Euroconference 98 on Quantum Chromodynamics, Montpellier, July 1998, I described the discovery of Asymptotic Freedom and the emergence of QCD. 
  Making combined use of the Matrix and Maldacena conjectures, the relation between various thermodynamic transitions in super Yang-Mills (SYM) and supergravity is clarified. The thermodynamic phase diagram of an object in DLCQ M-theory in four and five non-compact space dimensions is constructed; matrix strings, matrix black holes, and black $p$-branes are among the various phases. Critical manifolds are characterized by the principles of correspondence and longitudinal localization, and a triple point is identified. The microscopic dynamics of the Matrix string near two of the transitions is studied; we identify a signature of black hole formation from SYM physics. 
  The Aharonov Bohm scattering for spinless, isospin 1/2, particles interacting through a nonabelian Chern-Simons field is studied. Starting from the relativistic quantum field theory and using a Coulomb gauge formulation, the one loop renormalization program is implemented. Through the introduction of an intermediary cutoff, separating the regions of high and low integration momentum, the nonrelativistic limit is derived. The next to leading relativistic approximation is also determined. In this approach quantum field theory vacuum polarization effects are automatically incorporated. 
  We study the system of D6+D0 branes at sub-stringy scale. We show that the proper description of the system, for large background field associated with the D0-branes, is via spinning chargeless black holes in five dimensions. The repulsive force between the D6-branes and the D0-branes is understood through the centrifugal barrier. We discuss the implication on the stability of the D6+D0 solution. 
  The topic of 4D, N = 1 supersymmetry is introduced for the reader with a prior background in relativistic quantum field theory. The presentation is designed to be a useful primer for those who plan to later engage in serious investigation of the area or as an overview for the generally interested. 
  We present a new 1/8 supersymmetric intersecting M-brane solution of D=11 supergravity with two independent rotation parameters. The metric has a non-singular event horizon and the near-horizon geometry is $adS_3\times S^3\times S^3\times\bE^2$ (just as in the non-rotating case). We also present a method of determining the isometry supergroup of supergravity solutions from the Killing spinors and use it to show that for the near horizon solution it is $D(2|1,\alpha)\times D(2|1,\alpha)$ where $\alpha$ is the ratio of the two 3-sphere radii. We also consider various dimensional reductions of our solution, and the corresponding effect of these reductions on the Killing spinors and the isometry supergroups. 
  An antiferromagnetic S=1/2 Heisenberg chain is mapped to the two-flavor massless Schwinger model at \theta=\pi. The electromagnetic coupling constant and velocity of light in the Schwinger model are determined in terms of the Heisenberg coupling and lattice spacing in the spin chain system. 
  In a string calculation to order $\alpha'^3$, we compute an eight-derivative four-dilaton term in the type IIB effective action. Following the AdS prescription, we compute the order $(g_{YM}^2N_c)^{-3/2}$ correction to the four-point correlation function involving the operator $tr F^2$ in four dimensional N=4 super Yang-Mills using the string corrected type IIB action extending the work of Freedman et al. (hep-th/9808006). In the limit where two of the Yang-Mills operators approach each other, we find that our correction to the four-point correlation functions develops a logarithmic singularity. We discuss the possible cancellation of this logarithmic singularities by conjecturing new terms in the type IIB effective action. 
  Universal Lax pairs (the root type and the minimal type) are presented for Calogero-Moser models based on simply laced root systems, including E_8. They exist with and without spectral parameter and they work for all of the four choices of potentials: the rational, trigonometric, hyperbolic and elliptic. For the elliptic potential, the discrete symmetries of the simply laced models, originating from the automorphism of the extended Dynkin diagrams, are combined with the periodicity of the potential to derive a class of Calogero-Moser models known as the `twisted non-simply laced models'. For untwisted non-simply laced models, two kinds of root type Lax pairs (based on long roots and short roots) are derived which contain independent coupling constants for the long and short roots. The BC_n model contains three independent couplings, for the long, middle and short roots. The G_2 model based on long roots exhibits a new feature which deserves further study. 
  These lecture notes give an introduction to the algebraic renormalization of the Standard Model. We start with the construction of the tree approximation and give the classical action and its defining symmetries in functional form. These are the Slavnov-Taylor identity, Ward identities of rigid symmetry and the abelian local Ward identity. The abelian Ward identity ensures coupling of the electromagnetic current in higher orders of perturbation theory, and is the functional form of the Gell-Mann-Nishijima relation. In the second part of the lectures we present in simple examples the basic properties of renormalized perturbation theory: scheme dependence of counterterms and the quantum action principle. Together with an algebraic characterization of the defining symmetry transformations they are the ingredients for a scheme independent unique construction of Green's functions to all orders of perturbation theory. 
  We employ the LSZ reduction formula for Matrix Theory introduced in our earlier work to compute the t-pole S-matrix for three form-three form scattering. The result agrees completely with tree level D=11 SUGRA. Taken together with previous results on graviton-graviton scattering this shows that Matrix Theory indeed reproduces the bosonic sector of the D=11 SUGRA action including the Chern-Simons term. Furthermore we provide a detailed account of our framework along with the technology to compute any Matrix Theory one-loop t-pole scattering amplitude at vanishing p^- exchange. 
  Gauge invariant conservation laws for the linear and angular momenta are studied in a certain 2+1 dimensional first order dynamical model of vortices in superconductivity. In analogy with fluid vortices it is possible to express the linear and angular momenta as low moments of vorticity. The conservation laws are compared with those obtained in the moduli space approximation for vortex dynamics. 
  Complex monopole configurations dominate in the path integral in the Georgi-Glashow-Chern-Simons model and disorder the Higgs vacuum. No cancellation is expected among Gribov copies of the monopole configurations. 
  We present briefly the deformation philosophy and indicate, with references, how it was applied to the quantization of Nambu mechanics and to particle physics in anti De Sitter space. 
  We apply the method of angular quantization to calculation of the wave function renormali- zation constants in $D_{l}^{(1)}$ affine Toda quantum field theories. A general formula for the wave function renormalization constants in ADE Toda field theories is proposed. We also calculate all one-particle form factors and some of the two-particle form factors of an exponential field. 
  We discuss non-relativistic scattering by a Newtonian potential. We show that the gray-body factors associated with scattering by a black hole exhibit the same functional dependence as scattering amplitudes in the Newtonian limit, which should be the weak-field limit of any quantum theory of gravity. This behavior arises independently of the presence of supersymmetry. The connection to two-dimensional conformal field theory is also discussed. 
  We present a supersymmetric and $\kappa$-symmetric D-string action on $AdS_5 \times S^5$ in supercoset construction. As in the previous work of the super D-string action in the flat background, the super D-string action on $AdS_5 \times S^5$ can be transformed to a form of the IIB Green-Schwarz superstring action with the $SL(2,Z)$ covariant tension on $AdS_5 \times S^5$ through a duality transformation. In order to understand a part of the duality transformation as SO(2) rotation of N=2 spinor coordinates, it seems to be necessary to fix the $\kappa$-symmetry in a gauge condition which simplifies the classical action. This is the article showing for the first time that there exists S-duality in type IIB superstring theory in a curved background whose validity has been conjectured in the past but not shown so far in an explicit way. 
  We give a definition and study Hopf structures in ternary (and n-ary) Nambu-Lie algebra. The fundamental concepts of 3-coalgebra, 3-bialgebra and Hopf 3- algebra are introduced. Some examples of Hopf structures are analyzed. 
  W_4 gravity is treated algebraically, represented by a set of transformations on classical fields. The Ward identities of the theory are determined by requiring the algebra to close. The general forms for the anomalies are found by looking for solutions to the Wess-Zumino consistency conditions, and some specific cases are considered. 
  We introduce a systematic approach for treating the large N limit of matrix field theories. 
  We study bosonisation in the massive Thirring and sine-Gordon models at finite temperature and nonzero fermion chemical potential. Both canonical operator and path integral approaches are used to prove the equality of the partition functions of the two models at finite $T$ and zero chemical potential, as it has been recently shown. This enables the relationship between thermal normal ordering and path-integral renormalisation to be specified. Furthermore, we prove that thermal averages of zero-charge operators can also be identified. At nonzero chemical potential and temperature we show, in perturbation theory around the massless case, that the bosonised theory is the sine-Gordon model plus an additional topological term, accounting for the existence of zero charge excitations (the fermions or the kinks) in the thermal bath. This result is the 2D version of the low-energy lagrangian at finite baryon density. 
  We study the problem of a Dirac field in the background of an Aharonov-Bohm flux string. We exclude the origin by imposing spectral boundary conditions at a finite radius then shrinked to zero. Thus, we obtain a behaviour of eigenfunctions which is compatible with the self-adjointness of the radial Hamiltonian and the invariance under integer translations of the reduced flux. After confining the theory to a finite region, we check the consistency with the index theorem, and evaluate its vacuum fermionic number and Casimir energy. 
  We describe in superspace a classical theory of two dimensional $(1,1)$ dilaton supergravity with a cosmological constant, both with and without coupling to a massive superparticle. We give general exact non-trivial superspace solutions for the compensator superfield that describes the supergravity in both cases. We then use these compensator solutions to construct models of two-dimensional supersymmetric cosmological black holes. 
  We examine the relation between the damping rate of a chiral fermion mode propagating in a hot plasma and the rate at which the mode approaches equilibrium. We show how these two quantities, obtained from the imaginary part of the fermion self-energy, are equal when the reaction rate is defined using the appropriate wave function of the mode in the medium. As an application, we compute the production rate of hard axions by Compton-like scattering processes in a hot QED plasma starting from both, the axion self-energy and the electron self-energy. We show that the latter rate coincides with the former only when this is computed using the corresponding medium spinor modes. 
  I review some work done in the past four years concerning the transition of Yang-Mills theories from 1+3 to 1+1 dimensions. The problem is considered both in a perturbative context and in exact solutions when available. Several interesting features are discussed, mainly in relation to the phenomenon of confinement, and some controversial issues are clarified. 
  We introduce a new class of duality symmetries amongst quantum field theories. The new class is based upon global spacetime symmetries, such as Poincare invariance and supersymmetry, in the same way as the existing duality transformations are based on global internal symmetries.  We illustrate these new duality transformations by dualizing several scalar and spin-half theories in 1+1 spacetime dimensions, involving nonsupersymmetric as well as (1,1) and (2,2) supersymmetric models. For  (2,2) models the new duality transformations can interchange chiral and twisted-chiral multiplets. 
  Three topics, the self-consistent resummation of the perturbative expansion for thermal Yang-Mills (YM) theory, nonperturbative analysis of Yang-Mills theories in (2+1) dimensions and modification of the Bose distribution for gluons, are discussed. These topics are related to the magnetic mass or magnetic screening in the quark-gluon plasma. The value of the magnetic mass is argued to be close to $e^2N/2\pi=g^2TN/2\pi$ for an SU(N) gauge theory. An analytic calculation of the string tension for $YM_{2+1}$, which agrees with numerical simulations to within 3%, is another result. 
  Using a gauge-invariant matrix parametrization of the gauge fields, we present an analysis of how the mass gap arises in (2+1)-dimensional Yang-Mills theory. We further derive an analytical continuum expression for the vacuum wavefunction and based on this we calculate the string tension which is in excellent agreement with Monte Carlo simulations of the corresponding lattice gauge theory. 
  For the kappa-symmetric super IIA D-brane action by the canonical approach we construct an equivalent effective action which is characterized by an auxiliary scalar field. By analyzing the canonical equations of motion for the kappa-symmetry-gauge-fixed action we find a suitable conformal-like covariant gauge fixing of reparametrization symmetry to obtain a simplified effective action where the non-linear square root structure is removed. We discuss how the two effective actions are connected. 
  We consider the nilpotent additions to classical trajectories in supersymmetric and nonsupersymmetric theories. The condition of anilpotence of action on some generalized solutions leads to the Witten supersymmetric Lagrangian. The condition of anilpotence of topological charge is the same as one of superpotential with spontaneous broken supersymmetry. We should vanish half of Grassmann constants of integration, because in this case only we obtain the same number of normalized bosonic and fermionic zero modes. 
  We determine the structure of geometrical superconformal anomalies for N=1 supersymmetric quantum field theories on curved superspace to all orders in h bar. For the massless Wess-Zumino model we show how these anomalies contribute to the local Callan-Symanzik equation which expresses the breakdown of superconformal symmetry in terms of the usual beta and gamma functions. 
  We present the so-called Liouville-Neumann (LN) approach to nonequilibrium quantum fields. The LN approach unifies the functional Schr\"{o}dinger equation and the LN equation for time-independent or time-dependent quantum systems and for equilibrium or nonequilibrium quantum systems. The LN approach is nonperturbative in that at the lowest order of coupling constant it gives the same results as those of the Gaussian effective potential at the zero and finite temperature in a Minkowski spacetime. We study a self-interacting quantum field in an expanding Friedmann-Robertson-Walker Universe. By studying a toy model of anharmonic oscillator and finding the underlying algebraic structure we propose a scheme to go beyond the Gaussian approximation. 
  Using the path-integral bosonization procedure at Finite Temperature we study the equivalence between a massive Thirring model with non-local interaction between currents (NLMT) and a non-local extension of the sine-Gordon theory (NLSG). To this end we make a perturbative expansion in the mass parameter of the NLMT model and in the cosine term of the NLSG theory in order to obtain explicit expressions for the corresponding partition functions. We conclude that for certain relationship between NLMT and NLSG potentials both the fermionic and bosonic expansions are equal term by term. This result constitutes a generalization of Coleman's equivalence at T=0, when considering a Thirring model with bilocal potentials in the interaction term at Finite Temperature. The study of this model is relevant in connection with the physics of strongly correlated systems in one spatial dimension. Indeed, in the language of many-body non-relativistic systems, the relativistic mass term can be shown to represent the introduction of backward-scattering effects. 
  We consider classical Teichmuller theory and the geodesic flow on the cotangent bundle of the Teichmuller space. We show that the corresponding orbits provide a canonical description of certain (2+1) gravity systems in which a set of point-like particles evolve in universes with topology S_g x R where S_g is a Riemann surface of genus g >1. We construct an explicit York's slicing presentation of the associated spacetimes, we give an interpretation of the asymptotic states in terms of measured foliations and discuss the structure of the phase spaces. 
  We show that the mean-field time dependent equations in the Phi^4 theory can be put into a classical non-canonical hamiltonian framework with a Poisson structure which is a generalization of the standard Poisson bracket. The Heisenberg invariant appears as a structural invariant of the Poisson tensor. (To be pubished in Annals of Physics) 
  We quantize pure 2d Yang-Mills theory on a torus in the gauge where the field strength is diagonal. Because of the topological obstructions to a global smooth diagonalization, we find string-like states in the spectrum similar to the ones introduced by various authors in Matrix string theory. We write explicitly the partition function, which generalizes the one already known in the literature, and we discuss the role of these states in preserving modular invariance. Some speculations are presented about the interpretation of 2d Yang-Mills theory as a Matrix string theory. 
  We study the operator product expansion (OPE) of the auxiliary scalar field \lambda(x) with itself, in the conformally invariant O(N) Vector Model for 2<d<4, to leading order in 1/N in a strip-like geometry with one finite dimension of length L. We show that consistency of the finite-geometry OPE with bulk OPE calculations requires the physical conditions of, either finite-size scaling at criticality, or finite-temperature phase transition. 
  The classification of rational conformal field theories is reconsidered from the standpoint of boundary conditions. Solving Cardy's equation expressing the consistency condition on a cylinder is equivalent to finding integer valued representations of the fusion algebra. A complete solution not only yields the admissible boundary conditions but also gives valuable information on the bulk properties. 
  The relativistic complex scalar field at finite temperature and in presence of a net conserved charge is studied in reference to recent developments on the multiplicative anomaly. This quantity, overlooked until now, is computed and it is shown how it could play a role for this system. Other possible applications are also mentioned. 
  We compute the quadratic part of the thermal effective action for real scalar fields which are initially in thermal equilibrium and vary slowly in time using a generalised real-time formalism proposed by Le Bellac and Mabilat \cite{belmab}. We derive both Real Time and Imaginary Time Formalisms and find that the result is analytic at the limits of zero external four-momenta when using our full time contour. We expand the fields in time up to the second derivative and discuss the initial time dependence of our result before and after the expansion in terms of equilibrium. 
  We examine in detail various string scattering amplitudes in order to extract the world-volume interactions of massless fields on a Dirichlet brane. We find that the leading low-energy interactions are consistent with the Born-Infeld and Chern-Simons actions. In particular, our results confirm that the background closed string fields appearing in these actions must be treated as functionals of the non-abelian scalar fields describing transverse fluctuations of the D-brane. 
  We simplify and generalize an approach proposed by Di Vecchia and Ravndal to describe a massive Dirac particle in external vector and scalar fields. Two different path integral representations for the propagator are derived systematically without the usual five-dimensional extension and shown to be equivalent due to the supersymmetry of the action. They correspond to a projection on the mass of the particle either continuously or at the end of the time evolution. It is shown that the supersymmetry transformations are generated by shifting and scaling the supertimes and the invariant difference of two supertimes is given for the general case. A nonrelativistic reduction of the relativistic propagator leads to a three-dimensional path integral with the usual Pauli Hamiltonian. By integrating out the photons we obtain the effective action for quenched QED and use it to derive the gauge-transformation properties of the general Green function of the theory. 
  The construction of a $covariant$ Loop Wave functional equation in a 4D spacetime is attained by introducing a generalized $eleven$ dimensional categorical {\bf C}-space comprised of $8\times 8$ antisymmetric matrices. The latter matrices encode the generalized coordinates of the histories of points, loops and surfaces $combined$. Spacetime Topology change and the Holographic principle are natural consequences of imposing the principle of $covariance$ in {\bf C}-space. The Planck length is introduced as a necessary rescaling parameter to establish the correspondence limit with the physics of point-histories in ordinary Minkowski space, in the limit $l_P\to 0$. Spacetime quantization should appear in discrete units of Planck length, area, volume ,....All this seems to suggest that the generalized principle of covariance, representing invariance of proper $area$ intervals in {\bf C}-space, under matrix-coordinate transformations, could be relevant in discovering the underlying principle behind the origins of $M$ theory. We construct an ansatz for the $SU(\infty)$ Yang-Mills vacuum wavefunctional as a solution of the Schroedinger Loop Wave equation associated with the Loop Quantum Mechanical formulation of the Eguchi-Schild String . The Strings/Loops ($SU(\infty)$ gauge field) correspondence implements one form of the Bulk/Boundary duality conjecture in this case. 
  In these lectures, we review the D=11 supermembrane and supersymmetric matrix models at an introductory level. We also discuss some more recent developments in connection with non-perturbative string theory. 
  The first quantized theory of N=2, D=3 massive superparticles with arbitrary fixed central charge and (half)integer or fractional superspin is constructed. The quantum states are realized on the fields carrying a finite dimensional, or a unitary infinite dimensional representation of the supergroups OSp(2|2) or SU(1,1|2). The construction originates from quantization of a classical model of the superparticle we suggest. The physical phase space of the classical superparticle is embedded in a symplectic superspace $T^\ast({R}^{1,2})\times{L}^{1|2}$, where the inner K\"ahler supermanifold ${L}^{1|2}=OSp(2|2)/[U(1)\times U(1)]=SU(1,1|2)/[U(2|2)\times U(1)]$ provides the particle with superspin degrees of freedom. We find the relationship between Hamiltonian generators of the global Poincar\'e supersymmetry and the ``internal'' SU(1,1|2) one. Quantization of the superparticle combines the Berezin quantization on ${L}^{1|2}$ and the conventional Dirac quantization with respect to space-time degrees of freedom. Surprisingly, to retain the supersymmetry, quantum corrections are required for the classical N=2 supercharges as compared to the conventional Berezin method. These corrections are derived and the Berezin correspondence principle for ${L}^{1|2}$ underlying their origin is verified. The model admits a smooth contraction to the N=1 supersymmetry in the BPS limit. 
  We discuss the semi-classical perturbation spectra produced in the massless fields of the low energy string action in a pre big bang type scenario. Axion fields may possess an almost scale-invariant spectrum on large scales dependent upon the evolution of the dilaton and moduli fields to which they are coupled. As an example we calculate the spectra for three axion fields present in a truncated type IIB model and show that they are related with at least one of the fields having a scale-invariant or red perturbation spectrum. In the simplest pre big bang scenario this may be inconsistent with the observed isotropy of the microwave background. More generally, relations between the perturbation spectra in low energy string cosmologies reflect the symmetries of the theory. 
  We study dynamical aspects of large N QCD_4 via supergravity on Anti de Sitter black hole geometry. We compute the mass spectrum, the topological susceptibility and the gluon condensate and make a comparison to lattice simulations. We present analogous results for QCD_3. 
  In this paper we find the general (i.e. valid for arbitrary values of the winding number) form of the gauge zero-modes, in the adjoint representation, for theories living on manifolds of the ALE type. 
  We use the harmonic space technique to construct explicitly a quaternionic extension of the Taub-NUT metric. It depends on two parameters, the first being the Taub-NUT `mass' and the second one the cosmological constant. 
  We propose a new "Hamiltonian inspired" covariant formula to define (without harmful ambiguities) the superpotential and the physical charges associated to a gauge symmetry. The criterion requires the variation of the Noether current not to contain any derivative terms in $\partial_{\mu}\delta \f$.   The examples of Yang-Mills (in its first order formulation) and 3-dimensional Chern-Simons theories are revisited and the corresponding charge algebras (with their central extensions in the Chern-Simons case) are computed in a straightforward way.   We then generalize the previous results to any (2n+1)-dimensional non-abelian Chern-Simons theory for a particular choice of boundary conditions. We compute explicitly the superpotential associated to the non-abelian gauge symmetry which is nothing but the Chern-Simons Lagrangian in (2n-1) dimensions. The corresponding charge algebra is also computed. However, no associated central charge is found for $n \geq 2$.   Finally, we treat the abelian p-form Chern-Simons theory in a similar way. 
  We present a novel procedure for calculating non-equilibrium two-point Green's functions in the $O(N) \phi^{4}$ theory at large $N$. The non-equilibrium density matrix $\rho$ is constructed via the Jaynes-Gibbs principle of maximal entropy and it is directly implemented into the Dyson-Schwinger equations through initial value conditions. In the large $N$ limit we perform an explicit evaluation of two-point Green's functions for two illustrative choices of $\rho$. 
  In a previous paper (hep-th/9808141) we showed that the type I string theory contains a stable non-BPS D-particle carrying SO(32) spinor charge. In this paper we formulate the rules for computing the spectrum and interaction of open strings with one or both ends lying on this D-particle. 
  We show that the chiral Gross-Neveu model in $2+ \epsilon$ dimensions has for a small number $N$ of fermions two phase transitions corresponding to pair formation and pair condensation.   In the first transition, fermions and antifermions acquire spontaneously a mass and are bound to pairs which behave like a Bose liquid in a chirally symmetric state. In the second transition, the Bose liquid condenses into a coherent state which breaks chiral symmetry. This suggests the possibility that in particle physics, the generation of quark masses may also happen separately from the breakdown of chiral symmetry. 
  We present a geometric formulation of type-IIA and -IIB superstring theories in which the Wess-Zumino term is second order in the supersymmetric currents. The currents are constructed using supergroup manifolds corresponding to superalgebras: the IIA superalgebra derived from M-algebra and the IIB superalgebra obtained by a T-duality transformation of the IIA superalgebra. We find that a slight modification of the IIB superalgebra is needed to describe D-string theories, in which the U(1) gauge field on the worldsheet is explicitly constructed in terms of D-string charges. A unification of the superalgebras in a (10+1)-dimensional N=2 superalgebra is discussed too. 
  The coherent-state path-integral representation for the propagator of fermionic systems subjected to first-class constraints is constructed. As in the bosonic case the importance of path-integral measures for Lagrange multipliers is emphasized. One example is discussed in some detail. 
  A path integral representation is given for the solutions of the 3+1 dimensional Dirac equation. The regularity of the trajectories, the non-relativistic limit and the semiclassical approximation are briefly mentioned. 
  A c-number path integral representation is constructed for the solution of the Dirac equation. The integration is over the real trajectories in the continuous three-space and other two canonical pairs of compact variables controlling the spin and the chirality flips. 
  Quantum field theory is constructed upon the assumption of stabilities of the vacuum and of the one-particle state. For finite temperature, the one-particle state becomes unstable because of thermal fluctuations, whereas the thermal vacuum is still stable. In non-equilibrium situation, both the vacuum and the one-particle state lose their stability. Proposed is the introduction of the {\it reference vacuum} which takes care of thermal and non-adiabatic time-evolution of a system, and produces a time-dependent Fock's representation space. This may provide us with an extension of the concept of {\it dynamical mapping} where a migration among unitary inequivalent representation spaces can be handled for non-equilibrium and dissipative systems. 
  The light-cone Hamiltonian approach is applied to the super D2-brane, and the equivalent area--preserving and U(1) gauge-invariant effective Lagrangian, which is quadratic in the U(1) gauge field, is derived. The latter is recognised to be that of the three-dimensional U(1) gauge theory, interacting with matter supermultiplets, in a special external induced supergravity metric and the gravitino field, depending on matter fields. The duality between this theory and 11d supermembrane theory is demonstrated in the light-cone gauge. 
  The projector onto gauge invariant physical states was recently constructed for arbitrary constrained systems. This approach, which does not require gauge fixing nor any additional degrees of freedom beyond the original ones---two characteristic features of all other available methods for quantising constrained dynamics---is put to work in the context of a general class of quantum mechanical gauge invariant systems. The cases of SO(2) and SO(3) gauge groups are considered specifically, and a comprehensive understanding of the corresponding physical spectra is achieved in a straightforward manner, using only standard methods of coherent states and group theory which are directly amenable to generalisation to other Lie algebras. Results extend by far the few examples available in the literature from much more subtle and delicate analyses implying gauge fixing and the characterization of modular space. 
  Some results from arguments of research dealt with R. Raczka are exposed and extended. In particular new arguments are brought in favor of the conjecture, formulated with him, that both space-time and momentum may be conformally compactified, building up a compact phase space of automorphism for the conformal group, where conformal reflections determine a convolution between space-time and momentum space which may have consequences of interest for both classical and quantum physics. 
  A manifestly N=2 supersymmetric completion of the four-dimensional Nambu-Goto-Born-Infeld action, which is self-dual with respect to electric-magnetic duality, is constructed in terms of the abelian N=2 superfield strength W in the conventional N=2 superspace. A relation to the known N=1 supersymmetric Born-Infeld action in N=1 superspace is established. The action found can be considered either as the Goldstone action associated with partial breaking of N=4 supersymmetry down to N=2, with the N=2 vector superfield being a Goldstone field, or, equivalently, as the gauge-fixed superfield action of a D-3-brane in flat six-dimensional ambient spacetime. 
  We discuss string theories with small numbers of non-compact moduli and describe constructions of string theories whose low-energy limit is described by various pure supergravity theories. We also construct a D=4,N=4 compactification of type II string theory with 34 vector fields. 
  We present several Galileo invariant Lagrangians, which are invariant against Poincare transformations defined in one higher (spatial) dimension. Thus these models, which arise in a variety of physical situations, provide a representation for a dynamical (hidden) Poincare symmetry. The action of this symmetry transformation on the dynamical variables is nonlinear, and in one case involves a peculiar field-dependent diffeomorphism. Some of our models are completely integrable, and we exhibit explicit solutions. 
  A new framework for solving the hierarchy problem was recently proposed which does not rely on low energy supersymmetry or technicolor. The fundamental Planck mass is at a $\tev$ and the observed weakness of gravity at long distances is due the existence of new sub-millimeter spatial dimensions. In this picture the standard model fields are localized to a $(3+1)$-dimensional wall or ``3-brane''. The hierarchy problem becomes isomorphic to the problem of the largeness of the extra dimensions. This is in turn inextricably linked to the cosmological constant problem, suggesting the possibility of a common solution. The radii of the extra dimensions must be prevented from both expanding to too great a size, and collapsing to the fundamental Planck length $\tev^{-1}$. In this paper we propose a number of mechanisms addressing this question. We argue that a positive bulk cosmological constant $\bar\Lambda$ can stabilize the internal manifold against expansion, and that the value of $\bar\Lambda$ is not unstable to radiative corrections provided that the supersymmetries of string theory are broken by dynamics on our 3-brane. We further argue that the extra dimensions can be stabilized against collapse in a phenomenologically successful way by either of two methods: 1) Large, topologically conserved quantum numbers associated with higher-form bulk U(1) gauge fields, such as the naturally occurring Ramond-Ramond gauge fields, or the winding number of bulk scalar fields. 2) The brane-lattice-crystallization of a large number of 3-branes in the bulk. These mechanisms are consistent with theoretical, laboratory, and cosmological considerations such as the absence of large time variations in Newton's constant during and after primordial nucleosynthesis, and millimeter-scale tests of gravity. 
  The Quantum Stationary HJ Equation (QSHJE) that we derived from the equivalence principle, gives rise to initial conditions which cannot be seen in the Schroedinger equation. Existence of the classical limit leads to a dependence of the integration constant $\ell=\ell_1+i\ell_2$ on the Planck length. Solutions of the QSHJE provide a trajectory representation of quantum mechanics which, unlike Bohm's theory, has a non-trivial action even for bound states and no wave guide is present. The quantum potential turns out to be an intrinsic potential energy of the particle which, similarly to the relativistic rest energy, is never vanishing. 
  A basic aspect of the recently proposed approach to quantum mechanics is that no use of any axiomatic interpretation of the wave function is made. In particular, the quantum potential turns out to be an intrinsic potential energy of the particle, which, similarly to the relativistic rest energy, is never vanishing. This is related to the tunnel effect, a consequence of the fact that the conjugate momentum field is real even in the classically forbidden regions. The quantum stationary Hamilton-Jacobi equation is defined only if the ratio psi^D/psi of two real linearly independent solutions of the Schroedinger equation, and therefore of the trivializing map, is a local homeomorphism of the extended real line into itself, a consequence of the Moebius symmetry of the Schwarzian derivative. In this respect we prove a basic theorem relating the request of continuity at spatial infinity of psi^D/psi, a consequence of the q - 1/q duality of the Schwarzian derivative, to the existence of L^2(R) solutions of the corresponding Schroedinger equation. As a result, while in the conventional approach one needs the Schroedinger equation with the L^2(R) condition, consequence of the axiomatic interpretation of the wave function, the equivalence principle by itself implies a dynamical equation that does not need any assumption and reproduces both the tunnel effect and energy quantization. 
  The Equivalence Principle (EP), stating that all physical systems are connected by a coordinate transformation to the free one with vanishing energy, univocally leads to the Quantum Stationary HJ Equation (QSHJE). Trajectories depend on the Planck length through hidden variables which arise as initial conditions. The formulation has manifest p-q duality, a consequence of the involutive nature of the Legendre transform and of its recently observed relation with second-order linear differential equations. This reflects in an intrinsic psi^D-psi duality between linearly independent solutions of the Schroedinger equation. Unlike Bohm's theory, there is a non-trivial action even for bound states. No use of any axiomatic interpretation of the wave-function is made. Tunnelling is a direct consequence of the quantum potential which differs from the usual one and plays the role of particle's self-energy. The QSHJE is defined only if the ratio psi^D/psi is a local self-homeomorphism of the extended real line. This is an important feature as the L^2 condition, which in the usual formulation is a consequence of the axiomatic interpretation of the wave-function, directly follows as a basic theorem which only uses the geometrical gluing conditions of psi^D/psi at q=\pm\infty as implied by the EP. As a result, the EP itself implies a dynamical equation that does not require any further assumption and reproduces both tunnelling and energy quantization. Several features of the formulation show how the Copenhagen interpretation hides the underlying nature of QM. Finally, the non-stationary higher dimensional quantum HJ equation and the relativistic extension are derived. 
  We present a 3D approximation of the three-fermion Bethe-Salpeter equation. Our 3D equation is covariantly cluster separable and the two-fermion cluster separated limits are exact equivalents of the corresponding two-fermion Bethe-Salpeter equations. The potentials include positive free energy projectors in order to avoid continuum dissolution. 
  The evaluation of the 4-point Green functions in the 1+1 Schwinger model is presented both in momentum and coordinate space representations. The crucial role in our calculations play two Ward identities: i) the standard one, and ii) the chiral one. We demonstrate how the infinite set of Dyson-Schwinger equations is simplified, and is so reduced, that a given n-point Green function is expressed only through itself and lower ones. For the 4-point Green function, with two bosonic and two fermionic external `legs', a compact solution is given both in momentum and coordinate space representations. For the 4-fermion Green function a selfconsistent equation is written down in the momentum representation and a concrete solution is given in the coordinate space. This exact solution is further analyzed and we show that it contains a pole corresponding to the Schwinger boson. All detailed considerations given for various 4-point Green functions are easily generizable to higher functions. 
  We evaluate quasi-classically the Ramond partition function of Euclidean D=10 U(N) super-Yang--Mills theory reduced to a two-dimensional torus. The result can be interpreted in terms of free strings wrapping the space-time torus, as expected from the point of view of Matrix string theory. We demonstrate that, when extrapolated to the ultraviolet limit (small area of the torus), the quasi-classical expressions reproduce exactly the recently obtained expression for the partition of the completely reduced SYM theory, including the overall numerical factor. This is an evidence that our quasi-classical calculation might be exact. 
  We write a 3D equation for three fermions by combining the three two-body potentials obtained by the reductions of the corresponding two-fermion Bethe-Salpeter equations to equivalent 3D equations, putting the spectator fermion on the mass shell. In this way, the cluster-separated limits are still exact, and the Lorentz invariance / cluster separability requirement is automatically satisfied, provided no supplementary approximation, like the Born approximation, is made. The use of positive free-energy projectors in the chosen reductions of the two-fermion Bethe-Salpeter equations prevents continuum dissolution in our 3D three-fermion equation. The potentials are hermitian and depend only slowly on the total three-fermion energy. The one high-mass limits are approximately exact. In view of a possible perturbation calculation, correcting the remaining discrepancies with the three-fermion Bethe-Salpeter equation, we succeeded in deriving our 3D equation from an approximation of the three-fermion Bethe-Salpeter equation, in which the three-body kernel is neglected and the two-body kernels approached by positive-energy instantaneous expressions, with the spectator fermion on the mass shell. The neglected terms are transformed into corrections to the 3D equation. A comparison is made with Gross' spectator model. 
  The generating functional of correlation functions of the currents corresponding to general massless $p$-form potential is calculated in $AdS/CFT$ correspondence of Maldacena. For this we construct the boundary-to-bulk Green's functions of $p$-form potentials. The proportional constant of the current-current correlation function, which is related to the central charge of the operator product expansion, is shown to be $c=(d-p\over 2\pi^{d/2}) (\Gamma (d-p) \over \Gamma ({d\over 2}-p)).$ The result agrees with the known cases such as $p=1$ or 2. 
  We compute the M theory corrections to the confining linear potential between a quark and an anti-quark in N=1 Super Yang-Mills theory. We find a constant term, and a term exponentially small with characteristic length of $\Lambda_{QCD}^{-1}$. The potential in the MQCD setup that corresponds to softly broken N=1 SYM is found to have a similar behavior. 
  The quantization of charged matter system coupled to Chern-Simons gauge fields is analyzed in a covariant gauge fixing, and gauge invariant physical anyon operators satisfying fractional statistics are constructed in a symmetric phase, based on Dirac's recipe performed on QED. This method provides us a definite way of identifying physical spectrums free from gauge ambiguity and constructing physical anyon operators under a covariant gauge fixing. We then analyze the statistical spin phase transition in a symmetry-broken phase and show that the Higgs mechanism transmutes an anyon satisfying fractional statistics into a canonical boson, a spin 0 Higgs boson or a topologically massive photon. 
  A canonical action describing the interaction of chiral gauge fields in D=6 Minkowski space-time is constructed. In a particular partial gauge fixing it reduces to the action found by Perry and Schwarz. The additional gauge symmetries are used to show the off-shell equivalence of the dimensional reduction to D=5 Minkowski space-time of the chiral gauge field canonical action and the Born-Infeld canonical action describing an interacting D=5 Abelian vector field. Its extension to improve the on-shell equivalence arguments of dual D-brane actions to off-shell ones is discussed. 
  We study the two-fold dimensional dependence of the electromagnetic duality groups. We introduce the dual projection operation that systematically discloses the presence of an internal space of potentials where the group operation is defined. A two-fold property of the kernel in the projection is shown to define the dimensional dependence of the duality groups. The dual projection is then generalized to reveal another hidden two-dimensional structure. The new unifying concept of the external duality space remove the dimensional dependence of the kernel, allowing the presence of both $Z_2$ and SO(2) duality groups in all even dimensions. This result, ultimately unifies the notion of selfduality to all D=2k+2 dimensions. Finally, we show the presence of an unexpected duality between the internal and external spaces leading to a duality of the duality groups. 
  We present extremal stationary solutions that generalize the Israel-Wilson-Perjes class for the d+3-dimensional low-energy limit of heterotic string theory with n >= d+1 U(1) gauge fields compactified on a d-torus. A rotating axisymmetric dyonic solution is obtained using the matrix Ernst potential formulation and expressed in terms of a single d+1 X d+1-matrix harmonic function. By studying the asymptotic behaviour of the field configurations we define the physical charges of the field system. The extremality condition makes the charges to saturate the Bogomol'nyi-Prasad-Sommerfield (BPS) bound. The gyromagnetic ratios of the corresponding field configurations appear to have arbitrary values. A subclass of rotating dyonic black hole-type solutions arises when the NUT charges are set to zero. In the particular case d=1, n=6, which correspond to N=4, D=4 supergravity, the found dyon reproduces the supersymmetric dyonic solution constructed by Bergshoeff et al. 
  If constituent quarks are to emerge from QCD they must have well defined colour and be energetically favoured. After reviewing the general properties of charges in gauge theories, a method for constructing charges is presented and applied to the infra-red structure of the theory and to the interquark potential. Both of these applications supply a physical interpretation of the structures found in the construction of charges. We will see that constituent structures arise in QCD. 
  We construct super AdS black holes in 2+1 dimensions in terms of Chern Simons gauge theory of N=(1,1) super AdS group coupled to a (super)source. We take the source to be a super AdS state specified by its Casimir invariants. We show that the corresponding space-time is a supermultiplet of AdS space-times related to each other by supersymmetry transformations. We give explicit expressions for the masses and the angular momenta of the black holes in a supermultiplet. With one exception, for N=(1,1) one pair of extremal black holes can be accommodated in such all-black hole supermultiplets. The requirement that the source be a unitary representation leads to a discrete tower of excited states which provide a microscopic model for the super black hole. 
  We give a new integrable boundary condition in affine Toda theory which is soliton-preserving in the sense that a soliton hitting the boundary is reflected as a soliton. All previously known integrable boundary conditions forced a soliton to be converted into an antisoliton upon reflection. We prove integrability of our boundary condition using a generalization of Sklyanin's formalism. 
  The auxiliary field method, defined through introducing an auxiliary (also called as the Hubbard-Stratonovich or the Mean-) field and utilizing a loop-expansion, gives an excellent result for a wide range of a coupling constant. The analysis is made for Anharmonic-Oscillator and Double-Well examples in 0-(a simple integral) and 1-(quantum mechanics)dimension. It is shown that the result becomes more and more accurate by taking a higher loop into account in a weak coupling region, however, it is not the case in a strong coupling region. The 2-loop approximation is shown to be still insufficient for the Double-Well case in quantum mechanics. 
  In this paper we study three point functions of the Type II superstring involving one graviton and two massive states, focusing in particular on the spin-7/2 fermions at the first mass level. Defining a gravitational quadrupole ``h-factor'', we find that the non-minimal interactions of string states in general are parametrized by $h\ne1$, in contrast to the preferred field theory value of h=1 (for tree-level unitarity). This difference arises from the fact that consistent gravitational interactions of strings are related to the presence of a complete tower of massive states, not present in the ordinary field theory case. 
  The technique of Hamiltonian flow equations is applied to the canonical Hamiltonian of quantum electrodynamics in the front form and 3+1 dimensions.   The aim is to generate a bound state equation in a quantum field theory, particularly to derive an effective Hamiltonian which is practically solvable in Fock-spaces with reduced particle number. The effective Hamiltonian, obtained as a solution of flow eqautions to the second order, is solved numerically for positronium spectrum. The impact of different similarity functions is explicitly studied.   The approach discussed can ultimately be used to address to the same problem for quantum chromodynamics. 
  We investigate the propagation of arbitrarily coupled scalar fields on the $N$-dimensional hyperbolic space ${\mathbb H}^N$. Using the $\zeta$-function regularization we compute exactly the one loop effective action. The vacuum expectation value of quadratic field fluctuations and the one loop renormalized stress tensor are then computed using the recently proposed direct $\zeta$-function technique. Our computation tests the validity of this approach in presence of a continuous spectrum. Our results apply as well to the $N$-dimensional anti-de Sitter spacetime, whose appropriate euclidean section is the hyperbolic space ${\mathbb H}^N$. 
  We describe the type IIB supergravity background on AdS_{3} x S_{3} x T^{4} using the potentials of AdS_{3|4} x S_{3} x T^{4} and we use the supersolvable algebra associated to AdS_{3|4} to compute the kappa gauge fixed type IIB string action. From the explicit form of the action we can clearly see how passing from pure NSNS backgrounds to pure RR backgrounds the WZW term disappears. 
  The exceptional superalgebra $D(2,1;\alpha)$ has been classified as a candidate conformal supersymmetry algebra in two dimensions. We propose an alternative interpretation of it as extended BFV-BRST quantisation superalgebras in 2D ($D(2,1;1) \simeq osp(2,2|2)$). A superfield realization is presented wherein the standard extended phase space coordinates can be identified. The physical states are studied via the cohomology of the BRST operator. It is conjectured that the underlying model giving rise to this `quantisation' is that of a scalar relativistic particle in 1+1 dimensions, for which the light cone coordinates $x_R$, $x_L$ transform under worldline diffeomorphisms as scalar densities of appropriate weight. 
  In string theory, the brane world scenario, where the Standard Model gauge and matter fields live inside some branes while gravitons live in the bulk, can be a viable description of our universe. In this note we argue that the brane world actually is a likely description of nature. Our discussion includes a revisit of certain issues, namely, coupling unification, dilaton stabilization and supersymmetry breaking, in the context of the brane world scenario. In particular, we discuss various possible string scenarios and their phenomenological implications in the brane world framework. 
  The definition of the spectral action involves the trace operator over states in the physical Hilbert space. We show that in the presence of chiral fermions there are consistency conditions on the fermionic representations. These conditions are identical to the conditions for absence of gauge and gravitational anomalies obtained in the path integral formalism. 
  In the preceding paper(hep-th/9806084), we constructed submodels of nonlinear Grassmann sigma models in any dimension and, moreover, an infinite number of conserved currents and a wide class of exact solutions.   In this paper, we first construct almost all conserved currents for the submodels and all ones for the one of ${\bf C}P^1$-model. We next review the Smirnov and Sobolev construction for the equations of ${\bf C}P^1$-submodel and extend the equations, the S-S construction and conserved currents to the higher order ones. 
  We introduce a consistent gauge extension of the SL(2,R) WZNW system, defined by a difference of two simple WZNW actions. By integrating out some dynamical variables in the functional integral, we show that the resulting effective theory coincides with the induced gravity in 2D. General solutions of both theories are found and related to each other. 
  We find the general solution of the equations of motion for the WZNW system in curved space-time for arbitrary external gauge fields. Using the connection between the WZNW system for $SL(2,R)$ group and 2D induced gravity we obtain the general solution of the equations of motion for 2D induced gravity in curved space-time from that of the WZNW system. We independently presented the direct solution of 2D induced gravity equations of motion and obtain the same result. 
  We develop a constructive method to derive exactly solvable quantum mechanical models of rational (Calogero) and trigonometric (Sutherland) type. This method starts from a linear algebra problem: finding eigenvectors of triangular finite matrices. These eigenvectors are transcribed into eigenfunctions of a selfadjoint Schr\"odinger operator. We prove the feasibility of our method by constructing a new "$AG_3$ model" of trigonometric type (the rational case was known before from Wolfes 1975). Applying a Coxeter group analysis we prove its equivalence with the $B_3$ model. In order to better understand features of our construction we exhibit the $F_4$ rational model with our method. 
  We present some consequences of non-anomalous propagation requirements on various massless fields. Among the models of nonlinear electrodynamics we show that only Maxwell and Born-Infeld also obey duality invariance. Separately we show that, for actions depending only on the $F_\mn^2$ invariant, the permitted models have $L \sim \sqrt{1 + F^2}$. We also characterize acceptable vector-scalar systems. Finally we find that wide classes of gravity models share with Einstein the null nature of their characteristic surfaces. 
  We report the results of a very high statistics Monte Carlo study of the continuum limit of the two dimensional O(3) non-linear $\sigma$ model. We find a significant discrepancy between the continuum extrapolation of our data and the form factor prediction of Balog and Niedermaier, inspired by the Zamolodchikovs' S-matrix ansatz. On the other hand our results for the O(3) and the dodecahedron model are consistent with our earlier finding that the two models possess the same continuum limit. 
  Although the whole conformal group $SO(4,2)$ can be considered as a symmetry in a classical massless field theory, the subgroup of special conformal transformations (SCT), usually related to transitions to uniformly accelerated frames, causes vacuum radiation in the corresponding quantum field theory, in analogy to the Fulling-Unruh effect. The spectrum of the outgoing particles can be calculated exactly and proves to be a generalization of the Planckian one. 
  This paper reviews some recent work on (s)pin structures and the Dirac operator on hypersurfaces (in particular, on spheres), on real projective spaces and quadrics. Two approaches to spinor fields on manifolds are compared. The action of space and time reflections on spinors is discussed, also for two-component (chiral) spinors. 
  A fermionic analogue of the Hodge star operation is shown to have an explicit operator representation in models with fermions, in spacetimes of any dimension. This operator realizes a conjugation (pairing) not used explicitly in field-theory, and induces a metric in the space of wave-function(al)s just as in exterior calculus. If made real (Hermitian), this induced metric turns out to be identical to the standard one constructed using Hermitian conjugation; the utility of the induced complex bilinear form remains unclear. 
  In this note we show that rigid N=2 superconformal hypermultiplets must have target manifolds which are cones over tri-Sasakian metrics. We comment on the relation of this work to cone-branes and the AdS/CFT correspondence. 
  By globally embedding curved spaces into higher dimensional flat ones, we show that Hawking thermal properties map into their Unruh equivalents: The relevant curved space detectors become Rindler ones, whose temperature and entropy reproduce the originals. Specific illustrations include Schwarzschild, Schwarzschild-(anti)deSitter, Reissner-Nordstrom and BTZ spaces. 
  We present an ansatz for all one-loop amplitudes in pure Einstein gravity for which the n external gravitons have the same outgoing helicity. These loop amplitudes, which are rational functions of the momenta, also arise in the quantization of self-dual gravity in four-dimensional Minkowski space. Our ansatz agrees with explicit computations via D-dimensional unitarity cuts for n less than or equal to 6. It also has the expected analytic behavior, for all n, as a graviton becomes soft, and as two momenta become collinear. The gravity results are closely related to analogous amplitudes in (self-dual) Yang-Mills theory. 
  We describe the application of methods from the study of discrete dynanmical systems to the problem of the continuum limit of evolving spin networks. These have been found to describe the small scale structure of quantum general relativity and extensions of them have been conjectured to give background independent formulations of string theory. We explain why the the usual equilibrium critical phenomena may not be relevant for the problem of the continuum limit of such theories and why, instead, the continuum limits of such theories are likely to be governed by non-equilibrium critical phenomena such as directed percolation. The fact that such non-equilibrium critical phenomena can be self-organized implies the possibility that the classical limit of quantum theories of gravity may exist without fine tuning of parameters. 
  We show that three-dimensional N=8 $Sp(N)$ and $SO(2N+1)$ gauge theories flow to the same strong coupling fixed point. As a consequence, the corresponding orientifold two-planes in type IIA string theory are described at strong coupling and low-energies by the same M theory background. In the large $N$ limit, these assertions are confirmed by studying discrete torsion in the supergravity theory corresponding to membranes on $R^8/Z_2$. 
  We discuss nontrivial examples illustrating that perturbative gravity is in some sense the `square' of gauge theory. This statement can be made precise at tree-level using the Kawai, Lewellen and Tye relations between open and closed string tree amplitudes. These relations, when combined with modern methods for computing amplitudes, allow us to obtain loop-level relations, and thereby new supergravity loop amplitudes. The amplitudes show that N=8 supergravity is less ultraviolet divergent than previously thought. As a different application, we show that the collinear splitting amplitudes of gravity are essentially squares of the corresponding ones in QCD. 
  We derive the classical kappa-symmetric Type IIB string action on AdS(3) x S(3) by employing the SU(1,1|2)^2 algebra. We then gauge fix kappa-symmetry in the background adapted Killing spinor gauge and present the action in a very simple form. 
  In recent years a supersymmetric form of discrete light-cone quantization (hereafter `SDLCQ') has emerged as a very powerful tool for solving supersymmetric field theories. In this scheme, one calculates the light-cone supercharge with respect to a discretized light-cone Fock basis, instead of working with the light-cone Hamiltonian. This procedure has the advantage of preserving supersymmetry even in the discretized theory, and eliminates the need for explicit renormalizations in 1+1 dimensions. In order to compare the usual DLCQ prescription with the supersymmetric prescription, we consider two dimensional SU(N) Yang-Mills theory coupled to a massive adjoint Majorana fermion, which is known to be supersymmetric at a particular value of the fermion mass. After studying how singular-valued amplitudes and intermediate zero momentum modes are regularized in both schemes, we are able to establish a precise connection between conventional DLCQ and its supersymmetric extension, SDLCQ. In particular, we derive the explicit form of the (irrelevant) interaction that renders the DLCQ formulation of the theory exactly supersymmetric for any light-cone compactification. We check our analytical results via a numerical procedure, and discuss the relevance of this interaction when supersymmetry is explicitly broken. 
  I compute the cohomology of a non-commutative complex underlying the notion of the gauge field on the fuzzy sphere. 
  The Geroch-Wald-Jang-Huisken-Ilmanen approach to the positive energy problem to may be extended to give a negative lower bound for the mass of asymptotically Anti-de-Sitter spacetimes containing horizons with exotic topologies having ends or infinities of the form $\Sigma_g \times {\Bbb R}$, in terms of the cosmological constant. We also show how the method gives a lower bound for for the mass of time-symmetric initial data sets for black holes with vectors and scalars in terms of the mass, $|Z(Q,P)|$ of the double extreme black hole with the same charges. I also give a lower bound for the area of an apparent horizon, and hence a lower bound for the entropy in terms of the same function $|Z(Q,P)|$. This shows that the so-called attractor behaviour extends beyond the static spherically symmetric case. and underscores the general importance of the function $|Z(Q,P)|$. There are hints that higher dimensional generalizations may involve the Yamabe conjectures. 
  General 2d dilaton theories, containing spherically symmetric gravity and hence the Schwarzschild black hole as a special case, are quantized by an exact path integral of their geometric (Cartan-) variables. Matter, represented by minimally coupled massless scalar fields is treated in terms of a systematic perturbation theory. The crucial prerequisite for our approach is the use of a temporal gauge for the spin connection and for light cone components of the zweibeine which amounts to an Eddington Finkelstein gauge for the metric. We derive the generating functional in its most general form which allows a perturbation theory in the scalar fields. The relation of the zero order functional to the classical solution is established. As an example we derive the effective (gravitationally) induced 4-vertex for scalar fields. 
  In this paper, the dependence of the Einstein gravity with the cosmological constant as well as of this theory in the first-order formalism on the gauge and parametrization is been analyzed. The one-loop counterterms off the mass shell have been plainly calculated in arbitrary gauge and parametrization. The tensor package of analytic calculations, written in REDUCE, allowed all the calculations to be carried out. A method of renormalization group functions calculations off shell is discussed. 
  An instantaneous temperature path of a point particle in the space - temperature manifold can be represented as a string of length L=1/kT (thermostring). The thermostring swepts a surface in the space-time-temperature manyfold at its temporal evolution. The thermostring is closed, its points can be rearranged and the charge is distributed along the length. Some consequences of this method for statistical mechanics and string theories are discussed. 
  In this talk we will study the partial breaking of supersymmetry in flat and anti de Sitter space. We will see that partial breaking in flat space can be accomplished using either of two representations for the massive N=1 spin-3/2 multiplet. We will "unHiggs" each representation and find a new N=2 supergravity and a new N=2 supersymmetry algebra. We will also see that partial supersymmetry breaking in AdS space can give rise to a new N=2 supersymmetry algebra, one that is necessarily nonlinearly realized. 
  We resolve the entropy problem in the AdS$_3$/CFT correspondence by introducing both the normalizable and non-normalizable bulk modes. On the boundary, the normalizable Liouville states gives us $c=1$ conformal field theory(CFT), whereas the non-normalizable Liouville states provide $c = {3 \ell \over 2 G}$ CFT. Such (boundary) non-normalizable modes come from non-normalizable bulk modes which serve as classical, non-fluctuating bulk background and encode the choice of local operator insertion on the boundary. Since the non-normalizable bulk modes can transfer information from bulk to boundary, it suggests that counting of non-normalizable states on the boundary at infinity leads to the entropy($S={2 \pi r_+ \over 4 G}$) of (2+1)-dimensional gravity with $ \Lambda= -1/\ell^2$. 
  We use the fivebrane of M theory to study the $\theta$ dependence of four dimensional $SU(N_c)$ super Yang-Mills and super QCD softly broken by a gaugino mass. We compute the energy of the vacuum in the supergravity approximation. The results obtained are in qualitative agreement with field theory. We also study the $\theta$ dependence of the QCD string tension via the fivebrane. 
  We consider the classical solution of the type IIB supergravity spontaneously compactified on S^5, whose metric depends only on the radial coordinate and whose asymptotic geometry is locally that of AdS_5, i.e., R \times S^1 \times T^2. We solve the equations of motion to obtain the general solutions satisfying these conditions, and find that the only naked-singularity-free solutions are the AdS black holes and AdS solitons. The other solutions, that smoothly interpolate these two solutions, are shown to have naked singularities even though their Ricci tensor is proportional to the metric with a negative constant. Thus, among the possible solutions of this type, the AdS solitons are the unique lowest energy solution; this result is consistent with the recently proposed positive energy conjecture for the IIB AdS supergravity on S^5. 
  We obtain the characteristic equation for the nonlinear Born-Infeld electrodynamics. This equation has the form of the characteristic equation for the linear electrodynamics in some effective Riemann space. The effective metric include the energy-momentum tensor components of electromagnetic field. We study a distortion of light beams by the action of some distant solitons. This distortion corresponds to attraction with the solitons and looks like the gravitational distortion. 
  String dynamics near the photon sphere in Schwarzschild spacetime is considered on the basis of a perturbative approach with respect to a rescaled string tension as a small parameter. The perturbative string solution in the zeroth and first approximation is presented. The perturbative solution corresponds to a small deformations of the photon sphere in Schwarzschild spacetime. 
  In this paper, we give a simple diagrammatic identification of the unique combination of the causal n-point vertex functions in the real time formalism that would coincide with the corresponding functions obtained in the imaginary time formalism. Furthermore, we give a simple calculational method for evaluating the temperature dependent parts of the retarded vertex functions, to one loop, by identifying them with the forward scattering amplitudes of on-shell thermal particles. As an application of the method, we calculate and show that the temperature dependent parts of all the retarded functions vanish at one loop order for 1+1 dimensional massless QED. We further point out that, in this model, in fact, the temperature dependent parts of all the retarded vertex functions vanish to all orders in perturbation theory. 
  We study the quantum integrability of the O(N) nonlinear $\sigma$ (nls) model and the O(N) Gross-Neveu (GN) model on the half-line. We show that the \nls model is integrable with Neumann, Dirichlet and a mixed boundary condition, and that the GN model is integrable if $\psi_+^a\x=\pm\psi_-^a\x$. We also comment on the boundary condition found by Corrigan and Sheng for the O(3) nls model. 
  We study the amplitude for exchange of massless gauge bosons between pairs of massive scalar fields in Anti-de Sitter space. In the AdS/CFT correspondence this amplitude describes the contribution of conserved flavor symmetry currents to 4-point functions of scalar operators in the boundary conformal theory. A concise, covariant (Y2K-compatible) derivation of the gauge boson propagator in $\AdS_{d+1}$ is given. Techniques are developed to calculate the two bulk integrals over AdS space leading to explicit expressions or convenient, simple integral representations for the amplitude. The amplitude contains leading power and sub-leading logarithmic singularities in the gauge boson channel and leading logarithms in the crossed channel. The new methods of this paper are expected to have other applications in the study of the Maldacena conjecture. 
  Differential equations for scaling relation of prepotential in N=2 supersymmetric SU(2) Yang-Mills theory coupled with massive matter hypermultiplet are proposed and are explicitly demonstrated in one flavour ($N_f =1$) theory. By applying Whitham dynamics, the first order derivative of the prepotential over the $T_0$ variable corresponding to the mass of the hypermultiplet, which has a line integral representation, is found to satisfy a differential equation. As the result, the closed form of this derivative can be obtained by solving this equation. In this way, the scaling relation of massive prepotential is established. Furthermore, as an application of another differential equation for the massive scaling relation, the massive prepotential in strong coupling region is derived. 
  We show that the brane configuration describing the Izawa-Yanagida-Intriligator-Thomas (IYIT) model with gauged U(1) subgroup of the global symmetry contains inconsistent geometry, implying that there exists a stable vacuum where supersymmetry is dynamically broken. 
  We recover a general representation for the quantum state of a relativistic closed line (loop) in terms of string degrees of freedom.The general form of the loop functional splits into the product of the Eguchi functional, encoding the holographic quantum dynamics, times the Polyakov path integral, taking into account the full Bulk dynamics, times a loop effective action, which is needed to renormalize boundary ultraviolet divergences. The Polyakov string action is derived as an effective actionfrom a phase space,covariant,Schild action, by functionally integrating out the world-sheet coordinates.The area coordinates description of the boundary quantum dynamics, is shown to be induced by the ``zero mode'' of the bulk quantum fluctuations. Finally, we briefly comment about a ``unified, fully covariant'' description of points, loops and strings in terms of Matrix Coordinates. 
  A dual Ginzburg-Landau model corresponding to SU(3) gluodynamics in abelian projection is studied. A string theory describing QCD string dynamics is obtained in this model. The interaction of static quarks in mesons and baryons is investigated in an approximation to leading order. 
  In the (non-supersymmetric) Yang-Mills theory in the large N limit there exists an infinite set of non-degenerate vacua. The distinct vacua are separated by domain walls whose tension determines the decay rate of the false vacua. I discuss the phenomenon from a field-theoretic point of view, starting from supersymmetric gluodynamics and then breaking supersymmetry, by introducing a gluino mass. By combining previously known results, the decay rate of the excited vacua is estimated, \Gamma \sim \exp (-const \times N^4). The fourth power of N in the exponent is a consequence of the fact that the wall tension is proportional to N. 
  The inhomogeneities associated with massless Kalb-Ramond axions can be amplified not only in isotropic (four-dimensional) string cosmological models but also in the fully anisotropic case. If the background geometry is isotropic, the axions (which are not part of the homogeneous background) develop, outside the horizon, growing modes leading, ultimately, to logarithmic energy spectra which are "red" in frequency and increase at large distance scales. We show that this conclusion can be evaded not only in the case of higher dimensional backgrounds with contracting internal dimensions but also in the case of string cosmological scenarios which are completely anisotropic in four dimensions. In this case the logarithmic energy spectra turn out to be "blue" in frequency and, consequently, decreasing at large distance scales. We elaborate on anisotropic dilaton-driven models and we argue that, incidentally, the background models leading to (or flat) logarithmic energy spectra for axionic fluctuations are likely to be isotropized by the effect of string tension corrections. 
  In this article we consider the local supersymmetry breaking and the broken SU(5) symmetry permisible by dilaton vacuum configuration in supergravity theories. We establish the parameter relation of spontaneuos breaking of supersymmetry and of the GUT theory. 
  The $R^2 F^{2g-2}$ terms of Type IIA strings on Calabi-Yau 3-folds, which are given by the corresponding topological string amplitudes (a worldsheet instanton sum for all genera), are shown to have a simple M-theory interpretation.   In particular, a Schwinger one-loop computation in M-theory with wrapped M2 branes and Kaluza-Klein modes going around the loop reproduces the all genus string contributions from constant maps and worldsheet instanton corrections. In the simplest case of an isolated M2 brane with the topology of the sphere, we obtain the contributions of small worldsheet instantons (sphere ``bubblings'') which extends the results known or conjectured for low genera. Surprisingly, the 't Hooft expansion of large $N$ Chern-Simons theory on $S^3$ can also be used in a novel way to compute these gravitational terms at least in special cases. 
  The operator product expansion for ``small'' Wilson loops in {\cal N}=4, d=4 SYM is studied. The OPE coefficients are calculated in the large N and g_{YM}^2 N limit by exploiting the AdS/CFT correspondence. We also consider Wilson surfaces in the (0,2), d=6 superconformal theory. In this case, we find that the UV divergent terms include a term proportional to the rigid string action. 
  The effective classical/quantum dynamics of a particle constrained on a closed line embedded in a higher dimensional configuration space is analyzed. By considering explicit examples it is shown how different reduction mechanisms produce unequivalent dynamical behaviors. The relation with a formal treatment of the constraint is discussed. While classically it is always possible to strictly enforce the constraint by setting to zero the energy stored in the motion normal to the constraint surface, the quantum description is far more sensitive to the reduction mechanism. Not only quantum dynamics is plagued by the usual ambiguities inherent to the quantization procedure, but also in some cases the constraint's equations do not contain all the necessary information to reconstruct the effective motion. 
  We describe the partial breaking of $N=1 D=10$ supersymmetry down to $(1,0) d=6$ supersymmetry within the non-linear realization approach. The basic Goldstone superfield associated with this breaking is shown to be the $(1,0) d=6$ hypermultiplet superfield $q^{ia}$ subjected to a non-linear generalization of the standard hypermultiplet superfield constraint. The dynamical equations implied by this constraint are identified as the manifestly worldvolume supersymmetric equations of the Type I super 5-brane in D=10. We give arguments in favour of existence of an appropriate brane extension of off-shell hypermultiplet actions in harmonic superspace. Some related problems, in particular, the issue of utilizing other $(1,0) d=6$ supermultiplets as Goldstone ones, are shortly discussed. 
  Local symmetries of the action for a relativistic particle with curvature and torsion of its world curve in the (2+1)-dimensional space-time are studied. With the help of the method, worked out recently by the authors (Phys.Rev., D56, 1135, 1142 (1997)), first the local-symmetry transformations are obtained both in the phase and configuration space. At the classical level, the dependence of the particle mass on the parameters of curvature and torsion and the Regge trajectory are obtained. It is shown that the tachyonic sector can be removed by a proper gauge choice. 
  We determine the spectrum of currents generated by the operator product expansion of the energy-momentum tensor in N=4 super-symmetric Yang-Mills theory. Up to the regular terms and in addition to the multiplet of the stress tensor, three current multiplets appear, Sigma, Xi and Upsilon, starting with spin 0, 2 and 4, respectively. The OPE's of these new currents generate an infinite tower of current multiplets, one for each even spin, which exhibit a universal structure, of length 4 in spin units, identified by a two-parameter rational family. Using higher spin techniques developed recently for conformal field theories, we compute the critical exponents of Sigma, Xi and Upsilon in the TT OPE and prove that the essential structure of the algebra holds at arbitrary coupling. We argue that the algebra closes in the strongly coupled large-$N_c$ limit. Our results determine the quantum conformal algebra of the theory and answer several questions that previously remained open. 
  The greybody factors in BTZ black holes are evaluated from 2D CFT in the spirit of AdS$_3$/CFT correspondence. The initial state of black holes in the usual calculation of greybody factors by effective CFT is described as Poincar\'{e} vacuum state in 2D CFT. The normalization factor which cannot be fixed in the effective CFT without appealing to string theory is shown to be determined by the normalized bulk-to-boundary Green function. The relation among the greybody factors in different dimensional black holes is exhibited. Two kinds of $(h,{\bar h})=(1,1)$ operators which couple with the boundary value of massless scalar field are discussed. 
  We consider QCD_3 with an odd number of flavors in the mesoscopic scaling region where the field theory finite-volume partition function is equivalent to a random matrix theory partition function. We argue that the theory is parity invariant at the classical level if an odd number of masses are zero. By introducing so-called pseudo-orthogonal polynomials we are able to relate the kernel to the kernel of the chiral unitary ensemble in the sector of topological charge $\nu={1/2}$. We prove universality and are able to write the kernel in the microscopic limit in terms of field theory finite-volume partition functions. 
  The cosmological solutions of Horava-Witten theory discovered by Lukas, Ovrut and Waldram are generalized to allow non vanishing spatial curvature. The solution with closed spatial sections has initial and final curvature singularities. We find two solutions with open spatial sections, both of which evolve from an initial curvature singularity to the supersymmetric domain wall solution at late times. We also present a solution with open spatial sections and a non-zero Ramond-Ramond scalar. The behaviour of the solutions in eleven dimensions is discussed. 
  We briefly review the Whitham hierarchies and their applications to integrable systems of the Seiberg-Witten type. The simplest example of the N=2 supersymmetric SU(2) pure gauge theory is considered in detail and the corresponding Whitham solutions are found explicitely. 
  Extending early work, we formulate the large N matrix mechanics of general bosonic, fermionic and supersymmetric matrix models, including Matrix theory: The Hamiltonian framework of large N matrix mechanics provides a natural setting in which to study the algebras of the large N limit, including (reduced) Lie algebras, (reduced) supersymmetry algebras and free algebras. We find in particular a broad array of new free algebras which we call symmetric Cuntz algebras, interacting symmetric Cuntz algebras, symmetric Bose/Fermi/Cuntz algebras and symmetric Cuntz superalgebras, and we discuss the role of these algebras in solving the large N theory. Most important, the interacting Cuntz algebras are associated to a set of new (hidden) local quantities which are generically conserved only at large N. A number of other new large N phenomena are also observed, including the intrinsic nonlocality of the (reduced) trace class operators of the theory and a closely related large N field identification phenomenon which is associated to another set (this time nonlocal) of new conserved quantities at large N. 
  The existence of fluctuations together with interactions leads to scale-dependence in the couplings of quantum field theories for the case of quantum fluctuations, and in the couplings of stochastic systems when the fluctuations are of thermal or statistical nature. In both cases the effects of these fluctuations can be accounted for by solutions of the corresponding renormalization group equations. We show how the renormalization group equations are intimately connected with the effective action: given the effective action we can extract the renormalization group equations; given the renormalization group equations the effects of these fluctuations can be included in the classical action by using what is known as improved perturbation theory (wherein the bare parameters appearing in tree-level expressions are replaced by their scale-dependent running forms). The improved action can then be used to reconstruct the effective action, up to finite renormalizations, and gradient terms. 
  Values for the vacuum energy of scalar fields under Dirichlet and Neuman boundary conditions on an infinite clylindrical surface are found, and they happen to be of opposite signs. In contrast with classical works, a complete zeta function regularization scheme is here applied. These fields are regarded as interesting both by themselves and as the key to describing the electromagnetic (e.m.) case. With their help, the figure for the e.m. Casimir effect in the presence of this surface, found by De Raad and Milton, is now confirmed. 
  The quantum theory of a free particle on a portion of two-dimensional Euclidean space bounded by a circle and subject to non-local boundary conditions gives rise to bulk and surface states. Starting from this well known property, a counterpart for gravity is here considered. In particular, if spatial components of metric perturbations are set to zero at the boundary, invariance of the full set of boundary conditions under infinitesimal diffeomorphisms is compatible with non-local boundary conditions on normal components of metric perturbations if and only if both the gauge-field operator and the ghost operator are pseudo-differential operators in one-loop quantum gravity. 
  We present a q-analogue of Nahm's formalism for the BPS monopole, which gives self-dual gauge fields with a deformation parameter q. The theory of the basic hypergeometric series is used in our formalism. In the limit q -> 1 the gauge fields approach the BPS monopole and Nahm's result is reproduced. 
  In the framework of special Kahler geometry we consider the supergravity-matter system which emerges on a K3-fibered Calabi-Yau manifold. By applying the rigid limit procedure in the vicinity of a conifold singularity we compute the Kahler potential of the scalars and the kinetic matrix of the vectors to first order in the gravitational coupling. 
  Dynamical symmetry breaking is investigated for a four-fermion Nambu-Jona-Lasinio model in external electromagnetic and gravitational fields. An effective potential is calculated in the leading order of the large-N expansion using the proper-time Schwinger formalism.   Phase transitions accompanying a chiral symmetry breaking in the Nambu-Jona-Lasinio model are studied in detail. A magnetic calalysis phenomenon is shown to exist in curved spacetime but it turns out to lose its universal character because the chiral symmetry is restored above some critical positive value of the spacetime curvature. 
  We give a brief history of the passage from strings to branes and we review some aspects of the following topics in M-theory: (a) an extended brane scan, (b) superembedding approach to the dynamics of superbranes and (c) supermembranes in anti de Sitter space, singletons and massless higher spin field theories. 
  The term $\Theta\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}$, when added to the electromagnetic Lagrangian $-{1\over 16\pi}F^{\mu\nu}F_{\mu\nu}$, does not change the signature of the Lagrangian. Actually, it increases the part with negative kinetic energy term at the spatial infinity. For this reason it does not change the conclusion, that at the spatial infinity the magnetic part of the electromagnetic field should be absent. 
  We demonstrate how to extract all the one-loop renormalization group equations for arbitrary quantum field theories from knowledge of an appropriate Seeley--DeWitt coefficient. By formally solving the renormalization group equations to one loop, we renormalization group improve the classical action, and use this to derive the leading-logarithms in the one-loop effective action for arbitrary quantum field theories. 
  We use variational methods to calculate quasilocal energy quantum corrections. A comparison with the effective potential calculated at quadratic order is made by means of gaussian wave functionals. The method is a particular case of the effective action for composite operators used in quantum field theory. In pure gravity the method is applied for the first time. Implications on the foam-like scenario are discussed. 
  The recently presented quantum antibrackets are generalized to quantum Sp(2)-antibrackets. For the class of commuting operators there are true quantum versions of the classical Sp(2)-antibrackets. For arbitrary operators we have a generalized bracket structure involving higher Sp(2)-antibrackets. It is shown that these quantum antibrackets may be obtained from generating operators involving operators in arbitrary involutions. A recently presented quantum master equation for operators, which was proposed to encode generalized quantum Maurer-Cartan equations for arbitrary open groups, is generalized to the Sp(2) formalism. In these new quantum master equations the generalized Sp(2)-brackets appear naturally. 
  We consider Yang-Mills theories with general gauge groups $G$ and twists on the four torus. We find consistent boundary conditions for gauge fields in all instanton sectors. An extended Abelian projection with respect to the Polyakov loop operator is presented, where $A_0$ is independent of time and in the Cartan subalgebra. Fundamental domains for the gauge fixed $A_0$ are constructed for arbitrary gauge groups. In the sectors with non-vanishing instanton number such gauge fixings are necessarily singular. The singularities can be restricted to Dirac strings joining magnetically charged defects. The magnetic charges of these monopoles take their values in the co-root lattice of the gauge group. We relate the magnetic charges of the defects and the windings of suitable Higgs fields about these defects to the instanton number. 
  Recently it has been shown that antibrackets may be expressed in terms of Poisson brackets and vice versa for commuting functions in the original bracket. Here we also introduce generalized brackets involving higher antibrackets or higher Poisson brackets where the latter are of a new type. We give generating functions for these brackets for functions in arbitrary involutions in the original bracket. We also give master equations for generalized Maurer-Cartan equations. The presentation is completely symmetric with respect to Poisson brackets and antibrackets. 
  We consider the four dimensional Euclidean Maxwell theory with a Chern-Simons term on the boundary. The corresponding gauge invariant boundary conditions become dependent on tangential derivatives. Taking the four-sphere as a particular example, we calculate explicitly a number of the first heat kernel coefficients and obtain the general formulas that yields any desired coefficient. A remarkable observation is that the coefficient $a_2$, which defines the one-loop counterterm and the conformal anomaly, does not depend on the Chern-Simons coupling constant, while the heat kernel itself becomes singular at a certain (critical) value of the coupling. This could be a reflection of a general property of Chern-Simons theories. 
  We study the classical and quantum motion of a relativistic charged particle on the spacetime produced by a global monopole. The self-potential, which is present in this spacetime, is considered as an external electrostatic potential. We obtain classical orbits and quantum states for a spin-1/2 and spin-0 particles. 
  To construct actions for describing superbranes propagating in AdS x S superbackgrounds we propose a coset space realization of these superbackgrounds which results in a short polynomial fermionic dependence (up to the sixth power in Grassmann coordinates) of target superspace supervielbeins and superconnections. Gauge fixing kappa-symmetry in a way compatible with a static brane solution further reduces the fermionic dependence down to the fourth power. Subtleties of consistent gauge fixing worldvolume diffeomorphisms and kappa-symmetry of the superbrane actions are discussed. 
  We compute the influence of an external magnetic field on the Casimir energy of a massive charged scalar field confined between two parallel infinite plates. For this case the obtained result shows that the magnetic field inhibits the Casimir effect. 
  The influence of an external constant uniform magnetic field on the Casimir energy density of a Dirac field under antiperiodic (and periodic) boundary condition is computed by applying Schwinger's proper time method. The result thus obtained shows that in principle, under suitable conditions, the magnetic field can enhance the fermionic Casimir energy density. 
  We compute the influence of boundary conditions on the Euler-Heisenberg effective Lagrangian scalar QED scalar for the case of a pure magnetic field. The boundary conditions constrain the quantum scalar field to vanish on two parallel planes separeted by a distance $a$ and the magnetic field is assumed to be constant, uniform and perpendicular to the planes. The effective Lagrangian is obtained using Schwinger's proper-time representation and exhibits new contributions generated by the boundary condition much in the same way as a material pressed between two plates exhibits new magnetic properties. The confined bosonic vacuum presents the expected diamagnetic properties and besides the new non-linear $a$-dependent contributions to the susceptibility we show that there exists also a new $a$-dependent contribution for the vacuum permeability in the linear approximation. 
  A modified version of triplectic quantization, first introduce by Batalin and Martnelius, is proposed which makes use of two independent master equations, one for the action and one for the gauge functional such that the initial classical action also obeys that master equation. 
  I consider the problem of generalising the Abelian Coulomb gauge condition to the non-Abelian Yang-Mills theory, with an arbitrary compact and semi-simple gauge group. It is shown that a straightforward generalisation exists, which reduces the Gauss law into a form involving the gauge potentials only, but not their time derivatives. The existence and uniqueness of the generalised Coulomb gauge is shown to depend on an elliptic linear partial differential equation for a Lie-algebra valued quantity, which defines the gauge transform by means of which the generalised Coulomb gauge condition is realised. Thus the Gribov problem is actually non-existent in this case. 
  I consider the problem of defining canonical coordinates and momenta in pure Yang-Mills theory, under the condition that Gauss' law is identically satisifed. This involves among other things particular boundary conditions for certain dependent variables. These boundary conditions are not postulated a priori, but arise as consistency conditions related to the equations of motion. It is shown that the theory indeed has a canonical structure, provided one uses a special gauge condition, which is a natural generalisation to Yang-Mills theory of the Coulomb gauge condition in electrodynamics. The canonical variables and Hamiltonian are explicitly constructed. Quantisation of the theory is briefly discussed. 
  We consider two Dirichlet p-branes with lower dimensional brane charges and their scattering. We first calculate the cylinder amplitude of open string with suitable boundary conditions. We compare this result with that in the IIB matrix model. We find the agreement between them in the long distance, low velocity, or large field limit. We also find a way to investigate more general boundary conditions for open string. 
  We present details of the harmonic space construction of a quaternionic extension of the four-dimensional Taub-NUT metric. As the main merit of the harmonic space approach, the metric is obtained in an explicit form following a generic set of rules. It exhibits $SU(2)\times U(1)$ isometry group and depends on two parameters, Taub-NUT `mass' and the cosmological constant. We consider several limiting cases of interest which correspond to special choices of the involved parameters. 
  There has been some debate about the validity of quantum affine Toda field theory at imaginary coupling, owing to the non-unitarity of the action, and consequently of its usefulness as a model of perturbed conformal field theory. Drawing on our recent work, we investigate the two simplest affine Toda theories for which this is an issue - a2(1) and a2(2). By investigating the S-matrices of these theories before RSOS restriction, we show that quantum Toda theory, (with or without RSOS restriction), indeed has some fundamental problems, but that these problems are of two different sorts. For a2(1), the scattering of solitons and breathers is flawed in both classical and quantum theories, and RSOS restriction cannot solve this problem. For a2(2) however, while there are no problems with breather-soliton scattering there are instead difficulties with soliton-excited soliton scattering in the unrestricted theory. After RSOS restriction, the problems with kink-excited kink may be cured or may remain, depending in part on the choice of gradation, as we found in [12]. We comment on the importance of regradations, and also on the survival of R-matrix unitarity and the S-matrix bootstrap in these circumstances. 
  The aim of this paper is to consider a possibility of constructing for arbitrary dynamical systems with first-class constraints a generalized canonical quantization method based on the osp(1,2) supersymmetry principle. This proposal can be considered as a counterpart to the osp(1,2)-covariant Lagrangian quantization method introduced recently by Geyer, Lavrov and M\"ulsch. The gauge dependence of Green's functions is studied. It is shown that if the parameter m^2 of the osp(1,2) superalgebra is not equal to zero then the vacuum functional and S-matrix depend on the gauge. In the limit $m\to 0$ the gauge independence of vacuum functional and S - matrix are restored. The Ward identities related to the osp(1,2) symmetry are derived. 
  Trace anomaly for dilaton coupled conformal theories on curved background with non-zero dilaton is found from supergravity side as an IR effect using AdS/CFT correspondence. For $d=2$ it coincides with the conformal anomaly for dilaton coupled scalar (up to total derivative term which is known to be ambiguous). In four-dimensional case we get conformal anomaly for ${\cal N}=4$ super YM theory interacting with conformal supergravity. In the same way the calculation of dilaton dependent conformal anomaly in higher dimensions seems to be much easier than using standard QFT methods. 
  Invariance principles determine many key properties in quantum field theory, including, in particular, the appropriate form of the boundary conditions. A crucial consistency check is the proof that the resulting boundary-value problem is strongly elliptic. In Euclidean quantum gravity, the appropriate principle seems to be the invariance of boundary conditions under infinitesimal diffeomorphisms on metric perturbations, and hence their BRST invariance. However, if the operator on metric perturbations is then chosen to be of Laplace type, the boundary-value problem for the quantized gravitational field fails to be strongly elliptic. A detailed proof is presented, and the corresponding open problems are discussed. 
  We consider the generation of Fayet-Iliopoulos terms at one string loop in some recently found N=1 open string orbifolds with anomalous U(1) factors with nonvanishing trace of the charge. Low-energy field theory arguments lead one to expect a one loop quadratically divergent Fayet-Iliopoulos term. We show that a one loop Fayet-Iliopoulos term is not generated, due to a cancellation between contributions of worldsheets of different topology. The vanishing of the one loop Fayet-Iliopoulos term in open string compactifications is related to the cancellation of twisted Ramond-Ramond tadpoles. 
  The problem of the classification of the extensions of the Virasoro algebra is discussed. It is shown that all $H$-reduced $\hat{\cal G}_{r}$-current algebras belong to one of the following basic algebraic structures: local quadratic $W$-algebras, rational $U$-algebras, nonlocal $V$-algebras, nonlocal quadratic $WV$-algebras and rational nonlocal $UV$-algebras. The main new features of the quantum $V$-algebras and their heighest weight representations are demonstrated on the example of the quantum $V_{3}^{(1,1)}$-algebra. 
  We show that a necessary condition, for the partition function of four-dimensional Yang-Mills theory to satisfy a S-duality property, is that certain functional determinants, generated by the dual change of variables, cancel each other. This result holds up to non-topological boundary terms in the dual action and modulo the problem of field-strength copies for the Bianchi identity constraint. 
  We transform, by means of a fiberwise duality, the partition function of QCD on a product of two two-tori, into a four-dimensional sigma-model, whose target space is the cotangent space of unitary connections on the fiber torus fiberwise. 
  We examine N=1 supersymmetric gauge theories which confine in the presence of a tree-level superpotential. We show the confining spectra which satisfy the 't Hooft anomaly matching conditions and give a simple method to find the confining superpotential. Using this method we fix the confining superpotentials in the simplest cases, and show how these superpotentials are generated by multi-instanton effects in the dual theory. These new type of confining theories may be useful for model building, since the size of the matter content is not restricted by an index constraint. Therefore, one expects that a large variety of new confining spectra can be obtained using such models. 
  By means of a certain exact non-abelian duality transformation, we show that there is a natural embedding, dense in the sense of the distributions in the large-N limit, of parabolic Higgs bundles of rank N on a fiber two-dimensional torus into the QCD functional integral, fiberwise over the base two-dimensional torus of the trivial elliptic fibration on which the four-dimensional theory is defined. The moduli space of parabolic Higgs bundles of rank N is an integrable Hamiltonian system, that admits a foliation by the moduli of holomorphic line bundles over N-sheeted spectral covers (or, what is the same, over a space of N gauge-invariant polynomials), the Hitchin fibration. According to Hitchin, the Higgs bundles can be recovered from the spectral covers and the line bundles. If the N invariant polynomials together with the abelian connection on the line bundles are chosen as the N+1 collective fields of the Hitchin fibration, all the entropy of the functional integration over the moduli of the Higgs bundles is absorbed, in the large-N limit, into the Jacobian determinant of the change of variables to the collective fields of the Hitchin fibration. Hence, the large-N limit is dominated by the saddle-point of the effective action as in vector-like models. 
  We show how to systematically derive the complete set of the gauge transformations of different types of the gauge invariant models, which are the chiral Schwinger and CP$^1$ with Chern-Simons term, in the Lagrangian Formalism. 
  Starting from lagrangian field theory and the variational principle, we show that duality in equations of motion can also be obtained by introducing explicit spacetime dependence of the lagrangian. Poincare invariance is achieved precisely when the duality conditions are satisfied in a particular way. The same analysis and criteria are valid for both abelian and nonabelian dualities. We illustrate how (1)Dirac string solution (2)Dirac quantisation condition (3)t'Hooft-Polyakov monopole solutions and (4)a procedure emerges for obtaining {\it new} classical solutions of Yang-Mills (Y-M) theory. Moreover, these results occur in a way that is strongly reminiscent of the {\it holographic principle}. 
  Odd-dimensional Riemannian spaces that are non-orientable, but have a pin structure, require the consideration of the twisted adjoint representation of the corresponding pin group. It is shown here how the Dirac operator should be modified, also on even-dimensional spaces, to make it equivariant with respect to the action of that group when the twisted adjoint representation is used in the definition of the pin structure. An explicit description of a pin structure on a hypersurface, defined by its immersion in a Euclidean space, is used to derive a "Schroedinger" transform of the Dirac operator in that case. This is then applied to obtain - in a simple manner - the spectrum and eigenfunctions of the Dirac operator on spheres and real projective spaces. 
  A new rigorous approach to conformal field theory is presented. The basic objects are families of complex-valued amplitudes, which define a meromorphic conformal field theory (or chiral algebra) and which lead naturally to the definition of topological vector spaces, between which vertex operators act as continuous operators. In fact, in order to develop the theory, M\"obius invariance rather than full conformal invariance is required but it is shown that every M\"obius theory can be extended to a conformal theory by the construction of a Virasoro field.   In this approach, a representation of a conformal field theory is naturally defined in terms of a family of amplitudes with appropriate analytic properties. It is shown that these amplitudes can also be derived from a suitable collection of states in the meromorphic theory. Zhu's algebra then appears naturally as the algebra of conditions which states defining highest weight representations must satisfy. The relationship of the representations of Zhu's algebra to the classification of highest weight representations is explained. 
  Seiberg duality in supersymmetric gauge theories is the claim that two different theories describe the same physics in the infrared limit. However, one cannot easily work out physical quantities in strongly coupled theories and hence it has been difficult to compare the physics of the electric and magnetic theories. In order to gain more insight into the equivalence of two theories, we study the ``e+ e-'' cross sections into ``hadrons'' for both theories in the superconformal window. We describe a technique which allows us to compute the cross sections exactly in the infrared limit. They are indeed equal in the low-energy limit and the equality is guaranteed because of the anomaly matching condition. The ultraviolet behavior of the total ``e+ e-'' cross section is different for the two theories. We comment on proposed non-supersymmetric dualities. We also analyze the agreement of the ``\gamma\gamma'' and ``WW'' scattering amplitudes in both theories, and in particular try to understand if their equivalence can be explained by the anomaly matching condition. 
  It has recently been shown that the Matrix model and supergravity give the same predictions for three graviton scattering. This contradicts an earlier claim in the literature (hep-th/9710174). We explain the error in this earlier work, and go on to show that certain terms in the $n$-graviton scattering amplitude involving $v^{2n}$ are given correctly by the Matrix model. The Matrix model also generates certain $v^6$ terms in four graviton scattering at three loops, which do not seem to have any counterparts in supergravity. The connection of these results with nonrenormalization theorems is discussed. 
  Using set-theoretic considerations, we show that the forest formula for overlapping divergences comes from the Hopf algebra of rooted trees. 
  We investigate BPS properties of the charged soliton solutions of D-brane worldvolume theory, which is described by the supersymmetric Dirac-Born-Infeld action, by means of the N=2 target-space supersymmetry algebra. Our results agree with those obtained previously. We also extend our BPS analysis to the case where axion background exists. 
  It is shown that a supersymmetric and $\kappa$-symmetric D3-brane action on $AdS_5 \times S^5$ is mapped into itself by a duality transformaion, thereby verifying the $SL(2,Z)$ invariance of the D3-brane action in the $AdS_5 \times S^5$ background as in the flat background. To this end, we fix the $\kappa$-symmetry in a gauge which simplifies the classical action in order to perform an SO(2) rotation of the N=2 spinor index in a manifest way, though this may not be necessary. This situation is the same as the case of a super D-string on $AdS_5 \times S^5$ where it was shown that the super D-string action is transformed to a form of the IIB Green-Schwarz superstring action with the $SL(2,Z)$ covariant tension in the $AdS_5 \times S^5$ background through a duality transformation. These results strongly suggest that various duality relations originally found in the flat background may be independent of background geometry, in other words, the duality transformations in string and p-brane theories may exist even in general curved space-time. 
  The massive non-relativistic free particle in d-1 space dimensions has an action with a surprizing non-linearly realized SO(d,2) symmetry. This is the simplest example of a host of diverse one-time-physics systems with hidden SO(d,2) symmetric actions. By the addition of gauge degrees of freedom, they can all be lifted to the same SO(d,2) covariant unified theory that includes an extra spacelike and an extra timelike dimension. The resulting action in d+2 dimensions has manifest SO(d,2) Lorentz symmetry and a gauge symmetry Sp(2,R) and it defines two-time-physics. Conversely, the two-time action can be gauge fixed to diverse one-time physical systems. In this paper three new gauge fixed forms that correspond to the non-relativistic particle, the massive relativistic particle, and the particle in AdS_(d-n) x S^n spacetime will be discussed. The last case is discussed at the first quantized and field theory levels as well. For the last case the popularly known symmetry is SO(d-n-1,2) x SO(n+1), but yet we show that it is symmetric under the larger SO(d,2). In the field theory version the action is symmetric under the full SO(d,2) provided it is improved with a quantized mass term that arises as an anomaly from operator ordering ambiguities. The anomalous cosmological term vanishes for AdS_2 x S^0 and AdS_n x S^n (i.e. d=2n). The strikingly larger symmetry could be significant in the context of the proposed AdS/CFT duality. 
  Notes on exact S-matrices in 1+1 dimensions, based on lectures given at the 1996 Eotvos Graduate School, Budapest, and at the Institut Henri Poincare, Paris. 
  This Ph.D. thesis pursues two goals: The study of the geometrical structure of two-dimensional quantum gravity and in particular its fractal nature. To address these questions we review the continuum formalism of quantum gravity with special focus on the scaling properties of the theory. We discuss several concepts of fractal dimensions which characterize the extrinsic and intrinsic geometry of quantum gravity. This work is partly based on work done in collaboration with Jan Ambj{\o}rn, Dimitrij Boulatov, Jakob L. Nielsen and Yoshiyuki Watabiki (1997).   The other goal is the discussion of the discretization of quantum gravity and to address the so called quantum failure of Regge calculus. We review dynamical triangulations and show that it agrees with the continuum theory in two dimensions. Then we discuss Regge calculus and prove that a continuum limit cannot be taken in a sensible way and that it does not reproduce continuum results. This work is partly based on work done in collaboration with Jan Ambj{\o}rn, Jakob L. Nielsen and George Savvidy (1997). 
  A dynamical theory of hypersurface deformations is presented. It is shown that a (n+1)-dimensional space-time can be always foliated by pure deformations, governed by a non zero Hamiltonian. Quantum deformations states are defined by Schroedinger's equation constructed with the corresponding deformation Hamiltonian operator, interpreted as the generator of the deformation diffeomorphism group. Applications to quantum gravity and to a modified Kaluza-Klein theory are proposed. 
  The massive Schwinger model in bosonic representation is quantized on the light front using the Dirac--Bergmann method. The non-perturbative theta- vacuum in terms of coherent states of the gauge-field zero mode is derived and found to coincide with the massless case. On the other hand, the mass term becomes highly non-linear due to the constrained zero mode of the scalar field. A non-trivial mixing between the normal-mode and zero-mode sectors of the model is crucial for the correct calculation of the theta-dependence of the leading order mass correction to the chiral condensate. 
  Statistical method is applied to Neveu and Schwartz model for obtaining the average characteristics of the heavy resonances: spin distribution, decay widths etc. The properties of the dual model spectrum of states constructed of both commuting and anticommuting operators are considered. In this case the spin characteristics coincide with the results of the statistical bootstrap model. 
  Review of some old and relatively new ideas surrounding the subjects of AdS/CFT correspondence, generalized tau-functions and possible equivalences between a priori different quantum field theories. 
  This is a very elementary introduction to the Heisenberg (XXX) quantum spin chain, the Yang-Baxter equation, and the algebraic Bethe Ansatz. 
  The Casimir effect for Dirac as well as for scalar charged particles is influenced by external magnetic fields. It is also influenced by finite temperature. Here we consider the Casimir effect for a charged scalar field under the combined influence of an external magnetic field and finite temperature. The free energy for such a system is computed using Schwinger's method for the calculation of the effective action in the imaginary time formalism. We consider both the limits of strong and weak magnetic field in which we compute the Casimir free energy and pressure. 
  We formulate the notion of parity for the periodic XXZ spin chain within the Quantum Inverse Scattering Method. We also propose an expression for the eigenvalues of the charge conjugation operator. We use these discrete symmetries to help classify low-lying S^z=0 states in the critical regime, and we give a direct computation of the S matrix. 
  We study the regularized correlation functions of the light-like coordinate operators in the reduction to zero dimensions of the matrix model describing $D$-particles in four dimensions. We investigate in great detail the related matrix model originally proposed and solved in the planar limit by J. Hoppe. It also gives the solution of the problem of 3-coloring of planar graphs. We find interesting strong/weak 't Hooft coupling dependence. The partition function of the grand canonical ensemble turns out to be a tau-function of KP hierarchy. As an illustration of the method we present a new derivation of the large-N and double-scaling limits of the one-matrix model with cubic potential. 
  We work out the constraints imposed by SL(2C) invariance for sphere topology and modular invariance for torus topology, on the discretized form of Liouville action in Polyakov's non local covariant form. These are sufficient to completely fix the discretized action except for the overall normalization constant and a term which in the continuum limit goes over to a topological invariant. The treatment can be extended to the supersymmetric case. 
  We calculate the probability of electron-positron pair creation in vacuum in 3+1 dimensions by an external electromagnetic field composed of a constant uniform electric field and a constant uniform magnetic field, both of arbitrary magnitudes and directions. The same problem is also studied in 2+1 and 1+1 dimensions in appropriate external fields and similar results are obtained. 
  We present the general solution of the system of coupled nonlinear equations describing dynamics of $D$--dimensional bosonic string in the geometric (or embedding) approach. The solution is parametrized in terms of two sets of the left- and right-moving Lorentz harmonic variables providing a special coset space realization of the product of two (D-2)- dimensional spheres $S^{D-2}={SO(1,D-1) \over SO(1,1) \times SO(D-2) \semiprod K_{D-2}}.$ 
  We criticize and generalize some properties of Noether charges presented in a paper by V. Iyer and R. M. Wald and their application to entropy of black holes. The first law of black holes thermodynamics is proven for any gauge-natural field theory. As an application charged Kerr-Newman solutions are considered. As a further example we consider a (1+2) black hole solution. 
  We review the background field method for general N = 2 super Yang-Mills theories formulated in the N = 2 harmonic superspace. The covariant harmonic supergraph technique is then applied to rigorously prove the N=2 non-renormalization theorem as well as to compute the holomorphic low-energy action for the N = 2 SU(2) pure super Yang-Mills theory and the leading non-holomorphic low-energy correction for N = 4 SU(2) super Yang-Mills theory. 
  The maximal SO(5) gauged D=7 supergravity is dimensionally reduced to six dimensions yielding a new SO(5) gauged D=6 model. It is shown that, unlike in D=7, the SO(5) gauge coupling constant can be taken to zero to yield the maximally extended supergravity in six dimensions. It is also shown that the limit of D=5 N=4 SU(2)xU(1) gauged supergravity in which the U(1) coupling constant is turned off can be obtained. 
  We construct eight-dimensional gravitational instantons by solving appropriate self-duality equations for the spin-connection. The particular gravitational instanton we present has $Spin(7)$ holonomy and, in a sense, it is the eight-dimensional analog of the Eguchi-Hanson 4D space. It has a removable bolt singularity which is topologically S^4 and its boundary at infinity is the squashed S^7. We also lift our solutions to ten and eleven dimensions and construct fundamental string and membrane configurations that preserve 1/16 of the original supersymmetries. 
  This paper describes perturbative framework, on the basis of closed-time-path formalism, for studying quasiuniform relativistic quantum field systems near equilibrium and nonequilibrium quasistationary systems. At the first part, starting from first principles, we construct perturbative schemes for relativistic complex-scalar-field theory. We clarify what assumption is involved in arriving at a standard perturbative framework and to what extent the $n (\geq 4)$-point initial correlation functions that are usually discarded in the standard framework can in fact be discarded. Two calculational schemes are introduced, the one is formulated on the basis of the initial particle distribution function and the one is formulated on the basis of the `` physical'' particle distribution function. Both schemes are equivalent and lead to a generalized relativistic kinetic or Boltzmann equation. At the second part, using the perturbative loop-expansion scheme for an $O (N)$ linear $\sigma$ model, we analyze how the chiral phase transition proceeds through disoriented chiral condensates. The system of coupled equations that governs the spacetime evolution of the condensate or order-parameter fields is derived. The region where the curvature of the ``potential'' is negative is dealt with by introducing the random-force fields. Application to simple situations is made. 
  We show that when the induced parity breaking part of the effective action for the low-momentum region of U(1) x ... x U(1) Maxwell gauge field theory with massive fermions in 3 dimensions is coupled to a \phi^4 scalar field theory, it is not possible to eliminate the screening of the long-range Coulomb interactions and get external charges confined in the broken Higgs phase. This result is valid for non-zero temperature as well. 
  First examples of quasi-exactly solvable models describing spin-orbital interaction are constructed. In contrast with other examples of matrix quasi-exactly solvable models discussed in the literature up to now, our models admit (but still incomplete) sets of exact (algebraic) solutions. 
  We generalize the dressing symmetry construction in mKdV hierarchy. This leads to non-local vector fields (expressed in terms of vertex operators) closing a Virasoro algebra. We argue that this algebra realization should play an important role in the study of 2D integrable field theories and in particular should be related to the Deformed Virasoro Algebra (DVA) when the construction is perturbed out of the critical theory. 
  We desribe the minimal configurations of the bosonic membrane potential, when the membrane wraps up in an irreducible way over $S^{1}\times S^{1}$. The membrane 2-dimensional spatial world volume is taken as a Riemann Surface of genus $g$ with an arbitrary metric over it. All the minimal solutions are obtained and described in terms of 1-forms over an associated U(1) fiber bundle, extending previous results. It is shown that there are no infinite dimensional valleys at the minima. 
  We present a (1+1)-dimensional fermionic QFT with non-local couplings between currents. This model describes an ensemble of spinless fermions interacting through forward, backward and umklapp scattering processes. We express the vacuum to vacuum functional in terms of a non trivial fermionic determinant. Using path-integral methods we find a bosonic representation for this determinant. Thus we obtain an effective action depending on three scalar fields, two of which correspond to the physical collective excitations whereas the third one is an auxiliary field that is left to be integrated by means of an approximate technique. 
  We consider the most general loop integral that appears in non-relativistic effective field theories with no light particles. The divergences of this integral are in correspondence with simple poles in the space of complex space-time dimensions. Integrals related to the original integral by subtraction of one or more poles in dimensions other than D=4 lead to nonminimal subtraction schemes. Subtraction of all poles in correspondence with ultraviolet divergences of the loop integral leads naturally to a regularization scheme which is precisely equivalent to cutoff regularization. We therefore recover cutoff regularization from dimensional regularization with a nonminimal subtraction scheme. We then discuss the power-counting for non-relativistic effective field theories which arises in these alternative schemes. 
  We consider the Matrix theory proposal describing eleven-dimensional Schwarzschild black holes. We argue that the Newtonian potential between two black holes receives a genuine long range quantum gravity correction, which is finite and can be computed from the supergravity point of view. The result agrees with Matrix theory up to a numerical factor which we have not computed. 
  We compute the imaginary part of scalar four-point functions in the AdS/CFT correspondence relevant to N=4 super Yang-Mills theory. Unitarity of the AdS supergravity demands that the imaginary parts of the correlation functions factorize into products of lower-point functions. We include the exchange diagrams for scalars as well as gravitons and find explicit expressions for the imaginary parts of these correlators. In momentum space these expressions contain only rational functions and logarithms of the kinematic invariants, in such a manner that the correlator is not a free-field result. The simplicity of these results, however, indicate the possibility of additional symmetry structures in N=4 super Yang-Mills theory in the large $N_c$ limit at strong effective coupling. The complete expressions may be computed from the integral results derived here. 
  After presenting a survey of theoretical results concerning the structure of two-dimensional QCD, we present a numerical study related to the mass eigenstates and the decay amplitudes of higher mesonic states. We discuss in detail the fate of important dynamical points such as stability of the spectrum and the problem of screening versus confinement in this context. We point out differences in the large distance behaviour of the potential, which can be responsible for the question of stability of the spectrum, as well as whether it is finite. 
  We argue that the large N strong/weak phase transition is a generic phenomenon in a finite temperature supersymmetric Yang-Mills theory of maximal supersymmetry. ${\cal N}=4$, D=4 SYM is the canonical example, where we also argue that the large N Hawking-Page phase transition disappears for a sufficiently small coupling. The Hawking-Page transition temperature is lowered by the first $\ap$ correction. Physically, the strong/weak phase transition is identified with the correspondence point of Horowitz and Polchinski. We also try to construct toy models to demonstrate the large N phase transitions, with limited success. 
  In the light of $\phi$-mapping method and topological current theory, the topological structure and the topological quantization of topological linear defects are obtained under the condition that the Jacobian $J(\phi/v) \neq 0$. When $J(\phi/v) = 0$, it is shown that there exist the crucial case of branch process. Based on the implicit function theorem and the Taylor expansion, the origin and bifurcation of the linear defects are detailed in the neighborhoods of the limit points and bifurcation points of $\phi$-mapping, respectively. 
  The two-dimensional anti-de Sitter space(AdS_2) is constructed in terms of the CGHS model. The geometric solutions are composed of the AdS vacuum and the AdS black hole which are locally equivalent but distinguishable by their mass. The infalling classical fields do not play any role but the quantum back reaction is crucial in the formation of the AdS vacuum and AdS black hole. In the presence of the AdS black hole, there does not exist any radiation, which is consistent with the constraint equations. Therefore the transition from the AdS black hole to the AdS vacuum is impossible, and they are quantum mechanically stable . We discuss the reason why the vanishing Hawking radiation appears in the AdS_2 black hole in contrast to asymptotically flat black holes. 
  This is an elementary introduction to Wilson renormalization group and continuum effective field theories. We first review the idea of Wilsonian effective theory and derive the flow equation in a form that allows multiple insertion of operators in Green functions. Then, based on this formalism, we prove decoupling and heavy-mass factorization theorems, and discuss how the continuum effective field theory is formulated in this approach. 
  The algebraic conditions that specific gauged G/H-WZW model have to satisfy in order to give rise to Non-Abelian Toda models with singular metric with or without torsion are found. The classical algebras of symmetries corresponding to grade one rank 2 and 3 singular NA-Toda models are derived. 
  The spectrum of collective fermionic excitations in a finite temperature QED_{3+1} is studied in different regimes. It is shown that within the standard perturbation approach the one-loop dispersion equation, besides the ordinary one-particle excitation, has four new solutions. The additional excitations are gauge-dependent and two of them have nonphysical signs of residues in the propagator poles. The temperature evolution of the solutions is investigated and it is shown that the use of effective propagators leaves no more than one additional mode which becomes propagating at $T \bolpor 10M$, when the gauge invariance is restored. The other three modes, including those with nonphysical residues in the propagator poles, are always strongly damped, thus the thermal effects do not produce pathologies in QED_{3+1}. 
  We construct free-field resolutions of unitary representations of the N=2 superconformal algebra. The irreducible representations are singled out from free-field spaces as the cohomology of fermionic screening operators. We construct and evaluate the cohomology of the resolution associated with one fermionic screening (which is related to the representation theory picture of ``gravitational descendants''), and a {\it butterfly} resolution associated with two fermionic screenings. 
  The principle of equivalence is translated into the language of the world-volume field theories that define matrix and string theories. This idea leads to explore possible matrix descriptions of M-theory compactifications. An interesting case is the relationship between D=6 N=1 U(M) SYM and Matrix Theory on K3. 
  We study in a systematic way all static solutions of the Goldstone model in 1+1 dimension with a periodicity condition on the spatial coordinate. The solutions are presented in terms of the standard trigonometric functions and of Jacobi elliptic functions. Their stability analysis is carried out, and the complete list of classical stable quasi-topological solitons is given. 
  We show that the Casimir, or zero-point, energy of a dilute dielectric ball, or of a spherical bubble in a dielectric medium, coincides with the sum of the van der Waals energies between the molecules that make up the medium. That energy, which is finite and repulsive when self-energy and surface effects are removed, may be unambiguously calculated by either dimensional continuation or by zeta function regularization. This physical interpretation of the Casimir energy seems unambiguous evidence that the bulk self-energy cannot be relevant to sonoluminescence. 
  It is argued that N=6 supergravity on $AdS_5$, with gauge group $SU(3)\times U(1)$ corresponds, at the classical level, to a subsector of the ``chiral'' primary operators of N=4 Yang-Mills theories. This projection involves a ``duality transformation'' of N=4 Yang-Mills theory and therefore can be valid if the coupling is at a self-dual point, or for those amplitudes that do not depend on the coupling constant. 
  To define a consistent perturbative geometric heterotic compactification the bundle is required to satisfy a subtle constraint known as ``stability,'' which depends upon the Kahler form. This dependence upon the Kahler form is highly nontrivial---the Kahler cone splits into subcones, with a distinct moduli space of bundles in each subcone---and has long been overlooked by physicists. In this article we describe this behavior and its physical manifestation. 
  We compute the effective actions for the 0+1 dimensional scalar field interacting with an Abelian gauge background, as well as for its supersymmetric generalization at finite temperature. 
  The large $N_c$ limit of Yang-Mills gauge theory is the dynamics of a closed gluonic chain, but this fact does not obviate the inherently strong coupling nature of the dynamical problem. However, we suggest that a single link in such a chain might be reasonably described in the quasi-perturbative language of gluons and their interactions. To implement this idea, we use the MIT bag to model the physics of a nearest neighbor bond. 
  We construct local zero curvature representations for non-linear sigma models on homogeneous spaces, defined on a space-time of any dimension, following a recently proposed approach to integrable theories in dimensions higher than two. We present some sufficient conditions for the existence of integrable submodels possessing an infinite number of local conservation laws. Examples involving symmetric spaces and group manifolds are given. The $CP^N$ models are discussed in detail. 
  We reexamine the issue of the soliton mass in two-dimensional models with N =1 supersymmetry. The superalgebra has a central extension, and at the classical level the soliton solution preserves 1/2 of supersymmetry which is equivalent to BPS saturation. We prove that the property of BPS saturation, i.e. the equality of the soliton mass to the central charge, remains intact at the quantum level in all orders of the weak coupling expansion. Our key finding is an anomaly in the expression for the central charge. The classical central charge, equal to the jump of the superpotential, is amended by an anomalous term proportional to the second derivative of the superpotential. The anomaly is established by various methods in explicit one-loop calculations. We argue that this one-loop result is not affected by higher orders. We discuss in detail how the impact of the boundary conditions can be untangled from the soliton mass calculation. In particular, the soliton profile and the energy distribution are found at one loop. A "supersymmetry" in the soliton mass calculations in the non-supersymmetric models is observed. 
  A connection between the Hagedorn transition in string theory and the deconfinement transition in (non-supersymmetric) Yang-Mills theory is made using the AdS/CFT correspondence. We modify the model of zero temperature QCD proposed by Witten by compactifying an additional space-time coordinate with supersymmetry breaking boundary conditions thus introducing a finite temperature in the boundary theory. There is a Hagedorn-like transition associated with winding modes around this coordinate, which signals a topology changing phase transition to a new AdS/Schwarzschild blackhole where this coordinate is the time coordinate. In the boundary gauge theory time like Wilson loops acquire an expectation value above this temperature. 
  We present a review of gravitating particle-like and black hole solutions with non-Abelian gauge fields. The emphasis is given to the description of the structure of the solutions and to the connection with the results of flat space soliton physics. We describe the Bartnik-McKinnon solitons and the non-Abelian black holes arising in the Einstein-Yang-Mills theory, and their various generalizations. These include axially symmetric and slowly rotating configurations, solutions with higher gauge groups, $\Lambda$-term, dilaton, and higher curvature corrections. The stability issue is discussed as well. We also describe the gravitating generalizations for flat space monopoles, sphalerons, and Skyrmions. 
  Chern-Simons gauge theory, since its inception as a topological quantum field theory, has proved to be a rich source of understanding for knot invariants. In this work the theory is used to explore the definition of the expectation value of a network of Wilson lines - an embedded graph invariant. Using a slight generalization of the variational method, lowest-order results for invariants for arbitrary valence graphs are derived; gauge invariant operators are introduced; and some higher order results are found. The method used here provides a Vassiliev-type definition of graph invariants which depend on both the embedding of the graph and the group structure of the gauge theory. It is found that one need not frame individual vertices. Though, without a global projection of the graph, there is an ambiguity in the relation of the decomposition of distinct vertices. It is suggested that framing may be seen as arising from this ambiguity - as a way of relating frames at distinct vertices. 
  Noncommutative torus compactification of Matrix model is shown to be a direct consequence of quantization of the open strings attached to a D-membrane with a non-vanishing background $B$ field. We calculate the BPS spectrum of such a brane system using both string theory results and DBI action. The DBI action leads to a new transformation property of the compactification radii under the $SL(2,Z)_N$ transformations. 
  We derive a first order formalism for solving the scattering of point sources in (2+1) gravity with negative cosmological constant. We show that their physical motion can be mapped, with a polydromic coordinate transformation, to a trivial motion, in such a way that the point sources move as time-like geodesics (in the case of particles) or as space-like geodesics (in the case of BTZ black holes) of a three-dimensional hypersurface immersed in a four-dimensional Minkowskian space-time, and that the two-body dynamics is solved by two invariant masses, whose difference is simply related to the total angular momentum of the system. 
  For nonrelativistic Hamiltonians which are shape invariant, analytic expressions for the eigenvalues and eigenvectors can be derived using the well known method of supersymmetric quantum mechanics. Most of these Hamiltonians also possess spectrum generating algebras and are hence solvable by an independent group theoretic method. In this paper, we demonstrate the equivalence of the two methods of solution by developing an algebraic framework for shape invariant Hamiltonians with a general change of parameters, which involves nonlinear extensions of Lie algebras. 
  The consistency of Matrix theory with supergravity requires that in the large N_c limit terms of order v^4 in the SU(N_c) Matrix effective potential are not renormalized beyond one loop in perturbation theory. For SU(2) gauge group, the required non-renormalization theorem was proven recently by Paban, Sethi and Stern. In this paper we consider the constraints supersymmetry imposes on these terms for groups SU(N_c) with N_c>2. Non-renormalization theorems are proven for certain tensor structures, including the structures that appear in the one-loop effective action. However it is expected other tensor structures can in general be present, which may suffer renormalization at three loops and beyond. 
  The WDVV equations of associativity in 2-d topological field theory are completely integrable third order Monge-Amp\`ere equations which admit bi-Hamiltonian structure. The time variable plays a distinguished role in the discussion of Hamiltonian structure whereas in the theory of WDVV equations none of the independent variables merits such a distinction. WDVV equations admit very different alternative Hamiltonian structures under different possible choices of the time variable but all these various Hamiltonian formulations can be brought together in the framework of the covariant theory of symplectic structure. They can be identified as different components of the covariant Witten-Zuckerman symplectic 2-form current density where a variational formulation of the WDVV equation that leads to the Hamiltonian operator through the Dirac bracket is available. 
  A theory recently proposed by the author aims to explain decoherence and the thermodynamical behaviour of closed systems within a conservative, unitary, framework for quantum gravity by assuming that the operators tied to the gravitational degrees of freedom are unobservable and equating physical entropy with matter-gravity entanglement entropy. Here we obtain preliminary results on the extent of decoherence this theory predicts. We treat first a static state which, if one were to ignore quantum gravitational effects, would be a quantum superposition of two spatially displaced states of a single classically well describable ball of uniform mass density in empty space. Estimating the quantum gravitational effects on this system within a simple Newtonian approximation, we obtain formulae which predict e.g. that as long as the mass of the ball is considerably larger than the Planck mass, such a would-be-coherent static superposition will actually be decohered whenever the separation of the centres of mass of the two ball-states excedes a small fraction (which decreases as the mass of the ball increases) of the ball radius. We then obtain a formula for the quantum gravitational correction to the would-be-pure density matrix of a non-relativistic many-body Schroedinger wave function and argue that this formula predicts decoherence between configurations which differ (at least) in the "relocation" of a cluster of particles of Planck mass. We estimate the entropy of some simple model closed systems, finding a tendency for it to increase with "matter-clumping" suggestive of a link with existing phenomenological discussions of cosmological entropy increase. 
  We continue our study of non-Abelian gauge theories in the framework of Epstein-Glaser approach to renormalisation theory. We consider the case when massive spin-one Bosons are present into the theory and we modify appropriately the analysis of the origin of gauge invariance performed in a preceding paper in the case of null-mass spin-one Bosons. Then we are able to extend a result of D\"utsch and Scharf concerning the uniqueness of the standard model consistent with renormalisation theory. In fact we consider the most general case i.e. the consistent interaction of $r$ spin-one Bosons and we do not impose any restriction on the gauge group and the mass spectrum of the theory. We show that, beside the natural emergence of a group structure (like in the massless case) we obtain, new conditions of group-theoretical nature, namely the existence of a certain representation of the gauge group associated to the Higgs fields. Some other mass relations connecting the structure constants of the gauge group and the masses of the Bosons emerge naturally. The proof is done using Epstein-Glaser approach to renormalisation theory. 
  Convergence of the Schwinger --- DeWitt expansion for the evolution operator kernel for special class of potentials is studied. It is shown, that this expansion, which is in general case asymptotic, converges for the potentials considered (widely used, in particular, in one-dimensional many-body problems), and besides, convergence takes place only for definite discrete values of the coupling constant. For other values of the charge divergent expansion determines the kernels having essential singularity at origin (beyond usual $\delta$-function). If one consider only this class of potentials then one can avoid many problems, connected with asymptotic expansions, and one get the theory with discrete values of the coupling constant that is in correspondence with discreteness of the charge in the nature. This approach can be transmitted into the quantum field theory. 
  The present state of QFT is analysed from a new viewpoint whose mathematical basis is the modular theory of von Neumann algebras. Its physical consequences suggest new ways of dealing with interactions, symmetries, Hawking-Unruh thermal properties and possibly also extensions of the scheme of renormalized perturbation theory. Interactions are incorporated by using the fact that the S-matrix is a relative modular invariant of the interacting- relative to the incoming- net of wedge algebras. This new point of view allows many interesting comparisions with the standard quantization approach to QFT and is shown to be firmly rooted in the history of QFT. Its radical ``change of paradigm'' aspect becomes particularily visible in the quantum measurement problem. Key words: Quantum Field Theory, S-matrix Theory, Tomita-Takesaki Modular Theory. 
  The superembedding approach to $p$-branes is used to study a class of $p$-branes which have linear multiplets on the worldvolume. We refer to these branes as L-branes. Although linear multiplets are related to scalar multiplets (with 4 or 8 supersymmetries) by dualising one of the scalars of the latter to a $p$-form field strength, in many geometrical situations it is the linear multiplet version which arises naturally. Furthermore, in the case of 8 supersymmetries, the linear multiplet is off-shell in contrast to the scalar multiplet. The dynamics of the L-branes are obtained by using a systematic procedure for constructing the Green-Schwarz action from the superembedding formalism. This action has a Dirac-Born-Infeld type structure for the $p$-form. In addition, a set of equations of motion is postulated directly in superspace, and is shown to agree with the Green-Schwarz equations of motion. 
  The field-theoretical renormalization group approach in three dimensions is used to estimate the universal critical values of renormalized coupling constants g_6 and g_8 for the O(n)-symmetric model. The RG series for g_6 and g_8 are calculated in the four-loop and three-loop approximations respectively and then resummed by means of the Pade-Borel-Leroy technique. Under the optimal value of the shift parameter b providing the fastest convergence of the iteration procedure numerical estimates for the universal critical values g_6^*(n) are obtained for n = 1, 2, 3,...40 with the accuracy no worse than 0.3%. The RG expansion for g_8 demonstrates stronger divergence and results in considerably cruder numerical estimates. They are found to be consistent with the values of g_8^* deduced from the exact RG equations and, for n > 8, with those given by a constrained analysis of corresponding \epsilon-expansion. 
  In Strominger's proposal for the computation of the statistical entropy of black holes based on the asymptotic symmetry analysis of Brown and Henneaux a fundamental role is played by the asymptotic conditions that the considered metric must satisfy at infinity. Here it is shown that $T$-duality does not preserve such conditions. This observation is used to discuss a possible reformulation of the proposal. 
  A simple argument against the existence of magnetic monopoles is given. The argument is an important part of the quantum theory of the electric charge developed by the author. 
  The theory of the quantum Coulomb field associates with each Lorentz frame, i. e., with each unit, future oriented time-like vector $u$, the operator of the number of transversal infrared photons $N(u)$ and the phase $S(u)$ which is the coordinate canonically conjugated with the total charge $Q: [Q, S(u)] = ie, e$ being the elementary charge. It is shown that the operators $N(u), Q/e S(u)$ and $Q^2$ form an infinite Lie algebra. One can conclude from this algebra that $\DeltaN(u) = (4/\pi) Q^2$, where $\Delta$ is the Laplace operator in the Lobachevsky space of four-velocities $u$, thus relating the total charge $Q$ with the number of infrared photons. 
  A conceptual summary is given of a deterministic unified field and particle theory (the metron model) developed in more mathematical detail in a four-part paper published in Physics Essays (1996/97). The model is developed from Einsteins vacuum gravitational equations, Ricci tensor $R_{LM}=0$, in a higher dimensional space. It is postulated that the equations support soliton-type solutions (metrons) which reproduce all the basic field equations of quantum field theory, including not only the Maxwell-Dirac-Einstein system, but also all fields and symmetries of the Standard Model. Bell's theorem on the non-existence of hidden-variable models of quantum phenomena is circumvented through the time-reversal symmetry of all interactions on microphysical scales. The model, when completed, should yield all particle properties and universal physical constants from first principles. 
  It was recently shown that the renormalization of quantum field theory is organized by the Hopf algebra of decorated rooted trees, whose coproduct identifies the divergences requiring subtraction and whose antipode achieves this. We automate this process in a few lines of recursive symbolic code, which deliver a finite renormalized expression for any Feynman diagram. We thus verify a representation of the operator product expansion, which generalizes Chen's lemma for iterated integrals. The subset of diagrams whose forest structure entails a unique primitive subdivergence provides a representation of the Hopf algebra ${\cal H}_R$ of undecorated rooted trees. Our undecorated Hopf algebra program is designed to process the 24,213,878 BPHZ contributions to the renormalization of 7,813 diagrams, with up to 12 loops. We consider 10 models, each in 9 renormalization schemes. The two simplest models reveal a notable feature of the subalgebra of Connes and Moscovici, corresponding to the commutative part of the Hopf algebra ${\cal H}_T$ of the diffeomorphism group: it assigns to Feynman diagrams those weights which remove zeta values from the counterterms of the minimal subtraction scheme. We devise a fast algorithm for these weights, whose squares are summed with a permutation factor, to give rational counterterms. 
  We express the Pontryagin index in Polyakov gauge completely in terms of magnetically charged gauge fixing defects, namely magnetic monopoles, lines, and domain walls. Open lines and domain walls are topologically equivalent to monopoles, which are the genuine defects. The emergence of non-genuine magnetically charged closed domain walls can be avoided by choosing the temporal gauge field smoothly. The Pontryagin index is then exclusively determined by the magnetic monopoles. 
  In this article we investigate charged particles in gauge theories. After reviewing the physical and theoretical problems, a method to construct charged particles is presented. Explicit solutions are found in the Abelian theory and a physical interpretation is given. These solutions and our interpretation of these variables as the true degrees of freedom for charged particles, are then tested in the perturbative domain and are demonstrated to yield infra-red finite, on-shell Green's functions at all orders of perturbation theory. The extension to collinear divergences is studied and it is shown that this method applies to the case of massless charged particles. The application of these constructions to the charged sectors of the standard model is reviewed and we conclude with a discussion of the successes achieved so far in this programme and a list of open questions. 
  We find solutions of the bosonic sector of gauged N=4, D=4 SU(2) $\times$ SU(2) supergravity, which represent dilaton black holes with toroidal or spherical event horizons. The axion is consistently truncated, and the gauge group is broken to U(1) $\times$ U(1). The spherical black holes carry two electric and two magnetic abelian charges, whereas the toroidal holes have vanishing magnetic charges. The spacetime metrics are warped products, and the manifolds turn out to be globally hyperbolic, in contrast to standard gauged supergravity ground states. It is shown that in the toroidal case, there are solutions preserving one quarter or one half of the supersymmetries, while for spherical topologies all supersymmetries are broken. In general, the toroidal BPS states represent naked singularities, but there is also a supersymmetric black hole with vanishing Hawking temperature. The 1/2 supersymmetric case arises for vanishing charges and mass, and represents the known domain wall solution of the Freedman-Schwarz model. It provides the background in which the black holes live. Finally, we use Chamseddine's and Volkov's Kaluza-Klein interpretation of gauged N=4, D=4 SU(2) $\times$ SU(2) supergravity to lift our solutions to ten and eleven dimensions and to consider them as solutions to the leading order equations of motion of the string-/M-theory effective action. 
  The review is devoted to the integrable properties of the Generalized Kontsevich Model which is supposed to be an universal matrix model to describe the conformal field theories with $c<1$. It is shown that the deformations of the "monomial" phase to "polynomial" one have the natural interpretation in context of so-called equivalent hierarchies. The dynamical transition between equivalent integrable systems is exactly along the flows of the dispersionless Kadomtsev-Petviashvili hierarchy; the coefficients of the potential are shown to be directly related with the flat (quasiclassical) times arising in N=2 Landau-Ginzburg topological model. The Virasoro constraint for solution with an arbitrary potential is shown to be a standard $\L_{-p}$-constraint of the (equivalent) $p$-reduced hierarchy with the times additively corrected by the flat coordinates. 
  We construct BPS-exact solutions of the worldvolume Born-Infeld plus WZW action of a D5-brane in the background of N D3-branes. The non-trivial background metric and RR five-form field strength play a crucial role in the solution. When a D5-brane is dragged across a stack of N D3-branes a bundle of N fundamental strings joining the two types of branes is created, as in the Hanany-Witten effect. Our solutions give a detailed description of this bundle in terms of a D5-brane wrapped on a sphere. We discuss extensions of these solutions which have an interpretation in terms of gauge theory multi-quark states via the AdS/CFT correspondence. 
  A representation of the Lorentz group is given in terms of 4 X 4 matrices defined over a simple non-division algebra. The transformation properties of the corresponding four component spinor are studied, and shown to be equivalent to the transformation properties of the usual complex Dirac spinor. As an application, we show that there exists an algebra of automorphisms of the complex Dirac spinor that leave the transformation properties of its eight real components invariant under any given Lorentz transformation. Interestingly, the representation of the Lorentz group presented here has a natural embedding in SO(3,3) instead of the conformal symmetry SO(2,4). 
  A recent investigation of the SU(3) Yang-Mills field equations found several classical solutions which exhibited a type of confinement due to gauge fields which increased without bound as $r \to \infty$. This increase of the gauge fields gave these solutions an infinite field energy, raising questions about their physical significance. In this paper we apply some ideas of Heisenberg about the quantization of strongly interacting, non-linear fields to this classical solution and find that at large $r$ this quantization procedure softens the unphysical behaviour of the classical solution, while the interesting short distance behaviour is still maintained. This quantization procedure may provide a general method for approximating the quantum corrections to certain classical field configurations. 
  Using the background-metric independence for the traceless mode as well as the conformal mode, 4D quantum gravity is described as a quantum field theory defined on a non-dynamical background-metric. The measure then induces an action with 4 derivatives. So we think that 4-th order gravity is essential and the Einstein-Hilbert term should be treated like a mass term. We introduce the dimensionless self-coupling constant t for the traceless mode. In this paper we study a model where the measure can be evaluated in the limit $t \to 0$ exactly, using the background-metric independence for the conformal mode. The t-dependence of the measure is determined perturbatively using the background-metric independence for the traceless mode. 
  We derive the operator content of the closed SU(2)_q invariant quantum chain for generic values of the deformation parameter q. 
  The charged black hole is considered from the viewpoint of D0-brane in the Matrix theory. It can be obtained from the Kaluza-Klein mechanism by boosting the Schwarzschild black hole in a circle, which is the compactified one dimensional space. Especially, how the extremal limit is realized by the Boltzmann gas of D0-brane, has been shown. In the course of our discussion, the Virial theorem for the statistical average plays an important role. 
  We make a complete pole analysis of the reflection factors of the boundary scaling Lee-Yang model. In the process we uncover a number of previously unremarked mechanisms for the generation of simple poles in boundary reflection factors, which have implications for attempts to close the boundary bootstrap in more general models. We also explain how different boundary conditions can sometimes share the same fundamental reflection factor, by relating the phenomenon to potential ambiguities in the interpretation of certain poles. In the case discussed, this ambiguity can be lifted by specifying the sign of a bulk-boundary coupling. While the recipe we employ for the association of poles with general on-shell diagrams is empirically correct, we stress that a justification on the basis of more fundamental principles remains a challenge for future work. 
  We investigate in the Matrix theory framework, the subgroup of dualities of the DLCQ of M-theory compactified on three-tori, which corresponds to T-duality in the auxiliary Type II string theory. We show how these dualities are realized in the supersymmetric Yang-Mills gauge theories on dual noncommutative three-tori. 
  We revisit Wigner's question about the admissible commutation relations for coordinate and velocity operators given their equations of motion (EOM). In more general terms we want to consider the question of how to quantize dynamically equivalent Hamiltonian structures. A unique answer can presumably be given in those cases, where we have a dynamical symmetry. In this case arbitrary deformations of the symmetry algebra should be dynamically equivalent. We illustrate this for the linear as well as the singular 1d-oscillator. In the case of nonlinear EOM quantum corrections have to be taken into account. We present some examples thereof New phenomena arise in case of more then one degree of freedom, where sometimes the interaction can be described either by the Hamiltonian or by nonstandard commutation relations. This may induce a noncommutative geometry (for example the 2d-oscillator in a constant magnetic field). Also some related results from nonrelativistic quantum field theory applied to solid state physics are briefly discussed within this framework 
  Gogilidze and Surovtsev have claimed recently (hep-th/9809191) that the tachyonic sector can be removed from the spectrum of the relativistic particle with curvature and torsion by a proper gauge choice. We show that the mass-spin dependence obtained by them is incorrect and point out that their gauge surface does not cross all the gauge orbits. We discuss the nature of the tachyonic sector of the model and argue why it cannot be removed by any gauge fixing procedure. 
  We prove a c-theorem for holographic theories. 
  The Rarita-Schwinger equations are generalised for the delta baryon having spin 3/2 and isospin 3/2.  The coupling of the nucleon and the delta fields is studied. A possible generalisation of the Walecka model is proposed. 
  In these lectures we describe the construction of a gauge invariant renormalization group equation for pure non-Abelian gauge theory. In the process, a non-perturbative gauge invariant continuum Wilsonian effective action is precisely defined. The formulation makes sense without gauge fixing and thus manifest gauge invariance may be preserved at all stages. In the large N limit (of SU(N) gauge theory) the effective action simplifies: it may be expressed through a path integral for a single particle whose trajectory describes a Wilson loop. Regularization is achieved with the help of a set of Pauli-Villars fields whose formulation follows naturally in this picture. Finally, we show how the one loop beta function was computed, for the first time without any gauge fixing. 
  The wave functions of the Haldane-Rezayi paired Hall state have been previously described by a non-unitary conformal field theory with central charge c=-2. Moreover, a relation with the c=1 unitary Weyl fermion has been suggested. We construct the complete unitary theory and show that it consistently describes the edge excitations of the Haldane-Rezayi state. Actually, we show that the unitary (c=1) and non-unitary (c=-2) theories are related by a local map between the two sets of fields and by a suitable change of conjugation. The unitary theory of the Haldane-Rezayi state is found to be the same as that of the 331 paired Hall state. Furthermore, the analysis of modular invariant partition functions shows that no alternative unitary descriptions are possible for the Haldane-Rezayi state within the class of rational conformal field theories with abelian current algebra. Finally, the known c=3/2 conformal theory of the Pfaffian state is also obtained from the 331 theory by a reduction of degrees of freedom which can be physically realized in the double-layer Hall systems. 
  In 2+1 dimensions, for low momenta, using dimensional renormalization we study the effect of a Chern-Simons field on the perturbative expansion of fermions self interacting through a Gross Neveu coupling. For the case of just one fermion field, we verify that the dimension of operators of canonical dimension lower than three decreases as a function of the Chern-Simons coupling. 
  The relative entropy in two-dimensional field theory is studied on a cylinder geometry, interpreted as finite-temperature field theory. The width of the cylinder provides an infrared scale that allows us to define a dimensionless relative entropy analogous to Zamolodchikov's $c$ function. The one-dimensional quantum thermodynamic entropy gives rise to another monotonic dimensionless quantity. I illustrate these monotonicity theorems with examples ranging from free field theories to interacting models soluble with the thermodynamic Bethe ansatz. Both dimensionless entropies are explicitly shown to be monotonic in the examples that we analyze. 
  Interquark confinement potential is calculated in the dual Monopole Nambu-Jona-Lasinio model with dual Dirac strings as a functional of a dual Dirac string length. The calculation is carried out by the explicit integration over quantum fluctuations of a dual-vector field (monopole-antimonopole collective excitation) around the Abrikosov flux line and string shape fluctuations. The contribution of the scalar field (monopole-antimonopole collective excitation) exchange is taken into account in the tree approximation due to the London limit regime. 
  In this talk we use nonlinear realizations to study the spontaneous partial breaking of rigid and local supersymmetry. 
  We show that the BPS configurations of uniform field strength can be interpreted as those for sheets of infinite number of BPS magnetic monopoles, and found that the number of normalizable zero modes per each magnetic monopole charge is four. We identify monopole sheets as the intersecting planes of D3 branes. Similar analysis on self-dual instanton configurations is worked out and the number of zero modes per each instanton number is found to match that of single isolated instanton. 
  In the light of $\phi$-mapping method and topological current theory, the topological structure and the topological quantization of arbitrary dimensional topological defects are obtained under the condition that the Jacobian $J(\phi/v) \neq 0$. When $J(\phi/v)=0$, it is shown that there exist the crucial case of branch process. Based on the implicit function theorem and the Taylor expansion, we detail the bifurcation of generalized topological current and find different directions of the bifurcation. The arbitrary dimensional topological defects are found splitting or merging at the degenerate point of field function $\vec \phi$ but the total charge of the topological defects is still unchanged. 
  A path-integral representation for the kernel of the evolution operator of general Hamiltonian systems is reviewed. We study the models with bosonic and fermionic degrees of freedom. A general scheme for introducing boundary conditions in the path-integral is given. We calculate the path-integral for the systems with quadratic first class constraints and present an explicit formula for the heat kernel in this case. These results may be applied to many quantum systems which can be reduced to the Hamiltonian systems with quadratic constraints (confined quarks, Calogero type models, string and $p$-brain theories, etc.). 
  We systematically study the exclusion statistics for quasi-particles for Conformal Field Theory spectra by employing a method based on recursion relations for truncated spectra. Our examples include generalized fermions in c<1 unitary minimal models, Z_k parafermions, and spinons for the su(n)_1, so(n)_1 and sp(2n)_1 Wess-Zumino-Witten models. For some of the latter examples we present explicit expressions for finitized affine characters and for the N-spinon decomposition of affine characters. 
  We consider a quantization of relativistic wave equations which allows to treat quantum fields together with interacting particles at a finite time. We discuss also a dissipative interaction with the environment. We introduce a stochastic wave function whose dynamics is determined by a non-linear Schr\"odinger-type evolution equation in an additional time parameter.   The correct classical limit requires the proper time interpretation of the time parameter. An average over the proper time leads to the conventional quantum field theory of particles which are free at an infinite space separation. We consider models with scalar and vector fields on a pseudoriemannian manifold. A quantization of the Einstein gravity in this approach is briefly discussed. 
  We consider quantum-mechanical path integrals for non-linear sigma models on a circle defined by the string-inspired method of Strassler, where one considers periodic quantum fluctuations about a center-of-mass coordinate. In this approach one finds incorrect answers for the local trace anomalies of the corresponding $n$-dimensional field theories in curved space. The quantum field theory approach to the quantum-mechanical path-integral, where quantum fluctuations are not periodic but vanish at the endpoints, yields the correct answers. We explain these results by a detailed analysis of general coordinate invariance in both methods. Both approaches can be derived from the same operator expression and the integrated trace anomalies in both schemes agree. In the string-inspired method the integrands are not invariant under general coordinate transformations and one is therefore not permitted to use Riemann normal coordinates. 
  We discuss Born-Infeld on the noncommutative two-torus as a description of compactified string theory. We show that the resulting theory, including the fluctuations, is manifestly invariant under the T-duality group SO(2,2;Z). The BPS mass even has a full SL(3,Z)xSL(2,Z) U-duality symmetry. The direct identification of the noncommutative parameter \theta with the B-field modulus however seems to be problematic at finite volume. 
  We give a wilsonian formulation of non-abelian gauge theories explicitly consistent with axial gauge Ward identitities. The issues of unitarity and dependence on the quantization direction are carefully investigated. A wilsonian computation of the one-loop QCD beta function is performed. 
  The simplest supersymmetry (SUSY) algebra in four dimensional Euclidean space ($4dE$) has been shown to closely resemble the $N = 2$ SUSY algebra in four dimensional Minkowski space ($4dM$). The structure of the former algebra is examined in greater detail in this paper. We first present its Clifford algebra structure. This algebra shows that the momentum Casimir invariant of physical states has an upper bound which is fixed by the central charges. Secondly, we use reduction of the $N = 1$ SUSY algebra in six dimensional Minkowski space ($6dM$) to $4dE$; this reproduces our SUSY algebra in $4dE$. Moreover, this same reduction of supersymmetric Yang-Mills theory (SSYM) in $6dM$ reproduces Zumino's SSYM in $4dE$. We demonstrate how this dimensional reduction can be used to introduce additional generators into the SUSY algebra in $4dE$. 
  The path integral representation of the transition amplitude for a particle moving in curved space has presented unexpected challenges since the introduction of path integrals by Feynman fifty years ago. In this paper we discuss and review mode regularization of the configuration space path integral, and present a three loop computation of the transition amplitude to test with success the consistency of such a regularization. Key features of the method are the use of the Lee-Yang ghost fields, which guarantee a consistent treatment of the non-trivial path integral measure at higher loops, and an effective potential specific to mode regularization which arises at two loops. We also perform the computation of the transition amplitude using the regularization of the path integral by time discretization, which also makes use of Lee-Yang ghost fields and needs its own specific effective potential. This computation is shown to reproduce the same final result as the one performed in mode regularization. 
  We show equivalence between the massive Thirring model and the sine-Gordon theory by gauge fixing a wider gauge invariant theory in two different ways. The exact derivation of the equivalence hinges on the existence of an underlying conformal symmetry. Previous derivations were all perturbative in mass (althought to all orders). 
  We review a new prescription for calculating the Lagrangian path integral measure directly from the Hamiltonian Schwinger-Dyson equations. The method agrees with the usual way of deriving the measure in which one has to perform the path integration over momenta. 
  In a recently developed approximation technique for quantum field theory the standard one-loop result is used as a seed for a recursive formula that gives a sequence of improved Gaussian approximations for the generating functional. In this paper we work with the generic $\phi^3+\phi^4$ model in $d=0$ dimensions. We compare the first, and simplest, approximation in the above sequence with the one-loop and two-loop approximations, as well as the exact results (calculated numericaly). 
  We study the spacetimes of the near-horizon regions in D3-brane, M2-brane and M5-brane configurations, in cases where there is a pp-wave propagating along a direction in the world-volume. While non-extremal configurations of this kind locally have the same Carter-Novotny-Horsky-type metrics as those without the wave, taking the BPS limit results instead in Kaigorodov-type metrics, which are homogeneous, but preserve 1/4 of the supersymmetry, and have global and local structures that are quite different from the corresponding anti-de Sitter spacetimes associated with solutions where there is no pp-wave. We show that the momentum density of the system is non-vanishing and held fixed under the gravity decoupling limit. In view of the AdS/CFT correspondence, M-theory and type IIB theory in the near-horizon region of these boosted BPS-configurations specifies the corresponding CFT on the boundary in an infinitely-boosted frame with constant momentum density. We model the microstates of such boosted configurations (which account for the microscopic counting of near-extremal black holes in D=7, D=9 and D=6) by those of a boosted dilute massless gas in a d=4, d=3 and d=6 spacetime respectively. Thus we obtain a simple description for the entropy of 2-charge black holes in D=7,9 and 6 dimensions. The paper includes constructions of generalisations of the Kaigorodov and Carter-Novotny-Horsky metrics in arbitrary spacetime dimensions, and an investigation of their properties. 
  We construct quantum deformations of the integrals of motion of the generalized mKdV equations for $\hat {\SL}_2$. For this, we give the relevant vertex operator algebra and prove quantum Serre relations for vertex operators, it allows to construct a $q$-BGG resolution and to deform the classical integrals of motion in a commutativ family. 
  There is an elaborated abstract form of BRST quantization on inner product spaces within the operator formalism which leads to BRST invariant states of the form |ph>=e^{[Q,\psi]} |\phi> where \psi is a gauge fixing fermion, and where |\phi> is a BRST invariant state determined by simple hermitian conditions. These state representations are closely related to the path integral formulation. Here we analyse the basics of this approach in detail. The freedom in the choice of \psi and |\phi> as well as their properties under gauge transformations are explicitly determined for simple abelian models. In all considered cases SL(2,R) is shown both to be a natural extended gauge symmetry and to be useful to determine |ph>. The results are also applied to nonabelian models. 
  We find new, local, non-supersymmetric conformal field theories obtained by relevant deformations of the N=4 super Yang Mills theory in the large $N$ limit. We contruct interpolating supergravity solutions that naturally represent the flow from the N=4 super Yang Mills UV theory to these non-supersymmetric IR fixed points. We also study the linearization around the N=4 superconformal point of N=1 supersymmetric, marginal deformations. We show that they give rise to N=1 superconformal fixed points, as expected from field-theoretical arguments. 
  We study the effects of quenching in Super-Yang-Mills theory. While supersymmetry is broken, the lagrangian acquires a new flavour U(1 | 1) symmetry. The anomaly structure thus differs from the unquenched case. We derive the corresponding low-energy effective lagrangian. As a consequence, we predict the mass splitting expected in numerical simulations for particles belonging to the lowest-lying supermultiplet. An estimate of the systematic error due to quenching follows. 
  We study the dilaton/axion configuration near D-instantons in type IIB superstring theory. In the field theory limit, the metric near the instantons becomes flat in the string frame as well as in the Einstein frame. In the large N limit, the string coupling constant becomes zero except near the origin. The supersymmetry of this configuration is analyzed. An implication of this result to the IIB Matrix Model is discussed. 
  It has recently been suggested that in certain special nonsupersymmetric type II string compactifications, at least the first two perturbative contributions to the cosmological constant $\Lambda$ vanish. Support for perturbative vanishing beyond 1-loop (as well as evidence for the absence of some nonperturbative contributions) has come from duality arguments. There was also a direct 2-loop computation which was incomplete; in this note we explain the deficiency of the previous 2-loop calculation and discuss the complete 2-loop computation in two different models. The corrected analysis yields a vanishing 2-loop contribution to $\Lambda$ in these models. 
  An heuristic derivation of the tranformation law for the Berezin integration measure in noncompact supermanifolds, obtained by Roshstein \cite{Ro}, is presented. 
  We compute the expectations of the squares of the electric and magnetic fields in the vacuum region outside a half-space filled with a uniform non-dispersive dielectric. This gives predictions for the Casimir-Polder force on an atom in the `retarded' regime near a dielectric. We also find a positive energy density due to the electromagnetic field. This would lead, in the case of two parallel dielectric half-spaces, to a positive, separation-independent contribution to the energy density, besides the negative, separation-dependent Casimir energy. Rough estimates suggest that for a very wide range of cases, perhaps including all realizable ones, the total energy density between the half-spaces is positive. 
  We consider Type IIB Neveu-Schwarz five-branes transverse to C^2/Z_n orbifolds and conjecture that string theory on the near horizon geometry is dual to the decoupled theory on the branes. We analyze the conformal field theory describing the near horizon region and the world volume non-critical string theory. The modular invariance consistency condition of string theory is exactly reproduced as the gauge anomaly cancellation condition in the little string theories. We comment on aspects of the holographic nature of this duality. 
  Some new expressions are found, concerning the effective action as a regularized path-integral for Dirac's spinors in {\it 3+1} dimensions, in the presence of general uniform (i.e., constant and homogeneous) electric and magnetic fields. The rate of $e^+e^-$ pairs production is computed and briefly discussed. 
  Scale factor duality, a truncated form of time dependent T-duality, is a symmetry of string effective action in cosmological backgrounds interchanging small and large scale factors. The symmetry suggests a cosmological scenario ("pre-big-bang") in which two duality related branches, an inflationary branch and a decelerated branch are smoothly joined into one non-singular cosmology. The use of scale factor duality in the analysis of the higher derivative corrections to the effective action, and consequences for the nature of exit transition, between the inflationary and decelerated branches, are outlined. A new duality symmetry is obeyed by the lowest order equations for inhomogeneity perturbations which always exist on top of the homogeneous and isotropic background. In some cases it corresponds to a time dependent version of S-duality, interchanging weak and strong coupling and electric and magnetic degrees of freedom, and in most cases it corresponds to a time dependent mixture of both S-, and T-duality.   The energy spectra obtained by using the new symmetry reproduce known results of produced particle spectra, and can provide a useful lower bound on particle production when our knowledge of the detailed dynamical history of the background is approximate or incomplete. 
  We investigate the connection between the BTZ black holes and the near-horizon geometry of higher-dimensional black holes. Under mild conditions, we show that (i) if a black hole has a global structure of the type of the non-extremal Reissner-Nordstrom black holes, its near-horizon geometry is $AdS_2$ times a sphere, and further (ii) if such a black hole is obtained from a boosted black string by dimensional reduction, the near-horizon geometry of the latter contains a BTZ black hole. Because of these facts, the calculation of the Bekenstein-Hawking entropy and the absorption cross-sections of scalar fields is essentially reduced to the corresponding calculation in the BTZ geometry under appropriate conditions. This holds even if the geometry is not supersymmetric in the extremal limit. Several examples are discussed. We also discuss some generalizations to geometries which do not have $AdS$ near the horizon. 
  We show that the super D3-brane action on $AdS_5 \times S^5$ background recently constructed by Metsaev and Tseytlin is exactly invariant under the combination of the electric-magnetic duality transformation of the worldvolume gauge field and the SO(2) rotaion of N=2 spinor coordinates. The action is shown to satisfy the Gaillard-Zumino duality condition, which is a necessary and sufficient condition for a action to be self-dual. Our proof needs no gauge fixing for the $\kappa$-symmetry. 
  We consider additional properties of CNM (chiral-nonminimal) models. We show how 4D, N = 2 nonlinear sigma-models can be described solely in terms of N = 1 superfield CNM doublets. These actions are described by a Kahler potential together with an infinite number (in the general case) of terms involving its successively higher derivatives. We briefly discuss how N = 2 supersymmetric extension of the previously proposed N = 1 CNM low-energy QCD effective action can be achieved 
  We put forward the following, physically motivated premise for constructing a theory that underlies the standard model in four-dimensional space-time: The Euler-Lagrange equations of such a theory formally resemble some equations of motion underlying fluid-dynamics equations in the kinetic theory of gases. Following this premise, we point out Lorentz-invariant Lagrangians whose Euler-Lagrange equations contain a subsystem equivalent to the Euler-Lagrange equations of the standard model with covariantly regularized propagators. 
  A new resonance version of NLS equation is found and embedded to the reaction-diffusion system, equivalent to the anti-de Sitter valued Heisenberg model, realizing a particular gauge fixing condition of the Jackiw-Teitelboim gravity. The space-time points where dispersion change the sign correspond to the event horizon, and the soliton solutions to the AdS black holes. The soliton with velocity bounded above describes evolution on the hyperboloid with nontrivial winding number and create under collisions the resonance states with a specific life time. 
  We analyse the world-sheet perturbations of string theory formulated around $AdS_3$ background. We identify a set of operators that, while added to the world-sheet action, generate the boundary fluctuations of $AdS_3$. The effect of these operators can be collectively defined in terms of Liouville field living on the $AdS_3$ boundary. We then study various deformations of $AdS_3$ generated by boundary fluctuations by turning on suitable world-sheet couplings. We also discuss certain small fluctuations around the BTZ black hole. 
  We investigate the possibility of semigroup extensions of the isometry group of an identification space, in particular, of a compactified spacetime arising from an identification map $p: \RR^n_t \to \RR^n_t / \Gamma$, where $\RR^n_t$ is a flat pseudo-Euclidean covering space and $\Gamma$ is a discrete group of primitive lattice translations on this space. We show that the conditions under which such an extension is possible are related to the index of the metric on the subvector space spanned by the lattice vectors: If this restricted metric is Euclidean, no extensions are possible. Furthermore, we provide an explicit example of a semigroup extension of the isometry group of the identification space obtained by compactifying a Lorentzian spacetime over a lattice which contains a lightlike basis vector. The extension of the isometry group is shown to be isomorphic to the semigroup $(\ZZ^{\times},\cdot)$, i.e. the set of nonzero integers with multiplication as composition and 1 as unit element. A theorem is proven which illustrates that such an extension is obstructed whenever the metric on the covering spacetime is Euclidean. 
  We review the status of microscopic counting of rotating black hole degrees of freedom. We present two complementary approaches which both utilize the near-horizon geometry and precisely reproduce the Bekenstein Hawking entropy for near-extreme rotating black holes in D=4 and D=5. The first one, proposed by Strominger, is applicable for the Ramond-Ramond sector and relies on the correspondence between the near-horizon geometry and the conformal field theory on its boundary. The second one (somewhat more heuristic), employs the conformal sigma-model in the Neveu-Schwarz-Neveu-Schwarz sector and accounts for the black hole microstates by counting the small scale oscillations of the dyonic string there. We also present the wave equation for the minimally coupled scalars for such rotating black hole background, and discuss its implication for the greybody factors. The results are illustrated for the prototype D=5 rotating black hole. 
  The proper definition and evaluation of the configuration space path integral for the motion of a particle in curved space is a notoriously tricky problem. We discuss a consistent definition which makes use of an expansion in Fourier sine series of the particle paths. Salient features of the regularization are the Lee-Yang ghosts fields and a specific effective potential to be added to the classical action. The Lee-Yang ghost fields are introduced to exponentiate the non-trivial path integral measure and make the perturbative loop expansion finite order by order, whereas the effective potential is necessary to maintain the general coordinate invariance of the model. We also discuss a three loop computation which tests the mode regularization scheme and reproduces consistently De Witt's perturbative solution of the heat kernel. 
  The effective potential for the composite fields responsible for chiral symmetry breaking in weakly coupled QED in a magnetic field is derived. The global minimum of the effective potential is found to acquire a non-vanishing expectation value of the composite fields that leads to generating the dynamical fermion mass by an external magnetic field. The results are compared with those for the Nambu-Jona-Lasinio model. 
  A manifestly Lorentz-covariant calculus based on two matrix-coordinates and their associated derivatives is introduced. It allows formulating relativistic field theories in any even-dimensional spacetime. The construction extends a single-coordinate matrix formalism based on coupling spacetime coordinates with the corresponding Gamma-matrices. A 2D matrix-calculus can be introduced for each one of the structures, adjoint, complex and transposed acting on Gamma-matrices. The adjoint structure works for spacetimes with (n,n) signature only. The complex structure requires an even number of timelike directions. The transposed structure is always defined. A further structure which can be referred as "spacetime-splitting" is based on a fractal property of the Gamma-matrices. It is present in spacetimes with dimension D=4n+2. The conformal invariance in the matrix-approach is analyzed. A complex conjugation is present for the complex structure, therefore in euclidean spaces, or spacetimes with (2,2), (2,4) signature and so on. As a byproduct it is here introduced an index which labels the classes of inequivalent Gamma-structures under conjugation performed by real and orthogonal matrices. At least two timelike directions are necessary to get more than one classes of equivalence. Furthermore an algorithm is presented for iteratively computing D-dimensional Gamma-matrices from the p and q dimensional ones where D=p+q+2. Possible applications of the 2D-matrix calculus concern the investigation of higher-dimensional field theories with techniques borrowed from 2D-physics. 
  We study in detail the extension of the generalized conformal symmetry proposed previously for D-particles to the case of supersymmetric Yang-Mills matrix models of Dp-branes for arbitrary p. It is demonstrated that such a symmetry indeed exists both in the Yang-Mills theory and in the corresponding supergravity backgrounds produced by Dp-branes. On the Yang-Mills side, we derive the field-dependent special conformal transformations for the collective coordinates of Dp-branes in the one-loop approximation, and show that they coincide with the transformations on the supergravity side. These transformations are powerful in restricting the forms of the effective actions of probe D-branes in the fixed backgrounds of source D-branes. Furthermore, our formalism enables us to extend the concept of (generalized) conformal symmetry to arbitrary configurations of D-branes, which can still be used to restrict the dynamics of D-branes. For such general configurations, however, it cannot be endowed a simple classical space-time interpretation at least in the static gauge adopted in the present formulation of D-branes. 
  Motivated by the recently observed relation between the physics of $D$-branes in the background of $B$-field and the noncommutative geometry we study the analogue of Nahm transform for the instantons on the noncommutative torus. 
  We study how local symmetry transformations of (p, q) anti de Sitter supergravities in three dimensions act on fields on the two-dimensional boundary. The boundary transformation laws are shown to be the same as those of two-dimensional (p, q) conformal supergravities for p, q \leq 2. Weyl and super Weyl transformations are generated from three-dimensional general coordinate and super transformations. 
  Talk given at ICM '98, Berlin, reviewing some of the recent developments in understanding of string theory for a mathematical audience (to appear in Documenta Mathematica). 
  Using the D3-brane as the fundamental tool, we adress two aspects of D-branes physics. The first regards the interaction between two electromagnetic dual D-branes in 10 dimensions. In particular, we give a meaning to {\it both} even and odd spin structure contributions, the latter being non vanishing for non zero relative velocity $v$ (and encoding the Lorentz-like contribution). The second aspect regards the D-brane/black holes correspondence. We show how the 4 dimensional configuration corresponding to a {\it single} D3-brane wrapped on the orbifold T^6/Z_3 represents a regular Reissner-Nordstrom solution of d=4 N=2 supergravity 
  Using the AdS-CFT correspondence we calculate the two point function of CFT energy momentum tensors. The AdS gravitons are considered by explicitly solving the Dirichlet boundary value problem for $x_0=\epsilon$. We consider this treatment as complementary to existing work, with which we make contact. 
  The full low energy effective action of N=4 SYM is believed to be self-dual. Starting with the first two leading terms in a momentum expansion of this effective action, we perform a duality transformation and find the conditions for self-duality. These determine some of the higher order terms. We compare the effective action of N=4 SYM with the probe-source description of type II_B D3-branes in the AdS_5 \times S_5 background. We find agreement up to six derivative terms if we identify the separation of the 3-branes with a redefinition of the gauge scalar that involves the gauge field strength. 
  The SL(2,Z) duality transformations of type IIB supergravity are shown to be anomalous in generic F-theory backgrounds due to the anomalous transformation of the phase of the chiral fermion determinant. The anomaly is partially cancelled provided the ten-dimensional type IIB theory lagrangian contains a term that is a ten-form made out of the composite U(1) field strength and four powers of the curvature. A residual anomaly remains uncancelled, and this implies a certain topological restriction on consistent backgrounds of the euclidean theory. A similar, but slightly stronger, restriction is also derived from an explicit F-theory compactification on K3 x M8 (where M8 is an eight-manifold with a nowhere vanishing chiral spinor) where the cancellation of tadpoles for Ramond--Ramond fields is only possible if M8 has an Euler character that is a positive multiple of 24. The interpretation of this restriction in the dual heterotic theory on T2 x M8 is also given. 
  Utilizing coset superspace approach, dual actions of super D1-and D3-branes on $AdS_5 \times S^5$ are constructed by carrying out duality transformation of world-volume U(1) gauge field. Resulting world-volume actions are shown to possess expected SL(2,{\bf Z}) properties. Crucial ingredient for deriving SL(2, {\bf Z}) transformation property of the D-brane actions is covariance of SU(2,2|4) coset superspace algebra under SO(2) rotation between two ten-dimensional Type IIB Majorana-Weyl spinors. 
  We show that in a general hidden sector model, supersymmetry breaking necessarily generates at one-loop a scalar and gaugino mass as a consequence of the super-Weyl anomaly. We study a scenario in which this contribution dominates. We consider the Standard Model particles to be localized on a (3+1)-dimensional subspace or ``3-brane'' of a higher dimensional spacetime, while supersymmetry breaking occurs off the 3-brane, either in the bulk or on another 3-brane. At least one extra dimension is assumed to be compactified roughly one to two orders of magnitude below the four-dimensional Planck scale. This framework is phenomenologically very attractive; it introduces new possibilities for solving the supersymmetric flavor problem, the gaugino mass problem, the supersymmetric CP problem, and the mu-problem. Furthermore, the compactification scale can be consistent with a unification of gauge and gravitational couplings. We demonstrate these claims in a four-dimensional effective theory below the compactification scale that incorporates the relevant features of the underlying higher dimensional theory and the contribution of the super-Weyl anomaly. Naturalness constraints follow not only from symmetries but also from the higher dimensional origins of the theory. We also introduce additional bulk contributions to the MSSM soft masses. This scenario is very predictive: the gaugino masses, squark masses, and $A$ terms are given in terms of MSSM renormalization group functions. 
  We study caustics in classical and quantum mechanics for systems with quadratic Lagrangians. We derive a closed form of the transition amplitude on caustics and discuss their physical implications in the Gaussian slit (gedanken-)experiment. Application to the quantum mechanical rotor casts doubt on the validilty of Jevicki's correspondence hypothesis which states that in quantum mechanics, stationary points(instantons) arise as simple poles. 
  Fivebranes are non-perturbative objects in string theory that generalize two-dimensional conformal field theory and relate such diverse subjects as moduli spaces of vector bundles on surfaces, automorphic forms, elliptic genera, the geometry of Calabi-Yau threefolds, and generalized Kac-Moody algebras. 
  The Discrete Light-Cone Quantization (DLCQ) of a supersymmetric gauge theory in 1+1 dimensions is discussed, with particular attention given to the inclusion of the gauge zero mode. Interestingly, the notorious `zero-mode' problem is now tractable because of special supersymmetric cancellations. In particular, we show that anomalous zero-mode contributions to the currents are absent, in contrast to what is observed in the non-supersymmetric case. An analysis of the vacuum structure is provided by deriving the effective quantum mechanical Hamiltonian of the gauge zero mode. It is shown that the inclusion of the zero modes of the adjoint scalars and fermions is crucial for probing the phase properties of the vacua. We find that the ground state energy is zero and thus consistent with unbroken supersymmetry, and conclude that the light-cone Fock vacuum is unchanged with or without the presence of matter fields. 
  We study the duality relationship between M-theory and heterotic string theory at the classical level, emphasising the transformations between the Kaluza-Klein reductions of these two theories on the K3 and T^3 manifolds. Particular attention is devoted to the corresponding structures of sigma-model cosets and the correspondence between the p-brane charge lattices. We also present simple and detailed derivations of the global symmetries and coset structures of the toroidally-compactified heterotic theory in all dimensions D \ge 3, making use of the formalism of solvable Lie algebras. 
  After having developed a method that measures real time evolution of quantum systems at a finite temperature, we present here the simplest field theory where this scheme can be applied to, namely the 1+1 Ising model.   We will compute the probability that if a given spin is up, some other spin will be up after a time $t$, the whole system being at temperature $T$. We can thus study spatial correlations and relaxation times at finite $T$. The fixed points that enable the continuum real time limit can be easily found for this model.   The ultimate aim is to get to understand real time evolution in more complicated field theories, with quantum effects such as tunneling at finite temperature. 
  We report on the status of the string-inspired world line path integral formalism, a recently developed powerful tool for the reorganisation of standard perturbative amplitudes in quantum field theory. The method is outlined and the present range of its applicability surveyed. The emphasis is on QED and QCD photon/gluon amplitudes, with a short discussion of axial couplings. 
  The non-perturbative validity of covariant BRST-quantization of gauge theories on compact Euclidean space-time manifolds is reviewed. BRST-quantization is related to the construction of a Topological Quantum Field Theory (TQFT) of Witten type on the gauge group. The criterion for the non-perturbative validity of the quantization is that the partition function of the corresponding TQFT does not vanish and that its (equivariant) BRST-algebra is free of anomalies. I sketch the construction of a TQFT whose partition function is proportional to the generalized Euler-characteristic of the coset space $SU(n)_{gauge}/SU(n)_{global}$ with an associated equivariant BRST-algebra that manifestly preserves translational symmetry. Some non-perturbative consequences of this approach are discussed. 
  I revisit the solution of Born-Infeld theory which corresponds to a 3-brane and anti-brane joined by a (fundamental) string. The global instability of this configuration makes possible the semiclassical tunneling into a wide, short tube which keeps expanding out, thus annihilating the brane . This tunneling is suppressed exponentially as $\exp(- S_{cl}/g)$. The attraction between the branes causes them to approach and at some point to tunnel, because the action of the bounce solution goes to zero. The energy of the solution at the top of the barrier, the sphaleron, goes like $\sim D^3$ for large separarions D, while the energy of the string is proportional to its length D. 
  Exact characteristic trajectories are specified for the time-propagating Wigner phase-space distribution function. They are especially simple---indeed, classical---for the quantized simple harmonic oscillator, which serves as the underpinning of the field theoretic Wigner functional formulation introduced. Scalar field theory is thus reformulated in terms of distributions in field phase space. Applications to duality transformations in field theory are discussed. 
  A quantization of field theory based on the DeDonder-Weyl covariant Hamiltonian formulation is discussed. A hypercomplex extension of quantum mechanics, in which the space-time Clifford algebra replaces that of the complex numbers, appears as a result of quantization of Poisson brackets of differential forms put forward for the DeDonder-Weyl formulation earlier. The proposed covariant hypercomplex Schr\"odinger equation is shown to lead in the classical limit to the DeDonder-Weyl Hamilton-Jacobi equation and to obey the Ehrenfest principle in the sense that the DeDonder-Weyl canonical field equations are satisfied for the expectation values of properly chosen operators. 
  We investigate the canonical structure of the (2+1)-dimensional non-linear $\sigma$ model in a $polynomial$ formulation. A current density defined in the non-linear $\sigma$ model is a vector field which satisfies a $formal$ flatness (or pure gauge) condition. It is the polynomial formulation in which the vector field is regarded as a dynamical variable on which the flatness condition is imposed as a constraint condition by introducing a Lagrange multiplier field. The model so formulated has gauge symmetry under a transformation of the Lagrange multiplier field. We construct the generalized Hamiltonian formalism of the model explicitly by using the Dirac method for constrained systems. We derive three types of the pre-gauge-fixing Hamiltonian systems: In the first system, the current algebra is realized as the fundamental Dirac Brackets. The second one manifests the similar canonical structure as the Chern-Simons or BF theories. In the last one, there appears an interesting interaction as the dynamical variables are coupled to their conjugate momenta via the covariant derivative. 
  We give another derivation of quark confinement in QCD from the viewpoint of the low-energy effective Abelian gauge theory of QCD obtained via Abelian projection. It is based on the recently discovered Berezinskii-Kosterlitz-Thouless phase transition in the four-dimensional Abelian gauge theory. Moreover, we show that there exists a critical gauge coupling constant in QCD, above which confinement set in and below which there is no confinement. In a SU(N) gauge theory with $N_f$ flavored fermions, we argue that this leads to a critical value of fermion flavors $N_f^c$ for the confinement as well as the chiral symmetry breaking, which separates the deconfinement and chiral symmetric phase. A finite-temperature deconfinement transition is also discussed briefly. 
  We study BPS states which arise in compactifications of M-theory on Calabi-Yau manifolds. In particular, we are interested in the spectrum of the particles obtained by wrapping M2-brane on a two-cycle in the CY manifold X. We compute the Euler characteristics of the moduli space of genus zero curves which land in a holomorphic four-cycle $S \subset X$. We use M. Kontsevich's method which reduces the problem to summing over trees and observe the discrepancy with the predictions of local mirror symmetry. We then turn this discrepancy into a supporting evidence in favor of existence of extra moduli of M2-branes which consists of the choice of a flat U(1) connection recently suggested by C. Vafa and partially confirm this by counting of the arbitrary genus curves of bi-degree (2,n) in $\IP^1 \times \IP^1$ (this part has been done together with Barak Kol). We also make a conjecture concerning the counting of higher genus curves using second quantized Penner model and discuss possible applications to the string theory of two-dimensional QCD. 
  We calculate the statistical entropy of a scalar field on the background of three-dimensional De Sitter space in terms of the brick wall method and finally derive the perimeter law of the entropy. 
  We examine the matrix theory representation of D0-brane dynamics at finite temperature. In this case, violation of supersymmetry by temperature leads to a non-trivial static potential between D0-branes at any finite temperature. We compute the static potential in the 1-loop approximation and show that it is short-ranged and attractive. We compare the result with the computations in superstring theory. We show that thermal states of D0-branes can be reproduced by matrix theory only when certain care is taken in integration over the moduli space of classical solutions in compactified time. 
  We propose a new string model by adding a higher-order gradient term to the rigid string, so that the stiffness can be positive or negative without loosing stability. In the large-D approximation, the model has three phases, one of which with a new type of generalized "antiferromagnetic" orientational correlations. We find an infrared-stable fixed point describing world-sheets with vanishing tension and Hausdorff dimension D_H=2. Crumpling is prevented by the new term which suppresses configurations with rapidly changing extrinsic curvature. 
  Characters of $E_8\otimes E_8$ and SO(32) heterotic strings involving the full internal symmetry Cartan subalgebra generators are defined after circle compactification so that they are T dual. The novel point, as compared with an earlier study of the type II case (hep-th/9707107), is the appearence of Wilson lines. Using SO(17,1) transformations between the weight lattices reveals the existence of an intermediate theory where T duality transformations are disentangled from the internal symmetry. This intermediate theory corresponds to a sort of twisted compactification of a novel type. Its modular invariance follows from an interesting interplay between three representations of the modular group. 
  We study two constrained scalar models. While there seems to be equivalence when the partially integrated Feynman path integral is expanded graphically, the dynamical behaviour of the two models are different when quantization is done using Dirac constraint analysis. 
  We discuss a path integral formalism to introduce noncommutative generalizations of spacetime manifold in even dimensions, which have been suggested to be reasonable effective pictures at very small length scales, of the order of Planck length. 
  A non-linear sigma model mimicking the Skyrme model on S_3 is proposed and a family of classical solutions to the equations are constructed numerically. The solutions terminate into catastrophe-like spikes at critical values of the Skyrme coupling constant and, when this constant is zero, they coincide with the series of Harmonic maps on S_3 constructed some years ago by P. Bizon. 
  The canonical transformation and its unitary counterpart which relate the rational Calogero-Moser system to the free one are constructed. 
  In the language of Feynman path integrals the quantization of gauge theories is most easily carried out with the help of the Schr\"odinger Functional (SF). Within this formalism the essentially unique gauge fixing condition is $A_{\circ} = 0$ (temporal gauge), as any other rotationally invariant gauge choice can be shown to be functionally equivalent to the former. In the temporal gauge Gauss' law is automatically implemented as a constraint on the states. States not annihilated by the Gauss operator describe the situation in which external (infinitely heavy) colour sources interact with the gauge field. The SF in the presence of an arbitrary distribution of external colour sources can be expressed in an elegant and concise way. 
  It has recently been shown how the effect of the divergent part of the gravitational self interaction for a classical string model in 4 dimensions can be allowed for by a renormalisation of its stress energy tensor and in the elastic case a corresponding renormalisation of the off shell action. It is shown here that that it is possible to construct a new category of elastic string models for which this effect is describable as a renormalisation in the stricter ``formal'' sense, meaning that it only requires a rescaling of one of the fixed parameters characterising the model. 
  Considering the scattering of massless open strings attached to a D2-brane living in the $B$ field background, we show that corresponding scattering upto the order of $\a'^2$ is exactly given by the gauge theory on noncommutative background, which is characterized by the Moyal bracket. 
  We study aspects of the new phases of M-theory recently conjectured using generalised dualities such as timelike T-duality. Our focus is on brane solutions. We derive the intersection rules in a general framework and then specialise to the new phases of M-theory. We discuss under which conditions a configuration with several branes leads to a regular extremal black hole under compactification. We point out that the entropy seems not to be constant when the radius of the physical timelike direction is varied. This could be interpreted as a non-conservation of the entropy (and the mass) under at least some of the new dualities. 
  We study the unscreened Coulomb interaction in a one-dimensional electron system at low-energy. We use renormalization group methods and a GW approximation, in order to analyze the model. This yields both a strong wavefunction renormalization and a renormalization of the Fermi velocity. The significance of the effects depends on the filling level of the Fermi system. Despite the long-range character of the interaction, the system still falls into the Luttinger liquid universality class, since the effective couplings remain bounded at arbitrarily low energies. 
  The properties of a nonrelativistic charged particle in two dimensions in the presence of an arbitrary number of nonquantized magnetic fluxes are investigated in free space as well as in a uniform magnetic field. The fluxes are represented mathematically as branch points in one of the complex coordinates. It is found that in order to construct solutions, the fluxes have to be treated in general as dynamical objects dual to the charges. A medium made up of fluxes acts like an anti-magnetic field and tends to expel the charges. 
  Nonlinear sigma models arise in supergravity theories with or without matter couplings in various dimensions and they are important in understanding the duality symmetries of M-theory. With this motivation in mind, we review the salient features of gauged and ungauged nonlinear sigma models with or without Wess-Zumino terms for general target spaces in a minimal as well as lifted formulation. Relevant to the question of finding interesting vacua of gauged supergravity theories is the highly constrained potential which arises naturally in these theories. Motivated by this fact, we derive a general and simple formula for a gauge invariant potential of this kind. 
  The integrability of the Bukhvostov-Lipatov four-fermion model is investigated. It is shown that the classical model possesses a current of Lorentz spin 3, conserved both in the bulk and on the half-line for specific types of boundary actions. It is then established that the conservation law is spoiled at the quantum level -- a fact that might indicate that the quantum Bukhvostov-Lipatov model is not integrable, contrary to what was previously believed. 
  We calculate, using noncommutative supersymmetric Yang-Mills gauge theory, the part of the spectrum of the toroidally compactified Matrix theory which corresponds to quantized electric fluxes. 
  We study large N SU(N) Yang-Mills theory in three and four dimensions using a one-parameter family of supergravity models which originate from non-extremal rotating D-branes. We show explicitly that varying this "angular momentum" parameter decouples the Kaluza-Klein modes associated with the compact D-brane coordinate, while the mass ratios for ordinary glueballs are quite stable against this variation, and are in good agreement with the latest lattice results. We also compute the topological susceptibility and the gluon condensate as a function of the "angular momentum" parameter. 
  We consider solutions to the $\mu$-problem originating in the effective low energy theories, of N=1 orbifold compactifications of the heterotic string, after supersymmetry breaking. They are consistent with the invariance of the one loop corrected effective action in the linear representation of the dilaton. The proposed $\mu$-terms naturally generalize solutions proposed previously, in the literature, in the context of minimal low energy supergravity models. They emanate from the connection of the non-perturbative superpotential to the determinant of the mass matrix of the chiral compactification modes. Within this approach we discuss the lifting of our solutions to their M-theory compactification counterparts. 
  By exploiting recent arguments about stable nonsupersymmetric D-brane states, we argue that D-brane charge takes values in the K-theory of spacetime, as has been suspected before. In the process, we gain a new understanding of some novel objects proposed recently - such as the Type I zerobrane - and we describe some new objects - such as a -1-brane in Type I superstring theory. 
  Locality is analyzed for Toda field theories by noting novel chiral description in the conventional nonchiral formalism. It is shown that the canonicity of the interacting to free field mapping described by the classical solution is automatically guaranteed by the locality. Quantum Toda theories are investigated by applying the method of free field quantization. We give Toda exponential operators associated with fundamental weight vectors as bilinear forms of chiral fields satisfying characteristic quantum exchange algebra. It is shown that the locality leads to nontrivial relations among the ${\cal R}$-matrix and the expansion coefficients of the exponential operators. The Toda exponentials are obtained for $A_2$-system by extending the algebraic method developed for Liouville theory. The canonical commutation relations and the operatorial field equations are also examined. 
  A novel technique based on Schwinger's proper time method is applied to the Casimir problem of the M.I.T. bag model. Calculations of the regularized vacuum energies of massless scalar and Dirac spinor fields confined to a static and spherical cavity are presented in a consistent manner. While our results agree partly with previous calculations based on asymptotic methods, the main advantage of our technique is that the numerical errors are under control. Interpreting the bag constant as a vacuum expectation value, we investigate potential cancellations of boundary divergences between the canonical energy and its bag constant counterpart in the fermionic case. It is found that such cancellations do not occur. 
  Recently it was shown by Cho and Kimm that the gauged $CP^1$ model, obtained by gauging the global $SU(2)$ group and adding a corresponding Chern-Simons term, has got its own soliton. These solitons are somewhat distinct from those of pure $CP^1$ model as they cannot always be characterised by $\pi_2(CP^1)=Z$. In this paper, we first carry out a detailed Hamiltonian analysis of this gauged $CP^1$ model. This reveals that the model has only $SU(2)$ as the gauge invariance, rather than $SU(2) \times U(1)$. The $U(1)$ gauge invariance of the original (ungauged) $CP^1$ model is actually contained in the $SU(2)$ group itself. Then we couple the Hopf term associated to these solitons and again carry out its Hamiltonian analysis. The symplectic structures, along with the structures of the constraints of these two models (with or without Hopf term) are found to be essentially the same. The model with a Hopf term is shown to have fractional spin which, when computed in the radiation gauge, is found to depend not only on the soliton number $N$, but also on the nonabelian charge. We then carry out a reduced (partially) phase space analysis in a different physical sector of the model where the degrees of freedom associated with the $CP^1$ fields are transformed away. The model now reduces to a $U(1)$ gauge theory with two Chern-Simons gauge fields getting mass-like terms and one remaining massless. In this case the fractional spin is computed in terms of the dynamical degrees of freedom and shown to depend purely on the charge of the surviving abelian symmetry. Although this reduced model is shown to have its own solitonic configuration, it turns out to be trivial. 
  In both QCD and supersymmetric QCD (SQCD) with N_f flavors there are conformal windows where the theory is asymptotically free in the ultraviolet while the infrared physics is governed by a non-trivial fixed-point. In SQCD, the lower N_f boundary of the conformal window, below which the theory is confining is well understood thanks to duality. In QCD there is just a sufficient condition for confinement based on superconvergence. Studying the Banks-Zaks expansion and analyzing the conditions for the perturbative coupling to have a causal analyticity structure, it is shown that the infrared fixed-point in QCD is perturbative in the entire conformal window. This finding suggests that there can be no analog of duality in QCD. On the other hand in SQCD the infrared region is found to be strongly coupled in the lower part of the conformal window, in agreement with duality. Nevertheless, we show that it is possible to interpolate between the Banks-Zaks expansions in the electric and magnetic theories, for quantities that can be calculated perturbatively in both. This interpolation is explicitly demonstrated for the critical exponent that controls the rate at which a generic physical quantity approaches the fixed-point. 
  Path integrals have played a fundamental role in emphasizing the profound analogies between Quantum Field Theory (QFT), and Classical as well as Quantum Statistical Physics. Ideas coming from Statistical Physics have then led to a deeper understanding of Quantum Field Theory and open the way for a wealth of non-perturbative methods. Conversely QFT methods are become essential for the description of the phase transitions and critical phenomena beyond mean field theory. This is the point we want to illustrate here. We therefore review the methods, based on renormalized phi^4_3 quantum field theory and renormalization group, which have led to an accurate determination of critical exponents of the N-vector model, and more recently of the equation of state of the 3D Ising model. The starting point is the perturbative expansion for RG functions or the effective potential to the order presently available. Perturbation theory is known to be divergent and its divergence has been related to instanton contributions. This has allowed to characterize the large order behaviour of perturbation series, an information that can be used to efficiently "sum" them. Practical summation methods based on Borel transformation and conformal mapping have been developed, leading to the most accurate results available probing field theory in a non perturbative regime. We illustrate the methods with a short discussion of the scaling equation of state of the 3D Ising model. Compared to exponents its determination involves a few additional (non-trivial) technical steps, like the use of the parametric representation, and the order-dependent mapping. 
  In this paper it is shown how the AdS/CFT correspondence extends to a more general situation in which the first theory is defined on (d+1)-dimensional manifold $\tilde M$ defined as the filling in of a compact d-dimensional manifold M. The stability of the spectral correspondence mass/conformal-weight under such geometry changes is also proven. 
  Recently, Douglas and Taylor proposed to identify the Higgs vev of the Coulomb branch of the 4 dimensional N=4 super Yang-Mills theory with the positions of D3 branes in the AdS_5 x S^5 string theory. We extend this identification to more general configurations that preserve less supersymmetry. We show that a single D3 brane in AdS_5 x S^5 string theory can break the spacetime supersymmetry to 1/4 or even 1/8. Such configurations, interpretated as backgrounds of the SYM, also break the N=4 superconformal symmetry of SYM to 1/4 or 1/8, giving N=2 or N=1. We discuss the implications of this correspondence. We also present other BPS D3-brane configurations that preserve 1/8 supersymmetry corresponding to N=1 on the SYM side. 
  Making use of a recursive approach, derivative dispersion relations are generalized for an arbitrary number of subtractions. The results for both cross even and odd amplitudes are theoretically consistent at sufficiently high energies and in the region of small momentum transfer. 
  In this brief report, we analyze a generalized theory of massless scalar QED_2 and show that, unlike the conventional scalar QED_2, it is free from infrared divergence problems. The model is exactly soluble and may describe, in an 1+1 dimensional space-time, noninteracting spin-one tachyons. 
  In these lecture notes prepared for the 11th Taiwan Spring School, Taipei 1997}, and updated for the Saalburg summer school 1998, we review the solutions of O(N) or U(N) models in the large N limit and as 1/N expansions, in the case of vector representations. The general idea is that invariant composite fields have small fluctuations for N large. Therefore the method relies on constructing effective field theories for these composite fields after integration over the initial degrees of freedom. We illustrate these ideas by showing that the large N expansion allows to relate the phib^2^2 theory and the non-linear sigma-model, models which are renormalizable in different dimensions. In the same way large N techniques allow to relate the Gross--Neveu, an example of a theory with four-fermi self-interaction, with a Yukawa-type theory renormalizable in four dimensions, a topic relevant for four dimensional field theory. Among other issues for which large N methods are also useful we will briefly discuss finite size effects and finite temperature field theory, because they involve a crossover between different dimensions.\par Finally we consider the case of a general scalar V(phib^2) field theory, explain how the large N techniques can be generalized, and discuss some connected issues like tricritical behaviour and double scaling limit. Some sections in these notes are directly adapted from the work Zinn-Justin J., 1989, Quantum Field Theory and Critical Phenomena, Clarendon Press (Oxford third ed. 1996). 
  We calculate the exact parity odd part of the effective action ($\Gamma_{odd}^{2d+1}$) for massive Dirac fermions in 2d+1 dimensions at finite temperature, for a certain class of gauge field configurations. We consider first Abelian external gauge fields, and then we deal with the case of a non-Abelian gauge group containing an Abelian U(1) subgroup. For both cases, it is possible to show that the result depends on topological invariants of the gauge field configurations, and that the gauge transformation properties of $\Gamma_{odd}^{2d+1}$ depend only on those invariants and on the winding number of the gauge transformation. 
  Banks, Douglas, Horowitz and Martinec [hep-th/9808016] recently argued that in the microcanonical ensemble for string theory on AdS_m xS^n, there is a phase transition between a black hole solution extended over the S^n and a solution localized on the S^n. If we think of this AdS_m x S^n geometry as arising from the near-horizon limit of a black m-2 brane, the existence of this phase transition is puzzling. We present a resolution of this puzzle, and discuss its significance from the point of view of the dual m-1 dimensional field theory. We also discuss multi-black hole solutions in AdS. 
  We formulate an extension of Maldacena's AdS/CFT conjectures to the case of branes located at singular points in the ambient transverse space. For singularities which occur at finite distance in the moduli space of M or F theory models with spacetime-filling branes, the conjectures identify the worldvolume theory on the p-branes with a compactification of M or IIB theory on $AdS_{p+2} \times H^{D-p-2}$. We show how the singularity determines the horizon H, and demonstrate the relationship between global symmetries on the worldvolume and gauge symmetries in the AdS model. As a first application, we study some singularities relevant to the D3-branes required in four-dimensional F-theory. For these we are able to explicitly derive the low-energy field theory on the worldvolume and compare its properties to predictions from the dual AdS model. In particular, we examine the baryon spectra of the models and the fate of the Abelian factors in the gauge group. 
  We point out some problems with the previously-proposed phase diagram of the $Z_6$ spin models. Consideration of the diagram near to the decoupling surface using both exact and approximate arguments suggests a modification which remedies these deficiencies. With the aid of a new parametrisation of the phase space, we study the models numerically, with results which support our conjectures. 
  It is shown in a quantum-mechanically exact manner that a supersymmetric and $\kappa$-symmetric D-string action in a general type IIB supergravity background is transformed to a form of the type IIB Green-Schwarz superstring action with the SL(2,Z) covariant tension through an S-duality transformation. This result precisely proves a conjecture mentioned previously that the SL(2,Z) S-duality of a super D-string action in a flat background is also valid even in a curved IIB background geometry. We point further out the validity of the more generalized conjecture that various duality relations of super D-brane and M-brane actions originally found in a flat background also hold true in general ten dimensional type II supergravity and eleven dimensional supergravity background geometries by applying the present formalism to those cases. 
  We discuss some general features of black holes of five-dimensional supergravity, such as the first law of black hole mechanics. We also discuss some special features of rotating supersymmetric black holes. In particular, we show that the horizon is a non-singular, and {\sl non-rotating}, null hypersurface whose intersection with a Cauchy surface is a squashed 3-sphere. We find the Killing spinors of the near-horizon geometry and thereby determine the near-horizon isometry supergroup. 
  We show that the five-dimensional general relativity with a negative cosmological constant allows the solutions of the form M_3 \times M_g where M_3 is the three-dimensional BTZ black hole and M_g is a higher genus (g>1) Riemann surface with a fixed size. It is shown that this type of spontaneous compactification on a Riemann surface is possible only for the genus larger than one. From type IIB string theory point of view, certain near horizon geometry of D three-branes wrapped on the compact Riemann surface (g>1) is the BTZ (or AdS_3) space-time tensored with the Riemann surface and a constant size five-sphere. The relevance of our analysis to the positive energy conjecture of Horowitz and Myers is discussed. 
  We describe new non-supersymmetric conformal field theories in three and four dimensions, using the CFT/AdS correspondence. In order to believe in their existence at large N_c and strong 't Hooft coupling, we explicitly check the stability of the corresponding non-supersymmetric anti-de Sitter backgrounds. Cases of particular interest are the relevant deformations of the N=4 SCFT in SU(3) and SO(5) invariant directions. It turns out that the former is a stable, and the latter an unstable non-supersymmetric type IIB background. 
  A variant of the divergence theory for vacuum-condensation developed in a previous communication is analyzed from the viewpoint of a 'time' asymmetric law in vacuum. This law is found to establish a substantial distinction between dynamically allowed vacuum-configurations related by signature changing duality transformations. 
  Starting from the well-known quantum Miura-like transformation for the non simply-laced Lie algebras B(3),we give an explicit construction of the Casimir WB(3) algebras.We reserve the notation WB(N) for the Casimir W algebras of type W(2,4,6,...,2N,N+1/2) which contains one fermionic field. It is seen that WB(3) algebra is closed an associative for all values of the central element c. 
  The generalized Weierstrass representation for surfaces in $\Bbb{R}^{3}$ is used to study quantum effects for strings governed by Polyakov-Nambu-Goto action. Correlators of primary fields are calculated exactly in one-loop approximation for the pure extrinsic Polyakov action. Geometrical meaning of infrared singularity is discussed. The Nambu-Goto and spontaneous curvature actions are treated perturbatively. 
  We discuss two-dimensional sigma models on moduli spaces of instantons on K3 surfaces. These N=(4,4) superconformal field theories describe the near-horizon dynamics of the D1-D5-brane system and are dual to string theory on AdS_3. We derive a precise map relating the moduli of the K3 type IIB string compactification to the moduli of these conformal field theories and the corresponding classical hyperkaehler geometry. We conclude that, in the absense of background gauge fields, the metric on the instanton moduli spaces degenerates exactly to the orbifold symmetric product of K3. Turning on a self-dual NS B-field deforms this symmetric product to a manifold that is diffeomorphic to the Hilbert scheme. We also comment on the mathematical applications of string duality to the global issues of deformations of hyperkaehler manifolds. 
  We construct generalizations of the Calogero-Sutherland-Moser system by appropriately reducing a classical Calogero model by a subset of its discrete symmetries. Such reductions reproduce all known variants of these systems, including some recently obtained generalizations of the spin-Sutherland model, and lead to further generalizations of the elliptic model involving spins with SU(n) non-invariant couplings. 
  After defining cohomologically higher order BRST and anti-BRST operators for a compact simple algebra {\cal G}, the associated higher order Laplacians are introduced and the corresponding supersymmetry algebra $\Sigma$ is analysed. These operators act on the states generated by a set of fermionic ghost fields transforming under the adjoint representation. In contrast with the standard case, for which the Laplacian is given by the quadratic Casimir, the higher order Laplacians $W$ are not in general given completely in terms of the Casimir-Racah operators, and may involve the ghost number operator. The higher order version of the Hodge decomposition is exhibited. The example of su(3) is worked out in detail, including the expression of its higher order Laplacian W. 
  The SL(2,Z) anomaly recently derived for type IIB supergravity in 10 dimensions is shown to be a consequence of T-duality with the type IIA string, after compactification to 2 dimensions on an 8-fold. This explains the identity of the gravitational 8-forms appearing in different contexts in the effective actions of type IIA and IIB string theories. In this framework, constraints on the compactification manifold arise from modular invariance of the 2d theory. Related issues in 6 dimensions are examined, and it is argued that similar anomalies are present on the worldvolumes of M-theory 5-branes and orientifold 5-planes. 
  We use the crosscap constraint to construct open descendants of the 0B string compactified on $T^6 /Z_3$ and on $T^4/Z_2$ free of tachyons both in the closed and in the open unoriented sectors. In four dimensions the construction results in a Chan-Paton gauge group $U(8)\otimes U(12)\otimes U(12)$ with three generations of chiral fermions in the representations $(\overline{8},1,\overline{12}) +(8,12,1)+(1,\overline{66},1)+(1,1,66)$. 
  The set of space-time short-distance structures which can be described through linear operators is limited to a few basic cases. These are continua, lattices and a further short-distance structure which implies an ultraviolet cut-off. Under certain conditions, these cut-off degrees of freedom can reappear as internal degrees of freedom. We review the current status of the classification and present new conjectures. 
  An exact one monopole solution in a uniform self-dual background field is obtained in the BPS limit of the SU(2) Yang-Mills-Higgs theory by using the inverse scattering method. 
  We apply an improved version of Batalin-Fradkin-Tyutin (BFT) Hamiltonian method to the a=1 chiral Schwinger Model, which is much more nontrivial than the a>1.$ one. Furthermore, through the path integral quantization, we newly resolve the problem of the non-trivial $\delta$ function as well as that of the unwanted Fourier parameter $\xi$ in the measure. As a result, we explicitly obtain the fully gauge invariant partition function, which includes a new type of Wess-Zumino (WZ) term irrelevant to the gauge symmetry as well as usual WZ action. 
  We present a new perturbative approach to QCD based on the use of quark composites with hadronic quantum numbers as fundamental variables. We apply it to the case of the nucleons by performing a nonlinear change of variables in the Berezin integral which defines the partition function of QCD. The nucleon composites are thereby assumed as new integration variables. We evaluate the jacobian and certain transformation functions which appear in the change of variables. We show that the free action of the nucleon composites is the Dirac action, and we evaluate the first perturbative contributions to their electroweak effective action, which turn out to be a pure renormalization. Our expansion is compatible with a perturbative as well as nonperturbative regime of the gluons and it has the characteristic feature that the confinement of the quarks is buit in. 
  In this paper, we describe non-abelian gauge bundles with magnetic and electric fluxes on higher dimensional noncommutative tori. We give an explicit construction of a large class of bundles with nonzero magnetic 't Hooft fluxes. We discuss Morita equivalence between these bundles. The action of the duality is worked out in detail for the four-torus. As an application, we discuss Born-Infeld on this torus, as a description of compactified string theory. We show that the resulting theory, including the fluctuations, is manifestly invariant under the T-duality group SO(4,4;Z). The U-duality invariant BPS mass-formula is discussed shortly. We comment on a discrepancy of this result with that of a recent calculation. 
  A novel algebra underlying integrable systems is shown to generate and unify a large class of quantum integrable models with given $R$-matrix, through reductions of an ancestor Lax operator and its different realizations. Along with known discrete and field models a new class of inhomogeneous and impurity models are obtained. 
  QED vacua under the influence of external conditions (background fields, finite temperature, boundary conditions) can be considered as dispersive media whose complex behaviour can no longer be described in terms of a single universal vacuum velocity of light c. Beginning in the early 1950's (J.S. Toll), quantum field theoretic investigations have led to considerable insight into the relation between the vacuum structure and the propagation of light. Recent years have witnessed a significant growth of activity in this field of research. After a short overview, two characteristic situations are discussed: the propagation of light in a constant homogeneous magnetic field and in a Casimir vacuum. The latter appears to be particularly interesting because the Casimir vacuum has been found to exhibit modes of the propagation of light with phase and group velocities larger than c in the low frequency domain omega<<m where m is the electron mass. The impact of this result on the front velocity of light in a Casimir vacuum is discussed by means of the Kramers-Kronig relation. 
  The modification of classical dynamics of abelian confining theory by virtue of quantum Abrikosov-Nielsen-Olesen strings is discussed taking D=4 abelian Higgs model as an example. The form of string corrections to the Wilson loop correlators, gauge boson propagator, effective potential is presented and possible relations to abelian projected QCD are outlined. 
  As in two and four dimensions, supersymmetric conformal field theories in three dimensions can have exactly marginal operators. These are illustrated in a number of examples with N=4 and N=2 supersymmetry. The N=2 theory of three chiral multiplets X,Y,Z and superpotential W=XYZ has an exactly marginal operator; N=2 U(1) with one electron, which is mirror to this theory, has one also. Many N=4 fixed points with superpotentials W \sim Phi Q_i \tilde Q^i have exactly marginal deformations consisting of a combination of Phi^2 and (Q_i \tilde Q^i)^2. However, N=4 U(1) with one electron does not; in fact the operator Phi^2 is marginally irrelevant. The situation in non-abelian theories is similar. The relation of the marginal operators to brane rotations is briefly discussed; this is particularly simple for self-dual examples where the precise form of the marginal operator may be guessed using mirror symmetry. 
  The complete phase diagram of objects in M-theory compactified on tori $T^p$, $p=1,2,3$, is elaborated. Phase transitions occur when the object localizes on cycle(s) (the Gregory-Laflamme transition), or when the area of the localized part of the horizon becomes one in string units (the Horowitz-Polchinski correspondence point). The low-energy, near-horizon geometry that governs a given phase can match onto a variety of asymptotic regimes. The analysis makes it clear that the matrix conjecture is a special case of the Maldacena conjecture. 
  Spinning black three-branes in type IIB supergravity are thermodynamically stable up to a critical value of the angular momentum density. Inside the region of thermodynamic stability, the free energy from supergravity is roughly reproduced by a naive model based on free N=4 super-Yang-Mills theory on the world-volume. The field theory model correctly predicts a limit on angular momentum density, but near this limit it does not reproduce the critical exponents one can compute from supergravity. Analogies with Bose condensation and modified matrix models are discussed, and a mean field theory improvement of the naive model is suggested which corrects the critical exponents. 
  We continue our study of the unitary supermultiplets of the N=8, d=5 anti-de Sitter (AdS_5) superalgebra SU(2,2|4), which is also the N=4 extended conformal superalgebra in d=4. We show explicitly how to go from the compact SU(2)XSU(2)XU(1) basis to the non-compact SL(2,C)XD basis of the positive energy unitary representations of the conformal group SU(2,2) in d=4. The doubleton representations of the AdS_5 group SU(2,2), which do not have a smooth Poincare limit in d=5, are shown to represent fields with vanishing masses in four dimensional Minkowski space. The unique CPT self-conjugate irreducible doubleton supermultiplet of SU(2,2|4)is simply the N=4 Yang-Mills supermultiplet in d=4. We study some novel short non-doubleton supermultiplets of SU(2,2|4) that have spin range 2 and that do not appear in the Kaluza-Klein spectrum of type IIB supergravity or in tensor products of the N=4 Yang-Mills supermultiplet with itself. These novel supermultiplets can be obtained from tensoring chiral doubleton supermultiplets, some of which we expect to be related to the massless limits of 1/4 BPS states. Hence, these novel supermultiplets may be relevant to the solitonic sector of IIB superstring and/or (p,q) superstrings over AdS_5 X S^5. 
  We derive and analyse the full set of equations of motion for non-extreme static black holes (including examples with the spatial curvatures k=-1 and k=0) in D=5 N=2 gauged supergravity by employing the techniques of "very special geometry". These solutions turn out to differ from those in the ungauged supergravity only in the non-extremality function, which has an additional term (proportional to the gauge coupling g), responsible for the appearance of naked singularities in the BPS-saturated limit. We derive an explicit solution for the STU model of gauged supergravity which is incidentally also a solution of D=5 N=4 and N=8 gauged supergravity. This solution is specified by three charges, the asymptotic negative cosmological constant (minimum of the potential) and a non-extremality parameter. While its BPS-saturated limit has a naked singularity, we find a lower bound on the non-extremality parameter (or equivalently on the ADM mass) for which the non-extreme solutions are regular. When this bound is saturated the extreme (non-supersymmetric) solution has zero Hawking temperature and finite entropy. Analogous qualitative features are expected to emerge for black hole solutions in D=4 gauged supergravity as well. 
  We study the interaction of gauge fields of arbitrary integer spins with the constant electromagnetic field. We reduce the problem of obtaining the gauge-invariant Lagrangian of integer spin fields in the external field to purely algebraic problem of finding a set of operators with certain features using the representation of the high-spin fields in the form of vectors in a pseudo-Hilbert space. We consider such a construction up to the second order in the electromagnetic field strength and also present an explicit form of interaction Lagrangian for a massive particle of spin $s$ in terms of symmetrical tensor fields in linear approximation. The result obtained does not depend on dimensionality of space-time. 
  We propose an approach to the theory of higher order anisotropic field interactions and curved spaces (in brief, ha-field, ha-interactions and ha-spaces). The concept of ha-space generalises various types of Lagrange and Finsler spaces and higher dimension (Kaluza-Klein) spaces. This work consists from two parts. In the first we outline the theory of Yang-Mills ha-fields and two gauge models of higher order anisotropic gravity are analyzed. The second is devoted to the theory of nearly autoparallel maps (na-maps) of locally anisotropic spaces (la-spaces) and to the problem of formulation of conservation laws for la-field interactions.   By defining invariants of na-map transforms we present a systematic classification of la-spaces. 
  The OSp(2|2)-invariant planar dynamics of a D=4 superparticle near the horizon of a large mass extreme black hole is described by an N=2 superconformal mechanics, with the SO(2) charge being the superparticle's angular momentum. The {\it non-manifest} superconformal invariance of the superpotential term is shown to lead to a shift in the SO(2) charge by the value of its coefficient, which we identify as the orbital angular momentum. The full SU(1,1|2)-invariant dynamics is found from an extension to N=4 superconformal mechanics. 
  Arbitrary spin free massless bosonic fields propagating in even $d$ - dimensional anti-de Sitter spacetime are investigated. Free wave equations of motion, subsidiary conditions and the corresponding gauge transformations for such fields are proposed. The lowest eigenvalues of the energy operator for the massless fields and the gauge parameter fields are derived. The results are formulated in $SO(d-1,2)$ covariant form as well as in terms of intrinsic coordinates. An inter-relation of two definitions of masslessness based on gauge invariance and conformal invariance is discussed. 
  Following Feynman's treatment of the non-relativistic polaron problem, similar techniques are used to study relativistic field theories: after integrating out the bosonic degrees of freedom the resulting effective action is formulated in terms of particle trajectories (worldlines) instead of field operators. The Green functions of the theory are then approximated variationally on the pole of the external particles by using a retarded quadratic trial action. Application to a scalar theory gives non-perturbative, covariant results for vertex functions and scattering processes. Recent progress in dealing with the spin degrees of freedom in fermionic systems, in particular Quantum Electrodynamics, is discussed. We evaluate the averages needed in the Feynman variational principle for a general quadratic trial action and study the structure of the dressed fermion propagator. 
  We formulate Yang-Mills theory in terms of the large-N limit, viewed as a classical limit, of gauge-invariant dynamical variables, which are closely related to Wilson loops, via deformation quantization. We obtain a Poisson algebra of these dynamical variables corresponding to normal-ordered quantum (at a finite value of $\hbar$) operators. Comparing with a Poisson algebra one of us introduced in the past for Weyl-ordered quantum operators, we find, using ideas closly related to topological graph theory, that these two Poisson algebras are, roughly speaking, the same. More precisely speaking, there exists an invertible Poisson morphism between them. 
  We present BRST gauge fixing approach to quantum mechanics in phase space. The theory is obtained by $\hbar$-deformation of the cohomological classical mechanics described by d=1, N=2 model. We use the extended phase space supplied by the path integral formulation with $\hbar$-deformed symplectic structure. 
  We give a superfield formulation of the path integral on an arbitrary curved phase space, with or without first class constraints. Canonical tranformations and BRST transformations enter in a unified manner. The superpartners of the original phase space variables precisely conspire to produce the correct path integral measure, as Pfaffian ghosts. When extended to the case of second-class constraints, the correct path integral measure is again reproduced after integrating over the superpartners. These results suggest that the superfield formulation is of first-principle nature. 
  We use the light-front Hamiltonian of transverse lattice gauge theory to compute from first principles the glueball spectrum and light-front wavefunctions in the leading order of the 1/N_c colour expansion. We find 0^{++}, 2^{++}, and 1^{+-} glueballs having masses consistent with N_c=3 data available from Euclidean lattice path integral methods. The wavefunctions exhibit a light-front constituent gluon structure. 
  The nonabelian Berry phase is computed in the T dualized quantum mechanics obtained from the USp(2k) matrix model. Integrating the fermions, we find that each of the spacetime points X_{\nu}^{(i)} is equipped with a pair of su(2) Lie algebra valued pointlike singularities located at a distance m_{(f)} from the orientifold surface. On a four dimensional paraboloid embedded in the five dimensional Euclidean space, these singularities are recognized as the BPST instantons. 
  A method to perform bosonization of a fermionic theory in (1+1) dimensions in a path integral framework is developed. The method relies exclusively on the path integral property of allowing variable shifts, and does not depend on the explicit form of Greens functions. Two examples, the Schwinger model and the massless Thirring model, are worked out. 
  We compute the one-loop non-holomorphic effective potential for the N=4 SU(n) supersymmetric Yang-Mills theory with the gauge symmetry broken down to the maximal torus. Our approach remains powerful for arbitrary gauge groups and is based on the use of N=2 harmonic superspace formulation for general N=2 Yang-Mills theories along with the superfield background field method. 
  This is the written version of a set of introductory lectures on string theory. 
  The Bethe-Salpeter equation provides the most widely used technique to extract bound states and resonances in a relativistic Quantum Field Theory. Nevertheless a thorough discussion how to identify its solutions with physical states is still missing. The occurrence of complex eigenvalues of the homogeneous Bethe-Salpeter equation complicates this issue further. Using a perturbative expansion in the mass difference of the constituents we demonstrate for scalar fields bound by a scalar exchange that the underlying mechanism which results in complex eigenvalues is the crossing of a normal (or abnormal) with an abnormal state. Based on an investigation of the renormalization of one-particle properties we argue that these crossings happen beyond the applicability region of the ladder Bethe-Salpeter equation. The implications for a fermion-antifermion bound state in QED are discussed, and a consistent interpretation of the bound state spectrum of QED is proposed. 
  We discuss a simple system which has a central charge in its Poincare algebra. We show that this system is exactly solvable after quantization and that the algebra holds without anomalies. 
  We examine certain n-point functions G_n in {\cal N}=4 supersymmetric SU(N) gauge theory at the conformal point. In the large-N limit, we are able to sum all leading-order multi-instanton contributions exactly. We find compelling evidence for Maldacena's conjecture: (1) The large-N k-instanton collective coordinate space has the geometry of AdS_5 x S^5. (2) In exact agreement with type IIB superstring calculations, at the k-instanton level, $G_n = \sqrt{N} g^8 k^{n-7/2} e^{-8\pi^2 k/g^2}\sum_{d|k} d^{-2} \times F_n(x_1,...,x_n)$, where F_n is identical to a convolution of n bulk-to-boundary SUGRA propagators. 
  Recently it was conjectured that parallel branes at conical singularities are related to string/M theory on $AdS \times X$ where $X$ is an Einstein manifold. In this paper we consider coincident M2 branes near a conifold singularity when M theory is compactified on $AdS_4 \times Q^{1,1,1}$ where $Q^{1,1,1} = (SU(2) \times SU(2) \times SU(2))/(U(1) \times U(1))$ is a seven dimensional Sasaki-Einstein manifold. We argue that M theory on $AdS_4 \times Q^{1,1,1}$ can be described in terms of a three dimensional superconformal field theory.We use the fact that the three dimensional self-mirror duality is preserved by exact marginal operators, as observed by Strassler. 
  Starting with the 2+1 Einstein--Maxwell--Dilaton system with a cosmological constant and assuming two commuting Killing symmetries we derive the corresponding $1+0 \sigma$--model. It is shown that, for general values of the coupling parameters, the $T$--duality group is $SL(2,R)$, which coincides with the group of linear coordinate transformations along the Killing orbits. This duality, along with a suitable parameter choice, is applied to obtain some new spinning solutions in the alternative gravity theories: Einstein--Maxwell, Brans--Dicke and Einstein--Maxwell--Dilaton. 
  The structure of the QFT expansion is studied in the framework of a new "Invariant analytic" version of the perturbative QCD. Here, an invariant (running) coupling $a(Q^2/\Lambda^2)=\beta_1\alpha_s(Q^2)/4\pi$ is transformed into a "$Q^2$--analytized" invariant coupling $a_{\rm an}(Q^2/\Lambda^2) \equiv {\cal A}(x)$ which, by constuction, is free of ghost singularities due to incorporating some nonperturbative structures.   Meanwhile, the "analytized" perturbation expansion for an observable $F$, in contrast with the usual case, may contain specific functions ${\cal A}_n(x)= [a^n(x)]_{\rm an}$, the "n-th power of $a(x)$ analytized as a whole", instead of $({\cal A}(x))^n$. In other words, the pertubation series for $F(x)$, due to analyticity imperative, may change its form turning into an {\it asymptotic expansion \`a la Erd\'elyi over a nonpower set} $\{{\cal A}_n(x)\}$.   We analyse sets of functions $\{{\cal A}_n(x)\}$ and discuss properties of non-power expansion arising with their relations to feeble loop and scheme dependence of observables.   The issue of ambiguity of the invariant analytization procedure and of possible inconsistency of some of its versions with the RG structure is also discussed. 
  In the method of thermostring quantization the time evolution of point particles at finite temperature kT is described in a geometric manner. The temperature paths of particles are represented as closed (thermo)strings, which are swept surfaces in space-time-temperature manifold. The method makes it possible a new physical interpretation of superstrings IIA and heterotic strings as point particles in a thermal bath with Planck temperature. 
  Universality in unitary invariant random matrix ensembles with complex matrix elements is considered. We treat two general ensembles which have a determinant factor in the weight. These ensembles are relevant, e.g., for spectra of the Dirac operator in QCD. In addition to the well established universality with respect to the choice of potential, we prove that microscopic spectral correlators are unaffected when the matrix in the determinant is replaced by an expansion in powers of the matrix. We refer to this invariance as logarithmic universality. The result is used in proving that a simple random matrix model with Ginsparg-Wilson symmetry has the same microscopic spectral correlators as chiral random matrix theory. 
  A full quantum description of global vortex strings is presented in the framework of a pure Higgs system with a broken global U(1) symmetry in 3+1D. An explicit expression for the string creation operator is obtained, both in terms of the Higgs field and in the dual formulation where a Kalb-Ramond antisymmetric tensor gauge field is employed as the basic field. The quantum string correlation function is evaluated and from this, the string energy density is obtained. Potential application in cosmology (cosmic strings) and condensed matter (vortices in superfluids) are discussed. 
  This is an expository paper describing the geometry of certain Sasakian-Einstein manifolds. Such manifolds have recently become of interest due to Maldacena's AdS/CFT conjecture. They describe near-horizon geometries of branes at conical singularities. 
  We present a microscopical derivation of the entropy of the black hole solutions of the Jackiw-Teitelboim theory. We show that the asymptotic symmetry of two-dimensional (2D) Anti-de Sitter space is generated by a central extension of the Virasoro algebra. Using a canonical realization of this symmetry and Cardy's formula we calculate the statistical entropy of 2D black holes, which turns out to agree, up to a factor $\sqrt 2$, with the thermodynamical result. 
  The method of the BRST quantization is considered for the system of constraints, which form a Lie algebra. When some of the Cartan generators do not imply any conditions on the physical states, the system contains the first and the second class constraints. After the introduction auxiliary bosonic degrees of freedom for these cases, the corresponding BRST charges with the nontrivial structure of nonlinear terms in ghosts are constructed. 
  We consider models of dynamical supersymmetry breaking in which the extremization of a tree-level superpotential conflicts with a quantum constraint. We show that in such models the low-energy effective theory near the origin of moduli space is an O'Raifeartaigh model, and the sign of the mass-squared for the pseudo-flat direction at the origin is calculable. We analyze vector-like models with gauge groups SU(N) and Sp(2N) with and without global symmetries. In all cases there is a stable minimum at the origin with an unbroken U(1)_R symmetry. 
  In this paper we discuss the strong coupling limit of chiral N=1 supersymmetric gauge theory via their embedding into M-theory. In particular we focus on the brane box models of Hanany and Zaffaroni and show that after a T-duality transformation their M-theory embedding is described by supersymmetric 3-cycles; its geometry will encode the holomorphic non-perturbative information about the gauge theory. 
  The Calogero model with external harmonic oscillator potential is discussed from sL(2,R) algebra point of view. Explicit formulae for functions with exponential time behaviour are given; in particular, the integrals of motion are constructed and their involutiveness demonstrated. The superintegrability of the model appears to be a simple consequence of the formalism. 
  An algorithm to obtain the Weyl anomaly in higher dimensions is presented. It is based on the heat-kernel method. Feynman rules, such as the vertex rule and the propagator rule, are given in (regularized) coordinate space. Graphical calculation is introduced. The 6 dimensional scalar-gravity theory is taken as an example, and its explicit result is obtained. 
  We present nontrivial examples of d=3 gauge theories with sixteen and eight supercharges which are infrared dual at special points in the moduli space. This duality is distinct from mirror symmetry. To demonstrate duality we construct the gauge theories of interest using D2-branes and orientifolds and then consider their lift to M-theory. We also discuss the strong coupling limit of orientifold two-planes and orbifolds of orientifold six-planes. 
  We look at the equivalence of the massive Thirring and sine-Gordon models. Previously, this equivalence was derived perturbatively in mass (though to all orders). Our calculation goes beyond that and uncovers an underlying conformal symmetry. 
  We study a supersymmetric extension of the Virasoro algebra on the boundary of the anti-de Sitter space-time AdS_{3}. Using the free field realization of the currents, we show that the world-sheet affine Lie superalgebras osp(1|2)^{(1)}, sl(1|2)^{(1)} and sl(2|2)^{(1)} provide the boundary N=1,2 and 4 extended superconformal algebras, respectively. 
  The general model of an arbitrary spin massive particle in any dimensional space-time is derived on the basis of Kirillov - Kostant - Souriau approach.   Keywords: spinning particles, Poincar\'e group, orbit method, constrained dynamics, geometric quantization. 
  We study the T duality between a set of type IIB D3 branes atnon-orbifold threefold singularities, and type IIA configurations of D4 branes stretched between relatively rotated NS fivebranes. The four-dimensional N=1 field theories on the D3 brane world-volume can be easily described using the IIA brane configuration. These models include families of chiral theories continuously connected to the theories appearing in brane box models (or D3 branes at orbifold singularities). We propose that phase transitions in the K\"ahler moduli space of the singularity are related to the crossing of rotated NS fivebranes in the T dual picture, and thus to Seiberg's duality in one of the gauge factors. We also comment on the inclusion of orientifold planes in the IIA brane picture. 
  We study AdS/CFT correspondence in the case of AdS3. We obtain the statistical entropy of the BTZ black hole in terms of the correct central charge and the conformal dimensions for the states corresponding to the BTZ black hole. We point out the difference between our method and the old fashioned approaches based on SL(2,R) Wess-Zumino-Witten model or Liouville theory. 
  We consider the conformally-invariant coupling of topologically massive gravity to a dynamical massless scalar field theory on a three-manifold with boundary. We show that, in the phase of spontaneously broken Lorentz and Weyl symmetries, this theory induces the target space zero mode of the vertex operator for the string dilaton field on the boundary of the three-dimensional manifold. By a further coupling to topologically massive gauge fields in the bulk, we demonstrate directly from the three-dimensional theory that this dilaton field transforms in the expected way under duality transformations so as to preserve the mass gaps in the spectra of the gauge and gravitational sectors of the quantum field theory. We show that this implies an intimate dynamical relationship between T-duality and S-duality transformations of the quantum string theory. The dilaton in this model couples bulk and worldsheet degrees of freedom to each other and generates a dynamical string coupling. 
  We construct a one-parametric family of the double-scaling limits in the hermitian matrix model $\Phi^6$ for 2D quantum gravity. The known limit of Bresin, Marinari and Parisi belongs to this family. The family is represented by the Gurevich-Pitaevskii solution of the Korteveg-de Vries equation which describes the onset of nondissipative shock waves in media with small dispersion. Numerical simulation of the universal Gurevich-Pitaevskii solution is made. 
  String cosmology is revisited from cosmological viewpoint of holographic principle put forward by `t Hooft, and by Fischler and Susskind. It is shown that the holography requires existence of a `graceful exit' mechanism, which renders the Universe nonsingular by connecting pre- and post-big bang phases smoothly. It is proven that flat Universe is consistent with the holography only if it starts with an absolutely cold and vacuous state. Particle entropy produced during the `graceful exit' apparently saturates the holography bound. Open Universe can always satisfy the holography no matter what initial state of the Universe is. 
  We consider two different versions of gauged WZW theories with the exceptional groups and gauged with any of theirs null subgroups. By constructing suitable automorphism, we establish the equivalence of these two theories. On the other hand our automorphism, relates the two dual irreducible Riemannian globally symmetric spaces with different characters based on the corresponding exceptional Lie groups. 
  We study M theory on $AdS_4 \times \RP^7$ corresponding to 3 dimensional ${\cal N}=8$ superconformal field theory which is the strong coupling limit of 3 dimensional super Yang-Mills theory. For SU(N) theory, a wrapped M5 brane on $\RP^5$ can be interpreted as baryon vertex. For $SO(N)/Sp(2N)$ theory, by using the property of (co-)homology of $\RP^7$, we classify various wrapping branes and consider domain walls and the baryon vertex. 
  We construct a quantum mechanical model of the Calogero type for the icosahedral group as the structural group. Exact solvability is proved and the spectrum is derived explicitly. 
  In a space-time with torsion, the action for the gravitational field can be extended with a parity-violating piece. We show how to obtain such a piece from geometry itself, by suitably modifying the affine connection so as to include a pseudo-tensorial part. A consistent method is thus suggested for incorporating parity-violation in the Lagrangians of all matter fields with spin in a space-time background with torsion. 
  We determine the spectrum of D-string bound states in various classes of generalized type I vacuum configurations with sixteen and eight supercharges. The precise matching of the BPS spectra confirms the duality between unconventional type IIB orientifolds with quantized NS-NS antisymmetric tensor and heterotic CHL models in D=8. A similar analysis puts the duality between type II (4,0) models and type I strings {\it without open strings} on a firmer ground. The analysis can be extended to type II (2,0) asymmetric orbifolds and their type I duals that correspond to unconventional K3 compactifications. Finally we discuss BPS-saturated threshold corrections to the correponding low-energy effective lagrangians. In particular we show how the exact moduli dependence of some F^4 terms in the eight-dimensional type II (4,0) orbifold is reproduced by the infinite sum of D-instanton contributions in the dual type I theory. 
  By rigorous application of the Hamiltonian methods we show that the ABC-formulation of the Siegel superparticle admits consistent minimal coupling to external supergravity. The consistency check proves to involve all the supergravity constraints. 
  We analyse the ultraviolet divergencies in the ground state energy for a penetrable sphere and a dielectric ball. We argue that for massless fields subtraction of the ``empty space'' or the ``unbounded medium'' contribution is not enough to make the ground state energy finite whenever the heat kernel coefficient $a_2$ is not zero. It turns out that $a_2\ne 0$ for a penetrable sphere, a general dielectric background and the dielectric ball. To our surprise, for more singular configurations, as in the presence of sharp boundaries, the heat kernel coefficients behave to some extend better than in the corresponding smooth cases, making, for instance, the dilute dielectric ball a well defined problem. 
  Polymomentum canonical theories, which are manifestly covariant multi-parameter generalizations of the Hamiltonian formalism to field theory, are considered as a possible basis of quantization. We arrive at a multi-parameter hypercomplex generalization of quantum mechanics to field theory in which the algebra of complex numbers and a time parameter are replaced by the space-time Clifford algebra and space-time variables treated in a manifestly covariant fashion. The corresponding covariant generalization of the Schroedinger equation is shown to be consistent with several aspects of the correspondence principle such as a relation to the De Donder-Weyl Hamilton-Jacobi theory in the classical limit and the Ehrenfest theorem. A relation of the corresponding wave function (over a finite dimensional configuration space of field and space-time variables) to the Schroedinger wave functional in quantum field theory is examined in the ultra-local approximation. 
  Following the work of Dine and Seiberg for SU(2), we study the leading irrelevant operators on the moduli space of N=4 supersymmetric SU(N) gauge theory. These operators are argued to be one-loop exact, and are explicitly computed. 
  We discuss the formal structure of a functional measure for Gauge Theories preserving the Slavnov-Taylor identity in the presence of Gribov horizons. Our construction defines a gauge-fixed measure in the framework of the lattice regularization by dividing the configuration space into patches with different gauge-fixing prescriptions. Taking into account the bounds described by Dell'Antonio and Zwanziger we discuss the behaviour of the measure in the continuum limit for finite space-time volume. 
  This is an introduction to some recent developments in string theory and M theory. We try to concentrate on the main physical aspects, and often leave more technical details to the original literature. 
  We present the Hamilton-Jakobi method for the classical mechanics with constrains in Grassmann algebra. In the frame of this method the solution for the classical system characterized by the SUSY Lagrangian is obtained. 
  The massive IIA string theory whose low energy limit is the massive supergravity theory constructed by Romans is obtained from M-theory compactified on a 2-torus bundle over a circle in a limit in which the volume of the bundle shrinks to zero. The massive string theories in 9-dimensions given by Scherk-Schwarz reduction of IIB string theory are interpreted as F-theory compactified on 2-torus bundles over a circle. The M-theory solution that gives rise to the D8-brane of the massive IIA theory is identified. Generalisations of Scherk-Schwarz reduction are discussed. 
  We consider firstly simple D=4 superalgebra with six real tensorial central charges $Z_{\mu\nu}$, and discuss its possible realizations in massive and massless cases. Massless case is dynamically realized by generalized Ferber-Shirafuji (FS) model with fundamental bosonic spinor coordinates. The Lorentz invariance is not broken due to the realization of central charges generators in terms of bosonic spinors. The model contains four fermionic coordinates and possesses three kappa-symmetries thus providing the BPS configuration preserving 3/4 of the target space supersymmetries. We show that the physical degrees of freedom (8 real bosonic and 1 real Grassmann variable) of our model can be described by OSp(8|1) supertwistor. The relation with recent superparticle model by Rudychev and Sezgin is pointed out. Finally we propose a higher dimensional generalization of our model with one real fundamental bosonic spinor. D=10 model describes massless superparticle with composite tensorial central charges and in D=11 we obtain 0-superbrane model with nonvanishing mass which is generated dynamically. 
  In this paper I complete the solution of the Bukhvostov Lipatov model by computing the physical excitations and their factorized S matrix. I also explain the paradoxes which led in recent years to the suspicion that the model may not be integrable. 
  We study the conditions for the fivebrane worldvolume theory in D=11 to admit supersymmetric solitons with non-vanishing self-dual three-form. We construct some new soliton solutions consisting of ``superpositions'' of calibrated surfaces, self-dual strings and instantons. 
  We study the BPS states of the M-fivebrane which correspond to monopoles of N=2 SU(2) gauge theory. Far away from the centres of the monopoles these states may be viewed as solitons in the Seiberg-Witten effective action. It is argued that these solutions are smooth and some properties of their moduli space are discussed. 
  BRST formulation of cohomological Hamiltonian mechanics is presented. In the path integral approach, we use the BRST gauge fixing procedure for the partition function with trivial underlying Lagrangian to fix symplectic diffeomorphism invariance. Resulting Lagrangian is BRST and anti-BRST exact and the Liouvillian of classical mechanics is reproduced in the ghost-free sector. The theory can be thought of as a topological phase of Hamiltonian mechanics and is considered as one-dimensional cohomological field theory with the target space a symplectic manifold. Twisted (anti-)BRST symmetry is related to global N=2 supersymmetry, which is identified with an exterior algebra. Landau-Ginzburg formulation of the associated $d=1$, N=2 model is presented and Slavnov identity is analyzed. We study deformations and perturbations of the theory. Physical states of the theory and correlation functions of the BRST invariant observables are studied. This approach provides a powerful tool to investigate the properties of Hamiltonian systems. 
  Relation between Bopp-Kubo formulation and Weyl-Wigner-Moyal symbol calculus, and non-commutative geometry interpretation of the phase space representation of quantum mechanics are studied. Harmonic oscillator in phase space via creation and annihilation operators, both the usual and $q$-deformed, is investigated. We found that the Bopp-Kubo formulation is just non-commuting coordinates representation of the symbol calculus. The Wigner operator for the $q$-deformed harmonic oscillator is shown to be proportional to the 3-axis spherical angular momentum operator of the algebra $su_{q}(2)$. The relation of the Fock space for the harmonic oscillator and double Hilbert space of the Gelfand-Naimark-Segal construction is established. The quantum extension of the classical ergodiicity condition is proposed. 
  The AdS/CFT correspondence of elementary string theory has been recently suggested as a ``microscopic'' approach to QCD string theory in various dimensions. We use the microscopic theory to show that the ultraviolet regime on the string world-sheet is mapped to the ultraviolet effects in QCD. In the case of QCD_2, a world-sheet path integral representation of QCD strings is known, in terms of a topological rigid string theory whose world-sheet supersymmetry is reminiscent of Parisi-Sourlas supersymmetry. We conjecture that the supersymmetric rigid string theory is dual to the elementary Type IIB string theory in the singular AdS background that corresponds to the large-N limit of QCD_2. We also generalize the rigid string with world-sheet Parisi-Sourlas supersymmetry to dimensions greater than two, and argue that the theory is asymptotically free, a non-zero string tension is generated dynamically through dimensional transmutation, and the theory is topological only asymptotically in the ultraviolet. 
  We attempt to construct new superstring actions with a $D$-plet of Majorana fermions $\psi^{\cal B}_A$, where ${\cal B}$ is the $D$ dimensional space-time index and $A$ is the two dimensional spinor index, by deforming the Schild action. As a result, we propose three kinds of actions: the first is invariant under N=1 (the world-sheet) supersymmetry transformation and the area-preserving diffeomorphism. The second contains the Yukawa type interaction. The last possesses some non-locality because of bilinear terms of $\psi^{\cal B}_A$. The reasons why completing a Schild type superstring action with $\psi^{\cal B}_A$ is difficult are finally discussed. 
  We study the interaction of gauge fields of arbitrary half-integer spins with the homogeneous electromagnetic field. We reduce the problem of obtaining the gauge-invariant Lagrangian and transformations of the half-integer spin fields in the external field to an algebraic problem of search for a set of operators with certain algebraical features using the representation of the higher-spin fields as vectors in a pseudo-Hilbert space. We consider such construction at linear order in the external electromagnetic field and also present an explicit form of interaction Lagrangians and gauge transformations for the massive particles of spins 3/2 and 5/2 in terms of symmetric spin-tensor fields. The obtained result is valid for space-time of arbitrary even dimension. 
  A simple systematic method for calculating derivative expansions of the one-loop effective action is presented. This method is based on using symbols of operators and well known deformation quantization theory. To demonstrate its advantages we present several examples of application for scalar theory, Yang-Mills theory, and scalar electrodynamics. The superspace formulation of the method is considered for K\"ahlerian and non-K\"ahlerian quantum corrections for Wess-Zumino and for Heisenberg-Euler lagrangians in super QED models. 
  Non--minimal repulsive singularities (``repulsons'') in extended supergravity theories are investigated. The short distance antigravity properties of the repulsons are tested at the classical and the quantum level by a scalar test--particle. Using a partial wave expansion it is shown that the particle gets totally reflected at the origin. A high frequency incoming particle undergoes a phase shift of $\frac{\pi}{2}$. However, the phase shift for a low--frequency particle depends upon the physical data of the repulson. The curvature singularity at a finite distance $r_h$ turns out to be transparent for the scalar test--particle and the coordinate singularity at the origin serves as a repulsive barrier at which particles bounce off. 
  I consider the classical Kac-Moody algebra and Virasoro algebra in Chern-Simons theory with boundary within the Dirac's canonical method and Noether procedure. It is shown that the usual (bulk) Gauss law constraint becomes a second-class constraint because of the boundary effect. From this fact, the Dirac bracket can be constructed explicitly without introducing additional gauge conditions and the classical Kac-Moody and Virasoro algebras are obtained within the usual Dirac method. The equivalence to the symplectic reduction method is presented and the connection to the Ba\~nados's work is clarified. It is also considered the generalization to the Yang-Mills-Chern-Simons theory where the diffeomorphism symmetry is broken by the (three-dimensional) Yang-Mills term. In this case, the same Kac-Moody algebras are obtained although the two theories are sharply different in the canonical structures. The both models realize the holography principle explicitly and the pure CS theory reveals the correspondence of the Chern-Simons theory with boundary/conformal field theory, which is more fundamental and generalizes the conjectured anti-de Sitter/conformal field theory correspondence. 
  We present a geometric construction of a new class of hyper-Kahler manifolds with torsion. This involves the superposition of the four-dimensional hyper-Kahler geometry with torsion associated with the NS-5-brane along quaternionic planes in quaternionic k-space, $\bH^k$. We find the moduli space of these geometries and show that it can be constructed using the bundle space of the canonical quaternionic line bundle over a quaternionic projective space. We also investigate several special cases which are associated with certain classes of quaternionic planes in $\bH^k$. We then show that the eight-dimensional geometries we have found can be constructed using quaternionic calibrations. We generalize our construction to superpose the same four-dimensional hyper-Kahler geometry with torsion along complex planes in $\bC^{2k}$. We find that the resulting geometry is Kahler with torsion. The moduli space of these geometries is also investigated. In addition, the applications of these new geometries to M-theory and sigma models are presented. In particular, we find new solutions of IIA supergravity with the interpretation of intersecting NS-5-branes at Sp(2)-angles on a string and show that they preserve 3/32, 1/8, 5/32 and 3/16 of supersymmetry. We also show that two-dimensional sigma models with target spaces the above manifolds have (p,q) extended supersymmetry. 
  We consider the type 0 theories, obtained from the closed NSR string by a diagonal GSO projection which excludes space-time fermions, and study the D-branes into these theories. The low-energy dynamics of N coincident D-branes is governed by a U(N) gauge theory coupled to adjoint scalar fields. It is tempting to look for the type 0 string duals of such bosonic gauge theories in the background of the R-R charged p-brane classical solutions. This results in a picture analogous to the one recently proposed by Polyakov (hep-th/9809057). One of the serious problems that needs to be resolved is the closed string tachyon mode which couples to the D-branes and appears to cause an instability. We study the tachyon terms in the type 0 effective action and argue that the background R-R flux provides a positive shift of the (mass)^2 of the tachyon. Thus, for sufficiently large flux, the tachyonic instability may be cured, removing the most basic obstacle to constructing the type 0 duals of non-supersymmetric gauge theories. We further find that the tachyon acquires an expectation value in presence of the R-R flux. This effect is crucial for breaking the conformal invariance in the dual description of the 3+1 dimensional non-supersymmetric gauge theory. 
  The realization of ${\cal N}=2$ four dimensional super Yang-Mills theories in terms of a single Dirichlet fivebrane in type IIB string theory is considered. A classical brane computation reproduces the full quantum low energy effective action. This result has a simple explanation in terms of mirror symmetry. 
  The 256 dimensional M2-brane multiplet contains solitons of many different intrinsic spins. Using the broken supersymmetry transformations of the M2-brane, we find supergravity solutions which explicitly display these spins. This amounts to quantizing the fermionic zero modes and computing the back reaction on the metric and gauge potential. These spacetime fields are therefore operator valued and acquire a conventional classical meaning only after taking expectations in given BPS states. Our spinning spacetimes are not of the standard Kerr form -- there is a non-vanishing gravitino. Nevertheless, the solutions have angular momentum and magnetic dipole moments with a g-factor of 2. We use probe techniques to study scattering of spinning BPS M2-branes. The static interactions cancel between like-sign branes at leading order, but there are static spin-spin forces between branes and anti-branes. The general probe-background Lagrangian contains gravitational spin-spin and magnetic dipole-dipole forces, as well as gravitino exchanges which allow branes to change fermion number. 
  We connect the discrete and continuous Bogomolny equations. There exists one-parameter algebra relating two equations which is the deformation of the extended conformal algebra. This shows that the deformed algebra plays the role of the link between the matrix valued model and the model with one more space dimension higher. 
  It is shown that the multiplicative anomaly in the vector-axial-vector model, which apparently has nothing to do with the breaking of classical current symmetries, nevertheless is strictly related to the well known consistent and covariant anomalies. 
  Conformal field theories based on $g/u(1)^d$ coset constructions where $g$ is a reductive algebra are studied.It is shown that the theories are equivalent to constrained WZNW models for $g.$ Generators of extended symmetry algebras and primary fields are constructed. 
  Following two different tracks, we arrive at a definition of Nahm's transformation valid for self-dual fields on the 4-dimensional torus with non-zero twist tensor.The transform is again a self-dual gauge field defined on a new torus and with non-zero twist tensor. It preserves the property of being an involution. 
  In this article we consider some properties of the (0,2) theory using AdS/ CFT correspondence. We also consider the "string baryonic state" of the theory. We will show that stable baryon states for $k$ string exist provided $2/3 N \leq k \leq N$. The (0,2) theory in the finite temperature is also considered using Schwarzshild geometry, giving information about the five dimensional theory obtained at higher temperature in this background. One can also see that there are two descriptions of a four dimensional gauge theory which become equivalent at strong coupling and approach this five dimensional theory. 
  A complete set of Feynman rules is derived, which permits a perturbative description of the nonequilibrium dynamics of a symmetry-breaking phase transition in $\lambda\phi^4$ theory in an expanding universe. In contrast to a naive expansion in powers of the coupling constant, this approximation scheme provides for (a) a description of the nonequilibrium state in terms of its own finite-width quasiparticle excitations, thus correctly incorporating dissipative effects in low-order calculations, and (b) the emergence from a symmetric initial state of a final state exhibiting the properties of spontaneous symmetry breaking, while maintaining the constraint $<\phi>\equiv 0$. Earlier work on dissipative perturbation theory and spontaneous symmetry breaking in Minkowski spacetime is reviewed. The central problem addressed is the construction of a perturbative approximation scheme which treats the initial symmetric state in terms of the field $\phi$, while the state that emerges at later times is treated in terms of a field $\zeta$, linearly related to $\phi^2$. The connection between early and late times involves an infinite sequence of composite propagators. Explicit one-loop calculations are given of the gap equations that determine quasiparticle masses and of the equation of motion for $<\phi^2(t)>$ and the renormalization of these equations is described. The perturbation series needed to describe the symmetric and broken-symmetry states are not equivalent, and this leads to ambiguities intrinsic to any perturbative approach. These ambiguities are discussed in detail and a systematic procedure for matching the two approximations is described. 
  We use the graded eigenvalue method, a variant of the supersymmetry technique, to compute the universal spectral correlations of the QCD Dirac operator in the presence of massive dynamical quarks. The calculation is done for the chiral Gaussian unitary ensemble of random matrix theory with an arbitrary Hermitian matrix added to the Dirac matrix. This case is of interest for schematic models of QCD at finite temperature. 
  We describe some single-sided BPS domain wall configurations in M-theory. These are smooth non-singular resolutions of Calabi--Yau orbifolds obtained by identifying the two sides of the wall under reflection. They may thus be thought of as domain walls at the end of the universe. We also describe related domain wall type solutions with a negative cosmological constant. 
  We introduce the chemical potential in a system of two-flavored massless fermions in a chiral bag by imposing boundary conditions in the Euclidean time direction. We express the fermionic mean number in terms of a functional trace involving the Green function of the boundary value problem, which is studied analytically. Numerical evaluations for the fermionic number are presented. 
  General conjectures about the SL(2,Z) modular transformation properties of N=4 super-Yang-Mills correlation functions are presented. It is shown how these modular transformation properties arise from the conjectured duality with IIB string theory on AdS_5 x S^5. We discuss in detail a prediction of the AdS duality: that N=4 field theory, in an appropriate limit, must exhibit bonus symmetries, corresponding to the enhanced symmetries of IIB string theory in its supergravity limit. 
  A generalization of the Maldacena conjecture asserts that Type IIB string theory on $\S^5/\Z_3$ is equivalent to a certain supersymmetric $SU(N)^3$ gauge theory with bifundamental matter. To test this assertion, we analyze the wrapped branes on $\S^5/\Z_3$ and their interpretation in terms of gauge theory. The wrapped branes are interpreted in some cases as baryons or dibaryons of the gauge theory and in other cases as strings around which there is a global monodromy. In order to successfully match the brane analysis with field theory, we must uncover some aspects of $S$-duality which are novel even in the case of four-dimensional free field theory. 
  We study the effect of the gravitational Chern-Simons term (GCST) in the (2+1)-dimensional anti-de Sitter (AdS$_{2+1}$) geometry. In the context of the gauge gravity, we obtain black hole solution and its boundary WZW theory. The BTZ black hole solution can still be retrieved but its gravitational mass and angular momentum become different from their inherent values. The deformation on these quantities due to the GCST can be summarized as $SO(1,1)$ times rescaling. The boundary WZW theory is found to be chiral, i.e., composed of the right moving part and the left moving part with different Kac-Moody levels. The statistical entropy is proportional to the area only for the large levels and vanishing GCST limit, but its coefficient is not the correct order in the Newton constant $G$. Some related physics are discussed. 
  The instanton contributions to the partition function and to homologically trivial Wilson loops for a U(N) Yang-Mills theory on a torus $T^2$ are analyzed. An exact expression for the partition function is obtained as a sum of contributions localized around the classical solutions of Yang-Mills equations, that appear according to the general classification of Atiyah and Bott. Explicit expressions for the exact Wilson loop averages are obtained when N=2, N=3. For general $N$ the contribution of the zero-instanton sector has been carefully derived in the decompactification limit, reproducing the sum of the perturbative series on the plane, in which the light-cone gauge Yang-Mills propagator is prescribed according to Wu-Mandelstam-Leibbrandt (WML). Agreement with the results coming from $S^2$ is therefore obtained, confirming the truly perturbative nature of the WML computations. 
  The consequences of holography hypothesis are investigated for the Pre-big-bang string cosmological models. The evolution equations are obtained from the tree level string effective action. It is shown that $S/A$ is bounded by a constant in each case, $S$ being the entropy within the volume bounded by the horizon of area $A$. 
  We consider the most general diffeomorphism invariant action in 1+1 spacetime dimensions that contains a metric, dilaton and Abelian gauge field, and has at most second derivatives of the fields. Our action contains a topological term (linear in the Abelian field strength) that has not been considered in previous work. We impose boundary conditions appropriate for a charged black hole confined to a region bounded by a surface of fixed dilaton field and temperature. By making some simplifying assumptions about the quantum theory, the Hamiltonian partition function is obtained. This partition function is analyzed in some detail for the Reissner-Nordstrom black hole and for the rotating BTZ black hole. 
  We have studied numerically Faddeev-Hopf knots, which are defined as those unit-vector fields in $R^3$ that have a nontrivial Hopf charge and minimize Faddeev's Lagrangian. A given initial configuration was allowed to relax into a (local) minimum using the first order dissipative dynamics corresponding to the steepest descent method. A linked combination of two un-knots was seen to relax into different minimum energy configurations depending on their charges and their relative handedness and direction. In order to visualize the results we plot certain gauge-invariant iso-surfaces. 
  Julian Schwinger became interested in the Casimir effect in 1975. His original impetus was to understand the quantum force between parallel plates without the concept of zero point fluctuations of field quanta, in the language of source theory. He went on to consider applications to dielectrics and to spherical geometries in 1977. Although he published nothing on the subject in the following decade, he did devote considerable effort to understanding the connection between acceleration and temperature in the mid 1980s. During the last four years of his life, he became fascinated with sonoluminescence, and proposed that the dynamical Casimir effect could be responsible for the copious emission of photons by collapsing air bubbles in water. 
  In these notes the exact renormalization group formulation of the scalar theory is briefly reviewed. This regularization scheme is then applied to supersymmetric theories. In case of a supersymmetric gauge theory it is also shown how to recover gauge invariance, broken by the introduction of the infrared cutoff. 
  In asymptotically flat space a rotating black hole cannot be in thermodynamic equilibrium because the thermal radiation would have to be co-rotating faster than light far from the black hole. However in asymptotically anti-de Sitter space such equilibrium is possible for certain ranges of the parameters. We examine the relationship between conformal field theory in rotating Einstein universes of dimensions two to four and Kerr anti-de Sitter black holes in dimensions three to five. The five dimensional solution is new. We find similar divergences in the partition function of the conformal field theory and the action of the black hole at the critical angular velocity at which the Einstein rotates at the speed of light. This should be an interesting limit in which to study large $N$ Yang-Mills. 
  The greybody factors for spin 1/2 particles in the BTZ black holes are discussed from 2D CFT in bulk-boundary correspondence. It is found that the initial state of spin 1/2 particle in the BTZ black holes can be described by the Poincar\'{e} vacuum state in boundary 2D CFT, and the nonlinear coordinate transformation causes the thermalization of the Poincar\'{e} vacuum state. For special case, our results for the greybody factors agree with the semiclassical calculation. 
  General two-dimensional pure dilaton-gravity can be discussed in a unitary way by introducing suitable field redefinitions. The new fields are directly related to the original spacetime geometry and in the canonical picture they generalize the well-known geometrodynamical variables used in the discussion of the vacuum dilatonic black hole. So the model can be quantized using the techniques developed for the latter case. The resulting quantum theory coincides with the quantum theory obtained imposing the Birkhoff theorem at the classical level. 
  Two-dimensional matterless dilaton gravity with arbitrary dilatonic potential can be discussed in a unitary way, both in the Lagrangian and canonical frameworks, by introducing suitable field redefinitions. The new fields are directly related to the original spacetime geometry and in the canonical picture they generalize the well-known geometrodynamical variables used in the discussion of the Schwarzschild black hole. So the model can be quantized using the techniques developed for the latter case. The resulting quantum theory exhibits the Birkhoff theorem at the quantum level. 
  We report on the computation of the one-instanton contribution to the 16-point Green function of fermionic composite operators in N=4 Super YM theory. The remarkable agreement, initially found in the case of an SU(2) gauge group, with the one D-instanton contribution to the corresponding type IIB superstring amplitude on $AdS_5\times S^5$ is reviewed here. The recent extension by other authors of this result to any SU(N) gauge group and to multi-instantons in the large N limit is briefly discussed. We also argue that for the AdS/SCFT correspondence under consideration to work for any $N\geq 2$, at least for some protected interactions, string effects should require a truncation of the Kaluza--Klein spectrum for finite $S^5$ radius much in the same way as worldsheet unitarity restricts the allowed isospins in SU(2) WZW models. Finally, we briefly comment on the logarithmic behavior of some four-point Green functions of scalar composite operators. 
  Using reflection positivity as the main tool, we establish a connection between the existence of a critical point in classical spin models and the triviality of a certain local cohomology class related to the Noether current of the model in the continuum limit. Furthermore we find a relation between the location of the critical point and the momentum space autocorrelation function of the Noether current. 
  We consider the photon field between an unusual configuration of infinite parallel plates: a perfectly conducting plate $(\epsilon\to\infty)$ and an infinitely permeable one $\mu\to\infty)$. After quantizing the vector potential in the Coulomb gauge, we obtain explicit expressions for the vacuum expectation values of field operators of the form $<{\hat E}_i{\hat E}_j>_0$ and $<{\hat B}_i{\hat B}_j>_0$. These field correlators allow us to reobtain the Casimir effect for this set up and to discuss the light velocity shift caused by the presence of plates (Scharnhorst effect \cite{Scharnhorst,Barton,BarScharn}) for both scalar and spinor QED. 
  Let X=Sl(3,Z)\Sl(3,R)/SO(3,R). Let N(lambda) denote the dimension of the space of cusp forms with Laplace eigenvalue less than lambda. We prove that N(lambda)=C lambda^(5/2)+O(lambda^2) where C is the appropriate constant establishing Weyl's law with a good error term for the noncompact space X. The proof uses the Selberg trace formula in a form that is modified from the work of Wallace and also draws on results of Stade and Wallace and techniques of Huntley and Tepper. We also, in the course of the proof, give an upper bound on the number of cusp forms that can violate the Ramanujan conjecture. 
  As a novel application of string junctions, we provide evidence for the existence of stable non-BPS dyons with magnetic charge greater than 1 in (the semiclassical regime of) N=2 SU(2) Super-Yang-Mills theory. In addition, we find a new curve of marginal stability. Moduli space is therefore divided into four regions, each containing a different stable particle spectrum. 
  An example is given of a plane topological defect solution of linearized Einstein-Cartan (EC) field equation representing a cosmic wall boundary of spinning matter. The source of Cartan torsion is composed of two orthogonal lines of static polarized spins bounded by the cosmic plane wall. The Kopczy\'{n}ski- Obukhov - Tresguerres (KOT) spin fluid stress-energy current coincides with thin planar matter current in the static case. Our solution is similar to Letelier solution of Einstein equation for multiple cosmic strings. Due to this fact we suggest that the lines of spinning matter could be analogous to multiple cosmic spinning string solution in EC theory of gravity. When torsion is turned off a pure Riemannian cosmic wall is obtained. 
  In the framework of Dirac quantization, the SU(2) Skyrmion is canonically quantized to yield the modified predictions of the static properties of baryons. We show that the energy spectrum of this Skyrmion obtained by the Dirac method with the suggestion of generalized momenta is consistent with the result of the Batalin-Fradkin-Tyutin formalism. 
  A propagation torsion model for quantized vortices is proposed.The model is applied to superfluids and liquid Helium II. 
  General results on the structure of the bosonization of fermionic systems in $(2+1)$d are obtained. In particular, the universal character of the bosonized topological current is established and applied to generic fermionic current interactions. The final form of the bosonized action is shown to be given by the sum of two terms. The first one corresponds to the bosonization of the free fermionic action and turns out to be cast in the form of a pure Chern-Simons term, up to a suitable nonlinear field redefinition. We show that the second term, following from the bosonization of the interactions, can be obtained by simply replacing the fermionic current by the corresponding bosonized expression. 
  Algebraic realizations of supersymmetry through SU(m,n) type superalgebras are developed. We show their applications to a bilocal quark-antiquark or a quark-diquark systems. A new scheme based on SU(6/1) is fully exploited and the bilocal approximation is shown to get carried unchanged into it. Color algebra based on octonions allows the introduction of a new larger algebra that puts quarks, diquarks and exotics in the same supermultiplet as hadrons and naturally suppresses quark configurations that are symmetrical in color space and antisymmetrical in remaining flavor, spin and position variables. A preliminary work on the first order relativistic formulation through the spin realization of Wess-Zumino super-Poincare algebra is presented. 
  The unconstrained classical system equivalent to spatially homogeneous SU(2) Yang-Mills theory with theta angle is obtained and canonically quantized. The Schr\"odinger eigenvalue problem is solved approximately for the low lying states using variational calculation. The properties of the groundstate are discussed, in particular its electric and magnetic properties, and the value of the "gluon condensate" is calculated. Furthermore it is shown that the energy spectrum of SU(2) Yang-Mills quantum mechanics is independent of the theta angle. Explicit evaluation of the Witten formula for the topological susceptibility gives strong support for the consistency of the variational results obtained. 
  We show that any solution of the SU(2) Skyrme model can be used to give a topologically trivial solution of the SU(4) one. In addition, we extend the method introduced by Houghton et al. and use harmonic maps from S2 to CP(N-1) to construct low energy configurations of the SU(N) Skyrme models. We show that one of such maps gives an exact, topologically trivial, solution of the SU(3) model. We study various properties of these maps and show that, in general, their energies are only marginally higher than the energies of the corresponding SU(2) embeddings. Moreover, we show that the baryon (and energy) densities of the SU(3) configurations with baryon number B=2-4 are more symmetrical than their SU(2) analogues. We also present the baryon densities for the B=5 and B=6 configurations and discuss their symmetries. 
  We discuss the theoretical basis for the search of the possible experimental manifestations of the torsion field at low energies. First, the quantum field theory in an external gravitational field with torsion is reviewed. The renormalizability requires the nonminimal interaction of torsion with spinor and scalar (Higgs) fields. The Pauli-like equation contains new torsion-dependent terms which have a different structure as compared with the standard electromagnetic ones. The same concerns the nonrelativistic equations for spin-${1}/{2}$ particle in an external torsion and electromagnetic fields. Second, we discuss the propagating torsion. For the Dirac spinor coupled to the electromagnetic and torsion field there is some additional softly broken local symmetry associated with torsion. As a consequence of this symmetry, in the framework of effective field theory, the torsion action is fixed with accuracy to the values of the coupling constant of the torsion-spinor interaction, mass of the torsion and higher derivative terms. The introduction of the Higgs field spoils the consistency of this scheme, and the effective quantum field theory for torsion embedded into the Standard Model is not possible. The phenomenological consequences of the torsion-fermion interaction are drown and the case of the torsion mass of the Planck order is discussed. 
  The chiral and scale anomalies of a very general class of non local Dirac operators are computed using the $\zeta$-function definition of the fermionic determinant. For the axial anomaly all new terms introduced by the non locality are shown to be removable by counterterms and such counterterms are also explicitly computed. It is verified that the non local Dirac operators have the standard minimal anomaly in Bardeen's form. 
  We investigate the connection between Abelian bosonization in the Minkowski and Euclidean formalisms. The relation is best seen in the complex time formalism of S. A. Fulling and S. N. M. Ruijsenaars \cite{fulling}. 
  We extend a recent analysis of gravitational perturbations on Dirac-Nambu-Goto strings, membranes and higher dimensional branes. In an arbitrary gauge, it is shown that the relevant first order equations governing the displacement vector of the worldsheet and metric perturbation are obtainable from a variational principle whose Lagrangian is constructed as a second order perturbation of the standard Dirac-Goto-Nambu action density. A symplectic current functional is obtained as a by-product that is potentially useful for the derivation of conservation laws in particular circumstances. 
  Improved expansion in width is applied to a curved domain wall in nonrelativistic dissipative $\lambda(\Phi^2 - v^2)^2 $ model with real scalar order parameter $\Phi$. Approximate analytic description of such a domain wall to second order in the width is presented. 
  Using numerical simulations of the full nonlinear equations of motion we investigate topological solitons of a modified O(3) sigma model in three space dimensions, in which the solitons are stabilized by the Hopf charge. We find that for solitons up to charge five the solutions have the structure of closed strings, which become increasingly twisted as the charge increases. However, for higher charge the solutions are more exotic and comprise linked loops and knots. We discuss the structure and formation of these solitons and demonstrate that the key property responsible for producing such a rich variety of solitons is that of string reconnection. 
  The one-loop quantum corrections to the free energy associated with scalar field in a higher dimensional static curved space-time is investigated making use of the conformal transformation method. For a space-time with bifurcate horizon, horizon divergences are accounted for choosing the Planck length as natural cutoff. The leading term in the high temperature quantum correction satifies holographically the "area law", like the tree level Bekenstein-Hawking term. Furthermore it is stressed that only for the asymptotically AdS black holes one may have a microscopic interpretation of the entropy also at quantum level. 
  We describe a (first, to the best of our knowledge) essentially soluble example of dynamical symmetry breaking phenomenon in a 3+1 dimensional gauge theory without fundamental scalar fields: QED in a constant magnetic field. 
  We consider the 5-brane placed at one end of the world in the Heterotic $E_8 \times E_8$ theory. The low energy theory is a 6 dimensional $(1,0)$ superconformal theory with $E_8$ as a global symmetry. We calculate the two-point correlator of the $E_8$ current in 6 dimensions and in 4 dimensions after compactification on $\MT{2}$. This correlator is derived in 3 different ways: From field theory, from 11 dimensional supergravity and from F-theory. 
  I review some recent results showing that the physics of negative energy densities, as predicted by relativistic quantum field theories, is more complicated than has generally been appreciated. On the one hand, in external potentials where there is a time--dependence, however slight, the Hamiltonians are unbounded below. On the other, there are limitations of quantum measurement in detecting or utilizing these negative energies. 
  A general condition for the existence of fermion zero modes is derived for the M-5-brane, the M-2-brane and the D=4, N=2 Majumdar-Papapetrou 0-brane. The fermion zero modes of these p-branes do not exist if the supersymmetry spinor generator goes to a constant at the horizon and they exist only if it vanishes there. In particular it is shown that the fermion zero mode of the M-2-brane in D=11 can be forbidden from existence if Rarita-Schwinger gamma tracelessness condition is imposed on the gravitino field. Non-existence of fermion zero mode is interpreted, in analogy to the three dimensional example of Becker et.al., as a world with zero cosmological constant without supersymmetric excited states. Also derived are the spin of the M-5-brane and its 3-form electric and magnetic dipole moments. 
  We consider three dimensional SU(N) N=1 super-Yang-Mills compactified on the space-time R X S^1 X S^1. In particular, we compactify the light-cone coordinate x^- on a light-like circle via DLCQ, and wrap the remaining transverse coordinate on a spatial circle. By retaining only the first few excited modes in the transverse direction, we are able to solve for bound state wave functions and masses numerically by diagonalizing the discretized light-cone supercharge. This regularization of the theory is shown to preserve supersymmetry. We plot bound state masses as a function of the coupling, showing the transition in particle masses as we move from a weakly to a strongly-coupled theory. We analyze both numerically and analytically massless states which exist only in the limit of strong or weak gauge coupling. In addition, we find massless states that persist for all values of the gauge coupling. An analytical treatment of these massless states is provided. Interestingly, in the strong coupling limit, these massless states become string-like. 
  In this work we investigate the role of the symmetry of the Lagrangian on the existence of defects in systems of coupled scalar fields. We focus attention mainly on solutions where defects may nest defects. When space is non-compact we find topological BPS and non-BPS solutions that present internal structure. When space is compact the solutions are nontopological sphalerons, which may be nested inside the topological defects. We address the question of classical stability of these topological and nontopological solutions and investigate how the thermal corrections may modify the classical scenario. 
  Vortex solutions to the classical field equations in a massive, renormalizable U(1) gauge model are considered in (2+1) dimensions. A vector field whose kinetic term consists of a Chern-Simons term plus a Stuekelberg mass term is coupled to a scalar field. If the classical scalar field is set equal to zero, then there are classical configurations of the vector field in which the magnetic flux is non-vanishing and finite. In contrast to the Nielsen-Olesen vortex, the magnetic field vanishes exponentially at large distances and diverges logarithmicly at short distances. This divergence, although not so severe as to cause the flux to diverge, results in the Hamiltonian becoming infinite. If the classical scalar field is no longer equal to zero, then the magnetic flux is not only finite, but quantized and the asymptotic behaviour of the field is altered so that the Hamiltonian no longer suffers from a divergence due to the field configuration at the origin. Furthermore, the asymptotic behaviour at infinity is dependent on the magnitude of the Stuekelberg mass term. 
  By making use of the decomposition theory of gauge potential, the inner structure of SU(2) and SO(4) gauge theory is discussed in detail. We find the SO(4) monopole can be given via projecting the SO(4) gauge field onto an antisymmetric tensor. This projection fix the coset $% SU(2)/U(1)\bigotimes SU(2)/U(1)$ of SO(4) gauge group. The generalized Hopf map is given via a Dirac spinor. Further we prove that this monopole can be consider as a new topological invariant. Which is composed of two monopole structures. Local topological structure of the SO(4) monopole is discussed in detail, which is quantized by winding number. The Hopf indices and Brouwer degree labels the local property of the monopoles. 
  We discuss BPS states preserving 1/4 supersymmetries of N=4 supersymmetric Yang-Mills theory as M2-branes holomorphically embedded and ending on M5-branes. We use techniques in electrodynamics to find the M2-brane configurations, and give some explicit examples. In case the M2-brane worldsheet has handles, the worldsheet moduli of the M2-brane is constrained in a discrete manner. Several aspects of multi-pronged strings in type IIB string theory are beautifully reproduced in the M-theory description. We also discuss the relation between the above construction and the D2-brane dynamics in type IIA string theory. 
  We study two systems of BPS solitons in which spin-spin interactions are important in establishing the force balances which allow static, multi-soliton solutions to exist. Solitons in the Israel-Wilson-Perjes (IWP) spacetimes each carry arbitrary, classical angular momenta. Solitons in the Aichelburg-Embacher "superpartner" spacetimes carry quantum mechanical spin, which originates in the zero-modes of the gravitino field of N=2 supergravity in an extreme Reissner-Nordstrom background. In each case we find a cancellation between gravitational spin-spin and magnetic dipole-dipole forces, in addition to the usual one between Newtonian gravitational attraction and Coulombic electrostatic repulsion. In both cases, we analyze the forces between two solitons by treating one of the solitons as a probe or test particle, with the appropriate properties, moving in the background of the other. In the IWP case, the equation of motion for a spinning test particle, originally due to Papapetrou, includes a coupling between the background curvature and the spin of the test particle. In the superpartner case, the relevant equation of motion follows from a kappa-symmetric superparticle action. 
  D-branes, topological defects in string theory on which string endpoints can live, may give new insight into the understanding of the cosmological evolution of the Universe at early epochs. We analyze the dynamics of D-branes in curved backgrounds and discuss the parameter space of M-theory as a function of the coupling constant and of the curvature of the Universe. We show that D-branes may be efficiently produced by gravitational effects. Furthermore, in curved spacetimes the transverse fluctuations of the D-branes develop a tachyonic mode and when the fluctuations grow larger than the horizon the branes become tensionless and break up. This signals a transition to a new regime. We discuss the implications of our findings for the singularity problem present in string cosmology, suggesting the existence of a limiting value for the curvature which is in agreement with the value suggested by the cosmological version of the holography principle. We also comment on possible implications for the so-called brane world scenario, where the Standard Model gauge and matter fields live inside some branes while gravitons live in the bulk. 
  The quantisation of the two-dimensional Liouville field theory is investigated using the path integral, on the sphere, in the large radius limit. The general form of the $N$-point functions of vertex operators is found and the three-point function is derived explicitly. In previous work it was inferred that the three-point function should possess a two-dimensional lattice of poles in the parameter space (as opposed to a one-dimensional lattice one would expect from the standard Liouville potential). Here we argue that the two-dimensionality of the lattice has its origin in the duality of the quantum mechanical Liouville states and we incorporate this duality into the path integral by using a two-exponential potential. Contrary to what one might expect, this does not violate conformal invariance; and has the great advantage of producing the two-dimensional lattice in a natural way. 
  We examine wave propagation and the formation of shocks in strongly magnetized plasmas by applying a variational technique and the method of characteristics to the coupled magnetohydrodynamic (MHD) and quantum-electrodynamic (QED) equations of motion. In sufficiently strong magnetic fields such as those found near neutron stars, not only is the plasma extremely relativistic but the effects of QED must be included to understand processes in the magnetosphere. As Thompson & Blaes [1] find, the fundamental modes in the extreme relativistic limit of MHD coupled with QED are two oppositely directed Alfv\'{e}n modes and the fast mode. QED introduces nonlinear couplings which affect the propagation of the fast mode such that waves traveling in the fast mode evolve as vacuum electromagnetic ones do in the presence of an external magnetic field [2] (Heyl & Hernquist 1998). The propagation of a single Alfv\'{e}n mode is unaffected but QED does alter the coupling between the Alfv\'{e}n modes. 
  The effective one-loop action for general dilaton theories with arbitrary dilaton-dependent measure and nonminimal coupling to scalar matter is computed. As an application we determine the Hawking flux to infinity from black holes in d-dimensions. We resolve the recently resurrected problem of an apparent negative flux for nonminimally coupled scalars: For any $D \geq 4$ Black Hole the complete flux turns out to be precisely the one of minimal coupling. This result is obtained from a Christensen-Fulling type argument involving the (non-)conservation of energy-momentum. It is compared with approaches using the effective action. 
  Massless BPS-saturated states in toroidally compactified heterotic string were discussed first. Then, I constructed a large class of BPS-saturated states of toroidally compactified type II string theories. They were parametrized by four independent parameters. Both orthogonal and non-orthogonal intersections of non-perturbative states in the corresponding ten dimensional theories were discussed. Phenomenologically, I considered the string effect in the form of threshold corrections on dilatonic charged black holes, and the superpotential of N=1 supergravity theory 
  We give an interpretation to the issue of the chiral determinant in the heat-kernel approach. The extra dimension (5-th dimension) is interpreted as (inverse) temperature. The 1+4 dim Dirac equation is naturally derived by the Wick rotation for the temperature. In order to define a ``good'' temperature, we choose those solutions of the Dirac equation which propagate in a fixed direction in the extra coordinate. This choice fixes the regularization of the fermion determinant. The 1+4 dimensional Dirac mass ($M$) is naturally introduced and the relation: $|$4 dim electron momentum$|$ $\ll$ $|M|$ $\ll$ ultraviolet cut-off, naturally appears. The chiral anomaly is explicitly derived for the 2 dim Abelian model. Typically two different regularizations appear depending on the choice of propagators. One corresponds to the chiral theory, the other to the non-chiral (hermitian) theory. 
  We compute the combined effect of confinement, an external magnetic field and temperature on the vacuum of the charged scalar field using Schwinger's formula for the effective action in the imaginary time formalism. The final result reproduces an effective Lagrangian similar to the Heisenberg-Euler one in the limit of no confinement, in the case of confinement it provides the necessary corrections to this Lagrangian at each order of magnitude of the magnetic field. The results show a finite temperature contribution to the vacum permeability constant apart from the one due to confinement alone. 
  We investigate the non-perturbative equivalence of some heterotic/type IIA dual pairs with N=2 supersymmetry. We compute R2-like corrections, both on the type IIA and on the heterotic side. The coincidence of their perturbative part provides a test of duality. The type IIA result is then used to predict the full, non-perturbative correction to the heterotic effective action. We determine the instanton numbers and the Olive-Montonen duality groups. 
  By solving the equations of motion of massive $p$-form potential in Anti-de-Sitter space and using the $AdS/CFT$ correspondence of Maldacena, the generating functional of two-point correlation functions of the currents is obtained. When the mass parameter vanishes the result agrees with the known massless case. 
  Kaluza-Klein monopole and H-monopole solutions, which are T-dual to each other, are the well-known solutions of string theory compactified on $T^6$. Since string theory in this case has an S-duality symmetry, we explicitly construct the corresponding dyonic solutions by expressing the $D = 4$ string effective action in a manifestly $SL(2, R)$ invariant form with an $SL(2, R)$ invariant constraint. The Schwarz-Sen charge spectrum, the BPS saturated mass formula as well as the stability of these states are discussed briefly. 
  Dynamical spontaneous breaking of some discrete symmetries including special parities and time reversal and their restoration at finite temperature T are researched in 3D Gross-Neveu model by means of Schwinger-Dyson equation in the real-time thermal field theory in the fermion bubble diagram approximation. When the momentum cut-off $\Lambda$ is large enough, the equation of critical chemical potential $\mu_c$ and critical temperature $T_c$ will be $\Lambda$-independent and identical to the one obtained by auxialiary scalar field approach. The dynamical fermion mass m, as the order parameter of symmetry breaking, has the same $(T_c-T)^{1/2}$ behavior as one in 4D NJL-model when T is less than and near $T_c$ and this shows the second-order phase transition feature of the symmetry restoration at $T>T_c$. It is also proven that no scalar bound state could exist in this model. 
  For each lattice one can define a free boson theory propagating on the corresponding torus. We give an alternative definition where one employs any automorphism of the group $M^*/M$. This gives a wealth of conformal data, which we realize as some bosonic theory, in all the `regular' cases. We discuss the generalization to affine theories. As a byproduct, we compute the gauss sum for any lattice and any diagonal automorphism. 
  In this review we show that a Clifford algebra possesses a unique irreducible representation; the spinor representation. We discuss what types of spinors can exist in Minkowski space-times and we explain how to construct all the supersymmetry algebras that contain a given space-time Lie algebra. After deriving the irreducible representations of the superymmetry algebras, we explain how to use them to systematically construct supergravity theories. We give the maximally supersymmetric supergravity theories in ten and eleven dimensions and discuss their properties. We find which superbranes can exist for a given supersymmetry algebra and we give the dynamics of the superbranes that occur in M theory. Finally, we discuss how the properties of supergravity theories and superbranes provide evidence for string duality.    In effect, we present a continuous chain of argument that begins with Clifford algebras and leads via supersymmetry algebras and their irreducible representations to supergravity theories, string duality, brane dynamics and M theory. 
  An SO(4) gauge invariant model with extended field transformations is examined in four dimensional Euclidean space. The gauge field is $(A^\mu)^{\alpha\beta} = 1/2 t^{\mu\nu\lambda} (M^{\nu\lambda})^{\alpha\beta}$ where $M^{\nu\lambda}$ are the SO(4) generators in the fundamental representation. The SO(4) gauge indices also participate in the Euclidean space SO(4) transformations giving the extended field transformations. We provide the decomposition of the reducible field $t^{\mu\nu\lambda}$ in terms of fields irreducible under SO(4). The SO(4) gauge transformations for the irreducible fields mix fields of different spin. Reducible matter fields are introduced in the form of a Dirac field in the fundamental representation of the gauge group and its decomposition in terms of irreducible fields is also provided. The approach is shown to be applicable also to SO(5) gauge models in five dimensional Euclidean space. 
  We analyse the structure of N=1 and N=2 supersymmetric non-linear sigma-models built up with a pair of real superfields defined in the superspace of Atiyah-Ward space-time. The geometry arising has new features such as the existence of a locally product structure (N=1 case) and a set of automorphisms of the tangent space that is isomorphic to the split-quaternionic algebra (N=2 case). 
  We study the manifestly covariant three-dimensional symmetric Chern-Simons action in terms of the Batalin-Vilkovisky quantization method. We find that the Lorentz covariant gauge fixed version of this action is reduced to the usual Chern-Simons type action after a proper field redefinition. Furthermore, the renormalizability of the symmetric Chern-Simons theory turns out to be the same as that of the original Chern-Simons theory. 
  We investigate the issue of electromagnetic duality on the light front. We work with Zwanziger's theory of electric and magnetic sources which is appropriate for treating duality. When quantized on the light-front in the light front gauge, this theory yields two independent phase space degrees of freedom, namely the two transverse field components, the right number to describe the gauge field sector of normal light-front QED and also the appropriate commutator between them. The electromagnetic duality transformation formulated in terms of them is similar in form to the Susskind transformation proposed for the free theory, provided one identifies them as the dynamical field components of the photon on the light-front in the presence of magnetic sources. The Hamiltonian density written in terms of these components is invariant under the duality transformation. 
  The quadratic action for physical fields of type IIB supergravity model on $AdS_5\times S^5$ is derived starting from the recently found covariant action. All boundary terms that have to be added to the action to be used in the AdS/CFT correspondence are determined. 
  We study the space-times of non-extremal intersecting p-brane configurations in M-theory, where one of the components in the intersection is a ``NUT,'' i.e. a configuration of the Taub-NUT type. Such a Taub-NUT configuration corresponds, upon compactification to D=4, to a Gross-Perry-Sorkin (GPS) monopole. We show that in the decoupling limit of the CFT/AdS correspondence, the 4-dimensional transverse space of the NUT configuration in D=5 is foliated by surfaces that are cyclic lens spaces S^3/Z_N, where N is the quantised monopole charge. By contrast, in D=4 the 3-dimensional transverse space of the GPS monopole is foliated by 2-spheres. This observation provides a straightforward interpretation of the microscopics of a D=4 string-theory black hole, with a GPS monopole as one of its constituents, in terms of the corresponding D=5 black hole with no monopole. Using the fact that the near-horizon region of the NUT solution is a lens space, we show that if the effect of the Kaluza-Klein massive modes is neglected, p-brane configurations can be obtained from flat space-time by means of a sequence of dimensional reductions and oxidations, and U-duality transformations. 
  It is shown that there exists an on-shell light cone gauge where half of the fermionic components of the super vector potential vanish, so that part of the superspace flatness conditions becomes linear. After reduction to $(1+1)$ space-time dimensions, the general solution of this subset of equations is derived. The remaining non-linear equations are written in a form which is analogous to Yang equations, albeit with superderivatives involving sixteen fermionic coordinates. It is shown that this non-linear part may, nevertheless, be solved by methods similar to powerful technics previously developed for the (purely bosonic) self-dual Yang Mills equations in four dimensions. 
  We consider a large-N Chern-Simons theory for the attractive bosonic matter (Jackiw-Pi model) in the Hamiltonian collective-field approach based on the 1/N expansion. We show that the dynamics of low-lying density excitations around the ground-state vortex configuration is equivalent to that of the Sutherland model. The relationship between the Chern-Simons coupling constant lambda and the Calogero-Sutherland statistical parameter lambda_s signalizes some sort of statistical transmutation accompanying the dimensional reduction of the initial problem. 
  We analyze $S^1$ equivariant cohomology from the supergeometrical point of view. For this purpose we equip the external algebra of given manifold with equivariant even super(pre)symplectic structure, and show, that its Poincare-Cartan invariant defines equivariant Euler classes of surfaces. This allows to derive localization formulae by use of superanalog of Stockes theorem. 
  The unification of general relativity and standard model for strong and electro-weak interactions is considered on the base of the conformal symmetry principle. The Penrose-Chernikov-Tagirov Lagrangian is used to describe the Higgs scalar field modulus and gravitation. We show that the procedure of the Hamiltonian reduction converts the homogeneous part of the Higgs field into the dynamical parameter of evolution of the equivalent reduced system. The equation of dynamics of the "proper time" of an observer with respect to the evolution parameter reproduces the Friedmann-like equation, which reflects the cosmological evolution of elementary particle masses. The value of the Higgs field is determined, at the present time, by the values of mean density of matter and the Hubble parameter in satisfactory agreement with the data of cosmological observations. 
  We present the exact expression for the Nahm gauge field associated to a SU(N) charge one self-dual gauge field on T^3XR. The result implies that the size of the instanton is determined by the ``distance'' between its two flat connections at t plus or minus infinty. 
  This is the first of two articles devoted to a comprehensive exposition of the generating-function method for computing fusion rules in affine Lie algebras. The present paper is entirely devoted to the study of the tensor-product (infinite-level) limit of fusions rules. We consider thus in detail the problem of constructing tensor-product generating functions in finite Lie algebras. From the beginning, the problem is recast in terms of the concept of a model, which is an algebra whose Poincar\'e series is the generating function under study. We start by reviewing Sharp's character method. Simple examples are worked out in detail, illustrating thereby its intrinsic limitations. An alternative approach is then presented; it is based on the reformulation of the problem of calculating tensor products in terms of the solution of a set of linear and homogeneous Diophantine equations whose elementary solutions represent ``elementary couplings''. Grobner bases provide a tool for generating the complete set of relations between elementary couplings and, most importantly, as an algorithm for specifying a complete, compatible set of ``forbidden couplings''. This machinery is then applied to the construction of various tensor-product generating functions. 
  In this work we consider the 2-point Green's functions in (1+1) dimensional quantum electrodynamics and show that the correct implementation of analytic regularization gives a gauge invariant result for the vaccum polarization amplitude and the correct coefficient for the axial anomaly. 
  The method of reduction of a non-Abelian gauge theory to the corresponding unconstrained system is exemplified for SU(2) Yang-Mills field theory. The reduced Hamiltonian which describes the dynamics of the gauge invariant variables is presented in the form of a strong coupling expansion. The physical variables are separated into fields, which are scalars under spatial rotations, and rotational degrees of freedom. It is shown how in the infrared limit an effective nonlinear sigma model type Lagrangian can be derived which out of the six physical fields involves only one of three scalar fields and two rotational fields summarized in a unit vector. Its possible relation to the effective Lagrangian proposed recently by Faddeev and Niemi is discussed. 
  A worldsheet approach to the study of non-abelian D-particle dynamics is presented based on viewing matrix-valued D-brane coordinate fields as coupling constants of a deformed sigma-model which defines a logarithmic conformal field theory. The short-distance structure of spacetime is shown to be naturally captured by the Zamolodchikov metric on the corresponding moduli space which encodes the geometry of the string interactions between D-particles. Spacetime quantization is induced directly by the string genus expansion and leads to new forms of uncertainty relations which imply that general relativity at very short-distance scales is intrinsically described by a non-commutative geometry. The indeterminancies exhibit decoherence effects suggesting the natural incorporation of quantum gravity by short-distance D-particle probes. Some potential experimental tests are briefly described. 
  Spherical field theory is a new non-perturbative method for studying quantum field theories. It uses the spherical partial wave expansion to reduce a general d-dimensional Euclidean field theory into a set of coupled one-dimensional systems. The coupled one-dimensional systems are then converted to partial differential equations and solved numerically. We demonstrate the methods of spherical field theory by analyzing Euclidean phi^4 theory in two dimensions. 
  Entanglement measures based on a logarithmic functional form naturally emerge in any attempt to quantify the degree of entanglement in the state of a multipartite quantum system. These measures can be regarded as generalizations of the classical Shannon-Wiener information of a probability distribution into the quantum regime. In the present work we introduce a previously unknown approach to the Shannon-Wiener information which provides an intuitive interpretation for its functional form as well as putting all entanglement measures with a similar structure into a new context: By formalizing the process of information gaining in a set-theoretical language we arrive at a mathematical structure which we call ''tree structures'' over a given set. On each tree structure, a tree function can be defined, reflecting the degree of splitting and branching in the given tree. We show in detail that the minimization of the tree function on, possibly constrained, sets of tree structures renders the functional form of the Shannon-Wiener information. This finding demonstrates that entropy-like information measures may themselves be understood as the result of a minimization process on a more general underlying mathematical structure, thus providing an entirely new interpretational framework to entropy-like measures of information and entanglement. We suggest three natural axioms for defining tree structures, which turn out to be related to the axioms describing neighbourhood topologies on a topological space. The same minimization that renders the functional form of the Shannon-Wiener information from the tree function then assigns a preferred topology to the underlying set, hinting at a deep relation between entropy-like measures and neighbourhood topologies. 
  In this paper we investigate N=1 supersymmetric gauge theories with a product gauge group. By using smoothly confining dynamics, we can find new dualities which include higher-rank tensor fields, and in which the dual gauge group is simple, not a product. Some of them are dualities between chiral and non-chiral gauge theories. We also discuss some applications to dynamical supersymmetry breaking phenomena and new confining theories with a tree-level superpotential. 
  At zero temperature the Coulomb Branch of ${\cal N}=4$ super Yang-Mills theory is described in supergravity by multi-center solutions with D3-brane charge. At finite temperature and chemical potential the vacuum degeneracy is lifted, and minima of the free energy are shown to have a supergravity description as rotating black D3-branes.   In the extreme limit these solutions single out preferred points on the moduli space that can be interpreted as simple distributions of branes --- for instance, a uniformly charged planar disc. We exploit this geometrical representation to study the thermodynamics of rotating black D3-branes. The low energy excitations of the system appear to be governed by an effective string theory which is related to the singularity in spacetime. 
  We discuss the relation between the $c$--theorem and the the way various symmetries are realized in quantum field theory. We review our recent proof of the $c$--theorem in four dimensions. Based on this proof and further evidence, we conjecture that the realization of chiral symmetry be irreversible, flowing from the Wigner-Weyl realization at short distance to the Nambu--Goldstone realization at long distance. We argue that three apparently independent constraints based on the renormalization group, namely anomaly matching, the c-theorem and the conjectured symmetry realization theorem, are particular manifestations of a single underlying principle. 
  It is pointed out that the entanglement entropy of quantum fields near the horizon of a two-dimensional black hole can be derived by means of the conformal field theory. This can be done in a way analogous to the computation of the entropy of BTZ black holes. The important feature of the considered case is that the degrees of freedom of the conformal theory are states localized in the physical space-time. 
  In the solution of the superintegrable chiral Potts model special polynomials related to the representation theory of the Onsager algebra play a central role. We derive approximate analytic formulae for the zeros of particular polynomials which determine sets of low-lying energy eigenvalues of the chiral Potts quantum chain. These formulae allow the analytic calculation of the leading finite-size corrections to the energy eigenvalues without resorting to a numerical determination of the zeros. 
  In this paper, we study the structure of the quantum cohomology ring of a projective hypersurface with non-positive 1st Chern class. We prove a theorem which suggests that the mirror transformation of the quantum cohomology of a projective Calabi-Yau hypersurface has a close relation with the ring of symmetric functions, or with Schur polynomials. With this result in mind, we propose a generalized mirror transformation on the quantum cohomology of a hypersurface with negative first Chern class and construct an explicit prediction formula for three point Gromov-Witten invariants up to cubic rational curves. We also construct a projective space resolution of the moduli space of polynomial maps, which is in a good correspondence with the terms that appear in the generalized mirror transformation. 
  A new approach to summation of divergent field-theoretical series is suggested. It is based on the Borel transformation combined with a conformal mapping and does not imply the knowledge of the exact asymptotic parameters. The method is tested on functions expanded in their asymptotic power series and applied to estimating the ground state energy of simple quantum mechanical problems including anisotropic oscillators and caclulating the critical exponents for certain comformal field models. It can be expected that the new approach to summation may be used to obtaining numerical estimates for important physical quantities represented by divergent series in two- and three-dimensional field models. 
  We present new rotating black brane solutions which solve Einstein's equations with cosmological constant $\Lambda$ in arbitrary dimension $d$. For negative $\Lambda$, the branes naturally appear in AdS supergravity compactifications, and should therefore play some role in the AdS/CFT correspondence. The spacetimes are warped products of a four-dimensional part and an Einstein space of dimension $d-4$, which is not necessarily of constant curvature. As a special subcase, the solutions contain the higher dimensional generalization of the Kerr-AdS metric recently found by Hawking et al. 
  A relation between Wilson action and the Cornwall-Jackiw-Tomboulis effective action, which was recently suggested by Periwal, is here derived using path-integral techniques. 
  The computation of higher derivative corrections to the low energy effective actions of ${\cal N}=2$ gauge theories is considered. In particular, higher derivative corrections are computed for four dimensional ${\cal N}=2$ super Yang-Mills theory with gauge group SU(2) and $N_f=4$ hypermultiplets in the fundamental representation. The four derivative terms computed in an approach which realizes the gauge theory as the world volume theory of three branes in F theory are in agreement with the field theory result. 
  We consider four dimensional heterotic compactifications on smooth elliptic Calabi-Yau threefolds. Using spectral cover techniques, we study bundle cohomology groups corresponding to charged matter multiplets. The analysis shows that in generic situations, the resulting charged matter spectrum is stable under deformations of the vector bundle. 
  The N=1 superspace generalization of the 3-brane action in 6 dimensions with partially broken N=2 supersymmetry can be constructed using N=1 chiral, complex linear, or real linear superfields. The physical scalars of these multiplets give equivalent descriptions of the two transverse coordinates. The second supersymmetry is realized nonlinearly in all these actions. We derive the superspace brane actions and their nonlinear supersymmetry for both kinds of linear superfields when we break N=2 supersymmetry spontaneously. This breaking is realized in the free action of hypermultiplets that live in N=2 projective superspace by constraining the N=2 multiplets to reduce them to a pure Goldstone multiplet. For the chiral superfield, the superspace brane action and nonlinear supersymmetry can be deduced by dualizing the brane action of either linear superfield. We find that the dual action is unique up to field redefinitions that introduce arbitrariness in the dependence on the auxiliary fields. 
  The 't Hooft expansion of SU(N) Chern-Simons theory on $S^3$ is proposed to be exactly dual to the topological closed string theory on the $S^2$ blow up of the conifold geometry. The $B$-field on the $S^2$ has magnitude $Ng_s=\lambda$, the 't Hooft coupling. We are able to make a number of checks, such as finding exact agreement at the level of the partition function computed on {\it both} sides for arbitrary $\lambda$ and to all orders in 1/N. Moreover, it seems possible to derive this correspondence from a linear sigma model description of the conifold. We propose a picture whereby a perturbative D-brane description, in terms of holes in the closed string worldsheet, arises automatically from the coexistence of two phases in the underlying U(1) gauge theory. This approach holds promise for a derivation of the AdS/CFT correspondence. 
  We analyze the structure of the moduli space of a supersymmetric SU(5) chiral gauge theory with two matter fields in the 10 representation, and two fields in the \bar{5} representation. Inspection of the exact Kahler potential of the classical moduli space shows that the symmetry group of the moduli space is larger than the global symmetry group of the underlying gauge theory. As a consequence, the gauge theory has classical inequivalent vacua which yield identical low energy theories. 
  We extend the formulation of gauged supergravity in five dimensions, as obtained by compactification of $M$~theory on a deformed Calabi-Yau manifold, to include non-universal matter hypermultiplets. Even in the presence of this gauging, only the graviton supermultiplets and matter hypermultiplets can couple to supersymmetry breaking sources on the walls, though these mix with vector supermultiplets in the bulk. Whatever the source of supersymmetry breaking on the hidden wall, that on the observable wall is in general a combination of dilaton- and moduli-dominated scenarios. 
  We show that supersymmetric and $\kappa$-symmetric Dp-brane actions in type II supergravity background have the same duality transformation properties as those in a flat Minkowskian background. Specially, it is shown that the super D-string transforms in a covariant way while the super D3-brane is self-dual under the $SL(2,Z)$ duality. Also, the D2-brane and the D4-brane transform in ways expected from the relation between type IIA superstring theory and M-theory. The present study proves that various duality symmetries, which were originally found in the flat background field, are precisely valid even in the curved background geometry. 
  Generalized Yang-Mills theories are constructed, that can use fields other than vector as gauge fields. Their geometric interpretation is studied. An application to the Glashow-Weinberg-Salam model is briefly review, and some related mathematical and physical considerations are made. 
  We study a connection between duality and topological field theories. First, 2d Kramers-Wannier duality is formulated as a simple 3d topological claim (more or less Poincar\'e duality), and a similar formulation is given for higher-dimensional cases. In this form they lead to simple TFTs with boundary coloured in two colours. Classical models (Poisson-Lie T-duality) suggest a non-abelian generalization in the 2d case, with abelian groups replaced by quantum groups. Amazingly, the TFT formulation solves the problem without computation: quantum groups appear in pictures, independently of the classical motivation. Connection with Chern-Simons theory appears at the symplectic level, and also in the pictures of the Drinfeld double: Reshetikhin-Turaev invariants of links in 3-manifolds, computed from the double, are included in these TFTs. All this suggests nice phenomena in higher dimensions. 
  We formulate a complete path integral bosonization procedure for any fermionic theory in two dimensions. The method works equally well for massive and massless fermions, and is a generalization of an approach suggested earlier by Andrianov. The classical action of the bosons in the bosonized theory is identified with -i times the logarithm of the Jacobian of a local chiral transformation, with the boson fields as transformation parameters. Three examples, the Schwinger model, the massive Thirring model and massive non-Abelian bosonization, are worked out. 
  We introduce two massive versions of the anisotropic spin 1/2 Kondo model and discuss their integrability. The two models have the same bulk sine-Gordon interactions, but differ in their boundary interactions. At the Toulouse free fermion point each of the models can be understood as two decoupled Ising models in boundary magnetic fields. Reflection S-matrices away from the free fermion point are conjectured. 
  We show that a set of parallel 3-brane probes near a conifold singularity can be mapped onto a configuration of intersecting branes in type IIA string theory. The field theory on the probes can be explicitly derived from this formulation. The intersecting-brane metric for our model is obtained using various dualities and related directly to the conifold metric. The M-theory limit of this model is derived and turns out to be remarkably simple. The global symmetries and counting of moduli are interpreted in the M-theory picture. 
  By exploiting relations between gravity and gauge theories, we present two infinite sequences of one-loop n-graviton scattering amplitudes: the `maximally helicity-violating' amplitudes in N=8 supergravity, and the `all-plus' helicity amplitudes in gravity with any minimally coupled massless matter content. The all-plus amplitudes correspond to self-dual field configurations and vanish in supersymmetric theories. We make use of the tree-level Kawai-Lewellen-Tye (KLT) relations between open and closed string theory amplitudes, which in the low-energy limit imply relations between gravity and gauge theory tree amplitudes. For n < 7, we determine the all-plus amplitudes explicitly from their unitarity cuts. The KLT relations, applied to the cuts, allow us to extend to gravity a previously found `dimension-shifting' relation between (the cuts of) the all-plus amplitudes in gauge theory and the maximally helicity-violating amplitudes in N=4 super-Yang-Mills theory. The gravitational version of the relation lets us determine the n < 7 N=8 supergravity amplitudes from the all-plus gravity amplitudes. We infer the two series of amplitudes for all n from their soft and collinear properties, which can also be derived from gauge theory using the KLT relations. 
  We consider some aspects of inflation in models with large internal dimensions. If inflation occurs on a 3D wall after the stabilization of internal dimensions in the models with low unification scale (M ~ 1 TeV), the inflaton field must be extremely light. This problem may disappear In models with intermediate (M ~10^{11} GeV) to high (M ~ 10^{16} GeV) unification scale. However, in all of these cases the wall inflation does not provide a complete solution to the horizon and flatness problems. To solve them, there must be a stage of inflation in the bulk before the compactification of internal dimensions. 
  Observational diagnostics are constructed which reflect the underlying topology of Planckian space-time, and are directly related to phenomena on much larger scales. Specific predictions are made for the masses of elementary particles, coupling constants, quark confinement, black hole states, and the cosmological constant. In particular, a mass of 131.6 GeV is found for the Higgs boson. The main presented result is a discrete spectral signature at inverse integer multiples of the zero point frequency $\nu_0=857.3588$ MHz. That is, each photon of frequency $\nu_0m$, for an integer $m$, is paired with an otherwise identical photon $\nu_0/m$, produced by the vacuum, but not vice versa. The reader interested in the latter result only, should proceed to Section 6 after the Introduction. 
  We present a geometric formulation of $(p,q)$-strings in which the $Sl(2;Z)$-doublet of the two-form gauge potentials is constructed as second order in the supersymmetric currents. The currents are constructed using a supergroup manifold corresponding to the $(p,q)$-string superalgebra, which contains fermionic generators in addition to the supercharges and transforms under the $Sl(2;Z)$. The properties of the superalgebra and the generalizations to higher $p$-branes are discussed. 
  From the Dirac action on the world sheet, an effective action is obtained by integrating over the 4-dimensional fermion fields pulled back to the world sheet. This action consists of the Nambu-Goto area term with right dimensionful constant in front, extrinsic curvature action and the topological Euler characteristic term. 
  In the context of brane solutions of supergravity, we discuss a general method to introduce collective modes of any spin by exploiting a particular way of breaking symmetries. The method is applied to the D3, M2 and M5 branes and we derive explicit expressions for how the zero-modes enter the target space fields, verify normalisability in the transverse directions and derive the corresponding field equations on the brane. In particular, the method provides a clear understanding of scalar, spinor, and rank r tensorial Goldstone modes, chiral as well as non-chiral, and how they arise from the gravity, Rarita-Schwinger, and rank r+1 Kalb-Ramond tensor gauge fields, respectively. Some additional observations concerning the chiral tensor modes on the M5 brane are discussed. 
  The AdS/CFT correspondence is established for the case of AdS$_3$ space compactified on a filled rectangular torus with the CFT field on the boundary. 
  We study the Kaluza-Klein spectrum of D=5 simple supergravity on $S^2$ with special interest in the relation to a two-dimensional N=4 superconformal field theory. The spectrum is obtained around the maximally supersymmetric Freund-Rubin-like background $AdS_3\times S^2$ by closely following the well-known techniques developed in D=11 supergravity. All the vector excitations turn out to be ``(anti-)self-dual'', having only one dynamical degree of freedom. The representation theory for the Lie superalgebra $SU(1,1|2)$ is developed by means of the oscillator method. We calculate the conformal weight of the boundary operator by estimating the asymptotic behavior of the wave function for each Kaluza-Klein mode. All the towers of particles are shown to fall into four infinite series of chiral primary representations of $SU(1,1|2)\times SL(2,{\bf R})$ (direct product), or $OSp(2,2|2;-1)\cong SU(1,1|2)\times SL(2,{\bf R})$ (semi-direct product). 
  We study a simple system that comprises all main features of critical gravitational collapse, originally discovered by Choptuik and discussed in many subsequent publications. These features include universality of phenomena, mass-scaling relations, self-similarity, symmetry between super-critical and sub-critical solutions, etc.    The system we consider is a stationary membrane (representing a domain wall) in a static gravitational field of a black hole. For a membrane that spreads to infinity, the induced 2+1 geometry is asymptotically flat. Besides solutions with Minkowski topology there exists also solutions with the induced metric and topology of a 2+1 dimensional black hole. By changing boundary conditions at infinity, one finds that there is a transition between these two families. This transition is critical and it possesses all the above-mentioned properties of critical gravitational collapse. It is remarkable that characteristics of this transition can be obtained analytically. In particular, we find exact analytical expressions for scaling exponents and wiggle-periods.    Our results imply that black hole formation as a critical phenomenon is far more general than one might expect. 
  We investigate the quantum conformal algebras of N=2 and N=1 supersymmetric gauge theories. Phenomena occurring at strong coupling are analysed using the Nachtmann theorem and very general, model-independent, arguments. The results lead us to introduce a novel class of conformal field theories, identified by a closed quantum conformal algebra. We conjecture that they are the exact solution to the strongly coupled large-N_c limit of the open conformal field theories. We study the basic properties of closed conformal field theory and work out the operator product expansion of the conserved current multiplet T. The OPE structure is uniquely determined by two central charges, c and a. The multiplet T does not contain just the stress-tensor, but also R-currents and finite mass operators. For this reason, the ratio c/a is different from 1. On the other hand, an open algebra contains an infinite tower of non-conserved currents, organized in pairs and singlets with respect to renormalization mixing. T mixes with a second multiplet T* and the main consequence is that c and a have different subleading corrections. The closed algebra simplifies considerably at c=a, where it coincides with the N=4 one. 
  We follow recent work and study the relativistic d-brane system in (d+1,1) dimensions and its connection with a Galileo invariant system in (d,1) dimensions. In particular, we solve d-brane systems in (2,1), (3,1) and (4,1) dimensions and show that their solutions solve the corresponding Galileo invariant systems in (1,1), (2,1), and (3,1) dimensions. The results are extended to higher dimensions. 
  We investigate a $Z_2$-symmetric scalar field theory in two dimensions using the Polchinski exact renormalization group equation expanded to second order in the derivative expansion. We find preliminary evidence that the Polchinski equation is able to describe the non-perturbative infinite set of fixed points in the theory space, corresponding to the minimal unitary series of 2D conformal field theories. We compute the anomalous scaling dimension $\eta$ and the correlation length critical exponent $\nu$ showing that an accurate fit to conformal field theory selects particular regulating functions. 
  We develop a formalism to evaluate generic scalar exchange diagrams in AdS_{d+1} relevant for the calculation of four-point functions in AdS/CFT correspondence. The result may be written as an infinite power series of functions of cross-ratios. Logarithmic singularities appear in all orders whenever the dimensions of involved operators satisfy certain relations. We show that the AdS_{d+1} amplitude can be written in a form recognisable as the conformal partial wave expansion of a four-point function in CFT_{d} and identify the spectrum of intermediate operators. We find that, in addition to the contribution of the scalar operator associated with the exchanged field in the AdS diagram, there are also contributions of some other operators which may possibly be identified with two-particle bound states in AdS. The CFT interpretation also provides a useful way to ``regularize'' the logarithms appearing in AdS amplitude. 
  A discrete field formalism exposes the physical meaning and the origins of gauge fields and of their symmetries and singularities. 
  In this work we revisit questions recently raised in the literature associated to relevant but divergent amplitudes in the gauged NJL model. The questions raised involve ambiguities and symmetry violations which concern the model's predictive power at one loop level. Our study shows by means of an alternative prescription to handle divergent amplitudes, that it is possible to obtain unambiguous and symmetry preserving amplitudes. The procedure adopted makes use solely of {\it general} properties of an eventual regulator, thus avoiding an explicit form. We find, after a thorough analysis of the problem that there are well established conditions to be fulfiled by any consistent regularization prescription in order to avoid the problems of concern at one loop level. 
  We find that the first-order correction to the free-field result for the four-point function of the conformal operator $\tr(\phi^i\phi^j)$ is nonvanishing and survives in the limit $N_c \rar \infty$. 
  We derive WKB expressions for glueball masses of various finite temperature supergravity models. The results are very close to recent numerical computations. We argue that the spectra has some universality that depends only on the dimension of the AdS space and the singularity structure of the horizon. This explains the stability of the $0^{++}$ glueball mass ratios between various models. We also consider the recently proposed nonsupersymmetric model arising from the type 0 string. In the supergravity limit of this model, the heavy quark potential has an effective coupling with 1/(log u) behavior in the UV. Unfortunately, the supergravity solution implies that the heavy quark potential is still coulombic in the infrared, with an effective coupling of order 1. We also argue that the type 0 supergravity background solution does not have normalizable glueball solutions. 
  We propose an implicit regularisation scheme. The main advantage is that since no explicit use of a regulator is made, one can in principle avoid undesirable symmetry violations related to its choice. The divergent amplitudes are split into basic divergent integrals which depend only on the loop momenta and finite integrals. The former can be absorbed by a renormalisation procedure whereas the latter can be evaluated without restrictions. We illustrate with the calculation of the $QED$ and $\phi^4_4$-theory $\beta$-function to one and two-loop order, respectively. 
  We introduce a novel method for the renormalization of the Hamiltonian operator in Quantum Field Theory in the spirit of the Wilson renormalization group. By a series of unitary transformations that successively decouples the high-frequency degrees of freedom and partially diagonalizes the high-energy part, we obtain the effective Hamiltonian for the low energy degrees of freedom. We successfully apply this technique to compute the 2-loop renormalized Hamiltonian in scalar $\lambda \phi^4$ theory. 
  We study gauge theories on noncommutative tori. It was proved in [5] that Morita equivalence of noncommutative tori leads to a physical equivalence (SO(d,d| Z)-duality) of the corresponding gauge theories. We calculate the energy spectrum of maximally supersymmetric BPS states in these theories and show that this spectrum agrees with the SO(d,d| Z)-duality. The relation of our results with those of recent calculations is discussed. 
  The paper contains successive description of the strong-coupling perturbation theory. Formal realization of the idea is based on observation that the path-integrals measure for absorption part of amplitudes $\R$ is Diracian ($\d$-like). New form of the perturbation theory achieved mapping the quantum dynamics in the space $W_G$ of (action, angle) type collective variables. It is shown that the transformed perturbation theory contributions are accumulated on the boundary $\pa W_G$, i.e. are sensible to the $topology$ of factor space $W_G$ and,therefore, to the theory symmetry group $G$. The abilities of our perturbation theory are illustrated by examples from quantum mechanics and field theory. Considering the Coulomb potential $1/|x|$ the total reduction of quantum degrees of freedom is demonstrated mapping the dynamics in the (angle, angular momentum, Runge-Lentz vector) space. To solve the (1+1)-dimensional sin-Gordon model the theory is considered in the space (coordinates, momenta) of solitons. It is shown the total reduction of quantum degrees of freedom and, in result, there is not multiple production of particles. This result we interpret as the $S$-matrix form of confinement. The scalar $O(4,2)$-invariant field theoretical model is quantized in the $W_O=O(4,2)/O(4) {\times} O(2)$ factor manifold. It is shown that the corresponding $\R$ is nontrivial because of the scale invariance breaking. 
  We construct a consistent set of monopole equations on eight-manifolds with Spin(7) holonomy. These equations are elliptic and admit non-trivial solutions including all the 4-dimensional Seiberg-Witten solutions as a special case. 
  We consider embeddings of the Virasoro algebra into other Virasoro algebras with different central charges. A Virasoro algebra with central charge c (assumed to be a positive integer) and zero mode operator L_0 can be embedded into another Virasoro algebra with central charge one and zero mode operator c L_0. We point out that this provides a new route to investigate the black hole entropy problem in 2+1 dimensions. 
  Two different systems of the generalized coherent states for the de Sitter group are constructed. These systems have the physical sense of particles and antiparticles over de Sitter space and obeys the Klein-Gordon equation. Using these systems the de Sitter-invariant propagators for massive spin~0 and~1/2 fields over de Sitter space are constructed. 
  We calculate the zero point energy of a massive scalar field in the background of an infinitely thin spherical shell given by a potential of the delta function type. We use zeta functional regularization and express the regularized ground state energy in terms of the Jost function of the related scattering problem. Then we find the corresponding heat kernel coefficients and perform the renormalization, imposing the normalization condition that the ground state energy vanishes when the mass of the quantum field becomes large. Finally the ground state energy is calculated numerically. Corresponding plots are given for different values of the strength of the background potential, for both attractive and repulsive potentials. 
  A simple WKB approximation gives explicit information about D0brane boundstate wavefunctions, suggesting that at large $N$ each individual D0brane has a wavefunction $\exp(- c r^{9/2}N^{-1/2}).$ Thus the velocity dependent interaction energy $v^{4}r^{-7}$ leads to an effective confining potential that grows as $r^{7}.$ 
  Starting from determinants at finite temperature obeying an intermediate boundary condition between the periodic (bosonic) and antiperiodic (fermionic) cases, we find results which can be mapped onto the ones obtained from anyons for the second virial coefficient. Using this approach, we calculate the corresponding higher virial coefficients and compare them with the results known in the literature. 
  We consider multicenter supergravity solutions corresponding to Higgs phases of supersymmetric Yang-Mills theories with Z_N symmetric vacua. In certain energy regimes, we find a description in terms of a generalized wormhole solution that corresponds to the SL(2,R)/U(1) \times SU(2)/U(1) exact conformal field theory. We show that U-dualities map these backgrounds to purely gravitational ones and comment on the relation to the black holes arising from intersecting D1 and D5 branes. We also discuss supersymmetric properties of the various solutions and the relation to 2-dim solitons, on flat space, of the reduced axion-dilaton-gravity equations. Finally, we address the problem of understanding other supergravity solutions from the multicenter ones. As prototype examples we use rotating D3 branes and NS5 and D5 branes associated to non-Abelian duals of 4-dim hyper-Kahler metrics with SO(3) isometry. 
  In this letter, we introduce a general theory for the construction of particle physics theories, with three families and realistic gauge groups, within the context of heterotic M-theory. This is achieved using semi-stable holomorphic gauge bundles over elliptically fibered Calabi-Yau three-folds. Construction of realistic theories is facilitated by the appearance of non-perturbative five-branes in the vacuum. The complete moduli space of these five-branes is computed and their worldvolume gauge theory discussed. It is shown, within the context of holomorphic gauge bundles, how grand unified gauge groups can be spontaneously broken to the gauge group of the standard model. These ideas are illustrated in an explicit SU(5) three-family example. 
  Monopoles and solitons have important topological aspects like quantized fluxes, winding numbers and curved target spaces. Naive discretizations which substitute a lattice of points for the underlying manifolds are incapable of retaining these features in a precise way. We study these problems of discrete physics and matrix models and discuss mathematically coherent discretizations of monopoles and solitons using fuzzy physics and noncommutative geometry. A fuzzy sigma-model action for the two-sphere fulfilling a fuzzy Belavin-Polyakov bound is also put forth. 
  In this work we use the Matrix Model of Strings in order to extract some non-perturbative information on how the Hagedorn critical temperature arises from eleven-dimensional physics. We study the thermal behavior of M and Matrix theories on the compactification backgrounds that correspond to string models. We obtain some information that allows us to state that the Hagedorn temperature is not unique for all Matrix String models and we are also able to sketch how the $S$-duality transformation works in this framework. 
  The existence of ring-like and knotted solitons in O(3) non-linear sigma model is analysed. The role of isotopy of knots/links in classifying such solitons is pointed out. Appearance of torus knot solitons is seen. 
  Four-point functions of gauge-invariant operators in D=4, N=4 supersymmetric Yang-Mills theory are studied using N=2 harmonic superspace perturbation theory. The results are expressed in terms of differential operators acting on a scalar two loop integral. The leading singular behaviour is obtained in the limit that two of the points approach one another. We find logarithmic singularities which do not cancel out in the sum of all diagrams. It is confirmed that Green's functions of analytic operators are indeed analytic at this order in perturbation theory. 
  We identify 998 closed hyperbolic 3-manifolds whose volumes are rationally related to Dedekind zeta values, with coprime integers $a$ and $b$ giving $a/b vol(M)=(-D)^{3/2}/(2\pi)^{2n-4} (\zeta_K(2))/(2\zeta(2))$ for a manifold M whose invariant trace field $K$ has a single complex place, discriminant $D$, degree $n$, and Dedekind zeta value $\zeta_K(2)$. The largest numerator of the 998 invariants of Hodgson-Weeks manifolds is, astoundingly, $a=2^4\times23\times37\times691 =9,408,656$; the largest denominator is merely b=9. We also study the rational invariant a/b for single-complex-place cusped manifolds, complementary to knots and links, both within and beyond the Hildebrand-Weeks census. Within the censi, we identify 152 distinct Dedekind zetas rationally related to volumes. Moreover, 91 census manifolds have volumes reducible to pairs of these zeta values. Motivated by studies of Feynman diagrams, we find a 10-component 24-crossing link in the case n=2 and D=-20. It is one of 5 alternating platonic links, the other 4 being quartic. For 8 of 10 quadratic fields distinguished by rational relations between Dedekind zeta values and volumes of Feynman orthoschemes, we find corresponding links. Feynman links with D=-39 and D=-84 are missing; we expect them to be as beautiful as the 8 drawn here. Dedekind-zeta invariants are obtained for knots from Feynman diagrams with up to 11 loops. We identify a sextic 18-crossing positive Feynman knot whose rational invariant, a/b=26, is 390 times that of the cubic 16-crossing non-alternating knot with maximal D_9 symmetry. Our results are secure, numerically, yet appear very hard to prove by analysis. 
  It has been suggested by various authors that the `dynamical Casimir effect' might prove responsible for the production of visible-light photons in the bubble collapse which occurs in sonoluminescence. Previously, I have argued against this point of view based on energetic considerations, in the adiabatic approximation. Those arguments have recently been strengthened by the demonstration of the equivalence between van der Waals and Casimir energies. In this note I concentrate on the other extreme possibility, that of the validity of the `sudden approximation' where in effect the bubble instantaneously ceases to exist. Previous estimates which seemed to support the relevance of the Casimir effect are shown to be unconvincing because they require macroscopic changes on excessively small time scales, involving the entire volume of the bubble at maximum radius. 
  The model of D4 open string with non-Grassmann spinning variables is considered. The non-linear gauge, which is invariant both Poincar\'e and scale transformations of the space-time, is used for subsequent studies. It is shown that the reduction of the canonical Poisson structure from the original phase space to the surface of constraints and gauge conditions gives the degenerated Poisson brackets. Moreover it is shown that such reduction is non-unique. The conseption of the adjunct phase space is introduced. The consequences for subsequent quantization are discussed. Deduced dependence of spin $J$ from the square of mass $\mu^2$ of the string generalizes the ''Regge spectrum`` for conventional theory. 
  The Skyrme-Faddeev system, a modified O(3) sigma model in three space dimensions, admits topological solitons with nonzero Hopf number. One may learn something about these solitons by considering the system on the 3-sphere of radius R. In particular, the Hopf map is a solution which is unstable for R > \sqrt{2}. 
  We review the relation between the classical dynamics of the M-fivebrane and the quantum low energy effective action for N=2 Yang-Mills theories. We also discuss some outstanding issues in this correspondence. 
  We consider A-series modular invariant Virasoro minimal models on the upper half plane. From Lewellen's sewing constraints a necessary form of the bulk and boundary structure constants is derived. Necessary means that any solution can be brought to the given form by rescaling of the fields. All constants are expressed essentially in terms of fusing (F-) matrix elements and the normalisations are chosen such that they are real and no square roots appear. It is not shown in this paper that the given structure constants solve the sewing constraints, however random numerical tests show no contradiction and agreement of the bulk structure constants with Dotsenko and Fateev. In order to facilitate numerical calculations a recursion relation for the F-matrices is given. 
  In this paper, the metric on the moduli space of the k=1 SU(n) periodic instanton -or caloron- with arbitrary gauge holonomy at spatial infinity is explicitly constructed. The metric is toric hyperKaehler and of the form conjectured by Lee and Yi. The torus coordinates describe the residual U(1)^{n-1} gauge invariance and the temporal position of the caloron and can also be viewed as the phases of n monopoles that constitute the caloron. The (1,1,..,1) monopole is obtained as a limit of the caloron. The calculation is performed on the space of Nahm data, which is justified by proving the isometric property of the Nahm construction for the cases considered. An alternative construction using the hyperKaehler quotient is also presented. The effect of massless monopoles is briefly discussed. 
  We construct N=1 supersymmetric versions of four-dimensional Freedman-Townsend models and generalizations thereof found recently by Henneaux and Knaepen, with couplings between 1-form and 2-form gauge potentials. The models are presented both in a superfield formulation with linearly realized supersymmetry and in WZ gauged component form. In the latter formulation the supersymmetry transformations are nonlinear and do not commute with all the gauge transformations. Among others, our construction yields N=1 counterparts of recently found N=2 supersymmetric gauge theories involving vector-tensor multiplets with gauged central charge. 
  A redesigned starting point for covariant \phi^4_n, n\ge 4, models is suggested that takes the form of an alternative lattice action and which may have the virtue of leading to a nontrivial quantum field theory in the continuum limit. The lack of conventional scattering for such theories is understood through an interchange of limits. 
  A residual gauge symmetry, exhibited by light-front gauge theories quantized in a finite volume, is analyzed at the quantum level. Unitary operators, which implement the symmetry, transform the trivial Fock vacuum into an infinite set of degenerate coherent-state vacua. A fermionic component of the vacuum emerges naturally without the need to introduce a Dirac sea. The vacuum degeneracy along with the derivation of the theta-vacuum is discussed within the massive Schwinger model. A possible generalization of the approach to more realistic gauge theories is suggested. 
  We study orbifolds of ${\cal N} = 4$ U(n) super-Yang-Mills theory given by discrete subgroups of SU(2) and SU(3). We have reached many interesting observations that have graph-theoretic interpretations. For the subgroups of SU(2), we have shown how the matter content agrees with current quiver theories and have offered a possible explanation. In the case of SU(3) we have constructed a catalogue of candidates for finite (chiral) ${\cal N}=1$ theories, giving the gauge group and matter content. Finally, we conjecture a McKay-type correspondence for Gorenstein singularities in dimension 3 with modular invariants of WZW conformal models. This implies a connection between a class of finite ${\cal N}=1$ supersymmetric gauge theories in four dimensions and the classification of affine SU(3) modular invariant partition functions in two dimensions. 
  In this paper we study dynamical chiral symmetry breaking in dimensionally regularized quenched QED within the context of Dyson-Schwinger equations. In D < 4 dimensions the theory has solutions which exhibit chiral symmetry breaking for all values of the coupling. To begin with, we study this phenomenon both numerically and, with some approximations, analytically within the rainbow approximation in the Landau gauge. In particular, we discuss how to extract the critical coupling alpha_c = pi/3 relevant in four dimensions from the D dimensional theory. We further present analytic results for the chirally symmetric solution obtained with the Curtis-Pennington vertex as well as numerical results for solutions exhibiting chiral symmetry breaking. For these we demonstrate that, using dimensional regularization, the extraction of the critical coupling relevant for this vertex is feasible. Initial results for this critical coupling are in agreement with cut-off based work within the currently achievable numerical precision. 
  We show that in field systems with U(1)-symmetry, first-order transitions are nucleated by vortex lines, not bubbles, thus calling for a reinvestigation of the Kibble mechanism for the phase transition of the early universe. 
  We study the supersymmetric extension of the gauged $ O(3) $ sigma model in $ 2+1 $ dimensions and find the supersymmetry algebra. We also discuss soliton solutions in case the Maxwell term is replaced by the Born-Infeld term. We show that by appropriate choice of the potential, the self-dual equations in the Born-Infeld case coincide with those of the Maxwell's case. 
  A simple model of spacetime foam, made by $N$ wormholes in a semiclassical approximation, is taken under examination. We show that the qualitative behaviour of the fluctuation of the metric conjectured by Wheeler is here reproduced. 
  The nonlinear Schrodinger equation on the half line with mixed boundary condition is investigated. After a brief introduction to the corresponding classical boundary value problem, the exact second quantized solution of the system is constructed. The construction is based on a new algebraic structure, which is called in what follows boundary algebra and which substitutes, in the presence of boundaries, the familiar Zamolodchikov-Faddeev algebra. The fundamental quantum field theory properties of the solution are established and discussed in detail. The relative scattering operator is derived in the Haag-Ruelle framework, suitably generalized to the case of broken translation invariance in space. 
  Using the Ernst potential formulation we construct all it finite symmetry transformations which preserve asymptotics of the bosonic fields of the (d+3)--dimensional low--energy heterotic string theory compactified on a d--torus. We combine all the dynamical variables into a single (d+1)X(d+1+n)--dimensional matrix potential which linearly transforms under the action of these symmetry transformations in a transparent SO(2,d-1) X SO(2,d-1+n) way, where n is the number of Abelian vector fields. We formulate the most general solution generation technique based on the use of these symmetries and show that they form an invariance group of the general Israel--Wilson--Perj'es class of solutions. 
  The path integral of a gauge theory is studied in Coulomb-like gauges. The Christ-Lee terms of operator ordering are reproduced {\it{within}} the path integration framework. In the presence of fermions, a new operator term, in addition to that of Christ-Lee, is discovered. Such kind of terms is found to be instrumental in restoring the invariance of the effective Lagrangian under a field dependent gauge transformation, which underlies the BRST symmetry. A unitary regularization scheme which maintains manifest BRST symmetry and is free from energy divergences is proposed for a nonabelian gauge field. 
  The BRST cohomology of the N=2 supersymmetric Yang-Mills theory in four dimensions is discussed by making use of the twisted version of the N=2 algebra. By the introduction of a set of suitable constant ghosts associated to the generators of N=2, the quantization of the model can be done by taking into account both gauge invariance and supersymmetry. In particular, we show how the twisted N=2 algebra can be used to obtain in a straightforward way the relevant cohomology classes. Moreover, we shall be able to establish a very useful relationship between the local gauge invariant polynomial $tr\phi^2$ and the complete N=2 Yang-Mills action. This important relation can be considered as the first step towards a fully algebraic proof of the one-loop exactness of the N=2 beta function. 
  We compute the leading behaviour of the quark anti-quark potential from a generalized Nambu-Goto action associated with a curved space-time having an "extra dimension". The extra dimension can be the radial coordinate in the AdS/CFT correspondence, the Liouville field in Polyakov's approach, or an internal dimension in MQCD. In particular, we derive the condition for confinement, and in the case it occurs we find the string tension and the correction to the linear potential. 
  Motivated by the coupling unification problem, we propose a novel extension of the Minimal Supersymmetric Standard Model. One of the predictions of this extension is existence of new states neutral under SU(3)_c X SU(2)_w but charged under U(1)_Y. The mass scale for these new states can be around the mass scale of the electroweak Higgs doublets. This suggests a possibility of their detection in the present or near future collider experiments. Unification of gauge couplings in this extension is as precise (at one loop) as in the MSSM, and can occur in the TeV range. 
  We show that a subgroup of the modular group of M-theory compactified on a ten torus, implies the Lorentzian structure of the moduli space, that is usually associated with naive discussions of quantum cosmology based on the low energy Einstein action. This structure implies a natural division of the asymptotic domains of the moduli space into regions which can/cannot be mapped to Type II string theory or 11D Supergravity (SUGRA) with large radii. We call these the safe and unsafe domains. The safe domain is the interior of the future light cone in the moduli space while the unsafe domain contains the spacelike region and the past light cone. Within the safe domain, apparent cosmological singularities can be resolved by duality transformations and we briefly provide a physical picture of how this occurs. The unsafe domains represent true singularities where all field theoretic description of the physics breaks down. They violate the holographic principle. We argue that this structure provides a natural arrow of time for cosmology. All of the Kasner solutions, of the compactified SUGRA theory interpolate between the past and future light cones of the moduli space. We describe tentative generalizations of this analysis to moduli spaces with less SUSY. 
  We study BPS saturated domain walls in the supersymmetric SU(2) gauge theory. For a theory with a very light adjoint scalar (mass <~ Lambda/400) we use the perturbed N=2 Seiberg-Witten theory to calculate the actual field configuration of the domain wall. The wall has a sandwich-like five-layer structure of three distinct phases -- electric confinement, Coulomb and oblique confinement -- separated by two separate transition regions. For larger scalar masses, the three-phase structure disappears and the Seiberg-Witten theory becomes inadequate because of two major problems: First, the higher-derivative interactions between the light fields become relevant and second, both the magnetic monopole condensate and the dyon condensate show up in the same region of space, a phenomenon indescribable in terms of a local field theory. Nevertheless, we argue that the BPS saturated domain wall continues to exist in this regime and give a qualitative description of the scalar and gaugino condensates. Finally, we discuss the domain walls in MQCD and translate the BPS conditions into coupled non-linear differential equations. 
  The neutral kaon system can be effectively described by non-unitary, dissipative, completely positive dynamics that extend the usual treatment. In the framework of open quantum systems, we show how the origin of these non-standard time evolutions can be traced to the interaction of the kaon system with a large environment. We find that D-branes, effectively described by a heat-bath of quanta obeying infinite statistics, could constitute a realistic example of such an environment. 
  The Nakamura's algorithm is applied to compute the Hilbert scheme for the D-brane model 1/13(1,2,10). 
  We describe topological gauge theories for which duality properties are encoded by construction. We study them for compact manifolds of dimensions four, eight and two. The fields and their duals are treated symmetrically, within the context of field--antifield unification. Dual formulations correspond to different gauge-fixings of the topological symmetry. We also describe novel features in eight-dimensional theories, and speculate about their possible "Abelian" descriptions. 
  The path integral generalization of the Casson invariant as developed by Rozansky and Witten is investigated. The path integral for various three manifolds is explicitly evaluated. A new class of topological observables is introduced that may allow for more effective invariants. Finally it is shown how the dimensional reduction of these theories corresponds to a generalization of the topological B sigma model. 
  We investigate the two-logarithm matrix model with the potential $X\Lambda+\alpha\log(1+X)+\beta\log(1-X)$ related to an exactly solvable Kazakov-Migdal model. In the proper normalization, using Virasoro constraints, we prove the equivalence of this model and the Kontsevich-Penner matrix model and construct the 1/N-expansion solution of this model. 
  We investigate the classical moduli space of D-branes on a nonabelian Calabi-Yau threefold singularity and find that it admits topology-changing transitions. We construct a general formalism of worldvolume field theories in the language of quivers and give a procedure for computing the enlarged Kahler cone of the moduli space. The topology changing transitions achieved by varying the Fayet-Iliopoulos parameters correspond to changes of linearization of a geometric invariant theory quotient and can be studied by methods of algebraic geometry. Quite surprisingly, the structure of the enlarged Kahler cone can be computed by toric methods. By using this approach, we give a detailed discussion of two low-rank examples. 
  Quantum mechanics on manifolds is not unique and in general infinite number of inequivalent quantizations can be considered. They are specified by the induced spin and the induced gauge structures on the manifold. The configuration space of collective mode in the Skyrme model can be identified with $S^{3}$ and thus the quantization is not unique. This leads to the different predictions for the physical observables. 
  In the quantum Teichmuller theory, based on Penner coordinates, the mapping class groups of punctured surfaces are represented projectively. The case of a genus three surface with one puncture is worked out explicitly. The projective factor is calculated. It is given by the exponential of the Liouville central charge. 
  I briefly review recent work on the comparison between two and three graviton scattering in supergravity and matrix theory 
  We calculate the Hausdorff dimension, $d_H$, and the correlation function exponent, $\eta$, for polymerized two dimensional quantum gravity models. If the non-polymerized model has correlation function exponent $\eta_0 >3$ then $d_H=\gamma^{-1}$ where $\gamma$ is the susceptibility exponent. This suggests that these models may be in the same universality class as certain non-generic branched polymer models. 
  The fermion propagator and the 4-fermion Green function in the massless QED2 are explicitly found with topological effects taken into account. The corrections due to instanton sectors k=+1,-1, contributing to the propagator, are shown to be just the homogenous terms admitted by the Dyson-Schwinger equation for S. In the case of the 4-fermion function also sectors k=+2,-2 are included into consideration. The quark condensates are then calculated and are shown to satisfy cluster property. The theta-dependence exhibited by the Green functions corresponds to and may be removed by performing certain chiral gauge transformation. 
  We find the dyonic worldvolume solitons due to parallel (p,q) strings ending on a D-3-brane. These solutions preserve 1/4 of bulk supersymmetry. Then we investigate the scattering of well-separated dyons and find that their moduli space is a toric hyper-K\"ahler manifold. In addition, we present the worldvolume solitons of the D-3-brane which are related by duality to the M-theory configuration of two orthogonal membranes ending on a M-5-brane. We show that these solitons preserve 1/8 of supersymmetry and compute their effective action. 
  We investigate Polyakov's proposal of constructing Yang-Mills theories by using non critical type 0 strings. We break conformal invariance by putting the system at finite temperature and find that the entropy of the cosmological solutions for these theories matches that of a gas of weakly interacting Yang-Mills bosons, up to a numerical constant. The computation of the entropy using the effective action approach presents some novelties in that the whole contribution comes from the RR fields. We also find an area law and a mass gap in the theory and show that such behavior persists for $p>4$. We comment on the possible physical meaning of this result. 
  Superconformal Ward identities for N=1 supersymmetric quantum field theories in four dimensions are convenienty obtained in the superfield formalism by combining diffeomorphisms and Weyl transformations on curved superspace. Using this approach we study the superconformal transformation properties of Green functions with one or more insertions of the supercurrent to all orders in perturbation theory. For the case of two insertions we pay particular attention to fixing the additional counterterms present, as well as to the purely geometrical anomalies which contribute to the transformation behaviour. Moreover we show in a scheme-independent way how the quasi-local terms in the Ward identities are related to similar terms which contribute to the supercurrent two and three point functions.   Furthermore we relate our superfield approach to similar studies which use the component formalism by discussing the implications of our approach for the components of the supercurrent and of the supergravity prepotentials. 
  The energy eigenvalues of the quantum particle constrained in a surface of the sphere of D dimensions embedded in a $R^{D+1}$ space are obtained by using two different procedures: in the first, we derive the Hamiltonian operator by squaring the expression of the momentum, written in cartesian components, which satisfies the Dirac brackets between the canonical operators of this second class system. We use the Weyl ordering prescription to construct the Hermitian operators. When D=2 we verify that there is no constant parameter in the expression of the eigenvalues energy, a result that is in agreement with the fact that an extra term would change the level spacings in the hydrogen atom; in the second procedure it is adopted the non-abelian BFFT formalism to convert the second class constraints into first class ones. The non-abelian first class Hamiltonian operator is symmetrized by also using the Weyl ordering rule. We observe that their energy eigenvalues differ from a constant parameter when we compare with the second class system. Thus, a conversion of the D-dimensional sphere second class system for a first class one does not reproduce the same values. 
  In two-dimensional conformal field theory, we analyze conformally invariant boundary conditions which break part of the bulk symmetries. When the subalgebra that is preserved by the boundary conditions is the fixed algebra under the action of a finite group G, orbifold techniques can be used to determine the structure of the space of such boundary conditions. We present explicit results for the case when G is abelian. In particular, we construct a classifying algebra which controls these symmetry breaking boundary conditions in the same way in which the fusion algebra governs the boundary conditions that preserve the full bulk symmetry. 
  We derive the microscopic spectral density of the Dirac operator in $SU(N_c\geq 3)$ Yang-Mills theory coupled to $N_f$ fermions in the fundamental representation. An essential technical ingredient is an exact rewriting of this density in terms of integrations over the super Riemannian manifold $Gl(N_f+1|1)$. The result agrees exactly with earlier calculations based on Random Matrix Theory. 
  We investigate T-duality of toroidally compactified Matrix model with arbitrary Ramond-Ramond backgrounds in the framework of noncommutative super Yang-Mills gauge theory. 
  We consider Donaldson-Witten theory on four-manifolds of the form $X=Y \times {\bf S}^1$ where $Y$ is a compact three-manifold. We show that there are interesting relations between the four-dimensional Donaldson invariants of $X$ and certain topological invariants of $Y$. In particular, we reinterpret a result of Meng-Taubes relating the Seiberg-Witten invariants to Reidemeister-Milnor torsion. If $b_1(Y)>1$ we show that the partition function reduces to the Casson-Walker-Lescop invariant of $Y$, as expected on formal grounds. In the case $b_1(Y)=1$ there is a correction. Consequently, in the case $b_1(Y)=1$, we observe an interesting subtlety in the standard expectations of Kaluza-Klein theory when applied to supersymmetric gauge theory compactified on a circle of small radius. 
  We use the exact scattering description of the scaling Ashkin-Teller model in two dimensions to compute the two-particle form factors of the relevant operators. These provide an approximation for the correlation functions whose accuracy is tested against exact sum rules. 
  We develop a method to obtain the large N renormalization group flows for matrix models of 2 dimensional gravity plus branched polymers. This method gives precise results for the critical points and exponents for one matrix models. We show that it can be generalized to two matrices models and we recover the Ising critical points. 
  We use local mirror symmetry in type IIA string compactifications on Calabi-Yau n+1 folds $X_{n+1}$ to construct vector bundles on (possibly singular) elliptically fibered Calabi-Yau n-folds Z_n. The interpretation of these data as valid classical solutions of the heterotic string compactified on Z_n proves F-theory/heterotic duality at the classical level. Toric geometry is used to establish a systematic dictionary that assigns to each given toric n+1-fold $X_{n+1}$ a toric n fold Z_n together with a specific family of sheafs on it. This allows for a systematic construction of phenomenologically interesting d=4 N=1 heterotic vacua, e.g. on deformations of the tangent bundle, with grand unified and SU(3)\times SU(2) gauge groups. As another application we find non-perturbative gauge enhancements of the heterotic string on singular Calabi-Yau manifolds and new non-perturbative dualities relating heterotic compactifications on different manifolds. 
  An analytic form for the crossover of the conductivity tensor between two Hall plateaux, as a function of the external magnetic field, is proposed. The form of the crossover is obtained from the action of a symmetry group, a particular subgroup of the modular group, on the upper-half complex conductivity plane, by assuming that the beta-function describing the crossover is a holomorphic function of the conductivity. The group action also leads to a selection rule, |p_1q_2-p_2q_1|=1, for allowed transitions between Hall plateaux with filling factors p_1/q_1 and p_2/q_2, where q_1 and q_2 are odd. 
  The Casimir energy for the transverse oscillations of a piecewise uniform closed string is calculated. The string is relativistic in the sense that the velocity of transverse waves is always equal to c. The great adaptibility of this string model with respect to various regularization methods is pointed out. We survey several regularization methods: the cutoff method, the complex contour integration method, and the zeta-function method. The most powerful method in the present case is the contour integration method. The Casimir energy turns out to be negative, and the more so the larger is the number of pieces in the string. The thermodynamic free energy F is calculated for a two-piece string in the limit when the tension ratio x = T_I/T_II approaches zero. For large values of the length ratio s = L_II/L_I, the Hagedorn temperature becomes proportional to the square root of s. 
  We study the large N reduced model of D-dimensional Yang-Mills theory with special attention to dynamical aspects related to the eigenvalues of the N by N matrices, which correspond to the space-time coordinates in the IIB matrix model. We first put an upper bound on the extent of space time by perturbative arguments. We perform a Monte Carlo simulation and show that the upper bound is actually saturated. The relation of our result to the SSB of the U(1)^D symmetry in the Eguchi-Kawai model is clarified. We define a quantity which represents the uncertainty of the space-time coordinates and show that it is of the same order as the extent of space time, which means that a classical space-time picture is maximally broken. We develop a 1/D expansion, which enables us to calculate correlation functions of the model analytically. The absence of an SSB of the Lorentz invariance is shown by the Monte Carlo simulation as well as by the 1/D expansion. 
  The spontaneous generation of uniform magnetic condensate in $QED_3$ gives rise to ferromagnetic domain walls at the electroweak phase transition. These ferromagnetic domain walls are caracterized by vanishing effective surface energy density avoiding, thus, the domain wall problem. Moreover we find that the domain walls generate a magnetic field $B \simeq 10^{24} Gauss$ at the electroweak scale which account for the seed field in the so called dynamo mechanism for the cosmological primordial magnetic field. We find that the annihilation processes of walls with size $R \simeq 10^5 Km$ could release an energy of order $10^{52} erg$ indicating the invisible ferromagnetic walls as possible compact sources of Gamma Ray Bursts. 
  We consider the problem of evolving a quantum field between any two (in general, curved) Cauchy surfaces. Classically, this dynamical evolution is represented by a canonical transformation on the phase space for the field theory. We show that this canonical transformation cannot, in general, be unitarily implemented on the Fock space for free quantum fields on flat spacetimes of dimension greater than 2. We do this by considering time evolution of a free Klein-Gordon field on a flat spacetime (with toroidal Cauchy surfaces) starting from a flat initial surface and ending on a generic final surface. The associated Bogolubov transformation is computed; it does not correspond to a unitary transformation on the Fock space. This means that functional evolution of the quantum state as originally envisioned by Tomonaga, Schwinger, and Dirac is not a viable concept. Nevertheless, we demonstrate that functional evolution of the quantum state can be satisfactorily described using the formalism of algebraic quantum field theory. We discuss possible implications of our results for canonical quantum gravity. 
  A linear degenerate odd Poisson bracket (antibracket) realized solely on Grassmann variables is presented. It is revealed that this bracket has at once three nilpotent $\Delta$-like differential operators of the first, the second and the third orders with respect to the Grassmann derivatives. It is shown that these $\Delta$-like operators together with the Grassmann-odd nilpotent Casimir function of this bracket form a finite-dimensional Lie superalgebra. 
  The strong coupling behavior of finite temperature free energy in N=4 supersymmetric SU(N) Yang-Mills theory has been recently discussed by Gubser, Klebanov and Tseytlin in the context of AdS-SYM correspondence. In this note, we focus on the weak coupling behavior. As a result of a two-loop computation we obtain, in the large N 't Hooft limit, $F(g^2N\to 0)\approx -\frac{\pi^2}{6}N^2V_3T^4(1-\frac{3}{2\pi^2}g^2N)$. Comparison with the strong coupling expansion provides further indication that free energy is a smooth monotonic function of the coupling constant. 
  The bosonized Chiral Schwinger model (CSM) is quantized on the light-front (LF). The physical Hilbert space of CSM is obtained directly once the constraints on the LF phase space are eliminated. The discussion of the degenerate vacua and the absence in the CSM of the $\theta$-vacua, as found in the Schwinger model (SM), becomes straightforward. The differences in the structures of the the mass excitations and the vacua in these gauge theories are displayed transparently. The procedure followed is the one used successfully in the previous works for describing the spontaneous symmetry breaking (SSB) and the SM on the LF. The physical contents following from the LF quantized theory agree with those known in the conventional treatment. The LF hyperplane is argued to be equally appropriate as the conventional equal-time one for the canonical quantization. Some comments on the irrelevance, in quantized field theory, of the fact that the hyperplanes $x^{\pm}=0$ constitute characteristic surfaces of hyperbolic partial differential equation are also made. 
  A topological phase transition in two-dimensional nonlinear sigma-models on tori, connected with self-dual (unimodular) 24-dimensional Niemeier lattices, is considered. It is shown that critical properties of these transitions are determined by corresponding Coxeter numbers of lattices. A case of general integer-valued lattices with minimal norm equal 1 or 2 and a possible application to string theory are discussed. 
  We review the heuristic arguments suggesting that any thermal quantum field theory, which can be interpreted as a quantum statistical mechanics of (interacting) relativistic particles, obeys certain restrictions on its number of local degrees of freedom. As in the vacuum representation, these restrictions can be expressed by a `nuclearity condition'. If a model satisfies this nuclearity condition, then the net of von Neumann algebras representing the local observables in the thermal representation has the split property. 
  We show how certain F^4 couplings in eight dimensions can be computed using the mirror map and K3 data. They perfectly match with the corresponding heterotic one-loop couplings, and therefore this amounts to a successful test of the conjectured duality between the heterotic string on T^2 and F-theory on K3. The underlying quantum geometry appears to be a 5-fold, consisting of a hyperk"ahler 4-fold fibered over a IP^1 base. The natural candidate for this fiber is the symmetric product Sym^2(K3). We are lead to this structure by analyzing the implications of higher powers of E_2 in the relevant Borcherds counting functions, and in particular the appropriate generalizations of the Picard-Fuchs equations for the K3. 
  We consider a possibility to describe spin one-half and higher spins of massive relativistic particles by means of commuting spinors. We present two classical gauge models with the variables $x^\mu,\xi_\alpha,\chi_\alpha$, where $\xi,\chi$ are commuting Majorana spinors. In course of quantization both models reproduce Dirac equation. We analyze the possibility to introduce an interaction with an external electromagnetic background into the models and to generalize them to higher spin description. The first model admits a minimal interaction with the external electromagnetic field, but leads to reducible representations of the Poincare group being generalized for higher spins. The second model turns out to be appropriate for description of the massive higher spins. However, it seams to be difficult to introduce a minimal interaction with an external electromagnetic field into this model. We compare our approach with one, which uses Grassman variables, and establish a relation between them. 
  This paper lays groundwork for the detailed study of the non-trivial renormalization group flow connecting supersymmetric fixed points in four dimensions using string theory on AdS spaces. Specifically, we consider D3-branes placed at singularities of Calabi-Yau threefolds which generalize the conifold singularity and have an ADE classification. The $\CN=1$ superconformal theories dictating their low-energy dynamics are infrared fixed points arising from deforming the corresponding ADE $\CN=2$ superconformal field theories by mass terms for adjoint chiral fields. We probe the geometry with a single $D3$-brane and discuss the near-horizon supergravity solution for a large number $N$ of coincident $D3$-branes. 
  We review our recent work, hep-th/9803030, on the constraints imposed by global or local symmetries on perturbative quantum field theories. The analysis is performed in the Bogoliubov-Shirkov-Epstein-Glaser formulation of perturbative quantum field theory. In this formalism the S-matrix is constructed directly in the asymptotic Fock space with only input causality and Poincare invariance. We reformulate the symmetry condition proposed in our earlier work in terms of interacting Noether currents. 
  We show that the connection between partial breaking of supersymmetry and nonlinear actions is not accidental and has to do with constraints that lead directly to nonlinear actions of the Born-Infeld type. We develop a constrained superfield approach that gives a universal way of deriving and using these actions. In particular, we find the manifestly supersymmetric form of the action of the 3-brane in 6-dimensional space in terms of N=1 superfields by using the tensor multiplet as a tool. We explain the relation between the Born-Infeld action and the model of partial N=2 supersymmetry breaking by a dual D-term. We represent the Born-Infeld action in a novel form quadratic in the gauge field strengths by introducing two auxiliary complex scalar fields; this makes duality covariance and the connection with the N=1 supersymmetric extension of the action very transparent. We also suggest a general procedure for deriving manifestly duality symmetric actions, explaining in a systematic way relations between previously discussed Lorentz covariant and noncovariant actions. 
  In this article a non--technical survey is given of the present status of Axiomatic Quantum Field Theory and interesting future directions of this approach are outlined. The topics covered are the universal structure of the local algebras of observables, their relation to the underlying fields and the significance of their relative positions. Moreover, the physical interpretation of the theory is discussed with emphasis on problems appearing in gauge theories, such as the revision of the particle concept, the determination of symmetries and statistics from the superselection structure, the analysis of the short distance properties and the specific features of relativistic thermal states. Some problems appearing in quantum field theory on curved spacetimes are also briefly mentioned.   (Talk given at Ringberg Symposium on Quantum Field Theory, Ringberg Castle, June 1998) 
  A two dimensional anomaly cancellation argument is used to construct the SO(32) heterotic and type IIB membranes. By imposing different boundary conditions at the two boundaries of a membrane, we shift all of the two dimensional anomaly to one of the boundaries. The topology of these membranes is that of a 2-dimensional cone propagating in the 11-dimensional target space. Dimensional reduction of these membranes yields the SO(32) heterotic and type IIB strings. 
  We study the implications of the index theorem and chiral Jacobian in lattice gauge theory, which have been formulated by Hasenfratz, Laliena and Niedermayer and by L\"{u}scher, on the continuum formulation of the chiral Jacobian and anomaly. We take a continuum limit of the lattice Jacobian factor without referring to perturbative expansion and recover the result of continuum theory by using only the general properties of the lattice Dirac operator. This procedure is based on a set of well-defined rules and thus provides an alternative approach to the conventional analysis of the chiral Jacobian and related anomaly in continuum theory. By using an explicit form of the lattice Dirac operator introduced by Neuberger, which satisfies the Ginsparg-Wilson relation, we illustrate our calculation in some detail. We also briefly comment on the index theorem with a finite cut-off from the present viewpoint. 
  We calculate the off-equilibrium hydrostatic pressure for the O(N) Phi^{4} theory to the leading order in 1/N. The present paper, the first of a series, concentrates on the calculation of pressure in the non-equilibrium but translationally invariant medium. The Jaynes-Gibbs principle of maximal entropy is used to introduce the relevant density matrix which is then directly implemented into dynamical equations through generalised Kubo-Martin-Schwinger (KMS) conditions. We show that in the large N limit use of Ward identities enables the pressure to be expressed in terms of two point Green's functions. These satisfy the Kadanoff-Baym equations which are exactly solvable, and we explicitly calculate the pressure for three illustrative choices of \rho. 
  Boundary conformal field theory is the suitable framework for a microscopic treatment of D-branes in arbitrary CFT backgrounds. In this work, we develop boundary deformation theory in order to study the changes of boundary conditions generated by marginal boundary fields. The deformation parameters may be regarded as continuous moduli of D-branes. We identify a large class of boundary fields which are shown to be truly marginal, and we derive closed formulas describing the associated deformations to all orders in perturbation theory. This allows us to study the global topology properties of the moduli space rather than local aspects only. As an example, we analyse in detail the moduli space of c=1 theories, which displays various stringy phenomena. 
  We study the structure of minimal-energy solutions of the baby Skyrme models for any topological charge n; the baby multi-skyrmions. Unlike in the (3+1)D nuclear Skyrme model, a potential term must be present in the (2+1)D Skyrme model to ensure stability. The form of this potential term has a crucial effect on the existence and structure of baby multi-skyrmions. The simplest holomorphic model has no known stable minimal-energy solution for n greater than one. The other baby Skyrme model studied in the literature possesses non-radially symmetric minimal-energy configurations that look like `skyrmion lattices' formed by skyrmions with n=2. We discuss a baby Skyrme model with a potential that has two vacua. Surprisingly, the minimal-energy solution for every n is radially-symmetric and the energy grows linearly for large n. Further, these multi-skyrmions are tighter bound, have less energy and the same large r behaviour than in the model with one vacuum. We rely on numerical studies and approximations to test and verify this observation. 
  Modulo the ideal generated by the derivative fields, the normal ordered product of holomorphic fields in two-dimensional conformal field theory yields a commutative and associative algebra. The zero mode algebra can be regarded as a deformation of the latter. Alternatively, it can be described as an associative quotient of the algebra given by a modified normal ordered product. We clarify the relation of these structures to Zhu's product and Zhu's algebra of the mathematical literature. 
  We study, as hypersurfaces in toric varieties, elliptic Calabi-Yau fourfolds for F-theory compactifications dual to E8xE8 heterotic strings compactified to four dimensions on elliptic Calabi-Yau threefolds with some choice of vector bundle. We describe how to read off the vector bundle data for the heterotic compactification from the toric data of the fourfold. This map allows us to construct, for example, Calabi-Yau fourfolds corresponding to three generation models with unbroken GUT groups. We also find that the geometry of the Calabi-Yau fourfold restricts the heterotic vector bundle data in a manner related to the stability of these bundles. Finally, we study Calabi-Yau fourfolds corresponding to heterotic models with fivebranes wrapping curves in the base of the Calabi-Yau threefolds. We find evidence of a topology changing extremal transition on the fourfold side which corresponds, on the heterotic side, to fivebranes wrapping different curves in the same homology class in the base. 
  The finite-volume QED$_{1+1}$ is formulated in terms of Dirac variables by an explicit solution of the Gauss constraint with possible nontrivial boundary conditions taken into account. The intrinsic nontrivial topology of the gauge group is thus revealed together with its zero-mode residual dynamics. Topologically nontrivial gauge transformations generate collective excitations of the gauge field above Coleman's ground state, that are completely decoupled from local dynamics, the latter being equivalent to a free massive scalar field theory. 
  We study the problem of a Dirac field in the background of an Aharonov-Bohm flux string. We exclude the origin by imposing spectral boundary conditions at a finite radius then shrinked to zero. Thus, we obtain a behaviour of the eigenfunctions which is compatible with the self-adjointness of the radial Hamiltonian and the invariance under integer translations of the reduced flux. After confining the theory to a finite region, we check the consistency with the index theorem, and discuss the vacuum fermionic number and Casimir energy. 
  Holography suggests a considerable reduction of degrees of freedom in theories with gravity. However it seems to be difficult to understand how holography could be realized in a closed re--contracting universe. In this letter we claim that a scenario which achieves that goal will eliminate all spatial degrees of freedom. This would require a different concept of quantum mechanics and would imply an intriguing increase of power for the natural laws. 
  We describe the spontaneous partial breaking of $N=1 D=10$ supersymmetry to $N=(1,0) d=6$ and its dimensionally-reduced versions in the framework of the nonlinear realizations method. The basic Goldstone superfield is $N=(1,0) d=6$ hypermultiplet superfield satisfying a nonlinear generalization of the standard hypermultiplet constraint. We interpret the generalized constraint as the manifestly worldvolume supersymmetric form of equations of motion of the Type I super 5-brane in D=10. The related issues we address are a possible existence of brane extension of off-shell hypermultiplet actions, the possibility to utilize vector $N=(1,0) d=6$ supermultiplet as the Goldstone one, and the description of 1/4 breaking of $N=1 D=11$ supersymmetry. 
  Spacetime properties of superstrings on AdS_3 x S^3 x S^3 x S^1 are studied. The boundary theory is a two dimensional superconformal field theory with a large N=(4,4) supersymmetry. 
  By using a classical Liouville-type model of two dimensional dilaton gravity we show that the one-loop theory implies that the fate of a black hole depends on the conformal frame. There is one frame for which the evaporation process never stops and another one leading to a complete disappearance of the black hole. This can be seen as a consequence of the fact that thermodynamic variables are not conformally invariant. In the second case the evaporation always produces the same static and regular end-point geometry, irrespective of the initial state. 
  1d Bose gas interacting through delta, delta' and double-delta function potentials is shown to be equivalent to a delta anyon gas allowing exact Bethe ansatz solution. In the noninteracting limit it describes an ideal gas with generalized exclusion statistics and solves some recent controversies. 
  We investigate the Abelian projection with respect to the Polyakov loop operator for SU(N) gauge theories on the four torus. The gauge fixed $A_0$ is time-independent and diagonal. We construct fundamental domains for $A_0$. In sectors with non-vanishing instanton number such gauge fixings are always singular. The singularities define the positions of magnetically charged monopoles, strings or walls. These magnetic defects sit on the Gribov horizon and have quantized magnetic charges. We relate their magnetic charges to the instanton number. 
  There is a natural way to study the long distance interactions of gauge theories in the electric (momentum) representation. Here, the main ideas are presented for the Abelian and Yang-Mills gauge theories emphasizing on the structure and the advantages of this approach. 
  I study the response of a detector that is coupled non-linearly to a quantized complex scalar field in different types of classical electromagnetic backgrounds. Assuming that the quantum field is in the vacuum state, I show that, when in {\it inertial} motion, the detector responds {\it only} when the electromagnetic background produces particles. However, I find that the response of the detector is {\it not} proportional to the number of particles produced by the background. 
  In the context of N=8 supergravity we consider BPS black-holes that preserve 1/8 supersymmetry. It was shown in a previous paper that, modulo U-duality transformations of E_{7(7)} the most general solution of this type can be reduced to a black-hole of the STU model. In this paper we analize this solution in detail, considering in particular its embedding in one of the possible Special K\"ahler manifold compatible with the consistent truncations to N=2 supergravity, this manifold being the moduli space of the T^6/Z^3 orbifold, that is: SU(3,3)/SU(3)*U(3). This construction requires a crucial use of the Solvable Lie Algebra formalism. Once the group-theoretical analisys is done, starting from a static, spherically symmetric ans\"atz, we find an exact solution for all the scalars (both dilaton and axion-like) and for gauge fields, together with their already known charge-dependent fixed values, which yield a U-duality invariant entropy. We give also a complete translation dictionary between the Solvable Lie Algebra and the Special K\"ahler formalisms in order to let comparison with other papers on similar issues being more immediate. Although the explicit solution is given in a simplified case where the equations turn out to be more manageable, it encodes all the features of the more general one, namely it has non-vanishing entropy and the scalar fields have a non-trivial radial dependence. 
  A linear odd Poisson bracket (antibracket) realized solely in terms of Grassmann variables is suggested. It is revealed that the bracket, which corresponds to a semi-simple Lie group, has at once three Grassmann-odd nilpotent $\Delta$-like differential operators of the first, the second and the third orders with respect to Grassmann derivatives, in contrast with the canonical odd Poisson bracket having the only Grassmann-odd nilpotent differential $\Delta$-operator of the second order. It is shown that these $\Delta$-like operators together with a Grassmann-odd nilpotent Casimir function of this bracket form a finite-dimensional Lie superalgebra. 
  Abelian Lagrangians containing Phi^4-type vertices are regularized by means of a suitable point-splitting scheme combined with generalized gauge transformations.. The calculation is developed in details for a general Lagrangean, whose fields (gauge and matter ones) satisfy usual conditions. We illustrate our results by considering some special cases, such as the ($\overline{\psi}\psi)^2 and a modified version of the Avddev-Chizhov models. Possible application of our results to the Abelian Higgs model, whenever spontaneous symmetry breaking is considered, is also discussed. We also pay attention to a number of features of the point-split action such as the regularity and non-locality of its new ``interacting terms''. 
  The Discrete Light-Cone Quantization (DLCQ) of a supersymmetric SU(N) gauge theory in 1+1 dimensions is discussed, with particular emphasis given to the inclusion of all dynamical zero modes. Interestingly, the notorious `zero-mode problem' is now tractable because of special supersymmetric cancellations. In particular, we show that anomalous zero-mode contributions to the currents are absent, in contrast to what is observed in the non-supersymmetric case. We find that the supersymmetric partner of the gauge zero mode is the diagonal component of the fermion zero mode. An analysis of the vacuum structure is provided and it is shown that the inclusion of zero modes is crucial for probing the phase properties of the vacua. In particular, we find that the ground state energy is zero and N-fold degenerate, and thus consistent with unbroken supersymmetry. We also show that the inclusion of zero modes for the light-cone supercharges leaves the supersymmetry algebra unchanged. Finally, we remark that the dependence of the light-cone Fock vacuum in terms of the gauge zero is unchanged in the presence of matter fields. 
  We show that the logarithmically rising static potential between opposite-charged sources in two dimensions is screened by dynamical fields even if the probe charges are fractional, in units of the charge of the dynamical fields. The effect is due to quantum mechanics: the wave functions of the screening charges are superpositions of two bumps localized both near the opposite- and the same-charge sources, so that each of them gets exactly screened. 
  The Hamiltonian formalism offers a natural framework for discussing the notion of Poisson Lie T-duality. This is because the duality is inherent in the Poisson structures alone and exists regardless of the choice of Hamiltonian. Thus one can pose alternative dynamical systems possessing nonabelian T-duality. As an example, we find a dual Hamiltonian formulation of the O(3) nonlinear sigma-model. In addition, starting from a general quadratic Hamiltonian, we easily recover the known dynamical systems having Poisson Lie T-duality. 
  The scalar field exchange diagram for the correlation function of four scalar operators is evaluated in anti-de Sitter space, $AdS_{d+1}$. The conformal dimensions $\Delta_i$, $i=1,...,4$ of the scalar operators and the dimension $\Delta$ of the exchanged field are arbitrary, constrained only to obey the unitarity bound. Techniques similar to those developed earlier for gauge boson exchange are used, but results are generally more complicated. However, for integer $\Delta_i, \Delta$, the amplitude can be presented as a multiple derivative of a simple universal function. Results simplify if further conditions hold, such as the inequalities, $\Delta< \Delta_1+\Delta_3$ or $\Delta<\Delta_2+\Delta_4$. These conditions are satisfied, with $<$ replaced by $\le$, in Type IIB supergravity on $AdS_5\times S_5$ because of selection rules from SO(6) symmetry. A new form of interaction is suggested for the marginal case of the inequalities. The short distance asymptotics of the amplitudes are studied. In the direct channel the leading singular term agrees with the double operator product expansion. Logarithmic singularities occur at sub-leading order in the direct channel but at leading order in the crossed channel. When the inequalities above are violated, there are also $(\log)^2$ singularities in the direct channel. 
  We study D-branes on a three complex dimensional nonabelian orbifold ${\bf C}^3/\Gamma$ with $\Gamma$ a finite subgroup of SU(3). We present general formulae necessary to obtain quiver diagrams which represent the gauge group and the spectrum of the D-brane worldvolume theory for dihedral-like subgroups $\Delta(3n^2)$ and $\Delta(6n^2)$. It is found that the quiver diagrams have a similar structure to webs of branes. 
  We show how to systematically derive the exact form of local symmetries for the abelian Proca and CS models, which are converted into first class constrained systems by the BFT formalism, in the Lagrangian formalism. As results, without resorting to a Hamiltonian formulation we obtain the well-known U(1) symmetry for the gauge invariant Proca model, while showing that for the CS model there exist novel symmetries as well as the usual symmetry transformations. 
  An attempt is made at a systematic approach to anomaly matching problem in non-Abelian electric-magnetic duality in N=1 supersymmetric QCD. A strategy we employ is somewhat analogous to anomaly analyses in grand unified models where the anomaly cancellation becomes more transparent if one embeds SU(5) multiplets into a multiplet of (anomaly-free) SO(10). A complication arises in the treatment of $U^{AF}_{R}(1)^{3}$ matching where $U^{AF}_{R}(1)$ is anomaly-free $R$ symmetry. It is noted that a relatively systematic analysis of the anomaly matching is possible if one considers the formal breaking sequence of color gauge symmetry: $SU(N_{f})_{c}\to SU(N_{c})_{c}\times SU(\tilde{N}_{c})_{c}$ with $N_{f}= N_{c}+ \tilde{N}_{c}$, where $N_{f}$ stands for the number of massless quarks. 
  Sen has shown that tachyon condensation in Dbrane-anti-Dbrane configurations can lead to remarkable connections between string theories. A consequence of his results is that there is a minimal value of the radial coordinate $r_{c} \propto \sqrt{\alpha'} (gN)^{1/(9-p)}$ such that N units of D$p$brane charge cannot be localized to values smaller than $r_{c}.$ At this value of $r_{c}$ the curvature and the gradient of the Ramond-Ramond field strength are of order $1/\alpha',$ and the vacuum, regarded as a Dbrane-anti-Dbrane configuration with a tachyon condensate, is rendered unstable, leading to a separation of the Dbrane and the anti-Dbrane. This value of $r_{c}$ lies in the region intermediate between the near-horizon regime and the asymptotic regime for Dbrane classical solutions for small $g$. This vacuum stability bound on the curvature can be interpreted as an uncertainty relation for Dbrane charge. 
  We propose explicit recipes to construct the euclidean Green functions of gauge-invariant charged, monopole and dyon fields in four-dimensional gauge theories whose phase diagram contains phases with deconfined electric and/or magnetic charges. In theories with only either abelian electric or magnetic charges, our construction is an euclidean version of Dirac's original proposal, the magnetic dual of his proposal, respectively. Rigorous mathematical control is achieved for a class of abelian lattice theories. In theories where electric and magnetic charges coexist, our construction of Green functions of electrically or magnetically charged fields involves taking an average over Mandelstam strings or the dual magnetic flux tubes, in accordance with Dirac's flux quantization condition. We apply our construction to 't Hooft-Polyakov monopoles and Julia-Zee dyons. Connections between our construction and the semiclassical approach are discussed. 
  We study the propagation of gauge fields with arbitrary integer spins in the symmetrical Einstein space of any dimensionality. We reduce the problem of obtaining a gauge-invariant Lagrangian of integer spin fields in such background to an purely algebraic problem of finding a set of operators with certain features using the representation of high-spin fields in the form of some vectors of pseudo-Hilbert space. We consider such construction in the linear order in the Riemann tensor and scalar curvature and also present an explicit form of interaction Lagrangians and gauge transformations for massive particles with spins 1 and 2 in terms of symmetrical tensor fields. 
  We study the topological structure of the symmetry group of the standard model, $G_{SM}=U(1)\times SU(2)\times SU(3)$. Locally, $G_{SM}\cong S^1\times (S^3)^2\times S^5$. For SU(3), which is an $S^3$ bundle over $S^5$ (and therefore a local product of these spheres) we give a canonical gauge i.e. a canonical set of local trivializations. These formulae give the matrices of SU(3) in terms of points of spheres. Globally, we prove that the characteristic function of SU(3) is the suspension of the Hopf map $h: S^3 \to S^2$. We also study the case of SU(n) for arbitrary $n$, in particular the cases of SU(4), a flavour group, and of SU(5), a candidate group for grand unification. We show that the 2-sphere is also related to the fundamental symmetries of nature due to its relation to $SO^0(3,1)$, the identity component of the Lorentz group, a subgroup of the symmetry group of several gauge theories of gravity. 
  In this paper we investigate the relation between the bulk and boundary in AdS/CFT. We first discuss the relation between the Poincare and the global vacua, and then study various probes of the bulk from the boundary theory point of view. We derive expressions for retarded propagators and note that objects in free fall look like expanding bubbles in the boundary theory. We also study several Yang-Mills theory examples where we investigate thermal screening and confinement using propagators. In the case of confinement we also calculate the profile of a flux tube and provide an alternative derivation of the tension. 
  The IKKT matrix model was proposed to be a non-perturbative formulation of type IIB superstring theory. One of its important consistency criteria is that the leading one-loop $1/r^8$ effective interaction between a cluster of type IIB D-objects should not receive any corrections from higher loop effects for it to describe accurately the type IIB supergravity results. In analogy with the BFSS matrix model {\it versus} the eleven-dimensional supergravity example, we show in this work that the one-loop effective potential in the IKKT matrix model is also not renormalized at the two-loop order. 
  We study the scattering theory for the Gross-Neveu model on the half-line. We find the reflection matrices for the elementary fermions, and by fusion we compute the ones for the two-particle bound-states, showing that they satisfy non-trivial bootstrap consistency conditions. We also compute more general reflection matrices for the Gross-Neveu model and the nonlinear sigma model, and argue that they correspond to the integrable boundary conditions we identified in our previous paper hep-th/9809178. 
  Dvali and Shifman have proposed a field-theoretic mechanism for localizing gauge fields to "branes" in higher dimensional spaces using confinement in a bulk gauge theory. The resulting objects have a number of qualitative features in common with string theory D-branes; they support a gauge field and flux strings can end on them. In this letter, we explore this analogy further, by considering what happens when N of these "branes" approach each other. Unlike in the case of D-branes, we find a reduction of the gauge symmetry as the "branes" overlap. This can be attributed to a tachyonic instability of the flux string stretching between the branes. 
  This talk reviews our recent work on the construction of SL(2,Z) multiplets of type IIB superfivebranes. We here pay particular attention to the methods employed and some salient features of the solutions. 
  In ${\cal N}=4$ super Yang-Mills theory on a four-manifold $M$, one can specify a discrete magnetic flux valued in $H^2(M,\Z_N)$. This flux is encoded in the AdS/CFT correspondence in terms of a five-dimensional topological field theory with Chern-Simons action. A similar topological field theory in seven dimensions governs the space of ``conformal blocks'' of the six-dimensional $(0,2)$ conformal field theory. 
  Restricted to a black hole horizon, the ``gauge'' algebra of surface deformations in general relativity contains a Virasoro subalgebra with a calculable central charge. The fields in any quantum theory of gravity must transform accordingly, i.e., they must admit a conformal field theory description. Applying Cardy's formula for the asymptotic density of states, I use this result to derive the Bekenstein-Hawking entropy. This method is universal---it holds for any black hole, and requires no details of quantum gravity---but it is also explicitly statistical mechanical, based on counting microscopic states. 
  We derive superalgebras in many types of supersymmetric M-brane backgrounds. The backgrounds examined here include the cases of the M-wave and the M-Kaluza-Klein monopole. On the basis of the obtained algebras, we deduce all the supersymmetric non-orthogonal intersections of the M-Kaluza-Klein monopole and the M-5-brane at angles. In addition, we present a 1/4 supersymmetric worldvolume 3-brane soliton on the M-5-brane in the M-5-brane background as an extended solution of the 3-brane solitons of the M-5-brane by Howe, Lambert and West. This soliton can be interpreted as a certain intersection of three M-5-branes. 
  The Molien function counts the number of independent group invariants of a representation. For chiral superfields, it is invariant under duality by construction. We illustrate how it calculates the spectrum of supersymmetric gauge theories. 
  We calculate the absorption cross section by studying the spin--dependent wave equation in three-dimensional anti-de Sitter space(AdS$_3$). Here the AdS/CFT correspondence is used. It turns out that the new gauge bosons coupled to (2,0) and (0,2) operators on the boundary at infinity receive logarithmic corrections. This shows that the gauge bosons may play the role of singletons in AdS$_5$. On the other hand, test fields including the intermediate scalars($\eta, \xi$) and fixed scalar($\lambda$) do not receive any logarithmic correction in the first-order approximation. 
  We discuss dilatonic gravity (bulk theory) from the point of view of (generalized) AdS/CFT correspondence. Self-consistent dilatonic background is considered. It may be understood as two boundaries space where AdS boundary appears as infinite boundary and new (singular) boundary occurs at short distances. The two-point correlation function and conformal dimension for minimal and dilaton coupled scalar are found. Even for minimal scalar, the conformal dimension is found to be different on above two boundaries. Hence, new CFT appears in AdS/CFT correspondence at short distances. AdS/CFT correspondence may be understood as interpolating bulk theory between two different CFTs. 
  Points of conflict between the principles of general relativity and quantum theory are highlighted. I argue that the current language of QFT is inadequete to deal with gravity and review attempts to identify some of the features which are likely to present in the correct theory of quantum gravity. 
  The chiral Jacobian, which is defined with Neuberger's overlap Dirac operator of the lattice fermion, is explicitly evaluated in the continuum limit without expanding it in the gauge coupling constant. Our calculational scheme is simple and straightforward. We determine a coefficient of the chiral anomaly for general values of the mass parameter and the Wilson parameter of the overlap Dirac operator. 
  As an alternative to the covariant Ostrogradski method, we show that higher-derivative relativistic Lagrangian field theories can be reduced to second differential-order by writing them directly as covariant two-derivative theories involving Lagrange multipliers and new fields. Notwithstanding the intrinsic non-covariance of the Dirac's procedure used to deal with the constraints, the Lorentz invariance is recovered at the end. We develope this new setting for a simple scalar model and then its applications to generalized electrodynamics and higher-derivative gravity are outlined. This method is better suited than Ostrogradski's for a generalization to 2n-derivative theories. 
  Evidence is given that some particles with large charges in gauge theories, including 4d N=4 SU(3) and 5d theories with N=1 supersymmetry, have macroscopic entropy. The ground state entropy is found to be extensive in the charges. The mass gap for excitation is estimated and is of the same order as the binding energy. The question of the decay of excited states through Hawking radiation is raised. The analysis is based on the model of (p,q) string webs and on various other results. The evidence points to an explanation in terms of a string model. 
  Wilson's approach to renormalization group is reanalyzed for supersymmetric Yang-Mills theory. Usual demonstration of exact renormalization group equation must be modified due to the presence of the so called Konishi anomaly under the rescaling of superfields. We carry out the explicit computation for N=1 SUSY Yang-Mills theory with the simpler, gauge invariant regularization method, recently proposed by Arkani-Hamed and Murayama. The result is that the Wilsonian action S_M consists of two terms, i.e. the non anomalous term, which obeys Polchinski's flow equation and Fujikawa-Konishi determinant contribution. This latter is responsible for Shifman-Vainshtein relation of exact beta-function. 
  Considering the simple chiral fermion meson model when the chiral symmetry is explicitly broken, we show the validity of a trace identity -- to all orders of perturbation theory -- playing the role of a Callan-Symanzik equation and which allows us to identify directly the breaking of dilatations with the trace of the energy-momentum tensor. More precisely, by coupling the quantum field theory considered to a classical curved space background, represented by the non-propagating external vielbein field, we can express the conservation of the energy-momentum tensor through the Ward identity which characterizes the invariance of the theory under the diffeomorphisms. Our ``Callan-Symanzik equation'' then is the anomalous Ward identity for the trace of the energy-momentum tensor, the so-called ``trace identity''. 
  We study 2+1D toroidal compactifications of M-theory with twists in the U-duality group. These compactifications realize many symmetric-manifolds from the classification of 2+1D extended supergravity moduli-spaces. We then focus on the moduli-space $SU(2,1)/U(2)$ obtained by dimensional reduction of pure N=2 supergravity in 3+1D. This space is realized with an explicit example. Assuming that there are no quantum corrections, we conjecture that the classical discrete duality group has to be augmented with an extra strong/weak coupling duality. This implies the existence of new phases of the theory in which the original 8 compactification radii are all fixed at the Planck scale. 
  We obtain N=1 SU(N)^k gauge theories with bifundamental matter and a quartic superpotential as the low energy theory on D3-branes at singular points. These theories generalize that on D3-branes at a conifold point, studied recently by Klebanov and Witten. For k=3 the defining equation of the singular point is that of an isolated D_4 singularity. For k>3 we obtain a family of multimodular singularities. The considered SU(N)^k theories flow in the infrared to a non-trivial fixed point. We analyze the AdS/CFT correspondence for our examples. 
  We extend the notion of central charge superspace to the case of local supersymmetry. Gauged central charge transformations are identified as diffeomorphisms at the same footing as space-time diffeomorphisms and local supersymmetry transformations. Given the general structure we then proceed to the description of a particular vector-tensor supergravity multiplet of 24+24 components, identified by means of rather radical constraints. 
  We discuss string propagation in the near-horizon geometry generated by Neveu-Schwarz fivebranes, Kaluza-Klein monopoles and fundamental strings. When the fivebranes and KK monopoles are wrapped around a compact four-manifold $\MM$, the geometry is $AdS_3\times S^3/\ZZ_N\times \MM$ and the spacetime dynamics is expected to correspond to a local two dimensional conformal field theory. We determine the moduli space of spacetime CFT's, study the spectrum of the theory and compare the chiral primary operators obtained in string theory to supergravity expectations. 
  We begin a classification of the symmetry algebras arising on configurations of type IIB [p,q] 7-branes. These include not just the Kodaira symmetries that occur when branes coalesce into a singularity, but also algebras associated to other physically interesting brane configurations that cannot be collapsed. We demonstrate how the monodromy around the 7-branes essentially determines the algebra, and thus 7-brane gauge symmetries are classified by conjugacy classes of the modular group SL(2,Z). Through a classic map between the modular group and binary quadratic forms, the monodromy fixes the asymptotic charge form which determines the representations of the various (p,q) dyons in probe D3-brane theories. This quadratic form also controls the change in the algebra during transitions between different brane configurations. We give a unified description of the brane configurations extending the D_N, E_N and Argyres-Douglas H_N series beyond the Kodaira cases. We anticipate the appearance of affine and indefinite infinite-dimensional algebras, which we explore in a sequel paper. 
  This is a short summary of the phase structure of vector O(N) symmetric quantum field theories in a singular limit, the double scaling limit.It is motivated by the fact that summing up dynamically triangulated random surfaces using Feynman graphs of the O(N) matrix model results in a genus expansion and it provides,in some sense, a nonperturbative treatment of string theory when the double scaling limit is enforced. The main point emphasized here is that this formal singular limit, recently discussed mainly in d=0 O(N) matrix models, has an intriguing physical meaning in d>2 O(N) vector theories. In this limit all orders in {1\over N} are of equal importance since at each order infrared divergences compensate for the decrease in powers of {1\over N}. The infrared divergences are due to the tuning of the strength of the force {g \to g_c} between the O(N) quanta so that a massless O(N) singlet appears in the spectrum. At the critical dimension an interesting phase structure is revealed, the massless excitation has the expected physical meaning: it is the Goldston boson of spontaneous breaking of scale invariance - the dilaton. 
  We consider the monopole-antimonopole static potential in the confining phase of the Abelian Higgs model and in particular the corrections to the Coulomb-like potential at small distances r. By minimizing numerically the classical energy functional we observe a linear in r, stringy correction even at distances much smaller than the apparent physical scales. We argue that this term is a manifestation of the condition that the monopoles are connected by a mathematically thin line along which the scalar field vanishes. These short strings modify the operator product expansion as well. Implications for QCD are discussed. 
  In certain regions of the moduli space of K3 and Calabi-Yau manifolds, D-branes wrapped on non-supersymmetric cycles may give rise to stable configurations. We show that in the orbifold limit, some of these stable configurations can be described by solvable boundary conformal field theories. The world-volume theory of N coincident branes of this type is described by a non-supersymmetric U(N) gauge theory. At the boundary of the region of stability, there are marginal deformations connecting the non-supersymmetric brane to a pair of D-branes wrapped on supersymmetric cycles. We also discuss various relationships between BPS and non-BPS D-branes of type II string theories. 
  We review our calculation of the Weyl anomaly for d-dimensional conformal field theories that have a description in terms of a (d + 1)-dimensional gravity theory. 
  We derive the supersymmetric structure present in W-gravities which has been already observed in various contexts as Yang-Mills theory, topological field theories, bosonic string and chiral W_{3}-gravity. This derivation which is made in the geometrical framework of Zucchini, necessitates the introduction of an appropriate new basis of variables which replace the canonical fields and their derivatives. This construction is used, in the W_{2}-case, to deduce from the Chern-Simons action the Wess-Zumino-Polyakov action. 
  We argue that the large $n$ limit of the $n$-particle $SU(1,1|2)$ superconformal Calogero model provides a microscopic description of the extreme Reissner-Nordstr{\"o}m black hole in the near-horizon limit. 
  We analyze a particular SU(2) invariant sector of the scalar manifold of gauged N=8 supergravity in five dimensions, and find all the critical points of the potential within this sector. The critical points give rise to Anti-de Sitter vacua, and preserve at least an SU(2) gauge symmetry. Consistent truncation implies that these solutions correspond to Anti-de Sitter compactifications of IIB supergravity, and hence to possible near-horizon geometries of 3-branes. Thus we find new conformal phases of softly broken N=4 Yang--Mills theory. One of the critical points preserves N=2 supersymmetry in the bulk and is therefore completely stable, and corresponds to an N=1 superconformal fixed point of the Yang--Mills theory. The corresponding renormalization group flow from the N=4 point has c_{IR}/c_{UV} = 27/32. We also discuss the ten-dimensional geometries corresponding to these critical points. 
  A representation of the Lorentz group is given in terms of 4 X 4 matrices defined over the hyperbolic number system. The transformation properties of the corresponding four component spinor are studied, and shown to be equivalent to the transformation properties of the complex Dirac spinor. As an application, we show that there exists an algebra of automorphisms of the complex Dirac spinor that leaves the transformation properties of its eight real components invariant under any given Lorentz transformation. Interestingly, the representation of the Lorentz algebra presented here is naturally embedded in the Lie algebra of a group isomorphic to SO(3,3;R) instead of the conformal group SO(2,4;R). 
  An introductory survey of some of the developments that have taken place in superstring theory in the past few years is presented. The main focus is on three particular dualities. The first one is the appearance of an 11th dimension in the strong coupling limit of the type IIA theory, which give rise to M theory. The second one is the duality between the type IIB theory compactified on a circle and M theory on a two-torus. The final topic is an introduction to the recently proposed duality between superstring theory or M theory on certain anti de Sitter space backgrounds and conformally invariant quantum field theories. 
  D-instantons are considered as a probe of coinciding $N$ D3-branes. They can feel an external metric via the commutator terms in their effective action. We show that when the D-instantons are separated from the D3-branes, the metric which is probed at the one loop level, {\it exactly} coincides with that of the BPS R-R 3-brane. Interesting connection of this result to the possible explanation of the AdS/CFT correspondence within IKKT M-atrix theory is discussed. 
  Spin interactions beteween two moving Dp-branes are analyzed using the Green-Schwarz formalism of boundary states. This approach turns out to be extremely efficient to compute all the spin effects related by supersymmetry to the leading v^4/r^7-p term. All these terms are shown to be scale invariant, supporting a matrix model description of supergravity interactions. By employing the LSZ reduction formula for matrix theory and the mentioned supersymmetric effective potential for D0-branes, we compute the t-pole of graviton-graviton and three form-three form scattering in matrix theory. The results are found to be in complete agreement with tree level supergravity in the corresponding kinematical regime and provide, moreover, an explicit map between these degrees of freedom in both theories. 
  We give a simple and elegant proof of the Equivalence Theorem, stating that two field theories related by nonlinear field transformations have the same S matrix. We are thus able to identify a subclass of nonrenormalizable field theories which are actually physically equivalent to renormalizable ones. Our strategy is to show by means of the BRS formalism that the "nonrenormalizable" part of such fake nonrenormalizable theories, is a kind of gauge fixing, being confined in the cohomologically trivial sector of the theory. 
  In (2+1)-dimensional QED with a Chern-Simons term, we show that spontaneous magnetization occurs in the context of finite density vacua, which are the lowest Landau levels fully or half occupied by fermions. Charge condensation is shown to appear so as to complement the fermion anti-fermion condensate, which breaks the flavor U(2N) symmetry and causes fermion mass generation. The solutions to the Schwinger-Dyson gap equation show that the fermion self-energy contributes to the induction of a finite fermion density and/or fermion mass. The magnetization can be supported by charge condensation for theories with the Chern-Simons coefficient $\kappa=N e^2/2 \pi$, and $\kappa=N e^2/4 \pi$, under the Gauss law constraint. For $\kappa=N e^2/4 \pi$, both the magnetic field and the fermion mass are simultaneously generated in the half-filled ground state, which breaks the U(2N) symmetry as well as the Lorentz symmetry. 
  We review some properties of the field equations of six-dimensional (1,0) supergravity coupled to tensor and vector multiplets, and in particular their relation to covariant and consistent anomalies and a peculiar Noether identity for the energy-momentum tensor. We also describe a lagrangian formulation for this system, obtained applying the Pasti-Sorokin-Tonin prescription. 
  We apply the holographic principle during the inflationary stage of our universe. Where necessary, we illustrate the analysis in the case of new and extended inflation which, together, typify generic models of inflation. We find that in the models of extended inflation type, and perhaps of new inflation type also, the holographic principle leads to a lower bound on the density fluctuations. 
  We discuss the NSR formulation of the superstring action on AdS_5 X S^5 proposed recently by Kallosh and Tseytlin in the Green-Schwarz formalism.We show that the stress-energy tensor corresponding to the NSR action for AdS superstring contains the branelike terms, corresponding to exotic massless vertex operators (refered to as the branelike vertices). The corresponding sigma-model action has the manifest SO(1,3) X SO(6) invariance of superstring theory on AdS_5 X S^5. We argue that adding the branelike terms is equivalent to curving the space-time to obtain the AdS_5 X S^5 background. We commence the study of the proposed NSR sigma-model by analyzing the scattering amplitudes involving the branelike vertex operators.The analysis shows quite an unusual momentum dependence of these scattering amplitudes. 
  We construct a partition function for fields obeying a quasiperiodic boundary condition at finite temperature, $\psi(0;\vec x)= e^{i\theta} \psi(\beta;\vec x)$, which interpolate continously that ones corresponding to bosons and fermions and discuss the possibility of condensation for these fields. 
  It was shown by Brown and Henneaux that the classical theory of gravity on AdS_3 has an infinite-dimensional symmetry group forming a Virasoro algebra. More recently, Giveon, Kutasov and Seiberg (GKS) constructed the corresponding Virasoro generators in the first-quantized string theory on AdS_3. In this paper, we explore various aspects of string theory on AdS_3 and study the relation between these two works. We show how semi-classical properties of the string theory reproduce many features of the AdS/CFT duality. Furthermore, we examine how the Virasoro symmetry of Brown and Henneaux is realized in string theory, and show how it leads to the Virasoro Ward identities of the boundary CFT. The Virasoro generators of GKS emerge naturally in this analysis. Our work clarifies several aspects of the GKS construction: why the Brown-Henneaux Virasoro algebra can be realized on the first-quantized Hilbert space, to what extent the free-field approximation is valid, and why the Virasoro generators act on the string worldsheet localized near the boundary of AdS_3. On the other hand, we find that the way the central charge of the Virasoro algebra is generated is different from the mechanism proposed by GKS. 
  As a toy model to search for Hamiltonian formalism of the $AdS/CFT$ correspondence, we examine a Hamiltonian formulation of the $AdS_2/CFT_1$ correspondence emphasizing unitary representation theory of the symmetry. In the course of a canonical quantization of the bulk scalars, a particular isomorphism between the unitary irreducible representations in the bulk and boundary theories is found. This isomorphism defines the correspondence of field operators. It states that field operators of the bulk theory are field operators of the boundary theory by taking their boundary values in a due way. The Euclidean continuation provides an operator formulation on the hyperbolic coordinates system. The associated Fock vacuum of the bulk theory is located at the boundary, thereby identified with the boundary CFT vacuum. The correspondence is interpreted as a simple mapping of the field operators acting on this unique vacuum. Generalization to higher dimensions is speculated. 
  Poisson-Lie T-duality in quantum N=2 superconformal WZNW models is considered. The Poisson-Lie T-duality transformation rules of the super-Kac-Moody algebra currents are found from the conjecture that, as in the classical case, the quantum Poisson-Lie T-duality is given by an automorphism which interchanges the isotropic subalgebras of the underlying Manin triple of the model. It is shown that quantum Poisson-Lie T-duality acts on the generators of the N=2 super-Virasoro algebra of the quantum models as a mirror symmetry acts: in one of the chirality sectors it is trivial transformation while in another chirality sector it changes the sign of the U(1) current and interchanges the spin-3/2 currents. A generalization of Poisson-Lie T-duality for the Kazama-Suzuki models is proposed. It is shown that quantum Poisson-Lie T-duality acts in these models as a mirror symmetry also. 
  We study the attractor mechanism in low energy effective D=4, N=2 Yang-Mills theory weakly coupled to gravity, obtained from the effective action of type IIB string theory compactified on a Calabi-Yau manifold. Using special K\"{a}hler geometry, the general form of the leading gravitational correction is derived, and from this the attractor equations in the weak gravity limit. The effective Newton constant turns out to be spacetime-dependent due to QFT loop and nonperturbative effects. We discuss some properties of the attractor solutions, which are gravitationally corrected dyons, and their relation with the BPS spectrum of quantum Yang-Mills theory. Along the way, we obtain a satisfying description of Strominger's massless black holes, moving at the speed of light, free of pathologies encountered in some earlier proposals. 
  The non-perturbative renormalization group (NPRG) is applied to analysis of tunnelling in quantum mechanics. The vacuum energy and the energy gap of anharmonic oscillators are evaluated by solving the local potential approximated Wegner-Houghton equation (LPA W-H eqn.). We find that our results are very good in a strong coupling region, but not in a very weak coupling region, where the dilute gas instanton calculation works very well. So it seems that in analysis of quantum tunnelling, the dilute gas instanton and LPA W-H eqn. play complementary roles to each other. We also analyze the supersymmetric quantum mechanics and see if the dynamical supersymmetry (SUSY) breaking is described by NPRG method. 
  The supersymmetric Born-Infeld actions describing gauge-fixed D-5- and D-3-branes in ambient six-dimensional (6d) spacetime are constructed in superspace. A new 6d action is the (1,0) supersymmetric extension of the 6d Born-Infeld action. It is related via dimensional reduction to another remarkable 4d action describing the N=2 supersymmetric extension of the Born-Infeld-Nambu-Goto action with two real scalars. Both actions are the Goldstone actions associated with partial (1/2) spontaneous breaking of extended supersymmetry having 16 supercharges down to 8 supercharges. Both actions can be put into the `non-linear sigma-model' form by using certain non-linear superfield constraints. The unbroken supersymmetry is always linearly realised in our construction. 
  In these lectures, we present cosmological vacuum solutions of Horava-Witten theory and discuss their physical properties. We begin by deriving the five-dimensional effective action of strongly coupled heterotic string theory by performing a reduction, on a Calabi-Yau three-fold, of M-theory on S1/Z2. The effective theory is shown to be a gauged version of five-dimensional N=1 supergravity coupled, for simplicity, to the universal hypermultiplet and four-dimensional boundary theories with gauge and universal gauge matter fields. The static vacuum of the theory is a pair of BPS three-brane domain walls. We show that this five-dimensional theory, together with the domain wall vacuum solution, provides the correct starting point for early universe cosmology in Horava-Witten theory. Relevant cosmological solutions are those associated with the BPS domain wall vacuum. Such solutions must be inhomogeneous, depending on the orbifold coordinate as well as on time. We present two examples of this new type of cosmological solution, obtained by separation of variables. The first example represents the analog of a rolling radii solution with the radii specifying the geometry of the domain wall pair. This is generalized in the second example to include a nontrivial Ramond-Ramond scalar. 
  The D-3 brane is examined from the point of view of the wrapped M-theory five brane on a torus. In particular, the S-dual versions of the 3-brane are identified as coming from different gauge choices of the auxiliary field that is introduced in the PST description of the five brane world volume theory. 
  A generalization of the Wigner function for the case of a free particle with the ``relativistic'' Hamiltonian $\sqrt{{\bf p}^2+m^2}$ is given. 
  The correlation functions of supersymmetric gauge theories on a four-manifold X can sometimes be expressed in terms of topological invariants of X. We show how the existence of superconformal fixed points in the gauge theory can provide nontrivial information about four-manifold topology. In particular, in the example of gauge group SU(2) with one doublet hypermultiplet, we derive a theorem relating classical topological invariants such as the Euler character and signature to sum rules for Seiberg-Witten invariants. 
  The existence of black hole horizon is considered as a boundary condition to be imposed on the fluctuating metrics. The coordinate invariant form of the condition for class of spherically symmetric metrics is formulated. The diffeomorphisms preserving this condition act in (arbitrary small) vicinity of the horizon and form the group of conformal transformations of two-dimensional space ($r-t$ sector of the total space-time). The corresponding algebra recovered at the horizon is one copy of the Virasoro algebra. For general relativity in $d$ dimensions we find an effective two-dimensional theory governing the conformal dynamics at the horizon universally for any $d\geq 3$. The corresponding Virasoro algebra has central charge $c$ proportional to the Bekenstein-Hawking entropy. Identifying the zero-mode configuration we calculate $L_0$. The counting of states of this horizon's conformal field theory by means of Cardy's formula is in complete agreement with the Bekenstein-Hawking expression for the entropy of black hole in $d$ dimensions. 
  Abelian potentials of pointlike moving sources are obtained from the nonstandard theory of Yang--Mills field. They are used for the construction of the time-symmetric and time-asymmetric Fokker-type action integrals describing the dynamics of two-particle system with confinement interaction. The time-asymmetric model is reformulated in the framework of the Hamiltonian formalism. The corresponding two-body problem is reduced to quadratures. The behaviour of Regge trajectories is estimated within the semiclassical consideration. 
  An interesting tool for investigating the quantum features of a field theory is the introduction of compensating fields. For instance, the anomalous divergence of the chiral current can be calculated in the field-antifield formalism from an extended form of QCD with compensating fields. The interpretation of this procedure from the bosonized point of view, in the two dimensional case, crucially depends on the possibility of defining a bosonized version for the extended theory. We show, by using some recent results on the soldering of bosonized actions corresponding to chiral fermions, how is the mapping between bosonic and fermionic representations of this extended $QCD_2$. In the bosonic formulation the anomalous divergence of the chiral current shows up from the equations of motion of the compensating fields. 
  A simple method is proposed to construct the spectral zeta functions required for calculating the electromagnetic vacuum energy with boundary conditions given on a sphere or on an infinite cylinder. When calculating the Casimir energy in this approach no exact divergencies appear and no renormalization is needed. The starting point of the consideration is the representation of the zeta functions in terms of contour integral, further the uniform asymptotic expansion of the Bessel function is essentially used. After the analytic continuation, needed for calculating the Casimir energy, the zeta functions are presented as infinite series containing the Riemann zeta function with rapidly falling down terms. The spectral zeta functions are constructed exactly for a material ball and infinite cylinder placed in an uniform endless medium under the condition that the velocity of light does not change when crossing the interface. As a special case, perfectly conducting spherical and cylindrical shells are also considered in the same line. In this approach one succeeds, specifically, in justifying, in mathematically rigorous way, the appearance of the contribution to the Casimir energy for cylinder which is proportional to $\ln (2\pi)$. 
  We develop a formalism for the calculation of the ground state energy of a spinor field in the background of a cylindrically symmetric magnetic field. The energy is expressed in terms of the Jost function of the associated scattering problem. Uniform asymptotic expansions needed are obtained from the Lippmann-Schwinger equation. The general results derived are applied to the background of a finite radius flux tube with a homogeneous magnetic field inside and the ground state energy is calculated numerically as a function of the radius and the flux. It turns out to be negative, remaining smaller by a factor of $\alpha$ than the classical energy of the background except for very small values of the radius which are outside the range of applicability of QED. 
  We discuss aspects of the heterotic string effective field theories in orbifold constructions of the heterotic string. We calculate the moduli dependence of threshold corrections to gauge couplings in (2,2) symmetric orbifold compactifications. We perform the calculation of the threshold corrections for a particular class of abelian (2,2) symmetric non-decomposable orbifold models... internal twist is realized as generalized Coxeter automorphism. We define the limits for the existence of states causing singularities in the moduli space in the perturbative regime for a generic vacuum of the heterotic string. The 'proof' provides evidence for the explanation of the stringy 'Higgs effect'. Furthermore, we calculate the moduli dependence of threshold corrections as target space invariant free energies for non-decomposable orbifolds, identifying the Hauptmodul' functions for the relevant congruence subgroups. The required solutions provide for the \mu mass term generation in the effective low energy theory and affect the induced sypersymmetry breaking by gaugino condensation. In addition, we discuss the one loop gauge and gravitational couplings in (0,2) non-decomposable orbifold compactifications. In the second part of the Thesis the one loop correction to the Kahler metric for a generic N=2 orbifold compactification of the heterotic string is calculated... In this way, with the use of the one loop string amplitudes, the prepotential of the vector multiplets of the N=2 effective low-energy heterotic string is calculated in decomposable toroidal compactifications of the heterotic string ... This method provides the solution for the one loop correction to the prepotential of the vector multiplets of the heterotic string compactified on the K_3 \times T^2... 
  Green-Schwarz action of Type IIB string on $AdS_3 \times S^3$ is constructed via coset superspace approach. The construction relies exclusively on symplectically Majorana-Weyl spinor formalism, thus permitting it easier to prove $\kappa$-symmetry for on-shell Type IIB supergravity backgrounds. 
  We describe three ways of modifying the relativistic Heisenberg algebra - first one not linked with quantum symmetries, second and third related with the formalism of quantum groups. The third way is based on the identification of generalized deformed phase space with the semidirect product of two dual Hopf algebras describing quantum group of motions and the corresponding quantum Lie algebra. As an example the $\kappa$-deformation of relativistic Heisenberg algebra is given, determined by $\kappa$-deformed D=4 Poincar\'{e} symmetries. 
  A spatially discrete version of the general kink-bearing nonlinear Klein-Gordon model in (1+1) dimensions is constructed which preserves the topological lower bound on kink energy. It is proved that, provided the lattice spacing h is sufficiently small, there exist static kink solutions attaining this lower bound centred anywhere relative to the spatial lattice. Hence there is no Peierls-Nabarro barrier impeding the propagation of kinks in this discrete system. An upper bound on h is derived and given a physical interpretation in terms of the radiation of the system. The construction, which works most naturally when the nonlinear Klein-Gordon model has a squared polynomial interaction potential, is applied to a recently proposed continuum model of polymer twistons. Numerical simulations are presented which demonstrate that kink pinning is eliminated, and radiative kink deceleration greatly reduced in comparison with the conventional discrete system. So even on a very coarse lattice, kinks behave much as they do in the continuum. It is argued, therefore, that the construction provides a natural means of numerically simulating kink dynamics in nonlinear Klein-Gordon models of this type. The construction is compared with the inverse method of Flach, Zolotaryuk and Kladko. Using the latter method, alternative spatial discretizations of the twiston and sine-Gordon models are obtained which are also free of the Peierls-Nabarro barrier. 
  The left-right symmetric model (LRSM) with gauge group $SU(2)_{L} \times SU(2)_{R} \times U(1)_{B-L}$ is reconstructed from the geometric formulation of gauge theory in $M_4 \times Z_2 \times Z_2$ where $M_4$ is the four-dimensional Minkowski space and $Z_2 \times Z_2$ the discrete space with four points. The geometrical structure of this model becomes clearer compared with other works based on noncommutative geometry. As a result, the Yukawa coupling terms and the Higgs potential are derived in more restricted forms than in the standard LRSM. 
  We show how the anti-de Sitter isometries of a brane solution of supergravity theory produce superconformal invariance of their world-volume action. In this way linear as well as non-linear superconformal actions are obtained in various dimensions. Two particular examples are a conformal action with the antisymmetric tensor in 6 dimensions in Pasti-Sorokin-Tonin formulation, and superconformal mechanics. 
  We analyze short and long multiplets which appear in the OPE expansion of ``chiral'' primary operators in N=4 Super Yang--Mills theory. Among them, higher spin long and new short multiplets appear, having the interpretation, in the AdS/CFT correspondence, of string states and supergravity multiparticle states respectively. We also analyze the decomposition of long multiplets under N=1 supersymmetry, as a possible tool to explore other supersymmetric deformations of IIB string on AdS_5 x S_5. 
  The perturbative prepotential and the K\"ahler metric of the vector multiplets of the N=2 effective low-energy heterotic strings is calculated directly in N=1 six-dimensional toroidal compactifications of the heterotic string vacua. This method provides the solution for the one loop correction to the N=2 vector multiplet prepotential for compactifications of the heterotic string for any rank three and four models, as well for compactifications on $K_3 \times T^2$. In addition, we complete previous calculations, derived from string amplitudes, by deriving the differential equation for the third derivative of the prepotential with respect of the usual complex structure U moduli of the $T^2$ torus. Moreover, we calculate the one loop prepotential, using its modular properties, for N=2 compactifications of the heterotic string exhibiting modular groups similar with those appearing in N=2 sectors of N=1 orbifolds based on non-decomposable torus lattices and on N=2 supersymmetric Yang-Mills. 
  An Abelian gauge model, with vector and 2-form potential fields linked by a topological mass term that mixes the two Abelian factors, is shown to exhibit Dirac-like magnetic monopoles in the presence of a matter background. In addition, considering a "non-minimal coupling" between the fermions and the tensor field, we obtain a generalised quantisation condition that involves, among others, the mass parameter. Also, it is explicitly shown that 1_loop (finite) corrections do not shift the value of such a mass parameter. 
  Relativistic effects in the thermodynamical properties of interacting particle systems are investigated within the framework of the relativistic direct interaction theory in various forms of dynamics. In the front form of relativistic dynamics an exactly solvable model of a one-dimensional hard spheres gas is formulated and an equation of state and thermodynamical potentials for such a gas are found. Weakly-relativistic corrections to the thermodynamical functions of the dilute gas with short-range interactions are discussed on the basis of the approximately relativistic Hamiltonian function in the instant form of dynamics. 
  We study anomalous Wess-Zumino couplings of D-branes and O-planes in a general background and derive them from a direct string computation by factorizing in the RR channel various one-loop amplitudes. In particular, we find that Op-planes present gravitational anomalous couplings involving the Hirzebruch polynomial L, similarly to the roof genus A encoding Dp-brane anomalous couplings. We determine, in each case, the precise dependence of these couplings on the curvature of the tangent and normal bundles. 
  I review certain aspects of Hanany-Witten setups and other approaches used to embed (and solve) gauge theories in string theory. Applications covered include dualities in 4 and 3 dimensions, fixed points in 6 dimensions, phase transitions between different geometric backgrounds and dualities between branes and geometry. 
  Low-energy, near-horizon scaling limits of black holes which lead to string theory on AdS_2 x S^2 are described. Unlike the higher-dimensional cases, in the simplest approach all finite-energy excitations of AdS_2 x S^2 are suppressed. Surviving zero-energy configurations are described. These can include tree-like structures in which the AdS_2 x S^2 throat branches as the horizon is approached, as well as disconnected AdS_2 x S^2 universes. In principle, the black hole entropy counts the quantum ground states on the moduli space of such configurations. In a nonsupersymmetric context AdS_D for general D can be unstable against instanton-mediated fragmentation into disconnected universes. Several examples are given. 
  We consider the superparticle models invariant under the supersymmetries with tensorial central charges, which were not included in D=4 Haag-Lopuszanski-Sohnius (HLS) supersymmetry scheme. We present firstly a generalization of D=4 Ferber-Shirafuji (FS) model with fundamental bosonic spinors and tensorial central charge coordinates. The model contains four fermionic coordinates and possesses three kappa- symmetries thus providing the BPS configuration preserving 3/4 of the target space supersymmetries. We show that the physical degrees of freedom (8 real bosonic and 1 real Grassmann variable) of our model can be described by OSp(8|1) supertwistor. Then we propose a higher dimensional generalization of our model with one real fundamental bosonic spinor. D=10 model describes massless superparticle with composite tensorial central charges and in D=11 we obtain 0-superbrane model with nonvanishing mass which is generated dynamically. The introduction of D=11 Lorentz harmonics provides the possibility to construct massless D=11 superparticle model which can be formulated in a way preserving 1/2, 17/32, 18/32,..., 31/32 supersymmetries. In a special case we obtain the twistor-like formulation of the usual massless D=11 superparticle proposed recently by Bergshoeff and Townsend. 
  A relativistically invariant scheme for the description of excited states in a one-kink sector is formulated. The normal oscillations of fluctuations against the background of a moving kink are determined. Zero mode of these oscillations is excluded automatically due to the properties of the integral equation describing normal oscillations. A Hamiltonian for elementary excitations is obtained reflecting the relativistic nature of the problem considered. (35 kb) 
  The large N limit of the Gross-Neveu model is here studied on manifolds with constant curvature, at zero and finite temperature. Using the zeta-function regularization, the phase structure is investigated for arbitrary values of the coupling constant. The critical surface where the second order phase transition takes place is analytically found for both the positive and negative curvature cases. For negative curvature, where the symmetry is always broken at zero temperature, the mass gap is calculated. The free energy density is evaluated at criticality and the zero curvature and zero temperature limits are discussed. 
  We consider the simplest geometrical particle model associated with light-like curves in (2+1)-dimensions. The action is proportional to the pseudo-arc length of the particle's path. We show that under quantization it yields the (2+1)-dimensional anyonic field equation supplemented with a Majorana-like relation on mass and spin, i.e., $mass \times spin =\alpha^2$, with $\alpha$ the coupling constant in front of the action. 
  The prepotential and spectral curve are described for a smooth interpolation between an enlarged N=4 SUSY and ordinary N=2 SUSY Yang-Mills theory in four dimensions, obtained by compactification from five dimensions with non-trivial (periodic and antiperiodic) boundary conditions. This system provides a new solution to the generalized WDVV equations. We show that this exhausts all possible solutions of a given functional form. 
  A relation between confinement and Maldacena conjecture is briefly discussed. The gauge symmetry enhancement for two coincident D-6-branes is analyzed from the viewpoint of the hypermultiplet low-energy effective action given by the N=2 supersymmetric non-linear sigma-model with the Eguchi-Hanson (ALE) target space. 
  Hamiltonian light-front field theory can be used to solve for hadron states in QCD. To this end, a method has been developed for systematic renormalization of Hamiltonian light-front field theories, with the hope of applying the method to QCD. It assumed massless particles, so its immediate application to QCD is limited to gluon states or states where quark masses can be neglected. This paper builds on the previous work by including particle masses non-perturbatively, which is necessary for a full treatment of QCD. We show that several subtle new issues are encountered when including masses non-perturbatively. The method with masses is algebraically and conceptually more difficult; however, we focus on how the methods differ. We demonstrate the method using massive phi^3 theory in 5+1 dimensions, which has important similarities to QCD. 
  We study a three-dimensional gauge theory obtained from the dimensional reduction of a D4-brane worldvolume theory in the background of space-time moduli. An SL(3) symmetry in this theory, which acts on fields as well as coupling constants, is identified. By comparing the energies with the string tensions, we show that certain 1/2 supersymmetric classical solutions of this theory can be identified as SL(3,Z) multiplets of type II strings in eight dimensions. Results are then generalized to the non-linear Born-Infeld action. We also discuss the possibility of 1/8 BPS states in this theory and their representations in terms of string networks. 
  We determine the corrections to the entropy of extremal black holes arising from terms quadratic in the Riemann tensor in $N=2, D=4$ supergravity theories. We follow Wald's proposal to modify the Bekenstein-Hawking area law. The new entropy formula, whose value only depends on the electric/magnetic charges, is expressed in terms of a single holomorphic function and is consistent with electric-magnetic duality. For string effective field theories arising from Calabi-Yau compactifications, our result for the entropy of a certain class of extremal black-hole solutions fully agrees with the counting of microstates performed some time ago by Maldacena, Strominger, Witten and by Vafa. 
  Three dimensional Euclidean gravity in the dreibein-spin connection formalism is investigated. We use the monopole-instanton ansatz for the dreibein and the spin connection. The equations of motion are solved. We point out a two dimensional solution with a vanishing action. 
  It was recently pointed out by E. Witten that for a D-brane to consistently wrap a submanifold of some manifold, the normal bundle must admit a Spin^c structure. We examine this constraint in the case of type II string compactifications with vanishing cosmological constant, and argue that in all such cases, the normal bundle to a supersymmetric cycle is automatically Spin^c. 
  In general, Picard-Fuchs systems in N=2 supersymmetric Yang-Mills theories are realized as a set of simultaneous partial differential equations. However, if the QCD scale parameter is used as unique independent variable instead of moduli, the resulting Picard-Fuchs systems are represented by a single ordinary differential equation (ODE) whose order coincides with the total number of independent periods. This paper discusses some properties of these Picard-Fuchs ODEs. In contrast with the usual Picard-Fuchs systems written in terms of moduli derivatives, there exists a Wronskian for this ordinary differential system and this Wronskian produces a new relation among periods, moduli and QCD scale parameter, which in the case of SU(2) is reminiscent of scaling relation of prepotential. On the other hand, in the case of the SU(3) theory, there are two kinds of ordinary differential equations, one of which is the equation directly constructed from periods and the other is derived from the SU(3) Picard-Fuchs equations in moduli derivatives identified with Appell's $F_4$ hypergeometric system, i.e., Burchnall's fifth order ordinary differential equation published in 1942. It is shown that four of the five independent solutions to the latter equation actually correspond to the four periods in the SU(3) gauge theory and the closed form of the remaining one is established by the SU(3) Picard-Fuchs ODE. The formula for this fifth solution is a new one. 
  D=11 supergravity possesses D=5 Calabi-Yau compactified solutions that may be identified with the vacua of the Horava-Witten orbifold construction for M--theory/heterotic duality. The simplest of these solutions naturally involves two 3-brane domain walls, which may be identified with the orbifold boundary planes; this solution also possesses an unbroken $\Z_2$ symmetry. Consideration of nearby excited solutions, truncated to the zero-mode and $\Z_2$ invariant sector, yields an effective D=4 heterotic theory displaying chirality and N=1, D=4 supersymmetry. 
  We find the superisometry of the near-horizon superspace, forming the superconformal algebra. We present here the explicit form of the transformation of the bosonic and fermionic coordinates (as well as the compensating Lorentz-type transformation) which keeps the geometry invariant. We comment on  i) the BPS condition of the branes in adS background and  ii) significant simplification of superisometries of the gauge-fixed Green-Schwarz action in adS(5)*S(5) background. 
  An explicit calculation is performed to check all the tangent bundle gravitational couplings of Dirichlet branes and Orientifold planes by scattering $q$ gravitons with a $p+1$ form Ramond-Ramond potential in the world-volume of a $D(p+2q)$-brane. The structure of the D-brane Wess-Zumino term in the world-volume action is confirmed, while a different O-plane Wess-Zumino action is obtained. 
  In a recent paper we considered the type 0 string theories, obtained from the ten-dimensional closed NSR string by a GSO projection which excludes space-time fermions, and studied the low-energy dynamics of N coincident D-branes. This led us to conjecture that the four-dimensional SU(N) gauge theory coupled to 6 adjoint massless scalars is dual to a background of type 0 theory carrying N units of R-R 5-form flux and involving a tachyon condensate. The tachyon background leads to a ``soft breaking'' of conformal invariance, and we derived the corresponding renormalization group equation. Minahan has subsequently found its asymptotic solution for weak coupling and showed that the coupling exhibits logarithmic flow, as expected from the asymptotic freedom of the dual gauge theory. We study this solution in more detail and identify the effect of the 2-loop beta function. We also demonstrate the existence of a fixed point at infinite coupling. Just like the fixed point at zero coupling, it is characterized by the AdS_5\times S^5 Einstein frame metric. We argue that there is a RG trajectory extending all the way from the zero coupling fixed point in the UV to the infinite coupling fixed point in the IR. 
  Recently we have proposed a set of variables for describing the infrared limit of four dimensional SU(2) Yang-Mills theory. here we extend these variables to the general case of four dimensional SU(N) Yang-Mills theory. We find that the SU(N) connection A decomposes according to irreducible representations of SO(N-1) and the curvature two-form F is related to the symplectic Kirillov two forms that characterize irreducible representations of SU(N). We propose a general class of nonlinear chiral models that may describe stable, soliton-like configurations with nontrivial topological numbers. 
  We prove that certain nonequilibrium expectation values in the boundary sine-Gordon model coincide with associated equilibrium-state expectation values in the systems which differ from the boundary sine-Gordon in that certain extra boundary degrees of freedom (q-oscillators) are added. Applications of this result to actual calculation of nonequilibrium characteristics of the boundary sine-Gordon model are also discussed. 
  We explore subleading contributions to the two basic central charges c and a of four-dimensional conformal field theories in the AdS/CFT scheme. In particular we probe subleading corrections to the difference c-a from the string-theory side. In the N=4 CFT, c-a vanish identically consistently with the string-theory expectations. However, for N=1 and N=2 CFTs, the U_R(1) anomaly, which is proportional to c-a, is subleading in the large N limit for theories in the AdS/CFT context and one expects string one-loop R^2 and B \wedge R \wedge R terms in the low energy effective action. We identify these terms as coming from the R^4 terms. Similar considerations apply to the U_R(1)^3 anomaly which is, however, subleading only for N=2 theories. As a result, a string one-loop term B \wedge F \wedge F should exist in the low energy effective action of the N=4 five-dimensional supergravity. The U_R(1)^3 term is leading for the N=1 CFT and it is indeed present in the N=2 five-dimensional supergravity. 
  If the fundamental type-I string scale is of the order of few TeV, the problem of the gauge hierarchy is that of understanding why some dimensions transverse to our brane-world are so large. The technical aspect of this problem, as usually formulated, is `why quantum corrections do not modify drastically the masses and other parameters of the Standard Model'. We argue that within type-I perturbation theory, the technical hierarchy problem is solved (a) if all massless tadpoles cancel locally over distances of order the string length in the transverse space, or (b) if the massless fields with uncancelled local tadpoles propagate `effectively' in $d_\perp \ge 2$ large transverse dimensions. These restrictions ensure that loop corrections to the Standard Model parameters decouple from the four-dimensional Planck scale, except when there are uncancelled tadpoles in $d_\perp =2$ in which case the dependence on $M_P$ is logarithmic. This latter case is thus singled out as the only one in which the origin of the hierarchy would not be attributed entirely to `out of this world' bulk physics. The role of the renormalization group equations in summing the leading large logs is replaced by the classical 2d supergravity equations in the transverse space. 
  Using the natural curvature invariants as building blocks in a superfield construction, we show that the use of a symmetric trace is mandatory if one is to reproduce the square root structure of the non-Abelian Dirac-Born-Infeld Lagrangian in the bosonic sector. We also discuss the BPS relations in connection with our supersymmetry construction. 
  T-duality realized on D-brane effective actions is studied from a pure worldvolume point of view. It is proved that invariance in the form of the Dirac-Born-Infeld and Wess-Zumino terms fixes the T-duality transformations of the NS-NS and R-R background fields, respectively. The analysis is extended to uncover the mapping of global symmetries of the corresponding pair of D-branes involved in the transformation. 
  We discuss the relation between canonical and Schrodinger quantization of the CGHS model. We also discuss the situation when background charges are added to cancel the Virasoro anomaly. New physical states are found when the square of the background charges vanishes. 
  We show that a system of parallel D3 branes near a conifold singularity can be mapped onto an intersecting configuration of orthogonal branes in type IIA string theory. Using this brane configuration, we analyze the Higgs moduli space of the associated field theory. The dimension of the Higgs moduli space is computed from a geometrical analysis of the conifold singularity. Our results provide evidence for an extended s-rule. In addition, a discrepancy between the prediction of the brane configuration and the result obtained from a geometrical anaysis is noted. This discrepancy is traced back to worldsheet instanton effects. 
  We continue to explore the possibility that the graviton in two dimensions is related to a quadratic differential that appears in the anomalous contribution of the gravitational effective action for chiral fermions. A higher dimensional analogue of this field might exist as well. We improve the defining action for this diffeomorphism tensor field and establish a principle for how it interacts with other fields and with point particles in any dimension. All interactions are related to the action of the diffeomorphism group. We discuss possible interpretations of this field. 
  Biconformal gauging of the conformal group has a scale-invariant volume form, permitting a single form of the action to be invariant in any dimension. We display several 2n-dim scale-invariant polynomial actions and a dual action. We solve the field equations for the most general action linear in the curvatures for a minimal torsion geometry. In any dimension n>2, the solution is foliated by equivalent n-dim Ricci-flat Riemannian spacetimes, and the full 2n-dim space is symplectic. Two fields defined entirely on the Riemannian submanifolds completely determine the solution: a metric, and a symmetric tensor. 
  We explore the superstring theory on AdS_3 x S^3 x T^4 in the framework given in hep-th/9806194. We argue on the Hilbert space of "space-time CFT", and especially construct a suitable vacuum of this CFT from the physical degrees of freedom of the superstring theory in bulk. We first construct it explicitly in the case of p=1, and then present a proposal for the general cases of p>1.   After giving some completion of the GKS's constructions of the higher mode operators (in particular, of those including spin fields), we also make some comparison between the space-time CFT and T^{4kp}/S_{kp} SCFT, namely, with respect to the physical spectrum of chiral primaries and some algebraic structures of bosonic and fermionic oscillators in both theories. We also observe how our proposal about the Hilbert space of space-time CFT leads to a satisfactory correspondence between the spectrum of chiral primaries of both theories in the cases of p>1. 
  We use finite field-dependent BRS transformations (FFBRS) to connect the Green functions in a set of two otherwise unrelated gauge choices. We choose the Lorentz and the axial gauges as examples. We show how the Green functions in axial gauge can be written as a series in terms of those in Lorentz gauges. Our method also applies to operator Green's functions. We show that this process involves another set of related FFBRS transfomations that is derivable from infinitesimal FBRS. We suggest possible applications. 
  Several issues concerning the self-dual solutions of the Chern-Simons-Higgs model are addressed. The topology of the configuration space of the model is analysed when the space manifold is either the plane or an infinite cylinder. We study the local structure of the moduli space of self-dual solitons in the second case by means of an index computation. It is shown how to manage the non-integer contribution to the heat-kernel supertrace due to the non-compactness of the base space. A physical picture of the local coordinates parametrizing the non-topological soliton moduli space arises . 
  The low energy dynamics of the vortices of the Abelian Chern-Simons-Higgs system is investigated from the adiabatic approach. The difficulties involved in treating the field evolution as motion on the moduli space in this system are shown. Another two generalized Abelian Higgs systems are discusssed with respect to their vortex dynamics at the adiabatic limit. The method works well and we find bound states in the first model and scattering at right angles in the second system. 
  I describe our understanding of physics near the Planck length, in particular the great progress in the last four years in string theory. These are lectures presented at the 1998 SLAC Summer Institute. 
  We summarize the progress made during the last few years on the study of Vassiliev invariants from the point of view of perturbative Chern-Simons gauge theory. We argue that this approach is the most promising one to obtain a combinatorial universal formula for Vassiliev invariants. The combinatorial expressions for the two primitive Vassiliev invariants of order four, recently obtained in this context, are reviewed and rewritten in terms of Gauss diagrams. 
  In a recent paper it was shown that the response of an integrable QFT under variation of the Unruh temperature can be computed from a S-matrix preserving deformation of the form factor approach. We give explicit expressions for the deformed two-particle formfactors for various integrable models: The Sine-Gordon and SU(2) Thirring model, several perturbed minimal CFTs and the real coupling affine Toda series. A uniform pattern is found to emerge when both the S-matrix and the deformed form factors are expressed in terms Barnes' multi-periodic functions. 
  Collective center-of-mass variables are introduced in the Lagrangian formalism of the relativistic classical mechanics of directly interacting particles. It is shown that the transition to the Hamiltonian formalism leads to the Bakamjian-Thomas model. The quantum-mechanical system consisting of two spinless particles is investigated. Quasi-relativistic corrections to the discrete energy spectrum are calculated for some Coulomb-like interactions having field theoretical analogues. 
  In this letter we implement a recently proposed {\it spacetime duality} approach to dualize a two dimensional, Abelian, gauge field theory, which has no dual version under $p$--duality. Our result suggests that spacetime duality spans a new, wider, class of dual theories, which cannot be related one to another by $p$--duality transformations. 
  We give a short introduction to and a partial review of the work on the calculation of Wilson loops and $Q\bar Q$-potentials via the conjectured AdS/CFT duality. Included is a discussion of the relative weight of the stringy correction to the target space background versus the correction by the quantum fluctuations of the string world sheet. 
  In this paper we further develop the theory of $\alpha$-induction for nets of subfactors, in particular in view of the system of sectors obtained by mixing the two kinds of induction arising from the two choices of braiding. We construct a relative braiding between the irreducible subsectors of the two ``chiral'' induced systems, providing a proper braiding on their intersection. We also express the principal and dual principal graphs of the local subfactors in terms of the induced sector systems. This extended theory is again applied to conformal or orbifold embeddings of SU(n) WZW models. A simple formula for the corresponding modular invariant matrix is established in terms of the two inductions, and we show that it holds if and only if the sets of irreducible subsectors of the two chiral induced systems intersect minimally on the set of marked vertices i.e. on the ``physical spectrum'' of the embedding theory, or if and only if the canonical endomorphism sector of the conformal or orbifold inclusion subfactor is in the full induced system. We can prove either condition for all simple current extensions of SU(n) and many conformal inclusions, covering in particular all type I modular invariants of SU(2) and SU(3), and we conjecture that it holds also for any other conformal inclusion of SU(n) as well. As a by-product of our calculations, the dual principal graph for the conformal inclusion $SU(3)_5 \subset SU(6)_1$ is computed for the first time. 
  A new force on the magnetic dipole, which exists in the presence of both electric and magnetic fields, is described. Its origin due to the `hidden momentum', implications and possible experimental tests are discussed. 
  Representations of the Abelian-projected SU(2)- and SU(3)-gluodynamics in terms of the magnetic monopole currents are derived. Besides the quadratic part, the obtained effective actions contain interactions of these currents with the world-sheets of electric strings in 4D or electric vortex lines in 3D. Next, we illustrate that 3D compact QED is a small gauge boson mass limit of 3D Abelian Higgs model with external monopoles and give a physical interpretation to the confining string theory as the integral over the monopole densities. Finally, we derive the bilocal field strength correlator in the weak-field limit of 3D compact QED, which turns out to be in line with the one predicted by the Stochastic Vacuum Model. 
  We present a new framework for a Lagrangian description of conformal field theories in various dimensions based on a local version of d+2-dimensional conformal space. The results include a true gauge theory of conformal gravity in d=(1,3) and any standard matter coupled to it. An important feature is the automatic derivation of the conformal gravity constraints, which are necessary for the analysis of the matter systems. 
  We study the boundary limit of the bulk isometries of AdS x S. The superconformal symmetry is realized on the coordinates of the AdS boundary, the fermionic superspace coordinates, and the harmonics on the sphere. We show how these may be related to the coordinates of an off-shell harmonic superspace of SCFT living on the boundary. In the special case of d=4, N=2 super Yang-Mills theory, a truncation of the N=4 SYM dual to Type IIB string theory compactified on AdS_5 x S^5, we identify the bosonic space SU(2)/U(1) of the N=2 harmonic superspace (known before as auxiliary) with the S^2 submanifold of the S^5. 
  We demonstrate that certain Virasoro characters (and their linear combinations) in minimal and non-minimal conformal models which admit factorized forms are manifestly related to the ADE series. This permits to extract quasi-particle spectra of a Lie algebraic nature which resembles the features of Toda field theory. These spectra possibly admit a construction in terms of the $W_n$-generators. In the course of our analysis we establish interrelations between the factorized characters related to the parafermionic models, the compactified boson and the minimal models. 
  We consider a classical field configuration, corresponding to intersection of two domain walls in a supersymmetric model, where the field profile for two parallel walls at a finite separation is known explicitly. An approximation to the solution for intersecting walls is constructed for a small angle at the intersection. We find a finite effective length of the intersection region and also an energy, associated with the intersection. 
  Compactification of Type IIB superstring on an $AdS_5 \times S^5/\Gamma$ background leads to SU(N) gauge field theories with prescribed matter representations. In the 't Hooft limit of large N such theories are conformally finite. For finite N and broken supersymmetry ($\cal N$ = 0) I derive the constraints to be two-loop conformal and examine the consequences for a wide choice of $\Gamma$ and its embedding $\Gamma \subset {\cal C}^3 (\supset S^5)$. 
  We study total and partial supersymmetry breaking by freely acting orbifolds, or equivalently by Scherk-Schwarz compactifications, in type I string theory. In particular, we describe a four-dimensional chiral compactification with spontaneously broken N=1 supersymmetry, some models with partial $N=4\to N=2$ and $N=4\to N=1$ supersymmetry breaking and their heterotic and M-theory duals. A generic feature of these models is that in the gravitational sector and in the spectrum of D-branes parallel to the breaking coordinate, all mass splittings are proportional to the compactification scale, while global (extended) supersymmetry remains unbroken at tree level for the massless excitations of D-branes transverse to the breaking direction. 
  The $S^N\R^8$ supersymmetric orbifold sigma model is expected to describe the IR limit of the Matrix string theory. In the framework of the model the type IIA string interaction is governed by a vertex which was recently proposed by R.Dijkgraaf, E.Verlinde and H.Verlinde. By using this interaction vertex we derive all four particle scattering amplitudes directly from the orbifold model in the large $N$ limit. 
  It is here explained how the Green-Schwarz superstring theory arises from Matrix String Theory. This is obtained as the strong YM-coupling limit of the theory expanded around its BPS instantonic configurations, via the identification of the interacting string diagram with the spectral curve of the relevant configuration. Both the GS action and the perturbative weight $g_s^{-\chi}$, where $\chi$ is the Euler characteristic of the world-sheet surface and $g_s$ the string coupling, are obtained. 
  The low energy excitation of the rotating D3-branes is thermodynamically stable up to a critical angular momentum density. This indicates that there is a corresponding phase transition of the ${\cal N}$=4 large $N$ super Yang-Mills theory at finite temperature. On the side of supergravity, we investigate the phase transition in the grand canonical ensemble and canonical ensemble. Some critical exponents of thermodynamic quantities are calculated. They obey the static scaling laws. Using the scaling laws related to the correlation length, we get the critical exponents of the correlation function of gauge field. The thermodynamic stability of low energy excitations of the rotating M5-branes and rotating M2-branes is also studied and similar critical behavior is observed. We find that the critical point is shifted in the different ensembles and there is no critical point in the canonical ensemble for the rotating M2-branes. We also discuss the Hawking-Page transition for these rotating branes. In the grand canonical ensemble, the Hawking-Page transition does not occur. In the canonical ensemble, however, the Hawking-Page transition may appear for the rotating D3- and M5-branes, but not for the rotating M2-branes. 
  We show that the effective potential for local composite operators is a useful object in studing dynamical symmetry breaking by calculating the effective potential for the local composite operators $\bar{\psi} \psi$ and $\phi^2$ in the Gross-Neveu (GN) and O(N) models, respectively. Since the effective potential for local composite operators can be calculated by using the Cornwall-Jackiw-Tomboulis (CJT) effective potential in theory with additional bare mass terms, we show that divergences in the effective potential for local composite operators are the same as in the CJT effective potential. We compare the results obtained with the results give by the auxiliary field method. 
  The harmonic oscillator in pseudo euclidean space is studied. A straightforward procedure reveals that although such a system may have negative energy, it is stable. In the quantized theory the vacuum state has to be suitably defined and then the zero-point energy corresponding to a positive-signature component is canceled by the one corresponding to a negative-signature component. This principle is then applied to a system of scalar fields. The metric in the space of fields is assumed to have signature (+ + + ... - - -) and it is shown that the vacuum energy, and consequently the cosmological constant, are then exactly zero. The theory also predicts the existence of stable, negative energy field excitations (the so called "exotic matter") which are sources of repulsive gravitational fields, necessary for construction of the time machines and Alcubierre's hyperfast warp drive. 
  The short survey of computation and properties of effective Lagrange function of intensive field in two-loop approximation accounting for radiative interaction of virtual electrons is given. The renormalization of field, charge and mass is completely defined by the weak field behaviour of the exact Lagrange function: its real part must be Maxwellian and imaginary part must be quasiclassical $\propto \exp (-\pi m^2c^3/\hbar e\epsilon)$. For weak electric field radiative interaction manifests itself as electron mass shift. Using the renorminvariance at strong field and exponentiation at weak field it is possible to obtain information about contributions of high order in $\alpha$. 
  The Lagrangian relativistic direct interaction theory in the various forms of dynamics is formulated and its connections with the Fokker-type action theory and with the constrained Hamiltonian mechanics are established. The motion of classical two-particle system with relativistic direct interaction is analysed within the framework of isotropic forms of dynamics in the two- and four-dimensional space-time. Some relativistic exactly solvable quantum-mechanical models are also discussed. 
  We show the equivalence of three different realisations of gauge theory in string theory. These are the Hanany-Witten brane constructions, the use of branes as probes and geometric engineering. We illustrate the equivalence via T- and S-dualities with the simplest non-trivial examples in four dimensions: N = 2 SYM with gauge groups SU(N_1) x SU(N_2) x ..... 
  It is shown how the topological string amplitudes encode the BPS structure of wrapped M2 branes in M-theory compactification on Calabi-Yau threefolds. This in turn is related to a twisted supersymmetric index in 5 dimensions which receives contribution only from BPS states. The spin dependence of BPS states in 5 dimensions is captured by the string coupling constant dependence of topological string amplitudes. 
  We suggest that the correspondence between gauge theories strongly coupled in the infrared and their low energy effective theories may be probed by introducing topologically non-trivial background scalar fields. We argue that one loop expressions for the global charges induced in vacuum by these background fields are in some cases exact in the fundamental theory, and hence should be matched in the effective theory. These matching conditions are sometimes inequivalent to 't Hooft ones. A few examples of induced charge matching are presented. 
  It is shown that a simple model for 4-dimensional quantum gravity based on a 3-dimensional generalization of anyon superconductivity can be regarded as a discrete form of Polyakov's string theory. This suggests that there is a universal negative pressure that is on the order of the string tension divided by the square of the Robertson-Walker scale factor. This is in accord with recent observations of the brightness of distant supernovae, which suggest that at the present time there is a vacuum energy whose magnitude is close to the mass density of an Einstein-de Sitter universe. 
  It is shown that the introduction of an upper limit to the proper acceleration of a particle can smooth the problem of ultraviolet divergencies in local quantum field theory. For this aim, the classical model of a relativistic particle with maximal proper acceleration is quantized canonically by making use of the generalized Hamiltonian formalism developed by Dirac. The equations for the wave function are treated as the dynamical equations for the corresponding quantum field. Using the Green's function connected to these wave equations as propagators in the Feynman integrals leads to an essential improvement of their convergence properties. 
  We consider theories characterized by a set of Ward operators which do not form a closed algebra. We impose the Slavnov--Taylor identity built out of the Ward operators and we derive the acceptable breaking of the algebra and the general form of the classical action. The 1PI generating functional is expressed in terms of the known quantities characterizing the theory and of a nontrivial integrability condition. As a nontrivial application of our formalism, we discuss the N=4 supersymmetric nonlinear sigma model. 
  We calculate by the method of dimensional regularization and derivative expansion the one-loop effective action for a Dirac fermion with a Lorentz-violating and CPT-odd kinetic term in the background of a gauge field. We show that this term induces a Chern-Simons modification to Maxwell theory. Some related issues are also discussed. 
  A short survey of some aspects of harmonic superspace is given. In particular, the $d=3, N=8$ scalar supermultiplet and the $d=6, N=(2,0)$ tensor multiplet are described as analytic superfields in appropriately defined harmonic superspaces. 
  We show that it is possible to define Majorana (s)pinor fields on M-branes which have been identified under the action of the antipodal map on the adS factor of the throat geometry, or which have been wrapped on two-cycles of arbitrary genus. This is an important consistency check, since it means that one may still take the generators of supertranslations in superspace to transform as Majorana fermions under the adjoint action of $Spin(10,1)$, even though the antipodally identified M2-brane is {\it not} space-orientable. We point out that similar conclusions hold for any p-branes which have the generic (adS)$~{\times}~$(Sphere) throat geometry. 
  We show that all supersymmetric Type IIA D-branes can be constructed as bound states of a certain number of unstable non-supersymmetric Type IIA D9-branes. This string-theoretical construction demonstrates that D-brane charges in Type IIA theory on spacetime manifold $X$ are classified by the higher K-theory group $K^{-1}(X)$, as suggested recently by Witten. In particular, the system of $N$ D0-branes can be obtained, for any $N$, in terms of sixteen Type IIA D9-branes. This suggests that the dynamics of Matrix theory is contained in the physics of magnetic vortices on the worldvolume of sixteen unstable D9-branes, described at low energies by a U(16) gauge theory. 
  We construct a local on-shell invariant in D=11 supergravity from the nonlocal four-point tree scattering amplitude.Its existence, together with earlier arguments, implies non-renormalizability of the theory at lowest possible, two loop, level. This invariant, whose leading bosonic terms are exhibited, may also express the leading, "zero-slope", M$-$theory corrections to its D=11 supergravity limit. 
  We settle the ``apparent'' paradox present in thermal QED_3 that the perturbative series is not invariant, as manifested by the temperature dependence of the induced Chern-Simons term, by showing that large (unlike small) transformations and hence their Ward identities, are not perturbative order-preserving. Instead the thermal effective gauge field actions induced by charged fermions in QED_3 can be made invariant under both small and large gauge transformations by suitable regularization of the Dirac operator determinant, at the usual price of parity anomalies. Our result is illustrated by a concrete example. 
  Phase transition of decay rate from quantum tunneling to thermal activity regimes is investigated in (3+1)-dimensional field theories with symmetry breaking term $f\phi$. By applying the two independent criteria for the sharp first-order transition to the same model, the upper and lower bounds of critical value of the symmetry breaking parameter are obtained. Unlike two dimensional case continuum states of the fluctuation operator near sphaleron solution play an important role to determine the type of transition. 
  Based on our work hep-th/9809039, we discuss how U-duality arises as an exact symmetry of M-theory from T-duality and 11D diffeomorphism invariance. A set of Weyl generators are shown to realize the Weyl group of SO(d,d,Z) and E_{d(d)}(Z), while Borel generators extend these finite groups into the full T- and U-duality groups. We discuss how the BPS states fall into various representations, and obtain duality invariant mass formulae, relevant for the computation of exact string amplitudes. The realization of U-duality symmetry in Matrix gauge theory is also considered. 
  The general solution of the antifield-independent Wess-Zumino consistency condition is worked out for models involving exterior form gauge fields of arbitrary degree. We consider both the free theory and theories with Chapline-Manton couplings. Our approach relies on solving the full set of descent equations by starting from the last element down ("bottom"). 
  We study the elliptic fibrations of some Calabi-Yau three-folds, including the $Z_2\times Z_2$ orbifold with $(h_{1,1},h_{2,1})=(27,3)$, which is equivalent to the common framework of realistic free-fermion models, as well as related orbifold models with $(h_{1,1},h_{2,1})=(51,3)$ and (31,7). However, two related puzzles arise when one considers the $(h_{1,1},h_{2,1})=(27,3)$ model as an F-theory compactification to six dimensions. The condition for the vanishing of the gravitational anomaly is not satisfied, suggesting that the F-theory compactification does not make sense, and the elliptic fibration is well defined everywhere except at four singular points in the base. We speculate on the possible existence of N=1 tensor and hypermultiplets at these points which would cancel the gravitational anomaly in this case. 
  We study four-dimensional black hole configurations which result from wrapping M5-branes on a Calabi-Yau manifold, as well as U-dual realizations. Our aim is to understand the microscopic degrees of freedom responsible for the existence of bound states of multiple branes. The details depend on the chosen U-frame; in some cases, they are massless string junctions. We also identify a perturbative description in which these states correspond to twisted strings of intersecting D3-branes at an orbifold singularity. In each case, these are the preponderant states of the spacetime infrared conformal field theory and account for the entropy of the blackhole. 
  Generalizing previous results for orbifolds, in this paper we describe the compactification of Matrix model on an orientifold which is a quotient space as a Yang-Mills theory living on a quantum space. The information of the compactification is encoded in the action of the discrete symmetry group G on Euclidean space and a projective representation U of G. The choice of Hilbert space on which the algebra of U is realized as an operator algebra corresponds to the choice of a physical background for the compactification. All these data are summarized in the spectral triple of the quantum space. 
  We consider brane solutions where the tensor degrees of freedom are excited. Explicit solutions to the full non-linear supergravity equations of motion are given for the M5 and D3 branes, corresponding to finite selfdual tensor or Born-Infeld field strengths. The solutions are BPS-saturated and half-supersymmetric. The resulting metric space-times are analysed. 
  In this lecture we review some non-perturbative results obtained in globally supersymmetric theories and show how they can be obtained in the framework of topological theories. 
  Universal low-energy behaviour ${2 m c}\over{\ln |s-4m^2|}$ of the scattering function of particles of positive mass m near the threshold $s=4m^2$, and ${\pi} \over {\ln |s-4m^2|}$ for the corresponding S-wave phase-shift, is established for weakly coupled field theory models with a positive mass m in space-time dimension 3; c is a numerical constant independent of the model and couplings. This result is a non-perturbative property based on an exact analysis of the scattering function in terms of a two-particle irreducible (or Bethe-Salpeter) structure function. It also appears as generic by the same analysis in the framework of general relativistic quantum field theory. 
  We derive the spherical field formalism for fermions. We find that the spherical field method is free from certain difficulties which complicate lattice calculations, such as fermion doubling, missing axial anomalies, and computational problems regarding internal fermion loops. 
  In this review we present a pedagogical introduction to recent, more mathematical developments in the Skyrme model. Our aim is to render these advances accessible to mainstream nuclear and particle physicists. We start with the static sector and elaborate on geometrical aspects of the definition of the model. Then we review the instanton method which yields an analytical approximation to the minimum energy configuration in any sector of fixed baryon number, as well as an approximation to the surfaces which join together all the low energy critical points. We present some explicit results for B=2. We then describe the work done on the multibaryon minima using rational maps, on the topology of the configuration space and the possible implications of Morse theory. Next we turn to recent work on the dynamics of Skyrmions. We focus exclusively on the low energy interaction, specifically the gradient flow method put forward by Manton. We illustrate the method with some expository toy models. We end this review with a presentation of our own work on the semi-classical quantization of nucleon states and low energy nucleon-nucleon scattering. 
  The direct string computation of anomalous D-brane and orientifold plane couplings is extended to include the curvature of the normal bundle. The normalization of these terms is fixed unambiguously. New, non-anomalous gravitational couplings are found. 
  It is proven that for each given two-field channel - called the ``t-channel''- with (off-shell) ``scattering angle'' $\Theta_t$, the four-point Green's function of any scalar Quantum Fields satisfying the basic principles of locality, spectral condition together with temperateness admits a Laplace-type transform in the corresponding complex angular momentum variable $\lambda_t$, dual to $\Theta_t$. This transform enjoys the following properties: a) it is holomorphic in a half-plane of the form $Re \lambda_t > m$, where m is a certain ``degree of temperateness'' of the fields considered, b) it is in one-to-one (invertible) correspondence with the (off-shell) ``absorptive parts'' in the crossed two-field channels, c) it extrapolates in a canonical way to complex values of the angular momentum the coefficients of the (off-shell) t-channel partial-wave expansion of the Euclidean four-point function of the fields. These properties are established for all space-time dimensions d+1 with $d \ge 2$. 
  We study the superconformally covariant pseudodifferential symbols defined on N=2 super Riemann surfaces. This allows us to construct a primary basis for N=2 super W_KP^(n)-algebras and, by reduction, for N=2 super W_n-algebras. 
  We give, in the framework of the bosonic string theory, simple prescriptions for computing, at tree and one-loop levels, off-shell string amplitudes for open and closed string massless states. In particular we obtain a tree amplitude for three open strings that in the field theory limit coincides with the three-gluon vertex in the usual covariant gauge and two-string one-loop amplitudes satisfying the property of transversality. 
  Using the manifestly spacetime-supersymmetric version of open superstring field theory, we construct the free action for the first massive states of the open superstring compactified to four dimensions. This action is in N=1 D=4 superspace and describes a massive spin-2 multiplet coupled to two massive scalar multiplets. 
  We discuss heterotic corrections to quartic internal U(1) gauge couplings and check duality by calculating one-loop open string diagrams and identifying the D-instanton sum in the dual type I picture. We also compute SO(8)^4 threshold corrections and finally R^2 corrections in type I theory. 
  We expand on an idea of Seiberg that an N=1 supersymmetric gauge theory shows confinement without breaking of chiral symmetry when the gauge symmetry of its magnetic dual is completely broken by the Higgs effect. This has recently been applied to some models involving tensor fields and an appropriate tree-level superpotential. We show how the confining spectrum of a supersymmetric gauge theory can easily be derived when a magnetic dual is known and we determine it explicitly for many models containing fields in second rank tensor representations. We also give the form of the confining superpotential for most of these models. 
  In this work we define a scalar product ``weighted'' with the scalar factor $R$ and show how to find a normalized wave function for the supersymmetric quantum FRW cosmological model using the idea of supersymmetry breaking selection rules under local n=2 conformal supersymmetry. We also calculate the expectation value of the scalar factor R in this model and its corresponding behaviour. 
  Starting from a one-particle quasi-exactly solvable system, which is characterized by an intrinsic sl(2) algebraic structure and the energy-reflection symmetry, we construct a daughter N-body Hamiltonian presenting a deformation of the Calogero model. The features of this Hamiltonian are (i) it reduces to a quadratic combination of the generators of sl(N+1); (ii) the interaction potential contains two-body terms and interaction with the force center at the origin; (iii) for quantized values of a certain cohomology parameter n it is quasi-exactly solvable, the multiplicity of states in the algebraic sector is (N+n)!/(N!n!); (iv) the energy-reflection symmetry of the parent system is preserved. 
  We consider certain four dimensional supersymmetric and non-supersymmetric asymmetric orbifolds with vanishing cosmological constant up to two loops and gauge the world sheet parity transformation. This leads to new string vacua, in which Dp and D(p-4) branes or Dp and anti-D(p-4) are identified. Moreover, it is shown that different degrees of supersymmetry can be realized in the bulk and on the brane. We show that for non-supersymmetric models the cosmological constant still vanishes at one loop order. 
  We construct an explicit supergravity solution for a configuration of localized D4-brane ending on a D6-brane, restricted to the near horizon region of the latter. We generate this solution by dimensionally reducing the supergravity solution for a flat M5-brane in $R^{1,7} \times C^2/Z_N$ with the M5-brane partially embedded in $C^2/Z_N$. We describe the general class of localized intersections and overlaps whose supergravity solutions are constructible in this way. 
  In these lectures we explain the concept of supergravity p-branes and BPS black holes. Introducing an audience of general relativists to all the necessary geometry related with extended supergravity (special geometry, symplectic embeddings and the like) we describe the general properties of N=2 black holes, the structure of central charges in extended supergravity and the description of black hole entropy as an invariant of the U duality group. Then, after explaining the concept and the use of solvable Lie algebras we present the detailed construction of 1/2, 1/4 and 1/8 supersymmetry preserving black holes in the context of N=8 supergravity. The Lectures are meant to be introductory and self contained for non supersymmetry experts but at the same time fully detailed and complete on the subject. 
  We construct an Sp(2,R) gauge invariant particle action which possesses manifest space-time SO(d,2) symmetry, global supersymmetry and kappa supersymmetry. The global and local supersymmetries are non-abelian generalizations of Poincare type supersymmetries and are consistent with the presence of two timelike dimensions. In particular, this action provides a unified and explicit superparticle representation of the superconformal groups OSp(N/4), SU(2,2/N) and OSp(8*/N) which underlie various AdS/CFT dualities in M/string theory. By making diverse Sp(2,R) gauge choices our action reduces to diverse one-time physics systems, one of which is the ordinary (one-time) massless superparticle with superconformal symmetry that we discuss explicitly. We show how to generalize our approach to the case of superalgebras, such as OSp(1/32), which do not have direct space-time interpretations in terms of only zero branes, but may be realizable in the presence of p-branes. 
  We study a topological field theory in four dimensions on a manifold with boundary. A bulk-boundary interaction is introduced through a novel variational principle rather than explicitly. Through this scheme we find that the boundary values of the bulk fields act as external sources for the boundary theory. Furthermore, the full quantum states of the theory factorize into a single bulk state and an infinite number of boundary states labeled by loops on the spatial boundary. In this sense the theory is purely holographic. We show that this theory is dual to Chern-Simons theory with an external source. We also point out that the holographic hypothesis must be supplemented by additional assumptions in order to take into account bulk topological degrees freedom, since these are apriori invisible to local boundary fields. 
  Recently we proposed a TeV-scale Supersymmetric Standard Model in which the gauge coupling unification is as precise (at one loop) as in the MSSM, and occurs in the TeV range. One of the key ingredients of this model is the presence of new states neutral under $SU(3)_c\otimes SU(2)_w$ but charged under $U(1)_Y$ whose mass scale is around that of the electroweak Higgs doublets. In this paper we show that introduction of these states allows to gauge novel anomaly free discrete (as well as continuous) symmetries (similar to ``lepton'' and ``baryon'' numbers) which suppress dangerous higher dimensional operators and stabilize proton. Moreover, we argue that these gauge symmetries are essential for successfully generating small neutrino masses via a recently proposed higher dimensional mechanism. Furthermore, the mass hierarchy between the up and down quarks (e.g., $t$ vs. $b$) can be explained without appealing to large $\tan\beta$, and the $\mu$-term for the electroweak Higgs doublets (as well as for the new states) can be generated. We also discuss various phenomenological implications of our model which lead to predictions testable in the present or near future collider experiments. In particular, we point out that signatures of scenarios with high vs. low unification (string) scale might be rather different. This suggest a possibility that the collider experiments may distinguish between these scenarios even without a direct production of heavy Kaluza-Klein or string states. 
  Using one-loop effective action in large N and s-wave approximation we discuss the possibility to induce primordial wormholes at the early Universe. An analytical solution is found for self-consistent primordial wormhole with constant radius. Numerical study gives the wormhole solution with increasing throat radius and increasing red-shift function. There is also some indication to the possibility of a topological phase transition. 
  Multicenter supergravity solutions corresponding to Higgs phases of supersymmetric Yang-Mills theories are considered. For NS5 branes we identify three cases where there is a description in terms of exact conformal field theories. Other supergravity solutions, such as D3-branes with angular momentum, are understood as special limits of multicenter ones. Within our context we also consider 4-dim gravitational multi-instantons. 
  We study the O(N) vector model and the U(N) Gross-Neveu model with fixed total fermion number, in three dimensions. Using non-trivial polylogarithmic identities, we calculate the large-N renormalized free-energy density of these models, at their conformal points in a ``slab'' geometry with one finite dimension of length L. We comment on the possible implications of our results. 
  We discuss the application of the method of the gaugeless Hamiltonian reduction to general relativity. This method is based on explicit resolving the global part of the energy constraint and on identification of one of the metric components with the evolution parameter of the equivalent unconstrained (reduced) system. The Hamiltonian reduction reveals a possibility to unify General Relativity and Standard Model of strong and electro-weak interactions with the modulus of the Higgs field identified with the product of the determinant of 3D metric and the Planck constant. We give the geometrical foundation of the scalar field, derive and discuss experimental consequences of this unified model: the cosmic Higgs vacuum, the Hoyle-Narlikar cosmology, a $\sigma$-model version of Standard Model without Higgs particle excitations and inflation. 
  We will show that gauge theory can be described by an almost product structure, which is a certain type of endomorphism of the tangent bundle. We will recover the gauge field strength as the Nijenhuis tensor of this endomorphism. We discuss a generalization to the case of a general Kaluza-Klein theory. Furthermore, we will look at the classification of these almost product structures in the case where we have a manifold with metric, and fit the M-brane solutions into this classification scheme. In this analysis certain algebraic properties of the space of differential forms and multivectors are obtained. All analysis is global but we will give local expressions where we find it suitable. 
  The 3-d BTZ black hole represents an orbifold of $AdS_3$ gravity. The UV as well as the IR region of the CFT is governed by a gauged SL(2, R) WZW model. In the UV it corresponds to a light-cone gauging (Liouville model) whereas in the IR it is a space-like gauging (2-d black hole). 
  We review the construction of a manifestly covariant, supersymmetric and SL(2R) invariant action for IIB supergravity in D=10. 
  We study a possible symmetrical behavior of the effective charges defined in the Euclidean and Minkowskian regions and prove that such symmetry is inconsistent with the causality principle. 
  We show that the moduli space of the $(2,0)$ and little-string theories compactified on $T^3$ with R-symmetry twists is equal to the moduli space of U(1) instantons on a non-commutative $T^4$. The moduli space of $U(q)$ instantons on a non-commutative $T^4$ is obtained from little-string theories of NS5-branes at $A_{q-1}$ singularities with twists. A large class of gauge theories with ${\cal N}=4$ SUSY in 2+1D and ${\cal N}=2$ SUSY in 3+1D are limiting cases of these theories. Hence, the moduli spaces of these gauge theories can be read off from the moduli spaces of instantons on non-commutative tori. We study the phase transitions in these theories and the action of T-duality. On the purely mathematical side, we give a prediction for the moduli space of 2 U(1) instantons on a non-commutative $T^4$. 
  The relation between connections on 2-dimensional manifolds and holomorphic bundles provides a new perspective on the role of classical gauge fields in quantum field theory in two, three and four dimensions. In particular we show that there is a close relation between unstable bundles and monopoles, sphalerons and instantons. Some of these classical configurations emerge as nodes of quantum vacuum states in non-confining phases of \qft which suggests a relevant role for those configurations in the mechanism of quark confinement in QCD. 
  We introduce two maximal non-abelian gauge fixing conditions on the space of gauge orbits M for gauge theories over spaces with dimensions d < 3. The gauge fixings are complete in the sense that describe an open dense set M_0 of the space of gauge orbits M and select one and only one gauge field per gauge orbit in M_0. There are not Gribov copies or ambiguities in these gauges. M_0 is a contractible manifold with trivial topology. The set of gauge orbits which are not described by the gauge conditions M \ M_0 is the boundary of M_0 and encodes all non-trivial topological properties of the space of gauge orbits. The gauge fields configurations of this boundary M \ M_0 can be explicitly identified with non-abelian monopoles and they are shown to play a very relevant role in the non-perturbative behaviour of gauge theories in one, two and three space dimensions. It is conjectured that their role is also crucial for quark confinement in 3+1 dimensional gauge theories. 
  In the context of field theory two elements seem to be necessary to search for strong-weak coupling duality. First, a gauge theory formulation and second, supersymmetry. For gravitation these two elements are present in MacDowell-Mansouri supergravity. The search for an "effective duality" in this theory presents technical and conceptual problems that we discuss. Nevertheless, by means of a field theoretical approach, which in the abelian case coincides with $S$-duality, we exhibit a dual theory, with inverted couplings. This results in a supersymmetric non-linear sigma model of the Freedman-Townsend type. 
  The O(n) nonlinear sigma model in 1+1 dimensions is examined as quantum mechanics on an infinite-dimensional configuration space. Two metrics are defined in this space. One of these metrics is the same as Feynman's distance, but we show his conclusions concerning potential energy versus distance from the classical vacuum are incorrect. The potential-energy functional is found to have barriers; the configurations on these barriers are solitons of an associated sigma model with an external source. The tunneling amplitude is computed for the O(2) model and soliton condensation is shown to drive the phase transition at a critical coupling. We find the tunneling paths in the configuration space of the O(3) model and argue that these are responsible for the mass gap at $\theta=0$. These tunneling paths have half-integer topological charge, supporting the conjecture due to Affleck and Haldane that there is a massless phase at $\theta=\pi$. 
  We derive the Schwinger-Dyson/loop equations for the USp(2k) matrix model which close among the closed and open Wilson loop variables. These loop equations exhibit a complete set of the joining and splitting interactions required for the nonorientable Type I superstrings. The open loops realize the SO(2n_f) Chan-Paton factor and their linearized loop equations derive the mixed Dirichlet/Neumann boundary conditions. 
  Phase-space path-integrals are used in order to illustrate various aspects of a recently proposed interpretation of quantum mechanics as a gauge theory of metaplectic spinor fields. 
  The implementation of supersymmetry transformations by Hilbert space operators is discussed in the framework of supersymmetric C$^*$--dynamical systems. It is shown that the only states admitting such an implementation are pure supersymmetric ground states or mixtures and elementary excitations thereof. Faithful states, such as KMS--states, are never supersymmetric. 
  We study properties of a scalar quantum field theory on two-dimensional noncommutative space-times. Contrary to the common belief that noncommutativity of space-time would be a key to remove the ultraviolet divergences, we show that field theories on a noncommutative plane with the most natural Heisenberg-like commutation relations among coordinates or even on a noncommutative quantum plane with $E_q(2)$-symmetry have ultraviolet divergences, while the theory on a noncommutative cylinder is ultraviolet finite. Thus, ultraviolet behaviour of a field theory on noncommutative spaces is sensitive to the topology of the space-time, namely to its compactness. We present general arguments for the case of higher space-time dimensions and as well discuss the symmetry transformations of physical states on noncommutative space-times. 
  The suggested operator manifold formalism enables to develop an approach to the unification of the geometry and the field theory. We also elaborate the formalism of operator multimanifold yielding the multiworld geometry involving the spacetime continuum and internal worlds, where the subquarks are defined implying the Confinement and Gauge principles. This formalism in Part II  (hep-th/9812182) is used to develop further the microscopic approach to some key problems of particle physics. 
  Within the operator manifold approach (part I, hep-th/9812181) we derive the Gell-Mann-Nishijima relation and flavour group, whereas the leptons are particles with integer electric and leptonic charges and free of confinement, while quarks carry fractional electric and baryonic charges and imply the confinement. We consider the unified electroweak interactions with small number of free parameters, exploit the background of the local expanded symmetry $SU(2)\otimes U(1)$ and P-violation. The Weinberg mixing angle is shown to have fixed value at $30^{o}$. The Higgs bosons arise on an analogy of the Cooper pairs in superconductivity. Within the present microscopic approach we predict the Kobayashi-Maskawa quark flavour mixing; the appearance of the CP-violation phase; derive the mass-spectrum of leptons and quarks, as well as other emerging particles, and also some useful relations between their masses. 
  The equation for QCD string proposed earlier is reviewed. This equation appears when we examine the gonihedric string model and the corresponding transfer matrix. Arguing that string equation should have a generalized Dirac form we found the corresponding infinite-dimensional gamma matrices as a symmetric solution of the Majorana commutation relations. The generalized gamma matrices are anticommuting and guarantee unitarity of the theory at all orders of $v/c$. In the second quantized form the equation does not have unwanted ghost states in Fock space. In the absence of Casimir mass terms the spectrum reminds hydrogen exitations. On every mass level $r=2,4,..$ there are different charged particles with spin running from $j=1/2$ up to $j_{max}=r-1/2$, and the degeneracy is equal to $d_{r}=2r-1 = 2j_{max}$. This is in contrast with the exponential degeneracy in superstring theory. 
  We construct a matrix representation of compact membranes analytically embedded in complex tori. Brane configurations give rise, via Bergman quantization, to U(N) gauge fields on the dual torus, with almost-anti-self-dual field strength. The corresponding U(N) principal bundles are shown to be non-trivial, with vanishing instanton number and first Chern class corresponding to the homology class of the membrane embedded in the original torus. In the course of the investigation, we show that the proposed quantization scheme naturally provides an associative star-product over the space of functions on the surface, for which we give an explicit and coordinate-invariant expression. This product can, in turn, be used the quantize, in the sense of deformation quantization, any symplectic manifold of dimension two. 
  The consistency condition of the Faddeev-Niemi ansatz for the gauge-fixed massless SU(2) gauge field is discussed. The generality of the ansatz is demonstrated by obtaining a sufficient condition for the existence of the three-component field introduced by Faddeev and Niemi. It is also shown that the consistency conditions determine this three-component field as a functional of two arbitrary functions. The consistency conditions corresponding to the Periwal ansatz for the SU(N) gauge field with N larger than 2 are also obtained. It is shown that the gauge field obeying the Periwal ansatz must satisfy extra (N-1)(N-2)/2 conditions. 
  We show an algebra morphism between Yangians and some finite W-algebras. This correspondence is nicely illustrated in the framework of the Non Linear Schrodinger hierarchy. For such a purpose, we give an explicit realization of the Yangian generators in terms of deformed oscillators. 
  We summarise what lattice simulations have to say about the physical properties of continuum SU(N) gauge theories in 3+1 dimensions. The quantities covered are: the glueball mass spectrum, the confining string tension, the temperature at which the theory becomes deconfined, the topological susceptibility, the value of the scale Lambda{MS-bar} that governs the rate at which the coupling runs and the r0 parameter that characterises the static quark potential at intermediate distances. 
  We construct the effective action of the KK-monopole in a massive Type IIA background. We follow two approaches. First we construct a massive M-theory KK-monopole from which the IIA monopole is obtained by double dimensional reduction. This eleven dimensional monopole contains two isometries: one under translations of the Taub-NUT coordinate and the other under massive transformations of the embedding coordinates. Secondly, we construct the massive T-duality rules that map the Type IIB NS-5-brane onto the massive Type IIA KK-monopole. This provides a check of the action constructed from eleven dimensions. 
  Several quantum mechanical wave equations for $p$-branes are proposed based on the role that the volume-preserving diffeomorphisms group has on the physics of extended objects. The $p$-brane quantum mechanical wave equations determine the quantum dynamics involving the creation/destruction of $p$-branes in a $D$ dimensional spacetime background with a given world-volume measure configuration in a given quantum state $\Psi$. 
  We review the foundations as well as a number of important applications of light-cone dynamics. Topics covered are: relativistic particle dynamics, reparametrization invariance, Dirac's front form, light-cone quantization of fields via Schwinger's quantum action principle or Faddeev-Jackiw reduction, light-cone vacuum and wave functions, zero modes and spontaneous symmetry breaking, `t Hooft and Schwinger model, and the light-cone wave function of the pion. 
  Monopole ``superpartner'' solutions are constructed by acting with finite, broken supersymmetry transformations on a bosonic N=2 BPS monopole. The terms beyond first order in this construction represent the backreaction of the the fermionic zero-mode state on the other fields. Because of the quantum nature of the fermionic zero-modes, the superpartner solution is necessarily operator valued. We extract the electric dipole moment operator and show that it is proportional to the fermion zero-mode angular momentum operator with a gyroelectric ratio g=2. The magnetic quadrupole operator is shown to vanish identically on all states. We comment on the usefulness of the monopole superpartner solution for a study of the long-range spin dependent dynamics of BPS monopoles. 
  The principal admissible representations of affine Kac-Moody algebras are studied, with a view to their use in conformal field theory. We discuss the generation of the set of principal admissible highest weights, concentrating mainly on $A_r^{(1)}$ at rational level $k$. A related algorithm is described that produces the Malikov-Feigen-Fuchs null vectors of these representations. With the principal admissible description of the highest weights, we are able to prove that field identifications (including maverick ones) lead to the canonical description of the primary fields of the nonunitary diagonal coset theories. 
  It is stated in the literature that D-branes in the WZW-model associated with the gluing condition J = - \bar{J} along the boundary correspond to branes filling out the whole group volume. We show instead that the end-points of open strings are rather bound to stay on `integer' conjugacy classes. In the case of SU(2) level k WZW model we obtain k-1 two dimensional Euclidean D-branes and two D particles sitting at the points e and -e. 
  We give a group-theoretic interpretation of the AdS/CFT correspondence as relation of representation equivalence between representations of the conformal group describing the bulk AdS fields $\phi$ and the coupled boundary fields $\phi_0$ and ${\cal O}$. We use two kinds of equivalences. The first kind is equivalence between bulk fields and boundary fields and is established here. The second kind is the equivalence between coupled boundary fields. Operators realizing the first kind of equivalence for special cases were given by Witten and others - here they are constructed in a more general setting from the requirement that they are intertwining operators. The intertwining operators realizing the second kind of equivalence are provided by the standard conformal two-point functions. Using both equivalences we find that the bulk field has in fact two boundary fields, namely, the coupled boundary fields. Thus, from the viewpoint of the bulk-boundary correspondence the coupled fields are on an equal footing. Our setting is more general since our bulk fields are described by representations of the Euclidean conformal group $G=SO(d+1,1)$, induced from representations $\tau$ of the maximal compact subgroup $SO(d+1)$ of $G$. From these large reducible representations we can single out representations which are equivalent to conformal boundary representations labelled by the conformal weight and by arbitrary representations $\mu$ of the Euclidean Lorentz group $M=SO(d)$, such that $\mu$ is contained in the restriction of $\tau$ to $M$. Thus, our boundary-to-bulk operators can be compared with those in the literature only when for a fixed $\mu$ we consider a 'minimal' representation $\tau=\tau(\mu)$ containing $\mu$. 
  Theories in more than ten dimensions play an important role in understanding nonperturbative aspects of string theory. Consistent compactifications of such theories can be constructed via Calabi-Yau fourfolds. These models can be analyzed particularly efficiently in the Landau-Ginzburg phase of the linear sigma model, when available. In the present paper we focus on those sigma models which have both a Landau-Ginzburg phase and a geometric phase described by hypersurfaces in weighted projective five-space. We describe some of the pertinent properties of these models, such as the cohomology, the connectivity of the resulting moduli space, and mirror symmetry among the 1,100,055 configurations which we have constructed. 
  This is a short arrangement of notes on D-branes, offered as an embellishment of five lectures which were presented at the 1998 Trieste Spring School entitled ``Non-Perturbative Aspects of String Theory and Supersymmetric Gauge Theory''. There is a good number of collections of pedagogical notes on D-branes in the literature, and since space here is limited, no attempt will be made to cover all of the introductory material again. Instead, the notes cover selected topics and themes in string theory and M-theory, emphasizing how certain technical aspects of D-branes play a role. The subject is developed mainly from the perspective of non-perturbative string theory, touching on aspects of the old matrix model, string duality, the new matrix model, the AdS/CFT correspondence and other gauge theory/geometry correspondences. 
  Using strong-coupling quantum field theory we calculate highly accurate critical exponents nu, eta from new seven-loop expansions in three dimensions. Our theoretical value for the critical exponent alpha of the specific heat near the lambda-point of superfluid helium is alpha =-0.01294+-0.00060, in excellent agreement with the space shuttle experimental value alpha =-0.01285+-0.00038. 
  A generic ansatz is introduced which provides families of exact solutions to the equations of motion and constraints for null-strings in Bianchi type I cosmological models. This is achieved irrespective of the form of the metric. Within classes of dilaton cosmologies a backreaction mapping relation is established where the null string leads to more or less anisotropic members of the family. The equations of motion and constraints for the generic model are casted in their first order form and integrated both analytically and numerically. 
  We consider dynamics of massless particle in 2d spacetimes with constant curvature. We analyze different examples of spacetime. Dynamical integrals are constructed from spacetime symmetry related to $sl(2.{\bf R})$ algebra. Mass-shell condition restricts dynamical integrals to a cone (without vertex) which defines physical-phase space. We parametrize the cone by canonical coordinates. Canonical quantization with definite choice of operator ordering leads to unitary irreducible representations of $SO_\uparrow (2.1)$ group. 
  The description of D-branes as boundary states for type II string theories (in the covariant formulation) requires particular care in the R-R sector. Also the vertices for R-R potentials that can couple to D-branes need a careful handling. As an illustration of this, the example of the D0-D8 system is reviewed, where a ``microscopic'' description of the interaction via exchange of R-R potentials becomes possible. 
  This is the written version of a short talk given at the University of Leipzig in December 1998. It reviews some general aspects of string theory from the viewpoint of the search for an unifying theory. Here, special emphasis lies on the motivation to consider string theory not only as the leading candidate for the unification of gravity and the other fundamental forces of nature, but also as a possible step towards a new understanding of nature and its description within the framework of physical models. Without going into details, some recent developments, including duality symmetries and the appearance of $M$--theory, are reviewed. 
  The effective actions of a scalar and massless spin-half field are determined as functions of the deformation of a symmetrically squashed three-sphere. The extreme oblate case is particularly examined as pertinant to a high temperature statistical mechanical interpretation that may be relevant for the holographic principle. Interpreting the squashing parameter as a temperature, we find that the effective `free energies' on the three-sphere are mixtures of thermal two-sphere scalars and spinors which, in the case of the spinor on the three-sphere, have the `wrong' thermal periodicities. However the free energies do have the same leading high temperature forms as the standard free energies on the two-sphere. The next few terms in the high-temperature expansion are also explicitly calculated and briefly compared with the Taub-Bolt-AdS bulk result. 
  This is an account of the author's recollections of the turbulent days preceding the establishment of the Standard Model as an accurate description of all known elementary particles and forces.  
  Absolute confinement of its color charges is a natural property of gauge theories such as quantum chromodynamics. On the one hand, it can be attributed to the existence of color-magnetic monopoles, a topological feature of the theory, but one can also maintain that all non-Abelian gauge theories confine. It is illustrated how ``confinement'' works in the SU(2) sector of the Standard Model, and why for example the electron and its neutrino can be viewed as SU(2)-hadronic bound states rather than a gauge doublet.   The mechanism called `Abelian projection' then puts the Abelian sector of any gauge theory on a separate footing. 
  We geometrically engineer d=4 N=1 supersymmetric Yang-Mills theories by considering M theory on various Joyce orbifolds. We argue that the superpotential of these models is generated by fractional membrane instantons. The relation of this superpotential to membrane anomalies is also discussed. 
  We show how a theorem of Sullivan provides a precise mathematical statement of a 3d holographic principle, that is, the hyperbolic structure of a certain class of 3d manifolds is completely determined in terms of the corresponding Teichmuller space of the boundary. We explore the consequences of this theorem in the context of the Euclidean BTZ black hole in three dimensions. 
  We present a convenient method for deriving the transformation of the dilaton under T-duality in the path-integral approach. Subtleties arising in performing the integral over the gauge fields are carefully analysed using Pauli-Villars regularization, thereby clarifying existing ambiguities in the literature. The formalism can not only be applied to the abelian case, but, and this for the first time, to the non-abelian case as well. Furthermore, by choosing a particular gauge, we directly obtain the target-space covariant expression for the dual geometry in the abelian case. Finally it is shown that the conditions for gauging non-abelian isometries are weaker than those generally found in the literature. 
  A simplified version of Higher Covariant Derivative regularization for Yang-Mills theory is constructed. This may make Higher Covariant Derivative method more attractive for practical calculations. 
  In a previous paper we explored how conjugacy classes of the modular group classify the symmetry algebras that arise on type IIB [p,q] 7-branes. The Kodaira list of finite Lie algebras completely fills the elliptic classes as well as some parabolic classes. Loop algebras of E_N fill additional parabolic classes, and exotic finite algebras, hyperbolic extensions of E_N and more general indefinite Lie algebras fill the hyperbolic classes. Since they correspond to brane configurations that cannot be made into strict singularities, these non-Kodaira algebras are spectrum generating and organize towers of massive BPS states into representations. The smallest brane configuration with unit monodromy gives rise to the loop algebra \hat{E}_9 which plays a central role in the theory. We elucidate the patterns of enhancement relating E_8, E_9, \hat{E}_9 and E_10. We examine configurations of 24 7-branes relevant to type IIB compactifications on a two-sphere, or F-theory on K3. A particularly symmetric configuration separates the 7-branes into two groups of twelve branes and the massive BPS spectrum is organized by E_10 + E_10. 
  The SO(32) heterotic string can be obtained from the type IIB string by gauging a discrete symmetry that acts as $(-1)^{F_L}$ on the perturbative string states and reverses the parity of the D-string. Consistency requires the presence of 32 NS 9-branes -- the S-duals of D9-branes -- which give SO(32) Chan-Paton factors to open D-strings. At finite string coupling, there are SO(32) charges tethered to the heterotic string world-sheet by open D-strings. At zero-coupling, the D-string tension becomes infinite and the SO(32) charges are pulled onto the world-sheet, and give the usual SO(32) world-sheet currents of the heterotic string. 
  The spectral determinant $D(E)$ of the quartic oscillator is known to satisfy a functional equation. This is mapped onto the $A_3$-related $Y$-system emerging in the treatment of a certain perturbed conformal field theory, allowing us to give an alternative integral expression for $D(E)$. Generalising this result, we conjecture a relationship between the $x^{2M}$ anharmonic oscillators and the $A_{2M-1}$ TBA systems. Finally, spectral determinants for general $|x|^{\alpha}$ potentials are mapped onto the solutions of nonlinear integral equations associated with the (twisted) XXZ and sine-Gordon models. 
  Beginning with the Legendre transform construction of hyperk\"ahler metrics, we analyze the ALF version of the D_n metrics. We determine the constraint equation obtained from extremizing the $w$ coordinate of the generating function F(z,\bar{z},u,\bar{u},w) and study its behavior as we send two of the mass parameters of the $D_n$ metric to zero. We find that the constraint equation enforces the limit that the metric becomes that of multi-Taub-NUT. 
  We further develop the new approach, proposed in part I (hep-th/9807072), to computing the heat kernel associated with a Fermion coupled to vector and axial vector fields. We first use the path integral representation obtained for the heat kernel trace in a vector-axialvector background to derive a Bern-Kosower type master formula for the one-loop amplitude with $M$ vectors and $N$ axialvectors, valid in any even spacetime dimension. For the massless case we then generalize this approach to the full off-diagonal heat kernel. In the D=4 case the SO(4) structure of the theory can be broken down to $SU(2) \times SU(2)$ by use of the 't Hooft symbols. Various techniques for explicitly evaluating the spin part of the path integral are developed and compared. We also extend the method to external fermions, and to the inclusion of isospin. On the field theory side, we obtain an extension of the second order formalism for fermion QED to an abelian vector-axialvector theory. 
  The gauge equivalent counterparts of the some (1+1)-, or (2+0)-dimensional sigma models with potentials are found. The gauge equivalence between the some soliton equations of spin-phonon systems and the Yajima-Oikawa and Ma equations are also established. 
  The soliton solution of the integrable coupled nonlinear Schrodinger equation (NLS) of Manakov type is investigated by using Zakharov-Shabat (ZS) scheme. We get the bright N-solitons solution by solving the integrable uncoupled NLS of Manakov type. We also find that there is an elastic collision of the bright N-solitons. 
  Light-cone gauge quantization procedures are given, for superstring theory on $AdS_3$ space charged with NS-NS background, both in NSR and GS formalism. The spacetime (super)conformal algebras are constructed in terms of the transversal physical degrees of freedom. The spacetime conformal anomaly agrees with that of covariant formalism, provided that the worldsheet conformal anomaly $c$ equals 26 or 15 for bosonic string or superstring, respectively. The spacetime (super)conformal field theory is found to correspond to orbifold construction on symmetric product space $\it{Sym_p} {\cal{M}}/Z_p$. 
  The canonical quantization on a single light front is performed for the Abelian gauge fields with the Weyl gauge coupled with fermion field currents. The analysis is carried separately for 1+1 dimensions and for higher dimensions. The Gauss law, implemented weakly as the condition on states, selects physical subspace with the Poincare covariance recovered. The perturbative gauge field propagators are found with the ML prescription for their spurious poles. The LF Feynman rules are found and their equivalence with the usual equal-time perturbation for the S-matrix elements is studied for all orders. 
  We study the large N limit of matrix models of M5-branes, or (2,0) six-dimensional superconformal field theories, by making use of the Bulk/Boundary correspondence. Our emphasis is on the relation between the near-horizon limit of branes and the light-like limit of M-theory. In particular we discuss a conformal symmetry in the D0 + D4 system, and interpret it as a conformal symmetry in the discrete light-cone formulation of M5-branes. We also compute two-point functions of scalars by applying the conjecture for the AdS/CFT correspondence to the near-horizon geometry of boosted M5-branes. We find an expected result up to a point subtle, but irrelevant to the IR behavior of the theory. Our analysis matches with the Seiberg and Sen's argument of a justification for the matrix model of M-theory. 
  In this paper we consider the quantization of open strings ending on D-branes with a background B field. We find that spacetime coordinates of the open string end-points become noncommutative, and correspondingly the D-brane worldvolume also becomes noncommutative. This provides a string theory derivation and generalization of the noncommutativity obtained previously in the Matrix model compactification. For Dp-branes with p>=2 our results are new and agree with that of Matrix theory for the case of A=0 (where $A$ is the worldvolume gauge field) if the T-duality radii are used. 
  We present a construction of the superspace of maximally supersymmetric adS_{p+2} x S^{d-p-2} near-horizon geometry based entirely on the supergravity constraints of which the bosonic space is a solution. Besides the geometric superfields, i.e. the vielbeine and the spinconnection, we also derive the isometries of the superspace together with the compensating tangent space transformations to all orders in anticommuting superspace coordinates. 
  Exact constant solutions of field equations in the classical nonabelian SU(2) field theory with the Chern-Simons topological mass in 2+1-dimensional space-time have been obtained. One-loop contributions of boson and fermion fluctuations to the gauge field energy have been calculated. 
  In this work we apply the Lie group representation method introduced in the real time formalism for finite-temperature quantum-field theory, thermofield dynamics, to derive a spinorial density matrix equation. Symmetry properties of such equation are analysed, and as a basic result it is shown that one solution is the generalised density matrix operator proposed by Heinz, to deal with gauge covariant kinetic equations. In the same context, preliminary aspects of a Lagrangian formalism to derive kinetic equations, as well as quantum density matrix equations in curved space-time, are discussed. 
  A vector space G is introduced such that the Galilei transformations are considered linear mappings in this manifold. The covariant structure of the Galilei Group (Y. Takahashi, Fortschr. Phys. 36 (1988) 63; 36 (1988) 83) is derived and the tensor analysis is developed. It is shown that the Euclidean space is embedded the (4,1) de Sitter space through in G. This is an interesting and useful aspect, in particular, for the analysis carried out for the Lie algebra of the generators of linear transformations in G. 
  We discuss branes whose worldvolume dimension equals the target spacetime dimension, i.e. ``spacetime-filling branes''. In addition to the D9-branes, there are 9-branes in the NS-NS sectors of both the IIA and IIB strings. The worldvolume actions of these branes are constructed, via duality, from the known actions of branes with codimension larger than zero. Each of these types of branes is used in the construction of a string theory with sixteen supercharges by modding out a type II string by an appropriate discrete symmetry and adding 32 9-branes. These constructions are related by a web of dualities and each arises as a different limit of the Horava-Witten construction. 
  An effective action for the M9-brane is proposed. We study its relation with other branes via dualities. Among these, we find actions for branes which are not suggested by the central charges of the Type II superalgebras. 
  The study of brane-antibrane configurations in string theory leads to the understanding of supersymmetric D$p$-branes as the bound states of higher dimensional branes. Configurations of pairs brane-antibrane do admit in a natural way their description in terms of K-theory. We analyze configurations of brane-antibrane at fixed point orbifold singularities in terms of equivariant K-theory as recently suggested by Witten. Type I and IIB fivebranes and small instantons on ALE singularities are described in K-theoretic terms and their relation to Kronheimer-Nakajima construction of instantons is also provided. Finally the D-brane charge formula is reexamined in this context. 
  We study the BPS spectrum of four dimensional N=2 SU(2) theory with massive fundamental matters using the D3-brane probe. Since the BPS states are realized by string webs subject to the BPS conditions, we determine explicitly the configurations of such webs. It is observed that there appear BPS string webs with multiple of junctions corresponding to the fact that the curves of marginal stability in massive theory are infinitely nested. In terms of the string configurations, various properties of the curves of marginal stability are explained intuitively. 
  Different models with nonabelian homogeneous condensate fields are considered in the one-loop approximation. Effective action in a model of gluodynamics in curved space is calculated. Free energy and its minimum in a (2+1)-dimensional model of QCD are investigated. Photon polarization operator (PO) is obtained. 
  Based on the world-line formalism with a sewing method, we derive the Yang-Mills effective action in a form useful to generate the Bern-Kosower-type master formulae for gluon scattering amplitudes at the two-loop level. It is shown that four-gluon ($\Phi^4$ type sewing) contributions can be encapsulated in the action with three-gluon ($\Phi^3$ type) vertices only, the total action thus becoming a simple expression. We then derive a general formula for a two-loop Euler-Heisenberg type action in a pseudo-abelian $su(2)$ background. The ghost loop and fermion loop cases are also studied. 
  We investigate the dynamics of an arbitrary Dirichlet (D-) string in presence of general curved backgrounds following a path-integral formalism. In particular, we consider the interaction of D-string with the massless excitations of closed string in open bosonic string theory. The background fields induce invariant curvatures on the D-string manifold and the extrinsic curvature can be seen to contain a divergence at the disk boundary. The re-normalization of D-string coordinates, next to the leading order in its derivative expansion, is performed to handle the divergence. Then we obtain the generalized Dirac-Born-Infeld action representing the effective dynamics of D-string in presence of the non-trivial backgrounds. On the other hand, D-string acts as a source for the Ramond-Ramond two-form which induces an additional (lower) form due to its coupling to the U(1) gauge invariant fields on the D-string. These forms are reviewed in this formalism for an arbitrary D-string and is encoded in the Wess-Zumino action. Quantization of the D-string collective coordinates, in the U(1) gauge sector, is performed by taking into account the coupling to the lower form and the relevant features of D-string are analyzed in presence of the background fields. 
  It is shown that Connes' generalized gauge field in non-commutative geometry is derived by simply requiring that Dirac lagrangian be invariant under local transformations of the unitary elements of the algebra, which define the gauge group. The spontaneous breakdown of the gauge symmetry is guaranteed provided the chiral fermions exist in more than one generations as first observed by Connes-Lott. It is also pointed out that the most general gauge invariant lagrangian in the bosonic sector has two more parameters than in the original Connes-Lott scheme. 
  Universal Lax pairs of the root type with spectral parameter and independent coupling constants for twisted non-simply laced Calogero-Moser models are constructed. Together with the Lax pairs for the simply laced models and untwisted non-simply laced models presented in two previous papers, this completes the derivation of universal Lax pairs for all of the Calogero-Moser models based on root systems. As for the twisted models based on B_n, C_n and BC_nroot systems, a new type of potential term with independent coupling constants can be added without destroying integrability. They are called extended twisted models. All of the Lax pairs for the twisted models presented here are new, except for the one for the F_4 model based on the short roots. The Lax pairs for the twisted G_2 model have some novel features. Derivation of various functions, twisted and untwisted, appearing in the Lax pairs for elliptic potentials with the spectral parameter is provided. The origin of the spectral parameter is also naturally explained. The Lax pairs with spectral parameter, twisted and untwisted, for the hyperbolic, the trigonometric and the rational potential models are obtained as degenerate limits of those for the elliptic potential models. 
  When an electrically charged source is capable of both emitting the electromagnetic waves and creating charged particles from the vacuum, its radiation gets so much amplified that only the backreaction of the vacuum makes it finite. The released energy and charge are calculated in the high-frequency approximation. The technique of expectation values is advanced and employed. 
  This paper introduces the idea of pseudo-group. Applications of pseudo-groups in Group Theory and Symmetry Breaking in Particle Physics and Cosmology are considered. 
  The structure of heterotic string target space compactifications is studied using the formalism of the noncommutative geometry associated with lattice vertex operator algebras. The spectral triples of the noncommutative spacetimes are constructed and used to show that the intrinsic gauge field degrees of freedom disappear in the low-energy sectors of these spacetimes. The quantum geometry is thereby determined in much the same way as for ordinary superstring target spaces. In this setting, non-abelian gauge theories on the classical spacetimes arise from the K-theory of the effective target spaces. 
  In a space of arbitrary dimensions, the effect of an external magnetic field on the vacuum of a quantized charged scalar field is studied for the field configuration in the form of a singular vortex. The zeta-function technique is used to regularize ultraviolet divergences. The expression for the effective action is derived. It is shown that the energy density and current induced in the vacuum decrease exponentially at large distances from the vortex. The analytic properties of vacuum features as functions of the complex-valued space dimension are discussed. 
  We observe that the existence of black holes limits the extent to which M Theory (or indeed any quantum theory of gravity) can be described by conventional quantum mechanics. Although there is no contradiction with the fundamental properties of quantum mechanics, one can prove that expectation values of Heisenberg operators at fixed times cannot exist in an ordinary asymptotic Lorentz frame. Only operators whose matrix elements between the vacuum and energy eigenstates with energy greater than the Planck scale are artificially cut off, can have conventional Green's functions. This implies a Planck scale cutoff on the possible localization of measurements in time. A similar behavior arises also in ``little string theories''. We argue that conventional quantum mechanics in light cone time is compatible with the properties of black holes if there are more than four non-compact flat dimensions, and also with the properties of ``little string theories''. We contrast these observations with what is known about M Theory in asymptotically Anti-de Sitter spacetimes. 
  We consider a massive Rarita-Schwinger field on the Anti-de Sitter space and solve the corresponding equations of motion. We show that appropriate boundary terms calculated on-shell give two-point correlation functions for spin-3/2 fields of the conformal field theory on the boundary. The relation between Rarita-Schwinger field masses and conformal dimensions of corresponding operators is established. 
  The leading terms in the long-range interaction potential between an arbitrary pair of matrix theory objects are calculated at one-loop order. This result generalizes previous calculations by including arbitrary fermionic background field configurations. The interaction potential at orders 1/r^7 and 1/r^8 is shown to correspond precisely with the leading terms expected from linearized supergravity interactions between arbitrary objects in M-theory. General expressions for the stress tensor, membrane current and 5-brane current of an arbitrary matrix configuration are derived, including fermionic contributions. Supergravity effects which are correctly reproduced include membrane/5-brane interactions, 0-brane/6-brane interactions, supercurrent/supercurrent interactions and the spin contributions to moments of the supergravity currents. The matrix theory description of the supergravity stress tensor, membrane current and 5-brane current are used to propose an explicit formulation of matrix theory in an arbitrary background metric and 3-form field. 
  According to one of Maldacena's dualities, type IIB string theory on AdS_3 X S^3 X K3 is equivalent to a certain N=(4,4) superconformal field theory. In this note we compute the elliptic genus of the boundary theory in the supergravity approximation. A finite quantity is obtained once we introduce a particular exclusion principle. In the regime where the supergravity approximation is reliable, we find exact agreement with the elliptic genus of a sigma model with target space K3^N/S_N. 
  A product of two Riemann surfaces of genuses p_1 and p_2 solves the Seiberg-Witten monopole equations for a constant Weyl spinor that represents a monopole condensate. Self-dual electromagnetic fields require p_1=p_2=p and provide a solution of the euclidean Einstein-Maxwell-Dirac equations with p-1 magnetic vortices in one surface and the same number of electric vortices in the other. The monopole condensate plays the role of cosmological constant. The virtual dimension of the moduli space is zero, showing that for given p_1 and p_2, the solutions are unique. 
  This paper is a letter-type version of hep-th/9806236. We discuss properties of non-linear equations of motion which describe higher-spin gauge interactions for massive spin-0 and spin-1/2 matter fields in 2+1 dimensional anti-de Sitter space. The model is shown to have N=2 supersymmetry and to describe higher-spin interactions of d3 N=2 massive hypermultiplets. An integrating flow is found which reduces the full non-linear system to the free field equations via a non-local B\"acklund-Nicolai-type mapping. 
  We propose to study the infrared behaviour of polymerised (or tethered) random manifolds of dimension D interacting via an exclusion condition with a fixed impurity in d-dimensional Euclidean space in which the manifold is embedded. We prove rigorously, via methods of Wilson's renormalization group, the convergence to a non Gaussian fixed point for suitably chosen physical parameters. 
  We discuss alternative descriptions of four-dimensional self-dual Yang-Mills fields in harmonic space with additional commuting spinor coordinates. In particular, the linear analyticity equation and nonlinear covariant harmonic-field equations are studied. A covariant harmonic field can be treated as an infinite set of ordinary four-dimensional fields with higher spins. We analyze different constructions of invariant harmonic-field actions corresponding to the self-dual harmonic equations. 
  The calculation of the interaction energy in pure QED and Maxwell-Chern-Simons gauge theory is re-examined by exploiting the path dependence of the gauge-invariant variables formalism. In particular, we consider a spacelike straight line which leads to the Poincar\'{e} gauge. Subtleties related to the problem of exhibiting explicitly the interaction energies are illustrated. 
  This paper has been withdrawn by the author due to inconsistency of the considered working hypothesis. The consistent treatment is presented in the last publications of the author. 
  Relation between the vacuum eigenvalues of CFT Q-operators and spectral determinants of one-dimensional Schroedinger operator with homogeneous potential, recently conjectured by Dorey and Tateo for special value of Virasoro vacuum parameter p, is proven to hold, with suitable modification of the Schroedinger operator, for all values of p. 
  A general condition for sharp transition of decay rate from quantum to thermal regimes is derived in dissipative tunneling models when position dependent mass is involved. It is shown that the effect of dissipation in general changes the order of the phase transition. Especially, for the models with constant mass the Ohmic dissipation enlarges the range of parameters for first-order phase transitions. In the case of second-order phase transition the Ohmic dissipation suppresses the decay rate near the transition temperature(T_c). For the super-Ohmic case the dissipation yields an opposite effects to the Ohmic dissipation within exponential approximation. 
  We consider D-dimensional supersymmetric gauge theories with 8 supercharges (D<6,$~\cN=8$) in the framework of harmonic superspaces. The effective Abelian low-energy action for D=5 contains the free and Chern-Simons terms. Effective $\cN=8$ superfield actions for D<4 can be written in terms of the superpotentials satisfying the superfield constraints and (6-D)-dimensional Laplace equations. The role of alternative harmonic structures is discussed. 
  We suggest that RG flows in the N=2 SUSY YM theories are governed by the pair of the integrable systems. The main dynamical ingredient amounts from the interaction of the small size instantons with the regulator degrees of freedom. The relation with the bulk/boundary correspondence is discussed. 
  We extend the previously introduced constructive modular method to nonperturbative QFT. In particular the relevance of the concept of ``quantum localization'' (via intersection of algebras) versus classical locality (via support properties of test functions) is explained in detail, the wedge algebras are constructed rigorously and the formal aspects of double cone algebras for d=1+1 factorizing theories are determined. The well-known on-shell crossing symmetry of the S-Matrix and of formfactors (cyclicity relation) in such theories is intimately related to the KMS properties of new quantum-local PFG (one-particle polarization-free generators) of these wedge algebras. These generators are ``on-shell'' and their Fourier transforms turn out to fulfill the Zamolodchikov-Faddeev algebra. As the wedge algebras contain the crossing symmetry informations, the double cone algebras reveal the particle content of fields. Modular theory associates with this double cone algebra two very useful chiral conformal quantum field theories which are the algebraic versions of the light ray algebras. 
  A brief review of string theory on group manifolds is given, and comparisons are then drawn between Minkowski space, SU(2), and SU(1,1) = AdS_3. The proof of the no-ghost theorem is outlined, assuming a certain restriction on the representation content for bosonic and fermionic strings on SU(1,1). Some possible connections with the AdS/CFT correspondence are mentioned. (Based on invited talk by JME at Trieste Conference on Super 5-branes and Physics in (5+1)-dimensions.) 
  We explore a relation between four-dimensional N=2 heterotic vacua induced by Mirror Symmetry via Heterotic/Type II duality. It allows us to compute the \alpha' corrections to the hypermultiplet moduli space of heterotic compactifications on K3xT^2 in the limit of large base of the elliptic K3. We concentrate on the case of point-like instantons on orbifold singularities leading to low-dimensional hypermultiplet moduli spaces. 
  The Nelson stochastic mechanics of inhomogeneous quantum diffusion in flat spacetime with a tensor of diffusion can be described as a homogeneous one in a Riemannian manifold where this tensor of diffusion plays the role of a metric tensor. It is shown that the such diffusion accelerates both a sample particle and a local frame such that their mean accelerations do not depend on their masses. This fact, explaining the principle of equivalence, allows one to represent the curvature and gravitation as consequences of the quantum fluctuations. In this diffusional treatment of gravitation it can be naturally explained the fact that the energy density of the instantaneous Newtonian interaction is negative defined. 
  We explicitly give the correspondence between spectra of heterotic string theory compactified on $T^2$ and string junctions in type IIB theory compactified on $S^2$. 
  We show that the minimally coupled massless scalar wave equation in the background of an six-dimensional extremal dyonic string (or D1-D5 brane intersection) is exactly solvable, in terms of Mathieu functions. Using this fact, we calculate absorption probabilities for these scalar waves, and present the explicit results for the first few low energy corrections to the leading-order expressions. For a specific tuning of the dyonic charges one can reach a domain where the low energy absorption probability goes to zero with inverse powers of the logarithm of the energy. This is a dividing domain between the regime where the low energy absorption probability approaches zero with positive powers of energy and the regime where the probability is an oscillatory function of the logarithm of the energy. By the conjectured AdS/CFT correspondence, these results shed novel light on the strongly coupled two-dimensional field theory away from its infrared conformally invariant fixed point (the strongly coupled ``non-critical'' string). 
  $N=1 D=4$ supermembrane (in a flat background) and super D2-brane dual to it are described within the nonlinear realizations approach as theories of the partial supersymmetry breaking $N=1 D=4 \to N=1 d=3$. We construct the relevant invariant off-shell Goldstone superfield actions and demonstrate them to be dual to each other. Their bosonic cores are, respectively, the static-gauge Nambu-Goto and $d=3$ Born-Infeld actions. The supermembrane superfield equation of motion admits a transparent geometric interpretation suggesting an extension of the standard superembedding constraints. We briefly discuss the 1/4 breaking of $N=1 D=5$ supersymmetry along similar lines. 
  We provide evidence for the validity of AdS/CFT correspondence in the Coulomb branch by comparing the Yang-Mills effective action with the potential between waves on two separated test 3-branes in the presence of a large number of other 3-branes. For constant gauge fields excited on the branes, this requires that the supergravity potential in a $AdS_5 \times S^5$ background is the {\it same} as that in flat space, despite the fact that both propagators and couplings of some relevant supergravity modes are different. We show that this is indeed true, due to a subtle cancellation. With time-dependent gauge fields on the test branes, the potential is sensitive to retardation effects of causal propagation in the bulk. We argue that this is reflected in higher derivative (acceleration) terms in the Yang-Mills effective action. We show that for two 3-branes separated in flat space the structure of lowest order acceleration terms is in agreement with supergravity expectations. 
  A brief review of some aspects of heterotic (0,2) compactifications in the framework of exactly solvable superconformal field theories and gauged linear sigma models is presented. 
  We study the Seiberg-Witten-Whitham equations in the strong coupling regime of the N=2 super Yang-Mills theory in the vicinity of the maximal singularities. In the case of SU(2) the Seiberg-Witten-Whitham equations fix completely the strong coupling expansion. For higher rank SU(N) they provide a set of non-trivial constraints on the form of this expansion. As an example, we study the off-diagonal couplings at the maximal point for which we propose an ansatz that fulfills all the equations. 
  We extend a recently proposed non-local version of Coleman's equivalence between the Thirring and sine-Gordon models to the case in which the original fermion fields interact with fixed impurities. We explain how our results can be used in the context of one-dimensional strongly correlated systems (the so called Tomonaga-Luttinger model) to study the dependence of the charge-density oscillations on the range of the fermionic interactions. 
  We study explicit solutions for orientifolds of Matrix theory compactified on noncommutative torus. As quotients of torus, cylinder, Klein bottle and M\"obius strip are applicable as orientifolds. We calculate the solutions using Connes, Douglas and Schwarz's projective module solution, and investigate twisted gauge bundle on quotient spaces as well. They are Yang-Mills theory on noncommutative torus with proper boundary conditions which define the geometry of the dual space. 
  We review the spectral cover formalism for constructing both U(n) and SU(n) holomorphic vector bundles on elliptically fibered Calabi-Yau three-folds which admit a section. We discuss the allowed bases of these three-folds and show that physical constraints eliminate Enriques surfaces from consideration. Relevant properties of the remaining del Pezzo and Hirzebruch surfaces are presented. Restricting the structure group to SU(n), we derive, in detail, a set of rules for the construction of three-family particle physics theories with phenomenologically relevant gauge groups. We show that anomaly cancellation generically requires the existence of non-perturbative vacua containing five-branes. We illustrate these ideas by constructing four explicit three-family non-perturbative vacua. 
  An abstract formulation of quantum dynamics in the presence of a general set of quantum constraints is developed. Our constructive procedure is such that the relevant projection operator onto the physical Hilbert space is obtained with a single, common integration procedure over the original Lagrange multiplier variables that is completely independent of the general nature of the constraints. In the associated lattice-limit formulation it is demonstrated that expansion of the constraint operator contribution to second order in the lattice spacing is necessary while, as usual, only a first-order expansion is needed for the dynamical operator contribution. Among various possibilities, coherent state path integrals are used to illustrate a completely functional representation of the abstract quantization procedure. 
  Zero-point fluctuations in quantum fields give rise to observable forces between material bodies, the so-called Casimir forces. In these lectures I present the theory of the Casimir effect, primarily formulated in terms of Green's functions. There is an intimate relation between the Casimir effect and van der Waals forces. Applications to conductors and dielectric bodies of various shapes will be given for the cases of scalar, electromagnetic, and fermionic fields. The dimensional dependence of the effect will be described. Finally, we ask the question: Is there a connection between the Casimir effect and the phenomenon of sonoluminescence? 
  We find the expectation value of the energy-momentum tensor in the CFT corresponding to a moving black hole in AdS. Boosting the black hole to the speed of light, keeping the total energy fixed, yields a gravitational shock wave in AdS. The analogous procedure on the field theory side leads to ``light cone'' states, i.e., states with energy-momentum tensor localized on the light cone. The correspondence between the gravitational shock wave and these light cone states provides a useful tool for testing causality. We show, in several examples, how the CFT reproduces the causal relations in AdS. 
  The Nelson stochastic mechanics is derived as a consequence of the basic physical principles such as the principle of relativity of observations and the invariance of the action quantum. The unitary group of quantum mechanics is represented as the transformations of the systems of perturbing devices. It is argued that the physical spacetime has a stochastic nature, and that quantum mechanics in Nelson's formulation correctly describes this stochasticity. 
  The relation between some perturbative non-BPS states of the heterotic theory on T^4 and non-perturbative non-BPS states of the orbifold limit of type IIA on K3 is exhibited. The relevant states include a non-BPS D-string, and a non-BPS bound state of BPS D-particles (`D-molecule'). The domains of stability of these states in the two theories are determined and compared. 
  We establish the Schlieder and the Borchers property for thermal field theories. In addition, we provide some information on the commutation and localization properties of projection operators. 
  Index theorems for the Dirac operator allow one to study spinors on manifolds with boundary and torsion. We analyse the modifications of the boundary Chern-Simons correction and APS eta invariant in the presence of torsion. The bulk contribution must also be modified and is computed using a supersymmetric quantum mechanics representation. Here we find agreement with existing results which employed heat kernel and Pauli-Villars techniques. Nonetheless, this computation also provides a stringent check of the Feynman rules of de Boer et al. for the computation of quantum mechanical path integrals. Our results can be verified via a duality relation between manifolds admitting a Killing-Yano tensor and manifolds with torsion. As an explicit example, we compute the indices of Taub-NUT and its dual constructed using this method and find agreement for any finite radius to the boundary. We also suggest a resolution to the problematic appearance of the Nieh-Yan invariant multiplied by the regulator mass^2 in computations of the chiral gravitational anomaly coupled to torsion. 
  Supersymmetry breaking via gaugino condensation is studied in vacua of heterotic M-theory with five-branes. We show that supersymmetry is still broken by a global mechanism and that the non-perturbative superpotential takes the standard form. When expressed in terms of low energy fields, a modification arises due to a threshold correction in the gauge kinetic function that depends on five-brane moduli. We also determine the form of the low energy matter field Kahler potential. These results are used to discuss the soft supersymmetry breaking parameters, in particular the question of universality. 
  This is a brief introductory review of the AdS/CFT correspondence and of the ideas that led to its formulation. Emphasis is placed on dualities between conformal large $N$ gauge theories in 4 dimensions and string backgrounds of the form $AdS_5\times X_5$. Attempts to generalize this correspondence to asymptotically free theories are also included. 
  A new computational idea for continuum quantum field theories is outlined. This approach is based on the lattice source Galerkin methods developed by Garcia, Guralnik and Lawson. The method has many promising features including treating fermions on a relatively symmetric footing with bosons. As a spinoff of the technology developed for ``exact'' solutions, the numerical methods used have a special case application to perturbation theory. We are in the process of developing an entirely numerical approach to evaluating graphs to high perturbative order. 
  We calculate F^4 and R^4T^(4g-4) couplings in d=8 heterotic and type I string vacua (with gauge and graviphoton field strengths F,T, and Riemann curvature R). The holomorphic piece F_g of the heterotic one-loop coupling R^4T^(4g-4) is given by a polylogarithm of index 5-4g and encodes the counting of genus g curves with g nodes on the K3 of the dual F-theory side. We present closed expressions for world-sheet tau-integrals with an arbitrary number of lattice vector insertions. Furthermore we verify that the corresponding heterotic one-loop couplings sum up perturbative open string and non-perturbative D-string contributions on the type I side. Finally we discuss a type I one-loop correction to the R^2 term. 
  We analyze the conformal invariance of submanifold observables associated with $k$-branes in the AdS/CFT correspondence. For odd $k$, the resulting observables are conformally invariant, and for even $k$, they transform with a conformal anomaly that is given by a local expression which we analyze in detail for $k=2$ 
  Some time ago, conformal data with affine fusion rules were found. Our purpose here is to realize some of these conformal data, using systems of free bosons and parafermions. The so constructed theories have an extended $W$ algebras which are close analogues of affine algebras. Exact character formulae is given, and the realizations are shown to be full fledged unitary conformal field theories. 
  We demonstrate an unambiguous and robust method for computing fermionic corrections to the energies of classical background field configurations. We consider the particular case of a sequence of background field configurations that interpolates continuously between the trivial vacuum and a widely separated soliton/antisoliton pair in 1+1 dimensions. Working in the continuum, we use phase shifts, the Born approximation, and Levinson's theorem to avoid ambiguities of renormalization procedure and boundary conditions. We carry out the calculation analytically at both ends of the interpolation and numerically in between, and show how the relevant physical quantities vary continuously. In the process, we elucidate properties of the fermionic phase shifts and zero modes. 
  A review is made on some recent studies which support the point of view that the relativistic field theory quantized on the light-front (LF) is more transparent compared to the conventional equal-time one. The discussion may be of relevance in the context of the quantization of gravitation theory. The LF quantization is argued to be equally appropriate as the conventional equal-time one. The description on the LF of the spontaneous symmetry breaking and the (tree level) Higgs mechanism, the emergence of the $\theta$-vacua in the Schwinger model, the absence of such vacua in the Chiral SM, the BRS-BFT quantization of the latter on the LF are among the topics discussed. Comments on the irrelevance, in the quantized theory, of the fact that the hyperplanes $x^{\pm}=0$ constitute characteristic surfaces of the hyperbolic partial differential equation are also made. The LF theory quantized on, say, the $x^{+}=const.$ hyperplanes seems to already contain in it the information on the equal-$x^{-}$ commutators as well. A theoretical reaffirmation of that the experimental data is to be confronted with the predictions of a classical theory model only after it has been upgraded through its quantization seems to emerge. 
  In a chiral $U_L(N)\times U_R(N)$ fermion model of NJL-form, we prove that, if all the fermions are assumed to have equal masses and equal chemical potentials, then at the finite temperature $T$ below the symmetry restoration temperature $T_c$, there will be $N^2$ massive scalar composite particles and $N^2$ massless pseudoscalar composite particles (Nambu-Goldstone bosons). This shows that the Goldstone Theorem at finite temperature for spontaneous symmetry breaking $U_L(N)\times U_R(N) \to U_{L+R}(N)$ is consistent with the real-time formalism of thermal field theory in this model. 
  Evidence for the possible existence of a quantum process opposite to the famous Hawking radiation (evaporation) of black holes is presented. This new phenomenon could be very relevant in the case of exotic multiple horizon Nariai black holes and in the context of common grand unified theories. This is clearly manifested in the case of the SO(10) GUT, that is here investigated in detail. The remarkable result is obtained, that anti-evaporation can occur there only in the SUSY version of the theory. It is thus concluded that the existence of primordial black holes in the present Universe might be considered as an evidence for supersymmetry. 
  We present a superspace formulation for super eightbrane theory based on massive type IIA supergravity in ten-dimensions. Remarkably, in addition to the 10-form superfield strength originally needed for super eightbrane, we also need an `over-ranked' 11-form superfield strength $H_{A_1... A_{11}}$ with identically vanishing purely bosonic component, in order to satisfy all the Bianchi identities. As a natural super p-brane formulation for the 11-form superfield strength, we present a super ninebrane action on 10-dimensional super-worldvolume invariant under a local fermionic $\k$-symmetry. We also show that we can formulate such a superspace with an `over-ranked' superfield strength also in eleven-dimensions, and possibly in other lower dimensions as well. 
  We examine how to construct explicit heterotic string models dual to F-theory in eight dimensions. In doing so we learn about where the moduli spaces of the two theories overlap, and how non-perturbative features leave traces on a purely perturbative level. We also propose a modified T-duality linking M and F-theories based on non-perturbative structures of string theories in eight dimensions. 
  Canonical formulation of quantum field theory on the Light Front (LF) is reviewed. The problem of constructing the LF Hamiltonian which gives the theory equivalent to original Lorentz and gauge invariant one is considered. We describe possible ways of solving this problem: (a) the limiting transition from the equal-time Hamiltonian in a fastly moving Lorentz frame to LF Hamiltonian, (b) the direct comparison of LF perturbation theory in coupling constant and usual Lorentz-covariant Feynman perturbation theory. Gauge invariant regularization of LF Hamiltonian via introducing a lattice in transverse coordinates and imposing periodic boundary conditions in LF coordinate $x^-$ for gauge fields on the interval $|x^-|< L$ is considered. We find that LF canonical formalism for this regularization avoid usual most complicated constraints connecting zero and nonzero modes of gauge fields. 
  We show how the gauge symmetry representations of the massless particle content of gauged supergravities that arise in the AdS/CFT correspondences can be derived from symmetric subgroups to be carried by string theory vertex operators in these compactified models, although an explicit vertex operator construction of IIB string and M theories on AdSxS remains elusive. Our symmetry mechanism parallels the construction of representations of the Monster group and affine algebras in terms of twisted conformal field theories, and may serve as a guide to the perturbative description of the IIB string on AdSxS. 
  In thermal states of chiral theories, as recently investigated by H.-J. Borchers and J. Yngvason, there exists a rich group of hidden symmetries. Here we show that this leads to a radical converse of of the Hawking-Unruh observation in the following sense. The algebraic commutant of the algebra associated with a (heat bath) thermal chiral system can be used to reprocess the thermal system into a ground state system on a larger algebra with a larger localization space-time. This happens in such a way that the original system appears as a kind of generalized Unruh restriction of the ground state sytem and the thermal commutant as being transmutated into newly created ``virgin space-time region'' behind a horizon. The related concepts of a ``chiral conformal core'' and the possibility of a ``blow-up'' of the latter suggest interesting ideas on localization of degrees of freedom with possible repercussion on how to define quantum entropy of localized matter content in Local Quantum Physics. 
  We explicitly construct BPS domain walls interpolating between neighboring chirally asymmetric vacua in a model for large N pure supersymmetric QCD. The BPS equations for the corresponding ${\bf Z}_N$ symmetric order parameter effective Lagrangian reduce to those in the $A_N$ Landau-Ginsburg model assuming that the higher derivative terms in the K{\"a}hler potential are suppressed in the large N limit. These BPS domain walls, which have vanishing width in the large N limit, can be viewed as supermembranes embedded in a (3+1)-dimensional supersymmetric QCD background. The supermembrane couples to a three-form supermultiplet whose components we identify with the composite fields of supersymmetric QCD. We also discuss certain aspects of chromoelectric flux tubes (open strings) ending on these walls which appear to support their interpretation as D-branes. 
  We give arguments in the support of a relation between M-atrix theory and Maldacena's conjecture. M-atrix theory conjecture implies the equivalence of 11-D light-cone supergravity and strongly-coupled (0+1)-D SYM. Maldacena's SUGRA/SYM duality conjecture implies, in the one dimensional SYM case, the equivalence between strongly-coupled (0+1)-D SYM and 11-D supergravity compactified on a spatial circle in the formal Seiberg-Sen limit. Using the classical equivalence between 11-D supergravity on a light-like circle and on a spatial circle in the formal Seiberg-Sen limit, we argue that in the (0+1)-D SYM case, the large-N M-atrix theory in the supergravity regime is equivalent to SUGRA/SYM duality. 
  Recently, it was shown by Jevicki, Kazama and Yoneya (JKY) that the Super-Yang-Mills theories (SYM) in $D\leq 4$ can reproduce the conformal symmetry of the near-horizon geometry of the D-brane solutions. However, the eikonal approximation they used is not sufficient for analyzing the conformal symmetry in SYM. We carry out the 1-loop calculation beyond the eikonal approximation for D=1 SYM (i.e., Matrix model) to confirm the claim of JKY. 
  In a one-generation fermion condensate scheme of electroweak symmetry breaking, it is proven based on Schwinger-Dyson equation in the real-time thermal field theory in the fermion bubble diagram approximation that, at finite temperature $T$ below the symmetry restoration temperature $T_c$, a massive Higgs boson and three massless Nambu-Goldstone bosons could emerge from the spontaneous breaking of electroweak group $SU_L(2)\times U_Y(1) \to U_Q(1)$ if the two fermion flavors in the one generation are mass-degenerate, thus Goldstone Theorem is rigorously valid in this case. However, if the two fermion flavors have unequal masses, owing to "thermal flactuation", the Goldstone Theorem will be true only approximately for a very large momentum cut-off $\Lambda$ in zero temperature fermion loop or for low energy scales. All possible pinch singularities are proven to cancel each other, as is expected in a real-time thermal field theory. 
  It is shown how one can define vector topological charges for topological exitations of non-linear sigma-models on compact homogeneous spaces T_G and G/T_G (where G is a simple compact Lie group and T_G is its maximal commutative subgroup). Explicit solutions for some cases, their energies and interaction of different topological charges are found. A possibility of the topological interpretation of the quantum numbers of groups and particles is discussed. 
  Lecture delivered at XXXVI Summer School on Subnuclear Physics at Erice in September 1998. It contains a brief reflection on the development of QFT in late sixties and a discussion of the role of solitons in realistic field theoretic models. 
  Superfields in 2-dimensional (2,2)-superspacetime which are independent of (some) half of the fermionic coordinates are discussed in a hopefully both comprehensive and comprehensible manner. An embarrassing abundance of these simplest `building blocks' makes it utterly impossible to write down the `most general Lagrangian'. With some ad hoc but perhaps plausible restrictions, a rather general Lagrangian is found, which exhibits many of the phenomena that have been studied recently, and harbors many more. In particular, it becomes patently obvious that the (2,2)-supersymmetric 2-dimensional field theory target space geometries (many of which are suitable for (super)string propagation) are far more general than Kahler manifolds with holomorphic bundles. 
  In a gauge theory with no Higgs fields the mechanism for confinement is by center vortices, but in theories with adjoint Higgs fields and generic symmetry breaking, such as the Georgi-Glashow model, Polyakov showed that in d=3 confinement arises via a condensate of 't Hooft-Polyakov monopoles. We study the connection in d=3 between pure-gauge theory and the theory with adjoint Higgs by varying the Higgs VEV v. As one lowers v from the Polyakov semi- classical regime v>>g (g is the gauge coupling) toward zero, where the unbroken theory lies, one encounters effects associated with the unbroken theory at a finite value v\sim g, where dynamical mass generation of a gauge-symmetric gauge- boson mass m\sim g^2 takes place, in addition to the Higgs-generated non-symmetric mass M\sim vg. This dynamical mass generation is forced by the infrared instability (in both 3 and 4 dimensions) of the pure-gauge theory. We construct solitonic configurations of the theory with both m,M non-zero which are generically closed loops consisting of nexuses (a class of soliton recently studied for the pure-gauge theory), each paired with an antinexus, sitting like beads on a string of center vortices with vortex fields always pointing into (out of) a nexus (antinexus); the vortex magnetic fields extend a transverse distance 1/m. An isolated nexus with vortices is continuously deformable from the 't Hooft-Polyakov (m=0) monopole to the pure-gauge nexus-vortex complex (M=0). In the pure-gauge M=0 limit the homotopy $\Pi_2(SU(2)/U(1))=Z_2$ (or its analog for SU(N)) of the 't Hooft monopoles is no longer applicable, and is replaced by the center-vortex homotopy $\Pi_1(SU)N)/Z_N)=Z_N$. 
  In this work (PartI) the qualitative analysis of statics and dynamics of defects and textures in liquid crystals is performed with help of meanders and train tracks. It is argued that similar analysis can be applied to 2+1 gravity. More rigorous justifications are presentedin the companion paper (PartII). Meanders were recently introduced by V.Arnold (Siberian J.of Math. Vol.29,36(1988)). Train tracks were originally introduced by W.Thurston in 1979 in his Princeton U. Lecture Notes (http://www.msri.org/gt3m/) in connection with description of self-homeomorphisms of 2 dimensional surfaces. Using train tracks the master equation is obtained which could be used alternatively to the Wheeler-DeWitt equation for 2+1 gravity. Since solution of this equation requires a large scale numerical work, in this paper we resort to the approximation of train tracks by the meandritic labyrinths. This allows us to analyse possible phases (and phase transitions)in both liquid crystals and gravity using Peierls- like arguments. 
  In this note we construct a new class of superconformal field theories as mass deformed N=4 super Yang-Mills theories. We will argue that these theories correspond to the fixed points which were recently found by Khavaev, Pilch and Warner studying the deformations of the dual IIB string theory on AdS_5\times S^5. 
  We use equivariant K-theory to classify charges of new (possibly non-supersymmetric) states localized on various orientifolds in Type II string theory. We also comment on the stringy construction of new D-branes and demonstrate the discrete electric-magnetic duality in Type I brane systems with p+q=7, as proposed by Witten. 
  We review the developments in the past twenty years (which are based on our deformation philosophy of physical theories) dealing with elementary particles composed of singletons in anti De Sitter space-time. The study starts with the kinematical aspects (especially for massless particles) and extends to the beginning of a field theory of composite elementary particles and its relations with conformal field theory (including very recent developments). 
  An important element in a model of non-singular string cosmology is a phase in which classical corrections saturate the growth of curvature in a deSitter-like phase with a linearly growing dilaton (an `algebraic fixed point'). As the form of the classical corrections is not well known, here we look for evidence, based on a suggested symmetry of the action, scale factor duality and on conformal field theory considerations, that they can produce this saturation. It has previously been observed that imposing scale factor duality on the $O(\alpha')$ corrections is not compatible with fixed point behavior. Here we present arguments that these problems persist to all orders in $\alpha'$. We also present evidence for the form of a solution to the equations of motion using conformal perturbation theory, examine its implications for the form of the effective action and find novel fixed point structure. 
  The low energy effective action describing the standard Kaluza-Klein reduction of heterotic string theory on a d-torus possesses a manifest O(d,d+16) symmetry. We consider generalized Scherk-Schwarz reductions of the heterotic string to construct massive gauged supergravities. We show that the resulting action can still be written in a manifestly O(d,d+16) invariant form, however, the U-duality transformations also act on the mass parameters. The latter play the dual role of defining the scalar potential and the nonabelian structure constants. We conjecture that just as for the standard reduction, a subgroup of this symmetry corresponds to an exact duality symmetry of the heterotic string theory. 
  We consider the pair of degenerate compatible antibrackets satisfying a generalization of the axioms imposed in the triplectic quantization of gauge theories. We show that this actually encodes a Lie group structure, with the antibrackets being related to the left- and right- invariant vector fields on the group. The standard triplectic quantization axioms then correspond to Abelian groups. 
  The theory of measured foliations which is discussed in PartI(hep-th/9901040) in connection with train tracks and meanders is shown to be related to the theory of Jenkins-Strebel quadratic differentials by Hubbard and Masur (Acta Math.Vol.142,221(1979)). Use of quadratic differentials not only provides an adequate description of defects and textures in liquid crystals but also is ideally suited for study of 2+1 classical gravity which was initiated in the seminal paper by Deser, Jackiw and 't Hooft (Ann.Phys.Vol.152,220(1984)). In this paper not only their results are reproduced but, in addition, many new results are obtained. In particular, using the results of Rivin (Ann.Math.Vol.139,553(1994)) the restriction on the total mass of the 2+1 Universe is removed. It is shown that the masses can have only discrete values and, moreover, the theoretically obtained sum rules forbid the existence of some of these values. The dynamics of 2+1 gravity which is associated with the dynamics of train tracks is being reinterpreted in terms of the emerging hyperbolic 3-manifolds. The existence of knots and links associated with complements of these 3-manifolds is highly nontrivial and requires careful proofs. The paper provides a concise introduction into this topic. A brief discussion of connections with related physical problems, e.g.string theory, classical and quantum billiards, dynamics of fracture, protein folding, etc. is also provided. 
  We study the large N operator spectrum of the (1,0) superconformal chiral six-dimensional theory with E_8 global symmetry. This spectrum is dual to the Kaluza-Klein spectrum of supergravity on AdS_7 X S^4/Z_2 with a ten-dimensional E_8 theory at its singular locus. We identify those operators in short multiplets of OSp(6,2|2), whose dimensions are exact for any N. We also discuss more general issues concerning AdS/CFT duality on orbifold supergravity backgrounds. 
  Dynamical mass generation of a two-component fermion in $QED_3$ with a Chern-Simons term is investigated by solving the Schwinger-Dyson equation formulated in the lowest ladder approximation. Dependence of the dynamical fermion mass on a gauge-fixing parameter, a gauge coupling constant, and a topological mass is examined by approximated analytical and also numerical methods. The inclusion of the Chern-Simons term makes impossible to choose a peculiar gauge in which a wave function renormalization is absent. The numerical evaluation shows that the wave function renormalization is fairly close to 1 in the Landau gauge. It means that this gauge is still a specific gauge where the Ward-Takahashi identity is satisfied approximately. We also find that the dynamical mass is almost constant if the topological mass is larger than the coupling constant, while it decreases when the topological mass is comparable to or smaller than the coupling constant and tends to the value in $QED_3$ without the Chern-Simons term. 
  This is a set of introductory lecture notes on black holes in string theory. After reviewing some aspects of string theory such as dualities, brane solutions, supersymmetric and non-extremal intersection rules, we analyze in detail extremal and non-extremal 5d black holes. We first present the D-brane counting for extremal black holes. Then we show that 4d and 5d non-extremal black holes can be mapped to the BTZ black hole (times a compact manifold) by means of dualities. The validity of these dualities is analyzed in detail. We present an analysis of the same system in the spirit of the adS/CFT correspondence. In the ``near-horizon'' limit (which is actually a near inner-horizon limit for non-extremal black holes) the black hole reduces again to the BTZ black hole. A state counting is presented in terms of the BTZ black hole. 
  Bosonization of the gauged, massive Thirring model in 2+1-dimensions produces a Maxwell-Chern-Simons gauge theory, coupled to a dynamical, massive vector field. Exploiting the Master Lagrangian formalism, two dual theories are constructed, one of them being a gauge theory. The full two-point functions of both the interacting fields are computed in the path integral quantization scheme. Furthermore, some new dual models, derived from the original master lagrangian and valid for different regimes of coupling parameters, are constructed and analysed in details. 
  We introduce thermal superspace and show how it can be used to reconcile the superfield formulation of supersymmetry with finite temperature environments 
  Quantization of two-dimensional Yang-Mills theory on a torus in the gauge where the field strength is diagonal leads to twisted sectors that are completely analogous to the ones that originate long string states in Matrix String Theory. If these sectors are taken into account the partition function is different from the standard one found in the literature and the invariance of the theory under modular transformations of the torus appears to hold in a stronger sense. The twisted sectors are in one-to-one correspondence with the coverings of the torus without branch points, so they define by themselves a string theory. A possible duality between this string theory and the Gross-Taylor string is discussed, and the problems that one encounters in generalizing this approach to interacting strings are pointed out. This talk is based on a previous paper by the same authors, but it contains some new results and a better interpretation of the results already obtained. 
  The global additive and multiplicative properties of the Laplacian on j-forms and related zeta functions are analyzed. The explicit form of zeta functions on a product of closed oriented hyperbolic manifolds \Gamma\backslash{\Bbb H}^d and of the multiplicative anomaly are derived. We also calculate in an explicit form the analytic torsion associated with a connected sum of such manifolds. 
  We consider two inequivalent truncations of the super D9--brane: the ``Heterotic'' and the ``Type I'' truncation. Both of them lead to an N=1 nonlinear supersymmetrization of the D=10 cosmological constant. The propagating degrees of freedom in the Heterotic and Type I truncation are given by the components of a D=10 vector multiplet and a single Majorana-Weyl spinor, respectively. As a by-product we find that, after the Type I truncation, the Ramond-Ramond super ten-form provides an interesting reformulation of the Volkov-Akulov action. These results can be extended to all dimensions in which spacetime filling D-branes exist, i.e. D=3,4,6 and 10. 
  We investigate the rotating D3-brane solution with maximum number of angular momentum parameters. After determining the angular velocities, Hawking temperature, ADM mass and entropy, we use this geometry to construct general three-parameter models of non-supersymmetric pure SU(N) Yang-Mills theories in 2+1 dimensions. We calculate glueball masses in the WKB approximation and obtain closed analytic expressions for generic values of the parameters. We also determine the masses of Kaluza--Klein states associated with internal parts of the ten-dimensional metric and investigate the parameter region where some of these states are decoupled. To leading order in 1/\lambda and 1/N (where \lambda is the 't Hooft coupling) we find a global U(1)^3 symmetry and states with masses comparable to glueball masses, which have no counterpart in the more familiar (finite \lambda, N) Yang-Mills theories. 
  We consider contributions to the heavy quark potential, in the AdS/CFT approach to SU(N) gauge theory, which arise from first order fluctuations of the associated worldsheet in anti-deSitter space. The gaussian fluctuations occur around a classical worldsheet configuration resembling an infinite square well, with the bottom of the well lying at the AdS horizon. The eigenvalues of the corresponding Laplacian operators can be shown numerically to be very close to those in flat space. We find that two of the transverse world sheet fields become massive, which may have implications for the existence of a L{\"u}scher term in the heavy quark potential. It is also suggested that these massive degrees of freedom may relate to extrinsic curvature in an effective D=4 string theory. 
  The paper is devoted to quantization of polynomial momentum observables in the cotangent bundle of a smooth manifold. A quantization procedure is proposed allowing to quantize a wide class of functions which are polynomials of any order in momenta. In the last part of the paper the quantum mechanics of scalar particle in curved space-time is studied with the use of proposed approach. 
  We study the interaction of two D-particles in the space-time of the shock wave. We first write the amplitude in string theory and find that, at large distances from the shock-wave source, the O(v^4) term in the relative velocity v is an \alpha'-independent function of the interbrane separation b. The amplitude is therefore that of supergravity--for large b, only closed-string massless modes contribute. We then show how the same result is obtained in the matrix model (at small b) by setting up the formulation of the dimensionally reduced super Yang-Mills theory in the curved background of the shock wave. 
  The BPS method is used to find BPS solutions of the worldvolume theory of a D5-brane in the near horizon geometry of a D3-brane. The BPS bound is interpreted in terms of the `maximally extended' D5 worldvolume supersymmetry algebra in the corresponding curved background, which is OSp(1|16). This algebra is an extension of the worldvolume superalgebra OSp(4^*|4). The analysis is generalized to the non-near horizon case. 
  It is well known that any three-manifold can be obtained by surgery on a framed link in $S^3$. Lickorish gave an elementary proof for the existence of the three-manifold invariants of Witten using a framed link description of the manifold and the formalisation of the bracket polynomial as the Temperley-Lieb Algebra. Kaul determined three-manifold invariants from link polynomials in SU(2) Chern-Simons theory. Lickorish's formula for the invariant involves computation of bracket polynomials of several cables of the link. We describe an easier way of obtaining the bracket polynomial of a cable using representation theory of composite braiding in SU(2) Chern-Simons theory. We prove that the cabling corresponds to taking tensor products of fundamental representations of SU(2). This enables us to verify that the two apparently distinct three-manifold invariants are equivalent for a specific relation of the polynomial variables. 
  I briefly describe a new class of soliton configurations in field theories. These consist of topological defects which can end when they intersect other defects of equal or higher dimensionality. Such configurations may be termed ``Dirichlet topological defects'', in analogy with the D-branes of string theory. I provide a specific example - cosmic strings that terminate on domain walls - and discuss some new directions for this work, including an interesting and qualitatively different extension to supersymmetric theories. 
  We investigate non-Abelian gauge theories within a Wilsonian Renormalisation Group approach. Our main question is: How close can one get to a gauge invariant flow, despite the fact that a Wilsonian coarse-graining seems to be incompatible with gauge invariance? We discuss the possible options in the case of quantum fluctuations, and argue that for thermal fluctuations a fully gauge invariant implementation can be obtained. 
  We review briefly generalized Freedman-Townsend models found recently by Henneaux and Knaepen, and provide supersymmetric versions of such models in four dimensions which couple 2-form gauge potentials and ordinary gauge fields in a gauge invariant and supersymmetric manner. The latter models have the unusual feature that, in a WZ gauge, the supersymmetry transformations do not commute with all the gauge transformations. 
  On the basis of the Weyl equations of congruent transference, we consider a possible influence of the Weyl non-Abelian gauge field defining the transference on the precession of a gyroscope. Plane-wave solutions to the equations of the non-Abelian gauge field are derived. 
  We study anti-de Sitter black holes in 2+1 dimensions in terms of Chern Simons gauge theory of the anti-de Sitter group coupled to a Source. Taking the source to be an anti-de Sitter state specified by its Casimir invariants, we show how all the relevant features of the black hole are accounted for. Enlarging the gauge symmetry to super AdS group, we obtain a supermultiplet of AdS black holes. We give explicit expressions for the masses and the angular momenta of the members of the multiplet. 
  The electromagnetic interaction of massive superparticles with N=2 extended Maxwell supermultiplet is studied. It is proved that the minimal coupling breaks the k-symmetry. A non-minimal k-symmetric action is built and it is established that the k-symmetry uniquely fixes the value of the superparticle's anomalous magnetic moment 
  We show that an essential assumption in the Vafa and Witten's theorem on P and CT realization in vector-like theories, the existence of a free energy density in Euclidean space in the presence of any external hermitian symmetry breaking source, does not apply if the symmetry is spontaneously broken. The assumption that the free energy density is well defined requires the previous assumption that the symmetry is realized in the vacuum. Even if Vafa and Witten's conjecture is plausible, actually a theorem is still lacking. 
  It is well known that, in the first-order formalism, pure three-dimensional gravity is just the BF theory. Similarly, four-dimensional general relativity can be formulated as BF theory with an additional constraint term added to the Lagrangian. In this paper we show that the same is true also for higher-dimensional Einstein gravity: in any dimension gravity can be described as a constrained BF theory. Moreover, in any dimension these constraints are quadratic in the B field. After describing in details the structure of these constraints, we scketch the ``spin foam'' quantization of these theories, which proves to be quite similar to the spin foam quantization of general relativity in three and four dimensions. In particular, in any dimension, we solve the quantum constraints and find the so-called simple representations and intertwiners. These exhibit a simple and beautiful structure that is common to all dimensions. 
  We describe the investigation of spontaneous mass-generation and chiral symmetry breaking in supersymmetric QED3 using numerical solutions of the Dyson-Schwinger equation together with the CJT effective action and supersymmetric Ward identities. We find that, within the quenched bare approximation, the chirally symmetric solution is favoured. 
  Spacetime is modelled as a homogeneous manifold given by the classes of unitary $\U(2)$ operations in the general complex operations $\GL(\C^2)$. The residual representations of this noncompact symmetric space of rank two are characterized by two continuous real invariants, one invariant interpreted as a particle mass for a positive unitary subgroup and the second one for an indefinite unitary subgroup related to nonparticle interpretable interaction ranges. Fields represent nonlinear spacetime $\GL(\C^2)/\U(2)$ by their quantization and include necessarily nonparticle contributions in the timelike part of their flat space Feynman propagator. 
  Various aspects of the Nahm equations in 3 and 7 dimensions are investigated. The residues of the variables at simple poles in the 7-dimensional case form an algebra. A large class of matrix representations of this algebra is constructed. The large $N$ limit of these equations is taken by replacing the commutators by Moyal Brackets, and a set of non trivial solutions in a generalised form of Wigner distribution functions is obtained. 
  We investigate monopole solutions for the Born-Infeld Higgs system. We analyze numerically these solutions and compare them with the standard 't Hooft-Polyakov monopoles. We also discuss the existence of a critical value of beta (the Born-Infeld "absolute field parameter") below which no regular solution exists. 
  We propose a method to derive the low-energy efective action of QCD assuming that the long-distance properties of strong interactions can be described by a string theory. We bypass the usual problems related to the existence of the tachyon and absence of the adequate Adler zero by using a sigma model approach where excitations above the correct (chirally non-invariant) QCD vacuum are included. Two-dimensional conformal invariance then implies the vanishing of the O(p^4) effective lagrangian coefficients. We discuss ways to go beyond this limit. 
  By explicit construction of the ADHM data, we prove the existence of a charge seven instanton with icosahedral symmetry. By computing the holonomy of this instanton we obtain a Skyrme field which approximates the minimal energy charge seven Skyrmion. We also present a one parameter family of tetrahedrally symmetric instantons whose holonomy gives a family of Skyrme fields which models a Skyrmion scattering process, where seven well-separated Skyrmions collide to form the icosahedrally symmetric Skyrmion. 
  In the large-N limit of d=4, N=4 gauge theory, the dual AdS spacetime becomes flat. We identify a gauge theory correlator whose large-N limit is the flat spacetime S-matrix. 
  We consider 10-dimensional super Yang-Mills theory with topological terms compactified on a noncommutative torus. We calculate supersymmetry algebra and derive BPS energy spectra from it. The cases of d-dimensional tori with d=2,3,4 are considered in full detail. SO(d,d|Z)-invariance of the BPS spectrum and relation of new results to the previous work in this direction are discussed. 
  We present two new solutions of Romans' massive type IIA supergravity characterized by the two non-trivial massive potentials of Romans' theory: the NSNS 2-form and the RR 7-form. They can be interpreted respectively as the intersection of a fundamental string and a D8-brane over a D0-brane and the intersection of a D6-brane with a D8-brane over a NSNS5-brane. The D8-brane manifests itself through the mass parameter and in the massless limit one recovers the standard fundamental string and D6-brane solutions. Although these solutions do not have the usual form for BPS bound states at threshold and each of them involves 3 objects, both of them preserve 1/4 of the supersymmetries. 
  Matrix theory and the AdS/CFT correspondence provide nonperturbative holographic formulations of string theory. In both cases the finite N theories can be thought of as infrared regulated versions of flat space string theory in which removing the cutoff is equivalent to letting N go to infinity.   In this paper we consider the nature of this limit. In both cases the holographic mapping becomes completely nonlocal. In matrix theory this corresponds to the growth of D0-brane bound states with N. For the AdS/CFT correspondence there is a similar delocalization of the holographic image of a system as N increases. In this case the limiting theory seems to require a number of degrees of freedom comparable to large N matrix quantum mechanics. 
  Studying dynamics of open strings attached to a D2-brane in a NS two form field background, we find that these open strings act as dipoles of U(1) gauge field of the brane. This provides an string theoretic description of the flux modifications needed for the DBI action on noncommutative torus. 
  A regular method for constructing vortex-like solutions with cylindrical symmetry to the equations of the SU(2) Skyrme chiral model is proposed. A numerical estimate for the length density of mass is given. 
  The Floreanini-Jackiw formulation of the chiral quantum-mechanical system oscillator is a model of constrained theory with only second-class constraints. in the Dirac's classification.The covariant quantization needs infinite number of auxiliary variables and a Wess-Zumino term. In this paper we investigate the path integral quatization of this model using $G\ddot{u}ler's$ canonical formalism. All variables are gauge variables in this formalism. The Siegel's action is obtained using Hamilton-Jacobi formulation of the systems with constraints. 
  The Hamiltonian treatment of constrained systems in $G\ddot{u}ler's$ formalism leads us to the total differential equations in many variables. These equations are integrable if the corresponding system of partial differential equations is a Jacobi system. The main aim of this paper is to investigate the quantization of the finite dimensional systems with constraints using the canonical formalism introduced by $G\ddot{u}ler$. This approach is applied for two systems with constraints and the results are in agreement with those obtained by Dirac's canonical quatization method and path integral quantization method. 
  It is shown, that recently constructed PST Lagrangians for chiral supergravities follow directly from earlier Kavalov-Mkrtchyan Lagrangians by an Ansatz for the $\theta $ tensor by expressing this in terms of the PST scalar. The susy algebra which included earlier $\alpha$-symmetry in the commutator of supersymmetry transformations, is now shown to include both PST symmetries, which arise from the single $\alpha$-symmetry term. The Lagrangian for the 5-brane is not described by this correspondence, and probably can be obtained from more general Lagrangians, posessing $\alpha$-symmetry. 
  We derive fully covariant expressions for all two-point scattering amplitudes involving closed string tachyon and massless strings from Dirichlet brane in type 0 theories. The amplitude for two massless D-brane fluctuations to produce closed string tachyon is also evaluated. We then examine in detail these string scattering amplitudes in order to extract world-volume couplings of the tachyon with itself and with massless fields on a D-brane. We find that the tachyon appears as an overall coupling function in the Born-Infeld action. 
  The Casimir surface force density F on a dielectric dilute spherical ball of radius a, surrounded by a vacuum, is calculated at zero temperature. We treat (n-1) (n being the refractive index) as a small parameter. The dispersive properties of the material are taken into account by adopting a simple dispersion relation, involving a sharp high frequency cutoff at omega = omega_0. For a nondispersive medium there appears (after regularization) a finite, physical, force F^{nondisp} which is repulsive. By means of a uniform asymptotic expansion of the Riccati-Bessel functions we calculate F^{nondisp} up to the fourth order in 1/nu. For a dispersive medium the main part of the force F^{disp} is also repulsive. The dominant term in F^{disp} is proportional to (n-1)^2{omega_0}^3/a, and will under usual physical conditions outweigh F^{nondisp} by several orders of magnitude. 
  In this paper we describe a Liouville gravity which is induced by a set of quantum fields (constituents) and represents a two-dimensional analog of Sakharov's induced gravity. The important feature of the considered theory is the presence of massless constituents which are responsible for the appearance of the induced Liouville field. The role of the massive constituents is only to induce the cosmological constant. We consider the instanton solutions of the Euclidean Liouville gravity with negative and zero cosmological constants, some instantons being interpreted as two-dimensional anti-de Sitter $AdS_2$ black holes. We study thermodynamics of all the solutions and conclude that their entropy is completely determined by the statistical-mechanical entropy of the massless constituents. This shows, in particular, that the constituents of the induced gravity are the true degrees of freedom of $AdS_2$ black holes. Special attention is also paid to the induced Liouville gravity with zero cosmological constant on a torus. We demonstrate the equivalence of its thermodynamics to the thermodynamics of BTZ black holes and comment on computations of the BTZ black hole entropy. 
  We construct the supergravity solution for the intersecting D1-D5 brane system in Type I String Theory. The solution encodes the dependence on all the electric charges of the SO(32) gauge group. We discuss the near horizon geometry of the solution and a proposed dual (0,4) superconformal field theory. 
  We explain Sklyanin's separation of variables in geometrical terms and construct it for Hitchin and Mukai integrable systems. We construct Hilbert schemes of points on $T^{*}\Sigma$ for $\Sigma = {\IC}, {\IC}^{*}$ or elliptic curve, and on ${\bf C}^{2}/{\Gamma}$ and show that their complex deformations are integrable systems of Calogero-Sutherland-Moser type. We present the hyperk\"ahler quotient constructions for Hilbert schemes of points on cotangent bundles to the higher genus curves, utilizing the results of Hurtubise, Kronheimer and Nakajima. Finally we discuss the connections to physics of $D$-branes and string duality. 
  Bilocal light-ray operators which are Lorentz scalars, vectors or antisymmetric tensors, and which appear in various hard scattering QCD processes, are decomposed into operators of definite twist. These operators are harmonic tensor functions and their Taylor expansion consists of (traceless) local light-cone operators with span irreducible representations of the Lorentz group with definite spin j and common geometric twist (= dimension - spin). Some applications concerning the nonforward matrix elements of these operators and the generalization fo conformal light-cone operators of definite twist is considered. The group theoretical background of the method has been made clear. 
  We use toric geometry to investigate the recently proposed relation between a set of D3 branes at a generalized conifold singularity and type IIA configurations of D4 branes stretched between a number of relatively rotated NS5 branes. In particular we investigate how various resolutions of the singularity corresponds to moving the NS branes and how Seiberg's duality is realized when two relatively rotated NS-branes are interchanged. 
  We show that the decay spectrum of massive excitations in perturbative string theories is thermal when averaged over the (many) initial degenerate states. We first compute the inclusive photon spectrum for open strings at the tree level showing that a black body spectrum with the Hagedorn temperature emerges in the averaging. A similar calculation for a massive closed string state with winding and Kaluza-Klein charges shows that the emitted graviton spectrum is thermal with a "grey-body" factor, which approaches one near extremality. These results uncover a simple physical meaning of the Hagedorn temperature and provide an explicit microscopic derivation of the black body spectrum from a unitary $S$ matrix. 
  The correspondence between Matrix String Theory in the strong coupling limit and IIA superstring theory can be shown by means of the instanton solutions of the former. We construct the general instanton solutions of Matrix String Theory which interpolate between given initial and final string configurations. Each instanton is characterized by a Riemann surface of genus h with n punctures, which is realized as a plane curve. We study the moduli space of such plane curves and find out that, at finite N, it is a discretized version of the moduli space of Riemann surfaces: instead of 3h-3+n its complex dimensions are 2h-3+n, the remaining h dimensions being discrete. It turns out that as $N$ tends to infinity, these discrete dimensions become continuous, and one recovers the full moduli space of string interaction theory. 
  We develop classical globally supersymmetric theories. As much as possible, we treat various dimensions and various amounts of supersymmetry in a uniform manner. We discuss theories both in components and in superspace. Throughout we emphasize geometric aspects. The beginning chapters give a general discussion about supersymmetric field theories; then we move on to detailed computations of lagrangians, etc. in specific theories. An appendix details our sign conventions. This text will appear in a two-volume work "Quantum Fields and Strings: A Course for Mathematicians" to be published soon by the American Mathematical Society. Some of the cross-references may be found at http://www.math.ias.edu/~drm/QFT/ 
  The main focus of this lecture is on extended objects in adS*S bosonic backgrounds with unbroken supersymmetry. The backgrounds are argued to be exact, special consideration are given to the non-maximal supersymmetry case. The near horizon superspace construction is explained. The superconformal symmetry appears in the worldvolume actions as the superisometry of the near horizon superspace, like the superPoincare symmetry of GS superstring and BST supermembrane in the flat superspace. The issues in gauge fixing of local kappa-symmetry are reviewed. We describe the features of the gauge-fixed IIB superstring in adS(5)*S(5) background with RR 5-form. From a truncated boundary version of it we derive an analytic N=2 off shell harmonic superspace of Yang-Mills theory. The reality condition of the analytic subspace, which includes the antipodal map on the sphere, has a simple meaning of the symmetry of the string action in the curved space. The relevant issues of black holes and superconformal mechanics are addressed. 
  The regularity of static axially symmetric solutions in SU(2) Yang-Mills-dilaton theory is examined. We show that the solutions obtained previously within a singular Ansatz for the non-abelian gauge field can be gauge transformed into a regular form. The local form of the gauge transformation is given on the singular axis and at the origin. 
  In this paper the $Guler's$ formalism for the systems with finite degrees of freedom is applied to the field theories with constraints. The integrability conditions are investigated and the path integral quantization is performed using the action given by Hamilton-Jacobi formulation. The Proca's model is investigated in details. 
  In the context of the free-fermionic formulation of the heterotic superstring, we construct a three generation N=1 supersymmetric SU(4)xSU(2)LxSU(2)R model supplemented by an SU(8) hidden gauge symmetry and five Abelian factors. The symmetry breaking to the standard model is achieved using vacuum expectation values of a Higgs pair in (4bar,2R)+(4,2R) at a high scale. One linear combination of the Abelian symmetries is anomalous and is broken by vacuum expectation values of singlet fields along the flat directions of the superpotential. All consistent string vacua of the model are completely classified by solving the corresponding system of F- and D-flatness equations including non-renormalizable terms up to sixth order. The requirement of existence of electroweak massless doublets further restricts the phenomenologically viable vacua. The third generation fermions receive masses from the tree-level superpotential. Further, a complete calculation of all non-renormalizable fermion mass terms up to fifth order shows that in certain string vacua the hierarchy of the fermion families is naturally obtained in the model as the second and third generation fermions earn their mass from fourth and fifth order terms. Along certain flat directions it is shown that the ratio of the SU(4) breaking scale and the reduced Planck mass is equal to the up quark ratio m_c/m_t at the string scale. An additional prediction of the model, is the existence of a U(1) symmetry carried by the fields of the hidden sector, ensuring thus the stability of the lightest hidden state. It is proposed that the hidden states may account for the invisible matter of the universe. 
  The recently discovered formalism underlying renormalization theory, the Hopf algebra of rooted trees, allows to generalize Chen's lemma. In its generalized form it describes the change of a scale in Green functions, and hence relates to the operator product expansion. Hand in hand with this generalization goes the generalization of the ordinary factorial $n!$ to the tree factorial $t^!$. Various identities on tree-factorials are derived which clarify the relation between Connes-Moscovici weights and Quantum Field Theory. 
  We introduce the spherical field formalism for free gauge fields. We discuss the structure of the spherical Hamiltonian for both general covariant gauge and radial gauge and point out several new features not present in the scalar field case. We then use the evolution equations to compute gauge-field and field-strength correlators. 
  We show that type 0B theory has a classical AdS_5 x S^5 solution and argue that it is stable at the string-theory level for small enough radius. The dual 4-d conformal field theory is the infrared limit of the theory on N electric D3-branes coincident with N magnetic D3-branes. We explicitly construct this SU(N) x SU(N) gauge theory with global SO(6) symmetry and verify that the one-loop term in the beta function vanishes exactly, while the two-loop term vanishes in the large N limit. We find that this theory is related by a certain projection to the maximally supersymmetric Yang-Mills theory suggesting its large N conformal invariance to all orders in perturbation theory. 
  I review a recently proposed method for determining the symmetry superalgebra of a supergravity configuration from its Killing spinors, and its application to the `near-horizon' limits of various rotating and intersecting branes. 
  The effects of extra space-time dimensions on the Wilsonian effective K\"ahler potential and the perturbative one loop effective K\"ahler potential are determined within the framework of an Abelian gauge theory with N=2 supersymmetric field content. The relation between the K\"ahler metric and the effective gauge couplings which leads to the absence of radiative corrections to the K\"ahler potential is expressed as a function of the radius of compactification of a fifth dimension. In general, the quantum corrections to the low energy K\"ahler potential are shown to grow with this radius reflecting the underlying higher dimensional nature of the theory. 
  Starting with the ordinary ten-dimensional supersymmetric Yang-Mills theory for the gauge group U(N), we obtain a twelve-dimensional supersymmetric gauge theory as the large N limit. The two symplectic canonical coordinates parametrizing the unitary N X N matrices for U(N) are identified with the extra coordinates in twelve dimensions in the $N\to\infty$ limit. Applying further a strong/weak duality, we get the `decompactified' twelve-dimensional theory. The resulting twelve-dimensional theory has peculiar gauge symmetry which is compatible also with supersymmetry. We also establish a corresponding new superspace formulation with the extra coordinates. By performing a dimensional reduction from twelve dimensions directly into three dimensions, we see that the Poisson bracket terms which are needed for identification with supermembrane action arises naturally. This result indicates an universal duality mechanism that the 't Hooft limit of an arbitrary supersymmetric theory promotes the original supersymmetric theory in (D-1,1) dimensions into a theory in (D,2) dimensions with an additional pair of space-time coordinates. This also indicates interesting dualities between supermembrane theory, type IIA superstring with D0-branes, and the recently-discovered twelve-dimensional supersymmetric theories. 
  Membrane source-probe dynamics is investigated in the framework of the finite N-sector DLCQ M theory compactified on a transverse two-torus for an arbitrary size of the longitudinal dimension. The non-perturbative two fermion terms in the effective action of the matrix theory, the (2+1)-dimensional supersymmetric Yang-Mills theory, that are related to the four derivative F^4 terms by the supersymmetry transformation are obtained, including the one-loop term and full instanton corrections. On the supergravity side, we compute the classical probe action up to two fermion terms based on the classical supermembrane formulation in an arbitrary curved background geometry produced by source membranes satisfying the BPS condition; two fermion terms correspond to the spin-orbit couplings for membranes. We find precise agreement between two approaches when the background space-time is chosen to be that of the DLCQ M theory, which is asymptotically locally Anti-de Sitter. 
  The low-energy theory on the world volume of parallel static D3-branes of type 0 strings is the Yang-Mills theory with six scalar fields in the adjoint representation. One-loop corrections in this theory induce Coleman-Weinberg effective potential, which can be interpreted as an interaction energy of D3-branes. The potential is repulsive at short distances and attractive at large ones. In the equilibrium, a large number of D3-branes forms a spherical shell with the radius proportional to the characteristic energy scale of the world-volume theory. 
  The numbers of bosonic and fermionic zero modes of multi-pronged strings are counted in ${\cal N}=4$ super-Yang-Mills theory and compared with those of the IIB string theory. We obtain a nice agreement for the fermionic zero modes, while our result for the bosonic zero modes differs from that obtained in the IIB string theory. The possible origin of the discrepancy is discussed 
  We consider the dynamics of a relativistic Dirac particle constrained to move in the interior of a twisted tube by confining boundary conditions, in the approximation that the curvature of the tube is small and slowly varying. In contrast with the nonrelativistic theory, which predicts that a particle's spin does not change as the particle propagates along the tube, we find that the angular momentum eigenstates of a relativistic spin-1/2 particle may behave nontrivially. For example, a particle with its angular momentum initially polarized in the direction of propagation may acquire a nonzero component of angular momentum in the opposite direction on turning through 2 \pi radians. Also, the usual nonrelativistic effective potential acquires an additional factor in the relativistic theory. 
  We propose a new constraint on the structure of strongly coupled field theories. The constraint takes the form of an inequality limiting the number of degrees of freedom in the infrared description of a theory relative to the number of underlying, ultraviolet degrees of freedom. We apply the inequality to a variety of theories (both supersymmetric and nonsupersymmetric), where it agrees with all known results and leads to interesting new constraints on low energy spectra. We discuss the relation of this constraint to Renormalization Group c-theorems. 
  The relationship between the perturbation theory in light-front coordinates and Lorentz-covariant perturbation theory is investigated. A method for finding the difference between separate terms of the corresponding series without their explicit evaluation is proposed. A procedure of constructing additional counter-terms to the canonical Hamiltonian that compensate this difference at any finite order is proposed. For the Yukawa model, the light-front Hamiltonian with all of these counter-terms is obtained in a closed form. Possible application of this approach to gauge theories is discussed. 
  In supersymmetric models with the run-away vacua or with the stable but non-supersymmetric ground state there exist stable field configurations (vacua) which restore one half of supersymmetry and are characterized by constant positive energy density. The formal foundation for such vacua is provided by the central extension of the N =1 superalgebra with the infinite central charge. 
  It is shown that the Poisson bracket with boundary terms recently proposed by Bering (hep-th/9806249) can be deduced from the Poisson bracket proposed by the present author (hep-th/9305133) if one omits terms free of Euler-Lagrange derivatives ("annihilation principle"). This corresponds to another definition of the formal product of distributions (or, saying it in other words, to another definition of the pairing between 1-forms and 1-vectors in the formal variational calculus). We extend the formula (initially suggested by Bering only for the ultralocal case with constant coefficients) onto the general non-ultralocal brackets with coefficients depending on fields and their spatial derivatives. The lack of invariance under changes of dependent variables (field redefinitions) seems a drawback of this proposal. 
  Generalized Weierstrass formulae for surfaces in four-dimensional space $\Bbb{R}^{4}$ are used to study (anti)self-dual rigid string configurations. It is shown that such configurations are given by superminimal immersions into $\Bbb{R}^{4}$. Explicit formulae for generic (anti)instantons are presented. Particular classes of surfaces are also analyzed. 
  A geometric definition for a magnetic charge of Abelian monopoles in SU(N) lattice gauge theories with Higgs fields is presented. The corresponding local monopole number defined for almost all field configurations does not require gauge fixing and is stable against small perturbations. Its topological content is that of a 3-cochain. A detailed prescription for calculating the local monopole number is worked out. Our method generalizes a magnetic charge definition previously invented by Phillips and Stone for SU(2). 
  We obtain the closed-form absorption probabilities for minimally-coupled massless scalars propagating in the background of D=5 single-charge and D=4 two-charge black holes. These are the only two examples of extremal black holes with non-vanishing absorption probabilities that can be solved in closed form for arbitrary incident frequencies. In both cases, the absorption probability vanishes when the frequency is below a certain threshold, and we discuss the connection between this phenomenon and the behaviour of geodesics in these black hole backgrounds. We also obtain leading-order absorption cross-sections for generic extremal p-branes, and show that the expression for the cross-section as a function of frequency coincides with the leading-order dependence of the entropy on the temperature in the corresponding near-extremal p-branes. 
  The covariant operator quantization of the ordinary free spinning BDH string modified by adding the supersymmetric Liouville sector is analysed in the even target space dimensions $d=2,4,6,8$. The spectrum generating algebra for this model is constructed and a general version of the no-ghost theorem is proven. A counterpart of the GSO projection leads to a family of tachyon free unitary free string theories. One of these models is equivalent to the non-critical Rammond-Neveu-Schwarz spinning string truncated in the Neveu-Schwarz sector to the tachyon free eigenspace of the fermion parity operator. 
  The non-perturbative equivalence of four-dimensional N=2 superstrings with three vector multiplets and four hypermultiplets is analysed. These models are obtained through freely acting orbifold compactifications from the heterotic, the symmetric and the asymmetric type II strings. The heterotic scalar manifolds are (SU(1,1) / U(1))^3 for the S,T,U moduli sitting in the vector multiplets and SO(4,4)/ (SO(4) X SO(4)) for those in the hypermultiplets. The type II symmetric duals correspond to a self-mirror Calabi-Yau threefold compactification with Hodge numbers h(1,1)=h(2,1)=3, while the type II asymmetric construction corresponds to a spontaneous breaking of the N=(4,4) supersymmetry to N=(2,0). Both have already been considered in the literature. The heterotic construction instead is new and we show that there is a weak/strong coupling S-duality relation between the heterotic and the asymmetric type IIA ground state with S(Het)=-1/S(As); we also show that there is a partial restoration of N=8 supersymmetry in the heterotic strong-coupling regime. We compute the full (non-)perturbative R2 and F2 corrections and determine the prepotential. 
  Space-time N=2 and N=4 superconformal algebras can be built using world-sheet free fields or, equivalently, free field representations of osp(1|2) and sl(2|1), respectively. A prescription for the calculation of space-time correlators is given. As applications we compute one and two-point correlation functions of Virasoro (superVirasoro) generators. We also present a possible scenario for taking into account of NS fivebranes in the framework of our construction for description of superstring on AdS_3. It is nothing but a simplified version of Green's ``world-sheets for world-sheets''. 
  We study quantum dilaton coupled spinors in two and four dimensions. Making classical transformation of metric, dilaton coupled spinor theory is transformed to minimal spinor theory with another metric and in case of 4d spinor also in the background of the non-trivial vector field. This gives the possibility to calculate 2d and 4d dilaton dependent conformal (or Weyl) anomaly in easy way. Anomaly induced effective action for such spinors is derived. In case of 2d, the effective action reproduces, without any extra terms, the term added by hands in the quantum correction for RST model, which is exactly solvable. For 4d spinor the chiral anomaly which depends explicitly from dilaton is also found. As some application we discuss SUSY Black Holes in dilatonic supergravity with WZ type matter and Hawking radiation in the same theory. As another application we investigate spherically reduced Einstein gravity with 2d dilaton coupled fermion anomaly induced effective action and show the existence of quantum corrected Schwarszchild-de Sitter (SdS) (Nariai) BH with multiple horizon. 
  Gorsky et al. presented an explicit construction of Whitham deformations of the Seiberg-Witten curve for the $SU(N+1)$ $\calN = 2$ SUSY Yang-Mills theory. We extend their result to all classical gauge groups and some other cases such as the spectral curve of the $A^{(2)}_{2N}$ affine Toda Toda system. Our construction, too, uses fractional powers of the superpotential $W(x)$ that characterizes the curve. We also consider the $u$-plane integral of topologically twisted theories on four-dimensional manifolds $X$ with $b_2^{+}(X) = 1$ in the language of these explicitly constructed Whitham deformations and an integrable hierarchy of the KdV type hidden behind. 
  We compute the 3-point function of the stress-energy tensor in the d-dimensional CFT from the AdS_{d+1} gravity. For d=4 the coefficients of the three linearly independent conformally covariant forms entering the 3-point function are exactly the same as given by the free field computations in the ${\cal N}=4$ SYM just as expected from the known renormalization theorems. For d=3 and d=6 our results give the value of the corresponding 3-point function in the theories of strongly coupled ${\cal N}=8$ superconformal scalar and (2,0) tensor multiplets respectively. 
  We study the possible D-brane configurations in an AdS_3 x S^3 x T^4 background with a NS-NS B field. We use its WZW model description and the boundary state formalism, and we analyze the bosonic and the N=1 supersymmetric cases separately. We also discuss the corresponding classical open string sigma model. We determine the spacetime supersymmetry preserved by the supersymmetric D-brane configurations. 
  Using free world-sheet fermions, we construct and classify all the N=2, Z2 X Z2 four-dimensional orbifolds of the type IIA/B strings for which the orbifold projections act symmetrically on the left and right movers. We study the deformations of these models out of the fermionic point, deriving the partition functions at a generic point in the moduli of the internal torus T6=T2 X T2 X T2. We investigate some of their perturbative and non-perturbative dualities and construct new dual pairs of type IIA/type II asymmetric orbifolds, which are related non-perturbatively and allow us to gain insight into some of the non-perturbative properties of the type IIA/B strings in four dimensions. In particular, we consider some of the (non-)perturbative gravitational corrections. 
  One-instanton predictions for the prepotential are obtained from the Seiberg-Witten curve for the Coulomb branch of N=2 supersymmetric gauge theory for the product group \prod_{n=1}^{m} SU(N_n) with a massless matter hypermultiplet in the bifundamental representation (N_n,\bar N_{n+1}) of SU(N_n) x SU(N_{n+1}) for n=1 to m-1, together with N_0 and N_{m+1} matter hypermultiplets in the fundamental representations of SU(N_1) and SU(N_m) respectively. The derivation uses a generalization of the systematic perturbation expansion about a hyperelliptic curve developed by us in earlier work. 
  We study the higher-derivative extensions of the D=3 Abelian Chern--Simons topological invariant that would appear in a perturbative effective action's momentum expansion. The leading, third-derivative, extension I_ECS turns out to be unique. It remains parity-odd but depends only on the field strength, hence no longer carries large gauge information, nor is it topological because metric dependence accompanies the additional covariant derivatives, whose positions are seen to be fixed by gauge invariance. Viewed as an independent action, I_ECS requires the field strength to obey the wave equation. The more interesting model, adjoining I_ECS to the Maxwell action, describes a pair of excitations. One is massless, the other a massive ghost, as we exhibit both via the propagator and by performing the Hamiltonian decomposition. We also present this model's total stress tensor and energy. Other actions involving I_ECS are also noted. 
  We analyze the hypermultiplet moduli space describing the universal sector of type IIA theory compactified on a Calabi-Yau threefold. The classical moduli space is described in terms of the coset $SU(2,1)/U(2)$. The flux quantization condition of the antisymmetric tensor field of M-theory implies discrete identifications for the scalar fields describing this four- dimensional quaternionic geometry. Non-perturbative corrections of the classical theory are described in terms of a membrane- fivebrane instanton action which we construct herein. 
  We describe a class of diffeomorphism invariant SU(N) gauge theories in N^2 dimensions, together with some matter couplings. These theories have (N^2-3)(N^2-1) local degrees of freedom, and have the unusual feature that the constraint associated with time reparametrizations is identically satisfied. A related class of SU(N) theories in N^2-1 dimensions has the constraint algebra of general relativity, but has more degrees of freedom. Non-perturbative quantization of the first type of theory via SU(N) spin networks is briefly outlined. 
  We present a self-contained study of ADHM multi-instantons in SU(N) gauge theory, especially the novel interplay with supersymmetry and the large-N limit. We give both field- and string-theoretic derivations of the N=4 supersymmetric multi-instanton action and collective coordinate integration measure. As a central application, we focus on certain n-point functions G_n, n=16, 8 or 4, in N=4 SU(N) gauge theory at the conformal point (as well as on related higher-partial-wave correlators); these are correlators in which the 16 exact supersymmetric and superconformal fermion zero modes are saturated. In the large-N limit, for the first time in any 4-dimensional theory, we are able to evaluate all leading-order multi-instanton contributions exactly. We find compelling evidence for Maldacena's conjecture: (1) The large-N k-instanton collective coordinate space has the geometry of a single copy of AdS_5 x S^5. (2) The integration measure on this space includes the partition function of 10-dimensional N=1 SU(k) gauge theory dimensionally reduced to 0 dimensions, matching the description of D-instantons in Type IIB string theory. (3) In exact agreement with Type IIB string calculations, at the k-instanton level, G_n = \sqrt{N} g^8 k^{n-7/2} e^{2\pi ik\tau} \sum_{d|k} d^{-2} F_n(x_1,...,x_n), where F_n is identical to a convolution of n bulk-to-boundary supergravity propagators. 
  We study the Ward-Takahashi identities in the standard model with the gauge fixing terms given by (1.1) and (1.2). We find that the isolated singularities of the propagators for the unphysical particles are poles of even order, not the simple poles people have assumed them to be. Furthermore, the position of these poles are ultraviolet divergent. Thus the standard model in the alpha gauge in general, and the Feynman gauge in particular, is not renormalizable. We study also the case with the gauge fixing terms (1.3), and find that the propagators remain non-renormalizable. The only gauge without these difficulties is the Landau gauge. One therefore has to make a distinction between the renormalizability of the Green functions and that of the physical scattering amplitudes. 
  We give an explicit formulation of the three-dimensional $SL(4,R)/SO(2,2) \sigma$-model representing the five-dimensional Einstein gravity coupled to the dilaton and the three-form field for spacetimes with two commuting Killing vector fields. New matrix representation is obtained which is similar to one found earlier in the four-dimensional Einstein-Maxwell-Dilaton-Axion (EMDA) theory. The SL(4,R) symmetry joins a variety of 5D solutions of different physical types including strings, 0-branes, KK monopoles etc. interpreting them as duals to the four-dimensional Kerr metric translated along the fifth coordinate. The symmetry transformations are used to construct new rotating strings and composite 0-1-branes endowed with a NUT parameter. 
  We propose a world-sheet realization of the zigzag-invariant strings as a perturbed WZNW model at large negative level. We argue that the gravitational dressing produces additional fixed points of the dressed renormalization beta function. One of these new critical points can be interpreted as a zigzag-invariant string model. 
  The graviton self-energy at finite temperature depends on fourteen structure functions. We show that, in the absence of tadpoles, the gauge invariance of the effective action imposes three non-linear relations among these functions. The consequences of such constraints, which must be satisfied by the thermal graviton self-energy to all orders, are explicitly verified in general linear gauges to one loop order. 
  The SU(2) collective coordinates quantization of the Born-Infeld Skyrmions Lagrangean is performed. The obtainment of the classical Hamiltonian from this special Lagrangean is made by using an approximate way: it is derived from the expansion of this non-polynomial Lagrangean up to second-order variable in the collective coordinates, using for this some causality arguments. Because this system presents constraints, we use the Dirac Hamiltonian method and the Faddeev-Jackiw Lagrangean approach to quantize this model. 
  We test the AdS/CFT correspondence in the case of a d=4 N=2 SCFT by comparing chiral anomalies which are of order N in the 't Hooft large N limit. These include corrections of order 1/N to the conformal anomaly, thus testing the correspondence beyond the extreme large N limit. The field theory anomalies are reproduced by terms in the 7-brane effective action in the bulk. 
  The phase diagram for Dp-branes in M-theory compactified on $T^4$, $T^4/Z_2$, $T^5$, and $T^6$ is constructed. As for the lower-dimensional tori considered in our previous work (hep-th/9810224), the black brane phase at high entropy connects onto matrix theory at low entropy; we thus recover all known instances of matrix theory as consequences of the Maldacena conjecture. The difficulties that arise for $T^6$ are reviewed. We also analyze the D1-D5 system on $T^5$; we exhibit its relation to matrix models of M5-branes, and use spectral flow as a tool to investigate the dependence of the phase structure on angular momentum. 
  We present some new supersymmetric solutions of massive IIA supergravity involving D0-branes, a D8-brane and a string. For the bosonic fields we use a general ansatz with SO(8) symmetry. 
  Quantum corrections in the hypermultiplet moduli space of type IIA string theories compactified on Calabi-Yau threefolds are investigated. 
  A new attempt is demonstrated that QFTs can be UV finite if they are viewed as the low energy effective theories of a fundamental underlying theory (complete and well-defined in all respects) according to the modern standard point of view. This approach is much simpler in principle and in technology comparing to any known renormalization program. Some subtle and difficult issues can be easily resolved. The importance of the procedure for defining the ambiguities is fully appreciated in the new approach. Some simple but important nonperturbative examples are discussed to show the power and plausibility of the new approach. 
  We demonstrate that any scale-invariant mechanics of one variable exhibits not only 0+1 conformal symmetry, but also the symmetries of a full Virasoro algebra. We discuss the implications for the adS/CFT correspondence. 
  The concept of perturbative gauge invariance formulated exclusively by means of asymptotic fields is used to construct massive gauge theories. We consider the interactions of $r$ massive and $s$ massless gauge fields together with $(r+s)$ fermionic ghost and anti-ghost fields. First order gauge invariance requires the introduction of unphysical scalars (Goldstone bosons) and fixes their trilinear couplings. At second order additional physical scalars (Higgs fields) are necessary, their coupling is further restricted at third order. In case of one physical scalar all couplings are determined by gauge invariance, including the Higgs potential. For three massive and one massless gauge field the $SU(2)\times U(1)$ electroweak theory comes out as the unique solution. 
  A quantization condition due to the boundary conditions and the compatification of the light cone space-time coordinate $x^-$ is identified at the level of the classical equations for the right-handed fermionic field in two dimensions. A detailed analysis of the implications of the implementation of this quantization condition at the quantum level is presented. In the case of the Thirring model one has selection rules on the excitations as a function of the coupling and in the case of the Schwinger model a double integer structure of the vacuum is derived in the light-cone frame. Two different quantized chiral Schwinger models are found, one of them without a $\theta$-vacuum structure. A generalization of the quantization condition to theories with several fermionic fields and to higher dimensions is presented. 
  We study the strong coupling limit of the 2-flavor massless Schwinger model on a lattice using staggered fermions and the Hamiltonian approach to lattice gauge theories. Using the correspondence between the low-lying states of the 2-flavor strongly coupled lattice Schwinger model and the antiferromagnetic Heisenberg chain established in a previous paper, we explicitly compute the mass gaps of the other excitations in terms of vacuum expectation values (v.e.v.'s) of powers of the Heisenberg Hamiltonian and spin-spin correlation functions. We find a satisfactory agreement with the results of the continuum theory already at the second order in the strong coupling expansion. We show that the pattern of symmetry breaking of the continuum theory is well reproduced by the lattice theory; we see indeed that in the lattice theory the isoscalar and isovector chiral condensates are zero to every order in the strong coupling expansion. In addition, we find that the chiral condensate $<\bar{\psi}_{L}^{(2)}\bar{\psi}_{L}^{(1)}\psi_{R}^{(1)}\psi_{R}^{(2)}>$ is non zero also on the lattice; this is the only relic in this lattice model of the axial anomaly in the continuum theory. We compute the v.e.v.'s of the spin-spin correlators of the Heisenberg model which are pertinent to the calculation of the mass spectrum and obtain an explicit construction of the lowest lying states for finite size Heisenberg Hamiltonian chains. 
  We compute the O(alpha'^3) corrections to the AdS_5 Black hole metric in type IIB string theory. Contrary to previous work in this direction we keep the Black Hole radius finite. Thus the topology of the boundary is S^3 x S^1. We find the corrections to the free energy and the critical temperature of the phase transition. 
  The general correlator of composite operators of N=4 supersymmetric gauge field theory is divergent. We introduce a means for renormalizing these correlators by adding a boundary theory on the AdS space correcting for the divergences. Such renormalizations are not equivalent to the standard normal ordering of current algebras in two dimensions. The correlators contain contact terms that contribute to the OPE; we relate them diagrammatically to correlation functions of compound composite operators dual to multi-particle states. 
  The constraints implied by analyticity in two-dimensional factorised S-matrix theories are reviewed. Whenever the theory is not time-reversal invariant, it is argued that the familiar condition of `Real analyticity' for the S-matrix amplitudes has to be superseded by a different one known as `Hermitian analyticity'. Examples are provided of integrable quantum field theories whose (diagonal) two-particle S-matrix amplitudes are Hermitian analytic but not Real analytic. It is also shown that Hermitian analyticity is consistent with the bootstrap equations and that it ensures the equivalence between the notion of unitarity in the quantum group approach to factorised S-matrices and the genuine unitarity of the S-matrix. 
  This is a write-up of lectures given at the 1998 Spring School at the Abdus Salam ICTP. We give a conceptual introduction to D-geometry, the study of geometry as seen by D-branes in string theory, and to noncommutative geometry as it has appeared in D-brane and Matrix theory physics. 
  De Sitter spacetime is known to have a cosmological horizon that enjoys thermodynamic-like properties similar to those of a black hole horizon. In this note we show that a universal argument can be given for the entropy of de Sitter spacetime in arbitrary dimensions, by generalizing a recent near horizon symmetry plus conformal field theory argument of Carlip for black hole entropy. The implications of this argument are also discussed. 
  We review some aspects of three-dimensional quantum gravity with emphasis in the `CFT -> Geometry' map that follows from the Brown-Henneaux conformal algebra. The general solution to the classical equations of motion with anti-de Sitter boundary conditions is displayed. This solution is parametrized by two functions which become Virasoro operators after quantisation. A map from the space of states to the space of classical solutions is exhibited. Some recent proposals to understand the Bekenstein-Hawking entropy are reviewed in this context. The origin of the boundary degrees of freedom arising in 2+1 gravity is analysed in detail using a Hamiltonian Chern-Simons formalism. 
  We present new anti-de Sitter black hole solutions of gauged N=8, SO(8) supergravity, which is the massless sector of the AdS_4\times S^7 vacuum of M-theory. By focusing on the U(1)^4 Cartan subgroup, we find non-extremal 1, 2, 3 and 4 charge solutions. In the extremal limit, they may preserve up to 1/2, 1/4, 1/8 and 1/8 of the supersymmetry, respectively. In the limit of vanishing SO(8) coupling constant, the solutions reduce to the familiar black holes of the M_4\times T^7 vacuum, but have very different interpretation since there are no winding states on S^7 and no U-duality. In contrast to the T^7 compactification, moreover, we find no static multi-center solutions. Also in contrast, the S^7 fields appear "already dualized" so that the 4 charges may be all electric or all magnetic rather than 2 electric and 2 magnetic. Curiously, however, the magnetic solutions preserve no supersymmetries. We conjecture that a subset of the extreme electric black holes preserving 1/2 the supersymmetry may be identified with the S^7 Kaluza-Klein spectrum, with the non-abelian SO(8) quantum numbers provided by the fermionic zero modes. 
  We construct new F-theory vacua in 8-dimensions. They are coming by projective realizations of F-theory on $K_3$ surfaces admitting double covers onto $\P^2$, branched along a plane sextic curve, the so called double sextics. The new vacua are associated with singular $K_3$ surfaces. In this way the stable picture of the heterotic string is mapped at the triple points of the sextic. We argue that this formulation naturally incorporates the $Sp(4,Z)$ invariance that the extrapolating four dimensional vector multiplet sector of all heterotic vacua may possess. In addition, we describe the way that the 4D g=2 description of (0,2) moduli dependence of N=1 gauge coupling constants may be connected to Riemann surfaces, with natural Sp(4,Z) duality invarinace. Here we recover a novel way to break space-time supersymmetry and fix the moduli parameters in the presence of Wilson lines. In the context of arithmetic of torsion points on elliptic curves, we describe in detail, the derivation of elliptic fibrations in Weierstrass form. We also consider the heterotic duals to compactifications of F-theory in four dimensions belonging to isomorphic classes of elliptic curves with point-cusps of order two. For the latter theories, we calculate the $\cal N}=2$ 4D heterotic prepotential $f_{TTT}$ corresponding to ${\Gamma_o(2)}_T \times {\Gamma_o(2)}_U$, classical perturbative duality group, and their conjugate modular theories. 
  We consider the generating function (prepotential) for Gromov-Witten invariants of rational elliptic surface. We apply the local mirror principle to calculate the prepotential and prove a certain recursion relation, holomorphic anomaly equation, for genus 0 and 1. We propose the holomorphic anomaly equation for all genera and apply it to determine higher genus Gromov-Witten invariants and also the BPS states on the surface. Generalizing G\"ottsche's formula for the Hilbert scheme of $g$ points on a surface, we find precise agreement of our results with the proposal recently made by Gopakumar and Vafa(hep-th/9812127). 
  The classical probe dynamics of the eleven-dimensional massless superparticles in the background geometry produced by N source M-momenta is investigated in the framework of N-sector DLCQ supergravity. We expand the probe action up to the two fermion terms and find that the fermionic contributions are the spin-orbit couplings, which precisely agree with the matrix theory calculations. We comment on the lack of non-perturbative corrections in the one-loop matrix quantum mechanics effective action and its compatibility with the supergravity analysis. 
  We consider a model for tensionless (null) p-branes with N=1 global supersymmetry in 10-dimensional Minkowski space-time. We give an action for the model and show that it is reparametrization and kappa-invariant. We also find some solutions of the classical equations of motion. In the case of null superstring (p=1), we obtain the general solution in arbitrary gauge. 
  We propose a construction of non-trivial vacua for Yang-Mills theories on the 3-torus. Although we consider theories with periodic boundary conditions, twisted boundary conditions play an essential auxiliary role in our construction. In this article we will limit ourselves to the simplest case, based on twist in SU(2) subgroups. These reproduce the recently constructed new vacua for SO(N) and G_2 theories on the 3-torus. We show how to embed the results in the other exceptional groups F_4 and E_{6,7,8} and how to compute the relevant unbroken subgroups. In a subsequent article we will generalise to SU(N > 2) subgroups. The number of vacua found this way exactly matches the number predicted by the calculation of the Witten index in the infinite volume. 
  The vacuum expectation value of the Wilson loop functional in pure Yang-Mills theory on an arbitrary two-dimensional orientable manifold is studied. We consider the calculation of this quantity for the abelian theory in the continuum case and for the arbitrary gauge group and arbitrary lattice action in the lattice case. A classification of topological sectors of the theory and the related classification of the principal fibre bundles over two-dimensional surfaces are given in terms of a cohomology group. The contribution of SU(2)-invariant connections to the vacuum expectation value of the Wilson loop variable is also analyzed and is shown to be similar to the contribution of monopoles. 
  The quantum effective theory of general relativity, independent of the eventual full theory at high energy, expresses graviton-graviton scattering at one loop order O(E^4) with only one parameter, Newton's constant. Dunbar and Norridge have calculated the one loop amplitude using string based techniques. We complete the calculation by showing that the 1/(d-4) divergence which remains in their result comes from the infrared sector and that the cross section is finite and model independent when the usual bremsstrahlung diagrams are included. 
  Massless elementary-particle propagation is represented historically (cosmologically) through 3-scale ``towers of quartet rings'' within a lattice of magneto-electrodynamically communicating ``pre-events''. The lightlike intervals within a ring of 4 pre-events (discrete ``closed string'') display transverse GUT-scale and longitudinal ``particle scale''. The lightlike (longitudinal) spacing between successive rings of a tower is at Planck scale. Ratio between GUT scale and Planck scale relates quantum-dynamically to elementary magnetic charge. Permutations of a ring quartet, in conjunction with Lorentz-group representations, control elementary-particle quantum numbers. 
  We discuss the ten dimensional black holes made of D0-branes in the regime where the effective coupling is large, and yet the 11D geometry is unimportant. We suggest that these black holes can be interpreted as excitations over the threshold bound state. Thus, the entropy formula for the former is used to predict a scaling region of the wave function of the latter. The horizon radius and the mass gap predicted in this picture agree with the formulas derived from the classical geometry. 
  We argue that M2 brane is realized as a topological soliton on a coincident pair of M5 and anti-M5 branes, as the two five-branes annihilate each other. Topology and quantum numbers of this world-volume soliton are discussed in some detail, and its formation is explained qualitatively. It follows from a compactification that a D4-anti-D4 pair annihilate and produce type II fundamental strings. The phenomenon is best described as the confinement of a world-volume U(1) gauge field on D4-anti-D4, where the confined electric flux string is identified as the fundamental string. This generalizes to other D$p$-anti-D$p$ systems, and solves a puzzle recently pointed out by Witten. 
  We consider a model for tensionless (null) super p-branes with N chiral supersymmetries in ten dimensional flat space-time. After establishing the symmetries of the action, we give the general solution of the classical equations of motion in a particular gauge. In the case of a null superstring (p=1) we find the general solution in an arbitrary gauge. Then, using a harmonic superspace approach, the initial algebra of first and second class constraints is converted into an algebra of Lorentz-covariant, BFV-irreducible, first class constraints only. The corresponding BRST charge is as for a first rank dynamical system. 
  We present a summary of the progress made in the last few years on topological quantum field theory in four dimensions. In particular, we describe the role played by duality in the developments which led to the Seiberg-Witten invariants and their relation to the Donaldson invariants. In addition, we analyze the fruitful framework that this connection has originated. This analysis involves the study of topological quantum field theories which contain twisted N=2 supersymmetric matter fields as well as theories obtained after twisting N=4 supersymmetry. In the latter case, we present some recent results including the generalization of the partition function of the Vafa-Witten theory for gauge group SU(N) with prime N. 
  We investigate the prepotential that describes certain F^4 couplings in eight dimensional string compactifications, and show how they can be computed from the solutions of inhomogenous differential equations. These appear to have the form of the Picard-Fuchs equations of a fibration of Sym^2(K3) over P^1. Our findings give support to the conjecture that the relevant geometry which underlies these couplings is given by a five-fold. 
  On manifolds with non-trivial Killing tensors admitting a square root of the Killing-Yano type one can construct non-standard Dirac operators which differ from, but commute with, the standard Dirac operator. We relate the index problem for the non-standard Dirac operator to that of the standard Dirac operator. This necessitates a study of manifolds with torsion and boundary and we summarize recent results obtained for such manifolds. 
  By carefully analysing the picture-dependence of the BRST cohomology an infinite set of symmetry charges of the closed N=2 string is identified. The transformation laws of the physical vertex operators are shown to coincide with the linearised non-local symmetries of the Plebanski equation (which is the effective field theory of the closed N=2 string). Moreover, the corresponding Ward identities are powerful enough to allow for a rederivation of the well known vanishing theorem for the tree-level correlation functions with more than three external legs. 
  Based on a generalization of the stochastic quantization scheme recently a modified Faddeev-Popov path integral density for the quantization of Yang-Mills theory was derived, the modification consisting in the presence of specific finite contributions of the pure gauge degrees of freedom. Due to the Gribov problem the gauge fixing can be defined only locally and the whole space of gauge potentials has to be partitioned into patches. We propose a global path integral density for the Yang-Mills theory by summing over all patches, which can be proven to be manifestly independent of the specific local choices of patches and gauge fixing conditions, respectively. In addition to the formulation on the whole space of gauge potentials we discuss the corresponding global path integral on the gauge orbit space relating it to the original Parisi-Wu stochastic quantization scheme and to a proposal of Stora, respectively. 
  We present a complete proof that solutions of the WDVV equations in Seiberg-Witten theory may be constructed from root systems. A generalization to weight systems is proposed. 
  We apply stochastic quantization method to Kostov's matrix-vector models for the second quantization of orientable strings with Chan-Paton like factors, including both open and closed strings. The Fokker-Planck hamiltonian deduces an orientable open-closed string field theory at the double scaling limit. There appears an algebraic structure in the continuum F-P hamiltonian including a Virasoro algebra and a SU(r) current algebra. 
  We evaluate the fermionic determinant for massless QED_2 at finite temperature, in the imaginary time formalism. By using a decoupling transformation of the fermionic fields, we show that the determinant factorizes into the usual, temperature independent expression, times an extra factor which depends on the temperature and on the constant component of the gauge field. 
  A new family of supergravity theories in odd dimensions is presented. The Lagrangian densities are Chern-Simons forms for the connection of a supersymmetric extension of the anti-de Sitter algebra. The superalgebras are the supersymmetric extensions of the AdS algebra for each dimension, thus completing the analysis of van Holten and Van Proeyen, which was valid for N=1 and for D=2,3,4,mod 8. The Chern-Simons form of the Lagrangian ensures invariance under the gauge supergroup by construction and, in particular, under local supersymmetry. Thus, unlike standard supergravity, the local supersymmetry algebra closes off-shell and without requiring auxiliary fields. The Lagrangian is explicitly given for D=5, 7 and 11. In all cases the dynamical field content includes the vielbein, the spin connection, N gravitini, and some extra bosonic ``matter'' fields which vary from one dimension to another. The superalgebras fall into three families: osp(m|N) for D=2,3,4, mod 8, osp(N|m) for D=6,7,8, mod 8, and su(m-2,2|N) for D=5 mod 4, with m=2^{[D/2]}. The possible connection between the D=11 case and M-Theory is also discussed. 
  It is known that the noncommutativity of D-brane coordinate is responsible for describing the higher-dimensional D-branes in terms of more fundamental ones such as D-particles or D-instantons, while considering a noncommutative torus as a target space is conjectured to be equivalent to introducing the background antisymmetric tensor field in matrix models. In the present paper we clarify the dual nature of both descriptions. Namely the noncommutativity of conjugate momenta of the D-brane coordinates realizes the target space structure, whereas noncommutativity of the coordinates themselves realizes world volume structure. We explicitly construct a boundary state for the Dirichlet boundary condition where the string boundary is adhered to the D-brane on the noncommutative torus. There are non-trivial relations between the parameters appeared in the algebra of the coordinates and that of the momenta. 
  We study classical solutions of N=4 super Yang-Mills theories that are invariant under 1/4 of the supersymmetry generators. Expressions for the mass and electric charge of the configurations are derived as functions on the monopole moduli space. These functions also provide a method of determining the number of normalisable bosonic zero modes. 
  Following the subtraction procedure for manifolds with boundaries, we calculate by variational methods, the Schwarzschild and Flat space energy difference. The one loop approximation for TT tensors is considered here. An analogy between the computed energy difference in momentum space and the Casimir effect is illustrated. We find a singular behaviour in the UV-limit, due to the presence of the horizon when $r=2m.$ When $r>2m$ this singular behaviour disappears, which is in agreement with various other models previously presented. 
  The motion of a particle near the RN black hole horizon is described by conformal mechanics. Models of this type have no ground state with vanishing energy. This problem was resolved in past by a redefinition of the Hamiltonian which breaks translational time invariance but gives a normalizable ground state. We show that this change of the Hamiltonian is a quantum mechanical equivalent of the change of coordinates near the black hole horizon removing the singularity. The new Hamiltonian of quantum mechanics is identified as an operator of a rotation between 2 time-like coordinates of the adS hypersurface which translates global time. Therefore conformal quantum mechanics may eventually help to resolve the puzzles of the classical black hole physics. 
  Local conserved charges in principal chiral models in 1+1 dimensions are investigated. There is a classically conserved local charge for each totally symmetric invariant tensor of the underlying group. These local charges are shown to be in involution with the non-local Yangian charges. The Poisson bracket algebra of the local charges is then studied. For each classical algebra, an infinite set of local charges with spins equal to the exponents modulo the Coxeter number is constructed, and it is shown that these commute with one another. Brief comments are made on the evidence for, and implications of, survival of these charges in the quantum theory. 
  The master equation is quantized. This is an example of quantization of a gauge theory with nilpotent generators. No ghosts are needed for a generation of the gauge algebra. The point about the nilpotent generators is that one can't write down a single functional integral for this theory. One has to write down a product of two coupled functional integrals and take a square root. In the special gauge where the gauge conditions are commuting, the functional integrals decouple, and one recovers the known result. 
  It is shown that, under certain conditions, the existence of a Killing spinor on a bosonic background of a supergravity theory implies that the Einstein equations are also satisfied. As an application of the theorem, we obtain a new black fivebrane solution of D=11 supergravity, which has $K3\times R$ topology and preserves 1/4'th supersymmetries of the theory. 
  We investigate the thermodynamic Bethe ansatz (TBA) equations for a system of particles which dynamically interacts via the scattering matrix of affine Toda field theory and whose statistical interaction is of a general Haldane type. Up to the first leading order, we provide general approximated analytical expressions for the solutions of these equations from which we derive general formulae for the ultraviolet scaling functions for theories in which the underlying Lie algebra is simply laced. For several explicit models we compare the quality of the approximated analytical solutions against the numerical solutions. We address the question of existence and uniqueness of the solutions of the TBA-equations, derive precise error estimates and determine the rate of convergence for the applied numerical procedure. A general expression for the Fourier transformed kernels of the TBA-equations allows to derive the related Y-systems and a reformulation of the equations into a universal form. 
  String vacua for non critical strings satisfying the requirements of Zig-Zag invariance are constructed. The Liouville mode is shown to play the r\^ole of scale in the Renormalization Group operation. Differences and similarities with the D-brane near horizon approach to non supersymmetric gauge theories are discussed as well. 
  The paper is devoted to the formulation of quantum field theory for an early universe in General Relativity considered as the Dirac general constrained system. The main idea is the Hamiltonian reduction of the constrained system in terms of measurable quantities of the observational cosmology: the world proper time, cosmic scale factor, and the density of matter. We define " particles" as field variables in the holomorphic representation which diagonalize the measurable density. The Bogoliubov quasiparticles are determined by diagonalization of the equations of motion (but not only of the initial Hamiltonian) to get the set of integrals of motion (or conserved quantum numbers, in quantum theory). This approach is applied to describe particle creation in the models of the early universe where the Hubble parameter goes to infinity. 
  It has been shown in literature that a possible mechanism of mass generation for gauge fields is through a topological coupling of vector and tensor fields. After integrating over the tensor degrees of freedom, one arrives at an effective massive theory that, although gauge invariant, is nonlocal. Here we quantize this nonlocal resulting theory both by path integral and canonical procedures. This system can be considered as equivalent to one with an infinite number of time derivatives and consequently an infinite number of momenta. This means that the use of the canonical formalism deserves some care. We show the consistency of the formalism we use in the canonical procedure by showing that the obtained propagators are the same as those of the (Lagrangian) path integral approach. The problem of nonlocality appears in the obtainment of the spectrum of the theory. This fact becomes very transparent when we list the infinite number of commutators involving the fields and their velocities. 
  For two decades it was believed that chiral symmetries cannot be realized in lattice field theory but this has changed now. Highlights of these new developments will be presented with emphasis on the mathematical structure of the so called ``overlap''. 
  The link of the Division Algebras to 10-dimensional spacetime and one leptoquark family is extended to 26-dimensional spacetime and three leptoquark families. 
  We propose a modification of the Faddeev-Popov procedure to construct a path integral representation for the transition amplitude and the partition function for gauge theories whose orbit space has a non-Euclidean geometry. Our approach is based on the Kato-Trotter product formula modified appropriately to incorporate the gauge invariance condition, and thereby equivalence to the Dirac operator formalism is guaranteed by construction. The modified path integral provides a solution to the Gribov obstruction as well as to the operator ordering problem when the orbit space has curvature. A few explicit examples are given to illustrate new features of the formalism developed. The method is applied to the Kogut-Susskind lattice gauge theory to develop a nonperturbative functional integral for a quantum Yang-Mills theory. Feynman's conjecture about a relation between the mass gap and the orbit space geometry in gluodynamics is discussed in the framework of the modified path integral. 
  An introduction to the instanton formalism in supersymmetric gauge theories is given. We explain how the instanton calculations, in conjunction with analyticity in chiral parameters and other general properties following from supersymmetry, allow one to establish exact results in the weak and strong coupling regimes. Some key applications are reviewed, the main emphasis is put on the mechanisms of the dynamical breaking of supersymmetry. 
  The one-loop effective action in QED at zero and finite temperature is obtained by using the worldline approach. The Feynman rules for the perturbative expansion of the action in the number of derivatives are derived. The general structure of the temperature dependent part of the effective action in an arbitrary external inhomogeneous magnetic field is established. The two-derivative term in the effective action for spinor and scalar QED in a static magnetic background at $T\neq 0$ is calculated. 
  A recent experiment with squeezed light has shown that two-photon absorption by an atom can occur with a linear intensity dependence. We point out that this result verifies a prediction made by us more than a decade ago from an analysis of a nonlocal model of QED. This model had earlier been proposed by us in an ad hoc manner to interpret certain features of multiphoton double ionization and above-threshold ionization in an atom placed in a strong laser field ; in this paper we show that the model can be obtained field- theoretically by demanding covariance of the field Lagrangian under a nonlocal U(1) gauge transformation. The model also makes direct contact with squeezed light, and thus allows us to describe these two completely different scenarios from a unified point of view.      We obtain a fundamentally new result from our nonlocal QED, namely that only the past, but not the future, can influence the present - thus establishing a non-thermodynamic arrow of time at the quantum level. We also show that correlations within a quantum system should necessarily be of the EPR-type, a result that agrees with Bell's theorem. These results follow from the simple requirement of energy conservation in matter-radiation interaction. Furthermore, we also predict new and experimentally verifiable results on the basis of our model QED. 
  The Dirac equation for a massive spin-1/2 field in a central potential V in three dimensions is studied without fixing a priori the functional form of V. The second-order equations for the radial parts of the spinor wave function are shown to involve a squared Dirac operator for the free case, whose essential self-adjointness is proved by using the Weyl limit point-limit circle criterion, and a `perturbation' resulting from the potential. One then finds that a potential of Coulomb type in the Dirac equation leads to a potential term in the above second-order equations which is not even infinitesimally form-bounded with respect to the free operator. Moreover, the conditions ensuring essential self-adjointness of the second-order operators in the interacting case are changed with respect to the free case, i.e. they are expressed by a majorization involving the parameter in the Coulomb potential and the angular momentum quantum number. The same methods are applied to the analysis of coupled eigenvalue equations when the anomalous magnetic moment of the electron is not neglected. 
  This is a write-up of two lectures on AdS/CFT correspondance given by the authors at the 1998 Spring School at the Abdus Salam ICTP 
  We show that certain excitations of the F-string/D3-brane system can be shown to obey Neumann boundary conditions by considering Born-Infeld dynamics of the F-string (viewed as a 3-brane cylindrically wrapped on an $S_2$). Excitations which are coming down the string with a polarization along a direction parallel to the brane are almost completely reflected just as in the case of all-normal excitations, but the end of the string moves freely on the 3-brane, thus realizing Polchinski's open string Neumann boundary condition dynamically. In the low energy limit w -> 0, i.e. for wavelengths much larger than the string scale only a small fraction ~w^4 of the energy escapes in the form of dipole radiation. The physical interpretation is that a string attached to the 3-brane manifests itself as an electric charge, and waves on the string cause the end point of the string to freely oscillate and therefore produce electromagnetic dipole radiation in the asymptotic outer region. 
  Various fluid mechanical systems enjoy a hidden, higher-dimensional dynamical Poincare symmetry, which arises owing to their descent from a Nambu-Goto action. Also, for the same reason, there are equivalence transformations between different models. These interconnections are discussed in our paper. 
  We describe a new formalism which expresses asymtotically free thories in a manifestly finite way, after renormalization and dimensional transmutation. The time evolution is NOT differentiable in these systems, so the hamiltonian does not exist. Instead, there is a new operator, the `Principal Operator', (which is roughly speaking the logarithm of the hamiltonian) which is finite (cut-off independent). We construct the Principal operator in several examples, including the Many body Problem of bosons in two dimensions with a short range attractive interaction. This allows us to estimate the ground state energy of a two-dimensional Bose condensate (with an attractive interaction). The ground state energy depends exponentially on the number of particles. 
  Two classes of Conformal Field Theories have been proposed to describe the Hierarchical Quantum Hall Effect:the multi-component bosonic theory, characterized by the symmetry U(1)xSU(m)_1 and the W_{1+\infty} minimal models with central charge c=m. In spite of having the same spectrum of edge excitations, they manifest differences in the degeneracy of the states and in the quantum statistics, which call for a more detailed comparison between them. Here, we describe their detailed relation for the general case, c=m and extend the methods previously published for c < 4. Specifically, we obtain the reduction in the number of degrees of freedom from the multi-component Abelian theory to the minimal models by decomposing the characters of the U(1)xSU(m)_1 representations into those of the c=m W_{1+\infty} minimal models. Furthermore, we find the Hamiltonian whose renormalization group flow interpolates between the two models, having the W_{1+\infty} minimal models as infra-red fixed point. 
  Gauge field configurations appropriate for the infrared region of QCD are proposed. Using the usual QCD action, confinement is realized as in the London theory of Meissner effect. 
  The quantum mechanics and thermodynamics of SU(2) non-Abelian Chern-Simons particles (non-Abelian anyons) in an external magnetic field are addressed. We derive the N-body Hamiltonian in the (anti-)holomorphic gauge when the Hilbert space is projected onto the lowest Landau level of the magnetic field. In the presence of an additional harmonic potential, the N-body spectrum depends linearly on the coupling (statistics) parameter. We calculate the second virial coefficient and find that in the strong magnetic field limit it develops a step-wise behavior as a function of the statistics parameter, in contrast to the linear dependence in the case of Abelian anyons. For small enough values of the statistics parameter we relate the N-body partition functions in the lowest Landau level to those of SU(2) bosons and find that the cluster (and virial) coefficients dependence on the statistics parameter cancels. 
  We consider the pure supersymmetric Yang--Mills theories placed on a small 3-dimensional spatial torus with higher orthogonal and exceptional gauge groups. The problem of constructing the quantum vacuum states is reduced to a pure mathematical problem of classifying the flat connections on 3-torus. The latter problem is equivalent to the problem of classification of commuting triples of elements in a connected simply connected compact Lie group which is solved in this paper. In particular, we show that for higher orthogonal SO(N), N > 6, and for all exceptional groups the moduli space of flat connections involves several distinct connected components. The total number of vacuumstates is given in all cases by the dual Coxeter number of the group which agrees with the result obtained earlier with the instanton technique. 
  Integrable systems underlying the Seiberg-Witten solutions for the N=2 SQCD with gauge groups SO(n) and Sp(n) are proposed. They are described by the inhomogeneous XXX spin chain with specific boundary conditions given by reflection matrices. We attribute reflection matrices to orientifold planes in the brane construction and briefly discuss its possible deformations. 
  In the leading log approximation and at large $N_C$ the interaction of two fermionic and one gluonic reggeons is described by an integrable system corresponding to an open spin chain. 
  We derive a universal thermal effective potential, which describes all possible high-temperature instabilities of the known N=4 superstrings, using the properties of gauged N=4 supergravity. These instabilities are due to three non-perturbative thermal dyonic modes, which become tachyonic in a region of the thermal moduli space. The latter is described by three moduli, s,t,u, which are common to all non-perturbative dual-equivalent strings with N=4 supersymmetry in five dimensions: the heterotic on T^4xS^1, the type IIA on K3xS^1, the type IIB on K3xS^1 and the type I on T^4x S^1. The non-perturbative instabilities are analysed. These strings undergo a high-temperature transition to a new phase in which five-branes condense. This phase is described in detail, using both the effective supergravity and non-critical string theory in six dimensions. In the new phase, supersymmetry is perturbatively restored but broken at the non-perturbative level. In the infinite-temperature limit the theory is topological with an N=2 supersymmetry based on a topologically non-trivial hyper-Kahler manifold. 
  We present an identity relating the partition function of N=4 supersymmetric QED to that of its dual under mirror symmetry. The identity is a generalized Fourier transform. Many known properties of abelian theories can be derived from this formula, including the mirror transforms for more general gauge and matter content. We show that N=3 Chern-Simons QED and N=4 QED with BF-type couplings are conformal field theories with exactly marginal couplings. Mirror symmetry acts on these theories as strong-weak coupling duality. After identifying the mirror of the gauge coupling (sometimes called the ``magnetic coupling'') we construct a theory which is exactly mirror -- at all scales -- to N=4 SQED. We also study vortex-creation operators in the large $N_f$ limit. 
  We present a new bound for the worldvolume actions of branes with a Wess-Zumino term. For this we introduce a generalization of calibrations for which the calibration form is not closed. We then apply our construction to find the M-5-brane worldvolume solitons in an AdS background that saturate this bound. We show that these worldvolume solitons are supersymmetric and that they satisfy differential equations which generalize those of standard calibrations. 
  Using s-wave and large N approximation the one-loop effective action for 2d dilaton coupled scalars and spinors which are obtained by spherical reduction of 4d minimal matter is found. Quantum effective equations for reduced Einstein gravity are written. Their analytical solutions corresponding to 4d Kantowski-Sachs (KS) Universe are presented. For quantum-corrected Einstein gravity we get non-singular KS cosmology which represents 1) quantum-corrected KS cosmology which existed on classical level or 2)purely quantum solution which had no classical limit. The analogy with Nariai BH is briefly mentioned. For purely induced gravity (no Einstein term) we found general analytical solution but all KS cosmologies under discussion are singular. The corresponding equations of motion are reformulated as classical mechanics problem of motion of unit mass particle in some potential V. 
  In this thesis, I use the strong coupling expansion to investigate the multiflavor lattice Schwinger models in the hamiltonian formalism using staggered fermions. In particular, I am interested in analysing the mapping of these gauge theories onto quantum spin-1/2 antiferromagnetic Heisenberg chains. Exploting this mapping, the chiral symmetry breaking patterns are studied and the spectra are computed. The extrapolation to zero lattice spacing of the results compare favorably qualitatively and quantitatively with the weak coupling studies of the gauge models in the continuum. 
  We describe the local D=4 field theory on $\kappa$--deformed Minkowski space as nonlocal relativistic field theory on standard Minkowski space--time. For simplicity the case of $\kappa$-deformed scalar field $\phi$ with the interaction $\lambda \phi^{4}$ is considered, and the $\kappa$--deformed interaction vertex is described. It appears that fundamental mass parameter $\kappa$ plays a role of regularizing imaginary Pauli--Villars mass in $\kappa$--deformed propagator. 
  Models of interactions of D-dimensional hypermultiplets and supersymmetric gauge multiplets with $\cN=8$ supercharges $(D{\leq} 6)$ can be formulated in the framework of harmonic superspaces. The effective Coulomb low-energy action for D=5 includes the free and Chern-Simons terms. We consider also the non-Abelian superfield D=5 Chern-Simons action. The biharmonic $D=3,\cN=8$ superspace is introduced for a description of l and r supermultiplets and the mirror symmetry. The D=2,(4,4) gauge theory and hypermultiplet interactions are considered in the triharmonic superspace. Constraints for $D{=}1,\cN{=}8$ supermultiplets are solved with the help of the $SU(2){\times}Spin(5)$ harmonics. Effective gauge actions in the full $D{\leq}3,\cN{=}8$ superspaces contain constrained (harmonic) superpotentials satisfying the (6-D) Laplace equations for the gauge group U(1) or corresponding (6-D)p-dimensional equations for the gauge groups $[U(1)]^p$. Generalized harmonic representations of superpotentials connect equivalent superfield structures of these theories in the full and analytic superspaces. The harmonic approach simplifies the proofs of non-renormalization theorems. 
  This paper discusses in a systematical way exact retarded solutions to the classical SU(N) Yang-Mills equations with the source composed of several colored point particles. A new method of finding such solutions is reviewed. Relying on features of the solutions, a toy model of quark binding is suggested. According to this model, quarks forming a hadron are influenced by no confining force in spite of the presence of a linearly rising term of the potential. The large-N dynamics of quarks conforms well with Witten's phenomenology. On the semiclassical level, hadrons are color neutral in the Gauss law sense. Nevertheless, a specific multiplet structure is observable in the form of the Regge sequences related to infinite-dimensional unitary representations of SL(4,R) which is shown to be the color gauge group of the background field generated by any hadron. The simultaneous consideration of SU(N), SO(N), and Sp(N) as gauge groups offers a plausible explanation of the fact that clusters containing two or three quarks are more stable than multiquark clusters. 
  We present a detailed discussion of the asymptotic symmetries of Anti-de Sitter space in two dimensions and their relationship with the conformal group in one dimension. We use this relationship to give a microscopical derivation of the entropy of 2d black holes that have asymptotically Anti-de Sitter behaviour. The implications of our results for the conjectured AdS_2/CFT_1 duality are also discussed. 
  We construct a topological field theory which, on the one hand, generalizes BF theories in that there is non-trivial coupling to `topological matter fields'; and, on the other, generalizes the three-dimensional model of Carlip and Gegenberg to arbitrary dimensional manifolds. Like the three dimensional model, the theory can be considered to describe a gravitational field interacting with topological matter. In particular, in two dimensions, the model is that of gravity on a torus. In four dimensions, the model is shown to admit constant curvature black hole solutions. 
  We construct the gauge field and graviton propagators in Euclidean AdS(d+1) space-time by two different methods. In the first method the gauge invariant Maxwell or linearized Ricci operator is applied directly to bitensor ansatze for the propagators which reflect their gauge structure. This leads to a rapid determination of the physical part of the propagators in terms of elementary functions. The second method is a more traditional approach using covariant gauge fixing which leads to a solution for both physical and gauge parts of the propagators. The gauge invariant parts agree in both methods. 
  We present a calculation of the matrix theory 2-loop effective action for a D0-brane in the background of the recently discussed D0-D6 bound state configuration. The effective DBI action of a D0-brane probe in the background of the corresponding 4-dimensional non-supersymmetric black hole solution to low-energy type IIA string theory compactified on a 6-torus is known to agree with the matrix theory calculation at 1-loop order, in the limit in which the ratio of the D0-brane to the D6-brane charges carried by the black hole is large. Agreement at 2-loop between the supergravity description and a conjectured nonabelian BDI effective superYang-Mills description has also been recently reported. However, we find uncanceled ultraviolet divergences in our direct matrix theory calculation of the 2-loop effective action. This is consistent with the expected nondecoupling of massive open string states from the 6-brane. 
  We introduce a suitable adapted ordering for the twisted N=2 superconformal algebra (i.e. with mixed boundary conditions for the fermionic fields). We show that the ordering kernels for complete Verma modules have two elements and the ordering kernels for G-closed Verma modules just one. Therefore, spaces of singular vectors may be two-dimensional for complete Verma modules whilst for G-closed Verma modules they can only be one-dimensional. We give all singular vectors for the levels 1/2, 1, and 3/2 for both complete Verma modules and G-closed Verma modules. We also give explicit examples of degenerate cases with two-dimensional singular vector spaces in complete Verma modules. General expressions are conjectured for the relevant terms of all (primitive) singular vectors, i.e. for the coefficients with respect to the ordering kernel. These expressions allow to identify all degenerate cases as well as all G-closed singular vectors. They also lead to the discovery of subsingular vectors for the twisted N=2 superconformal algebra. Explicit examples of these subsingular vectors are given for the levels 1/2, 1, and 3/2. Finally, the multiplication rules for singular vector operators are derived using the ordering kernel coefficients. This sets the basis for the analysis of the twisted N=2 embedding diagrams. 
  The system of D2 branes localized on or near D6 branes is considered. The world-volume theory on the D2 branes is investigated, using its conjectured relation to the near-horizon geometry. The results are in agreement with known facts and expectations for the corresponding field theory and a rich phase structure is obtained as a function of the energy scale and the number of branes. In particular, for an intermediate range of the number of D6 branes, the IR geometry is that of an AdS_4 space fibered over a compact space. This D2-D6 system is compared to other systems, related to it by compactification and duality and it is shown that the qualitative differences have compatible explanations in the geometric and field-theoretic descriptions. Another system -- that of NS5 branes located at D6 branes -- is also briefly studied, leading to a similar phase structure. 
  We discuss the quantization of the Green-Schwarz string action on $AdS_5 \times S^5$. We construct consistent, globally well-defined, gauge fixing choices for kappa symmetry and worldsheet diffeomorphism invariance. We then proceed to quantize the theory in a perturbation series in the inverse radius of curvature, in a background field expansion. We discuss vertex operators and correlation functions, and demonstrate agreement with supergravity results in the appropriate limit. 
  The finiteness properties of the N=4 supersymmetric Yang-Mills theory are reanalyzed both in the component formulation and using N=1 superfields, in order to discuss some subtleties that emerge in the computation of gauge dependent quantities. The one-loop corrections to various Green functions of elementary fields are calculated. In the component formulation it is shown that the choice of the Wess-Zumino gauge, that is standard in supersymmetric gauge theories, introduces ultraviolet divergences in the propagators at the one-loop level. Such divergences are exactly cancelled when the contributions of the fields that are put to zero in the Wess-Zumino gauge are taken into account. In the description in terms of N=1 superfields infrared divergences are found for every choice of gauge different from the supersymmetric generalization of the Fermi-Feynman gauge. Two-, three- and four-point functions of N=1 superfields are computed and some general features of the infrared problem are discussed. We also examine the effect of the introduction of mass terms for the (anti) chiral superfields in the theory, which break supersymmetry from N=4 to N=1. It is shown that in the mass deformed model no ultraviolet divergences appear in two-point functions. It argued that this result can be generalized to n-point functions, supporting the proposal of a possible of use of this modified model as a supersymmetry-preserving regularization scheme for N=1 theories. 
  A gauge invariant formulation for the massive axion is considered. The axion acquires mass through a topological term which couples a (pseudo)scalar and a third rank antisymmetric tensor. Duality, local and canonical equivalences with the non-gauge invariant proposal are established. The supersymmetric version of the gauge invariant model is constructed. 
  We define gravitational mass in asymptotically de Sitter space-times with compactified dimension. It was shown that the mass can be negative for space-time with matter spreading beyond the cosmological horizon scale or large outward `momentum' in four dimension. We give simple examples with negative energy in higher dimensions even if the matter is not beyond horizon or system does not have large `momentum'. They do not have the lower bound on the mass. We also give a positive energy argument in higher dimensions and realise that elementary fermion cannot exist in our examples. 
  A derivation of the standard symmetry and leptoquark family structure is presented that is more straightforward than a previous derivation. 
  We review recent developments in the theory of supermembranes and their relation to matrix models. 
  The AdS/CFT conjecture relates quantum gravity on Anti-de Sitter (AdS) space to a conformal field theory (CFT) defined on the spacetime boundary. We interpret the CFT in terms of natural analogues of the bulk S-matrix. Our first approach finds the bulk S-matrix as a limit of scattering from an AdS bubble immersed in a space admitting asymptotic states. Next, we show how the periodicity of geodesics obstructs a standard LSZ prescription for scattering within global AdS. To avoid this subtlety we partition global AdS into patches within which CFT correlators reconstruct transition amplitudes of AdS states. Finally, we use the AdS/CFT duality to propose a large N collective field theory that describes local, perturbative supergravity. Failure of locality in quantum gravity should be related to the difference between the collective 1/N expansion and genuine finite N dynamics. 
  We prove one of conjectures, raised by Dorey and Tateo in the connection among the spectral determinant of anharmonic oscillator and vacuum eigenvalues of transfer matrices in field theory and statistical mechanics. The exact sequence of $U_q(\hat{sl}_2)$ plays a fundamental role in the proof. 
  In this paper we investigate a generation mechanism of the non-Abelian gauge fields in the SU(N) gauge theory. It is shown that the SU(N) gauge fields ensuring the local invariance of the theory are generated at the quantum level only due to nonsmoothness of the scalar phases of the fundamental spinor fields. The expression for the gauge fields are obtained in terms of the nonsmooth scalar phases. 
  We study large dimensions and low string scale in four-dimensional compactifications of type II theories of closed strings at weak coupling. We find that the fundamental string scale, together with all compact dimensions, can be at the TeV, while the smallness of the string coupling accounts for the weakness of gravitational interactions. This is in contrast to the situation recently studied in type I theories, where the string scale can be lowered only at the expense of introducing large transverse dimensions felt by gravity only. As a result, in these type II strings, there are no strong gravity effects at the TeV, and the main experimental signature is the production of Kaluza-Klein excitations with gauge interactions. In the context of type IIB theories, we find a new possibility providing a first instance of large non-transverse dimensions at weak coupling: two of the internal dimensions seen by gauge interactions can be at the TeV, with the string scale and all other dimensions at intermediate energies of the order of $10^{11}$ GeV, where gravity becomes also strong. Finally, using duality, we provide a perturbative description for the generic case of large dimensions in the heterotic string. In particular, we show that the two type II theories above describe the cases of one and two heterotic large dimensions. A new M-theory derivation of heterotic-type II duality is instrumental for this discussion. 
  A particularly simple class of nonselfdual solutions are obtained for gauge fields in Schwarzschild and deSitter backgrounds. For Lorentz signature these have finite energy and finite action for Euclidean signature. In each case one obtains either real or a pair of complex conjugate solutions. The actions are easily computed for any dimension d. Numerical values are given for d= 4,6,7,8,9,10. It is explained why d=5 is a very special case. Possible continuations and generalizations of the results obtained are indicated. A particular solution for AdS_4 background is presented in the Appendix. 
  Calculations of the Casimir energy for spherical geometries which are based on integrations of the stress tensor are critically examined. It is shown that despite their apparent agreement with numerical results obtained from mode summation methods, they contain a number of serious errors. Specifically, these include (1) an improper application of the stress tensor to spherical boundaries, (2) the neglect of pole terms in contour integrations, and (3) the imposition of inappropriate boundary conditions upon the relevant propagators. A calculation which is based on the stress tensor and which avoids such problems is shown to be possible. It is, however, equivalent to the mode summation method and does not therefore constitute an independent calculation of the Casimir energy. 
  We discuss the thermal properties of string gases propagating in various D-brane backgrounds in the weak-coupling limit, and at temperatures close to the Hagedorn temperature. We determine, in the canonical ensemble, whether the Hagedorn temperature is limiting or non-limiting. This depends on the dimensionality of the D-brane, and the size of the compact dimensions. We find that in many cases the non-limiting behaviour manifest in the canonical ensemble is modified to a limiting behaviour in the microcanonical ensemble and show that, when there are different systems in thermal contact, the energy flows into open strings on the `limiting' D-branes of largest dimensionality. Such energy densities may eventually exceed the D-brane intrinsic tension. We discuss possible implications of this for the survival of Dp-branes with large values of p in an early cosmological Hagedorn regime. We also discuss the general phase diagram of the interacting theory, as implied by the holographic and black-hole/string correspondence principles. 
  The exclusion principle of Maldacena and Strominger is seen to follow from deformed Heisenberg algebras associated with the chiral rings of S_N orbifold CFTs. These deformed algebras are related to quantum groups at roots of unity, and are interpreted as algebras of space-time field creation and annihilation operators. We also propose, as space-time origin of the stringy exclusion principle, that the $ADS_3 \times S^3$ space-time of the associated six-dimensional supergravity theory acquires, when quantum effects are taken into account, a non-commutative structure given by $SU_q(1,1) \times SU_q (2)$. Both remarks imply that finite N effects are captured by quantum groups $SL_q(2)$ with $q= e^{{i \pi \over {N + 1}}}$. This implies that a proper framework for the theories in question is given by gravity on a non-commutative spacetime with a q-deformation of field oscillators. An interesting consequence of this framework is a holographic interpretation for a product structure in the space of all unitary representations of the non-compact quantum group $SU_q(1,1)$ at roots of unity. 
  A simple algebraic technique is developed to obtain deformed energy spectra for the P\"oschl-Teller potentials. 
  We study a massive Thirring-like model in 2-dimensional space-time, which contains fermions with arbitrary number (N) of different species. This model is an extension of that of a previous paper, where we have considered two-species case. By this extension we expect that we can expose more general structures of this kind of model. We obtain the equivalent boson model with N species to our fermion model. We find that the coupling constant must be set in some regions in order for the model to be physically sensible. It seems hard to find such regions from direct obsavation of the original fermion model. We also find that for specific values of the coupling constant some of the boson fields disappear from the system. Therefore, the N-species fermion model is described by the boson model with fewer species. 
  A regular method is suggested for constructing vortex-like solutions with cylindrical symmetry in the Skyrme-Einstein chiral model. The method is based on the expansion of metric and field functions in power series with respect to the two small parameters entering the model. The length mass density of the vortex is estimated. 
  The N=1 self-dual supergravity has SL(2.C) symmetry. This symmetry results in SU(2) charges as the angular-momentum. As in the non-supersymmetric self-dual gravity, the currents are also of their potentials and are therefore identically conserved. The charges are generally invariant and gauge covariant under local SU(2) transforms approaching to be rigid at spatial infinity. The Poisson brackets constitute su(2) algebra and hence can be interpreted as the generally covariant conservative angular-momentum. 
  We give an explicit demonstration that the derivative expansion of the QED effective action is a divergent but Borel summable asymptotic series, for a particular inhomogeneous background magnetic field. A duality transformation B\to iE gives a non-Borel-summable perturbative series for a time dependent background electric field, and Borel dispersion relations yield the non-perturbative imaginary part of the effective action, which determines the pair production probability. Resummations of leading Borel approximations exponentiate to give perturbative corrections to the exponents in the non-perturbative pair production rates. Comparison with a WKB analysis suggests that these divergence properties are general features of derivative expansions and effective actions. 
  We study some dynamical properties of a Dirac field in 2+1 dimensions with spacetime dependent domain wall defects. We show that the Callan and Harvey mechanism applies even to the case of defects of arbitrary shape, and in a general state of motion. The resulting chiral zero modes are localized on the worldsheet of the defect, an embedded curved two dimensional manifold. The dynamics of these zero modes is governed by the corresponding induced metric and spin connection. Using known results about determinants and anomalies for fermions on surfaces embedded in higher dimensional spacetimes, we show that the chiral anomaly for this two dimensional theory is responsible for the generation of a current along the defect. We derive the general expression for such a current in terms of the geometry of the defect, and show that it may be interpreted as due to an "inertial" electric field, which can be expressed entirely in terms of the spacetime curvature of the defects. We discuss the application of this framework to fermionic systems with defects in condensed matter. 
  We present details of a geometric method to associate a Lie superalgebra with a large class of bosonic supergravity vacua of the type AdS x X, corresponding to elementary branes in M-theory and type II string theory. 
  The most general black M5-brane solution of eleven-dimensional supergravity (with a flat R^4 spacetime in the brane and a regular horizon) is characterized by charge, mass and two angular momenta. We use this metric to construct general dual models of large-N QCD (at strong coupling) that depend on two free parameters. The mass spectrum of scalar particles is determined analytically (in the WKB approximation) and numerically in the whole two-dimensional parameter space. We compare the mass spectrum with analogous results from lattice calculations, and find that the supergravity predictions are close to the lattice results everywhere on the two dimensional parameter space except along a special line. We also examine the mass spectrum of the supergravity Kaluza-Klein (KK) modes and find that the KK modes along the compact D-brane coordinate decouple from the spectrum for large angular momenta. There are however KK modes charged under a U(1)xU(1) global symmetry which do not decouple anywhere on the parameter space. General formulas for the string tension and action are also given. 
  We study the potential between a string and an anti-string source in M5-theory by using the adS/CFT duality conjecture. We find that the next to leading order corrections in a saddle point approximation renormalize the classical result. 
  We propose a mechanism to break the translational invariance of compactified space spontaneously. As a simple model, we study a real $\phi^4$ model compactified on $M^{D-1}\otimes S^1$ in detail, where we impose a nontrivial boundary condition on $\phi$ for the $S^1$-direction. It is shown that the translational invariance for the $S^1$-direction is spontaneously broken when the radius $R$ of $S^1$ becomes larger than a critical radius $R^*$ and also that the model behaves like a $\phi^4$ model on a single kink background for $R \to \infty$. It is pointed out that spontaneous breakdown of translational invariance is accompanied by that of some global symmetries, in general, in our mechanism. 
  We propose a new spontaneous supersymmetry breaking mechanism, in which extra compact dimensions play an important role. To illustrate our mechanism, we study a simple model consisting of two chiral superfields, where one spatial dimension is compactified on a circle $S^1$. It is shown that supersymmetry is spontaneously broken irrespective of the radius of the circle, and also that the translational invariance for the $S^1$-direction and a global symmetry are spontaneously broken when the radius becomes larger than a critical radius. These results are expected to be general features of our mechanism. We further discuss that our mechanism may be observed as the O'Raifeartaigh type of supersymmetry breaking at low energies. 
  Inflationary solutions are constructed in a specific five-dimensional model with boundaries motivated by heterotic M-theory. We concentrate on the case where the vacuum energy is provided by potentials on those boundaries. It is pointed out that the presence of such potentials necessarily excites bulk Kaluza-Klein modes. We distinguish a linear and a non-linear regime for those modes. In the linear regime, inflation can be discussed in an effective four-dimensional theory in the conventional way. We lift a four-dimensional inflating solution up to five dimensions where it represents an inflating domain wall pair. This shows explicitly the inhomogeneity in the fifth dimension. We also demonstrate the existence of inflating solutions with unconventional properties in the non-linear regime. Specifically, we find solutions with and without an horizon between the two boundaries. These solutions have certain problems associated with the stability of the additional dimension and the persistence of initial excitations of the Kaluza-Klein modes. 
  Using the properties of gauged N=4 supergravity, we show that it is possible to derive a universal thermal effective potential that describes all possible high-temperature instabilities of the known N=4 superstrings. These instabilities are due to non-perturbative dyonic modes, which become tachyonic in a region of the thermal moduli space M={s,t,u}; M is common to all non-perturbative dual-equivalent N=4 superstrings in five dimensions. We analyse the non-perturbative thermal potential and show the existence of a phase transition at high temperatures corresponding to a condensation of 5-branes. This phase is described in detail, using an effective non-critical string theory. 
  We discuss the gauge theory mechanisms which are responsible for the causal structure of the dual supergravity. For D-brane probes we show that the light cone structure and Killing horizons of supergravity emerge dynamically. They are associated with the appearance of new light degrees of freedom in the gauge theory, which we explicitly identify. This provides a picture of physics at the horizon of a black hole as seen by a D-brane probe. 
  We argue that there are generic solutions to the type 0 gravity equations of motion that are confining in the infrared and have log scaling in the ultraviolet. The background curvature generically diverges in the IR. Nevertheless, there exist solutions where higher order string corrections appear to be exponentially suppressed in the IR with respect to the leading type 0 gravity terms. For these solutions the tachyon flows to a fixed value.  We show that the generic solutions lead to a long range linear quark potential, magnetic screening and a discrete glueball spectrum. We also estimate some WKB glueball mass ratios and compare them to ratios found using finite temperature models and lattice computations. 
  Finite temperature correlation functions in integrable quantum field theories are formulated only in terms of the usual, temperature-independent form factors, and certain thermodynamic filling fractions which are determined from the thermodynamic Bethe ansatz. Explicit expressions are given for the one and two-point functions. 
  Spherically and cylindrically symmetric solutions of SU(3) Yang-Mills theory are found, whose gauge potentials have confining properties. The spherically symmetric solutions give field distributions which have a spherical surface on which the gauge fields become infinite (which is similar to bag models of confinement), and the other solution has a potential which increases at large distances. The cylindrically symmetric solution describes a classical field "string" (flux tube) of the kind which is expected to form between quarks in the dual superconductor picture of confinement. These solutions with classical confining behaviour appear to be typical solutions for the classical SU(3) Yang-Mills equations. This implies that the confining properties of the classical SU(3) Yang-Mills theory are general properties of this theory. 
  We propose a method to construct quantum theory of matter fields in a topology changing universe. Analytic continuation of the semiclassical gravity of a Lorentzian geometry leads to a non-unitary Schr\"{o}dinger equation in a Euclidean region of spacetime, which does not have a direct interpretation of quantum theory of the Minkowski spacetime. In this Euclidean region we quantize the Euclidean geometry, derive the time-dependent Schr\"{o}dinger equation and find the quantum states using the Liouville-Neumann method. The Wick rotation of these quantum states provides the correct Hilbert space of matter field in the Euclidean region of the Lorentzian geometry. It is found that the direct quantization of a scalar field in the Lorentzian geometry involves an unusual commutation rule in the Euclidean region. Finally we discuss the interpretation of the periodic solution of the semiclassical gravity equation in the Euclidean geometry as a finite temperature solution for the gravity-matter system in the Lorentzian geometry. 
  The correspondence between the ground states of the BFSS matrix model and the type IIA string is investigated through the 11th direction. We derive the type IIA string from 11 dimensional supermembrane wrapped around the 11th direction of periodicity without a procedure of the double dimensional reduction, but by taking the large limit of radius R in the 11th direction. It is shown that the center of mass of this string as a function on the 11th coordinate has the same matrix description as the BFSS matrix model in the flat direction. The fact shows that the matrix model at the strong string coupling in the flat direction is directly connected with the 11 dimensional supermembrane theory in the transverse light-cone frame. 
  We study a Wess-Zumino-Witten model with target space AdS_3 x (S^3 x S^3 x S^1)/Z_2. This allows us to construct space-time N=3 superconformal theories. By combining left-, and right-moving parts through a GSO and a Z_2 projections, a new asymmetric (N,\bar{N})=(3,1) model is obtained. It has an extra gauge (affine) SU(2) symmetry in the target space of the type IIA string. An associated configuration is realized as slantwise intersecting M5-M2 branes with a Z_2-fixed plane in the M-theory viewpoint. 
  Recently a TeV-scale Supersymmetric Standard Model (TSSM) was proposed in which the gauge coupling unification is as precise (at one loop) as in the MSSM, and occurs in the TeV range. Proton stability in the TSSM is due to an anomaly free Z_3 X Z_3 discrete gauge symmetry, which is also essential for successfully generating neutrino masses in the desirable range. In this paper we show that the TSSM admits anomaly free non-Abelian discrete flavor gauge symmetries (based on a left-right product tetrahedral group) which together with a ``vector-like'' Abelian (discrete) flavor gauge symmetry suppresses dangerous higher dimensional operators corresponding to flavor changing neutral currents (FCNCs) to an acceptable level. Discrete flavor gauge symmetries are more advantageous compared with continuous flavor gauge symmetries as the latter must be broken, which generically results in unacceptably large gauge mediated flavor violation. In contrast, in the case of discrete flavor gauge symmetries the only possibly dangerous sources of flavor violation either come from the corresponding ``bulk'' flavon (that is, flavor symmetry breaking Higgs) exchanges, or are induced by flavon VEVs. These sources of flavor violation, however, are adequately suppressed by the above flavor gauge symmetries for the string scale \sim 10-100 TeV 
  A new type of non-Abelian generalization of the Born-Infeld action is proposed, in which the spacetime indices and group indices are combined. The action is manifestly Lorentz and gauge invariant. In its power expansion, the lowest order term is the Yang-Mills action and the second term corresponds to the bosonic stringy correction to this action. Solutions of the Euler-Lagrange equation for the SU(2) case are considered and we show that there exists an instanton-like solution which has winding number one and finite action. 
  The N=1 self-dual supergravity has SL(2,C) and the left-handed and right -handed local supersymmetries. These symmetries result in SU(2) charges as the angular-momentum and the supercharges. The model possesses also the invariance under the general translation transforms and this invariance leads to the energy-momentum. All the definitions are generally covariant . As the SU(2) charges and the energy-momentum we obtained previously constituting the 3-Poincare algebra in the Ashtekar's complex gravity, the SU(2) charges, the supercharges and the energy-momentum here also restore the super-Poincare algebra, and this serves to support the reasonableness of their interpretations. 
  In this paper, we extend our earlier one loop analysis to two loops and give a simple diagrammatic description for the retarded Greens functions at finite temperature, in terms of forward scattering amplitudes of on-shell thermal particles. We present a simple discussion, which can be easily generalized to any field theory, of the temperature dependent parts of the retarded two and three point functions in scalar field theory and QED. As an application of our result at two loops, we show how the infrared singularities in the thermal part of the retarded photon self-energy, cancel in QED_2 in the limit of vanishing electron mass. 
  We apply the formalism of extended BRS symmetry to the investigation of the gauge dependence of the effective potential in a spontaneously symmetry broken gauge theory. This formalism, which includes a set of Grassmann parameters defined as the BRS variations of the gauge-fixing parameters, allows us to derive in a quick and unambiguous way the related Nielsen identities, which express the physical gauge independence, in a class of generalized 't Hooft gauges, of the effective potential. We show in particular that the validity of the Nielsen identities does not require any constraint on the gauge-fixing parameters, to the contrary of some claims found in the literature. We use the method of algebraic renormalization, which leads to results independent of the particular renormalization scheme used. 
  The full set of polynomial solutions of the nested Bethe Ansatz is constructed for the case of A_2 rational spin chain. The structure and properties of these associated solutions are more various then in the case of usual XXX (A_1) spin chain but their role is similar. 
  In this letter we shall discuss a description of non-supersymmetric four-dimensional Yang-Mills theory based on Type 0 strings recently proposed by Klebanov and Tseytlin. The three brane near-horizon geometry allows one to study the UV behaviour of the gauge theory. Following Minahan and Klebanov and Tseytlin we shall discuss how the gravity solution reproduces logarithmic renormalization of coupling constant $g_e$ extracted from quark-antiquark potential and then show that effective coupling constant $g_m$ describing monopole-antimonopole interactions is of zero-charge type and Dirac condition $g_e g_m = 1$ is scale invariant in logarithmic approximation. 
  The high temperature limit of a system of two D-0 branes is investigated. The partition function can be expressed as a power series in $\beta$ (inverse temperature). The leading term in the high temperature expression of the partition function and effective potential is calculated exactly. Physical quantities like the mean square separation can also be exactly determined in the high temperature limit. 
  We propose that in time dependent backgrounds the holographic principle should be replaced by the generalized second law of thermodynamics. For isotropic open and flat universes with a fixed equation of state, the generalized second law agrees with the cosmological holographic principle proposed by Fischler and Susskind. However, in more complicated spacetimes the two proposals disagree. A modified form of the holographic bound that applies to a post-inflationary universe follows from the generalized second law. However, in a spatially closed universe, or inside a black hole event horizon, there is no simple relationship that connects the area of a region to the maximum entropy it can contain. 
  We introduce a general method for transforming the equations of motion following from a Das-Jevicki-Sakita Hamiltonian, with boundary conditions, into a boundary value problem in one-dimensional quantum mechanics. For the particular case of a one-dimensional chain of interacting NxN Hermitean matrices, the corresponding large N boundary value problem is mapped into a linear Fredholm equation with Hilbert-Schmidt type kernel. The equivalence of this kernel, in special cases, to a second order differential operator allows us recover all previously known explicit solutions for the matrix eigenvalues. In the general case, the distribution of eigenvalues is formally derived through a series of saddle-point approximations. The critical behaviour of the system, including a previously observed Kosterlitz-Thouless transition, is interpreted in terms of the stationary points. In particular we show that a previously conjectured infinite series of sub-leading critical points are due to expansion about unstable stationary points and consequently not realized. 
  Canonical quantization of three dimensional gravity in the first order formalism suggests that one should allow singular solutions. This paper addresses the importance of singular solutions in the path integral approach to quantum gravity. Using a simple ansatz for the dreibein and the spin connection in the de-Sitter and the anti-de-Sitter spaces we propose that the sum over the 3D manifolds in the path integral should be extended to include 2D surfaces. 
  It is shown using both conventional and algebraic approach to quantum field theory that it is impossible to perform quantization on Unruh modes in Minkowski spacetime. Such quantization implies setting boundary condition for the quantum field operator which changes topological properties and symmetry group of spacetime and leads to field theory in two disconnected left and right Rindler spacetimes. It means that "Unruh effect" does not exist. 
  M-theory suggests the study of 11-dimensional space-times compactified on some 7-manifolds. From its intimate relation to superstrings, one possible class of such 7-manifolds are those that have Calabi-Yau threefolds as boundary. In this article, we construct a special class of such 7-manifolds, named as {\it K3-Thurston} (K3T) 7-manifolds. The factor from the K3 part of the deformation space of these K3T 7-manifolds admits a K\"{a}hler structure, while the factor of the deformation space from the Thurston part admits a special K\"{a}hler structure. The latter rings with the nature of the scalar manifold of a vector multiplet in an N=2 $d=4$ supersymmetric gauge theory. Remarks and examples on more general K3T 7-manifolds and issues to possible interfaces of K3T to M-theory are also discussed. 
  We calculate the absorption coefficient of scalar field on the background of the two-dimensional AdS black hole, which is of relevance to Hawking radiation. For the massless scalar field, we find that there does not exist any massless radiation. 
  We construct a topological invariant of the renormalization group trajectories of a large class of 2D quantum integrable models, described by the thermodynamic Bethe ansatz approach. A geometrical description of this invariant in terms of triangulations of three-dimensional manifolds is proposed and associated dilogarithm identities are proven. 
  We study some aspects of short-distance interaction between parallel D3-branes in type 0 string theory as described by the corresponding world-volume gauge theory. We compute the one-loop effective potential in the non-supersymmetric SU(N) x SU(N) gauge theory (which is a Z_2 projection of the U(2N) n=4 SYM theory) representing dyonic branes composed of N electric and N magnetic D3-branes. The branes of the same type repel at short distances, but an electric and a magnetic brane attract, and the forces between self-dual branes cancel. The self-dual configuration (with the positions of the electric and the magnetic branes, i.e. the diagonal entries of the adjoint scalar fields, being the same) is stable against separation of one electric or one magnetic brane, but is unstable against certain modes of separation of several same-type branes. This instability should be suppressed in the large N limit, i.e. should be irrelevant for the large N CFT interpretation of the gauge theory suggested in hep-th/9901101. 
  We obtain a generally covariant conservation law of energy-momentum for gravitational anyons by the general displacement transform. The energy-momentum currents have also superpotentials and are therefore identically conserved. It is shown that for Deser's solution and Clement's solution, the energy vanishes. The reasonableness of the definition of energy-momentum may be confirmed by the solution for pure Einstein gravity which is a limit of vanishing Chern-Simons coulping of gravitational anyons. 
  Classical and quantum gravitational instabilities, can, respectively, inflate and warm up a primordial Universe satisfying a superstring-motivated principle of "Asymptotic Past Triviality". A physically viable big bang is thus generated without invoking either large fine-tunings or a long period of post-big bang inflation. Properties of the pre-bangian Universe can be probed through its observable relics, which include: i) a (possibly observable) stochastic gravitational-wave background; ii) a (possible) new mechanism for seeding the galactic magnetic fields; iii) a (possible) new source of large-scale structure and CMB anisotropy. 
  We review a formalism of superstring quantization with manifest six-dimensional spacetime supersymmetry, and apply it to AdS_3 x S^3 backgrounds with Ramond-Ramond flux. The resulting description is a conformal field theory based on a sigma model whose target space is a certain supergroup SU'(2|2). 
  The superstring is quantized in a manner which manifestly preserves a U(5) subgroup of the (Wick-rotated) ten-dimensional super-Poincar\'e invariance. This description of the superstring contains critical N=2 worldsheet superconformal invariance and is a natural covariantization of the U(4)-invariant light-cone Green-Schwarz description. 
  We begin by presenting the superparticle action in the background of N=2, D=4 supergravity coupled to n vector multiplets interacting via an arbitrary special Kahler geometry. Our construction is based on implementing kappa-supersymmetry. In particular, our result can be interpreted as the source term for N=2 BPS black holes with a finite horizon area. When the vector multiplets can be associated to the complex structure moduli of a Calabi-Yau manifold, then our 0-brane action can be derived by wrapping 3-branes around 3-cycles of the 3-fold. Our result can be extended to the case of higher supersymmetry; we explicitly construct the kappa supersymmetric action for a superparticle moving in an arbitrary N=8 supergravity background with 1/2, 1/4 or 1/8 residual supersymmetry. 
  We study dynamical aspects of holographic correspondence between d=5 anti-de Sitter supergravity and $d=4$ super Yang-Mills theory. We probe causality and locality of ambient spacetime from super Yang-Mills theory by studying transmission of low-energy brane waves via an open string stretched between two D3-branes in Coulomb branch. By analyzing two relevant physical threshold scales, we find that causality and locality is encoded in the super Yang-Mills theory provided infinite tower of long supermultiplet operators are added. Massive W-boson and dual magnetic monopole behave more properly as extended, bilocal objects. We also study causal time-delay of low-energy excitation on heavy quark or meson and find an excellent agreement between anti-de Sitter supergravity and super Yang-Mills theory descriptions. We observe that strong `t Hooft coupling dynamics and holographic scale-size relation thereof play a crucial role to the agreement of dynamical processes. 
  We show that the transformation of D-branes under T-duality on four-torus is represented by Nahm transform of instantons. The argument for this allows us to generalize Nahm transform to the case of orthogonal and symplectic gauge groups as well as to instantons on Z_2 orbifold of four-torus. In addition, we identify the isomorphism of K-theory groups that realizes the transformation of D-brane charges under T-duality on torus of arbitrary dimensions. By the isomorphism we are lead to identify the correct K-theory group that classifies D-brane charges in Type II orientifold. 
  The method of flow equations is applied to QED on the light front. Requiring that the particle number conserving terms in the Hamiltonian are considered to be diagonal and the other terms off-diagonal an effective Hamiltonian is obtained which reduces the positronium problem to a two-particle problem, since the particle number violating contributions are eliminated.   Using an effective electron-positron Hamiltonian, obtained in the second order in coupling, we analyze the positronium bound state problem analytically and numerically. The results obtained for Bohr spectrum and hyperfine splitting coincide to a high accuracy with experimental values. The rotational invariance, that is not manifest symmetry on the light-front, is recovered for positronium mass spectrum.   Except for the longitudinal infrared divergences, that are special for the light-front gauge calculations, no infrared divergences appear. The ultraviolet renormalization in the second order in coupling constant is performed simultaneously. To preserve boost invariance we take into account the diagrams arising from the normal ordering of instantaneous interactions. Using flow equations and coupling coherence we obtain the counterterms for electron and photon masses, which are free from longitudinal infrared divergences. 
  By using the approach of non-commutative geometry, we study spinors and scalars on the two layers AdS$_{d+1}$ space. We have found that in the boundary of two layers AdS$_{d+1}$ space, by using the AdS/CFT correspondence, we have a logarithmic conformal field theory. This observation propose a way to get the quantum field theory in the context of non-commutative geometry. 
  We show that the tachyonic kink solution on a pair of D-p-branes in the bosonic string theory can be identified with the D-(p-1)-brane of the same theory. We also speculate on the possibility of obtaining the D-(p-1)-brane as a tachyonic lump on a single D-p-brane. We suggest a possible reinterpretation of the first result which unifies these two apparently different descriptions of the D-(p-1) brane. 
  The recent proposal by Polchinski and Susskind for the holographic flat space S-matrix is discussed. By using Feynman diagrams we argue that in principle all the information about the S-matrix in the interacting field theory in the bulk of the anti-de Sitter space is encoded into the data on the timelike boundary. The problem of locality of interpolating field is discussed and it is suggested that the interpolating field lives in a quantum Boltzmannian Hilbert space. 
  A universal effective supergravity Lagrangian describing the thermal phases of heterotic strings on T^4 x S^1, IIA and IIB strings on K^3 x S^1 is constructed. The resulting non-perturbative phase structure is discussed. 
  The boundary charges which constitute the Virasoro algebra in 2+1 dimensional anti-de Sitter gravity are derived by way of Noether theorem and diffeomorphic invariance. It shows that the boundary charges under discussion recently exhaust all the independent nontrivial charges available. Therefore, the state counting via the Virasoro algebra is complete. 
  A free fermionic string quantum model is constructed directly in the light-cone variables in the range of dimensions $1<d<10$. It is shown that after the GSO projection this model is equivalent to the fermionic massive string and to the non-critical Rammond-Neveu-Schwarz string. The spin spectrum of the model is analysed. For $d=4$ the character generating functions is obtained and the particle content of first few levels is numerically calculated. 
  We show that the $\CN=0$ theories on the self-dual D3-branes of Type 0 string theory are in the class of the previously considered tadpole-free orbifolds of $\CN = 4$ theory (although they have SO(6) global symmetry) and hence have vanishing beta function in the planar limit to all orders in 't Hooft coupling. Also, all planar amplitudes in this theory are equal to those of $\CN = 4$ theory, up to a rescaling of the coupling. 
  The metric on the moduli space of charge (2,1) SU(3) Bogomolny-Prasad-Sommerfield monopoles is calculated and investigated. The hyperKahler quotient construction is used to provide an alternative derivation of the metric. Various properties of the metric are derived using the hyperKahler quotient construction and the correspondence between BPS monopoles and rational maps. Several interesting limits of the metric are also considered. 
  Using the recently proposed non-linear gauge condition, we show the area law behavior of the Wilson loop and the linear dependence of the instantaneous gluon propagator. The field configurations responsible for confinement are those in the non-linear sector of the gauge-fixing condition (the linear sector being the Coulomb gauge). The non-linear sector is actually composed of "Gribov horizons" on the surfaces parallel to the Coulomb surface. In this sector, the gauge field can be expressed in terms of a scalar field and a new vector field. The effective dynamics of the scalar field suggests non-perturbative effects. This was confirmed by showing that all spherically symmetric (in 4-D Euclidean) scalar fields are classical solutions and averaging these solutions using a qaussian distribution (thereby treating these fields as random) lead to confinement. In essence the confinement mechanism is not quantum mechanical in nature but simply a statistical treatment of classical spherically symmetric fields on the "horizons" of the surfaces parallel to the Coulomb surface. 
  We investigate one-matrix correlation functions for finite SU(N) Yang-Mills integrals with and without supersymmetry. We propose novel convergence conditions for these correlators which we determine from the one-loop perturbative effective action. These conditions are found to agree with non-perturbative Monte Carlo calculations for various gauge groups and dimensions. Our results yield important insights into the eigenvalue distributions rho(lambda) of these random matrix models. For the bosonic models, we find that the spectral densities rho(lambda) possess moments of all orders as N -> Infinity. In the supersymmetric case, rho(lambda) is a wide distribution with an N-independent asymptotic behavior rho(lambda) ~ lambda^(-3), lambda^(-7), lambda^(-15) for dimensions D=4,6,10, respectively. 
  In this paper we present a renormalizability proof for spontaneously broken SU(2) gauge theory. It is based on Flow Equations, i.e. on the Wilson renormalization group adapted to perturbation theory. The power counting part of the proof, which is conceptually and technically simple, follows the same lines as that for any other renormalizable theory. The main difficulty stems from the fact that the regularization violates gauge invariance. We prove that there exists a class of renormalization conditions such that the renormalized Green functions satisfy the Slavnov-Taylor identities of SU(2) Yang-Mills theory on which the gauge invariance of the renormalized theory is based. 
  Lectures at the 1998 Les Houches Summer School: Topological Aspects of Low Dimensional Systems. These lectures contain an introduction to various aspects of Chern-Simons gauge theory: (i) basics of planar field theory, (ii) canonical quantization of Chern-Simons theory, (iii) Chern-Simons vortices, and (iv) radiatively induced Chern-Simons terms. 
  In this paper we describe how Grothendieck groups of coherent sheaves and locally free sheaves can be used to describe type II D-branes, in the case that all D-branes are wrapped on complex varieties and all connections are holomorphic. Our proposal is in the same spirit as recent discussions of K-theory and D-branes; within the restricted class mentioned, Grothendieck groups encode a choice of connection on each D-brane worldvolume, in addition to information about the smooth bundles. We also point out that derived categories can also be used to give insight into D-brane constructions, and analyze how a Z_2 subset of the T-duality group acting on D-branes on tori can be understood in terms of a Fourier-Mukai transformation. 
  Quark spectra in QCD are linked to fundamental properties of the theory including the identification of pions as the Goldstone bosons of spontaneously broken chiral symmetry. The lattice Overlap-Dirac operator provides a nonperturbative, ultraviolet-regularized description of quarks with the correct chiral symmetry. Properties of the spectrum of this operator and their relation to random matrix theory are studied here. In particular, the predictions from chiral random matrix theory in topologically non-trivial gauge field sectors are tested for the first time. 
  We study brane configurations for four dimensional N=1 supersymmetric gauge theories with quartic superpotentials which flow in the infrared to manifolds of interacting superconformal fixed points. We enumerate finite N=2 theories, from which a large class of marginal N=1 theories descend. We give the brane descriptions of these theories in Type IIA and Type IIB string theory. The Type IIB descriptions are in terms of D3 branes in orientifold and generalized conifold backgrounds. We calculate the Weyl and Euler anomalies in these theories, and find that they are equal in elliptic models and unequal in a large class of finite N=2 and marginal N=1 non-elliptic theories. 
  We consider four dimensional supersymmetric gauge field theories from brane configurations with the matter content given by semi-infinite D4 branes ending on both sides of NS branes. In M theory configuration, we discuss the splitting of the M5 brane into infinite cylindrical M5 branes (which decouple) and transversal M5 brane. The splitting condition appears naturally from the consistency of the different projections of the Seiberg-Witten curve. 
  In this paper D-brane boundary states constructed in Gepner models are used to analyze some aspects of the dynamics of D0-branes in Calabi-Yau compactifications of type II theories to four dimensions. It is shown that the boundary states correspond to BPS objects carrying dyonic charges. By analyzing the couplings to closed string fields a correspondence between the D0-branes and extremal charged black holes in N=2 supergravity is found. 
  We propose a procedure for computing the boundary stress tensor associated with a gravitating system in asymptotically anti-de Sitter space. Our definition is free of ambiguities encountered by previous attempts, and correctly reproduces the masses and angular momenta of various spacetimes. Via the AdS/CFT correspondence, our classical result is interpretable as the expectation value of the stress tensor in a quantum conformal field theory. We demonstrate that the conformal anomalies in two and four dimensions are recovered. The two dimensional stress tensor transforms with a Schwarzian derivative and the expected central charge. We also find a nonzero ground state energy for global AdS_5, and show that it exactly matches the Casimir energy of the dual N=4 super Yang-Mills theory on S^3 x R. 
  The spacetime superalgebra via the supermembrane probe in the background of AdS_4 \times S^7 is discussed to the lowest order in the spinor coordinate $\t$. To obtain the correct spacetime superalgebras, all $\t^2$ order corrections for supervielbein and super 3-form gauge potential have to be included. The central extension of the superalgebra OSp(8|4) of the super isometries for AdS_4 \times S^7 is found. 
  We consider the AdS/CFT correspondence for theories with a Chern-Simons term in three dimensions. We find the two-point functions of the boundary conformal field theories for the Proca-Chern-Simons theory and the Self-Dual model. We also discuss particular limits where we find the two-point function of the boundary conformal field theory for the Maxwell-Chern-Simons theory. In particular our results are consistent with the equivalence between the Maxwell-Chern-Simons theory and the Self-Dual model. 
  We show the presence of a topological (Berry) phase in the time evolution of a mixed state. For the case of mixed neutrinos, the Berry phase is a function of the mixing angle only. 
  We construct an explicit solution of type-IIB supergravity describing the strong coupling regime of a non-supersymmetric gauge theory. The latter has a running coupling with an ultraviolet stable fixed point corresponding to the N=4 SU(N) super-Yang-Mills theory at large N. The running coupling has a power law behaviour, argued to be universal, that is consistent with holography. Around the critical point, our solution defines an asymptotic expansion for the gauge coupling beta-function. We also calculate the first correction to the Coulombic quark-antiquark potential. 
  I argue that, in the chaotic version of string cosmology proposed recently, classical and quantum effects generate, at the time of exit to radiation, the correct amount of entropy to saturate a Hubble (or holography) entropy bound (HEB) and to identify, within our own Universe, the arrow of time. Demanding that the HEB be fulfilled at all times forces a crucial "branch change" to occur, and the so-called string phase to end at a critical value of the effective Planck mass, in agreement with previous conjectures. 
  The field-theoretical description of quantum fluctuations on the background of a tunneling field $\sigma$ is revisited in the case of a functional Schrodinger approach. We apply this method in the case when quantum fluctuations are coupled to the $\sigma$ field through a mass-squared term, which is 'time-dependent' since we include the dynamics of $\sigma$ . The resulting mode functions of the fluctuation field, which determine the quantum state after tunneling, display a previously unseen resonance effect when their mode number is comparable to the curvature scale of the bubble. A detailed analysis of the relation between the excitations of the field about the true vacuum (interpreted as particle creation) and the phase shift coming from tunneling is presented. 
  We present a class of two-charged intersecting brane solutions of the D=11 supergravity, which contain the M2-brane, M5-brane, Kaluza-Klein monopole or Brinkmann wave as their building blocks. These solutions share the common feature that one charge is smeared out or uniform over all spatial directions occupied by the branes or waves, while the other charge is localized in them. 
  In this paper we construct several supersymmetric theories, including SU(N) gauge theory, on AdS_5 background. We discuss the proper definition of the Killing equation for the symplectic Majorana spinors required in AdS_5 supersymmetric theories. We find that the symplectic Killing spinor equation involves a matrix M in the USp(2N) indices whose role was not recognized previously. Using the correct Killing spinors we explicitly confirm that the particle masses in the constructed theories agree with the predictions of the AdS/CFT correspondence. Finally, we establish correct O(d-1,2) isometry transformations required to keep the Lagrangian invariant on AdS_d. 
  Mass and decay spectra are calculated for quantum massive excitations of a piecewise uniform bosonic string. The physical meaning of the critical temperatures characterizing the radiation in the decay of a massive microstate in string theory is discussed. 
  These lectures do not at all provide a general review of this rapidly growing field. Instead a rather detailed account is presented of a number of the most elementary aspects. 
  We study conformally invariant boundary conditions that break part of the bulk symmetries. A general theory is developped for those boundary conditions for which the preserved subalgebra is the fixed algebra under an abelian orbifold group. We explicitly construct the boundary states and reflection coefficients as well as the annulus amplitudes. Integrality of the annulus coefficients is proven in full generality. 
  Nonsingularity conditions are established for the BFV gauge-fixing fermion which are sufficient for it to lead to the correct path integral for a theory with constraints canonically quantized in the BFV approach. The conditions ensure that anticommutator of this fermion with the BRST charge regularises the path integral by regularising the trace over non-physical states in each ghost sector. The results are applied to the quantization of a system which has a Gribov problem, using a non-standard form of the gauge-fixing fermion. 
  We study N=(2,2) supersymmetric abelian gauge theories in two dimensions. The exact BPS spectrum of these models is shown to coincide with the spectrum of massive hypermultiplets of certain N=2 supersymmetric gauge theories in four dimensions. A special case of these results involves a surprising connection between four-dimensional N=2 SQCD with N colours and N_{f}>N flavours at the root of the baryonic Higgs branch and the supersymmetric CP^{2N-N_{f}-1} sigma-model in two dimensions. This correspondence implies a new prediction for the strong-coupling spectrum of the four-dimensional theory. 
  A charged superconductiong cosmic string produces an extremely large electric field in its vicinity. This leads to vacuum instability and to the formation of a charged vacuum condensate which screens the electric charge of the string. We analyze the structure of this condensate using the Thomas-Fermi method. 
  We put forward an example of local, covariant Lagrangians where the Feynman rules result in diagrams of QED but with regularized propagators. Following 't Hooft and Veltman, these diagrams may be taken to define a quantum field theory of electrodynamic phenomena that requires no regularization and is realistic, because: (i) The corresponding Green's functions are causal. (ii) Its S-matrix is unitary. (iii) The theory does not imply the existence of particles with wrong metric and/or wrong statistics. (iv) It contains the experimental predictions of QED. No such Lagrangians were known before 
  The requirement that the action be stationary for solutions of the Dirac equations in anti-de Sitter space with a definite asymptotic behaviour is shown to fix the boundary term (with its coefficient) that must be added to the standard Dirac action in the AdS/CFT correspondence for spinor fields. 
  We give a noncommutative version of the complex projective space CP^2 and show that scalar QFT on this space is free of UV divergencies. The tools necessary to investigate Quantum fields on this fuzzy CP^2 are developed and possibilities to introduce spinors and Dirac operators are discussed. 
  Kinks, vortices, monopoles are extended objects, or defects, of quantum origin with topologically non-trivial properties and macroscopic behavior. They are described in Quantum Field Theory in terms of non-homogeneous boson condensation. I will review the related QFT formalism, the spontaneous breakdown of symmetry framework in which the defects appear and discuss finite temperature effects, also in connection with phase transition problematics. 
  We study the geometry of interacting knotted solitons. The interaction is local and advances either as a three-body or as a four-body process, depending on the relative orientation and a degeneracy of the solitons involved. The splitting and adjoining is governed by a four-point vertex in combination with duality transformations. The total linking number is preserved during the interaction. It receives contributions both from the twist and the writhe, which are variable. Therefore solitons can twine and coil and links can be formed. 
  We analyze the integrability properties of models defined on the symmetric space SU(2)/U(1) in 3+1 dimensions, using a recently proposed approach for integrable theories in any dimension. We point out the key ingredients for a theory to possess an infinite number of local conservation laws, and discuss classes of models with such property. We propose a 3+1-dimensional, relativistic invariant field theory possessing a toroidal soliton solution carrying a unit of topological charge given by the Hopf map. Construction of the action is guided by the requirement that the energy of static configuration should be scale invariant. The solution is constructed exactly. The model possesses an infinite number of local conserved currents. The method is also applied to the Skyrme-Faddeev model, and integrable submodels are proposed. 
  A special class of solutions to the generalised WDVV equations related to a finite set of covectors is investigated. Some geometric conditions on such a set which guarantee that the corresponding function satisfies WDVV equations are found (check-conditions). These conditions are satisfied for all root systems and their special deformations discovered in the theory of the Calogero-Moser systems by O.Chalykh, M.Feigin and the author. This leads to the new solutions for the generalized WDVV equations. 
  It is shown that dynamics of D+2 elements of the (super)diffeomorphism group in one (1+1 for the super) dimension describes the D - dimensional (spinning) massless relativistic particles. The coordinates of this elements (D+2 einbeins, D+2 connections and 1 additional common coordinate of higher dimensionality) play the role of coordinates, momenta and Lagrange multiplier, needed for the manifestly conformal and reparametrization invariant description of the D - dimensional (spinning) particle in terms of the D+2 - dimensional spacetime. 
  The renormalization group method is a successive integration over the fluctuations which are ordered according to their length scale, a parameter in the external space. A different procedure is described, where the fluctuations are treated in a successive manner, as well, but their order is given by an internal space parameter, their amplitude. The differential version of the renormalization group equation is given which is the functional generalization of the Callan-Symanzik equation in one special case and resums the loop expansion in another one. 
  The Newtonian character of gauge theories on a light front requires that the longitudinal momentum P^+, which plays the role of Newtonian mass, be conserved. This requirement conflicts with the standard definition of the force between two sources in terms of the minimal energy of quantum gauge fields in the presence of a quark and anti-quark pinned to points separated by a distance R. We propose that, on a light front, the force be defined by minimizing the energy of gauge fields in the presence of a quark and an anti-quark pinned to lines (1-branes) oriented in the longitudinal direction singled out by the light front and separated by a transverse distance R. Such sources will have a limited 1+1 dimensional dynamics. We study this proposal for weak coupling gauge theories by showing how it leads to the Coulomb force law. For QCD we also show how asymptotic freedom emerges by evaluating the S-matrix through one loop for the scattering of a particle in the N_c representation of color SU(N_c) on a 1-brane by a particle in the \bar N_c representation of color on a parallel 1-brane separated from the first by a distance R<<1/Lambda_{QCD}. Potential applications to the problem of confinement on a light front are discussed. 
  We discuss some aspects of cohomological properties of a two-dimensional free Abelian gauge theory in the framework of BRST formalism. We derive the conserved and nilpotent BRST- and co-BRST charges and express the Hodge decomposition theorem in terms of these charges and a conserved bosonic charge corresponding to the Laplacian operator. It is because of the topological nature of free U(1) gauge theory that the Laplacian operator goes to zero when equations of motion are exploited. We derive two sets of topological invariants which are related to each-other by a certain kind of duality transformation and express the Lagrangian density of this theory as the sum of terms that are BRST- and co-BRST invariants. Mathematically, this theory captures together some of the key features of Witten- and Schwarz type of topological field theories. 
  We consider an extension of Lipatov's conjecture about the deep relation between amplitudes in the high-energy limit of QCD and XXX Heisenberg chains with non-compact spins. 
  We analyze p-brane black hole solutions with `block-orthogonal' intersection rules. The post-Newtonian parameters beta and gamma corresponding to 4-dimensional section of the metric are calculated. A family of solutions with gamma=1 is singled out. Some examples of solutions (e.g. in D=11 supergravity) are considered. 
  We give a description of supermembranes in nontrivial target-space geometries. A special class are the $AdS_4\times S^7$ and $AdS_7\times S^4$ spaces that have the maximal number of 32 supersymmetries. 
  By the use of an effective superpotential in supersymmetric quantum chromodynamics (SQCD) with N_f flavors and N_c colors of quarks for N_f>=N_c+2, the influence of soft supersymmetry (SUSY) breakings is examined to clarify dynamics of chiral symmetry breakings near the SUSY limit. In case that SQCD triggers spontaneous chiral symmetry breakings, it is possible to show that our superpotential dynamically favors the successive formation of condensates, leaving either SU(N_f-N_c) or SU(N_f-N_c+1) unbroken as a chiral nonabelian symmetry. 
  We discuss the determination of the lowest Form Factors relative to the trace operators of N=1 Super Sinh-Gordon Model. Analytic continuations of these Form Factors as functions of the coupling constant allows us to study a series of models in a uniform way, among these the latest model of the Roaming Series and a class of minimal supersymmetric models. 
  Absolute scaling electrical with gravitational forces has remained unsuccessful until today. Using recent results on scaling spectroscopic constants, we now scale the internal electrical potential of a two particle system by its gravitational potential at the surface of the earth. This relative scaling is based upon classical equilibrium conditions. Compactifying earth-bound gravitaion (the Newton-scale) to the Fermi-scale in particle interacions is of the same order as going down from the latter to the Plank-scale. The unification scale factor for gravitational and electromagnetic forces is of the order of 2pi. The unification scale factor for two vibrating particles is the reciprocal of Sommerfeld's fine structure constant (deviation < 0.2 %)and gravitational forces are weak electromagnetic forces. The earth-bound bravitational energy asymptote is the natural unit for the relativistic fine structure and the magnetic spin-orbit interaction in atoms. Maybe, in the Standard Model and beyond, one has been looking in the wrong direction for compactification to attempt superunification. 
  We compute the two and three-point correlation functions of chiral primary operators in the large N limit of the (0,2), d=6 superconformal theory. We also consider the operator product expansion of Wilson surfaces in the (0,2) theory and compute the OPE coefficients of the chiral primary operators at large N from the correlation functions of surfaces. 
  The abelian Chern-Simons theory is perturbed by introducing local gauge-invariant interaction terms depending on the curvature. The computation of the correlation function of two Wilson lines for two smooth closed nonintersecting curves is reported up to four loops and is shown to be unaffected by radiative corrections. This result ensures the stability of the linking number of the two curves with respect to the local perturbations which may be added to the Chern-Simons action. 
  We derive a solution of type IIB supergravity which is asymptotic to AdS_5 x S^5, has SO(6) symmetry, and exhibits some of the features expected of geometries dual to confining gauge theories. At the linearized level, the solution differs from pure AdS_5 x S^5 only by a dilaton profile. It has a naked singularity in the interior. Wilson loops follow area law behavior, and there is a mass gap. We suggest a field theory interpretation in which all matter fields of N=4 gauge theory acquire a mass and the infrared theory is confining. 
  We discuss the entropy for the extremal BTZ black hole and the extremal EBTZ black hole. The EBTZ black hole means the BTZ black hole embedded in a five-dimensional(5D) black hole. The six-dimensional(6D) black string with traveling waves is introduced as a concrete model for realizing the AdS/CFT correspondence. The traveling waves carry the momentum distribution $p(u)$ which plays an important role in counting the entropy and establishing the correspondence. It turns out that the EBTZ black hole is consistent with the AdS/CFT correspondence. 
  An exposition of the different definitions and approaches to quantum statistics is given, with emphasis in one-dimensional situations. Permutation statistics, scattering statistics and exclusion statistics are analyzed. The Calogero model, matrix model and spin chain models constitute specific realizations. 
  We study a membrane -- anti-membrane system in Matrix theory. It in fact exhibits the tachyon instability. By suitably representing this configuration, we obtain a (2+1)-dimensional U(2) gauge theory with a 't Hooft's twisted boundary condition. We identify the tachyon field with a certain off-diagonal element of the gauge fields in this model. Taking into account the boundary conditions carefully, we can find vortex solutions which saturate the Bogomol'nyi-type bound and manifest the tachyon condensation. We show that they can be interpreted as gravitons in Matrix theory. 
  Lienard-Wiechert potentials of the relativistic spinning particle with anomalous magnetic moment in pseudoclassical theory are constructed. General expressions for the Lienard-Wiechert potentials are used for investigation of some specific cases of the motion of the spinning particle. In particular the spin dependence of the intensity of the synchrotron radiation of the transversely polarized particle performing uniform circular motion is considered. When the movement of the particle in external homogeneous magnetic field the obtained formulae coincide with those known from quantum theory of radiation. The dependence of the polarization of the synchrotron radiation on the spin of the particle is investigated. 
  Using the relation between D-brane charges and K-theory, we study non-BPS D-branes and their behavior under T-duality. We point out that in general compactifications, D-brane charges are classified by relative K-theory groups. T-duality is found to act as a symmetry between the relative K-theory groups in Type II and Type I/IA theories. We also study Type \tilde\IA theory (which contains an O8^- plane and an O8^+ plane), using K-theory and T-duality to identify its stable D-branes. Comparison with string theory constructions reveals two interesting effects. One of them involves the transfer of branes between O-planes, while in the other, a D-brane charge which seems conserved near one O-plane in fact decays due to the presence of another type of O-plane. 
  We review recent work on the study of N=2 super Yang-Mills theory with gauge group SU(N) from the point of view of the Whitham hierarchy, mainly focusing on three main results: (i) We develop a new recursive method to compute the whole instanton expansion of the low-energy effective prepotential; (ii) We interpret the slow times of the hierarchy as additional couplings and promote them to spurion superfields that softly break N=2 supersymmetry down to N=0 through deformations associated to higher Casimir operators of the gauge group; (iii) We show that the Seiberg-Witten-Whitham equations provide a set of non-trivial constraints on the form of the strong coupling expansion in the vicinity of the maximal singularities. We use them to check a proposal that we make for the value of the off-diagonal couplings at those points of the moduli space. 
  We re-examine perturbative and nonperturbative aspects of the beta function in N=1 and N=2 supersymmetric gauge theories, make comments on the recent literature on the subject and discuss the exactness of several known results such as the NSVZ beta function. 
  We investigate nonperturbative effects in N=1 supersymmetric theories and propose a new expression for the effective action, which correctly reproduces quantum anomalies and agrees with the transformation law of instanton measure. Actually the result is a nonperturbative extension of Veneziano-Yankielowitch effective Lagrangian. The possibility of integrating out the gluino condensate is discussed. 
  In the framework of osp(1,2)-symmetric quantization of irreducible massive gauge theories the background field method is studied for the simplest case of a linear splitting of the gauge field into a background configuration and the quantum fluctuations. The symmetries of that approach - including three types of background-dependent gauge transformations - are expressed by Ward identities. From these identities together with the equations of motion of the auxiliary field the background dependence of the vertex functions and of the Greens functions is determined. It is proven that the introduction of a background field does not change the ultraviolet asymptotics of the theory. 
  We propose a lattice version of Chern-Simons gravity and show that the partition function coincides with Ponzano-Regge model and the action leads to the Chern-Simons gravity in the continuum limit. The action is explicitly constructed by lattice dreibein and spin connection and is shown to be invariant under lattice local Lorentz transformation and gauge diffeomorphism. The action includes the constraint which can be interpreted as a gauge fixing condition of the lattice gauge diffeomorphism. 
  We present a class of spacetime rotations that preserve a proportion of spacetime supersymmetry. We then give the rules for superposing these rotations with various branes to construct rotating brane solutions which preserve exotic fractions of supersymmetry. We also investigate the superposition of rotations with intersecting branes at angles and we find new rotating intersecting branes at angles configurations. We demonstrate this with two examples of such solutions one involving intersecting NS-5-branes on a string at $Sp(2)$ angles superposed with fundamental strings and pp-waves, and the other involving intersecting M-5-branes on a string at $Sp(2)$ angles superposed with membranes and pp-waves. We find that the geometry of some of these solutions near the intersection region of every pair of 5-branes is $AdS_3\times S^3\times S^3\times \bE$ and $AdS_3\times S^3\times S^3\times\bE^2$, respectively. We also present a class of solutions that can be used for null string and M-theory compactifications preserving supersymmetry. 
  In this paper,based on the available mathematical works on geometry and topology of hyperbolic manifolds and discrete groups, some results of Freedman et al (hep-th/9804058) are reproduced and broadly generalized. Among many new results the possibility of extension of work of Belavin, Polyakov and Zamolodchikov to higher dimensions is investigated. Known in physical literature objections against such extension are removed and the possibility of an extension is convincingly demonstrated. 
  Non-supersymmetric d=4 gauge theories which arise from superstring duality on a manifold $AdS_5 \times S_5/Z_p$ are cataloged for a range $2 \leq p \leq 41$. A number have vanishing two-loop gauge \beta-function, a necessary but not sufficient condition to be a conformal field theory. 
  We suggest that the boundary cosmological constant \zeta in c<1 unitary string theory be regarded as the one-dimensional complex coordinate of the target space on which the boundaries of world-sheets can live. From this viewpoint we explicitly construct analogues of D-instantons which satisfy Polchinski's ``combinatorics of boundaries.'' We further show that our operator formalism developed in the preceding articles is powerful in evaluating D-instanton effects, and also demonstrate for simple cases that these effects exactly coincide with the stringy nonperturbative effects found in the exact solutions of string equations. 
  We compute the properties of a class of charged black holes in anti-de Sitter space-time, in diverse dimensions. These black holes are solutions of consistent Einstein-Maxwell truncations of gauged supergravities, which are shown to arise from the inclusion of rotation in the transverse space. We uncover rich thermodynamic phase structures for these systems, which display classic critical phenomena, including structures isomorphic to the van der Waals-Maxwell liquid-gas system. In that case, the phases are controlled by the universal `cusp' and `swallowtail' shapes familiar from catastrophe theory. All of the thermodynamics is consistent with field theory interpretations via holography, where the dual field theories can sometimes be found on the world volumes of coincident rotating branes. 
  The kappa symmetry of an open M2-brane ending on an M5-brane requires geometrical constraints on the embedding of the system in target superspace. These constraints lead to the M5-brane equations of motion, which we review both in superspace and in component (i.e. in Green-Schwarz) formalism. We also describe the embedding of the chiral M5-brane theory in a non-chiral theory where the equations of motion follow from an action that involves a non-chiral 2-form potential, upon the imposition of a non-linear self-duality condition. In this formulation, we find a simplified form of the second order field equation for the worldvolume 2-form potential, and we derive the nonlinear holomorphicity condition on the partition function of the chiral M5-brane. 
  The fields nonlinear modes quantization scheme is discussed. New form of the perturbation theory achieved by unitary mapping the quantum dynamics in the space $W_G$ of (action, angle)-type collective variables. It is shown why the transformed perturbation theory contributions may accumulated exactly on the boundary $\pa W_G$. Abilities of the developed formalism are illustrated by examples from quantum mechanics and field theory. 
  A new version of holographic principle for cosmology is proposed, which dictates that particle entropy within `cosmological apparent horizon' should not exceed gravitational entropy associated with the apparent horizon. It is shown that, in the Friedman-Robertson-Walker (FRW) cosmology, the open Universe as well as a restricted class of flat cases are compatible with the principle, whereas closed Universe is not. It is also found that inflationary Universe after the big-bang is incompatible with the cosmic holography. 
  An algebraic proof of the Gluing Theorem at tree level of perturbation theory in String Field Theory is given. Some applications of the theorem to closed string non-polynomial action are briefly discussed 
  New clues for the best understanding of the nature of the symmetry-breaking mechanism are revealed in this paper. A revision of the standard gauge transformation properties of Yang-Mills fields, according to a group approach to quantization scheme, enables the gauge group coordinates to acquire dynamical content outside the null mass shell. The corresponding extra (internal) field degrees of freedom are transferred to the vector potentials to conform massive vector bosons. 
  In is shown explicitly that the Witten's interaction 3-vertex is a solution to the comma overlap equations; hence establishing the equivalence between the conventional and the "comma" formulation of interacting string theory at the level of vertices. 
  By making use of the Abelian projection method, a dual version of the SU(2)-gluodynamics with manifest monopole-like excitations, arising from the integration over singular gauge transformations, is formulated in the continuum limit. The resulting effective theory emerges due to the summation over the grand canonical ensemble of these excitations in the dilute gas approximation. As a result, the dual Abelian gauge boson acquires a nonvanishing (magnetic) mass due to the Debye screening effects in such a gas. The obtained theory is then used for the construction of the corresponding effective potential of monopole loop currents and the string representation. Finally, by virtue of this representation, confining properties of the SU(2)-gluodynamics are emphasized. 
  We write the BRST operator of the N=1 superstring as $Q= e^{-R} (\oint dz \gamma^2 b)e^R$ where $\gamma$ and $b$ are super-reparameterization ghosts. This provides a trivial proof that $Q$ is nilpotent. 
  We generalize the (p,q) 5-brane web construction of five-dimensional field theories by introducing (p,q) 7-branes, and apply this construction to theories with a one-dimensional Coulomb branch. The 7-branes render the exceptional global symmetry of these theories manifest. Additionally, 7-branes allow the construction of all E_n theories up to n=8, previously not possible in 5-brane configurations. The exceptional global symmetry in the field theory is a subalgebra of an affine symmetry on the 7-branes, which is necessary for the existence of the system. We explicitly determine the quantum numbers of the BPS states of all E_n theories using two simple geometrical constraints. 
  We discuss the sigma model on the $PSL(n|n)$ supergroup manifold. We demonstrate that this theory is exactly conformal. The chiral algebra of this model is given by some extension of the Virasoro algebra, similar to the $W$ algebra of Zamolodchikov. We also show that all group invariant correlation functions are coupling constant independent and can be computed in the free theory. The non invariant correlation functions are highly nontrivial and coupling dependent. At the end we compare two and three-point correlation functions of the $PSL(1,1|2)$ sigma model with the correlation functions in the boundary theory of $AdS_3 \times S^3$ and find a qualitative agreement. 
  We compute the effective potential of a system composed by a Dp brane and a separated anti-Dp brane at tree level in string theory. We show explicitly that the tachyon condenses and that the scalars which describe transverse fluctuations acquire a VEV proportional to the distance. 
  See hep-th/9903228. 
  I study the global structure of de Sitter space in the semi-classical and one-loop approximations to quantum gravity. The creation and evaporation of neutral black holes causes the fragmentation of de Sitter space into disconnected daughter universes. If the black holes are stabilized by a charge, I find that the decay leads to a necklace of de Sitter universes (`beads') joined by near-extremal black hole throats. For sufficient charge, more and more beads keep forming on the necklace, so that an unbounded number of universes will be produced. In any case, future infinity will not be connected. This may have implications for a holographic description of quantum gravity in de Sitter space. 
  We consider quantum \cal{N} = 4 super Yang-Mills theory interacting in a covariant way with \cal{N} = 4 conformal supergravity. The induced large N effective action for such a theory is calculated on a dilaton-gravitational background using the conformal anomaly found via AdS/CFT correspondence. Considering such an effective action as a quantum correction to the classical gravity action we study quantum cosmology. In particular, the effect from dilaton to the scale factor (which without dilaton corresponds to the inflationary universe) is investigated. It is shown that, dependent on the initial conditions for the dilaton, the dilaton may slow down, or accelerate, the inflation process. At late times, the dilaton is decaying exponentially. 
  We show that the regularization of the Standard Model proposed by Frolov and Slavnov describes a nonlocal theory with quite simple Lagrangian. 
  We continue the construction of non-trivial vacua for gauge theories on the 3-torus, started in hep-th/9901154. Application of constructions based on twist in SU(N) with N > 2 produce more extra vacua in theories with exceptional groups. We calculate the relevant unbroken subgroups, and their contribution to the Witten index. We show that the extra vacua we find in the exceptional groups are sufficient to solve the Witten index problem for these groups. 
  We describe how the presence of the antisymmetric tensor (torsion) on the world sheet action of string theory renders the size of the target space a gauge non invariant quantity. This generalizes the R <--> 1/R symmetry in which momenta and windings are exchanged, to the whole O(d,d,Z). The crucial point is that, with a transformation, it is possible always to have all of the lowest eigenvalues of the Hamiltonian to be momentum modes. We interpret this in the framework of noncommutative geometry, in which algebras take the place of point spaces, and of the spectral action principle for which the eigenvalues of the Dirac operator are the fundamental objects, out of which the theory is constructed. A quantum observer, in the presence of many low energy eigenvalues of the Dirac operator (and hence of the Hamiltonian) will always interpreted the target space of the string theory as effectively uncompactified. 
  Using the low energy effective action of the N=2 supersymmetric SU(2) Yang-Mills theory we calculate the free energy at finite temperature, both in the semiclassical region and in the dual monopole/dyon theory. In all regions the free energy depends on both the temperature T and the appropriate moduli parameter, and is thus minimized only for specific values of the moduli parameter, in contrast to the T=0 case where the energy vanishes all over the moduli space. Within the validity of perturbation theory, we find that the finite temperature Yang-Mills theory is stable only at definite points in the moduli space, i.e. for a specific value of the monopole/dyon mass or when the scalar field expectation value goes to infinity. 
  A deformation of the wave equation on a two-dimensional black hole is considered as a toy-model for possible gravitational or stringy nonlocal effects. The deformed wave-equation allows for an initial-value problem despite being nonlocal. The classical singularity present in the classical geometry is resolved by the deformation, so that propagation of a wave-packet can be continued through the classically singular region, ultimately reaching another asymptotically ``flat'' region. 
  We discuss the thermodynamics of a gas of black holes in five-dimensional anti-de-Sitter (AdS) space, showing that they are described by a van der Waals equation of state. Motivated by the Maldacena conjecture, we relate the energy density and pressure of this non-ideal AdS black-hole gas to those of four-dimensional gauge theory in the unconfined phase. We find that the energy density rises rapidly above the deconfinement transition temperature, whilst the pressure rises more slowly towards its asymptotic high-temperature value, in qualitative agreement with lattice simulations. 
  The theta dependence of the vacuum energy in large N Yang-Mills theory has been studied some time ago by Witten using a duality of large N gauge theories with string theory compactified on a certain space-time. We show that within the field theory context vacuum fluctuations of the topological charge give rise to the vacuum energy consistent with the string theory computation. Furthermore, we calculate 1/N suppressed corrections to the string theory result. The reconciliation of the string and field theory approaches is based on the fact that the gauge theory instantons carry zerobrane charge in the corresponding D-brane construction of Yang-Mills theory. Given the formula for the vacuum energy we study certain aspects of stability of the false vacua of the model for different realizations of the initial conditions. The vacuum structure appears to be different depending on whether N is infinite or, alternatively, large but finite. 
  We consider here XXZ spin chain perturbed by the operator sigma^x (``in transverse field'') which is a lattice regularization of the sine-Gordon model. This can be shown using conformal perturbation theory. We calculated mass ratios of particles which lie in a discrete part of the spectrum and obtained results in accord with the DHN formula and in disagreement with recent calculations in literature based on numerical Bethe Ansatz and infinite momentum frame methods. We also analysed a short distance behavior of this states (UV or conformal limit). Our result for conformal dimension of the second breather state is different from the conjecture in [Klassen and Melzer, Int. J. Mod. Phys. A8, 4131 (1993)] and is consistent with this paper for other states. 
  Charret et. al. applied the properties of the Grassmann generators to develop a new method to calculate the coefficients of the high temperature expansion of the grand canonical partition function of self-interacting fermionic models in any d-dimensions (d>=1). The method explores the anti-commuting nature of fermionic fields and avoids the calculation of the fermionic path integral. We apply this new method to the relativistic free Dirac fermions and recover the known results in the literature. 
  In this paper we present a method of deforming to first order the stress-tensor and the supercurrent of the superstring corresponding to turning on NS-NS bosonic fields. Furthermore we discuss the difficulties associated with turning on spacetime fermions and R-R bosons. We also derive the gauge symmetries of the massless spacetime fields. 
  We study the thermodynamic stability of charged black holes in gauged supergravity theories in D=5, D=4 and D=7. We find explicitly the location of the Hawking-Page phase transition between charged black holes and the pure anti-de Sitter space-time, both in the grand-canonical ensemble, where electric potentials are held fixed, and in the canonical ensemble, where total charges are held fixed. We also find the explicit local thermodynamic stability constraints for black holes with one non-zero charge. In the grand-canonical ensemble, there is in general a region of phase space where neither the anti-de Sitter space-time is dynamically preferred, nor are the charged black holes thermodynamically stable. But in the canonical ensemble, anti-de Sitter space-time is always dynamically preferred in the domain where black holes are unstable.   We demonstrate the equivalence of large R-charged black holes in D=5, D=4 and D=7 with spinning near-extreme D3-, M2- and M5-branes, respectively. The mass, the charges and the entropy of such black holes can be mapped into the energy above extremality, the angular momenta and the entropy of the corresponding branes. We also note a peculiar numerological sense in which the grand-canonical stability constraints for large charge black holes in D=4 and D=7 are dual, and in which the D=5 constraints are self-dual. 
  Using boundary states we derive the presence of (chiral) fermions on the intersection of type 0 D-branes. The corresponding anomalous couplings on the branes are then computed. Furthermore, we discuss systems of branes sitting at $A_n$ singularities. In particular, the massless spectrum on the branes is derived, and a boundary state description is given. 
  We study baryon configurations in large N non-supersymmetric SU(N) gauge theories, applying the AdS/CFT correspondence. Using the D5-brane worldvolume theory in the near-horizon geometry of non-extremal D3-branes, we find embeddings which describe baryonic states in three-dimensional QCD. In particular, we construct solutions corresponding to a baryon made of N quarks, and study what happens when some fraction $\nu$ of the total number of quarks are bodily moved to a large spatial separation from the others. The individual clumps of quarks are represented by Born-Infeld string tubes obtained from a D5-brane whose spatial section has topology $R \times S^4$. They are connected by a confining color flux tube, described by a portion of the fivebrane that runs very close and parallel to the horizon. We find that this flux tube has a tension with a nontrivial $\nu$-dependence (not previously obtained by other methods). A similar picture is presented for the four-dimensional case. 
  We present a theta function representation of the twisted characters for the rational N=2 superconformal field theory, and discuss the Jacobi-form like functional properties of these characters for a fixed central charge under the action of a finite Heisenberg group and modular transformations. 
  The two-dimensional chiral anomaly is calculated using differential regularization. It is shown that the anomaly emerges naturally in the vector and axial Ward identities on the same footing as the four-dimensional case. The vector gauge symmetry can be achieved by an appropriate choice of the mass scales without introducing the seagull term. We have analyzed the reason why such a universal result can be obtained in differential regularization. 
  The recent developments towards the possible non-perturbative formulation of string/M theory using supersymmetric Yang-Mills matrix models (SYMs) are discussed. In the first part, we give a critical review on the status of our present understanding, focusing on the connection of the D0-brane matrix models to supergravity and its relevance to the so-called Matrix-theory conjecture. We also discuss some problems concerning the conjectured relation between supergravity in AdS background and SYM from the viewpoint of D-brane interactions. We present a qualitative argument showing how the boundary condition at AdS boundary dictates the correlators on the large N system of source D-branes. Then, in the final part, we turn to the question how to formulate the condensation of graviton in matrix models, taking the simplest example of type IIB matrix model. We argue the emergence of a hidden symmetry GL(10, R), beyond the manifest Lorentz symmetry SO(9,1), by embedding U(N) model into models with higher N and by treating the whole recursive series of models simultaneously. This suggests a possible approach toward background independent formulations of matrix models. 
  We discuss the ways of constructing the exact superpotential for N=1 supersymmetric theories and propose a new approach. As a consequence, a new structure of the superpotential is found. 
  We study ${\cal F}^4$-threshold corrections in an eight dimensional S-dual pair of string theories, as a prototype of dual string vacua with sixteen supercharges. We show that the orbifold CFT description of D-string instantons gives rise to a perturbative expansion similar to the one appearing on the fundamental string side. By an explicit calculation, using the Nambu-Goto action in the static gauge, we show that the first subleading term agrees precisely on the two sides. We then give a general argument to show that the agreement extends to all orders. 
  An expression for the exact (nonperturbative) effective action of $N$=1 supersymmetric gauge theories is proposed, supposing, that all particles except for the gauge bosons are massive. Analysis of its form shows, that instanton effects in the supersymmetric theories can lead to the quark confinement. The typical scale of confinement in MSSM QCD, calculated from the first principles, is in agreement with the experimental data. The proposed explanation is quite different from the dual Higgs mechanism. 
  Properties of the Dirac-Born-Infeld Lagrangian analogous to those of the Nambu-Goto String are analysed. In particular the Lagrangian is shown to be constant or zero on the space of solutions of the equations of motion if the Lagrangian is taken to any power other than 1/2. 
  The compactification of five dimensional N=2 SUSY Yang-Mills (YM) theory onto a circle provides a four dimensional YM model with N=4 SUSY. This supersymmetry can be broken down to N=2 if non-trivial boundary conditions in the compact dimension, \phi(x_5 +R) = e^{2\pi i\epsilon}\phi(x_5), are imposed on half of the fields. This two-parameter (R,\epsilon) family of compactifications includes as particular limits most of the previously studied four dimensional N=2 SUSY YM models with supermultiplets in the adjoint representation of the gauge group. The finite-dimensional integrable system associated to these theories via the Seiberg-Witten construction is the generic elliptic Ruijsenaars-Schneider model. In particular the perturbative (weak coupling) limit is described by the trigonometric Ruijsenaars-Schneider model. 
  We review the construction of the Kaluza-Klein monopole of the Type IIA theory in the most general case of a massive background, as well as its relation via T-duality with the Type IIB NS-5-brane. This last effective action is shown to be related by S-duality to the D5-brane effective action. 
  Gauging of space translations for nonrelativistic point particles in one dimension leads to general coordinate transformations with fixed Newtonian time. The minimal gauge invariant extension of the particle velocity requires the introduction of two gauge fields whose minimal self interaction leads to a Maxwellian term in the Lagrangean. No dilaton field is introduced. We fix the gauge such that the residual symmetry group is the Galilei group. In case of a line the two-particle reduced Lagrangean describes the motion in a Newtonian gravitational potential with strength proportional to the energy. For particles on a circle with certain initial conditions we only have a collective rotation with constant angular velocity. 
  We present the explicit forms of supergravity solutions for the various intersecting two BPS branes in eleven and ten dimensions, where one brane is localized at the delocalized other brane. Our partially localized supergravity solutions describe brane configurations in the near horizon region of the delocalized branes, where the two constituent branes meet in the overall transverse space and the delocalized branes coincide. We also give the brane worldvolume interpretations for some of such supergravity solutions. 
  We study N=1 supersymmetric SU(N) Yang-Mills theory on the lattice at strong coupling. Our method is based on the hopping parameter expansion in terms of random walks, resummed for any value of the Wilson parameter r in the small hopping parameter region. Results are given for the mesonic (2-gluino) and fermionic (3-gluino) propagators and spectrum. 
  We write supersymmetry preserving conditions for infinite M5-branes intersecting on a (3+1)-dimensional space. In contrast to previously known solutions, these intersections are completely localized. We solve the equations for a particular class of configurations which in the near-horizon decoupling limit are dual to N_f = 2N_c Seiberg-Witten superconformal field theories with gauge group SU(N) and generalisations to SU(N)^n. We also discuss the relationship to D3-branes in the presence of an A_k singularity. 
  Continuing the investigation of CNM (chiral-nonminimal) hypermultiplet nonlinear sigma-models, we propose extensions of the concept of the c-map which relate holomorphic functions to hyper-Kahler geometries. In particular, we show that a whole series of hyper-Kahler potentials can be derived by replacing the role of the 4D, N = 1 tensor multiplet in the original c-map by 4D, N = 1 non-minimal multiplets and auxiliary superfields. The resulting N = 2 models appear to have interesting connections to Calabi-Yau manifolds and algebraic varieties. These models also emphasize the fact that special hyper-Kahler manifolds (the analogs of special Kahler manifolds) without isometries exist. 
  In this paper, we study the renormalizability of the Standard Model in the Landau gauge. On the basis of the Ward-Takahashi identities, we derive exact expressions for the physical masses of the W and Z as well as the renormalized coupling constants in the theory. We show that it is impossible to make all these renormalized quantities finite. Thus the quantum theory of the Standard Model with the divergent amplitudes obeying the Ward-Takahashi identities is not renormalizable. 
  We obtain new intersecting 5-brane, string and pp-wave solutions in the heterotic string on a torus and on a K3 manifold. In the former case the 5-brane is supported by Yang-Mills instantons, and in the latter case both the 5-brane and the string are supported by the instantons. The instanton moduli are parameterised by the sizes and locations of the instantons. We exhibit two kinds of phase transition in which, for suitable choices of the instanton moduli, a 5-brane and/or a string can be created. One kind of phase transition occurs when the size of an instanton vanishes, while the other occurs when a pair of Yang-Mills instantons coalesce. We also study the associated five-dimensional black holes and the implications of these phase transitions for the black-hole entropy. Specifically, we find that the entropy of the three-charge black holes is zero when the instantons are separated and of non-zero scale size, but becomes non-zero (which can be counted miscrospically) after either of the phase transitions. 
  We summarize recent results on the construction of Lax pairs with spectral parameter for the twisted and untwisted elliptic Calogero-Moser systems associated with arbitrary simple Lie algebras, their scaling limits to Toda systems, and their role in Seiberg-Witten theory. We extend part of this work by presenting a new parametrization for the spectral curves for elliptic spin Calogero-Moser systems associated with SL(N). 
  We discuss the field theory limit of Dp-branes. In this limit, the black Dp-brane solution approaches a solution which is conformal to adS_{p+2} \times S^{8-p}. We argue that the frame in which the conformal factor is equal to one, the dual frame, is a `holographic' frame. The radial coordinate of adS_{p+2} provides a UV/IR connection as in the case of the D3 brane. The gravitational description involves gauged supergravities, typically with non-compact gauge groups. The near-horizon Dp-brane solution becomes a domain-wall solution of the latter. 
  We present some comments concerning the validity of the Nielsen identities for renormalizable theories quantized in general linear covariant gauges in a context of compact gauge Lie groups. 
  In N=1 super Yang-Mills theory in three spacetime dimensions, with a simple gauge group $G$ and a Chern-Simons interaction of level $k$, the supersymmetric index $\Tr (-1)^F$ can be computed by making a relation to a pure Chern-Simons theory or microscopically by an explicit Born-Oppenheimer calculation on a two-torus. The result shows that supersymmetry is unbroken if $|k|\geq h/2$ (with $h$ the dual Coxeter number of $G$) and suggests that dynamical supersymmetry breaking occurs for $|k|<h/2$. The theories with large $|k|$ are massive gauge theories whose universality class is not fully described by the standard criteria. 
  A consistent N=1 supersymmetric $\sigma$-model can be constructed, given a K\"ahler manifold by adding chiral matter multiplets. Their scalar components are covariant tensors on the underlying K\"ahler manifold. The K\"ahler U(1)-charges can be adjusted such that the anomalies cancel, using the holomorphic functions in which the K\"ahler potential transforms. The arbitrariness of the U(1)-charges of matter multiplets is related to their Weyl-weights in superconformal gravity, before it is reduced to supergravity. The covariance of the K\"ahler potential forces the superpotential to be covariant as well. This relates the cut-off, the Planck scale and the matter charges to each other. A non-vanishing VEV of the covariant superpotential breaks the K\"ahler U(1) spontaneously. If this VEV vanishes, the gravitino is massless and depending on the above mentioned parameters there may be additional internal symmetry breaking. The separation of the different representations of chiral multiplets can be achieved by covariantizations of derivatives and fermions. Using non-holomorphic transformations, the full K\"ahler metric can be block-diagonalized and the necessary covariantizations come out naturally. Various aspects are illustrated by applying them to Grassmannian coset models. In particular a consistent model of coset $SU(5)/SU(2)\times U(1) \times SU(3)$ with the field content of the standard model is constructed. The phenomenology of this model is analyzed. 
  This is an introduction to the Maldacena conjecture on the equivalence between ${\cal{N}}=4$ super Yang-Mills in Minkowski space-time and type IIB string theory compactified on $AdS_5 \otimes S_5 $. 
  We examine the effect of perturbative string loops on the cosmological pre-big-bang evolution. We study loop corrections derived from heterotic string theory compactified on a $Z_N$ orbifold and we consider the effect of the all-order loop corrections to the Kahler potential and of the corrections to gravitational couplings, including both threshold corrections and corrections due to the mixed Kahler-gravitational anomaly. We find that string loops can drive the evolution into the region of the parameter space where a graceful exit is in principle possible, and we find solutions that, in the string frame, connect smoothly the superinflationary pre-big-bang evolution to a phase where the curvature and the derivative of the dilaton are decreasing. We also find that at a critical coupling the loop corrections to the Kahler potential induce a ghost-like instability, i.e. the kinetic term of the dilaton vanishes. This is similar to what happens in Seiberg-Witten theory and signals the transition to a new regime where the light modes in the effective action are different and are related to the original ones by S-duality. In a string context, this means that we enter a D-brane dominated phase. 
  We evaluate both the vacuum decay rate at zero temperature and the finite temperature nucleation rate for the $(\lambda\phi^4/4! + \sigma\phi^6/6!)_{3D}$ model. Using the thin-wall approximation, we obtain the bounce solution for the model and we were also able to give the approximate eigenvalue equations for the bounce. 
  In this note a simple calculation of one loop threshold corrections for the SO(32) heterotic string is performed. In particular the compactification on T^2 with a Wilson line breaking the gauge group to SO(16) x SO(16) is considered. Using heterotic type I duality, these corrections can be related to quantities appearing in the quantum mechanics of type I' D0 particles. 
  We describe the construction of quantum gravity, i.e. of a theory of self-interacting massless spin-2 quantum gauge fields, the gravitons, on flat space-time, in the framework of causal perturbation theory. 
  We analyse the dipole solution of heterotic string theory in four dimensions. It has the structure of monopole and anti-monopole connected by flux line (string). Due to growing coupling near the poles, the length of the string diverges. However, exploiting the self-duality of heterotic string theory in four dimension, we argue that this string is correctly described in terms of dual variables. 
  We review the construction of 4D, N =2 globally supersymmetric off-shell nonlinear sigma models whose target spaces are the cotangent bundles of K\"ahler manifolds. 
  We study discrete fluxes in four dimensional SU(N) gauge theories with a mass gap by using brane compactifications which give ${\cal{N}} = 1$ or ${\cal{N}} = 0$ supersymmetry. We show that when such theories are compactified further on a torus, the t'Hooft magnetic flux $m$ is related to the NS two-form modulus $B$ by $B = 2\pi {m\over N}$. These values of $B$ label degenerate brane vacua, giving a simple demonstration of magnetic screening. Furthermore, for these values of $B$ one has a conventional gauge theory on a commutative torus, without having to perform any T-dualities. Because of the mass gap, a generic $B$ does not give a four dimensional gauge theory on a non-commutative torus. The Kaluza-Klein modes which must be integrated out to give a four dimensional theory decouple only when $B=2\pi {m\over N}$. Finally we show that $2\pi {m\over N}$ behaves like a two form modulus of the QCD string. This confirms a previous conjecture based on properties of large $N$ QCD suggesting a T-duality invariance. 
  In this paper, we investigate about two physically distinct classes of the `one-dimensional' worldvolume solutions describing the status of an arbitrary brane in the presence of another arbitrary (flat) brane which supplies the required supergravity background. One of these classes concerns with a relative (transverse) motion of two parallel flat branes, while the other class is related to a static configuration in which one of the branes is flat and the other is curved as a cylindrical hypersurface. Global symmetries of the worldvolume theory are used to show that both types of these solutions are described by some sort of `planar orbits' which are specified by their `energy' and `angular momentum' $(E,l)$ parameters. We find that various phases of the `motion' along these orbits, for different values of $(E,l)$, are easily deduced from the curve of an $E$-dependent function of the relative `distance' between the two branes, which is somehow related to their mutual `effective potential'. 
  A gauge formulation for the Proca model quantum theory in an open path functional space representation is revisited. The path dependent vacuum state is obtained. Starting from this one, other excited states can be obtained too. Additionally, the functional integration measure needed to define an internal product in the state space is constructed. 
  We consider constructing a canonical quantum theory of the light-cone gauge ($A_-$=0) Schwinger model in the light-cone representation. Quantization conditions are obtained by requiring that translational generators $P_+$ and $P_-$ give rise to Heisenberg equations which, in a physical subspace, are consistant with the field equations. A consistent operator solution with residual gauge degrees of freedom is obtained by solving initial value problems on the light-cones. The construction allows a parton picture although we have a physical vacuum with nontrivial degeneracies in the theory. 
  We continue McCartor and Robertson's recent demonstration of the indispensability of ghost fields in the light-cone gauge quantization of gauge fields. It is shown that the ghost fields are indispensable in deriving well-defined antiderivatives and in regularizing the most singular component of gauge field propagator. To this end it is sufficient to confine ourselves to noninteracting abelian fields. Furthermore to circumvent dealing with constrained systems, we construct the temporal gauge canonical formulation of the free electromagnetic field in auxiliary coordinates $x^{\mu}=(x^-,x^+,x^1,x^2)$ where $x^-=x^0 cos{\theta}-x^3 sin{\theta}, x^+=x^0 sin{\theta}+x^3 cos{\theta}$ and $x^-$ plays the role of time. In so doing we can quantize the fields canonically without any constraints, unambiguously introduce "static ghost fields" as residual gauge degrees of freedom and construct the light-cone gauge solution in the light-cone representation by simply taking the light-cone limit (${\theta}\to \pi/4$). As a by product we find that, with a suitable choice of vacuum the Mandelstam-Leibbrandt form of the propagator can be derived in the ${\theta}=0$ case (the temporal gauge formulation in the equal-time representation). 
  Recently, a number of authors have challenged the conventional assumption that the string scale, Planck mass, and unification scale are roughly comparable. It has been suggested that the string scale could be as low as a TeV. In this note, we explore constraints on these scenarios. We argue that the most plausible cases have a fundamental scale of at least 10 TeV and five dimensions of inverse size 10 MeV. We show that a radial dilaton mass in the range of proposed millimeter scale gravitational arises naturally in these scenarios. Most other scenarios require huge values of flux and may not be realizable in M Theory. Existing precision experiments put a conservative lower bound of 6-10 TeV on the fundamental energy scale. We note that large dimensions with bulk supersymmetry might be a natural framework for quintessence, and make some other tentative remarks about cosmology. 
  We review the basic structure of the higher spin extension of D=4, N=8 AdS supergravity. The theory is obtained by gauging the higher spin superalgebra shs^E(8|4) by a procedure pioneered by Vasiliev. The algebra shs^E(8|4) is a subalgebra of the enveloping algebra of OSp(8|4). The physical states of the theory are in one to one correspondence with the symmetric product of two OSp(8|4) singletons. This singleton theory, which may be viewed in a certain limit as the supermembrane theory on AdS_4 x S^7, is expected to describe the dynamics of the higher spin theory. Thus, the higher spin N=8 supergravity on AdS_4 is conjectured to describe the field theory limit of M-theory on AdS_4 x S^7. 
  We argue that non-supersymmetric large N QCD compactified on T^2 exhibits properties characteristic of an SL(2,Z) T-duality. The kahler structure on which this SL(2,Z) acts is given by $m/N + i\Lambda^2 A$, where A is the area of the torus, m is the 't Hooft magnetic flux on the torus, and $\Lambda^2$ is the QCD string tension. 
  Via supersymmetry argument, we determine the effective action of the SU(2) supersymmetric Yang-Mills quantum mechanics up to two constants, which results from the full supersymmetric completion of the F^4 term. The effective action, consisting of zero, two, four, six and eight fermion terms, agrees with the known perturbative one-loop calculations from the type II string theory and the matrix theory. Our derivation thus demonstrates its non-renormalization properties, namely, the one-loop exactness of the aforementioned action and the absence of the non-perturbative corrections. We briefly discuss generalizations to other branes and the comparison to the DLCQ supergravity analysis. In particular, our results show that the stringent constraints from the supersymmetry are responsible for the agreement between the matrix theory and supergravity with sixteen supercharges. 
  In the microcanonical ensemble for string theory on $ AdS_m \times S^n$, there is a phase transition between a black hole solution extended over the $S^n$ and a solution localized on the $S^n$ if the $AdS_m$ has the topology $R^2 \times S^{m-2}$. The phase transition will not appear if the $AdS_m$ has the topology $R^2 \times T^{m-2}$, that is, when the $AdS_m\times S^n$ geometry is regarded as arising from the near-horizon limit of a black $m-2$ brane. In this paper, we argue that when the black branes are rotating, the localization phase transition will occur between some rotating branes and corresponding Kerr black holes when the angular momentum reaches its critical value. 
  We consider N=4 supersymmetric Yang-Mills theory formulated in terms of N=2 superfields in harmonic superspace. Using the background field method we define manifestly gauge invariant and N=2 supersymmetric effective action depending on N=2 strength superfields and develop a general procedure for its calculation in one-loop approximation. Explicit form for this effective action is found for the case of SU(2) gauge group broken down to U(1). 
  The one-loop effective action for general trajectories of D-particles in Matrix theory is calculated in the expansion with respect to the number of derivatives up to six, which gives the equation of motion consistently. The result shows that the terms with six derivatives vanish for straight-line trajectories, however, they do not vanish in general. This provides a concrete example that non-renormalization of twelve-fermion terms does not necessarily imply that of six-derivative terms. 
  We discuss a solution of the equations of motion of five-dimensional gauged type IIB supergravity that describes confining SU(N) gauge theories at large N and large 't Hooft parameter. We prove confinement by computing the Wilson loop, and we show that our solution is generic, independent of most of the details of the theory. In particular, the Einstein-frame metric near its singularity, and the condensates of scalar, composite operators are universal. Also universal is the discreteness of the glueball mass spectrum and the existence of a mass gap. The metric is also identical to a generically confining solution recently found in type 0B theory. 
  We give a construction of off-shell tree bosonic string amplitudes, based on the operatorial formalism of the $N$-string Vertex, with three external massless states both for open and closed strings by requiring their being projective invariant. In particular our prescription leads, in the low-energy limit, to the three-gluon amplitude in the usual covariant gauge. 
  A systematic program is developed for analyzing and cancelling local anomalies on networks of intersecting orbifold planes in the context of M-theory. Through a delicate balance of factors, it is discovered that local anomaly matching on the lower-dimensional intersection of two orbifold planes may require twisted matter on those planes which do not conventionally support an anomaly (such as odd-dimensional planes). In this way, gravitational anomalies can, in principle, tell us about (twisted) gauge groups on subspaces which are not necessarily ten-, six- or two-dimensional. An example is worked out for the case of an $S^1/{\bf Z}_2\times T^4/{\bf Z}_2$ orbifold and possible implications for four-dimensional physics are speculated on. 
  We construct solutions of type IIB supergravity dual to N=2 super Yang-Mills theories. By considering a probe moving in a background with constant coupling and an AdS_{5} component in its geometry, we are able to reproduce the exact low energy effective action for the theory with gauge group SU(2) and N_{f}=4 massless flavors. After turning on a mass for the flavors we find corrections to the AdS_{5} geometry. In addition, the coupling shows a power law dependence on the energy scale of the theory. The origin of the power law behaviour of the coupling is traced back to instanton corrections. Instanton corrections to the four derivative terms in the low energy effective action are correctly obtained from a probe analysis. By considering a Wilson loop in this geometry we are also able to compute the instanton effects on the quark-antiquark potential. Finally we consider a solution corresponding to an asymptotically free field theory. Again, the leading form of the four derivative terms in the low energy effective action are in complete agreement with field theory expectations. 
  The interaction between static D0-branes at finite temperature is considered in the matrix theory and the superstring theory. The results agree in both cases to the leading order in the supersymmetry violation by temperature, where the one-loop approximation is reliable. The effective static potential is short-ranged and attractive.   Talk at the 32nd International Symposium Ahrenshoop on the Theory of Elementary Particles, Buckow, Germany, September 1-5, 1998. 
  We derive D-brane gauge theories for C^3/Z_n x Z_n orbifolds with discrete torsion and study the moduli space of a D-brane at a point. We show that, as suggested in previous work, closed string moduli do not fully resolve the singularity, but the resulting space -- containing n-1 conifold singularities -- is somewhat surprising. Fractional branes also have unusual properties.   We also define an index which is the CFT analog of the intersection form in geometric compactification, and use this to show that the elementary D6-brane wrapped about T^6/Z_n x Z_n must have U(n) world-volume gauge symmetry. 
  Non-trivial supergeneralization of the Kerr-Newman solution is considered as representing a combined model of the Kerr-Newman spinning particle and superparticle.   We show that the old problem of obtaining non-trivial super black hole solutions can be resolved in supergravity broken by Goldstone fermion. Non-linear realization of broken N=2 supersymmetry specific for the Kerr geometry is considered and some examples of the super-Kerr geometries generated by Goldstone fermion are analyzed. The resulting geometries acquire torsion, Rarita-Schwinger field and extra wave contributions to metric and electromagnetic field caused by Grassmann variables.   One family of the self-consistent super-Kerr-Newman solutions to broken N=2 supergravity is selected, and peculiarities of these solutions are discussed. In particular, the appearance of extra `axial' singular line and traveling waves concentrated near `axial' and ring-like singularities. 
  We follow to Witten proposal in the calculation of conformal anomaly from d+1-dimensional higher derivative gravity via AdS/CFT correspondence. It is assumed that some d-dimensional conformal field theories have a description in terms of above d+1-dimensional higher derivative gravity which includes not only Einstein term and cosmological constant but also curvature squared terms. The explicit expression for two-dimensional and four-dimensional anomalies is found, it contains higher derivative corrections. In particular, it is shown that not only Einstein gravity but also theory with the Lagrangian $L=aR^2+bR_{\mu\nu}R^{\mu\nu} + \Lambda$ (even when a=0 or b=0) is five-dimensional bulk theory for d=4 ${\cal N}=4$ super Yang-Mills theory in AdS/CFT correspondence. Similar d+1=3 theory with (or without) Einstein term may describe d=2 scalar or spinor CFTs. That gives new versions of bulk side which may be useful in different aspects. As application of our general formalism we find next-to-leading corrections to the conformal anomaly of ${\cal N}=2$ supersymmetric theory from d=5 AdS higher derivative gravity (low energy string effective action). 
  Several energy-momentum "tensors" of gravitational field are considered and compared in the lowest approximation. Each of them together with energy-momentum tensor of point-like particles satisfies the conservation laws when equation of motion of particles are the same as in general relativity. It is shown that in Newtonian approximation the considered tensors differ one from the other in the way their energy density is distributed between energy density of interection (nonzero only at locations of particles) and energy density of gravitational field.   Starting from Lorentz invariance the Lagrangians for spin-2, mass-0 field are constracted. They differ only by divergences. From these Lagrangians by Belinfante-Rosenfeld procedure the energy-momentum tensors are build. Only one of them is suitable for explaining the perihelion shift. This tensor does not coincide with Weinberg`s one (directly obtainable from Einstein equation).   It is noted that phenomenological field-theoretical approach (utilizing only vertices and propagators) can lead to modification of theory in the region of strong field, where till now no observational data are available. 
  We compare two opposite ways of performing a 3D reduction of the two-fermion Bethe-Salpeter equation beyond the instantaneous approximation and Salpeter's equation. The more usual method consists in performing an expansion around an instantaneous approximation of the product of the free propagators (propagator-approximated reduction). The second method starts with an instantaneous approximation of the Bethe-Salpeter kernel (kernel-approximated reduction). In both reductions the final 3D potential can be obtained by following simple modified Feynman rules. The kernel-approximated reduction, however, does not give the correct scattering amplitudes, and must thus be limited to the computation of bound states. Our 3D reduction of the three-fermion Bethe-Salpeter equation is inspired by these results. We expand this equation around positive-energy instantaneous approximations of the three two-body kernels, but these starting approximations are given by propagator-approximated reductions at the two-body level. The three-fermion Bethe-Salpeter equation is first transformed into a set of three coupled equations for three wave functions depending each on one two-fermion total energy, then into a set of three 3D equations and finally into a single 3D equation. This last equation is rather complicated, as it combines three series expansions, but we use it to write a manageable expression of the first-order corrections to the energy spectrum. 
  We reconsider the Kaluza Klein compactifications of D=11 supergravity on AdS_4x(G/H)_7 manifolds that were classified in the eighties, in the modern perspective of AdS_4/CFT_3 correspondence. We focus on one of the three N=2 cases: (G/H)_7=M^{111}=SU(3)xSU(2)xU(1)/SU(2)xU(1)'xU(1)''. Relying on the systematic use of the harmonic analysis techniques developed in the eighties by one of us (P. Fre') with R. D'Auria, we derive the complete spectrum of long, short and massless Osp(2|4)xSU(3)xSU(2) unitary irreducible representations obtained in this compactification. Our result also provides a general scheme for the other N=2 compactifications. Furthermore, it is a necessary comparison term in the AdS_4/CFT_3 correspondence: the complete AdS/CFT match of the spectra that we obtain will provide a much more stringent proof of the AdS/CFT correspondence than in the S^7 case, since the structure of the superconformal field theory on the M2-brane world volume must be such as to reproduce, at the level of composite operators, the flavor group representations, the conformal dimensions and the hypercharges that we obtain in the present article. The investigation of the match is left to future publications. Here we provide an exhaustive construction of the Kaluza Klein side of our spectroscopy. 
  We propose a unitary matrix formulation of type IIB matrix model. One-loop effective action of the model exhibits supersymmetry for BPS background in the large N limit. 
  We construct SUGRA solutions of brane configurations of intersecting Dp and NS branes. These solutions are semi-localized in the sense that one of the intersecting branes is smeared along the world volume of the other, while the second is localized. We examine the gauge theory that lives on the localized brane, and the various descriptions possible via T and S dualities, and M-theory. 
  We study the microcanonical description of string gases in the presence of D-branes. We obtain exact expressions for the single string density of states and draw the regions in phase space where asymptotic approximations are valid. We are able to describe the whole range of energies including the SYM phase of the D-branes and we remark the importance of the infrared cut-off used in the high energy approximations. With the complete expression we can obtain the density of states of the multiple string gas and study its thermal properties, showing that the Hagedorn temperature is maximum for every system and there is never a phase transition whenever there is thermal contact among the strings attached to different D-branes. 
  We investigate zero modes of the Dirac operator coupled to an Abelian gauge field in three dimensions. We find that the existence of a certain class of zero modes is related to a specific topological property precisely when the requirement of finite Chern--Simons action is imposed. 
  We derive the free Osp(8|4) singleton action by sending the M2brane to the Minkowski boundary of an AdS_4x{\cal M}_7 background. We do this by means of the solvable Lie algebra parametrization of the coset space. We also give some comments on singleton actions from membranes on AdS_4xG/H backgrounds. 
  We study stringy fluctuations as a source for corrections to the Wilson loop as obtained from the superstrings on (adS_5 x S^5)/ N=4 SYM correspondence. We give a formal expression in terms of determinants of two dimensional operators for the leading order correction. 
  The M(atrix) model has a dual realization as IIA superstring theory in the near-horizon geometry of the supergravity D0-brane. The role of $adS_2$ in this correspondence is reviewed, and some aspects of holography that it suggests are discussed. 
  In quantum field theory radiative corrections can be finite but undetermined. 
  Within the context of a super-critical (Liouville) string, we discuss (target-space) two-dimensional string cosmology. A numerical analysis indicates that the identification of time with the Liouville mode results in an expanding universe with matter which exhibits an inflationary phase, and `graceful exit' from it, tending asymptotically to a flat-metric fixed point.This fixed point is characterized by a dilaton configuration which, depending on the initial conditions, either decreases linearly with the cosmic time, or is a finite constant. This implies that, in contrast to the critical string case, the string coupling remains bounded during the exit from the inflationary phase, and, thus, the pertinent dynamics can be reliably described in terms of a tree-level string effective action. The r\^ole of matter in inducing such phenomena is emphasized. It is also interesting to note that the asymptotic value of the vacuum energy, which in the $\sigma$-model framework is identified with the `running' central charge deficit, depends crucially on the set of initial conditions. Thus, although preliminary, this toy model seems to share all the features expected to characterize a phenomenologically acceptable cosmological string model. 
  BPS electric and magnetic black hole solutions which break half of supersymmetry in the theory of N=2 five-dimensional supergravity are discussed. For models which arise as compactifications of M-theory on a Calabi-Yau manifold, these solutions correspond, respectively, to the two and five branes wrapping around the homology cycles of the Calabi-Yau compact space. The electric solutions are reviewed and the magnetic solutions are constructed. The near-horizon physics of these solutions is examined and in particular the phenomenon of the enhancement of supersymmetry. The solutions for the supersymmetric Killing spinor of the near horizon geometry, identified as $AdS_{3}\times S^{2}$ and $AdS_{2} \times S^{3}$ are also given. 
  New non-perturbatives excitations in the massless Thirring and Schwinger models are discussed. 
  An S-matrix analog is defined for anti-de Sitter space by constructing ``in'' and ``out'' states that asymptote to the timelike boundary. A derivation parallel to that of the LSZ formula shows that this ``boundary S-matrix'' is given directly by correlation functions in the boundary conformal theory. This provides a key entry in the AdS to CFT dictionary. 
  We study the effective action of quantum mechanical SU(N) Yang-Mills theories with sixteen supersymmetries and N>2. We show that supersymmetry requires that the eight fermion terms in the supersymmetric completion of the $v^4$ terms be one-loop exact. We also show that the twelve fermion terms in the supersymmetric completion of the $v^6$ terms are two-loop exact for N=3. For N>3, this no longer seems to be true; we are able to find non-renormalization theorems for only certain twelve fermion structures. We call these structures `generalized F-terms.' We argue that as the rank of the gauge group is increased, there can be more generalized F-terms at higher orders in the derivative expansion. 
  We discuss the Matrix Model aspect of configurations saturating a fixed number of fermionic zero modes. This number is independent of the rank of the gauge group and the instanton number. This will allow us to define a large-$N_c$ limit of the embeddeding of $K$ D-instantons in the Matrix Model and make contact with the leading term (the measure factor) of the supergravity computations of D-instanton effects. We show that the connection between these two approaches is done through the Abelian modes of the Matrix variables. 
  We study the blowing-up of the four-dimensional Z_3 orientifold of Angelantonj, Bianchi, Pradisi, Sagnotti and Stanev (ABPSS) by giving nonzero vacuum expectation values (VEV's) to the twisted sector moduli blowing-up modes. The blowing-up procedure induces a Fayet-Iliopoulos (FI) term for the ``anomalous'' U(1), whose magnitude depends linearly on the VEV's of the blowing-up modes. To preserve the N=1 supersymmetry, non-Abelian matter fields are forced to acquire nonzero VEV's, thus breaking (some of) the non-Abelian gauge structure and decoupling some of the matter fields. We determine the form of the FI term, construct explicit examples of (non-Abelian) D and F flat directions, and determine the surviving gauge groups of the restabilized vacua. We also determine the mass spectra, for which the restabilization reduces the number of families. 
  We show that it is possible to construct supersymmetric three-generation models of Standard Model gauge group in the framework of non-simply-connected elliptically fibered Calabi-Yau, without section but with a bi-section. The fibrations on a cover Calabi-Yau, where the model has 6 generations of SU(5) and the bundle is given via the spectral cover description, use a different description of the elliptic fibre which leads to more than one global section. We present two examples of a possible cover Calabi-Yau with a free involution: one is a fibre product of rational elliptic surfaces $dP_9$; another example is an elliptic fibration over a Hirzebruch surface. There we give the necessary amount of chiral matter by turning on in the bundles a further parameter, related to singularities of the fibration and the branching of the spectral cover. 
  We describe local mirror symmetry from a mathematical point of view and make several A-model calculations using the mirror principle (localization). Our results agree with B-model computations from solutions of Picard-Fuchs differential equations constructed form the local geometry near a Fano surface within a Calabi-Yau manifold. We interpret the Gromov-Witten-type numbers from an enumerative point of view. We also describe the geometry of singular surfaces and show how the local invariants of singular surfaces agree with the smooth cases when they occur as complete intersections. 
  We investigate the close relationship between the potential and absorption cross section for test fields in an AdS$_3$ bubble(a 5D black hole) and an exact AdS$_3$. There are two solutions in type IIB string theory: an AdS$_3$ bubble corresponds to the dilatonic solution, while an exact AdS$_3$ is the non-dilatonic solution. In order to obtain the cross section for an AdS$_3$ bubble, we introduce the \{out\}-state scattering picture with the AdS$_3$-AFS matching procedure. For an exact AdS$_3$, one considers the \{in\}-state scattering picture with the AdS$_3$-AdS$_3$ matching. Here the non-normalizable modes are crucially taken into account for the matching procedure. It turns out that the cross sections for the test fields in an AdS$_3$ bubble take the same forms as those in an exact AdS$_3$. This suggests that in the dilute gas and the low energy limits, the S-matrix for an AdS$_3$ bubble can be derived from an exact AdS$_3$ space. 
  In the framework of causal perturbation theory we analyze the gauge structure of a massless self-interacting quantum tensor field. We look at this theory from a pure field theoretical point of view without assuming any geometrical aspect from general relativity. To first order in the perturbation expansion of the $S$-matrix we derive necessary and sufficient conditions for such a theory to be gauge invariant, by which we mean that the gauge variation of the self-coupling with respect to the gauge charge operator $Q$ is a divergence in the sense of vector analysis. The most general trilinear self-coupling of the graviton field turns out to be the one derived from the Einstein-Hilbert action plus divergences and coboundaries. 
  The ubiquitous ADE classification has induced many proposals of often mysterious correspondences both in mathematics and physics. The mathematics side includes quiver theory and the McKay Correspondence which relates finite group representation theory to Lie algebras as well as crepant resolutions of Gorenstein singularities. On the physics side, we have the graph-theoretic classification of the modular invariants of WZW models, as well as the relation between the string theory nonlinear $\sigma$-models and Landau-Ginzburg orbifolds. We here propose a unification scheme which naturally incorporates all these correspondences of the ADE type in two complex dimensions. An intricate web of inter-relations is constructed, providing a possible guideline to establish new directions of research or alternate pathways to the standing problems in higher dimensions. 
  The gauge transformations of p-form fields in supergravity theories acquire a non-commuting character when one introduces potentials both for the theory's original field strengths and for their duals. This has previously been shown in the ``doubled'' formalism for maximal supergravities, where a generalised duality relation between original and dual field strengths replaces the equations of motion. In the doubled formalism, the gauge transformations generate a superalgebra, and the corresponding symmetries have accordingly been called ``superdualities.'' The corresponding Noether charges form a representation of the cohomology ring on the spacetime manifold. In this paper, we show that the gauge symmetry superalgebra implies certain non-trivial relations among the various p-brane tensions, which can straightforwardly be read off from the superalgebra commutation relations. This provides an elegant derivation of the brane-tension relations purely within a given theory, without the need to make use of duality relations between different theories, such as the type IIA/IIB T-duality, although the results are consistent with such dualities. We present the complete set of brane-tension relations in M-theory, in the type IIA and type IIB theories, and in all the lower-dimensional maximal supergravities. We also construct a doubled formalism for massive type IIA supergravity, and this enables us to obtain the brane-tension relations involving the D8-brane, purely within the framework of the massive IIA theory. We also obtain explicit transformations for the nine-dimensional T-duality between the massive type IIA theory and the Scherk-Schwarz reduced type IIB theory. 
  We consider the entropy problem of AdS_3 black holes using the conformal field theory at the horizon. We observe that the supersymmetry is enhanced at the horizon of massless AdS_3 black hole. This allows us to determine the vacuum of the modular invariant conformal field theory to be the NS-ground state (which corresponds to AdS_3 spacetime). This is smoothly related to the R-ground state (corresponding to massless black hole) by a spectral flow, which can be understood as a superconformal transformation. 
  The trace anomaly in external gravity is the sum of three terms at criticality: the square of the Weyl tensor, the Euler density and Box R, with coefficients, properly normalized, called c, a and a', the latter being ambiguously defined by an additive constant. Considerations about unitarity and positivity properties of the induced actions allow us to show that the total RG flows of a and a' are equal and therefore the a'-ambiguity can be consistently removed through the identification a'=a. The picture that emerges clarifies several long-standing issues. The interplay between unitarity and renormalization implies that the flux of the renormalization group is irreversible. A monotonically decreasing a-function interpolating between the appropriate values is naturally provided by a'. The total a-flow is expressed non-perturbatively as the invariant (i.e. scheme-independent) area of the graph of the beta function between the fixed points. We test this prediction to the fourth loop order in perturbation theory, in QCD with Nf ~< 11/2 Nc and in supersymmetric QCD. There is agreement also in the absence of an interacting fixed point (QED and phi^4-theory). Arguments for the positivity of a are also discussed. 
  The finite gauge transformation, in the submanifold of su(3) introduced in hep-th/9902027, is found to be linear in the gauge functions ${\omega}^a$'s. In this manner, a parameterization of SU(3) in terms of a single real octet vector ${\omega}^a$ is achieved. Under this finite transformation, those gauge fields satisfying $D^{ab}_{\mu}{\omega}^b = 0$, remain unchanged. A geometric meaning to the submanifold in terms of rotations in E8 is presented. 
  Duality between N=1 supersymmetric gauge theories(Seiberg's duality) is geometrized, in the framework of AdS/CFT correspodences. It is shown that Seiberg's duality corresponds to monodromy of wrapped D5 branes on the homology cycles of a generalized conifold where D3 branes are located. The celebrated \tilde{N}_c=N_f-N_c, \tilde{N}_f=N_f rule is reproduced and a braid group structure in a sequence of dualities, is found. 
  We present a simple and systematic method to calculate the Rindler noise, which is relevant to the analysis of the Unruh effect, by using the fluctuation-dissipative theorem. To do this, we calculate the dissipative coefficient explicitly from the equations of motion of the detector and the field. This method gives not only the correct answer but also a hint as to the origin of the apparent statistics inversion effect. Moreover, this method is generalized to the Dirac field, by using the fermionic fluctuation-dissipation theorem. We can thus confirm that the fermionic fluctuation-dissipation theorem is working properly. 
  We show that the solutions of SU(2) Yang-Mills-dilation and Einstein-Yang-Mills-dilaton theories described in a sequence of papers by Kleihaus and Kunz are not regular in the gauge field part. 
  An effective supergravity description of all instabilities of N_4=4 superstrings is derived. The construction is based on the N_4=4 BPS mass formula at finite temperature and uses the properties of N_4=4 gauged supergravity. It provides the boundaries of the various thermal phases in the non-perturbative moduli space. It also draws a precise picture of the dynamics in the high-temperature heterotic phase. This brief contribution summarizes results obtained in collaboration with I. Antoniadis and C. Kounnas. 
  A conformal field theory on the boundary of three-dimensional asymptotic anti-de Sitter spaces which appear as near horizon geometry of D-brane bound states is discussed. It is shown that partition functions of gravitational instantons appear as high and low temperature limits of the partition function of the conformal field theory. The result reproduces phase transition between the anti-de Sitter space and the BTZ black hole in the bulk gravity. 
  We formulate a systematic Bethe-Ansatz approach for computing bound-state (``breather'') S matrices for integrable quantum spin chains. We use this approach to calculate the breather boundary S matrix for the open XXZ spin chain with diagonal boundary fields. We also compute the soliton boundary S matrix in the critical regime. 
  Using the spectral representation approach to the Zamolodchikov's c-function and the Maldacena conjecture for the D1-branes, we compute the entropy of type IIB strings. An agreement, up to a numerical constant which cannot be determined using this approach, with the Bekenstein-Hawking entropy is found. 
  We summarize recent results on the resolution of two intimately related problems, one physical, the other mathematical. The first deals with the resolution of the non-perturbative low energy dynamics of certain N=2 supersymmetric Yang-Mills theories. We concentrate on the theories with one massive hypermultiplet in the adjoint representation of an arbitrary gauge algebra G. The second deals with the construction of Lax pairs with spectral parameter for certain classical mechanics Calogero-Moser integrable systems associated with an arbitrary Lie algebra G. We review the solution to both of these problems as well as their interrelation. 
  Recently proposed new gauge invariant formulation of the Chern-Simons gauge theory is considered in detail. This formulation is consistent with the gauge fixed formulation. Furthermore it is found that the canonical (Noether) Poincar\'e generators are not gauge invariant even on the constraints surface and do not satisfy the Poincar\'e algebra contrast to usual case. It is the improved generators, constructed from the symmetric energy-momentum tensor, which are (manifestly) gauge invariant and obey the quantum as well as classical Poincar\'e algebra. The physical states are constructed and it is found in the Schr\"odinger picture that unusual gauge invariant longitudinal mode of the gauge field is crucial for constructing the physical wavefunctional which is genuine to (pure) Chern-Simons theory. In matching to the gauge fixed formulation, we consider three typical gauges, Coulomb, axial and Weyl gauges as explicit examples. Furthermore, recent several confusions about the effect of Dirac's dressing function and the gauge fixings are clarified. The analysis according to old gauge independent formulation a' la Dirac is summarized in an appendix. 
  We investigate a relationship between the number of the negative modes around periodic instanton solution and the type of the decay-rate transition. It is shown that for the case of first-order decay-rate transition the lowest positive mode at low energy periodic instanton becomes additional negative mode at high energy regime, while in the second-order case there is only one negative mode in the full range of energy. This kind of analysis on the negative modes makes it possible to derive the criterion for the first-order transition. 
  The current algebra generated by fermions coupled to external gauge potentials and metrics on a manifold with boundary is discussed. It is shown that the previous methods, based on index theory arguments and used in the case without boundaries, carry over to the present problem. The resulting current algebra is the same as obtained from a quantization of bosonic Chern-Simons theories on space-times with nonempty boundaries. This means that also in the fermionic setting the construction is 'holographic', the Schwinger terms reside on the boundary. 
  We consider the simplest bosonic-type S-matrix which is usually regarded as unphysical due to the complex values of the finite volume ground state energy. While a standard quantum field theory interpretation of such a scattering theory is precluded, we argue that the physical situation described by this S-matrix is of a massive Ising model perturbed by a particular set of irrelevant operators. The presence of these operators drastically affects the stability of the original vacuum of the massive Ising model and its ultraviolet properties. 
  This text follows the line of a talk on Ringberg symposium dedicated to Wolfhart Zimmermann 70th birthday. The historical overview (Part 1) partially overlaps with corresponding text of my previous commemorative paper. At the same time second part includes some recent results in QFT and summarize an impressive progress of the "QFT renormalization group" application in mathematical physics. 
  We study the evolution, the transverse spreading and the subsequent thermalization of string states in the Weyl static axisymmetric spacetime. This possesses a singular event horizon on the symmetry axis and a naked singularity along the other directions. The branching diffusion process of string bits approaching the singular event horizon provides the notion of temperature that is calculated for this process. We find that the solution of the Fokker-Planck equation in the phase space of the transverse variables of the string, can be factored as a product of two thermal distributions, provided that the classical conjugate variables satisfy the uncertainty principle. We comment on the possible physical significance of this result. 
  We formulate the soliton equations on the lattice in terms of the reduced Moyal algebra which includes one parameter. The vanishing limit of the parameter leads to the continuous soliton equations. 
  Negative dimensional integration method (NDIM) is revealing itself as a very useful technique for computing Feynman integrals, massless and/or massive, covariant and non-covariant alike. Up to now, however, the illustrative calculations done using such method are mostly covariant scalar integrals, without numerator factors. Here we show how those integrals with tensorial structures can also be handled with easiness and in a straightforward manner. However, contrary to the absence of significant features in the usual approach, here the NDIM also allows us to come across surprising unsuspected bonuses. In this line, we present two alternative ways of working out the integrals and illustrate them by taking the easiest Feynman integrals in this category that emerges in the computation of a standard one-loop self-energy diagram. One of the novel and as yet unsuspected bonus is that there are degeneracies in the way one can express the final result for the referred Feynman integral. 
  We show that U(1) Yang-Mills theory on noncommutative R^4 can be renormalized at the one-loop level by multiplicative dimensional renormalization of the coupling constant and fields of the theory. We compute the beta function of the theory and conclude that the theory is asymptotically free. We also show that the Weyl-Moyal matrix defining the deformed product over the space of functions on R^4 is not renormalized at the one-loop level. 
  We consider black holes in anti-de Sitter space AdS_{p+2} (p = 2,3,5), which have hyperbolic, flat or spherical event horizons. The $O(\alpha'^3)$ corrections (or the leading corrections in powers of the eleven-dimensional Planck length) to the black hole metrics are computed for the various topologies and dimensions. We investigate the consequences of the stringy or M-theory corrections for the black hole thermodynamics. In particular, we show the emergence of a stable branch of small spherical black holes. We obtain the corrected Hawking-Page transition temperature for black holes with spherical horizons, and show that for p=3 this phase transition disappears at a value of $\alpha'$ considerably smaller than that estimated previously by Gao and Li. Using the AdS/CFT correspondence, we determine the $S^1 x S^3$ N=4 SYM phase diagram for sufficiently large `t Hooft coupling, and show that the critical point at which the Hawking-Page transition disappears (the correspondence point of Horowitz-Polchinski), occurs at $g_{YM}^2N \approx 20.5$. The d=4 and d=7 black hole phase diagrams are also determined, and connection is made with the corresponding boundary CFTs. Finally, for flat and hyperbolic horizons, we show that the leading stringy or M-theory corrections do not give rise to any phase transition. For horizons compactified to a torus $T^p$ or to a quotient of hyperbolic space, $H^p/\Gamma$, we comment on the effects of light winding modes. 
  We review the construction of the unitary supermultiplets of the ${\cal{N}}=8$ d=5 anti-de Sitter (AdS_5) superalgebra SU(2,2|4), which is the symmetry group of the type IIB superstring theory on $AdS_5 \times S^5$, using the oscillator method. 
  We consider a version of the $AdS_{d+1}/CFT_{d}$ correspondence, in which the bulk space is taken to be the quotient manifold $AdS_{d+1} /\Gamma$ with a fairly generic discrete group $\Gamma$ acting isometrically on $AdS_{d+1}$. We address some geometrical issues concerning the holographic principle and the UV/IR relations. It is shown that certain singular structures on the quotient boundary ${\bf S}^{d}/\Gamma$ can affect the underlying physical spectrum. In particular, the conformal dimension of the most relevant operators in the boundary CFT can increase as $\Gamma$ becomes ``large''. This phenomenon also has a natural explanation in terms of the bulk supergravity theory. The scalar two-point function is computed using this quotient version of the AdS/CFT correspondence, which agrees with the expected result derived from conformal invariance of the boundary theory. 
  The production of string charge during a crossing of certain oriented D-branes is studied. We compute the string charge in the system of a probe D2-brane and a background D6-brane by use of the equations of motion in the ten-dimensions. We confirm the creation of string charge as inflow from the background D6-brane. 
  We consider the two-dimensional dilute q-state Potts model on its first order phase transition surface for 0<q\leq 4. After determining the exact scattering theory which describes the scaling limit, we compute the two-kink form factors of the dilution, thermal and spin operators. They provide an approximation for the correlation functions whose accuracy is illustrated by evaluating the central charge and the scaling dimensions along the tricritical line. 
  This paper discusses the symmetry of the wave field that lies to the right and left of a two-sided accelerated mirror in 1 + 1 space and satisfies a single condition on it. The symmetry is accumulated in the Bogolyubov matrix coefficients $\alpha$ and $\beta$ that connect the two complete sets of solutions of the wave equations. The amplitudes of the quantum processes in the right and left half-spaces are expressed in terms of $\alpha$ and $\beta$ and are related to each other by transformation (12). Coefficient $\beta_{\omega'\omega}^*$ plays the role of the source amplitude of a pair of particles that are directed to opposite sides with frequencies $\omega$ and $\omega'$ but that are in either the left or the right half-space as a consequence of the reflection of one of them. Such an interpretation makes $\beta_{\omega'\omega}^*$ observable and explains the equalities, given by Eq. (1) and found earlier by Nikishov and author [Zh. Eksp. Teor. Fiz. 108, 1121 (1995)] and by author [Zh. Eksp. Teor. Fiz. 110, 526 (1996)] that the radiation spectra of a mirror in 1+1 space coincide with those of charges in 3 + 1 space by the fact that the moment of the pair emitted by the mirror coincide with the spin of the single particle emitted by the charge. 
  In this work a field theoretical model is constructed to describe the statistical mechanics of an arbitrary number of topologically linked polymers in the context of the analytical approach of Edwards. As an application, the effects of the topological interactions are studied in the one loop approximation. A natural way to include in the treatment also more sophisticated link invariants than the Gauss linking number is outlined. 
  Recently, it has been conjectured that supergravity solutions with two asymptotically AdS regions describe the RG flow of a 4-d field theory from a UV fixed point to an interacting IR fixed point. In this paper we lend support to this conjecture by showing that, in the UV (IR) limit, the two-point function of a minimally-coupled scalar field depends only on the UV (IR) region of the metric, asymptotic to AdS_5. This result is consistent with the interpretation of the radial coordinate of Anti de Sitter space as an energy scale, and it may provide an analog of the Callan-Symanzik equation for supergravity duals of strongly coupled field theories. 
  We quantize the type IIB GS string action on AdS3 X S3 with pure NSNS background and we show that it is equivalent to Liouville with periodic boundary condition plus free spacetime fermions. 
  Review of the theory of effective actions and the hypothetical origins of integrability in Seiberg-Witten theory. 
  We present a review of the results on the associativity algebras and WDVV equations associated with the Seiberg-Witten solutions of N=2 SUSY gauge theories. It is mostly based on the integrable treatment of these solutions. We consider various examples of the Seiberg-Witten solutions and corresponding integrable systems and discuss when the WDVV equations hold. We also discuss a covariance of the general WDVV equations. 
  We consider general properties of central charges of zero branes and associated duality invariants, in view of their double role, on the bulk and on the world volume (quantum-mechanical) theory. A detailed study of the BPS condition for the mass spectrum arising from toroidal compactifications is given for 1/2, 1/4 and 1/8 BPS states in any dimensions. As a byproduct, we retreive the U-duality invariant conditions on the charge (zero mode) spectrum and the orbit classification of BPS states preserving different fractions of supersymmetry. The BPS condition for 0-branes in theories with 16 supersymmetries in any dimension is also discussed. 
  We study five dimensional non critical type 0 string theory and its correspondence to non supersymmetric Yang Mills theory in four dimensions. We solve the equations of motion of the low energy effective action and identify a class of solutions that translates into a confining behavior in the IR region of the dual gauge theories. In particular we identify a setup which is dual to pure SU(N) Yang-Mills theory. Possible flows of the solutions to the UV region are discussed. The validity of the solutions and potential sub-leading string corrections are also discussed. 
  We consider SO(4) x SO(6) invariant type IIB string solution describing D3-branes superposed with D-instantons homogeneously distributed over D3-brane world-volume. In the near D3-brane horizon limit this background interpolates between AdS_5 x S^5 space in UV and flat space (with non-constant dilaton and RR scalar) in IR. Generalizing the AdS/CFT conjecture we suggest that type IIB string in this geometry is dual to N=4 SYM theory in a state with a constant self-dual gauge field background. The semiclassical string representation for the Wilson factor implies confinement with effective string tension depending on constant D-instanton density parameter. This provides a simple example of type IIB string -- gauge theory duality with clear D-brane and gauge theory interpretation. 
  The general method of the reduction in the number of coupling parameters is discussed. Using renormalization group invariance, theories with several independent couplings are related to a set of theories with a single coupling parameter. The reduced theories may have particular symmetries, or they may not be related to any known symmetry. The method is more general than the imposition of invariance properties. Usually, there are only a few reduced theories with an asymptotic power series expansion corresponding to a renormalizable Lagrangian. There also exist `general' solutions containing non-integer powers and sometimes logarithmic factors. As an example for the use of the reduction method, the dual magnetic theories associated with certain supersymmetric gauge theories are discussed. They have a superpotential with a Yukawa coupling parameter. This parameter is expressed as a function of the gauge coupling. Given some standard conditions, a unique, isolated power series solution of the reduction equations is obtained. After reparametrization, the Yukawa coupling is proportional to the square of the gauge coupling parameter. The coefficient is given explicitly in terms of the numbers of colors and flavors. `General' solutions with non-integer powers are also discussed. A brief list is given of other applications of the reduction method. 
  We study local mirror symmetry on non-compact Calabi-Yau manifolds (conifold type of singularities) in the presence of D3 brane probes. Using an intermediate brane setup of NS 5-branes `probed' by D4 resp. D5 branes, we can explicitly T-dualize three isometry directions to relate a non-compact Calabi-Yau manifold to its local mirror. The intermediate brane setup is the one that is best suited to read off the gauge theory on the probe. Both intervals and boxes of NS 5-branes appear as brane setups. Going from one to the other is equivalent to performing a conifold transition in the dual geometry. One result of our investigation is that the brane box rules as they have been discussed so far should be generalized. Our new rules do not need diagonal fields localized at the intersection. The old rules reappear on baryonic branches of the theory. 
  Certain two and three point functions of gauge invariant primary operators of ${\cal N}=4$ SYM are computed in ${\cal N}=1$ superspace keeping all the $\th$-components. This allows one to read off many component descendent correlators. Our results show the only possible $g^2_{YM}$ corrections to the free field correlators are contact terms. Therefore they vanish for operators at separate points, verifying the known non-renormalization theorems. This also implies the results are consistent with ${\cal N}=4$ supersymmetry even though the Lagrangian we use has only ${\cal N}=1$ manifest supersymmetry. We repeat some of the calculations using supersymmetric Landau gauge and obtain, as expected, the same results as those of supersymmetric Feynman gauge. 
  We investigate the role of massless magnetic monopoles in the N=4 supersymmetric Yang-Mills Higgs theories. They can appear naturally in the 1/4-BPS dyonic configurations associated with multi-pronged string configurations. Massless magnetic monopoles can carry nonabelian electric charge when their associated gauge symmetry is unbroken. Surprisingly, massless monopoles can also appear even when the gauge symmetry is broken to abelian subgroups. 
  The interplay between between gauge-field winding numbers, theta-vacua, and the Dirac operator spectrum in finite-volume gauge theories is reconsidered. To assess the weight of each topological sector, we compare the mass-dependent chiral condensate in gauge field sectors of fixed topological index with the answer obtained by summing over the topological charge. Also the microscopic Dirac operator spectrum in the full finite-volume Yang-Mills theory is obtained in this way, by summing over all topological sectors with the appropriate weight. 
  Complete constraint analysis and choice of gauge conditions consistent with equations of motion is done for non-abelian Chern-Simons field interacting with N-component complex scalar field. Dirac-Schwinger condition is satisfied by the reduced phase-space Hamiltonian density with respect to the Dirac bracket. 
  Complete constraint analysis and choice of gauge conditions consistent with equations of motion is done for Abelian Chern Simons field interacting minimally with a complex scalar field. The Dirac-Schwinger consistency condition is satisfied by the reduced phase space Hamiltonian density with respect to the the Dirac bracket. It is shown that relativistic invariance under boosts can be obtained only if gauge conditions were chosen consistent with the equations of motion. Moreover all gauge invariant quantities are shown to be free of transformation anomaly. 
  We briefly review the computation of graviton and antisymmetric tensor scattering amplitudes in Matrix Theory from a diagramatic S-Matrix point of view. 
  A direct consequence of the occurrence of fermion families is the invariance of currents under certain groups of (universality) transformations. We show how these universality groups can themselves be used to find and study grand family unification models. Identifying two independent - weak and strong - universality groups and assuming that the grand unification group is SU(8N), its subgroup respecting either weak or strong universality is shown to be G = SU(2)xU(1)xSU(3). The fundamental representation of SU(8N) decomposes as N families of leptons and quarks. In the G-invariant limit, all fermions are left-handed. A mechanism for generating the correct number of right-handed fermions with the correct couplings so as to give pure vector colour and electromagnetic currents is exhibited. Universality is shown to result most naturally from a preonic structure of fermions. In such a preonic picture there are no ultraheavy gauge bosons and no anomaly or hierarchy problem. 
  Compact canonical quantization on the light cone (DLCQ) is examined in the limit of infinite periodicity lenth L. Pauli Jordan commutators are found to approach continuum expressions with marginal non causal terms of order $L^{-3/4}$ traced back to the handling of IR divergence through the elimination of zero modes. In contrast direct quantization in the continuum (CLCQ) in terms of field operators valued distributions is shown to provide the standard causal result while at the same time ensuring consistent IR and UV renormalization. 
  We construct the model with light-like world-lines for the massive 4D spinning particles and 3D anyons. It is obtained via the formal bosonization of pseudoclassical model for the massive Dirac particle with subsequent reduction to the light-like curves. The peculiarity of the light-like trajectories produced due to the Zitterbewegung is explained from the viewpoint of reduction and reparametrization invariance. 
  We compare two approaches in supersymmetric confinement, the Seiberg formulation and the superconvergence rule. For the latter, the critical point is $\gamma_{00}=0$ in the Landau gauge. We find $4\gamma_{00}=\beta_0$ is the critical point for most of confining theories without a tree-level superpotential in the Seiberg formulation, in particular, in the large $N_c$ and $N_f$ limit. We show how confining theories with a discrete symmetry and a tree-level superpotential connect these two critical points: the large and small discrete symmetry limits correspond to the critical points $4\gamma_{00}=\beta_0$ and $\gamma_{00}=0$, respectively. 
  We use the method of stable degenerations to study the local geometry of Calabi-Yau fourfolds for F-theory compactifications dual to heterotic compactifications on a Calabi-Yau threefold with fivebranes wrapping holomorphic curves in the threefold. When fivebranes wrap intersecting curves, or when many fivebranes wrap the same curve, the dual fourfolds degenerate in interesting ways. We find that some of these can be usefully described in terms of degenerations of the base of the elliptic fibrations of these fourfolds. We use Witten's criterion to determine which of the fivebranes can lead to the generation of a non-perturbative superpotential. 
  Ward identities in the case of scattering of antisymmetric three form RR gauge fields off a D2-brane target has been studied in type-IIA theory. 
  The possibility of non-trivial representations of the gauge group on wavefunctionals of a gauge invariant quantum field theory leads to a generation of mass for intermediate vector and tensor bosons. The mass parameters m show up as central charges in the algebra of constraints, which then become of second-class nature. The gauge group coordinates acquire dynamics outside the null-mass shell and provide the longitudinal field degrees of freedom that massless bosons need to form massive bosons. This is a `non-Higgs' mechanism that could provide new clues for the best understanding of the symmetry breaking mechanism in unified field theories. A unified quantization of massless and massive non-Abelian Yang-Mills, linear Gravity and Abelian two-form gauge field theories are fully developed from this new approach, where a cohomological origin of mass is pointed out. 
  We argue that Yang-Mills theory on noncommutative torus, expressed in the Fourrier modes, is described by a gauge theory in a usual commutative space, the gauge group being a generalization of the area-preserving diffeomorphisms to the noncommutative case. In this way, performing the loop calculation in this gauge theory in the continuum limit we show that this theory is {\it one loop renormalizable}, and discuss the UV and IR limits. The moduli space of the vacua of the noncommutative super Yang-Mills theories in (2+1) dimensions is discussed. 
  In this paper we investigate about several configurations of two intersecting branes at arbitrary angles. We choose the viewpoint of a brane source and a brane probe and use the low-energy dynamics of p-branes. For each p-brane this dynamics is governed by a generic DBI action including a WZ term, which couples to the SUGRA background of the other brane. The analysis naturally reveals two types of configurations: the ``marginal'' and the ``non-marginal'' ones. We specify possible configurations for a pair of similar or non-similar branes in either of these two categories. In particular, for two similar branes at angles, this analysis reveals that all the marginal configurations are specified by SU(2) angles while the non-marginal configurations are specified by $Sp(2)$ angles. On the other hand, we find that no other configuration of two intersecting branes at non-trivial angles can be constructed out of flat p-branes. So in particular, two non-similar branes can only be found in an orthogonal configuration. In this case the intersection rules for either of the marginal or non-marginal configurations are derived, which thereby provide interpretations for the known results from supergravity. 
  We present a new type-IIB supergravity vacuum that describes the strong coupling regime of a non-supersymmetric gauge theory. The latter has a running coupling such that the theory becomes asymptotically free in the ultraviolet. It also has a running theta angle due to a non-vanishing axion field in the supergravity solution. We also present a worm-hole solution, which has finite action per unit four-dimensional volume and two asymptotic regions, a flat space and an AdS^5\times S^5. The corresponding N=2 gauge theory, instead of being finite, has a running coupling. We compute the quark-antiquark potential in this case and find that it exhibits, under certain assumptions, an area-law behaviour for large separations. 
  We propose that the structure of gauge theories, the $(2,0)$ and little-string theories is encoded in a unique function on the real group manifold $E_{10}(R)$. The function is invariant under the maximal compact subgroup $K$ acting on the right and under the discrete U-duality subgroup $E_{10}(Z)$ on the left. The manifold $E_{10}(Z)\backslash E_{10}(R) /K$ contains an infinite number of periodic variables. The partition function of U(n), N=4 Super-Yang-Mills theory on $T^4$, with generic SO(6) R-symmetry twists, for example, is derived from the $n^{th}$ coefficient of the Fourier transform of the function with respect to appropriate periodic variables, setting other variables to the R-symmetry twists and the radii of $T^4$. In particular, the partition function of nonsupersymmetric Yang-Mills theory is a special case, obtained from the twisted $(2,0)$ or little-string theories. The function also seems to encode the answer to questions about M-theory on an arbitrary $T^8$. The second conjecture that we wish to propose is that this function is harmonic with respect to the $E_{10}(R)$ invariant metric. In a similar fashion, we propose that there exists a function on the infinite Kac-Moody group $DE_{18}$ that encodes the twisted partition functions of the $E_8$ 5+1D theories as well as answers to questions about the heterotic string on $T^7$. 
  We present the algebraic framework for the quantization of the classical bosonic charge algebra of maximally extended (N=16) supergravity in two dimensions, thereby taking the first steps towards an exact quantization of this model. At the core of our construction is the Yangian algebra $Y(e_8)$ whose RTT presentation we discuss in detail. The full symmetry algebra is a centrally extended twisted version of the Yangian double $DY(e_8)_c$. We show that there exists only one special value of the central charge for which the quantum algebra admits an ideal by which the algebra can be divided so as to consistently reproduce the classical coset structure $E_{8(8)}/SO(16)$ in the limit $\hbar\to 0$. 
  It is shown that noncommutative geometry is a nonperturbative regulator which can manifestly preserve a space supersymmetry and a supergauge symmetry while keeping only a finite number of degrees of freedom in a theory. The simplest N=1 case of an U(1) supergauge theory on the sphere is worked out in detail. 
  We investigate the relevance of Eisenstein series for representing certain $G(Z)$-invariant string theory amplitudes which receive corrections from BPS states only. $G(Z)$ may stand for any of the mapping class, T-duality and U-duality groups $Sl(d,Z)$, $SO(d,d,Z)$ or $E_{d+1(d+1)}(Z)$ respectively. Using $G(Z)$-invariant mass formulae, we construct invariant modular functions on the symmetric space $K\backslash G(R)$ of non-compact type, with $K$ the maximal compact subgroup of $G(R)$, that generalize the standard non-holomorphic Eisenstein series arising in harmonic analysis on the fundamental domain of the Poincar\'e upper half-plane. Comparing the asymptotics and eigenvalues of the Eisenstein series under second order differential operators with quantities arising in one- and $g$-loop string amplitudes, we obtain a manifestly T-duality invariant representation of the latter, conjecture their non-perturbative U-duality invariant extension, and analyze the resulting non-perturbative effects. This includes the $R^4$ and $R^4 H^{4g-4}$ couplings in toroidal compactifications of M-theory to any dimension $D\geq 4$ and $D\geq 6$ respectively. 
  The Hilbert spaces of supersymmetric systems admit symmetries which are often related to the topology and geometry of the (target) field-space. Here, we study certain (2,2)-supersymmetric systems in 2-dimensional spacetime which are closely related to superstring models. They all turn out to posess some hitherto unexploited and geometrically and topologically unobstructed symmetries, providing new tools for studying the topology and geometry of superstring target spacetimes, and so the dynamics of the effective field theory in these. 
  In models inspired by non-commutative geometry, patterns of gauge symmetry breaking are analyzed, and SU(5) models are found to naturally favor a vacuum preserving SU(3) X SU(2) X U(1). A more realistic model is presented, and the possibility of D-brane interpretation is discussed. 
  The quantized magnetic flux $\Phi=-4\pi N\sk, N=0,\pm1,...$ of non-topological vortices in the non-relativistic Chern-Simons theory is related to the topological degree of the $S^2\to S^2$ mapping defined by lifting the problem to the Riemann spheres. Regular solutions with finite degree only arise for rational functions, whose topological degree, $N$, is the commun number of their zeros and poles on the Riemann sphere, also called their algebraic order. 
  Compactification of Matrix Model on a Noncommutative torus is obtained from strings ending on D-branes with background B field. The BPS spectrum of the system and a novel SL(2,Z) symmetry are discussed. 
  In this paper, we use only the equation of motion for an interacting system of gravity, dilaton and antisymmetric tensor to study the soliton solutions by making use of a Poincar\'e invariant ansatz. We show that the system of equations are completly integrable and the solution is unique with appropriate boundary conditions. Some new class of solutions are also given explicitly. 
  The correlations of the QCD Dirac eigenvalues are studied with use of an extended chiral random matrix model. The inclusion of spatial dependence which the original model lacks enables us to investigate the effects of diffusion modes. We get analytical expressions of level correlation functions with non-universal behavior caused by diffusion modes which is characterized by Thouless energy. Pion mode is shown to be responsible for these diffusion effects when QCD vacuum is considered a disordered medium. 
  We study the proposed duality between the 5-dimensional supergravity/superstring on $AdS_3\times S^2$ and the 2-dimensional N=(0,4) SCFT defined on the boundary of AdS-space. We construct explicitly the N=(0,4) SCFT by imposing the `quiver projection' developed by Douglas-Moore on the N=(4,4) SCFT of symmetric orbifold, which is proposed to be the dual of the 6-dimensional supergravity/superstring on $AdS_3\times S^3$.   We explore in detail the spectrum of chiral primaries in this `quiver $SCFT_2$'. We compare it with the Kaluza-Klein spectrum on $AdS_3\times S^2$ and check the consistency between them. We further emphasize that orbifolding of bulk theory should {\em not} correspond to orbifolding of the boundary CFT in the usual sense of two dimensional CFT, rather corresponds to the quiver projection. We observe that these are not actually equivalent with each other when we focus on the multi-particle states. 
  We discuss the BRST cohomology and exhibit a connection between the Hodge decomposition theorem and the topological properties of a two dimensional free non-Abelian gauge theory having no interaction with matter fields. The topological nature of this theory is encoded in the vanishing of the Laplacian operator when equations of motion are exploited. We obtain two sets of topological invariants with respect to BRST and co-BRST charges on the two dimensional manifold and show that the Lagrangian density of the theory can be expressed as the sum of terms that are BRST- and co-BRST invariants. 
  We develop two N=2 superfield formulations of free equations of motion for the joint model of all D=4 massless higher-superspin fields in generating form. The explicit Osp(2|4) supersymmetry is achieved without exploiting the harmonic superspace, and with adding no auxiliary component fields to those of N=1 superfields. The formulations are developed in two different Osp(2|4) homogeneous superspaces which have a structure of a fibre bundle over the standard D=4 AdS superspace, with dimensions (7|4) and (7|8). The N=2 covariant derivatives in these spaces are expressed in terms of N=1 ones which gives simple rules for component analysis. 
  We use the boundary state formalism to study, from the closed string point of view, superpositions of branes and anti-branes which are relevant in some non-perturbative string dualities. Treating the tachyon instability of these systems as proposed by A. Sen, we show how to incorporate the effects of the tachyon condensation directly in the boundary state. In this way we manage to show explicitly that the D1 -- anti-D1 pair of Type I is a stable non-BPS D-particle, and compute its mass. We also generalize this construction to describe other non-BPS D-branes of Type I. By requiring the absence of tachyons in the open string spectrum, we find which configurations are stable and compute their tensions. Our classification is in complete agreement with the results recently obtained using the K-theory of space-time. 
  These lecture notes begin with a review of the first nonleading contributions to the derivative expansion of the M theory effective action compactified on a two-torus. The form of these higher-derivative interactions is shown to follow from ten-dimensional type IIB supersymmetry as well as from one-loop quantum corrections to classical eleven-dimensional supergravity. The detailed information concerning D-instanton effects encoded in these terms is related to the problem of evaluating the Witten index for $N$ D-particles in the type IIA theory. Using the AdS/CFT conjecture, it also leads to very specific predictions of multi-instanton contributions in $\cal N=4$ supersymmetric SU(N) Yang--Mills theory in the limit of strong 't Hooft coupling. [Extended version of lectures given at 22nd Johns Hopkins Workshop (Gothenberg, August 20-22 1998); `Quantum Aspects of Gauge Theories, Supersymmetry and Unification', TMR meeting (Corfu, September 20-26 1998); Andrjewski lectures (Berlin, November 1-6 1998).] 
  We present the evidence that $(-1)^{F_L}$ is a nonperturbative symmetry of Type II string theory. We argue that $(-1)^{F_L}$ is a symmetry of string theory as much as the $SL(2, Z)$ of the Type II string is and how the branes are mapped under the $(-1)^{F_L}$ modding. NS branes are mapped into the NS branes with the same world volume dimensions but with the different chiral structure. Supersymmetric Dp-branes (bound with anti-Dp branes) are mapped to unstable nonsupersymmetric Dp-branes, which has D(p-1)-branes as kink solutions according to Horava. 
  Considering a class of (2,0)-super-Yang-Mills multiplets that accommodate a pair of independent gauge potentials in connection with a single symmetry group, we present here their coupling to ordinary matter and to non-linear $\sigma$-models in (2,0)-superspace. The dynamics and the couplings of the gauge potentials are discussed and the interesting feature that comes out is a sort of ``chirality'' for one of the gauge potentials once light-cone coordinates are chosen. 
  We study and discuss some of the consequences of the inclusion of torsion in 3D Einstein-Chern-Simons gravity. Torsion may trigger the excitation of non-physical modes in the spectrum. Higher-derivative terms are then added up and tree-level unitarity is contemplated. 
  We study intersecting D-branes in type 0 string theories and show that the D$p_{\pm}$-brane bound states obey similar intersecting rules as the D$p$-branes of the type II theories. The D$5_{\pm}$-D$1_{\pm}$ brane system is studied in detail. We show that the corresponding near-horizon geometry is the $AdS_3\times S^3\times T^4$ space and that there is no tachyon instability in this background. The Bekenstein-Hawking entropy is calculated. The worldvolume theory on the D$5_{\pm}$-D$1_{\pm}$ system is also studied. This theory contains both bosons and fermions and it is seen to arise as a projection of the supersymmetric gauge theory related to the D5-D1 system of the type IIB theory. The Bekenstein-Hawking entropy formula is reproduced exactly using the dual conformal field theory. 
  Examples of stable, non-BPS M-theory membrane configuration are constructed via M(atrix) theory. The stable membranes are localized on O4 or O8 orientifolds, which projects Chan-Paton gauge bundle of M(atrix) zero-brane partons to USp-type. The examples are shown to be consistent with prediction based on K-theory analysis. 
  The existence of intimate relation between generalized statistics and supersymmetry is established by observation of hidden supersymmetric structure in pure parabosonic systems. This structure is characterized generally by a nonlinear superalgebra. The nonlinear supersymmetry of parabosonic systems may be realized, in turn, by modifying appropriately the usual supersymmetric quantum mechanics. The relation of nonlinear parabosonic supersymmetry to the Calogero-like models with exchange interaction and to the spin chain models with inverse-square interaction is pointed out. 
  We find an infinite dimensional free algebra which lives at large N in any SU(N)-invariant action or Hamiltonian theory of bosonic matrices. The natural basis of this algebra is a free-algebraic generalization of Chebyshev polynomials and the dual basis is closely related to the planar connected parts. This leads to a number of free-algebraic forms of the master field including an algebraic derivation of the Gopakumar-Gross form. For action theories, these forms of the master field immediately give a number of new free-algebraic packagings of the planar Schwinger-Dyson equations. 
  We determine the thermodynamic stability conditions for near-extreme rotating D3, M5, and M2-branes with multiple angular momenta. Critical exponents near the boundary of stability are discussed and compared with a naive field theory model. From a partially numerical computation we conclude that outside the boundary of stability, the angular momentum density tends to become spatially inhomogeneous.   Periodic Euclidean spinning brane solutions have been studied as models of QCD. We explain how supersymmetry is restored in the world-volume field theory in the limit of large spin and discuss the hierarchy of energy scales that develops as this limit is approached. 
  We analytically and numerically investigate the 't Hooft-Bergknoff-Eller equations, the lowest order mesonic Light-Front Tamm-Dancoff equations for U(N_C) and SU(N_C) gauge theories. We find the wavefunction can be well approximated by new basis functions and obtain an analytic formula for the mass of the lightest bound state. Its value is consistent with the precedent results. 
  In this paper we compute the scaling functions of the effective central charges for various quantum integrable models in a deep ultraviolet region $R\to 0$ using two independent methods. One is based on the ``reflection amplitudes'' of the (super-)Liouville field theory where the scaling functions are given by the conjugate momentum to the zero-modes. The conjugate momentum is quantized for the sinh-Gordon, the Bullough-Dodd, and the super sinh-Gordon models where the quantization conditions depend on the size $R$ of the system and the reflection amplitudes. The other method is to solve the standard thermodynamic Bethe ansatz (TBA) equations for the integrable models in a perturbative series of $1/(const. - \ln R)$. The constant factor which is not fixed in the lowest order computations can be identified {\it only when} we compare the higher order corrections with the quantization conditions. Numerical TBA analysis shows a perfect match for the scaling functions obtained by the first method. Our results show that these two methods are complementary to each other. While the reflection amplitudes are confirmed by the numerical TBA analysis, the analytic structures of the TBA equations become clear only when the reflection amplitudes are introduced. 
  A universal Lagrangian that defines various four-dimensional massive Yang-Mills theories without Higgs bosons is presented. Each of the theories is characterized by a constant k contained in the Lagrangian. For k=0, the Lagrangian reduces to one defining the topologically massive Yang-Mills theory, and for k=1, the Lagrangian reduces to one defining the Freedman-Townsend model. New massive Yang-Mills theories are obtained by choosing k to be real numbers other than 0 and 1. 
  In this paper we continue the study of the path-integral formulation of classical mechanics and in particular we better clarify, with respect to previous papers, the geometrical meaning of the variables entering this formulation. With respect to the first paper with the same title, we {\it correct} here the set of transformations for the auxiliary variables $\lambda_{a}$. We prove that under this new set of transformations the Hamiltonian ${\widetilde{\cal H}}$, appearing in our path-integral, is an exact scalar and the same for the Lagrangian. Despite this different transformation, the variables $\lambda_{a}$ maintain the same operatorial meaning as before but on a different functional space. Cleared up this point we then show that the space spanned by the whole set of variables ($\phi, c, \lambda,\bar c$) of our path-integral is the cotangent bundle to the {\it reversed-parity} tangent bundle of the phase space ${\cal M}$ of our system and it is indicated as $T^{\star}(\Pi T{\cal M})$. In case the reader feel uneasy with this strange {\it Grassmannian} double bundle, we show in this paper that it is possible to build a different path-integral made only of {\it bosonic} variables. These turn out to be the coordinates of $T^{\star}(T^{\star}{\cal M})$ which is the double cotangent bundle of phase-space. 
  We investigate the integrability of the SO(N) principal chiral model on a half-line, and find that mixed Dirichlet/Neumann boundary conditions (as well as pure Dirichlet or Neumann) lead to infinitely many conserved charges classically in involution. We use an anomaly-counting method to show that at least one non-trivial example survives quantization, compare our results with the proposed reflection matrices, and, based on these, make some preliminary remarks about expected boundary bound-states. 
  We have constructed nonrelativistic fermion and scalar field theories coupled to a Maxwell-Chern-Simons gauge field which admit static multi-vortex solutions. This is achieved by introducing a magnetic coupling term in addition to the usual minimal coupling. 
  D-branes in curved backgrounds can be treated with techniques of boundary conformal field theory. We discuss the influence of scalar condensates on such branes, i.e. perturbations of boundary conditions by marginal boundary operators. A general criterion is presented that guarantees a boundary perturbation to be truly marginal in all orders of perturbation theory. Our results on boundary deformations have several interesting applications which are sketched at the end of this note. 
  We examine the possibility that gauge field configurations on stacks of parallel Dp branes support topological solitons. We give an exhaustive list of possible soliton charges for p<7. We also discuss how configurations carrying the soliton charges can be constructed from intersecting branes. 
  The unitary implementation of a symmetry group $G$ of a classical system in the corresponding quantum theory entails unavoidable deformations $\TG$ of $G$, namely, central extensions by the typical phase invariance group U(1). The appearance of central charges in the corresponding Lie-algebra quantum commutators, as a consequence of non-trivial responses of the phase of the wave function under symmetry transformations, lead to a quantum generation of extra degrees of freedom with regard to the classical counterpart. In particular, symmetries of the Hall effect, Yang-Mills and conformally invariant classical field theories are affected when passing to the quantum realm. 
  We review the calculation of the spectrum of glueball masses in non-supersymmetric Yang-Mills theory using the conjectured duality between supergravity and large N gauge theories. The glueball masses are obtained by solving the supergravity wave equations in a black hole geometry. The glueball masses found this way are in unexpected agreement with the available lattice data. We also show how to use a modified version of the duality based on rotating branes to calculate the glueball mass spectrum with some of the Kaluza-Klein states of the supergravity theory decoupled from the spectrum. 
  Electrically charged solutions breaking half of the supersymmetry in Anti-De Sitter four dimensional N=2 supergravity coupled to vector supermultiplets are constructed. These static black holes live in an asymptotic $AdS_4$ space-time. The Killing spinor, i. e., the spinor for supersymmetry variation is explicitly constructed for these solutions. 
  We construct vacua of heterotic M-theory with general gauge bundles and five-branes. Some aspects of the resulting low-energy effective theories are discussed. 
  We derive the general form of the anomaly for chiral spinors and self-dual antisymmetric tensors living on D-brane and O-plane interesections, using both path-integral and index theorem methods. We then show that the anomalous couplings to RR forms of D-branes and O-planes in a general background are precisely those required to cancel these anomalies through the inflow mechanism. This allows, for instance, for local anomaly cancellation in generic orientifold models, the relevant Green-Schwarz term being given by the sum of the anomalous couplings of all the D-branes and O-planes in the model. 
  We propose an uncertainty relation of space-time. This relation is characterized by GhT \lesssim \delta V, where T and \delta V denote a characteristic time scale and a spatial volume, respectively. Using this uncertainty relation, we give qualitative estimations for the entropies of a black hole and our universe. We obtain qualitative agreements with the known results. The holographic principle of 't Hooft and Susskind is reproduced. We also discuss cosmology and give a relation to the cosmic holographic principle of Fischler and Susskind. However, as for the maximal entropy of a system with an energy E, we obtain the formula \sqrt{EV/Gh^2}, with V denoting the volume of the system, which is distinct from the Bekenstein entropy formula ER/h with R denoting the length scale of the system. 
  An expression for the curvature of the "covariant" determinant line bundle is given in even dimensional space-time. The usefulness is guaranteed by its prediction of the covariant anomaly and Schwinger term. It allows a parallel derivation of the consistent anomaly and Schwinger term, and their covariant counterparts, which clarifies the similarities and differences between them. In particular, it becomes clear that in contrary to the case for anomalies, the difference between the consistent and covariant Schwinger term can not be extended to a local form on the space of gauge potentials. 
  Z_2-graded Schwinger terms for neutral particles in 1 and 3 space dimensions are considered. 
  Motivated by the search for a space-time supersymmetric extension of the N=2 string, we construct a particle model which, upon quantization, describes (abelian) self-dual super Yang-Mills in 2+2 dimensions. The local symmetries of the theory are shown to involve both world-line supersymmetry and kappa symmetry. 
  Current superalgebras and corresponding Schwinger terms in 1 and 3 space dimensions are studied. This is done by generalizing the quantization of chiral fermions in an external Yang-Mills potential to the case of a Z_2-graded potential coupled to bosons and fermions. 
  We study the applicability of Pade Approximants (PA) to estimate a "sum" of asymptotic series of the type appearing in QCD. We indicate that one should not expect PA to converge for positive values of the coupling constant and propose to use PA for the Borel transform of the series. If the latter has poles on the positive semiaxis, the Borel integral does not exist, but we point out that the Cauchy pricipal value integral can exist and that it represents one of the possible "sums" of the original series, the one that is real on the positive semiaxis. We mention how this method works for Bjorken sum rule, and study in detail its application to series appearing for the running coupling constant for the Richardson static QCD potential. We also indicate that the same method should work if the Borel transform has branchpoints on the positive semiaxis and support this claim by a simple numerical experiment. 
  All Bianchi bialgebras have been obtained. By introducing a non-degenerate adjoint invariant inner product over these bialgebras the associated Drinfeld doubles have been constructed, then by calculating the coupling matrices for these bialgebras several sigma-models with Poisson-Lie symmetry have been obtained. Two simple examples as prototypes of Poisson-Lie dual models have been given. 
  We review and elaborate on properties of the string tension in two-dimensional gauge theories. The first model we consider is massive QED in the $m\ll e$ limit. We evaluate the leading string tension both in the fermionic and bosonized descriptions. We discuss the next to leading corrections in $m/e$. The next-to-leading terms in the long distance behavior of the quark-antiquark potential, are evaluated in a certain region of external versus dynamical charges. The finite temperature behavior is also determined. In $QCD_2$ we review the results for the string tension of quarks in cases with dynamical quarks in the fundamental, adjoint, symmetric and antisymmetric representations. The screening nature of $SYM_2$ is re-derived. 
  It is shown that the T-duality in \sigma-model with Kaluza-Klein metric, without or with a torsion term, can be interpreted as electric-magnetic duality for some of their solitonic solutions. Actually Buscher's duality transformation interchanges the topological and Noether charges. 
  It is shown that global fermionic charges induced in vacuum by slowly varying, topologically non-trivial background scalar fields are not renormalized provided that expansion in momenta of background fields is valid. This suggests that strongly coupled theories obey induced charge matching conditions which are analogous, but generally not equivalent, to 't Hooft anomaly matching conditions. We give a few examples of induced charge matching. In particular, the corresponding constraints in softly broken supersymmetric QCD suggest non-trivial low energy mass pattern, in full accord with the results of direct analyses. 
  It is shown that the worldvolume field theory of a single D3-brane in a supergravity D3-brane background admits finite energy, and non-singular, abelian monopoles and dyons preserving 1/2 or 1/4 of the ${\cal N}=4$ supersymmetry and saturating a Bogomol'nyi-type bound. The 1/4 supersymmetric solitons provide a worldvolume realisation of string-junction dyons. We also discuss the dual M-theory realisation of the 1/2 supersymmetric dyons as finite tension self-dual strings on the M5-brane, and of the 1/4 supersymmetric dyons as their intersections. 
  By working with the free energy for the type II supergravity near-horizon solution of N coincident non-extremal Dp-branes we study the transitions among the non-conformal Dp-brane system, the perturbative super Yang-Mills theory and a certain system associated with M theory. We derive a relation between this free energy and the action of a Dp-brane probe in the N Dp-brane background. Constructing the free energy for the five dimensional black hole labeled by the D1-brane and D5-brane charges we find the similar relation between it and the action of a D1 or D5 brane probe in the D1 + D5 brane background. These relations are explained by the massive open strings stretched between the relevant D-branes 
  We propose a simple method to reduce a general p-form electrodynamics with respect to the standard Gauss constraints. The canonical structure of the reduced theory displays a p-dependent sign which makes the essential difference between theories with different parities of p. This feature was observed recently in the corresponding quantization condition for p-brane dyons. It suggests that these two structures are closely related. 
  We find topological defect solutions to the equations of motion of a generalised Higgs model with antisymmetric tensor fields. These solutions are direct higher dimensional analogues of the Nielsen-Olesen vortex solution for a gauge field in four dimensions. 
  Anomalous U(1) gauge symmetries in type II orientifold theories show some unexpected properties. In contrast to the heterotic case, the masses of the gauge bosons are in general of order of the string scale even in the absence of large Fayet-Iliopoulos terms. Despite this fact, the notion of heterotic-type II orientifold duality remains a useful concept, although this symmetry does not seem to hold in all cases considered. We analyse the status of this duality symmetry, clarify the properties of anomalous U(1) gauge symmetry in the orientifold picture and comment on the consequences for phenomenological applications of such anomalous gauge symmetries. 
  We determine from 11D supergravity the quadratic bulk action for the physical bosonic fields relevant for the computation of correlation functions of normalized chiral operators in D=6, N=(0,2) and D=3, N=8 supersymmetric CFT in the large N limit, as dictated by the AdS/CFT duality conjecture. 
  We discuss a possible relation between singletons in $AdS$ space and logarithmic conformal field theories at the boundary of $AdS$. It is shown that the bulk Lagrangian for singleton field (singleton dipole) induces on the boundary the two-point correlation function for logarithmic pair. Bulk interpretation of mixing between logarithmic operator $D$ and zero mode operator $C$ under the scale transformation is discussed as well as some other issues. 
  We derive a $U$-duality invariant formula for the degeneracies of BPS multiplets in a D1-D5 system for toroidal compactification of the type II string. The elliptic genus for this system vanishes, but it is found that BPS states can nevertheless be counted using a certain topological partition function involving two insertions of the fermion number operator. This is possible due to four extra toroidal U(1) symmetries arising from a Wigner contraction of a large $\CN=4$ algebra $\CA_{\kappa,\kappa'}$ for $\kappa' \to \infty$. We also compare the answer with a counting formula derived from supergravity on $AdS_3\times S^3 \times T^4$ and find agreement within the expected range of validity. 
  We determine, to the first order in the radius of Anti-de-Sitter, the realisation of the $OSp(6,2|2)$ superconformal algebra on vector fields. We then calculate, to this order, the superspace metric describing the background of $AdS_7\times S^4$. The coordinates we work with are adapted to a 6+5 splitting of the eleven dimensional superspace. Finally, we deduce in a manifestly supersymmetric form the equations governing the dynamics of the fivebrane near the boundary of $AdS_7$. 
  I discuss D0-brane quantum mechanics as a nonperturbative formulation of string theory, in particular the relation between the Banks-Fischler-Shenker-Susskind matrix model, the Maldacena conjecture for D0-branes, and type IIA/M-theory duality. Some features of the quantum mechanics of D0-branes are also discussed. Lecture presented at the Nishinomiya-Yukawa Memorial Symposium, Nov. 1998. 
  A class of background independent membrane field theories are studied, and several properties are discovered which suggest that they may play a role in a background independent form of M theory. The bulk kinematics of these theories are described in terms of the conformal blocks of an algebra G on all oriented, finite genus, two-surfaces. The bulk dynamics is described in terms of causal histories in which time evolution is specified by giving amplitudes to certain local changes of the states. Holographic observables are defined which live in finite dimensional states spaces associated with boundaries in spacetime. We show here that the natural observables in these boundary state spaces are, when G is chosen to be Spin(D) or a supersymmetric extension of it, generalizations of matrix model coordinates in D dimensions. In certain cases the bulk dynamics can be chosen so the matrix model dynamics is recoverd for the boundary observables. The bosonic and supersymmetric cases in D=3 and D=9 are studied, and it is shown that the latter is, in a certain limit, related to the matrix model formulation of M theory. This correspondence gives rise to a conjecture concerning a background independent form of M theory in terms of which excitations of the background independent membrane field theory that correspond to strings and D0 branes are identified. 
  We study possible relations between the full Green's functions of softly broken supersymmetric theories and the full Green's functions of rigid supersymmetric theories on the example of the supersymmetric quantum mechanics and find that algebraic relations can exist and can be written in a simple form. These algebraic relations between the Green's functions have been derived by transforming the path integral of the rigid theory. In this approach soft terms appear as the result of general changes of coordinates in the superspace. 
  We review the developments of a recently proposed approach to study integrable theories in any dimension. The basic idea consists in generalizing the zero curvature representation for two-dimensional integrable models to space-times of dimension $d+1$ by the introduction of a $d$-form connection. The method has been used to study several theories of physical interest, like self-dual Yang-Mills theories, Bogomolny equations, non-linear sigma models and Skyrme-type models. The local version of the generalized zero curvature involves a Lie algebra and a representation of it, leading to a number of conservation laws equal to the dimension of that representation. We discuss the conditions a given theory has to satisfy in order for its associated zero curvature to admit an infinite dimensional (reducible) representation. We also present the theory in the more abstract setting of the space of loops, which gives a deeper understanding and a more simple formulation of integrability in any dimension. 
  We constructed canonical non-highest weight unitary irreducible representation of $\hat{so}(1,n)$ current algebra as well as canonical non-highest weight non-unitary representations, We constructed certain Laplacian operators as elements of the universal enveloping algebra, acting in representation space. We speculated about a possible relation of those Laplacians with the loop operator for the Yang-Mills. 
  We construct dual Lagrangians for $G/H$ models in two space-time dimensions for arbitrary Lie groups $G$ and $H\subset G$. Our approach does not require choosing coordinates on $G/H$, and allows for a natural generalization to Lie-Poisson duality. For the case where the target metric on $G/H$ is induced from the invariant metric on $G$, the dual system is a gauged Higgs model, with a nonconstant metric and a coupling to an antisymmetric tensor. The dynamics for the gauge connection is governed by a $BF$-term. Lie-Poisson duality is relevant once we allow for a more general class of target metrics, as well as for couplings to an antisymmetric tensor, in the primary theory. Then the dual theory is written on a group $\tilde G$ dual to $G$, and the gauge group $H$ (which, in general, is not a subgroup of $\tilde G$) acts nonlinearly on $\tilde G$. The dual system therefore gives a nonlinear realization of a gauge theory. All dual descriptions are shown to be canonically equivalent to the corresponding primary descriptions, at least at the level of the current algebra. 
  The infinite dimensional generalization of the quantum mechanics of extended objects, namely, the quantum field theory of extended objects is presented. The paradigm example studied in this paper is the Euclidean scalar field with a $\lambda (\phi^{6})/6!$ interaction in four spacetime dimensions. The theory is found to be finite when the virtual particle intermediate states are characterized by fuzzy particles instead of ordinary pointlike particles. Causality, Lorentz invariance, and unitarity (verified up to fourth order in the coupling constant) are preserved in the theory. In addition, the Kallen-Lehmann spectral representation for the propagator is discussed. 
  We derive the partition function for the Vafa-Witten twist of the $\cn=4$ supersymmetric gauge theory with gauge group SU(N) (for prime $N$) and arbitrary values of the 't Hooft fluxes $v\in H^{2}(X,\IZ_{N})$ on K\"ahler four-manifolds with $b^{+}_2>1$. 
  We study the derivative expansion for the effective action in the framework of the Exact Renormalization Group for a single component scalar theory. By truncating the expansion to the first two terms, the potential $U_k$ and the kinetic coefficient $Z_k$, our analysis suggests that a set of coupled differential equations for these two functions can be established under certain smoothness conditions for the background field and that sharp and smooth cut-off give the same result. In addition we find that, differently from the case of the potential, a further expansion is needed to obtain the differential equation for $Z_k$, according to the relative weight between the kinetic and the potential terms. As a result, two different approximations to the $Z_k$ equation are obtained. Finally a numerical analysis of the coupled equations for $U_k$ and $Z_k$ is performed at the non-gaussian fixed point in $D<4$ dimensions to determine the anomalous dimension of the field. 
  We have developed N=1 supersymmetric nonlinear realization methods, which realize global symmetry breaking in N=1 supersymmetric theories. The target space of nonlinear sigma models with a linear model origin is a G^C-orbit, which is a non-compact non-homogeneous K"ahler manifold. We show that, if and only if the orbit is open, it includes a compact homogeneous K"ahler manifold as a submanifold, and a class of strictly G-invariant K"ahler potentials reduces to a K"ahler potential G-invariant up to a K"ahler transformation on the submanifold. Hence, in the case of an open orbit, the most general low-energy effective K"ahler potential can be written as the sum of those of the compact submanifolds and an arbitrary function of strictly G-invariants. 
  Chiral superfields have been used, and extensively, almost ever since supersymmetry has been discovered. Complex linear superfields afford an alternate representation of matter, but are widely misbelieved to be 'physically equivalent' to chiral ones. We prove the opposite is true. Curiously, this re-enables a previously thwarted interpretation of the low-energy (super)field limit of superstrings. 
  In this paper we further elaborate on the notion of fractional exclusion statistics, as introduced by Haldane, in two-dimensional conformal field theory, and its connection to the Universal Chiral Partition Function as defined by McCoy and collaborators. We will argue that in general, besides the pseudo-particles introduced recently by Guruswamy and Schoutens, one needs additional `null quasi-particles' to account for the null-states in the quasi-particle Fock space. We illustrate this in several examples of WZW-models. 
  Description of two three-dimensional topological quantum field theories of Witten type as twisted supersymmetric theories is presented. Low-energy effective action and a corresponding topological invariant of three-dimensional manifolds are considered. 
  Twisted sectors --solutions to the equations of motion with non-trivial monodromies-- of three dimensional Euclidean gravity are studied. We argue that upon quantization this new sector of the theory provides the necessary (and no more) degrees of freedom to account for the Bekenstein-Hawking entropy. 
  The quantum field theory of extended objects is employed to address the hitherto nonrenormalizable Pauli interaction. This is achieved by quantizing the Dirac field using the infinite dimensional generalization of the extended object formulation. The order $\alpha$ contribution to the anomalous magnetic moment of the electron (and of the muon) arising from the Pauli term is calculated. 
  A summary of the successes of and obstacles to the gauge technique (a non-perturbative method of solving Dyson-Schwinger equations in gauge theories) is given, as well as an outline of how progress may be achieved in this field. 
  We study toric singularities of the form of $\C^4/\Ga$ for finite abelian groups $\Ga \subset SU(4)$. In particular, we consider the simplest case $\Ga=\Z_2 \times \Z_2 \times \Z_2$ and find explicitly charge matrices for partial resolutions of this orbifold by extending the method by Morrison and Plesser. We obtain three kinds of algebraic equations, $z_1 z_2 z_3 z_4=z_5^2, z_1 z_2 z_3=z_4^2 z_5 $ and $z_1 z_2 z_5 = z_3 z_4$ where $z_i$'s parametrize $\C^5$. When we put $N$ D1 branes at this singularity, it is known that the field theory on the worldvolume of $N$ D1 branes is T-dual to $2 \times 2 \times 2 $ brane cub model. We analyze geometric interpretation for field theory parameters and moduli space. 
  This note reviews the progress on the low energy dynamics of N=2 supersymmetric Yang-Mills theories after the works of Seiberg and Witten. Specifically, the theory of prepotential for non-specialists is reviewed. 
  Recently Jarvis has proved a correspondence between SU(N) monopoles and rational maps of the Riemann sphere into flag manifolds. Furthermore, he has outlined a construction to obtain the monopole fields from the rational map. In this paper we examine this construction in some detail and provide explicit examples for spherically symmetric SU(N) monopoles with various symmetry breakings. In particular we show how to obtain these monopoles from harmonic maps into complex projective spaces. The approach extends in a natural way to monopoles in hyperbolic space and we use it to construct new spherically symmetric SU(N) hyperbolic monopoles. 
  See hep-th/9907179. 
  Domain wall - type solution with oscillating thickness in a real, scalar field model is investigated with the help of a polynomial approximation. We propose a simple extension of the polynomial approximation method. In this approach we calculate higher order corrections to the planar domain wall solution, find that the domain wall with oscillating thickness radiates, and compute dumping of oscillations of the domain wall. 
  We consider Type 0B D3-branes placed at conical singularities and analyze in detail the conifold singularity. We study the non supersymmetric gauge theories on their worldvolume and their conjectured dual gravity descriptions. In the ultraviolet the solutions exhibit a logarithmic running of the gauge coupling. In the infrared we find confining solutions and IR fixed points. 
  Using standard field theoretical techniques, we survey pure Yang-Mills theory on the noncommutative torus, including Feynman rules and BRS symmetry. Although in general free of any infrared singularity, the theory is ultraviolet divergent. Because of an invariant regularization scheme, this theory turns out to be renormalizable and the detailed computation of the one loop counterterms is given, leading to an asymptoticaly free theory. Besides, it turns out that non planar diagrams are overall convergent when $\theta$ is irrational. 
  We apply the DMRG method to the 2 dimensional delta function potential which is a simple quantum mechanical model with asymptotic freedom and formation of bound states. The system block and the environment block of the DMRG contain the low energy and high energy degrees of freedom, respectively. The ground state energy and the lowest excited states are obtained with very high accuracy. We compare the DMRG method with the Similarity RG method and propose its generalization to field theoretical models in high energy physics. 
  When the theory of Quantum Chromodynamics was introduced, it was to explain the observed phenomena of quark confinement and scaling. It was then discovered that the emergence of instantons is an essential consequence of this theory. This led to unanticipated explanations not only for the anomalously high masses of the $\eta$ and the $\eta'$ particles, but also for the remarkable differences that had been observed in the mixing angles for the pseudoscalar mesons and the vector mesons. 
  Anti-de Sitter (AdS) space can be foliated by a family of nested surfaces homeomorphic to the boundary of the space. We propose a holographic correspondence between theories living on each surface in the foliation and quantum gravity in the enclosed volume. The flow of observables between our ``interior'' theories is described by a renormalization group equation. The dependence of these flows on the foliation of space encodes bulk geometry. 
  We compute the boundary entropies for the allowed boundary conditions of the SU(2)-invariant principal chiral model at level k=1. We used the reflection factors determined in a previous work. As a by-product we obtain some miscellaneous results such as the ground-state energy for mixed boundary conditions as well as the degeneracies of the Kondo model in the underscreened and exactly screened cases. All these computations are in perfect agreement with known results. 
  We show how spin networks can be described and evaluated as Feynman integrals over an internal space. This description can, in particular, be applied to the so-called simple SO(D) spin networks that are of importance for higher-dimensional generalizations of loop quantum gravity. As an illustration of the power of the new formalism, we use it to obtain the asymptotics of an amplitude for the D-simplex and show that its oscillatory part is given by the Regge action. 
  Simple derivation of the Hamilton-Jacobi equation for bosonic strings and p-branes is given. The motion of classical strings and p-branes is described by two and p+1 local fields, respectively. A variety of local field equations which reduce to the Hamilton-Jacobi equation in the classical limit are given. They are essentially nonlinear, having no linear term. 
  We consider the non-interacting massive Wess-Zumino model in four-dimensional anti-de Sitter space and show that the conformal dimensions of the corresponding boundary fields satisfy the relations expected from superconformal invariance. In some cases the irregular mode must be used for one of the scalar fields. 
  We review some of the problems associated with deriving field theoretic results from nonsupersymmetric AdS, focusing on how to control the behavior of the field theory along the flat directions. We discuss an example in which the origin of the moduli space remains a stable vacuum at finite $N$, and argue that it corresponds to an interacting CFT in three dimensions. Associated to this fixed point is a statement of nonsupersymmetric duality. Because 1/N corrections may change the global picture of the RG flow, the statement of duality is much weaker than in the supersymmetric case. 
  The graviton exchange diagram for the correlation function of arbitrary scalar operators is evaluated in anti-de Sitter space, AdS(d+1). This enables us to complete the computation of the 4-point amplitudes of dilaton and axion fields in IIB supergravity on AdS5 X S5. By the AdS/CFT correspondence, we obtain the 4-point functions of the marginal operators Tr (F^2+...) and Tr(F F~ +...) in N=4, d=4 SU(N) SYM at large N, large g^2 N. The short distance asymptotics of the amplitudes are studied. We find that in the direct channel the leading power singularity agrees with the expected contribution of the stress-energy tensor in a double OPE expansion. Logarithmic singularities occur in the complete 4-point functions at subleading orders. 
  A self-consistent random phase approximation (RPA) is proposed as an effective Hamiltonian method in Light-Front Field Theory (LFFT). We apply the general idea to the light-front massive Schwinger model to obtain a new bound state equation and solve it numerically. 
  The general method of the supersymmetrization of the soliton equations with the odd (bi) hamiltonian structure is established. New version of the supersymmetric N=2,4 (Modified) Korteweg de Vries equation is given, as an example. The second odd Hamiltonian operator of the SUSY KdV equation generates the odd N=2,4 SUSY Virasoro - like algebra. 
  We consider \caln=1 supersymmetric SO(N) gauge theory with a symmetric traceless tensor. This theory saturates 't Hooft matching conditions at the origin of the moduli space. This naively suggests a confining phase, but Brodie, Cho, and Intriligator have conjectured that the origin of the moduli space is in a Non-Abelian Coulomb phase. We construct a dual description by the deconfinement method, and also show that the theory indeed has an infrared fixed point for certain values of N. This result supports their argument. 
  In a recent paper, Ekstrand proposed a simple expression from which covariant anomaly, covariant Schwinger term and higher covariant chain terms may be computed. We comment on the relation of his result to the earlier work of Tsutsui. 
  We construct the most general solution in type-II string theory that represents N coincident non-extremal rotating NS5-branes and determine the relevant thermodynamic quantities. We show that in the field theory limit, it has an exact description. In particular, it can be obtained by an O(3,3) duality transformation on the exact string background for the coset model SL(2,R)_{-N}/U(1) \times SU(2)_N. In the extreme supersymmetric limit we recover the multicenter solution, with a ring singularity structure, that has been discussed recently. 
  A noncommutative associative algebra of N=2 fuzzy supersphere is introduced. It turns out to possess a nontrivial automorphism which relates twisted chiral to twisted anti-chiral superfields and hence makes possible to construct noncommutative nonlinear $\si$-models with extended supersymmetry. 
  We discuss various approaches to extracting the full stress-energy tensor of the conformal field theory from the corresponding supergravity solutions, within the framework of the Maldacena conjecture. This provides a more refined probe of the AdS/CFT correspondence. We apply these techniques in considering the Casimir energy of the conformal field theory on a torus. It seems that either generically the corresponding supergravity solutions are singular (i.e., involve regions of large string-scale curvatures), or that they are largely insensitive to the boundary conditions of the CFT on the torus. 
  The dynamical content of local AdS supergravity in five dimensions is discussed. The bosonic sector of the theory contains the vielbein ($e^{a}$), the spin connection ($\omega ^{ab}$) and internal SU(N) and U(1) gauge fields. The fermionic fields are complex Dirac spinors ($\psi ^{i}$) in a vector representation of SU(N). All fields together form a connection 1-form in the superalgebra SU(2,2|N). For N=4, the symplectic matrix has maximal rank in a locally AdS background in which the dynamical degrees of freedom can be identified. The resulting efective theory have different numbers of bosonic and fermionic degrees of freedom. 
  In this note we explain how world-volume geometries of D-branes can be reconstructed within the microscopic framework where D-branes are described through boundary conformal field theory. We extract the (non-commutative) world-volume algebras from the operator product expansions of open string vertex operators. For branes in a flat background with constant non-vanishing B-field, the operator products are computed perturbatively to all orders in the field strength. The resulting series coincides with Kontsevich's presentation of the Moyal product. After extending these considerations to fermionic fields we conclude with some remarks on the generalization of our approach to curved backgrounds. 
  We complete our study of non-Abelian gauge theories in the framework of Epstein-Glaser approach to renormalization theory including in the model an arbitrary number of Dirac Fermions. We consider the consistency of the model up to the third order of the perturbation theory. In the second order we obtain pure group theoretical relations expressing a representation property of the numerical coefficients appearing in the left and right handed components of the interaction Lagrangian. In the third order of the perturbation theory we obtain the the condition of cancellation of the axial anomaly. 
  The BPS states of N=2 super Yang-Mills theory with gauge group SU(2) are constructed as non-trivial finite-energy solutions of the worldvolume theory of a threebrane probe in F theory. The solutions preserve 1/2 of N=2 supersymmetry and provide a worldvolume realization of strings stretching from the probe to a sevenbrane. The positions of the sevenbranes correspond to singularities in the field theory moduli space and to curvature singularities in the supergravity background. We explicitly show how the UV cut off of the effective field theory is mapped into an IR cut off in the supergravity. Finally, we discuss some features of the moduli spaces of these solutions. 
  A recently proposed model of confining strings has a non-local world-sheet action induced by a space-time Kalb-Ramond tensor field. Here we show that, in the large-D approximation, an infinite set of ghost- and tachyon-free truncations of the derivative expansion of this action all lead to c=1 models. Their infrared limit describes smooth strings with world-sheets of Hausdorff dimension D_H=2 and long-range orientational order, as expected for QCD strings. 
  The AdS/CFT correspondence is established for the AdS_3 space compactified on a solid torus with the CFT field on the boundary. Correlation functions that correspond to the bulk theory at finite temperature are obtained in the regularization a'la Gubser, Klebanov, and Polyakov. The BTZ black hole solutions in AdS_3 are T-dual to the solution in the AdS_3 space without singularity. 
  We derive the complete $(curvature)^2$ terms of effective D-brane actions, for arbitrary ambient geometries and world-volume embeddings, at lowest order (disk-level) in the string-loop expansion. These terms reproduce the $o(\alpha'^2)$ corrections to string scattering amplitudes, and are consistent with duality conjectures. In the particular case of the D3-brane with trivial normal bundle, considerations of $SL(2,\IZ)$ invariance lead to a complete sum of D-instanton corrections for both the parity-conserving and the parity-violating parts of the effective action. These corrections are required for the cancellation of the modular anomalies of massless modes, and are consistent with the absence of chiral anomalies in the intersection domain of pairs of D-branes. We also show that the parity-conserving part of the non-perturbative R^2 action follows from a one-loop quantum calculation in the six-dimensional world-volume of the M5-brane compactified on a two-torus. 
  The computation of mode sums of the types encountered in basic quantum field theoretic applications is addressed with an emphasis on their expansions into functions of distance that can be interpreted as potentials. We show how to regularize and calculate the Casimir energy for the continuum Nambu-Goto string with massive ends as well as for the discrete Isgur-Paton non-relativistic string with massive ends. As an additional example, we examine the effect on the interquark potential of a constant Kalb-Ramond field strength interacting with a QCD string. 
  Recently, a number of authors have challenged the conventional assumption that the string scale, Planck mass, and unification scale are roughly comparable. It has been suggested that the string scale could be as low as a TeV. The greatest obstacle to developing a string phenomenology is our lack of understanding of the ground state. We explain why the dynamics which determines this state is not likely to be accessible to any systematic approximation. We note that the racetrack scheme, often cited as a counterexample, suffers from similar difficulties. We stress that the weakness of the gauge couplings, the gauge hierarchy, and coupling unification suggest that it may be possible to extract some information in a systematic approximation. We review the ideas of Kahler stabilization, an attempt to reconcile these facts. We consider whether the system is likely to sit at extremes of the moduli space, as in recent proposals for a low string scale. Finally we discuss the idea of Maximally Enhanced Symmetry, a hypothesis which is technically natural, compatible with basic facts about cosmology, and potentially predictive. 
  The search for intersecting brane solutions in supergravity is a large and profitable industry. Recently, attention has focused on finding localized forms of known `delocalized' solutions. However, in some cases, a localized version of the delocalized solution simply does not exist. Instead, localized separated branes necessarily delocalize as the separation is removed. This phenomenon is related to black hole no-hair theorems, i.e. `baldness.' We continue the discussion of this effect and describe how it can be understood, in the case of Dirichlet branes, in terms of the corresponding intersection field theory. When it occurs, it is associated with the quantum mixing of phases and lack of superselection sectors in low dimensional field theories. We find surprisingly wide agreement between the field theory and supergravity both with respect to which examples delocalize and with respect to the rate at which this occurs. 
  We construct the non-linear Kaluza-Klein ans\"atze describing the embeddings of the U(1)^3, U(1)^4 and U(1)^2 truncations of D=5, D=4 and D=7 gauged supergravities into the type IIB string and M-theory. These enable one to oxidise any associated lower dimensional solutions to D=10 or D=11. In particular, we use these general ans\"atze to embed the charged AdS_5, AdS_4 and AdS_7 black hole solutions in ten and eleven dimensions. The charges for the black holes with toroidal horizons may be interpreted as the angular momenta of D3-branes, M2-branes and M5-branes spinning in the tranverse dimensions, in their near-horizon decoupling limits. The horizons of the black holes coincide with the worldvolumes of the branes. The Kaluza-Klein ans\"atze also allow the black holes with spherical or hyperbolic horizons to be reinterpreted in D=10 or D=11. 
  We study mass deformations of N=2 superconformal field theories with ADE global symmetries on a D3-brane. The N=2 Seiberg-Witten curves with ADE symmetries are determined by the Type IIB 7-brane backgrounds which are probed by a D3-brane. The Seiberg-Witten differentials $\lambda$ for these ADE theories are constructed. We show that the poles of $\lambda$ with residues are located on the global sections of the bundle in an elliptic fibration. It is then clearly seen how the residues transform in an irreducible representation of the ADE groups. The explicit form of $\lambda$ depends on the choice of a representation of the residues. Nevertheless the physics results are identical irrespective of the representation of $\lambda$. This is considered as the global symmetry version of the universality found in N=2 Yang-Mills theory with local ADE gauge symmetries. 
  We consider random superstrings of type IIB in $d$-dimensional space. The discretized action is constructed from the supersymetric matrix model, which has been proposed as a constructive definition of superstring theory. Our action is invariant under the local N=2 super transformations, and doesn't have any redundant fermionic degrees of freedom. 
  We pursue the study of the type IIB matrix model as a constructive definition of superstring. In this paper, we justify the interpretation of space-time as distribution of eigenvalues of the matrices by showing that some low energy excitations indeed propagate in it. In particular, we show that if the distribution consists of small clusters of size $n$, low energy theory acquires local SU(n) gauge symmetry and a plaquette action for the associated gauge boson is induced, in addition to a gauge invariant kinetic term for a massless fermion in the adjoint representation of the SU(n). We finally argue a possible identification of the diffeomorphism symmetry with permutation group acting on the set of eigenvalues, and show that the general covariance is realized in the low energy effective theory even though we do not have a manifest general covariance in the IIB matrix model action. 
  In a recent paper (hep-th/9811108), Saveliev and the author showed that there exits an on-shell light cone gauge where the non-linear part of the field equations reduces to a (super) version of Yang's equations which may be solved by methods inspired by the ones previously developed for self-dual Yang-Mills equations in four dimensions. Here, the analogy between these latter theories and the present ones is pushed further by writing down a set of super partial linear differential equations whose consistency conditions may be derived from the SUSY Y-M equations in ten dimensions, and which are the analogues of the Lax pair of Belavin and Zakharov. On the simplest example of the two pole ansatz, it is shown that the same solution-generating techniques are at work, as for the derivation of the celebrated multi-instanton solutions carried out in the late seventies. The present Lax representation, however, is only a consequence of (instead of being equivalent to) the field equations, in contrast with the Belavin Zakharov Lax pair. 
  We clarify a number of issues regarding the worldsheet and spacetime descriptions of string propagation on AdS_3. We construct the vertex operators of spacetime current algebra and spacetime (super) Virasoro generators in the full interacting SL(2) WZW theory and study their Ward identities. We also explain the relation between the analysis in this note and some recent work on this subject. 
  The polysymplectic phase space of covariant Hamiltonian field theory can be provided with the current algebra bracket. 
  We make use of product integrals to provide an unambiguous mathematical representation of Wilson line and Wilson loop operators. Then, drawing upon various properties of product integrals, we discuss such properties of these operators as approximating them with partial sums, their convergence, and their behavior under gauge transformations. We also obtain a surface product integral representation for the Wilson loop operator. The result can be interpreted as the non-abelian version of Stokes theorem. 
  We study the modular transformation properties of Euclidean solutions of 3D gravity whose asymptotic geometry has the topology of a torus. These solutions represent saddle points of the grand canonical partition function with an important example being the BTZ black hole, and their properties under modular transformations are inherited from the boundary conformal field theory encoding the asymptotic dynamics. Within the Chern Simons formulation, classical solutions are characterised by specific holonomies describing the wrapping of the gauge field around cycles of the torus. We find that provided these holonomies transform in an appropriate manner, there exists an associated modular invariant grand canonical partition function and that the spectrum of saddle points naturally includes a thermal bath in $AdS_3$ as discussed by Maldacena and Strominger. Indeed, certain modular transformations can naturally be described within classical bulk dynamics as mapping between different foliations with a "time" coordinate along different cycles of the asymptotic torus. 
  By comparision with numerical results in the maximal Abelian projection of lattice Yang-Mills theory, it is argued that the nonperturbative dynamics of Yang Mills theory can be described by a set of fields that take their values in the coset space SU(2)/U(1). The Yang-Mills connection is parameterized in a special way to separate the dependence on the coset field. The coset field is then regarded as a collective variable, and a method to obtain its effective action is developed. It is argued that the physical excitations of the effective action may be knot solitons. A procedure to calculate the mass scale of knot solitons is discussed for lattice gauge theories in the maximal Abelian projection. The approach is extended to the SU(N) Yang-Mills theory. A relation between the large N limit and the monopole dominance is pointed out. 
  We study the conformal field theory of the D1/D5 system compactified on X (X is T^4 or K3). It is described by a sigma model whose target space is the moduli space of instantons on X. For values of the parameters where the branes can separate, the spectrum of dimensions in the conformal field theory exhibits a continuum above a gap. This continuum leads to a pathology of the conformal field theory, which explains a variety of problems in various systems. In particular, we explain the apparent discrepancy between different methods of finding the spectrum of chiral fields at certain points in the moduli space of the system. 
  Motivated by the ``universe as a brane'' idea, we investigate the motion of a $(D-2)$-brane (or domain wall) that couples to bulk matter. Usually one would expect the spacetime outside such a wall to be time dependent however we show that in certain cases it can be static, with consistency of the Israel equations yielding relationships between the bulk metric and matter that can be used as ans\"atze to solve the Einstein equations. As a concrete model we study a domain wall coupled to a bulk dilaton with Liouville potentials for the dilaton both in the bulk and on the wall. The bulk solutions we find are all singular but some have black hole or cosmological horizons, beyond which our solutions describe domain walls moving in time dependent bulks. A significant period of world volume inflation occurs if the potential on the wall is not too steep; in some cases the bulk also inflates (with the wall comoving) while in others the wall moves relative to a non-inflating bulk. We apply our method to obtain cosmological solutions of Ho\v{r}ava-Witten theory compactified on a Calabi-Yau space. tive to a non-inflating bulk. We apply our method to obtain cosmological solutions of Ho\v{r}ava-Witten theory compactified on a Calabi-Yau space. 
  We propose the existence of a non-supersymmetric conformal field theory softly broken at the TeV scale as a new mechanism for solving the hierarchy problem. We find the imposition of conformal invariance to be very restrictive with many predictive consequences, including severe restrictions on the field content, the number of families as well as on the structure of inter-family Yukawa couplings. A large class of potentially conformal non-supersymmetric theories are considered and some general predictions are made about the existence of a rich spectrum of color and weak multiplets in the TeV range. 
  We describe a scheme for constructing the holographic dual of the full D3-brane geometry with charge $K$ by embedding it into a large anti-de Sitter space of size $N$. Such a geometry is realized in a multi-center anti-de Sitter geometry which admits a simple field theory interpretation as $SU(N+K)$ gauge theory broken to $SU(N) \times SU(K)$. We find that the characteristic size of the D3-brane geometry is of order $(K/N)^{1/4} U^0$ where $U^0$ is the scale of the Higgs. By choosing $N$ to be much larger than $K$, the scale of the D3-brane metric can be well separated from the Higgs scale in the radial coordinate. We generalize the holographic energy-distance relation and estimate the characteristic energy scale associated with these radial scales, and find that the $E/U$ relation becomes effectively $U$ independent in the range $(K/N)^{1/2} U^0 < U < U^0$. This implies that all detailed structure of the D3-brane geometry is encoded in the fine structure of the boundary gauge theory at around the Higgs scale. 
  The holographic connection between large $N$ Super Yang Mills theory and gravity in anti deSitter space requires unfamiliar behavior of the SYM theory in the limit that the curvature of the AdS geometry becomes small. The paradoxical behavior includes superluminal oscillations and negative energy density. These effects typically occur in the SYM description of events which take place far from the boundary of AdS when the signal from the event arrives at the boundary. The paradoxes can be resolved by assuming a very rich collection of hidden degrees of freedom of the SYM theory which store information but give rise to no local energy density. These degrees of freedom, called precursors, are needed to make possible sudden apparently acausal energy momentum flows. Such behavior would be impossible in classical field theory as a consequence of the positivity of the energy density. However we show that these effects are not only allowed in quantum field theory but that we can model them in free quantum field theory. 
  I show that gravitational entropy can be ascribed to spacetimes containing Misner strings (the gravitational analogues of Dirac strings), even in the absence of any other event horizon (or bolt) structures. This result follows from an extension of proposals for evaluating the stress-energy of a gravitational system which are motivated by the AdS/CFT correspondence. 
  Four-dimensional N-extended superconformal symmetry and correlation functions of quasi-primary superfields are studied within the superspace formalism. A superconformal Killing equation is derived and its solutions are classified in terms of supertranslations, dilations, Lorentz transformations, R-symmetry transformations and special superconformal transformations. In general, due to the invariance under supertranslations and special superconformal transformations, superconformally invariant n-point functions reduce to one unspecified (n-2)-point function which must transform homogeneously under the remaining rigid transformations, i.e. dilations, Lorentz transformations and R-symmetry transformations. Based on this result, we are able to identify all the superconformal invariants and obtain the general form of n-point functions for scalar superfields. In particular, as a byproduct, a selection rule for correlation functions is derived, the existence of which in N=4 super Yang-Mills theory was previously predicted in the context of AdS/CFT correspondence. Superconformally covariant differential operators are also discussed. 
  The Kaluza-Klein spectrum of D=5 simple supergravity compactified on S^3 is studied. A classical background solution which preserves maximal supersymmetry is fulfilled by the geometry of AdS_2\times S^3. The physical spectrum of the fluctuations is classified according to SU(1,1|2)\times SU(2) symmetry, which has a very similar structure to that in the case of compactification on AdS_3\times S^2. 
  We review interactions between certain 7-planes which are composed out of mutually non-local (p,q) 7-branes and which correspond to specific Kodaira singularities. We discuss in particular how to compute certain moduli-dependent terms in the effective action. These do not only probe the local Chern-Simons terms on the world-volumina, but also certain global aspects of plane interactions that can be attributed to ``torsion'' (or \ZZ_N-valued) D-brane charges. 
  The large N Matrix model is studied with attention to the quantum fluctuations around a given diagonal background. Feynman rules are explicitly derived and their relation to those in usual Yang-Mills theory is discussed. Background D-instanton configuration is naturally identified as a discretization of momentum space of a corresponding QFT. The structure of large N divergence is also studied on the analogy of UV divergences in QFT. 
  We propose a type IIB super-Poincare algebra with SO(2,1) covariant central extension. Together with SO(2,1) and SO(9,1) generators, a SO(2,1) triplet (momenta), a Majorana-spinor doublet (supercharges) and a Rarita-Schwinger central charge generate a group, G. We consider a coset G/H where H=(SO(2) x Lorentz), and the SL(2,R) 2-form doublet is obtained by the coset construction. It is shown that U(1) connections, whose strengths are associated with 2-forms, are recognized as coordinates of the enlarged space. We suggest that this is the fundamental algebra governing the superstring theories which explains the IIB SL(2,R) duality and geometrical origin of U(1) fields. 
  We point out that the statements in [hep-th/9903063] concerning the regularity of static axially symmetric solutions in Yang-Mills-dilaton (YMD) [1] and Einstein-Yang-Mills(-dilaton) (EYMD) theory [2,3] are incorrect, and that the non-singular local gauge potential of the YMD solutions [4] is twice differentiable. 
  We provide a complete picture of the self-gravitating non-relativistic gas at thermal equilibrium using Monte Carlo simulations (MC), analytic mean field methods (MF) and low density expansions. The system is shown to possess an infinite volume limit, both in the canonical (CE) and in the microcanonical ensemble (MCE) when N, V \to \infty, keeping N/ V^{1/3} fixed. We {\bf compute} the equation of state (we do not assume it as is customary), the entropy, the free energy, the chemical potential, the specific heats, the compressibilities, the speed of sound and analyze their properties, signs and singularities. The MF equation of state obeys a {\bf first order} non-linear differential equation of Abel type. The MF gives an accurate picture in agreement with the MC simulations both in the CE and MCE. The inhomogeneous particle distribution in the ground state suggest a fractal distribution with Haussdorf dimension D with D slowly decreasing with increasing density, 1 \lesssim D < 3. 
  The information paradox in the quantum evolution of black holes is studied within the framework of the AdS/CFT correspondence. The unitarity of the CFT strongly suggests that all information about an initial state that forms a black hole is returned in the Hawking radiation. The CFT dynamics implies an information retention time of order the black hole lifetime. This fact determines many qualitative properties of the non-local effects that must show up in a semi-classical effective theory in the bulk. We argue that no violations of causality are apparent to local observers, but the semi-classical theory in the bulk duplicates degrees of freedom inside and outside the event horizon. Non-local quantum effects are required to eliminate this redundancy. This leads to a breakdown of the usual classical-quantum correspondence principle in Lorentzian black hole spacetimes. 
  We examine the recently proposed technique of adding boundary counterterms to the gravitational action for spacetimes which are locally asymptotic to anti-de Sitter. In particular, we explicitly identify higher order counterterms, which allow us to consider spacetimes of dimensions d<=7. As the counterterms eliminate the need of ``background subtraction'' in calculating the action, we apply this technique to study examples where the appropriate background was ambiguous or unknown: topological black holes, Taub-NUT-AdS and Taub-Bolt-AdS. We also identify certain cases where the covariant counterterms fail to render the action finite, and we comment on the dual field theory interpretation of this result. In some examples, the case of vanishing cosmological constant may be recovered in a limit, which allows us to check results and resolve ambiguities in certain asymptotically flat spacetime computations in the literature. 
  The propagator is calculated on a noncommutative version of the flat plane and the Lobachevsky plane with and without an extra (euclidean) time parameter. In agreement with the general idea of noncommutative geometry it is found that the limit when the two `points' coincide is finite and diverges only when the geometry becomes commutative. The flat 4-dimensional case is also considered. This is at the moment less interesting since there has been no curved case developed with which it can be compared. 
  We describe the Brown-Henneaux asymptotic symmetry of the general black holes in the Chern-Simons gauge theory of the gauge group $SL(2;{\bf R})_L\times SL(2;{\bf R})_R$. We make it clear that the vector-like subgroup $SL(2; {\bf R})_{L+R}$ plays an essential role in describing the asymptotic symmetry consistently. We find a quite general black hole solution in the $AdS_3$ gravity theory. The solution is specified by an infinite number of conserved quantities which constitute a family of mapping from $S^1$ to the gauge group. The BTZ black hole is one of the simplest case. 
  We consider the inclusion of brane charges in AdS_5 superalgebras that contain the maximal central extension of the super-Poincar\'e algebra on the boundary of AdS_5. For theories with N supersymmetries on the boundary, the maximal extension is OSp(1/8N,R), which contains the group $Sp(8N,R)\supset U(2N,2N) \supset SU(2,2)\times U(N)$ as extension of the conformal group. An ``intermediate'' extension to U(2N,2N/1) is also discussed, as well as the inclusion of brane charges in AdS_7 and AdS_4 superalgebras. BPS conditions in the presence of brane charges are studied in some details. 
  We discuss issues related to orientifolds and the brane realization for gauge theories with orthogonal and symplectic groups. We specifically discuss the case of theories with (hidden) global SO(2n) symmetry, from three to six dimensions. We analyze mirror symmetry for three dimensional N=4 gauge theories, Brane Box Models and six-dimensional gauge theories. We also discuss the issue of T-duality for D_n space-time singularities. Stuck D branes on ON^0 planes play an interesting role. 
  We study the decoupling effects in one-loop corrected N=1 supersymmetric theory with gauge neutral chiral superfields, by calculating the one-loop corrected effective Lagrangian that involves light and heavy fields with the mass scale M, and subsequently eliminating heavy fields by their equations of motion. In addition to new non-renormalizable couplings, we determine the terms that grow as log(M) and renormalize the fields and couplings in the effective field theory, in accordance with the decoupling theorem. However, in a theory derived from superstring theory, these terms can significantly modify low energy predictions for the effective couplings of light fields. For example, in a class of heterotic superstring vacua with an anomalous U(1) the vacuum restabilization introduces such decoupling effects which in turn correct the low energy predictions for certain couplings by 10-50%. 
  These notes are the written version of two lectures delivered at the VIII Mexican School on Particles and Fields on November 1998. The level of the notes is basic assuming only some knowledge on Statistical Mechanics, General Relativity and Yang-Mills theory. After a brief introduction to the classical and semiclassical aspects of black holes, we review some relevant results on 2+1 quantum gravity. These include the Chern-Simons formulation and its affine Kac-Moody algebra, the asymptotic algebra of Brown and Henneaux, and the statistical mechanics description of 2+1 black holes. Hopefully, this contribution will be complementary with the review paper hep-th/9901148 by the same author, and perhaps, a shortcut to some recent developments in three dimensional gravity. 
  We describe new theories of composite vector and tensor (p-form) gauge fields made out of zero-dimensional constituent scalar fields (``primitives''). The local gauge symmetry is replaced by an infinite-dimensional global Noether symmetry -- the group of volume-preserving (symplectic) diffeomorphisms of the target space of the scalar primitives. We find additional non-Maxwell and non-Kalb-Ramond solutions describing topologically massive tensor gauge field configurations in odd space-time dimensions. Generalization to the supersymmetric case is also sketched. 
  We show that, in the limit of zero string coupling, $g_s \to 0$, the thermodynamic partition function of matrix string theory is identical to that of the finite temperature, discrete light-cone quantised (DLCQ) type IIA superstring. We discuss how the superstring is recovered in the decompactified $R^+\to\infty$ limit. 
  The path-integral of the fermionic oscillator with a time-dependent frequency is analyzed. We give the exact relation between the boundary condition to define the domain in which the path-integral is performed and the transition amplitude that the path-integral calculates. According to this relation, the amplitude suppressed by a zero mode does not indicate any special dynamics, unlike the analogous situation in field theories. It simply says the path-integral picks up a combination of the amplitudes that vanishes. The zero mode that is often neglected in the reason of not being normalizable is necessary to obtain the correct answer for the propagator and to avoid an anomaly on the fermion number. We give a method to obtain the fermionic determinant by the determinant of a simple (2\times 2) matrix, which enables us to calculate it for a variety of boundary conditions. 
  We analyse the canonical structure of AdS_3 gravity in terms of the coadjoint orbits of the Virasoro group. There is one subset of orbits, associated to BTZ black hole solutions, that can be described by a pair of chiral free fields with a background charge. There is also a second subset of orbits, associated to point-particle solutions, that are described by two pairs of chiral free fields obeying a constraint. All these orbits admit K\"ahler quantization and generate a Hilbert space which, despite of having $\Delta_0(\bar{\Delta}_0)=0$, does not provide the right degeneracy to account for the Bekenstein-Hawking entropy due to the breakdown of modular invariance. Therefore, additional degrees of freedom, reestablishing modular invariance, are necessarily required to properly account for the black hole entropy. 
  We employ the recently discovered Hopf algebra structure underlying perturbative Quantum Field Theory to derive iterated integral representations for Feynman diagrams. We give two applications: to massless Yukawa theory and quantum electrodynamics in four dimensions. 
  This paper is concerned with the formulation of a non-pertubative theory of the bosonic string. We introduce a formal group $G$ which we propose as the ``universal moduli space'' for such a formulation. This is motivated because $G$ establishes a natural link between representations of the Virasoro algebra and the moduli space of curves. Among other properties of $G$ it is shown that a ``local'' version of the Mumford formula holds on $G$. 
  Gauge field configurations appropriate for the infrared region of QCD are proposed in a submanifold of $su(3)$. Some properties of the submanifold are presented. Using the usual action for QCD, in the absense of quarks, confinement of these configurations is realized as in the London theory of Meissner effect. Choosing a representation for the monopole field strength, a string action corresponding to the effective gauge theory action in the infrared region, is obtained. This confining string action contains the Nambu-Goto term, extrinsic curvature action and the Euler characteristic of the string world sheet. 
  This talk gives an introduction into the subject of Seiberg-Witten curves and their relation to integrable systems. We discuss some motivations and origins of this relation and consider explicit construction of various families of Seiberg-Witten curves in terms of corresponding integrable models. 
  A singular configuration of external static magnetic field in the form of a pointlike vortex polarizes the vacuum of quantized massless spinor field in 2+1-dimensional space-time. This results in an analogue of the Bohm-Aharonov effect: the chiral symmetry breaking condensate, energy density and current emerge in the vacuum even in the case when the spatial region of nonvanishing external field strength is excluded. The dependence of the vacuum characteristics both on the value of the vortex flux and on the choice of the boundary condition at the location of the vortex is determined. 
  In this lecture, a limited introduction of gauge invariance in phase-space is provided, predicated on canonical transformations in quantum phase-space. Exact characteristic trajectories are also specified for the time-propagating Wigner phase-space distribution function: they are especially simple - indeed, classical - for the quantized simple harmonic oscillator. This serves as the underpinning of the field theoretic Wigner functional formulation introduced. Scalar field theory is thus reformulated in terms of distributions in field phase-space. This is a pedagogical selection from work published in J Phys A32 (1999) 771 and Phys Rev D58 (1998) 025002, reported at the Yukawa Institute Workshop "Gauge Theory and Integrable Models", 26-29 January, 1999. 
  One route towards understanding both fractional charges and chiral anomalies delves into Dirac's negative energy sea. Usually we think of Dirac's negative energy sea as an unphysical construct, invented to render quantum field theory physically acceptable by hiding the negative energy solutions. I suggest that in fact physical consequences can be drawn from Dirac's construction. 
  We outline the construction of the Atiyah-Hitchin metric on the moduli space of SU(2) BPS monopoles with charge 2, first as an algebraic curve in C^3 following Donaldson and then as a solution of the Toda field equations in the continual large N limit. We adopt twistor methods to solve the underlying uniformization problem, which by the generalized Legendre transformation yield the Kahler coordinates and the Kahler potential of the metric. We also comment on the connection between twistors and the Seiberg-Witten construction of quantum moduli spaces, as they arise in three dimensional supersymmetric gauge theories, and briefly address the uniformization of algebraic curves in C^3 in the context of large N Toda theory. (Based on talks delivered in September 1998 at the 32nd International Symposium Ahrenshoop on the Theory of Elementary Particles, Buckow; the 21st Triangular Meeting on Quantum Field Theory, Crete and the TMR meeting on Quantum Aspects of Gauge Theories, Supersymmetry and Unification, Corfu; to be published in the proceedings in Fortschritte der Physik.) 
  A nonlinear charged version of the (2+1)-anti de Sitter black hole solution is derived. The source to the Einstein equations is a Born-Infeld electromagnetic field, which in the weak field limit becomes the (2+1)-Maxwell field. The obtained Einstein-Born-Infeld solution for certain range of the parameters (mass, charge, cosmological and Born-Infeld constants) represent a static circularly symmetric black hole. Although the covariant metric components and the electric field do not exhibit a singular behavior at r=0 the curvature invariants are singular at that point. 
  We have investigated the perturbative ambiguity of the radiatively induced Chern-Simons term in differential regularization. The result obtained in this method contains all those obtained in other regularization schemes and the ambiguity is explicitly characterized by an indefinite ratio of two renormalization scales. It is argued that the ambiguity can only be eliminated by either imposing a physical requirement or resorting to a more fundamental principle. Some calculation techniques in coordinate space are developed in the appendices. 
  A quantization of (2+1)-dimensional gravity with negative cosmological constant is presented and quantum aspects of the (2+1)-dimensional black holes are studied thereby. The quantization consists of two procedures. One is related with quantization of the asymptotic Virasoro symmetry. A notion of the Virasoro deformation of 3-geometry is introduced. For a given black hole, the deformation of the exterior of the outer horizon is identified with a product of appropriate coadjoint orbits of the Virasoro groups $\hat{diff S^1}_{\pm}$. Its quantization provides unitary irreducible representations of the Virasoro algebra, in which state of the black hole becomes primary. To make the quantization complete, holonomies, the global degrees of freedom, are taken into account. By an identification of these topological operators with zero modes of the Liouville field, the aforementioned unitary representations reveal, as far as $c \gg 1$, as the Hilbert space of this two-dimensional conformal field theory. This conformal field theory, living on the cylinder at infinity of the black hole and having continuous spectrums, can recognize the outer horizon only as a it one-dimensional object in $SL_2({\bf R})$ and realize it as insertions of the corresponding vertex operator. Therefore it can not be a conformal field theory on the horizon. Two possible descriptions of the horizon conformal field theory are proposed. 
  We discuss instantons in dimensions higher than four. A generalized self-dual or anti-self-dual instanton equation in n-dimensions can be defined in terms of a closed (n-4) form $\Omega$ and it was recently employed as a topological gauge fixing condition in higher dimensional generalizations of cohomological Yang-Mills theory. When $\Omega$ is a calibration which is naturally introduced on the manifold of special holomony, we argue that higher dimensional instanton may be locally characterized as a family of four dimensional instantons over a supersymmetric (n-4) cycle $\Sigma$ with respect to the calibration $\Omega$. This is an instanton configuration on the total space of the normal bundle $N(\Sigma)$ of the submanifold $\Sigma$ and regarded as a natural generalization of point-like instanton in four dimensions that plays a distinguished role in a compactification of instanton moduli space. 
  Paper withdrawn. The argument was based on a misconception 
  Various definitions of chiral observables in a given Moebius covariant two-dimensional theory are shown to be equivalent. Their representation theory in the vacuum Hilbert space of the 2D theory is studied. It shares the general characteristics of modular invariant partition functions, although SL(2,Z) transformation properties are not assumed. First steps towards classification are made. 
  Using the natural extension of the notion of the generalized coherent states the scalar and spinor ones for the de Sitter group SO(4,1) are constructed. These systems of coherent states obey the de Sitter--invariant Klein-Gordon and Dirac equations and correspond to the massive spin~0 and~1/2 particles over de Sitter space. These coherent states are used for the construction of the invariant scalar and spinor propagators over de Sitter space. 
  The slow motion of a self-gravitating CP^1 lump is investigated in the approximation of geodesic flow on the moduli space of unit degree static solutions M_1. It is found that moduli which are frozen in the absence of gravity, parametrizing the lump's width and internal orientation, may vary once gravitational effects are included. If gravitational coupling is sufficiently strong, the presence of the lump shrinks physical space to finite volume, and the moduli determining the boundary value of the CP^1 field thaw also. Explicit formulae for the metric on M_1 are found in both the weak and strong coupling regimes. The geodesic problem for weak coupling is studied in detail, and it is shown that M_1 is geodesically incomplete. This leads to the prediction that self-gravitating lumps are unstable. 
  The possibility that the expansion rate of the Universe, as reflected by the Red Shift, could be produced by the existence of the dilaton field is explored. The analysis starts from previously studied solutions of the Einstein equations for gravity interacting with a massive scalar field. It is firstly underlined that such solutions can produce the observed values of the Hubble constant. Since the Einstein-Klein-Gordon lagrangian could be expected to appear as an effective one for the dilaton in some approximation, the mentioned solutions are applied to study this field. Therefore, the vacuum expectation value for the dilaton is selected to be of the order of the Planck mass, as it is frequently fixed in string phenomenology. Then, it follows that the value of its effective mass should be as low as m=3.9 10^(-29) cm^(-1) in order produce the observed expansion rate. The discussion can also predict a radius of the Universe of the order of 10^(29) cm. Finally, after adopting the view advanced ina previous work, in which these mentioned solutions are associated to interior configurations of collapsed scalar fields, a picture of our Universe as a black hole interior is suggested. 
  In the framework of causal perturbation theory renormalization consists of the extension of distributions. We give the explicit form of a Lorentz invariant extension of a scalar distribution, depending on one difference of space time coordinates. 
  There has been some speculation about relations of D-brane models of black holes to arithmetic. In this note we point out that some of these speculations have implications for a circle of questions related to the generalized Riemann hypothesis on the zeroes of Dirichlet $L$-functions. 
  I describe our understanding of physics near the planck length, in particular the great progress of the last four years in string theory. Superstring theory, and a recent extension called M theory, are leading candidates for a quantum theory that unifies gravity with the other forces. As such, they are certainly not ordinary quantum field theories. However, recent duality conjectures suggest that a more complete definition of these theories can be provided by the large N limits of suitably chosen U(N) gauge theories associated to the asymptotic boundary of spacetime. 
  We investigate maximal gauged supergravity in seven dimensions and some of its solitonic solutions. By focusing on a truncation of the gauged SO(5) R-symmetry group to its U(1)^2 Cartan subgroup, we construct general two charge black holes that are asymptotically anti-de Sitter. We demonstrate that 1- and 2-charge black holes preserve 1/2 and 1/4 of the supersymmetries respectively. Additionally, we examine the odd-dimensional self-duality equation governing the three-form potential transforming as the 5 of SO(5), and provide some insight on the construction of membrane solutions in anti-de Sitter backgrounds. 
  We introduce a new invariant for foliations using the even pairing between K-homology and cyclic homology and discuss some possible applications in physics. 
  We determine the exact quantum particle reflection amplitudes for all known vacua of a_n^(1) affine Toda theories on the half-line with integrable boundary conditions. (Real non-singular vacuum solutions are known for about half of all the classically integrable boundary conditions.) To be able to do this we use the fact that the particles can be identified with the analytically continued breather solutions, and that the real vacuum solutions are obtained by analytically continuing stationary soliton solutions. We thus obtain the particle reflection amplitudes from the corresponding breather reflection amplitudes. These in turn we calculate by bootstrapping from soliton reflection matrices which we obtained as solutions of the boundary Yang-Baxter equation (reflection equation).   We study the pole structure of the particle reflection amplitudes and uncover an unexpectedly rich spectrum of excited boundary states, created by particles binding to the boundary. For a_2^(1) and a_4^(1) Toda theories we calculate the reflection amplitudes for particle reflection off all these excited boundary states. We are able to explain all physical strip poles in these reflection factors either in terms of boundary bound states or a generalisation of the Coleman-Thun mechanism. 
  Straightforward application of the standard Noether method in supergravity theories yields an incorrect superpotential for local supersymmetry transformations, which gives only half of the correct supercharge. We show how to derive the correct superpotential through Lagrangian methods, by applying a criterion proposed recently by one of us. We verify the equivalence with the Hamiltonian formalism. It is also indicated why the first-order and second-order formalisms lead to the same superpotential. We rederive in particular the central extension by the magnetic charge of the ${\cal N}_4 =2$ algebra of SUGRA asymptotic charges. 
  Feynman integrals in the physical light-cone gauge are harder to solve than their covariant counterparts. The difficulty is associated with the presence of unphysical singularities due to the inherent residual gauge freedom in the intermediate boson propagators constrained within this gauge choice. In order to circumvent these non-physical singularities, the headlong approach has always been to call for mathematical devices --- prescriptions --- some successful ones and others not so much so. A more elegant approach is to consider the propagator from its physical point of view, that is, an object obeying basic principles such as causality. Once this fact is realized and carefully taken into account, the crutch of prescriptions can be avoided altogether. An alternative third approach, which for practical computations could dispense with prescriptions as well as prescinding the necessity of careful stepwise watching out of causality would be of great advantage. And this third option is realizable within the context of negative dimensions, or as it has been coined, negative dimensional integration method, NDIM for short. 
  We review modifications of the Bekenstein-Hawking area law for black hole entropy in the presence of higher-derivative interactions. In four-dimensional N=2 compactifications of string theory or M-theory these modifications are crucial for finding agreement between the macroscopic entropy obtained from supergravity and the microscopic entropy obtained by counting states in string or M-theory. Our discussion is based on the effective Wilsonian action, which in the context of N=2 supersymmetric theories is defined in terms of holomorphic quantities. At the end we briefly indicate how to incorporate non-holomorphic corrections. 
  Arguments presented in an earlier paper, demonstrating the breakdown of global supersymmetry in Hawking radiation from a generic four dimensional black hole with infalling massless scalar and spinor particles, are reexamined. Careful handling of the Grassmann-valued spinorial parameter is shown to lead to a situation wherein supersymmetry may not actually break. A comparative analysis in flat spacetime at a finite temperature, is also presented. 
  We consider non-perturbative six and four dimensional N=1 space-time supersymmetric orientifolds. Some states in such compactifications arise in ``twisted'' open string sectors which lack world-sheet description in terms of D-branes. Using Type I-heterotic duality we are able to obtain the massless spectra for some of such orientifolds. The four dimensional compactification we discuss in this context is an example of a chiral N=1 supersymmetric string vacuum which is non-perturbative in both orientifold and heterotic pictures. In particular, it contains both D9- and D5-branes plus non-perturbative ``twisted'' open string sector states as well. 
  We study the asymptotic limits of the heterotic string theories compactified on tori. We find a bilinear form uniquely determined by dualities which becomes Lorentzian in the case of one spacetime dimension. For the case of the SO(32) theory, the limiting descriptions include SO(32) heterotic strings, type I, type IA and other T-duals, M-theory on K3, type IIA theory on K3 and type IIB theory on K3 and possibly new limits not understood yet. 
  Quantum causal histories are defined to be causal sets with Hilbert spaces attached to each event and local unitary evolution operators. The reflexivity, antisymmetry, and transitivity properties of a causal set are preserved in the quantum history as conditions on the evolution operators. A quantum causal history in which transitivity holds can be treated as ``directed'' topological quantum field theory. Two examples of such histories are described. 
  Anomalies of N = (4,4) superconformal field theories coupled to a conformal supergravity background in two dimensions are computed by using the AdS/CFT correspondence. We find that Weyl, axial gauge and super Weyl transformations are anomalous, while general coordinate, local Lorentz, vector gauge and local super transformations are not. The coefficients of the anomalies show that the superconformal field theories have the central charge expected in the AdS/CFT correspondence. 
  We perform the spherical symmetric dimensional reduction $4d\to2d$ of heterotic string theory. We find a class of two-dimensional (2d) dilaton gravity models that gives a general description of the near-horizon, near-extremal behavior of four-dimensional (4d) heterotic string black holes. We show that the duality group of the 4d theory is realized in two dimensions in terms of Weyl transformations of the metric. We use the 2d dilaton gravity theory to compute the statistical entropy of the near-extremal 4d, $a=1/\sqrt3$, black hole. 
  We provide compelling evidence that a previously introduced model of non-perturbative 2d Lorentzian quantum gravity exhibits (two-dimensional) flat-space behaviour when coupled to Ising spins. The evidence comes from both a high-temperature expansion and from Monte Carlo simulations of the combined gravity-matter system. This weak-coupling behaviour lends further support to the conclusion that the Lorentzian model is a genuine alternative to Liouville quantum gravity in two dimensions, with a different, and much `smoother' critical behaviour. 
  In this paper we continue the study of the non-critical type 0 string and its field theory duals. We begin by reviewing some facts and conjectures about these theories. We move on to our proposal for the type 0 effective action in any dimension, its RR fields and their Chern--Simons couplings. We then focus on the case without compact dimensions and study its field theory duals. We show that one can parameterize all dual physical quantities in terms of a finite number of unknown parameters. By making some further assumptions on the tachyon couplings, one can still make some ``model independent'' statements. 
  A connection between the algebra of rooted trees used in renormalization theory and Runge-Kutta methods is pointed out. Butcher's group and B-series are shown to provide a suitable framework for renormalizing a toy model of field the ory, following Kreimer's approach. Finally B-series are used to solve a class of non-linear partial differential equations. 
  The density of mass levels \rho(m) and the critical temperature for strings in de Sitter space-time are found. QFT and string theory in de Sitter space are compared. A `Dual'-transform is introduced which relates classical to quantum string lengths, and more generally, QFT and string domains. Interestingly, the string temperature in De Sitter space turns out to be the Dual transform of the QFT-Hawking-Gibbons temperature. The back reaction problem for strings in de Sitter space is addressed selfconsistently in the framework of the `string analogue' model (or thermodynamical approach), which is well suited to combine QFT and string study.We find de Sitter space-time is a self-consistent solution of the semiclassical Einstein equations in this framework. Two branches for the scalar curvature R(\pm) show up: a classical, low curvature solution (-), and a quantum high curvature solution (+), enterely sustained by the strings. There is a maximal value for the curvature R_{\max} due to the string back reaction. Interestingly, our Dual relation manifests itself in the back reaction solutions: the (-) branch is a classical phase for the geometry with intrinsic temperature given by the QFT-Hawking-Gibbons temperature.The (+) is a stringy phase for the geometry with temperature given by the intrinsic string de Sitter temperature. 2 + 1 dimensions are considered, but conclusions hold generically in D dimensions. 
  We define supersymmetric spin networks, which provide a complete set of gauge invariant states for supergravity and supersymmetric gauge theories. The particular case of Osp(1/2) is studied in detail and applied to the non-perturbative quantization of supergravity. The supersymmetric extension of the area operator is defined and partly diagonalized. The spectrum is discrete as in quantum general relativity, and the two cases could be distinguished by measurements of quantum geometry. 
  We obtain first order equations that determine a supersymmetric kink solution in five-dimensional N=8 gauged supergravity. The kink interpolates between an exterior anti-de Sitter region with maximal supersymmetry and an interior anti-de Sitter region with one quarter of the maximal supersymmetry. One eighth of supersymmetry is preserved by the kink as a whole. We interpret it as describing the renormalization group flow in N=4 super-Yang-Mills theory broken to an N=1 theory by the addition of a mass term for one of the three adjoint chiral superfields. A detailed correspondence is obtained between fields of bulk supergravity in the interior anti-de Sitter region and composite operators of the infrared field theory. We also point out that the truncation used to find the reduced symmetry critical point can be extended to obtain a new N=4 gauged supergravity theory holographically dual to a sector of N=2 gauge theories based on quiver diagrams.   We consider more general kink geometries and construct a c-function that is positive and monotonic if a weak energy condition holds in the bulk gravity theory. For even-dimensional boundaries, the c-function coincides with the trace anomaly coefficients of the holographically related field theory in limits where conformal invariance is recovered. 
  We review the construction and theoretical implications of the USp(2k) matrix model in zero dimension introduced in ref. \cite{IT1,IT2}. It is argued that the model provides a constructive approach to Type I superstrings and is at the same time dynamical theory of spacetime points. Three subjects are discussed : semiclassical pictures and series of degenerate perturbative vacua associated with the worldvolume representation of the model, the formation of extended (D-)objects from the fermionic integrations via the (non-)abelian Berry phase, and the Schwinger-Dyson/loop equations which exhibit the joining-splitting interactions required. Lectures presented at the 13th Nishinomiya-Yukawa Memorial Symposium ``Dynamics of Fields and Strings'' (November 12-13,1998) and at the YITP workshop (November 16-18, 1998). 
  We examine duality transformations of supersymmetric and $\kappa$-symmetric Dp-brane actions in a general type II supergravity background where in particular the dilaton and the axion are supposed to not be zero or a constant but a general superfield. Due to non-constant dilaton and axion, we can explicitly show that the dilaton and the axion as well as the two 2-form gauge potentials transform as doublets under the $SL(2,R)$ transformation from the point of view of the world-volume field theory. 
  Energy bounds are derived for planar and compactified M2-branes in a hyper-K\"ahler background. These bounds are saturated, respectively, by lump and Q-kink solitons, which are shown to preserve a half of the worldvolume supersymmetry. The Q-kinks have a dual IIB interpretation as strings that migrate between fivebranes. 
  We discuss theories in which the standard-model particles are localized on a brane embedded in space-time with large compact extra dimensions, whereas gravity propagates in the bulk. In addition to the ground state corresponding to a straight infinite brane, such theories admit a (one parameter) family of stable configurations corresponding to branes wrapping with certain periodicity around the extra dimension(s) when one moves along a noncompact coordinate (tilted walls). In the effective four-dimensional field-theory picture, such walls are interpreted as one of the (stable) solutions with the constant gradient energy, discussed earlier. In the cosmological context their energy "redshifts" by the Hubble expansion and dissipates slower then the one in matter or radiation. The tilted wall eventually starts to dominate the Universe. The upper bound on the energy density coincides with the present critical energy density. Thus, this mechanism can become significant any time in the future. The solutions we discuss are characterized by a tiny spontaneous breaking of both the Lorentz and rotational invariances. Small calculable Lorentz noninvariant terms in the standard-model Lagrangian are induced. Thus, the tilted walls provide a framework for the spontaneous breaking of the Lorentz invariance. 
  The formulation of gravity in 3+1 dimensions in which the spin connection is the basic field ($\omega $-frame) leads to a theory with first and second class constraints. Here, the Dirac brackets for the second class constraints are evaluated and the Dirac algebra of first class constraints is found to be the usual algebra associated to space-time reparametrizations and tangent space rotations. This establishes the classical equivalence with the vierbein approach (e-frame). The explicit form of the path integral for this theory is given and the quantum equivalence with the e-frame is also established. 
  It has been assumed that it is possible to approximate the interactions of quantized BPS solitons by quantising a dynamical system induced on a moduli space of soliton parameters. General properties of the reduction of quantum systems by a Born-Oppenheimer approximation are described here and applied to sigma models and their moduli spaces in order to learn more about this approximation. New terms arise from the reduction proceedure, some of them geometrical and some of them dynamical in nature. The results are generalised to supersymmetric sigma models, where most of the extra terms vanish. 
  String backgrounds of the form AdS_3 x N that give rise to two dimensional spacetime superconformal symmetry are constructed. 
  We review versions of the Generalized Uncertainty Principle (GUP) obtained in string theory and in gedanken experiments carried out in Quantum Gravity. We show how a GUP can be derived from a measure gedanken experiment involving micro-black holes at the Planck scale of spacetime. The model uses only Heisenberg principle and Schwarzschild radius and it is independent from particular versions of Quantum Gravity. 
  We demonstrate that QCD gluon amplitudes can be used to construct a Lagrangian for gravity. This procedure makes use of perturbative `squaring' relations between gravity and gauge theory that follow from string theory. We explicitly carry out the construction for up to five-point interactions and discuss a set of field variables in the Einstein-Hilbert Lagrangian for interpreting the Lagrangian obtained from QCD. A spin-off from our analysis is that it can be used to provide simpler tree-level gravity Feynman rules than for conventional gauges. 
  We consider a two dimensional SU(N) gauge theory coupled to an adjoint Majorana fermion, which is known to be supersymmetric for a particular value of fermion mass. We investigate the `soft' supersymmetry breaking of the discrete light cone quantization (DLCQ) of this theory. There are several DLCQ formulations of this theory currently in the literature and they naively appear to behave differently under `soft' supersymmetry breaking at finite resolution. We show that all these formulations nevertheless yield identical bound state masses in the decompactification limit of the light-like circle. Moreover, we are able to show that the supersymmetry-inspired version of DLCQ (so called `SDLCQ') provides the best rate of convergence of DLCQ bound state masses towards the actual continuum values, except possibly near or at the critical fermion mass. In this last case, we discuss improved extrapolation schemes that must supplement the DLCQ algorithm in order to obtain correct continuum bound state masses. Interestingly, when we truncate the Fock space to two particles, the SDLCQ prescription presented here provides a scheme for improving the rate of convergence of the massive t'Hooft model. Thus the supersymmetry-inspired SDLCQ prescription is applicable to theories without supersymmetry. 
  An exact formula for the solutions to the WDVV equation in terms of horizontal sections of the corresponding flat connection is found. 
  We apply a boost-invariant similarity renormalization group procedure to a light-front Hamiltonian of a scalar field phi of bare mass mu and interaction term g phi^3 in 6 dimensions using 3rd order perturbative expansion in powers of the coupling constant g. The initial Hamiltonian is regulated using momentum dependent factors that approach 1 when a cutoff parameter Delta tends to infinity. The similarity flow of corresponding effective Hamiltonians is integrated analytically and two counterterms depending on Delta are obtained in the initial Hamiltonian: a change in mu and a change of g. In addition, the interaction vertex requires a Delta-independent counterterm that contains a boost invariant function of momenta of particles participating in the interaction. The resulting effective Hamiltonians contain a running coupling constant that exhibits asymptotic freedom. The evolution of the coupling with changing width of effective Hamiltonians agrees with results obtained using Feynman diagrams and dimensional regularization when one identifies the renormalization scale with the width. The effective light-front Schroedinger equation is equally valid in a whole class of moving frames of reference including the infinite momentum frame. Therefore, the calculation described here provides an interesting pattern one can attempt to follow in the case of Hamiltonians applicable in particle physics. 
  The QED renormalization is restudied by using a mass-dependent subtraction which is performed at a time-like renormalization point. The subtraction exactly respects necessary physical and mathematical requirements such as the gauge symmetry, the Lorentz- invariance and the mathematical convergence. Therefore, the renormalized results derived in the subtraction scheme are faithful and have no ambiguity. Especially, it is proved that the solution of the renormalization group equation satisfied by a renormalized wave function, propagator or vertex can be fixed by applying the renormalization boundary condition and of the form as given in the Feynman rules and, thus, an exact S-matrix element can be expressed in the form as written in the tree diagram approximation provided that the coupling constant and the fermion mass are replaced by their effective ones. In the one-loop approximation, the effective coupling constant and the effective fermion mass obtained by solving their renormalization group equations are given in rigorous and explicit expressions which are suitable in the whole range of distance. and exhibit physically reasonable asymptotic behaviors. 
  The QCD one-loop renormalization is restudied in a mass-dependent subtraction scheme in which the quark mass is not set to vanish and the renormalization point is chosen to be an arbitrary timelike momentum. The correctness of the subtraction is ensured by the Ward identities which are respected in all the processes of subtraction. By considering the mass effect, the effective coupling constant and the effective quark mass are given in improved expressions which are different from the previous results. 
  We exactly calculated the parity-odd term of the effective action induced by the fermions in 2+1 dimensions at finite chemical potential and finite temperature. It shows that gauge invariance is still respected. A more gerneral class of background configurations is considered. The knowledge of the reduced 1+1 determinant is required in order to draw exact conclusions about the gauge invariance of the parity-odd term in this latter case. 
  These lectures intend to give a pedagogical introduction into some of the developments in string theory during the last years. They include perturbative T-duality and non perturbative S- and U-dualities, their unavoidable demand for D-branes, an example of enhanced gauge symmetry at fixed points of the T-duality group, a review of classical solitonic solutions in general relativity, gauge theories and tendimensional supergravity, a discussion of their BPS nature, Polchinski's observations that allow to view D-branes as RR charged states in the non perturbative string spectrum, the application of all this to the computation of the black hole entropy and Hawking radiation and finally a brief survey of how everything fits together in M-theory. 
  We propose the principle that the scale of the glueball masses in the AdS/CFT approach to QCD should be set by the square root of the string tension. It then turns out that the strong bare coupling runs logarithmically with the ultraviolet cutoff T if first order world sheet fluctuations are included. We also point out that in the end, when all corrections are included, one should obtain an equation for the coupling running with T which has some similarity with the equation for the strong bare coupling. 
  We propose a field theory argument, which rests on the non-renormalization of the two point function of the energy-momentum tensor, why the ratio between the entropies of strongly coupled and weakly coupled N=4 is of order one. 
  The running gauge coupling and quark-antiquark potential in d-dimensions are calculated from the explicit solution of d+1-dimensional dilatonic gravity. This background interpolates between usual AdS in UV and flat space with singular dilaton in IR and it realizes two-boundaries AdS/CFT correspondence. The behaviour of running coupling and potential is consistent with results following from IIB supergravity. 
  We show that in the functional integral formalism the (finite) coefficient of the induced, Lorentz- and CPT-violating Chern-Simons term, arising from the Lorentz- and CPT-violating fermion sector, is undetermined. 
  The restricted class of Natanzon potentials with two free parameters is studied within the context of Supersymmetric Quantum Mechanics. The hierarchy of Hamiltonians is indicated, where the first members of the superfamily are explicitly evaluated and a general form for the superpotential is proposed. 
  The prescription of the AdS/CFT correspondence is refined by using a regularization procedure, which makes is possible to calculate the divergent local terms in the CFT two-point function. We present the procedure for the example of the scalar field. 
  We construct string target spacetimes with AdS_3 x X geometry, which have an N=2 spacetime superconformal algebra. X is found to be a U(1) fibration over a manifold which is a target for an N=2 worldsheet conformal field theory. We emphasize theories with free field realizations where in principle it is possible to compute the full one-particle string spectrum. 
  We discuss the processes of brane bubble nucleation induced by the external branes. The quasiclassical solution for the nucleation by the single external brane has been found in the case when the brane junctions are possible. Exponential factor in the production rate has been calculated. The process induced by the fundamental or D string in the background of two D3 branes is analyzed and its interpretation from the D3 worldvolume theory viewpoint is described. 
  We generalize the background gauge in the Matrix model to propose a new gauge which is useful for discussing the conformal symmetry. In this gauge, the special conformal transformation (SCT) as the isometry of the near-horizon geometry of the D-particle solution is directly reproduced with the correct coefficient as the quantum correction to the SCT in the Matrix model. We also present a general argument for the relation between the gauge choice and the field redefinition in the Matrix model. 
  By taking into account the effect of the would be Chern-Simons term, we calculate the quantum correction to the Chern-Simons coefficient in supersymmetric Chern-Simons Higgs theories with matter fields in the fundamental representation of SU(n). Because of supersymmetry, the corrections in the symmetric and Higgs phases are identical. In particular, the correction is vanishing for N=3 supersymmetric Chern-Simons Higgs theories. The result should be quite general, and have important implication for the more interesting case when the Higgs is in the adjoint representation. 
  We discuss the prominence of Hopf algebras in recent progress in Quantum Field Theory. In particular, we will consider the Hopf algebra of renormalization, whose antipode turned out to be the key to a conceptual understanding of the subtraction procedure. We shall then describe several occurences of this or closely related Hopf algebras in other mathematical domains, such as foliations, Runge Kutta methods, iterated integrals and multiple zeta values. We emphasize the unifying role which the Butcher group, discovered in the study of numerical integration of ordinary differential equations, plays in QFT. 
  By making use of the background field method, we derive a novel reformulation of the Yang-Mills theory which was proposed recently by the author to derive quark confinement in QCD. This reformulation identifies the Yang-Mills theory with a deformation of a topological quantum field theory. The relevant background is given by the topologically non-trivial field configuration, especially, the topological soliton which can be identified with the magnetic monopole current in four dimensions. We argue that the gauge fixing term becomes dynamical and that the gluon mass generation takes place by a spontaneous breakdown of the hidden supersymmetry caused by the dimensional reduction. We also propose a numerical simulation to confirm the validity of the scheme we have proposed. Finally we point out that the gauge fixing part may have a geometric meaning from the viewpoint of global topology where the magnetic monopole solution represents the critical point of a Morse function in the space of field configurations. 
  In this work we consider a coupled system of Schwinger-Dyson equations for self-energy and vertex functions in QED_3. Using the concept of a semi-amputated vertex function, we manage to decouple the vertex equation and transform it in the infrared into a non-linear differential equation of Emden-Fowler type. Its solution suggests the following picture: in the absence of infrared cut-offs there is only a trivial infrared fixed-point structure in the theory. However, the presence of masses, for either fermions or photons, changes the situation drastically, leading to a mass-dependent non-trivial infrared fixed point. In this picture a dynamical mass for the fermions is found to be generated consistently. The non-linearity of the equations gives rise to highly non-trivial constraints among the mass and effective (`running') gauge coupling, which impose lower and upper bounds on the latter for dynamical mass generation to occur. Possible implications of this to the theory of high-temperature superconductivity are briefly discussed. 
  The winding number transition in the Mottola-Wipf model with and without Skyrme term is examined. For the model with Skyrme term the number of discrete modes of the fluctuation operator around sphaleron is shown to be dependent on the value of $\lambda m^2$. Following Gorokhov and Blatter we derive a sufficient condition for the sharp first-order transition, which indicates that first-order transition occurs when $0< \lambda m^2 < 0.0399$ and $2.148 < \lambda m^2$. In the intermediate region of $\lambda m^2$ the winding number transition is conjectured to be smooth second order. For the model without Skyrme term the winding number transition is always first order regardless of the value of parameter. 
  The Abelian Higgs model and the Georgi-Glashow model in 2 and 3 Euclidean dimensions respectively, support both finite size instantons and sphalerons. The instantons are the familiar Nielsen-Oleson vortices and the 't Hooft-Polyakov monopole solutions respectively. We have constructed the sphaleron solutions and calculated the Chern-Simons charges N_cs for sphalerons of both models and have constructed two types of noncontractible loops between topologically distinct vacuua. In the 3 dimensional model, the sphaleron and the vacuua have zero magnetic and electric flux while the configurations on the loops have non vanishing magnetic flux. 
  We show that the Reeh-Schlieder property w.r.t. the KMS-vector is a direct consequence of locality, additivity and the relativistic KMS-condition. The latter characterises the thermal equilibrium states of a relativistic quantum field theory. The statement remains vaild even if the given equilibrium state breaks spatial translation invariance. 
  We construct an N=1 superconformal field theory using branes of type IIA string theory. The IIA construction is related via T-duality to a IIB configuration with 3-branes in a background generated by two intersecting O7-planes and 7-branes. The IIB background can be viewed as a local piece of an F-theory compactification previously studied by Sen in connection with the Gimon-Polchinski orientifold. We discuss the deformations of the IIA and IIB constructions and describe a new supersymmetric configuration with curving D6-branes. Starting from the IIB description we find the supergravity dual of the large N field theory and discuss the matching of operators and KK states. The matching of non-chiral primaries exhibits some interesting new features. We also discuss a relevant deformation of the field theory under which it flows to a line of strongly coupled N=1 fixed points in the infrared. For these fixed points we find a partial supergravity description. 
  Generalizing the geometry of the gauge covariant variables in Yang-Mills theory proposed by Johnson and Haagensen, the 4-d geometry associated with a monopole is defined for SU(2). There are three relevant geometries: AdS$_2\times S^2$, $R^2\times S^2$ and $H_+\times S^2$, depending on the asymptotic behavior of the torsion. Using this geometry, the Wilson loop average is computed {\it \`{a} la} Nambu-Goto action. In case of AdS$_2\times S^2$, it satisfies the area law. 
  Differential realizations in coordinate space for deformed Lie algebras with three generators are obtained using bosonic creation and annihilation operators satisfying Heisenberg commutation relations. The unified treatment presented here contains as special cases all previously given coordinate realizations of $so(2,1),so(3)$ and their deformations. Applications to physical problems involving eigenvalue determination in nonrelativistic quantum mechanics are discussed. 
  The velocity basis of the Poincare group is used in the direct product space of two irreducible unitary representations of the Poincare group. The velocity basis with total angular momentum j will be used for the definition of relativistic Gamow vectors. 
  We present a general method for calculating the moduli spaces of fivebranes wrapped on holomorphic curves in elliptically fibered Calabi-Yau threefolds, in particular, in the context of heterotic M theory. The cases of fivebranes wrapped purely on a fiber curve, purely on a curve in the base and, generically, on a curve with components both in the fiber and the base are each discussed in detail. The number of irreducible components of the fivebrane and their properties, such as their intersections and phase transitions in moduli space, follow from the analysis. Even though generic curves have a large number of moduli, we show that there are isolated curves that have no moduli associated with the Calabi-Yau threefold. We present several explicit examples, including cases which correspond to potentially realistic three family models with grand unified gauge group SU(5). 
  A tentative proposal is demonstrated that there is a natural strategy to get rid of unphysical (UV) infinities in QFTs if one adopts the modern standard point of view that a fundamental theory that is complete and well-defined in all respects underlies the QFTs. This simple strategy works in principle for any interaction model and space-time dimension. It provides a physical rationality behind the UV divergence and the conventional renormalization programs and improves the latter in several important aspects. 
  We study the relationship between the gauge boson coupled to spin 2 operator and the singleton in three-dimensional anti-de Sitter space(AdS$_3$). The singleton can be expressed in terms of a pair of dipole ghost fields $A$ and $B$ which couple to $D$ and $C$ operators on the boundary of AdS$_3$. These operators form the logarithmic conformal field theory(LCFT$_2$). Using the correlation function for logarithmic pair, we calculate the greybody factor for the singleton. In the low temperature limit of $\omega \gg T_{\pm}$, this is compared with the result of the bulk AdS$_3$ calculation of the gauge boson. We find that the gauge boson cannot be realized as a model of the AdS$_3$/LCFT$_2$ correspondence. 
  In the conformal-gauge two-dimensional quantum gravity, the solution obtained by the perturbative or path-integral approach is compared with the one obtained by the operator-formalism approach. Treatments of the anomaly problem in both approaches are different. This difference is found to be essentially caused by the fact that the perturbative or path-integral approach is based on the T*-product (covariantized T-product), which generally violates field equations. Indeed, this fact induces some extra one-loop Feynman diagrams, which would not exist unless a nonzero contribution arose from a zero field. Some demerits of the path-integral approach are explicitly demonstrated. 
  In this letter, we show how one can solve easily the Potts-3 + branching interactions and Potts-\infty matrix models, by the means of the equations of motion (loop equations). We give an algebraic equation for the resolvents of these models, and their scaling behaviour. This shows that the equations of motion can be a useful tool for solving such models. 
  Characters and linear combinations of characters that admit a fermionic sum representation as well as a factorized form are considered for some minimal Virasoro models. As a consequence, various Rogers-Ramanujan type identities are obtained. Dilogarithm identities producing corresponding effective central charges and secondary effective central charges are derived. Several ways of constructing more general fermionic representations are discussed. 
  A description of the one-loop approximation formula for the partition function of a three-dimensional abelian version of the Donaldson-Witten theory is proposed. The one-loop expression is shown to contain such topological invariants of a three-dimensional manifold M like the Reidemeister-Ray-Singer torsion and Betti numbers. 
  We calculate in dimensions $D=2+\epsilon$ and in light-cone gauge (LCG) the perturbative ${\cal O}(g^4)$ contribution to a rectangular Wilson loop in the (t,x)-plane coming from diagrams with a self-energy correction in the vector propagator. In the limit $\epsilon \to 0$ the result is finite, in spite of the vanishing of the triple vector vertex in LCG, and provides the expected agreement with the analogous calculation in Feynman gauge. 
  We study the relationship between the equations of first order Lagrangian field theory on fiber bundles and the covariant Hamilton equations on the finite-dimensional polysymplectic phase space of covariant Hamiltonian field theory. The main peculiarity of these Hamilton equations lies in the fact that, for degenerate systems, they contain additional gauge fixing conditions. We develop the BRST extension of the covariant Hamiltonian formalism, characterized by a Lie superalgebra of BRST and anti-BRST symmetries. 
  M-theory has different global supersymmetry structures in its various dual incarnations, as characterized by the M-algebra in 11D, the type IIA, type-IIB, heterotic, type-I extended supersymmetries in 10D, and non-Abelian supersymmetries in the AdS_n x S^m backgrounds. We show that all of these supersymmetries are unified within the supersymmetry OSp(1/64), thus hinting that the overall global spacetime symmetry of M-theory is OSp(1/64). We suggest that the larger symmetries contained within OSp(1/64) which go beyond the familiar symmetries, are non-linearly realized hidden symmetries of M-theory. These can be made manifest by lifting 11D M-theory to the formalism of two-time physics in 13D by adding gauge degrees of freedom. We illustrate this idea by constructing a toy M-model on the worldline in 13D with manifest OSp(1/64) global supersymmetry, and a number of new local symmetries that remove ghosts. Some of the local symmetries are bosonic cousins of kappa supersymmetries. The model contains 0-superbrane and p-forms (for p=3,6) as degrees of freedom. The gauge symmetries can be fixed in various ways to come down to a one time physics model in 11D, 10D, AdS_n x S^m, etc., such that the linearly realized part of OSp(1/64) is the global symmetry of the various dual sectors of M-theory. 
  A review of the applications of noncommutative geometry to a systematic formulation of duality symmetries in string theory is presented. The spectral triples associated with a lattice vertex operator algebra and the corresponding Dirac-Ramond operators are constructed and shown to naturally incorporate target space and discrete worldsheet dualities as isometries of the noncommutative space. The target space duality and diffeomorphism symmetries are shown to act as gauge transformations of the geometry. The connections with the noncommutative torus and Matrix Theory compactifications are also discussed. 
  We study static spherically symmetric dyonic black holes in Einstein-Yang-Mills-Higgs theory. As for the magnetic non-abelian black holes, the domain of existence of the dyonic non-abelian black holes is limited with respect to the horizon radius and the dimensionless coupling constant $\alpha$, which is proportional to the ratio of vector meson mass and Planck mass. At a certain critical value of this coupling constant, $\hat \alpha$, the maximal horizon radius is attained. We derive analytically a relation between $\hat \alpha$ and the charge of the black hole solutions and confirm this relation numerically. Besides the fundamental dyonic non-abelian black holes, we study radially excited dyonic non-abelian black holes and globally regular gravitating dyons. 
  We present a formulation of the coupling of vector multiplets to N=2 supergravity which is symplectic covariant (and thus is not based on a prepotential) and uses superconformal tensor calculus. We do not start from an action, but from the combination of the generalised Bianchi identities of the vector multiplets in superspace, a symplectic definition of special Kahler geometry, and the supersymmetric partners of the corresponding constraints. These involve the breaking to super-Poincare symmetry, and lead to on-shell vector multiplets. This symplectic approach gives the framework to formulate vector multiplet couplings using a weaker defining constraint for special Kahler geometry, which is an extension of older definitions of special Kahler manifolds for some cases with only one vector multiplet. 
  The algebraic consistency of spin and isospin at the level of an unbroken SU(2) gauge theory suggests the existence of an additional angular momentum besides the spin and isospin and also produces a full quaternionic spinor operator. The latter corresponds to a vector boson in space-time, interpreted as a SU(2) gauge field. The existence of quaternionic spinor fields implies in a quaternionic Hilbert space and its necessary mathematical analysis. It is shown how to obtain a unique representation of a quaternion function by a convergent positive power series. 
  We construct the anti-de Sitter supergravity in three dimensions associated with the supergroup $SU(1,1|2)\times SU(1,1|2)$. The field content and the action are inferred using the fact that $AdS$ supergravity theories in three dimensions are Chern-Simons theories. 
  We construct non-tachyonic, non-supersymmetric orientifolds of Type 0B strings in ten, six and four space-time dimensions. Typically, these models have unitary gauge groups with charged massless fermionic and bosonic matter fields. However, generically there remains an uncancelled dilaton tadpole. 
  We consider one-loop quantum corrections to the thermodynamics of a black hole in generic 2-dimensional dilaton gravity. The classical action is the most general diffeomorphism invariant action in 1+1 space-time dimensions that contains a metric, dilaton, and Abelian gauge field, and having at most second derivatives of the fields. Quantum corrections are introduced by considering the effect of matter fields conformally coupled to the metric and non-minimally coupled to the dilaton. Back reaction of the matter fields (via non-vanishing trace conformal anomaly) leads to quantum corrections to the black hole geometry. Quantum corrections also lead to modifications in the gravitational action and hence in expressions for thermodynamic quantities. One-loop corrections to both geometry and thermodynamics (energy, entropy) are calculated for the generic dilaton theory. This formalism is applied to a charged black hole in spherically symmetric gravity and to a rotating BTZ black hole. 
  It is often stated in the literature concerning D=4,6 compact Type IIB orientifolds that tadpole cancellation conditions i) uniquely fix the gauge group (up to Wilson lines and/or moving of branes) and ii) are equivalent to gauge anomaly cancellation. We study the relationship between tadpole and anomaly cancellation conditions and qualify both statements. In general the tadpole cancellation conditions imply gauge anomaly cancellation but are stronger than the latter conditions in D=4, N=1 orientifolds. We also find that tadpole cancellation conditions in Z_N D=4,6 compact orientifolds do not completely fix the gauge group and we provide new solutions different from those previously reported in the literature. 
  Gamow vectors in non-relativistic quantum mechanics are generalized eigenvectors (kets) of self-adjoint Hamiltonians with complex eigenvalues. Like the Dirac kets, they are mathematically well defined in the Rigged Hilbert Space. Gamow kets are derived from the resonance poles of the S-matrix. They have a Breit-Wigner energy distribution, an exponential decay law, and are members of a basis vector expansion whose truncation gives the finite dimensional effective theories with a complex Hamiltonian matrix. They also have an asymmetric time evolution described by a semigroup generated by the Hamiltonian, which expresses a fundamental quantum mechanical arrow of time. These Gamow kets are generalized to relativistic Gamow vectors by extrapolating from the Galilei group to the Poincare group. This leads to semigroup representations of the Poincare group which are characterized by spin j and complex invariant mass square. In these non-unitary representations the Lorentz subgroup is unitarily represented and the four-momenta are "minimally complex" in the sense that the four-velocity is real. The relativistic Gamow vectors have all the properties listed above for the non-relativistic Gamow vectors and are therefore ideally suited to describe relativistic resonances and quasistable particles. 
  We study the duality between string theory on AdS_3 X S^3 X S^3 and two-dimensional conformal theories with large N=4 superconformal algebra A_gamma. We discuss configurations of intersecting branes which give rise to such near-horizon geometries. We compute the Kaluza-Klein spectrum and propose that the boundary superconformal theory can be described by a sigma model on a suitable symmetric product space with a particular choice of anti-symmetric two-form. 
  The quantum stress tensor in the Unruh state for a conformal scalar propagating in a 4D Schwarzschild black hole spacetime is reconstructed in its leading behaviour at infinity and near the horizon by means of an effective action derived by functionally integrating the trace anomaly. 
  It is shown that a large class of higher-order (i.e. non-quadratic) scalar kinetic terms can, without the help of potential terms, drive an inflationary evolution starting from rather generic initial conditions. In many models, this kinetically driven inflation (or "k-inflation" for short) rolls slowly from a high-curvature initial phase, down to a low-curvature phase and can exit inflation to end up being radiation-dominated, in a naturally graceful manner. We hope that this novel inflation mechanism might be useful in suggesting new ways of reconciling the string dilaton with inflation. 
  By solving the Ward identities in a superconformal field theory we find the unique three-point Green's functions composed of chiral superfields for N = 1,2,3,4 supersymmetry. We show that the N=1 four-point function with R-charge equal to one is uniquely determined by the Ward identities up to the specification of four constants. We discuss why chiral Green's functions above three-points, with total R-charge greater than N, are not uniquely determined. 
  Multidimensional model describing the cosmological evolution of n Einstein spaces in the theory with l scalar fields and forms is considered. When electro-magnetic composite p-brane ansatz is adopted, and certain restrictions on the parameters of the model are imposed, the dynamics of the model near the singularity is reduced to a billiard on the (N-1)-dimensional Lobachevsky space, N = n+l. The geometrical criterion for the finiteness of the billiard volume and its compactness is used. This criterion reduces the problem to the problem of illumination of (N-2)-dimensional sphere by point-like sources. Some examples with billiards of finite volume and hence oscillating behaviour near the singularity are considered. Among them examples with square and triangle 2-dimensional billiards (e.g. that of the Bianchi-IX model) and a 4-dimensional billiard in ``truncated'' D = 11 supergravity model (without the Chern-Simons term) are considered. It is shown that the inclusion of the Chern-Simons term destroys the confining of a billiard. 
  The one-instanton contribution to the prepotential for N=2 supersymmetric gauge theories with classical groups exhibits a universality of form. We extrapolate the observed regularity to SU(N) gauge theory with two antisymmetric hypermultiplets and N_f \leq 3 hypermultiplets in the defining representation. Using methods developed for the instanton expansion of non-hyperelliptic curves, we construct an effective quartic Seiberg-Witten curve that generates this one-instanton prepotential. We then interpret this curve in terms of an M-theoretic picture involving NS 5-branes, D4-branes, D6-branes, and orientifold sixplanes, and show that for consistency, an infinite chain of 5-branes and orientifold sixplanes is required, corresponding to a curve of infinite order. 
  A generalization of scalar electrodynamics called fluid electrodynamics is presented. In this theory a fluid replaces the Higgs scalar field. Fluid electrodynamics might have application to the theory of low temperature Helium superfluids, but here it is argued that it provides an alternative method of approaching symmetry breaking in particle physics. The method of constructing fluid electrodynamics is to start with the velocity decomposition of a perfect fluid as in general relativity. A unit vector tangent to the flow lines of an isentropic fluid can be written in terms of scalar potentials: $V_a=h^{-1}(\ph_a+\al\bt_a-\th S)$. A novel interacting charged fluid can be obtained by applying the covariant derivative: $D_a=\p_a+ieA_a$ to these scalar potentials. This fluid is no longer isentropic and there are choices for which it either obeys the second law of thermodynamics or not. A mass term of the correct sign occurs for the $A$ term in the stress, and this mass term depends on the potentials in the above vector. The charged fluid can be reduced to scalar electrodynamics and the standard approach to symmetry breaking applied; alternatively a mass can be induced by the fluid by using just the thermodynamic potentials and then fixing at a critical point, if this is taken to be the Bose condensation point then the induced mass is negligible. 
  The Higgs model is generalized so that in addition to the radial Higgs field there are fields which correspond to the themasy and entropy. The model is further generalized to include state and sign parameters. A reduction to the standard Higgs model is given and how to break symmetry using a non-zero VEV (vacuum expectation value) is shown. A 'fluid rotation' can be performed on the standard Higgs model to give a model dependant on the entropy and themasy and with a constant mass. 
  The effective Hamiltonian, as obtained from applying the Hamiltonian flow equations to front form QED, are solved numerically for positronium. Both the exchange and the annihilation channels are included. The impact of different similarity functions is explicitly studied. Perfect numerical agreement with other methods is found. 
  The goal of this paper is to analyse the method of angular quantization for the Sine-Gordon model at the free fermion point, which is one of the most investigated models of the two-dimensional integrable field theories. The angular quantization method (see hep-th/9707091) is a continuous analog of the Baxter's corner transfer matrix method. Investigating the canonical quantization of the free massive Dirac fermions in one Rindler wedge we identify this quantization with a representation of the infinite-dimensional algebra introduced in the paper q-alg/9702002 and specialized to the free fermion point. We construct further the main ingredients of the SG theory in terms of the representation theory of this algebra following the approach by M.Jimbo, T.Miwa et al. 
  It is shown that the effective theory of D-particles has conformal symmetry with field-dependent parameters. This is a consequence of the supersymmetry. The string coupling constant is not transformed in contrast with the recent proposal of generalized conformal symmtery by Jevicki et al. This conformal symmetry can also be generalized to other Dp-brane systems. 
  We look at and compare two different methods developed earlier for inducing gauge invariances in systems with second class constraints. These two methods, the Batalin-Fradkin method and the Gauge Unfixing method, are applied to a number of systems. We find that the extra field introduced in the Batalin-Fradkin method can actually be found in the original phase space itself. 
  We construct the supercurrent multiplet that contains the energy-momentum tensor of the (2,0) tensor multiplet. By coupling this multiplet of currents to the fields of conformal supergravity, we first construct the linearized superconformal transformations rules of the (2,0) Weyl multiplet. Next, we construct the full non-linear transformation rules by gauging the superconformal algebra OSp(8^*|4). We then use this result to construct the full equations of motion of the tensor multiplet in a conformal supergravity background. Coupling N+5 copies of the tensor multiplet to conformal supergravity and imposing a geometrical constraint on the scalar fields which fixes the conformal symmetry, we obtain the coupling of (2,0) Poincare supergravity to N tensor multiplets in which the physical scalars parametrize the coset SO(N,5)/(SO(N) x SO(5)). 
  We study the generalized conformal quantum mechanics of the probe D0-brane in the near horizon background of the bound state of source D0-branes. We elaborate on the relationship of such model to the M theory in the light cone frame. 
  We compute instanton corrections to the low energy effective prepotential of N=2 supersymmetric theories in a variety of cases, including all classical gauge groups and even number of fundamental matter hypermultiplets. To this end, we take profit of a set of first- and second-order equations for the logarithmic derivatives of the prepotential with respect to the dynamical scale expressed in terms of Riemann's theta-function. These equations emerge in the context of the Whitham hierarchy approach to the low-energy Seiberg--Witten solution of supersymmetric gauge theories. Our procedure is recursive and allows to compute the effective prepotential to arbitrary order in a remarkably straightforward way. General expressions for up to three-instanton corrections are given. We illustrate the method with explicit expressions for several cases. 
  The partition function with boundary conditions for various two-dimensional Ising models is examined and previously unobserved properties of conformal invariance and universality are established numerically. 
  Non-standard overview on the possible formulation towards a unified model on the lattice is presented. It is based on the generalized gauge theory which is formulated by differential forms and thus expected to fit in a simplicial manifold. We first review suggestive known results towards this direction. As a small step of concrete realization of the program, we propose a lattice Chern-Simons gravity theory which leads to the Chern-Simons gravity in the continuum limit via Ponzano-Regge model. We then summarize the quantization procedure of the generalized gauge theory and apply the formulation to the generalized topological Yang-Mills action with instanton gauge fixing. We find N=2 super Yang-Mills theory with Dirac-K{\"a}hler fermions which are generated from ghosts via twisting mechanism. The Weinberg-Salam model is formulated by the generalized Yang-Mills action which includes Connes's non-commutative geometry formulation as a particular case. In the end a possible scenario to realize the program is proposed. The formulations given here are by far incomplete towards the final goal yet include hopeful evidences. This summary of the overview is the extended version of the talk given at Nishinomiya-Yukawa memorial symposium (Nishinomiya-city Japan, Nov. 1998). 
  This paper has been withdrawn by the author 
  The quantum Anti-de Sitter (AdS) group and quantum AdS space is discussed. Ways of getting the quantum AdS group from real forms of quantum orthogonal group are presented. Differential calculus on the quantum AdS space are also introduced. In particular, reality of differential calculus are given. We set up explicit relationships between quantum group and quantum algebra, which can be refereed as the quantum counterpart of the classical exponential. By this way, quantum AdS algebra is deduced from conjugation on the quantum AdS group. 
  We study open descendants of non-supersymmetric type IIB asymmetric (freely acting) orbifolds with zero cosmological constant. A generic feature of these models is that supersymmetry remains unbroken on the brane at all mass levels, while it is broken in the bulk in a way that preserves Fermi-Bose degeneracy in both the massless and massive (closed string) spectrum. This property remains valid in the heterotic dual of the type II model but only for the massless excitations. A possible application of these constructions concerns scenarios of low-energy supersymmetry breaking with large dimensions. 
  We consider scalar field theories in dimensions lower than four in the context of the Wegner-Houghton renormalization group equations (WHRG). The renormalized trajectory makes a non-perturbative interpolation between the ultraviolet and the infrared scaling regimes. Strong indication is found that in two dimensions and below the models with polynomial interaction are always non-perturbative in the infrared scaling regime. Finally we check that these results do not depend on the regularization and we develop a lattice version of the WHRG in two dimensions. 
  A study of the thermodynamics in IIA Matrix String Theory is presented. The free string limit is calculated and seen to exactly reproduce the usual result. When energies are enough to excite non-perturbative objects like D-particles and specially membranes, the situation changes because they add a large number of degrees of freedom that do not appear at low energies. There seems to be a negative specific heat (even in the Microcanonical Ensemble) that moves the asymptotic temperature to zero. Besides, the mechanism of interaction and attachment of open strings to D-particles and D-membranes is analyzed. 
  We investigate further our recent proposal for the form of the matrix theory action in weak background fields. Using Seiberg's scaling argument we relate the matrix theory action to a low-energy system of many D0-branes in an arbitrary but weak NS-NS and R-R background. The resulting multiple D0-brane action agrees with the known Born-Infeld action in the case of a single brane and gives an explicit formulation of many additional terms which appear in the multiple brane action. The linear coupling to an arbitrary background metric satisfies the nontrivial consistency condition suggested by Douglas that the masses of off-diagonal fields are given by the geodesic distance between the corresponding pair of D0-branes. This agreement arises from combinatorial factors which depend upon the symmetrized trace ordering prescription found earlier for higher moments of the matrix theory stress-energy tensor. We study the effect of a weak background metric on two graviton interactions and find that our formalism agrees with the results expected from supergravity. The results presented here can be T-dualized to give explicit formulae for the operators in any D-brane world-volume theory which couple linearly to bulk gravitational fields and their derivatives. 
  We present a direct field theoretical calculation of the consistent gauge anomaly in the superfield formalism, on the basis of a definition of the effective action through the covariant gauge current. The scheme is conceptually and technically simple and the gauge covariance in intermediate steps reduces calculational labors considerably. The resultant superfield anomaly, being proportional to the anomaly $d^{abc}=\tr T^a\{T^b,T^c\}$, is minimal without supplementing any counterterms. Our anomaly coincides with the anomaly obtained by Marinkovi\'c as the solution of the Wess-Zumino consistency condition. 
  We construct the energy momentum tensor for the bosonic fields of the covariant formulation of the M-theory fivebrane within that formalism. We then obtain the energy for various solitonic solutions of the fivebrane equations of motion. 
  In this letter the boundary problem for massless and massive Rarita-Schwinger field in the AdS/CFT correspondence is considered. The considerations are along the lines of a paper by Henneaux (hep-th/9902137) and are based on the requirement the solutions to be a stationary point for the action functional. It is shown that this requirement, along with a definite asymptotic behavior of the solutions, fixes the boundary term that must be added to the initial Rarita-Schwinger action. It is also shown that the boundary term reproduce the known two point correlation functions of certain local operators in CFT living on the boundary. 
  New relations involving curvature components for the various connections appearing in the theory of almost product manifolds are given and the conformal behaviour of these connections are studied. New identities for the irreducible parts of the deformation tensor are derived. Some direct physical applications in Kaluza-Klein and gauge theory are discussed. 
  Conformally invariant boundary conditions for minimal models on a cylinder are classified by pairs of Lie algebras $(A,G)$ of ADE type. For each model, we consider the action of its (discrete) symmetry group on the boundary conditions. We find that the invariant ones correspond to the nodes in the product graph $A \otimes G$ that are fixed by some automorphism. We proceed to determine the charges of the fields in the various Hilbert spaces, but, in a general minimal model, many consistent solutions occur. In the unitary models $(A,A)$, we show that there is a unique solution with the property that the ground state in each sector of boundary conditions is invariant under the symmetry group. In contrast, a solution with this property does not exist in the unitary models of the series $(A,D)$ and $(A,E_6)$. A possible interpretation of this fact is that a certain (large) number of invariant boundary conditions have unphysical (negative) classical boundary Boltzmann weights. We give a tentative characterization of the problematic boundary conditions. 
  Exponential regularization of orthogonal and Anti-de Sitter (AdS) space is presented based on noncommutative geometry. We show that an adequately adopted noncommutative deformation of geometry makes the holography of higher dimensional quantum theory of gravity and low dimensional theory possible. Detail counting for observable degrees of freedom of quantum system of gravity in the bulk of noncommutative space SO_q(3) and the classical limit of its boundary surface S^2 is discussed. Taking noncommutivity effect into account, we get the desired form of entropy for our world, which is consistent with the physical phenomena associated with gravitational collapse. Conformally invariant symmetry is obtained for the equivalent theory of the qunatum gravity living on the classical limit of boundary of the noncommutative AdS space. This is the basis of the AdS/CFT correspondence in string theory. 
  A loop space formulation of Yang-Mills theory high-lighting the significance of monopoles for the existence of gauge potentials is used to derive a generalization of electric-magnetic duality to the nonabelian theory. The result implies that the gauge symmetry is doubled from SU(N) to $SU(N) \times \widetilde{SU}(N)$, while the physical degrees of freedom remain the same, so that the theory can be described in terms of either the usual Yang-Mills potential $A_\mu(x)$ or its dual $\tilde{A}_\mu(x)$. Nonabelian `electric' charges appear as sources of $A_\mu$ but as monopoles of $\tilde{A}_\mu$, while their `magnetic' counterparts appear as monopoles of $A_\mu$ but sources of $\tilde{A}_\mu$. Although these results have been derived only for classical fields, it is shown for the quantum theory that the Dirac phase factors (or Wilson loops) constructed out of $A_\mu$ and $\tilde{A}_\mu$ satisfy the 't Hooft commutation relations, so that his results on confinement apply. Hence one concludes, in particular, that since colour SU(3) is confined then dual colour $\widetilde{SU}(3)$ is broken. Such predictions can lead to many very interesting physical consequences which are explored in a companion paper. 
  The simplest non-trivial solutions of WDVV equations are A_n and B_n-potentials, which describe metrics of K.Saito on spaces of versal deformation of A_n and B_n-singularities. These are some polynomials, which were known for $n\leqslant$ 4. We find some recurrence relations, which give a possibility to find all A_n and B_n-potentials. In passing we give recurrence formulas for coefficients of dispersionless KP hierarchy. 
  We consider perturbative Type II superstring theory in the covariant NSR formalism in the presence of NSNS and RR backgrounds. A concrete example that we have in mind is the geometry of D3-branes which in the near-horizon region is AdS_5 x S_5, although our methods may be applied to other backgrounds as well. We show how conformal invariance of the string path integral is maintained order by order in the number of holes. This procedure makes uses of the Fischler-Susskind mechanism to build up the background geometry. A simple formal expression is given for a \sigma-model Lagrangian. This suggests a perturbative expansion in 1/g^2N and 1/N. As applications, we consider at leading order the mixing of RR and NSNS states, and the realization of the spacetime supersymmetry algebra. 
  We obtain the orbifold Virasoro master equation (OVME) at integer order lambda, which summarizes the general Virasoro construction on orbifold affine algebra. The OVME includes the Virasoro master equation when lambda=1 and contains large classes of stress tensors of twisted sectors of conventional orbifolds at higher lambda. The generic construction is like a twisted sector of an orbifold (with non-zero ground state conformal weight) but new constructions are obtained for which we have so far found no conventional orbifold interpretation. 
  We formulate a superspace field theory which is shown to be equivalent to the $c-\bar{c}$ symmetric BRS/Anti-BRS invariant Yang-Mills action. The theory uses a 6-dimensional superspace and one OSp(3,1|2) vector multiplet of unconstrained superfields. We establish a superspace WT identity and show that the formulation has an asymptotic OSp(3,1|2) invariance as the gauge parameter goes to infinity. We give a physical interpretation of this asymptotic OSp(3,1|2) invariance as a symmetry transformation among the longitudinal/time like degrees of freedom of $A_\mu$ and the ghost degrees of freedom. 
  We use the earlier results on the correlations of axial gauge Green's functions and the Lorentz gauge Green's functions obtained via finite field-dependent BRS transformations to study the question of the correct treatment of 1/(eta\cdot k)^p- type singularities in the axial gauge boson propagator. We show how the known treatment of the 1/(k^2)^n-type singularity in the Lorentz-type gauges can be used to write down the axial propagator via field transformation. We examine the singularity structure of the latter and find that the axial propagator so constructed has $no$ spurious poles, but a complex structure near $\eta\cdot k=0$. 
  The solutions of the arbitrary-spin massless wave equations over ${\bf R}^1 \times H^3$ space are obtained using the generalized coherent states for the Lorentz group. The use of these solutions for the construction of invariant propagators of quantized massless fields with an arbitrary spin over the ${\bf R}^1 \times H^3$ space is considered. The expression for the scalar propagator is obtained in the explicit form. 
  A generalization of the Ferber-Shirafuji formulation of superparticle mechanics is considered. The generalized model describes the dynamics of a superparticle in a superspace extended by tensorial central charge coordinates and commuting twistor-like spinor variables. The D=4 model contains a continuous real parameter $a\geq 0$ and at a=0 reduces to the SU(2,2|1) supertwistor Ferber-Shirafuji model, while at a=1 one gets an OSp(1|8) supertwistor model of ref. [1] (hep-th/9811022) which describes BPS states with all but one unbroken target space supersymmetries. When 0<a<1 the model admits an OSp(2|8) supertwistor description, and when a>1 the supertwistor group becomes OSp(1,1|8). We quantize the model and find that its quantum spectrum consists of massless states of an arbitrary (half)integer helicity. The independent discrete central charge coordinate describes the helicity spectrum. We also outline the generalization of the a=1 model to higher space-time dimensions and demonstrate that in D=3,4,6 and 10, where the quantum states are massless, the extra degrees of freedom (with respect to those of the standard superparticle) parametrize compact manifolds. These compact manifolds can be associated with higher-dimensional helicity states. In particular, in D=10 the additional ``helicity'' manifold is isomorphic to the seven-sphere. 
  A closed universe containing pressureless dust, more generally perfect fluid matter with pressure-to-density ratio w in the range (1/3, - 1/3), violates holographic principle applied according to the Fischler-Susskind proposal. We show, first for a class of two-fluid solutions and then for the general multifluid case, that the closed universe will obey the holographic principle if it also contains matter with w < - 1/3, and if the present value of its total density is sufficiently close to the critical density. It is possible that such matter can be realised by some form of `quintessence', much studied recently. 
  We discuss aspects of recent novel approaches towards understanding the large N limit of matrix field theories with local or global non-abelian symmetry. 
  An (m, n)-string bound state (with m, n relatively prime integers) in type IIB string theory can be interpreted from the D-string worldsheet point of view as n D-strings carrying m units of quantized electric flux or quantized electric field. We argue, from the D-brane worldvolume point of view, that similar Dp-brane bound states should also exist for $2 \le p \le 8$ in both type IIA (when $p$ is even) and type IIB (when $p$ is odd) string theories. As in $p = 1$ case, these bound states can each be interpreted as $n$ Dp-branes carrying m units of quantized constant electric field. In particular, they all preserve one half of the spacetime supersymmetries. 
  We find, from the toric description of the moduli space of D3-branes on non-compact six-dimensional singularities $\C^3/\Z_3$ and $\C^3/\Z_5$ in the blown-down limit, the four-dimensional bases on which these singular spaces are complex cones, and prove the existence of K\"ahler-Einstein metrics on these four-dimensional bases. This shows, in particular, that one can use the horizons obtained from these base spaces by a U(1)-foliation as compact parts of the target space for Type-IIB string theory with $\ads{5}$ in the context of the AdS-CFT correspondence. 
  We use a recently proposed formulation of stable holomorphic vector bundles $V$ on elliptically fibered Calabi--Yau n-fold $Z_n$ in terms of toric geometry to describe stability conditions on $V$. Using the toric map $f: W_{n+1} \to (V,Z_n)$ that identifies dual pairs of F-theory/heterotic duality we show how stability can be related to the existence of holomorphic sections of a certain line bundle that is part of the toric construction. 
  We review a systematic construction of $\cx N=1$ supersymmetric heterotic string vacua using mirror symmetry. The method provides a large class of explicit solutions for stable, holomorphic vector bundles on Calabi-Yau n-folds Z_n in terms of toric geometry. Phenomenologically interesting compactifications as well as non-perturbative dynamics of the heterotic string are discussed within this framework. 
  A longstanding puzzle concerns the calculation of the gluino condensate <{tr\lambda^2\over 16\pi^2}> = c\Lambda^3 in N=1 supersymmetric SU(N) gauge theory: so-called weak-coupling instanton (WCI) calculations give c=1, whereas strong-coupling instanton (SCI) calculations give, instead, c=2[(N-1)!(3N-1)]^{-1/N}. By examining correlators of this condensate in arbitrary multi-instanton sectors, we cast serious doubt on the SCI calculation of <{tr\lambda^2\over 16\pi^2}> by showing that an essential step --- namely cluster decomposition --- is invalid. We also show that the addition of a so-called Kovner-Shifman vacuum (in which <{tr\lambda^2\over 16\pi^2}> = 0) cannot straightforwardly resolve this mismatch. 
  We explore the possibility of constructing p-brane world-volume actions with the requirements of kappa-symmetry and gauge invariance as the only input. In the process, we develop a general framework which leads to actions interpolating between Poincare-dual descriptions of the world-volume theories. The method does not require any restrictions on the on-shell background configurations or on the dimensions of the branes. After some preliminary studies of low-dimensional cases we apply the method to the type IIB five-branes and, in particular, construct a kappa-symmetric action for the type IIB NS5-brane with a world-volume field content reflecting the fact that the D1-, D3- and D5-branes can end on it. 
  We study the correspondence between the moduli space of vacua of three-dimensional supersymmetric Yang-Mills (Maxwell) Chern-Simons theories and brane configurations with (p,q)5-brane. For Coulomb branches, the number of the massless adjoint scalar fields in various supersymmetric theories exactly coincides with the number of the freely moving directions of D3-branes stretched between two 5-branes. When we include a matter superfield into the supersymmetric Chern-Simons theory two distinct symmetric and asymmetric phase appear. The symmetric phase is peculiar to this Chern-Simons Higgs system. We find the corresponding brane configuration for these phases. We also identify the stringy counterpart of the topological vortex state in the asymmetric phase. 
  A family of connections on the space of couplings for a renormalizable field theory is defined. The connections are obtained from a Levi-Civita connection, for a metric which is a generalisation of the Zamolodchikov metric in two dimensions, by adding a family of tensors which are solutions of the renormalization group equation for the operator product expansion co-efficients. The connections are torsion free, but not metric compatible in general. The renormalization group flows of N=2 supersymmetric Yang-Mills theory in four dimensions and the O(N)-model in three dimensions, in the large $N$ limit, are analysed in terms of parallel transport under these connections. 
  The most radical version of the holographic principle asserts that all information about physical processes in the world can be stored on its surface. This formulation is at odds with inflationary cosmology, which implies that physical processes in our part of the universe do not depend on the boundary conditions. Also, there are some indications that the radical version of the holographic theory in the context of cosmology may have problems with unitarity and causality. Another formulation of the holographic principle, due to Fischler and Susskind, implies that the entropy of matter inside the post-inflationary particle horizon must be smaller than the area of the horizon. Their conjecture was very successful for a wide class of open and flat universes, but it did not apply to closed universes. Bak and Rey proposed a different holographic bound on entropy which was valid for closed universes of a certain type. However, as we will show, neither proposal applies to open, flat and closed universes with matter and a small negative cosmological constant. We will argue, in agreement with Easther, Lowe, and Veneziano, that whenever the holographic constraint on the entropy inside the horizon is valid, it follows from the Bekenstein-Hawking bound on the black hole entropy. These constraints do not allow one to rule out closed universes and other universes which may experience gravitational collapse, and do not impose any constraints on inflationary cosmology. 
  Prepotentials in N=2 supersymmetric Yang-Mills theories are known to obey non-linear partial differential equations called Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations. In this paper, the prepotentials at one-instanton level in N=2 supersymmetric SU(4) Yang-Mills theory are studied from the standpoint of WDVV equations. Especially, it is shown that the one-instanton prepotentials are obtained from WDVV equations by assuming the perturbative prepotential and by using the scaling relation as a subsidiary condition but are determined without introducing Seiberg-Witten curve. In this way, various one-instanton prepotentials which satisfy both WDVV equations and scaling relation can be derived, but it turns out that among them there exist one-instanton prepotentials which coincide with the instanton calculus. 
  In this paper we consider IIA and IIB matrix string theories which are defined by two-dimensional and three-dimensional super Yang-Mills theory with the maximal supersymmetry, respectively. We exactly compute the partition function of both of the theories by mapping to a cohomological field theory. Our result for the IIA matrix string theory coincides with the result obtained in the infra-red limit by Kostov and Vanhove, and thus gives a proof of the exact quasi classics conjectured by them. Further, our result for the IIB matrix string theory coincides with the exact result of IKKT model by Moore, Nekrasov and Shatashvili. It may be an evidence of the equivalence between the two distinct IIB matrix models arising from different roots. 
  We present a new general topological tensor current of $\tilde{p}$-branes by making use of the $\phi $-mapping theory. It is shown that the current is identically conserved and behave as $\delta (\vec{\phi}),$ and every isolated zero of the vector field $\vec{\phi}(x)$ corresponds to a `magnetic' $\tilde{p}$-brane. Using this topological current, the generalized Nambu action for multi $\tilde{p}$-branes is given, and the field strength $F$ corresponding to this topological tensor current is obtained. It is also shown that the `magnetic' charges carried by $\tilde{p}$-branes are topologically quantized and labeled by Hopf index and Brouwer degree, the winding number of the $\phi$-mapping. 
  We present a systematic attempt at classification of supersymmetric M-theory vacua with zero flux; that is, eleven-dimensional lorentzian manifolds with vanishing Ricci curvature and admitting covariantly constant spinors. We show that there are two distinct classes of solutions: static spacetimes generalising the Kaluza-Klein monopole, and non-static spacetimes generalising the supersymmetric wave. The classification can be further refined by the holonomy group of the spacetime. The static solutions are organised according to the holonomy group of the spacelike hypersurface, whereas the non-static solutions are similarly organised by the (lorentzian) holonomy group of the spacetime. These are subgroups of the Lorentz group which act reducibly yet indecomposably on Minkowski spacetime. We present novel constructions of non-static vacua consisting of warped products of d-dimensional pp-waves with (11-d)-dimensional manifolds admitting covariantly constant spinors. Our construction yields local metrics with a variety of exotic lorentzian holonomy groups. In the process, we write down the most general local metric in d<6 dimensions describing a pp-wave admitting a covariantly constant spinor. Finally, we also discuss a particular class of supersymmetric vacua with nonzero four-form obtained from the previous ones without modifying the holonomy of the metric. This is possible because in a lorentzian spacetime a metric which admits parallel spinors is not necessarily Ricci-flat, hence supersymmetric backgrounds need not satisfy the equations of motion. 
  Finding solutions to non-linear field theories, such as Yang-Mills theories or general relativity, is usually difficult. The field equations of Yang-Mills theories and general relativity are known to share some mathematical similarities, and this connection can be used to find solutions to one theory using known solutions of the other theory. For example, the Schwarzschild solutions of general relativity can be shown to have a mathematically similar counterpart in Yang-Mills theory. In this article we will discuss several solutions to the Yang-Mills equations which can be found using this connection between general relativity and Yang-Mills theory. Some comments about the possible physical meaning of these solutions will be discussed. In particular it will be argued that some of these analog solutions of Yang-Mills theory may have some connection with the confinement phenomenon. To this end we will briefly look at the motion of test particles moving in the background potential of the Schwarzschild analog solution. 
  A generalization of super-Lie algebras is presented. It is then shown that all known examples of fractional supersymmetry can be understood in this formulation. However, the incorporation of three dimensional fractional supersymmetry in this framework needs some care. The proposed solutions lead naturally to a formulation of a fractional supersymmetry starting from any representation D of any Lie algebra g. This involves taking the Fth-roots of D in an appropriate sense. A fractional supersymmetry in any space-time dimension is then possible. This formalism finally leads to an infinite dimensional extension of g, reducing to the centerless Virasoro algebra when g=sl(2,R). 
  In $D$-dimensional dilaton gravitational model with the central charge deficit the generalized Friedmann-type cosmological solutions (spatially homogeneous and isotropic) are obtained and classified. 
  Eigenvalue repulsion can explain the holographic growth of black holes in Matrix theory. The resulting picture is essentially the same as the Boltzman gas picture but avoids any assumption about the effective potential between the D0 branes. Further, eigenvalue repulsion extends the Boltzman gas picture past the BFKS point to N >> S. The use of Boltzman statistics is natural in this picture. 
  In the previous paper [hep-th/9904112], we argued that there exist BPS bound states of Dp branes carrying certain units of quantized constant electric field for every $p$ (with $1 \le p \le 8$). Each of these bound states preserves one half of the spacetime supersymmetries. In this paper, we construct these bound state configurations explicitly for $2 \le p \le 7$ from Schwarz's $(m,n)$-string or (F, D1) bound state in type IIB string theory by T-dualities along the transverse directions. We calculate the charge per $(2\pi)^{p -1} \alpha'^{(p - 1)/2}$ of $(p -1)$-dimensional area for F-strings in (F, Dp) and the tension for each of these bound states. The results agree precisely with those obtained previously from the worldvolume study. We study the decoupling limit for the (F, D3) bound state and find that Maldacena's $AdS_5/CFT_4$ correspondence may hold true even with respect to thisbound state but now with an effective string coupling rather than the usual string coupling. This coupling is quantized and can be independent of the usual string coupling in certain limit. 
  The second-order differential equation describes harmonic oscillators, as well as currents in LCR circuits. This allows us to study oscillator systems by constructing electronic circuits. Likewise, one set of closed commutation relations can generate group representations applicable to different branches of physics. It is pointed out that polarization optics can be formulated in terms of the six-parameter Lorentz group. This allows us to construct optical instruments corresponding to the subgroups of the Lorentz groups. It is shown possible to produce combinations of optical filters that exhibit transformations corresponding to Wigner rotations and Iwasawa decompositions, which are manifestations of the internal space-time symmetries of massive and massless particles. 
  Compactifications of type IIB string theory on AdS5 x X5, where X5 is an Einstein space, can have one-fourth or half maximal supersymmetry for certain choices of X5. Some of these theories admit exotic domain walls arising from 5-branes wrapping 2-cycles in X5. We explore the relationship among these domain walls, fractional branes and branes stretched on intervals. World-volume fluxes in the wrapped branes play an important role in the analysis. We draw some parallels between the AdS background with exotic domain walls and N=1 supersymmetric MQCD, and identify other extended objects on the AdS side in the dual brane construction. The process of brane creation is used to give an alternate derivation of the relationship between fractional branes and branes on intervals. 
  We study properties of a scalar quantum field theory on the two-dimensional noncommutative plane with $E_q(2)$ quantum symmetry. We start from the consideration of a firstly quantized quantum particle on the noncommutative plane. Then we define quantum fields depending on noncommutative coordinates and construct a field theoretical action using the $E_q(2)$-invariant measure on the noncommutative plane. With the help of the partial wave decomposition we show that this quantum field theory can be considered as a second quantization of the particle theory on the noncommutative plane and that this field theory has (contrary to the common belief) even more severe ultraviolet divergences than its counterpart on the usual commutative plane. Finally, we introduce the symmetry transformations of physical states on noncommutative spaces and discuss them in detail for the case of the $E_q(2)$ quantum group. 
  By quantizing an open string ending on a D-brane in a nontrivial supergravity background, we argue that there is a new kind of uncertainty relation on a D-brane worldvolume. Furthermore, we fix the form of the uncertainty relations and their dependence on the string coupling constant by requiring them to be consistent with various string theory and M theory dualities. In this way we find a web of uncertainties of spacetime for all kinds of brane probes, including fundamental strings, D-branes of all dimensions as well as M theory membranes and fivebranes. 
  Clifford algebras and Majorana conditions are analyzed in any spacetime. An index labeling inequivalent $\Gamma$-structures up to orthogonal conjugations is introduced. Inequivalent charge-operators in even-dimensions, invariant under Wick rotations, are considered. The hermiticity condition on free-spinors lagrangians is presented. The constraints put by the Majorana condition on the free-spinors dynamics are analyzed. Tables specifying which spacetimes admit lagrangians with non-vanishing kinetic, massive or pseudomassive terms (for both charge-operators in even dimensions) are given. The admissible free lagrangians for free Majorana-Weyl spinors are fully classified. 
  We study the absorption probability of minimally-coupled massive scalars by extremal p-branes. In particular, we find that the massive scalar wave equation under the self-dual string background has the same form as the massless scalar wave equation under the dyonic string background. Thus it can be cast into the form of a modified Mathieu equation and solved exactly. Another example that we can solve exactly is that of the D=4 two-charge black hole with equal charges, for which we obtain the closed-form absorption probability. We also obtain the leading-order absorption probabilities for D3-, M2- and M5-branes. 
  We give explicit and inductive formulas for the construction of a Lorentz covariant renormalization in the EG approach. This automatically provides for a covariant BPHZ subtraction at totally spacelike momentum useful for massless theories. 
  We argue that a description of supersymmetric extended objects from a unified geometric point of view requires an enlargement of superspace. To this aim we study in a systematic way how superspace groups and algebras arise from Grassmann spinors when these are assumed to be the only primary entities. In the process, we recover generalized spacetime superalgebras and extensions of supersymmetry found earlier. The enlargement of ordinary superspace with new parameters gives rise to extended superspace groups, on which manifestly supersymmetric actions may be constructed for various types of p-branes, including D-branes (given by Chevalley-Eilenberg cocycles) with their Born-Infeld fields. This results in a field/extended superspace democracy for superbranes: all brane fields appear as pull-backs from a suitable target superspace. Our approach also clarifies some facts concerning the origin of the central charges for the different p-branes. 
  The paper has been temporarily withdrawn. 
  In (2+1) dimensions, we consider the model of a $N$ flavor, two-component fermionic field interacting through a Chern-Simons field besides a four fermion self-interaction which consists of a linear combination of the Gross-Neveu and Thirring like terms. The four fermion interaction is not perturbatively renormalizable and the model is taken as an effective field theory in the region of low momenta. Using Zimmerman procedure for reducing coupling constants, it is verified that, for small values of the Chern-Simons parameter, the origin is an infrared stable fixed point but changes to ultraviolet stable as $\alpha$ becomes bigger than a critical $\alpha_c$. Composite operators are also analyzed and it is shown that a specific four fermion interaction has an improved ultraviolet behavior as $N$ increases. 
  We perform a two-loop calculation in light-front phi^4 theory to determine the effective mass renormalization of the light-front Hamiltonian. The renormalization scheme adopted here is manifestly boost invariant, and yields results that are in perfect agreement with the explicitly covariant Feynman diagram approach. 
  We consider an extension of the Coulomb gas picture which is motivated by Liouville theory and contains negative powers of screening operators on the same footing as positive ones. The braiding problem for chiral vertex operators in this extended framework is analyzed. We propose explicit expressions for the R-matrix with general integer screening numbers, which are given in terms of 4F3 q-hypergeometric functions through natural analytic continuations of the well-known expression for positive integer screenings. These proposals are subsequently verified using a subset of the Moore-Seiberg equations that is obtained by simple manipulations in the operator approach. Interesting new relations for q-hypergeometric functions (particularly of type 4F3) arise on the way. 
  We study $N$ coincident IIA NS5-branes at large $N$ using supergravity. We show that the absorption cross section for gravitons in this background does not vanish at zero string coupling for energies larger than $m_s/\sqrt{N}$ ($m_s$ is the string scale). Using a holographic description of the intrinsic theory of the IIA NS5-branes, we find an expression for the two point function of the stress energy tensor, and comment on its structure. 
  An AdS_2 black hole spacetime is an AdS_2 spacetime together with a preferred choice of time. The Boulware, Hartle-Hawking and SL(2,R) invariant vacua are constructed, together with their Green functions and stress tensors, for both massive and massless scalars in an AdS_2 black hole. The classical Bekenstein-Hawking entropy is found to be independent of the temperature, but at one loop a non-zero entanglement entropy arises. This represents a logarithmic violation of finite-temperature decoupling for AdS_2 black holes which arise in the near-horizon limit of an asymptotically flat black hole. Correlation functions of the SL(2,R) invariant boundary quantum mechanics are computed as functions of the choice of AdS_2 vacuum. 
  We derive the complete set of supersymmetric Ward identities involving only two- and three- point proper vertices in supersymmetric QED. We also present the most general form of the proper vertices consistent with both the supersymmetric and U(1) gauge Ward identities. These vertices are the supersymmetric equivalent of the non supersymmetric Ball-Chiu vertices. 
  This is a set of introductory lecture notes devoted to the Wess-Zumino-Witten model of two-dimensional conformal field theory. We review the construction of the exact solution of the model from the functional integral point of view. The boundary version of the theory is also briefly discussed. 
  The reduction of 4d matter-gravity theory to $S_2$, $H_2$ or $R_2$ leads to effective 2d dilatonic gravity with dilaton coupled matter. Spinors give the exceptional example of the theory which is conformally invariant in 4d as well as in 2d, after reduction. We find 4d and 2d conformal anomaly induced effective action (EA) for Majorana spinor. It is expected for some time that s-wave EA (i.e. the one for dilaton coupled 2d matter) is some (s-wave) approximation to 4d EA. We compare such 2d and 4d spinor EAs on the same gravitational background and argue that s-wave EA indeed qualitatively corresponds to no higher derivatives approximation for 4d EA. 
  We improve the heavy quark potential, extracted from the Wilson loop average in the Ads/CFT approach by taking the quantum fluctuation of the only radial coordinate of $Ads_5$ which is transverse to the world-sheet of the classical Nambu-Goto string in the static gauge, and obtain the universal L{\"u}scher-Symanzik-Weisz/L{\"u}scher term. 
  Using a boundary counterterm prescription motivated by the AdS/CFT conjecture, I evaluate the energy, entropy and angular momentum of the class of Kerr-NUT/bolt-AdS spacetimes. As in the non-rotating case, when the NUT charge is nonzero the entropy is no longer equal to one-quarter of the area due to the presence of the Misner string. When the cosmological constant is also non-zero, the entropy is bounded from above. 
  The AdS/CFT correspondence predicts a phase transition in Wilson loop correlators in the strong coupling N=4, D=4 SYM theory which arises due to instability of the classical string stretched between the loops. We study this transition in detail by solving equations of motion for the string in the particular case of two circular Wilson loops. The transition is argued to be smoothened at finite `t Hooft coupling by fluctuations of the string world sheet and to be promoted to a sharp crossover. Some general comments about Wilson loop correlators in gauge theories are made. 
  We describe a class of supersymmetric gauged linear sigma-model, whose target space is the infinite dimensional space of bundles on a Calabi-Yau 3- or 2-fold. This target space can be considered the configuration space of D-branes wrapped around the Calabi-Yau. We propose that this model can be used to define matrix string theory compactifications. In the infrared limit the model flows to a superconformal non-linear sigma-model whose target space is the moduli space of BPS configurations of branes on the compact space, containing the moduli space of semi-stable bundles. We argue that the bulk degrees of freedom decouple in the infrared limit if semi-stability implies stability. We study topological versions of the model on Calabi-Yau 3-folds. The resulting B-model is argued to be equivalent to the holomorphic Chern-Simons theory proposed by Witten. The A-model and half-twisted model define the quantum cohomology ring and the elliptic genus, respectively, of the moduli space of stable bundles on a Calabi-Yau 3-fold. 
  We investigate the connections between four-dimensional, N=2 M-theory vacua constructed as orbifolds of type II, heterotic, and type I strings. All these models have the same massless spectrum, which contains an equal number of vector multiplets and hypermultiplets, with a gauge group of the maximal rank allowed in a perturbative heterotic string construction. We find evidence for duality between two type I compactifications recently proposed and a new heterotic construction that we present here. This duality allows us to gain insight into the non-perturbative properties of these models. In particular we consider gravitational corrections to the effective action. 
  The classification of the regularization ambiguity of 2D fermionic determinant in three different classes according to the number of second-class constraints, including the new faddeevian regularization, is examined and extended. We found a new and important result that the faddeevian class, with three second-class constraints, possess a free continuous one parameter family of elements. The criterion of unitarity restricts the parameter to the same range found earlier by Jackiw and Rajaraman for the two-constraints class. We studied the restriction imposed by the interference of right-left modes of the chiral Schwinger model ($\chi QED_{2}$) using Stone's soldering formalism. The interference effects between right and left movers, producing the massive vectorial photon, are shown to constrain the regularization parameter to belong to the four-constraints class which is the only non-ambiguous class with a unique regularization parameter. 
  The various descent and duality relations among BPS and non-BPS D-branes are classified using topological K-theory. It is shown how the descent procedures for producing type-II D-branes from brane-antibrane bound states by tachyon condensation and $\klein$ projections arise as natural homomorphisms of K-groups generating the brane charges. The transformations are generalized to type-I theories and type-II orientifolds, from which the complete set of vacuum manifolds and field contents for tachyon condensation is deduced. A new set of internal descent relations is found which describes branes over orientifold planes as topological defects in the worldvolumes of brane-antibrane pairs on top of planes of higher dimension. The periodicity properties of these relations are shown to be a consequence of the fact that all fundamental bound state constructions and hence the complete spectrum of brane charges are associated with the topological solitons which classify the four Hopf fibrations. 
  We study the non relativistic limit of a Model of Fermions interacting through a Chern-Simons Field, from a perspective that resembles the Wilson's Renormalization Group approach, instead of the more usual approach found in most texts of Field Theory. The solution of some difficulties, and a new understanding of non relativistic models is given. 
  We discuss the possibility of the extension of the duality between the webs of heterotic string and the type IIA string to Calabi-Yau 3-folds with another K3 fiber by comparing the dual polyhedron of Calabi-Yau 3-folds given by Candelas, Perevalov and Rajesh. 
  We study the gauged Landau-Lifshitz model in which a suitably chosen triplet of background scalar fields is included. It is shown that the model admits solitonic configurations. 
  In this letter we present the complete explicit brane solution in D-dimensional coupled gravity system. 
  A recently proposed topological mechanism for the quantization of the charge gives the value e_0=\sqrt{\hbar c} for both the fundamental electric and magnetic charges. It is argued that the corresponding fine structure constant \alpha_0=1/4pi could be interpreted as its value at the unification scale. 
  In this work we consider an Abelian Chern-Simons-Higgs model coupled non-minimally to matter fields. This coupling is implemented by means of a Pauli-type coupling. We show that the Pauli term is sufficient to gives rise to fractional spin. 
  Recently we have presented in hep-th/9811071 an ansatz which allows us to construct skyrmion fields from the harmonic maps of $S\sp2$ to $CP\sp{N-1}$. In this paper we examine this construction in detail and use it to construct, in an explicit form, new static spherically symmetric solutions of the SU(N) Skyrme models. We also discuss some properties of these solutions. 
  We obtain local parametrizations of classical non-compact Lie groups where adjoint invariants under maximal compact subgroups are manifest. Extension to non compact subgroups is straightforward. As a by-product parametrizations of the same type are obtained for compact groups. They are of physical interest in any theory gauge invariant under the adjoint action, typical examples being the two dimensional gauged Wess-Zumino-Witten-Novikov models where these coordinatizations become of extreme usefulness to get the background fields representing the vacuum expectation values of the massless modes of the associated (super) string theory. 
  The characteristic features of $<T_{\mu\nu}>$ in the Boulware, Unruh and Hartle-Hawking states for a conformal massless scalar field propagating in the Schwarzschild space-time are obtained by means of effective actions deduced by the trace anomaly. The actions are made local by the introduction of auxiliary fields and boundary conditions are carefully imposed on them in order to select the different quantum states. 
  We present the explicit global realization of the isometries of anti-de Sitter like spaces of signature $(d_-,d_+)$, and their algebras in the space of functions on the pseudo-Riemannian symmetric space $SO(d_- +1,d_+) / SO(d_-,d_+)$. The process of going to the invariant boundaries leads to the realization of the corresponding conformal groups and algebras. We compute systematically the geodesics in these spaces by considering the coset representation of them. 
  We investigate the consequence of the energy-momentum conservation law for the holographic S-matrix from AdS/CFT correspondence. It is shown that the conservation law is not a natural consequence of conformal invariance in the large N limit. We predict a new singularity for the four point correlation function of a marginal operator. Only the two point scattering amplitude is explicitly calculated, and the result agrees with what is expected. 
  An intense study of the relationship between certain quantum theories of gravity realized on curved backgrounds and suitable gauge theories, has been originated by a remarkable conjecture put forward by Maldacena almost one year ago. Among the possible curved vacua of superstring or M-theory, spaces having the form of an Anti-de Sitter space-time times a compact Einstein manifold, have been playing a special role in this correspondence, since the quantum theory realized on them, in the original formulation of the conjecture, was identified with the effective superconformal theory on the world volume of parallel p-branes set on the boundary of such a space (holography). An important step in order to verify such a conjecture and eventually generalize it, consists in a precise definition of the objects entering both sides of the holographic correspondence. In the most general case indeed it turns out that important features of the field theory on the boundary of the curved background, identified with the quantum theory of gravity in the bulk, are encoded in the dynamics of the coinciding parallel p-branes set on the boundary of the same space. The study of p-brane dynamics in curved space-times which are vacua of superstring of M-theory, turns out therefore to be a relevant issue in order to verify the existence of the holographic correspondence. In the present paper, besides providing a hopefully elementary introduction to Maldacena's duality,   I shall deal in a tentatively self contained way with a particular aspect of the problem of p-brane dynamics in Anti-de Sitter space-time, discussing some recent results. 
  The quantum mechanics of a spin 1/2 particle on a locally spatial constant curvature part of a (2+1)- spacetime in the presence of a constant magnetic field of a magnetic monopole has been investigated. It has been shown that these 2-dimensional Hamiltonians have the degeneracy group of SL(2,c), and para-supersymmetry of arbitrary order or shape invariance. Using this symmetry we have obtained its spectrum algebraically. The Dirac's quantization condition has been obtained from the representation theory. Also, it is shown that the presence of angular deficit suppresses both the degeneracy and shape invariance. 
  By using the fact that Polychronakos-like models can be obtained through the `freezing limit' of related spin Calogero models, we calculate the exact spectrum as well as partition function of SU(m|n) supersymmetric Polychronakos (SP) model. It turns out that, similar to the non-supersymmetric case, the spectrum of SU(m|n) SP model is also equally spaced. However, the degeneracy factors of corresponding energy levels crucially depend on the values of bosonic degrees of freedom (m) and fermionic degrees of freedom (n). As a result, the partition functions of SP models are expressed through some novel q-polynomials. Finally, by interchanging the bosonic and fermionic degrees of freedom, we obtain a duality relation among the partition functions of SP models. 
  Let us consider a Lie (super)algebra $G$ spanned by $T_{\alpha}$ where $T_{\alpha}$ are quantum observables in BV-formalism. It is proved that for every tensor $c^{\alpha_1...\alpha_k}$ that determines a homology class of the Lie algebra $G$ the expression $c^{\alpha_1...\alpha_k}T_{\alpha _1}...T_{\alpha_k}$ is again a quantum observables. This theorem is used to construct quantum observables in BV sigma-model. We apply this construction to explain Kontsevich's results about the relation between homology of the Lie algebra of Hamiltonian vector fields and topological invariants of manifolds. 
  Using a sigma model formulation of the field equations as on a two-dimensional manifold we provide the proof of a black hole uniqueness solution in N=4, d=4 supergravity subject to certain boundary conditions. We considered the black hole solutions both in SU(4) and SO(4) versions of the underlying theory. 
  A representation of the quadratic Dirac equation and the Maxwell equations in terms of the three-dimensional universal complex Clifford algebra $\bar{\bfa{C}}_{3,0}$ is given. The investigation considers a subset of the full algebra, which is isomorphic to the Baylis algebra. The approach is based on the two Casimir operators of the Poincar\'e group, the mass operator and the spin operator, which is related to the Pauli-Lubanski vector. The extension to spherical symmetries is discussed briefly. The structural difference to the Baylis algebra appears in the shape of the hyperbolic unit, which plays an integral part in this formalism. 
  In this work we examine generalized Connes-Lott models on the two-sphere. The Hilbert space of the continuum spectral triple is taken as the space of sections of a twisted spinor bundle, allowing for nontrivial topological structure (magnetic monopoles). The finitely generated projective module over the full algebra is also taken as topologically non-trivial, which is possible over $S^2$. We also construct a real spectral triple enlarging this Hilbert space to include "particle" and "anti-particle" fields. 
  Recently, a new constraint on the structure of a wide class of strongly coupled field theories has been proposed. It takes the form of an inequality limiting the number of degrees of freedom in the infrared description of a theory to be no larger than the number of underlying, ultraviolet degrees of freedom. Here we apply this inequality to chiral gauge theories. For some models we find that it is always satisfied, while for others we find that the assumption of the validity of the inequality implies a strong additional restriction on the spectrum of massless composite particles. 
  We investigate and contrast the non-perturbative infra red structure of N=1 and N=2 supersymmetric non-compact U(1) gauge field theory in three space-time dimensions with N matter flavours. We study the Dyson-Schwinger equations in a general gauge using superfield formalism; this ensures that supersymmetry is kept manifest, though leads to spurious infra red divergences which we have to avoid carefully. In the N=1 case the superfield formalism allows us to choose a vertex which satisfies the U(1) Ward identity exactly, and we find the expected critical behaviour in the wavefunction renormalization and strong evidence for the existence of a gauge independent dynamically generated mass, but with no evidence for a critical flavour number. We study the N=2 model by dimensional reduction from four dimensional N=1 electrodynamics, and we refine the old gauge dependence argument that there is no dynamical mass generation. We recognize that the refinement only holds after dimensional reduction. 
  A static, spherically symmetric and purely magnetic solution of the Einstein-Yang-Mills-Dilaton theory, found previously by numerical integration is shown to obey a system of first order Bogomol'nyi equations. As common for such equations, there is a tight relation to supersymmetry, in the present case to the N=4 gauged SU(2)$\times$SU(2) supergravity of Freedman and Schwarz. Specifically, the dilaton potential of the latter can be avoided by choosing one of the two gauge coupling constants to be imaginary. It is argued that this corresponds to a hitherto unknown N=4 gauged SU(2)$\times$SU(1,1) supergravity in four Euclidean dimensions leading to Bogomol'nyi equations with asymptotically flat solutions. 
  We derive manifestly locally supersymmetric extensions of the Born-Infeld action with $p=2$. The construction is based on a first order bosonic action for $Dp$-branes with a generalized Weyl invariance. 
  We extend the theory of cosmological perturbations to the case when the ``matter'' Lagrangian is an arbitrary function of the scalar field and its first derivatives. In particular, this extension provides a unified description of known cases such as the usual scalar field and the hydrodynamical perfect fluid. In addition, it applies to the recently proposed k-inflation, which is driven by non-minimal kinetic terms in the Lagrangian. The spectrum of quantum fluctuations for slow-roll and power law k-inflation is calculated. We find, for instance, that the usual ``consistency relation'' between the tensor spectral index and the relative amplitude of scalar and tensor perturbations is modified. Thus, at least in principle, k-inflation is phenomenologically distinguishable from standard inflation. 
  We classify the spectrum, family structure and stability of Nielsen-Olesen vortices embedded in a larger gauge group when the vacuum manifold is related to a symmetric space. 
  We analyse symmetry breaking in the Weinberg-Salam model paying particular attention to the underlying geometry of the theory. In this context we find two natural metrics upon the vacuum manifold: an isotropic metric associated with the scalar sector, and a squashed metric associated with the gauge sector. Physically, the interplay between these metrics gives rise to many of the non-perturbative features of Weinberg-Salam theory. 
  In the context of the AdS/CFT correspondence, we perform a direct computation in AdS_5 supergravity of the trace anomaly of a d=4, N=2 SCFT. We find agreement with the field theory result up to next to leading order in the 1/N expansion. In particular, the order N gravitational contribution to the anomaly is obtained from a Riemann tensor squared term in the 7-brane effective action deduced from heterotic - type I duality. We also discuss, in the AdS/CFT context, the order N corrections to the trace anomaly in d=4, N=4 SCFTs involving SO or Sp gauge groups. 
  By using the Nambu-Jona-Lasinio model, we study dynamical symmetry breaking in spaces with constant negative curvature. We show that the physical reason for zero value of critical coupling value $g_c = 0$ in these spaces is connected with the effective reduction of dimension of spacetime $1 + D \to 1 + 1$ in the infrared region, which takes place for any dimension $1 + D$. Since the Laplace-Beltrami operator has a gap in spaces with constant negative curvature, such an effective reduction for scalar fields is absent and there are not problems with radiative corrections due to scalar fields. Therefore, dynamical symmetry breaking with the effective reduction of the dimension of spacetime for fermions in the infrared region is consistent with the Mermin-Wagner-Coleman theorem, which forbids spontaneous symmetry breaking in (1 + 1)-dimensional spacetime. 
  We study various three-dimensional supersymmetric Maxwell Chern-Simons solitons by using type IIB brane configurations. We give a systematic classification of soliton spectra such as topological BPS vortices and nontopological vortices in $\cn=2,3$ supersymmetric Maxwell Chern-Simons system via the branes of type IIB string theory. We identify the brane configurations with the soliton spectra of the field theory and obtain a nice agreement with field theory aspects. We also discuss possible brane constructions for BPS domain wall solutions. 
  We derive the power law decay, and asymptotic form, of SU(2) x Spin(d) invariant wave-functions which are zero-modes of all s_d=2(d-1) supercharges of reduced (d+1)-dimensional supersymmetric SU(2) Yang Mills theory, resp. of the SU(2)-matrix model related to supermembranes in d+2 dimensions. 
  We start with a new first order gauge non-invariant formulation of massive spin-one theory and map it to a reducible gauge theory viz; abelian $B{\wedge}F$ theory by the Hamiltonian embedding procedure of Batalin, Fradkin and Tyutin(BFT). This equivalence is shown from the equations of motion of the embedded Hamiltonian. We also demonstrate that the original gauge non-invariant model and the topologically massive gauge theory can both be obtained by suitable choice of gauges, from the phase space partition function of the emebedded Hamiltonian, proving their equivalence. Comparison of the first order formulation with the other known massive spin-one theories is also discussed. 
  Found is a general form of static solutions in exactly solvable models of 2d dilaton gravity at finite temperature. We reveal a possibility for the existence of everywhere regular solutions including black holes, semi-infinite throats and star-like configurations. In particular, we consider the Bose-Parker-Peleg (BPP) model which possesses a semi-infinite throat and analyze it at finite temperature. We also suggest generalization of the BPP model in which the appearance of semi-infinite throat has a generic character and does not need special fine tuning between parameters of the solution. 
  We discuss the ultraviolet finiteness of the two-dimensional BF model coupled to topological matter quantized in the axial gauge. This noncovariant gauge fixing avoids the infrared problem in the two-dimensional space-time. The BF model together with the matter coupling is obtained by dimensional reduction of the ordinary three-dimensional BF model. This procedure furnishes the usual linear vector supersymmetry and an additional scalar supersymmetry. The whole symmetry content of the model allows to apply the standard algebraic renormalization procedure which we use to prove that this model is ultraviolet finite and anomaly free to all orders of perturbation theory. 
  We find static spherically symmetric monopoles in Einstein-Born-Infeld-Higgs model in 3+1 dimensions. The solutions exist only when a parameter $\a $ (related to the strength of Gravitational interaction) does not exceed certain critical value. We also discuss magnetically charged non Abelian black holes in this model. We analyse these solutions numerically. 
  We study \kk spectrum of type IIB string theory compactified on $AdS_5 \times T^{nn'}$ in the context of $AdS/CFT$ correspondence. We examine some of the modes of the complexified 2 form potential as an example and show that for the states at the bottom of the \kk tower the corresponding $d=4$ boundary field operators have rational conformal dimensions. The masses of some of the fermionic modes in the bottom of each tower as functions of the $R$ charge in the boundary conformal theory are also rational. Furthermore the modes in the bottom of the towers originating from $q$ forms on $T^{11}$ can be put in correspondence with the BRS cohomology classes of the $c=1$ non critical string theory with ghost number $q$. However, a more detailed investigation is called for, to clarify further the relation of this supergravity background with the $c=1$ strings. 
  We construct a new class of two-dimensional field theories with target spaces that are finite multiparameter deformations of the usual coset G/H-spaces. They arise naturally, when certain models, related by Poisson-Lie T-duality, develop a local gauge invariance at specific points of their classical moduli space. We show that canonical equivalences in this context can be formulated in loop space in terms of parafermionic-type algebras with a central extension. We find that the corresponding generating functionals are non-polynomial in the derivatives of the fields with respect to the space-like variable. After constructing models with three- and two-dimensional targets, we study renormalization group flows in this context. In the ultraviolet, in some cases, the target space of the theory reduces to a coset space or there is a fixed point where the theory becomes free. 
  In [Carey, A.L., J. Mickelsson, and M. K. Murray: Comm. Math. Phys. 183, 707 (1997)] Schwinger terms in hamiltonian quantization of chiral fermions coupled to vector potentials were computed, using some ideas from the theory of gerbes, with the help of the family index theorem for a manifold with boundary. Here, we generalize this method to include gravitational Schwinger terms. 
  We consider aspects of instanton dynamics in the large-N limit using the AdS/CFT duality for D0/D4 bound states. In the supergravity picture of wrapped D0-brane world-lines on D4-branes, we find the single-instanton measure and discuss its dependence on compactification finite-size effects, as well as its matching to perturbative results. In the non-supersymmetric case, the same dynamical effects that produce the theta-angle dependence perturbatively in 1/N, render the instantons unstable, although approximate instantons of very small size still exist. The smeared D0/D4 black-brane supergravity solution can be interpreted as dual to a field theory configuration of an instanton condensate in the vacuum. In this case, we derive a holographic relation between the bare theta angle and the topological charge density of the instanton condensate. 
  The AdS/CFT correspondence suggests that the Wilson loop of the large N gauge theory with N=4 supersymmetry in 4 dimensions is described by a minimal surface in AdS_5 x S^5. We examine various aspects of this proposal, comparing gauge theory expectations with computations of minimal surfaces. There is a distinguished class of loops, which we call BPS loops, whose expectation values are free from ultra-violet divergence. We formulate the loop equation for such loops. To the extent that we have checked, the minimal surface in AdS_5 x S^5 gives a solution of the equation. We also discuss the zig-zag symmetry of the loop operator. In the N=4 gauge theory, we expect the zig-zag symmetry to hold when the loop does not couple the scalar fields in the supermultiplet. We will show how this is realized for the minimal surface. 
  The consequences of coupling of the torsion (highest curvature) term to the Lagrangian of a massive spinless particle in four-dimensional space-time are studied. It is shown that the modified system remains spinless and possesses extended gauge invariance. Though the torsion term does not generate spin, it provides the system with a nontrivial mass spectrum, described by one-dimensional conformal mechanics. Under an appropriate choise of characteristic constants the system has solutions with a discrete mass spectrum. 
  We present an approach to studying the Casimir effects by means of the effective theory. An essential point of our approach is replacing the mirror separation into the size of space S^1 in the adiabatic approximation. It is natural to identify the size of space S^1 with the scale factor of the Robertson-Walker-type metric. This replacement simplifies the construction of a class of effective models to study the Casimir effects. To check the validity of this replacement we construct a model for a scalar field coupling to the two-dimensional gravity and calculate the Casimir effects by the effective action for the variable scale factor. Our effective action consists of the classical kinetic term of the mirror separation and the quantum correction derived by the path-integral method. The quantum correction naturally contains both the Casimir energy term and the back-reaction term of the dynamical Casimir effect, the latter of which is expressed by the conformal anomaly. The resultant effective action describes the dynamical vacuum pressure, i.e., the dynamical Casimir force. We confirm that the force depends on the relative velocity of the mirrors, and that it is always attractive and stronger than the static Casimir force within the adiabatic approximation. 
  In angular quantization approach a perturbation theory for the Massive Thirring Model (MTM) is developed, which allows us to calculate Vacuum Expectation Values of exponential fields in sin-Gordon theory near the free fermion point in first order of MTM coupling constant $g$. The Hankel-transforms play an important role when carrying out this calculations. The expression we have found coincides with that of the direct expansion over $g$ of the exact formula conjectured by S.Lukyanov and A.Zamolodchikov. 
  The well-known $D$-dimensional Feynman integrals were shown, by Halliday and Ricotta, to be capable of undergoing analytic continuation into the domain of negative values for the dimension of space-time. Furthermore, this could be identified with Grassmannian integration in positive dimensions. From this possibility follows the concept of negative dimensional integration for loop integrals in field theories. Using this technique, we evaluate three two-loop three-point scalar integrals, with five and six massless propagators, with specific external kinematic configurations (two legs on-shell), and four three-loop two-point scalar integrals. These results are given for arbitrary exponents of propagators and dimension, in Euclidean space, and the particular cases compared to results published in the literature. 
  We use N=2 harmonic and projective superspaces to formulate the most general `Ansatz' for the SU(2)_R invariant hypermultiplet low-energy effective action (LEEA) in four dimensions, which describes the two-parametric family of the hyper-K"ahler metrics generalizing the Atiyah-Hitchin metric. We then demonstrate in the very explicit and manifestly N=2 supersymmetric way that the (magnetically charged, massive) single hypermultiplet LEEA in the underlying non-abelian N=2 supersymmetric quantum field theory can receive both perturbative (e.g., in the Coulomb branch) and non-perturbative (e.g., in the Higgs branch) quantum corrections. The manifestly N=2 supersymmetric Feynman rules in harmonic superspace can be used to calculate the perturbative corrections described by the Taub-NUT metric. The non-perturbative corrections (due to instantons and anti-instantons) can be encoded in terms of an elliptic curve, which is very reminiscent to the Seiberg-Witten theory. Our four-dimensional results agree with the three-dimensional Seiberg-Witten theory and instanton calculations. 
  The physical properties of Reissner-Nordstrom black holes in (n+1)-dimensional anti-de Sitter spacetime are related, by a holographic map, to the physics of a class of n-dimensional field theories coupled to a background global current. Motivated by that fact, and the recent observations of the striking similarity between the thermodynamic phase structure of these black holes (in the canonical ensemble) and that of the van der Waals-Maxwell liquid-gas system, we explore the physics in more detail. We study fluctuations and stability within the equilibrium thermodynamics, examining the specific heats and electrical permittivity of the holes, and consider the analogue of the Clayperon equation at the phase boundaries. Consequently, we refine the phase diagrams in the canonical and grand canonical ensembles. We study the interesting physics in the neighbourhood of the critical point in the canonical ensemble. There is a second order phase transition found there, and that region is characterized by a Landau-Ginzburg model with A_3 potential. The holographically dual field theories provide the description of the microscopic degrees of freedom which underlie all of the thermodynamics, as can be seen by examining the form of the microscopic fluctuations. 
  We make some comments on the derivation of N=2 super-conformal field theories with smooth gauge group from M2-branes placed at conifold singularities, giving a detailed prescription for two specific examples: the singular cones over the Q^{111} and M^{110} manifolds. 
  We study the spectrum of the QCD Dirac operator for two colors with fermions in the fundamental representation and for two or more colors with adjoint fermions. For $N_f$ flavors, the chiral flavor symmetry of these theories is spontaneously broken according to $SU(2N_f)\to Sp(2N_f)$ and $SU(N_f)\to O(N_f)$, respectively, rather than the symmetry breaking pattern $SU(N_f) \times SU(N_f) \to SU(N_f)$ for QCD with three or more colors and fundamental fermions. In this paper we study the Dirac spectrum for the first two symmetry breaking patterns. Following previous work for the third case we find the Dirac spectrum in the domain $\lambda \ll \Lambda_{\rm QCD}$ by means of partially quenched chiral perturbation theory. In particular, this result allows us to calculate the slope of the Dirac spectrum at $\lambda = 0$. We also show that for $\lambda \ll 1/L^2 \Lambda_{QCD}$ (with $L$ the linear size of the system) the Dirac spectrum is given by a chiral Random Matrix Theory with the symmetries of the Dirac operator. 
  We point out that the concept of Abelian projection gives us a physical interpretation of the role that the Hitchin fibration of parabolic K(D) pairs plays in the large-N limit of four-dimensional QCD. This physical interpretation furnishes also a simple criterium for the confinement of electric fluxes in the large-N limit of QCD. There is also an alternative, compatible interpretation, based on the QCD string. 
  We show that, as a consequence of a physical interpretation based on the Abelian projection and on the QCD string, four-dimensional QCD confines the electric flux if and only if the functional integral in the fiberwise-dual variables admits a hyper-Kahler reduction under the action of the gauge group. 
  The canonical decomposition of a real Klein-Gordon field in collective and relative variables proposed by Longhi and Materassi is reformulated on spacelike hypersurfaces. This allows to obtain the complete canonical reduction of the system on Wigner hyperplanes, namely in the rest-frame Wigner-covariant instant form of dynamics. From the study of Dixon's multipoles for the energy-momentum tensor on the Wigner hyperplanes we derive the definition of the canonical center-of-mass variable for a Klein-Gordon field configuration: it turns out that the Longhi-Materassi global variable should be interpreted as a center of phase of the field configuration. A detailed study of the kinematical "external" and "internal" properties of the field configuration on the Wigner hyperplanes is done. The construction is then extended to charged Klein-Gordon fields: the centers of phase of the two real components can be combined to define a global center of phase and a collective relative variable describing the action-reaction between the two Feshbach-Villars components of the field with definite sign of energy and charge. The Dixon multipoles for both the energy-momentum and the electromagnetic current are given. Also the coupling of the Klein-Gordon field to scalar relativistic particles is studied and it is shown that in the reduced phase space, besides the particle and field relative variables, there is also a collective relative variable describing the relative motion of the particle subsytem with respect to the field one. 
  Using the recently developed soldering formalism we highlight certain features of quantum mechanical models. The complete correspondence between these models and self dual field theoretical models in odd dimensions is established. The distinction between self duality and self dual factorisation in these dimensions is clarified. 
  We study the ultraviolet and the infrared behavior of 2D topological BF-Theory coupled to vector and scalar fields. This model is equivalent to 2D gravity coupled to topological matter. Using techniques of the algebraic renormalization program we show that this model is anomaly free and ultraviolet as well as infrared finite at all orders of perturbation theory. 
  Using the BRS techniques, we prove the existence of a local and nonlinear symmetry of the gauge fixed action of the antisymmetric tensor field model in curved background. 
  We present results for infinite series appearing in Feynman diagram calculations, many of which are similar to the Euler series. These include both one-, two- and three-dimensional series. All these series can be expressed in terms of zeta(2) and zeta(3). 
  We review the recent developments in our understanding of non-BPS states and branes in string theory. The topics include 1) construction of unstable non-BPS D-branes in type IIA and type IIB string theories, 2) construction of stable non-BPS D-branes on various orbifolds and orientifolds of type II string theories, 3) description of BPS and non-BPS D-branes as tachyonic soliton solutions on brane-antibrane pair of higher dimension, and 4) study of the spectrum of non-BPS states and branes on a system of coincident D-brane - orientifold plane system. Some other related results are also discussed briefly. 
  We consider null bosonic p-branes in curved space-times. Some exact solutions of the classical equations of motion and of the constraints for the null membrane in general stationary, axially symmetrical, four dimensional, gravity background are found. 
  The process of collision of two parallel domain walls in a supersymmetric model is studied both in effective Lagrangian approximation and by numerical solving of the exact classical field problem. For small initial velocities we find that the walls interaction looks like elastic reflection with some delay. It is also shown that in such approximation internal parameter of the wall may be considered as a time-dependent dynamical variable. 
  In this talk I review and generalize an idea of Seiberg that an N=1 supersymmetric gauge theory shows confinement without breaking of chiral symmetry when the gauge symmetry of its magnetic dual is completely broken by the Higgs effect. It is shown how the confining spectrum of a supersymmetric gauge theory can easily be derived when a magnetic dual is known and this method is applied to many models containing fields in second rank tensor representations and an appropriate tree-level superpotential. 
  We consider non-perturbative four dimensional N=1 space-time supersymmetric orientifolds corresponding to Type I compactifications on (generalized) Voisin-Borcea orbifolds. Some states in such compactifications arise in ``twisted'' open string sectors which lack world-sheet description in terms of D-branes. Using Type I-heterotic duality as well as the map between Type IIB orientifolds and F-theory we are able to obtain the massless spectra of such orientifolds. The four dimensional compactifications we discuss in this context are examples of chiral N=1 supersymmetric string vacua which are non-perturbative from both orientifold and heterotic points of view. In particular, they contain both D9- and D5-branes as well as non-perturbative ``twisted'' open string sector states. We also explain the origins of various inconsistencies arising in such compactifications for certain choices of the gauge bundle. 
  It is pointed out that the Mersenne primes $M_p=(2^p-1)$ and associated perfect numbers ${\cal M}_p=2^{p-1}M_p$ play a significant role in string theory; this observation may suggest a classification of consistent string theories. 
  The partial breaking of supersymmetry in flat space can be accomplished using any one of three dual representations for the massive N=1 spin-3/2 multiplet. Each of the representations can be ``unHiggsed'', which gives rise to a set of dual N=2 supergravities and supersymmetry algebras. 
  To obtain some exact results of U(1) gauge theory (QED), we construct the low energy effective action of N=2 supersymmetric QED with a massless matter and Fayet-Iliopoulos term, assuming no confinement. The harmonic superspace formalism for N=2 extended supersymmetry makes the construction easy. We analyze the vacuum structure and find no vacuum. It suggests the confinement in non-supersymmetric QED at low energies. 
  We propose and develop a new calculus for local variational differential operators. The main difference of the new formalism with the canonical differential calculus is that the image of higher order operators on local functionals does not contain indefinite quantities like $\delta(0)$. We apply this formalism to BV formulation of general gauge field theory and to its Sp(2)-symmetric generalization. Its relation to a quasiclassical expansion is also discussed. 
  We construct the N=1 supergravity analog of the Green-Schwarz and heterotic superstring theories in 10D. We find the SO(8) theory previously found by compactification of 10D on $K3$. We also find eleven $G_2\times G_2$ theories, two with symmetric matter content. It is not obvious how ten of the $G_2\times G_2$ theories can be gotten from 10D. 
  When an M-theory fivebrane wraps a holomorphic surface $\CP$ in a Calabi-Yau 3-fold $X$ the low energy dynamics is that of a black string in 5 dimensional $\CN=1$ supergravity. The infrared dynamics on the string worldsheet is an $\CN = (0,4)$ 2D conformal field theory. Assuming the 2D CFT can be described as a nonlinear sigma model, we describe the target space geometry of this model in terms of the data of $X$ and $\CP$. Variations of weight two Hodge structures enter the construction of the model in an interesting way. 
  We discuss how to obtain an N=(2,2) supersymmetric SU(3) gauge theory in two dimensions via geometric engineering from a Calabi-Yau 4-fold and compute its non-perturbative twisted chiral potential. The relevant compact part of the 4-fold geometry consists of two intersecting P^1's fibered over P^2. The rigid limit of the local mirror of this geometry is a complex surface that generalizes the Seiberg-Witten curve and on which there exist two holomorphic 2-forms. These stem from the same meromorphic 2-form as derivatives w.r.t. the two moduli, respectively. The middle periods of this meromorphic form give directly the twisted chiral potential. The explicit computation of these and of the four-point Yukawa couplings allows for a non-trivial test of the analogue of rigid special geometry for a 4-fold with several moduli. 
  Regularization of quantum field theories introduces a mass scale which breaks axial rotational and scaling invariances. We demonstrate from first principles that axial torsion and torsion trace modes have non-transverse vacuum polarization tensors, and become massive as a result. The underlying reasons are similar to those responsible for the Adler-Bell-Jackiw (ABJ) and scaling anomalies. Since these are the only torsion components that can couple minimally to spin 1/2 particles, the anomalous generation of masses for these modes, naturally of the order of the regulator scale, may help to explain why torsion and its associated effects, including CPT violation in chiral gravity, have so far escaped detection. As a simpler manifestation of the reasons underpinning the ABJ anomaly than triangle diagrams, the vacuum polarization demonstration is also pedagogically useful. 
  We consider some unitary representations of infinite dimensional Lie algebras motivated by string theory on AdS_3. These include examples of two kinds: the A,D,E type affine Lie algebras and the N=4 superconformal algebra. The first presents a new construction for free field representations of affine Lie algebras. The second is of a particular physical interest because it provides some hints that a hybrid of the NSR and GS formulations for string theory on AdS_3 exists. 
  Extended BRST invariance (BRST plus anti-BRST invariances) provides in principle a natural way of introducing the complete gauge fixing structure associated to a gauge field theory in the minimum representation of the algebra. However, as it happens in topological Yang Mills theory, not all gauge fixings can be obtained from a symmetrical extended BRST algebra, where antighosts belong to the same representation of the Lorentz group of the corresponding ghosts. We show here that, at non interacting level, a simple field redefinition makes it possible to start with an extended BRST algebra with symmetric ghost antighost spectrum and arrive at the gauge fixing action of topological Yang Mills theory. 
  We analyze the Coulomb branch of Matrix string theory in the presence of NS5-branes. If we regard the components of U(1) gauge fields as the dualized longitudinal coordinates, we obtain the symmetric product of AdS_3 x S^3 x R^4 as the geometry of Coulomb branch. We observe that the absence or presence of the nonzero electric flux determines whether the string propagates in bulk as an ordinary closed string or is forced to live near the boundary.    We further discuss the issues of the physical spectrum from the viewpoint of Matrix string theory. We show that the twisted sectors of CFT on the symmetric orbifold, which correspond to glued strings, turn out to yield many chiral primaries that were hitherto considered to be missing. We also comment on the threshold energy in Liouville sector where continuous spectrum begins. 
  Some recent ideas are generalized from four dimensions to the general dimension n. In quantum field theory, two terms of the trace anomaly in external gravity, the Euler density G_n and Box^{n/2-1}R, are relevant to the problem of quantum irreversibility. By adding the divergence of a gauge-invariant current, G_n can be extended to a new notion of Euler density, linear in the conformal factor. We call it pondered Euler density. This notion relates the trace-anomaly coefficients a and a' of G_n and Box^{n/2-1}R in a universal way (a=a') and gives a formula expressing the total RG flow of a as the invariant area of the graph of the beta function between the fixed points. I illustrate these facts in detail for n=6 and check the prediction to the fourth-loop order in the phi^3-theory. The formula of quantum irreversibility for general n even can be extended to n odd by dimensional continuation. Although the trace anomaly in external gravity is zero in odd dimensions, I show that the odd-dimensional formula has a predictive content. 
  We present a pedagogical review of the stable non-BPS states in string theory which have recently attracted some attention in the literature. In particular, following the analysis of A. Sen, we discuss in detail the case of the stable non-BPS D-particle of Type I theory whose existence is predicted (and required) by the heterotic/Type I duality. We show that this D-particle originates from an unstable bound state formed by a D1/anti-D1 pair of Type IIB in which the tachyon field acquires a solitonic kink configuration. The mechanism of tachyon condensation is discussed first at a qualitative level and then with an exact conformal field theory analysis. 
  A recent study shows that Hawking radiation of the massless scalar field does not appear on the two-dimensional AdS$_2$ black hole background. We study this issue by investigating absorption and reflection coefficients under dilaton coupling with the matter field. If the scalar field does not couple to the dilaton, then it is fully absorbed into the black hole without any outgoing mode. On the other hand, once it couples to the dilaton field, the outgoing mode of the massless scalar field exists and the nontrivial Hawking radiation is obtained. Finally, we comment on this dilaton dependence of Hawking radiation in connection with a three-dimensional black hole. 
  This paper was originally designated as Comment to the paper by R. Jackiw and V. Alan Kostelecky (hep-ph/9901358). We provide an example of the fermionic system, the superfluid 3He-A, in which the CPT-odd Chern-Simons terms in the effective action are unambiguously induced by chiral fermions. In this system the Lorentz and gauge invariances both are violated at high energy, but the behavior of the system beyond the cut-off is known. This allows us to construct the CPT-odd action, which combines the conventional 3+1 Chern-Simons term and the mixed axial-gravitational Chern-Simons term discussed in hep-ph/9905460. The influence of Chern-Simons term on the dynamics of the effective gauge field has been experimentally observed in rotating 3He-A. 
  The Atiyah-Drinfeld-Hitchin-Manin matrix corresponding to a tetrahedrally symmetric 3-instanton is calculated. Some small variations of the matrix correspond to vibrations of the instanton-generated 3-Skyrmion. These vibrations are decomposed under tetrahedral symmetry and this decomposition is compared to previous knowledge of the 3-Skyrmion vibration spectrum. 
  In this paper we prove that for a large and physically relevant class of smooth complete metrics on open four-manifolds the set of L^2-norms of (anti)instantons is a discrete set for physically relevant Lie-groups. The proof is based on the Atiyah-Patodi-Singer Index Theorem. To demonstrate our result we consider the Euclidean Schwarzschild manifold. This metric is significantly non-cylindrical hence standard techniques are not applicable for studying the moduli space of instantons in this physically relavant case. In this case the L^2-norm of an (anti)instanton is an integer similarly to the flat R^4-case. 
  Calogero-Moser models can be generalised for all of the finite reflection groups. These include models based on non-crystallographic root systems, that is the root systems of the finite reflection groups, H_3, H_4, and the dihedral group I_2(m), besides the well-known ones based on crystallographic root systems, namely those associated with Lie algebras. Universal Lax pair operators for all of the generalised Calogero-Moser models and for any choices of the potentials are constructed as linear combinations of the reflection operators. The consistency conditions are reduced to functional equations for the coefficient functions of the reflection operators in the Lax pair. There are only four types of such functional equations corresponding to the two-dimensional sub-root systems, A_2, B_2, G_2, and I_2(m). The root type and the minimal type Lax pairs, derived in our previous papers, are given as the simplest representations. The spectral parameter dependence plays an important role in the Lax pair operators, which bear a strong resemblance to the Dunkl operators, a powerful tool for solving quantum Calogero-Moser models. 
  We consider ``brane-universes'', where matter is confined to four-dimensional hypersurfaces (three-branes) whereas one extra compact dimension is felt by gravity only. We show that the cosmology of such branes is definitely different from standard cosmology and identify the reasons behind this difference. We give a new class of exact solutions with a constant five-dimensional radius and cosmologically evolving brane. We discuss various consequences. 
  The production of electron-positron pairs in a time-dependent magnetic field is estimated in the hypotheses that the magnetic field is uniform over large distances with respect to the pair localization and it is so strong that the spacing of the Landau levels is larger than the rest mass of the particles. This calculation is presented since it has been suggested that extremely intense and varying magnetic fields may be found around some astrophysical objects. 
  In the previous paper [hep-th/9904129], we constructed a general explicit BPS solution for (F, D3) non-threshold bound state. By using the SL(2,Z) symmetry of type IIB string theory, we here construct from (F, D3) a more general BPS configuration for a D3 brane with certain units of quantized 5-form flux and an infinite number of parallel (F, D1)-strings. We study its decoupling limit and find that given Maldacena's $AdS_5/CFT_4$ correspondence with respect to simple D3 branes and with the usual string coupling, we should have a similar correspondence with respect to this bound state but now with an effective string coupling. We discuss possible descendants of this bound state by T-dualities along its longitudinal or transverse directions. In particular, we present explicit configurations for ((F, Dp), D(p + 2)) bound states for $2 \le p \le 5$. All these configurations preserve one half of the spacetime supersymmetries. 
  We examine supersymmetric SU(N) gauge theories on R^3*S^1 with a circle of circumference beta. These theories interpolate between four-dimensional N=1 pure gauge theory for beta=infinity and three-dimensional N=2 gauge theory for beta=0. The dominant field configurations of the R^3*S^1 SU(N) theories in the semi-classical regime arise from N varieties of monopole. Periodic instanton configurations correspond to mixed configurations of N single monopoles of the N different types. We semi-classically evaluate the non-perturbatively generated superpotential of the R^3*S^1 theory and hence determine its vacuum structure. We then calculate monopole contributions to the gluino condensate in these theories and take the decompactification limit beta=infinity. In this way we obtain a value for the gluino condensate in the four-dimensional N=1 supersymmetric SU(N) Yang-Mills theory, which agrees with the previously known `weak coupling' expression but not with the `strong coupling' expression derived in the early literature solely from instanton considerations. Moreover, we discover that the superpotential gives a mass to the dual (magnetic) photon, which implies confinement of the original electric photon and disappearance of all the massless modes. 
  We give the analytic result for the fermion zero-mode of the SU(2) calorons with non-trivial holonomy. It is shown that the zero-mode is supported on ONLY ONE of the constituent monopoles. We discuss some of its implications. 
  Before turning to the new result that D=11 supergravity is 2-loop nonrenormalizable, we give a very brief history of the ultraviolet problems of ordinary quantum gravity and of supergravities in general D. 
  We find novel bound states of NS5, D6 and D8-branes in massive Type IIA string theory. As the NS gauge transformations can change the Chern class of the RR field these configurations should be thought of as nonlocal objects called gerbes. We develop a global formalism for theories that involve massive tensor fields in general, and apply it in massive Type IIA supergravity. We check the results by investigating the T-dual NS5/D5-brane configurations in Type IIB, and relate them to an F-theory compactification on a CY3. We comment on the implications to consistency conditions for brane-wrapping and the classification of D-brane charges in terms of K-theory classes. 
  We determine a manifestly SL(2,Z)-covariant kappa-symmetric action for the type IIB (p,q) five-branes as a perturbative expansion in the world-volume field strengths within the framework where the brane tension is generated by a world-volume field. In this formulation the Lagrangian is expected to be polynomial; we construct the kappa-invariant action to fourth order in the world-volume field strengths. 
  The superconformal group of N=4 super-Yang-Mills has two types of operator representations: short and long. We conjecture that operator product expansions for which at least two of the three operators are short exactly respect a bonus U(1)_Y R-symmetry, which acts as an automorphism of the superconformal group. This conjecture is for arbitrary gauge group G and gauge coupling g_{YM}. A consequence is that n\leq 4-point functions involving only short operators exactly respect the U(1)_Y symmetry, as has been previously conjectured based on AdS duality. This, in turn, would imply that all n\leq 3 -point functions involving only short operators are not renormalized, as has also been previously conjectured and subjected to perturbative checks. It is argued that instantons are compatible with our conjecture. Some perturbative checks of the conjecture are presented and SL(2,Z) modular transformation properties are discussed. 
  The well known operator ordering ambiguity could motivate the existence of generations. This possibility is explored by exploiting the relationship between ordering and discretization rules. Context is drawn from lattice theory and non commutative geometry. 
  Perturbative quantum gauge field theory seen within the perspective of physical gauge choices such as the light-cone entails the emergence of troublesome poles of the type $(k\cdot n)^{-\alpha}$ in the Feynman integrals, and these come from the boson field propagator, where $\alpha = 1,2,...$ and $n^{\mu}$ is the external arbitrary four-vector that defines the gauge proper. This becomes an additional hurdle to overcome in the computation of Feynman diagrams, since any graph containing internal boson lines will inevitably produce integrands with denominators bearing the characteristic gauge-fixing factor. How one deals with them has been the subject of research for over decades, and several prescriptions have been suggested and tried in the course of time, with failures and successes.   However, a more recent development in this front which applies the negative dimensional technique to compute light-cone Feynman integrals shows that we can altogether dispense with prescriptions to perform the calculations. An additional bonus comes attached to this new technique in that not only it renders the light-cone prescriptionless, but by the very nature of it, can also dispense with decomposition formulas or partial fractioning tricks used in the standard approach to separate pole products of the type $(k\cdot n)^{-\alpha}[(k-p)\cdot n]^{-\beta}$, $(\beta = 1,2,...)$.   In this work we demonstrate how all this can be done. 
  We study an extension of the procedure to construct duality transformations among abelian gauge theories to the non abelian case using a path space formulation. We define a pre-dual functional in path space and introduce a particular non local map among Lie algebra valued 1-form functionals that reduces to the ordinary Hodge-* duality map of the abelian theories. Further, we establish a full set of equations on path space representing the ordinary Yang Mills equations and Bianchi identities of non abelian gauge theories of 4-dimensional euclidean space. 
  We construct D-brane states on an asymmetric orbifold of type IIA on a four-torus, which is modded out by T-duality. We find explicit boundary states charged under the twisted sector gauge fields. Unlike other cases, the boundary states involve an explicit dependence on the twist fields. The D-brane spectrum is consistent with the model being equivalent to type IIA on a four-torus. 
  The effect of fermionic string conductivity by purely right (or purely left) moving ``zero modes'' is shown to be governed by a simple Lagrangian characterising a certain ``chiral'' (null current carrying) string model whose dynamical equations of motion turn out to be explicitly integrable in a flat spacetime background. 
  The conformal affine sl(2) Toda model coupled to matter field is treated as a constrained system in the context of Faddeev-Jackiw and the (constrained) symplectic schemes. We recover from this theory either, the sine-Gordon or the massive Thirring model, through a process of Hamiltonian reduction, considering the equivalence of the Noether and topological currrents as a constraint and gauge fixing the conformal symmetry. 
  The worldline casting of a gauge field system with spin-1/2 matter fields has provided a, particle-based, first quantization formalism in the framework of which the Bern-Kosower algorithms for efficient computations in QCD acquire a simple interpretation. This paper extends the scope of applicability of the worldline scheme so as to include open fermionic paths. Specific algorithms are established which address themselves to the fermionic propagator and which are directly applicable to any other process involving external fermionic states. It is also demonstrated that in this framework the sole agent of dynamics operating in the system is the Wilson line (loop) operator, which makes a natural entrance in the worldline action; everything else is associated with geometrical properties of particle propagation, of which the most important component is Polyakov's spin factor. 
  This paper is almost an exercise in which the Hamiltonian scheme is developed for Polyakov's classical string, by following the usual framework suggested by Dirac and Bergman for the reduction of gauge theories to their essential physical degrees of freedom. The results collected here will be useful in some forthcoming papers, where strings will be studied in the unusual context of Wigner-covariant rest frame theory.   After a short introduction outlining the work, the Lagrangean scheme is presented in Section II, where the classical equivalence between Polyakov and Nambu-Goto string is rederived. In Section III the Hamiltonian framework is worked out, primary and secondary constraints are deduced. Then Lagrange multipliers are introduced; finally the Hamiltonian equations of motion are presented.   In Section IV gauge symmetries are treated, by constructing their canonical generators; then some gauge degrees of freedom are eliminated by Dirac-Bergmann fixing procedure. In this paper only the gauge-freedom coming from primary constraints is fixed, while secondary constraints will need a deeper analysis. At the end of   Section IV the classical version of Virasoro algebra is singled out as that of the secondary constraints, surviving to the first gauge fixing. 
  We calculate from M-theory the two-dimensional low energy effective dynamics of various brane configurations. In the first part we study configurations that have a dual description in type IIA string theory as two-dimensional (4,0) Yang-Mills theories with gauge group SU(N_1)xSU(N_2) and chiral fermions in the bi-fundamental representation. In the second part we derive related equations of motion which describe the low energy internal dynamics of a supersymmetric black hole in four-dimensional N=1 supergravity, obtained as an M-fivebrane wrapped on a complex four-cycle. 
  The thermodynamics of supersymmetric Yang-Mills theories is studied by computing the two-loop correction to the canonical free energy and to the equation of state for theories with 16, 8 and 4 supercharges in any dimension $4\leq d\leq 10$, and in two dimensions at finite volume. In the four-dimensional case we also evaluate the first non-analytic contribution in the 't Hooft coupling to the free energy, arising from the resummation of ring diagrams. To conclude, we discuss some applications to the study of the Hagedorn transition in string theory in the context of Matrix strings and speculate on the possible physical meaning of the transition. 
  We give a basic account of supersymmetric open strings and D-branes using the Green-Schwarz formalism, obtaining a manifestly spacetime supersymmetric description of their spectrum. In addition we discuss a mechanism whereby some of the D-brane states are projected out and which can lead to chiral quantum field theories on the brane. 
  Various exact two-dimensional conformal field theories with AdS_{2d+1} target space are constructed. These models can be solved using bosonization techniques. Some examples are presented that can be used in building perturbative superstring theories with AdS backgrounds, including AdS_5. 
  We consider non-perturbative six dimensional N=1 space-time supersymmetric orientifolds of Type IIB on K3 with non-trivial NS-NS B-flux. All of these models are non-perturbative in both orientifold and heterotic pictures. Thus, some states in such compactifications arise in ``twisted'' open string sectors which lack world-sheet description in terms of D-branes. We also discuss their dual F-theory compactifications on certain Voisin-Borcea orbifolds. In particular, the explicit construction of non-perturbative K3 orientifolds with NS-NS B-flux gives additional evidence for the conjectured extension of Nikulin's classification in the context of Voisin-Borcea orbifolds. 
  We study the relation between the D8-branes wrapped on an orientable compact manifold $W$ in a massive Type IIA supergravity background and the M9-branes wrapped on a compact manifold $Z$ in a massive d=11 supergravity background from the K-theoretic point of view. By speculating on the use of the dimensional reduction to relate the two theories in different dimensions and by interpreting the D8-brane charges as elements of $K_0 (C(W))$ and the (inequivalent classes of) spaces of gauge fields on the M9-branes as the elements of $K_0(C(Z)\times_{\bar{k}^*}G}$ a connection between charges and gauge fields is argued to exist. This connection is realized as a map between the corresponding algebraic K-theory groups. 
  Non-equilibrium quantum field theory studies time dependence of processes which are not available for the S-matrix description. One of the new methods of investigation in non-equilibrium quantum theory is the stochastic limit method. This method is an extension of the works by Bogoliubov, van Hove and Prigogine and it permits to study not only the system but also the reservoir degrees of freedom. We consider the stochastic limit of translation invariant Hamiltonians in quantum field theory and show that the master field satisfies a new type of commutation relations, the so called entangled (or interacting) commutation relations. These relations extend the interacting Fock relations established earlier in non-relativistic QED and the free (or Boltzmann) commutation relations which have been found in the large N limit of QCD. As an application of the stochastic limit method we consider the photon splitting cascades in magnetic field and show that photons in cascades form entangled states ("triphons") and they obey not Bose but a new type of statistics corresponding to the entangled commutation relations. 
  We show that T-duality can be broken by nonperturbative effects in string coupling. The T-duality in question is that of the 2-torus when the heterotic string is compactified on K3 x T2. This case is compared carefully to a situation where T-duality appears to work. A holonomy argument is presented to show that T-dualities (and general U-dualities) should only be expected for large amounts of supersymmetry. This breaking of R <-> 1/R symmetry raises some interesting questions in string theory which we discuss. Finally we discuss how the classical modular group of a 2-torus appears to be broken too. 
  If the AdS/CFT correspondence is valid in the Coulomb branch, the potential between waves on a pair of test branes in the bulk should be reproduced by the relevant Yang-Mills theory effective action on the boundary. Earlier work has provided evidence for this in the case of constant gauge field brane waves. In this paper we provide concrete evidence for an earlier proposal that the effects of exchange of supergravity modes with nonzero momentum in the brane directions are encoded in certain terms involving derivatives of the field strength in the gauge theory effective action. We explicitly calculate the force quadratic in the field strengths coming from the exchange of non-zero momentum two form fields between two 3-branes in $AdS_5 \times S^5$ to lowest nontrivial order in the momentum. We show that this is exactly the same as that between the branes living in flat space. The result is in agreement with the gauge theory effective action and consistent with the non-renormalization property of this term. We comment on the relationship of other ``acceleration'' terms in the SYM effective action with quantities in supergravity. 
  Symmetry breaking boundary conditions for WZW theories are discussed. We derive explicit formulae for the reflection coefficients in the presence of boundary conditions that preserve only an orbifold subalgebra with respect to an involutive automorphism of the chiral algebra. The characters and modular transformations of the corresponding orbifold theories are computed. Both inner and outer automorphisms are treated. 
  The zero-point energy of a massless fermion field in the interior of two parallel plates in a D-dimensional space-time at zero temperature is calculated. In order to regularize the model, a mix between dimensional and zeta-function regularization procedure is used and it is found that the regularized zero-point energy density is finite for any number of space-time dimensions. We present a general expression for the Casimir energy for the fermionic field in such a situation. 
  Instead of the infinitesimal extrinsic and intrinsic perturbations on strings, considered so far, we discuss the evolution and propagation of finite-amplitude perturbations. Those intrinsic perturbations may result in appearance of stable discontinuities similar to the shock waves. 
  New string dynamics is formulated on the basis of the extended set of supergauge constraints including not only supergauge Virasoro conditions but also nilpotent supercurrent constraints . This approach arises from a natural generalization of the classical operator many-string vertices. The formulation of this model leads to three two-dimensional surfaces for description of hadron strings. It gives some dynamical generalization of Chan-Paton factor for string amplitudes in terms of operator vertices. Supersymmetry on the 2-d world surface is present but ten-dimensional supersymmetry is absent.   In this approach two-dimensional fermion string fields make it possible to give a unified description of hadron and lepton degrees of freedom and of its dynamics. This model allows to solve the problem of elimination of the most part of parity twins in the baryon spectrum.   One-loop (and many-loops perhaps) amplitudes in this model are finite due to the extended set of supergauge constraints and to the significant excess of the total number of fermion two-dimensional fields over the number of boson 2-d fields. 
  It has been recently proposed to embed the standard model in a conformal gauge theory to resolve the hierarchy problem, and to avoid assuming either grand unification or low-energy supersymmetry. By model building based on string-field duality we show how to maintain the successful prediction of an electroweak mixing angle with $sin^2\theta \simeq 0.231$ in conformal gauge theories with three chiral families. 
  We calculate one-loop 2-point tachyon amplitudes in unoriented open-closed string field theory, and determine all the coupling constants of the interaction vertices in the theory. It is shown that the planar, nonplanar and nonorientable amplitudes are all reproduced correctly, and nontrivial consistencies between the determined coupling constants are observed. The necessity for the gauge group to be SO(2^{13}=8192) is also reconfirmed. 
  By a toroidal compactification with a vortex like configuration of tachyon fields, the unoriented bosonic string in 26 dimensions becomes equivalent to 10 dimensional string theory with gauge group $SO(32)\times G$ where $G$ has rank 16. The reduction from SO(8192) to SO(32) is induced by the action of non-abelian Wilson lines which twists the tachyon. The additional enhanced gauge symmetry $G$ appears from closed string sector by Frenkel-Kac construction. We also examine the consistency of the tachyon condensation. 
  Conformally compactified phase space is conceived as an automorphism space for the global action of the extended conformal group. Space time and momentum space appear then as conformally dual, that is conjugate with respect to conformal reflections. If now the former, as generally agreed, is appropriate for the description of classical mechanics in euclidean geometrical form, then the latter results appropriate for the description of quantum mechanics in spinor geometrical form. In such description, fermion multiplets will naturally appear as consequence of higher symmetries and furthermore, the euclidean geometry, bilinearly resulting from that of spinors, will a priori guarantee the absence of ultraviolet divergences when dealing with quantum field theories. Some further possible consequences of conformal reflections of interest for physics, are briefly outlined. 
  We illustrate the Dirichlet prescription of the AdS/CFT correspondence using the example of a massive scalar field and argue that it is the only entirely consistent regularization procedure known so far. Using the Dirichlet prescription, we then calculate the divergent terms for gravity in the cases $d=2,4,6$, which give rise to the Weyl anomaly in the boundary conformal field theory. 
  Recent indications of neutrino oscillations raise the question of the possibility of incorporating massive neutrinos in the formulation of the Standard Model (SM) within noncommutative geometry (NCG). We find that the NCG requirement of Poincare duality constrains the numbers of massless quarks and neutrinos to be unequal unless new fermions are introduced. Possible scenarios in which this constraint is satisfied are discussed. 
  The two-point Green function of a local operator in CFT corresponding to a massive symmetric tensor field on the AdS background is computed in the framework of the AdS/CFT correspondence. The obtained two-point function is shown to coincide with the two-point function of the graviton in the limit when the mass vanishes. 
  A new method is discussed which vastly simplifies one of the two integrals over AdS(d+1) required to compute exchange graphs for 4-point functions of scalars in the AdS/CFT correspondence. The explicit form of the bulk-to-bulk propagator is not required. Previous results for scalar, gauge boson and graviton exchange are reproduced, and new results are given for massive vectors. It is found that precisely for the cases that occur in the AdS(5) X S(5) compactification of Type IIB supergravity, the exchange diagrams reduce to a finite sum of graphs with quartic scalar vertices. The analogous integrals in n-point scalar diagrams for n>4 are also evaluated. 
  In this note we derive an explicit modular invariant formula for the two loop 4-point amplitude in superstring theory in terms of a multiple integral (7 complex integration variables) over the complex plane which is shown to be convergent. We consider in particular the case of the leading term for vanishing momenta of the four graviton amplitude, which would correspond to the two-loop correction of the R^4 term in the effective Action. The resulting expression is not positive definite and could be zero, although we cannot see that it vanishes. 
  We study the problem of decoupling fermion fields in 1+1 and 2+1 dimensions, in interaction with a gauge field, by performing local transformations of the fermions in the functional integral. This could always be done if singular (large) gauge transformations were allowed, since any gauge field configuration may be represented as a singular pure gauge field. However, the effect of a singular gauge transformation of the fermions is equivalent to the one of a regular transformation with a non-trivial action on the spinorial indices. For example, in the two dimensional case, singular gauge transformations lead naturally to chiral transformations, and hence to the usual decoupling mechanism based on Fujikawa Jacobians. In 2+1 dimensions, using the same procedure, different transformations emerge, which also give rise to Fujikawa Jacobians. We apply this idea to obtain the v.e.v of the fermionic current in a background field, in terms of the Jacobian for an infinitesimal decoupling transformation, finding the parity violating result. 
  Some N=1 gauge theories, including SQED and N_F=1 SQCD have the property that, for arbitrary superpotentials, all stationary points of the potential V = F+D are D-flat. For others, stationary points of V are complex gauge transformationss of D-flat configurations. As an implication, the technique to parametrize the moduli space of supersymmetric vacua in terms of a set of basic holomorphic G invariants can be extended to non-supersymmetric vacua. A similar situation is found in non-gauge theories with a compact global symmetry group. 
  We review some of the uses of Whitham hierarchies in the context of the theory of the prepotential in N=2 supersymmetric gauge theories. We focus on the structure of the contact terms in the twisted topological theory, and on the connection between Whitham hierarchies and the u-plane integrals for higher rank gauge groups, trying to put together the different approaches involved in this connection. We also review two other uses of the Whitham hierarchies: the interpretation of the slow times as supersymmetry breaking parameters, and the new techniques to extract instanton corrections using the RG equations written in terms of theta functions. 
  A solution to the long-standing problem of identifying the conformal field theory governing the transition between quantized Hall plateaus of a disordered noninteracting 2d electron gas, is proposed. The theory is a nonlinear sigma model with a Wess-Zumino-Novikov-Witten term, and fields taking values in a Riemannian symmetric superspace based on H^3 x S^3. Essentially the same conformal field theory appeared in very recent work on string propagation in AdS_3 backgrounds. We explain how the proposed theory manages to obey a number of tight constraints, two of which are constancy of the partition function and noncriticality of the local density of states. An unexpected feature is the existence of a truly marginal deformation, restricting the extent to which universality can hold in critical quantum Hall systems. The marginal coupling is fixed by matching the short-distance singularity of the conductance between two interior contacts to the classical conductivity sigma_xx = 1/2 of the Chalker-Coddington network model. For this value, perturbation theory predicts a critical exponent 2/pi for the typical point-contact conductance, in agreement with numerical simulations. The irrational exponent is tolerated by the fact that the symmetry algebra of the field theory is Virasoro but not affine Lie algebraic. 
  It is known that if we compactify D0-branes on a torus with constant B-field, the resulting theory becomes SYM theory on a noncommutative dual torus. We discuss the extension to the case of a H-field background. In the case of constant H-field on a three-torus, we derive the constraints to realize this compactification by considering the correspondence to string theory. We carry out this work as a first step to examine the possibility to describe transverse M5-branes in Matrix theory. 
  Recently the space-time configurations of a set of non-threshold bound states, called the (F, Dp) bound states, have been constructed explicitly for every $p$ with $2 \le p \le 7$ in both type IIA (for $p$ even) and type IIB (for $p$ odd) string theories by the present authors. By making use of the SL(2, Z) symmetry of type IIB theory we construct a more general SL(2, Z) invariant bound state of the type ((F, D1), (NS5, D5)) in this theory from the (F, D5) bound state. There are actually an infinite number of $(m,n)$ strings forming bound states with $(m',n')$ 5-branes, where strings lie along one of the spatial directions of the 5-branes. By applying T-duality along one of the transverse directions we also construct the bound state ((F, D2), (KK, D6)) in type IIA string theory. Then we give a list of possible bound states which can be obtained from these newly constructed bound states by applying T-dualities along the longitudinal directions as well as S-dualities to those in type IIB theory. 
  A brief review on the progress made in the study of Chern-Simons gauge theory since its relation to knot theory was discovered ten years ago is presented. Emphasis is made on the analysis of the perturbative study of the theory and its connection to the theory of Vassiliev invariants. It is described how the study of the quantum field theory for three different gauge fixings leads to three different representations for Vassiliev invariants. Two of these gauge fixings lead to well known representations: the covariant Landau gauge corresponds to the configuration space integrals while the non-covariant light-cone gauge to the Kontsevich integral. The progress made in the analysis of the third gauge fixing, the non-covariant temporal gauge, is described in detail. In this case one obtains combinatorial expressions, instead of integral ones, for Vassiliev invariants. The approach based on this last gauge fixing seems very promising to obtain a full combinatorial formula. We collect the combinatorial expressions for all the Vassiliev invariants up to order four which have been obtained in this approach. 
  The topological structure of the Nieh-Yan form in 4-dimensional manifold is given by making use of the decomposition of spin connection. The case of the generalized Nieh-Yan form on $2^d$-dimensional manifold is discussed with an example of 8-dimensional case studied in detail. The chiral anomaly with nonvanishing torsion is studied also. The further contributions from torsional part to chiral anomaly are found coming from the zeroes of some fields under pure gauge condition. 
  By means of the heterotic/type IIB duality, we study properties of junctions on backgrounds with a positively charged orientifold seven-plane and D-branes, which are expected to give seven dimensional Sp(r) gauge theories. We give a modified BPS condition for the junctions and show that it reproduces the adjoint representation of the Sp(r) group. 
  The infinite dimensional generalization of the quantum mechanics of extended objects, namely, the quantum field theory of extended objects is employed to address the hitherto nonrenormalizable gravitational interaction following which the cosmological constant problem is addressed. The response of an electron to a weak gravitational field (linear approximation) is studied and the order $\alpha$ correction to the magnetic gravitational moment is computed. 
  Radiative corrections arising from the axial coupling of charged fermions to a constant vector b_\mu can induce a Lorentz- and CPT-violating Chern-Simons term in the QED action. We calculate the exact one-loop correction to this term keeping the full b_\mu dependence, and show that in the physically interesting cases it coincides with the lowest-order result. The effect of regularization and renormalization and the implications of the result are briefly discussed. 
  We study a holomorphic effective potential $W_{eff}(\Phi)$ in chiral superfield model defined in terms of arbitrary k\"{a}hlerian potential $K(\bar{\Phi},\Phi)$ and arbitrary chiral potential $W(\Phi)$. Such a model naturally arises as an ingredient of low-energy limit of superstring theory and it is called here the general chiral superfield model. Generic procedure for calculating the chiral loop corrections to effective action is developed. We find lower two-loop correction in the form $W^{(2)}_{eff}(\Phi)= 6/(4\pi)^4 \bar{W}^{'''2}(0){(\frac{W^{''}(\Phi)}{K^2_{\Phi\bar{\Phi}(0,\Phi)}})}^3$ where $K_{\Phi\bar{\Phi}}(0,\Phi)=\frac{\partial^2 K(\bar{\Phi},\Phi)} {\partial\Phi\partial\bar{\Phi}}|_{\bar{\Phi}=0}$ and $\zeta(x)$ be Riemannian zeta-function. This correction is finite at any $K(\bar{\Phi},\Phi), W(\Phi)$. 
  The Kac determinant for the Topological N=2 superconformal algebra is presented as well as a detailed analysis of the singular vectors detected by the roots of the determinants. In addition we identify the standard Verma modules containing `no-label' singular vectors (which are not detected directly by the roots of the determinants). We show that in standard Verma modules there are (at least) four different types of submodules, regarding size and shape. We also review the chiral determinant formula, for chiral Verma modules, adding new insights. Finally we transfer the results obtained to the Verma modules and singular vectors of the Ramond N=2 algebra, which have been very poorly studied so far. This work clarifies several misconceptions and confusing claims appeared in the literature about the singular vectors, Verma modules and submodules of the Topological N=2 superconformal algebra. 
  The decoupling limit of the D1-D5 system compactified on T^4\times S^1 has a rich spectrum of U(1) charged excitations. Even though these states are not BPS in the limit, BPS considerations determine the mass and the semiclassical entropy for a given charge vector. The dependence of the mass formula on the compactification moduli situates the symmetric orbifold Sym^N(T^4) x T^4 conformal field theory in the moduli space. A detailed analysis of the global identifications of the moduli space yields a picture of multiple weak-coupling limits - one for each factorization of N into D1 and D5 charges d1 and d5=N/d1 - joined through regions of strong coupling in the CFT moduli space. 
  The paper was withdrawn by the authors. 
  The moduli space of charge k SU(2) BPS monopoles is diffeomorphic to the moduli space of degree k rational maps between Riemann spheres. In this note we describe a numerical algorithm to compute the monopole fields and energy density from the rational map. The results for some symmetric examples are presented. 
  As a test of the conjectured QCD/supergravity duality, we consider mass gaps in the supergravity construction of QCD_2. We find a mass gap in the dual field theory both when using non-rotating and rotating black D2-branes as backgrounds in the supergravity construction of QCD_2. So, since pure QCD_2 does not have a mass gap, the dual field theory of the supergravity construction of QCD_2 cannot be pure QCD_2. Considering the mass scales in the dual field theory of the supergravity construction of QCD_2, we find that this is explainable both in the case of the non-rotating background and of the rotating background. In particular, the mass gap in the case of the rotating background can be explained using results of the large angular momentum limit of euclidean rotating branes, obtained recently by Cvetic and Gubser. We furthermore remark on the possible implications for the mass gaps in the supergravity constructions of QCD_3 and QCD_4. 
  The Born-Infeld theory of a toroidal D3-brane has an SL(5,Z) U-duality symmetry. We investigate how this symmetry is reflected in the supersymmetry algebra. We propose an action of the group on the gauge theory fields in the BPS sector by introducing an extra field together with an additional symmetry, and argue for the U-invariance of the degeneracies of the BPS spectrum. 
  The central charge for the Seiberg-Witten low-energy effective Action is computed using Noether supercharges. A reliable method to construct supersymmetric Noether currents is presented. 
  A parametrization of the Hamiltonian of the generalized Witten model of the SUSY QM by a single arbitrary function in d=1 has been obtained for an arbitrary number of the supersymmetries N. Possible applications of this formalism have been discussed. It has been shown that the N=1 and 2 conformal SUSY QM is generalized for any N. 
  We compute part of the superfield in terms of the component fields of 11-dimensional on-shell supergravity by using `Gauge completion' in 2nd-order formalism. The result is the same as was derived recently in 1.5-order formalism by B. de Wit, K. Peeters and J. Plefka. We use 2nd-order formalism because in order to hold $\kappa $-invariance generally 2nd-order formalism is more hopeful and simpler than 1.5-order formalism. 
  We solve a long-standing problem in particle physics: that of deriving the Deep Inelastic structure functions of the proton from the fundamental theory of strong interactions, Quantum ChromoDynamics (QCD). In the Bjorken limit, the momenta of the constituents of the proton (the partons) can be assumed to be in a two-dimensional plane in Minkowski space: a dimensional reduction of QCD to two space-time dimensions. Two dimensional QCD is then shown to be equivalent for all energies and values of number of colors \m{N} to a new theory of hadrons, Quantum HadronDynamics (QHD). The phase space of QHD is the Grassmannian (set of subspaces) of the complex Hilbert space L^2(R). The natural symplectic form along with a hamiltonian define a classical dynamical system, which is equivalent to the large N limit of QCD. 't Hooft's planar limit is the linear approximation to our theory: we recover his integral equation for the meson spectrum but also all the interactions of the mesons. The Grassmannian is a union of connected components labelled by an integer (the renormalized dimension of the subspace) which has the physical meaning of baryon number. The proton is the topological soliton: the minimum of the energy in the sector with baryon number one gives the structure functions of the proton. We solve the resulting integral equations numerically; the agreement with experimental data is quite good for values of the Bjorken variable x>0.2. 
  We propose a method to investigate the conservation of brane charges at the intersection of two or more branes using the Thom classes of their normal bundles. In particular we find a relation between the charge of the branes involved in the configuration and the charge of the defects on the branes due to the intersection. We also explore the applications of our method for various brane intersections in type II strings and M-theory. 
  The vortex-like solution to the non-linear field equations in a two-dimensional SU(2) gauge theory with the Chern-Simons mass term is found at high temperature. It is derived from the effective Lagrangian including the leading order finite temperature corrections. The discovered field configuration possesses the finite energy and the quantized magnetic flux. At the centre of the vortex the point charge is located which is surrounded by the distributed charge of the opposite sign and the vortex is neutral as a whole. At high temperature the energy of the vortex is negative and it corresponds to the ground state. The derived solution is considered to be a result of heating the lattice vacuum structure formed at zero temperature. 
  We show that there exists a consistent truncation of 11 dimensional supergravity to the 'massless' fields of maximal (N=4) 7 dimensional gauged supergravity. We find the complete expressions for the nonlinear embedding of the 7 dimensional fields into the 11 dimensional fields, and check them by reproducing the d=7 susy transformation laws from the d=11 laws in various sectors. In particular we determine explicitly the matrix U which connects the Killing spinors to the gravitinos in the KK ansatz, and the dependence of the 4-index field strength on the scalars. This is the first time a complete nonlinear KK reduction of the original d=11 supergravity on a nontrivial compact space has been explicitly given. We need a first order formulation for the 3 index tensor field $A_{\Lambda\Pi\Sigma}$ in d=11 to reproduce the 7 dimensional result. The concept of 'self-duality in odd dimensions' is thus shown to originate from first order formalism in higher dimensions. For the AdS-CFT correspondence, our results imply that one can use 7d gauged supergravity (without further massive modes) to compute certain correlators in the d=6 (0,2) CFT at leading order in N. This eliminates an ambiguity in the formulation of the correspondence. 
  The one-loop renormalization of a general chiral gauge theory without scalar and Majorana fields is fully worked out within Breitenlohner and Maison dimensional renormalization scheme. The coefficients of the anomalous terms introduced in the Slavnov-Taylor equations by the minimal subtraction algorithm are calculated and the asymmetric counterterms needed to restore the BRS symmetry, if the anomaly cancellation conditions are met, are computed. The renormalization group equation and its coefficients are worked out in the anomaly free case. The computations draw heavily from the existence of action principles and BRS cohomology theory. 
  We derive several results concerning non-perturbative renormalization in the spherical field formalism. Using a small set of local counterterms, we are able to remove all ultraviolet divergences in a manner such that the renormalized theory is finite and translationally invariant. As an explicit example we consider massless phi^4 theory in four dimensions. 
  In this note we discuss D-branes on T^4/Z_2 using the boundary states formalism. Explicit formulas for the untwisted boundary states inherited from the underlying T^4 and twisted states corresponding to branes wrapping collapsed 2-cycles at the orbifold singularities are given. The exact CFT description of the orbifold makes it possible to study how the boundary states transform under R_i -> 1/R_i transformation on all directions of the underlying T^4. We compare their transformation law with results obtained from world volume considerations. 
  In this paper we construct explicitly an infinite number of Hopfions (static, soliton solutions with non-zero Hopf topological charges) within the recently proposed 3+1-dimensional, integrable and relativistically invariant field theory. Two integers label the family of Hopfions we have found. Their product is equal to the Hopf charge which provides a lower bound to the soliton's finite energy. The Hopfions are constructed explicitly in terms of the toroidal coordinates and shown to have a form of linked closed vortices. 
  The decay of rotated brane configurations and the corresponding condensation of tachyons is discussed. In a certain IIB orbifold case a heuristic argument about the mass of the state living on the fixed plane is made. When the rotation angle is $\pi$ this mass agrees with that obtained by Sen. 
  We investigate a family of solutions of Type IIb supergravity which asymptotically approach AdS_5 X S^5 but contain a non-constant dilaton and volume scalar for the five-sphere. These solutions preserve an SO(1,3) X SO(6) symmetry. We discuss the solution in the context of the AdS/CFT correspondence, and we find that as well as running coupling from the nontrivial dilaton, the corresponding field theory has no supersymmetry and displays confinement at least for a certain range of parameters. 
  We compute the spin of both the topological and nontopological solitons of the Chern - Simons - Higgs model by using our approach based on constrained analysis. We also propose an extension of our method to the non - relativistic Chern - Simons models. The spin formula for both the relativistic and nonrelativistic theories turn out to be structurally identical. This form invariance manifests the topological origin of the Chern - Simons term responsible for inducing fractional spin. Also, some comparisons with the existing results are done. 
  The recently introduced quantum antibracket is further generalized allowing for the defining odd operator Q to be arbitrary. We give exact formulas for higher quantum antibrackets of arbitrary orders and their generalized Jacobi identities. Their applications to BV-quantization and BFV-BRST quantization are then reviewed including some new aspects. 
  We calculate the energy of a Yang-Mills vortex as function of its magnetic flux or, else, of the Wilson loop surrounding the vortex center. The calculation is performed in the 1-loop approximation. A parallel with a potential as function of the Polyakov line at nonzero temperatures is drawn. We find that quantized Z(2) vortices are dynamically preferred though vortices with arbitrary fluxes cannot be ruled out. 
  It is shown that there are no nilpotent invariants in N=4 analytic superspace for $n\leq4$ points. It is argued that there is (at least) one such invariant for n=5 points which is not invariant under U(1)_Y. The consequences of these results are that the n=2 and 3 point correlation functions of the N=4 gauge-invariant operators which correspond to KK multiplets in AdS supergravity are given exactly by their tree level expressions, the 4 point correlation functions of such operators are invariant under U(1)_Y and correlation functions with $n\geq 4$ points have non-trivial dependence on the Yang-Mills coupling constant. 
  Using the method of renormalization group, we improve the two-loop effective potential of the massive $\phi^4$ theory to obtain the next-next-to-leading logarithm correction in the $\bar{MS}$ scheme. Our result well reproduces the next-next-to-leading logarithm parts of the ordinary loop expansion result known up to the four-loop order. 
  High energy fixed angle scattering is studied in matrix string theory. The saddle point world sheet configurations, which give the dominant contributions to the string theory amplitude, are taken as classical backgrounds in matrix string theory. A one loop fluctuation analysis about the classical background is performed. An exact treatment of the fermionic and bosonic zero modes is shown to lead to all of the expected structure of the scattering amplitude. The ten dimensional Lorentz invariant kinematical structure is obtained from the fermion zero modes, and the correct factor of the string coupling constant is obtained from the abelian gauge field zero modes. Up to a numerical factor we reproduce, from matrix string theory, the high energy limit of the tree level, four graviton scattering amplitude. 
  The renormalization group (RG) method as a powerful tool for reduction of evolution equations is formulated in terms of the notion of invariant manifolds. We start with derivation of an exact RG equation which is analogous to the Wilsonian RG equations in statistical physics and quantum field theory. It is clarified that the perturbative RG method constructs invariant manifolds successively as the initial value of evolution equations, thereby the meaning to set $t_0=t$ is naturally understood where $t_0$ is the arbitrary initial time. We show that the integral constants in the unperturbative solution constitutes natural coordinates of the invariant manifold when the linear operator $A$ in the evolution equation has no Jordan cell; when $A$ has a Jordan cell, a slight modification is necessary because the dimension of the invariant manifold is increased by the perturbation. The RG equation determines the slow motion of the would-be integral constants in the unperturbative solution on the invariant manifold. We present the mechanical procedure to construct the perturbative solutions hence the initial values with which the RG equation gives meaningful results. The underlying structure of the reduction by the RG method as formulated in the present work turns out to completely fit to the universal one elucidated by Kuramoto some years ago. We indicate that the reduction procedure of evolution equations has a good correspondence with the renormalization procedure in quantum field theory; the counter part of the universal structure of reduction elucidated by Kuramoto may be the Polchinski's theorem for renormalizable field theories. We apply the method to interface dynamics such as kink-anti-kink and soliton-soliton interactions in the latter of which a linear operator having a Jordan-cell structure appears. 
  We discuss the quantum corrections to thermodynamics (and geometry) of S(A)dS BHs using large $N$ one-loop anomaly induced effective action for dilaton coupled matter (scalars and spinors). It is found the temperature, mass and entropy with account of quantum effects for multiply horizon SdS BH and SAdS BH what also gives the corresponding expressions for their limits: Schwarzschild and de Sitter spaces. In the last case one can talk about quantum correction to entropy of expanding Universe.   The anomaly induced action under discussion corresponds to 4d formulation (s-wave approximation, 4d quantum matter is minimal one) as well as 2d formulation (complete effective action, 2d quantum matter is dilaton coupled one). Hence, most of results are given for the same gravitational background with interpretation as 4d quantum corrected BH or 2d quantum corrected dilatonic BH. Quantum aspects of thermodynamics of 4d 't Hooft BH model are also considered. 
  This paper describes perturbative framework, on the basis of the closed-time-path formalism, in terms of quasiparticle picture for studying quasiuniform relativistic quantum field systems near equilibrium and nonequilibrium quasistationary systems. Two calculational schemes are introduced, the one is formulated on the basis of the initial-particle distribution function and the one is formulated on the basis of the ``physical''-particle distribution function. It is shown that both schemes are equivalent and lead to a generalized kinetic or Boltzmann equation. Concrete procedure of computing a generic amplitude is presented. 
  We study the scattering of three gravitons in M-atrix theory at finite N. With a specific choice of the background we obtain the complete result up to two loops. The contributions from three-body forces agree with the ones presented in recent papers. We extend the calculation and evaluate the two-body exchanges as well. Such terms, somewhat difficult to isolate and compute, had been neglected so far in the existing literature. We show that the result we have obtained from M-atrix theory precisely matches the result from one-particle reducible tree diagrams in eleven-dimensional supergravity . 
  We extend the results found for Matrix String Theory to Heterotic Matrix String Theory, i.e. to a 2d O(N) SYM theory with chiral (anomaly free) matter and N=(8,0) supersymmetry. We write down the instanton equations for this theory and solve them explicitly. The solutions are characterized by branched coverings of the basis cylinder, i.e. by compact Riemann surfaces with punctures. We show that in the strong coupling limit the action becomes the heterotic string action plus a free Maxwell action. Moreover the amplitude based on a Riemann surface with p punctures and h handles is proportional to g^{2-2h-p}, as expected for the heterotic string interaction theory with string coupling g_s=1/g. 
  A systematic method to calculate the low energy spectrum of SU(2) Yang-Mills quantum mechanics with high precision is given and applied to obtain the energies of the groundstate and the first few excited states. 
  We study fully localized BPS brane solutions in classical supergravity using a perturbative approach to the coupled Born-Infeld/bulk supergravity system. We derive first order bulk supergravity fields for world-volume solitons corresponding to intersecting M2-branes and to a fundamental string ending on a D3-brane. One interesting feature is the appearance of certain off-diagonal metric components and corresponding components of the gauge potentials. Making use of a supersymmetric ansatz for the exact fields, we formulate a perturbative expansion which applies to $M2 \perp M2 (0), M5 \perp M5 (3)$ and $Dp \perp Dp (p-2)$ intersections. We find that perturbation theory qualitatively distinguishes between certain of these cases: perturbation theory breaks down at second order for intersecting M2-branes and D$p$-branes with $p\le 3$ while it is well behaved, at least to this order, for the remaining cases. This indicates that the behavior of the full non-linear intersecting Dp-brane solutions may be qualitatively different for $p \le 3$ than for $p \ge 4$, and that fully localized asymptotically flat solutions for $p \le 3$ may not exist. We discuss the consistency of these results with world-volume field theory properties. 
  We calculate the induced Lorentz- and CPT-violating Chern-Simons term arising from the Lorentz- and CPT-violating sector of quantum electrodynamics with a $b_\mu\bar{\psi}\gamma^\mu\gamma_5\psi$ term. The result to all orders in $b$ coincides with the previous linear-in-$b$ calulation by Chung and Oh [hep-th/9812132] as well as Jackiw and Kosteleck\'{y} [Phys. Rev. Lett. {\bf 82}, 3572 (1999)], since all higher order terms vanish. 
  Kaluza-Klein sphere reductions of supergravities that admit AdS x Sphere vacuum solutions are believed to be consistent. The examples include the S^4 and S^7 reductions of eleven-dimensional supergravity, and the S^5 reduction of ten-dimensional type IIB supergravity. In this paper we provide evidence that sphere reductions of supergravities that admit instead Domain-wall x Sphere vacuum solutions are also consistent, where the background can be viewed as the near-horizon structure of a dilatonic p-brane of the theory. The resulting lower-dimensional theory is a gauged supergravity that admits a domain wall, rather than AdS, as a vacuum solution. We illustrate this consistency by taking the singular limits of certain modulus parameters, for which the original S^n compactifying spheres (n=4,5 or 7) become S^p x R^q, with p=n-q<n. The consistency of the S^4, S^7 and S^5 reductions then implies the consistency of the S^p reductions of the lower-dimensional supergravities. In particular, we obtain explicit non-linear ansatze for the S^3 reduction of type IIA and heterotic supergravities, restricting to the U(1)^2 subgroup of the SO(4) gauge group of S^3. We also study the black hole solutions in the lower-dimensional gauged supergravities with domain-wall backgrounds. We find new domain-wall black holes which are not the singular-modulus limits of the AdS black holes of the original theories, and we obtain their Killing spinors. 
  We list the spectrum generating algebras for string theory and M-theory compactified on various backgrounds of the form $AdS_{d+1} \times S^n$. We identify the representations of these algebras which make up the classical supergravity spectra and argue for the presence of these spectrum generating algebras in the classical string/M-theory. We also discuss the role of the spectrum generating algebras on the conformal field theory side. 
  Compact Type IIB D=4, N=1 orientifolds have certain U(1) sigma model symmetries at the level of the effective Lagrangian. These symmetries are generically anomalous. We study the particular case of $Z_N$ orientifolds and find that these anomalies may be cancelled by a generalized Green-Schwarz mechanism. This mechanism works by the exchange of twisted RR-fields associated to the orbifold singularities and it requires the mixing between twisted and untwisted moduli of the orbifold. As a consequence, the Fayet-Iliopoulos terms which are present for the gauged anomalous U(1)'s of the models get an additional untwisted modulus dependent piece at the tree level. 
  We study the near horizon limit of a four dimensional extreme rotating black hole. The limiting metric is a completely nonsingular vacuum solution, with an enhanced symmetry group SL(2,R) x U(1). We show that many of the properties of this solution are similar to the AdS_2 x S^2 geometry arising in the near horizon limit of extreme charged black holes. In particular, the boundary at infinity is a timelike surface. This suggests the possibility of a dual quantum mechanical description. A five dimensional generalization is also discussed. 
  We study dynamical gauge symmetry breaking via compactified space in the framework of SU(N) gauge theory in M^{d-1}\times S^1 (d=4,5,6) space-time. In particular, we study in detail the gauge symmetry breaking in SU(2) and SU(3) gauge theories when the models contain both fundamental and adjoint matter. As a result, we find that any pattern of gauge symmetry breaking can be realized by selecting an appropriate set of numbers (Nf,Nad) in these cases. This is achieved without tuning boundary conditions of the matter fields. As a by-product, in some cases we obtain an effective potential which has no curvature at the minimum, thus leading to massless Higgs scalars, irrespective of the size of the compactified space. 
  It is known that the quantization of a system defined on a topologically non-trivial configuration space is ambiguous in that many inequivalent quantum systems are possible. This is the case for multiply connected spaces as well as for coset spaces. Recently, a new framework for these inequivalent quantizations approach has been proposed by McMullan and Tsutsui, which is based on a generalized Dirac approach. We employ this framework for the quantization of the Yang-Mills theory in the simplest fashion. The resulting inequivalent quantum sectors are labelled by quantized non-dynamical topological charges. 
  It is shown that the modulus of any graded or, more generally, twisted KMS functional of a C*-dynamical system is proportional to an ordinary KMS state and the twist is weakly inner in the corresponding GNS-representation. If the functional is invariant under the adjoint action of some asymptotically abelian family of automorphisms, then the twist is trivial. As a consequence, such functionals do not exist for supersymmetric C*-dynamical systems. This is in contrast with the situation in compact spaces where super KMS functionals occur as super-Gibbs functionals. 
  An infinite series of Grassmann-odd and Grassmann-even flow equations is defined for a class of supersymmetric integrable hierarchies associated with loop superalgebras. All these flows commute with the mutually commuting bosonic ones originally considered to define these hierarchies and, hence, provide extra fermionic and bosonic symmetries that include the built-in N=1 supersymmetry transformation. The corresponding non-local conserved quantities are also constructed. As an example, the particular case of the principal supersymmetric hierarchies associated with the affine superalgebras with a fermionic simple root system is discussed in detail. 
  We study, using the dual AdS description, the vacua of field theories where some of the gauge symmetry is broken by expectation values of scalar fields. In such vacua, operators built out of the scalar fields acquire expectation values, and we show how to calculate them from the behavior of perturbations to the AdS background near the boundary. Specific examples include the ${\cal N}=4$ SYM theory, and theories on D3 branes placed on orbifolds and conifolds. We also clarify some subtleties of the AdS/CFT correspondence that arise in this analysis. In particular, we explain how scalar fields in AdS space of sufficiently negative mass-squared can be associated with CFT operators of {\it two} possible dimensions. All dimensions are bounded from below by $(d-2)/2$; this is the unitarity bound for scalar operators in $d$-dimensional field theory. We further argue that the generating functional for correlators in the theory with one choice of operator dimension is a Legendre transform of the generating functional in the theory with the other choice. 
  We study the Yangian symmetry of the multicomponent Quantum Nonlinear Schrodinger hierarchy in the framework of the Quantum Inverse Scattering Method. We give an explicit realization of the Yangian generators in terms of the deformed oscillators algebra which naturally occurs in this framework. 
  Generalising a result of classical mechanics an infinite set of conserved quantities can be found for the bare equations of motion describing the evolution of a scalar field in out of equilibrium quantum field theory, in the large N approximation, with initial conditions corresponding to a thermal system of the free Hamiltonian. Using these new conserved quantities, sum-rules relating integrals over the mode-functions (momenta) can be derived. More, the corresponding renormalised quantities can also be computed out thus giving information about the evolution of the already known renormalised equations; finally it is also possible to write a renormalised version of the sum-rules. 
  The microscopic spectrum of the QCD Dirac operator is shown to obey random matrix model statistics in the bulk region of the spectrum close to the origin using finite-volume partition functions. 
  The superfield models with the partial spontaneous breaking of the global D=3, N=2 supersymmetry are discussed. The abelian gauge model describes low-energy interactions of the real scalar field with the 3D vector and fermion fields. We introduce the new Goldstone-Maxwell representation of the 3D gauge superfield and show that the partial spontaneous breaking N=2 to N=1 is possible for the non-minimal self-interaction of this modified gauge superfield including the linear Fayet-Iliopoulos term. The dual description of the partial breaking in the model of the self- interacting Goldstone chiral superfield is also considered. These models have the constant vacuum solutions and describe, respectively, the interactions of the N=1 Goldstone multiplets of the D2-brane or supermembrane with the additional massive multiplets. 
  We show that all spatial gluon connected correlation functions in SU(N) or SO(N) QCD vanish at finite temperature and zero momentum in lattice Landau or Coulomb gauges, due to the proximity of the Gribov horizon. These observations also apply to QCD with two colors and an even number of flavors at large chemical potential. These nonperturbative results may have consequences on the nature of the thermal magnetic mass and the character of the magnetic color superconductivity. 
  In this paper, fermions are minimally coupled to 3D-gravity where a dynamical torsion is introduced. A Kalb-Ramond field is non-minimally coupled to these fermions in a gauge-invariant way. We show that a 1-loop mass generation mechanism takes place for both the 2-form gauge field and the torsion. As for the fermions, no mass is dynamically generated: at 1-loop, there is only a mass shift proportional to the Yukawa coupling whenever the fermions have a non-vanishing tree-level mass. 
  We review the holographic correspondence between field theories and string/M theory, focusing on the relation between compactifications of string/M theory on Anti-de Sitter spaces and conformal field theories. We review the background for this correspondence and discuss its motivations and the evidence for its correctness. We describe the main results that have been derived from the correspondence in the regime that the field theory is approximated by classical or semiclassical gravity. We focus on the case of the N=4 supersymmetric gauge theory in four dimensions, but we discuss also field theories in other dimensions, conformal and non-conformal, with or without supersymmetry, and in particular the relation to QCD. We also discuss some implications for black hole physics. 
  The GS superstring on AdS_5 x S^5 has a nonlinearly realized, spontaneously broken SU(2,2|4) symmetry. Here we introduce a two-dimensional model in which the unbroken SU(2,2|4) symmetry is linearly realized. The basic variables are supertwistors, which transform in the fundamental representation of this supergroup.   The quantization of this supertwistor model leads to the complete oscillator construction of the unitary irreducible representations of the centrally extended SU(2,2|4). They include the states of d=4 SYM theory, massless and KK states of AdS_5 supergravity, and the descendants on AdS_5 of the standard massive string states, which form intermediate and long supermultiplets. We present examples of such multiplets and discuss possible states of solitonic and (p,q) strings. 
  The quantum properties of solitons at one loop can be related to phase shifts of waves on the soliton background. These can be combined with heat kernel methods to calculate various parameters. The vacuum energy of a CP(1) soliton in 2+1 dimensions is calculated as an example. 
  It has been shown that space-time coordinates can exhibit only very few types of short-distance structures, if described by linear operators: they can be continuous, discrete or "unsharp" in one of two ways. In the literature, various quantum gravity models of space-time at short distances point towards one of these two types of unsharpness. Here, we investigate the properties of fields over such unsharp coordinates. We find that these fields are continuous - but possess only a finite density of degrees of freedom, similar to fields on lattices. We observe that this type of unsharpness is technically the same as the aperture induced unsharpness of optical images. It is also of the same type as the unsharpness of the time-resolution of bandlimited electronic signals. Indeed, as a special case we recover the Shannon sampling theorem of information theory. 
  In the first lecture, we derive the five-dimensional effective action of strongly coupled heterotic string theory for the complete (1,1) sector of the theory by performing a reduction, on a Calabi-Yau three-fold, of M-theory on S1/Z2. The supersymmetric ground state of the theory is a multi-charged BPS three-brane domain wall, which we construct in general. In this first lecture, we assume the ``standard'' embedding of the spin connection into the E8 gauge connection on one orbifiold fixed plane. In the second lecture, we generalize these results to ``non-standard'' embeddings. That is, we allow for general E8 X E8 gauge bundles and for the presence of five-branes. Properties of these ``non-perturbative''vacua, as well as of the resulting low-energy theories, are discussed. In the last lecture, we review the spectral cover formalism for constructing both U(n) and SU(n) holomorphic vector bundles on elliptically fibered Calabi-Yau three-folds which admit a section. Restricting the structure group to SU(n), we derive, in detail, a set of rules for the construction of three-family particle physics theories with phenomenologically relevant gauge groups. We illustrate these ideas by constructing several explicit three-family non-perturbative vacua. 
  The hypothesis that the magnetic catalysis of chiral symmetry breaking is due to interactions of massless fermions in their lowest Landau level is examined in the context of chirally symmetric models with short ranged interactions. It is argued that, when the magnetic field is sufficiently large, even an infinitesimal attractive interaction in the appropriate channel will break chiral symmetry. 
  We perform the gauge-fixing of the theory of a chiral two-form boson in six dimensions starting from the action given by Pasti, Sorokin and Tonin. We use the Batalin-Vilkovisky formalism, introducing antifields and writing down an extended action satisfying the classical master equation. Then we gauge-fix the three local symmetries of the extended action in two different ways. 
  We study the embedding of D(8-p)-branes in the background geometry of parallel Dp-branes for $p\le 6$. The D(8-p)-brane is extended along the directions orthogonal to the Dp-branes of the background. The D(8-p)-brane configuration is determined by its Dirac-Born-Infeld plus Wess-Zumino-Witten action. We find a BPS condition which solves the equation of motion. The analytical solution of the BPS equation for the near-horizon and asymptotically flat geometries is given. The embeddings we obtain represent branes joined by tubes. By analyzing the energy of these tubes we conclude that they can be regarded as bundles of fundamental strings. Our solution provides an explicit realization of the Hanany-Witten effect. When $p=6$ the solution of the BPS equation must be considered separately and, in general, the embeddings of the D2-branes do not admit the same interpretation as in the $p<6$ case. 
  The Klein-Gordon system describing three scalar particles without interaction is cast into a new form, by transformation of the momenta. Two redundant degrees of freedom are eliminated; we are left with a covariant equation for a reduced wave function with three-dimensional arguments. This new formulation of the mass-shell constraints is equivalent to the original KG system in a sector characterized by positivity of the energies and, if the mass differences are not too large, by a moderately relativistic regime. Then, mutual interaction is introduced and produces a tractable model. 
  We develop a theory of non-relativistic bosons in two spatial dimensions with a weak short range attractive interaction. In the limit as the range of the interaction becomes small, there is an ultra-violet divergence in the problem. We devise a scheme to remove this divergence and produce a completely finite formulation of the theory. This involves reformulating the dynamics in terms of a new operator whose eigenvalues give the {\it logarithm} of the energy levels. Then, a mean field theory is developed which allows us to describe the limit of a large number of bosons. The ground state is a new kind of condensate (soliton) of bosons that breaks translation invariance spontaneously. The ground state energy is negative and its magnitude grows {\it exponentially} with the number of particles, rather than like a power law as for conventional many body systems. 
  We study supersymmetry breaking by hidden-sector gaugino condensation in N=1 D=4 supergravity models with multiple dilaton-like moduli fields. Our work is motivated by Type I string theory, in which the low-energy effective Lagrangian can have different dilaton-like fields coupling to different sectors of the theory. We construct the effective Lagrangian for gaugino condensation and use it to compute the visible-sector gaugino masses. We find that the gaugino masses can be of order the gravitino mass, in stark contrast to heterotic string models with a single dilaton field. 
  We calculate gaugino masses in string-derived models with hidden-sector gaugino condensation. The linear multiplet formulation for the dilaton superfield is used to implement perturbative modular invariance. The contribution arising from quantum effects in the observable sector includes the term recently found in generic supergravity models. A much larger contribution is present if matter fields with Standard Model gauge couplings also couple to the Green-Schwarz counter term. We comment on the relation of our K\"ahler U(1) superspace formalism to other calculations. 
  In order to obtain a consistent formulation of octonionic quantum mechanics (OQM), we introduce left-right barred operators. Such operators enable us to find the translation rules between octonionic numbers and $8\times 8$ real matrices (a translation is also given for $4\times 4$ complex matrices). The use of a complex geometry allows us to overcome the hermiticity problem and define an appropriate momentum operator within OQM. As an application of our results, we develop an octonionic relativistic free wave equation, linear in the derivatives. Even if the wave functions are only one-component we show that four independent solutions, corresponding to those of the Dirac equation, exist. 
  Complex geometry represents a fundamental ingredient in the formulation of the Dirac equation by the Clifford algebra. The choice of appropriate complex geometries is strictly related to the geometric interpretation of the complex imaginary unit $i=\sqrt{-1}$. We discuss {\em two} possibilities which appear in the multivector algebra approach: the $\sigma_{123}$ and $\sigma_{21}$ complex geometries. Our formalism permits to perform a set of rules which allows an immediate translation between the complex standard Dirac theory and its version within geometric algebra. The problem concerning a double geometric interpretation for the complex imaginary unit $i=\sqrt{-1}$ is also discussed. 
  Due to the non-commutative nature of quaternions we introduce the concept of left and right action for quaternionic numbers. This gives the opportunity to manipulate appropriately the $H$-field. The standard problems arising in the definitions of transpose, determinant and trace for quaternionic matrices are overcome. We investigate the possibility to formulate a new approach to Quaternionic Group Theory. Our aim is to highlight the possibility of looking at new quaternionic groups by the use of left and right operators as fundamental step toward a clear and complete discussion of Unification Theories in Physics. 
  This paper studies the dual form of Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations in N=2 supersymmetric Yang-Mills theory by applying a duality transformation to WDVV equations. The dual WDVV equations called in this paper are non-linear differential equations satisfied by dual prepotential and are found to have the same form with the original WDVV equations. However, in contrast with the case of weak coupling calculus, the perturbative part of dual prepotential itself does not satisfy the dual WDVV equations. Nevertheless, it is possible to show that the non-perturbative part of dual prepotential can be determined from dual WDVV equations, provided the perturbative part is given. As an example, the SU(4) case is presented. The non-perturbative dual prepotential derived in this way is consistent to the dual prepotential obtained by D'Hoker and Phong. 
  It has been conjectured that the Abelian projection of QCD is responsible for the confinement of color. Using a gauge independent definition of the Abelian projection which does {\it not} employ any gauge fixing, we provide a strong evidence for the Abelian dominance in Wilson loop integral. In specific we prove that the gauge potential which contributes to the Wilson loop integral is precisely the one restricted by the Abelian projection. 
  This is a short and simple introduction into perturbiners, that is, the solutions of field equations which are generating functions for tree amplitudes. The perturbiners have been constructed in Yang-Mills, in SUSY Yang-Mills, in gravity, and in sin(h)-Gordon. 
  The one loop gluon - W-meson amplitude is calculated by means of the gauge-invariant generalized Pauli-Villars regularization and with the help of dimensional regularization. It is shown that in the former case the amplitude satisfies Generalized Ward Identities, whereas in the latter case the amplitude differs from the first one by the constant. 
  We explicitly calculate a Witten diagram with general spinor field exchange on $(d+1)$-dimensional Euclidean Anti-de Sitter space, which is necessary to evaluate four-point correlation functions with spinor fields when we make use of the AdS/CFT correspondence, especially in supersymmetric cases. We also show that the amplitude can be reduced to a scalar exchange amplitude. We discuss the operator product expansion of the dual conformal field theory by interpreting the short distance expansion of the amplitude according to the AdS/CFT correspondence. 
  Classical description of relativistic pointlike particle with intrinsic degrees of freedom such as isospin or colour is proposed. It is based on the Lagrangian of general form defined on the tangent bundle over a principal fibre bundle. It is shown that the dynamics splits into the external dynamics which describes the interaction of particle with gauge field in terms of Wong equations, and the internal dynamics which results in a spatial motion of particle via integrals of motion only. A relevant Hamiltonian description is built too. 
  Using superfield Dyson-Schwinger equations, we compute the infrared dynamics of the semi-amputated full vertex, corresponding to the effective running gauge coupling, in N-flavour {\mathcal N}=1 supersymmetric QED(3). It is shown that the presence of a supersymmetry-preserving mass for the matter multiplet stabilizes the infrared gauge coupling against oscillations present in the massless case, and we therefore infer that the massive vacuum is thus selected at the level of the (quantum) effective action. We further demonstrate that such a mass can indeed be generated dynamically in a self-consistent way by appealing to the superfield Dyson-Schwinger gap equation for the full matter propagator. 
  The 2-dimensional space-time sine-Gordon field theory is extended algebraically within the n-dimensional space of extended complex numbers. This field theory is constructed in terms of an adapted extension of standard vertex operators. A whole set of non-local conserved charges is constructed and studied in this framework. Thereby, an algebraic non-perturbative description is possible for this n-1 parameters family of quantum field theories. Known results are obtained for specific values of the parameters, especially in relation to affine Toda field theories. Different (dual)-models can then be described in this formalism. 
  In this paper we fill a necessary gap in order to realize the explicit comparison between the Kaluza Klein spectra of supergravity compactified on AdS_4 x X^7 and superconformal field theories living on the world volume of M2-branes. On the algebraic side we consider the superalgebra Osp(N|4) and we study the double interpretation of its irreducible representations either as supermultiplets of particle states in the bulk or as conformal superfields on the boundary. On the lagrangian field theory side we construct, using rheonomy rather than superfield techniques, the generic form of an N=2, d=3 gauge theory. Indeed the superconformal multiplets are supposed to be composite operators in a suitable gauge theory. 
  The breaking of supersymmetry due to singular potentials in supersymmetric quantum mechanics is critically analyzed. It is shown that, when properly regularized, these potentials respect supersymmetry, even when the regularization parameter is removed. 
  The topology of orientable (2 + 1)d spacetimes can be captured by certain lumps of non-trivial topology called topological geons. They are the topological analogues of conventional solitons. We give a description of topological geons where the degrees of freedom related to topology are separated from the complete theory that contains metric (dynamical) degrees of freedom. The formalism also allows us to investigate processes of quantum topology change. They correspond to creation and annihilation of quantum geons. Selection rules for such processes are derived. 
  We discuss higher loop corrections to gauge coupling renormalization in the context of gauge coupling unification via Kaluza-Klein thresholds. We show that in the case N=1 supersymmetric compactifications the one-loop threshold contributions are dominant, while the higher loop correction are subleading. This is due to the fact that at heavy Kaluza-Klein levels the spectrum as well as the interactions are N=2 supersymmetric. In particular, we give two different arguments leading to this result - one is field theoretic, while the second one utilizes the power of string perturbation techniques. To illustrate our discussions we perform explicit two-loop computations of various corrections to gauge couplings within this framework. We also remark on phenomenological applications of our discussions in the context of TeV-scale brane world. 
  We give a unified description of self-dual SU(2) gauge fields on tori of size lt x ls^3 based on a mixture of analytical and numerical methods using the Nahm transformation, extended to the case of twisted boundary conditions. We show how torus calorons (lt/ls small) are Nahm dual to the torus instantons (lt/ls large). Holonomies are dual to the locations of constituents, this duality becoming exact in the limiting cases ls or lt --> infinity. Implications for the moduli spaces are discussed. 
  A compactification of 11-dimensional supergravity with two (or more) walls is considered. The whole tower of massive Kaluza-Klein modes along the fifth dimension is taken into account. With the sources on the walls, an explicit composition in terms of Kaluza-Klein modes of massless gravitino (in the supersymmetry preserving case) and massive gravitino (in the supersymmetry breaking case) is obtained. The super-Higgs effect is discussed in detail. 
  We illustrate a method for improving Renormalisation Group improved perturbation theory by calculating the infrared central charge of a perturbed conformal field theory. The additional input is a dispersion relation that exploits analyticity of the energy-momentum tensor correlator. 
  Whenever the N=(2,2) supersymmetry algebra of non-linear sigma-models in two dimensions does not close off-shell, a holomorphic two-form can be defined. The only known superfields providing candidate auxiliary fields to achieve an off-shell formulation are semi-chiral fields. Such a semi-chiral description is only possible when the two-form is constant. Using an explicit example, hyper-Kahler manifolds, we show that this is not always the case. Finally, we give a concrete construction of semi-chiral potentials for a class of hyper-Kahler manifolds using the duality exchanging a pair consisting of a chiral and a twisted-chiral superfield for one semi-chiral multiplet. 
  The purpose of the present thesis is the implementation of symmetries in the Wilsonian Exact Renormalization Group (ERG) approach. After recalling how the ERG can be introduced in a general theory (i.e. containing both bosons and fermions, scalars and vectors) and having applied it to the massless scalar theory as an example of how the method works, we discuss the formulation of the Quantum Action Principle (QAP) in the ERG and show that the Slavnov-Taylor identities can be directly derived for the cutoff effective action at any momentum scale. Firstly the QAP is exploited to analyse the breaking of dilatation invariance occurring in the scalar theory in this approach. Then we address SU(N) Yang-Mills theory and extensively treat the key issue of the boundary conditions of the flow equation which, in this case, have also to ensure restoration of symmetry for the physical theory. In case of a chiral gauge theory, we show how the chiral anomaly can be obtained in the ERG. Finally, we extend the ERG formulation to supersymmetric (gauge) theories. It is emphasized regularization is implemented in such a way that supersymmetry is preserved. 
  In this paper we construct the 5 parameter generating solution of N=8 BPS regular supergravity black holes as a five parameter solution of the N=2 STU model. Our solution has a simpler form with respect to previous constructions already appeared in the literature and moreover, through the embedding [SL(2)]^3\subset SU(3,3)\subset E_{(7)7} discussed in previous papers, the action of the U-duality group is well defined. This allows to reproduce via U-duality rotations any other solution, like those corresponding to R-R black holes whose microscopic description is given by intersecting D-branes. 
  We propose a method to obtain a manifestly supersymmetric action functional for interacting brane systems. It is based on the induced map of the worldvolume of low-dimensional branes into the worldvolume of the space-time filling brane ((D-1)-brane), which may be either dynamical or auxiliary, and implies an identification of Grassmann coordinate fields of lower dimensional branes with an image of the Grassmann coordinate fields of that (D-1)-brane. With this identification the covariant current distribution forms with support on the superbrane worldvolumes become invariant under the target space supersymmetry and can be used to write the coupled superbrane action as an integral over the D-dimensional manifolds ((D-1)-brane worldvolume). We compare the equations derived from this new ('Goldstone fermion embedded') action with the ones produced by a more straightforward generalization of the free brane actions based on the incorporation of the boundary terms with Lagrange multipliers ('superspace embedded' action). We find that both procedures produce the same equations of motion and thus justify each other. Both actions are presented explicitly for the coupled system of a D=10 super-D3-brane and a fundamental superstring which ends on the super-D3-brane. 
  We extend superspace by introducing an antisymmetric tensorial coordinate. The resulting theory presents a supersymmetry with central charge. After integrating over the tensorial coordinate, an effective action describing massive bosons and fermions is explicitely derived for the spacetime dimension D=2. The adopted procedure is simpler than the Kaluza-Klein one and can suggest an alternative for string compactifications. 
  In this paper, we use only the equation of motion for an interacting system of gravity, dilaton and antisymmetric tensor to study the black brane solutions. By making use of the property of Schwarzian derivative, we obtain the complete solution of this system of equations. For some special values we obtain the well-known BPS brane and black brane solutions. 
  We employ a heat kernel expansion to derive an effective action that describes four dimensional SU(2) Yang-Mills theory in the infrared limit. Our result supports the proposal that at large distances the theory is approximated by the dynamics of knotted string-like fluxtubes which appear as solitons in the effective theory. 
  We construct dual Type I' string descriptions to five dimensional supersymmetric fixed points with $E_{N_f+1}$ global symmetry. The background is obtained as the near horizon geometry of the D4-D8 brane system in massive Type IIA supergravity. We use the dual description to deduce some properties of the fixed points. 
  We show that the single-mode parafermionic type systems possess supersymmetry, which is based on the symmetry of characteristic functions of the parafermions related to the generalized deformed oscillator of Daskaloyannis et al. The supersymmetry is realized in both unbroken and spontaneously broken phases. As in the case of parabosonic supersymmetry observed recently by one of the authors, the form of the associated superalgebra depends on the order of the parafermion and can be linear or nonlinear in the Hamiltonian. The list of supersymmetric parafermionic systems includes usual parafermions, finite-dimensional q-deformed oscillator, q-deformed parafermionic oscillator and parafermionic oscillator with internal $Z_2$ structure. 
  We analyse the highest weight representations of the N=1 Ramond algebra and show that their structure is richer than previously suggested in the literature. In particular, we show that certain Verma modules over the N=1 Ramond algebra contain degenerate (2-dimensional) singular vector spaces and that in the supersymmetric case they can even contain subsingular vectors. After choosing a suitable ordering for the N=1 Ramond algebra generators we compute the ordering kernel, which turns out to be two-dimensional for complete Verma modules and one-dimensional for G-closed Verma modules. These two-dimensional ordering kernels allow us to derive multiplication rules for singular vector operators and lead to expressions for degenerate singular vectors. Using these multiplication rules we study descendant singular vectors and derive the Ramond embedding diagrams for the rational models. We give all explicit examples for singular vectors, degenerate singular vectors, and subsingular vectors until level 3. We conjecture the ordering kernel coefficients of all (primitive) singular vectors and therefore identify these vectors uniquely. 
  The general static solutions of the scalar field equation for the potential $V(\phi)= -1/2 M^2\phi^2 + \lambda/4 \phi^4$ are determined for a finite domain in $(1+1)$ dimensional space-time. A family of real solutions is described in terms of Jacobi Elliptic Functions. We show that the vacuum-vacuum boundary conditions can be reached by elliptic cn-type solutions in a finite domain, such as of the Kink, for which they are imposed at infinity. We proved uniqueness for elliptic sn-type solutions satisfying Dirichlet boundary conditions in a finite interval (box) as well the existence of a minimal mass corresponding to these solutions in a box. We define expressions for the ``topological charge'', ``total energy'' (or classical mass) and ``energy density'' for elliptic sn-type solutions in a finite domain. For large length of the box the conserved charge, classical mass and energy density of the Kink are recovered. Also, we have shown that using periodic boundary conditions the results are the same as in the case of Dirichlet boundary conditions. In the case of anti-periodic boundary conditions all elliptic sn-type solutions are allowed. 
  The problem of gauge independent definition of the effective gauge field is considered. The Slavnov identities corresponding to a system of interacting quantum gauge and classical matter fields, playing the role of a measuring device, are obtained. With their help, in the case of power-counting renormalizable theories, gauge independence of the effective device action is proved in the low-energy limit, which allows to introduce the gauge independent notion of the effective gauge field. 
  We review the method of constructing dipole and string loop solutions from the higher-dimensional (Euclidean) Kerr black hole. We analyse the results in various dimensions, finding solutions earlier given in the literature. Then we construct a new heterotic dipole with non-trivial dilaton and gauge fields. This can, in turn, be describes as a brane/anti-brane pair which interpolates between the KK-dipole and the H-dipole. Finally an argument is presented on the tachyonic instability by analysing the string fluctuations on the dipole background. 
  In pure Einstein theory, Ricci flat Lorentzian 4-metrics of Petrov types III or N have vanishing counter terms up to and including two loops. Moreover for pp-waves and type-N spacetimes of Kundt's class which admit a non-twisting, non expanding, null congruence all possible invariants formed from the Weyl tensor and its covariant derivatives vanish. Thus these Lorentzian metrics suffer no quantum corrections to all loop orders. By contrast for complete non-singular Riemannian metrics the two loop counter term vanishes only if the metric is flat. 
  We construct partially localized supergravity counterpart solutions to the 1/2 supersymmetric non-threshold and the 1/4 supersymmetric threshold bound state BI dyons in the D3-brane Dirac-Born-Infeld theory. Such supergravity solutions have all the parameters of the BI dyons. By applying the IIA/IIB T-duality transformations to these supergravity solutions, we obtain the supergravity counterpart solutions to 1/2 and 1/4 supersymmetric BIons carrying electric and magnetic charges of the worldvolume U(1) gauge field in the Dirac-Born-Infeld theory in other dimensions. 
  A static minimal energy configuration of a super p-brane in a supersymmetric (n+1)-dimensional spacetime is shown to be a `generalized calibrated' submanifold. Calibrations in $\bE^{(1,n)}$ and $AdS_{n+1}$ are special cases. We present several M-brane examples. 
  We study how non-BPS type II D-branes couple to R-R potentials. Upon tachyon condensation the couplings we find give rise to the Wess-Zumino action of BPS D-branes. 
  We study the influence of boundary conditions on energy levels of interacting fields in a box and discuss some consequences when we change the size of the box. In order to do this we calculate the energy levels of bound states of a scalar massive field $\chi$ interacting with another scalar field $\phi$ through the lagrangian ${\cal L}_{int} = 3/2 g\phi^{2}\chi^{2}$ in an one-dimensional box, on which we impose Dirichlet boundary conditions. We have found that the gap between the bound states changes with the size of the box in a non-trivial way. For the case the masses of the two fields are equal and for large box the energy levels of Dashen-Hasslacher-Neveu (DHN model) (Dashen et al, 1974) are recovered and we have a kind of boson condensate for the ground state. Below to a critical box size $L\sim 2.93 2\sqrt{2}/M$ the ground state level splits, which we interpret as particle-antiparticle production under small perturbations of box size. Below another critical sizes $(L\sim 6/10 2\sqrt{2}/M)$ and $(L\sim 1.71 2\sqrt{2}/M)$ of the box, the ground state and first excited state merge in the continuum part of the spectrum. 
  We check some consistency conditions for the D9-anti-D9 system in type I string theory. The gravitational anomaly and gauge anomaly for SO(n) x SO(m) gauge symmetry are shown to be cancelled when n-m=32. In addition, we find that a string theory with USp(n) x USp(m) gauge symmetry also satisfies the anomaly cancellation conditions. After tachyon condensation, the theory reduces to a tachyon-free USp(32) string theory, though there is no spacetime supersymmetry. 
  In this paper, a new approach to string dynamics is proposed. String coordinates are identified with a non-commuting set of operators familiar from free string quantization, and the dynamics follows from the Virasoro algebra. There is a very large gauge group operating on the non-commuting coordinates. The gauge has to be fixed suitably to make contact with the standard string picture. 
  The usual T-duality that relates the type IIA and IIB theories compactified on circles of inversely-related radii does not operate if the dimensional reduction is performed on the time direction rather than a spatial one. This observation led to the recent proposal that there might exist two further ten-dimensional theories, namely type IIA^* and type IIB^*, related to type IIB and type IIA respectively by a timelike dimensional reduction. In this paper we explore such dimensional reductions in cases where time is the coordinate of a non-trivial U(1) fibre bundle. We focus in particular on situations where there is an odd-dimensional anti-de Sitter spacetime AdS_{2n+1}, which can be described as a U(1) bundle over \widetilde{CP}^n, a non-compact version of CP^n corresponding to the coset manifold SU(n,1)/U(n). In particular, we study the AdS_5\times S^5 and AdS_7\times S^4 solutions of type IIB supergravity and eleven-dimensional supergravity. Applying a timelike Hopf T-duality transformation to the former provides a new solution of the type IIA^* theory, of the form \widetilde{CP}^2\times S^1\times S^5. We show how the Hopf-reduced solutions provide further examples of ``supersymmetry without supersymmetry.'' We also present a detailed discussion of the geometrical structure of the Hopf-fibred metric on AdS_{2n+1}, and its relation to the horospherical metric that arises in the AdS/CFT correspondence. 
  We investigate various properties of classical configurations of the Born-Infeld theory in a uniform electric field. This system is involved with dynamics of (F,Dp) bound states, which are bound states of fundamental strings and Dp-branes. The uniform electric field can be treated as a constraint on the asymptotic behavior of the fields on the brane. BPS configurations in this theory correspond to fundamental strings attached to the (F,Dp) bound state, and are found to be stable due to force balance. Fluctuations around these stable configurations are subject to appropriate Dirichlet and Neumann boundary conditions which are identical with the ones deduced from the ordinary worldsheet description of the attached string. Additionally, non-BPS solutions are studied and related physics are discussed. 
  We consider a free two-form in six dimensions and calculate the conformal anomaly associated with a Wilson surface observable. 
  Pseudo conformal field theories are theories with the same fusion rules, but with different modular matrix as some conventional field theory. One of the authors defined these and conjectured that, for bosonic systems, they can all be realized by some actual RCFT, which is of free bosons. We complete the proof here by treating the non diagonal automorphism case. It is shown that for characteristics $p\neq2$ they are all equivalent to a diagonal case, fully classified in our previous publication. For $p=2^n$ we realize the non diagonal case, establishing this theorem. 
  We consider domain walls that appear in supersymmetric QCD with Nf < Nc massive flavours. In particular, for 2 Nf < Nc we explicitly construct the domain walls that interpolate between vacua labeled by i and (i+ N_f). We show that these solutions are Bogomol'nyi-Prasad-Sommerfield (BPS) saturated for any value of the mass of the matter fields. This fact allows us to evaluate the large mass limit of these domain walls. We comment on the relevance of these solutions for supersymmetric gluodynamics. 
  Topological terms in the O(3) nonlinear sigma model in (1+1) and (2+1) dimensions are re-examined based on the description of the SU(2)-valued field $g$. We first show that the topological soliton term in (1+1) dimensions arises from the unitary representations of the group characterizing the global structure of the symmetry inherent in the description, in a manner analogous to the appearance of the $\theta$-term in Yang-Mills theory in (3+1) dimensions. We then present a detailed argument as to why the conventional Hopf term, which is the topological counterpart in (2+1) dimensions and has been widely used to realize fractional spin and statistics, is ill-defined unless the soliton charge vanishes. We show how this restriction can be lifted by means of a procedure proposed recently, and provide its physical interpretation as well. 
  We apply a superspace formulation to the four-dimensional gauge theory of a massless Abelian antisymmetric tensor field of rank 2. The theory is formulated in a six-dimensional superspace using rank-2 tensor, vector and scalar superfields and their associated supersources. It is shown that BRS transformation rules of fields are realized as Euler-Lagrange equations without assuming the so-called horizontality condition and that a generating functional $\bar{W}$ constracted in the superspace reduces to that for the ordinary gauge theory of Abelian rank-2 antisymmetric tensor field. The WT identity for this theory is derived by making use of the superspace formulation and is expressed in a neat and compact form $\partial\bar{W}/\partial\theta=0$. 
  Discrete analogs of the Darboux-Egoroff metrics are considered. It is shown that the corresponding lattices in the Euclidean space are described by discrete analogs of the Lame equations. It is proved that up to a gauge transformation these equations are necessary and sufficient for discrete analogs of rotation coefficients to exist. Explicit examples of the Darboux-Egoroff lattices are constructed by means of algebro-geometric methods. 
  For N>2 we present static monopole solutions of the second order SU(N) BPS Yang-Mills-Higgs equations which are not solutions of the first order Bogomolny equations. These spherically symmetric solutions may be interpreted as monopole anti-monopole configurations and their construction involves harmonic maps into complex projective spaces. 
  The dynamics of a string near a Kaluza-Klein black hole are studied. Solutions to the classical string equations of motion are obtained using the world sheet velocity of light as an expansion parameter. The electrically and magnetically charged cases are considered separately. Solutions for string coordinates are obtained in terms of the world-sheet coordinate $\tau$. It is shown that the Kaluza-Klein radius increases/decreases with $\tau$ for electrically/magnetically charged black hole. 
  We study the behavior of bound energy levels for the case of two classical interacting fields $\phi$ and $\chi$ in a finite domain (box) in (1 + 1) dimension on which we impose Dirichlet boundary conditions (DBC). The total Lagrangian contain a $\frac{\lambda}{4}\phi^4$ self-interaction and an interaction term given by $g \phi^2 \chi^2$. We calculate the energy eigenfunctions and its correspondent eigenvalues and study their dependence on the size of the box (L) as well on the free parameters of the Lagrangian: mass ratio $\beta = \frac{M^{2}_{\chi}}{M^{2}_{\phi}}$, and interaction coupling constants $\lambda$ and $g$. We show that for some configurations of the above parameters, there exists critical sizes of the box for which instability points of the field $\chi$ appear. 
  We investigate two-loop quantum corrections to non-minimally coupled Maxwell-Chern-Simons theory. The non-minimal gauge interaction represents the magnetic moment interaction between the charged scalar and the electromagnetic field. We show that the one-loop renormalizability of the theory found in previous work does not survive to the two-loop level. However, with an appropriate choice of the non-minimal coupling constant, it is possible to renormalize the two-loop effective potential and hence render it potentially useful for a detailed analysis of spontaneous symmetry breaking induced by radiative corrections. 
  D-instanton effects are studied for the IIB orientifold T^2/I\Omega(-1)^{F_L} of Sen using type I/heterotic duality. An exact one loop threshold calculation of t_8 \tr F^4 and t_8(\tr F^2)^2 terms for the heterotic string on T^2 with Wilson lines breaking SO(32) to SO(8)^4 is related to D-instanton induced terms in the worldvolume of D7 branes in the orientifold. Introducing D3 branes and using the AdS/CFT correspondence in this case, these terms are used to calculate Yang-Mills instanton contributions to four point functions of the large N_c limit of N=2 USp(2N_c) SYM with four fundamental and one antisymmetric tensor hypermultiplets. 
  We calculate the leading term in the low-energy absorption cross section for an arbitrary partial wave of the dilaton field by a stack of many coincident D3-branes. We find that it precisely reproduces the semiclassical absorption cross section of a 3-brane geometry, including all numerical factors. The crucial ingredient in making the correspondence is the identification of the precise operators on the D3-brane world-volume which couple to the dilaton field and all its derivatives. The needed operators are related through T-duality and the IIA/M-theory correspondence to the recently determined M(atrix) theory expressions for multipole moments of the 11D supercurrent. These operators have a characteristic symmetrized trace structure which plays a key combinatorial role in the analysis for the higher partial waves. The results presented here give new evidence for an infinite family of non-renormalization theorems which are believed to exist for two-point functions in ${\cal N} = 4$ gauge theory in four dimensions. 
  We consider the theory obtained by adding to the usual string frame dilaton gravity action specially constructed higher derivative terms motivated by the limited curvature construction of [MukhanovET1992a] and determine the spatially homogeneous and isotropic solutions to the resulting equations of motion. All solutions of the resulting theory of gravity with these symmetries are nonsingular and all curvature invariants are bounded. For initial conditions inspired by the pre-big-bang scenario solutions exist which correspond to a spatially flat Universe starting in a dilaton-dominated superinflationary phase with ${\dot H} > 0$ and having a smooth transition to an expanding Friedmann Universe with ${\dot H} < 0$. Hence, the graceful exit problem of pre-big-bang cosmology is solved in a natural way. 
  We construct the three point function involving an axial vector current and two energy-momentum tensors for four dimensional conformal field theories. Conformal symmetry determines the form of this three point function uniquely up to a constant factor if the necessary conservation conditions are imposed. The gravitational axial anomaly present on a curved space background leads to a non-zero contribution for the divergence of the axial current in this three point function even on flat space. Using techniques related to differential regularisation which guarantee that the energy-momentum tensor is conserved and traceless, we calculate the anomaly in the three point function directly. In this way we relate the overall coefficient of the three point function to the scale of the gravitational axial anomaly. We check our results by applying them to the examples of the fermion and photon axial currents. 
  We conjecture the following entropy bound to be valid in all space-times admitted by Einstein's equation: Let A be the area of any two-dimensional surface. Let L be a hypersurface generated by surface-orthogonal null geodesics with non-positive expansion. Let S be the entropy on L. Then S does not exceed A/4.   We present evidence that the bound can be saturated, but not exceeded, in cosmological solutions and in the interior of black holes. For systems with limited self-gravity it reduces to Bekenstein's bound. Because the conjecture is manifestly time reversal invariant, its origin cannot be thermodynamic, but must be statistical. Thus it places a fundamental limit on the number of degrees of freedom in nature. 
  If the vacuum is passive for uniformly accelerated observers in anti-de Sitter space-time (i.e. cannot be used by them to operate a "perpetuum mobile"), they will (a) register a universal value of the Hawking-Unruh temperature, (b) discover a TCP symmetry, and (c) find that observables in complementary wedge-shaped regions are commensurable (local) in the vacuum state. These results are model independent and hold in any theory which is compatible with some weak notion of space-time localization. 
  A rigorous (and simple) proof is given that there is a one-to-one correspondence between causal anti-deSitter covariant quantum field theories on anti-deSitter space and causal conformally covariant quantum field theories on its conformal boundary. The correspondence is given by the explicit identification of observables localized in wedge regions in anti-deSitter space and observables localized in double-cone regions in its boundary. It takes vacuum states into vacuum states, and positive-energy representations into positive-energy representations. 
  The embedding of a Taub-NUT space in the directions transverse to the world volume of branes describes branes at (spherical) orbifold singularities. Similarly, the embedding of a pp-wave in the brane world volume yields an AdS orbifold. In case of the D1-D5--brane system, the AdS3 orbifolds yields a BTZ black hole; as we will show, the same holds for D3-branes corresponding to AdS5. In addition we will show that the AdS orbifolds and the spherical orbifolds are U-dual to each other. However in contrast to spherical orbifolds the AdS orbifolds lead to a running coupling, which is in the IR inverse to the coupling of the spherical orbifold. A discussion of the general pp-wave solution in AdS space is added. 
  See hep-th/0002188 
  Supersymmetric String theories find occurrences of extremal Black Holes with gravitational mass M=Q where Q is the charge (G=c=1). Thus, for the chargeless cases, they predict M=0. We show that General Theory of Relativity, too, demands a unique BH mass M=0. And we do this by considering the problem from various angles, both mathematically and physically. Our analysis may be understandable even to the readers not having sufficient background in GTR. Physically, this M=0 result means that, continued gravitational collapse indeed continues for infinite proper time as the system hopelessly tries to radiate its entire original mass energy to attain the lowest energy state M=0. Since no event horizon is formed, the problem of loss of information associated with creation of finite mass BHs or their evaporation ceases to exist. 
  The leading eikonal S-matrix for three graviton scattering in d=11 supergravity and Matrix Theory are shown to precisely agree. The result unifies the source-probe plus recoil approach of Okawa and Yoneya and relaxes the restriction imposed by those authors that all D-particle impact parameters and velocities are mutually perpendicular. Furthermore, the unified S-matrix approach facilitates a clean-cut study of M-theoretic R^4 curvature corrections to the low energy supergravity effective action. In particular, the leading R^4 correction to the three graviton S-matrix is computed and compared to the corresponding next to leading order two loop U(3) amplitude in Matrix Theory. We find a clear disagreement of the two resulting tensor structures. 
  We investigate a model for a real scalar field in bidimensional space-time, described in terms of a positive semi-definite potential that presents no vacuum state. The system presents topological solutions of the BPS type, with energy density that follows a Lorentzian law. These BPS solutions differ from the standard $\tanh$-type kink, but they also support bosonic and fermionic zero modes. 
  The bare bones of a theory of quantum gravity are exposed. It may have the potential to solve the cosmological constant problem. Less certain is its behavior in the Newtonian limit. 
  In the Poincare patch of Minkovski AdS_5 we explicitly construct local bulk fields from the boundary operators, to leading order in 1/N. We also construct the Green's function implicitly defined by this procedure. We generalize the construction of local fields for near horizon geometries of Dp branes. We try to expand the procedure to the interacting case, with partial success. 
  We study an extreme non-static limit of 2+1-dimensional QED obtained by making a dimensional reduction so that all fields are spatially uniform but time dependent. This dimensional reduction leads to a 0+1-dimensional field theory that inherits many of the features of the 2+1-dimensional model, such as Chern-Simons terms, time-reversal violation, an analogue of parity violation, and global U(2) flavor symmetry. At one-loop level, interactions induce a Chern-Simons term at finite T with coefficient tanh(beta m_F/2), where m_F is the fermion mass. The finite temperature two loop self-energies are also computed, and are non-zero for all temperatures. 
  We discuss the spontaneous breakdown of chiral symmetry in Quantum Chromodynamics by considering gluonic instanton configurations in the partition function. It is shown that in order to obtain nontrivial fermionic correlators in a two dimensional gauge theory for the strong interactions among quarks, a regular instanton background has to be taken into account. We work over massless quarks in the -fundamental- representation of SU(N_c). For large N_c, massive quarks are also considered. 
  We study quantum caustics in $d$-dimensional systems with quadratic Lagrangians. Based on Schulman's procedure in the path-integral we derive the transition amplitude on caustics in a closed form for generic multiplicity $f$, and thereby complete the previous analysis carried out for the maximal multiplicity case $f=d$. Multiplicity dependence of the caustics phenomena is illusrated by examples of a particle interacting with external electromagnetic fields. 
  We present a new point of view on the quantization of the gravitational field, namely we use exclusively the quantum framework of the second quantization. More explicitly, we take as one-particle Hilbert space, $H_{graviton}$ the unitary irreducible representation of the Poincar\'e group corresponding to a massless particle of helicity 2 and apply the second quantization procedure with Einstein-Bose statistics. The resulting Hilbert space ${\cal F}^{+}(H_{graviton})$ is, by definition, the Hilbert space of the gravitational field. Then we prove that this Hilbert space is canonically isomorphic to a space of the type $Ker(Q)/Im(Q)$ where $Q$ is a supercharge defined in an extension of the Hilbert space ${\cal F}^{+} (H}_{graviton})$ by the inclusion of ghosts: some Fermion ghosts $u_{\mu}, \tilde{u}_{\mu}$ which are vector fields and a Bosonic ghost $\Phi$ which is a scalar field. This has to be contrasted to the usual approaches where only the Fermion ghosts are considered. However, a rigorous proof that this is, indeed, possible seems to be lacking from the literature. 
  This paper has been withdrawn for revision, due to misinterpretations of the results. 
  A purely algebraic method is devised in order to recover Slavnov-Taylor identities (STI), broken by intermediate renormalization. The counterterms are evaluated order by order in terms of finite amplitudes computed at zero external momenta. The evaluation of the breaking terms of the STI is avoided and their validity is imposed directly on the vertex functional. The method is applied to the abelian Higgs-Kibble model. An explicit mass term for the gauge field is introduced, in order to check the relevance of nilpotency. We show that, since there are no anomalies, the imposition of the STI turns out to be equivalent to the solution of a linear problem. The presence of ST invariants implies that there are many possible solutions, corresponding to different normalization conditions. Moreover, we find more equations than unknowns (over-determined problem). This leads us to the consideration of consistency conditions, that must be obeyed if the restoration of STI is possible. 
  Open groups whose generators are in arbitrary involutions may be quantized within a ghost extended framework in terms of a nilpotent BFV-BRST charge operator. Previously we have shown that generalized quantum Maurer-Cartan equations for arbitrary open groups may be extracted from the quantum connection operators and that they also follow from a simple quantum master equation involving an extended nilpotent BFV-BRST charge and a master charge. Here we give further details of these results. In addition we establish the general structure of the solutions of the quantum master equation. We also construct an extended formulation whose properties are determined by the extended BRST charge in the master equation. 
  The central charge in the N=2 Super-Yang-Mills theory plays an essential role in the work of Seiberg and Witten as it gives the mass spectrum of the BPS states of the quantum theory. Our aim in this note is to present a direct computation of this central charge for the leading order (in a momentum expansion) of the effective action. We will consider the N=2 Super-Yang-Mills theory with gauge group SU(2). The leading order of the effective action is given by the same holomorphic function F appearing in the low energy U(1) effective action. 
  Couplings between a closed string RR field and open strings are calculated in a system of coincident branes and antibranes of type II theory. The result can be written cleanly using the curvature of the superconnection. 
  We argue that the Wess-Zumino model with quartic superpotential admits static solutions in which three domain walls intersect at a junction. We derive an energy bound for such junctions and show that configurations saturating it preserve 1/4 supersymmetry. 
  Boundary operators and boundary ground states in sine-Gordon model with a fixed boundary condition are studied using bosonization and q-deformed oscillators.We also obtain the form-factors of this model. 
  We conjecture that a T-dual form of pure QCD describes dynamics of point-like monopoles. T-duality transforms the QCD Lagrangian into a matrix quantum mechanics of zerobranes which we identify with monopoles. At generic points of the monopole moduli space the SU(N) gauge group is broken down to $U(1)^{N-1}$ reproducing the key feature of 't Hooft's Abelian projection. There are certain points in the moduli space where monopole positions coincide, gauge symmetry is enhanced and gluons emerge as massless excitations. We show that there is a linearly rising potential between zerobranes. This indicates the presence of a stretched flux tube between monopoles. The lowest energy state is achieved when monopoles are sitting on top of each other and gauge symmetry is enhanced. In this case they behave as free massive particles and can condense. In fact, we find a constant eigenfunction of the corresponding Hamiltonian which describes condensation of monopoles. Using the monopole quantum mechanics, we argue that large $N$ QCD in this T-dual picture is a theory of a closed bosonic membrane propagating in {\em five} dimensional space-time. QCD point-like monopoles can be regarded in this approach as constituents of the membrane. 
  Using the auxiliary field representation of the simplicial chiral models on a (d-1)-dimensional simplex, the simplicial chiral models are generalized through replacing the term Tr$(AA^{\d})$ in the Lagrangian of these models by an arbitrary class function of $AA^{\d}$; $V(AA^{\d})$. This is the same method used in defining the generalized two-dimensional Yang-Mills theories (gYM_2) from ordinary YM_2. We call these models, the ``generalized simplicial chiral models''. Using the results of the one-link integral over a U(N) matrix, the large-N saddle-point equations for eigenvalue density function $\ro (z)$ in the weak ($\b >\b_c$) and strong ($\b <\b_c$) regions are computed. In d=2, where the model is in some sense related to the gYM_2 theory, the saddle-point equations are solved for $\ro (z)$ in the two regions, and the explicit value of critical point $\b_c$ is calculated for $V(B)$=Tr$B^n$ $(B=AA^{\d})$. For $V(B$)=Tr$B^2$,Tr$B^3$, and Tr$B^4$, the critical behaviour of the model at d=2 is studied, and by calculating the internal energy, it is shown that these models have a third order phase transition. 
  We construct IIB supergravity (viewed as dilatonic gravity) background with non-trivial dilaton and with curved four-dimensional space. Such a background may describe another vacuum of maximally supersymmetric Yang-Mills theory or strong coupling regime of (non)-supersymmetric gauge theory with (power-like) running gauge coupling which depends on curvature. Curvature dependent quark-antiquark potential is calculated where the geometry type of hyperbolic (or de Sitter universe) shows (or not) the tendency of the confinement. Generalization of IIB supergravity background with non-constant axion is presented. Quark-antiquark potential being again curvature-dependent has a possibility to produce the standard area law for large separations. 
  The one-loop effective action of the abelian and nonabelian Higgs models has been studied in various gauges, in the context of instanton and sphaleron transition, bubble nucleation and most recently in nonequilibrium dynamics. Gauge invariance is expected on account of Nielsen' s theorem, if the classical background field is an extremum of the classical action, i.e., a solution of the classical equation of motion. We substantiate this general statement for the one-loop effective action, as computed using mode functions. We show that in the gauge-Higgs sector there are two types of modes that satisfy the same equation of motion as the Faddeev-Popov modes. We apply the general analysis to the computation of the fluctuation determinant for bubble nucleation in the SU(2) Higgs model in the 't Hooft gauge with general gauge parameter $\xi$. We show that due to the cancellation of the modes mentioned above the fluctuation determinant is independent of $\xi$. 
  In this paper, we give the general crossing relation for boundary reflection matrix $R(\beta)$, which is the extension of the work given by Ghoshal and Zamolodchikov .We also use the first non-trivial extended crossing relation to determine the scaler factor of $R(\beta)$ which is the rational diagonal solution to the boundary Yang-Baxter equation in the case of l=2 and n=3. 
  We show that the degeneracy of topological solitons in the gauged O(3) non-linear sigma model with Chern-Simons term may be removed by chosing a self-interaction potential with symmetry - breaking minima. The topological solitons in the model has energy,charge,flux and angular momentum quantised in each topological sector. 
  The vacuum energy of a scalar field in a spherically symmetric background field is considered. Based on previous work [hep-th/9608070], the numerical procedure is refined further and applied to several examples. We provide numerical evidence that repulsive potentials lead to positive contributions to the vacuum energy. Furthermore, the crucial role played by bound-states is clearly seen. 
  Thermodynamics of d=4, N=4 supersymmetric SU(N) Yang-Mills theory is studied with particular attention on perturbative expansion at weak `t Hooft coupling regime and interpolation to strong coupling regime thereof. Non-ideal gas effect to free-energy is calculated and found that leading- and next-to-leading-order corrections contribute with relative opposite sign. Pade approximant method is adopted to improve fixed-order, perturbative series and is found to decrease free-energy monotonically as `t Hooft coupling parameter is increased. This may be regarded as an indication of smooth interpolation of thermodynamics between weak and strong `t Hooft coupling regimes, as suggested by Maldacena's AdS/CFT correspondence. 
  Renormalization group flow equations for scalar lambda Phi^4 are generated using three classes of smooth smearing functions. Numerical results for the critical exponent nu in three dimensions are calculated by means of a truncated series expansion of the blocked potential. We demonstrate how the convergence of nu as a function of the order of truncation can be improved through a fine tuning of the smoothness of the smearing functions. 
  Exact relations between the QCD thermal pressure and the trace anomaly are derived. These are used, first, to prove the equivalence of the thermodynamic and the hydrodynamic pressure in equilibrium in the presence of the trace anomaly, closing a gap in previous arguments. Second, in the temporal axial gauge a formula is derived which expresses the thermal pressure in terms of a Dyson-resummed two-point function. This overcomes the infrared problems encountered in the conventional perturbation-theory approach. 
  A simple formal computation, and a variation on an old thought experiment, both indicate that QCD with light quarks may confine fundamental color magnetic charges, giving an explicit as well as elegant resolution to the `global color' paradox, strengthening Vachaspati's SU(5) electric-magnetic duality, opening new lines of inquiry for monopoles in cosmology, and suggesting a class of geometrically large QCD excitations -- loops of Z(3) color magnetic flux entwined with light-quark current. The proposal may be directly testable in lattice gauge theory or supersymmetric Yang-Mills theory. Recent results in deeply-inelastic electron scattering, and future experiments both there and in high-energy collisions of nuclei, could give evidence on the existence of Z(3) loops. If confirmed, they would represent a consistent realization of the bold concept underlying the Slansky-Goldman-Shaw `glow' model -- phenomena besides standard meson-baryon physics manifest at long distance scales -- but without that model's isolable fractional electric charges. 
  We consider the multi-instanton collective coordinate integration measure in N=2 supersymmetric SU(N) gauge theory with N_F fundamental hypermultiplets. In the large-N limit, at the superconformal point where N_F=2N and all VEVs are turned off, the k-instanton moduli space collapses to a single copy of AdS_5*S^1. The resulting k-instanton effective measure is proportional to N^{1/2} g^4 Z_k^(6), where Z_k^(6) is the partition function of N=(1,0) SYM theory in six dimensions reduced to zero dimensions. The multi-instanton can in fact be summed in closed form. As a hint of an AdS/CFT duality, with the usual relation between the gauge theory and string theory parameters, this precisely matches the normalization of the charge-k D-instanton measure in type IIB string theory compactified to six dimensions on K3 with a vanishing two-cycle. 
  We consider domain walls embedded in curved backgrounds as an approximation for braneworld scenarios. We give a large class of new exact solutions, exhausting the possibilities for describing one and two walls for the cases where the curvature of both the bulk and the wall is locally constant. In the case of two walls, we find solutions where each wall has positive tension. An interesting property of these solutions is that the curvature of the walls can be much smaller than the tension, leading to a significant cancellation of the effective cosmological constant, which however is still much larger than the observational limits. We further discuss some aspects of inflation in models based on wall solutions. 
  M/string theory on noncompact, negatively curved, cosets which generalize $AdS_{D+1}=SO(D,2)/SO(D,1)$ is considered. Holographic descriptions in terms of a conformal field theory on the boundary of the spacetime are proposed. Examples include $SU(2,1)/U(2)$, which is a Euclidean signature (4,0) space with no supersymmetry, and $SO(2,2)/SO(2)$ and $SO(3,2)/SO(3)$, which are Lorentzian signature (4,1) and (6,1) spaces with eight supersymmetries. Qualitatively new features arise due to the degenerate nature of the conformal boundary metric. 
  We here present, in modern notation, the classification of the discrete finite subgroups of SU(4) as well as the character tables for the exceptional cases thereof (Cf. http://pierre.mit.edu/~yhe/su4.ct). We hope this catalogue will be useful to works on string orbifold theories, quiver theories, WZW modular invariants, Gorenstein resolutions, nonlinear sigma-models as well as some recently proposed inter-connections among them. 
  Motivated by the debate of possible definitions of mass and width of resonances for $Z$-boson and hadrons, we suggest a definition of unstable particles by ``minimally complex'' semigroup representations of the Poincar\'e group characterized by $(j,{\mathsf s}=(m-i\Gamma/2)^{2})$ in which the Lorentz subgroup is unitary. This definition, though decidedly distinct from those based on various renormalization schemes of perturbation theory, is intimately connected with the first order pole definition of the $S$-matrix theory in that the complex square mass $(m-i\Gamma/2)^{2}$ characterizing the representation of the Poincar\'e semigroup is exactly the position ${\mathsf s}_R$ at which the $S$-matrix has a simple pole. Wigner's representations $(j,m)$ are the limit case of the complex representations for $\Gamma=0$. These representations have generalized vectors (Gamow kets) which have, in addition to the $S$-matrix pole at ${\mathsf s}=(m-i\Gamma/2)^{2}$, all the other properties that heuristically the unstable states need to possess: a Breit-Wigner distribution in invariant square mass and a lifetime $\tau=\frac{1}{\Gamma}$ defined by the exactly exponential law for the decay probability ${\cal P}(t)$ and rate $\dot{\cal P}(t)$ given by an exact Golden Rule which becomes Dirac's Golden Rule in the Born-approximation. In addition and unintended, they have an asymmetric time evolution. 
  The integrability of the N-cosine model, a N-field generalization of the sine-Gordon model, is investigated. We establish to first order in conformal perturbation theory that, for arbitrary N, the model possesses a quantum conserved current of Lorentz spin 3 on a submanifold of the parameter space where the interaction becomes marginal. The integrability of the model on this submanifold is further studied using renormalization techniques. It is shown that for N = 2, 3, and 4 there exist special points on the marginal manifold at which the N-cosine model is equivalent to models of Gross-Neveu type and therefore is integrable. In the 2-field case we further argue that the points mentioned above exhaust all integrable cases on the marginal submanifold. 
  We derive a generalized Skyrme-Faddeev action as the effective action of QCD in the low energy limit. Our result demonstrates the existence of a mass gap in QCD which triggers the confinement of color. 
  We write the BRST operator of the N=2 superstring as $Q= e^{-R} (\oint \frac{dz}{2\pi i} ~ b \gamma_+ \gamma_-)e^R$ where $b$ and $gamma_\pm$ are super-reparameterization ghosts. This provides a trivial proof of the nilpotence of this operator. 
  We study N=1 SUSY theories in four dimensions with multiple discrete vacua, which admit solitonic solutions describing segments of domain walls meeting at one-dimensional junctions. We show that there exist solutions preserving one quarter of the underlying supersymmetry -- a single Hermitian supercharge. We derive a BPS bound for the masses of these solutions and construct a solution explicitly in a special case. The relevance to the confining phase of N=1 SUSY Yang-Mills and the M-theory/SYM relationship is discussed. 
  We present an Sp(2n,R) duality invariant Born-Infeld U(1)^2n gauge theory with scalar fields. To implement this duality we had to introduce complex gauge fields and as a result the rank of the duality group is only half as large as that of the corresponding Maxwell gauge theory with the same number of gauge fields. The latter is self-dual under Sp(4n,R), the largest allowed duality group. A special case appears for n=1 when one can also write an SL(2,R) duality invariant Born-Infeld theory with a real gauge field. We also describe the supersymmetric version of the above construction. 
  The question of how infalling matter in a pure state forms a Schwarzschild black hole that appears to be at non-zero temperature is discussed in the context of the AdS/CFT connection. It is argued that the phenomenon of self-thermalization in non-linear (chaotic) systems can be invoked to explain how the boundary theory, initially at zero temperature self thermalizes and acquires a finite temperature. Yang-Mills theory is known to be chaotic (classically) and the imaginary part of the gluon self-energy (damping rate of the gluon plasma) is expected to give the Lyapunov exponent. We explain how the imaginary part would arise in the corresponding supergravity calculation due to absorption at the horizon of the black hole. 
  Recently it has been suggested by S. Carlip that black hole entropy can be derived from a central charge of the Virasoro algebra arising as a subalgebra in the surface deformations of General Relativity in any dimension. Here it is shown that the argumentation given in Section 2 of hep-th/9812013 and based on the Regge-Teitelboim approach is unsatisfactory. The functionals used are really ``non-differentiable'' under required variations and also the standard Poisson brackets for these functionals are exactly zero so being unable to get any Virasoro algebra with a central charge. Nevertheless Carlip's calculations will be correct if we admit another definition for the Poisson bracket. This new Poisson bracket differs from the standard one in surface terms only and allows to work with ``non-differentiable'' functionals. 
  We introduce and study one parameter family of integrable quantum field theories. This family has a Lagrangian description in terms of massive Thirring fermions $\psi,\psi^{\dagger}$ and charged bosons $\chi,\bar{\chi}$ of complex sinh-Gordon model coupled with $BC_n$ affine Toda theory. Perturbative calculations, analysis of the factorized scattering theory and the Bethe ansatz technique are applied to show that under duality transformation, which relates weak and strong coupling regimes of the theory the fermions $\psi,\psi^{\dagger}$ transform to bosons and $\chi,\bar{\chi}$ and vive versa. The scattering amplitudes of neutral particles in this theory coincide exactly with S-matrix of particles in pure $BC_n$   Toda theory, i.e. the contribution of charged bosons and fermions to these amplitudes exactly cancel each other. We describe and discuss the symmetry responsible for this compensation property. 
  We investigate the generalized gauge theory which has been proposed previously and show that in two dimensions the instanton gauge fixing of the generalized topological Yang-Mills action leads to a twisted N=2 supersymmetric action. We have found that the R-symmetry of N=2 supersymmetry can be identified with the flavour symmetry of Dirac-Kahler fermion formulation. Thus the procedure of twist allows topological ghost fields to be interpreted as the Dirac-Kahler matter fermions. 
  We present new analytic and numerical results for self-gravitating SU(2)-Higgs magnetic monopoles approaching the black hole threshold. Our investigation extends to large Higgs self-coupling, lambda, a regime heretofore unexplored. When lambda is small, the critical solution where a horizon first appears is extremal Reissner-Nordstrom outside the horizon but has a nonsingular interior. When lambda is large, the critical solution is an extremal black hole with non-Abelian hair and a mass less than the extremal Reissner-Nordstrom value. The transition between these two regimes is reminiscent of a first-order phase transition. We analyze in detail the approach to these critical solutions as the Higgs expectation value is varied, and compare this analysis with the numerical results. 
  We review new aspects of integrable systems discovered recently in N=2 supersymmetric gauge theories and their topologically twisted versions. The main topics are (i) an explicit construction of Whitham deformations of the Seiberg-Witten curves for classical gauge groups, (ii) its application to contact terms in the u-plane integral of topologically twisted theories, and (iii) a connection between the tau functions and the blowup formula in topologically twisted theories. 
  We present a regularization scheme which respects the supersymmetry and the maximal background gauge covariance in supersymmetric chiral gauge theories. When the anomaly cancellation condition is satisfied, the effective action in the superfield background field method automatically restores the gauge invariance without counterterms. The scheme also provides a background gauge covariant definition of composite operators that is especially useful in analyzing anomalies. We present several applications: The minimal consistent gauge anomaly; the super-chiral anomaly and the superconformal anomaly; as the corresponding anomalous commutators, the Konishi anomaly and an anomalous supersymmetric transformation law of the supercurrent (the ``central extension'' of N=1 supersymmetry algebra) and of the R-current. 
  We derive the full Kaluza-Klein spectrum of type IIB supergravity compactified on AdS_5 x T^{11} with T^{11} = (SU(2) x SU(2))/U(1). From the knowledge of the spectrum and general multiplet shortening conditions, we make a refined test of the AdS/CFT correspondence, by comparison between various shortenings of SU(2,2|1) supermultiplets on AdS_5 and different families of boundary operators with protected dimensions. Additional towers of long multiplets with rational dimensions, that are not protected by supersymmetry, are also predicted from the supergravity analysis. 
  We investigate the collapse of a spherical shell of matter in an anti-de Sitter space. We search for a holographic description of the collapsing shell in the boundary theory. It turns out that in the boundary theory it is possible to find information about the radial size of the shell. The shell deforms the background spacetime, and the deformed background metric enters into the action of a generic bulk field. As a consequence, the correlators of operators coupling to the bulk field are modified. By studying the analytic structure of the correlators, we find that in the boundary theory there are unstable excitations ("shellons") whose masses are multiples of a scale set by the radius of the shell. We also comment on the relation between black hole formation in the bulk and thermalization in the boundary. 
  We examine the dynamics induced on the four dimensional boundary of a five dimensional anti-deSitter spacetime by the five dimensional Chern-Simons theory with gauge group the direct product of SO(4,2) with U(1). We show that, given boundary conditions compatible with the geometry of 5d AdS spacetime in the asymptotic region, the induced surface theory is the $WZW_4$ model. 
  Fractons are anyons classified into equivalence classes and they obey a specific fractal statistics. The equivalence classes are labeled by a fractal parameter or Hausdorff dimension $h$. We consider this approach in the context of the Fractional Quantum Hall Effect (FQHE) and the concept of duality between such classes, defined by $\tilde{h}=3-h$ shows us that the filling factors for which the FQHE were observed just appear into these classes. A connection between equivalence classes $h$ and the modular group for the quantum phase transitions of the FQHE is also obtained. A $\beta-$function is defined for a complex conductivity which embodies the classes $h$. The thermodynamics is also considered for a gas of fractons $(h,\nu)$ with a constant density of states and an exact equation of state is obtained at low-temperature and low-density limits. We also prove that the Farey sequences for rational numbers can be expressed in terms of the equivalence classes $h$. 
  We study brane configurations corresponding to D-branes on complex three-dimensional orbifolds ${\bf C}^3/\Gamma$ with $\Gamma=\Delta(3n^2)$ and $\Delta(6n^2)$, nonabelian finite subgroups of SU(3). We first construct a brane configuration for ${\bf C}^3/{\bf Z}_n \times {\bf Z}_n$ by using D3-branes and a web of (p,q) 5-branes of type IIB string theory. Brane configurations for the nonabelian orbifolds are obtained by performing certain quotients on the configuration for ${\bf C}^3/{\bf Z}_n \times {\bf Z}_n$. Structure of the quiver diagrams of the groups $\Delta(3n^2)$ and $\Delta(6n^2)$ can be reproduced from the brane configurations. We point out that the brane configuration for ${\bf C}^3/\Gamma$ can be regarded as a physical realization of the quiver diagram of $\Gamma$. Based on this observation, we discuss that three-dimensional McKay correspondence may be interpreted as T-duality. 
  According to the positive energy conjecture of Horowitz and Myers, there is a specific supergravity solution, AdS soliton, which has minimum energy among all asymptotically locally AdS solutions with the same boundary conditions. Related to the issue of semiclassical stability of AdS soliton in the context of pure gravity with a negative cosmological constant, physical boundary conditions are determined for an instanton solution which would be responsible for vacuum decay by barrier penetration. Certain geometric properties of instantons are studied, using Hermitian differential operators. On a $d$-dimensional instanton, it is shown that there are $d-2$ harmonic functions. A class of instanton solutions, obeying more restrictive boundary conditions, is proved to have $d-1$ Killing vectors which also commute. All but one of the Killing vectors are duals of harmonic one-forms, which are gradients of harmonic functions, and do not have any fixed points. 
  It is pointed out that the Parke-Taylor or maximally helicity violating amplitudes in the pure Yang-Mills can, after some specifications, be interpreted as amplitudes of scattering of massive vector bosons in the Higgs-Yang-Mills system. 
  The pahse transition from instanton-dominated quantum tunneling regime to sphaleron-dominated classical crossover regime is explored in (1+1)-dimensional scalar field theory when spatial coordinate is compactified. It is shown that the type of sphaleron transition is critically dependent on the circumference of the spatial coordinate. 
  Quantization of 4-dimensional Nambu-Goto theory of open string in light cone gauge, related in Lorentz-invariant way with the world sheet, is performed. Obtained quantum theory has no anomalies in Lorentz group. Determined spin-mass spectra of the theory have Regge-like behavior and do not contain the tachyon. Vertex operators of interaction theory, acting in the physical subspace, are constructed. 
  Kaluza--Klein compactification in quantum field theory is analysed from the perturbation theory viewpoint. Renormalisation group analysis for compactification size dependence of the coupling constant is proposed. 
  The problem of 1-dimensional ultra-relativistic scattering of 2 identical charged particles in classical electrodynamics with retarded and advanced interactions is investigated. 
  We consider cosmological solutions of string and M-theory compactified to four dimensions by giving a general prescription to construct four-dimensional modular cosmologies with two commuting Killing vectors from vacuum solutions. By lifting these solutions to higher dimensions we analyze the existence of cosmological singularities and find that, in the case of non-closed Friedmann-Robertson-Walker universes, singularities can be removed from the higher-dimensional model when only one of the extra dimensions is time-varying. By studying the moduli space of compactifications of M-theory resulting in homogeneous cosmologies in four dimensions we show that U-duality transformations map singular cosmologies into non-singular ones. 
  Four different types of free energies are computed by both thermodynamical Bethe Ansatz (TBA) techniques and by weak coupling perturbation theory in an integrable one-parameter deformation of the O(4) principal chiral sigma-model (with SU(2)xU(1) symmetry). The model exhibits both `fermionic' and `bosonic' type free energies and in all cases the perturbative and the TBA results are in perfect agreement, strongly supporting the correctness of the proposed S matrix. The mass gap is also computed in terms of the Lambda parameters of the modified minimal substraction scheme and a lattice regularized version of the model. 
  The 'tHooft's 5N-parametric multiinstanton solution is generalized to curvilinear coordinates. Expressions can be simplified by a gauge transformation that makes $\eta$-symbols constant in the vierbein formalism. This generates the compensating addition to the gauge potential of pseudoparticles. Typical examples (4-spherical, 2+2- and 3+1-cylindrical coordinates) are studied and explicit formulae presented for reference. Singularities of the compensating field are discussed. They are irrelevant for physics but affect gauge dependent quantities. 
  The quantum string emission by Black Holes is computed in the framework of the `string analogue model' (or thermodynamical approach), which is well suited to combine QFT and string theory in curved backgrounds (particulary here, as black holes and strings posses intrinsic thermal features and temperatures). The QFT-Hawking temperature T_H is upper bounded by the string temperature T_S in the black hole background. The black hole emission spectrum is an incomplete gamma function of (T_H - T_S). For T_H << T_S, it yields the QFT-Hawking emission. For T_H \to T_S, it shows highly massive string states dominate the emission and undergo a typical string phase transition to a microscopic `minimal' black hole of mass M_{\min} or radius r_{\min} (inversely proportional to T_S) and string temperature T_S. The semiclassical QFT black hole (of mass M and temperature T_H) and the string black hole (of mass M_{min} and temperature T_S) are mapped one into another by a `Dual' transform which links classical/QFT and quantum string regimes. The string back reaction effect (selfconsistent black hole solution of the semiclassical Einstein equations with mass M_+ (radius r_+) and temperature T_+) is computed. Both, the QFT and string black hole regimes are well defined and bounded: r_{min} leq r_+ \leq r_S, M_{min} \leq M_+ \leq M, T_H \leq T_+ \leq T_S. The string `minimal' black hole has a life time tau_{min} \simeq \frac{k_B c}{G \hbar} T^{-3}_S. 
  We calculate the vacuum expectation values of local fields for the two-parameter family of integrable field theories introduced and studied by Fateev. Using this result we propose an explicit expression for the vacuum expectation values of local operators in parafermionic sine-Gordon models and in integrable perturbed SU(2) coset conformal field theories. 
  An exact superpotential is derived for the N=1 theories which arise as massive deformations of N=4 supersymmetric Yang-Mills (SYM) theory. The superpotential of the SU(N) theory formulated on R^{3}\times S^{1} is shown to coincide with the complexified potential of the N-body elliptic Calogero-Moser Hamiltonian. This superpotential reproduces the vacuum structure predicted by Donagi and Witten for the corresponding four-dimensional theory and also transforms covariantly under the S-duality group of N=4 SYM. The analysis yields exact formulae with interesting modular properties for the condensates of gauge-invariant chiral operators in the four-dimensional theory. 
  We consider D3 branes at orbifolded conifold singularities which are not quotient singularities. We use toric geometry and gauged linear sigma model to study the moduli space of the gauge theories on the D3 branes. We find that topologically distinct phases are related by a flop transition. It is also shown that an orbifold singularity can occur in some phases if we give expectation values to some of the chiral fields. 
  The super 0-brane and GS superstring actions on AdS$_2 \times S^2$ background with 2-form flux are constructed by supercoset approach. We find the super 0-brane action contains two parameters which are interpreted as the electric and magnetic charges of the super 0-brane. The obtained super 0-brane action describes the BPS saturated dyonic superparticle moving on AdS$_2 \times S^2$ background. The WZ action contains the required coupling with 2-form flux. For GS superstring, we find the string action on AdS$_2 \times S^2$ takes the same form as those in AdS$_3 \times S^3$ and AdS$_5 \times S^5$ with RR field background. 
  Using a sequence of Bogoliubov transformations, we obtain an exact expression for the vacuum state of a Dirac field in 2+1 dimensions in presence of a constant magnetic field. This expression reveals a peculiar two level pairing structure for any value of the mass m greater than zero. This calculation clarifies the nature of the condensate in the lowest Landau level whose existence has been emphasized recently by a several authors. 
  In a nonrelativistic contact four-fermion model we show that simple regularisation prescriptions together with a definite fine-tuning of the cut-off-parameter dependence of ``bare'' quantities give the exact solutions for the two-particle sector and Goldstone modes. Their correspondence with the self-adjoint extension into Pontryagin space is established leading to self-adjoint semi-bounded Hamiltonians in three-particle sectors as well. Renormalized Faddeev equations for the bound states with Fredholm properties are obtained and analysed. 
  The spectrum of massless bosonic and fermionic fluids satisfying the equation of state $p=(\gamma-1)\rho$ is derived using elementary statistical methods. As a limiting case, the Lorentz invariant spectrum of the vacuum ($\gamma=0, p=-\rho$) is deduced. These results are in agreement with our earlier derivation for bosons using thermodynamics and semiclassical considerations. 
  It is shown in the framework of the N-component scalar model that the saddle point structure may generate non-trivial renormalization group flow. The spinodal phase separation can be described in this manner and a flat action is found as an exact result which is valid up to any order of the loop expansion. The correlation function is computed in a mean-field approximation. 
  In these lecture notes duality tests and instanton effects in supersymmetric vacua of string theory are discussed.  A broad overview of BPS-saturated terms in the effective actions is first given. Their role in testing the consistency of duality conjectures as well as discovering the rules of instanton calculus in string theory is discussed. The example of heterotic/type-I duality is treated in detail. Thresholds of F^4 and R^4 terms are used to test the duality as well as to derive rules for calculated with D1-brane instantons. We further consider the case of R^2 couplings in N=4 ground-states. Heterotic/type II duality is invoked to predict the heterotic NS5-brane instanton corrections to the R^2 threshold. The R^4 couplings of type-II string theory with maximal supersymmetry are also analysed and the D-instanton contributions are described Other applications and open problems are sketched. 
  The embedding procedure of Batalin, Fradkin, and Tyutin, which allows to convert a second-class system into first-class, is pushed beyond the formal level. We explicitly construct, in all cases, the variables of the converted first-class theory in terms of those of the corresponding second-class one. Moreover, we only conclude about the equivalence between these two different kind of theories after compairing their respective spectra of excitations. 
  We construct three-dimensional N=1 QED with N_f flavors using branes of type IIB string theory. This theory has a mirror, which can be realized using the S-dual brane configuration. As in examples with more supersymmetry, the Higgs branch of the original theory gets mapped into the Coulomb branch of the mirror. We use parity invariance to argue that these branches cannot be lifted by quantum corrections. 
  We study the behaviour of Wilson and 't Hooft loop operators for the 2+1 dim. Abelian-Higgs model with Chern-Simons term. The phase of topological symmetry breaking where the vortex field condenses, found by Samuel for the model in the absence of the Chern-Simons term, persists in its presence. In this phase, the topological linking of instantons, which are configurations of closed vortex loops, with the Wilson loop on one hand and with the 't Hooft loop on the other hand gives rise to a long-range, logarithmic, confining potential between electric charges and magnetic flux tubes. This is surprising since the gauge field is short range due to the Chern-Simons term. Gauss' law forces the concomitance of charge and magnetic flux, hence the confinement is actually of anyons. 
  We provide a background-independent formulation of the holographic principle. It permits the construction of embedded hypersurfaces (screens) on which the entire bulk information can be stored at a density of no more than one bit per Planck area. Screens are constructed explicitly for AdS, Minkowski, and de Sitter spaces with and without black holes, and for cosmological solutions. The properties of screens provide clues about the character of a manifestly holographic theory. 
  We study the low-energy effective theory of N=2 supersymmetric Yang-Mills theory with ADE gauge groups in view of the spectral curves of the periodic Toda lattice and the A-D-E singularity theory. We examine the exact solutions by using the Picard-Fuchs equations for the period integrals of the Seiberg-Witten differential. In particular, we find that the superconformal fixed point in the strong coupling region of the Coulomb branch is characterized by the ADE superpotential. We compute the scaling exponents, which agree with the previous results. 
  In this paper we discuss two approaches to anomaly-free quantization of a two-dimensional string. The first approach is based on the canonical Dirac prescription of quantization of degenerated systems. At the second approach we "weaken" the Dirac quantization conditions requiring the solving of first class constraints only in the sense of mean values. At both approaches there are no states with the indefinite metrics. 
  A {\cal R} "dual" transform is introduced which relates Quantum Field Theory and String regimes, both in a curved background with D-non compact dimensions. This operation maps the characteristic length of one regime into the other (and, as a consequence, mass domains as well). The {\cal R}-transform is not an assumed or {\it a priori} imposed symmetry but is revealed by the QFT and String dynamics in curved backgrounds. The Hawking-Gibbons temperature and the string maximal or critical temperature are {\cal R}-mapped one into the other. If back reaction of quantum matter is included, Quantum Field Theory and String phases appear, and {\cal R}-relations between them manifest as well. These {\cal R}-transformations are explicitly shown in two relevant examples: Black Hole and de Sitter space times. 
  A broad class of contour gauges is shown to be determined by admissible contractions of the geometrical region considered and a suitable equivalence class of curves is defined. In the special case of magnetostatics, the relevant electromagnetic potentials are directly related to the ponderomotive forces. Schwinger's method of extracting a gauge invariant factor from the fermion propagator could, it is argued, lead to incorrect results. Dirac brackets of both Maxwell and Yang-Mills theories are given for arbitrary admissible space-like paths. It is shown how to define a non-abelian flux and local charges which obey a local charge algebra. Fields associated with the charges differ from the electric fields of the theory by singular topological terms; to avoid this obstruction to the Gauss law it is necessary to exclude a single, gauge fixing curve from the region considered. 
  We present a brief account of a series of recent results on twisted and untwisted elliptic Calogero-Moser systems, and on their fundamental role in the Seiberg-Witten solution of gauge theories with one massive hypermultiplet in the adjoint representation of an arbitrary gauge algebra G. 
  In this paper we find automorphic functions of coset manifolds with special K\"ahler geometry. We use \zeta-functions to regularize an infinite product over integers which belong to a duality-invariant lattice, this product is known to produce duality-invariant functions. In turn these functions correspond to Eisenstein series which can be understood as string theory amplitudes that receive contributions from BPS states. The Ansatz is constructed using the coset manifold SU(1,n)\over SU(n) \times U(1) as an example but it can be generalized. Automorphic functions play an important role in the calculation of threshold corrections to gauge coupling and other stringy phenomena. We also find some connections between the theory of Abelian varieties and moduli spaces of Calabi-Yau manifolds 
  We review the Seiberg-Witten construction of low-energy effective actions and BPS spectra in SUSY gauge theories and its formulation in terms of integrable systems. It is also demonstrated how this formulation naturally appears from the compactified version of the theory with partially broken supersymmetry so that the integrable structures arise from the relation between bare and quantum variables and superpotentials of SUSY gauge theories. The Whitham integrable systems, literally corresponding to the uncompactified theory, are then restored by averaging over fast variables in the decompactification limit. 
  We provide a non-renormalization theorem for the coefficients of the conformal anomaly associated with operators with vanishing anomalous dimensions. Such operators include conserved currents and chiral operators in superconformal field theories. We illustrate the theorem by computing the conformal anomaly of 2-point functions both by a computation in the conformal field theory and via the adS/CFT correspondence. Our results imply that 2- and 3-point functions of chiral primary operators in N=4 SU(N) SYM will not renormalize provided that a ``generalized Adler-Bardeen theorem'' holds. We further show that recent arguments connecting the non-renormalizability of the above mentioned correlation functions to a bonus U(1)_Y symmetry are incomplete due to possible U(1)_Y violating contact terms. The tree level contribution to the contact terms may be set to zero by considering appropriately normalized operators. Non-renormalizability of the above mentioned correlation functions, however, will follow only if these contact terms saturate by free fields. 
  An example of a non-Abelian Brane Box Model, namely one corresponding to a $Z_k \times D_{k'}$ orbifold singularity of $\C^3$, is constructed. Its self-consistency and hence equivalence to geometrical methods are subsequently shown. It is demonstrated how a group-theoretic twist of the non-Abelian group circumvents the problem of inconsistency that arise from na\"{\i}ve attempts at the construction. 
  In the London limit of the Ginzburg-Landau theory (Abelian Higgs model), vortex dipoles (small vortex loops) are treated as a grand canonical ensemble in the dilute gas approximation. The summation over these objects with the most general rotation- and translation invariant measure of integration over their shapes leads to effective sine-Gordon theories of the dual fields. The representations of the partition functions of both grand canonical ensembles are derived in the form of the integrals over the vortex dipoles and the small vortex loops, respectively. By virtue of these representations, the bilocal correlator of the vortex dipoles (loops) is calculated in the low-energy limit. It is further demonstrated that once the vortex dipoles (loops) are considered as such an ensemble rather than individual ones, the London limit of the Ginzburg-Landau theory (Abelian Higgs model) with external monopoles is equivalent up to the leading order in the inverse UV cutoff to the compact QED in the corresponding dimension with the charge of Cooper pairs changed due to the Debye screening. 
  The relation between SU(2) Yang-Mills mechanics, originated from the 4-dimensional SU(2) Yang-Mills theory under the supposition of spatial homogeneity of the gauge fields, and the Euler-Calogero-Moser model is discussed in the framework of Hamiltonian reduction. Two kinds of reductions of the degrees of freedom are considered: due to the gauge invariance and due to the discrete symmetry. In the former case, it is shown that after elimination of the gauge degrees of freedom from the SU(2) Yang-Mills mechanics the resulting unconstrained system represents the ID_3 Euler-Calogero-Moser model with an external fourth-order potential. Whereas in the latter, the IA_6 Euler-Calogero-Moser model embedded in an external potential is derived whose projection onto the invariant submanifold through the discrete symmetry coincides again with the SU(2) Yang-Mills mechanics. Based on this connection, the equations of motion of the SU(2) Yang-Mills mechanics in the limit of the zero coupling constant are presented in the Lax form. 
  The mechanism of dynamical chiral symmetry breaking is studied in the Abelian version of the gauged Nambu-Jona-Lasinio model in four dimensions. The most interesting feature of the gauged Nambu-Jona-Lasinio model is the appearance of relevant (renormalizable) four-fermion interactions near a critical curve separating a chiral symmetric and a dynamically chiral symmetry broken phase. The first three chapters of the thesis are introductory. Chapter 4 is based on hep-th/9712123. In an attempt to go beyond standard mean field approximations for four-fermion interactions, the 1/N expansion is utilized in chapter 5. Within the 1/N expansion, where N is the number of fermion flavors, it is shown that the renormalization group beta function of the U(1) gauge coupling has ultra-violet stable fixed points for sufficiently large N. This implies that the gauged Nambu-Jona-Lasinio model is a rare example of a nontrivial nonasymptotically free gauge field theory in four dimensions. 
  In this letter we show that the ``preferred'' Klein-Gordon Quantum Field Theories (QFT's) on a d-dimensional de Sitter spacetime can be obtained from a Klein-Gordon QFT on a (d+1)-dimensional ``ambient'' Minkowski spacetime satisfying the spectral condition and, conversely, that a Klein-Gordon QFT on a (d+1)-dimensional ``ambient'' Minkowski spacetime satisfying the spectral condition can be obtained as superposition of d-dimensional de Sitter Klein-Gordon fields in the preferred vacuum. These results establish a correspondence between QFT's living on manifolds having different dimensions. The method exposed here can be applied to study other situations and notably QFT on Anti de Sitter spacetime. 
  We show that two-dimensional sigma models are equivalent to certain perturbed conformal field theories. When the fields in the sigma model take values in a space G/H for a group G and a maximal subgroup H, the corresponding conformal field theory is the $k\to\infty$ limit of the coset model $(G/H)_k$, and the perturbation is related to the current of G. This correspondence allows us for example to find the free energy for the "O(n)" (=O(n)/O(n-1)) sigma model at non-zero temperature. It also results in a new approach to the CP^{n} model. 
  The canonical quantization is performed at a light-front surface for the SU(N) Yang-Mills theory. The Weyl gauge is imposed as a gauge condition. The suitable parameterization is chosen for the transverse gauge field components in order to have Dirac brackets independent of interactions. The generating functional is defined for the perturbation theory and it is shown to coincide with its equal-time counterpart. 
  We present a model for high energy two body scattering in a quantum theory of gravity. The model is applicable for center of mass energies higher than the relevant Planck scale. At impact parameters smaller than the Schwarzchild radius appropriate to the center of mass energy and total charge of the initial state, the cross section is dominated by an inelastic process in which a single large black hole is formed. The black hole then decays by Hawking radiation. The elastic cross section is highly suppressed at these impact parameters because of the small phase space for thermal decay into a high energy two body state. For very large impact parameter the amplitude is dominated by eikonalized single graviton exchange. At intermediate impact parameters the scattering is more complicated, but since the Schwarzchild radius grows with energy, we speculate that a more sophisticated eikonal calculation which uses the nonlinear classical solutions of the field equations may provide a good approximation at all larger impact parameters. We discuss the extent to which black hole production will be observable in theories with low scale quantum gravity and large dimensions. 
  We compute threshold effects to gauge couplings in four-dimensional $Z_N$ orientifold models of type I strings with ${\cal N}=2$ and ${\cal N}=1$ supersymmetry, and study their dependence on the geometric moduli. We also compute the tree-level (disk) couplings of the open sector gauge fields to the twisted closed string moduli of the orbifold in various models and study their effects and that of the one-loop threshold corrections on gauge coupling unification. We interpret the results from the (supergravity) effective theory point of view and comment on the conjectured heterotic-type I duality 
  The horizon of a static black hole in Anti-deSitter space can be spherical, planar, or hyperbolic. The microscopic dynamics of the first two classes of black holes have been extensively discussed recently within the context of the AdS/CFT correspondence. We argue that hyperbolic black holes introduce new and fruitful features in this respect, allowing for more detailed comparisons between the weak and strong coupling regimes. In particular, by focussing on the stress tensor and entropy of some particular states, we identify unexpected increases in the entropy of Super-Yang-Mills theory at strong coupling that are not accompanied by increases in the energy. We describe a highly degenerate state at zero temperature and zero energy density. We also find that the entanglement entropy across a Rindler horizon in exact AdS_5 is larger than might have been expected from the dual SYM theory. Besides, we show that hyperbolic black holes can be described as thermal Rindler states of the dual conformal field theory in flat space. 
  A supersymmetric action functional describing the interaction of the fundamental superstring with the D=10, type IIB Dirichlet super-9-brane is presented. A set of supersymmetric equations for the coupled system is obtained from the action principle. It is found that the interaction of the string endpoints with the super-D9-brane gauge field requires some restrictions for the image of the gauge field strength. When those restrictions are not imposed, the equations imply the absence of the endpoints, and the equations coincide either with the ones of the free super-D9-brane or with the ones for the free closed type IIB superstring. Different phases of the coupled system are described. A generalization to an arbitrary system of intersecting branes is discussed. 
  The duality symmetric but not manifestly covariant action proposed by Schwarz-Sen is canonically quantized in the Coulomb gauge. The resulting theory turns out to be, nevertheless, relativistically invariant. It is shown, afterwards, that the Schwarz-Sen model naturally emerges when duality is implemented as a local symmetry of sourceless electrodynamics. This implies in the equivalence of these theories at the quantum level. 
  We conjecture that the neutral black hole pair production is related to the vacuum fluctuation of pure gravity via the Casimir-like energy. A generalization of this process to a multi-black hole pair is considered. Implications on the foam-like structure of spacetime and on the cosmological constant are discussed. 
  This is a thesis/review article that combines some of the results of hep-th/9809061, hep-th/9810224 and hep-th/9901135 with a short discussion of introductory background material; an attempt has been made to present the work in a self-contained manner. The first chapter mostly targets readers who are vaguely familiar with traditional and contemporary string theory. Chapter two discusses in detail the thermodynamics of the 0+1 dimensional Super Yang-Mills (SYM) theory as an illustrative example of the main ideas of the work. The third chapter outlines the phase structures of p+1 dimensional SYM theories on tori for 1<=p<=5, and that of the D1D5 system; we avoid presenting the technical details of the construction of these phase diagrams, focusing instead on the physics of the final results. The last chapter discusses the dynamics of the formation of boosted black holes in strongly coupled SYM theory. 
  We introduce a Lagrangian density for M-Theory which is purely topological using Gelfand-Fuchs cohomology. Next we calculate the partition function which indeed gives topological invariants that can be expressed via the Ray-Singer analytic torsion. 
  We discuss local mirror symmetry for higher-genus curves. Specifically, we consider the topological string partition function of higher-genus curves contained in a Fano surface within a Calabi-Yau. Our main example is the local P^2 case. The Kodaira-Spencer theory of gravity, tailored to this local geometry, can be solved to compute this partition function. Then, using the results of Gopakumar and Vafa and the local mirror map, the partition function can be rewritten in terms of expansion coefficients, which are found to be integers. We verify, through localization calculations in the A-model, many of these Gromov-Witten predictions. The integrality is a mystery, mathematically speaking. The asymptotic growth (with degree) of the invariants is analyzed. Some suggestions are made towards an enumerative interpretation, following the BPS-state description of Gopakumar and Vafa. 
  We initiate a study of cosmology within the framework of Maldacena's AdS/CFT correspondence. We present a comprehensive analysis of the classical motion of a charged domain wall that separates an external Reissner-Nordstrom region of spacetime (with small or vanishing cosmological constant) from an internal de-Sitter region. The possible associated spacetime diagrams are drawn, although in the classical case, an unambiguous prediction of what occurs at late times in the interior region is not possible, since singularities and Cauchy horizons form. We argue that, when the asymptotic region is anti-de Sitter, the AdS/CFT correspondence gives a prescription for resolving the curvature singularities and evolving solutions across the expected Cauchy horizon. Some of our solutions contain inflating interiors, and we provide evidence these can be patched onto solutions with smooth initial data, circumventing an obstacle found by Farhi and Guth to creating an inflating universe in the laboratory. 
  We discuss the dynamics and thermodynamics of particle and D-brane probes moving in non-extremal black hole/brane backgrounds. When a probe falls from asymptotic infinity to the horizon, it transforms its potential energy into heat, $TdS$, which is absorbed by the black hole in a way consistent with the first law of thermodynamics. We show that the same remains true in the near-horizon limit, for BPS probes only, with the BPS probe moving from AdS infinity to the horizon. This is a quantitative indication that the brane-probe reaching the horizon corresponds to thermalization in gauge theory. It is shown that this relation provides a way to reliably compute the entropy away from the extremal limit (towards the Schwarzschild limit). 
  Powerful methods based on supersymmetry allow one to find exact solutions to certain problems in strong coupling gauge theories. The inception of some of these methods (holomorphy in the gauge coupling and other chiral parameters, in conjunction with instanton calculations) dates back to the 1980's. I describe the early exact results -- the calculation of the beta function and the gluino condensate -- and their impact on the subsequent developments. A brief discussion of the recent breakthrough discoveries where these results play a role is given. 
  Solutions of classical string theory, correspondent to the world sheets, mapped in Minkowsky space with a fold, are considered. Typical processes for them are creation of strings from vacuum, their recombination and annihilation. These solutions violate positiveness of square of mass and Regge condition. In quantum string theory these solutions correspond to physical states |DDF>+|sp> with non-zero spurious component. 
  The superconformal Ward identities combined with N=2 harmonic analyticity are used to evaluate two-loop four-point correlation functions of gauge-invariant operators in D=4, N=4 supersymmetric Yang-Mills theory in terms of the well-known one-loop box integral. The result is confirmed by a direct numerical computation. 
  Marchesini showed that the Fokker-Planck Hamiltonian for Yang-Mills theories is the loop operator. Jevicki and Rodrigues showed that the Fokker-Planck Hamiltonian of some matrix models co\"\i ncides with temporal gauge non-critical string field theory Hamiltonians constructed by Ishibashi and Kawai (and their collaborators). Thus the loop operator for Yang-Mills theory is the temporal gauge Hamiltonian for a noncritical string field theory, in accord with Polyakov's conjecture. The consistency condition of the string interpretation is the zigzag symmetry emphasized by Polyakov. Several aspects of the noncritical string theory are considered, relating the string field theory Hamiltonian to the worldsheet description. 
  We consider N = 2 super Yang-Mills theory with SU(2) gauge group and a single quark hypermultiplet in the fundamental representation. For a specific value of the quark bare mass and at a certain point in the moduli space of vacua, the central charges corresponding to two mutually non-local electro-magnetic charges vanish simultaneously, indicating the possibility of massless such states in the spectrum. By realizing the theory as an M-theory configuration, we show that these states indeed exist in the spectrum near the critical point. 
  We compute the conformal anomalies of boundary CFTs for scalar and fermionic fields propagating in AdS spacetime at one-loop. The coefficients are quantized, with values related to the mass-spectra for Kaluza-Klein compactifications of Supergravity on AdS5xS5 and AdS7xS4. Our approach interprets the boundary partition function of fields propagating on AdS spacetime in terms of wave-functionals that satisfy the functional Schrodinger equation. 
  It is conjectured that the two closed bosonic string theories, Type 0A and Type 0B, correspond to certain supersymmetry breaking orbifold compactifications of M-theory. Various implications of this conjecture are discussed, in particular the behaviour of the tachyon at strong coupling and the existence of non-perturbative fermionic states in Type 0A. The latter are shown to correspond to bound states of Type 0A D-particles, thus providing further evidence for the conjecture. We also give a comprehensive description of the various Type 0 closed and open string theories. 
  The Kaluza-Klein spectrum of N=2, D=4 supergravity compactified on AdS_2 x S^2 is found and shown to consist of two infinite towers of SU(1,1|2) representations. In addition to `pure gauge' modes living on the boundary of AdS which are familiar from higher dimensional cases, in two dimensions there are modes (e.g. massive gravitons) which enjoy no gauge symmetry yet nevertheless have no on-shell degrees of freedom in the bulk. We discuss these two-dimensional subtleties in detail. 
  We examine a class of gauge theories obtained by projecting out certain fields from an N=4 supersymmetric SU(N) gauge theory. These theories are non-supersymmetric and in the large N limit are known to be conformal. Recently it was proposed that the hierarchy problem could be solved by embedding the standard model in a theory of this kind with finite N. In order to check this claim one must find the conformal points of the theory. To do this we calculate the one-loop beta functions for the Yukawa and quartic scalar couplings. We find that with the beta functions set to zero the one-loop quadratic divergences are not canceled at sub-leading order in N; thus the hierarchy between the weak scale and the Planck scale is not stabilized unless N is of the order 10^28 or larger. We also find that at sub-leading orders in N renormalization induces new interactions, which were not present in the original Lagrangian. 
  We develop an algorithm which can be used to exclude the existence of classical breathers (periodic finite energy solutions) in scalar field theories, and apply it to several cases of interest. In particular, the technique is used to show that a pair of potentially periodic solutions of the 3+1 Sine-Gordon Lagrangian, found numerically in earlier work, are not breathers. These ``pseudo-breather states'' do have a signature in our method, which we suggest can be used to find similar quasi-bound state configurations in other theories. We also discuss the results of our algorithm when applied to the 1+1 Sine-Gordon model (which exhibits a well-known set of breathers), and $\phi ^4$ theory. 
  In discussions of the T-duality between the two heterotic string theories, the duality is actually implemented through the "common" SO(16) x SO(16) subgroup of "SO(32)" and E_8 x E_8. In fact, however, a global investigation shows that no such "common" subgroup exists. This paper is a survey of the relevant global properties of Spin(32)/Z_2 and its subgroups. 
  This is a review article in which we will introduce, in a unifying fashion and with more intermediate steps in some difficult calculations, two infinite-dimensional Lie algebras of quantum matrix models, one for the open string sector and one for the closed string sector. Physical observables of quantum matrix models in the large-N limit can be expressed as elements of these Lie algebras. We will see that both algebras arise as quotient algebras of a larger Lie algebra. We will also discuss some properties of these Lie algebras not published elsewhere yet, and briefly review their relationship with well-known algebras like the Cuntz algebra, the Witt algebra and the Virasoro algebra. We will also review how Yang--Mills theory, various low energy effective models of string theory, quantum gravity, string-bit models, and quantum spin chain models can be formulated as quantum matrix models. Studying these algebras thus help us understand the common symmetry of these physical systems. 
  A field theoretic description of monopole condensation in strongly coupled gauge theories is given by actions involving antisymmetric tensors B_{\mu\nu} of rank 2. We rederive the corresponding action for 4d compact QED, summing explicitly over all possible monopole configurations. Its gauge symmetries and Ward identities are discussed. Then we consider the Wilsonian RGs for Yang-Mills theories in the presence of collective fields (again tensors B_{\mu\nu}) for the field strengths F_{\mu \nu} associated to the U(1) subgroups. We show that a ``vector-like'' Ward identity for the Wilsonian action involving B_{\mu\nu}, whose validity corresponds to monopole condensation, constitutes a fixed point of the Wilsonian RG flow. 
  The QCD-vacuum is characterized by the Higgs phenomenon for colored scalar fields. In this dual picture the gluons appear as the octet of vector mesons. Also quarks and baryons are identified. Gluon-meson and quark-baryon duality can account in a simple way for realistic masses of all low-mass hadrons and for their interactions. 
  The finite energy vibrational normal modes of the baryon number B=7 Skyrme soliton are computed. The structure of the spectrum obtained displays considerable similarity to those previously calculated for baryon numbers 2, 3 and 4. All modes expected from an approximate correspondence between skyrmions and BPS monopoles are found to be present. However, in contrast to earlier calculations, they do not all have energies below the pion mass. The remaining `breather-type' modes also conform to predictions, except that one predicted multiplet is not observed. 
  Conventional wisdom states that Newton's force law implies only four non-compact dimensions. We demonstrate that this is not necessarily true in the presence of a non-factorizable background geometry. The specific example we study is a single 3-brane embedded in five dimensions. We show that even without a gap in the Kaluza-Klein spectrum, four-dimensional Newtonian and general relativistic gravity is reproduced to more than adequate precision. 
  We define a special matrix multiplication among a special subset of $2N\x 2N$ matrices, and study the resulting (non-associative) algebras and their subalgebras. We derive the conditions under which these algebras become alternative non-associative and when they become associative. In particular, these algebras yield special matrix representations of octonions and complex numbers; they naturally lead to the Cayley-Dickson doubling process. Our matrix representation of octonions also yields elegant insights into Dirac's equation for a free particle. A few other results and remarks arise as byproducts. 
  The character of the exceptional series of representations of SU(1,1) is determined by using Bargmann's realization of the representation in the Hilbert space $H_\sigma$ of functions defined on the unit circle. The construction of the integral kernel of the group ring turns out to be especially involved because of the non-local metric appearing in the scalar product with respect to which the representations are unitary. Since the non-local metric disappears in the `momentum space' $i.e.$ in the space of the Fourier coefficients the integral kernel is constructed in the momentum space, which is transformed back to yield the integral kernel of the group ring in $H_\sigma$. The rest of the procedure is parallel to that for the principal series treated in a previous paper. The main advantage of this method is that the entire analysis can be carried out within the canonical framework of Bargmann. 
  A magnetic monopole is placed at the centre of a 3-ball whose surface, S, is tiled by the symmetry group, G, of a regular solid. The quantum mechanics on the two-dimensional quotient, S/G, is developed and the monopole charge is found to be quantized in the expected manner. The heat-kernels and zeta-functions are evaluated and the Casimir energies computed as examples of the formalism. Numerical approaches to the calculation of the derivative of the Barnes zeta-function are presented. 
  Based on the generating functional method with an external source function, a useful constraint on the source function is proposed for analyzing the one- and two-loop world-line Green functions. The constraint plays the same role as the momentum conservation law of a certain nontrivial form, and transforms ambiguous Green functions into the uniquely defined Green functions. We also argue reparametrizations of the Green functions defined on differently parameterized world-line diagrams. 
  We compute by Bethe Ansatz both bulk and boundary hole scattering matrices for the critical A_{N-1}^(1) quantum spin chain. The bulk S matrix coincides with the soliton S matrix for the A_{N-1}^(1) Toda field theory with imaginary coupling. We verify our result for the boundary S matrix using a generalization of the Ghoshal-Zamolodchikov boundary crossing relation. 
  We consider F/M/Type IIA theory compactified to four, three, or two dimensions on a Calabi-Yau four-fold, and study the behavior near an isolated singularity in the presence of appropriate fluxes and branes. We analyze the vacuum and soliton structure of these models, and show that near an isolated singularity, one often generates massless chiral superfields and a superpotential, and in many instances in two or three dimensions one obtains nontrivial superconformal field theories. In the case of two dimensions, we identify some of these theories with certain Kazama-Suzuki coset models, such as the N=2 minimal models. 
  Some of the consequences of Eyvind Wichmann's contributions to modular theory and the QFT phase-space structure are presented. In order to show the power of those ideas in contemporary problems, I selected the issue of algebraic holography as well as a new nonperturbative constructive approach (based on the modular structur of wedge-localized algebras and modular inclusions) and show that these ideas are recent consequences of the pathbreaking work which Wichmann together with his collaborator Bisognano initiated in the mid 70$^{ies}. 
  We reconsider the problem of finding all local symmetries of a Lagrangian. Our approach is completely Hamiltonian without any reference to the associated action. We present a simple algorithm for obtaining the restrictions on the gauge parameters entering in the definition of the generator of gauge transformations. 
  We derive a generalised concavity condition for potentials between static sources obtained from Wilson loops coupling both to gauge bosons and a set of scalar fields. It involves the second derivatives with respect to the distance in ordinary space as well as with respect to the relative orientation in internal space. In addition we discuss the use of this field theoretical condition as a nontrivial consistency check of the AdS/CFT duality. 
  A simple model of spacetime foam, made by N wormholes in a semiclassical approximation, is taken under examination. We show that the Bekenstein-Hawking entropy is here ``quantized'' in agreement with the heuristic calculation of Bekenstein. 
  In this thesis I present applications of the Born-Infeld action to the description of D-branes, mostly in the context of type IIB string theory. First I present the Callan-Maldacena construction of a fundamental string attached to a threebrane, whereby the string is made out of the wrapped brane. This interpretation withstands a rather detailed scrutiny, including the magnitude of the static tension and of the B-I charge; and Polchinski's mixed Dirichlet/Neumann boundary conditions for small perturbations of the string. The latter arise dynamically from the full non-linear equations of motion. Also, the Born-Infeld charge is shown to produce the correct form of dipole radiation under the influence of harmonic oscillations of the string. The second part comprises of the application of these ideas to the AdS/CFT correspondence. Using the essentially non-perturbative nature of the B-I action we were able to construct in the near-horizon geometry of N D3-branes the color-singlet baryon vertex of the SU(N) gauge theory as a fivebrane wrapped on the $S^5$ sphere, with B-I strings playing the role of the quarks. Also, in the asymptotically flat background we obtain a smooth classical description of the Hanany-Witten phenomenon. In the case of non-extremal background we construct baryons of the three-dimensional non-supersymmetric theory, the low energy dynamics of which was conjectured to be equivalent to ordinary QCD. Our approach allows to exhibit confining properties of color flux tubes, i.e. the dependence of the tension of the flux tube on its color content. We do this by building an extended baryon, where a fraction $\nu$ of the total of N quarks is removed to a finite distance. 
  A common assumption in quantum field theory is that the energy-momentum 4-vector of any quantum state must be timelike. It will be proven that this is not the case for a Dirac-Maxwell field. In this case quantum states can be shown to exist whose energy-momentum is spacelike. This implies that there must exist quantum states with less energy than the vacuum state. 
  The complete solution to the massive Rarita-Schwinger field equation in anti-de Sitter space is constructed, and used in the AdS/CFT correspondence to calculate the correlators for the boundary conformal field theory. It is found that when no condition is imposed on the field solution, there appear two different boundary conformal field operators, one coupling to a Rarita-Schwinger field and the other to a Dirac field. These two operators are seen to have different scaling dimensions, with that of the spinor-coupled operator exhibiting non-analytic mass dependence. 
  We realize the two dimensional anti-de Sitter ($AdS_2$) space as a Kaluza-Klein reduction of the $AdS_3$ space in the framework of the discrete light cone quantization (DLCQ). Introducing DLCQ coordinates which interpolate the original (unboosted) coordinates and the light cone coordinates, we discuss that $AdS_2/CFT$ correspondence can be deduced from the $AdS_3/CFT$. In particular, we elaborate on the deformation of WZW model to obtain the boundary theory for the $AdS_2$ black hole. This enables us to derive the entropy of the $AdS_2$ black hole from that of the $AdS_3$ black hole. 
  We present a manifestly Lorentz- and SO(2)-Duality-invariant local Quantum Field Theory of electric charges, Dirac magnetic monopoles and dyons. The manifest invariances are achieved by means of the PST-mechanism. The dynamics for classical point particles is described by an action functional living on a circle, if the Dirac-Schwinger quantization condition for electric and magnetic charges holds. The inconsistent classical field theory depends on an arbitrary, but fixed, external vector field, a generalization of the Dirac-string. Nevertheless, the Quantum Field Theory, obtained from this classical action via a functional integral approach, turns out to be independent of the particular vector field chosen, and thus consistent, if the Dirac-Schwinger quantization condition holds. We provide explicit expressions for the generating functionals of observables, proving that they are Dirac-string independent. Since Lorentz-invariance is manifest at each step, the quantum theory admits also a manifestly diffeomorphism invariant coupling to external gravity. Relations with previous formulations, and with SO(2)--non invariant theories are clarified. 
  We present a brief historical overview of the classical theory of a radiating point charge, described by the Lorentz-Dirac equation. A recent development is the discovery of tunnelling of a charge through a potential barrier, in a completely classical context. Also, a concrete example is discussed of the existence of several physically acceptable solutions for a range of initial data. We end by pointing out some open problems in connection with D-brane and monopole physics. 
  We use dyonic brane configurations of type 0 string theory to study large N non-supersymmetric 4d gauge theories. The brane configurations define theories similar to the supersymmetric ones which arise in type II. We find the non-SUSY analogues of N=2 and N=1. In particular we suggest new non-SUSY CFT's and a brane realization of a non-SUSY Seiberg duality. 
  We present a general review about the N=1 supersymmetric SU(Nc) Yang-Mills on the lattice focusing our attention on the quenched approximation in supersymmetry. Finally we analyse and discuss the recent results obtained at strong coupling and large Nc for the mesonic and fermionic propagators and spectrum. 
  Developing recently proposed constructions for the description of particles in the $(1/2,0)\oplus (0,1/2)$ representation space, we derive the second-order equations. The similar ones were proposed in the sixties and the seventies in order to understand the nature of various mass and spin states in the representations of the $O(4,2)$ group. We give some additional insights into this problem. The used procedure can be generalized for {\it arbitrary} number of lepton families. 
  We propose a (generalized) ``mass formula'' for a fundamental string described as a BPS solution of a D-brane's worldvolume. The mass formula is obtained by using the Hamiltonian density on the worldvolume, based on transformation properties required for it. Its validity is confirmed by investigating the cases of point charge solutions of D-branes in a D-8-brane (i.e. curved) background, where the mass of each of the corresponding strings is proportional to the geodesic distance from the D-brane to the point parametrized by the (regularized) value of a transverse scalar field. It is also shown that the mass of the string agrees with the energy defined on the D-brane's worldvolume only in the flat background limit, but the agreement does not always hold when the background is curved. 
  We discuss the free-energy density of bosonic and fermionic theories possessing strongly coupled critical points in D=3. We construct a stationary renormalization group trajectory which interpolates between the free massless theory of N scalars and a class of interacting theories including both bosons and fermions. At a special point of this trajectory the free-energy density is 5/4 times the free-energy density of the O(N) vector model at its nontrivial critical point. Our method could in principle be useful in the study of other theories with strongly coupled fixed points, such as {\cal N}=4 SYM in D=4. 
  I consider global transformations of a Dirac fermion field, that are generated by the generators of Poincar'e transformations, but with a \gamma_5 appended. Such chiral translations and chiral Lorentz transformations are usually not symmetries of the Lagrangian, but naively they are symmetries of the fermionic measure. However, by using proper time regularization in Minkowski space, I find that they in general give rise to a nontrivial Jacobian. In this sense they have "anomalies". I calculate these anomalies in a theory of a massive fermion coupled to an external Abelian vector field. My motivation for considering chiral Poincar'e transformations is the possibility that they are relevant to bosonization in four dimensions. 
  We consider the Maldacena conjecture applied to the near horizon geometry of a D1-brane in the supergravity approximation and consider the possibility of testing the conjecture against the boundary field theory calculation using DLCQ. We propose the two point function of the stress energy tensor as a convenient quantity that may be computed on both sides of the correspondence. On the supergravity side, we may invoke the methods of Gubser, Klebanov, Polyakov, and Witten. On the field theory side, we derive an explicit expression for the two point function in terms of data that may be extracted from a DLCQ calculation at a given harmonic resolution. This gives rise to a well defined numerical algorithm for computing the two point function, which we test in the context of free fermions and the 't Hooft model. For the supersymmetric Yang-Mills theory with 16 supercharges that arises in the Maldacena conjecture, the algorithm is perfectly well defined, although the size of the numerical computation grows too fast to admit any detailed analysis at present, and our results are only preliminary. We are, however, able to present more detailed results on the supersymmetric DLCQ computation of the stress energy tensor correlators for two dimensional Yang Mills theories with (1,1) and (2,2) supersymmetries. 
  The purpose of this work is to present some basic concepts about the non-linear sigma model in a simple and direct way. We start with showing the bosonic model and the Wess-Zumino-Witten term, making some comments about its topological nature, and its association with the torsion. It is also shown that to cancel the quantum conformal anomaly the model should obey the Einstein equations. We provide a quick introduction about supersymmetry in chapter 2 to help the understanding the supersymmetric extension of the model. In the last chapter we present the supersymmetric model and its equations of motion. Finally we work-out the two-supersymmetry case, introducing the chiral as well as the twisted chiral fields, expliciting the very specific $SU(2)\otimes U(1)$ case. 
  We show that the requirements of renormalizability and physical consistency imposed on perturbative interactions of massive vector mesons fix the theory essentially uniquely. In particular physical consistency demands the presence of at least one additional physical degree of freedom which was not part of the originally required physical particle content. In its simplest realization (probably the only one) these are scalar fields as envisaged by Higgs but in the present formulation without the ``symmetry-breaking Higgs condensate''. The final result agrees precisely with the usual quantization of a classical gauge theory by means of the Higgs mechanism. Our method proves an old conjecture of Cornwall, Levin and Tiktopoulos stating that the renormalization and consistency requirements of spin=1 particles lead to the gauge theory structure (i.e. a kind of inverse of 't Hooft's famous renormalizability proof in quantized gauge theories) which was based on the on-shell unitarity of the $S$-matrix. We also speculate on a possible future ghostfree formulation which avoids ''field coordinates'' altogether and is expected to reconcile the on-shell S-matrix point of view with the off-shell field theory structure. 
  We derive relations between type 0 and type II D-brane configurations under the T-duality suggested by Bergman and Gaberdiel and confirm that the massless fields on D-branes are identical to those on the dual D-brane configurations. Furthermore, we discuss dualities of type 0 and type II NS5-branes and find that the dual of an unwrapped type 0 NS5-brane is a Kaluza-Klein monopole with non-supersymmetric blow up modes. 
  The Berezinsky-Kosterlitz-Thouless (BKT) type phase transitions in two-dimensional systems with internal abelian continuous symmetries are investigated. The necessary conditions for they can take place are: 1) conformal invariance of the kinetic part of the model action, 2) vacuum manifold must be degenerated with abelian discrete homotopy group pi_1. Then topological excitations have a logarithmically divergent energy and they can be described by effective field theories generalizing the two-dimensional euclidean sine-Gordon theory, which is an effective theory of the initial XY-model. In particular, the effective actions for the two-dimensional chiral models on maximal abelian tori T_G of simple compact groups G are found. Critical properties of possible effective theories are determined and it is shown that they are characterized by the Coxeter number h_G of lattices from the series A,D,E,Z and can be interpreted as those of conformal field theories with integer central charge C=n, where n is a rank of the groups pi_1 and G. A possibility of restoration of full symmetry group G in massive phase is also dicussed. 
  The invariant integration method for Chern-Simons theory defined on the compact hyperbolic manifold {\Gamma}\H^3 is verified in the semiclassical approximation. The semiclassical limit for the partition function is presented. We discuss briefly L^2 - analytic torsion and the eta invariant of Atiyah-Patodi-Singer for compact hyperbolic 3-manifolds. 
  A systematic treatment is given of the Dirac quantisation condition for electromagnetic fluxes through two-cycles on a four-manifold space-time which can be very complicated topologically, provided only that it is connected, compact, oriented and smooth. This is sufficient for the quantised Maxwell theory on it to satisfy electromagnetic duality properties. The results depend upon whether the complex wave function needed for the argument is scalar or spinorial in nature. An essential step is the derivation of a "quantum Stokes' theorem" for the integral of the gauge potential around a closed loop on the manifold. This can only be done for an exponentiated version of the line integral (the "Wilson loop") and the result again depends on the nature of the complex wave functions, through the appearance of what is known as a Stiefel-Whitney cohomology class in the spinor case. A nice picture emerges providing a physical interpretation, in terms of quantised fluxes and wave functions, of mathematical concepts such as spin structures, spin-C structures, the Stiefel-Whitney class and Wu's formula. Relations appear between these, electromagnetic duality and the Atiyah-Singer index theorem. Possible generalisations to higher dimensions of space-time in the presence of branes is mentioned. 
  In four-dimensional N=2 compactifications of string theory or M-theory, modifications of the Bekenstein-Hawking area law for black hole entropy in the presence of higher-derivative interactions are crucial for finding agreement between the macroscopic entropy obtained from supergravity and subleading corrections to the microscopic entropy obtained via state counting. Here we compute the modifications to the area law for various classes of black holes, such as heterotic black holes, stemming from certain higher-derivative gravitational Wilsonian coupling functions. We consider the extension to heterotic N=4 supersymmetric black holes and their type-II duals and we discuss its implications for the corresponding micro-state counting. In the effective field theory approach the Wilsonian coupling functions are known to receive non-holomorphic corrections. We discuss how to incorporate such corrections into macroscopic entropy formulae so as to render them invariant under duality transformations, and we give a concrete example thereof. 
  Modifications of the area law are crucial in order to find agreement with microscopic entropy calculations based on string theory, when including contributions that are subleading for large charges. The deviations of the area law are in accord with Wald's proposal for the entropy based on a Noether charge. We discuss this for the case of four-dimensional N=2 supersymmetric black holes. 
  Classification of N=4 superconformal symmetries in two dimensions is re-examined. It is proposed that apart from SU(2) and $SU(2)\times SU(2)\times U(1)$ their Kac-Moody symmetry can also be $SU(2)\times(U(1))^4$. These superconformal symmetries and corresponding algebras are named small, large and middle ones respectively. Operator product expansions for the middle algebra are derived. Complete free field realizations of large and middle superconformal symmetries are obtained. 
  In this work we derive certain topological theories of transverse vector fields whose amplitudes reproduce topological invariants involving the interactions among the trajectories of three and four random walks. This result is applied to the construction of a field theoretical model which describes the statistical mechanics of an arbitrary number of topologically linked polymers in the context of the analytical approach of Edwards. With respect to previous attempts, our approach is very general, as it can treat a system involving an arbitrary number of polymers and the topological states are not only specified by the Gauss linking number, but also by higher order topological invariants. 
  The geodesics of the rotating extreme black hole in five spacetime dimensions found by Breckenridge, Myers, Peet and Vafa are Liouville integrable and may be integrated by additively separating the Hamilton-Jacobi equation. This allows us to obtain the St\"ackel-Killing tensor. We use these facts to give the maximal analytic extension of the spacetime and discuss some aspects of its causal structure. In particular, we exhibit a `repulson'-like behaviour occuring when there are naked closed timelike curves. In this case we find that the spacetime is geodesically complete (with respect to causal geodesics) and free of singularities. When a partial Cauchy surface exists, we show, by solving the Klein-Gordon equation, that the absorption cross-section for massless waves at small frequencies is given by the area of the hole. At high frequencies a dependence on the angular quantum numbers of the wave develops. We comment on some aspects of `inertial time travel' and argue that such time machines cannot be constructed by spinning up a black hole with no naked closed timelike curves. 
  We analyse the leading logarithmic singularities in direct and crossed channel limit of the four-point functions in dilaton-axion sector of type IIB supergravity on $AdS_{5}$ in AdS/CFT correspondence. Logarithms do not cancel in the full correlator in both channels. 
  An exact expression for the Green function of a purely fermionic system moving on the manifold $\Re \times \Sigma^{D-1}$, where $\Sigma^{D-1}$ is a $(D-1)$-torus, is found. This expression involves the bosonic analog of $\chi_n = e^{in\theta}$ corresponding to the irreducible representation for the n-th class of homotopy and in the fermionic case for D=2 and 3, $\chi_n$ is a measure of the statistics of the particles. For higher dimensions ($D \geq 4$), there is no analogue interpretation however this could, presumably, indicate a generation of mass as in quantum field theories at finite temperature. 
  We construct non-supersymmetric four dimensional gauge theories arising as effective theories of D-branes placed on various singularities in Type 0B string theory. We mostly focus on models which are conformal in the large N limit and present both examples appearing on self-dual D3-branes on orbifold singularities and examples including orientifold planes. Moreover, we derive type 0 Hanany-Witten setups with NS 5-branes intersected by D-branes and the corresponding rules for determining the massless spectra. Finally, we discuss possible duality symmetries (Seiberg-duality) for non-supersymmetric gauge theories within this framework. 
  An initial step is taken in investigating the duality between the near horizon region of a four dimensional extremal Reissner-Nordstrom black hole and the n-particle, N=4 Calogero model as conjectured by Gibbons and Townsend. Specifically we compute the mass spectrum of d=4, N=8 supergravity about the Bertotti-Robinson solution and find the corresponding set of conformal dimensions of states in the dual conformal quantum mechanics. We find that the dual states fill irreducible representations of the supergroup SU(1,1|2), and furthermore transform under various irreducible representations of the group SU(2) \times SU(6) spontaneously broken down from the E_{7(7)} duality group of N=8 supergravity. 
  We find general non-linear lagrangians of a U(1) field invariant under electric-magnetic duality. They are characterized by an arbitrary function and go to the Maxwell theory in the weak field limit. We give some explicit examples which are generalizations of the Born-Infeld theory. 
  The self-dual Chern-Simons solitons under the influence of the quantum potential are considered. The single-valuedness condition for an arbitrary integer number $N \ge 0$ of solitons leads to quantization of Chern-Simons coupling constant $\kappa = m {e^{2} \over g}$, and the integer strength of quantum potential $s = 1 - m^{2}$. As we show, the Jackiw-Pi model corresponds to the first member (m = 1) of our hierarchy of the Chern-Simons gauged nonlinear Schr\"odinger models, admitting self-dual solitons. New type of exponentially localized Chern-Simons solitons for the Bloch electrons near the hyperbolic energy band boundary are found. 
  We compute the Kaluza-Klein mass spectrum of the D=11 supergravity compactified on AdS2 x S2 x T7 and arrange it into representations of the SU(1,1|2) superconformal algebra. This geometry arises in M theory as the near horizon limit of a D=4 extremal black-hole constructed by wrapping four groups of M-branes along the T7. Via AdS/CFT correspondence, our result gives a prediction for the spectrum of the chiral primary operators in the dual conformal quantum mechanics yet to be formulated. 
  Invited talk at the conference "Fundamental Interactions: From Symmetries to Black Holes" in honor of Francois Englert, 24 - 27 March, Universite Libre de Bruxelles, Belgium 
  The notion of the integral over the anticommuting Grassmann variables is applied to analyze the fermionic structure of the 2D Ising model with quenched site dilution. In the $N$-replica scheme, the model is explicitly reformulated as a theory of interacting fermions on a lattice. For weak dilution, the continuum-limit approximation implies the log-log singularity in the specific heat near $T_c$. 
  In these lectures we give a brief introduction to perturbative and non-perturbative string theory. The outline is the following:  1. Introduction to perturbative string theory   1.1 From point particle to extended objects   1.2 Free closed and open string spectrum   1.3 Compactification on a circle and T-duality   1.4 The Superstring: type IIA and IIB   1.5 Heterotic string and orbifold compactifications   1.6 Type I string theory   1.7 Effective field theories   References  2. Introduction to non-perturbative string theory   2.1 String solitons   2.2 Non-perturbative string dualities   2.3 M-theory   2.4 Effective field theories and duality tests   References 
  In the orbifold limit of K3, one can give exact conformal field theory description of D-branes wrapped on certain non-supersymmetric cycles of K3. We study the effect of switching on the `non-geometric blow up modes' corresponding to anti-symmetric tensor gauge field flux through the 2-cycles on these D-branes. Working to first order in the blow up parameter, we determine the region of the moduli space in which these D-branes are stable. Across the boundary of this region, the D-brane wrapped on the non-supersymmetric cycle decays to a pair of D-branes, each wrapped on a supersymmetric cycle, via a second order phase transition. 
  Under very general assumptions we show that Vafa-Witten theorem on vector symmetries in vector-like theories can be extended to some physically relevant gauge theories with non-positive definite integration measure as QCD with a theta-vacuum term. 
  The solution of some equations involving functional derivatives is given as a series indexed by planar binary trees. The terms of the series are given by an explicit recursive formula. Some algebraic properties of these series are investigated. Several examples are treated in the case of quantum electrodynamics: the complete fermion and photon propagators, the two-body Green function, and the one-body Green function in the presence of an external source, the complete vacuum polarization, electron self-energy and irreducible vertex. 
  We discuss, in terms of the AdS_3-CFT_2 correspondence, a one-parameter family of (asymptotically AdS_3) conical geometries which are generated by point masses and interpolate between AdS_3 and BTZ spacetimes. We show that these correspond to spectral flow in N= (4,4) SCFT_2 which interpolate between NS and R sectors. Our method involves representing the conical spaces as solutions of three-dimensional supergravity based on the supergroup SU(1,1|2) \times SU(1,1|2). The boundary CFT we use is based on the D1/D5 system. The correspondence includes comparing the Euclidean free energies between supergravity and SCFT for the family of conical spaces including BTZ black holes. 
  We perform a precise analytic test of the instanton approximation by comparing the exact band spectrum of the periodic Lam\'e potential to the tight-binding, instanton and WKB approximations. The instanton result gives the correct leading behavior in the semiclassical limit, while the tight-binding approximation does even better. WKB is off by an overall factor of $\sqrt{e/\pi}$. 
  Thermodynamic Bethe ansatz equations are coupled non-linear integral equations which appear frequently when solving integrable models. Those associated with models with N=2 supersymmetry can be related to differential equations, among them Painleve III and the Toda hierarchy. In the simplest such case the massless limit of these non-linear integral equations can be solved in terms of the Airy function. This is the only known closed-form solution of thermodynamic Bethe ansatz equations, outside of free or classical models. This turns out to give the spectral determinant of the Schrodinger equation in a linear potential. 
  The evolution of ideas which has led from the first proofs of the renormalizability of non-abelian gauge theories, based on Slavnov--Taylor identities, to the modern proof based on the BRS symmetry and the master equation is recalled. This lecture has been delivered at the {\bf Symposium in the Honour of Professor C. N. Yang}, Stony-Brook, May 21-22 1999. 
  The origin of the rather mysterious duality symmetry found in quantum Liouville theory is investigated by considering the Liouville theory as the reduction of a WZW-like theory in which the form of the potential for the Cartan field is not fixed a priori. It is shown that in the classical theory conformal invariance places no condition on the form of the potential, but the conformal invariance of the classical reduction requires that it be an exponential. In contrast, the quantum theory requires that, even before reduction, the potential be a sum of two exponentials. The duality of these two exponentials is the fore-runner of the Liouville duality. An interpretation for the reflection symmetry found in quantum Liouville theory is also obtained along similar lines. 
  The nature of the transition from the quantum tunneling regime at low temperatures to the thermal hopping regime at high temperatures is investigated analytically in scalar field theory. An analytical bounce solution is presented, which reproduces the action in the thin-wall as well as thick-wall limits. The transition is first order for the case of a thin wall while for the thick wall case it is second order. 
  The SO(4,2) isometries of AdS_5 are realized non-linearly on its horospherical coordinates (x^m,\rho). On the other hand, Penrose twistors have long been known to linearly realize these symmetries on 4-dimensional Minkowski space, the boundary of AdS_5, parametrized by x^m. Here we extend the twistor construction and define a pair of twistors, allowing us to include a radial coordinate in the construction. The linear action of SO(4,2) on the twistors induces the correct isometries of AdS_5. We apply this new construction to the study of the dynamics of a massive particle in AdS_5. We show that in terms of the twistor variables the action takes a simple form of a 1-dimensional gauge theory. Our result might open up the possibility to find a simple worldvolume action also for the string propagating on AdS_5. 
  Classical 1/4 BPS configurations consist of 1/2 BPS dyons which are positioned by competing static forces from electromagnetic and Higgs sectors. These forces do not follow the simple inverse square law, but can be encoded in some low energy effective potential between fundamental monopoles of distinct types. In this paper, we find this potential, by comparing the exact 1/4 BPS bound from a Yang-Mills field theory with its counterpart derived from low energy effective dynamics of monopoles. Our method is generalized to arbitrary gauge groups and to arbitrary BPS monopole/dyon configurations. The resulting effective action for 1/4 BPS states is written explicitly, and shown to be determined entirely by the geometry of multi-monopole moduli spaces. We also explore its natural supersymmetric extension. 
  We study the strong coupling limit of the Bethe ansatz solutions in the massive Thirring model. We find analytical expressions for the energy eigenvalues for the vacuum state as well as n-particle n- hole states. This formula is compared with the numerical results and is found to achieve a very good agreement.   Also, it is found that the 2-particle 2- hole and higher particle-hole states describe n- free bosons states in this limit. The behaviors of the strong coupling limit of the boson mass for various model calculations are examined. We discuss an ambiguity of the coupling constant normalization due to the current regularization. 
  We determine all the terms that are gauge-invariant up to a total spacetime derivative ("semi-invariant terms") for gauged non-linear sigma models. Assuming that the isotropy subgroup $H$ of the gauge group is compact or semi-simple, we show that (non-trivial) such terms exist only in odd dimensions and are equivalent to the familiar Chern-Simons terms for the subgroup $H$. Various applications are mentioned, including one to the gauging of the Wess-Zumino-Witten terms in even spacetime dimensions. Our approach is based on the analysis of the descent equation associated with semi-invariant terms. 
  In this talk, based on work done in collaboration with G. Landi and R.J Szabo, I will review how string theory can be considered as a noncommutative geometry based on an algebra of vertex operators. The spectral triple of strings is introduced, and some of the string symmetries, notably target space duality, are discussed in this framework. 
  The equivalence between the Chern-Simons gauge theory on a three-dimensional manifold with boundary and the WZNW model on the boundary is established in a simple and general way using the BRST symmetry. Our approach is based on restoring gauge invariance of the Chern-Simons theory in the presence of a boundary. This gives a correspondence to the WZNW model that does not require solving any constraints, fixing the gauge or specifying boundary conditions. 
  By dimensional reduction of a massive BF theory, a new topological field theory is constructed in (2+1) dimensions. Two different topological terms, one involving a scalar and a Kalb-Ramond fields and another one equivalent to the four-dimensional BF term, are present. We constructed two actions with these topological terms and show that a topological mass generation mechanism can be implemented. Using the non-Chern-Simons topological term, an action is proposed leading to a classical duality relation between Klein-Gordon and Maxwell actions. We also have shown that an action in (2+1) dimensions with the Kalb-Ramond field is related by Buscher's duality transformation to a massive gauge-invariant Stuckelberg-type theory. 
  The exact expression for Wilson loop averages winding n times on a closed contour is obtained in two dimensions for pure U(N) Yang-Mills theory and, rather surprisingly, it displays an interesting duality in the exchange $n \leftrightarrow N$. The large-N limit of our result is consistent with previous computations. Moreover we discuss the limit of small loop area ${\cal A}$, keeping $n^2 {\cal A}$ fixed, and find it coincides with the zero-instanton approximation. We deduce that small loops, both at finite and infinite "volume", are blind to instantons. Next we check the non-perturbative result by resumming 't Hooft-CPV and Wu-Mandelstam-Leibbrandt (WML)-prescribed perturbative series, the former being consistent with the exact result, the latter reproducing the zero-instanton contribution. A curious interplay between geometry and algebraic invariants is observed. Finally we compute the spectral density of the Wilson loop operator, at large $N$, via its Fourier representation, both for 't Hooft and WML: for small area they exhibit a gap and coincide when the theory is considered on the sphere $S^2$. 
  I present cosmological arguments which point towards a Horava-Witten like picture of the universe, with the unification scale of order the fundamental gravitational scale. The SUSY breaking scale is determined by the dynamics of gauge fields which are weakly coupled at the fundamental scale. Bulk moduli whose potential originates at short distances are the inflatons, while bulk moduli whose potential originates from SUSY breaking are the origin of the energy density in the present era. The latter decay just before nucleosynthesis, and a consistent theory of baryogenesis requires that there be renormalizable baryon number violating interactions at the TeV scale. The dark matter is a boundary modulus, perhaps the QCD axion, and the temperature of matter radiation equality is related to the ratio between the fundamental and effective four dimensional Planck scales. The same ratio determines the amplitude of fluctuations in the microwave background. 
  The divergences of the gravitational action are analyzed for spacetimes that are asymptotically anti-de Sitter and asymptotically flat. The gravitational action is rendered finite using a local counterterm prescription, and the relation of this method to the traditional reference spacetime is discussed. For AdS, an iterative procedure is devised that determines the counterterms efficiently. For asymptotically flat space, we use a different method to derive counterterms which are sufficient to remove divergences in most cases. 
  We give a simple argument for the cancellation of the log(-k^2) terms (k is the gluon momentum) between the zero-temperature and the temperature-dependent parts of the thermal self-energy. 
  We derive a new version of SU(3) non-Abelian Stokes theorem by making use of the coherent state representation on the coset space $SU(3)/(U(1)\times U(1))=F_2$, the flag space. Then we outline a derivation of the area law of the Wilson loop in SU(3) Yang-Mills theory in the maximal Abelian gauge (The detailed exposition will be given in a forthcoming article). This derivation is performed by combining the non-Abelian Stokes theorem with the reformulation of the Yang-Mills theory as a perturbative deformation of a topological field theory recently proposed by one of the authors. Within this framework, we show that the fundamental quark is confined even if $G=SU(3)$ is broken by partial gauge fixing into $H=U(2)$ just as $G$ is broken to $H=U(1) \times U(1)$. An origin of the area law is related to the geometric phase of the Wilczek-Zee holonomy for U(2). Abelian dominance is an immediate byproduct of these results and magnetic monopole plays the dominant role in this derivation. 
  In this article, we study the q-state Potts random matrix models extended to branched polymers, by the equations of motion method. We obtain a set of loop equations valid for any arbitrary value of q. We show that, for q=2-2 \cos {l \over r} \pi (l, r mutually prime integers with l < r), the resolvent satisfies an algebraic equation of degree 2 r -1 if l+r is odd and r-1 if l+r is even. This generalizes the presently-known cases of q=1, 2, 3. We then derive for any 0 \leq q \leq 4 the Potts-q critical exponents and string susceptibility. 
  A 3d topological sigma model describing maps from a 3-manifold Y to a Calabi-Yau 3-fold M is introduced. As the model is topological, we can choose an arbitrary metric on M. Upon scaling up the metric, the path integral by construction localizes on the moduli space of special Lagrangian submanifolds of M. We couple the theory to dynamical gauge fields and discuss the case where M has a mirror and the gauge group is U(1). 
  A model, based on a noncommutative geometry, unifying general relativity with quantum mechanics, is further develped. It is shown that the dynamics in this model can be described in terms of one-parameter groups of random operators. It is striking that the noncommutative counterparts of the concept of state and that of probability measure coincide. We also demonstrate that the equation describing noncommutative dynamics in the quantum gravitational approximation gives the standard unitary evolution of observables, and in the "space-time limit" it leads to the state vector reduction. The cases of the spin and position operators are discussed in details. 
  It has been demonstrated in a recent paper (Mod.Phys.Lett. A13, 1265 (1998); hep-th/9902020) that the existence of a non-thermodynamic arrow of time at the atomic level is a fundamental requirement for conservation of energy in matter-radiation interaction. Since the universe consists of two things only --- energy and massive matter --- we argue that as a consequence of this earlier result, particles and antiparticles must necessarily move in opposite directions in time. Our result further indicates that the CPT theorem can be extended to cover non-local gauge fields. 
  The thermal dynamics of D-branes and of open superstrings in background gauge fields is studied. It is shown that D-brane dynamics forbids constant velocity motion at finite temperature. T-duality is used to interpret this feature as a consequence of the absence of an equilibrium state of charged strings at finite temperature in a constant background electric field, as a result of Debye screening of electric fields. The effective action for the Polyakov loop operator is computed and the corresponding screening solutions are described. The finite temperature theory is also used to illustrate the importance of carefully incorporating Wu-Yang terms into the string path integral for compact target spaces. 
  The straightforward description of q-deformed systems leads to transition amplitudes that are not numerically valued. To give physical meaning to these expressions without introducing {\it ad hoc} remedies, one may exploit an "internal" Fock space already defined by the q-algebra. This internal space may be interpreted in terms of internal degrees of freedom of the deformed system or alternatively in terms of non-locality. It is shown that the $q$-deformation may give stringy characteristics to a Yang-Mills theory. 
  Examples are given of q-deformed systems that may be interpreted by the standard rules of quantum theory in terms of new degrees of freedom and supplementary quantum numbers. 
  We study the thermodynamics of the D1-D5 system on a five-torus, focussing on the roles of different scales. One can take a decoupling limit such that the tension of the `little string' inside the fivebrane remains finite and the physics is 5+1 dimensional. The dual black geometry exhibits a boosted Hagedorn phase, as well as a phase describing a boosted fivebrane gas. The dependence on the boost yields information about the nature of the fivebrane modes and their interactions. In particular, the form of the equations of state suggests a description in terms of $k=Q_1 Q_5$ degrees of freedom, which may lead to an explanation of the $(Q_5)^3$ growth in the fivebrane density of states below the Hagedorn transition. 
  A scattering scattering description is proposed for a boundary perturbation of a c=1 SL(2,Z) invariant conformal field theory. The bulk massless S-matrices are of the form of Zamolodchikov's staircase model. Using the boundary version of the thermodynamic Bethe ansatz, we show that the boundary free energy goes through a series of integer valued plateaux as a function of system size. 
  We find the solution of the $\hat{sl}(3)_k$ singular vector decoupling equations on 3-point functions for the particular case when one of the fields is of weight $w_0\cdot k\Lambda_0$. The result is a function with non-trivial singularities in the flag variables, namely a linear combination of 2F1 hypergeometric functions. This calculation fills in a gap in [1] and confirms the $\hat{sl}(3)_k$ fusion rules determined there both for generic $\kappa \not \in \IQ$ and fractional levels.   We have also analysed the fusion in $\hat{sl}(3)_k$ using algebraic methods generalising those of Feigin and Fuchs and again find agreement with [1]. In the process we clarify some details of previous treatments of the fusion of $\hat{sl}(2)_k$ fractional level admissible representations. 
  We derive expressions for Poincar\'e group generators using preturbative similarity renormalization group procedure for Hamiltonians. We show that generators obtained in second-order perturbation theory satisfy required commutation relations in weak sense, i.e. in matrix elements between states of finite invariant masses. 
  We study the decoupling effects in N=1 (global) supersymmetric theories with chiral superfields at the one-loop level. The examples of gauge neutral chiral superfields with the minimal (renormalizable) as well as non-minimal (non- renormalizable) couplings are considered, and decoupling in gauge theories with U(1) gauge superfields that couple to heavy chiral matter is studied. We calculate the one-loop corrected effective Lagrangians that involve light fields and heavy fields with mass of order M. The elimination of heavy fields by equations of motion leads to decoupling effects with terms that grow logarithmically with M. These corrections renormalize light fields and couplings in the theory (in accordance with the "decoupling theorem"). When the field theory is an effective theory of the underlying fundamental theory, like superstring theory, where the couplings are calculable, such decoupling effects modify the low-energy predictions for the effective couplings of light fields. In particular, for the class of string vacua with an "anomalous" U(1) the vacuum restabilization triggers the decoupling effects, which can significantly modify the low energy predictions for the couplings of the surviving light fields. We also demonstrate that quantum corrections to the chiral potential depending on massive background superfields and corresponding to supergraphs with internal massless lines and external massive lines can also arise at the two-loop level. 
  We review the geometrical approach to the description of the dynamics of super-p-branes, Dirichlet branes and the M5-brane, which is based on a generalization of the elements of surface theory to the description of the embedding of supersurfaces into target superspaces. Being manifestly supersymmetric in both, the superworldvolume of the brane and the target superspace, this approach unifies the Neveu-Schwarz-Ramond and the Green-Schwarz formulation and provides the fermionic kappa-symmetry of the Green-Schwarz-type superbrane actions with a clear geometrical meaning of standard worldvolume local supersymmetry. We describe the properties of doubly supersymmetric (superembedding) brane actions and show how they are related to the standard Green-Schwarz formulation. In the second part of the article basic geometrical grounds of the (super)embedding approach are considered and applied to the description of the M2-brane and the M5-brane. Various applications of the superembedding approach are reviewed. 
  We show that on three-dimensional Riemannian manifolds without boundaries and with trivial first real de Rham cohomology group (and in no other dimensions) scalar field theory and Maxwell theory are equivalent: the ratio of the partition functions is given by the Ray-Singer torsion of the manifold. On the level of interaction with external currents, the equivalence persists provided there is a fixed relation between the charges and the currents. 
  We calculate the coefficient $a_5$ of the heat kernel asymptotics for an operator of Laplace type with mixed boundary conditions on a general compact manifold. 
  We review a detailed investigation of the perturbative part of the low-energy effective action of N=2 supersymmetric Yang-Mills theory in a conventional effective field theory approach. With the restriction that the effective action should contain at most two derivatives and not more than four-fermion couplings, the features of the low-energy effective action obtained by Seiberg based on $U(1)_R$ anomaly and non-perturbative $\beta$-function arguments are shown to emerge. 
  A new topological model is proposed in three dimensions as an extension of the BF-model. It is a three-dimensional counterpart of the two-dimensional model introduced by Chamseddine and Wyler ten years ago. The BFK-model, as we shall call it, shows to be quantum scale invariant at all orders in perturbation theory. The proof of its full finiteness is given in the framework of algebraic renormalization. 
  We study the perturbation expansion of the free energy of N=4 supersymmetric SU(N) Yang-Mills at finite temperature in powers of 't Hooft's coupling g^2 N in the large N limit. Infrared divergences are controlled by constructing a hierarchy of two 3 dimensional effective field theories. This procedure is applied to the calculation of the free energy to order (g^2 N)^(3/2), but it can be extended to higher order corrections. 
  We discuss supersymmetry breaking in the field-theoretical limit of the strongly-coupled heterotic string compactified on a Calabi-Yau manifold, from the different perspectives of four and five dimensions. The former applies to light degrees of freedom below the threshold for five-dimensional Kaluza-Klein excitations, whereas the five-dimensional perspective is also valid up to the Calabi-Yau scale. We show how, in the latter case, two gauge sectors separated in the fifth dimension are combined to form a consistent four-dimensional supergravity. In the lowest order of the $\kappa^{2/3}$ expansion, we show how a four-dimensional supergravity with gauge kinetic function $f_{1,2}=S$ is reproduced, and we show how higher-order terms give rise to four-dimensional operators that differ in the two gauge sectors. In the four-dimensional approach, supersymmetry is seen to be broken when condensates form on one or both walls, and the goldstino may have a non-zero dilatino component. As in the five-dimensional approach, the Lagrangian is not a perfect square, and we have not identified a vacuum with broken supersymmetry and zero vacuum energy. We derive soft supersymmetry-breaking terms for non-standard perturbative embeddings, that are relevant in more general situations such as type I/type IIB orientifold models. 
  We study conserved currents of any integer or half integer spin built from massless scalar and spinor fields in $AdS_3$. 2-forms dual to the conserved currents in $AdS_3$ are shown to be exact in the class of infinite expansions in higher derivatives of the matter fields with the coefficients containing inverse powers of the cosmological constant. This property has no analog in the flat space and may be related to the holography of the AdS spaces. 
  To formulate two-dimensional Yang-Mills theory with adjoint matter fields in the large-N limit as classical mechanics, we derive a Poisson algebra for the color-invariant observables involving adjoint matter fields. We showed rigorously in J. Math. Phys. 40, 1870 (1999) that different quantum orderings of the observables produce essentially the same Poisson algebra. Here we explain, in a less precise but more pedagogical manner, the crucial topological graphical observations underlying the formal proof. 
  We address the question of AdS/CFT correspondence in the case of the 3-point function <O_4 O_4 O_8>. O_4 and O_8 are particular primary states represented by F^2 + ... and F^4 + ... operators in \N=4 SYM theory and dilaton \phi and massive `fixed' scalar \nu in D=5 supergravity. While the value of <O_4 O_4 O_8> computed in large N weakly coupled SYM theory is non-vanishing, the D=5 action of type IIB supergravity compactified on S^5 does not contain \phi\phi\nu coupling and thus the corresponding correlator seems to vanish on the AdS_5 side. This is in obvious contradiction with arguments suggesting non-renormalization of 2- and 3-point functions of states from short multiplets and implying agreement between the supergravity and SYM expressions for them. We propose a natural resolution of this paradox which emphasizes the 10-dimensional nature of the correspondence. The basic idea is to treat the constant mode of the dilaton as a part of the full S^5 Kaluza-Klein family of dilaton modes. This leads to a non-zero result for the <O_4 O_4 O_8> correlator on the supergravity side. 
  In this talk I summarize several recent results concerning the four-dimensional effective supergravity obtained using a Calabi-Yau compactification of the $E_8\times E_8$ heterotic string from M-theory. A simple macroscopic study is provided expanding the theory in powers of two dimensionless variables. Higher order terms in the K\"ahler potential are identified and matched with the heterotic string corrections. In the context of this M-theory expansion, I discuss several phenomenological issues: universality of soft scalar masses, relations between the different scales of the theory (eleven-dimensional Planck mass, compactification scale and orbifold scale) in order to obtain unification at $3\times 10^{16}$ GeV or lower values, soft supersymmetry-breaking terms, and finally charge and colour breaking minima. The above analyses are also carried out in the presence of (non-perturbative) five-branes. 
  We study some dynamical aspects of the correspondence between strings in AdS space and external heavy quarks in N=4 SYM. Specifically, by examining waves propagating on such strings, we make some plausible (and some surprising) inferences about the time-dependent fields produced by oscillating quarks in the strongly-coupled gauge theory. We point out a puzzle regarding energy conservation in the SYM theory. In addition, we perform a similar analysis of the gauge fields produced by a baryon (represented as a D5-brane with string-like extension in AdS space) and compare and contrast with the gauge fields produced by a quark-antiquark pair (represented as a string looping through AdS space). 
  We study 2+1 Chern-Simons gravity at the classical action level. In particular we rederive the linear combinations of the ``standard'' and ``exotic'' Einstein actions, from the (anti) self-duality of the ``internal'' Lorentzian indices. The relation to a genuine four-dimensional (anti)self-dual topological theory greatly facilitates the analysis and its relation to hyperbolic three-dimensional geometry. Finally a non-abelian vector field ``dual'' action is also obtained. 
  We expand on Klebanov and Witten's recent proposal for formulating the AdS/CFT correspondence using irregular boundary conditions. The proposal is shown to be correct to any order in perturbation theory. 
  We demonstrate the reparametrization invariance of perturbatively defined one-dimensional functional integrals up to the three-loop level for a path integral of a quantum-mechanical point particle in a box. We exhibit the origin of the failure of earlier authors to establish reparametrization invariance which led them to introduce, superfluously, a compensating potential depending on the connection of the coordinate system. We show that problems with invariance are absent by defining path integrals as the epsilon-> 0 -limit of 1+ epsilon -dimensional functional integrals. 
  The Davey-Stewartson 1(DS1) system[9] is an integrable model in two dimensions. A quantum DS1 system with 2 colour-components in two dimensions has been formulated. This two-dimensional problem has been reduced to two one-dimensional many-body problems with 2 colour-components. The solutions of the two-dimensional problem under consideration has been constructed from the resulting problems in one dimensions. For latters with the $\delta $-function interactions and being solved by the Bethe ansatz, we introduce symmetrical and antisymmetrical Young operators of the permutation group and obtain the exact solutions for the quantum DS1 system. The application of the solusions is discussed. 
  This article gives a review of the topic of regularising chiral gauge theories and is aimed at a general audience. It begins by clarifying the meaning of chirality and goes on to discussing chiral projections in field theory, parity violation and the distinction between vector and chiral field theories. It then discusses the standard model of electroweak interactions from the perspective of chirality.  It also reviews at length the phenomenon of anomalies in quantum field theories including the intuitive understanding of anomalies based on the Dirac sea picture as given by Nielsen and Ninomiya. It then raises the issue of a non-perturbative and constructive definition of the standard model as well as the importance of such formulations. The second Nielsen-Ninomiya theorem about the impossibility of regularising chiral gauge theories under some general assumptions is also discussed. After a brief review of lattice regularisation of field theories, it discusses the issue of fermions on the lattice with special emphasis on the problem of species doubling. The implications of these problems to introducing chiral fermions on the lattice as well as the interpretations of anomalies within the lattice formulations and the lattice Dirac sea picture are then discussed. Finally the difficulties of formulating the standard model on the lattice are illustrated through detailed discussions of the Wilson-Yukawa method, the domain wall fermions method and the recently popular Ginsparg-Wilson method. 
  A pedagogical presentation of integrable models with special reference to the Toda lattice hierarchy has been attempted. The example of the KdV equation has been studied in detail, beginning with the infinite conserved quantities and going on to the Lax formalism for the same. We then go on to symplectic manifolds for which we construct the Lax operator. This formalism is applied to Toda Lattice systems. The Zakharov Shabat formalism aimed at encompassing all integrable models is also covered after which the zero curvature condition and its fallout are discussed. We then take up Toda Field Theories and their connection to W algebras via the Hamiltonian reduction of the WZNW model. Finally, we dwell on the connection between four dimensional Yang Mills theories and the KdV equation along with a generalization to supersymmetry. 
  We present and analyze exact solutions of the Einstein-Maxwell and Einstein-Maxwell-Dilaton equations that describe static pairs of oppositely charged extremal black holes, i.e., black diholes. The holes are suspended in equilibrium in an external magnetic field, or held apart by cosmic strings. We comment as well on the relation of these solutions to brane-antibrane configurations in string and M-theory. 
  We apply the Dirac bracket quantization to open strings attached to branes in the presence of background antisymmetric field and recover an inherent noncommutativity in the internal coordinates of the brane. 
  By analysing supersymmetry transformations we derive new BPS equations for the D=11 fivebrane propagating in flat space that involve the world-volume three-form. The equations generalise those of 2,3,4 and 5 dimensional special Lagrangian submanifolds and are relevant for describing membranes ending on these submanifolds. 
  The domain of applicability of the Poisson-Lie T-duality is enlarged to include the gauged WZNW models. 
  We find static spherically symmetric dyons in Einstein-Born-Infeld-Higgs model in 3+1 dimensions. The solutions share many features with the gravitating monopoles in the same model. In particular, they exist only up to some critical value of a parameter $\a$ related to the strength of the gravitational interaction. We also study dyonic non-Abelian black holes. We analyse these solutions numerically. 
  We respond to the comment by Kreimer et. al. about the torsional contribution to the chiral anomaly in curved spacetimes. We discuss their claims and refute its main conclusion. 
  We investigate the convergence of the derivative expansion of the exact renormalization group, by using it to compute the beta function of scalar field theory. We show that the derivative expansion of the Polchinski flow equation converges at one loop for certain fast falling smooth cutoffs. The derivative expansion of the Legendre flow equation trivially converges at one loop, but also at two loops: slowly with sharp cutoff (as a momentum-scale expansion), and rapidly in the case of a smooth exponential cutoff. Finally, we show that the two loop contributions to certain higher derivative operators (not involved in beta) have divergent momentum-scale expansions for sharp cutoff, but the smooth exponential cutoff gives convergent derivative expansions for all such operators with any number of derivatives. 
  Various formulas for currents with arbitrary spin are worked out in general space-time dimension, in the free field limit and, at the bare level, in presence of interactions. As the n-dimensional generalization of the (conformal) vector field, the (n/2-1)-form is used. The two-point functions and the higher-spin central charges are evaluated at one loop. As an application, the higher-spin hierarchies generated by the stress-tensor operator-product expansion are computed in supersymmetric theories. The results exhibit an interesting universality. 
  We obtain the explicit and complete bosonic non-linear Kaluza-Klein ansatz for the consistent S^4 reduction of D=11 supergravity to N=1, D=7 gauged supergravity. This provides a geometrical interpretation of the lower dimensional solutions from the eleven-dimensional point of view. 
  Recent work in the literature has proposed the use of non-local boundary conditions in Euclidean quantum gravity. The present paper studies first a more general form of such a scheme for bosonic gauge theories, by adding to the boundary operator for mixed boundary conditions of local nature a two-by-two matrix of pseudo-differential operators with pseudo-homogeneous kernels. The request of invariance of such boundary conditions under infinitesimal gauge transformations leads to non-local boundary conditions on ghost fields. In Euclidean quantum gravity, an alternative scheme is proposed, where non-local boundary conditions and the request of their complete gauge invariance are sufficient to lead to gauge-field and ghost operators of pseudo-differential nature. The resulting boundary conditions have a Dirichlet and a pseudo-differential sector, and are pure Dirichlet for the ghost. This approach is eventually extended to Euclidean Maxwell theory. 
  We consider the scattering of relativistic electrons from a thin magnetic flux tube and perturbatively calculate the order $\alpha$, radiative correction, to the first order Born approximation. We show also that the second order Born amplitude vanishes, and obtain a finite inclusive cross section for the one-body scattering which incorporates soft photon bremsstrahlung effects. Moreover, we determine the radiatively corrected Aharonov-Bohm potential and, in particular, verify that an induced magnetic field is generated outside of the flux tube. 
  A derivative expansion technique is developed to compute functional determinants of quadratic operators, non diagonal in spacetime indices. This kind of operators arise in general 't Hooft gauge fixed Lagrangians. Elaborate applications of the developed derivative expansion are presented. 
  It has recently been claimed that the inclusion of a Pauli term in (2+1) dimensions gives rise to a new type of anomalous spin term. The form of that term is shown to contradict the structure relations for the inhomogeneous Lorentz group. 
  This paper has been withdrawn by the author due to inconsistency of the considered working hypothesis. The consistent treatment is presented in the last publications of the author. 
  It is well-known that is spite of sharing some properties with conventional particles, topological geons in general violate the spin-statistics theorem. On the other hand, it is generally believed that in quantum gravity theories allowing for topology change, using pair creation and annihilation of geons, one should be able to recover this theorem. In this paper, we take an alternative route, and use an algebraic formalism developed in previous work. We give a description of topological geons where an algebra of "observables" is identified and quantized. Different irreducible representations of this algebra correspond to different kinds of geons, and are labeled by a non-abelian "charge" and "magnetic flux". We then find that the usual spin-statistics theorem is indeed violated, but a new spin-statistics relation arises, when we assume that the fluxes are superselected. This assumption can be proved if all observables are local, as is generally the case in physical theories. Finally, we also show how our approach fits into conventional formulations of quantum gravity. 
  We propose that the spectrum of light mesons (the $\pi$ and $\rho$, together with their radial excitations) can be calculated in the limit of vanishing light quark masses by studying gauge theory (open string theory) on suitable higher dimensional background geometries. Using the metric proposed by Witten for glueball calculations as a paradigmatic example, we find a spectrum which is in startlingly good agreement with the masses tabulated by the Particle Data Group. These calculations have only one free parameter, corresponding to the overall mass scale. We make predictions for the next several particles in the spectrum. 
  Conformally invariant sigma models in $D=2n$ dimensions with target non-compact O(2n,1) groups are studied. It is shown that despite the non-compact nature of the O(2n,1) groups, the classical action and Hamiltonian are positive definite. Instanton field configurations are found to correspond geometrically to conformal ``stereographic'' mappings of $R^{2n}$ into the Euclidean signature $AdS_{2n}$ spaces. Zaikov's relationship between Self Dual $p$-branes and Chern-Simons $p'$-branes, provided $p=p'+1$ and the embedding $D=p+1$-dimensional manifold is Euclidean, is elaborated further. Branes actions can be obtained also from a Moyal deformation quantization of Generalized Yang Mills Theories. Using this procedure, we show how four dimensional SU(N) YM theories contain Chern-Simons membranes and hadronic bags in the large $N$ limit. Since Chern-Simons $p'$-branes have an underlying infinite dimensional algebra containing $W_{1+\infty}$, as shown by Zaikov, we discuss the importance that $W$ geometry should have in the final formulation of $M$ theory. 
  Following the Papapetrou-Dixon-Wald procedure we derive the equation of motion for a dilatonic test body(probe) with the dilaton coupling $\alpha = {\sqrt {p/(2+p)}}$ in four dimension. Since the dilatonic freedom sometimes comes from extra dimensions, it is best to derive the EOM by a dimensional reduction from $(p+4)-$dimensions. We discuss about the force balance up to the gravitational spin-spin interactions via the probe technique. The force balance condition yields the saturation of a Bogomol'nyi bound and the gyromagnetic ratio of the test body. 
  I briefly review the outcomes of two very different old questions -- where global SUSY improves on ordinary QFT -- when they are in turn posed in the local, SUGRA, context. The first concerns the unexpectedly powerful role of the "Dirac square root" graded algebra relation $E=Q^2$, originally found in D=4 SUSY by Gol'fand and Likhtman, in proving positive energy theorems both in SUGRA and in ordinary gravity theories, where $E$ and $Q$ have very different -- gauge generator -- definitions. The second seeks SUGRA counterparts of the boson/fermion loop ultraviolet cancellations in SUSY models. Here the result is negative: local supersymmetry cannot overcome the infinities associated with the dimensional gravitational $\kappa^2$ even, as recently shown, in maximal D=11 SUGRA. 
  The massless degrees of freedom of type 0 NS5-branes are derived. A non-chiral, purely bosonic spectrum is found in both type 0A and 0B. This non-chirality is confirmed by a one-loop computation in the bulk. Some puzzles concerning type 0B S-duality are pointed out in this context. An interpretation of the spectra in terms of ``type 0 little strings'' is proposed. 
  This Ph.D. thesis reaches two main results. The first one is represented by a detailed study, in Feynman gauge, of the perturbative ${\cal O}(g^4)$ contribution to a space-time Wilson loop, with respect to its (expected) Abelian-like time exponentiation when the temporal side goes to infinity. As soon as we are in dimensions greater than two, the expected behavior is found. But if we proceed first to the dimensional limit $D \to 2$, the exponentiation is not recovered. The limits $T \to \infty$ and $D \to 2$ do not commute. The other result is the computation in dimensions $D=2+\epsilon$ and in light-cone gauge with Mandelstam-Leibbrandt prescription of the perturbative ${\cal O}(g^4)$ contribution to the same Wilson loop, coming from diagrams with a self-energy correction in the vector propagator. In the limit $\epsilon \to 0$ the result is finite, in spite of the vanishing of the triple vector vertex in light-cone gauge, and provides the expected agreement with the analogous calculation in Feynman gauge. Consequences of these results concerning two and higher-dimensional gauge theories are pointed out. 
  We have analyzed the Unruh problem in the frame of quantum field theory and have shown that the Unruh quantization scheme is valid in the double Rindler wedge rather than in Minkowski spacetime. The double Rindler wedge is composed of two disjoint regions ($R$- and $L$-wedges of Minkowski spacetime) which are causally separated from each other. Moreover the Unruh construction implies existence of boundary condition at the common edge of $R$- and $L$-wedges in Minkowski spacetime. Such boundary condition may be interpreted as a topological obstacle which gives rise to a superselection rule prohibiting any correlations between $r$- and $l$- Unruh particles. Thus the part of the field from the $L$-wedge in no way can influence a Rindler observer living in the $R$-wedge and therefore elimination of the invisible "left" degrees of freedom will take no effect for him. Hence averaging over states of the field in one wedge can not lead to thermalization of the state in the other. This result is proved both in the standard and algebraic formulations of quantum field theory and we conclude that principles of quantum field theory does not give any grounds for existence of the "Unruh effect". 
  Following a recent suggestion by Randall and Sundrum, we consider string compactification scenarios in which a compact slice of AdS-space arises as a subspace of the compactification manifold. A specific example is provided by the type II orientifold equivalent to type I theory on (orbifolds of) $T^6$, upon taking into account the gravitational backreaction of the D3-branes localized inside the $T^6$. The conformal factor of the four-dimensional metric depends exponentially on one of the compact directions, which, via the holographic correspondence, becomes identified with the renormalization group scale in the uncompactified world. This set-up can be viewed as a generalization of the AdS/CFT correspondence to boundary theories that include gravitational dynamics. A striking consequence is that, in this scenario, the fundamental Planck size string and the large N QCD string appear as (two different wavefunctions of) one and the same object. 
  We propose a modification of the gauge-fixing procedure in the Lagrangian method of superfield BRST quantization for general gauge theories which simultaneously provides a natural generalization of the well-known BV quantization scheme as far as gauge-fixing is concerned. A superfield form of BRST symmetry for the vacuum functional is found. The gauge-independence of the S-matrix is established. 
  The canonical structure of pure Yang-Mills theory is analysed in the case when Gauss' law is satisfied identically by construction. It is shown that the theory has a canonical structure in this case, provided one uses a special gauge condition, which is a natural generalisation of the Coulomb gauge condition of electrodynamics. The emergence of a canonical structure depends critically also on the boundary conditions used for the relevant field variables. Possible boundary conditions are analysed in detail. A comparison of the present formulation in the generalised Coulomb gauge with the well known Weyl gauge ($A_{0} = 0$) formulation is made. It appears that the Hamiltonians in these two formulations differ from one another in a non-trivial way. It is still an open question whether these differences give rise to truly different structures upon quantisation. An extension of the formalism to include coupling to fermionic fields is briefly discussed. 
  We propose a new approach to the study of the correlation functions of W-algebras. The conformal blocks (chiral correlation functions), for fixed arguments, are defined to be those linear functionals on the product of the highest weight (h.w.) representation spaces which satisfy the Ward identities. First we investigate the dimension of the chiral correlation functions in the case when there is no singular vector in any of the representations. Then we pass to the analysis of the completely degenerate representations. A special subspace of the h.w. representation spaces, introduced by Nahm, plays an important role in the considerations. The structure of these subspaces shows a deep connection with the quantum and classical Toda models and relates certain completely degenerate representations of the \wg algebra to representations of $G$. This is confirmed by an analysis for the Virasoro, \wa2 and \wbc2 algebra. We also relate our work to Nahms, Feigen-Fuchs' and Watts' results. 
  The reduced SL(2,R) WZW quantum mechanics is analysed in the framework of geometric quantization. The spectrum of the Hamiltonian is determined, and it is found, that contrary to the previous approaches, there is a unique, physically preferred quantisation of the system. 
  We analyse an integrable model of two-dimensional gravity which can be reduced to a pair of Liouville fields in conformal gauge. Its general solution represents a pair of ``mirror'' black holes with the same temperature. The ground state is a degenerate constant dilaton configuration similar to the Nariai solution of the Schwarzschild-de Sitter case. The existence of $\phi=const.$ solutions and their relation with the solution given by the 2D Birkhoff's theorem is then investigated in a more general context. We also point out some interesting features of the semiclassical theory of our model and the similarity with the behaviour of AdS$_2$ black holes. 
  We show that the logarithmic behaviour seen in perturbative and non perturbative contributions to Green functions of gauge-invariant composite operators in N=4 SYM with SU(N) gauge group can be consistently interpreted in terms of anomalous dimensions of unprotected operators in long multiplets of the superconformal group SU(2,2|4). In order to illustrate the point we analyse the short-distance behaviour of a particularly simple four-point Green function of the lowest scalar components of the N=4 supercurrent multiplet. Assuming the validity of the Operator Product Expansion, we are able to reproduce the known value of the one-loop anomalous dimension of the single-trace operators in the Konishi supermultiplet. We also show that it does not receive any non-perturbative contribution from the one-instanton sector. We briefly comment on double- and multi-trace operators and on the bearing of our results on the AdS/SCFT correspondence. 
  We describe the generalization of spherical field theory to other modal expansion methods. The main approach remains the same, to reduce a d-dimensional field theory into a set of coupled one-dimensional systems. The method we discuss here uses an expansion with respect to periodic-box modes. We apply the method to phi^4 theory in two dimensions and compute the critical coupling and critical exponents. We compare with lattice results and predictions via universality and the two-dimensional Ising model. 
  We study M-theory fivebranes wrapped on Special Lagrangian submanifolds ($\S_n$) in Calabi-Yau three- and fourfolds. When the M5 wraps a four-cycle, the resulting theory is a two-dimensional domain wall embedded in three-dimensional bulk with four supercharges. The theory on the wall is specified in terms of the geometry of the CY manifold and the cycle $\S_4$. It is chiral and anomalous, however the presence of a three-dimensional gravitational Chern-Simons terms with a coefficient that jumps when crossing the wall allows to cancel the anomaly by inflow. Kahler manifolds of special type, where the potential depends only on the real part of the complex coordinate, are shown to emerge as the target spaces of two-dimensional sigma-models when the M5 is wrapped on $\S_3 \times S^1$, thus providing a physical realization of some recent symplectic construction by Hitchin. 
  We consider bosonic strings propagating on Euclidean adS_3, and study in particular the realization of various worldsheet symmetries. We give a proper definition for the Brown-Henneaux asymptotic target space symmetry, when acting on the string action, and derive the Giveon-Kutasov-Seiberg worldsheet contour integral representation simply by using Noether's theorem. We show that making identifications in the target space is equivalent to the insertion of an (exponentiated) graviton vertex operator carrying the corresponding charge. Finally, we point out an interesting relation between 3D gravity and the dynamics of the worldsheet on adS_3. Both theories are described by an SL(2,C)/SU(2) WZW model, and we prove that the reduction conditions determined on one hand by worldsheet diffeomorphism invariance, and on the other by the Brown-Henneaux boundary conditions, are the same. 
  In a previous paper we provided a consistent quantization of open strings ending on D-branes with a background $B$ field. In this letter, we show that the same result can also be obtained using the more traditional method of Dirac's constrained quantization. We also extend the discussion to the fermionic sector. 
  Linear recursion relations for the instanton corrections to the effective prepotential of N=2 supersymmetric gauge theories with an arbitrary number of hypermultiplets in the fundamental representation of an arbitrary classical gauge group are dervied. The construction proceeds from the Seiberg-Witten solutions and the renormalization group type equations for the prepotential. Successive iterations of these recursion relations allow us to simple obtain instanton corrections to arbitrarily high order, which we exhibit explicitly up to 6-th order. For gauge groups SU(2) and SU(3), our results agree with previous ones. 
  States on the Coulomb branch of N=4 super-Yang-Mills theory are studied from the point of view of gauged supergravity in five dimensions. These supersymmetric solutions provide examples of consistent truncation from type IIB supergravity in ten dimensions. A mass gap for states created by local operators and perfect screening for external quarks arise in the supergravity approximation. We offer an interpretation of these surprising features in terms of ensembles of brane distributions. 
  We perform the quantization of a massive particle propagating on AdS_5. We use the twistor formulation in which the action can be brought into a quadratic form. We construct the BRST operator which commutes with AdS_5 isometries forming SU(2,2). The condition of a consistent BRST quantization requires that the AdS energy E is quantized in units of the AdS_5 radius R, E=\frac{1}{2R}(N_a +N_b+4), with N_a, N_b being some non-negative integers. We also argue that the mass operator will be identified with the moduli of the U(1) central extension Z of the SU(2,2|4) algebra in the supersymmetric case. The spectrum of physical states with vanishing ghost number contains a particular subset of `massless' SU(2,2) multiplets (including the bosonic part of the `novel short' supermultiplets). We hope that our results will help to quantize also the string on AdS_5. 
  We find the absorption probability of dilaton field in type 0B string theory. Since the background solutions are of the form $AdS_5 \times S^5$ on both regions, we use the semiclassical formalism adopted in type IIB theory to find the absorption cross section. The background tachyon field solution was used as a reference to relate the solutions of the two regions. We also consider the possible corrections to absorption probability and the $\ln(\ln z)$ form of the correction is expected as in the calculation of the confinement solution. 
  We consider Chern-Simons gauged nonlinear sigma model with boundary which has a manifest bulk diffeomorphism invariance. We find that the Gauss's law can be solved explicitly when the nonlinear sigma model is defined on the Hermitian symmetric space, and the original bulk theory completely reduces to a boundary nonlinear sigma model with the target space of Hermitian symmetric space. We also study the symplectic structure, compute the diffeomorphism algebra on the boundary, and find an (enlarged) Virasoro algebra with classical central term. 
  We present a strong evidence for the magnetic confinement in QCD by demonstrating that the one loop effective action of SU(2) QCD induces a dynamical symmetry breaking thorugh the monopole condensation, which could induce the dual Meissner effect and guarantee the confinement of color in the non-Abelian gauge theory. The result is obtained by separating the topological degrees which describes the non-Abelian monopoles from the dynamical degrees of the potential, and integrating out all the dynamical degrees of QCD. 
  The well-known physical equivalence drawn from hole theory is applied in this article. The author suggests to replace, in the part of Feynman diagram which cannot be fixed by experiments, each fermion field operator, and hence fermion propagator, by pairs of equivalent fermion field operators and propagators. The formulation of this article thus yields additional terms which reveal characteristic effects that have not been explored previously; such characteristic effects lead to the appearence of logarithmic running terms and that finite radiative corrections are directly obtained in calculations. 
  We study D-branes on the quintic CY by combining results from several directions: general results on holomorphic curves and vector bundles, stringy geometry and mirror symmetry, and the boundary states in Gepner models recently constructed by Recknagel and Schomerus, to begin sketching a picture of D-branes in the stringy regime. We also make first steps towards computing superpotentials on the D-brane world-volumes. 
  Within the AdS/CFT correspondence we use multicentre D3-brane metrics to investigate Wilson loops and compute the associated heavy quark-antiquark potentials for the strongly coupled SU(N) super-Yang-Mills gauge theory, when the gauge symmetry is broken by the expectation values of the scalar fields. For the case of a uniform distribution of D3-branes over a disc, we find that there exists a maximum separation beyond which there is no force between the quark and the antiquark, i.e. the screening is complete. We associate this phenomenon with the possible existence of a mass gap in the strongly coupled gauge theory. In the finite-temperature case, when the corresponding supergravity solution is a rotating D3-brane solution, there is a class of potentials interpolating between a Coulombic and a confining behaviour. However, above a certain critical value of the mass parameter, the potentials exhibit a behaviour characteristic of statistical systems undergoing phase transitions. The physical path preserves the concavity property of the potential and minimizes the energy. Using the same rotating-brane solutions, we also compute spatial Wilson loops, associated with the quark-antiquark potential in models of three-dimensional gauge theories at zero temperature, with similar results. 
  We consider a 1+1 dimensional field theory constrained to a finite box of length L. Traditionally, calculations in a box are done by replacing the integrals over the spatial momenta by discrete sums and then evaluating sums and doing analytic continuations. We show that it is also possible to do such calculations using an analogy to finite temperature field theory. We develop a formalism that is similar to the closed time path formulation of finite temperature field theory. Our technique can be used to calculate spatially retarded green functions, without evaluating sums or doing analytic continuations. We calculate the self energy in a simple scalar theory as an example. 
  Supergravity solution describing two intersecting M5-branes is presented. The branes are fixed in the relative transverse directions and are delocalized along the overall transverse ones. The intersection can be smoothed, so that the M5-branes present one holomorphic cycle. We also obtain a solution corresponding to an M5-brane on a holomorphic cycle of multi-Taub-NUT space. All these solutions preserve 1/4 of supersymmetry. 
  We examine how a d-dimensional mass hierarchy can be generated from a d+1-dimensional set up. We consider a d+1--dimensional scalar, the hierarchon, which has a potential as in gauged supergravities. We find that when it is in its minimum, there exist solutions of Horava-Witten topology R^d X S^1/Z^2 with domain walls at the fixed points and anti-de Sitter geometry in the bulk. We show that while standard Poincare supergravity leads to power-law hierarchies, (e.g. a power law dependence of masses on the compactification scale), gauged supergravity produce an exponential hierarchy as recently proposed by Randall and Sundrum. 
  An operator formalism for bosonization at finite temperature and density is developed. We treat the general case of anyon statistics. The exact $n$-point correlation functions, satisfying the Kubo-Martin-Schwinger condition, are explicitly constructed. The invariance under both vector and chiral transformations allows to introduce two chemical potentials. Investigating the exact momentum distribution, we discover anyon condensation in certain range of the statistical parameter. Another interesting feature is the occurrence of a non-vanishing persistent current. As an application of the general formalism, we solve the massless Thirring model at finite temperature, deriving the charge density and the persistent current. 
  A D3-brane probe in the context of AdS/CFT correspondence at finite temperature is considered. The supergravity predictions for the physical effective couplings of the world-volume gauge theory of the probe brane are compared to those calculated in one-loop perturbation theory in the thermal gauge theory. It is argued that when the Higgs expectation value is much larger than the temperature, the supergravity result must agree with perturbative thermal Yang-Mills. This provides a perturbative test of the Maldacena conjecture. Predictions for the running electric and magnetic effective couplings, beyond perturbation theory are also obtained. Phenomenological applications for universe-branes are discussed. In particular mechanisms are suggested for reducing the induced cosmological constant and naturally obtaining a varying speed of light and a consequent inflation on the universe brane. 
  In general, Whitham dynamics involves infinitely many parameters called Whitham times, but in the context of N=2 supersymmetric Yang-Mills theory it can be regarded as a finite system by restricting the number of Whitham times appropriately. For example, in the case of SU(r+1) gauge theory without hypermultiplets, there are r Whitham times and they play an essential role in the theory. In this situation, the generating meromorphic 1-form of the Whitham hierarchy on Seiberg-Witten curve is represented by a finite linear combination of meromorphic 1-forms associated with these Whitham times, but it turns out that there are various differential relations among these differentials. Since these relations can be written only in terms of the Seiberg-Witten 1-form, their consistency conditions are found to give the Picard-Fuchs equations for the Seiberg-Witten periods. 
  A subjective and incomplete list of interesting and unique features of the deconfinement phase transition is presented. Furthermore a formal similarity of the density matrix of the Aharonov-Bohm system and QCD is mentioned, as well. 
  I point out an unexpected relation between the BV (Batalin-Vilkovisky) and the BFV (Batalin-Fradkin-Vilkovisky) formulations of the same pure gauge (topological) theory. The nonminimal sector in the BV formulation of the topological theory allows one to construct the Poisson bracket and the BRST charge on some Lagrangian submanifold of the BV configuration space; this Lagrangian submanifold can be identified with the phase space of the BFV formulation of the same theory in the minimal sector of ghost variables. The BFV Poisson bracket is induced by a natural even Poisson bracket on the stationary surface of the master action, while the BRST charge originates from the BV gauge-fixed BRST transformation defined on a gauge-fixing surface. The inverse construction allows one to arrive at the BV formulation of the topological theory starting with the BFV formulation. This correspondence gives an intriguing geometrical interpretation of the nonminimal variables and clarifies the relation between the Hamiltonian and Lagrangian quantization of gauge theories.   This is an extended version of the talk given at the QFTHEP-99 workshop in Moscow, May 27 -June 2, 1999. 
  The Type 0 string theory is considered as a dual model of a non-supersymmetric gauge theory. A background geometry with N electric D3-branes is calculated in UV/IR regions. In this paper, we study a D5-brane around N D3-branes from the D5-brane world volume action as in the Type IIB case, and we obtain some baryon configurations at UV/IR regions. 
  Correlators of Wilson loop operators with O_4=Tr(F_{\mu\nu}^2+...) are computed in N=4 super-Yang-Mills theory using the AdS/CFT correspondence. The results are compared with the leading order perturbative computations. As a consequence of conformal invariance, these correlators have identical forms in the weak and strong coupling limits for circular loops. They are essentially different for contours not protected by conformal symmetry. 
  Supersymmetric $\sigma$-models obtained by constraining linear supersymmetric field theories are ill defined. Well defined subsectors parametrising Kahler manifolds exist but are not believed to arise directly from constrained linear ones. A counterexample is offered using improved understanding of membranes in superstring theories leading to crucial central terms modifying the algebra of supercharge densities 
  We construct the finite energy path between topologically distinct vacua of a 4 dimensional SO(4) Higgs model which is known to support an instanton, and show that there is a sphaleron with Chern-Simons number N_CS=1/2 at the top of the energy barrier. This is carried out using the original geometric loop construction of Manton. 
  We use the boundary state formalism to provide the full conformal description of (F,Dp) bound states. These are BPS configurations that arise from a superposition of a fundamental string and a Dp brane, and are charged under both the NS-NS antisymmetric tensor and the (p+1)-form R-R potential. We construct the boundary state for these bound states by switching on a constant electric field on the world-volume of a Dp brane and fix its value by imposing the Dirac quantization condition on the charges. Using the operator formalism we also derive the Dirac-Born-Infeld action and the classical supergravity solutions corresponding to these configurations. 
  Precise descriptions are given for the operator product expansion of generic primary fields as well as the factorization of four point functions as sum over intermediate states. The conjecture underlying the recent derivation of the space-time current algebra for string theory on $ADS_3$ by Kutasov and Seiberg is thereby verified. The roles of microscopic and macroscopic states are further clarified. The present work provides the conformal field theory prerequisites for a future study of factorization of amplitudes for string theory on $ADS_3$ as well as operator product expansion in the corresponding conformal field theory on the boundary. 
  We discuss the non-constant dilaton deformed ${\rm AdS}_5\times{\rm S}_5$ solutions of IIB supergravity where AdS sector is described by black hole. The investigation of running gauge coupling (exponent of dilaton) of non-SUSY gauge theory at finite temperature is presented for different regimes (high or low T, large radius expansion). Running gauge coupling shows power-like behavior on temperature with stable fixed point. The quark-antiquark potential at finite T is found and possibility of confinement is established. It is shown that non-constant dilaton affects the potential, sometimes reversing its behavior  if we compare it with the constant dilaton case (${\cal N}=4$ super Yang-Mills theory). Thermodynamics of obtained backgrounds is studied. In particular, next-to-leading term to free energy F is evaluated as $F=-{\tilde V_3 \over 4\pi^2}({N^2 (\pi T)^4 \over 2} + {5c^2 \over 768 g_{YM}^6 N{\alpha'}^6 (\pi T)^4})$. Here $\tilde V_3$ is the volume of the space part in the boundary of AdS, c is the parameter coming from the non-constant dilaton and N is the number of the coincident D3-branes. 
  Light-cone form of field dynamics in anti-de Sitter space-time is developed. Using field theoretic and group theoretic approaches the light-cone representation for generators of anti-de Sitter algebra acting as differential operators on bulk fields is found. We also present light-cone reformulation of the boundary conformal field theory representations. Making use of these explicit representations of AdS algebra as isometry algebra in the bulk and the algebra of conformal transformations at the boundary a precise correspondence between the bulk fields and the boundary operators is established. 
  Non-singular two and three dimensional string cosmologies are constructed using the exact conformal field theories corresponding to SO(2,1)/SO(1,1) and SO(2,2)/SO(2,1). All semi-classical curvature singularities are canceled in the exact theories for both of these cosets, but some new quantum curvature singularities emerge. However, considering different patches of the global manifolds, allows the construction of non-singular spacetimes with cosmological interpretation. In both two and three dimensions, we construct non-singular oscillating cosmologies, non-singular expanding and inflationary cosmologies including a de Sitter (exponential) stage with positive scalar curvature as well as non-singular contracting and deflationary cosmologies. We analyse these cosmologies in detail with respect to the behaviour of the scale factors, the scalar curvature and the string-coupling. The sign of the scalar curvature is changed by the quantum corrections in oscillating cosmologies and evolves with time in the non-oscillating cases. Similarities between the two and three dimensional cases suggest a general picture for higher dimensional coset cosmologies: Anisotropy seems to be a generic unavoidable feature, cosmological singularities are generically avoided and it is possible to construct non-singular cosmologies where some spatial dimensions are experiencing inflation while the others experience deflation. 
  The vacuum expectation values of the so-called Q-operators of certain integrable quantum field theories have recently been identified with spectral determinants of particular Schrodinger operators. In this paper we extend the correspondence to the T-operators, finding that their vacuum expectation values also have an interpretation as spectral determinants. As byproducts we give a simple proof of an earlier conjecture of ours, proved by another route by Suzuki, and generalise a problem in PT symmetric quantum mechanics studied by Bender and Boettcher. We also stress that the mapping between Q-operators and Schrodinger equations means that certain problems in integrable quantum field theory are related to the study of Regge poles in non-relativistic potential scattering. 
  We consider duality between type 0B string theory on $AdS_5\times S^5$ and the planar CFT on $N$ electric D3-branes coincident with $N$ magnetic D3-branes. It has been argued that this theory is stable up to a critical value of the `t Hooft coupling but is unstable beyond that point. We suggest that from the gauge theory point of view the development of instability is associated with singularity in the dimension of the operator corresponding to the tachyon field via the AdS/CFT map. Such singularities are common in large $N$ theories because summation over planar graphs typically has a finite radius of convergence. Hence we expect transitions between stability and instability for string theories in AdS backgrounds that are dual to certain large $N$ gauge theories: if there are tachyons for large AdS radius then they may be stabilized by reducing the radius below a critical value of order the string scale. 
  We obtain the complete non-linear Kaluza-Klein ansatz for the reduction of the bosonic sector of massive type IIA supergravity to the Romans F(4) gauged supergravity in six dimensions. The latter arises as a consistent warped S^4 reduction. 
  In this note we discuss dynamical mechanisms for brane stabilization in the brane world context. In particular, we consider supersymmetry preserving brane stabilization, and also brane stabilization accompanied by supersymmetry breaking. These mechanisms are realized in some four dimensional N=1 supersymmetric orientifold models. For illustrative purposes we consider two explicit orientifold models previously constructed in hep-th/9806008. In both of these models branes are stabilized at a finite distance from the orientifold planes. In the first model brane stabilization occurs via supersymmetry preserving non-perturbative gauge dynamics. In the second model supersymmetry is dynamically broken, and brane stabilization is due to an interplay between non-perturbatively generated superpotential and tree-level Kahler potential. 
  We generalize the ideas and formalism of Two-Time Physics from particle dynamics to some specific examples of string and p-brane (p >= 1) dynamics. The two-time string or p-brane action can be gauge fixed to produce various one-time string or p-brane actions that are dual to each other under gauge transformations. We discuss the particular gauges that correspond to tensionless strings and p-branes in flat (d-1)+1 spacetime, rigid strings and p-branes in flat (d-1)+1 spacetime, and tensionless strings and p-branes propagating in the AdS_{d-n} x S^n backgrounds. 
  We compute certain two-point functions in D=4, ${\cal N}=4$, SU(N) SYM theory on the Coulomb branch using SUGRA/SYM duality and find an infinite set of first order poles at masses of order $({\rm Higgs~scale})/(g_{YM} \sqrt{N})$. 
  We develop a general framework for quantum field theory on noncommutative spaces, i.e., spaces with quantum group symmetry. We use the path integral approach to obtain expressions for $n$-point functions. Perturbation theory leads us to generalised Feynman diagrams which are braided, i.e., they have non-trivial over- and under-crossings. We demonstrate the power of our approach by applying it to $\phi^4$-theory on the quantum 2-sphere. We find that the basic divergent diagram of the theory is regularised. 
  In anti-de Sitter (AdS) space, classical supergravity solutions are represented "holographically" by conformal field theory (CFT) states in which operators have expectation values. These 1-point functions are directly related to the asymptotic behaviour of bulk fields. In some cases, distinct supergravity solutions have identical asymptotic behaviour; so dual expectation values are insufficient to distinguish them. We argue that non-local objects in the gauge theory can resolve the ambiguity, and explicitly show that collections of point particles in AdS_3 can be detected by studying kinks in dual CFT Green functions. Three dimensional black holes can be formed by collision of such particles. We show how black hole formation can be detected in the holographic dual, and calculate CFT quantities that are sensitive to the distribution of matter inside the event horizon. 
  We find a universal SO(2) symmetry of a p-form Maxwell theory for both odd and even p. For odd p it corresponds to the duality rotations but for even p it defines a new set of transformations which is not related to duality rotations. In both cases a symmetry group defines a subgroup of the O(2,1) group of R-linear canonical transformations which has also a natural representation on the level of quantization condition for p-brane dyons. 
  A general calculation of Casimir energies --in an arbitrary number of dimensions-- for massless quantized fields in spherically symmetric cavities is carried out. All the most common situations, including scalar and spinor fields, the electromagnetic field, and various boundary conditions are treated with care. The final results are given as analytical (closed) expressions in terms of Barnes zeta functions. A direct, straightforward numerical evaluation of the formulas is then performed, which yields highly accurate numbers of, in principle, arbitrarily good precision. 
  The concept of determinant for a linear operator in an infinite-dimensional space is addressed, by using the derivative of the operator's zeta-function (following Ray and Singer) and, eventually, through its zeta-function trace. A little play with operators as simple as $\pm I$ ($I$ being the identity operator) and variations thereof, shows that the presence of a non-commutative anomaly (i.e., the fact that det $(AB) \neq$ det $A$ det $B$), is unavoidable, even for commuting and, remarkably, also for almost constant operators. In the case of Dirac-type operators, similarly basic arguments lead to the conclusion ---contradicting common lore--- that in spite of being $\det (\slash D +im) = \det (\slash D -im)$ (as follows from the symmetry condition of the $\slash D$-spectrum), it turns out that these determinants may {\it not} be equal to $\sqrt{\det (\slash D^2 +m^2)}$, simply because $\det [(\slash D +im) (\slash D -im)] \neq \det (\slash D +im) \det (\slash D -im)$. A proof of this fact is given, by way of a very simple example, using operators with an harmonic-oscillator spectrum and fulfilling the symmetry condition. This anomaly can be physically relevant if, in addition to a mass term (or instead of it), a chemical potential contribution is added to the Dirac operator. 
  The N=1 supersymmetric version of generalized 2d dilaton gravity can be cast into the form of a Poisson Sigma Model, where the target space and its Poisson bracket are graded. The target space consists of a 1+1 superspace and the dilaton, which is the generator of Lorentz boosts therein. The Poisson bracket on the target space induces the invariance of the worldsheet theory against both diffeomorphisms and local supersymmetry transformations (superdiffeomorphisms). The machinery of Poisson Sigma Models is then used to find the general local solution to the field equations. As a byproduct, classical equivalence between the bosonic theory and its supersymmetric extension is found. 
  The article gives explicit calculation and interpretation of the additional locally anisotropic effects. Double role of the resulted gauge like fields discussed. 
  We study the dynamics of the probe fundamental string in the field background of the partially localized supergravity solution for the fundamental string ending on Dp-brane. We separately analyze the probe dynamics for its motion along the worldvolume direction and the transverse direction of the source Dp-brane. We compare the dynamics of the probe along the Dp-brane worldvolume direction to the BIon dynamics. 
  We give the Lax representations for for the elliptic, hyperbolic and homogeneous second order Monge-Ampere equations. The connection between these equations and the equations of hydrodynamical type give us a scalar dispersionless Lax representation. A matrix dispersive Lax representation follows from the correspondence between sigma models, a two parameter equation for minimal surfaces and Monge-Ampere equations. Local as well nonlocal conserved densities are obtained. 
  We generalize the construction of four dimensional non-tachyonic orientifolds of type 0B string theory to non-supersymmetric backgrounds. We construct a four dimensional model containing self-dual D3 and D9-branes and leading to a chiral anomaly-free massless spectrum. Moreover, we discuss a further tachyon-free six dimensional model with only D5 branes. Eventually, we speculate about strong coupling dual models of the ten-dimensional orientifolds of type 0B. 
  We discuss various dualities, relating integrable systems and show that these dualities are explained in the framework of Hamiltonian and Poisson reductions. The dualities we study shed some light on the known integrable systems as well as allow to construct new ones, double elliptic among them. We also discuss applications to the (supersymmetric) gauge theories in various dimensions. 
  We show that in certain limits the (1+1)-dimensional massive Thirring model at finite temperature $T$ is equivalent to a one-dimensional Coulomb gas of charged particles at the same $T$. This equivalence is then used to explore the phase structure of the massive Thirring model. For strong coupling and $T>>m$ (the fermion mass) the system is shown to behave as a free gas of "molecules" (charge pairs in the Coulomb gas terminology) made of pairs of chiral condensates. This binding of chiral condensates is responsible for the restoration of chiral symmetry as $T\to\infty$. In addition, when a fermion chemical potential $\mu\neq 0$ is included, the analogy with a Coulomb gas still holds with $\mu$ playing the role of a purely imaginary external electric field. For small $T$ and $\mu$ we find a typical massive Fermi gas behaviour for the fermion density, whereas for large $\mu$ it shows chiral restoration by means of a vanishing effective fermion mass. Some similarities with the chiral properties of low-energy QCD at finite $T$ and baryon chemical potential are discussed. 
  We compare oscillations of a fundamental string ending on a D3-brane in two different settings: (1) a test-string radially threading the horizon of an extremal black D3-brane and (2) the spike soliton of the DBI effective action for a D3-brane. Previous work has shown that overall transverse modes of the test-string appear as l=0 modes of the transverse scalar fields of the DBI system. We identify DBI world-volume degrees of freedom that have dynamics matching those of the test-string relative transverse modes. We show that there is a map, resembling T-duality, between relative and overall transverse modes for the test-string that interchanges Neumann and Dirichlet boundary conditions and implies equality of the absorption coefficients for both modes. We give general solutions to the overall and relative transverse parts of the DBI coupled gauge and scalar system and calculate absorption coefficients for the higher angular momentum modes in the low frequency limit. We find that there is a nonzero amplitude for l>0 modes to travel out to infinity along the spike, demonstrating that the spike remains effectively 3+1-dimensional. 
  We confirm the intuition that a string theory which is perturbatively infrared finite is automatically perturbatively ultraviolet finite. Our derivation based on the asymptotics of the Selberg trace formula for the Greens function on a Riemann surface holds for both open and closed string amplitudes and is independent of modular invariance and supersymmetry. The mass scale for the open strings stretched between Dbranes suggests a natural world-sheet ultraviolet regulator in the string path integral, preserving both T-duality and open-closed string world-sheet duality. Note added (Jan 2005): Comments and related references added. 
  We calculate one-loop corrections to the effective Lagrangian for the D3 brane. We perform the gauge-fixing of the kappa-symmetric Born-Infeld D3 brane action in the flat background using Killing gauge. The linearized supersymmetry of the gauge-fixed action coincides with that of the N=4 Yang-Mills theory. We use the helicity amplitude and unitarity technique to calculate the one-loop amplitudes at order alpha^4. The counterterms and the finite 1-loop corrections are of the form (dF)^4 and their supersymmetric generalization. This is to be contrasted with the Born-Infeld action which contains (F)^4 and other terms which do not depend on derivatives of the vector field strength. 
  We construct a two parameter family of 2-particle Hamiltonians closed under the duality operation of interchanging the (relative) momentum and coordinate. Both coordinate and momentum dependence are elliptic, and the modulus of the momentum torus is a non-trivial function of the coordinate. This model contains as limiting cases the standard Ruijsenaars-Calogero and Toda family of Hamiltonians, which are at most elliptic in the coordinates, but not in the momenta. 
  Solution is presented to the simplest problem about the vacuum backreaction on a pair creating source. The backreaction effect is nonanalytic in the coupling constant and restores completely the energy conservation law. The vacuum changes the kinematics of motion like relativity theory does and imposes a new upper bound on the velocity of the source. 
  We study the D-brane spectrum of N=2 string orbifold theories using the boundary state formalism. The construction is carried out for orbifolds with isolated singularities, non-isolated singularities and orbifolds with discrete torsion. Our results agree with the corresponding K-theoretic predictions when they are available and generalize them when they are not. This suggests that the classification of boundary states provides a sort of "quantum K-theory" just as chiral rings in CFT provide "quantum" generalizations of cohomology. We discuss the identification of these states with D-branes wrapping holomorphic cycles in the large radius limit of the CFT moduli space. The example C^3/Z_3 is worked out in full detail using local mirror symmetry techniques. We find a precise correspondence between fractional branes at the orbifold point and configurations of D-branes described by vector bundles on the exceptional P^2 cycle. 
  We study the mechanism of confinement via formation of Abrikosov-Nielsen-Olesen vortices on the Higgs branch of N=2 supersymmetric SU(2) gauge theory with massive fundamental matter. Higgs branch represents a limiting case of superconductor of type I with vanishing Higgs mass. We show that in this limit vortices still exist although they become logarithmically "thick". Because of this the confining potential is not linear any longer. It behaves as $L/\log L$ with a distance $L$ between confining heavy charges (monopoles). This new confining regime can occur only in supersymmetric theories. We also address the problem of quantum stability of vortices. To this end we develop string representation for a vortex and use it to argue that vortices remain stable. 
  We present dyon solutions to an SU(2) Dirac-Born-Infeld (DBI) gauge theory coupled to a Higgs triplet. We consider different non-Abelian extensions of the DBI action and study the resulting solutions numerically, comparing them with the standard Julia-Zee dyons. We discuss the existence of a critical value of $\beta$, the Born-Infeld absolute field parameter, below which the solution ceases to exist. We also analyse the effect of modifying the DBI action so as to include the analogous of the $\theta$ term, showing that Witten formula for the dyon charge also holds in DBI theories. 
  Some generalized BRS transformations are developed for the pure Yang-Mills theory, and a form of quantum gravity. Unlike the usual BRS transformations: these are nonlocal; may be infinite formal power series in the gauge fields; and do not leave the action invariant, but only the product exp(-S) with the Jacobian. Similar constructions should exist for many other field theory situations. 
  There are only a small number of ideas for stabilizing the moduli of string theory. One of the most appealing of these is the racetrack mechanism, in which a delicate interplay between two strongly interacting gauge groups fixes the value of the coupling constant. In this note, we explore this scenario. We find that quite generally, some number of discrete tunings are required in order that the mechanism yield a small gauge coupling. Even then, there is no sense in which a weak coupling approximation is valid. On the other hand, certain holomorphic quantities can be computed, so such a scheme is in principle predictive. Searching for models which realize this mechanism is thus of great interest. We also remark on cosmology in these schemes. 
  Three-String junctions are allowed configurations in II B string theory which preserve one-fourth supersymmetry. We obtain the 11-dimensional supergravity solution for curved membranes corresponding to these three-string junctions. 
  We present several non-trivial examples of the three-dimensional quantum Nambu bracket which involve square matrices or three-index objects. Our examples satisfy two fundamental properties of the classical Nambu bracket: they are skew-symmetric and they obey the Fundamental Identity. We contrast our approach to the existing literature on the quantum deformations of Nambu mechanics. We also discuss possible applications of our results in M-theory. 
  We present a short and direct derivation of Hawking radiation as a tunneling process, based on particles in a dynamical geometry. The imaginary part of the action for the classically forbidden process is related to the Boltzmann factor for emission at the Hawking temperature. Because the derivation respects conservation laws, the exact spectrum is not precisely thermal. We compare and contrast the problem of spontaneous emission of charged particles from a charged conductor. 
  This thesis addresses some classical and semi-classical aspects of black holes using an effective membrane representation of the event horizon. The classical "membrane paradigm" equations are derived from an underlying action formulation, even though the equations are dissipative. It is shown how the membrane action can also account for the hole's entropy, including the numerical factor. Two short derivations of Hawking radiation are then presented, the first based on particles in a dynamical geometry and the second on the membrane. The calculations obey conservation laws and lead to a correction to the thermal spectrum. Finally, a possible Penrose diagram for the formation and evaporation of a charged black hole is given, with the property that the post-evaporation spacetime is predictable from initial conditions, provided that the dynamics of the time-like singularity can be calculated. 
  We probe the AdS/CFT correspondence by comparing the thermodynamics of a rotating black hole in five-dimensional anti-de Sitter space with that of a conformal field theory on $S^3$, whose parameters come from the boundary of spacetime. In the high temperature limit, we find agreement between gauge theory and gravity in all thermodynamic quantities upto the same factor of 4/3 that appears for nonrotating holes. 
  Using the Gaussian wave-functional approach with the normal-ordering renormalization prescription, we show that for the (3+1)-dimensional massive lambda phi^4 theory, ``precarious'' and ``autonomous'' phi^4s can exist if and only if the normal-ordering mass is equal to the classical masses at the symmetrc and asymmetric vacua, respectively. 
  We investigate the dimensional, the dynamical and the topological structures of four dimensional Einstein and Yang-Mills theories. It is shown that these theories are constructed from two dimensional quantities, so that they possess always a distinguished two dimensional substructure. In this sense the four dimensional field theories are equivalent to related two dimensional field theories. 
  We investigate domain-wall/quantum field theory correspondences in various dimensions. Our general analysis does not only cover the well-studied cases in ten and eleven dimensions but also enables us to discuss new cases like a Type I/Heterotic 6-brane in ten dimensions and domain wall dualities in lower than ten dimensions. The examples we discuss include `d-branes' in six dimensions preserving 8 supersymmetries and extreme black holes in various dimensions. In the latter case we construct the quantum mechanics Hamiltonian and discuss several limits. 
  We study nonperturbative aspects of asymmetric orbifolds of type IIA, focussing on models that allow a dual perturbative heterotic description. In particular we derive the boundary states that describe the nonsupersymmetric D-branes of the untwisted sector and their zero mode spectra. These we use to demonstrate, how some special non BPS multiplets are identified under the duality map, and give some indications, how the mismatch of bosons and fermions in the perturbative heterotic spectrum is to be interpreted in terms of the nonperturbative degrees of freedom on the type IIA side. 
  We give an explicit expression for classical 1/4-BPS fields in supersymmetric Yang-Mills theory on noncommutative tori. We use it to study quantum 1/4-BPS states. In particular we calculate the degeneracy of 1/4-BPS energy levels. 
  Pure spinor formalism and non-integrable exponential factors are used for constructing the conformal-invariant wave equation and Lagrangian density for massive fermion. It is proved that canonical Dirac Lagrangian for massive fermion is invariant under induced projective conformal transformations. 
  A constant homogeneous magnetic field is applied to a composite system made of two scalar particles with opposite charges. Motion is described by a pair of coupled Klein-Gordon equations that are written in closed form with help of a suitable representation. The relativistic symmetry associated with the magnetic field is carefully respected. Considering eigenstates of the pseudo momentum four-vector, we separate out collective variables and obtain a 3- dimensional reduced equation, posing a nonconventional eigenvalue problem. The velocity of the system as a whole generates "motional" terms in the formulas these terms are taken into account within a manifestly covariant framework. 
  The SO(2,10) covariant extension of M-theory superalgebra is considered, with the aim to construct a correspondingly generalized M-theory, or 11d supergravity. For the orbit, corresponding to the $11d$ supergravity multiplet, the simplest unitary representations of the bosonic part of this algebra, with sixth-rank tensor excluded, are constructed on a language of field theory in 66d space-time. The main peculiarities are the presence of more than one equation of motion and corresponding Lagrangians for a given field and that the gauge and SUSY invariances of the theory mean that the sum of variations of these Lagrangians (with different variations of the same field) is equal to zero. 
  After discussing some old (and not-so-old) entropy bounds both for isolated systems and in cosmology, I will argue in favour of a "Hubble entropy bound" holding in the latter context. I will then apply this bound to recent developments in string cosmology, show that it is naturally saturated throughout pre-big bang inflation, and claim that its fulfilment at later times has interesting implications for the exit problem of string cosmology. 
  We attempt to define a new invariant I of (almost) Calabi-Yau 3-folds M, by counting special Lagrangian rational homology 3-spheres N in M in each 3-homology class, with a certain weight w(N) depending on the topology of N. This is motivated by the Gromov-Witten invariants of a symplectic manifold, which count the J-holomorphic curves in each 2-homology class.   In order for this invariant to be interesting, it should either be unchanged by deformations of the underlying (almost) Calabi-Yau structure, or else transform according to some rigid set of rules as the periods of the almost Calabi-Yau structure pass through some topologically determined hypersurfaces in the cohomology of M.   As we deform the underlying almost Calabi-Yau 3-fold, the collection of special Lagrangian homology 3-spheres only change when they become singular. Thus, to determine the stability of the invariant under deformations we need know about the singular behaviour of special Lagrangian 3-folds, which is not well understood.   We describe two kinds of singular behaviour of special Lagrangian 3-folds, and derive identities on the weight function w(N) for I to be unchanged or transform well under them. The weight function w(N)=|H_1(N,Z)| satisfies these identities. We conjecture that an invariant I defined with this weight is independent of the Kahler class, and changes in certain ways as the holomorphic 3-form passes through some real hypersurfaces in H^3(M,C).   Finally we consider connections with String Theory. We argue that our invariant I counts isolated 3-branes, and that it should play a part in the Mirror Symmetry story for Calabi-Yau 3-folds. 
  We show that there exist finite energy, non-singular instanton solutions for five-dimensional theories with broken gauge symmetry. The soliton is supported against collapse by a non-zero electric charge. The low-energy dynamics of these solutions is described by motion on the ADHM moduli space with potential. 
  The correspondence between the 't Hooft limit of N=4 super Yang-Mills theory and tree-level IIB superstring theory on AdS(5)xS(5) in a Ramond-Ramond background at values of lambda=g^2 N ranging from infinity to zero is examined in the context of unitarity. A squaring relation for the imaginary part of the holographic scattering of identical string fields in the two-particle channels is found, and a mismatch between weak and strong 't Hooft coupling is pointed out within the correspondence. Several interpretations and implications are proposed. 
  We treat the action for a bosonic membrane as a sigma model, and then compute quantum corrections by integrating out higher membrane modes. As in string theory, where the equations of motion of Einstein's theory emerges by setting $\beta = 0$, we find that, with certain assumptions, we can recover the equations of motion for the background fields. Although the membrane theory is non-renormalizable on the world volume by power counting, the investigation of the ultra-violet behavior of membranes may give us insight into the supersymmetric case, where we hope to obtain higher order M-theory corrections to 11 dimensional supergravity. 
  We directly derive the classical equation of motion, which governs the centre of mass of a test string, from the string action. In a certain case, the equation is basically same as one derived by Papapetrou, Dixon and Wald for a test extended body. We also discuss the force balance using a stringy probe particle for an exact spinning multi-soliton solution of Einstein-Maxwell-Dilaton-Axion theory. It is well known that the force balance condition yields the saturation of the Bogomol'nyi type bound in the lowest order. In the present formulation the gyromagnetic ratio of the stringy probe particle is automatically determined to be $g=2$ which is the same value as the background soliton. As a result we can confirm the force balance via the gravitational spin-spin interaction. 
  Supersymmetric CPN models based on underlying bosonic Kahler manifolds have not been thought to arise directly from constrained linear ones. A counterexample for N=4 is presented using improved understanding of membranes in superstring theories leading to crucial central terms modifying the algebra of supercharge densities. The example has an immediate extension to all higher N. 
  Solutions in multidimensional gravity with m p-branes related to Toda-like systems (of general type) are obtained. These solutions are defined on a product of n+1 Ricci-flat spaces M_0 x M_1 x...x M_n and are governed by one harmonic function on M_0. The solutions are defined up to the solutions of Laplace and Toda-type equations and correspond to null-geodesics of the (sigma-model) target-space metric. Special solutions relating to A_m Toda chains (e.g. with m =1,2) are considered. 
  We study conserved currents of any integer or half integer spin built from massless scalar and spinor fields in $AdS_3$. 2-forms dual to the conserved currents in $AdS_3$ are shown to be exact in the class of infinite expansions in higher derivatives of the matter fields with the coefficients containing inverse powers of the cosmological constant. This property has no analog in the flat space and may be related to the holography of the AdS spaces. `Improvements' to the physical currents are described as the trivial local current cohomology class. A complex of spin $s$ currents $(T^s, {\cal D})$ is defined and the cohomology group $H^1(T^s, {\cal D}) = {\bf C}^{2s+1}$ is found. This paper is an extended version of hep-th/9906149. 
  We perform the derivation of the two-point correlation function in N=4 D=4 super Yang-Mills theory from superstring theory by computing the dilaton scattering amplitude in the NSR formulation of superstring theory on $AdS_5\times{S^5}$ 
  We derive an energy bound for a `baryonic' D5-brane probe in the $adS_5\times S^5$ background near the horizon of $N$ D3-branes. Configurations saturating the bound are shown to be 1/4 supersymmetric $S^5$-wrapped D5-branes, with a total Born-Infeld charge $N$. Previous results are recovered as a special case. We derive a similar energy bound for a `baryonic' M5-brane probe in the background of $N$ M5-branes. Configurations saturating the bound are again 1/4 supersymmetric and, in the $adS_7\times S^4$ near-horizon limit, provide a worldvolume realization of the `baryon string' vertex of the (2,0)-supersymmetric six-dimensional conformal field theory on coincident M5-branes. For the full M5-background we find a worldvolume realization of the Hanany-Witten effect in M-theory. 
  I review the issue of string and compactification scales in the weak-coupling regimes of string theory. I explain how in the Brane World scenario a (effectively) two-dimensional transverse space that is hierarchically larger than the string length may replace the conventional `energy desert' described by renormalizable supersymmetric QFT. I comment on the puzzle of unification in this context. 
  In the type 0B string theory, we discuss the role of tachyon($T$) and fixed scalars($\nu,\lambda$). The issue is to explain the difference between tachyon and fixed scalars in the D$5_{\pm}$-D$1_{\pm}$ black hole background. For this purpose, we perform the semiclassical calculation. Here one finds a mixing between ($\nu, \lambda, T$) and the other fields. Using the decoupling procedure, one finds the linearized equation for the tachyon. From the potential analysis, it turns out that $\nu$ plays a role of test field well, while the tachyon induces an instability of Minkowski space vacuum. But the roles of $\nu$ and $T$ are the same in the near-horizon geometry. Finally we discuss the stability problem. 
  It has been argued that the extremal dilaton black holes exhibit a flux expulsion of Abelian-Higgs vortices. We re-examine carefully the problem and give analytic proofs for the flux expulsion always takes place. We also conduct numerical analysis of the problem using three initial data sets on the horizon of an extreme dilatonic black hole, namely, core, vacuum and sinusoidal initial conditions. We also show that an Abelian-Higgs vortex can end on the extremal dilaton black hole. Concluding, we calculate the backreaction of the Abelian-Higgs vortex on the geometry of the extremal black hole and drew a conclusion that a straight cosmic string and the extreme dilaton black hole hardly knew their presence. 
  We investigate nonperturbative effects in M-theory compactifications arising from wrapped membranes. In particular, we show that in $d=4, \CN=1$ compactifications along manifolds of $G_2$ holonomy, membranes wrapped on rigid supersymmetric 3-cycles induce nonzero corrections to the superpotential. Thus, membrane instantons destabilize many M-theory compactifications. Our computation shows that the low energy description of membrane physics is usefully described in terms of three-dimensional topological field theories, and the superpotential is expressed in terms of topological invariants of the 3-cycle. We discuss briefly some applications of these results. For example, using mirror symmetry we derive a counting formula for supersymmetric three-cycles in certain Calabi-Yau manifolds. 
  The non-trivial ultraviolet fixed point in quantum gravity is calculated by means of the exact renormalization group equation in d-dimensions $(2\simeq d\leq4)$. It is shown that the ultraviolet non-Gaussian fixed point which is expected from the perturbatively $\epsilon$-expanded calculations in $2+\epsilon$ gravity theory remains in d=4. Hence it is possible that quantum gravity is an asymptotically safe theory and renormalizable in 2<d. 
  Solutions, exactly expressed in terms of elementary functions (unique Laughlin states), of the correlated motion problem for a pair of 2D-electrons in a constant and uniform magnetic field have been shown to exist for a certain relation between the magnetic field induction and the electron charge. Arguments that can help to understand the physical meaning of these remarkable magnetic field values have been provided. The special interest to this problem is justified by the importance of the new state of matter recently observed. 
  Guided by the generalized conformal symmetry, we investigate the extension of AdS-CFT correspondence to the matrix model of D-particles in the large N limit. We perform a complete harmonic analysis of the bosonic linearized fluctuations around a heavy D-particle background in IIA supergravity in 10 dimensions and find that the spectrum precisely agrees with that of the physical operators of Matrix theory. The explicit forms of two-point functions give predictions for the large $N$ behavior of Matrix theory with some special cutoff. We discuss the possible implications of our results for the large N dynamics of D-particles and for the Matrix-theory conjecture. We find an anomalous scaling behavior with respect to the large N limit associated to the infinite momentum limit in 11 dimensions, suggesting the existence of a screening mechanism for the transverse extension of the system. 
  The configuration of typical highly excited (M >> M_s ~ (alpha')^{-1/2}) string states is considered as the string coupling g is adiabatically increased. The size distribution of very massive single string states is studied and the mass shift, due to long-range gravitational, dilatonic and axionic attraction, is estimated. By combining the two effects, in any number of spatial dimensions d, the most probable size of a string state becomes of order l_s = sqrt{2 alpha'} when g^2 M / M_s ~ 1. Depending on the dimension d, the transition between a random-walk-size string state (for low g) and a compact (~ l_s) string state (when g^2 M / M_s ~ 1) can be very gradual (d=3), fast but continuous (d=4), or discontinuous (d > 4). Those compact string states look like nuggets of an ultradense state of string matter, with energy density rho ~ g^{-2} M_s^{d+1}. Our results extend and clarify previous work by Susskind, and by Horowitz and Polchinski, on the correspondence between self-gravitating string states and black holes. 
  We study the spectral properties of the transfer matrix for a gonihedric random surface model on a three-dimensional lattice. The transfer matrix is indexed by generalized loops in a natural fashion and is invariant under a group of motions in loop-space. The eigenvalues of the transfer matrix can be evaluated exactly in terms of the partition function, the internal energy and the correlation functions of the two-dimensional Ising model and the corresponding eigenfunctions are explicit functions on loop-space. 
  A generalized second law in string cosmology accounts for geometric and quantum entropy in addition to ordinary sources of entropy. The proposed generalized second law forbids singular string cosmologies, under certain conditions, and forces a graceful exit transition from dilaton-driven inflation by bounding curvature and dilaton kinetic energy. 
  Starting with Green-Schwarz superstring action, we construct a type IIB matrix model. We fix the local $\kappa$ symmetry in the Killing spinor gauge and then perform the world-sheet duality transformation. A matrix model obtained from this gauge-fixed action is shown to be equivalent to the type IIB matrix model constructed by Ishibashi et al. Our construction does not make use of an analytic continuation of spinor variable. Moreover, it seems that our construction is applicable to that of more general type IIB matrix models in a curved background such as $AdS_5 \times S^5$. 
  This is a slightly extended version of a seminar given the 8th of June at the TASI 99 at Colorado University in Boulder. The motivations behind two time theory are explained and the theory is introduced via one of the theory's easier gauges of a particle on a black hole background. Important results that should be interesting as well in the light of the recent AdS mania will be summarized. 
  A systematic numerical study of the classical solutions to the combined system consisting of the Georgi-Glashow model and the SO(3) gauged Skyrme model is presented. The gauging of the Skyrme system permits a lower bound on the energy, so that the solutions of the composite system can be topologically stable. The solutions feature some very interesting bifurcation patterns, and it is found that some branches of these solutions are unstable. 
  In the present paper we calculate the statistical partition function for any number of extended objects in Matrix theory in the one loop approximation. As an application, we calculate the statistical properties of K clusters of D0 branes and then the statistical properties of K membranes which are wound on a torus. 
  An action for supersymmetric D0-branes in curved backgrounds is obtained by dimensional reduction of N=1 ten-dimensional supergravity coupled to super Yang-Mills system to 0+1 dimensions. The resultant action exhibits the coset-space symmetry $\frac{SO(9,9+n)}{SO(9)\times SO(9+n)}\times U(1)$ where $n=N^{2}-1$ is the dimension of the SU(N) gauge group. 
  We discuss the fate of certain tachyonic closed string theories from two perspectives. In both cases our approach involves studying directly configurations with finite negative tree-level cosmological constant. Closed string analogues of orientifolds, which carry negative tension, are argued to represent the minima of the tachyon potential in some cases. In other cases, we make use of the fact, noted in the early string theory literature, that strings can propagate on spaces of subcritical dimension at the expense of introducing a tree-level cosmological constant. The form of the tachyon vertex operator in these cases makes it clear that a subcritical-dimension theory results from tachyon condensation. Using results of Kutasov, we argue that in some Scherk-Schwarz models, for finely-tuned tachyon condensates, a minimal model CFT times a subcritical dimension theory results. In some instances, these two sets of ideas may be related by duality. 
  We study an integrable quantum field theory of a single stable particle with an infinite number of resonance states. The exact $S$--matrix of the model is expressed in terms of Jacobian elliptic functions which encode the resonance poles inherently. In the limit $l \to 0$, with $l$ the modulus of the Jacobian elliptic function, it reduces to the Sinh--Gordon $S$--matrix. We address the problem of computing the Form Factors of the model by studying their monodromy and recursive equations. These equations turn out to possess infinitely many solutions for any given number of external particles. This infinite spectrum of solutions may be related to the irrational nature of the underlying Conformal Field Theory reached in the ultraviolet limit. We also discuss an elliptic version of the thermal massive Ising model which is obtained by a particular value of the coupling constant. 
  In two-dimensional lattice fermion model a determinant representation for the two-point correlation function of the twist field in the disorder phase is obtained. This field is defined by twisted boundary conditions for lattice fermion field. The large distance asymptotics of the correlation function is calculated at the critical point and in the scaling region. The result is compared with the vacuum expectation values of exponential fields in the sine-Gordon model conjectured by S.Lukyanov and A.Zamolodchikov. 
  We use a D-instanton or physical gauge approach to re-derive the heterotic string worldsheet instanton contribution to the superpotential in Calabi-Yau compactification. We derive an analogous formula for worldsheet instanton corrections to the moduli space metric in heterotic string or Type I compactification on a K3 surface. In addition, we give a global analysis of the phase of the worldsheet path integral of the heterotic string, showing precisely how the B-field must be interpreted. 
  Given a symmetry group acting on a principal fibre bundle, symmetric states of the quantum theory of a diffeomorphism invariant theory of connections on this fibre bundle are defined. These symmetric states, equipped with a scalar product derived from the Ashtekar-Lewandowski measure for loop quantum gravity, form a Hilbert space of their own. Restriction to this Hilbert space yields a quantum symmetry reduction procedure in the framework of spin network states the structure of which is analyzed in detail. Three illustrating examples are discussed: Reduction of 3+1 to 2+1 dimensional quantum gravity, spherically symmetric electromagnetism and spherically symmetric gravity. 
  Utilizing the previously established general formalism for quantum symmetry reduction in the framework of loop quantum gravity the spectrum of the area operator acting on spherically symmetric states in 4 dimensional pure gravity is investigated. The analysis requires a careful treatment of partial gauge fixing in the classical symmetry reduction and of the reinforcement of SU(2)-gauge invariance for the quantization of the area operator. The eigenvalues of that operator applied to spherically symmetric spin network states have the form A_n propor. sqrt{n(n+2)}, n=0,1,2..., giving A_n propor. n for large n. The result clarifies (and reconciles!) the relationship between the more complicated spectrum of the general (non-symmetric) area operator in loop quantum gravity and the old Bekenstein proposal that A_n propor. n. 
  We present a method for a recursive graphical construction of Feynman diagrams with their correct multiplicities in quantum electrodynamics. The method is first applied to find all diagrams contributing to the vacuum energy from which all n-point functions are derived by functional differentiation with respect to electron and photon propagators, and to the interaction. Basis for our construction is a functional differential equation obeyed by the vacuum energy when considered as a functional of the free propagators and the interaction. Our method does not employ external sources in contrast to traditional approaches. 
  In recent papers it has been observed that non-Hermitian Hamiltonians, such as those describing $ig\phi^3$ and $-g\phi^4$ field theories, still possess real positive spectra so long as the weaker condition of ${\cal PT}$ symmetry holds. This allows for the possibility of new kinds of quantum field theories that have strange and quite unexpected properties. In this paper a technique based on truncating the Schwinger-Dyson equations is presented for renormalizing and solving such field theories. Using this technique it is argued that a $-g\phi^4$ scalar quantum field theory in four-dimensional space-time is renormalizable, is asymptotically free, has a nonzero value of $<0|\phi|0>$, and has a positive definite spectrum. Such a theory might be useful in describing the Higgs boson. 
  We investigate the relation between supersymmetry and geometry for two dimensional sigma models with target spaces of arbitrary signature, and Lorentzian or Euclidean world-sheets. In particular, we consider twisted forms of the two-dimensional $(p,q)$ supersymmetry algebra. Superspace formulations of the $(p,q)$ heterotic sigma-models with twisted or untwisted supersymmetry are given. For the twisted (2,1) and the pseudo-K\"{a}hler sigma models, we give extended superspace formulations. 
  We use the AdS/CFT correspondence to calculate three point functions of chiral primary operators at large N in d=3, N=8 and d=6, N=(2,0) superconformal field theories. These theories are related to the infrared fixed points of world-volume descriptions of N coincident M2 and M5 branes, respectively. The computation can be generalized by employing a gravitational action in arbitrary dimensions D, coupled to a (p+1)-form and appropriately compactified on AdS(D-p-2)xS(p+2). We note a surprising coincidence: this generalized model reproduces for D=10, p=3 the three point functions of d=4, N=4 SYM chiral primary operators at large N. 
  We discuss the derivation of the path integral representation over gauge degrees of freedom for Wilson loops in SU(N) gauge theory and 4-dimensional Euclidean space-time by using well-known properties of group characters. A discretized form of the path integral is naturally provided by the properties of group characters and does not need any artificial regularization. We show that the path integral over gauge degrees of freedom for Wilson loops derived by Diakonov and Petrov (Phys. Lett. B224 (1989) 131) by using a special regularization is erroneous and predicts zero for the Wilson loop. This property is obtained by direct evaluation of path integrals for Wilson loops defined for pure SU(2) gauge fields and Z(2) center vortices with spatial azimuthal symmetry. Further we show that both derivations given by Diakonov and Petrov for their regularized path integral, if done correctly, predict also zero for Wilson loops. Therefore, the application of their path integral representation of Wilson loops cannot give "a new way to check confinement in lattice" as has been declared by Diakonov and Petrov (Phys. Lett. B242 (1990) 425). From the path integral representation which we consider we conclude that no new non-Abelian Stokes theorem can exist for Wilson loops except the old-fashioned one derived by means of the path-ordering procedure. 
  A universal disentangling formula (such as the Baker-Campbell-Hausdorff one) for coherent states of Perelomov's type ($ |z \ra = \exp (z\Adag -\bar{z}A) |0 \ra $) which are defined for generalized oscillator algebras is given. 
  The precise relationship between the arbitrary monodromy dependent 2-form appearing in the chiral WZNW symplectic form and the `exchange r-matrix' that governs the corresponding Poisson brackets is established. Generalizing earlier results related to diagonal monodromy, the exchange r-matrices are shown to satisfy a new dynamical generalization of the classical modified Yang-Baxter equation, which is found to admit an interpretation in terms of (new) Poisson-Lie groupoids. Dynamical exchange r-matrices for which right multiplication yields a classical or a Poisson-Lie symmetry on the chiral WZNW phase space are presented explicitly. 
  If the standard model is embedded in a conformal theory, what is the simplest possibility? We analyse all abelian orbifolds for discrete symmetry $Z_p$ with $p \leq 7$, and find that the simplest such theory is indeed $SU(3)^7$. Such a theory predicts the correct electroweak unification (sin$^2 \theta \simeq 0.231$). A color coupling $\alpha_C(M) \simeq 0.07$ suggests a conformal scale $M$ near to 10 TeV. 
  Two-dimensional matterless dilaton gravity is a topological theory and can be classically reduced to a (0+1)-dimensional theory with a finite number of degrees of freedom. If quantization is performed, a simple gauge invariant quantum mechanics is obtained. The properties of the gauge invariant operators and of the Hilbert space of physical states can be determined. In particular, for N-dimensional pure gravity with (N-2)-dimensional spherical symmetry, the square of the ADM mass operator is self-adjoint, not the mass itself. 
  We propose that the effective field theories of certain wrapped D-branes are given by topological actions based on Born-Infeld theory. In particular, we present a Born-Infeld version of the Abelian Donaldson-Witten theory. We then consider wrapping D3 branes on calibrated submanifolds and for the Calabi-Yau four-fold case, discuss how the resulting theory could give rise to a Born-Infeld version of the ampicheiral twisted N=4 super Yang-Mills topological field theory. 
  We review the construction of non-supersymmetric open string vacua in various dimensions. They can be obtained either projecting the (compactified) non-supersymmetric 0B theory, or applying the Scherk-Schwarz mechanism to open strings. Generically, these vacua generate a non-vanishing cosmological constant. However, one can construct particular kinds of Scherk-Schwarz compactifications with vanishing cosmological constant, at least for low orders, based on asymmetric orbifolds. A generic feature of these models is that supersymmetry remains unbroken on the branes at all mass level, while it is broken in the bulk in a way that preserves Fermi-Bose degeneracy at each mass level in the perturbative string spectrum. 
  In this article we show that boundary conditions can be treated as Lagrangian and Hamiltonian constraints. Using the Dirac method, we find that boundary conditions are equivalent to an infinite chain of second class constraints which is a new feature in the context of constrained systems. Constructing the Dirac brackets and the reduced phase space structure for different boundary conditions, we show why mode expanding and then quantizing a field theory with boundary conditions is the proper way. We also show that in a quantized field theory subjected to the mixed boundary conditions, the field components are noncommutative. 
  A reparametrization invariant model, introduced by Montesinos, Rovelli and Thiemann, possessing an SL(2,R) gauge symmetry is treated along the guidelines of an algebraic constraint quantization scheme that translates the vanishing of the constraints into representation conditions for the algebra of observables. The application of this algebraic scheme to the SL(2,R) model yields an unambiguous identification of the physical representation of the algebra of observables. 
  Gauged WZNW models are integrable conformal field theories. We integrate the classical \slu{} theory with periodic boundary conditions, which describes closed strings moving in a curved target-space geometry. We calculate its Poisson bracket structure by solving an initial state problem. The results differ from previous field-theoretic calculations due to zero modes. For a future exact canonical quantization the physical fields are (non-locally) transformed onto canonical free fields. 
  We study the quantum mechanical model obtained as a dimensional reduction of N=1 super Yang-Mills theory to a periodic light-cone "time". After mapping the theory to a cohomological field theory, the partition function (with periodic boundary conditions) regularized by a massive term appears to be equal to the partition function of the twisted matrix oscillator. We show that this partition function perturbed by the operator of the holonomy around the time circle is a tau function of Toda hierarchy. We solve the model in the large N limit and study the universal properties of the solution in the scaling limit of vanishing perturbation. We find in this limit a phase transition of Gross-Witten type. 
  The principle of invariance of the c-number symmetric bracket is used to derive both the quantum operator commutator relation $[\hat q, \hat p]=i\hbar$ and the time-dependent Schr\"odinger equation. A c-number dynamical equation is found which leads to the second quantized field theory of bosons and fermions. 
  In these notes we explain how the CFT description of random matrix models can be used to perform actual calculations. Our basic example is the hermitian matrix model, reformulated as a conformal invariant theory of free fermions. We give an explicit operator construction of the corresponding collective field theory in terms of a bosonic field on a hyperelliptic Riemann surface, with special operators associated with the branch points. The quasiclassical expressions for the spectral kernel and the joint eigenvalue probabilities are then easily obtained as correlation functions of current, fermionic and twist operators. The result for the spectral kernel is valid both in macroscopic and microscopic scales. At the end we briefly consider generalizations in different directions. 
  The T-dual formulation of Type I superstring theory, sometimes called Type I' theory, has a number of interesting features. Here we review some of them including the role of D0-branes and D8-branes in controlling possible gauge symmetry enhancement. 
  Recently Bousso conjectured the entropy crossing a certain light-like hypersurface is bounded by the surface area. We point out a number of difficulties with this conjecture. 
  A formula constituting the non-Abelian Stokes theorem for general semi-simple compact gauge groups is presented. The formula involves a path integral over a group space and is applicable to Wilson loop variables irrespective of the topology of loops. Some simple expressions analogous to the 't Hooft tensor of a magnetic monopole are given for the 2-form of interest. A special property in the case of the fundamental representation of the group SU(N) is pointed out. 
  We study a system of nonabelian anyons in the lowest Landau level of a strong magnetic field. Using diagrammatic techniques, we prove that the virial coefficients do not depend on the statistics parameter. This is true for all representations of all nonabelian groups for the statistics of the particles and relies solely on the fact that the effective statistical interaction is a traceless operator. 
  In this paper we re-express the Schouten-Nijenhuis, the Fr\"olicher-Nijenhuis and the Nijenhuis-Richardson brackets on a symplectic space using the extended Poisson brackets structure present in the path-integral formulation of classical mechanics. 
  In the light of recent studies of BPS triple junctions in the Wess-Zumino model we describe techniques to construct infinite lattices using similar junctions. It is shown that whilst these states are only approximately locally BPS they are nevertheless stable to small perturbations, giving a metastable tiling of the plane. 
  This is a contracted version of a series of lectures for graduate and undergraduate students given at the {\sl ``VI Seminario Nazionale di Fisica Teorica"} (Parma, September 1997), at the Second Int. Conference {\sl ``Around VIRGO"} (Pisa, September 1998), and at the Second {\sl SIGRAV} School on {\sl "Gravitational Waves in Astrophysics, Cosmology and String Theory"} (Center "A. Volta", Como, April 1999). The aim is to provide an elementary, self-contained introduction to string cosmology and, in particular, to the background of relic cosmic gravitons predicted in the context of the so-called "pre-big bang" scenario. No special preparation is required besides a basic knowledge of general relativity and of standard (inflationary) cosmology. All the essential computations are reported in full details either in the main text or in the Appendices. For a deeper and more complete approach to the pre-big bang scenario the interested reader is referred to the updated collection of papers available at {\tt http://www.to.infn.it/\~{}gasperin/}. 
  We study the Lund-Regge equation that governs the motion of strings in a constant background antisymmetric tensor field by using the duality between the Lund-Regge equation and the complex sine-Gordon equation. Similar to the cases of vortex filament configurations in fluid dynamics, we find various exact solitonic string configurations which are the analogue of the Kelvin wave, the Hasimoto soliton and the smoke ring. In particular, using the duality relation, we obtain a completely new type of configuration which corresponds to the breather of the complex sine-Gordon equation. 
  We generalize the Lund-Regge model for vortex string dynamics in 4-dimensional Minkowski space to the arbitrary n-dimensional case. The n-dimensional vortex equation is identified with a nonabelian sine-Gordon equation and its integrability is proven by finding the associated linear equations of the inverse scattering. An explicit expression of vortex coordinates in terms of the variables of the nonabelian sine-Gordon system is derived. In particular, we obtain the n-dimensional vortex soliton solution of the Hasimoto-type from the one soliton solution of the nonabelian sine-Gordon equation. 
  Employing the AdS/CFT correspondence, we give an explicit supergravity picture for the renormalization group flow of couplings 4-d super Yang Mills with four supercharges. The solution represents a domain wall of 5-d, N=2 supergravity, that interpolates between two (different) AdS_5 vacua and is obtained by gauging a U(1) subgroup of the SU(2) R-symmetry. On the supergravity side the domain wall couples only to scalar fields from vector mulitplets, but not to scalars from hyper multiplets. We discuss the c-theorem, the beta-functions and consider two examples: one is the sugra solution related to Z_k orbifolds (corresponding to N=2 SYM) and the other is an orientifold construction for an elliptically fibered CY with F_1 basis (corresponding to N=1 SYM). 
  Dynamical chiral symmetry breaking in the DLCQ method is investigated in detail using a chiral Yukawa model closely related to the Nambu-Jona-Lasinio model. By classically solving three constraints characteristic of the light-front formalism, we show that the chiral transformation defined on the light front is equivalent to the usual one when bare mass is absent. A quantum analysis demonstrates that a nonperturbative mean-field solution to the ``zero-mode constraint'' for a scalar boson (sigma) can develop a nonzero condensate while a perturbative solution cannot. This description is due to our identification of the ``zero-mode constraint'' with the gap equation. The mean-field calculation clarifies unusual chiral transformation properties of fermionic field, which resolves a seemingly inconsistency between triviality of the null-plane chiral charge Q_5|0>=0 and nonzero condensate. We also calculate masses of scalar and pseudoscalar bosons for both symmetric and broken phases, and eventually derive the PCAC relation and nonconservation of Q_5 in the broken phase. 
  We study the ultraviolet asymptotics in affine Toda theories. These models are considered as perturbed non-affine Toda theories. We calculate the reflection amplitudes, which relate different exponential fields with the same quantum numbers. Using these amplitudes we derive the quantization condition for the vacuum wave function, describing zero-mode dynamics, and calculate the UV asymptotics of the effective central charge. These asymptotics are in a good agreement with thermodynamic Bethe ansatz results. 
  A field theoretical framework for the recently proposed photon condensation effect in a nonlinear Fabry-Perot cavity is discussed. The dynamics of the photon gas turns out to be described by an effective 2D Hamiltonian of a complex massive scalar field. Finite size effects are shown to be relevant for the existence of the photon condensate. 
  We consider four-dimensional N=1 field theories realized by type IIA brane configurations of NS-branes and D4-branes, in the presence of orientifold six-planes and D6-branes. These configurations are known to present interesting effects associated to the appearance of chiral symmetries and chiral matter in the four-dimensional field theory. We center on models with one compact direction (elliptic models) and show that, under T-duality, the configurations are mapped to a set of type IIB D3-branes probing N=1 orientifolds of C^2/Z_N singularities. We explicitly construct these orientifolds, and show the field theories on the D3-brane probes indeed reproduces the field theories constructed using the IIA brane configurations. This T-duality map allows to understand the type IIB realization of several exotic brane dynamics effects on the type IIA side: Flavour doubling, the splitting of D6-branes and O6-planes in crossing a NS-brane and the effect of a non-zero type IIA cosmological constant turn out to have surprisingly standard type IIB counterparts. 
  We construct marginal operators of the orbifold SCFT corresponding to all twenty near-horizon moduli in supergravity, including operators involving twist fields which correspond to the blowing up modes. We identify the operators with the supergravity moduli in a 1-1 fashion by inventing a global SO(4) algebra in the SCFT. We analyze the gauge dynamics of the D1/D5 system relevant to the splitting $(Q_1,Q_5)\to (Q'_1,Q'_5)+ (Q''_1,Q''_5)$ with the help of a linear sigma model. We show in supergravity as well as in SCFT that the absorption cross-section for minimal scalars is the same all over the near-horizon moduli space. 
  We study the heat kernel for an operator of Laplace type with a $\delta$-function potential concentrated on a closed surface. We derive the general form of the small $t$ asymptotics and calculate explicitly several first heat kernel coefficients. 
  Two time theory is derived via localization of the global Sp(2) [or Osp(1/2), Osp(N/2), Sp(2N),...] symmetry in phase space in order to give a self contained introduction to two time theory. Then it is shown that from the two-times physics point of view theories of point particles on many known black hole backgrounds are Sp(2) gauge duals of one another and of course also gauge dual to all other equal dimensional gauges from earlier two time related publications (hydrogen atom, ...). We reproduce the free (quantum) relativistic particle on 1+1 dimensional black hole backgrounds and 2+1 dimensional BTZ ones. Other 2+1 black holes and n+1 ones are touched on but explicitely found only as cross sections of complicated (n+1)+1 backgrounds. Further we give near horizon solutions (e.g. n+1 Robertson-Bertotti). Since two time physics can reproduce these backgrounds all particle actions have hidden symmetries that have not been noticed before. 
  A Hamiltonian analysis of Yang-Mills (YM) theory in (2+1) dimensions with a level $k$ Chern-Simons term is carried out using a gauge invariant matrix parametrization of the potentials. The gauge boson states are constructed and the contribution of the dynamical mass gap to the gauge boson mass is obtained. Long distance properties of vacuum expectation values are related to a Euclidean two-dimensional YM theory coupled to $k$ flavors of Dirac fermions in the fundamental representation. We also discuss the expectation value of the Wilson loop operator and give a comparison with previous results. 
  We consider the role of boundary conditions in the $AdS_{d+1}/CFT_{d}$ correspondence for the scalar field theory. Also a careful analysis of some limiting cases is presented. We study three possible types of boundary conditions, Dirichlet, Neumann and mixed. We compute the two-point functions of the conformal operators on the boundary for each type of boundary condition. We show how particular choices of the mass require different treatments. In the Dirichlet case we find that there is no double zero in the two-point function of the operator with conformal dimension $\frac{d}{2}$. The Neumann case leads to new normalizations for the boundary two-point functions. In the massless case we show that the conformal dimension of the boundary conformal operator is precisely the unitarity bound for scalar operators. We find a one-parameter family of boundary conditions in the mixed case. There are again new normalizations for the boundary two-point functions. For a particular choice of the mixed boundary condition and with the mass squared in the range $-d^2/4<m^2<-d^2/4+1$ the boundary operator has conformal dimension comprised in the interval $[\frac{d-2}{2}, \frac{d}{2}]$. For mass squared $m^2>-d^2/4+1$ the same choice of mixed boundary condition leads to a boundary operator whose conformal dimension is the unitarity bound. 
  We argue that the brane-world picture with matter-fields confined to 4-d domain walls and with gravitational interactions across the bulk disallows adding an arbitrary constant to the low-energy, 4-d effective theory -- which finesses the usual cosmological constant problem. The analysis also points to difficulties in stabilizing moduli fields; as an alternative, we suggest scenarios in which the moduli motion is heavily damped by various cosmological mechanisms and varying ultra-slowly with time. 
  A review is given of the status and developments of the research program aiming to reformulate the physics of the four interactions at the classical level in a unified way in terms of Dirac-Bergmann observables with special emphasis on the open mathematical, physical and interpretational problems. 
  We give an off-shell formulation of N=2 Poincare supergravity in five dimensions 
  A matrix model for type 0 strings is proposed. It consists in making a non-supersymmetric orbifold projection in the Yang-Mills theory and identifying the infrared configurations of the system at infinite coupling with strings. The correct partition function is calculated. Also, the usual spectrum of branes is found. Both type A and B models are constructed. The model in a torus contains all the degrees of freedom and interpolates between the four string theories (IIA, IIB, 0A, 0B) and the M theory as different limits are taken. 
  We introduce and solve a family of discrete models of 2D Lorentzian gravity with higher curvature, which possess mutually commuting transfer matrices, and whose spectral parameter interpolates between flat and curved space-times. We further establish a one-to-one correspondence between Lorentzian triangulations and directed Random Walks. This gives a simple explanation why the Lorentzian triangulations have fractal dimension 2 and why the curvature model lies in the universality class of pure Lorentzian gravity. We also study integrable generalizations of the curvature model with arbitrary polygonal tiles. All of them are found to lie in the same universality class. 
  All cubic couplings in type IIB supergravity on $AdS_5\times S^5$ that involve two scalar fields $s^I$ that are mixtures of the five form field strength on $S^5$ and the trace of the graviton on $S^5$ are derived by using the covariant equations of motion and the quadratic action for type IIB supergravity on $AdS_5\times S^5$. All corresponding three-point functions in SYM$_4$ are calculated in the supergravity approximation. It is pointed out that the scalars $s^I$ correspond not to the chiral primary operators in the ${\cal N}=4$ SYM but rather to a proper extension of the operators. 
  In this paper we study the N=1 supersymmetric field theories realized on the world-volume of type IIB D3-branes sitting at orientifolds of non-orbifold singularities (conifold and generalizations). Several chiral models belong to this family of theories. These field theories have a T-dual realization in terms of type IIA configurations of relatively rotated NS fivebranes, D4-branes and orientifold six-planes, with a compact $x^6$ direction, along which the D4-branes have finite extent. We compute the spectrum on the D3-branes directly in the type IIB picture and match the resulting field theories with those obtained in the type IIA setup, thus providing a non-trivial check of this T-duality. Since the usual techniques to compute the spectrum of the model and check the cancellation of tadpoles, cannot be applied to the case orientifolds of non-orbifold singularities, we use a different approach, and construct the models by partially blowing-up orientifolds of C^3/(Z_2 x Z_2) and C^3/(Z_2 x Z_3) orbifolds. 
  We discuss string theory on AdS(3)xS(3)xM(4) with particular emphasis on unitarity and state-operator correspondence. The AdS-CFT correspondence, in the Minkowski signature, is re-examined by taking into account the only allowed unitary representation: the principal series module of the affine current algebra SL(2,R) supplemented with zero modes. Zero modes play an important role in the description of on-shell states as well as of windings in space-time at the AdS(3) boundary. The theory is presented as part of the supersymmetric WZW model that includes the supergroup SU(2/1,1) or OSp(4/2) or D(2,1;\alpha) with central extension k. A free field representation is given and the vertex operators are constructed in terms of free fields in SL(2,R) principal series representation bases that are labeled by position space or momentum space at the boundary of AdS(3). The vertex operators have the correct operator products with the currents and stress tensor, all of which are constructed from free fields, including the subtle zero modes. It is shown that as k goes to infinity, AdS(3) tends to flat 3D-Minkowski space and the AdS(3) vertex operators in momentum space tend to the vertex operators of flat 3D-string theory (furthermore the theory readjusts smoothly in the rest of the dimensions in this limit). 
  Multi-trace scalar operators in the (0,k,0) representations of SU(4)_R share many properties with their single-trace analogs. These multi-trace operators are primary fields of short representations of the N=4 superconformal group. Thus, they have protected dimensions. We show that two- and three-point correlator functions of such operators do not receive radiative corrections at order g^2 in perturbation theory for an arbitrary gauge group. 
  The chiral Gross-Neveu model is one of the most popular toy models for QCD. In the past, it has been studied in detail in the large-N limit. In this paper we study its small-N behavior at finite temperature in 2+1 dimensions. We show that at small N the phase diagram of this model is {\it principally} different from its behavior at $N\to \infty$. We show that for a small number $N$ of fermions the model possesses two characteristic temperatures $T_{KT}$ and $T^*$. That is, at small N, along with a quasiordered phase $0<T<T_{KT}$ the system possesses a very large region of precursor fluctuations $T_{KT}<T<T^*$ which disappear only at a temperature $T^*$, substantially higher than the temperature $T_{KT}$ of Kosterlitz-Thouless transition. 
  Classical properties of 1/4 BPS dyons were previously well understood both in field theory context and in string theory context. Its quantum properties, however, have been more difficult to probe, although the elementary information of the supermultiplet structures is known from a perturbative construction. Recently, a low energy effective theory of monopoles was constructed and argued to contain these dyons as quantum bound states. In this paper, we find these dyonic bound states explicitly in the N=4 supersymmetric low energy effective theory. After identifying the correct angular momentum operators, we motivate an anti-self-dual ansatz for all BPS bound states. The wavefunctions are found explicitly, whose spin contents and degeneracies match exactly the expected results. 
  Three dimensional (abelian) gauged massive Thirring model is bosonized in the large fermion mass limit. A further integration of the gauge field results in a non-local theory. A truncated version of that is the Maxwell Chern Simons (MCS) theory with a conventional mass term or MCS Proca theory. This gauge invariant theory is completely solved in the Hamiltonian and Lagrangian formalism, with the spectra of the modes determined. Since the vector field constituting the model is identified (via bosonization) to the fermion current, the charge current algebra, including the Schwinger term is also computed in the MCS Proca model. 
  The aim of this paper is to give a firm and clear proof of the existence in the background field framework of a gauge invariant effective action for any gauge theory ({\it background gauge equivalence}). Here by effective action we mean a functional whose Legendre transform restricted to the physical shell generates the matrix elements of the connected $S$-matrix. We resume and clarify a former argument due to Abbott, Grisaru and Schaefer based on the gauge-artifact nature of the background fields and on the identification of the gauge invariant effective action with the generator of the proper, background field, vertices. 
  In a recent paper Klebanov and Witten (hep-th/9905104) proposed to formulate the AdS/CFT correspondence principle by taking an "irregular boundary condition" for a scalar field. In this paper we generalize this idea to the case of spinor field with interaction. The action functional following from the choice of irregular boundary conditions and which must be used in the AdS/CFT correspondence is related to the usual action by a Legendre transform. For the new theory we found the modified Green's function that must be used for internal lines in calculating higher order graphs. It is proved that the considerations are valid to all orders in perturbation theory. 
  We construct D0-branes in SO(32)$\times$SO(32) open type 0 string theory using the same method as the one used to construct non-BPS D0-brane in type I string theory. It has been proposed that this theory is S-dual to bosonic string theory compactified on SO(32) lattice, which has SO(32)$\times$SO(32) spinor states as excited states of fundamental string. One of these states seems to correspond to the D0-brane, and by the requirement that other states which do not have corresponding states must be removed, we can determine the way of truncation of the spectrum. This result supports the conjecture. 
  A theorem of differential geometry is employed to locally embed a wide class of superstring backgrounds that admit a covariantly constant null Killing vector field in eleven-dimensional, Ricci-flat spaces. Included in this class are exact type IIB superstring backgrounds with non-trivial Ramond-Ramond fields and a class of supersymmetric string waves. The embedding spaces represent exact solutions to eleven-dimensional, vacuum Einstein gravity. A solution of eleven-dimensional supergravity is also embedded in a twelve--dimensional, Ricci-flat space. 
  We consider the parity-invariant Dirac operator with a mass term in three-dimensional QCD for $N_c=2$ and quarks in the fundamental representation. We show that there exists a basis in which the matrix elements of the Euclidean Dirac operator are real. Assuming there is spontaneous breaking of flavor and/or parity, we read off from the fermionic action the flavor symmetry-breaking pattern $Sp(4N_f) \to Sp(2N_f) \times Sp(2N_f)$ that might occur in such a theory. We then construct a random matrix theory with the same global symmetries as two-color QCD$_3$ with fundamental fermions and derive from here the finite-volume partition function for the latter in the static limit. The expected symmetry breaking pattern is confirmed by the explicit calculation in random matrix theory. We also derive the first Leutwyler-Smilga-like sum rule for the eigenvalues of the Dirac operator. 
  Three-photon vertex in a dense degenerated plasma is calculated. It is discovered the polarization tensor has the longitudinal part which makes possible an interaction between transverse and longitudinal modes in the medium. Using dispersion relations it is shown that the only kinematically permitted type of the photon splitting is the decay of the transverse photon to two plasmons. 
  We study two-dimensional SQED viewed as the world-volume theory of a D-string in the presence of D5-branes with non-zero background fields that induce attractive forces between the branes. In various approximations, the low-energy dynamics is given by a hyperKahler, or hyperKahler with torsion, massive sigma-model. We demonstrate the existence of kink solutions corresponding to the string interpolating between different D5-branes. Bound states of the D-string with fundamental strings are identified with Q-kinks which, in turn, are identified with dyonic instanton strings on the D5-brane world-volume. 
  We address the issue of the worldsheet and spacetime covariant formulation for matrix strings. The problem is solved in the limit of vanishing string coupling. To go beyond the g_s = 0 limit, we propose a topological quantum field theory as a twisted candidate. Our model involves the generalized octonionic or SU(4) instanton equations defined in eight dimensions for a supersymmetric U(N) Yang--Mills field living on a special holonomy manifold. The question of untwisting this matrix model into an anomaly free theory enlightens the need for an "extended" 2d-gravity sector, that we suggest could be (partially twisted) W-gravity. 
  We discuss the relation between the Matrix theory definitions of a class of decoupled theories and their AdS/CFT description in terms of the corresponding near-horizon geometry. The near horizon geometry, naively part of the Coulomb branch, is embedded in the Higgs branch via a natural change of variables. The principles of the map apply to all DLCQ descriptions in terms of hyper-K\"ahler quotients, such as the ADHM quantum mechanics for the D1-D5 system. We then focus the (2,0) field theory, and obtain an explicit mapping from all states in the $N_0=1$ momentum sector of $N_4$ M5-branes to states in (a DLCQ version of) $AdS_7\times S^4$. We show that, even for a single D0-brane, the space-time coordinates become non-commuting variables, suggesting an inherent non-commutativity of space-time in the presence of field strengths even for theories with gravity. 
  By assuming the existence of a novel multipronged string state for D-particles interacting with D-brane intersections in type IIA string theory, we are able to derive a quantum mechanical description of supersymmetric Reissner-Nordstrom black holes. A supersymmetric index calculation provides evidence for this conjecture. The quantum mechanical system becomes two decoupled conformal quantum mechanical systems in the low energy limit. The conformal quantum mechanics has expected properties of a dual description of string theory on $AdS_2\times S^2$. 
  Calogero-Moser models and Toda models are well-known integrable multi-particle dynamical systems based on root systems associated with Lie algebras. The relation between these two types of integrable models is investigated at the levels of the Hamiltonians and the Lax pairs. The Lax pairs of Calogero-Moser models are specified by the representations of the reflection groups, which are not the same as those of the corresponding Lie algebras. The latter specify the Lax pairs of Toda models. The Hamiltonians of the elliptic Calogero-Moser models tend to those of Toda models as one of the periods of the elliptic function goes to infinity, provided the dynamical variables are properly shifted and the coupling constants are scaled. On the other hand most of Calogero-Moser Lax pairs, for example, the root type Lax pairs, do not a have consistent Toda model limit. The minimal type Lax pairs, which corresponds to the minimal representations of the Lie algebras, tend to the Lax pairs of the corresponding Toda models. 
  In this paper we count the numbers of real and complex solutions to Bethe constraints in the two particle sector of the XXZ model. We find exact number of exceptions to the string conjecture and total number of solutions which is required for completeness. 
  We construct the Georgi-Glashow Lagrangian for gauge group SU_q(n). Breaking this symmetry spontaneously gives q-dependent masses of gauge field and vacuum manifold. It turned out that the vacuum manifold is parameterized by the non-commutative quantities. We showed that the monopole solutions exist in this model, which is indicated by the presence of the BPS states. 
  In this paper we investigate complex solutions of the Bethe equations in the two-particle sector both for arbitrary finite number of sites and for the thermodynamic limit . We find the number of complex solutions (strings) and compare it with the string conjecture prediction. Some simple properties of these solutions like position in the spectrum, crossing of levels, connection to the ground state and transformation to the real solutions are discussed. Counting both real and complex solutions we find expected number of highest weight Bethe states. 
  According to the AdS/CFT correspondence,the d=4, N=4 SU(N) SYM is dual to the Type IIB string theory compactified on AdS_5xS^5. A mechanism was proposed previously that the chiral anomaly of the gauge theory is accounted for to the leading order in N by the Chern-Simons action in the AdS_5 SUGRA. In this paper, we consider the SUGRA\string action at one loop and determine the quantum corrections to the Chern-Simons action. While gluon loops do not modify the coefficient of the Chern-Simons action, spinor loops shift the coefficient by an integer. We find that for finite N, the quantum corrections from the complete tower of Kaluza-Klein states reproduce exactly the desired shift N^2 ---> N^2- 1 of the Chern-Simons coefficient, suggesting that this coefficient does not receive corrections from the other states of the string theory. We discuss why this is plausible. 
  Using a manifestly supersymmetric formalism, we determine the general structure of two- and three- point functions of the supercurrent and the flavour current of N = 2 superconformal field theories. We also express them in terms of N = 1 superfields and compare to the generic N = 1 correlation functions. A general discussion of the N = 2 supercurrent superfield and the multiplet of anomalies and their definition as derivatives with respect to the supergravity prepotentials is also included. 
  We study the exchange diagrams in the computation of four-point functions of all chiral primary operators in D=4, $\CN=4$ Super-Yang-Mills using AdS/CFT correspondence. We identify all supergravity fields that can be exchanged and compute the cubic couplings. As a byproduct, we also rederive the normalization of the quadratic action of the exchanged fields. The cubic couplings computed in this paper and the propagators studied extensively in the literature can be used to compute almost all the exchange diagrams explicitly. Some issues in computing the complete four-point function in the massless sector is discussed. 
  The most general theory of gravity in d-dimensions which leads to second order field equations for the metric has [(d-1)/2] free parameters. It is shown that requiring the theory to have the maximum possible number of degrees of freedom, fixes these parameters in terms of the gravitational and the cosmological constants. In odd dimensions, the Lagrangian is a Chern-Simons form for the (A)dS or Poincare groups. In even dimensions, the action has a Born-Infeld-like form. Torsion may occur explicitly in the Lagrangian in the parity-odd sector and the torsional pieces respect local (A)dS symmetry for d=4k-1 only. These torsional Lagrangians are related to the Chern-Pontryagin characters for the (A)dS group. The additional coefficients in front of these new terms in the Lagrangian are shown to be quantized. 
  Quantum group Fourier transform methods are applied to the study of processes on noncommutative Minkowski spacetime $[x^i,t]=\imath\lambda x^i$. A natural wave equation is derived and the associated phenomena of {\it in vacuo} dispersion are discussed. Assuming the deformation scale $\lambda$ is of the order of the Planck length one finds that the dispersion effects are large enough to be tested in experimental investigations of astrophysical phenomena such as gamma-ray bursts. We also outline a new approach to the construction of field theories on the noncommutative spacetime, with the noncommutativity equivalent under Fourier transform to non-Abelianness of the `addition law' for momentum in Feynman diagrams. We argue that CPT violation effects of the type testable using the sensitive neutral-kaon system are to be expected in such a theory. 
  We develop the canonical ADM approach to 2+1 dimensional gravity in presence of point particles. The instantaneous York gauge can be applied for open universes or universes with the topology of the sphere. The sequence of canonical ADM equations is solved in terms of the conformal factor. A simple derivation is given for the solution of the two body problem. A geometrical characterization is given for the apparent singularities occurring in the N-body problem and it is shown how the Garnier hamiltonian system arises in the ADM treatment by considering the time development of the conformal factor at the locations where the extrinsic curvature tensor vanishes. 
  We study in detail the pattern of anomaly cancellation in D=6 Type IIB Z_N orientifolds, occurring through a generalized Green-Schwarz mechanism involving several RR antisymmetric tensors and scalars fields. The starting point is a direct string theory computation of the inflow of anomaly arising from magnetic interaction of D-branes, O-planes and fixed-points, which are encoded in topological one-loop partition functions in the RR odd spin-structure. All the RR anomalous couplings of these objects are then obtained by factorization. They are responsible for a spontaneous breaking of U(1) factors through a Higgs mechanism involving the corresponding hypermultiplets. Some of them are also related by supersymmetry to gauge couplings involving the NSNS scalars sitting in the tensor multiplets. We also comment on the possible occurrence of tensionless strings when these couplings diverge. 
  We construct simple twistor-like actions describing superparticles propagating on a coset superspace OSp(1|4)/SO(1,3) (containing the D=4 anti-de-Sitter space as a bosonic subspace), on a supergroup manifold OSp(1|4) and, generically, on OSp(1|2n). Making two different contractions of the superparticle model on the OSp(1|4) supermanifold we get massless superparticles in Minkowski superspace without and with tensorial central charges. Using a suitable parametrization of OSp(1|2n) we obtain even Sp(2n)-valued Cartan forms which are quadratic in Grassmann coordinates of OSp(1|2n). This result may simplify the structure of brane actions in super-AdS backgrounds. For instance, the twistor-like actions constructed with the use of the even OSp(1|2n) Cartan forms as supervielbeins are quadratic in fermionic variables. We also show that the free bosonic twistor particle action describes massless particles propagating in arbitrary space-times with a conformally flat metric, in particular, in Minkowski space and AdS space. Applications of these results to the theory of higher spin fields and to superbranes in AdS superbackgrounds are mentioned. 
  It is shown by Connes, Douglas and Schwarz that gauge theory on noncommutative torus describes compactifications of M-theory to tori with constant background three-form field. This indicates that noncommutative gauge theories on more general manifolds also can be useful in string theory. We discuss a framework to noncommutative quantum gauge theory on Poisson manifolds by using the deformation quantization. The Kontsevich formula for the star product was given originally in terms of the perturbation expansion and it leads to a non-renormalizable quantum field theory. We discuss the nonperturbative path integral formulation of Cattaneo and Felder as a possible approach to construction of noncommutative quantum gauge theory on Poisson manifolds. Some other aspects of classical and quantum noncommutative field theory are also discussed. 
  We reconsider the options for cosmological holography. We suggest that a global and time--symmetric version of the Fischler-Susskind bound is the most natural generalization of the holographic bound encountered in AdS and De Sitter space. A consistent discussion of cosmological holography seems to imply an understanding of the notion of ``number of degrees of freedom'' that deviates from its simple definition as the entropy of the current state. The introduction of a more adequate notion of degree of freedom makes the suggested variation of the Fischler-Susskind bound look like a stringent and viable bound in all 4--dimensional cosmologies without a cosmological constant. 
  We study the emission rates of scalar, spinor and vector particles from a 5 dimensional black hole for arbitrary partial waves. The solution is lifted to 6 dimensions, and the near horizon $ BTZ \times S^3$ geometry of the black hole solution is probed to determine the greybody factors. We show that the exact decay rates can be reproduced from a $(1+1)$-dimensional conformal field theory which lies on the boundary of the near horizon geometry. The AdS/CFT correspondence is used to determine the dimension of the CFT operators corresponding to the bulk fields. These operators couple to plane waves incident on the CFT from infinity to produce emission in the bulk. 
  We investigate the effective dynamics of an arbitrary Dirichlet p-brane, in a path-integral formalism, by incorporating the massless excitations of closed string modes in open bosonic string theory. It is shown that the closed string background fields in the bosonic sector of type II theories induce invariant extrinsic curvature on the world-volume. In addition, the curvature can be seen to be associated with a divergence at the boundary of string world-sheet. The re-normalization of the collective coordinates, next to leading order in its derivative expansion, is performed to handle the divergence and the effective dynamics is encoded in Dirac-Born-Infeld action. Furthermore, the collective dynamics is generalized to include appropriate fermionic partners in type I super-string theory. The role of string modes is reviewed in terms of the collective coordinates and the gauge theory on the world-volume is argued to be non-local in presence of the U(1) invariant field strength. 
  The two different approaches - K\"{a}llen's and Brandt's methods - for the calculations of the Schwinger terms in the 1+1 dimensional Abelian and non-Abelian free current algebras are discussed. These methods are applied to calculation of the single and double commutators. The validity of the Jacobi identities is examined in 1+1 and 3+1 dimensions and in this way is given natural restriction on the regularization. It is shown that the Jacobi identity cannot be broken in 1+1 dimensions even using the regularization which fails in the 3+1 dimensional case. A connection between the Schwinger term and anomaly is shown in the simplest case of the Schwinger model. 
  Topological quantum field theories can be used as a powerful tool to probe geometry and topology in low dimensions. Chern-Simons theories, which are examples of such field theories, provide a field theoretic framework for the study of knots and links in three dimensions. These are rare examples of quantum field theories which can be exactly (non-perturbatively) and explicitly solved. Abelian Chern-Simons theory provides a field theoretic interpretation of the linking and self-linking numbers of a link. In non-Abelian theories, vacuum expectation values of Wilson link operators yield a class of polynomial link invariants; the simplest of them is the famous Jones polynomial. Other invariants obtained are more powerful than that of Jones. Powerful methods for completely analytical and non-perturbative computation of these knot and link invariants have been developed. In the process answers to some of the open problems in knot theory are obtained. From these invariants for unoriented and framed links in $S^3$, an invariant for any three-manifold can be easily constructed by exploiting the Lickorish-Wallace surgery presentation of three-manifolds. This invariant up to a normalization is the partition function of the Chern-Simons field theory. Even perturbative analysis of the Chern-Simons theories are rich in their mathematical structure; these provide a field theoretic interpretation of Vassiliev knot invariants. Not only in mathematics, Chern-Simons theories find important applications in three and four dimensional quantum gravity also. 
  The non-renormalization theorems of chiral vertex functions are derived on the basis of an algebraic analysis. The property, that the interaction vertex is a second supersymmetry variation of a lower dimensional field monomial, is used to relate chiral Green functions to superficially convergent Green functions by extracting the two supersymmetry variations from an internal vertex and transforming them to derivatives acting on external legs. The analysis is valid in the massive as well as in the massless model and can be performed irrespective of properties of the superpotential at vanishing momentum. 
  It is shown that the commutation relations of W-algebras can be recovered from the singular vectors of their simplest nontrivial, completely degenerate highest weight representation. 
  We perform a 3D reduction of the two-fermion Bethe-Salpeter equation based on Sazdjian's explicitly covariant propagator, combined with a covariant substitute of the projector on the positive-energy free states. We use this combination in the two fermions in an external potential and in the three-fermion problems. The covariance of the two-fermion propagators insures the covariance of the two-body equations obtained by switching off the external potential, or by switching off all interactions between any pair of two fermions and the third one, even if the series giving the 3D potential is limited to the Born term or more generally truncated. The covariant substitute of the positive energy projector preserves the equations against continuum dissolution without breaking the covariance. 
  This thesis presents a thorough analysis of the links between the N=4 supersymmetric gauge theory in four dimensions and its three topological twisted counterparts. Special emphasis is put in deriving explicit results in terms of the vacuum structure and low-energy effective description of four-dimensional supersymmetric gauge theories. A key ingredient is the realization of Montonten-Olive duality in the twisted theories, which is discussed in detail from different viewpoints. 
  We investigate the production of gravitinos in a cosmological background. Gravitinos can be produced during preheating after inflation due to a combined effect of interactions with an oscillating inflaton field and absence of conformal invariance. In order to get insight on conformal properties of gravitino we reformulate phenomenological supergravity in SU(2,2|1)-symmetric way. The Planck mass and F- and D-terms appear via the gauge-fixed value of a superfield that we call conformon. We find that in general the probability of gravitino production is not suppressed by the small gravitational coupling. This may lead to a copious production of gravitinos after inflation. Efficiency of the new non-thermal mechanism of gravitino production is very sensitive to the choice of the underlying theory. This may put strong constraints on certain classes of inflationary models. 
  By exploiting the properties of q-deformed Coxeter elements, the scattering matrices of affine Toda field theories with real coupling constant related to any dual pair of simple Lie algebras may be expressed in a completely generic way. We discuss the governing equations for the existence of bound states, i.e. the fusing rules, in terms of q-deformed Coxeter elements, twisted q-deformed Coxeter elements and undeformed Coxeter elements. We establish the precise relation between these different formulations and study their solutions. The generalized S-matrix bootstrap equations are shown to be equivalent to the fusing rules. The relation between different versions of fusing rules and quantum conserved quantities, which result as nullvectors of a doubly q-deformed Cartan like matrix, is presented. The properties of this matrix together with the so-called combined bootstrap equations are utilised in order to derive generic integral representations for the scattering matrix in terms of quantities of either of the two dual algebras. We present extensive case-by-case data, in particular on the orbits generated by the various Coxeter elements. 
  Using the AdS/CFT correspondence and the eikonal approximation, we evaluate the elastic parton-parton scattering amplitude at large $N$ and strong coupling $g^2N$ in N=4 SYM. We obtain a scattering amplitude with a Regge behavior that unitarizes at large $\sqrt{s}$. 
  Lattice calculations performed in Abelian gauges give strong evidence that confinement is realized as a dual Meissner effect, implying that the Yang-Mills vacuum consists of a condensate of magnetic monopoles. Alternative lattice caluclations performed in the maximum center gauge give strong support that center vortex configurations are the relevant infrared degrees of freedom responsible for confinement and that the magnetic monopoles are mostly sitting on vortices. In this talk I study the continuum Yang-Mills-theory in Abelian and center gauges. In Polyakov gauge the Pontryagin index of the gauge field is expressed by the magnetic monopole charges. The continuum analogues of center vortices and the continuum version of the maximum center gauge are presented. It is shown that the Pontryagin index of center vortices is given by their self-intersection number, which vanishes unless magnetic monopole currents are flowing on the vortices. 
  We study a configuration in matrix theory carying longitudinal fivebrane charge, i.e. a D0-D4 bound state. We calculate the one-loop effective potential between a D0-D4 bound state and a D0-anti-D4 bound state and compare our results to a supergravity calculation. Next, we identify the tachyonic fluctuations in the D0-D4 and D0-anti-D4 system. We analyse classically the action for these tachyons and find solutions to the equations of motion corresponding to tachyon condensation. 
  The large radius limit in the AdS/CFT correspondence is expected to provide a holographic derivation of flat-space scattering amplitudes. This suggests that questions of locality in the bulk should be addressed in terms of properties of the S-matrix and their translation into the conformal field theory. There are, however, subtleties in this translation related to generic growth of amplitudes near the boundary of anti de-Sitter space. Flat space amplitudes are recovered after a delicate projection of CFT correlators onto normal-mode frequencies of AdS. Once such amplitudes are obtained from the CFT, possible criteria for approximate bulk locality include bounds on growth of amplitudes at high energies and reproduction of semiclassical gravitational scattering at long distances. 
  Quantum matrix models in the large-N limit arise in many physical systems like Yang-Mills theory with or without supersymmetry, quantum gravity, string-bit models, various low energy effective models of string theory, M(atrix) theory, quantum spin chain models, and strongly correlated electron systems like the Hubbard model. We introduce, in a unifying fashion, a hierachy of infinite-dimensional Lie superalgebras of quantum matrix models. One of these superalgebras pertains to the open string sector and another one the closed string sector. Physical observables of quantum matrix models like the Hamiltonian can be expressed as elements of these Lie superalgebras. This indicates the Lie superalgebras describe the symmetry of quantum matrix models. We present the structure of these Lie superalgebras like their Cartan subalgebras, root vectors, ideals and subalgebras. They are generalizations of well-known algebras like the Cuntz algebra, the Virasoro algebra, the Toeplitz algebra, the Witt algebra and the Onsager algebra. Just like we learnt a lot about critical phenomena and string theory through their conformal symmetry described by the Virasoro algebra, we may learn a lot about quantum chromodynamics, quantum gravity and condensed matter physics through this symmetry of quantum matrix models described by these Lie superalgebras. 
  We consider D-branes wrapped around supersymmetric cycles of Calabi-Yau manifolds from the viewpoint of N=2 Landau-Ginzburg models with boundary as well as by consideration of boundary states in the corresponding Gepner models. The Landau-Ginzburg approach enables us to provide a target space interpretation for the boundary states. The boundary states are obtained by applying Cardy's procedure to combinations of characters in the Gepner models which are invariant under spectral flow. We are able to relate the two descriptions using the common discrete symmetries of the two descriptions. We are thus able to provide an extension to the boundary of the bulk correspondence between Landau-Ginzburg orbifolds and the corresponding Gepner models. 
  The type IIA/IIB effective actions compactified on T^d are known to be invariant under the T-duality group SO(d, d; Z) although the invariance of the R-R sector is not so direct to see. Inspired by a work of Brace, Morariu and Zumino,we introduce new potentials which are mixture of R-R potentials and the NS-NS 2-form in order to make the invariant structure of R-R sector more transparent. We give a simple proof that if these new potentials transform as a Majorana-Weyl spinor of SO(d, d; Z), the effective actions are indeed invariant under the T-duality group. The argument is made in such a way that it can apply to Kaluza-Klein forms of arbitrary degree. We also demonstrate that these new fields simplify all the expressions including the Chern-Simons term. 
  We compute certain 2K+4-point, one-loop couplings in the type IIA string compactified on K3 x T^2, which are related to a topological index on this manifold. Their special feature is that they are sensitive to only short and intermediate BPS multiplets. The couplings derive from underlying prepotentials G[K](T,U), which can be nicely summed up into a fundamental generating function. In the dual heterotic string on T^6, the amplitudes describe non-perturbative gravitational corrections to K-loop amplitudes due to bound states of fivebrane instantons with heterotic world-sheet instantons. We argue, as a consequence, that our results also give information about instanton configurations in six dimensional Sp(2k) gauge theories on T^6. 
  Elliptic curves for the 7-brane configurations realizing the affine Lie algebras $\wh E_n$ $(1 \leq n \leq 8)$ and $\wh{\wt E}_n$ $(n=0,1)$ are systematically derived from the cubic equation for a rational elliptic surface. It is then shown that the $\wh E_n$ 7-branes describe the discriminant locus of the elliptic curves for five-dimensional (5D) N=1 $E_n$ theories compactified on a circle. This is in accordance with a recent construction of 5D N=1 $E_n$ theories on the IIB 5-brane web with 7-branes, and indicates the validity of the D3 probe picture for 5D $E_n$ theories on $\bR^4 \times S^1$. Using the $\wh E_n$ curves we also study the compactification of 5D $E_n$ theories to four dimensions. 
  We have shown that self interaction effects in massive quantum electrodynamics can lead to the formation of bound states of quark antiquark pairs. A current-current fermion coupling term is introduced, which induces a well in the potential energy profile. Explicit expressions of the effective potential and renormalized parameters are provided. 
  Based on results for real deformation parameter q we introduce a compact non- commutative structure covariant under the quantum group SOq(3) for q being a root of unity. To match the algebra of the q-deformed operators with necesarry conjugation properties it is helpful to define a module over the algebra genera- ted by the powers of q. In a representation where X is diagonal we show how P can be calculated. To manifest some typical properties an example of a one-di- mensional q-deformed Heisenberg algebra is also considered and compared with non-compact case. 
  We introduce a general method in order to construct the non chiral fusion rules which determine the operator content of the operator product algebra for rational conformal field theories. We are particularly interested in the models of the complementary D-like solutions of modular invariant partition functions with cyclic center Zn. We find that the non chiral fusion rules have a Zn-grading structure. 
  Generalized BRS transformations such as introduced in Part I (hep-th/9906245) are applied to a model of quantum gravity. This development is technically complex; but at the least should illustrate how much less rigid and more general of application are the new BRS transformations. 
  The maximal center gauge, combined with center projection, is a means to associate Yang-Mills lattice gauge configurations with closed center vortex world-surfaces. This technique allows to study center vortex physics in lattice gauge experiments. In the present work, the continuum analogue of the maximal center gauge is constructed. This sheds new light on the meaning of the procedure on the lattice and leads to a sketch of an effective vortex theory in the continuum. Furthermore, the manner in which center vortex configurations generate the Pontryagin index is investigated. The Pontryagin index is built up from self-intersections of the vortex world-surfaces, where it is crucial that the surfaces be globally non-oriented. 
  A detailed review of recent developments in the topological classification of D-branes in superstring theory is presented. Beginning with a thorough, self-contained introduction to the techniques and applications of topological K-theory, the relationships between the classic constructions of K-theory and the recent realizations of D-branes as tachyonic solitons, coming from bound states of higher dimensional systems of unstable branes, are described. It is shown how the K-theory formalism naturally reproduces the known spectra of BPS and non-BPS D-branes, and how it can be systematically used to predict the existence of new states. The emphasis is placed on the new interpretations of D-branes as conventional topological solitons in other brane worldvolumes, how the mathematical formalism can be used to deduce the gauge field content on both supersymmetric and non-BPS branes, and also how K-theory predicts new relationships between the various superstring theories and their D-brane spectra. The implementations of duality symmetries as natural isomorphisms of K-groups are discussed. The relationship with the standard cohomological classification is presented and used to derive an explicit formula for D-brane charges. Some string theoretical constructions of the K-theory predictions are also briefly described. 
  We examine the effect of light-cone broadening induced by quantum-gravity foam in the context of theories with ``large'' extra dimensions stretching between two parallel brane worlds. We consider the propagation of photon probes on one of the branes, including the response to graviton fluctuations, from both field- and string-theoretical viewpoints. In the latter approach, the dominant source of light-cone broadening may be the recoil of the D-brane, which scales linearly with the string coupling. Astrophysical constraints then place strong restrictions on consistent string models of macroscopic extra dimensions. The broadening we find in the field-theoretical picture seems to be close to the current sensitivity of gravity-wave interferometers, and therefore could perhaps be tested experimentally in the foreseeable future. 
  Type IIB string theory expanded around D3 brane backgrounds describes the dynamics of D3 branes as solitonic objects. On the other hand, there is a fundamental description of them via Polchinski's open strings with Dirichlet boundary conditions. Since these two descriptions describe the dynamics of the same objects, D3 branes, it is natural to believe that they are dual. Therefore at this level, we have a string-string duality as opposed to a string-field theory duality. Once we take the same limits in both descriptions, Maldacena Conjecture in its weaker form follows. We try to make this viewpoint precise and study the implication of it for the stronger form of Maldacena Conjecture. 
  The correspondence of the non-critical string theory and the Yang-Mills theory is examined according to the recent Polyakov's proposal, and two possible solutions of the bulk equations are addressed near the fixed points of the pure Yang-Mills theory: (i) One solution asymptotically approaches to the AdS space at the ultraviolet limit where the conformally invariant field theory is realized. (ii) The second one approaches to the flat space in both the infrared and the ultraviolet limits. The area law of the Wilson-loop and the asymptotic freedom with logarithmic behaviour are seen in the respective limit. 
  The orbifold CFT dual to string theory on $ADS_3 \times S^3$ allows a construction of gravitational actions based on collective field techniques. We describe a fundamental role played by a Lie algebra constructed from chiral primaries and their CFT conjugates. The leading terms in the algebra at large $N$ are derived from the computation of chiral primary correlation functions. The algebra is argued to determine the dynamics of the theory, its representations provide free and interacting hamiltonians for chiral primaries. This dynamics is seen to be given by an effective one plus one dimensional field theory. The structure of the algebra and its representations shows qualitatively new features associated with thresholds at $L_0= N$, $ L_0 = N/2$ and $L_0=N/4$, which are related to the stringy exclusion principle and to black holes. We observe relations between fusion rules of $SU_q(2|1,1)$ for $q = e^{i \pi / {N+1}} $, and the correlation functions, which provide further evidence for a non-commutative spacetime. 
  The truncation in the number of single-trace chiral primary operators of $\N=4$ SYM and its conjectured connection with gravity on quantum spacetimes are elaborated. The model of quantum spacetime we use is $AdS^5_q \times S^5_q$ for $q$ a root of unity. The quantum sphere is defined as a homogeneous space with manifest $SU_q(3)$ symmetry, but as anticipated from the field theory correspondence, we show that there is a hidden $SO_q(6)$ symmetry in the constrution. We also study some properties of quantum space quotients as candidate models for the quantum spacetime relevant for some $Z_n$ quiver quotients of the $\N=4$ theory which break SUSY to $\N=2$. We find various qualitative agreements between the proposed models and the properties of the corresponding finite $N$ gauge theories. 
  It is argued that quantum traveling of D-particles presents the ``joining-splitting'' processes of field theory Feynman graphs. The amplitudes in $d$ dimensions can be corresponded with those of a $d+2$ dimensional theory in the Light-Cone frame. It is shown that this Light-Cone formulation enables to study processes with arbitrary longitudinal momentum transfers. It is discussed that a massless sector exists which can be taken as the low energy limit. By taking the constant relative distance in the bound-states we find a spectrum for the intermediatory fields. 
  The response of an integrable QFT under variation of the Unruh temperature has recently been shown to be computable from an S-matrix preserving (`replica') deformation of the form factor approach. We show that replica-deformed form factors of the SU(2)-invariant Thirring model can be found among the solutions of the rational $sl_2$-type quantum Knizhnik-Zamolodchikov equation at generic level. We show that modulo conserved charge solutions the deformed form factors are in one-to-one correspondence to the ones at level zero and use this to conjecture the deformed form factors of the Noether current in our model. 
  Majorana-Weyl spacetimes offer a rich algebraic setup and new types of space-time dualities besides those discussed by Hull. The triality automorphisms of Spin(8) act non-trivially on Majorana-Weyl representations and Majorana-Weyl spacetimes with different signatures. In particular relations exist among the (1+9)-(5+5)-(9+1) spacetimes, as well as their transverse coordinates spacetimes (0+8)-(4+4)-(8+0). Larger dimensional spacetimes such as (2+10)-(6+6)-(10+2) also show dualities induced by triality. A precise three-languages dictionary is here given. It furnishes the exact translations among, e.g., the three different versions (one in each signature) of the ten-dimensional N=1 superstring and superYang-Mills theories. Their dualities close the six-element permutation group S_3. Bilinear and trilinear invariants allowing to formulate theories with a manifest space-time symmetry are constructed. 
  After some recalls on the standard (non)-linear $\sigma$ model, we discuss the interest of B.R.S. symmetry in non-linear $\sigma$ models renormalisation. We also emphasise the importance of a correct definition of a theory through physical constraints rather than as given by a particular Lagrangian and discuss some ways to enlarge the notion of renormalisability. 
  Recently much work has been done in lowering the Planck threshold of quantum gravitational effects (sub-millimeter dimension(s), Horava-Witten fifth dimension, strings or branes low energy effects, etc.). Working in the framework of 4-dim gravity, with semi-classical considerations based on Hawking evaporation of planckian micro-black holes, I shall show here as quantum gravity effects could occur also near GUT energies. 
  The algebraic formulation of Large N matrix mechanics recently developed by Halpern and Schwartz leads to a practical method of numerical computation for both action and Hamiltonian problems. The new technique posits a boundary condition on the planar connected parts X_w, namely that they should decrease rapidly with increasing order. This leads to algebraic/variational schemes of computation which show remarkably rapid convergence in numerical tests on some many- matrix models. The method allows the calculation of all moments of the ground state, in a sequence of approximations, and excited states can be determined as well. There are two unexpected findings: a large d expansion and a new selection rule for certain types of interaction. 
  We obtain the T-duality transformations of space-time spinors (the supersymmetry transformation parameters, gravitinos and dilatinos) of type-II theories in curved backgrounds with an isometry. The transformation of the spinor index is shown to be a consequence of the twist that T-duality introduces between the left and right-moving local Lorentz frames. The result is then used to derive the T-duality action on Ramond-Ramond field strengths and potentials in a simple way. We also discuss the massive IIA theory and, using duality, give a short derivation of ``mass''-dependent terms in the Wess-Zumino actions on the D-brane worldvolumes. 
  The ADHM construction, which yields (anti-)selfdual configurations in classical Yang-Mills theories, is applied to an infinite dimensional l^2 vector space, and as a consequence, a family of (anti-)selfdual configurations with a parameter q is obtained for SU(2) Yang-Mills theory. This l^2 formulation can be seen as a q-analog of Nahm's monopole construction, so that the configuration approaches the BPS monopole at q->1 limit. 
  Boulatov and Ooguri have generalized the matrix models of 2d quantum gravity to 3d and 4d, in the form of field theories over group manifolds. We show that the Barrett-Crane quantum gravity model arises naturally from a theory of this type, but restricted to the homogeneous space S^3=SO(4)/SO(3), as a term in its Feynman expansion. From such a perspective, 4d quantum spacetime emerges as a Feynman graph, in the manner of the 2d matrix models. This formalism provides a precise meaning to the ``sum over triangulations'', which is presumably necessary for a physical interpretation of a spin foam model as a theory of gravity. In addition, this formalism leads us to introduce a natural alternative model, which might have relevance for quantum gravity. 
  The linearized interactions of eleven-dimensional supergravity are obtained in a manifestly supersymmetric light-cone gauge formalism. These vertices are used to calculate certain one-loop processes involving external gravitini, antisymmetric three-form potentials and gravitons, thereby determining some protected terms in the effective action of M-theory compactified on a two-torus. 
  We show that CPn sigma model solitons solve the field equations of a Dirac-Born-Infeld (DBI) action and, furthermore, we prove that the non-BPS soliton/anti-soliton solutions of the sigma model also solve the DBI equations. Using the moduli space approximation we compare the dynamics of the BPS sigma model solitons with that of the associated DBI solitons. We find that for the CP1 case the metric on the moduli space of sigma model solitons is identical to that of the moduli space of DBI solitons, but for CPn with n>1 we show that the two metrics are not equal. We also consider the possibility of similar non-BPS solitons in other DBI theories. 
  We construct non-BPS dyon solutions of N=4 SU(n) supersymmetric Yang-Mills theory. These solutions are the worldvolume solitons which describe non-BPS Type IIB non-planar string junctions connecting n parallel D3-branes. The solutions are smooth deformations of the 1/4 BPS states which describe planar string junctions. 
  A candidate for the confining string of gauge theories is constructed via a representation of the ultraviolet divergences of quantum field theory by a closed string dilaton insertion, computed through the soft dilaton theorem. The resulting (critical) confining string is conformally invariant, singles out naturally $d=4$ dimensions, and can not be used to represent theories with Landau poles. 
  We discuss the functional representation of fermions, and obtain exact expressions for wave-functionals of the Schwinger model. Known features of the model such as bosonization and the vacuum angle arise naturally. Contrary to expectations, the vacuum wave-functional does not simplify at large distances, but it may be reconstructed as a large time limit of the corresponding Schrodinger functional, which has an expansion in local terms. The functional Schrodinger equation reduces to a set of algebraic equations for the coefficients of these terms. These ideas generalize to a numerical approach to QCD in higher dimensions. 
  In quantum theory, real degrees of freedom are usually described by operators which are self-adjoint. There are, however, exceptions to the rule. This is because, in infinite dimensional Hilbert spaces, an operator is not necessarily self-adjoint even if its expectation values are real. Instead, the operator may be merely symmetric. Such operators are not diagonalizable - and as a consequence they describe real degrees of freedom which display a form of "unsharpness" or "fuzzyness". For example, there are indications that this type of operators could arise with the description of space-time at the string or at the Planck scale, where some form of unsharpness or fuzzyness has long been conjectured.   A priori, however, a potential problem with merely symmetric operators is the fact that, unlike self-adjoint operators, they do not generate unitaries - at least not straightforwardly. Here, we show for a large class of these operators that they do generate unitaries in a well defined way, and that these operators even generate the entire unitary group of the Hilbert space. This shows that merely symmetric operators, in addition to describing unsharp physical entities, may indeed also play a r{\^o}le in the generation of symmetries, e.g. within a fundamental theory of quantum gravity. 
  In this work we study how the infrared sector of the interaction Hamiltonian can affect the construction of the S matrix operator of QED in (2+1) dimensions. 
  The role of the contribution from the fermion mass term in the axial vector Ward identity in generating the U(1) axial anomaly, both local and global, is elucidated. Gauge invariance requires the fermion to decouple from the gauge field if it is very heavy. This identifies the Adler-Bell-Jackiw (ABJ) anomaly with the asymptotic limit of the sign reversed mass term. In an instanton background, the chiral limit $(m = 0)$ of the mass term does not vanish but consists of contributions from fermion zero modes. Space time integral of these zero mode contributions exactly cancels, thanks to the Atiyah-Singer index theorem, the integral of the ABJ anomaly and suggests that the Jacobian for global U(1) chiral transformation is trivial even in an instanton background. This can be realised in the representation of the fermion partition function in a Weyl basis. The resolution of the strong CP problem is thus achieved in an axionless physical world. In chiral gauge theories the fermion partition function admits of a gauge invariant representation but only at the cost of locality. Implementation of fermion averaging of the gauge current with the invariant partition function yields the current whose covariant derivative is the covariant anomaly. With the covariant current as input one can derive an integrable current whose covariant derivative is the minimal consistent anomaly obeying the Wess-Zumino consistency condition. The distinction between the two currents disappears if either the covariant or the consistent anomaly vanishes. This is realised only if the fermion belongs to an anomaly-free representation of the gauge group. 
  The thermodynamic properties of Schwarzschild-anti-de Sitter black holes confined within finite isothermal cavities are examined. In contrast to the Schwarzschild case, the infinite cavity limit may be taken which, if suitably stated, remains double valued. This allows the correspondence between non-existence of negative modes for classical solutions and local thermodynamic stability of the equilibrium configuration of such solutions to be shown in a well defined manner. This is not possible in the asymptotically flat case. Furthermore, the non-existence of negative modes for the larger black hole solution in Schwarzschild-anti-de Sitter provides strong evidence in favour of the recent positive energy conjecture by Horowitz and Myers. 
  F-theory on K3 admits non-BPS states that are represented as string junctions extending between 7-branes. We classify the non-BPS states which are guaranteed to be stable on account of charge conservation and the existence of a region of moduli space where the 7-branes supporting the junction can be isolated from the rest of the branes. We find three possibilities; the 7-brane configurations carrying: (i) the D_1 algebra representing a D7-brane near an orientifold O7-plane, whose stable non-BPS state was identified before, (ii) the exotic affine E_1 algebra, whose stable non-BPS state seems to be genuinely non-perturbative, and, (iii) the affine E_2 algebra representing a D7-brane near a pair of O7-planes. As a byproduct of our work we construct explicitly all 7-brane configurations that can be isolated in a K3. These include non-collapsible configurations of affine type. 
  Using the u-plane integral of Moore and Witten, we derive a simple expression for the Donaldson invariants of $\Sigma_g \times S^2$, where $\Sigma_g$ is a Riemann surface of genus g. This expression generalizes a theorem of Morgan and Szabo for g=1 to any genus g. We give two applications of our results: (1) We derive Thaddeus' formulae for the intersection pairings on the moduli space of rank two stable bundles over a Riemann surface. (2) We derive the eigenvalue spectrum of the Fukaya-Floer cohomology of $\Sigma_g \times S^1$. 
  We study the non-commutative supersymmetric Yang-Mills theory at strong coupling using the AdS/CFT correspondence. The supergravity description and the UV/IR relation confirms the expectation that the non-commutativity affects the ultra-violet but not the infra-red of the Yang-Mills dynamics. We show that the supergravity solution dual to the non-commutative N=4 SYM in four dimensions has no boundary and defines a minimal scale. We also show that the relation between the B field and the scale of non-commutativity is corrected at large coupling and determine its dependence on the 't Hooft coupling \lambda. 
  Recently D. Buchholz and R. Verch have proposed a method for implementing in algebraic quantum field theory ideas from renormalization group analysis of short-distance (high energy) behavior by passing to certain scaling limit theories. Buchholz and Verch distinguish between different types of theories where the limit is unique, degenerate, or classical, and the method allows in principle to extract the `ultraparticle' content of a given model, i.e. to identify particles (like quarks and gluons) that are not visible at finite distances due to `confinement'. It is therefore of great importance for the physical interpretation of the theory. The method has been illustrated in a simple model in with some rather surprising results.   This paper will focus on the question how the short distance behavior of models defined by euclidean means is reflected in the corresponding behavior of their Minkowski counterparts. More specifically, we shall prove that if a euclidean theory has some short distance limit, then it is possible to pass from this limit theory to a theory on Minkowski space, which is a short distance limit of the Minkowski space theory corresponding to the original euclidean theory. 
  The free energy of a field theory can be considered as a functional of the free correlation function. As such it obeys a nonlinear functional differential equation which can be turned into a recursion relation. This is solved order by order in the coupling constant to find all connected vacuum diagrams with their proper multiplicities. The procedure is applied to a multicomponent scalar field theory with a phi^4-self-interaction and then to a theory of two scalar fields phi and A with an interaction phi^2 A. All Feynman diagrams with external lines are obtained from functional derivatives of the connected vacuum diagrams with respect to the free correlation function. Finally, the recursive graphical construction is automatized by computer algebra with the help of a unique matrix notation for the Feynman diagrams. 
  We derive the superconformal transformation properties of the supercurrent for N=1 supersymmetric QED in four dimensions within the superfield formalism. Superconformal Ward identities for Green functions involving insertions of the supercurrent are conveniently derived by coupling the supercurrent to the appropriate prepotential of a classical curved superspace background, and by combining superdiffeomorphisms and super Weyl transformations. We determine all superconformal anomalies of SQED on curved superspace within an all-order perturbative approach and derive a local Callan-Symanzik equation. Particular importance is given to the issue of gauge invariance. 
  We construct a covariant bound on the energy-momentum of the M-fivebrane which is saturated by all supersymmetric configurations. This leads to a generalised notion of a calibrated geometry for M-fivebranes when the worldvolume gauge field is non-zero. The generalisation relevant for Dp-branes is also given. 
  We discover a realisation of the affine Lie superalgebra sl(2|1) and of the exceptional affine superalgebra D(2|1;alpha) as vertex operator extensions of two affine sl(2) algebras with dual levels (and an auxiliary level 1 sl(2) algebra). The duality relation between the levels is (k+1)(k'+1)=1. We construct the representation of sl(2|1) at level k' on a sum of tensor products of sl(2) at level k, sl(2) at level k' and sl(2) at level 1 modules and decompose it into a direct sum over the sl(2|1) spectral flow orbit. This decomposition gives rise to character identities, which we also derive. The extension of the construction to the affine D(2|1;k') at level k is traced to properties of sl(2)+sl(2)+sl(2) embeddings into D(2|1;alpha) and their relation with the dual sl(2) pairs. Conversely, we show how the level k' sl(2) representations are constructed from level k sl(2|1) representations. 
  We calculate (using zeta function regularization) the Casimir energy of the rotating Nambu-Goto string with the Gauss-Bonnet term in the action and point-like masses at the ends. The resulting value turns out to be negative for all values of the parameters of the model. 
  We derive a perturbative scheme to treat the interaction between point sources and AdS-gravity. The interaction problem is equivalent to the search of a polydromic mapping $X^A= X^A(x^\mu)$, endowed with 0(2,2) monodromies, between the physical coordinate system and a Minkowskian 4-dimensional coordinate system, which is however constrained to live on a hypersurface. The physical motion of point sources is therefore mapped to a geodesic motion on this hypersuface. We impose an instantaneous gauge which induces a set of equations defining such a polydromic mapping. Their consistency leads naturally to the Einstein equations in the same gauge. We explore the restriction of the monodromy group to O(2,1), and we obtain the solution of the fields perturbatively in the cosmological constant. 
  We discuss a formalism for solving (2+1) AdS gravity on Riemann surfaces. In the torus case the equations of motion are solved by two functions f and g, solutions of two independent O(2,1) sigma models, which are distinct because their first integrals contain a different time dependent phase factor. We then show that with the gauge choice $k = \sqrt{\Lambda}/ tg (2 \sqrt{\Lambda}t)$ the same couple of first integrals indeed solves exactly the Einstein equations for every Riemann surface. The $X^A=X^A(x^mu)$ polydromic mapping which extends the standard immersion of a constant curvature three-dimensional surface in a flat four-dimensional space to the case of external point sources or topology, is calculable with a simple algebraic formula in terms only of the two sigma model solutions f and g. A trivial time translation of this formalism allows us to introduce a new method which is suitable to study the scattering of black holes in (2+1) AdS gravity. 
  This is a review of some basic features on the relation between supergravity and pure gauge theories with special emphasis on the relation between T-duality and supersymmetry. Some new results concerning the interplay between T-duality and near horizon geometries are presented 
  We investigate solutions to a nonlinear integral equation which has a central role in implementing the non-Abelian Gauss's Law and in constructing gauge-invariant quark and gluon fields. Here we concern ourselves with solutions to this same equation that are not operator-valued, but are functions of spatial variables and carry spatial and SU(2) indices. We obtain an expression for the gauge-invariant gauge field in two-color QCD, define an index that we will refer to as the ``winding number'' that characterizes it, and show that this winding number is invariant to a small gauge transformation of the gauge field on which our construction of the gauge-invariant gauge field is based. We discuss the role of this gauge field in determining the winding number of the gauge-invariant gauge field. We also show that when the winding number of the gauge field is an integer $\ell{\neq}0$, the gauge-invariant gauge field manifests winding numbers that are not integers, and are half-integers only when $\ell=0$. 
  We consider small angle and large impact parameter high energy scattering of colourless states in SYM using the AdS/CFT correspondence. The gauge theory scattering amplitude is linked with a correlation function of tilted Wilson loops, which can be calculated by the exchange of bulk supergravity fields between the two corresponding string worldsheets. We identify the dominant contributions, which all correspond to real phase shifts. In particular, we find a contribution of the bulk graviton which gives an unexpected `gravity-like' $s^1$ behaviour of the gauge theory phase shift in a specific range of energies and (large) impact parameters. 
  We argue that a class of ``non-critical superstring'' vacua is holographically related to the (non-gravitational) theory obtained by studying string theory on a singular Calabi-Yau manifold in the decoupling limit $g_s\to 0$. In two dimensions, adding fundamental strings at the singularity of the CY manifold leads to conformal field theories dual to a recently constructed class of $AdS_3$ vacua. In four dimensions, special cases of the construction correspond to the theory on an NS5-brane wrapped around a Riemann surface. 
  We extend Polchinski's evaluation of the measure for the one-loop closed string path integral to open string tree amplitudes with boundaries and crosscaps embedded in Dbranes. We explain how the nonabelian limit of near-coincident Dbranes emerges in the path integral formalism. We give a careful path integral derivation of the cylinder amplitude including the modulus dependence of the volume of the conformal Killing group. 
  Recently we have proposed a set of variables for describing the physical parameters of SU(N) Yang--Mills field. Here we propose an off-shell generalization of our Ansatz. For this we envoke the Darboux theorem to decompose arbitrary one-form with respect to some basis of one-forms. After a partial gauge fixing we identify these forms with the preimages of holomorphic and antiholomorphic forms on the coset space $ SU(N)/U(1)^{N-1}$, identified as a particular coadjoint orbit. This yields an off-shell gauge fixed decomposition of the Yang-Mills connection that contains our original variables in a natural fashion. 
  We show that, in the framework of covariant Hamiltonian field theory, a degenerate almost regular quadratic Lagrangian $L$ admits a complete set of non-degenerate Hamiltonian forms such that solutions of the corresponding Hamilton equations, which live in the Lagrangian constraint space, exhaust solutions of the Euler--Lagrange equations for $L$. We obtain the characteristic splittings of the configuration and momentum phase bundles. Due to the corresponding projection operators, the Koszul-Tate resolution of the Lagrangian constraints for a generic almost regular quadratic Lagrangian is constructed in an explicit form. 
  A general procedure to reveal an Abelian structure of Yang-Mills theories by means of a (nonlocal) change of variables, rather than by gauge fixing, in the space of connections is proposed. The Abelian gauge group is isomorphic to the maximal Abelian subgroup of the Yang-Mills gauge group, but not its subgroup. A Maxwell field of the Abelian theory contains topological degrees of freedom of original Yang-Mills fields which generate monopole-like and flux-like defects upon an Abelian projection. 't Hooft's conjecture that ``monopole'' dynamics is projection independent is proved for a special class of Abelian projections. A partial duality and a dynamical regime in which the theory may have massive excitations being knot-like solitons are discussed. 
  Recently M. Kontsevich found a combinatorial formula defining a star-product of deformation quantization for any Poisson manifold. Kontsevich's formula has been reinterpreted physically as quantum correlation functions of a topological sigma model for open strings as well as in the context of D-branes in flat backgrounds with a Neveu-Schwarz B-field. Here the corresponding Kontsevich's formula for the dual of a Lie algebra is derived in terms of the formalism of D-branes on group manifolds. In particular we show that that formula is encoded at the two-point correlation functions of the Wess-Zumino-Witten effective theory with Dirichlet boundary conditions. The B-field entering in the formalism plays an important role in this derivation. 
  We discuss Z_2 \times Z_2 orientifolds where the orbifold twists are accompanied by shifts on momentum or winding lattice states. The models contain variable numbers of D5 branes, whose massless (and, at times, even massive) modes have variable numbers of supersymmetries. We display new type-I models with partial supersymmetry breaking N=2 \to N=1, N=4 \to N=1 and N=4 \to N=2. The geometry of these models is rather rich: the shift operations create brane multiplets related by orbifold transformations that support gauge groups of reduced rank. Some of the models are deformations of six-dimensional supersymmetric type-I models, while others have dual M-theory descriptions. 
  We find the spectrum P(w)dw of the gravitational wave background produced in the early universe in string theory. We work in the framework of String Driven Cosmology, whose scale factors are computed with the low-energy effective string equations as well as selfconsistent solutions of General Relativity with a gas of strings as source. The scale factor evolution is described by an early string driven inflationary stage with an instantaneous transition to a radiation dominated stage and successive matter dominated stage. This is an expanding string cosmology always running on positive proper cosmic time. A careful treatment of the scale factor evolution and involved transitions is made. A full prediction on the power spectrum of gravitational waves without any free-parameters is given. We study and show explicitly the effect of the dilaton field, characteristic to this kind of cosmologies. We compute the spectrum for the same evolution description with three differents approachs. Some features of gravitational wave spectra, as peaks and asymptotic behaviours, are found direct consequences of the dilaton involved and not only of the scale factor evolution. A comparative analysis of different treatments, solutions and compatibility with observational bounds or detection perspectives is made. 
  We investigate field theories on the worldvolume of a D3-brane transverse to partial resolutions of a $\Z_3\times\Z_3$ Calabi-Yau threefold quotient singularity. We deduce the field content and lagrangian of such theories and present a systematic method for mapping the moment map levels characterizing the partial resolutions of the singularity to the Fayet-Iliopoulos parameters of the D-brane worldvolume theory. As opposed to the simpler cases studied before, we find a complex web of partial resolutions and associated field-theoretic Fayet-Iliopoulos deformations. The analysis is performed by toric methods, leading to a structure which can be efficiently described in the language of convex geometry. For the worldvolume theory, the analysis of the moduli space has an elegant description in terms of quivers. As a by-product, we present a systematic way of extracting the birational geometry of the classical moduli spaces, thus generalizing previous work on resolution of singularities by D-branes. 
  We investigate the properties of non-linear electromagnetism based on the Born-Infeld Lagrangian in multi-dimensional theories of Kaluza-Klein type. We consider flat space-time solutions only, which means that the space-time metric is constant, and the only supplementary variable is the dilaton field, a scalar. We show that in the case of Kaluza-Klein theory, the Born-Infeld Lagrangian describes an interesting interaction between the electromagnetic and scalar fields, whose propagation properties are modified in a non-trivial manner. 
  We compute the high-temperature limit of the free energy for four-dimensional N=4 supersymmetric SU(N_c) Yang-Mills theory. At weak coupling we do so for a general ultrastatic background spacetime, and in the presence of slowly-varying background gauge fields. Using Maldacena's conjectured duality, we calculate the strong-coupling large-N_c expression for the special case that the three-space has constant curvature. We compare the two results paying particular attention to curvature corrections to the leading order expressions. 
  We analyze global anomalies for elementary Type II strings in the presence of D-branes. Global anomaly cancellation gives a restriction on the D-brane topology. This restriction makes possible the interpretation of D-brane charge as an element of K-theory. 
  We study the divergence of large-order perturbation theory in the worldline expression for the two-loop Euler-Heisenberg QED effective Lagrangian in a constant magnetic field. The leading rate of divergence is identical, up to an overall factor, to that of the one-loop case. From this we deduce, using Borel summation techniques, that the leading behaviour of the imaginary part of the two-loop effective Lagrangian for a constant E field, giving the pair-production rate, is proportional to the one-loop result. This also serves as a test of the mass renormalization, and confirms the earlier analysis by Ritus. 
  N-particle quantum mechanics described by a sigma model with an N-dimensional target space with torsion is considered. It is shown that an SL(2,R) conformal symmetry exists if and only if the geometry admits a homothetic Killing vector $D^a$ whose associated one-form $D_a$ is closed. Further, the SL(2,R) can always be extended to Osp(1|2) superconformal symmetry, with a suitable choice of torsion, by the addition of N real fermions. Extension to SU(1,1|1) requires a complex structure I and a holomorphic U(1) isometry $D^a I_a{^b} \partial_b$. Conditions for extension to the superconformal group D(2,1;\alpha), which involve a triplet of complex structures and SU(2) x SU(2) isometries, are derived. Examples are given. 
  The Casimir energy of an infinite compact cylinder placed in a uniform unbounded medium is investigated under the continuity condition for the light velocity when crossing the interface. As a characteristic parameter in the problem the ratio $\xi^2=(\epsilon_1-\epsilon_2)^2/ (\epsilon_1+\epsilon_2)^-2 = (\mu_1-\mu_2)^2/(\mu_1+ \mu_2)^2 \le 1$ is used, where $\epsilon_1$ and $\mu_1$ are, respectively, the permittivity and permeability of the material making up the cylinder and $\epsilon_2$ and $\mu_2$ are those for the surrounding medium. It is shown that the expansion of the Casimir energy in powers of this parameter begins with the term proportional to $\xi^4$. The explicit formulas permitting us to find numerically the Casimir energy for any fixed value of $\xi^2$ are obtained. Unlike a compact ball with the same properties of the materials, the Casimir forces in the problem under consideration are attractive. The implication of the calculated Casimir energy in the flux tube model of confinement is briefly discussed. 
  In this note we present an application of the Schwarzian derivative. By exploiting some properties of the Schwarzian derivative, we solve the equation appearing in the gravity-dilaton-antisymmetric tensor system. We also mention that this method can also be used to solve some other equations. 
  We define a non-local gauge-invariant Green's function which can distinguish between the symmetric (confinement) and broken (Higgs) phases of the hot SU(2)xU(1) electroweak theory to all orders in the perturbative expansion. It is related to the coupling of the Chern-Simons number to a massless Abelian gauge field. The result implies either that there is a way to distinguish between the phases, even though the macroscopic thermodynamical properties of the system have been observed to be smoothly connected, or that the perturbative Coleman-Hill theorem on which the argument is based, is circumvented by non-perturbative effects. We point out that this question could in principle be studied with three-dimensional lattice simulations. 
  Possible generalizations of the topological (or Berezinskii-Kosterlitz-Thouless) phase transition on multicomponent 2D systems with nontrivial vector homotopic group pi_1 are considered. Relations between Ginzburg-Landau like theories, non-linear sigma-models on maximal Cartan subgroups of simple compact Lie groups and generalized sine-Gordon type theories are discussed. D-dimensional non-linear sigma-model admitting topological excitations with logarithmic energies are constructed. 
  A theory of closed bosonic string in time-like gauge, related in Lorentz-invariant way with the world sheet, is considered. Absence of quantum anomalies in this theory is shown. 
  We study classical solutions of a particular version of the modified Skyrme model in (3+1) dimensions. The model possesses Skyrmion solutions as well as stable domain walls that connect different vacua of the theory. We show that there is an attractive interaction between Skyrmions and domain walls. Thus Skyrmions can be captured by the domain walls. We show also that, when the mass term is of a special type, the model possesses bound states of Skyrmions and of the domain wall. They look like deformed 2-dimensional Skyrmions captured by the wall. The field configurations of these solutions can interpreted as having come from the evolution of the 3-dimensional Skyrmions captured by the domain wall. For more conventional choices of the mass term of the model in the model the attraction between the Skyrmions and the wall leads to the capture of the Skyrmions which are then turned into topological waves which spread out on the wall. We have observed, numerically, such captures and the emission of the waves.   We speculate that this observation may be useful in the explanation of the problem of baryogenesis and baryon-antibaryon asymmetry of the Universe. 
  We extend the concept of quintessence to a flat nonminimally coupled scalar - tensor theories of gravity. By means of Noether's symmetries for the cosmological pointlike Lagrangian L, it is possible to exhibit exact solutions for a class of models depending on a free parameter s. This parameter comes out in the relationship existing between the coupling F(\phi) and the potential V(\phi) because of such a symmetry for L. When inverse power law potentials are taken in account, a whole family of exact solutions parametrized by such an s is proposed as a class of tracker fields, and some considerations are made about them. 
  The Casimir energy is the first-order-in-\hbar correction to the energy of a time-independent field configuration in a quantum field theory. We study the Casimir energy in a toy model, where the classical field is replaced by a separable potential. In this model the exact answer is trivial to compute, making it a good place to examine subtleties of the problem. We construct two traditional representations of the Casimir energy, one from the Greens function, the other from the phase shifts, and apply them to this case. We show that the two representations are correct and equivalent in this model. We study the convergence of the Born approximation to the Casimir energy and relate our findings to computational issues that arise in more realistic models. 
  We quantize the superstring on the AdS_2 x S^2 background with Ramond-Ramond flux using a PSU(1,1|2)/U(1) x U(1) sigma model with a WZ term. One-loop conformal invariance of the model is guaranteed by a general mechanism which holds for coset spaces G/H where G is Ricci-flat and H is the invariant locus of a Z_4 automorphism of G. This mechanism gives conformal theories for the PSU(1,1|2) x PSU(2|2)/SU(2) x SU(2) and PSU(2,2|4)/SO(4,1) x SO(5) coset spaces, suggesting our results might be useful for quantizing the superstring on AdS_3 x S^3 and AdS_5 x S^5 backgrounds. 
  We discuss AdS_{2+1} (BTZ) black holes arising in type 0 string theory corresponding to D1-D5 and F1-NS5 bound states. In particular we describe a new family of non-dilatonic solutions with only Dp_{+}, that is ``electric'' branes. These solutions are distinguished by the absence of fermions in the world volume theory which is an interacting CFT. They can not be obtained as a projection of a type II BPS-configuration. As for previous type 0 backgrounds linear stability is guaranteed only if the curvature is of the order of the string scale where alpha' corrections cannot be excluded. Some problems concerning the counting of states are discussed. 
  We derive a component-field expansion of the Green-Schwarz action for the type IIA string, in an arbitrary background of massless NS-NS and R-R bosonic fields, up to quadratic order in the fermionic coordinates \theta. Using this action, we extend the usual derivation of Buscher T-duality rules to include not only NS-NS, but also R-R fields. Our implementation of the T-duality transformation rules makes use of adapted background-field parametrizations, which provide a more geometrically natural and elegant description for the duality maps than the ones previously presented. These T-duality rules allow us to derive the Green-Schwarz action for the type IIB string in an arbitrary background of massless NS-NS and R-R bosonic fields, up to O(\theta^2). Implemention of another T-duality transformation on this type IIB action then allows us also to derive the Green-Schwarz action for the massive IIA string. By further considering T-duality transformations for backgrounds with the two U(1) isometries of a 2-torus, we give a string-theoretic derivation of the direct T-duality relation between the massless and massive type IIA strings. In addition, we give an explicit construction of the D=8 SL(3,R)xSL(2,R) invariant supergravity with two mass parameters that form a doublet under the SL(2,R) factor. 
  We consider anomaly cancelations in Type IIB orientifolds on T^4/Z_N with quantized NS-NS sector background B-flux. For a rank b B-flux on T^4 (b is always even) and when N is even, the cancelation requires a 2^{b/2} multiplicity of states in the 59-open string sector. We identify the twisted sector R-R scalars and tensor multiplets which are involved in the Green-Schwarz mechanism. We give more details of the construction of these models and argue that consistency with the 2^{b/2} multiplicity of 59-sector states requires a modification of the relation between the open string 1-loop channel modulus and the closed string tree channel modulus in the 59-cylinder amplitudes. 
  We use the AdS/CFT duality to study the special point on the Coulomb branch of ${\cal N}=4$ SU(N) gauge theory which corresponds to a spherically symmetric shell of D3-branes. This point is of interest both because the spacetime region inside the shell is flat, and because this configuration gives a very simple example of the transition between D-branes in the perturbative string regime and the non-perturbative regime of black holes. We discuss how this geometry is described in the dual gauge theory, through its effect on the two-point functions and Wilson loops. In the calculation of the two-point function, we stress the importance of absorption by the branes. 
  The infinite tension limit of string amplitudes is examined with some care, identifying the part responsible of diamagnetic behaviour as well as a peculiar paramagnetic {\em tachyon magnifying} responsible of asymptotic freedom. The way string theory represents abelian gauge theories is connected with the non-planar reggeon/pomeron amplitude and a nontrivial beta function is found in the low energy limit of a single D-brane. 
  We consider N=2 superconformal theories defined on a 3+1 dimensional hyperplane intersection of two sets of M5 branes. These theories have (tensionless) BPS string solitons. We use a dual supergravity formulation to deduce some of their properties via the AdS/CFT correspondence. 
  We join quintessence cosmological scenarios with the duality simmetry existing in string dilaton cosmologies. Actually, we consider the tracker potential type $V = V_0/{\phi}^{\alpha}$ and show that duality is only established if $\alpha = - 2$. 
  We present a simple algorithm to implement the generalized derivative expansion introduced previously by L-H. Chan, and apply it to the calculation of the one-loop mass correction to the classical soliton mass in the 1+1 dimensional Jacobi model. We then show how this derivative expansion approach implies that the total (bosonic plus fermionic) mass correction in an N=1 supersymmetric soliton model is determined solely by the asymptotic values (and derivatives) of the fermionic background potential. For a static soliton the total mass correction is $-m/(2\pi)$, in agreement with recent analyses using phase-shift methods. 
  We construct intersecting brane configurations in Anti-de-Sitter space localizing gravity to the intersection region, with any number $n$ of extra dimensions. This allows us to construct two kinds of theories with infinitely large new dimensions, TeV scale quantum gravity and sub-millimeter deviations from Newton's Law. The effective 4D Planck scale $M_{Pl}$ is determined in terms of the fundamental Planck scale $M_*$ and the $AdS$ radius of curvature $L$ via the familiar relation $M_{Pl}^2 \sim M_{*}^{2+n} L^n$; $L$ acts as an effective radius of compactification for gravity on the intersection. Taking $M_* \sim $ TeV and $L \sim $ sub-mm reproduces the phenomenology of theories with large extra dimensions. Alternately, taking $M_* \sim L^{-1} \sim M_{Pl}$, and placing our 3-brane a distance $\sim 100 M_{Pl}^{-1}$ away from the intersection gives us a theory with an exponential determination of the Weak/Planck hierarchy. 
  It has been conjectured that Little String Theories in six dimensions are holographic to critical string theory in a linear dilaton background. We test this conjecture for theories arising on the worldvolume of heterotic fivebranes. We compute the spectrum of chiral primaries in these theories and compare with results following from Type I-heterotic duality and the AdS/CFT correspondence. We also construct holographic duals for heterotic fivebranes near orbifold singularities. Finally we find several new Little String Theories which have Spin(32)/Z_2 or E_8 \times E_8 global symmetry but do not have a simple interpretation either in heterotic or M-theory. 
  We conjecture that the Sen-Seiberg limit of the Type IIA D2-brane action in a flat spacetime background can be resummed, at all orders in \alpha', to define an associative star product on the membrane. This star product can be independently constrained from the equivalent Matrix theory description of the corresponding M2-brane, by carefully analyzing the known BPS conditions. Higher derivative corrections to the Born-Infeld action on the IIA side are reinterpreted, after the Sen-Seiberg limit, as higher derivative corrections to a field theory on the membrane, which itself can be resummed to yield the known Matrix theory quantum mechanics action. Conversely, given the star product on the membrane as a formal power series in \alpha', one can constrain the higher derivative corrections to the Born-Infeld action, in the Sen-Seiberg limit. This claim is explicitly verified to first order. Finally, we also comment on the possible application of this method to the derivation of the Matrix theory action for membranes in a curved background. 
  The quantum theory of a free particle in two dimensions with non-local boundary conditions on a circle is known to lead to surface and bulk states. Such a scheme is here generalized to the quantized Maxwell field, subject to mixed boundary conditions. If the Robin sector is modified by the addition of a pseudo-differential boundary operator, gauge-invariant boundary conditions are obtained at the price of dealing with gauge-field and ghost operators which become pseudo-differential. A good elliptic theory is then obtained if the kernel occurring in the boundary operator obeys certain summability conditions, and it leads to a peculiar form of the asymptotic expansion of the symbol. The cases of ghost operator of negative and positive order are studied within this framework. 
  The study of the spectrum of coupled random matrices has received rather little attention. To the best of our knowledge, coupled random matrices have been studied, to some extent, by Mehta. In this work, we explain how the integrable technology can be brought to bear to gain insight into the nature of the distribution of the spectrum of coupled Hermitean random matrices and the equations the associated probabilities satisfy. In particular, the two-Toda lattice, its algebra of symmetries and its vertex operators will play a prominent role in this interaction. Namely, the method is to introduce time parameters, in an artificial way, and to dress up a certain matrix integral with a vertex integral operator, for which we find Virasoro-like differential equations. These methods lead to very simple nonlinear third-order partial differential equations for the joint statistics of the spectra of two coupled Gaussian random matrices. 
  We consider the N=2 supersymmetric Born-Infeld action and compute one-loop divergences quantizing the theory in N=1 superspace. We find that in the presence of non constant curvature the theory is not renormalizable. The structure of the $(\alpha')^4$ counterterm, proportional to derivatives of the curvature, is consistent with effective action calculations from superstring theory. 
  We examine string-theory orientifold planes of various types including the Sp and SO-odd planes, and deduce the gravitational Chern-Simons couplings on their world-volumes. Consistency checks are carried out in different spacetime dimensions using various dualities, including those relating string theory with F-theory and M-theory. It is shown that when an orientifold 3-plane crosses a 5-brane, the jump in the charge is accompanied by a corresponding change in the gravitational couplings. 
  We give full details for the computation of the Kaluza--Klein mass spectrum of Type IIB Supergravity on AdS_5 x T^{11}, with T^{11}=SU(2)xSU(2)/U(1), that has recently lead to both stringent tests and interesting predictions on the AdS_5/CFT_4 correspondence for N=1 SCFT's (hep-th/9905226). We exhaustively explain how KK states arrange into SU(2,2|1) supermultiplets, and stress some relevant features of the T^{11} manifold, such as the presence of topological modes in the spectrum originating from the existence of non-trivial 3-cycles. The corresponding Betti vector multiplet is responsible for the extra baryonic symmetry in the boundary CFT. More generally, we use the simple T^{11} coset as a laboratory to revive the technique and show the power of KK harmonic expansion, in view of the present attempts to probe along the same lines also M-theory compactifications and the AdS_4/CFT_3 map. 
  Using purely Hamiltonian methods we derive a simple differential equation for the generator of the most general local symmetry transformation of a Lagrangian. The restrictions on the gauge parameters found by earlier approaches are easily reproduced from this equation. We also discuss the connection with the purely Lagrangian approach. The general considerations are applied to the Yang-Mills theory. 
  We present a self-contained formulation of the Nonlinear Schrodinger hierarchy and its Yangian symmetry in terms of deformed oscilator algebra (Z.F. algebra). The link between Yangian Y(gl(N)) and finite W(gl(pN),N.gl(p)) algebras is also illustrated in this framework. 
  In this paper we discuss candidate superconformal N=2 gauge theories that realize the AdS/CFT correspondence with M--theory compactified on the homogeneous Sasakian 7-manifolds M^7 that were classified long ago. In particular we focus on the two cases M^7=Q^{1,1,1} and M^7=M^{1,1,1}, for the latter the Kaluza Klein spectrum being completely known. We show how the toric description of M^7 suggests the gauge group and the supersingleton fields. The conformal dimensions of the latter can be independently calculated by comparison with the mass of baryonic operators that correspond to 5-branes wrapped on supersymmetric 5-cycles and are charged with respect to the Betti multiplets. The entire Kaluza Klein spectrum of short multiplets agrees with these dimensions. Furthermore, the metric cone over the Sasakian manifold is a conifold algebraically embedded in some C^p. The ring of chiral primary fields is defined as the coordinate ring of C^p modded by the ideal generated by the embedding equations; this ideal has a nice characterization by means of representation theory. The entire Kaluza Klein spectrum is explained in terms of these vanishing relations. We give the superfield interpretation of all short multiplets and we point out the existence of many long multiplets with rational protected dimensions, whose presence and pattern seem to be universal in all compactifications. 
  The two-by-two representation of the SL(2,c) group is for spin-1/2 particles. Starting from this two-by-two representation, it is possible to construct the four-by-four matrices for spin-1 particles. For massless particles, it is possible to construct four-potentials from two-component SL(2,c) spinors. Four potentials are subject to gauge transformations and are gauge-dependent. Then this gauge dependence necessarily comes from the two-component spinors which make up the four-potential. Then there must be a gauge-dependent spinor. This gauge-dependent spinor is discussed in detail. It is shown that neutrino polarization is a consequence of gauge invariance applicable to the two-by-two representation of the SL(2,c) group. 
  The mathematical framework of superbundles suggests that one considers the Higgs field as a natural constituent of a superconnection. I propose to take as superbundle the exterior algebra obtained from a Hermitian vector bundle of rank 5 for the Standard Model. 
  The paper is withdrawn because the analysis appeared to be incomplete. 
  We show that the vacuum of the quantized massless spinor field in 2+1-dimensional space-time is polarized in the presence of a singular magnetic vortex. Depending on the choice of the boundary condition at the location of the vortex, either chiral symmetry or parity is broken; the formation of the appropriate vacuum condensates is comprehensively studied. In addition, we find that current, energy and other quantum numbers are induced in the vacuum. 
  The multidimensional N=4 supersymmetric quantum mechanics (SUSY QM) is constructed using the superfield approach. As a result, the component form of the classical and quantum Lagrangian and Hamiltonian is obtained. In the considered SUSY QM both classical and quantum N=4 algebras include central charges, and this opens various possibilities for partial supersymmetry breaking. It is shown that quantum mechanical models with one quarter, one half and three quarters of unbroken(broken) supersymmetries can exist in the framework of the multidimensional N=4 SUSY QM, while the one-dimensional N=4 SUSY QM, constructed earlier, admits only one half or total supersymmetry breakdown. We illustrate the constructed general formalism, as well as all possible cases of the partial SUSY breaking taking as an example a direct multidimensional generalization of the one-dimensional N=4 superconformal quantum mechanical model. Some open questions and possible applications of the constructed multidimensional N=4 SUSY QM to the known exactly integrable systems and problems of quantum cosmology are briefly discussed. 
  Supersymmetry is one of the most plausible and theoretically motivated frameworks for extending the Standard Model. However, any supersymmetry in Nature must be a broken symmetry. Dynamical supersymmetry breaking (DSB) is an attractive idea for incorporating supersymmetry into a successful description of Nature. The study of DSB has recently enjoyed dramatic progress, fueled by advances in our understanding of the dynamics of supersymmetric field theories. These advances have allowed for direct analysis of DSB in strongly coupled theories, and for the discovery of new DSB theories, some of which contradict early criteria for DSB. We review these criteria, emphasizing recently discovered exceptions. We also describe, through many examples, various techniques for directly establishing DSB by studying the infrared theory, including both older techniques in regions of weak coupling, and new techniques in regions of strong coupling. Finally, we present a list of representative DSB models, their main properties, and the relations between them. 
  N=4 Super Yang-Mills theory supplies us with a non-Abelian 4D gauge theory with a meaningful perturbation expansion, both in the UV and in the IR. We calculate the free energy on a 3-sphere and observe a deconfinement transition for large N at zero coupling. The same thermodynamic behaviour is found for a wide class of toy models, possibly also including the case of non-zero coupling. Below the transition we also find Hagedorn behaviour, which is identified with fluctuations signalling the approach to the deconfined phase. The Hagedorn and the deconfinement temperatures are identical. Application of the AdS/CFT correspondence gives a connection between string Hagedorn behaviour and black holes. 
  A covariant spinor representation of $iosp(d,2/2)$ is constructed for the quantization of the spinning relativistic particle. It is found that, with appropriately defined wavefunctions, this representation can be identified with the state space arising from the canonical extended BFV-BRST quantization of the spinning particle with admissible gauge fixing conditions after a contraction procedure. For this model, the cohomological determination of physical states can thus be obtained purely from the representation theory of the $iosp(d,2/2)$ algebra. 
  We investigate the effective potential for a scalar $\Phi^{4}$ theory with spontaneous symmetry breaking at finite temperature. All 'daisy' and 'super daisy' diagrams are summed up and the properties of the corresponding gap eqation are investigated. It is shown exactly that the phase transition is first order. 
  We explore the interplay between black holes in supergravity and quantum field theories on the world-volumes of D-branes. A brief summary of black hole entropy calculations for D-brane black holes is followed by a detailed study of particle absorption by black holes whose string theory description involves D-branes intersecting along a string. A conformal field theory with large central charge describes the low-energy excitations of this string. The absorption cross-sections give rise to greybody factors in Hawking radiation processes which are characteristic of conformal field theory at finite temperature.   Particle absorption by extremal three-branes is examined next, with particular attention to the implications for supersymmetric gauge theory in four dimensions. A fascinating duality between supergravity and gauge theory emerges from the study of these processes. Fields of supergravity are dual to local operators in the gauge theory. A non-renormalization theorem of N=4 gauge theory helps explain why certain aspects of the duality can be explored perturbatively. Anomalous dimensions of a large class of local operators in the gauge theory are shown to become large at strong 't Hooft coupling, signaling a possible simplification of N=4 gauge theory in this limit. 
  We give a pedagogical introduction to certain aspects of supersymmetric field theories in anti-de Sitter space. Among them are the presence of masslike terms in massless wave equations, irreducible unitary representations and the phenomenon of multiplet shortening. 
  We study the 3D field theory on one D3-brane stretched between (r,s) and (p,q)5-branes. The boundary conditions are determined from the analysis of NS5 and D5 charges of the two 5-branes. We carry out the mode expansions for all the fields and identify the field theory as Maxwell-Chern-Simons theory. We examine the mass spectrum to determine the conditions for unbroken supersymmetry (SUSY) in this field theory and compare the results with those from the brane configurations. The spectrum is found to be invariant under the Type IIB SL(2,{\bf Z})-transformation. We also discuss the theory with matters and its S-dual configuration. The result suggests that the equivalence under S-duality may be valid if we include all the higher modes in the theories with matters. We also find an interesting phenomenon that SUSY enhancement happens in the field theory after dimensional reduction from 3D to 2D. 
  We study compactifications of type IIB supergravity on Calabi-Yau threefolds. The resulting low energy effective Lagrangian is displayed in the large volume limit and its symmetry properties - with specific emphasis on the SL(2,Z) - are discussed. The explicit map to type IIA string theory compactified on a mirror Calabi-Yau is derived. We argue that strong coupling effects on the worldsheet break the SL(2,Z). 
  Using N=2 Landau-Ginzburg theories, we examine the recent conjectures relating the SU(3) WZW modular invariants, finite subgroups of SU(3) and Gorenstein singularities. All isolated three-dimensional Gorenstein singularities do not appear to be related to any known Landau-Ginzburg theories, but we present some curious observations which suggest that the SU(3)_n/SU(2)xU(1) Kazama-Suzuki model may be related to a deformed geometry of C^3/Z_{n+3}xZ_{n+3}. The toric resolution diagrams of those particular singularities are also seen to be classifying the diagonal modular invariants of the SU(3)_n as well as the SU(2)_{n+1} WZW models. 
  We notice that, for branes wrapped on complex analytic subvarieties, the algebraic-geometric version of K-theory makes the identification between brane-antibrane pairs and lower-dimensional branes automatic. This is because coherent sheaves on the ambient variety represent gauge bundles on subvarieties, and they can be put in exact sequences (projective resolutions) with sheaves corresponding to vector bundles on the pair; this automatically gives a D(p-2) as a formal difference of bundles on the Dp - D\bar p pair, both belonging to the Grothendieck group of coherent sheaves of the ambient. 
  According to the AdS/CFT correspondence, the strong coupling limit of large n, N=4 supersymmetric gauge theory at finite temperature is described by asymptotically anti de Sitter black holes. These black holes exist with planar, spherical and hyperbolic horizon geometries. We concentrate on the hyperbolic and spherical cases and probe the associated gauge theories with D3-branes and Wilson loops. The D3-brane probe reproduces the coupling of the scalars in the gauge theory to the background geometry and we find thermal stabilization in the hyperbolic case. We investigate the vacuum expectation value of Wilson loops with particular emphasis on the screening length at finite temperature. We find that the thermal phase transition of the theory on the sphere is not related to screening phenomena. 
  For linear scalar field theories, I characterize those classical Hamiltonian vector fields which have self-adjoint operators as their quantum counterparts. As an application, it is shown that for a scalar field in curved space-time (in a Hadamard representation), a self-adjoint Hamiltonian for evolution along the unit timelike normal to a Cauchy surface exists only if the second fundamental form of the surface vanishes identically. 
  For linear bose field theories, I show that if a classical Hamiltonian function is strictly positive, then there is a canonical transformation making the evolution orthogonal. This structure theorem is used to analyze the corresponding quantum theories. It is shown that there is an intimate connection between boundedness-below and self-adjoint implementability.   Finally, it is shown that there is a broad class of "quantum inequalities:" any timelike component of the four-momentum density operator, averaged over a compact region in curved space-time, must be bounded below. 
  In M-theory on $S^1/Z_2$, we point out that to be consistant, we should keep the scale, gauge couplings and soft terms at next order, and obtain the soft term relations: $M_{1/2} = -A$, $|{{M_{0}}/{M_{1/2}}}| \leq {1/{\sqrt 3}}$ in the standard embedding and $M_{1/2}=-A$ in the non-standard embedding with five branes and $K_{5,n}=0$. We construct a toy compactification model which includes higher order terms in 4-dimensional Lagrangian in standard embedding, and discuss its scale, gauge couplings, soft terms, and show that the higher order terms do affect the scale, gauge couplings and especially the soft terms if the next order correction was not small. We also construct a toy compactification model in non-standard embedding with five branes and discuss its phenomenology. We argue that one might not push the physical Calabi-Yau manifold's volume to zero at any point along the eleventh dimension. 
  I identify the class of even-dimensional conformal field theories that is most similar to two-dimensional conformal field theory. In this class the formula, elaborated recently, for the irreversibility of the renormalization-group flow applies also to massive flows. This implies a prediction for the ratio between the coefficient of the Euler density in the trace anomaly (charge a) and the stress-tensor two-point function (charge c). More precisely, the trace anomaly in external gravity is quadratic in the Ricci tensor and the Ricci scalar and contains a unique central charge. I check the prediction in detail in four, six and eight dimensions, and then in arbitrary even dimension. 
  We propose a new method to generate the internal isospin degree of freedom by non-local bound states. This can be seen as motivated by Bargmann-Wigner like considerations, which originated from local spin coupling. However, our approach is not of purely group theoretical origin, but emerges from a geometrical model. The rotational part of the Lorentz group can be seen to mutate into the internal iso-group under some additional assumptions. The bound states can thereafter be characterized by either a triple of spinors (\xi_1, \xi_2, \eta) or a pair of an average spinor and a ``gauge'' transformation (\phi, R). Therefore, this triple can be considered to be an isospinor. Inducing the whole dynamics from the covariant gauge coupling we arrive at an isospin gauge theory and its Lagrangian formulation. Clifford algebraic methods, especially the Hestenes approach to the geometric meaning of spinors, are the most useful concepts for such a development. The method is not restricted to isospin, which served as an example only. 
  Recently it was pointed out that in the TeV-scale brane world there is a logical possibility where the electroweak Higgs can be identified with a fourth generation slepton. In this paper we address various issues in this four-generation TeV-scale Supersymmetric Standard Model with Higgs as a slepton. In particular we discuss how to achieve proton stability by suppressing the corresponding baryon number violating operators via gauging (a discrete subgroup of) the baryon number U(1) symmetry. Dimension five lepton number violating operators which would result in unacceptably large neutrino masses can be similarly suppressed via gauging a discrete subgroup of the lepton number U(1) symmetry. In fact, the four generation feature allows for a novel higher dimensional mechanism for generating small Majorana neutrino masses. We also discuss how to achieve gauge coupling unification, which can be as precise at one loop as in the MSSM, and point out a possible geometric embedding of the corresponding matter content in the brane world context. Finally, we discuss adequate suppression of flavor changing neutral currents in this model, and also point out a novel possibility for supersymmetry breaking via a non-zero F-term of the fourth generation lepton superfield. 
  We study D=4, N=1, type IIA orientifold with orbifold group $Z_N$ and $Z_N \times Z_M$. We calculate one-loop vacuum amplitudes for Klein bottle, cylinder and Mobius strip and extract the tadpole divergences. We find that the tadpole cancellation conditions thus obtained are satisfied by the $Z_4$, $Z_8$, $Z'_8$, $Z'_{12}$ orientifolds while there is no solution for $Z_3$, $Z_7$, $Z_6$, $Z'_6$, $Z_{12}$. The $Z_4 \times Z_4$ type IIA orientifold is also constructed by introducing four different configurations of 6-branes. We argue about perturbative versus non-perturbative orientifold vacua under T- duality between the type IIA and the type IIB $Z_N$ orientifolds in four dimensions. 
  Up to overall harmonic factors, the D8-brane solution of the massive type IIA supergravity theory is the product of nine-dimensional Minkowski space (the worldvolume) with the real line (the transverse space). We show that the equations of motion allow for the worldvolume metric to be generalised to an arbitrary Ricci-flat one. If this nine-dimensional Ricci-flat manifold admits Killing spinors, then the resulting solutions are supersymmetric and satisfy the usual Bogomol'nyi bound, although they preserve fewer than the usual one half of the supersymmetries. We describe the possible choices of such manifolds, elaborating on the connection between the existence of Killing spinors and the self-duality condition on the curvature two-form. Since the D8-brane is a domain wall in ten dimensions, we are led to consider the general case: domain walls in any supergravity theory. Similar considerations hold here also. Moreover, it is shown that the worldvolume of any magnetic brane --- of which the domain walls are a specific example --- can be generalised in precisely the same way. The general class of supersymmetric solutions have gravitational instantons as their spatial sections. Some mention is made of the worldvolume solitons of such branes. 
  T-duality in M(atrix) theory has been argued to be realized as Morita equivalence in Yang-Mills theory on a non-commutative torus (NCSYM). Even though the two have the same structure group, they differ in their action since Morita equivalence makes crucial use of an additional modulus on the NCSYM side, the constant Abelian magnetic background. In this paper, we reanalyze and clarify the correspondence between M(atrix) theory and NCSYM, and provide two resolutions of this puzzle. In the first of them, the standard map is kept and the extra modulus is ignored, but the anomalous transformation is offset by the M(atrix) theory ``rest term''. In the second, the standard map is modified so that the duality transformations agree, and a $SO(d)$ symmetry is found to eliminate the spurious modulus. We argue that this is a true symmetry of supersymmetric Born-Infeld theory on a non-commutative torus, which allows to freely trade a constant magnetic background for non-commutativity of the base-space. We also obtain a BPS mass formula for this theory, invariant under T-duality, U-duality, and continuous $SO(d)$ symmetry. 
  The series of perturbative fluctuations around a multi-instanton contribution to a specific class of correlation functions of supercurrents in $\cal N=4$ supersymmetric SU(N) Yang-Mills theory is examined in the light of the AdS/CFT correspondence. Subject to certain plausible assumptions, we argue that a given term in the 1/N expansion in such a background receives only a finite number of perturbative corrections in the 't Hooft limit. Such instanton non-renormalisation theorems would explain, for example, the exact agreement of certain weak coupling Yang-Mills instanton calculations with the strong coupling predictions arising from D-instanton effects in string theory amplitudes. These non-renormalisation theorems essentially follow from the assumption of a well defined derivative $(\alpha^{\prime})$ expansion in the string theory dual of the Yang-Mills theory. 
  Recent ideas on modular localization in local quantum physics are used to clarify the relation between on- and off-shell quantities in particle physics; in particular the relation between on-shell crossing symmetry and off-shell Einstein causality. Among the collateral results of this new nonperturbative approach are profound relations between crossing symmetry of particle physics and Hawking-Unruh like thermal aspects (KMS property, entropy attached to horizons) of quantum matter behind causal horizons, aspects which hitherto were exclusively related with Killing horizons in curved spacetime rather than with localization aspects in Minkowski space particle physics. The scope of this modular framework is amazingly wide and ranges from providing a conceptual basis for the d=1+1 bootstrap-formfactor program for factorizable d=1+1 models to a decomposition theory of QFT's in terms of a finite collection of unitarily equivalent chiral conformal theories placed a specified relative position within a common Hilbert space (in d=1+1 a holographic relation and in higher dimensions more like a scanning). The new framework gives a spacetime interpretation to the Zamolodchikov-Faddeev algebra and explains its thermal aspects. 
  We study the thermodynamics of four-dimensional Kerr-Newman-AdS black holes both in the canonical and the grand-canonical ensemble. The stability conditions are investigated, and the complete phase diagrams are obtained, which include the Hawking-Page phase transition in the grand-canonical ensemble. In the canonical case, one has a first order transition between small and large black holes, which disappears for sufficiently large electric charge or angular momentum. This disappearance corresponds to a critical point in the phase diagram. Via the AdS/CFT conjecture, the obtained phase structure is also relevant for the corresponding conformal field theory living in a rotating Einstein universe, in the presence of a global background U(1) current. An interesting limit arises when the black holes preserve some supersymmetry. These BPS black holes correspond to highly degenerate zero temperature states in the dual CFT, which lives in an Einstein universe rotating with the speed of light. 
  We show how to construct chiral tachyon-free perturbative orientifold models, where supersymmetry is broken at the string scale on a collection of branes while, to lowest order, the bulk and the other branes are supersymmetric. In higher orders, supersymmetry breaking is mediated to the remaining sectors, but is suppressed by the size of the transverse space or by the distance from the brane where supersymmetry breaking primarily occurred. This setting is of interest for orbifold models with discrete torsion, and is of direct relevance for low-scale string models. It can guarantee the stability of the gauge hierarchy against gravitational radiative corrections, allowing an almost exact supergravity a millimeter away from a non-supersymmetric world. 
  We compute the leading-order back-reaction to dilaton-driven inflation, due to graviton, dilaton and gauge-boson production. The one-loop effect turns out to be non-vanishing (unlike the case for pure de-Sitter and for power-law inflation), to be of relative order $\ell_P^2H^2(t)$, and to have the correct sign for favouring the exit to a FRW phase. 
  Various structural properties of the space of symmetry breaking boundary conditions that preserve an orbifold subalgebra are established. To each such boundary condition we associate its automorphism type. It is shown that correlation functions in the presence of such boundary conditions are expressible in terms of twisted boundary blocks which obey twisted Ward identities. The subset of boundary conditions that share the same automorphism type is controlled by a classifying algebra, whose structure constants are shown to be traces on spaces of chiral blocks. T-duality on boundary conditions is not a one-to-one map in general. These structures are illustrated in a number of examples. Several applications, including the construction of non-BPS boundary conditions in string theory, are exhibited. 
  We derive a formula for the large N behaviour of the expectation values of an arbitrary product of Wilson loops in the adjoint representation. We show the consequences of our formula for the study of the large N strong coupling behaviour of SU(N)/Z_N pure gauge theories, and theories with matter fields in the adjoint representation. This allows us to calculate large N corrections to the gluino condensate and meson propagators in the lattice version of Supersymmetric Yang-Mills. Applications to the strong coupling behaviour of the Kazakov-Migdal model are also given. 
  The question of anyons and fractional statistics in field theories in 2+1 dimensions with Chern-Simons (CS) term is discussed in some detail. Arguments are spelled out as to why fractional statistics is only possible in two space dimensions. This phenomenon is most naturally discussed within the framework of field theories with CS term, hence as a prelude to this discussion I first discuss the various properties of the CS term. In particular its role as a gauge field mass term is emphasized. In the presence of the CS term, anyons can appear in two different ways i.e. either as soliton of the corresponding field theory or as a fundamental quanta carrying fractional statistics and both approaches are elaborated in some detail. 
  We investigate the problem of derivation of consistent equations of motion for the massive spin 2 field interacting with gravity within both field theory and string theory. In field theory we derive the most general classical action with non-minimal couplings in arbitrary spacetime dimension, find the most general gravitational background on which this action describes a consistent theory and generalize the analysis for the coupling with background scalar dilaton field. We show also that massive spin 2 field allows in principle consistent description in arbitrary background if one builds its action in the form of an infinite series in the inverse mass square. Using sigma-model description of string theory in background fields we obtain in the lowest order in $\alpha'$ the explicit form of effective equations of motion for the massive spin 2 field interacting with gravity from the requirement of quantum Weyl invariance and demonstrate that they coincide with the general form of consistent equations derived in field theory. 
  Non-BPS type II D-branes couple to R-R potentials via an action that, upon tachyon condensation, gives rise to the Wess-Zumino action of BPS D-branes. 
  Closed type 0 string theories and their D-branes are introduced. The full Wess-Zumino action of these D-branes is derived. The analogy with type II is emphasized throughout the argument. 
  We argue that the quantized non-Abelian gauge theory can be obtained as the infrared limit of the corresponding classical gauge theory in a higher dimension. We show how the transformation from classical to quantum dynamics emerges and calculate Planck's constant from quantities defined in the underlying classical field theory. 
  Application of the geometrically-inspired representations to the epsilon-expansion of the two-point function with different masses is considered. Explicit result for an arbitrary term of the expansion is obtained in terms of log-sine integrals. Construction of the epsilon-expansion in the three-point case is also discussed. 
  Grand canonical ensemble of small vortex loops emerging in the London limit of the effective Abelian-projected theory of the SU(3)-gluodynamics is investigated in the dilute gas approximation. An essential difference of this system from the SU(2)-case is the presence of two interacting gases of vortex loops. Two alternative representations for the partition function of such a grand canonical ensemble are derived, and one of them, which is a representation in terms of the integrals over vortex loops, is employed for the evaluation of the correlators of both kinds of loops in the low-energy limit. 
  We study some properties of a dimensional reduction mechanism for fermions in an odd number D+1 of spacetime dimensions. A fermionic field is equipped with a mass term with domain wall like defects along one of the spacelike dimensions, which is moreover compactified. We show that there is a regime such that the only relevant degrees of freedom are massless fermionic fields in D dimensions. For any fixed gauge field configuration, the extra modes may be decoupled, since they can be made arbitrarily heavy. This decoupling combines the usual Kaluza-Klein one, due to the compactification, with a mass enhancement for the non-zero modes provided by the domain wall mechanism. We obtain quantitative results on the contribution of the massive modes in the cases D=2 and D=4. 
  We consider ADHM instantons in product group gauge theories that arise from D3-branes located at points in the orbifold R^6/Z_p. At finite N we argue that the ADHM construction and collective coordinate integration measure can be deduced from the dynamics of D-instantons in the D3-brane background. For the large-N conformal field theories of this type, we compute a saddle-point approximation of the ADHM integration measure and show that it is proportional to the partition function of D-instantons in the dual AdS_5 x S^5/Z_p background, in agreement with the orbifold AdS/CFT correspondence. Matching the expected behaviour of D-instantons, we find that when S^5/Z_p is smooth a saddle-point solution only exists in the sector where the instanton charges in each gauge group factor are the same. However, when S^5/Z_p is singular, the instanton charges at large N need not be the same and the space of saddle-point solutions has a number of distinct branches which represent the possible fractionations of D-instantons at the singularity. For the theories with a type 0B dual the saddle-point solutions manifest two types of D-instantons. 
  We develop further the theory of Rational Conformal Field Theories (RCFTs) on a cylinder with specified boundary conditions emphasizing the role of a triplet of algebras: the Verlinde, graph fusion and Pasquier algebras. We show that solving Cardy's equation, expressing consistency of a RCFT on a cylinder, is equivalent to finding integer valued matrix representations of the Verlinde algebra. These matrices allow us to naturally associate a graph $G$ to each RCFT such that the conformal boundary conditions are labelled by the nodes of $G$. This approach is carried to completion for $sl(2)$ theories leading to complete sets of conformal boundary conditions, their associated cylinder partition functions and the $A$-$D$-$E$ classification. We also review the current status for WZW $sl(3)$ theories. Finally, a systematic generalization of the formalism of Cardy-Lewellen is developed to allow for multiplicities arising from more general representations of the Verlinde algebra. We obtain information on the bulk-boundary coefficients and reproduce the relevant algebraic structures from the sewing constraints. 
  The Chern-Simons invariants of irreducible U(n)- flat connections on compact hyperbolic 3-manifolds of the form {\Gamma}\H^3 are derived. The explicit formula for the Chern-Simons functional is given in terms of Selberg type zeta functions related to the twisted eta invariants of Atiyah-Patodi-Singer. 
  We review our proposal for a constructive definition of superstring, type IIB matrix model. The IIB matrix model is a manifestly covariant model for space-time and matter which possesses N=2 supersymmetry in ten dimensions. We refine our arguments to reproduce string perturbation theory based on the loop equations. We emphasize that the space-time is dynamically determined from the eigenvalue distributions of the matrices. We also explain how matter, gauge fields and gravitation appear as fluctuations around dynamically determined space-time. 
  In a recent work (hep-th/9905198) we argued that a certain matrix quantum mechanics may describe 't Hooft's monopoles which emerge in QCD when the theory is projected to its maximal Abelian subgroup. In this note we find further evidence which supports this interpretation. We study the theory with a non-zero theta-term. In this case, 't Hooft's QCD monopoles become dyons since they acquire electric charges due to the Witten effect. We calculate a potential between a dyon and an anti-dyon in the matrix quantum mechanics, and find that the attractive force between them grows as the theta angle increases. 
  The geometry of D-branes can be probed by open string scattering. If the background carries a non-vanishing B-field, the world-volume becomes non-commutative. Here we explore the quantization of world-volume geometries in a curved background with non-zero Neveu-Schwarz 3-form field strength H = dB. Using exact and generally applicable methods from boundary conformal field theory, we study the example of open strings in the SU(2) Wess-Zumino-Witten model, and establish a relation with fuzzy spheres or certain (non-associative) deformations thereof. These findings could be of direct relevance for D-branes in the presence of Neveu-Schwarz 5-branes; more importantly, they provide insight into a completely new class of world-volume geometries. 
  A sigma model action with N=2 D=6 superspace variables is constructed for the Type II superstring compactified to six curved dimensions with Ramond-Ramond flux. The action can be quantized since the sigma model is linear when the six-dimensional spacetime is flat. When the six-dimensional spacetime is $AdS_3\times S^3$, the action reduces to one found earlier with Vafa and Witten. 
  The critical dynamics of the chiral symmetry breaking induced by gauge interaction is examined in the Wilson renormalization group framework in comparison with the Schwinger-Dyson approach. We derive the beta functions for the four-fermi couplings in the sharp cutoff renormalzation group scheme, from which the critical couplings and the anomalous dimensions of the fermion composite operators near criticality are immediately obtained. It is also shown that the beta functions lead to the same critical behavior found by solving the so-called ladder Schwinger-Dyson equation, if we restrict the radiative corrections to a certain limited type. 
  The non-perturbative renormalization group equation for the Wilsonian effective potential is given in a certain simple approximation scheme in order to study chiral symmetry breaking phenomena dynamically induced by strong gauge interactions. The evolving effective potential is found to be non-analytic in infrared, which indicates spontaneous generation of the fermion mass. It is also shown that the renormalization group equation gives the identical effective fermion mass with that obtained by solving the Schwinger-Dyson equation in the (improved) ladder approximation. Moreover introduction of the collective field corresponding to the fermion composite into the theory space is found to offer an efficient method to evaluate the order parameters; the dynamical mass and the chiral condensate. The relation between the renormalization group equation incorporating the collective field and the Schwinger-Dyson equation is also clarified. 
  The quantum mechanics of N slowly-moving charged BPS black holes in five-dimensional ${\cal N}=1$ supergravity is considered. The moduli space metric of the N black holes is derived and shown to admit 4 supersymmetries. A near-horizon limit is found in which the dynamics of widely separated black holes decouples from that of strongly-interacting, near-coincident black holes. This decoupling suggests that the quantum states supported in the near-horizon moduli space can be interpreted as internal states of a single composite black hole carrying all of the charge. The near-horizon theory is shown to have an enhanced D(2,1;0) superconformal symmetry. Eigenstates of the Hamiltonian H of the near-horizon theory are ill-defined due to noncompact regions of the moduli space corresponding to highly redshifted near-coincident black holes. It is argued that one should consider, instead of H eigenstates, eigenstates of $2 L_0 = H+K$, where K is the generator of special conformal transformations. The result is a well-defined Hilbert space with a discrete spectrum describing the N-black hole dynamics. 
  The nature of $\kappa$-symmmetry transformations is examined for p-branes embedded in a class of coset superspaces G/H, where G is an appropriate supergroup and H is the Lorentz subgroup. It is shown that one of the conditions $\delta Z^M E_M{}^a = 0$ which characterizes $\kappa$-symmetry transformations arises very naturally if they are implemented in terms of a right action of a subgroup of the supergroup G on the supergroup elements which represent the coset. Unlike the global left action of G on G/H (which gives rise to supersymmetry on the coset superspace), there is no canonically defined right action of G on G/H. However, an interpretation of this right action involving an enlargement of the isotropy group from the Lorentz subgroup to a subgroup of G with generators which include some of the fermionic generators of G is suggested. Closure of the generators of this larger subgroup under commutation leads to the usual `brane scan' for p-branes which have bosonic degrees of freedom which are worldvolume scalars. 
  In this paper expressions are given for the bulk and boundary structure constants of D-series Virasoro minimal models on the upper half plane. It is the continuation of an earlier work on the A-series. The solution for the boundary theory is found first and then extended to the bulk. The modular invariant bulk field content is recovered as the maximal set of bulk fields consistent with the boundary theory. It is found that the structure constants are unique up to redefinition of the fields and in the chosen normalisation exhibit a manifest Z_2-symmetry associated to the D-diagram. The solution has been subjected to random numerical tests against the constraints it has to fulfill. 
  We study the D8-branes of the Romans massive IIA supergravity theory using the coupled supergravity and worldvolume actions. D8 branes can be regarded as domain walls with the jump in the extrinsic curvature at the brane given by the Israel matching conditions. We examine the restrictions that these conditions place on extreme and non-extreme solutions and find that they rule out some of the supersymmetric solutions given by Bergshoeff {\em et al}. We consider what happens when the dilaton varies on the worldvolume of the brane, which implies that the brane is no longer static. We obtain a family of D8-brane solutions parametrized by a non-extremality term on each side of the brane and the asymptotic values of the 10-form field. The non-extremality parameters can be related to the velocity of the brane. We also study 8-brane solutions of a massive IIA supergravity theory introduced by Howe, Lambert and West. This theory also admits a 10-form formulation, but the 10-form is not a R-R sector field and so these 8-branes are not D-branes. 
  In this note, we review various seemingly different ways of obtaining Eguchi-Hanson metric with or without a cosmological constant term. Interestingly, the conformal class of metric corresponding to hyperbolic $n$-monopole solution obtained from the generalized Gibbons-Hawking ansatz, reduces to the Eguchi-Hanson metric in a particular limit. These results, though known from an algebraic geometry point of view, are useful while dealing with rotational killing symmetry of self-dual metrics in general theory of relativity as well as in the context of duality symmetry in string theory. 
  In this thesis we will discuss various aspects of noncommutative geometry and compactified Little-String theories. First we will give an introduction to the use of noncommutative geometry in string theory. Thereafter we will present a proof of the connection between D-brane dynamics and noncommutative geometry. Then we will explain the concept of instantons in noncommutative gauge theories. The last chapters shift the focus to Little-String- and $(2,0)$-theories. We study compactifications of these theories on tori with twists. First we study the case of two coinciding branes in detail. Afterwards we study the case of an arbitrary number of coinciding branes. The main result here is that the moduli spaces of vacua for the twisted compactifications are equal to moduli spaces of instantons on a noncommutative torus. A special case of this is that a large class of gauge theories with $\SUSY{2}$ supersymmetry in D=4 or $\SUSY{4}$ in D=3 has moduli spaces which are moduli spaces of instantons on noncommutative tori. 
  We consider the integrable XXZ model with the special open boundary conditions. We perform Quantum Group reduction of this model in roots of unity and use it for the definition Minimal Models of Interable lattice theory. It is shown that after this Quantum Group reduction Sklyanin's transfer-matrices satisfy the closed system of the truncated functional relations. We solve these equations for the simplest case. 
  Canonical quantization of abelian BF-type topological field theory coupled to extended sources on generic d-dimensional manifolds and with curved line bundles is studied. Sheaf cohomology is used to construct the appropriate topological extension of the action and the topological flux quantization conditions, in terms of the Cech cohomology of the underlying spatial manifold, as required for topological invariance of the quantum field theory. The wavefunctions are found in the Hamiltonian formalism and are shown to carry multi-dimensional representations of various topological groups of the space. Expressions for generalized linking numbers in any dimension are thereby derived. In particular, new global aspects of motion group presentations are obtained in any dimension. Applications to quantum exchange statistics of objects in various dimensionalities are also discussed. 
  It has been proposed that the noncommutative geometry of the "fuzzy" 2-sphere provides a nonperturbative regularization of scalar field theories. This generalizes to compact Kaehler manifolds where simple field theories are regularized by the geometric quantization of the manifold.   In order to permit actual calculations and the comparison with other regularizations, I describe the perturbation theory of these regularized models and propose an approximation technique for evaluation of the Feynman diagrams. I present example calculations of the simplest diagrams for the $\phi^4$ model on the 2-sphere, a 2-sphere times a 2-sphere, and 2-dimensional complex projective space.   This regularization fails for noncompact spaces; I give a brief dimensional analysis argument as to why this is so. I also discuss the relevance of the topology of Feynman diagrams to their ultra-violet and infra-red divergence behavior in this model. 
  We construct a tensionless string model in a four-dimensional space-time ${\bf R}^{2,2}$ with the (2,2) signature and solve the string equations. We find that the signature change radically changes the structure of the holonomy group of the null worldsheet. As a result the introduction of the worldsheet Dirac operator invariant under the holonomy group becomes possible. We show that the such possibility is absent in the tensionless string model in the (3+1)-dimensional Minkowski space. 
  We compute the one-loop holomorphic effective action of the massless Cartan sector of N=2 SYM theory in the Coulomb branch, taking into account the contributions both from the charged hypermultiplets and off-diagonal components of the gauge superfield. We use the manifestly supersymmetric harmonic superfields diagram techniques adapted to N=2 supersymmetry with the central charges induced by Cartan generators. The (anti)holomorphic part proves to be proportional to the central charges and it has the generic form of Seiberg's action obtained by integrating U(1) R-anomaly. It vanishes for N=4 SYM theory, i.e. the coupled system of N=2 gauge superfield and hypermultiplet in the adjoint representation. 
  Within the framework of the Gaussian wave-functional approach, we investigate the influences of quantum and finite-temperature effects on the Z_2-symmetry(\phi \to -\phi) of the (1+1)-dimensional massive sine-Gordon field theory. It is explicitly demonstrated that by quantum effects the Z_2-symmetry can be restored in one region of the parameter space and dynamically spontaneously broken in another region. Moreover, a finite-temperature effect can further restore the Z_2-symmetry only. 
  We construct a simple physical model of a particle moving on the infinite noncommutative 2-plane. The model consists of a pair of opposite charges moving in a strong magnetic field. In addition, the charges are connected by a spring. In the limit of large magnetic field, the charges are frozen into the lowest Landau level. Interaction of such particles include Moyal bracket phases characteristics of field theory on noncommutative space. The simple system arises in lightcone quantization of open strings attached to D-branes in a.s. tensor background. We use the model to work out the general form of lightcone vertices from string splitting. We then consider Feynman diagrams in uncompactified NC YM theories and find that for all planar diagrams the comm. and noncomm. theories are the same. This means large N theories are equivalent in the 't Hooft limit. Non planar diagrams convergence is improved. 
  We construct the transversal and longitudinal 4-vector potentials and strengths F^{\mu\nu} in the momentum representation on using the second-type 4-spinors recently proposed by Ahluwalia. Questions of relevance of this kind of theories to the correct photon statistics are briefly discussed. 
  Using the new regularization and renormalization scheme recently proposed by Yang and used by Ni et al, we analyse the sine-Gordon and sinh-Gordon models within the framework of Gaussian effective potential in D+1 dimensions. Our analysis suffers no divergence and so does not suffer from the manipulational obscurities in the conventional analysis of divergent integrals. Our main conclusions agree exactly with those of Ingermanson for D=1,2 but disagree for D=3: the D=3 sinh(sine)-Gordon model is non-trivial. Furthermore, our analysis shows that for D=1,2, the running coupling constant (RCC)has poles for sine-Gordon model($\gamma^2<0$) and the sinh-Gordon model ($\gamma^2>0$) has a possible critical point $\gamma^2_c$ while for D=3, the RCC has poles for both $\gamma^2>0$ and $\gamma^2<0$. 
  We evaluate exactly both the non-relativistic and relativistic fermion determinant in 2+1 dimensions in a constant background field at finite temperature. The effect of finite chemical potential is also considered. In both cases, the systems are decoupled into an infinite number of 1+1 fermions by Fourier transformation in the $\beta$-variable. The total effective actions demonstrate non-extensiveness in the $\beta$ dimension. 
  The spectrum of open strings on various non-BPS D-brane configurations in type II string theory on a K3 orbifold is analysed. At a generic point in the corresponding moduli space the spectrum of open strings does not have any degeneracy between bosonic and fermionic states. However, there exist special values for these moduli for which many non-BPS D-brane configurations have an exactly bose-fermi degenerate open string spectrum at all mass levels. In this case the closed string exchange interaction between a pair of such D-brane configurations vanishes at all distances. 
  We present a simple derivation of an upper bound on the average size of the true vacuum bubbles at the end of inflation, in models of extended inflation type. The derivation uses the inequality that the total energy inside a given volume must be less than its linear dimensions. The above bound is the same as that obtained earlier, by applying the holographic principle according to Fischler-Susskind prescription. Such a bound leads to a lower bound on the denisty fluctuations. 
  Non-perturbative renormalizability, or non-triviality, of the gauged Nambu-Jona-Lasinio (NJL) model in four dimensions is examined by using non-perturbative (exact) renormalization group in large $N_c$ limit. When running of the gauge coupling is asymptotically free and slow enough, the two dimensional renormalized trajectory (subspace) spanned by the four fermi coupling and the gauge coupling is found to exist, which implies renormalizability of the gauged NJL model. In the case of fixed gauge coupling, renormalizability of the model turns out to be guaranteed by the line of the UV fixed points. We discuss also non-triviality of the gauged NJL model extended to include higher dimensional operators and correspondence with the gauge-Higgs-Yukawa system. 
  We study a novel five-dimensional, {\it N}=5 supergravity in the context of Lie superalgebra SU(5/2). The possible successive superalgebraic truncations from {\it N}=5 theory to the lower supersymmetric {\it N}=4,3,2, and 1 supergravity theories are systematically analyzed as a sub-superalgebraic chain of SU(5/2)$\supset$ SU(4/2) $\supset$ SU(3/2) $\supset$ SU(2/2) $\supset$ SU(1/2) by using the Kac-Dynkin weight techniques. 
  We study the effect of a non-vanishing flux for the NS-NS antisymmetric tensor in open-string orbifolds. As in toroidal models, the total dimension of the Chan-Paton gauge group is reduced proportionally to the rank of $B_{ab}$, both on D9 and on D5-branes, while the Moebius amplitude involves some signs that, in the $Z_2$ case, allow one to connect continuously ${\rm U} (n)$ groups to $Sp(n) \otimes Sp(n)$ groups on each set of D-branes. In this case, non-universal couplings between twisted scalars and gauge vectors arise, as demanded by the generalised Green-Schwarz mechanism. We also comment on the role of the NS-NS antisymmetric tensor in a recently proposed type scenario, where supersymmetry is broken on the D-branes, while it is preserved in the bulk. 
  We study the role of higher derivative terms (Riemann curvature squared ones) in thermodynamics of SCFTs via AdS/CFT correspondence. Using IIB string effective action (d5 AdS gravity) with such HD terms deduced from heterotic string via duality we calculate strong coupling limit of ${\cal N}=2$ SCFT free energy with the account of next to leading term in large $N$ expansion. It is compared with perturbative result following from boundary QFT. Considering modification of such action where HD terms form Weyl squared tensor we found (strong coupling limit) free energy in such theory. It is interesting that leading and next to leading term of large $N$ expanded free energy may differ only by factor 3/4 if compare with perturbative result. Considering HD gravity as bosonic sector of some (compactified) HD supergravity we suggest new version of AdS/CFT conjecture and successfully test it on the level of free energies for ${\cal N}=2,4$ SCFTs. 
  We discuss general multidimensional axion-dilatonic AdS gravity which may correspond to bosonic sector of Gibbons-Green-Perry (compactified) IIB supergravity with RR-scalar (axion). Using AdS/CFT correspondence the 4d conformal anomaly on axion-dilaton-gravitational background is found from SG side. It is shown that for IIB SG with axion such conformal anomaly coincides with the one obtained from QFT calculation in ${\cal N}=4$ super Yang-Mills theory conformally coupled with ${\cal N}=4$ conformal SG. Brief discussion on possibility to apply these results for gauged SGs is also presented. 
  We study the duality between M-theory compactified on Calabi-Yau fourfolds and the heterotic string compactified on Calabi-Yau threefolds times a circle. Our analysis is based on a comparison of the low energy effective actions in three dimensions. 
  Global canonical transformations to free chiral fields are constructed for DG models minimally coupled to scalar fields. The boundary terms for such canonical transformations are shown to vanish in asymptotically static coordinates if there is no scalar field. 
  At high temperatures the A_0 component of the Yang--Mills field plays the role of the Higgs field, and the 1-loop potential V(A_0) plays the role of the Higgs potential. We find a new stable vortex solution of the Abrikosov-Nielsen-Olesen type, and discuss its properties and possible implications. 
  Bousso has conjectured that in any spacetime satisfying Einstein's equation and satisfying the dominant energy condition, the "entropy flux" S through any null hypersurface L generated by geodesics with non-positive expansion starting from some spacelike 2 surface of area A must satisfy S<=A/4. This conjecture reformulates earlier conjectured entropy bounds of Bekenstein and also of Fischler and Susskind, and can be interpreted as a statement of the so-called holographic principle. We show that Bousso's entropy bound can be derived from either of two sets of hypotheses. The first set of hypotheses is (i) associated with each null surface L in spacetime there is an entropy flux 4-vector s^a_L whose integral over L is the entropy flux through L, and (ii) along each null geodesic generator of L, we have $|s^a_L k_a| \le \pi (\lambda_\infty - \lambda) T_{ab} k^a k^b$, where $T_{ab}$ is the stress-energy tensor, $\lambda$ is an affine parameter, $k^a = (d / d\lambda)^a$, and $\lambda_\infty$ is the value of affine parameter at the endpoint of the geodesic. The second (purely local) set of hypotheses is (i) there exists an absolute entropy flux 4-vector s^a such that the entropy flux through any null surface L is the integral of s^a over L, and (ii) this entropy flux 4-vector obeys the pointwise inequalities $(s_a k^a)^2 \le T_{ab} k^a k^b / (16 \pi)$ and $|k^a k^b \nabla_a s_b| \le \pi T_{ab} k^a k^b /4$ for any null vector k^a. Under the first set of hypotheses, we also show that a stronger entropy bound can be derived, which directly implies the generalized second law of thermodynamics. 
  We consider the construction of a general tree level amplitude for the interactions between dynamical D-branes where the configurations have non-zero odd spin structure. Using Riemann Theta Identities we map the conditions for the preservation of some supersymmetry to a set of integer matrices satisfying a simple but non-trivial equation. We also show how the regularization of the RR zero modes plays an important role in determining which configurations are permitted. 
  We derive the rules to construct type IIB compact orientifolds in six and four dimensions including D-branes and anti-D-branes. Even though the models are non-supersymmetric due to the presence of the anti-D-branes, we show that it is easy to construct large classes of models free of tachyons. Brane-antibrane annihilation can be prevented for instance by considering models with branes and antibranes stuck at different fixed points in the compact space. We construct several anomaly-free and tachyon-free six-dimensional orientifolds containing D9-branes and anti-D5-branes. This setup allows to construct four-dimensional chiral models with supersymmetry unbroken in the bulk and in some D-brane sectors, whereas supersymmetry is broken (at the string scale) in some `hidden' anti-D-brane sector. We present several explicit models of this kind. We also comment on the role of the non-cancelled attractive brane-antibrane forces and the non-vanishing cosmological constant, as providing interesting dynamics for the geometric moduli and the dilaton, which may contribute to their stabilization. 
  We here consider a generalization of the Klein-Gordon scalar wave equation which involves a single arbitrary function. The quantization may be viewed as allowing $\hbar$ to be a function of the momentum or wave vector rather than a constant. The generalized theory is most easily viewed in the wave vector space analog of the Lagrangian. We need no reference to spacetime. In the generalized theory the de Broglie relation between wave vector and momentum is generalized, as are the canonical commutation relations and the uncertainty principle. The generalized uncertainty principle obtained is the same as has been derived from string theory, or by a general consideration of gravitational effects during the quantum measurement process. The propagator of the scalar field is also generalized, and an illustrative example is given in which it factors into the usual propagator times a "propagator form factor." 
  We show that supersymmetry Ward identities contain an anomalous term which takes the form of a surface term in Hilbert space. In the one-instanton sector the anomalous term is the integral of a total rho-derivative where rho is the instanton's size. There are cases where the anomalous term is non-zero, and cannot be modified by subtractions. This constitutes a supersymmetry anomaly. The derivation is based on Feynman rules suitable for any non-perturbative sector of a weakly-coupled, renormalizable gauge theory. 
  It is shown that supersymmetry is spontaneously broken in certain three-dimensional supersymmetric gauge theories, by using the s-rule in their string theory realization as brane configurations. In particular, supersymmetry is broken in N=3 supersymmetric Yang-Mills-Chern-Simons theory with gauge group SU(n) and CS coefficient k, as well as in its N=2 and N=1 deformations, when n>|k|. In addition, supersymmetry is broken in the N=1 mass deformation of N=2 supersymmetric Yang-Mills theory with gauge group SU(n) and one matter multiplet when n>1. In the latter case the breaking is induced by an instanton-generated repulsive potential. 
  In a nontrivial background geometry with extra dimensions, gravitational effects will depend on the shape of the Kaluza-Klein excitations of the graviton. We investigate a consistent scenario of this type with two positive tension three-branes separated in a five-dimensional Anti-de Sitter geometry. The graviton is localized on the ``Planck'' brane, while a gapless continuum of additional gravity eigenmodes probe the {\it infinitely} large fifth dimension. Despite the background five-dimensional geometry, an observer confined to either brane sees gravity as essentially four-dimensional up to a position-dependent strong coupling scale, no matter where the brane is located. We apply this scenario to generate the TeV scale as a hierarchically suppressed mass scale. Arbitrarily light gravitational modes appear in this scenario, but with suppressed couplings. Real emission of these modes is observable at future colliders; the effects are similar to those produced by {\it six} large toroidal dimensions. 
  We investigate Yang-Mills theory on a spatial torus at finite temperature in the presence of discrete electric and magnetic fluxes using the AdS/CFT correspondence. We calculate the leading dependence of the partition function on the fluxes using the dual supergravity theory and comment upon the interpretation of these fluxes as discrete quantum hair for black holes in AdS spacetime. 
  We show that the calculation of L-loop Feynman integrals in D dimensions can be reduced to a series of matrix multiplications in D times L dimensions. This gives rise to a new type of expansions for the critical exponents in three dimensions in which all coefficients can be calculated exactly. 
  A mechanism for the dynamical mass generation of a non-Abelian gauge field which is based on taking into account the contributions of the gauge field vacuum configurations into the formation of the physical vacuum is considered. For a model of the physical vacuum as a superposition of Abelian configurations the gauge field propagator is calculated in the leading order of $1/d$-expansion ($d$ is a space-time dimension). One-particle spectrum of the model corresponds to the gauge sector of SU(2) Georgi-Glashow model. 
  A relativistic quantized particle model avoids difficulties through (1) a Hamiltonian undecomposable into H=H(0)+H(I), (2) a separation of the evolution parameter s from dynamics, (3) "leptons" and "hadrons" composed of "quarks," and (4) the absence of background reference frames. The stringlike Lagrangian is L=-{-[F(Q)]2 [dQ/ds]2+[FdQ/ds]2}1/2. Q(s) defines quark positions; the form of F(Q) determines the interaction. The "strong" Lagrangian is symmetric under quark exchange. Transformation to new quark coordinates "hides" the symmetry. A variational principle for the parametrically invariant action in terms of these coordinates supplies natural boundary conditions (n.b.c.). The resulting symmetry breaking yields "lepton" and "hadron" quarks that behave differently. However, both become "strings" asymptotically. The n.b.c. produce composite mass-shell constraints and suppress time-oscillations. "Strong" scattering between composites is calculated. The "leptons" behave as free particles. A second choice of F(Q) produces unified "electroweak" interactions. First-order perturbation theory is applied to "lepton-lepton" scattering. Unperturbed states are asymptotic solutions from separate "strong interaction" clusters. Transforms between position and momentum representations, determined by the n.b.c., eliminate advanced potentials. Scattering amplitudes obey Feynman rules. 
  The chiral symmetry breaking in the 4-dimensional QED with the chirally invariant four-fermion interaction is discussed by using a novel path integral expression in terms of the field-strength tensor. In the local potential approximation, we find that the chiral symmetry is spontaneously broken for any nonzero gauge and four-fermion couplings on the tree level of an auxiliary field $\sigma$. The present approach allows us to easily include higher orders of the gauge coupling so that the effective potential up to the sixth order is obtained. 
  We develop a graphical representation of polynomial invariants of unitary gauge groups, and use it to find the algebraic curve corresponding to a hyperkahler quotient of a linear space. We apply this method to four dimensional ALE spaces, and for the A_k, D_k, and E_6 cases, derive the explicit relation between the deformations of the curves away from the orbifold limit and the Fayet-Iliopoulos parameters in the corresponding quotient construction. We work out the orbifold limit of E_7, E_8, and some higher dimensional examples. 
  We present the general method to introduce the generalized Chern-Simons form and the descent equation which contain the scalar field in addition to the gauge fields. It is based on the technique in a noncommutative differential geometry (NCG) which extends the $N$-dimensional Minkowski space $M_N$ to the discrete space such as $M_N\times Z_2$ with two point space $Z_2$. However, the resultant equations do not depend on NCG but are justified by the algebraic rules in the ordinary differential geometry. 
  The partial breaking of supersymmetry in anti-de Sitter (AdS) space can be accomplished using two of four dual representations for the massive OSp(1,4) spin-3/2 multiplet. The representations can be ``unHiggsed,'' which gives rise to a set of dual N=2 supergravities and supersymmetry algebras. 
  We give a self-contained discussion of recent progress in computing the non-perturbative effects of small non-holomorphic soft supersymmetry breaking, including a simple new derivation of these results based on an anomaly-free gauged U(1)_R background. We apply these results to N = 1 theories with deformed moduli spaces and conformal fixed points. In an SU(2) theory with a deformed moduli space, we completely determine the vacuum expectation values and induced soft masses. We then consider the most general soft breaking of supersymmetry in N = 2 SU(2) super-Yang-Mills theory. An N = 2 superfield spurion analysis is used to give an elementary derivation of the relation between the modulus and the prepotential in the effective theory. This analysis also allows us to determine the non-perturbative effects of all soft terms except a non-holomorphic scalar mass, away from the monopole points. We then use an N = 1 spurion analysis to determine the effects of the most general soft breaking, and also analyze the monopole points. We show that naive dimensional analysis works perfectly. Also, a soft mass for the scalar in this theory forces the theory into a free Coulomb phase. 
  In this note we display an observation about the geometrical properties of the gauge group manifold of the standard electroweak theory, and the values of the gauge coupling constants. Heuristically obtained stucture relates the value 137.036... to the low energy U(1) coupling. The constuction resembles techniques recently used within the framework of the string and D-brane theory. We consider this as an indication that the string theory has direct relevance for the phenomenological QFT and particle physics. 
  A relativistic quantum field theory is presented for finite density problems based on the principle of locality. It is found that, in addition to the conventional ones, a local approach to the relativistic quantum field theories at both zero and finite density consistent with the violation of Bell like inequalities should contain, and provide solutions to at least three additional problems, namely, 1) the statistical gauge invariance 2) the dark components of the local observables and 3) the fermion statistical blocking effects, base upon an asymptotic non-thermo ensemble. An application to models are presented to show the importance of the discussions. 
  We study the properties of M and F theory compactifications to three and four dimensions with background fluxes. We provide a simple construction of supersymmetric vacua, including some with orientifold descriptions. These vacua, which have warp factors, typically have fewer moduli than conventional Calabi-Yau compactifications. The mechanism for anomaly cancellation in the orientifold models involves background RR and NS fluxes. We consider in detail an orientifold of $K3\times T^2$ with background fluxes. After a combination of T and S-dualities, this type IIB orientifold is mapped to a compactification of the SO(32) heterotic string on a non-Kahler space with torsion. 
  We couple n copies of N=(2,0) scalar multiplets to a gauged N=(2,0) supergravity in 2+1 dimensions which admits AdS_3 as a vacuum. The scalar fields are charged under the gauged R-symmetry group U(1) and parametrize certain Kahler manifolds with compact or non-compact isometries. The radii of these manifolds are quantized in the compact case, but arbitrary otherwise. In the compact case, we find half-supersymmetry preserving and asymptotically Minkowskian black string solutions. For a particular value of the scalar manifold radius, the solution coincides with that of Horne and Horowitz found in the context of a string theory in 2+1 dimensions. In the non-compact case, we find half-supersymmetry preserving and asymptotically AdS_3 string solutions which have naked singularities. We also obtain two distinct AdS_3 supergravities coupled to n copies of N=(1,0) scalar multiplets either by the truncation of the (2,0) model or by a direct construction. 
  We present a way of tiling the plane with a regular hexagonal network of defects. The network is stable and follows in consequence of the three-junctions that appear in a model of two real scalar fields that presents $Z_3$ symmetry. The $Z_3$ symmetry is effective in both the vacuum and defect sectors, and no supersymmetry is required to build the network. 
  Following Sen's discovery of various stable non-BPS D-branes, K-theory has been shown to be the appropriate mathematical framework for classifying conserved D-brane charges. The classification accounts for known D-branes and predicts some new ones including a D8-brane in type I superstring theory. After briefly reviewing these developments, we discuss certain issues pertaining to the D8-brane, which is unstable. 
  We obtain the BTZ black hole (AdS$_3 \times$S$^3$) as a non-dilatonic solution from type 0B string theory. Analyzing the perturbation around this black hole background, we show that the tachyon is not a propagating mode. 
  We consider a recent generalisation by Bousso of an earlier holography proposal by Fischler and Susskind. We demonstrate that in general inhomogeneous universes such a proposal would involve extremely complicated - possibly fractal - light sheets. Furthermore, in general such a light sheet cannot be known a priori on the basis of theory and moreover, the evolution of the universe makes it clear that in general such bounds cannot remain invariant under time reversal and will change with epoch.   We propose a modified version of this proposal in which the light sheets end on the boundary of the past, and hence avoid contact with the caustics. In this way the resulting light sheets and projections can be made much simpler. We discuss the question of operational definability of these sheets within the context of both proposals and conclude that in both cases the theoretical existence of such sheets must be clearly distinguished from their complexity and the difficulty of their construction in practice. This puts into perspective the likely practical difficulties one would face in applying the holographic principle to the real cosmos. These issues may also be of relevance in debates regarding the applications of the holographic principle to other settings such as string theory. 
  We construct the effective action of certain exotic branes in the Type II theories which are not predicted by their spacetime supersymmetry algebras. We analyze in detail the case of the NS-7B brane, S-dual to the D7-brane, and connected by T-duality to other exotic branes in Type IIA: the KK-6A brane and the KK-8A brane (obtained by reduction of the M-theory Kaluza-Klein monopole and M9-brane, respectively). The NS-7B brane carries charge with respect to the S-dual of the RR 8-form, which we identify as a non-local combination of the electric-magnetic duals of the axion and the dilaton. The study of its effective action agrees with previous results in the literature showing that it transforms as an SL(2,Z) triplet together with the D7-brane. We discuss why this brane is not predicted by the Type IIB spacetime supersymmetry algebra. In particular we show that the modular transformation relating the D7 and NS-7B brane solutions can be undone by a simple coordinate transformation in the two dimensional transverse space, equivalent to choosing a different region to parametrize the SL(2,Z) moduli space. We discuss a similar relation between the D6 and KK-6A branes and the D8 and KK-8A branes. 
  I review some aspects of BPS magnetic monopoles and of electric-magnetic duality in theories with arbitrary gauge groups. When the symmetry is maximally broken to a U(1)^r subgroup, all magnetically charged configurations can be understood in terms of r species of massive fundamental monopoles. When the unbroken group has a non-Abelian factor, some of these fundamental monopoles become massless and can be viewed as the duals to the massless gauge bosons. Rather than appearing as distinct solitons, these massless monopoles are manifested as clouds of non-Abelian field surrounding one or more massive monopoles. I describe in detail some examples of solutions with such clouds. 
  We study the question of existence and the number of normalized vacuum states in N = 4 super-Yang-Mills quantum mechanics for any gauge group. The mass deformation method is the simplest and clearest one. It allowed us to calculate the number of normalized vacuum states for all gauge groups. For all unitary groups, #(vac) = 1, but for the symplectic groups [starting from Sp(6) ], for the orthogonal groups [starting from SO(8)] and for all the exceptional groups, it is greater than one. We also discuss at length the functional integral method. We calculate the ``deficit term'' for some non-unitary groups and predict the value of the integral giving the ``principal contribution''. The issues like the Born-Oppenheimer procedure to derive the effective theory and the manifestation of the localized vacua for the asymptotic effective wave functions are also discussed. 
  Magnetic monopole solutions naturally arise in the context of spontaneously broken gauge theories. When the unbroken symmetry includes a non-Abelian subgroup, investigation of the low-energy monopole dynamics by means of the moduli space approximation reveals degrees of freedom that can be attributed to massless monopoles. These do not correspond to distinct solitons, but instead are manifested as a cloud of non-Abelian field surrounding one or more massive monopoles. In these talks I explain how one is led to these solutions and then describe them in some detail. 
  From the algebraic treatment of the quasi-solvable systems, and a q-deformation of the associated $su(2)$ algebra, we obtain exact solutions for the q-deformed Schrodinger equation with a 3-dimensional q-deformed harmonic oscillator potential. 
  Scalar field theory on a lattice falling freely into a 1+1 dimensional black hole is studied using both WKB and numerical approaches. The outgoing modes are shown to arise from incoming modes by a process analogous to a Bloch oscillation, with an admixture of negative frequency modes corresponding to the Hawking radiation. Numerical calculations show that the Hawking effect is reproduced to within 0.5% on a lattice whose proper spacing where the wavepacket turns around at the horizon is $\sim0.08$ in units where the surface gravity is 1. 
  Symmetric vacua of heterotic M-theory, characterized by vanishing cohomology classes of individual sources in the three-form Bianchi identity, are analyzed on smooth Calabi-Yau three-folds. We show that such vacua do not exist for elliptically fibered Calabi-Yau spaces. However, explicit examples are found for Calabi-Yau three-folds arising as intersections in both unweighted and weighted projective space. We show that such symmetric vacua can be combined with attractive phenomenological features such as three generations of quarks and leptons. Properties of the low energy effective actions associated with symmetric vacua are discussed. In particular, the gauge kinetic functions receive no perturbative threshold corrections, there are no corrections to the matter field Kahler metric and the associated five-dimensional effective theory admits flat space as its vacuum. 
  In this paper we sum over the spherical modes appearing in the expression for the Casimir energy of a conducting sphere and of a dielectric ball (assuming the same speed of light inside and outside), before doing the frequency integration. We derive closed integral expressions that allow the calculations to be done to all orders, without the use of regularization procedures. The technique of mode summation using a contour integral is critically examined. 
  A complete global analysis of spatially-flat, four-dimensional cosmologies derived from the type IIA string and M-theory effective actions is presented. A non--trivial Ramond-Ramond sector is included. The governing equations are written as a dynamical system. Asymptotically, the form fields are dynamically negligible, but play a crucial role in determining the possible intermediate behaviour of the solutions (i.e. the nature of the equilibrium points). The only past-attracting solution (source in the system) may be interpreted in the eleven-dimensional setting in terms of flat space. This source is unstable to the introduction of spatial curvature. 
  We calculate the chiral anomaly in the neighbourhood of the fixed point space M_h which is constructed by the group action of a discrete symmetry h on a compact manifold M. The Feynman diagrams approach for the corresponding supersymmetric quantum mechanical system with twisted boundary conditions is used. The result we derive in this way agrees with the generalization of the ordinary index theorem (the G-index theorem) on the spin complex. 
  We construct a new model with exponential mass hierarchy by starting with the Einstein-Hilbert action with the cosmological constant in five dimensions plus an action describing many domain walls in four dimensions. The model includes many hidden sectors and one visible sector, and each four-dimensional domain wall, that is, 3-brane, interacts with one another through only a gravitational interaction and realizes many universe cosmology inspired by D-brane perspective. It is shown that in the present model only even numbers of domain walls are allowed to locate in five dimensional space-time and the validity of Randall-Sundrum scenario, which explains mass hierarchy between the Planck mass and the electro-weak scale in our world, depends on a relative relation between our world and hidden worlds. 
  We review and elaborate on some aspects of Born-Infeld action and its supersymmetric generalizations in connection with string theory. Contents: BI action from string theory; some properties of bosonic D=4 BI action; N=1 and N=2 supersymmetric BI actions with manifest linear D=4 supersymmetry; four-derivative terms in N=4 supersymmetric BI action; BI actions with `deformed' supersymmetry from D-brane actions; non-abelian generalization of BI action; derivative corrections to BI action in open superstring theory. 
  This is a short note on the relation of the Matrix model with the non-commutative geometry of the 11-dimensional supermembrane. We put forward the idea that M-theory is described by the 't Hooft topological expansion of the Matrix model in the large N-limit where all topologies of membranes appear. This expansion can faithfully be represented by the Moyal Yang-Mills theory of membranes. We discuss this conjecture in the case of finite N, where the non-commutative geometry of the membrane is given be the finite quantum mechanics. The use of the finite dimensional representations of the Heisenberg group reveals the cellular structure of a toroidal supermembrane on which the Matrix model appears as a non-commutatutive Yang-Mills theory. The Moyal star product on the space of functions in the case of rational values of the Planck constant \hbar represents exactly this cellular structure. We also discuss the integrability of the instanton sector as well as the topological charge and the corresponding Bogomol'nyi bound. 
  We consider a free (2 k)-form gauge-field on a Euclidean (4 k + 2)-manifold. The parameters needed to specify the action and the gauge-invariant observables take their values in spaces with natural complex structures. We show that the correlation functions can be written as a finite sum of terms, each of which is a product of a holomorphic and an anti-holomorphic factor. The holomorphic factors are naturally interpreted as correlation functions for a chiral (2 k)-form, i.e. a (2 k)-form with a self-dual (2 k + 1)-form field strength, after Wick rotation to a Minkowski signature. 
  It is found that the 2-index potential in nonabelian theories does not behave geometrically as a connection but that, considered as an element of the second de Rham cohomology group twisted by a flat connection, it fits well with all the properties assigned to it in various physical contexts. We also prove some results on the Euler characteristic of the twisted de Rham complex. Finally, provided that some conditions are satisfied, we propose a non-Abelian generalisation of S-duality. 
  Black hole solutions that are asymptotic to $ AdS_5 \times S^5$ or $ AdS_4 \times S^7$ can rotate in two different ways. If the internal sphere rotates then one can obtain a Reissner-Nordstrom-AdS black hole. If the asymptotically AdS space rotates then one can obtain a Kerr-AdS hole. One might expect superradiant scattering to be possible in either of these cases. Superradiant modes reflected off the potential barrier outside the hole would be re-amplified at the horizon, and a classical instability would result. We point out that the existence of a Killing vector field timelike everywhere outside the horizon prevents this from occurring for black holes with negative action. Such black holes are also thermodynamically stable in the grand canonical ensemble. The CFT duals of these black holes correspond to a theory in an Einstein universe with a chemical potential and a theory in a rotating Einstein universe. We study these CFTs in the zero coupling limit. In the first case, Bose-Einstein condensation occurs on the boundary at a critical value of the chemical potential. However the supergravity calculation demonstrates that this is not to be expected at strong coupling. In the second case, we investigate the limit in which the angular velocity of the Einstein universe approaches the speed of light at finite temperature. This is a new limit in which to compare the CFT at strong and weak coupling. We find that the free CFT partition function and supergravity action have the same type of divergence but the usual factor of 4/3 is modified at finite temperature. 
  Utilizing AdS/CFT correspondence in M-theory, an example of interacting d=3 conformal field theories and renormalization group flow between them is presented. Near-horizon geometry of N coincident M2-branes located on a conical singularity on eight-dimensional hyperk\"ahler manifold or manifold with Spin(7) holonomy is, in large-N limit, AdS4*X7, where X7 is seven-sphere with squashing. Deformation from round $\S_7$ to squashed one is known to lead to spontaneous breaking of N=8 local supersymmetry in gauged AdS4 supergravity to N=1, 0. Via AdS/CFT correspondence, it is interpreted as renormalization group flow from SO(5)*SO(3) symmetric UV fixed point to SO(8) symmetric IR fixed point. Evidences for the interpretation are found both from supergravity scalar potential and existence of interpolating static domain-wall thereof, and from conformal dimensions of relevant chiral primary operator at each fixed point. 
  New equations governing the scale transformation behaviors of a QFT with underlying structures are derived. These equations, with their several equivalent versions, can yield some new and significant insights and results that are difficult to see in the conventional renormalization programs. Among the several equivalent versions, one is similar to the usual Callan-Symanzik equation and renormalization group equation but with different meanings. From another version, the anomalous equation of energy-momentum tensor trace can be easily obtained. It can be shown that with the new versions one could partially fix the scheme dependence and hence some renormalization schemes like momentum space subtractions are questionable. The asymptotic freedom for QCD is easily reproduced at one-loop level in our new version CSE. Finally the decoupling theorem a la Appelquist and Carazzone are discussed within our strategy. 
  We newly apply the improved Batalin-Fradkin-Tyutin(BFT) Hamiltonian method to the O(3) nonlinear sigma model, and directly obtain the compact form of nontrivial first class Hamiltonian by introducing the BFT physical fields. Furthermore, following the BFV formalism, we derive the BRST invariant gauge fixed Lagrangian through the standard path-integral procedure. Finally, by introducing collective coordinates, we also study a semi-classical quantization of soliton background to conclude that the spectrum of zero modes are unchanged through the BFT procedure. 
  We derive a new form of loop equations for light-like Wilson loops. In bosonic theories those loop equations close only for straight light-like Wilson lines. In the case of N=1 in ten dimensions they close for any light-like Wilson loop. Upon dimensional reduction to N=4 SYM in four dimensions, these loops become exactly the chiral loops which can be evaluated semiclassically, in the strong coupling limit, by a minimal surface in anti de-Sitter space. We show that the AdS calculation satisfies those loop equations. We also find a new fermionic loop equation derived from the gauge theory fermionic equation of motion. 
  Light cone gauge manifestly supersymmetric formulations of type IIB 10-dimensional supergravity in $AdS_5 \times S^5$ background and related boundary conformal field theory representations are developed. A precise correspondence between the bulk fields of IIB supergravity and the boundary operators is established. The formulations are given entirely in terms of light cone scalar superfields, allowing us to treat all component fields on an equal footing. 
  It is shown that a Moyal deformation quantization of the SO(4k) Generalized Yang-Mills (GYM) theory action in D=4k dimensions, for spacetime independent field configurations, in the $\hbar \to 0$ limit, yields the Dirac-Nambu-Goto p-brane actions (obtained from the conformally invariant Dolan-Tchrakian p-brane actions after elimination of the auxiliary world volume metrics), in the orthonormal gauge, for p+1=4k world volumes embedded in a D=4k target spacetime background. The gauge fields/target spacetime coordinates correspondence is required but no large N limit is necessary. The equivalence between Moyal SDYM and Self Dual p-branes is proposed without choosing the orthonormal gauge. 
  The four-dimensional Minkowski space-time is considered as a three-brane embedded in five dimensions, using solutions of five-dimensional supergravity. These backgrounds have a string theoretical interpretation in terms of D3-brane distributions. By studying linear fluctuations of the graviton we find a zero-mode representing the massless graviton in four-dimensional space-time. The novelty of our models is that the graviton spectrum has a genuine mass gap (independent of the position of the world-brane) above the zero-mode or it is discrete. Hence, an effective four-dimensional theory on a brane that includes the massless graviton mode is well defined. The gravitational force between point particles deviates from the Newton law by Yukawa-type corrections, which we compute explicitly. We show that the parameters of our solutions can be chosen such that these corrections lie within experimental bounds. 
  Coherent states have three main properties: coherence, overcompleteness and intrinsic geometrization. These unique properties play fundamental roles in field theory, especially, in the description of classical domains and quantum fluctuations of physical fields, in the calculations of physical processes involving infinite number of virtual particles, in the derivation of functional integrals and various effective field theories, also in the determination of long-range orders and collective excitations, and finally in the exploration of origins of topologically nontrivial gauge fields and associated gauge degrees of freedom. 
  We define a semi-Hopf algebra which is more general than a Hopf algebra. Then we construct the supersymmetry algebra via the adjoint action on this semi-Hopf algebra. As a result we have a supersymmetry theory with quantum gauge group, i.e., quantised enveloping algebra of a simple Lie algebra. For the example, we construct the Lagrangian N=1 and N=2 supersymmetry. 
  The freedom one has in constructing locally gauge invariant charged fields in gauge theories is analyzed in full detail and exploited to construct, in QED, an electron field whose two-point function W(p), up to the fourth order in the coupling constant, is normalized with on-shell normalization conditions and is, nonetheless, infra-red finite; as a consequence the radiative corrections vanish on the mass shell $p^2=\mu^2$ and the free field singularity is dominant, although, in contrast to quantum field theories with mass gap, the eigenvalue $\mu^2$ of the mass operator is not isolated. The same construction, carried out for the quark in QCD, is not sufficient for cancellation of infra-red divergences to take place in the fourth order. The latter divergences, however, satisfy a simple factorization equation. We speculate on the scenario that could be drawn about infra-red asymptotic dynamics of QCD, should this factorization equation be true in any order of perturbation theory. 
  We investigate the supersymmetric index of N=2,3 SU(n) supersymmetric Yang-Mills Chern Simons theories at level k by using the brane configuration with a (p,q)5-brane. We can explain that the supersymmetry breaking occurs when k<n in terms of the s-rule for Type IIB branes. The supersymmetric index coincides with the number of the possible supersymmetric brane configurations. We also discuss a construction of a family of theories which have the same supersymmetric index. 
  We consider non-BPS D8 (and D7) branes in type 0 open string theory and describe under which circumstances these branes are stable. We find stable non-BPS D7 and D8 in type 0 with and without D9-branes in the background. By extending the descent relations between D-branes to type 0 theories, the non-BPS D8-brane is considered as the result of a tachyon condensation of a D9 anti-D9 pair in type 0. We study the condensation of the open string tachyons in type 0 with generic gauge groups giving rise to different configurations involving non-BPS D8-branes and discuss the stability in each case. The results agree with the topological analysis of the vacuum manifold of the tachyon potential for each case. 
  We discuss aspects of the non-standard version of F-theory based on the arithmetic of torsion points on elliptic curves. We construct new F-theory vacua in 8 dimensions... 
  Features of screening and confinement are reviewed in two-dimensional quantum electrodynamics (QED2). Our discussion is carried out using the gauge-invariant but path-dependent variables formalism. This alternative and useful approach exploits the rich structure of the electromagnetic cloud or dressing around static fermions in a straightforward and simple way. 
  We calculate a renormalized Hamiltonian for pure-glue QCD and diagonalize it. The renormalization procedure is designed to produce a Hamiltonian that will yield physical states that rapidly converge in an expansion in free-particle Fock-space sectors. To make this possible, we use light-front field theory to isolate vacuum effects, and we place a smooth cutoff on the Hamiltonian to force its free-state matrix elements to quickly decrease as the difference of the free masses of the states increases. The cutoff violates a number of physical principles of light-front pure-glue QCD, including Lorentz covariance and gauge covariance. This means that the operators in the Hamiltonian are not required to respect these physical principles. However, by requiring the Hamiltonian to produce cutoff-independent physical quantities and by requiring it to respect the unviolated physical principles of pure-glue QCD, we are able to derive recursion relations that define the Hamiltonian to all orders in perturbation theory in terms of the running coupling. We approximate all physical states as two-gluon states, and use our recursion relations to calculate to second order the part of the Hamiltonian that is required to compute the spectrum. We diagonalize the Hamiltonian using basis-function expansions for the gluons' color, spin, and momentum degrees of freedom. We examine the sensitivity of our results to the cutoff and use them to analyze the nonperturbative scale dependence of the coupling. We investigate the effect of the dynamical rotational symmetry of light-front field theory on the rotational degeneracies of the spectrum and compare the spectrum to recent lattice results. Finally, we examine our wave functions and analyze the various sources of error in our calculation. 
  We study, using ADHM construction, instanton effects in an ${\CN}=2$ superconformal $Sp(N)$ gauge theory, arising as effective field theory on a system of $N$ D-3-branes near an orientifold 7-plane and 8 D-7-branes in type I' string theory. We work out the measure for the collective coordinates of multi-instantons in the gauge theory and compare with the measure for the collective coordinates of $(-1)$-branes in the presence of 3- and 7-branes in type I' theory. We analyse the large-N limit of the measure and find that it admits two classes of saddle points: In the first class the space of collective coordinates has the geometry of $AdS_5\times S^3$ which on the string theory side has the interpretation of the D-instantons being stuck on the 7-branes and therefore the resulting moduli space being $AdS_5\times S^3$, In the second class the geometry is $AdS_5\times S^5/Z_2$ and on the string theory side it means that the D-instantons are free to move in the 10-dimensional bulk. We discuss in detail a correlator of four O(8) flavour currents on the Yang-Mills side, which receives contributions from the first type of saddle points only, and show that it matches with the correlator obtained from $F^4$ coupling on the string theory side, which receives contribution from D-instantons, in perfect accord with the AdS/CFT correspondence. In particular we observe that the sectors with odd number of instantons give contribution to an O(8)-odd invariant coupling, thereby breaking O(8) down to SO(8) in type I' string theory. We finally discuss correlators related to $R^4$, which receive contributions from both saddle points. 
  A brief introduction to the boundary state approach to Dirichlet branes is given. The example of the non-BPS D-string of Type IIA on K3 is analysed in some detail, and its dual heterotic state is identified. 
  A method is developed to construct the solutions of one and many variable, linear differential equations of arbitrary order. Using this, the $N$-particle Sutherland model, with pair-wise inverse sine-square interactions among the particles, is shown to be equivalent to free particles on a circle. Applicability of our method to many other few and many-body problems is also illustrated. 
  Starting with the curvature 2-form a recursive construction of totally antisymmetrised 2p-forms is introduced, to which we refer as p-Riemann tensors. Contraction of indices permits a corresponding generalisation of the Ricci tensor. Static, spherically symmetric ``$p$-Ricci flat'' Schwarzschild like metrics are constructed in this context for d>2p+1, d being the spacetime dimension. The existence of de Sitter type solutions is pointed out. Our 2p-forms vanish for $d<2p$ and the limiting cases d=2p and d=2p+1 exhibit special features which are discussed briefly. It is shown that for d=4p our class of solutions correspond to double-selfdual Riemann 2p-form (or p-Riemann tensor). Topological aspects of such generalised gravitational instantons and those of associated (through spin connections) generalised Yang-Mills instantons are briefly mentioned. The possibility of a study of surface deformations at the horizons of our class of ``p-black holes'' leading to Virasoro algebras with a p-dependent hierarchy of central charges is commented on. Remarks in conclusion indicate directions for further study and situate our formalism in a broader context. 
  We discuss the possibility that the electro-weak and strong interactions arise as the low energy effective description of branes in M-theory. As a step towards constructing such a model we show how one can naturally obtain SU(N_1)\times SU(N_2)\times U(1) gauge theories from branes, including matter in the bi-fundamental representation of SU(N_1)\times SU(N_2) which are fractionally charged under U(1). 
  We study a new class of N=1 supersymmetric orientifolds in six space-time dimensions. The world-sheet parity transformation is combined with a permutation of the internal complex coordinates. In contrast to ordinary orientifolds the twisted sectors contribute to the Klein bottle amplitude leading to new tadpoles to be cancelled by twisted open string sectors. They arise from open strings stretched between D7-branes intersecting at non-trivial angles. We study in detail the Z_3, Z_4 and Z_6 permutational orientifolds obtaining in all cases anomaly free massless spectra. 
  We use the massless Thirring model to demonstrate a new approach to non-perturbative fermion calculations based on the spherical field formalism. The methods we present are free from the problems of fermion doubling and difficulties associated with integrating out massless fermions. Using a non-perturbative regularization, we compute the two-point correlator and find agreement with the known analytic solution. 
  New solution to the six-dimensional vacuum Einstein's equations is constructed as a non-linear superposition of two five-dimensional solutions representing the Melvin-Gibbons-Maeda Universe and its S-dual. Then using duality between D=8 vacuum and a certain class of D=11 supergravity configurations we generate M2 and M5 fluxbranes as well as some of their intersections also including waves and KK-monopoles. 
  A non-local classical duality between the three-block truncated 11D supergravity and the 8D vacuum gravity with two commuting Killing symmetries is established. The supergravity four-form field is generated via an inverse dualisation of the corresponding Killing two-forms in six dimensions. 11D supersymmetry condition is shown to be equivalent to existence of covariantly constant spinors in eight dimensions. Thus any solution to the vacuum Einstein equations in eight dimensions depending on six coordinates and admitting Killing spinors have supersymmetric 11D-supergravity counterparts. Using this duality we derive some new brane solutions to 11D-supergravity including 1/4 supersymmetric intersecting M-branes with a NUT parameter and a dyon solution joining the M2 and M5-branes intersecting at a point. 
  Using the correspondence between gauge theories and string theory in curved backgrounds, we investigate aspects of the large $N$ limit of non-commutative gauge theories by considering gravity solutions with $B$ fields. We argue that the total number of physical degrees of freedom at any given scale coincides with the commutative case. We then compute a two-point correlation function involving momentum components in the directions of the $B$-field. In the UV regime, we find that the two-point function decays exponentially with the momentum. A calculation of Wilson lines suggests that strings cannot be localized near the boundary. We also find string configurations that are localized in a finite region of the radial direction. These are worldsheet instantons. 
  We describe physical phenomena associated with a class of transitions that occur in the study of supersymmetric three-cycles in Calabi-Yau threefolds. The transitions in question occur at real codimension one in the complex structure moduli space of the Calabi-Yau manifold. In type IIB string theory, these transitions can be used to describe the evolution of a BPS state as one moves through a locus of marginal stability: at the transition point the BPS particle becomes degenerate with a supersymmetric two particle state, and after the transition the lowest energy state carrying the same charges is a non-supersymmetric two particle state. In the IIA theory, wrapping the cycles in question with D6-branes leads to a simple realization of the Fayet model: for some values of the CY modulus gauge symmetry is spontaneously broken, while for other values supersymmetry is spontaneously broken. 
  We present component and superspace formulations for the recently-proposed Type IIA* (or so-called `star') supergravity theory, which is timelike dual to the conventional Type IIB theory. First, within the component approach, all terms in the action are fixed up to the quartic fermionic ones. As desired, the kinetic terms for Ramond-Ramond fields have signs opposite to the conventional case. Consistency of these are then insured by the construction of a superspace description of this theory. As a by-product, we find that a single signature parameter $s = \pm 1$ can interpolate the Type IIA and Type IIA* theories in superspace. This superspace result naturally allows us to present a Green-Schwarz action, that possesses $\kappa$-symmetry, consistent with such backgrounds. We also give general algebraic descriptions of such `star' theories, so that they can be identified as representatives of some of the equivalence classes of $\kappa$-invariant Green-Schwarz actions. 
  We obtain the non-perturbative effective potential for the dual five-dimensional N=4 strings in the context of finite-temperature regarded as a breaking of supersymmetry into four space-time dimensions. Using the properties of gauged N=4 supergravity we derive the universal thermal effective potential describing all possible high-temperature instabilities of the known N=4 superstrings. These strings undergo a high-temperature transition to a new phase in which five-branes condense. This phase is described in detail, using both the effective supergravity and non-critical string theory in six dimensions. In the new phase, supersymmetry is perturbatively restored but broken at the non-perturbative level. 
  The Arnold conjecture yields a lower bound to the number of periodic classical trajectories in a Hamiltonian system. Here we count these trajectories with the help of a path integral, which we inspect using properties of the spectral flow of a Dirac operator in the background of a $\Sp(2N)$ valued gauge field. We compute the spectral flow from the Atiyah-Patodi-Singer index theorem, and apply the results to evaluate the path integral using localization methods. In this manner we find a lower bound to the number of periodic classical trajectories which is consistent with the Arnold conjecture. 
  We propose a new mechanism for the formation of conical singularities on D-branes by means of recoil resulting from scattering of closed string states propagating in the (large) transverse dimensions. By viewing the (spatial part of the) four-dimensional world as a 3-brane with large transverse dimensions the above mechanism can lead to supersymmetry obstruction at the TeV scale. The vacuum remains supersymmetric while the mass spectrum picks up a supersymmetry obstructing mass splitting. The state with ``broken'' supersymmetry is not an equilibrium ground state, but is rather an excited state of the D-brane which relaxes to the supersymmetric ground state asymptotically in (cosmic) time. 
  We provide a new general setting for scalar interacting fields on the covering of a d+1-dimensional AdS spacetime. The formalism is used at first to construct a one-paramater family of field theories, each living on a corresponding spacetime submanifold of AdS, which is a cylinder $R\times S_{d-1}$. We then introduce a limiting procedure which directly produces Luescher-Mack CFT's on the covering of the AdS asymptotic cone. Our AdS/CFT correspondence is generally valid for interacting fields, and is illustrated by a complete treatment of two-point functions, the case of Klein-Gordon fields appearing as particularly simple in our context.   We also show how the Minkowskian representation of these boundary CFT's can be directly generated by an alternative limiting procedure involving Minkowskian theories in horocyclic sections (nowadays called (d-1)-branes, 3-branes for AdS_5). These theories are restrictions to the brane of the ambient AdS field theory considered. This provides a more general correspondence between the AdS field theory and a Poincare' invariant QFT on the brane, satisfying all the Wightman axioms. The case of two-point functions is again studied in detail from this viewpoint as well as the CFT limit on the boundary. 
  We show that twisted reduced models can be interpreted as noncommutative Yang-Mills theory. Based upon this correspondence, we obtain noncommutative Yang-Mills theory with D-brane backgrounds in IIB matrix model. We propose that IIB matrix model with D-brane backgrounds serve as a concrete definition of noncommutative Yang-Mills. We investigate D-instanton solutions as local excitations on D3-branes. When instantons overlap, their interaction can be well described in gauge theory and AdS/CFT correspondence. We show that IIB matrix model gives us the consistent potential with IIB supergravity when they are well separated. 
  We extend earlier ideas about the appearance of noncommutative geometry in string theory with a nonzero B-field. We identify a limit in which the entire string dynamics is described by a minimally coupled (supersymmetric) gauge theory on a noncommutative space, and discuss the corrections away from this limit. Our analysis leads us to an equivalence between ordinary gauge fields and noncommutative gauge fields, which is realized by a change of variables that can be described explicitly. This change of variables is checked by comparing the ordinary Dirac-Born-Infeld theory with its noncommutative counterpart. We obtain a new perspective on noncommutative gauge theory on a torus, its T-duality, and Morita equivalence. We also discuss the D0/D4 system, the relation to M-theory in DLCQ, and a possible noncommutative version of the six-dimensional (2,0) theory. 
  We study the non-perturbative, instanton-corrected effective action of the N=2 SU(2) x SU(2) supersymmetric Yang-Mills theory with a massless hypermultiplet in the bifundamental representation. Starting from the appropriate hyperelliptic curve, we determine the periods and the exact holomorphic prepotential in a certain weak coupling expansion. We discuss the dependence of the solution on the parameter q=L2^2/L1^2 and several other interesting properties. 
  In this set of lectures various properties of D-branes are discussed. After reviewing the basics, we discuss unstable D-brane/anti-D-brane systems, a subject pioneered by Sen. Following him, we discuss the construction of the non-BPS D0-brane in type I theory. This state is stable since it carries a conserved Z_2 charge. The general classification of D-brane charges using K-theory is discussed. The results for the type I theory, and the T-dual type I' theory, are emphasized. Compactification of type I on a circle or torus gives a theory with 16 supersymmetries in 9d or 8d. In each case the moduli space has three branches. The spectrum of non-BPS D-branes are different for each of these branches. We conclude by pointing out some problems with the type I D7-brane and D8-brane predicted by K-theory. 
  It is shown from a simple scaling invariance that the ultra-relativistic Hamiltonian (mu=0) does not have bound states when the potential is Coulombic. This supplements the application of the relativistic virial theorem derived by Lucha and Schoeberl which shows that bound states do not exist for potentials more singular than the Coulomb potential. 
  In theories with TeV string scale and sub-millimeter extra dimensions the attractive picture of logarithmic gauge coupling unification at $10^{16}$ GeV is seemingly destroyed. In this paper we argue to the contrary that logarithmic unification {\it can} occur in such theories. The rationale for unification is no longer that a gauge symmetry is restored at short distances, but rather that a geometric symmetry is restored at large distances in the bulk away from our 3-brane. The apparent `running' of the gauge couplings to energies far above the string scale actually arises from the logarithmic variation of classical fields in (sets of) two large transverse dimensions. We present a number of N=2 and N=1 supersymmetric D-brane constructions illustrating this picture for unification. 
  We investigate the formation of dynamical gaugino condensates and supersymmetry breaking in the compactifications of Horava-Witten theory with perturbative nonstandard embeddings. Specific models are considered where the underlying massless charged states of the condensing sector are determined by the spectra of $Z_2 \times Z_2 $ and $Z_4 $ orbifolds with nonstandard embeddings. We find among them viable examples where gaugino condensation is triggered on the wall with the weakest gauge coupling at $M_{GUT} $. In all these cases the magnitude of the condensate formed is below the energy scales at which extra dimensions are resolved, and so justifies the analysis of condensation in an effective 4-dimensional framework. We make some comments concerning the size of the largest extra dimension in the models considered. We discuss racetrack scenarios in the framework of perturbative nonstandard embeddings. 
  In the first part of these lectures we will review the main aspects of large N QCD and the explicit results obtained from it. Then, after a review of the properties of N=4 super Yang-Mills, type IIB string theory and of AdS space, we briefly discuss the Maldacena conjecture. Finally in the last part of these lectures we will discuss the finite temperature case and we show how "hadronic" quantities as the string tension, the mass gap and the topological susceptibility can be computed in this approach. 
  We analyze the high temperature (or classical) limit of the Casimir effect. A useful quantity which arises naturally in our discussion is the ``relative Casimir energy", which we define for a configuration of disjoint conducting boundaries of arbitrary shapes, as the difference of Casimir energies between the given configuration and a configuration with the same boundaries infinitely far apart. Using path integration techniques, we show that the relative Casimir energy vanishes exponentially fast in temperature. This is consistent with a simple physical argument based on Kirchhoff's law. As a result the ``relative Casimir entropy", which we define in an obviously analogous manner, tends, in the classical limit, to a finite asymptotic value which depends only on the geometry of the boundaries. Thus the Casimir force between disjoint pieces of the boundary, in the classical limit, is entropy driven and is governed by a dimensionless number characterizing the geometry of the cavity. Contributions to the Casimir thermodynamical quantities due to each individual connected component of the boundary exhibit logarithmic deviations in temperature from the behavior just described. These logarithmic deviations seem to arise due to our difficulty to separate the Casimir energy (and the other thermodynamical quantities) from the ``electromagnetic'' self-energy of each of the connected components of the boundary in a well defined manner. Our approach to the Casimir effect is not to impose sharp boundary conditions on the fluctuating field, but rather take into consideration its interaction with the plasma of ``charge carriers'' in the boundary, with the plasma frequency playing the role of a physical UV cutoff. This also allows us to analyze deviations from a perfect conductor behavior. 
  We study the formation of BTZ black holes by the collision of point particles. It is shown that the Gott time machine, originally constructed for the case of vanishing cosmological constant, provides a precise mechanism for black hole formation. As a result, one obtains an exact analytic understanding of the Choptuik scaling. 
  We develop the general theory of Noether symmetries for constrained systems. In our derivation, the Dirac bracket structure with respect to the primary constraints appears naturally and plays an important role in the characterization of the conserved quantities associated to these Noether symmetries. The issue of projectability of these symmetries from tangent space to phase space is fully analyzed, and we give a geometrical interpretation of the projectability conditions in terms of a relation between the Noether conserved quantity in tangent space and the presymplectic form defined on it. We also examine the enlarged formalism that results from taking the Lagrange multipliers as new dynamical variables; we find the equation that characterizes the Noether symmetries in this formalism. The algebra of generators for Noether symmetries is discussed in both the Hamiltonian and Lagrangian formalisms. We find that a frequent source for the appearance of open algebras is the fact that the transformations of momenta in phase space and tangent space only coincide on shell. Our results apply with no distinction to rigid and gauge symmetries; for the latter case we give a general proof of existence of Noether gauge symmetries for theories with first and second class constraints that do not exhibit tertiary constraints in the stabilization algorithm. Among some examples that illustrate our results, we study the Noether gauge symmetries of the Abelian Chern-Simons theory in $2n+1$ dimensions. An interesting feature of this example is that its primary constraints can only be identified after the determination of the secondary constraint. The example is worked out retaining all the original set of variables. 
  We consider a two-point spatial lattice approximation to an open string moving in a flat background with B field. It gives a constrained dipole system under the influence of a vector potential. Solving and quantizing this system recover all the essential features of a noncommutative space. In particular, open string interactions induce a canonical product structure on the Hilbert space of the dipole system. It coincides with the usual star product, even though the position operators can be thought of as mutually commuting. Modification of gauge transformations in this noncommutative space also naturally emerges. 
  The large N behavior of Matrix theory is discussed on the basis of the previously proposed generalized conformal symmetry. The concept of `oblique' AdS/CFT correspondence, in which the conformal symmetry involves both the space-time coordinates and the string coupling constant, is proposed. Based on the explicit predictions for two-point correlators, possible implications for the Matrix-theory conjecture are discussed. 
  We examine the solutions of world-volume action for a D3-brane being put near other D3-brane which is replaced by the background configuration of bulk space. It is shown that the BPS solutions are not affected by the D3-brane background, and they are interpreted as dyonic strings connecting two branes. On the contrary, the non-BPS configurations are largely influenced by the background D-brane, and we find that the solutions with pure electric charge cannot connect two branes. These solutions are corresponding to the bound state of brane and anti-brane which has been found by Callan and Maldacena 
  A construction of compact tachyon-free orientifolds of the non-supersymmetric Type 0B string theory is presented. Moreover, we study effective non-supersymmetric gauge theories arising on self-dual D3-branes in Type 0B orbifolds and orientifolds. 
  We present a new regularization method, for d dim (Euclidean) quantum field theories in the continuum formalism, based on the domain wall configuration in (1+d) dim space-time. It is inspired by the recent progress in the chiral fermions on the lattice. The wall "height" is given by 1/M, where M is a regularization mass parameter and appears as a 1+d dim Dirac fermion mass. The present approach gives a thermodynamic view to the domain wall or the overlap formalism in the lattice field theory. We will show qualitative correspondence between the present continuum results and those of the lattice. The extra dimension is regarded as the (inverse) temperature t. The domains are defined by the directions of the "system movement", not by the sign of M as in the original overlap formalism. Physically the parameter M controls both the chirality selection and the dimensional reduction to d dimension. From the point of regularization, the limit $Mt\ra 0$ regularize the infra-red behaviour whereas the condition on the momentum ($k^\m$) integral, $|k^\m|\leq M$, regularize the ultra-violet behaviour.   To check the new regularization works correctly, we take the 4 dim QED and 2 dim chiral gauge theory as examples. Especially the consistent and covariant anomalies are correctly obtained. The choice of solutions of the higher dim Dirac equation characterize the two anomalies. The projective properties of the positive and negative energy free solutions are exploited in calculation. Some integral functions, the incomplete gamma functions and the generalized hypergeometric functions characteristically appear in the regularization procedure. 
  The quantum mechanical transition between a free particle Lagrangian and the Klein Gordon field description of a free particle (particle wave duality) is conjectured to extend to an analogous construction of relativistically invariant wave equations associated with strings and branes, which we propose to call brane-wave duality. Electromagnetic interactions in the two systems are discussed. It is emphasised that all integrable free field theories, including those of Dirac-Born-Infeld type, are associated with Lagrangians equivalent to divergences on the space of solutions of the equations of motion. 
  We study a particular class of Abelian gauged Nambu-Jona-Lasinio models with global U_L(N)xU_R(N) symmetry, where N is the number of fermion flavors. We show, by treating the gauge interaction in the ladder approximation and four-fermion interactions in the leading order of the 1/N expansion, that the renormalization-group beta function of the U(1) gauge coupling has ultraviolet stable fixed points for sufficiently large N. This implies the existence of a nontrivial continuum limit. 
  In this letter we study 1+1 anyon fields at finite temperature and density with non-vanishing chemical potentials. Our approach is based on an operator formalism for bosonization at finite temperature; the correlation functions for the system are given in an explicit form. Two are the main results of this construction: we point out the existence of persistent currents in 1+1 anyon systems; from the analysis of 2-point anyon field correlation function, a remarkable and new condensation phenomenon in momentum space is discovered. As a concrete example, the above formalism is applied to the Thirring model. 
  The non-renormalization of the 3-point functions $tr X^{k_1} tr X^{k_2} tr X^{k_3}$ of chiral primary operators in N=4 super-Yang-Mills theory is one of the most striking facts to emerge from the AdS/CFT correspondence. A two-fold puzzle appears in the extremal case, e.g. k_1 = k_2 + k_3. First, the supergravity calculation involves analytic continuation in the k_i variables to define the product of a vanishing bulk coupling and an infinite integral over AdS. Second, extremal correlators are uniquely sensitive to mixing of the single-trace operators $tr X^k$ with protected multi-trace operators in the same representation of SU(4). We show that the calculation of extremal correlators from supergravity is subject to the same subtlety of regularization known for the 2-point functions, and we present a careful method which justifies the analytic continuation and shows that supergravity fields couple to single traces without admixture. We also study extremal n-point functions of chiral primary operators, and argue that Type IIB supergravity requires that their space-time form is a product of n-1 two-point functions (as in the free field approximation) multiplied by a non-renormalized coefficient. This non-renormalization property of extremal n-point functions is a new prediction of the AdS/CFT correspondence. As a byproduct of this work we obtain the cubic couplings $t \phi \phi$ and $s \phi \phi$ of fields in the dilaton and 5-sphere graviton towers of Type IIB supergravity on $AdS_5 \times S^5$. 
  We develop a new systematic approach to quantum field theory that is designed to lead to physical states that rapidly converge in an expansion in free-particle Fock-space sectors. To make this possible, we use light-front field theory to isolate vacuum effects, and we place a smooth cutoff on the Hamiltonian to force its free-state matrix elements to quickly decrease as the difference of the free masses of the states increases. The cutoff violates a number of physical principles of light-front field theory, including Lorentz covariance and gauge covariance. This means that the operators in the Hamiltonian are not required to respect these physical principles. However, by requiring the Hamiltonian to produce cutoff-independent physical quantities and by requiring it to respect the unviolated physical principles of the theory, we are able to derive recursion relations that define the Hamiltonian to all orders in perturbation theory in terms of the fundamental parameters of the field theory. We present two applications of this method. First we work in massless phi-cubed theory in six dimensions. We derive the recursion relations that determine the Hamiltonian and demonstrate how they are used by computing and analyzing some of its second- and third-order matrix elements. Then we apply our method to pure-glue quantum chromodynamics. After deriving the recursion relations for this theory, we use them to calculate to second order the part of the Hamiltonian that is required to compute the spectrum. We diagonalize the Hamiltonian using basis-function expansions for the gluons' color, spin, and momentum degrees of freedom. We analyze our results for the spectrum, compare them to recent lattice results, and discuss the various sources of error in our calculation. 
  Coincident M2 branes at a conical singularity are related to M theory on AdS_4 x X_7 for an appropriate 7 dimensional Sasaki-Einstein manifold X_7. For X_7=Q^{1,1,1}=(SU(2) x SU(2) x SU(2))/(U(1) x U(1)) which was found sometime ago, the infrared limit of the theory on N M2 branes was constructed recently. It is the SU(N) x SU(N) x SU(N) gauge theories with three series of chiral fields A_i, i=1,2 transforming in the (N, \bar{N},1) representation, B_j, j=1,2 transforming in the (1,N, \bar{N}) representation and C_k, k=1,2 transforming in the (\bar{N},1,N) representation. From the scalar Laplacian of X_7 on the supergravity side, we discuss the spectrum of chiral primary operators of dual N=2 superconformal field theory in 3 dimensions. We study M5 branes wrapped over 5-cycle of X_7 which were identified as (three types of) baryon like operators made out of N chiral fields recently. We consider M5 brane wrapped over 3-cycle of X_7 which plays the role of domain wall in AdS_4. The new aspect arises when baryon like operators(M5 branes wrapped over 5-cycle) cross a domain wall(M5 brane wrapped over 3-cycle), M2 brane between them must be created. 
  We review the unitarity bounds and the multiplet shortening of UIR's of 4 dimensional superconformal algebras $SU(2,2|N)$, ($N=1,2,4$) in view of their dual role in the AdS/SCFT correspondence. Some applications to KK spectra, non-perturbative states and stringy states are given. 
  We discuss the moduli space of flat connections of Yang-Mills theories formulated on T^3 x R, with periodic boundary conditions. When the gauge group is SO(N>=7), G_2, F_4, E_6, E_7 or E_8, the moduli space consists of more than one component. 
  In the first part of this talk I discuss two somewhat different supergravity approaches to calculating correlation functions in strongly coupled Yang-Mills theory. The older approach relates two-point functions to cross-sections for absorption of certain incident quanta by threebranes. In this approach the normalization of operators corresponding to the incident particles is fixed unambiguously by the D3-brane DBI action. By calculating absorption cross-sections of all partial waves of the dilaton we find corresponding two-point functions at strong `t Hooft coupling and show that they are identical to the weak coupling results. The newer approach to correlation functions relates them to boundary conditions in AdS space. Using this method we show that for a certain range of negative mass-squared there are two possible operator dimensions corresponding to a given scalar field in AdS, and indicate how to calculate correlation functions for either of these choices. In the second part of the talk I discuss an example of AdS/CFT duality which arises in the context of type 0 string theory. The CFT on N coincident electric and magnetic D3-branes is argued to be stable for sufficiently weak `t Hooft coupling. It is suggested that its transition to instability at a critical coupling is related to singularity of planar diagrams. 
  Majumder and Sen have given an explicit construction of a first order phase transition in a non-supersymmetric system of Dbranes that occurs when the B field is varied. We show that the description of this transition in terms of K-theory involves a bundle of K groups of non-commutative algebras over the Kahler cone with nontrivial monodromy. Thus the study of monodromy in K groups associated with quantized algebras can be used to predict the phase structure of systems of (non-supersymmetric) Dbranes. 
  In this article I first give an abbreviated history of string theory and then describe the recently-conjectured field-string duality. This suggests a class of nonsupersymmetric gauge theories which are conformal (CGT) to leading order of 1/N and some of which may be conformal for finite N. These models are very rigid since the gauge group representations of not only the chiral fermions but also the Higgs scalars are prescribed by the construction. If the standard model becomes conformal at TeV scales the GUT hierarchy is nullified, and model-building on this basis is an interesting direction. Some comments are added about the dual relationship to gravity which is absent in the CGT description. 
  Topological properties of quantum system is directly associated with the wave function. Based on the decomposition theory of gauge potential, a new comprehension of topological quantum mechanics is discussed. One shows that a topological invariant, the first Chern class, is inherent in the Schr\"odinger system, which is only associated with the Hopf index and Brouwer degree of the wave function. This relationship between the first Chern class and the wave function is the topological source of many topological effects in quantum system. 
  The connection between IIA superstring theory compactified on a circle of radius R and IIB theory compactified on a circle of radius 1/R is reexamined from the perspective of N=2, D=9 space-time supersymmetry. We argue that the consistency of IIA/B duality requires the BPS states corresponding to momentum and winding of either of the type-II superstrings to transform as inequivalent supermultiplets. We show that this is indeed the case for any finite compactification radius, thus providing a nontrivial confirmation of IIA/B duality. From the point of view of N=2, D=9 supergravity, one is naturally led to an SL(2,Z) invariant field theory that encompasses both the M-theory torus and the Kaluza-Klein states of the IIB theory. 
  Three different methods to quantize the spherically symmetric sector of electromagnetism are presented: First, it is shown that this sector is equivalent to Abelian BF-theory in four spacetime dimensions with suitable boundary conditions. This theory, in turn, is quantized by both a reduced phase space quantization and a spin network quantization. Finally, the outcome is compared with the results obtained in the recently proposed general quantum symmetry reduction scheme. In the magnetically uncharged sector, where all three approaches apply, they all lead to the same quantum theory. 
  This dissertation reviews various aspects of the N=4 supersymmetric Yang--Mills theory in particular in relation with the AdS/CFT correspondence. The first two chapters are introductory. The first one contains a description of the general properties of rigid supersymmetric theories in four dimensions both at the classical and at the quantum level. The second chapter is a review of the main properties of the N=4 SYM theory under consideration. Original results are reported in chapters 3, 4 and 5. A systematic re-analysis of the perturbative properties of the theory is presented in the third chapter. Two-, three- and four-point Green functions of elementary fields are computed using the component formulation and/or the superfield approach and subtleties related to the gauge-fixing are pointed out. In the fourth chapter, after an introduction to instanton calculus in supersymmetric gauge theories, the computation of the one-instanton contributions to Green functions of gauge invariant composite operators in the semiclassical approximation is reported. The calculations of four-, eight- and sixteen-point Green functions of operators in the supercurrent multiplet are reviewed in detail. The final chapter is devoted to the AdS/SCFT correspondence. Some general aspects are discussed. Then the attention is focused on the relation between instantons in N=4 SYM and D-instanton effects in type IIB string theory. The comparison between instanton contributions to Green functions of composite operators in the boundary field theory and D-instanton generated terms in the amplitudes computed in type IIB string theory is performed and agreement between these two sources of non-perturbative effects is shown. 
  The free energy of a multi-component scalar field theory is considered as a functional W[G,J] of the free correlation function G and an external current J. It obeys non-linear functional differential equations which are turned into recursion relations for the connected Greens functions in a loop expansion. These relations amount to a simple proof that W[G,J] generates only connected graphs and can be used to find all such graphs with their combinatoric weights. A Legendre transformation with respect to the external current converts the functional differential equations for the free energy into those for the effective energy Gamma[G,Phi], which is considered as a functional of the free correlation function G and the field expectation Phi. These equations are turned into recursion relations for the one-particle irreducible Greens functions. These relations amount to a simple proof that Gamma[G,J] generates only one-particle irreducible graphs and can be used to find all such graphs with their combinatoric weights. The techniques used also allow for a systematic investigation into resummations of classes of graphs. Examples are given for resumming one-loop and multi-loop tadpoles, both through all orders of perturbation theory. Since the functional differential equations derived are non-perturbative, they constitute also a convenient starting point for other expansions than those in numbers of loops or powers of coupling constants. We work with general interactions through four powers in the field. 
  We demonstrate that front form quantisation with periodicity in a compact light-like direction (discretized light-cone quantisation) violates microcausality. 
  If the universe is (or a slice of) $AdS_5$ space with 3-branes, the 5-dimensional GUT scale on each brane can be indetified as the 5-dimensional Planck scale, but, the 4-dimensional Planck scale is generated from the low 4-dimensional GUT scale exponentially in our world.   The 4-dimensional GUT scales and Planck scale are related to the 5-dimensional GUT scales and Planck scale by exponential factors, respectively.   One of such scenarios was suggested by Randall and Sundrum recently. We give another scenario that the 4-dimensional Planck scale is generated from the low five-dimensional Planck scale by an exponential hierarchy, and the mass scale in the Standard Model is not rescaled from the 5-dimensional metric to the 4-dimensional metric. We also argue that the additional constant in the solution might exist, which will rescale the 5-dimensional Planck scale and affect the physical scale picture. Finally, we embed those compactifications to the general compactification on $AdS_5$ space and discuss the origin of the additional constant. 
  We determine the spectrum of graviton excitations in the background geometry of the AdS soliton in p+2 dimensions. Via the AdS/CFT correspondence this corresponds to determining the spectrum of spin two excitations in the dual effective p-dimensional field theories For the cases of D3- and M5-branes these are the spin two glueballs of QCD_3 and QCD_4 respectively. For all values of p we find an exact degeneracy of the spectra of these tensor states and certain scalar excitations. Our results also extend the perturbative proof of a positive energy conjecture for asymptotically locally AdS spacetimes (originally proposed for p=3) to an arbritrary number of dimensions. 
  We investigate a class of models in 1+1 dimensions with four fermion interaction term. At each order of the perturbation expansion, the models are ultraviolet finite and Lorentz non-invariant. We show that for certain privileged values of the coupling constants, Lorentz symmetry is restored, and indeed the model turns out to be conformally invariant. This phenomenon is both quantum mechanical and non-perturbative. 
  We describe constructing solutions of the field equations of Chern-Simons and topological BF theories in terms of deformation theory of locally constant (flat) bundles. Maps of flat connections into one another (dressing transformations) are considered. A method of calculating (nonlocal) dressing symmetries in Chern-Simons and topological BF theories is formulated. 
  We inspect the excitation energy spectrum of a confining string in terms of solitons in an effective field theory model. The spectrum can be characterized by a spectral function, and twisting and bending of the string is manifested by the invariance of this function under a duality transformation. Both general considerations and numerical simulations reveal that the spectral function can be approximated by a simple rational form, which we propose becomes exact in the Yang-Mills theory. 
  We discuss the construction of Baxter's Q-operator. The suggested approach leads to the one-parametric family of Q-operators, satisfying to the wronslian-type relations. Also we have found the generalization of Baxter operators, with defines the nondiagonal part of the monodromy. 
  We study the interaction between a tensionless (null) string and an antisymmetric background field B_{ab} using a 2-component spinor formalism. A geometric condition for the absence of such an interaction is formulated. We show that only one gauge-invariant degree of freedom of the field B_{ab} does not satisfy this condition. Identification of this degree of freedom with the notoph field \phi of Ogievetskii-Polubarinov-Kalb-Ramond is suggested. Application of a two-component spinor formalism allows us a reduction of the complete system of non-linear partial differential equations and constraints governing the interacting null string dynamics to a system of linear differential equations for the basis spinors of the spin-frame. We find that total effect of the interaction is contained in a single derivation coefficient which is identified with the notoph field. 
  We study spin one particles interacting through a Chern-Simons field. In the Born approximation, we calculate the two body scattering amplitude considering three possible ways to introduce the interaction: (a) a Proca like model minimally coupled to a Chern-Simons field, (b) the model obtained from (a) by replacing the Proca's mass by a Chern-Simons term and (c) a complex Maxwell-Chern-Simons model minimally coupled to a Chern-Simons field. In the low energy regime the results show similarities with the Aharonov-Bohm scattering for spin 1/2 particles. We discuss the one loop renormalization program for the Proca's model. In spite of the bad ultraviolet behavior of the matter field propagator, we show that, up to one loop the model is power counting renormalizable thanks to the Ward identities satisfied by the interaction vertices. 
  We obtain various solutions for D=4 dipoles and their bound states whose U(1) fields originate from various form fields in the effective string theories. We oxidize such dipole solutions to D=10 to obtain delocalized supergravity solutions for the brane/anti-brane pairs and their bound states. We speculate on generalized harmonic superposition rules for supergravity solutions for (intersecting) brane/anti-brane pairs. 
  I review the computation of the conformal anomaly of a Wilson surface observable in free two-form gauge theory in six dimensions. 
  We note that S-matrix/conserved charge identities in affine Toda field theories of the type recently noted by Khastgir can be put on a more systematic footing. This makes use of a result first found by Ravanini, Tateo and Valleriani for theories based on the simply-laced Lie algebras (A,D and E) which we extend to the nonsimply-laced case. We also present the generalisation to nonsimply-laced cases of the observation - for simply-laced situations - that the conserved charges form components of the eigenvectors of the Cartan matrix. 
  We present new examples of maverick coset conformal field theories. They are closely related to conformal embeddings and exceptional modular invariants. 
  We present static solutions to Einstein's equations corresponding to branes at various angles intersecting in a single 3-brane. Such configurations may be useful for building models with localized gravity via the Randall-Sundrum mechanism. We find that such solutions may exist only if the mechanical forces acting on the junction exactly cancel. In addition to this constraint there are further conditions that the parameters of the theory have to satisfy. We find that at least one of these involves only the brane tensions and cosmological constants, and thus can not have a dynamical origin. We present these conditions in detail for two simple examples. We discuss the nature of the cosmological constant problem in the framework of these scenarios, and outline the desired features of the brane configurations which may bring us closer towards the resolution of the cosmological constant problem. 
  We find new duality transformations which allow us to construct the stress tensors of all the twisted sectors of any orbifold A(H)/H, where A(H) is the set of all current-algebraic conformal field theories with a finite symmetry group H \subset Aut(g). The permutation orbifolds with H = Z_\lambda and H = S_3 are worked out in full as illustrations but the general formalism includes both simple and semisimple g. The motivation for this development is the recently-discovered orbifold Virasoro master equation, whose solutions are identified by the duality transformations as sectors of the permutation orbifolds A(D_\lambda)/Z_\lambda. 
  We study the renormalization of non-semisimple gauge models quantized in the `t Hooft-background gauge to all orders. We analyze the normalization conditions for masses and couplings compatible with the Slavnov-Taylor and Ward-Takahashi Identities and with the IR constraints. We take into account both the problem of renormalization of CKM matrix elements and the problem of CP violation and we show that the Background Field Method (BFM) provides proper normalization conditions for fermion, scalar and gauge field mixings. We discuss the hard and the soft anomalies of the Slavnov-Taylor Identities and the conditions under which they are absent. 
  String theory on curved backgrounds has received much attention on account of both its own interest, and of its relation with gauge theories. Despite the progress made in various directions, several quite elementary questions remain unanswered, in particular in the very simple case of three-dimensional anti-de Sitter space. I will very briefly review these problems, discuss in some detail the important issue of constructing a consistent spectrum for a string propagating on ADS3 plus torsion background, and comment on potential solutions. 
  A systematic algorithm to derive superpropagators in the case of either explicitly or spontaneously broken supersymmetric three-dimensional theories is presented. We discuss how the explicit breaking terms that are introduced at tree-level induce 1-loop radiative corrections to the effective action. We also point out that the renormalisation effects and the breaking-inducing-breaking mechanism become more immediate whenever we adopt the shifted superpropagators discussed in this letter. 
  We show how to use the method of orthogonal polynomials for integrating, in the planar approximation, the partition function of one-matrix models with a potential with even or odd vertices, or any combination of them. 
  We examine the behaviour of Killing spinors on AdS5 under various discrete symmetries of the spacetime. In this way we discover a number of supersymmetric orbifolds, reproducing the known ones and adding a few novel ones to the list. These orbifolds break the SO(4,2) invariance of AdS5 down to subgroups. We also make some comments on the non-compact Stiefel manifold W{4,2}. 
  We give an indirect argument for the matching $G^2=-\pi_* \gamma^2$ of four-flux and discrete twist in the duality between N=1 heterotic string and $F$-theory. This treats in detail the Euler number computation for the physically relevant case of a Calabi-Yau fourfold with singularities. 
  We apply the concept of helicity from classical hydrodynamics to elucidate two problematical issues in cosmic string physics. Helicity, the space integral of the scalar product of a velocity-like field with its vorticity field (curl), can be defined for a complex scalar field in analogy with fluids. We dwell on the topological interpretation of helicity as related to the linking of field lines of the vorticity field. Earlier works failed to fully implement this interpretation for cosmic strings by missing a term connected with the linking of these lines inside the strings. As a result paradoxical conclusions were drawn: global cosmic string loops may not take on certain simple shapes, and baryon number is not quantized in integers in the presence of local cosmic strings in gauge theory. We show that both paradoxes are removed when internal contributions to helicity are properly taken into account. In particular, quantization of baryon number can be understood within a special case of the Glashow-Weinberg-Salam model if cosmic strings are the unique mechanism for baryosynthesis. In addition, we find a new constraint on the permitted linkages of cosmic strings in a string tangle. 
  In these notes, the transverse ($\perp$) lattice approach is presented as a means to control the $k^+\to 0$ divergences in light-front QCD. Technical difficulties of both the canonical compact formulation as well as the non-compact formulation of the $\perp$ lattice motivate the color-dielectric formulation, where the link fields are linearized. 
  The discrete spectrum of fluctuations of the metric about an $AdS^5$ black hole background are found. These modes are the strong coupling limit of so called glueball states in a dual 3-d Yang-Mills theory with quantum numbers $J^{PC} = 2^{++}, 1^{-+}, 0^{++}$. For the ground state modes, we find the mass relation: $m(0^{++}) < m(2^{++}) < m(1^{-+})$. Contrary to expectation, the mass of our new $0^{++}$ state ($m^2=5.4573$) associated with the graviton is smaller than the mass of the $0^{++}$ state ($m^2=11.588$) from the dilaton. In fact the dilatonic excitations are exactly degenerate with our tensor $2^{++}$ states. We find that variational methods gives remarkably accurate mass estimates for all three low-lying levels while a WKB treatment describes the higher modes well. 
  Based on a canonically derived path integral formalism, we demonstrate that the perturbative calculation of the matrix element for gauge dependent operators has crucial difference from that for gauge invariant ones. For a gauge dependent operator ${\cal O}(\phi)$ what appears in the Feynman diagrams is not ${\cal O} (\phi)$ itself, but the gauge-transformed one ${\cal O}(^\omega \phi)$, where $\omega$ characterizes the specific gauge transformation which brings any field variable into the particular gauge which we have adopted to quantize the gauge theory using the canonical method. The study of the matrix element of gauge dependent operators also reveals that the formal path integral formalism for gauge theory is not always reliable. 
  We consider the $AdS_5$ solution deformed by a non-constant dilaton interpolating between the standard AdS (UV region) and flat boundary background (IR region). We show that this dilatonic solution can be generalized to the case of a non-flat boundaries provided that the metric of the boundaries satisfies the vacuum Einstein field equations. As an example, we describe the case when the four-dimensional boundaries represent the Kerr space-time. 
  This thesis discusses various aspects of duality in quantum field theory and string theory. In the first part we consider duality in topological quantum field theories, concentrating on the Donaldson and Seiberg-Witten theories as (dual) approaches to the study of four-manifolds. Lower-dimensional variants of these theories are also discussed. In the second part of the thesis we discuss duality in 2D sigma models by studying the interplay between renormalization group flows - as generated by the beta functions - and T-duality. The hypothesis put forward in the thesis is that for these to be mutually consistent, they should commute as operators in parameter space. The last part of the thesis bring these subjects into perspective by reviewing some nonperturbative dualities in string theory. 
  Recently Daviau showed the equivalence of ordinary matrix based Dirac theory -formulated within a spinor bundle S_x \simeq C^4_x-, to a Clifford algebraic formulation within space Clifford algebra CL(R^3,delta) \simeq M_2(C) \simeq P \simeq Pauli algebra (matrices) \simeq H \oplu H \simeq biquaternions. We will show, that Daviau's map theta : C^4 \mapsto M_2(C) is an isomorphism. Furthermore it is shown that Hestenes' and Parra's formulations are equivalent to Daviau's space Clifford algebra formulation, which however uses outer automorphisms. The connection between such different formulations is quite remarkable, since it connects the left and right action on the Pauli algebra itself viewed as a bi-module with the left (resp. right) action of the enveloping algebra P^e \simeq P\otimes P^T on P. The isomorphism established in this article and given by Daviau's map does clearly show that right and left actions are of similar type. This should be compared with attempts of Hestenes, Daviau and others to interprete the right action as the iso-spin freedom. 
  We examine ADHM multi-instantons in the conformal N=2 supersymmetric Sp(N) gauge theory with one anti-symmetric tensor and four fundamental hypermultiplets. We argue that the ADHM construction and measure can also be deduced from purely field theoretic considerations and also from the dynamics of D-instantons in the presence of D3-branes, D7-branes and an orientifold O7-plane. The measure then admits a large-N saddle-point approximation where the D3-branes disappear but the background is changed to AdS_5 x S^5/Z_2, as expected on the basis of the AdS/CFT correspondence. The large-N measure displays the fractionation of D-instantons at the singularity S^3 in S^5/Z_2 and is described for instanton number k by a certain O(k) matrix model. 
  We study the supersymmetric quantum mechanical systems that arise from discrete light cone quantization of theories with minimal supersymmetry in various dimensions. These systems, which have previously arisen in the study of black hole moduli spaces, are distinguished by having fewer fermionic fields than the familiar K\"ahler and hyper-K\"ahler models. 
  Polyakov has conjectured that Yang-Mills theory should be equivalent to a noncritical string theory. I pointed out, based on the work of Marchesini, Ishibashi, Kawai and collaborators, and Jevicki and Rodrigues, that the loop operator of the Yang-Mills theory is the temporal gauge string field theory Hamiltonian of a noncritical string theory. In the present note I explicitly show how this works for the one--plaquette model, providing a consistent direct string interpretation of the unitary matrix model for the first time. 
  We study instanton-corrected renormalization group flow in the two dimensional sigma models and four dimensional gauge theory. In two dimensions we do that by replacing the non-linear supersymmetric ${\IC\IP}^{N-1}$ model by the gauged linear sigma model which is in the same universality class. We compare the moduli spaces of the instantons in the non-linear model and that of BPS field configurations in the linear model. We reduce the problem of matching of the parameters of the two systems to the intersection theory on the compact moduli space of the latter model. Both cases (2d and 4d) are unified by the notion of the {\it freckled instantons}. We also put an end to the discussion of the nature of the superpotentials $W \sim {\s} {\rm log} {\s}$ in 2d and 4d and discover the surprising disconnectnessness of the effective target space. 
  Some physics models have 10 dimensions that are usually decomposed into: 4 spacetime dimensions with local Lorentz Spin(1,3) symmetry plus a 6-dimensional compact space related to internal symmetries. A possibly useful alternative decomposition is into: 6 spacetime dimensions with local Conformal symmetry of the Conformal Group C(1,3) = Spin(2,4) = SU(2,2) plus a 4-dimensional compact Internal Symmetry Space that can be taken to be complex projective 2-space CP2 which, since CP2 = SU(3) / U(2), is a natural representation space for SU(3) and is a natural local representation space for U(2) = SU(2) x U(1). 
  It is shown that the model of 2d dilaton gravity is equivalent to the dynamical system of massless particles in the Liouville field. 
  Using properties of the determinant line bundle for a family of elliptic boundary value problems, we explain how the Fock space functor defines an axiomatic quantum field theory which formally models the Fermionic path integral. The 'sewing axiom' of the theory arises as an algebraic pasting law for the determinant of the Dirac operator. We show how representations of the boundary gauge group fit into this description and that this leads to a Fock functor description of certain gauge anomalies. 
  After a brief historical survey that emphasizes the role of the algebra obeyed by the Dirac operator, we examine an algebraic Dirac operator associated with Lie algebras and Lie algebra cosets. For symmetric cosets, its ``massless'' solutions display non-relativistic supersymmetry, and can be identified with the massless degrees of freedom of some supersymmetric theories: N=1 supergravity in eleven dimensions (M-theory), type IIB string theory in ten and four dimensions, and in four dimensions, N=8 supergravity, N=4 super-Yang-Mills, and the N=1 Wess-Zumino multiplet. By generalizing this Dirac operator to the affine case, we generate superconformal algebras associated with cosets ${\bf g}/\bf h$, where $\bf h$ contains the {\it space} little group. Only for eleven dimensional supergravity is $\bf h$ simple. This suggests, albeit in a non-relativistic setting, that these may be the limit of theories with underlying two-dimensional superconformal structure. 
  We consider the problem of creating locally supersymmetric theories in signature (10,2). The most natural algebraic starting point is the F-algebra, which is the de Sitter-type (10,2) extension of the super-Poincare algebra. We derive the corresponding geometric group curvatures and evaluate the transformations of the associated gauge fields under the action of an infinitesimal group element. We then discuss the formation of locally supersymmetric actions using these quantities. Due to the absence of any vielbein terms there is no obvious way to define spacetime as such. In addition, there is also no way in which we may naturally construct an action which is linear in the twelve dimensional curvatures. We consider the implications of the simplest possible quadratic theories. We then investigate the relationship between the twelve dimensional theories and Lorentz signature theories in lower dimensions. We argue that in this context the process of dimensional reduction must be replaced by that of group theoretic contraction. Upon contraction a regular spacetime emerges and we find that the twelve dimensional curvature constraint reduces to an Einstein-type equation in which a quadratic non-linearity in the Ricci scalar is suppressed by a factor of the same magnitude as the cosmological constant. Finally, we discuss the degrees of freedom of multi-temporal variables and their relation to ultra-hyperbolic wave equations. 
  In non-Abelian field theories with q-symmetry groups the massive particles have a non-local interpretation with a stringlike spectrum. It is shown that a massless vector similarly acquires a tower of masses by spontaneous symmetry breaking. 
  I construct solutions to Einstein's equations in 6 dimensions with bulk cosmological constant and intersecting 4-branes. Solutions exist for a continuous range of 4-brane tension, with long distance gravity localized to a 3+1 dimensional Minkowski intersection, provided that the additional tension of the intersection satisfies one condition. 
  The general properties of the ordinary and generalized parafermionic algebras are discussed. The generalized parafermionic algebras are proved to be polynomial algebras. The ordinary parafermionic algebras are shown to be connected to the Arik-Coon oscillator algebras. 
  In this short note we review the main features of open-string orbifolds with a quantised flux for the NS-NS antisymmetric tensor in the context of the open descendants of non-supersymmetric asymmetric orbifolds with a vanishing cosmological constant. 
  A relation between the total instanton number and the quantum-numbers of magnetic monopoles that arise in general Abelian gauges in SU(2) Yang-Mills theory is established. The instanton number is expressed as the sum of the `twists' of all monopoles, where the twist is related to a generalized Hopf invariant. The origin of a stronger relation between instantons and monopoles in the Polyakov gauge is discussed. 
  The general criteria for finding bosonic supersymmetric worldvolume solitons is reviewed. We concentrate on D-branes, discussing in particular, bion/dyon solutions and D3 branes on NS5 backgrounds. 
  We propose a 5-dimensional definition for the physical 4D-Yang-Mills theory. The fifth dimension corresponds to the Monte-Carlo time of numerical simulations of QCD_4. The 5-dimensional theory is a well-defined topological quantum field theory that can be renormalized at any given finite order of perturbation theory. The relation to non-perturbative physics is obtained by expressing the theory on a lattice, a la Wilson. The new fields that must be introduced in the context of a topological Yang-Mills theory have a simple lattice expression. We present a 5-dimensional critical limit for physical correlation functions and for dynamical auto-correlations, which allows new Monte-Carlo algorithm based on the time-step in lattice units given by $\e = g_0^{-13/11}$ in pure gluodynamics. The gauge-fixing in five dimensions is such that no Gribov ambiguity occurs. The weight is strictly positive, because all ghost fields have parabolic propagators and yield trivial determinants. We indicate how our 5-dimensional description of the Yang-Mills theory may be extended to fermions. 
  We discuss a (10+2)D N=(1,1) superalgebra and its projections to M-theory, type IIA and IIB algebras. From the complete classification of a second-rank central term valued in the so(10,2) algebra, we find all possible BPS states coming from this term. We show that, among them, there are two types of 1/2-susy BPS configurations; one corresponds to a super (2+2)-brane while another one arises from a nilpotent element in so(10,2). 
  Topological property in a spinning system should be directly associated with its wavefunction. A complete decomposition formula of SU(2) gauge potential in terms of spinning wavefunction is established rigorously. Based on the $\phi $-mapping theory and this formula, one proves that the second Chern class is inherent in the spinning system. It is showed that this topological invariant is only determined by the Hopf index and Brouwer degree of the spinning wavefunction. 
  The quantization of General Relativity invariant with respect to time-reparametrizations is considered. We construct the Faddeev-Popov generating functional for the unitary perturbation theory in terms of invariants of the kinemetric group of diffeomorphisms of a frame of reference as a set of Einstein's observers with the equivalent Hamiltonian description ($t'=t'(t)$, $x'^i=x'^i(t,x^1,x^2,x^3)$). The algebra of the kinemetric group has other dimensions than the constraint algebra in the conventional Dirac-Faddeev-Popov (DFP) approach to quantization. To restore the reparametrization invariance broken in the DFP approach, the invariant dynamic evolution parameter is introduced as the zero Fourier harmonic of the space metric determinant. The unconstrained version of the reparametrization invariant GR is obtained. We research the infinite space-time limit of the Faddeev-Popov generating functional in the theory and discuss physical consequences of the considered quantization. 
  We study open descendants of four dimensional Z_N x Z_M orbifolds of the non-supersymmetric type 0B string theory. An exhaustive analysis shows, that using the crosscap constraint the only model for which one can project out the tachyon is the Z_2 x Z_2 orbifold. For this case we explicitly construct the open string amplitudes. The gauge group corresponding to the various inequivalent Klein bottle projections turns out to be either symplectic or unitary. 
  The local action of an SU(2) gauge theory in general covariant Abelian gauges and the associated equivariant BRST symmetry that guarantees the perturbative renormalizability of the model are given. I show that a global SL(2,R) symmetry of the model is spontaneously broken by ghost-antighost condensation at arbitrarily small coupling and leads to propagators that are finite at Euclidean momenta for all elementary fields except the Abelian ``photon''. The Goldstone states form a BRST-quartet. The mechanism eliminates the non-abelian infrared divergences in the perturbative high-temperature expansion of the free energy. 
  We derive model independent, non-perturbative supersymmetric sum rules for the gravitational quadrupole moments of arbitrary-spin particles in any N=1 supersymmetric theory. These sum rules select a ``preferred'' value of h=1 where the ``h-factor'' is the gravitational quadrupole analog of the gyromagnetic ratio or g-factor. This value of h=1 corresponds identically to the preferred field theory value obtained by tree-level unitarity considerations. The presently derived h-factor sum rule complements and generalizes previous work on electromagnetic moments where g=2 was shown to be preferred by both supersymmetric sum rule and tree-level unitarity arguments. 
  There is substantial evidence that string theory on AdS_5 x S_5 is a holographic theory in which the number of degrees of freedom scales as the area of the boundary in Planck units. Precisely how the theory can describe bulk physics using only surface degrees of freedom is not well understood. A particularly paradoxical situation involves an event deep in the interior of the bulk space. The event must be recorded in the (Schroedinger Picture) state vector of the boundary theory long before a signal, such as a gravitational wave, can propagate from the event to the boundary. In a previous paper with Polchinski, we argued that the "precursor" operators which carry information stored in the wave during the time when it vanishes in a neighborhood of the boundary are necessarily non-local. In this paper we argue that the precursors cannot be products of local gauge invariant operators such as the energy momentum tensor. In fact gauge theories have a class of intrinsically non-local operators which cannot be built from local gauge invariant objects. These are the Wilson loops. We show that the precursors can be identified with Wilson loops whose spatial size is dictated by the UV-IR connection. 
  In the framework of a gauge invariant continuous and non-perturbative regularization scheme based on the smearing of point like interactions by means of cutoff functions, we show that the axial anomaly, though cutoff independent, depends on the shape of the cutoff functions. The standard value for the strength of the axial anomaly is recovered if we assume that the regularized gauge invariant axial current is in addition local. 
  We study the central extensions of the N=1 superalgebras relevant to the soliton solutions with the axial geometry - strings, wall junctions, etc. A general expression valid in any four-dimensional gauge theory is obtained. We prove that the only gauge theory admitting BPS strings at weak coupling is supersymmetric electrodynamics with the Fayet-Iliopoulos term. The problem of ambiguity of the (1/2,1/2) central charge in the generalized Wess-Zumino models and gauge theories with matter is addressed and solved. A possibility of existence of the BPS strings at strong coupling in N=2 theories is discussed. A representation of different strings within the brane picture is presented. 
  The Yuri Golfand Memorial Volume commemorates Thirty Years of Supersymmetry. It will be published soon by World Scientific. The participants of the project are: D. Brace, L. Brink, S. Deser, G. Dvali, B. Feng, D. Freedman, G.-L. Gervais, G. Gabadadze, M. Grisaru, A. Hanany, Y.-H. He, S. Hellerman, E. D'Hoker, P. Fayet, V. Kac, I. Klebanov, N. Koretz-Golfand, D. Kutasov, E. Likhtman, A. Losev, M. Marinov, S. Mathur, A. Matusis, B. Morariu, N. Nekrasov, J. Polchinski, E. Rabinovici, L. Rastelli, P. Ramond, J. Schwarz, N. Seiberg, A. Semikhatov, G. Senjanovic, S. Shatashvili, M. Shifman, A. Smilga, M. Strassler, A. Tseytlin, M. Vasiliev, J. Wess, P. West, E. Witten, B. Zumino. 
  We construct a Dirichlet boundary state for linear dilaton backgrounds. The state is conformally invariant and satisfies Cardy's conditions. We apply this construction to two dimensional string theory. 
  We formulate the field equations for $SU(\infty)$ Einstein-Yang-Mills theory, and find spherically symmetric black-hole solutions. This model may be motivated by string theory considerations, given the enormous gauge symmetries which characterize string theory. The solutions simplify considerably in the presence of a negative cosmological constant, particularly for the limiting cases of a very large cosmological constant or very small gauge field. The situation of an arbitrarily small gauge field is relevant for holography and we comment on the AdS/CFT conjecture in this light. The black holes possess infinite amounts of gauge field hair, and we speculate on possible consequences of this for quantum decoherence, which, however, we do not tackle here. 
  I discuss and extend the recent proposal of Leclair and Mussardo for finite temperature correlation functions in integrable QFTs. I give further justification for its validity in the case of one point functions of conserved quantities. I also argue that the proposal is not correct for two (and higher) point functions, and give some counterexamples to justify that claim. 
  We generalize the proof of the non-renormalization of the four derivative operators in ${\cal N}=4$ Yang Mills theory with gauge group SU(2) to show that certain terms with 2N derivatives are not renormalized in the theory with gauge group SU(N). These terms may be determined exactly by a simple perturbative computation. Similar results hold for finite ${\cal N}=2$ theories. We comment on the implications of these results. 
  We show that the Born-Infeld theory with n complex abelian gauge fields written in an auxiliary field formulation has a U(n,n) duality group. We conjecture the form of the Lagrangian obtained by eliminating the auxiliary fields and then introduce a new reality structure leading to a Born-Infeld theory with n real fields and an Sp(2n,R) duality symmetry. The real and complex constructions are extended to arbitrary even dimensions. The maximal noncompact duality group is U(n,n) for complex fields. For real fields the duality group is Sp(2n,R) if half of the dimension of space-time is even and O(n,n) if it is odd. We also discuss duality under the maximal compact subgroup, which is the self-duality group of the theory obtained by fixing the expectation value of a scalar field. Supersymmetric versions of self-dual theories in four dimensions are also discussed. 
  We discuss deformation quantization of the covariant, light-cone and conformal gauge-fixed p-brane actions (p>1) which are closely related to the structure of the classical and quantum Nambu brackets. It is known that deformation quantization of the Nambu bracket is not of the usual Moyal type. Yet the Nambu bracket can be quantized using the Zariski deformation quantization (discovered by Dito, Flato, Sternheimer and Takhtajan) which is based on factorization of polynomials in several real variables. We discuss a particular application of the Zariski deformed quantization in M-theory by considering the problem of a covariant formulation of Matrix theory. We propose that the problem of a covariant formulation of Matrix theory can be solved using the formalism of Zariski deformed quantization of the triple Nambu bracket. 
  The 3D vector van der Waals (or conformal) nonlinear sigma-model is proposed. It is shown that it has the "hedgehog"-like topological excitations with logarithmic energy. Their "neutral" configurations have nontrivial topological structures described by Hopf invariant. A possible influence of these excitations on the properties of the model are discussed. 
  We start with a simple introduction into the renormalization group (RG) in quantum field theory and give an overview of the renormalization group method. The third section is devoted to essential topics of the renorm-group use in the QFT. Here, some fresh results are included.   Then we turn to the remarkable proliferation of the RG ideas into various fields of physics. The last section summarizes an impressive recent progress of the "QFT renormalization group" application in mathematical physics. 
  We consider typeIIA supergravity solution of D2-branes and D3-branes localized within D6-branes in the near-core region of D6-branes. With these solutions we can calculate the spectrum of the glueball mass in QCD3 and QCD4. The equation of motion describing the dilaton has the same eigenvalues and the same glueball masses in QCD3 and QCD4. Glueball mass spectrum is the same in the near core region of D6-branes of their M-theory counterpart is KK monopole. We conclude that the glueball mass spectrum is the same in QCD3 and QCD4 by considering the `near-core' limit of D6-branes of which M-theory counterpart (KK monopole background) becomes an ALE space with an $A_{N-1}$ singularity times 7 dimensional Minkowski space $M^{(6,1)}$. 
  Noncommutative geometry(NCG) on the discrete space successfully reproduces the Higgs mechanism of the spontaneously broken gauge theory, in which the Higgs boson field is regarded as a kind of gauge field on the discrete space. We could construct the generalized differential geometry(GDG) on the discrete space $M_4\times Z_N$ which is very close to NCG in case of $M_4\times Z_2$. GDG is a direct generalization of the differential geometry on the ordinary manifold into the discrete one. In this paper, we attempt to construct the BRST invariant formulation of spontaneously broken gauge theory based on GDG and obtain the BRST invariant Lagrangian with the t'Hooft-Feynman gauge fixing term. 
  In this paper, we generalize a boundary state to the one incorporating non-constant gauge field strength as an external background coupled to the boundary of a string worldsheet in bosonic string theory. This newly defined boundary state satisfies generalized nonlinear boundary conditions with non-constant gauge field strength, and is BRST invariant. The divergence immanent in this boundary state coincide with the one calculated in a string sigma model. We extract the relevant massless part of this generalized boundary state, and give a part of the D-brane action with the non-constant gauge field strength, that is, derivative corrections to the D-brane action. 
  We identify the lift to M theory of the four types of orientifold points, and show that they involve a chiral fermion on an orbifold fixed circle. From this lift, we compute the number of normalizable ground states for the SO(N) and $Sp(N)$ supersymmetric quantum mechanics with sixteen supercharges. The results agree with known results obtained by the mass deformation method. The mass of the orientifold is identified with the Casimir energy. 
  We investigate (2+1)-dimensional QED coupled with Dirac fermions both at zero and finite temperature. We discuss in details two-components (P-odd) and four-components (P-even) fermion fields. We focus on P-odd and P-even Dirac fermions in presence of an external constant magnetic field. In the spontaneous generation of the magnetic condensate survives even at infinite temperature. We also discuss the spontaneous generation of fermion mass in presence of an external magnetic field. 
  The structures in target space geometry that correspond to conformally invariant boundary conditions in WZW theories are determined both by studying the scattering of closed string states and by investigating the algebra of open string vertex operators. In the limit of large level, we find branes whose world volume is a regular conjugacy class or, in the case of symmetry breaking boundary conditions, a `twined' version thereof. In particular, in this limit one recovers the commutative algebra of functions over the brane world volume, and open strings connecting different branes disappear. At finite level, the branes get smeared out, yet their approximate localization at (twined) conjugacy classes can be detected unambiguously.   As a by-product, it is demonstrated how the pentagon identity and tetrahedral symmetry imply that in any rational conformal field theory the structure constants of the algebra of boundary operators coincide with specific entries of fusing matrices. 
  We describe an extension of the nonlinear integral equation (NLIE) method to Virasoro minimal models perturbed by the relevant operator $\Phi_{(1,3)$. Along the way, we also complete our previous studies of the finite volume spectrum of sine-Gordon theory by considering the attractive regime and more specifically, breather states. For the minimal models, we examine the states with zero topological charge in detail, and give numerical comparison to TBA and TCS results. We think that the evidence presented strongly supports the validity of the NLIE description of perturbed minimal models. 
  We find the Seiberg-Witten geometry for four dimensional N=2 supersymmetric E_6 gauge theories with massless fundamental hypermultiplets, by geometrically embedding them in type II string theories compactified on Calabi-Yau threefolds. The resulting geometry completely agrees with that of recent works, which are based on the technique of N=1 confining phase superpotentials. We also derive the Seiberg-Witten geometry for E_7 gauge theories with massive fundamental hypermultiplets. 
  The quantisation of gauge invariant systems usually proceeds through some gauge fixing procedure of one type or another. Typically for most cases, such gauge fixings are plagued by Gribov ambiguities, while it is only for an admissible gauge fixing that the correct dynamical description of the system is represented, especially with regards to non perturbative phenomena. However, any gauge fixing procedure whatsoever may be avoided altogether, by using rather a recently proposed new approach based on the projection operator onto physical gauge invariant states only, which is necessarily free on any such issues. These different aspects of gauge invariant systems are explicitely analysed within a solvable U(1) gauge invariant quantum mechanical model related to the dimensional reduction of Yang-Mills theory. 
  We present unique solutions of the Seiberg-Witten Monopole Equations in which the U(1) curvature is covariantly constant, the monopole Weyl spinor consists of a single constant component, and the 4-manifold is a product of two Riemann surfaces of genuses p_1 and p_2. There are p_1 -1 magnetic vortices on one surface and p_2 - 1 electric ones on the other, with p_1 + p_2 \geq 2 p_1 = p_2= 1 being excluded). When p_1 = p_2, the electromagnetic fields are self-dual and one also has a solution of the coupled euclidean Einstein-Maxwell-Dirac equations, with the monopole condensate serving as cosmological constant. The metric is decomposable and the electromagnetic fields are covariantly constant as in the Bertotti-Robinson solution. The Einstein metric can also be derived from a K\"{a}hler potential satisfying the Monge-Amp\`{e}re equations. 
  We rederive the nonrelativistic Lagrangian for the low energy dynamics of 1/4 BPS dyons by considering the time dependent fluctuations around classical 1/4 BPS configurations. The relevant fluctuations are the zero modes of the underlying 1/2 BPS monopoles. 
  It is argued that a low energy string may be an Aristotelian top, i.e. a rigid body which however cannot be rotationally excited, but in an external electromagnetic field exhibits a sort of precession with a Larmor type angular velocity. On the basis of this observation a proposal is made for a new two steps derivation of the electroweak standard model from the string dynamics. In the first step a theory of Aristotelian top is formulated and studied in more detail and in the second step an attempt is made to derive the electroweak standard model from the top dynamics. Before the symmetry breaking fermions are represented by straight frozen strings - rotators, whose symmetry under the rotation around their axes is interpreted as the group $U(1)_Y$. The emergence of $SU(2)_L$ group is somewhat less transparent and is supposed to be connected with the new degree of freedom of relativistic rotators, which leads to the up and down type fermions. The symmetry breaking is associated with the bending of the rotators under the influence of Higgs field and with their subsequent transformation into the curved frozen strings - tops. In this new picture of electroweak interaction the chirality of the theory has a simple and natural explanation, the weak isospin and hypercharge are inherent properties of relativistic rotators/tops and the superselection rule associated with electric charge is a consequence of the accepted distinction between up and down type fermions. 
  As shown by Taubes, in the Bogomol'nyi-Prasad-Sommerfield limit the SU(2) Yang-Mills-Higgs model possesses smooth finite energy solutions, which do not satisfy the first order Bogomol'nyi equations. We construct numerically such a non-Bogomol'nyi solution, corresponding to a monopole-antimonopole pair, and extend the construction to finite Higgs potential. 
  A model for the fundamental structure of nature is presented. It is based on two fundamental fermions moving with the velocity of light and differing from each other by the projection of the spin on the momentum vector. The energy of both fermions is proportional to the momentum, which is scaled inversely with the size of a length quantum. All the known forces are a manifestation of one elementary interaction, the spin exchange or spin flip-flop which takes place when two different elementary fermions come together in the same space cell. At this stage the model can explain the properties of the photon as a two-fermion particle and it can be shown that the Dirac theory for relativistic fermions could be deduced from this model. The model predicts that particles like the electron or the quark are stable combinations of a large number of the fundamental fermions, but proof of that prediction has not been given. 
  We obtain, for a subclass of structure functions characterizing a first class Hamiltonian system, recursive relations from which the general form of the local symmetry transformations can be constructed in terms of the independent gauge parameters. We apply this to a non-trivial Hamiltonian system involving two primary constraints, as well as two secondary constraints of the Nambu-Goto type. 
  Talk presented at Strings '99 in Potsdam, Germany (July 19 - 24, 1999). 
  We consider the sector of N=8 five-dimensional gauged supergravity with non-trivial scalar fields in the coset space SL(6,R)/SO(6), plus the metric. We find that the most general supersymmetric solution is parametrized by six real moduli and analyze its properties using the theory of algebraic curves. In the generic case, where no continuous subgroup of the original SO(6) symmetry remains unbroken, the algebraic curve of the corresponding solution is a Riemann surface of genus seven. When some cycles shrink to zero size the symmetry group is enhanced, whereas the genus of the Riemann surface is lowered accordingly. The uniformization of the curves is carried out explicitly and yields various supersymmetric configurations in terms of elliptic functions. We also analyze the ten-dimensional type-IIB supergravity origin of our solutions and show that they represent the gravitational field of a large number of D3-branes continuously distributed on hyper-surfaces embedded in the six-dimensional space transverse to the branes. The spectra of massless scalar and graviton excitations are also studied on these backgrounds by casting the associated differential equations into Schrodinger equations with non-trivial potentials. The potentials are found to be of Calogero type, rational or elliptic, depending on the background configuration that is used. 
  Exploiting the quantum integrability condition we construct an ancestor model associated with a new underlying quadratic algebra. This ancestor model represents an exactly integrable quantum lattice inhomogeneous anisotropic model and at its various realizations and limits can generate a wide range of integrable models. They cover quantum lattice as well as field models associated with the quantum $R$-matrix of trigonometric type or at the undeformed $q \to 1$ limit similar models belonging to the rational class. The classical limit likewise yields the corresponding classical discrete and field models. Thus along with the generation of known integrable models in a unifying way a new class of inhomogeneous models including variable mass sine-Gordon model, inhomogeneous Toda chain, impure spin chains etc. are constructed. 
  An analysis of one and two point functions of the energy momentum tensor on homogeneous spaces of constant curvature is undertaken. The possibility of proving a $c$-theorem in this framework is discussed, in particular in relation to the coefficients $c,a$, which appear in the energy momentum tensor trace on general curved backgrounds in four dimensions. Ward identities relating the correlation functions are derived and explicit expressions are obtained for free scalar, spinor field theories in general dimensions and also free vector fields in dimension four. A natural geometric formalism which is independent of any choice of coordinates is used and the role of conformal symmetries on such constant curvature spaces is analysed. The results are shown to be constrained by the operator product expansion. For negative curvature the spectral representation, involving unitary positive energy representations of $O(d-1,2)$, for two point functions of vector currents is derived in detail and extended to the energy momentum tensor by analogy. It is demonstrated that, at non coincident points, the two point functions are not related to $a$ in any direct fashion and there is no straightforward demonstration obtainable in this framework of irreversibility under renormalisation group flow of any function of the couplings for four dimensional field theories which reduces to $a$ at fixed points. 
  A crucial element of scattering theory and the LSZ reduction formula is the assumption that the coupling vanishes at large times. This is known not to hold for the theories of the Standard Model and in general such asymptotic dynamics is not well understood. We give a description of asymptotic dynamics in field theories which incorporates the important features of weak convergence and physical boundary conditions. Applications to theories with three and four point interactions are presented and the results are shown to be completely consistent with the results of perturbation theory. 
  Neutral fermions of spin 1/2 with magnetic moment can interact with electromagnetic fields through nonminimal coupling. In 2+1 dimensions the electromagnetic field strength plays the same role to the magnetic moment as the vector potential to the electric charge. This duality enables one to obtain physical results for neutral particles from known ones for charged particles. We give the probability of neutral particle-antiparticle pair creation in the vacuum by non-uniform electromagnetic fields produced by constant uniform charge and current densities. 
  We renormalize to three loops a version of the Thirring model where the fermion fields not only lie in the fundamental representation of a non-abelian colour group SU(N_c) but also depend on the number of flavours, N_f. The model is not multiplicatively renormalizable in dimensional regularization due to the generation of evanescent operators which emerge at each loop order. Their effect in the construction of the true wave function, mass and coupling constant renormalization constants is handled by considering the projection technique to a new order. Having constructed the MSbar renormalization group functions we consider other massless independent renormalization schemes to ensure that the renormalization is consistent with the equivalence of the non-abelian Thirring model with other models with a four-fermi interaction. One feature to emerge from the computation is the establishment of the fact that the SU(N_f) Gross Neveu model is not multiplicatively renormalizable in dimensional regularization. An evanescent operator arises first at three loops and we determine its associated renormalization constant explicitly. 
  We find an exact, N=1 supersymmetric kink solution of 5d gauged supergravity. We associate this solution with the RG flow from N=4 super Yang-Mills theory, deformed by a relevant operator, to pure N=1 super Yang-Mills in the IR. We test this identification by computing various QFT quantities using the supergravity dual: the tension of electric and magnetic strings and the gaugino condensate. As demanded by our identification, our kink solution is a true deformation of N=4, that exhibits confinement of quarks, magnetic screening, and spontaneous chiral symmetry breaking. 
  We construct a model consisting of many D3-branes with only positive tension in a five-dimensional anti-de Sitter space-time geometry. It is shown that this type of model naturally realizes not only exponential mass hierarchy between the Planck scale and the electroweak scale but also trapping of the graviton on the D3-branes. It is pointed out that our model may have a flexibility to explain the existence of more than one disparate mass scales, such as the electroweak scale and the GUT scale, on the same D3-brane. 
  Introductory lectures on the relations between the thermodynamics of gauge theory on branes and black holes, including the correspondence principle of Horowitz and Polchinski, the AdS/CFT conjecture, and matrix theory. 
  A non-abelian topological quantum field theory describing the scattering of self-dual field configurations over topologically non-trivial Riemann surfaces, arising from the reduction of 4-dim self-dual Yang-Mills fields, is introduced. It is shown that the phase space of the theory can be exactly quantized in terms of the space of holomorphic structures over stable vector bundles of degree zero over Riemann surfaces. The Dirac monopoles are particular static solutions of the field equations. Its relation to topological gravity is discussed. 
  The hamiltonian formulation of the supersymmetric closed 2-brane dual to the double compactified D=11 closed supermembrane is presented. The formulation is in terms of two U(1) vector fields related by the area preserving constraint of the SUSY 2-brane. Stable solutions of the field equations, which are local minima of the hamiltonian, are found. In the semiclassical approximation around the stable solutions the action becomes the reduction of D=10 Super-Maxwell to the worldvolume. The solutions carry RR charges as a type of magnetic charges associated with the worldvolume vector field. The geometrical interpretation of the solution in terms of U(1) line bundles over the worldvolume is obtained. 
  We briefly review the nonlinear sigma model approach for the subject of increasing interest: "two-step" phase transitions in the Gross-Neveu and the modified Nambu-Jona-Lasinio models at low $N$ and condensation from pseudogap phase in strong-coupling superconductors. Recent success in describing "Bose-type" superconductors that possess two characterstic temperatures and a pseudogap above $T_c$ is the development approximately comparable with the BCS theory. One can expect that it should have influence on high-energy physics, similar to impact of the BCS theory on this subject. Although first generalizations of this concept to particle physics were made recently, these results were not systematized. In this review we summarize this development and discuss similarities and differences of the appearence of the pseudogap phase in superconductors and the Gross-Neveu and Nambu-Jona-Lasinio - like models. We discuss its possible relevance for chiral phase transition in QCD and color superconductors. This paper is organized in three parts: in the first section we briefly review the separation of temperatures of pair formation and pair condensation in strong - coupling and low carrier density superconductors (i.e. the formation of the {\it pseudogap phase}).   Second part is a review of nonlinear sigma model approach to an analogous phenomenon in the Chiral Gross-Neveu model at small N. In the third section we discuss the modified Nambu-Jona-Lasinio model where the chiral phase transition is accompanied by a formation of a phase analogous to the pseudogap phase. 
  We obtain the general inflationary solutions for the slab of five-dimensional AdS spacetime where the fifth dimension is an orbifold $S^1/Z_2$ and two three-branes reside at its boundaries, of which the Randall-Sundrum model corresponds to the static limit. The investigation of the general solutions and their static limit reveals that the RS model recasts both the cosmological constant problem and the gauge hierarchy problem into the balancing problem of the bulk and the brane cosmological constants. 
  We review open descendants of non-supersymmetric type IIB asymmetric orbifolds with zero cosmological constant. We find that supersymmetry remains unbroken on the branes at all mass levels, whereas it is broken in the bulk. 
  We establish a direct correspondence between certain higher-rank p-form Chern-Simons topological type theories in the bulk of a manifold with boundary and particular sectors of supergravity models on the boundary, provided that certain boundary conditions are satisfied. The cases we investigate include eleven-dimensional supergravity and both of the type II theories in ten dimensions. 
  We investigate the decay of a scalar field outside a Schwarzschild anti de Sitter black hole. This is determined by computing the complex frequencies associated with quasinormal modes. There are qualitative differences from the asymptotically flat case, even in the limit of small black holes. In particular, for a given angular dependence, the decay is always exponential - there are no power law tails at late times. In terms of the AdS/CFT correspondence, a large black hole corresponds to an approximately thermal state in the field theory, and the decay of the scalar field corresponds to the decay of a perturbation of this state. Thus one obtains the timescale for the approach to thermal equilibrium. We compute these timescales for the strongly coupled field theories in three, four, and six dimensions which are dual to string theory in asymptotically AdS spacetimes. 
  A particular initial state for the construction of the perturbative expansion of QCD is investigated. It is formed as a coherent superposition of zero momentum gluon pairs and shows Lorentz as well as global SU(3) symmetries. It follows that the gluon and ghost propagators determined by it, coincides with the ones used in an alternative of the usual perturbation theory proposed in a previous work. Therefore, the ability of such a procedure of producing a finite gluon condensation parameter already in the first orders of perturbation theory is naturally explained. It also follows that this state satisfies the physicality condition of the BRST procedure in its Kugo and Ojima formulation. The BRST quantization is done for the value alpha=1 of the gauge parameter where the procedure is greatly simplified. Therefore, after assuming that the adiabatic connection of the interaction does not take out the state from the interacting physical space, the predictions of the perturbation expansion, at the value alpha=1, for the physical quantities should have meaning. The validity of this conclusion solves the gauge dependence indeterminacy remained in the proposed perturbation expansion. 
  We address a supersymmetric embedding of domain walls with asymptotically anti-deSitter (AdS) space-times in five-dimensional simple, N=2 U(1) gauged supergravity theory constructed by Gunaydin, Townsend and Sierra. These conformally flat solutions interpolate between supersymmetric AdS vacua, satisfy the Killing spinor (first order) differential equations, and the four-dimensional world on the domain wall is a flat world with N=1 supersymmetry. Regular solutions in this class have the energy density related to the cosmological constants of the supersymmetric AdS vacua. An analysis of such solutions is given for the example of one (real, neutral) vector supermultiplet with the most general form of the prepotential. There are at most two supersymmetric AdS vacua that are in general separated by a singularity in the potential. Nevertheless the supersymmetric domain wall solution exists with the scalar field interpolating continuously across the singular region. 
  Closed string dynamics in the presence of noncommutative Dp-branes is investigated. In particular, we compute bulk closed string two-point scattering amplitudes; the bulk space-time geometries encoded in the amplitudes are shown to be consistent with the recently proposed background space-time geometries dual to noncommutative Yang-Mills theories. Three-point closed string absorption/emission amplitudes are obtained to show some features of noncommutative Dp-branes, such as modified pole structures and exponential phase factors linearly proportional to the external closed string momentum. 
  We show that correlation functions for branched polymers correspond to those for $\phi^3$ theory with a single mass insertion, not those for the $\phi^3$ theory themselves, as has been widely believed. In particular, the two-point function behaves as 1/p^4, not as 1/p^2. This behavior is consistent with the fact that the Hausdorff dimension of the branched polymer is four. 
  Aspects of superstring cosmology are reviewed with an emphasis on the cosmological implications of duality symmetries in the theory. The string effective actions are summarized and toroidal compactification to four dimensions reviewed. Global symmetries that arise in the compactification are discussed and the duality relationships between the string effective actions are then highlighted. Higher-dimensional Kasner cosmologies are presented and interpreted in both string and Einstein frames, and then given in dimensionally reduced forms. String cosmologies containing both non-trivial Neveu-Schwarz/Neveu-Schwarz and Ramond-Ramond fields are derived by employing the global symmetries of the effective actions. Anisotropic and inhomogeneous cosmologies in four-dimensions are also developed. The review concludes with a detailed analysis of the pre-big bang inflationary scenario. The generation of primordial spectra of cosmological perturbations in such a scenario is discussed. Possible future directions offered in the Horava-Witten theory are outlined. 
  We construct the world-volume action for non-BPS D-branes in type II string theories. This action is invariant under all the unbroken supersymmetries in the bulk, but these symmetries are realised as spontaneously broken symmetries in the world-volume theory. Coupling of this action to background supergravity fields is straightforward. We also discuss the fate of the U(1) gauge field on the D-brane world-volume after tachyon condensation. 
  String theory on curved backgrounds has received much attention on account of both its own interest, and of its relation with gauge theories. Despite the progress made in various directions, several quite elementary questions remain unanswered, in particular in the very simple case of three-dimensional anti-de Sitter space. I briefly review these problems. 
  A superspace formulation is proposed for the osp(1,2)-covariant Lagrangian quantization of general massive gauge theories. The superalgebra os0(1,2) is considered as subalgebra of sl(1,2); the latter may be considered as the algebra of generators of the conformal group in a superspace with two anticommuting coordinates. The mass-dependent (anti)BRST symmetries of proper solutions of the quantum master equations in the osp(1,2)-covariant formalism are realized in that superspace as invariance under translations combined with mass-dependent special conformal transformations. The Sp(2) symmetry - in particular the ghost number conservation - and the "new ghost number" conservation are realized as invariance under symplectic rotations and dilatations, respectively. The transformations of the gauge fields - and of the full set of necessarily required (anti)ghost and auxiliary fields - under the superalgebra sl(1,2) are determined both for irreducible and first-stage reducible theories with closed gauge algebra. 
  We study the spin-spin, spin-energy and energy-energy correlators in the 2d Ising model perturbed by a magnetic field. We compare the results of a set of high precision Montecarlo simulations with the predictions of two different approximations: the Form Factor approach, based on the exact S-matrix description of the model, and a short distance perturbative expansion around the conformal point. Both methods give very good results, the first one performs better for distances larger than the correlation length, while the second one is more precise for distances smaller than the correlation length. In order to improve this agreement we extend the perturbative analysis to the second order in the derivatives of the OPE constants. 
  We review recent results on the derivation of a global path integral density for Yang-Mills theory. Based on a generalization of the stochastic quantization scheme and its geometrical interpretation we first recall how locally a modified Faddeev-Popov path integral density for the quantization of Yang-Mills theory can be derived, the modification consisting in the presence of specific finite contributions of the pure gauge degrees of freedom. Due to the Gribov problem the gauge fixing can be defined only locally and the whole space of gauge potentials has to be partitioned into patches. We discuss a global extension of the path integral by summing over all patches, which can be proven to be manifestly independent of the specific local choices of patches and gauge fixing conditions, respectively. In addition to the formulation on the whole space of gauge potentials we discuss the corresponding path integral on the gauge orbit space. 
  The modular transformation properties of admissible characters of the affine superalgebra sl(2|1;C) at fractional level k=1/u-1, u=2,3,... are presented. All modular invariants for u=2 and u=3 are calculated explicitly and an A-series and D-series of modular invariants emerge. 
  We use logarithmic conformal field theory techniques to describe recoil effects in the scattering of two Dirichlet branes in D dimensions. In the particular case that a D1 brane strikes a D3 brane perpendicularly, thereby folding it, we find that the recoil space-time is maximally symmetric, with AdS_3 x E_{D-3} geometry. We comment on the possible applications of this result to the study of transitions between different background metrics. 
  Some general aspects of the AdS_2/CFT_1 correspondence, previously discussed in hep-th/9812073, are summarized. The majority of this summary is devoted to the question of where the CFT_1 should live. Almost as a byproduct, the decay---or fragmentation---of AdS_2 is also discussed. 
  First-order `Bogomol'nyi' equations are found for dilaton domain walls of D-dimensional gravity with the general dilaton potential admitting a stable anti-de Sitter vacuum. Implications for renormalization group flow in the holographically dual field theory are discussed. 
  We discuss absorption of scalars by a distribution of spinning D3-branes. The D3-branes are multi-center solutions of supergravity theory. We solve the wave equation in various cases of supergravity backgrounds in which the equation becomes separable. We show that the absorption coefficients exhibit a universal behavior as functions of the angular momentum quantum number and the Hawking temperature. This behavior is similar to the form of the gray-body factors one encounters in scattering by black-holes. Our discussion includes the problematic case of spherically symmetric distributions of D-branes, where resonances arise. We obtain the same universal form for the absorption coefficients, if the region enclosed by the D-branes is excluded from physical considerations. Non-extremal D-branes are also discussed. The results are similar to the extremal cases, albeit at half the Hawking temperature. We speculate that new degrees of freedom enter as one moves away from extremality. 
  We propose a simple universal formula for the tension of a D-brane in terms of a regularized dimension of the associated conformal field theory statespace. 
  We consider instanton contributions to chiral correlators, such as <0| Tr \lambda^2 (x) Tr \lambda^2(x') |0>, in N=1 supersymmetric Yang-Mills theory with either light adjoint or fundamental matter. Within the former model, extraction of the gluino condensate from a connected 1-instanton diagram, evaluated at strong coupling, can be contrasted with expectations from the Seiberg-Witten solution perturbed to an N=1 vacuum. We observe a numerical discrepancy, coinciding with that observed previously in N=1 SQCD. Moreover, since knowledge of the vacuum structure is complete for softly broken N=2 Yang-Mills, this model serves as a counterexample to the hypothesis of Amati et al. that 1-instanton calculations at strong coupling can be interpreted as averaging over vacua. Within N=1 SQCD, we point out that the connected contribution to the relevant correlators actually vanishes in the weakly coupled Higgs phase, despite having a nonzero value through infra-red effects when calculated in the unbroken phase. 
  In the present article we analyze the phenomenon of particle creation in a cosmological anisotropic universe when a constant electric field is present. We compute, via the Bogoliubov transformations, the density number of particles created. 
  We study the expectation value of (the product) of the one-particle projector (s) in the reduced matrix model and matrix quantum mechanics in general. This quantity is given by the nonabelian Berry phase: we discuss the relevance of this with regard to the spacetime structure. The case of the USp matrix model is examined from this respect. Generalizing our previous work, we carry out the complete computation of this quantity which takes into account both the nature of the degeneracy of the fermions and the presence of the space time points belonging to the antisymmetric representation. We find the singularities as those of the SU(2) Yang monopole connection as well as the pointlike singularities in 9+1 dimensions coming from its SU(8) generalization. The former type of singularities, which extend to four of the directions lying in the antisymmetric representations, may be regarded as seeds of our four dimensional spacetime structure and is not shared by the IIB matrix model. From a mathematical viewpoint, these connections can be generalizable to arbitrary odd space dimensions due to the nontrivial nature of the eigenbundle and the Clifford module structure. 
  Recently, Randall and Sundrum proposed a static solution to Einstein's equations in five spacetime dimensions with two 3-branes located at the fixed points of $S^1/Z_2$ to solve the hierarchy problem. We extend the solution and construct static and also inflationary solutions to Einstein's equations in five spacetime dimensions, one of which is compactified on $S^1$, with any number of 3-branes whose locations are taken to be arbitrary. We discuss how the hierarchy problem can be explained in our model. 
  We study the dynamics and thermodynamics of a probe D3-brane in the rotating D3-brane background and in its extremal limit, which is a multicenter configuration of D3-branes distributed uniformly on a disc. In the extremal background, if the angular momentum of the probe does not vanish, the probe is always bounced back at some turning point. When its angular momentum vanishes, in the disc plane, the probe will be captured at the edge of the disc; in the hyperplane orthogonal to the disc, the probe will be absorbed at the center of the disc. In the non-extremal background, if the probe is in the hyperplane orthogonal to the disc, it will be captured at the horizon; if the probe is restricted in the disc plane, the probe will be bounced back at a turning point, which is just the infinite red-shift hyperplane of the rotating background, even when the angular momentum of the probe vanishes. The thermodynamics of a relative static D3-brane probe is also investigated to the rotating D3-brane source. Two critical points are found. One is just the thermodynamically stable boundary of the source rotating D3-branes; the other is related to the distance between the probe and the source, which can be regarded as the mass scale in the corresponding super Yang-Mills theory. If the probe is static, the second critical point occurs as the probe is at the infinite red-shift hyperplane of the background. The relevance to the thermodynamics of the super Yang-Mills theory is discussed briefly. 
  We introduce an explicit form of the multi-instanton weight including also instanton--anti-instanton interactions for arbitrary $N_c$ in the two-dimensional $CP^{N_c-1}$ model. To that end, we use the parametrization of multi-instantons in terms of instanton `constituents' which we call `zindons' for short. We study the statistical mechanics of zindons analytically (by means of the Debye-H\"uckel approximation) and numerically (running a Metropolis algorithm). Though the zindon parametrization allows for a complete `melting' of instantons we find that, through a combination of dynamical and purely geometric factors, a dominant portion of topological charge is residing in well-separated instantons and anti-instantons. 
  In this article the interaction of branes at angles with respect to each other with non-zero internal gauge fields are calculated by construction of the boundary states in spacetime in which some of its directions are compact on tori. The interaction depends on both angle and fields. 
  We study the significance of T-duality in the context of the gravitational description of gauge theories. We found that T-duality relates the deferents points of the moduli of a given gauge theory always far from the conformal fixed point. Also the described gauge theories seems to flow naturally to the able conformal points, those that naturally saturate all the possible known examples of near horizon geometries. Supersymmetry properties and T-duality breaking of it are discuss. 
  We discuss the physics of a single Dp-brane in the presence of a background electromagnetic field B_{ij}. It has recently been shown \cite{SW} that, in a specific \alpha ' \to 0 limit, the physics of the brane is correctly described by noncommutative Yang-Mills theory, where the noncommutative gauge potential is given explicitly in terms of the ordinary U(1) field. In a previous paper \cite{SC} the physics of a D2-brane was analyzed in the Sen-Seiberg limit of M(atrix) theory by considering a specific coordinate change on the brane world-volume. We show in this note that the limit considered in \cite{SC} is the same as the one described in \cite{SW}, in the specific case p=2, rk B_{ij} = 2. Moreover we show that the coordinate change in \cite{SC} can be reinterpreted, in the spirit of \cite{SW}, as a field redefinition of the ordinary Yang-Mills field, and we prove that the transformations agree for large backgrounds. The results are finally used to considerably streamline the proof of the equivalence of the standard Born-Infeld action with noncommutative Yang-Mills theory, in the large wave-length regime. 
  We discuss four dimensional renormalization group flows which preserve sixteen supersymmetries. In the infra-red, these can be viewed as deformations of the N=4 superconformal fixed points by special, irrelevant operators. It is argued that the gauge coupling beta function continues to vanish identically, for all coupling constants and energy scales, for such RG flows. In addition, the dimensions of all operators in short supersymmetry representations are constant along such flows. It is conjectured that there is a generalization of the AdS/CFT holography correspondence which describes such flows, e.g. the D3 brane vacuum before taking the near-horizon limit, at all energy scales. RG flows in three and six dimensions, preserving 16 supersymmetries, are also briefly discussed, including a conjectured generalized AdS/CFT duality for the M2 and M5 brane cases. Finally, we discuss maximally supersymmetric RG flows associated with non-commutative geometry. 
  I review a recent progress towards solution of the Almost Mathieu equation (A.G. Abanov, J.C. Talstra, P.B. Wiegmann, Nucl. Phys. B 525, 571, 1998), known also as Harper's equation or Azbel-Hofstadter problem. The spectrum of this equation is known to be a pure singular continuum with a rich hierarchical structure. Few years ago it has been found that the almost Mathieu operator is integrable. An asymptotic solution of this operator became possible due analysis the Bethe Ansatz equations. 
  In the old papers of Ogievetskii and Polubarinov, Hayashi, Kalb and Ramond the {\it notoph} concept, the longitudinal field originated from the antisymmetric tensor, has been proposed. In our work we analyze the theory of antisymmetric tensor field of the second rank from a viewpoint of the normalization problem. We obtain the 4-potentials and field strengths, which coincide with those which have been previously obtained in the works of Ahluwalia and Dvoeglazov. Slightly modifying the Bargmann-Wigner field function we conclude that it is possible to describe explicitly the degrees of freedom of the photon and of the {\it notoph} by the same equation. The physical consequences, such as parity properties of field functions, are discussed, relations to the previous works are discussed as well. Moreover, we derive equations for {\it symmetric} tensor of the second rank on the basis of the same modification of the Bargmann-Wigner formalism, i.e. the equations which describe dynamical behavior of the fields of maximal spin 2. 
  In this note a lately proposed gravity dual of noncommutative Yang-Mills theory is derived from the relations, recently suggested by Seiberg and Witten, between closed string moduli and open string moduli. The only new input one needs is a simple form of the running string tension as a function of energy. This derivation provides convincing evidence that string theory integrates with the holographical principle, and demonstrates a direct link between noncommutative Yang-Mills theory and holography. 
  In a wide class of D-dimensional spacetimes which are direct or semi-direct sums of a (D-n)-dimensional space and an n-dimensional homogeneous ``internal'' space, a field can be decomposed into modes. As a result of this mode decomposition, the main objects which characterize the free quantum field, such as Green functions and heat kernels, can effectively be reduced to objects in a (D-n)-dimensional spacetime with an external dilaton field. We study the problem of the dimensional reduction of the effective action for such spacetimes. While before renormalization the original D-dimensional effective action can be presented as a ``sum over modes'' of (D-n)-dimensional effective actions, this property is violated after renormalization. We calculate the corresponding anomalous terms explicitly, illustrating the effect with some simple examples. 
  We rewrite the SL(2,Z) covariant worldsheet action for the IIB string proposed by Townsend in a Polyakov form. In a flat background the formalism yields separate (p,q) sectors. In each one the action is that of the IIB string action with the string slope parameter \alp\pr replaced by its SL(2,Z) analogue \alp_{pq}\pr. SL(2,Z) invariant graviton scattering amplitudes are obtained from those of the fundamental (1,0) string by summing over the different sectors. The tree-level four-graviton amplitude in this formalism differs from a previously conjectured non-perturbative form; both yield the same expansion in order \alpha\pr^3. 
  We consider black strings in five dimensions and their description as a (4,0) CFT. The CFT moduli space is described explicitly, including its subtle global structure. BPS conditions and global symmetries determine the spectrum of charged excitations, leading to an entropy formula for near-extreme black holes in four dimensions with arbitrary charge vector. In the BPS limit, this formula reduces to the quartic E(7,7) invariant. The prospects for a description of the (4,0) theory as a solvable CFT are explored. 
  We study global worldsheet anomalies for open strings ending on several coincident D-branes in the presence of a B-field. We show that cancellation of anomalies is made possible by a correlation between the t'Hooft magnetic flux on the D-branes and the topological class of the B-field. One application of our results is a proper understanding of the geometric nature of the gauge field living on D-branes: rather than being a connection on a vector bundle, it is a connection on a module over a certain noncommutative algebra. Our argument works for a general closed string background. We also explain why in the presence of a topologically nontrivial B-field whose curvature is pure torsion D-branes represent classes in a twisted K-theory, as conjectured by E. Witten. 
  In [hep-th/9907222] Hannibal claims to exclude the existence of particle-like static axially symmetric non-abelian solutions in SU(2) Einstein-Yang-Mills-dilaton theory. His argument is based on the asymptotic behaviour of such solutions. Here we disprove his claim by giving explicitly the asymptotic form of non-abelian solutions with winding number n=2. 
  We consider the entropy of a black hole which has zero area horizon. The microstates appear as monopole solutions of the effective theory on the corresponding brane configurations.The resulting entropy formula coincides with the one expected from stringy calculation and agrees with U-duality invariance of entropy. 
  We consider null bosonic p-branes moving in curved space-times. Some exact solutions of the classical equations of motion and of the constraints for the null string and the null membrane in Demianski-Newman background are found. 
  We review some aspects of Polyakov's proposal for constructing nonsupersymmetric field theories from non-critical Type 0 string theory. 
  Motivated by a system consisting of a number of parallel M5-branes, we study possible local deformations of a chiral two-form in six dimensions. Working to first order in the coupling constant, this reduces to the study of the local BRST cohomological group at ghost number zero. We obtain an exhaustive list of all possible deformations. None of them allows for a satisfactory formulation of the M5-branes system leading to the conclusion that no local field theory can describe such a system. 
  Following our previous paper (hep-th/9909027), we generalize a supersymmetric boundary state so that arbitrary configuration of the gauge field coupled to the boundary of the worldsheet is incorpolated. This generalized boundary state is BRST invariant and satisfy the non-linear boundary conditions with non-constant gauge field strength. This boundary state contains divergence which is identical with the loop divergence in a superstring sigma model. Hence vanishing of the beta function in the superstring sigma model corresponds to a well-defined boundary state with no divergence. The coupling of a single closed superstring massless mode with multiple open string massless modes is encoded in the boundary state, and we confirm that derivative correction to the D-brane action in this sector vanishes up to the first non-trivial order O(alpha'(derivative)^2). Combining T-dualities, we incorpolate also general configurations of the scalar fields on the D-brane, and construct boundary states representing branes stuck to another D-brane, with use of BIon configuration. 
  We present a generalization of calibrations in which the calibration form is not closed. We use this to examine a class of supersymmetric p-brane worldvolume solitons.As an example we consider M5-brane worldvolume solitons in an AdS background. 
  The Chern-Simons formulation of $AdS_3$ supergravity is considered. Asymptotic conditions on the Rarita-Schwinger fields are given. Together with the known boundary conditions on the bosonic fields, these ensure that the asymptotic algebra is the superconformal algebra, with the same central charge as the one of pure gravity. It is also indicated that the asymptotic dynamics is described by super-Liouville. 
  A state saturating a BPS bound derived from a supersymmetry algebra preserves some fraction of the supersymmetry. This fraction of supersymmetry depends on the charges carried by the system, and we show that in general there are configurations of charges for which a BPS state would preserve more than half the original supersymmetry. We investigate configurations that could preserve 3/4 supersymmetry in string theory, M-theory and supersymmetric field theories and discuss whether states saturating these bounds actually occur in these theories. 
  It is well known that the toroidal dimensional reduction of supergravities gives rise in three dimensions to theories whose bosonic sectors are described purely in terms of scalar degrees of freedom, which parameterise sigma-model coset spaces. For example, the reduction of eleven-dimensional supergravity gives rise to an E_8/SO(16) coset Lagrangian. In this paper, we dispense with the restrictions of supersymmetry, and study all the three-dimensional scalar sigma models G/H where G is a maximally-non-compact simple group, with H its maximal compact subgroup, and find the highest dimensions from which they can be obtained by Kaluza-Klein reduction. A magic triangle emerges with a duality between rank and dimension. Interesting also are the cases of Hermitean symmetric spaces and quaternionic spaces. 
  I investigate bosonization in four dimensions, using the smooth bosonization scheme. I argue that generalized chiral ``phases'' of the fermion field corresponding to chiral phase rotations and ``chiral Poincare transformations'' are the appropriate degrees of freedom for bosonization. Smooth bosonization is then applied to an Abelian fermion coupled to an external vector. The result is an exact rewriting of the theory, including the fermion, the bosonic fields, and ghosts. Exact bosonization is therefore not achieved since the fermion and the ghosts are not completely eliminated. The action for the bosons is given by the Jacobian of a change of variables in the path integral, and I calculate parts of this. The action describes a nonlinear field theory, and thus static, topologically stable solitons may exist in the bosonic sector of the theory, which become the fermions of the original theory after quantization. 
  We analyze the superconformal theories (SCFTs) which arise in the low-energy limit of N=(4,4) supersymmetric gauge theories in two dimensions, primarily the Higgs branch SCFT. By a direct field theory analysis we find a continuum of "throat"-like states localized near the singularities of the Higgs branch. The "throat" is similar to the "throat" found in the Coulomb branch of the same theories, but the full superconformal field theories of the two branches are different. A particular example is the SCFT of the R^4/Z_2 sigma model with zero theta angle. In the application of the Higgs branch SCFTs to the DLCQ description of "little string theories" (LSTs), the "throat" continuum is identified with the continuum of "throat" states in the holographic description of the LSTs. We also match the descriptions of the string interactions (in the "throat" region) in the DLCQ and holographic descriptions of the N=(2,0) LSTs. 
  All regular four-dimensional black holes are constructed in the theory obtained by Kaluza-Klein reduction of five-dimensional Einstein gravity. They are interpreted in string theory as rotating bound states of D0- and D6-branes. Conservation of angular momentum is important for the stability of the bound states. The thermodynamics, the duality symmetries, and the near-horizon limit are explored. 
  We derive the analogue of the vanishing of the cosmological constant in 3+1 dimensions, T_0^0 = 0, in terms of an integral over components of the energy-momentum tensor of a 4+1 dimensional universe with parallel three-branes, and an additional constraint local to the branes. The basic ingredients are the existence of a static solution of the Einstein equations, and the compactness of the 5th dimension. The corresponding constraints are applied to a general action of scalar fields with arbitrary potentials in the bulk and on the branes. The equations of motion are solved in a linearized approximation in the 5th dimension, whereupon they require the search for extrema of an ``effective potential'', which depends nonlinearly on the action in the bulk and on the branes. The previous constraints then turn into the vanishing of this ``effective potential'' at the extremum. 
  The hidden E_{7} (E_{6}) structure has been conjectured for the minimal model ${\cal M}_{4,5} ({\cal M}_{6,7}$) perturbed by $\Phi_{1,2}$ in the context of conformal field theory(CFT). Motivated by this, we examine the dilute A_{4, 6} models, which are expected to be corresponding lattice models. Thermodynamics of the equivalent one dimensional quantum systems is analyzed via the quantum transfer matrix approach. Appropriate auxiliary functions, related to kinks in the theory, play a role in constructing functional relations among transfer matrices. We successfully recover the universal Y- systems and thereby Thermodynamic Bethe Ansatz equations for E_{6,7} from the dilute A_{6,4} model, respectively. 
  In this short note we review the construction and role of Wess-Zumino couplings of Dirichlet branes and Orientifold planes, and show how these combine to give the Green-Schwarz anomaly cancelling terms. 
  The oscillating inflation model recently proposed by Damour and Mukhanov is investigated with a non-minimal coupling. Numerical study confirms an inflationary behavior and the density perturbation is obtained. A successful inflation requires the gravity-dilaton coupling to be small. 
  We study D1-branes on the fourfold $\C^4/(\Z_2\times\Z_2\times\Z_2)$, in the presence of discrete torsion. Discrete torsion is incorporated in the gauge theory of the D1-branes by considering a projective representation of the finite group $\Z_2\times\Z_2\times\Z_2$. The corresponding orbifold is then deformed by perturbing the F-flatness condition of the gauge theory. The moduli space of the resulting gauge theory retains a stable singularity of codimension three. 
  In this technical note we give a purely geometric understanding of discrete torsion, as an analogue of orbifold Wilson lines for two-form tensor field potentials. In order to introduce discrete torsion in this context, we describe gerbes and the description of certain type II supergravity tensor field potentials as connections on gerbes. Discrete torsion then naturally appears in describing the action of the orbifold group on (1-)gerbes, just as orbifold Wilson lines appear in describing the action of the orbifold group on the gauge bundle. Our results are not restricted to trivial gerbes -- in other words, our description of discrete torsion applies equally well to compactifications with nontrivial H-field strengths. We also describe a speciric program for rigorously deriving analogues of discrete torsion for many of the other type II tensor field potentials, and we are able to make specific conjectures for the results. 
  We study a ${{\IZ}}_2$ orbifold of Type 0B string theory by reflection of six of the coordinates (this theory may also be thought of as a ${{\IZ}}_4$ orbifold of Type IIB string theory by a rotation by $\pi$ in three independent planes). We show that the only massless mode localized on the fixed fourplane $\IR^{3,1}$ is a U(1) gauge field. After introducing D3-branes parallel to the fixed fourplane we find non-supersymmetric non-abelian gauge theories on their worldvolume. One of our results is that the theory on equal numbers of electric and magnetic D3-branes placed at the fourplane is the ${{\IZ}}_4$ orbifold of ${\cal N}=4$ supersymmetric Yang-Mills theory by the center of its R-symmetry group. 
  A double scaling limit can be defined in string theory on a Calabi-Yau (CY) manifold by approaching a point in moduli space where the CY space develops an isolated singularity and at the same time taking the string coupling to zero, while keeping a particular combination of the two parameters fixed. This leads to a decoupled theory without gravity which has a weak coupling expansion, and can be studied using a holographically dual non-critical superstring description. The usual ``Little String Theory'' corresponds to the strong coupling limit of this theory. We use holography to compute two and three point functions in weakly coupled double scaled little string theory, and study the spectrum of the theory in various dimensions. We find a discrete spectrum of masses which exhibits Hagedorn growth. 
  It is shown that the charged non-diagonal BTZ (2+1)-spacetime is not a solution of the Einstein-Maxwell field equations with cosmological constant. 
  A five-dimensional Chern-Simons gravity theory based on the anti-de Sitter group SO(4,2) is argued to be a useful model in which to understand the details of holography and the relationship between generally covariant and dual local quantum field theories. Defined on a manifold with boundary, conformal geometry arises naturally as a gauge invariance preserving boundary condition. By matching thermodynamic quantities for a particular background geometry, the dimensionless coupling constant of the Chern-Simons theory is directly related to the number of fields in a putative dual theory at high temperature. As a consistency check, the semiclassical factorization of Wilson line observables in the gravity theory is shown to induce a factorization in dual theory observables as expected by general arguments of large N gauge theory. 
  A procedure for reducing the functional integral of QED to an integral over bosonic gauge invariant fields is presented. Next, a certain averaging method for this integral, giving a tractable effective quantum field theory, is proposed. Finally, the current-current propagator and the chiral anomaly are calculated within this new formulation. These results are part of our programme of analyzing gauge theories with fermions in terms of local gauge invariants. 
  The standard Klein bottle coefficient in the construction of open descendants is shown to equal the Frobenius-Schur indicator of a conformal field theory. Other consistent Klein bottle projections are shown to correspond to simple currents. These observations enable us to generalize the standard open string construction from C-diagonal parent theories to include non-standard Klein bottles. Using (generalizations of) the Frobenius-Schur indicator we prove positivity and integrality of the resulting open and closed string state multiplicities for standard as well as non-standard Klein bottles. 
  By using the field-antifield formalism, we show that the method of Batalin, Fradkin, Fradkina and Tyutin to convert Hamiltonian systems submitted to second class constraints introduces compensating fields which do not belong to the BRST cohomology at ghost number one. This assures that the gauge symmetries which arise from the BFFT procedure are not obstructed at quantum level. An example where massive electrodynamics is coupled to chiral fermions is considered. We solve the quantum master equation for the model and show that the respective counterterm has a decisive role in extracting anomalous expectation values associated with the divergence of the Noether chiral current. 
  We present a Z_2 x Z_2 four dimensional orientifold in which supersymmetry is broken by a temperature-like Scherk-Schwarz mechanism. As usual in this type of models, at the tree-level the breaking affects only the branes tangent to the direction involved by the SS deformation, and it can propagate only through radiative corrections to other sectors, where the gauge group is broken. The result is a non-chiral model with gauge group Usp(16)^2 x Usp(8)^4. 
  The extended Yang-Mills gauge theory in Euclidean space is a renormalizable (by power counting) gauge theory describing a local interacting theory of scalar, vector, and tensor gauge fields (with maximum spin 2). In this article we study the quantum aspects and various generalizations of this model in Euclidean space. In particular the quantization of the pure gauge model in a common class of covariant gauges is performed. We generalize the pure gauge sector by including matter fermions in the adjoint representation of the gauge group and analyze its N=1 and N=2 supersymmetric extensions. We show that the maximum half-integer spin contained in these fermion fields in dimension 4 is 3/2. Moreover we develop an extension of this theory so as to include internal gauge symmetries and the coupling to bosonic matter fields. The spontaneous symmetry breaking of the extended gauge symmetry is also analyzed. 
  We consider a two-dimensional integrable and conformally invariant field theory possessing two Dirac spinors and three scalar fields. The interaction couples bilinear terms in the spinors to exponentials of the scalars. Its integrability properties are based on the sl(2) affine Kac-Moody algebra, and it is a simple example of the so-called conformal affine Toda theories coupled to matter fields. We show, using bosonization techniques, that the classical equivalence between a U(1) Noether current and the topological current holds true at the quantum level, and then leads to a bag model like mechanism for the confinement of the spinor fields inside the solitons. By bosonizing the spinors we show that the theory decouples into a sine-Gordon model and free scalars. We construct the two-soliton solutions and show that their interactions lead to the same time delays as those for the sine-Gordon solitons. The model provides a good laboratory to test duality ideas in the context of the equivalence between the sine-Gordon and Thirring theories. 
  We find the explicit expression of the supercharges of eleven dimensional supergraviton on the background geometry of gravitational waves in asymptotically light-like compactified spacetime. We perform the calculations order by order in the fermions $\p$, while retaining all orders in bosonic degrees of freedom, and get the closed form up to $\p^5$ order. This should correspond to the supercharge of the effective action of (0+1)-dimensional matrix quantum mechanics for, at least, $v^4$ and $v^6$ order terms and their superpartners. 
  In a previous paper we outlined how discrete torsion can be understood geometrically as an analogue of orbifold U(1) Wilson lines. In this paper we shall prove the remaining details. More precisely, in this paper we describe gerbes in terms of objects known as stacks (essentially, sheaves of categories), and develop much of the basic theory of gerbes in such language. Then, once the relevant technology has been described, we give a first-principles geometric derivation of discrete torsion. In other words, we define equivariant gerbes, and classify equivariant structures on gerbes and on gerbes with connection. We prove that in general, the set of equivariant structures on a gerbe with connection is a torsor under a group which includes H^2(G,U(1)), where G is the orbifold group. In special cases, such as trivial gerbes, the set of equivariant structures can furthermore be canonically identified with the group. 
  A class of conformally flat and asymptotically anti-de Sitter geometries involving profiles of scalar fields is studied from the point of view of gauged supergravity. The scalars involved in the solutions parameterise the SL(N,R)/SO(N) submanifold of the full scalar coset of the gauged supergravity, and are described by a symmetric potential with a universal form. These geometries descend via consistent truncation from distributions of D3-branes, M2-branes, or M5-branes in ten or eleven dimensions. We exhibit analogous solutions asymptotic to AdS_6 which descend from the D4-D8-brane system. We obtain the related six-dimensional theory by consistent reduction from massive type IIA supergravity. All our geometries correspond to states in the Coulomb branch of the dual conformal field theories. We analyze linear fluctuations of minimally coupled scalars and find both discrete and continuous spectra, but always bounded below. 
  We analyze the structure of singularities, Mordell-Weil lattices and torsions of a rational elliptic surface using string junctions in the background of 12 7-branes. The classification of the Mordell-Weil lattices due to Oguiso-Shioda is reproduced in terms of the junction lattice. In this analysis an important role played by the global structure of the surface is observed. It is then found that the torsions in the Mordell-Weil group are generated by the fraction of loop junctions which represent the imaginary roots of the loop algebra $\hat E_9$. From the structure of the Mordell-Weil lattice we find 7-brane configurations which support non-BPS junctions carrying conserved Abelian charges. 
  The propagators in axial-type, light-cone and planar gauges contain 1/(\eta\cdot k)^p-type singularities. These singularities have generally been treated by inventing prescriptions for them. In this work, we propose an alternative procedure for treating these singularities in the path integral formalism using the known way of treating the singularities in Lorentz gauges. To this end, we use a finite field-dependent BRS transformation that interpolates between Lorentz-type and the axial-type gauges. We arrive at the $\epsilon$-dependent tree propagator in the axial-type gauges. We examine the singularity structure of the propagator and find that the axial gauge propagator so constructed has {\it no} spurious poles (for real $k$). It however has a complicated structure in a small region near $\eta\cdot k=0$. We show how this complicated structure can effectively be replaced by a much simpler propagator. 
  We use the AdS/CFT correspondence to study near forward scattering of colourless objects in gauge theory in the high energy limit. We find an unexpected from the gauge theory perspective `gravity-like' s^1 behaviour of the amplitudes coming from bulk graviton exchange. The details of the calculations are presented in hep-th/9907177. 
  Generalising ideas of an earlier work \cite{Bo-Han}, we address the problem of constructing Brane Box Models of what we call the Z-D Type from a new point of view, so as to establish the complete correspondence between these brane setups and orbifold singularities of the non-Abelian G generated by Z_k and D_d under certain group-theoretic constraints to which we refer as the BBM conditions. Moreover, we present a new class of ${\cal N}=1$ quiver theories of the ordinary dihedral group d_k as well as the ordinary exceptionals E_{6,7,8} which have non-chiral matter content and discuss issues related to brane setups thereof. 
  We show that renormalization in quantum field theory is a special instance of a general mathematical procedure of multiplicative extraction of finite values based on the Riemann-Hilbert problem. Given a loop $\gamma(z), | z |=1$ of elements of a complex Lie group G the general procedure is given by evaluation of $ \gamma_{+}(z)$ at z=0 after performing the Birkhoff decomposition $ \gamma(z)=\gamma_{-}(z)^{-1} \gamma_{+}(z)$ where $ \gamma_{\pm}(z) \in G$ are loops holomorphic in the inner and outer domains of the Riemann sphere (with $\gamma_{-}(\infty)=1$). We show that, using dimensional regularization, the bare data in quantum field theory delivers a loop (where z is now the deviation from 4 of the complex dimension) of elements of the decorated Butcher group (obtained using the Milnor-Moore theorem from the Kreimer Hopf algebra of renormalization) and that the above general procedure delivers the renormalized physical theory in the minimal substraction scheme. 
  We discuss the conditions under which 4D gravity is localized on domain walls in 5D anti-de Sitter spaces. Our approach is based on considering the limits in which the localized gravity decouples. We find that gravity is localized if the wall is located a finite distance from the boundary of the anti-de Sitter space and has a finite tension. In addition, it has to be a $\delta$-function source of gravity. 
  Soliton time-delays and the semiclassical limit for soliton S-matrices are calculated for non-simply laced Affine Toda Field Theories. The phase shift is written as a sum over bilinears on the soliton conserved charges. The results apply to any two solitons of any Affine Toda Field Theory. As a by-product, a general expression for the number of bound states and the values of the coupling in which the S-matrix can be diagonal are obtained. In order to arrive at these results, a vertex operator is constructed, in the principal gradation, for non-simply laced affine Lie algebras, extending the previous constructions for simply laced and twisted affine Lie algebras. 
  We formulate an effective Schroedinger wave equation describing the quantum dynamics of a system of D0-branes by applying the Wilson renormalization group equation to the worldsheet partition function of a deformed sigma-model describing the system, which includes the quantum recoil due to the exchange of string states between the individual D-particles. We arrive at an effective Fokker-Planck equation for the probability density with diffusion coefficient determined by the total kinetic energy of the recoiling system. We use Galilean invariance of the system to show that there are three possible solutions of the associated non-linear Schroedinger equation depending on the strength of the open string interactions among the D-particles. When the open string energies are small compared to the total kinetic energy of the system, the solutions are governed by freely-propagating solitary waves. When the string coupling constant reaches a dynamically determined critical value, the system is described by minimal uncertainty wavepackets which describe the smearing of the D-particle coordinates due to the distortion of the surrounding spacetime from the string interactions. For strong string interactions, bound state solutions exist with effective mass determined by an energy-dependent shift of the static BPS mass of the D0-branes. 
  We show that smooth domain wall spacetimes supported by a scalar field separating two anti-de-Sitter like regions admit a single graviton bound state. Our analysis yields a fully non-linear supergravity treatment of the Randall-Sundrum model. Our solutions describe a pp-wave propagating in the domain wall background spacetime. If the latter is BPS, our solutions retain some supersymmetry. Nevertheless, the Kaluza-Klein modes generate ``pp curvature'' singularities in the bulk located where the horizon of AdS would ordinarily be. 
  This paper is a shortened version of the previous work hep-th/9907099: We propose a topological quantum field theory as a twisted candidate to formulate covariant matrix strings. The model relies on the octonionic or complexified instanton equations defined on an eight dimensional manifold with reduced holonomy. To allow untwisting of the model without producing an anomaly, we suggest (partially twisted) W-gravity as an "extended" 2d-gravity sector. 
  We consider the constraints on the effective Lagrangian of the rank-one gauge field on D-branes imposed by the equivalence between the description by ordinary gauge theory and that by non-commutative gauge theory in the presence of a constant B field. It is shown that we can consistently construct the two-derivative corrections to the Dirac-Born-Infeld Lagrangian up to the quartic order of field strength and the most general form which satisfies the constraints up to this order is derived. 
  We suggest that proper variables for the description of non-Abelian theories are those gauge invariant ones which keep the invariance of the winding number functional with respect to topologically nontrivial (large) gauge transformations. We present a model for these variables using the zero mode of the Gauss law constraint and investigate their physical consequences for hadron spectrum and confinement on the level of the generating functional for two-color QCD. 
  A method for obtaining solutions to the classical equations for scalars plus gravity in five dimensions is applied to some recent suggestions for brane-world phenomenology. The method involves only first order differential equations. It is inspired by gauged supergravity but does not require supersymmetry. Our first application is a full non-linear treatment of a recently studied stabilization mechanism for inter-brane spacing. The spacing is uniquely determined after conventional fine-tuning to achieve zero four-dimensional cosmological constant. If the fine-tuning is imperfect, there are solutions in which the four-dimensional branes are de Sitter or anti-de Sitter spacetimes. Our second application is a construction of smooth domain wall solutions which in a well-defined limit approach any desired array of sharply localized positive-tension branes. As an offshoot of the analysis we suggest a construction of a supergravity c-function for non-supersymmetric four-dimensional renormalization group flows.   The equations for fluctuations about an arbitrary scalar-gravity background are also studied. It is shown that all models in which the fifth dimension is effectively compactified contain a massless graviton. The graviton is the constant mode in the fifth dimension. The separated wave equation can be recast into the form of supersymmetric quantum mechanics. The graviton wave-function is then the supersymmetric ground state, and there are no tachyons. 
  We introduce a new class of gauge field theories in any complex dimension, based on algebra-valued (p,q)-forms on complex n-manifolds. These theories are holomorphic analogs of the well-known Chern-Simons and BF topological theories defined on real manifolds. We introduce actions for different special holomorphic BF theories on complex, Kahler and Calabi-Yau manifolds and describe their gauge symmetries. Candidate observables, topological invariants and relations to integrable models are briefly discussed. 
  Recently the long-standing puzzle about counting the Witten index in N=1 supersymmetric gauge theories was resolved. The resolution was based on existence (for higher orthogonal $SO(N), N \geq 7$ and exceptional gauge groups) of flat connections on $T^{3}$ which have commuting holonomies but cannot be gauged to a Cartan torus. A number of papers has been published which studied moduli spaces and some topological characteristics of those flat connections. In the present letter an explicit description of such flat connection for the basic case of $Spin(7)$ is given. 
  We derive the invariant operators of the zero-form, the one-form, the two-form and the spinor from which the mass spectrum of Kaluza Klein of eleven-dimensional supergravity on AdS_4 x N^{010} can be derived by means of harmonic analysis. We calculate their eigenvalues for all representations of SU(3)xSO(3). We show that the information contained in these operators is sufficient to reconstruct the complete N=3 supersymmetry content of the compactified theory. We find the N=3 massless graviton multiplet, the Betti multiplet and the SU(3) Killing vector multiplet. 
  We discuss the two-dimensional dilaton gravity with a scalar field as the source matter where the coupling with the gravity is given, besides the minimal one, through an external field. This coupling generalizes the conformal anomaly in the same way as those found in recent literature, but with a diferent motivation. The modification to the Hawking radiation is calculated explicity and shows an additional term that introduces a dependence on the (effective) mass of the black-hole. 
  We investigate the transformation from ordinary gauge field to noncommutative one which was introduced by N.Seiberg and E.Witten (hep-th/9908142). It is shown that the general transformation which is determined only by gauge equivalence has a path dependence in `\theta-space'. This ambiguity is negligible when we compare the ordinary Dirac-Born-Infeld action with the noncommutative one in the U(1) case, because of the U(1) nature and slowly varying field approximation. However, in general, in the higher derivative approximation or in the U(N) case, the ambiguity cannot be neglected due to its noncommutative structure. This ambiguity corresponds to the degrees of freedom of field redefinition. 
  We present a general construction of all correlation functions of a two-dimensional rational conformal field theory, for an arbitrary number of bulk and boundary fields and arbitrary topologies. The correlators are expressed in terms of Wilson graphs in a certain three-manifold, the connecting manifold. The amplitudes constructed this way can be shown to be modular invariant and to obey the correct factorization rules. 
  We look back at early efforts to approximate the large N_c Feynman diagrams of QCD as very large fishnet diagrams. We consider more carefully the uniqueness of rules for discretizing P^+ and ix^+ which fix the fishnet model in the strong 't Hooft coupling limit, and we offer some refinements that allow more of the crucial QCD interactions to be retained in the fishnet approximation. This new discretization has a better chance to lead to a physically sensible `bare QCD string' model. Not surprisingly the resulting fishnet diagrams are both richer in structure and harder to evaluate than those considered in older work. As warm-ups we analyze arbitrarily large fishnets of a paradigm scalar cubic theory and very small fishnets of QCD. 
  The periodic bounce configurations responsible for quantum tunneling are obtained explicitly and are extended to the finite energy case for minisuperspace models of the Universe. As a common feature of the tunneling models at finite energy considered here we observe that the period of the bounce increases with energy monotonically. The periodic bounces do not have bifurcations and make no contribution to the nucleation rate except the one with zero energy. The sharp first order phase transition from quantum tunneling to thermal activation is verified with the general criterions. 
  We provide a classification of the IIB D$p$- and NS$p$-branes in which the brane action exists due to a non-trivial class of the Chevalley-Eilenberg cohomology of free differential algebras. We then present a new geometric formulation of the IIB D$p$- and NS$p$-branes ($p\leq 5$) in which the manifestly superinvariant Wess-Zumino terms are constructed in terms of the supersymmetric currents. The supercurrents are obtained by using supergroup manifolds corresponding to the IIB-brane superalgebra, which is characterized by the generators of D3-, D5-, NS5- and KK5-branes in addition to the previously introduced generators of supertranslations, F- and D-strings. The charges of D1-, F1- and D3-branes are related to those of the M-algebra, but some charges of D5- and NS5-branes are not. The S-duality of the type-IIB theory is realized as transformations of the supercurrents generalizing the SO(2) R-symmetry of the superalgebra. We thus find that the superalgebra is lifted into twelve-dimensions with signature (11,1). 
  We present some multiplets of N=2 off-shell supergravity in five dimensions. One is the Super Yang-Mills multiplet, another one is the linear multiplet. The latter one is used to establish a general action formula from which we derive an action for the Super Yang-Mills multiplet. The Super Yang-Mills multiplet is used to construct the nonlinear multiplet with gauged SU(2). This nonlinear multiplet and the action formula for the Yang-Mills multiplet enable us to write down an SU(2) gauged supergravity which we finally truncate to arrive at gauged supergravity with gauge group SO(2). 
  We present an investigation of the boundary breather states of the sinh-Gordon model restricted to a half-line. The classical boundary breathers are presented for a two parameter family of integrable boundary conditions. Restricting to the case of boundary conditions which preserve the \phi --> -\phi symmetry of the bulk theory, the energy spectrum of the boundary states is computed in two ways: firstly, by using the bootstrap technique and subsequently, by using a WKB approximation. Requiring that the two descriptions of the spectrum agree with each other allows a determination of the relationship between the boundary parameter, the bulk coupling constant, and the parameter appearing in the reflection factor derived by Ghoshal to describe the scattering of the sinh-Gordon particle from the boundary. 
  We study the topologically twisted string theory on the general back-ground $AdS_3\times {\cal N}$ which is compatible with the world-sheet N=2 superconformal symmetry and is extensively discussed in the recent works (hep-th/9904024, hep-th/9904040). After summarizing the algebraic structure of the world-sheet topological theory, we show that the space-time (boundary) conformal theory should be also topological. We directly construct the space-time topological conformal algebra (twisted N=2 superconformal algebra) from the degrees of freedom in the world-sheet topological theory. Firstly, we work on the world-sheet of the string propagating near boundary, in which we can safely make use of the Wakimoto free field representation. Secondly, we present a more rigid formulation of space-time topological conformal algebra which is still valid far from the boundary along the line of (hep-th/9903219). We also discuss about the relation between this space-time topological theory and the twisted version of the space-time N=2 superconformal field theory given in (hep-th/9904024, hep-th/9904040). 
  We show that conformal maps of simply connected domains with an analytic boundary to a unit disk have an intimate relation to the dispersionless 2D Toda integrable hierarchy. The maps are determined by a particular solution to the hierarchy singled out by the conditions known as "string equations". The same hierarchy locally solves the 2D inverse potential problem, i.e. reconstruction of the domain out of a set of its harmonic moments. This is the same solution which is known to describe 2D gravity coupled to c=1 matter. We also introduce a concept of the $\tau$-function for analytic curves. 
  The radiation emitted by accelerated fundamental strings and D-branes is studied within the linear approximation to the supergravity limit of string theory. We show that scalar, gauge field and gravitational radiation is generically emitted by such branes. In the case where an external scalar field accelerates the branes, we derive a Larmor-type formula for the emitted scalar radiation and study the angular distribution of the outgoing energy flux. The classical radii of the branes are calculated by means of the corresponding Thompson scattering cross sections. Within the linear approximation, the interaction of the external scalar field with the velocity fields of the branes gives a contribution to the observed gauge field and gravitational radiation. 
  We consider string theory on AdS_3 in terms of the Wakimoto free field representation. The scattering amplitudes for N unitary tachyons are analysed in the factorization limit and the poles corresponding to the mass-shell conditions for physical states are extracted. The vertex operators for excited levels are obtained from the residues and their properties are examined. Negative norm states are found at the second mass level. 
  We give a complete set of generators for the discrete exceptional U-duality groups of toroidal compactified type II theory and M-theory in d>2. For this, we use the DSZ quantization in d=4 as originally proposed by Hull and Townsend, and determine the discrete group inducing integer shifts on the charge lattice. It is generated by fundamental unipotents, which are constructed by exponentiating the Chevalley generators of the corresponding Lie algebra. We then extend a method suggested by the above authors and used by Sen for the heterotic string to get the discrete U-duality group in d=3, thereby obtaining a quantized symmetry in d=3 from a d=4 quantization condition. This is studied first in a toy model, corresponding to d=5 simple supergravity, and then applied to M-theory. It turns out that, in the toy model, the resulting U-duality group in d=3 is strictly smaller than the one generated by the fundamental unipotents corresponding to all Chevalley generators. However, for M-theory, both groups agree. We illustrate the compactification to d=3 by an embedding of d=4 particle multiplets into the d=3 theory. 
  A discrete field formalism exposes the physical meaning and origins of gauge fields, their symmetries and singularities. They represent a lack of a stricter field-source coherence. 
  We exhibit a nonperturbative background independent dynamical truncation of the string spectrum and a quantization of the string coupling constant directly from the Hamiltonian governing the dynamics of strings constructed from Yang-Mills theories. 
  Using the methods of the "form factor program" exact expressions of all matrix elements are obtained for several operators of the quantum sine Gordon model: all powers of the fundamental bose field, general exponentials of it, the energy momentum tensor and all higher currents. It is found that the quantum sine-Gordon field equation holds with an exact relation between the "bare" mass and the renormalized mass. Also a relation for the trace of the energy momentum is obtained. The eigenvalue problem for all higher conserved charges is solved. All results are compared with Feynman graph expansions and full agreement is found. 
  We discuss some aspects of the generalization of the Born-Infeld action to non-abelian gauge groups and show how the discrepancy between Tseytlin's symmetrized trace proposal and string theory can be corrected at order $F^6$. We also comment on the possible quadratic order fermionic terms. 
  Recently it was proposed that matching of global charges induced in vacuum by slowly varying, topologically non-trivial scalar fields provides consistency conditions analogous to the 't Hooft anomaly matching conditions. We study matching of induced charges in supersymmetric SU(N) gauge theories with quantum modified moduli space. We find that the Wess-Zumino term should be present in the low energy theory in order that these consistency conditions are satisfied. We calculate the lowest homotopy groups of the quantum moduli space, and show that there are no obstructions to the existence of the Wess-Zumino term at arbitrary N. The explicit expression for this term is given. It is shown that neither vortices nor topological solitons exist in the model. The case of softly broken supersymmetry is considered as well. We find that the possibility of global flavor symmetric vacuum is strongly disfavored. 
  We review the string representations of Abelian-projected SU(2)- and SU(3)-gauge theories and their application to the evaluation of bilocal field strength correlators. The large distance asymptotic behaviours of the latter ones are shown to be in agreement with the Stochastic Vacuum Model of QCD and existing lattice data. 
  Generators for the discrete U-duality groups of toroidally compactified M-theory in d>3 are presented and used to determine the d=3 U-duality group. This contribution summarizes the results of hep-th/9909150. 
  Color confinement is a consequence of an unbroken non-Abelian gauge symmetry and the resulting asymptotic freedom inherent in quantum chromodynamics. A qualitative sketch of its proof is presented. 
  In connection with the question of color confinement the origin of the Goto-Imamura-Schwinger term has been studied with the help of renormalization group. An emphasis has been laid on the difference between theories with and without a cut-off. 
  We discuss the asymptotic form of the static axially symmetric, globally regular and black hole solutions, obtained recently in Einstein-Yang-Mills and Einstein-Yang-Mills-dilaton theory. 
  In this article, it is shown how to obtain objects called Eichler integrals in the mathematical literature that can be used for calculating scattering amplitudes in String Theory. These Eichler integrals are also new examples of Eichler integrals with poles. 
  It is shown how to sew string vertices with ghosts at tree level in order to produce new tree vertices using the Group Theoretic approach to String Theory. It is then verified the BRST invariance of the sewn vertex and shown that it has the correct ghost number. 
  In this paper we re-examine the geometric interpretation of gluing conditions in WZW models and the possible D-brane configurations that they give rise to. We show how the boundary conditions are encoded in the gluing conditions imposed on the chiral currents. We analyse two special classes of gluing conditions: the first, which preserves the affine symmetry of the bulk theory, describes D-branes whose worldvolumes are given by `twisted' conjugacy classes; the second class describes configurations which include subgroups and cosets. 
  We re-examine the problem of determining the possible D-branes in the Nappi-Witten background. In addition to the known branes, we find that there are also D-instantons, flat euclidean D-strings and curved D-membranes admitting parallel spinors, all of which can be interpreted as (twisted) conjugacy classes in the Nappi-Witten group. 
  Assuming dynamical spontaneous breakdown of chiral symmetry for massless gauge theory without scalar fields, we find a method how to construct an effective action of the dynamical Nambu-Goldstone bosons and elemetary fermions by using auxiliary fields. Here dynamical particles are asssumed to be composed of elementary fermions. Various quantities including decay constants are calculated from this effective action. 
  We establish the existence of the topological vector supersymmetry in the six dimensional topological field theory for two-form fields introduced by Baulieu and West. We investigate the relation of these symmetries to the twist operation for the (2,0) supersymmetry and comment on their resemblance to the analogous symmetries in topological Yang-Mills theory. 
  Interaction of branes in presence of internal gauge fields is considered by using the boundary state formalism. This approach enables us to consider the problems that are not easily accessible to the canonical approach via open strings. The effects of compactification of some of the dimensions on tori are also discussed. Also we study the massless state contribution on this interaction. 
  We discuss the explicit formulation of the transcendental constraints defining spectral curves of SU(2) BPS monopoles in the twistor approach of Hitchin, following Ercolani and Sinha. We obtain an improved version of the Ercolani-Sinha constraints, and show that the Corrigan-Goddard conditions for constructing monopoles of arbitrary charge can be regarded as a special case of these. As an application, we study the spectral curve of the tetrahedrally symmetric 3-monopole, an example where the Corrigan-Goddard conditions need to be modified. A particular 1-cycle on the spectral curve plays an important role in our analysis. 
  One of us got spins and charges of not only scalars and vectors but also of spinors out of fields, which are antisymmetric tensor fields. Kahler got spins of spinors out of differential forms, which again are antisymmetric tensor fields. Using our simple Grassmann formulation of spins and charges of either spinors or vectors and comparing it to the Dirac-Kahler formulation of spinors, we generalize the Dirac-Kahler approach to vector internal degrees of freedom and to charges of either spinors or vectors and tenzors and point out how at all spinors can appear in both approaches. 
  Suppose a regularised functional integral depends holomorphically on a parameter that receives only a finite renormalization. Can one expect the correlation functions to retain the analyticity in the parameter after removal of the cutoff(s)? We examine the issue in the Sinh-Gordon theory by computing the intrinsic 4-point coupling as a function of the Lagrangian coupling \beta. Drawing on the conjectured triviality of the model in its functional integral formulation for \beta^2 > 8\pi, and the weak-strong coupling duality in the bootstrap formulation on the other hand, we conclude that the operations: ``Removal of the cutoff(s)'' and ``analytic continuation in \beta'' do not commute. 
  We consider the elastic scattering of two open strings living on two D-branes separated by a distance $r$. We compute the high-energy behavior of the amplitude, to leading order in string coupling, as a function of the scattering angle $\phi$ and of the dimensionless parameter $v= r/(\pi\alpha^\prime\sqrt{s})$ with $\sqrt{s}$ the center-of-mass energy. The result exhibits an interesting phase diagram in the $(v,\phi)$ plane, with a transition at the production threshold for stretched strings at $v=1$. We also discuss some more general features of the open-string semiclassical world-sheets, and use T-duality to give a quantum tunneling interpretation of the exponential suppression at high-energy. 
  We construct D=4 Type I vacua with massless content remarkably close to that of the standard model of particle physics. They are tachyon-free non-supersymmetric models which are obtained starting with a standard D=4, N=1 compact Type IIB orientifold and adding the same number of Dp-branes and anti-Dp-branes distributed at different points of the underlying orbifold. Supersymmetry-breaking is felt by the observable world either directly, by gravity mediation or gauge mediation, depending on the brane configuration. We construct several simple three generation examples with the gauge group of the standard model or its left-right symmetric extensions. The models contain a number of U(1) gauge groups whose anomalies are cancelled by a generalized Green-Schwarz mechanism. These U(1)'s are broken but may survive as global symmetries providing for a flavour structure to the models. The value of the string scale may be lowered down to the intermediate scale (as required in the gravity mediation case) or down to 1-100 TeV for the non-SUSY models. Thus the present models are the first semirealistic string vacua realizing the possibility of a low string scale. The unbalanced force between the pairs of Dp- and anti-Dp-branes provides for an effect which tends to compactify some of the extra dimensions but no others. This could provide a new mechanism for radius stabilization. 
  In this paper we investigate massless scalar field theory on non-degenerate algebraic curves. The propagator is written in terms of the parameters appearing in the polynomial defining the curve. This provides an alternative to the language of theta functions. The main result is a derivation of the third kind differential normalized in such a way that its periods around the homology cycles are purely imaginary. All the physical correlation functions of the scalar fields can be expressed in terms of this object. This paper contains a detailed analysis of the techniques necessary to study field theories on algebraic curves. A simple expression of the scalar field propagator is found in a particular case in which the algebraic curves have $Z_n$ internal symmetry and one of the fields is located at a branch point. 
  In this paper we study various aspects of classical solutions to the affine Toda equations on a half-line with integrable boundary conditions. We begin by finding conditions that the theory has a stable vacuum by finding a Bogomolny bound on the energy, and analysing the possible singularities of the field at the boundary. Using these constraints and extensive numerical investigations we classify the vacuum configurations and reflection factors for A_r^(1) Toda theories up to r=5. 
  We have done a study of the zero-dimensional $\lambda\phi^{4}$ model. Firstly, we exhibit the partition function as a simple exact expression in terms of the Macdonald's function for $Re(\lambda)>0$. Secondly, an analytic continuation of the partition function for $Re(\lambda)<0$ is performed, and we obtain an expression defined in the complex coupling constant plane $\lambda$, for $|arg \lambda|<\pi$. Consequently, the partition function understood as an analytic continuation is defined for all values of $\lambda$, except for a branch cut along the real negative $\lambda$ axis. We also evaluate the partition function on perturbative grounds, using the Borel summation technique and we found that in the common domain of validity for $Re(\lambda)>0$, it coincides precisely with the exact expression. 
  In string theory, D-branes can be expressed as a configuration of infinitely many lower dimensional D-branes. Using this relation, the worldvolume theory of D-branes can be regarded as the worldvolume theory of the infinitely many lower dimensional branes. In the description in terms of the lower dimensional branes, some of the worldvolume coordinates become noncommutative. Actually this noncommutative theory can be regarded as noncommutative Yang-Mills theory. Therefore the worldvolume theory of D-branes have two equivalent descriptions, namely the usual static gauge description using ordinary Yang-Mills theory and the noncommutative description using noncommutative Yang-Mills theory. It will be shown that these two descriptions correspond to two different ways of gauge fixing of the reparametrization invariance and its generalization. We will give an explicit relation between commutative gauge field and noncommutative gauge field in semiclassical approximation, when the gauge group is U(1). 
  We consider the general hypermultiplet Low-Energy Effective Action (LEEA) that may appear in quantized, four-dimensional, N=2 supersymmetric, gauge theories, e.g. in the Coulomb and Higgs branches. Our main purpose is a description of the exact LEEA of n magnetically charged hypermultiplets. The hypermultiplet LEEA is given by the N=2 supersymmetric Non-Linear Sigma-Model (NLSM) with a 4n-dimensional hyper-K"ahler metric, subject to non-anomalous symmetries. Harmonic Superspace (HSS) and the NLSM isometries are very useful to constrain the hyper-K"ahler geometry of the LEEA. We use N=2 supersymmetric projections of HSS superfields to N=2 linear (tensor) O(2) and O(4) multiplets in N=2 Projective Superspace (PSS) to deduce the explicit form of the LEEA in some particular cases. As the by-product, a simple new classification of all multi-monopole moduli space metrics having su(2)_R symmetry is proposed in terms of real quartic polynomials of 2n variables, modulo Sp(n) transformations. The 4d hypermultiplet LEEA for n=2 can be encoded in terms of an elliptic curve. 
  Explicit relations among moduli of the Heterotic and Type IIB string theories in 8 dimensions are obtained. We identify the BPS states responsible for gauge enhancements in the type IIB theory and their dual partners in the Heterotic theory compactified with and without Wilson lines. The masses of BPS states in Type IIB string theory compactified on the base space of a elliptically fibred K3 are computed explicitly for the special cases in which the complex structure of the fibre is constant, ie, for constant scalar fields backgrounds. 
  We present a calculation of three point functions for a class of chiral operators, including the primary ones, in d = 3, N = 8; d = 6, N = (2,0) and d = 4, N = 4 superconformal field theories at large N. These theories are related to the infrared world-volume descriptions of N coincident M2, M5 and D3 branes, respectively. The calculation is done in the framework of the AdS/CFT correspondence and can be given a unified treatment employing a gravitational action in arbitrary dimensions D, coupled to a p+1 form and suitably compactified on AdS(D-2-p) x S(2+p). The interesting cases are obtained setting (D,p) to the values (11,5), (11,2) and (10,3). 
  We study the relation between the (generalized) conformal quantum mechanics of 0-branes and the two-dimensional dilaton gravity. The two-dimensional actions obtained from the supergravity effective actions for the (dilatonic) 0-branes through the compactification on a sphere are related to known two-dimensional dilaton gravity models. The SL(2,R) symmetry of the (generalized) conformal quantum mechanics is realized within such two-dimensional models. The two-dimensional dilatonic gravity model derived from the non-dilatonic 0-brane action is related to the Liouville theory and therefore is conformal, whereas the two-dimensional model derived from the dilatonic 0-brane action does not have the conformal symmetry. 
  Using a one-loop approximation for the effective potential in the Higgs model of electrodynamics for a charged scalar field, we argue for the existence of a triple point for the renormalized (running) values of the selfinteraction $\lambda$ and the "charge" g given by $(\lambda_{run}, g^2) = (-{10/9} \pi^2,{4/3}\sqrt{{5/3}}{\pi^2}) \approx(-11, 17)$. Considering the beta-function as a typical quantity we estimate that the one-loop approximation is valid with accuracy of deviations not more than 30% in the region of the parameters: $0.2 \stackrel{<}{\sim}{\large \alpha, \tilde{\alpha}} \stackrel{<}{\sim}1.35.$ The phase diagram given in the present paper corresponds to the above-mentioned region of $\alpha, \tilde \alpha$. Under the point of view that the Higgs particle is a monopole with a magnetic charge g, the obtained electric fine structure constant turns out to be $\alpha_{crit}\approx{0.18_5}$ by the Dirac relation. This value is very close to the $\alpha_{crit}^{lat}\approx{0.20}$ which in a U(1) lattice gauge theory corresponds to the phase transition between the "Coulomb" and confinement phases. Such a result is very encouraging for the idea of an approximate "universality" (regularization independence) of gauge couplings at the phase transition point. This idea was suggested by the authors in their earlier papers. 
  We analyse perturbatively, whether a flat background with vanishing G-flux in Horava-Witten supergravity represents a vacuum state, which is stable with respect to interactions between the ten-dimensional boundaries, mediated through the D=11 supergravity bulk fields. For this, we consider fluctuations in the graviton, gravitino and 3-form around the flat background, which couple to the boundary $E_8$ gauge-supermultiplet. They give rise to exchange amplitudes or forces between both boundary fixed-planes. In leading order of the D=11 gravitational coupling constant $\kappa$, we find an expected trivial vanishing of all three amplitudes and thereby stability of the flat vacuum in the static limit, in which the centre-of-mass energy $\sqrt{s}$ of the gauge-multiplet fields is zero. For $\sqrt{s}>0$, however, which could be regarded a vacuum state with excitations on the boundary, the amplitudes neither vanish nor cancel each other, thus leading to an attractive force between the fixed-planes in the flat vacuum. A ground state showing stability with regard to boundary excitations, is therefore expected to exhibit a non-trivial metric. Ten-dimensional Lorentz-invariance requires a warped geometry. Finally, we extrapolate the amplitudes to the case of coinciding boundaries and compare them to the ones resulting from the weakly coupled $E_8 \times E_8$ heterotic string theory at low energies. 
  We study the structure of holomorphic effective action for hypermultiplet models interacting with background super Yang-Mills fields. A general form of holomorphic effective action is found for hypermultiplet belonging to arbitrary representation of any semisimple compact Lie group spontaneously broken to its maximal abelian subgroup. The applications of obtained results to hypermultiplets in fundamental and adjoint representations of the SU(n), SO(n), Sp(n) groups are considered. 
  We obtain new solutions where a string and a pp-wave lie in the common worldvolume directions of the non-standard intersection of two gauge 5-branes in the heterotic string. The two 5-branes are supported by independent SU(2) Yang-Mills instantons in their respective (non-overlapping) transverse spaces. We present a detailed study of the unbroken supersymmetry, focusing especially on a comparison between a direct construction of Killing spinors and a counting of zero eigenvalues in the annticommutator of supercharges. The results are in agreement with some previous arguments, to the effect that additional zero eigenvalues resulting from a ``fine-tuning'' between positive-energy and negative-energy contributions from different components in an intersection are spurious, and should not be taken as an indication of supersymmetry enhancements. These observations have a general applicability that goes beyond the specific example we study in this paper. 
  A dimensionally continued background-field method makes the rationality of the 4-loop quenched QED beta function far more reasonable than had previously appeared. After 33 years of quest, dating from Rosner's discovery of 3-loop rationality, one finally sees cancellation of zeta values by the trace structure of individual diagrams. At 4-loops, diagram-by-diagram cancellation of $\zeta(5)$ does not even rely on the values of integrals at d=4. Rather, it is a property of the rational functions of $d$ that multiply elements of the full d-dimensional basis. We prove a lemma: the basis consists of slices of wheels. We explain the previously mysterious suppression of $\pi^4$ in massless gauge theory. The 4-loop QED result $\beta_4=-46$ is obtained by setting d=4 in a precisely defined rational polynomial of d, with degree 11. The other 5 rational functions vanish at d=4. 
  We construct the retarded Green function and the Hadamard function in the Lorentzian (d+1)-dimensional anti-de Sitter spacetime for the Poincar\'e coordinate by performing the mode integration directly. We explore the structure of singularities for the position-space Feynman propagator derived from them. The boundary scaling limits of the bulk Feynman propagator yield the bulk-boundary propagator and the boundary conformal correlation function with an extra factor. 
  We analyse field strength configurations in U(N) Yang-Mills theory on T^{2n} that are diagonal and constant, extending early work of Van Baal on T^4. The spectrum of fluctuations is determined and the eigenfunctions are given explicitly in terms of theta functions on tori. We show the relevance of the analysis to higher dimensional D-branes and discuss applications of the results in string theory. 
  In this paper, relying on previous results of one of us on harmonic analysis, we derive the complete spectrum of Osp(3|4) X SU(3) multiplets that one obtains compactifying D=11 supergravity on the unique homogeneous space N^{0,1,0} that has a tri-sasakian structure, namely leads to N=3 supersymmetry both in the four-dimensional bulk and on the three-dimensional boundary. As in previously analyzed cases the knowledge of the Kaluza Klein spectrum, together with general information on the geometric structure of the compact manifold is an essential ingredient to guess and construct the corresponding superconformal field theory. This is work in progress. As a bonus of our analysis we derive and present the explicit structure of all unitary irreducible representations of the superalgebra Osp(3|4) with maximal spin content s_{max}>=2. 
  We study an Abelian Chern-Simons and Fermion system in three dimensions. In the presence of a fixed prescribed background magnetic field we find an infinite number of fully three-dimensional solutions. These solutions are related to Hopf maps and are, therefore, labelled by the Hopf index. Further we discuss the interpretation of the background field. 
  We reconsider the problem of BRST quantization of a mechanics with infinitely reducible first class constraints. Following an earlier recipe [Phys. Lett. B 381, 105, (1996)], the original phase space is extended by purely auxiliary variables, the constraint set in the enlarged space being first stage of reducibility. The BRST charge involving only a finite number of ghost variables is explicitly constructed. 
  We show that the requirement of $S_3\times Spin(d)$ invariance for a ``asymptotically free'' SU(3)-Cartan subalgebra wave-function does {\em not} give a unique candidate for a SU(3)-invariant zero energy state of the d=9 supersymmetric matrix model - nor does it rule out the existence of such a state in the case d=3. For d=9 we explicitly construct various $S_3\times Spin(9)$-invariant wave-functions.} 
  It is shown that the auxiliary field in the low-energy effective theory of the supersymmetric QCD (SU(N_c) gauge symmetry with flavors N_f < N_c) can be understood as a physical degree of freedom, once the supersymmetry is explicitly broken. Although the vacuum expectation value of the auxiliary field is just a measure of the supersymmetry breaking in the perturbative treatment of the supersymmetry breaking, it can be the vacuum expectation value of the quark bilinear operator in the non-perturbative treatment of the supersymmetry breaking. We show that the vacuum expectation value remains finite in the limit of the infinite supersymmetry-breaking mass of the squark. We have to take the large N_c limit simultaneously to keep the low-energy effective Kaehler potential being in good approximation. 
  Responding to the recent claim that the origin of moduli space may be unstable in "magnetic" supersymmetric quantum chromodynamics (SQCD) with N_f <= 3N_c/2 (N_c>2) for N_f flavors and N_c colors of quarks, we explore the possibility of finding nonperturbative physics for "electric" SQCD. We present a recently discussed effective superpotential for "electric" SQCD with N_c+2 <= N_f <= 3N_c/2 (N_c>2) that generates chiral symmetry breaking with a residual nonabelian symmetry of SU(N_c)_{L+R}xSU(N_f-N_c)_Lx SU(N_f-N_c)_R. Holomorphic decoupling property is shown to be respected. For massive N_f-N_c quarks, our superpotential with instanton effect taken into account produces a consistent vacuum structure for SQCD with N_f=N_c compatible with the holomorphic decoupling. 
  In this paper we would like to show simple mechanisms how from the action for non-BPS D-brane we can obtain action describing BPS D(p-1)-brane in Type IIA theory. 
  We point out some misconceptions in a recent paper by H. Aoki et al. [hep-th/9909060]. In particular, the claim that the two-point function of branched polymers behaves as 1/p^4 instead of 1/p^2 for large p is mistaken and in no way a precondition for the Hausdorff dimension of branched polymers having the well known value four. 
  Instanton correction of prepotential of one-dimensional SL(2) Ruijsenaars model is presented with the help of Picard-Fuchs equation of Pakuliak-Perelomov type. It is shown that the instanton induced prepotential reduces to that of the SU(2) gauge theory coupled with a massive adjoint hypermultiplet. 
  The recently proposed technique to regularize the divergences of the gravitational action on non-compact space by adding boundary counterterms is studied. We propose prescription for constructing the boundary counterterms which are polynomial in the boundary curvature. This prescription is efficient for both asymptotically Anti-de Sitter and asymptotically flat spaces. Being mostly interested in the asymptotically flat case we demonstrate how our procedure works for known examples of non-compact spaces: Eguchi-Hanson metric, Kerr-Newman metric, Taub-NUT and Taub-bolt metrics and others. Analyzing the regularization procedure when boundary is not round sphere we observe that our counterterm helps to cancel large $r$ divergence of the action in the zero and first orders in small deviations of the geometry of the boundary from that of the round sphere. In order to cancel the divergence in the second order in deviations a new quadratic in boundary curvature counterterm is introduced. We argue that cancelation of the divergence for finite deviations possibly requires infinite series of (higher order in the boundary curvature) boundary counterterms. 
  We investigate the stability of the hidden sector gaugino condensate in a SL(2,Z)-invariant supergravity model inspired by the E8*E8 heterotic string, using the chiral superfield formalism. We calculate the Planck-suppressed corrections to the ``truncated approximation'' for the condensate value and the scalar potential. A transition to a phase with zero condensate occurs near special points in moduli space and at large compactification radius. We discuss the implications for the T-modulus dependence of supersymmetry-breaking. 
  We study the solutions of the Einstein equations in (d+2)-dimensions, describing parallel p-branes (p=d-1) in a space with two transverse dimensions of positive gaussian curvature. These solutions generalize the solutions of Deser and Jackiw of point particle sources in (2+1)-dimensional gravity with cosmological constant. Determination of the metric is reduced to finding the roots of a simple algebraic equation. These roots also determine the nontrivial "warp factors" of the metric at the positions of the branes. We discuss the possible role of these solutions and the importance of "warp factors" in the context of the large extra dimensions scenario. 
  It is shown how to obtain the consistent light front form quantization of a non-Abelian pure Yang-Mills theory (gluondynamics) in the framework of the standard perturbative approach. After a short review of the previous attempts in the light cone gauge $A_-=0$, it is explained how the difficulties can be overcome after turning to the anti light cone gauge $A_+=0$. In particular, the generating functional of the renormalized Green's functions turns out to be the same as in the conventional instant form approach, leading to the Mandelstam-Leibbrandt prescription for the free gluon propagator. 
  We show that the recently formulated Equivalence Principle (EP) implies a basic cocycle condition both in Euclidean and Minkowski spaces, which holds in any dimension. This condition, that in one-dimension is sufficient to fix the Schwarzian equation [6], implies a fundamental higher dimensional Moebius invariance which in turn univocally fixes the quantum version of the Hamilton-Jacobi equation. This holds also in the relativistic case, so that we obtain both the time-dependent Schroedinger equation and the Klein-Gordon equation in any dimension. We then show that the EP implies that masses are related by maps induced by the coordinate transformations connecting different physical systems. Furthermore, we show that the minimal coupling prescription, and therefore gauge invariance, arises quite naturally in implementing the EP. Finally, we show that there is an antisymmetric two-tensor which underlies Quantum Mechanics and sheds new light on the nature of the Quantum Hamilton-Jacobi equation. 
  Taking advantage of the equivalence between supersymmetric Yang-Mills theory on non-commutative spaces and the field theory limit of D3-branes in the background of NSNS 2-form field, we investigate the static properties of magnetic monopoles and dyons using brane construction techniques. When parallel D3-branes are separated by turning on a Higgs vacuum expectation value, D-strings will stretch between them at an angle which depends on the value of the background 2-form potential. These states preserve half of the supersymmetries and have the same masses as their commutative counterparts in the field theory limit. We also find stable (p,q)-dyons and string junctions. We find that they do not preserve any supersymmetry but have the same masses as their commutative counterparts. In the field theory limit, the (p,q)-dyons and the string junctions restore 1/2 and 1/4 of the 16 supersymmetries, respectively. 
  We construct the complete and explicit non-linear Kaluza-Klein ansatz for deriving the bosonic sector of N=4 SU(2)\times U(1) gauged five-dimensional supergravity from the reduction of type IIB supergravity on S^5. This provides the first complete example of such an S^5 reduction that includes non-abelian gauge fields, and it allows any bosonic solution of the five-dimensional N=4 gauged theory to be embedded in D=10. 
  Effects of fermion-vacuum polarization by a singular configuration of an external static vector field are considered in (2 + 1)-dimensional spacetime. Expressions for the induced vacuum charge and magnetic flux are obtained. 
  Gravitational collapse of matter trapped on a brane will produce a black hole on the brane. We discuss such black holes in the models of Randall and Sundrum where our universe is viewed as a domain wall in five dimensional anti-de Sitter space. We present evidence that a non-rotating uncharged black hole on the domain wall is described by a ``black cigar'' solution in five dimensions. 
  We study the deformation quantization of scalar and abelian gauge classical free fields. Stratonovich-Weyl quantizer, star-products and Wigner functionals are obtained in field and oscillator variables. Abelian gauge theory is particularly intriguing since Wigner functional is factorized into a physical part and other one containing the constraints only. Some effects of non-trivial topology within deformation quantization formalism are also considered. 
  Generalizing the 't Hooft and Veltman method of unitary regulators, we demonstrate for the first time the existence of local, Lorentz-invariant, physically motivated Lagrangians of quantum-electrodynamic phenomena such that: (i) Feynman diagrams are finite and equal the diagrams of QED but with regularized propagators. (ii) N-point Green functions are causal. (iii) S-matrix relates only electrons, positrons and photons, is unitary and Lorentz-invariant, and conserves charge and total four-momentum. 
  We outline a derivation of area law of the Wilson loop in SU(N) Yang-Mills theory in the maximal Abelian gauge. This is based on a new version of non-Abelian Stokes theorem and the novel reformulation of the Yang-Mills theory. Abelian dominance and monopole dominance of the string tension in SU(N) QCD are immediate consequences of this derivation. 
  Recent progress achieved in the solution of the problem of confinement in various (non-)Abelian gauge theories by virtue of a derivation of their string representation is reviewed. The theories under study include QCD within the so-called Method of Field Correlators, QCD-inspired Abelian-projected theories, and compact QED in three and four space-time dimensions. Various nonperturbative properties of the vacua of the above mentioned theories are discussed. The relevance of the Method of Field Correlators to the study of confinement in Abelian models, allowing for an analytical description of this phenomenon, is illustrated by an evaluation of field correlators in these models. 
  We apply general formalism of quantum field theory and addition theorem for Bessel functions to derive formula for the Casimir-Polder energy of interaction between a polarizable particle and a dilute dielectric ball and Casimir energy of a dilute dielectric ball. The correspondence between the Casimir-Polder formula and Casimir energy of a dilute dielectric ball is shown. Different approaches to the problem of Casimir energy of a dielectric ball are reviewed and analysed by use of addition theorem for Bessel functions. 
  Applying the time-dependent variational principle of Balian and V\'en\'eroni, we derive variational approximations for multi-time correlation functions in $\Phi^4$ field theory. We assume first that the initial state is given and characterized by a density operator equal to a Gaussian density matrix. Then, we study the more realistic situation where only a few expectation values are given at the initial time and we perform an optimization with respect to the initial state. We calculate explicitly the two-time correlation functions with two and four field operators at equilibrium in the symmetric phase. 
  I review the relations between mass scales in various string theories and in M-theory. I discuss physical motivations and possible consistent realizations of large volume compactifications and low string scale. Large longitudinal dimensions, seen by Standard Model particles, imply in general that string theory is strongly coupled unless its tension is close to the compactification scale. Weakly coupled, low-scale strings can in turn be realized only in the presence of extra large transverse dimensions, seen through gravitational interactions, or in the presence of infinitesimal string coupling. In the former case, quantum gravity scale is also low, while in the latter, gravitational and string interactions remain suppressed by the four-dimensional Planck mass. There is one exception in this general rule, allowing for large longitudinal dimensions without low string scale, when Standard Model is embedded in a six-dimensional fixed-point theory described by a tensionless string. Extra dimensions of size as large as TeV$^{-1}\simeq 10^{-16}$ cm are motivated from the problem of supersymmetry breaking in string theory, while TeV scale strings offer a solution to the gauge hierarchy problem, as an alternative to softly broken supersymmetry or technicolor. I discuss these problems in the context of the above mentioned string realizations, as well as the main physical implications both in particle accelerators and in experiments that measure gravity at sub-millimeter distances. 
  We propose an approximate wavefunction of the bound state of $N$ $D0$-branes. Its spread grows as $N^{1\over 3}$ per particle, i.e. it saturates the Polchinski's bound. 
  By integrating the Seiberg-Witten differential equation in a special path, we write ordinary gauge fields in terms of their non-commutative counterparts up to three non-commutative gauge fields. We then use this change of variables to write ordinary abelian Dirac-Born-Infeld action in terms of non-commutative fields. The resulting action is then compared with various low energy contact terms of world-sheet perturbative string scattering amplitudes from non-commutative D$p$-brane. We find completely agreement between the field theory and string theory results. Hence, it shows that perturbative string theory knows the solution of the Seiberg-Witten differential equation. 
  We consider systems of Dp branes in the presence of a nonzero B field. We study the corresponding supergravity solutions in the limit where the branes worldvolume theories decouple from gravity. These provide dual descriptions of large N noncommutative field theories. We analyse the phase structure of the theories and the validity of the different description. We provide evidence that in the presence of a nonzero B field the worldvolume theory of D6 branes decouples from gravity. We analyse the systems of M5 branes and NS5 branes in the presence of a nonzero C field and nonzero RR fields, respectively. Finally, we study the Wilson loops (surfaces) using the dual descriptions. 
  The g-function was introduced by Affleck and Ludwig as a measure of the ground state degeneracy of a conformal boundary condition. We consider this function for perturbations of the conformal Yang-Lee model by bulk and boundary fields using conformal perturbation theory, the truncated conformal space approach and the thermodynamic Bethe Ansatz (TBA). We find that the TBA equations derived by LeClair et al describe the massless boundary flows, up to an overall constant, but are incorrect when one considers a simultaneous bulk perturbation; however the TBA equations do correctly give the `non-universal' linear term in the massive case, and the ratio of g-functions for different boundary conditions is also correctly produced. This ratio is related to the Y-system of the Yang-Lee model and by comparing the perturbative expansions of the Y-system and of the g-functions we obtain the exact relation between the UV and IR parameters of the massless perturbed boundary model. 
  It is shown that there exists a soluble four parameter model in (1+1) dimensions all of whose propagators can be determined in terms of the corresponding known propagators of the vector coupling theory. Unlike the latter case, however, the limit of zero bare mass is nonsingular and yields a nontrivial theory with a rigorously unbroken gauge invariance. 
  The mass of 1/4-BPS dyonic configurations in N=4 D=4 supersymmetric Yang-Mills theories is calculated within the Nahm formulation. The SU(3) example, with two massive monopoles and one massless monopole, is considered in detail. In this case, the massless monopole is attracted to the massive monopoles by a linear potential. 
  It is emphasized that compactified little string theories have compact moduli spaces of vacua, which globally probe compact string geometry. Compactifying various little string theories on T^3 leads to 3d theories with exact, quantum Coulomb branch given by: an arbitrary T^4 of volume M_s^2, an arbitrary K3 of volume M_s^2, and moduli spaces of G=SU(N), SO(2N), or E_{6,7,8} instantons on an arbitrary T^4 or K3 of fixed volume. Compactifying instead on a T^2 leads to 4d theories with a compact Coulomb branch base which, when combined with the exact photon gauge coupling fiber, is a compact, elliptically-fibered space related to the above spaces. 
  The issue of defining discrete light-cone quantization (DLCQ) in field theory as a light-like limit is investigated. This amounts to studying quantum field theory compactified on a space-like circle of vanishing radius in an appropriate kinematical setting. While this limit is unproblematic at the tree-level, it is non-trivial for loop amplitudes. In one-loop amplitudes, when the propagators are written using standard Feynman $\alpha$-parameters we show that, generically, in the limit of vanishing radius, one of the $\alpha$-integrals is replaced by a discrete sum and the (UV renormalized) one-loop amplitude has a finite light-like limit. This is analogous to what happens in string theory. There are however exceptions and the limit may diverge in certain theories or at higher loop order. We give a rather detailed analysis of the problems one might encounter. We show that quantum electrodynamics at one loop has a well-defined light-like limit. 
  The recently proposed physical projector approach to the quantisation of gauge invariant systems is applied to the U(1) Chern-Simons theory in 2+1 dimensions as one of the simplest examples of a topological quantum field theory. The physical projector is explicitely demonstrated to be capable of effecting the required projection from the initially infinite number of degrees of freedom to the finite set of gauge invariant physical states whose properties are determined by the topology of the underlying manifold. 
  We show how some features of the AdS/CFT correspondence for AdS_3 can easily be understood via standard world-sheet methods and 2d gravity like scaling arguments. To do this, we propose a stringy way for perturbing two-dimensional CFT's around their critical points. Our strategy is to start from a stringy (world-sheet) representation of 2d CFT in space-time. Next we perturb a world-sheet action by some marginal operators such that the space-time symmetry becomes finite dimensional. As a result, we get a massive FT in space-time with a scale provided by two-dimensional coupling constant. It turns out that there exists a perturbation that leads to string theory on AdS_3. In this case the scale is equivalently provided by the radial anti-de-Sitter coordinate. 
  Previously we have shown that open groups whose generators are in arbitrary involutions may be quantized within a ghost extended framework in terms of the nilpotent BFV-BRST charge operator. Here we show that they may also be quantized within an Sp(2)-frame in which there are two odd anticommuting operators called Sp(2)-charges. Previous results for finite open group transformations are generalized to the Sp(2)-formalism. We show that in order to define open group transformations on the whole ghost extended space we need Sp(2)-charges in the nonminimal sector which contains dynamical Lagrange multipliers. We give an Sp(2)-version of the quantum master equation with extended Sp(2)-charges and a master charge of a more involved form, which is proposed to represent the integrability conditions of defining operators of connection operators and which therefore should encode the generalized quantum  Maurer-Cartan equations for arbitrary open groups. General solutions of this master equation are given in explicit form. A further extended Sp(2)-formalism is proposed in which the group parameters are quadrupled to a supersymmetric set and from which all results may be derived. 
  The action-angle variables for N-particle Hamiltonian system with the Hamiltonian $H=\sum_{n=0}^{N-1} \ln sh^{-2}(p_n/2)+\ln(\wp(x_n-x_{n+1})- \wp(x_n+x_{n+1})), x_N=x_0,$ are constructed, and the system is solved in terms of the Riemann $\theta$-functions. It is shown that this system describes pole dynamics of the elliptic solutions of 2D Toda lattice corresponding to spectral curves defined by the equation $w^2-P_{N}^{el}(z)w+\Lambda^{2N}=0$, where $P_{N}^{el}(z)$ is an elliptic function with pole of order N at the point z=0. 
  We construct families of Hamiltonians extending the Calogero model and such that a finite number of eigenvectors can be computed algebraically. 
  We consider monopole and dyon classical solutions of the Yang-Mills-Higgs system coupled to gravity in asymptotically anti-de Sitter space. We discuss both singular and regular solutions to the second order equations of motion showing that singular Wu-Yang like dyons can be found, the resulting metric being of the Reissner-N\"ordstrom type (with cosmological constant). Concerning regular solutions, we analyze the conditions under which they can be constructed discussing, for vanishing coupling constant, the main distinctive features related to the anti-de Sitter asymptotic condition; in particular, we find in this case that the v.e.v. of the Higgs scalar, $|\vec H(\infty)|$, should be quantized in units of the natural mass scale $1/e r_0$ (related to the cosmological constant) according to $|\vec H(\infty)|^2 = m(m+1) (e r_0 )^{-2}$, with $m \in Z$. 
  This PhD-thesis reviews matrix string theory and recent developments therein. Emphasis is put on symmetries, interactions and scattering processes in the matrix model. We start with an introduction to matrix string theory and a review of the orbifold model that flows out of matrix string theory in the strong YM coupling limit. Then we turn our attention to the appearance of U-duality symmetry in gauge models, after a (very) short summary of string duality, D-branes and M-theory. The last chapter reviews matrix string interactions and scattering processes in the high energy limit. Also, pair production of D-particles is studied in detail. D-pair production is expected to give important corrections to high energy scattering processes in string theory. 
  We present theories of N=2 hypermultiplets in four spacetime dimensions that are invariant under rigid or local superconformal symmetries. The target spaces of theories with rigid superconformal invariance are (4n)-dimensional {\it special} hyper-K\"ahler manifolds. Such manifolds can be described as cones over tri-Sasakian metrics and are locally the product of a flat four-dimensional space and a quaternionic manifold. The latter manifolds appear in the coupling of hypermultiplets to N=2 supergravity. We employ local sections of an Sp$(n)\times{\rm Sp}(1)$ bundle in the formulation of the Lagrangian and transformation rules, thus allowing for arbitrary coordinatizations of the hyper-K\"ahler and quaternionic manifolds. 
  We analyze the behavior of the heterotic string near an A-D-E singularity without small instantons. This problem is governed by a strongly coupled worldsheet conformal field theory, which, by a combination of O(alpha') corrections and worldsheet instantons, smooths out the singularities present in the classical geometry. 
  We advocate a set of approximations for studying the finite temperature behavior of strongly-coupled theories in 0+1 dimensions. The approximation consists of expanding about a Gaussian action, with the width of the Gaussian determined by a set of gap equations. The approximation can be applied to supersymmetric systems, provided that the gap equations are formulated in superspace. It can be applied to large-N theories, by keeping just the planar contribution to the gap equations. We analyze several models of scalar supersymmetric quantum mechanics, and show that the Gaussian approximation correctly distinguishes between a moduli space, mass gap, and supersymmetry breaking at strong coupling. Then we apply the approximation to a bosonic large-N gauge theory, and argue that a Gross-Witten transition separates the weak-coupling and strong-coupling regimes. A similar transition should occur in a generic large-N gauge theory, in particular in 0-brane quantum mechanics. 
  We investigate a bifurcation of periodic instanton in Euclidean action-temperature diagram in quantum mechanical models. It is analytically shown that multiple zero modes of fluctuation operator should be arised at bifurcation points. This fact is used to derive a condition for the appearance of bifurcation points in action-temperature diagram. This condition enables one to compute the number of bifurcation points for a given quantum mechanical system and hence, to understand the whole behaviour of decay rate. It is explicitly shown that the previous criterion derived by nonlinear perturbation or negative-mode consideration is special limit of our case. 
  The total space of the spinor bundle on the four dimensional sphere S^4 is a quaternionic line bundle that admits a metric of Spin(7) holonomy. We consider octonionic Yang-Mills instanton on this eight dimensional gravitational instanton. This is a higher dimensional generalization of (anti-)self-dual instanton on the Eguchi-Hanson space.   We propose an ansatz for Spin(7) Yang-Mills field and derive a system of non-linear ordinary differential equations. The solutions are classified according to the asymptotic behavior at infinity. We give a complete solution, when the gauge group is reduced to a product of SU(2) subalgebras in Spin(7). The existence of more general Spin(7) valued solutions can be seen by making an asymptotic expansion. 
  We study the correlation functions of the Wilson loops in noncommutative Yang-Mills theory based upon its equivalence to twisted reduced models. We point out that there is a crossover at the noncommutativity scale. At large momentum scale, the Wilson loops in noncommmutative Yang-Mills represent extended objects. They coincide with those in ordinary Yang-Mills theory in low energy limit. The correlation functions on D-branes in IIB matrix model exhibit the identical crossover behavior. It is observed to be consistent with the supergravity description with running string coupling. We also explain that the results of Seiberg and Witten can be simply understood in our formalism. 
  The bi-local model of hadrons is studied from the viewpoint of non-commutative geometry formulated so that Higgs-like scalar fields play the role of a bridge, the bi-local fields, connecting different spacetime points. We show that the resultant action for Higgs-like scalar fields has a structure similar to that of the linear sigma model. According to this formalism, we can deduce the dual nature of meson fields as the Nambu-Goldstone bosons associated with chiral symmetry breaking and bound states of quarks. 
  The statistical origin of the entropy of charged black holes in models of induced Einstein-Maxwell gravity is investigated. The constituents inducing the Einstein-Maxwell action are charged and interact with an external gauge potential. This new feature, however, does not change divergences of the statistical-mechanical entropy of the constituents near the horizon. It is demonstrated that the mechanism of generation of the Bekenstein-Hawking entropy in induced gravity is universal and it is basically the same for charged and neutral black holes. The concrete computations are carried out for induced Einstein-Maxwell gravity with a negative cosmological constant in three space-time dimensions. 
  We present a systematic and 'from the ground up' analysis of the 'minimal coupling' type of gauging of Yang-Mills symmetries in (2,2)-supersymmetric 1+1-dimensional spacetime. Unlike in the familiar 3+1-dimensional N=1 supersymmetric case, we find several distinct types of minimal coupling symmetry gauging, and so several distinct types of gauge (super)fields, some of which entirely novel. Also, we find that certain (quartoid) constrained superfields can couple to no gauge superfield at all, others (haploid ones) can couple only very selectively, while still others (non-minimal, i.e., linear ones) couple universally to all gauge superfields. 
  We re-examine Quantum Electrodynamics (QED) with massless electron as a finite quantum field theory as advocated by Gell-Mann-Low, Baker-Johnson, Adler, Jackiw and others. We analyze the Dyson-Schwinger equation satisfied by the massless electron in finite QED and conclude that the theory admits no nontrivial eigenvalue for the fine structure constant. 
  We utilize Coleman's theorem and show that quantum chromodynamics based on asymptotic freedom and confinement must have chiral symmetry realized as a spontaneously broken symmetry. 
  We rewrite the standard 4-dimensional Dirac equation in terms of quaternionic 2-component spinors, leading to a formalism which treats both massive and massless particles on an equal footing. The resulting unified description has the correct particle spectrum to be a generation of leptons, with the correct number of spin/helicity states. Furthermore, precisely three such generations naturally combine into an octonionic description of the 10-dimensional massless Dirac equation, as discussed in previous work. 
  Some aspects of correlation functions in N=4 SYM are discussed. Using N=4 harmonic superspace we study two and three-point correlation functions which are of contact type and argue that these contact terms will not affect the non-renormalisation theorem for such correlators at non-coincident points. We then present a perturbative calculation of a five-point function at two loops in N=2 harmonic superspace and verify that it reproduces the derivative of the previously found four-point function with respect to the coupling. The calculation of this four-point function via the five-point function turns out to be significantly simpler than the original direct calculation. This calculation also provides an explicit construction of an N=2 component of an N=4 five-point nilpotent covariant that violates U(1)_Y symmetry. 
  In this talk we describe the application of discrete light cone quantization (DLCQ) to supersymmetric field theories. We find that it is possible to formulate DLCQ so that supersymmetry is exactly preserved in the discrete approximation and call this formulation of DLCQ, SDLCQ. It combines the power of DLCQ with all of the beauty of supersymmetry. We have applied SDLCQ to several interesting supersymmetric theories and discussed zero modes, vacuum degeneracy, massless states, mass gaps, and theories in higher dimensions. Most recently we have used it to discuss the Maldacena conjecture. 
  We investigate systematically the asymptotic dynamics and symmetries of all three-dimensional extended AdS supergravity models. First, starting from the Chern-Simons formulation, we show explicitly that the (super)anti-de Sitter boundary conditions imply that the asymptotic symmetry algebra is the extended superconformal algebra with quadratic nonlinearies in the currents. We then derive the super-Liouville action by solving the Chern-Simons theory and obtain a realization of the superconformal algebras in terms of super-Liouville fields. Finally, we discuss the possible periodic conditions that can be imposed on the generators of the algebra and generalize the spectral flow analysed previously in the context of the $N$-extended linear superconformal algebras with $N \leq 4$. The $(2+1)$-AdS/2-CFT correspondence sheds a new light on the properties of the nonlinear superconformal algebras. It also provides a general and natural interpretation of the spectral flow. 
  We present a representation independent solution to the continuum Schwinger model in light-cone ($A^+ = 0$) gauge. We then discuss the problem of finding that solution using various quantization schemes. In particular we shall consider equal-time quantization and quantization on either characteristic surface, $x^+ = 0$ or $x^- = 0$. 
  We determine the instanton corrections to the effective coupling in SU(2), N=2 Yang-Mills theory with four flavours to all orders. Our analysis uses the S(2,Z)-invariant curve and the two instanton contribution obtained earlier to fix the higher order contributions uniquely. 
  We show that extremal correlators of chiral primary operators in N=4 supersymmetric Yang-Mills theory with SU(N) gauge group are neither renormalised at first order in perturbation theory nor receive contribution from any instanton sector at leading order in the semiclassical expansion. This lends support to the strongest version of a new prediction recently put forward on the basis of the AdS/SCFT correspondence. 
  We consider the AdS space formulation of the classical dynamics deriving from the Stueckelberg Lagrangian. The on-shell action is shown to be free of infrared singularities as the vector boson mass tends to zero. In this limit the model becomes Maxwell theory formulated in an arbitrary covariant gauge. Then we use the AdS/CFT correspondence to compute the two-point correlation functions on the boundary. It is shown that the gauge dependence concentrates on the contact terms. 
  A unified and fully relativistic treatment of the interaction of the electric and magnetic dipole moments of a particle with the electromagnetic field is given. New forces on the particle due to the combined effect of electric and magnetic dipoles are obtained. Four new experiments are proposed, three of which would observe topological phase shifts. 
  We study thermodynamical aspects of string theory in the limit in which it corresponds to Noncommutative Yang-Mills. We confirm, using the AdS/CFT correspondence, that for general Dp branes the entropy in the planar approximation depends neither on the value of the background magnetic field B nor on its rank. We find 1/N corrections to the planar entropy in the WKB approximation. For all appropriate values of p these corrections are much softer than the corresponding corrections for the B=0 case, and vanish altogether in the high temperature limit. 
  We present the most general family of stationary point-like solutions of pure N=4, d=4 Supergravity characterized by completely independent electric and magnetic charges, mass, angular momentum and NUT charge plus the asymptotic values of the scalars. It includes, for specific values of the charges all previously known BPS and non-BPS, extreme and non-extreme black holes and Taub-NUT solutions. As a family of solutions, it is manifestly invariant under T and S duality transformations and exhibits a structure related to the underlying special geometry structure of the theory. Finally, we study briefly the black-hole-type subfamily of metrics and give explicit expressions for their entropy and temperature. 
  We explain that supersymmetric attractors in general have several critical points due to the algebraic nature of the stabilization equations. We show that the critical values of the cosmological constant of the adS_5 vacua are given by the topological (moduli independent) formulae analogous to the entropy of the d=5 supersymmetric black holes. We present conditions under which more than one critical point is available (for black hole entropy as well as to the cosmological constant) so that the system tends to its own locally stable attractor point. We have found several families of Z_2-symmetric critical points where the central charge has equal absolute values but opposite signs in two attractor points. We present examples of interpolating solutions and discuss their generic features. 
  We show that the moduli space of supersymmetric black holes that arise in the five-dimensional N=2 supergravity theory with any number of vector multiplets is a weak HKT manifold. The moduli metric is expressed in terms of a HKT potential which is determined by the associated very special geometry of the supergravity theory. As an example, we give explicitly the black hole moduli metric for the STU model. 
  We obtain an Einstein metric of constant negative curvature given an arbitrary boundary metric in three dimensions, and a conformally flat one given an arbitrary conformally flat boundary metric in other dimensions. In order to compute the on-shell value of the gravitational action for these solutions, we propose to integrate the radial coordinate from the boundary till a critical value where the bulk volume element vanishes. The result, which is a functional of the boundary metric, provides a sector of the quantum effective action common to all conformal field theories that have a gravitational description. We verify that the so-defined boundary effective action is conformally invariant in odd (boundary) dimensions and has the correct conformal anomaly in even (boundary) dimensions. In three dimensions and for arbitrary static boundary metric the bulk metric takes a rather simple form. We explicitly carry out the computation of the corresponding effective action and find that it equals the non-local Polyakov action. 
  A general form of multi-channel Bethe-Salpeter equation is considered. In contradistinction to the hitherto applied approaches, our coupled system of equations leads to the simultaneous solutions for all relativistic four-point Green functions (elastic and inelastic) appearing in a given theory. A set of relations which may be helpful in approximate treatments is given. An example of extracting useful information from the equations is discussed: we consider the most general trilinear coupling of N different scalar fields and obtain - in the ladder approximation - closed expressions for the Regge trajectories and their couplings to different channels in the vicinity of l = -1. Sum rules and an example containing non-obvious symmetry are discussed. 
  Momentum space Ward identities are derived for the amputated n-point Green's functions in 3+1 dimensional non-relativistic conformal field theory. For n=4 and 6 the implications for scattering amplitudes (i.e. on-shell amputated Green's functions) are considered. Any scale invariant 2-to-2 scattering amplitude is also conformally invariant. However, conformal invariance imposes constraints on off-shell Green's functions and the three particle scattering amplitude which are not automatically satisfied if they are scale invariant. As an explicit example of a conformally invariant theory we consider non-relativistic particles in the infinite scattering length limit. 
  We study the Thermodynamic Bethe Ansatz equations for a one-parameter quantum field theory recently introduced by V.A.Fateev. The presence of chemical potentials produces a kink condensate that modifies the excitation spectrum. For some combinations of the chemical potentials an additional gapless mode appears. Various energy scales emerge in the problem. An effective field theory, describing the low energy excitations, is also introduced. 
  The approach which relates thermal Green functions to forward scattering amplitudes of on-shell thermal particles is applied to the calculation of the gluon self-energy, in a class of temporal gauges. We show to all orders that, unlike the case of covariant gauges, the exact self-energy of the gluon is transverse at finite temperature. The leading T^2 and the sub-leading ln(T) contributions are obtained for temperatures T high compared with the external momentum. The logarithmic contributions have the same structure as the ultraviolet pole terms which occur at zero temperature. 
  One-dimensional sigma-models with N supersymmetries are considered. For conventional supersymmetries there must be N-1 complex structures satisfying a Clifford algebra and the constraints on the target space geometry can be formulated in terms of these. In the cases in which the complex structures are simultaneously integrable, a conventional extended superspace formulation is given, with the geometry determined by a 2-form potential for N=2, by a 1-form potential for N=3 and a scalar potential for N=4; for N>4 it is given by a scalar potential satisfying differential constraints. This gives explicit constructions of models with N=3 but not N=4 supersymmetry and of N=4 models in which the complex structures do not satisfy a quaternionic algebra. Generalisations with central terms in the superalgebra are also considered. 
  A manifestly Lorentz invariant effective action for Yang-Mills theory depending only on commuting fields is constructed. This action posesses a bosonic symmetry, which plays a role analogous to the BRST symmetry in the standard formalism. 
  This is an elementary introduction to basic tools of supersymmetry: the spacetime symmetries, gauge theory and its application in gravity, spinors and superalgebras. Special attention is devoted to conformal and anti-de Sitter algebras. 
  We suggest that orbifold field theories which are obtained from non-commutative N=4 SYM are UV finite. In particular, non-supersymmetric orbifold truncations might be finite even at finite values of Nc. 
  By making use of the Langevin equation with a kernel, it was shown that the Feynman measure exp(-S) can be realized in a restricted sense in a diffusive stochastic process, which diverges and has no equilibrium, for bottomless systems. In this paper, the dependence on the initial conditions and the temporal behavior are analyzed for 0-dim bottomless systems. Furthermore, it is shown that it is possible to find stationary quantities. 
  It is shown that the Calogero-Moser models based on all root systems of the finite reflection groups (both the crystallographic and non-crystallographic cases) with the rational (with/without a harmonic confining potential), trigonometric and hyperbolic potentials can be simply supersymmetrised in terms of superpotentials. There is a universal formula for the supersymmetric ground state wavefunction. Since the bosonic part of each supersymmetric model is the usual quantum Calogero-Moser model, this gives a universal formula for its ground state wavefunction and energy, which is determined purely algebraically. Quantum Lax pair operators and conserved quantities for all the above Calogero-Moser models are established. 
  The relativistic two-body system in (1+1)-dimensional quantum electrodynamics is studied. It is proved that the eigenvalue problem for the two-body Hamiltonian without the self-interaction terms reduces to the problem of solving an one-dimensional stationary Schr\"odinger type equation with an energy-dependent effective potential which includes the delta-functional and inverted oscillator parts. The conditions determining the metastable energy spectrum are derived, and the energies and widths of the metastable levels are estimated in the limit of large particle masses. The effects of the self-interaction are discussed. 
  Non-perturbative effects of constant magnetic fields in a Higgs-Yukawa gauge model are studied using the extremum equations of the effective action for composite operators. It is found that the magnetic field induces a Higgs condensate, a fermion-antifermion condensate, and a fermion dynamical mass, hence breaking the discrete chiral symmetry of the theory. The results imply that for a non-simple group extension of the present model, the external magnetic field would either induce or reinforce gauge symmetry breaking. Possible cosmological applications of these results in the electroweak phase transition are suggested. 
  We present a general analysis of the thermodynamics of spinning black p-branes of string and M-theory. This is carried out both for the asymptotically-flat and near-horizon case, with emphasis on the latter. In particular, we use the conjectured correspondence between the near-horizon brane solutions and field theories with 16 supercharges in various dimensions to describe the thermodynamic behavior of these field theories in the presence of voltages under the R-symmetry. Boundaries of stability are computed for all spinning branes both in the grand canonical and canonical ensemble, and the effect of multiple angular momenta is considered. A recently proposed regularization of the field theory is used to compute the corresponding boundaries of stability at weak coupling. For the D2, D3, D4, M2 and M5-branes the critical values of Omega/T in the weak and strong coupling limit are remarkably close. Finally, we also show that for the spinning D3-brane the tree level R^4 correction supports the conjecture of a smooth interpolating function between the free energy at weak and strong coupling. 
  An algebra of functions on q-deformed Anti-de Sitter space AdS_q^D is defined which is covariant under U_q(so(2,D-1)), for q a root of unity. The star-structure is studied in detail. The scalar fields have an intrinsic high-energy cutoff, and arise most naturally as fields on orbifolds AdS_q^D \times S^D/G if D is odd, and AdS_q^D \times S_{\chi}^{2D-1}/G if D is even. Here G is a finite abelian group, and S_{\chi} is a certain ``chiral sector'' of the classical sphere. Hilbert spaces of square integrable functions are discussed. Analogous results are found for the q-deformed sphere S_q^D. 
  The zeta and eta-functions associated with massless and massive Dirac operators, in a D-dimensional (D odd or even) manifold without boundary, are rigorously constructed. Several mathematical subtleties involved in this process are stressed, as the intrisic ambiguity present in the definition of the associated fermion functional determinant in the massless case and, also, the unavoidable presence (in some situations) of a multiplicative anomaly, that can be conveniently expressed in terms of the noncommutative residue. The ambiguity is here seen to disappear in the massive case, giving rise to a phase of the Dirac determinant - that agrees with very recent calculations appeared in the mathematical literature - and to a multiplicative anomaly - also in agreement with other calculations, in the coinciding situations. After explicit, nontrivial resummation of the mass series expansions involving zeta and eta functions, the results are expressed in terms of quite simple formulas. 
  We calculate the equal-time commutator of two fermionic currents within the framework of the 1+1 dimensional fully quantized theory, describing the interaction of massive fermions with a massive vector boson. It is shown that the interaction does not change the result obtained within the theory of free fermions. 
  We use the counterterm subtraction method to calculate the action and stress-energy-momentum tensor for the (Kerr) rotating black holes in AdS_{n+1}, for n=2, 3 and 4. We demonstrate that the expressions for the total energy for the Kerr-AdS_3 and Kerr-AdS_5 spacetimes, in the limit of vanishing black hole mass, are equal to the Casimir energies of the holographically dual n-dimensional conformal field theories. In particular, for Kerr-AdS_5, dual to the case of four dimensional N=4 supersymmetric Yang-Mills theory on the rotating Einstein universe, we explicitly verify the equality of the zero mass stress tensor from the two sides of the correspondence, and present the result for general mass as a prediction from gravity. Amusingly, it is observed in four dimensions that while the trace of the stress tensor defined using the standard counterterms does not vanish, its integral does, thereby keeping the action free of ultraviolet divergences. Using a different regularisation scheme another stress tensor can be defined, which is traceless. 
  Black hole generalized p-brane solutions for a wide class of intersection rules are obtained. The solutions are defined on a manifold that contains a product of n - 1 Ricci-flat ``internal'' spaces. They are defined up to a set of functions H_s obeying a non-linear differential equations (equivalent to Toda-type equations) with certain boundary conditions imposed. A conjecture on polynomial structure of governing functions H_s for intersections related to semisimple Lie algebras is suggested. This conjecture is proved for Lie algebras: A_m, C_{m+1}, m = 1,2,... Explicit formulas for A_2-solution are obtained. Two examples of A_2-dyon solutions (e.g. dyon in D = 11 supergravity and Kaluza-Klein dyon) are considered. Post-Newtonian parameters "beta" and "gamma" corresponding to 4-dimensional section of the metric are calculated. It is shown that "beta" does not depend upon intersections of p-branes. Extremal black hole configurations are also considered. 
  In the first part of this paper I review the construction of the realistic free fermionic models, as well as current attempts to study aspects of these models in the nonperturbative framework of M- and F-theories. I discuss the recent demonstration of a Minimal Superstring Standard Model, which contains in the observable sector, below the string scale, solely the MSSM charged spectrum, and provides further support to the assertion that the true string vacuum is connected to the Z_2 X Z_2 orbifold in the vicinity of the free fermionic point in the Narain moduli space. In the second part I review the recent formulation of quantum mechanics from an equivalence postulate, which offers a new perspective on the synthesis of gravity and quantum mechanics, and contemplate possible relations with string theory and beyond. 
  In this article we present a general description of two moving branes in presence of the $B_{\mu \nu}$ field and gauge fields $A^{(1)}_{\alpha_1}$ and $A^{(2)}_{\alpha_2}$ on them, in spacetime in which some of its directions are compact on tori. Some examples are considered to elucidate this general description. Also contribution of the massless states to the interaction is extracted. Boundary state formalism is a useful tool for these considerations. 
  Conventional non-Abelian SO(4) gauge theory is able to describe gravity provided the gauge field possesses a specific polarized vacuum state in which the instantons have a preferred orientation. Their orientation plays the role of the order parameter for the polarized phase of the gauge field. The interaction of a weak and smooth gauge field with the polarized vacuum is described by an effective long-range action which is identical to the Hilbert action of general relativity. In the classical limit this action results in the Einstein equations of general relativity. Gravitons appear as the mode describing propagation of the gauge field which strongly interacts with the oriented instantons. The Newton gravitational constant describes the density of the considered phase of the gauge field. The radius of the instantons under consideration is comparable with the Planck radius. 
  The structure of spinning particle suggested by the rotating Kerr-Newman (black hole) solution, super-Kerr-Newman solution and the Kerr-Sen solution to low energy string theory is considered. Main peculiarities of the Kerr spinning particle are discussed: a vortex of twisting principal null congruence, singular ring and the Kerr source representing a rotating relativistic disk of the Compton size. A few stringy structures can be found in the real and complex Kerr geometry.   Low-energy string theory predicts the existence of a heterotic string placed on the sharp boundary of this disk. The obtained recently supergeneralization of the Kerr-Newman solution suggests the existence of extra axial singular line and fermionic traveling waves concentrating near these singularities.   We discuss briefly a possibility of experimental test of these predictions. 
  The chiral WZNW symplectic form $\Omega^{\rho}_{chir}$ is inverted in the general case. Thereby a precise relationship between the arbitrary monodromy dependent 2-form appearing in $\Omega^{\rho}_{chir}$ and the exchange r-matrix that governs the Poisson brackets of the group valued chiral fields is established. The exchange r-matrices are shown to satisfy a new dynamical generalization of the classical modified Yang-Baxter (YB) equation and Poisson-Lie (PL) groupoids are constructed that encode this equation analogously as PL groups encode the classical YB equation. For an arbitrary simple Lie group G, exchange r-matrices are found that are in one-to-one correspondence with the possible PL structures on G and admit them as PL symmetries. 
  We study the hybrid formulation of Green-Schwarz superstrings on AdS_3 with NS flux and the boundary N=4 superconformal algebra. We show the equivalence between the NSR and GS superstrings by a field redefinition. The boundary N=4 superconformal algebra is realized by the free fields of the affine Lie superalgebra A(1|1)^{(1)}. We also consider the light-cone gauge and obtain the N=4 super-Liouville theory which describes the effective theory of the single long string near the singularities of the D1-D5 system. 
  As superstring solitons that carry Neuveu-Schwarz charge can be described in terms of gerbes, one expects non-Abelian gerbes to appear e.g. in the exotic six-dimensional world-volume theories of coinciding NS5 branes. We consider open bosonic strings on a space-time that is branched in such a way that the B-field is provided with the same Lie algebra structure as the world-volume gauge field on a D-brane. These considerations motivate a generalization of the cocycle conditions and the transformation rules of an Abelian gerbe in hypercohomology. The resulting system incorporates in a natural way the NS two-form, the RR gauge field, the Chan-Paton gauge field, the relevant gauge transformations and the holonomies associated to Wilson surface observables. 
  We develope the analogue of S-duality for linearized gravity in (3+1)-dimensions. Our basic idea is to consider the self-dual (anti-self-dual) curvature tensor for linearized gravity in the context of the Macdowell-Mansouri formalism. We find that the strong-weak coupling duality for linearized gravity is an exact symmetry and implies small-large duality for the cosmological constant. 
  A number of arguments exists that the "minimal" BPS wall width in large N supersymmetric gluodynamics vanishes as 1/N. There is a certain tension between this assertion and the fact that the mesons coupled to the gluino condensate have masses which do not scale with N. To reconcile these facts we argue that there should exist additional soliton-like states with masses scaling as N. The BPS walls must be "made" predominantly of these heavy states which are coupled to the gluino condensate stronger than the conventional mesons. The tension of the BPS wall junction scales as N^2, which serves as an additional argument in favor of the 1/N scaling of the wall width. The heavy states can be thought of as solitons of the corresponding closed string theory. They are related to certain fivebranes in the M-theory construction. We study the issue of the wall width in toy models which capture some features of supersymmetric gluodynamics. We speculate that special hadrons with mass scaling as N should also exist in the large N limit of non-supersymmetric gluodynamics. 
  It is shown that quantum chromodynamics based on asymptotic freedom and confinement exhibits the vector mode of chiral symmetry conjectured by Georgi. 
  We find the terms in the nonabelian world-volume action of a system of many Dp-branes which describe the leading coupling to all type II supergravity background fields. These results are found by T-dualizing earlier results for D0-branes, which in turn were determined from calculations of the M(atrix) theory description of the supercurrent of 11D supergravity. Our results are compatible with earlier results on the supersymmetric Born-Infeld action for a single D-brane in a general background and with Tseytlin's symmetrized trace proposal for extending the abelian Born-Infeld action to a nonabelian theory. In the case p = 3, the operators we find on the D-brane world-volume are closely related to those which couple to supergravity fields in the AdS_5 * S^5 IIB supergravity background. This gives an explicit construction, including normalization, of some of the operators used in the celebrated AdS/CFT correspondence for 3-branes. We also discuss the S-duality of the action in the case p = 3, finding that the S-duality of the action determines how certain operators in the 4D N = 4 SYM theory transform under S-duality. These S-duality results give some new insight into the puzzle of the transverse 5-brane in M(atrix) theory. 
  We extend the usual world-volume action for a Dp-brane to the case of N coincident Dp-branes where the world-volume theory involves a U(N) gauge theory. The guiding principle in our construction is that the action should be consistent with the familiar rules of T-duality. The resulting action involves a variety of potential terms, i.e., nonderivative interactions, for the nonabelian scalar fields. This action also shows that Dp-branes naturally couple to RR potentials of all form degrees, including both larger and smaller than p+1. We consider the dynamics resulting from this action for Dp-branes moving in nontrivial background fields, and illustrate how the Dp-branes are ``polarized'' by external fields. In a simple example, we show that a system of D0-branes in an external RR four-form field expands into a noncommutative two-sphere, which is interpreted as the formation of a spherical D2-D0 bound state. 
  A map between string junctions in the affine 7-brane backgrounds and vector bundles on del Pezzo surfaces is constructed using mirror symmetry. It is shown that the lattice of string junctions with support on an affine 7-brane configuration is isomorphic to the K-theory group of the corresponding del Pezzo surface. This isomorphism allows us to construct a map between the states of the N=2, D=4 theories with E_N global symmetry realized in two different ways in Type IIB and Type IIA string theory. A subgroup of the SL(2,Z) symmetry of the \hat{E}_9 7-brane background appears as the Fourier-Mukai transform acting on the D-brane configurations realizing vector bundles on elliptically fibered B_9. 
  The two-loop Feynman diagram contribution to the four-graviton amplitude of eleven-dimensional supergravity compactified on a two-torus, T^2, is analyzed in detail. The Schwinger parameter integrations are re-expressed as integration over the moduli space of a second torus, \hat T^2, which enables the leading low-momentum contribution to be evaluated in terms of maps of \hat T^2 into T^2. The ultraviolet divergences associated with boundaries of moduli space are regularized in a manner that is consistent with the expected duality symmetries of string theory. This leads to an exact expression for terms of order D^4 R^4 in the effective M theory action (where R^4 denotes a contraction of four Weyl tensors), thereby extending earlier results for the R^4 term that were based on the one-loop eleven-dimensional amplitude. Precise agreement is found with terms in type IIA and IIB superstring theory that arise from the low energy expansion of the tree-level and one-loop string amplitudes and predictions are made for the coefficients of certain two-loop string theory terms as well as for an infinite set of D-instanton contributions. The contribution at the next order in the derivative expansion, D^6 R^4, is problematic, which may indicate that it mixes with higher-loop effects in eleven-dimensional supergravity. 
  The one-loop four-graviton amplitude in either of the type II superstring theories is expanded in powers of the external momenta up to and including terms of order s^4 log s R^4, where R^4 denotes a specific contraction of four linearized Weyl tensors and s is a Mandelstam invariant. Terms in this series are obtained by integrating powers of the two-dimensional scalar field theory propagator over the toroidal world-sheet as well as the moduli of the torus. The values of these coefficients match expectations based on duality relations between string theory and eleven-dimensional supergravity. 
  It is pointed out that it is very unlikely that we can find analytical solutions built up from elementary functions for the domain wall junctions problem in Wess-Zumino models possessing a trivial Kahler potential. 
  A manifestly gauge invariant continuous renormalization group flow equation is constructed for pure SU(N) gauge theory. The formulation makes sense without gauge fixing and manifestly gauge invariant calculations may thus be carried out. The flow equation is naturally expressed in terms of fluctuating Wilson loops, with the effective action appearing as an integral over a `gas' of Wilson loops. At infinite N, the effective action collapses to a path integral over the trajectory of a single particle describing one Wilson loop. We show that further regularization of these flow equations is needed. (This is introduced in part II.) 
  In this brief note we construct the propagator for the antisymmetric tensor in AdS_{d+1}. We check our result using the Poincare duality between the antisymmetric tensor and the gauge boson in AdS_5. This propagator was needed for a computation which turned out to be too hard. It can be used for computing various other things in AdS. 
  I discuss some issues of perturbative quantum gravity, namely of a theory of self-interacting massless spin-2 quantum gauge fields, the gravitons, on flat space-time, in the framework of causal perturbation theory. The central aspects of this approach lie in the construction of the scattering matrix by means of causality and Poincare covariance and in the analysis of the gauge structure of the theory. For this purpose, two main tools will be used: the Epstein-Glaser inductive and causal construction of the perturbation series for the scattering matrix and the concept of perturbative operator quantum gauge invariance borrowed from non-Abelian quantum gauge theories. The first method deals with the ultraviolet problem of quantum gravity and the second one ensures gauge invariance at the quantum level, formulated by means of a gauge charge, in each order of perturbation theory. The gauge charge leads to a characterization of the physical subspace of the graviton Fock space. Aspects of quantum gravity coupled to scalar matter fields are also discussed. 
  We analyse symmetry breaking in general gauge theories paying particular attention to the underlying geometry of the theory. In this context we find two natural metrics upon the vacuum manifold: a Euclidean metric associated with the scalar sector, and another generally inequivalent metric associated with the gauge sector. Physically, the interplay between these metrics gives rise to many of the non-perturbative features of symmetry breaking. 
  By adding gauge fields to the D-string classical solution, which have non-zero contribution to commutators in continuum limit (extreme large $N$), we introduced $(m,n)$-strings in IIB matrix model. It is found that the size of matrices depends on the value of the electric field. The tension of these strings appears in $SL(2,Z)$ invariant form. The interaction for parallel and angled strings are found in agreement with the string theory for small electric fields. 
  We discuss the use of derivative expansion techniques for the construction of thermal effective potentials. We present a theory for which the thermal bubble is analytic at the origin of the momentum-frequency space, although the internal propagators in the loop have the same mass. This means that, for this theory, the thermal effective potential is uniquely defined. We then examine a slightly different theory for which the thermal bubble displays the usual non-analyticity at the origin and the thermal effective potential is not uniquely defined. For this latter theory we compare our results with those of other works in the literature which employ the derivative expansion but find a uniquely defined thermal effective potential. We raise several questions concerning the interchange of the order of the perturbative and the derivative expansions, the thermal generalization of some non-perturbative zero temperature methods and the use of the periodicity of the external bosonic field. Finally, we re-examine the physical interpretation given to the imaginary part of the thermal bubble in the literature. 
  The consequences of noncommutativity of space coordinates of string theory in the proposed large extra dimension solution to the hierarchy problem are explored; in particular the large dimension stabilization and the graviton reabsorption in the brane are considered. 
  We discuss BPS solitons in gauged ${\cal N}=2$, D=4 supergravity. The solitons represent extremal black holes interpolating between different vacua of anti-de Sitter spaces. The isometry superalgebras are determined and the motion of a superparticle in the extremal black hole background is studied and confronted with superconformal mechanics. We show that the Virasoro symmetry of conformal mechanics, which describes the dynamics of the superparticle near the horizon of the extremal black hole under consideration, extends to a symmetry under the $w_{\infty}$ algebra of area-preserving diffeomorphisms. We find that a Virasoro subalgebra of $w_{\infty}$ can be associated to the Virasoro algebra of the asymptotic symmetries of $AdS_2$. In this way spacetime diffeomorphisms of $AdS_2$ translate into diffeomorphisms in phase space: our system offers an explicit realization of the $AdS_2/CFT_1$ correspondence. Using the dimensionally reduced action, the central charge is computed. Finally, we also present generalizations of superconformal mechanics which are invariant under ${\cal N} =1$ and ${\cal N} =2$ superextensions of $w_{\infty}$. 
  We report on tests of the AdS/CFT correspondence that are made possible by complete knowledge of the Kaluza-Klein mass spectrum of type IIB supergravity on $AdS_5 \times T^{11}$ with T^{11}=SU(2)^2/U(1). After briefly discussing general multiplet shortening conditions in SU(2,2|1) and PSU(2,2|4), we compare various types of short SU(2,2|1) supermultiplets on AdS_5 and different families of boundary operators with protected dimensions. The supergravity analysis predicts the occurrence in the SCFT at leading order in N and g_s N, of extra towers of long multiplets whose dimensions are rational but not protected by supersymmetry. 
  How to gauge fix $\k$-symmetry for the super 0-brane action on $AdS_2 \times S^2$ in Killing gauge properly is discussed in order to find the superconformal mechanics which describes super 0-brane probes moving on $AdS_2 \times S^2$. The dependence on the coordinate frame for the proper Killing gauge is considered and the subtleties of gauge-fixing $\k$-symmetry in Killing gauge are analysed explicitly. It is found that the Killing gauge works indeed without the imcompatibility if the magnetic charge of the super 0-brane is nonzero. 
  We consider the $\frac{\lambda}{4!}(\phi^{4}_{1}+\phi^{4}_{2})$ model on a d-dimensional Euclidean space, where all but one of the coordinates are unbounded. Translation invariance along the bounded coordinate, z, which lies in the interval [0,L], is broken because of the boundary conditions (BC's) chosen for the hyperplanes z=0 and z=L. Two different possibilities for these BC's boundary conditions are considered: DD and NN, where D denotes Dirichlet and N Newmann, respectively. The renormalization procedure up to one-loop order is applied, obtaining two main results. The first is the fact that the renormalization program requires the introduction of counterterms which are surface interactions. The second one is that the tadpole graphs for DD and NN have the same z dependent part in modulus but with opposite signs. We investigate the relevance of this fact to the elimination of surface divergences. 
  The topological antisymmetric tensor field theory in n-dimensions is perturbed by the introduction of local metric dependent interaction terms in the curvatures. The correlator describing the linking number between two surfaces in n-dimensions is shown to be not affected by the quantum corrections. 
  We present a technique to construct, for $D_{m}$ unitary minimal models, the non-chiral fusion rules which determines the operator content of the operator product algebra. Using these rules we solve the bootstrap equations and therefore determine the structure constants of these models. Through this approach we emphasize the role played by some discrete symmetries in the classification of minimal models. 
  We discuss topologically stable solitons in two-dimensional theories with the extended supersymmetry assuming that the spatial coordinate is compact. This problem arises in the consideration of the domain walls in the popular theories with compactified extra dimensions. Contrary to naive expectations, it is shown that the solitons on the cylinder can be BPS saturated. In the case of one chiral superfield, a complete theory of the BPS saturated solitons is worked out. We describe the classical solutions of the BPS equations. Depending on the choice of the Kahler metric, the number of such solutions can be arbitrarily large. Although the property of the BPS saturation is preserved order by order in perturbation theory, nonperturbative effects eliminate the majority of the classical BPS states upon passing to the quantum level. The number of the quantum BPS states is found. It is shown that the N=2 field theory includes an auxiliary N=1 quantum mechanics, Witten's index of which counts the number of the BPS particles. 
  In the present article we solve the Dirac-Pauli and Klein Gordon equations in an asymptotically uniformly accelerated frame when a constant magnetic field is present. We compute, via the Bogoliubov coefficients, the density of scalar and spin 1/2 particles created. We discuss the role played by the magnetic field and the thermal character of the spectrum. 
  By means of the geometric algebra the general decomposition of SU(2) gauge potential on the sphere bundle of a compact and oriented 4-dimensional manifold is given. Using this decomposition theory the SU(2) Chern density has been studied in detail. It shows that the SU(2) Chern density can be expressed in terms of the $\delta -$function $\delta (\phi) $. And one can find that the zero points of the vector fields $\phi$ are essential to the topological properties of a manifold. It is shown that there exists the crucial case of branch process at the zero points. Based on the implicit function theorem and the taylor expansion, the bifurcation of the Chern density is detailed in the neighborhoods of the bifurcation points of $\phi$. It is pointed out that, since the Chren density is a topological invariant, the sum topological chargers of the branches will remain constant during the bifurcation process. 
  The AdS-CFT correspondence is established as a re-assignment of localization to the observables which is consistent with locality and covariance. 
  The scalars of an N = 1 supersymmetric sigma-model in 4 dimensions parameterize a Kaehler manifold. The transformations of their fermionic superpartners under the isometries are often anomalous. These anomalies can be canceled by introducing additional chiral multiplets with appropriate charges. To obtain the right charges a non-trivial singlet compensating multiplet can be used. However when the topology of the underlying Kaehler manifold is non-trivial, the consistency of this multiplet requires that its charge is quantized. This singlet can be interpreted as a section of a line bundle. We determine the Kaehler potentials corresponding to the minimal non-trivial singlet chiral superfields for any compact Kaehlerian coset space G/H. The quantization condition may be in conflict with the requirement of anomaly cancelation. To illustrate this, we discuss the consistency of anomaly free models based on the coset spaces E_6/SO(10)xU(1) and SU(5)/SU(2)xU(1)xSU(3). 
  We provide a realization of the AdS$_2$/CFT$_1$ correspondence in terms of asymptotic symmetries of the AdS$_2\times$S$^1$ and AdS$_2\times$S$^2$ geometries arising in near-extremal BTZ and Reissner-Nordstr\"om black holes. Cardy's formula exactly accounts for the deviation of the Bekenstein-Hawking entropy from extremality. We also argue that this result can be extended to more general black holes near extremality. 
  We construct p-brane solutions with non-trivial world volume metrics and show that applied to supergravity theories, they will lead to threshold BPS bound states of intersecting solutions. However applied to certain specific values of the couplings in cosmological (a,b)-models non-trivial solutions can be constructed. 
  The excitations of the vortex in Abelian Higgs model with small ratio of vector and Higgs particle masses are considered. Three main modes encountered in numerical computations are described in detail. They are also compared to analytic results obtained recently by Arodz and Hadasz. 
  A general theory of permutation orbifolds is developed for arbitrary twist groups. Explicit expressions for the number of primaries, the partition function, the genus one characters, the matrix elements of modular transformations and for fusion rule coefficients are presented, together with the relevant mathematical concepts, such as Lambda-matrices and twisted dimensions. The arithmetic restrictions implied by the theory for the allowed modular representations in CFT are discussed. The simplest nonabelian example with twist group S_3 is described to illustrate the general theory. 
  The leading long-distance 1-loop quantum corrections to the Coulomb potential are derived for scalar QED and their gauge-independence is explicitly checked. The potential is obtained from the direct calculation of the 2-particle scattering amplitude, taking into account all relevant 1-loop diagrams. Our investigation should be regarded as a first step towards the same programme for effective Quantum Gravity. In particular, with our calculation in the framework of scalar QED, we are able to demonstrate the incompleteness of some previous studies concerning the Quantum Gravity counterpart. 
  We propose a supergravity inspired derivation of a Randall-Sundrum's type action as an effective description of the dynamics of a brane coupled to the bulk through gravity only. The cosmological constants in the bulk and on the brane appear at the classical level when solving the equations of motion describing the bosonic sector of supergravities in ten and eleven dimensions coupled to the brane. They are related to physical quantities like the brane electric charge and thus inherit some of their physical properties. The most appealing property is their quantization: in d_\perp extra dimensions, Lambda_brane goes like N and Lambda_bulk like N^{2/(2-d_perp)}. This dynamical origin also explains the apparent fine-tuning required in the Randall-Sundrum scenario. In our approach, the cosmological constants are derived parameters and cannot be chosen arbitrarily; instead they are determined by the underlying Lagrangian. Some of the branes we construct that support cosmological constant in the bulk have supersymmetric properties: D3-branes of type IIB superstring theory provide an explicit example. 
  A very brief review is given of some of the developments leading to our current understanding of black holes in string theory. This is followed by a discussion of two possible misconceptions in this subject - one involving the stability of small black holes and the other involving scale radius duality. Finally, I describe some recent results concerning quasinormal modes of black holes in anti de Sitter spacetime, and their implications for strongly coupled conformal field theories (in various dimensions). 
  We consider an easy way to get the noncommutative spacetime in Minkowski space. This corresponds to introducing a magnetic field ${\rm\bf B} = B \hat {\rm\bf k}$ in the plane. We construct a green's function in coordinate space which includes a Moyal phase factor. The projection to the lowest Landau level(LLL) is necessary for a simple calculation. Using this green's function and a second quantized formalism, we study the thermodynamic property of the anyons on the noncommutative geometry. It turns out that the Moyal phase factors contribute to the thermodynamic potential $\Omega$ as opposed to the free-particle nature. 
  From the topological properties of a three dimensional vector order parameter, the topological current of point defects is obtained. One shows that the charge of point defects is determined by Hopf indices and Brouwer degrees. The evolution of point defects is also studied. One concludes that there exist crucial cases of branch processes in the evolution of point defects when the Jacobian $D(\frac \phi x)=0$. 
  For the description of space-time fermions, Dirac-K\"ahler fields (inhomogeneous differential forms) provide an interesting alternative to the Dirac spinor fields. In this paper we develop a similar concept within the symplectic geometry of phase-spaces. Rather than on space-time, symplectic Dirac-K\"ahler fields can be defined on the classical phase-space of any Hamiltonian system. They are equivalent to an infinite family of metaplectic spinor fields, i.e. spinors of Sp(2N), in the same way an ordinary Dirac-K\"ahler field is equivalent to a (finite) mulitplet of Dirac spinors. The results are interpreted in the framework of the gauge theory formulation of quantum mechanics which was proposed recently. An intriguing analogy is found between the lattice fermion problem (species doubling) and the problem of quantization in general. 
  Certain supergravity solutions (including domain walls and the magnetic fivebrane) have recently been generalised by Brecher and Perry by relaxing the condition that the brane worldvolume be flat. In this way they obtain examples in which the brane worldvolume is a static spacetime admitting parallel spinors. In this note we simply point out that the restriction to static spacetimes is unnecessary, and in this way exhibit solutions where the brane worldvolume is an indecomposable Ricci-flat lorentzian manifold admitting parallel spinors. We discuss more Ricci-flat fivebranes and domain walls, as well as new Ricci-flat D3-branes. 
  We point out that in (open) string compactifications with non-zero NS-NS B-field we can have large Kaluza-Klein thresholds even in the small volume limit. In this limit the corresponding gauge theory description is in terms of a compactification on a non-commutative space (e.g., a torus or an orbifold thereof). Based on this observation we discuss a brane world scenario of non-commutative unification via Kaluza-Klein thresholds. In this scenario, the unification scale can be lowered down to the TeV-range, yet the corresponding compactification radii are smaller than the string length. We discuss a potential application of this scenario in the context of obtaining mixing between different chiral generations which is not exponentially suppressed - as we point out, such mixing is expected to be exponentially suppressed in certain setups with large volume compactifications. We also point out that T-duality is broken by certain non-perturbative twisted open string sectors which are supposed to give rise to chiral generations, so that in the case of a small volume compactification with a rational B-field we cannot T-dualize to a large volume description. In this sense, the corresponding field theoretic picture of unification via Kaluza-Klein thresholds in this setup is best described in the non-commutative language. 
  I propose a class of D\geq{2} lattice SU(N) gauge theories dual to certain vector models endowed with the local [U(N)]^{D} conjugation-invariance and Z_{N} gauge symmetry. In the latter models, both the partitition function and Wilson loop observables depend nontrivially only on the eigenvalues of the link-variables. Therefore, the vector-model facilitates a master-field representation of the large N loop-averages in the corresponding induced gauge system. As for the partitition function, in the limit N->{infinity} it is reduced to the 2Dth power of an effective one-matrix eigenvalue-model which makes the associated phase structure accessible. In particular a simple scaling-condition, that ensures the proper continuum limit of the induced gauge theory, is proposed. We also derive a closed expression for the large N average of a generic nonself-intersecting Wilson loop in the D=2 theory defined on an arbitrary 2d surface. 
  We present a theorem that determines the value of the Wilson loop associated with a Nambu-Goto action which generalizes the action of the $AdS_5\times S_5$ model. In particular we derive sufficient conditions for confining behavior. We then apply this theorem to various string models. We go beyond the classical string picture by incorporating quadratic quantum fluctuations. We show that the bosonic determinant of $D_p$ branes with 16 supersymmetries yields a Luscher term.  We confirm that the free energy associated with a BPS configuration of a single quark is free from divergences. We show that unlike for a string in flat space time in the case of $AdS_5\times S_5$ the fermionic determinant does not cancel the bosonic one. For a setup that corresponds to a confining gauge theory the correction to the potential is attractive. We determine the form of the Wilson loop for actions that include non trivial $B_{\mu\nu}$ field. The issue of an exact determination of the value of the stringy Wilson loop is discussed. Talk presented in string 99 Potsdam. 
  We consider heterotic string compactifications to four dimensions when instantons shrink to zero size. If the standard model gauge group originates from the new gauge symmetry associated with the small instantons singularity, then the weakly or strongly coupled heterotic string scales can be taken to be arbitrarily low. The SO(32) and E_8\times E_8 gauge groups can then be very weakly coupled even at the string scale and behave as non-abelian global symmetries. We comment on a possible role of small instantons in supersymmetry breaking. 
  We discuss the quantum mechanics of a particle in a magnetic field when its position x^{\mu} is restricted to a periodic lattice, while its momentum p^{\mu} is restricted to a periodic dual lattice. Through these considerations we define non-commutative geometry on the lattice. This leads to a deformation of the algebra of functions on the lattice, such that their product involves a ``diamond'' product, which becomes the star product in the continuum limit. We apply these results to construct non-commutative U(1) and U(M) gauge theories, and show that they are equivalent to a pure U(NM) matrix theory, where N^{2} is the number of lattice points. 
  We study the thermodynamics of the large N noncommutative super Yang-Mills theory in the strong 't Hooft coupling limit in the spirit of AdS/CFT correspondence. It has already been noticed that some thermodynamic quantities of near-extremal D3-branes with NS B fields, which are dual gravity configurations of the noncommutative ${\cal N}$=4 super Yang-Mills theory, are the same as those without B fields. In this paper, (1) we examine the $\alpha'^3 R^4$ corrections to the free energy and find that the part of the tree-level contribution remains unchanged, but the one-loop and the non-perturbative D-instanton corrections are suppressed, compared to the ordinary case. (2) We consider the thermodynamics of a bound state probe consisting of D3-branes and D-strings in the near-extremal D3-brane background with B field, and find the thermodynamics of the probe is the same as that of a D3-brane probe in the D3-brane background without B field. (3) The stress-energy tensor of the noncommutative super Yang-Mills theory is calculated via the AdS/CFT correspondence. It is found that the tensor is not isotropic and its trace does not vanish, which confirms that the super Yang-Mills is not conformal even in four dimensions due to the noncommutative nature of space. Our results render further evidence for the argument that the large N noncommutative and ordinary super Yang-Mills theories are equivalent not only in the weak coupling limit, but also in the strong coupling limit. 
  We discuss an exotic class of Kaluza-Klein models in which the internal space is neither compact nor even of finite volume. Rather than using the usual compact internal space we consider the case where particles are gravitationally trapped near a four-dimensional submanifold of the higher dimensional spacetime. A specific model exhibiting this phenomenon is constructed in five dimensions.   Physics Letters B159, 22-25 (1985).   Note: This rather old paper has recently been subject of renewed interest due to the explosion of activity in the ``non-compact extra dimensions'' variant of the Kaluza-Klein model. It is not available elsewhere on the Internet, and in the interests of easy access I am placing it on hep-th. The body of the paper is identical to the published version. A small separate note has been added at the end of the paper. 
  We consider freely acting orbifold compactifications, which interpolate in two possible decompactification limits between the supersymmetric type II string and the non-supersymmetric type 0 string. In particular we discuss how D-branes are incorporated into these orbifold models. Investigating the open string spectrum on D3-branes, we will show that one can interpolate in this way between N=4 supersymmetric U(N) respectively U(2N) Yang-Mills theories and non-supersymmetric U(N)*U(N) gauge theories with adjoint massless scalar fields plus bifundamental massless fermions in a smooth way. Finally, by lifting the orbifold construction to M-theory, we conjecture some duality relations and show that in particular a new supersymmetric branch of gauge like theories emanate for the non-supersymmetric model. 
  An exact solution of domain wall junction is obtained in a four-dimensional N=1 supersymmetric U(1) X U(1)' gauge theory with three pairs of chiral superfields which is motivated by the N=2 SU(2) gauge theory with one flavor perturbed by an adjoint scalar mass. The solution allows us to evaluate various quantities including a new central charge Y_k associated with the junction besides Z_k which appears already in domain walls. We find that the new central charge Y_k gives a negative contribution to the mass of the domain wall junction whereas the central charge Z_k gives a dominant positive contribution. One has to be cautious to identify the central charge Y_k alone as the mass of the junction. 
  We review the theory of higher spin gauge fields in 2+1 and 3+1 dimensional anti-de Sitter space and present some new results on the structure of higher spin currents and explicit solutions of the massless equations. A previously obtained d=3 integrating flow is generalized to d=4 and is shown to give rise to a perturbative solution of the d=4 nonlinear higher spin equations. A particular attention is paid to the relationship between the star-product origin of the higher spin symmetries, AdS geometry and the concept of space-time locality. 
  Spectral zeta functions $\zeta(s)$ for the massless scalar fields obeying the Dirichlet and Neumann boundary conditions on a surface of an infinite cylinder are constructed. These functions are defined explicitly in a finite domain of the complex plane s containing the closed interval of real axis $-1\le$ Re $s \le 0$. Proceeding from this the spectral zeta functions for the boundary conditions given on a circle (boundary value problem on a plane) are obtained without any additional calculations. The Casimir energy for the relevant field configurations is deduced. 
  Marginally bound systems of two types of branes are considered, such as the prototypical case of Dp+4 branes and Dp branes. As the transverse separation between the two types of branes goes to zero, different behaviour occurs in the supergravity solutions depending on p; no-hair theorems result for p<=1 only. Within the framework of the AdS/CFT correspondence, these supergravity no-hair results are understood as dual manifestations of the Coleman-Mermin-Wagner theorem. Furthermore, the rates of delocalization for p<=1 are matched in a scaling analysis. Talk given at ``Strings '99''; based on hep-th/9903213 with D. Marolf. 
  We propose a covariant geometrical expression for the c-function for theories which admit dual gravitational descriptions. We state a c-theorem with respect to this quantity and prove it. We apply the expression to a class of geometries, from domain walls in gauged supergravities, to extremal and near extremal Dp branes, and the AdS Schwarzschild black hole. In all cases, we find agreement with expectations. 
  When the gauge groups of the two heterotic string theories are broken, over tori, to their "SO(16)x SO(16)" subgroups, the winding modes correspond to representations which are spinorial with respect to those subgroups. Globally, the two subgroups are isomorphic neither to SO(16)x SO(16) nor to each other. Any attempt to formulate the T-duality of the two theories on any manifold more complicated than the product of a circle with a Euclidean space must therefore take into account the possible non-existence of the relevant "generalised spin structure". We give here a global formulation of T-duality in this case, and discuss examples where the duality seems to be obstructed. 
  The quantum field theory describing electric and magnetic charges and revealing a dual symmetry was developed in the Zwanziger formalism. The renormalization group (RG) equations for both fine structure constants - electric $\alpha$ and magnetic $\tilde \alpha$ - were obtained. It was shown that the Dirac relation is valid for the renormalized $\alpha $ and $\tilde \alpha$ at the arbitrary scale, but these RG equations can be considered perturbatively only in the small region: $0.25 \stackrel{<}{\sim} \alpha, \tilde \alpha \stackrel{<}{\sim} 1$ with $\tilde \alpha$ given by the Dirac relation: $\alpha {\tilde \alpha}$ = 1/4. 
  We exhibit a relationship between the massless $a_2^{(2)}$ integrable quantum field theory and a certain third-order ordinary differential equation, thereby extending a recent result connecting the massless sine-Gordon model to the Schr\"odinger equation. This forms part of a more general correspondence involving $A_2$-related Bethe ansatz systems and third-order differential equations. A non-linear integral equation for the generalised spectral problem is derived, and some numerical checks are performed. Duality properties are discussed, and a simple variant of the nonlinear equation is suggested as a candidate to describe the finite volume ground state energies of minimal conformal field theories perturbed by the operators $\phi_{12}$, $\phi_{21}$ and $\phi_{15}$. This is checked against previous results obtained using the thermodynamic Bethe ansatz. 
  We introduce a spectral approach to non-perturbative field theory within the periodic field formalism. As an example we calculate the real and imaginary parts of the propagator in 1+1 dimensional phi^4 theory, identifying both one-particle and multi-particle contributions. We discuss the computational limits of existing diagonalization algorithms and suggest new quasi-sparse eigenvector methods to handle very large Fock spaces and higher dimensional field theories. 
  In the light of $\phi$-mapping method and the topological tensor current theory, the topological structure and the topological quantization of topological defects are obtained under the condition that Jacobian $J(\phi/v)\neq0$. When $J(\phi/v)=0$, it is shown that there exists the crucial case of branch process. Based on the implicit function theorem and the Taylor expansion, the generation, annihilation and bifurcation of the linear defects are detailed in the neighborhoods of the limit points and bifurcation points of $\phi$-mapping, respectively. 
  Strominger has derived the Bekenstein-Hawking entropy of the BTZ black hole using asymptotic Virasoro algebra. We apply Strominger's method to a black hole solution found by Martinez and Zanelli (MZ). This is a solution of three-dimensional gravity with a conformal scalar field. The solution is not $AdS_3$, but it is asymptotically $AdS_3$; therefore, it has the asymptotic Virasoro algebra. We compute the central charge for the theory and compares Cardy's formula with the Bekenstein-Hawking entropy. It turns out that the functional form does agree, but the overall numerical coefficient does not. This is because this approach gives the "maximum possible entropy" for the numerical coefficient. 
  The holographic ratio in Pre-big bang string cosmology is obtained in the presence string sources. An iterative procedure is adopted to solve the equations of motion and derive the ratio in four dimensional world. First the zeroth order ratio is computed in the remote past, i.e. at $t=-\infty$, then the holographic ratio is obtained taking into account the evolution of the backgrounds following the iterative procedure. The corrections to the zeroth order value of the ratio depends on the form of the initial number distribution of the strings chosen. Moreover, we estimate the holographic ratio in the recent past (i.e. when $\gamma=-\frac{1}{d}$) and in the remote past (i.e. when $\gamma= 0$), $\gamma\equiv\frac{p}{\varrho}$, in different dimensions in the Einstein frame and in the string frame. We find that in the first case it has similar time dependences in both the frames, especially in four dimensions the ratio is explicitly computed to be the same in the two cases, whereas for $\gamma=0$ case the time dependence is different. 
  Recent theoretical and numerical developments show analogies between quantum chromodynamics (QCD) and disordered systems in condensed matter physics. We study the spectral fluctuations of a Dirac particle propagating in a finite four dimensional box in the presence of gauge fields. We construct a model which combines Efetov's approach to disordered systems with the principles of chiral symmetry and QCD. To this end, the gauge fields are replaced with a stochastic white noise potential, the gauge field disorder. Effective supersymmetric non-linear sigma-models are obtained. Spontaneous breaking of supersymmetry is found. We rigorously derive the equivalent of the Thouless energy in QCD. Connections to other low-energy effective theories, in particular the Nambu-Jona-Lasinio model and chiral perturbation theory, are found. 
  Issues related with microcausality violation and continuum limit in the context of (1+1) dimensional scalar field theory in discretized light-cone quantization (DLCQ) are addressed in parallel with discretized equal time quantization (DETQ) and the fact that Lorentz invariance and microcausality are restored if one can take the continuum limit properly is emphasized. In the free case, it is shown with numerical evidence that the continuum results can be reproduced from DLCQ results for the Pauli-Jordan function and the real part of Feynman propagator. The contributions coming from $k^+$ near zero region in these cases are found to be very small in contrast to the common belief that $k^+=0$ is an accumulation point. In the interacting case, aspects related to the continuum limit of DLCQ results in perturbation theory are discussed. 
  The D-brane spectrum of a class of $\Zop_2$ orbifolds of toroidally compactified Type IIA and Type IIB string theory is analysed systematically. The corresponding K-theory groups are determined and complete agreement is found. The charge densities of the various branes are also calculated. 
  We study the unitary supermultiplets of the N=4 d=7 anti-de Sitter (AdS_7) superalgebra OSp(8^*|4), with the even subalgebra SO(6,2) X USp(4), which is the symmetry superalgebra of M-theory on AdS_7 X S^4. We give a complete classification of the positive energy doubleton and massless supermultiplets of OSp(8^*|4) . The ultra-short doubleton supermultiplets do not have a Poincar\'{e} limit in AdS_7 and correspond to superconformal field theories on the boundary of AdS_7 which can be identified with d=6 Minkowski space. We show that the six dimensional Poincare mass operator vanishes identically for the doubleton representations. By going from the compact U(4) basis of SO^*(8)=SO(6,2) to the noncompact basis SU^*(4)XD (d=6 Lorentz group times dilatations) one can associate the positive (conformal) energy representations of SO^*(8) with conformal fields transforming covariantly under the Lorentz group in d=6. The oscillator method used for the construction of the unitary supermultiplets of OSp(8^*|4) can be given a dynamical realization in terms of chiral super-twistor fields. 
  Six dimensional supergravities on $ADS_3 \times S^3$ present interest due to the role they play in the $AdS/CFT$ correspondence. The correspondence in this case states the equivalence between supergravity on the given background and a still unknown conformal field theory. The conformal field theory in question is expected to appear by deforming of the free conformal field theory on $S^N(T^4)$ in a way which preserves the superconformal symmetry. The purpose of this paper is to compute the first nontrivial corrections to the equations of motion for the chiral primary fields coming from supergravity. Using the methods already developed which involve nontrivial redefinitions of fields, we compute three-point correlation functions for scalar chiral primaries and notice similarities between their expressions and those obtained in the orbifold conformal field theory. 
  It is well-known that the chiral WZNW Bloch waves satisfy a quadratic classical exchange algebra which implies the affine Kac-Moody algebra for the corresponding currents. We here obtain a direct derivation of the exchange algebra by inverting the symplectic form on the space of Bloch waves, and give a completely algorithmic construction of its generalized free field realizations that extend the classical Wakimoto realizations of the current algebra. 
  We calculate the conformal anomaly from 5d Weyl gravity (with broken conformal symmetry) which is conjectured to be supergravity dual to ${\cal N}=2$ superconformal field theory via AdS/CFT correspondence. Its comparison with ${\cal N}=2$ SCFT conformal anomaly (UV calculation) suggests that such duality may exist subject to presence of sub-leading 1/N corrections to cosmological and gravitational constants. 
  Light-Front (LF) Hamiltonian for QED in (1+1)-dimensions is constructed using the boson form of this model with additional Pauli-Villars type ultraviolet regularization. Perturbation theory, generated by this LF Hamiltonian, is proved to be equivalent to usual covariant chiral perturbation theory. The obtained LF Hamiltonian depends explicitly on chiral condensate parameters which enter in a form of some coupling constants. 
  We discuss the relevance of Eisenstein series for representing certain G(Z)-invariant string theory amplitudes which receive corrections from BPS states only. The Eisenstein series are constructed using G(Z)-invariant mass formulae and are manifestly invariant modular functions on the symmetric space K\G(R) of non-compact type, with K the maximal compact subgroup of G(R). In particular, we show how Eisenstein series of the T-duality group SO(d,d,Z) can be used to represent one- and g-loop amplitudes in compactified string theory. We also obtain their non-perturbative extensions in terms of the Eisenstein series of the U-duality group E_{d+1(d+1)}(Z). 
  The N=4 gauged SU(2)$\times$SU(1,1) supergravity in four-dimensional Euclidean space is obtained via a consistent dimensional reduction of the N=1, D=10 supergravity on $S^3\times AdS_3$. The dilaton potential in the theory is proportional to the difference of the two gauge coupling constants, which is due to the opposite signs of the curvatures of $S^3$ and $AdS_3$. As a result, the potential can be positive, negative, or zero-depending on the values of the constants. A consistent reduction of the fermion supersymmetry transformations is performed at the linearized level, and special attention is paid to the Euclidean Majorana condition. A further reduction of the D=4 theory is considered to the static, purely magnetic sector, where the vacuum solutions are studied. The Bogomol'nyi equations are derived and their essentially non-Abelian monopole-type and sphaleron-type solutions are presented. Any solution in the theory can be uplifted to become a vacuum of string or M-theory. 
  Asymptotically anti-de Sitter spacetimes with Poincare invariance along the boundary can describe, via the AdS/CFT correspondence, either relevant deformations of a conformal field theory or non-conformal vacuum states. I consider examples of both types constructed in the framework of five-dimensional gauged supergravity. I explain the proof and motivation of a gravitational ``c-theorem'' which is independent of dimension. I show how one class of examples can be elevated to ten-dimensional geometries involving distributions of parallel D3-branes. For these cases some peculiar properties of two-point functions emerge, and I close with speculations on their physical origin. 
  Following work on theories with SU(N) gauge groups, we perform a large-N saddle-point approximation of the measure for ADHM multi-instantons in N=4 supersymmetric gauge theories with symplectic or orthogonal gauge groups. For Sp(N) we find that a saddle-point only exists in the even instanton charge sector. For either Sp(N) or SO(N) the saddle-point solution parametrizes AdS_5 x RP^5, the dual supergravity geometry in the AdS/CFT correspondence for these theories. The instanton measure at large-N has the form of the partition function of ten-dimensional N=1 supersymmetric gauge theory with a unitary gauge group dimensionally reduced to zero dimensions. 
  We extend equal rank embedding of reductive Lie algebras to that of basic Lie superalgebras. The Kac character formulas for equal rank embedding are derived in terms of subalgebras and Kostant's cubic Dirac operator for equal rank embedding of Lie superalgebras is constructed from both even and odd generators and their related structure constants. 
  A renormalization scheme is suggested where QCD input parameters - quark mass and coupling constant - are expressed in terms of gauge invariant and infrared stable quantities. For the renormalization of coupling constant the quark anomalous electromagnetic moment is used; the latter is calculated in a two loop approximation. Examination of the renormalized S matrix indicated confinement phenomenon already in the framework of perturbation theory. 
  Subsingular vectors of the N=2 superconformal algebras were discovered, and examples given, in 1996. Shortly afterwards Semikhatov and Tipunin claimed to have obtained a complete classification of the N=2 subsingular vectors in the paper `The Structure of Verma Modules over the N=2 Superconformal algebra', hep-th/9704111, published in CMP 195 (1998) 129. Surprisingly, the only explicit examples of N=2 subsingular vectors known at that time did not fit into their classification. All the results presented in that paper, including the classification of subsingular vectors, were based on the following assumptions: i) The authors claimed that there are only two different types of submodules in N=2 Verma modules, overlooking from the very beginning indecomposable `no-label' singular vectors, that had been discovered a few months before, and clearly do not fit into their two types of submodules, and ii) The authors claimed to have constructed `non-conventional' singular vectors with the property of generating the two types of submodules maximally, i.e. with no subsingular vectors left outside. In this note we prove that both assumptions are incorrect. These facts also affect profoundly the results presented in several other publications, especially the papers: `On the Equivalence of Affine sl(2) and N=2 ....', by Semikhatov, hep-th/9702074, `Embedding Diagrams of N=2 Verma Modules ....', by Semikhatov and Sirota, hep-th/9712102, and `All Singular Vectors of the N=2 ....', by Semikhatov and Tipunin, hep-th/9604176 (last revised version in September 98). 
  In a recent essay, we discussed the possibility of using polymer sizing to model the collapse of a single, long excited string to a black hole. In this letter, we apply this idea to bring further support to string/black hole correspondence. In particular, we reproduce Horowitz and Polchinki's results for self-gravitating fundamental strings and speculate on the nature of the quantum degrees of freedom of black holes in string theory. 
  The algebraic structure and the relationships between the eigenspaces of the Calogero-Sutherland model (CSM) and the Sutherland model (SM) on a circle are investigated through the Cherednik operators. We find an exact connection between the simultaneous non-symmetric eigenfunctions of the $A_{N-1}$ Cherednik operators, from which the eigenfunctions of the CSM and SM are constructed, and the monomials. This construction, not only, allows one to write down a harmonic oscillator algebra involving the Cherednik operators, which yields the raising and lowering operators for both of these models, but also shows the connection of the CSM with free oscillators and the SM with free particles on a circle. We also point out the subtle differences between the excitations of the CSM and the SM. 
  Counterterm actions are constructed along the ADM formalism. It is shown that the counterterm action can be intrinsically written in terms of intrinsic boundary geometry. Using the expression of counterterm action, we obtain a general form of the counterterm action available for any $d$-dimensional spherical boundary. In the description, we also derive {\it arbitrary} dimensional holographic conformal anomaly. It is also shown that counterterm actions for AF spaces can be obtained from the AdS description just as taking the limit of $\ell \to \infty$. An asymptotically flat spacetime with non-spherical boundary is speculated. In the example, additional counterterms to eliminate (leading) divergent terms due to deviation of boundary from round sphere are imagined by observing (4-dimensional) holographic anomaly proportional to $\Box R$. Argument of the deceptive-like anomaly is given by comparing with the holographic description of 5-dimensional Kerr-AdS spacetime. 
  New S-dualities in a scale invariant N=2 gauge theory with SU(2) x SU(2) gauge group are derived from embeddings of the theory in two different larger asymptotically free theories. The true coupling space of the scale invariant theory is a 20-fold identification of the coupling space found in the M- and string-theory derivations of the low energy effective action, implying a larger S-duality group. Also, this coupling space is different from the naively expected direct product of two SL(2,Z) fundamental domains, as it contains a different topology of fixed points. 
  Supergravity on $AdS_3\times S^3\times {\bf T}^4$ has a dual description as a conformal sigma-model with the target space being the moduli space of instantons on the noncommutative torus. We derive the precise relation between the parameters of this noncommutative torus and the parameters of the near-horizon geometry. We show that the low energy dynamics of the system of $D1D5$ branes wrapped on the torus of finite size is described in terms of the noncommutative geometry. As a byproduct, we give a prediction on the dependence of the moduli space of instantons on the noncommutative ${\bf T}^4$ on the metric and the noncommutativity parameter. We give a compelling evidence that the moduli space of stringy instantons on ${\bf R}^4$ with the $B$ field does not receive $\alpha'$-corrections. We also study the relation between the $D1D5$ sigma-model instantons and the supergravity instantons. 
  By performing the canonical quantization of the Abelian Chern-Simons model on the light-front (as suggested by Dirac), we clarify some controversies appearing in recent papers that discuss the relation between the existence of excitations carrying fractional spin and statistics (anyons) and this model. Properties of the Chern-Simons model on the light-front are investigated in detail, following the Dirac method for constrained dynamical systems, both for a coupled complex scalar field as well as for a spinor field. 
  A perturbation theory for Massive Thirring Model (MTM) in radial quantization approach is developed. Investigation of the twisted sector in this theory allows us to calculate the vacuum expectation values of exponential fields $ exp iaphi (0) $ of the sine-Gordon theory in first order over Massive Thirring Models coupling constant. It appears that the apparent difficulty in radial quantization of massive theories, namely the explicite ''time'' dependence of the Hamiltonian, may be successfully overcome. The result we have obtained agrees with the exact formula conjectured by Lukyanov and Zamolodchikov and coincides with the analogous calculations recently carried out in dual angular quantization approach by one of the authors. 
  In continuum physics, there are important topological aspects like instantons, theta-terms and the axial anomaly. Conventional lattice discretizations often have difficulties in treating one or the other of these aspects. In this paper, we develop discrete quantum field theories on fuzzy manifolds using noncommutative geometry. Basing ourselves on previous treatments of instantons and chiral fermions (without fermion doubling) on fuzzy spaces and especially fuzzy spheres, we present discrete representations of theta-terms and topological susceptibility for gauge theories and derive axial anomaly on the fuzzy sphere. Our gauge field action for four dimensions is bounded by the modulus of the instanton number as in the continuum. 
  A model of SU(2) gauge theory in the space-time $R\times S^3$ is constructed in terms of local gauge-invariant variables. A metric tensor $g_{\mu \nu}$ is defined starting with the components of the strength tensor $F_{\mu \nu}^k$ and of its dual $\widetilde{F}_{\mu \nu}^k$. It is shown that the components $g_{\mu \nu}$ determine the gauge field equations if some supplementary constraints are imposed. Two families of analitical solutions of the field equations are also obtained. 
  In fundamental physics, this has been the century of quantum mechanics and general relativity. It has also been the century of the long search for a conceptual framework capable of embracing the astonishing features of the world that have been revealed by these two ``first pieces of a conceptual revolution''. I discuss the general requirements on the mathematics and some specific developments towards the construction of such a framework. Examples of covariant constructions of (simple) generally relativistic quantum field theories have been obtained as topological quantum field theories, in nonperturbative zero-dimensional string theory and its higher dimensional generalizations, and as spin foam models. A canonical construction of a general relativistic quantum field theory is provided by loop quantum gravity. Remarkably, all these diverse approaches have turn out to be related, suggesting an intriguing general picture of general relativistic quantum physics. 
  We show that gravitational effects of global cosmic 3-branes can be responsible for compactification from six to four space-time dimensions, naturally producing the observed hierarchy between electroweak and gravitational forces. The finite radius of the transverse dimensions follows from Einstein's equation, and is exponentially large compared with the scales associated with the 3-brane. The space-time ends on a mild naked singularity at the boundary of the transverse dimensions; nevertheless unitary boundary conditions render the singularity harmless. 
  Nonlinear gauge theory is a gauge theory based on a nonlinear Lie algebra (finite W algebra) or a Poisson algebra, which yields a canonical star product for deformation quantization as a correlator on a disk. We pursue nontrivial deformation of topological gauge theory with conjugate scalars in two dimensions. This leads to two-dimensional nonlinear gauge theory exclusively, which implies its essential uniqueness. We also consider a possible extension to higher dimensions. 
  It is shown that the AdS_3 gravity action with boundary terms is non invariant under diffeomorphisms and that its Lie derivative has the form of the Weyl anomaly in two dimensions. This variation is compensated by a Weyl transformation of the boundary metric when the radial derivative of the metric on the boundary is expressed in terms of the stress tensor of a Liouville field. The obtained invariance of the action under the combined transformation of a diffeomorphism and a Weyl transformation allows to interpret the computed Lie derivative as minus the Weyl anomaly of the two-dimensional effective action. 
  A deformed Nahm equation for the BPS equation in the noncommutative N=4 supersymmetric U(2) Yang-Mills theory is obtained. Using this, we constructed explicitly a monopole solution of the noncommutative BPS equation to the linear order of the noncommutativity scale. We found that the leading order correction to the ordinary SU(2) monopole lies solely in the overall U(1) sector and that the overall U(1) magnetic field has an expected long range component of magnetic dipole moment. 
  Two kinds of realizations of symmetry on classical domains or the Euclidean version of AdS space are used to study AdS/CFT correspondence. Mass of free particles is defined as an AdS group invariant, the Klein-Gordon and Dirac equations for relativistic particles in the AdS space are set up as a mimic in the case of Minkowskian space. The bulk-boundary propagator on the AdS space is given by the Poisson kernel. Theorems on the Poisson kernel guarantee the existence and sole of the bulk-boundary propagator. The propagator is used to calculate correlators of the theories that live on the boundary of the AdS space and show conformal invariance, which is desired by the AdS/CFT correspondence. 
  Effective action of an SU(2) gauge model with a vortex in 4-dimensional space time is calculated in the 1-loop approximation. The minimum of the effective potential is found. 
  The world-volume theory on a D-brane in a constant B-field background can be described by either commutative or noncommutative Yang-Mills theories. These two descriptions correspond to two different gauge fixing of the diffeomorphism on the brane. Comparing the boundary states in the two gauges, we derive a map between commutative and noncommutative gauge fields in a path integral form, when the gauge group is U(1). 
  One of the key properties of Dirac operators is the possibility of a degeneracy of zero modes. For the Abelian Dirac operator in three dimensions the question whether such multiple zero modes may exist has remained unanswered until now. Here we prove that the feature of zero mode degeneracy indeed occurs for the Abelian Dirac operator in three dimensions, by explicitly constructing a class of Dirac operators together with their multiple zero modes. Further, we discuss some implications of our results, especially a possible relation to the topological feature of Hopf maps. 
  This note looks at the possibility of a system of free particles presenting decoherence in the total momentum when tracing upon their relative momenta if we take into account a relativistic correction to the expression of the kinetic energy. 
  We suggest a closed form expression for the path integral of quantum transition amplitudes. We introduce a quantum action with renormalized parameters. We present numerical results for the $V \sim x^{4}$ potential. The renormalized action is relevant for quantum chaos and quantum instantons. 
  The use of the physical variables in the fashion of Dirac in the three-dimensional Chern-Simons theories is presented. Our previous results are reinterpreted in a new aspect. 
  It has been proposed that gauge and Kaehler anomalies in four-dimensional type IIB orientifolds are cancelled by a generalized Green-Schwarz mechanism involving exchange of twisted RR-fields. We explain how this can be understood using the well-known duality between linear and chiral multiplets. We find that all the twisted fields associated to the N=1 sectors and some of the fields associated to the N=2 sectors reside in linear multiplets. But there are no linear multiplets associated to order-two twists. Only the linear multiplets contribute to anomaly cancellation. This suffices to cancel all U(1) anomalies. In the case of Kaehler symmetries the complete SL(2,R) can be restored at the quantum level for all planes that are not fixed by an order-two twist. 
  We express the infinite sum of D-fivebrane instanton corrections to ${\cal R}^2$ couplings in ${\cal N}=4$ type I string vacua, in terms of an elliptic index counting 1/2-BPS excitations in the effective $Sp(N)$ brane theory. We compute the index explicitly in the infrared, where the effective theory is argued to flow to an orbifold CFT. The form of the instanton sum agrees completely with the predicted formula from a dual one-loop computation in type IIA theory on $K3\times T^2$. The proposed CFT provides a proper description of the whole spectrum of masses, charges and multiplicities for 1/2- and 1/4- BPS states, associated to bound states of D5-branes and KK momenta. These results are applied to show how fivebrane instanton sums, entering higher derivative couplings which are sensitive to 1/4-BPS contributions, also match the perturbative results in the dual type IIA theory. 
  We present a perturbative study of Ramond-Ramond backgrounds in the NSR formalism. We show how to perform sigma-model computations and discuss in detail the structure of the BRST charge and picture-changing operators. Contact terms play a vital role in the analysis. We also give evidence for a two loop non-renormalization theorem for the background beta functions. 
  We propose a formulation of the holographic principle, suitable for a background independent quantum theory of cosmology. It is stated as a relationship between the flow of quantum information and the causal structure of a quantum spacetime. Screens are defined as sets of events at which the observables of a holographic cosmological theory may be measured, and such that information may flow across them in two directions. A discrete background independent holographic theory may be formulated in terms of information flowing in a causal network of such screens. Geometry is introduced by defining the area of a screen to be a measure of its capacity as a channel of quantum information from its null past to its null future. We call this a ``weak'' form of the holographic principle, as no use is made of a bulk theory. 
  The on-shell regularization of the one-loop divergences of supergravity theories is generalized to include a dilaton of the type occurring in effective field theories derived from superstring theory, and the superfield structure of the one-loop corrections is given. Field theory anomalies and quantum contributions to soft supersymmetry breaking are discussed. The latter are sensitive to the precise choice of couplings that generate Pauli-Villars masses, which in turn reflect the details of the underlying theory above the scale of the effective cut-off. With a view to the implementation the Green-Schwarz and other mechanisms for canceling field theory anomalies under a U(1) gauge transformation and under the T-duality group of modular transformations, we show that the K\"ahler potential renormalization for the untwisted sector of orbifold compactification can be made invariant under these groups. 
  We show that the $Y_{ab}^c$ of Pradisi-Sagnotti-Stanev are indeed integers, and we prove a conjecture of Borisov-Halpern-Schweigert. We indicate some of the special features which arise when the order of the modular matrix T is odd. Our arguments are general, applying to arbitrary ``parent'' RCFT assuming only that T has odd order. 
  We study solutions corresponding to moving domain walls in the Randall-Sundrum universe. The bulk geometry is given by patching together black hole solutions in AdS$_5$, and the motion of the wall is determined from the junction equations. Observers on the wall interpret the motion as cosmological expansion or contraction. We describe the possible wall trajectories, and examine the consequences for localized gravity on the wall. 
  It is shown that certain extremal correlators in four-dimensional N=2 superconformal field theories (including N=4 super-Yang-Mills as a special case) have a free-field functional form. It is further argued that the coupling constant dependence receives no correction beyond the lowest order. These results hold for any finite value of N_c. 
  Winding number transitions from quantum to classical behavior are studied in the case of the {1+1} dimensional Mottola-Wipf model with the space coordinate on a circle for exploring the possibility of obtaining transitions of second order. The model is also studied as a prototype theory which demonstrates the procedure of such investigations. In the model at hand we find that even on a circle the transitions remain those of first order. 
  Equivalence in physics is discussed on the basis of experimental data accompanied by experimental errors. The introduction of the equivalence being consistent with the mathematical definition is possible only in theories constructed on non-standard number spaces by taking the experimental errors as infinitesimal numbers of the non-standard spaces. Following the idea for the equivalence (the physical equivalence), a new description of space-time in terms of infinitesimal-lattice points on non-standard real number space $\SR$ is proposed. The infinitesimal-lattice space, $^*{\cal L}$, is represented by the set of points on $\SR$ which are written by $l_n=n\SE$, where the infinitesimal lattice-spacing $\SE$ is determined by a non-standard natural number $^*N$ such that $\SE\equiv ^*N^{-1}$. By using infinitesimal neighborhoos ($\MON$) of real number $r$ on $\SL$ we can make a space $\SM$ which is isomorphic to $\RE$ as additive group. Therefore, every point on $(\SM)^N$ automatically has the internal confined-subspace $\MON$. A field theory on $\SL$ is proposed. To determine a projection from $\SL$ to $\SM$, a fundamental principle based on the physical equivalence is introduced. The physical equivalence is expressed by the totally equal treatment for indistinguishable quantities in our observations. Following the principle, we show that U(1) and SU(N) symmetries on the space $(\SM)^N$ are induced from the internal substructure $(\MON)^N$. Quantized state describing configuration space is constructed on $(\SM)^N$. We see that Lorentz and general relativistic transformations are also represented by operators which involve the U(1) and SU(N) internal symmetries. 
  We investigate the superconformal transformation properties of the supercurrent as well as of the superconformal anomalies themselves in d=4, N=1 supersymmetric quantum field theory. Matter supercurrent and anomalies are coupled to a classical background of minimal supergravity fields. On flat superspace, there exist two different types of the superconformal Ward identity (called S and B) which correspond to the flat space limits of old resp. new minimal background supergravity fields. In the present publication we give particular importance to the new minimal case. A general formalism is set up which is then applied to the massless Wess-Zumino model. 
  We study the effect of a background flux string on the vacuum energy of massive Dirac fermions in 2+1 dimensions confined to a finite spatial region through MIT boundary conditions. We treat two admissible self-adjoint extensions of the Hamiltonian and compare the results. In particular, for one of these extensions, the Casimir energy turns out to be discontinuous at integer values of the flux. 
  We discuss the consistency conditions of a novel orientifold projection of type IIB string theory on C^2/Z_N singularities, in which one mods out by the combined action of world-sheet parity and a geometric operation which exchanges the two complex planes. The field theory on the world-volume of D5-brane probes defines a family of six-dimensional RG fixed points, which had been previously constructed using type IIA configurations of NS-branes and D6-branes in the presence of O6-planes. Both constructions are related by a T-duality transforming the set of NS-branes into the C^2/Z_N singularity. We also construct additional models, where both the standard and the novel orientifold projections are imposed. They have an interesting relation with orientifolds of D_K singularities, and provide the T-duals of certain type IIA configurations containing both O6- and O8-planes. 
  A comprehensive introduction to two-dimensional conformal field theory is given. 
  Without a gauge fixing, canonical variables for the light-front SU(2) gluodynamics are determined. The Gauss law is written in terms of the canonical variables. The system is qualified as a generalized dynamical system with first class constraints. Abeliazation is a specific feature of the formulation (most of the canonical variables transform nontrivially only under the action of an Abelian subgroup of the gauge transformations). At finite volume, a discrete spectrum of the light-front Hamiltonian $P_+$ is obtained in the sector of vanishing $P_-$. We obtain, therefore, a quantized form of the classical solutions previously known as non-Abelian plane waves. Then, considering the infinite volume limit, we find that the presence of the mass gap depends on the way the infinite volume limit is taken, which may suggest the presence of different ``phases'' of the infinite volume theory. We also check that the formulation obtained is in accord with the standard perturbation theory if the latter is taken in the covariant gauges. 
  In a recent letter, Carlip proposed a generalization of the Brown-Henneaux-Strominger construction to any dimension. We present two criticisms about his formulation. 
  For background geometries whose metric contain a scale $\gamma$ we reformulate the Born-Infeld  D-brane action in terms of $\epsilon \equiv \gamma /(2\pi \alpha ')$.  This may be taken as a starting point for various perturbative treatments of the theory. We study two limits that arise at zeroth order of such perturbations. In the first limit, that corresponds to the $g_s\to\infty$ with $\epsilon$ fix, we find a "string parton" picture, also in the presense of some background  RR-fields. In the second limit, $\epsilon\to 0$, we find a topological model. 
  We study N=4 supersymmetric quantum-mechanical many-body systems with M bosonic and 4M fermionic degrees of freedom. We also investigate the further restrictions of conformal and superconformal invariance. In particular, we construct conformal N=4 extensions of the A_{M-1} Calogero models, which for generic values of the coupling constant are not SU(1,1|2) superconformal. This class of models is also extended to arbitrary (even) N. We give both hamiltonian and (classical) lagrangean formulations. In the latter case we use both component and N=4 superfield formulations. 
  We present a new solution in the heterotic M-theory in which the metric depends on (cosmic) time. The solution preserves N=1 supersymmetry in 4 dimensions in the leading order of the $\kappa^{2/3}$ expansion. It is the first example of the time-dependent supersymmetric solution in M-theory on $S^1/Z_2$. It describes expanding 4-dimensional space-time with shrinking orientifold interval and static Calabi-Yau internal space. 
  We consider the quantum mechanical analog of the nonlinear sigma model. There are difficulties to completely embed this theory by directly using the Batalin, Fradkin, Fradkina, and Tyutin (BFFT) formalism. We show in this paper how the BFFT method can be conveniently adapted in order to achieve a gauge theory that partially embeds the model. 
  We consider relativistic and non-relativistic particles and strings in spaces (or space-times) with a degenerate metric. We show that the resulting dynamics is described by a rich structure of constraints. All constraints are classified and the dynamics depends strongly on the parity of the difference between the dimension of the space (or space-time) and the rank of the degenerate metric. For a particular class of degenerate metrics we can identify the null eigenvectors of the metric with its Killing vectors. We also give the first steps towards the quantization of the non-relativistic particle using the Senjanovic path integral quantization and the Batalin-Fradkin-Tyutin conversion method. 
  We construct the matrix description for a twisted version of the IIA string theory on S^1 with fermions antiperiodic around a spatial circle. The result is a 2+1-dimensional U(N) x U(N) nonsupersymmetric Yang-Mills theory with fermionic matter transforming in the (N,Nbar). The two U(N)'s are exchanged if one goes around a twisted circle of the worldvolume. Relations with Type 0 theories are explored and we find Type 0 matrix string limits of our gauge theory. We argue however that most of these results are falsified by the absence of SUSY nonrenormalization theorems and that the models do not in fact have a sensible Lorentz invariant space time interpretation. 
  We generalize effective energy variational techniques to study appropriately quantized solitonic field configurations. Our approach rests on collective quantization ideas and is specifically designed for the numerical evaluation of soliton parameters. We employ this method to obtain the one-loop quantum corrections to the soliton mass and form factor. Special attention is given to the regularization of the physical observables in the solitonic sector of the theory. The numerical implementation of the method is demonstrated for a simple one-dimensional scalar field example. 
  Following our earlier investigations we examine the quantum-classical winding number transition in the Abelian-Higgs system. It is demonstrated that the sphaleron transition in this system is of the smooth second order type in the full range of parameter space. Comparison of the action of classical vortices with that of the sphaleron supports our finding. 
  The renormalization of the periodic potential is investigated in the framework of the Euclidean one-component scalar field theory by means of the differential RG approach. Some known results about the sine-Gordon model are recovered in an extremely simple manner. There are two phases, an ordered one with asymptotical freedom and a disordered one where the model is non-renormalizable and trivial. The order parameter of the periodicity, the winding number, indicates spontaneous symmetry breaking in the ordered phase where the fundamental group symmetry is broken and the solitons acquire dynamical stability. It is argued that the periodicity and the convexity are so strong constraints on the effective potential that it always becomes flat. This flattening is reproduced by integrating out the RG equation. 
  We discuss the possibility to construct an effective quantum field theory for an axial vector coupled to a Dirac spinor field. A massive axial vector describes antisymmetric torsion. The consistency conditions include unitarity and renormalizability in the low-energy region. The investigation of the Ward identities and the one- and two-loop divergences indicate serious problems arising in the theory. The final conclusion is that torsion may exist as a string excitation, but there are very severe restrictions for the existence of a propagating torsion field, subject to the quantization procedure, at low energies. 
  We re-examine the graceful exit problem in the pre-big bang scenario of string cosmology, by considering the most general time-dependent classical correction to the Lagrangian with up to four derivatives. By including possible forms for quantum loop corrections we examine the allowed region of parameter space for the coupling constants which enable our solutions to link smoothly the two asymptotic low-energy branches of the pre-big bang scenario, and observe that these solutions can satisfy recently proposed entropic bounds on viable singularity free cosmologies. 
  We discuss the general theory of D-branes on Calabi-Yaus, recent results from the theory of boundary states, and new results on the spectrum of branes on the quintic CY. (Contribution to the proceedings of Strings '99 in Potsdam, Germany.) 
  It is shown that the Born-Infeld-type modification of the quadratic Yang-Mills action suggested by the superstring theory gives rise to classical particle-like solutions prohibited in the standard Yang-Mills theory. This becomes possible due to the scale invariance breaking by the Born-Infeld non-linearity. New classical glueballs are sphaleronic in nature and exhibit a striking similarity with the Bartnik-McKinnon solutions of the Yang-Mills theory coupled to gravity. 
  We study the D-brane spectrum on a two-parameter Calabi-Yau model. The analysis is based on different tools in distinct regions of the moduli space: wrapped brane configurations on elliptic fibrations near the large radius limit, and SCFT boundary states at the Gepner point. We develop an explicit correspondence between these two classes of objects, suggesting that boundary states are natural quantum generalizations of bundles. We also find interesting D-brane dynamics in deep stringy regimes. The most striking example is, perhaps, that nonsupersymmetric D6-D0 and D4-D2 large radius configurations become stable BPS states at the Gepner point. 
  We make use of the properties of product integrals to obtain a surface product integral representation for the Wilson loop operator. The result can be interpreted as the non-abelian version of Stokes' theorem. 
  A brane universe moving in a curved higher dimensional bulk space is considered. The motion induces a cosmological evolution on the universe brane that is indistiguishable from a similar one induced by matter density on the brane. The phenomenological implications of such an idea are discussed. Various mirage energy densities are found, corresponding to dilute matter driving the cosmological expansion, many having superluminal properties $|w|>1$ or violating the positive energy condition. It is shown that energy density due to the world-volume fields is nicely incorporated into the picture. It is also pointed out that the initial singularity problem is naturally resolved in this context. 
  We present the action for a self-dual tensor in six dimensions, coupled to a (2,0) conformal supergravity background. This action gives rise to the expected equations of motion. An alternative look upon one of the gauge symmetries clarifies its role in the supersymmetry transformation rules and the realisation of the algebra. 
  We give a cosmological spinning multi-`black-hole' solution in the Einstein-Maxwell-Dilaton-Axion theory with a positive cosmological constant. This solution is the cosmological dilatonic Israel-Wilson-Perjes solution and describes the collision of several spinning `black-holes'. 
  We construct non-standard interactions between exterior form gauge fields by gauging a particular global symmetry of the Einstein-Maxwell action for such fields. Furthermore we discuss generalizations of such interactions by adding couplings to gravitational Chern-Simons forms and to fields arising through dimensional reduction. The construction uses an appropriate tensor calculus. 
  We apply the antifield quantization method of Batalin and Vilkovisky to the calculation of the path integral for the Poisson-Sigma model in a general gauge. For a linear Poisson structure the model reduces to a nonabelian gauge theory, and we obtain the formula for the partition function of two-dimensional Yang-Mills theory for closed two-dimensional manifolds. 
  Modifications of the Bekenstein-Hawking area law for black holes are crucial in order to find agreement between the microscopic entropy based on state counting and the macroscopic entropy based on an effective field theory computation. We discuss this and related issues for the case of four-dimensional N=2 supersymmetric black holes. We also briefly comment on the state counting for N=4 and N=8 black holes. 
  Starting from the Dirac equation in external electromagnetic and torsion fields we derive a path integral representation for the corresponding propagator. An effective action, which appears in the representation, is interpreted as a pseudoclassical action for a spinning particle. It is just a generalization of Berezin-Marinov action to the background under consideration. Pseudoclassical equations of motion in the nonrelativistic limit reproduce exactly the classical limit of the Pauli quantum mechanics in the same case. Quantization of the action appears to be nontrivial due to an ordering problem, which needs to be solved to construct operators of first-class constraints, and to select the physical sector. Finally the quantization reproduces the Dirac equation in the given background and, thus, justifies the interpretation of the action. 
  We consider M-theory compactification on Calabi-Yau threefolds. The recently discovered connection between the BPS states of wrapped M2 branes and the topological string amplitudes on the threefold is used both as a tool to compute topological string amplitudes at higher genera as well as to unravel the degeneracies and quantum numbers of BPS states. Moduli spaces of $k$-fold symmetric products of the wrapped M2 brane play a crucial role. We also show that the topological string partition function is the Calabi-Yau version of the elliptic genus of the symmetric product of $K3$'s and use the macroscopic entropy of spinning black holes in 5 dimensions to obtain new predictions for the asymptotic growth of the topological string amplitudes at high genera. 
  We study geometric engineering of Argyres-Douglas superconformal theories realized by type IIB strings propagating in singular Calabi-Yau threefolds. We use this construction to count the degeneracy of light BPS states under small perturbations away from the conformal point, by computing the degeneracy of D3-branes wrapped around supersymmetric 3-cycles in the Calabi-Yau. We find finitely many BPS states, the number of which depends on how this deformation is done, similarly to the degeneracy of kink solutions for the deformation of N=2 Landau-Ginzburg superconformal theories in two dimensions. Also, some aspects of worldsheet theories near general Calabi-Yau singularities are discussed. 
  We give a path integral prescription for the pair correlation function of Wilson loops lying in the worldvolume of Dbranes in the bosonic open and closed string theory. The results can be applied both in ordinary flat spacetime in the critical dimension d or in the presence of a generic background for the Liouville field. We compute the potential between heavy nonrelativistic sources in an abelian gauge theory in relative collinear motion with velocity v = tanh(u), probing length scales down to r_min^2 = 2 \pi \alpha' u. We predict a universal -(d-2)/r static interaction at short distances. We show that the velocity dependent corrections to the short distance potential in the bosonic string take the form of an infinite power series in the dimensionless variables z = r_min^2/r^2, uz/\pi, and u^2. 
  We show that there are two different dualities of two dimensional gauge theories with N=(2,2) supersymmetry. One is basically a consequence of 3d mirror symmetry. The non-linear sigma model with Calabi-Yau target space on the Higgs branch of the gauge theory is mapped into an equivalent non-linear sigma model on the Coulomb branch of the dual theory, realizing a T-dual target space with torsion. The second duality is genuine to two dimensions. In addition to swapping Higgs and Coulomb branch it trades twisted for untwisted multiplets, implying a sign flip of the left moving U(1)_R charge. Succesive application of both dualities leads to geometric mirror symmetry for the target space Calabi-Yau. 
  The cosmic holographic principle suggested by Fischler and Susskind has been examined in (2+1)-dimensional cosmological models. Analogously to the (3+1)-dimensional counterpart, the holographic principle is satisfied in all flat and open universes. For (2+1)-dimensional closed universes the holographic principle cannot be realized in general. The principle cannot be maintained, neither introducing negative pressure matter nor matter with highly unconventional equation of state. 
  We obtain a series expansion for the one loop fermion contribution to the effective potential for constant A^a_\mu fields in SU(2) theory with a massive fermionic doublet. The series converges for bounded electric fields in terms of the magnetic fields and the gauge potentials. One finds that spontaneous fermion pair creation may be absent for arbitrary strong pure electric fields, with an appropriate choice of the classical currents. 
  In the framework of the Closed-Time-Path formalism, we show how topological defects may arise in Quantum Field Theory as result of a localized (inhomogeneous) condensation of particles. We demonstrate our approach on two examples; kinks in the $2D \lambda \psi^{4}$ theory (both at zero and finite temperature) and vortices in the complex $4D \lambda \psi^{4} $ theory. 
  We investigate the problems of consistency and causality for the equations of motion describing massive spin two field in external gravitational and massless scalar dilaton fields in arbitrary spacetime dimension. From the field theoretical point of view we consider a general classical action with non-minimal couplings and find gravitational and dilaton background on which this action describes a theory consistent with the flat space limit. In the case of pure gravitational background all field components propagate causally. We show also that the massive spin two field can be consistently described in arbitrary background by means of the lagrangian representing an infinite series in the inverse mass. Within string theory we obtain equations of motion for the massive spin two field coupled to gravity from the requirement of quantum Weyl invariance of the corresponding two dimensional sigma-model. In the lowest order in $\alpha'$ we demonstrate that these effective equations of motion coincide with consistent equations derived in field theory. 
  Open systems acquire time-dependent coupling constants through interaction with an external field or environment. We generalize the Lewis-Riesenfeld invariant theorem to open system of quantum fields after second quantization. The generalized invariants and thereby the quantum evolution are found explicitly for time-dependent quadratic fermionic systems. The pair production of fermions is computed and other physical implications are discussed. 
  We prove a Mahoux-Mehta--type theorem for finite-volume partition functions of SU(N_c\geq 3) gauge theories coupled to fermions in the fundamental representation. The large-volume limit is taken with the constraint V << 1/m_{\pi}^4. The theorem allows one to express any k-point correlation function of the microscopic Dirac operator spectrum entirely in terms of the 2-point function. The sum over topological charges of the gauge fields can be explicitly performed for these k-point correlation functions. A connection to an integrable KP hierarchy, for which the finite-volume partition function is a $\tau$-function, is pointed out. Relations between the effective partition functions for these theories in 3 and 4 dimensions are derived. We also compute analytically, and entirely from finite-volume partition functions, the microscopic spectral density of the Dirac operator in SU(N_c) gauge theories coupled to quenched fermions in the adjoint representation. The result coincides exactly with earlier results based on Random Matrix Theory. 
  Recent developments concerning canonical quantisation and gauge invariant quantum mechanical systems and quantum field theories are briefly discussed. On the one hand, it is shown how diffeomorphic covariant representations of the Heisenberg algebra over curved manifolds of non trivial topology involve topology classes of flat U(1) bundles. On the other hand, through some examples, the recently proposed physical projector approach to the quantisation of general gauge invariant systems is shown to avoid the necessity of any gauge fixing - hence also avoiding the possibility of Gribov problems which usually ensue any gauge fixing procedure - and is also capable to provide the adequate description of the physical content of gauge invariant systems. 
  It is shown that local symmetry transformations of the maximal AdS supergravity in seven-dimensional anti de Sitter space induce those of the N=(2,0) conformal supergravity on the six-dimensional boundary at infinity. Boundary values of the AdS supergravity fields form a supermultiplet of the conformal supergravity. 
  String theory compactified on a three-torus possesses an SL(5,Z) U-duality group. We investigate the realisation of this symmetry on the Born-Infeld theory on a three-brane, and discuss a U-duality covariant formulation of the BPS sector of the theory where the rank of the gauge group is treated on an equal footing with the fluxes. 
  The D1/D5 system is considered in the presence of the NS B field. An explicit supergravity solution in the asymptotically flat and near horizon limits is presented. Explicit mass formulae are presented in both cases. This solution has no D3 source branes and represents a true bound state of the D1/D5 system. We study the motion of a separated D1-brane in the background geometry described above and reproduce the Liouville potential that binds the D1 brane. A gauge theory analysis is also presented in the presence of Fayet-Iliopoulos (FI) parameters which can be identified with the self-dual part of the NS B field. In the case of a single D5-brane and an arbitrary number of D1 branes we can demonstrate the existence of a bound state in the Higgs branch. We also point out the connection of the SCFT on the resolved Sym$_{Q_1Q_5}(\tilde T^4)$ with recent developments in non-commutative Yang-Mills theory. 
  We describe a simple lattice model of higher-curvature quantum gravity in two dimensions and study the phase structure of the theory as a function of the curvature coupling. It is shown that the ensemble of flat graphs is entropically unstable to the formation of baby universes. In these simplified models the growth in graphs exhibits a branched polymer behaviour in the phase directly before the flattening transition. 
  We study the structure of the monopole configuration in U(2) non-commutative super Yang-Mills theory. Our analysis consists of two steps: solving the BPS equation and then the eigenvalue equation in the non-commutative space. Calculation to the first non-trivial order in the non-commutativity parameter theta shows that the monopole exhibits a certain non-locality. This structure is precisely the one expected from the recent predictions by the brane-configuration technique. 
  We compute two-point functions of chiral operators Tr \Phi^3 in {\cal N}=4 SU(N) supersymmetric Yang-Mills theory to the order g^4 in perturbation theory. We perform explicit calculations using {\cal N}=1 superspace techniques and find that perturbative corrections to the correlators vanish for all N. While at order g^2 the cancellations can be ascribed to the nonrenormalization theorem valid for correlators of operators in the same multiplet as the stress tensor, at order g^4 this argument no longer applies and the actual cancellation occurs in a highly nontrivial way. Our result is obtained in complete generality, without the need of additional conjectures or assumptions. It gives further support to the belief that such correlators are not renormalized to all orders in g and to all orders in N. 
  We revise the calculation of the one-loop effective action for scalar and spinor fields coupled to the dilaton in two dimensions. Applying the method of covariant perturbation theory for the heat kernel we derive the effective action in an explicitly covariant form that produces both the conformally invariant and the conformally anomalous terms.For scalar fields the conformally invariant part of the action is nonlocal. The obtained effective action is proved to be infrared finite. We also compute the one-loop effective action for scalar fields at finite temperature. 
  Three-dimensional N-extended superconformal symmetry is studied within the superspace formalism. A superconformal Killing equation is derived and its solutions are classified in terms of supertranslations, dilations, Lorentz transformations, R-symmetry transformations and special superconformal transformations. Superconformal group is then identified with a supermatrix group, OSp(N|2,R), as expected from the analysis on simple Lie superalgebras. In general, due to the invariance under supertranslations and special superconformal transformations, superconformally invariant n-point functions reduce to one unspecified (n-2)-point function which must transform homogeneously under the remaining rigid transformations, i.e. dilations, Lorentz transformations and R-symmetry transformations. After constructing building blocks for superconformal correlators, we are able to identify all the superconformal invariants and obtain the general form of n-point functions. Superconformally covariant differential operators are also discussed. 
  We give a very simple proof that the renormalization of the Chern-Simons coupling in the Wilsonian effective action is exhausted at one-loop. Our proof can apply to arbitrary 2+1-dimensional abelian as well as nonabelian gauge theories without a bare Chern-Simons coupling, including any non-renormalizable interactions and non-minimal couplings. Our proof reveals that small (but not large) gauge invariance is enough to ensure the absence of higher order corrections. 
  The relation between the gauge-invariant local BRST cohomology involving the antifields and the gauge-fixed BRST cohomology is clarified. It is shown in particular that the cocycle conditions become equivalent once it is imposed, on the gauge-fixed side, that the BRST cocycles should yield deformations that preserve the nilpotency of the (gauge-fixed) BRST differential. This shows that the restrictions imposed on local counterterms by the Quantum Noether condition in the Epstein--Glaser construction of gauge theories are equivalent to the restrictions imposed by BRST invariance on local counterterms in the standard Lagrangian approach. 
  We analyze in detail a specific 5-dimensional realization of a "brane-universe" scenario where the visible and hidden sectors are localized on spatially separated 3-branes coupled only by supergravity, with supersymmetry breaking originating in the hidden sector. Although general power counting allows order 1/M_{Planck}^2 contact terms between the two sectors in the 4-dimensional theory from exchange of supergravity Kaluza-Klein modes, we show that they are not present by carefully matching to the 5-dimensional theory. We also find that the radius modulus corresponding to the size of the compactified dimension must be stabilized by additional dynamics in order to avoid run-away behavior after supersymmetry breaking and to understand the communication of supersymmetry breaking. We stabilize the radius by adding two pure Yang--Mills sectors, one in the bulk and the other localized on a brane. Gaugino condensation in the 4-dimensional effective theory generates a superpotential that can naturally fix the radius at a sufficiently large value that supersymmetry breaking is communicated dominantly by the recently-discovered mechanism of anomaly mediation. The mass of the radius modulus is large compared to m_{3/2}. The stabilization mechanism requires only parameters of order one at the fundamental scale, with no fine-tuning except for the cosmological constant. 
  It is shown that the vacuum of QED$_2$ in Minkowski spacetime does not favour a periodic electric mean field. The projected effective action exhibiting a genuine dependence on the non-vanishing background field has been introduced. The functional dependence of the energy density of the vacuum on the assumed periodic vacuum expectation value of the vector potential is determined from the component $T^{00}$ of the energy-momentum tensor at one-loop order. Treating the background field non-perturbatively, the energy of the vacuum in the presence of a periodic mean field is found not be equal to the negative of the effective action. 
  The goal of this paper is to give a representation-theoretic interpretation of the sine-Gordon equation. We consider a vertex operator representation of affine Kac-Moody algebra \hat{sl_2} on the space of differential operators. In this formulation, the tau-function becomes a function of non-commuting variables. Using the skew Casimir operators, we obtain a hierarchy of equations in Hirota form that contains sine-Gordon, KdV and mKdV equations and construct their soliton solutions. 
  In order to study vertex operators for the Type IIB superstring on AdS space, we derive supersymmetric constraint equations for the vertex operators in AdS3xS3 backgrounds with Ramond-Ramond flux, using Berkovits-Vafa-Witten variables. These constraints are solved to compute the vertex operators and show that they satisfy the linearized D=6, N=(2,0) equations of motion for a supergravity and tensor multiplet expanded around the AdS3xS3 spacetime. 
  The left and right zero modes of the level k SU(n) WZNW model give rise to a pair of isomorphic (left and right) mutually commuting quantum matrix algebras. For a deformation parameter q being an even (2h-th, h = k + n) root of unity each of these matrix algebras admits an ideal such that the corresponding factor algebra is finite dimensional. The structure of superselection sectors of the (diagonal) 2D WZNW model is then reduced to a finite dimensional problem of a gauge theory type. For n=2 this problem is solved using a generalized BRS formalism. 
  We review quantitative tests on the duality between the heterotic string on T^2 and F-theory on K3. On the heterotic side, certain threshold corrections to the effective action can be exactly computed at one-loop order, and the issue is to reproduce these from geometric quantities pertaining to the K3 surface. In doing so we learn about certain non-perturbative interactions of 7-branes. 
  We consider domain walls that appear in supersymmetric SU(N) with one massive flavour. In particular, for N > 3 we explicitly construct the elementary domain wall that interpolates between two contiguous vacua. We show that these solutions are BPS saturated for any value of the mass of the matter fields. We also comment on their large N limit and their relevance for supersymmetric gluodynamics. 
  This paper is devoted to a general and self-contained approach to any cohomological field theory with K\"{a}hler structure. 
  Nontrivial twisted boundary conditions associated with extra compact dimensions produce an ambiguity in the value of the four dimensional coupling constants of the renormalizable interactions of the twisted fields' zero modes. Resolving this indeterminancy would require a knowledge of the exact form of the higher dimensional action including the coefficients of higher dimensional operators. For the case of moderately sized extra dimensions, the uncertainty in the coupling constants can be of order one and may lead to modifications in the stability of the model. 
  In the framework of Matrix theory we show that Wilson loops can serve as interpolating fields to define string scattering amplitudes as gauge theory observables. 
  In the generalized Legendre transform construction the Kaehler potential is related to a particular function. Here, the form of this function appropriate to the monopole metric is calculated from the known twistor theory of monopoles. 
  In a recent letter, Cadoni and Mignemi proposed a formulation for the statistical computation of the 2D black holes entropy. We present a criticism about their formulation. 
  The most general 2+1 dimensional spinning particle model is considered. The action functional may involve all the possible first order Poincare invariants of world lines, and the particular class of actions is specified thus the corresponding gauge algebra to be unbroken by inhomogeneous external fields. Nevertheless, the consistency problem reveals itself as a requirement of the global compatibility between first and second class constraints. These compatibility conditions, being unnoticed before in realistic second class theories, can be satisfied for a particle iff the gyromagnetic ratio takes the critical value g=2. The quantization procedure is suggested for a particle in the generic background field by making use of a Darboux co-ordinates, being found by a perturbative expansion in the field multipoles and the general procedure is described for constructing of the respective transformation in any order. 
  Recently, Dorey and Tateo have investigated functional relations among Stokes multipliers for a Schr{\"o}dinger equation (second order differential equation) with a polynomial potential term in view of solvable models. Here we extend their studies to a restricted case of n+1-th order linear differential equations. 
  The study of curved D-brane geometries in type II strings implies a general relation between local singularities $\cx W$ of Calabi-Yau manifolds and gravity free supersymmetric QFT's. The minimal supersymmetric case is described by F-theory compactifications on $\cx W$ and can be used as a starting point to define minimal supersymmetric heterotic string compactifications on compact Calabi-Yau manifolds with holomorphic, stable gauge backgrounds. The geometric construction generalizes to non-perturbative vacua with five-branes and provides a framework to study non-perturbative dynamics of the heterotic theory. 
  There are two families of non-BPS bi-spinors in the perturbative spectrum of the nine dimensional heterotic string charged under the gauge group $SO(16)\times SO(16)$. The relation between these perturbative non-BPS states and certain non-perturbative non-BPS D-brane states of the dual type I$^\prime$ theory is exhibited. The relevant branes include a $\Zop_2$ charged non-BPS D-string, and a bound state of such a D-string with a fundamental string. The domains of stability of these states as well as their decay products in both theories are determined and shown to agree with the duality map. 
  A new family of S-matrix theories with resonance poles is constructed and conjectured to correspond to the Homogeneous sine-Gordon theories associated with simply laced compact Lie groups, where some of the resonance poles can be traced to the presence of unstable particles in the spectrum. These theories are unitary in the usual S S^\dagger =1 sense, they are not parity invariant, and they exhibit continuous coupling constants that determine both the mass spectrum of stable particles and the masses and the position of the resonance poles. 
  We consider the cosmology of a ``3-brane universe'' in a five dimensional (bulk) space-time with a cosmological constant. We show that Einstein's equations admit a first integral, analogous to the first Friedmann equation, which governs the evolution of the metric in the brane, whatever the time evolution of the metric along the fifth dimension. We thus obtain the cosmological evolution in the brane for any equation of state describing the matter in the brane, without needing the dependence of the metric on the fifth dimension. In the particular case $p = w \rho$, $(w = constant)$, we give explicit expressions for the time evolution of the brane scale factor, which show that standard cosmological evolution can be obtained (after an early non conventional phase) in a scenario \`a la Randall and Sundrum, where a brane tension compensates the bulk cosmological constant. We also show that a tiny deviation from exact compensation leads to an effective cosmological constant at late time. Moreover, when the metric along the fifth dimension is static, we are able to extend the solution found on the brane to the whole spacetime. 
  In this paper we propose a new supersymmetric extension of conformal mechanics. The Grassmannian variables that we introduce are the basis of the forms and of the vector-fields built over the symplectic space of the original system. Our supersymmetric Hamiltonian itself turns out to have a clear geometrical meaning being the Lie-derivative of the Hamiltonian flow of conformal mechanics. Using superfields we derive a constraint which gives the exact solution of the supersymmetric system in a way analogous to the constraint in configuration space which solved the original non-supersymmetric model. Besides the supersymmetric extension of the original Hamiltonian, we also provide the extension of the other conformal generators present in the original system. These extensions have also a supersymmetric character being the square of some Grassmannian charge. We build the whole superalgebra of these charges and analyze their closure. The representation of the even part of this superalgebra on the odd part turns out to be integer and not spinorial in character. 
  We introduce a model designed to describe charged particles as stable topological solitons of a field with values on the internal space S^3. These solitons behave like particles with relativistic properties like Lorentz contraction and velocity dependence of mass. This mass is defined by the energy of the soliton. In this sense this model is a generalisation of the sine-Gordon model from 1+1 dimensions to 3+1 dimensions, from S^1 to S^3. (We do not chase the aim to give a four-dimensional generalisation of Coleman's isomorphism between the Sine-Gordon model and the Thirring model which was shown in 2-dimensional space-time.) For large distances from the center of solitons this model tends to a dual U(1)-theory with freely propagating electromagnetic waves. Already at the classical level it describes important effects, which usually have to be explained by quantum field theory, like particle-antiparticle annihilation and the running of the coupling. 
  In these lectures we discuss the application of discrete light cone quantization (DLCQ) to supersymmetric field theories. We will see that it is possible to formulate DLCQ so that supersymmetry is exactly preserved in the discrete approximation. We call this formulation of DLCQ, SDLCQ and it combines the power of DLCQ with all of the beauty of supersymmetry. In these lecture we will review the application of SDLCQ to several interesting supersymmetric theories. We will discuss two dimensional theories with (1,1), (2,2) and (8,8) supersymmetry, zero modes, vacuum degeneracy, massless states, mass gaps, theories in higher dimensions, and the Maldacena conjecture among other subjects. 
  The linear particle-antiparticle conjugation $\ty C$ and position space reflection $\ty P$ as well as the antilinear time reflection $\ty T$ are shown to be inducable by the selfduality of representations for the operation groups $\SU(2)$, $\SL(\C^2)$ and $\R$ for spin, Lorentz transformations and time translations resp. The definition of a colour compatible linear $\ty{CP}$-reflection for quarks as selfduality induced is impossible since triplet and antitriplet $\SU(3)$-representations are not linearly equivalent. 
  Using the collective field theory approach of large-N generalized two-dimensional Yang-Mills theory on cylinder, it is shown that the classical equation of motion of collective field is a generalized Hopf equation. Then, using the Itzykson-Zuber integral at the large-N limit, it is found that the classical Young tableau density, which satisfies the saddle-point equation and determines the large-N limit of free energy, is the inverse of the solution of this generalized Hopf equation, at a certain point. 
  We show that the arguments proposed by Park and Yee against our recent derivation of the statistical entropy of 2D black holes do not apply to the case under consideration 
  In this lecture we discuss `beyond CFT' from symmetry point of view. After reviewing the Virasoro algebra, we introduce deformed Virasoro algebras and elliptic algebras. These algebras appear in solvable lattice models and we study them by free field approach. 
  An open bosonic string is considered with the aim to construct a general gauge invariant, being a polynomial of Fubini-Veneziano (FV) fields. The FV fields are transformed as 1-forms on $S^1$, that allows to formulate the problem in geometric terms. We introduce a most general anzats for these invariants and explicitly resolve the invariance conditions in the framework of the anzats. The invariants are interpreted as integrals of n-form over a gauge invariant domains in an n-dimensional torus, where the invariance of these domains is considered with respect to the action of the diagonal of the group $\times (Diff~S^1)^n$. We also discuss a possibility to get a complete set of gauge invariants which allow an actual dependence on the string zero modes. We find that the complete set can't be restricted by polynomial invariants only. The classical polynomial invariants, being directly defined in the string Fock space, turn out to break the structure of the respective BRST cohomology even in the critical dimension. We discuss a possibility to restore the BRST invariance of the corresponding operator algebra by a non-trivial quantum deformation of the original invariants. 
  We discuss some Z_N^L x Z_N^R orbifold compactifications of the type IIB superstring to D= 4,6 dimensions and their type I descendants. Although the Z_N^L x Z_N^R generators act asymmetrically on the chiral string modes, they result into left-right symmetric models that admit sensible unorientable reductions. We carefully work out the phases that appear in the modular transformations of the chiral amplitudes and identify the possibility of introducing discrete torsion. We propose a simplifying ansatz for the construction of the open-string descendants in which the transverse-channel Klein-bottle, annulus and Moebius-strip amplitudes are numerically identical in the proper parametrization of the world-sheet. A simple variant of the ansatz for the Z_2^L x Z_2^R orbifold gives rise to models with supersymmetry breaking in the open-string sector. 
  It is argued that there are states (quasiparticles) with masses ranging over the scales $\Lambda N_{c}^{1/3} \div \Lambda N_{c}$ in N=1 supersymmetric multicolor gluodynamics. These states exist in the form of quantum bubbles made out of the BPS domain walls. Analogous states are likely to exist in non-supersymmetric case as well. 
  The anomalies of a very general class of non local Dirac operators are computed using the $\zeta$-function definition of the fermionic determinant and an asymmetric version of the Wigner transformation. For the axial anomaly all new terms introduced by the non locality can be brought to the standard minimal Bardeen's form. Some extensions of the present techniques are also commented. 
  We presented the proof of the positive mass theorem for black holes in Einstein-Maxwell axion-dilaton gravity being the low-energy limit of the heterotic string theory. We show that the total mass of a spacetime containing a black hole is greater or equal to the square root of the sum of squares of the adequate dilaton-electric and dilaton-axion charges. 
  In two space-time dimensions, there is a theory of Lorentzian quantum gravity which can be defined by a rigorous, non-perturbative path integral and is inequivalent to the well-known theory of (Euclidean) quantum Liouville gravity. It has a number of appealing features: i) its quantum geometry is non-fractal, ii) it remains consistent when coupled to matter, even beyond the c=1 barrier, iii) it is closer to canonical quantization approaches than previous path-integral formulations, and iv) its construction generalizes to higher dimensions. 
  We discuss how the variational principle can be used as a criterion for choosing, among scalar field actions implying the same equation of motion, the appropriate one for the AdS/CFT correspondence. 
  The multi-instanton solutions by 'tHooft and Jackiw, Nohl & Rebbi are generalized to curvilinear coordinates. Expressions can be notably simplified by the appropriate gauge transformation. This generates the compensating addition to the gauge potential of pseudoparticles. Singularities of the compensating connection are irrelevant for physics but affect gauge dependent quantities. 
  In a recent paper (hep-th/9811108), Saveliev and the author showed that there exits an on-shell light cone gauge where the non-linear part of the field equations reduces to a (super) version of Yang's equations which may be solved by methods inspired by the ones previously developed for self-dual Yang-Mills equations in four dimensions. Later on (hep-th/9903218), the analogy between these latter theories and the present ones was pushed further by writing down a set of super partial linear differential equations which are the analogues of the Lax pair of Belavin and Zakharov. Using this Lax representation, it is shown in the present article that solution-generating techniques are at work, which are similar to the ones developed for four dimensional self-dual Yang-Mills theories in the late seventies. 
  In a previous paper we conjectured that the structure of various gauge theories as well as M-theory on $T^8$ is encoded in a unique function $\Xi$ on the coset $E_{10}(Z)\backslash E_{10}(R)/K$ and that this function is harmonic with respect to the $E_{10}(R)$ invariant metric. In this paper we elaborate on the conjecture. We discuss various mass deformations of the D-instanton integral and their realizations in $\Xi$. We then present a conjectured prescription for extracting partition functions of the twisted little-string theory out of $\Xi$. We also study various effects of combinations of branes such as D0-branes near D4-branes with 2-form flux, D-instantons near Taub-NUT metrics, and more, in terms of harmonic functions on $E_d(R)/K$. We propose tests of the conjecture that are related to BPS states of global symmetries in gauge theories. 
  We show in a precise group theoretical fashion how the generating solution of regular BPS black holes of N=8 supergravity, which is known to be a solution also of a simpler N=2 STU model truncation, can be characterized as NS-NS or R-R charged according to the way the corresponding STU model is embedded in the original N=8 theory. Of particular interest is the class of embeddings which yield regular BPS black hole solutions carrying only R-R charge and whose microscopic description can possibly be given in terms of bound states of D-branes only. The microscopic interpretation of the bosonic fields in this class of STU models relies on the solvable Lie algebra (SLA) method. In the present article we improve this mathematical technique in order to provide two distinct descriptions for type IIA and type IIB theories and an algebraic characterization of S*T--dual embeddings within the N=8,d=4 theory. This analysis will be applied to the particular example of a four parameter (dilatonic) solution of which both the full macroscopic and microscopic descriptions will be worked out. 
  We establish a relation, conjectured recently by E. Witten, between the hypermultiplet moduli space in compactifications of the heterotic string on an A-D-E singularities, and the moduli spaces of three dimensional pure gauge theories with the corresponding A-D-E gauge groups. It is possible to add a bounded number of heterotic fivebranes sitting in the singularity, while (in leading order in $\alpha'$) keeping the heterotic string perturbative. The corresponding hypermultiplet moduli space is given by the moduli space of a three dimensional gauge theory with matter. 
  We present a new generalized topological current in terms of the order parameter field $\vec \phi$ to describe the arbitrary dimensional topological defects. By virtue of the $% \phi$-mapping method, we show that the topological defects are generated from the zero points of the order parameter field $\vec \phi$, and the topological charges of these topological defects are topological quantized in terms of the Hopf indices and Brouwer degrees of $\phi$-mapping under the condition that the Jacobian $% J(\frac \phi v)\neq 0$. When $J(\frac \phi v)=0$, it is shown that there exist the crucial case of branch process. Based on the implicit function theorem and the Taylor expansion, we detail the bifurcation of generalized topological current and find different directions of the bifurcation. The arbitrary dimensional topological defects are found splitting or merging at the degenerate point of field function $\vec \phi $ but the total charge of the topological defects is still unchanged. 
  The Parisi-Sourlas mechanism is exhibited in pure Yang-Mills theory. Using the new scalar degrees of freedom derived from the non-linear gauge condition, we show that the non-perturbative sector of Yang-Mills theory is equivalent to a 4D O(1,3) sigma model in a random field. We then show that the leading term of this equivalent theory is invariant under supersymmetry transformations where (x^{2}+\thetabar\theta) is unchanged. This leads to dimensional reduction proving the equivalence of the non-perturbative sector of Yang-Mills theory to a 2D O(1,3) sigma model. 
  We construct unstable system D8-brane+D8-antibrane as a kink solution on world-volume of non-BPS D9-brane in Type IIA theory. Further we will make other checks confirming validity of our appro. 
  We present a minimal model for the Universe evolution fully extracted from effective String Theory. By linking this model with a minimal but well established observational information, we prove that it gives realistic predictions on early and current energy density and its results are compatible with General Relativity. Interestingly enough, the predicted current energy density is found $\Omega = 1$ and in anycase with lower limit $\Omega \geq {4/9}$. On the other hand, the energy density at the exit of inflationary stage gives also $\Omega|_{inf}=1$. This result shows an agreement with General Relativity (spatially flat metric gives critical energy density) within an unequivalent Non-Einstenian context (string low energy effective equations). The order of magnitude of the energy density-dilaton coupled term at the beginning of radiation dominated stage agrees with GUT scale. Without solving the known problems about higher order corrections and graceful exit of inflation, we find this model closer to the observational Universe properties than the current available string cosmology scenarii. At a more fundamental level, this model is by its construction close to the standard cosmological evolution, and it is driven selfconsistently by the evolution of the string equation of state itself. The inflationary String Driven stage is able to reach an enough amount of inflation, describing a Big Bang like evolution for the metric. 
  We discuss the status and some perspectives of relativistic quantum physics. 
  We relate various black hole solutions in the near-horizon region to black hole solutions in two-dimensional dilaton gravity theories in order to argue that thermodynamics of black holes in D>=4 can be effectively described by thermodynamics of black holes in two-dimensional dilaton gravity theories. We show that the Bekenstein-Hawking entropies of single-charged dilatonic black holes and dilatonic p-branes with an arbitrary dilaton coupling parameter in arbitrary spacetime dimensions are exactly reproduced by the Bekenstein-Hawking entropy of the two-dimensional black hole in the associated two-dimensional dilaton gravity model. We comment that thermodynamics of non-extreme stringy four-dimensional black hole with four charges and five-dimensional black hole with three charges may be effectively described by thermodynamics of the black hole solutions with constant dilaton field in two-dimensional dilaton gravity theories. 
  Let $M$ be a complete Einstein manifold of negative curvature, and assume that (as in the AdS/CFT correspondence) it has a Penrose compactification with a conformal boundary $N$ of positive scalar curvature. We show that under these conditions, $H_n(M;Z)=0$ and in particular $N$ must be connected. These results resolve some puzzles concerning the AdS/CFT correspondence. 
  In this note we describe the most general coupling of {\it abelian} vector and tensor multiplets to six-dimensional $(1,0)$ supergravity. As was recently pointed out, it is of interest to consider more general Chern-Simons couplings to abelian vectors of the type $H^{r}=d B^{r}-1/2 c^{rab} A^{a}d A^{b}$, with $c^{r}$ matrices that may not be simultaneously diagonalized. We show that these couplings can be related to Green-Schwarz terms of the form $B^r c_r^{ab} F^a F^b$, and how the complete local Lagrangian, that embodies factorized gauge and supersymmetry anomalies (to be disposed of by fermion loops) is uniquely determined by Wess-Zumino consistency conditions, aside from an arbitrary quartic coupling for the gauginos. 
  This is a reply to a recent comment (hep-th/9910158) by Park and Ho on a paper (hep-th/9812013) that describes a derivation of black hole entropy from horizon conformal field theory. The criticism of Park and Ho is partially correct - the original paper did not give the complete surface term in the Hamiltonian - but it does not affect the conclusions, which have been checked using a somewhat different, manifestly covariant formalism (gr-qc/9906126). 
  We introduce a method of using the a dual type IIA string to compute alpha'-corrections to the moduli space of heterotic string compactifications. In particular we study the hypermultiplet moduli space of a heterotic string on a K3 surface. One application of this machinery shows that type IIB strings compactified on a Calabi-Yau space suffer from worldsheet instantons, spacetime instantons and, in addition, "mixed" instantons which in a sense are both worldsheet and spacetime. As another application we look at the hyperkaehler limit of the moduli space in which the K3 surface becomes an ALE space. This is a variant of the "geometric engineering" method used for vector multiplet moduli space and should be applicable to a wide range of examples. In particular we reproduce Sen and Witten's result for the heterotic string on an A1 singularity and a trivial bundle and generalize this to a collection of E8 point-like instantons on an ALE space. 
  We consider the spectrum of open strings for non-BPS D-brane configuration in type II string theory on a Calabi-Yau threefold. In general, there is no degeneracy between bosonic and fermionic states. However we find special values for the moduli space of Calabi-Yau threefolds there are non-BPS brane configurations which have an exact degeneracy between bosonic and fermionic states. For these values there is no force between pairs of non-BPS D-branes. This gives rise to a possibility of building diverse non-supersymmetric gauge field theories on the brane world-volume. We use the approach recently elaborated by Gaberdiel and Sen. 
  Four-dimensional N=1 supersymmetric gauge theories with two adjoints and a quartic superpotential are believed, from AdS/CFT duality, to have SL(2,Z) invariance. In this note we review an old, unpublished argument for this property, based solely on field theory. The technique involves a complexified flavor rotation which deforms an N=2 supersymmetric gauge theory with matter to an N=1 theory, leaving all holomorphic invariants unchanged. We apply this to the N=1 gauge theory with two massless adjoints and show that it has the same auxiliary torus as that of N=4 gauge theory, from which SL(2,Z) invariance follows. In an appendix, we check that our arguments are consistent with earlier work on the SU(2) case. Our technique is general and applies to many other N=1 theories. 
  Sigma model actions are constructed for the Type II superstring compactified to four and six dimensional curved backgrounds which can contain non-vanishing Ramond-Ramond fields. These actions are N=2 worldsheet superconformally invariant and can be covariantly quantized preserving manifest spacetime supersymmetry. They are constructed using a hybrid version of superstring variables which combines features of the Ramond-Neveu-Schwarz and Green-Schwarz formalisms. For the $AdS_2\times S^2$ and $AdS_3\times S^3$ backgrounds, these actions differ from the classical Greeen-Schwarz actions by a crucial kinetic term for the fermions. Parts of this work have been done in collaborations with M. Bershadsky, T. Hauer, W. Siegel, C. Vafa, E. Witten, S. Zhukov and B. Zwiebach. 
  We construct the complete and explicit non-linear Kaluza-Klein Ansatz for deriving the bosonic sector of the standard N=4 SO(4) gauged four-dimensional supergravity from the reduction of D=11 supergravity on S^7. This provides a way of interpreting all bosonic solutions of the four-dimensional gauged theory as exact solutions in eleven-dimensional supergravity. We discuss certain limiting forms of the Kaluza-Klein reduction, and compare them with related forms in the Freedman-Schwarz N=4 SU(2)xSU(2) gauged theory. This leads us to the result that the Freedman-Schwarz model is in fact a singular limiting case of the standard SO(4) gauged supergravity. We show that in this limit, our Ansatz for getting the SO(4) gauged theory as an S^7 reduction from D=11 indeed reduces to an S^3 x S^3 reduction from D=10, which makes contact with previous results in the literature. We also show that there is no distinction to be made between having equal or unequal values for the gauge coupling constants $g$ and $\tilde g$ of the two SU(2) gauge-group factors in the standard N=4 SO(4) gauged supergravity, whilst by contrast the ratio of $g$ to $\tilde g$ is a non-trivial parameter of the Freedman-Schwarz model. 
  We derive a formula that expresses the local spin and field operators of fundamental graded models in terms of the elements of the monodromy matrix. This formula is a quantum analogue of the classical inverse scattering transform. It applies to fundamental spin chains, such as the XYZ chain, and to a number of important exactly solvable models of strongly correlated electrons, such as the supersymmetric t-J model or the the EKS model. 
  The Variational Method is applied within the context of Supersymmetric Quantum Mechanics to provide information about the energy and eigenfunction of the lowest levels of a Hamiltonian. The approach is illustrated by the case of the Morse potential applied to several diatomic molecules and the results are compared with stablished results. 
  I review recent results on conformal field theories in four dimensions and quantum field theories interpolating between conformal fixed points, supersymmetric and non-supersymmetric. The talk is structured in three parts: i) central charges, ii) anomalous dimensions and iii) quantum irreversibility. 
  We propose a gauge invariant formulation of the exact renormalization group equation for nonsupersymmetric pure U(N) Yang-Mills theory, based on the construction by Tim Morris. In fact we show that our renormalization group equation amounts to a regularized version of the loop equation, thereby providing a direct relation between the exact renormalization group and the Schwinger-Dyson equations. We also discuss a possible implication of our formulation to the holographic correspondence of the bulk gravity and the boundary gauge theory. 
  We consider an abelian gauge theory with spontaneously broken symmetry containing a scalar-fermion coupling which is non-linear in the Higgs field. Although in the unitary gauge it reduces to a pure Yukawa term, suggesting that the theory is renormalizable, the one loop divergence structure in this gauge in the fermion-fermion elastic scattering amplitude shows this is not so. Comparison is made with the theory with a conventional coupling, for which cancellation of the non-renormalizable divergences occurs. 
  In this note we will show that the $\Lambda$ symmetry, namely the U(1) symmetry of the open string sigma model which relates the B-field and the U(1) gauge field of a brane to each other, is deformed to a noncommutative version in a constant B-field background. 
  We concern with various aspects of equilibrium and non-equilibrium quantum field theory. 
  Via T-duality, a stack of unwrapped type 0 NS5-branes is transformed into a Kaluza-Klein monopole with A_n type singularity at its center. The spectrum of twisted modes at the singularity contains tachyonic modes. We show that, in certain parameter region, this tachyonic spectrum is completely reproduced as modes of the bulk tachyon field localized on a classical NS5-brane solution. In passing, we show how twisted modes at the singularity reproduce gauge fields on stacks of NS5-branes. 
  Study of magnetic monopoles in the original version of Born-Infeld (BI) electrodynamics is performed. It then is realized that interesting new physics emerge and they include exotic behavior of radial electric monopole field such as its regularity as $r\to 0$ and its changing behavior with the absence or presence of the radial magnetic monopole field. This last point has been interpreted as the manifestation of the existence of point-like dyons in abelian BI theory. Two pieces of clear evidences in favor of this dyon interpretation are provided. It is also demonstrated that despite these unique features having no analogues in standard Maxwell theory, the cherished Dirac quantisation condition remains unchanged. Lastly, comments are given concerning that dyons found here in the original version of BI electrodynamics should be distinguished from the ones with the same name or BIons being studied in the recent literature on D-brane physics. 
  We study the thermodynamic behavior of branched polymers. We first study random walks in order to clarify the thermodynamic relation between the canonical ensemble and the grand canonical ensemble. We then show that correlation functions for branched polymers are given by those for $\phi^3$ theory with a single mass insertion, not those for the $\phi^3$ theory themselves. In particular, the two-point function behaves as $1/p^4$, not as $1/p^2$, in the scaling region. This behavior is consistent with the fact that the Hausdorff dimension of the branched polymer is four. 
  We study the generic $p-p^\prime$ system in the presence of constant NS 2-form $B_{ij}$ field. We derive properties concerning with the noncommutativity of D-brane worldvolume, the Green functions and the spectrum of this system. In the zero slope limit, a large number of light states appear as the lowest excitations in appropriate cases. We are able to relate the energies of the lowest states after the GSO projection with the configurations of branes at angles. Through analytic continuation, the system is compared with the branes with relative motion. 
  We introduce the basics of the nonabelian duality transformation of SU(N) or U(N) vector-field models defined on a lattice. The dual degrees of freedom are certain species of the integer-valued fields complemented by the symmetric groups' \otimes_{n} S(n) variables. While the former parametrize relevant irreducible representations, the latter play the role of the Lagrange multipliers facilitating the fusion rules involved. As an application, I construct a novel solvable family of SU(N) D-matrix systems graded by the rank 1\leq{k}\leq{(D-1)} of the manifest [U(N)]^{\oplus k} conjugation-symmetry. Their large N solvability is due to a hidden invariance (explicit in the dual formulation) which allows for a mapping onto the recently proposed eigenvalue-models \cite{Dub1} with the largest k=D symmetry. Extending \cite{Dub1}, we reconstruct a D-dimensional gauge theory with the large N free energy given (modulo the volume factor) by the free energy of a given proposed 1\leq{k}\leq{(D-1)} D-matrix system. It is emphasized that the developed formalism provides with the basis for higher-dimensional generalizations of the Gross-Taylor stringy representation of strongly coupled 2d gauge theories. 
  Integral representation for the eigenfunctions of quantum periodic Toda chain is constructed for N-particle case. The multiple integral is calculated using the Cauchy residue formula. This gives the representation which reproduces the particular results obtained by Gutzwiller for N=2,3 and 4-particle chain. Our method to solve the problem combines the ideas of Gutzwiller and R-matrix approach of Sklyanin with the classical results in the theory of the Whittaker functions. In particular, we calculate Sklyanin's invariant scalar product from the Plancherel formula for the Whittaker functions derived by Semenov-Tian-Shansky thus obtaining the natural interpretation of the Sklyanin measure in terms of the Harish-Chandra function. 
  We discuss in a systematic way all the possible realisations of branes of M and type II theories as topological solitons of a brane-antibrane system. The classification of all the possibilities, consistent with the structure of the theory, is achieved by studying the Wess-Zumino terms in the worldvolume effective actions of the branes of M-theory and their reductions. 
  Using the relation between diffeomorphisms in the bulk and Weyl transformations on the boundary we study the Weyl transformation properties of the bulk metric on shell and of the boundary action. We obtain a universal formula for one of the classes of trace anomalies in any even dimension in terms of the parameters of the gravity action. 
  According to a recent conjecture, the moduli space of the heterotic conformal field theory on a $G\subset$ ADE singularity of an ALE space is equivalent to the moduli space of a pure $\cx N=4$ supersymmetric three-dimensional gauge theory with gauge group G. We establish this relation using geometric engineering of heterotic strings and generalize it to theories with non-trivial matter content. A similar equivalence is found between the moduli of heterotic CFT on isolated Calabi--Yau 3-fold singularities and two-dimensional Kazama-Suzuki coset theories. 
  Black hole production in the collision of ultra-relativistic particles in the brane-world approach is considered. In particular, stability of the brane under collision with ultra-relativistic particles is discussed. As a toy model we consider the 3 dimensional version of the Randall and Sundrum solution and show that stability of the brane depends on a choice of continuation of the solution across the horizon. In the unstable case black holes can be produced in the collision of a particle with the brane. 
  Quantum mechanics on the moduli space of N supersymmetric Reissner-Nordstrom black holes is shown to admit 4 supersymmetries using an unconventional supermultiplet which contains 3N bosons and 4N fermions. A near-horizon limit is found in which the quantum mechanics of widely separated black holes decouples from that of strongly-interacting, near-coincident black holes. This near-horizon theory is shown to have an enhanced D(2,1;0) superconformal symmetry. The bosonic symmetries are SL(2,R) conformal symmetry and SU(2)xSU(2) R-symmetry arising from spatial rotations and the R-symmetry of N=2 supergravity. 
  We aim to establish the holographic principle as a universal law, rather than a property only of static systems and special space-times. Our covariant formalism yields an upper bound on entropy which applies to both open and closed surfaces, independently of shape or location. It reduces to the Bekenstein bound whenever the latter is expected to hold, but complements it with novel bounds when gravity dominates. In particular, it remains valid in closed FRW cosmologies and in the interior of black holes. We give an explicit construction for obtaining holographic screens in arbitrary space-times (which need not have a boundary). This may aid the search for non-perturbative definitions of quantum gravity in space-times other than AdS. 
  This thesis aims to make precise the microscopic understanding of Hawking radiation from the D1/D5 black hole. We present an explict construction of all the shortmultiplets of the ${\cal N}=(4,4)$ SCFT on the symmetric product $\tilde{T}^4/S(Q_1Q_5)$. An investigation of the symmerties of this SCFT enables us to make a one-to-one correspondence beween the supergravity moduli and the marginal opeerators of the SCFT. We analyse the gauge theory dynamics of the splitting of the D1/D5 system into subsystems and show that it agrees with supergravity. We have shown that the fixed scalars of the D1/D5 system couple only to (2,2) operators thus removing earlier discrepancies between D-brane calculations and semiclassical calculations. The absorption cross-section of the minimal scalars is determined from first principles upto a propotionality constant. We show that the absorption cross-section of the minimal scalars computed in supergravity and the SCFT is independent of the moduli. 
  Thermodynamical aspects of string theory are reviewed and discussed. 
  It is shown that three-dimensional charged black holes can approach the extreme state at nonzero temperature. Unlike even dimensional cases, the entropy for the extreme three-dimensional charged black hole is uniquely described by the Bekenstein-Hawking formula, regardless of different treatments of preparing the extreme black hole, namely, Hawking's treatment and Zaslavskii's treatment. 
  We show that a new soliton mode is generated in a gauged sigma model with interpolating potential when the interpolation parameter decreases below a critical value. 
  We construct new configurations of oppositely charged, static black hole pairs (diholes) in four dimensions which are solutions of low energy string/M-theory. The black holes are extremal and have four different charges. We also consider diholes in other theories with an arbitrary number of abelian gauge fields and scalars, where the black holes can be regarded as composite objects. We uplift the four-charge solutions to higher dimensions in order to describe intersecting brane-antibrane systems in string and M-theory. The properties of the strings and membranes stretched inbetween these branes and antibranes are studied. Several other generic features of these solutions are discussed. 
  This is a continuation of hep-th/9811108, hep-th/9903218, hep-th/9910235, on exact integration technics for modified dynamical equations in ten dimensional supersymmetric gauge theory. A B\"acklund transformation is derived for the Yang type (super) equations previously derived (hep-th/9811108) by M. Saveliev and the author, from the ten dimensional super Yang-Mills field equations in an on-shell light cone gauge. It is shown to be based upon a particular gauge transformation satisfying nonlinear conditions which ensure that the particular form of the equations is retained. These Yang type field equations are shown to be precisely such that they automatically provide a solution of these conditions. This B\"acklund transformation is similar to the one proposed by A. Lesnov for self-dual Yang-Mills in four dimensions. In the introduction a personal recollection on the birth of supersymmetry is given. 
  According to conventional theory, the annihilation reaction p-bar p --> pi_0 pi_0 cannot occur from a p-bar p atomic S state. However, this reaction occurs so readily for antiprotons stopping in liquid hydrogen, that it would require 30% P-wave annihilations. Experimental results from other capture and p-bar p annihilation channels show that the fraction of P-wave annihilations is less than 6% in agreement with theoretical expectations. An experimental test to determine whether this reaction can occur from an atomic S state is suggested. If indeed this reaction is occurring from an atomic S state, then certain neutral vector mesons should exhibit a pi_0 pi_0 decay mode, and this can also be tested experimentally. 
  We study massless and massive Hawking radiations on a two-dimensional AdS spacetime. For the massless case, the quantum stress-energy tensor of a massless scalar field on the AdS background is calculated, and the expected null radiation is obtained. However, for the massive case, the scattering analysis is performed in order to calculate the absorption and reflection coefficients which are related to statistical Hawking temperature. On the contrary to the massless case, we obtain a nonvanishing massive radiation. 
  In this paper we study several issues related to the generation of superpotential induced by background Ramond-Ramond fluxes in compactification of Type IIA string theory on Calabi-Yau four-folds. Identifying BPS solitons with D-branes wrapped over calibrated submanifolds in a Calabi-Yau space, we propose a general formula for the superpotential and justify it comparing the supersymmetry conditions in D=2 and D=10 supergravity theories. We also suggest a geometric interpretation to the supersymmetric index in the two-dimensional effective theory in terms of topological invariants of the Calabi-Yau four-fold, and estimate the asymptotic growth of these invariants from BTZ black hole entropy. Finally, we explicitly construct new supersymmetric vacua for Type IIA string theory compactification on a Calabi-Yau four-fold with Ramond-Ramond fluxes. 
  The cubic scalar field theory admits the bell-shaped solitary wave solutions which can be interpreted as massive Bose particles. We rule out the nonminimal p-brane action for such a solution as the point particle with curvature. When quantizing it as the theory with higher derivatives, it is shown that the corresponding quantum equation has SU(2) dynamical symmetry group realizing the exact spin-coordinate correspondence. Finally, we calculate the quantum corrections to the mass of the bell boson which can not be obtained by means of the perturbation theory starting from the vacuum sector. 
  The effective p-brane action approach is generalized for arbitrary scalar field and applied for the Liouville theory near a particle-like solution. It was established that this theory has the remarkable features discriminating it from the theories studied earlier. Removing zero modes we obtain the effective action describing the solution as a point particle with curvature, quantize it as the theory with higher derivatives and calculate the quantum corrections to mass. 
  The Schr\"odinger equation is investigated in the Euclidean Taub-NUT geometry. The bound states are degenerate and an extra degeneracy is due to the conserved Runge-Lenz vector. The existence of the extra conserved quantities, quadratic in four-velocities implies the possibility of separating variables in two different coordinate systems. The eigenvalues and the eigenvectors are given in both cases in explicit, closed form. 
  We consider an exotic `compactification' of spacetime in which there are two infinite extra dimensions, using a global string instead of a domain wall. By having a negative cosmological constant we prove the existence of a nonsingular static solution using a dynamical systems argument. A nonsingular solution also exists in the absence of a cosmological constant with a time-dependent metric. We compare and contrast this solution with the Randall-Sundrum universe and the Cohen-Kaplan spacetime, and consider the options of using such a model as a realistic resolution of the hierarchy problem. 
  Light cone form of field dynamics in anti-de Sitter spacetime is described. We also present light cone reformulation of the boundary conformal field theory representations. AdS/CFT correspondence between the bulk fields and the boundary operators is discussed. 
  Successive toro\"\i dal compactifications of a closed bosonic string are studied and some Lie groups solutions are derived. 
  We sketch the main steps of old covariant quantization of bosonic open strings in a constant $B$ field background. We comment on its space-time symmetries and the induced effective metric. The low-energy spectrum is evaluated and the appearance of a new non-commutative gauge symmetry is addressed. 
  We consider, in a string theory framework, physical processes of phenomenological interest in models with a low string scale. The amplitudes we study involve tree-level virtual gravitational exchange, divergent in a field-theoretical treatment, and massive gravitons emission, which are the main signatures of this class of models. First, we discuss the regularization of summations appearing in virtual gravitational (closed string) Kaluza-Klein exchanges in Type I strings. We argue that a convenient manifestly ultraviolet convergent low energy limit of type I string theory is given by an effective field theory with an arbitrary cutoff $\Lambda$ in the closed (gravitational) channel and a related cutoff $M_s^2/\Lambda$ in the open (Yang-Mills) channel. We find the leading string corrections to the field theory results. Second, we calculate exactly string tree-level three and four-point amplitudes with gauge bosons and one massive graviton and examine string deviations from the field-theory result. 
  We analyze the possibility of description of D-dimensional massless particles by the Lagrangians linear on world-line curvatures k_i,  {\cal S}=\sum_{i=1}^Nc_i\int k_i d{\tilde s}.   We show, that the nontrivial classical solutions of this model are given by space-like curves with zero 2N-th curvature for N\leq[(D-2)/2]. Massless spinning particles correspond to the curves with constant k_{N+a}/k_{N-a} ratio.   It is shown that only the system with action {\cal S}=c\int k_N d{\tilde s} leads to irreducible representation of Poincar\'e group. This system has maximally possible number (N+1) of gauge degrees of freedom. Its classical solutions obey the conditions k_{N+a}=k_{N-a}, a=1,..., N-1, while first N curvatures k_i remain arbitrary. This solution is specified by coinciding N weights of the massless representation of little Lorentz group, while the remaining weights vanish. 
  We address the implementation of non-Abelian Wilson lines in D=4 N=1 Type IIB orientifold constructions. We present an explicit three-family example with the gauge group (U(4)xU(2)xSU(2)xSU(2))^2x(U(6)xSp(4))^2 and give the particle spectrum and the trilinear superpotential. Emphasizing the new subtleties associated with the introduction of non-Abelian Wilson lines, we show that the Abelian gauge anomalies are cancelled by the Green-Schwarz-type mechanism, and calculate the Fayet-Iliopoulos terms and gauge coupling corrections. The analysis thus sets a stage for further investigations of the phenomenological implications of this model. 
  Methods are reviewed for computing the instanton expansion of the prepotential for N=2 Seiberg-Witten (SW) theory with non-hyperelliptic curves. These results, if compared with the instanton expansion obtained from the microscopic Lagrangian, will provide detailed tests of M-theory. We observe group-theoretic regularities of the one-instanton prepotential which allow us to "reverse engineer" a SW curve for SU(N) gauge theory with two hypermultiplets in the antisymmetric representation and $N_f\leq 3$ hypermultiplets in the fundamental representations, a result not yet available by other methods. Consistency with M-theory requires a curve of infinite order, which we identify as a decompactified version of elliptic models of the type described by Donagi and Witten, Uranga, and others. This leads us to a brief discussion of some elliptic models that relate to our work. 
  We solve exactly the 6-vertex model on a dynamical random lattice, using its representation as a large N matrix model. The model describes a gas of dense nonintersecting oriented loops coupled to the local curvature defects on the lattice. The model can be mapped to the c=1 string theory, compactified at some length depending on the vertex coupling. We give explicit expression for the disk amplitude and evaluate the fractal dimension of its boundary, the average number of loops and the dimensions of the vortex operators, which vary continuously with the vertex coupling. 
  For arbitrary spacetime dimension a systematic procedure is carried on to uniquely decompose nonlocal light-cone operators into harmonic operators of well defined twist. Thereby, harmonic tensor polynomials up to rank 2 are introduced. Symmetric tensor operators of rank 2 are considered as an example. 
  We trace the origin of theta-terms in non-linear sigma-models as a nonperturbative anomaly of current algebras. The non-linear sigma-models emerge as a low energy limit of fermionic sigma-models. The latter describe Dirac fermions coupled to chiral bosonic fields. We discuss the geometric phases in three hierarchies of fermionic sigma-models in spacetime dimension (d+1) with chiral bosonic fields taking values on d-, d+1-, and d+2-dimensional spheres. The geometric phases in the first two hierarchies are theta-terms. We emphasize a relation between theta-terms and quantum numbers of solitons. 
  Motivated by the form of the noncommutative *-product in a system of open strings and Dp-branes with constant nonzero Neveu-Schwarz 2-form, we define a deformed multiplication operation on a quasitriangular Hopf algebra in terms of its R-matrix, and comment on some of its properties. We show that the noncommutative string theory *-product is a particular example of this multiplication, and comment on other possible Hopf algebraic properties which may underlie the theory. 
  The Future Tube $T^+_n$ of n-dimensional Minkowski spacetime may be identified with the reduced phase space or "space of motions" of a particle moving in (n+1)-dimensional Anti-de-Sitter spacetime. Both are isomorphic to a bounded homogeneous domain in ${\bf C}^{n}$ whose Shilov boundary may be identified with $n$-dimensional conformally compactified Minkowski spacetime. 
  We study the question of the Ward identity for "large" gauge invariance in 0+1 dimensional theories. We derive the relevant Ward identities for a single flavor fermion and a single flavor complex scalar field interacting with an Abelian gauge field. These identities are nonlinear. The Ward identity for any other complicated theory can be derived from these basic sets of identities. However, the structure of the Ward identity changes since these are nonlinear identities. In particular, we work out the "large" gauge Ward identity for a supersymmetric theory involving a single flavor of fermion as well as a complex scalar field. Contrary to the effective action for the individual theories, the solution of the Ward identity in the supersymmetric theory involves an infinity of Fourier component modes. We comment on which features of this analysis are likely/unlikely to generalize to the 2+1 dimensional theory. 
  We consider the effects of anomalies on the supersymmetry-breaking parameters in supergravity theories. We construct a supersymmetric expression for the anomaly-induced terms in the 1PI effective action; we use this result to compute the complete one-loop formula for the anomaly-induced gaugino mass. The mass receives contributions from the super-Weyl, Kahler, and sigma-model anomalies of the supergravity theory. We point out that the anomaly-mediated gaugino mass can be affected by local counterterms that cancel the super-Weyl-Kahler anomaly. This implies that the gaugino mass cannot be predicted unless the full high-energy theory is known. 
  A general method for solving the so-called quantum inverse scattering problem (namely the reconstruction of local quantum (field) operators in term of the quantum monodromy matrix satisfying a Yang-Baxter quadratic algebra governed by an R-matrix) for a large class of lattice quantum integrable models is given. The principal requirement being the initial condition (R(0) = P, the permutation operator) for the quantum R-matrix solving the Yang-Baxter equation, it applies not only to most known integrable fundamental lattice models (such as Heisenberg spin chains) but also to lattice models with arbitrary number of impurities and to the so-called fused lattice models (including integrable higher spin generalizations of Heisenberg chains). Our method is then applied to several important examples like the sl(n) XXZ model, the XYZ spin-1/2 chain and also to the spin-s Heisenberg chains. 
  We propose the two formalisms for obtaining the noncommutative spacetime in a magnetic field. One is the first-order formalism and the other is the second-order formalism. Although the noncommutative spacetime is realized manifestly in the first-order formalism, the second-order formalism would be more useful for calculating the physical quantities in the noncommutative geometry than the first-order one. Several interesting points for string theory and fractional quantum Hall effect are discussed. In particular, we point out that the noncommutative geometry is closely related to the fractional quantum Hall effect(FQHE). 
  We compare the way one of us got spinors out of fields, which are a priori antisymmetric tensor fields, to the Dirac-K\"ahler rewriting. Since using our Grassmann formulation is simple it may be useful in describing the Dirac-K\"ahler formulation of spinors and in generalizing it to vector internal degrees of freedom and to charges. The ``cheat'' concerning the Lorentz transformations for spinors is the same in both cases and is put clearly forward in the Grassmann formulation. Also the generalizations are clearly pointed out. The discrete symmetries are discussed, in particular the appearance of two kinds of the time-reversal operators as well as the unavoidability of four families. 
  A new approach proposed recently by author for the calculation of Green functions in quantum field theory and quantum mechanics is briefly reviewed. The method is applied to nonperturbative calculations for anharmonic oscillator, $\phi^4$-theory, quantum electrodynamics and other models. 
  In contrast with the classical gauge group cases, any method to prove exactly the scaling relation which relates moduli and prepotential is not known in the case of exceptional gauge groups. This paper provides a direct method to establish this relation by using Picard-Fuchs equations. In particular, it is shown that the scaling relation found by Ito in N=2 supersymmetric G_2 Yang-Mills theory actually holds exactly. 
  Supergravities are usually presented in a so-called 1.5 order formulation. Here we present a general scheme to derive pure 1^{st} order formulations of supergravities from the 1.5 order ones. The example of N_4=1 supergravity will be rederived and new results for N_4=2 and N_11=1 will be presented. It seems that beyond four dimensions the auxiliary fields introduced to obtain first order formulations of SUGRA theories do not admit supergeometrical transformation laws at least before a full superfield treatment. On the other hand first order formalisms simplify eventually symmetry analysis and the study of dimensional reductions. 
  In these notes Yang-Mills theories in 1+1 dimensions are reviewed. Instantons on a sphere prove to be -in the decompactification limit- the key issue to clarify an old controversy between equal-time and light-front quantization. 
  We discuss dualities of the integrable dynamics behind the exact solution to the N=2 SUSY YM theory. It is shown that T duality in the string theory is related to the separation of variables procedure in dynamical system. We argue that there are analogues of S duality as well as 3d mirror symmetry in the many-body systems of Hitchin type governing low-energy effective actions. 
  This is a thesis submitted for the cand.scient. degree at the University of Oslo. It is meant to give a thorough presentation of two methods for deriving tensionless limits of strings, and the analogue in other models. Also, the applicability of these methods are investigated by explicitly going through the calculations for a variety of models. An important part of the thesis is the study of constrained Hamiltonian systems. 
  We study little string theory in a weak coupling limit defined in \gk. 
  We study the SU(3)-invariant relevant deformation of D=4 N=4 SU(N) gauge theory at large N using the AdS/CFT correspondence. At low energies, we obtain a nonsupersymmetric gauge theory with three left-handed quarks in the adjoint of SU(N). In terms of the five dimensional gauged supergravity, there is an unstable critical point in the scalar potential for fluctuations of some fields in a nontrivial representation of the symmetry group SU(3). On the field theory side, this corresponds to dynamical breaking of the SU(3) chiral symmetry down to SO(3). We compute the condensate of the quark bilinear and the two-point correlation function of the spontaneously broken currents from supergravity and find a nonzero `pion' decay constant, f_pi. 
  We describe a unitary matrix model which is constructed from discrete analogs of the usual projective modules over the noncommutative torus and use it to construct a lattice version of noncommutative gauge theory. The model is a discretization of the noncommutative gauge theories that arise from toroidal compactification of Matrix theory and it includes a recent proposal for a non-perturbative definition of noncommutative Yang-Mills theory in terms of twisted reduced models. The model is interpreted as a manifestly star-gauge invariant lattice formulation of noncommutative gauge theory, which reduces to ordinary Wilson lattice gauge theory for particular choices of parameters. It possesses a continuum limit which maintains both finite spacetime volume and finite noncommutativity scale. We show how the matrix model may be used for studying the properties of noncommutative gauge theory. 
  We present a formal supersymmetric solution of type IIB supergravity generalizing previously known solutions corresponding to D3 branes to geometries without an orthogonal split between parallel and transverse directions. The metric is given implicitly as one with respect to which a certain connection is compatible. The case of the deformed conifold is discussed in detail. 
  To test recent ideas about lower dimensional gravity bound to a brane, we construct exact solutions describing black holes on two-branes in four dimensions. We find that 2+1 gravity is indeed recovered at large distances along the brane, although there are significant deviations at smaller scales. Large black holes appear as flattened "pancakes", with a relatively small extent off the brane. Black hole thermodynamics is discussed from both the standpoint of the brane and the bulk. We comment on the analogous black holes bound to higher dimensional branes. 
  Withdrawn by arXiv administration because authors have forged affiliations and acknowledgements, and have not adequately responded to charges [hep-th/9912039] of unattributed use of verbatim material. 
  Withdrawn by arXiv administration because authors have forged affiliations and acknowledgements, and have not adequately responded to charges [hep-th/9912039] of unattributed use of verbatim material. 
  Withdrawn by arXiv administration because authors have forged affiliations and acknowledgements, and have not adequately responded to charges [hep-th/9912039] of unattributed use of verbatim material. 
  Withdrawn by arXiv administration because authors have forged affiliations and acknowledgements, and have not adequately responded to charges [hep-th/9912039] of unattributed use of verbatim material. 
  Withdrawn by arXiv administration because authors have forged affiliations and acknowledgements, and have not adequately responded to charges [hep-th/9912039] of unattributed use of verbatim material. 
  Withdrawn by arXiv administration because authors have forged affiliations and acknowledgements, and have not adequately responded to charges [hep-th/9912039] of unattributed use of verbatim material. 
  I show that the method of finding if the ground state energy is nonzero or not by saturating the functional integral directly with instanton-anti-instanton type fluctuations is not reliable either for the double or for the triple well potential models in supersymmetric quantum mechanics. 
  We study the vacuum partition functional Z [J] for a system of closed, bosonic p-branes coupled to p-forms in the limiting case: p+1 = space-time dimension. We suggest an extension of the duality transformation which can be applied to the limiting case even though no dual gauge potential exists in the conventional sense. The dual action thus obtained describes a current-current, static interaction within the bulk volume bounded by the d-1-brane. Guided by these results, we then construct a general expression for the parent Lagrangian that allows for a unified treatment of p-duality, even in the presence of external currents, using a first order formalism instead of the Bianchi identities. Finally, we show how this generalized dualization approach can accommodate the inclusion of a massive topological term in the parent action of an Abelian gauge theory. 
  We review a recently proposed method for constructing super-p-brane world-volume actions. In this approach, starting from a democratic choice of world-volume gauge-fields guided by p-brane intersection rules, the requirements of kappa-symmetry and gauge invariance can be used to determine the corresponding action. We discuss the application of the method to some cases of interest, notably the (p,q)-5-branes of type-IIB string theory in a manifestly S-duality covariant formulation. 
  Quantization of a system constrained to move on a sphere is considered by taking a square root of the ``on sphere condition''. We arrive at the fibre bundle structure of the Hopf map in the cases of $S^{2} $and $S^{4}$. This leads to more geometrical understanding of monopole and instanton gauge structures that emerge in the course of quantization. 
  We show that the motion on the n-dimensional ellipsoid is complete integrable by exhibiting n integrals in involution. The system is separable at classical and quantum level, the separation of classical variables being realized by the inverse of the momentum map. This system is a generic one in a new class of n-dimensional complete integrable Hamiltonians defined by an arbitrary function f(q,p) invertible with respect to momentum p and rational in the coordinate q. 
  We discuss the weak gravitational field created by isolated matter sources in the Randall-Sundrum brane-world. In the case of two branes of opposite tension, linearized Brans-Dicke (BD) gravity is recovered on either wall, with different BD parameters. On the wall with positive tension the BD parameter is larger than 3000 provided that the separation between walls is larger than 4 times the AdS radius. For the wall of negative tension, the BD parameter is always negative but greater than -3/2. In either case, shadow matter from the other wall gravitates upon us. For equal Newtonian mass, light deflection from shadow matter is 25 % weaker than from ordinary matter. Hence, the effective mass of a clustered object containing shadow dark matter would be underestimated if naively measured through its lensing effect. For the case of a single wall of positive tension, Einstein gravity is recovered on the wall to leading order, and if the source is stationary the field stays localized near the wall. We calculate the leading Kaluza-Klein corrections to the linearized gravitational field of a non-relativistic spherical object and find that the metric is different from the Schwarzschild solution at large distances. We believe that our linearized solution corresponds to the field far from the horizon after gravitational collapse of matter on the brane. 
  This is one of the `New Talents' seminars at `Erice International School of Subnuclear Physics 1999' and looks at numerical studies of (2+1)D topological Skyrme-like solitons; the baby skyrmions. We explain the concept of integrable and topological solitons. Then we introduce the non-linear sigma model in 2D in order to discuss the 3D nuclear Skyrme model. We explain that the baby Skyrme models can be viewed as a (2+1)D version of the Skyrme model. We describe the numerical methods needed to study the baby skyrmions. Finally, we present some results on skyrmion-skyrmion scattering and our work on skyrmions with higher topological charge. 
  Non-commutative supersymmetric Yang-Mills with rational Theta is dual to an ordinary supersymmetric Yang-Mills with a 't Hooft flux. It is believed that the simplest description is via the ordinary supersymmetric Yang-Mills. We claim, however, that the two descriptions form a hierarchy. The SYM description is the proper description in the ultra violet while the non-commutative description takes over in the infra red. 
  We calculate the partition functions of the affine Pasquier models on the cylinder in the continuum limit. We show that the partition function of any affine model may be expressed in terms of the orbit structure of the affine Coxeter element of the Weyl group associated with the defining graph of the model. Some of the consequences of this geometric relationship are explored. 
  Gamow vectors are generalized eigenvectors (kets) of self-adjoint Hamiltonians with complex eigenvalues $(E_{R}\mp i\Gamma/2)$ describing quasistable states. In the relativistic domain this leads to Poincar\'e semigroup representations which are characterized by spin $j$ and by complex invariant mass square ${\mathsf{s}}={\mathsf{s}}_{R}=(M_{R}-\frac{i}{2}\Gamma_{R})^{2}$. Relativistic Gamow kets have all the properties required to describe relativistic resonances and quasistable particles with resonance mass $M_{R}$ and lifetime $\hbar/\Gamma_{R}$. 
  We calculate, in the context of higher dimensional gravity, the stress-energy tensor and Weyl anomaly associated with anti-de Sitter and anti-de Sitter black hole solutions. The boundary counter-term method is used to regularize the action and the resulting stress-energy tensor yields both the correct black hole energies as well as a vacuum energy contribution which is interpreted as a Casimir energy. This calculation is done up to d = 8 (d being the boundary dimension). We confirm some results for d < 8 as well as comment on some new results (some of which are relevant to (2,0) theory). All results for d=8 are new. 
  Recent progress in understanding (2+1)-dimensional Yang-Mills (YM_{2+1}) theory via the use of gauge-invariant variables is reviewed. Among other things, we discuss the vacuum wavefunction, an analytic calculation of the string tension and the propagator mass for gluons and its relation to the magnetic mass for YM_{3+1} at nonzero temperature. 
  We propose that Strominger's method to derive the BTZ black hole entropy is in fact applicable to other asymptotically AdS_3 black holes and gives the correct functional form of entropies. We discuss various solutions in the Einstein-Maxwell theory, dilaton gravity, Einstein-scalar theories, and Einstein-Maxwell-dilaton theory. In some cases, solutions approach AdS_3 asymptotically, but their entropies do not have the form of Cardy's formula. However, it turns out that they are actually not "asymptotically $AdS_3$" solutions. On the other hand, for truly asymptotically AdS_3 solutions, their entropies have the form of Cardy's formula. In this sense, all known solutions are consistent with our proposal. 
  A formula describing finite renormalizations is derived in the Epstein-Glaser formalism and an explicit calculation of finite counterterms in $\Phi ^4$-theory is performed. The Zimmermann identities and the action principle for changes of parameters in the interaction are presented independent of the adiabatic limit. 
  We show that in a two-dimensional sigma-model whose fields only depend on one target space co-ordinate, the O(d,d) invariance of the conformal invariance conditions observed at one loop is preserved at two loops (in the general case with torsion) and at three loops (in the case without torsion). 
  We study static spherically symmetric monopole solutions in non-Abelian Einstein-Born-Infeld-Higgs model with normal trace structure. These monopoles are similar to the corresponding solution with symmetrised trace structure and are existing only up to some critical value of the strength of the gravitational interaction. In addition, similar to their flat space counterpart, they also admit a critical value of the Born-Infeld parameter  $\b$. 
  This contribution to the proceedings of the 1999 NATO ASI on Quantum Geometry at Akureyri, Iceland, is based on notes of lectures given by A. Strominger. Topics include N-particle conformal quantum mechanics, extended superconformal quantum mechanics and multi-black hole moduli spaces. 
  This is a series of lectures on M Theory for cosmologists. After summarizing some of the main properties of M Theory and its dualities I show how it can be used to address various fundamental and phenomenological issues in cosmology. 
  This is a summary of key issues in Matrix Theory and its compactifications. It is emphasized that Matrix Theory is a valid Discrete Light Cone Quantization of M Theory with at least 6 noncompact asymptotically flat dimensions and 16 or 32 Supersymmetry Charges. The background dependence of the quantum mechanics of M Theory, and the necessity of working in light cone frame in asymptotically flat spacetimes are explained in terms of the asymptotic density of states of the theory, which follows from the Bekenstein-Hawking entropy formula. In four noncompact dimensions one is led to expect a Hagedorn spectrum in light cone energy. This suggests the possible relevance of ``little string theories'' (LSTs) to the quantum description of four dimensional compactifications, because one can argue that their exact high energy spectrum has the Hagedorn form. Some space is therefore devoted to a discussion of the properties of LSTs, which were first discovered as the proper formulation of Matrix Theory on the five torus. 
  The electrogravity transformation is applied to the three-dimensional Einstein field equations to obtain new multi-parameter families of black hole solutions. The Ba\~{n}ados-Teitelboim-Zanelli black hole is shown to be a special case of one of these families. The causal structure, associated matter, as well as the mechanical and thermodynamical properties of some of the solutions are discussed. 
  We explore the (non)-universality of Martinez's conjecture, originally proposed for Kerr black holes, within and beyond general relativity. The conjecture states that the Brown-York quasilocal energy at the outer horizon of such a black hole reduces to twice its irreducible mass, or equivalently, to \sqrt{A} /(2\sqrt{pi}), where `A' is its area. We first consider the charged Kerr black hole. For such a spacetime, we calculate the quasilocal energy within a two-surface of constant Boyer-Lindquist radius embedded in a constant stationary-time slice. Keeping with Martinez's conjecture, at the outer horizon this energy equals the irreducible mass. The energy is positive and monotonically decreases to the ADM mass as the boundary-surface radius diverges. Next we perform an analogous calculation for the quasilocal energy for the Kerr-Sen spacetime, which corresponds to four-dimensional rotating charged black hole solutions in heterotic string theory. The behavior of this energy as a function of the boundary-surface radius is similar to the charged Kerr case. However, we show that in this case it does not approach the expression conjectured by Martinez at the horizon. 
  A lesson for the new millennium from quantum field theory: Not all field-theoretic infinities are bad. Some give rise to finite, symmetry-breaking effects, whose consequences are observed in Nature. 
  Projecting a non-Abelian SU(2) vacuum gauge field - a pure gauge constructed from the group element U - onto a fixed (electromagnetic) direction in isospace gives rise to a nontrivial magnetic field, with nonvanishing magnetic helicity, which coincides with the winding number of U. Although the helicity is not conserved under Maxwell (vacuum) evolution, it retains one-half its initial value at infinite time. 
  Using the known propagator equations for 0,1 and 2 forms in AdS_{d+1}, we find the p-form field propagator equations in dimensions where the forms are Poincare dual. Since the general equation obeyed by the propagators is linear in dimension, this gives us the equation obeyed by the propagators for any d. Furthermore, based on the Poincare duality formulas for 0,1,2 and 3 forms we conjecture the general form of the Poincare duality formulas, and check them against the previously found propagator equations. The whole structure is self-consistent. Once we have the equations, we can easily obtain all the p-form field propagators in AdS_{d+1}. The generalization to massive p-forms can also be easily done. 
  In this paper we construct non-Abelian gauge theories with fermions and scalars that nevertheless possess asymptotic freedom.The scalars are taken to be in a chiral multiplet transforming as $(2,2)$ under $SU(2)_L\otimes SU(2)_R$ and transforming as singlets under the colour SU(3) group. We consider two distinct scenarios, one in which the additional scalars are light and another in which they are heavier than half the Z-boson mass. It is shown that asymptotic freedom is obtained without requiring that all additional couplings keep fixed ratios with each other. It is also shown that both scenarios can not be ruled out by what are considered standard tests of QCD like R- parameter, g-2 for muons or deep inelastic phenomena. The light mass scenario is however ruled out by high precision Z-width data (and only by that one data).The heavy mass scenario is still viable and is shown to naturally pass the test of flavour changing neutral currents. It also is not ruled out by precision electroweak oblique parameters. Many distinctive experimental signatures of these models are also discussed. 
  A consistent definition of high dimensional compactified quantum field theory without breaking the Kaluza-Klein tower is proposed. It is possible in the limit when the size of compact dimensions is of the order of the cut off. This limit is nontrivial and depends on the geometry of compact dimensions. Possible consequences are discussed for the scalar model. 
  Applying the distributional formalism to study the dynamics of thin shells in general relativity, we regain the junction equations for matching of two spherically symmetric spacetimes separated by a singular hypersurface. In particular, we have shown how to define and insert the relevant sign functions in the junction equations corresponding to the signs of the extrinsic curvature tensor occurred in the Darmois-Israel method. 
  It is shown that static solutions with a finite curvature at the horizon may exist in dilaton gravity at temperatures $T\neq T_{H}$ (including T=0) where $T_{H} $is the Hawking one. Hawking radiation is absent and the state of a system represents thermal excitation over the Boulware vacuum. The horizon remains unattainable for a observer because of thermal divergences in the stress-energy of quantum fields there. However, the curvature at the horizon is finite, when measured from outside, since these divergences are compensated by those in gradients of a dilaton field. Spacetimes under consideration are geodesically incomplete and the coupling between dilaton and gravity diverges at the horizon, so we have ''singularity without singularity''. 
  We propose a connection between conformal field theory (CFT) and the exact solution and integrability of the reduced BCS model of superconductivity. The relevant CFT is given by the $SU(2)_k$-WZW model in the singular limit when the level k goes to -2. This theory has to be perturbed by an operator proportional to the inverse of the BCS coupling constant. Using the free field realization of this perturbed Wess-Zumino-Witten model, we derive the exact Richardson's wave function and the integrals of motion of the reduced BCS model in the saddle point approximation. The construction is reminiscent of the CFT approach to the Fractional Quantum Hall effect. 
  We formulate the renormalization procedure using the domain wall regularization that is based on the heat-kernel method. The quantum effects of both fermions and bosons (gauge fields) are taken into account. The background field method is quite naturally introduced. With regard to the treatment of the loop-momentum integrals, an interesting contrast between the fermion-determinant part and other parts is revealed. These points are elucidated by considering some examples. The Weyl anomalies for 2D QED and 4D QED are correctly obtained. It is found that the ``chiral solution'' produces (1/2)$^{d/2}$ $\times$ ``correct values'', where $d$ is the spatial dimension. Considering the model of 2D QED, both Weyl and chiral anomalies are directly obtained from the effective action. The mass and wave function renormalization are explicitly performed in 4D QED. We confirm the multiplicative (not additive) renormalization, which demonstrates the advantage of no fine-tuning. The relation with the recently popular higher-dimensional approach, such as the Randall- Sundrum model, is commented on. 
  Duality symmetries in M--theory and string theory are reviewed, with particular emphasis on the way in which string winding modes and brane wrapping modes can lead to new spatial dimensions. Brane world-volumes wrapping around Lorentzian tori can give rise to extra time dimensions and in this way dualities can change the number of time dimensions as well as the number of space dimensions. This suggests that brane wrapping modes and spacetime momenta should be on an equal footing and M--theory should not be formulated in a spacetime of definite dimension or signature. 
  We show how chiral type I models whose tadpole conditions have no supersymmetric solution can be consistently defined introducing antibranes with non-supersymmetric world volumes. At tree level, the resulting stable non-BPS configurations correspond to tachyon-free spectra, where supersymmetry is broken at the string scale on some (anti)branes but is exact in the bulk, and can be further deformed by the addition of brane-antibrane pairs of the same type. As a result, a scalar potential is generated, that can stabilize some radii of the compact space. This setting has the novel virtue of linking supersymmetry breaking to the consistency requirements of an underlying fundamental theory. 
  Duality transformations involving compactifications on timelike as well as spacelike circles link M-theory, the 10+1-dimensional strong coupling limit of IIA string theory, to other 11-dimensional theories in signatures 9+2 and 6+5 and to type II string theories in all 10-dimensional signatures. These theories have BPS branes of various world-volume signatures, and here we construct the world-volume theories for these branes, which in each case have 16 supersymmetries. For the generalised D-branes of the various type II string theories, these are always supersymmetric Yang-Mills theories with 16 supersymmetries, and we show that these all arise from compactifications of the supersymmetric Yang-Mills theories in 9+1 or 5+5 dimensions. We discuss the geometry of the brane solutions and, for the cases in which the world-volume theories are superconformally invariant, we propose holographically dual string or M theories in constant curvature backgrounds. For product space solutions $X\times Y$, there is in general a conformal field theory associated with the boundary of $X$ and another with the boundary of $Y$. 
  We consider models of scalar fields coupled to gravity which are higher-dimensional generalizations of four dimensional supergravity. We use these models to describe domain wall junctions in an anti-de Sitter background. We derive Bogomolnyi equations for the scalar fields from which the walls are constructed and for the metric. From these equations a BPS-like formula for the junction energy can be derived. We demonstrate that such junctions localize gravity in the presence of more than one uncompactified extra dimension. 
  We give a simple proof of the known S-duality of Heterotic String theory compactified on a T^6. Using this S-duality we calculate the tensions for a class of BPS 5-branes in Heterotic String theory on a circle. One of these, the Kaluza-Klein monopole, becomes tensionless when the radius of the circle is equal to the string length. We study the question of stability of the Heterotic NS5-brane with a transverse circle. For large radii the NS5-brane is absolutely stable. However for small radii it is only marginally stable. We also study the moduli space of 2 Kaluza-Klein monopoles and show that it is equal to the moduli space of a Heterotic A_1 singularity. 
  We study Dirac commutators of canonical variables on D-branes with a constant Neveu-Schwarz 2-form field by using the Dirac constraint quantization method, and point out some subtleties appearing in previous works in analyzing constraint structure of the brane system. Overcoming some ad hoc procedures, we obtain desirable noncommutative coordinates exactly compatible with the result of the conformal field theory in recent literatures. Furthermore, we find interesting commutator relations of other canonical variables. 
  A complete qualitative study of the dynamics of string cosmologies is presented for the class of isotopic curvature universes. These models are of Bianchi types I, V and IX and reduce to the general class of Friedmann-Robertson-Walker universes in the limit of vanishing shear isotropy. A non-trivial two-form potential and cosmological constant terms are included in the system. In general, the two-form potential and spatial curvature terms are only dynamically important at intermediate stages of the evolution. In many of the models, the cosmological constant is important asymptotically and anisotropy becomes dynamically negligible. There also exist bouncing cosmologies. 
  We propose a resolution for the fermion doubling problem in discrete field theories based on the fuzzy sphere and its Cartesian products. 
  We compute the leading order perturbative correction to the static potential in ${\cal N}=4$ supersymmetric Yang-Mills theory. We show that the perturbative expansion contains infrared logarithms which, when resummed, become logarithms of the coupling constant. The resulting correction goes in the right direction to match the strong coupling behavior obtained from the AdS/CFT correspondence. We find that the strong coupling extrapolation of the sum of ladder diagrams goes as $\sqrt{g^2N}$, as in the supergravity approach. 
  We show that dualization of Stueckelberg-like massive gauge theories and $B\wedge F$ models, follows form a general p-dualization of interacting theories in d spacetime dimensions. This is achieved by a particular choice of the external current. 
  We write down charged macroscopic string solutions in type II string theories, compactified on torii, and present an explicit solution of the spinor Killing equations to show that they preserve 1/2 of the type II supersymmetries. The S-duality symmetry of the type IIB string theory in ten-dimensions is used to write down the SL(2,Z) multiplets of such strings and the corresponding 1/2 supersymmetry conditions. Finally we present examples of planar string networks, using charged macroscopic (p,q)-strings. An interesting feature of some of these networks, which preserve 1/4 supersymmetry, is a required alignment among three parameters, namely the orientation of strings, a U(1) phase associated with the maximal compact subgroup of SL(2, Z), and an (angular) parameter associated with a solution generating transformation, which is responsible for creating charges and currents on the strings. 
  We show that a realization of the correspondence AdS_2/CFT_1 for near extremal Reissner-Nordstrom black holes in arbitrary dimensional Einstein-Maxwell gravity exactly reproduces, via Cardy's formula, the deviation of the Bekenstein-Hawking entropy from extremality. We also show that this mechanism is valid for Schwarzschild-de Sitter black holes around the degenerate solution dS_2xS^n. These results reinforce the idea that the Bekenstein-Hawking entropy can be derived from symmetry principles. 
  Using a superfield formulation of extended phase space, we propose a new form of the Hamiltonian action functional. A remarkable feature of this construction is that it directly leads to the BV master action on phase space. Conversely, superspace can be used to construct nilpotent BRST charges directly from solutions to the classical Lagrangian Master Equation. We comment on the relation between these constructions and the specific master action proposal of Alexandrov, Kontsevich, Schwarz and Zaboronsky. 
  Born-Infeld nonlinear electrodynamics with point singularities having both electric and magnetic charges are considered. Problem of interaction between the associated soliton dyon solutions is investigated. For the case of long-range interaction at first order by a small field of distant solitons we obtain that the generalized Lorentz force is acted on a dyon under consideration. Short-range interaction between two dyons having identical electric and opposite magnetic charges is investigated for an initial approximation. We consider the case when the velocities of the dyons have equal modules and opposite directions on a common line. It is shown that the associated field configuration has a constant full angular momentum which is independent of the interdyonic distance and their speed. This property permits a consideration of this bidyon configuration as an electromagnetic model of charged particle with spin. We numerically investigate movement of the dyons in this configuration for the case when the full electric charge equals the electron charge and the full angular momentum equals the electron spin. It is shown that for this case the absolute value of relation between electric and magnetic charges of the dyons equals the fine structure constant. The calculation gives that the bidyon may behave as nonlinear oscillator. Associated dependence of frequency on the full energy is obtained for the initial approximation. In the limits of the electrodynamic model we obtain that the quick-oscillating wave packet may behave like massive gravitating particle when it move in high background field. We discuss the possible electrodynamic world with the oscillating bidyons as particles. 
  We study three dimensional gauge theories with N=2 supersymmetry. We show that the Coulomb branches of such theories may be rendered compact by the dynamical generation of Chern-Simons terms and present a new class of mirror symmetric theories in which both Coulomb and Higgs branches have a natural description in terms of toric geometry. 
  AdS/CFT-correspondence establishes a relationship between supersymmetric gravity (SUGRA) on Anti-de-Sitter (AdS) space and supersymmetric Yang-Mills (SYM) theory which is conformaly invariant (CFT). The AdS space is the solution of the Einstein-Hilbert equations with a constant negative curvature. Why is this relationship important? What kind of relationship is this? How does one find it? The purpose of this text is to answer these questions. We try to present the main ideas and arguments underlying this relationship, starting with a brief sketch of old string theory statements and proceeding with the definition of D-branes and a description of their main features. We finish with the observation of the correspondence in question and the arguments that favor it. 
  We study fractional branes in ${\CN}=2$ orbifold and ${\CN}=1$ conifold theories. Placing a large number $N$ of regular D3-branes at the singularity produces the dual ${\bf AdS}_5\times X^5$ geometry, and we describe the fractional branes as small perturbations to this background. For the orbifolds, $X^5={\bf S}^5/\Gamma$ and fractional D3-branes excite complex scalars from the twisted sector which are localized on the fixed circle of $X^5$. The resulting solutions are given by holomorphic functions and the field-theoretic beta-function is simply reproduced. For $N$ regular and $M$ fractional D3-branes at the conifold singularity we find a non-conformal ${\cal N}=1$ supersymmetric $SU(N+M)\times SU(N)$ gauge theory. The dual Type $\II$B background is ${\bf AdS}_5\times {\bf T}^{1,1}$ with NS-NS and R-R 2-form fields turned on. This dual description reproduces the logarithmic flow of couplings found in the field theory. 
  We identify two distinct, complementary gauge field configurations for QCD with SU(2) gauge group, one (instanton-like configurations) having to do with chiral symmetry breaking but not with confinement, the other (regularized Wu-Yang monopoles) very likely responsible for confinement but unrelated to chiral symmetry breaking. Our argument is based on a semiclassical analysis of fermion zero modes in these backgrounds, made by use of a gauge field decomposition recently introduced by Faddeev and Niemi. Our result suggests that the two principal dynamical phenomena in QCD, confinement and chiral symmetry breaking, are distinct effects, caused by two competing classes of gauge field configurations. 
  A noncommutative Feynman graph is a ribbon graph and can be drawn on a genus $g$ 2-surface with a boundary. We formulate a general convergence theorem for the noncommutative Feynman graphs in topological terms and prove it for some classes of diagrams in the scalar field theories. We propose a noncommutative analog of Bogoliubov-Parasiuk's recursive subtraction formula and show that the subtracted graphs from a class $\Omega_d$ satisfy the conditions of the convergence theorem. For a generic scalar noncommutative quantum field theory on $\re^d$, the class $\Omega_d$ is smaller than the class of all diagrams in the theory. This leaves open the question of perturbative renormalizability of noncommutative field theories. We comment on how the supersymmetry can improve the situation and suggest that a noncommutative analog of Wess-Zumino model is renormalizable. 
  Field theory with instantons can be partially regularized by adding degrees of freedom at some scale. These extra degrees of freedom lead to the appearence of the new topological defects. These defects which we call freckles have some characteristic size depending on the scale at which the extra degrees of freedom revive.   The examples of two dimensional sigma model, four dimensional gauge theory are studied. The compactification of the four dimensional supersymmetric gauge theory down to two dimensions is also considered and the new phenomena are found. 
  It is natural to analyse the AdS$_{d+1}$-CQFT$_{d}$ correspondence in the context of the conformal- compactification and covering formalism. In this way one obtains additional inside about Rehren's rigorous algebraic holography in connection with the degree of freedom issue which in turn allows to illustrates the subtle but important differences beween the original string theory-based Maldacena conjecture and Rehren's theorem in the setting of an intrinsic field-coordinatization-free formulation of algebraic QFT. I also discuss another more generic type of holography related to light fronts which seems to be closer to 't Hooft's original ideas on holography. This in turn is naturally connected with the generic concept of ``Localization Entropy'', a quantum pre-form of Bekenstein's classical black-hole surface entropy. 
  We consider travelling waves propagating on the anti-de Sitter (AdS) background. It is pointed out that for any dimension d, this space of solutions has a Virasoro symmetry with a non-zero central charge. This result is a natural generalization to higher dimensions of the three-dimensional Brown-Henneaux symmetry. 
  Recent astrophysical observations seem to indicate that the cosmological constant is small but nonzero and positive. The old cosmological constant problem asks why it is so small; we must now ask, in addition, why it is nonzero (and is in the range found by recent observations), and why it is positive. In this essay, we try to kill these three metaphorical birds with one stone. That stone is the unimodular theory of gravity, which is the ordinary theory of gravity, except for the way the cosmological constant arises in the theory. We argue that the cosmological constant becomes dynamical, and eventually, in terms of the cosmic scale factor $R(t)$, it takes the form $\Lambda(t) = \Lambda(t_0)(R(t_0)/R(t))^2$, but not before the epoch corresponding to the redshift parameter $z \sim 1$. 
  We consider the (2+1)-dimensional gauged Thirring model in the Heisenberg picture. In this context we evaluate the vacuum polarization tensor as well as the corrected gauge boson propagator and address the issues of generation of mass and dynamics for the gauge boson (in the limits of QED$_3$ and Thirring model as a gauge theory, respectively) due to the radiative corrections. 
  We study junctions of supersymmetric domain walls in N=1 supergravity theories in four dimensions, coupled to a chiral superfield with quartic superpotential having $Z_3$ symmetry. After deriving a BPS equation of the domain wall junction, we consider a stable hexagonal configuration of network of brane junctions, which are only approximately locally BPS. We propose a model for a mechanism of supersymmetry breaking without loss of stability, where a messenger for the SUSY breaking comes from the neighboring anti-BPS junction world, propagating along the domain walls connection them. 
  We study a two-dimensional disordered system consisting of Dirac fermions coupled to a scalar potential. This model is closely related to a more general disordered system that has been introduced in conjunction with the integer quantum Hall transition. After disorder averaging, the interaction can be written as a marginal osp(2|2) current-current perturbation. The osp(2|2) current-current model in turn can be viewed as the fully renormalized version of an osp(2|2)^(1) Toda-type system (at the marginal point). We build non-local charges for the Toda system satisfying the U_q[osp(2|2)^(1)] quantum superalgebra. The corresponding quantum group symmetry is used to construct a Toda S-matrix for the vector representation. We argue that in the marginal (or rational) limit, this S-matrix gives the exact (Yangian symmetric) physical S-matrix for the fundamental "solitons" of the osp(2|2) current-current model. 
  A string in four dimensions is constructed by supplementing it with forty four Majorana fermions. The central charge is 26. The fermions are grouped in such a way that the resulting action is supersymmetric. The super-Virasoro algebra is constructed and closed by the use of Jacobi identity. The tachyonic ground state decouples from the physical states. GSO projections are necessary for proving modular invariance and space-time supersymmetry is shown to exist for modes of zero mass. The symmetry group of the model desends to the low energy group SU(3)x SU(2)x U(1)x U(1). 
  A proof of renormalizability of the theory of the dynamical non-Abelian two-form is given using the Zinn-Justin equation. Two previously unknown symmetries of the quantum action, different from the BRST symmetry, are needed for the proof. One of these is a gauge fermion dependent nilpotent symmetry, while the other mixes different fields with the same transformation properties. The BRST symmetry itself is extended to include a shift transformation by use of an anticommuting constant. These three symmetries restrict the form of the quantum action up to arbitrary order in perturbation theory. The results show that it is possible to have a renormalizable theory of massive vector bosons in four dimensions without a residual Higgs boson. 
  A field model on fibre bundles can be extended in a standard way to the BRS-invariant SUSY field model which possesses the Lie supergroup ISp(2) of symmetries. 
  Two issues regarding the interactions of the chiral two-forms are reviewed. First, the problem of constructing Lorentz-invariant self-couplings of a single chiral two-form is investigated in the light of the Dirac-Schwinger condition on the energy-momentum tensor commutation relations. We show how the Perry-Schwarz condition follows from the Dirac-Schwinger criterion and point out that consistency of the gravitational coupling is automatic. Secondly, we study the possible local deformations of chiral two-forms. This problem reduces to the study of the local BRST cohomological group at ghost number zero. We proof that the only consistent deformations of a system of free chiral two-forms are (up to redefinitions) deformations that do not modify the abelian gauge symmetries of the free theory. The consequence of this result for a system consisting of a number of parallel M5-branes is explained. 
  The purpose of this short note is to announce results that amount to a verification of the bootstrap for Liouville theory in the generic case under certain assumptions concerning existence and properties of fusion transformations. Under these assumptions one may characterize the fusion and braiding coefficients as solutions of a system of functional equations that follows from the combination of consistency requirements and known results. This system of equations has a unique solution for irrational central charge c>25. The solution is constructed by solving the Clebsch-Gordan problem for a certain continuous series of quantum group representations and constructing the associated Racah-coefficients. This gives an explicit expression for the fusion coefficients. Moreover, the expressions can be continued into the strong coupling region 1<c<25, providing a solution of the bootstrap also for this region. 
  The relative entropy of the massive free bosonic field theory is studied on various compact Riemann surfaces as a universal quantity with physical significance, in particular, for gravitational phenomena. The exact expression for the sphere is obtained, as well as its asymptotic series for large mass and its Taylor series for small mass. One can also derive exact expressions for the torus but not for higher genus. However, the asymptotic behaviour for large mass can always be established-up to a constant-with heat-kernel methods. It consists of an asymptotic series determined only by the curvature, hence common for homogeneous surfaces of genus higher than one, and exponentially vanishing corrections whose form is determined by the concrete topology. The coefficient of the logarithmic term in this series gives the conformal anomaly. 
  Certain exact results in supersymmetric gauge theories are generated by non-perturbative effects different from instantons. In supersymmetric QCD with N colours and Nf fundamental flavours we examine the Affleck-Dine-Seiberg (ADS) superpotential using controlled semi-classical analysis. We show how for Nf < N-1 the ADS superpotential arises from monopole contributions to the path integral of the supersymmetric gauge theory compactified on R^3*S^1. These are the monopole effects leading to gaugino condensation and confinement of the low-energy SU(N-Nf) supersymmetric gauge theory. 
  A realization of E_{n+1} monopoles in string theory is given. The NS five brane stuck to an Orientifold eight plane is identified as the 't Hooft Polyakov monopole. Correspondingly, the moduli space of many such NS branes is identified with the moduli space of SU(2) monopoles. These monopoles transform in the spinor representation of an SO(2n) gauge group when n D8 branes are stacked upon the orientifold plane. This leads to a realization of E_{n+1} monopole moduli spaces. Charge conservation leads to a dynamical effect which does not allow the NS branes to leave the orientifold plane. This suggests that the monopole moduli space is smooth for n<8. Odd n>8 obeys a similar condition. Using a chain of dualities, we also connect our system to an Heterotic background with Kaluza-Klein monopoles. 
  D-brane probes, Hanany-Witten setups and geometrical engineering stand as a trichotomy of the currently fashionable techniques of constructing gauge theories from string theory. Meanwhile, asymptotic freedom, finitude and IR freedom pose as a trichotomy of the beta-function behaviour in quantum field theories. Parallel thereto is a trichotomy in set theory of finite, tame and wild representation types. At the intersection of the above lies the theory of quivers. We briefly review some of the terminology standard to the physics and to the mathematics. Then we utilise certain results from graph theory and axiomatic representation theory of path algebras to address physical issues such as the implication of graph additivity to finiteness of gauge theories, the impossibility of constructing completely IR free string orbifold theories and the unclassifiability of N<2 Yang-Mills theories in four dimensions. 
  We embed the Seiberg-Witten solution for the low energy dynamics of N=2 super Yang-Mills theory with an even number of massive hypermultiplets into the Whitham hierarchy. Expressions for the first and second derivatives of the prepotential in terms of the Riemann theta function are provided which extend previous results obtained by Gorsky, Marshakov, Mironov and Morozov. Checks in favour of the new equations involve both their behaviour under duality transformations and the consistency of their semiclassical expansions. 
  Using string field theory, we argue that the tachyon potential on a D-brane anti-D-brane system in type II string theory in arbitrary background has a universal form, independent of the boundary conformal field theory describing the brane. This implies that if at the minimum of the tachyon potential the total energy of the brane antibrane system vanishes in a particular background, then it vanishes in any other background. Similar result holds for the tachyon potential of the non-BPS D-branes of type II string theory, and the D-branes of bosonic string theory. 
  We study the effect of the Dp-brane gas in string cosmology. When one kind of Dp-brane gas dominates, we find that the cosmology is equivalent to that of the Brans-Dicke theory with the perfect fluid type matter. We obtain $\gamma$, the equation of state parameter, in terms of p and the space-time dimension. 
  In this paper we lay the foundations of causal quantum gravity (CQG), i.e. of a quantum theory of self-interacting symmetric massless rank-2 tensor gauge fields, the gravitons, on flat space-time, in the framework of causal perturbation theory. The causal inductive construction of the S-matrix for quantum gravity leads to a very satisfactory treatment of the ultraviolet problem. Here we concentrate on some main fundamental issues that concern the quantization of the gravitational interactions: the quantization of the free graviton field, the role of the fermionic ghost vector fields in preserving perturbative gauge invariance, the construction of the Fock space for the graviton and the consequent characterization of the subspace for the physical graviton states. This last point is necessary, together with perturbative gauge invariance, in order to prove unitary of the S-matrix restricted to the physical subspace. 
  Non-BPS dyon solutions to D3-brane actions are constructed when one or more scalar fields describing transverse fluctuations of the brane, are considered. The picture emerging from such non-BPS configurations is analysed, in particular the response of the D-brane-string system to small perturbations. 
  Using sigma-model approach, we study a class of coset spaces with torsion which compactify the D=26 closed bose-string theory. Requiring also that massless chiral fermions arise from the geometry/topology of coset space, we are left with the unique possibility: it implies D=14 subcritical dimensions and the isometry group G_2 X G_2. 
  Starting from the Fierz transform of the two-flavour 't Hooft interaction (a four-fermion Lagrangian with antisymmetric Lorentz tensor interaction terms augmented by an NJL type Lorentz scalar inetraction responsible for dynamical symmetry breaking and quark mass generation), we show that: (1) antisymmetric tensor Nambu-Goldstone bosons appear provided that the scalar and tensor couplings stand in the proportion of two to one, which ratio appears naturally in the Fierz transform of the two-flavour 't Hooft interaction; (2) non-Abelian vector gauge bosons coupled to this system acquire a non-zero mass. Axial-vector fields do not mix with antisymmetric tensor fields, so there is no mass shift there. 
  An algebraic method is devised to look for non-local symmetries of the pseudopotential type of nonlinear field equations. The method is based on the use of an infinite-dimensional subalgebra of the prolongation algebra $L$ associated with the equations under consideration. Our approach, which is applied by way of example to the Dym and the Korteweg-de Vries equations, allows us to obtain a general formula for the infinitesimal operator of the non-local symmetries expressed in terms of elements of $L$. The method could be exploited to investigate the symmetry properties of other nonlinear field equations possessing nontrivial prolongations. 
  We discuss the impact of quadratic quantum fluctuations on the Wilson loop extracted from classical string theory. We show that a large class of models, which includes the near horizon limit of D_p branes with 16 supersymmetries, admits a L\"{u}scher type correction to the classical potential. We confirm that the quantum determinant associated with a BPS configuration of a single quark in the AdS_5 \times S^5 model is free from divergences. We find that for the Wilson loop in that model, unlike the situation in flat space-time, the fermionic determinant does not cancel the bosonic one. For string models that correspond to gauge theories in the confining phase, we show that the correction to the potential is of a L\"{u}scher type and is attractive. 
  String or M-theory in the background of Kerr-AdS black holes is thought to be dual to the large n limit of certain conformal field theories on a rotating sphere at finite temperature. The five dimensional black hole is associated to N=4 supersymmetric Yang-Mills theory on a rotating three-sphere and the four dimensional one to the superconformal field theory of coinciding M2 branes on a rotating two-sphere. The thermodynamic potentials can be expanded in inverse powers of the radius of the sphere. We compute the leading and subleading terms of this expansion in the field theory at one loop and compare them to the corresponding supergravity expressions. The ratios between these terms at weak and strong coupling turns out not to depend on the rotation parameters in the case of N=4 SYM. For the field theory living on one M2 brane we find a subleading logarithmic term. No such term arises from the supergravity calculation. 
  It has been remarked in several previous works that the combination of center vortices and nexuses (a nexus is a monopole-like soliton whose world line mediates certain allowed changes of field strengths on vortex surfaces) carry topological charge quantized in units of 1/N for gauge group SU(N). These fractional charges arise from the interpretation of the standard topological charge integral as a sum of (integral) intersection numbers weighted by certain (fractional) traces. We show that without nexuses the sum of intersection numbers gives vanishing topological charge (since vortex surfaces are closed and compact). With nexuses living as world lines on vortices, the contributions to the total intersection number are weighted by different trace factors, and yield a picture of the total topological charge as a linking of a closed nexus world line with a vortex surface; this linking gives rise to a non-vanishing but integral topological charge. This reflects the standard 2\pi periodicity of the theta angle. We argue that the Witten-Veneziano relation, naively violating 2\pi periodicity, scales properly with N at large N without requiring 2\pi N periodicity. This reflects the underlying composition of localized fractional topological charge, which are in general widely separated. Some simple models are given of this behavior. Nexuses lead to non-standard vortex surfaces for all SU(N) and to surfaces which are not manifolds for N>2. We generalize previously-introduced nexuses to all SU(N) in terms of a set of fundamental nexuses, which can be distorted into a configuration resembling the 't Hooft-Polyakov monopole with no strings. The existence of localized but widely-separated fractional topological charges, adding to integers only on long distance scales, has implications for chiral symmetry breakdown. 
  A necessary condition that a St\"ackel-Killing tensor of valence 2 be the contracted product of a Killing-Yano tensor of valence 2 with itself is re-derived for a Riemannian manifold. This condition is applied to the generalized Euclidean Taub-NUT metrics which admit a Kepler type symmetry. It is shown that in general the St\"ackel-Killing tensors involved in the Runge-Lenz vector cannot be expressed as a product of Killing-Yano tensors. The only exception is the original Taub-NUT metric. 
  We show the existence of a renormalizable local supersymmetry for the gauge fixed action of the four dimensional antisymmetric tensor field model in a curved background quantized in a generalized axial gauge. By using the technique of the algebraic renormalization procedure, we prove the ultraviolet finiteness of the model to all orders of perturbation theory. 
  We develop in a systematic fashion the idea of gauging 1d-space translations with fixed Newtonian time for nonrelativistic matter (particles and fields). By starting with a nonrelativistic free theory we obtain its minimal gauge invariant extension by introducing two gauge fields with a Maxwellian self interaction. We fix the gauge so that the residual symmetry group is the Galilei group and construct a representation of the extended Galilei algebra. The reduced N-particle Lagrangian describes geodesic motion in a (N-1)-dimensional (Pseudo-) Riemannian space. The singularity of the metric for negative gauge coupling leads in classical dynamics to the formation of geometric bags in the case of two or three particles. The ordering problem within the quantization scheme for $N$-particles is solved by canonical quantization of a pseudoclassical Schroedinger theory obtained by adding to the continuum generalization of the point-particle Lagrangian an appropriate quantum correction. We solve the two-particle bound state problem for both signs of the gauge coupling. At the end we speculate on the possible physical relevance of the new interaction induced by the gauge fields. 
  I present, in any D$\geq$4, closed-form type B conformal anomaly effective actions incorporating the logarithmic scaling cutoff dependence that generates these anomalies. Their construction is based on a novel class of Weyl-invariant tensor operators. The only known type A actions in D$\geq$4 are extensions of the Polyakov integral in D=2; despite contrary appearances, we show that their nonlocality does not conflict with general anomaly requirements. They are, however, physically unsatisfactory, prompting a brief attempt at better versions. 
  A study of the partition function of a 3-dimensional scalar-vector model formally related via duality to the Rozansky-Witten topological sigma-model is presented. The partition function is shown to consist of such topological quantities of a 3-dimensional manifold M like a lattice sum, the Reidemeister-Ray-Singer torsion and the Massey product. 
  We consider the effective action for massive two-dimensional QED in flat Euclidean space-time in the background of a general square-integrable magnetic field with finite range. It is shown that its small mass limit is controlled by the chiral anomaly. New results for the low-energy scattering of electrons in 2+1 dimensions in static, inhomogenous magnetic fields are also presented. 
  We give a general decomposition of SU(N) connection and derive a generalized Skyrme-Faddeev action as the effective action of SU(N) QCD in the low energy limit. The result is obtained by separating the topological degrees which describes the non-Abelian monopoles from the dynamical degree of gauge potential, and integrating all the dynamical degrees of SU(N) QCD. 
  We consider the Non-Abelian Chern-Simons term coupled to external particles, in a gauge and diffeomorphism invariant form. The classical equations of motion are perturbativelly studied, and the on-shell action is shown to produce knot-invariants associated with the sources. The first contributions are explicitly calculated, and the corresponding knot-invariants are recognized. We conclude that the interplay between Knot Theory and Topological Field Theories is manifested not only at the quantum level, but in a classical context as well. 
  The possibility of neutral, brane-like solutions in a higher dimensional setting is discussed. In particular, we describe a supersymmetric solution in six dimensions which can be interpreted as a "three-brane" with a non-compact transverse space of finite volume. The construction can be generalized to n+4-dimensions and the result is a n+3-brane compactified on a n-1-dimensional Einstein manifold with a non-compact extra dimension. We find that there always exists a massless graviton trapped in four-dimensions while a bulk abelian gauge field gives rise to a unique four-dimensional massless photon. Moreover, all massless modes are accompanied by massive KK states and we show that it is possible in such a scenario the masses of the KK states to be at the TeV scale without hierarchically large extra dimensions. 
  We consider free superconformal theories of n=8 scalar multiplet in d=3 and (2,0) tensor multiplet in d=6 and compute 2-point and 3-point correlators of their stress tensors. The results for the 2-point and 3-point correlators for a single d=3 and d=6 multiplet differ from the "strong-coupling" AdS_4 and AdS_7 supergravity predictions by the factors $4\sqrt2 \over 3\pi}N^{3/2}$ and $4N^3$ respectively. These are the same factors as found earlier in hep-th/9703040 in the comparison of the brane free field theory and the 11-d supergravity predictions for the absorption cross-sections of longitudinally polarized gravitons by multiple M2 and M5 branes. While the correspondence of the results for the cross-sections and 2-point functions was expected on the basis of unitarity, the fact that the same coefficients appear in the ratio of the free-theory and supergravity 3-point functions is non-trivial. Thus, like in the d=4 SYM case, in both d=3 and d=6 theories the ratio of the 3-point and 2-point correlators <TTT>/<TT> is exactly the same in the free field theory and in the interacting CFT as described (to leading order in large $N$) by the 11-d supergravity on AdS_{d+1} x S^{10-d}. 
  We examine noncommutative solutions of the nonabelian theory on the world-volume of N coincident D-strings. These solutions can be interpreted in terms of noncommutative geometry as funnels describing the nonabelian D-string expanding out into an orthogonal D3-brane. These configurations are `dual' to the bion solutions in the abelian world-volume theory of the D3-brane. In the latter, a charge N magnetic monopole describes N D-strings attached to the D3-brane with a spike deformation of the world-volume. The noncommutative D-string solutions give a reliable account of physics at the core of the monopole, where the bion description is expected to breakdown. In the large N limit, we find good agreement between the two points of view, including the energy, couplings to background fields, and the shape of the funnel. We also study fluctuations traveling along the D-string, again obtaining agreement in the large N limit. At finite N, our results give a limit on the number of modes that can travel to infinity along the N D-strings attached to the D3-brane. 
  We study Type IIB supergravity in the presence of (euclidean) D3 branes and nonzero self-dual B-fields. We point out that the Einstein frame metric is identical to the full geometry for D3 branes without B fields turned on. Furthermore, in a decoupling limit in which the theory is conjectured to be dual to noncommutative Yang-Mills theory, the entire Einstein metric remains intact, and in particular, is asymptotically flat. We construct D-instanton solutions in this geometry. We show that in the decoupling limit the D-instanton action agrees with the action of the corresponding instanton in the noncommutative Yang-Mills theory and is expressed in terms of the open string coupling. Some other aspects of this correspondence, which have unusual features because the underlying metric is asymptotically flat, are explored. 
  Freed, Harvey, Minasian and Moore have proposed a mechanism to cancel the gravitational anomaly of the M-theory fivebrane coming from diffeomorphisms acting on the normal bundle. This procedure is based on a modification of the conventional M-theory Chern-Simons term. We compactify this space-time interaction to the ten-dimensional type IIA theory. We then analyze the relation to the anomaly cancellation mechanism for the type IIA fivebrane proposed by Witten. 
  Supersymmetric nonlinear sigma models are obtained from linear sigma models by imposing supersymmetric constraints. If we introduce auxiliary chiral and vector superfields, these constraints can be expressed by D-terms and F-terms depending on the target manifolds. Auxiliary vector superfields appear as gauge fields without kinetic terms. If there are no D-term constraints, the target manifolds are always non-compact manifolds. When all the degrees of freedom in these non-compact directions are eliminated by gauge symmetries, the target manifold becomes compact. All supersymmetric nonlinear sigma models, whose target manifolds are the hermitian symmetric spaces, are successfully formulated as gauge theories. 
  We perform canonical quantization of open strings in the $D$-brane background with a $B$-field. Treating the mixed boundary condition as a primary constraint, we get a set of secondary constraints. Then these constraints are shown to be equivalent to orbifold conditions to be imposed on normal string modes. These orbifold conditions are a generalization of the familiar orbifold conditions which arise when we describe open strings in terms of closed strings. Solving the constraints explicitly, we obtain a simple Hamiltonian for the open string, which reveals the nature of noncommutativity transparently. 
  The irreducible antifield formalism for $p$-form gauge theories with gauge invariant interaction terms is exposed. The ghosts of ghosts do not appear. The acyclicity of the Koszul-Tate operator is ensured without introducing antifields at resolution degrees higher that two. 
  Using the Chern-Simon formulation of (2+1) gravity, we derive, for the general asymptotic metrics given by the Fefferman-Graham-Lee theorems, the emergence of the Liouville mode associated to the boundary degrees of freedom of (2+1) dimensional anti de Sitter geometries. 
  p-form gauge theories with Stueckelberg coupling are quantized in an irreducible antifield-BRST way. As a consequence, neither the ghosts of ghosts nor their antifields appear. Some irreducible gauge conditions are inferred naturally within our formalism. In the end we briefly discuss the interacting case. 
  Spin-5/2 gauge fields are quantized in an irreducible way within both the BRST and BRST-anti-BRST manners. To this end, we transform the reducible generating set into an irreducible one, such that the physical observables corresponding to these two formulations coincide. The gauge-fixing procedure emphasizes on the one hand the differences among our procedure and the results obtained in the literature, and on the other hand the equivalence between our BRST and BRST-anti-BRST approaches. 
  The vortex picture of confinement is studied. The deconfinement phase transition is explained as a transition from a phase in which vortices percolate to a phase of small vortices. Lattice results are presented in support of this scenario. Furthermore the topological properties of magnetic monopoles and center vortices arising, respectively, in Abelian and center gauges are studied in continuum Yang-Mills-theory. For this purpose the continuum analog of the maximum center gauge is constructed. 
  We present a general description of two mixed branes interactions. For this we consider two mixed branes with dimensions p_1 and p_2, in external field B_{\mu\nu} and arbitrary gauge fields A^1_{\alpha_1} and A^2_{\alpha_2} on the world volume of them, in spacetime in which some of its directions are compactified on circles with different radii. Some examples are considered to clear these general interactions. Finally contribution of the massless states on the interactions is extracted. Closed string with mixed boundary conditions and boundary state formalism, provide useful tools for calculation of these interactions. 
  This is a brief review of the current state of knowledge on "little string theories", which are non-gravitational theories having several string-like properties. We focus on the six dimensional maximally supersymmetric "little string theories" and describe their definition, some of their simple properties, the motivations for studying them, the DLCQ and holographic constructions of these theories and their behaviour at finite energy density. (Contribution to the proceedings of Strings '99 in Potsdam, Germany.) 
  Self-gravitating SU(2) Higgs magnetic monopoles exist up to a critical value of the ratio of the vector meson mass to the Planck mass, which depends on the Higgs boson mass. At the critical value a critical solution with a degenerate horizon is reached. As pointed out by Lue and Weinberg, there are two types of critical solutions, with a transition at an intermediate Higgs boson mass. Here we investigate this transition for black holes, and reconsider it for the case of gravitating monopoles. 
  We study the spectrum of SU(2) x SO(2) matrix supersymmetric quantum mechanics. We use angular coordinates that allow us to find an explicit solution of the Gauss law constraints and single out the quantum number n (the Lorentz angular momentum). Energy levels are four-fold degenerate with respect to n and are labeled by n_q, the largest n in a quartet. The Schr\"odinger equation is reduced to two different systems of two-dimensional partial differential equations. The choice of a system is governed by n_q. We present the asymptotic solutions for the systems deriving thereby the asymptotic formula for the spectrum. Odd n_q are forbidden, for even n_q the spectrum has a continuous part as well as a discrete one, meanwhile for half-integer n_q the spectrum is purely discrete. Taking half-integer n_q one can cure the model from instability caused by the presence of continuous spectrum. 
  Despite the nice geometrical properties of higher dimensional Chern-Simons (CS) supergravity theories these actions suffer from one major drawback, namely, their connection with the real world. After some quick remarks on three-dimensional gravity, we consider five-dimensional CS supergravity and study to what extend this theory reproduces the standard low energy description of gravitons and gauge fields. We point out that if one deforms the CS action by changing the value of the cosmological constant by a small amount (thus breaking the CS symmetry), propagation around AdS becomes non-trivial, asymptotically (AdS) Schwarzschild solutions exist, and the gauge field acquires its standard quadratic propagator. 
  In the frame of the scalar theory $g \phi ^{4}$, we explore the occurrence of thermal renormalons, i. e. temperature dependent singularities in the Borel plane. The discussion of a particular renormalon type diagram at finite temperature, using Thermofield Dynamics, allows us to establish that these singularities actually get a temperature dependence. This dependence appears in the residues of the poles, remaining their positions unchanged with temperature. 
  We generalize the local surface counterterm prescription suggested in Einstein gravity for higher derivative (HD) and Weyl gravities. Explicitly, the surface counterterm is found for three- and five-dimensional HD gravities. As a result, the gravitational action for asymptotically AdS spaces is finite and gravitational energy-momentum tensor is well-defined. The holographic trace anomaly for d2 and d4 boundary (gauge) QFT dual to above HD gravity is calculated from gravitational energy-momentum tensor. The calculation of AdS black hole mass in HD gravity is presented within above prescrition. The comparison with the standard prescription (using reference spacetime) is done. 
  D0-branes moving in a constant antisymmetric C field are found to be described by quantum mechanics of the supersymmetric matrix model with a similarity transformation. Sometimes this similarity transformation is singular or ill-defined and cannot be ignored. As an example, when there are non-vanishing C_{-ij} components, we obtain the theory for Dp-branes which is effectively the noncommutative super Yang-Mills theory. We also briefly discuss the effects of other non-vanishing components such as C_{+ij} and C_{ijk}. 
  The relevance of calibrations, and `generalized' calibrations, to supersymmetric M-brane configurations, and their associated field theories, is reviewed, with emphasis on applications to domain walls and domain wall junctions of D=4 N=1 supersymmetric field theories. 
  We employ the Dirac-like equation for the gauge field proposed by Majorana to obtain an action that is symmetric under duality transformation. We also use the equivalence between duality and chiral symmetry in this fermion-like formulation to show how the Maxwell action can be seen as a mass term. 
  We study the constraints on five-dimensional N=1 heterotic M-theory imposed by a consistent anomaly-free coupling of bulk and boundary theory. This requires analyzing the cancellation of triangle gauge anomalies on the four-dimensional orbifold planes due to anomaly inflow from the bulk. We find that the semi-simple part of the orbifold gauge groups and certain U(1) symmetries have to be free of quantum anomalies. In addition there can be several anomalous U(1) symmetries on each orbifold plane whose anomalies are cancelled by a non-trivial variation of the bulk vector fields. The mixed U(1) non-abelian anomaly is universal and there is at most one U(1) symmetry with such an anomaly on each plane. In an alternative approach, we also analyze the coupling of five-dimensional gauged supergravity to orbifold gauge theories. We find a somewhat generalized structure of anomaly cancellation in this case which allows, for example, non-universal mixed U(1) gauge anomalies. Anomaly cancellation from the perspective of four-dimensional N=1 effective actions obtained from E_8xE_8 heterotic string- or M-theory by reduction on a Calabi-Yau three-fold is studied as well. The results are consistent with the ones found for five-dimensional heterotic M-theory. Finally, we consider some related issues of phenomenological interest such as model building with anomalous U(1) symmetries, Fayet-Illiopoulos terms and threshold corrections to gauge kinetic functions. 
  The Fischler-Susskind entropy bound has been studied in (2+1)-dimensional universes with negative cosmological constant. As in all contracting universes, that bound is not satisfied. Furthermore, we found that the Fischler-Susskind bound is not compatible with a generalized second law of thermodynamics in (2+1)-dimensional cosmology, neither the classical nor the quantum version. On the other hand, the Hubble entropy bound has been constructed in (2+1)-dimensional cosmology and it is shown compatible with the generalized second law of thermodynamics. 
  We compute the one-loop divergences of D=10, N=1 supergravity and of its reduction to D=8. We study the tensor structure of the counterterms appearing in D=8 and D=10 and compare these to expressions previously found in the low energy expansion of string theory. The infinities have the primitive Yang-Mills tree amplitude as a common factor. 
  We discuss three-dimensional $ \lambda\phi^4+\eta\phi^6 $ theory in the context of the 1/N expansion at finite temperature. We use the method of the composite operator (CJT) for summing a large sets of Feynman graphs. We analyse the behavior of the thermal square mass and the thermal coupling constant in the low and high temperature limit. The existent of the tricritical point at some temperature is found using this non-pertubative method. 
  In the present paper we study SU(N)->S(U(N/2)xU(N/2)) symmetry breaking in N=4 SYM via AdS/CFT correspondence. We consider two stacks of N/2 parallel D3 branes separated by a distance 2\vec d. In this case there is mixing between the different l-wave dilatonic KK modes. We calculate certain the two point correlation functions in the dual gauge theory. Due to mode mixing, the diagonal correlation functions have 1/N conformal-like correction as well as deformation terms. The off-diagonal correlators are also non-vanishing and their leading order is 1/N. We discuss briefly the spectrum of the glueball exitations. 
  We study brane configurations that give rise to large-N gauge theories with eight supersymmetries and no hypermultiplets. These configurations include a variety of wrapped, fractional, and stretched branes or strings. The corresponding spacetime geometries which we study have a distinct kind of singularity known as a repulson. We find that this singularity is removed by a distinctive mechanism, leaving a smooth geometry with a core having an enhanced gauge symmetry. The spacetime geometry can be related to large-N Seiberg-Witten theory. 
  Generalized Yang-Mills theories have a covariant derivative that employs both scalar and vector bosons. Here we show how grand unified theories of the electroweak and strong interactions can be constructed with them. In particular the SU(5) GUT can be obtained from SU(6) with SU(5)xU(1) as a maximal subgroup. The choice of maximal subgroup also determines the chiral structure of the theory. The resulting Lagrangian has only two terms, and only two irreducible representations are needed, one for fermions and another for bosons. 
  We discuss the relation between singletons in AdS_3 and logarithmic operators in the CFT on the boundary. In 2 dimensions there can be more logarithmic operators apart from those which correspond to singletons in AdS, because logarithmic operators can occur when the dimensions of primary fields differ by an integer instead of being equal. These operators may be needed to account for the greybody factor for gauge bosons in the bulk. 
  There appears to be no natural explanation for the cosmological constant's small size within the framework of local relativistic field theories. We argue that the recently-discussed framework for which the observable universe is identified with a p-brane embedded within a higher-dimensional `bulk' spacetime, has special properties that may help circumvent the obstacles to this understanding. This possibility arises partly due to several unique features of the brane proposal. These are: (1) the potential such models introduce for partially breaking supersymmetry, (2) the possibility of having low-energy degrees of freedom which are not observable to us because they are physically located on a different brane, (3) the fundamental scale may be much smaller than the Planck scale. Furthermore, although the resulting cosmological constant in the scenarios we outline is naturally suppressed by weak coupling constants of gravitational strength, it need not be exactly zero, raising the possibility it could be in the range favoured by recent cosmological observations. 
  New exact solutions of brane-world cosmology are given. These solutions include an arbitrary constant C, which is determined by the geometry outside the brane and which affects the cosmological evolution in the brane-world. If C is zero, then the standard cosmology governs the brane-world as a low-energy effective cosmological theory. However, if C is not zero, then even in low-energy the brane-world cosmology gives predictions different from the standard one. The difference can be understood as ``dark radiation'', which is not real radiation but alters cosmological evolutions. 
  We consider the modified superfield constraints with constant terms for the D=3, N=2 Goldstone-Maxwell gauge multiplet which contains Goldstone fermions, real scalar and vector fields. The partial spontaneous breaking N=2 to N=1 is possible for the non-minimal self-interaction of this modified gauge superfield including the linear Fayet-Iliopoulos term. The dual description of the partial breaking in the model of the self-interacting Goldstone chiral superfield is also discussed. 
  The finiteness requirement for Euclidean Einstein gravity is shown to be so stringent that only the flat metric is allowed. We examine counterterms in 4D and 6D Ricci-flat manifolds from general invariance arguments. 
  It is found that the existence of spacetime foam leads to a situation in which the number of fundamental quantum bosonic fields is a variable quantity. The general aspects of an exact theory that allows for a variable number of fields are discussed, and the simplest observable effects generated by the foam are estimated. It is shown that in the absence of processes related to variations in the topology of space, the concept of an effective field can be reintroduced and standard field theory can be restored. However, in the complete theory the ground state is characterized by a nonvanishing particle number density. From the effective-field standpoint, such particles are "dark". It is assumed that they comprise dark matter of the universe. The properties of this dark matter are discussed, and so is the possibility of measuring the quantum fluctuation in the field potentials. 
  General spinning brane bound states are constructed, along with their near-horizon limits which are relevant as dual descriptions of non-commutative field theories. For the spinning D-brane world volume theories with a B-field a general analysis of the gauge coupling phase structure is given, exhibiting various novel features, already at the level of zero angular momenta. We show that the thermodynamics is equivalent to the commutative case at large N and we discuss the possibility and consequences of finite N. As an application of the general analysis, the range of validity of the thermodynamics for the NCSYM is discussed. In view of the recently conjectured existence of a 7-dimensional NCSYM, the thermodynamics of the spinning D6-brane theory, for which a stable region can be found, is presented in detail. Corresponding results for the spinning M5-M2 brane bound state, including the near-horizon limit and thermodynamics, are given as well. 
  SU(N) Yang-Mills integrals form a new class of matrix models which, in their maximally supersymmetric version, are relevant to recent non-perturbative definitions of 10-dimensional IIB superstring theory and 11-dimensional M-theory. We demonstrate how Monte Carlo methods may be used to establish important properties of these models. In particular we consider the partition functions as well as the matrix eigenvalue distributions. For the latter we derive a number of new exact results for SU(2). We also report preliminary computations of Wilson loops.   (Based on talk presented by M. Staudacher at Strings '99, Potsdam, July 19-24 1999) 
  The structure constants for Moyal brackets of an infinite basis of functions on the algebraic manifolds M of pseudo-unitary groups U(N_+,N_-) are provided. They generalize the Virasoro and W_\infty algebras to higher dimensions. The connection with volume-preserving diffeomorphisms on M, higher generalized-spin and tensor operator algebras of U(N_+,N_-) is discussed. These centrally-extended, infinite-dimensional Lie-algebras provide also the arena for non-linear integrable field theories in higher dimensions, residual gauge symmetries of higher-extended objects in the light-cone gauge and C^*-algebras for tractable non-commutative versions of symmetric curved spaces. 
  We explicitly show the area law behavior of a circular Wilson loop in confining theories from supergravity. We calculate the correlator of two Wilson loops from supergravity in confining backgrounds. We find that it is dominated by an exchange of a ``scalarball'' that is lighter than the glueballs. We interpret these results in terms of the meson-meson potential in such theories. 
  We present a description of the type IIB NS-NS p-branes in terms of topological solitons in systems of spacetime-filling brane-antibrane pairs. S-duality implies that these spacetime-filling branes are NS9-branes, S-dual to the D9-branes of the type IIB theory. The possible vortex-like solutions in an NS9,anti-NS9 configuration are identified by looking at its worldvolume effective action. Finally we discuss the implications of these constructions in the description of BPS and non-BPS states in the strongly coupled Heterotic SO(32) theory. 
  Using a nonlinear electrodynamics coupled to General Relativity a new regular exact black hole solution is found. The nonlinear theory reduces to the Maxwell one in the weak limit, and the solution corresponds to a charged black hole for |q| \leq 2s_c m \approx 1.05 m, with metric, curvature invariants, and electric field regular everywhere. 
  We show that the De Donder-Weyl (DW) covariant Hamiltonian field equations of any field can be written in Duffin-Kemmer-Petiau (DKP) matrix form. As a consequence, the (modified) DKP beta-matrices (5 X 5 in four space-time dimensions) are of universal significance for all fields admitting the DW Hamiltonian formulation, not only for a scalar field, and can be viewed as field theoretic analogues of the symplectic matrix, leading to the so-called ``k-symplectic'' (k=4) structure. We also briefly discuss what could be viewed as the covariant Poisson bracket given by beta-matrices and the corresponding Poisson bracket formulation of DW Hamiltonian field equations. 
  A short rewiev of covariant quantization methods based on BRST-antiBRST symmetry is given. In particular problems of correct definition of Sp(2) symmetric quantization scheme known as triplectic quantization are considered. 
  We study the thermodynamics of the 2+1 dimensional Gross-Neveu model in the presence of a chemical potential by introducing a representation for the canonical partition function which encodes both real and imaginary chemical potential cases. It is pointed out that the latter case probes the thermodynamics of the possible anyon-like excitations in the spectrum. It is also intimately connected to the breaking of the discrete Z-symmetry of a U(1) gauge theory coupled to the Gross-Neveu model at finite temperature, which we interpret as signaling anyon deconfinement. Finally, the chiral properties of the model in the presence of an imaginary chemical potential are discussed and analytical results for the free-energy density at the transition points are presented. 
  Extending previous work on $a_2^{(1)}$, we present a set of reflection matrices, which are explicit solutions to the $a_n^{(1)}$ boundary Yang-Baxter equation. Unlike solutions found previously these are multiplet-changing $K$-matrices, and could therefore be used as soliton reflection matrices for affine Toda field theories on the half-line. 
  We calculate the Casimir force between two parallel plates if the boundary conditions for the photons are modified due to presence of the Chern-Simons term. We show that this effect should be measurable within the present experimental technique. 
  The theory of SU(2) gauged seven-dimensional supergravity is obtained by compactifying ten dimensional N=1 supergravity on the group manifold SU(2). 
  In this paper a general solution is found for a five dimensional orbifold spacetime that induces a $k=0$ cosmology on a three-brane. Expressions for the energy density and pressure on the brane in terms of the brane metric are derived. Given a metric on the brane it is possible to find five dimensional spacetimes that contain the brane. This calculation is carried out for an inflationary universe and for a metric that corresponds to a radiation dominated universe in standard cosmology. It is also shown that any $k=0$ cosmology can be embedded in a flat five dimensional orbifold spacetime and the equation of the three-brane surface is derived. For an inflationary universe it is shown that the surface is the usual hyperboloid representation of de Sitter space, although it is embedded in an orbifold spacetime. 
  We construct propagators in Euclidean AdS(d+1) space-time for massive p-forms and massive symmetric tensors. 
  The classical cross section for low energy absorption of the RR-scalar by a stack of noncommutative D3-branes in the large NS B-field limit is calculated. In the spirit of AdS/CFT correspondence, this cross section is related to two point function of a certain operator in noncommutative Yang-Mills theory. Compared at the same gauge coupling, the result agrees with that of obtained from ordinary D3-branes. This is consistent with the expectation that ordinary and noncommutative Yang-Mills theories are equivalent below the noncommutativity scale, but it is a nontrivial prediction above this scale. 
  We investigate Yang-Mills instantons on a 7-dimensional manifold of G_2 holonomy. By proposing a spherically symmetric ansatz for the Yang-Mills connection, we have ordinary differential equations as the reduced instanton equation, and give some explicit and numerical solutions. 
  A macroscopic universe may emerge naturally from a Planck cell fluctuation by unfolding through a stage of exponential expansion towards a homogeneous cosmological background. Such primordial inflation requires a large and presumably infinite degeneracy at the Planck scale, rooted in the unbounded negative gravitational energy stored in conformal classes. This complex Planck structure is consistent with a quantum tunneling description of the transition from the Planck scale to the inflationary era and implies, in the limit of vanishing Planck size, the Hartle-Hawking no-time boundary condition. On the other hand, string theory give credence to the holographic principle and the concomitant depletion of states at the Planck scale. The apparent incompatibility of primordial inflation with holography either invalidates one of these two notions or relegates the nature of the Planck size outside the realm of quantum physics, as we know it. 
  We explore quantum states of instanton solitons in five dimensional noncommutative Yang-Mills theories. We start with maximally supersymmetric U(N) theory compactified on a circle S^1, and derive the low energy dynamics of instanton solitons, or calorons, which is no longer singular. Quantizing the low energy dynamics, we find N physically distinct ground states with a unit Pontryagin number and no electric charge. These states have a natural D-string interpretation. The conclusion remains unchanged as we decompactify S^1, as long as we stay in the Coulomb phase by turning on adjoint Higgs expectation values. 
  These lecture notes are a brief introduction to Wess-Zumino-Witten models, and their current algebras, the affine Kac-Moody algebras. After reviewing the general background, we focus on the application of representation theory to the computation of 3-point functions and fusion rules. 
  No positive result has been obtained on the magnetic monopoles search. This allows to consider different theoretical approaches as the proposed in this paper, developed in the framework of the Einstein General Relativity. The properties of second rank skew-symmetrical fields are the basis of electromagnetic theories. In the space-time the Hodge duality of these fields is narrowly related with the rotations in the SO(2) group. An axiomatic approach to a dual electromagnetic theory is presented. The main result of this paper is that the stress-energy tensor can be decomposed on two parts: the parallel and the perpendicular. The parallel part is easily integrated on the Lagrangian approach, while some problems appears with the perpendicular part. A solution with the parallel part alone is found, it generates a non-standard model of magnetic monopoles neutral to the electric charges. 
  We show how to determine the unknown functions arising when the peeling decomposition is applied to multi-critical matter coupled to two-dimensional quantum gravity and compute the loop-loop correlation functions. The results that $\eta=2+2/(2K-3)$ and $\nu=1-3/2K$ agree with the slicing decomposition, and satisfy Fisher scaling. 
  We study supersymmetric orientifolds where the world-sheet parity transformation is combined with a conjugation of some compact complex coordinates. We investigate their T-duality relation to standard orientifolds and discuss the origin of continuous and discrete moduli. In contrast to standard orientifolds, the antisymmetric tensor describes a continuous deformation, while the off-diagonal part of the metric is frozen to quantized values and is responsible for the rank reduction of the gauge group. We also give a geometrical interpretation of some recently constructed six-dimensional permutational orientifolds. 
  In this paper we first give arguments supporting the idea that a B.T.Z black hole can face a transplankian problem even when its mass is small. K.M.M quantum theory is applied to the Hawking evaporation of the Schwarzchild and B.T.Z black holes. Working in the physically safe quasi position representation, we argue that the oscillating term present in a previous analysis is removed so that actually one doesn't need an average procedure. We expand the s wave function as the exponential of a series in the minimal length of the new quantum theory. This reduces an infinite order differential equation to a numerable set of finite order ones. We obtain the striking result that the infinity of arbitrary constants induced by the order of the wave equation has no physical meaning due to normalization. We finally construct gaussian wave pacqets and study their trajectories. We suggest a quantitative description of the non locality zone and its dependance on the K.M.M energy scale. Potential incidences on unitarity are briefly evoqued. 
  The study of the scaling limit of two-dimensional models of statistical mechanics within the framework of integrable field theory is illustrated through the example of the RSOS models. Starting from the exact description of regime III in terms of colliding particles, we compute the correlation functions of the thermal, $\phi_{1,2}$ and (for some cases) spin operators in the two-particle approximation. The accuracy obtained for the moments of these correlators is analysed by computing the central charge and the scaling dimensions and comparing with the exact results. We further consider the (generally non-integrable) perturbation of the critical points with both the operators $\phi_{1,3}$ and $\phi_{1,2}$ and locate the branches solved on the lattice within the associated two-dimensional phase diagram. Finally we discuss the fact that the RSOS models, the dilute $q$-state Potts model at and the O(n) vector model are all described by the same perturbed conformal field theory. 
  We calculate the effect of noncommutative spacetime on the greybody factor on the supergravity side. For this purpose we introduce a system of D3-branes with a constant NS $B$-field along their world volume directions ($x_2, x_3$). Considering the propagation of minimally coupled scalar with non-zero momentum along($x_2, x_3$), we derive an exact form of the greybody factor in $B$ field. It turns out that $\sigma^{B\ne0}_l > \sigma^{B=0}_l$. This means that the presence of $B$-field (the noncommutativity) suppresses the potential barrier surrounding the black hole. As a result, it comes out the increase of greybody factor. 
  We review the relations that have been found between multi-loop scattering amplitudes in gauge theory and gravity, and their implications for ultraviolet divergences in supergravity. 
  Magnetic BPS string solutions preserving quarter of supersymmetry are obtained for all abelian gauged d=5 N=2 supergravity theories coupled to vector supermultiplets. Due to a ``generalised Dirac quantization'' condition satisfied by the minimized magnetic central charge, the string metric takes a universal form for all five dimensional gauged theories. 
  The renormalization group method is applied to the three-loop effective potential of the massive $\phi^4$ theory in the $\bar{\rm MS}$ scheme in order to obtain the next-next-next-to-leading logarithm resummation. For this, we exploit four-loop parts of the renormalization group functions $\beta_\lambda$, $\gamma_m$, $\gamma_\phi$, and $\beta_\Lambda$, which were already given to five-loop order via the renormalization of the zero-, two-, and four-point one-particle-irreducible Green's functions, to solve evolution equations for the parameters $\lambda$, $m^2$, $\phi$, and $\Lambda$ within the accuracy of the three-loop order. 
  In an earlier paper, Alvarez, Alvarez-Gaume, Barbon and Lozano pointed out, that the only way to "flatten" negative curvature by means of a T-duality is by introducing an appropriate, non-constant NS-NS B-field. In this paper, we are investigating this further and ask, whether it is possible to T-dualize AdS_d space to flat space with some suitably chosen B. To answer this question, we derive a relationship between the original curvature tensor and the one of the T-dualized metric involving the B-field. It turns out that there is one particular component, which is independent of B. By inspection of this component, we then show, that it is not possible to dualize AdS_d to flat space irrespective of the choice of B. Finally, we examine the extension of AdS to an AdS_5 x S^5 geometry and propose a chain of S- and T-dualities together with an SL(2,Z) coordinate transformation, leading to a dual D9-brane geometry. 
  The effective Hamiltonian introduced many years ago by Bloch and generalized later by Wilson, appears to be the ideal starting point for Hamiltonian perturbation theory in quantum field theory. The present contribution derives the Bloch-Wilson Hamiltonian from a generalization of the Gell-Mann-Low theorem, thereby enabling a diagrammatic analysis of Hamiltonian perturbation theory in this approach. 
  Near-horizon geometry of coincident M2-branes at a conical singularity is related to M-theory on AdS4 times an appropriate seven-dimensional manifold X7. For X_7=N^{0,1,0}, squashing deformation is known to lead to spontaneous (super) symmetry breaking from N=(3, 0) to N=(0, 1) in gauged AdS4 supergravity. Via AdS/CFT correspondence, it is interpreted as renormalization group flow of strongly coupled three-dimensional field theory with SU(3)*SU(2) global symmetry. The flow interpolates between N=(0,1) fixed point in the UV to N=(3,0)fixed point in the IR. Evidences for the interpretation are found both from critical points of the supergravity scalar potential and from conformal dimension of relevant chiral primary operators at each fixed point. We also analyze cases with X7=SO(5)/SO(3)_{max}, V_{5,2}(R), M^{1,1,1}, Q^{1,1,1} and find that there is no nontrivial renormalization group flows. We extend the analysis to Englert type vacua of M-theory. By analyzing de Wit-Nicolai potential, we find that deformation of S7 gives rise to renormalization group flow from N=8, SO(8) invariant UV fixed point to N=1, G_2 invariant IR fixed point. For AdS_7 supergravity relevant for near-horizon geometry of coincident M5-branes, we also point out a nontrivial renormalization group flow from N=1 superconformal UV fixed point to non-supersymmetric IR fixed point. 
  The purpose of this short review is to present progresses in string theory in the recent past. There have been very important developments in our understanding of string dynamics, especially in the nonperturbative aspects. In this context, dualities play a cardinal role. The string theory provides a deeper understanding of the physics of special class of black holes from a microscopical point of view and has resolved several important questions. It is also recognized that M-theory provides a unified description of the five perturbatively consistent string theories. The article covers some of these aspects and highlights important progress made in string theory. 
  We consider a supersymmetric extension of the SL(2;Z)-covariant D3-brane action proposed by Nurmagambetov, and prove its kappa-symmetry in an on-shell type-IIB supergravity background. 
  A string background, which is in some precise sense {\em universal} (i.e., incorporating all orders in the Feynman diagram expansion), is proposed to represent pure gauge theories. S-duality at the level of the string metric is considered as well as the vacuum expectation values of 't Hooft and Wilson loops in semiclassical approximation. 
  In this paper we study T-duality for open strings ending on branes with non-zero B-field on them from the point of view of canonical transformations. For the particular case of type II strings on the two torus we show that the $Sl(2,Z)_N$ transformations can be understood as a sub-class of canonical transformations on the open strings in the B-field background. 
  We critically examine a recent suggestion that "ambiguous" statistics is equivalent to infinite quon statistics and that it describes a dilute, nonrelativistics ideal gas of extremal black holes. We show that these two types of statistics are different and that the description of extremal black holes in terms of "ambiguous" statistics cannot be applied. 
  We review physical motivations and possible realizations of string vacua with large internal volume and/or low string scale and discuss the issue of supersymmetry breaking. In particular, we describe the key features of Scherk-Schwarz deformations in type I models and conclude by reviewing the phenomenon of ``brane supersymmetry breaking'': the tadpole conditions of some type-I models require that supersymmetry be {\it broken at the string scale} on a collection of branes, while being exact, to lowest order, in the bulk and on other branes. 
  Supersymmetric configurations of type II D-branes with nonzero gauge field strengths in general supersymmetric backgrounds with nonzero B fields are analyzed using the kappa-symmetric worldvolume action. It is found in dimension four or greater that the usual instanton equation for the gauge field obtains a nonlinear deformation. The deformation is parameterized by the topological data of the B-field, the background geometry and the cycle wrapped by the brane. In the appropriate dimensions, limits and settings these equations reduce to deformed instanton equations recently found in the context of noncommutative geometry as well as those following from Lagrangians based on Bott-Chern forms. We further consider instantons comprised of M5-branes wrapping a Calabi-Yau space with non-vanishing three-form field strengths. It is shown that the instanton equations for the three-form are related to the Kodaira-Spencer equations. 
  The complete two-loop correction to the quark propagator, consisting of the spider, rainbow, gluon bubble and quark bubble diagrams, is evaluated in the noncovariant light-cone gauge (lcg). (The overlapping self-energy diagram had already been computed.) The chief technical tools include the powerful matrix integration technique, the n^*-prescription for the spurious poles of 1/qn, and the detailed analysis of the boundary singularities in five- and six-dimensional parameter space. It is shown that the total divergent contribution to the two-loop correction Sigma_2 contains both covariant and noncovariant components, and is a local function of the external momentum p, even off the mass-shell, as all nonlocal divergent terms cancel exactly. Consequently, both the quark mass and field renormalizations are local. The structure of Sigma_2 implies a quark mass counterterm of the form $\delta m (lcg) = m\tilde\alpha_s C_F(3+\tilde\alpha_sW) + {\rm O} (\tilde\alpha_s^3)$, $\tilde\alpha_s = g^2\Gamma(\eps)(4\pi)^{\eps -2}$, with W depending only on the dimensional regulator epsilon, and on the numbers of colors and flavors. It turns out that \delta m(lcg) is identical to the mass counterterm in the general linear covariant gauge. Our results are in agreement with the Bassetto-Dalbosco-Soldati renormalization scheme. 
  Two different scenarios (light-front and equal-time) are possible for Yang-Mills theories in two dimensions. The exact $\bar q q$-potential can be derived in perturbation theory starting from the light-front vacuum, but requires essential instanton contributions in the equal-time formulation. In higher dimensions no exact result is available and, paradoxically, only the latter formulation (equal-time) is acceptable, at least in a perturbative context. 
  We find monopole solutions for a spontaneously broken SU(2)-Higgs system coupled to gravity in asymptotically anti-de Sitter space. We present new analytic and numerical results discussing,in particular, how the gravitational instability of self-gravitating monopoles depends on the value of the cosmological constant. 
  We consider null bosonic p-branes moving in curved space-times and develop a method for solving their equations of motion and constraints, which is suitable for string theory backgrounds. As an application, we give an exact solution for such background in ten dimensions. 
  We evaluate the coefficients of the leading poles of the complete two-loop quark self-energy \Sigma(p) in the Coulomb gauge. Working in the framework of split dimensional regularization, with complex regulating parameters \sigma and n/2-\sigma for the energy and space components of the loop momentum, respectively, we find that split dimensional regularization leads to well-defined two-loop integrals, and that the overall coefficient of the leading pole term for \Sigma(p) is strictly local. Extensive tables showing the pole parts of one- and two-loop Coulomb integrals are given. We also comment on some general implications of split dimensional regularization, discussing in particular the limit \sigma \to 1/2 and the subleading terms in the epsilon-expansion of noncovariant integrals. 
  We construct operator representation of Moyal algebra in the presence of fermionic fields. The result is used to describe the matrix model in Moyal formalism, that treat gauge degrees of freedom and outer degrees of freedom equally. 
  Based on the rational R-matrix of the supersymmetric sl(2,1) matrix difference equations are solved by means of a generalization of the nested algebraic Bethe ansatz. These solutions are shown to be of highest-weight with respect to the underlying graded Lie algebra structure. 
  We present an extremely simple solution to the renormalization of quantum electrodynamics based on Epstein-Glaser approach to renormalization theory. 
  The renormalization group approach towards the string representation of non abelian gauge theories translates, in terms of the string sigma model beta function equations, the renormalization group evolution of the gauge coupling constant and Zamolodchikov`s $c$ function. Tachyon stability, glueball mass gap, renormalization group evolution of the $c$ function and the area law for the Wilson loop are studied for a critical bosonic string vacuum corresponding to a non abelian gauge theory in four dimensional space-time. We prove that the same intrinsic geometry for the string vacuum is universal in some sense, reproducing the Yang-Mills beta function to arbitrary loop order in perturbation theory. 
  We compute the spectrum and several critical amplitudes of the two dimensional Ising model in a magnetic field with the transfer matrix method. The three lightest masses and their overlaps with the spin and the energy operators are computed on lattices of a width up to L=21. In extracting the continuum results we also take into account the corrections to scaling due to irrelevant operators. In contrast with previous Monte Carlo simulations our final results are in perfect agreement with the predictions of S-matrix and conformal field theory. We also obtain the amplitudes of some of the subleading corrections, for which no S-matrix prediction has yet been obtained. 
  Bianchi type I and type IX ('Mixmaster') geometries are investigated within the framework of Ho\v{r}ava-Witten cosmology. We consider the models for which the fifth coordinate is a $S^1/Z_2$ orbifold while the four coordinates are such that the 3-space is homogeneous and has geometry of Bianchi type I or IX while the rest six dimensions have already been compactified on a Calabi-Yau space. In particular, we study Kasner-type solutions of the Bianchi I field equations and discuss Kasner asymptotics of Bianchi IX field equations. We are able to recover the isotropic 3-space solutions found by Lukas {\it et al}. Finally, we discuss if such Bianchi IX configuration can result in chaotic behaviour of these Ho\v{r}ava-Witten cosmologies. 
  We study some aspects of dilatonic domain walls in relation to the idea on the noncompact internal space. We find that the warp factor in the spacetime metric increases as one moves away from the domain wall for all the supersymmetric dilatonic domain wall solutions obtained from the (intersecting) BPS branes in string theories through toroidal compactifications, unlike the case of the Randall-Sundrum model. On the other hand, when the dilaton coupling parameter a for the D-dimensional extreme dilatonic domain wall takes the values |a|<2/(D-2), the Kaluza-Klein spectrum of graviton has the same structure as that of the Randall-Sundrum model (and the warp factor decreases in the finite interval around the dilatonic domain wall), thereby implying the possibility of extending the Randall-Sundrum model to the |a|<2/(D-2) case. We construct fully localized solutions describing extreme dilatonic branes within extreme dilatonic domain walls and the supersymmetric branes within the supersymmetric domain walls of string theories. These solutions are valid in any region of spacetime, not just in the region close to the domain walls. 
  A strict proof of equivalence between Duffin-Kemmer-Petiau (DKP) and Klein-Gordon (KG) theories is presented for physical S-matrix elements in the case of charged scalar particles interacting in minimal way with an external or quantized electromagnetic field. First, Hamiltonian canonical approach to DKP theory is developed in both component and matrix form. The theory is then quantized through the construction of the generating functional for Green functions (GF) and the physical matrix elements of S-matrix are proved to be relativistic invariants. The equivalence between both theories is then proved using the connection between GF and the elements of S-matrix, including the case of only many photons states, and for more general conditions - so called reduction formulas of Lehmann, Symanzik, Zimmermann. 
  We discuss a possible generalization of the Calabi-Yau/Landau-Ginzburg correspondence to a more general class of manifolds. Specifically we consider the Fermat type hypersurfaces $M_N^k$: $\sum_{i=1}^N X_i^k =0$ in ${\bf CP}^{N-1}$ for various values of k and N. When k<N, the 1-loop beta function of the sigma model on $M_N^k$ is negative and we expect the theory to have a mass gap. However, the quantum cohomology relation $\sigma^{N-1}={const.}\sigma^{k-1}$ suggests that in addition to the massive vacua there exists a remaining massless sector in the theory if k>2. We assume that this massless sector is described by a Landau-Ginzburg (LG) theory of central charge $c=3N(1-2/k)$ with N chiral fields with U(1) charge $1/k$. We compute the topological invariants (elliptic genera) using LG theory and massive vacua and compare them with the geometrical data. We find that the results agree if and only if k=even and N=even.   These are the cases when the hypersurfaces have a spin structure. Thus we find an evidence for the geometry/LG correspondence in the case of spin manifolds. 
  We study some aspects of low-energy effective actions in 4-d superconformal gauge theories on the Coulomb branch. We describe superconformal invariants constructed in terms of N=2 abelian vector multiplet which play the role of building blocks for the N=2,4 supersymmetric low-energy effective actions. We compute the one-loop effective actions in constant N=2 field strength background in N=4 SYM theory and in N=2 SU(2) SYM theory with four hypermultiplets in fundamental representation. Using the classification of superconformal invariants we then find the manifestly N=2 superconformal form of these effective actions. While our explicit computations are done in the one-loop approximation, our conclusions about the structure of the effective actions in N=2 superconformal theories are general. We comment on some applications to supergravity - gauge theory duality in the description of D-brane interactions. 
  We give a detailed Operator Product Expansion interpretation of the results for conformal 4-point functions computed from supergravity through the AdS/CFT duality. We show that for an arbitrary scalar exchange in AdS(d+1) all the power-singular terms in the direct channel limit (and only these terms) exactly match the corresponding contributions to the OPE of the operator dual to the exchanged bulk field and of its conformal descendents. The leading logarithmic singularities in the 4-point functions of protected N=4 super-Yang Mills operators (computed from IIB supergravity on AdS(5) X S(5) are interpreted as O(1/N^2) renormalization effects of the double-trace products appearing in the OPE. Applied to the 4-point functions of the operators Ophi ~ tr F^2 + ... and Oc ~ tr FF~ + ..., this analysis leads to the prediction that the double-trace composites [Ophi Oc] and [Ophi Ophi - Oc Oc] have anomalous dimension -16/N^2 in the large N, large g_{YM}^2 N limit. We describe a geometric picture of the OPE in the dual gravitational theory, for both the power-singular terms and the leading logarithms. We comment on several possible extensions of our results. 
  We review the anomaly inflow mechanism on D-branes and O-planes. In particular, we compute the one-loop world-volume anomalies and derive the RR anomalous couplings required for their cancellation. 
  A complete derivation, from first principles, of the reaction-rate formula for a generic reaction taking place in an out of equilibrium quantum-field system is given. It is shown that the formula involves no finite-volume correction. Each term of the reaction-rate formula represents a set of physical processes that contribute to the reaction under consideration. 
  Quantum corrections to Legendre transformations are shown to cancel to all orders in supersymmetric theories in path integral formalism. Using this result, lagrangians for auxiliary fields are generalized to non-quadratic forms. In supersymmetric effective nonlinear lagrangians, the arbitrariness due to the existence of quasi Nambu-Goldstone bosons is shown to disappear when local auxiliary gauge fields are introduced. 
  We show that quantized matter fields in the presence of background metrics with Horizon exhibit spontaneous time asymmetry. All quantized matter fields have to vanish at the horizon. Some phenemenological applications of this in the context of black holes and early universe are considered. 
  Different models of field theories in two dimensions can be described by the action $Tr\int \vf F$. In the presence of a curved background, we construct a local supersymmetry-like transformations under which the action is invariant. Furthermore, by analysing the cohomology of the theory we show the absence of anomalies. Also the ultraviolet as well as the infrared finiteness of the theory are proven to all orders of perturbation theory. 
  We propose the lattice version of $BF$ gravity action whose partition function leads to the product of a particular form of 15-$j$ symbol which corresponds to a 4-simplex. The action is explicitly constructed by lattice $B$ field defined on triangles and link variables defined on dual links and is shown to be invariant under lattice local Lorentz transformation and Kalb-Ramond gauge transformation. We explicitly show that the partition function is Pachner move invariant and thus topological. The action includes the vanishing holonomy constraint which can be interpreted as a gauge fixing condition. This formulation of lattice $BF$ theory can be generalized into arbitrary dimensions. 
  The open descendants of simple current automorphism invariants are constructed. We consider the case where the order of the current is two or odd. We prove that our solutions satisfy the completeness conditions, positivity and integrality of the open and closed sectors and the Klein bottle constraint (apart from an interesting exception). In order to do this, we derive some new relations between the tensor Y and the fixed point conformal field theory. Some non-standard Klein bottle projections are considered as well. 
  Asymptotically anti-de Sitter space-times are considered in a general dimension $d\ge 4$. As one might expect, the boundary conditions at infinity ensure that the asymptotic symmetry group is the anti-de Sitter group (although there is an interesting subtlety if d=4). Asymptotic field equations imply that, associated with each generator $\xi$ of this group, there is a quantity $Q_\xi$ which satisfies the expected `balance equation' if there is flux of physical matter fields across the boundary $\I$ at infinity and is absolutely conserved in absence of this flux. Irrespective of the dimension d, all these quantities vanish if the space-time under considerations is (globally) anti-de Sitter. Furthermore, this result is required by a general covariance argument. However, it contradicts some of the recent findings based on the conjectured ADS/CFT duality. This and other features of our analysis suggest that, if a consistent dictionary between gravity and conformal field theories does exist in fully non-perturbative regimes, it would have to be more subtle than the one used currently. 
  By making use of the path-integral duality transformation, string representation of the Abelian-projected SU(3)-QCD with the $\Theta$-term is derived. Besides the short-range (self-)interactions of quarks (which due to the $\Theta$-term acquire a nonvanishing magnetic charge, i.e. become dyons) and electric Abrikosov-Nielsen-Olesen strings, the resulting effective action contains also a long-range topological interaction of dyons with strings. This interaction, which has the form of the 4D Gauss linking number of the trajectory of a dyon with the world-sheet of a closed string, is shown to become nontrivial at $\Theta$ not equal to $3\pi$ times an integer. At these values of $\Theta$, closed electric Abrikosov-Nielsen-Olesen strings in the model under study can be viewed as solenoids scattering dyons, which is the 4D analogue of the Aharonov-Bohm effect. 
  Corresponding to the Izergin-Korepin (A_2^(2)) R matrix, there are three diagonal solutions (``K matrices'') of the boundary Yang-Baxter equation. Using these R and K matrices, one can construct transfer matrices for open integrable quantum spin chains. The transfer matrix corresponding to the identity matrix K=1 is known to have U_q(o(3)) symmetry. We argue here that the transfer matrices corresponding to the other two K matrices also have U_q(o(3)) symmetry, but with a nonstandard coproduct. We briefly explore some of the consequences of this symmetry. 
  The use of proper time as a tool for causality implementation in field theory is clarified and extended to allow a manifestly covariant definition of discrete fields proper to be applied in field theory and quantum mechanics. It implies on a constraint between a radiation field and its sources, valid in principle for all fundamental interactions and with a solid experimental confirmation for the electromagnetic one. Some results of its applications to an abelian classical theory (electrodynamics taken as a first example), and with the discrete field being regarded as a classical representation of the field quantum (photon) are anticipated in order to illuminate the physical meaning and the origins of gauge fields and of their symmetries and singularities. They are associated to a loss of field-source coherence. 
  We construct the general model with only parallel positive tension 3-branes on $M^4 \times R^1$ and $M^4 \times R^1/Z_2$. The cosmology constant is sectional constant along the fifth dimension. In this general scenario, the 5-dimensional GUT scale on each brane can be indentified as the 5-dimensional Planck scale, but, the 4-dimensional Planck scale is generated from the low 4-dimensional GUT scale exponentially in our world. We also give two simple models to show explicitly how to solve the gauge hierarchy problem. 
  Various fluid mechanical systems, governed by nonlinear differential equations, enjoy a hidden, higher-dimensional dynamical Poincar\'e symmetry, which arises owing to their descent from a Nambu-Goto action. Also, for the same reason, there are equivalence transformations between different models. These interconnections are discussed in this lecture, and are summarized in Fig. 3. 
  The evolution of the distribution-theoretic methods in perturbative quantum field theory is reviewed starting from Bogolyubov's pioneering 1952 work with emphasis on the theory and calculations of perturbation theory integrals. 
  We consider the Newton's force law for brane world consisting of periodic configuration of branes. We show that it supports a massless graviton. Furthermore, this massless mode is well separated from the Kaluza-Klein spectrum by a mass gap. Thus most of the problems in phenomenology coming from continuum of Kaluza-Klein modes without mass gap are potentially cured in such a model. 
  We discuss the full nonlinear Kaluza-Klein (KK) reduction of the original formulation of d=11 supergravity on $AdS_7\times S_4$ to gauged maximal ({\cal N}=4) supergravity in 7 dimensions. We derive the full nonlinear embedding of the d=7 fields in the d=11 fields (``the ansatz'') and check the consistency of the ansatz by deriving the d=7 supersymmetry laws from the d=11 transformation laws in the various sectors. The ansatz itself is nonpolynomial but the final d=7 results are polynomial. The correct d=7 scalar potential is obtained. For most of our results the explicit form of the matrix U connecting the d=7 gravitino to the Killing spinor is not needed, but we derive the equation which U has to satisfy and present a solution. Requiring that the expression $\delta F=d\delta A$ in d=11 can be written as $\delta d(fields in d=7)$, we find the ansatz for the 4-form F. It satisfies the Bianchi identities. The corresponding ansatz for A modifies the geometrical proposal by Freed et al. by including d=7 scalar fields. A first order formulation for the three index photon $A_{\Lambda\Pi\Sigma}$ in d=11 is needed to obtain the d=7 supersymmetry laws and the action for the nonabelian selfdual antisymmetric tensor field $S_{\alpha\beta\gamma,A}$. Therefore selfduality in odd dimensions originates from a first order formalism in higher dimensions. 
  A two dimensional gauge theory is canonically associated to every Drinfeld double. For particular doubles, the theory turns out to be e.g. the ordinary Yang-Mills theory, the G/G gauged WZNW model or the Poisson $\sigma$-model that underlies the Kontsevich quantization formula. We calculate the arbitrary genus partition function of the latter. The result is the $q$-deformation of the ordinary Yang-Mills partition function in the sense that the series over the weights is replaced by the same series over the $q$-weights. For $q$ equal to a root of unity the series acquires the affine Weyl symmetry and its truncation to the alcove coincides with the Verlinde formula. 
  The use of gauged ${\cal N} = 8$ supergravity as a tool in studying the AdS/CFT correspondence for ${\cal N} = 4$ Yang-Mills theory is reviewed. The supergravity potential implies a non-trivial, supersymmetric IR fixed point, and the flow to this fixed point is described in terms of a supergravity kink. The results agree perfectly with earlier, independent field theory results. A supergravity inspired $c$-function, and corresponding $c$-theorem is discussed for general flows, and the simplified form for supersymmetric flows is also given. Flows along the Coulomb branch of the Yang-Mills theory are also described from the five-dimensional perspective. 
  The first half is a rapid review of 30 years of work on physics in anti-De Sitter space, with heavy emphasis on singletons. Principal topics are the kinematical basis for regarding singletons as the constituents of massless particles, and the effect of (negative) curvature in the infrared domain. Ideas that lead to an alternative to Big Bang cosmology are merely sketched. The second half presents new ideas inspired by experimental results on neutrino oscillations. Since leptons are massless before symmetry breaking it is natural to view them as composite states consisting of one Bose singleton (the Rac) and one Fermi singleton (the Di). This gives rise to a particular formulation of the phenomenology of electroweak physics, and strong suggestions for an expansion of the Standard model. An expansion of the Higgs sector seems inevitable, and flavor changing symmetry, complete with a new set of heavy vector mesons, is a very attractive possibility. 
  We derive a new version of the non-Abelian Stokes theorem for the Wilson loop in the SU(N) case by making use of the coherent state representation on the coset space $SU(N)/U(1)^{N-1}=F_{N-1}$, the flag space. We consider the SU(N) Yang-Mills theory in the maximal Abelian gauge in which SU(N) is broken down to $U(1)^{N-1}$. First, we show that the Abelian dominance in the string tension follows from this theorem and the Abelian-projected effective gauge theory that was derived by one of the authors. Next (but independently), combining the non-Abelian Stokes theorem with a novel reformulation of the Yang-Mills theory recently proposed by one of the authors, we proceed to derive the area law of the Wilson loop in four-dimensional SU(N) Yang-Mills theory in the maximal Abelian gauge. Owing to dimensional reduction, the planar Wilson loop at least for the fundamental representation in four-dimensional SU(N) Yang-Mills theory can be estimated by the diagonal (Abelian) Wilson loop defined in the two-dimensional $CP^{N-1}$ model. This derivation shows that the fundamental quarks are confined by a single species of magnetic monopole. The origin of the area law is related to the geometric phase of the Wilczek-Zee holonomy for $U(N-1)$. The calculations are performed using the instanton calculus (in the dilute instanton-gas approximation) and using the large $N$ expansion (in the leading order). 
  We consider supersymmetric Yang-Mills theory on R x S^1 x S^1. In particular, we choose one of the compact directions to be light-like and another to be space-like. Since the SDLCQ regularization explicitly preserves supersymmetry, this theory is totally finite, and thus we can solve for bound state wave functions and masses numerically without renormalizing. We present the masses as functions of the longitudinal and transverse resolutions and show that the masses converge rapidly in both resolutions. We also study the behavior of the spectrum as a function of the coupling and find that at strong coupling there is a stable, well defined spectrum which we present. We also find several unphysical states that decouple at large transverse resolution. There are two sets of massless states; one set is massless only at zero coupling and the other is massless at all couplings. Together these sets of massless states are in one-to-one correspondence with the full spectrum of the dimensionally reduced theory. 
  We analyse the equivalence between topologically massive gauge theory (TMGT) and different formulations of non-topologically massive gauge theories (NTMGTs) in the canonical approach. The different NTMGTs studied are St\"uckelberg formulation of (A) a first order formulation involving one and two form fields, (B) Proca theory, and (C) massive Kalb-Ramond theory. We first quantise these reducible gauge systems by using the phase space extension procedure and using it, identify the phase space variables of NTMGTs which are equivalent to the canonical variables of TMGT and show that under this the Hamiltonian also get mapped. Interestingly it is found that the different NTMGTs are equivalent to different formulations of TMGTs which differ only by a total divergence term. We also provide covariant mappings between the fields in TMGT to NTMGTs at the level of correlation function. 
  We derive a BPS-type bound for four-dimensional Born-Infeld action with constant B field background. The supersymmetric configuration saturates this bound and is regarded as an analog of instanton in U(1) gauge theory. Furthermore, we find the explicit solutions of this BPS condition. These solutions have a finite action proportional to the instanton number and represent D(p-4)-branes within a Dp-brane although they have a singularity at the origin. Some relations to the noncommutative U(1) instanton are discussed. 
  There is a close relation between classical supergravity and quantum SYM descriptions of interactions between separated branes. In the case of D3 branes, the equivalence of leading-order potentials is due to non-renormalization of the F^4 term in N=4 SYM theory. Here we point out the existence of another special non-renormalized term in quantum SYM effective action. This term reproduces the interaction potential between electric charge of a D3-brane probe and magnetic charge of a D3-brane source, represented by the Chern-Simons part of the D-brane action. This unique Wess-Zumino term depends on all six scalar fields and originates from a phase of the euclidean fermion determinant in SYM theory. It is manifestly scale invariant (i.e. is the same for large and small separations between branes) and can not receive higher loop corrections in gauge theory. Maximally supersymmetric SYM theories in D=p+1>4 contain mixed WZ terms which depend on both scalar and gauge field backgrounds, and which reproduce the corresponding CS terms in the supergravity interaction potentials between separated Dp-branes for p>3. Purely scalar WZ terms appear in other cases, e.g., in N=2 gauge theories in various dimensions describing magnetic interactions between Dp and D(6-p) branes. 
  The K\"ahler-Dirac equation is derived on the Weitzenb\"ock space-time, which has a quadruplet of parallel vector fields as the fundamental structure. A consistent system of equations for the K\"ahler fields and parallel vector fields is obtained.   Key words: teleparallelism, fermions, internal symmetry, equations. 
  The Casimir energy of a non-uniform string built up from two pieces with different speed of sound is calculated. A standard procedure of subtracting the energy of an infinite uniform string is applied, the subtraction being interpreted as the renormalization of the string tension. It is shown that in the case of a homogeneous string this method is completely equivalent to the zeta renormalization. 
  We quantize pure 2d Yang-Mills theory on an arbitrary Riemann surface in the gauge where the field strength is diagonal. Twisted sectors originate, as in Matrix string theory, from permutations of the eigenvalues around homotopically non-trivial loops. These sectors, that must be discarded in the usual quantization due to divergences occurring when two eigenvalues coincide, can be consistently kept if one modifies the action by introducing a coupling of the field strength to the space-time curvature. This leads to a generalized Yang-Mills theory whose action reduces to the usual one in the limit of zero curvature. After integrating over the non-diagonal components of the gauge fields, the theory becomes a free string theory (sum over unbranched coverings) with a U(1) gauge theory on the world-sheet. This is shown to be equivalent to a lattice theory with a gauge group which is the semi-direct product of S_N and U(1)^N. By using well known results on the statistics of coverings, the partition function on arbitrary Riemann surfaces and the kernel functions on surfaces with boundaries are calculated. Extensions to include branch points and non-abelian groups on the world-sheet are briefly commented upon. 
  In general relativity and electrodynamics fields are always generated from static monopoles (like mass or electric charge) or their corresponding currents by surrounding them in a spherical configuration. We investigate a generation of fields from primary fields by a scalar coupling. The generated secondary fields fulfill the condition of source-freedom and therefore cannot occur in a spherical configuration. The coupling strength depends on the energies of the primary fields. In most cases these fields can be approximately considered as dipole fields. We discuss two applications of couplings for electromagnetic and gravitational spin-1 fields and for electric and magnetic fields. We calculate for both applications the threshold values of field energy for the maximum coupling strength. The proposed approach yields to a further step towards an unification of electromagnetism and gravitation and has important consequences for the discrete symmetries. 
  The non-commutative geometry is revisited from the perspective of a generalized D p-brane. In particular, we analyze the open bosonic string world-sheet description and show that an effective non-commutative description on a D p-brane corresponds to a re-normalized world-volume. The world-volume correlators are analyzed to make a note on the non-commutative geometry. It is argued that the U(1) gauge symmetry can be viewed in a homomorphic image space for the non-commutative space of coordinates with an SO(p) symmetry. In the large B-field limit, the non-commutativity reduces to that among the zero modes and the world-volume description is precisely in agreement with the holographic principle. 
  We discuss some possible relationships in gauge theories, string theory and M theory in the light of some recent results obtained in gauge invariant supersymmetric quantum mechanics. In particular this reveals a new relationship between the gauge group E_8 and 11-dimensional space. 
  We describe a simple method for generating new string solutions for which the brane worldvolume is a curved space. As a starting point we use solutions with NS-NS charges combined with 2-d CFT's representing different parts of space-time. We illustrate our method with many examples, some of which are associated with conformally invariant sigma models. Using U-duality, we also obtain supergravity solutions with RR charges which can be interpreted as D-branes with non-trivial worldvolume geometry. In particular, we discuss the case of a D5-brane wrapped on AdS_3 x S^3, a solution interpolating between AdS_3 x S^3 x R^5 and AdS_3 x S^3 x S^3 x R, and a D3-brane wrapped over S^3 x R or AdS_2 x S^2. Another class of solutions we discuss involves NS5-branes intersecting over a 3-space and NS5-branes intersecting over a line. These solutions are similar to D7-brane or cosmic string backgrounds. 
  Two problems relative to the electromagnetic coupling of Duffin-Kemmer-Petiau (DKP) theory are discussed: the presence of an anomalous term in the Hamiltonian form of the theory and the apparent difference between the Interaction terms in DKP and Klein-Gordon (KG) Lagrangians. For this, we first discuss the behavior of DKP field and its physical components under gauge transformations. From this analysis, we can show that these problems simply do not exist if one correctly analyses the physical components of DKP field. 
  We discuss the general issues and ambiguities involved in matching the exact results for the low energy effective action of scale invariant N=2 supersymmetric QCD to those obtained by instanton methods. We resolve the reported disagreements and verify agreement between an infinite series of coefficients in the low energy effective actions calculated in the two approaches. In particular, we show that the exact low-energy effective couplings for SU(N) for all N with 2N fundamental hypermultiplets agree at a special vacuum on the Coulomb branch where a large unbroken discrete global symmetry makes the matching of parameters relatively straightforward. 
  The radiation-dominated k=0 FRW cosmology emerges as the induced metric on a codimension one hypersurface of constant extrinsic curvature in the five-dimensional AdS-Schwarzschild solution. That we should get FRW cosmology in this way is an expected result from AdS/CFT in light of recent comments regarding the coupling of gravity to "boundary" conformal field theories. I remark on how this calculation bears on the understanding of Randall and Sundrum's "alternative to compactification." A generalization of the AdS/CFT prescription for computing Green's functions is suggested, and it is shown how gravity emerges from it with a strength G_4 = 2 G_5/L. Some numerical bounds are set on the radius of curvature L of AdS_5. One of them comes from estimating the rate of leakage of visible sector energy into the CFT. That rate is connected via a unitarity relation to deviations from Newton's force law at short distances. The best bound on L obtained in this paper comes from a match to the parameters of string theory. It is L < 1 nm if the string scale is 1 GeV. Higher string scales imply a tighter bound on L. 
  We examine the attractor mechanism for extremal black holes in the context of five dimensional N = 2 supergravity and show that attractor points are unique in the extended vector multiplet moduli space. Implications for black hole entropy are discussed. 
  We consider the canonical quantization of fermions on an odd dimensional manifold with boundary, with respect to a family of elliptic hermitean boundary conditions for the Dirac hamiltonian. We show that there is a topological obstruction to a smooth quantization as a function of the boundary conditions. The obstruction is given in terms of a gerbe and its Dixmier-Douady class is evaluated. 
  The action changes (and thus the vacuum conservation amplitudes) in the proper-time representation are found for an accelerated mirror interacting with scalar and spinor vacuum fields in 1+1 space. They are shown to coincide to within the multiplier e^2 with the action changes of electric and scalar charges accelerated in 3+1 space. This coincidence is attributed to the fact that the Bose and Fermi pairs emitted by a mirror have the same spins 1 and 0 as do the photons and scalar quanta emitted by charges. It is shown that the propagation of virtual pairs in 1+1 space can be described by the causal Green's function \Delta_f(z,\mu) of the wave equation for 3+1 space. This is because the pairs can have any positive mass and their propagation function is represented by an integral of the causal propagation function of a massive particle in 1+1 space over mass which coincides with \Delta_f(z,\mu). In this integral the lower limit \mu is chosen small, but nonzero, to eliminate the infrared divergence. It is shown that the real and imaginary parts of the action change are related by dispersion relations, in which a mass parameter serves as the dispersion variable. They are a consequence of the same relations for \Delta_f(z,\mu). Therefore, the appearance of the real part of the action change is a direct consequence of the causality, according to which real part of \Delta_f(z,\mu) is nonzero only for timelike and zero intervals. 
  We show how to calculate the quantum mass correction to (1+1)D solitonic field theories using numerical methods. This is essential if we want to find the corrections to non-integrable models. We start with a review of the standard derivation of the first order quantum correction. Then, we re-derive a trace formula which allows us to compute the mass correction mode by mode. Specifically, we are interested in the extent to which the lowest modes from both, the soliton and the vacuum, sectors give the leading contribution. We apply the technique to both the Sine-Gordon and the $\phi^4$-kink model. Then, we compute all the modes numerically and hence the first order quantum contribution to the mass of the Sine-Gordon and $\phi^4$ soliton. 
  Instantons in massless theories do not carry over to massive theories due to Derrick's theorem. This theorem can, however, be circumvented, if a constraint that restricts the scale of the instanton is imposed on the theory. Constrained instantons are considered in four dimensions in phi^4 theory and SU(2) Yang-Mills-Higgs theory. In each of these theories a calculational sceme is set up and solved in the lowest few orders in the mass parameter in such a way that the need for a constraint is exhibited clearly. Constrained instantons are shown to exist as finite action solutions of the field equations with exponential fall off only for specific constraints that are unique in lowest order in the mass parameter in question. 
  We present the analysis of all possible shortenings which occur for composite gauge invariant conformal primary superfields in SU(2,2/N) invariant gauge theories. These primaries have top-spin range N/2 \leq J_{max} < N with J_{max} = J_1 + J_2, (J_1,J_2) being the SL(2,C) quantum numbers of the highest spin component of the superfield. In Harmonic superspace, analytic and chiral superfields give J_{max}= N/2 series while intermediate shortenings correspond to fusion of chiral with analytic in N=2, or analytic with different analytic structures in N=3,4. In the AdS/CFT language shortenings of UIR's correspond to all possible BPS conditions on bulk states. An application of this analysis to multitrace operators, corresponding to multiparticle supergravity states, is spelled out. 
  Recent progress on a constructive approach to QFT which is based on modular theory is reviewed and compared with the standard quantization approaches. Talk given at ``Quantum Theory and Symmetries'', Goslar, Germany, July 1999 
  We examine the structure of the potential energy of 2+1-dimensional Yang-Mills theory on a torus with gauge group SU(2). We use a standard definition of distance on the space of gauge orbits. A curve of extremal potential energy in orbit space defines connections satisfying a certain partial differential equation. We argue that the energy spectrum is gapped because the extremal curves are of finite length. Though classical gluon waves satisfy our differential equation, they are not extremal curves. We construct examples of extremal curves and find how the length of these curves depends on the dimensions of the torus. The intersections with the Gribov horizon are determined explicitly. The results are discussed in the context of Feynman's ideas about the origin of the mass gap. 
  The system of D1 and D5 branes with a Kaluza-Klein momentum is re-investigated using the five-dimensional U-duality group E_{6(+6)}(Z). We show that the residual U-duality symmetry that keeps this D1-D5-KK system intact is generically given by a lift of the Weyl group of F_{4(+4)}, embedded as a finite subgroup in E_{6(+6)}(Z). We also show that the residual U-duality group is enhanced to F_{4(+4)}(Z) when all the three charges coincide. We then apply the analysis to the AdS(3)/CFT(2) correspondence, and discuss that among 28 marginal operators of CFT(2) which couple to massless scalars of AdS(3) gravity at boundary, 16 would behave as exactly marginal operators for generic D1-D5-KK systems. This is shown by analyzing possible three-point couplings among 42 Kaluza-Klein scalars with the use of their transformation properties under the residual U-duality group. 
  Using the ADHM instanton calculus, we evaluate the one-instanton contribution to the low-energy effective prepotential of N=2 supersymmetric SU(N) Yang-Mills theory with N_F flavors of hypermultiplets in the fundamental representation and a hypermultiplet in the symmetric rank two tensor representation. For N_F<N-2, when the theory is asymptotically free, our result is compared with the exact solution that was obtained using M-theory and we find complete agreement. 
  We propose a direct correspondence between the classical evolution equations of 5-d supergravity and the renormalization group (RG) equations of the dual 4-d large $N$ gauge theory. Using standard Hamilton-Jacobi theory, we derive first order flow equations for the classical supergravity action $S$, that take the usual form of the Callan-Symanzik equations, including the corrections due to the conformal anomaly. This result gives direct support for the identification of $S$ with the quantum effective action of the gauge theory. In addition we find interesting new relations between the beta-functions and the counterterms that affect the 4-d cosmological and Newton constant. 
  For spaces which are not asymptotically anti-de Sitter where the asymptotic behavior is deformed by replacing the cosmological constant by a dilaton scalar potential, we show that it is possible to have well-defined boundary stress-energy tensors and finite Euclidean actions by adding appropriate surface counterterms. We illustrate the method by the examples of domain-wall black holes in gauged supergravities, three-dimensional dilaton black holes and topological dilaton black holes in four dimensions. We calculate the boundary stress-energy tensor and Euclidean action of these black configurations and discuss their thermodynamics. We find new features of topological black hole thermodynamics. 
  We provide a gauge covariant formalism of the canonically quantized theory of spin-3/2 Rarita-Schwinger gauge field. The theory admits a quantum gauge transformation by which we can shift the gauge fixing parameter. The quantum gauge transformation does not change the BRST charge. Thus, the physical Hilbert space is trivially independent of the gauge fixing parameter. 
  We investigate several models of coupled scalar fields that present discrete Z_2, Z_2 x Z_2, Z_3 and other symmetries. These models support topological domain wall solutions of the BPS and non-BPS type. The BPS solutions are stable, but the stability of the non-BPS solutions may depend on the parameters that specify the models. The BPS and non-BPS states give rise to bags, and also to three-junctions that may allow the presence of networks of topological defects. In particular, we show that the non-BPS defects of a specific model that engenders the Z_3 symmetry give rise to a stable regular hexagonal network of domain walls. 
  The existence of a ground ring of ghost number zero operators in the chiral BRST cohomology of the N=2 string is used to derive an infinite set of Ward identities for the closed-string scattering amplitudes at arbitrary genus. These identities are sufficient to rederive the well known vanishing theorem for loop amplitudes with more than three external legs. 
  We find a new d-parameter family of ultra-local boundary Poisson brackets that satisfy the Jacobi identity. The two already known cases (hep-th/9305133, hep-th/9806249 and hep-th/9901112) of ultra-local boundary Poisson brackets are included in this new continuous family as special cases. 
  We study the low energy effective action $S$ of gravity, induced by integrating out gauge and matter fields, in a general class of Randall-Sundrum type string compactification scenarios with exponential warp factors. Our method combines dimensional reduction with the holographic map between between 5-d supergravity and 4-d large $N$ field theory. Using the classical supergravity approximation, we derive a flow equation of the effective action $S$ that controls its behavior under scale transformations. We find that as a result each extremum of $S$ automatically describes a complete RG trajectory of classical solutions. This implies that, provided the cosmological constant is canceled in the high energy theory, classical flat space backgrounds naturally remain stable under the RG-flow. The mechanism responsible for this stability is that the non-zero vacuum energy generated by possible phase transitions, is absorbed by a dynamical adjustment of the contraction rate of the warp factor. 
  We show that a U(1) instanton on non-commutative R^4 corresponds to a supersymmetric non-singular U(1) gauge field on a commutative Kahler manifold X which is a blowup of C^2 at a finite number of points. For instanton charge k the manifold X can be viewed as a space-time foam. A direct connection with integrable systems of Calogero-Moser type is established. We also make some comments on the non-abelian case. 
  We consider the question of gauge invariance of the Wilson loop in the light of a new treatment of axial gauge propagator proposed recently based on a finite field-dependent BRS (FFBRS) transformation. We remark that as under the FFBRS transformation the vacuum expectation value of a gauge invariant observable remains unchanged, our prescription automatically satisfies the Wilson loop criterion. Further, we give an argument for {\it direct} verification of the invariance of Wilson loop to O(g^4) using the earlier work by Cheng and Tsai. We also note that our prescription preserves the thermal Wilson loop to O(g^2). 
  In this thesis, we compute within the field-antifield formalism the local BRST cohomology of various theories involving p-form gauge fields. A particular emphasis is put on the cohomology groups corresponding to the consistent local interactions, counterterms and candidate anomalies. 
  We consider non perturbative effects in M-theory compactifications on a seven-manifold of G_2 holonomy arising from membranes wrapped on supersymmetric three-cycles. When membranes are wrapped on associative submanifolds they induce a superpotential that can be calculated using calibrated geometry. This superpotential is also derived from compactification on a seven-manifold, to four dimensional Anti-de Sitter spacetime, of eleven dimensional supergravity with non vanishing expectation value of the four-form field strength. 
  Gravitating t'Hooft-Polyakov magnetic monopoles can be constructed when coupling the Georgi-Glashow model to gravitation. For a given value of the Higgs boson mass, these gravitating solitons exist up to a critical value of the ratio of the vector meson mass to the Planck mass. The critical solution is characterized by a degenerate horizon of the metric. As pointed out recently by Lue and Weinberg, two types of critical solutions can occur, depending on the value of the Higgs boson mass. Here we investigate this transition for dyons and show that the Lue and Weinberg phenomenon is favorized by the presence of the electric-charge degree of freedom. 
  Vacuum spherically symmetric Einstein gravity in $N\ge 4$ dimensions can be cast in a two-dimensional conformal nonlinear sigma model form by first integrating on the $(N-2)$-dimensional (hyper)sphere and then performing a canonical transformation. The conformal sigma model is described by two fields which are related to the Arnowitt-Deser-Misner mass and to the radius of the $(N-2)$-dimensional (hyper)sphere, respectively. By quantizing perturbatively the theory we estimate the quantum corrections to the ADM mass of a black hole. 
  We consider the heterotic string on an elliptic Calabi-Yau three-fold with five-branes wrapping curves in the base ('horizontal' curves) of the Calabi-Yau as well as some elliptic fibers ('vertical' curves). We show that in this generalized set-up, where the association of the heterotic side with the $F$-theory side is changed relative to the purely vertical situation, the number of five-branes wrapping the elliptic fibers still matches the corresponding number of $F$-theory three-branes. 
  A general calculational method is applied to investigate symmetry relations among divergent amplitudes in a free fermion model. A very traditional work on this subject is revisited. A systematic study of one, two and three point functions associated to scalar, pseudoscalar, vector and axial-vector densities is performed. The divergent content of the amplitudes are left in terms of five basic objects (external momentum independent). No specific assumptions about a regulator is adopted in the calculations. All ambiguities and symmetry violating terms are shown to be associated with only three combinations of the basic divergent objects. Our final results can be mapped in the corresponding Dimensional Regularization calculations (in cases where this technique could be applied) or in those of Gertsein and Jackiw which we will show in detail. The results emerging from our general approach allow us to extract, in a natural way, a set of reasonable conditions (e.g. crucial for QED consistency) that could lead us to obtain all Ward Identities satisfied. Consequently, we conclude that the traditional approach used to justify the famous triangular anomalies in perturbative calculations could be questionable. An alternative point of view, dismissed of ambiguities, which lead to a correct description of the associated phenomenology, is pointed out. 
  We study the general gaugings of N=2 Maxwell-Einstein supergravity theories (MESGT) in five dimensions, extending and generalizing previous work. The global symmetries of these theories are of the form SU(2)_R X G, where SU(2)_R is the R-symmetry group of the N=2 Poincare superalgebra and G is the group of isometries of the scalar manifold that extend to symmetries of the full action. We first gauge a subgroup K of G by turning some of the vector fields into gauge fields of K while dualizing the remaining vector fields into tensor fields transforming in a non-trivial representation of K. Surprisingly, we find that the presence of tensor fields transforming non-trivially under the Yang-Mills gauge group leads to the introduction of a potential which does not admit an AdS ground state. Next we give the simultaneous gauging of the U(1)_R subgroup of SU(2)_R and a subgroup K of G in the presence of K-charged tensor multiplets. The potential introduced by the simultaneous gauging is the sum of the potentials introduced by gauging K and U(1)_R separately. We present a list of possible gauge groups K and the corresponding representations of tensor fields. For the exceptional supergravity we find that one can gauge the SO^*(6) subgroup of the isometry group E_{6(-26)} of the scalar manifold if one dualizes 12 of the vector fields to tensor fields just as in the gauged N=8 supergravity. 
  We present what we believe is the minimal three-family $AdS/CFT$ model compactified on a nonabelian orbifold $S^{5}/(Q\times Z_{3})$. Nontrivial irreps of the discrete nonabelian group $Q\times Z_{3}$ are identified with the $4$ of $SU(4)$ $R$ symmetry to break all supersymmetries, and the scalar content of the model is sufficient to break the gauge symmetry to the standard model. According to the conformality hypothesis the progenitor $SU(4)^3 \times SU(2)^{12}$ theory becomes conformally invariant at an infra-red fixed point of the renormalization group. 
  We investigate the instanton effects of non-BPS D0-brane in type I string theory. We argue the general properties of these instanton effects and consider these on $R^9\times S^1$ as a simple example using the effective action of D0-brane. 
  We discuss some properties of an M-9-brane in ``massive 11D theory'' proposed by Bergshoeff, Lozano and Ortin. A 10-form gauge potential is consistently introduced into the massive 11D supergravity, and an M-9-brane Wess-Zumino action is constructed as that of a gauged $\sigma$-model. Using duality relations is crucial in deriving the action, which we learn from the study of a 9-form potential in 10D massive IIA theory. A target space solution of an M-9-brane with a non-vanishing 10-form gauge field is also obtained, whose source is shown to be the M-9-brane effective action. 
  Recent progress in mathematical theory of random processes provides us with non-Fock product systems (continuous tensor products of Hilbert spaces) used here for constructing a toy model for fermions. Some state vectors describe infinitely many particles in a finite region; the particles accumulate to a point. Electric charge can be assigned to the particles, the total charge being zero. Time dynamics is not considered yet, only kinematics (a single time instant). 
  We discuss the role of conformal matter quantum effects (using large $N$ anomaly induced effective action) to creation-annihilation of an Anti-de Sitter Universe. The arbitrary GUT with conformally invariant content of fields is considered. On a purely gravitational (supersymmetric) AdS background, the quantum effects act against an (already existing) AdS Universe. The annihilation of such a Universe occurs, what is common for any conformal matter theory. On a dilaton-gravitational background, where there is dilatonic contribution to the induced effective action, the quantum creation of an AdS Universe is possible assuming fine-tuning of the dilaton. 
  Using relationships between open and closed strings, we present a construction of tree-level scattering amplitudes for gravitons minimally coupled to matter in terms of gauge theory partial amplitudes. In particular, we present examples of amplitudes with gravitons coupled to vectors or to a single fermion pair. We also present two examples with massive graviton exchange, as would arise in the presence of large compact dimensions. The gauge charges are represented by flavors of dynamical scalars or fermions. This also leads to an unconventional decomposition of color and kinematics in gauge theories. 
  Based on the renormalisability of the SU(n) theory with massive gauge bosons, we start with the path integral of the generating functional for the renormalized Green functions and develop a method to construct the scattering matrix so that the unitarity is evident. By using as basical variables the renormalized field functions and defining the unperturbed Hamiltonian operator $H_0$ that, under the Lorentz condition, describes the free particles of the initial and final states in scattering processes, we form an operator description with which the renormalized Green functions can be expressed as the vacuum expectations of the time ordered products of the Heisenberg operators of the renormalized field functions, that satisfy the usual equal time commutation or anticommutation rules. From such an operator description we find a total Hamiltonian $\widetilde{H}$ that determine the time evolution of the Heisenberg operators of the renormalized field functions. The scattering matrix is nothing but the matrix of the operator $U(\infty, -\infty)$, which describes the time evolution from $-\infty$ to $\infty$ in the interaction picture specified by $\widetilde{H}$ and $H_0$, respect to a base formed by the physical eigen states of $H_0$. We also explain the asymptotic field viewpoint of constructing the scattering matrix within our operator description. Moreover, we find a formular to express the scattering matrix elements in terms of the truncated renormalized Green functions. 
  We review results of the last two years concerning caloron solutions of unit charge with non-trivial holonomy, revealing the constituent monopole nature of these instantons. For SU(n) there are n such BPS constituents. New is the presentation of the exact values for the Polyakov loop at the three constituent locations for the SU(3) caloron with arbitrary holonomy. At these points two eigenvalues coincide, extending earlier results for SU(2) to a situation more generic for general SU(n). 
  A U(1) BF-Yang-Mills theory on noncommutative ${\mathbb{R}}^4$ is presented and in this formulation the U(1) Yang-Mills theory on noncommutative ${\mathbb{R}}^4$ is seen as a deformation of the pure BF theory. Quantization using BRST symmetry formalism is discussed and Feynman rules are given. Computations at one-loop order have been performed and their renormalization studied. It is shown that the U(1) BFYM on noncommutative ${\mathbb{R}}^4$ is asymptotically free and its UV-behaviour in the computation of the $\beta$-function is like the usual SU(N) commutative BFYM and Yang Mills theories. 
  We investigate a string/M theoretic realization of the varying speed of light scenario. We consider a 3+1 dimensional probe-brane universe in the background of a black hole in the bulk formed by a stack of branes, in the spirit of Kiritsis (hep-th/9906206). We generalize the dynamics of the system at hand by including rotation and Hubble damping of the bulk space-time and show that this may lead to a mechanism to stabilize the brane-universe and hence fix the speed of light at late times. 
  We derive the noncommutative Dirac-Born-Infeld action for the $D$-brane, which governs dynamics of $D$-brane with a NS-NS $B$-field in the low energy regime. Depending on some details of the path integral prescriptions, both ordinary Dirac-Born-Infeld action and noncommutative one can be obtained by evaluating the same Polyakov string path integral for the open string ending on the $D$-brane. Thus, it establishes the equivalence of the noncommutative Dirac-Born-Infeld action and the ordinary one. 
  It is pointed out that the essential parts of some recent papers by Mebarki {\it et al.} (hep-th/9911045, hep-th/9911046, hep-th/9911048, hep-th/9911049, dated 6 Nov.1999) are taken from a book written by Nakanishi and Ojima, published in 1990. 
  We consider vacuum quantum effects in the Early Universe, which may lead to inflation. The inflation is a direct consequence of the supposition that, at high energies, all the particles can be described by the weakly interacting, massless, conformally invariant fields. We discuss, from the effective field theory point of view, the stability of inflation, transition to the FRW solution, and also possibility to study metric and density perturbations. 
  Based on a generalization of the stochastic quantization scheme we recently proposed a generalized, globally defined Faddeev-Popov path integral density for the quantization of Yang-Mills theory. In this talk first our approach on the whole space of gauge potentials is shortly reviewed; in the following we discuss the corresponding global path integral on the gauge orbit space relating it to the original Parisi-Wu stochastic quantization scheme. 
  We consider the free field approach or bosonization technique for the Wess-Zumino-Novikov-Witten model with arbitrary Kac-Moody algebra on Riemann surface of genus zero. This subject was much studied previously, and the paper can be partially taken as a brief survey. The way to obtain well-known Schechtman-Varchenko solutions of the Knizhnik-Zamolodchikov equations as certain correlators in free chiral theory is revisited. This gives rise to simple description of space of the WZNW conformal blocks. The general $N$-point correlators of the model are constructed from the conformal blocks using non-chiral action for free fields perturbed by exactly marginal terms. The method involved generalizes the Dotsenko-Fateev prescription for minimal models. As a consequence of this construction we obtain new integral identities. 
  D0 brane (D-particle) and D1 brane actions possess first and second class constraints that result in local $\kappa$ symmetry. The $\kappa$ symmetry of the D-particle and the D1 brane is extended here into a larger symmetry ($\kappa_-$ and $\kappa_+$) in a larger phase space by turning second class constraints into first class. Different gauge fixings of these symmetries result in different presentations of these systems while a "unitary" gauge fixing of the new $ \kappa_+$ symmetry retrieves the original action with $\kappa_- = \kappa $ symmetry. For D1 brane our extended phase space makes all constraints into first class in the case of vanishing world sheet electric field (namely $(0, 1)$ string). 
  We propose that Little String Theories in six dimensions are quasilocal quantum field theories. Such field theories obey a modification of Wightman axioms which allows Wightman functions (i.e. vacuum expectation values of products of fundamental fields) to grow exponentially in momentum space. Wightman functions of quasilocal fields in x-space violate microlocality at short distances. With additional assumptions about the ultraviolet behavior of quasilocal fields, one can define approximately local observables associated to big enough compact regions. The minimum size of such a region can be interpreted as the minimum distance which observables can probe. We argue that for Little String Theories this distance is of order {\sqrt N}/M_s. 
  Mass and other conserved Noether charges are discussed for solutions of gravity theories with locally Anti-de Sitter asymptotics in 2n dimensions. The action is supplemented with a boundary term whose purpose is to guarantee that it reaches an extremum on the classical solutions, provided the spacetime is locally AdS at the boundary. It is also shown that if spacetime is locally AdS at spatial infinity, the conserved charges are finite and properly normalized without requiring subtraction of a reference background. In this approach, Noether charges associated to Lorentz and diffeomorphism invariance vanish identically for constant curvature spacetimes. The case of zero cosmological constant is obtained as a limit of AdS, where $\Lambda $ plays the role of a regulator. 
  The problem of renormalisability of the SU(n) theory with massive gauge bosons is reinverstigated in the present work. We expound that the quantization under the Lorentz condition caused by the mass term of the gauge fields leads to a ghost action which is the same as that of the usual SU(n) Yang-Mills theory in the Landau gauge. Furthermore, we clarify that the mass term of the gauge fields cause no additional complexity to the Slavnov-Taylor identity of the generating functional for the regular vertex functions and does not change the equations satisfied by the divergent part of this generating functional. Finally, we prove that the renormalisability of the theory can be deduced from the renormalisability of the Yang-Mills theory. 
  I carefully study noncommutative version of ADHM construction of instantons, which was proposed by Nekrasov and Schwarz. Noncommutative ${\bf R}^4$ is described as algebra of operators acting in Fock space. In ADHM construction of instantons, one looks for zero-modes of Dirac-like operator. The feature peculiar to noncommutative case is that these zero-modes project out some states in Fock space. The mechanism of these projections is clarified when the gauge group is U(1). I also construct some zero-modes when the gauge group is U(N) and demonstrate that the projections also occur, and the mechanism is similar to the U(1) case. A physical interpretation of the projections in IIB matrix model is briefly discussed. 
  We describe a general method for calculating the infra-red limit of physical quantities in unitary quantum field theories. Using analyticity of Green functions in a complex scale parameter, the infra-red limit is expressed as a contour integral entirely in the ultra-violet region. The infra-red limit is shown to be the limit of the Borel transform of the physical quantity. The method is illustrated by calculating the central charge of the perturbed unitary minimal models and the critical exponents of phi^4 theory in three dimensions. We obtain approximate values for the central charge which are very close to the exact values using only a one loop perturbative calculation. For phi^4 theory we obtain estimates which are within the errors of other more elaborate approaches. 
  We show how non-near horizon p-brane theories can be obtained from two embedding constraints in a flat higher dimensional space with 2 time directions. In particular this includes the construction of D3 branes from a flat 12-dimensional action, and M2 and M5 branes from 13 dimensions. The worldvolume actions are determined by constant forms in the higher dimension, reduced to the usual expressions by Lagrange multipliers. The formulation affords insight in the global aspects of the spacetime geometries and makes contact with recent work on two-time physics. 
  We describe the scalar and spinor fields on noncommutative sphere starting from canonical realizations of the enveloping algebra ${\cal A}={\cal U}{u(2))}$. The gauge extension of a free spinor model, the Schwinger model on a noncommutative sphere, is defined and the model is quantized. The noncommutative version of the model contains only a finite number of dynamical modes and is non-perturbatively UV-regular. An exact expresion for the chiral anomaly is found. In the commutative limit the standard formula is recovered. 
  We show that in the supersymmetry framework described by a Poincar\'{e} superalgebra with tensorial central charges the role of generalized superconformal symmetry which contains all these central charges is played by $OSp(1|2^{k})$, where k=3 for D=4. Following [1,2] we describe the free supertwistor model for $OSp(1|8)$. It appears that in such a scheme the tensorial central charges satisfy additional relations and the model describes the tower of supersymmetric massless states with an arbitrary (integer and half--integer) helicity spectrum. 
  The complete proof of a theorem announced in [1] on the consistent interactions for (non-chiral) exterior form gauge fields is given. The theorem can be easily generalized to the analysis of anomalies. Its proof amounts to computing the local BRST cohomology H^0(s|d) in the space of local n-forms depending on the fields, the ghosts, the antifields and their derivatives. 
  The supersymmetric flow equations describing the flow of moduli from infinity to the black hole horizon, and vice versa, are derived in the five-dimensional theories where the moduli space of the very special geometry has disjoint branches. The multiple solutions are derived from the `off the horizon' attractor equation. Within each branch, the black hole entropy, as usual, depends only on the near horizon attractor values of moduli, i.e. the entropy depends on the charges and on coefficients of the cubic polynomial. It does not depend on the values of the moduli fields at infinity. However, the entropy, as well as the near horizon values of the moduli fields, are shown to depend on the choice of the branch specified by the choice of the set of moduli at infinity. We present examples of BPS black hole solutions with the same Q_I and C_{IJK}, whose entropies differ significantly. 
  We compute the Pauli-Jordan, Hadamard and Feynman propagators for the massive metrical perturbations on de Sitter space. They are expressed both in terms of mode sums and in invariant forms. 
  The identification of a causal-connection scale motivates us to propose a new covariant bound on entropy within a generic space-like region. This "causal entropy bound", scaling as the square root of EV, and thus lying around the geometric mean of Bekenstein's S/ER and holographic S/A bounds, is checked in various "critical" situations. In the case of limited gravity, Bekenstein's bound is the strongest while naive holography is the weakest. In the case of strong gravity, our bound and Bousso's holographic bound are stronger than Bekenstein's, while naive holography is too tight, and hence typically wrong. 
  The Kalb-Ramond monopole, as discussed by Nepomechie, is identical with the (singular) Dirac monopole in d=3 dimensions. The latter can be described by the (regular) 't Hooft-Polyakov monopole, via the 't Hooft tensor construction. This construction is extended to arbitrary odd dimensions by performing the d=5 case explicitly, exploiting the (regular) `monopoles' of generalised Georgi-Glashow models and identifying their 't Hooft tensors as the Kalb-Ramond fields. The relevant `magnetic charges' are expressed as topological invariants. 
  Duality symmetric electromagnetic action a la Schwarz-Sen is shown to appear naturally in a chain of equivalent actions which interchange equations of motion with Bianchi identities. Full symmetry of the electromagnetic stress tensor is exploited by generalizing this duality symmetric action to allow for a space-time dependent mixing angle between electric and magnetic fields. The rotated fields are shown to satisfy Maxwell-like equations which involve the mixing angle as a parameter, and a generalized gauge invariance of the new action is established. 
  The AdS/CFT correspondence implies that the effective action of certain strongly coupled large $N$ gauge theories satisfy the Hamilton-Jacobi equation of 5d gravity. Using an analogy with the relativistic point particle, I construct a low energy effective action that includes the Einstein action and obeys a Callan-Symanzik-type RG-flow equation. It follows from the flow equation that under quite general conditions the Einstein equations admit a flat space-time solution, but other solutions with non-zero cosmological constant are also allowed. I discuss the geometric interpretation of this result in the context of warped compactifications. 
  We introduce the spinor parallel propagator for maximally symmetric spaces in any dimension. Then, the Dirac spinor Green's functions in the maximally symmetric spaces R^n, S^n and H^n are calculated in terms of intrinsic geometric objects. The results are covariant and coordinate-independent. 
  We consider an especially simple version of a thick domain wall in $AdS$ space and investigate how four-dimensional gravity arises in this context. The model we consider has the advantage, that the equivalent quantum mechanics problem can be stated in closed form. The potential in this Schr\"odinger equation suggests that there could be resonances in the spectrum of the continuum modes. We demonstrate that there are no such resonances in the model we consider. 
  We consider a (2+1)-dimensional mechanical system with the Lagrangian linear in the torsion of a light-like curve. We give Hamiltonian formulation of this system and show that its mass and spin spectra are defined by one-dimensional nonrelativistic mechanics with a cubic potential. Consequently, this system possesses the properties typical of resonance-like particles. 
  The analogy between General Relativity and monopole physics is pointed out and the presence of a 3-cocycle which corresponds to a source leads to discretization of field momentum. This is analogous to the same phenomena in monopole physics. 
  The field-to-particle transition formalizm based on the effective zero-brane action approach is generalized for arbitrary multiscalar fields. As a fruitful example, by virtue of this method we derive the non-minimal particle action for the Jackiw-Teitelboim gravity at fixed gauge in the vicinity of the black hole solution as an instanton-dilaton doublet. When quantizing it as the theory with higher derivatives, it is shown that the appearing quantum equation has SU(2) dynamical symmetry group realizing the exact spin-coordinate correspondence. Finally, we calculate the quantum corrections to the mass of the JT black hole. 
  We study the properties of the topologically nontrivial doublet solution arisen in the biscalar theory with a fourth-power potential introducing an example of the spontaneous breaking of symmetry. We rule out the zero-brane (non-minimal point particle) action for this doublet as a particle with curvature. When quantizing it as the theory with higher derivatives, we calculate the quantum corrections to the mass of the doublet which could not be obtained by means of the perturbation theory. 
  In this paper we investigate the form of calibrated M5-branes in the presence of a nonvanishing 3-form field H. We discuss the influence of the H-field on the deformation of supersymmetric n-cycles (in particular SLAG submanifolds in R^n). In addition we argue for a construction which relates calibrated M5-branes of different curved dimensions to each other. 
  The duality symmetries of various chiral boson actions are investigated using D=2 and D=6 space-time dimensions as examples. These actions involve the Siegel, Floreanini-Jackiw, Srivastava and Pasti-Sorokin-Tonin formulations. We discover that the Siegel, Floreanini-Jackiw and Pasti-Sorokin-Tonin actions have self-duality with respect to a common anti-dualization of chiral boson fields in D=2 and D=6 dimensions, respectively, while the Srivastava action is self-dual with respect to a generalized dualization of chiral boson fields. Moreover, the action of the Floreanini-Jackiw chiral bosons interacting with gauge fields in D=2 dimensions also has self-duality but with respect to a generalized anti-dualization of chiral boson fields. 
  We study the moduli space ${\cal M}$ of N=(4,4) superconformal field theories with central charge c=6. After a slight emendation of its global description we find the locations of various known models in the component of ${\cal M}$ associated to K3 surfaces. Among them are the Z_2 and Z_4 orbifold theories obtained from the torus component of ${\cal M}$. Here, SO(4,4) triality is found to play a dominant role. We obtain the B-field values in direction of the exceptional divisors which arise from orbifolding. We prove T-duality for the Z_2 orbifolds and use it to derive the form of ${\cal M}$ purely within conformal field theory. For the Gepner model (2)^4 and some of its orbifolds we find the locations in ${\cal M}$ and prove isomorphisms to nonlinear sigma models. In particular we prove that the Gepner model (2)^4 has a geometric interpretation with Fermat quartic target space. 
  A relationship between the action-angle variables and the canonical transformation relating the rational Calogero-Moser system to the free one is discussed. 
  We discuss the formulation of the CGHS model in terms of a topological BF theory coupled to particles carrying non-Abelian charge. 
  We consider the ordinary and noncommutative Dirac-Born-Infeld theories within the open string sigma-model. First, we propose a renormalization scheme, hybrid point splitting regularization, that leads directly to the Seiberg-Witten description including their two-form. We also show how such a form appears within the standard renormalization scheme just by some freedom in changing variables. Second, we propose a Wilson factor which has the noncommutative gauge invariance on the classical level and then compute the sigma-model partition function within one of the known renormalization scheme that preserves the noncommutative gauge invariance. As a result, we find the noncommutative Yang-Mills action. 
  The electroweak phase transition in the magnetic and hypermagnetic fields is studied in the Standard Model on the base of investigation of symmetry behaviour within the consistent effective potential of the scalar and magnetic fields at finite temperature. It includes the one-loop and daisy diagram contributions. All discovered fundamental fermions and bosons are taken into consideration with their actual masses. The Higgs boson mass is chosen to be in the energy interval 75 GeV $\le m_H \le$ 115 GeV. The effective potential calculated is real at sufficiently high temperatures due to mutual cancellation of the imaginary terms entering the one-loop and the daisy diagram parts. Symmetry behaviour shows that neither the magnetic nor the hypermagnetic field does not produce the sufficiently strong first order phase transition. For the field strengths $H, H_Y$ $\ge 10^{23}$ G the electroweak phase transition is of second order at all. Therefore, baryogenesis does not survive in the Standard Model in smooth magnetic fields. The problems on generation of the fields at high temperature and their stabilization are also discussed in a consistent way. In particular, it is determined that the nonabelian component of the magnetic field $ (gH)^{1/2} \sim g^{4/3}T$ has to be produced spontaneously. To investigate the stability problem the $W$-boson mass operator in the magnetic field at high temperature is calculated in one-loop approximation. The comparison with results obtained in other approaches is done. 
  We study the perturbative dynamics of noncommutative field theories on R^d, and find an intriguing mixing of the UV and the IR. High energies of virtual particles in loops produce non-analyticity at low momentum. Consequently, the low energy effective action is singular at zero momentum even when the original noncommutative field theory is massive. Some of the nonplanar diagrams of these theories are divergent, but we interpret these divergences as IR divergences and deal with them accordingly. We explain how this UV/IR mixing arises from the underlying noncommutativity. This phenomenon is reminiscent of the channel duality of the double twist diagram in open string theory. 
  We calculate the classical cross-section for absorption of a minimally coupled scalar in the double-centered D3-brane geometry. The dual field theory has gauge symmetry broken to S(U(N_1)*U(N_2)) and is on the Coulomb branch of N=4 Super Yang-Mills theory. Our analysis is valid at energy scales much smaller than the W particles mass, giving logarithmic corrections to the cross section calculated at the IR conformal fixed points. These corrections are associated with deformations of the N=4 Super Yang-Mills theory by irrelevant operators that break conformal invariance and correspond to processes where a virtual pair of gauge particles or a virtual pair of W bosons interact with the incident wave to create a pair of gauge particles. 
  We reanalyse the computation of the cosmological constant $\Lambda$ at two loops in recently proposed Superstring models without massless gravitini, both in the theta-function based formalism and by a detailed computation in the more explicit hyperelliptic description of the underlying genus two Riemann surface. $\Lambda$ is expressed as the integral over the surface moduli of an amplitude which is zero if susy is not completely broken, but we find it to be nonvanishing in the susy breaking models which can be given an explicitly workable fermionic formulation. Thus unfortunately the issue of getting realistic and perturbatively viable models from Superstring Theory remains open. 
  Explicit two-loop calculations in noncommutative $\phi^4_4$ theory are presented. It is shown that the model is two-loop renormalizable. 
  We demonstrate how actions for interacting superconformal field theories in (p+1)-dimensions arise as a result of gauge fixing worldvolume diffeomorphisms and fermionic kappa-symmetry in actions for super-p-branes propagating in superbackgrounds of AdS(p+2)xS(D-p-2) geometry. The method of nonlinear realizations and coset spaces is used for getting an explicit form of supervielbeins and superconnections of the AdSxS superbackgrounds, which are required for the construction of the superconformal theories. Subtleties of consistent gauge fixing worldvolume symmetries of the branes are discussed. 
  No-go theorems on gauge-symmetry-deforming interactions of chiral p-forms are reviewed. We consider the explicit case of p=4, D=10 and show that the only symmetry-deforming consistent vertex for a system of one chiral 4-form and two 2-forms is the one that occurs in the type II B supergravity Lagrangian. 
  In this ``experimental'' research, we use known topological recursion relations in genera-zero, -one, and -two to compute the n-point descendant Gromov-Witten invariants of P^1 for arbitrary degrees and low values of n. The results are consistent with the Virasoro conjecture and also lead to explicit computations of all Hodge integrals in these genera. We also derive new recursion relations for simple Hurwitz numbers similar to those of Graber and Pandharipande. 
  We study conformal field theory correlation functions relevant for string diagrams with open strings that stretch between several parallel branes of different dimensions. In the framework of conformal field theory, they involve boundary condition changing twist fields which intertwine between Neumann and Dirichlet conditions. A Knizhnik-Zamolodchikov-like differential equation for correlators of such boundary twist fields and ordinary string vertex operators is derived, and explicit integral formulas for its solutions are provided. 
  Theories of partial supersymmetry breaking N=2 -> N=1 in four dimensions are derived by coupling the N=2 massless gravitino multiplet to N=2 supergravity in five dimensions and performing a generalized dimensional reduction on S^1/Z_2 with the Scherk-Schwarz mechanism. These theories agree with results that were previously derived from four dimensions. 
  We investigate domain wall junctions in a generalized Wess-Zumino model with a Z(N) symmetry. We present a method to identify the junctions which are potentially BPS saturated. We then use a numerical simulation to show that those junctions indeed saturate the BPS bound for N=4. In addition, we study the decay of unstable non-BPS junctions. 
  We determine the low energy dynamics of monopoles in pure N=2 Yang-Mills theories for points in the vacuum moduli space where the two Higgs fields are not aligned. The dynamics is governed by a supersymmetric quantum mechanics with potential terms and four real supercharges. The corresponding superalgebra contains a central charge but nevertheless supersymmetric states preserve all four supercharges. The central charge depends on the sign of the electric charges and consequently so does the BPS spectrum. We focus on the SU(3) case where certain BPS states are realised as zero-modes of a Dirac operator on Taub-NUT space twisted by the tri-holomorphic Killing vector field. We show that the BPS spectrum includes hypermultiplets that are consistent with the strong- and weak-coupling behaviour of the Seiberg-Witten theory. 
  We find the most general low energy dynamics of 1/2 BPS monopoles in the N=4 supersymmetric Yang-Mills theories (SYM) when all six adjoint Higgs expectation values are turned on. When only one Higgs is turned on, the Lagrangian is purely kinetic. When all six are turned on, however, this moduli space dynamics is augmented by five independent potential terms, each in the form of half the squared norm of a Killing vector field on the moduli space. A generic stationary configuration of the monopoles can be interpreted as stable non BPS dyons, previously found as non-planar string webs connecting D3-branes. The supersymmetric extension is also found explicitly, and gives the complete quantum mechanics of monopoles in N=4 SYM theory. We explore its supersymmetry algebra. 
  We apply the new orbifold duality transformations to discuss the special case of cyclic coset orbifolds in further detail. We focus in particular on the case of the interacting cyclic coset orbifolds, whose untwisted sectors are Z_\lambda(permutation)-invariant g/h coset constructions which are not \lambda copies of coset constructions. Because \lambda copies are not involved, the action of Z_\lambda(permutation) in the interacting cyclic coset orbifolds can be quite intricate. The stress tensors and ground state conformal weights of all the sectors of a large class of these orbifolds are given explicitly and special emphasis is placed on the twisted h subalgebras which are generated by the twisted (0,0) operators of these orbifolds. We also discuss the systematics of twisted (0,0) operators in general coset orbifolds. 
  We address problems associated with compactification near and on the light front. In perturbative scalar field theory we illustrate and clarify the relationships among three approaches: (1) quantization on a space-like surface close to a light front; (2) infinite momentum frame calculations; and (3) quantization on the light front. Our examples emphasize the difference between zero modes in space-like quantization and those in light front quantization. In particular, in perturbative calculations of scalar field theory using discretized light cone quantization there are well-known ``zero-mode induced'' interaction terms. However, we show that they decouple in the continuum limit and covariant answers are reproduced. Thus compactification of a light-like surface is feasible and defines a consistent field theory. 
  We explore two different problems in string theory in which duality relates an ordinary p-form in one theory to a self-dual (p+1)-form in another theory. One problem involves comparing D4-branes to M5-branes, and the other involves comparing the Ramond-Ramond forms in Type IIA and Type IIB superstring theory. In each case, a subtle topological effect involving the p-form can be recovered from a careful analysis of the quantum mechanics of the self-dual (p+1)-form. 
  The partition function of a two dimensional Abelian gauge model reproducing magnetic vortices is discussed in the harmonic approximation. Classical solutions exhibit conformal invariance, that is broken by statistical fluctuations, apart from an exceptional case. The corresponding ``anomaly'' has been evaluated. Zero modes of the thermal fluctuation operator have been carefully discussed. 
  Elementary MAPLE calculations are used to support the claim of hep-th/9906240 that the ratios of theta-functions, associated with the Seiberg-Witten complex curves, provide Poisson-commuting Hamiltonians which describe the dual of the original Seiberg-Witten integrable system. 
  The superspace flatness conditions which are equivalent to the field equations of supersymmetric Yang-Mills theory in ten dimensions have not been useful so far to derive non trivial classical solutions. Recently, modified flatness conditions were proposed, which are explicitly integrable (hep-th/9811108), and are based on the breaking of symmetry SO(9,1) -> SO(2,1)xSO(7). In this article, we investigate their physical content. To this end, group-algebraic methods are developed which allow to derive the set of physical fields and their equations of motion from the superfield expansion of the supercurl, systematically.   A set of integrable superspace constraints is identified which drastically reduces the field content of the unconstrained superfield but leaves the spectrum including the original Yang-Mills vector field completely off-shell. A weaker set of constraints gives rise to additional fields obeying first order differential equations. Geometrically, the SO(7) covariant superspace constraints descend from a truncation of Witten's original linear system to particular one-parameter families of light-like rays. 
  We study the world-volume effective action of stable non-BPS branes present in Type II theories compactified on K3. In particular, by exploiting the conformal description of these objects available in the orbifold limit, we argue that their world-volume effective theory can be chiral. The resulting anomalies are cancelled through the usual inflow mechanism provided there are anomalous couplings, similar to those of BPS branes, to the twisted R-R fields. We also show that this result is in agreement with the conjectured interpretation of these non-BPS configurations as BPS branes wrapped on non-supersymmetric cycles of the K3. 
  We construct a deformation of the quantum algebra Fun(T^*G) associated with Lie group G to the case where G is replaced by a quantum group G_q which has a bicovariant calculus. The deformation easily allows for the inclusion of the current algebra of left and right invariant one forms. We use it to examine a possible generalization of the Gauss law commutation relations for gauge theories based on G_q. 
  This paper gives a complete selfcontained proof of our result announced in hep-th/9909126 showing that renormalization in quantum field theory is a special instance of a general mathematical procedure of extraction of finite values based on the Riemann-Hilbert problem.   We shall first show that for any quantum field theory, the combinatorics of Feynman graphs gives rise to a Hopf algebra $\Hc$ which is commutative as an algebra. It is the dual Hopf algebra of the envelopping algebra of a Lie algebra $\ud G$ whose basis is labelled by the one particle irreducible Feynman graphs. The Lie bracket of two such graphs is computed from insertions of one graph in the other and vice versa. The corresponding Lie group $G$ is the group of characters of $\Hc$.   We shall then show that, using dimensional regularization, the bare (unrenormalized) theory gives rise to a loop $$ \g (z) \in G \qquad z \in C $$ where $C$ is a small circle of complex dimensions around the integer dimension $D$ of space-time. Our main result is that the renormalized theory is just the evaluation at $z = D$ of the holomorphic part $\g_+$ of the Birkhoff decomposition of $\g$.   We begin to analyse the group $G$ and show that it is a semi-direct product of an easily understood abelian group by a highly non-trivial group closely tied up with groups of diffeomorphisms. The analysis of this latter group as well as the interpretation of the renormalization group and of anomalous dimensions are the content of our second paper with the same overall title. 
  It is easy to sum chain-free self-energy rainbows, to obtain contributions to anomalous dimensions. It is also easy to resum rainbow-free self-energy chains. Taming the combinatoric explosion of all possible nestings and chainings of a primitive self-energy divergence is a much more demanding problem. We solve it in terms of the coproduct $\Delta$, antipode S, and grading operator Y of the Hopf algebra of undecorated rooted trees. The vital operator is $S\star Y$, with a star product effected by $\Delta$. We perform 30-loop Pad\'e-Borel resummation of 463 020 146 037 416 130 934 BPHZ subtractions in Yukawa theory, at spacetime dimension d=4, and in a trivalent scalar theory, at d=6, encountering residues of $S\star Y$ that involve primes with up to 60 digits. Even with a very large Yukawa coupling, g=30, the precision of resummation is remarkable; a 31-loop calculation suggests that it is of order $10^{-8}$. 
  Here we examine the noncommutative counterpart of QED, which is called as noncommutative QED. The theory is obtained by examining the consistent minimal coupling to noncommutative U(1) gauge field. The *-product admits the coupling of the matter with only three varieties of charges, i.e., 0, +1 and -1. Ultraviolet divergence can be absorbed into the rescaling of the fields and the parameters at least at one loop level. To examine the infrared aspect of the theory the anomalous magnetic dipole moment is calculated. The dependence on the direction of photon momentum reflects the Lorentz symmetry violation of the system. The explicit calculation of the finite part of the photon vacuum polarization shows the singularity ln({q C^TC q}) (C^{\mu\nu} is a noncommutative parameter.) in the infrared side which also exists in noncommutative Yang-Mills theory. It is associated with the ultraviolet behavior of the theory. We also consider the extension to chiral gauge theory in the present context, but the requirement of anomaly cancellation allows only noncommutative QED. 
  I study variations of the fermionic determinant for a nonabelian Dirac fermion with external vector and axial vector sources. I consider different regularizations, leading to different chiral anomalies when the variations are chiral transformations. For these different regularizations, I then consider variations associated with Poincare transformations. I find that both Lorentz and translational invariance are anomalously violated in general, but that they are respected when the variations of the determinant are regularized to give a Wess-Zumino consistent anomaly (the Bardeen anomaly). If the variations are regularized to give a covariant anomaly, then Poincare invariance is not respected. Following Manohar in an investigation of Poincare anomalies in a chiral gauge theory, this gives an alternative way to understand the need for a consistent regularization of the variations of the fermionic determinant. 
  We discuss the microscopic states of the extremal BTZ black holes. Degeneracy of the primary states corresponding to the extremal BTZ black holes in the boundary N=(4,4) SCFT is obtained by utilizing the elliptic genus and the unitary representation theory of N=4 SCA. The degeneracy is consistent with the Bekenstein-Hawking entropy. 
  In this paper we show that the one-loop graviton self-energy contribution is ultraviolet finite, without introducing counterterms, and cutoff-free in the framework of causal perturbation theory. In addition, it satisfies the gravitational Slavnov-Ward identities for the two-point connected Green function. The condition of perturbative gauge invariance to second order for loop graphs is proved. Corrections to the Newtonian potential are also derived. 
  We present evidence that there is a 4D model that satisfies the conditions of renormalizability and diffeomorphism invariance simultaneously at the 2-loop level. The traceless mode is treated perturbatively, while the conformal mode can be managed exactly. The two conditions constrain the theory strongly and determine the measure of the gravitational field uniquely. Quantum corrections of the cosmological constant are computed in part to 3-loop diagrams. The method to remove the negative-metric states is also discussed from the viewpoint of diffeomorphism invariance in analogy to the $R_{\xi}$ gauge in spontaneously broken gauge theory. The model may be a candidate for a continuum version of 4D simplicial quantum geometry realized in recent numerical simulations. 
  We write a 3D equation for three fermions by combining the three two-body potentials obtained in 3D reductions (based on a series expansion around a relative-energy fixing "approximation" of the free propagators) of the corresponding two-fermion Bethe-Salpeter equations to equivalent 3D equations, putting the third fermion on its positive-energy mass shell. In this way, the cluster-separated limits are exact, and the Lorentz invariance / cluster separability requirement is automatically satisfied, provided no supplementary approximation, like the Born approximation, is made. The use of positive free-energy projectors in the chosen reductions of the two-fermion Bethe-Salpeter equations prevents continuum dissolution in our 3D three-fermion equation. The potentials are hermitian below the inelastic threshold and depend only slowly on the total three-fermion energy. This "hand-made" three-fermion 3D equation is also obtained by starting with an approximation of the three-fermion Bethe-Salpeter equation, in which the three-body kernel is neglected and the two-body kernels approached by positive-energy instantaneous expressions, with the spectator fermion on the mass shell. The neglected terms are then transformed into corrections to the 3D equation, in three steps implying each a series expansion. The result is of course complicated, but the lowest-order contributions of these correction terms to the energy spectrum remain manageable.We also present some other 3D reduction procedures and compare them to our's: use of Sazdjian's covariant approximation of the free propagator, 3D reductions performed by a series expansion around instantaneous approximations of the kernels instead of "approximations" of the propagators, Gross' spectator model. 
  We study the mass and different RR charge distributions of the BPS (p,p-2)-brane bound states in the closed string brane-like $\sigma$-model. We show that such brane bound states can be realized by introducing a constant B field in the closed string theory. In addition we show that the worldvolume coordinates of these brane bound states turn out to be noncommutative. 
  In this note we point out that a symmetric product orbifold CFT can be twisted by a unique nontrivial two-cocycle of the permutation group. This discrete torsion changes the spins and statistics of corresponding second-quantized string theory making it essentially ``supersymmetric.'' The long strings of even length become fermionic (or ghosts), those of odd length bosonic. The partition function and elliptic genus can be described by a sum over stringy spin structures. The usual cubic interaction vertex is odd and nilpotent, so this construction gives rise to a DLCQ string theory with a leading quartic interaction. 
  The Casimir force between parallel plates of arbitrary kind is shown to be simply related to the plates transmission and reflection coefficient. A trivial application of this general relation leads to the known Lifshitz force between dielectrics as well as its generalizations. 
  We study matrix string scattering amplitudes and matrix string instantons on a marked Riemann surface in the limit of a vanishing string coupling constant. We give an explicit parameterization of the moduli space of such instantons. We also give a description of the set of fermionic supermoduli. The integration over the supermoduli leads to the inclusion of picture changing operators at the interaction points. Finally we investigate the large N limit of the measure on the instanton moduli space and show its convergence to the Weil-Petersson measure on the moduli space of marked Riemann surfaces. 
  In these notes we review the role played by the quantum mechanics and sigma models of symmetric product spaces in the light-cone quantization of quantum field theories, string theory and matrix theory. Lectures given at the Institute for Theoretical Physics, UC Santa Barbara, January 1998 and the Spring School on String Theory and Mathematics, Harvard University, May 1998. 
  We show that the conformal invariance conditions for a general sigma-model with torsion are invariant under T-duality through two loops. 
  We consider type IIB (p,q) 5-branes in constant non-zero background tensor potentials, or equivalently, with finite constant field strength on the brane. At linear level, zero-modes are introduced and the physical degrees of freedom are found to be parametrised by a real 2- or 4-form field strength on the brane. An exact, SL(2;Z)-covariant solution to the full non-linear supergravity equations is then constructed. The resulting metric space-times are analysed, with special emphasis on the limiting cases with maximal values of the tensor. The analysis provides an answer to how the various background tensor fields are related to Born-Infeld degrees of freedom and to non-commutativity parameters. 
  We compute the mass and multiplet spectrum of M theory compactified on the product of AdS(4) spacetime by the Stiefel manifold V(5,2)=SO(5)/SO(3), and we use this information to deduce via the AdS/CFT map the primary operator content of the boundary N=2 conformal field theory. We make an attempt for a candidate supersymmetric gauge theory that, at strong coupling, should be related to parallel M2-branes on the singular point of the non-compact Calabi-Yau four-fold $\sum_{a=1}^5 z_a^2 = 0$, describing the cone on V(5,2). 
  The cancellation of U(1)-gauge and U(1)-gravitational anomalies in certain D=4 N=1 Type IIB orientifolds is analyzed in detail, from a string theory point of view. We verify the proposal that these anomalies are cancelled by a Green-Schwarz mechanism involving only twisted Ramond-Ramond fields. By factorizing one-loop partition functions, we also get the anomalous couplings of D-branes, O-planes and orbifold fixed-points to these twisted fields. Twisted sectors with fixed-planes participate to the inflow mechanism in a peculiar way. 
  We develop a new, systematic approach towards studying the integrability of the ordinary Calogero-Moser-Sutherland models as well as the elliptic Calogero models associated with arbitrary (semi-)simple Lie algebras and with symmetric pairs of Lie algebras. It is based on the introduction of a function F, defined on the relevant root system and with values in the respective Cartan subalgebra, satisfying a certain set of combinatoric identities that ensure, in one stroke, the existence of a Lax representation and of a dynamical R-matrix, given by completely explicit formulas. It is shown that among the simple Lie algebras, only those belonging to the A-series admit such a function F, whereas the AIII-series of symmetric pairs of Lie algebras, corresponding to the complex Grassmannians SU(p,q)/S(U(p) x U(q)), allows non-trivial solutions when |p-q| <= 1. Apart from reproducing all presently known dynamical R-matrices for Calogero models, our approach provides new ones, namely for the ordinary models when |p-q| = 1 and for the elliptic models when |p-q| = 1 or p = q. 
  We dispute a recent claim for a nonperturbative nonrenormalisation theorem stating that mass cannot be spontaneously generated in supersymmetric QED. We also extend a long-standing perturbative result, namely that the effective potential is zero to all orders of perturbation theory, to the nonperturbative regime for arbitrary numbers of flavours. 
  Cylindrical gravitational waves of Einstein gravity are described by an integrable system (Ernst system) whose quantization is a long standing problem. We propose to bootstrap the quantum theory along the following lines: The quantum theory is described in terms of matrix elements e.g. of the metric operator between spectral-transformed multi-vielbein configurations. These matrix elements are computed exactly as solutions of a recursive system of functional equations, which in turn is derived from an underlying quadratic algebra. The Poisson algebra emerging in its classical limit links the spectral-transformed vielbein and the non-local conserved charges and can be derived from first principles within the Ernst system. Among the noteworthy features of the quantum theory are: (i) The issue of (non-)renormalizability is sidestepped and (ii) there is an apparently unavoidable ``spontaneous'' breakdown of the SL(2,R) symmetry that is a remnant of the 4D diffeomorphism invariance in the compactified dimensions. 
  The reductions of conformal field theories which lead to generalized abelian cosets are studied. Primary fields and correlation functions of arbitrary abelian coset conformal field theory are explicitly expressed in terms of those of the original theory. The coset theory has global abelian symmetry. 
  This report is an extension of previous one hep-th/9812189. Several quantum mechanical wave equations for $p$-branes are proposed. The most relevant $p$-brane quantum mechanical wave equations determine the quantum dynamics involving the creation/destruction of $p$-dimensional loops of topology $S^p$, moving in a $D$ dimensional spacetime background, in the quantum state $\Phi$. To implement full covariance we are forced to enlarge the ordinary Relativity principle to a $new$ Relativity principle, suggested earlier by the author based on the construction of {\bf C}-space, and also by Pezzaglia's Polydimensional Relativity, where all dimensions and signatures of spacetime should be included on the same footing. 
  One-loop calculations in quantum gravity coupled to U(1)-Abelian fields (photon fields) are ultraviolet finite and cutoff-free in the framework of causal perturbation theory. We compute the photon loop correction to the graviton propagator and the photon self-energy in second order perturbation theory. Perturbative gauge invariance to second order is shown and generates the gravitational Slavnov-Ward identities. 
  It is argued that the formal rules of correspondence between local observation procedures and observables do not exhaust the entire physical content of generally covariant quantum field theory. This result is obtained by expressing the distinguishing features of the local kinematical structure of quantum field theory in the generally covariant context in terms of a translocal structure which carries the totality of the nonlocal kinematical informations in a local region. This gives rise to a duality principle at the dynamical level which emphasizes the significance of the underlying translocal structure for modelling a minimal algebra around a given point. We discuss the emergence of classical properties from this point of view. 
  The condensation of a dual-vector field is investigated in the dual Monopole Nambu-Jona-Lasinio model with dual Dirac strings. The condensate of a dual-vector field is calculated as a functional of a shape of a dual Dirac string. The obtained result is compared with the gluon condensate calculated in a QCD sum rules approach and on lattice. 
  The $D$-brane spectra in the Type IIB string theory compactified on $AdS_{p+2}\times S^{8-p}$ are computed using the K-theory approach. The results signal the existence of the mirror-symmetry-analogue for $D$-branes, analogous to that rised in the context of derived categories. 
  An inflating brane-world can be created from ``nothing'' together with its Anti-de Sitter (AdS) bulk. The resulting space-time has compact spatial sections bounded by the brane. During inflation, the continuum of KK modes is separated from the massless zero mode by the gap $m=(3/2) H$, where $H$ is the Hubble rate. We consider the analog of the Nariai solution and argue that it describes pair production of ``Black cigars'' attached to the inflating brane. In the case when the size of the instantons is much larger than the AdS radius, the 5-dimensional action agrees with the 4-dimensional one. Hence, the 5D and 4D gravitational entropies are the same in this limit. We also consider thermal instantons with an AdS black hole in the bulk. These may be interpreted as describing the creation of a hot universe from nothing, or the production of AdS black holes in the vicinity of a pre-existing inflating brane-world. The Lorentzian evolution of the brane-world after creation is briefly discussed. An additional "integration constant" in the Friedmann equation -accompanying a term which dilutes like radiation- describes the tidal force in the fifth direction and arises from the mass of a spherical object inside the bulk. This could be a 5-dimensional black hole or a "parallel" brane-world of negative tension concentrical with our brane-world. In the case of thermal solutions, and in the spirit of the $AdS/CFT$ correspondence, one may attribute the additional term to thermal radiation in the boundary theory. Then, for temperatures well below the AdS scale, the entropy of this radiation agrees with the entropy of the black hole in the AdS bulk. 
  In gr-qc/9902061 it was shown that (n+1)-dimensional asymptotically anti-de-Sitter spacetimes obeying natural causality conditions exhibit topological censorship. We use this fact in this paper to derive in arbitrary dimension relations between the topology of the timelike boundary-at-infinity, $\scri$, and that of the spacetime interior to this boundary. We prove as a simple corollary of topological censorship that any asymptotically anti-de Sitter spacetime with a disconnected boundary-at-infinity necessarily contains black hole horizons which screen the boundary components from each other. This corollary may be viewed as a Lorentzian analog of the Witten and Yau result hep-th/9910245, but is independent of the scalar curvature of $\scri$. Furthermore, the topology of V', the Cauchy surface (as defined for asymptotically anti-de Sitter spacetime with boundary-at-infinity) for regions exterior to event horizons, is constrained by that of $\scri$. In this paper, we prove a generalization of the homology results in gr-qc/9902061 in arbitrary dimension, that H_{n-1}(V;Z)=Z^k where V is the closure of V' and k is the number of boundaries $\Sigma_i$ interior to $\Sigma_0$. As a consequence, V does not contain any wormholes or other compact, non-simply connected topological structures. Finally, for the case of n=2, we show that these constraints and the onto homomorphism of the fundamental groups from which they follow are sufficient to limit the topology of interior of V to either B^2 or $I\times S^1$. 
  An open superstring field theory action has been proposed which does not suffer from contact term divergences. In this paper, we compute the on-shell four-point tree amplitude from this action using the Giddings map. After including contributions from the quartic term in the action, the resulting amplitude agrees with the first-quantized prescription. 
  I review the construction of an action for open superstring field theory which does not suffer from the contact term problems of other approaches. I also discuss a possible generalization of this action for closed superstring field theory. 
  I discuss the properties of the central charges c and a for higher-derivative and higher-spin theories (spin 2 included). Ordinary gravity does not admit a straightforward identification of c and a in the trace anomaly, because it is not conformal. On the other hand, higher-derivative theories can be conformal, but have negative c and a. A third possibility is to consider higher-spin conformal field theories. They are not unitary, but have a variety of interesting properties. Bosonic conformal tensors have a positive-definite action, equal to the square of a field strength, and a higher-derivative gauge invariance. There exists a conserved spin-2 current (not the canonical stress tensor) defining positive central charges c and a. I calculate the values of c and a and study the operator-product structure. Higher-spin conformal spinors have no gauge invariance, admit a standard definition of c and a and can be coupled to Abelian and non-Abelian gauge fields in a renormalizable way. At the quantum level, they contribute to the one-loop beta function with the same sign as ordinary matter, admit a conformal window and non-trivial interacting fixed points. There are composite operators of high spin and low dimension, which violate the Ferrara-Gatto-Grillo theorem. Finally, other theories, such as conformal antisymmetric tensors, exhibit more severe internal problems. This research is motivated by the idea that fundamental quantum field theories should be renormalization-group (RG) interpolations between ultraviolet and infrared conformal fixed points, and quantum irreversibility should be a general principle of nature. 
  We find further evidence for the conjecture relating large N Chern-Simons theory on S^3 with topological string on the resolved conifold geometry by showing that the Wilson loop observable of a simple knot on S^3 (for any representation) agrees to all orders in N with the corresponding quantity on the topological string side. For a general knot, we find a reformulation of the knot invariant in terms of new integral invariants, which capture the spectrum (and spin) of M2 branes ending on M5 branes embedded in the resolved conifold geometry. We also find an intriguing link between knot invariants and superpotential terms generated by worldsheet instantons in N=1 supersymmetric theories in 4 dimensions. 
  Duality in the integrable systems arising in the context of Seiberg-Witten theory shows that their tau-functions indeed can be seen as generating functions for the mutually Poisson-commuting hamiltonians of the {\em dual} systems. We demonstrate that the $\Theta$-function coefficients of their expansion can be expressed entirely in terms of the co-ordinates of the Seiberg-Witten integrable system, being, thus, some set of hamiltonians for a dual system. 
  We derive crystal braneworld solutions, comprising of intersecting families of parallel $n+2$-branes in a $4+n$-dimensional $AdS$ space. Each family consists of alternating positive and negative tension branes. In the simplest case of exactly orthogonal families, there arise different crystals with unbroken 4D Poincare invariance on the intersections, where our world can reside. A crystal can be finite along some direction, either because that direction is compact, or because it ends on a segment of $AdS$ bulk, or infinite, where the branes continue forever. If the crystal is interlaced by connected 3-branes directed both along the intersections and orthogonal to them, it can be viewed as an example of a Manyfold universe proposed recently by Arkani-Hamed, Dimopoulos, Dvali and the author. There are new ways for generating hierarchies, since the bulk volume of the crystal and the lattice spacing affect the 4D Planck mass. The low energy physics is sensitive to the boundary conditions in the bulk, and has to satisfy the same constraints discussed in the Manyfold universe. Phenomenological considerations favor either finite crystals, or crystals which are infinite but have broken translational invariance in the bulk. The most distinctive signature of the bulk structure is that the bulk gravitons are Bloch waves, with a band spectrum, which we explicitly construct in the case of a 5-dimensional theory. 
  The spin of a glueball is usually taken as coming from the spin (and possibly the orbital angular momentum) of its constituent gluons. In light of the difficulties in accounting for the spin of the proton from its constituent quarks, the spin of glueballs is reexamined. The starting point is the fundamental QCD field angular momentum operator written in terms of the chromoelectric and chromomagnetic fields. First, we look at the restrictions placed on the structure of glueballs from the requirement that the QCD field angular momentum operator should satisfy the standard commutation relationships. This can be compared to the electromagnetic charge/monopole system, where the quantization of the field angular momentum places restrictions (i.e. the Dirac condition) on the system. Second, we look at the expectation value of this operator under some simplifying assumptions. 
  A correspondence between the three-block truncated 11D supergravity and the 8D pure Einstein gravity with two commuting Killing symmetries is discussed. The Kaluza-Klein two-forms of the 6D theory obtained after dimensional reduction along the Killing orbits generate the four-form field of supergravity via an inverse dualization. Thus any solution to the vacuum Einstein equations in eight dimensions depending on six coordinates have 11D-supergravity counterparts with the non-trivial four-form field. Using this proposed duality we derive a new dyon solution of 11D supergravity describing the M2 and M5-branes intersecting at a point. 
  Quantum gravity coupled to scalar massive matter fields is investigated within the framework of causal perturbation theory. One-loop calculations include matter loop graviton self-energy and matter self-energy and yield ultraviolet finite and cutoff-free expressions. Perturbative gauge invariance to second order implies the usual Slavnov-Ward identities for the graviton self-energy in the loop graph sector and generates the correct quartic graviton-matter interaction in the tree graph sector. The mass zero case is also discussed. 
  The problem of gauge independent definition of effective gravitational field is considered from the point of view of the process of measurement. Under assumption that dynamics of the measuring apparatus can be described by the ordinary classical action, effective Slavnov identities for the generating functionals of Green functions corresponding to a system of arbitrary gravitational field measured by means of scalar particles are obtained. With the help of these identities, the total gauge dependence of the non-local part of the one-loop effective apparatus action, describing the long-range quantum corrections, is calculated. The value of effective gravitational field inferred from the effective apparatus action is found to be gauge-dependent. A probable explanation of this result, referring to a peculiarity of the gravitational interaction, is given. 
  We describe how and to what extent the noncommutative two-torus can be approximated by a tower of finite-dimensional matrix geometries. The approximation is carried out for both irrational and rational deformation parameters by embedding the algebra of the noncommutative torus into an approximately finite algebra. The construction is a rigorous derivation of the recent discretizations of noncommutative gauge theories using finite dimensional matrix models, and it shows precisely how the continuum limits of these models must be taken. We clarify various aspects of Morita equivalence using this formalism and describe some applications to noncommutative Yang-Mills theory. 
  The freedom in choosing finite renormalizations in quantum field theories (QFT) is characterized by a set of parameters $\{c_i \}, i = 1 ..., n >...$, which specify the renormalization prescriptions used for the calculation of physical quantities. For the sake of simplicity, the case of a single $c$ is selected and chosen mass-independent if masslessness is not realized, this with the aim of expressing the effect of an infinitesimal change in $c$ on the computed observables. This change is found to be expressible in terms of an equation involving a vector field $V$ on the action's space $M$ (coordinates x). This equation is often referred to as ``evolution equation'' in physics. This vector field generates a one-parameter (here $c$) group of diffeomorphisms on $M$. Its flow $\sigma_c (x)$ can indeed be shown to satisfy the functional equation $$ \sigma_{c+t} (x) = \sigma_c (\sigma_t (x)) \equiv \sigma_c \circ \sigma_t $$ $$\sigma_0 (x) = x,$$ so that the very appearance of $V$ in the evolution equation implies at once the Gell-Mann-Low functional equation. The latter appears therefore as a trivial consequence of the existence of a vector field on the action's space of renormalized QFT. 
  We provide an algebraic classification of all supersymmetric domain wall solutions of maximal gauged supergravity in four and seven dimensions, in the presence of non-trivial scalar fields in the coset SL(8,R)/SO(8) and SL(5,R)/SO(5) respectively. These solutions satisfy first-order equations, which can be obtained using the method of Bogomol'nyi. From an eleven-dimensional point of view they correspond to various continuous distributions of M2- and M5-branes. The Christoffel-Schwarz transformation and the uniformization of the associated algebraic curves are used in order to determine the Schrodinger potential for the scalar and graviton fluctuations on the corresponding backgrounds. In many cases we explicitly solve the Schrodinger problem by employing techniques of supersymmetric quantum mechanics. The analysis is parallel to the construction of domain walls of five-dimensional gauged supergravity, with scalar fields in the coset SL(6,R)/SO(6), using algebraic curves or continuous distributions of D3-branes in ten dimensions. In seven dimensions, in particular, our classification of domain walls is complete for the full scalar sector of gauged supergravity. We also discuss some general aspects of D-dimensional gravity coupled to scalar fields in the coset SL(N,R)/SO(N). 
  We give a unified analysis of four-dimensional elliptic models with N=2 supersymmetry and a simple gauge group, and their relation to M-theory. Explicit calculations of the Seiberg-Witten curves and the resulting one-instanton prepotential are presented. The remarkable regularities that emerge are emphasized. In addition, we calculate the prepotential in the Coulomb phase of the (asymptotically-free) Sp(2N) gauge theory with N_f fundamental hypermultiplets of arbitrary mass. 
  The weak-field expansion of the charged fermion propagator under a uniform magnetic field is studied. Starting from Schwinger's proper-time representation, we express the charged fermion propagator as an infinite series corresponding to different Landau levels. This infinite series is then reorganized according to the powers of the external field strength $B$. For illustration, we apply this expansion to $\gamma\to \nu\bar{\nu}$ and $\nu\to \nu\gamma$ decays, which involve charged fermions in the internal loop. The leading and subleading magnetic-field effects to the above processes are computed. 
  We extend our recent discussion of four-dimensional black holes bound to a two-brane to include a negative cosmological constant on the brane. We find that for large masses, the solutions are precisely BTZ black holes on the brane, and BTZ `black strings' in the bulk. For smaller masses, there are localized black holes which look like BTZ with corrections that fall off exponentially. We compute when the maximum entropy configuration changes from the black string to the black hole. We also present exact solutions describing rotating black holes on two-branes which are either asymptotically flat or asymptotically $AdS_3$. The mass and angular momentum on the brane agree with that in the bulk. 
  We extend the gauge invariant variational approach of Phys. Rev. D52 (1995) 3719, hep-th/9408081, to theories with fermions. As the simplest example we consider the massless Schwinger model in 1+1 dimensions. We show that in this solvable model the simple variational calculation gives exact results. 
  A review is given of recent work aimed at constructing a quantum theory of cosmology in which all observables refer to information measurable by observers inside the universe. At the classical level the algebra of observables should be modified to take into account the fact that observers can only give truth values to observables that have to do with their backwards light cone. The resulting algebra is a Heyting rather than a Boolean algebra. The complement is non-trivial and contains information about horizons and topology change. Representation of such observables quantum mechanically requires a many-Hilbert space formalism, in which different observers make measurements in different Hilbert spaces. I describe such a formalism, called "quantum causal histories"; examples include causally evolving spin networks and quantum computers. 
  We test the renormalization of Wilson operators and the Mandelstam- Leibbrandt gauge in the case when the sides of the loop are parallel to the n, n* vectors used in the M-L gauge. Graphs which in the Feynman gauge are free of ultra-violet divergences, in the M-L gauge show double divergences and single divergences with non-local Si and Ci functions. These non-local functions cancel out when we add all graphs together and the constraints of gauge invariance are satisfied. In Appendix C we briefly discuss the problems of the M-L gauge for loops containing spacelike lines. 
  According to the holographic principle all information in the bulk of a space is coded at its border. We will check this statement in three situations involving the AdS/CFT correspondence. There is a well known equivalence between the Maxwell-Chern-Simons theory and the self-dual model in 3 dimensions when the parameters of both theories are related in a given way. We will show that when this relation holds the corresponding CFT's at the border are the same. Then we will study scalar fields. There are two quantum theories for the scalar field in AdS space. The usual prescription of the AdS/CFT correspondence which takes Dirichlet boundary conditions at the border corresponds to one of the quantum theories. We will show that changing boundary conditions will allow us to get the other quantum theory. Finally we consider an Abelian gauge theory in AdS. We will show that the corresponding CFT is independent of the gauge choice and that the gauge dependence stays only in the contact terms at the border. 
  The thermodynamics of gauge theories on the noncommutative plane is studied in perturbation theory. For U(1) noncommutative Yang-Mills we compute the first quantum correction to the ideal gas free energy density and study their behavior in the low and high temperature regimes. Since the noncommutativity scale effectively cutoff interactions at large distances, the theory is regular in the infrared. In the case of U(N) noncommutative Yang-Mills we evaluate the two-loop free energy density and find that it depends on the noncommutativity parameter through the contribution of non-planar diagrams. 
  By considering the gradient expansion for the wilsonian effective action S_k of a single component scalar field theory truncated to the first two terms, the potential U_k and the kinetic term Z_k, I show that the recent claim that different expansion of the fluctuation determinant give rise to different renormalization group equations for Z_k is incorrect. The correct procedure to derive this equation is presented and the set of coupled differential equations for U_k and Z_k is definitely established. 
  Abelian theories in three dimensions can have linearly confining phases as a result of monopole-instantons, as shown, for SU(2) Yang-Mills theory broken to its abelian subgroup, by Polyakov. In this article the generalization of this phase for N=2 supersymmetric abelian theories is identified, using a dual description. Topologically stable BPS-saturated and unsaturated particle and string solitons play essential roles. A plasma of chiral monopoles of charge 1 and -1 (along with their antichiral conjugates) are required for a stable confining vacuum. N=2 SU(2) Yang-Mills theory broken to U(1) lacks this phase because its chiral monopoles all have the same charge, leading to a runaway instability. The possibility of analogous confining phases of string theory, and a dual field theoretic model thereof, are briefly discussed. 
  We show that $SL(2;C)/SU(2)$ model which had been recently proposed to describe the behaviour of the local densities of states at the plateau transition in Integer Quantum Hall effect, has logarithmic operators. They unusual properties are studied in this letter. 
  The heterotic $E_8\times E_8$ string compactified on an orbifold $T^4/\IZ_N$ has gauge group $G\times G'$ with (massless) states in its twisted sectors which are charged under both gauge group factors. In the dual M-theory on $(T^4/\IZ_N)\otimes(S^1/\IZ_2)$ the two group factors are separated in the eleventh direction and the G and G' gauge fields are confined to the two boundary planes, respectively. We present a scenario which allows for a resolution of this apparent paradox and assigns all massless matter multiplets locally to the different six-dimensional boundary fixed planes. The resolution consists of diagonal mixing between the gauge groups which live on the connecting seven-planes (6d and the eleventh dimension) and one of the gauge group factors. We present evidence supporting this mixing by considering gauge couplings and verify local anomaly cancellation. We also discuss open problems which arise in the presence of U_1 factors. 
  We consider several properties of a set of anti-D$p$-branes in the presence of orientifold $p$-planes in type II theory. This system breaks all the supersymmetries of the theory, but is free of tachyons. In particular, we center on the case of a single anti-D$p$-brane stuck at a negatively charged orientifold $p$-plane, and study its strong coupling behaviour for $p=2,3,4$. Interestingly enough, as the coupling increases the system undergoes a phase transition where an additional antibrane is created. We conclude with some remarks on the limit of large number of antibranes on top of orientifold planes. 
  Exploiting Virasoro constraints on the effective finite-volume partition function, we derive generalized Leutwyler-Smilga spectral sum rules of the Dirac operator to high order. By introducing $N_v$ fermion species of equal masses, we next use the Virasoro constraints to compute two (low-mass and large-mass) expansions of the partially quenched chiral condensate through the replica method of letting $N_v \to 0$. The low-mass expansion can only be pushed to a certain finite order due to de Wit-'t Hooft poles, but the large-mass expansion can be carried through to arbitrarily high order. Results agree exactly with earlier results obtained through both Random Matrix Theory and the supersymmetric method. 
  B-type D-branes are constructed on two different K3-fibrations over IP_1 using boundary conformal field theory at the rational Gepner points of these models. The microscopic CFT charges are compared with the Ramond charges of D-branes wrapped on holomorphic cycles of the corresponding Calabi-Yau manifold. We study in particular D4-branes and bundles localized on the K3 fibers, and find from CFT that each irreducible component of a bundle on K3 gains one modulus upon fibration over IP_1. This is in agreement with expectations and so provides a further test of the boundary CFT. 
  In this comment we discuss some serious inconsistencies presented by Gomes, Malacarne and da Silva in their paper, Phys.Rev. D60 (1999) 125016 (hep-th/9908181). 
  We present an analytic calculation of the first transcendental in phi^4-Theory that is not of the form zeta(2n+1). It is encountered at 6 loops and known to be a weight 8 double sum. Here it is obtained by reducing multiple zeta values of depth <= 4. We give a closed expression in terms of a zeta-related sum for a family of diagrams that entails a class of physical graphs. We confirm that this class produces multiple zeta values of weights equal to the crossing numbers of the related knots. 
  We carefully re-examine the issues of solving the modified Bianchi identity, anomaly cancellations and flux quantization in the S^1/Z_2 orbifold of M-theory using the boundary-free "upstairs" formalism, avoiding several misconceptions present in earlier literature. While the solution for the four-form G to the modified Bianchi identity appears to depend on an arbitrary parameter b, we show that requiring G to be globally well-defined, i.e. invariant under small and large gauge and local Lorentz transformations, fixes b=1. This value also is necessary for a consistent reduction to the heterotic string in the small-radius limit. Insisting on properly defining all fields on the circle, we find that there is a previously unnoticed additional contribution to the anomaly inflow from the eleven-dimensional topological term. Anomaly cancellation then requires a quadratic relation between b and the combination lambda^6/kappa^4 of the gauge and gravitational coupling constants lambda and kappa. This contrasts with previous beliefs that anomaly cancellation would give a cubic equation for b. We observe that our solution for G automatically satisfies integer or half-integer flux quantization for the appropriate cycles. We explicitly write out the anomaly cancelling terms of the heterotic string as inherited from the M-theory approach. They differ from the usual ones by the addition of a well-defined local counterterm. We also show how five-branes enter our analysis. 
  We study the F-terms in N=1 supersymmetric, d=4 gauge theories arising from D(p+3)-branes wrapping supersymmetric p-cycles in a Calabi-Yau threefold. If p is even the spectrum and superpotential for a single brane are determined by purely classical ($\alpha^\prime \to 0$) considerations. If p=3, superpotentials for massless modes are forbidden to all orders in $\alpha^\prime$ and may only be generated by open string instantons. For this latter case we find that such instanton effects are generically present. Mirror symmetry relates even and odd p and thus perturbative and nonperturbative superpotentials; we provide a preliminary discussion of a class of examples of such mirror pairs. 
  We show how turning on Flux for RR (and NS-NS) field strengths on non-compact Calabi-Yau 3-folds can serve as a way to partially break supersymmetry from N=2 to N=1 by mass deformation. The freezing of the moduli of Calabi-Yau in the presence of the flux is the familiar phenomenon of freezing of fields in supersymmetric theories upon mass deformations. 
  We consider the open superstring ending on a D-brane in the presence of a constant NS-NS B field, using the Green-Schwarz formalism. Quantizing in the light-cone gauge, we find that the anti-commutation relations for the fermionic variables of superspace remain unmodified. We also derive the unbroken supersymmetry algebra living on the D-brane. This establishes how the Moyal product is extended in a superspace formulation of non-commutative field theories. The superfield formulation of non-commutative supersymmetric field theories is briefly considered. 
  We study the action of picture-changing and spectral flow operators on a ground ring of ghost number zero operators in the chiral BRST cohomology of the closed N=2 string and describe an infinite set of symmetry charges acting on physical states. The transformations of physical string states are compared with symmetries of self-dual gravity which is the effective field theory of the closed N=2 string. We derive all infinitesimal symmetries of the self-dual gravity equations in 2+2 dimensional spacetime and introduce an infinite hierarchy of commuting flows on the moduli space of self-dual metrics. The dependence on moduli parameters can be recovered by solving the equations of the SDG hierarchy associated with an infinite set of abelian symmetries generated recursively from translations. These non-local abelian symmetries are shown to coincide with the hidden abelian string symmetries responsible for the vanishing of most scattering amplitudes. Therefore, N=2 string theory "predicts" not only self-dual gravity but also the SDG hierarchy. 
  An analogy is noted between RG flow equations in 4-dimensional gauge theory, as derived from the AdS/CFT correspondence, and the RG flow equations in 4-dimensional field theory coupled to a particular limit of Weyl supergravity. This suggests a possible theory of dynamical 3-branes with fluctuating 4-dimensional conformal factor. The argument involves a map from flows in 4-dimensional gauge theories to flows in a class of 2-dimensional sigma models. 
  The hypothesis is discussed that our universe is really 5--dimensional with a nonzero cosmological constant that produces a large negative curvature. In this scenario, the observable flat 4--dimensional universe is identified with the holographic projection of the 5--dimensional world onto its own boundary. 
  We investigate the anomalous creation of fundamental strings using the boundary state formalism of fractional D-branes on ALE spaces in the orbifold limit. The open string Witten index plays a crucial role in this calculation and so the result remains unchanged even if we blow up the orbifold geometrically, matching the anomaly inflow argument. Further we consider the quiver gauge theories on such fractional D3-branes and see that the string creation mechanism determines 1-loop logarithmic monodromy of these gauge theories. Also we comment on the relation of D(-1)-D3 amplitude to the 1-loop beta function. 
  Using the {\em cutting and sewing} procedure we show how to get Feynman diagrams, up to two-loop order, of $\Phi^{4}$-theory with an internal SU(N) symmetry group, starting from tachyon amplitudes of the open bosonic string theory. In a properly defined field theory limit, we easily identify the corners of the string moduli space reproducing the correctly normalized field theory amplitudes expressed in the Schwinger parametrization. 
  The S-duality transformations in type IIB string theory can be seen as local U(1) transformations in type IIB supergravity. We use this approach to construct the $SL(2,Z)$ multiplets associated to supersymmetric backgrounds of type IIB string theory and the transformation laws of their corresponding Killing spinors. 
  We calculate the linearized metric perturbation corresponding to a massless four-dimensional scalar field, the radion, in a five-dimensional two-brane model of Randall and Sundrum. In this way we obtain relative strengths of the radion couplings to matter residing on each of the branes. The results are in agreement with the analysis of Garriga and Tanaka of gravitational and Brans--Dicke forces between matter on the branes. We also introduce a model with infinite fifth dimension and ``almost'' confined graviton, and calculate the radion properties in that model. 
  In these lectures we present a detailed description of the origin and of the construction of the boundary state that is now widely used for studying the properties of D branes. (Lectures given at NATO-ASI on "Quantum Geometry" in Akureyri, Iceland, August 1999). 
  We study BPS saturated objects with axial geometry (wall junctions, vortices) in generalized Wess-Zumino models. It is observed that the tension of such objects is negative in general (although ``exceptional'' models are possible). We show how an ambiguity in the definition of central charges does not affect physical quantities, and we comment on the stability of the junctions and vortices. We illustrate these issues in two classes of models with Z_N symmetry. On the basis of analytical large N calculations and numerical calculations at finite N, we argue that the domain wall junctions in these models are indeed BPS saturated, and we calculate the junction tensions explicitly. 
  Some ideas are presented concerning the question which of the harmonic wavefunctions constructed in [hep-th/9909191] may be annihilated by all supercharges. 
  In the light of the duality between physics in the bulk of anti-de Sitter space and a conformal field theory on the boundary, we review the M2, D3 and M5 branes and how their near-horizon geometry yields the compactification of D=11 supergravity on S^{7}, Type IIB supergravity on S^{5} and D=11 supergravity on S^{4}, respectively. We discuss the ``Membrane at the End of the Universe'' idea and its relation to the corresponding superconformal singleton theories that live on the boundary of the AdS_{4}, AdS_{5} and AdS_{7} vacua. The massless sectors of these compactifications are described by the maximally supersymmetric D=4, D=5 and D=7 gauged supergravities. We construct the non-linear Kaluza-Klein ans\"atze describing the embeddings of the U(1)^4, U(1)^3 and U(1)^2 truncations of these supergravities, which admit 4-charge AdS_{4}, 3-charge AdS_{5} and 2-charge AdS_{7} black hole solutions. These enable us to embed the black hole solutions back in ten and eleven dimensions and reinterpret them as M2, D3 and M5 branes spinning in the transverse dimensions with the black hole charges given by the angular momenta of the branes. A comprehensive Appendix lists the field equations, symmetries and transformation rules of D=11 supergravity, Type IIB supergravity, and the M2, D3 and M5 branes. 
  We consider systems of non-threshold bound states (D(p$-$2), Dp), for $2\le p \le 6$, in type II string theories. Each of them can be viewed as Dp branes with a nonzero (rank two) Neveu-Schwarz $B$ field. We study the noncommutative effects in the gravity dual descriptions of noncommutative gauge theories for these systems in the limit where the brane worldvolume theories decouple from gravity. We find that the noncommutative effects are actually due to the presence of infinitely many D(p$-$2) branes in the (D(p$-$2), Dp) system which play the dominant role over the Dp branes in the large $B$-field limit. Our study indicates that Dp branes with a constant $B$-field represents dynamically the system of infinitely many D(p$-$2) branes without $B$-field in the decoupling limit. This implies an equivalence between the noncommutative Yang-Mills in $(p + 1)$-dimensions and an ordinary Yang-Mills with gauge group $U (\infty)$ in $(p - 1)$-dimensions. We provide a physical explanation for the new scale which measures the noncommutativity. 
  We investigate the semi-classical instability of vacuum domain walls to processes where the domain walls decay by the formation of closed string loop boundaries on their worldvolumes. Intuitively, a wall which is initially spherical may `pop', so that a hole corresponding to a string boundary component on the wall, may form. We find instantons, and calculate the rates, for such processes. We show that after puncture, the hole grows exponentially at the same rate that the wall expands. It follows that the wall is never completely thermalized by a single expanding hole; at arbitrarily late times there is still a large, thin shell of matter which may drive an exponential expansion of the universe. We also study the situation where the wall is subjected to multiple punctures. We find that in order to completely annihilate the wall by this process, at least four string loops must be nucleated. We argue that this process may be relevant in certain brane-world scenarios, where the universe itself is a domain wall. 
  Here we discuss the ultraviolet and infrared aspects of the noncommutative counterpart of QED, which is called as noncommutative QED, as well as some infrared dynamics of noncommutative Yang-Mills (NCYM) theory. First we demonstrate that the divergence in the theory can be subtracted by the similar counterterms as in ordinary theory at one loop level. Then the anomalous magnetic moment is calculated to see the infrared aspect of the theory which reflects the violation of Lorentz symmetry. The evaluation of the finite part of the photon vacuum polarization shows that the logarithmically singular term in the infrared limit appears with the same weight as UV logarithmic divergence, showing the correlation between the UV and infrared dynamics in NCYM theory. NC-QED theory does not show such a property. We also consider the extension to chiral gauge theory in the present context, but the requirement of anomaly cancellation allows only noncommutative QED. 
  We consider the harmonic superspaces associated to SU(2,2/N) superconformal algebras. For arbitrary N, we show that massless representations, other than the chiral ones, correspond to [N/2] ``elementary'' ultrashort analytic superfields whose first component is a scalar in the k antisymmetric irrep of SU(N) (k=1... [N/2]) with top spin $J_{\rm\scriptsize top}= (N/2-k/2,0)$. For N=2n we analyze UIR's obtained by tensoring the self-conjugate ultrashort multiplet $J_{\rm\scriptsize top}$= (n/2,0) and show that N-1 different basic products give rise to all possible UIR's with residual shortening. 
  We consider chiral gauge theories defined over a four-dimensional spacetime manifold with a Cartesian product structure for at least one compact spatial dimension. For a simple setup, we calculate the effective gauge field action by integrating out the chiral fermions, while maintaining gauge invariance. Due to a combination of infrared and ultraviolet effects, there appears a CPT-odd term in the effective gauge field action. This CPT anomaly could occur in chiral gauge theories relevant to elementary particle physics, provided the spacetime manifold has the appropriate topology. Two possible applications for cosmology are discussed. 
  We give a new formula for the antipode of the algebra of rooted trees, directly in terms of the bialgebra structure. The equivalence, proved in this paper, among the three available formulae for the antipode, reflects the equivalence among the Bogoliubov-Parasiuk-Hepp, Zimmermann, and Dyson-Salam renormalization schemes. 
  We examine some recent developments in noncommutative geometry, including spin geometries on noncommutative tori and their quantization by the Shale-Stinespring procedure, as well as the emergence of Hopf algebras as a tool linking index theory and renormalization calculations 
  Three-dimensional bicovariant differential calculus on the quantum group SU_q(2) is constructed using the approach based on global covariance under the action of the stabilizing subgroup U(1). Explicit representations of possible q-deformed Lie algebras are obtained in terms of differential operators. The consistent gauge covariant differential calculus on SU_q(2) is uniquely defined. A non-standard Leibnitz rule is proposed for the exterior differential. The minimal gauge theory with SU_q(2) quantum group symmetry is considered. 
  This is a review of recent work on the chiral extensions of the WZNW phase space describing both the extensions based on fields with generic monodromy as well as those using Bloch waves with diagonal monodromy. The symplectic form on the extended phase space is inverted in both cases and the chiral WZNW fields are found to satisfy quadratic Poisson bracket relations characterized by monodromy dependent exchange r-matrices. Explicit expressions for the exchange r-matrices in terms of the arbitrary monodromy dependent 2-form appearing in the chiral WZNW symplectic form are given. The exchange r-matrices in the general case are shown to satisfy a new dynamical generalization of the classical modified Yang-Baxter (YB) equation and Poisson-Lie (PL) groupoids are constructed that encode this equation analogously as PL groups encode the classical YB equation. For an arbitrary simple Lie group $G$, exchange r-matrices are exhibited that are in one-to-one correspondence with the possible PL structures on $G$ and admit them as PL symmetries. 
  Background independence of noncommutative Yang-Mills theory on $\mathbb R^n$ is discussed. The quantity $\theta \hat F \theta - \theta$ is found to be background dependent at subleading order, and it becomes background independent only when the ordinary gauge field strength $F$ is constant. It is shown that, at small values of $B$, the noncommutative Dirac-Born-Infeld action possesses $\Lambda$-symmetry at least to subleading order in $\theta$ if $F$ damps fast enough at infinity. 
  We study dynamics of a probe p-brane and a test particle in the field background of fully localized solutions describing the source p-brane within the worldvolume of the source domain wall. We find that the probe dynamics in the background of the source p-brane in one lower dimensions is not reproduced, indicating that p-branes within the worldvolume of domain walls perhaps describe an exotic phase of p-branes in brane worlds. We speculate therefore that a (p+1)-brane where one of its longitudinal directions is along the direction transverse to the domain wall is the right description of the p-brane in the brane world with the expected properties. 
  In the $(\epsilon_1-\epsilon_2)^2$--approximation the Casimir energy of a dilute dielectric ball is derived using a simple and clear method of the mode summation. The addition theorem for the Bessel functions enables one to present in a closed form the sum over the angular momentum before the integration over the imaginary frequencies. The linear in $(\epsilon_1-\epsilon_2)$ contribution into the vacuum energy is removed by an appropriate subtraction. The role of the contact terms used in other approaches to this problem is elucidated. 
  We analyze the world-volume solitons of a D3-brane probe in the background of parallel (p,q) five-branes. The D3-brane is embedded along the directions transverse to the five-branes of the background. By using the S-duality invariance of the D3-brane, we find a first-order differential equation whose solutions saturate an energy bound. The SO(3) invariant solutions of this equation are found analytically. They represent world-volume solitons which can be interpreted as formed by parallel (-q,p) strings emanating from the D3-brane world-volume. It is shown that these configurations are 1/4 supersymmetric and provide a world-volume realization of the Hanany-Witten effect. 
  A gauge transformation in quantum electrodynamics involves the product of field operators at the same space-time point and hence does not have a well-defined meaning. One way to avoid this difficulty is to generalize the gauge transformation by using different space-time points in the spirit of Dirac's point splitting. Such a generalization indeed exists and the resulting infinitesimal gauge transformation takes the form of an infinite series in the coupling constant. In this text I will present two examples of generalized gauge transformations. 
  We study in the Hamiltonian framework the local transformations $\delta_\epsilon q^A(\tau)=\sum^{[k]}_{k=0}\partial^k_\tau\epsilon^a{} R_{(k)a}{}^A(q^B, \dot q^C)$ which leave invariant the Lagrangian action: $\delta_\epsilon S=div$. Manifest form of the symmetry and the corresponding Noether identities is obtained in the first order formalism as well as in the Hamiltonian one. The identities has very simple form and interpretation in the Hamiltonian framework. Part of them allows one to express the symmetry generators which correspond to the primarily expressible velocities through the remaining one. Other part of the identities allows one to select subsystem of constraints with a special structure from the complete constraint system. It means, in particular, that the above written symmetry implies an appearance of the Hamiltonian constraints up to at least $([k]+1)$ stage. It is proved also that the Hamiltonian symmetries can always be presented in the form of canonical transformation for the phase space variables. Manifest form of the resulting generating function is obtained. 
  We construct the integrable model corresponding to the $\N=2$ supersymmetric SU(N) gauge theory with matter in the antisymmetric representation, using the spectral curve found by Landsteiner and Lopez through M Theory. The model turns out to be the Hamiltonian reduction of a $N+2$ periodic spin chain model, which is Hamiltonian with respect to the universal symplectic form we had constructed earlier for general soliton equations in the Lax or Zakharov-Shabat representation. 
  We study heterotic vacua with four supercharges in three and four space-time dimensions and their duals obtained as M/F-theory compactified on Calabi-Yau fourfolds. We focus on their respective moduli spaces and derive the Kahler potential for heterotic vacua obtained as circle compactifications of four-dimensional N=1 heterotic theories. The Kahler potential of the dual theory is computed by compactifying 11-dimensional supergravity on Calabi-Yau fourfolds. The duality between these theories is checked for K3-fibred fourfolds and an appropriate F-theory limit is discussed. 
  If the fifth dimension is one-dimensional connected manifold, up to diffeomorphic, the only possible space-time will be $M^4 \times R^1$, $M^4 \times R^1/Z_2$, $M^4 \times S^1$ and $M^4 \times S^1/Z_2$. And there exist two possibilities on cosmology constant along the fifth dimension: the cosmology constant is constant, and the cosmology constant is sectional constant. We construct the general models with parallel 3-branes and with constant/sectional constant cosmology constant along the fifth dimension on those kinds of the space-time, and point out that for compact fifth dimension, the sum of the brane tensions is zero, for non-compact fifth dimension, the sum of the brane tensions is positive. We assume the observable brane which includes our world should have positive tension, and obtain that the gauge hierarchy problem can be solved in those scenarios. We also discuss some simple models. 
  We describe in detail the techniques needed to compute scattering amplitudes for colored scalars from the infinite tension limit of bosonic string theory, up to two loops. These techniques apply both to cubic and quartic interactions, and to planar as well as non-planar diagrams. The resulting field theories are naturally defined in the space-time dimension in which they are renormalizable. With a careful analysis of string moduli space in the Schottky representation we determine the region of integration for the moduli, which plays a crucial role in the derivation of the correct combinatorial and color factors for all diagrams. 
  We study a family of BPS solutions of type IIA supergravity that can be interpreted as describing the `transmutation' of a Neveu-Schwarz five-brane into a D4-brane in the presence of a D6-brane. The D4-brane, which terminates on the D6-brane, can be equally well interpreted as a `pure multipole' configuration of NS5-brane wrapped tightly around the D6-brane. Such a transmutation is a "near-core" version (i.e., near the D6-brane) of the brane-creation that can occur when two branes pass through each other, as in the Hanany-Witten construction. The work below highlights certain charge non-conservation features of type IIA supergravity. 
  It was shown by A. Connes, M. Douglas and A. Schwarz that noncommutative tori arise naturally in consideration of toroidal compactifications of M(atrix) theory. A similar analysis of toroidal Z_{2} orbifolds leads to the algebra B_{\theta} that can be defined as a crossed product of noncommutative torus and the group Z_{2}. Our paper is devoted to the study of projective modules over B_{\theta} (Z_{2}-equivariant projective modules over a noncommutative torus). We analyze the Morita equivalence (duality) for B_{\theta} algebras working out the two-dimensional case in detail. 
  We construct a class of solutions to the Einstein's equations for dimensions greater than or equal to six. These solutions are characterized by a non-trivial warp factor and possess a non-compact extra dimension. We study in detail a simple model in six dimensions containing two four branes. One of each brane's four spatial directions is compactified. The hierarchy problem is resolved by the enormous difference between the warp factors at the positions of the two branes, with the standard model fields living on the brane with small warp factor. Both branes can have positive tensions. Their positions, and the size of the compact dimension are determined in terms of the fundamental parameters of the theory by a combination of two independent and comparable effects---an anisotropic contribution to the stress tensor of each brane from quantum fields living on it and a contribution to the stress tensor from a bulk scalar field. One overall fine tuning of the parameters of the theory is required ---that for the cosmological constant. 
  We study local and global gravitational effects of (D-2)-brane configurations (domain-walls) in the vacuum of D-dimensional space-time. We focus on infinitely thin vacuum domain walls with arbitrary cosmological constants on either side of the wall. In the comoving frame of the wall we derive a general metric Ansatz, consistent with the homogeneity and isotropy of the space-time intrinsic to the wall, and employ Israel's matching conditions at the wall. The space-time, intrinsic to the wall, is that of (D-1)-dimensional Freedman-Lemaitre-Robertson-Walker universe (with k=-1,0,1) which has a (local) description as either anti-deSitter, Minkowski or deSitter space-time. For each of these geometries, we provide a systematic classification of the local and global space-time structure transverse to the walls, for those with both positive and negative tension; they fall into different classes according to the values of their energy density relative to that of the extreme (superysmmetric) configurations. We find that in any dimension D, both local and global space-time structure for each class of domain-walls is universal. We also comment on the phenomenological implications of these walls in the special case of D=5. 
  D-branes on one-parameter Calabi-Yau spaces and two-parameter K3-fibered Calabi-Yau manifolds are analyzed from both the Gepner model point of view and the geometric perspective. We compute part of the spectrum of the boundary states and comment on the appearance of the D0-brane as well as on nonsupersymmetric large volume configurations becoming supersymmetric at the Gepner point. 
  The purpose of this paper is to investigate the gauge symmetry of classical field theories in integral formalism. A gauge invariant theory is defined in terms of the invariance of the physical observables under the coordinate transformations in principal bundle space. Through the detailed study on the property of non-Abelian parallel transportor under gauge transformations, we show that it is not generally a two-point spinor, i.e. an operator to be affected only by the gauge group elements at the two end points of the parallel transport path, except for the pure gauge situation, and therefore the local gauge symmetry for non-Abelian models is found to be broken in non-perturbative domain. However, an Abelian gauge theory is proved to be strictly invariant under local gauge transformation, as it is illustrated by the invariance of the interference pattern of electrons in Aharonov-Bohm effect. The related issues of the phenomenon are discussed. 
  See hep-th/0005054. 
  Using AdS/CFT correspondence we found the conformal anomaly from d3 and d5 gauged supergravity with single scalar (dilaton) and the arbitrary scalar potential on AdS-like scalar-gravitational background. Such dilatonic gravity action describes the special RG flows in extended gauged SG when scalars lie in one-dimensional submanifold of complete scalars space. This dilaton-dependent conformal anomaly corresponds to dual non-conformal (gauge) QFT (which is classically conformally invariant) with account of radiative corrections. Equations of motion in d5 gauged supergravity put some restrictions to the dilatonic potential on the conformal boundary. Using these restrictions we propose the candidate c-functions away from exact conformity. These c-functions are positively defined and monotonic, expressed in terms of dilatonic potential and have the fixed points in asymptotically AdS region. 
  The dynamics and evolution of Bianchi type I space-times is considered in the framework of the four-dimensional truncation of a reduced theory obtained from the N=2,D=5 supergravity. The general solution of the gravitational field equations can be represented in an exact parametric form. All solutions have a singular behavior at the initial/final moment, except when the space-time geometry reduces to the isotropic flat case. Generically the obtained cosmological models describe an anisotropic, expanding or collapsing, singular Universe with a non-inflationary evolution for all times. 
  We argue that the familiar gauge hierarchy between the fundamental Planck scale M_{Pl} and the electroweak scale M_{W}, can be naturally explained in higher dimensional theories with relatively large radii (R_c > 1/M_{Pl}) extra dimensions. In particular, we show that it is possible that the electroweak Higgs mass at high energies is of the order of M_{Pl}, but radiative corrections drive it to an infrared stable fixed-point $\sim M_{W}$ at low energies thus inducing a large hierarchy without any fine tuning of parameters. 
  In this paper we apply a variant of Heisenberg's quantization method for strongly interacting, non-linear fields, to solutions of the classical Yang-Mills field equations which have bad asymptotic behavior. After quantization we find that the bad features (i.e. divergent fields and energy densities) of these solutions are moderated. From these results we argue that in general the n-point Green's functions for Yang-Mills theories can have non-perturbative pieces which can not be represented as the sum of Feynman diagrams. A formalism for dealing with these non-Feynman pieces via nonassociative field operators is suggested. These methods may also find some application in dealing with high-$T_c$ superconductors. 
  We compute the CP-odd part of the finite temperature effective action for massive Dirac fermions in the presence of a Dirac monopole. We confirm that the induced charge is temperature dependent, and in the effective action we find an infinite series of CP-violating terms that generalize the familiar zero temperature $F\tilde{F}$ term. These results are analogous to recent results concerning finite temperature induced Chern-Simons terms. 
  We apply the thermodynamic Bethe Ansatz to investigate the high energy behaviour of a class of scattering matrices which have recently been proposed to describe the Homogeneous sine-Gordon models related to simply laced Lie algebras. A characteristic feature is that some elements of the suggested S-matrices are not parity invariant and contain resonance shifts which allow for the formation of unstable bound states. From the Lagrangian point of view these models may be viewed as integrable perturbations of WZNW-coset models and in our analysis we recover indeed in the deep ultraviolet regime the effective central charge related to these cosets, supporting therefore the S-matrix proposal. For the $SU(3)_k$-model we present a detailed numerical analysis of the scaling function which exhibits the well known staircase pattern for theories involving resonance parameters, indicating the energy scales of stable and unstable particles. We demonstrate that, as a consequence of the interplay between the mass scale and the resonance parameter, the ultraviolet limit of the HSG-model may be viewed alternatively as a massless ultraviolet-infrared-flow between different conformal cosets. For $k=2$ we recover as a subsystem the flow between the tricritical Ising and the Ising model. 
  An irreducible antifield BRST quantization method for reducible gauge theories is proposed. The general formalism is illustrated in the case of the Freedman-Townsend model. 
  In this paper we develop an irreducible antifield BRST-anti-BRST formalism for reducible gauge theories. 
  The irreducible BRST symmetry for the Freedman-Townsend model is derived. The comparison with the standard reducible approach is also addressed. 
  The irreducible Hamiltonian BRST symmetry for p-form gauge theories with Stueckelberg coupling is derived. The cornerstone of our approach is represented by the construction of an irreducible theory that is equivalent from the point of view of the BRST formalism with the original system. The equivalence makes permissible the substitution of the BRST quantization of the reducible model by that of the irreducible theory. Our procedure maintains the Lorentz covariance of the irreducible path integral. 
  An irreducible Hamiltonian BRST approach to topologically coupled p- and (p+1)-forms is developed. The irreducible setting is enforced by means of constructing an irreducible Hamiltonian first-class model that is equivalent from the BRST point of view to the original redundant theory. The irreducible path integral can be brought to a manifestly Lorentz covariant form. 
  It is shown how $W$-algebras emerge from very peculiar canonical transformations with respect to the canonical symplectic structure on a compact Riemann surface. The action of smooth diffeomorphisms of the cotangent bundle on suitable generating functions is written in the BRS framework while a $W$-symmetry is exhibited. Subsequently, the complex structure of the symmetry spaces is studied and the related BRS properties are discussed. The specific example of the so-called $W_3$-algebra is treated in relation to some other different approaches. 
  Talk given at the International Workshop ``Physical Variables in Gauge Theories", Dubna 1999 
  We study a certain class of four-dimensional N=1 supersymmetric orientifolds for which the world-sheet parity transformation is combined with a complex conjugation in the compact directions. We investigate in detail the orientifolds of the Z_3, Z_4, Z_6 and Z_6' toroidal orbifolds finding solutions to the tadpole cancellation conditions for all models. Generically, all the massless spectra turn out to be non-chiral. 
  The first free comprehensive textbook on quantum (and classical) field theory. The approach is pragmatic, rather than traditional or artistic: It includes practical techniques, such as the 1/N expansion (color ordering) and spacecone (spinor helicity), and diverse topics, such as supersymmetry and general relativity, as well as introductions to supergravity and strings. The PDF version can be more convenient than paper books, with Web links and a clickable outline (contents) window. 
  We study the implications of target-space duality symmetries for low-energy effective actions of various four-dimensional string theories. In the heterotic case such symmetries can be incorporated in simple orbifold examples. At present a similar statement cannot be made about the simplest type IIB orientifolds due to an obstruction at the level of gravitational anomalies. This fact confirms previous doubts concerning a conjectured heterotic-type IIB orientifold duality and shows that target-space symmetries can be a powerful tool in studying relations between various string theories at the level of the effective low-energy action. Contraints on effective Lagrangians from these symmetries are discussed in detail. In particular, we consider ways of extending T-duality to include additional corrections to the Kaehler potential in heterotic string models with N=2 subsectors. 
  In this paper we complete the derivations of finite volume partition functions for QCD using random matrix theories by calculating the effective low-energy partition function for three-dimensional QCD in the adjoint representation from a random matrix theory with the same global symmetries. As expected, this case corresponds to Dyson index $\beta =4$, that is, the Dirac operator can be written in terms of real quaternions. After discussing the issue of defining Majorana fermions in Euclidean space, the actual matrix model calculation turns out to be simple. We find that the symmetry breaking pattern is $O(2N_f) \to O(N_f) \times O(N_f)$, as expected from the correspondence between symmetric (super)spaces and random matrix universality classes found by Zirnbauer. We also derive the first Leutwyler--Smilga sum rule. 
  We present a class of N=1 supersymmetric models of particle physics, derived directly from heterotic M-theory, that contain three families of chiral quarks and leptons coupled to the gauge group $SU(3)_C\times SU(2)_{L}\times U(1)_{Y}$. These models are a fundamental form of ``brane-world'' theories, with an observable and hidden sector each confined, after compactification on a Calabi-Yau threefold, to a BPS threebrane separated by a five-dimensional bulk space with size of the order of the intermediate scale. The requirement of three families, coupled to the fundamental conditions of anomaly freedom and supersymmetry, constrains these models to contain additional fivebranes wrapped around holomorphic curves in the Calabi-Yau threefold. These fivebranes ``live'' in the bulk space and represent new, non-perturbative aspects of these particle physics vacua. We discuss, in detail, the relevant mathematical structure of a class of torus-fibered Calabi-Yau threefolds with non-trivial first homotopy groups and construct holomorphic vector bundles over such threefolds, which, by including Wilson lines, break the gauge symmetry to the standard model gauge group. Rules for constructing phenomenological particle physics models in this context are presented and we give a number of explicit examples. 
  We investigate black hole formation by a spherically collapsing thin shell of matter in AdS space. This process has been suggested to have a holographic interpretation as thermalization of the CFT on the boundary of the AdS space. The AdS/CFT duality relates the shell in the bulk to an off-equilibrium state of the boundary theory which evolves towards a thermal equilibrium when the shell collapses to a black hole. We use 2-point functions to obtain information about the spectrum of excitations in the off-equilibrium state, and discuss how it characterizes the approach towards thermal equilibrium. The full holographic interpretation of the gravitational collapse would require a kinetic theory of the CFT at strong coupling. We speculate that the kinetic equations should be interpreted as a holographic dual of the equation of motion of the collapsing shell. 
  All quartic couplings of scalar fields $s^I$ that are dual to extended chiral primary operators in ${\cal N}=4$ SYM$_4$ are derived by using the covariant equations of motion for type IIB supergravity on $AdS_5\times S^5$. It is shown that despite some expectations if one keeps the structure of the cubic terms untouched, the quartic action obtained contains terms with two and four derivatives. It is shown that the quartic action vanishes on shell in the extremal case, e.g. k_1=k_2+k_3+k_4. Consistency of the truncation of the quartic couplings to the massless multiplet of the ${\cal N}=8$, d=5 supergravity is proven and the explicit values of the couplings are found. It is argued that the consistency of the KK reduction implies non-renormalization of $n$-point functions of $n-1$ operators dual to the fields from the massless multiplet and one operator dual to a field from a massive multiplet. 
  We study the time evolution of configurations in the form of two parallel domain walls moving towards each other in a supersymmetric field model. The configurations involved are not BPS-saturated. It is found that for such collisions there exists some critical value $v_{cr}\approx0.9120$ of the initial velocity v_i of the walls. At v_i<v_{cr} we observed reflection, that was not followed by change of vacuum states sequence. In collisions with v_i>v_{cr} the sequence of vacuum states changes. The results of the numerical simulations are in agreement with "potential" consideration. 
  We introduce a notion of Q-algebra that can be considered as a generalization of the notion of Q-manifold (a supermanifold equipped with an odd vector field obeying {Q,Q} =0). We develop the theory of connections on modules over Q-algebras and prove a general duality theorem for gauge theories on such modules. This theorem contains as a simplest case SO(d,d,{\bf Z})-duality of gauge theories on noncommutative tori. 
  Our motivation is to find the relationship between the commutator of coordinates and uncertainty relation involving only the coordinates. The boundary condition with constant background field is connected with the rotation of D-brane at general angle. And the mode expansions of D-brane we found is more reasonable than those appeared in literature. The partition functions and scattering amplitudes are also discussed. 
  We deal with the problem of diffeomorphism anomaly in theories with branes. In particular we thoroughly analyze the problem of the residual chiral anomaly of a five-brane immersed in M-theory, paying attention to its global formulation in the five-brane world-volume. We conclude that the anomaly can be canceled by a {\it local} counterterm in the five-brane world-volume. 
  The chiral phase structure of the Nambu-Jona-Lasinio/Gross-Neveu model at finite temperature T and finite chemical potential \mu is investigated using (Wilsonian) Non-Perturbative Renormalization Group (NPRG). In the large N_c limit, the solutions of NPRG with various cutoff schemes are shown. For a sufficiently large ultra-violet cutoff, NPRG results coincide with those of Schwinger-Dyson equation and have little cutoff scheme dependence. Next, to improve the approximation, we incorporate the mesonic fluctuations. We introduce the auxiliary fields for mesons, and then derive NPRG equation for finite N_c. The chiral phase structure on (T,\mu) plane beyond the leading of 1/N_c expansion is investigated in the sharp cutoff limit. N_c dependence of chiral phase diagram is obtained. 
  The gauged WZNW model has been derived as an effective action, whose Poisson bracket algebra of the constraints is isomorphic to the commutator algebra of operators in quantized fermionic theory. As a consequence, the hamiltonian as well as usual lagrangian non-abelian bosonization rules have been obtained, for the chiral currents and for the chiral densities. The expression for the anomaly has been obtained as a function of the Schwinger term, using canonical methods. 
  Following the general principles of noncommutative geometry, it is possible to define a metric on the space of pure states of the noncommutative algebra generated by the coordinates. This metric generalizes the usual Riemannian one. We investigate some general properties of this metric in the finite commutative case which corresponds to a metric on a finite set, and also give some examples of computations in both commutative and noncommutative cases. 
  We discuss the open descendants of diagonal irrational $Z_3$ orbifolds, starting from the $c=2$ case and analyzing six-dimensional and four-dimensional models. As recently argued, their consistency is linked to the presence of geometric discrete moduli. The different classes of open descendants, related to different resolutions of the fixed-point ambiguities, are distinguished by the number of geometric fixed points surviving the unoriented projection. 
  The construction of the known interacting quantum field theory models is mostly based on euclidean techniques. The expectation values of interesting quantities are usually given in terms of euclidean correlation functions from which one should be able to extract information about the behavior of the correlation functions of the Minkowskian counterpart.   We think that the C*-algebraic approach to euclidean field theory gives an appropriate setup in order to study structural aspects model independently. A previous paper deals with a construction scheme which relates to each euclidean field theory a Poincar\'e covariant quantum field theory model in the sense of R. Haag and D. Kastler.   Within the framework of R. Haag and D. Kastler, the physical concept of PCT symmetry and spin and statistics is related to the Tomita-Takesaki theory of von Neumann algebras and this important aspects has been studied by several authors.   We express the PCT symmetry in terms of euclidean reflexions and we explicitly identify the corresponding modular operator and the modular conjugation of the related Tomita-Takesaki theory. Locality, wedge duality, and a geometric action of the modular group of the von Neumann algebra of observables, localized within a wedge region in Minkowski space, are direct consequences. 
  We review Kreimer's construction of a Hopf algebra associated to the Feynman graphs of a perturbative quantum field theory. 
  We review the appearance of Hopf algebras in the renormalization of quantum field theories and in the study of diffeomorphisms of the frame bundle important for index computations in noncommutative geometry. 
  We construct the worldline superfield massive superparticle actions which preserve 1/4 portion of the underlying higher-dimensional supersymmetry. The rest of supersymmetry is spontaneously broken and realized by nonlinear transformations. We consider the cases of N=4 to N=1 and N=8 to N=2 partial breaking. In the first case we present the corresponding Green-Schwarz type target superspace action with one $\kappa$-supersymmetry. It is related to the superfield action via a field redefinition. In the second case we find out two possible models, one of which is a direct generalization of the N=4 to N=1 case, while another is essentially different. For the first model we formulate Green-Schwarz type action with two kappa supersymmetries. We elaborate on the bosonic part of the superfield action for the second model and find that only in two special limits it takes the standard Nambu-Goto form. In the general case it is determined by a fourth-order algebraic equation. The characteristic common feature of these new superparticle models is that the algebras of their spontaneously broken supersymmetries are non-trivial truncations of the general extensions of N=1 and N=2 Poincare D=4 superalgebras by tensorial central charges. 
  The Casimir energy of a dilute dielectric cylinder, with the same light-velocity as in its surrounding medium, is evaluated exactly to first order in $\xi^2$ and numerically to higher orders in $\xi^2$. The first part is carried out using addition formulas for Bessel functions, and no Debye expansions are required. 
  I review some recent work (done in collaboration with G. Veneziano) which clarifies the existence of a correspondence between self-gravitating fundamental string states and Schwarzschild black holes. The main result is a detailed calculation showing that self-gravity causes a typical string state of mass M to shrink, as the string coupling g^2 increases, down to a compact string state whose mass, size, entropy and luminosity match (for the critical value g_c^2 = (M sqrt{alpha'})^{-1}) those of a Schwarzschild black hole. This confirms the idea that the entropy of black holes can be accounted for by counting string states, and suggests that the level spacing of the quantum states of Schwarzschild black holes is exponentially small, and very much blurred by radiative effects. 
  We study BPS dominated loop amplitudes in M-theory on T^2. For this purpose we generalize the concept of helicity supertraces to nine spacetime dimensions. These traces distinguish between various massive supermultiplets and appear as coefficients in their one-loop contributions to n-graviton scattering amplitudes. This can be used to show that only ultrashort BPS multiplets contribute to the R^4 term in the effective action, which was first computed by Green, Gutperle and Vanhove. There are two inequivalent ultrashort BPS multiplets which describe the Kaluza-Klein states and the wrapped membranes that cover the torus a number of times. From the perspective of the type-II strings they correspond to momentum and winding states and D0 or D1 branes. 
  We show that the fully covariant equations of motion for the M-theory fivebrane can be interpreted as charge conservation equations. The associated charges induce `shift'-symmetries of the scalar, spinor and gauge-fields of the fivebrane, so allowing an interpretation of all these fields as Goldstone fields. We also find that the fivebrane possesses a new symmetry that is part of the GL(32) automorphism group of the eleven dimensional supersymmetry algebra. 
  We study classical solutions (ic-instantons) in N=4 SYM in 4D which, in the strong coupling limit, correspond to complex two-dimensional manifolds. Asymptotically in time the latter have boundaries represented by compact real three-manifolds. Therefore they lend themselves to an interpretation in terms of 3-brane scattering. We suggest that these solutions may represent scattering of D3-branes of type IIB theory in 10D. In particular we show that the world-volume theory on complex two-dimensional manifolds is the correct one for D3-branes. 
  The foundations of time asymmetric quantum theory are reviewed and are applied to the construction of relativistic Gamow vectors. Relativistic Gamow vectors are obtained from the resonance pole of the S-matrix and furnish an irreducible representation of the Poincare' semigroup. They have all the properties needed to represent relativistic quasistable particles and can be used to fix the definition of mass and width of relativistic resonances like the Z-boson. Most remarkably, they have only a semigroup time evolution into the forward light cone---expressing time asymmetry on the microphysical level. 
  We propose a new mechanism of spontaneous supersymmetry breaking. The existence of extra dimensions with nontrivial topology plays an important role. We investigate new features resulted from the mechanism in two simple supersymmetric Z_2 and U(1) models. One of remarkable features is that there exists a phase in which the translational invariance for the compactified directions is broken spontaneously, accompanying the breakdown of the supersymmetry. The mass spectrum of the models appeared in reduced dimensions is a full of variety, reflecting the highly nontrivial vacuum structure of the models. The Nambu-Goldstone bosons (fermions) associated with breakdown of symmetries are found in the mass spectrum. Our mechanism also yields quite different vacuum structures if models have different global symmetries. 
  Using conformal field theory methods we construct a metric that describes the distortion of space-time surrounding a D(irichlet)-brane (solitonic) defect after being struck by another D-brane. By viewing our four-dimensional universe as such a struck brane, embedded in a five-dimensional space-time, we argue on the appearance of a band of massive Kaluza-Klein excitations for the bulk graviton which is localized in a region of the fifth dimension determined by the inverse size of the band. The band incorporates the massless mode (ordinary graviton) and its thickness is determined essentially by the width of the Gaussian distribution describing the (target-space) quantum fluctuations of the intersecting-brane configuration. 
  A geometric approach to the standard model in terms of the Clifford algebra $% C\ell_{7}$ is advanced. The gauge symmetries and charge assignments of the fundamental fermions are seen to arise from a simple geometric model involving extra space-like dimensions. The bare coupling constants are found to obey $g_{s}/g=1$ and $g^{\prime}/g=\sqrt{3/5}$, consistent with SU(5) grand unification but without invoking the notion of master groups. In constructing the Lagrangian density terms, it is found that the Higgs isodoublet field emerges in a natural manner. A matrix representation of $% C\ell_{7}$ is included as a computational aid. 
  We study two issues, the localization of various spin fields, and the problem of the cosmological constant on a brane in five-dimensional anti de Sitter space. We find that spin-zero fields are localized on a positive-tension brane. In addition to the localized zero-mode there is a continuous tower of states with no mass gap. Spin one-half and three-half states can be localized on a brane with ``negative tension''. Their localization can be achieved on the positive-tension brane as well, if additional interactions are introduced. The necessary ingredient of the scenario with localized gravity is the relation between the bulk cosmological constant and the brane tension. In the absence of supersymmetry this implies fine-tuning between the parameters of the theory. To deal with this issue we introduce a four-form gauge field. This gives an additional arbitrary contribution to the bulk cosmological constant. As a result, the model gives rise to a continuous family of brane Universe solutions for generic values of the bulk cosmological constant and the brane tension. Among these solutions there is one with a zero four-dimensional cosmological constant. 
  We carefully investigate the gravitational perturbation of the Randall-Sundrum (RS) single brane-world solution [hep-th/9906064], based on a covariant curvature tensor formalism recently developed by us. Using this curvature formalism, it is known that the `electric' part of the 5-dimensional Weyl tensor, denoted by $E_{\mu\nu}$, gives the leading order correction to the conventional Einstein equations on the brane. We consider the general solution of the perturbation equations for the 5-dimensional Weyl tensor caused by the matter fluctuations on the brane. By analyzing its asymptotic behaviour in the direction of the 5th dimension, we find the curvature invariant diverges as we approach the Cauchy horizon. However, in the limit of asymptotic future in the vicinity of the Cauchy horizon, the curvature invariant falls off fast enough to render the divergence harmless to the brane-world. We also obtain the asymptotic behavior of $E_{\mu\nu}$ on the brane at spatial infinity, assuming the matter perturbation is localized. We find it falls off sufficiently fast and will not affect the conserved quantities at spatial infinity. This indicates strongly that the usual conservation law, such as the ADM energy conservation, holds on the brane as far as asymptotically flat spacetimes are concerned. 
  The SU(2) Skyrme model,expanding in the collective coordinates variables, gives rise to second-class constraints. Recently this system was embedded in a more general Abelian gauge theory using the BFFT Hamiltonian method. In this work we quantize this gauge theory computing the Noether current anomaly using for this two different methods: an operatorial Dirac first class formalism and the non-local BV quantization coupled with the Fujikawa regularization procedure. 
  This paper is concerned with a structural analysis of euclidean field theories on the euclidean sphere. In the first section we give proposal for axioms for a euclidean field theory on a sphere in terms of C*-algebras.   Then, in the second section, we investigate the short-distance behavior of euclidean field theory models on the sphere by making use of the concept of {\em scaling algebras}, which has first been introduced by D. Buchholz, and R. Verch and which has also be applied to euclidean field theories on flat euclidean space in a previous paper. We establish the expected statement that that scaling limit theories of euclidean field theories on a sphere are euclidean field theories on flat euclidean space.   Keeping in mind that the minkowskian analogue of the euclidean sphere is the de Sitter space, we develop a Osterwalder-Schrader type construction scheme which assigns to a given euclidean field theory on the sphere a quantum field theory on de Sitter space. We show that the constructed quantum field theoretical data fulfills the so called geodesic KMS condition in the sense of H. J. Borchers and D. Buchholz, i.e. for any geodesic observer the system looks like a system within a thermal equilibrium state. 
  We explicitly construct the SO(d,d) transformations of Ramond-Ramond field strengths and potentials, along with those of the space-time supersymmetry parameters, the gravitinos and the dilatinos in type-II theories. The results include the case when the SO(d,d) transformation involves the time direction. The derivation is based on the compatibility of SO(d,d) transformations with space-time supersymmetry, which automatically guarantees compatibility with the equations of motion. It involves constructing the spinor representation of a twist that an SO(d,d) action induces between the local Lorentz frames associated with the left- and right-moving sectors of the worldsheet theory. The relation to the transformation of R-R potentials as SO(d,d) spinors is also clarified. 
  The three-phase version of the hybrid chiral bag model, containing the phase of asymptotic freedom, the hadronization phase as well as the intermediate phase of constituent quarks, is proposed. For this model the self-consistent solution, which takes into account the fermion vacuum polarization effects, is found in (1+1) D. Within this solution the total energy of the bag, including the one-loop contribution from the Dirac's sea, is studied as the function of the bag geometry under condition of nonvanishing boson condensate density in the interior region. The existence and uniqueness of the ground state bag configuration, which minimizes the total energy and contains all the three phases, are shown. 
  The geometric picture of the star-product based on its Fourier representation kernel is utilized in the evaluation of chains of star-products and the intuitive appreciation of their associativity and symmetries. Such constructions appear even simpler for a variant asymmetric product, and carry through for the standard star-product supersymmetrization. 
  We give a general construction of correlation functions in rational conformal field theory on a possibly non-orientable surface with boundary in terms of 3-dimensional topological quantum field theory. The construction applies to any modular category. It is proved that these correlation functions obey modular and factorization rules. Structure constants are calculated and expressed in terms of the data of the modular category. 
  In this work quantum electrodynamics at T > 0 is considered. For this purpose we use thermo field dynamics and the causal approach to quantum field theory according to Epstein and Glaser, the latter being a rigorous method to avoid the well-known ultraviolet divergencies of quantum field theory. It will be shown that the theory is infrared divergent if the usual scattering states are used. The same is true if we use more general mixed states. This is in contradiction to the results established in the literature, and we will point out why these earlier approaches fail to describe the infrared behaviour correctly. We also calculate the thermal corrections to the electron magnetic moment in the low temperature approximation k_B T << m_e. This is done by investigating the scattering of an electron on a C-number potential in third order in the limit of small momentum transfer p -> q. We reproduce one of the different results reported up to now in literature. In the low temperature approximation infrared finiteness is recovered in a very straightforward way: In contrast to the literature we do not have to introduce a thermal Dirac equation or thermal spinors. 
  For the purpose of better understanding the AdS/CFT correspondence it is useful to have a description of the theory for all values of the 't Hooft coupling, and for all $N$. We discuss such a description in the framework of Matrix theory for SYM on D4-branes, which is given in terms of quantum mechanics on the moduli space of solutions of the Nahm equations. This description reduces to both SYM perturbation theory and to closed string perturbation theory, each in its appropriate regime of validity, suggesting a way of directly relating the variables in the two descriptions. For example, it shows explicitly how holes in the world-sheets of the 't Hooft expansion close to give closed surfaces. 
  Recently a realization of the four-dimensional gravity on a brane in five-dimensional spacetime has been discussed. Randall and Sundrum have shown that the equation for the longitudinal components of the metric fluctuations admit a normalizable zero mode solution, which has been interpreted as the localized gravity on the brane. We point out that the equation for the transverse components of the metric fluctuations has a solutions which is not localized on the brane. This indicates that probably the effective theory is unstable or, in other words, actually it is not four-dimensional but five-dimensional. Perhaps a modification of the proposal by using matter fields can lead to the trapping of gravity to the brane. 
  The off-shell structure of the string sigma model is investigated. In the open bosonic string, nonperturbative effects are shown to depend crucially on the regularization scheme. A scheme retaining the notion of string width reproduces the structure of Witten's string field theory and the associated unconventional gauge transformations. 
  We study renormalization-group flow patterns in theories arising on D1-branes in various supersymmetry-breaking backgrounds. We argue that the theory of N D1-branes transverse to an orbifold space can be fine-tuned to flow to the corresponding orbifold conformal field theory in the infrared, for particular values of the couplings and theta angles which we determine using the discrete symmetries of the model. By calculating various nonplanar contributions to the scalar potential in the worldvolume theory, we show that fine-tuning is in fact required at finite N, as would be generically expected. We further comment on the presence of singular conformal field theories (such as those whose target space includes a ``throat'' described by an exactly solvable CFT) in the non-supersymmetric context. Throughout the analysis two applications are considered: to gauge theory/gravity duality and to linear sigma model techniques for studying worldsheet string theory. 
  This brief report presents analytical solutions to the equations of motion of a null string. The background spacetime is a magnetically charged Kaluza-Klein black hole. The string coordinates are expanded with the world-sheet velocity of light as an expansion parameter. It is shown that the zeroth order solutions can be expressed in terms of elementary functions in an appropriate large distance approximation. In addition, a class of exact solutions corresponding to the Pollard-Gross-Perry-Sorkin monopole case is also obtained. 
  A model for quantized gravitation based on the simplicial lattice discretization is studied in detail using a comprehensive finite size scaling analysis combined with renormalization group methods. The results are consistent with a value for the universal critical exponent for gravitation $\nu=1/3$, and suggest a simple relationship between Newton's constant, the gravitational correlation length and the observable average space-time curvature. Some perhaps testable phenomenological implications of these results are discussed. To achieve a high numerical accuracy in the evaluation of the lattice path integral a dedicated parallel machine was assembled. 
  The relevance of the BRST cohomology of the extended antifield formalism is briefly discussed along with standard homological tools needed for its computation. 
  In the 't Hooft-Veltman dimensional regularization scheme it is necessary to introduce finite counterterms to satisfy chiral Ward identities. It is a non-trivial task to evaluate these counterterms even at two loops. We suggest the use of Wilsonian exact renormalization group techniques to reduce the computation of these counterterms to simple master integrals. We illustrate this method by a detailed study of a generic Yukawa model with massless fermions at two loops. 
  It has been conjectured that at a stationary point of the tachyon potential for the D-brane of bosonic string theory, the negative energy density exactly cancels the D-brane tension. We evaluate this tachyon potential by off-shell calculations in open string field theory. Surprisingly, the condensation of the tachyon mode alone into the stationary point of its cubic potential is found to cancel about 70% of the D-brane tension. Keeping relevant scalars up to four mass levels above the tachyon, the energy density at the shifted stationary point cancels 99% of the D-brane tension. 
  We show that next-to-extremal correlators of chiral primary operators in N=4 SYM theory do not receive quantum corrections to first order in perturbation theory. Furthermore we consider next-to-extremal correlators within AdS supergravity. Here the exchange diagrams contributing to these correlators yield results of the same free-field form as obtained within field theory. This suggests that quantum corrections vanish at strong coupling as well. 
  In this work we derive the Lorentz-PCT-violating effective action for a fermion in a constant and uniform electromagnetic field using the Fock-Schwinger proper time method and extract the exact value of the coefficient of the nonperturbatively induced Chern-Simons term. 
  We find an analytic solution of the Bethe Ansatz equations (BAE) for the special case of a finite XXZ spin chain with free boundary conditions and with a complex surface field which provides for $U_q(sl(2))$ symmetry of the Hamiltonian. More precisely, we find one nontrivial solution, corresponding to the ground state of the system with anisotropy parameter $\Delta = {1/2}$ corresponding to $q^3 = -1$. 
  In the absence of Gribov complications, the modified gauge fixing in gauge theory $ \int{\cal D}A_{\mu}\{\exp[-S_{YM}(A_{\mu})-\int f(A_{\mu})dx] /\int{\cal D}g\exp[-\int f(A_{\mu}^{g})dx]\}$ for example, $f(A_{\mu})=(1/2)(A_{\mu})^{2}$, is identical to the conventional Faddeev-Popov formula $\int{\cal D}A_{\mu}\{\delta(D^{\mu}\frac{\delta f(A_{\nu})}{\delta A_{\mu}})/\int {\cal D}g\delta(D^{\mu}\frac{\delta f(A_{\nu}^{g})} {\delta A_{\mu}^{g}})\}\exp[-S_{YM}(A_{\mu})]$ if one takes into account the variation of the gauge field along the entire gauge orbit. Despite of its quite different appearance,the modified formula defines a local and BRST invariant theory and thus ensures unitarity at least in perturbation theory. In the presence of Gribov complications, as is expected in non-perturbative Yang-Mills theory, the modified formula is equivalent to the conventional formula but not identical to it:Both of the definitions give rise to non-local theory in general and thus the unitarity is not obvious. Implications of the present analysis on the lattice regularization are briefly discussed. 
  In this paper we consider Witten's bosonic open string field theory in the presence of a constant background of the second-rank antisymmetric tensor field $B_{ij}$. We extend the operator formulation of Gross and Jevicki in this situation and construct the overlap vertices explicitly. As a result we find a noncommutative structure of the Moyal type only in the zero-mode sector, which is consistent with the result of the correlation functions among vertex operators in the world sheet formulation. Furthermore we find out a certain unitary transformation of the string field which absorbs the Moyal type noncommutative structure. It can be regarded as a microscopic origin of the transformation between the gauge fields in commutative and noncommutative gauge theories discussed by Seiberg and Witten. 
  We argue that orientable open string theory without GSO projection has N=2 space-time supersymmetry in a spontaneously broken phase. The arguments are presented both in Neveu-Schwarz-Ramond and Green-Schwarz formulations. The formal but explicit supersymmetry transformation law for string field is given in the framework of Witten's open string field theory. Our results support a fundamental assumption which lies behind the topological construction of stable D-branes starting from the unstable systems of D9-branes. 
  The zero-hamiltonian problem, present in reparametrization invariant systems, is solved for the 2-D induced gravity model. Working with methods developed by Henneaux et al. we find systematically the reduced phase-space physics, generated by an {\it effective} hamiltonian obtained after complete gauge fixing. 
  We discuss gauge theories on D3 branes embedded in special non-tachyonic orientifolds of the 0B string theory. In general, they correspond to non-supersymmetric SU(N) gauge theories with scalars in the adjoint representation and spinors in the (anti-)symmetric representation. We study these theories via the AdS/CFT correspondence and present evidence of their relation to N=4 SYM in the planar limit. We also discuss finite N properties, focusing in particular on the renormalization group flow. Up to two loops, the logarithmic running of the gauge coupling is described by the dilaton tadpole and the cosmological constant that naturally emerge on the string theory side. 
  We consider IIB matrix model with D1-D5-brane backgrounds. Using the fact that noncommutative gauge theory on the D-branes can be obtained as twisted reduced model in IIB matrix model, we study two-dimensional gauge theory on D1-branes and D5-branes. Especially the spectrum of the zero modes in the off-diagonal parts is examined. We also consider the description of D1-branes as local excitations of gauge theory on D5-branes. Relation to supergravity solution is also discussed. 
  The generalization of the black hole in three-dimensional spacetime to include an electric charge Q in addition to the mass M and the angular momentum J is given. The field equations are first solved explicitly when Q is small and the general form of the field at large distances is established. The total ``hairs'' M, J and Q are exhibited as boundary terms at infinity. It is found that the inner horizon of the rotating uncharged black hole is unstable under the addition of a small electric charge. Next it is shown that when Q=0 the spinning black hole may be obtained from the one with J=0 by a Lorentz boost in the $\phi -t$ plane. This boost is an ``illegitimate coordinate transformation'' because it changes the physical parameters of the solution. The extreme black hole appears as the analog of a particle moving with the speed of light. The same boost may be used when $Q\neq 0$ to generate a solution with angular momentum from that with J=0, although the geometrical meaning of the transformation is much less transparent since in the charged case the black holes are not obtained by identifying points in anti-de Sitter space. The metric is given explicitly in terms of three parameters, $\widetilde{M}$, $ \widetilde{Q}$ and $\omega $ which are the ``rest mass'' and ``rest charge'' and the angular velocity of the boost. These parameters are related to M, J and Q through the solution of an algebraic cubic equation. Altogether, even without angular momentum, the electrically charged 2+1 black hole is somewhat pathological since (i) it exists for arbitrarily negative values of the mass, and (ii) there is no upper bound on the electric charge. 
  Exact solutions to the Dirac-Born-Infeld equation, which describes scatterings of localized wave packets in the presence of constant background fields, are derived in this paper. 
  We construct the effective theory of intersecting branes and investigate the BPS monopoles in the theory. The monopoles obtained are the generalization of Nielsen-Olesen vortex. We study the properties of the solutions and interpret them as the D0-branes on the brane-intersections. 
  We show that symmetries are preserved exactly along the (Wilsonian) renormalization group flow, though the IR cutoff deforms concrete forms of the transformations. For a gauge theory the cutoff dependent Ward-Takahashi identity is written as the master equation in the antifield formalism: one may read off the renormalized BRS transformation from the master equation. The Maxwell theory is studied explicitly to see how it works. The renormalized BRS transformation becomes non-local but keeps off-shell nilpotency. Our formalism is applicable for a generic global symmetry. The master equation considered for the chiral symmetry provides us with the continuum analog of the Ginsparg-Wilson relation and the L{\" u}scher's symmetry. 
  The canonical ADM equations are solved in terms of the conformal factor in the instantaneous York gauge. A simple derivation is given for the solution of the two body problem. A geometrical characterization is given for the apparent singularities occurring in the N-body problem and it is shown how the Garnier hamiltonian system arises in the ADM treatment by considering the time development of the conformal factor at the locations where the extrinsic curvature tensor vanishes. The equations of motion for the position of the particles and of the apparent singularities and also the time dependence of the linear residues at such singularities are given by the transformation induced by an energy momentum tensor of a conformal Liouville theory. Such an equation encodes completely the dynamics of the system. 
  We discuss the dynamics of a superparticle in a superspace whose isometry is generated by the superalgebra OSp(1|4) or its central-charge contraction. Extra coordinates of the superspace associated with tensorial central charges are shown to describe spin degrees of freedom of the superparticle, so quantum states form an infinite tower of (half)-integer helicities. A peculiar feature of the model is that it admits BPS states which preserve 3/4 of target-space supersymmetries. 
  We propose a new star pruduct which interpolates the Berezin and Moyal quantization. A multiple of this product is shown to reduce to a path-integral quantization in the continuous time limit. In flat space the action becomes the one of free bosonic strings. Relation to Kontsevich prescription is also discussed. 
  We study the relation between intersecting NS5-branes whose intersection is smoothed out and the deformed conifold in terms of the supergravity solution. We solve the condition of preserved supersymmetry on a metric inspired by the deformed conifold metric and obtain a solution of the NS5-branes which is delocalized except for one of the overall transverse directions. The solution has consistent properties with other configurations obtained by string dualities. 
  Starting from 2D Euclidean quantum gravity, we show that one recovers 2D Lorentzian quantum gravity by removing all baby universes. Using a peeling procedure to decompose the discrete, triangulated geometries along a one-dimensional path, we explicitly associate with each Euclidean space-time a (generalized) Lorentzian space-time. This motivates a map between the parameter spaces of the two theories, under which their propagators get identified. In two dimensions, Lorentzian quantum gravity can therefore be viewed as a ``renormalized'' version of Euclidean quantum gravity. 
  We perform the dual transformation of the Yang-Mills theory in d=3 dimensions using the Wilson action on the cubic lattice. The dual lattice is made of tetrahedra triangulating a 3-dimensional curved manifold but embedded into a flat 6-dimensional space (for the SU(2) gauge group). In the continuum limit the theory can be reformulated in terms of 6-component gauge-invariant scalar fields having the meaning of the external coordinates of the dual lattice sites. These 6-component fields induce a metric and a curvature of the 3-dimensional dual colour space. The Yang-Mills theory can be identically rewritten as a quantum gravity theory with the Einstein-Hilbert action but purely imaginary Newton constant, plus a homogeneous `matter' term. Interestingly, the theory can be formulated in a gauge-invariant and local form without explicit colour degrees of freedom. 
  We calculate the effective action for Quantum Electrodynamics (QED) in D=2,3 dimensions at the quadratic approximation in the gauge fields. We analyse the analytic structure of the corresponding nonlocal boson propagators nonperturbatively in k/m. In two dimensions for any nonzero fermion mass, we end up with one massless pole for the gauge boson . We also calculate in D=2 the effective potential between two static charges separated by a distance L and find it to be a linearly increasing function of L in agreement with the bosonized theory (massive Sine-Gordon model). In three dimensions we find nonperturbatively in k/m one massive pole in the effective bosonic action leading to screening. Fitting the numerical results we derive a simple expression for the functional dependence of the boson mass upon the dimensionless parameter e^{2}/m . 
  We obtain in a closed form the 1/N^2 contribution to the free energy of the two Hermitian N\times N random matrix model with non symmetric quartic potential. From this result, we calculate numerically the Yang-Lee zeros of the 2D Ising model on dynamical random graphs with the topology of a torus up to n=16 vertices. They are found to be located on the unit circle on the complex fugacity plane. In order to include contributions of even higher topologies we calculated analytically the nonperturbative (sum over all genus) partition function of the model Z_n = \sum_{h=0}^{\infty} \frac{Z_n^{(h)}}{N^{2h}} for the special cases of N=1,2 and graphs with n\le 20 vertices. Once again the Yang-Lee zeros are shown numerically to lie on the unit circle on the complex fugacity plane. Our results thus generalize previous numerical results on random graphs by going beyond the planar approximation and strongly indicate that there might be a generalization of the Lee-Yang circle theorem for dynamical random graphs. 
  We present a series of four self-contained lectures on the following topics:  (I) An introduction to 4-dimensional 1\leq N \leq 4 supersymmetric Yang-Mills theory, including particle and field contents, N=1 and N=2 superfield methods and the construction of general invariant Lagrangians;  (II) A review of holomorphicity and duality in N=2 super-Yang-Mills, of Seiberg-Witten theory and its formulation in terms of Riemann surfaces;  (III) An introduction to mechanical Hamiltonian integrable systems, such as the Toda and Calogero-Moser systems associated with general Lie algebras; a review of the recently constructed Lax pairs with spectral parameter for twisted and untwisted elliptic Calogero-Moser systems;  (IV) A review of recent solutions of the Seiberg-Witten theory for general gauge algebra and adjoint hypermultiplet content in terms of the elliptic Calogero-Moser integrable systems. 
  We construct twisted quantum bundles and adjoint sections on noncommutative $T^4$, and investigate relevant D-brane bound states with non-Abelian backgrounds. We also show that the noncommutative $T^4$ with non-Abelian backgrounds exhibits SO$(4,4|Z)$ duality and via this duality we get a Morita equivalent $T^4$ on which only D0-branes exist. For a reducible non-Abelian background, the moduli space of D-brane bound states in Type II string theory takes the form $\prod_a (T^4)^{q_a}/S_{q_a}$. 
  We study D-branes on three-dimensional orbifolds ${\bf C}^3/\Gamma$ where $\Gamma$ are finite subgroups of SU(3). The quiver diagram of $\ZnZn \in SU(3)$ can be expressed in three-dimensional form. According to the correspondence between quiver diagrams and brane configurations, we construct a brane configuration for ${\bf C}^3/\ZnZn$ which has essentially three-dimensional structrue. Brane configurations for nonabelian orbifolds $\C^3/\Delta(3n^2)$ and $\C^3/\Delta(6n^2)$ are obtained from that for $\C^3/\ZnZn$ by certain quotienting procedure. 
  We study Witten's open string field theory in the presence of a constant B field. We construct the string field theory in the operator formalism and find that, compared to the ordinary theory with no B field, the vertices in the resulting theory has an additional factor. The factor makes the zero modes of strings noncommutative. This is in agreement with the results in the first-quantized formulation. We also discuss background independence of the purely cubic action derived from the above string field theory and then find a redefinition of string fields to remove the additional factor from the vertex. Furthermore, we briefly discuss the supersymmetric extension of our string field theory. 
  In these lectures we review the properties of a boosted and rotated boundary state and of a boundary state with an abelian gauge field deriving from it the Dirac-Born-Infeld action and a newly constructed class of classical solutions. We also review the construction of the boundary state for the stable non-BPS state of type I theory corresponding to the perturbative state present at the first excited level of the SO(32) heterotic string and transforming according to the spinor representation of SO(32) (Lectures presented at the YITP Workshop on ``Developments in Superstring and M-theory'', Kyoto, Japan, October 1999). 
  It has recently been proposed that the hierarchy problem can be solved by considering the warped fifth dimension compactified on $S^{1}/Z_{2}$. Many studies in the context have assumed a particular choice for an integration constant $\sigma_{0}$ that appears when one solves the five-dimensional Einstein equation. Since $\sigma_{0}$ is not determined by the boundary condition of the five-dimensional theory, $\sigma_{0}$ may be regarded as a gauge degree of freedom in a sense. To this time, all indications are that the four-dimensional Planck mass depends on $\sigma_{0}$. In this paper, we carefully investigate the properties of the geometry in the Randall-Sundrum model, and consider in which location $y$ the four-dimensional Planck mass is measured. As a result, we find a $\sigma_{0}$-independent relation between the four-dimensional Planck mass $M_{\rm Pl}$ and five- dimensional fundamental mass scale $M$, and remarkably enough, we can take $M$ to TeV region when we consider models with the Standard Model confined on a distant brane. We also confirm that the physical masses on the distant brane do not depend on $\sigma_{0}$ by considering a bulk scalar field as an illustrative example. The resulting mass scale of the Kaluza-Klein modes is on the order of $M$. 
  The Riemannian geometry of coset spaces is reviewed, with emphasis on its applications to supergravity and M-theory compactifications. Formulae for the connection and curvature of rescaled coset manifolds are generalized to the case of nondiagonal Killing metrics.   The example of the N^{010} spaces is discussed in detail. These are a subclass of the coset manifolds N^{pqr}=G/H = SU(3) x U(1)/U(1) x U(1), the integers p,q,r characterizing the embedding of H in G. We study the realization of N^{010} as G/H=SU(3) x SU(2)/U(1) x SU(2) (with diagonal embedding of the SU(2) \in H into G). For a particular G-symmetric rescaling there exist three Killing spinors, implying N=3 supersymmetry in the AdS_4 \times N^{010} compactification of D=11 supergravity. This rescaled N^{010} space is of particular interest for the AdS_4/CFT_3 correspondence, and its SU(3) x SU(2) isometric realization is essential for the OSp(4|3) classification of the Kaluza-Klein modes. 
  We propose a treatment of $\gamma^5$ in dimensional regularization which is based on an algebraically consistent extension of the Breitenlohner-Maison-'t Hooft-Veltman (BMHV) scheme; we define the corresponding minimal renormalization scheme and show its equivalence with a non-minimal BMHV scheme. The restoration of the chiral Ward identities requires the introduction of considerably fewer finite counterterms than in the BMHV scheme. This scheme is the same as the minimal naive dimensional renormalization in the case of diagrams not involving fermionic traces with an odd number of $\gamma^5$, but unlike the latter it is a consistent scheme. As a simple example we apply our minimal subtraction scheme to the Yukawa model at two loops in presence of external gauge fields. 
  Just as D-brane charge of Type IIA and Type IIB superstrings is classified, respectively, by K^1(X) and K(X), Ramond-Ramond fields in these theories are classified, respectively, by K(X) and K^1(X). By analyzing a recent proposal for how to interpret quantum self-duality of RR fields, we show that the Dirac quantization formula for the RR p-forms, when properly formulated, receives corrections that reflect curvature, lower brane charges, and an anomaly of D-brane world-volume fermions. The K-theory framework is important here, because the term involving the fermion anomaly cannot be naturally expressed in terms of cohomology and differential forms. 
  A coordinate-free action principle for the N=2 supersymmetric model of spin 0 and spin 1/2 matter fields interacting via Chern-Simons higher spin gauge fields in AdS3 is formulated in terms of star-product algebra of two oscillators. Although describing proper relativistic dynamics the action does not contain space-time coordinates. It is given by a supertrace of a forth-order polynomial in the 3d higher spin superalgebra. The theory admits a free parameter characterizing the inequivalent vacua associated with the parameter of mass m of the matter fields. For the case of the massless matter, the model has a form of the noncommutative 2d Yang-Mills theory with some infinite-dimensional gauge group. The limit m\to\infty corresponds to the SU(\infty) 2d Yang-Mills theory. 
  Open and closed strings with two worldsheet supersymmetries in 2+2 dimensional spacetime are reviewed in the NSR formulation. I briefly discuss their quantization, mutual and self-interactions, classical spacetime dynamics and interpretation in terms of self-dual Yang-Mills and gravity. A stringy origin of the infinite self-dual gravity hierarchy is presented. An outlook to the loop level concludes. 
  In the present work we consider a time-dependent Schr\"odinger equation for systems invariant under the reparametrization of time. We develop the two-stage procedure of construction such systems from a given initial ones, which is not invariant under the time reparametrization. One of the first-class constraints of the systems in such description becomes the time-dependent Schr\"odinger equation. The procedure is applicable in the supersymmetric theories as well. The $n=2$ supersymmetric quantum mechanics is coupled to world-line supergravity, and the local supersymmetric action is constructed leading to the square root representation of the time-dependent Schr\"odinger equation. 
  We show that quantum effects can stabilize a soliton in a model with no soliton at the classical level. The model has a scalar field chirally coupled to a fermion in 1+1 dimensions. We use a formalism that allows us to calculate the exact one loop fermion contribution to the effective energy for a spatially varying scalar background. This energy includes the contribution from counterterms fixed in the perturbative sector of the theory. The resulting energy is therefore finite and unambiguous. A variational search then yields a fermion number one configuration whose energy is below that of a single free fermion. 
  We explore all the possible ways of fixing the kappa symmetry for both branes and strings by means of a constant projector. We find that they can be classified according to their behaviour under Dirac conjugation and conjugation. This latter controls the maximum power of the fermionic variables in which the (super)vielbein can be expanded while the former determines which states cannot be described in the gauge. In particular there exists an interesting class of projectors for which vielbein are at most quadratic in the fermionic variables. As an example we compute the action for the type IIb on a AdS_{5} X S_{5} background with a lightcone-like projector; this action reduces to the usual lightcone GS string action in the flat limit. 
  We study some consequences of dimensionally reducing systems with massless fermions and Abelian gauge fields from 3+1 to 2+1 dimensions. We first consider fermions in the presence of an external Abelian gauge field. In the reduced theory, obtained by compactifying one of the coordinates `a la Kaluza-Klein, magnetic flux strings are mapped into domain wall defects. Fermionic zero modes, localized around the flux strings of the 3+1 dimensional theory, become also zero modes in the reduced theory, via the Callan and Harvey mechanism, and are concentrated around the domain wall defects. We also study a dynamical model: massless $QED_4$, with fermions confined to a plane, deriving the effective action that describes the `planar' system. 
  We develop a method for computing exact one-loop quantum corrections to the energies of static classical backgrounds in renormalizable quantum field theories. We use a continuum density of states formalism to construct a regularized Casimir energy in terms of phase shifts and their Born approximations. This method unambiguously incorporates definite counterterms fixed in the standard way by physical renormalization conditions. The result is a robust computation that can be efficiently implemented both numerically and analytically. We carry out such calculations in models of bosons and fermions in one and three dimensions. 
  We find the explicit coordinate transformation which links two exact cosmological solutions of the brane world which have been recently discovered. This means that both solutions are exactly the same with each other. One of two solutions is described by the motion of a domain wall in the well-known 5-dimensional Schwarzshild-AdS spacetime. Hence, we can easily understand the region covered by the coordinate used by another solution. 
  On the supergravity side, we study the propagation of the RR scalar and the dilaton in the D3-branes with NS $B$-field. To obtain the noncommutative effect, we consider the case of $B\to \infty(\theta \to\pi/2)$. We approximate this as the smeared D1-brane background with $F_5=H=0$. In this background, the RR scalar induces an instability of the near-horizon geometry. However, it turns out that the RR scalar is nonpropagating, while the dilaton is a physically propagating mode. We calculate the s-wave absorption cross section of the dilaton. One finds $\sigma_0^\phi |_{B\to\infty} \sim (\tilde \omega \tilde R_{\pi \over 2})^{8.9} / \omega^5$ in the leading-order while $\sigma_0^\phi|_{B=0} \sim (\tilde \omega R_0)^{8}/\omega^5$ in the D3-branes without $B$-field. This means that although the dilaton belongs to a minimally coupled scalar in the absence of $B$-field, it becomes a sort of fixed scalar in the limit of $B \to \infty$. 
  In a recent work we showed that for a Hamiltonian system with constraints, the set of constraints can be investigated in first and second class constraint chains. We show here that using this "chain by chain" method for an arbitrary system one can fix the gauges in the most economical and consistent way. We show that it is enough to assume some gauge fixing conditions conjugate to last elements of first class chains. The remaining necessary conditions would emerge from consistency conditions. 
  We discuss shuffle identities between Feynman graphs using the Hopf algebra structure of perturbative quantum field theory. For concrete exposition, we discuss vertex function in massless Yukawa theory. 
  We show a simple way of deriving the Casimir Polder interaction, present some general arguments on the finiteness and sign of mutual Casimir interactions and finally we derive a simple expression for Casimir radiation from small accelerated objects. 
  We construct the worldline superfield massive superparticle actions which preserve 1/4 portion of the underlying higher-dimensional supersymmetry. We consider the cases of N=4/N=1 and N=8/N=2 partial breaking. In the first case we present the corresponding Green-Schwarz type target superspace action with one kappa supersymmetry. In the second case we find out two possibilities, one of which is a direct generalization of the N=4/N=1 case, while another is essentially different. 
  In a recent paper Seiberg and Witten have argued that the full action describing the dynamics of coincident branes in the weak coupling regime is invariant under a specific field redefinition, which replaces the group of ordinary gauge transformations with the one of noncommutative gauge theory. This paper represents a first step towards the classification of invariant actions, in the simpler setting of the abelian single brane theory. In particular we consider a simplified model, in which the group of noncommutative gauge transformations is replaced with the group of symplectic diffeomorphisms of the brane world volume. We carefully define what we mean, in this context, by invariant actions, and rederive the known invariance of the Born-Infeld volume form. With the aid of a simple algebraic tool, which is a generalization of the Poisson bracket on the brane world volume, we are then able to describe invariant actions with an arbitrary number of derivatives. 
  Schrodinger equation with two-component wave function which describes a relativistic spin 1/2 particle in a weak electromagnetic field is obtained. In the same approximation Schrodinger equation with traditional norm condition and one-component wave function for a spinless particle is obtained as well. To construct it Foldy-Wouthuysen procedure with the electron charge value as the small parameter is used. 
  These are notes based on lectures given at TASI99. We review the geometry of the moduli space of N=2 theories in four dimensions from the point of view of superstring compactification. The cases of a type IIA or type IIB string compactified on a Calabi-Yau threefold and the heterotic string compactified on K3xT2 are each considered in detail. We pay specific attention to the differences between N=2 theories and N>2 theories. The moduli spaces of vector multiplets and the moduli spaces of hypermultiplets are reviewed. In the case of hypermultiplets this review is limited by the poor state of our current understanding. Some peculiarities such as ``mixed instantons'' and the non-existence of a universal hypermultiplet are discussed. 
  We point out that massive gauged supergravity potentials, for example those arising due to the massive breathing mode of sphere reductions in M-theory or string theory, allow for supersymmetric (static) domain wall solutions which are a hybrid of a Randall-Sundrum domain wall on one side, and a dilatonic domain wall with a run-away dilaton on the other side. On the anti-de Sitter (AdS) side, these walls have a repulsive gravity with an asymptotic region corresponding to the Cauchy horizon, while on the other side the runaway dilaton approaches the weak coupling regime and a non-singular attractive gravity, with the asymptotic region corresponding to the boundary of spacetime. We contrast these results with the situation for gauged supergravity potentials for massless scalar modes, whose supersymmetric AdS extrema are generically maxima, and there the asymptotic regime transverse to the wall corresponds to the boundary of the AdS spacetime. We also comment on the possibility that the massive breathing mode may, in the case of fundamental domain-wall sources, stabilize such walls via a Goldberger-Wise mechanism. 
  Recently, Ivanov and Volovich (hep-th/9912242) claimed that the perturbation of $h_{\mu\nu}$ with nonvanishing transverse components $h_{5\mu}$ is not localized on the brane because $h_{\mu\nu}$ depends on the fifth coordinate $z$ linearly. Consequently, it may indicate that the effective theory is unstable. However, we point out that such linear dependence on $z$ can be {\it gauged away}. Hence the solution does not belong to the physical one. Therefore, even if one includes $h_{5\mu}$, Randall and Sundrum's argument for the localized gravity on the brane remains correct. 
  Quantum fields responding to "moving mirrors" have been predicted to give rise to thermodynamic paradoxes. I show that the assumption in such work that the mirror can be treated as an external field is invalid: the exotic energy-transfer effects necessary to the paradoxes are well below the scales at which the model is credible. For a first-quantized point-particle mirror, it appears that exotic energy-transfers are lost in the quantum uncertainty in the mirror's state. An accurate accounting of these energies will require a model which recognizes the mirror's finite reflectivity, and almost certainly a model which allows for the excitation of internal mirror modes, that is, a second-quantized model. 
  Various aspects of spaces of chiral blocks are discussed. In particular, conjectures about the dimensions of irreducible sub-bundles are reviewed and their relation to symmetry breaking conformal boundary conditions is outlined. 
  An assessment of the present status of the theory, some immediate tasks which are suggested thereby and some questions whose answers may require a longer breath since they relate to significant changes in the conceptual and mathematical structure of the theory. 
  By making use of the complete decomposition of SO(3) spin connection, the topological defect in 3-dimensional Euclidean gravity is studied in detail. The topological structure of disclination is given as the combination of a monopole structure and a vortex structure. Furthermore, the Kac-Moody algebra generated by the monopole and vortex is discussed in three dimensional Chern-Simons theory. 
  We study some of the algebraic properties of the non-relativistic monopole. We find that we can construct theories that possess an exotic conserved fermionic charge that squares to the Casimir of the rotation group, yet do not possess an ordinary supersymmetry. This is in contrast to previous known examples with such exotic fermionic charges. We proceed to show that the presence of the exotic fermionic charge in the non-supersymmetric theory can nonetheless be understood using supersymmetric techniques, providing yet another example of the usefulness of supersymmetry in understanding non-supersymmetric theories. 
  We compute the probabilty for the processes A -> q \bar{q}, gg via vacuum polarization in the presence of a classical space-time dependent non-abelian field A by applying the background field method of QCD. The processes we consider are leading order in gA and are simillar to A -> e^+e^- in QED. Gluons are treated like a matter field and gauge transformations of the quantum gluonic field are different from those of the classical chromofield. To obtain the correct physical polarization of gluons we deduct the corresponding ghost contributions. We find that the expression for the probability of the leading order process A -> gg is transverse with respect to the momentum of the field. We observe that the contributions from higher order processes to gluon pair production need to be added to the this leading order process. Our result presented here is a part of the expression for the total probability for gluon pair production from a space-time dependent chromfield. The result for q \bar{q} pair production is similar to that of the e^+e^- pair production in QED. Quark and gluon production from a space-time dependent chromofield will play an important role in the study of the production and equilibration of the quark-gluon plasma in ultra relativistic heavy-ion collisions. 
  We discuss the reality properties of the fermionic collective coordinates in Euclidean space in an instanton background and construct hermitean actions for N = 4 and N = 2 supersymmetric Euclidean Yang-Mills theories. 
  It is known that when there are several D-branes, their space-time coordinates in general become noncommutative. From the point of view of noncommutative geometry, it reflects noncommutativity of the world volume of the D-branes. On the other hand, as we showed in the previous work, in the presence of the constant antisymmetric tensor field the momentum operators of the D-branes have noncommutative structure. In the present paper, we investigate a relation between these noncommutativities and the description of D-branes in terms of the noncommutative Yang-Mills theory recently proposed by Seiberg and Witten. It is shown that the noncommutativity of the Yang-Mills theory, which implies that of the world volume coordinates, originates from both noncommutativities of the transverse coordinates and momenta from the viewpoint of the lower-dimensional D-branes. Moreover, we show that this noncommutativity is transformed by coordinate transformations on the world volume and thereby can be chosen in an arbitrary fixed value. We also make a brief comment on a relation between this fact and a hidden symmetry of the IIB matrix models. 
  Liouville conformal field theory is considered with conformal boundary. There is a family of conformal boundary conditions parameterized by the boundary cosmological constant, so that observables depend on the dimensional ratios of boundary and bulk cosmological constants. The disk geometry is considered. We present an explicit expression for the expectation value of a bulk operator inside the disk and for the two-point function of boundary operators. We comment also on the properties of the degenrate boundary operators. Possible applications and further developments are discussed. In particular, we present exact expectation values of the boundary operators in the boundary sin-Gordon model. 
  We point out that the gravitational evolution equations in the Randall-Sundrum model appear in a different form than hitherto assumed. As a consequence, the model yields a correct Newtonian limit in a novel manner. 
  In this thesis quantum gauge theories are considered in the framework of local, causal perturbation theory. Gauge invariance is described in terms of the BRS formalism. Local interacting field operators are constructed perturbatively and field equations are established. A nilpotent BRS transformation is defined on the local algebra of fields. It allows the definition of the algebra of local observables as an operator cohomology. This algebra of local observables can be represented in a Hilbert space.   The interacting field operators are defined in terms of time ordered products of free field operators. For the results above to hold the time ordered products must satisfy certain normalization conditions. To formulate these conditions also for field operators that contain a spacetime derivative a suitable mathematical description of time ordered products is developed.   Among the normalization conditions are Ward identities for the ghost current and the BRS current. The latter are generalizations of a normalization condition that is postulated by D"utsch, Hurth, Krahe and Scharf for Yang-Mills theory. It is not yet proven that this condition has a solution in every order. All other normalization conditions can be accomplished simultaneously.   A principle for the correspondence between interacting quantum fields and interacting classical fields is established. Quantum electrodynamics and Yang-Mills theory are examined and the results are compared with the literature. 
  The canonical quantization of the topological particle is described; it is shown that BRST quantization of the model gives the supersymmetric quantum mechanical model considered by Witten when investigating Morse theory, and the rigorous path integral method appropriate for this model is discussed. Possibilities for the extension of this work to two dimensional models are briefly considered. 
  The equivalence between the holographic renormalization group and the soft dilaton theorem is shown for a class of wrapped metrics solutions of the string beta function equations for the bosonic string. 
  This Letter deals with topological solitons in an O(3) sigma model in three space dimensions (with a Skyrme term to stabilize their size). The solitons are classified topologically by their Hopf number N. The N=2 sector is studied; in particular, for two solitons far apart, there are three ``attractive channels''. Viewing the solitons as dipole pairs enables one to predict the force between them. Relaxing in the attractive channels leads to various static 2-soliton solutions. 
  We study the solution describing a non-extreme dilatonic (p+1)-brane intersecting a D-dimensional extreme dilatonic domain wall, where one of its longitudinal directions is along the direction transverse to the domain wall, in relation to the Randall-Sundrum type model. The dynamics of the probe (p+1)-brane in such source background reproduces that of the probe p-brane in the background of the (D-1)-dimensional source p-brane. However, as for a probe test particle, the dynamics in one lower dimensions is reproduced, only when the source (p+1)-brane is uncharged. 
  The Casimir energy, free energy and Casimir force are evaluated, at arbitrary finite temperature, for a dilute dielectric ball with uniform velocity of light inside the ball and in the surrounding medium. In particular, we investigate the classical limit at high temperature. The Casimir force found is repulsive, as in previous calculations. 
  In this talk I shall try to give an elementary introduction to certain areas of mathematical physics where the idea of moduli space is used to help solve problems or to further our understanding. In the wide area of gauge theory, I shall mention instantons, monopoles and duality. Then, under the general heading of string theory, I shall indicate briefly the use of moduli space in conformal field theory and $M$-theory. 
  We study the two loop quantum equivalence of sigma models related by Buscher's T-duality transformation. The computation of the two loop perturbative free energy density is performed in the case of a certain deformation of the SU(2) principal sigma model, and its T-dual, using dimensional regularization and the geometric sigma model perturbation theory. We obtain agreement between the free energy density expressions of the two models. 
  We take advantage of the superspace formalism and explicitly find the N=2 supersymmetric extension of the Maxwell Chern-Simons model. In our construction a special form of a potential term and indispensability of an additional neutral scalar field arise naturally. By considering the algebra of supersymmetric charges we find Bogomol'nyi equations for the investigated model. 
  The String Uncertainty Relations have been known for some time as the stringy corrections to the original Heisenberg's Uncertainty principle. In this letter the Stringy Uncertainty relations, and corrections thereof, are explicitly derived from the New Relativity Principle that treats all dimensions and signatures on the same footing and which is based on the postulate that the Planck scale is the minimal length in Nature in the same vein that the speed of light was taken as the maximum velocity in Einstein's theory of Special Relativity. The Regge behaviour of the string's spectrum is also a natural consequence of this New Relativity Principle. 
  We find the combinations of momentum and domain-wall charges corresponding to BPS states preserving 1/4, 1/2 or 3/4 of D=4 N=1 supersymmetry, and we show how the supersymmetry algebra implies their stability. These states form the boundary of the convex cone associated with the Jordan algebra of $4\times 4$ real symmetric matrices, and we explore some implications of the associated geometry. For the Wess-Zumino model we derive the conditions for preservation of 1/4 supersymmetry when one of two parallel domain-walls is rotated and in addition show that this model does not admit any classical configurations with 3/4 supersymmetry. Our analysis also provides information about BPS states of N=1 D=4 anti-de Sitter supersymmetry. 
  We study the quantum volume of D-branes wrapped around various cycles in Calabi-Yau manifolds, as the manifold's moduli are varied. In particular, we focus on the behaviour of these D-branes near phase transitions between distinct low energy physical descriptions of the resulting string theory. Whereas previous studies have solely considered quantum volumes in the context of two-cycles in perturbative string theory or D-branes in the specific example of the quintic hypersurface, we work more generally and find qualitatively new features. On the mathematical side, as we briefly note, our work has some interesting implications for certain issues in arithmetics. 
  We investigate the Poisson-Sigma model on the classical and quantum level. In the classical analysis we show how this model includes various known two-dimensional field theories. Then we perform the calculation of the path integral in a general gauge, and demonstrate that the derived partition function reduces to the familiar form in the case of 2d Yang-Mills theory. 
  We propose a bi-local representation in noncommutative field theory. It provides a simple description for high momentum degrees of freedom. It also shows that the low momentum modes can be well approximated by ordinary local fields. Long range interactions are generated in the effective action for the lower momentum modes after integrating out the high momentum bi-local fields. The low momentum modes can be represented by diagonal blocks in the matrix model picture and the high momentum bi-local fields correspond to off-diagonal blocks. This block-block interaction picture simply reproduces the infrared singular behaviors of nonplanar diagrams in noncommutative field theory. 
  We exploit the tree level bosonic 4-particle scattering amplitudes in D=11 supergravity to construct the bosonic part of a linearized supersymmetry-, coordinate- and gauge-invariant. By differentiation, this invariant can be promoted to be the natural lowest (two-loop) order counterterm. Its existence implies that the perturbative supersymmetry does not protect this ultimate supergravity from infinities, given also the recently demonstrated divergence of its 4-graviton amplitude. 
  We revisit the duality between type I' and heterotic strings in 9 dimensions. We resolve a puzzle about the validity of type I' perturbation theory and show that there are regions in moduli which are not within the reach of type I' perturbation theory. We find however, that all regions of moduli are described by a special class of real elliptic $K3$'s in the limit where the $K3$ shrinks to a one dimensional interval. We find a precise map between the geometry of dilaton and branes of type I' on the one hand and the geometry of real elliptic $K3$ on the other. We also argue more generally that strong coupling limits of string compactifications generically do not have a weakly coupled dual in terms of any known theory (as is exemplified by the strong coupling limit of heterotic strings in 9 dimensions for certain range of parameters). 
  We give a detailed general description of a recent geometrical discretisation scheme and illustrate, by explicit numerical calculation, the scheme's ability to capture topological features. The scheme is applied to the Abelian Chern-Simons theory and leads, after a necessary field doubling, to an expression for the discrete partition function in terms of untwisted Reidemeister torsion and of various triangulation dependent factors. The discrete partition function is evaluated computationally for various triangulations of $S^3$ and of lens spaces. The results confirm that the discretisation scheme is triangulation independent and coincides with the continuum partition function 
  Generic solution of free equations for massless fields of an arbitrary spin in $AdS_4$ is built in terms of the star-product algebra with spinor generating elements. A class of "plane wave" solutions is described explicitly. 
  It is shown that the transformation between ordinary and noncommutative Yang-Mills theory as formulated by Seiberg and Witten is due to the equivalence of certain star products on the D-brane world-volume. 
  We study gravity in backgrounds that are smooth generalizations of the Randall-Sundrum model, with and without scalar fields. These generalizations include three-branes in higher dimensional spaces which are not necessarily Anti-de Sitter far from the branes, intersecting brane configurations and configurations involving negative tension branes. We show that under certain mild assumptions there is a universal equation for the gravitational fluctuations. We study both the graviton ground state and the continuum of Kaluza-Klein modes and we find that the four-dimensional gravitational mode is localized precisely when the effects of the continuum modes decouple at distances larger than the fundamental Planck scale. The decoupling is contingent only on the long-range behaviour of the metric from the brane and we find a universal form for the corrections to Newton's Law. We also comment on the possible contribution of resonant modes. Given this, we find general classes of metrics which maintain localized four-dimensional gravity. We find that three-brane metrics in five dimensions can arise from a single scalar field source, and we rederive the BPS type conditions without any a priori assumptions regarding the form of the scalar potential. We also show that a single scalar field cannot produce conformally-flat locally intersecting brane configurations or a p-brane in greater than (p+2)-dimensions. 
  The form of the vacuum stress-tensor for the quantized scalar field at a dielectric to vacuum interface is studied. The dielectric is modeled to have an index of refraction that varies with frequency. We find that the stress-tensor components, derived from the mode function expansion of the Wightman function, are naturally regularized by the reflection and transmission coefficients of the mode at the boundary. Additionally, the divergence of the vacuum energy associated with a perfectly reflecting mirror is found to disappear for the dielectric mirror at the expense of introducing a new energy density near the surface which has the opposite sign. Thus the weak energy condition is always violated in some region of the spacetime. For the dielectric mirror, the mean vacuum energy density per unit plate area in a constant time hypersurface is always found to be positive (or zero) and the averaged weak energy condition is proven to hold for all observers with non-zero velocity along the normal direction to the boundary. Both results are found to be generic features of the vacuum stress-tensor and not necessarily dependent of the frequency dependence of the dielectric. 
  Using pure spinors, the superstring is covariantly quantized. For the first time, massless vertex operators are constructed and scattering amplitudes are computed in a manifestly ten-dimensional super-Poincar\'e covariant manner. Quantizable non-linear sigma model actions are constructed for the superstring in curved backgrounds, including the $AdS_5\times S^5$ background with Ramond-Ramond flux. 
  We study dynamical flavor symmetry breaking in the context of a class of N=1 supersymmetric SU(n_c) and USp(2 n_c) gauge theories, constructed from the exactly solvable N=2 theories by perturbing them with small adjoint and generic bare hypermultiplet (quark) masses. We find that the flavor U(n_{f}) symmetry in SU(n_{c}) theories is dynamically broken to $U(r)\times U(n_{f}-r)$ groups for $n_f \leq n_c$. In the r=1 case the dynamical symmetry breaking is caused by the condensation of monopoles in the $\underline{n_{f}}$ representation. For general r, however, the monopoles in the $\underline{{}_{n_{f}}C_{r}}$ representation, whose condensation could explain the flavor symmetry breaking but would produce too-many Nambu--Goldstone multiplets, actually ``break up'' into ``magnetic quarks'' which condense and induce confinement and the symmetry breaking. In USp(2n_c) theories with $n_f \leq n_c + 1$, the flavor SO(2n_f) symmetry is dynamically broken to U(n_f), but with no description in terms of a weakly coupled local field theory. In both SU(n_c) and USp(2 n_c) theories, with larger numbers of quark flavors, besides the vacua with these properties, there exist also vacua with no flavor symmetry breaking. 
  We extend the construction of the boundary states in Gepner models to the non-diagonal modular invariant theories, and derive the same supersymmetric conditions as the diagonal theories. We also investigate the relation between the microscopic charges of the boundary states and Ramond charges of the B-type D-branes on the Calabi-Yau threefolds with one K\"ahler modulus in the large volume limit. 
  The non-perturbative part of the vacuum energy density for static configuration in pure SU(2) Y-M theory is described. The vacuum state is constructed. 
  The effect of quantum corrections to a conformally invariant field theory for a self-interacting scalar field on a curved manifold with boundary is considered. The analysis is most easily performed in a space of constant curvature the boundary of which is characterised by constant extrinsic curvature. An extension of the spherical formulation in the presence of a boundary is attained through use of the method of images. Contrary to the consolidated vanishing effect in maximally symmetric space-times the contribution of the massless "tadpole" diagram no longer vanishes in dimensional regularisation. As a result, conformal invariance is broken due to boundary-related vacuum contributions. The evaluation of one-loop contributions to the two-point function suggests an extension, in the presence of matter couplings, of the simultaneous volume and boundary renormalisation in the effective action. 
  A system of generalized coherent states for the de Sitter group obeying Klein-Gordon equation and corresponding to the massive spin zero particles over the de Sitter space is considered. This allows us to construct the quantized scalar field by the resolution over these coherent states; the corresponding propagator can be computed by the method of analytic continuation to the complexified de Sitter space and coincides with expressions obtained previously by other methods. We show that this propagator possess the de Sitter-invariance and causality properties. 
  We compute the conformal anomaly of free d=6 superconformal (2,0) tensor multiplet on generic curved background. Up to a trivial covariant total-derivative term, it is given by the sum of the type A part proportional to the 6-d Euler density, and the type B part containing three independent Weyl invariants. Multiplied by factor 4N^3, the type B part of the anomaly reproduces exactly the corresponding part of the conformal anomaly of large N multiple M5-brane (2,0) theory as predicted (hep-th/9806087) by the AdS/CFT correspondence. The coefficients of the type A anomaly differ by the factor 4/7 x 4 N^3, so that the free tensor multiplet anomaly does not vanish on a Ricci-flat background. The coefficient 4N^3 is the same as found (hep-th/9703040) in the comparison of the tensor multiplet theory and the d=11 supergravity results for the absorption cross-sections of gravitons by M5 branes, and in the comparison (hep-th/9911135) of 2- and 3-point stress tensor correlators of the free tensor multiplet with the AdS_7 supergravity predictions. The reason for this coincidence is that the three Weyl-invariant terms in the anomaly are related to the $h^2$ and $h^3$ terms in the near flat space expansion of the corresponding non-local effective action, and thus to the 2-point and 3-point stress tensor correlators in flat background. At the same time, the type A anomaly is related to the $h^4$ term in the non-local part of the effective action, i.e. to a certain structure in the 4-point correlation function of stress tensors. 
  We describe key elements of the perturbative similarity renormalization group procedure for Hamiltonians using two, third-order examples: phi^3 interaction term in the Hamiltonian of scalar field theory in 6 dimensions and triple-gluon vertex counterterm in the Hamiltonian of QCD in 4 dimensions. These examples provide insight into asymptotic freedom in Hamiltonian approach to quantum field theory. The renormalization group procedure also suggests how one may obtain ultraviolet-finite effective Schr\"odinger equations that correspond to the asymptotically free theories, including transition from quark and gluon to hadronic degrees of freedom in case of strong interactions. The dynamics is invariant under boosts and allows simultaneous analysis of bound state structure in the rest and infinite momentum frames. 
  In this paper, we put forth a new massive spin-1 field theory. In contrast to the quantization of traditional vector field, the quantization of the new vector field is carried out in a natural way. The Lorentz invariance of the theory is discussed, where owing to an interesting feature of the new vector field, the Lorentz invariance has a special meaning. In term of formalism analogical to QED(i. e. spinor QED), we develop the quantum electrodynamics concerning the new spin-1 particles, say, vector QED, where the Feynman rules are given. The renormalizability of vector QED is manifest without the aid of Higgs mechanism. As an example, the polarization cross section $\sigma_{polar}$ for $e^+ e^-\to f^+ f^-$ is calculated in the lowest order. It turns out that $\sigma_{polar}\sim 0$ and the momentum of $f^+$ and $f^-$ is purely longitudinal. 
  We discuss tachyon configuration for the unoriented bosonic string theory which produces a bosonic string theory with SO(32) gauge symmetry in ten dimensions. It is closely related to the tachyon condensation scenario proposed by A. Sen. We also give the boundary state description of the tachyon condensation process, with some emphasis on the r\^ole of orbifold conformal field theory. 
  We argue that the exactly computable, angle dependent, Casimir force between parallel plates with different directions of conductivity can be measured. 
  In the semiclassical approximation in which the electric charges of scalar particles are described by Grassmann variables ($Q_i^2=0, Q_iQ_j\ne 0$), it is possible to re-express the Lienard-Wiechert potentials and electric fields in the radiation gauge as phase space functions, because the difference among retarded, advanced, and symmetric Green functions is of order Q_i^2. By working in the rest-frame instant form of dynamics, the elimination of the electromagnetic degrees of freedom by means of suitable second classs contraints leads to the identification of the Lienard-Wiechert reduced phase space containing only N charged particles with mutual action-at-a-distance vector and scalar potentials. A Darboux canonical basis of the reduced phase space is found. This allows one to re-express the potentials for arbitrary N as a unique effective scalar potential containing the Coulomb potential and the complete Darwin one, whose 1/c^2 component agrees for with the known expression. The effective potential gives the classical analogue of all static and non-static effects of the one-photon exchange Feynman diagram of scalar electrodynamics. 
  Four pedagogical Lectures at the NATO-ASI on "Quantum Geometry" in Akureyri, Iceland, August 1999. Contents: 1. O(N) Vector Models, 2. Large-N QCD, 3. QCD in Loop Space, 4. Large-N Reduction 
  Combining a local gauge principle, the Pauli Exclusion Principle, and the principles of General Relativity in a particular way, we obtain the mathematical framework for the formulation of a new type of variational principle in space-time. The postulate that physics can be formulated within this framework is called the "principle of the fermionic projector."   The principle of the fermionic projector is introduced and discussed. We describe a limiting process with which our variational principles can be analyzed in the setting of relativistic quantum mechanics and classical field theory. 
  We study the gauge and gravitational interactions of the stable non-BPS D-particles of the type I string theory. The gravitational interactions are obtained using the boundary state formalism while the SO(32) gauge interactions are determined by evaluating disk diagrams with suitable insertions of boundary changing (or twist) operators. In particular the gauge coupling of a D-particle is obtained from a disk with two boundary components produced by the insertion of two twist operators. We also compare our results with the amplitudes among the non-BPS states of the heterotic string which are dual to the D-particles. After taking into account the known duality and renormalization effects, we find perfect agreement, thus confirming at a non-BPS level the expectations based on the heterotic/type I duality. 
  We present the proof of the equivalence theorem in quantum field theory which is based on a formulation of this problem in the field-antifield formalism. As an example, we consider a model in which a different choices of natural finite counterterms is possible, leading to physically non-equivalent quantum theories while the equivalent theorem remains valid. 
  Using the graphical method developed in hep-th/9908082, we obtain the full curve corresponding to the hyperk\"ahler quotient from the extended E_7 Dynkin diagram. As in the E_6 case discussed in the same paper above, the resulting curve is the same as the one obtained by Minahan and Nemeschansky. Our results seem to indicate that it is possible to define a generalized Coulomb branch such that four dimensional mirror symmetry would act by interchanging the generalized Coulomb branch with the Higgs branch of the dual theory. To understand these phenomena, we discuss mirror symmetry and F-theory compactifications probed by D3 branes. 
  We obtain a time-dependent Schrodinger equation for the Friedmann - Robertson - Walker (FRW) model interacting with a homogeneous scalar matter field. We show that for this purpose it is necesary to include an additional action invariant under the reparametrization of time. The last one does not change the equations of motion of the system, but changes only the constraint which at the quantum level becomes time-dependent Schrodinger equation. The same procedure is applied to the supersymmetric case and the supersymmetric quantum constraints are obtained, one of them is a square root of the Schrodinger operator. 
  In this paper we study the spectrum of bosonic string theory on AdS_3. We study classical solutions of the SL(2,R) WZW model, including solutions for long strings with non-zero winding number. We show that the model has a symmetry relating string configurations with different winding numbers. We then study the Hilbert space of the WZW model, including all states related by the above symmetry. This leads to a precise description of long strings. We prove a no-ghost theorem for all the representations that are involved and discuss the scattering of the long string. 
  Type II string theory compactified on a Calabi-Yau manifold, with a singularity modeled by a hypersurface in an orbifold, is considered. In the limit of vanishing string coupling, one expects a non gravitational theory concentrated at the singularity. It is proposed that this theory is holographicly dual to a family of ``non-critical'' superstring vacua, generalizing a previous proposal for hypersurfaces in flat space. It is argued that a class of such singularities is relevant for the study of non-trivial IR fixed points that appear in the moduli space of four-dimensional N=2 SQCD: SU(N_c) gauge theory with matter in the fundamental representation. This includes the origin in the moduli space of the SU(N_c) gauge theory with N_f=2N_c fundamentals. The 4D IR fixed points are studied using the anti-holographic description and the results agree with information available from gauge theory. 
  We analyse D-branes on orbifolds with discrete torsion, extending earlier results. We analyze certain Abelian orbifolds of the type C^3/ \Gamma, where \Gamma is given by Z_m x Z_n, for the most general choice of discrete torsion parameter. By comparing with the AdS/CFT correspondence, we can consider different geometries which give rise to the same physics. This identifies new mirror pairs and suggests new dualities at large N. As a by-product we also get a more geometric picture of discrete torsion. 
  It was recently shown, using the AdS/CFT correspondence, that the low energy effective action of a large $N$ open string theory satisfies a holographic RG flow equation closely related to the Hamilton-Jacobi equation of 5-d supergravity. In this paper we re-obtain the same flow equation in the dual regime of small 't Hooft coupling $\lambda\ll 1$. Our derivation makes use of the conformal equivalence between planar open string diagrams and closed string tree diagrams. This equivalence can be viewed as a microscopic explanation of the open/closed string duality that underlies the AdS/CFT correspondence. 
  We consider a $(3+1)$-dimensional QCD model using a dual supergravity description with a non-extremal $D0$-$D4$ brane background. We calculate the spectrum of glueball masses and Wilson loops in the background. The mass spectrum is shown to coincide with one in non-extremal $D4$-brane systems, and an area low of spatial Wilson loops is established. We show that there is a region that Kaluza-Klein modes of the Euclidean time direction are decoupled without decoupling glueball masses. 
  This paper presents the operator form of the effective potential V governing the time evolution in 2 and 3 and n dimensional subspace of states. The general formula for the n dimensional case is considered the starting point for the calculation of the explicit formulae for 2 and 3 dimensional degenerate and non-degenerate cases. We relate the 2 and 3 dimensional cases to some physical systems which are currently investigated. 
  The basic principles of the quantum mechanics in the K-field formalism (see author's papers) are stated in the paper. K-field formalism arises from geometric generalization of de Broglie postulate. So, the quantum theory equations (including well-known Schrodinger, Klein-Gordon and quadratic Dirac equations) are obtained as the free wave equations on a manifold metrizing force interactions of particles. In this paper, describing wave properties of particles we will restricted ourself with construction special geometric formulation of force interactions. 
  A new approach to study the scaling behavior of the scalar theory near the Gaussian fixed point in $d$-dimensions is presented. For a class of initial data an explicit use of the Green's function of the evolution equation is made. It is thus discussed under which conditions non-polynomial relevant interactions can be generated by the renormalization group flow. 
  We extend 2n-dim biconformal gauge theory by including Lorentz-scalar matter fields of arbitrary conformal weight. We show that for a massless scalar field of conformal weight zero in a torsion-free biconformal geometry, the solution is determined by the Einstein equation on an n-dim submanifold, with the stress-energy tensor of the scalar field as source. The matter field satisfies the n-dim Klein-Gordon equation. 
  We discuss the appearance of modular functions at the one-loop gauge and gravitational couplings in (0,2) non-decomposable N=1 four dimensional orbifold compactifications of the heterotic string. We define the limits for the existence of states causing singularities in the moduli space in the perturbative regime for a generic vacuum of the heterotic string. The "proof" provides evidence for the explanation of the stringy Higgs effect. 
  We discuss the modular invariance of the SL(2,R) WZW model. In particular, we discuss in detail the modular invariants using the \hat{sl}(2,R) characters based on the discrete unitary series of the SL(2,R) representations. The explicit forms of the corresponding characters are known when no singular vectors appear. We show, for example, that from such characters modular invariants can be obtained only when the level k < 2 and infinitely large spins are included. In fact, we give a modular invariant with three variables Z(z,\tau, u) in this case. We also argue that the discrete series characters are not sufficient to construct a modular invariant compatible with the unitarity bound, which was proposed to resolve the ghost problem of the SL(2,R) strings. 
  Recently it was shown how to regularize the Batalin-Vilkovisky (BV) field-antifield formalism of quantization of gauge theories with the non-local regularization (NLR) method. The objective of this work is to make an analysis of the behaviour of this NLR formalism, connected to the BV framework, using two different regulators: a simple second order differential regulator and a Fujikawa-like regulator. This analysis has been made in the light of the well known fact that different regulators can generate different expressions for anomalies that are related by a local couterterm, or that are equivalent after a reparametrization. This has been done by computing precisely the anomaly of the chiral Schwinger model. 
  We investigate domain-wall/quantum field theory correspondences in various dimensions. We give particular emphasis to the special case of the quantum mechanics of 0--branes. 
  In type II string theories, we examine intersecting brane constructions containing brane-antibrane pairs suspended between 5-branes, and more general non-BPS constructions. The tree-level spectra are obtained in each case. We identify various models with distinct physics: parallel brane-antibrane pairs, adjacent pairs, non-adjacent pairs, and configurations which break all supersymmetry even though any pair of branes preserves some supersymmetry. In each case we examine the possible decay modes. Some of these configurations turn out to be tachyon-free, stable non-BPS states. We use T-duality to map some of our brane constructions to brane-antibrane pairs at ALE singularities. This enables us to explicitly derive the spectra by the analogue of the quiver construction, and to compute the sign of the brane-antibrane force in each case. 
  We construct new relativistic linear differential equation in $d$ dimensions generalizing Dirac equation by employing the Clifford algebra of the cubic polynomial associated to Klein-Gordon operator multiplied by the mass parameter. Unlike the Dirac case where the spin content is unique and Lorentz covariance is manifest, here the spin as well as Lorentz covariance of the theory are related to the choice of representation of the Clifford algebra. One of the considered explicit matrix representations gives rise to anyon-like fields in $d=1+1$. Coupling to a U(1) gauge field is discussed and compared with Dirac theory. 
  We derive N = 1, 2 superfield equations as the conditions for a (nonlinear) theory of one abelian N = 1 or N = 2 vector multiplet to be duality invariant. The N = 1 super Born-Infeld action is a particular solution of the corresponding equation. A family of duality invariant nonlinear N = 1 supersymmetric theories is described. We present the solution of the N = 2 duality equation which reduces to the N = 1 Born-Infeld action when the (0,1/2) part of N = 2 vector multiplet is switched off. We also propose a constructive perturbative scheme to compute duality invariant N = 2 superconformal actions. 
  Connection of the invariant Dirac equation over the de Sitter space with irreducible representations of the de Sitter group is ascertained. The set of solutions of this equation is obtained in the form of the product of two different systems of generalized coherent states for the de Sitter group. Using these solutions the quantized Dirac field over de Sitter space is constructed and its propagator is found. It is a result of action of some de Sitter invariant spinor operator onto the spin zero propagator with an imaginary shift of a mass. 
  Non-perturbative effect of the formation of a chiral symmetry breaking condensate <\bar\psi\psi> and of a dynamically generated fermion mass in QED in the presence of an external magnetic field is considered. The dynamical mass of a fermion (energy gap in the fermion spectrum) is shown to depend essentially nonanalytically on the renormalized coupling constant \alpha in a strong magnetic field. Possible applications of this effect are discussed. 
  We investigate the possibility of gravity localization on the brane in the context of supersymmetric theories. To realize this scenario one needs to find a theory with the supersymmetric flow stable in IR at two critical points, one with positive and the other with negative values of the superpotential. We perform a general study of the supersymmetric flow equations of gauged massless supergravity interacting with arbitrary number of vector multiplets and demonstrate that localization of gravity does not occur. The same conclusion remains true when tensor multiplets are included. We analyze all recent attempts to find a BPS brane-world and conclude that localization of gravity on the brane in supersymmetric theories remains a challenging but unsolved problem. 
  Particle and string actions on coset spaces typically lack a quadratic kinetic term, making their quantization difficult. We define a notion of twistors on these spaces, which are hypersurfaces in a vector space that transform linearly under the isometry group of the coset. By associating the points of the coset space with these hypersurfaces, and the internal coordinates of these hypersurfaces with momenta, it is possible to construct manifestly symmetric actions with leading quadratic terms. We give a general algorithm and work out the case of a particle on AdS_p explicitly. In this case, the resulting action is a world-line gauge theory with sources, (the gauge group depending on p) which is equivalent to a nonlocal world-line sigma-model. 
  We consider static U(1) monopole in non-commutative space. Up to the second order in the non-commutativity scale $\theta$, we find no non-trivial corrections to the Dirac solution, the monopole mass remains infinite. We argue the same holds for any arbitrary higher order. Some speculation about the nature of non-commutative spacetime and its relation to the cosmological constant is made. 
  Calogero-Moser systems can be generalized for any root system (including the non-crystallographic cases). The algebraic linearization of the generalized Calogero-Moser systems and of their quadratic (resp. quartic) perturbations are discussed. 
  We investigate the sigma and baby Skyrme models with an RP^2 target space. We compare these models to models with an S^2 target space. We investigate the interactions between solitons and defects in the RP^2 sigma model. 
  We consider two-dimensional Yang-Mills theories on arbitrary Riemann surfaces. We introduce a generalized Yang-Mills action, which coincides with the ordinary one on flat surfaces but differs from it in its coupling to two-dimensional gravity. The quantization of this theory in the unitary gauge can be consistently performed taking into account all the topological sectors arising from the gauge-fixing procedure. The resulting theory is naturally interpreted as a Matrix String Theory, that is as a theory of covering maps from a two-dimensional world-sheet to the target Riemann surface. 
  I review the salient features of three classes of open-string models with broken supersymmetry. These suffice to exhibit, in relatively simple settings, the two phenomena of ``brane supersymmetry'' and ``brane supersymmetry breaking''. In the first class of models, to lowest order supersymmetry is broken both in the closed and in the open sectors. In the second class of models, to lowest order supersymmetry is broken in the closed sector, but is {\it exact} in the open sector, at least for the low-lying modes, and often for entire towers of string excitations. Finally, in the third class of models, to lowest order supersymmetry is {\it exact} in the closed (bulk) sector, but is broken in the open sector. Brane supersymmetry breaking provides a natural solution to some old difficulties met in the construction of open-string vacua. 
  The goal of this note is to provide a recursive algorithm that allows one to calculate the expansion of the metric tensor up to the desired order in Riemann normal coordinates. We test our expressions up to fourth order and predict results up to sixth order. For an arbitrary number of symmetric partial derivatives acting on the components of the metric tensor subtle treatment is required since the degree of complication increases rapidly. 
  This paper has been withdrawn, because the argument for generating Goldstone modes at a Lifshitz point is incorrect. 
  Starting from a WZWN action in the ISO(2,1) Poincare' group which describes a bosonized spinning string in 2+1 Minkowski space-time, we show that a sequence of non-trivial compactifications leads to the description of a spinless string which moves in a (linear dilaton) vacuum, AdS_3 or BTZ black hole background. Other solutions are also obtained and their T-duals analyzed. 
  In the BRST quantization of gauge theories, the zero locus $Z_Q$ of the BRST differential $Q$ carries an (anti)bracket whose parity is opposite to that of the fundamental bracket. We show that the on-shell BFV/BV gauge symmetries are in a 1:1 correspondence with Hamiltonian vector fields on $Z_Q$, and observables of the BRST theory are in a 1:1 correspondence with characteristic functions of the bracket on $Z_Q$. By reduction to the zero locus, we obtain relations between bracket operations and differentials arising in different complexes (the Gerstenhaber, Schouten, Berezin-Kirillov, and Sklyanin brackets); the equation ensuring the existence of a nilpotent vector field on the reduced manifold can be the classical Yang-Baxter equation. We also generalize our constructions to the bi-QP-manifolds which from the BRST theory viewpoint corresponds to the BRST-anti-BRST-symmetric quantization. 
  Compactification of M theory in the presence of G-fluxes yields N=2 five-dimensional gauged supergravity with a potential that lifts all supersymmetric vacua. We derive the effective superpotential directly from the Kaluza-Klein reduction of the eleven-dimensional action on a Calabi-Yau three-fold and compare it with the superpotential obtained by means of calibrations. We discuss an explicit domain wall solution, which represents five-branes wrapped over holomorphic cycles. This solution has a ``running volume'' and we comment on the possibility that quantum corrections provide a lower bound allowing for an AdS_5 vacuum of the 5-dimensional supergravity. 
  A detailed understanding of instanton effects for half-BPS couplings is pursued in theories with 16 supersymmetries. In particular, we investigate the duality between heterotic string on $T^4$ and type IIA on $K_3$ at the $T^4/Z_2$ orbifold point, as well as their higher and lower dimensional versions. We present a remarkably clean quantitative test of the duality at the level of $F^4$ couplings, by completely matching a purely one-loop heterotic amplitude to a purely tree-level type II result. The triality of SO(4,4) and several other miracles are shown to be crucial for the duality to hold. Exact non-perturbative new results for type I', F on $K_3$, M on $K_3$, and IIB on $K_3$ are found, and the general form of D-instanton contributions in type IIA or B on $T^4/Z_2$ is obtained. We also analyze the NS5-brane contributions in type II on $K_3\times T^2$, and predict the value $\mu (N)=\sum_{d|N} (1/d^3)$ for the bulk contribution to the index of the NS5-brane world-volume theory on $K_3 \times T^2$. 
  A classical action for open superstring field theory has been proposed which does not suffer from contact term problems. After generalizing this action to include the non-GSO projected states of the Neveu-Schwarz string, the pure tachyon contribution to the tachyon potential is explicitly computed. The potential has a minimum of $V = -{1\over{32 g^2}}$ which is 60% of the predicted exact minimum of $V=-{1\over{2\pi^2 g^2}}$ from D-brane arguments. 
  Trans-Planckian redshifts in cosmology and outside black holes may provide windows on a hypothetical short distance cutoff on the fundamental degrees of freedom. In cosmology, such a cutoff seems to require a growing Hilbert space, but for black holes, Unruh's sonic analogy has given rise to both field theoretic and lattice models demonstrating how such a cutoff in a fixed Hilbert space might be compatible with a low energy effective quantum field theory of the Hawking effect. In the lattice case, the outgoing modes arise via a Bloch oscillation from ingoing modes. A short distance cutoff on degrees of freedom is incompatible with local Lorentz invariance, but may nevertheless be compatible with general covariance if the preferred frame is defined non-locally by the cosmological background. Pursuing these ideas in a different direction, condensed matter analogs may eventually allow for laboratory observations of the Hawking effect. This paper introduces and gives a fairly complete but brief review of the work that has been done in these areas, and tries to point the way to some future directions. 
  The fundamental laws of physics can be derived from the requirement of invariance under suitable classes of transformations on the one hand, and from the need for a well-posed mathematical theory on the other hand. As a part of this programme, the present paper shows under which conditions the introduction of pseudo-differential boundary operators in one-loop Euclidean quantum gravity is compatible both with their invariance under infinitesimal diffeomorphisms and with the requirement of a strongly elliptic theory. Suitable assumptions on the kernel of the boundary operator make it therefore possible to overcome problems resulting from the choice of purely local boundary conditions. 
  Using the commutativity of a general variation with the time differentiation we discuss both global and local (gauge) symmetries of a lagrangian from a unified point of view. The Noether considerations are thereby applicable for both cases. A complete equivalence between the hamiltonian and lagrangian formulations is established. 
  Mesoscopic effects associated with wave propagation in spacetime with metric stochasticity are studied. We show that the scalar and spinor waves in a stochastic spacetime behave similarly to the electrons in a disordered system. Viewing this as the quantum transport problem, mesoscopic fluctuations in such a spacetime are discussed. The conductance and its fluctuations are expressed in terms of a nonlinear sigma model in the closed time path formalism. We show that the conductance fluctuations are universal, independent of the volume of the stochastic region and the amount of stochasticity. 
  In this paper we consider D=4 NCSYM theories with 8 supercharges. We study these theories through a proper type IIA (and M-theory) brane configuration. We find the one loop beta function of these theories and show that there is an elliptic curve describing the moduli space of the theory, which is in principle the same as the curve for the commutative counter-part of our theory. We study some other details of the dynamics by means of this brane configuration. 
  It is argued that in certain 2d dilaton gravity theories there exist self-consistent solutions of field equations with quantum terms which describe extreme black holes at nonzero temperature. The curvature remains finite on the horizon due to cancelation of thermal divergencies in the stress-energy tensor against divergencies in the classical part of field equations. The extreme black hole solutions under discussion are due to quantum effects only and do not have classical counterparts. 
  The dual relationship between two n-1 parameter families of quantum field theories based on extended complex numbers is investigated in two dimensions. The non-local conserved charges approach is used. The lowest rank affine Toda field theories are generated and identified as integrability submanifolds in parameter space. A truncation of the model leads to a conformal field theory in extended complex space. Depending on the projection over usual complex space chosen, a parametrized central charge is calculated. 
  We present a complete scheme to discuss linear perturbations in the two-branes model of the Randall and Sundrum scenario with the stabilization mechanism proposed by Goldberger and Wise. We confirm that under the approximation of zero mode truncation the induced metric on the branes reproduces that of the usual 4-dimensional Einstein gravity. We also present formulas to evaluate the mass spectrum and the contribution to the metric perturbations from all the Kaluza-Klein modes. We also conjecture that the model has tachyonic modes unless the background configuration for the bulk scalar field introduced to stabilize the distance between the two branes is monotonic in the fifth dimension. 
  I review the arguments in favor of/against the traditional hypothesis that the Planck, string and compactification scales are all within a couple of orders of magnitude from each other. I explain how the extreme brane-world scenario, with TeV type I scale and two large (near millimetric) transverse dimensions, creates conditions analogous to those of the energy desert and is thus naturally singled out. I comment on the puzzle of gauge coupling unification in this context. 
  We review the construction of minimally bosonized supersymmetric quantum mechanics and its relation to hidden supersymmetries in pure parabosonic (parafermionic) systems. 
  We present a model in which the breackdown of conformal symmetry of a quantum stress-tensor due to the trace anomaly is related to a cosmological effect in a gravitational model. This is done by characterizing the traceless part of the quantum stress-tensor in terms of the stress-tensor of a conformal invariant classical scalar field. We introduce a conformal frame in which the anomalous trace is identified with a cosmological constant. In this conformal frame we establish the Einstein field equations by connecting the quantum stress-tensor with the large scale distribution of matter in the universe. 
  The Hadamard state condition is used to analyze the local constraints on the two-point function of a quantum field conformally coupled to a background geometry. Using these constraints we develop a scalar tensor theory which controls the coupling of the stress-tensor induced by the two-point function of the quantum field to the conformal class of the background metric. It is then argued that the determination of the state-dependent part of the two-point function is connected with the determination of a conformal frame. We comment on a particular way to relate the theory to a specific conformal frame (different from the background frame) in which the large scale properties are brought into focus. 
  The intrinsic 4-point coupling, defined in terms of a truncated 4-point function at zero momentum, provides a well-established measure for the interaction strength of a QFT. We show that this coupling can be computed non-perturbatively and to high accuracy from the form factors of an (integrable) QFT. The technique is illustrated and tested with the Ising model, the XY-model and the O(3) nonlinear sigma-model. The results are compared to those from high precision lattice simulations. 
  I emphasize analogy between Dp-branes in string theories and solitons in gauge theories comparing their common properties and showing differences. We will show that for certain excitations of the string/D3-brane system Neumann boundary conditions emerge from the Born-Infeld dynamics. The excitations which are coming down the string with a polarization along a direction parallel to the brane are almost completely reflected. For the wavelengths much larger than the string scale only a small fraction of the energy escapes in the form of dipole radiation. The physical interpretation is that a string attached to the 3-brane manifests itself as an electric charge, and waves on the string cause the end point of the string to freely oscillate and produce electromagnetic dipole radiation in the asymptotic outer region. The magnitude of emitted power is in fact exactly equal to the one given by Thompson formula in ordinary electrodynamics. 
  Closed bosonic string with different normal ordering constants $a \ne \bar a$ for the right and the left moving sectors is considered. One immediate consequence of this choice is absence of tachyon in the physical state spectrum. Selfconsistency of the resulting model in the "old covariant quantization" (OCQ) framework is studyed. The model is manifestly Poincare invariant, it has non trivial massless sector and is ghost free for $D=26, ~ a=1, ~\bar a=0$. A possibility to obtain the light-cone formulation for the model is also discussed. 
  The standard model of electroweak interactions is minimally coupled to gravity and the response of the spherically symmetric solutions -the sphaleron and the bisphaleron- to gravity is emphasized. For a given value of the Higgs mass $M_H$, several branches of solutions exist which terminate into cusp-catastrophy at some ($M_H$-depending) critical value of the parameter $\alpha$ defined by the ratio of the vector-boson mass to the Planck mass. A given branch either bifurcates from another one at an intermediate value of $\alpha$ or persists in the limit $\alpha \to 0$ where it terminates into a flat sphaleron or bisphaleron or into a Bartnik-McKinnon solution. These bifurcation patterns are studied in some details. 
  We present a class of N=1 supersymmetric ``standard'' models of particle physics, derived directly from heterotic M-theory, that contain three families of chiral quarks and leptons coupled to the gauge group SU(3)_C X SU(2)_L X U(1)_Y. These models are a fundamental form of ``brane world'' theories, with an observable and hidden sector each confined, after compactification on a Calabi--Yau threefold, to a BPS three-brane separated by a higher dimensional bulk space with size of the order of the intermediate scale. The requirement of three families, coupled to the fundamental conditions of anomaly freedom and supersymmetry, constrains these models to contain additional five-branes located in the bulk space and wrapped around holomorphic curves in the Calabi--Yau threefold. 
  The possibility that a static magnetic field may decay through production of electron positron pairs is studied. The conclusion is that this decay cannot happen through production of single pairs, as in the electric case, but only through the production of a many-body state, since the mutual magnetic interactions of the created pairs play a relevant role. The investigation is made in view of the proposed existence of huge magnetic field strengths around some kind of neutron stars. 
  We calculate the exact values of the holomorphic observables of {\cal N}=4 supersymmetric SU(N) Yang-Mills theory deformed by mass terms which preserve {\cal N}=1 SUSY. These include the chiral condensates in each massive vacuum of the theory as well as the central charge which determines the tension of BPS saturated domain walls interpolating between these vacua. Several unexpected features emerge in the large-N limit, including anomalous modular properties under an SL(2,Z) duality group which acts on a complexification of the 't Hooft coupling \lambda=g^{2}N/4\pi. We discuss our results in the context of the AdS/CFT correspondence. 
  The role of the (dynamical) dilaton in the vortices associated with the spontaneous breaking of an anomalous U(1) from heterotic string theory is examined. We demonstrate how the anomaly (and the coupling to the dilaton/axion) can appear in the Lagrangian and associated field equations as a controlled perturbation about the standard Nielsen-Olesen equations. In such a picture, the additional field equation for the dilaton becomes a series of corrections to a constant dilaton vev as the anomaly is turned on. In particular we find that even the first nontrivial correction to a constant dilaton generically leads to a (positive) logarithmic divergence of the heterotic dilaton near the vortex core. Since the dilaton field governs the strength of quantum fluctuations in string theory, this runaway behaviour implies that anomalous U(1) vortices in string theory are intrinsically quantum mechanical objects. 
  Via T-duality a theory of open strings on a D1-brane wrapped along a cycle of slanted torus is described by a U(1) gauge theory on a D2-brane in the B-field background. It is also known that there is another dual description of the D1-brane configuration by a non-commutative gauge theory on a D2-brane. Therefore, these two gauge theories on D2-branes are equivalent. Recently, the existence of a continuous set of equivalent gauge theories including these two was suggested. We give a dual D1-brane configuration for each theory in this set, and show that the relation among parameters for equivalent gauge theories can be easily reproduced by rotation of the D1-brane configuration. We also discuss a relation between this duality and Morita equivalence. 
  We present a simple derivation of the supersymmetric one-loop effective action of SU(2) Matrix theory by expressing it in a compact exponential form whose invariance under supersymmetry transformations is obvious. This result clarifies the one-loop exactness of the leading v^4 interactions and the absence of non-perturbative corrections. 
  We discuss the Randall-Sundrum (RS) choice for $h_{MN}$ in the brane-world. We begin with the de Donder gauge (transverse-tracefree) including scalar($h_{55}$), vector($h_{5\mu}$) and tensor($h_{\mu\nu}$) in five dimensions for comparison. One finds that $h_{55}=0$ and $h_{5\mu}=0$. This leads to the RS choice. It appears that the RS choice is so restrictive for the five massless states, whereas it is unique for describing the massive states. Furthermore, one can establish the stability of the RS solution with the RS choice only. 
  We study the impact of Aharonov-Bohm solenoid on the radiation of a charged particle moving in a constant uniform magnetic field. With this aim in view, exact solutions of Klein-Gordon and Dirac equations are found in the magnetic-solenoid field. Using such solutions, we calculate exactly all the characteristics of one-photon spontaneous radiation both for spinless and spinning particle. Considering non-relativistic and relativistic approximations, we analyze cyclotron and synchrotron radiations in detail. Radiation peculiarities caused by the presence of the solenoid may be considered as a manifestation of Aharonov-Bohm effect in the radiation. In particular, it is shown that new spectral lines appear in the radiation spectrum. Due to angular distribution peculiarities of the radiation intensity, these lines can in principle be isolated from basic cyclotron and synchrotron radiation spectra 
  We use the manifestly N=2 supersymmetric, off-shell, harmonic (or twistor) superspace approach to solve the constraints implied by four-dimensional N=2 superconformal symmetry on the N=2 non-linear sigma-model target space, known as the special hyper-K"ahler geometry. Our general solution is formulated in terms of a homogeneous (of degree two) function of unconstrained (analytic) Fayet-Sohnius hypermultiplet superfields. We also derive the improved (N=2 superconformal) actions for the off-shell (constrained) N=2 projective hypermultiplets, and relate them (via non-conformal deformations) to the asymptotically locally-flat (ALF) A_k and D_k series of the gravitational instantons. The same metrics describe Kaluza-Klein monopoles in M-theory, while they also arise in the quantum moduli spaces of N=4 supersymmetric gauge field theories with SU(2) gauge group and matter hypermultiplets in three spacetime dimensions. We comment on rotational isometries versus translational isometries in the context of N=2 NLSM in terms of projective hypermultiplets. 
  We analyze the smallest Dirac eigenvalues by formulating an effective theory for the QCD Dirac spectrum. We find that in a domain where the kinetic term of the effective theory can be ignored, the Dirac eigenvalues are distributed according to a Random Matrix Theory with the global symmetries of the QCD partition function. The kinetic term provides information on the slope of the average spectral density of the Dirac operator. In the second half of this lecture we interpret quenched QCD Dirac spectra at nonzero chemical potential (with eigenvalues scattered in the complex plane) in terms of an effective low energy theory. 
  Recently Seiberg and Witten have proposed that noncommutative gauge theories realized as effective theories on D-brane are equivalent to some ordinary gauge theories. This proposal has been proved, however, only for the Dirac-Born-Infeld action in the approximation of neglecting all derivative terms. In this paper we explicitly construct general forms of the 2n-derivative terms which satisfy this equivalence under their assumption in the approximation of neglecting (2 n+2)-derivative terms. We also prove that the D-brane action computed in the superstring theory is consistent with the equivalence neglecting the fourth and higher order derivative terms. 
  In this paper, we show how the generic coupling of moduli to the kinetic energy of ordinary matter fields results in a cosmological mechanism that influences the evolution and stability of moduli. As an example, we reconsider the problem of stabilizing the dilaton in a non-perturbative potential induced by gaugino condensates. A well-known difficulty is that the potential is so steep that the dilaton field tends to overrun the correct minimum and to evolve to an observationally unacceptable vacuum. We show that the dilaton coupling to the thermal energy of matter fields produces a natural mechanism for gently relaxing the dilaton field into the correct minimum of the potential without fine-tuning of initial conditions. The same mechanism is potentially relevant for stabilizing other moduli fields. 
  Here we place the TeX-typeset version of the old preprint SMC-PHYS-66 (1982), which was published in K. Akama, "Pregeometry", in Lecture Notes in Physics, 176, Gauge Theory and Gravitation, Proceedings, Nara, 1982, edited by K. Kikkawa, N. Nakanishi and H. Nariai, (Springer-Verlag) 267--271. In the paper, we presented the picture that we live in a "brane world" (in the present-day terminology) i.e. in a dynamically localized 3-brane in a higher dimensional space. We adopt, as an example, the dynamics of the Nielsen-Olesen vortex type in six dimensional spacetime to localize our space-time within a 3-brane. At low energies, everything is trapped in the 3-brane, and the Einstein gravity is induced through the fluctuations of the 3-brane. The idea is important because it provides a way basically distinct from the "compactification" to hide the extra dimensions which become necessary for various theoretical reasons. 
  It is shown, at the level of the classical action, that the Wess-Zumino-Witten-Novikov model is equivalent to a combined BF theory and a Chern-Simons action in the presence of a unique boundary term. This connection relies on the techniques of non-Abelian T-duality in non-linear sigma models. We derive some consistency conditions whose various solutions lead to different dual theories. Particular attention is paid to the cases of the Lie algebras SO(2,1) and SO(2,1)*SO(2,1). These are shown to yield three dimensional gravity only if the BF term is ignored. 
  Using the Dirac constraint method we show that the pure fourth-order Pais-Uhlenbeck oscillator model is free of observable negative norm states. Even though such ghosts do appear when the fourth order theory is coupled to a second order one, the limit in which the second order action is switched off is found to be a highly singular one in which these states move off shell. Given this result, construction of a fully unitary, renormalizable, gravitational theory based on a purely fourth order action in 4 dimensions now appears feasible. 
  The massive SU(2) gauge field theory coupled with fermions is considered in 2+1 dimensions. Quark energy spectrum and radiative shift in constant external nonabelian field, being exact solution of the gauge field equations with the Chern-Simons term, are calculated. Under the condition $m = \theta/4$ the quark state is shown to be supersymmetric. 
  The concept of the space (space-time) of the formless finite fundamental elements (FFFE) is suggested. This space can be defined as a set of coverings of the continual space by non-overlapping simply connected regions of any form and arbitrary sizes with some probability measure. The average sizes of each fundamental element are equal to the fundamental length. This definition enables to describe correctly the passage from the space of the formless finite fundamental elements to the continual space in the limit of zero value of the fundamental length. FFFE space-time functional integral construction is suggested. A wave function of a separate FFFE and the overall wave function of a manifold are introduced. It is shown that many other constructions of the discrete space-time (the Regge coverings, the lattice space-time etc.) are the special cases of this space-time. A vacuum action problem is analyzed. One term of this action is proportional to the volume of a fundamental element. It is possible to direct the way for this term to yield the Nambu-Goto action in consideration the string as one-dimensional excitation of a number of FFFEs. Fermionic and bosonic fields in the space-time of FFFEs are excited states of elements. Space-time supersymmetry leads to supposition that the maximal possible number of fermionic excitations at one FFFE is equal to the number of elements in all space-time. The compactification in this space-time means the condition of the neighbour elements absence in compactificated dimensions. 
  We consider the open string sigma-model in the presence of a constant Neveu-Schwarz B-field on the world-sheet that is topologically equivalent to a disk with n holes. First, we compute the sigma-model partition function. Second, we make a consistency check of ideas about the appearance of noncommutative geometry within open strings. 
  The electrogravity transformation is defined by an interchange of the ``active'' and ``passive'' electric parts of the Riemann tensor. Such a transformation has been used to find new solutions that are ``dual'' to the Kerr family of black hole spacetimes in general relativity. In such a case, the dual solution is a similar black hole spacetime endowed with a global monopole charge. Here, we extend this formalism to obtain solutions dual to the static, spherically symmetric solutions of two different scalar-tensor gravity theories. In particular, we first study the duals of the charged black hole solutions of a four-dimensional low-energy effective action of heterotic string theory. Next, we study dual of the Xanthopoulos-Zannias solution in Brans-Dicke theory, which contains a naked singularity. We show that, analogous to general relativity, in these scalar-tensor gravity theories the dual solutions are similar to the original spacetimes, but with a global monopole charge. 
  The thermodynamic properties of an ideal gas of charged vector bosons (with mass m and charge e) is studied in a strong external homogeneous magnetic field no greater than the critical value B_{cr}=m^2/e. The thermodynamic potential, after appropriate analytic continuation, is then used in the study of the spontaneous production of charged spin-one boson pairs from vacuum in the presence of a supercritical homogeneous magnetic field at finite temperature. 
  We consider a general d=4 N=1 globally supersymmetric lagrangian involving chiral and vector superfields, with arbitrary superpotential, Kahler potential and gauge kinetic function. We compute perturbative quantum corrections by employing a component field approach that respects supersymmetry and background gauge invariance. In particular, we obtain the full one-loop correction to the Kahler potential in supersymmetric Landau gauge. Two derivations of this result are described. The non-renormalization of the superpotential and the quadratic correction to the Fayet-Iliopoulos terms are further checks of our computations. 
  Gauged supergravity (SG) with single scalar (dilaton) and arbitrary scalar potential is considered. Such dilatonic gravity describes special RG flows in extended SG where scalars lie in one-dimensional submanifold of total space. The surface counterterm and finite action for such gauged SG in three-, four- and five-dimensional asymptotically AdS space are derived. Using finite action and consistent gravitational stress tensor (local surface counterterm prescription) the regularized expressions for free energy, entropy and mass of d4 dilatonic AdS black hole are found. The same calculation is done within standard reference background subtraction.   The dilaton-dependent conformal anomaly from d3 and d5 gauged SGs is calculated using AdS/CFT correspondence. Such anomaly should correspond to two- and four-dimensional dual quantum field theory which is classically (not exactly) conformally invariant, respectively. The candidate c-functions from d3 and d5 SGs are suggested. These c-functions which have fixed points in asymptoticaly AdS region are expressed in terms of dilatonic potential and they are positively defined and monotonic for number of potentials. 
  In this paper we will construct all BPS and non-BPS D-branes in Type IIA and Type IIB theories from tachyon condensation. We also propose form of Wess-Zumino term for non-BPS D-brane and we will show that tachyon condensation in this term leads to standard Wess-Zumino term for BPS D-brane. 
  We review some recent attempts to extract information about the nature of quantum gravity, with and without matter, by quantum field theoretical methods. More specifically, we work within a covariant lattice approach where the individual space-time geometries are constructed from fundamental simplicial building blocks, and the path integral over geometries is approximated by summing over a class of piece-wise linear geometries. This method of ``dynamical triangulations'' is very powerful in 2d, where the regularized theory can be solved explicitly, and gives us more insights into the quantum nature of 2d space-time than continuum methods are presently able to provide. It also allows us to establish an explicit relation between the Lorentzian- and Euclidean-signature quantum theories. Analogous regularized gravitational models can be set up in higher dimensions. Some analytic tools exist to study their state sums, but, unlike in 2d, no complete analytic solutions have yet been constructed. However, a great advantage of our approach is the fact that it is well-suited for numerical simulations. In the second part of this review we describe the relevant Monte Carlo techniques, as well as some of the physical results that have been obtained from the simulations of Euclidean gravity. We also explain why the Lorentzian version of dynamical triangulations is a promising candidate for a non-perturbative theory of quantum gravity. 
  In the derivation of the Born-Infeld action for the case with a nontrivial boundary of the string world sheet the appearance of a new term changes the conformal anomaly. This may have many consequences, especially also in the study of generalized interacting brane systems. 
  The combined Einstein equations and scalar equation of motion in the Horava-Witten scenario of the strongly coupled heterotic string compactified on a Calabi-Yau manifold are solved in the presence of additional matter densities on the branes. We take into account the universal Calabi-Yau modulus phi with potentials in the 5-d bulk and on the 3-branes, and allow for an arbitrary coupling of the additional matter to phi and an arbitrary equation of state. No ad hoc stabilization of the five dimensional radius is assumed. The matter densities are assumed to be small compared to the potential for phi on the branes; in this approximation we find solutions in the bulk which are exact in y and t. Depending on the coupling of the matter to phi and its equation of state, various solutions for the metric on the branes and in the 5-d bulk are obtained: Solutions corresponding to a ``rolling radius'', and solutions with a static 5-d radius, which reproduce the standard cosmological evolution. 
  The nondeterminantal forms of the Born-Infeld and related brane actions in which the gauge fields couple to both an induced metric and an intrinsic metric are generalised by letting either or both metrics be dynamical. The resulting actions describe ` brane world' and cosmological scenarios in which the gauge fields are confined to the brane, while gravity propagates in both the world-volume and the bulk. In particular, for actions involving a nonsymmetric ` metric', nonsymmetric gravity propagates on the worldvolume. For 3-branes with a symmetric metric, conformal (Weyl) gravity propagates on the worldvolume and has conformally invariant couplings to the gauge fields. 
  We describe a general construction principle which allows to add colour values to a coupling constant dependent scattering matrix. As a concrete realization of this mechanism we provide a new type of S-matrix which generalizes the one of affine Toda field theory, being related to a pair of Lie algebras. A characteristic feature of this S-matrix is that in general it violates parity invariance. For particular choices of the two Lie algebras involved this scattering matrix coincides with the one related to the scaling models described by the minimal affine Toda S-matrices and for other choices with the one of the Homogeneous sine-Gordon models with vanishing resonance parameters. We carry out the thermodynamic Bethe ansatz and identify the corresponding ultraviolet effective central charges. 
  The perturbative treatment of quantum field theory is formulated within the framework of algebraic quantum field theory. We show that the algebra of interacting fields is additive, i.e. fully determined by its subalgebras associated to arbitrary small subregions of Minkowski space. We also give an algebraic formulation of the loop expansion by introducing a projective system ${\cal A}^{(n)}$ of observables ``up to $n$ loops'' where ${\cal A}^{(0)}$ is the Poisson algebra of the classical field theory. Finally we give a local algebraic formulation for two cases of the quantum action principle and compare it with the usual formulation in terms of Green's functions. 
  Modular invariance is a necessary condition for the consistency of any closed string theory. In particular, it imposes stringent constraints on the spectrum of orbifold theories, and in principle determines their spectrum uniquely up to discrete torsion classes. In practice, however, there are often ambiguities in the construction of orbifolds that are a consequence of the fact that the action of the orbifold elements on degenerate ground states is not unambiguous. We explain that there exists an additional consistency condition, related to the spectrum of D-branes in the theory, which eliminates these ambiguities. For supersymmetric orbifolds this condition turns out to be equivalent to the condition that supersymmetry is unbroken in the twisted sectors, but for non-supersymmetric orbifolds it appears to be a genuinely new consistency condition. 
  Supersymmetry of five dimensional string solutions is examined in the context of gauged D=5, N=2 supergravity coupled to abelian vector multiplets. We find magnetic black strings preserving one quarter of supersymmetry and approaching the half-supersymmetric product space AdS_3\times H^2 near the event horizon. The solutions thus exhibit the phenomenon of supersymmetry enhancement near the horizon, like in the cases of ungauged supergravity theories, where the near horizon limit is fully supersymmetric. Finally, product space compactifications are studied in detail, and it is shown that only for negative curvature (hyperbolic) internal spaces, some amount of supersymmetry can be preserved. Among other solutions, we find that the extremal rotating BTZ black hole tensored by H^2 preserves one quarter of supersymmetry. 
  We study the Gepner model description of D-branes in Calabi-Yau manifolds with singular curves. From a geometrical point of view, the resolution of singularities leads to additional homology cycles around which branes can wrap. Using techniques from conformal field theory we address the construction of boundary states for branes wrapping additional 3-cycles on the resolved Calabi-Yau manifold. Explicit formulas are provided for Z_2 singular curves. 
  We discuss non-perturbative phase transitions, within the context of heterotic M-theory, which occur when all, or part, of the wrapped five-branes in the five-dimensional bulk space come into direct contact with a boundary brane. These transitions involve the transformation of the five-brane into a ``small instanton'' on the Calabi-Yau space at the boundary brane, followed by the ``smoothing out'' of the small instanton into a holomorphic vector bundle. Small instanton phase transitions change the number of families, the gauge group or both on the boundary brane, depending upon whether a base component, fiber component or both components of the five-brane class are involved in the transition. We compute the conditions under which a small instanton phase transition can occur and present a number of explicit, phenomenologically relevant examples. 
  The Stringy Uncertainty relations, and corrections thereof, were explicitly derived recently from the New Relativity Principle that treats all dimensions and signatures on the same footing and which is based on the postulate that the Planck scale is the minimal length in Nature in the same vein that the speed of light was taken as the maximum velocity in Einstein's theory of Special Relativity. A simple numerical argument is presented which suggests that Quantum Spacetime may very well be $infinite$ dimensional. A discussion of the repercusions of this new paradigm in Physics is given. A truly remarkably simple and plausible solution of the cosmological constant problem results from the New Relativity Principle : The cosmological constant is $not$ constant, in the same vein that Energy in Einstein's Special Relativity is observer dependent. Finally, following El Naschie, we argue why the observed D=4 world might just be an $average$ dimension over the infinite possible values of the Quantum Spacetime and why the compactification mechanisms from higher to four dimensions in String theory may not be actually the right way to look at the world at Planck scales. 
  We construct a 1/4 BPS soliton solution in N=4 non-commutative super Yang-Mills theory to the first order in the non-commutativity parameter \theta_{ij}. We then solve the non-commutative eigenvalue equations for the scalar fields. The Callan-Maldacena interpretation of the eigenvalues precisely reproduces the expected string junction picture: the string junction is tilted against the D3-branes with angle \theta_{ij}. 
  The force densities exerted on a localised material system by linearised interaction with fields of axionic and dilatonic type are shown to be describable very generally by relatively simple expressions that are well behaved for fields of purely external origin, but that will be subject to ultraviolet divergences requiring regularisation for fields arising from self interaction in submanifold supported ``brane'' type systems. In the particular case of 2-dimensionally supported, i.e. string type, system in an ordinary 4-dimensional background it is shown how the result of this regularisation is expressible in terms of the worldsheet curvature vector $K^\mu$, and more particularly that (contrary to what was suggested by early work on this subject) for a string of Nambu Goto type the divergent contribution from the dilatonic self action will always be directed oppositely to its axionic counterpart. This makes it possible for the dilatonic and axionic divergences to entirely cancel each other out (so that there is no need of a renormalisation to get rid of ``infinities'') when the relevant coupling coefficents are related by the appropriate proportionality condition provided by the low energy limit of superstring theory. 
  In a deep-infrared (ergodic) regime, QCD coupled to massive pseudoreal and real quarks are described by chiral orthogonal and symplectic ensembles of random matrices. Using this correspondence, general expressions for the QCD partition functions are derived in terms of microscopically rescaled mass variables. In limited cases, correlation functions of Dirac eigenvalues and distributions of the smallest Dirac eigenvalue are given as ratios of these partition functions. When all masses are degenerate, our results reproduce the known expressions for the partition functions of zero-dimensional sigma models. 
  Four-point correlation functions of hypermultiplet bilinear composites are analysed in N=2 superconformal field theory using the superconformal Ward identities and the analyticity properties of the composite operator superfields. It is shown that the complete amplitude is determined by a single arbitrary function of the two conformal cross-ratios of the space-time variables. 
  We point out the existence of some singular, radial, spin-0 instantons for curvature-quadratic gravity theories. They are complex. 
  Although nonsingular spacetimes and those containing black holes are qualitatively quite different, there are continuous families of configurations that connect the two. In this paper we use self-gravitating monopole solutions as tools for investigating the transition between these two types of spacetimes. We show how causally distinct regions emerge as the black hole limit is achieved, even though the measurements made by an external observer vary continuously. We find that near-critical solutions have a naturally defined entropy, despite the absence of a true horizon, and that this has a clear connection with the Hawking-Bekenstein entropy. We find that certain classes of near-critical solutions display naked black hole behavior, although they are not truly black holes at all. Finally, we present a numerical simulation illustrating how an incident pulse of matter can induce the dynamical collapse of a monopole into an extremal black hole. We discuss the implications of this process for the third law of black hole thermodynamics. 
  In this note we study systems with a closed algebra of second class constraints. We describe a construction of the reduced theory that resembles the conventional treatment of first class constraints. It suggests, in particular, to compute the symplectic form on the reduced space by a fiber integral of the symplectic form on the original space. This approach is then applied to a class of systems with loop group symmetry. The chiral anomaly of the loop group action spoils the first class character of the constraints but not their closure. Proceeding along the general lines described above, we obtain a 2-form from a fiber (path)integral. This form is not closed as a relict of the anomaly. Examples of such reduced spaces are provided by D-branes on group manifolds with WZW action. 
  We study a problem of low-energy effective action in N=4 super Yang-Mills theories. Using harmonic superspace approach we consider N=4 SYM in terms of unconstrained N=2 superfield and apply N=2 background field method to finding effective action for N=4 SU(n) SYM broken down to U(n)$^{n-1}$. General structure of leading low-energy corrections to effective action is discussed and calculational procedure for their explicit finding is presented. 
  We show that unstable D-branes play the role of ``D-sphalerons'' in string theory. Their existence implies that the configuration space of Type II string theory has a complicated homotopy structure, similar to that of an infinite Grassmannian. In particular, the configuration space of Type IIA (IIB) string theory on $\R^{10}$ has non-trivial homotopy groups $\pi_k$ for all $k$ even (odd). 
  We review the quantization of open string in NS-NS background and demonstrate that its endpoint becomes noncommutative. The same approach allows us to determine the noncommutativity that arises for a charged open string in background gauge fields. While NS-NS background is relevant for ``worldvolume'' noncommutativity, a simple argument suggests that RR background is likely to be relevant for ``spacetime'' noncommutativity. 
  We point out that for a large class of universes, holography implies that the most probable value of the cosmological constant is zero. In four spacetime dimensions, the probability distribution takes the Baum-Hawking form, $dP\sim\exp(cM_p^2/\Lambda)d\Lambda$. 
  Rapidly moving sources create pairs in the vacuum and lose energy. In consequence of this, the velocity of a charged body cannot approach the speed of light closer than a certain limit which depends only on the coupling constant. The vacuum back-reaction secures the observance of the conservation laws. A source can lose up to 50% of energy and charge because of the vacuum instability. 
  By analysing the work of Campolattaro we argue that the second Seiberg-Witten equation over the Spin^c_4 manifold, i.e., F^+_{ij}=< M,S_ij M >, is the generalization of the Campolattaro's description of the electromagnetic field tensor F^{\mu\nu} in the bilinear form F^{\mu\nu}=\bar{\Psi} S^{\mu\nu}\Psi. It turns out that the Seiberg-Witten equations (also the perturbed Seiberg-Witten equations) can be well understood from this point of view. We suggest that the second Seiberg-Witten equation can be replaced by a nonlinear Dirac-like Equation. We also derive the spinor representation of the connection on the associated unitary line bundle over the Spin^c_4 manifold. 
  We first apply Connes' noncommutative geometry to a finite point space. The explicit form of the action functional of U(1) gauge field on this n-point space is obtained. We then consider the case when the n-point space is replaced by {space-time}\times{n-point space}. This action is shown to relate the Hamiltonian of the continuous-spin formulation of the Potts model. We argue that U(1) gauge theory on the discrete space-time determines the geometric origin of a class of phase transitions. 
  We present an action functional and derive equations of motion for a coupled system of a bosonic Dp--brane and an open string ending on the Dp-brane. With this example we address the key issues of the recently proposed method (hep-th/9905144, hep-th/9906041) for the construction of manifestly supersymmetric action functionals for interacting superbrane systems. We clarify, in particular, how the arbitrariness in sources localized on the intersection is related to the standard description of the flat D-branes as rigid planes where the string for endpoints 'live'. 
  We present a supersymmetric action functional for the coupled system of an open fundamental superstring and super-D0-branes attached to (identified with) the string endpoints. As a preliminary step the geometrical actions for a free super-D0-brane and a free type IIA superstring have been built. The pure bosonic limits of the action for the coupled system and of the equations of motion are discussed in some detail. 
  The knowledge of {\it non usual} and sometimes {\it hidden} symmetries of (classical) integrable systems provides a very powerful setting-out of solutions of these models. Primarily, the understanding and possibly the quantisation of intriguing symmetries could give rise to deeper insight into the nature of field spectrum and correlation functions in quantum integrable models. With this perspective in mind we will propose a general framework for discovery and investigation of local, quasi-local and non-local symmetries in classical integrable systems. We will pay particular attention to the structure of symmetry algebra and to the r\^ole of conserved quantities. We will also stress a nice unifying point of view about KdV hierarchies and Toda field theories with the result of obtaining a Virasoro algebra as exact symmetry of Sine-Gordon Model. 
  For the Becchi-Rouet-Stora-Tyutin (BRST) invariant extended action for any gauge theory, there exists another off-shell nilpotent symmetry. For linear gauges, it can be elevated to a symmetry of the quantum theory and used in the construction of the quantum effective action. Generalizations for nonlinear gauges and actions with higher order ghost terms are also possible. 
  We consider a class of (2+1) dimensional baby Skyrme models with potentials that have more than one vacum. These potentials are generalisation of old and new baby Skyrme models;they involve more complicated dependence on phi_3.We find that when the potential is invariant under phi_3 -> -phi_3 the configuration corresponding to the baby skyrmions lying "on top of each other" are the minima of the energy. However when the potential breaks this symmetry the lowest field configurations correspond to separated baby skyrmions. We compute the energy distributions for skyrmions of degrees between one and eight and discuss their geometrical shapes and binding energies. We also compare the 2-skyrmion states for these potentials. Most of our work has been performed numerically with the model being formulated in terms of three real scalar fields (satisfying one constraint). 
  Recently it has been shown that the Reeh-Schlieder property w.r.t. thermal equilibrium states is a direct consequence of locality, additivity and the relativistic KMS condition. Here we extend this result to ground states. 
  We analyse a supersymmetric mechanical model derived from (1+1)-dimensional field theory with Yukawa interaction, assuming that all physical variables take their values in a Grassmann algebra B. Utilizing the symmetries of the model we demonstrate how for a certain class of potentials the equations of motion can be solved completely for any B. In a second approach we suppose that the Grassmann algebra is finitely generated, decompose the dynamical variables into real components and devise a layer-by-layer strategy to solve the equations of motion for arbitrary potential. We examine the possible types of motion for both bosonic and fermionic quantities and show how symmetries relate the former to the latter in a geometrical way. In particular, we investigate oscillatory motion, applying results of Floquet theory, in order to elucidate the role that energy variations of the lower order quantities play in determining the quantities of higher order in B. 
  The dynamical symmetry breaking in a two-field model is studied by numerically solving the coupled effective field equations. These are dissipative equations of motion that can exhibit strong chaotic dynamics. By choosing very general model parameters leading to symmetry breaking along one of the field directions, the symmetry broken vacua make the role of transitory strange attractors and the field trajectories in phase space are strongly chaotic. Chaos is quantified by means of the determination of the fractal dimension, which gives an invariant measure for chaotic behavior. Discussions concerning chaos and dissipation in the model and possible applications to related problems are given. 
  We review some aspects of moduli in string theory. We argue that one should focus on {\it approximate moduli spaces}, and that there is evidence that such spaces exist non-perturbatively. We ask what it would mean for string theory to predict low energy supersymmetry. Aspects of two proposed mechanisms for fixing the moduli are discussed, and solutions to certain cosmological problems associated with moduli are proposed. 
  In a remarkable variety of contexts appears the modular data associated to finite groups. And yet, compared to the well-understood affine algebra modular data, the general properties of this finite group modular data has been poorly explored. In this paper we undergo such a study. We identify some senses in which the finite group data is similar to, and different from, the affine data. We also consider the data arising from a cohomological twist, and write down, explicitly in terms of quantities associated directly with the finite group, the modular S and T matrices for a general twist, for what appears to be the first time in print. 
  We investigate supersymmetric extrema of Abelian gauged supergravity theories with non-trivial vector multiplets and 8 supercharges in four and five dimensions. The scalar fields of these models parameterize a manifold consisting of disconnected branches and restricting to the case where this manifold has a non-singular metric we show that on every branch there can be at most one extremum, which is a local maximum (for W>0) or a minimum (for W<0) of the superpotential W. Therefore, these supergravity models do not allow for regular domain wall solutions interpolating between different extrema of the superpotential and the space-time transverse to the wall asymptotically always approaches the boundary of AdS (UV-fixed points in a dual field theory). 
  The reducible K\"ahler coset space G/S\otimesU(1)^k is discussed in a geometrical approach. We derive the formula which expresses the Riemann curvature of the reducible K\"ahler coset space in terms of its Killing vectors. The formula manifests the group structure of G. On the basis of this formula we establish an algebraic method to evaluate the four-fermi coupling of the supersymmetric non-linear sigma-model on G/S\otimesU(1)^k at the low-energy limit. As an application we consider the supersymmetric non-linear sigma-model on E_7/SU(5)\otimesU(1)^3 which contains the three families of {10} + {5^*} + {1} of SU(5) as the pseudo NG fermions. The four-fermi coupling constants among diffferent families of the fermions are explicitly given at the low-energy limit. 
  We discuss a Lorentz covariant space-time uncertainty relation, which agrees with that of Karolyhazy-Ng-van Dam when an observational time period delta t is larger than the Planck time lp. At delta t < lp, this uncertainty relation takes roughly the form delta t delta x > lp^2, which can be derived from the condition prohibiting the multi-production of probes to a geometry. We show that there exists a minimal area rather than a minimal length in the four-dimensional case. We study also a three-dimensional free field theory on a non-commutative space-time realizing the uncertainty relation. We derive the algebra among the coordinate and momentum operators and define a positive-definite norm of the representation space. In four-dimensional space-time, the Jacobi identity should be violated in the algebraic representation of the uncertainty relation. 
  QCD_2 with fundamental quarks on a cylinder is solved to leading order in the 1/N expansion, including the zero mode gluons. As a result of the non-perturbative dynamics of these gauge degrees of freedom, the compact space-time direction gets effectively decompactified. In a thermodynamic interpretation, this implies that there is no pressure of order N and that the chiral condensate of order N is temperature independent. These findings are consistent with confinement of quarks, rule out both chiral and deconfining phase transitions in the finite temperature 't Hooft model, and help to resolve some controversial issues in the literature 
  We generalise recent results on Hopf instantons in a Chern--Simons and Fermion theory in a fixed background magnetic field. We find that these instanton solutions have to obey the Liouville equation in target space. As a consequence, these solutions are given by a class of Hopf maps that consist of the composition of the standard Hopf map with an arbitrary rational map. 
  One of the key properties of Dirac operators is the possibility of a degeneracy of zero modes. For the Abelian Dirac operator in three dimensions the construction of multiple zero modes has been sucessfully carried out only very recently. Here we generalise these results by discussing a much wider class of Dirac operators together with their zero modes. Further we show that those Dirac operators that do admit zero modes may be related to Hopf maps, where the Hopf index is related to the number of zero modes in a simple way. 
  Starting from the most general harmonic superspace action of self-interacting Q^+ hypermultiplets in the background of N=2 conformal supergravity, we derive the general action for the bosonic sigma model with a generic 4n dimensional quaternionic-Kahler (QK) manifold as the target space. The action is determined by the analytic harmonic QK potential. We find out this action to have two flat limits. One gives the hyper-Kahler sigma model with a 4n dimensional target manifold, while another yields a conformally invariant sigma model with 4(n+1) dimensional hyper-Kahler target. We work out the harmonic superspace version of the QK quotient construction and use it to give a new derivation of QK extensions of Taub-NUT and Eguchi-Hanson metrics. We analyze in detail the geometrical and symmetry structure of the second metric. The QK sigma model approach allows us to reveal the enhancement of its isometry group to SU(3) or SU(2,1) at the special relations between its parameters : the Einstein constant and the "mass". 
  We obtain the zero mode effective action for gravitating objects in the bulk of dilatonic domain walls. Without additional fields included in the bulk action, the zero mode effective action reproduces the action in one lower dimensions obtained through the ordinary Kaluza-Klein (KK) compactification, only when the transverse (to the domain wall) component of the bulk metric does not have non-trivial term depending on the domain wall worldvolume coordinates. With additional fields included in the bulk action, non-trivial dependence of the transverse metric component on the domain wall worldvolume coordinates appears to be essential in reproducing the lower-dimensional action obtained via the ordinary KK compactification. We find, in particular, that the effective action for the charged (p+1)-brane in the domain wall bulk reproduces the action for the p-brane in one lower dimensions. 
  In this paper we study the invariance of the noncmmutative gauge theories under C, P and T transformations. For the noncommutative space (when only the spatial part of $\theta$ is non-zero) we show that NCQED is Parity invariant. In addition, we show that under charge conjugation the theory on noncommutative $R^4_{\theta}$ is transformed to the theory on $R^4_{-\theta}$, so NCQED is a CP violating theory. The theory remains invariant under time reversal if, together with proper changes in fields, we also change $\theta$ by $-\theta$. Hence altogether NCQED is CPT invariant. Moreover we show that the CPT invariance holds for general noncommutative space-time. 
  This talk presents a list of problems related to the double-elliptic (Dell) integrable systems with elliptic dependence on both momenta and coordinates. As expected, in the framework of Seiberg-Witten theory the recently discovered explicit self-dual family of 2-particle Dell Hamiltonians provides a perturbative period matrix which is a logarithm of the ratio of the (momentum-space) theta-functions. 
  We study codimension one branes, i.e. p-branes in (p+2)-dimensions, in the superembedding approach for the cases where the worldvolume superspace is embedded in a minimal target superspace with half supersymmetry breaking. This singles out the cases p=1,2,3,5,9. For p=3,5,9 the superembedding geometry naturally involves a fundamental super 2-form potential on the worldvolume whose generalised field strength obeys a constraint deducible from considering an open supermembrane ending on the p-brane. This constraint, together with the embedding constraint, puts the system on-shell for p=5 but overconstrains the 9-brane in D=11 such that the Goldstone superfield is frozen. For p=3 these two constraints give rise to an off-shell linear multiplet on the worldvolume. An alternative formulation of this case is given in which the linear multiplet is dualised to an off-shell scalar multiplet. Actions are constructed for both cases and are shown to give equivalent equations of motion. After gauge fixing a local Sp(1) symmetry associated with shifts in the Sp(1)_R Goldstone modes, we find that the auxiliary fields in the scalar multiplet parametrise a two-sphere. For completeness we also discuss briefly the cases p=1,2 where the equations of motion (for off-shell multiplets) are obtained from an action principle. 
  We discuss how the BPS branes of M-theory could be described as bound states of non-BPS M10-branes. This conjectured M10-brane is constructed as an unstable spacetime-filling brane in the massive eleven dimensional supergravity defined with a Killing direction, such that 1) the BPS M9-brane is obtained after the tachyonic mode of an M2-brane ending on it condenses, and 2) it gives the non-BPS D9-brane of the type IIA theory upon reduction. The existence of other non-BPS M-branes is also discussed, together with their possible stabilisation within the Horava-Witten construction. 
  Motivated by several recent papers on string-inspired calculations in QED, we here present our own use of world-line techniques in order to calculate the vacuum polarization and effective action in scalar and spinor QED with external arbitrary constant electromagnetic field configuration. 
  In this thesis, we review recent progresses on Nonlinear Integral Equation approach to finite size effects in two dimensional integrable quantum field theories, with emphasis to Sine-Gordon/Massive Thirring model and restrictions to minimal models perturbed by $\Phi_{1,3}$. Exact calculations of the dependence of energy levels on the size are presented for vacuum and many excited states. 
  The "orbifold covariance principle", or OCP for short, is presented to support a conjecture of Pradisi, Sagnotti and Stanev on the expression of the Klein- bottle amplitude. 
  We study the Slavnov-Taylor Identities (STI) breaking terms, up to the second order in perturbation theory. We investigate which requirements are needed for the first order Wess-Zumino consistency condition to hold true at the next order in perturbation theory. We find that: a) If the cohomologically trivial contributions to the first order ST breaking terms are not removed by a suitable choice of the first order normalization conditions, the Wess-Zumino consistency condition is modified at the second order. b) Moreover, if one fails to recover the cohomologically trivial part of the first order STI breaking local functional, the second order anomaly actually turns out to be a non-local functional of the fields and the external sources. By using the Wess-Zumino consistency condition and the Quantum Action Principle, we show that the cohomological analysis of the first order STI breaking terms can actually be performed also in a model (the massive Abelian Higgs-Kibble one) where the BRST transformations are not nilpotent. 
  We calculate the Faddeev-Popov operator corresponding to the maximally Abelian gauge for gauge group SU(N). Specializing to SU(2) we look for explicit zero modes of this operator. Within an illuminating toy model (Yang-Mills mechanics) the problem can be completely solved and understood. In the field theory case we are able to find an analytic expression for a normalizable zero mode in the background of a single `t Hooft instanton. Accordingly, such an instanton corresponds to a horizon configuration in the maximally Abelian gauge. Possible physical implications are discussed. 
  The quantum fluctuations of a homogeneous, isotropic, open pre-big bang model are discussed. By solving exactly the equations for tensor and scalar perturbations we find that particle production is negligible during the perturbative Pre-Big Bang phase. 
  We discuss the AdS/CFT correspondence for negative curvature Einstein manifolds whose conformal boundary is degenerate in the sense that it is of codimension greater than one. In such manifolds, hypersurfaces of constant radius do not blow up uniformly as one increases the radius; examples include products of hyperbolic spaces and the Bergman metric. We find that there is a well-defined correspondence between the IR regulated bulk theory and conformal field theory defined in a background whose degenerate geometry is regulated by the same parameter. We are hence able to make sense of supergravity in backgrounds such as $AdS_{3} \times H^{2}$. 
  We construct unitary representations of (1,0) and (2,0) superconformal algebras in six dimensions by using superfields defined on harmonic superspaces with coset manifolds USp(2n)/[U(1)]^n, n=1,2. In the spirit of the AdS_7/CFT_6 correspondence massless conformal fields correspond to "supersingletons" in AdS_7. By tensoring them we produce all short representations corresponding to 1/2 and 1/4 BPS anti-de Sitter bulk states of which "massless bulk" representations are particular cases. 
  It has recently been shown that the equation of motion of a massless scalar field in the background of some specific p branes can be reduced to a modified Mathieu equation. In the following the absorption rate of the scalar by a D3 brane in ten dimensions is calculated in terms of modified Mathieu functions of the first kind, using standard Mathieu coefficients. The relation of the latter to Dougall coefficients (used by others) is investigated. The S-matrix obtained in terms of modified Mathieu functions of the first kind is easily evaluated if known rapidly convergent low energy expansions of these in terms of products of Bessel functions are used. Leading order terms, including the interesting logarithmic contributions, can be obtained analytically. 
  In this paper we make the connection between semi-classical string quantization and exact conformal field theory quantization of strings in 2+1 Anti de Sitter spacetime. More precisely, considering the WZWN model corresponding to SL(2,R) and its covering group, we construct quantum {\it coherent} string states, which generalize the ordinary coherent states of quantum mechanics, and show that in the classical limit they correspond to oscillating circular strings. After quantization, the spectrum is found to consist of two parts: A continuous spectrum of low mass states (partly tachyonic) fulfilling the standard spin-level condition necessary for unitarity |j|< k/2, and a discrete spectrum of high mass states with asymptotic behaviour m^2\alpha'\propto N^2 (N positive integer). The quantization condition for the high mass states arises from the condition of finite positive norm of the coherent string states, and the result agrees with our previous results obtained using semi-classical quantization. In the k\to\infty limit, all the usual properties of coherent or {\it quasi-classical} states are recovered. It should be stressed that we consider the circular strings only for simplicity and clarity, and that our construction can easily be used for other string configurations too. We also compare our results with those obtained in the recent preprint hep-th/0001053 by Maldacena and Ooguri. 
  We discuss the spectrum of the three dimensional phi^4 theory in the broken symmetry phase. In this phase the effective potential between the elementary quanta of the model is attractive and bound states of two or more of them may exist. We give theoretical and numerical evidence for the existence of these bound states. Looking in particular at the Ising model realization of the phi^4 theory we show, by using duality, that these bound states are in one-to-one correspondence with the glueball states of the gauge Ising model. We discuss some interesting consequences of this identification. 
  We prove by explicit calculation that Feynman graphs in noncommutative Yang-Mills theory made of repeated insertions into itself of arbitrarily many one-loop ghost propagator corrections are renormalizable by local counterterms. This provides a strong support for the renormalizability conjecture of that model. 
  We consider a three-dimensional brane-universe moving in a Type 0 String background. The motion induces on the brane a cosmological evolution which, for some range of the parameters, exhibits an inflationary phase. 
  We study the electric flux tubes that undertake color confinement in N=2 supersymmetric Yang-Mills theories softly broken down to N=1 by perturbing with the first two Casimir operators. The relevant Abelian Higgs model is not the standard one due to the presence of an off-diagonal coupling among different magnetic U(1) factors. We perform a preliminary study of this model at a qualitative level. BPS vortices are explicitely obtained for particular values of the soft breaking parameters. Generically however, even in the ultrastrong scaling limit, vortices are not critical but live in a "hybrid" type II phase. Also, ratios among string tensions are seen to follow no simple pattern. We examine the situation at the half Higgsed vacua and find evidence for solutions with the behaviour of superconducting strings. In some cases they are solutions to BPS equations. 
  We present evidence for the existence of infinitely-many new families of renormalisation group flows between the nonunitary minimal models of conformal field theory. These are associated with perturbations by the $\phi_{21}$ and $\phi_{15}$ operators, and generalise a family of flows discovered by Martins. In all of the new flows, the finite-volume effective central charge is a non-monotonic function of the system size. The evolution of this effective central charge is studied by means of a nonlinear integral equation, a massless variant of an equation recently found to describe certain massive perturbations of these same models. We also observe that a similar non-monotonicity arises in the more familiar $\phi_{13}$ perturbations, when the flows induced are between nonunitary minimal models. 
  Supersymmetric orbifold projection of N=1 SQCD with relatively small number of flavors (not larger than the number of colors) is considered. The purpose is to check whether orbifolding commutes with the infrared limit. On the one hand, one considers the orbifold projection of SQCD and obtains the low-energy description of the resulting theory. On the other hand, one starts with the low-energy effective theory of the original SQCD, and only then perfoms orbifolding. It is shown that at finite N_c the two low-energy theories obtained in these ways are different. However, in the case of stabilized run-away vacuum these two theories are shown to coincide in the large N_c limit. In the case of quantum modified moduli space, topological solitons carrying baryonic charges are present in the orbifolded low-energy theory. These solitons may restore the correspondence between the two theories provided that the soliton mass tends to zero in the large N_c limit. 
  In order to eliminate gauge variant degrees of freedom we study the way to introduce gauge invariant fields in pure non-Abelian Yang-Mills theory. Our approach is based on the use of the gauge-invariant but path-dependent variables formalism. It is shown that for a special class of paths these fields coincide with the usual ones in some definite gauges. The interquark potential is discussed by exploiting the rich structure of the gluonic cloud or dressing around static fermions. 
  We show that integrable structure of chiral random matrix models incorporating global symmetries of QCD Dirac operators (labeled by the Dyson index beta=1,2, and 4) leads to emergence of a connection relation between the spectral statistics of massive and massless Dirac operators. This novel link established for beta-fold degenerate massive fermions is used to explicitly derive (and prove the random matrix universality of) statistics of low--lying eigenvalues of QCD Dirac operators in the presence of SU(2) massive fermions in the fundamental representation (beta=1) and SU(N_c >= 2) massive adjoint fermions (beta=4). Comparison with available lattice data for SU(2) dynamical staggered fermions reveals a good agreement. 
  We consider quantum mechanical Yang-Mills theories with eight supercharges and a $Spin(5) \times SU(2)_R$ flavor symmetry. We show that all normalizable ground states in these gauge theories are invariant under this flavor symmetry. This includes, as a special case, all bound states of D0-branes and D4-branes. As a consequence, all bound states of D0-branes are invariant under the $Spin(9)$ flavor symmetry. When combined with index results, this implies that the bound state of two D0-branes is unique. 
  S and U-duality dictate that graviton scattering amplitudes in IIB superstring theory be automorphic functions on the appropriate fundamental domain which describe the inequivalent vacua of (compactified) theories. A constrained functional form of graviton scattering is proposed using Eisenstein series and their generalizations compatible with: a) two-loop supergravity, b) genus one superstring theory, c) the perturbative coupling dependence of the superstring, and d) with the unitarity structure of the massless modes. The form has a perturbative truncation in the genus expansion at a given order in the derivative expansion. Comparisons between graviton scattering S-matrices and effective actions for the first quantized superstring are made at the quantum level. Possible extended finiteness properties of maximally extended quantum supergravity theories in different dimensions is implied by the perturbative truncation of the functional form of graviton scattering in IIB superstring theory. 
  We extend 2n-dim biconformal gauge theory by including Lorentz-scalar matter fields of arbitrary conformal weight. For a massless scalar field of conformal weight zero in a torsion-free biconformal geometry, the solution is determined by the Einstein equation on an n-dim submanifold, with the stress-energy tensor of the scalar field as source. The matter field satisfies the n-dim Klein-Gordon equation. 
  Stability of dilatonic AdS spaces due to quantum effects of dilaton coupled conformal matter is considered. When such spaces do not exist on classical level, their dynamical generation occurs. Explicit examples corresponding to quantum creation of d4 dilatonic AdS Universe and of d2 dilatonic AdS Black Hole (BH) are presented. Motivated by holographic RG, in the similar approach the complete d5 effective action is discussed. The intermediate region where it is the sum of two parts: bulk (classical gravity) and boundary quantum action is investigated. The effective equations solution representing d5 AdS Universe with warp scale factor is found. Four-dimensional de Sitter or AdS world is generated on the boundary of such Universe as a result of quantum effects. 
  We clarify the relation between the recently formulated holographic renormalization group equation and Polchinski's exact renormalization group equation. 
  The multidimensional N=4 supersymmetric quantum mechanics (SUSY QM) is constructed and the various possibilities for partial supersymmetry breaking are discussed. It is shown that quantum mechanical models with one quarter, one half and three quarters of unbroken(broken) supersymmetries can exist in the framework of the multidimensional N=4 SUSY QM. 
  The method of construction of auxiliary representations for a given Lie algebra is discussed in the framework of the BRST approach. The corresponding BRST charge turns out to be non -- hermitian. This problem is solved by the introduction of the additional kernel operator in the definition of the scalar product in the Fock space. The existence of the kernel operator is proven for any Lie algebra. 
  Polynomials in Grassmann space can be used to describe all the internal degrees of freedom of spinors, scalars and vectors, that is their spins and charges. It was shown that K\"ahler spinors, which are polynomials of differential forms, can be generalized to describe not only spins of spinors but also spins of vectors as well as spins and charges of scalars, vectors and spinors. If the space (ordinary and noncommutative) has 14 dimensions or more, the appropriate spontaneous break of symmetry leads gravity in $d$ dimensions to manifest in four dimensional subspace as ordinary gravity and all needed gauge fields as well as the Yukawa couplings. Both approaches, the K\"ahler's one (if generalized) and our, manifest four generations of massless fermions, which are left handed SU(2) doublets and right handed SU(2) singlets. In this talk a possible way of spontaneously broken symmetries is pointed out on the level of canonical momentum. 
  We propose a new approach to the Cosmological Constant Problem which makes essential use of an extra dimension. A model is presented in which the Standard Model vacuum energy ``warps'' the higher-dimensional spacetime while preserving 4D flatness. We argue that the strong curvature region of our solutions may effectively cut off the size of the extra dimension, thereby giving rise to macroscopic 4D gravity without a cosmological constant. In our model, the higher-dimensional gravity dynamics is treated classically with carefully chosen couplings. Our treatment of the Standard Model is however fully quantum field-theoretic, and the 4D flatness of our solutions is robust against Standard Model quantum loops and changes to Standard Model couplings. 
  Some applications of simple current techniques and fixed point resolution to theories of open strings are discussed. In addition to a brief review of work presented in two recent papers with L. Huiszoon and N. Sousa, some new results concerning uniqueness of crosscap coefficients are presented, as well as a strange sum rule for the modular matrix implied by the existence of crosscaps. 
  We analyse the possible D-brane configurations in an AdS_3xS^3xS^3xS^1 background with a NS-NS B field, by using the boundary state formalism. We study their geometry and we determine the fraction of spacetime supersymmetry preserved by these solutions. 
  We find the orbifold analog of the topological relation recently found by Freed and Witten which restricts the allowed D-brane configurations of Type II vacua with a topologically non-trivial flat $B$-field. The result relies in Douglas proposal -- which we derive from worldsheet consistency conditions -- of embedding projective representations on open string Chan-Paton factors when considering orbifolds with discrete torsion. The orbifold action on open strings gives a natural definition of the algebraic K-theory group -- using twisted cross products -- responsible for measuring Ramond-Ramond charges in orbifolds with discrete torsion. We show that the correspondence between fractional branes and Ramond-Ramond fields follows in an interesting fashion from the way that discrete torsion is implemented on open and closed strings. 
  Open string field theory is considered as a tool for deriving the effective action for the massless or tachyonic fields living on D-branes. Some simple calculations are performed in open bosonic string field theory which validate this approach. The level truncation method is used to calculate successive approximations to the quartic terms \phi^4, (A^\mu A_\mu)^2 and [A_\mu, A_\nu]^2 for the zero momentum tachyon and gauge field on one or many bosonic D-branes. We find that the level truncation method converges for these terms within 2-4% when all massive fields up to level 20 are integrated out, although the convergence is slower than exponential. We discuss the possibility of extending this work to determine the structure of the nonabelian Born-Infeld theory describing the gauge field on a system of many parallel bosonic or supersymmetric D-branes. We also describe a brane configuration in which tachyon condensation arises in both the gauge theory and string field theory pictures. This provides a natural connection between recent work of Sen and Zwiebach on tachyon condensation in string field theory and unstable vacua in super Yang-Mills and Born-Infeld field theory. 
  The renormalization of quantum field theory twists the antipode of a noncocommutative Hopf algebra of rooted trees, decorated by an infinite set of primitive divergences. The Hopf algebra of undecorated rooted trees, ${\cal H}_R$, generated by a single primitive divergence, solves a universal problem in Hochschild cohomology. It has two nontrivial closed Hopf subalgebras: the cocommutative subalgebra ${\cal H}_{\rm ladder}$ of pure ladder diagrams and the Connes-Moscovici noncocommutative subalgebra ${\cal H}_{\rm CM}$ of noncommutative geometry. These three Hopf algebras admit a bigrading by $n$, the number of nodes, and an index $k$ that specifies the degree of primitivity. In each case, we use iterations of the relevant coproduct to compute the dimensions of subspaces with modest values of $n$ and $k$ and infer a simple generating procedure for the remainder. The results for ${\cal H}_{\rm ladder}$ are familiar from the theory of partitions, while those for ${\cal H}_{\rm CM}$ involve novel transforms of partitions. Most beautiful is the bigrading of ${\cal H}_R$, the largest of the three. Thanks to Sloane's {\tt superseeker}, we discovered that it saturates all possible inequalities. We prove this by using the universal Hochschild-closed one-cocycle $B_+$, which plugs one set of divergences into another, and by generalizing the concept of natural growth beyond that entailed by the Connes-Moscovici case. We emphasize the yet greater challenge of handling the infinite set of decorations of realistic quantum field theory. 
  We introduce a formulation of gauge theory on noncommutative spaces based on the concept of covariant coordinates. Some important examples are discussed in detail. A Seiberg-Witten map is established in all cases. 
  A systematic approach to the study of semiclassical fluctuations of strings in AdS_5 x S^5 based on the Green-Schwarz formalism is developed. We show that the string partition function is well defined and finite. Issues related to different gauge choices are clarified. We consider explicitly several cases of classical string solutions with the world surface ending on a line, on a circle or on two lines on the boundary of AdS. The first example is a BPS object and the partition function is one. In the third example the determinants we derive should give the first corrections to the Wilson loop expectation value in the strong coupling expansion of the n=4 SYM theory at large N. 
  We point out that the low energy theory of 6d, N=(2,0) field theories, when away from the origin of the moduli space of vacua, necessarily includes a new kind of Wess-Zumino term. The form of this term is related to the Hopf invariant associated with \pi_7 (S^4). The coefficient of the Wess-Zumino term is fixed by an anomaly matching relation for a global flavor symmetry. For example, in the context of a single M5 brane probe in the background of N distant M5 branes, the probe must have the Hopf-WZ term with coefficient proportional to N(N+1). Various related checks and observations are made. We also point out that there are skyrmionic strings, and propose that they are the W-boson strings. 
  We present Poincare invariant domain wall (``3-brane'') solutions to some 5-dimensional effective theories which can arise naturally in string theory. In particular, we find theories where Poincare invariant solutions exist for arbitrary values of the brane tension, for certain restricted forms of the bulk interactions. We describe examples in string theory where it would be natural for the quantum corrections to the tension of the brane (arising from quantum fluctuations of modes with support on the brane) to maintain the required form of the action. In such cases, the Poincare invariant solutions persist in the presence of these quantum corrections to the brane tension, so that no 4d cosmological constant is generated by these modes. 
  I consider the implications for brane-world scenarios of the rather robust quantum-gravity expectation that there should be a quantum minimum limit on the uncertainty of all physical length scales. In order to illustrate the possible significance of this issue, I observe that, according to a plausible estimate, the quantum limit on the length scales that characterize the bulk geometry could affect severely the phenomenology of a recently-proposed brane-world scenario. 
  We study issues related to F-theory on Calabi-Yau fourfolds and its duality to heterotic theory for Calabi-Yau threefolds. We discuss principally fourfolds that are described by reflexive polyhedra and show how to read off some of the data for the heterotic theory from the polyhedron. We give a procedure for constructing examples with given gauge groups and describe some of these examples in detail. Interesting features arise when the local pieces are fitted into a global manifold. An important issue is how to compute the superpotential explicitly. Witten has shown that the condition for a divisor to contribute to the superpotential is that it have arithmetic genus 1. Divisors associated with the short roots of non-simply laced gauge groups do not always satisfy this condition while the divisors associated to all other roots do. For such a `dissident' divisor we distinguish cases for which the arithmetic genus is greater than unity corresponding to an X that is not general in moduli (in the toric case this corresponds to the existence of non-toric parameters). In these cases the `dissident' divisor D does not remain an effective divisor for general complex structure. If however the arithmetic genus is less than or equal to 0, then the divisor is general in moduli and there is a genuine instability. 
  We provide the cosmological solutions for Randall-Sundrum models with the bulk energy-momentum $\hat T^5_5$ incorporated. It alters the Friedmann equation for the brane scale factor. We make a specific choice of $\hat T^5_5$ which is adjusted to stabilize the extra dimension. This makes it possible to compactify the extra dimension with a single positive tension brane, and this model provides a RS-type solution to the cosmological constant problem. When the same idea is applied to the RS model with two branes, the wrong sign of Friedmann equation for the negative tension brane can be resolved and usual FRW cosmology is reproduced for the brane. 
  Evolution of the concept known in the theoretical physics as the Renormalization Group (RG) is presented. The corresponding symmetry, that has been first introduced in QFT in mid-fifties, is a continuous symmetry of a solution with respect to transformation involving parameters (e.g., of boundary condition) specifying some particular solution.   After short detour into Wilson's discrete semi-group, we follow the expansion of QFT RG and argue that the underlying transformation, being considered as a reparameterisation one, is closely related to the self-similarity property. It can be treated as its generalization, the Functional Self-similarity (FS).   Then, we review the essential progress during the last decade of the FS concept in application to boundary value problem formulated in terms of differential equations. A summary of a regular approach recently devised for discovering the RG = FS symmetries with the help of the modern Lie group analysis and some of its applications are given.   As a main physical illustration, we give application of new approach to solution for a problem of self-focusing laser beam in a non-linear medium. 
  In the dilatonic domain wall model, we study the Schwarzschild black hole as a solution to the Kaluza-Klein (KK) zero mode effective action which is equivalent to the Brans-Dicke (BD) model with a potential. This can describe the large Randall-Sundrum (RS) black hole whose horizon is to be the intersection of the black cigar with the brane.   The black cigar located far from the AdS$_5$-horizon is known to be stable, but any explicit calculation for stability of the RS black hole at $z=0$ is not yet performed. Here its stability is investigated against the $z$-independent perturbations composed of odd, even parities of graviton ($h_{\mu\nu}$) and BD scalar($h_{44} = 2\phi $). It seems that the RS black hole is classically unstable because it has a potential instability at wavelength with $\lambda > 1/(2k)$. However, this is not allowed inside an AdS$_5$-box of the size with $1/(2k)$. Thus the RS black hole becomes stable. The RS black hole can be considered as a stable remnant at $z=0 $ of the black cigar. 
  We discuss geometry underlying orientifolds with non-trivial NS-NS B-flux. If D-branes wrap a torus with B-flux the rank of the gauge group is reduced due to non-commuting Wilson lines whose presence is implied by the B-flux. In the case of D-branes transverse to a torus with B-flux the rank reduction is due to a smaller number of D-branes required by tadpole cancellation conditions in the presence of B-flux as some of the orientifold planes now have the opposite orientifold projection. We point out that T-duality in the presence of B-flux is more subtle than in the case with trivial B-flux, and it is precisely consistent with the qualitative difference between the aforementioned two setups. In the case where both types of branes are present, the states in the mixed (e.g., 59) open string sectors come with a non-trivial multiplicity, which we relate to a discrete gauge symmetry due to non-zero B-flux, and construct vertex operators for the the mixed sector states. Using these results we revisit K3 orientifolds with B-flux (where K3 is a T^4/Z_M orbifold) and point out various subtleties arising in some of these models. For instance, in the Z_2 case the conformal field theory orbifold does not appear to be the consistent background for the corresponding orientifolds with B-flux. This is related to the fact that non-zero B-flux requires the presence of both O5^- as well as O5^+ planes at various Z_2 orbifold fixed points, which appears to be inconsistent with the presence of the twisted B-flux in the conformal field theory orbifold. We also consider four dimensional N=2 and N=1 supersymmetric orientifolds. We construct consistent four dimensional models with B-flux which do not suffer from difficulties encountered in the K3 cases. 
  We study properties of (D$(p-2)$, D$p$) nonthreshold bound states ($2 \le p \le 6 $) in the dual gravity description. These bound states can be viewed as D$p$-branes with a nonzero NS $B$ field of rank two. We find that in the decoupling limit, the thermodynamics of the $N_p$ coincident D$p$-branes with $B$ field is the same not only as that of $N_p$ coincident D$p$-branes without $B$ field, but also as that of the $N_{p-2}$ coincident D$(p-2)$-branes with two smeared coordinates and no $B$ field, for $N_{p-2}/N_p= \tilde{V}_2/[(2\pi)^2 \tilde{b}]$ with $\tilde{V}_2$ being the area of the two smeared directions and $\tilde{b}$ a noncommutativity parameter. We also obtain the same relation from the thermodynamics and dynamics by probe methods. This suggests that the noncommutative super Yang-Mills with gauge group $U(N_p)$ in ($p+1$) dimensions is equivalent to an ordinary one with gauge group $U(\infty)$ in ($p-1$) dimensions in the limit $\tilde{V}_2 \to \infty$. We also find that the free energy of a D$p$-brane probe with $B$ field in the background of D$p$-branes with $B$ field coincides with that of a D$p$-brane probe in the background of D$p$-branes without $B$ field. 
  The Green-Siegel central extension of superalgebras for BPS branes is studied. In these cases commutators of usual bosonic brane charges only with the broken supersymmetry charges allow this central extension. We present an interpretation of these fermionic central charges as fermionic brane charges, and show that they take nonzero values for a nontrivial fermionic boundary condition. Such fermionic coordinates solutions are determined by equations of motion in a suitable gauge condition which manifests Nambu-Goldstone fermionic modes as well as bosonic modes in the static gauge. We also show that some modes of dilatino fields couple with the fermionic brane currents. 
  Noncommutative quantum field theory of a complex scalar field is considered. There is a two-coupling noncommutative analogue of U(1)-invariant quartic interaction $(\phi^*\phi)^2$, namely $A\phi^*\star\phi\star\phi^*\star\phi+ B\phi^*\star\phi^*\star\phi\star\phi$. For arbitrary values of $A$ and $B$ the model is nonrenormalizable. However, it is one-loop renormalizable in two special cases: B=0 and $A=B$. Furthermore, in the case B=0 the model does not suffer from IR divergencies at least at one-loop insertions level. 
  The theory of non-linear realizations is used to derive the dynamics of the branes of M theory. A crucial step in this procedure is to use the enlarged automorphism group of the supersymmetry algebra recently introduced. The field strengths of the worldvolume gauge fields arise as some of the Goldstone fields associated with this automorphism group. The relationship to the superembedding approach is given. 
  We discuss the calculation of threshold corrections to gauge coupling constants for the, only, non-decomposable class of abelian (2, 2) symmetric N=1 four dimensional heterotic orbifold models, where the internal twist is realized as a generalized Coxeter automorphism. The latter orbifold was singled out in earlier work as the only N=1 heterotic $Z_N$ orbifold that satisfy the phenomenological criteria of correct minimal gauge coupling unification and cancellation of target space modular anomalies. 
  Constrained hamiltonian structure of noncommutative gauge theory for the gauge group U(1) is discussed. Constraints are shown to be first class, although, they do not give an Abelian algebra in terms of Poisson brackets. The related BFV-BRST charge gives a vanishing generalized Poisson bracket by itself due to the associativity of *-product. Equivalence of noncommutative and ordinary gauge theories is formulated in generalized phase space by using BFV-BRST charge and a solution is obtained. Gauge fixing is discussed. 
  Softly broken dual magnetic theory of N=1 supersymmetric SU(N_c) QCD with N_f flavours is investigated with the inclusion of trilinear coupling term of scalar fields in the case of N_f>N_c+1. It is found that the trilinear soft supersymmetric breaking term greatly change the phase and the vacuum structure. 
  We discuss the formulation of spin observables associated to a non-relativistic spinning particles in terms of grassmanian differential operators. We use as configuration space variables for the pseudo-classical description of this system the positions $x$ and a Grassmanian vector $\vec\epsilon$. We consider an explicit discretization procedure to obtain the quantum amplitudes as path integrals in this superspace. We compute the quantum action necessary for this description including an explicit expression for the boundary terms. Finally we shown how for simple examples, the path integral may be performed in the semi-classical approximation, leading to the correct quantum propagator. 
  We consider the model of a massless charged scalar field, in (2+1) dimensions, with a self interaction of the form $lambda (\phi^* \phi)^3$ and interacting with a Chern Simons field. We calculate the renormalization group $\beta$ functions of the coupling constants and the anomalous dimensions $\gamma$ of the basic fields. We show that the interaction with the Chern Simons field implies in a $\beta_{\lambda}$ which suggests that a dynamical symmetry breakdown occurs. We also study the effect of the Chern Simons field on the anomalous dimensions of the composite operators $(\phi^* \phi)^n$, getting the result that their operator dimensions are lowered. 
  Conserved and commuting charges are investigated in both bosonic and supersymmetric classical chiral models, with and without Wess-Zumino terms. In the bosonic theories, there are conserved currents based on symmetric invariant tensors of the underlying algebra, and the construction of infinitely many commuting charges, with spins equal to the exponents of the algebra modulo its Coxeter number, can be carried out irrespective of the coefficient of the Wess-Zumino term. In the supersymmetric models, a different pattern of conserved quantities emerges, based on antisymmetric invariant tensors. The current algebra is much more complicated than in the bosonic case, and it is analysed in some detail. Two families of commuting charges can be constructed, each with finitely many members whose spins are exactly the exponents of the algebra (with no repetition modulo the Coxeter number). The conserved quantities in the bosonic and supersymmetric theories are only indirectly related, except for the special case of the WZW model and its supersymmetric extension. 
  We propose relativistic second quantization of matrix model of D particles in a general framework of nonlocal field theory based on Snyder-Yang's quantized space-time. Second-quantized nonlocal field is in general noncommutative with quantized space-time, but conjectured to become commutative with light cone time $X^+$. This conjecture enables us to find second-quantized Hamiltonian of D particle system and Heisenberg's equation of motion of second-quantized {\bf D} field in close contact with Hamiltonian given in matrix model. We propose Hamilton's principle of Lorentz-invariant action of {\bf D} field and investigate what conditions or approximations are needed to reproduce the above Heisenberg's equation given in light cone time. Both noncommutativities appearing in position coordinates of D particles in matrix model and in quantized space-time will be eventually unified through second quantization of matrix model. 
  The counting of long strings in ADS3, in the context of Type IIB string theory on $ADS_3 \times S^3 \times T^4$, is used to exhibit the action of the duality group $O(5,5;Z)$, and in particular its Weyl Subgroup $S_5 \bowtie Z_2$, in the non-perturbative phenomena associated with continuous spectra of states in these backgrounds. The counting functions are related to states in Fock spaces. The symmetry groups also appear in the structure of compactifications of instanton moduli spaces on $T^4$. 
  We generalize the submodel of nonlinear CP^1 models. The generalized models include higher order derivatives. For the systems of higher order equations, we construct a B\"acklund-like transformation of solutions and an infinite number of conserved currents by using the Bell polynomials. 
  We finalize the study of collapsing D-branes in one-parameter models by completing the analysis of the associated hypergeometric hierarchy. This brings further evidence that the phenomenon of collapsing 6-branes at the mirror of the `conifold' point in IIA compactifications on one-parameter Calabi-Yau manifolds is generic. It also completes the reduction of the study of higher periods in one-parameter models to a few families which display characteristic behaviour. One of the models we consider displays an exotic form of small-large radius duality, which is a consequence of an ``accidental'' discrete symmetry of its moduli space. We discuss the implementation of this symmetry at the level of the associated type II string compactification and its action on D-brane states. We also argue that this model admits two special Lagrangian fibrations and that the symmetry can be understood as their exchange. 
  A possibility to represent the standard model of fundamental particles covariant derivatives by means of approximate generalized fractional Riemann-Liouville derivatives of multifractal time and space model is shown. 
  Type I string theory can be dimensionally reduced on group manifolds. The compactification on $S^3\times S^3$ leads to the N=4 gauged SU(2)$\times$SU(2) supergravity in four dimensions, which admits the BPS monopole type non-Abelian vacuum. The reduction on $S^3\times AdS_3$ gives the Euclidean N=4 gauged SU(2)$\times$SU(1,1) supergravity admitting a globally regular supersymmetric non-Abelian background. The latter can be analytically continued to the Lorentzian sector, which gives the regular, unstable particle-like configuration known as gravitational sphaleron. When lifted to D=10, the Euclidean vacuum describes a deformation of the D1--D5 brane system. 
  We investigate the conformal and superconformal properties of a non-relativistic spinning particle propagating in a curved background coupled to a magnetic field and with a scalar potential. We derive the conditions on the couplings for a large class of such systems which are necessary in order their actions admit conformal and superconformal symmetry. We find that some of these conditions can be encoded in the conformal and holomorphic geometry of the background. Several new examples of conformal and superconformal models are also given. 
  Light-cone gauge manifestly supersymmetric formulation of type IIB 10-dimensional supergravity in $AdS_5 \times S^5$ background is discussed. The formulation is given entirely in terms of light-cone scalar superfield, allowing us to treat all component fields on an equal footing. Discrete energy spectrum of field propagating in AdS space is explained within the framework of light-cone approach. Light-cone gauge formulation of self-dual fields propagating in AdS space is developed. An conjectured interrelation of higher spin massless fields theory in AdS space-time and superstring theory in Minkowski space is discussed. 
  We study a new class of matrix models, the simplest of which is based on an Sp(2) symmetry and has a compactification which is equivalent to Chern-Simons theory on the three-torus. By replacing Sp(2) with the super-algebra Osp(1|32), which has been conjectured to be the full symmetry group of M theory, we arrive at a supercovariant matrix model which appears to contain within it the previously proposed M theory matrix models. There is no background spacetime so that time and dynamics are introduced via compactifications which break the full covariance of the model. Three compactifications are studied corresponding to a hamiltonian quantization in D=10+1, a Lorentz invariant quantization in D=9+1 and a light cone gauge quantization in D=11=9+1+1. In all cases constraints arise which eliminate certain higher spin fields in terms of lower spin dynamical fields. In the SO(9,1) invariant compactification we argue that the one loop effective action reduces to the IKKT covariant matrix model. In the light cone gauge compactification the theory contains the standard M theory light cone gauge matrix model, but there appears an additional transverse five form field. 
  We derive properties of N-extended GR super Virasoro algebras. These include adding central extensions, identification of all primary fields and the action of the adjoint representation on its dual. The final result suggest identification with the spectrum of fields in supergravity theories and superstring/M-theory constructed from NSR N-extended supersymmetric ${\cal {GR}}$ Virasoro algebras. 
  Non-perturbative features of the derivative expansion of the effective action of a single D3-brane are obtained by considering scattering amplitudes of open and closed strings. This motivates expressions for the coupling constant dependence of world-volume interactions of the form $(\partial F)^4$ (where F is the Born-Infeld field strength), $(\partial^2\phi)^4$ (where $\phi$ are the normal coordinates of the D3-brane) and other interactions related by $\calN=4$ supersymmetry. These include terms that transform with non-trivial modular weight under Montonen-Olive duality. The leading D-instanton contributions that enter into these effective interactions are also shown to follow from an explicit stringy construction of the moduli space action for the D-instanton/D3-brane system in the presence of D3-brane open-string sources (but in the absence of a background antisymmetric tensor potential). Extending this action to include closed-string sources leads to a unified description of non-perturbative terms in the effective action of the form (embedding curvature)$^2$ together with open-string interactions that describe contributions of the second fundamental form. 
  The F-theory vacuum constructed from an elliptic Calabi-Yau threefold with section yields an effective six-dimensional theory. The Lie algebra of the gauge sector of this theory and its representation on the space of massless hypermultiplets are shown to be determined by the intersection theory of the homology of the Calabi-Yau threefold. (Similar statements hold for M-theory and the type IIA string compactified on the threefold, where there is also a dependence on the expectation values of the Ramond-Ramond fields.) We describe general rules for computing the hypermultiplet spectrum of any F-theory vacuum, including vacua with non-simply-laced gauge groups. The case of monodromy acting on a curve of A_even singularities is shown to be particularly interesting and leads to some unexpected rules for how 2-branes are allowed to wrap certain 2-cycles. We also review the peculiar numerical predictions for the geometry of elliptic Calabi-Yau threefolds with section which arise from anomaly cancellation in six dimensions. 
  The effective action in general chiral superfield model with arbitrary k\"{a}hlerian potential $K(\bar{\Phi},\Phi)$ and chiral (holomorphic) potential $W(\Phi)$ is considered. The one-loop and two-loop contributions to k\"{a}hlerian effective potential and two-loop (first non-zero) contribution to chiral effective potential are found for arbitrary form of functions $K(\bar{\Phi},\Phi)$ and $W(\Phi)$. It is found that despite the theory is non-renormalizable in general case two-loop contribution to holomorphic effective potential is always finite. 
  Static properties of SU(3) multiskyrmions with baryon number up to 6 (classical masses and momenta of inertia) are estimated. The calculations are based on the recently suggested generalization of the SU(2) rational map ansaetze applied to the SU(3) model. Both SU(2) embedded skyrmions and genuine SU(3) solutions are considered, and it is shown that although, at the classical level, the energy of embeddings is lower, the quantum corrections can alter this conclusion. This correction to the energy of lowest state, bilinear in the Wess-Zumino (WZ) term, is presented for the most general case as a convolution of the inverse tensor of inertia and the components of the WZ-term. 
  The formalism of Supersymmetric Quantum Mechanics supplies a trial wave function to be used in the Variational Method. The screened Coulomb potential is analysed within this approach. Numerical and exact results for energy eigenvalues are compared. 
  These lecture notes give a pedagogical and (mostly) self-contained review of some basic aspects of the Matrix model of M-theory. The derivations of the model as a regularized supermembrane theory and as the discrete light-cone quantization of M-theory are presented. The construction of M-theory objects from matrices is described, and gravitational interactions between these objects are derived using Yang-Mills perturbation theory. Generalizations of the model to compact and curved space-times are discussed, and the current status of the theory is reviewed. 
  The rate of pair production of open strings in general uniform electromagnetic fields is calculated in various space-time dimensions. The corrections with respect to the case of pure electric backgrounds are displayed. In particular, a contribution in the form of a Born-Infeld action is derived and its role in the present context emphasized 
  Negative norm Hilbert space state vectors can be BRST invariant, we show in a simplified Y-M model, which has gluons and ghost fields only, that such states can be created by starting with gluons only. 
  We study the Einstein-Chern-Simons gravity coupled to Yang-Mills-Higgs theory in three dimensional Euclidean space with cosmological constant. The classical equations reduce to Bogomol'nyi type first order equations in curved space. There are BPS type gauge theory instanton (monopole) solutions of finite action in a gravitational instanton which itself has a finite action. We also discuss gauge theory instantons in the vacuum (zero action) AdS space. In addition we point out to some exact solutions which are singular. 
  The U(1) gauge theory on a D3-brane with non-commutative worldvolume is shown to admit BIon-like solutions which saturate a BPS bound on the energy. The mapping of these solutions to ordinary fields is found exactly, namely non-perturbatively in the non-commutativity parameters. The result is precisely an ordinary supersymmetric BIon in the presence of a background B-field. We argue that the result provides evidence in favour of the exact equivalence of the non-commutative and the ordinary descriptions of D-branes. 
  We argue that different aspects of Light-Front QCD at confined phase can be recovered by the Matrix Quantum Mechanics of D0-branes. The concerning Matrix Quantum Mechanics is obtained from dimensional reduction of pure Yang-Mills theory to 0+1 dimension. The aspects of QCD dynamics which are studied in correspondence with D0-branes are: 1) phenomenological inter-quark potentials, 2) whiteness of hadrons and 3) scattering amplitudes. In addition, some other issues such as the large-N behavior, the gravity--gauge theory relation and also a possible justification for involving ``non-commutative coordinates'' in a study of QCD bound-states are discussed. 
  We apply Dirac's Hamiltonian approach to study the canonical structure of the teleparallel form of general relativity without matter fields. It is shown, without any gauge fixing, that the Hamiltonian has the generalized Dirac-ADM form, and constraints satisfy all the consistency requirements. The set of constraints involves some extra first class constraints, which are used to find additional gauge symmetries and clarify the gauge structure of the theory. 
  In this note we propose that D-brane charges, in the presence of a topologically non-trivial B-field, are classified by the K-theory of an infinite dimensional C^*-algebra. In the case of B-fields whose curvature is pure torsion our description is shown to coincide with that of Witten. 
  We review the interactions of massive fields of arbitrary integer spins with the constant electromagnetic field and symmetrical Einstein space in the gauge invariant framework. The problem of obtaining the gauge-invariant Lagrangians of integer spin fields in an external field is reduced to purely algebraic problem of finding a set of operators with certain features using the representation of the higher-spin fields in the form of vectors in a pseudo-Hilbert space. Such a construction is considered up to the second order for the electromagnetic field and at linear approximation for symmetrical Einstein space. The results obtained are valid for space-time of arbitrary dimensionality. 
  We discuss the analytic properties of AdS scalar exchange graphs in the crossed channel. We show that the possible non-analytic terms drop out by virtue of non-trivial properties of generalized hypergeometric functions. The absence of non-analytic terms is a necessary condition for the existence of an operator product expansion for CFT amplitudes obtained from AdS/CFT correspondence. 
  The role of one loop order corrections in the triplectic quantization is discussed in the case of W2 theory. This model illustrates the presence of anomalies and Wess Zumino terms in this quantization scheme where extended BRST invariance is represented in a completely anticanonical form. 
  A recent paper by Moore and Witten explained that Ramond-Ramond fields in Type II superstring theory have a global meaning in K-theory. In this note we amplify and generalize some points raised in that paper. In particular, we express the coupling of the Ramond-Ramond fields to D-branes in a K-theoretic framework and show that the anomaly in this coupling exactly cancels the anomaly from the fermions on the brane, both in Type IIA and Type IIB. 
  We address some issues related to the construction of general Kaluza-Klein (KK) ans\"atze for the compactification of a supergravity (sugra) theory on a sphere $S_m$. We first reproduce various ans\"atze for compactification to 7d from the ansatz for the full nonlinear KK reduction of 11d sugra on $AdS_7\times S_4$. As a side result, we obtain a lagrangian formulation of 7d ${\cal N}=2$ gauged sugra, which so far had only a on-shell formulation, through field equations and constraints. The $AdS_7\times S_4$ ansatz generalizes therefore all previous sphere compactifications to 7d. Then we consider the case when the scalars in the lower dimensional theory are in a coset $Sl(m+1)/SO(m+1)$, and we keep the maximal gauge group $SO(m+1)$. The 11-dimensional sugra truncated on $S_4$ fits precisely the case under consideration, and serves as a model for our construction. We find that the metric ansatz has a universal expression, with the internal space deformed by the scalar fluctuations to a conformally rescaled ellipsoid. We also find the ansatz for the dependence of the antisymmetric tensor on the scalars. We comment on the fermionic ansatz, which will contain a matrix $U$ interpolating between the spinorial $SO(m+1)$ indices of the spherical harmonics and the $R$-symmetry indices of the fermionic fields in the lower dimensional sugra theory. We derive general conditions which the matrix $U$ has to satisfy and we give a formula for the vielbein in terms of $U$. As an application of our methods we obtain the full ansatz for the metric and vielbein for 10d sugra on $AdS_5\times S_5$ (with no restriction on any fields). 
  As generalizations of the original Volkov-Akulov action in four-dimensions, actions are found for all space-time dimensions D invariant under N non-linear realized global supersymmetries. We also give other such actions invariant under the global non-linear supersymmetry. As an interesting consequence, we find a non-linear supersymmetric Born-Infeld action for a non-Abelian gauge group for arbitrary D and N, which coincides with the linearly supersymmetric Born-Infeld action in D=10 at the lowest order. For the gauge group U({\cal N}) for M(atrix)-theory, this model has {\cal N}^2-extended non-linear supersymmetries, so that its large {\cal N} limit corresponds to the infinitely many (\aleph_0) supersymmetries. We also perform a duality transformation from F_{\mu\nu} into its Hodge dual N_{\mu_1...\mu_{D-2}}. We next point out that any Chern-Simons action for any (super)groups has the non-linear supersymmetry as a hidden symmetry. Subsequently, we present a superspace formulation for the component results. We further find that as long as superspace supergravity is consistent, this generalized Volkov-Akulov action can further accommodate such curved superspace backgrounds with local supersymmetry, as a super p-brane action with fermionic kappa-symmetry. We further elaborate these results to what we call `simplified' (Supersymmetry)^2-models, with both linear and non-linear representations of supersymmetries in superspace at the same time. Our result gives a proof that there is no restriction on D or N for global non-linear supersymmetry. We also see that the non-linear realization of supersymmetry in `curved' space-time can be interpreted as `non-perturbative' effect starting with the `flat' space-time. 
  The kappa-invariant and supersymmetric actions of D1 and D5-branes in AdS_3 x S^3 are investigated, as well as the action of a D5-brane in an AdS_5 x S^5 background. The action of a D5-brane lying totally in an AdS_3 x S^3 background is found. Some progress was made towards finding the action for the D5-brane free to move in the whole AdS_3 x S^3 x T^4 space, however the supersymmetric action found here is not kappa-invariant and the reasons the method used did not find a kappa-invariant solution are discussed. 
  We study the thermodynamics of the confined and unconfined phases of superconformal Yang-Mills in finite volume and at large N using the AdS/CFT correspondence. We discuss the necessary conditions for a smooth phase crossover and obtain an N-dependent curve for the phase boundary. 
  String equations for the vertex operators of type IIB on $AdS_3\times S^3$ background with NS-NS flux are calculated using Berkovits-Vafa-Witten formalism. With suitable field definitions, the linearized field equations for six-dimensional supergravity and a tensor multiplet on $AdS_3\times S^3$ are recovered from these. We also discuss the massless degrees of freedom that survive the $S^3$ Kaluza-Klein compactification and how our vertex operators are related to the vertex operators introduced by Giveon, Kutasov and Seiberg. 
  In order to reconcile the non conventional character of brane cosmology with standard Friedmann cosmology, we introduce in this paper a slowly-varying quintessence scalar field in the brane and analyse the cosmological solutions corresponding to some equations of state for the scalar field. Different compensation mechanisms between the cosmological constant in the bulk and the constant tension resulting from the combined effect of ordinary matter and the quintessence scalar field are derived or assumed. It has been checked that the Randall-Sundrum approach is not necessarily the best procedure to reconcile brane and standard cosmologies, and that there exists at least another compensating mechanism that reproduces a rather conventional behaviour for an accelerating universe. 
  We critically review the use of the exact renormalization group equations (ERGE) in the framework of the scalar theory. We lay emphasis on the existence of different versions of the ERGE and on an approximation method to solve it: the derivative expansion. The leading order of this expansion appears as an excellent textbook example to underline the nonperturbative features of the Wilson renormalization group theory. We limit ourselves to the consideration of the scalar field (this is why it is an introductory review) but the reader will find (at the end of the review) a set of references to existing studies on more complex systems. 
  Hidden symmetries are the backbone of Integrable two-dimensional theories. They provide classical solutions of higher dimensional models as well, they seem to survive partially quantisation and their discrete remnants in M-theory called U-dualities, would provide a way to control infinities and nonperturbative effects in Supergravities and String theories. Starting from Einstein gravity we discuss the building blocks of these large groups of internal symmetries, and embed them in superalgebras of dynamical symmetries. The classical field equations for all bosonic matter fields of all toroidally compactified supergravities are invariant under such ``superdualities''. Possible extensions are briefly discussed. 
  We study the time-independent modes of a massless scalar field in various black hole backgrounds, and show that for these black holes, the time-independent mode is localised at the horizon. A similar analysis is done for time-independent equilibrium modes of the five dimensional plane AdS black hole. A self-adjointness analysis of this problem reveals that in addition to the modes corresponding to the usual glueball states, there is a discrete infinity of other equilibrium modes with imaginary mass for the glueball. We suggest these modes may be related to a Savvidy-Nielsen-Olesen-like vacuum instability in QCD. 
  We define the concept of Pi-stability, a generalization of mu-stability of vector bundles, and argue that it characterizes N=1 supersymmetric brane configurations and BPS states in very general string theory compactifications with N=2 supersymmetry in four dimensions. 
  The order parameters of dynamical chiral symmetry breaking in QCD, the dynamical mass of quarks and the chiral condensates, are evaluated by numerically solving the Non-Perturbative Renormalization Group (NPRG) equations. We employ an approximation scheme beyond ``the ladder'', that is, beyond the (improved) ladder Schwinger-Dyson equations. The chiral condensates are enhanced compared with the ladder ones, which is phenomenologically favorable. The gauge dependence of the order parameters is fairly reduced in this scheme. 
  Recently D'Hoker et al. (1999) computed 4-point correlation functions of axion and dilaton fields in type IIB SUGRA on AdS_5 x S^5. We reproduce from a CFT point of view all power law singular terms in these AdS 4-point amplitudes. We also calculate a corresponding 4-point function in the weak coupling limit, g_{YM}^2 N -> 0. Comparison reveals the existence of a primary operator that contributes to these same singular terms in the weak coupling limit but which does not contribute to the power law singular terms of the type IIB SUGRA 4-point function. We conclude that this new operator is not a chiral primary and hence acquires a large anomalous dimension in the strong coupling regime. 
  We discuss thick domain walls interpolating between spaces with naked singularities and give arguments based on the $AdS$/CFT correspondence why such singularities may be physically meaningful. Our examples include thick domain walls with Minkowski, de Sitter, and anti-de Sitter geometries on the four-dimensional slice. For flat domain walls we can solve the equivalent quantum mechanics problem exactly, which provides the spectrum of graviton states. In one of the examples we discuss, the continuum states have a mass gap. We compare the graviton spectra with expectations from the $AdS$/CFT correspondence and find qualitative agreement. We also discuss unitary boundary conditions and show that they project out all continuum states. 
  It is shown that 3D vector van der Waals (conformal) nonlinear $\sigma$-model (NSM) on a sphere $S^2$ has two types of topological excitations reminiscent vortices and instantons of 2D NSM. The first, the hedgehogs, are described by homotopic group $\pi_2(S^2) = \mathbb {Z}$ and have the logarithmic energies. They are an analog of 2D vortices. The energy and interaction of these excitations are found. The second, corresponding to 2D instantons, are described by hpmotopic group $\pi_3(S^2) = \mathbb {Z}$ or the Hopf invariant $H \in \mathbb {Z}$. A possibility of the topological phase transition in this model and its applications are briefly discussed. 
  We extend Randall-Sundrum dynamics to non-conformal metrics corresponding to non-constant dilaton. We study the appareance of space-time naked singularities and the renormalization group evolution of four-dimensional Newton constant. 
  The physical phase space in gauge systems is studied. Effects caused by a non-Euclidean geometry of the physical phase space in quantum gauge models are described in the operator and path integral formalisms. The projection on the Dirac gauge invariant states is used to derive a necessary modification of the Hamiltonian path integral in gauge theories of the Yang-Mills type with fermions that takes into account the non-Euclidean geometry of the physical phase space. The new path integral is applied to resolve the Gribov obstruction. Applications to the Kogut-Susskind lattice gauge theory are given. The basic ideas are illustrated with examples accessible for non-specialists. 
  These TASI lectures review the Holographic principle. The first lecture describes the puzzle of black hole information loss that led to the idea of Black Hole Complementarity and subsequently to the Holographic Principle itself. The second lecture discusses the holographic entropy bound in general space-times. The final two lectures are devoted to the ADS/CFT duality as a special case of the principle. The presentation is self contained and emphasizes the physical principles. Very little technical knowledge of string theory or supergravity is assumed. 
  By use of a special homotopy operator, we present an explicit, closed-form and simple expression for the left-right Bardeen-Gross-Jackiw anomalies described as the proper superspace integral of a superfunction. 
  We give a path integral derivation of the annulus diagram in a supersymmetric theory of open and closed strings with Dbranes. We compute the pair correlation function of Wilson loops in the generic weakly coupled supersymmetric flat spacetime background with Dbranes. We obtain a -u^4/r^9 potential between heavy nonrelativistic sources in a supersymmetric gauge theory at short distances. 
  There are at least two serious moduli problems in string cosmology. The first is the possibility that moduli dominate the energy density at the time of nucleosynthesis. The second is that they may not find their minima all together. After reviewing some previously proposed solutions to these problems, we propose another: all of the moduli but the dilaton sit at points of enhanced symmetry. The dilaton has a potential similar to those of racetrack models; it is very massive and its dynamics do not break supersymmetry. The dilaton is able to find the minimum of its potential because the energy is dominated by non-zero momentum modes. This energy need not be thermal. The effective potential for the dilaton is quite different from its flat space form. If certain conditions are satisfied, the dilaton settles into the desired minimum; if not, it is forced to weak coupling. 
  Can one make a Majorana field theory for fermions starting from the zero mass Weyl theory, then adding a mass term as an interaction? The answer to this question is: yes we can. We can proceed similarly to the case of the Dirac massive field theory. In both cases one can start from the zero mass Weyl theory and then add a mass term as an interacting term of massless particles with a constant (external) field. In both cases the interaction gives rise to a field theory for a free massive fermion field. We present the procedure for the creation of a mass term in the case of the Dirac and the Majorana field and we look for a massive field as a superposition of massless fields. 
  The finite temperature Casimir free energy is calculated for a dielectric ball of radius $a$ embedded in an infinite medium. The condition $\epsilon\mu=1$ is assumed for the inside/outside regions. Both the Green function method and the mode summation method are considered, and found to be equivalent. For a dilute medium we find, assuming a simple "square" dispersion relation with an abrupt cutoff at imaginary frequency $\hat \omega= \omega_0$, the high temperature Casimir free energy to be negative and proportional to $x_0 \equiv \omega_0 a$. Also, a physically more realistic dispersion relation involving spatial dispersion is considered, and is shown to lead to comparable results. 
  A well-defined regularized path integral for Lorentzian quantum gravity in three and four dimensions is constructed, given in terms of a sum over dynamically triangulated causal space-times. Each Lorentzian geometry and its associated action have a unique Wick rotation to the Euclidean sector. All space-time histories possess a distinguished notion of a discrete proper time. For finite lattice volume, the associated transfer matrix is self-adjoint and bounded. The reflection positivity of the model ensures the existence of a well-defined Hamiltonian. The degenerate geometric phases found previously in dynamically triangulated Euclidean gravity are not present. The phase structure of the new Lorentzian quantum gravity model can be readily investigated by both analytic and numerical methods. 
  We calculate the screening and anti-screening contributions to the inter-quark potential in 2+1 dimensions, which is relevant to the high temperature limit of QCD. We demonstrate that the relative strength of screening to anti-screening agrees with the 3+1 dimensional theory to better than one percent accuracy. 
  We use the AdS/SYM correspondence to study the relevant effects of compactified dimensions on the D-brane dynamics. We present a detailed picture of the T-duality transition between branes in type IIA and type IIB supergravity. An analysis of the renormalization scheme coming from the expectation values of background fields and the role of Wilson lines in it is given. We finally explore finite size effects and T-duality maps on the description of Wilson loops by supergravity. 
  Conformality is the idea that at TeV scales enrichment of the standard model particle spectrum leads to conformal invariance at a fixed point of the renormalization group. Some aspects of conformality in particle phenomenology and cosmology are discussed. 
  We study singular, supersymmetric domain-wall solutions supported by the massive breathing mode scalars of, for example, sphere reductions in M-theory or string theory. The space-time on one side of such a wall is asymptotic to the Cauchy horizon of the anti-de Sitter (AdS) space-time. However, on the other side there is a naked singularity. The higher-dimensional embedding of these solutions has the novel interpretation of (sphere compactified) brane configurations in the domain ``inside the horizon region,'' with the singularity corresponding to the sphere shrinking to zero volume. The naked singularity is the source of an infinite attractive gravitational potential for the fluctuating modes. Nevertheless, the spectrum is bounded from below, continuous and positive definite, with the wave functions suppressed in the region close to the singularity. The massless bound state is formally excluded due to the boundary condition for the fluctuating mode wave functions at the naked singularity. However, a regularisation of the naked singularity, for example by effects of the order of the inverse string scale, in turn regularises the gravitational potential and allows for precisely one (massless) bound-state spin-2 fluctuating mode. We also contrast spectra in these domain wall backgrounds with those of the domain walls due to the massless modes of sphere reductions. 
  In this paper we investigated the dynamics, at the quantum level, of the self-dual field minimally coupled to bosons. In this investigation we use the Dirac bracket quantization procedure to quantize the model. Also, the relativistic invariance is tested in connection with the elastic boson-boson scattering amplitudes. 
  In a recent paper, the complete (non-linear) Kaluza-Klein Ansatz for the consistent embedding of certain scalar plus gravity subsectors of gauged maximal supergravity in D=4, 5 and 7 was presented, in terms of sphere reductions from D=11 or type IIB supergravity. The scalar fields included in the truncations were the diagonal fields in the SL(N,R)/SO(N) scalar submanifolds of the full scalar sectors of the corresponding maximal supergravities, with N=8, 6 and 5. The embeddings were used for obtaining an interpretation of extremal D=4, 5 or 7 AdS domain walls in terms of distributed M-branes or D-branes in the higher dimensions. Although strong supporting evidence for the correctness of the embedding Ansatze was presented, a full proof of the consistency was not given. Here, we complete the proof, by showing explicitly that the full set of higher-dimensional equations of motion are satisfied if and only if the lower-dimensional fields satisfy the relevant scalar plus gravity equations. 
  Gauged supergravities (in four and five dimensions) with eight supercharges and with vector supermultiplets have a unique ultra-violet (UV) fixed point on a given physical domain \cal M of the space of the scalar fields. We show that in these models the infra-red (IR) fixed points are located on the boundary of \cal M, where the space-time metric becomes singular. 
  The search for solutions of field theories allowing for topological solitons requires that we find the field configuration with the lowest energy in a given sector of topological charge. The standard approach is based on the numerical solution of the static Euler-Lagrange differential equation following from the field energy. As an alternative, we propose to use a simulated annealing algorithm to minimize the energy functional directly. We have applied simulated annealing to several nonlinear classical field theories: the sine-Gordon model in one dimension, the baby Skyrme model in two dimensions and the nuclear Skyrme model in three dimensions. We describe in detail the implementation of the simulated annealing algorithm, present our results and get independent confirmation of the studies which have used standard minimization techniques. 
  The D-dimensional conformal nonlinear sigma-models (NSM) sre constructed. It is shown that the NSM on spaces with $\pi_{D-1} = \mathbb {Z}$ have the topological solutions of a "hedgehog" and "anti-hedgehog" types with logarithmic energies. For spaces with $\pi_D \ne 0$ they have also the topological excitations of instanton types with finite energies. 
  The self-duality of chiral p-forms was originally investigated by Pasti, Sorokin and Tonin in a manifestly Lorentz covariant action with non-polynomial auxiliary fields. The investigation was then extended to other chiral p-form actions. In this paper we point out that the self-duality appears in a wider context of theoretical models that relate to chiral p-forms. We demonstrate this by considering the interacting model of Floreanini-Jackiw chiral bosons and gauge fields, the generalized chiral Schwinger model (GCSM) and the latter's gauge invariant formulation, and discover that the self-duality of the GCSM corresponds to the vector and axial vector current duality. 
  We derive the action for the non-abelian field theory living on parallel non-BPS D3-branes in type IIA theory on the orbifold T^4/I_4(-1)^F_L. The classical moduli space for the massless scalars originating in the ``would be'' tachyonic sector shows an interesting structure. In particular, it contains non-abelian flat directions. At a generic point in this branch of the moduli space the scalars corresponding to the the separations of the branes acquire masses and the branes condense. Although these tree level flat directions are removed by quantum corrections we argue that within the loop approximation the branes still condense. 
  We propose the light-front Lagrangian and the corresponding Hamiltonian that produce a theory perturbatively equivalent to the conventional QCD in the Lorentz coordinates after the regularization is removed. The regularization used is nonstandard and breaks the gauge invariance. But after the regularization is removed, this invariance is restored by the introduction of a finite number of counterterms with coefficients dependent on the regularization parameters. 
  We use sets of trivial line bundles for the realization of gerbes. For 1-gerbes the structure arises naturally for the Weyl fermion vacuum bundle at a fixed time. The Schwinger term is an obstruction in the triviality of a 1-gerbe. 
  The Casimir energy for the transverse oscillations of a piecewise uniform closed string is calculated. The great adaptibility of this string model with respect to various regularization methods is pointed out. We survey several regularization methods: the cutoff method, the complex contour integration method, and the zeta-function method. The most powerful method in the present case is the contour integration method. The Casimir energy turns out to be negative, and more so the larger is the number of pieces in the string. The thermodynamic free energy F is calculated for a two-piece string in the limit when the tension ratio x=T_{I}/T_{II} approaches zero. 
  The sinh-Gordon model is restricted to a half-line by boundary conditions maintaining integrability. A perturbative calculation of the reflection factor is given to one loop order in the bulk coupling and to first order in the difference of the two parameters introduced at the boundary, providing a further verification of Ghoshal's formula. The calculation is consistent with a conjecture for the general dependence of the reflection factor on the boundary parameters and the bulk coupling. 
  The holographic RG flow in AdS/CFT correspondence naturally defines a holographic scheme in which the central charge c and the beta function are related by a universal formula. We perform some checks of that formula and we compare it with quantum field theory expectations. We discuss alternative definitions of the c-function. In particular, we compare, for a particular supersymmetric flow, the holographic c-function with the central charge computed directly from the two-point function of the stress-energy tensor. 
  We study noncommutative field theories at finite temperature to learn more about the degrees of freedom in the non-planar sector of these systems. We find evidence for winding states. At temperatures for which the thermal wavelength is smaller than the noncommutativity scale, there is a drastic reduction of the degrees of freedom in the non-planar sector. In this regime, the non-planar sector has thermodynamics resembling that of a 1+1 dimensional field theory. 
  We show that the grading of fields by conformal weight, when built into the initial group symmetry, provides a discrete, non-central conformal extension of any group containing dilatations. We find a faithful vector representation of the extended conformal group and show that it has a scale-invariant scalar product and satisfies a closed commutator algebra. The commutator algebra contains the infinite Heisenberg and Virasoro algebras. In contrast to the classical treatment of scale invariance, covariant derivatives and gauge transformations automatically incorporate the correct conformal weights when the extended symmetry is gauged. 
  We study the eleven-dimensional supergravity and the classical solutions corresponding to M2-branes ending on M5-branes. We obtain the generic BPS configuration representing two sets of parallel M5-branes with (1+1) commonly longitudinal directions and M2-branes stretching between them. We also discuss how the brane creation is described in supergravity. 
  We propose a picture, within the pre-big-bang approach, in which the universe emerges from a bath of plane gravitational and dilatonic waves. The waves interact gravitationally breaking the exact plane symmetry and lead generically to gravitational collapse resulting in a singularity with the Kasner-like structure. The analytic relations between the Kasner exponents and the initial data are explicitly evaluated and it is shown that pre-big-bang inflation may occur within a dense set of initial data. Finally, we argue that plane waves carry zero gravitational entropy and thus are, from a thermodynamical point of view, good candidates for the universe to emerge from. 
  On an explicit example of the Siegel superparticle we study an alternative to the harmonic superspace approach. The latter seems to be the only method for quantizing infinitely reducible first class constraints currently available. In an appropriately extended phase space, the infinite ghost tower is effectively canceled by that coming from the sector of auxiliary variables. After a proper BRST treatment the theory proves to be of rank two which correlates well with the results obtained earlier within the framework of the harmonic superspace approach. The advantage of the novel technique, however, is the existence of an explicit Lagrangian formulation and the standard spin--statistics relations which hold for all the variables involved. 
  The standard picture of viable higher-dimensional theories is that extra dimensions manifest themselves at short distances only, their effects being negligible at scales larger than some critical value. We show that this is not necessarily true in models with infinite extra dimensions. As an example, we consider a five-dimensional scenario with three 3-branes in which gravity is five-dimensional both at short {\it and} very long distance scales, with conventional four-dimensional gravity operating at intermediate length scales. A phenomenologically acceptable range of validity of four-dimensional gravity extending from microscopic to cosmological scales is obtained without strong fine-tuning of parameters. 
  We study a non-supersymmetric $E_8\times\bar E_8$ compactification of M-theory on $S^1/Z_2$, related to the supersymmetric $E_8\times E_8$ theory by a chirality flip at one of the boundaries. This system represents an M-theory analog of the D-brane anti-D-brane systems of string theory. Alternatively, this compactification can be viewed as a model of supersymmetry breaking in the ``brane-world'' approach to phenomenology. We calculate the Casimir energy of the system at large separations, and show that there is an attractive Casimir force between the $E_8$ and $\bar E_8$ boundary. We predict that a tachyonic instability develops at separations of order the Planck scale, and discuss the possibility that the M-theory fivebrane might appear as a topological defect supported by the $E_8\times\bar E_8$ system. Finally, we analyze the eventual fate of the configuration, in the semiclassical approximation at large separations: the two ends of the world annihilate by nucleating wormholes between the two boundaries. 
  p-Gerbes are a generalization of bundles that have (p+2)-form field strengths. We develop their properties and use them to show that every theory of p-gerbes can be reinterpreted as a gauge theory containing p-dimensional extended objects. In particular, we show that every closed (p+2)-form with integer cohomology is the field strength for a gerbe, and that every p-gerbe is equivalent to a bundle with connection on the space of p-dimensional submanifolds of the original space. We also show that p-gerbes are equivalent to sheaves of (p-1)-gerbes, and use this to define a K-theory of gerbes. This K-theory classifies the charges of (p+1)-form connections in the same way that bundle K-theory classifies 1-form connections. 
  Quantum field theory on non-commutative spaces does not enjoy the usual ultraviolet-infrared decoupling that forms the basis for conventional renormalization. The high momentum contributions to loop integrations can lead to unfamiliar long distance behavior which can potentially undermine naive expectations for the IR behavior of the theory. These "anomalies" involve non-analytic behavior in the noncommutativity parameter Theta making the limit Theta goes to zero singular.   In this paper we will analyze such effects in the one loop approximation to gauge theories on non-commutative space. We will see that contrary to expectations poles in Theta do occur and lead to large discrepancies between the expected and actual infrared behavior. We find that poles in Theta are absent in supersymmetric theories. The "anomalies" are generally still present, but only at the logarithmic level. A notable exception is non-commutative super Yang Mills theory with 16 real supercharges in which anomalous effects seem to be absent altogether. 
  We discuss the properties of black holes in brane-world scenarios where our universe is viewed as a four-dimensional sub-manifold of some higher-dimensional spacetime. We consider in detail such a model where four-dimensional spacetime lies at the junction of several domain walls in a higher dimensional anti-de Sitter spacetime. In this model there may be any number p of infinitely large extra dimensions transverse to the brane-world. We present an exact solution describing a black p-brane which will induce on the brane-world the Schwarzschild solution. This exact solution is unstable to the Gregory-Laflamme instability, whereby long-wavelength perturbations cause the extended horizon to fragment. We therefore argue that at late times a non-rotating uncharged black hole in the brane-world is described by a deformed event horizon in p+4 dimensions which will induce, to good approximation, the Schwarzschild solution in the four-dimensional brane world. When p=2, this deformed horizon resembles a black diamond and more generally for p>2, a polyhedron. 
  We propose a class of models in which extended objects are introduced in Chern-Simons supergravity in such a way that those objects appear on the same footing as the target space. This is motivated by the idea that branes are already first quantized object, so that it is desirable to have a formalism that treats branes and their target space in a similar fashion. Accordingly, our models describe interacting branes, as gauge systems for supergroups. We also consider the case in which those objects have boundaries, and discuss possible links to superstring theory and/or M-theory, by studying the fermionic kappa-symmetry of the action. 
  We review some applications of tree-level (classical) relations between gravity and gauge theory that follow from string theory. Together with $D$-dimensional unitarity, these relations can be used to perturbatively quantize gravity theories, i.e. they contain the necessary information for obtaining loop contributions. We also review recent applications of these ideas showing that N=1 D=11 supergravity diverges, and review arguments that N=8 D=4 supergravity is less divergent than previously thought, though it does appear to diverge at five loops. Finally, we describe field variables for the Einstein-Hilbert Lagrangian that help clarify the perturbative relationship between gravity and gauge theory. 
  Neutrino-photon interactions in the presence of an external homogeneous constant electromagnetic field are studied. The $\nu \nu \gamma $ amplitude is calculated in an electromagnetic field of the general type, when the two field invariants are nonzero. 
  We use the conformal Ward identities to study the structure of correlation functions in coset conformal field theories. For a large class of primary fields of arbitrary g/h theory a factorization anzatz is found.Corresponding correlation functions are explicitly expressed in terms of correlation functions of two independent WZNW theories for g and h. 
  In the last six years remarkable developments have taken place concerning the representation theory of N=2 superconformal algebras. Here we present the highlights of such developments. 
  The kinks of the (1+1)-dimensional Wess-Zumino model with polynomic superpotential are investigated and shown to be related to real algebraic curves. 
  A general model of nonlinear electrodynamics with dyon singularities is considered. We consider the field configuration having two dyon singularities with identical electric and opposite magnetic charges and we name it bidyon. We investigate the sum of two dyon solutions as an initial approximation to the bidyon solution. We consider the case when the velocities of the dyons have equal modules and opposite directions on a common line. It is shown that the associated field configuration has a constant full angular momentum which is independent of distance between the dyons and their speed. This property permits a consideration of this bidyon configuration as an electromagnetic model for charged particle with spin. We discuss the possible electrodynamic world with oscillating bidyons as particles. 
  We investigate some aspects of Moyal-Weyl deformations of superspace and their compatibility with supersymmetry. For the simplest case, when only bosonic coordinates are deformed, we consider a four dimensional supersymmetric field theory which is the deformation of the Wess-Zumino renormalizable theory of a chiral superfield. We then consider the deformation of a free theory of an abelian vector multiplet, which is a non commutative version of the rank one Yang-Mills theory. We finally give the supersymmetric version of the $\alpha'\mapsto 0$ limit of the Born-Infeld action with a B-field turned on, which is believed to be related to the non commutative U(1) gauge theory. 
  We generalize the standard product integral formalism to incorporate Grassmann valued matrices and show that the resulting supersymmetric product integrals provide a natural framework for describing supersymmetric Wilson Lines and Wilson Loops. We use this formalism to establish the supersymmetric version of the non-Abelian Stokes theorem. 
  We study supersymmetric intersecting configurations of D-branes with B-field backgrounds. Noncommutative D-brane or M-brane pairs can intersect supersymmetrically over (p-1)-brane, as well as over (p-2)-brane like ordinary branes. d=10 and d=11 supergravity solutions are obtained and the supersymmetry projection rule is examined. As an application we study a noncommutative D7-brane probe in noncommutative D3-brane background, intersecting at noncommutative plane, which describes BPS baryons of noncommutative gauge theory in the context of AdS/CFT correspondence. 
  If there is a single underlying "theory of everything" which in some limits of its "moduli space" reduces to the five weakly coupled string theories in 10D, and 11D SUGRA, then it is possible that all six of them have some common domain of validity and that they are in the same universality class, in the sense that the 4D low energy physics of the different theories is the same. We call this notion String Universality. This suggests that the true vacuum of string theory is in a region of moduli space equally far (in some sense) from all perturbative theories, most likely around the self-dual point with respect to duality symmetries connecting them. We estimate stringy non-perturbative effects from wrapped brane instantons in each perturbative theory, show how they are related by dualities, and argue that they are likely to lead to moduli stabilization only around the self-dual point. We argue that moduli stabilization should occur near the string scale, and SUSY breaking should occur at a much lower intermediate scale, and that it originates from different sources. We discuss the problems of moduli stabilization and SUSY breaking in currently popular scenarios, explain why these problems are generic, and discuss how our scenario can evade them. We show that String Universality is not inconsistent with phenomenology but that it is in conflict with some popular versions of brane world scenarios. 
  We consider a relativistic brane propagating in Minkowski spacetime described by any action which is local in its worldvolume geometry. We examine the conservation laws associated with the Poincar\'e symmetry of the background from a worldvolume geometrical point of the view. These laws are exploited to explore the structure of the equations of motion. General expressions are provided for both the linear and angular momentum for any action depending on the worldvolume extrinsic curvature. The conservation laws are examined in perturbation theory. It is shown how non-trivial solutions with vanishing energy-momentum can be constructed in higher order theories. Finally, subtleties associated with boundary terms are examined in the context of the brane Einstein-Hilbert action. 
  We consider N=2 SU(2) Seiberg-Witten duality theory for models with N_f=2 and N_f=3 quark flavors. We investigate arbitrary large bare mass ratios between the two or three quarks at the singular points. For N_f=2 we explore large bare mass ratios corresponding to a singularity in the strong coupling region. For N_f=3 we determine the location of both strong and weak coupling singularities that produce specific large bare mass ratios. 
  We study BPS Dirac monopole in U(1) gauge theory on non-commutative spacetime. The corresponding brane configuration is obtained in the equivalent ordinary gauge theory through the map proposed by Seiberg and Witten. This configuration coincides exactly with a tilted D-string as predicted. This study provides an interesting check of the equivalence of the non-commutative and ordinary gauge theories. 
  A treatment of linearized gravity is given in the Randall-Sundrum background. The graviton propagator is found in terms of the scalar propagator, for which an explicit integral expression is provided. This reduces to the four-dimensional propagator at long distances along the brane, and provides estimates of subleading corrections. Asymptotics of the propagator off the brane yields exponential falloff of gravitational fields due to matter on the brane. This implies that black holes bound to the brane have a "pancake"-like shape in the extra dimension, and indicates validity of a perturbative treatment off the brane. Some connections with the AdS/CFT correspondence are described. 
  Supersymmetric domain-wall solutions of maximal gauged supergravity are classified in 4, 5 and 7 dimensions in the presence of non-trivial scalar fields taking values in the coset SL(N, R)/SO(N) for N=8, 6 and 5 respectively. We use an algebro-geometric method based on the Christoffel-Schwarz transformation, which allows for the characterization of the solutions in terms of Riemann surfaces whose genus depends on the isometry group. The uniformization of the curves can be carried out explicitly for models of low genus and results into trigonometric and elliptic solutions for the scalar fields and the conformal factor of the metric. The Schrodinger potentials for the quantum fluctuations of the graviton and scalar fields are derived on these backgrounds and enjoy all properties of supersymmetric quantum mechanics. Special attention is given to a class of elliptic models whose quantum fluctuations are commonly described by the generalized Lame potential \mu(\mu+1)P(z) + \nu(\nu+1)P(z+\omega_1)+ \kappa(\kappa+1)P(z+\omega_2) + \lambda(\lambda+1)P(z+\omega_1 +\omega_2) for the Weierstrass function P(z) of the underlying Riemann surfaces with periods 2\omega_1 and 2\omega_2, for different half-integer values of the coupling constants \mu, \nu, \kappa, \lambda. 
  $QCD_2$ with fermions in the adjoint representation is invariant under $SU(N)/Z_N$ and thereby is endowed with a non-trivial vacuum structure (k-sectors). The static potential between adjoint charges, in the limit of infinite mass, can be therefore obtained by computing Wilson loops in the pure Yang-Mills theory with the same non-trivial structure. When the (Euclidean) space-time is compactified on a sphere $S^2$, Wilson loops can be exactly expressed in terms of an infinite series of topological excitations (instantons). The presence of k-sectors modifies the energy spectrum of the theory and its instanton content. For the exact solution, in the limit in which the sphere is decompactified, a k-sector can be mimicked by the presence of k-fundamental charges at $\infty$, according to a Witten's suggestion. However this property neither holds before decompactification nor for the genuine perturbative solution which corresponds to the zero-instanton contribution on $S^2$. 
  A review is attempted of physical motivations, theoretical and phenomenological aspects, as well as outstanding problems, of the pre-big bang scenario in string cosmology. 
  We prove that the massless neutral $\lambda\Phi^4$-theory does not possess a unique vacuum. Based on the Wightman axioms the non-existence of a state which preserves Poincar{\'e} and scale invariance is demonstrated non-perturbatively for a non-vanishing self-interaction. We conclude that it is necessary to break the scale invariance in order to define a vacuum state. The renormalized vacuum expectation value of the energy-momentum tensor is derived from the two-point Wightman function employing the point-splitting technique and its relation to the phionic and the scalar condensate is addressed. Possible implications to other self-interacting field theories and to different approaches in quantum field theory are pointed out. PACS: 11.10.Cd, 11.15.Tk, 11.30.Qc 
  Slavnov-Taylor identities have been applied to perform explicitly the renormalization procedure for the softly broken N=1 SYM. The result is in accordance with the previous results obtained at the level of supergraph technique. 
  We construct chiral N=1 gauge theories in 4D by compactifying the 6D Blum-Intriligator (1,0) theories of 5-branes at $A_k$ singularities on $T^2$ with a nontrivial bundle of the global U(1) symmetry of these theories. 
  A continuum of monopole, dyon and black hole solutions exist in the Einstein-Yang-Mills theory in asymptotically anti-de Sitter space. Their structure is studied in detail. The solutions are classified by non-Abelian electric and magnetic charges and the ADM mass. The stability of the solutions which have no node in non-Abelian magnetic fields is established. There exist critical spacetime solutions which terminate at a finite radius, and have universal behavior. The moduli space of the solutions exhibits a fractal structure as the cosmological constant approaches zero. 
  Several recent papers have made considerable progress in proving the existence of remarkable consistent Kaluza-Klein sphere reductions of D=10 and D=11 supergravities, to give gauged supergravities in lower dimensions. A proof of the consistency of the full gauged SO(8) reduction on S^7 from D=11 was given many years ago, but from a practical viewpoint a reduction to a smaller subset of the fields can be more manageable, for the purposes of lifting lower-dimensional solutions back to the higher dimension. The major complexity of the spherical reduction Ansatze comes from the spin-0 fields, and of these, it is the pseudoscalars that are the most difficult to handle. In this paper we address this problem in two cases. One arises in a truncation of SO(8) gauged supergravity in four dimensions to U(1)^4, where there are three pairs of dilatons and axions in the scalar sector. The other example involves the truncation of SO(6) gauged supergravity in D=5 to a subsector containing a scalar and a pseudoscalar field, with a potential that admits a second supersymmetric vacuum aside from the maximally-supersymmetric one. We briefly discuss the use of these emdedding Ansatze for the lifting of solutions back to the higher dimension. 
  We study the type II superstring theory on the background $\br^{d-1,1}\times X_n$, where $X_n$ is a Calabi-Yau $n$-fold ($2n+d=10$) with an isolated singularity, by making use of the holographically dual description proposed by Giveon-Kutasov-Pelc (hep-th/9907178). We compute the toroidal partition functions for each of the cases $d=6,4,2$, and obtain manifestly modular invariant solutions classified by the standard $A-D-E$ series corresponding to the type of singularities on $X_n$. Partition functions of these modular invariants all vanish due to theta function identities and are consistent with the presence of space-time supersymmetry. 
  We introduce the notion of a monodromy for gauge fields with vanishing curvature on the noncommutative torus. Similar to the ordinary gauge theory, traces of the monodromies define noncommutative Wilson lines. Our main result is that these Wilson lines are invariant under the Seiberg-Witten map changing the deformation parameter of the noncommutative torus. 
  We present an inductive algebraic approach to the systematic construction and classification of generalized Calabi-Yau (CY) manifolds in different numbers of complex dimensions, based on Batyrev's formulation of CY manifolds as toric varieties in weighted complex projective spaces associated with reflexive polyhedra. We show how the allowed weight vectors in lower dimensions may be extended to higher dimensions, emphasizing the roles of projection and intersection in their dual description, and the natural appearance of Cartan-Lie algebra structures. The 50 allowed extended four-dimensional vectors may be combined in pairs (triples) to form 22 (4) chains containing 90 (91) K3 spaces, of which 94 are distinct, and one further K3 space is found using duality. In the case of CY_3 spaces, pairs (triples) of the 10~270 allowed extended vectors yield 4242 (259) chains with K3 (elliptic) fibers containing 730 additional K3 polyhedra. A more complete study of CY_3 spaces is left for later work. 
  We show how discrete torsion can be implemented in D=4, N=1 type IIB orientifolds. Some consistency conditions are found from the closed string and open string spectrum and from tadpole cancellation. Only real values of the discrete torsion parameter are allowed, i.e. epsilon=+-1. Orientifold models are related to real projective representations. In a similar way as complex projective representations are classified by H^2(Gamma,C^*)=H^2(Gamma,U(1)), real projective representations are characterized by H^2(Gamma,R^*)=H^2(Gamma,Z_2). Four different types of orientifold constructions are possible. We classify these models and give the spectrum and the tadpole cancellation conditions explicitly. 
  Superstrings propagating on backgrounds of the form AdS_3 x G/H are studied using the coset CFT approach. We focus on seven dimensional cosets which have a semiclassical limit, and which give rise to N=3 superconformal symmetry in the dual CFT. This is realized for the two cases AdS_3 x SU(3)/U(1) and AdS_3 x SO(5)/SO(3), for which we present an explicit construction. We also provide sufficient conditions on a CFT background to enable a similar construction, and comment on the geometrical interpretation of our results. 
  We present some results about the interplay between the chiral and deconfinement phase transitions in parity-conserving QED3 (with N flavours of massless 4 component fermions) at finite temperature. Following Grignani et al (Phys. Rev. D53, 7157 (1996), Nucl. Phys. B473, 143 (1996)), confinement is discussed in terms of an effective Sine-Gordon theory for the timelike component of the gauge field A_0. But whereas in the references above the fermion mass m is a Lagrangian parameter, we consider the m=0 case and ask whether an effective S-G theory can again be derived with m replaced by the dynamically generated mass Sigma which appears below T_{ch}, the critical temperature for the chiral phase transition. The fermion and gauge sectors are strongly interdependent, but as a first approximation we decouple them by taking Sigma to be a constant, depending only on the constant part of the gauge field. We argue that the existence of a low-temperature confining phase may be associated with the generation of Sigma; and that, analogously, the vanishing of Sigma for T > T_{ch} drives the system to its deconfining phase. The effect of the gauge field dynamics on mass generation is also indicated. (38kb) 
  We explore gauge fields - strings duality by means of the loop equations and the zigzag symmetry. The results are striking and incomplete. Striking - because we find that the string ansatz proposed in [A.M. Polyakov, hep-th/9711002] satisfies gauge theory Schwinger-Dyson equations precisely at the critical dimension D=4. Incomplete - since we get these results only in the WKB approximation and only for a special class of contours. The ways to go beyond these limitations and in particular the OPE for operators defined on the loop are also discussed. 
  The asymptotic form of a SU(3) matrix theory groundstate is found by showing that a recent ansatz for a supersymmetric wavefunction is non-trivial (i.e. non-zero). 
  One of the axioms of quantum field theory is the property of unitarity of the evolution operator. However, if one considers the quantum electrodynamics in the external field in the leading order of perturbation theory, one will find that the evolution transformation is a non-unitary canonical transformation of creation and annihilation operators. This observation was one of the arguments for the hypothesis that one should choose different representations of the canonical commutation relations at different moments of time in the exact quantum field theory. In this paper the contradiction is analyzed for the case of a simple quantum mechanical model being an analog of the leading order of the large-N field theory. On the one hand, this model is renormalized with the help of the constructive field theory methods; the Hilbert space and unitary evolution operator are constructed. On the other hand, the leading order of the evolution transformation in the strong external field is shown to be non-unitary. Thus, unitarity of evolution in the exact theory is not in contradiction with non-unitarity of the approximate theory. 
  Infinite enlargements of finite pseudo-unitary symmetries are explicitly provided in this letter. The particular case of u(2,2)=so(4,2)+u(1) constitutes a (Virasoro-like) infinite-dimensional generalization of the 3+1-dimensional conformal symmetry, in addition to matter fields with all conformal spins. These algebras provide a new arena for integrable field models in higher dimensions; for example, Anti-de Sitter and conformal gauge theories of higher-so(4,2)-spin fields. A proposal for a non-commutative geometrical interpretation of space is also outlined. 
  In this letter we argue that instanton-dominated Green's functions in N=2 Super Yang-Mills theories can be equivalently computed either using the so-called constrained instanton method or making reference to the topological twisted version of the theory. Defining an appropriate BRST operator (as a supersymmetry plus a gauge variation), we also show that the expansion coefficients of the Seiberg-Witten effective action for the low-energy degrees of freedom can be written as integrals of total derivatives over the moduli space of self-dual gauge connections. 
  One of the most challenging technical aspects of the dualities between string theory on anti-de Sitter spaces and conformal field theories is understanding how location in the interior of spacetime is represented in the field theory. It has recently been argued that the interior of the spacetime can be directly probed by using intrinsically non-local quantities in the field theory. In addition, Balasubramanian and Ross [hep-th/9906226] argued that when the spacetime described the formation of an AdS_3 black hole, the propagator in the field theory probed the whole spacetime, including the region behind the horizon. We use the same approach to study the propagator for the BTZ black hole and a black hole solution with a single exterior region, and show that it reproduces the propagator associated with the natural vacuum states on these spacetimes. We compare our result with a toy model of the CFT for the single-exterior black hole, finding remarkable agreement. The spacetimes studied in this work are analytic, which makes them quite special. We also discuss the interpretation of this propagator in more general spacetimes, shedding light on certain issues involving causality, black hole horizons, and products of local operators on the boundary. 
  Supersymmetry and Yang-Mills type gauge invariance are two of the essential properties of most, and possibly the most important models in fundamental physics. Supersymmetry is nearly trivial to prove in the (traditionally gauge-noncovariant) superfield formalism, whereas the gauge-covariant formalism makes gauge invariance manifest. In 3+1-dimensions, the transformation from one into the other is elementary and essentially unique. By contrast, this transformation turns out to be impossible in the most general 1+1-dimensional case. In fact, only the (manifestly) gauge- and supersymmetry-covariant formalism guarantees both universal gauge-invariance and supersymmetry. 
  We investigate the presence of chaos in a system of two real scalar fields with discrete Z_2 x Z_2 symmetry. The potential that identify the system is defined with a real parameter r and presents distinct features for r>0 and for r<0. For static field configurations, the system supports two topological sectors for r>0, and only one for r<0. Under the assumption of spatially homogeneous fields, the system exhibts chaotic behavior almost everywhere in parameter space. In particular a more complex dynamics appears for r>0; in this case chaos can decrease for increasing energy, a fact that is absent for r<0. 
  In the present paper we introduce a hierarquical class of self-dual models in three dimensions, inspired in the original self-dual theory of Towsend-Pilch-Nieuwenhuizen. The basic strategy is to explore the powerful property of the duality transformations in order to generate a new field. The generalized propagator can be written in terms of the primitive one (first order), and also the respective order and disorder correlation functions. Some conclusions about the ``charge screening'' and magnetic flux were established. 
  An off-shell formulation for 6 and 10 dimensions simple supersymmetric Yang-Mills theories is presented. While the fermionic fields couple to left action of S^3 and S^7 respectively, the auxiliary ones couple to right action (and vice versa). To close the algebra off-shell, left and right actions must commute. For 6 dimensions quaternions work fine. The 10 dimensional case needs special care. Pure spinors and soft Lie algebra (algebra with structure functions instead of structure constants) are essential. Some tools useful for constructing the superspace are also derived. We show how to relate our results to the early works of Evans and Berkovits. 
  In this thesis I review various aspects of the AdS_4/CFT_3 correspondence, where AdS_4 supergravity arises from compactification of M-theory on a coset space G/H and preserves N<8 supersymmetries. One focal point of my review is that the complete spectrum of such N-extended supergravity can be determined by means of harmonic analysis on the homogeneous space G/H. This spectrum can be matched with the candidate conformal theory on the boundary, in this way providing very non-trivial checks of the AdS/CFT correspondence. Furthermore, this spectrum can be useful to study the representation theory of N-extended supersymmetry on AdS_4, namely representation theory for the superalgebra of Osp(N|4). I review Osp(N|4) representation theory, and derive the translation vocabulary between states of AdS_4 supergravity and conformal superfields on the boundary, by means of the double interpretation of Osp(N|4) unitary irreducible representations. In the cases of N=2,3, using results from harmonic analysis I give the complete structure of all supermultiplets. Harmonic analysis as a method to determine spectra of supergravity compactifications is explained. Calculations are explicitly performed in the case G/H=M^111=(SU(3)xSU(2)xU(1))/(SU(2)xU(1)xU(1)), preserving N=2 supersymmetries. For this manifold, and also for the case G/H=Q^111=(SU(2)x SU(2)xSU(2))/(U(1)xU(1)), I describe the construction of a candidate dual superconformal theory on the boundary. This construction is based on geometrical insight provided by the properties of the metric cone C(G/H) transverse to the M2-brane worldvolume. 
  We construct Dp-branes in bosonic string theory as unstable lumps in a truncated string field theory of open strings on a D25-brane. We find that the lowest level truncation gives good quantitative agreement with the predicted D-brane tension and low-lying spectrum of the D-brane for sufficiently large p and study the effect of the next level corrections for p=24. We show that a U(1) gauge field zero mode on the D-brane arises through a mechanism reminiscent of the Randall-Sundrum mechanism for gravity. 
  We study the spatial structure of 1/4 BPS solitons in 4 dimensional N=4 gauge theory. A weak binding approximation is used where the soliton is made of several "ingredient" particles. Some spatial moduli are described which are not accounted for in the (p,q) web picture. These moduli are counted and their effect on the solutions is demonstrated. The potential for off BPS configurations is estimated by a simple expression and is found to agree with previous expressions. We discuss the fermionic zero modes of the solitons, and find agreement with web predictions. 
  We consider the supersymmetric field theories on the noncommutative $R^4$ using the superspace formalism on the commutative space. The terms depending on the parameter of the noncommutativity $\Theta$ are regarded as the interactions. In this way we construct the N=1 supersymmetric action for the U(N) vector multiplets and chiral multiplets of the fundamental, anti-fundamental and adjoint representations of the gauge group. The action for vector multiplets of the products gauge group and its bi-fundamental matters is also obtained. We discuss the problem of the derivative terms of the auxiliary fields. 
  We consider holomorphic BF theories, their solutions and symmetries. The equivalence of Cech and Dolbeault descriptions of holomorphic bundles is used to develop a method for calculating hidden (nonlocal) symmetries of holomorphic BF theories. A special cohomological symmetry group and its action on the solution space are described. 
  We discuss maximally symmetric curved deformations of the flat domain wall solutions of five-dimensional dilaton gravity that appeared in a recent approach to the cosmological constant problem. By analyzing the bulk field configurations and the boundary conditions at a four-dimensional maximally symmetric curved domain wall, we obtain constraints on such solutions. For a special dilaton coupling to the brane tension that appeared in recent works, we find no curved deformations, confirming and extending slightly a result of Arkani-Hamed et al which was argued using a $Z_2$-symmetry of the solution. For more general dilaton-dependent brane tension, we find that the curvature is bounded by the Kaluza-Klein scale in the fifth dimension. 
  The q-state Potts model in two dimensions exhibits a first-order transition for q>4. As q->4+ the correlation length at this transition diverges. We argue that this limit defines a massive integrable quantum field theory whose lowest excitations are kinks connecting 4+1 degenerate ground states. We construct the S-matrix of this theory and the two-particle form factors, and hence estimate a number of universal amplitude ratios. These are in very good agreement with the results of extrapolated series in q^(-1/2) as well as Monte Carlo results for q=5. 
  The generalization of the Yang-Baxter equations (YBE) in the presence of Z_2 grading along both chain and time directions is presented. The XXZ model with staggered disposition along a chain of both, the anisotropy \pm\Delta, as well as shifts of the spectral parameters are considered and the corresponding integrable model is constructed. The Hamiltonian of the model is computed in fermionic and spin formulations. It involves three neighbour site interactions and therefore can be considered as a zig-zag ladder model. The Algebraic Bethe Ansatz technique is applied and the eigenstates, along with eigenvalues of the transfer matrix of the model are found. The model has a free fermionic limit at \Delta=0 and the integrable boundary terms are found in this case.   This construction is quite general and can be applied to other known integrable models. 
  Langlands recently constructed a map that factorizes the partition function of a free boson on a cylinder with boundary condition given by two arbitrary functions in the form of a scalar product of boundary states. We rewrite these boundary states in a compact form, getting rid of technical assumptions necessary in his construction. This simpler form allows us to show explicitly that the map between boundary conditions and states commutes with conformal transformations preserving the boundary and the reality condition on the scalar field. 
  The addition of boundary counterterms to the gravitational action of asymptotically anti-de Sitter spacetimes permits us to define the partition function unambiguously without background subtraction. We show that the inclusion of p-form fields in the gravitational action requires the addition of further counterterms which we explicitly identify. We also relate logarithmic divergences in the action dependent on the matter fields to anomalies in the dual conformal field theories. In particular we find that the anomaly predicted for the correlator of the stress energy tensor and two vector currents in four dimensions agrees with that of the ${\cal{N}} = 4$ superconformal SU(N) gauge theory. 
  We review the de Boer-Verlinde-Verlinde formalism for the renormalization group in the context of Dp brane vacua. We comment on various aspects of the dictionary between bulk and boundary and relate the discussion to the Randall-Sundrum scenario. We find that the gravitational coupling for the Randall-Sundrum gravity on the Dp brane worldvolume is proportional to the c-function of the Yang-Mills theory. We compute the beta function and find the expected uneventful flow prescribed by the classical dimension of the Yang-Mills operator. Finally, we argue for a dynamical mechanism for determining the cosmology on the brane. 
  Extending earlier results on the duality symmetries of three-brane probe theories we define the duality subgroup of SL(2,Z) as the symmetry group of the background 7-branes configurations. We establish that the action of Weyl reflections is implemented on junctions by brane transpositions that amount to exchanging branes that can be connected by open strings. This enables us to characterize duality groups of brane configurations by a map to the symmetry group of the Dynkin diagram. We compute the duality groups and their actions for all localizable 7-brane configurations. Surprisingly, for the case of affine configurations there are brane transpositions leaving them invariant but acting nontrivially on the charges of junctions. 
  The absorption in the extremal D3-brane background is studied for a class of massless fields whose linear perturbations leave the ten-dimensional background metric unperturbed, as well as the minimally-coupled massive scalar. We find that various fields have the same absorption probability as that of the dilaton-axion system, which is given exactly via the Mathieu equation. We analyze the features of the absorption cross-sections in terms of effective Schr\"odinger potentials, conjecture a general form of the dual effective potentials, and provide explicit numerical results for the whole energy range. As expected, all partial-wave absorption probabilities tend to zero (one) at low (large) energies, and exhibit an oscillatory pattern as a function of energy. The equivalence of absorption probabilities for various modes has implications for the correlation functions on the field, including subleading contributions on the field-theory side. In particular, certain half-integer and integer spin fields have identical absorption probabilities, thus providing evidence that the corresponding operator pairs on the field theory side belong to the same supermultiplets. 
  In this note the open string partition function is analyzed carefully in a way to reveal the group-theoretical aspects. For the simple cases of ADE orbifolds with regular Chan-Paton action a prescription for consistent boundary states is given. In the process of outlining how they encode McKay correspondence, we argue that this results from a non-trivial conspiracy of numerical factors in the string amplitudes. 
  We advocate that the orbifold Z_2 symmetry of the gravity trapping model proposed by Randall and Sundrum can be seen, in appropriate coordinates, as a symmetry that exchanges the short distances with the large ones. Using diffeomorphism invariance, we construct extensions defined by patch glued together. A singularity occurs at the junction and it is interpreted as a brane, the jump brane, of codimension one. We give explicit realization in ten and eleven dimensional supergravity and show that the lower dimensional Planck scale on the brane is finite. The standard model would be trapped on a supersymmetric brane located at the origin whereas the jump brane would surround it at a finite distance. The bulk interactions could transmit the supersymmetry breaking from the jump brane to the SM brane. 
  We derive a set of equations for the wavefunction describing the marginal bound state of a single D0-brane with a single D4-brane. These are equations determining the vacuum of an N=8 abelian gauge theory with a charged hypermultiplet. We then solve these equations for the most general possible zero-energy solution using a Taylor series. We find that there are an infinite number of such solutions of which only one must be normalizable. We explore the structure of a normalizable solution under the assumption of an asymptotic expansion. Even the leading terms in the asymptotic series, which should reflect the supergravity solution, are unusual. Through the $Spin(5)$ flavor symmetry, the modes which are massive at long distance actually influence the leading behavior. Lastly, we show that the vacuum equations can quite remarkably be reduced to a single equation involving one unknown function. The resulting equation has a surprisingly simple and suggestive form. 
  In brane world models our universe is considered as a brane imbedded into a higher dimensional space. We discuss the behaviour of geodesics in the Randall-Sundrum background and point out that free massive particles cannot move along the brane only. The brane is repulsive, and matter will be expelled from the brane into the extra dimension. This is rather undesirable, and hence we study an alternative model with a non-compact extra dimension, but with an attractive brane embedded into the higher dimensional space. We study the linearized gravity equations and show that Newton's gravitational law is valid on the brane also in the alternative background. 
  We find that the mixture of Ramond-Ramond fields and Neveu-Schwarz two form are transformed as Majorana spinors under the T-duality group $O(d,d)$. The Ramond-Ramond field transformation under the group $O(d,d)$ is realized in a simple form by using the spinor representation. The Ramond-Ramond field transformation rule obtained by Bergshoeff et al. is shown as a specific simple example. We also give some explicit examples of the spinor representation. 
  We find a manifestly N=3 supersymmetric generalization of the four-dimensional Euler-Heisenberg (four-derivative, or F^4) part of the Born-Infeld action in light-cone gauge, by using N=3 light-cone superspace. 
  We review recent results on the general structure of two- and three- point functions of the supercurrent and the flavor current of N = 2 superconformal field theories. 
  Renormalization group, and in particular its Quantum Field Theory implementation has provided us with essential tools for the description of the phase transitions and critical phenomena beyond mean field theory. We therefore review the methods, based on renormalized phi^4_3 quantum field theory and renormalization group, which have led to a precise determination of critical exponents of the N-vector model (R. Guida and J. Zinn-Justin, J. Phys. A31 (1998) 8103. cond-mat/9803240). and of the equation of state of the 3D Ising model (R. Guida and J. Zinn-Justin, Nucl. Phys. B489 [FS] (1997) 626, hep-th/9610223.). These results are among the most precise available probing field theory in a non-perturbative regime. 
  We continue our study of gravity described by the action density (-g)^(1/2)(R_ik^2+bR^2); and look for cosmological solutions of gravity coupled to dust, for the closed isotropic model. There is a solution that at t approaches zero has for the radius, a(t)=t/3^(1/2); in the absence of dust this solution holds for all time. 
  We construct noncommutative gauge theories based on the notion of the Weyl bundle, which appears in Fedosov's construction of deformation quantization on an arbitrary symplectic manifold. These correspond to D-brane worldvolume theories in non-constant B-field and curved backgrounds in string theory. All such theories are embedded into a "universal" gauge theory of the Weyl bundle. This shows that the combination of a background field and a noncommutative field strength has universal meaning as a field strength of the Weyl bundle. We also show that the gauge equivalence relation is a part of such a "universal" gauge symmetry. 
  We describe a method of writing down interacting equations for all the modes of the bosonic open string. It is a generalization of the loop variable approach that was used earlier for the free, and lowest order interacting cases. The generalization involves, as before, the introduction of a parameter to label the different strings involved in an interaction. The interacting string has thus becomes a ``band'' of finite width. The interaction equations expressed in terms of loop variables, has a simple invariance that is exact even off shell. A consistent definition of space-time fields requires the fields to be functions of all the infinite number of gauge coordinates (in addition to space time coordinates). The theory is formulated in one higher dimension, where the modes appear massless. The dimensional reduction that is needed to make contact with string theory (which has been discussed earlier for the free case) is not discussed here. 
  It is shown that all possible gravitational, gauge and other interactions experienced by particles in ordinary d-dimensions (one-time) can be described in the language of two-time physics in a spacetime with d+2 dimensions. This is obtained by generalizing the worldline formulation of two-time physics by including background fields. A given two-time model, with a fixed set of background fields, can be gauged fixed from d+2 dimensions to (d-1) +1 dimensions to produce diverse one-time dynamical models, all of which are dually related to each other under the underlying gauge symmetry of the unified two-time theory. To satisfy the gauge symmetry of the two-time theory the background fields must obey certain coupled differential equations that are generally covariant and gauge invariant in the target d+2 dimensional spacetime. The gravitational background obeys a null homothety condition while the gauge field obeys a differential equation that generalizes a similar equation derived by Dirac in 1936. Explicit solutions to these coupled equations show that the usual gravitational, gauge, and other interactions in d dimensions may be viewed as embedded in the higher d+2 dimensional space, thus displaying higher spacetime symmetries that otherwise remain hidden. 
  We derive massless and massive representations of all SU(2,2/N) superalgebras by using superfields defined in "harmonic superspace". This method allows one to easily construct "short superfields" which are relevant in the analysis of the AdS/CFT correspondence. 
  We consider logarithmic conformal field theories near a boundary and derive the general form of one and two point functions. We obtain results for arbitrary and two dimensions. Application to two dimensional magnetohydrodynamics is discussed. 
  The axial anomaly of the noncommutative U(1) gauge theory is calculated by a number of methods and compared with the commutative one. It is found to be given by the corresponding Chern class. 
  Physically relevant gauge and gravitational theories can be seen as special members of hierarchies of more elaborate systems. The Yang-Mills (YM) system is the first member of a hierarchy of Lagrangians which we will index by $p_1$, and the Einstein-Hilbert (EH) system of general relativity is the first member of another hierarchy which we index by $p_2$. In this paper, we study the classical equations of the $p_1 = 1,2$ YM hierarchy considered in the background of special geometries (Schwarzschild, deSitter,anti-deSitter) of the $p_2=1,2,3$ EH hierarchy. Solutions are obtained in various dimensions and lead to several examples of non-self-dual YM fields. When $p_1=p_2$ self-dual solutions exist in addition. Their action is equal to the Chern-Pontryagin charge and can be compared with that of the non-self-dual solutions. 
  A deSitter brane-world bounding regions of anti-deSitter space has a macroscopic entropy given by one-quarter the area of the observer horizon. A proposed variant of the AdS/CFT correspondence gives a dual description of this cosmology as conformal field theory coupled to gravity in deSitter space. In the case of two-dimensional deSitter space this provides a microscopic derivation of the entropy, including the one-quarter, as quantum entanglement of the conformal field theory across the horizon. 
  We describe a new class of supersymmetric orientifolds which combine the world-sheet parity transformation with a complex conjugation in the compact directions. As an example, we investigate in detail the orientifold of the Z_3 toroidal orbifold in six and four dimensions. We demonstrate how the solution to the tadpole cancellation conditions, the resulting gauge groups and the massless spectra depend on the choice of the complex structures on the tori, giving rise to a variety of inequivalent models. We also summarize the results for the orientifolds of the Z_4, Z_6 and Z_6' orbifolds in four and six dimensions. 
  We study the Kaluza-Klein zero modes of massless bulk fields with various spins in the background of dilatonic and self-tuning flat domain walls. We find that the zero modes of all the massless bulk fields in such domain wall backgrounds are normalizable, unlike those in the background of the non-dilatonic domain wall with infinite extra space of Randall and Sundrum. In particular, gravity in the bulk of dilatonic domain walls is effectively compactified to the Einstein gravity with vanishing cosmological constant and nonzero gravitational constant in one lower dimensions for any values of dilaton coupling parameter, provided the warp factor is chosen to decrease on both sides of the domain wall, in which case the tension of the domain wall is positive. However, unexpectedly, for the self-tuning flat domain walls, the cosmological constant of the zero mode effective gravity action in one lower dimensions does not vanish, indicating the need for additional ingredient or modification necessary in cancellation of the unexpected cosmological constant in the graviton zero mode effective action. 
  A random matrix model with a sigma-model like constraint, the restricted trace ensemble (RTE), is solved in the large-n limit. In the macroscopic limit the smooth connected two-point resolvent G(z,w) is found to be non-universal, extending previous results from monomial to arbitrary polynomial potentials. Using loop equation techniques we give a closed though non-universal expression for G(z,w), which extends recursively to all higher k-point resolvents. These findings are in contrast to the usual unconstrained one-matrix model. However, in the microscopic large-n limit, which probes only correlations at distance of the mean level spacing, we are able to show that the constraint does not modify the universal sine-law. In the case of monomial potentials $V(M)=M^{2p}$, we provide a relation valid for finite-n between the k-point correlation function of the RTE and the unconstrained model. In the microscopic large-n limit they coincide which proves the microscopic universality of RTEs. 
  We show that massless RR tadpoles in vacuum configurations with open and unoriented strings are always related to anomalies. RR tadpoles arising from sectors of the internal SCFT with non-vanishing Witten index are in one-to-one correspondence with conventional irreducible anomalies. The anomalous content of the remaining RR tadpoles can be disclosed by considering anomalous amplitudes with higher numbers of external legs. We then provide an explicit parametrization of the anomaly polynomial in terms of the boundary reflection coefficients, i.e. one-point functions of massless RR fields on the disk. After factorization of the reducible anomaly, we extract the relevant WZ couplings in the effective lagrangians. 
  Type II strings in D=5 contain particle-like 1/8 supersymmetric BPS states. In this note we give a string-network representation of such states by considering (periodic) non-planar $(p,q,r)$-string networks of eight dimensional type II string theory on $T^3$. We obtain the BPS mass formula of such states, in terms of charges and generating-vectors of the torus, and show its invariance under an $SL(3, Z)\times SL(3, Z)$ group of transformations. Results are then generalized to string-networks associated with the $SL(5, Z)$ $U$-duality in seven dimensions. We also discuss reinterpretation of the above $(D=5)$ mass formula in terms of BPS states in world-volume theories of $U2$-branes in D=8. 
  The simplest and the most straightforward new algorithm for generating solutions to (anti) self-dual Yang-Mills (YM) equation in the typical gravitational instanton backgrounds is proposed. When applied to the Taub-NUT and the Eguchi-Hanson metrics, the two best-known gravitational instantons, the solutions turn out to be the rather exotic type of instanton configurations carrying finite YM action but generally fractional topological charge values. 
  We review the recent results concerning the computation of cubic and quartic couplings of scalar fields in type IIB supergravity on AdS_5\times S^5 background that are dual to (extended) chiral primary operators in N=4 SYM_4. We discuss the vanishing of certain cubic and quartic couplings and non-renormalization property of corresponding correlators in the conformal field theory 
  The percolation properties of equatorial strips of the two dimensional O(3) nonlinear $\sigma$ model are investigated numerically. Convincing evidence is found that a sufficently narrow strip does not percolate at arbitrarily low temperatures. Rigorous arguments are used to show that this result implies both the presence of a massless phase at low temperature and lack of asymptotic freedom in the massive continuum limit. A heuristic estimate of the transition temperature is given which is consistent with the numerical data. 
  We present a detailed analysis of a scalar conformal four-point function obtained from AdS/CFT correspondence. We study the scalar exchange graphs in AdS and discuss their analytic properties. Using methods of conformal partial wave analysis, we present a general procedure to study conformal four-point functions in terms of exchanges of scalar and tensor fields. The logarithmic terms in the four-point functions are connected to the anomalous dimensions of the exchanged fields. Comparison of the results from AdS graphs with the conformal partial wave analysis, suggests a possible general form for the operator product expansion of scalar fields in the boundary CFT. 
  We argue that once octonions are formulated as soft Lie algebras, they may be safely used and the non-associativity can be overcame. The necessary points are: (a) Fixing the direction of action by introducing the \delta operator. (b) Closing the \delta algebra by using structure functions f_{ijk} (\phi). (c) Representation of the \delta algebra can be developed. The E or E(\phi) can be found and their structure functions can be computed easily. There may be different applications of soft seven sphere in physics. We have given two cases where the ring division algebras occupies a special position. Self-duality and Simple supersymmetric Yang-Mills theories are two promising places where soft seven sphere prove to be useful and essential. 
  Asymptotically anti-de Sitter space-times in pure gravity with negative cosmological constant are described, in all space-time dimensions greater than two, by classical degrees of freedom on the conformal boundary at space-like infinity. Their effective boundary action has a conformal anomaly for even dimensions and is conformally invariant for odd ones. These degrees of freedom are encoded in traceless tensor fields in the Fefferman-Graham asymptotic metric for any choice of conformally flat boundary and generate all Schwarzschild and Kerr black holes in anti-de Sitter space-time. We argue that these fields describe components of an energy-momentum tensor of a boundary theory and show explicitly how this is realized in 2+1 dimensions. There, the Fefferman-Graham fields reduce to the generators of the Virasoro algebra and give the mass and the angular momentum of the BTZ black holes. Their local expression is the Liouville field in a general curved background. 
  We find a new stable rotator solution in D0-brane matrix mechanics. The solution is interpreted as a D2/D0 brane bound state, constructed as a transversely rotating ellipsoidal membrane with N D0-branes pinned over it.   From the membrane point of view the attractive force of tension is exactly cancelled by the repulsive centrifugal force. Dynamical properties of the system are investigated, in particular the emission of spherical waves in the different massless sectors of supergravity. We compute the RR 1-form quadrupole, the RR 3-form dipole and the gravitational quadrupole radiation. Also, we show that our non-singular classical solution is stable against small perturbations in the initial conditions. Furthermore, we comment on a possible fundamental particle interpretation of our system. 
  We present a nonperturbative lattice formulation of noncommutative Yang-Mills theories in arbitrary even dimension. We show that lattice regularization of a noncommutative field theory requires finite lattice volume which automatically provides both an ultraviolet and an infrared cutoff. We demonstrate explicitly Morita equivalence of commutative U(p) gauge theory with (p*n_f) flavours of fundamental matter fields on a lattice of size L with twisted boundary conditions and noncommutative U(1) gauge theory with n_f species of matter on a lattice of size (p*L) with single-valued fields. We discuss the relation with twisted large N reduced models and construct observables in noncommutative gauge theory with matter. 
  The world volume theory on N regular and M fractional D3-branes at the conifold singularity is a non-conformal n=1 supersymmetric SU(N+M) x SU(N) gauge theory. In previous work the Type IIB supergravity dual of this theory was constructed to leading non-trivial order in M/N: it is the AdS_5 x T^{1,1} background with NS-NS and R-R 2-form fields turned on. Far in the UV this dual description was shown to reproduce the logarithmic flow of couplings found in the field theory. In this paper we study the supersymmetric RG flow at all scales. We introduce an ansatz for the 10-d metric and other fields and show that the equations of motion may be derived in first order form from a simple superpotential. This allows us to explicitly solve for the gravity dual of the RG trajectory. 
  Necessary conditions are proposed for the admissibility of singular classical solutions with 3+1-dimensional Poincare invariance to five-dimensional gravity coupled to scalars. Finite temperature considerations and examples from AdS/CFT support the conjecture that the scalar potential must remain bounded above for a solution to be physical. Having imposed some restrictions on naked singularities allows us to comment on a recent proposal for solving the cosmological constant problem. 
  We examine the behaviour of gravity in brane theories with extra dimensions in a non-factorizable geometry. We find that for metrics which are asymptotically flat far from the brane there is a resonant graviton mode at zero energy. The presence of this resonance ensures quasi-localization of gravity, whereby at intermediate scales the gravitational laws on the brane are approximately four dimensional. However, for scales larger than the lifetime of the graviton resonance the five dimensional laws of gravity will be reproduced due to the decay of the four dimensional graviton. We also present a simple classification of the possible types of effective gravity theories on the brane that can appear for general non-factorizable background theories. 
  It is shown how the the introduction of a suitably defined dilatonic auxiliary field, $\Phi$ say, makes it possible for the non-linear Lagrangian for a generic elastic string model, of the kind appropriate for representing superconducting cosmic strings, to be converted into a standardised form as the sum of a kinetic term that is just homogeneously quadratic in the relevant scalar phase gradient (as in a simple linear model) together with a potential energy term, $V$ say, that is specified as a generically non-linear function of $\Phi$. The explicit form of this function is derived for various noteworthy examples, of which the most memorable is that of the transonic string model, as characterised by a given mass scale, $m$ say, for which this potential energy density will be expressible in terms of the zero current limit value $\Phi_{_0}$ of $\Phi$ by $ V= {1\over 2} m \big(\Phi_{_0}^{-2} \Phi^2+ \Phi_{_0}^{2} \Phi^{-2} \big)$. 
  This manuscript has been withdrawn by the author. Some of the material has been included in the manuscript 'Embedded Monopoles' at hep-th/0106254. 
  In this note we elaborate on various five dimensional contributions to the effective 4D cosmological constant in brane systems. In solutions with vanishing 5D cosmological constant we describe a non-local mechanism of cancellation of vacuum energy between the brane and the singularities. We comment on a hidden fine tuning which is implied by this observation. 
  There has been spectacular progress in the development of string and superstring theories since its inception thirty years ago. Development in this area has never been impeded by the lack of experimental confirmation. Indeed, numerous bold and imaginative strides have been taken and the sheer elegance and logical consistency of the arguments have served as a primary motivation for string theorists to push their formulations ahead. In fact the development in this area has been so rapid that new ideas quickly become obsolete. On the other hand, this rapid development has proved to be the greatest hindrance for novices interested in this area. These notes serve as a gentle introduction to this topic. In these elementary notes, we briefly review the RNS formulation of superstring theory, GSO projection, $D$-branes, bosonic strings, dualities, dynamics of $D$-branes and the microscopic description of Bekenstein entropy of a black hole. 
  Non-supersymmetric black holes carrying both electric and magnetic charge with respect to a single Kaluza-Klein gauge field have much in common with supersymmetric black holes. Angular momentum conservation and other general physics principles underlies some of their basic features. Kaluza-Klein black holes are interpreted in string theory as bound states of D6-branes and D0-branes. The microscopic theory reproduces the full nonlinear mass formula of the extremal black holes. 
  We present a simple derivation of vector supersymmetry transformations for topological field theories of Schwarz- and Witten-type. Our method is similar to the derivation of BRST-transformations from the so-called horizontality conditions or Russian formulae. We show that this procedure reproduces in a concise way the known vector supersymmetry transformations of various topological models and we use it to obtain some new transformations of this type for 4d topological YM-theories in different gauges. 
  Link invariants, for 3-manifolds, are defined in the context of the Rozansky-Witten theory. To each knot in the link one associates a holomorphic bundle over a holomorphic symplectic manifold X. The invariants are evaluated for b_{1}(M) \geq 1 and X Hyper-Kaehler. To obtain invariants of Hyper-Kaehler X one finds that the holomorphic vector bundles must be hyper-holomorphic. This condition is derived and explained. Some results for X not Hyper-Kaehler are presented. 
  We continue to explore the conjectural expressions of the Gromov-Witten potentials for a class of elliptically and K3 fibered Calabi-Yau 3-folds in the limit where the base P^1 of the K3 fibration becomes infinitely large. At least in this limit we argue that the string partition function (=the exponential generating function of the Gromov-Witten potentials) can be expressed as an infinite product in which the Kahler moduli and the string coupling are treated somewhat on an equal footing. Technically speaking, we use the exponential lifting of a weight zero Jacobi form to reach the infinite product as in the celebrated work of Borcherds. However, the relevant Jacobi form is associated with a lattice of Lorentzian signature. A major part of this work is devoted to an attempt to interpret the infinite product or more precisely the Jacobi form in terms of the bound states of D2- and D0-branes using a vortex description and its suitable generalization. 
  We show that the recently found quartic action for the scalars from the massless graviton multiplet of type IIB supergravity compactified on AdS_5\times S^5 background coincides with the relevant part of the action of the gauged N=8 5d supergravity on AdS_5. We then use this action to compute the 4-point function of the lowest weight chiral primary operators $\tr(\phi^{(i}\phi^{j)})$ in N=4 SYM_4 at large $N$ and at strong `t Hooft coupling. 
  We discuss the noncommutative counterparts of chiral gauge theories and compute the associated anomalies. 
  We review some aspects of the AdS supergravity description of RG flows. The case of a flow to an IR CFT can be rigorously studied within the framework of supergravity. Here we discuss various central charges of the conformal theory (included the usually neglected ones) and we compare them with QFT expectations. The case of flows to non-conformal theories is more problematic in that one usually encounters a naked singularity. We mainly focus on the flow to an IR N=1 super Yang-Mills theory. We discuss the properties of the solution and we briefly comment on the fate of the singularity. We also compare the supergravity results with the expectations of an N=1 SYM at strong coupling. 
  A general construction of affine Non Abelian Toda models in terms of gauged two loop WZNW model is discussed. In particular we find the  Lie algebraic condition defining a subclass of {\it T-selfdual torsionless NA Toda models} and their zero curvature representation. 
  Brane world scenarios offer a way of ensuring that a Poincare invariant four dimensional world can emerge, without fine tuning, as a solution to the equations of motion of an effective action. We discuss the different ways in which this happens, and point out that the underlying reason is that there is a contribution to the effective cosmological constant which is a constant of integration, that maybe adjusted to ensure a flat space solution. Basically this is an old idea revived in a new context and we speculate that there may be string scenarios that provide a concrete realization of it. Finally we discuss to what extent this is a solution to the cosmological constant problem. 
  We describe a mechanism for localising branes in ambient space. When a 3-form flux is turned on in a Taub-NUT space, an M5-brane gets an effective potential that pins it to the center of the space. A similar effect occurs for M2-branes and D-branes with appropriate fluxes. In carefully chosen limits of the external parameters, this leads to new theories that are decoupled from gravity and appear to break Lorentz invariance. For example, we predict the existence of a new 5+1D theory that breaks Lorentz invariance at high-energy and has a low-energy description of N tensor multiplets with (1,0) supersymmetry. We also predict a new type of theory that, similarly to the little-string theory decouples from gravity by a dynamical (rather than kinematical) argument. 
  We study the possible BPS domain wall junction configurations for general polynomial superpotentials of N=1 supersymmetric Wess-Zumino models in D=4. We scan the parameter space of the superpotential and find different possible BPS states for different values of the deformation parameters and present our results graphically. We comment on the domain walls in F/M/IIA theories obtained from the Calabi-Yau fourfolds with isolated singularities and a background flux. 
  We study the behaviour of quantum field theories defined on a surface $S$ as it tends to a null surface $S_n$. In the case of a real, free scalar field theory the above limiting procedure reduces the system to one with a finite number of degrees of freedom. This system is shown to admit a one parameter family of inequivalent quantizations. A duality symmetry present in the model can be used to remove the quantum ambiguity at the self-dual point . In the case of the non-linear $\sigma$-model with the Wess-Zumino-Witten term a similar limiting behaviour is obtained. The quantization ambiguity in this case however cannot be removed by any means. 
  In 2+1 dimensions there exists a duality between a charged Dirac particle coupled minimally to a background vector potential and a neutral one coupled nonminimally to a background electromagnetic field strength. A constant uniform background electric current induces in the vacuum of the neutral particle a fermion current which is proportional to the background one. A background electromagnetic plane wave induces no current in the vacuum. For constant but nonuniform background electric charge, known results for charged particles can be translated to give the induced fermion number. Some new examples with infinite background electric charge are presented. The induced spin and total angular momentum are also discussed. 
  In this paper we investigate the vacuum expectation values of energy- momentum tensor for conformally coupled scalar field in the standard parallel plate geometry with Dirichlet boundary conditions and on background of planar domain wall case. First we calculate the vacuum expectation values of energy-momentum tensor by using the mode sums, then we show that corresponding properties can be obtained by using the conformal properties of the problem. The vacuum expectation values of energy-momentum tensor contains two terms which come from the boundary conditions and the the gravitational background. In the Minkovskian limit our results agree with those obtained in [3]. 
  Motivated by the results of Hashimoto and Taylor, we perform a detailed study of the mass spectrum of the non-abelian Born-Infeld theory, defined by the symmetrized trace prescription, on tori with constant magnetic fields turned on. Subsequently, we compare this for several cases to the mass spectrum of intersecting D-branes. Exact agreement is found in only two cases: BPS configurations on the four-torus and coinciding tilted branes. Finally we investigate the fluctuation dynamics of an arbitrarily wrapped Dp-brane with flux. 
  We introduce the notion of fractal index associated with the universal class $h$ of particles or quasiparticles, termed fractons, which obey specific fractal statistics. A connection between fractons and conformal field theory(CFT)-quasiparticles is established taking into account the central charge $c[\nu]$ and the particle-hole duality $\nu\longleftrightarrow\frac{1}{\nu}$, for integer-value $\nu$ of the statistical parameter. In this way, we derive the Fermi velocity in terms of the central charge as $v\sim\frac{c[\nu]}{\nu+1}$. The Hausdorff dimension $h$ which labelled the universal classes of particles and the conformal anomaly are therefore related. Following another route, we also established a connection between Rogers dilogarithm function, Farey series of rational numbers and the Hausdorff dimension. 
  We present a new view for duality in classical electromagnetic theory, based on the physical properties of a dual theory, eliminating the problems of the usual treatment of the subject. 
  We discuss general properties of the theory of higher spin gauge fields in $AdS_4$ focusing on the relationship between the star-product origin of the higher spin symmetries, AdS geometry and the concept of space-time locality. A full list of conserved higher spin currents in the flat space of arbitrary dimension is presented. 
  We review some aspects of the D1/D5 system of type IIB string theory and the associated five dimensional black hole. We include a pedagogical discussion of the construction of relevant classical solutions in supergravity. We discuss the gauge theory and the conformal field theory relevant to D-brane description of these systems. In order to discuss Hawking radiation we are automatically led to a discussion of near-horizon geometries and their relation to gauge theories and conformal field theories. We show how inputs from AdS/CFT correspondence resolve some earlier puzzles regarding Hawking radiation. Besides the D1/D5 system, we include a brief discussion of some nonsupersymmetric systems which show unexpected agreement between supergravity and perturbative brane/string computations. We also comment briefly on possible implications of the AdS/CFT relation for the correspondence principle and for the principle of black hole complementarity. 
  We provide analytical solutions to the thermodynamic Bethe ansatz equations in the large and small density approximations. We extend results previously obtained for leading order behaviour of the scaling function of affine Toda field theories related to simply laced Lie algebras to the non-simply laced case. The comparison with semi-classical methods shows perfect agreement for the simply laced case. We derive the Y-systems for affine Toda field theories with real coupling constant and employ them to improve the large density approximations. We test the quality of our analysis explicitly for the Sinh-Gordon model and the $(G_2^{(1)},D_4^{(3)})$-affine Toda field theory. 
  We analyze further the IR singularities that appear in noncommutative field theories on R^d. We argue that all IR singularities in nonplanar one loop diagrams may be interpreted as arising from the tree level exchanges of new light degrees of freedom, one coupling to each relevant operator. These exchanges are reminiscent of closed string exchanges in the double twist diagrams in open string theory. Some of these degrees of freedom are required to have propagators that are inverse linear or logarithmic. We suggest that these can be interpreted as free propagators in one or two extra dimensions respectively. We also calculate some of the IR singular terms appearing at two loops in noncommutative scalar field theories and find a complicated momentum dependence which is more difficult to interpret. 
  We consider the interaction between instantons and anti-instantons in four-dimensional N=4 super-Yang-Mills theory at large N and large 't Hooft coupling as described by D-instantons via AdS/CFT duality. We give an estimate of the strength of the interaction in various regimes. We discuss also the case of Non-Commutative super Yang-Mills theory where the interaction between instantons and anti-instantons can be used as a way to probe the locality properties of the theory in the supergravity picture, without explicit reference to the definition of local operators. 
  Motivated by recent progress on the correspondence between string theory on anti-de Sitter space and conformal field theory, we address the question of constructing space-time N extended superconformal algebras on the boundary of AdS_3. Based on a free field realization of an affine SL(2|N/2) current superalgebra residing on the world sheet, we construct explicitly the Virasoro generators and the N supercurrents. N is even. The resulting superconformal algebra has an affine SL(N/2) \otimes U(1) current algebra as an internal subalgebra. Though we do not complete the general superalgebra, we outline the underlying construction and present supporting evidence for its validity. Particular attention is paid to its BRST invariance. In the classical limit where the free field realization may be substituted by a differential operator realization, we discuss further classes of generators needed in the closure of the algebra. We find sets of half-integer spin fields, and for N>4 these include generators of negative weights. An interesting property of the construction is that for N>2 it treats the supercurrents in an asymmetric way. Thus, we are witnessing a new class of superconformal algebras not obtainable by conventional Hamiltonian reduction. The complete classical algebra is provided in the case N=4 and is of a new and asymmetric form. 
  The strong version of Maldacena's AdS/CFT conjecture implies that the large N expansion of free N=4 super-YM theory describes an interacting string theory in the extreme limit of high spacetime curvature relative to the string length. String states may then be understood as composed of SYM string bits. We investigate part of the low-lying spectrum of the tensionless (zero-coupling) limit and find a large number of states that are not present in the infinite tension (strong-coupling) limit, notably several massless spin two particles. We observe that all conformal dimensions are N-independent in the free SYM theory, implying that masses in the corresponding string theory are unchanged by string interactions. Degenerate string states do however mix in the interacting string theory because of the complicated N-dependence of general CFT two-point functions. Finally we verify the CFT crossing symmetry, which corresponds to the dual properties of string scattering amplitudes. This means that the SYM operator correlation functions define AdS dual models analogous to the Minkowski dual models that gave rise to string theory. 
  We address the issue of whether extra dimensions could have an infinite volume and yet reproduce the effects of observable four-dimensional gravity on a brane. There is no normalizable zero-mode graviton in this case, nevertheless correct Newton's law can be obtained by exchanging bulk gravitons. This can be interpreted as an exchange of a single {\it metastable} 4D graviton. Such theories have remarkable phenomenological signatures since the evolution of the Universe becomes high-dimensional at very large scales. Furthermore, the bulk supersymmetry in the infinite volume limit might be preserved while being completely broken on a brane. This gives rise to a possibility of controlling the value of the bulk cosmological constant. Unfortunately, these theories have difficulties in reproducing certain predictions of Einstein's theory related to relativistic sources. This is due to the van Dam-Veltman-Zakharov discontinuity in the propagator of a massive graviton. This suggests that all theories in which contributions to effective 4D gravity come predominantly from the bulk graviton exchange should encounter serious phenomenological difficulties. 
  In this paper we construct the BPS black hole generating solution of toroidally compactified string (and M) theory, giving for it both the macroscopic and microscopic description. Choosing a proper U-duality gauge the latter will be given by a bound state made solely of D-branes. The axionic nature of the supergravity solution will be directly related to non-trivial angles between the constituent D-branes (type IIB configuration) or, in a T-dual gauge, to the presence of magnetic flux on constituent D-brane world volumes (type IIA configuration). As expected, the four dimensional axion fields arise from the dimensional reduction of non-diagonal metric tensor components or Kalb-Ramond B field components for type IIB or type IIA cases, respectively. Thanks to this result it is then now possible to fill the full 56-dimensional U-duality orbit of N=8 BPS black holes and to have a macroscopic and microscopic description of all of them. 
  We present a new compactification of chiral, N=2 ten-dimensional supergravity down to five dimensions and show that it corresponds to the N=2 supersymmetric critical point of five-dimensional, N=8 gauged supergravity found in [KPW]. This solution presented here is of particular significance because it involves non-zero tensor gauge fields and, via the AdS/CFT correspondence, is dual to the non-trivial N=1 supersymmetric fixed point of N=4 Yang-Mills theory. 
  Recently N.Nekrasov and A.Schwarz proposed a modification of the ADHM construction of instantons which produces instantons on a noncommutative deformation of the 4-dimensional real affine space. In this paper we study the relation between their construction and algebraic bundles on noncommutative projective spaces. We exhibit one-to-one correspondences between three classes of objects: framed bundles on a noncommutative projective plane, certain complexes of sheaves on a noncommutative 3-dimensional projective space, and the modified ADHM data. The modified ADHM construction itself is interpreted in terms of a noncommutative version of the twistor transform. We also prove that the moduli space of framed bundles on the noncommutative projective plane has a natural hyperkahler metric and is isomorphic as a hyperkahler manifold to the moduli space of framed torsion free sheaves on the commutative projective plane. The natural complex structures on the two moduli spaces do not coincide but are related by an SO(3) rotation. Finally, we propose a construction of instantons on a more general noncommutative R^4 than the one considered by Nekrasov and Schwarz (a q-deformed R^4). 
  It was argued that there are two different descriptions of the effective Lagrangian of gauge fields on D-branes by non-commutative gauge theory and by ordinary gauge theory in the presence of a constant B field background. In the case of bosonic string theory, however, it was found in the previous works that the two descriptions are incompatible under the field redefinition which relates the non-commutative gauge field to the ordinary one found by Seiberg and Witten. In this paper we resolve this puzzle to observe the necessity of gauge-invariant but B-dependent correction terms involving metric in the field redefinition which have not been considered before. With the problem resolved, we establish a systematic method under the alpha' expansion to derive the constraints on the effective Lagrangian imposed by the compatibility of the two descriptions where the form of the field redefinition is not assumed. 
  We describe consequences of the chiral anomaly in the theory of quantum wires, the (quantum) Hall effect, and of a four-dimensional cousin of the Hall effect. We explain which aspects of conductance quantization are related to the chiral anomaly. The four-dimensional analogue of the Hall effect involves the axion field, whose time derivative can be interpreted as a (space-time dependent) difference of chemical potentials of left-handed and right-handed charged fermions. Our four-dimensional analogue of the Hall effect may play a significant role in explaining the origin of large magnetic fields in the (early) universe. 
  We study the quantum gravitational effects in spherically symmetric black hole spacetimes. The effective quantum spacetime felt by a point-like test mass is constructed by ``renormalization group improving'' the Schwarzschild metric. The key ingredient is the running Newton constant which is obtained from the exact evolution equation for the effective average action. The conformal structure of the quantum spacetime depends on its ADM-mass M and it is similar to that of the classical Reissner-Nordstrom black hole. For M larger than, equal to, and smaller than a certain critical mass $M_{\rm cr}$ the spacetime has two, one and no horizon(s), respectively. Its Hawking temperature, specific heat capacity and entropy are computed as a function of M. It is argued that the black hole evaporation stops when M approaches $M_{\rm cr}$ which is of the order of the Planck mass. In this manner a ``cold'' soliton-like remnant with the near-horizon geometry of $AdS_2\times S^2$ is formed. As a consequence of the quantum effects, the classical singularity at r=0 is either removed completely or it is at least much milder than classically; in the first case the quantum spacetime has a smooth de Sitter core which would be in accord with the cosmic censorship hypothesis even if $M<M_{\rm cr}$. 
  The Witten-Veneziano relation between the topological susceptibility of pure gauge theories without fermions and the main contribution of the complete theory and the corresponding formula of Seiler and Stamatescu with the so-called contact term are discussed for the Schwinger model on a circle. Using the (Euclidean) path integral and the canonical (Hamiltonian) approaches at finite temperatures we demonstrate that both formulae give the same result in the limit of infinite volume and (or) zero temperature. 
  In this letter we use Type II NS5-branes in the presence of an RR field at the decoupling limit to study a non-commutative version of little string theory. We shall see that the decoupling limit of NS5-branes in the presence of an RR field is completely different from the one we had in ordinary little string theory; in particular for Type IIA NS5-branes this decoupling limit can be defined only when the theory is wrapped on a circle, but nevertheless flows to the ordinary non-compact (0,2) theory at the IR limit. We also see that these theories, in the UV regime, where the non-commutative effects are important, can be described by smeared ordinary D3(D2)-branes in Type IIB(A) string theory. 
  We give an update on recent results about the matching between CFT operators and KK states in the AdS/CFT correspondence, and add some new comments on the realization of the baryonic symmetries from the supergravity point of view. 
  In multidimensional gravity with an arbitrary number of internal Ricci-flat factor spaces, interacting with electric and magnetic $p$-branes, spherically symmetric configurations are considered. It is shown that all single-brane black-hole solutions are stable under spherically symmetric perturbations, whereas similar solutions possessing naked singularities turn out to be catastrophically unstable. The black hole stability conclusion is extended to some classes of configurations with intersecting branes. These results do not depend on the particular composition of the $D$-dimensional space-time, on the number of dilatonic scalar fields $\phi^a$ and on the values of their coupling constants. Some examples from 11-dimensional supergravity are considered. 
  This paper studies the model of the quantum electrodynamics (QED) of a single nonrelativistic electron due to W. Pauli and M. Fierz and studied further by P. Blanchard. This model exhibits infrared divergence in a very simple context. The infrared divergence is associated with the inequivalence of the Hilbert spaces associated with the free Hamiltonian and with the complete Hamiltonian. Infrared divergences that are visible in the perturbative description disappear in the space of the clothed electrons. In this model when the Hamiltonian is expressed in terms of the ``physical'' fields that create the electron together with its cloud of soft photons the variational principle suggested earlier can be applied. At finite time the Heisenberg field of the model acts in the space of the perturbative electron together with a finite number of perturbative photons, while the ``physical'' field can be chosen to act in the space of the exact (``physical'') electron eigenstates together with a finite number of physical photons. The space of the physical (or clothed) electron states can be chosen to be a Fock space. 
  We study the infra-red dynamics of D1-branes at the conifold. We show using methods developed to study the infra-red dynamics of (4,4) theories, the infra-red degrees of freedom of the (2,2) theory of a single D1-brane at the conifold is that of a linear dilaton with background charge of $\sqrt{2}$ and a compact scalar. The gauge theory of $N$ D1-branes at the conifold is used to formulate the matrix string in the conifold background. 
  The Wilsonian renormalization group implies that an arbitrary four dimensional field theory with an ultraviolet cutoff is equivalent to a theory which is renormalizable by power counting at energy scales much below the cutoff. This applies to any theory including those with non-renormalizable interactions as long as we fine-tune the mass parameters. We analyze two simple models with current-current interactions but without elementary gauge fields from this viewpoint. We show how to tune the parameters of the models so that they become equivalent to QED at energies much below the cutoff. 
  This is a brief account of the approach to superbranes based upon the concept of Partial Breaking of Global Supersymmetry (PBGS). 
  We study the singularities of the Higgs branch of supersymmetric U(1)^r gauge theories with eight supercharges. We derive new solutions for the moduli space of vacua preserving manifestly the eight supercharges by using a geometric realization of the SU(2)_R symmetry and a separation procedure of the gauge and SU(2)_R charges, which allow us to put the hypermultiplet vacua in a form depending on a parameter $\gamma$. For $\gamma=1$, we obtain new models which flow in the infrared to 2d N=(4,4) conformal models and we show that the classical moduli spaces are given by intersecting cotangent weighted complex projective spaces containing the small instanton singularity, discussed in [17], as a leading special case. We also make comments regarding the d$2d N=4 conformal Liouville description of the Higgs branch throat by following the analysis of [18]. Other features are also discussed. 
  We discuss the approach to asymptotic dynamics due to Kulish and Faddeev. We show that there are problems in applying this method to theories with four point interactions. The source of the difficulties is identified and a more general method is constructed. This is then applied to various theories including some where the coupling does switch off at large times and some where it does not. 
  Using a Lax pair based on twisted affine $sl(2,R)$ Kac-Moody and Virasoro algebras, we deduce a r-matrix formulation of two dimensional reduced vacuum Einstein's equations. Whereas the fundamental Poisson brackets are non-ultralocal, they lead to pure c-number modified Yang-Baxter equations. We also describe how to obtain classical observables by imposing reasonable boundaries conditions. 
  We present and discuss, at a general level, new mathematical results on the spatial nonuniformity of thermal quantum fields coupled minimally to static background electromagnetic potentials. Two distinct examples are worked through in some detail: uniform (parallel and perpendicular) background electric and magnetic fields coupled to a thermal quantum scalar field. 
  Employing the nonabelian duality transformation, I derive the Gauge String form of certain D>=3 lattice Yang-Mills (YM_{D}) theories in the strong coupling (SC) phase. With the judicious choice of the actions, in D>=3 our construction generalizes the Gross-Taylor stringy reformulation of the continuous YM_{2} on a 2d manifold. Using the Eguchi-Kawai model as an example, we develope the algorithm to determine the weights w[\tilde{M}] for connected YM-flux worldsheets \tilde{M} immersed into the 2d skeleton of a D>=3 base-lattice. Owing to the invariance of w[\tilde{M}] under a continuous group of area-preserving worldsheet homeomorphism, the set of the weights {w[\tilde{M}]} can be used to define the theory of the smooth YM-fluxes which unambiguously refers to a particular continuous YM_{D} system. I argue that the latter YM_{D} models (with a finite ultraviolet cut-off) for sufficiently large bare coupling constant(s) are reproduced, to all orders in 1/N, by the smooth Gauge String thus associated. The asserted YM_{D}/String duality allows to make a concrete prediction for the 'bare' string tension \sigma_{0} which implies that (in the large N SC regime) the continuous YM_{D} systems exhibit confinement for $D\geq{2}$. The resulting pattern is qualitatively consistent (in the extreme D=4 SC limit) with the Witten's proposal motivated by the AdS/CFT correspondence. 
  Via descent equations we derive formulas for consistent gauge anomalies in noncommutative Yang-Mills theories. 
  It has been conjectured that at the stationary point of the tachyon potential for the D-brane-anti-D-brane pair or for the non-BPS D-brane of superstring theories, the negative energy density cancels the brane tensions. We study this conjecture using a Wess-Zumino-Witten-like open superstring field theory free of contact term divergences and recently shown to give 60% of the vacuum energy by condensation of the tachyon field alone. While the action is non-polynomial, the multiscalar tachyon potential to any fixed level involves only a finite number of interactions. We compute this potential to level three, obtaining 85% of the expected vacuum energy, a result consistent with convergence that can also be viewed as a successful test of the string field theory. The resulting effective tachyon potential is bounded below and has two degenerate global minima. We calculate the energy density of the kink solution interpolating between these minima finding good agreement with the tension of the D-brane of one lower dimension. 
  Deformation quantization of bosonic strings is considered. We show that the light-cone gauge is the most convenient classical description to perform the quantization of bosonic strings in the deformation quantization formalism. Similar to the field theory case, the oscillator variables greatly facilitates the analysis. The mass spectrum, propagators and the Virasoro algebra are finally described within this deformation quantization scheme. 
  We study the ultraviolet asymptotics in non-simply laced affine Toda theories considering them as perturbed non-affine Toda theories, which possess the extended conformal symmetry. We calculate the reflection amplitudes, in non-affine Toda theories and use them to derive the quantization condition for the vacuum wave function, describing zero-mode dynamics. The solution of this quantization conditions for the ground state energy determines the UV asymptotics of the effective central charge. These asymptotics are in a good agreement with Thermodynamic Bethe Ansatz(TBA) results. To make the comparison with TBA possible, we give the exact relations between parameters of the action and masses of particles as well as the bulk free energies for non-simply laced affine Toda theories. 
  We discuss the duality between two type I compactifications to four dimensions and an heterotic construction with spontaneous breaking of the N=4 supersymmetry to N=2. This duality allows us to gain insight into the non-perturbative properties of these models. Through the analysis of the gravitational corrections, we then investigate the connections between four-dimensional, N=2 M-theory vacua constructed as orbifolds of type II, heterotic, and type I strings. 
  We investigate four-dimensional cosmological vacuum solutions derived from an effective action invariant under global SL(n,R) transformations. We find the general solutions for linear axion field perturbations about homogeneous dilaton-moduli-vacuum solutions for an SL(4,R)-invariant action and find the spectrum of super-horizon perturbations resulting from vacuum fluctuations in a pre big bang scenario. We show that for SL(n,R)-invariant actions with n>3 there exists a regime of parameter space of non-zero measure where all the axion field spectra have positive spectral tilt, as required if light axion fields are to provide a seed for anisotropies in the microwave background and large-scale structure in the universe. 
  Time-like geodesics in AdS_4, AdS_5 and AdS_7 are constructed geometrically and independently of choice of AdS coordinates from division algebra spinors of the corresponding AdS groups, explaining and generalising the construction by Claus et al. of AdS_5 twistors. 
  The duality between the Cartesian coordinates on the Minkowski space-time and the Dirac field is investigated. Two distinct possibilities to define this duality are shown to exist. In both cases, the equations satisfied by prepotentials are of second order. 
  We extend the well-known method of calculating bulk correlation functions of the conformal Ising model via bosonisation to situations with boundaries. Oshikawa and Affleck have found the boundary states of two decoupled Ising models in terms of the orbifold of a single free boson compactified on a circle of radius r=1; we adapt their results to include disorder operators. Using these boundary states we calculate the expectation value of a single disorder field on a cylinder with free boundary conditions and show that in the appropriate limits we recover the standard and frustrated partition functions. We also show how to calculate Ising correlation functions on the upper half plane. 
  The perturbed conformal field theories corresponding to the massive Symmetric Space sine-Gordon soliton theories are identified by calculating the central charge of the unperturbed conformal field theory and the conformal dimension of the perturbation. They are described by an action with a positive-definite kinetic term and a real potential term bounded from below, their equations of motion are non-abelian affine Toda equations and, moreover, they exhibit a mass gap. The unperturbed CFT corresponding to the compact symmetric space G/G_0 is either the WZNW action for G_0 or the gauged WZNW action for a coset of the form G_0/U(1)^p. The quantum integrability of the theories that describe perturbations of a WZNW action, named Split models, is established by showing that they have quantum conserved quantities of spin +3 and -3. Together with the already known results for the other massive theories associated with the non-abelian affine Toda equations, the Homogeneous sine-Gordon theories, this supports the conjecture that all the massive Symmetric Space sine-Gordon theories will be quantum integrable and, hence, will admit a factorizable S-matrix. The general features of the soliton spectrum are discussed, and some explicit soliton solutions for the Split models are constructed. In general, the solitons will carry both topological charges and abelian Noether charges. Moreover, the spectrum is expected to include stable and unstable particles. 
  We study the formation and stability of regular black holes by employing a thin shell approximation to the dynamics of collapsing magnetic monopoles. The core deSitter region of the monopole is matched across the shell to a Reissner-Nordstrom exterior. We find static configurations which are nonsingular black holes and also oscillatory trajectories about these static points that share the same causal structure. In these spacetimes the shell is always hidden behind the black hole horizon. We also find shell trajectories that pass through the asymptotically flat region and model collapse of a monopole to form a regular black hole. In addition there are trajectories in which the deSitter core encompasses a deSitter horizon and hence undergoes topological inflation. However, these always yield singular black holes and never have the shell passing through the aymptotically flat region. Although the regular black hole spacetimes satisfy the strong energy condition, they avoid the singularity theorems by failing to satisfy the genericity condition on the Riemann tensor. The regular black holes undergo a change in spatial topology in accordance with a theorem of Borde's. 
  A brief summary of the development of perturbative Chern-Simons gauge theory related to the theory of knots and links is presented. Emphasis is made on the progress achieved towards the determination of a general combinatorial expression for Vassiliev invariants. Its form for all the invariants up to order four is reviewed, and a table of their values for all prime knots with ten crossings is presented. 
  We prove mirror symmetry for supersymmetric sigma models on Kahler manifolds in 1+1 dimensions. The proof involves establishing the equivalence of the gauged linear sigma model, embedded in a theory with an enlarged gauge symmetry, with a Landau-Ginzburg theory of Toda type. Standard R -> 1/R duality and dynamical generation of superpotential by vortices are crucial in the derivation. This provides not only a proof of mirror symmetry in the case of (local and global) Calabi-Yau manifolds, but also for sigma models on manifolds with positive first Chern class, including deformations of the action by holomorphic isometries. 
  A non-perturbative confinement mechanism has been proposed to explain the fate of the unbroken gauge group on the world-volume of annihilating D-brane-anti-D-brane pairs. In this paper, we examine this phenomenon closely from several different perspectives. Existence of the confinement mechanism is most easily seen by noticing that the fundamental string emerges as the confined electric flux string at the end of the annihilation process. After reviewing the confinement proposal in general, this is shown explicitly in the D2-anti-D2 case in the M-theory limit. Finally, we address the crucial issue of whether and how confinement occurs in the weakly coupled limit of string theory. 
  A string in four dimensions is constructed by supplementing it with forty four Majorana fermions. The central charge is 26. The fermions are grouped in such a way that the resulting action is supersymmetric. The energy momentum and current generators satisfy the super-Virasoro algebra. The tachyonic ground state decouples from the physical states. GSO projections are necessary for proving modular invariance. Space-time supersymmetry provides reasons to discard the tachyons and is substantiated for modes of zero mass. The symmetry group of the model descends to the low energy standard model group $SU (3) \times SU_L (2) \times U_Y (1)$ through the Pati-Salam group. Left right symmetry is broken spontaneously and the mass of the tau neutrino is calculated to be about 1/25 electron volt. 
  We consider the universality class of the two-dimensional Tricritical Ising Model. The scaling form of the free-energy naturally leads to the definition of universal ratios of critical amplitudes which may have experimental relevance. We compute these universal ratios by a combined use of results coming from Perturbed Conformal Field Theory, Integrable Quantum Field Theory and numerical methods. 
  We study the coupled equations describing fluctuations of scalars and the metric about background solutions of N=8 gauged supergravity which are dual to boundary field theories with renormalization group flow. For the case of a kink solution with a single varying scalar, we develop a procedure to decouple the equations, and we solve them in particular examples. However, difficulties occur in the calculation of correlation functions from the fluctuations, presumably because the AdS/CFT correspondence has not yet been properly implemented in the coupled scalar-gravity sector. Some new examples of correlators of operators dual to simpler uncoupled bulk scalars are given and are satisfactory. As byproducts of our study we make some observations relevant to the stability of domain walls in the brane-world scenario and to the Hamilton-Jacobi formulation of holographic RG flows. 
  We study the moduli space C^2 of unitary two-dimensional conformal field theories with central charge c=2. We construct all the 28 nonexceptional nonisolated irreducible components of C^2 that may be obtained by an orbifold procedure from toroidal theories. The parameter spaces and partition functions are calculated explicitly, and all multicritical points and lines are determined. We show that all but four of the 28 irreducible components of C^2 corresponding to nonexceptional orbifolds are directly or indirectly connected to the moduli space of toroidal theories in C^2. We relate our results to those by Dixon, Ginsparg, Harvey on the classification of c=3/2 superconformal field theories and thereby give geometric interpretations to all nonisolated orbifolds discussed there. 
  We give a detailed study of the critical points of the potentials of the simplest non-trivial N=2 gauged Yang-Mills/Einstein supergravity theories with tensor multiplets. The scalar field target space of these examples is SO(1,1)XSO(2,1)/SO(2). The possible gauge groups are SO(2)XU(1)_R and SO(1,1)XU(1)_R, where U(1)_R is a subgroup of the R-symmetry group SU(2)_R, and SO(2) and SO(1,1) are subgroups of the isometry group of the scalar manifold. The scalar potentials of these theories consist of a contribution from the U(1)_R gauging and a contribution that is due to the presence of the tensor fields. We find that the latter contribution can change the form of the supersymmetric extrema from maxima to saddle points. In addition, it leads to novel critical points not present in the corresponding gauged Yang-Mills/Einstein supergravity theories without the tensor multiplets. For the SO(2)XU(1)_R gauged theory these novel critical points correspond to anti-de Sitter ground states. For the non-compact SO(1,1)XU(1)_R gauging, the novel ground states are de Sitter. The analysis of the critical points of the potential carries over in a straightforward manner to the generic family of N=2 gauged Yang-Mills/Einstein supergravity theories coupled to tensor multiplets whose scalar manifolds are of the form SO(1,1)XSO(n-1,1)/SO(n-1). 
  We show how to map the Belavin-Polyakov instantons of the O(3)-nonlinear $\sigma-$model to a dual theory where they then appear as nontopological solitons. They are stationary points of the Euclidean action in the dual theory, and moreover, the dual action and the O(3)-nonlinear $\sigma-$model action agree on shell. 
  We develop a systematic method for renormalizing the AdS/CFT prescription for computing correlation functions. This involves regularizing the bulk on-shell supergravity action in a covariant way, computing all divergences, adding counterterms to cancel them and then removing the regulator. We explicitly work out the case of pure gravity up to six dimensions and of gravity coupled to scalars. The method can also be viewed as providing a holographic reconstruction of the bulk spacetime metric and of bulk fields on this spacetime, out of conformal field theory data. Knowing which sources are turned on is sufficient in order to obtain an asymptotic expansion of the bulk metric and of bulk fields near the boundary to high enough order so that all infrared divergences of the on-shell action are obtained. To continue the holographic reconstruction of the bulk fields one needs new CFT data: the expectation value of the dual operator. In particular, in order to obtain the bulk metric one needs to know the expectation value of stress-energy tensor of the boundary theory. We provide completely explicit formulae for the holographic stress-energy tensors up to six dimensions. We show that both the gravitational and matter conformal anomalies of the boundary theory are correctly reproduced. We also obtain the conformal transformation properties of the boundary stress-energy tensors. 
  The cutoff scheme dependence in the several formulations of the Exact Renormalization Group (ERG) is investigated. It is shown that the cutoff scheme dependence of the Wilsonian effective action is regarded as a certain coordinate transformation on the theory space. From this observation the Wilsonian effective actions are found to suffer from strong dependence on the schemes even in the infra-red asymptotic region for massive theories. However there is no such scheme dependence in the one particle irreducible parts of them, which is called the effective average actions. We also derive the explicit form of the Polchinski RG equation in the sharp cutoff limit. Finally this equation is shown to be identical with the Wegner-Houghton RG equation. 
  Multicomplex numbers of order n have an associated trigonometry (multisine functions with (n-1) parameters) leading to a natural extension of the sine-Gordon model. The parameters are constrained from the requirement of local current conservation. In two dimensions for n < 6 known integrable models (deformed Toda and non-linear sigma, pure affine Toda...) with dual counterparts are obtained in this way from the multicomplex space MC itself and from the natural embedding $\MC_n \subset \MMC_m, n < m$. For $ n \ge 6$ a generic constraint on the space of parametersis obtained from current conservation at first order in the interaction Lagragien. 
  Taking advantage of the two-parameter central extension of the planar Galilei group, we construct a non relativistic particle model in the plane. Owing to the extra structure, the coordinates do not commute. Our model can be viewed as the non-relativistic counterpart of the relativistic anyon considered before by Jackiw and Nair. For a particle moving in a magnetic field perpendicular to the plane, the two parameters combine with the magnetic field to provide an effective mass. For vanishing effective mass the phase space admits a two-dimensional reduction, which represents the condensation to collective ``Hall'' motions and justifies the rule called ``Peierls substitution''. Quantization yields the wave functions proposed by Laughlin to describe the Fractional Quantum Hall Effect. 
  A topological model in three dimensions is proposed. It combines the Chern-Simons action with a BFK-model which was investigated recently by the authors of hep-th/9906146. The finiteness of the model to all orders of perturbation theory is shown in the framework of algebraic renormalization procedure. 
  A comparison is made between the thermodynamics of weakly and strongly coupled Yang-Mills with fixed angular momentum. The free energy of the strongly coupled Yang-Mills is calculated by using a dual supergravity description corresponding to a rotating black hole in an Anti de Sitter (AdS) background. All thermodynamic quantities are shown have the same ratio of 3/4 (independent of angular momentum) between strong and weak coupling. 
  We study the effective action associated to the Dirac operator in two dimensional non-commutative Field Theory. Starting from the axial anomaly, we compute the determinant of the Dirac operator and we find that even in the U(1) theory, a Wess-Zumino-Witten like term arises. 
  The tachyonic instability of the open bosonic string is analyzed using the level truncation approach to string field theory. We have calculated all terms in the cubic action of the string field theory describing zero-momentum interactions of up to level 20 between scalars of level 10 or less. These results are used to study the tachyon effective potential and the nonperturbative stable vacuum. We find that the energy gap between the unstable and stable vacua converges much more quickly than the coefficients of the effective tachyon potential. By including fields up to level 10, 99.91% of the energy from the bosonic D-brane tension is cancelled in the nonperturbative stable vacuum. It appears that the perturbative expansion of the effective tachyon potential around the unstable vacuum has a small but finite radius of convergence. We find evidence for a critical point in the tachyon effective potential at a small negative value of the tachyon field corresponding to this radius of convergence. We study the branch structure of the effective potential in the vicinity of this point and speculate that the tachyon effective potential is globally nonnegative. 
  A geometrical interpretation of the consistent and covariant chiral anomaly is done in the space-time respective Hamiltonian framework. 
  One of the most efficient methods to obtain the vacuum expectation values for the physical observables in the Casimir effect is based on the using the Abel-Plana summation formula. This allows to derive the regularized quantities by manifestly cutoff independent way and to present them in the form of strongly converging integrals. However the applications of Abel- Plana formula in usual form is restricted by simple geometries when the eigenmodes have a simple dependence on quantum numbers. The author generalized the Abel-Plana formula which essentially enlarges its application range. Based on this generalization, formulae have been obtained for various types of series over the zeros of some combinations of Bessel functions and for integrals involving these functions. It have been shown that these results generalize the special cases existing in literature. Further the derived summation formulae have been used to summarize series arising in the mode summation approach to the Casimir effect for spherically and cylindrically symmetric boundaries. This allows to extract the divergent parts from the vacuum expectation values for the local physical observables in the manifestly cutoff independent way. The present paper reviews these results. Some new considerations are added as well. 
  Four dimensional reflexive polyhedra encode the data for smooth Calabi-Yau threefolds that are hypersurfaces in toric varieties, and have important applications both in perturbative and in non-perturbative string theory. We describe how we obtained all 473,800,776 reflexive polyhedra that exist in four dimensions and the 30,108 distinct pairs of Hodge numbers of the resulting Calabi-Yau manifolds. As a by-product we show that all these spaces (and hence the corresponding string vacua) are connected via a chain of singular transitions. 
  We compute the tree level 4-particle bosonic scattering amplitudes in D=11 supergravity. By construction, they are part of a linearized supersymmetry-, coordinate- and 3-form gauge-invariant. While this on-shell invariant is nonlocal, suitable SUSY-preserving differentiations turn it into a local one with correct dimension to provide a natural lowest (two-loop) order counterterm candidate. Its existence shows explicitly that no symmetries protect this ultimate supergravity from the nonrenormalizability of its lower-dimensional counterparts. 
  We propose a universal expression for the moduli metric of a class of four- and five-dimensional black holes which preserve at least four supersymmetries. These include the black holes that are associated with various intersecting branes in ten and eleven dimensions, the electrically charged black holes of N=2 D=5 and N=2 D=4 supergravities with any number of vector multiplets, and dyonic black holes of N=2 D=4 supergravity. The moduli metric of electrically charged N=2 D=4 black holes coupled to any number of vector multiplets is explicitly computed. We also investigate the superconformal symmetries of the black hole moduli spaces for small black hole separations. 
  The spherical field formalism---a nonperturbative approach to quantum field theory---was recently introduced and applied to phi^4 theory in two dimensions. The spherical field method reduces a quantum field theory to a finite-dimensional quantum mechanical system by expanding field configurations in terms of spherical partial wave modes. We extend the formalism to phi^4 theory in three dimensions and demonstrate the application of the method by analyzing the phase structure of this theory. 
  Supergravity solutions related to large N SU(N) pure gauge theories with eight supercharges have recently been shown to give rise to an ``enhancon'', a new type of hypersurface made of D-branes. We show that enhancons also arise in similar situations pertaining to SO(2N+1), USp(2N) and SO(2N) gauge theories, using orientifolds. Enhancons therefore appear to come in types A, B, C, and D. The latter three differ globally from type A by having an extra Z_2 identification, and are distinguished locally by their subleading behaviour in large N. We focus mainly on 2+1 dimensional gauge theory, where a relation to M-theory and the Atiyah-Hitchin and Taub-NUT manifolds enables the construction of the smooth supergravity solution and the study of some of the 1/N corrections. The role of the enhancon in eleven dimensional supergravity is also uncovered. There is a close relation to certain multi-monopole moduli space problems. 
  The general solution of the anomaly consistency condition (Wess-Zumino equation) has been found recently for Yang-Mills gauge theory. The general form of the counterterms arising in the renormalization of gauge invariant operators (Kluberg-Stern and Zuber conjecture) and in gauge theories of the Yang-Mills type with non power counting renormalizable couplings has also been worked out in any number of spacetime dimensions. This Physics Report is devoted to reviewing in a self-contained manner these results and their proofs. This involves computing cohomology groups of the differential introduced by Becchi, Rouet, Stora and Tyutin, with the sources of the BRST variations of the fields ("antifields") included in the problem. Applications of this computation to other physical questions (classical deformations of the action, conservation laws) are also considered. The general algebraic techniques developed in the Report can be applied to other gauge theories, for which relevant references are given. 
  This is an informal introduction to the concept of reflexive polyhedra and some of their most important applications in perturbative and non-perturbative string physics. Following the historical development, topics like mirror symmetry, gauged linear sigma models, and the geometrical structures relevant to string and F-theory dualities are discussed. Finally some recent developments concerning the classification of reflexive polyhedra are mentioned. 
  The general model of an arbitrary spin massive particle in any dimensional space-time is derived on the basis of Kirillov - Kostant - Souriau approach. It is shown that the model allows consistent coupling to an arbitrary background of electromagnetic and gravitational fields. 
  We consider the dual Yang-Mills theory which shows some kind of confinement at large distances. In the static system of the test color charges an analytic expression for the string tension is derived. 
  We review some applications of Type 0 string theory in the context of the AdS/CFT correspondence. 
  We discuss general spinning p-branes of string and M-theory and use their thermodynamics along with the correspondence between near-horizon brane solutions and field theories with 16 supercharges to describe the thermodynamic behavior of these theories in the presence of voltages under the R-symmetry. The thermodynamics is used to provide two pieces of evidence in favor of a smooth interpolation function between the free energy at weak and strong coupling of the field theory. (i) A computation of the boundaries of stability shows that for the D2, D3, D4, M2 and M5-branes the critical values of Omega/T in the two limits are remarkably close and (ii) The tree-level R^4 corrections to the spinning D3-brane generate a decrease in the free energy at strong coupling towards the weak coupling result. We also comment on the generalization to spinning brane bound states and their thermodynamics, which are relevant for non-commutative field theories. 
  We introduce a new diagonalization method called quasi-sparse eigenvector diagonalization which finds the most important basis vectors of the low energy eigenstates of a quantum Hamiltonian. It can operate using any basis, either orthogonal or non-orthogonal, and any sparse Hamiltonian, either Hermitian, non-Hermitian, finite-dimensional, or infinite-dimensional. The method is part of a new computational approach which combines both diagonalization and Monte Carlo techniques. 
  In this paper we review some properties of BPS black holes of supergravities with n=32,16 supersymmetries. The BPS condition, a condition on the eigenvalues of the central charge matrix, can be shown to be U-duality invariant. We explicitly work out D=4, N=8 and D=5, N=4 supergravities. 
  We construct Z_M, M= 2, 3, 4, 6 orbifold models of the N=2 superconformal field theories with central charge c=3. Then we check the description of the Z_3, Z_4 and Z_6 orbifolds by the N=2 superconformal Landau-Ginzburg models with c=3, by comparing the spectrum of chiral fields, in particular the Witten index Tr(-1)^F. 
  Using the local potential approximation of the exact renormalization group (RG) equation, we show the various domains of values of the parameters of the O(1)-symmetric scalar Hamiltonian. In three dimensions, in addition to the usual critical surface $S_{c}$ (attraction domain of the Wilson-Fisher fixed point), we explicitly show the existence of a first-order phase transition domain $S_{f}$ separated from $S_{c}$ by the tricritical surface $S_{t}$ (attraction domain of the Gaussian fixed point). $S_{f}$ and $S_{c}$ are two distinct domains of repulsion for the Gaussian fixed point, but $S_{f}$ is not the basin of attraction of a fixed point. $S_{f}$ is characterized by an endless renormalized trajectory lying entirely in the domain of negative values of the $\phi ^{4}$-coupling. This renormalized trajectory exists also in four dimensions making the Gaussian fixed point ultra-violet stable (and the $\phi_{4}^{4}$ renormalized field theory asymptotically free but with a wrong sign of the perfect action). We also show that very retarded classical-to-Ising crossover may exist in three dimensions (in fact below four dimensions). This could be an explanation of the unexpected classical critical behavior observed in some ionic systems. 
  We present the construction of the candidate conformal field theories dual to $AdS_4$ non-maximally supersymmetric compactifications of 11D supergravity. We compare the spectra of the two theories and discuss the realization of the baryonic symmetries. Finally we comment on the presence in the spectrum of long multiplets with rational energies, trying to explain their existence. 
  When two-dimensional Anti-de Sitter space (AdS_2) is endowed with a non-constant dilaton the origin of the central charge in the Virasoro algebra generating the asymptotic symmetries of AdS_2 can be traced back to the breaking of the SL(2,R) isometry group of AdS_2. We use this fact to clarify some controversial results appeared in the literature about the value of the central charge in these models. 
  This paper has been withdrawn. 
  String theory usually represents quantum black holes as systems whose statistical mechanics reproduces Hawking's thermodynamics in a very satisfactory way. Complicated brane theoretical models are worked out, as quantum versions of Supergravity solutions. These models are then assumed to be in thermal equilibrium: this is a little cheating, because one is looking for an explanation of the seeming thermodynamical nature of black holes, so they cannot be {\it assumed} to be finite temperature systems! In the model presented here, the black body spectrum arises with no statistical hypothesis as an approximation of the unitary evolution of microscopic black holes, which are always described by a 1+1 conformal field theory, characterized by some Virasoro algebra. At the end, one can state that {\it the Hawking-thermodynamics of the system is a by-product of the algebraic Virasoro-symmetric nature of the event horizon}. This is {\it the central result} of the present work. 
  We present a particular approach to the non-equilibrium dynamics of quantum field theory. This approach is based on the Jaynes-Gibbs principle of the maximal entropy and its implementation, throghout the initial value data, into the dynamical equations for Green's functions. 
  We discuss the confining features of the Schwinger model on the Poincare half plane. We show that despite the fact that the expectation value of the large Wilson loop of massless Schwinger model displays the perimeter behavior, the system can be in confining phase due to the singularity of the metric at horizontal axis. It is also shown that in the quenched Schwinger model, the area dependence of the Wilson loop, in contrast to the flat case, is a not a sign of confinement and the model has a finite energy even for large external charges separation. The presence of dynamical fermions can not modify the screening or the confining behavior of the system. Finally we show that in the massive Schwinger model, the system is again in screening phase. The zero curvature limit of the solutions is also discussed. 
  A brane world in the presence of a bulk black hole is constructed. The brane tension is fine tuned in terms of the black hole mass and cosmological constant. Gravitational perturbations localized on the brane world are discussed. 
  We construct a classical field theory action which upon quantization via the functional integral approach, gives rise to a consistent Dirac-string independent quantum field theory. The approach entails a systematic derivation of the correlators of all gauge invariant observables, and also of charged dyonic fields. Manifest SO(2)-duality invariance and Lorentz invariance are ensured by the PST-approach. 
  After a pedagogical overview of the present status of High-Energy Physics, some problems concerning physics at the Planck scale are formulated, and an introduction is given to a notion that became known as ``the holographic principle" in Planck scale physics, which is arrived at by studying quantum mechanical features of black holes. 
  Without invalidating quantum mechanics as a principle underlying the dynamics of a fundamental theory, it is possible to ask for even more basic dynamical laws that may yield quantum mechanics as the machinery needed for its statistical analysis. In conventional systems such as the Standard Model for quarks and leptons, this would lead to hidden variable theories, which are known to be plagued by problems such as non-locality. But Planck scale physics is so different from field theories in some flat background space-time that here the converse may be the case: we speculate that causality and locality can only be restored by postulating a deterministic underlying theory. A price to be paid may be that the underlying theory exhibits dissipation of information. 
  We give a survey of selected topics in noncommutative geometry, with some emphasis on those directly related to physics, including our recent work with Dirk Kreimer on renormalization and the Riemann-Hilbert problem. We discuss at length two issues. The first is the relevance of the paradigm of geometric space, based on spectral considerations, which is central in the theory. As a simple illustration of the spectral formulation of geometry in the ordinary commutative case, we give a polynomial equation for geometries on the four dimensional sphere with fixed volume. The equation involves an idempotent e, playing the role of the instanton, and the Dirac operator D. It expresses the gamma five matrix as the pairing between the operator theoretic chern characters of e and D. It is of degree five in the idempotent and four in the Dirac operator which only appears through its commutant with the idempotent. It determines both the sphere and all its metrics with fixed volume form.   We also show using the noncommutative analogue of the Polyakov action, how to obtain the noncommutative metric (in spectral form) on the noncommutative tori from the formal naive metric. We conclude on some questions related to string theory. 
  We consider noncommutative gauge theory with Dirac or Majorana fermions in odd dimensional spacetime and compute the induced noncommutative Chern-Simons action generated at 1-loop. We observe that the Chern-Simons term induced by a Dirac fermion has a smooth limit when $\th ---> 0$, but there is a finite jump for the Chern-Simons term induced by a Majorana spinor. The induced Chern-Simons action from a Majorana spinor is nonvanishing even in the $\th ---> 0$ limit, a discontinuity that shares a similar characteristic as the UV/IR singularity discovered originally by Minwalla, Raamsdonk and Seiberg. Properties of the noncommutative WZW action are also discussed. 
  The "measurability" of the non-minimal coupling is discussed in the context of the effective field theory of gravity. Although there is no obvious motive for excluding a non-minimal scalar coupling from the theory, we conclude that for reasonable values of the coupling constant it makes only a very small correction. 
  The zero momentum sectors in effective theories of QCD coupled to pseudoreal (two colors) and real (adjoint) quarks have alternative descriptions in terms of chiral orthogonal and symplectic ensembles of random matrices. Using this correspondence, we compute correlation functions of Dirac operator eigenvalues within a sector with an arbitrary topological charge in a presence of finite quark masses of the order of the smallest Dirac eigenvalue. These novel correlation functions, expressed in terms of Pfaffians, interpolate between known results for the chiral and quenched limits as quark masses vary. 
  We introduce "chain by chain" method for constructing the constraint structure of a system possessing both first and second class constraints. We show that the whole constraints can be classified into completely irreducible first or second class chains. We found appropriate redefinition of second class constraints to obtain a symplectic algebra among them. 
  We study the general non-minimally coupled charged massive spin 3/2 model both for its low energy phenomenological properties and for its unitarity, causality and degrees of freedom behaviour. When the model is viewed as an effective theory, its parameters (after ensuring the correct excitation count) are related to physical characteristics, such as the magnetic moment g factor, by means of low energy theorems. We also provide the corresponding higher spin generalisation. Separately, we consider both low and high energy unitarity, as well as the causality aspects of our models. None (including truncated N=2 supergravity) is free of the minimal model's acausality. 
  We study the notion of twisted bundles on noncommutative space. Due to the existence of projective operators in the algebra of functions on the noncommutative space, there are twisted bundles with non-constant dimension. The U(1) instanton solution of Nekrasov and Schwarz is such an example. As a mathematical motivation for not excluding such bundles, we find gauge transformations by which a bundle with constant dimension can be equivalent to a bundle with non-constant dimension. 
  We study the ultraviolet properties of the supersymmetric CP^{N-1} sigma model in three dimensions to next-to-leading order in the 1/N expansion. We calculate the \beta-function to this order and verify that it has no next-to-leading order corrections. 
  The quantum-mechanical D-dimensional inverse square potential is analyzed using field-theoretic renormalization techniques. A solution is presented for both the bound-state and scattering sectors of the theory using cutoff and dimensional regularization. In the renormalized version of the theory, there is a strong-coupling regime where quantum-mechanical breaking of scale symmetry takes place through dimensional transmutation, with the creation of a single bound state and of an energy-dependent s-wave scattering matrix element. 
  Kink dynamics in spatially discrete nonlinear Klein-Gordon systems is considered. For special choices of the substrate potential, such systems support continuous translation orbits of static kinks with no (classical) Peierls-Nabarro barrier. It is shown that these kinks experience, nevertheless, a lattice-periodic confining potential, due to purely quantum effects anaolgous to the Casimir effect of quantum field theory. The resulting ``quantum Peierls-Nabarro potential'' may be calculated in the weak coupling approximation by a simple and computationally cheap numerical algorithm, which is applied, for purposes of illustration, to a certain two-parameter family of substrates. 
  We compute the spectrum of primordial gravitational wave perturbations in open de Sitter spacetime. The background spacetime is taken to be the continuation of an O(5) symmetric instanton saddle point of the Euclidean no boundary path integral. The two-point tensor fluctuations are computed directly from the Euclidean path integral. The Euclidean correlator is then analytically continued into the Lorentzian region where it describes the quantum mechanical vacuum fluctuations of the graviton field. Unlike the results of earlier work, the correlator is shown to be unique and well behaved in the infrared. We show that the infrared divergence found in previous calculations is due to the contribution of a discrete gauge mode inadvertently included in the spectrum. 
  We discuss the two- and three-point functions of the SO(5) R-current of the (2,0) tensor multiplet in d=6, using AdS/CFT correspondence as well as a free field realization. The results obtained via AdS/CFT correspondence coincide with the ones from a free field calculation up to an overall 4N^3 factor. This is the same factor found recently in studies of the two- and three-point functions of the energy momentum tensor in the (2,0) theory. We also connect our results to the trace anomaly in d=6 in the presence of external vector fields and briefly discuss their implications. 
  We show that there is a duality exchanging noncommutativity and non-trivial statistics for quantum field theory on R^d. Employing methods of quantum groups, we observe that ordinary and noncommutative R^d are related by twisting. We extend the twist to an equivalence for quantum field theory using the framework of braided quantum field theory. The twist exchanges both commutativity with noncommutativity and ordinary with non-trivial statistics. The same holds for the noncommutative torus. 
  In these lectures we review the basic ideas of perturbative and non-perturbative string theory. On the non-perturbative side we give an introduction to D-branes and string duality. The elementary concepts of non-BPS branes and noncommutative gauge theories are also discussed. 
  We derive the graviton propagator on the brane for theories with quasi-localized gravity. In these models the ordinary 4D graviton is replaced by a resonance in the spectrum of massive Kaluza-Klein modes, which can decay into the extra dimension. We find that the effects of the extra polarization in the massive graviton propagator is exactly cancelled by the bending of the brane due to the matter sources, up to small corrections proportional to the width of the resonance. Thus at intermediate scales the classic predictions of Einstein's gravity are reproduced in these models to arbitrary precision. 
  Descriptions of the ground state in unbroken gauge theories with charged particles are discussed. In particular it is shown that the on-shell Green's functions and S-matrix elements corresponding to the scattering of these variables in QED are free of soft and phase infra red divergences and that these variables may be multiplicatively renormalised. 
  We review the boundary state description of the non-BPS D-branes in the type I string theory and show that the only stable configurations are the D-particle and the D-instanton. We also compute the gauge and gravitational interactions of the non-BPS D-particles and compare them with the interactions of the dual non-BPS particles of the heterotic string finding complete agreement. In this way we provide further dynamical evidence of the heterotic/type I duality. 
  We show how non-near horizon, non-dilatonic $p$-brane theories can be obtained from two embedding constraints in a flat higher dimensional space with 2 time directions. In particular this includes the construction of D3 branes from a flat 12-dimensional action, and M2 and M5 branes from 13 dimensions. The worldvolume actions are found in terms of fields defined in the embedding space, with the constraints enforced by Lagrange multipliers. 
  We investigate the D-brane contents of asymmetric orbifolds. Using T-duality we find that the consistent description of open strings in asymmetric orbifolds requires to turn on background gauge fields on the D-branes. We derive the corresponding noncommutative geometry arising on such D-branes with mixed Neumann-Dirichlet boundary conditions directly by applying an asymmetric rotation to open strings with pure Dirichlet or Neumann boundary conditions. As a concrete application of our results we construct asymmetric type I vacua requiring open strings with mixed boundary conditions for tadpole cancellation. 
  We find some lifts to M theory of orientifold and orbifold planes including the O1, O3 and O5 planes of Type IIB and their transformations under SL(2,Z). The possible discrete torsion variants (or K theory classes) are explored, and are interpreted as arising from brane intersections with planes. We find new variants of the O0 and of an orbifold line (OF1) and determine their tensions in some cases. A systematic review of orientifolds, M orientifolds, and known M lifts, with some new clarifications is included together with a discussion of the role of T duality. 
  We discuss recent results on two-point functions of chiral primary operators in {\cal N}=4 SU(N) supersymmetric Yang-Mills theory. Our results give further support to the belief that such correlators are not renormalized to all orders in g and to all orders in N. 
  In a symplectic framework, the infinitesimal action of symplectomorphisms together with suitable reparametrizations of the two dimensional complex base space generate some type of W-algebras. It turns out that complex structures parametrized by Beltrami differentials play an important role in this context. The construction parallels very closely two dimensional Lagrangian conformal models where Beltrami differentials are fundamental. 
  The existence of the vector supersymmetry is analysed within the context of the finite temperature Chern-Simons theory. 
  We study two-dimensional conformal field theories generated from a ``symplectic fermion'' - a free two-component fermion field of spin one - and construct the maximal local supersymmetric conformal field theory generated from it. This theory has central charge c=-2 and provides the simplest example of a theory with logarithmic operators.   Twisted states with respect to the global SL(2,C)-symmetry of the symplectic fermions are introduced and we describe in detail how one obtains a consistent set of twisted amplitudes. We study orbifold models with respect to finite subgroups of SL(2,C) and obtain their modular invariant partition functions. In the case of the cyclic orbifolds explicit expressions are given for all two-, three- and four-point functions of the fundamental fields. The C_2-orbifold is shown to be isomorphic to the bosonic local logarithmic conformal field theory of the triplet algebra encountered previously. We discuss the relation of the C_4-orbifold to critical dense polymers. 
  The local covariant continuum action of an SU(2) gauge theory in covariant Abelian gauges is investigated. It describes the critical limit of an Abelian Lattice Gauge Theory (LGT) with an equivariant BRST-symmetry. This Abelian LGT has previously been proven to be physically equivalent to the SU(2)-LGT. Renormalizability requires a quartic ghost interaction in these nonlinear gauges (also in maximal Abelian gauge). Arguments that a certain global SL(2,R) symmetry is dynamically broken by ghost-antighost condensation in a BCS-like mechanism are presented. The scenario can be viewed as a dynamical Higgs mechanism in the adjoint that gives massive off-diagonal gluons and a BRST quartet of Goldstone bosonsthat decouples from physical observables. The gap parameter is related to the expectation value of the trace anomaly and the consistency of this scenario with the Operator Product Expansion is discussed. 
  We describe numerical methods for constructing lump solutions in open string field theory. According to Sen, these lumps represent lower dimensional Dp-Branes and numerical evaluation of their energy can be compared with the expected value for the tension. We take particular care of all higher derivative terms inherent in Witten's version of open string field theory. The importance of these terms for off shell phenomena is argued in the text. Detailed numerical calculations done for the case of general $p$ brane show very good agreement with Sen's conjectured value. This gives credence to the conjecture itself and establishes further the usefulness of Witten's version of SFT . 
  In these lectures we describe the attempt to extract the expectation values of Wilson loops from the string/gauge correspondence. We start with the original calculation in $AdS_5$. It is then extended to the non-conformal background of $D_p$ in the near horizon limit. We discuss the computation at finite temperature. Supergravity models that admit confinement in 3d and 4d are described. A theorem that determines the classical values of loops associated with a generalized background is derived.In particular we determine sufficient conditions for confining behavior. We then apply the theorem to various string models including type 0 ones. We describe an attempt to go beyond the classical string picture by incorporating quadratic quantum fluctuations. In particular we address the BPS configuration of a single quark, the supersymmetric determinant of $AdS_5\times S^$ and a setup that corresponds to a confining gauge theory. We determine the form of the Wilson loop for actions that include non trivial $B_{\mu\nu}$ field. The issue of an exact determination of the value of the stringy Wilson loop is discussed. We end with a brief review of the baryons from the string/gauge correspondence Lectures presented in the ``Advanced School on Supersymmetry in the theories of fields, strings and branes'' Santiago de Compostela-99. 
  We investigate the classical geometry generated by a stable non-BPS D-particle. We consider the boundary state of a stable non-BPS D-particle in the covariant formalism in the type IIB theory orbifolded by (-1)^{F_L} I_4. We calculate the scattering amplitude between two D-particles in the non-compact and compact orbifold and analyse the short and long distance behaviour. At short distances we find no force at order $v^2$ for any radius, and at the critical radius we find a BPS-like behaviour up to $v^4$ corrections for long and short distances. Projecting the boundary state on the massless states of the orbifold closed string spectrum we obtain the large distance behaviour of the classical solution describing this non-BPS D-particle in the non-compact and compact cases. By using the non-BPS D-particle as a probe of the background geometry of another non-BPS D-particle, we recover the no-force condition at the critical radius and the $v^2$ behaviour of the probe. Moreover, assuming that the no-force persists for the complete geometry we derive part of the classical solution for the non-BPS D-particle. 
  In this paper we present a construction of a new class of explicit solutions to the WDVV (or associativity) equations. Our construction is based on a relationship between the WDVV equations and Whitham (or modulation) equations. Whitham equations appear in the perturbation theory of exact algebro-geometric solutions of soliton equations and are defined on the moduli space of algebraic curves with some extra algebro-geometric data. It was first observed by Krichever that for curves of genus zero the tau-function of a ``universal'' Whitham hierarchy gives a solution to the WDVV equations. This construction was later extended by Dubrovin and Krichever to algebraic curves of higher genus. Such extension depends on the choice of a normalization for the corresponding Whitham differentials. Traditionally only complex normalization (or the normalization w.r.t. a-cycles) was considered. In this paper we generalize the above construction to the real-normalized case. 
  We present a new and asymmetric N=4 superconformal algebra for arbitrary central charge, thus completing our recent work on its classical analogue with vanishing central charge. Besides the Virasoro generator and 4 supercurrents, the algebra consists of an internal SL(2) \otimes U(1) Kac-Moody algebra in addition to two spin 1/2 fermions and a bosonic scalar. The algebra is shown to be invariant under a linear twist of the generators, except for a unique value of the continuous twist parameter. At this value, the invariance is broken and the algebra collapses to the small N=4 superconformal algebra. In the context of string theory, the asymmetric N=4 superconformal algebra is provided with an explicit construction on the boundary of AdS_3, and is induced by an affine SL(2|2) current superalgebra residing on the world sheet. Substituting the world sheet SL(2|2) by the coset SL(2|2)/U(1) results in the small N=4 superconformal algebra on the boundary of AdS_3. 
  We discuss the classical solutions of the equations of motion and the possible boundary condition for a bosonic string with Kalb-Ramond background in AdS3. It turns out that there exists three different physical sectors and that it is also possible to describe the motion on an extremal black hole background. The existence of three sectors clearly shows how one of the spectra proposed is incomplete. We consider also the classical 'canonical' transformation which maps the string fields to the classical Wakimoto ones. It turns however out that the Wakimoto fields are not free because of the boundary conditions and in order to have the usual mode expansion with reasonable behaviour under complex conjugation it is necessary to consider the complexification of AdS3 and then add some constraints. Furthermore they cover only half AdS3 and we need different patches to cover the whole space and to make the above mentioned transformation really canonical. 
  We explain how D-branes on group manifolds are stabilized against shrinking by quantized worldvolume U(1) fluxes. Starting from the Born-Infeld action in the case of the SU(2) group manifold we derive the masses, multiplicities and spectrum of small fluctuations of these branes, and show that they agree exactly with the predictions of conformal field theory, to all orders in the $\alpha^\prime$ expansion. We discuss the generalization to other groups and comment on an apparent paradox: why are the `RR charges' of these branes not quantized? 
  It is shown that a nonlinear derivative-dependent transformation of gravity fields changes correlation functions in a boundary CFT, and, therefore, corresponds to a change of a basis of operators in the CFT. It is argued that only non-renormalized structures in correlation functions can be changed by such a field transformation, and that the study of the response of correlation functions to gravity field transformations allows one to find them. In the case of 4-point functions of CPOs in SYM_4 several non-renormalized structures are found, including the extremal and subextremal ones. It is also checked that quartic couplings of scalar fields s^I that are dual to extended chiral primary operators vanish in the subextremal case, as dictated by the non-renormalization theorem for the subextremal 4-point functions and the AdS/CFT correspondence. 
  The `strong-coupling' perturbation theory over the inverse interaction constant $1/g$ near the nontrivial solution of Lagrange equation is formulated. The ordinary `week-coupling' perturbation theory over $g$ is described also to compare both perturbation theories. The `strong-coupling' perturbation theory is developed by unitary mapping of the quantum dynamics into the space with local coordinates of (action, angle)-type. 
  If background fields are soft on the scale set by mass of the particle involved, a reliable approximation to the field-theoretic one-loop effective action is obtained by a systematic large mass expansion involving higher-order Seeley-DeWitt coefficients. Moreover, if the small mass limit of the effective action in a particular background has been found by some other means, the two informations may be used to infer the corresponding result for {\em arbitrary} mass values. This method is used to estimate the one-loop contribution to the QCD vacuum tunneling amplitude by quarks of arbitrary mass. 
  The U(1) gauged version of the Strominger-Vafa five dimensional N=2 supergravity with one vector multiplet is obtained via dimensional reduction from the N=1 ten dimensional supergavity. Using such explicit relation between the gauged supergravity theory and ten dimensional supergravity, all known solutions of the five dimensional theory can be lifted up to ten-dimensions. The eleven dimensional solutions can also obtained by lifting the ten-dimensional solutions. 
  Black hole no-hair theorems are proven using inequalities that govern the radial dependence of spherically symmetric configurations of matter fields. In this paper, we analyze the analogous inequalities for geometries dual to renormalization group flows via the AdS/CFT correspondence. These inequalities give much useful information about the qualitative properties of such flows. For Poincare invariant flows, we show that generic flows of relevant or irrelevant operators lead to singular geometries. For the case of irrelevant operators, this leads to an apparent conflict with Polchinski's decoupling theorem, and we offer two possible resolutions to this problem. 
  String representation of the Wilson loop in 3D Abelian-projected SU(3)-gluodynamics is constructed in the approximation that Abelian-projected monopoles form a gas. Such an assumption is much weaker than the standard one, demanding the monopole condensation. It is demonstrated that the summation over world sheets, bounded by the contour of the Wilson loop, is realized by the summation over branches of a certain effective multivalued potential of the monopole densities. Finally, by virtue of the so-constructed representation of the Wilson loop in terms of the monopole densities, this quantity is evaluated in the approximation of a dilute monopole gas, which makes confinement in the model under study manifest. 
  The {\em cutting and sewing} procedure is used for getting two-loop order Feynman diagrams of $\Phi^{4}$-theory with an internal SU(N) symmetry group, starting from tachyon amplitudes of the open bosonic string theory. In a suitably defined field theory limit, we reproduce the field theory amplitudes properly normalized and expressed in the Schwinger parametrization. 
  In the framework of a five-dimensional three-brane model with quasi-localized gravitons we evaluate metric perturbations induced on the positive tension brane by matter residing thereon. We find that at intermediate distances, the effective four-dimensional theory coincides, up to small corrections, with General Relativity. This is in accord with Csaki, Erlich and Hollowood and in contrast to Dvali, Gabadadze and Porrati. We show, however, that at ultra-large distances this effective four-dimensional theory becomes dramatically different: conventional tensor gravity changes into scalar anti-gravity. 
  In hep-th/9910245, Witten and Yau consider the AdS/CFT correspondence in the context of a Riemannian Einstein manifold $M^{n+1}$ of negative Ricci curvature which admits a conformal compactification with conformal boundary $N^n$. They prove that if the conformal class of the boundary contains a metric of positive scalar curvature, then $M$ and $N$ have several desirable properties: (1) $N$ is connected, (2) the $n$th homology of the compactified $M$ vanishes, and (3) the fundamental group of $M$ is "bounded by" that of $N$. Here it is shown that all of these results extend to the case where the conformal class of the boundary contains a metric of nonnegative scalar curvature. (The case of zero scalar curvature is of interest as it is borderline for the stability of the theory.) The proof method used here is different from, and in some sense dual to, that used by Witten and Yau. While their method involves minimizing the co-dimension one brane action on $M$, and requires the machinery of geometric measure theory, the main arguments presented here use only geodesic geometry. 
  A construction of $W$-symmetries is given only in terms of the nonlocal fields (parafermions ${\ps}_{\al}$), which take values on the homogeneous space $G/U(1)^r$, where $G$ is a simply connected compact Lie group manifold (its accompanying Lie algebra ${\cal G}$ is a simple one of rank $r$). Only certain restriction of the root set of Lie algebra on which the parafermionic fields take values are satisfied, then a consistent and non-trivial extension of the stress momentum tensor may exist. For arbitrary simple-laced algebras, i.e. the $A-D-E$ cases, a more detailed discussion is given. The OPE of spin three primary field are calculated, in which a primary field with spin four is emerging. 
  This article seeks to relate a recent proposal for the association of a covariant Field Theory with a string or brane Lagrangian to the Hamilton-Jacobi formalism for strings and branes. It turns out that since in this special case, the Hamiltonian depends only upon the momenta of the Jacobi fields and not the fields themselves, it is the same as a Lagrangian, subject to a constancy constraint. We find that the associated Lagrangians for strings or branes have a covariant description in terms of the square root of the same Lagrangian. If the Hamilton-Jacobi function is zero, rather than a constant, then it is in in one dimension lower, reminiscent of the `holographic' idea. In the second part of the paper, we discuss properties of these Lagrangians, which lead to what we have called `Universal Field Equations', characteristic of covariant equations of motion. 
  We compute the renormalized four-point coupling in the 2d Ising model using transfer-matrix techniques. We greatly reduce the systematic uncertainties which usually affect this type of calculations by using the exact knowledge of several terms in the scaling function of the free energy. Our final result is g4=14.69735(3). 
  We discuss the role of the dilaton tadpole in the holographic description of non-supersymmetric gauge theories that are conformal in the planar limit. On the string theory side, the presence of the dilaton tadpole modifies the AdS background inducing a logarithmic running for the radius and the dilaton. Using the holographic prescription we compute the Wilson loop on the gravity side and find a smooth interpolating potential between asymptotic freedom and confinement, as expected from field theory. 
  We derive short UIR's of the OSp(8/4,R) superalgebra of 3d N=8 superconformal field theories by the requirement that the highest weight states are annihilated by a subset of the super-Poincare odd generators. We then find a superfield realization of these BPS saturated UIR's as "composite operators" of the two basic ultrashort "supersingleton" multiplets. These representations are the AdS4 analogue of BPS states preserving different fractions of supersymmetry and are therefore suitable to classify perturbative and non-perturbative excitations of M-theory compactifications. 
  We study a Randall-Sundrum cosmological scenario consisting of a domain wall in anti-de Sitter space with a strongly coupled large $N$ conformal field theory living on the wall. The AdS/CFT correspondence allows a fully quantum mechanical treatment of this CFT, in contrast with the usual treatment of matter fields in inflationary cosmology. The conformal anomaly of the CFT provides an effective tension which leads to a de Sitter geometry for the domain wall. This is the analogue of Starobinsky's four dimensional model of anomaly driven inflation. Studying this model in a Euclidean setting gives a natural choice of boundary conditions at the horizon. We calculate the graviton correlator using the Hartle-Hawking ``No Boundary'' proposal and analytically continue to Lorentzian signature. We find that the CFT strongly suppresses metric perturbations on all but the largest angular scales. This is true independently of how the de Sitter geometry arises, i.e., it is also true for four dimensional Einstein gravity. Since generic matter would be expected to behave like a CFT on small scales, our results suggest that tensor perturbations on small scales are far smaller than predicted by all previous calculations, which have neglected the effects of matter on tensor perturbations. 
  Extending the seminal work of Bilal and Gervais, we construct a tachyon-free, modular invariant partition function for critical superstrings on four-dimensional Minkowski x two-dimensional black hole. This model may be thought of as an SL(2,R)/U(1) version of Gepner models and corresponds to a conifold point on the moduli space of Calabi-Yau compactifications. We directly deal with N=2, c=9 unitary superconformal characters. Modular invariance is achieved by requiring the string to have a momentum along an extra noncompact direction, in agreement with the picture of singular CFTs advocated by Witten. The four-dimensional massless spectrum coincides with that of the tensionless strings, suggesting a possible dual description of type II strings on a conifold in terms of two intersecting NS5-branes. An interesting relation to D=6, N=4 gauged supergravity is also discussed. 
  Theories with infinite volume extra dimensions open exciting opportunities for particle physics. We argued recently that along with attractive features there are phenomenological difficulties in this class of models. In fact, there is no graviton zero-mode in this case and 4D gravity is obtained by means of continuum bulk modes. These modes have additional degrees of freedom which do not decouple at low energies and lead to inconsistent predictions for light bending and the precession of Mercury's perihelion. In a recent papers, [hep-th/0003020] and [hep-th/0003045] the authors made use of brane bending in order to cancel the unwanted physical polarization of gravitons. In this note we point out that this mechanism does not solve the problem since it uses a {\it ghost} which cancels the extra degrees of freedom. In order to have a consistent model the ghost should be eliminated. As soon as this is done, 4D gravity becomes unconventional and contradicts General Relativity. New mechanisms are needed to cure these models. We also comment on the possible decoupling of the ghost at large distances due to an apparent flat-5D nature of space-time and on the link between the presence of ghosts and the violation of positive-energy conditions. 
  Perturbative computations of the expectation value of the Wilson loop in N=4 supersymmetric Yang-Mills theory are reported. For the two special cases of a circular loop and a pair of anti-parallel lines, it is shown that the sum of an infinite class of ladder-like planar diagrams, when extrapolated to strong coupling, produces an expectation value characteristic of the results of the AdS/CFT correspondence, $<W>\sim\exp((constant)\sqrt{g^2N})$. For the case of the circular loop, the sum is obtained analytically for all values of the coupling. In this case, the constant factor in front of $\sqrt{g^2N}$ also agrees with the supergravity results. We speculate that the sum of diagrams without internal vertices is exact and support this conjecture by showing that the leading corrections to the ladder diagrams cancel identically in four dimensions. We also show that, for arbitrary smooth loops, the ultraviolet divergences cancel to order $g^4N^2$. 
  We review the different proposals which have so far been made for the holographic principle and the related entropy bounds and classify them into the strong, null and weak forms. These are analyzed, with the aim of discovering which may hold at the level of the full quantum theory of gravity. We find that only the weak forms, which constrain the information available to observers on boundaries, are implied by arguments using the generalized second law. The strong forms, which go further and posit a bound on the entropy in spacelike regions bounded by surfaces, are found to suffer from serious problems, which give rise to counterexamples already at the semiclassical level. The null form, proposed by Fischler, Susskind, Bousso and others, in which the bound is on the entropy of certain null surfaces, appears adequate at the level of a bound on the entropy of matter in a single background spacetime, but attempts to include the gravitational degrees of freedom encounter serious difficulties. Only the weak form seems capable of holding in the full quantum theory.   The conclusion is that the holographic principle is not a relationship between two independent sets of concepts: bulk theories and measures of geometry vrs boundary theories and measures of information. Instead, it is the assertion that in a fundamental theory the first set of concepts must be completely reduced to the second. 
  Using properties of the DBI action we find D-branes on $S^3$ of the radius $Q_5$ corresponding to the conjugacy classes of SU(2). The branes are stable due to nonzero 2-form NSNS background. In the limit of large $Q_5$ the dynamics of branes is governed by the non-commutative Yang-Mills theory. The results partially overlap with those obtained in the recent paper hep-th/0003037. 
  A necessary condition for the validity of the holographic principle is the holographic bound: the entropy of a system is bounded from above by a quarter of the area of a circumscribing surface measured in Planck areas. This bound cannot be derived at present from consensus fundamental theory. We show with suitable {\it gedanken} experiments that the holographic bound follows from the generalized second law of thermodynamics for both generic weakly gravitating isolated systems and for isolated, quiescent and nonrotating strongly gravitating configurations well above Planck mass. These results justify Susskind's early claim that the holographic bound can be gotten from the second law. 
  We address the problem of computing scattering amplitudes related to the correlation function of two Wilson lines and/or loops elongated along light-cone directions in strongly coupled gauge theories. Using the AdS/CFT correspondence in the classical approximation, the amplitudes are shown to be related to minimal surfaces generalizing the {\em helicoid} in various $AdS_5$ backgrounds. Infra-red divergences appearing for Wilson lines can be factorized out or can be cured by considering the IR finite case of correlation functions of two Wilson loops. In non-conformal cases related to confining theories, reggeized amplitudes with linear trajectories and unit intercept are obtained and shown to come from the approximately flat metrics near the horizon, which sets the scale for the Regge slope. In the conformal case the absence of confinement leads to a different solution. A transition between both regimes appears, in a confining theory, when varying impact parameter. 
  We consider cosmological solutions to type II supergravity theories where the spacetime is split into a FRW universe and a K\"ahler space, which may be taken to be Calabi-Yau. The various 2-forms present in the theories are taken to be proportional to the K\"ahler form associated to the K\"ahler space. 
  We consider exact solutions for static black holes localized on a three-brane in five-dimensional gravity in the Randall-Sundrum scenario. We show that the Reissner-Nordstrom metric is an exact solution of the effective Einstein equations on the brane, re-interpreted as a black hole without electric charge, but with instead a tidal 'charge' arising via gravitational effects from the fifth dimension. The tidal correction to the Schwarzschild potential is negative, which is impossible in general relativity, and in this case only one horizon is admitted, located outside the Schwarzschild horizon. The solution satisfies a closed system of equations on the brane, and describes the strong-gravity regime. Current observations do not strongly constrain the tidal charge, and significant tidal corrections could in principle arise in the strong-gravity regime and for primordial black holes. 
  We renormalize at two loops the axial current and $F \tilde{F}$ in massless QED, using the recently proposed semi-naive dimensional renormalization scheme. We show that the results are in agreement with those in the Breitenlohner-Maison-'t Hooft-Veltman scheme, previously obtained indirectly by making a three-loop computation. 
  Supersymmetric, rotating, asymptotically flat black holes with a regular horizon are rare configurations in String Theory. One example is known in five spacetime dimensions, within the toroidal compactification of type IIB string theory. Such special solution is allowed by the existence of a Chern-Simons coupling in the Supergravity theory and by the possibility of imposing a self duality condition on the `rotation 2-form'. We explore three peculiar features of such black holes: 1) Oxidising to D=10 the five dimensional configuration may be interpreted as a system of $D1-D5$ branes with a Brinkmann wave propagating along their worldvolume. Unlike its five dimensional Kaluza-Klein compactification, the universal covering space of this manifold has no causality violations. In other words, causal anomalies can be solved in higher dimensions. From the dual SCFT viewpoint, the causality bound for the compactified spacetime arises as the unitarity bound; 2) The vanishing of the scattering cross section for uncharged scalars and sufficiently high angular momentum of the background is shown still to hold at the level of charged interactions. The same is verified when a non-minimal coupling to the geometry is used. Therefore, the `repulson' behaviour previously found is universal for non accelerated observers; 3) The solutions are shown to have a non-standard gyromagnetic ratio of $g=3$. In contrast, the superpartners of a static, BPS, five dimensional black hole have $g=1$. At the semi-classical level, we find that a Dirac fermion propagating in the rotating hole background has $g=2\pm1$, depending on the spinor direction of the fermion being parallel to Killing or `anti-Killing' spinors. 
  We review some old and new methods of reduction of the number of degrees of freedom from ~N^2 to ~N in the multi-matrix integrals. 
  A new formulation of four dimensional quantum field theories, such as scalar field theory, is proposed as a large N limit of a special NxN matrix model. Our reduction scheme works beyond planar approximation and applies for QFT with finite number of fields. It uses quenched coordinates instead of quenched momenta of the old Eguchi-Kawai reduction known to yield correctly only the planar sector of quantum field theory. Fermions can be also included. 
  We numerically study classical time evolutions of Kaluza-Klein bubble space-time which has negative energy after a decay of vacuum. As the zero energy Witten's bubble space-time, where the bubble expands infinitely, the subsequent evolutions of Brill and Horowitz's momentarily static initial data show that the bubble will expand in terms of the area. At first glance, this result may support Corley and Jacobson's conjecture that the bubble will expand forever as well as the Witten's bubble. The irregular signatures, however, can be seen in the behavior of the lapse function in the maximal slicing gauge and the divergence of the Kretchman invariant. Since there is no appearance of the apparent horizon, we suspect an appearance of a naked singularity as the final fate of this space-time. 
  We consider `brane universe' scenarios with standard-model fields localized on a 3-brane in 6 spacetime dimensions. We show that if the spacetime is rotationally symmetric about the brane, local quantities in the bulk are insensitive to the couplings on the brane. This potentially allows compactifications where the effective 4-dimensional cosmological constant is independent of the couplings on the 3-brane. We consider several possible singularity-free compactification mechanisms, and find that they do not maintain this property. We also find solutions with naked spacetime singularities, and we speculate that new short-distance physics can become important near the singularities and allow a compactification with the desired properties. The picture that emerges is that standard-model loop contributions to the effective 4-dimensional cosmological constant can be cut off at distances shorter than the compactification scale. At shorter distance scales, renormalization effects due to standard-model fields renormalize the 3-brane tension, which changes a deficit angle in the transverse space without affecting local quantities in the bulk. For a compactification scale of order 10^{-2} mm, this gives a standard-model contribution to the cosmological constant in the range favored by cosmology. 
  The Wilson loop in N=4 supersymmetric Yang-Mills theory admits a dual description as a macroscopic string configuration in the adS/CFT correspondence. We discuss the correction to the quark anti-quark potential arising from the fluctuations of the superstring. 
  We consider the five dimensional $USp(2k)$ gauge theory which consists of one antisymmetric and $n_{f}$ fundamental hypermultiplets. This gauge theory is a many-probe generalization of the SU(2) gauge theory in five dimensions considered by Seiberg in the context of probing type I superstring by a D4-brane. This gauge theory can also be obtained from the USp(2k) matrix model by matrix T-dual transformations in the large k limit. We exhibit the anomalous interaction associated with this five dimensional theory on the new phase, where the vacuum expectation values (vevs) of the scalars belonging to the antisymmetric hypermultiplet are also nonvanishing. On the Coulomb phase, the anomalous interaction has been computed in [1, 2]. 
  Within the framework of the world-line formalism we write down in detail a two-loop Euler-Heisenberg type action for gluon loops in Yang-Mills theory and discuss its divergence structure. We exactly perform all the world-line moduli integrals at two loops by inserting a mass parameter, and then extract divergent coefficients to be renormalized. 
  We extend the topological Kerr-Newman-aDS solutions by including NUT charge and find generalizations of the Robinson-Bertotti solution to the negative cosmological constant case with different topologies. We show how all these solutions can be obtained as limits of the general Plebanski-Demianski solution. We study the supersymmetry properties of all these solutions in the context of gauged N=2,d=4 supergravity. Generically they preserve at most 1/4 of the total supersymmetry. In the Plebanski-Demianski case, although gauged N=2,d=4 supergravity does not have electric-magnetic duality, we find that the family of supersymmetric solutions still enjoys a sort of electric-magnetic duality in which electric and magnetic charges and mass and Taub-NUT charge are rotated simultaneously. 
  It is shown that magnetic seven-branes previously considered as different objects are members of a one-parametric family of supersymmetric seven branes. We enlarge the class of seven-branes by constructing new magnetically and also electrically charged seven branes. The solutions display a kink-like behavior. We also construct a solution that naturally generalizes the D-instanton. 
  Regular and black-hole solutions of the spontaneously broken Einstein-Yang-Mills-Higgs theory with nonminimal coupling to gravity are shown to exist. The main characteristics of the solutions are presented and differences with respect to the minimally coupled case are studied. Since negative energy densities are found to be possible, traversable wormhole solutions might exist. We prove that they are absent. 
  We consider a modification of gravity at large distances in a Brane Universe which was discussed recently. In these models the modification of gravity at large distances is ultimately connected to existence of negative tension brane(s) and exponentially small tunneling factor. We discuss a general model which interpolates between Bi-gravity model and GRS model. We also discuss the possible mechanism of stabilization for negative tension branes in AdS background. Finally we show that extra degrees of freedom of massive gravitons do not lead to disastrous contradiction with General Relativity if the stabilization condition $\int dy \sqrt{-G^{(5)}} (T^\mu_\mu-2T^5_5)=0$ is implemented. 
  It has been known for some time that the AdS/CFT correspondence predicts a limit on the number of single particle states propagating on the compact spherical component of the AdS-times-sphere geometry. The limit is called the stringy exclusion principle. The physical origin of this effect has been obscure but it is usually thought of as a feature of very small distance physics. In this paper we will show that the stringy exclusion principle is due to a surprising large distance phenomenon. The massless single particle states become progressively less and less point-like as their angular momentum increases. In fact they blow up into spherical branes of increasing size. The exclusion principle is simply understood as the condition that the particle should not be bigger than the sphere that contains it. 
  We study the long distance behaviour of brane theories with quasi-localized gravity. The 5D effective theory at large scales follows from a holographic renormalization group flow. As intuitively expected, the graviton is effectively four dimensional at intermediate scales and becomes five dimensional at large scales. However in the holographic effective theory the essentially 4D radion dominates at long distances and gives rise to scalar anti-gravity. The holographic description shows that at large distances the GRS model is equivalent to the model recently proposed by Dvali, Gabadadze and Porrati (DGP), where a tensionless brane is embedded into 5D Minkowski space, with an additional induced 4D Einstein-Hilbert term on the brane. In the holographic description the radion of the GRS model is automatically localized on the tensionless brane, and provides the ghost-like field necessary to cancel the extra graviton polarization of the DGP model. Thus, there is a holographic duality between these theories. This analysis provides physical insight into how the GRS model works at intermediate scales; in particular it sheds light on the size of the width of the graviton resonance, and also demonstrates how the holographic RG can be used as a practical tool for calculations. 
  In this work, we suggest a view-point that leads to an intrinsic mass scale in Quantum Field Theories. This view-point is fairly independent of dynamical details of a QFT and does not rely on any particular framework to go beyond the standard Model. We use the setting of the nonlocal quantum field theories NLQFT with a finite scale parameter Lambda, which are unitary for finite Lambda. We propose that the condition 0 < Z < 1 [wherever proven] can be rigorously implemented/imposed in such theories and that it implies the existence of a mass scale Lambda that can be determined from this condition. We derive the nonlocal analogue of the above relation [which is a finite relation in NLQFT] and demonstrate that it can be arrived at only from general principles. We further propose that the nonlocal formulation should be looked as an effective field theory that incorporates the effect of dynamics beyond an energy scale and which breaks at the intrinsic scale Lambda so obtained. Beyond this scale it should be replaced by another [perhaps, a more fundamental] theory. We provide a heuristic justification for this view-point. 
  In the low-energy limit, M-theory compactified on S1/Z2 is formulated in terms of Bianchi identities with sources localized at orbifold singularities and anomaly-cancelling counterterms to the Wilson effective Lagrangian. Compactifying to four dimensions on a Calabi-Yau space leads to N=1 local supersymmetry. We derive a formulation of the effective supergravity which explicitly relates four-dimensional supergravity multiplets and field equations with these fundamental M-theory aspects. This formulation proves convenient for the introduction in the effective supergravity of non-perturbative M-theory contributions. It also applies to the universal sector of generic compactifications with N=1 supersymmetry. 
  We consider the dynamics governing the evolution of a many body system constrained by an nonabelian local symmetry. We obtain explicit forms of the global macroscopic condition assuring that at the microscopic level the evolution respects the overall symmetry constraint. We demonstrate the constraint mechanisms for the case of SU(2) system comprising particles in fundamental, and adjoint representations (`nucleons' and `pions'). 
  A group theoretical procedure, introduced earlier (hep-th/9901090), to decompose bilocal light-ray operators into (harmonic) operators of definite twist is applied to the case of arbitrary 2nd rank tensors. As a generic example the bilocal gluon operator is considered which gets contributions of twist-2 up to twist-6 from four different symmetry classes characterized by conrresponding Young tableaus; also the twist decomposition of the related vector and scalar operators is considered. In addition, we extend these reselts to various trilocal light-ray operators, like the Shuryak-Vainshtein, the three-gluon and the four-quark operators, which are required for the consideration of higher twist distribution amplitudes. The present results rely on the knowledge of harmonic tensor polynomials of any order n which habe been determined up to the case of 2nd rank symmetric tensors for arbitrary space-time dimension. 
  We present an exact one-loop calculation of the tunneling process in Euclidean quantum gravity describing creation of black hole pairs in a de Sitter universe. Such processes are mediated by $S^2\times S^2$ gravitational instantons giving an imaginary contribution to the partition function. The required energy is provided by the expansion of the universe. We utilize the thermal properties of de Sitter space to describe the process as the decay of a metastable thermal state. Within the Euclidean path integral approach to gravity, we explicitly determine the spectra of the fluctuation operators, exactly calculate the one-loop fluctuation determinants in the $\zeta$-function regularization scheme, and check the agreement with the expected scaling behaviour. Our results show a constant volume density of created black holes at late times, and a very strong suppression of the nucleation rate for small values of $\Lambda$. 
  The matrix elements of local operators such as the electromagnetic current, the energy momentum tensor, angular momentum, and the moments of structure functions have exact representations in terms of light-cone Fock state wavefunctions of bound states such as hadrons. We illustrate all of these properties by giving explicit light-cone wavefunctions for the two-particle Fock state of the electron in QED, thus connecting the Schwinger anomalous magnetic moment to the spin and orbital momentum carried by its Fock state constituents. We also compute the QED one-loop radiative corrections for the form factors for the graviton coupling to the electron and photon. Although the underlying model is derived from elementary QED perturbative couplings, it in fact can be used to simulate much more general bound state systems by applying spectral integration over the constituent masses while preserving all of the Lorentz properties, giving explicit realization of the spin sum rules and other local matrix elements. The role of orbital angular momentum in understanding the "spin crisis" problem for relativistic systems is clarified. We also prove that the anomalous gravitomagnetic moment B(0) vanishes for any composite system. This property is shown to follow directly from the Lorentz boost properties of the light-cone Fock representation and holds separately for each Fock state component. We show how the QED perturbative structure can be used to model bound state systems while preserving all Lorentz properties. We thus obtain a theoretical laboratory to test the consistency of formulae which have been proposed to probe the spin structure of hadrons. 
  We overview some attempts to find S-duality analogues of non-supersymmetric Yang-Mills theory, in the context of gravity theories. The case of MacDowell-Mansouri gauge theory of gravity is discussed. Three-dimensional dimensional reductions from the topological gravitational sector in four dimensions, enable to recuperate the 2+1 Chern-Simons gravity and the corresponding S-dual theory, from the notion of self-duality in the four-dimensional theory. 
  The article is a natural continuation of our paper {\em Quantum scalar field in FRW Universe with constant electromagnetic background}, Int. J. Mod. Phys. {\bf A12}, 4837 (1997). We generalize the latter consideration to the case of massive spinor field, which is placed in FRW Universe of special type with a constant electromagnetic field. To this end special sets of exact solutions of Dirac equation in the background under consideration are constructed and classified. Using these solutions representations for out-in, in-in, and out-out spinor Green functions are explicitly constructed as proper-time integrals over the corresponding contours in complex proper-time plane. The vacuum-to-vacuum transition amplitude and number of created particles are found and vacuum instability is discussed. The mean values of the current and energy-momentum tensor are evaluated, and different approximations for them are presented. The back reaction related to particle creation and to the polarization of the unstable vacuum is estimated in different regimes. 
  Via partial resolution of Abelian orbifolds we present an algorithm for extracting a consistent set of gauge theory data for an arbitrary toric variety whose singularity a D-brane probes. As illustrative examples, we tabulate the matter content and superpotential for a D-brane living on the toric del Pezzo surfaces as well as the zeroth Hirzebruch surface. Moreover, we discuss the non-uniqueness of the general problem and present examples of vastly different theories whose moduli spaces are described by the same toric data. Our methods provide new tools for calculating gauge theories which flow to the same universality class in the IR. We shall call it ``Toric Duality.'' 
  The purpose of this review is to discuss recent developments occurring at the interface of cosmology with string and M-theory. We begin with a short review of 1980s string cosmology and the Brandenberger-Vafa mechanism for explaining spacetime dimensionality. It is shown how this scenario has been modified to include the effects of p-brane gases in the early universe. We then introduce the Pre-Big-Bang scenario (PBB), Ho\v{r}ava-Witten heterotic M-theory and the work of Lukas, Ovrut and Waldram, and end with a discussion of large extra dimensions, the Randall-Sundrum model and Brane World cosmologies. 
  The particle Fock space of the matter fields in QED can be constructed using the free creation and annihilation operators. However, these particle operators are not, even at asymptotically large times, the modes of the matter fields that enter the QED Lagrangian. In this letter we construct the fields which do recover such particle modes at large times. We are thus able to demonstrate for the first time that, contrary to statements found in the literature, a relativistic description of charged particles in QED exists. 
  Using the ideas of effective field theory and dimensional reduction, we relate the parameters of two low energy models of QCD: the O(N) nonlinear sigma model in D=3+1, which describes the dynamics of cool pions, and the O(N) Heisenberg magnet in D=3+0, which is commonly argued to reproduce the correct critical behaviour of the chiral phase transition. As a result, we obtain a generalized expression for the finite temperature pion decay constant which reproduces, in certain limits, the available expressions in the literature. 
  A matter self-interacting model with N=1-supersymmetry in 3D is discussed in connection with the appearance of a central charge in the algebra of the supersymmetry generators. The result is extended to include gauge fields with a Chern-Simons term. The main result is that, for a simple supersymmetry, only the matter sector contributes to the central charge in contrast to what occurs in the N=2 case. 
  The SDLCQ regularization is known to explicitly preserve supersymmetry in 1+1 dimensions. To test this property in higher dimensions, we consider supersymmetric Yang-Mills theory on R x S^1 x S^1. In particular, we choose one of the compact directions to be light-like and another to be space-like. This theory is totally finite, and thus we can solve for bound state wave functions and masses numerically without renormalizing. We present the masses as functions of the longitudinal and transverse resolutions and show that the masses converge rapidly in both resolutions. We study the behavior of the spectrum as a function of the coupling and find that at strong coupling there is a stable, well-defined spectrum which we present. We discuss also the massless spectrum and find several unphysical states that decouple at large transverse resolution. 
  We present the new basis functions to investigate the 't Hooft equation, the lowest order mesonic Light-Front Tamm-Dancoff equation for $\rm SU(N_C)$ gauge theories. We find the wave function can be well approximated by new basis functions and obtain an analytic formula for the mass of the lightest bound state. Its value is consistent with the precedent results. 
  We show that the recently developed soldering formalism in the Lagrangian approach and canonical transformations in the Hamiltonian approach are complementary. The examples of gauged chiral bosons in two dimensions and self-dual models in three dimensions are discussed in details. 
  We study U(1) and U(2) instanton solutions on noncommutative R^4 based on the noncommutative version of ADHM equation proposed by Nekrasov and Schwarz. It is shown that the anti-self-dual gauge fields on self-dual noncommutative R^4 correctly give integer instanton numbers for all cases we consider. We also show that the completeness relation in the ADHM construction is generally satisfied even for noncommutative spaces. 
  Generalizing the 't Hooft and Veltman method of unitary regulators, we demonstrate for the first time the existence of local, Lorentz-invariant, physically motivated Lagrangians of quantum-electrodynamic phenomena such that: (i) Feynman diagrams are finite and equal the diagrams of QED but with regularized propagators. (ii) N-point Green functions are C-, P-, and T-invariant up to a phase factor, Lorentz-invariant and causal. (iii) No auxiliary particles or parameters are introduced. 
  For perfect fluids with equation of state $\rho = \rho (n,s)$, Brown gave an action principle depending only on their Lagrange coordinates $\alpha^i(x)$ without Clebsch potentials. After a reformulation on arbitrary spacelike hypersurfaces in Minkowski spacetime, the Wigner-covariant rest-frame instant form of these perfect fluids is given. Their Hamiltonian invariant mass can be given in closed form for the dust and the photon gas. The action for the coupling to tetrad gravity is given. Dixon's multipoles for the perfect fluids are studied on the rest-frame Wigner hyperplane. It is also shown that the same formalism can be applied to non-dissipative relativistic elastic materials described in terms of Lagrangian coordinates. 
  We explicitly compute the complete three-loop (O(g^4)) contribution to the four-point function of chiral primary current-like operators <(q)^2 q^2 (q)^2 q^2> in any finite N=2 SYM theory. The computation uses N=2 harmonic supergraphs in coordinate space. Dramatic simplifications are achieved by a double insertion of the N=2 SYM linearized action, and application of superconformal covariance arguments to the resulting nilpotent six-point amplitude. The result involves polylogarithms up to fourth order of the conformal cross ratios. It becomes particularly simple in the N=4 special case. 
  From a supersymmetry covariant source extension of N=2 SYM we study non-trivial thermodynamical limits thereof. Using an argument by one of us about the solution of the strong CP problem and the uniqueness of the QCD ground state we find that the dependence of the effective potential on the defining field operators is severely restricted. In contrast to the solution by Seiberg and Witten an acceptable infrared behavior only exists for broken supersymmetry while the gauge symmetry remains unbroken. 
  We provide a method to decompose the two-point function of a quantum field on a warped manifold in terms of fields living on a lower-dimensional manifold. We discuss explicit applications to Minkowski, de Sitter and anti-de Sitter quantum field theories. This decomposition presents a remarkable analogy with the holography principle, in the sense that physics in d+1 dimensions may be encoded into the physics in one dimension less. Moreover in a context a la Randall--Sundrum, the method outlined here allows a mechanism of generation of mass-spectra in the 3-brane (or more generally a d-1-brane). 
  We introduce non-linear $\sigma$-models in the framework of noncommutative geometry with special emphasis on models defined on the noncommutative torus. We choose as target spaces the two point space and the circle and illustrate some characteristic features of the corresponding $\sigma$-models. In particular we construct a $\sigma$-model instanton with topological charge equal to 1. We also define and investigate some properties of a noncommutative analogue of the Wess-Zumino-Witten model. 
  A field theory formulation of two-time physics in d+2 dimensions is obtained from the covariant quantization of the constraint system associated with the OSp(n|2) worldline gauge symmetries of two-time physics. Interactions among fields can then be included consistently with the underlying gauge symmetries. Through this process a relation between Dirac's work in 1936 on conformal symmetry in field theory and the more recent worldline formulation of two-time physics is established while providing a worldline gauge symmetry basis for the field equations in d+2 dimensions. It is shown that the field theory formalism goes well beyond Dirac's goal of linearizing conformal symmetry. In accord with recent results in the worldline approach of two-time physics, the d+2 field theory can be brought down to diverse d dimensional field theories by solving the subset of field equations that correspond to the ``kinematic'' constraints. This process embeds the one ``time'' in d-dimensions in different ways inside the d+2 dimensional spacetime. Thus, the two-time d+2 field theory appears as a more fundamental theory from which many one-time d dimensional field theories are derived. It is suggested that the hidden symmetries and relations among computed quantities in certain d-dimensional interacting field theories can be taken as the evidence for the presence of a higher unifying structure in a d+2 dimensional spacetime. These phenomena have similarities with ideas such as dualities, AdS-CFT correspondence and holography. 
  We study condensation of open string tachyons using renormalization group flow in the worldsheet field theory. This approach leads to a simple picture of the physics of the nontrivial condensate. 
  We present a new class of solutions of D=10, N=2 chiral supergravity. A nonvanishing background for the field strength G_{MNR} of the complex two-form triggers AdS_3 x M_7 compactifications, where M_7 is a 7-dimensional compact manifold. When M_7 is a nonsymmetric coset space G/H, we can always find a set of constants G_{MNR} covariantly conserved and thus satisfying the field equations. For example the structure constants of G with indices in the G/H directions are a covariantly conserved tensor. In some symmetric G/H, where these structure constants vanish, there may still exist conserved 3-forms, yielding AdS_3 x G/H solutions. The conditions for supersymmetry of the AdS_3 x M_7 compactifications are derived, and tested for the supersymmetric solutions AdS_3 x S^3 x T^4 and AdS_3 x S^3 x S^3 x S^1. Finally, we show that the AdS_3 x S^3 x T^4 supersymmetric background can be seen as the sigma = 0 limit in a one-parameter class of solutions of the form AdS_3 x S^3 x CP^2, the parameter sigma being the inverse "radius" of CP^2. For sigma different from 0 all supersymmetries are broken. 
  Type IIB supergravity can be consistently truncated to the metric and the self-dual 5-form. We obtain the complete non-linear Kaluza-Klein S^5 reduction Ansatz for this theory, giving rise to gravity coupled to the fifteen Yang-Mills gauge fields of SO(6) and the twenty scalars of the coset SL(6,R)/SO(6). This provides a consistent embedding of this subsector of N=8, D=5 gauged supergravity in type IIB in D=10. We demonstrate that the self-duality of the 5-form plays a crucial role in the consistency of the reduction. We also discuss certain necessary conditions for a theory of gravity and an antisymmetric tensor in an arbitrary dimension D to admit a consistent sphere reduction, keeping all the massless fields. We find that it is only possible for D=11, with a 4-form field, and D=10, with a 5-form. Furthermore, in D=11 the full bosonic structure of eleven-dimensional supergravity is required, while in D=10 the 5-form must be self-dual. It is remarkable that just from the consistency requirement alone one would discover D=11 and type IIB supergravities, and that D=11 is an upper bound on the dimension. 
  we show there exists a mathematically consistent framework in which the Renormalization Program can be understood in a natural manner. The framework does not require any violations of mathematical rigor usually associated with the Renormalization program. We use the framework of the non-local field theories [these carry a finite mass scale (\Lambda)]and set up a finite perturbative program. We show how this program leads to the perturbation series of the usual renormalization program [except one difference] if the series is restructured .We further show that the comparison becomes possible if there exists a finite mass scale (\Lambda), with certain properties, in the Quantum Field theory [which we take to be the scale present in the nonlocal theory]. We give a way to estimate the scale (\Lambda). We also show that the finite perturbation program differs from the usual renormalization program by a term; which we propose can also be used to put a bound on (\Lambda). 
  Naive light-front quantization, carried out by a light-front energy integration of covariant amplitudes, is not guaranteed to generate the corresponding Feynman amplitudes. In an explicit example we show that the nonvalence contribution to the minus-component of the EM current of a meson with fermion constituents has a persistent end-point singularity. Only after this term is subtracted, the result is covariant and satisfies current conservation. If the spin-1/2 constituents are replaced by spin zero ones, the singularity does not occur and the result is, without any adjustment, identical to the Feynman amplitude. Numerical estimates of valence and nonvalence contributions are presented for the cases of fermion and boson constituents. 
  We perform explicitly the toroidal compactification of eleven dimensional supergravity to six dimensions and present its action in a manifestly $SO(5,5)\over SO(5)\times SO(5)$ invariant form using the recently proposed covariant formulation of theories involving chiral fields. 
  In the paper a concept of a double symmetry is introduced, and its qualitative characteristics and rigorous definitions are given. We describe two ways to construct the double-symmetric field theories and present an example demonstrating the high efficiency of one of them. In noting the existing double-symmetric theories we draw attention at a dual status of the group $SU(2)_{L} \otimes SU(2)_{R}$ as a secondary symmetry group, and in this connexion we briefly discuss logically possible aspects of the $P$-violation in weak interactions. 
  We introduce the notion of a topological symmetry as a quantum mechanical symmetry involving a certain topological invariant. We obtain the underlying algebraic structure of the Z_2-graded uniform topological symmetries of type (1,1) and (2,1). This leads to a novel derivation of the algebras of supersymmetry and $p=2$ parasupersummetry. 
  In the framework of a five-dimensional model with one 3-brane and an infinite extra dimension, we discuss a process in which matter escapes from the brane and propagates into the bulk to arbitrarily large distances. An example is a decay of a particle of mass $2m$ residing on the brane into two particles of mass $m$ that leave the brane and accelerate away. We calculate, in the linearized theory, the metric induced by these particles on the brane. This metric does not obey the four-dimensional Einstein equations and corresponds to a spherical gravity wave propagating along the four-dimensional future light cone. The four-dimensional space-time left behind the spherical wave is flat, so the gravitational field induced in the brane world by matter escaping from the brane disappears in a causal way. 
  We consider unitary Virasoro minimal models on the disk with Cardy boundary conditions and discuss deformations by certain relevant boundary operators, analogous to tachyon condensation in string theory. Concentrating on the least relevant boundary field, we can perform a perturbative analysis of renormalization group fixed points. We find that the systems always flow towards stable fixed points which admit no further (non-trivial) relevant perturbations. The new conformal boundary conditions are in general given by superpositions of 'pure' Cardy boundary conditions. 
  We study the representation ${\cal D}$ of a simple compact Lie algebra $\g$ of rank l constructed with the aid of the hermitian Dirac matrices of a (${\rm dim} \g$)-dimensional euclidean space. The irreducible representations of $\g$ contained in ${\cal D}$ are found by providing a general construction on suitable fermionic Fock spaces. We give full details not only for the simplest odd and even cases, namely su(2) and su(3), but also for the next (${dim} \g$)-even case of su(5). Our results are far reaching: they apply to any $\g$-invariant quantum mechanical system containing ${\rm dim} \g$ fermions. Another reason for undertaking this study is to examine the role of the $\g$-invariant fermionic operators that naturally arise. These are given in terms of products of an odd number of gamma matrices, and include, besides a cubic operator, (l-1) fermionic scalars of higher order. The latter are constructed from the Lie algebra cohomology cocycles, and must be considered to be of theoretical significance similar to the cubic operator. In the (${\rm dim} \g$)-even case, the product of all l operators turns out to be the chirality operator $\gamma_q, q=({{\rm dim} \g+1})$. 
  We revise the problem of the quantization of relativistic particle models (spinless and spinning), presenting a modified consistent canonical scheme. One of the main point of the modification is related to a principally new realization of the Hilbert space. It allows one not only to include arbitrary backgrounds in the consideration but to get in course of the quantization a consistent relativistic quantum mechanics, which reproduces literally the behavior of the one-particle sector of the corresponding quantum field. In particular, in a physical sector of the Hilbert space a complete positive spectrum of energies of relativistic particles and antiparticles is reproduced, and all state vectors have only positive norms. 
  Starting from tree and one-loop tachyon amplitudes of open string theory in the presence of a constant B-field, we explore two problems. First we show that in the noncommutative field theory limit the amplitudes reduce to tree and one-loop diagrams of the noncommutative phi-three theory. Next, we check factorization of the one-loop amplitudes in the long cylinder limit. 
  The relationship is established between the Fedosov deformation quantization of a general symplectic manifold and the BFV-BRST quantization of constrained dynamical systems. The original symplectic manifold $\mathcal M$ is presented as a second class constrained surface in the fibre bundle ${{\mathcal T}^*_\rho}{\mathcal M}$ which is a certain modification of a usual cotangent bundle equipped with a natural symplectic structure. The second class system is converted into the first class one by continuation of the constraints into the extended manifold, being a direct sum of ${{\mathcal T}^*_\rho}{\mathcal M}$ and the tangent bundle $T {\mathcal M}$. This extended manifold is equipped with a nontrivial Poisson bracket which naturally involves two basic ingredients of Fedosov geometry: the symplectic structure and the symplectic connection. The constructed first class constrained theory, being equivalent to the original symplectic manifold, is quantized through the BFV-BRST procedure. The existence theorem is proven for the quantum BRST charge and the quantum BRST invariant observables. The adjoint action of the quantum BRST charge is identified with the Abelian Fedosov connection while any observable, being proven to be a unique BRST invariant continuation for the values defined in the original symplectic manifold, is identified with the Fedosov flat section of the Weyl bundle. The Fedosov fibrewise star multiplication is thus recognized as a conventional product of the quantum BRST invariant observables. 
  We present the analysis of the complete glueball spectrum for the $AdS^7$ black hole supergravity dual of $QCD_4$ in strong coupling limit: $g^2 N \to \infty$. The bosonic fields in the supergravity multiplet lead to 6 independent wave equations contributing to glueball states with $J^{PC} = 2^{++},1^{+-}, 1^{--}$, $0^{++}$ and $0^{-+}$. We study the spectral splitting and degeneracy patterns for both $QCD_4$ and $QCD_3$. Despite the expected limitations of a leading order strong coupling approximation, the pattern of spins, parities and mass inequalities bear a striking resemblance to the known $QCD_4$ glueball spectrum as determined by lattice simulations at weak coupling. 
  We study five-dimensional gravity models with non-vanishing background scalar fields which are dual to non-conformal boundary field theories. We develop a procedure to decouple the graviton fluctuations from the scalar ones and apply it to the simplest case of one scalar field. The quadratic action for the decoupled scalar fluctuations has a very simple form and can be used to compute two-point functions. We perform this computation for the two examples of background RG flow recently considered by DeWolfe and Freedman and find physically reasonable results. 
  We present the supersymmetric version of the minimal Randall-Sundrum model with two opposite tension branes. 
  We examine the evaporation of a small black hole on a brane in a world with large extra dimensions. Since the masses of many Kaluza-Klein modes are much smaller than the Hawking temperature of the black hole, it has been claimed that most of the energy is radiated into these modes. We show that this is incorrect. Most of the energy goes into the modes on the brane. This raises the possibility of observing Hawking radiation in future high energy colliders if there are large extra dimensions. 
  This is the write-up of a set of lectures on the comparison between Lattice Gauge Theories and AdS/CFT results for the non-perturbative behaviour of non-supersymmetric Yang Mills theories. These notes are intended for students which are assumed not to be experts in L.G.T. For this reason the first part is devoted to a pedagogical introduction to the Lattice regularization of QCD. In the second part we discuss some basic features of the AdS/CFT correspondence and compare the results obtained in the non-supersymmetric limit with those obtained on the Lattice. We discuss in particular the behaviour of the string tension and of the glueball spectrum. Lectures delivered at the School of Theoretical Physics (S.N.F.T.), Parma, September 1999. 
  Quantum field theory on d+1-dimensional anti-deSitter space-time admits a re-interpretation as a quantum field theory with conformal symmetry on d-dimensional Minkowski space-time. This conjecture originally emerged from string theory considerations. Here, it is proven in a general framework by an explicit identification between the local observables of the two corresponding theories. 
  The Wilson discretization of the dimensionally reduced supersymmetric Yang-Mills theory is constructed. This gives a lattice version of the matrix model of M-theory. An SU(2) model is studied numerically in the quenched approximation for D=4. The system shows canonical scaling in the continuum limit. A clear signal for a prototype of the "black hole to strings" phase transition is found. The pseudocritical temperature is determined and the temperature dependence of the total size of the system is measured in both phases. Further applications are outlined. 
  By explicit evaluation of certain disk S-matrix elements in the presence of background B-flux, we find coupling of two open string tachyons to gauge field, graviton, dilaton or Kalb-Ramond antisymmetric tensor on the world-volume of a single non-BPS D$p$-brane. We then propose an extension of the abelian Dirac-Born-Infeld action which naturally reproduces these couplings in field theory. This action includes non-linearly the dynamics of the tachyon field much like the other bosonic modes of the non-BPS D$p$-brane. On the general grounds of gauge and T-duality transformations and the symmetrized trace prescription, we then extend the abelian action to non-abelian cases. 
  We formulate the equations of motion of a free scalar field in the flat and $AdS$ space of an arbitrary dimension in the form of some "higher spin" covariant constancy conditions. Klein-Gordon equation is interpreted as a non-trivial cohomology of a certain "\sgm-complex". The action principle for a scalar field is formulated in terms of the "higher-spin" covariant derivatives for an arbitrary mass in $AdS_d$ and for a non-zero mass in the flat space. The constructed action is shown to be equivalent to the standard first-order Klein-Gordon action at the quadratic level but becomes different at the interaction level because of the presence of an infinite set of auxiliary fields which do not contribute at the free level. The example of Yang-Mills current interaction is considered in some detail. It is shown in particular how the proposed action generates the pseudolocally exact form of the matter currents in $AdS_d$. 
  It has been conjectured that the vortex solution on a D-brane - anti-D-brane system represents a D-brane of two lower dimension. We establish this result by first identifying a series of marginal deformations which create the vortex - antivortex pair on the brane - antibrane system, and then showing that under this series of marginal deformations the original D-brane - anti-D-brane system becomes a D-brane - anti-D-brane system with two lower dimensions. Generalization of this construction to the case of solitons of higher codimension is also discussed. 
  We show that a previous paper of Freund describing a solution to the Seiberg-Witten equations has a sign error rendering it a solution to a related but different set of equations. The non-$L^2$ nature of Freund's solution is discussed and clarified and we also construct a whole class of solutions to the Seiberg-Witten equations. 
  The class of the free relativistic covariant equations generated by the fractional powers of the D'Alambertian operator $(\square^{1/n})$ is studied. Meanwhile the equations corresponding to n=1 and 2 (Klein-Gordon and Dirac equations) are local in their nature, the multicomponent equations for arbitrary n>2 are non-local. It is shown, how the representation of generalized algebra of Pauli and Dirac matrices looks like and how these matrices are related to the algebra of SU(n) group. The corresponding representations of the Poincar\'e group and further symmetry transformations on the obtained equations are discussed. The construction of the related Green functions is suggested. 
  The possibility of non-trivial representations of the gauge group on wavefunctionals of a gauge invariant quantum field theory leads to a generation of mass for intermediate vector and tensor bosons. The mass parameters "m" show up as central charges in the algebra of constraints, which then become of second-class nature. The gauge group coordinates acquire dynamics outside the null-mass shell and provide the longitudinal field degrees of freedom that massless bosons need to form massive bosons. 
  A new local, covariant and nilpotent symmetry is shown to exist for the interacting BRST invariant U(1) gauge theory in two dimensions of space-time. Under this new symmetry, it is the gauge-fixing term that remains invariant and the corresponding transformations on the Dirac fields turn out to be the analogue of chiral transformations. The extended BRST algebra is derived for the generators of all the underlying symmetries, present in the theory. This algebra turns out to be the analogue of the algebra obeyed by the de Rham cohomology operators of differential geometry. Possible interpretations and implications of this symmetry are pointed out in the context of BRST cohomology and Hodge decomposition theorem. 
  According to the AdS/CFT correspondence, the maximally supersymmetric SU(N) Yang-Mills theory in 4 dimensions is dual to the type IIB string theory compactified on AdS_5 x S^5. Most of the tests performed so far are confined to the leading order at large N or equivalently string tree-level. To probe the correspondence beyond this leading order and obtain 1/N^2 corrections is difficult since string one-loop computations on an AdS_5 x S^5 background generally are beyond feasibility. However, we will show that the chiral SU(4)_R anomaly of the super YM theory provides an ideal testing ground to go beyond leading order in N. We review and develop further our previous results that the 1/N^2 corrections to the chiral anomaly on the super YM side can be exactly accounted for by the supergravity/string effective action induced at one loop. 
  We show that the second central extension of the Galilei group in (2+1) dimensions corresponds to spin, which can be any real number. 
  We compactify M(atrix) theory on Riemann surfaces Sigma with genus g>1. Following [1], we construct a projective unitary representation of pi_1(Sigma) realized on L^2(H), with H the upper half-plane. As a first step we introduce a suitably gauged sl_2(R) algebra. Then a uniquely determined gauge connection provides the central extension which is a 2-cocycle of the 2nd Hochschild cohomology group. Our construction is the double-scaling limit N\to\infty, k\to-\infty of the representation considered in the Narasimhan-Seshadri theorem, which represents the higher-genus analog of 't Hooft's clock and shift matrices of QCD. The concept of a noncommutative Riemann surface Sigma_\theta is introduced as a certain C^\star-algebra. Finally we investigate the Morita equivalence. 
  In this short note, we try to clarify a seemly trivial but often confusing question in relating a higher-dimensional physical gravitational constant to its lower-dimensional correspondence in Kaluza-Klein reduction. In particular, we re-derive the low-energy M-theory gravitational constant in terms of type IIA string coupling $g_s$ and constant $\alpha'$ through the metric relation between the two theories. 
  This work deals with the formation of black hole in bidimensional dilaton gravity coupled to scalar matter fields. We investigate two scalar matter systems, one described by a sixth power potential and the other defined with two scalar fields containing up to the fourth power in the fields. The topological solutions that appear in these cases allow the formation of black holes in the corresponding dilaton gravity models. 
  We show that the Pauli-Villars regularized action for a scalar field in a gravitational background in 1+1 dimensions has, for any value of the cutoff M, a symmetry which involves non-local transformations of the regulator field plus (local) Weyl transformations of the metric tensor. These transformations, an extension to the regularized action of the usual Weyl symmetry transformations of the classical action, lead to a new interpretation of the conformal anomaly in terms of the (non-anomalous) Jacobian for this symmetry. Moreover, the Jacobian is automatically regularized, and yields the correct result when the masses of the regulators tend to infinity. In this limit the transformations, which are non-local in a scale of 1/M, become the usual Weyl transformation of the metric. We also present the example of the chiral anomaly in 1+1 dimensions. 
  In the extended antifield formalism, a quantum BRST differential for anomalous gauge theories is constructed. Local BRST cohomological classes are characterized, besides the form degree and the ghost number, by the length of their descents and of their lifts, and this both in the standard and the extended antifield formalism. It is shown that during the BRST invariant renormalization of a local BRST cohomological class, the anomaly that can appear is constrained to be a local BRST cohomological class with a shorter descent and a longer lift than the given class. As an application of both results, a simple approach to the Adler-Bardeen theorem for the non abelian gauge anomaly is proposed. It applies independently of the gauge fixing, of power counting restrictions and does not rely on the use of the Callan-Symanzik equation. 
  We study N=1 gauge theories obtained by adding finite mass terms to N=4 Yang-Mills theory. The Maldacena dual is nonsingular: in each of the many vacua, there is an extended brane source, arising from Myers' dielectric effect. The source consists of one or more (p,q) 5-branes. In particular, the confining vacuum contains an NS5-brane; the confining flux tube is a fundamental string bound to the 5-brane. The system admits a simple quantitative description as a perturbation of a state on the N=4 Coulomb branch. Various nonperturbative phenomena, including flux tubes, baryon vertices, domain walls, condensates and instantons, have new, quantitatively precise, dual descriptions. We also briefly consider two QCD-like theories. Our method extends to the nonsupersymmetric case. As expected, the N=4 matter cannot be decoupled within the supergravity regime. 
  We investigate the noncommutative analogue of the spontaneously broken linear sigma model at the one-loop quantum level. In the commutative case, renormalization of a theory with a spontaneously broken continuous global symmetry depends on cancellations that enable the limited set of counterterms consistent with that symmetry to remove the divergences even after its spontaneous breaking, while preserving the masslessness of the associated Goldstone modes. In the noncommutative case, we find that these cancellations are violated, and the renormalized one-loop correction to the inverse pion propagator explicitly yields a mass shift which depends on the ultraviolet cutoff. Thus, we cannot naively take the ultraviolet cutoff to infinity first, and then take the external momentum to zero to verify Nambu-Goldstone symmetry realization. However, from the Wilsonian perspective where the cutoff is fixed and physical, the zero external momentum limit of the inverse pion propagator still vanishes, and implies the masslessness of the pion fields at one-loop. This is another demonstration of the failure of ultraviolet and infrared limits to commute in noncommutative field theories, and signals the incompatibility of Nambu-Goldstone symmetry realization with the continuum renormalization of these theories. 
  Magnetic monopoles are known to emerge as leading non-perturbative fluctuations in the lattice version of non-Abelian gauge theories in some gauges. In terms of the Dirac quantization condition, these monopoles have magnetic charge |Q_M|=2. Also, magnetic monopoles with |Q_M|=1 can be introduced on the lattice via the 't Hooft loop operator. We consider the |Q_M|=1,2 monopoles in the continuum limit of the lattice gauge theories. To substitute for the Dirac strings which cost no action on the lattice, we allow for singular gauge potentials which are absent in the standard continuum version. Once the Dirac strings are allowed, it turns possible to find a solution with zero action for a monopole--antimonopole pair. This implies equivalence of the standard and modified continuum versions in perturbation theory. To imitate the nonperturbative vacuum, we introduce then a nonsingular background. The modified continuum version of the gluodynamics allows in this case for monopoles with finite non-vanishing action. Using similar techniques, we construct the 't Hooft loop operator in the continuum and predict its behavior at small and large distances both at zero and high temperatures. 
  It is shown that the general solution near a spacelike singularity of the Einstein-dilaton-p-form field equations relevant to superstring theories and M-theory exhibits an oscillatory behaviour of the Belinskii-Khalatnikov-Lifshitz type. String dualities play a significant role in the analysis. 
  The RG functions of the 2D n-vector \phi^4 model are calculated in the five-loop approximation. Perturbative series for the \beta-function and critical exponents are resummed by the Pade-Borel and Pade-Borel-Leroy techniques, resummation procedures are optimized and an accuracy of the numerical results is estimated. In the Ising case n = 1 as well as in the others (n = 0, n = -1, n = 2, 3,...32) an account for the five-loop term is found to shift the Wilson fixed point location only briefly, leaving it outside the segment formed by the results of the corresponding lattice calculations; even error bars of the RG and lattice estimates do not overlap in the most cases studied. This is argued to reflect the influence of the singular (non-analytical) contribution to the \beta-function that can not be found perturbatively. The evaluation of the critical exponents for n = 1, n = 0 and n = -1 in the five-loop approximation and comparison of the numbers obtained with their known exact counterparts confirm the conclusion that non-analytical contributions are visible in two dimensions. For the 2D Ising model, the estimates \omega = 1.31(3) for the correction-to-scaling exponent is found that is close to the value 4/3 resulting from the conformal invariance. 
  Vacuum polarization in external fields is treated by way of calculating - exactly and then perturbatively - the phase of the quantum scattering matrix in the Shale-Stinespring approach to field theory. The link between the Shale-Stinespring method and the Epstein-Glaser renormalization procedure is highlighted. 
  I point out that standard two dimensional, asymptotically free, non-linear sigma models, supplemented with terms giving a mass to the would-be Goldstone bosons, share many properties with four dimensional supersymmetric gauge theories, and are tractable even in the non-supersymmetric cases. The space of mass parameters gets quantum corrections analogous to what was found on the moduli space of the supersymmetric gauge theories. I focus on a simple purely bosonic example exhibiting many interesting phenomena: massless solitons and bound states, Argyres-Douglas-like CFTs and duality in the infrared, and rearrangement of the spectrum of stable states from weak to strong coupling. At the singularities on the space of parameters, the model can be described by a continuous theory of randomly branched polymers, which is defined beyond perturbation theory by taking an appropriate double scaling limit. 
  We present a calculation of the ground state energy of massive spinor fields and massive scalar fields in the background of an inhomogeneous magnetic string with potential given by a delta function. The zeta functional regularization is used and the lowest heat kernel coefficients are calculated. The rest of the analytical calculation adopts the Jost function formalism. In the numerical part of the work the renormalized vacuum energy as a function of the radius $R$ of the string is calculated and plotted for various values of the strength of the potential. The sign of the energy is found to change with the radius. For both scalar and spinor fields the renormalized energy shows no logarithmic behaviour in the limit $R\to 0$, as was expected from the vanishing of the heat kernel coefficient $A_2$, which is not zero for other types of profiles. 
  We find static solitons stabilized by quantum corrections in a (1+1)-dimensional model with a scalar field chirally coupled to fermions. This model does not support classical solitons. We compute the renormalized energy functional including one-loop quantum corrections. We carry out a variational search for a configuration that minimizes the energy functional. We find a nontrivial configuration with fermion number whose energy is lower than the same number of free fermions quantized about the translationally invariant vacuum. In order to compute the quantum corrections for a given background field we use a phase-shift parameterization of the Casimir energy. We identify orders of the Born series for the phase shift with perturbative Feynman diagrams in order to renormalize the Casimir energy using perturbatively determined counterterms. Generalizing dimensional regularization, we demonstrate that this procedure yields a finite and unambiguous energy functional. 
  We explicitly evaluate one-loop (annulus) planar and nonplanar open string amplitudes in the presence of the background NS-NS two-form field. In the decoupling limit of Seiberg and Witten, we find that the nonplanar string amplitudes reproduce the UV/IR mixing of noncommutative field theories. In particular, the investigation of the UV regime of the open string amplitudes shows that certain IR closed string degrees of freedom survive the decoupling limit as previously predicted from the noncommutative field theory analysis. These degrees of freedom are responsible for the quadratic, linear and logarithmic IR singularities when the D-branes embedded in space-time have the codimension zero, one and two, respectively. The analysis is given for both bosonic and supersymmetric open strings. 
  In this paper, starting from pure group-theoretical point of view, we develop a regular approach to describing particles with different spins in the framework of a theory of scalar fields on the Poincare group. Such fields can be considered as generating functions for conventional spin-tensor fields. The cases of 2, 3, and 4 dimensions are elaborated in detail. Discrete transformations $C,P,T$ are defined for the scalar fields as automorphisms of the Poincare group. Doing a classification of the scalar functions, we obtain relativistic wave equations for particles with definite spin and mass. There exist two different types of scalar functions (which describe the same mass and spin), one related to a finite-dimensional nonunitary representation and another one related to an infinite-dimensional unitary representation of the Lorentz subgroup. This allows us to derive both usual finite-component wave equations for spin-tensor fields and positive energy infinite-component wave equations. 
  We show that if the beta functions of a field theory are given by the gradient of a certain potential on the space of couplings, a gravitational background in one more dimension can express the renormalization group (RG) flow of the theory. The field theory beta functions and the gradient flow constraint together reconstruct the second order spacetime equations of motion. The RG equation reduces to the conventional gravitational computation of the spacetime quasilocal stress tensor, and a c-theorem holds true as a consequence of the Raychaudhuri equation. Conversely, under certain conditions, if the RG evolution of a field theory possesses a monotonic c-function, the flow of couplings can be expressed in terms of a higher dimensional gravitational background. 
  The family of brane-world solutions of d+1-dimensional dilatonic gravity is presented. It includes flat brane with small cosmological constant and (anti) de Sitter brane, dilatonic brane-world black holes (Schwarzschild-(anti-) de Sitter, Kerr, etc). Gravitational and dilatonic perturbations around such branes are found. It is shown that near dilatonic brane-world black hole the gravity may be localized in a standard form. The brane corrections to Newton law are estimated. The proposal to take into account the dilaton coupled brane matter quantum effects is made. The corresponding effective action changes the structure of 4d de Sitter wall. RG flow of four-dimensional Newton constant in IR and UV is briefly discussed. 
  Since the SU(n) gauge theory with massive gauge bosons has been proven to be renormalisable we reinvestigate the renormalisability of the  SU$_L$(2) $\times$ U$_Y$(1) electroweak theory with massive W Z bosons. We expound that with the constraint conditions caused by the W Z mass term and the additional condition chosen by us we can performed the quantization and construct the ghost action in a way similar to that used for the massive SU(n) theory. We also show that when the $\delta-$ functions appearing in the path integral of the Green functions and representing the constraint conditions are rewritten as Fourier integrals with Lagrange multipliers $\lambda_a$ and $\lambda_y$, the BRST invariance is kept in the total effective action consisting of the Lagrange multipliers, ghost fields and the original fields. Furthermore, by comparing with the massless theory and with the massive SU(n) theory we find the general form of the divergent part of the generating functional for the regular vertex functions and prove the renormalisability of the theory. It is also clarified that the renormalisability of the theory with the W Z mass term is ensured by that of the massless theory and the massive SU(n) theory. 
  We study a simple model for which perturbation theory gives ultravioletly divergent results. We show that when the eigen-solution problem of the Hamiltonian of the model is treated nonperturbatively, it is possible for eigenenergies of the Hamiltonian to be finite. 
  We present an explicit study of the holographic renormalization group (RG) in six dimensions using minimal gauged supergravity. By perturbing the theory with the addition of a relevant operator of dimension four one flows to a non-supersymmetric conformal fixed point. There are also solutions describing non-conformal vacua of the same theory obtained by giving an expectation value to the operator. One such vacuum is supersymmetric and is obtained by using the true superpotential of the theory. We discuss the physical acceptability of these vacua by applying the criteria recently given by Gubser for the four dimensional case and find that those criteria give a clear physical picture in the six dimensional case as well. We use this example to comment on the role of the Hamilton-Jacobi equations in implementing the RG. We conclude with some remarks on AdS_4 and the status of three dimensional superconformal theories from squashed solutions of M-theory. 
  Anomalous U(1) gauge symmetries can appear both in heterotic and type I string theories. In the heterotic case we find a single anomalous U(1), while in open string theories several such symmetries can appear. Nonetheless, there is a conjectured duality symmetry that might connect these two theories. We review the properties of anomalous gauge symmetries in various string theories as well as the status of this heterotic-type I/II duality. We also comment on the possible phenomenological applications of anomalous gauge symmetries in string theory. 
  A D-dimensional induced gravity theory is studied carefully in a $4 + (D-4)$ dimensional Friedmann-Robertson-Walker space-time. We try to extract information of the symmetry breaking potential in search of an inflationary solution with non-expanding internal-space. We find that the induced gravity model imposes strong constraints on the form of symmetry breaking potential in order to generate an acceptable inflationary universe. These constraints are analyzed carefully in this paper. 
  We work out the basics of conformal $N=(4,4)$, 2D supergravity in the $N=(4,4)$, 2D analytic harmonic superspace with two independent sets of harmonic variables. We define the relevant most general analytic superspace diffeomorphism group and show that in the flat limit it goes over into the ``large'' $N=(4,4)$, 2D superconformal group. The basic objects of the supergravity considered are analytic vielbeins covariantizing two analyticity-preserving harmonic derivatives. For self-consistency they should be constrained in a certain way. We solve the constraints and show that the remaining irreducible field content in a WZ gauge amounts to a new short $N=(4,4)$ Weyl supermultiplet. As in the previously known cases, it involves no auxiliary fields and the number of remaining components in it coincides with the number of residual gauge invariances. We discuss various truncations of this ``master'' conformal supergravity group and its compensations via couplings to $N=(4,4)$ superconformal matter multiplets. Besides recovering the standard minimal off-shell $N=(4,4)$ conformal and Poincar\'e supergravity multiplets, we find, at the linearized level, several new off-shell gauge representations. 
  We investigate the electromagnetic duality properties of an abelian gauge theory on a compact oriented four-manifold by analysing the behaviour of a generalised partition function under modular transformations of the dimensionless coupling constants. The true partition function is invariant under the full modular group but the generalised partition function exhibits more complicated behaviour depending on topological properties of the four-manifold concerned. It is already known that there may be "modular weights" which are linear combinations of the Euler number and Hirzebruch signature of the four-manifold. But sometimes the partition function transforms only under a subgroup of the modular group (the Hecke subgroup). In this case it is impossible to define real spinor wave functions on the four-manifold. But complex spinors are possible provided the background magnetic fluxes are appropriately fractional rather that integral. This gives rise to a second partition function which enables the full modular group to be realised by permuting the two partition functions, together with a third. Thus the full modular group is realised in all cases. The demonstration makes use of various constructions concerning integral lattices and theta functions that seem to be of intrinsic interest. 
  I contemplate the possibility that the mismatch between the maximally symmetric point (the free fermionic point) and the strictly self-dual point in the Narain moduli space plays a role in the string vacuum selection. The role of self-duality in the recent formulation of quantum mechanics from an equivalence postulate, and the new perspective that it offers on the foundations of quantum gravity and the origin of mass, are discussed. 
  We present a class of static supersymmetric multi-center black hole solutions arising in four-dimensional N=2 supergravity theories with terms quadratic in the Weyl tensor. We also comment on possible corrections to the metric on the moduli space of these black holes solutions. 
  We apply the background field method and the effective action formalism to describe the four-dimensional dynamical Casimir effect. Our picture corresponds to the consideration of quantum cosmology for an expanding FRW universe (the boundary conditions act as a moving mirror) filled by a quantum massless GUT which is conformally invariant. We consider cases in which the static Casimir energy is repulsive and attractive. Inserting the simplest possible inertial term, we find, in the adiabatic (and semiclassical) approximation, the dynamical evolution of the scale factor and the dynamical Casimir stress analytically and numerically (for SU(2) super Yang-Mills theory). Alternative kinetic energy terms are explored in the Appendix. 
  We propose a random matrix model that interpolates between the chiral random matrix ensembles and the chiral Poisson ensemble. By mapping this model on a non-interacting Fermi-gas we show that for energy differences less than a critical energy $E_c$ the spectral correlations are given by chiral Random Matrix Theory whereas for energy differences larger than $E_c$ the number variance shows a linear dependence on the energy difference with a slope that depends on the parameters of the model. If the parameters are scaled such that the slope remains fixed in the thermodynamic limit, this model provides a description of QCD Dirac spectra in the universality class of critical statistics. In this way a good description of QCD Dirac spectra for gauge field configurations given by a liquid of instantons is obtained. 
  We find classically stable solitons (instantons) in odd (even) dimensional scalar noncommutative field theories whose scalar potential, $V(\ph)$, has at least two minima. These solutions are bubbles of the false vacuum whose size is set by the scale of noncommutativity. Our construction uses the correspondence between non-commutative fields and operators on a single particle Hilbert space. In the case of noncommutative gauge theories we note that expanding around a simple solution shifts away the kinetic term and results in a purely quartic action with linearly realised gauge symmetries. 
  Supersymmetric Ward identity for the low energy effective action in the standard background gauge is derived for {\it arbitrary} trajectories of supergravitons in Matrix Theory. In our formalism, the quantum-corrected supersymmetry transformation laws of the supergravitons are directly identified in closed form, which exhibit an intricate interplay between supersymmetry and gauge (BRST) symmetry. As an application, we explicitly compute the transformation laws for the source-probe configuration at 1-loop and confirm that supersymmetry fixes the form of the action completely, including the normalization, to the lowest order in the derivative expansion. 
  We discuss the role of the trace part of metric fluctuations $h_{MN}$ in the Randall-Sundrum scenario of gravity. Without the matter, this field ($h=\eta^{MN}h_{MN}$) is a gauge-dependent term, and thus it can be gauged away. But, including the uniform source $\tilde{T}_{MN}$, this field satisfies the linearized equation $\Box_4 h =16\pi G_5 T^{\mu}_{\mu}$. This may correspond to the scalar $\xi^5$ in the bending of the brane due to the localized source. Considering the case of longitudinal perturbations ($h_{5\mu} =h_{55}=0$), one finds the source relation $\tilde{T}^{\mu}_{\mu}=2\tilde{T}_{55}$, which leads to the ghost states in the massive modes. In addition, if one requires $T_{44}=2(T_{22}+T_{33})$, it is found that in the limit of $m^2_h \to 0$ we have the massless spin-2 propagation without the ghost state. This exactly corresponds to the same situation as in the intermediate scales of Gregory-Rubakov-Sibiryakov (GRS) model. 
  We demonstrate that under certain conditions a theory of conformal quantum mechanics will exhibit the symmetries of two half-Virasoro algebras. We further demonstrate the conditions under which these algebras combine to form a single Virasoro algebra, and comment on the connection between this result and the AdS/CFT correspondence. 
  We study a kaehler potential K of a one parameter family of Calabi-Yau d-fold embedded in CP^{d+1}. By comparing results of the topological B-model and the data of the CFT calculation at Gepner point, the K is determined unambiguously. It has a moduli parameter psi that describes a deformation of the CFT by a marginal operator. Also the metric, curvature and hermitian two-point functions in the neighborhood of the Gepner point are analyzed. We use a recipe of tt^{*} fusion and develop a method to determine the K from the point of view of topological sigma model. It is not restricted to this specific model and can be applied to other Calabi-Yau cases. 
  We study a kaehler potential K in the large radius region of a Calabi-Yau d-fold M embedded in CP^{d+1}. It has a kaehler parameter t that describes a deformation of the A-model moduli. Also the metric, curvature and hermitian two-point functions in the large volume region are analyzed. We use a result of our previous paper in the B-model of the mirror. We perform an analytic continuation of a parameter to the large complex structure region. By translating the result in the A-model side of M, we determine the K. The method is not restricted to this specific model and we apply the recipe to complete intersection Calabi-Yau cases. 
  We study a central charge Z of a one parameter family of Calabi-Yau d-fold embedded in CP^{d+1}. For a d-fold case, we construct the Z concretely and analyze charge vectors of D-branes and intersection forms of associated cycles. We find the charges are described as some kinds of Mukai vectors. They are represented as products of Chern characters of coherent sheaves restricted on the Calabi-Yau hypersurfaces and square roots of A-roof genera of the d-folds. By combining results of the topological sigma model and the data of the CFT calculations in the Gepner model, we find that the Z is determined and is specified by a set of integers. It labels boundary states in special classes where associated states are represented as tensor products of boundary states for constituent minimal models. The Z has a moduli parameter t that describes a deformation of a moduli space in the open string channel with B-type boundary conditions. Also monodromy matrices and homology cycles are investigated. 
  We extend the previous work and study the renormalisability of the SU$_L$(2) $\times$ U$_Y$(1) electroweak theory with massive W Z fields and massive matter fields. We expound that with the constraint conditions caused by the W Z mass term and the additional condition chosen by us we can still performed the quantization in the same way as before. We also show that when the $\delta-$ functions appearing in the path integral of the Green functions and representing the constraint conditions are rewritten as Fourier integrals with Lagrange multipliers $\lambda_a$ and $\lambda_y$, the total effective action consisting of the Lagrange multipliers, ghost fields and the original fields is BRST invariant. Furthermore, with the help of the the renormalisability of the theory without the the mass term of matter fields, we find the general form of the divergent part of the generating functional for the regular vertex functions and prove the renormalisability of the theory with the mass terms of the W Z fields and the matter fields. 
  We present a general `off-shell' description of the effective N=1 supergravity describing the low-energy limit of M-theory compactified on (Calabi-Yau) x S1/Z2. In our formulation, the M-theory Bianchi identities are imposed by the equations of motion of four-dimensional supermultiplets. Modifications of these identities (resulting for instance from contributions localized at orbifold singularities or non-perturbative sources like five-branes) can then easily be implemented. 
  The kappa--invariant worldvolume action for the NS5--brane in a D=10 type IIA supergravity background is obtained by carrying out the dimensional reduction of the M5--brane action. 
  We classify all the first-order vertices of gravity consistently coupled to a system of 2-form gauge fields by computing the local BRST cohomology H(s|d) in ghost number 0 and form degree n. The consistent deformations are at most linear in the undifferentiated two-form, confirming the previous results of [1] that geometrical theories constructed from a nonsymmetric gravity theory are physically inconsistent or trivial. No assumption is made here on the degree of homogeneity in the derivatives nor on the form of the gravity action. 
  We present intermediate results of an ongoing investigation which attempts a generalization of the well known one-loop Bern Kosower rules of Yang-Mills theory to higher loop orders. We set up a general procedure to extract the field theoretical limit of bosonic open string diagrams, based on the sewing construction of higher loop world sheets. It is tested with one- and two-loop scalar field theory, as well as one-loop and two-loop vacuum Yang-Mills diagrams, reproducing earlier results. It is then applied to two-loop two-point Yang-Mills diagrams in order to extract universal renormalization coefficients that can be compared to field theory. While developing numerous technical tools to compute the relevant contributions, we hit upon important conceptual questions: Do string diagrams reproduce Yang-Mills Feynman diagrams in a certain preferred gauge? Do they employ a certain preferred renormalization scheme? Are four gluon vertices related to three gluon vertices? Unfortunately, our investigations remained inconclusive up to now. 
  The gravitational anomalies in two dimensions, specifically the Einstein anomaly and the Weyl anomaly, are fully determined by means of dispersion relations. In this approach the anomalies originate from the peculiar infrared feature of the imaginary part of the relevant formfactor which approaches a $\delta$-function singularity at zero momentum squared when $m \to 0$. 
  The recently proposed remarkable mechanism explaining ``stringy exclusion principle" on an Anti de Sitter space is shown to be another beautiful manifestation of spacetime uncertainty principle in string theory as well as in M theory. Put in another way, once it is realized that the graviton of a given angular momentum is represented by a spherical brane, we deduce the maximal angular momentum directly from either the relation $\Delta t\Delta x^2>l_p^3$ in M theory or $\Delta t\Delta x>\ap$ in string theory. We also show that the result of hep-th/0003075 is similar to results on D2-branes in SU(2) WZW model. Using the dual D2-brane representation of a membrane, we obtain the quantization condition for the size of the membrane. 
  We study a general dilatonic p-brane solution in arbitrary dimensions in relation to the Randall-Sundrum scenario. When the p-brane is fully localized along its transverse directions, the Kaluza-Klein zero mode of bulk graviton is not normalizable. When the p-brane is delocalized along its transverse directions except one, the Kaluza-Klein zero mode of bulk graviton is normalizable if the warp factor is chosen to increase, in which case there are singularities at finite distance away from the p-brane. Such delocalized p-brane can be regarded as a dilatonic domain wall as seen in higher dimensions. This unusual property of the warp factor allows one to avoid a problem of dilatonic domain wall with decreasing warp factor that free massive particles are repelled from the domain wall and hit singularities, since massive particles with finite energy are trapped around delocalized p-branes with increasing warp factor by gravitational force and can never reach the singularities. 
  These lectures discuss some of the general issues in developing a phenomenology for Superstring Theory/M Theory. The focus is on the question: how might one obtain robust, generic predictions. For example, does the theory predict low energy supersymmetry breaking? In the course of these explorations, basics of supersymmetry and supersymmetry breaking, string moduli, cosmological issues, and other questions are addressed. The notion of approximate moduli and their possible role plays a central role in the discussion. 
  We consider noncommutative analogs of scalar electrodynamics and N=2 D=4 SUSY Yang-Mills theory. We show that one-loop renormalizability of noncommutative scalar electrodynamics requires the scalar potential to be an anticommutator squared. This form of the scalar potential differs from the one expected from the point of view of noncommutative gauge theories with extended SUSY containing a square of commutator. We show that fermion contributions restore the commutator in the scalar potential. This provides one-loop renormalizability of noncommutative N=2 SUSY gauge theory. We demonstrate a presence of non-integrable IR singularities in noncommutative scalar electrodynamics for general coupling constants. We find that for a special ratio of coupling constants these IR singularities vanish. Also we show that IR poles are absent in noncommutative N=2 SUSY gauge theory. 
  We develop a systematic framework for studying target space duality at the classical level. We show that target space duality between manifolds M and Mtilde arises because of the existence of a very special symplectic manifold. This manifold locally looks like M x Mtilde and admits a double fibration. We analyze the local geometric requirements necessary for target space duality and prove that both manifolds must admit flat orthogonal connections. We show how abelian duality, nonabelian duality and Poisson-Lie duality are all special cases of a more general framework. As an example we exhibit new (nonlinear) dualities in the case M = Mtilde = R^n. 
  We apply the framework developed in Target Space Duality I: General Theory. We show that both nonabelian duality and Poisson-Lie duality are examples of the general theory. We propose how the formalism leads to a systematic study of duality by studying few scenarios that lead to open questions in the theory of Lie algebras. We present evidence that there are probably new examples of irreducible target space duality. 
  We argue that the second-order gauge-invariant Schwinger-Dyson operator of a gauge theory is the Wheeler-DeWitt operator in the dual string theory. Using this identification, we construct a set of operators in the gauge theory that correspond to excitations of gravity in the bulk. We show that these gauge theory operators have the expected properties for describing the semiclassical local gravity theory. 
  By exploiting the boundary state formalism we obtain the string correlator between two internal points on the one loop open string world-sheet in the presence of a constant background $B$-field. From this derivation it is clear that there is an ambiguity when one tries to restrict the Green function to the boundary of the surface. We fix this ambiguity by showing that there is a {\em unique} form for the correlator between two points on the boundary which reproduces the one loop field theory results of different noncommutative field theories. In particular, we present the derivation of one loop diagrams for $\phi^3_6$ and $\phi^4_4$ scalar interactions and for Yang--Mills theory. From the 2-point function we are able to derive the one loop $\beta$-function for noncommutative gauge theory. 
  It is shown that the method of nonlinear realization of local supersymmetry being applied to the n=(1,1) superconformal symmetry allows one reduce the new version of the super-Liouville equation to the ordinary one owing to the relaxation of the auxiliary equation of motion fixing the gauge parameters. 
  It is shown that the method of the nonlinear realization of local supersymmetry previously developed in framework of supergravity being applied to the n=(2,2) superconformal symmetry allows one to get the new form of the exactly solvable n=(2,2) super-Liouville equation. The general advantage of this version as compared with the conventional one is that its bosonic part includes the complex Liouville equation. We obtain the suitable supercovariant constraints imposed on the corresponding superfields which provide the set of the resulting system of component equations be the same as that in model of N=2, D=4 Green-Schwarz superstring. The general solution of this system is derived from the corresponding solution of the bosonic string equation. 
  We consider D3-branes at an orbifolded conifold whose horizon ${X_5}$ resolves into a smooth Einstein manifold which joins several copies of ${\bf T}^{1,1}$. We describe in details the resolution of the singular horizon ${X_5}$ and describe different types of two-cycles appearing in the resolution. For a large number of D3 branes, the AdS/CFT conjecture becomes a duality between type IIB string theory on $AdS_5 \times {X_5} $ and the ${\cal N} = 1$ field theory living on the D3 branes. We study the fractional branes as small perturbations of the string background and we reproduce the logarithmic flow of field theory couplings by studying fluxes of NS-NS and R-R two forms through different 2-cycles of the resolved horizon. 
  We discuss the effect of boundaries in boundary logarithmic conformal field theory and show, with reference to both $c=-2$ and $c=0$ models, how they produce new features even in bulk correlation functions which are not present in the corresponding models without boundaries. We discuss the modification of Cardy's relation between boundary states and bulk quantities. 
  We study various aspects of the Kahler metric for matter fields in N=1,2 orientifold compactifications of type IIB string theory. The result has an infrared-divergent part which reproduces the field- theoretical anomalous dimensions, and a moduli-dependent part which comes from N=2 sectors of the orientifold. For the N=2 orientifolds, we also compute the disk amplitude for two matter fields on the boundary and a twisted closed string modulus in the bulk. Our results are in agreement with supersymmetry: the singlet under the SU(2)_R R-symmetry has vanishing coupling, while the coupling of the SU(2)_R triplet does not vanish. 
  Calculations of reaction rates for the third-order QED process of photon splitting in strong magnetic fields traditionally have employed either the effective Lagrangian method or variants of Schwinger's proper-time technique. Recently, Mentzel, Berg and Wunner (1994) presented an alternative derivation via an S-matrix formulation in the Landau representation. Advantages of such a formulation include the ability to compute rates near pair resonances above pair threshold. This paper presents new developments of the Landau representation formalism as applied to photon splitting, providing significant advances beyond the work of Mentzel et al. by summing over the spin quantum numbers of the electron propagators, and analytically integrating over the component of momentum of the intermediate states that is parallel to field. The ensuing tractable expressions for the scattering amplitudes are satisfyingly compact, and of an appearance familiar to S-matrix theory applications. Such developments can facilitate numerical computations of splitting considerably both below and above pair threshold. Specializations to two regimes of interest are obtained, namely the limit of highly supercritical fields and the domain where photon energies are far inferior to that for the threshold of single-photon pair creation. In particular, for the first time the low-frequency amplitudes are simply expressed in terms of the Gamma function, its integral and its derivatives. In addition, the equivalence of the asymptotic forms in these two domains to extant results from effective Lagrangian/proper-time formulations is demonstrated. 
  Branes in non-trivial backgrounds are expected to exhibit interesting dynamical properties. We use the boundary conformal field theory approach to study branes in a curved background with non-vanishing Neveu-Schwarz 3-form field strength. For branes on an $S^3$, the low-energy effective action is computed to leading order in the string tension. It turns out to be a field theory on a non-commutative `fuzzy 2-sphere' which consists of a Yang-Mills and a Chern-Simons term. We find a certain set of classical solutions that have no analogue for flat branes in Euclidean space. These solutions show, in particular, how a spherical brane can arise as bound state from a stack of D0-branes. 
  We showed in part I (hep-th/9912092) that the Hopf algebra ${\cal H}$ of Feynman graphs in a given QFT is the algebra of coordinates on a complex infinite dimensional Lie group $G$ and that the renormalized theory is obtained from the unrenormalized one by evaluating at $\ve=0$ the holomorphic part $\gamma_+(\ve)$ of the Riemann-Hilbert decomposition $\gamma_-(\ve)^{-1}\gamma_+(\ve)$ of the loop $\gamma(\ve)\in G$ provided by dimensional regularization. We show in this paper that the group $G$ acts naturally on the complex space $X$ of dimensionless coupling constants of the theory. More precisely, the formula $g_0=gZ_1Z_3^{-3/2}$ for the effective coupling constant, when viewed as a formal power series, does define a Hopf algebra homomorphism between the Hopf algebra of coordinates on the group of formal diffeomorphisms to the Hopf algebra ${\cal H}$. This allows first of all to read off directly, without using the group $G$, the bare coupling constant and the renormalized one from the Riemann-Hilbert decomposition of the unrenormalized effective coupling constant viewed as a loop of formal diffeomorphisms. This shows that renormalization is intimately related with the theory of non-linear complex bundles on the Riemann sphere of the dimensional regularization parameter $\ve$. It also allows to lift both the renormalization group and the $\beta$-function as the asymptotic scaling in the group $G$. This exploits the full power of the Riemann-Hilbert decomposition together with the invariance of $\gamma_-(\ve)$ under a change of unit of mass. This not only gives a conceptual proof of the existence of the renormalization group but also delivers a scattering formula in the group $G$ for the full higher pole structure of minimal subtracted counterterms in terms of the residue. 
  When the symmetry of a physical theory describing a finite system is deformed by replacing its Lie group by the corresponding quantum group, the operators and state function will lie in a new algebra describing new degrees of freedom. If the symmetry of a field theory is deformed in this way, the enlarged state space will again describe additional degrees of freedom, and the energy levels will acquire fine structure. The massive particles will have a stringlike spectrum lifting the degeneracy of the point-particle theory, and the resulting theory will have a non-local description. Theories of this kind naturally contain two sectors with one sector lying close to the standard theory while the second sector describes particles that should be more difficult to observe. 
  This document is an introduction to and review of two-dimensional mathematical physics. The reader is introduced to the subject matter primarily through problems, which are presented along with detailed worked solutions. For each chapter, there is a brief summary of important results and ideas that precedes the problems and the solutions. This document can be worked through independently by one wishing to learn this subject; alternatively, this could serve as a companion to a more conventional review article or textbook on the subject.   The current release contains Part I, and includes six chapters:   1. Curiosities In 2 Dimensions 2. General Principles Of Conformal Field Theory 3. Correlators In Critical Models 4. Other Models In Conformal Field Theory 5. Construction Of New Models 6. Modular Invariance 
  Consistent interactions between Yang-Mills gauge fields and an abelian 2-form are investigated by using a Hamiltonian cohomological procedure. It is shown that the deformation of the BRST charge and the BRST-invariant Hamiltonian of the uncoupled model generates the Yang-Mills Chern-Simons interaction term. The resulting interactions deform both the gauge transformations and their algebra, but not the reducibility relations. 
  Consistent interactions among a set of two-form gauge fields in four dimensions are derived along a Hamiltonian cohomological procedure. It is shown that the deformation of the BRST charge and BRST-invariant Hamiltonian for the free model leads to the Freedman-Townsend interaction vertex. The resulting interaction deforms both the gauge transformations and reducibility relations, but not the algebra of gauge transformations. 
  An irreducible cohomological derivation of the Freedman-Townsend vertex in four dimensions is given. 
  Supersymmetric ground state wave functions of a model of supersymmetric quantum mechanics on $S^1$ (supersymmetric simple pendulum) are studied. Supersymmetry can be broken due to the existence of an undetermined parameter, which is interpreted as a gauge field and appears as a firm consequence of quantization on a space with a nontrivial topology such as $S^1$. The breaking does not depend on the leading term of the superpotential, contrary to the usual case. The mechanism of supersymmetry breaking is similar to that through boundary conditions of fields in supersymmetric quantum field theory on compactified space. The supersymmetric harmonic oscillator is realized in the limit of the infinite radius of $S^1$ with the strength of the oscillator being constant. 
  We study two-dimensional, large $N$ field theoretic models (Gross-Neveu model, 't Hooft model) at finite baryon density near the chiral limit. The same mechanism which leads to massless baryons in these models induces a breakdown of translational invariance at any finite density. In the chiral limit baryonic matter is characterized by a spatially varying chiral angle with a wave number depending only on the density. For small bare quark masses a sine-Gordon kink chain is obtained which may be regarded as simplest realization of the Skyrme crystal for nuclear matter. Characteristic differences between confining and non-confining models are pointed out. 
  A geometric theory of brane-worlds with large or non-compact extra dimensions is presented. It is shown that coordinate gauge independent perturbations of the brane-world correspond to the Einstein-Hilbert dynamics derived from the embeddings of the brane-world. The quantum states of a perturbation are described by Schr\"odinger's equation with respect to the extra dimensions and the deformation Hamiltonian. A gauge potential with confined components is derived from the differentiable structure of the brane-world 
  We discuss the existance of smooth soliton solutions which interpolate between supersymmetric vacua in odd-dimensional theories. In particular we apply this analysis to a wide class of supergravities to argue against the existence of smooth domain walls interpolating between supersymmetric vacua. We find that if the superpotential changes sign then any Goldstino modes will diverge. 
  We use string duality to describe instanton induced spontaneous supersymmetry breaking in string compactifications with additional background fields. Dynamical supersymmetry breaking by space-time instantons in the heterotic string theory is mapped to a tree level breaking in the type II string which can be explicitly calculated by geometric methods. It is argued that the instanton corrections resurrect the no-go theorem on partial supersymmetry breaking. The point particle limit describes the non-perturbative scalar potential of a SYM theory localized on a hypersurface of space-time. The N=0 vacuum displays condensation of magnetic monopoles and confinement. The supersymmetry breaking scale is determined by $M_{str}$, which can be in the TeV range, and the geometry transverse to the gauge theory. 
  A Hamiltonian BRST deformation procedure for obtaining consistent interactions among fields with gauge freedom is proposed. The general theory is exemplified on the three-dimensional Chern-Simons model. 
  We consider the compactification M(atrix) theory on a Riemann surface Sigma of genus g>1. A natural generalization of the case of the torus leads to construct a projective unitary representation of pi_1(\Sigma), realized on the Hilbert space of square integrable functions on the upper half--plane. A uniquely determined gauge connection, which in turn defines a gauged sl_2(R) algebra, provides the central extension. This has a geometric interpretation as the gauge length of a geodesic triangle, and corresponds to a 2-cocycle of the 2nd Hochschild cohomology group of the Fuchsian group uniformizing Sigma. Our construction can be seen as a suitable double-scaling limit N\to\infty, k\to-\infty of a U(N) representation of pi_1(Sigma), where k is the degree of the associated holomorphic vector bundle, which can be seen as the higher-genus analog of 't Hooft's clock and shift matrices of QCD. We compare the above mentioned uniqueness of the connection with the one considered in the differential-geometric approach to the Narasimhan-Seshadri theorem provided by Donaldson. We then use our infinite dimensional representation to construct a C^\star-algebra which can be interpreted as a noncommutative Riemann surface Sigma_\theta. Finally, we comment on the extension to higher genus of the concept of Morita equivalence. 
  We compute the braiding for the `principal gradation' of $U_q(\hat{{\it sl}_2})$ for $|q|=1$ from first principles, starting from the idea of a rigid braided tensor category. It is not necessary to assume either the crossing or the unitarity condition from S-matrix theory. We demonstrate the uniqueness of the normalisation of the braiding under certain analyticity assumptions, and show that its convergence is critically dependent on the number-theoretic properties of the number $\tau$ in the deformation parameter $q=e^{2\pi i\tau}$. We also examine the convergence using probability, assuming a uniform distribution for $q$ on the unit circle. 
  The renormalized photon and electron propagators are expanded over planar binary trees. Explicit recurrence solutions are given for the terms of these expansions. In the case of massless Quantum Electrodynamics (QED), the relation between renormalized and bare expansions is given in terms of a Hopf algebra structure. For massive quenched QED, the relation between renormalized and bare expansions is given explicitly. 
  We compute four-point correlation functions of scalar composite operators in the N=4 supercurrent multiplet at order g^4 using the N=1 superfield formalism. We confirm the interpretation of short-distance logarithmic behaviours in terms of anomalous dimensions of unprotected operators exchanged in the intermediate channels and we determine the two-loop contribution to the anomalous dimension of the N=4 Konishi supermultiplet. 
  Based on the canonical quantization of open strings ending on D-branes with a background B field, we construct the open string propagator. We demonstrate the relation between the T duality of the underlying string theory and the Morita equivalence of the interpolating general Dirac-Born-Infeld theory on a noncommutative torus in the nonzero modulus \Phi sector. The general noncommutative Dirac-Born-Infeld action with the Wess-Zumino terms expressed by the background R-R fields is shown to be Morita invariant. 
  A non supersymmetric string background, directly derived from the string soft dilaton theorem, is used to compute, in the semiclassical approximation, the expectation value of Wilson loops in static gauge. The resulting potential shares common features with the one obtained through Schwarzschild-anti de Sitter spacetime metrics. In particular a linear confining potential appears naturally. 
  In an induced-gravity model, the stability condition of an inflationary slow-rollover solution is shown to be $\phi_0 \partial_{\phi_0}V(\phi_0)=4V(\phi_0)$. The presence of higher derivative terms will, however, act against the stability of this expanding solution unless further constraints on the field parameters are imposed. We find that these models will acquire a non-vanishing cosmological constant at the end of inflation. Some models are analyzed for their implication to the early universe. 
  In this work an extension of the Gaussian model of the stochastic vacuum is presented. It consists of including higher cumulants than just the second one in the cluster expansion in QCD. The influence of nonabelian fourth cumulants on the potential of a static quark-antiquarkpair is examined and the formation of flux tubes between a static $q\bar{q}$-pair is investigated. It is found that the fourth cumulants can contribute to chromomagnetic flux tubes. Furthermore the contribution of fourth cumulants to the total cross section of soft high energy hadron-hadron scattering is examined and it is found that the fourth cumulants do not change the general picture obtained in the Gaussian model. 
  We perform Monte Carlo simulations of a supersymmetric matrix model, which is obtained by dimensional reduction of 4D SU(N) super Yang-Mills theory. The model can be considered as a four-dimensional counterpart of the IIB matrix model. We extract the space-time structure represented by the eigenvalues of bosonic matrices. In particular we compare the large N behavior of the space-time extent with the result obtained from a low energy effective theory. We measure various Wilson loop correlators which represent string amplitudes and we observe a nontrivial universal scaling in N. We also observe that the Eguchi-Kawai equivalence to ordinary gauge theory does hold at least within a finite range of scale. Comparison with the results for the bosonic case clarifies the role of supersymmetry in the large N dynamics. It does affect the multi-point correlators qualitatively, but the Eguchi-Kawai equivalence is observed even in the bosonic case. 
  We construct the finite temperature field theory of the two-dimensional ghost-antighost system within the framework of thermo field theory. 
  The zero modes of the chiral SU(n) WZNW model give rise to an intertwining quantum matrix algebra A generated by an n x n matrix a=(a^i_\alpha) (with noncommuting entries) and by rational functions of n commuting elements q^{p_i}. We study a generalization of the Fock space (F) representation of A for generic q (q not a root of unity) and demonstrate that it gives rise to a model of the quantum universal enveloping algebra U_q(sl_n), each irreducible representation entering F with multiplicity 1. For an integer level k the complex parameter q is an even root of unity, q^h=-1 (h=k+n) and the algebra A has an ideal I_h such that the factor algebra A_h = A/I_h is finite dimensional. 
  T-duality realized on SuperD-brane effective actions probing in constant $G_{mn}$ and $b_{mn}$ backgrounds is studied from a pure world volume point of view. It is proved that requiring {\em T-duality covariance} of such actions ``fixes'' the T-duality transformations of the world volume dynamical fields, and consequently, of the NS-NS and R-R coupling superfields. The analysis is extended to uncover the mapping of the symmetry structure associated with these SuperD-brane actions. In particular, we determine the T-duality transformation properties of kappa symmetry and supersymmetry, which allow us to prove that bosonic supersymmetric world volume solitons of the original theory generate, through T-duality, the expected ones in the T-dual theory. The latter proof is generalized to arbitrary bosonic backgrounds. We conclude with some comments on extensions of our approach to arbitrary kappa symmetric backgrounds, non-BPS D-branes and non-abelian SuperD-branes. 
  We recall the concept of Baxterisation of an R-matrix, or of a monodromy matrix, which corresponds to build, from one point in the $ R$-matrix parameter space, the algebraic variety where the spectral parameter(s) live. We show that the Baxterisation, which amounts to studying the iteration of a birational transformation, is a ``win-win'' strategy: it enables to discard efficiently the non-integrable situations, focusing directly on the two interesting cases where the algebraic varieties are of the so-called ``general type'' (finite order iteration) or are Abelian varieties (infinite order iteration). We emphasize the heuristic example of the sixteen vertex model and provide a complete description of the finite order iterations situations for the Baxter model. We show that the Baxterisation procedure can be introduced in much larger frameworks where the existence of some underlying Yang-Baxter structure is not used: we Baxterise L-operators, local quantum Lax matrices, and quantum Hamiltonians. 
  Magnetic and Dyonic solutions are constructed for the theory of abelian gauged N=2 gauged four dimensional supergravity coupled to vector multiplets. The solutions found preserve 1/4 of the supersymmetry. 
  We generalize Witten's conjectured formula relating Donaldson and Seiberg-Witten invariants to manifolds of non-simple type, via equivariant localization techniques. This approach does not use the theory of non-abelian monopoles, but works directly on the Donaldson-Witten and Seiberg-Witten moduli spaces. We give a formal derivation of Witten's conjecture and its generalization, making use of an infinite dimensional version of the abelian localization theorem. 
  We show how string theory can be used to reproduce the one-loop two-point photon amplitude in noncommutative U(1) gauge theory. Using a simple realization of the gauge theory in bosonic string theory, we extract from a string cylinder computation in the decoupling limit the exact one loop field theory result. The result is obtained entirely from the region of moduli space where massless open strings dominate. Our computation indicates that the unusual IR/UV singularities of noncommutative field theory do not come from closed string modes in any simple way. 
  We extend a recent computation of the dependence of the free energy, F, on the noncommutative scale $\theta$ to theories with very different UV sensitivity. The temperature dependence of $F$ strongly suggests that a reduced number of degrees of freedom contributes to the free energy in the non-planar sector, $F_{\rm np}$, at high temperature. This phenomenon seems generic, independent of the UV sensitivity, and can be traced to modes whose thermal wavelengths become smaller than the noncommutativity scale. The temperature dependence of $F_{\rm np}$ can then be calculated at high temperature using classical statistical mechanics, without encountering a UV catastrophe even in large number of dimensions. This result is a telltale sign of the low number of degrees of freedom contributing to $F$ in the non-planar sector at high temperature. Such behavior is in marked contrast to what would happen in a field theory with a random set of higher derivative interactions. 
  We present the first computation of the thermodynamic properties of the complex su(3) Toda theory. This is possible thanks to a new string hypothesis, which involves bound states that are non self-conjugate solutions of the Bethe equations. Our method provides equivalently the solution of the su(3) generalization of the XXZ chain. In the repulsive regime, we confirm that the scattering theory proposed over the past few years - made only of solitons with non diagonal S-matrices - is complete. But we show that unitarity does not follow, contrary to early claims, eigenvalues of the monodromy matrix not being pure phases. In the attractive regime, we find that the proposed minimal solution of the bootstrap equations is actually far from being complete. We discuss some simple values of the couplings, where, instead of the few conjectured breathers, a very complex structure (involving E_6, or two E_8) of bound states is necessary to close the bootstrap. 
  We study near-extremal n-point correlation functions of chiral primary operators, in which the maximal scale dimension k is related to the others by k=\sum_i k_i-m with m equal to or smaller than n-3. Through order g^2 in field theory, we show that these correlators are simple sums of terms each of which factors into products of lower-point correlators. Terms which contain only factors of two- and three-point functions are not renormalized, but other terms have non-vanishing order g^2 corrections.   We then show that the contributing AdS exchange diagrams neatly match this factored structure. In particular, for n=4,5 precise agreement in form and coefficient is established between supergravity and the non-renormalized factored terms from field theory. On the other hand, contact diagrams in supergravity would produce a non-factored structure. This leads us to conjecture that the corresponding bulk couplings vanish, so as to achieve full agreement between the structure of these correlators in supergravity and weak-coupling field theory. 
  We describe a tachyon-free stable non-BPS brane configuration in type IIA string theory. The configuration is an elliptic model involving rotated NS5 branes, D4 branes and anti-D4 branes, and is dual to a fractional brane-antibrane pair placed at a conifold singularity. This configuration exhibits an interesting behaviour as we vary the radius of the compact direction. Below a critical radius the D4 and anti-D4 branes are aligned, but as the radius increases above the critical value the potential between them develops a minimum away from zero. This signals a phase transition to a configuration with finitely separated branes. 
  We compute the tachyon potential to level 4 in NS superstring field theory. We obtain 89% of the conjectured vacuum energy. 
  We employ T-duality to restrict the tachyon dependence of effective actions for non-BPS D-branes. For the Born-Infeld part the criteria of T-duality and supersymmetry are satisfied by a simple extension of the D-brane Born-Infeld action. 
  We study the gauge-fixing and symmetries (BRST-invariance and vector supersymmetry) of various six-dimensional topological models involving Abelian or non-Abelian 2-form potentials. 
  We study the IIB matrix model, which is conjectured to be a nonperturbative definition of superstring theory, by introducing an integer deformation parameter `nu' which couples to the imaginary part of the effective action induced by fermions. The deformed IIB matrix model continues to be well-defined for arbitrary `nu', and it preserves gauge invariance, Lorentz invariance, and the cluster property. We study the model at `nu' = infinity using a saddle-point analysis, and show that ten-dimensional Lorentz invariance is spontaneously broken at least down to an eight-dimensional one. We argue that it is likely that the remaining eight-dimensional Lorentz invariance is further broken, which can be checked by integrating over the saddle-point configurations using standard Monte Carlo simulation. 
  It is shown how a integrable mechanical system provides all the localized static solutions of a deformation of the linear O(N)-sigma model in two space-time dimensions. The proof is based on the Hamilton-Jacobi separability of the mechanical analogue system that follows when time-independent field configurations are being considered. In particular, we describe the properties of the different kinds of kinks in such a way that a hierarchical structure of solitary wave manifolds emerges for distinct N. 
  Using the formal analogy between the Dick superstring inspired model of ref.[6] and the problem of building of Eguchi Hanson metric in 4d N=2 harmonic superspace (hs), we derive a general formula for the quark-quark interaction potential V(r) including the Dick confining potential. The interquark potential V(r) depends on the dilaton-gluon coupling and may be related to the parameterization of confinement by the quark and gluon vacuum condensates. It is also shown how the axion field may be incorporated in agreement with 10d type IIB superstring requirements. Others features are also discussed. 
  Starting with the non-BPS D0-brane solution of IIB/$(-1)^{F_L}I_4$ constructed recently by Eyras and Panda we construct via T-duality the non-BPS D2-brane and D1-brane solutions of IIB/$(-1)^{F_L}I_4$ and IIA/$(-1)^{F_L}I_4$ predicted by Sen. The D2-brane couples magnetically to the vector field of the NS5B-brane living in the twisted sector of the Type IIB orbifold, whereas the D1-brane couples (electrically and magnetically) to the self-dual 2-form potential of the NS5A-brane that is present in the twisted sector of the Type IIA orbifold construction. Finally we discuss the eleven dimensional interpretation of these branes as originating from a non-BPS M1-brane solution of M-theory orientifolded by $\Omega_\rho I_5$. 
  We study the renormalization of non-commutative gauge theories with matter.  As in the scalar field theory cases, there are logarithmic infrared divergences resulting from integrating out high momentum modes. In order to reproduce the correct infrared behaviour, the Wilsonian effective action has to include certain ''closed string`` modes with prescribed couplings.   In the case of quiver gauge theories, realized in string theory on orbifolds, we identify the required modes with a set of twisted sector fields. These closed string modes have exactly the prescribed couplings to correct the Wilsonian effective action. This provides a concrete origin for the appearance of closed string modes in noncommutative field theories. 
  In this paper we prove a conjecture regarding the form of the Born-Infeld Lagrangian with a U(1)^2n gauge group after the elimination of the auxiliary fields. We show that the Lagrangian can be written as a symmetrized trace of Lorentz invariant bilinears in the field strength. More generally we prove a theorem regarding certain solutions of unilateral matrix equations of arbitrary order. For solutions which have perturbative expansions in the matrix coefficients, the solution and all its positive powers are sums of terms which are symmetrized in all the matrix coefficients and of terms which are commutators. 
  The vacuum energy-momentum tensor (EMT) and the vacuum energy corresponding to massive scalar field on $\Re_{t}\times [0,l] \times \Re^{D-2}$ space-time with boundary condition involving a dimensional parameter ($\delta$) are found. The dependent on the cavity size $l$ Casimir energy $\wt E_{C}$ is a uniquely determinable function of mass $m$, size $l$ and "skin-depth" $\delta$. This energy includes the "bulk" and the surface (potential energy) contributions. The latter dominates when $l \sim \delta$. Taking the surface potential energy into account is crucial for the coincidence between the derivative $-\d \wt E_{C}/\d l$ and the $ll$-component of the vacuum EMT. Casimir energy $\wt E_C$ and the bulk contribution to it are interconnected through Legendre transformation, in which the quantity $\delta^{-1}$ is conjugate to the vacuum surface energy multiplied by $\delta$. The surface singularities of the vacuum EMT do not depend on $l$ and, for even $D$, $\delta =0$ or $\infty$, possess finite interpretation. The corresponding vacuum energy is finite and retains known analytical dependence on the dimension $D$. 
  We compute all 3-point functions of the ``universal'' scalar operators contained in the interacting, maximally supersymmetric CFTs at large N by using the AdS/CFT correspondence. These SCFTs are related to the low energy description of M5, M2 and D3 branes, and the common set of universal scalars corresponds through the AdS/CFT relation to the fluctuations of the metric and the magnetic potential along the internal manifold. For the interacting (0,2) SCFT_6 at large N, which is related to M5 branes, this set of scalars is complete, while additional non-universal scalar operators are present in the d=4, N=4 super Yang-Mills theory and in the N=8 SCFT_3, related to D3 and M2 branes, respectively. 
  We solve the non-linear monopole equation of the Born-Infeld theory to all orders in the NS 2-form and give physical implications of the result. The solution is constructed by extending the earlier idea of rotating the brane configuration of the Dirac monopole in the target space. After establishing the non-linear monopole, we explore the non-commutative monopole by the Seiberg-Witten map. 
  We give a perturbative quantization of space-time $R^4$ in the case where the commutators $C^{{\mu}{\nu}}=[X^{\mu},X^{\nu}]$ of the underlying algebra generators are not central . We argue that this kind of quantum space-times can be used as regulators for quantum field theories . In particular we show in the case of the ${\phi}^4$ theory that by choosing appropriately the commutators $C^{{\mu}{\nu}}$ we can remove all the infinities by reproducing all the counter terms . In other words the renormalized action on $R^4$ plus the counter terms can be rewritten as only a renormalized action on the quantum space-time $QR^4$ . We conjecture therefore that renormalization of quantum field theory is equivalent to the quantization of the underlying space-time $R^4$ . 
  We present calculation of the anomaly cancellation in M-theory on orbifolds $S^1/Z_2$ and $T^5/Z_2$ in the upstairs approach. The main requirement that allows one to uniquely define solutions to the modified Bianchi identities in this case is that the field strength $G$ be globally defined on $S^1$ or $T^5$ and properly transforming under $Z_2$. We solve for general $G$ that satisfies these requirements and explicitly construct anomaly-free theories in the upstairs approach. We also obtain the solutions in the presence of five-branes. All these constructions show equivalence of the downstairs and upstairs approaches. For example in the $S^1/Z_2$ case the ten-dimensional gauge coupling and the anomaly cancellation at each wall are the same as in the downstairs approach. 
  We consider the derivatives which appear in the context of noncommutative string theory. First, we identify the correct derivations to use when the underlying structure of the theory is a quasitriangular Hopf algebra. Then we show that this is a specific case of a more general structure utilising the Drinfel'd twist. We go on to present reasons as to why we feel that the low-energy effective action, when written in terms of the original commuting coordinates, should explicitly exhibit this twisting. 
  We study theories generated by orbifolding the {\cal N}=4 super conformal U(N) Yang Mills theory with finite N, focusing on the r\^ole of the remnant U(1) gauge symmetries of the orbifold process. It is well known that the one loop beta functions of the non abelian SU(N) gauge couplings vanish in these theories. It is also known that in the large N limit the beta functions vanish to all order in perturbation theory. We show that the beta functions of the non abelian SU(N) gauge couplings vanish to two and three loop order even for finite N. This is the result of taking the abelian U(1) of U(N)=SU(N)xU(1) into account. However, the abelian U(1) gauge couplings have a non vanishing beta function. Hence, those theories are not conformal for finite N. We analyze the renormalization group flow of the orbifold theories, discuss the suppression of the cosmological constant and tackle the hierarchy problem in the non supersymmetric models. 
  We give a short review of a large class of warped string geometries, obtained via F-theory compactified on Calabi-Yau fourfolds, that upon reduction to 5 dimensions give consistent supersymmetric realizations of the RS compactification scenario. 
  We revive an old result, that one-loop corrections to the graviton propagator induce 1/r^3 corrections to the Newtonian gravitational potential, and compute the coefficient due to closed loops of the U(N) {\cal N}=4 super-Yang-Mills theory that arises in Maldacena's AdS/CFT correspondence. We find exact agreement with the coefficient appearing in the Randall-Sundrum brane-world proposal. This provides more evidence for the complementarity of the two pictures. 
  A new mathematical theory, non-associative geometry, providing a unified algebraic description of continuous and discrete spacetime, is introduced. 
  In this note we address the problem of finding Abelian instantons of finite energy on the Euclidean Schwarzschild manifold. This amounts to construct self-dual L^2 harmonic 2-forms on the space. Gibbons found a non-topological L^2 harmonic form in the Taub-NUT metric, leading to Abelian instantons with continuous energy. We imitate his construction in the case of the Euclidean Schwarzschild manifold and find a non-topological self-dual L^2 harmonic 2-form on it. We show how this gives rise to Abelian instantons and identify them with SU(2)-instantons of Pontryagin number 2n^2 found by Charap and Duff in 1977. Using results of Dodziuk and Hitchin we also calculate the full L^2 harmonic space for the Euclidean Schwarzschild manifold. 
  We discuss the relations of the M-9-brane with other branes via dimensional reductions, mainly focusing on their Wess-Zumino (WZ) actions. It is shown that on three kinds of dimensional reductions, the WZ action of the M-9-brane respectively gives those of the D-8-brane, the ``KK-8A brane'' (which we regard as a kind of D-8-brane) and the ``NS-9A brane'', the last two actions of which were obtained via dualities. Based on these results, we conclude that the relation of p-branes for $p\ge 8$, proposed previously, is consistent from the viewpoint of worldvolume actions. 
  We study the properties of branes in supergravity theory. We investigate a class of systems consisting of an M5-brane in the Kaluza-Klein monopole background with 1/4 supersymmetry in 11-dimensions. In the near core region of the KK-monopoles, the exact supergravity solution corresponding to each of these configurations is obtained. Then we argue the compactified 10-dimensional systems and suggest a way of unambiguous identification of branes in this background. Here the location of Dirac string type singularity accompanied by the D6-branes plays an important role. The method is essentially the same as that of (p,q)5-branes or (p,q)-strings within the 7-brane background in the IIB theory. We also argue the phenomena of D4-brane creation from D6-branes. 
  We consider Landau-Ginzburg (LG) models with boundary conditions preserving A-type N=2 supersymmetry. We show the equivalence of a linear class of boundary conditions in the LG model to a particular class of boundary states in the corresponding CFT by an explicit computation of the open-string Witten index in the LG model. We extend the linear class of boundary conditions to general non-linear boundary conditions and determine their consistency with A-type N=2 supersymmetry. This enables us to provide a microscopic description of special Lagrangian submanifolds in C^n due to Harvey and Lawson. We generalise this construction to the case of hypersurfaces in P^n. We find that the boundary conditions must necessarily have vanishing Poisson bracket with the combination (W(\phi)-\bar{W}(\bar{\phi})), where W(\phi) is the appropriate superpotential for the hypersurface. An interesting application considered is the T^3 supersymmetric cycle of the quintic in the large complex structure limit. 
  Polarization-free generators, i.e. ``interacting'' Heisenberg operators which are localized in wedge-shaped regions of Minkowski space and generate single particle states from the vacuum, are a novel tool in the analysis and synthesis of two-dimensional integrable quantum field theories. In the present article, the status of these generators is analyzed in a general setting. It is shown that such operators exist in any theory and in any number of spacetime dimensions. But in more than two dimensions they have rather delicate domain properties in the presence of interaction. If, for example, they are defined and temperate on a translation-invariant, dense domain, then the underlying theory yields only trivial scattering. In two-dimensional theories, these domain properties are consistent with non-trivial interaction, but they exclude particle production. Thus the range of applications of polarization-free generators seems to be limited to the realm of two-dimensional theories. 
  We Study instanton solutions in general relativity with a scalar field. The metric ansatz we use is composed of a particular warp product of general Einstein metrics, such as those found in a number of cosmological settings, including string cosmology, supergravity compactifications and general Kaluza Klein reductions. Using the Hartle-Hawking prescription the instantons we obtain determine whether metrics involving extra compact dimensions of this type are favoured as initial conditions for the universe. Specifically, we find that these product metric instantons, viewed as constrained instantons, do have a local minima in the action. These minima are then compared with the higher dimensional version of the Hawking-Turok instantons, and we argue that the latter always have lower action than those associated with these product metrics. 
  The N=2 superconformal Ward identities and their anomalies are discussed in N=2 superspace (including N=2 harmonic superspace), at the level of the low-energy effective action (LEEA) in four-dimensional N=2 supersymmetric field theories. The (first) chiral N=2 supergravity compensator is related to the known N=2 anomalous Ward identity in the N=2 (abelian) vector mulitplet sector. As regards the hypermultiplet LEEA given by the N=2 non-linear sigma-model (NLSM), a new anomalous N=2 superconformal Ward identity is found, whose existence is related to the (second) analytic compensator in N=2 supergravity. The celebrated solution of Seiberg and Witten is known to obey the (first) anomalous Ward identity in the Coulomb branch. We find a few solutions to the new anomalous Ward identity, after making certain assumptions about unbroken internal symmetries. Amongst the N=2 NLSM target space metrics governing the hypermultiplet LEEA are the SU(2)-Yang-Mills-Higgs monopole moduli-space metrics that can be encoded in terms of the spectral curves (Riemann surfaces), similarly to the Seiberg-Witten-type solutions. After a dimensional reduction to three spacetime dimensions (3d), our results support the mirror symmetry between the Coulomb and Higgs branches in 3d, N=4 gauge theories. 
  Using a gauge invariant reduction we directly integrate the SL(2,R)/U(1) WZNW theory. We prove that the conserved parafermions of this theory are coset currents. Quantum mechanically, the parafermion algebra, the energy-momentum tensor, and `auxiliary' parafermions are deformed in a self-consistent manner. 
  It is shown that in the two-exponential version of Liouville theory the coefficients of the three-point functions of vertex operators can be determined uniquely using the translational invariance of the path integral measure and the self-consistency of the two-point functions. The result agrees with that obtained using conformal bootstrap methods. Reflection symmetry and a previously conjectured relationship between the dimensional parameters of the theory and the overall scale are derived. 
  We present warped metrics which solve Einstein equations with arbitrary cosmological constants in both in upper and lower dimensions. When the lower-dimensional metric is the maximally symmetric one compatible with the chosen value of the cosmological constant, the upper-dimensional metric is also the maximally symmetric one and there is maximal unbroken supersymmetry as well. We then introduce brane sources and find solutions with analogous properties, except for supersymmetry, which is generically broken in the orbifolding procedure (one half is preserved in two special cases), and analyze metric perturbations in these backgrounds In analogy with the D8-brane we propose an effective $(\hat{d}-2)$-brane action which acts as a source for the RS solution. The action consists of a Nambu-Goto piece and a Wess-Zumino term containing a $(\hat{d}-1)$-form field. It has the standard form of the action for a BPS extended object, in correspondence with the supersymmetry preserved by the solution. 
  We consider the Maldacena conjecture applied to the near horizon geometry of a D1-brane in the supergravity approximation and present numerical results of a test of the conjecture against the boundary field theory calculation using DLCQ. We previously calculated the two-point function of the stress-energy tensor on the supergravity side; the methods of Gubser, Klebanov, Polyakov, and Witten were used. On the field theory side, we derived an explicit expression for the two-point function in terms of data that may be extracted from the supersymmetric discrete light cone quantization (SDLCQ) calculation at a given harmonic resolution. This yielded a well defined numerical algorithm for computing the two-point function. For the supersymmetric Yang-Mills theory with 16 supercharges that arises in the Maldacena conjecture, the algorithm is perfectly well defined; however, the size of the numerical computation prevented us from obtaining a numerical check of the conjecture. We now present numerical results with approximately 1000 times as many states as we previously considered. These results support the Maldacena conjecture and are within $10-15%$ of the predicted numerical results in some regions. Our results are still not sufficient to demonstrate convergence, and, therefore, cannot be considered to a numerical proof of the conjecture. We present a method for using a ``flavor'' symmetry to greatly reduce the size of the basis and discuss a numerical method that we use which is particularly well suited for this type of matrix element calculation. 
  We numerically examine the self-dual solutions of self-intersecting strings immersed in four dimensions. We find that open torus knots have topologies that can support monopole/anti-monopole as well as q-qbar production and annihilation. We give an estimate of the string tension from quark production rates, show that the intersection number for torus-knots is 4(p-q), and discuss a chiral symmetry breaking mechanism due to self-intersections. We supply a MAPLE program that can annimate the torus-knot solutions. These annimations can be found at http://www-hep.physics.uiowa.edu/~bacus/research.html under the ``Animation Control Panel''. 
  We discuss spacetime singularity resolution in the context of the gravity/gauge correspondence, for brane systems which give rise to gauge theories with eight supercharges and no hypermultiplets. The discussion is aimed at non-experts. Writeup of talk on hep-th/9911161 with C.V.Johnson and J.Polchinski, given in various forms at: PASCOS-99, Aspen winter conference `Way Beyond the Standard Models', Banff CIAR Gravity+Cosmology Programme Meeting. 
  We investigate the dynamics of Q-balls in one, two and three space dimensions, using numerical simulations of the full nonlinear equations of motion. We find that the dynamics of Q-balls is extremely complex, involving processes such as charge transfer and Q-ball fission. We present results of simulations which illustrate the salient features of 2-Q-ball interactions and give qualitative arguments to explain them in terms of the evolution of the time-dependent phases. 
  Following Feynman's successful treatment of the polaron problem we apply the same variational principle to quenched QED in the worldline formulation. New features arise from the description of fermions by Grassmann trajectories, the supersymmetry between bosonic and fermionic variables and the much more singular structure of a renormalizable gauge theory like QED in 3+1 dimensions. We take as trial action a general retarded quadratic action both for the bosonic and fermionic degrees of freedom and derive the variational equations for the corresponding retardation functions. We find a simple analytic, non-perturbative, solution for the anomalous mass dimension gamma_m(alpha) in the MS scheme. For small couplings we compare our result with recent four-loop perturbative calculations while at large couplings we find that gamma_m(alpha) becomes proportional to (alpha)^(1/2). The anomalous mass dimension shows no obvious sign of the chiral symmetry breaking observed in calculations based on the use of Dyson-Schwinger equations, however we find that a perturbative expansion of gamma_m(alpha) diverges for alpha > 0.7934. Finally, we investigate the behaviour of gamma_m(alpha) at large orders in perturbation theory. 
  We analyse 4-dimensional massive $\vp^4$ theory at finite temperature T in the imaginary-time formalism. We present a rigorous proof that this quantum field theory is renormalizable, to all orders of the loop expansion. Our main point is to show that the counterterms can be chosen temperature independent, so that the temperature flow of the relevant parameters as a function of $T$ can be followed. Our result confirms the experience from explicit calculations to the leading orders. The proof is based on flow equations, i.e. on the (perturbative) Wilson renormalization group. In fact we will show that the difference between the theories at T>0 and at T=0 contains no relevant terms. Contrary to BPHZ type formalisms our approach permits to lay hand on renormalization conditions and counterterms at the same time, since both appear as boundary terms of the renormalization group flow. This is crucial for the proof. 
  This is the first in a series of papers addressing the phenomenon of dimensional transmutation in nonrelativistic quantum mechanics within the framework of dimensional regularization. Scale-invariant potentials are identified and their general properties are derived. A strategy for dimensional renormalization of these systems in the strong-coupling regime is presented, and the emergence of an energy scale is shown, both for the bound-state and scattering sectors. Finally, dimensional transmutation is explicitly illustrated for the two-dimensional delta-function potential. 
  We study rolling radii solutions in the context of the four- and five-dimensional effective actions of heterotic M-theory. For the standard four-dimensional solutions with varying dilaton and T-modulus, we find approximate five-dimensional counterparts. These are new, generically non-separating solutions corresponding to a pair of five-dimensional domain walls evolving in time. Loop corrections in the four-dimensional theory are described by certain excitations of fields in the fifth dimension. We point out that the two exact separable solutions previously discovered are precisely the special cases for which the loop corrections are time-independent. Generically, loop corrections vary with time. Moreover, for a subset of solutions they increase in time, evolving into complicated, non-separating solutions. In this paper we compute these solutions to leading, non-trivial order. Using the equations for the induced brane metric, we present a general argument showing that the accelerating backgrounds of this type cannot evolve smoothly into decelerating backgrounds. 
  The transition from the quantum to the classical regime of the nucleation of the closed Robertson-Walker Universe with spacially homogeneous matter fields is investigated with a perturbation expansion around the sphaleron configuration. A criterion is derived for the occurrence of a first-order type transition, and the related phase diagram for scalar and vector fields is obtained. For scalar fields both the first and second order transitions can occur depending on the shape of the potential barrier. For a vector field, here that of an O(3) nonlinear $\sigma$-model, the transition is seen to be only of the first order. 
  We recalculate the two-loop beta functions in the two-dimensional Sine-Gordon model in a two-parameter expansion around the asymptotically free point. Our results agree with those of Amit et al., J. Phys. A13 (1980) 585. 
  We show the presence of Poincare anomaly in Maxwell-Chern-Simons theory with an explicit mass term, in 2+1-dimensions. 
  We investigate the algebraic structure of the most general neutrino mass Hamiltonian and place the see-saw mechanism in an algebraic framework. We show that this Hamiltonian can be written in terms of the generators of an Sp(4) algebra. The Pauli-Gursey transformation is an SU(2) rotation which is embedded in this Sp(4) group. This SU(2) also generates the see-saw mechanism. 
  We show that the complete superalgebra of symmetries, including central charges, that underlies F-theories, M-theories and type II string theories in dimensions 12, 11 and 10 of various signatures correspond to rewriting of the same OSp(1|32) algebra in different covariant ways. One only has to distinguish the complex and the unique real algebra. We develop a common framework to discuss all signatures theories by starting from the complex form of OSp(1|32). Theories are distinguished by the choice of basis for this algebra. We formulate dimensional reductions and dualities as changes of basis of the algebra. A second ingredient is the choice of a real form corresponding to a specific signature. The existence of the real form of the algebra selects preferred spacetime signatures. In particular, we show how the real d=10 IIA and IIB superalgebras for various signatures are related by generalized T-duality transformations that not only involve spacelike but also timelike directions. A third essential ingredient is that the translation generator in one theory plays the role of a central charge operator in the other theory. The identification of the translation generator in these algebras leads to the star algebras of Hull, which are characterized by the fact that the positive definite energy operator is not part of the translation generators. We apply our results to discuss different T-dual pictures of the D-instanton solution of Euclidean IIB supergravity. 
  We discuss a model in which our universe is pictured as a recoiling Dirichlet brane: we find that a proper treatment of the recoil leads naturally to supersymmetry obstruction on the four-dimensional world. An essential feature of our approach is the fact that the underlying worldsheet sigma model is non-critical, and the Liouville mode plays the role of the target time. Also, the extra bulk dimensions are viewed as sigma model couplings, and as such have to be averaged by appropriate summation over worldsheet genera. The recoiling brane is in an excited state rather than its ground state, to which it relaxes asymptotically in time, restoring supersymmetry. We also find that the excitation energy, which is considered as the observable effective cosmological `constant' on the brane, is naturally small and can accommodate upper bounds from observations. 
  We begin the study of the spectrum of BPS branes and its variation on lines of marginal stability on O_P^2(-3), a Calabi-Yau ALE space asymptotic to C^3/Z_3. We show how to get the complete spectrum near the large volume limit and near the orbifold point, and find a striking similarity between the descriptions of holomorphic bundles and BPS branes in these two limits. We use these results to develop a general picture of the spectrum. We also suggest a generalization of some of the ideas to the quintic Calabi-Yau. 
  Every classical sigma-model with target space a compact symmetric space $G/H$ (with $G$ classical) is shown to possess infinitely many local, commuting, conserved charges which can be written in closed form. The spins of these charges run over a characteristic set of values, playing the role of exponents of $G/H$, and repeating modulo an integer $h$ which plays the role of a Coxeter number. 
  We investigate the low-energy effective action in N=4 super Yang-Mills theory with gauge group SU(n) spontaneously broken down to its maximal torus. Using harmonic superspace technique we prove an absence of any three- and four-loop corrections to non-holomorphic effective potential depending on N=2 superfield strengths. A mechanism responsible for vanishing arbitrary loop corrections to low-energy effective action is discussed. 
  Shape Invariant potentials in the sense of [Gendenshte\"{\i}n L.\'E., JETP Lett. 38, (1983) 356] which depend on more than two parameters are not know to date. In [Cooper F., Ginocchio J.N. and Khare A., Phys. Rev. {\bf 36 D}, (1987) 2458] was posed the problem of finding a class of Shape Invariant potentials which depend on n parameters transformed by translation, but it was not solved. We analyze the problem using some properties of the Riccati equation and we find the general solution. 
  A thorough analysis is presented of the class of central fields of force that exhibit: (i) dimensional transmutation and (ii) rotational invariance. Using dimensional regularization, the two-dimensional delta-function potential and the $D$-dimensional inverse square potential are studied. In particular, the following features are analyzed: the existence of a critical coupling, the boundary condition at the origin, the relationship between the bound-state and scattering sectors, and the similarities displayed by both potentials. It is found that, for rotationally symmetric scale-invariant potentials, there is a strong-coupling regime, for which quantum-mechanical breaking of symmetry takes place, with the appearance of a unique bound state as well as of a logarithmic energy dependence of the scattering with respect to the energy. 
  The q-deformed super Virasoro algebra proposed by Chaichian and Presnajder is examined. Presented is the realizations by the FFZ algebra (the magnetic translation algebra) defined on a two-dimensional lattice with a supersymmetric Hamiltonian. 
  We consider Randall-Sundrum model with localized gravity, replacing the extra compact space-like dimension by a time-like one. In this way the solution to the hierarchy problem can be reconciled with a correct cosmological expansion of the visible universe, just as a trivial result of the sign flip of cosmological constants in the bulk and on the 3-branes relative to the case of extra space-like dimension. Some phenomenological aspects of the proposed scenario related to the tachyonic nature of Kaluza-Klein states of graviton are also discussed. 
  We consider a field theoretical model on the noncommutative cylinder which leads to a discrete-time evolution. Its Euclidean version is shown to be equivalent to a model on the complex $q$-plane. We reveal a direct link between the model on a noncommutative cylinder and the deformed Virasoro algebra constructed earlier on an abstract mathematical background. As it was shown, the deformed Virasoro generators necessarily carry a second index (in addition to the usual one), whose meaning, however, remained unknown. The present field theoretical approach allows one to ascribe a clear meaning to this second index: its origin is related to the noncommutativity of the underlying space-time. The problems with the supersymmetric extension of the model on a noncommutative super-space are briefly discussed. 
  Gravitation theories selected by requiring that they have a unique anti-de Sitter vacuum with a fixed cosmological constant are studied. For a given dimension d, the Lagrangians under consideration are labeled by an integer k=1,2,...,[(d-1)/2]. Black holes for each d and k are found and are used to rank these theories. A minimum possible size for a localized electrically charged source is predicted in the whole set of theories, except General Relativity. It is found that the thermodynamic behavior falls into two classes: If d-2k=1, these solutions resemble the three dimensional black hole, otherwise, their behavior is similar to the Schwarzschild-AdS_4 geometry. 
  The results obtained by Seiberg and Witten for the low-energy Wilsonian effective actions of N=2 supersymmetric theories with gauge group SU(2) are in agreement with instanton computations carried out for winding numbers one and two. This suggests that the instanton saddle point saturates the non-perturbative contribution to the functional integral. A natural framework in which corrections to this approximation are absent is given by the topological field theory built out of the N=2 Super Yang-Mills theory. After extending the standard construction of the Topological Yang-Mills theory to encompass the case of a non-vanishing vacuum expectation value for the scalar field, a BRST transformation is defined (as a supersymmetry plus a gauge variation), which on the instanton moduli space is the exterior derivative. The topological field theory approach makes the so-called "constrained instanton" configurations and the instanton measure arise in a natural way. As a consequence, instanton-dominated Green's functions in N=2 Super Yang-Mills can be equivalently computed either using the constrained instanton method or making reference to the topological twisted version of the theory. We explicitly compute the instanton measure and the contribution to $u=<\Tr \phi^2>$ for winding numbers one and two. We then show that each non-perturbative contribution to the N=2 low-energy effective action can be written as the integral of a total derivative of a function of the instanton moduli. Only instanton configurations of zero conformal size contribute to this result. Finally, the 8k-dimensional instanton moduli space is built using the hyperkahler quotient procedure, which clarifies the geometrical meaning of our approach. 
  Based on the concept of the partial breaking of global supersymmetry (PBGS), we derive the worldvolume superfield equations of motion for $N=1, D=4$ supermembrane, as well as for the space-time filling D2- and D3-branes, from nonlinear realizations of the corresponding supersymmetries. We argue that it is of no need to take care of the relevant automorphism groups when being interested in the dynamical equations. This essentially facilitates computations. As a by-product, we obtain a new polynomial representation for the $d=3,4$ Born-Infeld equations, with merely a cubic nonlinearity. 
  We evaluate the Casimir vacuum energy at finite temperature associated with the Maxwell field confined by a perfectly conducting rectangular cavity and show that an extended version of the temperature inversion symmetry is present in this system. 
  A solution of the Randall-Sundrum model for a simplified case (one wall) is obtained. It is given by the $1/k^2$-expansion (thin wall expansion) where $1/k$ is the {\it thickness} of the domain wall. The vacuum setting is done by the 5D Higgs potential and the solution is for a {\it family} of the Higgs parameters. The mass hierarchy problem is examined. Some physical quantities in 4D world such as the Planck mass, the cosmological constant, and fermion masses are focussed. Similarity to the domain wall regularization used in the chiral fermion problem is explained. We examine the possibility that the 4D massless chiral fermion bound to the domain wall in the 5D world can be regarded as the real 4D fermions such as neutrinos, quarks and other leptons. 
  Belavin's $\mathbb{Z}_n$-symmetric elliptic model with boundary reflection is considered on the basis of the boundary CTM bootstrap. We find non-diagonal $K$-matrices for $n>2$ that satisfy the reflection equation (boundary Yang--Baxter equation), and also find non-diagonal Boltzmann weights for the $A^{(1)}_{n-1}$-face model even for $n\geqq 2$. We derive difference equations of the quantum Knizhnik-Zamolodchikov type for correlation functions of the boundary model. The boundary spontaneous polarization is obtained by solving the simplest difference equations. The resulting quantity is the square of the spontaneous polarization for the bulk $\mathbb{Z}_n$-symmetric model, up to a phase factor. 
  A new model of supersymmetry between bosons and fermions is proposed. Its representation space is spanned by states with PT symmetry and real energies but the inter-related partner Hamiltonians themselves remain complex and non-Hermitian. The formalism admits vanishing Witten index. 
  It has been conjectured that an extremum of the tachyon potential of a bosonic D-brane represents the vacuum without any D-brane, and that various tachyonic lump solutions represent D-branes of lower dimension. We show that the tree level effective action of p-adic string theory, the expression for which is known exactly, provides an explicit realisation of these conjectures. 
  We find that sometimes the usual definition of functional integration over the gauge group through limiting process may have internal difficulties. 
  We derive the vertex operators that are expected to govern the emission of the massless d=11 supermultiplet from the supermembrane in the light cone gauge. We demonstrate that they form a representation of the supersymmetry algebra and reduce to the type IIA superstring vertex operators under double dimensional reduction, as well as to the vertices of the d=11 superparticle in the point-particle limit. As a byproduct, our results can be used to derive the corresponding vertex operators for matrix theory and to describe its linear coupling to an arbitrary d=11 supergravity background. Possible applications are discussed. 
  We describe string-theory and $d=11$ supergravity solutions involving symmetric spaces of constant negative curvature. Many examples of non-supersymmetric string compactifications on hyperbolic spaces $H_r$ of finite volume are given in terms of suitable cosets of the form $H_r/\Gamma $, where $\Gamma $ is a discrete group. We describe in some detail the cases of the non-compact hyperbolic spaces $F_2$ and $F_3$, representing the fundamental regions of $H_2$ and $H_3$ under $SL(2,Z)$ and the Picard group, respectively. By writing $AdS$ as a U(1) fibration, we obtain new solutions where $AdS_{2p+1}$ gets untwisted by T-duality to ${\bf R}\times SU(p,1)/(SU(p)\times U(1))$. Solutions with time-dependent dilaton field are also constructed by starting with a solution with NS5-brane flux over $H_3$. A new class of non-supersymmetric conformal field theories can be defined via holography. 
  We show that N=1, D=4 Einstein-frame supergravity is inconsistent at one loop because of an anomaly in local supersymmetry transformations. A Jacobian must be added to the Einstein-frame Lagrangian to cancel this anomaly. We show how the Jacobian arises from the super-Weyl field redefinition that takes the superspace Lagrangian to the Einstein frame. We present an explicit example which demonstrates that the Jacobian is necessary for one-loop scattering amplitudes to be frame independent. 
  Starting from Generating functional for Green Function (GF), constracted from Lagrangian action in Duffin-Kemmer-Petiau (DKP) theory (L-approach) we strictly prove that the physical matrix elements of S-matrix in DKP and Klein-Gordon-Fock (KGF) theories coincide in cases of interaction spin O particles with external and quantized Maxwell and Yang-Mills fields and in case of external gravitational field (without or with torsion). For the proof we use reduction formulas of Lehmann Symanzik, Zimmermann (LSZ). We prove that many photons and Yang-Mills particles GF coincide in the both theories, too. 
  We discuss non-extremal rotating D3-branes. We solve the wave equation for scalars in the supergravity background of certain distributions of branes and compute the absorption coefficients. The form of these coefficients is similar to the gray-body factors associated with black-hole scattering. They are given in terms of two different temperature parameters, indicating that fields (open string modes) do not remain in thermal equilibrium as we move off extremality. This should shed some light on the origin of the disagreement between the supergravity and conformal field theory results on the free energy of a system of non-coincident D-branes. 
  We describe a new first-order formulation of D=11 supergravity which shows that that theory can be understood to arise from a certain topological field theory by the imposition of a set of local constraints on the fields, plus a lagrange multiplier term. The topological field theory is of interest as the algebra of its constraints realizes the D=11 supersymmetry algebra with central charges. 
  We study the circumstances under which a Kaluza-Klein reduction on an n-sphere, with a massless truncation that includes all the Yang-Mills fields of SO(n+1), can be consistent at the full non-linear level. We take as the starting point a theory comprising a p-form field strength and (possibly) a dilaton, coupled to gravity in the higher dimension D. We show that aside from the previously-studied cases with (D,p)=(11,4) and (10,5) (associated with the S^4 and S^7 reductions of D=11 supergravity, and the S^5 reduction of type IIB supergravity), the only other possibilities that allow consistent reductions are for p=2, reduced on S^2, and for p=3, reduced on S^3 or S^{D-3}. We construct the fully non-linear Kaluza-Klein Ansatze in all these cases. In particular, we obtain D=3, N=8, SO(8) and D=7, N=2, SO(4) gauged supergravities from S^7 and S^3 reductions of N=1 supergravity in D=10. 
  The issue of field redefinition invariance of path integrals in quantum field theory is reexamined. A ``paradox'' is presented involving the reduction to an effective quantum-mechanical theory of a $(d+1)$-dimensional free scalar field in a Minkowskian spacetime with compactified spatial coordinates. The implementation of field redefinitions both before and after the reduction suggests that operator-ordering issues in quantum field theory should not be ignored. 
  The classical solution of bosonic d-brane in (d+1,1) space-time is studied. We work with light-cone gauge and reduce the problem into Chaplygin gas problem. The static equation is equivalent to vanishing of extrinsic mean curvature, which is similar to Einstein equation in vacuum. We show that the d-brane problem in this gauge is closely related to Plateau problem, and we give some non-trivial solutions from minimal surfaces. The solutions of d-1,d,d+1 spatial dimensions are obtained from d-dimensional minimal surfaces as solutions of Plateau problem. In addition we discuss on the relation to Hamiltonian-BRST formalism for d-branes. 
  We consider the N=4 SU(N) Super Yang Mills theory on the Coulomb branch with gauge symmetry broken to S(U(N_1) x U(N_2)). By integrating the W particles, the effective action near the IR SU(N_i) conformal fixed points is seen to be a deformation of the Super Yang Mills theory by a non-renormalized, irrelevant, dimension 8 operator. The correction to the two-point function of the dilaton field dual operator near the IR is related to a three-point function of chiral primary operators at the conformal fixed points and agrees with the classical gravity prediction, including the numerical factor. 
  In D-dimensional spacetimes which can be foliated by n-dimensional homogeneous subspaces, a quantum field can be decomposed in terms of modes on the subspaces, reducing the system to a collection of (D-n)-dimensional fields. This allows one to write bare D-dimensional field quantities like the Green function and the effective action as sums of their (D-n)-dimensional counterparts in the dimensionally reduced theory. It has been shown, however, that renormalization breaks this relationship between the original and dimensionally reduced theories, an effect called the dimensional-reduction anomaly. We examine the dimensional-reduction anomaly for the important case of spherically symmetric spaces. 
  In its Euclidean formulation, the AdS/CFT correspondence begins as a study of Yang-Mills conformal field theories on the sphere, S^4. It has been successfully extended, however, to S^1 X S^3 and to the torus T^4. It is natural to hope that it can be made to work for any manifold on which it is possible to define a stable Yang-Mills conformal field theory. We consider a possible classification of such manifolds, and show how to deal with the most obvious objection : the existence of manifolds which cannot be represented as boundaries. We confirm Witten's suggestion that this can be done with the help of a brane in the bulk. 
  We give an explicit realization of quantum Nambu bracket via matrix of multi-index, which reduces in the continunm limit to the classical Nambu bracket. 
  We construct explicit form of the anomalous effective action, in arbitrary even dimension, for Abelian vector and axial gauge fields coupled to Dirac fermions. It turns out to be a surprisingly simple extension of 2D Schwinger model effective action. 
  We show that the gauge groups SU(N), SO(N) and Sp(N) cannot be realized on a flat noncommutative manifold, while it is possible for U(N). 
  Complex frequencies associated with quasinormal modes for large Reissner-Nordstr$\ddot{o}$m Anti-de Sitter black holes have been computed. These frequencies have close relation to the black hole charge and do not linearly scale with the black hole temperature as in Schwarzschild Anti-de Sitter case. In terms of AdS/CFT correspondence, we found that the bigger the black hole charge is, the quicker for the approach to thermal equilibrium in the CFT. The properties of quasinormal modes for $l>0$ have also been studied. 
  We review recent developments in non-perturbative field theory using modal field methods. We discuss Monte Carlo results as well as a new diagonalization technique known as the quasi-sparse eigenvector method. 
  The present article is primarily a review of the projection-operator approach to quantize systems with constraints. We study the quantization of systems with general first- and second-class constraints from the point of view of coherent-state, phase-space path integration, and show that all such cases may be treated, within the original classical phase space, by using suitable path-integral measures for the Lagrange multipliers which ensure that the quantum system satisfies the appropriate quantum constraint conditions. Unlike conventional methods, our procedures involve no $\delta$-functionals of the classical constraints, no need for dynamical gauge fixing of first-class constraints nor any average thereover, no need to eliminate second-class constraints, no potentially ambiguous determinants, as well as no need to add auxiliary dynamical variables expanding the phase space beyond its original classical formulation, including no ghosts. Besides several pedagogical examples, we also study: (i) the quantization procedure for reparameterization invariant models, (ii) systems for which the original set of Lagrange mutipliers are elevated to the status of dynamical variables and used to define an extended dynamical system which is completed with the addition of suitable conjugates and new sets of constraints and their associated Lagrange multipliers, (iii) special examples of alternative but equivalent formulations of given first-class constraints, as well as (iv) a comparison of both regular and irregular constraints. 
  In string theory various projections have to be imposed to ensure supersymmetry. We study the consequences of these projections in the presence of world sheet boundaries. A-type boundary conditions come in several classes; only boundary fields that do not change the class preserve supersymmetry. Our analysis takes in particular properly into account the resolution of fixed points under the projections. Thus e.g. the compositeness of some previously considered boundary states of Gepner models follows from chiral properties of the projections. Our arguments are model independent; in particular, integrality of all annulus coefficients is ensured by model independent arguments. 
  We investigate (4+1)- and (5+0)-dimensional gravity coupled to a non-compact scalar field sigma-model and a perfect fluid within the context of the Randall-Sundrum scenario. We find cosmological solutions with a rolling fifth radius and a family of warp factors. Included in this family are both the original Randall-Sundrum solution and the self-tuning solution of Kachru, Schulz and Silverstein. Our solutions exhibit conventional cosmology. 
  Solutions of Einstein's equations are found for global defects in a higher-dimensional spacetime with a nonzero cosmological constant Lambda. The defect has a (p-1)-dimensional core (brane) and a `hedgehog' scalar field configuration in the n extra dimensions. For Lambda = 0 and n > 2, the solutions are characterized by a flat brane worldsheet and a solid angle deficit in the extra dimensions. For Lambda > 0, one class of solutions describes spherical branes in an inflating higher-dimensional universe. Instantons obtained by a Euclidean continuation of such solutions describe quantum nucleation of the entire inflating brane-world, or of a spherical brane in an inflating higher-dimensional universe. For Lambda < 0, one class of solutions exhibits an exponential warp factor. It is similar to spacetimes previously discussed by Randall and Sundrum for n = 1 and by Gregory for n = 2. 
  The nonrelativistic interpretation of quantum field theory achieved by quantization in an infinite momentum frame is spoiled by the inclusion of a mode of the field carrying p+=0. We therefore explore the viability of doing without such a mode in the context of spontaneous symmetry breaking (SSB), where its presence would seem to be most needed. We show that the physics of SSB in scalar quantum field theory in 1+1 space-time dimensions is accurately described without a zero-mode. 
  The Lorentz-covariant quantization performed in the Hamiltonian path-integral formalism for massless non-Abelian gauge fields has been achieved. In this quantization, the Lorentz condition, as a constraint, must be introduced initially and incorporated into the Yang-Mills Lagrangian by the Lagrange undetermined multiplier method. In this way, it is found that all Lorentz components of a vector potential have thier corresponding conjugate canonical variables. This fact allows us to define Lorentz-invariant poisson brackets and carry out the quantization in a Lorent-covariant manner. Key words: Non-Abelian gauge field, quantization, Hamiltonian path-integral formalism, Lorentz covariance. 
  BFYM on commutative and noncommutative ${\mathbb{R}}^4$ is considered and a Seiberg-Witten gauge-equivalent transformation is constructed for these theories. Then we write the noncommutative action in terms of the ordinary fields and show that it is equivalent to the ordinary action up to higher dimensional gauge invariant terms. 
  We propose that the UV/IR relation that underlies the AdS/CFT duality may provide a natural mechanism by which high energy supersymmetry can have large distance consequences. We motivate this idea via (a string realization of) the Randall-Sundrum scenario, in which the observable matter is localized on a matter brane separate from the Planck brane. As suggested via the holographic interpretation of this scenario, we argue that the local dynamics of the Planck brane - which determines the large scale 4-d geometry - is protected by the high energy supersymmetry of the dual 4-d theory. With this assumption, we show that the total vacuum energy naturally cancels in the effective 4-d Einstein equation. This cancellation is robust against changes in the low energy dynamics on the matter brane, which gets stabilized via the holographic RG without any additional fine-tuning. 
  We study the evaporation of black holes in space-times with extra dimensions of size L. We first obtain a potential which describes the expected behaviors of very large and very small black holes and then show that a (first order) phase transition, possibly signaled by an outburst of energy, occurs in the system when the horizon shrinks below L from a larger value. This is related to both a change in the topology of the horizon and the restoring of the translational symmetry along the extra dimensions. 
  The Gauss-Bonnet interaction is the only consistent quadratic interaction below the Planck scale in the Randall-Sundrum compactification. We study various static and inflationary solutions including this Gauss-Bonnet interaction. 
  We study the C_2 Ruijsenaars-Schneider(RS) model with interaction potential of trigonometric type. The Lax pairs for the model with and without spectral parameter are constructed. Also given are the involutive Hamiltonians for the system. Taking nonrelativistic limit, we obtain the Lax pair of C_2 Calogero-Moser model. 
  The axial-gauge boson propagator contains 1/(n.k)^p-type singularities. These singularities have generally been treated by inventing prescriptions for them. We propose an alternative procedere for treating these singularities in the path-integral formalism using the known way of treating the 1/k^{2n}-type singularities in Lorentz-type gauges. For this purpose we use a finite field-dependent BRS transformation that inerpolates between the Lorentz and axial-type gauges. We arrive at the \epsilon-dependent tree propagator in axial-type gauges. 
  It is shown that next-nearest-neighbor interactions may lead to unusual paramagnetic or ferromagnetic phases which physical content is radically different from the standard phases. Actually there are several particles described by the same quantum field in a manner similar to the species doubling of the lattice fermions. We prove the renormalizability of the theory at the one loop level. 
  In this talk we discuss two classes of examples of warped products of AdS spaces in the context of the AdS/CFT correspondence. The first class of examples appears in the construction of dual Type I' string descriptions to five dimensional supersymmetric fixed points with E_{N_f+1} global symmetry. The background is obtained as the near horizon geometry of the D4-D8 brane system in massive Type IIA supergravity. The second class of examples appears when considering the N=2 superconformal theories defined on a 3+1 dimensional hyperplane intersection of two sets of M5 branes. We use the dual string formulations to deduce properties of these field theories. 
  The $\epsilon$-expansion of several two-loop self-energy diagrams with different thresholds and one mass are calculated. On-shell results are reduced to multiple binomial sums which values are presented in analytical form. 
  We consider systems of D6 branes in the presence of a nonzero $B$ field of different ranks. We study the scattering of gravitons in the corresponding supergravity backgrounds. We show that the nonzero $B$ field does not modify the form of the scattering potential. The graviton scattering equation has two solutions one normalizable and one non-normalizable. The normalizable solution does not lead to an absorption, however the non-normalizable one does. We analyse the absorption of gravitons by the branes and show that it is nonzero in the decoupling limit. This result suggests that even in the presence of a $B$ field the D6 branes worldvolume theory does not decouple from the bulk gravity. For comparison we analyse the form of the scattering potential and absorption for Dp branes with $p <5$ and for NS5 branes. 
  We consider the quantum mechanical notion of the geometrical (Berry) phase in SU(2) gauge theory, both in the continuum and on the lattice. It is shown that in the coherent state basis eigenvalues of the Wilson loop operator naturally decompose into the geometrical and dynamical phase factors. Moreover, for each Wilson loop there is a unique choice of U(1) gauge rotations which do not change the value of the Berry phase. Determining this U(1) locally in terms of infinitesimal Wilson loops we define monopole-like defects and study their properties in numerical simulations on the lattice. The construction is gauge dependent, as is common for all known definitions of monopoles. We argue that for physical applications the use of the Lorenz gauge is most appropriate. And, indeed, the constructed monopoles have the correct continuum limit in this gauge. Physical consequences are briefly discussed. 
  Motivated by recent discussions of IR/UV mixing in noncommutative field theories, we perform a detailed analysis of the non-planar amplitudes of the bosonic open string in the presence of an external B-field at the one-loop level. We carefully isolate, at the string theory level, the contribution which is responsible for the IR/UV behavior in the field theory limit. We show that it is a pure open string effect by deriving it from the factorization of the one-loop amplitude into the disk amplitudes of intermediate open string insertions. We suggest that it is natural to understand IR/UV mixing as the creation of intermediate ``stretched strings''. 
  We present a metric solution in six dimensions where gravity is localized on a four-dimensional singular string-like defect. The corrections to four-dimensional gravity from the bulk continuum modes are suppressed by ${\cal O}(1/r^3)$. No tuning of the bulk cosmological constant to the brane tension is required in order to cancel the four-dimensional cosmological constant. 
  We extend the recent computation of the tachyon potential by Berkovits, Sen and Zwiebach by including level two fields and keeping up to level four terms in the action. We find 90.5% of the expected result. 
  The presence of cosmological perturbations affects the background metric and matter configuration in which the perturbations propagate. This effect, studied a long time ago for gravitational waves, also is operational for scalar gravitational fluctuations, inhomogeneities which are believed to be more important in inflationary cosmology. The back-reaction of fluctuations can be described by an effective energy-momentum tensor. The issue of coordinate invariance makes the analysis more complicated for scalar fluctuations than for gravitational waves. We show that the back-reaction of fluctuations can be described in a diffeomorphism-invariant way. In an inflationary cosmology, the back-reaction is dominated by infrared modes. We show that these modes give a contribution to the effective energy-momentum tensor of the form of a negative cosmological constant whose absolute value grows in time. We speculate that this may lead to a self-regulating dynamical relaxation mechanism for the cosmological constant. This scenario would naturally lead to a finite remnant cosmological constant with a magnitude corresponding to $\Omega_{\Lambda} \sim 1$. 
  The quantum mechanics of $N$ slowly-moving BPS black holes in five dimensions is considered. A divergent continuum of states describing arbitrarily closely bound black holes with arbitrarily small excitation energies is found. A superconformal structure appears at low energies and can be used to define an index counting the weighted number of supersymmetric bound states. It is shown that the index is determined from the dimensions of certain cohomology classes on the symmetric product of $N$ copies of $R^4$. An explicit computation for the case of N=2 with no angular momentum yields a finite nonzero result. 
  A straightforward two-line derivation of the Bekenstein-Hawking Area-Entropy relation for Black-Holes in {\bf any} dimension is shown based on Shannon's information theory and Clifford algebras required by the New Relativity Principle. 
  The dynamics of an N=4 spinning particle in a curved background is described using the N=4 superfield formalism. The $SU(2)_{local}\times SU(2)_{global}$ N=4 superconformal symmetry of the particle action requires the background to be a real "K\"ahler-like" manifold whose metric is generated by a sigma-model superpotential. The anti-de-Sitter spaces are shown to belong to this class of manifolds. 
  Let $P$ be an operator of Dirac type on a compact Riemannian manifold with smooth boundary. We impose spectral boundary conditions and study the asymptotics of the heat trace of the associated operator of Laplace type. 
  We find all possible static embeddings of a 4-brane in any dimension-6 spacetime with 4-dimensional Poincare symmetry and a negative cosmological constant, subject to orbifolding across the brane. Our new solutions allow intersecting branes at an angle determined by a new dynamical parameter. A collection of branes intersecting in one 3-brane allows an arbitrary excess angle that can be related to a vacuum density along the intersection. The 3-brane must be stabilized by additional fine-tuned interactions. We also note that localization of gravity is tied to the approximate fine tuning of the brane tensions. 
  Branes are ubiquitous elements of any low-energy limit of string theory. We point out that negative tension branes violate all the standard energy conditions of the higher-dimensional spacetime they are embedded in; this opens the door to very peculiar solutions of the higher-dimensional Einstein equations. Building upon the (3+1)-dimensional implementation of fundamental string theory, we illustrate the possibilities by considering a toy model consisting of a (2+1)-dimensional brane propagating through our observable (3+1)-dimensional universe. Developing a notion of "brane surgery", based on the Israel--Lanczos--Sen "thin shell" formalism of general relativity, we analyze the dynamics and find traversable wormholes, closed baby universes, voids (holes in the spacetime manifold), and an evasion (not a violation) of both the singularity theorems and the positive mass theorem. These features appear generic to any D-brane model that permits negative tension branes: This includes the Randall--Sundrum models and their variants. 
  We calculate all of vielbein superfields up to second order in anticommuting coordinates in terms of the component fields of 11-dimensional on-shell supergravity by using `Gauge completion'. This configuration of superspace holds the $\kappa $-symmetry for supermembrane Lagrangian and represents 11-dimensional on-shell supergravity. 
  We construct noncommutative extension of the Wess-Zumino-Witten (WZW) model and study its ultraviolet property. The \beta-function of the U(N) noncommutative WZW model resembles that of the ordinary WZW model. The U(1) noncommutative model has also a nontrivial fixed point. 
  Symmetric product orbifolds, i.e. permutation orbifolds of the full symmetric group S_{n} are considered by applying the general techniques of permutation orbifolds. Generating functions for various quantities, e.g. the torus partition functions and the Klein-bottle amplitudes are presented, as well as a simple expression for the discrete torsion coefficients. 
  We review several aspects of Yang-Mills theory (YMT) in two dimensions, related to its perturbative and topological properties. Consistency between light-front and equal-time formulations is thoroughly discussed. 
  The formation of near-extremal Reissner-Nordstrom black holes in the S-wave approximation can be described, near the event horizon, by an effective solvable model. The corresponding one-loop quantum theory remains solvable and allows to follow analytically the evaporation process which is shown to require an infinite amount of time. 
  We clarify some general issues in models where gravity is localized at intermediate distances. We introduce the radion mode, which is usually neglected, and we point out that its role in the model is crucial. We show that the brane bending effects discussed in the literature can be obtained in a formalism where the physical origin is manifest. The model violates positivity of energy due to a negative tension brane, which induces a negative kinetic term for the radion. The very same effect that violates positivity is responsible for the recovery of conventional Einstein gravity at intermediate distances. 
  J. F. van Diejen and H. Puschmann have recently shown that the dynamics of zeros of the n-solitonic solutions to the Schrodinger equation with the reflectionless potential is governed by a rational Ruijsenaars-Schneider system. We use the algebraic-geometrical construction of solutions to the Schrodinger equation to generalize this result to the elliptic case. 
  In this note we show how Ward identities may be derived for a quantum field theory dual of a string theory using the AdS/CFT correspondence. In particular associated with any gauge symmetry of the bulk supergravity theory there is a corresponding constraint equation. Writing this constraint in Hamilton-Jacobi form gives a generating functional for Ward identities in the dual QFT. We illustrate the method by considering various examples. 
  We describe the dynamics of a relativistic extended object in terms of the geometry of a configuration of constant time. This involves an adaptation of the ADM formulation of canonical general relativity. We apply the formalism to the hamiltonian formulation of a Dirac-Nambu-Goto relativistic extended object in an arbitrary background spacetime. 
  The topological nature of Chern-Simons term describing the interaction of a charge with magnetic monopole is manifested in two ways: it changes the plane dynamical geometry of a free particle for the cone dynamical geometry without distorting the free (geodesic) character of the motion, and in the limit of zero charge's mass it describes a spin system. This observation allows us to interpret the charge-monopole system alternatively as a free particle of fixed spin with translational and spin degrees of freedom interacting via the helicity constraint, or as a symmetric spinning top with dynamical moment of inertia and "isospin" U(1) gauge symmetry, or as a system with higher derivatives. The last interpretation is used to get the twistor formulation of the system. We show that the reparametrization and scale invariant monopole Chern-Simons term supplied with the kinetic term of the same invariance gives rise to the alternative description for the spin, which is related to the charge-monopole system in a spherical geometry. The relationship between the charge-monopole system and (2+1)-dimensional anyon is discussed in the light of the obtained results. 
  The idea of ``soft'' confinement when the lifetime of hadron with respect to quark-gluon channel of decay is greater or at least of the order of some characteristic time for our Universe is considered. Within the framework of a model of three nonlinearly interacting fields the explicit form of an effective potential is found. It provides the confinement of a massive particle within the limited region of space by means of constant component of the potential which arises as a result of reorganization of vacuum of one of the scalar fields. It is shown that the lifetime of hadron being equal to the age of the Universe leads to the Higgs boson with the mass m > 63.7 GeV and realistic self-constant. 
  In a model of nonlinear system of three scalar fields the problem on dynamics of a massive particle moving in effective potential provided by two relativistic fields is solving. The potentials for these fields are chosen in the form of anti-Higgs and Higgs potentials. It is shown that the effective potential has the shape of two-hump barrier localized in spacetime. It tends to constant attractive potential at spacetime infinity. The magnitude of this constant constituent is determined by the Higgs condensate. It is shown that nonlinear equation of motion of a particle has the solutions which describe the capture of a particle by the barrier and the scattering on the barrier. 
  Withdrawn due to the absence of an important shear term in Eq. (6), which cancels the other term. 
  We consider a model of a black hole consisting of a number of elementary components. Examples of such models occur in the Ashtekar's approach to canonical Quantum Gravity and in M-theory. We show that treating the elementary components as completely distinguishable leads to the area law for the black hole entropy. Contrary to previous results, we show that no Bose condensation occurs, the area has big local fluctuations and that in the framework of canonical Quantum Gravity the area of the black hole horizon is equidistantly quantized. 
  We study the thermodynamical and geometrical behaviour of the black holes that arise as solutions of the heterotic string action. We discuss the near-horizon scaling behaviour of the solutions that are described by two-dimensional Anti-de Sitter Space AdS(2). We find that finite-energy excitations of AdS(2) are suppressed only for scaling limits characterised by a dilaton which is constant near the horizon, whereas this suppression does not occur when the dilaton is non constant. 
  We identify the effective string scale of noncommutative Yang-Mills theory (NCYM) with the noncommutativity scale through its dual supergravity description. We argue that Newton's force law may be obtained with 4 dimensional NCYM with maximal SUSY. It provides a nonperturbative compactification mechanism of IIB matrix model. We can associate NCYM with the von Neumann lattice by the bi-local representation. We argue that it is superstring theory on the von Neumann lattice. We show that our identification of its effective string scale is consistent with exact T-duality (Morita equivalence) of NCYM. 
  Field theories that describe {\sl small} fluctuations of branes are limits of `brane theories' that describe {\sl large} fluctuations. In particular, supersymmetric sigma-models arise in this way. These lectures discuss the soliton solutions of the associated `brane theories' and their relation to calibrations. 
  We study the quark-antiquark interaction in the large N limit of the superconformal field theory on D-3branes at a Calabi-Yau conical singularity. We compute the Wilson loop in the AdS_5xT^{11} supergravity background for the SU(2N)x SU(2N) theory. We also calculate the Wilson loop for the Higgs phase where the gauge group is broken to SU(N)x SU(N)x SU_D(N). This corresponds to a two center configuration with some of the branes at the singularity and the rest of them at a smooth point. The calculation exhibits the expected Coulomb dependence for the interaction. The angular distribution of the BPS states is different than the one for a spherical horizon. 
  It is shown that recent criticism by C. R. Hagen (hep-th/9902057) questioning the validity of stress tensor treatments of the Casimir energy for space divided into two parts by a spherical boundary is without foundation. 
  We discuss the mapping of the conservative part of two-body electrodynamics onto that of a test charged particle moving in some external electromagnetic field, taking into account recoil effects and relativistic corrections up to second post-Coulombian order. Unlike the results recently obtained in general relativity, we find that in classical electrodynamics it is not possible to implement the matching without introducing external parameters in the effective electromagnetic field. Relaxing the assumption that the effective test particle moves in a flat spacetime provides a feasible way out. 
  We consider the classical equations of the gravitating Abelian-Higgs model in an axially symmetric ansatz. More properties of the solutions of these equations (the Melvin and the sting branches) are presented. These solutions are also constructed for winding numbers N=2. It is shown that these vortices exist in attractive and repulsive phases, separated by the value of the Higgs coupling constant parameter leading to self-dual equations. 
  We study the vacuum functional for a system of p-branes interacting with Maxwell fields of higher rank. This system represents a generalization of the usual electrodynamics of point particles, with one essential difference: namely, that the world-history of a p-brane, due to the spatial extension of the object, may possess a physical boundary. Thus, the objective of this study is twofold: first, we wish to exploit the breaking of gauge invariance due to the presence of a physical boundary, in order to generate mass as an alternative to the Higgs mechanism; second, we wish to investigate how the new mechanism of mass generation is affected by the duality transformation between electric and magnetic branes. The whole analysis is performed by using the path-integral method, as opposed to the more conventional canonical approach. The advantage of the path integral formulation is that it enables us to Fourier transform the field strength directly, rather than the gauge potential. To our knowledge, this field strength formulation represents a new application of the path integral method, and leads, in a straightforward way, to the dual representation of the vacuum functional. We find that the effect of the dual transformation is essentially that of exchanging the role of the gauge fields defined respectively on the " bulk'' and "boundary" of the p-brane history. 
  Reducible constrained Hamiltonian systems are quantized accordingly an irreducible BRST manner. Our procedure is based on the construction of an irreducible theory which is physically equivalent with the original one. The equivalence between the two systems makes legitimate the substitution of the BRST quantization for the reducible theory by that of the irreducible system. The general formalism is illustrated in detail on a model involving abelian one- and two-form gauge fields. 
  An irreducible Hamiltonian BRST quantization method for reducible first-class systems is proposed. The general theory is illustrated on a two-stage reducible model, the link with the standard reducible BRST treatment being also emphasized. 
  An irreducible canonical approach to reducible second-class constraints is given. The procedure is illustrated on gauge-fixed two-forms. 
  An algebraic proof of the nonrenormalization theorem for the perturbative beta function of the coupling constant of N=2 Super Yang-Mills theory is provided. The proof relies on a fundamental relationship between the N=2 Yang-Mills action and the local gauge invariant polynomial Tr phi^2, phi(x) being the scalar field of the N=2 vector gauge multiplet. The nonrenormalization theorem for the beta function follows from the vanishing of the anomalous dimension of Tr phi^2. 
  We systematically study deformations of chiral forms with applications to string theory in mind. To first order in the coupling constant, this problem can be translated into the calculation of the local BRST cohomological group at ghost number zero. We completely solve this cohomology and present detailed proofs of results announced in a previous letter. In particular, we show that there is no room for non-abelian, local, deformations of a pure system of chiral p-forms. 
  Issues that are specific for formulating fermions in light-cone quantization are discussed. Special emphasis is put on the use of parity invariance in the non-perturbative renormalization of light-cone Hamiltonians. 
  We compute the ground state energy of a massive scalar field in the background of a cylindrical shell whose potential is given by a delta function. The zero point energy is expressed in terms of the Jost function of the related scattering problem, the renormalization is performed with the help of the heat kernel expansion. The energy is found to be negative for attractive and for repulsive backgrounds as well. 
  CP violation by soft supersymmetry-breaking terms in orbifold compactifications is investigated. We include the universal part of the moduli-dependent threshold corrections in the construction of the non-perturbative effective potential due to gaugino-condensation. This allows interpolation of the magnitude of CP violating phases between the weakly and strongly coupled regimes. We find that the universal threshold corrections have a large effect on the CP violating phases in the weakly coupled regime. 
  The bosonic second invariant of SuperLiouville models in supersymmetric classical mechanics is described. 
  The energy radiated during the scattering of SU(3) monopoles is estimated as a function of their asymptotic velocity v. In a typical scattering process the total energy radiated is of order v^3 as opposed to v^5 for SU(2) monopoles. For charge (1,1) monopoles the dipole radiation produced is estimated for all geodesics on the moduli space. For charge (2,1) monopoles the dipole radiation is estimated for the axially symmetric geodesic. The power radiated appears to diverge in the massless limit. The implications of this for the case of non-Abelian unbroken symmetry are discussed. 
  We show how the threshold level of affine fusion, the fusion of Wess-Zumino-Witten (WZW) conformal field theories, fits into the Schubert calculus introduced by Gepner. The Pieri rule can be modified in a simple way to include the threshold level, so that calculations may be done for all (non-negative integer) levels at once. With the usual Giambelli formula, the modified Pieri formula deforms the tensor product coefficients (and the fusion coefficients) into what we call threshold polynomials. We compare them with the q-deformed tensor product coefficients and fusion coefficients that are related to q-deformed weight multiplicities. We also discuss the meaning of the threshold level in the context of paths on graphs. 
  We sketch a particularly simple and compelling version of D-brane cosmology. Inspired by the semi-phenomenological Randall--Sundrum models, and their cosmological generalizations, we develop a variant that contains a single (3+1)-dimensional D-brane which is located on the boundary of a single bulk (4+1)-dimensional region. The D-brane boundary is itself to be interpreted as our visible universe, with ordinary matter (planets, stars, galaxies) being trapped on this D-brane by string theory effects. The (4+1)-dimensional bulk is, in its simplest implementation, adS_{4+1}, anti-de Sitter space. We demonstrate that a k=+1 closed FLRW universe is the most natural option, though the scale factor could quite easily be so large as to make it operationally indistinguishable from a k=0 spatially flat universe. (With minor loss of elegance, spatially flat and hyperbolic FLRW cosmologies can also be accommodated.) We demonstrate how this model can be made consistent with standard cosmology, and suggest some possible observational tests. 
  We study some cases in D=4 and in infinite-volume high dimensional theories when unbroken supersymmetry in the vacuum cannot guarantee Fermi-Bose degeneracy among excited states. In 4D we consider an example in which both supersymmetry and $R$ symmetry are unbroken in the vacuum, and the cosmological constant vanishes. However, theory admits solitons that do not allow existence of conserved supercharges. These objects are magnetically charged global monopoles which in some respect behave as point-like particles, but create a solid angle deficit which eliminates asymptotically covariantly constant spinors and lifts Fermi-Bose degeneracy of the spectrum. The idea is that in some "dual" description monopoles with global topological number and gauge magnetic charge may be replaced by electrically chaged particles with global Noether charges, e.g., such as baryon number. Alternatively theories with infinite volume extra dimensions may support unbroken bulk supersymmetry without Fermi-Bose degeneracy in the brane spectrum. We suggest a scenario in which brane is a source similar to a global monopole embedded in 3 infinite extra dimensions. Brane produces a deficit angle at infinity and may localize a meta-stable 4D graviton. Although bulk cosmological constant is zero, conserved supercharges can not be defined on such a background. 
  We formulate a topological theory in six dimensions with gauge group SO(3,3) which reduces to gravity on a four dimensional defect if suitable boundary conditions are chosen. In such a framework we implement the reflection automorphism of SO(3,3) as a $\Z2$ symmetry which forbids the appearance of a gravitational cosmological constant. Some temptative speculations are presented also for the possible inclusion of the matter contribution at a full quantum level. 
  Thermalisation of configurations with initial white noise power spectrum is studied in numerical simulations of a classical one-component $\Phi^4$ theory in 2+1 dimensions, coupled to a small amplitude homogenous external field. The study is performed for energy densities corresponding to the broken symmetry phase of the system in equilibrium. The effective equation of the order parameter motion is reconstructed from its trajectory which starts from an initial value near the metastable point and ends in the stable ground state. This phenomenological theory quantitatively accounts for the decay of the false vacuum. The large amplitude transition of the order parameter between the two minima displays characteristics reflecting dynamical aspects of the Maxwell construction. 
  We perform a 3D reduction of the two-fermion Bethe-Salpeter equation, by series expansion around a positive-energy instantaneous approximation of the Bethe-Salpeter kernel, followed by another series expansion at the 3D level in order to get a manifestly hermitian 3D potential. It turns out that this potential does not depend on the choice of the starting approximation of the kernel anymore, and can be written in a very compact form. This result can also be obtained directly by starting with an approximation of the free propagator, based on integrals in the relative energies instead of the more usual delta-constraint. Furthermore, the method can be generalized to a system of N particles, consisting in any combination of bosons and fermions. As an example, we write the 3D equation for systems of two or three fermions exchanging photons, in Feynman or Coulomb's gauge. 
  We discuss the non-linear sigma model representing a NSR open string in a curved background with non-zero $B_{\mu\nu}$-field. With this coupling the theory is not automatically supersymmetric, due to boundary contributions. When B=0 supersymmetry is ensured by the conditions that follow as the boundary contribution to the field equations. We show that inclusion of a particular boundary term restores this state of affairs also in the presence of a $B$-field. The boundary conditions derived from the field equations in this case agree with those that have been proposed for constant $B$-field. A coupling to a boundary $A_\mu$-field will modify both the boundary conditions and affect the supersymmetry. It is shown that there is an $A$-coupling with non-standard fermionic part that respects both the supersymmetry and the shift symmetry (in the $B$ and $A$ fields), modulo the (modified) boundary conditions. 
  Description of the spectrum of fluctuations around a commutative vacuum solution, as well as around a solution with degenerate commutator in IIB matrix model is given in terms of supersymmetric Yang-Mills (YM) model. We construct explicitly the map from Hermitian matrices to YM fields and study the dependence of the spectrum and respective YM model on the symmetries of the solution. The gauge algebra of the YM model is shown to contain local reparameterisation algebra as well as Virasoro one. 
  We show that there is a non-trivial relationship between the dilaton of IIB supergravity, and the coset of scalar fields in five-dimensional, gauged N=8 supergravity. This has important consequences for the running of the gauge coupling in massive perturbations of the AdS/CFT correspondence. We conjecture an exact analytic expression for the ten-dimensional dilaton in terms of five-dimensional quantities, and we test this conjecture. Specifically, we construct a family of solutions to IIB supergravity that preserve half of the supersymmetries, and are lifts of supersymmetric flows in five-dimensional, gauged N=8 supergravity. Via the AdS/CFT correspondence these flows correspond to softly broken N=4, large N Yang-Mills theory on part of the Coulomb branch of N=2 supersymmetric Yang-Mills. Our solutions involve non-trivial backgrounds for all the tensor gauge fields as well as for the dilaton and axion. 
  It has recently been proposed that our universe is a three-brane embedded in a higher dimensional spacetime. Here I show that black holes on the brane, black strings intersecting the brane, and gravitational waves propagating in the bulk induce an effective energy-momentum tensor on the brane that contains negative energy densities. 
  The recurrent relations between the eigenfunctions for $GL(N,\RR)$ and $GL(N-1,\RR)$ quantum Toda chains is derived. As a corollary, the Mellin-Barnes integral representation for the eigenfunctions of a quantum open Toda chain is constructed for the $N$-particle case. 
  We show that each unitary representation of the N=2 superVirasoro algebra can be realized in terms of ``collective excitations'' over a filled Dirac sea of fermionic operators satisfying a generalized exclusion principle. These are semi-infinite forms in the modes of one of the fermionic currents. The constraints imposed on the fermionic operators have a counterpart in the form of a model one-dimensional lattice system, studying which allows us to prove the existence of a remarkable monomial basis in the semi-infinite space. This leads to a Rogers--Ramanujan-like character formula. We construct the N=2 action on the semi-infinite space using a filtration by finite-dimensional subspaces (the structure of which is related to the supernomial coefficients); the main technical tool is provided by the dual functional realization. As an application, we identify the coinvariants with the dual to a space of meromorphic functions on products of punctured Riemann surfaces with a prescribed behaviour on multiple diagonals. For products of punctured $CP^1$, such spaces are related to the unitary N=2 fusion algebra, for which we also give an independent derivation. 
  Gravitational perturbations of anti-deSitter spacetime play important roles in AdS/CFT correspondence and the brane world scenario. In this paper, we develop a gauge-invariant formalism of gravitational perturbations of maximally symmetric spacetimes including anti-deSitter spacetime. Existence of scalar-type master variables is shown and the corresponding master equations are derived. 
  We study the `enhancon', a spherical hypersurface apparently made of D-branes, which arises in string theory studies of large N SU(N) pure gauge theories with eight supercharges. When the gauge theory is 2+1 dimensional, the enhancon is an S^2. A relation to charge N BPS multi-monopoles is exploited to uncover many of its detailed properties. It is simply a spherical slice through an Atiyah-Hitchin-like submanifold of the charge $N$ BPS monopole moduli space. In the form of Nahm data, it is built from the N dimensional irreducible representation of SU(2). In this sense the enhancon is a non-commutative sphere, reminiscent of the spherical `dielectric' branes of Myers. 
  We investigate how a dynamical mass of a fermion is affected by a topological mass of a gauge field in a Maxwell-Chern-Simons $QED_3$ coupled with a two-component fermion. The dynamical mass and also a parity condensate are estimated by using a non-perturbative Schwinger-Dyson method. In particular, we study a limit of vanishing the topological mass in detail and clarify a linking between theories with and without a Chern-Simons term in a non-perturbative level. 
  We consider Z2, freely acting orbifolds of the type IIB string with 16 parallel D5-branes. When the string is compactified on T2 X T4 and the D5-branes are wrapped on T2, these systems possess N=2 supersymmetry, originating from the spontaneous, partial breaking of N=4. Extended supersymmetry allows us to investigate the duality with certain heterotic and type I constructions, and to obtain informations about their non-perturbative regime. 
  A U(N) Chern-Simons theory on noncommutative $\mathbb{R}^{3}$ is constructed as a $\q$-deformed field theory. The model is characterized by two symmetries: the BRST-symmetry and the topological linear vector supersymmetry. It is shown that the theory is finite and $\q_{\m\n}$-independent at the one loop level and that the calculations respect the restriction of the topological supersymmetry. Thus the topological $\q$-deformed Chern-Simons theory is an example of a model which is non-singular in the limit $\q \to 0$. 
  We show that the existent fuzzy S^2 and S^4 models are natural candidates for the quantum geometry on the corresponding spheres in AdS/CFT correspondence. These models fit nicely the data from the dipole mechanism for the stringy exclusion principle. In the AdS_2 X S^2 case, we show that a wrapped fractional membrane can be used to count for the large ground state degeneracy. We also propose a fuzzy AdS_2 model whose fundamental commutation relation may underlie the UV/IR connection. 
  It is argued that the T-dual of a crosscap is a combination of an O+ and an O- orientifold plane. Various theories with crosscaps and D-branes are interpreted as gauge-theories on tori obeying twisted boundary conditions. Their duals live on orientifolds where the various orientifold planes are of different types. We derive how to read off the holonomies from the positions of D-branes in the orientifold background. As an application we reconstruct some results from a paper by Borel, Friedman and Morgan for gauge theories with classical groups, compactified on a 2-- or 3--torus with twisted boundary conditions. 
  The notion of a space-time uncertainty principle in string theory is clarified and further developed. The motivation and the derivation of the principle are first reviewed in a reasonably self-contained way. It is then shown that the nonperturbative (Borel summed) high-energy and high-momentum transfer behavior of string scattering is consistent with the space-time uncertainty principle. It is also shown that, in consequence of the principle, string theories in 10 dimensions generically exhibit a characteristic length scale which is equal to the well-known 11 dimensional Planck length $g_s^{1/3}\ell_s$ of M-theory as the scale at which stringy effects take over the effects of classical supergravity, even without involving D-branes directly. The meanings of the space-time uncertainty relation in connection with D-branes and black holes are discussed and reinterpreted. Finally, we present a novel interpretation of the Schild-gauge action for strings from a viewpoint of noncommutative geometry, which conforms to the space-time uncertainty relation by manifestly exhibiting a noncommutativity of quantized string coordinates dominantly between space and time. We also discuss the consistency of the space-time uncertainty relation with S and T dualities. 
  This is a brief overview on the current status of string theory for non-specialists. The purpose is to give an aspect on the nature of string theory as a unified theory of all interactions including quantum gravity and to discuss future perspectives. Particular emphases are put on the mysteries why string theory contains gravity and why it resolves the ultraviolet problems. 
  We apply numerical and analytic techniques to the study of Yang-Mills integrals with orthogonal, symplectic and exceptional gauge symmetries. The main focus is on the supersymmetric integrals, which correspond essentially to the bulk part of the Witten index for susy quantum mechanical gauge theory. We evaluate these integrals for D=4 and group rank up to three, using Monte Carlo methods. Our results are at variance with previous findings. We further compute the integrals with the deformation technique of Moore, Nekrasov and Shatashvili, which we adapt to the groups under study. Excellent agreement with all our numerical calculations is obtained. We also discuss the convergence properties of the purely bosonic integrals. 
  We consider general black holes in D=5, N=2 supergravity coupled to vector multiplets, and discuss the issue of microstate counting from various viewpoints. The statistical entropy is computed for the near-extremal case using the central charge of the $AdS_2$ factor appearing in the near-horizon geometry. Furthermore, we explicitly construct the duality transformation connecting electrically charged black holes to magnetically charged black strings, under which the $AdS_2 \times S^3$ near horizon geometry becomes $AdS_3 \times S^2$. For $AdS_3$ the counting of microstates correctly reproduces the Bekenstein-Hawking entropy, thus resolving the discrepancy previously found for $AdS_2$. 
  A detailed study of the charge spectrum of three dimensional Abelian Topological Massive Gauge Theory (TMGT) is given. When this theory is defined on a manifold with two disconnected boundaries there are induced chiral Conformal Field Theories (CFT's) on the boundaries which can be interpreted as the left and right sectors of closed strings. We show that Narain constraints on toroidal compactification (integer, even, self-dual momentum lattice) have a natural interpretation in purely three dimensional terms. This is an important result which is necessary to construct toroidal compactification and heterotic string from Topological Membrane (TM) approach to string theory. We also derive the block structure of $c=1$ Rational Conformal Field Theory (RCFT) from the three dimensional gauge theory. 
  It is shown that the recent criticism of Brevik et al. (hep-th/0004041) is in error. 
  We show that thermal noncommutative field theories admit a version of `channel duality' reminiscent of open/closed string duality, where non-planar thermal loops can be replaced by an infinite tower of tree-level exchanges of effective fields. These effective fields resemble closed strings in three aspects: their mass spectrum is that of closed-string winding modes, their interaction vertices contain extra moduli, and they can be regarded as propagating in a higher-dimensional `bulk' space-time. In noncommutative models that can be embedded in a D-brane, we show the precise relation between the effective `winding fields' and closed strings propagating off the D-brane. The winding fields represent the coherent coupling of the infinite tower of closed-string oscillator states. We derive a sum rule that expresses this effective coupling in terms of the elementary couplings of closed strings to the D-brane. We furthermore clarify the relation between the effective propagating dimension of the winding fields and the true codimension of the D-brane. 
  The microscopic spectral correlations of the Dirac operator in Yang-Mills theories coupled to fermions in (2+1) dimensions can be related to three universality classes of Random Matrix Theory. In the microscopic limit the Orthogonal Ensemble (OE) corresponds to a theory with 2 colors and fermions in the fundamental representation and the Symplectic Ensemble (SE) corresponds to an arbitrary number of colors and fermions in the adjoint representation. Using a new method of Widom, we derive an expression for the two scalar kernels which through quaternion determinants give all spectral correlation functions in the Gaussian Orthogonal Ensemble (GOE) and in the the Gaussian Symplectic Ensemble (GSE) with all fermion masses equal to zero. The result for the GOE is valid for an arbitrary number of fermions while for the GSE we have results for an even number of fermions. 
  We evaluate the divergent part of the Vilkovisky-DeWitt effective action for Einstein gravity on even-dimensional Kaluza-Klein spacetimes of the form $M^{4}\times S^{N}$. Explicit results are given for $N$=2, 4, and 6. Trace anomalies for gravitons are also given for these cases. Stable Kaluza-Klein configurations are sought, unsuccessfully, assuming the divergent part of the effective action dominates the dynamics. 
  When anticommuting Grassmann variables are introduced into a fluid dynamical model with irrotational velocity and no vorticity, the velocity acquires a nonvanishing curl and the resultant vorticity is described by Gaussian potentials formed from the Grassmann variables. Upon adding a further specific interaction with the Grassmann degrees of freedom, the model becomes supersymmetric. 
  We propose a non-Abelian generalization of the Clebsch parameterization for a vector in three dimensions. The construction is based on a group-theoretical reduction of the Chern-Simons form on a symmetric space. The formalism is then used to give a canonical (symplectic) discussion of non-Abelian fluid mechanics, analogous to the way the Abelian Clebsch parameterization allows a canonical description of conventional fluid mechanics. 
  Motivated by the mixing of UV and IR effects, we test the OPE formula in noncommutative field theory. First we look at the renormalization of local composite operators, identifying some of their characteristic IR/UV singularities. Then we find that the product of two fields in general cannot be described by a series expansion of single local operator insertions. 
  BPS representations of 5-dimensional supersymmetry algebras are classified. For BPS states preserving 1/2 the supersymmetry, there are two distinct classes of multiplets for N=4 supersymmetry and three classes for N=8 supersymmetry. For N=4 matter theories, the two 1/2 supersymmetric BPS multiplets are the massive vector multiplet and the massive self-dual 2-form multiplet. Some applications to super-Yang-Mills, supergravity and little string theories are considered. 
  We consider chiral condensates in SU(2) gauge theory with broken N=2 supersymmetry. The matter sector contains an adjoint multiplet and one fundamental flavor. Matter and gaugino condensates are determined by integrating out the adjoint field. The only nonperturbative input is the Affleck-Dine-Seiberg (ADS) superpotential generated by one instanton plus the Konishi anomaly. These results are consistent with those obtained by the `integrating in' procedure, including a reproduction of the Seiberg-Witten curve from the ADS superpotential. We then calculate monopole, dyon, and charge condensates using the Seiberg-Witten approach. We show that the monopole and charge condensates vanish at the Argyres-Douglas point where the monopole and charge vacua collide. We interpret this phenomenon as a deconfinement of electric and magnetic charges at the Argyres-Douglas point. 
  We analyze the perturbation series for noncommutative eigenvalue problem $AX=X\lambda$ where $\lambda$ is an element of a noncommutative ring, $ A$ is a matrix and $X$ is a column vector with entries from this ring. As a corollary we obtain a theorem about the structure of perturbation series for Tr $x^r$ where $x$ is a solution of noncommutative algebraic equation (for $r=1$ this theorem was proved by Aschieri, Brace, Morariu, and Zumino, hep-th/0003228, and used to study Born-Infeld lagrangian for the gauge group $U(1)^k$). 
  We provide general determinant formulae for all n-particle form factors related to the trace of the energy momentum tensor and the analogue of the order and disorder operator in the $SU(3)_2$-homogeneous Sine-Gordon model. We employ the form factors related to the trace of the energy momentum tensor in the application of the c-theorem and find perfect agreement with the physical picture recently obtained by means of the thermodynamic Bethe ansatz. For finite resonance parameter we recover the expected WZNW-coset central charge and for infinite resonance parameter the theory decouples into two free fermions. 
  The conformal symmetry SO(d,2) of the massless particle in d dimensions, or superconformal symmetry OSp(N|4), SU(2,2|N), OSp(8|N) of the superparticle in d=3,4,6 dimensions respectively, had been previously understood as the global Lorentz symmetry and supersymmetries of 2T physics in d+2 dimensions. By utilising the gauge symmetries of 2T physics, it is shown that the dynamics can be cast in terms of superspace coordinates, momenta and theta variables or in terms of supertwistor variables a la Penrose and Ferber. In 2T physics these can be gauge transformed to each other. In the supertwistor version the quantization of the model amounts to the well known oscillator formalism for non-compact supergroups. 
  This talk will summarise the progress we have made in our programme to both characterise and construct charges in gauge theories. As an application of these ideas we will see how the dominant glue surrounding quarks, which is responsible for asymptotic freedom, emerges from a constituent description of the interquark potential. 
  We construct the three dimensional mirror theory of SO(2k) and SO(2k+1) gauge groups by using O3-planes. An essential ingredient in constructing the mirror is the splitting of a physical brane (NS-brane or D5-brane) on O3-planes. In particular, matching the dimensions of moduli spaces of mirror pair (for example, the SO(2k+1) and its mirror) there is a D3-brane creation or annihilation accompanying the splitting. This novel dynamical process gives a nontrivial prediction for strongly coupled field theories, which will be very interesting to check by Seiberg-Witten curves. Furthermore, applying the same idea, we revisit the mirror theory of Sp(k) gauge group and find new mirrors which differ from previously known results. Our new result for Sp(k) gives another example to a previously observed fact, which shows that different theories can be mirror to the same theory. We also discussed the phenomena such as "hidden FI-parameters" when the number of flavors and the rank of the gauge group satisfy certain relations, ``incomplete Higgsing'' for the mirror of SO(2k+1) and the ``hidden global symmetry''. After discussing the mirror for a single Sp or SO gauge group, we extend the study to a product of two gauge groups in two different models, namely the elliptic and the non-elliptic models. 
  We supersymmetrize a class of moduli dependent potentials living on branes with the help of additional bulk terms in 5d N=2 supergravity. The space of Poincare invariant vacuum solutions includes the Randall-Sundrum solution and the $M$-theoretical solution. After adding gauge sectors to the branes we discuss breakdown of low energy supersymmetry in this setup and hierarchy of physical scales. In the limit of large warp factors we find decoupling between effects stemming from different branes in the compactified theory. 
  We study inflation and entropy generation in a recently proposed pre-big-bang model universe produced in a collision of gravitational and dilaton waves. It is shown that enough inflation occurs provided the incoming waves are sufficiently weak. We also find that entropy in this model is dynamically generated as the result of the nonlinear interaction of the incoming waves, before the universe enters the phase of dilaton driven inflation. In particular, we give the scaling of the entropy produced in the collision in terms of the focusing lengths of the incoming waves. 
  We discuss Poincare three-brane solutions in D=5 M-Theory compactifications on Calabi-Yau (CY) threefolds with G-fluxes. We show that the vector moduli freeze at an attractor point. In the case with background flux only, the spacetime geometry contains a zero volume singularity with the three-brane and the CY space shrinking simultaneously to a point. This problem can be avoided by including explicit three-brane sources. We consider two cases in detail: a single brane and, when the transverse dimension is compactified on a circle, a pair of branes with opposite tensions. 
  We study the low energy string effective action with an exponential type dilaton potential and vanishing torsion in a Bianchi type I space-time geometry. In the Einstein and string frames the general solution of the gravitational field equations can be expressed in an exact parametric form. Depending on the values of some parameters the obtained cosmological models can be generically divided into three classes, leading to both singular and nonsingular behaviors. The effect of the potential on the time evolution of the mean anisotropy parameter is also considered in detail, and it is shown that a Bianchi type I Universe isotropizes only in the presence of a dilaton field potential or a central deficit charge. 
  We consider brane-world universe where an arbitrary large $N$ quantum CFT is living on the domain wall. This corresponds to implementing of Randall-Sundrum compactification within the context of AdS/CFT correspondence. Using anomaly induced effective action for domain wall CFT the possibility of self-consistent quantum creation of 4d de Sitter wall Universe (inflation) is demonstrated. In case of maximally SUSY Yang-Mills theory the exact correspondence with radius and effective tension found by Hawking-Hertog-Reall is obtained. The hyperbolic wall Universe may be induced by quantum effects only for exotic matter (higher derivatives conformal scalar) which has unusual sign of central charge. 
  We give an elementary introduction to black holes in supergravity and string theory. The focus is on BPS solutions in four- and higher-dimensional supergravity and string theory. Basic ideas and techniques are explained in detail, including exercises with solutions. 
  We consider here the massive Thirring model regularized with the XYZ spin chain. We numerically calculate the mass ratios of particles which lie in the discrete part of the spectrum and obtain results in accordance with the DHN formula and in disagreement with recent calculations in the literature based on the numerical Bethe ansatz and infinite momentum frame methods. We also analyze the short distance behavior of these states and evaluate the conformal dimensions. This paper, taken together with the previous one for the sine-Gordon model, confirms the duality relation between two models formulated by Klassen and Melzer [Int. J. Mod. Phys. A 8, 4131 (1993)]. 
  Open descendants with boundaries and crosscaps of non-trivial automorphism type are studied. We focus on the case where the bulk symmetry is broken to a Z_2 orbifold subalgebra. By requiring positivity and integrality for the open sector, we derive a unique crosscap of automorphism type g \in Z_2 and a corresponding g-twisted Klein bottle for a charge conjugation invariant. As a specific example, we use T-duality to construct the descendants of the true diagonal invariant with symmetry preserving crosscaps and boundaries. 
  We show how to extend the usual black string instability of vacuum or charged black p-branes to the anti-de Sitter background. The string fragments in an analogous fashion to the $\Lambda=0$ case, the main difference being that instead of a periodic array of black holes forming, an accumulation of ``mini'' black holes occurs towards the AdS horizon. In the case where the AdS space is of finite extent, such as an orbifold compactification, we show how the instability switches off below a certain compactification scale. 
  It is shown that certain extremal and next-to-extremal n-point correlators in four-dimensional N=2 superconformal field theories are free. These results hold for any gauge group. 
  We study M and F theory compactifications on Calabi-Yau four-folds in the presence of non-trivial background flux. The geometry is warped and belongs to the class of p-brane metrics. We solve for the explicit warp factor in the orbifold limit of these compactifications, compare our results to some of the more familiar recently studied warped scenarios, and discuss the effects on the low-energy theory. As the warp factor is generated solely by backreaction, we may use topological arguments to determine the massless spectrum. We perform the computation for the case where the four-fold equals K3 \times K3. 
  We present a supersymmetric version of Dirac-Born-Infeld (DBI) theory in noncommutative space-time, both for Abelian and non-Abelian gauge groups. We show, using the superfield formalism, that the definition of a certain ordering with respect to the * product leads naturally to a DBI action both in the U(1) as well as in the U(N) case. BPS equations are analysed in this context and properties of the resulting theory are discussed. 
  It was suggested in hep-th/0002106, that semiclassically, a partition function of a string theory in the 5 dimensional constant negative curvature space with a boundary condition at the absolute satisfy the loop equation with respect to varying the boundary condition, and thus the partition function of the string gives the expectation value of a Wilson loop in the 4 dimensional QCD. In the paper, we present the geometrical framework, which reveals that the equations of motion of such string theory are integrable, in the sense that they can be written via a Lax pair with a spectral parameter. We also show, that the issue of the loop equation rests solely on the properly posing the boundary condition. 
  In this short note we would like to propose a form for the action of a non-BPS D-brane, which will be manifestly supersymmetric invariant and T-duality covariant. We also explicitly show that tachyon condensation on the world-volume of this brane leads to the Dirac-Born-Infeld action for BPS D(p-1)-brane. 
  We discuss the possibility of realizing the infinite dimensional symmetries of conformal mechanics as time reparametrizations, generalizing the realization of the SL(2,R) symmetry of the de Alfaro, Fubini, Furlan model in terms of quasi--primary fields. We find that this is possible using an appropriate generalization of the transformation law for the quasi-primary fields. 
  We construct axially symmetric dyons in SU(2) Yang-Mills-Higgs theory. In the Prasad-Sommerfield limit, they are obtained via scaling relations from axially symmetric multimonopole solutions. For finite Higgs self-coupling they are constructed numerically. 
  We show that in brane-world scenarios with warped extra dimensions, the Casimir force due to bulk matter fields may be sufficient to stabilize the radion field $\phi$. In particular, we calculate one loop effective potential for $\phi$ induced by bulk gravitons and other possible massless bulk fields in the Randall-Sundrum background. This potential has a local extremum, which can be a maximum or a minimum depending on the detailed bulk matter content. If the parameters of the background are chosen so that the hierarchy problem is solved geometrically, then the radion mass induced by Casimir corrections is hierarchycally smaller than the $TeV$. Hence, in this important case, we must invoke an alternative mechanism (classical or nonperturbative) which gives the radion a sizable mass, to make it compatible with observations. 
  We study a similarity transformation to construct an effective Hamiltonian systematically, which does not contain particle-number-changing interactions, by means of Fukuda-Sawada-Taketani-Okubo's method. We show that such Hamiltonian can be constructed from Feynman diagrams and give rules for constructing it in the Light-Front Yukawa model. We prove that it is renormalized by the familiar covariant perturbative renormalization procedure. It is very advantageous that the effective Hamiltonian can be obtained from our rules {\em immediately}. We also numerically diagonalize it to the second order in the coupling constant as an exercise. 
  We give the full lagrangean and supersymmetry transformation rules for D=5, N=2 supergravity interacting with an arbitrary number of vector, tensor and hyper-multiplets, with gauging of the R-symmetry group SU(2)_R as well as a subgroup K of the isometries of the scalar manifold. Among the many possible applications, this theory provides the setting where a supersymmetric brane-world scenario could occur. We comment on the presence of AdS vacua and BPS solutions that would be relevant towards a supersymmetric smooth realization of the Randall-Sundrum "alternative to compactification". We also add some remarks on the connection between this most general 5D fully coupled supergravity model and type IIB theory on the T^{11} manifold. 
  We study the tachyon potential in the NS sector of Witten's cubic superstring field theory. In this theory, the pure tachyon contribution to the potential has no minimum. We find that this remains the case when higher modes up to level two are included. 
  In this note we obtain the discrete spectrum of the rotating ellipsoidal membrane, the solution to classical equations of motion in the matrix mechanics of N D0-branes. This solution has the interpretation of a closed D2-brane with the D0-branes bound to its surface. The semi-classical quantization is performed on the rotational modes with the result that both radii and energy are quantized. We also argue that the quantum mechanics of this system is well defined, with a unique ground state of positive energy in each sector with a non-zero angular momentum. The scaling of the size and energy of these states allows us to identify our rotating brane excited states with the previously conjectured resonances in the scattering of D0-branes. 
  A globally consistent treatment of linearized gravity in the Randall-Sundrum background with matter on the brane is formulated. Using a novel gauge, in which the transverse components of the metric are non-vanishing, the brane is kept straight. We analyze the gauge symmetries and identify the physical degrees of freedom of gravity. Our results underline the necessity for non-gravitational confinement of matter to the brane. 
  We study a non-commutative non-relativistic scalar field theory in 2+1 dimensions. The theory shows the UV/IR mixing typical of QFT on non-commutative spaces. The one-loop correction to the two-point function turns out to be given by a delta-function in momentum space. The one-loop correction to the four-point function is of logarithmic type. We also evaluate the thermodynamic potential at two-loop order. To avoid an IR-singularity we have to introduce a chemical potential. The suppression of the non-planar contribution with respect to the planar one turns out to depend crucially on the value of the chemical potential. 
  We manifestly verify that the miraculous cancellation of the infrared divergence of the topologically massive Yang-Mills theory in the Landau gauge is completely determined by a new vector symmetry existing in its large topological mass limit, the Chern-Simons theory. Furthermore, we show that the cancellation theorem proposed by Pisarski and Rao is an inevitable consequence of this new vector symmetry. 
  We show that certain five dimensional, N=2 Yang-Mills/Einstein supergravity theories admit the gauging of the full R-symmetry group, SU(2)_R, of the underlying N=2 Poincare superalgebra. This generalizes the previously studied Abelian gaugings of U(1)_R subgroup of SU(2)_R and completes the construction of the most general vector and tensor field coupled five dimensional N=2 supergravity theories with gauge interactions. The gauging of SU(2)_R turns out to be possible only in special cases, and leads to a new type of scalar potential. For a large class of these theories the potential does not have any critical points. 
  We give a new criterion of quark confinement/deconfinement by deriving a low-energy effective theory of QCD. The effective theory can explain Abelian dominance in low-energy physics of QCD, especially, quark confinement. Finally, we apply the above criterion to the large flavor QCD and discuss its phase structure. The result suggests the existence of conformal phase. 
  We investigate the universality of microscopic eigenvalue correlations for Random Matrix Theories with the global symmetries of the QCD partition function. In this article we analyze the case of real valued chiral Random Matrix Theories ($\beta =1$) by relating the kernel of the correlations functions for $\beta =1$ to the kernel of chiral Random Matrix Theories with complex matrix elements ($\beta = 2$), which is already known to be universal. Our proof is based on a novel asymptotic property of the skew-orthogonal polynomials: an integral over the corresponding wavefunctions oscillates about half its asymptotic value in the region of the bulk of the zeros. This result solves the puzzle that microscopic universality persists in spite of contributions to the microscopic correlators from the region near the largest zero of the skew-orthogonal polynomials. Our analytical results are illustrated by the numerical construction of the skew-orthogonal polynomials for an $x^4$ probability potential. 
  A set of on shell duality equations is proposed that leads to a map between strings moving on symmetric spaces with opposite curvatures. The transformation maps "waves" on a riemannian symmetric space to "waves" on its dual riemannian symmetric space. This transformation preserves the energy momentum tensor though it is not a canonical transformation. The preservation of the energy momentum tensor has a natural geometrical interpretation. The transformation maps "particle-like solutions" into static "soliton-like solutions". The results presented here generalize earlier results of E. Ivanov. 
  The one loop quantum corrections to the classical reflection factor of the sinh-Gordon model are calculated partially for general boundary conditions. The model is studied under boundary conditions which are compatible with integrability, and in the framework of the conventional perturbation theory generalized to the affine Toda field theory. It is found that the general form of the related quantum corrections are hypergeometric functions. 
  The torus and the Klein bottle amplitude coefficients are computed in permutation orbifolds of RCFT-s in terms of the same quantities in the original theory and the twist group. An explicit expression is presented for the number of self conjugate primaries in the orbifold as a polynomial of the total number of primaries and the number of self conjugate ones in the parent theory. The formulae in the $Z_2$ orbifold illustrate the general results. 
  We give arguments for a conjecture made in a previous paper, that one has to use only the gauged sugra action for the calculation of correlators of certain operators via the AdS-CFT correspondence. The existence of consistent truncations implies that the massive modes decouple, and gauged supergravity is sufficient for computing n-point functions of CFT operators coupled to the massless (sugra) sector. The action obtained from the linear ansatz, of the type $\phi(x,y)=\phi_I(x)Y^I(y)$ gives only part of the gauged sugra. This means that there is a difference for the correlators on the boundary of AdS space. We find, studying examples of correlators, that the right prescription is to use the full gauged sugra, which implies using the full nonlinear KK ansatz. To this purpose, we analyze 3 point functions of various gauge fields in 5 and 7 dimensions, and the R-current anomaly in the corresponding CFT. We also show that the nonlinear rotation in the tower of scalar fields of Lee et al., Corrado et al. and Bastianelli and Zucchini produces a consistent truncation to the massless level and coincides with the Taylor expansion of the nonlinear KK ansatz in massless scalar fluctuations. Finally, we speculate about the way to do the full nonlinear rotation for the massive tower. 
  We add a Gauss-Bonnet term to the Einstein-Hilbert action and study the recent proposal to solve the cosmological constant problem. We also consider the possibility of adding a dilaton potential to the action. In the absence of supersymmetry, we obtain first order Bogomol'nyi equation as a solution-generating method in our scenario. When the coefficient of the Gauss-Bonnet term is positive, the dilaton potential is bounded below. Assuming a simple double-well potential, the dilaton field is found to be a kink in the fifth dimension. 
  We argue that traditional methods of compactification of string theory make it very difficult to understand how the cosmological constant becomes zero. String inspired models can give zero cosmological constant after fine tuning but since string theory has no free parameters it is not clear that this is allowed. Brane world scenarios on the other hand while they do not answer the question as to why the cosmological constant is zero do actually allow a choice of integration constants that permit flat four space solutions. In this paper we discuss gauged supergravity realizations of such a world. To the extent that this starting point can be considered a low energy effective action of string theory (and there is some recent evidence supporting this) our model may be considered a string theory realization of this scenario. 
  The cosmology of the Randall-Sundrum scenario for a positive tension brane in a 5-D Universe with localized gravity has been studied extensively recently. Here we extend it to more general situations. We consider the time-dependent situation where the two sides of the brane are different AdS/Schwarzschild spaces. We show that the expansion rate in these models during inflation could be larger than in brane worlds with compactified extra dimensions of fixed size. The enhanced expansion rate could lead to the production of density perturbations of substantially larger amplitude. 
  We propose a geometrized Higgs mechanism based on the gravitational sector in the Connes-Lott formulation of the standard model, which has been constructed by Chamseddine, Fr$\ddot{o}$hlic and Grandjean. The point of our idea is that Higgs-like couplings depend on the local coordinates of the four-dimensional continuum,$M_4$. The localized couplings can be calculated by the Wilson loops of the $U(1)_{EM}$ gauge field and the connection, which is defined on $Z_2\times{M_4}$. 
  We study the nonlocal regularization for the non-abelian gauge theories for an arbitrary value of the gauge parameter (\xi). We show that the procedure for the nonlocalization of field theories established earlier by the original authors, when applied in that form to the Faddeev-Popov effective action in a linear gauge cannot lead to a (\xi)-independent result for the observables. We then show that an alternate procedure which is simpler can be used and that it leads to the S-matrix elements (where they exist) independent of (\xi). 
  I describe renewed efforts to establish a string description of large N_c QCD by summing large ``fishnet'' diagrams. Earlier work on fishnets indicated that the usual relativistic (zero thickness) string theory can arise at strong 't Hooft coupling, at best yielding a highly idealized description, which fails to incorporate such salient features of continuum QCD as asymptotic freedom and point-like constituents. The recently conjectured AdS/CFT correspondence is compatible with such limitations because it also gives a simple picture of large N_c gauge theory only at strong coupling. In order to better understand how string theory could emerge from large N_c QCD at strong coupling, Klaus Bering, Joel Rozowsky, and I have developed an improved implementation of my effort of the late seventies to digitize the planar diagrams of large N_c light-cone quantized QCD by discretizing both P+ and x+. This discretization allows a strong coupling limit of the sum of planar diagrams to be defined and studied. It also provides a natural framework to explore the possible dual relationship between QCD in light-cone gauge and string theory quantized on the light-cone. 
  We construct regular and black hole solutions in SU(2) Einstein-Born-Infeld theory. These solutions have many features in common with the corresponding SU(2) Einstein-Yang-Mills solutions. In particular, sequences of neutral non-abelian solutions tend to magnetically charged limiting solutions, related to embedded abelian solutions. Thermodynamic properties of the black hole solutions are addressed. 
  We construct unstable classical solutions of Yang-Mills theories and their dual unstable states of type IIB on AdS_5. An example is the unstable D0-brane of type IIB located at the center of AdS. This has a field theory dual which is a sphaleron in gauge theories on S^3 x R. We argue that the two are dual because both are sphalerons associated to the topology of the instanton/D-instanton. This agreement provides a non-supersymmetric test of the AdS/CFT duality. As an illustration, many aspects of Sen's hypothesis regarding the unstable branes can be seen easily in the weakly coupled dual field theory description. In Euclidean AdS the D0-branes are dual to gauge theory merons. This implies that the two ends of a D0-brane world-line carry half the charge of a D-instanton. Other examples involve unstable strings in the Coulomb phase. 
  We study exact solutions of Dirac and Klein-Gordon equations and Green functions in d-dimensional QED and in an external electromagnetic field with constant and homogeneous field invariants. The cases of even and odd dimensions are considered separately, they are essentially different. We direct attention to the asymmetry of the quasienergy spectrum, which appears in odd dimensions. The in and out classification of the exact solutions as well as the completeness and orthogonality relations is strictly proven. Different Green functions in the form of sums over the exact solutions are constructed. The Fock-Schwinger proper time integral representations of these Green functions are found. As physical applications we consider the calculations of different quantum effects related to the vacuum instability in the external field. For example, we present mean values of particles created from the vacuum, the probability of the vacuum remaining a vacuum, the effective action, and the expectation values of the current and energy-momentum tensor. 
  We study the dynamics of brane worlds coupled to a scalar field and gravity, and find that self-tuning of the cosmological constant is generic in theories with at most two branes or a single brane with orbifold boundary conditions. We demonstrate that singularities are generic in the self-tuned solutions compatible with localized gravity on the brane: we show that localized gravity with an infinitely large extra dimension is only consistent with particular fine-tuned values of the brane tension. The number of allowed brane tension values is related to the number of negative stationary points of the scalar bulk potential and, in the case of an oscillatory potential, the brane tension for which gravity is localized without singularities is quantized. We also examine a resolution of the singularities, and find that fine-tuning is generically re-introduced at the singularities in order to retain a static solution. However, we speculate that the presence of additional fields may restore self-tuning. 
  A four-form gauge flux makes a variable contribution to the cosmological constant. This has often been assumed to take continuous values, but we argue that it has a generalized Dirac quantization condition. For a single flux the steps are much larger than the observational limit, but we show that with multiple fluxes the allowed values can form a sufficiently dense `discretuum'. Multiple fluxes generally arise in M theory compactifications on manifolds with non-trivial three-cycles. In theories with large extra dimensions a few four-forms suffice; otherwise of order 100 are needed. Starting from generic initial conditions, the repeated nucleation of membranes dynamically generates regions with a cosmological constant in the observational range. Entropy and density perturbations can be produced. 
  A path integral description of an effective action of monopoles in Abelian projections of Yang-Mills theories is discussed and used to establish a projection independence of the effective action. A dynamic regime in which the effective dynamics may contain massive solitonic excitations is described. 
  We argue that MQCD admits intersecting domain walls that are realized as Cayley calibrations of the MQCD M5-brane. We discuss various dual realizations and comment on how branes can realise domain walls in N=1 supersymmetric theories in D=3. 
  We derive the vector supersymmetry and the \L-symmetry transformations for the fields of a generalized topological $p$-form model of Schwarz-type in $d$ space-time dimensions. 
  The Caswell-Wilczek analysis on the gauge dependence of the effective action and the renormalization group functions in Yang-Mills theories is generalized to generic, possibly power counting non renormalizable gauge theories. It is shown that the physical coupling constants of the classical theory can be redefined by gauge parameter dependent contributions of higher orders in $\hbar$ in such a way that the effective action depends trivially on the gauge parameters, while suitably defined physical beta functions do not depend on those parameters. 
  I shall discuss the regulation of the $P^+ = 0$ singularity and give some examples. Regulating the singularity induces new operators into the theory. This process seems rather different for the case of ultraviolet singularities than for the case of infrared singularities. 
  It is shown that ghost fields are indispensable in deriving well-defined antiderivatives in pure space-like axial gauge quantizations of gauge fields. To avoid inessential complications we confine ourselves to noninteracting abelian fields and incorporate their quantizations as a continuous deformation of those in light-cone gauge. We attain this by constructing an axial gauge formulation in auxiliary coordinates $x^{\mu}= (x^+,x^-,x^1,x^2)$, where $x^+=x^0{\rm sin}{\theta}+x^3{\rm cos}{\theta}, x^-=x^0{\rm cos}{\theta}-x^3{\rm sin}{\theta}$ and $x^+$ and $A_-=A^0{\rm cos} {\theta}+A^3{\rm sin}{\theta}=0$ are taken as the evolution parameter and the gauge fixing condition, respectively. We introduce $x^-$-independent residual gauge fields as ghost fields and accomodate them to the Hamiltonian formalism by applying McCartor and Robertson's method. As a result, we obtain conserved translational generators $P_{\mu}$, which retain ghost degrees of freedom integrated over the hyperplane $x^-=$ constant. They enable us to determine quantization conditions for the ghost fields in such a way that commutation relations with $P_{\mu}$ give rise to the correct Heisenberg equations. We show that regularizing singularities arising from the inversion of a hyperbolic Laplace operator as principal values, enables us to cancel linear divergences resulting from $({\partial}_-)^{-2}$ so that the Mandelstam- Leibbrandt form of gauge field propagator can be derived. It is also shown that the pure space-like axial gauge formulation in ordinary coordinates can be derived in the limit ${\theta}\to\frac{\pi}{2}-0$ and that the light-cone axial gauge formulation turns out to be the case of ${\theta}=\frac{\pi}{4}$. 
  This note focuses on the coupling of a type IIA D2-brane to a background B field. It is shown that the D0-brane charge arising from the integral over the D2-brane of the pullback of the B field is cancelled by bulk contributions, for a compact D2-brane wrapping a homotopically trivial cycle in space-time. In M-theory this cancellation is a straightforward consequence of momentum conservation. This result resolves a puzzle recently posed by Bachas, Douglas and Schweigert related to the quantization of R-R charges on stable spherical D2-branes on the group manifold SU(2). 
  In a recent paper, Polchinski and Strassler found a string theory dual of a gauge theory with reduced supersymmetry. Motivated by their approach, we perturb the $\N=8$ theory living on a set of N M2 branes to $\N=2$, by adding fermion mass terms. We obtain M-theory duals corresponding to M2 branes polarized into M5 branes, in $AdS_4 \times S_7$. In the course of doing this we come across an interesting feature of the M5 brane action, which we comment on. Depending on the fermion masses we obtain discrete or continuous vacua for our theories. We also obtain dual descriptions for domain walls, instantons and condensates. 
  Classical Maxwell and Maxwell-Chern-Simons (MCS) Electrodynamics in (2+1)D are studied in some details. General expressions for the potential and fields are obtained for both models, and some particular cases are explicitly solved. Conceptual and technical difficulties arise, however, for accelerated charges. The propagation of electromagnetic signals is also studied and their reverberation is worked out and discussed. Furthermore, we show that a Dirac-like monopole yields a (static) tangential electric field. We also discuss some classical and quantum consequences of the field created by such a monopole when acting upon an usual electric charge. In particular, we show that at large distances, the dynamics of one single charged particle under the action of such a potential and a constant (external) magnetic field as well, reduces to that of one central harmonic oscillator, presenting, however, an interesting angular sector which admits energy-eigenvalues. Among other peculiarities, both sectors, the radial and the angular one, present non-vanishing energy-eigenvalues for their lowest level. Moreover, those associated to the angle are shown to respond to discrete shifts of such a variable. We also raise the question whether the formation of bound states is possible in the system. 
  By carefully analyzing the geodesic motion of a test particle in the bulk of brane worlds, we identify an extra force which is recognized in spacetime of one lower dimensions as a non-gravitational force acting on the particle. Such extra force acts on the particle in such a way that the conventional particle mechanics in one lower dimensions is violated, thereby hinting at the higher-dimensional origin of embedded spacetime in the brane world scenario. We obtain the explicit equations describing the motion of the bulk test particle as observed in one lower dimensions for general gravitating configurations in brane worlds and identify the extra non-gravitational force acting on the particle measured in one lower dimensions. 
  We show that the BRST/anti-BRST invariant 3+1 dimensional 2-form gauge theory has further nilpotent symmetries (dual BRST /anti-dual BRST) that leave the gauge fixing term invariant. The generator for the dual BRST symmetry is analogous to the co-exterior derivative of differential geometry. There exists a bosonic symmetry which keeps the ghost terms invariant and it turns out to be the analogue of the Laplacian operator. The Hodge duality operation is shown to correspond to a discrete symmetry in the theory. The generators of all these continuous symmetries are shown to obey the algebra of the de Rham cohomology operators of differential geometry. We derive the extended BRST algebra constituted by six conserved charges and discuss the Hodge decomposition theorem in the quantum Hilbert space of states. 
  We consider two-dimensional nonlinear sigma model from the viewpoint of the holography, which has been applied to the study of the Yang-Mills theory, based on the non-critical string theory. We can see the renormalization group flows for both the nonlinear sigma model and the Yang-Mills theory at the same time, and the two theories are intimately related through a kind of dual relation of the coupling constants. We address the running behaviors of these coupling constants and also the asymptotic freedom of the Yang-Mills theory. 
  We present a lattice formulation of noncommutative Yang-Mills theory in arbitrary even dimensionality. The UV/IR mixing characteristic of noncommutative field theories is demonstrated at a completely nonperturbative level. We prove a discrete Morita equivalence between ordinary Yang-Mills theory with multi-valued gauge fields and noncommutative Yang-Mills theory with periodic gauge fields. Using this equivalence, we show that generic noncommutative gauge theories in the continuum can be regularized nonperturbatively by means of {\it ordinary} lattice gauge theory with 't~Hooft flux. In the case of irrational noncommutativity parameters, the rank of the gauge group of the commutative lattice theory must be sent to infinity in the continuum limit. As a special case, the construction includes the recent description of noncommutative Yang-Mills theories using twisted large $N$ reduced models. We study the coupling of noncommutative gauge fields to matter fields in the fundamental representation of the gauge group using the lattice formalism. The large mass expansion is used to describe the physical meaning of Wilson loops in noncommutative gauge theories. We also demonstrate Morita equivalence in the presence of fundamental matter fields and use this property to comment on the calculation of the beta-function in noncommutative quantum electrodynamics. 
  We study flows on the scalar manifold of N=8 gauged supergravity in five dimensions which are dual to certain mass deformations of N=4 super Yang--Mills theory. In particular, we consider a perturbation of the gauge theory by a mass term for the adjoint hyper-multiplet, giving rise to an N=2 theory. The exact solution of the 5-dim gauged supergravity equations of motion is found and the metric is uplifted to a ten-dimensional background of type-IIB supergravity. Using these geometric data and the AdS/CFT correspondence we analyze the spectra of certain operators as well as Wilson loops on the dual gauge theory side. The physical flows are parametrized by a single non-positive constant and describe part of the Coulomb branch of the N=2 theory at strong coupling. We also propose a general criterion to distinguish between `physical' and `unphysical' curvature singularities. Applying it in many backgrounds arising within the AdS/CFT correspondence we find results that are in complete agreement with field theory expectations. 
  This paper presents a review of the basic, model-independent differences between the pre-big bang scenario, arising naturally in a string cosmology context, and the standard inflationary scenario. We use an unconventional approach in which the introduction of technical details is avoided as much as possible, trying to focus the reader's attention on the main conceptual aspects of both scenarios. The aim of the paper is not to conclude in favour either of one or of the other scenario, but to raise questions that are left to the reader's meditation. Warnings: the paper does not contain equations, and is not intended as a complete review of all aspects of string cosmology. 
  We show that any Hamiltonian system with one degree of freedom is invariant under a $w_\infty$ algebra of symmetries. 
  We generalize the results of Polchinski and Strassler regarding the N=1-preserving mass deformation of N=4 SU(N) SYM theories and its string theory dual to SO(N) and USp(2N) gauge groups. The string theory duals involve 5-branes wrapped on RP^2. In order to match with the field theory classification of vacua, the 3-brane charge carried by such 5-branes must be shifted by a half, and this follows from a conjectured generalization of results of Freed and Witten. Our results provide an elegant physical picture for the classification of classically massive vacua in the mass-deformed N=4 theories. 
  A Cantorian fractal spacetime, a family member of von Neumann's noncommutative geometry is introduced as a geometry underlying a new relativity theory which is similar to the relation between general relativity and Riemannian geometry. Based on this model and the new relativity theory an ensemble distribution of all the dimensions of quantum spacetime is derived with the help of Fermat grand theorem. The calculated average dimension is very close to the value of $4+\phi^3 $ (where $\phi$ is the golden mean) obtained by El Naschie on the basis of a different approach. It is shown that within the framework of the new relativity the cosmological constant problem is nonexistent, since the Universe self-organizes and self-tunes according to the renormalization group (RG) flow with respect to a local scaling microscopic arrow of time. This implies that the world emerged as a result of a non-equilibrium process of self-organized critical phenomena launched by vacuum fluctuations in Cantorian fractal spacetime $\cal E^{\infty}$. It is shown that we are living in a metastable vacuum and are moving towards a fixed point ($ D$ = 4+$\phi^3$) of the RG. After reaching this point, a new phase transition will drive the universe to a quasi-crystal phase of the lower average dimension of $\phi^3$. 
  We find and prove relationships between Riemann zeta values and central binomial sums. We also investigate alternating binomial sums (also called Ap\'ery sums). The study of non-alternating sums leads to an investigation of different types of sums which we call multiple Clausen values. The study of alternating sums leads to a tower of experimental results involving polylogarithms in the golden ratio. In the non-alternating case, there is a strong connection to polylogarithms of the sixth root of unity, encountered in the 3-loop Feynman diagrams of {\tt hep-th/9803091} and subsequently in hep-ph/9910223, hep-ph/9910224, cond-mat/9911452 and hep-th/0004010. 
  A general two dimensional fractional supersymmetric conformal field theory is investigated. The structure of the symmetries of the theory is studied. Then, applying the generators of the closed subalgebra generated by $(L_{-1}, L_{0}, G_{-1/3})$ and $(\bar{L}_{-1}, \bar{L}_{0}, \bar{G}_{-1/3})$, the two point functions of the component fields of supermultiplets are calculated. 
  We consider the cosmological evolution of the three-brane in the background of type IIB string theory. For two different backgrounds which give nontrivial dilaton profile we have derived the Friedman-like equations. These give the cosmological evolution which is similar to the one by matter density on the universe brane. The effective density blows up as we move towards the singularity showing the initial singularity problem. The analysis shows that when there is axion field in the ambient space the recollapsing of the universe occurs faster compared with the case without axion field. 
  A modified Kaluza-Klein theory is proposed in which propagation takes place only at the speed of light. The propagation can be confined to a small volume, forming a particle with rest mass. The usual four space-time coordinates locate the confinement volume, and Kaluza's fifth coordinate is replaced by an internal degree of freedom. Electromagnetism corresponds to a gauge field on the phase of the internal motion. Self-gravity might create the confinement, as in a geon, but the particle would have a Planck mass. This large mass could be made closer to the masses of observed particles if gravity were allowed to increase in strength within the confinement volume. 
  We study noncommutative geometry in the framework of the Batalin-Fradkin-Tyutin(BFT) scheme, which converts second class constraint system into first class one. In an open string theory noncommutative geometry appears due to the mixed boundary conditions having second class constraints, which arise in string theory with $D$-branes under a constant Neveu-Schwarz $B$-field. Introduction of a new coordinate $y$ on $D$-brane through BFT analysis allows us to obtain the commutative geometry with the help of the first class constraints, and the resulting action corresponding to the first class Hamiltonian in the BFT Hamiltonian formalism has a new local symmetry. 
  We show that off-diagonal gluons and off-diagonal ghosts acquire their masses dynamically in QCD if the maximal Abelian gauge is adopted. This result strongly supports the Abelian dominance in low-energy region of QCD. The mass generation is shown to occur due to ghost-anti-ghost condensation caused by attractive quartic ghost interactions within the Abelian projected effective gauge theory (derived by one of the authors). In fact, the quartic ghost interaction is indispensable for the renormalizability due to nonlinearity of the maximal Abelian gauge. The ghost-anti-ghost condensation is associated with the spontaneous breaking of global SL(2,R) symmetry recently found by Schaden at least for SU(2) case. Moreover we write down a new extended BRS algebra in the maximal Abelian gauge which should be compared with that of Nakanishi-Ojima for the Lorentz gauge. Finally, we argue that the mass generation may be related to the spontaneous breaking of a supersymmetry $OSp(4|2)$ hidden in the maximal Abelian gauge. 
  We perform canonical quantization of the open Neveu-Schwarz-Ramond (NSR) superstrings in the background of a D-brane with the NS B-field. If we choose the mixed boundary condition as a primary constraint, it generates a set of secondary constraints. These constraints are easily solved and as a result, the noncommutative geometry in the bosonic string theory is extended to the superspace. Solving the constraint conditions we also find that the Hamiltonian for the superstring is equivalent to a free superstring Hamiltonian on the target space with the effective open string metric G. 
  In the present paper the gauge-invariant formalism is developed for perturbations of the brane-world model in which our universe is realized as a boundary of a higher-dimensional spacetime. For the background model in which the bulk spacetime is $(n+m)$-dimensional and has the spatial symmetry corresponding to the isometry group of a $n$-dimensional maximally symmetric space, gauge-invariant equations are derived for perturbations of the bulk spacetime. Further for the case corresponding to the brane-world model in which $m=2$ and the brane is a boundary invariant under the spatial symmetry in the unperturbed background, relations between the gauge-invariant variables describing the bulk perturbations and those for brane perturbations are derived from Israel's junction condition under the assumption of $\ZR_2$ symmetry. In particular, for the case in which the bulk spacetime is a constant-curvature spacetime, it is shown that the bulk perturbation equations reduce to a single hyperbolic master equation for a master variable, and that the physical condition on the gauge-invariant variable describing the intrinsic stress perturbation of the brane yield a boundary condition for the master equation through the junction condition. On the basis of this formalism it is pointed out that it seems to be difficult to suppress brane perturbations corresponding to massive excitations for a brane motion giving a realistic expanding universe model. 
  We describe a Riemannian space class where the second Dirac operator arises and prove that the operator is always equivalent to a standard Dirac one. The particle state in this gravitational field is degenerate to some extent and we introduce an additional value in order to describe a particle state completely. Some supersymmetry constructions are also discussed. As an example we study all Riemannian spaces with a five-dimentional motion group and find all metrics for which the second Dirac operator exists. On the basis of our discussed examples we hypothesize about the number of second Dirac operators in Riemannian space. 
  The relativistic scattering of a spin-1/2 particle from an infinitely long solenoid is considered in the framework of covariant perturbation theory. The first order term agrees with the corresponding term in the series expansion of the exact amplitude, and second order term vanishes, thus proving that Born approximation is consistent. 
  The dilation invariance is studied in the framework of Epstein-Glaser approach to renormalization theory. Some analogues of the Callan-Symanzik equations are found and they are applied to the scalar field theory and to Yang-Mills models. We find the interesting result that, if all the fields of the theory have zero masses, then from purely cohomological consideration, one can obtain the anomalous terms of logarithmic type. 
  The complex Monge-Amp\`ere equation admits covariant bi-symplectic structure for complex dimension 3, or higher. The first symplectic 2-form is obtained from a new variational formulation of complex Monge- Amp\`ere equation in the framework of the covariant Witten-Zuckerman approach to symplectic structure. We base our considerations on a reformulation of the Witten-Zuckerman theory in terms of holomorphic differential forms. The first closed and conserved Witten-Zuckerman symplectic 2-form for the complex Monge-Amp\`ere equation is obtained in arbitrary dimension and for all cases elliptic, hyperbolic and homogeneous.    The connection of the complex Monge-Amp\`ere equation with Ricci-flat K\"ahler geometry suggests the use of the Hilbert action. However, we point out that Hilbert's Lagrangian is a divergence for K\"ahler metrics. Nevertheless, using the surface terms in the Hilbert Lagrangian we obtain the second Witten-Zuckerman symplectic 2-form for complex dimension>2. 
  The tachyon-free nonsupersymmetric string theories in ten dimensions have dilaton tadpoles which forbid a Minkowski vacuum. We determine the maximally symmetric backgrounds for the $USp(32)$ Type I string and the $SO(16)\times SO(16)$ heterotic string. The static solutions exhibit nine dimensional Poincar\'e symmetry and have finite 9D Planck and Yang-Mills constants. The low energy geometry is given by a ten dimensional manifold with two boundaries separated by a finite distance which suggests a spontaneous compactification of the ten dimensional string theory. 
  In Randall-Sundrum-type brane-world cosmologies, the dynamical equations on the three-brane differ from the general relativity equations by terms that carry the effects of imbedding and of the free gravitational field in the five-dimensional bulk. We derive the covariant nonlinear dynamical equations for the gravitational and matter fields on the brane, and then linearize to find the perturbation equations on the brane. The local energy-momentum corrections are significant only at very high energies. The imprint on the brane of the nonlocal gravitational field in the bulk is more subtle, and we provide a careful decomposition of this effect into nonlocal energy density, flux and anisotropic stress. The nonlocal energy density determines the tidal acceleration in the off-brane direction, and can oppose singularity formation via the generalized Raychaudhuri equation. Unlike the nonlocal energy density and flux, the nonlocal anisotropic stress is not determined by an evolution equation on the brane. In particular, isotropy of the cosmic microwave background may no longer guarantee a Friedmann geometry. Adiabatic density perturbations are coupled to perturbations in the nonlocal bulk field, and in general the system is not closed on the brane. But on super-Hubble scales, density perturbations can be evaluated by brane observers. Tensor perturbations on the brane are suppressed by local bulk effects during inflation, while the nonlocal effects can serve as a source or a sink. Vorticity on the brane decays as in general relativity, but nonlocal bulk effects can source the gravito-magnetic field, so that vector perturbations can be generated in the absence of vorticity. 
  We search for an abelian description of the Yang-Mills instantons on certain eight dimensional manifolds with the special holonomies $Spin(7)$ and SU(4). By mimicing the Seiberg-Witten theory in four dimensions, we propose a set of monopole-like equations governing the 8-dimensional U(1) connections and spinors, which are supposed to be the dual theory of the nonabelian instantons. We also give a naive test of the generalized $S$-duality in the abelian sector of 8-dimensional Yang-Mills theory. Some problems in this approach are pointed out. 
  We study the low energy regime of the scattering of two fermionic particles carrying isospin 1/2 and interacting through a non-Abelian Chern-Simons field. We calculate the one-loop scattering amplitude for both the nonrelativistic and also for the relativistic theory. In the relativistic case we introduce an intermediate cutoff, separating the regions with low and high loop momenta integration. In this procedure purely relativistic field theory effects as the vacuum polarization and anomalous magnetic moment corrections are automatically incorporated. 
  Hydrodynamic turbulence is studied as a constrained system from the point of view of metafluid dynamics. We present a Lagrangian description for this new theory of turbulence inspired from the analogy with electromagnetism. Consequently it is a gauge theory. This new approach to the study of turbulence tends to renew the optimism to solve the difficult problem of turbulence. As a constrained system, turbulence is studied in the Dirac and Faddeev-Jackiw formalisms giving the Dirac brackets. An important result is that we show these brackets are the same in and out of the inertial range, giving the way to quantize turbulence. 
  We calculate the one-loop quantum contributions to soft supersymmetry breaking terms in the scalar potential in supergravity theories regulated \`a la Pauli-Villars. We find ``universal'' contributions, independent of the regulator masses and tree level soft supersymmetry breaking, that contribute gaugino masses and A-terms equal to the ``anomaly mediated'' contributions found in analyses using spurion techniques, as well as a scalar mass term not identified in those analyses. The universal terms are in general modified -- and in some cases canceled -- by model-dependent terms. Under certain restrictions on the couplings we recover the one-loop results of previous ``anomaly mediated'' supersymmetry breaking scenarios. We emphasize the model dependence of loop-induced soft terms in the potential, which are much more sensitive to the details of Planck scale physics then are the one-loop contributions to gaugino masses. We discuss the relation of our results to previous analyses. 
  Using pure spinors, the superstring was recently quantized in a manifestly ten-dimensional super-Poincar\'e covariant manner and a covariant prescription was given for tree-level scattering amplitudes. In this paper, we prove that this prescription is cyclically symmetric and, for the scattering of an arbitrary number of massless bosons and up to four massless fermions, it agrees with the standard Ramond-Neveu-Schwarz prescription. 
  We consider bosonic noncritical strings as QCD strings and we present a basic strategy to construct them in the context of Liouville theory. We show that Dirichlet boundary conditions play important roles in generalized Liouville theory. Specifically, they are used to stabilize the classical configuration of strings and also utilized to treat tachyon condensation in our model. We point out that Dirichlet boundary conditions for the Liouville mode maintains Weyl invariance, if an appropriate condition is satisfied, in the background with a (non-linear) dilaton. 
  Quantum mechanics on sphere $S^{n}$ is studied from the viewpoint that the Berry's connection has to appear as a topological term in the effective action. Furthermore we show that this term is the Chern-Simons term of gauge variables that correspond to the extra degrees of freedom of the enlarged space. 
  Atiyah and Manton have outlined a scheme to obtain approximations to the SU(2) skyrmions from instantons in $\R^4$. In this paper we apply this scheme to construct, in an explicit form, approximations to static spherically symmetric SU(N) skyrmions with various baryon numbers. In particular we show how to obtain the skyrmions from instantons using harmonic maps into complex projective spaces. 
  A certain notion of canonical equivalence in quantum mechanics is proposed. It is used to relate quantal systems with discrete ones. Discrete systems canonically equivalent to the celebrated harmonic oscillator as well as the quartic and the quasi-exactly-solvable anharmonic oscillators are found. They can be viewed as a translation-covariant discretization of the (an)harmonic oscillator preserving isospectrality. The notion of the $q-$deformation of the canonical equivalence leading to a dilatation-covariant discretization preserving polynomiality of eigenfunctions is also presented. 
  We find a large class of quantum gauge models with massless fermions where the coupling to the gauge fields is not chirally symmetric and which nevertheless do not suffer from gauge anomalies. To be specific we study two dimensional Abelian models in the Hamiltonian framework which can be constructed and solved by standard techniques. The general model describes $\Np$ photon fields and $\Nf$ flavors of Dirac fermions with $2\Nf\Np$ different coupling constants i.e. the chiral component of each fermion can be coupled to the gauge fields differently. We construct these models and find conditions so that no gauge anomaly appears. If these conditions hold it is possible to construct and solve the model explicitly, so that gauge- and Lorentz invariance are manifest. 
  (2+1)-regular static black hole solutions with a nonlinear electric field are derived. The source to the Einstein equations is an energy momentum tensor of nonlinear electrodynamics, which satisfies the weak energy conditions and in the weak field limit becomes the (2+1)-Maxwell field tensor. The derived class of solutions is regular; the metric, curvature invariants and electric field are regular everywhere. The metric becomes, for a vanishing parameter, the (2+1)-static charged BTZ solution. A general procedure to derive solutions for the static BTZ (2+1)-spacetime, for any nonlinear Lagrangian depending on the electric field is formulated; for relevant electric fields one requires the fulfillment of the weak energy conditions. 
  A class of generalized Wess-Zumino models with distinct vacua is investigated. These models allow for BPS saturated vacua interpolation along one compact spatial dimension. The properties of these interpolations are studied. 
  Construction of a five dimensional conformal supergravity (D=5 CSG) is attempted by applying the AdS/CFT correspondence to the F(4) AdS supergravity in six dimensions. As a first step, local transformation laws of D=5 CSG have been established, from which the Weyl weights of the various fields in D=5 can be predicted. 
  An $S$-matrix is proposed for the two dimensional O(3) $\sigma$-model with a dynamical $\theta$-term (axion model). Exploiting an Abelian T-duality transformation connecting the axion model to an integrable SU(2)$\times$U(1) symmetric principal $\sigma$-model, strong evidence is presented for the correctness of the proposed $S$-matrix by comparing the perturbatively calculated free energies with the ones based on the Thermodynamical Bethe Ansatz. This T-duality transformation also leads to a new Lax-pair for both models. The quantum non-integrability of the O(3) $\sigma$-model with a {\sl constant} $\theta$-term, in contradistinction to the axion model, is illustrated by calculating the $2\to3$ particle production amplitude to lowest order in $\theta$. 
  We consider the k-folded sine-Gordon model, obtained from the usual version by identifying the scalar field after k periods of the cosine potential. We examine (1) the ground state energy split, (2) the lowest lying multi-particle state spectrum and (3) vacuum expectation values of local fields in finite spatial volume, combining the Truncated Conformal Space Approach, the method of the Destri-de Vega nonlinear integral equation (NLIE) and semiclassical instanton calculations. We show that the predictions of all these different methods are consistent with each other and in particular provide further support for the NLIE method in the presence of a twist parameter. It turns out that the model provides an optimal laboratory for examining instanton contributions beyond the dilute instanton gas approximation. We also provide evidence for the exact formula for the vacuum expectation values conjectured by Lukyanov and Zamolodchikov. 
  It is proposed a new mechanism for the phenomenon of topological mass generation in three spacetime dimensions as the result of the interference of two opposite massless chiral modes. This mechanism, already used to produce the massive vectorial mode of the 2D Schwinger model, is here exploited to produce the gauge invariant massive mode of the Maxwell-Chern-Simons theory. Moreover the procedure is clearly dimensionally independent: a new chiral boson action is proposed for odd and even dimensional space-times to be used as the basic building blocks of the interference schemes. This is a new result that extends the two-dimensional Floreanini-Jackiw action to higher dimensional spaces and is in clear contrast with the twice odd dimensional chiral form extensions. 
  The multiloop amplitudes for the bosonic string in presence of a constant B-field are built by using the basic commutation relations for the open string zero modes and oscillators. The open string Green function on the annulus is obtained from the one loop scattering amplitude among N tachyons. For higher loops, it is necessary to use the so called three Reggeon vertex, which describes the emission from the open string of another string and not simply of a tachyon. We find that the modifications to the three (and multi) Reggeon vertex due to the B-field only affect the zero modes and can be written in a simple and elegant way. Therefore we can easily sew these vertices together and write the general expression for the multiloop N-Reggeon vertex, which contains any loop string amplitude, in presence of the B-field. The field theory limit is also considered in some examples at two loops and reproduces exactly the results of a noncommutative scalar field theory. 
  We discuss the phase structure of QCD_4 as a perturbative deformation of the Topological Quantum Field Theory (TQFT). When we choose a special Maximal Abelian gauge (MAG) as the gauge fixing, the TQFT sector is equivalent to a 2D O(3) non-linear sigma model (NLSM). We consider the finite temperature case, and investigate the effect of the boundary conditions and the phase structure of the TQFT sector. It can have a deconfining phase under the twisted boundary conditions. However, this phase is screened once pertubative parts are added. We conclude that the information about the phase structure is encoded in the U(1) background in the case of the MAG. 
  We study the Renormalization Group (RG) flow of critical bosonic background fields in the framework of the RG approach to string theory. In this approach quantum field theory beta-functions are the extra inputs in solving the string theory sigma-model equations. We study two different situations, the first one is the Yang-Mills theory where the coupling constant diverges in the infrared limit. The second case corresponds to a type of theories where the beta-function has a pole in the infrared limit and it changes sign through the pole (as in N=1 super-Yang-Mills theory). For this case in the strong coupling branch, in the infrared, there is an interval of values of the coupling in which the theory only leads to confinement. We have obtained this range. We also mention the theories with conformal-fixed points and their relation to theories with a pole in the beta-functions. We calculate the Wilson loops in these theories. 
  In these lectures, which are written at an elementary and pedagogical level, we discuss general aspects of (single) instantons in SU(N_c) Yang-Mills theory, and then specialize to the case of N = 4 supersymmetry and the large N_c limit. We show how to determine the measure of collective coordinates and compute instanton corrections to certain correlation functions. We then relate this to D-instantons in type IIB supergravity. By taking the D-instantons to live in an $AdS_5\times S^5$ background, we perform explicit checks of the AdS/CFT correspondence. 
  It has recently been conjectured that the AdS_5/SYM_4 correspondence can be generalized away from the conformal limit, to a duality between supergravity on the full asymptotically flat three-brane background and a theory characterized as N=4 SYM deformed in the IR by a specific dimension-eight operator. Assuming that this relation is valid, we derive a prescription for computing n-point correlation functions in the holographic theory, which reduces to the standard AdS/CFT recipe at low energies. One- and two-point functions are discussed in detail. The prescription follows from very simple considerations and appears to be applicable to any asymptotically flat background. We also compute the quark-antiquark potential and comment on the description of the baryon in the supergravity picture. We conclude with some comments on the possible relation between our work and recent results in non-commutative field theories. 
  Using an exact solution as a concrete example, Nambu-Goldstone modes on the BPS domain wall junction are worked out for N=1 supersymmetric theories in four dimensions. Their wave functions extend along the wall to infinity (not localized) and are not normalizable. It is argued that this feature is a generic phenomenon of Nambu-Goldstone modes on domain wall junctions in the bulk flat space in any dimensions. We formulate mode equations and show that fermion and boson with the same mass come in pairs except massless modes which can appear singly, in accordance with unitary representations of (1, 0) supersymmetry. 
  We obtain logarithmic behaviours of a four-point correlation function in the c=-2 conformal field theory by using the Feigin-Fuchs construction. It becomes an indeterminate form by a naive evaluation, but is obtained by introducing an appropriate regularization procedure. 
  The use of the mass term as a gauge fixing term has been studied by Zwanziger, Parrinello and Jona-Lasinio, which is related to the non-linear gauge $A_{\mu}^{2}=\lambda$ of Dirac and Nambu in the large mass limit. We have recently shown that this modified quantization scheme is in fact identical to the conventional {\em local} Faddeev-Popov formula {\em without} taking the large mass limit, if one takes into account the variation of the gauge field along the entire gauge orbit and if the Gribov complications can be ignored. This suggests that the classical massive vector theory, for example, is interpreted in a more flexible manner either as a gauge invariant theory with a gauge fixing term added, or as a conventional massive non-gauge theory. As for massive gauge particles, the Higgs mechanics, where the mass term is gauge invariant, has a more intrinsic meaning.  It is suggested to extend the notion of quantum gauge symmetry (BRST symmetry) not only to classical gauge theory but also to a wider class of theories whose gauge symmetry is broken by some extra terms in the classical action. We comment on the implications of this extended notion of quantum gauge symmetry. 
  We propose bosonised expressions for the chiral Schwinger models in four dimensions. Then, in complete analogy with the two dimensional case, we show the soldering of two bosonised chiral Schwinger models with opposite chiralities to yield the bosonised Schwinger model in four dimensions. The implications of the Schwinger model or its chiral version, as known for two dimensions, thereby get extended to four dimensions. 
  In this work we initiate a systematic investigation of the spin of a composite system in an arbitrary reference frame in QCD. After a brief review of the difficulties one encounters in equal-time quantization, we turn to light-front quantization. We show that, in spite of the complexities, light-front field theory offers a unique opportunity to address the issue of relativistic spin operators in an arbitrary reference frame since boost is kinematical in this formulation. Utilizing this symmetry, we show how to introduce transverse spin operators for massless particles in an arbitrary reference frame in analogy with those for massive particles. Starting from the manifestly gauge invariant, symmetric energy momentum tensor in QCD, we derive expressions for the interaction dependent transverse spin operators ${\cal J}^i$ ($i=1,2$) which are responsible for the helicity flip of the nucleon in light-front quantization. In order to construct ${\cal J}^i$, first we derive expressions for the transverse rotation operators $F^i$. In the gauge $A^+=0$, we eliminate the constrained variables. In the completely gauge fixed sector, in terms of the dynamical variables, we show that one can decompose ${\cal J}^i= {\cal J}^i_I + {\cal J}^i_{II} + {\cal J}^i_{III}$ where only ${\cal J}^i_{I}$ has explicit coordinate ($x^-, x^i$) dependence in its integrand. The operators ${\cal J}^i_{II}$ and ${\cal J}^i_{III}$ arise from the fermionic and bosonic parts respectively of the gauge invariant energy momentum tensor. We discuss the implications of our results. 
  In this paper we address various issues connected with transverse spin in light front QCD. The transverse spin operators, in $A^+ = 0$ gauge, expressed in terms of the dynamical variables are explicitly interaction dependent unlike the helicity operator which is interaction independent in the topologically trivial sector of light-front QCD. Although it cannot be separated into an orbital and a spin part, we have shown that there exists an interesting decomposition of the transverse spin operator. We discuss the physical relevance of such a decomposition. We perform a one loop renormalization of the full transverse spin operator in light-front Hamiltonian perturbation theory for a dressed quark state. We explicitly show that all the terms dependent on the center of mass momenta get canceled in the matrix element. The entire non-vanishing contribution comes from the fermion intrinsic -like part of the transverse spin operator as a result of cancellation between the gluonic intrinsic-like and the orbital-like part of the transverse spin operator. We compare and contrast the calculations of transverse spin and helicity of a dressed quark in perturbation theory. 
  We study the null compactification of type-IIA-string perturbation theory at finite temperature. We prove a theorem about Riemann surfaces establishing that the moduli spaces of infinite-momentum-frame superstring worldsheets are identical to those of branched-cover instantons in the matrix-string model conjectured to describe M-theory. This means that the identification of string degrees of freedom in the matrix model proposed by Dijkgraaf, Verlinde and Verlinde is correct and that its natural generalization produces the moduli space of Riemann surfaces at all orders in the genus expansion. 
  A strong coupling limit of theories whose low-energy effective field theory is 5-dimensional N=8 supergravity is proposed in which the gravitational coupling becomes large. It is argued that, if this limit exists, it should be a 6-dimensional theory with (4,0) supersymmetry compactified on a circle whose radius gives the 5-dimensional Planck length. The sector corresponding to the D=5 supergravity multiplet is a (4,0) D=6 superconformal field theory based on the (4,0) multiplet with 27 self-dual 2-forms, 42 scalars and, instead of a graviton, a fourth-rank tensor gauge field satisfying a self-duality constraint. The superconformal field theory has 32 supersymmetries and 32 conformal supersymmetries and its dimensional reduction gives the maximal supergravity in five dimensions. Electromagnetic duality generalises to a gravitational triality. 
  We make a precision test of a recently proposed conjecture relating Chern-Simons gauge theory to topological string theory on the resolution of the conifold. First, we develop a systematic procedure to extract string amplitudes from vacuum expectation values (vevs) of Wilson loops in Chern-Simons gauge theory, and then we evaluate these vevs in arbitrary irreducible representations of SU(N) for torus knots. We find complete agreement with the predictions derived from the target space interpretation of the string amplitudes. We also show that the structure of the free energy of topological open string theory gives further constraints on the Chern-Simons vevs. Our work provides strong evidence towards an interpretation of knot polynomial invariants as generating functions associated to enumerative problems. 
  In this work we apply Thompson's method (of the dimensions) to study the quantum electrodynamics (QED). This method can be considered as a simple and alternative way to the renormalisation group (R.G) approach and when applied to QED lagrangian is able to obtain the running coupling constant behavior $\alpha (\mu)$, namely the dependence of $\alpha$ on the energy scale. We also obtain the dependence of the mass on the energy scale. The calculations are evaluated just at $d_c=4$, where $d_c$ is the upper critical dimension of the problem, so that we obtain logarithmic behavior both for the coupling $\alpha$ and the mass $m$ on the energy scale $\mu$. 
  In this Ph.D. thesis, we study some aspects of strings and branes. We mainly focus on anomalous D-brane couplings and on non-supersymmetric branes.   In Chapter 2, we illustrate three key concepts in a field theory context: duality, supersymmetry and anomaly inflow.   Chapter 3 contains an introduction to string theory, emphasizing the concepts relevant to this thesis, like string scattering amplitudes and D-brane actions.   In Chapter 4, we introduce the boundary state formalism and use it to check the presence of anomalous couplings in the D-brane Wess-Zumino action. In addition, we show that this action also contains non-anomalous terms.   Chapter 5 gives a BPS analysis of D-branes in D-brane backgrounds, focusing on the baryon vertex in the AdS/CFT correspondence.   Chapter 6 is devoted to branes in type 0 string theory. We derive the D-brane Wess-Zumino action and the massless spectra of NS-fivebranes, and combine these ingredients to comment on type 0B S-duality.   In Chapter 7, we study non-BPS D-branes in type II string theory. We propose a Wess-Zumino action for these branes, check it by computing string scattering amplitudes and show that it is consistent with the interpretation of BPS D-branes as topologically non-trivial tachyon configurations on a non-BPS D-brane. 
  In a recent paper, we have pointed out a relation between the Killing spinor and Einstein equations. Using this relation, new brane solutions of D=11 and D=10 type IIB supergravity theories are constructed. It is shown that in a brane solution, the flat world-volume directions, the smeared transverse directions and the sphere located at a fixed radial distance can be replaced with any Lorentzian Ricci flat, Euclidean Ricci flat and Einstein manifolds, respectively. The solution obtained in this fashion is supersymmetric when the Ricci flat and Einstein manifolds have Killing spinors. We generalize intersecting brane solutions, in which M5, M2 and D3-branes also wrap over the cycles determined by the K\"{a}hler forms of Ricci flat K\"{a}hler manifolds. New, singular, Ricci flat manifolds as (generalized) cones over the U(1) bundles over Ricci flat K\"{a}hler spaces are constructed. These manifolds have covariantly constant spinors and give rise to new, supersymmetric, Ricci flat compactifications of non-gauged supergravity theories. We find M2 and D3-brane solutions, which asymptotically approach these singular vacua. 
  After elimination of the redundant variables, gauge theories may still exhibit symmetries associated with the gauge fields. The role of these residual gauge symmetries is discussed within the Abelian Higgs model and the Georgi-Glashow model. In the different phases of these models, these symmetries are realized differently. The characteristics of emergence and disappearance of the symmetries are studied in detail and the implications for the dynamics in Coulomb, Higgs, and confining phases are discussed. 
  We prove that any D-dimensional theory comprising gravity, an antisymmetric n-index field strength and a dilaton can be consistently reduced on S^n in a truncation in which just $n$ scalar fields and the metric are retained in (D-n)-dimensions, provided only that the strength of the couping of the dilaton to the field strength is appropriately chosen. A consistent reduction can then be performed for n\le 5; with D being arbitrary when n\le 3, whilst D\le 11 for n=4 and D\le 10 for n=5. (Or, by Hodge dualisation, $n$ can be replaced by (D-n) in these conditions.) We obtain the lower dimensional scalar potentials and construct associated domain wall solutions. We use the consistent reduction Ansatz to lift domain-wall solutions in the (D-n)-dimensional theory back to D dimensions, where we show that they become certain continuous distributions of (D-n-2)-branes. We also examine the spectrum for a minimally-coupled scalar field in the domain-wall background, showing that it has a universal structure characterised completely by the dimension n of the compactifying sphere. 
  We review the enveloping algebra of the 10 dimensional chiral sigma matrices. To facilitate the computation of the product of several chiral sigma matrices we have developed a symbolic program. Using this program one can reduce the multiplication of the sigma matrices down to linear combinations of irreducilbe elements. We are able to quickly derive several identities that are not restricted to traces. A copy of the program written in the Mathematica language is provided for the community. 
  We revisit the lattice formulation of the Abelian Chern-Simons model defined on an infinite Euclidean lattice. We point out that any gauge invariant, local and parity odd Abelian quadratic form exhibits, in addition to the zero eigenvalue associated with the gauge invariance and to the physical zero mode at p=0 due to traslational invariance, a set of extra zero eigenvalues inside the Brillouin zone. For the Abelian Chern-Simons theory, which is linear in the derivative, this proliferation of zero modes is reminiscent of the Nielsen-Ninomiya no-go theorem for fermions. A gauge invariant, local and parity even term such as the Maxwell action leads to the elimination of the extra zeros by opening a gap with a mechanism similar to that leading to Wilson fermions on the lattice. 
  A brief review of aspects of gravity gauge theory correspondance inspired by string theory is presented. 
  We examine the relation between Coulomb-gauge fields and the gauge-invariant fields constructed in the temporal gauge for two-color QCD by comparing a variety of properties, including their equal-time commutation rules and those of their conjugate chromoelectric fields. We also express the temporal-gauge Hamiltonian in terms of gauge-invariant fields and show that it can be interpreted as a sum of the Coulomb-gauge Hamiltonian and another part that is important for determining the equations of motion of temporal-gauge fields, but that can never affect the time evolution of ``physical'' state vectors. We also discuss multiplicities of gauge-invariant temporal-gauge fields that belong to different topological sectors and that, in previous work, were shown to be based on the same underlying gauge-dependent temporal-gauge fields. We argue that these multiplicities of gauge-invariant fields are manifestations of the Gribov ambiguity. We show that the differential equation that bases the multiplicities of gauge-invariant fields on their underlying gauge-dependent temporal-gauge fields has nonlinearities identical to those of the ``Gribov'' equation, which demonstrates the non-uniqueness of Coulomb-gauge fields. These multiplicities of gauge-invariant fields --- and, hence, Gribov copies --- appear in the temporal gauge, but only with the imposition of Gauss's law and the implementation of gauge invariance; they do not arise when the theory is represented in terms of gauge-dependent fields and Gauss's law is left unimplemented. 
  We introduce a new brane-world model in which the bulk solution consists of outgoing plane waves. This is an exact solution to string theory with no naked singularities. The recently discussed self-tuning mechanism to cancel the cosmological constant on a brane is naturally incorporated. We show that even if the vacuum energy on the brane changes, e.g. due to a phase transition, the brane geometry remains insensitive to the local vacuum energy. We also consider the static self-tuning branes introduced earlier, and find a solution in which the brane geometry starts to contract when the vacuum energy on the brane changes. 
  A solution to the cosmological constant problem has been proposed in which our universe is a 3-brane in a 5-dimensional spacetime. With a bulk scalar, the field equations admit a Poincare invariant brane solution regardless of the value of the cosmological constant (tension) on the brane. However, the solution does not include matter in the brane. We find new exact static solutions with matter density and pressure in the brane. We study small perturbations about these solutions. None seem consistent with observational cosmology. As a byproduct we find a class of new matterless static solutions and a non-static solution which, curiously, requires the string value for the dilaton coupling. 
  It is well-known that the standard no-ghost theorem can be extended straightforwardly to the general c=26 CFT on R^{d-1,1} \times K, where 2 \leq d \leq 26 and K is a compact unitary CFT of appropriate central charge. We prove the no-ghost theorem for d=1, i.e., when only the timelike direction is flat. This is done using the technique of Frenkel, Garland and Zuckerman. 
  We derive an exact expression for the Fourier coefficients of elliptic genera of Calabi-Yau manifolds. When applied to k-fold symmetric products of K3 surfaces the expression is well-suited to studying the AdS/CFT correspondence on AdS3 x S3. The expression also elucidates an SL(2,Z) invariant phase diagram for the D1/D5 system involving deconfining transitions in the limit as k goes to infinity. 
  By dimensional reduction of a massive supersymmetric B$\wedge $F theory, a manifestly N=1 supersymmetric completion of a massive antisymmetric tensor gauge theory is constructed in (2+1) dimensions. In the N=1-D=3 superspace, a new topological term is used to give mass for the Kalb-Ramond field. We have introduced a massive gauge invariant model using the Stuckelberg formalism and an abelian topologically massive theory for the Kalb-Ramond superfield. An equivalence of both massive models is suggested. Further, a component field analysis is performed, showing a second supersymmetry in the model. 
  A noncommutative gauge theory is associated to every Abelian gauge theory on a Poisson manifold. The semi-classical and full quantum version of the map from the ordinary gauge theory to the noncommutative gauge theory (Seiberg-Witten map) is given explicitly to all orders for any Poisson manifold in the Abelian case. In the quantum case the construction is based on Kontsevich's formality theorem. 
  When unstable Dp-branes in type II string theory are placed in a B-field, the resulting tachyonic world-volume theory becomes noncommutative. We argue that for large noncommutativity parameter, condensation of the tachyon as a noncommutative soliton leads to new decay modes of the Dp-brane into (p-2)-brane configurations, which we interpret as suitably smeared BPS D(p-1)-branes. Some of these configurations are metastable. We discuss various generalizations of this decay process. 
  A class of supergravity backgrounds have been proposed as dual descriptions of strong coupling large-N noncommutative Yang-Mills (NCYM) theories in 3+1 dimensions. However calculations of correlation functions in supergravity from an evaluation of relevant classical actions appear ambiguous. We propose a resolution of this ambiguity. Assuming that {\it some} holographic description exists - regardless of whether it is the NCYM theory - we argue that there should be operators in the holographic boundary theory which create normalized states of definite energy and momenta. An operator version of the dual correspondence then provides a calculation of correlators of these operators in terms of bulk Green's functions. We show that in the low energy limit the correlators reproduce expected answers of the ordinary Yang-Mills theory. 
  We study the dynamics of 5-dimensional gauge theory on $M_4\times S^1$ by compactifying type II/M theory on degenerate Calabi-Yau manifolds. We use the local mirror symmetry and shall show that the prepotential of the 5-dimensional SU(2) gauge theory without matter is given exactly by that of the type II string theory compactified on the local ${\bf F}_2$, i.e. Hirzebruch surface ${\bf F}_2$ lying inside a non-compact Calabi-Yau manifold. It is shown that our result reproduces the Seiberg-Witten theory at the 4-dimensional limit $R\to 0$ ($R$ denotes the radius of $S^1$) and also the result of the uncompactified 5-dimensional theory at $R\to \infty$. We also discuss SU(2) gauge theory with $1\le N_f\le 4$ matter in vector representations and show that they are described by the geometry of the local ${\bf F}_2$ blown up at $N_f$ points. 
  The idea of a companion Lagrangian associated with $p$-Branes is extended to include the presence of U(1) fields. The Brane Lagrangians are constructed with $F_{ij}$ represented in terms of Lagrange Brackets, which make manifest the reparametrisation invariance of the theory; these are replaced by Poisson Brackets in the companion Lagrangian, which is now covariant under field redefinition. The ensuing Lagrangians possess a similar formal structure to those in the absence of an anti-symmetric field tensor. 
  We discuss the notion of duality and selfduality in the context of the dual projection operation that creates an internal space of potentials. Contrary to the prevailing algebraic or group theoretical methods, this technique is applicable to both even and odd dimensions. The role of parity in the kernel of the Gauss law to determine the dimensional dependence is clarified. We derive the appropriate invariant actions, discuss the symmetry groups and their proper generators. In particular, the novel concept of duality symmetry and selfduality in Maxwell theory in (2+1) dimensions is analysed in details. The corresponding action is a 3D version of the familiar duality symmetric electromagnetic theory in 4D. Finally, the duality symmetric actions in the different dimensions constructed here manifest both the SO(2) and $Z_2$ symmetries, contrary to conventional results. 
  Four-dimensional Einstein gravity in the Palatini first order formalism is shown to possess a vector supersymmetry of the same type as found in the topological theories for Yang-Mills fields. A peculiar feature of the gravitational theory, characterized by diffeomorphism invariance, is a direct link of vector supersymmetry with the field equation of motion for the Faddeev-Popov ghost of diffeomorphisms. 
  Negative norm Hilbert space state vectors can be BRST invariant, we show in a simplified Y-M model that such states can be created by starting with gluons only. 
  We consider moving-brane-solutions in AdS type back ground. In the first Randall-Sundrum configuration, there are two branes at fixed points of the orbifold symmetry. We point out that if one brane is fixed and the other brane is moving, the configuration is still a solution provided the moving brane has a specific velocity determined by its tension and bulk cosmological constant. In the absence of the $\bf Z_2$ symmetry, we can construct multi-brane configurations by patching AdS-Schwarzshild solutions. In this case, we show that the 4-dimensional effective cosmological constant on the brane world is not well defined. We find a condition for a brane to be stationary. Using the brane scattering, we suggest a scenario of inflation on the brane universe during a finite time, i.e, a scenario of a graceful exit of inflation. 
  We study gravity localization in the context of a six-dimensional gravity model coupled with complex scalar fields. With a supergravity-motivated scalar potential, we show that the domain wall junction solutions localize a four-dimensional massless graviton under an assumption on the wall profile. We find that unlike the global supersymmetric model, contributions to the junction tension cancel locally with gravitational contributions. The wall tension vanishes due to the metric suppression. 
  Field theories based on non-commutative spacetimes exhibit very distinctive nonlocal effects which mix the ultraviolet with the infrared in bizarre ways. In particular if the time coordinate is involved in the non-commutativity the theory seems to be seriously acausal and inconsistent with conventional Hamiltonian evolution. To illustrate these effects we study the scattering of wave packets in a field theory with space/time non-commutativity. In this theory we find effects which seem to precede their causes and rigid rods which grow instead of Lorentz contract as they are boosted. These field theories are evidently inconsistent and violate causality and unitarity. On the other hand open string theory in a background electric field is expected to exhibit space/time non-commutativity. This raises the question of whether they also lead to acausal behavior. We show that this is not the case. Stringy effects conspire to cancel the acausal effects that are present for the non-commutative field theory. 
  We suggest a mechanism by which four-dimensional Newtonian gravity emerges on a 3-brane in 5D Minkowski space with an infinite size extra dimension. The worldvolume theory gives rise to the correct 4D potential at short distances whereas at large distances the potential is that of a 5D theory. We discuss some phenomenological issues in this framework. 
  We study the O(N) $\phi^4$ model compactified on $M^{D-1}\otimes S^1$, which allows to impose twisted boundary conditions for the $S^1$-direction. The O(N) symmetry can be broken to $H$ explicitly by the boundary conditions and further broken to $I$ spontaneously by vacuum expectation values of the fields. The symmetries $H$ and $I$ are completely classified and the model turns out to have unexpectedly a rich phase structure. The unbroken symmetry $I$ is shown to depend on not only the boundary conditions but also the radius of $S^1$, and the symmetry breaking patterns are found to be unconventional. The spontaneous breakdown of the translational invariance is also discussed. 
  We find a class of Fermion zero modes of Abelian Dirac operators in three dimensional Euclidean space where the gauge potentials and the related magnetic fields are nonzero only in a finite space region. 
  There exist two versions of the covariant Schwinger term in the literature. They only differ by a sign. However, we shall show that this is an essential difference. We shall carefully (taking all signs into account) review the existing quantum field theoretical computations for the covariant Schwinger term in order to determine the correct expression. 
  An example of the holographic correspondence between 2d, N=2 quantum field theories and classical 4d, N=2 supergravity theories is found. The constraints on the target space geometry of the 4d, N=2 non-linear sigma-models in N=2 supergravity background are interpreted as the renormalization flow equations in two dimensions. Our geometrical description of the renormalization flow is manifestly covariant under reparametrization of the 2d coupling constants. The proposed holography is described in terms of the (Weyl) anti-self-dual Einstein metrics, whose exact regular (Tod-Hitchin) solutions are governed by the Painleve VI equation. 
  Following the conjectured duality between near-horizon NS5-branes and little string theory, the string-corrected thermodynamics of near-horizon NS5-branes is studied and found to agree with the statistical thermodynamics of a 5+1 dimensional supersymmetric string theory near the Hagedorn temperature. Specifically, tree-level corrections to the temperature are argued to vanish, in accordance with the duality, while the one-loop string correction to the NS5-brane thermodynamics is shown to generate the correct temperature dependence of the entropy. 
  The gauge invariant method for calculation of the effective action of the local composite fields in QFT is proposed. The effective action of the local composite fields in QED is studied up to 2-loop level. The graph rules for the local composite fields are derived. On the basis of these rules the problem of one-particle irreducibility is discussed. 
  We show that the cohomology groups of the horizontal (total) differential on horizontal (local) exterior forms on the infinite-order jet manifold of an affine bundle coincide with the De Rham cohomology groups of the base manifold. This prevents one from the topological obstruction to the definition of global descent equations in BRST theory on an arbitrary affine bundle. 
  A nonlinear integral equation that is responsible for the implementation of the non-Abelian Gauss's law is applied to an investigation of the topological features of two-color QCD and to a discussion of their relation to QCD dynamics. We also draw a parallel between the nonuniqueness of the solutions of the equations that govern the gauge-invariant gauge field and Gribov copies. 
  We address the question of cosmological perturbations in the context of brane cosmology, where our Universe is a three-brane where matter is confined, whereas gravity lives in a higher dimensional spacetime. The equations governing the bulk perturbations are computed in the case of a general warped universe. The results are then specialized to the case of a five-dimensional spacetime, scenario which has recently attracted a lot of attention. In this context, we decompose the perturbations into `scalar', `vector' and `tensor' modes, which are familiar in the standard theory of cosmological perturbations. The junction conditions, which relate the metric perturbations to the matter perturbations in the brane, are then computed. 
  We investigate, in a certain decoupling limit, the effect of having a constant C-field on the M-theory five-brane using an open membrane probe. We define an open membrane metric for the five-brane that remains non-degenerate in the limit. The canonical quantisation of the open membrane boundary leads to a noncommutative loop space which is a functional analogue of the noncommutative geometry that occurs for D-branes. 
  Sklyanin's method of separation of variables is employed in a calculation of finite temperature expectation values. An essential element of the approach is Baxter's $Q$-function. We propose its explicit form corresponding to the ground state of the sinh-Gordon theory. With the method of separation of variables we calculate the finite temperature expectation values of the exponential fields to one-loop order of the semi-classical expansion. 
  We examine the cosmological effects of the Hagedorn phase in models where the observable universe is pictured as a D-brane. It is shown that, even in the absence of a cosmological constant, winding modes cause a negative `pressure' that can drive brane inflation of various types including both power law and exponential. We also find regimes in which the cosmology is stable but oscillating (a bouncing universe) with the Hagedorn phase softening the singular behavior associated with the collapse. 
  A comprehensive introduction to the boundary state approach to Dirichlet branes is given. Various examples of BPS and non-BPS Dirichlet branes are discussed. In particular, the non-BPS states in the duality of Type IIA on K3 and the heterotic string on T4 are analysed in detail. 
  We consider a Yang-Mills theory in loop space with an affine Lie gauge group. The Chapline-Manton coupling, the coupling between Yang-Mills fields and an abelian antisymmetric tensor field of second rank via the Chern-Simons term, is systematically derived within the framework of the Yang-Mills theory. The generalized Chapline-Manton couplings, the couplings among non-abelian tensor fields of second rank, Yang-Mills fields, and an abelian tensor field of third rank, are also derived by applying the non-linear realization method to the Yang-Mills theory. These couplings are accompanied by {\it BF}-like terms. 
  The non-commutative geometry of a large auxiliary $B$-field simplifies the construction of D-branes as solitons in open string field theory. Similarly, fundamental strings are constructed as localized flux tubes in the string field theory. Tensions are determined exactly using general properties of non-BPS branes, and the non-Abelian structure of gauge fields on coincident D-branes is recovered. 
  We develop a gauge-invariant formalism to describe metric perturbations in five-dimensional brane-world theories. In particular, this formalism applies to models originating from heterotic M-theory. We introduce a generalized longitudinal gauge for scalar perturbations. As an application, we discuss some aspects of the evolution of fluctuations on the brane. Moreover, we show how the five-dimensional formalism can be matched to the known four-dimensional one in the limit where an effective four-dimensional description is appropriate. 
  It has recently been suggested that our universe is a three-brane embedded in a higher dimensional spacetime. In this paper I examine static, spherically symmetric solutions that satisfy the effective Einstein field equations on a brane embedded in a five dimensional spacetime. The field equations involve a term depending on the five dimensional Weyl tensor, so that the solutions will not be Schwarzschild in general. This Weyl term is traceless so that any solution of $^{(4)}R=0$ is a possible four dimensional spacetime. Different solutions correspond to different five dimensional spacetimes and to different induced energy-momentum tensors on the brane. One interesting possibility is that the Weyl term could be responsible for the observed dark matter in the universe. 
  Higher dimensional Majorana-Weyl spacetimes present space-time dualities which are induced by the Spin(8) triality automorphisms. This corresponds to a very fundamental property of the supersymmetry in higher dimensions, i.e. that any given theory can be formulated in different signatures all interconnected by the S_3 permutation group. 
  Higher dimensional Majorana-Weyl spacetimes present space-time dualities which are induced by the Spin(8) triality automorphisms. Different signature versions of theories such as 10-dimensional SYM's, superstrings, five-branes, F-theory, are shown to be interconnected via the S_3 permutation group. Bilinear and trilinear invariants under space-time triality are introduced and their possible relevance in building models possessing a space-versus-time exchange symmetry is discussed. Moreover the Cartan's ``vector/chiral spinor/antichiral spinor" triality of SO(8) and SO(4,4) is analyzed in detail and explicit formulas are produced in a Majorana-Weyl basis. This paper is the extended version of hep-th/9907148. 
  It has been conjectured that the tachyonic lump solution of the open bosonic string field theory describing a D-brane represents a D-brane of one lower dimension. We place the lump on a circle of finite radius and develop a variant of the level expansion scheme that allows systematic account of all higher derivative terms in the string field theory action, and gives a calculational scheme that can be carried to arbitrary accuracy. Using this approach we obtain lump masses that agree with expected D-brane masses to an accuracy of about 1%. We find convincing evidence that in string field theory the lump representing a D-brane is an extended object with a definite profile. A gaussian fit to the lump gives a 6-sigma size of 9.3 \sqrt{\alpha'}. The level truncation scheme developed here naturally gives rise to an infrared and ultraviolet cut-off, and may be useful in the study of quantum string field theory. 
  The so-called conformal affine Toda theory coupled to the matter fields (CATM), associated to the $\hat{sl}(2)$ affine Lie algebra, is studied. The conformal symmetry is fixed by setting a connection to zero, then one defines an off-critical model, the affine Toda model coupled to the matter (ATM). The quantum version of this reduction process is discussed by means of the perturbative Lagrangian viewpoint, showing that the ATM theory is a spontaneously broken and reduced version of the CATM model. We show, using bosonization techniques that the off-critical theory decouples into a sine-Gordon model and a free scalar. Using the "dressing" transformation method we construct the explicit forms of the one and two-soliton classical solutions, and show that a physical bound soliton-antisoliton pair (breather) does not exist. Moreover, we verify that these solutions share some features of the sine- Gordon (massive Thirring) solitons, and satisfy the classical equivalence of topological and Noether currents in the ATM model. Imposing the Noether and topological currents equivalence as a constraint, one can show that the ATM model leads to a bag model like mechanism for the confinement of the U(1) "color" charge inside the sine-Gordon solitons (baryons). 
  The functional integral of pure Einstein 4D quantum gravity admits abnormally large and long-lasting "dipolar fluctuations", generated by virtual sources with the property Int d^4x Sqrt{g(x)} Tr T(x) = 0. These fluctuations would exist also at macroscopic scales, with paradoxical consequences. We set out their general features and give numerical estimates of possible suppression processes. 
  At higher energies the present complex quantum theory with its unitary group might expand into a real quantum theory with an orthogonal group, broken by an approximate $i$ operator at lower energies. Implementing this possibility requires a real quantum double-valued statistics. A Clifford statistics, representing a swap (12) by a difference $\gamma_1-\gamma_2$ of Clifford units, is uniquely appropriate. Unlike the Maxwell-Boltzmann, Fermi-Dirac, Bose-Einstein, and para- statistics, which are tensorial and single-valued, and unlike anyons, which are confined to two dimensions, Clifford statistics are multivalued and work for any dimensionality. Nayak and Wilczek proposed a Clifford statistics for the fractional quantum Hall effect. We apply them to toy quanta here. A complex-Clifford example has the energy spectrum of a system of spin-1/2 particles in an external magnetic field. This supports the proposal that the double-valued rotations --- spin --- seen at current energies might arise from double-valued permutations --- swap --- to be seen at higher energies. Another toy with real Clifford statistics illustrates how an effective imaginary unit $i$ can arise naturally within a real quantum theory. 
  Searching for space/time noncommutativity we reconsider open strings in a constant background electric field. The main difference between this situation and its magnetic counterpart is that here there is a critical electric field beyond which the theory does not make sense. We show that this critical field prevents us from finding a limit in which the theory becomes a field theory on a noncommutative spacetime. However, an appropriate limit toward the critical field leads to a novel noncritical string theory on a noncommutative spacetime. 
  In string theory, the consequences of replacing the measure of integration $\sqrt{-\gamma}d^2 x$ in the Polyakov's action by $\Phi d^2 x$ where $\Phi$ is a density built out of degrees of freedom independent of the metric $\gamma_{ab}$ defined in the string are studied. The string tension appears as an integration constant of the equations of motion. The string tension can change in different parts of the string due to the coupling of gauge fields and point particles living in the string. The generalization to higher dimensional extended objects is also studied. In this case there is no need of a fine tuned cosmological term, in sharp contrast to the standard formulation of the generalized Polyakov action for higher dimensional branes. 
  The breaking of supersymmetry due to singular potentials in supersymmetric quantum mechanics is critically analyzed. It is shown that, when properly regularized, these potentials respect supersymmetry, even when the regularization parameter is removed. 
  The entropy of anti-de Sitter Reissner-Nordstrom black hole is found to be stored in the material which gathers to form it and equals to $A/4$ regardless of material states. Extending the study to two kinds of extreme black holes, we find different entropy results for the first kind of extreme black hole due to different material states. However for the second kind of extreme black hole the results of entropy are uniform independently of the material states. Relations between these results and the stability of two kinds of extreme black holes have been addressed. 
  An N=2 supersymmetric self-interaction of the vector-tensor multiplet is presented, in which the vector provides the gauge field for local central charge transformations. The dual description in terms of a vector multiplet and an N=1 superspace formulation are given. 
  We explore the idea of a network of defects to live inside a domain wall in models of three real scalar fields, engendering the Z_2 x Z_3 symmetry. The field that governs the Z_2 symmetry generates a domain wall, and entraps the hexagonal network formed by the three-junctions of the model of two scalar fields that describes the remaining Z_3 symmetry. If the host domain wall bends to the spherical form, in the thin wall approximation there may appear non-topological structures hosting networks that accept diverse patterns. If Z_3 is also broken, the model may generate a buckyball containing sixty junctions, a fullerene-like structure. Applications to cosmology are outlined. 
  We study the generalization of S-duality to non-commutative gauge theories. For rank one theories, we obtain the leading terms of the dual theory by Legendre transforming the Lagrangian of the non-commutative theory expressed in terms of a commutative gauge field. The dual description is weakly coupled when the original theory is strongly coupled if we appropriately scale the non-commutativity parameter. However, the dual theory appears to be non-commutative in space-time when the original theory is non-commutative in space. This suggests that locality in time for non-commutative theories is an artifact of perturbation theory. 
  We analyze the thermodynamics of NS 5-branes as the temperature approaches the NS 5-branes' Hagedorn temperature, and conclude that the dynamics of ``Little String Theory'' is a new universality class of interacting strings. First we point out how to vary the temperature of the near extremal solution by taking into account $g_s$ corrections. The Hagedorn temperature is shown to be a limiting temperature for the theory. We then compare the thermodynamics to that of a toy model made of free strings and find basic discrepancies. This suggests a need for a new class of string interactions. We suggest that this new universality class is characterized by a strong attractive self-intersection interaction, which causes strings to be coiled. This model might also explain why ``Little String Theories'' exist in at most 5+1 dimensions. 
  It is conjectured that strongly coupled, spatially noncommutative $\CN=4$ Yang-Mills theory has a dual description as a weakly coupled open string theory in a near critical electric field, and that this dual theory is fully decoupled from closed strings. Evidence for this conjecture is given by the absence of physical closed string poles in the non-planar one-loop open string diagram. The open string theory can be viewed as living in a geometry in which space and time coordinates do not commute. 
  We investigate the correspondence between existence/stability of BPS states in type II string theory compactified on a Calabi-Yau manifold and BPS solutions of four dimensional N=2 supergravity. Some paradoxes emerge, and we propose a resolution by considering composite configurations. This in turn gives a smooth effective field theory description of decay at marginal stability. We also discuss the connection with 3-pronged strings, the Joyce transition of special Lagrangian submanifolds, and Pi-stability. 
  Classical relativistic system of point particles coupled with an electromagnetic field is considered in the three-dimensional representation. The gauge freedom connected with the chronometrical invariance of the four-dimensional description is reduced by use of the geometrical concept of the forms of relativistic dynamics. The remainder gauge degrees of freedom of the electromagnetic potential are analysed within the framework of Dirac's constrained Hamiltonian mechanics in the front form of dynamics. The results are implemented to the problems of relativistic statistical mechanics. Based on the corresponding Liouville equation the classical partition function of the system is written down in a gauge-invariant manner and an integration over field variables is performed. 
  We study Euclidean D=4, N=4 gauged $SU(2) \times SU(1,1)$ supergravity theory which has been obtained from dimensional reduction of N=1, D=10 supergravity on $S^3 \times AdS_3$. We obtain supersymmetric configurations like domain wall, electro-vac type of solutions with geometries $E^2 \times S^2$, $E^2 \times AdS_2$ and axio-vac type $E^1 \times S^3$ solution in this Euclidean Freedman-Schwarz (EFS) model. We also show that the Euclidean gravitational instantons with nontrivial (anti)self-dual U(1) gauge fields are stable vacua preserving one fourth of the original supersymmetry. 
  We study the dynamics of $D$-branes in the near-horizon geometry of $NS$ fivebranes. This leads to a holographically dual description of the physics of $D$-branes ending on and/or intersecting $NS5$-branes. We use it to verify some properties of such $D$-branes which were deduced indirectly in the past, and discuss some instabilities of non-supersymmetric brane configurations. Our construction also describes vacua of Little String Theory which are dual to open plus closed string theory in asymptotically linear dilaton spacetimes. 
  The Hopf algebra of Feynman diagrams, analyzed by A.Connes and D.Kreimer, is considered from the perspective of the theory of effective actions and generalized $\tau$-functions, which describes the action of diffeomorphism and shift groups in the moduli space of coupling constants. These considerations provide additional evidence of the hidden group (integrable) structure behind the standard formalism of quantum field theory. 
  In this paper we study non-commutative Yang-Mills theory (NCYM) through its gravity dual. First it is shown that the gravity dual of an NCYM with self-dual $\theta$-parameters has a Lagrangian in the form of five-dimensional dilatonic gravity. Then we use the de-Boer-Verlinde-Verlinde formalism for holographic renormalization group flows to calculate the coefficient functions in the Weyl anomaly of the NCYM at low energies under the assumption of potential dominance, and show that the $C$-theorem holds true in the present case. 
  It is of considerable importance to have a numerical method for solving supersymmetric theories that can support a non-zero central charge. The central charge in supersymmetric theories is in general a boundary integral and therefore vanishes when one uses periodic boundary conditions. One is therefore prevented from studying BPS states in the standard supersymmetric formulation of DLCQ (SDLCQ). We present a novel formulation of SDLCQ where the fields satisfy anti-periodic boundary conditions. The Hamiltonian is written as the anti-commutator of two charges, as in SDLCQ. The anti-periodic SDLCQ we consider breaks supersymmetry at finite resolution, but requires no renormalization and becomes supersymmetric in the continuum limit. In principle, this method could be used to study BPS states. However, we find its convergence to be disappointingly slow. 
  Classical and quantum dynamics of two distinct BPS monopoles in the case of non-aligned Higgs fields are studied on the basis of the recently determined low energy effective theory. Despite the presence of a specific potential together with a kinetic term provided by the metric of a Taub-NUT manifold, an O(4) or O(3,1) symmetry of the system allows for a group theoretical derivation of the bound-state spectrum and the scattering cross section. 
  The concept of particle weights has been introduced by Buchholz and the author in order to obtain a unified treatment of particles as well as (charged) infraparticles which do not permit a definition of mass and spin according to Wigner's theory. Particle weights arise as temporal limits of physical states in the vacuum sector and describe the asymptotic particle content. Following a thorough analysis of the underlying notion of localizing operators, we give a precise definition of this concept and investigate the characteristic properties. The decomposition of particle weights into pure components which are linked to irreducible representations of the quasi-local algebra has been a long-standing desideratum that only recently found its solution. We set out two approaches to this problem by way of disintegration theory, making use of a physically motivated assumption concerning the structure of phase space in quantum field theory. The significance of the pure particle weights ensuing from this disintegration is founded on the fact that they exhibit features of improper energy-momentum eigenstates, analogous to Dirac's conception, and permit a consistent definition of mass and spin even in an infraparticle situation. 
  Bound-state solutions are obtained numerically in the instantaneous approximation for a spin-0 and spin-1/2 constituent that interact via minimal electrodynamics. To solve the integral equations in momentum space, a method is developed for integrating over the logarithmic singularity in kernels, making it possible to use basis functions that essentially automatically satisfy the boundary conditions. For bound-state solutions that decrease rapidly at small and large values of momentum, accurate solutions are obtained with significantly fewer basis functions when the solution is expanded in terms of these more general basis functions. The presence of a derivative coupling in single-photon exchange complicates the construction of the Bethe-Salpeter equation in the instantaneous approximation and, in the nonrelativistic limit, gives rise to an additional electrostatic potential term that is second order in the coupling constant and decreases as the square of the distance between constituents. 
  We extend standard path-integral techniques of bosonization and duality to the setting of noncommutative geometry. We start by constructing the bosonization prescription for a free Dirac fermion living in the noncommutative plane R_\theta^2. We show that in this abelian situation the fermion theory is dual to a noncommutative Wess-Zumino-Witten model. The non-abelian situation is also constructed along very similar lines. We apply the techniques derived to the massive Thirring model on noncommutative R_\theta^2 and show that it is dualized to a noncommutative WZW model plus a noncommutative cosine potential (like in the noncommutative Sine-Gordon model). The coupling constants in the fermionic and bosonic models are related via strong-weak coupling duality. This is thus an explicit construction of S-duality in a noncommutative field theory. 
  We prove that a kind of averaging procedure for constructing gauge-invariant operators(or functionals) out of gauge-variant ones is erroneous and inapplicable for a large class of operators(or functionals). 
  The Penrose limit is generalized to show that, any leading order solution of the low-energy field equations in any one of the five string theories has a plane wave solution as a limit. This limiting procedure takes into account all the massless fields that may arise and commutes with the T-duality so that any dual solution has again a plane wave limit. The scaling rules used in the limit are unique and stem from the scaling property of the D=11 supergravity action. Although the leading order solutions need not be exact or supersymmetric, their plane wave limits always preserve some portion of the Poincare supersymmetry and solve the relevant field equations in all powers of the string tension parameter. Further properties of the limiting procedure are discussed. 
  The singularity present in cosmological instantons of the Hawking-Turok type is resolved by a conformal transformation, where the conformal factor has a linear zero of codimension one. We show that if the underlying regular manifold is taken to have the topology of $RP^4$, and the conformal factor is taken to be a twisted field so that the zero is enforced, then one obtains a one-parameter family of solutions of the classical field equations, where the minimal action solution has the conformal zero located on a minimal volume noncontractible $RP^3$ submanifold. For instantons with two singularities, the corresponding topology is that of a cylinder $S^3\times [0,1]$ with D=4 analogues of `cross-caps' at each of the endpoints. 
  In this note we give some remarks on the BRST formulation of a renormalizable and diffemorphism invariant 4D quantum gravity recently proposed by the author, which satisfies the integrability condition by Riegard, Fradkin and Tseytlin at the 2-loop level. Diffeomorphism invariance requires an addition of the Wess-Zumino action, from which the Weyl action can be induced by expanding around a vacuum expectation value of the conformal mode. This fact suggests the theory has in itself a mechanism to remove extra negative-metric states dynamically. 
  We study how instantons arise in the low energy effective theory of the SU(2) Yang-Mills theory in the context of the non-linear sigma model recently propose by Faddeev and Niemi. We find a simple relation between the instanton number $\nu$ and the charge m of the monopole that appears in the effective theory. It is given by $\nu = m \Phi/(2\pi)$, where $\Phi$ is the quantized flux associated with a U(1) gauge field passing through the loop formed by the singularity of the monopole. 
  We investigate (super) string theory on $AdS_3$ background based on an approach of free field realization. We demonstrate that this string theory can be reformulated as a string theory defined on a linear dilaton background along the transverse direction (``Liouville mode'') and compactified onto $S^1$ along a {\em light-like} direction.   Under this reformulation we analyze the physical spectrum as that of a free field system, and discuss the consequences when we turn on the Liouville potential. The discrete light-cone momentum in our framework is naturally interpreted as the ``winding number'' of the long string configuration and is identified with the spectral flow parameter that is introduced in the recent work by Maldacena and Ooguri \cite{MO}.   Moreover we show that there exist infinite number of the on-shell chiral primary states possessing the different light-cone momenta and the spectral flow consistently acts on the space of chiral primaries. We observe that they are also chiral primaries in the sense of space-time (or the conformal theory of long string) and the spectrum of space-time $U(1)_R$ charge is consistent with the expectation from the $AdS_3/CFT_2$-duality. We also clarify the correspondence between our framework and the symmetric orbifold theory of multiple long string system \cite{HS2}. 
  We suggest a modification of the Randall-Sundrum scenario which doesn't involve branes with the negative tensions. In our scenario four-dimensional brane world is produced by the external field. The probability of this process is calculated and the physical features of the model are discussed. 
  We propose a bottom-up approach to the building of particle physics models from string theory. Our building blocks are Type II D-branes which we combine appropriately to reproduce desirable features of a particle theory model: 1) Chirality ; 2) Standard Model group ; 3) N=1 or N=0 supersymmetry ; 4) Three quark-lepton generations. We start such a program by studying configurations of D=10, Type IIB D3-branes located at singularities. We study in detail the case of Z_N, N=1,0 orbifold singularities leading to the SM group or some left-right symmetricextension. In general, tadpole cancellation conditions require the presence of additional branes, e.g. D7-branes. For the N=1 supersymmetric case the unique twist leading to three quark-lepton generations is Z_3, predicting $\sin^2\theta_W=3/14=0.21$. The models obtained are the simplest semirealistic string models ever built. In the non-supersymmetric case there is a three-generation model for each Z_N, N>4, but the Weinberg angle is in general too small. One can obtain a large class of D=4 compact models by considering the above structure embedded into a Calabi Yau compactification. We explicitly construct examples of such compact models using Z_3 toroidal orbifolds and orientifolds, and discuss their properties. In these examples, global cancellation of RR charge may be achieved by adding anti-branes stuck at the fixed points, leading to models with hidden sector gravity-induced supersymmetry breaking. More general frameworks, like F-theory compactifications, allow completely $\NN=1$ supersymmetric embeddings of our local structures, as we show in an explicit example. 
  The effective action of N=2 supersymmetric 5-dimensional supergravity arising from compactifications of M-theory on Calabi-Yau threefolds receives non-perturbative corrections from wrapped Euclidean membranes and fivebranes. These contributions can be interpreted as instanton corrections in the 5 dimensional field theory. Focusing on the universal hypermultiplet, a solution of this type is presented and the instanton action is calculated, generalizing previous results involving membrane instantons. The instanton action is not a sum of membrane and fivebrane contributions: it has the form reminiscent of non-threshold bound states. 
  The linear $\delta$ expansion is used to obtain corrections up to O$(\delta^2)$ to the self-energy for a complex scalar field theory with a $\lambda (\phi^{\star}\phi)^2$ interaction at high temperature and non-zero charge density. The calculation is done in the imaginary-time formalism via the Hamiltonian form of the path integral. Nonperturbative results are generated by a systematic order by order variational procedure and the dependence of the critical temperature on the chemical potential $\mu$ is obtained. 
  Canonical BRST quantization of the topological particle defined by a Morse function h is described. Stochastic calculus, using Brownian paths which implement the WKB method in a new way providing rigorous tunnelling results even in curved space, is used to give an explicit and simple expression for the matrix elements of the evolution operator for the BRST Hamiltonian. These matrix elements lead to a representation of the manifold cohomology in terms of critical points of h along lines developed by Witten. 
  A formulation of Skyrme model as an embedded gauge theory with the constraint deformed away from the spherical geometry is proposed. The gauge invariant formulation is obtained firstly generalizing the intrinsic geometry of the model and then converting the constraint to first-class through an iterative Wess-Zumino procedure. The gauge invariant model is quantized via Dirac method for first-class system. A perturbative calculation provides new free parameters related to deformation that improve the energy spectrum obtained in earlier approaches. 
  Using AdS_7/CFT_6 correspondence we discuss computation of a subleading O(N) term in the scale anomaly of (2,0) theory describing N coincident M5 branes. While the leading O(N^3) contribution to the anomaly is determined by the supergravity action, the O(N) contribution comes from a particular R^4 term (8d Euler density) in the 11-dimensional effective action. This R^4 term is argued to be part of the same superinvariant as the P-odd C_3 R^4 term known to produce O(N) contribution to the R-symmetry anomaly of (2,0) theory. The known results for R-anomaly suggest that the total scale anomaly extrapolated to N=1 should be the same as the anomaly of a single free (2,0) tensor multiplet. A proposed explanation of this agreement is that the coefficient 4N^3 in the anomaly (which was found previously to be also the ratio of the 2-point and 3-point graviton correlators in the (2,0) theory and in the free tensor multiplet theory) is shifted to 4N^3 -3N. 
  We study aspects of obtaining field theories with noncommuting time-space coordinates as limits of open-string theories in constant electric-field backgrounds. We find that, within the standard closed-string backgrounds, there is an obstruction to decoupling the time-space noncommutativity scale from that of the string fuzziness scale. We speculate that this censorship may be string-theory's way of protecting the causality and unitarity structure. We study the moduli space of the obstruction in terms of the open- and closed-string backgrounds. Cases of both zero and infinite brane tensions as well as zero string couplings are obtained. A decoupling can be achieved formally by considering complex values of the dilaton and inverting the role of space and time in the light cone. This is reminiscent of a black-hole horizon. We study the corresponding supergravity solution in the large-N limit and find that the geometry has a naked singularity at the physical scale of noncommutativity. 
  In view of the recent interest in formulating a quantum theory of Ramond-Ramond p-forms, we exhibit an SL(10,Z) invariant partition function for the chiral four-form of Type IIB string theory on the ten-torus. We follow the strategy used to derive a modular invariant partition function for the chiral two-form of the M-theory fivebrane. We also generalize the calculation to self-dual quantum fields in spacetime dimension 2p=2+4k, and display the SL(2p,Z) automorphic forms for odd p>1. We relate our explicit calculation to a computation of the B-cycle periods, which are discussed in the work of Witten. 
  In this note we calculate the spectrum of two-dimensional QCD. We formulate the theory with SU(N_c) currents rather than with fermionic operators. We construct the Hamiltonian matrix in DLCQ formulation as a function of the harmonic resolution K and the numbers of flavors N_f and colors N_c. The resulting numerical eigenvalue spectrum is free from trivial multi-particle states which obscured previous results. The well-known 't Hooft and large N_f spectra are reproduced. In the case of adjoint fermions we present some new results. 
  We find the phase and flavor symmetry breaking pattern of each N=1 supersymmetric vacuum of SU(n_c) and USp(2 n_c) gauge theories, constructed from the exactly solvable N=2 theories by perturbing them with small adjoint and generic bare hypermultiplet (quark) masses. In SU(n_c) theories with n_f \leq n_c the vacua are labelled by an integer r, in which the flavor U(n_f) symmetry is dynamically broken to U(r) \times U(n_f-r) in the limit of vanishing bare hyperquark masses. In the r=1 vacua the dynamical symmetry breaking is caused by the condensation of magnetic monopoles in the n_f representation. For general r, however, the monopoles in the {}_{n_f}C_r representation, whose condensation could explain the flavor symmetry breaking but would produce too-many Nambu--Goldstone multiplets, actually "break up" into "magnetic quarks": the latter with nonabelian interactions condense and induce confinement and dynamical symmetry breaking. In USp(2n_c) theories with n_f \leq n_c + 1, the flavor SO(2n_f) symmetry is dynamically broken to U(n_f), but with no description in terms of a weakly coupled local field theory. In both SU(n_c) and USp(2 n_c) theories, with larger numbers of quark flavors, besides the vacua with these properties, there exist also vacua in free magnetic phase, with unbroken global symmetry. 
  The zero momentum sectors in effective theories of three dimensional QCD coupled to pseudoreal (two colors) and real (adjoint) quarks in a classically parity-invariant manner have alternative descriptions in terms of orthogonal and symplectic ensembles of random matrices. Using this correspondence, we compute finite-volume QCD partition functions and correlation functions of Dirac operator eigenvalues in a presence of finite quark masses of the order of the smallest Dirac eigenvalue. These novel correlation functions, expressed in terms of quaternion determinants, are reduced to conventional results for the Gaussian ensembles in the quenched limit. 
  In the space-dependent gauge, each mode of the Klein-Gordon equation in a strong electric field takes the form of a time-independent Schr\"{o}dinger equation with a potential barrier. We propose that the single- and multi-instantons of quantum tunneling may be related with the single- and multi-pair production of bosons and the relative probability for the no-pair production is determined by the total tunneling probability via instantons. In the case of a uniform electric field, the instanton interpretation recovers exactly the well-known pair production rate for bosons and when the Pauli blocking is taken into account, it gives the correct fermion production rate. The instanton is used to calculate the pair production rate even in an inhomogeneous electric field. Furthermore, the instanton interpretation confirms the fact that bosons and fermions can not be produced by a static magnetic field only. 
  We report on a detailed calculation of the anomaly coefficients for the odd and even parts of the $Z_2$-graded representation $\theta$ of the Lie algebra Lie$ G$ on the exterior algebra of dimension $2^n$ assuming that $G\subset U(n)$. The coefficients vanish provided $G\subset SU(n)$ and $n\ne3$. The singular role of the gauge group SU(3) is emphasized.   The Standard Model is covered by this result. 
  In this paper, we study open-closed string field theory in the background B-field in the so-called alpha=p^{+} formulation. The string field theory in the infrared gives noncommutative gauge theory in the open string sector. Since this theory includes closed string fields as dynamical variables, we can obtain another string field theory in the same background through the condensation of closed string fields, whose low-energy effective action does not show the noncommutativity of spacetime. Although we have two string field theories in the same background, we show that these theories are equivalent. In fact, we give the unitary transformation from string fields in one of them to string fields in the other. 
  We consider the D=4, N=8 supergravity on AdS2 x S2 space. We obtain the truncated Lagrangian for the bosonic chiral primary fields, and compute the tree level three-point correlation functions. 
  One of the main difficulties in studying Quantum Field Theory, in the perturbative regime, is the calculation of D-dimensional Feynman integrals. In general, one introduces the so-called Feynman parameters and associated with them the cumbersome parametric integrals. Solving these integrals beyond the one-loop level can be a difficult task. Negative dimensional integration method (\ndim{}) is a technique whereby such problem is dramatically reduced. In this work we present the calculation of two-loop integrals in three diferent cases: scalar ones with three diferent masses, massless with arbitrary tensor rank, with N-insertions of a 2-loop diagram. 
  We study a class of Wilsonian formulations of non-Abelian gauge theories in algebraic non-covariant gauges where the Wilsonian infrared cutoff $\Lambda$ is inserted as a mass term for the propagating fields. In this way the Ward-Takahashi identities are preserved to all scales. Nevertheless BRST-invariance in broken and the theory is gauge-dependent and unphysical at $\Lambda\neq0$. Then we discuss the infrared limit $\Lambda\to0$. We show that the singularities of the axial gauge choice are avoided in planar gauge and light-cone gauge. In addition the issue of infrared divergences is addressed in some explicit example. Finally the rectangular Wilson loop of size $2L\times 2T$ is evaluated at lowest order in perturbation theory and a non commutativity between the limits $\Lambda\to0$ and $T\to\infty$ is pointed out. 
  We investigate the distribution of instanton sizes in the framework of a simplified model for ensembles of instantons. This model takes into account the non-diluteness of instantons. The infrared problem for the integration over instanton sizes is dealt with in a self-consistent manner by approximating instanton interactions by a repulsive hard core potential. This leads to a dynamical suppression of large instantons. The characteristic features of the instanton size distribution are studied by means of analytic and Monte Carlo methods. In one dimension exact results can be derived. In any dimension we find a power law behaviour for small sizes, consistent with the semi-classical results. At large instanton sizes the distribution decays exponentially. The results are compared with those from lattice simulations. 
  The naive low energy effective action of the tachyon and the U(1) gauge field obtained from string field theory does not correspond to the world volume action of unstable branes in bosonic string theory. We show that there exists a field redefinition which relates the gauge field and the tachyon of the string field theory action to the fields in the world volume action of unstable branes. We identify a string gauge symmetry which corresponds to the U(1) gauge transformation. This is done to the first non-linear order in the fields. We examine the vector fluctuations at the tachyon condensate till level (4,8). 
  The double tensor multiplet of D=4, N=2 supersymmetry, relevant to type IIB superstring vacua, is derived and its gauge invariant and N=2 supersymmetric interactions are analysed, both self-interactions and interactions with vector multiplets and hypermultiplets. Using deformation theory, it is shown that the lowest dimensional nontrivial interaction vertices of this type have dimension 5 and all dimension 5 vertices are determined. They give rise to new N=2 supersymmetric gauge theories of the `exotic' type which are local but nonpolynomial in some of the fields and coupling constants. Explicit examples of such models are constructed. 
  We study marginal and relevant supersymmetric deformations of the N=4 super-Yang-Mills theory in four dimensions. Our primary innovation is the interpretation of the moduli spaces of vacua of these theories as non-commutative spaces. The construction of these spaces relies on the representation theory of the related quantum algebras, which are obtained from F-term constraints. These field theories are dual to superstring theories propagating on deformations of the AdS_5xS^5 geometry. We study D-branes propagating in these vacua and introduce the appropriate notion of algebraic geometry for non-commutative spaces. The resulting moduli spaces of D-branes have several novel features. In particular, they may be interpreted as symmetric products of non-commutative spaces. We show how mirror symmetry between these deformed geometries and orbifold theories follows from T-duality. Many features of the dual closed string theory may be identified within the non-commutative algebra. In particular, we make progress towards understanding the K-theory necessary for backgrounds where the Neveu-Schwarz antisymmetric tensor of the string is turned on, and we shed light on some aspects of discrete anomalies based on the non-commutative geometry. 
  The confinement scenario in N=2 supersymmetric gauge theory at the monopole point is reviewed. Basic features of this U(1) confinement are contrasted with those we expect in QCD. In particular, extra states in the hadron spectrum and non-linear Regge trajectories are discussed. Then another confinement scenario arising on Higgs branches of the theory with fundamental matter is also reviewed. Peculiar properties of the Abrikosov-Nielsen-Olesen string on the Higgs branch lead to a new confining regime with the logarithmic suppression of the linear rising potential. Motivations for a search for tensionless strings are proposed. 
  We extend the previous construction of loop transfer matrix to the case of nonzero self-intersection coupling constant $\kappa$. The loop generalization of Fourier transformation allows to diagonalize transfer matrices depending on symmetric difference of loops and express all eigenvalues of $3d$ loop transfer matrix through the correlation functions of the corresponding 2d statistical system. The loop Fourier transformation allows to carry out analogy with quantum mechanics of point particles, to introduce conjugate loop momentum P and to define loop quantum mechanics. We also consider transfer matrix on $4d$ lattice which describes propagation of memebranes. This transfer matrix can also be diagonalized by using generalized Fourier transformation, and all its eigenvalues are equal to the correlation functions of the corresponding $3d$ statistical system. 
  The partition function of Ramond-Ramond p-form fields in Type IIA supergravity on a ten-manifold X contains subtle phase factors that are associated with T-duality, self-duality, and the relation of the RR fields to K-theory. The analogous partition function of M-theory on X x S1 contains subtle phases that are similarly associated with E8 gauge theory. We analyze the detailed phase factors on the two sides and show that they agree, thereby testing M-theory/Type IIA duality as well as the K-theory formalism in an interesting way. We also show that certain D-brane states wrapped on nontrivial homology cycles are actually unstable, that (-1)^{F_L} symmetry in Type IIA superstring theory depends in general on a cancellation between a fermion anomaly and an anomaly of RR fields, and that Type IIA superstring theory with no wrapped branes is well-defined only on a spacetime with W_7=0. 
  We show how some aspects of the K-theory classification of RR fluxes follow from a careful analysis of the phase of the M-theory action. This is a shortened and simplified companion paper to ``E8 Gauge Theory, and a Derivation of K-Theory from M-Theory.'' 
  We consider the theory of closed $p$-branes propagating on $(p+1)$-dimensional space-time manifolds. This theory has no local degrees of freedom. Here we study its canonical and BRST structures of the theory. In the case of locally flat backgrounds one can show that the $p$-brane theory is related to another known topological field theory. In the general situation some equivalent actions can also be written for the topological $p$-brane theory. 
  We investigate properties of several string networks in $D < 10$ which carry electric currents as well as electrostatic charge densities. We show the electric-current conservations as well as the force-balance condition of the string tensions on 3-string junctions in these networks. We also show the consistency of the above string networks from their world-volume point of view by comparing the world-volume energy-density with the induced worldsheet energy density of the supergravity solution. Finally, we present new charged macroscopic string solutions in type II theories in D=8 and discuss certain issues related to their network construction. 
  We show that dilaton beta-function equation in the brane-like sigma-model (regarded as NSR analogue of string theory on $AdS_5\times{S^5}$) has the form of stochastic Langevin equation with non-Markovian noise. The worldsheet cutoff is identified with stochastic time and the $V_5$-operator plays the role of the noise. We derive the Fokker-Planck equation associated with this stochastic process and show that the Hamiltonian of the $AdS_5$ supergravity defines the distribution satisfying this Fokker-Planck equation. This means that the dynamical compactification of flat ten-dimensional space-time on $AdS_5\times{S^5}$ occurs as a result of the non-Markovian stochastic process, generated by the $V_5$-operator noise. This provides us with an insight into relation between holography principle and the concept of stochastic quantization from the point of view of critical string theory. 
  It is shown that the dual of the double compactified D=11 Supermembrane and a suitable compactified D=10 Super 4D-brane with nontrivial wrapping on the target space may be formulated as noncommutative gauge theories. The Poisson bracket over the world-volume is intrinsically defined in terms of the minima of the hamiltonian of the theory, which may be expressed in terms of a non degenerate 2-form. A deformation of the Poisson bracket in terms of the Moyal brackets is then performed. A noncommutative gauge theory in terms of the Moyal star bracket is obtained. It is shown that all these theories may be described in terms of symplectic connections on symplectic fibrations. The world volume being its base manifold and the (sub)group of volume preserving diffeomorphisms generate the symplectomorphisms which preserve the (infinite dimensional) Poisson bracket of the fibration. 
  Invariants for framed links in $S^3$ obtained from Chern-Simons gauge field theory based on an arbitrary gauge group (semi-simple) have been used to construct a three-manifold invariant. This is a generalization of a similar construction developed earlier for SU(2) Chern-Simons theory. The procedure exploits a theorem of Lickorish and Wallace and also those of Kirby, Fenn and Rourke which relate three-manifolds to surgeries on framed unoriented links. The invariant is an appropriate linear combination of framed link invariants which does not change under Kirby calculus. This combination does not see the relative orientation of the component knots. The invariant is related to the partition function of Chern-Simons theory. This thus provides an efficient method of evaluating the partition function for these field theories. As some examples, explicit computations of these manifold invariants for a few three-manifolds have been done. 
  It is shown that all bosonic and fermionic massive string models admit consistent light-cone formulations. This result is used to derive the spin generating functions of these models in four dimensions. 
  We construct two families of globally supersymmetric counterparts of standard Poincar\'e supersymmetric SYM theories on curved space-times admitting Killing spinors, in all dimensions less than six and eight respectively. The former differs from the standard theory only by mass terms for the fermions and scalars and modified supersymmetry transformation rules, the latter in addition has cubic Chern-Simons like couplings for the scalar fields. We partially calculate the supersymmetry algebra of these models, finding R-symmetry extensions proportional to the curvature. We also show that generically these theories have no continuous Coulomb branch of maximally supersymmetric vacua, but that there exists a half-BPS Coulomb branch approaching the standard Coulomb branch in the Ricciflat limit. 
  Recently it was shown that the Born-Infeld-type modification of the quadratic Yang-Mills action gives rise to classical particle-like solutions in the flat space which have a striking similarity with the Bartnik-McKinnon solutions known in the gravity coupled Yang-Mills theory. We show that both families are continuously related within the framework of the Einstein-Born-Infeld theory through interpolating sequences of parameters. We also investigate an internal structure of the associated black holes. It is found that the Born-Infeld non-linearity leads to a drastic modification of the black hole interior typical for the usual Yang-Mills theory. In the latter case a generic solution exhibits violent metric oscillations near the singularity. In the Born-Infeld case a generic interior solution is smooth, the metric has the standard Schwarzschild type singularity, and we did not observe internal horizons. Such smoothing of the 'violent' EYM singularity may be interpreted as a result of quantum effects. 
  On the basis of the result obtained by applying Baxter's exact perturbative approach to the dilute A_3 model to give the E_8 mass spectrum, the dilute A_L inversion relation was used to predict the eigenspectra in the L=4 and L=6 cases (corresponding to E_7 and E_6 respectively). In calculating the next-to-leading term in the correlation lengths, or equivalently masses, the inversion relation condition gives a surprisingly simple result in all three cases, and for all masses. 
  We study the noncommutative U(2) monopole solution at the second order in the noncommutativity parameter \theta^{ij}. We solve the BPS equation in noncommutative super Yang-Mills theory to O(\theta^2), transform the solution to the commutative description by the Seiberg-Witten (SW) map, and evaluate the eigenvalues of the scalar field. We find that, by tuning the free parameters in the SW map, we can make the scalar eigenvalues precisely reproduce the configuration of a tilted D-string suspended between two parallel D3-branes. This gives an example of how the ambiguities inevitable in the higher order SW map are fixed by physical requirements. 
  Using Kaluza-Klein technique we show that the singularity of Hawking-Turok type has a fixed point (bolt) contribution to the action in addition to the usual boundary contribution. Interestingly by adding this contribution we can obtain a simple expression for the total action which is feasible for both regular and singular instantons. Our result casts doubt on the constraint proposed by Turok in the recent calculation in which Vilenkin's instantons are regarded as a limit of certain constrained instantons. 
  The Chern isomorphism determines the free part of the K-groups from ordinary cohomology. Thus to really understand the implications of K-theory for physics one must look at manifolds with K-torsion. Unfortunately there are not many explicit examples, and usually for very symmetric spaces. Cartesian products of RP^n are examples where the order of the torsion part differs between K-theory and ordinary cohomology. The dimension of corresponding branes is also discussed. An example for a Calabi-Yau manifold with K-torsion is given. 
  Probabilities of crossing on same-spin clusters, seen as order parameters, have been introduced recently for the critical 2d Ising model by Langlands, Lewis and Saint-Aubin. We extend Cardy's ideas, introduced for percolation, to obtain an ordinary differential equation of order 6 for the horizontal crossing probability pih. Due to the identity pih(r)+pih(1/r)=1, the function pih must lie in a 3-dimensional subspace. New measurements of pih are made for 40 values of the aspect ratio r (r in [0.1443,6.928]). These data are more precise than those obtained by Langlands et al as the 95%-confidence interval is brought to 4x10^{-4}. A 3-parameter fit using these new data determines the solution of the differential equation. The largest gap between this solution and the 40 data is smaller than 4x10^{-4}. The probability pihv of simultaneous horizontal and vertical crossings is also treated. 
  In this note $D9$- and anti-$D9$-brane annihilation in type I string theory is probed by a $D1$-brane. We consider the covariant Green-Schwarz or twistor formulation of the probe theory. We expect the theory to be $\kappa$-invariant after the annihilation is completed. Conditions of the $\kappa$-invariance of the theory impose constraints on the background tachyon field. Solutions to the constraints define tachyon values which correspond to type I $D5$-branes as remnants of the annihilation. As a byproduct we get a theory which lies in the same universality class as the non-linear $\sigma$-model for the Atiyah-Drinfeld-Hitchin-Manin construction. 
  We study holography for asymptotically AdS spaces with an arbitrary genus compact Riemann surface as the conformal boundary. Such spaces can be constructed from the Euclidean AdS_3 by discrete identifications; the discrete groups one uses are the so-called classical Schottky groups. As we show, the spaces so constructed have an appealing interpretation of ``analytic continuations'' of the known Lorentzian signature black hole solutions; it is one of the motivations for our generalization of the holography to this case. We use the semi-classical approximation to the gravity path integral, and calculate the gravitational action for each space, which is given by the (appropriately regularized) volume of the space. As we show, the regularized volume reproduces exactly the action of Liouville theory, as defined on arbitrary Riemann surfaces by Takhtajan and Zograf. Using the results as to the properties of this action, we discuss thermodynamics of the spaces and analyze the boundary CFT partition function. Some aspects of our construction, such as the thermodynamical interpretation of the Teichmuller (Schottky) spaces, may be of interest for mathematicians working on Teichmuller theory. 
  There are two general irreversibility theorems for the renormalization group in more than two dimensions: the first one is of entropic nature, while the second one, by Forte and Latorre, relies on the properties of the stress-tensor trace, and has been recently questioned by Osborn and Shore. We start by establishing under what assumptions this second theorem can still be valid. Then it is compared with the entropic theorem and shown to be essentially equivalent. However, since the irreversible function of the (corrected) Forte-Latorre theorem is non universal (whereas the relative entropy of the other theorem is universal), it needs the additional step of renormalization. On the other hand, the irreversibility theorem is only guaranteed to be unambiguous if the integral of the stress-tensor trace correlator is finite, which happens for free theories only in dimension smaller than four. 
  The statistical entropy of a Schwarzschild black string in five dimensions is obtained by counting the black string states which form a representation of the near-horizon conformal symmetry with a central charge. The statistical entropy of the string agrees with its Bekenstein-Hawking entropy as well as that of the Schwarzschild black hole in four dimensions. The choice of the string length which gives the Virasoro algebra also reproduces the precise value of the Bekenstein-Hawking entropy and lies inside the stability bound of the string. 
  Following a brief review of known vortex solutions in SU(N) gauge-adjoint Higgs theories we show the existence of a new ``minimal'' vortex solution in SU(3) gauge theory with two adjoint Higgs bosons. At a critical coupling the vortex decouples into two abelian vortices, satisfying Bogomol'nyi type, first order, field equations. The exact value of the vortex energy (per unit length) is found in terms of the topological charge that equals to the N=2 supersymmetric charge, at the critical coupling. The critical coupling signals the increase of the underlying supersymmetry. 
  I show that a particle structure in conformal field theory is incompatible with interactions. As a substitute one has particle-like exitations whose interpolating fields have in addition to their canonical dimension an anomalous contribution. The spectra of anomalous dimension is given in terms of the Lorentz invariant quadratic invariant (compact mass operator) of a conformal generator $R_{\mu}$ with pure discrete spectrum. The perturbative reading of $R_{0\text{}}$as a Hamiltonian in its own right i.e. associated with an action in a functional integral setting naturally leads to the AdS formulation. The formal service role of AdS in order to access CQFT by a standard perturbative formalism (without being forced to understand first massive theories and then taking their scale-invariant limit) vastly increases the realm of conventionally accessible 4-dim. CQFT beyond those for which one had to use Lagrangians with supersymmetry in order to have a vanishing Beta-function. 
  We develop some basic properties of the open string on the symmetric product which is supposed to describe the open string field theory in discrete lightcone quantization (DLCQ). After preparing the consistency conditions of the twisted boundary conditions for Annulus/M\"obius/Klein Bottle amplitudes in generic non-abelian orbifold, we classify the most general solutions of the constraints when the discrete group is $S_N$. We calculate the corresponding orbifold amplitudes from two viewpoints -- from the boundary state formalism and from the trace over the open string Hilbert space. It is shown that the topology of the world sheet for the short string and that of the long string in general do not coincide. For example the annulus sector for the short string contains all the sectors (torus, annulus, Klein bottle, M\"obius strip) of the long strings. The boundary/cross-cap states of the short strings are classified into three categories in terms of the long string, the ordinary boundary and the cross-cap states, and the ``joint'' state which describes the connection of two short strings. We show that the sum of the all possible boundary conditions is equal to the exponential of the sum of the irreducible amplitude -- one body amplitude of long open (closed) strings. This is typical structure of DLCQ partition function. We examined that the tadpole cancellation condition in our language and derived the well-known gauge group $SO(2^{13})$. 
  We consider a nonstandard $D=2+1$ gravity described by a translational Chern--Simons action, and couple it to the nonrelativistic point particles. We fix the asymptotic coordinate transformations in such a way that the space part of the metric becomes asymptotically Euclidean. The residual symmetries are (local in time) translations and rigid rotations. The phase space Hamiltonian $H$ describing two-body interactions satisfies a nonlinear equation $H={\cal H}(\vec{x},\vec{p};H)$ what implies, after quantization, a nonstandard form of the Schr\"{o}dinger equation with energy-dependent fractional angular momentum eigenvalues. Quantum solutions of the two-body problem are discussed. The bound states with discrete energy levels correspond to a confined classical motion (for the planar distance between two particles $r\leq r_0$) and the scattering states with continuous energy correspond to classical motion for $r>r_0$. 
  We consider open strings ending on D-branes in the presence of constant metric, G, antisymmetric tensor, B and gauge field, A. The Hamiltonian is manifestly invariant under a global noncompact group; strikingly similar to toroidally compactified closed string Hamiltonian. The evolution equations for the string coordinates, $\{X^i \}$ and their dual partners, $\{Y_i \}$, are combined to obtain equations of motion invariant under the noncompact symmetry transformations. We show that one can start from a noncommutative theory, with nonvanishing G and B and mixed boundary conditions and then go over to a dual theory whose coordinates obey Dirichlet boundary conditions. It is possible to generate B-field by implementing the noncompact symmetry transformation. The connection between this duality transformation and Seiberg-Witten map is discussed. 
  We construct the explicit boundary state description of the vortex-type (codimension two) tachyon condensation in brane-antibrane systems generalizing the known result of the kink-type (Frau et al. hep-th/9903123). In this description we show how the RR-charge of the lower dimensional D-branes emerges. We also investigate the tachyon condensation in T^4/Z_2 orbifold and find that the twisted sector can be treated almost in the same way as the untwisted sector from the viewpoint of the boundary state. Further we discuss the higher codimension cases. 
  Recent conjectures of the c-theorem in four and higher dimensions have suggested that the coefficient of the Euler characteristic in the trace anomaly could measure the degrees of freedom in field theory and decrease along the renormalization-group flow. We compute this quantity for free massless scalar, fermion and antisymmetric tensor fields in any dimension, and analyse its dependence on spin and space-time dimension. In the limit of large number of dimensions, where the theories become semiclassical, we find that this quantity does not approach the classical number of field components, but is enhanced for spinful particles. This seemingly strange behaviour is found to be consistent with known renormalization-group patterns and a specific c-theorem conjecture. 
  We construct a model for dilatonic brane worlds with constant curvature on the brane, i.e. a non-zero four-dimensional cosmological constant, given in function of the dilaton coupling and the cosmological constant of the bulk. We compare this family of solutions to other known dilatonic domain wall solutions and apply a self-tunning mechanism to check the stability of our solutions under quantum fluctuations living on the brane. 
  It is shown that matroid theory may provide a natural mathematical framework for a duality symmetries not only for quantum Yang-Mills physics, but also for M-theory. Our discussion is focused in an action consisting purely of the Chern-Simons term, but in principle the main ideas can be applied beyond such an action. In our treatment the theorem due to Thistlethwaite, which gives a relationship between the Tutte polynomial for graphs and Jones polynomial for alternating knots and links, plays a central role. Before addressing this question we briefly mention some important aspects of matroid theory and we point out a connection between the Fano matroid and D=11 supergravity. Our approach also seems to be related to loop solutions of quantum gravity based in Ashtekar formalism. 
  First we argue in an informal, qualitative way that it is natural to enlarge space-time to five dimensions to be able to solve the problem of elementary particle masses. Several criteria are developed for the success of this program. Extending the Poincare group to the group C of all angle-preserving transformations of space-time is one such scheme which satisfies these criteria. Then we show that the field equation for spin 1/2 fermions coupled to a self-force gauge field predicts mass spectra of the desired type: for a certain range of a key parameter (Casimir invariant) a three-point mass spectrum which fits the ``down'' quarks d,s, and b to within their experimental bounds is obtained. Reasonable values of the coupling constant (of QCD magnitude) and the range of the spatial wave function (a few fermis) also result. Compatibility with the electroweak theory is also discussed. 
  We study the Abrikosov-Nielsen-Olesen string in N=2 supersymmetric QED with N=2-preserving superpotential, in which case the Abrikosov string is found to be 1/2-BPS saturated. Adding a quadratic small perturbation in the superpotential breaks N=2 supersymmetry to N=1 supersymmetry. Then the Abrikosov string is no longer BPS saturated. The difference between the string tensions for the non-BPS and BPS saturated situation is found to be negative to the first order of the perturbation parameter. 
  We consider two-dimensional supergravity theories with four supercharges constructed from compactification of Type II string theory on a generic Calabi-Yau four-fold. In Type IIA and Type IIB cases, respectively, new superspace formulations of N=(2,2) and N=(0,4) dilaton supergravities are found and their coupling to matter multiplets is discussed. 
  We present explicit methods for computing the discriminant curves and the associated Kodaira type fiber degeneracies of elliptically fibered Calabi-Yau threefolds. These methods are applied to a specific three-family, SU(5) grand unified theory of particle physics within the context of Heterotic M-Theory. It is demonstrated that there is always a region of moduli space where a bulk space five-brane is wrapped on a pure fiber in the Calabi-Yau threefold. Restricting the discussion to the smooth parts of the discriminant curve, we explore the properties of the N=2 BPS supermultiplets that arise on the worldvolume of this five-brane due to the degeneration of the elliptic fiber. The associated degenerating M membranes are shown to project to string junctions in the base space. We use string junction techniques to explicitly compute the light BPS hyper- and vector multiplet spectrum for each Kodaira type fiber near the smooth part of the discriminant curve in the SU(5) GUT theory. 
  It is shown that the generator of the nonstandard fermion-monopole supersymmetry uncovered by De Jonghe, Macfarlane, Peeters and van Holten, and the generator of its standard N=1/2 supersymmetry have to be supplemented by their product operator to be treated as independent supercharge. As a result, the fermion-monopole system possesses the nonlinear N=3/2 supersymmetry having the nature of the 3D spin-1/2 free particle's supersymmetry generated by the supercharges represented in a scalar form. Analyzing the supercharges' structure, we trace how under reduction of the fermion-monopole system to the spherical geometry the nonlinear N=3/2 superalgebra comprising the Hamiltonian and the total angular momentum as even generators is transformed into the standard linear N=1 superalgebra with the Hamiltonian to be the unique even generator. 
  We investigate the dynamics of open membrane boundaries in a constant C-field background. We follow the analysis for open strings in a B-field background, and take some approximations. We find that open membrane boundaries do show noncommutativity in this case by explicit calculations. Membrane boundaries are one dimensional strings, so we face a new type of noncommutativity, that is, noncommutative strings. 
  We study the bulk effective theory of a class of orbifolds of the type IIB string with D5-branes, compactified to four dimensions. These constructions are connected, in a region of their moduli space, to some orbifolds of the type I and heterotic string. We compare the effective actions through the coupling of the R^2 term, and we argue that these orbifolds provide non-perturbative deformations of the latter, in which the gauge group is entirely non-perturbative. 
  We show that an effective Abelian gauge theory can be obtained as a renormalizable theory from QCD in the maximal Abelian gauge. The derivation improves in a systematic manner the previous version that was obtained by one of the authors and was referred to as the Abelian-projected effective gauge theory. This result supports the view that we can construct an effective Abelian gauge theory from QCD without losing characteristic features of the original non-Abelian gauge theory. In fact, it is shown that the effective coupling constant in the resulting renormalizable theory has a renormalization-scale dependence governed by the $\beta$-function that is exactly the same as that of the original Yang-Mills theory, irrespective of the choice of gauge fixing parameters of the maximal Abelian gauge and the parameters used for identifying the dual variables. Moreover, we evaluate the anomalous dimensions of the fields and parameters in the resultant theory. By choosing the renormalized parameters appropriately, we can switch the theory into an electric or a magnetic theory. 
  I discuss the symmetry structure of the N=2 supersymmetric extension of the Born-Infeld action in four dimensions, and confirm its interpretation as the Goldstone-Maxwell action associated with partial breaking of N=4 extended supersymmetry down to N=2, by revealing a hidden invariance of the action with respect to two non-linearly realized supersymmetries and two spontaneously broken translations. I also argue about the uniqueness of supersymmetric extension of the Born-Infeld action, and its possible relation to noncommutative geometry. 
  5d dilatonic gravity (bosonic sector of gauged supergravity) with non-trivial bulk potential and with surface terms (boundary cosmological constant and trace anomaly induced effective action for brane quantum matter) is considered. For constant bulk potential and maximally SUSY Yang-Mills theory (CFT living on the brane) the inflationary brane-world is constructed. The bulk is singular asymptotically AdS space with non-constant dilaton and dilatonic de Sitter or hyperbolic brane is induced by quantum matter effects. At the same time, dilaton on the brane is determined dynamically. This all is natural realization of warped compactification in AdS/CFT correspondence. For fine-tuned toy example of non-constant bulk potential we found the non-singular dilatonic brane-world where bulk again represents asymptotically AdS space and de Sitter brane (inflationary phase of observable Universe) is induced exclusively by quantum effects. The radius of the brane and dilaton are determined dynamically. The analytically solvable example of exponential bulk potential leading to singular asymptotically AdS dilatonic bulk space with de Sitter (or hyperbolic) brane is also presented.In all cases under discussion the gravity on the brane is trapped via Randall-Sundrum scenario. It is shown that qualitatively the same types of brane-worlds occur when quantum brane matter is described by $N$ dilaton coupled spinors. 
  We study the statistical mechanics of random surfaces generated by NxN one-matrix integrals over anti-commuting variables. These Grassmann-valued matrix models are shown to be equivalent to NxN unitary versions of generalized Penner matrix models. We explicitly solve for the combinatorics of 't Hooft diagrams of the matrix integral and develop an orthogonal polynomial formulation of the statistical theory. An examination of the large N and double scaling limits of the theory shows that the genus expansion is a Borel summable alternating series which otherwise coincides with two-dimensional quantum gravity in the continuum limit. We demonstrate that the partition functions of these matrix models belong to the relativistic Toda chain integrable hierarchy. The corresponding string equations and Virasoro constraints are derived and used to analyse the generalized KdV flow structure of the continuum limit. 
  We study the perturbative unitarity of noncommutative scalar field theories. Field theories with space-time noncommutativity do not have a unitary S-matrix. Field theories with only space noncommutativity are perturbatively unitary. This can be understood from string theory, since space noncommutative field theories describe a low energy limit of string theory in a background magnetic field. On the other hand, there is no regime in which space-time noncommutative field theory is an appropriate description of string theory. Whenever space-time noncommutative field theory becomes relevant massive open string states cannot be neglected. 
  The conformal affine Toda model coupled to the matter field (CATM) is obtained through a classical reduction of the $sl(2)^{(1)}$ affine two-loop WZNW model. After spontaneously broken the conformal symmetry by means of BRST analysis, we end up with an effective theory, the so called affine Toda model coupled to the matter (ATM). Further, using a bosonization technique we recover from this theory the sine-Gordon model plus a free massless scalar field. The ATM model is considered as a QCD-motivated integrable field theory, since it describes various features in the baryonic sector of the low-energy effective Lagrangian of QCD in two dimensions with one flavor and two colors. Imposing the equivalence of the Noether and topological currrents as a constraint, it is shown that the intercharge ``quark''- ``anti-quark'' static potential reveals a linear confinement behavior for large intercharge separation. 
  A simple method of obtaining path-integral measures in higher-derivative gravities is presented. The measures are nothing but the generalized Lee-Yang terms. 
  We consider an action for a closed, bosonic, p-brane, where the brane tension is not an assigned parameter but rather it is induced by a maximal rank gauge p-form. This model is classically equivalent to the Nambu--Goto/Howe-Tucker model. We investigate how this classical equivalence can be implemented in the path integral framework. For this purpose we adopt a ``first order'' integration procedure over gauge p-forms and a ``shortened'' Fadeev-Popov procedure. 
  Spectral functions relevant in the context of quantum field theory under the influence of spherically symmetric external conditions are analysed. Examples comprise heat-kernels, determinants and spectral sums needed for the analysis of Casimir energies. First, we summarize that a convenient way of handling them is to use the associated zeta function. A way to determine all its needed properties is derived. Using the connection with the mentioned spectral functions, we provide: i.) a method for the calculation of heat-kernel coefficients of Laplace-like operators on Riemannian manifolds with smooth boundaries and ii.) an analysis of vacuum energies in the presence of spherically symmetric boundaries and external background potentials. 
  Using the previously gained insight about the particle/field relation in conformal quantum field theories which required interactions to be related to the existence of particle-like states associated with fields of anomalous scaling dimensions, we set out to construct a classification theory for the spectra of anomalous dimensions. Starting from the old observations on conformal superselection sectors related to the anomalous dimensions via the phases which appear in the spectral decomposition of the center of the conformal covering group $Z(\widetilde{SO(d,2)}),$ we explore the possibility of a timelike braiding structure consistent with the timelike ordering which refines and explains the central decomposition. We regard this as a preparatory step in a new construction attempt of interacting conformal quantum field theories in D=4 spacetime dimensions. Other ideas of constructions based on the $AdS_{5}$-$CQFT_{4}$ or the perturbative SYM approach in their relation to the present idea are briefly mentioned. 
  Representations of four-dimensional superconformal groups on harmonic superfields are discussed. It is shown how various short representations can be obtained by parabolic induction. It is also shown that such short multiplets may admit several descriptions as superfields on different superspaces. In particular, this is the case for on-shell massless superfields. This allows a description of short representations as explicit products of fundamental fields. Superconformal transformations of analytic fields in real harmonic superspaces are given explicitly. 
  Massless fields of generic Young symmetry type in $AdS_d$ space are analyzed. It is demonstrated that in contrast to massless fields in Minkowski space whose physical degrees of freedom transform in irreps of $o(d-2)$ algebra, $AdS$ massless mixed symmetry fields reduce to a number of irreps of $o(d-2)$ algebra. From the field theory perspective this means that not every massless field in flat space admits a deformation to $AdS_d$ with the same number of degrees of freedom, because it is impossible to keep all of the flat space gauge symmetries unbroken in the AdS space. An equivalent statement is that, generic irreducible AdS massless fields reduce to certain reducible sets of massless fields in the flat limit. A conjecture on the general pattern of the flat space limit of a general $AdS_d$ massless field is made. The example of the three-cell ``hook'' Young diagram is discussed in detail. In particular, it is shown that only a combination of the three-cell flat-space field with a graviton-like field admits a smooth deformation to $AdS_d$. 
  We construct a consistent reduction of type IIA supergravity on S^3, leading to a maximal gauged supergravity in seven dimensions with the full set of massless SO(4) Yang-Mills fields. We do this by starting with the known S^4 reduction of eleven-dimensional supergravity, and showing that it is possible to take a singular limit of the resulting standard SO(5)-gauged maximal supergravity in seven dimensions, whose eleven-dimensional interpretation involves taking a limit where the internal 4-sphere degenerates to RxS^3. This allows us to reinterpret the limiting SO(4)-gauged theory in seven dimensions as the S^3 reduction of type IIA supergravity. We also obtain the consistent S^4 reduction of type IIA supergravity, which gives an SO(5)-gauged maximal supergravity in D=6. 
  It is known that noncommutative Yang-Mills theory with periodical boundary conditions on torus at the rational value of the noncommutativity parameter is Morita equivalent to the ordinary Yang-Mills theory with twisted boundary conditions on dual torus. We present simple derivation of this fact. We describe one-to-one correspondence between and gauge invariant observables in these two theories. In particular, we show that under Morita map Polyakov loops in the ordinary YM theory go to the open noncommutative Wilson loops discovered by Ishibashi, Iso, Kawai and Kutazawa. 
  We provide the exact time-dependent cosmological solutions in the Randall-Sundrum (RS) setup with bulk matter, which may be smoothly connected to the static RS metric. In the static limit of the extra dimension, the solutions are reduced to the standard Friedmann equations. In view of our solutions, we also propose an explanation for how the extra dimension is stabilized in spite of a flat modulus potential at the classical level. 
  The time-reparametrization-invariant dynamics of a relativistic string is studied in the Dirac generalized Hamiltonian theory by resolving the first class constraints. The reparametrization-invariant evolution parameter is identified with the time-like coordinate of the "center of mass" of a string which is separated from local degrees of freedom by transformations conserving the group of diffeomorphisms of the generalized Hamiltonian formulation and the Poincare covariance of local constraints. To identify the "center of mass" time-like coordinate with the invariant proper time (measured by an observer in the comoving frame of reference), we apply the Levi-Civita - Shanmugadhasan canonical transformations which convert the global (mass-shell) constraint into a new momentum, so that the corresponding gauge is not needed for the Hamiltonian reduction. The resolving of local constraints leads to an "equivalent unconstrained system" in the reduced phase space with the Roehrlich-type Hamiltonian of evolution with respect to the proper time. 
  A general method based on the polynomial deformations of the Lie algebra sl(2,R) is proposed in order to exhibit the quasi-exactly solvability of specific Hamiltonians implied by quantum physical models. This method using the finite-dimensional representations and differential realizations of such deformations is illustrated on the sextic oscillator as well as on the second harmonic generation. 
  Lately, to provide a solid ground for quantization of the open string theory with a constant B-field, it has been proposed to treat the boundary conditions as hamiltonian constraints.   It seems that this proposal is quite general and should be applicable to a wide range of models defined on manifolds with boundaries. The goal of the present paper is to show how the boundary conditions can arise as constraints in a purely algebraic fashion within the Hamiltonian approach without any reference to the Lagrangian formulation of the theory. The construction of the boundary Dirac brackets is also given and some subtleties are pointed out.   We consider four examples of field theories with boundaries: the topological sigma model, the open string theory with and without a constant $B$-field and electrodynamics with topological term.   A curious result for electrodynamics on a manifold with boundaries is presented. 
  We investigate the evolution of a scalar field propagating in Reissner--Nordstr\"{o}m Anti--de Sitter spacetime. Due to the characteristic of spacetime geometry, the radiative tails associated with a massless scalar field propagation have an oscillatory exponential decay. The object--picture of the quasinormal ringing has also been obtained. For small charges, the approach to thermal equilibrium is faster for larger charges. However, after the black hole charge reaches a critical value, we get the opposite behavior for the imaginary frequencies of the quasinormal modes. Some possible explanations concerning the wiggle of the imaginary frequencies have been given. The picture of the quasinormal modes depending on the multipole index has also been illustrated. 
  Working in the axiomatic framework recently proposed by Gaberdiel and Goddard, we prove a generalized version of Zhu's Theorem; for any chiral bosonic conformal field theory on the sphere, our result characterizes the chiral blocks in terms of a certain quotient of the Fock space. We also establish, under a finiteness hypothesis closely related to rationality of the theory, that the relevant Knizhnik-Zamolodchikov-type equation admits solutions. 
  We investigate further the recent proposal for the form of the Matrix theory action in weak background fields. We perform DVV reduction to the multiple D0-brane action in order to find the Matrix string theory action for multiple fundamental strings in curved but weak NS-NS and R-R backgrounds. This matrix sigma model gives a definite prescription on how to deal with R-R fields with an explicit spacetime dependence in Type II string theory. We do this both via the 9-11 flip and the chain of T and S dualities, and further check on their equivalence explicitly. In order to do so, we also discuss the implementation of S-duality in the operators of the 2-dimensional world-volume supersymmetric gauge theory describing the Type IIB D-string. We compare the result to the known Green-Schwarz sigma model action (for one string), and use this comparison in order to discuss about possible, non-linear background curvature corrections to the Matrix string action (involving many strings), and therefore to the Matrix theory action. We illustrate the nonabelian character of our action with an example involving multiple fundamental strings in a non-trivial R-R flux, where the strings are polarized into a noncommutative configuration. This corresponds to a dielectric type of effect on fundamental strings. 
  We calculate the effective action for small velocity scattering of localized 1-branes and 5-branes. Momentum is allowed to flow in the direction along the 1-branes so that the moduli space has only 1/8 of the full supersymmetry. Relative to the more familiar case with the 1-branes delocalized along the 5-branes, this introduces new moduli associated with the motion of the 1-branes along the 5-branes. We consider in detail the moduli space metric for the associated two body problem. Even for motion transverse to the 5-brane, our results differ substantially from the delocalized case. However, this difference only appears when both the 1-brane charge and the momentum charge are localized. Despite the fact that, in a certain sense, 1-branes spontaneously delocalize near a 5-brane horizon, the moduli space metric in this limit continues to differ from the delocalized result. This fact may be of use in developing a new description of the associated BPS bound states. The new terms depend on the torus size $L$ in such a way that they give a finite contribution in the $L \to \infty$ limit. 
  The low-energy effective theory of the IIB matrix model developed by H. Aoki et al. is written down explicitly in terms of bosonic variables only. The effective theory is then studied by Monte Carlo simulations in order to investigate the possibility of a spontaneous breakdown of Lorentz invariance. The imaginary part of the effective action, which causes the so-called sign problem in the simulation, is dropped by hand. The extent of the eigenvalue distribution of the bosonic matrices shows a power-law large N behavior, consistent with a simple branched-polymer prediction. We observe, however, that the eigenvalue distribution becomes more and more isotropic in the ten-dimensional space-time as we increase N. This suggests that if the spontaneous breakdown of Lorentz invariance really occurs in the IIB matrix model, a crucial role must be played by the imaginary part of the effective action. 
  We discuss possible D-brane configurations on SU(2) group manifolds in the sigma model approach. When we turn the boundary conditions of the spacetime fields into the boundary gluing conditions of chiral currents, we find that for all D-branes except the spherical D2-branes, the gluing matrices R^a_{b} depend on the fields, so the chiral Kac-Moody symmetry is broken, but conformal symmetry is maintained. Matching the spherical D2-branes derived from the sigma model with those from the boundary state approach we obtain a U(1) field strength that is consistent with flux quantization. 
  In the gauge theory context, a definition of branching ratios and partial widths of unstable particles is proposed that satisfies the basic principles of additivity and gauge independence. A simpler definition, similar to the conventional one, is examined in the Z^0-boson case. In order to establish contact with experiment, we show that it leads to a peak cross section that justifies the expression used by the LEP Electroweak Working Group through next-to-next-to-leading order, provided that the pole rather than the on-shell mass and width of the Z^0 boson are employed. 
  We derive an effective action for Dirac fermions coupled to O(3) non-linear sigma-model (NLSM) through the Yukawa-type interaction. The nonperturbative (global) quantum anomaly of this model results in a Hopf term for the effective NLSM. We obtain this term using the ``embedding'' of the CP$^1$ model into the CP$^M$ generalization of the model which makes the quantum anomaly perturbative. This perturbative anomaly is calculated by means of a gradient expansion of a fermionic determinant and is given by the Chern-Simons term for an auxiliary gauge field. 
  We carry out a general analysis of the representations of the superconformal algebras SU(2,2/N), OSp(8/4,R) and OSp(8^*/4) and give their realization in superspace. We present a construction of their UIR's by multiplication of the different types of massless superfields ("supersingletons"). Particular attention is paid to the so-called "short multiplets". Representations undergoing shortening have "protected dimension" and correspond to BPS states in the dual supergravity theory in anti-de Sitter space. These results are relevant for the classification of multitrace operators in boundary conformally invariant theories as well as for the classification of AdS black holes preserving different fractions of supersymmetry. 
  Using the path integral approach, we discuss the correlation functions of the $SL(2,\bfC)/SU(2)$ WZW model, which corresponds to the string theory on the Euclidean $AdS_3$. We obtain the two- and three-point functions for generic primary fields in closed forms. By an appropriate change of the normalization of the primary fields, our results coincide with those by Teschner, which were obtained by using the bootstrap approach. The supergravity results are also obtained in the semi-classical limit. 
  The D-brane spectrum of a $\Zop_2\times\Zop_2$ Calabi-Yau three-fold orbifold of toroidally compactified Type IIA and Type IIB string theory is analysed systematically. The corresponding K-theory groups are determined and complete agreement is found. New kinds of stable non-BPS D-branes are found, whose stability regions are far more complicated than the previously discussed non-BPS D-branes. The decay channels of non-BPS D-branes beyond their stability regions are identified. Finally the T-dual orbifold is analysed and a suitable K-theory is found. 
  We establish a finite field-dependent BRS transformation that connects the Yang-Mills path-integrals with Faddeev-Popov effective actions for an arbitrary pair of gauges F and F'. We establish a result that relates an arbitrary Green's function [either a primary one or one that of an operator] in an arbitrary gauge F' to those in gauge F that are compatible to the ones in gauge F by its construction [in that the construction preserves expectation values of gauge-invariant observables]. We establish parallel results also for the planar gauge-Lorentz gauge connection. 
  A proof is given for the observation that the equations of motion for the companion Lagrangian without a square root, subject to some constraints, just reduce to the equations of motion for the companion Lagrangian with a square root in one less dimension. The companion Lagrangian is just an extension of the Klein-Gordon Lagrangian to more fields in order to provide a field description for strings and branes. 
  We present some arguments showing spectrum doubling of matrix models in the limit $N\to\infty$ which is connected with fermionic determinant behaviour. The problems are similar to ones encountered in the lattice gauge theories with chiral fermions. One may discuss the ``physical meaning'' of the doubling states or ways to eliminate them. We briefly consider both situations. 
  The bosonic IIB matrix model is studied using a numerical method. This model contains the bosonic part of the IIB matrix model conjectured to be a non-perturbative definition of the type IIB superstring theory. The large N scaling behavior of the model is shown performing a Monte Carlo simulation. The expectation value of the Wilson loop operator is measured and the string tension is estimated. The numerical results show the prescription of the double scaling limit. 
  We present a construction of a Virasoro symmetry of the sine-Gordon (SG) theory. It is a dynamical one and has nothing to do with the space-time Virasoro symmetry of 2D CFT. Although it is clear how it can be realized dyrectly in the SG field theory, we are rather concerned here with the corresponding N-soliton solutions. We present explicit expressions for their infinithesimal transformations and show that they are local in this case. Some preliminary stages about the quantization of the classical results presented in this paper are also given. 
  We show that when the field strength H of the NS-NS B field does not vanish, the coordinates X and momenta P of an open string endpoints satisfy a set of mixed commutation relations among themselves. Identifying X and P with the coordinates and derivatives of the D-brane world volume, we find a new type of noncommutative spaces which is very different from those associated with a constant B field background. 
  The Atiyah-Hitchin manifold arises in many different contexts, ranging from its original occurrence as the moduli space of two SU(2) 't Hooft-Polyakov monopoles in 3+1 dimensions, to supersymmetric backgrounds of string theory. In all these settings, (super)symmetries require the metric to be hyperk\"ahler and have an SO(3) transitive isometry, which in the four-dimensional case essentially selects out the Atiyah-Hitchin manifold as the only such smooth manifold with the correct topology at infinity. In this paper, we analyze the exponentially small corrections to the asymptotic limit, and interpret them as infinite series of instanton corrections in these various settings. Unexpectedly, the relevant configurations turn out to be bound states of $n$ instantons and $\bar n$ anti-instantons, with $|n-\bar n|=0,1$ as required by charge conservation. We propose that the semi-classical configurations relevant for the higher monopole moduli space are Euclidean open branes stretched between the monopoles. 
  We propose exact vacuum expectation values of local fields for a quantum group restriction of the $C_2^{(1)}$ affine Toda theory which corresponds to two coupled minimal models. The central charge of the unperturbed models ranges from $c=1$ to $c=2$, where the perturbed models correspond to two magnetically coupled Ising models and Heisenberg spin ladders, respectively. As an application, in the massive phase we deduce the leading term of the asymptotics of the two-point correlation functions. 
  It is shown that non-supersymmetric spacetime varying string vacua can lead to an exponential hierarchy between the electroweak and the gravitational scales. The hierarchy is naturally generated by a string coupling of O(1). 
  We use dielectric branes to find non singular string theory duals of a perturbed 2+1 dimensional gauge theory living on D2 branes. By adding fermion masses we obtain theories with reduced supersymmetry. The Higgs vacua of the perturbed theory correspond to polarization of the D2 branes into D4 branes. The confining vacua correspond to polarization of the D2 branes into NS5 branes. We consider different mass perturbations. Adding three equal masses preserves N=2 supersymmetry. In this case there are no confining vacua. By adding a fourth fermion mass we break all the supersymmetry, and find confining vacua. We also obtain duals for domain walls, condensates, baryon vertices, glueballs and flux tubes. We comment on the Kahler potentials for the Higgs and confining phases. In the course of the calculations we also find a nontrivial consistency check of the NS5 brane action in a D2 brane background. 
  It has been recently shown that F-theory on K3 with background B fields (NSNS and RR 2-forms) is dual to the CHL string in 8 dimensions. In this paper, we reexamine this duality in terms of string junctions in type IIB string theory. It is in particular stressed that certain 7-brane configurations produce Sp gauge groups in a novel way. 
  Standard methods of nonlinear dynamics are used to investigate the stability of particles, branes and D-branes of abelian Born-Infeld theory. In particular the equation of small fluctuations about the D-brane is derived and converted into a modified Mathieu equation and - complementing earlier low-energy investigations in the case of the dilaton-axion system - studied in the high-energy domain. Explicit expressions are derived for the S-matrix and absorption and reflection amplitudes of the scalar fluctuation in the presence of the D-brane. The results confirm physical expectations and numerical studies of others. With the derivation and use of the (hitherto practically unknown) high energy expansion of the Floquet exponent our considerations also close a gap in earlier treatments of the Mathieu equation. 
  We discuss examples of D-branes probing toric singularities, and the computation of their world-volume gauge theories from the geometric data of the singularities. We consider several such examples of D-branes on partial resolutions of the orbifolds ${\bf C^3/Z_2\times Z_2}$,${\bf C^3/Z_2\times Z_3}$ and ${\bf C^4/Z_2\times Z_2 \times Z_2}$. 
  We argue that a certain distribution of matter in higher dimensions can provide the correct behaviour of gravity in four dimensions. Some explicit examples illustrating the idea are considered. 
  I review the relationship between AdS/CFT (anti-de Sitter / conformal field theory) dualities and the general theory of positive energy unitary representations of non-compact space-time groups and supergroups. I show, in particular, how one can go from the manifestly unitary compact basis of the lowest weight (positive energy) representations of the conformal group (Wigner picture) to the manifestly covariant coherent state basis (Dirac picture). The coherent states labelled by the space-time coordinates correspond to covariant fields with a definite conformal dimension. These results extend to higher dimensional Minkowskian spacetimes as well as generalized spacetimes defined by Jordan algebras and Jordan triple systems. The second part of my talk discusses the extension of the above results to conformal supergroups of Minkowskian superspaces as well as of generalized superspaces defined by Jordan superalgebras. The (super)-oscillator construction of generalized (super)-conformal groups can be given a dynamical realization in terms of generalized (super)-twistor fields. 
  We discuss the string theory on AdS_3. In the first half of this talk, we review the SL(2,R) and the SL(2,C)/SU(2) WZW models which describe the strings on the Lorentzian and Euclidean AdS_3 without RR backgrounds, respectively. An emphasis is put on the fundamental issues such as the unitarity, the modular invariance and the closure of the OPE. In the second half, we discuss some attempts at clarifying such problems. In particular, we discuss the modular invariance of the SL(2,R) WZW model and the calculation of the correlation functions of the SL(2,C)/SU(2) WZW model using the path-integral approach. 
  The boundary supersymmetric sinh-Gordon model is an integrable quantum field theory in 1+1 dimensions with bulk N=1 supersymmetry, whose bulk and boundary S matrices are not diagonal. We present an exact solution of this model. In particular, we derive an exact inversion identity and the corresponding thermodynamic Bethe Ansatz equations. We also compute the boundary entropy, and find a rich pattern of boundary roaming trajectories corresponding to c < 3/2 superconformal models. 
  In this work we compare the quantization of a massless scalar field in an inertial frame with the quantization in a rotating frame. We used the Trocheries-Takeno mapping to relate measurements in the inertial and the rotating frames. An exact solution of the Klein-Gordon equation in the rotating coordinate system is found and the Bogolubov transformation between the inertial and rotating modes is calculated, showing that the rotating observer defines a vacuum state different from the Minkowski one. We also obtain the response function of an Unruh-De Witt detector coupled with the scalar field travelling in a uniformly rotating world-line. The response function is obtained for two different situations: when the quantum field is prepared in the usual Minkowski vacuum state and when it is prepared in the Trocheries-Takeno vacuum state. We also consider the case of an inertial detector interacting with the field in the rotating vacuum. 
  We make a detailed investigation of all spaces Q_{n_1... n_N}^{q_1... q_N} of the form of U(1) bundles over arbitrary products \prod_i CP^{n_i} of complex projective spaces, with arbitrary winding numbers q_i over each factor in the base. Special cases, including Q_{11}^{11} (sometimes known as T^{11}), Q_{111}^{111} and Q_{21}^{32}, are relevant for compactifications of type IIB and D=11 supergravity. Remarkable ``conspiracies'' allow consistent Kaluza-Klein S^5, S^4 and S^7 sphere reductions of these theories that retain all the Yang-Mills fields of the isometry group in a massless truncation. We prove that such conspiracies do not occur for the reductions on the Q_{n_1... n_N}^{q_1... q_N} spaces, and that it is inconsistent to make a massless truncation in which the non-abelian SU(n_i+1) factors in their isometry groups are retained. In the course of proving this we derive many properties of the spaces Q_{n_1... n_N}^{q_1... q_N} of more general utility. In particular, we show that they always admit Einstein metrics, and that the spaces where q_i=(n_i+1)/\ell all admit two Killing spinors. We also obtain an iterative construction for real metrics on CP^n, and construct the Killing vectors on Q_{n_1... n_N}^{q_1... q_N} in terms of scalar eigenfunctions on CP^{n_i}. We derive bounds that allow us to prove that certain Killing-vector identities on spheres, necessary for consistent Kaluza-Klein reductions, are never satisfied on Q_{n_1... n_N}^{q_1... q_N}. 
  We consider the strong coupling limit of conformal gauge theories in 4 dimensions. The action of the loop operator on the minimal area in the AdS space is analyzed, and the Schwinger-Dyson equations of gauge theory are checked. The general approach to the loop dynamics developed here goes beyond the special case of conformal theories. 
  A maximally supersymmetric configuration of super Yang-Mills living on a noncommutative torus corresponds to a constant curvature connection. On a noncommutative toroidal orbifold there is an additional constraint that the connection be equivariant. We study moduli spaces of (equivariant) constant curvature connections on noncommutative even-dimensional tori and on toroidal orbifolds. As an illustration we work out the cases of Z_{2} and Z_{4} orbifolds in detail. The results we obtain agree with a commutative picture describing systems of branes wrapped on cycles of the torus and branes stuck at exceptional orbifold points. 
  We consider the quadratic divergence of the Yang-Mills theory when we use the hybrid regularization method consisting of the higher covariant derivative terms and the Pauli-Villars fields. By the explicit calculation of the diagrams, we show that the higher derivative terms for the ghost fields are necessary for the complete cancellation of the quadratic divergence. 
  In this paper we construct path integral representations of the boundary states in some special backgrounds such as the U(1) gauge field background, the linear dilaton background and the open string tachyon background. The initial purpose of this paper is to construct a general solution of the boundary conformal field theory with the analytical approach, mainly for the constraint equations$(L_{n}-\tilde{L}_{-n}) |B > =0 $ are difficult to be solved to obtain the solution represented by string modes from the pure algebraic approach. However in the path integral representation it is easy transforming those algebraic equations into the differential equations which can be solved. Another purpose of this paper is to try to explore an open question. we do not know how to construct an exact theory of D-branes in the general background until now. However many recent researches show the boundary state description indeed seizes some fundamental features of D-branes in the rather special backgrounds. Since the general background field effects can be easily introduced in the path integral representation, we argue that path integral representation of the boundary state should provide an exact description of D branes in the general backgrounds. 
  We propose that the principles of relativistic quantum mechanics are incomplete for simultaneous measurement of non-commuting operators. Consistent joint measurement of incompatible observables at a single point in space-time requires that the system be in an entangled state with vacuum meters. We suggest that entagled simultaneous mesurement for noncommuting observables is the basis for the observed fermionic multiplets. This generalizes the standard spin representations for particles arising from Lorentz invariance. We show that operator entanglement for all quantum observables in the Poincare algebra, coupled with Fermi-Dirac statistics, mandates six fermions. We propose that the quark and lepton generations form a super-structure of the Poincare algebra based on the principles of entangled simultaneity. Mathematically, that super-structure is known as a Naimark extension. The required entanglement between particle generations for left-handed quarks is observed in the Cabbibo-Kobayashi-maskawa matrix. We show that the Naimark-extended von Neumann lattice is ditributive, thereby suggesting the principle of entangled simultaneity as a mechanism to avoid quantum non-locality. Keywords: enatangled simultaneous quantum measurement, Naimark extension, Lepton/quark generations. PACS Number: 03.65.BZ 
  We discuss scalar quantum field theories in a Lorentz-invariant three-dimensional noncommutative space-time. We first analyze the one-loop diagrams of the two-point functions, and show that the non-planar diagrams are finite and have infrared singularities from the UV/IR mixing. The scalar quantum field theories have the problem that the violation of the momentum conservation from the non-planar diagrams does not vanish even in the commutative limit. A way to obtain an exact translational symmetry by introducing an infinite number of tensor fields is proposed. The translational symmetry transforms local fields into non-local ones in general. We also discuss an analogue of thermodynamics of free scalar field theory in the noncommutative space-time. 
  We show that weak-coupled two-dimensional dilaton gravity on Anti-de Sitter space can be described by the dynamics of an open string. Neumann and Dirichlet boundary conditions for the string lead to two different realizations of the Anti-de Sitter/Conformal Field Theory correspondence. In particular, in the Dirichlet case the thermodynamical entropy of two-dimensional black holes can be exactly reproduced by counting the string states. 
  We discuss unifying features of topological field theories in 2, 3 and 4 dimensions. This includes relations among enumerative geometry (2d topological field theory) link invariants (3d Chern-Simons theory) and Donaldson invariants (4d topological theory). 
  Two implicit periodic structures in the solution of sinh-Gordon thermodynamic Bethe ansatz equation are considered. The analytic structure of the solution as a function of complex $\theta$ is studied to some extent both analytically and numerically. The results make a hint how the CFT integrable structures can be relevant in the sinh-Gordon and staircase models. More motivations are figured out for subsequent studies of the massless sinh-Gordon (i.e. Liouville) TBA equation. 
  We present a detailed analysis of the 4-point functions of the lowest weight chiral primary operators $O^{I} \sim \tr \phi^{(i}\phi^{j)}$ in $\N =4$ SYM$_4$ at strong coupling and show that their structure is compatible with the predictions of AdS/CFT correspondence. In particular, all power-singular terms in the 4-point functions exactly coincide with the contributions coming from the conformal blocks of the CPOs, the R-symmetry current and the stress tensor. Operators dual to string modes decouple at strong coupling. We compute the anomalous dimensions and the leading $1/N^2$ corrections to the normalization constants of the 2- and 3-point functions of scalar and vector double-trace operators with approximate dimensions 4 and 5 respectively. We also find that the conformal dimensions of certain towers of double-trace operators in the {\bf 105}, {\bf 84} and {\bf 175} irreps are non-renormalized. We show that, despite the absence of a non-renormalization theorem for the double-trace operator in the {\bf 20} irrep, its anomalous dimension vanishes. As by-products of our investigation, we derive explicit expressions for the conformal block of the stress tensor, and for the conformal partial wave amplitudes of a conserved current and of a stress tensor in $d$ dimensions. 
  We consider the one-loop partition function for Euclidean BTZ black hole backgrounds or equivalently thermal AdS_3 backgrounds which are quotients of H_3 (Euclidean AdS_3). The one-loop partition function is modular invariant and we can read off the spectrum which is consistent to that found in hep-th/0001053. We see long strings and discrete states in agreement with the expectations. 
  By using tensor analysis, we find a connection between normed algebras and the parallelizability of the spheres S$^1$, S$^3$ and S$^7.$ In this process, we discovered the analogue of Hurwitz theorem for curved spaces and a geometrical unified formalism for the metric and the torsion. In order to achieve these goals we first develope a proof of Hurwitz theorem based in tensor analysis. It turns out that in contrast to the doubling procedure and Clifford algebra mechanism, our proof is entirely based in tensor algebra applied to the normed algebra condition. From the tersor analysis point of view our proof is straightforward and short. We also discuss a possible connection between our formalism and the Cayley-Dickson algebras and Hopf maps. 
  We discuss domain wall solutions of 5-dimensional supergravity corresponding to a cosine-superpotential, which is derived by a gauging of the two Abelian isometries of the scalar coset SU(2,1)/U(2). We argue that this potential can be obtained from M-theory compactification in the presence of G-fluxes and an M5-brane instanton gas. If we decouple the volume scalar of the internal space, the superpotential allows for two extrema, which are either ultra-violet or infra-red attractive. Asymptotically we approach therefore either the boundary or the Killing horizon of an anti-deSitter space or flat spacetime for a vanishing cosmological constant. If the volume scalar does not decouple, we obtain a run-away potential corresponding to dilatonic domain walls, which always run towards a vanishing cosmological constant. 
  We discuss several aspects of three dimensional N=2 supersymmetric gauge theories coupled to chiral multiplets. The generation of Chern-Simons couplings at low-energies results in novel behaviour including compact Coulomb branches, non-abelian gauge symmetry enhancement and interesting patterns of dynamically generated potentials. We further show how, given any pair of mirror theories with N=4 supersymmetry, one may flow to a pair of mirror theories with N=2 supersymmetry by gauging a suitable combination of the R-symmetries. The resulting theories again have interesting properties due to Chern-Simons couplings. 
  Recently a construction was given for the stress tensors of all sectors of the general current-algebraic orbifold A(H)/H, where A(H) is any current-algebraic conformal field theory with a finite symmetry group H. Here we extend and further analyze this construction to obtain the mode formulation of each sector of each orbifold A(H)/H, including the twisted current algebra, the Virasoro generators, the orbifold adjoint operation and the commutator of the Virasoro generators with the modes of the twisted currents. As applications, general expressions are obtained for the twisted current-current correlator and ground state conformal weight of each twisted sector of any permutation orbifold A(H)/H, H \subset S_N. Systematics are also outlined for the orbifolds A(Lie h(H))/H of the (H and Lie h)-invariant conformal field theories, which include the general WZW orbifold and the general coset orbifold. Finally, two new large examples are worked out in further detail: the general S_N permutation orbifold A(S_N)/S_N and the general inner-automorphic orbifold A(H(d))/H(d). 
  In a previous paper we have shown how the Wilsonian renormalization group naturally leads to the equivalence of the standard QED with a matter-only theory. In this paper we give an improved explanation of the equivalence and discuss, as an example, the equivalence of a chiral QED in the Higgs phase with a matter-only theory. Ignoring the contributions suppressed by the negative powers of a UV cutoff, the matter-only theory is equivalent to the perturbatively renormalizable chiral QED with two complex Higgs fields. In the matter-only theory chiral anomaly arises without elementary gauge fields. 
  The quantisation of a scalar field in the five-dimensional model suggested by Randall and Sundrum is considered. Using the Kaluza-Klein reduction of the scalar field, discussed by Goldberger and Wise, we sum the infinite tower of modes to find the vacuum energy density. Dimensional regularisation is used and we compute the pole term needed for renormalisation, as well as the finite part of the energy density. Some comments are made concerning the possible self-consistent determination of the radius. 
  We present the exact solution of the Baxter's three-color problem on a random planar graph, using the random-matrix formulation of the problem, given by B. Eynard and C. Kristjansen. We find that the number of three-coloring of an infinite random graph is 0.9843 per vertex. 
  It should be of interest, whether Dirac's equation involves all 16 basis elements of his Clifford algebra $Cl_D.$ These include the 6 `tensorial' $\sigma^{\mu\nu}$ with which the `Pauli terms' are formed. We find that these violate a basic axiom of any *-algebra, when Dirac's $\Psi$ is canonical. Then the Dirac operator is spanned only by the 10 elements $1,i\gamma_5,\gamma^\mu,\gamma^\mu\gamma_5$ (which don't form a basis of $Cl_D$ because the $\sigma^{\mu\nu}$ are excluded). 
  We examine the dual correspondence between holographic IIB superstring theory and N=4 super Yang-Mills theory at finite values of the coupling constants. In particular we analyze a field theory strong-coupling expansion which is the S-dual of the planar expansion. This expansion arises naturally as the AdS/CFT dual of the IIB superstring scattering amplitudes given a genus truncation property due to modular invariance. The space-time structure of the contributions to the field theory four-point correlation functions obtained from the IIB scattering elements is investigated in the example of the product of four conserved stress tensors, and is expressed as an infinite sum of field theory triangle integrals. The OPE structure of these contributions to the stress tensor four-point function is analyzed and shown not to give rise to any poles. Quantization of the string in the background of a five-form field strength is performed through a covariantized background field approach, and relations to the N=4 topological string are found. 
  In the study of three-brane cosmological models, an unusual law of cosmological expansion on the brane has been reported. According to this law, the energy density of matter on the brane quadratically enters the right-hand side of the new equations for the brane world, in contrast with the standard cosmology, where it enters the similar equations linearly. However, this result is obtained in the absence of curvature-dependent terms in the action for the brane. In this paper, we derive the field equations for a brane world embedded into a five-dimensional spacetime in the case where such terms are present. We also discuss some cosmological solutions of the resulting equations. 
  By replacing the ordinary product with the so called $\star$-product, one can construct an analogue of the anti-self-dual Yang-Mills (ASDYM) equations on the noncommutative $\bbR^4$. Many properties of the ordinary ASDYM equations turn out to be inherited by the $\star$-product ASDYM equation. In particular, the twistorial interpretation of the ordinary ASDYM equations can be extended to the noncommutative $\bbR^4$, from which one can also derive the fundamental strutures for integrability such as a zero-curvature representation, an associated linear system, the Riemann-Hilbert problem, etc. These properties are further preserved under dimensional reduction to the principal chiral field model and Hitchin's Higgs pair equations. However, some structures relying on finite dimensional linear algebra break down in the $\star$-product analogues. 
  We propose a natural differential equation with respect to mass(es) to analyze the scheme dependence problem. It is shown that the vertex functions subtracted at an arbitrary Euclidean momentum (MOM) do not satisfy such differential equations, as extra unphysical mass dependence is introduced which is shown to lead to the violation of the canonical form of the Slavnov-Taylor identities, a notorious fact with MOM schemes. By the way, the traditional advantage of MOM schemes in decoupling issue is shown to be lost in the context of Callan-Symanzik equations. 
  We calculate a component of connection superfields and Lorentz superparameter at second order in anticommuting coordinates in terms of the component fields of 11-dimensional on-shell supergravity by using `Gauge completion'. This configuration of superspace holds the $\kappa $-symmetry for supermembrane Lagrangian and represents 11-dimensional on-shell supergravity. 
  d3 and d5 maximally SUSY gauged supergravity is considered in the parametrization (flow) of full scalar coset where the kinetic term for scalars takes the standard field theory form and the bulk potential is an arbitrary one subject to consistent parametrization. From such SG duals we calculate d2 and d4 holographic conformal anomaly which depends on bulk scalars potential. AdS/CFT correspondence suggests that such SG side conformal anomaly should be identified with (non-perturbative) QFT conformal anomaly (taking account of radiative corrections) for the theory living on the boundary of AdS space. In the limit of constant bulk potential and single scalar, d4 result reproduces the known exact conformal anomaly corresponding to maximally SUSY super Yang-Mills theory coupled to ${\cal N}=4$ conformal supergravity. 
  We investigate the possibility of lowering the string scale in four dimensional heterotic models possessing a non-perturbative extension of the gauge group. In particular, we consider a class of compactifications in which the perturbative gauge sector is massive, and all the gauge bosons are non-perturbative, with a coupling independent on the Planck and string scales. 
  Projections play crucial roles in the ADHM construction on noncommutative $\R^4$. In this article a framework for the description of equivalence relations between projections is proposed. We treat the equivalence of projections as ``gauge equivalence'' on noncommutative space. We find an interesting application of this framework to the study of U(2) instanton on noncommutative $\R^4$: A zero winding number configuration with a hole at the origin is ``gauge equivalent'' to the noncommutative analog of the BPST instanton. Thus the ``gauge transformation'' in this case can be understood as a noncommutative resolution of the singular gauge transformation in ordinary $\R^4$. 
  New approach to p-adic and adelic strings, which takes into account that not only world sheet but also Minkowski space-time and string momenta can be p-adic and adelic, is formulated. p-Adic and adelic string amplitudes are considered within Feynman's path integral formalism. The adelic Veneziano amplitude is calculated. Some discreteness of string momenta is obtained. Also, adelic coupling constant is equal to unity. 
  For the case of a relativistic scalar field at finite temperature with a chemical potential, we calculate an exact expression for the one-loop effective action using the full fourth order determinant and zeta-function regularisation. We find that it agrees with the exact expression for the factored operator and thus there appears to be no mulitplicative anomaly. The appearance of the anomaly for the fourth order operator in the high temperature limit is explained and we show that the multiplicative anomaly can be calculated as the difference between two zeta-regularised zero-point energies. This difference is a result of using a charge operator in the Hamiltonian which has not been normal ordered. 
  We study a bosonic open string coupled to a tachyonic background field $T[X]$ and find that the tachyon field can effectively be replaced by a configuration of D-branes placed at either the zeros or the critical points of $T[X]$, depending on our choice of boundary conditions. This dual picture of the open string tachyon is explored in detail for the explicit case when the tachyonic field is quadratic. 
  We analyze the relation between the Lagrangian and Hamiltonian BRST symmetry generators for a recently proposed two-dimensional symmetry. In particular it is shown that this symmetry may be obtained from a canonical transformation in the ghost sector in a gauge independent way. 
  We study some non-perturbative aspects of noncommutative gauge theories. We find analytic solutions of the equations of motion, for noncommutative U(1) gauge theory, that describe magnetic monopoles with a finite tension string attached. These solutions are non-singular, finite and sourceless. We identify the string with the projection of a D-string ending on a D3-brane in the presence of a constant B-field. 
  In this paper, we construct gauge bundles on a noncommutative toroidal orbifold $T^4_\theta/Z_2$. First, we explicitly construct a bundle with constant curvature connections on a noncommutative $T^4_\theta$ following Rieffel's method. Then, applying the appropriate quotient conditions for its $Z_2$ orbifold, we find a Connes-Douglas-Schwarz type solution of matrix theory compactified on $T^4_\theta/Z_2$. When we consider two copies of a bundle on $T^4_\theta$ invariant under the $Z_2$ action, the resulting Higgs branch moduli space of equivariant constant curvature connections becomes an ordinary toroidal orbifold $T^4/Z_2$. 
  We discuss the compatibility between the weaker energy condition and the stability of Gregory, Rubakov and Sibiryakov (GRS) model. Because the GRS spacetime violates the weak energy condition, it may cause the instability. In the GRS model, the four dimensional gravity can be described by the massive KK modes with the resonance. Hence, instead of considering the weaker energy condition, we require for the stability of this model: no tachyon and no ghost condition for graviton modes ($h_{\mu\nu}$). No tachyonic condition ($m^2_h \geq 0$) is satisfied because the lowest state $m_h=0$ is supersymmetric vacuum state. Further, no ghost state condition is achieved if one requires some relations for the matter source: $2T_{55}= T^{\mu}_{\mu}=3(T_{22}+T_{33})$. It turns out that, although the GRS spacetime does not satisfy the weaker energy condition, it is stable against small perturbation. 
  The transition from the instanton-dominated quantum regime to the sphaleron-dominated classical regime is studied in the $d=2$ abelian-Higgs model when the spatial coordinate is compactified to $S^1$. Contrary to the noncompactified case, this model allows both sharp first-order and smooth second-order transitions depending on the size of the circle. This finding may make the model a useful toy model for the analysis of baryon number violating processes. Since the model can to a large extent be treated analytically, it can also serve as a transparent prototype for the application of our method to more complicated cases, such as those in higher dimensions. 
  We study the U(N) non-commutative Yang-Mills theory at the one-loop approximation. We check renormalizability and gauge invariance of the model and calculate the one-loop beta function. The interaction of the SU(N) gauge bosons with the U(1) gauge boson plays an important role in the consistency check. In particular, the SU(N) theory by itself is not consistent. We also find that the theta --> 0 limit of the U(N) theory does not converge to the ordinary SU(N) x U(1) commutative theory, even at the planar limit. Finally, we comment on the UV/IR mixing. 
  In most current models of inflation based on a weakly self-coupled scalar matter field minimally coupled to gravity, the period of inflation lasts so long that, at the beginning of the inflationary period, the physical wavelengths of comoving scales which correspond to the present large-scale structure of the Universe were smaller than the Planck length. Thus, the usual computations of the spectrum of fluctuations in these models involve extrapolating low energy physics (both in the matter and gravitational sector) into regions where this physics is not applicable. In this paper we demonstrate that the usual predictions of inflation for the spectrum of cosmological fluctuations do indeed depend on the hidden assumptions about super-Planck scale physics. We introduce a class of modified dispersion relations to mimic possible effects of super-Planck scale physics, and show that in some cases important deviations from the usual predictions of inflation are obtained. Some implications of this result for the unification of fundamental physics and early Universe cosmology are discussed. 
  This review is based on two lectures given at the 2000 TMR school in Torino. We discuss two main themes: i) Moyal-type deformations of gauge theories, as emerging from M-theory and open string theories, and ii) the noncommutative geometry of finite groups, with the explicit example of Z_2, and its application to Kaluza-Klein gauge theories on discrete internal spaces. 
  The tree-level amplitude for the scattering of two gauge particles constrained to move on the two distinct boundaries of eleven-dimensional space-time in the Horava-Witten formulation of M-theory is constructed. At low momenta this reproduces the corresponding tree-level scattering amplitude of the E_8xE_8 heterotic string theory. After compactification to nine dimensions on a large circle with a suitable Wilson line to break the symmetry to SO(16)xSO(16) this amplitude is used to describe the scattering of two massive SO(16) spinor states - one from each factor of the unbroken symmetry group. The amplitude contains a component that is associated with the exchange of a Kaluza-Klein charge between the boundaries, which is interpreted as the exchange of a D-particle between orientifold planes in the Type IA theory. This is related by T-duality to the effect of a non-BPS D-instanton in the Type I theory which is only invariant under those elements of O(16)xSO(16) that are in SO(16)xSO(16). 
  Over the past decade it has become clear that fundamental strings are not the only fundamental degrees of freedom in string theory. D-branes are also part of the spectrum of fundamental states. In this paper we explore some possible effects of D-branes on early Universe string cosmology, starting with two key assumptions: firstly that the initial state of the Universe corresponded to a dense, hot gas in which all degrees of freedom were in thermal equilibrium, and secondly that the topology of the background space admits one-cycles. We argue by t-duality that in this context the cosmological singularities are not present. We derive the equation of state of the brane gases and apply the results to suggest that, in an expanding background, the winding modes of fundamental strings will play the most important role at late times. In particular, we argue that the string winding modes will only allow four space-time dimensions to become large. The presence of brane winding modes with $p > 1$ may lead to a hierarchy in the sizes of the extra dimensions. 
  Type 0B string theory has been proposed as the dual description of non-supersymmetric SU(N) Yang-Mills theory coupled to six scalars, in four dimensions. We study numerically and analytically the equations of motion of type 0B gravity and we find RG trajectories of the dual theory that flow from an asymptotically free UV regime to a confining IR regime. In the UV we find a one-parameter family of solutions that approach asymptotically $AdS_5\times S^5$ with a logarithmic flow of the coupling plus non-perturbative terms that correctly reproduce all UV and IR renormalon singularities. The first UV renormalon gives a contribution $\sim F_1(E)/E^2$ and we are able to predict also the form of the function $F_1(E)$, which, from the YM side, corresponds to summing all multiple-chain bubble graphs. The fact that the positions of the renormalon singularities in the Borel plane come out correctly is a non-trivial test of the conjectured duality. 
  The Symplectic Projector Method is applied to discuss quantisation aspects of an extended Abelian model with a pair of gauge potentials coupled by means of a mixed Chern-Simons term. We focuss on a field content that spans an N=2-D=3 supersymmetric theory whenever scalar and fermionic matter is suitably coupled to the family of gauge potentials. 
  We study the strong field limit of p-form Born-Infeld theory. It turns out that this limiting theory is a unique theory displaying the full symmetry group of the underlying canonical structure. Moreover, being a nonlinear theory, it possesses an infinite hierarchy of conservation laws. 
  We consider spectral problem for a free relativistic particle in p-adic and adelic quantum mechanics. In particular, we found p-adic and adelic eigenfunctions. Within adelic approach there exist quantum states that exhibit discrete structure of spacetime at the Planck scale. 
  We consider localization of gravity in domain wall solutions of Einstein's gravity coupled to a scalar field with a generic potential. We discuss conditions on the scalar potential such that domain wall solutions are non-singular. Such solutions even exist for appropriate potentials which have no minima at all and are unbounded below. Domain walls of this type have infinite tension, while usual kink type of solutions interpolating between two AdS minima have finite tension. Non-singular domain walls with infinite tension might a priori avoid recent ``no-go'' theorems indicating impossibility of supersymmetric embedding of kink type of domain walls in gauged supergravity. We argue that (non-singular) domain walls are stable even if they have infinite tension. This is essentially due to the fact that localization of gravity in smooth domain walls is a Higgs mechanism corresponding to a spontaneous breakdown of translational invariance. We point out that if the scalar potential has no minima and approaches finite negative values at infinity, then higher derivative terms are under control, and do not affect the cosmological constant on the brane which is vanishing for such backgrounds. Nonetheless, we also point out that higher curvature terms generically delocalize gravity, so that the desired lower dimensional Newton's law is no longer reproduced. 
  We study an ambiguity of the current regularization in the Thirring model. We find a new current definition which enables to make a comprehensive treatment of the current. Our formulation is simpler than Klaiber's formulation. We compare our result with other formulations and find a very good agreement with their result. We also obtain the Schwinger term and the general formula for any current regularization. 
  In this paper we investigate three-dimensional superconformal gauge theories with N=3 supersymmetry. Independently from specific models, we derive the shortening conditions for unitary representations of the Osp(3|4) superalgebra and we express them in terms of differential constraints on three dimensional N=3 superfields. We find a ring structure underlying these short representations, which is just the direct generalization of the chiral ring structure of N=2 theories. When the superconformal field theory is realized on the world-volume of an M2-brane such superfield ring is the counterpart of the ring defined by the algebraic geometry of the 8-dimensional cone transverse to the brane. This and other arguments identify the N=3 superconformal field theory dual to M-theory compactified on AdS_4 x N^{0,1,0}. It is an N=3 gauge theory with SU(N) x SU(N) gauge group coupled to a suitable set of hypermultiplets, with an additional Chern Simons interaction. The AdS/CFT correspondence can be directly verified using the recently worked out Kaluza Klein spectrum of N^{0,1,0} and we find a perfect match. We also note that besides the usual set of BPS conformal operators dual to the lightest KK states, we find that the composite operators corresponding to certain massive KK modes are organized into a massive spin 3/2 N=3 multiplet that might be identified with the super-Higgs multiplet of a spontaneously broken N=4 theory. We investigate this intriguing and inspiring feature in a separate paper. 
  We discuss a general pairing that occurs in compactifications of M-theory on AdS_4 x X^7 backgrounds between massless ultra short multiplets and their massive shadows, namely certain universal long multiplets with fixed protected dimensions. In particular we consider the shadow of the short graviton multiplet in N=3 compactifications. It turns out to be a massive spin 3/2 multiplet with scale dimension E_0=3 and with the quantum numbers of a super-Higgs multiplet. Hence each N=3 AdS_4 x X^7 vacuum is actually to be interpreted as a spontaneously broken phase of an N=4 theory. Comparison with standard gauged N=4 supergravity in 4 dimensions reveals the unexpected bound E_0<3 on the dimension of the broken gravitino multiplet. This hints to the existence of new versions of extended supergravities, in particular N=4 where such upper bounds are evaded and where all possible vacua have a reduced supersymmetry N_0<N. We name them shadow supergravities. In particular, using arguments based on the solvable Lie algebra parametrization of the scalar manifold, we discuss the possible structure of shadow N=4 supergravity. Using our previous results on the SCFT dual of the AdS_4 x N^{0,1,0} vacuum we discuss the SCFT realization of the universal N=3 shadow multiplet. RG flows from an N=4 to an N=3 phase are ruled out by the fact that the N=4 vacuum is at infinite distance in moduli space, denoting the presence of a topology change. 
  We apply the ADHM instanton construction to SU(2) gauge theory on T^n x R^(4-n)for n=1,2,3,4. To do this we regard instantons on T^n x R^(4-n) as periodic (modulo gauge transformations) instantons on R^4. Since the R^4 topological charge of such instantons is infinite the ADHM algebra takes place on an infinite dimensional linear space. The ADHM matrix M is related to a Weyl operator (with a self-dual background) on the dual torus tilde T^n. We construct the Weyl operator corresponding to the one-instantons on T^n x R^(4-n). In order to derive the self-dual potential on T^n x R^(4-n) it is necessary to solve a specific Weyl equation. This is a variant of the Nahm transformation. In the case n=2 (i.e. T^2 x R^2) we essentially have an Aharonov Bohm problem on tilde T^2. In the one-instanton sector we find that the scale parameter, lambda, is bounded above, (lambda)^2 tv<4 pi, tv being the volume of the dual torus tilde T^2. 
  The presence of a constant background antisymmetric tensor for open strings or D-branes forces the space-time coordinates to be noncommutative. This effect is equivalent to replacing ordinary products in the effective theory by the deformed star product. An immediate consequence of this is that all fields get complexified. The only possible noncommutative Yang-Mills theory is the one with U(N) gauge symmetry. By applying this idea to gravity one discovers that the metric becomes complex. We show in this article that this procedure is completely consistent and one can obtain complexified gravity by gauging the symmetry $U(1,D-1)$ instead of the usual $SO(1,D-1)$. The final theory depends on a Hermitian tensor containing both the symmetric metric and antisymmetric tensor. In contrast to other theories of nonsymmetric gravity the action is both unique and gauge invariant. The results are then generalized to noncommutative spaces. 
  We compute two-point functions of chiral operators Tr(\Phi^k) for any k, in {\cal N}=4 supersymmetric SU(N) Yang-Mills theory. We find that up to the order g^4 the perturbative corrections to the correlators vanish for all N. The cancellation occurs in a highly non trivial way, due to a complicated interplay between planar and non planar diagrams. In complete generality we show that this same result is valid for any simple gauge group. Contact term contributions signal the presence of ultraviolet divergences. They are arbitrary at the tree level, but the absence of perturbative renormalization in the non singular part of the correlators allows to compute them unambiguously at higher orders. In the spirit of the AdS/CFT correspondence we comment on their relation to infrared singularities in the supergravity sector. 
  We test the Maldacena conjecture for type IIB String Theory/ N=4 Yang-Mills by calculating the one-loop corrections in the bulk theory to the Weyl anomaly of the boundary CFT when the latter is coupled to a Ricci-flat metric. The contributions cancel within each supermultiplet, in agreement with the conjecture. 
  This report provides a pedagogical introduction to the description of the general Poincare supergravity/matter/Yang-Mills couplings using methods of Kahler superspace geometry. At a more advanced level this approach is generalized to include tensor field and Chern-Simons couplings in supersymmetry and supergravity, relevant in the context of weakly and strongly coupled string theories. 
  In this paper we explore some general aspects of the embeddings associated with brane-localized gravity. In particular we show that the consistency of such embeddings can require (or impose) very specific relations between all the involved bulk and brane matter source parameters. We specifically explore the embeddings of 3-branes with non-zero spatial 3-curvature $k$ into 5-dimensional spacetime bulks, and show that for such embeddings, a 5-dimensional bulk cosmological constant is not able to produce the exponential suppression of the geometry thought necessary to localize gravity to the brane. 
  We study soliton gauge states in the spectrum of bosonic string compatified on torus. The enhenced Kac-Moody gauge symmetry, and thus T-duality, is shown to be related to the existence of these soliton gauge states in some moduli points. 
  We study the mechanism of enhanced gauge symmetry of bosonic open string compatified on torus by analyzing the zero-norm soliton (nonzero winding of wilson line) gauge states in the spectrum. Unlike the closed string case, we find that the soliton gauge state exists only at massive levels. These soliton gauge states correspond to the existence of enhanced massive gauge symmetries with transformation parameters containing both Einstein and Yang-Mills indices. In the T-dual picture, these symmetries exist only at some discrete values of compatified radii when N D-branes are coincident. 
  We investigate the replica trick for the microscopic spectral density, $\rho_s(x)$, of the Euclidean QCD Dirac operator. Our starting point is the low-energy limit of the QCD partition function for $n$ fermionic flavors (or replicas) in the sector of topological charge $\nu$. In the domain of the smallest eigenvalues, this partition function is simply given by a U(n) unitary matrix integral. We show that the asymptotic behavior of $\rho_s(x)$ for $x \to \infty$ is obtained from the $n\to 0$ limit of this integral. The smooth contributions to this series are obtained from an expansion about the replica symmetric saddle-point, whereas the oscillatory terms follow from an expansion about a saddle-point that breaks the replica symmetry. For $\nu =0$ we recover the small-$x$ logarithmic singularity of the resolvent by means of the replica trick. For half integer $\nu$, when the saddle point expansion of the U(n) integral terminates, the replica trick reproduces the exact analytical result. In all other cases only an asymptotic series that does not uniquely determine the microscopic spectral density is obtained. We argue that bosonic replicas fail to reproduce the microscopic spectral density. In all cases, the exact answer is obtained naturally by means of the supersymmetric method. 
  We present an exact thick domain wall solution with naked sigularities to five dimensional gravity coupled with a scalar field with exponential potential. In our solution we found exactly the special coefficient of the exponent as coming from compactification of string theory with cosmological constant. We show that this solution is self-tuning when a 3-brane is included. In searching for solution with horizon we found a similar exact solution with fine-tuned exponent coefficient with an integration constant. Failed to find a solution with horizon we prove the non-existence of horizons. These naked sigularities actually can't be resolved by horizon. We also comment on the physical relevance of this solution.} 
  We compute the quantum vacuum polarization for a pure neutral scalar field theory within the context of single-particle quantum mechanics. The loop diagram is computed without ever encountering loop-momentum integrals. Our approach is based on standard Feynman path integrals. Contact is made to scalar QED. 
  Interesting non-linear functions on the phase spaces of classical field theories can never be quantized immediately because the basic fields of the theory become operator valued distributions. Therefore, one is usually forced to find a classical substitute for such a function depending on a regulator which is expressed in terms of smeared quantities and which can be quantized in a well-defined way. Namely, the smeared functions define a new symplectic manifold of their own which is easy to quantize. Finally one must remove the regulator and establish that the final operator, if it exists, has the correct classical limit.   In this paper we investigate these steps for diffeomorphism invariant quantum field theories of connections. We introduce a generalized projective family of symplectic manifolds, coordinatized by the smeared fields, which is labelled by a pair consisting of a graph and another graph dual to it. We show that there exists a generalized projective sequence of symplectic manifolds whose limit agrees with the symplectic manifold that one started from.   This family of symplectic manifolds is easy to quantize and we illustrate the programme outlined above by applying it to the Gauss constraint. The framework developed here is the classical cornerstone on which the semi-classical analysis developed in a new series of papers called ``Gauge Theory Coherent States'' is based. 
  In this article we outline a rather general construction of diffeomorphism covariant coherent states for quantum gauge theories.   By this we mean states $\psi_{(A,E)}$, labelled by a point (A,E) in the classical phase space, consisting of canonically conjugate pairs of connections A and electric fields E respectively, such that (a) they are eigenstates of a corresponding annihilation operator which is a generalization of A-iE smeared in a suitable way, (b) normal ordered polynomials of generalized annihilation and creation operators have the correct expectation value, (c) they saturate the Heisenberg uncertainty bound for the fluctuations of $\hat{A},\hat{E}$ and (d) they do not use any background structure for their definition, that is, they are diffeomorphism covariant.   This is the first paper in a series of articles entitled ``Gauge Field Theory Coherent States (GCS)'' which aim at connecting non-perturbative quantum general relativity with the low energy physics of the standard model. In particular, coherent states enable us for the first time to take into account quantum metrics which are excited {\it everywhere} in an asymptotically flat spacetime manifold. The formalism introduced in this paper is immediately applicable also to lattice gauge theory in the presence of a (Minkowski) background structure on a possibly {\it infinite lattice}. 
  In the preceding paper of this series of articles we established peakedness properties of a family of coherent states that were introduced by Hall for any compact gauge group and were later generalized to gauge field theory by Ashtekar, Lewandowski, Marolf, Mour\~ao and Thiemann.   In this paper we establish the ``Ehrenfest Property'' of these states which are labelled by a point (A,E), a connection and an electric field, in the classical phase space. By this we mean that i) The expectation value of {\it all} elementary quantum operators $\hat{O}$ with respect to the coherent state with label (A,E) is given to zeroth order in $\hbar$ by the value of the corresponding classical function O evaluated at the phase space point (A,E) and ii) The expectation value of the commutator between two elementary quantum operators $[\hat{O}_1,\hat{O}_2]/(i\hbar)$ divided by $i\hbar$ with respect to the coherent state with label (A,E) is given to zeroth order in $\hbar$ by the value of the Poisson bracket between the corresponding classical functions $\{O_1,O_2\}$ evaluated at the phase space point (A,E).   These results can be extended to all polynomials of elementary operators and to a certain non-polynomial function of the elementary operators associated with the volume operator of quantum general relativity. It follows that the infinitesimal quantum dynamics of quantum general relativity is to zeroth order in $\hbar$ indeed given by classical general relativity. 
  In the canonical approach to Lorentzian Quantum General Relativity in four spacetime dimensions an important step forward has been made by Ashtekar, Isham and Lewandowski some eight years ago through the introduction of an appropriate Hilbert space structure.   This Hilbert space, together with its generalization due to Baez and Sawin, is appropriate for semi-classical quantum general relativity if the spacetime is spatially compact. In the spatially non-compact case, however, an extension of the Hilbert space is needed in order to approximate metrics that are macroscopically nowhere degenerate.   For this purpose, in this paper we apply von Neumann's theory of the Infinite Tensor Product (ITP) of Hilbert Spaces to Quantum General Relativity. The cardinality of the number of tensor product factors can take the value of any possible Cantor aleph as is needed for our problem, where a Hilbert space is attached to each edge of an arbitrarily complicated, generally infinite graph.   The new framework opens a pandora's box full of techniques, appropriate to pose fascinating physical questions such as quantum topology change, semi-classical quantum gravity, effective low energy physics etc. from the universal point of view of the ITP. In particular, the study of photons and gravitons propagating on fluctuating quantum spacetimes is now in reach, the topic of the next paper in this series. 
  An anisotropic (Bianchi type I) cosmology is considered in the four-dimensional NS-NS sector of low-energy effective string theory coupled to a dilaton and an axion-like $H$-field within a de Sitter-Einstein frame background. The time evolution of this Universe is discussed in both the Einstein and string frames. 
  In this article we apply the methods outlined in the previous paper of this series to the particular set of states obtained by choosing the complexifier to be a Laplace operator for each edge of a graph. The corresponding coherent state transform was introduced by Hall for one edge and generalized by Ashtekar, Lewandowski, Marolf, Mour\~ao and Thiemann to arbitrary, finite, piecewise analytic graphs. However, both of these works were incomplete with respect to the following two issues : (a) The focus was on the unitarity of the transform and left the properties of the corresponding coherent states themselves untouched. (b) While these states depend in some sense on complexified connections, it remained unclear what the complexification was in terms of the coordinates of the underlying real phase space. In this paper we resolve these issues, in particular, we prove that this family of states satisfies all the usual properties : i) Peakedness in the configuration, momentum and phase space (or Bargmann-Segal) representation, ii) Saturation of the unquenched Heisenberg uncertainty bound. iii) (Over)completeness. These states therefore comprise a candidate family for the semi-classical analysis of canonical quantum gravity and quantum gauge theory coupled to quantum gravity, enable error-controlled approximations and set a new starting point for {\it numerical canonical quantum general relativity and gauge theory}. The text is supplemented by an appendix which contains extensive graphics in order to give a feeling for the so far unknown peakedness properties of the states constructed. 
  A numerical study of coupled to the dilaton field, static, spherically symmetric monopole solutions inspired by the Kaluza-Klein theory with large extra dimensions are presented. The generalized Prasad-Sommerfield solution is obtained. We show that monopole may have also the dilaton cloud configurations. 
  The evolution of the cosmological perturbations is studied in the context of the Randall-Sundrum brane world scenario, in which our universe is realized on a three-brane in the five dimensional Anti-de Sitter(AdS) spacetime. We develop a formalism to solve the coupled dynamics of the cosmological perturbations in the brane world and the gravitational wave in the AdS bulk. Using our formalism, the late time evolution of the cosmological scalar perturbations at any scales larger than the AdS curvature scale $l$ is shown to be identical with the one obtained in the conventional 4D cosmology, provided the effect of heavy graviton modes may be neglected. Here the late time means the epoch when the Hubble horizon $H^{-1}$ in the 4D brane world is sufficiently larger than the AdS curvature scale $l$. If the inflation occurs sufficiently lower than $l^{-1}$, the scalar temperature anisotropies in the Cosmic Microwave Background at large scales can be calculated using the constancy of the Bardeen parameter as is done in the 4D cosmology. The assumption of the result is that the effect of the massive graviton with mass $m e^{-\alpha_0}>l^{-1}$ in the brane world is negligible, where $e^{\alpha_0}$ is the scale factor of the brane world. We also discuss the effect of these massive gravitons on the evolution of the perturbations. 
  The Unruh effect can be correctly treated only by using the Minkowski quantization and a model of a "particle" detector, not by using the Rindler quantization. The energy produced by a detector accelerated only for a short time can be much larger than the energy needed to change the velocity of the detector. Although the measuring process lasts an infinite time, the production of the energy can be qualitatively explained by a time-energy uncertainty relation. 
  The possibility of using the quantum mechanics of D0-branes for the bound-states of quarks and QCD strings is investigated. Issues such as the inter D0-branes potential, the whiteness of the D0-branes bound-states and the large-N limit of D0-branes effective theory are studied. A possible role of the non-commutativity of relative distances of D0-branes in a study of ordinary QCD is discussed. 
  We construct supergravity solutions that correspond to N Dp-branes coinciding with \bar{N} \bar{Dp}-branes. We study the physical properties of the solutions and analyse the supergravity description of tachyon condensation. We construct an interpolation between the brane-antibrane solution and the Schwarzschild solution and discuss its possible application to the study of non-supersymmetric black holes. 
  Motivated by the dual standard model, we study the angular momentum spectrum of stable SU(5) dyons that can be transformed into purely electric states by a suitable duality rotation i.e. are dualizable. The problem reduces to solving a Diophantine equation for the holomorphic charges in each topological sector, but the solutions also have to satisfy certain constraints. We show that these equations can be solved and sets of dualizable, half-integer spin SU(5) dyons can be found, each of which corresponds to a single family of the standard model fermions. We then find two predictions of the dual standard model. First, the family of half-integer spin, dualizable dyons is accompanied by a set of dualizable, integer-spin partner states. Secondly, the dyon corresponding to the electron must necessarily contain non-trivial color internal structure. In addition, we provide other general results regarding the spectrum of dualizable dyons and their novel properties, and extend the stability analysis of SU(5) monopoles used in the dual standard model so far to discuss the stability of the half-integer spin dyons. 
  Generalization of the recent Taylor-Polchinski argument is presented, which helps to explain quantization of RR charges in IIA-like theories in the presence of cohomologically trivial H-fields. 
  A simple criterion to optimise coarse-grainings for exact renormalisation group equations is given. It is aimed at improving the convergence of approximate solutions of flow equations. The optimisation criterion is generic, as it refers only to the coarse-grained propagator at vanishing field. In physical terms, it is understood as an optimisation condition for amplitude expansions. Alternatively, it can be interpreted as the requirement to move poles of threshold functions away from the physical region. The link to expansions in field amplitudes is discussed as well. Optimal parameters are given explicitly for a variety of different coarse-grainings. As a by-product it is found that the sharp cut-off regulator does not belong to the class of such optimal coarse-grainings, which explains the poor convergence of amplitude expansions based on it. 
  We obtain a large class of AdS spacetimes warped with certain internal spaces in eleven-dimensional and type IIA/IIB supergravities. The warp factors depend only on the internal coordinates. These solutions arise as the near-horizon geometries of more general semi-localised multi-intersections of $p$-branes. We achieve this by noting that any sphere (or AdS spacetime) of dimension greater than 3 can be viewed as a foliation involving S^3 (or AdS_3). Then the S^3 (or AdS_3) can be replaced by a three-dimensional lens space (or a BTZ black hole), which arises naturally from the introduction of a NUT (or a pp-wave) to the M-branes or the D3-brane. We then obtain multi-intersections by performing a Kaluza-Klein reduction or Hopf T-duality transformation on the fibre coordinate of the lens space (or the BTZ black hole). These geometries provide further possible examples of the AdS/CFT correspondence and of consistent embeddings of lower-dimensional gauged supergravities in D=11 or D=10. 
  We study (2,2) supersymmetric field theories on two-dimensional worldsheet with boundaries. We determine D-branes (boundary conditions and boundary interactions) that preserve half of the bulk supercharges in non-linear sigma models, gauged linear sigma models, and Landau-Ginzburg models. We identify a mechanism for brane creation in LG theories and provide a new derivation of a link between soliton numbers of the massive theories and R-charges of vacua at the UV fixed point. Moreover we identify Lagrangian submanifolds that arise as the mirror of certain D-branes wrapped around holomorphic cycles of K\"ahler manifolds. In the case of Fano varieties this leads to the explanation of Helix structure of the collection of exceptional bundles and soliton numbers, through Picard-Lefshetz theory applied to the mirror LG theory. Furthermore using the LG realization of minimal models we find a purely geometric realization of Verlinde Algebra for SU(2) level k as intersection numbers of D-branes. This also leads to a direct computation of modular transformation matrix and provides a geometric interpretation for its role in diagonalizing the Fusion algebra. 
  We compute the 4--dimensional cosmological constant in string compactifications in which the Standard Model fields live on a non-supersymmetric brane inside a supersymmetric bulk. The cosmological constant receives contributions only from the vacuum energy of the bulk supergravity fields, but not from the vacuum energy of the brane fields. The latter is absorbed in a warp factor. Supersymmetry breaking on the brane at the TeV scale implies supersymmetry breaking in the bulk at the micrometer scale. This produces a tiny cosmological constant that agrees with experiment within a few orders of magnitude. Our argument predicts superpartners of the graviton with mass of order $10^{-3}$ eV. They should be observable in short-distance tests of Einstein Gravity. 
  We revise the problem of the quantization of relativistic particle, presenting a modified consistent canonical scheme, which allows one not only to include arbitrary backgrounds in the consideration but to get in course of the quantization a consistent relativistic quantum mechanics, which reproduces literally the behavior of the one-particle sector of the corresponding quantum field. At the same time this construction presents a possible solution of the well-known old problem how to construct a consistent quantum mechanics on the base of a relativistic wave equation. 
  We show that in general holographic stress tensor may contain a new term of divergence of a spacelike unit normal acceleration. Then, it is shown that in contrast to previous descriptions, a new stress tensor for Kerr-AdS solutions can be a traceless one. Interestingly, this prescription entails a local failure on the IR-UV connection. 
  In this paper we consider orbifold compactifications of M-theory on $S^1/{\bf Z}_2\times T^4/{\bf Z}_2$. We discuss solutions of the local anomaly matching conditions by twisted vector, tensor and hypermultiplets confined on the local orbifold six-planes. In addition we consider phase-transitions among different solutions which are mediated by M-theory fivebranes which touch the local orbifold planes and are converted there to gauge instantons. 
  The effective action for the Polyakov loop serving as an order parameter for deconfinement is obtained in one-loop approximation to second order in a derivative expansion. The calculation is performed in $d\geq 4$ dimensions, mostly referring to the gauge group SU(2). The resulting effective action is only capable of describing a deconfinement phase transition for $d>d_{\text{cr}}\simeq 7.42$. Since, particularly in $d=4$, the system is strongly governed by infrared effects, it is demonstrated that an additional infrared scale such as an effective gluon mass can change the physical properties of the system drastically, leading to a model with a deconfinement phase transition. 
  We consider the nonrelativistic field theory with a quartic interaction on a noncommutative plane. We compute the four point scattering amplitude within perturbative analysis to all orders and identify the beta function and the running of the coupling constant. Since the theory admits an equivalent description via the N particle Schrodinger equation, we regain the scattering amplitude by finding an exact scattering wavefunction of the two body equation. The wave function for the bound state is also identified. These wave functions unusually have two center positions in the relative coordinates. The separation of the centers is in the transverse direction of the total momentum and grows linearly with the noncommutativity scale and the total momentum, exhibiting the stringy nature of the noncommutative field theory. 
  We discuss the analytic properties of the Callan-Symanzik beta-function beta(g) associated with the zero-momentum four-point coupling g in the two-dimensional phi^4 model with O(N) symmetry. Using renormalization-group arguments, we derive the asymptotic behavior of beta(g) at the fixed point g^*. We argue that beta'(g) = beta'(g^*) + O(|g-g^*|^{1/7}) for N=1 and beta'(g) = beta'(g^*) + O(1/\log |g-g^*|) for N > 2. Our claim is supported by an explicit calculation in the Ising lattice model and by a 1/N calculation for the two-dimensional phi^4 theory. We discuss how these nonanalytic corrections may give rise to a slow convergence of the perturbative expansion in powers of g. 
  The increasing precision of many experiments in elementary particle physics leads to continuing interest in perturbative higher order calculations in the electroweak Standard Model or extensions of it. Such calculations are of increasing complexity because more loops and/or more legs are considered. Correspondingly efficient computational methods are mandatory for many calculations. One problem which affects the feasibility of higher order calculations is the problem with gamma(5) in dimensional regularization. Since the subject thirty years after its invention is still controversial I advocate here some ideas which seem not to be common knowledge but might shed some new light on the problem. I present arguments in favor of utilizing an anticommuting gamma(5) and a simple 4-dimensional treatment of the hard anomalies. 
  We study the trace anomaly of the (2,0) tensor multiplet in d=6 in the presence of a background SO(5) vector field acting as a source for the R-current. Using both a free-field theory calculation and AdS_7/CFT_6 correspondence, we find that only one of the two possible anomaly structures is non-zero and that its coefficient at strong-coupling differs by the well-known overall factor 4N^3 from the corresponding weak coupling result. We also discuss the relevance of our result to studies of the R-current anomaly in the (2,0) multiplet. 
  The Casimir energy of a semi-circular cylindrical shell is calculated by making use of the zeta function technique. This shell is obtained by crossing an infinite circular cylindrical shell by a plane passing through the symmetry axes of the cylinder and by considering only a half of this configuration. All the surfaces, including the cutting plane, are assumed to be perfectly conducting. The zeta functions for scalar massless fields obeying the Dirichlet and Neumann boundary conditions on the semi-circular cylinder are constructed exactly. The sum of these zeta functions gives the zeta function for electromagnetic field in question. The relevant plane problem is considered also. In all the cases the final expressions for the corresponding Casimir energies contain the pole contributions which are the consequence of the edges or corners in the boundaries. This implies that further renormalization is needed in order for the finite physical values for vacuum energy to be obtained for given boundary conditions. 
  We apply a soft version of the BPHZ subtraction scheme to the computation of two-loop corrections from an Abelian Chern-Simons field coupled to (massive) scalar matter with a $\lambda(\Phi^\dag\Phi)^2$ and $\nu(\Phi^\dag\Phi)^3$ self-interactions. The two-loop renormalization group functions are calculated. We compare our results with those in the literature. 
  We review the concept of $\tau$-function for simple analytic curves. The $\tau$-function gives a formal solution to the 2D inverse potential problem and appears as the $\tau$-function of the integrable hierarchy which describes conformal maps of simply-connected domains bounded by analytic curves to the unit disk. The $\tau$-function also emerges in the context of topological gravity and enjoys an interpretation as a large $N$ limit of the normal matrix model. 
  Spacetime, understood as a globally hyperbolic manifold, may be characterized by spectral data using a 3+1 splitting into space and time, a description of space by spectral triples and by employing causal relationships, as proposed earlier. Here, it is proposed to use the Hadamard condition of quantum field theory as a smoothness principle. 
  I study a class of interacting conformal field theories and conformal windows in three dimensions, formulated using the Parisi large-N approach and a modified dimensional-regularization technique. Bosons are associated with composite operators and their propagators are dynamically generated by fermion bubbles. Renormalization-group flows between pairs of interacting fixed points satisfy a set of non-perturbative g <-> 1/g dualities. There is an exact relation between the beta function and the anomalous dimension of the composite boson. Non-Abelian gauge fields have a non-renormalized and quantized gauge coupling, although no Chern-Simons term is present. A problem of the naive dimensional-regularization technique for these theories is uncovered and removed with a non-local, evanescent, non-renormalized kinetic term. The models are expected to be a fruitful arena for the study of odd-dimensional conformal field theory. 
  We show that a partition function of topological twisted N=4 Yang-Mills theory is given by Seiberg-Witten invariants on a Riemannian four manifolds under the condition that the sum of Euler number and signature of the four manifolds vanish. The partition function is the sum of Euler number of instanton moduli space when it is possible to apply the vanishing theorem. And we get a relation of Euler number labeled by the instanton number $k$ with Seiberg-Witten invariants, too. All calculation in this paper is done without assuming duality. 
  Participants of this workshop pursue the old Neutrino Theory of Light vigorously. Other physicists have long ago abandoned it, because it lacks gauge invariance. In the recent Quantum Induction (QI), all basic Bose fields ${\mathcal B}^{P}$ are local limits of quantum fields composed of Dirac's $\Psi$ (for leptons and quarks). The induced field equations of QI even determine all the interactions of those ${\mathcal B}^{P}$. Thus a precise gauge invariance and other physical consequences are unavoidable. They include the absence of divergencies, the exclusion of Pauli terms, a prediction of the Higgs mass and a `minimal' Quantum Gravity.   As we find in this paper, however, photons can't be bound states while Maxwell's potential $A_{\mu}$ contains all basic Dirac fields except those of neutrinos. 
  We deal with quantum field theory in the restriction to external Bose fields. Let $(i\gamma^\mu\partial_\mu - \mathcal{B})\psi=0$ be the Dirac equation. We prove that a non-quantized Bose field $\mathcal{B}$ is a functional of the Dirac field $\psi$, whenever this $\psi$ is strictly canonical. Performing the trivial verification for the $\mathcal{B} := m = $ constant which yields the free Dirac field, we also prepare the tedious verifications for all $\mathcal{B}$ which are non-quantized and static. Such verifications must not be confused, however, with the easy and rigorous proof of our formula, which is shown in detail. 
  New non-abelian supersymmetric generalizations of the four-dimensional Born-Infeld action are constructed in N=1 and N=2 superspace, to all orders in the gauge superfield strength. The proposed actions are dictated by simple (manifestly supersymmetric and gauge-covariant) non-linear constraints. 
  The quantum theory of near horizon regions of spacetimes with classical spatially flat, homogeneous and isotropic Friedman-Robertson-Walker geometry can be approximately described by a two dimensional conformal field theory. The central charge of this theory and expectation value of its Hamiltonian are both proportional to the horizon area in units of Newton's constant. The statistical entropy of horizon states, which can be calculated using two dimensional state counting methods, is proportional to the horizon area and depends on a numerical constant of order unity which is determined by Planck scale physics. This constant can be fixed such that the entropy is equal to a quarter of the horizon area in units of Newton's constant, in agreement with thermodynamic considerations. 
  We discuss the gauge invariance and "mass" of the Rarita-Schwinger field in a background spacetime which is assumed to be Einstein but not necessarily Ricci-flat. 
  We study the quantum analogue of primary fields and their descendants on fuzzy AdS_2, proposed in hep-th/0004072. Three-point vertices are calculated and shown to exhibit the conventional 1/N expansion as well as nonperturtive effects in large N, thus providing a strong consistency check of the fuzzy AdS_2 model. A few new physical motivations for this model are also presented. 
  We discuss a simple procedure for computing one-loop quantum energies of any static field configuration that depends non-trivially on only a single spatial coordinate. We specifically focus on domain wall-type field configurations that connect two distinct minima of the effective potential, and may or may not be the solutions of classical field equations. We avoid the conventional summation of zero-point energies, and instead exploit the relation between functional determinants and solutions of associated differential equations. This approach allows ultraviolet divergences to be easily isolated and extracted using any convenient regularization scheme. Two examples are considered: two-dimensional $\phi^4$ theory, and three-dimensional scalar electrodynamics with spontaneous symmetry breaking at the one-loop level. 
  The bosonic sectors of the eleven dimensional and IIA supergravity theories are derived as non-linear realisations. The underlying group includes the conformal group, the general linear group and as well as automorphisms of the supersymmetry algebra. We discuss the supersymmetric extension and argue that Osp(1/64) is a symmetry of M theory. We also derive the effective action of the closed bosonic string as a non-linear realisation. 
  We consider a conformal system of a string and a particle defined in D=10+2 space-time dimensions. The extra time-like dimension is a gauge artifact and can be eliminated by choosing a gauge in which the SO(10,1) Lorentz symmetry is manifest. The effective theory of string observables is the 11d supergravity. The same theory compactified on T^2 provides a non-perturbative unified picture of the Type IIA, Type IIB and 11d supergravity. This is confirmed by explicit determination of the R^4-terms which are finite and manifestly SL(2,Z) invariant as expected by the U-duality conjecture in nine non-compact dimensions with maximal supersymmetry. 
  We show that the noncommutative Wess-Zumino model is renormalizable to all orders of perturbation theory. The noncommutative scalar potential by itself is non-renormalizable but the Yukawa terms demanded by supersymmetry improve the situation turning the theory into a renormalizable one. As in the commutative case, there are neither quadratic nor linear divergences. Hence, the IR/UV mixing does not give rise to quadratic infrared poles. 
  We study the q-deformed fuzzy sphere, which is related to D-branes on SU(2) WZW models, for both real q and q a root of unity. We construct for both cases a differential calculus which is compatible with the star structure, study the integral, and find a canonical frame of one-forms. We then consider actions for scalar field theory, as well as for Yang-Mills and Chern-Simons-type gauge theories. The zero curvature condition is solved. 
  We suggest that quantum mechanics and gravity are intimately related. In particular, we investigate the quantum Hamilton-Jacobi equation in the case of two free particles and show that the quantum potential, which is attractive, may generate the gravitational potential. The investigation, related to the formulation of quantum mechanics based on the equivalence postulate, is based on the analysis of the reduced action. A consequence of this approach is that the quantum potential is always non-trivial even in the case of the free particle. It plays the role of intrinsic energy and may in fact be at the origin of fundamental interactions. We pursue this idea, by making a preliminary investigation of whether there exists a set of solutions for which the quantum potential can be expressed with a gravitational potential leading term which alone would remain in the limit hbar \to 0. A number of questions are raised for further investigation. 
  We count the supersymmetric bound states of many distinct BPS monopoles in N=4 Yang-Mills theories and in pure N=2 Yang-Mills theories. The novelty here is that we work in generic Coulombic vacua where more than one adjoint Higgs fields are turned on. The number of purely magnetic bound states is again found to be consistent with the electromagnetic duality of the N=4 SU(n) theory, as expected. We also count dyons of generic electric charges, which correspond to 1/4 BPS dyons in N=4 theories and 1/2 BPS dyons in N=2 theories. Surprisingly, the degeneracy of dyons is typically much larger than would be accounted for by a single supermultiplet of appropriate angular momentum, implying many supermutiplets of the same charge and the same mass. 
  We modify and extend an earlier proposal by Brown and Teitelboim to relax the effective cosmological term by nucleation of branes coupled to a three-index gauge potential. Microscopic considerations from string/M theory suggest two major innovations in the framework. First, the dependence of brane properties on the compactification of extra dimensions may generate a very small quantized unit for jumps in the effective cosmological term. Second, internal degrees of freedom for multiply coincident branes may enhance tunneling rates by exponentially large density of states factors. These new features essentially alter the relaxation dynamics. By requiring stability on the scale of the lifetime of the universe, rather than absolute stability, we derive a non-trivial relation between the supersymmetry breaking scale and the value of the cosmological term. It is plausibly, though not certainly, satisfied in Nature. 
  The issues related to the integrability of quantum Calogero-Moser models based on any root systems are addressed. For the models with degenerate potentials, i.e. the rational with/without the harmonic confining force, the hyperbolic and the trigonometric, we demonstrate the following for all the root systems: (i) Construction of a complete set of quantum conserved quantities in terms of a total sum of the Lax matrix (L), i.e. (\sum_{\mu,\nu\in{\cal R}}(L^n)_{\mu\nu}), in which ({\cal R}) is a representation space of the Coxeter group. (ii) Proof of Liouville integrability. (iii) Triangularity of the quantum Hamiltonian and the entire discrete spectrum. Generalised Jack polynomials are defined for all root systems as unique eigenfunctions of the Hamiltonian. (iv) Equivalence of the Lax operator and the Dunkl operator. (v) Algebraic construction of all excited states in terms of creation operators. These are mainly generalisations of the results known for the models based on the (A) series, i.e. (su(N)) type, root systems. 
  Liouville integrability of classical Calogero-Moser models is proved for models based on any root systems, including the non-crystallographic ones. It applies to all types of elliptic potentials, i.e. untwisted and twisted together with their degenerations (hyperbolic, trigonometric and rational), except for the rational potential models confined by a harmonic force. 
  We summarize the Hopf algebra structure on Feynman diagrams and emphasize the interest in further algebraic structures hidden in Feynman graphs. 
  The basics of RG equations for generic partition functions are briefly reviewed, keeping in mind an application to the Polyakov-de Boer-Verlindes description of the holomorphic RG flow. 
  We describe a simple volcano potential, which is supersymmetric and has an analytic, zero-energy, ground state. (The KK modes are also analytic.) It is an interior harmonic oscillator potential properly matched to an exterior angular momentum-like tail. Special cases are given to elucidate the physics, which may be intuitively useful in studies of higher-dimensional gravity. 
  We consider a bound state problem for a family of supersymmetric gauge theories with fundamental matter. These theories can be obtained by a dimensional reduction of supersymmetric QCD from three dimensions to 1+1 and subsequent truncation of some of the fields. We find that the models without adjoint scalar converge to well-defined continuum limits and calculate the resulting spectra of these theories. We also find the critical value of coupling at which an additional massless state is observed. By contrast, the models containing adjoint scalars, seem to have a continuous mass spectrum in the limit of infinite volume. 
  We study worldsheet and spacetime properties of the p-p' (p < p') open string system with constant B_{ij} field viewed from the Dp' brane. The description of this system in terms of the CFT with spin and twist fields leads us to consider the renormal ordering procedure from the SL(2,R) invariant vacuum to the oscillator vacuum. We compute the attendant two distinct superspace two-point functions as well as their difference (the subtracted two-point function). These bring us an integral (Koba-Nielsen) representation for the multiparticle tree scattering amplitudes consisting of N-2 vectors and two tachyons. We evaluate them explicitly for the N=3,4 cases. Several novel features are observed which include a momentum dependent multiplicative factor to each external vector leg and the emergence of a symplectic tensor multiplying the polarization vectors. In the zero slope limit, the principal parts of the amplitudes translate into a noncommutative field theory in p'+1 dimensions in which a scalar field decaying exponentially in (p'-p) dimensions and a noncommutative U(1) gauge field interact via the minimal coupling and a new interaction. A large number of nearly massless states noted before are shown to propagate in the t-channel. 
  Wigner's method of induced representations is applied to the N=1 super-Poincare group, and by using a state corresponding to the basic vector of the little group as a Clifford vacuum we show that the spin operator of a supersymmetric point particle obeys Wigner's constraints. As dynamical variables for the particle we use canonical coordinates on the symmetry group manifold. The physical phase space is then constructed using a vielbein formalism. We find that the Casalbuoni-Brink-Schwarz superparticle appears as a special case of our general construction. Finally, the theory is reformulated as a gauge theory where the gauge freedom corresponds to the choice of spin constraints or, equivalently, the free choice of relativistic center of mass. In a special case the gauge symmetry reduces to the well known kappa-symmetry. 
  Chiral symmetry breaking in the Nambu-Jona-Lasinio model in a constant magnetic field is studied in spacetimes of dimension D > 4. It is shown that a constant magnetic field can be characterized by [(D-1)/2] parameters. For the maximal number of nonzero field parameters, we show that there is an effective reduction of the spacetime dimension for fermions in the infrared region D $\to$ 1 + 1 for even-dimensional spacetimes and D $\to$ 0 + 1 for odd-dimensional spacetimes. Explicit solutions of the gap equation confirm our conclusions. 
  It has been known for some time that topological geons in quantum gravity may lead to a complete violation of the canonical spin-statistics relation : there may exist no connection between spin and statistics for a pair of geons. We present an algebraic description of quantum gravity in (2 + 1)d based on the first order formalism of general relativity and show that, although the usual spin-statistics theorem is not valid, statistics is completely determined by spin. Hence, a new spin-statistics theorem can be formulated. 
  Some problems related to construction of the epsilon-expansion of dimensionally regulated Feynman integrals are discussed. For certain classes of diagrams, an arbitrary term of the epsilon-expansion can be expressed in terms of log-sine integrals related to the polylogarithms. It is shown how the analytic continuation of these functions can be constructed in terms of the generalized Nielsen polylogarithms. 
  It is commonly believed that small black holes in AdS_5 x S^5 can be described by the ten dimensional Schwarzschild solution. This requires that the self-dual five-form (which is nonzero in the background) does not fall through the horizon and cause the black hole to grow. We verify that this is indeed the case: There are static solutions to the five-form field equations in a ten dimensional Schwarzschild spacetime. Similar results hold for other backgrounds AdS_p x S^q of interest in supergravity. 
  The Cattaneo-Felder path integral form of the perturbative Kontsevich deformation quantization formula is used to explicitly demonstrate the existence of nonperturbative corrections to na\"\i ve deformation quantization. 
  Inspiring ourselves by the assumption that the notion of symmetry itself is insufficient to construct the consistent physics of the Desert (the so-called region of energies beyond the Standard Model) and some additional insights are needed, we suggests that high-energy theories must take into account the higher-order variations of fields. With this in mind we propose the generalization of the concepts of kinetic energy and free particle. It is shown that the theory founded on such principles reveals major features of the genuinely high-energy one, first of all, it appears to be free from ultra-violet divergences, even the self-energy loop terms become finite. Also we discuss other arising interesting phenomena such as the high-gradient currents and charges, unified ``all-in-one'' multi-mass states, VR symmetry, regularization-without-renormalization of SM, etc. 
  A manifestly super-Poincar\'e covariant formalism for the superstring has recently been constructed using a pure spinor variable. Unlike the covariant Green-Schwarz formalism, this new formalism is easily quantized with a BRST operator and tree-level scattering amplitudes have been evaluated in a manifestly covariant manner.   In this paper, the cohomology of the BRST operator in the pure spinor formalism is shown to give the usual light-cone Green-Schwarz spectrum. Although the BRST operator does not directly involve the Virasoro constraint, this constraint emerges after expressing the pure spinor variable in terms of SO(8) variables. 
  We study the $C_{n}$ and $BC_{n}$ Ruijsenaars-Schneider(RS) models with interaction potential of trigonometric and rational types. The Lax pairs for these models are constructed and the involutive Hamiltonians are also given. Taking nonrelativistic limit, we also obtain the Lax pairs for the corresponding Calogero-Moser systems. 
  We discuss a relation between bicomplexes and integrable models, and consider corresponding noncommutative (Moyal) deformations. As an example, a noncommutative version of a Toda field theory is presented. 
  We construct quantum evolution operators on the space of states, that is represented by the vertices of the n-dimensional unit hypercube. They realize the metaplectic representation of the modular group SL(2,Z(2^n)). By construction this representation acts in a natural way on the coordinates of the non-commutative 2-torus,T^2, and thus is relevant for noncommutative field theories as well as theories of quantum space-time. 
  We discuss the evolution of gravitational waves in a brane-world cosmology embedded in five-dimensional anti-de Sitter spacetime. We show that during slow-roll inflation, modelled as a period of quasi-de Sitter expansion on the brane, there is a discrete normalizable massless graviton mode. There is a mass gap due to the expansion, above which there is a continuum of massive modes. Only the massless mode is `light' compared with the Hubble scale during inflation, leading to the production of classical perturbations on large scales from vacuum fluctuations on small scales. We calculate the amplitude of these fluctuations at horizon-crossing and show that the standard four-dimensional result is recovered at low energies, but the amplitude of the perturbations is enhanced at high energies. 
  We discuss the mass-deformed N=4 SU(N) supersymmetric Yang-Mills theory (also known as the N=1* theory). We analyze how the correlation functions of this theory transform under S-duality, and which correlation functions depend holomorphically on the complexified gauge coupling \tau. We provide exact modular-covariant expressions for the vacuum expectation values of chiral operators in the massive vacua of the N=1* theory. We exhibit a novel modular symmetry of the chiral sector of the theory in each vacuum, which acts on the coupling ${\tilde \tau}= (p\tau+k)/q$, where p, k and q are integers which label the different vacua. In the strong coupling limit, we compare our results to the results of Polchinski and Strassler in the string theory dual of this theory, and find non-trivial agreement after operator mixings are taken into account. In particular we find that their results are consistent with the predicted modular symmetry in ${\tilde \tau}$. Our results imply that certain singularities found in solutions to five dimensional gauged supergravity should not be resolvable in string theory, since there are no field theory vacua with corresponding vacuum expectation values in the large N limit. 
  We generalize, to any space-time dimension, the unitarity bounds of highest weight UIR's of the conformal groups with Lie algebras $so(2,d)$. We classify gauge theories invariant under $so(2,d)$, both integral and half-integral spins. A similar analysis is carried out for the algebras $so^*(2n)$. We study new unitary modules of the conformal algebra in $d>4$, that have no analogue for $d\leq 4$ as they cannot be obtained by "squaring" singletons. This may suggest the interpretation of higher dimensional non-trivial conformal field theories as theories of "tensionless" $p$-branes of which tensionless strings in $d=6$ are just particular examples. 
  This paper is devoted to a systematic discussion of the supersymmetric index Tr (-1)^F for the minimal supersymmetric Yang-Mills theory -- with any simple gauge group G -- primarily in four spacetime dimensions. The index has refinements that probe confinement and oblique confinement and the possible spontaneous breaking of chiral symmetry and of global symmetries, such as charge conjugation, that are derived from outer automorphisms of the gauge group. Predictions for the index and its refinements are obtained on the basis of standard hypotheses about the infrared behavior of gauge theories. The predictions are confirmed via microscopic calculations which involve a Born-Oppenheimer computation of the spectrum as well as mathematical formulas involving triples of commuting elements of G and the Chern-Simons invariants of flat bundles on the three-torus. 
  We examine the low-energy dynamics of four-dimensional supersymmetric gauge theories and calculate the values of the gluino condensate for all simple gauge groups. By initially compactifying the theory on a cylinder we are able to perform calculations in a controlled weakly-coupled way for small radius. The dominant contributions to the path integral on the cylinder arise from magnetic monopoles which play the role of instanton constituents. We find that the semi-classically generated superpotential of the theory is the affine Toda potential for an associated twisted affine algebra. We determine the supersymmetric vacua and calculate the values of the gluino condensate. The number of supersymmetric vacua is equal to c_2, the dual Coxeter number, and in each vacuum the monopoles carry a fraction 1/c_2 of topological charge. As the results are independent of the radius of the circle, they are also valid in the strong coupling regime where the theory becomes decompactified. In this way we obtain values for the gluino condensate which for the classical gauge groups agree with previously known ``weak coupling instanton'' expressions (but not with the ``strong coupling instanton'' calculations). This detailed agreement provides further evidence in favour of the recently advocated resolution of the the gluino condensate puzzle. We also make explicit predictions for the gluino condensate for the exceptional groups. 
  The present review aims both to offer some motivations and mathematical prerequisites for a study of NCG from the viewpoint of a theoretical physicist and to show a few applications to matrix theory and results obtained.   Lectures given by the author at the TMR School on contemporary string theory and brane physics, 26 Jan--2 Feb 2000, Torino. 
  In this paper we study the interplay of electric and magnetic backgrounds in determining the decoupling limit of coincident D-branes towards a noncommutative Yang-Mills (NCYM) or open string (NCOS) theory. No decoupling limit has been found for NCYM with space-time noncommutativity. It is suggested that there is a new duality, which we call V-duality, which acts on NCOS with both space-space and space-time noncommutativity, resulting from decoupling in Lorentz-boost related backgrounds. We also show that the holographic correspondence, previously suggested by Li and Wu, between NCYM and its supergravity dual can be generalized to NCOS as well. 
  We show that the one loop amplitudes of open and closed string theory in a constant background two-form tensor field are characterized by an effective string tension larger than the fundamental string tension, and by the appearance of antisymmetric and symmetric noncommutativity parameters. We derive the form of the phase functions normalizing planar and nonplanar tachyon scattering amplitudes in this background, verifying the decoupling of the closed string sector in the regime of infinite momentum transfer. We show that the functional dependence of the phase functions on the antisymmetric star product of external momenta permits interpretation as a finite wavefunction renormalization of vertex operators in the open string sector. Using world-sheet duality we clarify the regimes of finite and zero momentum transfer between boundaries, demonstrating the existence of poles in the nonplanar amplitude when the momentum transfer equals the mass of an on-shell closed string state. Neither noncommutativity parameter has any impact on the renormalizability of open and closed string theory in the Wilsonian sense. We comment on the relationship to noncommutative scalar field theory and the UV-IR correspondence. 
  We consider the non-relativistic effective field theory of ``extreme black holes'' in the Einstein-Maxwell-dilaton theory with an arbitrary dilaton coupling. We investigate finite-temperature behavior of gas of ``extreme black holes'' using the effective theory. The total energy of the classical many-body system is also derived. 
  In this paper, we investigate the beta-function of the gauge coupling constant ($e$) of the gauged four-fermi theory in the Exact Renormalization Group (ERG) framework. It seems that the presence of the four-fermi interaction strongly affects to the naive RG running of the gauge coupling constant. We show that this strong correction has no physical meaning since the vertex $\app A\SR\psi$ involves the contribution from a mixing as well as a pure gauge interaction due to the (anomalous) mixing among the photon and the vector composite field. By introducing the auxiliary field for the vector composite field, the situation turns to be rather clear. We adopt the counterterm to cancel the gauge non-invariant correction, and decompose the pure gauge interaction from the contribution of the mixing. We find the beta-function of the gauge coupling constant in the large $N$ limit. 
  We elaborate on a new N=2, d=3 supermultiplet (double-vector multiplet) with a non-trivial off-shell realization of the central charge. Its bosonic sector comprises two abelian gauge vector fields forming an SO(2) vector. We present a superfield formulation of this multiplet in the central-charge extended N=2, d=3 superspace and then employ it, in the framework of the nonlinear realizations approach, as the Goldstone one for the partial breaking N=4 -> N=2 in three dimensions. The covariant equations of motion for the self-interacting Goldstone superfield arise as a natural generalization of the free ones and are interpreted as the worldvolume supersymmetric form of the equations of motion of a N=4 D2-brane. For the vector fields we find a coupled nonlinear system of Born-Infeld type and demonstrate its dual equivalence to the d=5 membrane. The double-vector multiplet can be fused with some extra N=2, d=3 multiplet to form an off-shell N=4, d=3 supermultiplet. 
  We discuss the general structure of the non-abelian Born-Infeld action, together with all of the alpha-prime derivative corrections, in flat D-dimensional space-time. More specifically, we show how the connection between open strings propagating in background magnetic fields and gauge theories on non-commutative spaces can be used to constrain the form of the effective action for the massless modes of open strings at week coupling. In particular, we exploit the invariance in form of the effective action under a change of non-commutativity scale of space-time to derive algebraic equations relating the various terms in the alpha- prime expansion. Moreover, we explicitly solve these equations in the simple case D=2, and we show, in particular, how to construct the minimal invariant derivative extension of the NBI action. 
  We give a microscopic explanation for the recently observed equivalence among thermodynamics of supergravity solutions for Dp-branes with or without NS B-field and for D(p-2)-branes with vanishing B-field and two delocalized transverse directions by showing that these D-brane configurations are related to one another through T-duality transformations. This result also gives an evidence for the equivalence among the noncommutative and the ordinary Yang-Mills theories corresponding to the decoupling limits of the worldvolume theories of such D-brane configurations. 
  An approach to scattering theory in three dimensional AdS spaces is proposed. Firstly we consider the scattering of spinless relativistic particles by a three dimensional extremal black hole and compute the absorption cross section $\sigma_{abs} =J_{abs}/J_{\infty}$ without using {\it in} and {\it out} states. Secondly, we posit a reciprocal space ${\cal H}$ where {\it in} and {\it out} states and the scattering amplitude is defined as in usual scattering theory. We show that both descriptions are equivalent and ${\cal H}$ could be considered as the space where the scattering processes in AdS should be defined. 
  Extended BRS symmetry is used to prove gauge independence of the fermion renormalization constant $Z_2$ in on-shell QED renormalization schemes. A necessary condition for gauge independence of $Z_2$ in on-shell QCD renormalization schemes is formulated. Satisfying this necessary condition appears to be problematic at the three-loop level in QCD. 
  We discuss the properties of matter in a D-dimensional anti-de-Sitter-type space time induced dynamically by the recoil of a very heavy D(irichlet)-particle defect embedded in it. The particular form of the recoil geometry, which from a world-sheet view point follows from logarithmic conformal field theory deformations of the pertinent sigma-models, results in the presence of both infrared and ultraviolet (spatial) cut-offs. These are crucial in ensuring the presence of mass gaps in scalar matter propagating in the D-particle recoil space time. The analogy of this problem with the Liouville-string approach to QCD, suggested earlier by John Ellis and one of the present authors, prompts us to identify the resulting scalar masses with those obtained in the supergravity approach based on the Maldacena's conjecture, but without the imposition of any supersymmetry in our case. Within reasonable numerical uncertainties, we observe that agreement is obtained between the two approaches for a particular value of the ratio of the two cut-offs of the recoil geometry. Notably, our approach does not suffer from the ambiguities of the supergravity approach as regards the validity of the comparison of the glueball masses computed there with those obtained in the continuum limit of lattice gauge theories. 
  We study the non-critical space-time non-commutative open string (NCOS) theory using a dual supergravity description in terms of a certain near-horizon limit of the F1-Dp bound state. We find the thermodynamics of NCOS theory from supergravity. The thermodynamics is equivalent to Yang-Mills theory on a commutative space-time. We argue that this fact does not have to be in contradiction with the expected Hagedorn behaviour of NCOS theory. To support this we consider string corrections to the thermodynamics. We also discuss the relation to Little String Theory in 6 dimensions. 
  In addition to briefly reviewing recent progress in studying black hole physics in string/M theory, we describe several robust features pertaining to spacetime physics that one can glean by studying quantum physics of black holes. In particular, we review 't Hooft's S-matrix ansatz which results in a noncommutative horizon. A recent construction of fuzzy AdS2 is emphasized, this is a nice toy model for fuzzy black hole horizon. We demonstrate that this model captures some nonperturbative features of quantum gravity. 
  Supersymmetric nonlinear sigma models are formulated as gauge theories. Auxiliary chiral superfields are introduced to impose supersymmetric constraints of F-type. Target manifolds defined by F-type constraints are always non-compact. In order to obtain nonlinear sigma models on compact manifolds, we have to introduce gauge symmetry to eliminate the degrees of freedom in non-compact directions. All supersymmetric nonlinear sigma models defined on the hermitian symmetric spaces are successfully formulated as gauge theories. 
  We discuss quantum tunneling between classically BPS saturated solitons in two-dimensional theories with N=2 supersymmetry and a compact space dimension. Genuine BPS states form shortened multiplets of dimension two. In the models we consider there are two degenerate shortened multiplets at the classical level, but there is no obstruction to pairing up through quantum tunneling. The tunneling amplitude in the imaginary time is described by instantons. We find that the instanton is nothing but the 1/4 BPS saturated ``wall junction,'' considered previously in the literature in other contexts. Two central charges of the superalgebra allow us to calculate the instanton action without finding the explicit solution (it is checked, though, numerically, that the saturated solution does exist). We present a quantum-mechanical interpretation of the soliton tunneling. 
  The Riemann normal coordinate expansion method is generalized to a Kahler manifold. The Kahler potential and holomorphic coordinate transformations are used to define a normal coordinate preserving the complex structure. The existence of this Kahler normal coordinate is shown explicitly to all orders. The formalism is applied to background field methods in supersymmetric nonlinear sigma models. 
  We discuss the restructuring of the BPS spectrum which occurs on certain submanifolds of the moduli/parameter space -- the curves of the marginal stability (CMS) -- using quasiclassical methods. We argue that in general a `composite' BPS soliton swells in coordinate space as one approaches the CMS and that, as a bound state of two `primary' solitons, its dynamics in this region is determined by non-relativistic supersymmetric quantum mechanics. Near the CMS the bound state has a wave function which is highly spread out. Precisely on the CMS the bound state level reaches the continuum, the composite state delocalizes in coordinate space, and restructuring of the spectrum can occur. We present a detailed analysis of this behavior in a two-dimensional N=2 Wess-Zumino model with two chiral fields, and then discuss how it arises in the context of `composite' dyons near weak coupling CMS curves in N=2 supersymmetric gauge theories. We also consider cases where some states become massless on the CMS. 
  I consider infinitesimal translations $x'^{\alpha}=x^{\alpha}+\delta x^{\alpha}$ and demand that Noether's approach gives a symmetric electromagnetic energy-momentum tensor as it is required for gravitational sources. This argument determines the transformations of the electromagnetic potentials under infinitesimal translations to be $A'_{\gamma} (x') = A_{\gamma}(x)+\partial_{\gamma} [\delta x_{\beta} A^{\beta}(x)]$, which differs from the usually assumed invariance $A'_{\gamma} (x') = A_{\gamma}(x)$, by the gauge transformation $\partial_{\gamma} [\delta x_{\beta} A^{\beta}(x)]$. 
  We discuss the implications of multi-brane constructions involving combinations of positive and negative tension brane and show how anomalously light KK states emerge when negative tension ''-'' branes are sandwiched between ''+'' branes. We present a detailed study of a ''+--+'' brane assignment which interpolates between two models that have been previously proposed in which gravity is modified at large scales due to the anomalously light states. We show that it has the peculiar characteristic that gravity changes from four dimensional (4D) to 5D at large distances and returns to 4D at even larger scales. We also consider a crystalline universe which leads to a similar structure for gravity. The problems associated with intermediate negative tension branes are discussed and a possible resolution suggested. 
  This is a review of the constrained dynamical structure of Poincare gauge theory which concentrates on the basic canonical and gauge properties of the theory, including the identification of constraints, gauge symmetries and conservation laws. As an interesting example of the general approach we discuss the teleparallel formulation of general relativity. 
  We study string action with multiplet of $\Theta$-terms added, which turns out to be closely related with the bosonic sector of D=11 superstring action [3,4]. Alternatively, the model can be considered as describing class of special solutions of the membrane. An appropriate set of variables is find, in which the light-cone quantization turns out to be possible. It is shown that anomaly terms in the algebra of the light-cone Poincare generators are absent for the case D=27. 
  A new class of six-dimensional asymmetric orientifolds is considered where the orientifold operation is combined with T-duality. The models are supersymmetric in the bulk, but the cancellation of the tadpoles requires the introduction of brane configurations that break supersymmetry. These can be described by D7-brane anti-brane pairs, non-BPS D8-branes or D9-brane anti-brane pairs. The transition between these different configurations and their stability is analysed in detail. 
  We analyze the algebraic constraints of the generalized vielbein in SO(1,2) x SO(16) invariant d=11 supergravity, and show that the bosonic degrees of freedom of d=11 supergravity, which become the physical ones upon reduction to d=3, can be assembled into an E_8-valued vielbein already in eleven dimensions. A crucial role in the construction is played by the maximal nilpotent commuting subalgebra of E_8, of dimension 36, suggesting a partial unification of general coordinate and tensor gauge transformations. 
  We study the near-horizon AdS_2\timesS^2 geometry of evaporating near-extremal Reissner-Nordstrom black holes interacting with null matter. The non-local (boundary) terms t_{\pm}, coming from the effective theory corrected with the quantum Polyakov-Liouville action, are treated as dynamical variables. We describe analytically the evaporation process which turns out to be compatible with the third law of thermodynamics, i.e., an infinite amount of time is required for the black hole to decay to extremality. Finally we comment briefly on the implications of our results for the information loss problem. 
  A gauge theory with 4 physical dimensions can be consistently expressed as a renormalizable topological quantum field theory in 5 dimensions. We extend the symmetries in the 5-dimensional framework to include not only a topological BRST operator S that encodes the invisibility of the "bulk" (the fifth dimension), but also a gauge BRST operator W that encodes gauge-invariance and selects observables. These symmetries provide a rich structure of Ward identities which assure the renormalizability of the theory, including non-renormalization theorems. The 5-dimensional approach considerably simplifies conceptual questions such as for instance the Gribov phenomenon and fermion doubling. A confinement scenario in the 5-dimensional framework is sketched. We detail the five-dimensional mechanism of anomalies, and we exhibit a natural lattice discretization that is free of fermion doubling. 
  We present two examples of non-trivial field theories which are scale invariant, but not conformally invariant. This is done by placing certain field theories, which are conformally invariant in flat space, onto curved backgrounds of a specific type. We define this using the AdS/CFT correspondence, which relates the physics of gravity in asymptotically Anti-de Sitter (AdS) spacetimes to that of a conformal field theory (CFT) in one dimension fewer. The AdS rotating (Kerr) black holes in five and seven dimensions provide us with the examples, since by the correspondence we are able to define and compute the action and stress tensor of four and six dimensional field theories residing on rotating Einstein universes, using the ``boundary counterterm'' method. The rotation breaks conformal but not scale invariance. The AdS/CFT framework is therefore a natural arena for generating such examples of non-trivial scale invariant theories which are not conformally invariant. 
  We investigate geometrical structures and low-energy theorems of N=1 supersymmetric nonlinear sigma models in four dimensions. When a global symmetry spontaneously breaks down to its subgroup, the low-energy effective Lagrangian of massless particles is described by a supersymmetric nonlinear sigma model whose target manifold is parametrized by Nambu-Goldstone (NG) bosons and quasi-NG (QNG) bosons. The unbroken symmetry changes at each point in the target manifold and some QNG bosons change to NG bosons when unbroken symmetry become smaller. The QNG-NG change and their interpretation is shown in a simple example, the O(N) model. We investigate low-energy theorems at general points. 
  A model describing N particles on a line interacting pairwise via an elliptic function potential in the presence of an external field is partially solved in the quantum case in a totally algebraic way. As an example, the ground state and the lowest excitations are calculated explicitly for N=2. 
  When quantum fields are studied on manifolds with boundary, the corresponding one-loop quantum theory for bosonic gauge fields with linear covariant gauges needs the assignment of suitable boundary conditions for elliptic differential operators of Laplace type. There are however deep reasons to modify such a scheme and allow for pseudo-differential boundary-value problems. When the boundary operator is allowed to be pseudo-differential while remaining a projector, the conditions on its kernel leading to strong ellipticity of the boundary-value problem are studied in detail. This makes it possible to develop a theory of one-loop quantum gravity from first principles only, i.e. the physical principle of invariance under infinitesimal diffeomorphisms and the mathematical requirement of a strongly elliptic theory. It therefore seems that a non-local formulation of quantum field theory has some attractive features which deserve further investigation. 
  We study solutions of the one-loop beta-functions of the critical bosonic string theory in the framework of the Renormalization Group (RG) approach to string theory, considering explicitly the effects of the 21 extra dimensions. In the RG approach the 26-dimensional manifold is given in terms of a four dimensional Minkowski spacetime times R and a 21-dimensional hyper-plane. In calculating the Wilson loops, as it is wellknown for this kind of confining geometry, two phenomena appear: confinement and over-confinement. There is a critical minimal surface below of which it leads to confinement only. The role of the extra dimensions is understood in terms of a dimensionless scale l provided by them. Therefore the effective string tension in the area law, the length of the Wilson loops, as well as, the size of the critical minimal surface depend on this scale. When this confining geometry is used to study a field-theory beta-function with an infrared attractive point (as the Novikov-Shifman-Vainshtein-Zakharov beta-function) the range of the couplings where the field theory is confining depends on that scale. We have explicitly calculated the l-dependence of that range. 
  We give an explicit $L^2$-representation of chiral charged fermions using the Hardy-Lebesgue octant decomposition. In the " pure" case such a representation was already used by M. Sato in holonomic field theory. We study both "pure" and " mixed" cases. In the compact case we rigorously define unsmeared chiral charged fermion operators inside the unit circle. Using chiral fermions we orient our findings towards a functional analytic study of vertex algebras as one dimensional quantum field theory. 
  We derive an RG flow equation that is satisfied by the regularized partition function for noncritical strings in background fields. The flow refers to change in the position of a ``boundary'' in the liouville direction. The boundary is required to regularize the ultraviolet divergences in the partition function coming from integration over world-sheets of arbitrarily small area.   From the point of view of the target space effective gravitational action that the partition function evaluates on-shell, the boundary regularizes {\it infrared} divergences coming from the infinite volume of the liouville direction. The RG flow equation that we obtain looks very much like the Hamilton-Jacobi constraint equation that an on-shell gauge-fixed gravitational action must satisfy. 
  We describe spontaneous creation of the Brane World in $AdS_{5}$ with external field. The resulting Brane World consists of a flat 4d spatially finite expanding Universe and curved expanding "regulator" branes. No negative tension branes are involved. 
  I consider infinitesimal translations $x'^{\alpha}=x^{\alpha}+\delta x^{\alpha}$ and demand that Noether's approach gives a symmetric energy-momentum tensor as it is required for gravitational sources. This argument determines the transformations of non-abelian gauge fields under infinitesimal translations to differ from the usually assumed invariance by the gauge transformation, $A'^a_{\gamma} (x') - A^a_{\gamma}(x) = \partial_{\gamma} [ \delta x_{\beta} A^{a \beta}(x)] + C^a_{bc} \delta x_{\beta} A^{c \beta}(x) A^{b}_{\gamma}(x)$ where the $C^a_{bc}$ are the structure constants of the gauge group. 
  In a brane (domain wall) scenario with an infinite extra dimension and localized gravity, bulk fermions and scalars often have bound states with zero 4-dimensional mass. In this way massless matter residing on the brane may be obtained. We consider what happens when one tries to introduce small, but non-vanishing mass to these matter fields. We find that the discrete zero modes turn into quasi-localized states with finite 4-dimensional mass and finite width. The latter is due to tunneling of massive matter into extra dimension. We argue that this phenomenon is generic to fields that can have bulk modes. We also point out that in theories meant to describe massive scalars, the 4-dimensional scalar potential has, in fact, power-law behavior at large distances. 
  We discuss the generation of superpotentials in four-dimensional, N = 1 supersymmetric field theories arising from type IIA D6-branes wrapped on supersymmetric three-cycles of a Calabi-Yau threefold. In general, nontrivial superpotentials arise from sums over disc instantons. We find several examples of special Lagrangian three-cycles with nontrivial topology which are mirror to obstructed rational curves, conclusively demonstrating the existence of such instanton effects. In addition, we present explicit examples of disc instantons ending on the relevant three-cycles. Finally, we give a preliminary construction of a mirror map for the open string moduli, in a large-radius limit of the type IIA compactification. 
  We study the 5 dimensional SUGRA AdS duals of N=4, N=2 and N=1 Super-Yang-Mills theories. To sequentially break the N=4 theory mass terms are introduced that correspond, via the duality, to scalar VEVs in the SUGRA. We determine the appropriate scalar potential and study solutions of the equations of motion that correspond to RG flows in the field theories. Analysis of the potential at the end of the RG flows distinguishes the flows appropriate to the field theory expectations. As already identified in the literature, the dual to the N=2 theory has flows corresponding to the moduli space of the field theory. When the N=2 theory is broken to N=1 the single flow corresponding to the singular point on the N=2 moduli space is picked out as the vacuum. As the N=2 breaking mass scale is increased the vacuum deforms smoothly to previously analysed N=1 flows. 
  We investigate four-dimensional N=1 Type IIB orientifolds with continuous Wilson lines, and their T-dual realizations as orientifolds with moving branes. When continuous Wilson lines become discrete the gauge symmetry is enhanced and the T-dual orientifold corresponds to branes sitting at the orbifold fixed points. There is a field theoretic analog describing these phenomena as D- and F-flat deformations of the T-dual model, where the branes sit at the origin (original model without Wilson lines) as well as a deformation of the T-dual model where sets of branes sit at the fixed points (the model with discrete Wilson lines). We demonstrate these phenomena for the prototype Z_3 orientifold: we present an explicit construction of the general set of continuous Wilson lines as well as their explicit field theoretic realization. 
  We study Bogomolny equations on $R^2\times S^1$. Although they do not admit nontrivial finite-energy solutions, we show that there are interesting infinite-energy solutions with Higgs field growing logarithmically at infinity. We call these solutions periodic monopoles. Using Nahm transform, we show that periodic monopoles are in one-to-one correspondence with solutions of Hitchin equations on a cylinder with Higgs field growing exponentially at infinity. The moduli spaces of periodic monopoles belong to a novel class of hyperk\"ahler manifolds and have applications to quantum gauge theory and string theory. For example, we show that the moduli space of $k$ periodic monopoles provides the exact solution of ${\cal N}=2$ super Yang-Mills theory with gauge group $SU(k)$ compactified on a circle of arbitrary radius. 
  We present a gauge independent method to construct the effective action of QCD, and calculate the one loop effective action of $SU(2)$ QCD in an arbitrary constant background field. Our result establishes the existence of a dynamical symmetry breaking by demonstrating that the effective potential develops a unique and stable vacuum made of the monopole condensation in one loop approximation. This provides a strong evidence for the magnetic confinement of color through the dual Meissner effect in the non-Abelian gauge theory. The result is obtained by separating the topological degrees which describe the non-Abelian monopoles from the dynamical degrees of the gauge potential, and integrating out all the dynamical degrees of QCD. We present three independent arguments to support our result. 
  We construct a model describing BPS brane-systems using low energy effective theory of brane-antibrane system. Both parallel branes and intersecting branes can be treated by this model. After tachyon condensation, the dynamics of fluctuations around such brane-systems is supersymmetric if the degrees of freedom are restricted on the branes. The form of the tachyon potential and the application of this model to the black hole physics are discussed. 
  We develop the WKB expansion to relate Quantum Field Theory variables with those describing macroscopical matter. We find that, up to the first quantum correction, free scalar fields correspond to perfect fluids with pressure. We also find the law of motion for the associated particles which takes into account deviations from the classical trajectories, showing that they do not follow unaccelerated straight line trajectories.   Keywords: Quantum Field Theory, Fluid dynamics.   PACS number (s): 11.10.-z, 03.40.Gc 
  We discuss the mechanism of reheating in pre-big bang string cosmology and we calculate the amount of moduli and gravitinos produced gravitationally and in scattering processes of the thermal bath. We find that this abundance always exceeds the limits imposed by big-bang nucleosynthesis, and significant entropy production is required. The exact amount of entropy needed depends on the details of the high curvature phase between the dilaton-driven inflationary era and the radiation era. We show that the domination and decay of the zero-mode of a modulus field, which could well be the dilaton, or of axions, suffices to dilute moduli and gravitinos. In this context, baryogenesis can be accomodated in a simple way via the Affleck-Dine mechanism and in some cases the Affleck-Dine condensate could provide both the source of entropy and the baryon asymmetry. 
  We present a finite energy topological Z(2) vortex solution in a 2+1 dimensional SO(3) gauge field theory minimally coupled to a matrix valued Higgs field. The vortex carries a Z(2) magnetic charge and obeys a modulo two addition property. The core of this vortex has a structure similar to that of the Abrikosov vortex appearing in a type II superconductor. The implications of this solution for Wilson loops are quite interesting. In two Euclidean dimensions these vortices are instantons and a dilute gas of such vortices disorders Wilson loops producing an area law behaviour with an exponentially small string tension. In 2+1 dimensions the vortices are loops and they affect the same disordering in the phase having large loops. 
  It is widely believed that via the Seiberg-Witten map, the linearly realized BPS equation in the non-commutative space is related to the non-linearly realized BPS equation in the commutative space in the zero slope limit. We show that the relation also holds without taking the zero slope limit as is expected from the arguments of the BPS equation for the non-Abelian Born-Infeld theory. This is regarded as an evidence for the relation between the two BPS equations. As a byproduct of our analysis, the non-linear instanton equation is solved exactly. 
  The one-loop effective action of QED obtained by Euler and Heisenberg and by Schwinger has been expressed by an asymptotic perturbative series which is divergent. In this letter we present a non-perturbative but convergent series of the effective action. With the convergent series we establish the existence of the manifest electric-magnetic duality in the one loop effective action of QED. 
  In this paper we explicitly show the equivalence between the non-Abelian Born-Infeld action, which was proposed by Tseytlin as an effective action on several D-branes, and its noncommutative counterpart for slowly varying fields. This confirms the equivalence between the two descriptions of the D-branes using an ordinary gauge theory with a constant B field background and a noncommutative gauge theory, claimed by Seiberg and Witten. We also construct the general forms of the 2 n-derivative terms for non-Abelian gauge fields which are consistent with the equivalence in the approximation of neglecting (2 n+2)-derivative terms. 
  We consider a recent proposal to solve the cosmological constant problem within the context of brane world scenarios with infinite volume extra dimensions. In such theories bulk can be supersymmetric even if brane supersymmetry is completely broken. The bulk cosmological constant can therefore naturally be zero. Since the volume of the extra dimensions is infinite, it might appear that at large distances one would measure the bulk cosmological constant which vanishes. We point out a caveat in this argument. In particular, we use a concrete model, which is a generalization of the Dvali-Gabadadze-Porrati model, to argue that in the presence of non-zero brane cosmological constant at large distances such a theory might become effectively four dimensional. This is due to a mass gap in the spectrum of bulk graviton modes. In fact, the corresponding distance scale is set precisely by the brane cosmological constant. This phenomenon appears to be responsible for the fact that bulk supersymmetry does not actually protect the brane cosmological constant. 
  We study a Hanany-Witten set-up relevant to N = 2 superconformal field theories. We find the exact near-horizon solution for this 11-dimensional system which involves intersecting M5-branes. The metric describes a warped product of AdS_5 with a manifold with SU(2) x U(1) isometry. 
  We present a model for a spacetime dependent cosmological {\it constant}. We make a realization of this model based on a possible quantum aspects of the initial stage of the universe and relate the cosmological constant with the chiral anomaly. 
  Open string theories can be decoupled from closed strings and gravity by scaling to the critical electric field. We propose dual descriptions for the strong coupling limit of these NCOS (Non-Commutative Open String) theories in six or fewer spacetime dimensions. In particular, we conjecture that the five-dimensionsal NCOS theory at strong coupling, is a theory of light Open Membranes (OM), decoupled from gravity, on an M5-brane with a near-critical three-form field strength. The relation of OM theory to NCOS theories resembles that of M theory to Type II closed string theories. In two dimensions we conjecture that supersymmetric U($n$) gauge theory with a unit of electric flux is dual to the NCOS theory with string coupling $1 \over n$. A construction based on NS5-branes leads to new theories in six dimensions generalising the little string theory. A web of dualities relates all the above theories when they are compactified on tori. 
  We make some observations regarding string/black hole correspondence with a view to understanding the nature of the quantum degrees of freedom of a black hole in string theory. In particular, we compare entropy change in analogous string and black hole processes in order to support the interpretation of the area law entropy as arising from stringy constituents. 
  A manifestly gauge invariant and regularized renormalization group flow equation is constructed for pure SU(N) gauge theory in the large N limit. In this way we make precise and concrete the notion of a non-perturbative gauge invariant continuum Wilsonian effective action. Manifestly gauge invariant calculations may be performed, without gauge fixing, and receive a natural interpretation in terms of fluctuating Wilson loops. Regularization is achieved by covariant higher derivatives and by embedding in a spontaneously broken SU(N|N) supergauge theory; the resulting heavy fermionic vectors are Pauli-Villars fields. We prove the finiteness of this method to one loop and any number of external gauge fields. A duality is uncovered that changes the sign of the squared coupling constant. As a test of the basic formalism we compute the one loop beta function, for the first time without any gauge fixing, and prove its universality with respect to cutoff function. 
  Following a previous work on Abelian (2,0)-gauge theories, one reassesses here the task of coupling (2,0) relaxed Yang-Mills super-potentials to a (2,0)-nonlinear Sigma-model, by gauging the isotropy or the isometry group of the latter. One pays special attention to the extra "chiral-like" component-field gauge potential that comes out from the relaxation of constraints. 
  We consider N=1 supersymmetric renormalization group flows of N=4 Yang-Mills theory from the perspective of ten-dimensional IIB supergravity. We explicitly construct the complete ten-dimensional lift of the flow in which exactly one chiral superfield becomes massive (the LS flow). We also examine the ten-dimensional metric and dilaton configurations for the ``super-QCD'' flow (the GPPZ flow) in which all chiral superfields become massive. We show that the latter flow generically gives rise to a dielectric 7-brane in the infra-red, but the solution contains a singularity that may be interpreted as a ``duality averaged'' ring distribution of 5-branes wrapped on S^2. At special values of the parameters the singularity simplifies to a pair of S-dual branes with (p,q) charge (1,\pm 1). 
  We present a N=2 supersymmetric action for the Born Infeld theory in the non abelian case. We quantize the theory in N=1 superspace and compute divergences at one-loop. The result is discussed in the N=4 case. 
  We consider a model with two parallel (positive tension) 3-branes separated by a distance $L$ in 5-dimensional spacetime. If the interbrane space is anti-deSitter, or is not precisely anti-deSitter but contains no event horizons, the effective 4-dimensional cosmological constant seen by observers on one of the branes (chosen to be the visible brane) becomes exponentially small as $L$ grows large. 
  We consider the cosmology of a pair of domain walls bounding a five-dimensional bulk space-time with negative cosmological constant, in which the distance between the branes is not fixed in time. Although there are strong arguments to suggest that this distance should be stabilized in the present epoch, no such constraints exist for the early universe and thus non-static solutions might provide relevant inflationary scenarios. We find the general solution for the standard ansatz where the bulk is foliated by planar-symmetric hypersurfaces. We show that in all cases the bulk geometry is that of anti-de Sitter (AdS_5). We then present a geometrical interpretation for the solutions as embeddings of two de Sitter (dS_4) surfaces in AdS_5, which provide a simple interpretation of the physical properties of the solutions. A notable feature explained in the analysis is that two-way communication between branes expanding away from one another is possible for a finite amount of time, after which communication can proceed in one direction only. The geometrical picture also shows that our class of solutions (and related solutions in the literature) are not completely general, contrary to some claims. We then derive the most general solution for two walls in AdS_5. This includes novel cosmologies where the brane tensions are not constrained to have opposite signs. The construction naturally generalizes to arbitrary FRW cosmologies on the branes. 
  The near boundary limit of string theory in AdS_3 is analysed using the Wakimoto free field representation of SL(2,R). The theory is considered as a direct product of the SL(2,R)/U(1) coset and a free boson. Correlation functions are constructed generalizing to the non-compact case the integral representation of conformal blocks introduced by Dotsenko in the compact SU(2) CFT. Sectors of the theory obtained by spectral flow manifestly appear. The formalism naturally leads to consider scattering processes violating winding number conservation. The consistency of the procedure is verified in the factorization limit. 
  It has been shown recently that by turning on a large noncommutativity parameter, the description of tachyon condensation in string theory can be drastically simplified. We reconsider these issues from the standpoint of string field theory, showing that, from this point of view, the key fact is that in the limit of a large B-field, the string field algebra factors as the product of an algebra that acts on the string center of mass only and an algebra that acts on all other degrees of freedom carried by the string. 
  Assuming dynamical spontaneous breakdown of chiral symmetry for massless gauge theory without scalar fields, we present a method how to construct an effective action of the dynamical Nambu-Goldstone bosons and elemetary fermions by using auxiliary fields. Here dynamical particles are asssumed to be composed of elementary fermions. Various quantities including decay constants are calculated from this effective action. This technique is also applied to gauge symmetry breakdown, $SU(5)\to SU(4)$, to obtain massive gauge fields. 
  We investigate the open string modes, describing the world-volume of a D p-brane, for its cyclic symmetry in presence of a magnetic field. It is argued that the constant coordinate modes receive non-perturbative correction. We show that they introduce the notion of noncommutativity on the D p-brane world-volume and make it UV-renormalizable. An analogy between cyclic symmetry ($\a'$-corrections) and the noncommutative geometry ($\Theta_A$-corrections) is presented to explain some of the unusual IR phenomena often noticed in a noncommutative theory. 
  The Darboux transformation applied recurrently on a Schroedinger operator generates what is called a {\em dressing chain}, or from a different point of view, a set of supersymmetric shape invariant potentials. The finite-gap potential theory is a special case of the chain. For the finite-gap case, the equations of the chain can be expressed as a time evolution of a Hamiltonian system. We apply Sklyanin's method of separation of variables to the chain. We show that the classical equation of the separation of variables is the Baxter T-Q relation after quantization. 
  We reexamine physical causal propagators for scalar and pseudoscalar bound states at finite temperature in a chiral $U_L(1)\times U_R(1)$ NJL model, defined by four-point amputated functions subtracted through the gap equation, and prove that they are completely equivalent in the imaginary-time and real-time formalism by separating carefully the imaginary part of the zero-temperature loop integral. It is shown that the thermal transformation matrix of the matrix propagators for these bound states in the real-time formalism is precisely the one of the matrix propagator for an elementary scalar particle and this fact shows similarity of thermodynamic property between a composite and an elementary scalar particle. The retarded and advanced propagators for these bound states are also given explicitly from the imaginary-time formalism. 
  The Wess-Zumino action for generalized orientifold planes (GOp-planes) is presented and a series power expantion is realized from which processes that involves GOp-planes, RR-forms, gravitons and gaugeons, are obtained. Finally non-standard GOp-planes are showed. 
  Generalized Wess-Zumino models which admit topologically non-trivial BPS saturated configurations along one compact, spatial dimension are investigated in various dimensions of space-time. We show that, in a representative model and for sufficiently large circumference, there are BPS configurations along the compact dimension containing an arbitrary number of equidistant, well-separated domain walls. We analyze the spectrum of the bosonic and fermionic light and massless modes that are localized on these walls. The masses of the light modes are exponentially suppressed by the ratio of the distance between the walls and their width. States that are initially localized on one wall oscillate in time between all the walls. In (2+1) dimensions the ``chirality'' of localized, massless fermions is determined. In the (1+1)-dimensional case we show how the mass of certain classically BPS saturated solitons is lifted above the BPS bound by instanton tunneling. 
  Spontaneous symmetry breaking is studied within a simple version of the light-front sigma model with fermions. Its vacuum structure is derived by an implementation of global symmetries in terms of unitary operators in a finite volume with periodic fermi field. Due to the dynamical fermion zero mode, the vector and axial U(1) charges do not annihilate the light-front vacuum. The latter is transformed into a continuous set of degenerate vacuum states, leading to to the spontaneous breakdown of the axial symmetry. The existence of associated massless Nambu-Goldstone boson is demonstrated. 
  For superstrings, the consequences of replacing the measure of integration $\sqrt{-\gamma}d^2 x$ in the Polyakov's action by $\Phi d^2 x$ where $\Phi$ is a density built out of degrees of freedom independent of the metric $\gamma_{ab}$ defined in the string are studied. As in Siegel reformulation of the Green Schwarz formalism the Wess-Zumino term is the square of supersymmetric currents. As opposed to the Siegel case, the compensating fields needed for this do not enter into the action just as in a total derivative. They instead play a crucial role to make up a consistent dynamics. The string tension appears as an integration constant of the equations of motion. The generalization to higher dimensional extended objects is also studied using in this case the Bergshoeff and Sezgin formalism with the associated additional fields, which again are dynamically relevant unlike the standard formulation. Also unlike the standard formulation, there is no need of a cosmological term on the world brane. 
  We study gauge properties of the general teleparallel theory of gravity, defined in the framework of Poincare gauge theory. It is found that the general theory is characterized by two kinds of gauge symmetries: a specific gauge symmetry that acts on Lagrange multipliers, and the standard Poincare gauge symmetry. The canonical generators of these symmetries are explicitly constructed and investigated. 
  We generalise the fusion procedure for the $A_{\n-1}^{(1)}$ open spin chain ($\n>2$) and we show that the transfer matrix satisfies a crossing property. We use these results to solve the $A_{\n-1}^{(1)}$ open spin chain with $U_{q} (SU(\n))$ symmetry by means of the analytical Bethe ansatz method. Our results coincide with the known ones obtained by means of the nested Bethe ansatz. 
  A relativistic two-body wave equation, local in configuration space, is derived from the Bethe-Salpeter equation for two scalar particles bound by a scalar Coulomb interaction. The two-body bound-state wave equation is solved analytically, giving a two-body Bohr-Sommerfeld formula whose energies agree with the Bethe-Salpeter equation to order alpha^4 for all quantum states. From the Bohr-Sommerfeld formula, along with the expectation values of two remaining small corrections, the energy levels of the scalar Coulomb Bethe-Salpeter equation are worked out to order alpha^6 for all states. 
  Reminiscences of Bunji Sakita. 
  We discuss the quantization of a particle near an extreme Reissner-Nordstrom black hole in the canonical formalism. This model appears to be described by a Hamiltonian with no well-defined ground state. This problem can be circumvented by a redefinition of the Hamiltonian due to de Alfaro, Fubini and Furlan (DFF). We show that the Hamiltonian with no ground state corresponds to a gauge in which there is an obstruction at the boundary of spacetime requiring a modification of the quantization rules. The redefinition of the Hamiltonian a la DFF corresponds to a different choice of gauge. The latter is a good gauge leading to standard quantization rules. Thus, the DFF trick is a consequence of a standard gauge-fixing procedure in the case of black hole scattering. 
  We study some aspects of open string theories on D-branes with critical electric fields. We show that the massless open string modes that move in the direction of the electric field decouple. In the 1+1 dimensional case the dual theory is U(N) SYM with electric flux, and the decoupling of massless open strings is dual to the decoupling of the U(1) degrees of freedom. We also show that, if the direction along the electric field is compact, then there are finite energy winding closed string modes. They are dual to Higgs branch excitations of the SYM theory, and their energetics works accordingly. These properties provide new non-trivial evidence for the duality. 
  Several distinct mechanisms of confinement and dynamical symmetry breaking (DSB) are identified, in a class of supersymmetric $SU(n_c)$, $USp(2n_c)$ and $SO(n_c)$ gauge theories. In some of the vacua, the magnetic monopoles carrying nontrivial flavor quantum numbers condense, causing confinement and symmetry breaking simultaneously. In more general classes of vacua, however, the effective low-energy degrees of freedom are found to be constituents of the monopoles - dual (magnetic) quarks. These magnetic quarks condense and give rise to confinement and DSB. We find two more important classes of vacua, one is in various universality classes of nontrivial superconformal theories (SCFT), another in free-magnetic phase. 
  We consider two model field theories on a noncommutative plane that have smooth commutative limits. One is the single-component fermion theory with quartic interaction that vanishes identically in the commutative limit. The other is a scalar-fermion theory, which extends the scalar field theory with quartic interaction by adding a fermion. We compute the bound state energies and the two particle scattering amplitudes exactly. 
  Three-dimensional string theories with 16 supersymmetries are believed to possess a non-perturbative U-duality symmetry $SO(8,24,\Zint)$. By covariantizing the heterotic one-loop amplitude under U-duality, we propose an exact expression for the $(\partial \phi)^4$ amplitude, that reproduces known perturbative limits. The weak-coupling expansion in either of the heterotic, type II or type I descriptions exhibits the well-known instanton effects, plus new contributions peculiar to three-dimensional theories, including the Kaluza-Klein 5-brane, for which we extract the summation measure. This letter is a post-scriptum to hepth/0001083. 
  The temperature phase transition in scalar $\phi^4(x)$ field theory with spontaneous symmetry breaking is investigated in a partly resummed perturbative approach. The second Legendre transform is used and the resulting gap equation is considered in the extrema of the free energy functional. It is found that the phase transition is of first order in the super daisy as well as in a certain beyond super daisy resummations. No unwanted imaginary parts in the free energy are found but a loss of the smallness of the effective expansion parameter near the phase transition temperature is found in both cases. This means an insufficiency of the resummations or a deficit of the perturbative approach. 
  We introduce a calculus of the Lie algebra valued functions present in Tseytlin's proposal for the nonabelian DBI action, and apply it to show that the recently found dyonic instanton is a solution of the full nonabelian DBI action. The geometry of this solution exhibits a new effect of ``blowing-up'' of the brane, which was not present in the case of the brane realisation of monopoles. We interpret this solution as the superposition of the D0 brane and fundamental string F1 which connects two separated D4 branes. Both F1 and D0 are delocalised in the ``blown-up'' region between two separated D4 branes. 
  We study the generalization of noncommutative gauge theories to the case of orthogonal and symplectic groups. We find out that this is possible, since we are allowed to define orthogonal and symplectic subgroups of noncommutative unitary gauge transformations even though the gauge potentials and gauge transformations are not valued in the orthogonal and symplectic subalgebras of the Lie algebra of antihermitean matrices. Our construction relies on an antiautomorphism of the basic noncommutative algebra of functions which generalizes the charge conjugation operator of ordinary field theory. We show that the corresponding noncommutative picture from low energy string theory is obtained via orientifold projection in the presence of a non-trivial NSNS B-field. 
  The exact wave functions that describe scattering of a charged particle by a confined magnetic field (Aharonov-Bohm effect) and by a Coulomb field are analyzed. It is well known that the usual procedure of finding asymptotic forms of these functions which admit a separation into a superposition of an incident plane wave and a circular or spherical scattered wave is problematic near the forward direction. It thus appears to be impossible to express the conservation of probability by means of an optical theorem of the usual kind. Both the total cross section and the forward scattering amplitude appear to be infinite. To address these difficulties we find a new representation for the asymptotic form of the Aharonov-Bohm wave function that is valid for all angles. Rather than try to define a cross section at forward angles, however, we work instead with the probability current and find that it is quite well behaved. The same is true for Coulomb scattering. We trace the usual difficulties to a nonuniformity of limits. 
  We consider deformations of the singular "global cosmic string" compactifications, known to naturally generate exponentially large scales. The deformations are obtained by allowing a constant curvature metric on the brane and correspond to a choice of integration constant. We show that there exists a unique value of the integration constant that gives rise to a nonsingular solution. The metric on the brane is dS_4 with an exponentially small value of expansion parameter. We derive an upper bound on the brane cosmological constant. We find and investigate more general singular solutions---``dilatonic global string" compactifications---and show that they can have nonsingular deformations. We give an embedding of these solutions in type IIB supergravity. There is only one class of supersymmetry-preserving singular dilatonic solutions. We show that they do not have nonsingular deformations of the type considered here. 
  Integrable boundary conditions are studied for critical A-D-E and general graph-based lattice models of statistical mechanics. In particular, using techniques associated with the Temperley-Lieb algebra and fusion, a set of boundary Boltzmann weights which satisfies the boundary Yang-Baxter equation is obtained for each boundary condition. When appropriately specialized, these boundary weights, each of which depends on three spins, decompose into more natural two-spin edge weights. The specialized boundary conditions for the A-D-E cases are naturally in one-to-one correspondence with the conformal boundary conditions of sl(2) unitary minimal conformal field theories. Supported by this and further evidence, we conclude that, in the continuum scaling limit, the integrable boundary conditions provide realizations of the complete set of conformal boundary conditions in the corresponding field theories. 
  The Wess-Zumino actions for generalized orientifold planes (GOp-planes) and y-deformed orientifold planes (yOp-planes) are presented and two series power expantions are realized from whiches processes that involves GOp-planes,yOp-planes, RR-forms, gravitons and gaugeons, are obtained. Finally non-standard GOp-planes and y-Op-planes are showed. 
  We construct a multiple star product method and by using this method, show that integral forms of some star products can be written in terms of the path-integral. This method can be applied to some examples. Especially, the associativity of the skew-symmetrized Berezin star product proposed in \cite{SW}, is recovered in large $N$ limit of the multiple star product. We also derive the path integral form of the Kontsevich star product from the multiple Moyal star product. This paper includes some reviews about star products. 
  The interaction of electrically charged particles in a dilute gas of point--like magnetic dipoles is studied. We show that the interaction potential at small distances has a linear piece due to overlap of the dipole clouds gathered near electric sources. At large distances the potential becomes of the Coulomb type with non-perturbatively renormalized charge of the test particle. The physical applications of these results are discussed. 
  The requirements of N=1 superconformal invariance for the correlation functions of chiral superfields are analysed. Complete expressions are found for the three point function for the general spin case and for the four point function for scalar superfields for \sum q_i=3 where q_i is the scale dimension for the i'th superfield and is related to the U(1) R-charge. In the latter case the relevant Ward identities reduce to eight differential equations for four functions of u,v which are invariants when the superconformal symmetry is reduced to the usual conformal group. The differential equations have a general solution given by four linearly independent expressions involving a two variable generalisation of the hypergeometric function. By considering the behaviour under permutations, or crossing symmetry, the chiral four point function is shown to be determined up to a single overall constant. The results are in accord with the supersymmetric operator product expansion. 
  Motivated by an analogy with the conformal factor problem in gravitational theories of the $R+R^2$-type we investigate a $d$-dimensional Euclidean field theory containing a complex scalar field with a quartic self interaction and with a nonstandard inverse propagator of the form $-p^2+p^4$. Nonconstant spin-wave configurations minimize the classical action and spontaneously break the rotation symmetry to a lower-dimensional one. In classical statistical physics this corresponds to a spontaneous formation of layers. Within the effective average action approach we determine the renormalization group flow of the dressed inverse propagator and of a family of generalized effective potentials for nonzero-momentum modes. Already in the leading order of the semiclassical expansion we find strong ``instability induced'' renormalization effects which are due to the fact that the naive vacuum (vanishing field) is unstable towards the condensation of modes with a nonzero momentum. We argue that the (quantum) ground state of our scalar model indeed leads to spontaneous breaking of rotation symmetry. 
  We describe a simple method for determining the strong-coupling BPS spectrum of four dimensional N=2 supersymmetric Yang-Mills theory. The idea is to represent the magnetic monopoles and dyons in terms of D-brane boundary states of a non-compact d=2 N=2 Landau-Ginzburg model. In this way the quantum truncated BPS spectrum at the origin of the moduli space can be directly mapped to the finite number of primary fields of the superconformal minimal models. 
  We calculate the scalar potential of supersymmetric QCD (in the regime N_f < N_c) coupled to N=1 supergravity with moduli-dependent gauge kinetic function and masses. The gauge dynamics are described by the Taylor-Veneziano-Yankielowicz superpotential for composite effective fields. The potential can be expanded about the ``truncated'' point in the gaugino and matter condensate directions in order to find corrections to the globally supersymmetric minimum. The results are relevant to the phenomenology of supersymmetry-breaking in string-inspired supergravity models, and also to recent work on domain walls in SQCD. 
  We investigate the oblique vacua in the perturbed 2+1 dimensional gauge theory living on D2 branes. The string theory dual of these vacua is expected to correspond to polarizations of the D2 branes into NS5 branes with D4 brane charge. We perturb the gauge theory by adding fermions masses. In the nonsupersymmetric case, we also consider the effect of slight variations of the masses of the scalars. For certain ranges of scalar masses we find oblique vacua.   We show that D4 charge is an essential ingredient in understanding D2 -> NS5 polarizations. We find that some of the polarization states which appear as metastable vacua when D4 charge is not considered are in fact unstable. They decay by acquiring D4 charge, tilting and shrinking to zero size. 
  We present conjectures for the space-time form and leading large N dependence of extremal and near-extremal correlation functions in the \N=8 superconformal Yang-Mills theory in d=3 as well as in the (0,2) superconformal theory in d=6, using their gravity duals with M-theory on $AdS_4\times \S^7$ and $AdS_7 \times \S^4$ respectively. As a key part of the conjectures, we argue that the bulk couplings associated with extremal and near-extremal field configurations in the corresponding $AdS_4$ and $AdS_7$ gauged supergravities vanish. The vanishing of these couplings constitutes a generalization of the property of consistent truncation of the Kaluza-Klein modes. 
  It is shown that the application of the non-Abelian Stokes theorem to the computation of the operators constructed with Wilson loop will lead to ambiguity, if the gauge field under consideration is a non-trivial one. This point is illustrated by the specific examples of the computation of a non-local operator. 
  Most recently 't Hooft has postulated (G 't Hooft, Class. Quant. Grav. 16 (1999) 3263-3279) that quantum states at the ``atomic scale''can be understood as equivalence classes of primordial states governed by a dissipative deterministic theory underlying quantum theory at the ``Planck scale''. Defining invariant subspaces clearly for primordial states according to a given evolution, we mathematically re-formulate 't Hooft's theory as a quotient space construction with the time-reversible evolution operator induced naturally. With this observation and some analysis, 't Hooft's theory is generalized beyond his case where the evolution at the ``Planck scale'' is periodic or the time is discrete. We also give a novel illustration that the Fock space of quantum oscillator could follow from the quotient space construction for certain primordial states obeying non-reversible evolution governed by a non-Hermitian Hamiltonian. 
  Causal perturbative renormalization within the recursive Epstein-Glaser scheme involves extending, at each order, time-ordered operator-valued distributions to coinciding points. This is achieved by a generalized Taylor subtraction on test functions, which is transposed to distributions. We show how the Epstein-Glaser recursive construction can, by means of a slight modification of the Hopf algebra of Feynman graphs, be recast in terms of the new Connes-Kreimer algebraic setup for renormalization. This is illustrated for $\phi^4_4$-theory. 
  F(4) supergravity, the gauge theory of the exceptional six-dimensional Anti-de Sitter superalgebra, is coupled to an arbitrary number of vector multiplets whose scalar components parametrize the quaternionic manifold $SO(4,n)/SO(4)\times SO(n)$. By gauging the compact subgroup $SU(2)_d \otimes \cG$, where SU(2)_d is the diagonal subgroup of $SO(4)\simeq SU(2)_L\otimes SU(2)_R$ (the R-symmetry group of six-dimensional Poincar\'e supergravity) and $\cG$ is a compact group such that $dim\cG = n$, we compute the scalar potential which, besides the gauge coupling constants, also depends in non trivial way on the parameter m associated to a massive 2-form $B_{\mu\nu}$ of the gravitational multiplet. The potential admits an AdS background for g=3m, as the pure F(4)-supergravity. We compute the scalar squared masses (which are all negative) and retrieve the results dictated by AdS_6/CFT_5 correspondence from the conformal dimensions of boundary operators. The boundary F(4) superconformal fields are realized in terms of a singleton superfield (hypermultiplet) in harmonic superspace with flag manifold SU(2)/U(1)=S^2. We analize the spectrum of short representations in terms of superconformal primaries and predict general features of the K-K specrum of massive type IIA supergravity compactified on warped $AdS_6\otimes S^4$. 
  Rederiving the one-loop divergences for the most general coupling of the open string sigma model by the heat kernel technique, we distinguish the classical background field from the mean field of the effective action. The latter is arbitrary, i.e. does not fulfil the boundary conditions. As a consequence a new divergent counter term strongly suggests the introduction of another external one-form field (beside the usual gauge field), coupled to the normal derivative at the boundary. Actually such a field has been proposed in the literature for different reasons, but its full impact never seems to have thoroughly investigated before. The beta function for the resulting renormalizable model is calculated and the consequences are discussed, including the ones for the Born-Infeld action. The most exciting property of the new coupling is that it enters the coefficient in front of the normal derivative in Neumann boundary conditions. For certain values of the background fields this coefficient vanishes, leading to Dirichlet boundary conditions. This provides a natural mechanism for the emergence of D-branes. 
  The fermionic sector of the Standard Model of Elementary Particles emerges as the low energy limit of a single fermionic field freely propagating in a higher dimensional background. The local geometrical framework is obtained by enforcing at a space-time level the whole gauge group SO(1,3) x U(1) x SU(2) x SU(3) associated to fundamental interactions; equivalently, by assuming that internal gauge transformations are indeed local space-time transformations. The geometry naturally embodies freedoms corresponding to gravitational and non-gravitational gauge fields. As a consequence of the fact that the structural group is in part unitary, the motion of test particles gets automatically squeezed on an effective 1+3 space-time. Dimensional reduction takes place without compactification. In close analogy to the special relativistic mass-energy relation, the theory associates to every elementary particle an intrinsic energy presumably of the order of the Planck scale. The theory predicts the existence of a right-handed component of the neutrino and indicates the possibility of an extra U(1) gauge interaction. 
  Kaluza-Klein approach in an N(=1+3+D)-dimensional Friedmann-Robertson-Walker type space is often adopted in the literature. We derive a compact expression for the Friedmann equation in a (1+3+D)-dimensional space. The redundancy of the associated field equations due to the Bianchi identity is analyzed. We also study the dilaton gravity theory with higher-derivative gravitational couplings. It turns out that higher-order terms will not affect the Friedmann equation in a constant flat internal space. This is true only for the flat-De Sitter external space. The inflationary solution in an induced-gravity model is also discussed as an application. 
  Based on the exact relationship to Random Matrix Theory, we derive the probability distribution of the k-th smallest Dirac operator eigenvalue in the microscopic finite-volume scaling regime of QCD and related gauge theories. 
  The M5-brane is investigated near critical field-strength. We show that this limit on the M5-brane reduces to the noncommutative open string limit on the D4-brane. The reduction on a two-torus leads to both the noncommutative open string limit and the noncommutative Yang-Mills limit on the D3-brane. The decoupled noncommutative five-brane is identified with the strong coupling limit of the noncommutative open string theory on the D4-brane and S-duality on the noncommutative D3-brane is identified with a modular transformation on the five-brane. We argue that the open membrane metric defines a finite length scale on the worldvolume of the M5-brane in the decoupling limit. This length scale can be associated to the effective length scale of an open membrane. 
  We investigate blowup formulae in Donaldson-Witten theory with gauge group SU(N), using the theory of hyperelliptic Kleinian functions. We find that the blowup function is a hyperelliptic sigma-function and we describe an explicit procedure to expand it in terms of the Casimirs of the gauge group up to arbitrary order. As a corollary, we obtain a new expression for the contact terms and we show that the correlation functions involving the exceptional divisor are governed by the KdV hierarchy. We also show that, for manifolds of simple type, the blowup function becomes a tau-function for a multisoliton solution. 
  In this paper we study the one-loop shift in the coupling constant in a noncommutative pure U(N) Chern-Simons gauge theory in three dimensions. The one-loop shift is shown to be a constant proportional to $N$, independent of noncommutativity parameters, and non-vanishing for U(1) theory. Possible physical and mathematical implications of this result are discussed. 
  The role of bulk matter quantum effects (via the corresponding effective potential discussed on the example of conformal scalar) and of boundary matter quantum effects (via the conformal anomaly induced effective action) is considered in brane-world cosmology. Scenario is used where brane tension is not free parameter, and the initial bulk-brane classical action is defined by some considerations. The effective bulk-brane equations of motion are analyzed. The quantum creation of 4d de Sitter or Anti-de Sitter(AdS) brane Universe living in 5d AdS space is possible when quantum bulk and (or) brane matter is taken into account. The consideration of only conformal field theory (CFT) living on the brane admits the full analytical treatment. Then bulk gravitational Casimir effect leads to deformation of 5d AdS space shape as well as of shape of spherical or hyperbolic branes. The generalization of above picture for the dominant bulk quantum gravity naturally represents such scenario as self-consistent warped compactification within AdS/CFT set-up. 
  The reparametrization-invariant generating functional for the unitary and causal perturbation theory in general relativity in a finite space-time is obtained. The region of validity of the Faddeev-Popov-DeWitt functional is studied. It is shown that the invariant content of general relativity as a constrained system can be covered by two "equivalent" unconstrained systems: the "dynamic" (with "dynamic" evolution parameter as the metric scale factor) and "geometric" (given by the Levi-Civita type canonical transformation to the action-angle variables where the energy constraint converts into a new momentum). "Big Bang", the Hubble evolution, and creation of matter fields by the "geometric" vacuum are described by the inverted Levi-Civita (LC) transformation of the geomeric system into the dynamic one. The particular case of the LC transformations are the Bogoliubov ones of the particle variables (diagonalizing the dynamic Hamiltonian) to the quasiparticles (diagonalizing the equations of motion). The choice of initial conditions for the "Big Bang" in the form of the Bogoliubov (squeezed) vacuum reproduces all stages of the evolution of the Friedmann-Robertson-Walker Universe in their conformal (Hoyle-Narlikar) versions. 
  In theories with Chern-Simons terms or modified Bianchi identities, it is useful to define three notions of either electric or magnetic charge associated with a given gauge field. A language for discussing these charges is introduced and the properties of each charge are described. `Brane source charge' is gauge invariant and localized but not conserved or quantized, `Maxwell charge' is gauge invariant and conserved but not localized or quantized, while `Page charge' conserved, localized, and quantized but not gauge invariant. This provides a further perspective on the issue of charge quantization recently raised by Bachas, Douglas, and Schweigert. 
  The hidden supersymmetry of the monopole found by De Jonghe et al. is generalized to a spin $\2$ particle in the combined field of a Dirac monopole plus a $\lambda^2/r^2$ potential [considered before by D'Hoker and Vinet], and related to the operator introduced by Biedenharn a long time ago in solving the Dirac-Coulomb problem. Explicit solutions are obtained by diagonalizing the Biedenharn operator 
  We investigate the phase structure of non-commutative scalar field theories and find evidence for ordered phases which break translation invariance. A self-consistent one-loop analysis indicates that the transition into these ordered phases is first order. The phase structure and the existence of scaling limits provides an alternative to the structure of counter-terms in determining the renormalizability of non-commutative field theories. On the basis of the existence of a critical point in the closely related planar theory, we argue that there are renormalizable interacting non-commutative scalar field theories in dimensions two and above. We exhibit this renormalization explicitly in the large $N$ limit of a non-commutative O(N) vector model. 
  Deformations of topological open string theories are described, with an emphasis on their algebraic structure. They are encoded in the mixed bulk-boundary correlators. They constitute the Hochschild complex of the open string algebra -- the complex of multilinear maps on the boundary Hilbert space. This complex is known to have the structure of a Gerstenhaber algebra (Deligne theorem), which is also found in closed string theory. Generalising the case of function algebras with a B-field, we identify the algebraic operations of the bulk sector, in terms of the mixed correlators. This gives a physical realisation of the Deligne theorem. We translate to the language of certain operads (spaces of d-discs with gluing) and d-algebras, and comment on generalisations, notably to the AdS/CFT correspondence. The formalism is applied to the topological A- and B-models on the disc. 
  The Casimir effect at finite temperature is investigated for a dilute dielectric ball; i.e., the relevant internal and free energies are calculated. The starting point in this study is a rigorous general expression for the internal energy of a system of noninteracting oscillators in terms of the sum over the Matsubara frequencies. Summation over the angular momentum values is accomplished in a closed form by making use of the addition theorem for the relevant Bessel functions. For removing the divergences the renormalization procedure is applied that has been developed in the calculation of the corresponding Casimir energy at zero temperature. The behavior of the thermodynamic characteristics in the low and high temperature limits is investigated. 
  The non-abelian flat directions in the tachyon potential of stable non-BPS branes recently found are shown to persist to all orders in alpha' at tree level in the string coupling. We also obtain the non-abelian Born-Infeld action including the tachyon potential for a stack of stable non-BPS branes on a critical orbifold. Finally we discuss stable soliton states on the non-BPS brane. 
  We obtain, through zeta function methods, the one-loop effective action for massive Dirac fields in the presence of a uniform, but otherwise general, electromagnetic background. After discussing renormalization, we compare our zeta function result with Schwinger's proper time approach. 
  We propose a topological Chern-Simons term in D=5 dimensions coupled to Einstein Hilbert theory. Hartree approximation for topological Lagrangian and the Chern-Simons term in D=3 is considered. An effective model of Quantum Gravity in D=5 dimensions is presented here. The analysis of residues is considered and the unitarity is guaranteed at tree level. The propagator is ghost and tachyon free. 
  We propose a time-varying parameter $\underline{\alpha}$ for G\"{o}del metric and an energy momentum tensor corresponding to this geometry is found. To satisfy covariance arguments time-varying gravitational and cosmological term are introduced. The ``Einstein's equation'' for this special evolution for the Universe are written down where expansion, rotation and closed time-like curves appear as a combination between standard model, G\"{o}del and steady state properties are obtained. 
  We study the contribution to entropy of Black Holes in D=2+1 dimensions from an extension of the Chern Simons theory including higher derivative in a curved space-time. 
  We discuss here the possibility to write the Liouville-Vlasov equation for the Wigner-function of a spinor field coupled to a gauge field with field strength tensor $F^{\mu\nu}$ in a curved space-time versus a local Lorentz manifold (introduction of local Lorentz coordinates) with an appropriate definition of a covariant derivative carried out using a spin connection $B_{\mu}^{ab}(x)$. 
  The purpose of this paper is to study static solution configuration which describes the magnetic monopoles in a scenary where the gravitation is coupled with Higgs, Yang-Mills and fermions. We are looking for analysis of the energy functional and Bogomol'nyi equations. The Einstein equations now take into consideration the fermions' contribution for energy-momentum tensor. The interesting aspect here is to verify that the fermion field gives a contribution for non abelian magnetic field and for potential which minimise the energy functional. 
  The existence of an electromagnetic field with parallel electric and magnetic field components in the presence of a gravitational field is considered. A non-parallel solution is shown to exist. Next, we analyse the possibility of finding stationary gravitational waves in nature. Finally, we construct a D=4 effective quantum gravity model. Tree-level unitarity is verified. 
  At first, we discuss parallels electric and magnetic fields solutions in a gravitational background. Then, considering eletromagnetic and gravitational waves symmetries we show a particular solution for stationary gravitational waves. Finally we consider gravitation as a gauge theory (effective gravitational theory), evaluate the propagators of the model, analyze the corresponding quantum excitations and verify (confirm) the tree-level unitarity at many places of the model. 
  We introduce a 'quasi-topological` term [1] in D=1+1 dimensions and the entropy for black holes is calculated [2]. The source of entropy in this case is justified by a non-null stress-energy tensor. 
  This article, based on the Klein lecture, contains some new results and new speculations on various topics. They include discussion of open strings in the AdS space, unusual features of D-branes, conformal gauge theories in higher dimensions. We also comment on the infrared screening of the cosmological constant and on the "brane worlds" 
  In a gravitational field, we analyze the Maxwell equations, the correponding electromagnetic wave and continuity equations. A particular solution for parellel electric and magnetic fields in a gravitational background is presented. These solutions also satisfy the free-wave equations and the phenomenology suggested by plasma physics. 
  We study brane-world black holes from Randall-Sundrum(RS) models in ($D+1$)-dimensional anti-de Sitter spacetimes. The solutions are directly obtained by using a slightly modified RS metric ansatz in $D+1$ dimensions. The metric of the brane world can be described by the Schwarzschild solution promoted to the black cigar solution in $D+1$ dimensions, which is compatible with the recently suggested black cigar solution for D=4. Furthermore, we show that the Ricci flat condition for the brane can be easily derived from the effective gravity defined on the brane by using the RS dimensional reduction. Especially, it is shown that in two dimensions the effective gravity on the brane is described by the Polyakov action. 
  It is known that self-duality equations for multi-instantons on a line in four dimensions are equivalent to minimal surface equations in three dimensional Minkowski space. We extend this equivalence beyond the equations of motion and show that topological number, instanton moduli space and anti-self-dual solutions have representations in terms of minimal surfaces. The issue of topological charge is quite subtle because the surfaces that appear are non-compact. This minimal surface/instanton correspondence allows us to define a metric on the configuration space of the gauge fields. We obtain the minimal surface representation of an instanton with arbitrary charge. The trivial vacuum and the BPST instanton as minimal surfaces are worked out in detail. BPS monopoles and the geodesics are also discussed. 
  We show that the Hamiltonian dynamics of the self-interacting, abelian p-form theory in D=2p+2 dimensional space-time gives rise to the quasi-local structure. Roughly speaking, it means that the field energy is localized but on closed 2p-dimensional surfaces (quasi-localized). From the mathematical point of view this approach is implied by the boundary value problem for the corresponding field equations. Various boundary problems, e.g. Dirichlet or Neumann, lead to different Hamiltonian dynamics. Physics seems to prefer gauge-invariant, positively defined Hamiltonians which turn out to be quasi-local. Our approach is closely related with the standard two-potential formulation and enables one to generate e.g. duality transformations in a perfectly local way (but with respect to a new set of nonlocal variables). Moreover, the form of the quantization condition displays very similar structure to that of the symplectic form of the underlying p-form theory expressed in the quasi-local language. 
  We find evidence for a duality between the standard matrix formulations of M theory and a background independent theory which extends loop quantum gravity by replacing SU(2) with a supersymmetric and quantum group extension of SU(16). This is deduced from the recently proposed cubic matrix model for M theory which has been argued to have compactifications which reduce to the IKKT and dWHN-BFSS matrix models. Here we find new compactifications of this theory whose Hilbert spaces consist of SU(16) conformal blocks on compact two-surfaces. These compactifications break the SU(N) symmetry of the standard M theory compactifications, while preserving SU(16), while the BFSS model preserve the SU(N) but break SU(16) to the SO(9) symmetry of the 11 dimensional light cone coordinates. These results suggest that the supersymmetric and quantum deformed SU(16) extension of loop quantum gravity provides a dual, background independent description of the degrees of freedom and dynamics of the M theory matrix models. 
  The divergence found by Nesterenko, Lambiase and Scarpetta in the Casimir energy on a semi-circular cylinder is attributed to the existence of edges. 
  We point out several subtleties arising in brane-world scenarios of cosmological constant cancellation. We show that solutions with curvature singularities are inconsistent, unless the contribution to the effective four-dimentional cosmological constant of the physics that resolves the singularities is fine-tuned. This holds for both flat and curved branes. Irrespective of this problem, we then study an isolated class of flat solutions in models where a bulk scalar field with a vanishing potential couples to a 3-brane. We give an example where the introduction of a bulk scalar potential results in a nonzero cosmological constant. Finally we comment on the stability of classical solutions of the brane system with respect to quantum corrections. 
  We analyze the large N supergravity descriptions of the class of type IIB models T-dual to elliptic type IIA brane configurations containing two orientifold 6-planes and up to two NS 5-branes. The T-dual IIB configurations contain N D3-branes in the background of an orientifold 7-plane and, in some models, a Z_2 orbifold and/or D7-branes, which give rise to four-dimensional N=2 (or N=4) gauge theories with at most two factors. We identify the chiral primary states of the supergravity theories, and match them to gauge invariant operators of the corresponding superconformal theories using Maldacena's duality. 
  Methods are reviewed for computing the instanton expansion of the prepotential for N=2 Seiberg-Witten theory with non-hyperelliptic curves. These results, when compared with the instanton expansion obtained from the microscopic Lagrangian, provide detailed tests of M-theory. Group theoretic regularities of F_ 1-inst allow one to "reverse engineer" a Seiberg-Witten curve for SU(N) with two antisymmetric representations and N_f \leq 3 fundamental hypermultiplet representations, a result not yet available by other methods. Consistency with M-theory requires a curve of infinite order. 
  We write the Hamiltonian for a gravitational spherically symmetric scalar field collapse with massive scalar field source, and we discuss the application of Wheeler De Witt equation as well as the appearence of time in this context. Using an Ansatz for Wheeler De Witt equation, solutions are discussed including the appearence of time evolution. 
  The iterated BRST cohomology is studied by computing cohomology of the variational complex on the infinite order jet space of a smooth fibre bundle. This computation also provides a solution of the global inverse problem of the calculus of variations in Lagrangian field theory. 
  We investigate the structure of equations of motion and lagrangian constraints in a general theory of massive spin 2 field interacting with external gravity. We demonstrate how consistency with the flat limit can be achieved in a number of specific spacetimes. One such example is an arbitrary static spacetime though equations of motion in this case may lack causal properties. Another example is provided by external gravity fulfilling vacuum Einstein equations with arbitrary cosmological constant. In the latter case there exists one-parameter family of theories describing causal propagation of the correct number of degrees of freedom for the massive spin 2 field in arbitrary dimension. For a specific value of the parameter a gauge invariance with a vector parameter appears, this value is interpreted as massless limit of the theory. Another specific value of the parameter produces gauge invariance with a scalar parameter and this cannot be interpreted as a consistent massive or massless theory. 
  We show that a D-brane in a group manifold given by a (twisted) conjugacy class is characterised by a gauge invariant two-form field determined in terms of the matrix of gluing conditions. Using a quantisation argument based on the path integral one obtains the known quantisation condition for the corresponding D-branes. We find no evidence for the existence of a quantised U(1) gauge field flux. We propose an expression for the D0 charge of such  D-branes. 
  The junction condition across a singular surface in general relativity, formulated by Israel, has double covariance. In this paper, a general perturbation scheme of the junction condition around an arbitrary background is given in a doubly covariant way. After that, as an application of the general scheme, we consider perturbation of the junction condition around a background with the symmetry of a $(D-2)$-dimensional constant curvature space, where $D$ is the dimensionality of the spacetime. The perturbed junction condition is written in terms of doubly gauge-invariant variables only. Since the symmetric background includes cosmological solutions in the brane-world as a special case, the doubly gauge-invariant junction condition can be used as basic equations for perturbations in the brane-world cosmology. 
  This paper discusses multi-skyrmions on the 3-sphere with variable radius L using the rational map ansatz. For baryon number B = 3,...,9 this ansatz produces the lowest energy solutions known so far. By considering the geometry of the model we find an approximate analytic formula for the shape function. This provides an insight why skyrmions have a shell-like structure. 
  We construct static axially symmetric solutions of SU(2) Einstein-Yang-Mills-Higgs theory in the topologically trivial sector, representing gravitating monopole--antimonopole pairs, linked to the Bartnik-McKinnon solutions. 
  The Wess-Zumino action for y deformed and generalized orientifold planes (yGOp-planes) is presented and one power expantion is realized from which processes that involves yGOp-planes, RR-forms, gravitons and gaugeons, are obtained. Finally non-standard yGOp-planes are showed. 
  We formulate path integrals on any Riemannian manifold which admits the action of a compact Lie group by isometric transformations. We consider a path integral on a Riemannian manifold M on which a Lie group G acts isometrically. Then we show that the path integral on M is reduced to a family of path integrals on a quotient space Q=M/G and that the reduced path integrals are completely classified by irreducible unitary representations of G. It is not necessary to assume that the action of G on M is either free or transitive. Hence our formulation is applicable to a wide class of manifolds, which includes inhomogeneous spaces, and it covers all the inequivalent quantizations. To describe the path integral on inhomogeneous space, stratification geometry, which is a generalization of the concept of principal fiber bundle, is necessarily introduced. Using it we show that the path integral is expressed as a product of three factors; the rotational energy amplitude, the vibrational energy amplitude, and the holonomy factor. When a singular point arises in $ Q $, we determine the boundary condition of the path integral kernel for a path which runs through the singularity. 
  These pedagogical lectures present some material, classical or more recent, on (Rational) Conformal Field Theories and their general setting ``in the bulk'' or in the presence of a boundary. Two well posed problems are the classification of modular invariant partition functions and the determination of boundary conditions consistent with conformal invariance. It is shown why the two problems are intimately connected and how graphs -ADE Dynkin diagrams and their generalizations- appear in a natural way. 
  We present all NSR superstring and super-D-string actions invariant under a set of prescribed gauge transformations, and characterize completely their global symmetries. In particular we obtain locally supersymmetric Born-Infeld actions on general backgrounds in a formulation with extra target space dimensions. The nontrivial global symmetries of the superstring actions correspond to isometries of the background, whereas super-D-string actions can have additional symmetries acting nontrivially also on the coordinates of the extra dimensions. 
  Extending recent work on QED and the symmetric phase of the euclidean multicomponent scalar \phi^4-theory, we construct the vacuum diagrams of the free energy and the effective energy in the ordered phase of \phi^4-theory. By regarding them as functionals of the free correlation function and the interaction vertices, we graphically solve nonlinear functional differential equations, obtaining loop by loop all connected and one-particle irreducible vacuum diagrams with their proper weights. 
  The CPT anomaly, which was first seen in perturbation theory for certain four-dimensional chiral gauge theories, is also present in the exact result for a class of two-dimensional chiral U(1) gauge theories on the torus. Specifically, the chiral determinant for periodic fermion fields changes sign under a CPT transformation of the background gauge field. There is, in fact, an anomaly of Lorentz invariance, which allows for the CPT theorem to be circumvented. 
  The q-field theories are constructed by substituting quantum groups for the usual Lie groups. In earlier papers this construction was carried out for the quantum group SU_q(2). Here the investigation is extended to SL_q(3). The resulting theory describes two sectors, one sector lying close to the standard theory and accessible by perturbation theory, while the second sector describes particles that should be difficult to detect and become invisible in the q=1 limit. In this note we discuss these hypothetical particles: three quark-like spinor particles coupled to three gluon-like vector particles. 
  The quantum discrete Liouville model in the strongly coupled regime, 1<c<25, is formulated as a well defined quantum mechanical problem with unitary evolution operator. The theory is self-dual: there are two exponential fields related by Hermitean conjugation, satisfying two discrete quantum Liouville equations, and living in mutually commuting subalgebras of the quantum algebra of observables. 
  We study the mechanism of topological superconductivity in a hierarchical chain of chiral non-linear sigma-models (models of current algebra) in one, two, and three spatial dimensions. The models have roots in the 1D Peierls-Frohlich model and illustrate how the 1D Frohlich's ideal conductivity extends to a genuine superconductivity in dimensions higher than one. The mechanism is based on the fact that a point-like topological soliton carries an electric charge. We discuss a flux quantization mechanism and show that it is essentially a generalization of the persistent current phenomenon, known in quantum wires. We also discuss why the superconducting state is stable in the presence of a weak disorder. 
  We use the Israel condition to treat carefully the weak-field perturbations due to the presence of matter on a 3-brane embedded between two regions of anti-de Sitter (AdS) space with different curvature lengths. A four dimensional Newton's Law only emerges at distances that are large compared to the AdS lengths. When a scalar curvature is included in the brane action, however, it is possible to generate a four dimensional theory of gravity even when one or both of the AdS lengths is large compared to distances along the brane. In particular, we provide an example in which the AdS lengths can be larger than the millimeter experimental bound. 
  T-duality (Fourier-Mukai duality) and properties of classical instanton moduli spaces can be used to deduce some properties of $\alpha^{\prime}$-corrected moduli spaces of branes for Type IIA string theory compactified on $K3$ or $T^4$. Some interesting differences between the two compactifications are exhibited. 
  We study field theories on spaces with additional compact noncommutative dimensions. As an example, we study \phi^3 on R^{1,3}\times T^{2}_\theta using perturbation theory. The infrared divergences in the noncompact theory give rise to unusual dynamics for the mode of \phi which is constant along the torus. Correlation functions involving this mode vanish. Moreover, we show that the spectrum of Kaluza-Klein excitations can be very different from the analogous commuting theory. There is an additional contribution to the Kaluza-Klein mass formula that resembles the contribution of winding states in string theory. We also consider the effect of noncommutativity on the four dimensional Kaluza-Klein excitations of a six dimensional gauge field. 
  Starting from a recently proposed abelian topological model in (2+1) dimensions, we use the method of the consistent deformations to prove that a topologically massive model involving the Kalb-Ramond two form field does not admit a nonabelian generalization. The introduction of a connection-type one form field keeps the previous result. However we show that the goal is achieved if we introduce a vectorial auxiliary field, exhibiting a nonabelian topological mass generation mechanism in D=3, that provides mass for the Kalb-Ramond field. Further, we find the complete set of BRST and anti-BRST equations using the horizontality condition, suggesting a connection between this formalism and the method of the consistent deformations. 
  We construct a fourth-order derivative CP(N) model in 1+1 dimensions by incorporating the topological charge density squared term into the Lagrangian. We quantize the theory by reformulating with auxiliary fields and then performing the path integral explicitly. We discuss the renormalizability in the large N limit and relevance of the effective action with axion physics. 
  We consider higher derivative CP(N) model in 2+1 dimensions with the Wess-Zumino-Witten term and the topological current density squared term. We quantize the theory by using the auxiliary gauge field formulation in the path integral method and prove that the extended model remains renormalizable in the large N limit. We find that the Maxwell-Chern-Simons theory is dynamically induced in the large N effective action at a nontrivial UV fixed point. The quantization of the Chern-Simons term is also discussed. 
  We find two types [called (S) and (A)] of new vacuum solutions of open, flat, and closed universes which are inflating in the brane-world scenario. We show that the warp factor of the stabilized metric is universal for the three different kinds of universes. For (S) type solution, we show that one positive-tension brane universe solution is admitted as well as two positive tension brane solution even if we consider the vacuum solution. For (A) type solution, we find that the inflating bulk solutions have black hole like regions and that the full extended space is the R-S solution. 
  We develop a method of singularity analysis for conformal graphs which, in particular, is applicable to the holographic image of AdS supergravity theory. It can be used to determine the critical exponents for any such graph in a given channel. These exponents determine the towers of conformal blocks that are exchanged in this channel. We analyze the scalar AdS box graph and show that it has the same critical exponents as the corresponding CFT box graph. Thus pairs of external fields couple to the same exchanged conformal blocks in both theories. This is looked upon as a general structural argument supporting the Maldacena hypothesis. 
  This is an introduction for nonspecialists to the noncommutative geometric approach to Planck scale physics coming out of quantum groups. The canonical role of the `Planck scale quantum group' $C[x]\bicross C[p]$ and its observable-state T-duality-like properties are explained. The general meaning of noncommutativity of position space as potentially a new force in Nature is explained as equivalent under quantum group Fourier transform to curvature in momentum space. More general quantum groups $C(G^\star)\bicross U(g)$ and $U_q(g)$ are also discussed. Finally, the generalisation from quantum groups to general quantum Riemannian geometry is outlined. The semiclassical limit of the latter is a theory with generalised non-symmetric metric $g_{\mu\nu}$ obeying $\nabla_\mu g_{\nu\rho}-\nabla_\nu g_{\mu\rho}=0$ 
  Quantum groups emerged in the latter quarter of the 20th century as, on the one hand, a deep and natural generalisation of symmetry groups for certain integrable systems, and on the other as part of a generalisation of geometry itself powerful enough to make sense in the quantum domain. Just as the last century saw the birth of classical geometry, so the present century sees at its end the birth of this quantum or noncommutative geometry, both as an elegant mathematical reality and in the form of the first theoretical predictions for Planck-scale physics via ongoing astronomical measurements. Noncommutativity of spacetime, in particular, amounts to a postulated new force or physical effect called cogravity. 
  We review and extend recent work on the application of the non-commutative geometric framework to an interpretation of the moduli space of vacua of certain deformations of N=4 super Yang-Mills theories. We present a simple worldsheet calculation that reproduces the field theory results and sheds some light on the dynamics of the D-brane bubbles. Different regions of moduli space are associated with D5-branes of various topologies; singularities in the moduli space are associated with topology change. T-duality on toroidal topologies maps between mirror string realizations of the field theory. 
  We apply the Godbillon-Vey class to compute the transition amplitudes between some non-commutative solitons in M-Theory; our context is that of Connes-Douglas-Schwarz where they considered compactifications of matrix models on noncommutative tori. Two important consequences follow: we describe a new normalisation for the Abelian Chern-Simons theory using symplectic 4-manifolds as providing cobordisms for tight contact 3-manifolds and we construct a new(?) invariant for 3-manifolds. Moreover we modify the topological Lagrangian density suggested for M-Theory in a previous article to a \textsl{quadratic} one using the fact that the \emph{functor of immersions is a linearisation (or ``the differential'') of the functor of embeddings} 
  It is shown, by explicit calculation in the axial gauge, that the renormalization group flow for the Wilson loop in perturbation theory does exhibit singularities and consequently it can not eventually reproduce the gauge invariant result, when the infrared cut off is removed 
  Some spatially homogeneous Bianchi type I cosmological models filled with homogeneous ``electric'' $p$-form fields are shown to mimic the never-ending oscillatory behaviour of generic string cosmologies established recently. The validity of the ``Kasner-free-flights plus collisions-on-potential-walls'' picture is also illustrated in the case of known, non-chaotic, superstring solutions. 
  Recent advances in 11 dimensional Horava-Witten M-theory based on non-standard embeddings with torus fibered Calabi-Yau manifolds have allowed the construction of three generation models with Wilson line breaking to the Standard Model gauge symmetry. Central to these constructions is the existence of a set of 5-branes in the bulk. We examine within this framework the general structure of the matter Yukawa couplings and show that M-theory offers an alternate possible way of achieving the CKM and quark mass hierarchies without introducing undue fine tuning or (as in conventional analysis) small parameters raised to high powers. A phenomenological example is presented in accord with all CKM and quark mass data requiring mainly that the 5-branes cluster near the second orbifold plane, and that the instanton charges of the physical orbifold plane vanish. An explicit example of a three generation model with vanishing physical plane instanton charges based on a torus fibered Calabi-Yau three fold with a del Pezzo base $dP_7$ and Wilson line breaking is constructed. 
  The vector supersymmetry of the 2D topological BF model is extended to 2D Yang-Mills. The consequences of the corresponding Ward identity on the ultraviolet behavior of the theory are analyzed. 
  The cosmological term is assumed to be a function of time such as $\Lambda =Ba^{-2}$ where a(t) means the scale factor of standard cosmology. Analytical solutions for radiation dominated epoch and open universe are found. For closed universe, k=+1, and for flat universe, k=0, we show a numerical solution. In general the scenario of Big Bang is preserved in our case for the Friedmann-Robertson-Walker cosmology. 
  Any regular quantum mechanical system may be cast into an abelian gauge theory by simply reformulating it as a reparametrization invariant theory. We present a detailed study of the BRST quantization of such reparametrization invariant theories within a precise operator version of BRST. The treatment elucidates several intricate aspects of the BRST quantization of reparametrization invariant theories like the appearance of physical time. We propose general rules for how physical wave functions and physical propagators are to be projected from the BRST singlets and propagators in the ghost extended BRST theory. These projections are performed by boundary conditions which are precisely specified by the operator BRST. We demonstrate explicitly the validity of these rules for the considered class of models. The corresponding path integrals are worked out explicitly and compared with the conventional BFV path integral formulation. 
  We discuss the interpaly between IR and UV divergences in theories with open and unoriented strings in view of the AdS/CFT correspondence. We start by deriving general formulas for the computation of threshold corrections to gauge couplings in generic configurations with open and unoriented strings. These allow us to discuss the IR/UV correspondence between beta-function coefficients and ``dilaton'' tadpoles for several brane configurations probed by D3-branes. Finally we comment on the AdS supergravity descriptions of gauge theories that are (super)conformal in the large N limit. 
  The BRST-antiBRST invariant path integral formulation of classical mechanics of Gozzi et al is generalized to pseudomechanics. It is shown that projections to physical propagators may be obtained by BRST-antiBRST invariant boundary conditions. The formulation is also viewed from recent group theoretical results within BRST-antiBRST invariant theories. A natural bracket expressed in terms of BRST and antiBRST charges in the extended formulation is shown to be equal to the Poisson bracket. Several remarks on the operator formulation are made. 
  Gauge theory, which is the basis of all particle physics, is itself based on a few fundamental concepts, the consequences of which are often as beautiful as they are deep. In this short lecture course I shall try to give an introduction to these concepts, both from the physical and mathematical points of view. Then I shall show how these considerations lead to a nonabelian generalization of the well-known electric--magnetic duality in electromagnetism. I shall end by sketching some of the many consequences in quantum field theory that this duality engenders in particle physics.   These are notes from a lecture course given in the Summer School on {\em Geometric Methods in Quantum Field Theory}, Villa de Leyva, Colombia, July 1999, as well as a series of graduate lectures given in Oxford in Trinity Term of 1999 and 2000. 
  We introduce the general N=1 gauge theory superconformally coupled to supergravity. The theory has local SU(2,2|1) symmetry and no dimensional parameters. The superconformal origin of the Fayet-Iliopoulos terms is clarified. The phase of this theory with spontaneously broken conformal symmetry gives various formulations of N=1 supergravity interacting with matter, depending on the choice of the R-symmetry fixing.   We have found that the locally superconformal theory is useful for describing the physics of the early universe with a conformally flat FRW metric. Few applications of superconformal theory to cosmology include the study of i) particle production after inflation, particularly the non-conformal helicity 1/2 states of gravitino, ii) the super-Higgs effect in cosmology and the derivation of the equations for the gravitino interacting with any number of chiral and vector multiplets in the gravitational background with varying scalar fields, iii) the weak coupling limit of supergravity and gravitino-goldstino equivalence. This explains why gravitino production in the early universe is not suppressed in the limit of weak gravitational coupling.   We discuss the possible existence of an unbroken phase of the superconformal theories, interpreted as a strong coupling limit of supergravity. 
  Using the average action defined with a continuum analog of the block spin transformation, we show the presence of gauge symmetry along the Wilsonian renormalization group flow. As a reflection of the gauge symmetry, the average action satisfies the quantum master equation(QME). We show that the quantum part of the master equation is naturally understood once the measure contribution under the BRS transformation is taken into account. Furthermore an effective BRS transformation acting on macroscopic fields may be defined from the QME. The average action is explicitly evaluated in terms of the saddle point approximation up to one-loop order. It is confirmed that the action satisfies the QME and the flow equation. 
  We develop a systematic approach to boundary conditions that break bulk symmetries in a general way such that left and right movers are not necessarily connected by an automorphism. In the context of string compactifications, such boundary conditions typically include non-BPS branes. Our formalism is based on two dual fusion rings, one for the bulk and one for the boundary fields. Only in the Cardy case these two structures coincide. In general they are related by a version of alpha-induction. Symmetry breaking boundary conditions correspond to solitonic sectors. In examples, we compute the annulus amplitudes and boundary states. 
  Some exact static solutions of the SU(2) Yang-Mills-Higgs theory are presented. These solutions satisfy the first order Bogomol'nyi equations, and possess infinite energies. They are axially symmetric and could possibly represent monopoles and an antimonopole sitting on the z-axis. 
  We use the Wilson renormalization group (RG) formulation to solve the fine-tuning procedure needed in renormalization schemes breaking the gauge symmetry. To illustrate this method we systematically compute the non-invariant couplings of the ultraviolet action of the SU(2) pure Yang-Mills theory at one-loop order. 
  A brane world model is investigated, in which there are many branes that may intersect and self intersect. One of the branes, being a 3-brane, represents our spacetime, while the other branes, if they intersect our brane world, manifest themselves as matter in our 3-brane. It is shown that such a matter encompasses dust of point particles and higher dimensional p-branes, and all those objects follow "geodesics" in the world volume swept by our 3-brane. We also point point out that such a model can be formulated in a background independent way, and that the kinetic term for gravity arises from quantum fluctuation of the brane. 
  By considering the superembedding equation for the Type II superstring we derive the classical relation between the NSR string and the Type II GS superstring Grassmannian variables. The connection between the actions of these two models is also established. Then introducing the proper twistor-like Lorentz harmonic variables we fix $\kappa-$symmetry of the GS formulation in the manifestly SO(1,9) Lorentz covariant manner and establish the relation between the gauge-fixed variables of the NSR and the Type II GS models. 
  We study the principal sigma-models defined on any group manifold GL x GR/GD with breaking of GR, and their T-dual transforms. For abritary breaking we can express the torsion and Ricci tensor of the dual model in terms of the frame geometry of the initial principal model. Using these results, we give necessary and sufficient conditions for the dual model to be torsionless and prove that the one-loop renormalizability of a given principal model is inherited by its dual partner, who shares the same beta-functions. These results are shown to hold also if the principal model is endowed with torsion. As an application we compute the beta-functions for the full Bianchi family and show that for some choices of the breaking parameters the dilaton anomaly is absent: for these choices the dual torsion vanishes. For the dualized Bianchi V model (which is torsionless for any breaking), we take advantage of its simpler structure, to study its two-loops renormalizability. 
  This note is intended to emphasize the existence of estimated Feynman integrals in three dimensions for the free energy of the O(1) scalar theory up to five loops which may be useful for other work. We also correct some misprints of the published paper. 
  The classical moduli space M of a supersymmetric gauge theory with trivial superpotential can be stratified according to the unbroken gauge subgroup at different vacua. We apply known results about this stratification to obtain the W \neq 0 theory classical moduli space M^W \subset M, working entirely with the composite gauge invariant operators X that span M, assuming we do not known their elementary matter chiral field content. In this construction, the patterns of gauge symmetry breaking of the W \neq 0 theory are determined, Higgs flows in these theories show important differences from the W=0 case. The methods here introduced provide an alternative way to construct tree level superpotentials that lift all classical flat directions leaving a candidate theory for dynamical supersymmetry breaking, and are also useful to identify heavy composite fields to integrate out from effective superpotentials when the elementary field content of the composites is unknown. We also show how to recognize the massless singlets after Higgs mechanism at a vacuum X \in M^W among the moduli \delta X using the stratification of M, and establish conditions under which the space of non singlet massless fields after Higgs mechanism (unseen as moduli \delta X) is null. A small set of theories with so called "unstable" representations of the complexified gauge group is shown to exhibit unexpected properties regarding the dimension of their moduli space, and the presence of non singlet massless fields after Higgs mechanism at all of their vacua. 
  We discuss several aspects of D-brane moduli spaces and BPS spectra near orbifold points. We give a procedure to determine the decay products on a line of marginal stability, and we define the algebra of BPS states in terms of quivers. These issues are illustrated in detail in the case of type IIA theory on C^2/Z_N. We also show that many of these results can be extended to arbitrary points in the compactification moduli space using Pi-stability. 
  In these notes we review the theory of the microscopic modeling of the 5-dim. black hole of type IIB string theory in terms of the $D1-D5$ brane system. The emphasis here is more on the brane dynamics rather than on supergravity solutions. We present a discussion of the low energy brane dynamics and account for black hole thermodynamics and Hawking radiation rates. These considerations are valid in the regime of supergravity due to the non-renormalization of the low energy dynamics in this model. 
  We consider a brane-world construction which incorporates a finite region of flat space, ``the box,'' surrounded by a region of anti-de Sitter space. This hybrid construction provides a framework which interpolates between the scenario proposed by Arkani-Hamed, Dimopoulos and Dvali, and that proposed by Randall and Sundrum. Within this composite framework, we investigate the effects of resonant modes on four-dimensional gravity. We also show that, on a probe brane in the anti-de Sitter region, there is enhanced production of on-shell nonresonant modes. We compare our model to some recent attempts to incorporate the Randall-Sundrum scenario into superstring theory. 
  We discuss the possibility of having gravity ``localized'' in dimension d in a system where gauge bosons propagate in dimension d+1. In such a circumstance - depending on the rate of falloff of the field strengths in d dimensions - one might expect the gauge symmetry in d+1 dimensions to behave like a global symmetry in d dimensions, despite the presence of gravity. Naive extrapolation of warped long-wavelength solutions of general relativity coupled to scalars and gauge fields suggests that such an effect might be possible. However, in some basic realizations of such solutions in M theory, we find that this effect does not persist microscopically. It turns over either to screening or the Higgs mechanism at long distances in the d-dimensional description of the system. We briefly discuss the physics of charged objects in this type of system. 
  We study decoupling limits and S-dualities for noncommutative open string/ Yang-Mills theory in a gravity setup by considering an $SL(2,Z)$ invariant supergravity solution of the form ((F, D1), D3) bound state of type IIB string theory. This configuration can be regarded as D3-branes with both electric and magnetic fields turned on along one of the spatial directions of the brane and preserves half of the space-time supersymmetries of the string theory. Our study indicates that there exists a decoupling limit for which the resulting theory is an open string theory defined in a geometry with noncommutativity in both space-time and space-space directions. We study S-duality of this noncommutative open string (NCOS) and find that the same decoupling limit in the S-dual description gives rise to a space-space noncommutative Yang-Mills theory (NCYM). We also discuss independently the decoupling limit for NCYM in this D3 brane background. Here we find that S-duality of NCYM theory does not always give a NCOS theory. Instead, it can give an ordinary Yang-Mills with a singular metric and an infinitely large coupling. We also find that the open string coupling relation between the two S-duality related theories is modified such that S-duality of a strongly coupled open-string/Yang-Mills theory does not necessarily give a weakly coupled theory. The relevant gravity dual descriptions of NCOS/NCYM are also given. 
  We discuss supersymmetry breaking mechanism at the level of low energy N=1 effective superstring actions that exhibit $SL(2,Z)_T$ target space modular duality or $SL(2,Z)_S$ strong-weak coupling duality. The allowed superpotential forms use the assumption that the sourse of non-perturbative effects is not specified and as a result represent the most general parametrization of non-perturbative effects. We found that the allowed non-perturbative superpotential is severely constrained when we use the cusp forms of the modular group for its construction. By construction the poles of the superpotential are either inside the fundamental domain or beyond. We also found limits on the parameters of the superpotential by demanding that the truncated potential for the gaugino condensate never breaks down at finite values in the moduli space. The latter constitutes a criterion for avoiding poles in the fundamental domain. However, the potential in most of the cases avoids naturally singularities inside the fundamental domain, rendering the potential finite. The minimum values of the limits on the parameters in the superpotential may correspond to vacua with vanishing cosmological constant. 
  This paper has been withdrawn by the authors because the action has already been given in hep-th/9907202. 
  We develop a method for computing correlation functions of twist operators in the bosonic 2-d CFT arising from orbifolds M^N/S_N, where M is an arbitrary manifold. The path integral with twist operators is replaced by a path integral on a covering space with no operator insertions. Thus, even though the CFT is defined on the sphere, the correlators are expressed in terms of partition functions on Riemann surfaces with a finite range of genus g. For large N, this genus expansion coincides with a 1/N expansion. The contribution from the covering space of genus zero is `universal' in the sense that it depends only on the central charge of the CFT. For 3-point functions we give an explicit form for the contribution from the sphere, and for the 4-point function we do an example which has genus zero and genus one contributions. The condition for the genus zero contribution to the 3-point functions to be non--vanishing is similar to the fusion rules for an SU(2) WZW model. We observe that the 3-point coupling becomes small compared to its large N limit when the orders of the twist operators become comparable to the square root of N - this is a manifestation of the stringy exclusion principle. 
  We consider the case of an integrable quantum spin chain with "soliton non-peserving" boundary conditions. This is the first time that such boundary conditions have been considered in the spin chain framework. We construct the transfer matrix of the model, we study its symmetry and we find explicit expressions for its eigenvalues. Moreover, we derive a new set of Bethe ansatz equations by means of the analytical Bethe ansatz method. 
  A reliable method to construct Supersymmetric Noether currents is presented. As the most important application the central charge of the N=2 Supersymmetric Yang-Mills effective theory, known as Seiberg-Witten (SW) theory, is computed. The analisys is carried out in the SW low energy U(1) effective Sector, as well as in the SW high energy SU(2) effective Sector. 
  We propose that Kreimer's method of Feynman diagram renormalization via a Hopf algebra of rooted trees can be fruitfully employed in the analysis of block spin renormalization or coarse graining of inhomogeneous statistical systems. Examples of such systems include spin foam formulations of non-perturbative quantum gravity as well as lattice gauge and spin systems on irregular lattices and/or with spatially varying couplings. We study three examples which are Z_2 lattice gauge theory on irregular 2-dimensional lattices, Ising/Potts models with varying bond strengths and (1+1)-dimensional spin foam models. 
  We study a supergravity model of inflation with R-symmetry and a single scalar field, the inflaton, slowly rolling away from the origin. The scales of inflation can be as low as the supersymmetry breaking scale of 10^10 GeV or even the electroweak scale of 10^3 GeV which could be relevant in the context of theories with submillimiter dimensions. Exact analytical solutions are presented and a comparison with related models is given. 
  We investigate in the simplest compact D=4 N=1 Type IIB orientifold models the sigma-model symmetry suggested by the proposed duality of these models to heterotic orbifold vacua. This symmetry is known to be present at the classical level, and is associated to a composite connection involving untwisted moduli in the low-energy supergravity theory. In order to study possible anomalies arising at the quantum level, we compute potentially anomalous one-loop amplitudes involving gluons, gravitons and composite connections. We argue that the effective vertex operator associated to the composite connection has the same form as that for a geometric deformation of the orbifold. Assuming this, we are able to compute the complete anomaly polynomial, and find that all the anomalies are canceled through a Green-Schwarz mechanism mediated by twisted RR axions, as previously conjectured. Some questions about the field theory interpretation of our results remain open. 
  We investigate new compactifications of OM theory giving rise to a 3+1 dimensional open string theory with noncommutative $x^0$-$x^1$ and $x^2$-$x^3$ coordinates. The theory can be directly obtained by starting with a D3 brane with parallel (near critical) electric and magnetic field components, in the presence of a RR scalar field. The magnetic parameter permits to interpolate continuously between the $x^0$-$x^1$ noncommutative open string theory and the $x^2$-$x^3$ spatial noncommutative U(N) super Yang-Mills theory. We discuss $SL(2,Z)$ transformations of this theory. Using the supergravity description of the large $N$ limit, we also compute corrections to the quark-antiquark Coulomb potential arising in the NCOS theory. 
  After presenting string-like solutions with a warp factor to Einstein's equations, we study localization of various spin fields on a string-like defect in a general space-time dimension from the viewpoint of field theory. It is shown that spin 0 and 2 fields are localized on a defect with the exponentially decreasing warp factor. Spin 1 field can be also localized on a defect with the exponentially decreasing warp factor. On the other hand, spin one-half and three-half fields can be localized on a defect with the exponentially increasing warp factor, provided that additional interactions are not introduced. Thus, some mechanism of localization must be invoked for these fermionic fields. These results are very similar to those of a domain wall in five space-time dimensions except the case of spin 1 field. 
  We describe heterotic string and M-theory realizations of the Randall-Sundrum (RS) scenario with $\cx N=2$ and $\cx N=1$ supersymmetry in the bulk. Supersymmetry can be broken only on the world brane, a scenario that has been proposed to account for the smallness of the cosmological constant. An interesting prediction from string duality is the generation of a warp factor for conventional type II Calabi--Yau 3-fold compactifications. On the other hand we argue that an assumption that is needed in the RS explanation of the hierarchy is hard to satisfy in the string theory context. 
  Fermions on the lattice have bosonic excitations generated from the underlying periodic background. These, the lattice bosons, arise near the empty band or when the bands are nearly full. They do not depend on the nature of the interactions and exist for any fermion-fermion coupling. We discuss these lattice boson solutions for the Dirac Hamiltonian. 
  We show that the requirement of the relativistic invariance for any self-interacting, abelian p-form theory uniquely determines the form of the corresponding quantization condition. 
  I consider Lagrangians which depend nonlocally in time but in such a way that there is no mixing between times differing by more than some finite value $\Delta t$. By considering these systems as the limits of ever higher derivative theories I obtain a canonical formalism in which the coordinates are the dynamical variable from $t$ to $t + {\Delta t}$. A simple formula for the conjugate momenta is derived in the same way. This formalism makes apparent the virulent instability of this entire class of nonlocal Lagrangians. As an example, the formalism is applied to a nonlocal analog of the harmonic oscillator. 
  In this paper a new look on the electro-magnetic duality is presented and appropriately exploited. The duality analysis in the nonrelativistic and relativistic formulations is shown to lead to the idea the mathematical model field to be a differential form valued in the 2-dimensional vector space ${\cal R}^2$. A full ${\cal R}^2$ covariance is achieved through introducing explicitly the canonical complex structure ${\cal I}$ of ${\cal R}^2$ in the nonrelativistic equations. The connection of the relativistic Hodge * with ${\cal I}$ is shown and a complete coordinate free relativistic form of the equations and the conservative quantities is obtained. The duality symmetry is interpreted as invariance of the conservative quantities and conservation equations. 
  We show that the classic results of Schwinger on the exact propagation of particles in the background of constant field-strengths and plane waves can be readily extended to the case of non-commutative QED. It is shown that non-perturbative effects on constant backgrounds are the same as their commutative counterparts, provided the on-shell gauge invariant dynamics is referred to a non-perturbatively related space-time frame. For the case of the plane wave background, we find evidence of the effective extended nature of non-commutative particles, producing retarded and advanced effects in scattering. Besides the known `dipolar' character of non-commutative neutral particles, we find that charged particles are also effectively extended, but they behave instead as `half-dipoles'. 
  We study the dispersion relation for scalar excitations in supersymmetric, non-commutative theories at finite temperature. In N=4 Yang-Mills the low momenta modes have superluminous group velocity. In the massless Wess-Zumino model the minimum of the dispersion relation is at non zero momentum for temperatures above T_0 ~ (g \theta)^(-1\2). We briefly comment on N=2 Yang-Mills at finite density. 
  A new entropy bound, tighter than the standard holographic bound due to Bekenstein, is derived for spacetimes with non-rotating isolated horizons, from the quantum geometry approach in which the horizon is described by the boundary degrees of freedom of a three dimensional Chern Simons theory. 
  In the present paper we discuss arguments, favouring the view that massive fermions represent dislocations (i.e. topological solitons) in discrete space-time with Burgers vectors, parallel to an axis of time. If to put symmetrical parts of tensors of distortions (i.e. derivatives of atomic displacements on coordinates) and mechanical stresses equal zero, the equations of the field theory of dislocations get the form of the Maxwell equations. If to consider these tensors as symmetrical, we shall receive the equations of the theory of gravitation, and it turns out that the sum of tensor of distortions and pseudo-Euclidean metrical tensor is the analogue of metrical tensor. It is shown that we can also get Dirac equation with four-fermion interaction in the framework of the field theory of dislocations. This model explains quantization of electrical charge: it is proportional to the topological charge of dislocation, and this charge accepts quantized values because of discrete structure of the 4-dimensional lattice. 
  The theories in which our world presents a domain wall (brane) embedded in large extra dimensions predict new types of topological defects. These defects arise due to the fact that the brane on which we live spontaneously breaks isometries of the extra space giving mass to some graviphotons. In many cases the corresponding vacuum manifold has nontrivial homotopies -- this gives rise to topologically stable defects in four dimensions, such as cosmic strings and monopoles that carry gravimagnetic flux. The core structure of these defects is somewhat peculiar. Due to the fact that the translation invariance in the extra direction(s) is restored in their core, they act as "windows" to the extra dimensions. We also discuss the corresponding analog of the Alice strings. Encircling such an object one would get transported onto a parallel brane. 
  The physical meaning and the geometrical interpretation of causality implementation in classical field theories are discussed. Local causality are kinematical constraints dynamically implemented via solutions of the field equations, but in a limit of zero-distance from the field sources part of these constraints carries a dynamical content that explains old problems of classical electrodynamics away with deep implications to the nature of physical interactions. 
  We consider a recent proposal to solve the cosmological constant problem within the context of brane world scenarios with infinite volume extra dimensions. In such theories bulk can be supersymmetric even if brane supersymmetry is completely broken. We propose a setup where unbroken bulk supersymmetry appears to protect the brane cosmological constant. This is due to a non-trivial scalar potential in the bulk which implies a non-trivial profile for a bulk scalar field. In the presence of the latter bulk supersymmetry appears to be incompatible with non-vanishing brane cosmological constant. Moreover, in this setup the corresponding domain wall interpolates between an AdS and the Minkowski vacua, so that the weak energy condition is not violated. 
  In our previous paper (hep-th/9911087), we proposed a resolution for the fermion doubling problem in discrete field theories based on the fuzzy sphere and its cartesian products. In this paper after a review of that work, we bring out its relationship to the Ginsparg-Wilson approach. 
  Quantum fields on a stationary space-time in a rotating Killing reference frame are considered. Finding solutions of wave equations in this frame is reduced to a fiducial problem on a static background. The rotation results in a gauge connection in a way similar to appearance of gauge fields in Kaluza-Klein models. Such a Kaluza-Klein method in theory of rotating quantum fields enables one to simplify computations and get a number of new results similar to those established for static backgrounds. In particular, we find with its help functional form of free energy at high temperatures. Applications of these results to quantum fields near rotating black holes are briefly discussed. 
  Topological Yang-Mills theory is derived in the framework of Lagrangian BRST cohomology. 
  In this note we present a operator formulation of gauge theories in a quantum phase space which is specified by a operator algebra. For simplicity we work with the Heisenberg algebra. We introduce the notion of the derivative (transport) and Wilson line (parallel transport) which enables us to construct a gauge theory in a simple way. We illustrate the formulation by a discussion of the Higgs mechanism and comment on the large N masterfield. 
  The zero-point energy of a conducting spherical shell is studied by imposing the axial gauge via path-integral methods, with boundary conditions on the electromagnetic potential and ghost fields. The coupled modes are then found to be the temporal and longitudinal modes for the Maxwell field. The resulting system can be decoupled by studying a fourth-order differential equation with boundary conditions on longitudinal modes and their second derivatives. The exact solution of such equation is found by using a Green-function method, and is obtained from Bessel functions and definite integrals involving Bessel functions. Complete agreement with a previous path-integral analysis in the Lorenz gauge, and with Boyer's value, is proved in detail. 
  The zero-point energy of a conducting spherical shell is studied by imposing the axial gauge via path-integral methods, with boundary conditions on the electromagnetic potential and ghost fields. The coupled modes are then found to be the temporal and longitudinal modes for the Maxwell field. The resulting system can be decoupled by studying a fourth-order differential equation with boundary conditions on longitudinal modes and their second derivatives. Complete agreement is found with a previous path-integral analysis in the Lorenz gauge, and with Boyer's value. This investigation leads to a better understanding of how gauge independence is achieved in quantum field theory on backgrounds with boundary. 
  All high-temperature phases of the known N=4 superstrings in five dimensions can be described by the universal thermal potential of an effective four-dimensional supergravity. This theory, in addition to three moduli s, t, u, contains non-trivial winding modes that become massless in certain regions of the thermal moduli space, triggering the instabilities at the Hagedorn temperature. In this context, we look for exact domain wall solutions of first order BPS equations. These solutions preserve half of the supersymmetries, in contrast to the usual finite-temperature weak-coupling approximation, and as such may constitute a new phase of finite-temperature superstrings. We present exact solutions for the type-IIA and type-IIB theories and for a self-dual hybrid type-II theory. While for the heterotic case the general solution cannot be given in closed form, we still present a complete picture and a detailed analysis of the behaviour around the weak and strong coupling limits and around certain critical points. In all cases these BPS solutions have no instabilities at any temperature. Finally, we address the physical meaning of the resulting geometries within the contexts of supergravity and string theory. 
  It is argued that D-brane charge takes values in K-homology. For smooth manifolds with spin structure, this could explain why the phase factor $\Omega(x)$ calculated with a D-brane state x in IIB theory appears in Diaconescu, Moore and Witten's computation of the partition function of IIA string theory. 
  We argue that D-branes corresponding to rational B boundary states in a Gepner model can be understood as fractional branes in the Landau-Ginzburg orbifold phase of the linear sigma model description. Combining this idea with the generalized McKay correspondence allows us to identify these states with coherent sheaves, and to calculate their K-theory classes in the large volume limit, without needing to invoke mirror symmetry. We check this identification against the mirror symmetry results for the example of the Calabi-Yau hypersurface in $\WP^{1,1,2,2,2}$. 
  We study the compactification of OM-theory on tori and show a simple heuristic derivation of the $S$-duals of noncommutative open string theory in diverse dimensions from the OM-theoretical point of view. In particular, we identify the $S$-duality between noncommutative open string theory and noncommutative Yang-Mills theory in $(3+1)$ dimensions as the exchange of two circles of a torus on which OM-theory is compactified. Also, we briefly discuss $T$-duality between noncommutative open string theories. 
  We explore a mechanism to obtain the observational small value for the 4-dimensional vacuum energy through an exponential warp-factor suppression. Intriguingly the required suppression scale relates directly to the GUT scale. We demonstrate the mechanism explicitly in a 5-dimensional brane-world setup with warped geometry. Upon lifting the setup to 10-dimensional IIB string-theory, the relevance of the GUT scale becomes clear as the IIB string-theory description, which is based on D3-brane stacks, gives rise to a spontaneously broken SU(5) supersymmetric GUT theory with low-energy MSSM spectrum and Higgs doublet-triplet splitting. 
  The Batalin-Fradkin-Tyutin (BFT) scheme, which is an improved version of Dirac quantization, is applied to the $CP^1$ model, and the compact form of a nontrivial first-class Hamiltonian is directly obtained by introducing the BFT physical fields. We also derive a BRST-invariant gauge fixed Lagrangian through the standard path-integral procedure. Furthermore, performing collective coordinate quantization we obtain energy spectrum of rigid rotator in the $CP^1$ model. Exploiting the Hopf bundle, we also show that the $CP^1$ model is exactly equivalent to the O(3) nonlinear sigma model at the canonical level. 
  We derive the finite temperature description of bosonic D-branes in the thermo field approach. The results might be relevant to the study of thermical properties of D-brane systems. 
  We show that while the zero temperature induced fermion number in a chiral sigma model background depends only on the asymptotic values of the chiral field, at finite temperature the induced fermion number depends also on the detailed shape of the chiral background. We resum the leading low temperature terms to all orders in the derivative expansion, producing a simple result that can be interpreted physically as the different effect of the chiral background on virtual pairs of the Dirac sea and on the real particles of the thermal plasma. By contrast, for a kink background, not of sigma model form, the finite temperature induced fermion number is temperature dependent but topological. 
  We demonstrate that particle production for fermions coupled chirally to an Abelian gauge field like the hypercharge field is provided by the microscopic mechanism of level crossing. For this purpose we use recent results on zero modes of Dirac operators for a class of localized hypermagnetic knots. 
  We describe a class of six-dimensional conformal field theories that have some properties in common with and possibly are related to a subsector of the tensionless string theories. The latter theories can for example give rise to four-dimensional $N = 4$ superconformal Yang-Mills theories upon compactification on a two-torus. Just like the tensionless string theories, our theories have an $ADE$-classification, but no other discrete or continuous parameters. The Hilbert space carries an irreducible representation of the same Heisenberg group that appears in the tensionless string theories, and the `Wilson surface' observables obey the same superselection rules. When compactified on a two-torus, they have the same behaviour under $S$-duality as super Yang-Mills theory. Our theories are natural generalizations of the two-form with self-dual field strength that is part of the world-volume theory of a single five-brane in $M$-theory, and the $A_{N - 1}$ theory can in fact be seen as arising from $N$ non-interacting chiral two-forms by factoring out the collective `center of mass' degrees of freedom. 
  The general model of higher derivative (HD) gravity is considered. The search of brane-world cosmology in such theory is presented when bulk is d5 AdS and boundary is spherical, hyperbolic or flat (single) brane. It is found the wide range of theory parameters where such cosmology may be realized. Special attention is paid to the version of HD theory representing SG dual of ${\cal N}=2$ $Sp(N)$ SCFT (in next-to-leading order of large $N$ expansion). In particular, it is shown that inflationary brane Universe does not occur for SG dual while hyperbolic brane occurs (which was not possible in leading order).   The quantum effects of CFT living on the brane (via the corresponding conformal anomaly induced effective action) may qualitatively change the results of classical analysis. There appears inflationary (or hyperbolic) brane Universe induced by only quantum effects. In AdS/CFT correspondence (next-to-leading order) the addition of such CFT effective action (in some energy region) is naturally explained in terms of holographic renormalization group. It results in the possibility of quantum creation of inflationary brane Universe (with small rate) even for SG dual. 
  We study various aspects of parafermionic theories such as the precise field content, a description of a basis of states (that is, the counting of independent states in a freely generated highest-weight module) and the explicit expression of the parafermionic singular vectors in completely irreducible modules. This analysis culminates in the presentation of new character formulae for the $Z_N$ parafermionic primary fields. These characters provide novel field theoretical expressions for $\su(2)$ string functions. 
  This note addresses the question of the number of normalizable vacuum states in supersymmetric quantum mechanics with sixteen supercharges and arbitrary semi-simple compact gauge group, up to rank three. After evaluating certain contour integrals obtained by appropriately adapting BRST deformation techniques we propose novel rational values for the bulk indices. Our results demonstrate that an asymptotic method for obtaining the boundary contribution to the index, originally due to Green and Gutperle, fails for groups other than SU(N). We then obtain likely values for the number of ground states of these systems. In the case of orthogonal and symplectic groups our finding is consistent with recent conjectures of Kac and Smilga, but appears to contradict their result in the case of the exceptional group G_2. 
  Space-time non-commutative theories are non-local in time. We develop the Hamiltonian formalism for non-local field theories in d space-time dimensions by considering auxiliary d+1 dimensional field theories which are local with respect to the evolution time. The Hamiltonian path integral quantization is considered and the Feynman rules in the Lagrangian formalism are derived. The case of non-commutative \phi^3 theory is considered as an example. 
  We show that field theories with light-like noncommutativity, that is with $\theta^{0i}=-\theta^{1i}$, are unitary quantum theories, and that they can be obtained as decoupled field theory limits of string theory with D-branes in a background NS-NS $B$ field. For general noncommutativity parameters, we show that noncommutative field theories which are unitary can be obtained as decoupled field theory limits of string theory, while those that are not unitary cannot be obtained from string theory because massive open strings do not decouple. We study the different theories with light-like noncommutativity which arise from Type II D-branes. The decoupling limit of the D4-brane seems to lead to a noncommutative field theory deformation of the $(2,0)$ SCFT of M5-branes, while the D5-brane case leads to a noncommutative variation of ``little string theories''. We discuss the DLCQ description of these theories. 
  Conformal symmetry is taken as an attribute of theories of massless fields in manifolds with specific dimensionalities. This paper shows that this is not an absolute truth; it is a consequence of the mathematical representation used for the physical interactions. It introduces a new kind of representation where the propagation of massive (invariant mass) and massless interactions are unifiedly described by a single conformally symmetric Green's function. Sources and fields are treated at a same footing, symmetrically, as discrete fields - the fields in this new representation - fields defined with support on straight lines embedded in a (3+1)-Minkowski manifold. The discrete field turns out to be a point in phase space. It is finite everywhere. With a finite number of degrees of freedom it does not share the well known problems faced by the standard continuous formalism which can be retrieved from the discrete one by an integration over a hypersurface. The passage from discrete to continuous fields illuminates the physical meaning and origins of their properties and problems. The price for having massive discrete field with conformal symmetry is of hiding its mass and time-like velocity behind its non-constant proper-time. 
  Deformed Heisenberg algebra with reflection appeared in the context of Wigner's generalized quantization schemes underlying the concept of parafields and parastatistics of Green, Volkov, Greenberg and Messiah. We review the application of this algebra for the universal description of ordinary spin-$j$ and anyon fields in 2+1 dimensions, and discuss the intimate relation between parastatistics and supersymmetry. 
  We present direct arguments for non-commutativity of spheres in the AdS/CFT correspondence. The discussion is based on results for the $S_N$ orbifold SCFT. Concentrating on three point correlations (at finite $N$) we exhibit a comparison with correlations on a non-commutative sphere. In this manner an essential signature of non-commutativity is identified giving further support for the original proposal of hep-th/9902059. 
  We develop an efficient recursive method to evaluate the tachyon potential using the relevant universal subalgebra of the open string star algebra. This method, using off-shell versions of Virasoro Ward identities, avoids explicit computation of conformal transformations of operators and does not require a choice of background. We illustrate the procedure with a pedagogic computation of the level six tachyon potential in an arbitrary gauge, and the evaluation of a few simple star products. We give a background independent construction of the so-called identity of the star algebra, and show how it fits into family of string fields generating a commutative subalgebra. 
  We investigate the physical metric seen by a D0-brane probe in the background geometry of an N=2 sigma model. The metric is evaluated by calculating the Zamolodchikov metric for the disc two point function of the boundary operators corresponding to the displacement of the D0-brane boundary. At two loop order we show that the D0 metric receives an $R^2$ contribution. 
  We derive a closed expression for the SU(2) Born-Infeld action with the symmetrized trace for static spherically symmetric purely magnetic configurations. The lagrangian is obtained in terms of elementary functions. Using it, we investigate glueball solutions to the flat space NBI theory and their self-gravitating counterparts. Such solutions, found previously in the NBI model with the 'square root - ordinary trace' lagrangian, are shown to persist in the theory with the symmetrized trace lagrangian as well. Although the symmetrized trace NBI equations differ substantially from those of the theory with the ordinary trace, a qualitative picture of glueballs remains essentially the same. Gravity further reduces the difference between solutions in these two models, and, for sufficiently large values of the effective gravitational coupling, solutions tends to the same limiting form. The black holes in the NBI theory with the symmetrized trace are also discussed. 
  The generalization of the Yang-Baxter equations (YBE) in the presence of Z_2 grading along both chain and time directions is presented and an integrable model of t-J type with staggered disposition along a chain of shifts of the spectral parameter is constructed. The Hamiltonian of the model is computed in fermionic formulation. It involves three neighbour site interactions and therefore can be considered as a zig-zag ladder model. The Algebraic Bethe Ansatz technique is applied and the eigenstates, along with eigenvalues of the transfer matrix of the model are found. In the thermodynamic limit, the lowest energy of the model is formed by the quarter filling of the states by fermions instead of usual half filling. 
  The Seiberg-Witten analysis of the low-energy effective action of d=4 N=2 SYM theories reveals the relation between the Donaldson and Seiberg-Witten (SW) monopole invariants. Here we apply analogous reasoning to d=3 N=4 theories and propose a general relationship between Rozansky-Witten (RW) and 3-dimensional Abelian monopole invariants. In particular, we deduce the equality of the SU(2) Casson invariant and the 3-dimensional SW invariant (this includes a special case of the Meng-Taubes theorem relating the SW invariant to Milnor torsion). Since there are only a finite number of basic RW invariants of a given degree, many different topological field theories can be used to represent essentially the same topological invariant. This leads us to advocate using higher rank Abelian gauge theories to shed light on the higher (non-Abelian) RW invariants and we write down candidate higher rank SW equations. 
  The non commutative geometry is a possible framework to regularize Quantum Field Theory in a nonperturbative way. This idea is an extension of the lattice approximation by non commutativity that allows to preserve symmetries. The supersymmetric version is also studied and more precisely in the case of the Schwinger model on supersphere [14]. This paper is a generalization of this latter work to more general gauge groups. 
  An enveloping algebra valued gauge field is constructed, its components are functions of the Lie algebra valued gauge field and can be constructed with the Seiberg-Witten map. This allows the formulation of a dynamics for a finite number of gauge field components on non-commutative spaces. 
  The spectrum of D2-branes wrapped on an ALE space of general ADE type is determined, by representing them as boundary states of N=2 superconformal minimal models. The stable quantum states have RR charges which precisely represent the gauge fields of the corresponding Lie algebra. This provides a simple and direct physical link between the ADE classification of N=2 superconformal field theories, and the corresponding root systems. An affine extension of this structure is also considered, whose boundary states represent the D2-branes plus additional D0-branes. 
  We present an alternative derivation of the parity anomaly for a massless Dirac field in 2+1 dimensions coupled to a gauge field. The anomaly functional, a Chern-Simons action for the gauge field, is obtained from the non-trivial Jacobian corresponding to a non local symmetry of the Pauli-Villars regularized action. That Jacobian is well-defined, finite, and yields the standard Chern-Simons term when the cutoff tends to infinity. 
  We show that the class of functions of topologically nontrivial gauge transformations in QCD includes a zero-mode of the Gauss law constraint. The equivalent unconstrained system compatible with Feynman's integral is derived in terms of topological invariant variables, where the zero-mode is identified with the winding number collective variable and leads to the dominance of the Wu-Yang monopole. Physical consequences of Feynman's path integral in terms of the topological invariant variables are studied. 
  The physical meaning, the properties and the consequences of a discrete scalar field are discussed; limits for a continuous mathematical description of fundamental physics is a natural outcome of discrete fields with discrete interactions. The discrete scalar field is ultimately the gravitational field of general relativity, necessarily, and there is no place for any other fundamental scalar field, in this context. 
  We consider spherically symmetric higher-dimensional solutions of Einstein's equations with a bulk cosmological constant and n transverse dimensions. In contrast to the case of one or two extra dimensions we find no solutions that localize gravity when $n\geq 3$, for strictly local topological defects. We discuss global topological defects that lead to the localization of gravity and estimate the corrections to Newton's law. We show that the introduction of a bulk ``hedgehog'' magnetic field leads to a regular geometry and localizes gravity on the 3-brane with either a positive, zero or negative bulk cosmological constant. The corrections to Newton's law on the 3-brane are parametrically the same as for the case of one transverse dimension. 
  We show how the gravity, gauge, and matter fields are induced on dynamically localized brane world (domain wall, vortex etc.). Solitonic solutions localized on a curved brane in certain field theories in higher dimensions are given in terms of gravity, gauge and extrinsic curvature fields of the brane. Then we deduce the effective field theory on the brane in terms of those fields and the quantum fluctuations of the solitonic solution. The gravity, gauge and the extrinsic curvature fields should obey the Gauss-Codazzi-Ricci equation in addition to their own equations of motion. 
  A mechanism for localization of quantum fields on a $s$-brane, representing the boundary of a s+2 dimensional bulk space, is investigated. Minkowski and AdS bulk spaces are analyzed. Besides the background geometry, the relevant parameters controlling the theory are the mass M and a real parameter \eta, specifying the boundary condition on the brane. The importance of exploring the whole range of allowed values for these parameters is emphasized. Stability in Minkowski space requires \eta to be greater or equal to -M, whereas in the AdS background all real \eta are permitted. Both in the flat and in AdS case, the induced field on the brane is a non-canonical generalized free field. For a suitable choice of boundary condition, corresponding to the presence of a boundary state, the induced field on the brane mimics standard s+1 dimensional physics. In a certain range of \eta, the spectral function in the the AdS case is ominated by a massive excitation, which imitates the presence of massive particle on the brane. We show that the quantum field induced on the brane is stable. 
  The quantum space-time model which accounts material Reference Frames (RF) quantum effects considered for flat space-time and ADM canonical gravity. As was shown by Aharonov for RF - free material object its c.m. nonrelativistic motion in vacuum described by Schrodinger wave packet evolution which modify space coordinate operator of test particle in this RF and changes its Heisenberg uncertainty relations. In the relativistic case we show that Lorentz transformations between two RFs include the quantum corrections for RFs momentum uncertainty and in general can be formulated as the quantum space-time transformations. As the result for moving RF its Lorentz time boost acquires quantum fluctuations which calculated solving relativistic Heisenberg equations for the quantum clocks models. It permits to calculate RF proper time for the arbitrary RF quantum motion including quantum gravity metrics fluctuations. Space-time structure of canonical Quantum Gravity and its observables evolution for RF proper time discussed in this quantum space-time transformations framework. 
  A thin shell of light-like dust with its own gravitational field is studied in the special case of spherical symmetry. The action functional for this system due to Louko, Whiting, and Friedman is reduced to Kucha\v{r} form: the new variables are embeddings, their conjugate momenta, and Dirac observables. The concepts of background manifold and covariant gauge fixing, that underlie these variables, are reformulated in a way that implies the uniqueness and gauge invariance of the background manifold. The reduced dynamics describes motion on this background manifold. 
  The dynamics of a thin spherically symmetric shell of zero-rest-mass matter in its own gravitational field is studied. A form of action principle is used that enables the reformulation of the dynamics as motion on a fixed background manifold. A self-adjoint extension of the Hamiltonian is obtained via the group quantization method. Operators of position and of direction of motion are constructed. The shell is shown to avoid the singularity, to bounce and to re-expand to that asymptotic region from which it contracted; the dynamics is, therefore, truly unitary. If a wave packet is sufficiently narrow and/or energetic then an essential part of it can be concentrated under its Schwarzschild radius near the bounce point but no black hole forms. The quantum Schwarzschild horizon is a linear combination of a black and white hole apparent horizons rather than an event horizon. 
  According to 't Hooft (Class.Quantum.Grav. 16 (1999), 3263), quantum gravity can be postulated as a dissipative deterministic system, where quantum states at the ``atomic scale''can be understood as equivalence classes of primordial states governed by a dissipative deterministic dynamics law at the ``Planck scale''. In this paper, it is shown that for a quantum system to have an underlying deterministic dissipative dynamics, the time variable should be discrete if the continuity of its temporal evolution is required. Besides, the underlying deterministic theory also imposes restrictions on the energy spectrum of the quantum system. It is also found that quantum symmetry at the ``atomic scale'' can be induced from 't Hooft's Coarse Graining classification of primordial states at the "Planck scale". 
  We find some exact solutions of the Knizhnik-Zamolodchikov equation for the four point correlation functions that occur in the SL(2,R) WZNW model. They exhibit logarithmic behaviour in both the Kac-Moody and Virasoro parts. We discuss their implication for the operator product expansion. We also observe the appearance of several symmetries of the correlation functions. 
  We give the definition and explore the algebraic structure of a class of quantum symmetries, called topological symmetries, which are generalizations of supersymmetry in the sense that they involve topological invariants similar to the Witten index. A topological symmetry (TS) is specified by an integer n>1, which determines its grading properties, and an n-tuple of positive integers (m_1,m_2,...,m_n). We identify the algebras of supersymmetry, p=2 parasupersymmetry, and fractional supersymmetry of order n with those of the Z_2-graded TS of type (1,1), Z_2-graded TS of type (2,1), and Z_n-graded TS of type (1,1,...,1), respectively. We also comment on the mathematical interpretation of the topological invariants associated with the Z_n-graded TS of type (1,1,...,1). For n=2, the invariant is the Witten index which can be identified with the analytic index of a Fredholm operator. For n>2, there are n independent integer-valued invariants. These can be related to differences of the dimension of the kernels of various products of n operators satisfying certain conditions. 
  We construct an Anti-de Sitter(AdS) algebra in a nondegenerate superspace. Based on this algebra we construct a covariant kappa-symmetric superstring action, and we examine its dynamics: Although this action reduces to the usual Green-Schwarz superstring action in flat limit, the auxiliary fermionic coordinates of the nondegenerate superspace becomes dynamical in the AdS background. 
  We review several topics related to the diagonalization of quantum field Hamiltonians using the quasi-sparse eigenvector (QSE) method. 
  Dp-branes placed in a certain external RR (p+4)-form field expand into a transverse fuzzy two-sphere, as shown by Myers. We find that by changing the (p+4)-form background other fuzzy cosets can be obtained. Three new examples, S^2 X S^2, CP^2 and SU(3)/(U(1) X U(1)) are constructed. The first two are four-dimensional while the last is six-dimensional. The dipole and quadrupole moments which result in these configurations are discussed. Finally, the gravity backgrounds dual to these vacua are examined in a leading order approximation. These are multi-centered solutions containing (p+4)- or (p+6)-dimensional brane singularities. 
  There is a longstanding puzzle concerned with the existence of Op~-planes with p>=6, which are orientifold p-planes of negative charge with stuck Dp-branes. We study the consistency of configurations with various orientifold planes and propose a resolution to this puzzle. It is argued that O6~-planes are possible in massive IIA theory with odd cosmological constant, while O7~-planes and O8~-planes are not allowed. Various relations between orientifold planes and non-BPS D-branes are also addressed. 
  Logarithmic spin-1/3 superconformal field theories are investigated. the chiral and full two-point functions of two-(or more-) dimensional Jordanian blocks of arbitrary weights, are obtained. 
  Solitonic brane cosmologies are found where the world-volume is curved due to the evolution of the dilaton field on the brane. In many cases, these may be related to the solitonic Dp- and M5-branes of string and M-theory. An eleven-dimensional interpretation of the D8-brane cosmology of the massive type IIA theory is discussed in terms of compactification on a torus bundle. Braneworlds are also found in Horava-Witten theory compactified on a Calabi-Yau three-fold. The possibility of dilaton-driven inflation on the brane is discussed. 
  We apply a (Moyal) deformation quantization to a bicomplex associated with the classical nonlinear Schrodinger equation. This induces a deformation of the latter equation to noncommutative space-time while preserving the existence of an infinite set of conserved quantities. 
  The fermion generation puzzle has survived into this century as one of the great mysteries in particle physics. We consider here a possible solution within the Standard Model framework based on a nonabelian generalization of electric-magnetic duality. First, by constructing in loop space a nonabelian generalization of the abelian dual transform (Hodge *), one finds that a ``magnetic'' symmetry exists also in classical Yang-Mills theory dual to the original (``electric'') gauge symmetry. Secondly, from a result of 't Hooft's, one obtains that for confined colour SU(3), the dual symmetry $\widetilde{SU}(3)$ is spontaneously broken and can play the role of the ``horizontal symmetry'' for generations. Thirdly, such an identification not only offers an explanation why there should be three and apparently only three generations of fermions with the remarkable mass and mixing patterns seen in experiment, but allows even a calculation of the relevant parameters giving very sensible results. Other testible predictions follow ranging from rare hadron decays to cosmic ray air showers. 
  Recent developments in string and M theory rely heavily on supersymmetry suggesting that a revival of superspace techniques in ten and eleven dimensions may be advantageous. Here we discuss three topics of current interest where superspace is already playing an important role and where an improved understanding of superspace might provide additional insight into the issues involved. 
  In the first part of this paper we find supergravity solutions corresponding to branes on worldvolumes of the form $R^d \times \Sigma$ where $\Sigma$ is a Riemann surface. These theories arise when we wrap branes on holomorphic Riemann surfaces inside $K3$ or CY manifolds. In some cases the theory at low energies is a conformal field theory with two less dimensions. We find some non-singular supersymmetric compactifications of M-theory down to $AdS_5$. We also propose a criterion for permissible singularities in supergravity solutions.  In the second part of this paper, which can be read independently of the first, we show that there are no non-singular Randall-Sundrum or de-Sitter compactifications for large class of gravity theories. 
  We study non-linear electrodynamics in curved space from the viewpoint of dualities. After establishing the existence of a topological bound for self-dual configurations of Born-Infeld field in curved space, we check that the energy-momentum tensor vanishes. These properties are shown to hold for general duality-invariant non-linear electrodynamics. We give the dimensional reduction of Born-Infeld action to three dimensions in a general curved background admitting a Killing vector. The SO(2) duality symmetry becomes manifest but other symmetries present in flat space are broken, as is U-duality when one couples to gravity. We generalize our arguments on duality to the case of n U(1) gauge fields, and present a new Lagrangian possessing SO(n) X SO(2)_elemag duality symmetry. Other properties of this model such as Legendre duality and enhancement of the symmetry by adding dilaton and axion, are studied. We extend our arguments to include a background b-field in the curved space, and give new examples including almost Kaehler manifolds and Schwarzshild black holes with a $b$-field. 
  We calculate the genus-one three- and four-point amplitudes in the 2+2 dimensional closed N=(2,2) worldsheet supersymmetric string within the RNS formulation. Vertex operators are redefined with the incorporation of spinor helicity techniques, and the quantum scattering is shown to be manifestly gauge and Lorentz invariant after normalizing the string states. The continuous spin structure summation over the monodromies of the worldsheet fermions is carried out explicitly, and the field-theory limit is extracted. The amplitude in this limit is shown to be the maximally helicity violating amplitude in pure gravity evaluated in a two-dimensional setting, which vanishes, unlike the four-dimensional result. The vanishing of the genus-one N=2 closed string amplitude is related to the absence of one-loop divergences in dimensionally regulated IIB supergravity. Comparisons and contrasts between self-dual field theory and the N=2 string theory are made at the quantum level; they have different S-matrices. Finally, we point to further relations with self-dual field theory and two-dimensional models. 
  Conformal field theory of the D1/D5 system and superstrings on $AdS_3\times S^3\times T^4$ is studied with particular attention to the world-sheet fields corresponding to the $T^4$ part. A solution to the spacetime N=4 superconformal symmetry doubling and other problems is proposed. It is argued that the relevant spacetime symmetry should be based on the middle N=4 superconformal algebra. It is discussed as to why this superconformal structure has been missed so far. 
  We have recently proposed a dynamical mechanism that may realize a flat four-dimensional space time as a brane in type IIB superstring theory. A crucial role is played by the phase of the chiral fermion integral associated with the IKKT Matrix Theory, which is conjectured to be a nonperturbative definition of type IIB superstring theory. We demonstrate our mechanism by studying a simplified model, in which we find that a lower-dimensional brane indeed appears dynamically. We also comment on some implications of our mechanism on model building of the brane world. 
  We propose a new point of view to gauge theories based on taking the action of symmetry transformations directly on the coordinates of space. Via this approach the gauge fields are not introduced at the first step, and they can be interpreted as fluctuations around some classical solutions of the model. The new point of view is connected to the lattice formulation of gauge theories, and the parameter of noncommutativity of coordinates appears as the lattice spacing parameter. Through the statements concerning the continuum limit of lattice gauge theories, this suggestion arises that noncommutative spaces are the natural ones to formulate gauge theories at strong coupling. Via this point of view, a close relation between the large-N limit of gauge theories and string theory can be manifested. 
  We construct six- and four-dimensional toroidal compactifications of the Type I string with magnetic flux on the D-branes. The open strings in this background probe a noncommutative internal geometry. Phenomenologically appealing features such as chiral fermions and supersymmetry breaking in the gauge sector are naturally realized by these vacua. We investigate the spectra of such noncommutative string compactifications and in a bottom-up approach discuss the possibility to obtain the standard or some GUT like model. 
  We consider the dimensional reduction of a bulk scalar field in the Randall-Sundrum model. By allowing the scalar field to be non-minimally coupled to the spacetime curvature we show that it is possible to generate spontaneous symmetry breaking on the brane. 
  We calculate the R-R zero-norm states of type II string spectrum. To fit these states into the right symmetry charge parameters of the gauge transformations of the R-R tensor forms, one is forced to T-dualize some type I open string space-time coordinates and thus to introduce D-branes into the theory. We also demonstrate that the constant T-dual R-R 0-form zero-norm state, together with the NS-NS singlet zero-norm state are responsible for the SL(2,Z) S-duality symmetry of the type II B string theory. 
  We find that Euclidian or Minkowski gravity in d dimensions can be formally expressed as the restriction to a slice of a supersymmetric Yang-Mills theory in d+1 dimensions with SO(d+1), SO(d,1) or SO(d-1,2) internal symmetry. We suggest that renormalization effects in the bulk imply a contraction of the latter symmetry into the Poincare group ISO(d) or ISO(d-1,1). 
  We discuss the validity of the holographic principle in a $(4+n)$ dimensional universe in an asymmetric inflationary phase. 
  A five-dimensional solution to Einstein's equations coupled to a scalar field has been proposed as a partial solution to the cosmological constant problem: the effect of arbitrary vacuum energy (tension) of a 3-brane is cancelled; however, the scalar field becomes singular at some finite proper distance in the extra dimension. We show that in the original model with a vanishing bulk potential for the scalar, the solution is a saddle point which is unstable to expansion or contraction of the brane world. We construct exact time-dependent solutions which generalize the static solution, and demonstrate that they do not conserve energy on the brane; thus they do not have an effective 4-D field theoretic description. When a bulk scalar field potential is added, the boundary conditions on the brane cannot be trivially satisfied, raising hope that the self-tuning mechanism may still give some insight into the cosmological constant problem in this case. 
  Previously matrix model actions for ``fuzzy'' fields have been proposed using non-commutative geometry. They retained ``topological'' properties extremely well, being capable of describing instantons, $\theta$--states, the chiral anomaly, and even chiral fermions with no ``doubling''. Here, we demonstrate that the standard scalar and spinor actions on a $d$--dimensional manifold are recovered from such actions in the limit of large matrices if their normalizations are correctly scaled as the limit is taken. 
  It has been recently argued by some authors that is impossible to construct a Weyl invariant spinning membrane action, where the $S$-supersymmetry associated with the 3D superconformal algebra, is relinquished without gauge fixing. Contrary to those assertions, we show why it is possible to construct a Weyl-invariant spinning polynomial membrane action, without curvature terms,where $both$ the conformal boost symmetry and $S$-supersymmetry are explicitly broken by the action. It is shown that the gauge algebra $closes$ despite that the two latter symmetries are broken . For this to happen, a modifed $Q$-supersymmetry transformation, a sort of new $Q+K+S$ ``sum `` rule, is required that generates the compensating terms to cancel the spurious contributions fromthe $S$ and conformal boost anomalous transformations. A substantial discussion of the quantization of the spinning membrane and anomalies is given. We review briefly the role that this spinning membrane action may have in the theory of $D$-branes, Skyrmions and BPS monopoles in the large $N$-limit of SU(N) Yang-Mills . 
  A combinatorial formula of G.-C. Rota and J.A. Stein is taken to perform Wick re-ordering in quantum field theory. Wick's theorem becomes a Hopf algebraic identity called Cliffordization. The combinatorial method relying on Hopf algebras is highly efficient in computations and yields closed algebraic expressions.   AMS Subject Classifications 2000: 81R50; 16W30 
  By considering the area preserving geometric transformations in the configuration space of electrons moving in the lowest Landau level (LLL) we arrive at the Chern-Simons type Lagrangian. Imposing the LLL condition, we get a scheme with the complex gauge fields and transformations. Quantum theory for the matter field in LLL is considered and formal expressions for Read's operator and Laughlin wave function are presented in the second quantized form. 
  We consider the parity invariant (2+1)-dimensional QED where the matter is represented as a mixture of fermions with opposite spins. It is argued that the perturbative ground state of the system is unstable with respect to the formation of magnetized ground state. Carrying out the finite temperature analysis we show that the magnetic instability disappears in the high temperature regime. 
  In this paper, the low-energy effective dynamics of M-theory, eleven-dimensional supergravity, is taken off-shell in a manifestly supersymmetric formulation. We show that a previously proposed relaxation of the superspace torsion constraints does indeed accommodate a current supermultiplet which lifts the equations of motion corresponding to the ordinary second order derivative supergravity lagrangian. Whether the auxiliary fields obtained this way can be used to construct an off-shell lagrangian is not yet known. We comment on the relation and application of this completely general formalism to higher-derivative (R^4) corrections. Some details of the calculation are saved for a later publication. 
  As part of program to quantize superstrings in AdS_5 x S^5 background in light cone approach we find the explicit form of the corresponding Green-Schwarz action in fermionic light-cone kappa-symmetry gauge. The resulting action is quadratic and quartic in fermions. In the flat space limit it reduces to the standard light-cone Green-Schwarz action, and also has the correct superparticle limit. We discuss fixing the bosonic light-cone gauge and a reformulation of the action in terms of 2-d Dirac spinors. 
  Anomaly cancellation for M-theory fivebranes requires the introduction of a "bump-form" which smoothes out the five-brane source. We discuss the physical origin of this bump-form in the simpler case of axion strings in 3+1 dimensions and construct it in terms of the radial profile of the fermion zero modes. Our treatment allows for a clearer understanding of the role played by covariant rather than consistent anomalies when anomalies are canceled by inflow from the bulk. We briefly discuss the generalization of these results to fivebrane anomalies in M theory. 
  Duality arguments suggest the existence of massless magnetic monopoles in gauge theories with the symmetry broken to a non-Abelian subgroup. I discuss how these arise and show how they are manifested as clouds of massless fields surrounding massive monopoles. The dynamics of these clouds is discussed, and the scattering of massless monopole clouds and massive monopoles is described. 
  We present static solutions to Einstein's equations corresponding to the most general network of $N$ orthogonal families of $(2+N)$-branes in $(4+N)$-dimensional $AdS$ spacetimes. The bulk cosmological constant can take a different value in each cell enclosed by intersecting branes and the extra dimensions can be compact or noncompact. In each family of branes the inter-brane spacing is arbitrary. The extra dimensions may be any or all of the manifolds, $R^1$, $R^1/Z_2$, $S^1$ and $S^1/Z_2$. Only when the extra dimensions are $R^1$ or/and $R^1/Z_2$, can networks consisting solely of positive tension branes be constructed. Such configurations may find application in models with localized gravity and symmetry breaking by shining. 
  The integral representations for the eigenfunctions of $N$ particle quantum open and periodic Toda chains are constructed in the framework of Quantum Inverse Scattering Method (QISM). Both periodic and open $N$-particle solutions have essentially the same structure being written as a generalized Fourier transform over the eigenfunctions of the $N-1$ particle open Toda chain with the kernels satisfying to the Baxter equations of the second and first order respectively. In the latter case this leads to recurrent relations which result to representation of the Mellin-Barnes type for solutions of an open chain. As byproduct, we obtain the Gindikin-Karpelevich formula for the Harish-Chandra function in the case of $GL(N,\RR)$ group. 
  We develop furthur the correspondence between a d+1 dimensional theory and a d dimensional one with the "radial" (d+1)th corodinate \rho playing the role of an evolution parameter. We discuss the evolution of an effective action defined on a d dimensional surface charactarized by \rho by means of a new variational principle. The conditions under which the flow equations are valid are discussed in detail as is the choice of boundary conditions. It is explained how domain walls may be incorporated in the framework and some generalized junction conditions are obtained. The general principles are illustrated on the example of a supergravity theory on AdS_{d+1}. 
  Recently Magnea hep-th/9907096 , hep-th/9912207 [Phys.Rev.D61, 056005 (2000); Phys.Rev.D62, 016005 (2000)] claimed to have computed the first sum rules for Dirac operators in 3D gauge theories from 0D non-linear sigma models. I point out that these computations are incorrect, and that they contradict with the exact results for the spectral densities unambiguously derived from random matrix theory by Nagao and myself. 
  We derive nonperturbative classical solutions of noncommutative U(1) gauge theory, with or without a Higgs field, representing static magnetic flux tubes with arbitrary cross-section. The fields are nonperturbatively different from the vacuum in at least some region of space. The flux of these tubes is quantized in natural units. We also point out that magnetic monopole charge can be fractionized by embedding the monopoles in a constant magnetic field. 
  We develop the concept of supersymmetry in singular spaces, apply it in an example for 3-branes in D=5 and comment on 8-branes in D=10. The new construction has an interpretation that the brane is a sink for the flux and requires adding to the standard supergravity a (D-1)-form field and a supersymmetry singlet field. This allows a consistent definition of supersymmetry on a S_1/Z_2 orbifold, the bulk and the brane actions being separately supersymmetric. Randall-Sundrum brane-worlds can be reproduced in this framework without fine tuning. For fixed scalars, the doubling of unbroken supersymmetries takes place and the negative tension brane can be pushed to infinity. In more general BPS domain walls with 1/2 of unbroken supersymmetries, the distance between branes in some cases may be restricted by the collapsing cycles of the Calabi-Yau manifold. The energy of any static x^5-dependent bosonic configuration vanishes, E=0, in analogy with the vanishing of the Hamiltonian in a closed universe. 
  It is explained that the chiral WZNW phase space is a quasi-Poisson space with respect to the `canonical' Lie quasi-bialgebra which is the classical limit of Drinfeld's quasi-Hopf deformation of the universal enveloping algebra. This exemplifies the notion of quasi-Poisson-Lie symmetry introduced recently by Alekseev and Kosmann-Schwarzbach, and also permits us to generalize certain dynamical twists considered previously in this example. 
  We outline the description of Quantum Mechanics with noncommuting coordinates within the framework of star operation. We discuss simple cases of integrability. 
  For Connes' spectral triples, the group of automorphisms lifted to the Hilbert space is defined and used to fluctuate the metric. A few commutative examples are presented including Chamseddine and Connes' spectral unification of gravity and electromagnetism. One almost commutative example is treated: the full standard model. Here the lifted automorphisms explain O'Raifeartaigh's reduction $SU(2)\times U(3)/\zz_2.$ 
  It is advocated that the superembedding approach is a generic covariant method for the description of superbranes as models of (partial) spontaneous supersymmetry breaking. As an illustration we construct (in the framework of superembeddings) an n=1, d=3 worldvolume superfield action for a supermembrane propagating in N=1, D=4,5,7 and 11-dimensional supergravity backgrounds. We then show how in the case of an N=1, D=4 target superspace gauge fixing local worldvolume superdiffeomorphisms in the covariant supermembrane action results in an effective N=2, d=3 supersymmetric field theory with N=2 supersymmetry being spontaneously broken down to N=1. The broken part of N=2, d=3 supersymmetry is nonlinearly realized when acting on Goldstone N=1, d=3 superfields, which describe physical degrees of freedom of the model. As an introduction to the formalism, the procedure of getting effective field theories with partially broken supersymmetry by gauge fixing covariant superbrane actions is also demonstrated with a simpler example of a massive N=2, D=2 superparticle. 
  It is known for ten years that self-dual Yang-Mills theory is the effective field theory of the open N=2 string in 2+2 dimensional spacetime. We uncover an infinite set of abelian rigid string symmetries, corresponding to the symmetries and integrable hierarchy of the self-dual Yang-Mills equations. The twistor description of the latter naturally connects with the BRST approach to string quantization, providing an interpretation of the picture phenomenon in terms of the moduli space of string backgrounds. 
  We introduce the notion of superoperators on noncommutative R^4 and re-investigate in the framework of superfields the noncommutative Wess-Zumino model as a quantum field theory. In a highly efficient manner we are able to confirm the result that this model is renormalizable to all orders. 
  We develop an approximation scheme for the quantum mechanics of N D0-branes at finite temperature in the 't Hooft large-N limit. The entropy of the quantum mechanics calculated using this approximation agrees well with the Bekenstein-Hawking entropy of a ten-dimensional non-extremal black hole with 0-brane charge. This result is in accord with the duality conjectured by Itzhaki, Maldacena, Sonnenschein and Yankielowicz. Our approximation scheme provides a model for the density matrix which describes a black hole in the strongly-coupled quantum mechanics. 
  The dual supergravity description of the flow between (2,0) five-brane theory and the noncommutative five-brane (OM) theory is examined at critical five-brane field strength. The self-duality of the field strength is shown to arise as a consequence of the supergravity solution. Open membrane solutions are examined in the background of the five-brane giving rise to an M analogue of the noncommutative open string (NCOS) solution. 
  We solve the Einstein equations in the Randall-Sundrum framework with a static, spherically symmetric matter distribution on the {\it physical brane} and obtain an approximate expression for the gravitational field outside the source to second order in the gravitational coupling. This expression when confined on the {\it physical brane} coincides with the standard form of the Schwarzschild metric. Therefore, the Randall-Sundrum scenario is consistent with the Mercury precession test of General Relativity. 
  I discuss theories of violations of statistics, including intermediate statistics, parastatistics, parons, and quons. I emphasize quons, which allow small violations of statistics. I analyze the quon algebra and its representations, implications of the algebra including the observables allowed by the superselection rule separating inequivalent representations of the symmetric group, the conservation of statistics rules, and the rule for composite systems of quons. I conclude by raising the question of possible origins of violations of statistics and of the level at which violations should be expected if they exist. 
  We explore some aspects of causal time-delay in open string scattering studied recently by Seiberg, Susskind and Toumbas. By examining high-energy scattering amplitudes at higher order in perturbation theory, we argue that causal time-delay at G-th order is 1/(G+1) times smaller than the time-delay at tree level. We propose a space-time interpretation of the result by utilizing the picture of the high-energy open string scattering put forward by Gross and Manes. We argue that the phenomenon of reduced time-delay is attributed to the universal feature of the space-time string trajectory in high-energy scattering that string shape at higher order remains the same as that at tree level but overall scale is reduced. We also discuss implications to the space-time uncertainty principle and make brief comments on causal time-delay behavior in space/time noncommutative field theory. 
  We show that the recent world-sheet analysis of the quantum fluctuations of a short flux tube in type II string theory leads to a simple and precise description of a pair of stuck D0branes in an orientifold compactification of the type I' string theory. The existence of a stable type I' flux tube of sub-string-scale length is a consequence of the confinement of quantized flux associated with the scalar dualized ten-form background field strength *F_{10}, evidence for a -2brane in the BPS spectrum of M theory. Using heterotic-type I duality, we infer the existence of an M2brane of finite width O(\sqrt{\alpha'}) in M-theory, the strong coupling resolution of a spacetime singularity in the D=9 twisted and toroidally compactified E_8 x E_8 heterotic string. This phenomenon has a bosonic string analog in the existence of a stable short electric flux tube arising from the confinement of photons due to tachyon field dynamics. The appendix clarifies the appearance of nonperturbative states and enhanced gauge symmetry in toroidal compactifications of the type I' string. We account for all of the known disconnected components of the moduli space of theories with sixteen supercharges, in striking confirmation of heterotic-type I duality. 
  It is considered here the possibility of unitary spinor representations of the Virasoro and super-Virasoro algebras for conformal spin to be equal 1/k; k are integers. 
  We carry out a general analysis of the representations of the superconformal algebras OSp(8/4,R) and OSp(8*/2N) in terms of harmonic superspace. We present a construction of their highest-weight UIR's by multiplication of the different types of massless conformal superfields ("supersingletons"). Particular attention is paid to the so-called "short multiplets". Representations undergoing shortening have "protected dimension" and may correspond to BPS states in the dual supergravity theory in anti-de Sitter space. These results are relevant for the classification of multitrace operators in boundary conformally invariant theories as well as for the classification of AdS black holes preserving different fractions of supersymmetry. 
  Homogeneous Bethe-Salpeter equation for simplest Wick-Cutkosky model is studied in the case when the mass of the two-body system is more then the sum of constituent particles masses. It is shown that there is always a small attraction between the like-sigh charged particles as a pure relativistic effect. If the coupling constant exceeds some critical values there arise discrete levels.The situation here is analogous to the so-called "abnormal" solutions.The signature of the norm of these discrete states coincides with the "time-parity".The states with the negative norms can be excluded from the physical sector-the one-time (quasipotential) wave-function corresponding to them vanishes identically.However the positive norm states survive and contribute to the total Green function (and the S-matrix) with the proper sign. 
  We analyze the thermodynamical properties of brane-worlds, with a focus on the second model of Randall and Sundrum. We point out that during an inflationary phase on the brane, black holes will tend to be thermally nucleated in the bulk. This leads us to ask the question: Can the black hole - brane-world system evolve towards a configuration of thermal equilibrium? To answer this, we generalize the second Randall-Sundrum scenario to allow for non-static bulk regions on each side of the brane-world. Explicitly, we take the bulk to be a {\it Vaidya-AdS} metric, which describes the gravitational collapse of a spherically symmetric null dust fluid in Anti-de Sitter spacetime. Using the background subtraction technique to calculate the Euclidean action, we argue that at late times a sufficiently large black hole will relax to a point of thermal equilibrium with the brane-world environment. These results have interesting implications for early-universe cosmology. 
  We determine the AdS exchange diagrams needed for the computation of 4--point functions of chiral primary operators in the SCFT_2 dual to the D=6 N=4b supergravity on the AdS_3\times S^3 background and compute the corresponding cubic couplings. We also address the issue of consistent truncation. 
  We give a prescription for calculating the holographic Weyl anomaly in arbitrary dimension within the framework based on the Hamilton-Jacobi equation proposed by de Boer, Verlinde and Verlinde. A few sample calculations are made and shown to reproduce the results that are obtained to this time with a different method. We further discuss continuum limits, and argue that the holographic renormalization group may describe the renormalized trajectory in the parameter space. We also clarify the relationship of the present formalism to the analysis carried out by Henningson and Skenderis. 
  In his pioneering work on singular shells in general relativity, Lanczos had derived jump conditions across energy-momentum carrying hypersurfaces from the Einstein equation with codimension 1 sources. However, on the level of the action, the discontinuity of the connection arising from a codimension 1 energy-momentum source requires to take into account two adjacent space-time regions separated by the hypersurface.   The purpose of the present note is to draw attention to the fact that Lanczos' jump conditions can be derived from an Einstein action but not from an Einstein-Hilbert action. 
  In this paper we study the creation of brane-worlds in $AdS$ bulk. We first consider the simplest case of onebranes in $AdS_3$. In this case we are able to properly describe the creation of a spherically symmetric brane-world deriving a general expression for its wavefunction. Then, we sketch the $AdS_{d+1}$ set-up within the context of the WKB approximation. Finally, we comment on these scenarios in light of the $AdS/CFT$ correspondence. 
  We investigate the role of quantum fluctuations in the system composed of two branes bounding a region of AdS. It is shown that the modulus effective potential generated by quantum fluctuations of both brane and bulk fields is incapable of stabilizing the space naturally at the separation needed to generate the hierarchy. Consequently, a classical stabilization mechanism is required. We describe the proper method of regulating the loop integrals and show that, for large brane separation, the quantum effects are power suppressed and therefore have negligible affects on the bulk dynamics once a classical stabilization mechanism is in place. 
  A class of exact solutions to the Born-Infeld field equations, over manifolds of any even dimension, is constructed. They are an extension of the self-dual configurations. They are local minima of the action for riemannian base manifolds and local minima of the Hamiltonian for pseudo-riemannian ones. A general explicit expression for the Born-Infeld determinant is obtained, for any dimension of space-time. 
  We consider the following two problems: classical domain walls in the $N=1^*$ mass deformation of the maximally supersymmetric Yang Mills theory, and D-strings as external magnetic sources in the context of the AdS/CFT correspondence. We show that they are both described by Nahm's equations with unconventional boundary conditions, and analyze the relevant moduli space of solutions. We argue that general `fuzzy sphere' configurations of D-strings in AdS$_5$ correspond to Wilson-'t Hooft lines in higher representations of the dual SU(n) gauge theory. 
  The Wess-Zumino couplings for generalized sigma-orbifold fixed-points are presented and the generalized GS 6-form that encoding the complete sigma-standard gauge-gravitational-non standard gauge anomaly and its opposite inflow is derived. 
  We investigate a Gepner-like superstring model described by a combination of multiple minimal models and an N=2 Liouville theory. This model is thought to be equivalent to the superstring theory on a singular noncompact Calabi-Yau manifold. We construct the modular invariant partition function of this model, and confirm the validity of an appropriate GSO projection. We also calculate the elliptic genus and Witten index of the model. We find that the elliptic genus factorises into a rather trivial factor and a non-trivial one, and the non-trivial one has the information on the positively curved base manifold of the cone. 
  We calculate the NS-R fermionic zero-norm states of type II string spectrum. The massless and some possible massive zero-norm states are identified to be responsible for the space-time supersymmetry. The existence of other fermionic massive zero-norm states with higher spinor-tensor indices correspond to new enlarged boson-fermion symmetries of the theory at high energy. We also discuss the R-R charges and R-R zero-norm states and justify that perturbative string does not carry the massless R-R charges. However, the existence of some massive R-R zero-norm states make us speculate that string may carry some massive R-R charges. 
  We propose a monodromy invariant pairing $K_{hol}(X) \otimes H_3(X^\vee,\ZZ) \to \IQ$ for a mirror pair of Calabi-Yau manifolds, $(X,X^\vee)$. This pairing is utilized implicitly in the previous calculations of the prepotentials for Gromov-Witten invariants. After identifying the pairing explicitly we interpret some hypergeometric series from the viewpoint of homological mirror symmetry due to Kontsevich. Also we consider the local mirror symmetry limit to del Pezzo surfaces in Calabi-Yau 3-folds. 
  In his celebrated paper Kontsevich has proved a theorem which manifestly gives a quantum product (deformation quantization formula) and states that changing coordinates leads to gauge equivalent star products. To illuminate his procedure, we make an arbitrary change of coordinates in the Moyal product and obtain the deformation quantization formula up to the third order. In this way, the Poisson bi-vector is shown to depend on \hbar and not to satisfy the Jacobi identity. It is also shown that the values of coefficients in the formula obtained follow from associativity of the star product. 
  A general two-dimensional fractional supersymmetric conformal field theory is investigated. The structure of the symmetries of the theory is studied. Applying the generators of the closed subalgebra generated by $(L_{-1},L_{0},G_{-1/(F+1)})$ and $(\bar{L}_{-1},\bar{L}_{0},\bar{G}_{-1/(F+1)})$, the two point functions of the component-fields of supermultiplets are calculated. Then the logarithmic superconformal field theories are investigated and the chiral and full two-point functions are obtained. 
  We construct a deformation quantized version (ncKdV) of the KdV equation which possesses an infinite set of conserved densities. Solutions of the ncKdV are obtained from solutions of the KdV equation via a kind of Seiberg-Witten map. The ncKdV is related to a modified ncKdV equation by a noncommutative Miura transformation. 
  We study both A-type and B-type D-branes in the gauged linear sigma model by considering worldsheets with boundary. The boundary conditions on the matter and vector multiplet fields are first considered in the large-volume phase/non-linear sigma model limit of the corresponding Calabi-Yau manifold, where we also find that we need to add a contact term on the boundary. These considerations enable to us to derive the boundary conditions in the full gauged linear sigma model, including the addition of the appropriate boundary contact terms, such that these boundary conditions have the correct non-linear sigma model limit. Most of the analysis is for the case of Calabi-Yau manifolds with one Kahler modulus (including those corresponding to hypersurfaces in weighted projective space), though we comment on possible generalizations. 
  This paper presents a general covariant lagrangian framework for the dynamics of a system of closed n-branes and dual (D-n-4)-branes in D dimensions, interacting with a dynamical (n+1)-form gauge potential. The framework proves sufficiently general to include also a coupling of the branes to (the bosonic sector of) a dynamical supergravity theory. We provide a manifestly Lorentz-invariant and S-duality symmetric Lagrangian, involving the (n+1)-form gauge potential and its dual (D-n-3)-form gauge potential in a symmetric way. The corresponding action depends on generalized Dirac-strings. The requirement of string-independence of the action leads to Dirac-Schwinger quantization conditions for the charges of branes and dual branes, but produces also additional constraints on the possible interactions. It turns out that a system of interacting dyonic branes admits two quantum mechanically inequivalent formulations, involving inequivalent quantization conditions. Asymmetric formulations involving only a single vector potential are also given. For the special cases of dyonic branes in even dimensions known results are easily recovered. As a relevant application of the method we write an effective action which implements the inflow anomaly cancellation mechanism for interacting heterotic strings and five-branes in D=10. A consistent realization of this mechanism requires, in fact, dynamical p-form potentials and a systematic introduction of Dirac-strings. 
  We consider the one-point functions of bulk and boundary fields in the scaling Lee-Yang model for various combinations of bulk and boundary perturbations. The one-point functions of the bulk fields are analysed using the truncated conformal space approach and the form-factor expansion. Good agreement is found between the results of the two methods, though we find that the expression for the general boundary state given by Ghoshal and Zamolodchikov has to be corrected slightly. For the boundary fields we use thermodynamic Bethe ansatz equations to find exact expressions for the strip and semi-infinite cylinder geometries. We also find a novel off-critical identity between the cylinder partition functions of models with differing boundary conditions, and use this to investigate the regions of boundary-induced instability exhibited by the model on a finite strip. 
  We construct Nielsen-Olesen vortex solution in the noncommutative abelian Higgs model. We derive the quantized topological flux of the vortex solution. We find that the flux is integral by explicit computation in the large $\theta$ limit as well as in the small $\theta$ limit. In the context of a tachyon vortex on the brane-antibrane system we demonstrate that it is this topological charge that gives rise to the RR charge of the resulting BPS D-brane. We also consider the left-right-symmetric gauge theory which does not have a commutative limit and construct an exact vortex solution in it. 
  We examine the problem of gauge-field localization in higher-dimensional gauge theories. In particular, we study a five-dimensional U(1) by lattice techniques and we find that gauge fields can indeed be localized. Two models are considered. The first one has anisotropic couplings independent of each other and of the coordinates. It can be realized on a homogeneous but anisotropic flat Euclidean space. The second model has couplings depending on the extra transverse fifth direction. This model can be realized by a U(1) gauge theory on the Randall-Sundrum background. We find that in both models a new phase exists, the layer phase, in which a massless photon is localized on the four-dimensional layers. We find that this phase is separated from the strong coupling phase by a second order phase transition. 
  We discuss the scattering of relativistic spin zero particles by an infinitely long and arbitrarily thin solenoid. The exact solution of the first-quantized problem can be obtained as a mimic of the nonrelativistic case, either in the original Aharonov-Bohm way or by using the Berry's magnetization scheme. The perturbative treatment is developed in the Feshbach-Villars two-component formalism for the Klein-Gordon equation and it is shown that it also requires renormalization as in the Schrodinger counterpart. The results are compared with those of the field theoretical approach which corresponds to the two-body sector of the scalar Chern-Simons theory. 
  In light-cone quantization, the standard procedure to characterize the phases of a system by appropriate ground state expectation values fails. The light-cone vacuum is determined kinematically. We show that meaningful quantities which can serve as order parameters are obtained as expectation values of Heisenberg operators in the equal (light-cone) time limit. These quantities differ from the purely kinematical expectation values of the corresponding Schroedinger operators. For the Nambu--Jona-Lasinio and the Gross-Neveu model, we describe the spontaneous breakdown of chiral symmetry; we derive within light-cone quantization the corresponding gap equations and the values of the chiral condensate. 
  We study the SO(3)-invariant relevant deformations of N=4 SU(N) gauge theory using the methods of Polchinski and Strassler. We present the region of parameter space where the non-supersymmetric vacuum is still described by stable ``dielectric'' five branes within the supergravity approximation. 
  It is shown that the supergravity moduli spaces of D1-D5 and D2-D6 brane systems coincide with those of the Coulomb branches of the associated non-abelian gauge theories. We further discuss situations in which worldvolume brane actions include a potential term generated by probing certain supergravity backgrounds. We find that in many cases, the appearance of the potential is due to the application of the Scherk-Schwarz mechanism. We give some examples and discuss the existence of novel supersymmetric brane configurations. 
  We discuss some formal aspects of quantum anomalies with an emphasis on the regularization of field theory. We briefly review how ambiguities in perturbation theory have been resolved by various regularization schemes. To single out the true quantum anomaly among ambiguities, the combined ideas of PCAC, soft pion limit and renormalizability were essential. As for the formal treatment of quantum anomalies, we mainly discuss the path integral formulation both in continuum and lattice theories. In particular, we discuss in some detail the recent development in the treatment of chiral anomalies in lattice gauge theory. 
  In the Kaluza - Klein approach the (4+d)-dimensional Einstein--Hilbert gravity action is considered. The extra d-dimensional manifold V_d is a Riemann space with the d-parametric group of isometry $G_d$ which acts on V_d by the left shifts and with arbitrary nondegenerated left-invariant metric g_{ab}. The gauge fields A_{\mu} are introduced as the affine connection coefficients of the fibre bundle with V_d being the fibre. The effective Lagrangian as invariant integral over extra-dimensional manifold of the curvative scalar of mentioned structure is obtained. It is shown that such effective Lagrangian contains beside the square of gauge field strength tensor also quadratic form of A_{\mu} and all other fields have only pure gauge degrees of freedom when g_{ab}. satisfy some conditions. This conditions may be regarded as generalization of the General Relativity Principle to the extra dimensions. The eigenvalues of the quadratic form of A_{\mu} are calculated for the case of gauge group SO(3). It is shown that they are not equal to zero in the case when g_{ab} is not proportional to the unit matrix. 
  We consider a 2-d conformal theory based on (G x G')/ H coset sigma model introduced by Guadagnini, Martellini and Mintchev. It is shown that in the case of {SU(2) x SU(2)}/ U(1) the metric of the corresponding background is of T^{p,q} coset space form (but is not an Einstein one). Similar interpretation is possible for the Lorentzian coset space W_{4,2}= {SL(2,R) x SL(2,R)}/U(1). The resulting 10-d homogeneous space metric on W_{4,2} x T^{p,q} supplemented with 2-form field gives a critical NS-NS superstring background with conformal sigma model interpretation. 
  We construct compact type IIB orientifolds with discrete groups Z_4, Z_6, Z_6', Z_8, Z_12 and Z_12'. These models are N=1 supersymmetric in D=4 and have vector structure. The possibility of having vector structure in Z_N orientifolds with even N arises due to an alternative Omega-projection in the twisted sectors. Some of the models without vector structure are known to be inconsistent because of uncancelled tadpoles. We show that vector structure leads to a sign flip in the twisted Klein bottle contribution. As a consequence, all the tadpoles can be cancelled by introducing D9-branes and D5-branes. 
  Fusion rules for Wess-Zumino-Witten (WZW) models at fractional level can be defined in two ways, with distinct results. The Verlinde formula yields fusion coefficients that can be negative. These signs cancel in coset fusion rules, however. On the other hand, the fusion coefficients calculated from decoupling of singular vectors are non-negative. They produce incorrect coset fusion rules, however, when factorisation is assumed. Here we give two prescriptions that yield the correct coset fusion rules from those found for the WZW models by the decoupling method. We restrict to the Virasoro minimal models for simplicity, and because decoupling results are only complete in the $\su(2)$ case. 
  We examine several aspects of S-duality of four-dimensional noncommutative gauge theory. By making duality transformation explicit, we find that S-dual of noncommutative gauge theory is defined in terms of dual noncommutative deformation. In `magnetic' noncommutative U(1) gauge theory, we show that, in addition to gauge bosons, open D-strings constitute important low-energy excitations: noncritical open D-strings. Upon S-duality, they are mapped to noncritical open F-strings. We propose that, in dual `electric' noncommutative U(1) gauge theory, the latters are identified with gauge-invariant, open Wilson lines. We argue that the open Wilson lines are chiral due to underlying parity noninvariance and obey spacetime uncertainty relation. We finally argue that, at high energy-momentum limit, the `electric' noncommutative gauge theory describes correctly dynamics of delocalized multiple F-strings. 
  In the presence of internal magnetic fields, a D9 brane can acquire a D5 (or anti-D5) R-R charge, and can therefore contribute to the corresponding tadpole. In the resulting vacua, supersymmetry is generically broken and tachyonic instabilities are present. However, suitable choices for the magnetic fields, corresponding to self-dual configurations in the internal space, can yield new chiral supersymmetric vacua with gauge groups of reduced rank, where the magnetic energy saturates, partly or fully, the negative tension of the O5+ planes. These models contain Green-Schwarz couplings to untwisted R-R forms not present in conventional orientifolds. 
  A system of generalized coherent states for the de Sitter group obeying the Klein-Gordon equation and corresponding to the massive spin zero particles over the de Sitter space is considered. This allows us to construct the quantized scalar field by the resolution over these coherent states; the corresponding propagator is computed by the method of analytic continuation to the complex de Sitter space and coincides with expressions obtained previously by other methods. Considering the case of spin 1/2 we establish the connection of the invariant Dirac equation over the de Sitter space with irreducible representations of the de Sitter group. The set of solutions of this equation is obtained in the form of the product of two different systems of generalized coherent states for the de Sitter group. Using these solutions the quantized Dirac field over de Sitter space is constructed and its propagator is found. It is a result of action of some de Sitter invariant spinor operator onto the spin zero propagator with an imaginary shift of a mass. We show that the constructed propagators possess the de Sitter-invariance and causality properties. 
  Using the technique of finite field dependent BRST transformations we show that the classical massive Yang-Mills theory and the pure Yang-Mills theory whose gauge symmetry is broken by a gauge fixing term are identical from the view point of quantum gauge symmetry. The explicit infinitesimal transformations which leave the massive Yang-Mills theory BRST invariant are given. 
  The phase transition of the Gross-Neveu model with N fermions is investigated by means of a non-perturbative evolution equation for the scale dependence of the effective average action. The critical exponents and scaling amplitudes are calculated for various values of N in d=3. It is also explicitely verified that the Neveu-Yukawa model belongs to the same universality class as the Gross-Neveu model. 
  A simple model is introduced in which the cosmological constant is interpreted as a true Casimir effect on a scalar field filling the universe (e.g. $\mathbf{R} \times \mathbf{T}^p\times \mathbf{T}^q$, $\mathbf{R} \times \mathbf{T}^p\times \mathbf{S}^q, ...$). The effect is driven by compactifying boundary conditions imposed on some of the coordinates, associated both with large and small scales. The very small -but non zero- value of the cosmological constant obtained from recent astrophysical observations can be perfectly matched with the results coming from the model, by playing just with the numbers of -actually compactified- ordinary and tiny dimensions, and being the compactification radius (for the last) in the range $(1-10^3) l_{Pl}$, where $l_{Pl}$ is the Planck length. This corresponds to solving, in a way, what has been termed by Weinberg the {\it new} cosmological constant problem. Moreover, a marginally closed universe is favored by the model, again in coincidence with independent analysis of the observational results. 
  We classify the possible finite symmetries of conformal field theories with an affine Lie algebra su(2) and su(3), and discuss the results from the perspective of the graphs associated with the modular invariants. The highlights of the analysis are first, that the symmetries we found in either case are matched by the graph data in a perfect way in the case of su(2), but in a looser way for su(3), and second, that some of the graphs lead naturally to projective representations, both in su(2) and in su(3). 
  We consider the contribution of the B-field into the RR charge of a spherical D2-brane. Extending a recent analysis of Taylor, we show that the boundary and bulk contributions do not cancel in general. Instead, they add up to an integer as observed by Stanciu. The general formula is applied to compute the RR charges of spherical D-branes of the SU(2) WZW model at level k and it shows that these RR charges are only defined modulo k+2. We support this claim by studying bound state formation of D0-branes using boundary conformal field theory. 
  We study the classical geometry produced by a stack of stable (i.e. tachyon free) non-BPS D-branes present in K3 compactifications of type II string theory. This classical representation is derived by solving the equations of motion describing the low-energy dynamics of the supergravity fields which couple to the non-BPS state. Differently from what expected, this configuration displays a singular behaviour: the space-time geometry has a repulson-like singularity. This fact suggests that the simplest setting, namely a set of coinciding non-interacting D-branes, is not acceptable. We finally discuss the possible existence of other acceptable configurations corresponding to more complicated bound states of these non-BPS branes. 
  The embedding procedure of Batalin, Fradkin, and Tyutin, which allows to convert a second-class system into a first-class one, is employed to convert second-class interacting models. Two cases are considered. One, is the Self-Dual model minimally coupled to a Dirac fermion field. The other, the Self-Dual model minimally coupled to a charged scalar field. In both cases, they are found equivalent interacting Maxwell-Chern-Simons type field theories. These equivalences are pushed beyond the formal level, by analysing some tree level probability amplitudes associated to the models. 
  We derive explicit forms for the superisometries of a wide class of supercoset manifolds, including those with fermionic generators in the stability group. We apply the results to construct the action of SU(2,2|4) on three supercoset manifolds: (10|32)-dimensional AdS_5 x S^5 superspace, (4|16)-dimensional conformal superspace, and a novel (10|16)-dimensional conformal superspace. Using superembedding techniques, we show, to lowest non-trivial order in the fermions, that at the boundary of AdS_5, the superisometries of the AdS_5 x S^5$ superspace reduce to the standard N=4 superconformal transformations. In particular, half of the 32 fermionic coordinates decouple from the superisometries. 
  A D3 brane in the background of NS5 branes is studied semi-classically. The conditions for preserved supersymmetry are derived, leading to a differential equation for the shape of the D3 brane. The solutions of this equation are analyzed. For a D3 brane intersecting the NS5 branes, the angle of approach is known to be restricted to discrete values. Four different ways to obtain this quantization are described. In particular, it is shown that, assuming the D3 brane avoids intersecting a {\em single} NS5 brane, the above discrete values correspond to the different possible positions of the D3 branes among the NS5 branes. 
  In this article we study limits of models that contain a dimensionful parameter such as the mass of the relativistic point-particle. The limits are analogous to the massless limit of the particle and may be thought of as high energy limits. We present the ideas and work through several examples in a (hopefully) pedagogical manner. Along the way we derive several new results. 
  We consider the gauging of space translations with time-dependent gauge functions. Using fixed time gauge of relativistic theory, we consider the gauge-invariant model describing the motion of nonrelativistic particles. When we use gauge-invariant nonrelativistic velocities as independent variables the translation gauge fields enter the equations through a d\times (d+1) matrix of vielbein fields and their Abelian field strengths, which can be identified with the torsion tensors of teleparallel formulation of relativity theory. We consider the planar case (d=2) in some detail, with the assumption that the action for the dreibein fields is given by the translational Chern-Simons term. We fix the asymptotic transformations in such a way that the space part of the metric becomes asymptotically Euclidean. The residual symmetries are (local in time) translations and rigid rotations. We describe the effective interaction of the d=2 N-particle problem and discuss its classical solution for N=2. The phase space Hamiltonian H describing two-body interactions satisfies a nonlinear equation H={\cal H}(\vec x,\vec p;H) which implies, after quantization, a nonstandard form of the Schr\"odinger equation with energy dependent fractional angular momentum eigenvalues. Quantum solutions of the two-body problem are discussed. The bound states with discrete energy levels correspond to a confined classical motion (for the planar distance between two particles r\le r_0) and the scattering states with continuum energy correspond to the classical motion for r>r_0. We extend our considerations by introducing an external constant magnetic field and, for N=2, provide the classical and quantum solutions in the confined and unconfined regimes. 
  An extension of the Super KdV integrable system in terms of operator valued functions is obtained. Following the ideas of Gardner, a general algebraic approach for finding the infinitely many conserved quantities of integrable systems is presented. The approach is applied to the above described system and infinitely many conserved quantities are constructed. In a particular case they reduce to the corresponding conserved quantities of Super KdV. 
  The AdS/CFT correspondence identifies the coordinates of the conformal boundary of anti-de Sitter space with the coordinates of the conformal field theory. We generalize this identification to theories formulated in superspace. As an application of our results, we study a class of Wilson loops in N=4 SYM theory. A gauge theory computation shows that the expectation values of these loops are invariant under a local kappa-symmetry, except at intersections. We identify this with the kappa-invariance of the associated string worldsheets in the corresponding bulk superspace. 
  We use dimensional regularization to evaluate quantum mechanical path integrals in arbitrary curved spaces on an infinite time interval. We perform 3-loop calculations in Riemann normal coordinates, and 2-loop calculations in general coordinates. It is shown that one only needs a covariant two-loop counterterm (V_{DR} = R/8) to obtain the same results as obtained earlier in other regularization schemes. It is also shown that the mass term needed in order to avoid infrared divergences explicitly breaks general covariance in the final result. 
  We discuss the properties of the bound state (F1, D1, D3) in IIB supergravity in three different scaling limits and the $SL(2,{\bf Z})$ transformation of the resulting theories. In the simple decoupling limit with finite electric and magnetic components of NS $B$ field, the worldvolume theory is the ${\cal N}$=4 super Yang-Mills (SYM) and the supergravity dual is still the $AdS_5 \times S^5$. In the large magnetic field limit with finite electric field, the theory is the noncommutative super Yang-Mills (NCSYM), and the supergravity dual is the same as that without the electric background. We show how to take the decoupling limit of the closed string for the critical electric background and finite magnetic field, and that the resulting theory is the noncommutative open string (NCOS) with both space-time and space-space noncommutativities. It is shown that under the $SL(2, {\bf Z})$ transformation, the SYM becomes itself with a different coupling constant, the NCSYM is mapped to a NCOS, and the NCOS in general transforms into another NCOS and reduces to a NCSYM in a special case. 
  We discuss the characteristic properties of noncommutative solitons moving with constant velocity. As noncommutativity breaks the Lorentz symmetry, the shape of moving solitons is affected not just by the Lorentz contraction along the velocity direction, but also sometimes by additional `elongation' transverse to the velocity direction. We explore this in two examples: noncommutative solitons in a scalar field theory on two spatial dimension and `long stick' shaped noncommutative U(2) magnetic monopoles. However the elongation factors of these two cases are different, and so not universal. 
  Complete results were obtained by us in [Can. J. Phys. 71, 389 (1993)] for convergent series representations of both the real and the imaginary part of the QED effective action; these derivations were based on correct intermediate steps. In this comment, we argue that the physical significance of the "logarithmic correction term" found by Cho and Pak in [Phys. Rev. Lett. 86, 1947 (2001)] in comparison to the usual expression for the QED effective action remains to be demonstrated. Further information on related subjects can be found in Appendix A of hep-ph/0308223 and in hep-th/0210240. 
  We obtain new consistent Kaluza-Klein embeddings of the gauged supergravities with half of maximal supersymmetry in dimensions D=7, 6, 5 and 4. They take the form of warped embeddings in type IIA, type IIB, M-theory and type IIB respectively, and are obtained by performing Kaluza-Klein circle reductions or T-duality transformations on Hopf fibres in S^3 submanifolds of the previously-known sphere reductions. The new internal spaces are in some sense ``mirror manifolds'' that are dual to the original internal spheres. The vacuum AdS solutions of the gauged supergravities then give rise to warped products with these internal spaces. As well as these embeddings, which have singularities, we also construct new non-singular warped Kaluza-Klein embeddings for the D=5 and D=4 gauged supergravities. The geometry of the internal spaces in these cases leads us to study Fubini-Study metrics on complex projective spaces in some detail. 
  We apply the string-triality\cite{duffetal} to argue the existence of the string-networks of solitonic T and U-strings in the heterotic theory on T^6, S and T-strings in IIB on K3 x T^2 and, S and U-strings in IIA on K3 x T^2. We then show the existence of the above heterotic string networks by analyzing the supersymmetry property of the supergravity solutions. The consistency of these networks with supersymmetry in the case of IIA and IIB theories is also argued. Our results therefore give further evidence in favor of the string-triality in four dimensions. 
  We investigate superfluidity of the relativistic fermi-gas with gravitational interaction. The excitation spectrum is obtained within the linearized theory. While superfluidity may take place at a definite ratio of the Fermi momentum, rest mass and coupling constant, the metric coefficients play predominant role forming the gap of excitation spectrum. 
  New solutions to the abelian U(1) Higgs model, corresponding to vortices of integer and half-integer winding number bound onto the edges of domain walls and possibly surrounded by annular current flows, are described, based on a fine-grained analysis of the topology of such configurations in spacetime. The existence of these states, which saturate BPS bounds in specific limits and are quite reminiscent of D-branes and membranes in general, could have interesting and some important consequences in a wide range of physical contexts. For instance, they raise the possibility that for some regimes of couplings the usual vortex of unit winding number would split into two vortices each of one-half winding number bound by a domain wall. A similar approach may also be relevant to other known topological states of field theory. 
  We develop a perturbative expansion which allows the construction of non-abelian self-dual SU(2) Yang-Mills field configurations on the four-dimensional torus with topological charge 1/2. The expansion is performed around the constant field strength abelian solutions found by 't Hooft. Next to leading order calculations are compared with numerical results obtained with lattice gauge theory techniques. 
  We construct a q-deformed version of the conformal quantum mechanics model of de Alfaro, Fubini and Furlan for which the deformation parameter is complex and the unitary time evolution of the system is preserved. We also study differential calculus on the q-deformed quantum phase space associated with such system. 
  We present results from an inductive algebraic approach to the systematic construction and classification of the `lowest-level' CY3 spaces defined as zeroes of polynomial loci associated with reflexive polyhedra, derived from suitable vectors in complex projective spaces. These CY3 spaces may be sorted into `chains' obtained by combining lower-dimensional projective vectors classified previously. We analyze all the 4242 (259, 6, 1) two- (three-, four-, five-) vector chains, which have, respectively, K3 (elliptic, line-segment, trivial) sections, yielding 174767 (an additional 6189, 1582, 199) distinct projective vectors that define reflexive polyhedra and thereby CY3 spaces, for a total of 182737. These CY3 spaces span 10827 (a total of 10882) distinct pairs of Hodge numbers h_11, h_12. Among these, we list explicitly a total of 212 projective vectors defining three-generation CY3 spaces with K3 sections, whose characteristics we provide. 
  A general construction of affine Non Abelian (NA) - Toda models in terms of axial and vector gauged two loop WZNW model is discussed. They represent {\it integrable perturbations} of the conformal $\sigma$-models (with tachyons included) describing (charged) black hole type string backgrounds. We study the {\it off-critical} T-duality between certain families of axial and vector type of integrable models for the case of affine NA- Toda theories with one global U(1) symmetry. In particular we find the Lie algebraic condition defining a subclass of {\it T-selfdual} torsionless NA Toda models and their zero curvature representation. 
  I had the privilege of collaborating with Joel Scherk on three separate occasions: in 1970 at Princeton, in 1974 at Caltech, and in 1978-79 at the Ecole Normale Superieure. In this talk I give some reminiscences of these collaborations. 
  Lenny Susskind has made many important contributions to theoretical physics during the past 35 years. In this talk I will discuss the early history of string theory (1968-72) emphasizing Susskind's contributions. 
  We investigate the Laplacian Abelian gauge on the sphere S^4 in the background of a single `t Hooft instanton. To this end we solve the eigenvalue problem of the covariant Laplace operator in the adjoint representation. The ground state wave function serves as an auxiliary Higgs field. We find that the ground state is always degenerate and has nodes. Upon diagonalisation, these zeros induce toplogical defects in the gauge potentials. The nature of the defects crucially depends on the order of the zeros. For first-order zeros one obtains magnetic monopoles. The generic defects, however, arise from zeros of second order and are pointlike. Their topological invariant is the Hopf index S^3 -> S^2. These findings are corroborated by an analysis of the Laplacian gauge in the fundamental representation where similar defects occur. Possible implications for the confinement scenario are discussed. 
  We show that an earlier domain wall solution of type IIB supergravity provides a supersymmetric realization of the Randall-Sundrum brane-world, and give its ten-dimensional interpretation in terms of IIB 3-branes. We also explain how previous no-go theorems are circumvented. In particular, whereas D=5 supergravity scalars have AdS_5 energy E_0 <= 4 and are unable to support a D=5 positive tension brane, our scalar has E_0=8, and is the breathing mode of the S^5 compactification. Another essential element of the construction is the implementation of a Z_2 symmetry by patching together compactifications with opposite signs for their 5-form field strengths. This is thus a IIB analogue of a previous D=5 3-brane realization of the Horava-Witten orbifold. A mode-locking phenomenon avoids the appearance of negative energy zero-modes in spite of the necessity of a D=10 negative tension brane-source. 
  The Casimir energy for scalar field of two parallel conductor in two dimensional domain wall background, with Dirichlet boundary conditions, is calculated by making use of general properties of renormalized stress tensor.We show that vacuum expectation values of stress tensor contain two terms which come from the boundary conditions and the gravitational background. In two dimensions the minimal coupling reduces to the conformal coupling and stress tensor can be obtained by the local and non-local contribution of the anomalous trace. This work shows that there exists a subtle relation between Casimir effect and trace anomaly in curved space time. 
  Studied is the deformation of super Virasoro algebra proposed by Belov and Chaltikhian. Starting from abstract realizations in terms of the FFZ type generators, various connections of them to other realizations are shown, especially to deformed field representations, whose bosonic part generator is recently reported as a deformed string theory on a noncommutative world-sheet. The deformed Virasoro generators can also be expressed in terms of ordinary free fields in a highly nontrivial way. 
  Simple considerations about the fractal characteristic of the quantum-mechanical path give us the opportunity to derive the quantum black hole entropy in connection with the concept of fractal statistics. We show the geometrical origin of the numerical factor of four of the quantum black hole entropy expression and the statistics weight appears as a counting of the quanta of geometry. 
  The general scalar potential of D-dimensional massive sigma-models with eight supersymmetries is found for $D=3,4$. These sigma models typically admit 1/2 supersymmetric domain wall solutions and we find, for a particular hyper-K\"ahler target, exact 1/4 supersymmetric static solutions representing a non-trivial intersection of two domain walls. We also show that the intersecting domain walls can carry Noether charge while preserving 1/4 supersymmetry. We briefly discuss an application to the D1-D5 brane system. 
  We study supersymmetric domain walls in N=1 SU(N) gauge theory with 3 massive adjoint representation chiral multiplets. This theory, known as N=1^*, can be obtained as a massive deformation of N=4 Yang-Mills theory. Following Polchinski and Strassler, we consider the string dual of this theory in terms of spherical 5-branes and construct BPS domain walls interpolating between the many vacua. We compare our results to field theoretic domain walls and also find that this work is related to the physics of expanded ``dielectric'' branes near zero radius. 
  In this paper properties of D branes in a nine dimensional asymmetric orbifold are discussed, using a $(-1)^{F_L}\sigma_{1/2}$ projection, where $F_L$ is the leftmoving space-time fermion number and $\sigma_{1/2}$ is a freely acting shift of order two. There are two types of non BPS D branes, which are stable at $R>2$ and $R<2$ respectively. At R=2 there is a perturbative enhancement of gauge symmetry and the two types of branes are related by a global bulk symmetry transformation. At this point in the moduli space the associated boundary states are constructed using a free fermion representation of the theory. Some aspects of the enhancement of gauge symmetry in the S-dual type $\tilde I$ theory are discussed. 
  Here we construct a map from the algebra of fields in two-dimensional noncommutative of U(1) Yang-Mills fields interacting with Kaluza-Klein scalars to a D-dimensional one, as a solution in the two-dimensional model. This proves the equivalence of noncommutative models in various (even) dimensions. Physically this map describes condensation of D1-branes. 
  We calculate the two-loop renormalization group (RG) beta-function of a massless scalar field theory from the irreducible version of Polchinski's exact RG flow equation. To obtain the correct two-loop result within this method, it is necessary to take the full momentum-dependence of the irreducible four-point vertex and the six-point vertex into account. Although the same calculation within the orthodox field theory method is less tedious, the flow equation method makes no assumptions about the renormalizability of the theory, and promises to be useful for performing two-loop calculations for non-renormalizable condensed-matter systems. We pay particular attention to the problem of the field rescaling and the effect of the associated exponent eta on the RG flow. 
  The hybrid spectral problem where the field satisfies Dirichlet conditions (D) on part of the boundary of the relevant domain and Neumann (N) on the remainder is discussed in simple terms. A conjecture for the C_1 coefficient is presented and the conformal determinant on a 2-disc, where the D and N regions are semi-circles, is derived. Comments on higher coefficients are made. 
  Superstring theory has continued to develop at a rapid clip in the past few years. Following a quick review of some of the major discoveries prior to 1998, this talk focuses on a few of the more recent developments. The topics I have chosen to present are 1) the use of K-theory to classify conserved charges carried by D-branes; 2) tachyon condensation on unstable D-brane systems; and 3) an introduction to noncommutative field theories and their solitons. 
  U(1) gauge theory on non-commutative Minkowski space-time in the Feynman-'t Hooft background gauge is studied. In particular, UV divergences and non-commutative IR divergent contributions to the two, three and four-point functions are explicitly computed at one loop. We show that the negative sign of the beta function results from paramagnetism --producing UV charge anti-screening-- prevailing over diamagnetism --giving rise toUV charge screening. This dominance in the field theory setting corresponds to tachyon magnification dominance in the string theory framework. Our calculations provide an explicit realization of UV/IR mixing and lead to an IR renormalization of the coupling constant, where now paramagnetic contributions produce screening and diamagnetic contributions anti-screening. 
  It is shown that a Dirac particle of mass $m$ and arbitrarily small momentum will tunnel without reflection through a potential barrier $V=U_c(x)$ of finite range provided that the potential well $V=-U_c(x)$ supports a bound state of energy $E=-m.$ This is called a supercritical potential well. 
  Four-dimensional N=2 gauge theories may be obtained from configurations of D-branes in type IIA string theory. Unitary gauge theories with two-index representations, and orthogonal and symplectic gauge theories, are constructed from configurations containing orientifold planes. Models with two orientifold planes imply a compact dimension, and correspond to elliptic models. Lifting these configurations to M-theory allows one to derive the Seiberg-Witten curves for these gauge theories. We describe how the Seiberg-Witten curves, necessarily of infinite order, are obtained for these elliptic models. These curves are used to calculate the instanton expansion of the prepotential; we explicitly find the one-instanton prepotential for all the elliptic models considered. 
  We provide a class of nondilatonic solutions to the NS-NS sector of string theory. The solutions consist of products of Ricci-parallelizable spaces with adjusted radii. A representative of this class, AdS_3 x S^7, is presented in detail. Some comments on possible brane connections are made. 
  We review recent developments in understanding the physics of the magnetic monopoles in unbroken non-Abelian gauge theories. Since numerical data on the monopoles are accumulated in lattice simulations, the continuum theory is understood as the limiting case of the lattice formulation. In this review, written for a memorial volume dedicated to the memory of Academician A.B. Migdal, we emphasize physical effects related to the monopoles. In particular, we discuss the monopole-antimonopole potential at short and larger distances as well as a dual formulation of the gluodynamics, relevant to the physics of the confinement. 
  D4 N=1 SYM with an arbitrary chiral background superfield as the gauge coupling is considered. The solution to Slavnov-Taylor identities has been given. It has been shown that the solution is unique and allows us to restrict the gauge part of the effective action. Under the effective action in this paper we mean the 1PI diagram generator. 
  We develop the noncommutative harmonic space (NHS) analysis to study the problem of solving the non-linear constraint eqs of noncommutative Yang-Mills self-duality in four-dimensions. We show that this space, denoted also as NHS($\eta,\theta$), has two SU(2) isovector deformations $\eta^{(ij)}$ and $\theta^{(ij)}$ parametrising respectively two noncommutative harmonic subspaces NHS($\eta,0$) and NHS($0,\theta$) used to study the self-dual and anti self-dual noncommutative Yang-Mills solutions. We formulate the Yang-Mills self-dual constraint eqs on NHS($\eta,0$) by extending the idea of harmonic analyticity to linearize them. Then we give a perturbative self-dual solution recovering the ordinary one. Finally we present the explicit computation of an exact self-dual solution. 
  We show that the dissipation term in the Hamiltonian for a couple of classical damped-amplified oscillators manifests itself as a geometric phase and is actually responsible for the appearance of the zero point energy in the quantum spectrum of the 1D linear harmonic oscillator. We also discuss the thermodynamical features of the system. Our work has been inspired by 't Hooft proposal according to which information loss in certain classical systems may lead to ``an apparent quantization of the orbits which resembles the quantum structure seen in the real world". 
  We derive source terms for the production of quarks and gluons from the QCD vacuum in the presence of a space-time dependent external chromofield A_{cl} to the order of S^{(1)}. We found that the source terms for the parton production processes A_{cl} -> q\bar{q} and A_{cl},A_{cl}A_{cl} -> gg also include the annihilation processes q\bar{q} -> A_{cl} and gg -> A_{cl},A_{cl}A_{cl}. The source terms we derive are applicable for the description of the production of partons with momentum p larger rhan gA which itself must be larger than \Lambda_{QCD}. We observe that these source terms for the production of partons from a space-time dependent chromofield can be used to study the production and equilibration of the quark-gluon plasma during the very early stages of an ultrarelativistic heavy-ion collision. 
  We construct the consistent CP(n) model on noncommutative plane. The Bogomolny bound on the energy is saturated by (anti-)self-dual solitons with integer topological charge, which is independent of their scaling and orientation. This integer quantization is satisfied for our general solutions, which turns out regular everywhere. We discuss the possible implication of our result to the instanton physics in Yang-Mills theories on noncommutative R^4. 
  We construct the boundary state for the D-brane in the SU(2) group manifold directly in terms of the group variables. We propose a matching condition for the left- and the right-moving sectors including the zero modes that describes a D-brane of the Neumann-type. The free field realization of the WZW model is used to obtain the boundary state subject to the matching condition. We show that the resulting state coincides with Cardy's state. The structure of the BRST cohomology is realized by imposing the invariance of the state under the Weyl group of the current algebra. 
  Using recent results on string on $AdS_{3}\times N^d$, where N is a d-dimensional compact manifold, we re-examine the derivation of the non trivial extension of the (1+2) dimensional-Poincar\'e algebra obtained by Rausch de Traubenberg and Slupinsky, refs [1] and [29]. We show by explicit computation that this new extension is a special kind of fractional supersymmetric algebra which may be derived from the deformation of the conformal structure living on the boundary of $AdS_3$. The two so(1,2) Lorentz modules of spin $\pm{1\over k}$ used in building of the generalisation of the (1+2) Poincar\'e algebra are re-interpreted in our analysis as highest weight representations of the left and right Virasoro symmetries on the boundary of $AdS_3$. We also complete known results on 2d-fractional supersymmetry by using spectral flow of affine Kac-Moody and superconformal symmetries. Finally we make preliminary comments on the trick of introducing Fth-roots of g-modules to generalise the so(1,2) result to higher rank lie algebras g. 
  Using a geometric realization of the $SU(2)_R$ symmetry and a procedure of factorisation of the gauge and $SU(2)_R$ charges, we study the small instanton singularities of the Higgs branch of supersymmetric $U(1)^r$ gauge theories with eight supercharges. We derive new solutions for the moduli space of vacua preserving manifestly the eight supercharges. In particular, we obtain an extension of the ordinary ADE singularities for hyperKahler manifolds and show that the classical moduli space of vacua is in general given by cotangent bundles of compact weighted projective spaces. 
  A field theory is developed based on the idea that the effective action of yet unknown fundamental theory, at energy scale below M_{p} has the form of expansion in two measures: S=\intd^{4}x[\Phi L_{1}+\sqrt{-g}L_{2}] where the new measure \Phi is defined using the third-rank antisymmetric tensor. In the new variables (Einstein frame) all equations of motion take canonical GR form and therefore models are free of the well-known "defects" that distinguish the Brans-Dicke type theories from GR. All novelty is revealed only in an unusual structure of the effective potential U(\phi) and interactions which turns over intuitive ideas based on our experience in field theory. E.g. the greater \Lambda we admit in L_{2}, the smaller U(\phi) will be in the Einstein picture. Field theory models are suggested with explicitly broken global continuos symmetry which in the Einstein frame has the form \phi\to\phi+const. The symmetry restoration occurs as \phi\to\infty. A few models are presented where U is produced with the following shape: for \phi<-M_{p}, U has the form typical for inflation model, e. g. U=\lambda\phi^4 with \lambda\sim 10^{-14}; for\phi>-M_{p}, U has mainly exponential form U\sim e^{-a\phi/M_{p}} with variable a: a=14 for -M_{p}<\phi<M_{p} that admits nucleosynthesis; a=2 for \phi>M_{p} that implies quintessence era. There is no need in any fine tuning to prevent appearance of the CC term or any other terms that could violate flatness of U at \phi\ggM_{p}. \lambda\sim 10^{-14} is obtained without fine tuning as well. Quantized matter fields models, including gauge theories with SSB can be incorporated without altering mentioned above results. Direct fermion-inflaton coupling resembles Wetterich's model but it does not lead to any observable effect at present. SSB does not raise any problem with CC. 
  We investigate certain fixed points in the boundary conformal field theory representation of type IIA D-branes on Gepner points of K3. They correspond geometrically to degenerate brane configurations, and physically lead to enhanced gauge symmetries on the world-volume. Non-abelian gauge groups arise if the stabilizer group of the fixed points is realized projectively, which is similar to D-branes on orbifolds with discrete torsion. Moreover, the fixed point boundary states can be resolved into several irreducible components. These correspond to bound states at threshold and can be viewed as (non-locally free) sub-sheaves of semi-stable sheaves. Thus, the BCFT fixed points appear to carry two-fold geometrical information: on the one hand they probe the boundary of the instanton moduli space on K3, on the other hand they probe discrete torsion in D-geometry. 
  It is conjectured that M-theory in asymptotically flat spacetime must be supersymmetric, and that the observed SUSY breaking in the low energy world must be attributed to the existence of a nonzero cosmological constant. This would be consistent with experiment, if the {\it critical exponent} $\alpha$ in the relation $M_{SUSY} \sim M_P (\Lambda /M_P^4)^{\alpha}$ took on the value 1/8, rather than its classical value 1/4. We attribute this large renormalization to the effect of large virtual black holes via the UV/IR correspondence. 
  We conjecture the existence of two new non-gravitational six-dimensional string theories, defined as the decoupling limit of NS5-branes in the background of near-critical electrical two- and three-form RR fields. These theories are space-time non-commutative Little String Theories with open branes. The theory with (2,0) supersymmetry has an open membrane in the spectrum and reduces to OM theory at low energies. The theory with (1,1) supersymmetry has an open string in the spectrum and reduces to 5+1 dimensional NCOS theory for weak NCOS coupling and low energies. The theories are shown to be T-dual with the open membrane being T-dual to the open string. The theories therefore provide a connection between 5+1 dimensional NCOS theory and OM theory. We study the supergravity duals of these theories and we consider a chain of dualities that shows how the T-duality between the two theories is connected with the S-duality between 4+1 dimensional NCOS theory and OM theory. 
  We analyse in three space-time dimensions, the connection between abelian self dual vector doublets and their counterparts containing both an explicit mass and a topological mass. Their correspondence is established in the lagrangian formalism using an operator approach as well as a path integral approach. A canonical hamiltonian analysis is presented, which also shows the equivalence with the lagrangian formalism. The implications of our results for bosonisation in three dimensions are discussed. 
  A pure Yang-Mills theory extended by addition of a quartic term is considered in order to study the transition from the quantum tunneling regime to that of classical, i.e. thermal, behaviour. The periodic field configurations are found, which interpolate between the vacuum and sphaleron field configurations. It is shown by explicit calculation that only smooth second order transitions occur for all permissible values of the parameter $\L$ introduced with the quartic term. The theory is one of the rare cases which can be handled analytically. 
  Supersymmetry and super-Lie algebras have been consistently generalized previously. The so-called fractional supersymmetry and $F-$Lie algebras could be constructed starting from any representation $\D$ of any Lie algebra $g$. This involves taking the $F^{\mathrm th}$ root of $\D$ in some sense. We show, after having constructed differential realization(s) of any Lie algebra, how fractional supersymmetry can be explicitly realized in terms of appropriate homogeneous monomials. 
  We consider the integrable XXZ model with special open boundary conditions that renders its Hamiltonian ${SU(2)}_q$ symmetric, and the one-dimensional quantum Ising model with four different boundary conditions. We show that for each boundary condition the Ising quantum chain is exactly given by the Minimal Model of integrable lattice theory $LM(3, 4)$. This last theory is obtained as the result of the quantum group reduction of the XXZ model at anisotropy $\Delta=(q + q^{-1})/2=\sqrt{2}/2$, with a number of sites in the latter defined by the type of boundary conditions. 
  A brief review of the development of Chern-Simons gauge theory since its relation to knot theory was discovered in 1988 is presented. The presentation is done guided by a dictionary which relates knot theory concepts to quantum field theory ones. From the basic objects in both contexts the quantities leading to knot and link invariants are introduced and analyzed. The quantum field theory approaches that have been developed to compute these quantities are reviewed. Perturbative approaches lead to Vassiliev or finite type invariants. Non-perturbative ones lead to polynomial or quantum group invariants. In addition, a brief discussion on open problems and future developments is included. 
  We use the level truncation scheme to obtain accurate descriptions of open bosonic string field configurations corresponding to large marginal deformations such as background Wilson lines. To do so, we solve for all fields as functions of the massless string field, and confirm that the effective potential of the massless field becomes increasingly flat as the level of approximation is increased. Surprisingly, as a result of the merging of two branches of the solution - one originating at zero tachyon vev and the other originating at the tachyonic vacuum - this effective potential exists only for a finite range of values of the massless field. We use the D1 to D0 brane marginal transition on a circle to explore the possibility that this finite range corresponds to the infinite range of the conformal field theory parameter describing marginal deformations, but are unable to arrive at a definitive conclusion. 
  In AdS_n x S^m threaded by N-units of Ramond-Ramond flux, the dynamics of a Dm-brane wrapped on S^{m-1} embedded in S^m is investigated. Under the condition that flux of the dual gauge field on the D-brane worldvolume is quantized integrally, It is found that the wrapped D-brane is stable both locally and globally and, on AdS_n, behaves effectively as a fundamental string. It is also claimed that the semi-infinite partially wrapped D-brane `lasso' around the S^m. 
  Starting from the Moyal formulation of M-theory in the large N-limit, we propose to reexamine the associated membrane equations of motion in 10 dimensions formulated in terms of Poisson bracket. Among the results obtained, we rewrite the coupled first order Nahm's equations into a simple form leading in turn to their systematic relation with $SU(\infty)$ Yang Mills equations of motion. The former are interpreted as the vanishing condition of some conserved currents which we propose. We develop also an algebraic analysis in which an ansatz is considered and find an explicit form for the membrane solution of our problem. Typical solutions known in literature can also emerge as special cases of the proposed solution 
  Violation of unitarity for noncommutative field theory on compact space-times is considered. Although such theories are free of ultraviolet divergences, they still violate unitarity while in a usual field theory such a violation occurs when the theory is nonrenormalizable. The compactness of space-like coordinates implies discreteness of the time variable which leads to appearance of unphysical modes and violation of unitarity even in the absence of a star-product in the interaction terms. Thus, this conclusion holds also for other quantum field theories with discrete time. Violation of causality, among others, occurs also as the nonvanishing of the commutation relations between observables at space-like distances with a typical scale of noncommutativity. While this feature allows for a possible violation of the spin-statistics theorem, such a violation does not rescue the situation but makes the scale of causality violation as the inverse of the mass appearing in the considered model, i.e., even more severe. We also stress the role of smearing over the noncommutative coordinates entering the field operator symbols. 
  We study the polarization states of the D0-brane in type IIA string theory. In addition to states with angular momentum and magnetic dipole moments, there are polarization states of a single D0-brane with nonzero D2-brane dipole and magnetic H-dipole moments, as well as quadrupole and higher moments of various charges. These fundamental moments of the D0-brane polarization states can be determined directly from the linearized couplings of background fields to the D0-brane world-volume fermions. These couplings determine the long range supergravity fields produced by a general polarization state, which typically have non-zero values for all the bosonic fields of type IIA supergravity. We demonstrate the precise cancellation between spin-spin and magnetic dipole-dipole interactions and an analogous cancellation between 3-form and H-field dipole-dipole interactions for a pair of D0-branes. The first of these cancellations follows from the fact that spinning D0-brane states have gyromagnetic ratio g = 1, and the second follows from the fact that the ratio between the 3-form and H-field dipole moments is also 1 in natural units. Both of these relationships can be seen immediately from the couplings in the D0-brane world-volume action. 
  A system containing a pair of non-BPS D-strings of type IIA string theory on an orbifold, representing a single D2-brane wrapped on a nonsupersymmetric 2-cycle of a Calabi-Yau 3-fold with $(h^{(1,1)},h^{(1,2)})$ = (11,11), is analyzed. In certain region of the moduli space the configuration is stable. We show that beyond the region of stability the system can decay into a pair of non-BPS D3-branes. At one point on the boundary of the region of stability, there exists a marginal deformation which connects the system of non-BPS D-strings to the system of non-BPS D3-branes. Across any other point on the boundary of the region of stability, the transition from the system of non-BPS D-strings to the system of non-BPS D3-branes is first order. We discuss the phase diagram in the moduli space for these configurations. 
  Generalizing previous work, we study the collision of massless superstring plane waves in D space-time dimensions within an explicitly O(D-2,D-2)-invariant set of field equations. We discuss some general properties of the solutions, showing in particular that they always lead to the formation of a singularity in the future. Using the above symmetry, we obtain entire classes of new analytic solutions with non-trivial metric, dilaton and antisymmetric field, and discuss some of their properties of specific relevance to string cosmology. 
  A covariant formalism for Moyal deformations of gauge theory and differential equations which determine Seiberg-Witten maps is presented. Replacing the ordinary product of functions by the noncommutative Moyal product, noncommutative versions of integrable models can be constructed. We explore how a Seiberg-Witten map acts in such a framework. As a specific example, we consider a noncommutative extension of the principal chiral model. 
  A combinatorial formula to generate U(N) character expansions is presented. It is shown that the resulting character expansion formulas greatly simplify a number of problems where integrals over the group manifolds need to be calculated. Several examples are given, including direct and very quick calculations of the Itzykson-Zuber integral and the finite volume effective partition function of QCD in the sector with a given topological charge. 
  We show that the Yang-Baxter equation is equivalent to the associativity of the algebra generated by non-commuting link operators. Starting from these link operators we build out the (FFZ) algebras, the $s\ell_q (2)$ is derived by considering a special combination of the generators of (FFZ) algebra. 
  Following the recent paper by J. F. van Diejen and H. Puschmann we investigate the behavior of zeros of the finite-gap solutions to the Schrodinger equation. As a result, a new system of particles on a punctured Riemann surface is constructed. It is shown to be Hamiltonian and integrable. 
  It is shown that traces of mapping classes of finite order may be expressed by Verlinde-like formulae. The 3D topological argument is explained, and the resulting trace identities for modular matrix elements are presented. 
  The construction of anyonic operators and algebra is generalized by using quons operators. Therefore, the particular versionof fractional supersymmetry is constructed on the two-dimensional lattice by associating two generalized anyons of different kinds. The fractional supersymmetry Hamiltonian operator is obtained on the two-dimensional lattice and the quantum algebra $U_{q}(sl_{2})$ is realized. 
  We study the Quantum-Mechanics on the hyper-Kahler manifold that is the blow-up of an $A_1$-singularity. This system is relevant for M(atrix)-theory as it was conjectured to describe scattering in the "noncommutative" deformation of a free 5+1D tensor multiplet in the sector with two units of longitudinal light-like momentum. 
  The generalization of the Jaynes-Cummings (GJC) Model is proposed. In this model, the electromagnetic radiation is described by a Hamiltonian generalizing the harmonic oscillator to take into account some nonlinear effects which can occurs in the experimental situations. The dynamical superalgebra and supercoherent states of the related model are explicitly constructed. A relevant quantities (total number of particles, energy and atomic inversion) are computed. 
  We investigate how the four-dimensional noncommutative open string/Yang-Mills theory behaves under a general non-perturbative quantum $SL(2,Z)$ symmetry transformation. We discuss this by considering D3 branes in a constant background of axion, dilaton, and electric and magnetic fields (including both $ {\bf E} \perp {\bf B}$ and {\bf E}$||${\bf B} cases) in the respective decoupling limit. We find that the value of axion, whether rational or irrational, determines the nature of the resulting theory under $SL(2,Z)$ as well as its properties such as the coupling constant and the number of noncommutative directions. In particular, a strongly coupled theory with an irrational value of axion can never be physically equivalent to a weakly coupled theory while this is usually true for a theory with a rational value of axion. A noncommutative Yang-Mills (NCYM) (resulting from D3 branes with pure magnetic flux) is physically equivalent to a noncommutative open string (NCOS) but if the value of axion is irrational, we also have noncommutative space-space directions in addition to the usual noncommutative space-time directions for NCOS. We also find in general that a NCOS cannot be physically equivalent to a NCYM but to another NCOS if the value of axion is irrational. We find another new decoupling limit for possible light-like NCYM whose $SL(2,Z)$ duality is a light-like ordinary Yang-Mills if the value of the axion is rational. Various related questions are also discussed. 
  Two new families of T-Dual integrable models of dyonic type are constructed. They represent specific $A_n^{(1)}$ singular Non-Abelian Affine Toda models having U(1) global symmetry. Their 1-soliton spectrum contains both neutral and U(1) charged topological solitons sharing the main properties of 4-dimensional Yang-Mills-Higgs monopoles and dyons. The semiclassical quantization of these solutions as well as the exact counterterms and the coupling constant renormalization are studied. 
  Following is a collection of lecture notes on D-branes, which may be used by the reader as preparation for applications to modern research applications such as: the AdS/CFT and other gauge theory/geometry correspondences, Matrix Theory and stringy non-commutative geometry, etc. In attempting to be reasonably self-contained, the notes start from classical point-particles and develop the subject logically (but selectively) through classical strings, quantisation, D-branes, supergravity, superstrings, string duality, including many detailed applications. Selected focus topics feature D-branes as probes of both spacetime and gauge geometry, highlighting the role of world-volume curvature and gauge couplings, with some non-Abelian cases. Other advanced topics which are discussed are the (presently) novel tools of research such as fractional branes, the enhancon mechanism, D(ielectric)-branes and the emergence of the fuzzy/non-commutative sphere. 
  Two subtle aspects of brane intersections are investigated. The first concerns the `half-branes' that arise in discussions of the Hanany-Witten effect, often in the D0/D8 setting. The second involves the validity of seemingly singular classical BPS brane intersections. A study of holomorphic curves in the background of a Kaluza-Klein monopole and the associated reduction to type IIA supergravity sheds light on both issues. Many seemingly singular D2/D6 intersections are shown to lift to smooth configurations of M2-branes in 11-dimensions, and a mechanism is found for certain $Z_2$ confinement effects in type II string theories that eliminates any need for half-branes. 
  In this paper we analyse the quantum correction for Schwarzschild black hole in the Unruh state in the framework of spherically symmetric gravity (SSG) model. SSG is a two-dimensional dilaton model which is obtained by spherically symmetric reduction from the four-dimensional theory. We find the one-loop geometry of the (anti)-evaporating black hole and corrections for mass, entropy and apparent horizon. 
  Recent scenarios of phenomenologically realistic string compactifications involve the existence of gauge sectors localized on D-branes at singular points of Calabi-Yau threefolds. The spectrum and interactions in these gauge sectors are determined by the local geometry of the singularity, and can be encoded in quiver diagrams. We discuss the physical models arising for the simplest case of orbifold singularities, and generalize to non-orbifold singularities and orientifold singularities. Finally we show that relatively simple singularities lead to gauge sectors surprisingly close to the standard model of elementary particles. 
  Universal formulas for the boundary and crosscap coefficients are presented, which are valid for all symmetric simple current modifications of the charge conjugation invariant of any rational conformal field theory. 
  K-theory provides a framework for classifying Ramond-Ramond (RR) charges and fields. K-theory of manifolds has a natural extension to K-theory of noncommutative algebras, such as the algebra considered in noncommutative Yang-Mills theory or in open string field theory. In a number of concrete problems, the K-theory analysis proceeds most naturally if one starts out with an infinite set of D-branes, reduced by tachyon condensation to a finite set. This suggests that string field theory should be reconsidered for N=infinity. 
  It has been recently shown that solitons are fundamental classical solutions of non-commutative field theories. We reconsider this issue from the standpoint of the Hall effect and identify some solutions with known solutions in the integer Hall effect with no Zeeman coupling. 
  Starting from a completely general standpoint, we find the most general brane-Universe solutions for a three-brane in a five dimensional spacetime. The brane can border regions of spacetime with or without a cosmological constant. Making no assumptions other than the usual cosmological symmetries of the metric, we prove that the equations of motion form an integrable system, and find the exact solution. The cosmology is indeed a boundary of a (class II) Schwarzschild-AdS spacetime, or a Minkowski (class I) spacetime. We analyse the various cosmological trajectories focusing particularly on those bordering vacuum spacetimes. We find, not surprisingly, that not all cosmologies are compatible with an asymptotically flat spacetime branch. We comment on the role of the radion in this picture. 
  By generalizing the auxiliary field term in the Lagrangian of simplicial chiral models on a (d-1)-dimensional simplex, the generalized simplicial chiral models has been introduced in \c{Ali}. These models can be solved analytically only in d=0 and d=2 cases at large-N limit. In d=0 case, we calculate the eigenvalue density function in strong regime and show that the partition function computed from this density function is consistent with one calculated by path integration directly. In d=2 case, it is shown that all $V= {\rm Tr}(AA^{\d})^n$ models have a third order phase transition, same as the 2-dimensional Yang-Mills theory. 
  We summarize the work done in connecting Green's functions in a different classes of gauges and its applications to the problems in the axial gauges.The procedure adopted uses finite field-dependent BRS [FFBRS] transformations to connect axial and the Lorentz type gauges.These transformations preserve the vacuum expectation of gauge-invariant observables explicitly. We discuss the applications of these ideas to the axial gauge pole problem and to the preservation of the Wilson loop and the thermal Wilson loop. 
  The $\phi^4$ field model is generalized to the case when the field $\phi(x)$ is defined on a Lie group: $S[\phi]=\int_{x\in G} L[\phi(x)] d\mu(x)$, $d\mu(x)$ is the left-invariant measure on a locally compact group $G$. For the particular case of the affine group $G:x'=ax+b,a\in\R_+, x,b \in \R^n$ t he Feynman perturbation expansion for the Green functions is shown to have no ultra-violet divergences for certain choice of $\lambda(a) \sim a^\nu$. 
  The possible role of gravity in a noncommutative geometry is investigated. Due to the Moyal *-product of fields in noncommutative geometry, it is necessary to complexify the metric tensor of gravity. We first consider the possibility of a complex Hermitian, nonsymmetric $g_{\mu\nu}$ and discuss the problems associated with such a theory. We then introduce a complex symmetric (non-Hermitian) metric, with the associated complex connection and curvature, as the basis of a noncommutative spacetime geometry. The spacetime coordinates are in general complex and the group of local gauge transformations is associated with the complex group of Lorentz transformations CSO(3,1). A real action is chosen to obtain a consistent set of field equations. A Weyl quantization of the metric associated with the algebra of noncommuting coordinates is employed. 
  In order to understand QCD at the energies relevant to hadronic physics one requires analytical methods for dealing with relativistic gauge field theories at large couplings. Strongly coupled quenched QED provides an ideal laboratory for the development of such techniques, in particular as many calculations suggest that - like QCD - this theory has a phase with broken chiral symmetry. In this talk we report on a nonperturbative variational calculation of the electron propagator within quenched QED and compare results to those obtained in other approaches. We find surprising differences among these results. 
  A method is presented for the computation of the one-loop effective action at finite temperature and density. The method is based on an expansion in the number of spatial covariant derivatives. It applies to general background field configurations with arbitrary internal symmetry group and space-time dependence. Full invariance under small and large gauge transformations is preserved without assuming stationary or Abelian fields nor fixing the gauge. The method is applied to the computation of the effective action of spin zero particles in 2+1 dimensions at finite temperature and density and in presence of background gauge fields. The calculation is carried out through second order in the number of spatial covariant derivatives. Some limiting cases are worked out. 
  We consider BPS domain walls in the four dimensional N=1 supersymmetric models with continuous global symmetry. Since the BPS equation is covariant under the global transformation, the solutions of the BPS walls also have the global symmetry. The moduli space of the supersymmetric vacua in such models have non-compact flat directions, and the complex BPS walls interpolating between two disjoint flat directions can exist. We examine this possibility in two models with global O(2) symmetry, and construct the solution of such BPS walls. 
  Heat kernel expansion coefficients are calculated for vacuum fluctuations with distributional background potentials and field strengths. Terms up to and including t^5/2 are presented. 
  Motivated by a recent conjecture [1] that quantum corrections and the UV/IR connection modify the classical relation between SUSY breaking and the cosmological constant to the phenomenologically acceptable : $M_{SUSY} \sim M_P (\Lambda /M_P^4)^{1/8}$, we study SUSY breaking by boundary conditions in noncommutative field theories. In commutative field theory the violations of SUSY are finite and vanish as the inverse fourth power of the radius of the SUSY violating circle. We show that in the noncommutative theory, as a consequence of its UV/IR connection, the perturbative corrections to SUSY breaking are infinite. We have not yet performed the nonperturbative resummations to extract the true behavior of the system. 
  Dynamical R-matrix relations are derived for the group-valued chiral vertex operators in the SU(n) WZNW model from the KZ equation for a general four-point function including two step operators. They fit the exchange relations for the U_q(sl_n) covariant quantum matrix derived previously by solving the dynamical Yang-Baxter equation. As a byproduct, we extend the regular basis introduced earlier for SU(2) chiral fields to SU(n) step operators and display the corresponding triangular matrix representation of the braid group. 
  In earlier work we have given a Hamiltonian analysis of Yang-Mills theory in (2+1) dimensions showing how a mass gap could arise. In this paper, generalizing and covariantizing from the mass term in the Hamiltonian analysis, we obtain two manifestly covariant and gauge-invariant mass terms which can be used in a resummation of standard perturbation theory to study properties of the mass gap. 
  We compute fractional and integer fermion quantum numbers of static background field configurations using phase shifts and Levinson's theorem. By extending fermionic scattering theory to arbitrary dimensions, we implement dimensional regularization in a 1+1 dimensional gauge theory. We demonstrate that this regularization procedure automatically eliminates the anomaly in the vector current that a naive regulator would produce. We also apply these techniques to bag models in one and three dimensions. 
  Using perturbative QED we show that, while the retarded dispersive force between an electrically polarizable atom and a magnetically polarizable one is proportional to $1/r^{8}$, where $r$ is the distance between the atoms, the non-retarded force is proportiaonal to $1/r^{5}$. This is a rather surprising result that should be compared with the dispersive van der Waals force between two electrically polarizable atoms, where the retarded force is also proportional to $1/r^{8}$, but the non-retarded force is proportional to $1/r^{7}$. 
  We revisit the singular IIB supergravity solution describing M fractional 3-branes on the conifold [hep-th/0002159]. Its 5-form flux decreases, which we explain by showing that the relevant \NN=1 SUSY SU(N+M)xSU(N) gauge theory undergoes repeated Seiberg-duality transformations in which N -> N-M. Far in the IR the gauge theory confines; its chiral symmetry breaking removes the singularity of hep-th/0002159 by deforming the conifold. We propose a non-singular pure-supergravity background dual to the field theory on all scales, with small curvature everywhere if the `t Hooft coupling g_s M is large. In the UV it approaches that of hep-th/0002159, incorporating the logarithmic flow of couplings. In the IR the deformation of the conifold gives a geometrical realization of chiral symmetry breaking and confinement. We suggest that pure \NN=1 Yang-Mills may be dual to strings propagating at small g_s M on a warped deformed conifold. We note also that the standard model itself may lie at the base of a duality cascade. 
  We consider the Hamiltonian treatment of non-local theories and Ostrogradski's formalism. This has recently also been discussed by Woodard (hep-th/0006207) and by Gomis, Kamimura and Llosa (hep-th/0006235). In our approach we recast the second class constraints into first class constraints and invoke the boundary Poisson bracket. 
  This article shows that one can consistently incorporate nonunitary representations of at least one group into the ``ordinary'' nonrelativistic quantum mechanics. This group turns out to be Lorentz group thus giving us an alternative approach to QFT for combining the quantum mechanics and special theory of relativity which keeps the concept of wave function (belonging to some representation of Lorentz group) through the whole theory. Scalar product has been redefined to take into the account the nonunitarity of representations of Lorentz group. Understanding parity symmetry turns out to be the key ingredient throughout the process. Instead of trying to guess an equation or a set of equations for some wave functions or fields (or equivalently trying to guess a Lagrangian for the same), one derives them based only on the superposition principle and properties of wave functions under Lorentz transformations and parity. The resulting model has striking similarities with the standard quantum field theory and yet has no negative energy states, no zitterbewegung effects, symmetric energy momentum tensor and angular momentum density tensor for \emph{all} representations of Lorentz group (unifying the theoretical description of all particles), as well as clear physical interpretation. It also offers a possible interpretation why particles and antiparticles have opposite quantum numbers. 
  We calculate the genus 2 correlation functions of two-dimensional topological gravity, in a background with two primary fields, using the genus 2 topological recursion relations. 
  We review work done over the last years on the macroscopic and microscopic entropy of supersymmetric black holes in fourdimensional N=2 supergravity and in N=2 compactifications of string theory and M-theory. Particular emphasis is put on the crucial role of higher curvature terms and of modifications of the area law in obtaining agreement between the macroscopic entropy, which is a geometric property of black hole solutions and the microscopic entropy, which is computed by state counting in Calabi-Yau compactifications of string or M-theory. We also discuss invariance properties of the entropy under stringy T-duality and S-duality transformations in N=2,4 compactifications in presence of higher curvature terms.   In order to make the paper self-contained we review the laws of black hole mechanics in higher derivative gravity, the definition of entropy as a surface charge, the superconformal off-shell description of N=2 supergravity, special geometry, and N=2 compactifications of heterotic and type II string theory and of M-theory. 
  In this thesis we discuss recent new insights in the structure of the moduli space of flat connections of Yang-Mills theory on a 3-torus. Chapter 2 discusses the computation of Witten's index for 4-dimensional gauge theories, and the paradox that arises in comparing various computations. This was resolved by the discovery that for orthogonal and exceptional gauge groups, periodic flat connections exist that are contained in seperate, disconnected components of the moduli space. Chapter 3 and 4 discuss some aspects of the construction of holonomies parametrising vacua on such disconnected components. Chapter 5 demonstrates a construction of vacua and holonomies for gauge theories with classical groups, with non-periodic (twisted) boundary conditions, using an orientifold description. The new solutions with exceptional and orthogonal gauge groups also occur in string theory. Chapter 6, containing previously unpublished material, shows that they can be realised within heterotic string theories as asymmetric orbifolds. The presence of string winding states modifies the analysis for the gauge theory in a crucial way, eliminating many possibilities. The remaining ones are related by string dualities to known and new theories. 
  We investigate the zero mode dimensional reduction of the Kaluza-Klein unifications in the presence of a single brane in the infinite extra dimension. We treat the brane as fixed, not a dynamical object, and do not require the orbifold symmetry. It seems that, contrary to the standard Kaluza-Klein models, the 4D effective action is no longer invariant under the U(1) gauge transformations due to the explicit breaking of isometries in the extra dimension by the brane. Surprisingly, however, the linearized perturbation analysis around the RS vacuum shows that the Kaluza-Klein gauge field does possess the U(1) gauge symmetry at the linear level. In addition, the graviscalar also behaves differently from the 4D point of view. Some physical implications of our results are also discussed. 
  Motivated by recent work we study rotating ellipsoidal membranes in the framework of the light-cone supermembrane theory. We investigate stability properties of these classical solutions which are important for the quantization of super membranes. We find the stability modes for all sectors of small multipole deformations. We exhibit an isomorphism of the linearized membrane equation with that of the SU(N) matrix model for every value of $N$. The boundaries of the linearized stability region are at a finite distance and they appear for finite size perturbations. 
  It has recently been found that supernova explosions can be simulated in the laboratory by implosions induced in a plasma by intense lasers. A theoretical explanation is that the inversion transformation, ($\Sigma: t \to -1/t,~ {\bf x}\to {\bf x}/t$), leaves the Euler equations of fluid dynamics, with standard polytropic exponent, invariant. This implies that the kinematical invariance group of the Euler equations is larger than the Galilei group. In this paper we determine, in a systematic manner, the maximal invariance group ${\cal G}$ of general fluid dynamics and show that it is a semi-direct product ${\cal G} = SL(2,R) \wedge G$, where the $SL(2,R)$ group contains the time-translations, dilations and the inversion $\Sigma$, and $G$ is the static (nine-parameter) Galilei group. A subtle aspect of the inclusion of viscosity fields is discussed and it is shown that the Navier-Stokes assumption of constant viscosity breaks the $SL(2, R)$ group to a two-parameter group of time translations and dilations in a tensorial way. The 12-parameter group ${\cal G}$ is also known to be the maximal invariance group of the free Schr\"odinger equation. It originates in the free Hamilton-Jacobi equation which is central to both fluid dynamics and the Schr\"odinger equation. 
  We consider possible ambiguities in the holographic Weyl anomaly that are caused by local terms in the flow equation. We point out that such ambiguities actually do not give physically meaningful contributions to the Weyl anomaly. 
  We evaluate the quantum correlators associated with the Maxwell field vacuum distorted by the presence of plane parallel material surfaces. Regularization is performed through the generalized zeta funtion technique. Results are applied to a local analysis of the atractive and repulsive Casimir effect through Maxwell stress tensor. Surface divergences are shown to cancel out when stresses on both sides of the material surface are taken into account. Since an atom can be considered as a probe of the local distortion of the quantum vacuum, Casimir-Polder interactions between atoms and material surfaces are also considered. 
  We calculate the absorption probabilities for a class of massless fields whose linear perturbations leave the near-extremal D3-brane background metric unperturbed. It has previously been found that, for extremal D3-branes, these fields share the same absorption probability as that of the dilaton-axion. We find that these absorption probabilities diverge from each other as we move away from extremality. The form of the corresponding effective Schrodinger potentials leads us to conjecture that the absorption of various fields by nonextremal D3-branes depends on the polarization of angular momentum. 
  We numerically construct time-symmetric initial data sets of a black hole in the Randall-Sundrum brane world model, assuming that the black hole is spherical on the brane. We find that the apparent horizon is cigar-shaped in the 5D spacetime. 
  We continue our study of solitons in noncommutative gauge theories and present an extremely simple BPS solution of N=4 U(1) noncommutative gauge theory in 4 dimensions, which describes N infinite D1 strings that pierce a D3 brane at various points, in the presence of a background B-field in the Seiberg-Witten limit. We call this solution the N-fluxon. For N=1 we calculate the complete spectrum of small fluctuations about the fluxon and find three kinds of modes: the fluctuations of the superstring in 10 dimensions arising from fundamental strings attached to the D1strings, the ordinary particles of the gauge theory in 4 dimensions and a set of states with discrete spectrum, localized at the intersection point--- corresponding to fundamental strings stretched between the D1string and the D3 brane. We discuss the fluctuations about theN-fluxon as well and derive explicit expressions for the amplitudes of interactions between these various modes. We show that translations in noncommutative gauge theories are equivalent to gauge transformations (plus a constant shift of the gauge field) and discuss the implications for the translational zeromodes of our solitons. We also find the dyonic versions of N-fluxon, as well as of our previous string-monopole solution. 
  The model of d5 higher derivative (HD) gravity admitting Schwarzschild--anti de Sitter Black Hole (S-AdS BH) as exact solution is considered. The surface counterterms are added to the complete action, they are fixed by the condition of finiteness of bulk AdS spacetime when brane goes to infinity. As a result the brane (observable Universe) is defined dynamically in terms of d5 theory parameters (brane tension is fixed). Brane radius is always bigger than horizon radius and the 4d Universe itself could be static or inflationary. Natural ganeralization of this model gives warped compactification (with dynamically generated brane) in the next-to-leading order of AdS/CFT correspondence. 
  Motivated by recent work of Bousso and Polchinski (BP), we study theories which explain the small value of the cosmological constant using the anthropic principle. We argue that simultaneous solution of the gauge hierarchy problem is a strong constraint on any such theory. We exhibit three classes of models which satisfy these constraints. The first is a version of the BP model with precisely two large dimensions. The second involves 6-branes and antibranes wrapped on supersymmetric 3-cycles of Calabi-Yau manifolds, and the third is a version of the irrational axion model. All of them have possible problems in explaining the size of microwave background fluctuations. We also find that most models of this type predict that all constants in the low energy Lagrangian, as well as the gauge groups and representation content, are chosen from an ensemble and cannot be uniquely determined from the fundamental theory. In our opinion, this significantly reduces the appeal of this kind of solution of the cosmological constant problem. On the other hand, we argue that the vacuum selection problem of string theory might plausibly have an anthropic, cosmological solution. 
  We study backreaction effects in two-dimensional dilaton gravity. The backreaction comes from an $R^2$ term which is a part of the one-loop effective action arising from massive scalar field quantization in a certain approximation. The peculiarity of this term is that it does not contribute to the Hawking radiation of the classical black hole solution of the field equations. In the static case we examine the horizon and the physical singularity of the new black hole solutions. Studying the possibility of time dependence we see the generation of a new singularity. The particular solution found still has the structure of a black hole, indicating that non-thermal effects cannot lead, at least in this approximation, to black hole evaporation. 
  We show that the supersymmetric rational Calogero-Moser-Sutherland (CMS) model of A_{N+1}-type is equivalent to a set of free super-oscillators, through a similarity transformation. We prescribe methods to construct the complete eigen-spectrum and the associated eigen-functions, both in supersymmetry preserving as well as supersymmetry breaking phases, from the free super-oscillator basis. Further we show that a wide class of super-Hamiltonians realizing dynamical OSp(2|2) supersymmetry, which also includes all types of rational super-CMS as a small subset, are equivalent to free super-oscillators. We study BC_{N+1}-type super-CMS model in some detail to understand the subtleties involved in this method. 
  We review general domain-wall solutions supported by a delta-function source, together with a single pure exponential scalar potential in supergravity. These scalar potentials arise from a sphere reduction in M-theory or string theory. There are several examples of flat (BPS) domain walls that lead to a localisation of gravity on the brane, and for these we obtain the form of the corrections to Newtonian gravity. These solutions are lifted back on certain internal spheres to D=11 and D=10 as M-branes and D-branes. We find that the domain walls that can trap gravity yield M-branes or Dp-branes that have a natural decoupling limit, i.e. p\le 5, with the delta-function source providing an ultra-violet cut-off in a dual quantum field theory. This suggests that the localisation of gravity can generally be realised within a Domain-wall/QFT correspondence, with the delta-function domain-wall source providing a cut-off from the space-time boundary for these domain-wall solutions. We also discuss the form of the one-loop corrections to the graviton propagator from the boundary QFT that would reproduce the corrections to the Newtonian gravity on the domain wall. 
  We investigate how much a first-quantized charged bosonic test string gets excited after crossing a shock wave generated by a charged particle with mass $\tilde{M}$ and charge $\tilde{Q}$. On the basis of Kaluza-Klein theory, we pay attention to a closed string model where charge is given by a momentum along a compactified extra-dimension. The shock wave is given by a charged Aichelburg-Sexl (CAS) spacetime where $\tilde{Q}=0$ corresponds to the ordinary Aichelburg-Sexl one. We first show that the CAS spacetime is a solution to the equations of motion for the metric, the gauge field, and the axion field in the low-energy limit. Secondly, we compute the mass expectation value of the charged test string after passing through the shock wave in the CAS spacetime.   In the case of small $\tilde{Q}$, gravitational and Coulomb forces are canceled out each other and hence the excitation of the string remains very small. This is independent of the particle mass $\tilde{M}$ or the strength of the shock wave. In the case of large $\tilde{Q}$, however, every charged string gets highly excited by quantum fluctuation in the extra-dimension caused by both the gauge and the axion fields. This is quite different from classical "molecule", which consists of two electrically charged particles connected by a classical spring. 
  Exploring the role of conformal theories of gravity in string theory, we show that the minimal (N=2) gauged supergravities in five dimensions induce the multiplets and transformations of N=1 four dimensional conformal supergravity on the spacetime boundary. N=1 Poincare supergravity can be induced by explicitly breaking the conformal invariance via a radial cutoff in the 5d space. The AdS/CFT correspondence relates the maximal gauged supergravity in five dimensions to N=4 super Yang-Mills on the 4d spacetime boundary. In this context we show that the conformal anomaly of the gauge theory induces conformal gravity on the boundary of the space and that this theory, via the renormalization group, encapsulates the gravitational dynamics of the skin of asymptotically AdS spacetimes. Our results have several applications to the AdS/CFT correspondence and the Randall-Sundrum scenario. 
  Tadpole cancellation in F-theory on an elliptic Calabi-Yau fourfold $X\to B_3$ demands some spacetime-filling three-branes (points in $B_3$). If moved to the discriminant surface, which supports the gauge group, and dissolved into a finite size instanton, the second Chern class of the corresponding bundle $E$ is expected to give a compensating contribution. However the dependence of D-brane charge on the geometry of $W$ and on the embedding $i: W\to B_3$ gives a correction to $c_2(E)$. We show how this is reconciled by considering the torsion sheaf $i_*E$ and discuss some integrality issues related to global properties of $X$ as well as the moduli space of this object. 
  Various aspects of low energy M theory compactified to four dimensions are considered. If the supersymmetry parameter is parallel in the unwarped metric, then supersymmetry requires that the warp factor is trivial, the background four-form field strength is zero and that the internal 7-manifold has $G_2$ holonomy (we assume the absence of boundaries and other impurities). A proposal of Gukov - extended here to include M2-brane domain walls - for the superpotential of the compactified theory is shown to yield the same result. Finally, we make some speculative remarks concerning higher derivative corrections and supersymmetry breaking. 
  The supersymmetric sinh-Gordon model on a half-line with integrable boundary conditions is considered perturbatively to verify conjectured exact reflection factors to one loop order. Propagators for the boson and fermion fields restricted to a half-line contain several novel features and are developed as prerequisites for the calculations. 
  We construct dual supergravity descriptions of field theories and little string theories with light-like non-commutativity. The field theories are realized on the world-volume of Dp branes with light-like NS $B$ field and M5 branes with light-like $C$ field. The little string theories are realized on the world-volume of NS5 branes with light-like RR $A$ fields. The supergravity backgrounds are closely related to the $A=0,B=0,C=0$ backgrounds. We discuss the implications of these results. We also construct dual supergravity descriptions of ODp theories realized on the worldvolume of NS5 branes with RR backgrounds. 
  We study the symmetry behavior of the Gross-Neveu model in three and two dimensions with random chemical potential. This is equivalent to a four-fermion model with charge conjugation symmetry as well as Z_2 chiral symmetry. At high temperature the Z_2 chiral symmetry is always restored. In three dimensions the initially broken charge conjugation symmetry is not restored at high temperature, irrespective of the value of the disorder strength. In two dimensions and at zero temperature the charge conjugation symmetry undergoes a quantum phase transition from a symmetric state (for weak disorder) to a broken state (for strong disorder) as the disorder strength is varied. For any given value of disorder strength, the high-temperature behavior of the charge conjugation symmetry is the same as its zero-temperature behavior. Therefore, in two dimensions and for strong disorder strength the charge conjugation symmetry is not restored at high temperature. 
  We show that in the presence of U(1) noncommutative gauge interaction the noncommutative tachyonic system exhibits solitonic solutions for finite value of the noncommutativity parameter. 
  We report the statistical properties of classical particles in (2+1) gravity as resulting from numerical simulations. Only particle momenta have been taken into account. In the range of total momentum where thermal equilibrium is reached, the distribution function and the corresponding Boltzmann entropy are computed. In the presence of large gravity effects, different extensions of the temperature turn out to be inequivalent, the distribution function has a power law high-energy tail and the entropy as a function of the internal energy presents a flex. When the energy approaches the open universe limit, the entropy and the mean value of the particle kinetic energy seem to diverge. 
  The construction of the fusion ring of a quasi-rational CFT based on $\hat{sl}(3)_k$ at generic level $k\not \in {\Bbb Q}$ is reviewed. It is a commutative ring generated by formal characters, elements in the group ring ${\Bbb Z}[\tilde{W}]$ of the extended affine Weyl group $\tilde{W}$ of $\hat{sl}(3)_k$. Some partial results towards the $\hat{sl}(4)_k$ generalisation of this character ring are presented. 
  We investigate, in any spacetime dimension >=3, the problem of consistent couplings for a finite collection of massless, spin-2 fields described, in the free limit, by a sum of Pauli-Fierz actions. We show that there is no consistent (ghost-free) coupling, with at most two derivatives of the fields, that can mix the various "gravitons". In other words, there are no Yang-Mills-like spin-2 theories. The only possible deformations are given by a sum of individual Einstein-Hilbert actions. The impossibility of cross-couplings subsists in the presence of scalar matter. Our approach is based on the BRST-based deformation point of view and uses results on the so-called "characteristic cohomology" for massless spin-2 fields which are explained in detail. 
  Certain static soliton configurations of gauge fields in 4+1 dimensions correspond to the instanton in 4-Euclidean dimensions ``turned on its side,'' becoming a monopole in 4+1. The periodic instanton solution can be used with the method of images to construct solutions satisfying D-brane boundary conditions. The $\theta$-term on the brane becomes a topological current source, yielding an emission amplitude for monopoles into the bulk. Instantons have a novel reinterpretation in terms of monopole exchange between branes. 
  Conformally invariant quantum field theories develop trace anomalies when defined on curved backgrounds. We study again the problem of identifying all possible trace anomalies in d=6 by studying the consistency conditions to derive their 10 independent solutions. It is known that only 4 of these solutions represent true anomalies, classified as one type A anomaly, given by the topological Euler density, and three type B anomalies, made up by three independent Weyl invariants. However, we also present the explicit expressions of the remaining 6 trivial anomalies, namely those that can be obtained by the Weyl variation of local functionals. The knowledge of the latter is in general necessary to disentangle the universal coefficients of the type A and B anomalies from calculations performed on concrete models. 
  Brane world scenarios offer a way of setting the cosmological constant to zero after supersymmetry breaking provided there is a sufficient number of adjustable integration constants/parameters. In the case of the Horava-Witten theory compactified on a Calabi-Yau threefold, we argue that it is difficult to find enough freedom to get a zero (or small) cosmological constant after supersymmetry breaking. 
  A straightforward explanation of the Young's two-slit experiment of a quantum particle is obtained within the framework of the Noncommutative Geometric associated with El Naschie's Cantorian-Fractal transfinite Spacetime continuum. 
  A generalization of the Akulov-Volkov effective Lagrangian governing the self interactions of the Nambu-Goldstone fermions associated with spontaneously broken extended supersymmetry as well as their coupling to matter is presented and scrutinized. The resulting currents associated with R-symmetry, supersymmetry and space-time translations are constructed and seen to form a supermultiplet structure. 
  It has recently been argued that D-branes in bosonic string theory can be described as noncommutative solitons, outside whose core the tachyon is condensed to its ground state. We conjecture that, in addition, the local U(1) gauge symmetry is restored to a $U(\infty)$ symmetry in the vacuum outside this core. We present new solutions obeying this boundary condition. The tension of these solitons agrees exactly with the expected D-brane tension for arbitrary noncommutativity parameter $\t$, which effectively becomes a dynamical variable. The restored $U(\infty)$ eliminates unwanted extra modes which might otherwise appear outside the soliton core. 
  4D cylindrical gravitational waves with aligned polarizations (Einstein-Rosen waves) are shown to be described by a weight 1/2 massive free field on the double cover of AdS_2. Thorn's C-energy is one of the sl(2,R) generators, the reconstruction from the (timelike) symmetry axis is the CFT_1 holography. Classically the phase space is also invariant under a O(1,1) group action on the metric coefficients that is a remnant of the original 4D diffeomorphism invariance. In the quantum theory this symmetry is found to be spontaneously broken while the AdS_2 conformal invariance remains intact. 
  Static spherically symmetric asymptotically flat solutions to the U(1) Born-Infeld theory coupled to gravity and to a dilaton are investigated. The dilaton enters in such a way that the theory is $SL(2,R)$- symmetric for a unit value of the dilaton coupling constant. We find globally regular solutions for any value of the effective coupling which is the ratio of the gravitational and dilaton couplings; they form a continuous sequence labeled by the sole free parameter of the local solution near the origin. The allowed values of this parameter are bounded from below, and, as the limiting value is approached, the mass and the dilaton charge rise indefinitely and tend to a strict equality (in suitable units). Together with the electric charge they saturate the BPS bound of the corresponding linear U(1) theory. Beyond this boundary the solutions become compact (singular), while the limiting solution at the exact boundary value is non-compact and non-asymptotically flat. The black holes in this theory exist for any value of the event horizon radius and also form a sequence labelled by a continuously varying parameter restricted to a finite interval. The singularity inside the black hole exhibits a power-law mass inflation. 
  A model for the infrared sector of SU(2) Yang-Mills theory, based on magnetic vortices represented by (closed) random surfaces, is presented. The random surfaces, governed by an action penalizing curvature, are investigated using Monte Carlo methods on a hypercubic lattice. A low-temperature confining phase and a high-temperature deconfined phase are generated by this simple dynamics. After fixing the parameters of the model such as to reproduce the relation between deconfinement temperature and zero-temperature string tension found in lattice Yang-Mills theory, a surprisingly accurate prediction of the spatial string tension in the deconfined phase results. Furthermore, the Pontryagin index associated with the lattice random surfaces of the model is constructed. This allows to also predict the topological susceptibility; the result is compatible with measurements in lattice Yang-Mills theory. Thus, for the first time an effective model description of the infrared sector emerges which simultaneously and consistently reproduces both confinement and the topological aspects of Yang-Mills theory within a unified framework. Further details can be found in hep-lat/9912003 and hep-lat/0004013, both to appear in Nucl. Phys. B. 
  For many systems with second class constraints, the question posed in the title is answered in the negative. We prove this for a range of systems with two second class constraints. After looking at two examples, we consider a fairly general proof. It is shown that, to unravel gauge invariances in second class constrained systems, it is sufficient to work in the original phase space itself. Extension of the phase space by introducing new variables or fields is not required. 
  We review self-duality of nonlinear electrodynamics and its extension to several Abelian gauge fields coupled to scalars. We then describe self-duality in supersymmetric models, both N = 1 and N = 2. The self-duality equations, which have to be satisfied by the action of any self-dual system, are found and solutions are discussed. One important example is the Born-Infeld action. We explain why the N = 2 supersymmetric actions proposed so far are not the correct world-volume actions for D3 branes in d = 6. 
  Rotation symmetry is less constraining than space-time symmetry. The free electron propagator is a projection operator that we show can be constructed from rotation symmetric projection operators. Rotation-based identifications of time, space, energy, momentum, polarization matrices, and the positron hypothesis are determined by the constraints that turn rotation symmetric projection operators into the electron propagator.   PACS: 11.30.-j, 11.30.Cp, and 03.65.Fd 
  We include the backreaction on the warped geometry induced by non-finetuned parameters in a two domain-wall set-up to obtain an exponentially small Cosmological Constant $\Lambda_4$. The mechanism to suppress the Cosmological Constant involves one classical fine-tuning as compared to an infinity of finetunings at the quantum level in standard D=4 field theory. 
  We find the Holographic Renormalization Group equations for the holographic duals of generic gravity theories coupled to form fields and spin-1/2 fermions. Using Hamilton-Jacobi theory we discuss the structure of Ward identities, anomalies, and the recursive equations for determining the divergent terms of the generating functional. In particular, the Ward identity associated to diffeomorphism invariance contains an anomalous contribution that, however, can be solved either by a suitable counter term or by imposing a condition on the boundary fields. Consistency conditions for the existence of the dual arise, if one requires that a Callan-Symanzik type equation follows from the Hamiltonian constraint. Under mild assumptions we are able to find a class of solutions to the constraint equations. The structure of the fermionic phase space and constraints is treated extensively for any dimension and signature. 
  The D0/D4 system with a Neveu-Schwarz B-field in the spatial directions of the D4-brane has a tachyon in the spectrum of the (0,4) strings. The tachyon signals the instability of the system to form a bound state of the D0-brane with the D4-brane. We use the Wess-Zumino-Witten like open superstring field theory formulated by Berkovits to study the tachyon potential for this system. The tachyon potential lies outside the universality class of the D-brane anti-D-brane system. It is a function of the B-field. We calculate the tachyon potential at the zeroth level approximation. The minimum of the tachyon potential in this case is expected to reproduce the mass defect involved in the formation of the D0/D4 bound state. We compare the minimum of the tachyon potential with the mass defect in three cases. For small values of the B-field we obtain 70% of the expected mass defect. For large values of the B-field with Pf$(2\pi\alpha' B) >0$ the potential reduces to that of the D-brane anti-D-brane reproducing 62% of the expected mass defect. For large values of the B-field with Pf$(2\pi\alpha' B) <0$ the minimum of the tachyon potential gives 25% of the expected mass defect. At the tachyon condensate we show that the (0,4) strings decouple from the low energy dynamics. 
  We generalize the spectral-curve construction of moduli spaces of instantons on $\MT{4}$ and $K_3$ to noncommutative geometry. We argue that the spectral-curves should be constructed inside a twisted $\MT{4}$ or $K_3$ that is an elliptic fibration without a section. We demonstrate this explicitly for $T^4$ and to first order in the noncommutativity, for $K_3$. Physically, moduli spaces of noncommutative instantons appear as moduli spaces of theories with $\SUSY{4}$ supersymmetry in 2+1D. The spectral curves are related to Seiberg-Witten curves of theories with $\SUSY{2}$ in 3+1D. In particular, we argue that the moduli space of instantons of $U(q)$ Yang-Mills theories on a noncommutative $K_3$ is equivalent to the Coulomb branch of certain 2+1D theories with ${\cal N} = 4$ supersymmetry. The theories are obtained by compactifying the heterotic little-string theory on $T^3$ with global twists. This extends a previous result for noncommutative instantons on $\MT{4}$. We also briefly discuss the instanton equation on generic curved spaces. 
  If Bekenstein's conjectured bound on the microcanonical entropy, S < 2 pi E R, is applied to a closed subsystem of maximal linear size R and excitation energy up through E, it can be violated by an arbitrarily large factor by a scalar field having a symmetric potential allowing domain walls, and by the electromagnetic field modes in a coaxial cable. 
  Bekenstein's conjectured entropy bound for a system of linear size R and energy E, S < 2 pi E R, has counterexamples for many of the ways in which the ``system,'' R, E, and S may be defined. Here new ways are proposed to define these quantities for arbitrary nongravitational quantum field theories in flat spacetime, such as defining R as the smallest radius outside of which only vacuum expectation values occur. Difficulties of extending these definitions to gravitational quantum theories are noted. 
  Semiclassical Einstein's equation in five-dimension with a negative cosmological constant and conformally invariant bulk matter fields is examined in the brane world scenario with the S^1/Z_2 compactification. When numbers N_{b,f} of bosonic and fermionic fields satisfy I<\kappa^2l^{-3}(N_b-N_f)<32I, we obtain an exact semiclassical solution which has two static branes with positive tension and for which the warp factor can be arbitrarily large. Here, \kappa^2 is the five-dimensional gravitational constant, l is a length scale determined by the negative five-dimensional cosmological constant, and $I$ is a dimensionless positive constant of order unity. However, in order to obtain a large warp factor, fine tuning of brane tensions is required. Hence, in order to solve the hierarchy problem, we need to justify the fine tuning by another new mechanism. 
  Recently, developments in the understanding of low-energy N=1 supersymmetric gauge theory have revealed two important phenomena: the appearance of new four-dimensional superconformal field theories and a non-Abelian generalization of electric-magnetic duality at the IR fixed point. This report is a pedagogical introduction to these phenomena. After presenting some necessary background material, a detailed introduction to the low-energy non-perturbative dynamics of N=1 supersymmetric $SU(N_c)$ QCD is given. The emergence of new four-dimensional superconformal field theories and non-Abelian duality is explained. New non-perturbative phenomena in supersymmetric $SO(N_c)$ and $Sp(N_c=2n_c)$ gauge theories, such as the two inequivalent branches, oblique confinement and electric-magnetic-dyonic triality, are presented. Finally, some new features of these four-dimensional superconformal field theories are exhibited: the universal operator product expansion, evidence for a possible $c$-theorem in four dimensions and the critical behaviour of anomalous currents. The concluding remarks contain a brief history of electric-magnetic duality and a discussion on the possible applications of duality to ordinary QCD. 
  We present a general method for constructing supergravity solutions for intersecting branes. The solutions are written in terms of a single function, which is the solution to a nonlinear differential equation. We illustrate this procedure in detail for the case of M2-branes ending on M5-branes. We also present supergravity solutions for strings ending on Dp-branes. Unlike previous results in the literature, these branes are completely localized. 
  We study the Casimir effect for scalar fields with general curvature coupling subject to mixed boundary conditions $(1+\beta_{m}n^{\mu}\partial_{\mu})\phi =0$ at $x=a_{m}$ on one ($m=1$) and two ($m=1,2$) parallel plates at a distance $a\equiv a_{2}-a_{1}$ from each other. Making use of the generalized Abel-Plana formula previously established by one of the authors \cite{Sahrev}, the Casimir energy densities are obtained as functions of $\beta_{1}$ and of $\beta_{1}$,$\beta_{2}$,$a$, respectively. In the case of two parallel plates, a decomposition of the total Casimir energy into volumic and superficial contributions is provided. The possibility of finding a vanishing energy for particular parameter choices is shown, and the existence of a minimum to the surface part is also observed. We show that there is a region in the space of parameters defining the boundary conditions in which the Casimir forces are repulsive for small distances and attractive for large distances. This yields to an interesting possibility for stabilizing the distance between the plates by using the vacuum forces. 
  We revisit local mirror symmetry associated with del Pezzo surfaces in Calabi-Yau threefolds in view of five-dimensional N=1 E_N theories compactified on a circle. The mirror partner of singular Calabi-Yau with a shrinking del Pezzo four-cycle is described as the affine 7-brane backgrounds probed by a D3-brane. Evaluating the mirror map and the BPS central charge we relate junction charges to RR charges of D-branes wrapped on del Pezzo surfaces. This enables us to determine how the string junctions are mapped to D-branes on del Pezzo surfaces. 
  Static vortices close together are studied for two different models in 2-dimen- sional Euclidean space. In a simple model for one complex field an expansion in the parameters describing the relative position of two vortices can be given in terms of trigonometric and exponential functions. The results are then compared to those of the Ginzburg-Landau theory of a superconductor in a magnetic field at the point between type-I and type-II superconductivity. For the angular dependence a similar pattern emerges in both models. The differences for the radial functions are studied up to third order. 
  We continue to derive spacetime quantities and spin 1/2 propagators from rotations. Rotation-invariant projection operators are found for each element of a four element basis, i.e. a basis for four component quantities with specific transformation rules under rotations. With these four projection operators, we make two spacetime invariant projection operators, i.e. once space, time, energy, and momentum are identified. The spacetime invariant operators are propagators for free neutrinos. Except for the substitute basis, the process is the same as the one that gave electron propagators in Part I.   PACS number(s): 11.30.-j, 11.30.Cp, and 03.65.Fd 
  Fate of branes in external fields is reviewed with emphasis on a spontaneous creation of the Brane World. No negative tension brane is involved. 
  We consider noncommutative theory of a compact scalar field. The recently discovered projector solitons are interpreted as classical vacua in the model considered. Localized solutions to the projector equation are pointed out and their brane interpretation is discussed. An example of the noncommutative soliton interpolating between such vacua is given. No strong noncommutativity limit is assumed. 
  The Ising limit of a conventional Hermitian parity-symmetric scalar quantum field theory is a correlated limit in which two bare Lagrangian parameters, the coupling constant $g$ and the {\it negative} mass squared $-m^2$, both approach infinity with the ratio $-m^2/g=\alpha>0$ held fixed. In this limit the renormalized mass of the asymptotic theory is finite. Moreover, the limiting theory exhibits universal properties. For a non-Hermitian $\cal PT$-symmetric Lagrangian lacking parity symmetry, whose interaction term has the form $-g(i\phi)^N/N$, the renormalized mass diverges in this correlated limit. Nevertheless, the asymptotic theory still has interesting properties. For example, the one-point Green's function approaches the value $-i\alpha^{1/(N-2)}$ independently of the space-time dimension $D$ for $D<2$. Moreover, while the Ising limit of a parity-symmetric quantum field theory is dominated by a dilute instanton gas, the corresponding correlated limit of a $\cal PT$-symmetric quantum field theory without parity symmetry is dominated by a constant-field configuration with corrections determined by a weak-coupling expansion in which the expansion parameter (the amplitude of the vertices of the graphs in this expansion) is proportional to an inverse power of $g$. We thus observe a weak-coupling/strong-coupling duality in that while the Ising limit is a strong-coupling limit of the quantum field theory, the expansion about this limit takes the form of a conventional weak-coupling expansion. A possible generalization of the Ising limit to dimensions $D<4$ is briefly discussed. 
  We apply the 3D reduction method we recently proposed for the N-particle Bethe-Salpeter equation to the 4-particle case. We find that the writing of the Bethe-Salpeter equation is not a straightforward task when N is larger or equal to 4, owing to the presence of mutually unconnected interactions, which could lead to an overcounting of some diagrams in the resulting full propagator. We overcome this difficulty in the N=4 case by including three counterterms in the Bethe-Salpeter kernel. The application of our 3D reduction method to the resulting Bethe-Salpeter equation suggests us a modified 3D reduction method, which gives directly the 3D potential, without the need of writing the Bethe-Salpeter kernel explicitly. The modified reduction method is usable for all N. 
  We consider the supergravity dual of the N=1* theory at finite temperature by applying the Polchinski-Strassler construction to the black D3 brane solution of Type IIB supergravity. At finite temperature the 5-brane probe action is minimized when the probe falls to the horizon, although metastable minima with r>>r_H persist for a range of temperatures. Thermal effects on the 3-form source for the hypermultiplet mass m and its order m^2 back reaction on the other fields of the IIB theory are computed. We find unique solutions which are regular at the horizon and have the correct behavior on the boundary. For fixed temperature T, the horizon shrinks for increasing m^2 suggesting that there is a critical temperature separating the system into high and low temperature phases. In the high temperature phase 5-branes are unnecessary since there are no naked singularities. Using the order m^2 correction to the horizon area we calculate the correction to the entropy to be \Delta S =-0.1714N^2m^2T, which is less than the free field result. 
  This manuscript has been withdrawn by the author. For reasons please contact Klaus Kirsten at klaus@a35.ph.man.ac.uk or kirsten@itp.uni-leipzig.de 
  We supplement the discussion on localization of gravity on dilatonic domain walls in ``Solitons in Brane Worlds'' (Nucl. Phys. B576, 106, hep-th/9911218) by giving unified and coherent discussion which combines the result of this paper and the expanded results in the author's subsequent papers in an attempt to avoid misleading readers. We also discuss the possible string theory embeddings of the Randall-Sundrum type brane world scenarios through non-dilatonic and dilatonic domain walls, which straightforwardly follows from the author's previous works but was not elaborated explicitly. 
  The purpose of the present paper is twofold. In the first part, we provide an algebraic characterization of several families of $\nu= \frac{1}{2^n}$ $n\leq 5$ BPS states in M theory, at threshold and non-threshold, by an analysis of the BPS bound derived from the ${\cal N}=1$ D=11 SuperPoincar\'e algebra. We determine their BPS masses and their supersymmetry projection conditions, explicitly. In the second part, we develop an algebraic formulation to study the way BPS states transform under $GL(32,\bR)$ transformations, the group of automorphisms of the corresponding SuperPoincar\'e algebra. We prove that all $\nu={1/2}$ non-threshold bound states are SO(32) related with $\nu={1/2}$ BPS states at threshold having the same mass. We provide further examples of this phenomena for less supersymmetric $\nu={1/4},{1/8}$ non-threshold bound states. 
  We investigate constraints for including bulk and brane matter in the Randall- Sundrum model. In static configurations with two zero thickness branes, we find that no realistic brane matter is possible. We also consider the possibility that the radion has stabilized by dissipating its energy into the bulk in the form of some unspecified matter, and find the Randall-Sundrum cosmological solutions in the presence of bulk ideal fluid. We discover that the metric is necessarily in a static configuration. We also discover that there is only one allowed equation of state for the bulk fluid, $p=\rho $, corresponding to the stiff ideal fluid. We find the corresponding brane cosmologies and compare them with the Friedmann-Robertson-Walker model. 
  We investigate the behavior of the noncommutative scalar soliton solutions at finite noncommutative scale $\theta$. A detailed analysis of the equation of the motion indicates that fewer and fewer soliton solutions exist as $\theta$ is decreased and thus the solitonic sector of the theory exhibits an overall hierarchy structure. If the potential is bounded below, there is a finite $\theta_c$ below which all the solitons cease to exist even though the noncommutativity is still present. If the potential is not bounded below, for any nonzero $\theta$ there is always a soliton solution, which becomes singular only at $\theta = 0$. The $\phi^4$ potential is studied in detail and it is found the critical $(\theta m^2)_c =13.92$ ($m^2$ is the coefficient of the quadratic term in the potential) is universal for all the symmetric $\phi^4$ potential. 
  So far, there are described in the literature two ways to superize the Liouville equation: for a scalar field (for $N\leq 4$) and for a vector-valued field (analogs of the Leznov--Saveliev equations) for N=1. Both superizations are performed with the help of Neveu--Schwarz superalgebra. We consider another version of these superLiouville equations based on the Ramond superalgebra, their explicit solutions are given by Ivanov--Krivonos' scheme. Open problems are offered. 
  We find the gravity solution corresponding to a large number of NS or D fivebranes wrapped on a two sphere so that we have pure ${\cal N}=1$ super Yang-Mills in the IR. The supergravity solution is smooth, it shows confinement and it breaks the $U(1)_R$ chiral symmetry in the appropriate way. When the gravity approximation is valid the masses of glueballs are comparable to the masses of Kaluza Klein states on the fivebrane, but if we could quantize strings on this background it looks like we should be able to decouple the KK states. 
  Noncommutative \phi^3 field theory in six dimensions exhibits the logarithmic UV/IR mixing at the two-loop order. We show that open string theory in the presence of constant background NS-NS two-form field yields the same amplitude upon taking a decoupling limit. The stretched string picture proposed on the basis of one-loop analysis naturally generalizes to the two-loop amplitudes in consideration. Our string theory formulation can incorporate the closed string insertions as well as open string insertions. Furthermore, the analysis of the world-sheet partition function and propagators can be straightforwardly generalized to Riemann surfaces with genus zero but with an arbitrary number of boundaries. 
  In the imaginay-time formalism of thermal field theory, and also in the real-time formalism but by means of some redefined physical propagators for scalar bound states by diagonalization of four-point function matrices, we reexamine the Nambu-Goldstone mechanism of electroweak symmetry breaking in a one-generation fermion condensate scheme, based on the Schwinger-Dyson equation in the fermion bubble diagram approximation, and compare the obtained results. We have reached the conclusion that in both the formalisms, the Goldstone theorem of spontaneous electroweak symmetry breaking is rigorously true for the case of mass-degenerate two flavors of fermions and only approximately valid at low energy scales for the mass-nondegenerate case, in spite of existence of some difference between the two formalisms in the imaginary parts of the denominators of the propagators for scalar bound states. When the two flavors of fermions have unequal nonzero masses, the induced possible fluctuation effect for the Higgs particle is negligible if the momentum cut-off in the zero temperature loops is large enough. All the results show physical equivalence of the two formalisms in the present discussed problems. 
  The paper consists of two parts which seem to be independent of each other but at first glance only. In the Part one the field-to-particle transition formalism is applied to the sigma model (string), dilaton gravity and gauged supergravity 0-brane solutions. In addition to the fact that the field-to-particle transition is of interest itself it can be regarded also as the dynamical dimensional reduction which takes into account field fluctuations as well as one can treat it as the method of the consistent quantization in the vicinity of the nontrivial vacuum induced by a field solution. It is shown that in all the cases the end product of the approach is the so-called "non-minimal point particle". In view of this it is conjectured that point particles are not only the end product but also the underlying base of modern high-energy theory even more fundamental than the strings. The Part two is hence devoted to the formal axiomatics of the string-brane approach - there we deepen the above-mentioned conjecture. It turns out that a point particle can be regarded as an extended object from the viewpoint of a macroscopic measurement. The particle is actually observed as the ``cloud'' consisting of the virtual paths and nothing prevents their deviations from the classical trajectory. Therefore, for an observer the real particle can be no more viewed as a point object. However, it cannot be supposed also as a continuous extended object unless we average over all the deviations. Then we obtain the metabrane, which is the composite extended object consisting of the microscopical objects (strings) and cosmological-size ones (3-brane) where the size of a corresponding embedding is governed by the weight constants arising along with the decomposition. 
  We continue the discussion of our previous paper on writing down gauge invariant interacting equations for a bosonic string using the loop variable approach. In the earlier paper the equations were written down in one higher dimension where the fields are massless. In this paper we describe a procedure for dimensional reduction that gives interacting equations for fields with the same spectrum as in bosonic string theory. We also argue that the on-shell scattering amplitudes implied by these equations for the physical modes are the same as for the bosonic string. We check this explicitly for some of the simpler equations. The gauge transformation of space-time fields induced by gauge transformations of the loop variables are discussed in some detail. The unintegrated (i.e. before the Koba-Nielsen integration), regularized version of the equations, are gauge invariant off-shell (i.e. off the {\em free} mass shell). 
  We show how the classical string dynamics in $D$-dimensional curved background can be reduced to the dynamics of a massless particle constrained on a certain surface whenever there exists at least one Killing vector for the background metric. Then we obtain a number of sufficient conditions, which ensure the existence of exact solutions to the equations of motion and constraints. The results are also relevant to the null string case. Finally, we illustrate our considerations with an explicit example in four dimensions. 
  In Parts I and II we showed that e, $\nu$ propagators can be derived from rotation invariant projection operators, thereby providing examples of how quantities with spacetime symmetry can be obtained by constraining rotationally symmetric objects. One constraint is the restriction of the basis; only two kinds of bases were considered, one for the electron and one for the neutrino. In this part, we find that, of a wide range of bases each consistent with the constraint process, only the two kinds of bases considered in Parts I and II give spacetime symmetric propagators. We interpret the result geometrically. The spinor representation is unfaithful in four dimensional Euclidean space which explains why spin 1/2 wave functions have four, not two, components. Then we show how a basis relates to two planes in four dimensional Euclidean space. A pair of planes spanning two or three dimensions does not allow spacetime symmetry. Spacetime symmetry requires two planes that span four dimensions.   PACS: 11.30.-j, 11.30.Cp, and 03.65.Fd 
  We give a proof of the existence of $G=SU(5)$, stable holomorphic vector bundles on elliptically fibered Calabi--Yau threefolds with fundamental group $\bbz_2$. The bundles we construct have Euler characteristic 3 and an anomaly that can be absorbed by M-theory five-branes. Such bundles provide the basis for constructing the standard model in heterotic M-theory. They are also applicable to vacua of the weakly coupled heterotic string. We explicitly present a class of three family models with gauge group $SU(3)_C\times SU(2)_L\times U(1)_Y$. 
  This talk describes a proposal, due to Hull, for a conformally invariant limit of superstring theory in six dimensions. 
  A brief pedagogical survey of the star product is provided, through Groenewold's original construction based on the Weyl correspondence. It is then illustrated how simple Landau orbits in a constant magnetic field, through their Dirac Brackets, define a noncommutative structure since these brackets exponentiate to a star product---a circumstance rarely operative for generic Dirac Brackets. The geometric picture of the star product based on its Fourier representation kernel is utilized in the evaluation of chains of star products. The intuitive appreciation of their associativity and symmetries is thereby enhanced. This construction is compared and contrasted with the remarkable phase-space polygon construction of Almeida. 
  This is the abstract of the revised paper. The integrable XXZ model with a special open boundary condition is considered. We study Sklyanin transfer matrices after quantum group reduction in roots of unity. In this case Sklyanin transfer matrices satisfy a closed system of truncated functional equations. The algebraic reason for the truncation is found.The important role in proving of the result is performed by Zamolodchikov algebra introduced in the paper. 
  We study localization of bosonic bulk fields on a string-like defect with codimension 2 in a general space-time dimension in detail. We show that in cases of spin 0 scalar and spin 1 vector fields there are an infinite number of massless Kaluza-Klein (KK) states which are degenerate with respect to the radial quantum number, but only the massless zero mode state among them is coupled to fermion on the string-like defect. It is also commented on interesting extensions of the model at hand to various directions such as 'little' superstring theory, conformal field theory and a supersymmetric construction. 
  Using the construction of D-branes with nonzero $B$ field in the matrix model we give a physical interpretation of the known background independence in gauge theories on a noncommutative space. The background independent variables are identified as the degrees of freedom of the underlying matrix model. This clarifies and extends some recent results about the end point of tachyon condensation in D-branes with a $B$ field. We also explain the freedom in the description which is parametrized by a two form $\Phi$ from the points of view of the noncommutative geometry on the worldvolume of the branes, and of the first quantized string theory. 
  Using a special ansatz for the metric, by straightforward computation we prove that gravi-dilaton effective action in higher dimensions is reduced to the p-brane action. The dual symmetry of the generic type $%a\longleftrightarrow \frac 1a$ is an important symmetry of the reduced action. 
  We investigate the `giant gravitons' of McGreevy, Susskind and Toumbas [hep-th/0003075]. We demonstrate that these are BPS configurations which preserve precisely the same supersymmetries as a `point-like' graviton. We also show that there exist `dual' giant gravitons consisting of spherical branes expanding into the AdS component of the spacetime. Finally, we discuss the realization of the stringy exclusion principle within this expanded framework. 
  Recently it was suggested that a graviton in $AdS_5 \times S^5$ with a large momentum along the sphere can blow up into a spherical D-brane in $S^5$. In this paper we show that the same graviton can also blow up into a spherical D-brane in $AdS_5$ with exactly the same quantum numbers (angular momentum and energy). These branes are BPS, preserving 16 of the 32 supersymmetries. We show that there is a BPS {\it classical} solution for SYM on $S^3\times R$ with exactly the same quantum numbers. The solution has non-vanishing Higgs expectation values and hence is dual to the large brane in AdS. 
  A talk presented at International Conference ICMP-2000, London, England 
  A talk presented at International Conference ICMP-2000, London, England 
  We consider electric-magnetic duality(S-duality) in IIB matrix model with a D3-brane background. We propose the duality transformation by considering that of noncommutative Yang-Mills theory(NCYM) in four dimension. NCYM derived from the matrix model has a Yang-Mills coupling related to the noncommutativity of the spacetime. We argue that open strings bits as bi-local fields on the spacetime are decoupled from the bulk in NCOS decoupling limits as it is in string theory approach. We also discuss how our four dimensional spacetime relates to higher, by applying the decoupling and the commutative limits of the backgrounds of the matrix model. 
  In this paper we consider the worldsheet of superstring as a noncommutative space. Some additional terms can be added to the superstring action, such that for ordinary worldsheet they are zero. Expansion of this extended action up to the first order of the noncommutativity parameter, leads to the new supersymmetric action for string. For the closed superstring, we obtain the boundary state that describes a brane. From the open string point of view, the new boundary conditions on the worldsheet bosons, generalize the noncommutativity of spacetime. Finally, we suggest some definitions for the noncommutativity parameter of superstring worldsheet. 
  The energy-momentum conservation law is used to investigate the interaction of pulses in the framework of nonlinear electrodynamics with Lorentz-invariant constitutive relations. It is shown that for the pulses of the arbitrary shape the interaction results in phase shift only. 
  Chern-Simons couplings between Yang-Mills gauge fields and an abelian 2-form are derived by means of cohomological arguments. 
  We study a brane-antibrane system and a non-BPS D-brane in SU(2) WZW model. We first discuss the tachyon condensation using the vertex operator formalism and find the generation of codimension two D-branes after the condensation. Our result is consistent with the recent interpretation that a D2-brane is a bound state of D0-branes. Then we investigate the world volume effective theory on a non-BPS D-brane. It becomes a field theory on the ``fuzzy sphere'' when the level is sent to infinity. The most interesting feature is that there exist the noncommutative tachyonic solitons and we can identify them with D0-branes. We also discuss the brane-antibrane system from the world volume point of view and comment on the relation to the noncommutative version of the index theorem. 
  Invariant functions under the transformations of a compact linear group $G$ acting in $\real^n$ can be expressed in terms of functions defined in the orbit space of $G$. We develop a method to determine the isotropy classes of the orbit spaces of all the real linear groups whose integrity bases (IB) satisfy only one independent relation. The method is tested for IB's formed by 3 (independent) basic invariants. The result is obtained through a metric matrix $\hat P(p)$, defined only from the scalar products between the gradients of a minimal IB. We determine the matrices $\wP(p)$ from a universal differential equation, which satisfy new convenient additional conditions, which fit for the non-coregular case. Our results may be relevant in physical contexts where the study of covariant or invariant functions is important, like in the determination of patterns of spontaneous symmetry breaking in quantum field theory, in the analysis of phase spaces and structural phase transitions (Landau's theory), in covariant bifurcation theory, in crystal field theory and so on. 
  It was recently proposed that there exist stable supersymmetric phases for finite temperature superstings. This issue was investigated using an effective supergravity which takes into account massive winding modes. Such a theory admits BPS solutions that do not suffer from Hagedorn-type instabilities. We extend several aspects of this work. First we restrict to the real-field sector of the theory and allow, in general, for unequal right and left winding fields. Then, by further specializing to type-II theories (IIA, IIB and a self-dual hybrid) we construct the most general 1/2-BPS solution and reveal several new features arising in various consistent truncations. In the heterotic case we investigate the general properties of the solution which is presented in a closed form in the limit of infinitely large left-winding field. 
  The renormalization group flow in the theory space of a BRST invariant string $\sigma$-model is investigated. For the open bosonic string the non-perturbative off-shell effective action and its gauge symmetry properties are determined from $\beta$-functions defined by the local Weyl anomaly. The interactions are shown to explicitly break the free theory BRST invariance generating new non-linear gauge symmetries of the type present in Witten's string field theory. In the Feynman-Siegel gauge the $\sigma$-model is shown to generate Witten's structure of vertex couplings. 
  We consider the quantum mechanics of a particle on a noncommutative two-sphere with the coordinates obeying an SU(2)-algebra. The momentum operator can be constructed in terms of an $SU(2)\times SU(2)$-extension and the Heisenberg algebra recovered in the smooth manifold limit. Similar considerations apply to the more general SU(n) case. 
  We compute the conserved charges for Kerr anti-de Sitter spacetimes in various dimensions using the conformal and the counterterm prescriptions. We show that the conserved charge corresponding to the global timelike killing vector computed by the two methods differ by a constant dependent on the rotation parameter and cosmological constant in odd spacetime dimensions, whereas the charge corresponding to the rotational killing vector is the same in either approach. We comment on possible implications of our results to the AdS/CFT correspondence. 
  We present a Lorentz invariant lagrangian formulation for a supersymmetric Yang-Mills vector multiplet in eleven dimensions (11D). The Lorentz symmetry is broken at the field equation level, and therefore the breaking is spontaneous, as in other formulations of supersymmetric theories in 12D or higher dimensions. We introduce a space-like unit vector formed by the gradient of a scalar field, avoiding the problem of Lorentz non-invariance at the lagrangian level, which is also an analog of non-commutative geometry with constant field strengths breaking Lorentz covariance. The constancy of the space-like unit vector field is implied by the field equation of a multiplier field. The field equations for the physical fields are formally the same as those of 10D supersymmetric Yang-Mills multiplet, but now with some constraints on these fields for supersymmetric consistency. This formulation also utilizes the multiplier fields accompanied by the bilinear forms of constraints, such that these multiplier fields will not interfere with the physical field equations. Based on this component result, we also present a $\k$-symmetric supermembrane action with the supersymmetric Yang-Mills backgrounds. 
  T-duality of gauge theories on a noncommutative $T^d$ can be extended to include fields with twisted boundary conditions. The resulting T-dual theories contain novel nonlocal fields. These fields represent dipoles of constant magnitude. Several unique properties of field theories on noncommutative spaces have simpler counterparts in the dipole-theories. 
  We derive general low energy dynamics of monopoles and dyons in N=2 and N=4 supersymmetric Yang-Mills theories by utilising a collective coordinate expansion. The resulting new kind of supersymmetric quantum mechanics incorporates the effects of multiple Higgs fields, both in the N=2 vector multiplet and hypermultiplets, having non-vanishing expectation values. 
  An action with unconventional supersymmetry was introduced in an earlier paper. Here it is shown that this action leads to standard physics for fermions and gauge bosons at low energy, but to testable extensions of standard physics for fermions at high energy and for fundamental bosons which have not yet been observed. For example, the Lorentz-violating equation of motion for these bosons implies that they have spin 1/2. 
  The level truncation approach to string field theory is used to study the zero-momentum action for vector excitations on a bosonic D-brane which has been annihilated by tachyon condensation. It is shown that in the true vacuum the translation zero modes associated with transverse scalars on the D-brane are lifted by spontaneous generation of mass terms. Similarly, the U(1) gauge field on the brane develops a nonzero mass term. 
  We construct static axially symmetric black holes in SU(2) Einstein-Yang-Mills-Higgs theory. Located in between a monopole-antimonopole pair, these black holes possess magnetic dipole hair. The difference of their mass and their horizon mass equals the mass of the regular monopole-antimonopole solution, as expected from the isolated horizon framework. 
  We recall the non-Abelian Stokes theorem for the Wilson loop in the Yang-Mills theory and discuss its meaning. Then we move to `gravitational Wilson loops', i.e. to holonomies in curved d=2,3,4 spaces and derive non-Abelian Stokes theorems for these quantities as well, which are similar to our formula in the Yang-Mills theory. In particular we derive an elegant formula for the holonomy in the case of a constant-curvature background in three dimensions and a formula for small-area loops in any number of dimensions. 
  Through an $\hbar$-expansion of the confined Calogero model with spin exchange interactions, we extract a generating function for the involutive conserved charges of the Frahm-Polychronakos spin chain. The resulting conservation laws possess the spin chain yangian symmetry, although they are not expressible in terms of these yangians. 
  Formulating the QFT's as coarse grained 'low' energy sectors of a postulated complete quantum theory of everything with the 'high' energy modes integrated out or 'clustering' into 'low' energy objects, we can evaluate the Feynman amplitudes by solving a series of natural differential equations which automatically dissolves the necessity of infinity subtraction and the associated subtleties. This new strategy has direct implications to the scheme dependence problem. 
  We reconsider the problem of U(1) flux and D0-charge for D-branes in the WZW model and investigate the relationship between the different definitions that have been proposed recently. We identify the D0-charge as a particular reduction of a class in the relative cohomology of the group modulo the D-submanifold. We investigate under which conditions this class is equivalent to the first Chern class of a line bundle on the D-submanifold and we find that in general there is an obstruction given by the cohomology class of the NS 3-form. Therefore we conclude that for topologically nontrivial B-fields, there is strictly speaking no U(1) gauge field on the D-submanifold. Nevertheless the ambiguity in the flux is not detected by the D0-charge. This has a natural interpretation in terms of gerbes. 
  We show that SU(n) Bethe Ansatz equations with arbitrary `twist' parameters are hidden inside certain nth order ordinary differential equations, and discuss various consequences of this fact. 
  In this paper I shall discuss the way in which vacuum structure and condensates occur in the light-cone representation. I shall particularly emphasize the mechanism by which the mass squared of a composite such as the pion comes to depend linearly on the bare mass of its Fermion constituents. I shall give details in two dimensions then discuss the case of four dimensions more speculatively. 
  We consider a recently proposed setup where a codimension one brane is embedded in the background of a smooth domain wall interpolating between AdS and Minkowski minima. Since the volume of the transverse dimension is infinite, bulk supersymmetry is intact even if brane supersymmetry is completely broken. On the other hand, in this setup unbroken bulk supersymmetry is incompatible with non-zero brane cosmological constant, so the former appears to protect the latter. In this paper we point out that, to have a consistent coupling between matter localized on the brane and bulk gravity, in this setup generically it appears to be necessary that the brane world-volume theory be conformal. Thus, unbroken bulk supersymmetry appears to actually protect not only the cosmological constant but also conformal invariance on the brane. 
  We study the issue of gauge-invariant observables in d=4, N=4 noncommutative gauge theory and UV-IR relation therein. We show that open Wilson lines form a complete set of gauge invariant operators, which are local in momentum space and, depending on their size, exhibit two distinct behaviors of the UV-IR relation. We next study these properties in a proposed dual description in terms of supergravity and find agreement. 
  We show the equivalence between Stuckelberg and Wess-Zumino methods of restoration of gauge symmetries of the anomalous, Abelian, effective action, in arbitrary even dimensions D=2k. We present dual version of Wess-Zumino terms with the compensating field described by a Kalb-Ramond like p=2k-2 form. 
  We address the general question of how to reconstruct the field content of a quantum field theory from a given scattering theory in the context of the form factor program. For the $SU(3)_2$-homogeneous Sine-Gordon model we construct systematically all $n$-particle form factors for a huge class of operators in terms of general determinant formulae. We investigate how different operators are interrelated by the momentum space cluster property. Finally we compute several two-point correlation functions and carry out the ultraviolet limit in order to identify each operator with its corresponding partner in the underlying conformal field theory. 
  We extend dimensional regularization to the case of compact spaces. Contrary to previous regularization schemes employed for nonlinear sigma models on a finite time interval (``quantum mechanical path integrals in curved space'') dimensional regularization requires only a covariant finite two-loop counterterm. This counterterm is nonvanishing and given by R/8. 
  I construct a map from the Grothendieck group of coherent sheaves to $K$-homology. This results in explicit realizations of $K$-homology cycles associated with D-brane configurations. Non-Abelian degrees of freedom arise in this framework from the deformation quantization of $N$-tuple cycles. The large $N$ limit of the gauge theory on D-branes wrapped on a subvariety $V$ of some variety $X$ is geometrically interpreted as the deformation quantization of the formal completion of $X$ along $V.$ 
  The field strength superfield of IIB supergravity on $AdS_5\xz S^5$ is expanded in harmonics on $S^5$ with coefficients which are $D=5, N=8$ chiral superfields. On the boundary of $AdS_5$ these superfields map to $D=4,N=4$ chiral superfields and both sets of superfields obey additional fourth-order constraints. The constraints on the $D=4,N=4$ chiral fields are solved using harmonic superspace in terms of prepotential superfields which couple in a natural way to composite operator multiplets of the boundary $N=4,D=4$ superconformal field theory. 
  Some aspects of the $D=6, (2,0)$ tensor multiplet are discussed. Its formulation as an analytic superfield on a suitably defined superspace and its superconformal properties are reviewed. Powers of the field strength superfield define a series of superconformal fields which correspond to the KK multiplets of D=11 supergravity on an $AdS_7\xz S^4$ background. Correlation functions of these operators are briefly discussed. 
  We quantize the Maxwell Chern Simons theory in a geometric representation that generalizes the Abelian Loop Representation of Maxwell theory. We find that in the physical sector, the model can be seen as the theory of a massles scalar field with a topological interaction that enforces the wave functional to be multivalued. This feature allows to relate the Maxwell Chern Simons theory with the quantum mechanics of particles interacting through a Chern Simons field 
  We show that four-dimensional N=2 ungauged Einstein-Maxwell supergravity can be embedded on the Randall-Sundrum 3-brane, as a consistent Kaluza-Klein reduction of five-dimensional N=4 gauged supergravity. In particular, this allows us to describe four-dimensional Reissner-Nordstrom black holes within the Randall-Sundrum scenario. Using earlier results on the embedding of five-dimensional N=4 gauged supergravity in ten dimensions, we can then describe the four-dimensional Einstein-Maxwell supergravity on the 3-brane, and its solutions, from a type IIB viewpoint. We also show that the minimal ungauged supergravities in D=5 and D=6 can be consistently embedded in the half-maximally supersymmetric gauged supergravities in D=6 and D=7 respectively. These allow us to construct solutions including BPS black holes and strings living in "Randall-Sundrum 4-branes," and BPS self-dual strings living in "Randall-Sundrum 5-branes." We can also lift the embeddings to ten-dimensional massive type IIA and D=11 supergravity respectively. In particular, we obtain a solution describing the self-dual string living in the world-volume of an M5-brane, which can be viewed as an open membrane ending on the M5-brane. 
  We show that a Rademacher expansion can be used to establish an exact bound for the entropy of black holes within a conformal field theory framework. This convergent expansion includes all subleading corrections to the Bekenstein-Hawking term. 
  Fluctuations around a non-trivial solution of Born-Infeld theory have a limiting speed given not by the Einstein metric but the Boillat metric. The Boillat metric is S-duality invariant and conformal to the open string metric. It also governs the propagation of scalars and spinors in Born-Infeld theory. We discuss the potential clash between causality determined by the closed string and open string light cones and find that the latter never lie outside the former. Both cones touch along the principal null directions of the background Born-Infeld field. We consider black hole solutions in situations in which the distinction between bulk and brane is not sharp such as space filling branes and find that the location of the event horizon and the thermodynamic properties do not depend on whether one uses the closed or open string metric. Analogous statements hold in the more general context of non-linear electrodynamics or effective quantum-corrected metrics. We show how Born-Infeld action to second order might be obtained from higher-curvature gravity in Kaluza-Klein theory. Finally we point out some intriguing analogies with Einstein-Schr\"odinger theory. 
  The ratios of the masses of D(p-d) branes to the masses of Dp in open bosonic string field theory are computed within the modified level truncation approximation of Moeller, Sen and Zweibach. At the lowest non-trivial truncation requiring the addition of new primary states, we find evidence of rapid convergence to the expected result for 1<d<7 providing additional evidence for the consistency of this approximation. 
  We generalize the mechanism proposed in [hep-th/0005016] and show that a four-dimensional relativistic tensor theory of gravitation can be obtained on a delta-function brane in flat infinite-volume extra space. In particular, we demonstrate that the induced Ricci scalar gives rise to Einstein's gravity on a delta-function type brane if the number of space-time dimensions is bigger than five. The bulk space exhibits the phenomenon of infrared transparency. That is to say, the bulk can be probed by gravitons with vanishing four-dimensional momentum square, while it is unaccessible to higher modes. This provides an attractive framework for solving the cosmological constant problem. 
  We investigate the possibility of having the electrically charged Reissner-Nordstrom black hole in the gravity localized models in a brane world. It is shown that the Reissner-Nordstrom black hole exists as a solution in the 5D Randall-Sundrum domain wall model if there is the U(1) bulk gauge field. We find that the charged black hole is localized on a 3-brane even if zero mode of the bulk gauge field is not in general localized on the brane in the domain wall model. Moreover, we extend these observations to the higher-dimensional topological defect models with codimension more than one in a general space-time dimension such as the string-like defect model with codimension 2 in six dimensions and the monopole-like defect model with codimension 3 in seven dimensions e.t.c. 
  This paper has been withdrawn 
  We study perturbative aspects of noncommutative field theories. This work is arranged in two parts. First, we review noncommutative field theories in general and discuss both canonical and path integral quantization methods. In the second part, we consider the particular example of noncommutative $\Phi^4$ theory in four dimensions and work out the corresponding effective action and discuss renormalizability of the theory, up to two loops. 
  The boundary terms in the Hamiltonian, in the presence of horizons, are carefully analyzed in a simple 2D theory admitting AdS black holes. The agreement between the euclidean approach and Cardy's formula is obtained modulo certain assumptions on the spectrum of the Virasoro's algebra. There is no discrepancy factor "square root of 2" once the appropriate boundary conditions are properly recognized. The peculiar features of gravity, that the on-shell Hamiltonian is determined by boundary terms, is the reason of the mentioned agreement. 
  As a technical exercise with possible relevance to the holographic principle and string theory, the effective actions (functional determinants) for scalars and spinors on the squashed three-sphere identified under the action of a cyclic group, Z_m, are determined. Especially in the extreme oblate squashing limit, which has a thermodynamic interpretation, the high temperature behaviour is found as a function of m. Although the intermediate details for odd and even m are different, the final answers are the same. A thermodynamic interpretation for spinors is possible only for twisted periodicity conditions and m even. 
  We consider four-graviton scattering in Type II string theory on one-loop level in the large centre of mass energy M^2 limit. We extract from it an explicit integral expression for the full string theory corrections to the imaginary part of the mass-shift and the lifetime of a massive state with the highest allowed spin J=2M^2+2. We find a decay rate that is up to log M corrections of order 1 in string units, times g_s^2. We also find that the dominant decay mode corresponds to the emission of light particles, whereas the decay into two massive or two massless states is exponentially suppressed. We discuss the relation of our results to quantum gravity aspects of a Kerr Black Hole. 
  The issue related to the so-called dimensional reduction procedure is revisited within the Euclidean formalism. First, it is shown that for symmetric spaces, the local exact heat-kernel density is equal to the reduced one, once the harmonic sum has been succesfully performed. In the general case, due to the impossibility to deal with exact results, the short time heat-kernel asymptotics is considered. It is found that the exact heat-kernel and the dimensionally reduced one coincide up to two non trivial leading contributions in the short time expansion. Implications of these results with regard to dimensional-reduction anomaly are discussed. 
  We study the symmetry of the one-loop effective action of bosonic string theory under non-Abelian T-duality transformations. It is shown that the original Lagrangian and its dual are proportional. This result implies that the corresponding reduced low energy effective actions are equivalent and leads to a functional relation between the Weyl anomaly coefficients of the original and dual two-dimensional non-linear sigma models. {}Finally, we apply this formalism to some simple examples. 
  We present a nonlinear realization of E_8 on a space of 57 dimensions, which is quasiconformal in the sense that it leaves invariant a suitably defined ``light cone'' in 57 dimensions. This realization, which is related to the Freudenthal triple system associated with the unique exceptional Jordan algebra over the split octonions, contains previous conformal realizations of the lower rank exceptional Lie groups on generalized space times, and in particular a conformal realization of E_7 on a 27 dimensional vector space which we exhibit explicitly. Possible applications of our results to supergravity and M-Theory are briefly mentioned. 
  We show that the tensionless branes found recently on non-BPS D-branes using non-commutative field theory are in fact gauge equivalent to the vacuum under a discrete gauge symmetry. We also give a simple construction of the D(2p)-branes in IIA theory starting from a single non-BPS D9-brane. 
  By analyzing SU(3)xU(1) invariant stationary point, studied earlier by Nicolai and Warner, of gauged N=8 supergravity, we find that the deformation of S^7 gives rise to nontrivial renormalization group flow in a three-dimensional boundary super conformal field theory from N=8, SO(8) invariant UV fixed point to N=2, SU(3)xU(1) invariant IR fixed point. By explicitly constructing 28-beins u, v fields, that are an element of fundamental 56-dimensional representation of E_7, in terms of scalar and pseudo-scalar fields of gauged N=8 supergravity, we get A_1, A_2 tensors. Then we identify one of the eigenvalues of A_1 tensor with ``superpotential'' of de Wit-Nicolai scalar potential and discuss four-dimensional supergravity description of renormalization group flow, i.e. the BPS domain wall solutions which are equivalent to vanishing of variation of spin 1/2, 3/2 fields in the supersymmetry preserving bosonic background of gauged N=8 supergravity. A numerical analysis of the steepest descent equations interpolating two critical points is given. 
  The two-frequency sine-Gordon model is examined. The focus is mainly on the case when the ratio of the frequencies is 1/2, given the recent interest in the literature. We discuss the model both in a perturbative (form factor perturbation theory) and a nonperturbative (truncated conformal space approach) framework, and give particular attention to a phase transition conjectured earlier by Delfino and Mussardo. We give substantial evidence that the transition is of second order and that it is in the Ising universality class. Furthermore, we check the UV-IR operator correspondence and conjecture the phase diagram of the theory. 
  The phase structure of the generalized Yang--Mills theories is studied, and it is shown that {\it almost} always, it is of the third order. As a specific example, it is shown that all of the models with the interaction $\sum_j (n_j-j+N)^{2k}$ exhibit third order phase transition. ($n_j$ is the length of the $j$-th row of the Yang tableau corresponding to U($N$).) The special cases where the transition is not of the third order are also considered and, as a specific example, it is shown that the model $\sum_j (n_j-j+N)^2+g\sum_j (n_j-j+N)^{4}$ exhibits a third order phase transition, except for $g=27\pi^2/256$, where the order of the transition is 5/2. 
  We consider scattering of electromagnetic plane waves on a D3-brane spike which emanates normal to D3-barne in the extra space direction. We are interested in studying physical effects on D3-brane which are produced by a spike attached to D3-brane. We have observed that the spike sucks almost all electromagnetic radiation and therefore acts like a black hole. This is because absorption cross section for j=1 tends to a constant at low energy limit. This behaviour is appealing for a string interpretation of the spike soliton because the propagation of $j=1$ mode is indeed distinctive. Instead, the scattered part of the radiation on a D3-brane tends to zero demonstrating non-Thompson behaviour. 
  We apply the linear delta expansion to the quantum mechanical version of the slow rollover transition which is an important feature of inflationary models of the early universe. The method, which goes beyond the Gaussian approximation, gives results which stay close to the exact solution for longer than previous methods. It provides a promising basis for extension to a full field theoretic treatment. 
  In this paper the SU(2) Skyrme model will be reformulated as a gauge theory and the hidden symmetry will be investigated and explored in the energy spectrum computation. To this end we purpose a new constraint conversion scheme, based on the symplectic framework with the introduction of Wess-Zumino (WZ) terms in an unambiguous way. It is a positive feature not present on the BFFT constraint conversion. The Dirac's procedure for the first-class constraints is employed to quantize this gauge invariant nonlinear system and the energy spectrum is computed. The finding out shows the power of the symplectic gauge-invariant formalism when compared with another constraint conversion procedures present on the literature. 
  We find the spectrum of boundary bound states for the sine-Gordon model with Dirichlet boundary conditions, closing the bootstrap and providing a complete description of all the poles in the boundary reflection factors. The boundary Coleman-Thun mechanism plays an important role in the analysis. Two basic lemmas are introduced which should hold for any 1+1-dimensional boundary field theory, allowing the general method to be applied to other models. 
  Both, spin and statistics of a quantum system can be seen to arise from underlying (quantum) group symmetries. We show that the spin-statistics theorem is equivalent to a unification of these symmetries. Besides covering the Bose-Fermi case we classify the corresponding possibilities for anyonic spin and statistics. We incorporate the underlying extended concept of symmetry into quantum field theory in a generalised path integral formulation capable of handling general braid statistics. For bosons and fermions the different path integrals and Feynman rules naturally emerge without introducing Grassmann variables. We also consider the anyonic example of quons and obtain the path integral counterpart to the usual canonical approach. 
  We study the phases of the 1+1 dimensional Non-Commutative Open String theory on a circle. We find that the length scale of non-commutativity increases at strong coupling, the coupling in turn being dressed by a power of D-string charge. The system is stringy at around this length scale, with dynamics involving an interplay between the open and wrapped closed strings sectors. Above this energy scale and at strong coupling, and below it at weak coupling, the system acquires a less stringy character. The near horizon geometry of the configuration exhibits several intriguing features, such as a flip in the dilaton field and the curvature scale, reflecting UV-IR mixing in non-commutative dynamics. Two special points in the parameter measuring the size of the circle are also identified. 
  We consider the O(a') string effective action, with Gauss-Bonnet curvature-squared and fourth-order dilaton-derivative terms, which is derived by a matching procedure with string amplitudes in five space-time dimensions. We show that a non-factorizable metric of the Randall-Sundrum (RS) type, with four-dimensional conformal factor Exp(-2 k|z|), can be a solution of the pertinent equations of motion. The parameter k is found proportional to the string coupling g_s and thus the solution appears to be non-perturbative. It is crucial that the Gauss-Bonnet combination has the right (positive in our conventions) sign, relative to the Einstein term, which is the case necessitated by compatibility with string (tree) amplitude computations. We study the general solution for the dilaton and metric functions, and thus construct the appropriate phase-space diagram in the solution space. In the case of an anti-de-Sitter bulk, we demonstrate that there exists a continuous interpolation between (part of) the RS solution at z=infinity and an (integrable) naked singularity at z=0. This implies the dynamical formation of domain walls (separated by an infinite distance), thus restricting the physical bulk space time to the positive z axis. Some brief comments on the possibility of fine-tuning the four-dimensional cosmological constant to zero are also presented. 
  We construct gauge invariant operators in non-commutative gauge theories which in the IR reduce to the usual operators of ordinary field theories (e.g. F^2). We show that in the deep UV the two-point functions of these operators admit a universal exponential behavior which fits neatly with the dual supergravity results. We also consider the ratio between n-point functions and two-point functions to find exponential suppression in the UV which we compare to the high energy fixed angle scattering of string theory. 
  We investigate some properties of a recent supergravity solution of Pilch and Warner, which is dual to the N=4 gauge theory softly broken to N=2. We verify that a D3-brane probe has the expected moduli space and its effective action can be brought to N=2 form. The kinetic term for the probe vanishes on an enhancon locus, as in earlier work on large-n N=2 theories, though for the Pilch-Warner solution this locus is a line rather than a ring. On the gauge theory side we find that the probe metric can be obtained from a perturbative one-loop calculation; this principle may be useful in obtaining the supergravity dual at more general points in the N=2 gauge theory moduli space. We then turn on a B-field, following earlier work on the N=4 theory, to obtain the supergravity dual to the noncommutative N=2 theory. 
  An explicit canonical transformation is constructed to relate the physical subspace of Yang-Mills theory to the phase space of the ADM variables of general relativity. This maps 3+1 dimensional Yang-Mills theory to local evolution of metrics on 3 manifolds. 
  Polyakov's spin factor enters as a weight in the path-integral description of particle-like modes propagating in Euclidean space-times, accounting for particle spin. The Fock-Feynman-Schwinger path integral applied to QCD accomodates Polyakov's spin factor in a natural manner while, at the same time, it identifies Wilson line (loop) operators as sole agents of interaction dynamics among matter and gauge field quanta. A direct application of such a separation between spin content and dynamics is the emergence of master expressions for the perturbative series involving either open or closed fermionic lines which provide new, comprehensive approaches to perturbative QCD. 
  We study localization of bulk fermions on a brane with inclusion of Yang-Mills and scalar backgrounds in higher dimensions and give the conditions under which localized chiral fermions can be obtained. 
  We use the finite temperature AdS/CFT approach to demonstrate logarithmic broadening of the confining QCD3 flux tube as a function of quark separation. This behavior indicates that, unlike lattice QCD, there is no roughening transition in the AdS/CFT formulation, which raises the interesting possibility of extrapolating strong coupling results to weak couplings by the use of resummation techniques. In the zero-temperature non-confining limit, we find that this logarithmic broadening of the field strength distribution is absent. Our results are obtained numerically at strong couplings, in the supergravity approximation. 
  We study the family of ten dimensional type IIB supergravity solutions corresponding to renormalisation group flows from N=4 to N=2 supersymmetric Yang-Mills theory. Part of the solution set corresponds to a submanifold of the Coulomb branch of the gauge theory, and we use a D3-brane probe to uncover details of this physics. At generic places where supergravity is singular, the smooth physics of the probe yields the correct one-loop form of the effective low energy gauge coupling. The probe becomes tensionless on a ring at finite radius. Supergravity flows which end on this ``enhancon'' ring correspond to the vacua where extra massless degrees of freedom appear in the gauge theory, and the gauge coupling diverges there. We identify an SL(2,Z) duality action on the enhancon ring which relates the special vacua, and comment on the massless dyons within them. We propose that the supergravity solution inside the enhancon ring should be excised, since the probe's tension is unphysical there. 
  Using the Weyl transfomations induced by diffeomorphisms we set up a cohomological problem for the Fefferman-Graham coefficients. The cohomologically nontrivial solutions remove the ambiguity and give the nonlocal terms in the effective action responsible for the trace anomalies. 
  We compute the one loop Casimir energy of an interacting scalar field in a compact noncommutative space of $R^{1,d}\times T^2_\theta$, where we have ordinary flat $1+d$ dimensional Minkowski space and two dimensional noncommuative torus. We find that next order correction due to the noncommutativity still contributes an attractive force and thus will have a quantum instability. However, the case of vector field in a periodic boundary condition gives repulsive force for $d>5$ and we expect a stabilized radius. This suggests a stabilization mechanism for a senario in Kaluza-Klein theory, where some of the extra dimensions are noncommutative. 
  We present a detailed discussion of the duality between dilaton gravity on AdS_2 and open strings. The correspondence between the two theories is established using their symmetries and field theoretical, thermodynamic, and statistical arguments. We use the dual conformal field theory to describe two-dimensional black holes. In particular, all the semiclassical features of the black holes, including the entropy, have a natural interpretation in terms of the dual microscopic conformal dynamics. The previous results are discussed in the general framework of the Anti-de Sitter/Conformal Field Theory dualities. 
  In this paper we continue the study of the three-layer Zamolodchikov model started in our previous works. We analyse numerically the solutions to the Bethe ansatz equations. We consider two regimes I and II which differ by the signs of the spherical sides (a1,a2,a3)->(-a1,-a2,-a3). We accept the two-line hypothesis for the regime I and the one-line hypothesis for the regime II. In the thermodynamic limit we derive integral equations for distribution densities and solve them exactly. We calculate the partition function for the three-layer Zamolodchikov model and check a compatibility of this result with the functional relations. We also do some numerical checkings of our results. 
  In this lecture we discuss some exact results for the low-lying spectrum of the Dirac operator in adjoint QCD. In particular, we find an analytical expression for the slope of the average spectral density. These results are obtained by means of a generating function which is an extension of the QCD partition function with fermionic and bosonic ghost quarks. The low-energy limit of this generating function is completely determined by chiral (super-)symmetries. Our results for the slope of the average spectral density are consistent with the results for the scalar susceptibility which can be obtained from the usual chiral Lagrangian. 
  We present new solutions of higher dimensional Einstein's equations with a cosmological constant that localize gravity on branes which are transverse to Ricci-flat manifolds or to homogeneous spaces with topologically non-trivial solutions of gauge field equations. These solutions are relevant for the localization of chiral fermions on a brane. 
  It has been recently suggested that gravitons moving in $AdS_m \times S^n$ spacetimes along the $S^n$ blow up into spherical $(n-2)$ branes whose radius increases with increasing angular momentum. This leads to an upper bound on the angular momentum, thus ``explaining'' the stringy exclusion principle. We show that this bound is present only for states which saturate a BPS-like condition involving the energy $E$ and angular momentum $J$, $E \geq J/R$, where $R$ is the radius of $S^n$. Restriction of motion to such states lead to a noncommutativity of the coordinates on $S^n$. As an example of motions which do not obey the exclusion principle bound, we show that there are finite action instanton configurations interpolating between two possible BPS states. We suggest that this is consistent with the proposal that there is an effective description in terms of supergravity defined on noncommutative spaces and noncommutativity arises here because of imposing supersymmetry. 
  We study perturbative noncommutative quantum gravity by expanding the gravitational field about a fixed classical background. A calculation of the one loop gravitational self-energy graph reveals that only the non-planar graviton loops are damped by oscillating internal momentum dependent factors. The noncommutative quantum gravity perturbation theory is not renormalizable beyond one loop for matter-free gravity and all loops for matter interactions. Comments are made about the nonlocal gravitational interactions produced by the noncommutative spacetime geometry. 
  We present a rigorous proof of the convergence theorem for the Feynman graphs in arbitrary massive Euclidean quantum field theories on non-commutative R^d (NQFT). We give a detailed classification of divergent graphs in some massive NQFT and demonstrate the renormalizability of some of these theories. 
  We study D3 branes at orbifolded conifold singularities in the presence of discrete torsion. The vacuum moduli space of open strings becomes non-commutative due to a deformation of the superpotential and is studied via the representation theory of the moduli algebra. It is also shown that the center of the moduli algebra correctly describes the underlying orbifolded conifolds. The field theory can be obtained by a marginal deformation of the ${\cal N} = 1$ gauge theory on D3 branes at conifold singularity, the global symmetry being broken from $SU(2) \times SU(2)$ to $U(1) \times U(1)$. By using the AdS/CFT correspondence we argue that the marginal deformation is related to massless KK modes of NSNS and RR two form reduced on the compact space $T^{1,1}$. We build a $T^2$ fibration of $T^{1,1}$ and show that a D3 brane in the bulk correspond to a D5 brane on the $T^2$ fibre. We also discuss the possible brane construction of the system. 
  We employ the ADHM method to derive the moduli space of two instantons in U(1) gauge theory on a noncommutative space. We show by an explicit hyperK\"ahler quotient construction that the relative metric of the moduli space of two instantons on $R^4$ is the Eguchi-Hanson metric and find a unique threshold bound state. For two instantons on $R^3\times S^1$, otherwise known as calorons, we give the asymptotic metric and conjecture a completion. We further discuss the relationship of caloron moduli spaces of A, D and E groups to the Coulomb branches of three dimensional gauge theory. In particular, we show that the Coulomb branch of SU(2) gauge group with a single massive adjoint hypermultiplet coincides with the above two caloron moduli space. 
  Faddeev and Niemi have proposed a decomposition of SU(N) Yang-Mills theory in terms of new variables, appropriate for describing the theory in the infrared limit. We extend this method to SO(2N) Yang-Mills theory. We find that the SO(2N) connection decomposes according to irreducible representations of SO(N). The low energy limit of the decomposed theory is expected to describe soliton-like configurations with nontrivial topological numbers. How the method of decomposition generalizes for $SO(2N+1)$ Yang-Mills theory is also discussed. 
  We show that dualization of BF models to Stueckelberg-like massive gauge theories allows a non-Abelian extension. We obtain local Lagrangians which are straightforward extensions of the Abelian results. 
  We derive an exact duality transformation for pure non-Abelian gauge theory regularized on a lattice. The duality transformation can be applied to gauge theory with an arbitrary compact Lie group G as the gauge group and on Euclidean space-time lattices of dimension d >= 2. It maps the partition function as well as the expectation values of generalized non-Abelian Wilson loops (spin networks) to expressions involving only finite-dimensional unitary representations, intertwiners and characters of G. In particular, all group integrations are explicitly performed. The transformation maps the strong coupling regime of non-Abelian gauge theory to the weak coupling regime of the dual model. This dual model is a system in statistical mechanics whose configurations are spin foams on the lattice. 
  These lecture notes review the foundations and some applications of light-cone quantization. First I explain how to choose a time in special relativity. Inclusion of Poincare invariance naturally leads to Dirac's forms of relativistic dynamics. Among these, the front form, being the basis for light-cone quantization, is my main focus. I explain a few of its peculiar features such as boost and Galilei invariance or separation of relative and center-of-mass motion. Combining light-cone dynamics and field quantization results in light-cone quantum field theory. As the latter represents a first-order system, quantization is somewhat nonstandard. I address this issue using Schwinger's quantum action principle, the method of Faddeev and Jackiw, and the functional Schroedinger picture. A finite-volume formulation, discretized light-cone quantization, is analysed in detail. I point out some problems with causality, which are absent in infinite volume. Finally, the triviality of the light-cone vacuum is established. Coming to applications, I introduce the notion of light-cone wave functions as the solutions of the light-cone Schroedinger equation. I discuss some examples, among them nonrelativistic Coulomb systems and model field theories in two dimensions. Vacuum properties (like chiral condensates) are reconstructed from the particle spectrum obtained by solving the light-cone Schroedinger equation. In a last application, I make contact with phenomenology by calculating the pion wave function within the Nambu and Jona-Lasinio model. I am thus able to predict a number of observables like the pion charge and core radius, the r.m.s. transverse momentum, the pion structure function and the pion distribution amplitude. The latter turns out to be the asymptotic one. 
  This paper discusses a procedure for the consistent coupling of gauge- and matter superfields to supersymmetric sigma-models on symmetric coset spaces of Kaehler type. We exhibit the finite isometry transformations and the corresponding Kaehler transformations. These lead to the construction of a generalized type of Killing potentials. In certain cases a charge quantization condition needs to be imposed to guarantee the global existence of a line bundle on a coset space. The results are applied to the explicit construction of sigma-models on cosets SO(2N)/U(N). Only a finite number of these models can consistently incorporate matter in representations descending from the spinorial representations of SO(2N). We investigate in detail some aspects of the vacuum structure of the gauged SO(10)/U(5) theory, with surprising results: the fully gauged minimal anomaly-free model is shown be singular, as the kinetic terms of the quasi-Goldstone fermions vanish in the vacuum. Gauging only the linear isometry group SU(5)xU(1), or one of its subgroups, can give a physically well-behaved theory. With gauged U(1) this requires the Fayet-Iliopoulos term to take values in a specific limited range. 
  In this paper we consider the scalar sector of Duffin-Kemmer-Petiau theory in the framework of Epstein-Glaser causal method. We calculate the lowest order distributions for Compton scattering, vacuum polarization, self-energy and vertex corrections. By requiring gauge invariance of the theory we recover, in a natural way, the scalar propagator of the usual effective theory. 
  In this paper we quantize the free-particle on a D-dimensional sphere in an unambiguous way by converting the second-class constraint using St\"uckelberg field shiftting formalism. Further, we argument that this formalism is equivalent to the BFFT constraint conversion method and show that the energy spectrum is identical to the pure Laplace-Beltrami operator without additional terms arising from the curvature of the sphere. We work out the gauge symmetry generators with results consistent with those obtained through the nonlinear implementation of the gauge symmetry 
  I review some recent results on four-manifold invariants which have been obtained in the context of topological quantum field theory. I focus on three different aspects: (a) the computation of correlation functions, which give explicit results for the Donaldson invariants of non-simply connected manifolds, and for generalizations of these invariants to the gauge group SU(N); (b) compactifications to lower dimensions, and relations with three-manifold topology and with intersection theory on the moduli space of flat connections on Riemann surfaces; (c) four-dimensional theories with critical behavior, which give some remarkable constraints on Seiberg-Witten invariants and new results on the geography of four-manifolds. 
  We present some solutions for lumps in two dimensions in level-expanded string field theory, as well as in two tachyonic theories: pure tachyonic string field theory and pure $\phi^3$ theory. Much easier to handle, these theories might be used to help understanding solitonic features of string field theory. We compare lump solutions between these theories and we discuss some convergence issues. 
  In this paper we consider some constraints on brane-world cosmologies. In the first part we analyze different behaviors for the expansion of our universe by imposing constraints on the speed of sound. In the second part, we study the nature of matter on the brane world by means of the well-known energy conditions. We find that the strong energy condition must be completely violated at late stages of the universe. 
  We present two equivalent five-dimensional actions for six-dimensional (N,0) N=1,2 supersymmetric theories of self-dual tensor whose one spatial dimension is compactified on a circle. The Kaluza-Klein tower consists of a massless vector and infinite number of massive self-dual tensor multiplets living in five-dimensions. The self-duality follows from the equation of motion. Both actions are quadratic in field variables without any auxiliary field. When lifted back to six-dimensions, one of them gives a supersymmetric extension of the bosonic formulation for the chiral two-form tensor by Perry and Schwarz. 
  We generalize the method of superfield Lagrangian BRST quantization in the part of the gauge-fixing procedure and obtain a quantization method that can be considered as an alternative to the Batalin - Vilkovisky formalism. 
  The model of point particle in general external fields is considered and the generalized equivalence principle is suggested identifying all backgrounds which give rise to equivalent particle dynamics. The equivalence transformations for external fields are interpreted as gauge ones. The gauge group appears to be a semidirect product of all phase space canonical transformations to an abelian ideal of "hyperWeyl" transformations and includes U(1) and general coordinate symmetries as a subgroup. The implications of this gauge symmetry are considered and a connection of general backgrounds to the infinite collection of Fronsdal gauge fields is studied. Although the result is negative and no direct connection arises, it is discussed how higher spin fields could be found among general external fields if one relaxes somehow the equivalence principle. Besides, the particle action in general backgrounds is shown to reproduce the De Wit-Freedman point particle -- symmetric tensors first order interaction suggested many years ago, and generalizes their result to all orders in interaction. 
  We revisit the geometry representing l collinear Schwarzschild black holes. It is seen that the black holes' horizons are deformed by their mutual gravitational attraction. The geometry has a string like conical singularity that connects the holes but has nevertheless a well defined action. Using standard gravitational thermodynamics techniques we determine the Free energy for two black holes at fixed temperature and distance, their entropy and mutual force. When the black holes are far apart the results agree with Newtonian gravity expectations. This analyses is generalized to the case of charged black holes. Then we consider black holes embedded in String/M-theory as bound states of branes. Using the effective string description of these bound states and for large separation we reproduce exactly the semi-classical result for the entropy, including the correction associated with the interaction between the holes. 
  Following hep-th/0001204 we discuss the computation of quantum corrections near long IIB superstring configurations in AdS_5 x S^5 which are related to the Wilson loop expectation values in the strong coupling expansion of the dual n=4 SYM theory with large N. We use the Green-Schwarz description of superstrings in curved R-R backgrounds and demonstrate that it is well-defined and useful for developing perturbation theory near long string backgrounds. 
  We study solitons in three dimensional non-commutative scalar field theory at infinite non-commutativity parameter. We find the metric on the relative moduli space of all solitons of the form |n><n| and show that it is Kahler. We then find the geodesics of this metric and study the scattering of these solitons. In particular we find that the scattering is generally right angle for small values of the impact parameter. We can understand this behaviour in terms of a conical singularity at the origin of moduli space. 
  In this note we examine the supermembrane action on Calabi-Yau 3-folds. We write down the Dirac-Born-Infeld part of the action, and show that it is invariant under the rigid spacetime supersymmetry. 
  We discuss an ansatz for Skyrme fields in three dimensions which uses rational maps between Riemann spheres, and produces shell-like structures of polyhedral form. Houghton, Manton and Sutcliffe showed that a single rational map gives good approximations to the minimal energy Skyrmions up to baryon number of order ten. We show how the method can be generalised by using two or more rational maps to give a double-shell or multi-shell structure. Particularly interesting examples occur at baryon numbers twelve and fourteen. 
  We investigate warped compactification with an abelian gauge theory in six dimensions. The vanishing cosmological constant in four dimensions can generically be realized with a regular metric even in a 3-brane background without fine tuning of couplings. 
  We study BPS domain wall solutions of 5-dimensional N=2 supergravity where isometries of the hypermultiplet geometry have been gauged. We derive the corresponding supersymmetric flow equations and define an appropriate c-function. As an example we discuss a domain wall solution of Freedman, Gubser, Pilch and Warner which is related to a RG-flow in a dual superconformal field theory. 
  We build in this paper the algebra of q-deformed pseudo-differential operators shown to be an essential step towards setting a q-deformed integrability program. In fact, using the results of this q-deformed algebra, we derive the q-analogues of the generalized KdV hierarchy. We focus in particular on the first leading orders of this q-deformed hierarchy namely the q-KdV and q-Boussinesq integrable systems. We present also the q-generalization of the conformal transformations of the currents and discuss the primarity condition of the fields by using the Volterra gauge group transformations for the q-covariant Lax operators. An induced su(n)-Toda(su(2)-Liouville) field theory construction is discussed and other important features are presented. 
  The Kosterlitz-Thouless phase transition is described by the nonperturbative renormalization flow of the two dimensional $\phi^4$-model. The observation of essential scaling demonstrates that the flow equation incorporates nonperturbative effects which have previously found an alternative description in terms of vortices. The duality between the linear and nonlinear $\sigma$-model gives a unified description of the long distance behaviour for O(N)-models in arbitrary dimension $d$. We compute critical exponents in first order in the derivative expansion. 
  Methods developed for the analysis of non-linear integrable models are used in the harmonic superspace (HS) framework. These methods, when applied to the HS, can lead to extract more information about the meaning of integrability in non-linear physical problems. Among the results obtained, we give the basic ingredients towards building in the HS language the analogue of the G.D. algebra of pseudo-differential operators. Some useful convention notations and algebraic structures are also introduced to make the use of the harmonic superspace techniques more accessible. 
  We propose a consistently algebraic formulation of the extended KP (supersymmetric) integrable -hierarchy systems. We exploit the results already established in [14] and which consist in a framework suspected to unify in a fascinating way all the possible supersymmetric KP-hierarchies and then their underlying supergravity theories . This construction leads among other to built explicit non standard integrable Lax evolution equations suspected to reduce to the well known KP integrable equation. We present also a contribution of our construction to the subject of string equation and solitons. Other algebraic properties are also presented 
  This work consists in applying the analysis of integrable models to study the problem of Hyper-Kahler metrics building. In this context, we use the harmonic superspace language applied to D=2 N=4 SU(2) Liouville self interacting model and derive an explicit central extended Hyper-Kahler metric as well as the induced scalar potential. 
  We analyze the connection between Wess-Zumino-Witten and free fermion models in two-dimensional noncommutative space. Starting from the computation of the determinant of the Dirac operator in a gauge field background, we derive the corresponding bosonization recipe studying, as an example, bosonization of the U(N) Thirring model. Concerning the properties of the noncommutative Wess-Zumino-Witten model, we construct an orbit-preserving transformation that maps the standard commutative WZW action into the noncommutative one. 
  We show that combining the spatial noncommutative SYM limit and Lorentz transformation, one can obtain a well-behaved light-like noncommutative SYM limit. The light-like noncommutative SYM is unitary. When the boost velocity is finite, the resulting theory with space-time noncommutativity is unitary as well. The light-like noncommutative SYM limit can also be approached by combining the noncommutative open string theory limit and Lorentz transformation. Along this line, we obtain the supergravity dual for the light-like noncommutative SYM, which is the same as the one acquired using a different method. As a comparison, the supergravity duals for the ordinary SYM, spatial noncommutative SYM and the noncommutative open string theories are given as well, in an infinitely-boosted frame with finite momentum density, which are the decoupling limits of bound states (Dp, W), (D($p-2$), W, Dp), and (F1, W, Dp), respectively. 
  We investigate the Kac-Moody algebra of noncommutative Wess-Zumino-Witten model and find its structure to be the same as the commutative case. Various kinds of gauged noncommutative WZW models are constructed. In particular, noncommutative $U(2)/U(1)$ WZW model is studied and by integrating out the gauge fields, we obtain a noncommutative non-linear $\sigma$-model. 
  We give the realizations of the extended Weyl-Heisenberg (WH) algebra and the Rubakov-Spiridonov (RS) superalgebra in terms of anyons, characterized by the statistical parameter $\nu\in[0,1]$, on two-dimensional lattice. The construction uses anyons defined from usual fermionic oscillators (Lerda-Sciuto construction). The anyonic realization of the superalgebra $sl(1/1)$ is also presented. 
  The consequences of considering the measure of integration in the action to be defined by degrees of freedom independent of the metric are studied. Models without the cosmological constant problem, new ways of spontaneously breaking scale symmetry which have an interesting cosmology and theories of extended objects (string, branes) without a fundamental scale appear possible. 
  Scheme independence of exact renormalization group equations, including independence of the choice of cutoff function, is shown to follow from general field redefinitions, which remains an inherent redundancy in quantum field theories. Renormalization group equations and their solutions are amenable to a simple formulation which is manifestly covariant under such a symmetry group. Notably, the kernel of the exact equations which controls the integration of modes acts as a field connection along the flow. 
  The possible existence of a complex metric tensor field is studied. We show that an effective scalar field is induced by an overall phase component of the complex metric tensor. The corresponding gauge field is shown to be a tachyon. Possible implications of this scalar field to the no hair theorem in a spherically symmetric space is also analyzed. We also study its impact on the evolution of the early universe. 
  We study derivative corrections to the effective action for a single D-brane in type II superstring theory coupled to constant background fields. In particular, within this setting we determine the complete expression for the (disk level) four-derivative corrections to the Born-Infeld part of the action. We also determine 2n-form 2n-derivative corrections to the Wess-Zumino term. Both types of corrections involve all orders of the gauge field strength, F. The results are obtained via string sigma-model loop calculations using the boundary state operator language. The corrections can be succinctly written in terms of the Riemann tensor for a non-symmetric metric. 
  We discuss both the UV and IR origin of the one-loop triangle gauge anomalies for noncommutative nonabelian chiral gauge theories with fundamental, adjoint and bi-fundamental fermions for U(N) groups. We find that gauge anomalies only come from planar triangle diagrams, the non-planar triangle contributions giving rise to no breaking of the Ward identies. Generally speaking, theories with fundamental and bi-fundamental chiral matter are anomalous. Theories with only adjoint chiral fermions are anomaly free. 
  We use level truncated superstring field theory to obtain the effective potential for the Wilson line marginal deformation parameter which corresponds to the constant vacuum expectation value of the U(1) gauge field on the D-brane in a particular direction. We present results for both the BPS and the non-BPS D-brane. In the case of non-BPS D-brane the effective potential has branches corresponding to the extrema of the tachyon potential. In the branch with vanishing tachyon vev (M-branch), the effective potential becomes flatter as the level of the approximation is increased. The branch which corresponds to the stable vacuum after the tachyon has condensed (V-branch) exists only for a finite range of values of marginal deformation parameter. We use our results to find the mass of the gauge field in the stable tachyonic vacuum. We find this mass to be of a non-zero value which seems to stabilize as the level approximation is improved. 
  We point out some subtleties with gauge fixings (which sometimes include the so-called ``brane bending'' effects) typically used to compute the graviton propagator on the Randall-Sundrum brane. In particular, the brane, which has non-vanishing tension, explicitly breaks some part of the diffeomorphisms, so that there are subtleties arising in going to, say, the axial gauge or the harmonic gauge in the presence of (non-conformal) matter localized on the brane. We therefore compute the graviton propagator in the gauge where only the graviphoton fluctuations are set to zero (the diffeomorphisms necessary for this gauge fixing are intact), but the graviscalar component is untouched. We point out that in the Gaussian normal coordinates (where the graviscalar component vanishes on the brane) the graviton propagator blows up in the ultra-violet near the brane. In fact, the allowed gauge transformations, which do not lead to such ultra-violet behavior of the graviton propagator, are such that the coupling of the graviscalar to the brane matter cannot be gauged away in the ultra-violet. Because of this, at the quantum level, where we expect various additional terms to be generated in the brane world-volume action including those involving the graviscalar, fine-tuning (which is independent of that for the brane cosmological constant) is generically required to preserve consistent coupling between bulk gravity and brane matter. We also reiterate that in such warped backgrounds higher curvature terms in the bulk are generically expected to delocalize gravity. 
  The problem of a regular matter source for the Kerr spinning particle is discussed. A class of minimal deformations of the Kerr-Newman solution is considered obeying the conditions of regularity and smoothness for the metric and its matter source.   It is shown that for charged source corresponding matter forms a rotating bag-like core having (A)dS interior and smooth domain wall boundary. Similarly, the requirement of regularity of the Kerr-Newman electromagnetic field leads to superconducting properties of the core.   We further consider the U(I) x U'(I) field model (which was used by Witten to describe cosmic superconducting strings), and we show that it can be adapted for description of superconducting bags having a long range external electromagnetic field and another gauge field confined inside the bag. Supersymmetric version of the Witten field model given by Morris is analyzed, and corresponding BPS domain wall solution interpolating between the outer and internal supersymmetric vacua is considered. The charged bag bounded by this BPS domain wall represents an `ultra-extreme' state with a total mass which is lower than BPS bound of the wall. It is also shown that supergravity suggests the AdS vacuum state inside the bag.   Peculiarities of this model for the rotating bag-like source of the Kerr-Newman geometry are discussed. 
  We report our recent results concerning d5 gauged supergravity (dilatonic gravity) considered on AdS background. The finite action on such background as well as d4 holographic conformal anomaly (via AdS/CFT correspondence) are found. In such formalism the bulk potential is kept to be arbitrary, dilaton dependent function. Holographic RG in such theory is briefly discussed. d5 AdS brane-world Universe induced by quantum effects of brane CFT is constructed. Such brane is spherical, hyperbolic or flat one. Hence, the possibility of quantum creation of inflationary brane-world Universe is shown. 
  Superseded and extended in hep-th/0105110 and hep-th/0208112. 
  We study QED on noncommutative spaces, NCQED. In particular we present the detailed calculation for the noncommutative electron-photon vertex and show that the Ward identity is satisfied. We discuss that in the noncommutative case moving electron will show {\it electric} dipole effects. In addition, we work out the electric and magnetic dipole moments up to one loop level. For the magnetic moment we show that noncommutative electron has an intrinsic (spin independent) magnetic moment. 
  We show how the gravity, extrinsic curvature, and gauge field theories are induced on dynamically localized brane world. They should obey the Gauss-Codazzi-Ricci equation in addition to their own equations of motion. As an example, we derive the solitonic solution for curved domain wall in five dimensions in terms of gravity and extrinsic curvature fields of the brane, and then derive the effective action for their field theory on the brane. 
  We show how the spin 3/2 gravitino field can be localized on a brane in a general framework of supergravity theory. Provided that a scalar field coupled to the Rarita-Schwinger field develops an vacuum expectation value (VEV) whose phase depends on the 'radial' coordinate in extra internal space, the gravitino is localized on a brane with the exponentially decreasing warp factor by selecting an appropriate value of the VEV. 
  We show that a post-Riemannian spacetime can accommodate an internal symmetry structure of the Yang-Mills prototype in such a way that the internal symmetry becomes an integral part of the spacetime itself. The construction encrusts the internal degrees of freedom in spacetime in a manner that merges the gauging of these degrees of freedom with the frame geometrical gauges of spacetime. In particular, we prove that the three spacetime structural identities, which now become ``contaminated'' by internal degrees of freedom, remain invariant with respect to internal gauge transformations. In a Weyl Cartan spacetime, the theory regains the original form of Einstein's equations, in which gauge field sources on the r.h.s. determine on the l.h.s the geometry of spacetime and the fields it induces. In the more general case we identify new contributions of weak magnitude in the interaction between the Yang-Mills field and gravity. The merger of spacetime with internal degrees of freedom which we propose here is not constrained by the usual Coleman-Mandula considerations. 
  The extreme limit of a class of D-dimensional black holes is revisited. In the static limit, it is shown that well defined extremal limiting procedure exists and it leads to new solutions of the type AdS2 times constant curvature symmetric spaces. 
  The procedure of the dimensional reduction related to the partition function of a quantum field living in curved space-time which is the warp product of a symmetric space is investigated. 
  A (2+1)-static black hole solution with a nonlinear electric field is derived. The source to the Einstein equations is a nonlinear electrodynamics, satisfying the weak energy conditions, and it is such that the energy momentum tensor is traceless. The obtained solution is singular at the origin of coordinates. The derived electric field E(r) is given by $E(r)=q/r^2$, thus it has the Coulomb form of a point charge in the Minkowski spacetime. This solution describes charged (anti)--de Sitter spaces. An interesting asymptotically flat solution arises for $\Lambda=0$. 
  We study the locality properties of the vortex operators in compact U(1) Maxwell-Chern-Simons and SU(N) Yang-Mills-Chern-Simons theories in 2+1 dimensions. We find that these theories do admit local vortex operators and thus in the UV regularized versions should contain stable magnetic vortices. In the continuum limit however the energy of these vortex excitations generically is logarithmically UV divergent. Nevertheless the classical analysis shows that at small values of CS coefficient $\kappa$ the vortices become light. It is conceivable that they in fact become massless and condense due to quantum effects below some small $\kappa$. If this happens the magnetic symmetry breaks spontaneously and the theory is confining. 
  The holographic principle is studied in the context of a $n+1$ dimensional radiation dominated closed Friedman-Robertson-Walker (FRW) universe. The radiation is represented by a conformal field theory with a large central charge. Following recent ideas on holography, it is argued that the entropy density in the early universe is bounded by a multiple of the Hubble constant. The entropy of the CFT is expressed in terms of the energy and the Casimir energy via a universal Cardy formula that is valid for all dimensions. A new purely holographic bound is postulated which restricts the sub-extensive entropy associated with the Casimir energy. Unlike the Hubble bound, the new bound remains valid throughout the cosmological evolution. When the new bound is saturated the Friedman equation exactly coincides with the universal Cardy formula, and the temperature is uniquely fixed in terms of the Hubble parameter and its time-derivative. 
  We look at the string theory dual of the $\N=1^*$ theory, involving 5--branes, which was recently proposed by Polchinski and Strassler \cite{PolStr}. We argue that SUGRA alone is not enough in order to obtain the correct screening and confinement behaviour of the various massive field theory vacua, but that appropriate worldvolume phenomena of the 5--branes must be included. We therefore work within the SUGRA approximation, also taking into account the brane dynamics, and classify all the SUGRA configurations. In this level of analysis, we find multiple valid configurations for every given vacuum. We discuss some possible resolutions of this perplexing result. We also consider the spectrum of asymptotic states, and discuss the global symmetries of the SUGRA solution of the $\N=1^*$ theory and of the $\N=0^*$ theory obtained from it by explicit supersymmetry breaking. 
  We embed the large N Chern-Simons/topological string duality in ordinary superstrings. This corresponds to a large $N$ duality between generalized gauge systems with N=1 supersymmetry in 4 dimensions and superstrings propagating on non-compact Calabi-Yau manifolds with certain fluxes turned on. We also show that in a particular limit of the N=1 gauge theory system, certain superpotential terms in the N=1 system (including deformations if spacetime is non-commutative) are captured to all orders in 1/N by the amplitudes of non-critical bosonic strings propagating on a circle with self-dual radius. We also consider D-brane/anti-D-brane system wrapped over vanishing cycles of compact Calabi-Yau manifolds and argue that at large $N$ they induce a shift in the background to a topologically distinct Calabi-Yau, which we identify as the ground state system of the Brane/anti-Brane system. 
  We apply negative dimensional integration method (NDIM) to three outstanding gauges: Feynman, light-cone and Coulomb gauges. Our aim is to show that NDIM is a very suitable technique to deal with loop integrals, being them originated from any gauge choice. In Feynman gauge we perform scalar two-loop four-point massless integrals; in the light-cone gauge we calculate scalar two-loop integrals contributing for two-point functions without any kind of prescriptions, since NDIM can abandon such devices -- this calculation is the first test of our prescriptionless method beyond one-loop order; finally, for the Coulomb gauge we consider a four propagator massless loop integral, in the split dimensional regularization context. 
  In this note we discuss local gauge-invariant operators in noncommutative gauge theories. Inspired by the connection of these theories with the Matrix model, we give a simple construction of a complete set of gauge-invariant operators. We make connection with the recent discussions of candidate operators which are dual to closed strings modes. We also discuss large Wilson loops which in the limit of vanishing noncommutativity, reduce to the closed Wilson loops of the ordinary gauge theory. 
  After reviewing the Green-Schwarz superstring using the approach of Siegel, the superstring is covariantly quantized by constructing a BRST operator from the fermionic constraints and a bosonic pure spinor ghost variable. Physical massless vertex operators are constructed and, for the first time, N-point tree amplitudes are computed in a manifestly ten-dimensional super-Poincar\'e covariant manner. Quantization can be generalized to curved supergravity backgrounds and the vertex operator for fluctuations around $AdS_5\times S^5$ is explicitly constructed. This review is written in a self-contained manner and is based on talks given at the Fradkin Memorial Conference and Strings 2000. 
  We study the issue of defining the discrete light-cone quantization (DLCQ) in perturbative string theory as a light-like limit. While this limit is unproblematic at the classical level, it is non-trivial at the quantum level due to the divergences by the zero-mode loops. We reconsider this problem in bosonic string theory. We construct the multi-loop scattering amplitudes in both open and closed string theories by using the method of Kikkawa, Sakita and Virasoro (KSV), and then we show that these scattering amplitudes are perfectly well-defined in this limit. We also discuss the vacuum amplitudes of the string theory. They are, however, ill-defined in the light-like limit due to the zero-mode loop divergences, and hence we want supersymmetry to cure those pathological divergences even in string theory. 
  Using the Chern-Simon formulation of (2+1) gravity, we derive, for the general asymptotic metrics given by the Fefferman-Graham-Lee theorems, the emergence of the Liouville mode associated to the boundary degrees of freedom of (2+1) dimensional anti de Sitter geometries. Holonomies are described through multi-valued gauge and Liouville fields and are found to algebraically couple the fields defined on the disconnected components of spatial infinity. In the case of flat boundary metrics, explicit expressions are obtained for the fields and holonomies. We also show the link between the variation under diffeomorphisms of the Einstein theory of gravitation and the Weyl anomaly of the conformal theory at infinity. 
  The SU(3)/U(1) x U(1) harmonic variables are used in the harmonic-superspace representation of the D=4, N=3 SYM-equations. The harmonic superfield equations of motion in the simple non-covariant gauge contain the nilpotent harmonic analytic connections. It is shown that these harmonic SYM-equations are equivalent to the finite set of solvable linear iterative equations. 
  Non-trivial configurations of Yang-Mills fields and gravitational backgrounds induce charges on Dp-branes that couple them to lower and higher RR potentials. We show that these couplings can be described in a systematic and coordinate independent way by using Clifford multiplication. In the minimal formulation, D-brane charges and RR potentials combine into bispinors of an SO(1,9) which is defined with a flat metric and does not coincide with the space-time Lorentz group. In a non-minimal formulation, the RR potentials combine into SO(10,10) spinors while the space of charges is formally enlarged to construct SO(10,10) bispinors. The formalism suggests that the general form of the gravitational contribution to the D-brane charges is not modified, though the replacement of wedge product by Clifford multiplication gives rise to new couplings, consistent with T-duality. 
  In this highly speculative note we conjecture that it may be possible to understand features of coincident D-branes, such as the appearance of enhanced non-abelian gauge symmetry, in a purely geometric fashion, using a form of geometry known as scheme theory. We give a very brief introduction to some relevant ideas from scheme theory, and point out how these ideas work in special cases. 
  The dynamics of a stabilized radion in the Randall-Sundrum model (RS) with two branes is investigated, and the effects of the radion on electroweak precision observables are evaluated. The radius is assumed to be stabilized using a bulk scalar field as suggested by Goldberger and Wise. First the mass and the wavefunction of the radion is determined including the backreaction of the bulk stabilization field on the metric, giving a typical radion mass of order the weak scale. This is demonstrated by a perturbative computation of the radion wavefunction. A consequence of the background configuration for the scalar field is that after including the backreaction the Kaluza-Klein (KK) states of the bulk scalars couple directly to the Standard Model fields on the TeV brane. Some cosmological implications are discussed, and in particular it is found that the shift in the radion at late times is in agreement with the four-dimensional effective theory result. The effect of the radion on the oblique parameters is evaluated using an effective theory approach. In the absence of a curvature-scalar Higgs mixing operator, these corrections are small and give a negative contribution to S. In the presence of such a mixing operator, however, the corrections can be sizable due to the modified Higgs and radion couplings. 
  The microcanonical statistics of the Schwarzschild black holes as well as the Reissner-Nordstr$\sf \ddot{o}$m black holes are analyzed. In both cases we set up the inequalities in the microcanonical density of states.   These are then used to show that the most probable configuration in the gases of black holes is that one black hole acquires all of the mass and all of the charge at high energy limit. Thus the black holes obey the statistical bootstrap condition and, in contrast to the other investigation, we see that U(1) charge does not break the bootstrap property. 
  The thermodynamic potential of ideal gases described by the simplest non-abelian statistics is investigated. I show that the potential is the linear function of the element of the abelian-part statistics matrix. Thus, the factorizable property in the Haldane (abelian) fractional exclusion shown by the author [W. H. Huang, Phys. Rev. Lett. 81, 2392 (1998)] is now extended to the non-abelian case. The complete expansion of the thermodynamic potential is also given. 
  In this article we explain discrete torsion. Put simply, discrete torsion is the choice of orbifold group action on the B field. We derive the classification H^2(G, U(1)), we derive the twisted sector phases appearing in string loop partition functions, we derive M. Douglas's description of discrete torsion for D-branes in terms of a projective representation of the orbifold group, and we outline how the results of Vafa-Witten fit into this framework. In addition, we observe that additional degrees of freedom (known as shift orbifolds) appear in describing orbifold group actions on B fields, in addition to those classified by H^2(G, U(1)), and explain how these new degrees of freedom appear in terms of twisted sector contributions to partition functions and in terms of orbifold group actions on D-brane worldvolumes. This paper represents a technically simplified version of prior papers by the author on discrete torsion. We repeat here technically simplified versions of results from those papers, and have included some new material. 
  We review the thermodynamics of the confined and unconfined phases of superconformal Yang-Mills at large N on a three-sphere, focussing especially on the confinement-deconfinement transition. We determine an N-dependent phase boundary and point out some directions for future work. 
  The standard evaluation of the partition function $Z$ of Schwarz's topological field theory results in the Ray--Singer analytic torsion. Here we present an alternative evaluation which results in Z=1. Mathematically, this amounts to a novel perspective on analytic torsion: it can be formally written as a ratio of volumes of spaces of differential forms which is formally equal to 1 by Hodge duality. An analogous result for Reidemeister combinatorial torsion is also obtained. 
  In supersymmetric quantum chromodynamics with N_c-colors and N_f-flavors of quarks, our effective superpotential provides the alternative description to the Seiberg's N=1 duality at least for N_f>=N_c+2, where spontaneous breakdown of chiral symmetries leads to SU(N_c)_{L+R}xSU(N_f-N_c)_LXSU(N_f-N_c)_R as a nonabelian chiral symmetry. The anomaly-matching is ensured by the presence of Nambu-Goldstone superfields associated with this breaking and the instanton contributions are properly equipped in the effective superpotential. 
  The gauge parameter dependence of QED in the covariant gauge can be determined explicitly by introducing a Stueckelberg field, which is a non-interacting fictitious Goldstone boson field. Examples of QED with electrons or charged scalars are discussed. Our results generalize the long known results on the gauge parameter dependence of the wave function renormalization constant and the electron propagator. 
  We study the conservation laws associated with the asymptotic Poincare symmetry of spacetime in the general teleparallel theory of gravity. Demanding that the canonical Poincare generators have well defined functional derivatives in a properly defined phase space, we obtain the improved form of the generators, containing certain surface terms. These terms are shown to represent the values of the related conserved charges: energy-momentum and angular momentum. 
  The attempt to understand if AdS/CFT correspondence may be realized as the one between some AdS-like cosmological space and CFT living on the boundary is made. In order to obtain such cosmology we exchange the time and radial coordinates in d5 Schwarzschild-anti de Sitter (S-AdS) BH (with corresponding signature change). The test on proportionality of free energies from such d5 cosmological space (after AdS/CFT identification of parameters) and from ${\cal N}=4$ SU(N) super Yang-Mills quantum theory is successfully passed. 
  We use holomorphic factorization to find the partition functions of an abelian two-form chiral gauge-field on a flat six-torus. We prove that exactly one of these partition functions is modular invariant. It turns out to be the one that previously has been found in a hamiltonian formulation. 
  We describe the constraints imposed on quantum maximal supergravity theories in perturbation theory by the two duality frameworks: S-duality in superstring theory and the AdS/CFT holographic correspondence between IIB superstring theory and N=4 super Yang-Mills theory. 
  In a previous paper (G.Yoneda, Proc.R.Soc.London, A445,(1994),221), we proved the no-interaction theorem for four particles with the assumption that the (linear and angular) momentum on space-like planes is invariant. In this paper, we assume that the momentum on null cones is invariant and prove that there is no interaction for four particles. 
  Two-time physics (2T) is a general reformulation of one-time physics (1T) that displays previously unnoticed hidden symmetries in 1T dynamical systems and establishes previously unknown duality type relations among them. This may play a role in displaying the symmetries and constructing the dynamics of little understood systems, such as M-theory. 2T physics describes various 1T dynamical systems as different d-dimensional ``holographic'' views of the same 2T system in $d+2$ dimensions. The ``holography'' is due to gauge symmetries that tend to reduce the number of effective dimensions. Different 1T evolutions (i.e. different Hamiltonians) emerge from the same 2T theory when gauge fixing is done with different embeddings of d dimensions inside d+2 dimensions. Thus, in the 2T setting, the distinguished 1T which we call ``time'' is a gauge dependent concept. The 2T action has also a global SO(d,2) symmetry in flat spacetime, or a more general d+2 symmetry in curved spacetime, under which all dimensions are on an equal footing. This symmetry is observable in many 1T systems, but it remained unknown until discovered in the 2T formalism. 2T physics has mainly been developed in the context of particles, including spin and supersymmetry, but some advances have also been made with strings and p-branes, and insights for M-theory have already emerged. In the case of particles, there exists a general worldline formulation with background fields, as well as a field theory formulation, both described in terms of fields that depend on d+2 coordinates. The Standard Model of particle physics can be regarded as a gauge fixed form of a 2T theory in 4+2 dimensions. These facts already provide evidence for a new type of higher dimensional unification. 
  We study logarithmic conformal field theories (LCFTs) through the introduction of nilpotent conformal weights. Using this device, we derive the properties of LCFT's such as the transformation laws, singular vectors and the structure of correlation functions. We discuss the emergence of an extra energy momentum tensor, which is the logarithmic partner of the energy momentum tensor. 
  This paper presents a detailed investigation of the motion of a string near a Kaluza-Klein black hole, using the null string expansion. The zeroth-order string equations of motion are set up separately for electrically and magnetically charged black hole backgrounds. The case of a string falling head-on into the black hole is considered in detail. The equations reduce to quadratures for a magnetically charged hole, while they are amenable to numerical solution for an electrically charged black hole. The Kaluza-Klein radius seen by the string as it approaches the black hole decreases in the magnetic case and increases in the electric case. For magnetic backgrounds, analytical solutions can be obtained in terms of elliptical integrals. These reduce to elementary functions in special cases, including that of the well-known Pollard-Gross-Perry-Sorkin monopole. Here the string exhibits decelerated descent into the black hole. The results in the authors' earlier papers are substantiated here by presenting a detailed analysis. A preliminary analysis of first-order perturbations is also presented, and it is shown that the invariant string length receives a nonzero contribution in the first order. 
  We study one-loop effective action of hypermultiplet theory coupled to external N=2 vector multiplet. We formulate this theory in N=1 superspace and develop a general approach to constructing derivative expansion of the effective action based on an operator symbol technique adopted to N=1 supersymmetric field models. The approach under consideration allows to investigate on a unique ground a general structure of effective action and obtain both N=2 superconformal invariant (non-holomorphic) corrections and anomaly (holomorphic) corrections. The leading low-energy contributions to effective action are found in explicit form. 
  The $CP^1$ model with Hopf interaction is quantised following the Batalin-Tyutin (BT) prescription. In this scheme, extra BT fields are introduced which allow for the existence of only commuting first-class constraints. Explicit expression for the quantum correction to the expectation value of the energy density and angular momentum in the physical sector of this model is derived. The result shows, in the particular operator ordering that we have chosen to work with, that the quantum effect has a divergent contribution of $ {\cal O} (\hbar^2)$ in the energy expectation value. But, interestingly the Hopf term, though topological in nature, can have a finite ${\cal O} (\hbar)$ contribution to energy density in the homotopically nontrivial topological sector. The angular momentum operator, however, is found to have no quantum correction, indicating the absence of any fractional spin even at this quantum level. Finally, the extended Lagrangian incorporating the BT auxiliary fields is computed in the conventional framework of BRST formalism exploiting Faddeev-Popov technique of path integral method. 
  We review a recent progress in constructing low-energy effective action in N=4 super Yang-Mills theories. Using harmonic superspace approach we consider N=4 SYM in terms of unconstrained N=2 superfield and apply N=2 background field method to finding effective action for N=4 SU(n) SYM broken down to U(n)$^{n-1}$. General structure of leading low-energy corrections to effective action is discussed. 
  In this article we shall outline a derivation of the analogue of discrete torsion for the M-theory three-form potential. We find that some of the differences between orbifold group actions on the C field are classified by H^3(G, U(1)). We also compute the phases that the low-energy effective action of a membrane on T^3 would see in the analogue of a twisted sector, and note that they are invariant under the obvious SL(3,Z) action. 
  We consider the concept of fractons, i.e. particles or quasiparticles which obey specific fractal distribution function and for each universal class h of particles we obtain a fractal-deformed Heisenberg algebra. This one takes into account the braid group structure of these objects which live in two-dimensional multiply connected space. 
  In the first part of this letter, we analyse the supergravity dual descriptions of six-dimensional field theories realized on the worldvolume of (p,q) five-branes (OD5 theory). We show that in order for the low-energy gauge theory description to be valid the theta parameter must be rational. Irrational values of theta require a strongly coupled string description of the system at low-energy. We discuss the phase structure and deduce some properties of these theories. In the second part we construct and study the supergravity description of NS5-branes with two electric RR field, which provides a dual description of six-dimensional theories with several light open D-brane excitations. 
  We discuss compact four-dimensional Z_N x Z_M type IIB orientifolds. We take a systematic approach to classify the possible models and construct them explicitely. The supersymmetric orientifolds of this type have already been constructed some time ago. We find that there exist several consistent orientifolds for each of the discrete groups Z_2 x Z_2, Z_2 x Z_4, Z_4 x Z_4, Z_2 x Z_6, Z_2 x Z_6' and Z_6 x Z_6 if anti-D5-branes are introduced. Supersymmetry is broken by the open strings ending on antibranes. The rank of the gauge group is reduced by a factor two if the underlying orbifold space has discrete torsion. 
  This paper is replaced and superseded by nlin.SI/0106015 (Title: Structures in BC_N Ruijsenaars-Schneider models) 
  In certain 1+1 dimensional field theoretic toy models, one can go all the way from microscopic quarks via the hadron spectrum to the properties of hot and dense baryonic matter in an essentially analytic way. This "miracle" is illustrated through case studies of two popular large N models, the Gross-Neveu and the 't Hooft model - caricatures of the Nambu-Jona-Lasinio model and real QCD, respectively. The main emphasis will be on aspects related to spontaneous symmetry breaking (discrete or continuous chiral symmetry, translational invariance) and confinement. 
  Two-dimensional SU(N) Yang-Mills theory is endowed with a non-trivial vacuum structure (k-sectors). The presence of k-sectors modifies the energy spectrum of the theory and its instanton content, the (Euclidean) space-time being compactified on a sphere. For the exact solution, in the limit in which the sphere is decompactified, a k-sector can be mimicked by the presence of k-fundamental charges at infinity, according to a Witten's suggestion. However, this property neither holds before decompactification nor for the genuine perturbative solution which corresponds to the zero-instanton contribution on the sphere. 
  We study charged brane-world black holes in the model of Randall and Sundrum in which our universe is viewed as a domain wall in asymptotically anti-de Sitter space. Such black holes can carry two types of ``charge'', one arising from the bulk Weyl tensor and one from a gauge field trapped on the wall. We use a combination of analytical and numerical techniques to study how these black holes behave in the bulk. It has been shown that a Reissner-Nordstrom geometry is induced on the wall when only Weyl charge is present. However, we show that such solutions exhibit pathological features in the bulk. For more general charged black holes, our results suggest that the extent of the horizon in the fifth dimension is usually less than for an uncharged black hole that has the same mass or the same horizon radius on the wall. 
  We show that it is possible to construct a consistent model describing a current-carrying cosmic string endowed with torsion. The torsion contribution to the gravitational force and geodesics of a test-particle moving around the SCCS are analyzed. In particular, we point out two interesting astrophysical phenomena in which the higher magnitude force we derived may play a critical role: the dynamics of compact objects orbiting the torsioned SCCS and accretion of matter onto it. The deficit angle associated to the SCCS can be obtained and compared with data from the Cosmic Background Explorer (COBE) satellite. We also derived a value for the torsion contribution to matter density fluctuations in the early Universe. 
  Using the proposed AdS/CFT correspondence, we calculate the correlators of operators of conformal field theory at the boundary of AdS$_{d+1}$ corresponding to the sine-Gordon model in the bulk. 
  In this paper we continue the study of the model proposed in the previous paper hep-th/0002077. The model consist of a system of extended objects of diverse dimensionalities, with or without boundaries, with actions of the Chern-Simons form for a supergroup. We also discuss possible connections with Superstring/M-theory. 
  We investigate the SO(4,1) gauge-invariant theory of the Dirac fermions in the external field of the Kaluza-Klein monopole, pointing out that the quantum modes can be recovered from a Klein-Gordon equation analogous to the Schr\" odinger equation in the Taub-NUT background. Moreover, we show that there is a large collection of observables that can be directly derived from those of the scalar theory. These offer many possibilities of choosing complete sets of commuting operators which determine the quantum modes. In addition there are some spin- like and Dirac-type operators involving the covariantly constant Killing-Yano tensors of the hyper-K\" ahler Taub-NUT space. The energy eigenspinors of the central modes in spherical coordinates are completely evaluated in explicit, closed form. 
  We derive a new expansion of the Heisenberg equation of motion based on the projection operator method proposed by Shibata, Hashitsume and Shing\=u. In their projection operator method, a certain restriction is imposed on the initial state. As a result, one cannot prepare arbitrary initial states, for example a coherent state, to calculate the time development of quantum systems. In this paper, we generalize the projection operator method by relaxing this restriction. We explain our method in the case of a Hamiltonian both with and without explicit time dependence. Furthermore, we apply it to an exactly solvable model called the damped harmonic oscillator model and confirm the validity of our method. 
  We consider a spacetime formed by several pieces having common timelike boundary which plays the role of a junction between them. We establish junction conditions for fields of various spin and derive the resulting laws of wave propagation through the junction, which turn out to be quite similar for fields of all spins. As an application, we consider the case of branching four-dimensional spacetime that may arise in the context of the theory of quantum creation of a closed universe on the background of a big mother universe. The theory developed can also be applied to brane-world models and to the superstring theory. 
  In this paper we analyze discrete torsion in perturbative heterotic string theory. In previous work we have given a purely mathematical explanation of discrete torsion as the choice of orbifold group action on a B field, in the case that d H = 0; in this paper, we perform the analogous calculations in heterotic strings where d H is nonzero. 
  We present the 5-dimensional cosmological solutions in the Randall-Sundrum warped compactification scenario, using the Goldberger-Wise mechanism to stabilize the size of the extra dimension. Matter on the Planck and TeV branes is treated perturbatively, to first order. The back-reaction of the scalar field on the metric is taken into account. We identify the appropriate gauge-invariant degrees of freedom, and show that the perturbations in the bulk scalar can be gauged away. We confirm previous, less exact computations of the shift in the radius of the extra dimension induced by matter. We point out that the physical mass scales on the TeV brane may have changed significantly since the electroweak epoch due to cosmological expansion, independently of the details of radius stabilization. 
  We study relations between different kinds of non-commutative spheres which have appeared in the context of ADS/CFT correspondences recently, emphasizing the connections between spaces that have manifest quantum group symmetry and spaces that have manifest classical symmetry. In particular we consider the quotient $SU_q(2)/U(1)$ at roots of unity, and find its relations with the fuzzy sphere with manifest classical SU(2) symmetry. Deformation maps between classical and quantum symmetry, the $U_q(SU(2))$ module structure of quantum spheres and the structure of indecomposable representations of $U_q(SU(2))$ at roots of unity conspire in an interesting way to allow the relation between manifestly $U_q(SU(2)$ symmetric spheres and manifestly U(SU(2)) symmetric spheres. The relation suggests that a subset of field theory actions on the q-sphere are equivalent to actions on the fuzzy sphere. The results here are compatible with the proposal that quantum spheres at roots of unity appear as effective geometries which account for finite N effects in the ADS/CFT correspondence. 
  We attempt to evaluate the effective Lagrangian for a classical background field interacting with the vacuum of two quantum fields. The integration of one of the quantum fields in general leads to a non-local term in the effective Lagrangian and thus becomes intractable during the integration of the other quantum field. We show that $\phi F \tilde{K}$ interaction is an exception. We present the complete calculation for the evaluation of the effective Lagrangian for a pseudoscalar field interacting with photons in the presence of a background electromagnetic field. Expression for the probability of the vacuum breaking down into a pseudoscalar-photon pair is evaluated. We conclude that the gradient of an electric field beyond a certain threshold can give rise to pseudoscalar-photon pair production. 
  We derive the four-dimensional field equations for the induced metric and scalar field on the world-volume of a 3-brane in a five-dimensional bulk with Einstein gravity plus a self-interacting scalar field. We calculate the effective four-dimensional gravitational constant and cosmological constant for arbitrary forms of the brane tension and self-interaction potential for the scalar field in the bulk. In addition to the canonical energy-momentum tensor for the scalar field and ordinary matter on the brane, the effective four-dimensional Einstein equations include terms due to the scalar field and gravitational waves in the bulk. We present solutions corresponding to static Minkowski brane worlds and also dynamical Friedmann-Robertson-Walker brane world cosmologies. We discuss the induced coupling of the scalar field to ordinary matter on the brane. 
  Inspired by the concept of complementarity, we present a illustrative model for the weak interactions with unbroken gauge symmetry and unbroken supersymmetry. The observable particles are bound states of some more fundamental particles. Supersymmetry is broken at the macroscopic scale of the observable particles by a discrete symmetry but remains exact at the scale of the fundamental particle and is thus hidden. This provides a link between theories at very high energies and the observed particle physics. Supersymmetric particles are confined in usual matter. 
  We consider the theory of higher derivative gravity with non-factorizable Randall-Sundrum type space-time and obtain the metric solutions which characterize the $p$-brane world-volume as a curved or planar defect embedded in the higher dimensions. We consider the string inspired effective actions of the dilatonic Gauss-Bonnet type in the brane background and show its consistency with the RS brane-world scenario and the conformal weight of the dilaton couplings in the string theory with apppropriate choice of Regge slope ($\alpha'$) or Gauss-Bonnet coupling ($\alpha$) or both. We also discuss a time dependent dilaton solution for a string-inspired fourth-derivative gravity. 
  In this short note we briefly review some recent developments in understanding discrete torsion. Specifically, we give a short overview of the highlights of a group of recent papers which give the basic understanding of discrete torsion. Briefly, those papers observe that discrete torsion can be completely understood simply as the choice of action of the orbifold group on the B field. We summarize the main points of that work. 
  In this paper we study brane-world scenarios with a bulk scalar field, using a covariant formalism to obtain a 4D Einstein equation via projection onto the brane. We discuss, in detail, the effects of the bulk on the brane and how the scalar field contribute to the gravitational effects. We also discuss choice of conformal frame and show that the frame selected by the induced metric provides a natural choice. We demonstrate our formalism by applying it to cosmological scenarios of Randall-Sundrum and Horava-Witten type models. Finally we consider the cosmology of models where the scalar field couples non-minimally to the matter on the brane. This gives rise to a novel scenario where the universe expands from a finite scale factor with an initial period of accelerated expansion, thus avoiding the singularity and flatness problem of the standard big bang model. 
  Within the spirit of five-dimensional gravity in the Randall-Sundrum scenario, in this paper we consider cosmological and gravitational implications induced by forcing the spacetime metric to satisfy a Misner-like symmetry. We first show that in the resulting Misner-brane framework the Friedmann metric for a radiation dominated flat universe and the Schwarzschild or anti-de Sitter black holes metrics are exact solutions on the branes, but the model cannot accommodate any inflationary solution. The horizon and flatness problems can however be solved in Misner-brane cosmology by causal and noncausal communications through the extra dimension between distant regions which are outside the horizon. Based on a semiclassical approximation to the path-integral approach, we have calculated the quantum state of the Misner-brane universe and the quantum perturbations induced on its metric by brane propagation along the fifth direction. We have then considered testable predictions from our model. These include a scale-invariant spectrum of density perturbations whose amplitude can be naturally accommodated to the required value 10$^{-5}-10^{-6}$, and a power spectrum of CMB anisotropies whose acoustic peaks are at the same sky angles as those predicted by inflationary models, but having much smaller secondary-peak intensities. These predictions seem to be compatible with COBE and recent Boomerang and Maxima measurements 
  A large class of D-branes in Calabi-Yau spaces can be constructed at the Gepner points using the techniques of boundary conformal field theory. In this note we develop methods that allow to compute open string amplitudes for such D-branes. In particular, we present explicit formulas for the products of open string vertex operators of untwisted A-type branes. As an application we show that the boundary theories of the quintic associated with the special Lagrangian submanifolds Im \omega_i z_i = 0 where \omega_i^5=1 possess no continuous moduli. 
  We construct the {\cal N}=4 supersymmetric nonlinear sigma model in three dimensions which can be expanded in 1/N. We evaluate the effective action at leading order in the 1/N expansion and show the finiteness of the model to this order. 
  The purpose of this paper is twofold. The first purpose is to review a systematic construction of Noether currents for supersymmetric theories, especially effective supersymmetric theories. The second purpose is to use these currents to derive the mass-formula for the quantized Seiberg-Witten model from the supersymmetric algebra. We check that the mass-formula of the low-energy theory agrees with that of the full theory (in the broken phase). 
  Recently, the self-tuning mechanism of cancellation of vacuum energy has been proposed in which our universe is a flat 3-brane in a 5-dimensional spacetime. In this letter, the self-tuning mechanism of dark energy is proposed by considering the cosmological matter in the brane world. In our model, the bulk scalar field takes the role of the dark energy and its value is slowly varying in time. The claim is that even if the enormous amount of vacuum energy exists on the brane we can adjust the present value of the dark energy to be consistent with the current observations. In this self-tuning mechanism, the existence of the constant of integration associated with the bulk scalar is crucial. 
  We discuss the relevance of the Lee-Yang edge singularity to the finite-temperature Z_2-symmetry restoration transition of the Gross-Neveu model in three dimensions. We present an explicit result for its large-N free energy density in terms of Zeta(3) and the absolute maximum of Clausen's function. 
  We consider 2+1-dimensional classical noncommutative scalar field theory. The general ansatz for a radially symmetric solution is obtained. Some exact solutions are presented. Their possible physical meaning is discussed. The case of the finite $\theta$ is discussed qualitatively and illustrated by some numerical results. 
  The most prominent class of integrable quantum field theories in 1+1 dimensions is affine Toda theory. Distinguished by a rich underlying Lie algebraic structure these models have in recent years attracted much attention not only as test laboratories for non-perturbative methods in quantum field theory but also in the context of off-critical models. After a short introduction the mathematical preliminaries such as root systems, Coxeter geometry, dual algebras, q-deformed Coxeter elements and q-deformed Cartan matrices are introduced. Using this mathematical framework the bootstrap analysis of the affine Toda S-matrices with real coupling is performed and several universal Lie algebraic formulae proved. The Lie algebraic methods are then extended to define a new class of colour valued S-matrices and also here universal expressions are derived. The second part of the thesis presents a detailed analysis of the high-energy regime of the integrable models discussed in the first part. By means of the thermodynamic Bethe ansatz the central charges of the ultraviolet conformal field theories are calculated and in case of affine Toda theories also the first order term in the scaling function is analytically obtained. For the colour valued S-matrices the connection to WZNW coset models is discussed. A particular subclass of them, the so-called Homogeneous Sine-Gordon models, is investigated in some detail and it is found that the presence of unstable particles in these theories gives rise to a staircase pattern in the corresponding scaling function. This thesis summarizes the results of previously published articles listed in the introduction. 
  In a recent paper we have given the macroscopic and microscopic description of the generating solution of toroidally compactified string theory BPS black holes. In this note we compute its corresponding microscopic entropy. Since by definition the generating solution is the most general solution modulo U-duality transformations, this result allows for a description of the fundamental degrees of freedom accounting for the entropy of any regular BPS black holes of toroidally compactified string (or M) theory. 
  An outline of the conformality approach to the gauge hierarchy is given including the use of non-abelian orbifolds to give unified models of the left-right type. 
  We study the magnetic analogue of Myers' Dielectric Effect and, in some cases, relate it to the blowing up of particles into branes, first investigated by Greevy, Susskind and Toumbas. We show that $D0$ branes or gravitons in M theory, moving in a magnetic four-form field strength background expand into a non-commutative two sphere. Both examples of constant magnetic field and non-constant fields in curved backgrounds generated by branes are considered. We find, in all cases, another solution, consisting of a two-brane wrapping a classical two-sphere, which has all the quantum numbers of the $D0$ branes. Motivated by this, we investigate the blowing up of gravitons into branes in backgrounds different from $AdS_m \times S^n$. We find the phenomenon is quite general. In many cases with less or even no supersymmetry we find a brane configuration which has the same quantum numbers and the same energy as a massless particle in supergravity. 
  We consider the noncommutative Abelian-Higgs theory and construct new types of exact multi-vortex solutions that solve the static equations of motion. They in general do not follow from the BPS equations; only for some specific values of parameters, they satisfy the BPS equations saturating the Bogomol'nyi bound. We further consider the Abelian-Higgs theory with more complicated scalar potential allowing unstable minima and construct exact solutions of noncommutative false vacuum bubble with integer magnetic flux. The classical stability of the solutions is discussed. 
  We investigate N=4 noncommutative super Yang-Mills (SYM) theory. We compute the one-loop four gauge boson scattering amplitude on parallel Dp-branes, and find the corresponding contribution to the noncommutative SYM one-loop action in a momentum expansion. The result is somewhat surprising. We find that while the planar diagram can be written using the usual *-product, the contributions from nonplanar diagrams in general involve additional structure beyond the *-product, arising from the nontrivial worldsheet correlations surviving the field theory limit. To each nonplanar diagram, depending on the number n of external vertex operator insertions on each boundary, there is a corresponding *_n n-ary operation. We further find that it is no longer possible to write down an off-shell gauge invariant one-loop effective action using the noncommutative field strength defined at tree-level. 
  At the component-level we study the `beta-function-favored constraint' (bffc) formalism, suggested in 1988 as the most natural formulation for supergravity derived from more fundamental theories. We begin with the suggestion that \bffc supergravity be identified with new minimal supergravity together with an additional chiral compensator multiplet. After $U_{\rm A}(1)$symmetry breaking, the non-propagating axion 2-form of new minimal supergravity becomes the propagating axion 2-form required by string theory. The final form of the theory is seen to uniquely allow four simultaneous features: (i) local supersymmetry, (ii) implementation of the Green-Schwarz mechanism, (iii) a supersymmetry-breaking order parameter chiral superfield, and (iv) a dilaton superpotential. 
  Two thin conducting, electrically neutral, parallel plates forming an isolated system in vacuum exert attracting force on each other, whose origin is the quantum electrodynamical interaction. This theoretical hypothesis, known as Casimir effect, has been also confirmed experimentally. Despite long history of the subject, no completely convincing theoretical analysis of this effect appears in the literature. Here we discuss the effect (for the scalar field) anew, on a revised physical and mathematical basis. Standard, but advanced methods of relativistic quantum theory are used. No anomalous features of the conventional approaches appear. The Casimir quantitative prediction for the force is shown to constitute the leading asymptotic term, for large separation of the plates, of the full, model-dependent expression. 
  The renormalization group flow of an integrable two dimensional quantum field theory which contains unstable particles is investigated. The analysis is carried out for the Virasoro central charge and the conformal dimensions as a function of the renormalization group flow parameter. This allows to identify the corresponding conformal field theories together with their operator content when the unstable particles vanish from the particle spectrum. The specific model considered is the $SU(3)_{2}$-homogeneous Sine-Gordon model. 
  We give a short account of the recently constructed N=2 D=6 matter coupled supergravity based on the F(4) exceptional supergroup and of its 5D superconformal theory correspondent. 
  We analyse the recent controversy on a possible Chern-Simons like term generated through radiative corrections in QED with a CPT violating term : we prove that, if the theory is correctly defined through Ward identities and normalisation conditions, no Chern-Simons term appears, without any ambiguity. This is related to the fact that such a term is a kind of minor modification of the gauge fixing term, and then no renormalised. The past year literature on that subject is discussed, and we insist on the fact that any absence of an {\sl a priori} divergence should be explained by some symmetry or some non-renormalisation theorem. 
  Using the counterterm subtraction technique we calculatehe stress-energy tensor, action, and other physical quantities for Kerr-AdS black holes in various dimensions. For Kerr-AdS_5 with both rotation parameters non-zero, we demonstrate that stress-energy tensor, in the zero mass parameter limit, is equal to the stress tensor of the weakly coupled four dimensional dual field theory. As a result, the total energy of the generalKerr-AdS_5 black hole at zero mass parameter, exactly matches the Casimir energy of the dual field theory. We show that at high temperature, the general Kerr-AdS_5 and perturbative field theory stress-energy tensors are equal, up to the usual factor of 3/4. We also use the counterterm technique to calculate the stress tensors and actions for Kerr-AdS_6, and Kerr-AdS_7 black holes, with one rotation parameter, and we display the results. We discuss the conformal anomalies of the field theories dual to the Kerr-AdS_5 and Kerr-AdS_7 spacetimes. In these two field theories, we show that the rotation parameters break conformal invariance but not scale invariance, a novel result for a non-trivial field theory. For Kerr-AdS_7 the conformal anomalies calculated on the gravity side and the dual (0,2) tensor multiplet theory are equal up to 4/7 factor. We expect that the Casimir energy of the free field theory is the same as the energy of the Kerr-AdS_7 black hole (with zero mass parameter), up to that factor. 
  These are introductory lectures on the correspondence between SU(N) gauge theories and Superstring Theory in anti-de Sitter geometries (AdS). The subject combines a number of different topics, including supersymmetric field theory, classical and quantum physics of black holes, string theory, string dualities, conformal field theories (CFT), and quantum field theory in anti-de Sitter spaces. We also discuss applications of this AdS/CFT correspondence to the large $N$ dynamics of pure QCD. 
  We show that the NCOS (noncommutative open string) theories on torus $T^p$ ($p\leq 5$) are U-dual to matrix theory on torus with electric flux background. Under U-duality, the number of D-branes and the number of units of electric flux get interchanged. Furthermore under the same U-duality the decoupling limit taken in the NCOS theory maps to the decoupling limit taken in the matrix theory, thus ensure the U-duality between those two class of theories. We consider the energy needed for Higgsing process and some bound states with finite energy and find agreements in both theories. 
  On a noncommutative space of rank-1, we construct a codimension-one soliton explicitly and, in the context of noncommutative bosonic open string field theory, identify it with the D24-brane. We compute the tension of the proposed D24-brane, yielding an exact value and show that it is related to the tension of the codimension two D23-brane by the string T-duality. This resolves an apparent puzzle posed by the result of Harvey, Kraus, Larsen and Martinec and proves that the T-duality is a gauge symmetry; in particular, at strong noncommutativity, it is part of the U(oo) gauge symmetry on the worldvolume. We also apply the result to non-BPS D-branes in superstring theories and argue that the codimension-one soliton gives rise to new descent relation among the non-BPS D-branes in Type IIA and Type IIB string theories via T-duality. 
  The calculation of absorption cross sections for minimal scalars in supergravity backgrounds is an important aspect of the investigation of AdS/CFT correspondence and requires a matching of appropriate wave functions. The low energy case has attracted particular attention. In the following the dependence of the cross section on the matching point is investigated. It is shown that the low energy limit is independent of the matching point and hence exhibits universality. In the high energy limit the independence is not maintained, but the result is believed to possess the correct energy dependence. 
  We study (3+1)+D dimensional spacetime, where D extra dimensions are timelike. Compactification of the D timelike dimensions leads to tachyonic Kaluza-Klein gravitons. We calculate the gravitational self-energies of massive spherical bodies due to the tachyonic exchange, discuss their stability, and find that the gravitational force is screened in a certain number of the extra dimensions. We also derive the exact relationship between the Newton constants in the full 4+D dimensional spacetime with the D extra times and the ordinary Newton constant of our 4 dimensional world. 
  String theory is accused by some of its critics to be a purely abstract mathematical discipline, having lost the contact to the simple yet deeply rooted questions which physics provided until the beginning of this century. We argue that, in contrary, there are indications that string theory might be linked to a fundamental principle of a quantum computational character. In addition, the nature of this principle might be capable to provide some new insight into the question of universality of string theory. 
  We start from a quantum computational principle suggested for string theory in a previous paper and discuss how it might lead to a dynamical principle implying the correct classical gravitational limit. Besides this, we briefly look at some structural properties of moduli space. 
  The extremum of the Willmore-like functional for $m$-dimensional Riemannian surface immersed in $d$-dimensional Riemannian manifold under normal variations is studied and various cases of interest are examined. This study is used to relate the parameters of QCD string action, including the Polyakov-Kleinert extrinsic curvature action, with the geometric properties of the world sheet. The world sheet has been shown to have {\it{negative stiffness}} on the basis of geometric considerations. 
  In this talk I review some recent results concerning multi-instanton calculus in supersymmetric field theories. More in detail, I will show how these computations can be efficiently performed using the formalism of topological field theories. 
  Massive D=4 N=2 supersymmetric sigma models typically admit domain wall (Q-kink) solutions and string (Q-lump) solutions, both preserving 1/2 supersymmetry. We exhibit a new static 1/4 supersymmetric `kink-lump' solution in which a string ends on a wall, and show that it has an effective realization as a BIon of the D=4 super DBI-action. It is also shown to have a time-dependent Q-kink-lump generalization which reduces to the Q-lump in a limit corresponding to infinite BI magnetic field. All these 1/4 supersymmetric sigma-model solitons are shown to be realized in M-theory as calibrated, or `Q-calibrated', M5-branes in an M-monopole background. 
  The concept of fractal index is introduced in connection with the idea of universal class $h$ of particles or quasiparticles, termed fractons, which obey fractal statistics. We show the relation between fractons and conformal field theory(CFT)-quasiparticles taking into account the central charge $c[\nu]$ and the particle-hole duality $\nu\longleftrightarrow\frac{1}{\nu}$, for integer-value $\nu$ of the statistical parameter. The Hausdorff dimension $h$ which labelled the universal classes of particles and the conformal anomaly are therefore related. We also establish a connection between Rogers dilogarithm function, Farey series of rational numbers and the Hausdorff dimension. 
  We study, through zeta-function techniques, the vacuum energies for Dirac fields in a constant magnetic background. We consider the combined effect of the background and twisted boundary conditions. The required charge renormalization is discussed in each case. 
  The functional relation between interquark potential and interquark distance is explicitly derived by considering the Nambu--Goto action in the $AdS_5 \times S^5$ background. It is also shown that a similar relation holds in a general background. The implications of this relation for confinement are briefly discussed. 
  In this paper we revisit the arguments that have led to the proposal of a multi-instanton measure for supersymmetric Yang-Mills theories. We then recall how the moduli space of gauge connections on $\real^4$ can be built from a hyperk\"ahler quotient construction which we generalize to supermanifolds. The measure we are looking for is given by the supermetric of the supermoduli space thus introduced. To elucidate the construction we carry out explicit computations in the case of N=2 supersymmetric Yang-Mills theories. 
  We reconsider the four dimensional extremal black hole constructed in type IIB string theory as the bound state of D1-branes, D5-branes, momentum, and Kaluza-Klein monopoles. Specifically, we examine the case of an arbitrary number of monopoles. Consequently, the weak coupling calculation of the microscopic entropy requires a study of the D1-D5 system on an ALE space. We find that the complete expression for the Bekenstein-Hawking entropy is obtained by taking into account the massless open strings stretched between the fractional D-branes which arise in the orbifold limit of the ALE space. The black hole sector therefore arises as a mixed Higgs-Coulomb branch of an effective 1+1 dimensional gauge theory. 
  The lump solution of \phi^3 field theory provides a toy model for unstable D-branes of bosonic string theory. The field theory living on this lump is itself a cubic field theory involving a tachyon, two additional scalar fields, and a scalar field continuum. Its action can be written explicitly because the fluctuation spectrum of the lump turns out to be governed by a solvable Schroedinger equation; the \ell=3 case of a series of reflectionless potentials. We study the multiscalar tachyon potential both exactly and in the level expansion, obtaining insight into issues of convergence, branches of the solution space, and the mechanism for removal of states after condensation. In particular we find an interpretation for the puzzling finite domain of definition of string field marginal parameters. 
  The Poisson kernels and relations between them for a massive scalar field in a unit ball $B^n$ with Hua's metric and conformal flat metric are obtained by describing the $B^n$ as a submanifold of an $(n+1)$-dimensional embedding space. Global geometric properties of the AdS space are discussed. We show that the $(n+1)$-dimensional AdS space AdS$_{n+1}$ is isomorphic to $RP^1\times B^n$ and boundary of the AdS is isomorphic to $RP^1\times S^{n-1}$. Bulk-boundary propagator and the AdS/CFT like correspondence are demonstrated based on these global geometric properties of the $RP^1\times B^n$. 
  This paper proves that it is possible to build a Lagrangian for quantum electrodynamics which makes it explicit that the photon mass is eventually set to zero in the physical part on observational ground. Gauge independence is achieved upon considering the joint effect of gauge-averaging term and ghost fields. It remains possible to obtain a counterterm Lagrangian where the only non-gauge-invariant term is proportional to the squared divergence of the potential, while the photon propagator in momentum space falls off like 1 over (k-squared) at large k, which indeed agrees with perturbative renormalizability. The resulting radiative corrections to the Coulomb potential in QED are also shown to be gauge-independent. The experience acquired with quantum electrodynamics is used to investigate properties and problems of the extension of such ideas to non-Abelian gauge theories. 
  D-branes on orbifolds with and without discrete torsion are analysed in a unified way using the boundary state formalism. For the example of the Z2 x Z2 orbifold it is found that both the theory with and without discrete torsion possess D-branes whose world-volume carries conventional and projective representations of the orbifold group. The resulting D-brane spectrum is shown to be consistent with T-duality. 
  In hep-th/0008227, the unstable lump solution of \phi^3 theory was shown to have a spectrum governed by the solvable Schroedinger equation with the \ell=3 reflectionless potential and was used as a model for tachyon condensation in string theory. In this paper we study in detail an \ell\to \infty scalar field theory model whose lump solution mimics remarkably the string theory setup: the original field theory tachyon and the lump tachyon have the same mass, the spectrum of the lump consists of equally spaced infinite levels, there is no continuous spectrum, and nothing survives after tachyon condensation. We also find exact solutions for lumps with codimension \ge 2, and show that that their tensions satisfy (1/(2\pi)) (T_p/ T_{p+1})=e/(\sqrt{2\pi}) \approx 1.08. We incorporate gauge fixed couplings to a U(1) gauge field which preserve solvability and result in massless gauge fields on the lump. 
  We give necessary conditions for the existence of perturbative heterotic and common sector type II string warped compactifications preserving four and eight supersymmetries to four spacetime dimensions, respectively. In particular, we find that the only compactifications of heterotic string with the spin connection embedded in the gauge connection and type II strings are those on Calabi-Yau manifolds with constant dilaton. We obtain similar results for compactifications to six and to two dimensions. 
  The exact renormalisation group equation is studied for a two-dimensional theory with exponential interaction and a background charge at infinity. The motivation for studying this interaction is the flow between unitary minimal models perturbed by Phi_{(1,3)}, and their realisation in terms of a quantum group restricted sine-Gordon model. 
  Considering the theory of induced gravity coupled to matter fields, taking the $\phi ^6$ interaction potential model we evaluate the one-loop effective potential in a (3+1)dimensional Bianchi type-I spacetime. It is proved that the $\phi ^6$ theory can be regularised in (3+1)dimensional curved spacetime. We evaluate the finite temperature effective potential and study the temperature dependence of phase transitions. The nature of phase transitions in the early universe is clarified to be of first order. The effects of spacetime curvature and arbitrary field coupling on the phase transitions in the early universe are also discussed. 
  We discuss the O(4) invariant perturbation modes of cosmological instantons. These modes are spatially homogeneous in Lorentzian spacetime and thus not relevant to density perturbations. But their properties are important in establishing the meaning of the Euclidean path integral. If negative modes are present, the Euclidean path integral is not well defined, but may nevertheless be useful in an approximate description of the decay of an unstable state. When gravitational dynamics is included, counting negative modes requires a careful treatment of the conformal factor problem. We demonstrate that for an appropriate choice of coordinate on phase space, the second order Euclidean action is bounded below for normalized perturbations and has a finite number of negative modes. We prove that there is a negative mode for many gravitational instantons of the Hawking-Moss or Coleman-De Luccia type, and discuss the associated spectral flow. We also investigate Hawking-Turok constrained instantons, which occur in a generic inflationary model. Implementing the regularization and constraint proposed by Kirklin, Turok and Wiseman, we find that those instantons leading to substantial inflation do not possess negative modes. Using an alternate regularization and constraint motivated by reduction from five dimensions, we find a negative mode is present. These investigations shed new light on the suitability of Euclidean quantum gravity as a potential description of our universe. 
  Spontaneous chiral symmetry breaking and axial anomaly are studied in the light-front formulation. The existence of multiple vacua and a Nambu-Goldstone boson, both related to dynamical fermion zero modes, are demonstrated within a simple sigma model with fermions. The Weyl gauge formulation and a consistent gauge invariant point-splitting regularization, which includes the light front time, are crucial for obtaining the anomaly in the massive Schwinger model and QED(3+1). 
  The investigation of boundary breather states of the sinh-Gordon model restricted to a half-line is revisited. Properties of the classical boundary breathers for the two-parameter family of integrable boundary conditions are reviewed and extended. The energy spectrum of the quantized boundary states is computed, firstly by using a bootstrap technique and, subsequently using a WKB approximation. Requiring that the two descriptions of the spectrum agree with one another allows a determination of the relationship between the boundary parameters, the bulk coupling constant, and the two parameters appearing in the reflection factor describing the scattering of the sinh-Gordon particle from the boundary. These calculations had been performed previously for the case in which the boundary conditions preserve the bulk $Z_2$ symmetry of the model. The significantly more difficult case of general boundary conditions which violate the bulk symmetry is treated in this article. The results clarify the weak-strong coupling duality of the sinh-Gordon model with integrable boundary conditions. 
  The general structure of the matter Kahler metric in the $\kappa^{2/3}$ expansion of Horava-Witten M-theory with nonstandard embeddings is examined. It is shown that phenomenological models based on this structure can lead to Yukawa and V$_{\rm CKM}$ hierarchies (consistent with all data) without introducing ad hoc small parameters if the 5-branes lie near the distant orbifold plane and the instanton charges of the physical plane vanish. M-theory thus offers an alternate way of describing these hierarchies, different from the conventional models of Yukawa textures. 
  In AdS_{2p+1} we construct propagators for p-forms whose lagrangians contain terms of the form A / d A. In particular we explore the case of forms satisfying ``self duality in odd dimensions'', and the case of forms with a topological mass term. We point out that the ``complete'' set of maximally symmetric bitensors previously used in all the other propagator papers is incomplete - there exists another bitensor which can and does appear in the formulas for the propagators in this particular case. Nevertheless, its presence does not affect the other propagators computed so far.   On the AdS side of the correspondence we compute the 2 and 3 point functions involving the self-dual tensor of the maximal 7d gauged supergravity (sugra), S_{\mu\nu\rho}. Since the 7 dimensional antisymmetric self-dual tensor obeys first order field equations (S + * d S=0), to get a nonvanishing 2 point function we add a certain boundary term (to satisfy the variational principle on a manifold with boundary) to the 7d action. The 3 point functions we compute are of the type SSB and SBB, describing vertex interactions with the gauge fields B_{\mu}. 
  Two dimensional N=2 supersymmetric nonlinear sigma models on hermitian symmetric spaces are formulated in terms of the auxiliary superfields. If we eliminate auxiliary vector and chiral superfields, they give D- and F-term constraints to define the target manifolds. The integration over auxiliary vector superfields, which can be performed exactly, is equivalent to the elimination of the auxiliary fields by the use of the classical equations of motion. 
  This is a write-up of introductory lectures on black holes in string theory given at TASI-99. Topics discussed include: Black holes, thermodynamics and the Bekenstein-Hawking entropy, the information problem; supergravity actions, conserved quantum numbers, supersymmetry and BPS states, units and duality, dimensional reduction, solution-generating; extremal M-branes and D-branes, smearing, probe actions, nonextremal branes, the Gregory-Laflamme instability; breakdown of supergravity and the Correspondence Principle, limits in parameter space, singularity resolution; making black holes with branes, intersection-ology, explicit d=5,4 examples; string/brane computations of extremal black hole entropy in d=5,4, rotation, fractionation; non-extremality and entropy, the link to BTZ black holes, Hawking radiation and absorption cross-sections in the string/brane and supergravity pictures. 
  We construct an effective action describing brane-antibrane system containing N D-branes and N \bar{D}-branes. BPS equations for remaining D-branes after tachyon condensation are derived and their properties are investigated. The value of the D-brane tension and the number of brane bound states are discussed. 
  We show some results concerning the weak binding limit for J=0 states -- which turn out to strongly differ from the non relativistic case -- together with the construction of non zero angular momentum states. The calculation of such states in the Light-Front Dynamics (LFD) framework has some peculiarities which are absent in other approaches. They are related to the fact that the rotation generators contain interaction. We present here the construction of non zero angular momentum states in LFD and show how it leads to a restoration of rotational invariance. For this purpose, the use of Light-Front Dynamics in its explicitly covariant formulation is of crucial importance since the dependence of the wave function on the light-front plane is explicitly parametrized. 
  Light-front dynamics (LFD) is a powerful approach to the theory of relativistic composite systems (hadrons in the quark models and relativistic nucleons in nuclei). Its explicitly covariant version has been recently applied with success to describe the new CEBAF/TJNAF data on the deuteron electromagnetic form factors. The solutions used in were however not obtained from solving exactly the LFD equations but by means of a perturbative calculation with respect to the non relativistic wave function. Since, a consequent effort has been made to obtain exact solutions of LFD equations. The first results concerning J=0 states in a scalar model have been published in nucl-th/9912050. The construction of $J \ne 0$ states in LFD is complicated by the two following facts. First, the generators of the spatial rotations contain interaction and are thus difficult to handle. Second, one is always forced to work in a truncated Fock space, and consequently, the Poincar\'e group commutation relations between the generators -- ensuring the correct properties of the state vector under rotation -- are in practice destroyed. In the standard approach, with the light-front plane defined as $t+z=0$, this violation of rotational invariance manifests by the fact that the energy depends on the angular momentum projection on $z$-axis.   We present here a method to construct $J\ne0$ states in the explicitly covariant formulation of LFD and show how it leads to a restoration of rotational invariance. 
  An approach to find the field equation solution of the Randall-Sundrum model with the $S^1/Z_2$ extra axis is presented. We closely examine the infrared singularity. The vacuum is set by the 5 dimensional Higgs field. Both the domain-wall and the anti-domain-wall naturally appear, at the {\it ends} of the extra compact axis, by taking a {\it new infrared regularization}. The stability is guaranteed from the outset by the kink boundary condition. A {\it continuous} (infrared-)regularized solution, which is a truncated {\it Fourier series} of a {\it discontinuous} solution, is utilized.The ultraviolet-infrared relation appears in the regularized solution. 
  We employ the thermal forward scattering amplitudes technique in order to compute the gluon self-energy in a class of temporal gauges. The leading T^2 and the sub-leading ln(T) contributions are obtained for temperatures high compared with the external momentum. The logarithmic contributions have the same structure as the ultraviolet pole terms which occur at zero temperature (we have recently extended this result to the Coulomb gauge). We also show that the prescription poles, characteristic of temporal gauges, do not modify the leading and sub-leading high-temperature behavior. The one-loop calculation shows that the thermal self-energy is transverse. This result has also been extended to higher orders, using the BRS identities. 
  We obtain a BPS soliton of the effective theory of the M5-brane worldvolume with constant 3-form representing M2-branes ending on the M5-brane. The dimensional reduction of this solution agrees with the known results on D-branes. 
  We report here on the application of the perturbative renormalization-group to the Coulomb gauge in QCD. We use it to determine the high-momentum asymptotic form of the instantaneous color-Coulomb potential $V(\vec{k})$ and of the vacuum polarization $P(\vec{k}, k_4)$. These quantities are renormalization-group invariants, in the sense that they are independent of the renormalization scheme. A scheme-independent definition of the running coupling constant is provided by $\vec{k}^2 V(\vec{k}) = x_0 g^2(\vec{k}/\Lambda_{coul})$, and of $\alpha_s \equiv {{g^2(\vec{k} / \Lambda_{coul})} \over {4\pi}}$, where $x_0 = {{12N} \over {11N - 2N_f}}$, and $\Lambda_{coul}$ is a finite QCD mass scale. We also show how to calculate the coefficients in the expansion of the invariant $\beta$-function $\beta(g) \equiv |\vec{k}| {{\partial g} \over{\partial |\vec{k}|}} = -(b_0 g^3 + b_1 g^5 +b_2 g^7 + ...)$, where all coefficients are scheme-independent. 
  It is proved that a basic superembedding equation for the 2-dimensional worldsheet superspace $\S^{(2|8+8)}$ embedded into D=10 type IIB superspace $M^{(10|16+16)}$ provides a universal, S-duality invariant description of a fundamental superstring and super-D1-brane. We work out generalized action principle, obtain superfield equations of motion for both these objects and find how the S-duality transformations relate the superfield equations of superstring and super-D1-brane.   The superembedding of 6-dimensional worldsheet superspace $\S^{(6|16)}$ into the D=10 type IIB superspace will probably provide a similar universal description for the set of type IIB super-NS5-brane, super-D5-brane and a Kaluza-Klein monopole (super-KK5-brane). 
  We construct orientifolds of type IIA string theory. The theory is compactified on a T^6/Z_N times Z_M orbifold. In addition worldsheet parity in combination with a reflection of three compact directions is modded out. Tadpole cancellation requires to add D-6-branes at angles. The resulting four dimensional theories are N=1 supersymmetric and non-chiral. 
  We consider the vacuum energy for a scalar field subject to a frequency dependent boundary condition. The effect of a frequency cut-off is described in terms of an {\it incomplete} $\zeta$-function. The use of the Debye asymptotic expansion for Bessel functions allows to determine the dominant (volume, area, >...) terms in the Casimir energy. The possible interest of this kind of models for dielectric media (and its application to sonoluminescence) is also discussed. 
  Using analytical methods, a nonpertubative vacuum is constructed recursively in the field theory for the open bosonic string. Evidence suggests it corresponds to the Lorentz-invariant endpoint of tachyon condensation on a D25-brane. The corresponding string field is a twisted squeezed state. 
  This paper elaborates on the bulk/boundary relation between negative cosmological constant 3D gravity and Liouville field theory (LFT). We develop an interpretation of LFT non-normalizable states in terms of particles moving in the bulk. This interpretation is suggested by the fact that ``heavy'' vertex operators of LFT create conical singularities and thus should correspond to point particles moving inside AdS. We confirm this expectation by comparing the (semi-classical approximation to the) LFT two-point function with the (appropriately regularized) gravity action evaluated on the corresponding metric. 
  Using the observed time and spatial intervals defined originally by Einstein and the observation frame in the vierbein formalism, we propose that in curved spacetime, for a wave received in laboratories, the observed frequency is the changing rate of the phase of the wave relative to the local observable time scale and the momentum the changing rate of the phase relative to the local observable spatial length scale. The case of Robertson-Walker universe is especially considered. 
  The noncommutative soliton is characterized by the use of the projection operators in non-commutative space. By using the close relation with the K-theory of $C^*$-algebra, we consider the variations of projection operators along the commutative directions and identify their topological charges. When applied to the string theory, it gives the modification of the brane charges due to tachyon background. 
  The Compton effect in a two-dimensional world is compared with the same process in ordinary three-dimensional space. 
  Rationality of the Wightman functions is proven to follow from energy positivity, locality and a natural condition of global conformal invariance (GCI) in any number D of space-time dimensions. The GCI condition allows to treat correlation functions as generalized sections of a vector bundle over the compactification of Minkowski space and yields a strong form of locality valid for all non-isotropic intervals if assumed true for space-like separations. 
  Planar Quantum Electrodynamics is developed when charged fermions are under the influence of a constant and homogeneous external magnetic field. We compute the cross-length for the scattering of optical/ultraviolet photons by Dirac-Landau electrons. 
  The AdS/CFT correspondence is developed from classical solutions on AdS_5 with two boundaries. The corresponding limits and the reduction of degrees of freedom are discussed, as well as the required renormalization on the field theory side. The Hamiltonian first-order approach towards the solution of coupled gravitational/matter equations of motion is introduced, and the RG interpretation is exposed. Finally we discuss a recent approach towards a naturally vanishing cosmological constant which is based on the AdS/RG correspondence. 
  We write down the Yang-Mills partition function and the average Wilson loop in terms of local gauge-invariant variables being the six components of the metric tensor of dual space. The Wilson loop becomes the trace of the parallel transporter in curved space, else called the gravitational holonomy. We show that the external coordinates mapping the 3d curved space into a flat 6d space play the role of glueball fields, and there is a natural mechanism for the mass gap generation. 
  This survey intends to cover recent approaches to black hole entropy which attempt to go beyond the standard semiclassical perspective. Quantum corrections to the semiclassical Bekenstein-Hawking area law for black hole entropy, obtained within the quantum geometry framework, are treated in some detail. Their ramification for the holographic entropy bound for bounded stationary spacetimes is discussed. Four dimensional supersymmetric extremal black holes in string-based N=2 supergravity are also discussed, albeit more briefly. 
  As shown by Hashimoto and Itzhaki in hep-th/9911057, the perturbative degrees of freedom of a non-commutative Yang-Mills theory (NCYM) on a torus are quasi-local only in a finite energy range. Outside that range one may resort to a Morita equivalent (or T-dual) description appropriate for that energy. In this note, we study NCYM on a non-commutative torus with an irrational deformation parameter $\theta$. In that case, an infinite tower of dual descriptions is generically needed in order to describe the UV regime. We construct a hierarchy of dual descriptions in terms of the continued fraction approximations of $\theta$. We encounter different descriptions depending on the level of the irrationality of $\theta$ and the amount of non-locality tolerated. The behavior turns out to be isomorphic to that found for the phase structure of the four-dimensional Villain $Z_N$ lattice gauge theories, which we revisit as a warm-up. At large 't Hooft coupling, using the AdS/CFT correspondance, we find that there are domains of the radial coordinate $U$ where no T-dual description makes the derivative expansion converge. The radial direction obtains multifractal characteristics near the boundary of AdS. 
  In Randall-Sundrum-type brane-world cosmologies, density perturbations generate Weyl curvature in the bulk, which in turn backreacts on the brane via stress-energy perturbations. On large scales, the perturbation equations contain a closed system on the brane, which may be solved without solving for the bulk perturbations. Bulk effects produce a non-adiabatic mode, even when the matter perturbations are adiabatic, and alter the background dynamics. As a consequence, the standard evolution of large-scale fluctuations in general relativity is modified. The metric perturbation on large-scales is not constant during high-energy inflation. It is constant during the radiation era, except at most during the very beginning, if the energy is high enough. 
  Some aspects of light-like compactifications of superstring theory and their implications for the matrix model of M-theory are discussed. 
  Using y-deformed algebraic geometric techniques the y-deformed Mukay vector of RR-charges of the y-deformed BPS Dp-branes localized on a surface in a Calabi-Yau threefold. The formulas that are obtained here are generalizations of the formulas of the fourth section of the preprint hep-th/0007243 . 
  A left-unilateral matrix equation is an algebraic equation of the form $$ a_0+a_1 x+a_2 x^2+... +a_n x^n=0 $$ where the coefficients $a_r$ and the unknown $x$ are square matrices of the same order and all coefficients are on the left (similarly for a right-unilateral equation). Recently certain perturbative solutions of unilateral equations and their properties have been discussed. We present a unified approach based on the generalized Bezout theorem for matrix polynomials. Two equations discussed in the literature, their perturbative solutions and the relation between them are described. More abstractly, the coefficients and the unknown can be taken as elements of an associative, but possibly noncommutative, algebra. 
  An heuristic proof of the Riemman conjecture is proposed. It is based on the old idea of Polya-Hilbert. A discrete/fractal derivative self adjoint operator whose spectrum may contain the nontrivial zeroes of the zeta function is presented. To substantiate this heuristic proposal we show using generalized index-theory arguments, corresponding to the (fractal) spectral dimensions of fractal branes living in Cantorian-fractal space-time, how the required $negative$ traces associated with those derivative operators naturally agree with the zeta function evaluated at the spectral dimensions. The $\zeta (0) = - 1/2$ plays a fundamental role. Final remarks on the recent developments in the proof of the Riemann conjecture are made. 
  Bosonic sector of d5 gauged supergravity with specific parametrization of full scalar coset is considered (multi-dilaton gravity). Using holographic RG in the form suggested by de Boer-Verlinde-Verlinde the holographic d4 conformal anomaly (in the sector of curvature invariants) is found when bulk potential is not constant. Its comparison with the earlier calculation done in the scheme where expansion of metric and dilaton over radial coordinate of AdS space is used demonstrates scheme dependence of holographic conformal anomaly. In AdS/CFT correspondence where dilatons play role of coupling constants it coincides with multi-loop conformal anomaly which depends on regularization scheme in interacting dual QFT. Hence, scheme dependence of holographic conformal anomaly is consistent with dual QFT expectations. 
  We show that a generalised reduction of D=10 IIB supergravity leads, in a certain limit, to a maximally extended SO(2) gauged supergravity in D=9. We show the scalar potential of this model allows both Minkowski and a new type of domain wall solution to the Bogomol'nyi equations. We relate these vacua to type IIB D-branes. 
  We study the case of open string theory on a D1-brane in a critical electric field. It was argued in hep-th/0006085 that the massless open string modes of this theory decouple from the massive modes, corresponding to a decoupling of the U(1) degrees of freedom in the U(N) gauge theory dual. To provide further support for the decoupling, we present several examples of open string disk amplitudes. Because of the decoupling, many of the massive open string states are stable, indicating the presence of a number of correspondingly stable states in the gauge theory. We provide a lengthy list of these stable states. However, when the theory is compactified in the spatial direction of the electric field, we demonstrate that sufficiently massive such states may decay into wound closed strings. 
  We show that ${\cal N}=1$ supergravity with a cosmological constant can be expressed as constrained topological field theory based on the supergroup $Osp(1|4)$. The theory is then extended to include timelike boundaries with finite spatial area. Consistent boundary conditions are found which induce a boundary theory based on a supersymmetric Chern-Simons theory. The boundary state space is constructed from states of the boundary supersymmetric Chern-Simons theory on the punctured two sphere and naturally satisfies the Bekenstein bound, where area is measured by the area operator of quantum supergravity. 
  We examine the spectrum of small vibrations of giant gravitons, when the gravitons expand in the anti-de-Sitter space and when they expand on the sphere. For any given angular harmonic, the modes are found to have frequencies related to the curvature length scale of the background; these frequencies are independent of radius (and hence angular momentum) of the brane itself. This implies that the holographic dual theory must have, in a given R charge sector, low-lying non-BPS excitations with level spacings independent of the R charge. 
  In this paper we give a general introduction to supersymmetric spin networks. Its construction has a direct interpretation in context of the representation theory of the superalgebra. In particular we analyze a special kind of spin networks with superalgebra $Osp(1|2n)$. It turns out that the set of corresponding spin network states forms an orthogonal basis of the Hilbert space $\cal L\mit^2(\cal A\mit/\cal G)$, and this argument holds even in the q-deformed case. The $Osp(n|2)$ spin networks are also discussed briefly. We expect they could provide useful techniques to quantum supergravity and gauge field theories from the point of non-perturbative view. 
  We study D=3 field theories on the D3-branes stretched between two NS5-branes with NS B-field background. The theory is a noncommutative gauge theory. The mirror symmetry and S-duality of the theory are discussed. A new feature is that the mirror of the noncommutative gauge theory is not a field theory but an open string decoupled from the closed string. We also consider brane creation phenomena and use the result to discuss the analogue of the Seiberg duality. A noncommutative soliton is interpreted as a D1-brane induced on D3-brane. 
  We consider localization of gravity in smooth domain wall solutions of gravity coupled to a scalar field with a generic potential in the presence of the Gauss-Bonnet term. We discuss conditions on the scalar potential such that domain wall solutions are non-singular. We point out that the presence of the Gauss-Bonnet term does not allow flat solutions with localized gravity that violate the weak energy condition. We also point out that in the presence of the Gauss-Bonnet term infinite tension flat domain walls violate positivity. In fact, for flat solutions unitarity requires that on the solution the scalar potential be bounded below. 
  SUSY breaking without messenger fields is proposed. We assume that our world is on a wall and SUSY is broken only by the coexistence of another wall with some distance from our wall. The Nambu-Goldstone fermion is localized on the distant wall. Its overlap with the wave functions of physical fields on our wall gives the mass splitting of physical fields on our wall thanks to a low-energy theorem. We propose that this overlap provides a practical method to evaluate mass splitting in models with SUSY breaking due to the coexisting walls. 
  The description of the two sets of (4,0) supersymmetric models that are related by non-abelian duality transformations is given. The (4,0) supersymmetric WZNW is constructed and the formulation of the (4,0) supersymmetric sigma model dual to (4,0) supersymmetric WZNW model in the sense of Poisson-Lie T-duality is described. 
  We propose a Chern-Simons term for N coinciding non-BPS D-branes. Demanding full U(N) invariance and compatibility with T-duality, it is shown that it is necessary to introduce new interaction terms, through which the non-BPS D-branes couple to all p-form RR fields. 
  We discuss the problem of consistent description of higher spin massive fields coupled to external gravity. As an example we consider massive field of spin 2 in arbitrary gravitational field. Consistency requires the theory to have the same number of degrees of freedom as in flat spacetime and to describe causal propagation. By careful analysis of lagrangian structure of the theory and its constraints we show that there exist at least two possibilities of achieving consistency. The first possibility is provided by a lagrangian on specific manifolds such as static or Einstein spacetimes. The second possibility is realized in arbitrary curved spacetime by a lagrangian representing an infinite series in curvature. In the framework of string theory we derive equations of motion for background massive spin 2 field coupled to gravity from the requirement of quantum Weyl invariance. These equations appear to be a particular case of the general consistent equations obtained from the field theory point of view. 
  Phenomenological evidence suggests the existence of non-trivial background fields in the QCD vacuum. On the other hand SU(3) gauge theory possessses three different classes of both non-generic and non-trivial strata that may be used as classical backgrounds. It is suggested that this three-fold multiplicity of non-trivial vacua may be related to the existence of particle generations, which would then find an explanation in the framework of the standard model. 
  This is a short review of the relation between the Reduced Models and Noncommutative Yang-Mills Theories (NCYM) based on the work done in collaboration with J. Ambjorn, J. Nishimura and R. Szabo. Contents: 1.Twisted Eguchi-Kawai model (TEK), 2.Mapping onto NCYM, 3.Morita equivalence, 4.Fundamental matter, 5.Wilson loops in NCYM, 6.D-brane interpretation. Talk given at the 11th International Seminar "Quarks'2000", Pushkin, Russia, May 13-21, 2000 and the E.S.Fradkin Memorial Conference, Moscow, June 5-10, 2000. 
  In this note we prove the complete stability of the classical fluctuation modes of the rotating ellipsoidal membrane. The analysis is carried out in the full SU(N) setting, with the conclusion that the fluctuation matrix has only positive eigenvalues. This proves that the solution will remain close to the original one for all time, under arbitrary infinitesimal perturbations of the gauge fields. 
  We show that the relation between D-branes and noncommutative tachyons leads very naturally to the relation between D-branes and K-theory. We also discuss some relations between D-branes and K-homology, provide a noncommutative generalization of the ABS construction, and give a simple physical interpretation of Bott periodicity. In addition, a framework for constructing Neveu-Schwarz fivebranes as noncommutative solitons is proposed. 
  We study the radiative corrections to the Chern-Simons mass term at two loops in 2+1 dimensional quantum electrodynamics at finite temperature. We show that, in contrast to the behavior at zero temperature, thermal effects lead to a non vanishing contribution at this order. Using this result, as well as the large gauge Ward identity for the leading parity violating terms in the static limit, we determine the leading order parity violating effective action in this limit at two loops, which generalizes the one-loop effective action proposed earlier. 
  We consider the case of a generic braneworld geometry in the presence of one or more moduli fields (e.g., the dilaton) that vary throughout the bulk spacetime. Working in an arbitrary conformal frame, using the generalized junction conditions of gr-qc/0008008 and the Gauss--Codazzi equations, we derive the effective ``induced'' on-brane gravitational equations. As usual in braneworld scenarios, these equations do not form a closed system in that the bulk can exchange both information and stress-energy with the braneworld. We work with an arbitrary number of moduli fields described by an arbitrary sigma model, with arbitrary curvature couplings, arbitrary self interactions, and arbitrary dimension for the bulk. (The braneworld is always codimension one.) Among the novelties we encounter are modifications of the on-brane stress-energy conservation law, anomalous couplings between on-brane gravity and the trace of the on-brane stress-energy tensor, and additional possibilities for modifying the on-brane effective cosmological constant. After obtaining the general stress-energy ``conservation'' law and the ``induced Einstein equations'' we particularize the discussion to two particularly attractive cases: for a (n-2)-brane in ([n-1]+1) dimensions we discuss both the effect of (1) generic variable moduli fields in the Einstein frame, and (2) the effect of a varying dilaton in the string frame. 
  Recent results on finite open group transformations are reviewed. 
  We present a proof of the positivity of the Bondi energy in Einstein-Maxwell axion-dilaton gravity, being the low-energy limit of the heterotic string theory. We consider the spacelike hypersurface which asymptotically approaches a null cone and on which the equations of the theory under consideration are given. Next, we generalize the proof allowing the hypersurface having inner boundaries. 
  The state space and observables for the leading order of the large-N theory are constructed. The obtained model ("theory of infinite number of fields") is shown to obey Wightman-type axioms (including invariance under boost transformations) and to be nontrivial (there are scattering processes, bound states, unstable particles etc). The considered class of exactly solvable relativistic quantum models involves good examples of theories containing such difficulties as volume divergences associated with the Haag theorem, Stueckelberg divergences and infinite renormalization of the wave function. 
  The main difficulty of quantum field theory is the problem of divergences and renormalization. However, realistic models of quantum field theory are renormalized within the perturbative framework only. It is important to investigate renormalization beyond perturbation theory. However, known models of constructive field theory do not contain such difficulties as infinite renormalization of the wave function. In this paper an exactly solvable quantum mechanical model with such a difficulty is constructed. This model is a simplified analog of the large-N approximation to the $\Phi\phi^a\phi^a$-model in 6-dimensional space-time. It is necessary to introduce an indefinite inner product to renormalize the theory. The mathematical results of the theory of Pontriagin spaces are essentially used. It is remarkable that not only the field but also the canonically conjugated momentum become well-defined operators after adding counterterms. 
  In this paper we study QED on the noncommutative space in the constant electro-magnetic field background. Using the explicit solutions of the noncommutative version of Dirac equation in such background, we show that there are well-defined in and out -going asymptotic states and also there is a causal Green's function. We calculate the pair production rate in this case. We show that at tree level noncommutativity will not change the pair production and the threshold electric field. We also calculate the pair production rate considering the first loop corrections. In this case we show that the threshold electric field is decreased by the noncommutativity effects. 
  Techniques of non-commutative field theories have proven to be useful in describing D-branes as tachyonic solitons in open string theory. However, this procedure also leads to unwanted degeneracy of solutions not present in the spectrum of D-branes in string theories. In this paper we explore the possibility that this apparent multiplicity of solutions is due to the wrong choice of variables in describing the solutions, and that with the correct choice of variables the unwanted degeneracy disappears. 
  The quantization of a scalar field in anti de Sitter spacetime using Poincar\'e coordinates is considered. We find a discrete spectrum that is consistent with a possible mapping between bulk and boundary quantum states. 
  Equivalent relations between quantum mechanical systems in the Robertson-Walker (RW) background metric and quantum dynamics with an induced quadratic background potential are derived in this work. Two elementary applications, which include an algebraic derivation of the evolution operator for a simple harmonic oscillator without using any special function or the path integral technique, and a moving soliton solution of a free particle in an oscillating universe, are presented to illustrate the use of these equivalent relations. 
  A class of asymptotically free scalar theories with O(N) symmetry, defined via the eigenpotentials of the Gaussian fixed point (Halpern-Huang directions), are investigated using renormalization group flow equations. Explicit solutions for the form of the potential in the nonperturbative infrared domain are found in the large-N limit. In this limit, potentials without symmetry breaking essentially preserve their shape and undergo a mass renormalization which is governed only by the renormalization group distance parameter; as a consequence, these scalar theories do not have a problem of naturalness. Symmetry-breaking potentials are found to be ``fine-tuned'' in the large-N limit in the sense that the nontrivial minimum vanishes exactly in the limit of vanishing infrared cutoff: therefore, the O(N) symmetry is restored in the quantum theory and the potential becomes flat near the origin. 
  We analyze the D-branes of a type IIB string theory on an orbifold singularity including the possibility of discrete torsion following the work of Douglas et al. First we prove some general results about the moduli space of a point associated to the "regular representation" of the orbifold group. This includes some analysis of the "wrapped branes" which necessarily appear when the orbifold singularity is not isolated. Next we analyze the stringy homology of the orbifold using the McKay correspondence and the relationship between K-theory and homology. We find that discrete torsion and torsion in this stringy homology are closely-related concepts but that they differ in general. Lastly we question to what extent the D-1 brane may be thought of as being dual to a string. 
  We investigate asymptotic behaviors of the strong coupling limit in the N=2 supersymmetric non-commutative Yang-Mills theory. The strong coupling behavior is quite different from the commutative one since the non-commutative dual U(1) theory is asymptotic free, although the monodoromy is the same as that of the ordinary theory. Singularities are produced by infinitely heavy monopoles and dyons. Nonperturbative corrections may be determined by holomorphy. 
  A method for renormalization of the Casimir energy of confined fermion fields in (1+1)D is proposed. It is based on the extraction of singularities which appear as poles at the point of physical value of the regularization parameter, and subsequent compensation of them by means of redefinition of the "bare" constants. A finite ground state energy of the two-phase hybrid model of fermion bag with chiral boson-fermion interaction is calculated as the function of the bag's size. 
  For a general nonabelian group action and an arbitrary genus worldsheet we show that Vafa's old definition of discrete torsion coincides with Douglas's D-brane definition of discrete torsion associated to projective representations. 
  Motivated by recent works on the origin of inertial mass, we revisit the relationship between the mass of charged particles and zero-point electromagnetic fields. To this end we first introduce a simple model comprising a scalar field coupled to stochastic or thermal electromagnetic fields. Then we check if it is possible to start from a zero bare mass in the renormalization process and express the finite physical mass in terms of a cut-off. In scalar QED this is indeed possible, except for the problem that all conceivable cut-offs correspond to very large masses. For spin-1/2 particles (QED with fermions) the relation between bare mass and renormalized mass is compatible with the observed electron mass and with a finite cut-off, but only if the bare mass is not zero; for any value of the cut-off the radiative correction is very small. 
  We find solutions of 11-dimensional supergravity for M5-branes wrapped on Riemann surfaces. These solutions preserve ${\cal N} = 2$ four-dimensional supersymmetry. They are dual to ${\cal N} = 2$ gauge theories, including non-conformal field theories. We work out the case of ${\cal N} = 2$ Yang-Mills in detail. 
  We reformulate maximal D=5 supergravity in the consistent approach uniquely based on Free Differential Algebras and the solution of their Bianchi identities (= rheonomic method). In this approach the lagrangian is unnecessary since the field equations follow from closure of the supersymmetry algebra. This enables us to explicitly construct the non-compact gaugings corresponding to the non--semisimple algebras CSO(p,q,r), irrespectively from the existence of a lagrangian. The use of Free Differential Algebras is essential to clarify, within a cohomological set up, the dualization mechanism between one-forms and two-forms. Our theories contain 12-r self-dual two-forms and 15+r gauge vectors, r of which are abelian and neutral. These theories, whose existence is proved and their supersymmetry algebra constructed hereby, have potentially interesting properties in relation with domain wall solutions and the trapping of gravity. 
  We show that there is a class of regular charged black hole with a charged de Sitter core similar to the de Sitter-Schwarzschild black hole. One can show that the total energy momentum tensor for the static and spherically symmetric photon field can be made conserved even the Coulomb like energy momentum tensor for photon is not conserved alone. Possible impact and the numerical solutions of the charged black hole will be shown in this paper. 
  We consider the one-loop corrected geometry and thermodynamics of a rotating BTZ black hole by way of a dimensionally reduced dilaton model. The analysis begins with a comprehensive study of the non-extremal solution after which two different methods are invoked to study the extremal case. The first approach considers the extremal limit of the non-extremal calculations, whereas the second treatment is based on the following conjecture: extremal and non-extremal black holes ae qualitatively distinct entities. We show that only the latter method yields regularity and consistency at the one-loop level. This is suggestive of a generalized third law of thermodynamics that forbids continuous evolution from non-extremal to extremal black hole geometries. 
  By the AdS/CFT correspondence, the expectation value of certain local operators in the CFT is given by the asymptotic value of supergravity fields. We show that these local expectation values contain a remarkable amount of information about small sources deep inside AdS_p x S^q. In particular, they contain essentially all the multipole moments. More importantly, one can use them to determine the size of a spherical source. This is not a small effect: The size appears in an exponentially large contribution to the expectation values. This provides an easy way for the CFT to distinguish stars from black holes with the same mass, or to distinguish different "giant graviton" configurations. 
  Short distance structure of spacetime may show up in the form of high freqency dispersion. Although such dispersion is not locally Lorentz invariant, we show in a scalar field model how it can nevertheless be incorporated into a generally covariant metric theory of gravity provided the locally preferred frame is dynamical. We evaluate the resulting energy-momentum tensor and compute its expectation value for a quantum field in a thermal state. The equation of state differs at high temperatures from the usual one, but not by enough to impact the problems of a hot big bang cosmology. We show that a superluminal dispersion relation can solve the horizon problem via superluminal equilibration, however it cannot do so while remaining outside the Planck regime unless the dispersion relation is artificially chosen to have a rather steep dependence on wavevector. 
  Nonminimal coupling of the inflaton field to the Ricci curvature of spacetime is generally unavoidable, and the paradigm of inflation should be generalized by including the corresponding term in the Lagrangian of the inflationary theory. This paper reports on the status of the programme of generalizing inflation. First, the problem of finding the correct value (or set of values) of the coupling constant is analyzed; the result has important consequences for the success or failure of inflationary scenarios. Then, the slow-roll approximation to generalized inflation is studied. Both the unperturbed inflating universe models and scalar/tensor perturbations are discussed, and open problems are pointed out. 
  Five-branes lead in four dimensions to massless N=1 supermultiplets if M-theory is compactified on S1/Z2 x (a Calabi-Yau threefold). One of them describes the modulus associated with the position of the five-brane along the circle S1. We derive the effective four-dimensional supergravity of this multiplet and its coupling to bulk moduli and to Yang-Mills and charged matter multiplets located on Z2 fixed planes. The dynamics of the five-brane modes is obtained by reduction and supersymmetrization of the covariant five-brane bosonic action. Our construction respects all symmetries of M-theory, including the self-duality of the brane antisymmetric tensor. Corrections to gauge couplings are strongly constrained by this self-duality property. The brane contribution to the effective scalar potential is formally similar to a renormalization of the dilaton. The vacuum structure is not modified. Altogether, the impact of the five-brane modulus on the effective supergravity is reminiscent of string one-loop corrections produced by standard compactification moduli. 
  We show that the two dimensional Calogero-Marchioro Model (CMM) without the harmonic confinement can naturally be embedded into an extended SU(1,1|2) superconformal Hamiltonian. We study the quantum evolution of the superconformal Hamiltonian in terms of suitable compact operators of the N=2 extended de Sitter superalgebra with central charge and discuss the pattern of supersymmetry breaking. We also study the arbitrary D dimensional CMM having dynamical OSp(2|2) supersymmetry and point out the relevance of this model in the context of the low energy effective action of the dimensionally reduced Yang-Mills theory. 
  We investigate cosmological consequences arising from the interaction between a homogeneous and isotropic brane-universe and the bulk. A Friedmann equation is derived which incorporates both the brane and bulk matter contributions, which are both assumed to be of arbitrary fluid form. In particular, new terms arise which describe the energy flow onto (or away from) the brane, as well as changes of the equation of state in the bulk. We discuss Randall-Sundrum type models as well as dilatonic domain walls and carefully consider the conditions for stabilising the induced gravitational constant. Furthermore, consequences for cosmological perturbations are analysed. We show that, in general, super-horizon amplitudes are not constant. 
  We study the massive spectrum of fully wrapped branes in warped M-theory compactifications, including regimes where these states are parametrically lighter than the Planck scale or string scale. We show that many such states behave classically as extended objects in the noncompact directions in the sense that their mass grows with their size as measured along the Poincare slices making up the noncompact dimensions. On the other hand these states can be quantized in a nontrivial regime: in particular their spectrum of excitations in a limited regime can be obtained by a warped Kaluza-Klein reduction from ten dimensions. We briefly discuss scattering processes and loop effects involving these states, and also note the possibility of an exponential growth in the number of bound states of these objects as a function of energy. 
  Consideration of the Noether variational problem for any theory whose action is invariant under global and/or local gauge transformations leads to three distinct theorems. These include the familiar Noether theorem, but also two equally important but much less well-known results. We present, in a general form, all the main results relating to the Noether variational problem for gauge theories, and we show the relationships between them. These results hold for both Abelian and non-Abelian gauge theories. 
  d5 dilatonic gravity action with surface counterterms motivated by AdS/CFT correspondence and with contributions of brane quantum CFTs is considered around AdS-like bulk. The effective equations of motion are constructed. They admit two (outer and inner) or multi-brane solutions where brane CFTs may be different. The role of quantum brane CFT is in inducing of complicated brane dilatonic gravity. For exponential bulk potentials the number of AdS-like bulk spaces is found in analytical form.The correspondent flat or curved (de Sitter or hyperbolic) dilatonic two branes are created, as a rule, thanks to quantum effects. The observable early Universe may correspond to inflationary brane. The found dilatonic quantum two brane-worlds usually contain the naked singularity but in couple explicit examples the curvature is finite and horizon (corresponding to wormhole-like space) appears. 
  We consider the singular phases of the smooth finite-gap integrable systems arising in the context of Seiberg-Witten theory. These degenerate limits correspond to the weak and strong coupling regimes of SUSY gauge theories. The spectral curves in such limits acquire simpler forms: in most cases they become rational, and the corresponding expressions for coupling constants and superpotentials can be computed explicitly. We verify that in accordance with the computations from quantum field theory, the weak-coupling limit gives rise to precisely the "trigonometric" family of Calogero-Moser and open Toda models, while the strong-coupling limit corresponds to the solitonic degenerations of the finite-gap solutions. The formulae arising provide some new insights into the corresponding phenomena in SUSY gauge theories. Some open conjectures have been proven. 
  We consider Sen's effective action for unstable D-branes, and study its classical dynamics exactly. In the true vacuum, the Hamiltonian dynamics remains well-defined despite a vanishing action, and is that of massive relativistic string fluid of freely moving electric flux lines. The energy(tension) density equals the flux density in the local co-moving frame. Furthermore, a finite dual Lagrangian exists and is related to the Nielsen-Olesen field theory of ``dual'' strings, supplemented by a crucial constraint. We conclude with discussion on the endpoint of tachyon condensation. 
  We show that a class of spacetimes introduced by Fayyazuddin and Smith to describe intersecting M5-branes admit a generalized Kahler calibration. Equipped with this understanding, we are able to construct spacetimes corresponding to further classes of calibrated $p$-brane world-volume solitons. We note that these classes of spacetimes also describe the fields of $p$-branes wrapping certain supersymmetric cycles of Calabi-Yau manifolds. 
  It is suggested that the Minkowski vacuum of quantum field theories of a large number of fields N would be gravitationally unstable due to strong vacuum energy fluctuations unless an N dependent sub-Planckian ultraviolet momentum cutoff is introduced. We estimate this implied cutoff using an effective quantum theory of massless fields that couple to semi-classical gravity and find it (assuming that the cosmological constant vanishes) to be bounded by $M_Planck/N^1/4$. Our bound can be made consistent with entropy bounds and holography, but does not seem to be equivalent to either, and it relaxes but does not eliminate the implied bound on N inherent in entropy bounds. 
  We show that the thin wall limit of the thick domain wall associated with a sine-Gordon soliton in a single non-compactified patch of 5-dimensional spacetime explicitly yields the Randall-Sundrum localized gravity two patch brane, with its discrete $Z_2$ symmetry arising from the discrete symmetry of the potential, and with the thin Minkowski brane $\Lambda_5+\kappa^2_5\Lambda^2_b/6=0$ relation between bulk and brane cosmological constants arising naturally without any need for fine tuning. Additionally we show that for an embedded thin de Sitter brane, localization of gravity is again possible provided the 5-space is compactified, with the now non-zero net cosmological constant $\Lambda_5+\kappa^2_5\Lambda^2_b/6$ on the brane being found to vary inversely with the compactification radius. 
  We show that the embedding of either a static or a time dependent maximally 3-symmetric brane with non-zero spatial curvature $k$ into a non-compactified $AdS_5$ bulk does not yield exponential suppression of the geometry away from the brane. Implications of this result for brane-localized gravity are discussed. 
  We consider five-dimensional S(2,2|N) Chern-Simons supergravity on M_4 * R . By fine-tuning the Kaluza-Klein reduction to make the 4d cosmological constant equal zero, it is shown that selfdual curvatures on M_4 provide exact solutions to the equations of motion if N=2. 
  The renormalization of effective potential for the noncommutative scalar field theory is investigated to the two-loop approximation. It is seen that the nonplanar diagram does not appear in the one-loop potential. However, nonplanar diagram can become dominant in the two-loop level as the noncommutativity of geometry is sufficiently small. The result shows that the radiative corrections from the nonplanar diagrams have an inclination to induce the spontaneously symmetry breaking if it is not broken in the tree level, and have an inclination to restore the symmetry breaking if it has been broken in the tree level. 
  We develop a method in which it is possible to calculate one loop corrections to the noncommutativity parameter for open strings in a background F field. We reproduce known disk level results, and find that annulus corrections give the same structure multiplied by a constant due to the tachyon. 
  We consider generic N=1 supersymmetric matter coupled to linearized N=1 supergravity through the multiplet of currents. By completing the square, we find the effective action giving the leading supergravity induced correction to the matter dynamics, expressed explicitly as a quadratic form in the components of the current multiplet. The effective action is supersymmetry invariant through an interplay of the local terms arising from the auxiliary field couplings, and the nonlocal terms arising from graviton and gravitino exchange, neither of which is separately invariant. Having an explicit form for the supergravity induced effective action is a first step in studying whether supergravity corrections can lead to dynamical supersymmetry breaking in supersymmetric matter dynamics. In Appendices we give explicit expressions for the currents, in our notational conventions, in the Wess-Zumino and supersymmetric Yang Mills models. 
  We show that the noncommutative Wess-Zumino (NCWZ) Lagrangian with permutation terms in the interaction parts is renormalizable at one-loop level by only a wave function renormalization. When the non-commutativity vanishes, the logarithmic divergence of the wave function renormalization of the NCWZ theory is the same as that of the commutative one. Next the algebras of noncommutative field theories (NCFT's) are studied. From Neother currents, the field representation for the generators of NCFT's is extracted. Then based on this representation, the commutation relations between the generators are calculated for NCFT's. The symmetry properties of NCFT's inferred from these commutation relations are discussed and compared with those of the commutative ones. 
  The holographic principle is represented as the well-known de Alfaro, Fubini and Furlan correspondence between the generating functional for the Green functions of the Euclidean quantum field theory in $D$ dimensions and the Gibbs average for the classical statistical mechanics in $D+1$ dimensions. This correspondence is used to explain the origin of quantum anomalies and the irreversibility in the classical theory. The holographic mapping of a classical string field theory onto a local quantum field theory is outlined. 
  We study D-branes wrapping an exceptional four-cycle P(1,a,b) in a blown-up C^3/Z_m non-compact Calabi-Yau threefold with (m;a,b)=(3;1,1), (4;1,2) and (6;2,3). In applying the method of local mirror symmetry we find that the Picard-Fuchs equations for the local mirror periods in the Z_{3,4,6} orbifolds take the same form as the ones in the local E_{6,7,8} del Pezzo models, respectively. It is observed, however, that the orbifold models and the del Pezzo models possess different physical properties because the background NS B-field is turned on in the case of Z_{3,4,6} orbifolds. This is shown by analyzing the periods and their monodromies in full detail with the help of Meijer G-functions. We use the results to discuss D-brane configurations on P(1,a,b) as well as on del Pezzo surfaces. We also discuss the number theoretic aspect of local mirror symmetry and observe that the exponent which governs the exponential growth of the Gromov-Witten invariants is determined by the special value of the Dirichlet L-function. 
  The relationship between the classical and quantum theories of gravity is reexamined. The value of the gravitational potential defined with the help of the two-particle scattering amplitudes is shown to be in disagreement with the classical result of General Relativity given by the Schwarzschild solution. It is shown also that the potential so defined fails to describe whatever non-Newtonian interactions of macroscopic bodies. An alternative interpretation of the $\hbar^0$-order part of the loop corrections is given directly in terms of the effective action. Gauge independence of that part of the one-loop radiative corrections to the gravitational form factors of the scalar particle is proved, justifying the interpretation proposed. 
  We report the issues of localization of various bulk fields on a brane in a curved background from a local field-theoretic viewpoint where a special attention is paid to a warp geometry in a general space-time dimension. We point out that spin 0 scalar field and spin 2 graviton are naturally localized on a brane with the exponentially decreasing warp factor, while spin 1/2 and 3/2 fermionic fields can be localized on the brane by introducing a bulk mass term with a 'kink' profile. {}For the localization of spin 1 vector field on the brane with the exponentially decreasing warp factor, it is shown that there are essentially two ways. One way is to appeal to the Dvali-Shifman method which is based on a peculiar feature of the non-abelian gauge dynamics such as bulk confinement, whereas the other way is to look for a solution of Einstein equations in such a way that the geometry has a property of confining the vector field to the brane by a gravitational interaction as in six dimensions. We point out that the latter way is more universal. 
  The Coulomb gauge has at least two advantadges over other gauge choices in that bound states between quarks and studies of confinement are easier to understand in this gauge. However, perturbative calculations, namely Feynman loop integrations are not well-defined (there are the so-called energy integrals) even within the context of dimensional regularization. Leibbrandt and Williams proposed a possible cure to such a problem by splitting the space-time dimension into $D=\omega+\rho$, i.e., introducing a specific one parameter $\rho$ to regulate the energy integrals. The aim of our work is to apply negative dimensional integration method (NDIM) to the Coulomb gauge integrals using the recipe of split-dimension parameters and present complete results -- finite and divergent parts -- to the one and two-loop level for arbitrary exponents of propagators and dimension. 
  By using the background field method of QCD in a path integral approach, we derive the equation of motion for the classical chromofield and for the gluon in a system containing the gluon and the classical chromofield simultaneously. This inhomogeneous field equation contains a current term, which is the expectation value of a composite operator including linear, square and cubic terms of the gluon field. We also derive identities which the current should obey from the gauge invariance. We calculate the current at the leading order where the current induced by the gluon is opposite in sign to that induced by the quark. This is just the feature of the non-Abelian gauge field theory which has asymptotic freedom. Physically, the induced current can be treated as the 'displacement' current in the polarized vacuum, and its effect is equivalent to redefining the field and the coupling constant. 
  In this short note, inspired by much recent activity centered around attempts to formulate various correspondences between the classification of affine SU(k) WZW modular-invariant partition functions and that of discrete finite subgroups of SU(k), we present a small and perhaps interesting observation in this light. In particular we show how the groups generated by the permutation of the terms in the exceptional Affine SU(2)-WZW invariants encode the corresponding exceptional SU(2) subgroups. 
  Bosonic model inspired by D=11 superstring action is investigated. An appropriate set of variables is find, in which the light-cone quantization turns out to be possible. It is shown that anomaly terms in the algebra of the light-cone Poincare generators are absent for the case D=27. 
  Effects of the configuration of an external static magnetic field in the form of a singular vortex on the vacuum of a quantized massless spinor field are determined. The most general boundary conditions at the punctured singular point which make the twodimensional Dirac Hamiltonian to be self-adjoint are employed. 
  A massless spinor field is quantized in the background of a singular static magnetic vortex in 2+1-dimensional space-time. The method of self-adjoint extensions is employed to define the most general set of physically acceptable boundary conditions at the location of the vortex. Under these conditions, the vacuum energy density and effective potential in the vortex background are determined. 
  The problem of the axial anomaly in the presence of the Bohm-Aharonov gauge vector field is exactly solved. 
  We study BPS solutions for a self-dual string and a neutral string in M5-brane worldvolume theory with constant three-form field. We further generalize such solitons to superpose with a calibrated surface. We also study a traveling wave on a calibrated surface in the constant three-form field background. 
  Using N=2 off-shell supergravity in five dimensions, we supersymmetrize the brane world scenario of Randall and Sundrum. We extend their construction to include supersymmetric matter at the fixpoints. 
  We study discrete (duality) symmetries of functional determinants. An exact transformation of the effective action under the inversion of background fields $\beta (x) \to \beta^{-1}(x)$ is found. We show that in many cases this inversion does not change functional determinants. Explicitly studied models include a matrix theory in two dimensions, the dilaton-Maxwell theory in four dimensions on manifolds without a boundary, and a two-dimensional dilaton theory on manifolds with boundaries. Our results provide an exact relation between strong and weak coupling regimes with possible applications to string theory, black hole physics and dimensionally reduced models. 
  We study the geometry of a two-sheeted space-time within the framework of non-commutative geometry. As a prelude to the Standard Model in curved space-time, we present a model of a left- and a right- chiral field living on the two sheeted-space time and construct the action functionals that describe their interactions. 
  We suggest Clifford algebra as a useful simplifying language for present quantum dynamics. Clifford algebras arise from representations of the permutation groups as they arise from representations of the rotation groups. Aggregates using such representations for their permutations obey Clifford statistics. The vectors supporting the Clifford algebras of permutations and rotations are plexors and spinors respectively. Physical spinors may actually be plexors describing quantum ensembles, not simple individuals. We use Clifford statistics to define quantum fields on a quantum space-time, and to formulate a quantum dynamics-field-space-time unity that evades the compactification problem. The quantum bits of history regarded as a quantum computation seem to obey a Clifford statistics. 
  The background geometries of the AdS/CFT and the Randall-Sundrum theories are locally similar, and there is strong evidence for some kind of "complementarity" between them; yet the global structures of the respective manifolds are very different. We show that this apparent problem can be understood in the context of a more complete global formulation of AdS/CFT. In this picture, the brane-world arises within the AdS/CFT geometry as the inevitable consequence of recent results on the global structure of manifolds with "infinities". We argue that the usual coordinates give a misleading picture of this global structure, much as Schwarzschild coordinates conceal the global form of Kruskal-Szekeres space. 
  The classical first law of thermodynamics for a Kerr-Newman black hole (KNBH) is generalized to a law in quantum form on the event horizon. Then four quantum conservation laws on the KNBH equilibrium radiation process are derived. The Bekenstein-Hawking relation ${\cal{S}}={\cal{A}}/4$ is exactly established. It can be inferred that the classical entropy of black hole arises from the quantum entropy of field quanta or quasi-particles inside the hole. 
  We show that the dilaton and T-moduli can be stabilized by a single gaugino condensation mechanism in the four-dimensional effective field theory derived from Type IIB orientifolds. A crucial role is played by the mixing of the blowing-up modes Mk with the T-moduli in the Kahler metric, and by the presence of the Mk in the gauge kinetic functions. Supersymmetry breaking in these models is dominated by the auxiliary fields of the T moduli, and phenomenologically interesting patterns can emerge. 
  It has been conjectured that condensation of tachyons on a bosonic D-brane gives rise to vacuum / soliton solutions which are independent of the initial magnetic field on the D-brane. We present evidence for this conjecture using results from two dimensional conformal field theory. In particular we identify a continuous path in the configuration space of open string fields which interpolates between D-brane configurations with two different quantized magnetic flux. 
  We show how the gravity, gauge, and matter fields are induced on dynamically localized brane world. 
  The symmetries of the general Euler equations of fluid dynamics with polytropic exponent are determined using the Kaluza-Klein type framework of Duval et $\it{al}$. In the standard polytropic case the recent results of O'Raifeartaigh and Sreedhar are confirmed and generalized. Similar results are proved for polytropic exponent $\gamma=-1$, which corresponds to the dimensional reduction of $d$-branes. The relation between the duality transformation used in describing supernova explosion and Cosmology is explained. 
  We study, to one loop order, the behavior of the gluon self-energy in the non covariant Coulomb gauge at finite temperature. The cancellation of the peculiar energy divergences, which arise in such a gauge, is explicitly verified in the complete two point function of the Yang-Mills theory. At high temperatures, the leading T^2 term is determined to be transverse and nonlocal, in agreement with the results obtained in covariant gauges. The coefficient of the sub-leading ln(T) contribution, is non transverse but local and coincides (up to a multiplicative constant) with that of the ultraviolet pole term of the zero temperature amplitude. 
  A summary is given of a quantization of the multiflavour Schwinger model on a finite-temperature cylinder with chirality-breaking boundary conditions at its spatial ends, and it is shown that the analytic expression for the chiral condensate implies that the theory exhibits a second order phase transition with $T_c = 0$. 
  We study four-dimensional U(1) on a non-commutative T^2 with rational Theta. This theory has dual descriptions as ordinary SYM or as NCOS. We identify a set of massive non-interacting KK states in the SYM theory and track them through the various dualities. They appear as stretched strings in the non-commutative U(1) providing another example of the IR/UV mixing in non-commutative field theories. In the NCOS these states appear as D-strings with winding and momentum. They form an unconventional type of 1/4 BPS state with the 3-brane. To obtain a consistent picture of S-duality for compactified theories it is essential to keep track of both the NS and the RR B-fields. 
  The correspondences between logarithmic operators in the CFTs on the boundary of AdS_3 and on the world-sheet and dipole fields in the bulk are studied using the free field formulation of the SL(2,C)/SU(2) WZNW model. We find that logarithmic operators on the boundary are related to operators on the world-sheet which are in indecomposable representations of SL(2). The Knizhnik-Zamolodchikov equation is used to determine the conditions for those representations to appear in the operator product expansions of the model. 
  The use of the mass term of the gauge field as a gauge fixing term, which was discussed by Zwanziger, Parrinello and Jona-Lasinio in a large mass limit, is related to the non-linear gauge by Dirac and Nambu. We have recently shown that this use of the mass term as a gauge fixing term is in fact identical to the conventional local Faddeev-Popov formula without taking a large mass limit, if one takes into account the variation of the gauge field along the entire gauge orbit. This suggests that the classical massive vector theory, for example, could be re-interpreted as a gauge invariant theory with a gauge fixing term added in suitably quantized theory.  As for massive gauge particles, the Higgs mechanics, where the mass term is gauge invariant, has a more intrinsic meaning. We comment on several implications of this observation. 
  Perturbative aspects of ultraviolet and infrared dynamics of noncommutative quantum field theory is examined in detail. It is observed that high loop momentum contribution to the nonplanar diagram develops a new infrared singularity with respect to the external momentum. This singular behavior is closely related to that of ultraviolet divergence of planar diagram. It is also shown that such a relation is precise in noncommutative Yang-Mills theory, but the same feature does not persist in noncommutative generalization of QED. 
  We describe generally deformed Heisenberg algebras in one dimension. The condition for a generalized Leibniz rule is obtained and solved. We analyze conditions under which deformed quantum-mechanical problems have a Fock-space representation. One solution of these conditions leads to a q-deformed oscillator already studied by Lorek et al., and reduces to the harmonic oscillator only in the infinite-momentum frame. The other solution leads to the Calogero model in ordinary quantum mechanics, but reduces to the harmonic oscillator in the absence of deformation. 
  We construct explicitly the open descendants of some exceptional automorphism invariants of U(2N) orbifolds. We focus on the case N=pq, p and q prime, and on the automorphisms of the diagonal and charge conjugation invariants that exist for these values of N. These correspond to orbifolds of the circle with radius R^2=2p/q. For each automorphism invariant we find two consistent Klein bottles, and for each Klein bottle we find a complete (and probably unique) set of boundary states. The two Klein bottles are in each case related to each other by simple currents, but surprisingly for the automorphism of the charge conjugation invariant neither of the Klein bottle choices is the canonical (symmetric) one. 
  We write down couplings of the fields on a single BPS Dp-brane with noncommutative world-volume coordinates to the RR-forms in type II theories, in a manifestly background independent way. This generalises the usual Chern-Simons action for a commutative Dp-brane. We show that the noncommutative Chern-Simons terms can be mapped to Myers terms on a collection of infinitely many D-instantons. We also propose Chern-Simons couplings for unstable non-BPS branes, and show that condensation of noncommutative tachyons on these branes leads to the correct Myers terms on the decay products. 
  Using canonical method the Liouville theory has been obtained as a gravitational Wess-Zumino action of the Polyakov string. From this approach it is clear that the form of the Liouville action is the consequence of the bosonic representation of the Virasoro algebra, and that the coefficient in front of the action is proportional to the central charge and measures the quantum braking of the classical symmetry. 
  In these notes we revisit the tachyon lagrangian in the open string field theory using background independent approach of Witten from 1992. We claim that the tree level lagrangian (up to second order in derivatives and modulo some class of field redefinitions) is given by $L = e^{-T} (\partial T)^2 + (1+T)e^{-T}$. Upon obvious change of variables this leads to the potential energy $-\phi^2 \log {\phi^2 \over e}$ with canonical kinetic term. This lagrangian may be also obtained from the effective tachyon lagrangian of the p-adic strings in the limit $p\to 1$. Applications to the problem of tachyon condensation are discussed. 
  Inequivalent standard-like observable sector embeddings in $Z_3$ orbifolds with two discrete Wilson lines, as determined by Casas, Mondragon and Mu\~noz, are completed by examining all possible ways of embedding the hidden sector. The hidden sector embeddings are relevant to twisted matter in nontrivial representations of the Standard Model and to scenarios where supersymmetry breaking is generated in a hidden sector. We find a set of 175 models which have a hidden sector gauge group which is viable for dynamical supersymmetry breaking. Only four different hidden sector gauge groups are possible in these models. 
  Tachyon condensation on a bosonic D-brane was recently demonstrated numerically in Witten's open string field theory with level truncation approximation. This non-perturbative vacuum, which is obtained by solving the equation of motion, has to satisfy furthermore the requirement of BRST invariance. This is indispensable in order for the theory around the non-perturbative vacuum to be consistent. We carry out the numerical analysis of the BRST invariance of the solution and find that it holds to a good accuracy. We also mention the zero-norm property of the solution. The observations in this paper are expected to give clues to the analytic expression of the vacuum solution. 
  We show that, although the correlator of four stress-tensor multiplets in N=4 SYM is known to have radiative corrections, certain linear combinations of its components are protected from perturbative renormalisation and remain at their free-field values. This result is valid for weak as well as for strong coupling and for any gauge group. Our argument uses Intriligator's insertion formula, and includes a proof that the possible contact term contributions cannot change the form of the amplitudes. Combining this new non-renormalisation theorem with Maldacena's conjecture allows us to make a prediction for the structure of the corresponding correlator in AdS supergravity. This is verified by first considerably simplifying the strong coupling expression obtained by recent supergravity calculations, and then showing that it does indeed exhibit the expected structure. 
  We discuss the correlation functions of the SL(2,C)/SU(2) WZW model, or the CFT on the Euclidean AdS_3. We argue that their calculation is reduced to that of a free theory by taking into account the renormalization and integrating out a certain zero-mode, which is an analog of the zero-mode integration in Liouville theory. Based on the resultant free field picture, we give a simple prescription for calculating the correlation functions. The known exact two- and three-point functions of generic primary fields are correctly obtained, including numerical factors. We also obtain some four-point functions of primaries by solving the Knizhnik-Zamolodchikov equation, and verify that our prescription indeed gives them. 
  Generalized quantum statistics will be presented in the context of representation theory of Lie (super)algebras. This approach provides a natural mathematical framework, as is illustrated by the relation between para-Bose and para-Fermi operators and Lie (super)algebras of type B. Inspired by this relation, A-statistics is introduced, arising from representation theory of the Lie algebra A_n. The Fock representations for A_n=sl(n+1) provide microscopic descriptions of particular kinds of exclusion statistics, which may be called quasi-Bose statistics. It is indicated that A-statistics appears to be the natural statistics for certain lattice models in condensed matter physics. 
  We report a no-go theorem excluding consistent cross-couplings for a collection of massless, spin-2 fields described, in the free limit, by the sum of Pauli-Fierz actions (one for each field). We show that, in spacetime dimensions >2, there is no consistent coupling, with at most two derivatives of the fields, that can mix the various "gravitons". The only possible deformations are given by the sum of individual Einstein-Hilbert actions (one for each field) with cosmological terms. Our approach is based on the BRST-based deformation point of view. 
  We discuss gauge symmetry and Ward-Takahashi identities for Wilsonian flows in pure Yang-Mills theories. The background field formalism is used for the construction of a gauge invariant effective action. The symmetries of the effective action under gauge transformations for both the gauge field and the auxiliary background field are separately evaluated. We examine how the symmetry properties of the full theory are restored in the limit where the cut-off is removed. 
  The correlation functions of a two-dimensional rational conformal field theory, for an arbitrary number of bulk and boundary fields and arbitrary world sheets can be expressed in terms of Wilson graphs in appropriate three-manifolds. We present a systematic approach to boundary conditions that break bulk symmetries. It is based on the construction, by `alpha-induction', of a fusion ring for the boundary fields. Its structure constants are the annulus coefficients and its 6j-symbols give the OPE of boundary fields. Symmetry breaking boundary conditions correspond to solitonic sectors. 
  We propose a correspondence between brane-antibrane systems and stable triples (E_1,E_2,T), where E_1,E_2 are holomorphic vector bundles and the tachyon T is a map between them. We demonstrate that, under the assumption of holomorphicity, the brane-antibrane field equations reduce to a set of vortex equations, which are equivalent to the mathematical notion of stability of the triple. We discuss some examples and show that the theory of stable triples suggests a new notion of BPS bound states and stability, and curious relations between brane-antibrane configurations and wrapped branes in higher dimensions. 
  We consider simple CFT models which contain massless bosons, or massless fermions or a supersymmetric combination of the two, on the strip. We study the deformations of these models by relevant boundary operators. In particular, we work out the details for a boundary operator with a quadratic dependence on the fields and argue that some of our results can be extended to a more general situation. In the fermionic models, several subtleties arise due to the doubling of zero modes at the UV fixed point and a ``GSO projected'' RG flow. We attempt to resolve these issues and to discuss how bulk symmetries are realised along the flow. We end with some speculations on possible string theory applications of these results. 
  The explicit multi-instanton solutions by 'tHooft and Jackiw, Nohl & Rebbi are generalized to curvilinear coordinates. The idea is that a gauge transformation can notably simplify the expressions obtained after the change of variables. The gauge transform generates a compensating addition to the gauge potential of pseudoparticles. Singularities of the compensating field are irrelevant for physics but may affect gauge dependent quantities. 
  In this talk, I report on a work done in collaboration with R. Markazi and E.H. Saidi [1]. 
  By introducing a doublet of electromagnetic four dimensional vector potentials, we set up a manifestly Lorentz covariant and SO(2) duality invariant classical field theory of electric and magnetic charges. In our formulation one does not need to introduce the concept of Dirac string. 
  The question of whether the Kaluza-Klein (KK) graviphoton $h_{5\mu}$ and graviscalar $h_{55}$ are localized or not on the brane is one of the important issues. In this letter, we address this problem in five dimensions. Here we consider the massless (zero-mode) propagations without requiring the Z$_2$-symmetry on $h_{5\mu}$. We obtain the graviton $h_{\mu\nu}$, graviphoton, and graviscalar exchange amplitudes on shell. We find that the graviscalar has a tachyonic mass. It turns out that $h_{5\mu}$ admits the localized zero-modes on the brane while $h_{55}$ does not have a localized zero-mode. This is contrasted to the fact that the bulk spin-0 field has a localized zero-mode on the brane but the bulk spin-1 field does not have a localized solution in five dimensions. 
  Motivated by superstring theories on $AdS_3$, we construct spacetime superconformal algebras (SCAs) living on the $AdS_3$ boundary in terms of the transversal physical degrees of freedom. The SCAs constructed are N=4 large, middle algebras, and N=3 algebra, corresponding to superstring theories on $AdS_3 \times S ^3 \times S^3 \times S ^1$, $AdS_3 \times S ^3 \times T ^4 $ and $AdS_3 \times (S^3\times S^3\times S^1)/Z_2$ backgrounds respectively. 
  Several pieces of evidence have been recently brought up in favour of the c-theorem in four and higher dimensions, but a solid proof is still lacking. We present two basic results which could be useful for this search: i) the values of the putative c-number for free field theories in any even dimension, which illustrate some properties of this number; ii) the general form of three-point function of the stress tensor in four dimensions, which shows some physical consequences of the c-number and of the other trace-anomaly numbers. 
  We consider local field theory on $\kappa$-deformed Minkowski space which is an example of solvable Lie-algebraic noncommutative structure. Using integration formula over $\kappa$-Minkowski space and $\kappa$-deformed Fourier transform we consider for deformed local fields the reality conditions as well as deformation of action functionals in standard Minkowski space. We present explicite formulas for two equivalent star products describing CBH quantization of field theory on $\kappa$-Minkowski space. We express also via star product technique the noncommutative translations in $\kappa$-Minkowski space by commutative translations in standard Minkowski space. 
  Clifford number formalism for Maxwell equations is considered. The Clifford imaginary unit for space-time is introduced as coordinate independent form of fully antisymmetric fourth-rank tensor. The representation of Maxwell equations in massless Dirac equation form is considered; we also consider two approaches to the invariance of Dirac equation in respect of the Lorentz transformations. According to the first approach, the unknown column is invariant and according to the second approach it has the transformation properties known as spinorial ones. Clifford number representation for nonlinear electrodynamics equations is obtained. From this representation, we obtain the nonlinear like Dirac equation which is the form of nonlinear electrodynamics equations. As a special case we have the appropriate representations for  Born-Infeld nonlinear electrodynamics. 
  We consider a class of higher order corrections with arbitrary power $n$ of the curvature tensor to the standard gravity action in arbitrary space-time dimension $D$. The corrections are in the form of Euler densities and are unique at each $n$ and $D$. We present a generating functional and an explicit form of the corresponding conserved energy-momentum tensors. The case of conformally flat metrics is discussed in detail. We show that this class of corrections allows for domain wall solutions since, despite the presence of higher powers of the curvature tensor, the singularity structure at the wall is of the same type as in the standard gravity. However, models with higher order corrections have larger set of domain wall solutions and the existence of these solutions no longer depends on the presence of cosmological constants. We find for example that the Randall-Sundrum scenario can be realized without any need for bulk and/or brane cosmological constant. 
  We introduce the 2PPI expansion which sums the bubble graphs to all orders. We show that this expansion can be renormalised with the usual counterterms in a mass independent scheme. We discuss its application to the O(N) linear sigma model. 
  We derive new examples for algebraic relations of interacting fields in local perturbative quantum field theory. The fundamental building blocks in this approach are time ordered products of free (composed) fields. We give explicit formulas for the construction of Poincare covariant ones, which were already known to exist through cohomological arguments.   For a large class of theories the canonical energy momentum tensor is shown to be conserved. Classical theories without dimensionful couplings admit an improved tensor that is additionally traceless. On the example of phi^4-theory we discuss the improved tensor in the quantum theory. Its trace receives an anomalous contribution due to its conservation.   Moreover we define an interacting bilocal normal product for scalar theories. This leads to an operator product expansion of two time ordered fields. 
  We show that Bose-Einstein condensation of charged scalar fields interacting with a topological gauge field at finite temperature is inhibited except for special values of the topological field. We also show that fermions interacting with this topological gauge field can condense for some values of the gauge field. 
  We exhibit a tachyonic mode in a linearized analysis of perturbations of large anti-de Sitter Reissner-Nordstrom black holes in four dimensions. In the large black hole limit, and up to a 0.7% discrepancy which is probably round-off error in the numerical analysis, the tachyon appears precisely when the black hole becomes thermodynamically unstable. 
  We study the behavior of a general gravitational action, including quadratic terms in the curvature, supplemented by a compact scalar field in 4+1 dimensions. The generalized Einstein equation for this system admits solutions which are compact in one direction and Poincare invariant in the remaining directions. These solutions do not require any fine-tuning of the parameters in the action---including the cosmological constant---only that they should satisfy some mild inequalities. Some of these inequalities can be expressed in a universal form that does not depend on the number of extra compact dimensions when the scenario is generalized beyond 4+1 dimensions. 
  We discuss the quantization of a system of slowly-moving extreme Reissner-Nordstrom black holes. In the near-horizon limit, this system has been shown to possess an SL(2,R) conformal symmetry. However, the Hamiltonian appears to have no well-defined ground state. This problem can be circumvented by a redefinition of the Hamiltonian due to de Alfaro, Fubini and Furlan (DFF). We apply the Faddeev-Popov quantization procedure to show that the Hamiltonian with no ground state corresponds to a gauge in which there is an obstruction at the singularities of moduli space requiring a modification of the quantization rules. The redefinition of the Hamiltonian a la DFF corresponds to a different choice of gauge. The latter is a good gauge leading to standard quantization rules. Thus, the DFF trick is a consequence of a standard gauge-fixing procedure in the case of black hole scattering. 
  We use algebraic methods to compute the simple Hurwitz numbers for arbitrary source and target Riemann surfaces. For an elliptic curve target, we reproduce the results previously obtained by string theorists. Motivated by the Gromov-Witten potentials, we find a general generating function for the simple Hurwitz numbers in terms of the representation theory of the symmetric group S_n. We also find a generating function for Hodge integrals on the moduli space M_{g,2} of Riemann surfaces with two marked points, similar to that found by Faber and Pandharipande for the case of one marked point. 
  We consider self-duality in a 2+1 dimensional gauge theory containing both the Born-Infeld and the Chern-Simons terms. We introduce a Born-Infeld inspired generalization of the Proca term and show that the corresponding self dual equation is identical to that of the Born-Infeld-Chern-Simons theory. 
  We review an approach towards a covariant formulation of Matrix theory based on a discretization of the 11d membrane. Higher dimensional algebraic structures, such as the quantum triple Nambu bracket, naturally appear in this approach. We also discuss a novel geometric understanding of the space-time uncertainty relation which points towards a more geometric formulation of Matrix theory. 
  We discuss the general models with one time-like extra dimension and parallel 3-branes on the space-time $M^4 \times M^1$. We also construct the general brane models or networks with $n$ space-like and $m$ time-like extra dimensions and with constant bulk cosmological constant on the space-time $M^4\times (M^1)^{n+m}$, and point out that there exist two kinds of models with zero bulk cosmological constant: for static solutions, we have to introduce time-like and space-like extra dimensions, and for non-static solutions, we can obtain the models with only space-like extra dimension(s). In addition, we give two simplest models explicitly, and comment on the 4-dimensional effective cosmological constant. 
  Field theories with p-form gauge potentials can possess ``hidden'' symmetries leaving the field strengths invariant on-shell without being gauge symmetries on-shell. The relevance of such symmetries to supersymmetric models is discussed. They provide central charges of supersymmetry algebras, play a particular role in duality relations, and lead to peculiar interactions. A multiplet of N=2 supersymmetry in four dimensions with two hidden central charges is presented. 
  We study the problem of more general kinematics for the finite N M(atrix)-Model than the simple straight line motion that has been used before. This is supposed to be related to momentum transferring processes in the dual super-gravity description. We find a negative result for classical, perturbative processes and discuss briefly the possibility of instianton like quantum mechanical tunneling processes. 
  We discuss the construction of six- and four-dimensional Type IIB orientifolds with vanishing untwisted RR tadpoles, but generically non-zero twisted RR tadpoles. Tadpole cancellation requires the introduction of D-brane systems with zero untwisted RR charge, but non-zero twisted RR charges. We construct explicit models containing branes and antibranes at fixed points of the internal space, or non-BPS branes partially wrapped on it. The models are non-supersymmetric, but are absolutely stable against decay to supersymmetric vacua. For particular values of the compactification radii tachyonic modes may develop, triggering phase transitions between the different types of non-BPS configurations of branes, which we study in detail in a particular example. As an interesting spin-off, we show that the $\IT^6/\IZ_4$ orientifold without vector structure, previously considered inconsistent due to uncancellable twisted tadpoles, can actually be made consistent by introducing a set of brane-antibrane pairs whose twisted charge cancels the problematic tadpole. 
  After a brief review of the description of black holes on branes, we examine the evaporation of a small black hole on a brane in a world with large extra dimensions. We show that, contrary to previous claims, most of the energy is radiated into the modes on the brane. This raises the possibility of observing Hawking radiation in future high energy colliders if there are large extra dimensions. 
  The representation theory of the Virasoro algebra in the case of a logarithmic conformal field theory is considered. Here, indecomposable representations have to be taken into account, which has many interesting consequences. We study the generalization of null vectors towards the case of indecomposable representation modules and, in particular, how such logarithmic null vectors can be used to derive differential equations for correlation functions. We show that differential equations for correlation functions with logarithmic fields become inhomogeneous. 
  The bootstrap for Liouville theory with conformally invariant boundary conditions will be discussed. After reviewing some results on one- and boundary two-point functions we discuss some analogue of the Cardy condition linking these data. This allows to determine the spectrum of the theory on the strip, and illustrates in what respects the bootstrap for noncompact conformal field theories with boundary is richer than in RCFT. We briefly indicate some connections with $U_q(sl(2,R))$ that should help completing the bootstrap. 
  This is an introductory review of the AdS/CFT correspondence and of the ideas that led to its formulation. We show how comparison of stacks of D3-branes with corresponding supergravity solutions leads to dualities between conformal large $N$ gauge theories in 4 dimensions and string backgrounds of the form $AdS_5\times X_5$ where $X_5$ is an Einstein manifold. The gauge invariant chiral operators of the field theory are in one-to-one correspondence with the supergravity modes, and their correlation functions at strong `t Hooft coupling are determined by the dependence of the supergravity action on AdS boundary conditions. The simplest case is when $X_5$ is a 5-sphere and the dual gauge theory is the ${\cal N}=4$ supersymmetric SU(N) Yang-Mills theory. We also discuss D3-branes on the conifold corresponding to $X_5$ being a coset space $T^{1,1}=(SU(2)\times SU(2))/U(1)$. This background is dual to a certain ${\cal N}=1$ superconformal field theory with gauge group $SU(N)\times SU(N)$. 
  The Hagedorn transition in non-commutative open string theory (NCOS) is relatively simple because gravity decouples. For NCOS theories in no more than five spacetime dimensions, the Hagedorn transition is second order, and the high temperature phase involves long, nearly straight fundamental strings separating from the D-brane on which the NCOS theory is defined. Above five spacetime dimensions interaction effects become important below the Hagedorn temperature. Although this complicates studies of the transition, we believe that the high temperature phase again involves long strings liberated from the bound state. 
  We show that a four-parameter class of 3+1 dimensional NCOS theories can be obtained by dimensional reduction on a general 2-torus from OM theory. Compactifying two spatial directions of NCOS theory on a 2-torus, we study the transformation properties under the $SO(2,2;Z)$ T-duality group. We then discuss non-perturbative configurations of non-commutative super Yang-Mills theory. In particular, we calculate the tension for magnetic monopoles and (p,q) dyons and exhibit their six-dimensional origin, and construct a supergravity solution representing an instanton in the gauge theory. We also compute the potential for a monopole-antimonopole in the supergravity approximation. 
  We find a class of exact solutions of noncommutative gauge theories corresponding to unstable non-BPS solitons. In the two-dimensional euclidean (or 2+1 dimensional lorentzian) U(1) theory we find localized solutions carrying nonzero magnetic flux. In four euclidean dimensions we find non-BPS solutions with the same Pontrjagin charge but greater energy than the usual self-dual Yang-Mills instanton. We conjecture that these solutions and generalizations thereof correspond to nonsupersymmetric configurations of D-(p-2k) branes (or intersections thereof) in a D-p brane in the noncommutative scaling limit of large B-field. In the particular case of a 0-brane on a 2-brane the analysis of small fluctuations reveals an infinite tower of states which agrees exactly with that of the 0-2 CFT in the scaling limit. The spectrum contains a tachyon, and we show explicitly that the endpoint of tachyon condensation corresponds to the 0-brane dissolved in the 2-brane. 
  To understand the Abelian dominance and magnetic monopole dominance in low-energy QCD, we rewrite the non-Abelian Wilson loop into the form which is written in terms of its Abelian components or the 't Hooft-Polyakov tensor describing the magnetic monopole. This is peformed by making use of a version of non-Abelian Stokes theorem. We propose a modified version of the maximal Abelian (MA) gauge. By adopting the modified MA gauge in QCD, we show that the off-diagonal gluons and Faddeev-Popov ghosts acquire their masses through the ghost--anti-ghost condensation due to four ghost interaction coming from the gauge-fixing term of the modified MA gauge. The asymptotic freedom of the original non-Abelian gauge theory is preserved in this derivation. 
  We discuss the integrability of the Berkovits-Siegel open string field equations and derive an infinite set of their non-local (solution-generating) symmetries. The string field equations are embedded in an infinite system of overdetermined equations (BS hierarchy) associated with hidden string symmetries. The latter enforce the vanishing of most scattering amplitudes for the open N=2 string. 
  It is shown that all generalized two--dimensional dilaton theories with arbitrary matter content (with a curvature independent coupling to gravity) do not only obey a first law of black hole mechanics (which follows from Wald's general considerations, if the entropy S is defined appropriately), but also a second law: \delta S \ge 0 provided only that the null energy condition holds and that, loosely speaking, for late times a stationary state is assumed. Also any two-dimensional f(R)--theory is covered. This generalizes a previous proof of Frolov [1] to a much wider class of theories. 
  The extraction of Wilson loops of confining gauge systems from their SUGRA (string) duals is reviewed. I start with describing the basic classical setup. A theorem that determines the classical values of the loops associated with a generalized background is derived. In particular sufficient conditions for confining behavior are stated . I then introduce quadratic quantum fluctuations around the classical configurations. I discuss in details the following models of confining behavior: (i) Strings in flat space-time, (ii) $AdS_5$ black hole and its correspondence with pure YM theory in three dimensions. In particular an attractive Luscher term is shown to be the outcome of the quantum fluctuations. (iii) Type 0 string model (iv) The Polchinski Strassler $N=1*$ model. In the latter case we show that SUGRA alone is not enough to get the correct nature of the loops, and only by incorporating the worldvolume phenomena of the five branes a coherent qualitative picture can be derived. 
  We obtain the M5-M2-MW bound state solutions of 11-dimensional supergravity corresponding to the 1/2 supersymmetric vacua of the M5-brane equations with constant background fields. In the `near-horizon' case the solution interpolates between the $adS_7\times S^4$ Kaluza-Klein vacuum and D=11 Minkowski spacetime via a Domain Wall spacetime. We discuss implications for renormalization group flow of (2,0) D=6 field theories. 
  The study of open string tachyon condensation in string field theory can be drastically simplified by making an appropriate choice of coordinates on the space of string fields. We show that a very natural coordinate system is suggested by the connection between the worldsheet renormalization group and spacetime physics. In this system only one field, the tachyon, condenses while all other fields have vanishing expectation values. These coordinates are also well-suited to the study of D-branes as solitons. We use them to show that the tension of the D25-brane is cancelled by tachyon condensation and compute exactly the profiles and tensions of lower dimensional D-branes. 
  We discuss dynamics of N=2 supersymmetric SU(n_c) gauge theories with n_f quark hypermultiplets. Upon N=1 perturbation of introducing a finite mass for the adjoint chiral multiplet, we show that the flavor U(n_f) symmetry is dynamically broken to U(r) \times U(n_f-r), where r\leq [n_f/2] is an integer. This flavor symmetry breaking occurs due to the condensates of magnetic degrees of freedom which acquire flavor quantum numbers due to the quark zero modes. We briefly comment on the USp(2n_c) gauge theories. This talk is based on works with Giuseppe Carlino and Ken Konishi, hep-th/0001036 and hep-th/0005076. 
  We relate the moduli space of Yang-Mills instantons to quaternionic manifolds. For instanton number one, the Wolf spaces play an important role. We apply these ideas to instanton calculations in N=4 SYM theory. 
  Using a world-sheet covariant formalism, we derive the equations of motion for second order perturbations of a generic macroscopic string, thus generalizing previous results for first order perturbations. We give the explicit results for the first and second order perturbations of a contracting near-circular string; these results are relevant for the understanding of the possible outcome when a cosmic string contracts under its own tension, as discussed in a series of papers by Vilenkin and Garriga. In particular, second order perturbations are necessaary for a consistent computation of the energy.   We also quantize the perturbations and derive the mass-formula up to second order in perturbations for an observer using world-sheet time $\tau $. The high frequency modes give the standard Minkowski result while, interestingly enough, the Hamiltonian turns out to be non-diagonal in oscillators for low-frequency modes. Using an alternative definition of the vacuum, it is possible to diagonalize the Hamiltonian, and the standard string mass-spectrum appears for all frequencies. We finally discuss how our results are also relevant for the problems concerning string-spreading near a black hole horizon, as originally discussed by Susskind. 
  We show that the QCD vacuum (without dynamical quarks) is a dual superconductor at least in the low-energy region in the sense that monopole condensation does really occur. In fact, we derive the dual Ginzburg-Landau theory (i.e., dual Abelian Higgs model) directly from the SU(2) Yang-Mills theory by adopting the maximal Abelian gauge. The dual superconductor can be on the border between type II and type I, excluding the London limit. The masses of the dual Abelian gauge field is expressed by the Yang-Mills gauge coupling constant and the mass of the off-diagonal gluon of the original Yang-Mills theory. Moreover, we can rewrite the Yang-Mills theory into an theory written in terms of the Abelian magnetic monopole alone at least in the low-energy region. Magnetic monopole condensation originates in the non-zero mass of off-diagonal gluons. Finally, we derive the confining string theory describing the low-energy Gluodynamics. Then the area law of the large Wilson loop is an immediate consequence of these constructions. Three low-energy effective theories give the same string tension. 
  A deformation of Einstein Gravity is constructed based on gauging the noncommutative ISO(3,1) group using the Seiberg-Witten map. The transformation of the star product under diffeomorphism is given, and the action is determined to second order in the deformation parameter. 
  We show that the functional bosonization procedure can be generalized in such a way that, to any field theory with a conserved Abelian charge in (2+1) dimensions, there corresponds a dual Abelian gauge field theory. The properties of this mapping and of the dual theory are discussed in detail, presenting different explicit examples. In particular, the meaning and effect of the coefficient of the Chern-Simons term in the dual action is interpreted in terms of the spin and statistics connection. 
  In this talk we will discuss the low energy dynamics of non-BPS branes constructed as stable brane/anti-brane pairs at an orbifold. In particular we will determine the effective field theory and compare its predictions with those of the full open string theory. While the position and vector degrees of freedom have the familiar form found in supersymmetric gauge theories, the massless modes orginating in the tachyonic sector display novel non-commuting flat directions. We will show that these flat directions persist to al orders in alpha'. Finally we will briefly report on the open string loop corrections. 
  We derive and solve a subset of the fluctuation equations about two domain wall solutions of D=5, N=8 gauged supergravity. One solution is dual to D=4, N=4 SYM theory perturbed by an N=1, SO(3)-invariant mass term and the other to a Coulomb branch deformation. In the first case we study all SO(3)-singlet fields. These are assembled into bulk multiplets dual to the stress tensor multiplet and to the N=1 chiral multiplets Tr Phi^2 and Tr W^2, the former playing the role of anomaly multiplet. Each of these three multiplets has a distinct spectrum of "glueball" states. This behavior is contrasted with the Coulomb branch flow in which all fluctuations studied have a continuous spectrum above a common mass gap, and spontaneous breaking of conformal symmetry is driven by a bulk vector multiplet. R-symmetry is preserved in the field theory, and correspondingly the bulk vector is dual to a linear anomaly multiplet. Generic features of the fluctuation equations and solutions are emphasized. For example, the transverse traceless modes of all fields in the graviton multiplet can be expressed in terms of an auxiliary massless scalar, and gauge fields associated with R-symmetry have a universal effective mass. 
  We have studied the theory of gauged chiral bosons and proposed a general theory, a master action, that encompasses different kinds of gauge field couplings in chiral bosonized theories with first-class chiral constraints. We have fused opposite chiral aspects of this master action using the soldering formalism and applied the final action to several well known models. The Lorentz rotation permitted us to fix conditions on the parameters of this general theory in order to preserve the relativistic invariance. We also have established some conditions on the arbitrary parameter concerned in a chiral Schwinger model with a generalized constraint, investigating both covariance and Lorentz invariance. The results obtained supplements the one that shows the soldering formalism as a new method of mass generation. 
  The Hamiltonian (gauge) symmetry generators of non-local (gauge) theories are presented. The construction is based on the d+1 dimensional space-time formulation of d dimensional non-local theories. The procedure is applied to U(1) space-time non-commutative gauge theory. In the Hamiltonian formalism the Hamiltonian and the gauge generator are constructed. The nilpotent BRST charge is also obtained. The Seiberg-Witten map between non-commutative and commutative theories is described by a canonical transformation in the superphase space and in the field-antifield space. The solutions of classical master equations for non-commutative and commutative theories are related by a canonical transformation in the antibracket sense. 
  We consider the holographic dual of a general class of N=1* flows in which all three chiral multiplets have independent masses, and in which the corresponding Yang-Mills scalars can develop particular supersymmetry-preserving vevs. We also allow the gaugino to develop a vev. This leads to a six parameter subspace of the supergravity scalar action, and we show that this is a consistent truncation, and obtain a superpotential that governs the N=1* flows on this subspace. We analyse some of the structure of the superpotential, and check consistency with the asymptotic behaviour near the UV fixed point. We show that the dimensions of the six couplings obey a sum rule all along the N=1* flows. We also show how our superpotential describes part of the Coulomb branch of the non-trivial N=1 fixed point theory. 
  The inflationary scenario for the brane world driven by the bulk inflaton is proposed. The quantum fluctuations of the inflaton is calculated and compared to those of the conventional 4-dimensional inflationary scenario. It is shown that the deviation of the primordial spectrum of this model from that of the conventional one is too small to be observed even if $AdS$ radius is very large. Hence, it turns out that the inflation caused by the bulk inflaton is viable in the context of brane world cosmology. 
  In the previous paper, we generalized the method of Abelian decomposition to the case of SO(N) Yang-Mills theory. This method that was proposed by Faddeev and Niemi introduces a set of variables for describing the infrared limit of a Yang-Mills theory. Here, we extend the decomposition method further to the general case of four-dimensional Sp(2N) Yang-Mills theory. We find that the Sp(2N) connection decomposes according to irreducible representations of SO(N). 
  Various deformations of the position-momentum algebras operators have been proposed. Their implications for single systems like the hydrogen atom or the harmonic oscillator have been addressed. In this paper we investigate the consequences of some of these algebras for macroscopic systems. The key point of our analysis lies in the fact that the modification of the Heisenberg uncertainty relations present in these theories changes the volume of the elementary cell in the hamiltonian phase space and so the measure needed to compute partition functions.   The thermodynamics of a non interacting gas are studied for two members of the Kempf-Mangano-Mann (K.M.M.) deformations. It is shown that the theory which exhibits a minimal uncertainty in length predicts a new behavior at high temperature while the one with a minimal uncertainty in momentum displays unusual features for huge volumes. In the second model negative pressures are obtained and mixing two different gases does not necessarily increase the entropy . This suggests a possible violation of the second law of thermodynamics. Potential consequences of these models in the evolution of the early universe are briefly discussed.   Constructing the Einstein model of a solid for the q deformed oscillator, we find that the subset of eigenstates whose energies are bounded from above leads to a divergent partition function. 
  Following the formalism of enveloping algebras and star product calculus we formulate and analyze a model of gauge gravity on noncommutative spaces and examine the conditions of its equivalence to general relativity. The corresponding Seiberg-Witten maps are established which allow the definition of respective dynamics for a finite number of gravitational gauge field components on noncommutative spaces. 
  One of explanations of the black hole entropy implies that gravity is entirely induced by quantum effects. By using arguments based on the AdS/CFT correspondence we give induced gravity interpretation of the gravity in a brane world in higher dimensional anti-de Sitter (AdS) space. The underlying quantum theory is SU(N) theory where $N$ is related to the CFT central charge. The theory includes massless fields which correspond to degrees of freedom of the boundary CFT. In addition, on the brane there are massive degrees of freedom with masses proportional to $l^{-1}$ where $l$ is the radius of AdS. At the conformal boundary of AdS they are infinitely heavy and completely decouple. It is the massive fields which can explain the black hole entropy. We support our interpretation by a microscopic model of a 2D brane world in $AdS_3$. 
  A "Master" gauge theory is constructed in 2+1-dimensions through which various gauge invariant and gauge non-invariant theories can be studied. In particular, Maxwell-Chern-Simons, Maxwell-Proca and Maxwell-Chern-Simons -Proca models are considered here. The Master theory in an enlarged phase space is constructed both in Lagrangian (Stuckelberg) and Hamiltonian (Batalin-Tyutin) frameworks, the latter being the more general one, which includes the former as a special case. Subsequently, BRST quantization of the latter is performed. Lastly, the master Lagrangian, constructed by Deser and Jackiw (Phys. Lett. B139, (1984) 371), to show the equivalence between the Maxwell-Chern-Simons and the self-dual model, is also reproduced from our Batalin-Tyutin extended model. Symplectic quantization procedure for constraint systems is adopted in the last demonstration. 
  We construct the manifestly Lorenz-invariant formulation of the N=1 D=4 massive superparticle with tensorial central charges. The model contains a real parameter k and at $k\ne 0$ possesses one $\kappa$-symmetry while at k=0 the number of $\kappa$-symmetry is two. The equivalence of the formulations at all $k\ne 0$ is obtained. The local transformations of $\kappa$-symmetry are written out. It is considered the using of index spinor for construction of the tensorial central charges. It is obtained the equivalence at classical level between the massive D=4 superparticle with one $\kappa$-symmetry and the massive D=4 spinning particle 
  We present five dimensional gauged supergravity with universal hypermultiplet on M_4 \times S^1 / Z_2 coupled supersymmetrically to three-branes located at the fixed points. The construction is extended to the smooth picture with auxiliary singlet and four-form fields. The model admits the Randall-Sundrum solution as a BPS vacuum with vanishing energy. We give the form of all KK-tower modes for fields present in the model. 
  A quantizable action has recently been proposed for the superstring in an AdS_5 x S^5 background with Ramond-Ramond flux. In this paper, we construct physical vertex operators corresponding to on-shell fluctuations around the AdS_5 x S^5 background. The structure of these AdS_5 x S^5 vertex operators closely resembles the structure of the massless vertex operators in a flat background. 
  The requirement that the laws of physics must be invariant under point-dependent transformations of the units of length, time, and mass is used as a selection principle while studying different generic effective theories of gravity. Thereof theories with non-minimal coupling of the dilaton both to the curvature and to the Lagrangian of the matter fields seem to represent the most viable low-energy [and low-curvature] description of gravity. Consequently, the cosmological singularity problem is treated within the context of string cosmology with non-minimal coupling of the dilaton to a barotropic gas of solitonic p-brane. The results obtained are to be interpreted on the grounds of Weyl-integrable geometry. The implications of these results for the Mach's principle are briefly discussed. 
  We develop a general framework for effective equations of expectation values in quantum cosmology and pose for them the quantum Cauchy problem with no-boundary and tunneling wavefunctions. We apply this framework in the model with a big negative non-minimal coupling, which incorporates a recently proposed low energy (GUT scale) mechanism of the quantum origin of the inflationary Universe and study the effects of the quantum inflaton mode. 
  We consider fixing the bosonic light-cone gauge for string in AdS in the phase space framework, i.e. by choosing $x^+ = \tau$, and by choosing $\sigma$ so that $P^+$ is distributed uniformly (its density is independent of $\sigma$). We discuss classical bosonic string in AdS space and superstring in AdS_5 x S^5. In the latter case the starting point is the action found in hep-th/0007036 where the kappa-symmetry is fixed by a fermionic light cone gauge. We derive the light cone Hamiltonian in the AdS_5 x S^5 case and in the case of superstring in AdS_3 x S^3. We also obtain a realization of the generators of the basic symmetry superalgebra psu(2,2|4) in terms of the AdS_5 x S^5 superstring coordinate fields. 
  Static solutions of the higher dimensional Einstein-Hilbert gravity supplemented by quadratic curvature self-interactions are discussed in the presence of hedgehog configurations along the transverse dimensions. The quadratic part of the action is parametrized in terms of the (ghost-free) Euler-Gauss-Bonnet curvature invariant. Spherically symmetric profiles of the transverse metric admit exponentially decaying warp factors both for positive and negative bulk cosmological constants. 
  Zero-point fluctuations in quantum fields give rise to observable forces between material bodies, the so-called Casimir forces. In this lecture I present some results of the theory of the Casimir effect, primarily formulated in terms of Green's functions. There is an intimate relation between the Casimir effect and van der Waals forces. Applications to conductors and dielectric bodies of various shapes will be given for the cases of scalar, electromagnetic, and fermionic fields. The dimensional dependence of the effect will be described. Finally, we ask the question: Is there a connection between the Casimir effect and the phenomenon of sonoluminescence? 
  We show that the recently proposed formulation of noncommutative N=2 Super Yang-Mills theory implies that the commutative and noncommutative effective coupling constants \tau(u) and \tau_{nc}(u) coincide. We then introduce a key relation which allows to find a nontrivial solution of such equation, thus fixing the form of the low-energy effective action. The dependence on the noncommutative parameter arises from a rational function deforming the Seiberg-Witten differential. 
  We consider the motion of the brane universe moving in a background bulk space of tachyonic and non-tachyonic type 0B string theory. The effective densities are calculated for both cases and they show different power law behavior. Brane inflation for non-tachyonic type 0B background has the same power law behavior as that for type IIB background. The brane inflation under tachyonic background is less divergent than the one without tachyon. The role of tachyonic field in brane inflation scenario is different from that of the ordinary matter field. 
  We investigate aspects of the four-dimensional effective description of brane world scenarios based on warped compactification on anti-de Sitter space. The low-energy dynamics is described by visible matter gravitationally coupled to a ``dark'' conformal field theory. We give the linearized description of the 4d stress tensor corresponding to an arbitrary 5d matter distribution. In particular a 5d falling particle corresponds to a 4d expanding shell, giving a 4d interpretation of a trajectory that misses a black hole only by moving in the fifth dimension. Breakdown of the effective description occurs when either five-dimensional physics or strong gravity becomes important. In scenarios with a TeV brane, the latter can happen through production of black holes near the TeV scale. This could provide an interesting experimental window on quantum black hole dynamics. 
  We study the self-interaction effects for the Dirac particle moving in an external field created by static charges in (1+1)-dimensions. Assuming that the total electric charge of the system vanishes, we show that the asymptotically linearly rising part of the external potential responsible for nonexistence of bound states in the external field problem without self-interaction is cancelled by the self-potential of the zero mode of the Dirac particle charge density. We derive the Dirac equation which includes the self-potential of the non-zero modes and is nonlinear. We solve the spectrum problem in the case of two external positive charges of the same value and prove that the Dirac particle and external charges are confined in a stable system. 
  We propose that for non-localizable energy distribution the relevant energy condition is determined by the gravitational field energy which is negative for positive non-gravitational energy. That is negativity of the non-localized energy is the "positive" energy condition. This would have direct application and relevance for a black hole on the brane which would be sitting in a trace free stresses induced by the Weyl curvature of the bulk. 
  We study the presence of symmetry transformations in the Faddeev-Jackiw approach for constrained systems. Our analysis is based in the case of a particle submitted to a particular potential which depends on an arbitrary function. The method is implemented in a natural way and symmetry generators are identified. These symmetries permit us to obtain the absent elements of the sympletic matrix which complement the set of Dirac brackets of such a theory. The study developed here is applied in two different dual models. First, we discuss the case of a two-dimensional oscillator interacting with an electromagnetic potential described by a Chern-Simons term and second the Schwarz-Sen gauge theory, in order to obtain the complete set of non-null Dirac brackets and the correspondent Maxwell electromagnetic theory limit. 
  We observe that the main feature of the Randall-Sundrum model, used to solve the hierarchy problem, is already present in a class of Yang-Mills plus gravity theories inspired by noncommutative geometry. Strikingly the same expression for the Higgs potential is found in two models which have no apparent connection. Some speculations concerning the possible relationships are given. 
  We construct a Galilean invariant non-gravitational closed string theory whose excitations satisfy a non-relativistic dispersion relation. This theory can be obtained by taking a consistent low energy limit of any of the conventional string theories, including the heterotic string. We give a finite first order worldsheet Hamiltonian for this theory and show that this string theory has a sensible perturbative expansion, interesting high energy behavior of scattering amplitudes and a Hagedorn transition of the thermal ensemble. The strong coupling duals of the Galilean superstring theories are considered and are shown to be described by an eleven-dimensional Galilean invariant theory of light membrane fluctuations. A new class of Galilean invariant non-gravitational theories of light-brane excitations are obtained. We exhibit dual formulations of the strong coupling limits of these Galilean invariant theories and show that they exhibit many of the conventional dualities of M theory in a non-relativistic setting. 
  We examine the T-duality relation between 1+1 NCOS and the DLCQ limit of type IIA string theory. We show that, as long as there is a compact dimension, one can meaningfully define an `NCOS' limit of IIB/A string theory even in the absence of D-branes (and even if there is no B-field). This yields a theory of closed strings with strictly positive winding, which is T-dual to DLCQ IIA/B without any D-branes. We call this the Type IIB/A Wound String Theory. The existence of decoupled sectors can be seen directly from the energy spectrum, and mirrors that of the DLCQ theory. It becomes clear then that all of the different p+1 NCOS theories are simply different states of this single Wound IIA/B theory which contain D-branes. We study some of the properties of this theory. In particular, we show that upon toroidal compactification, Wound string theory is U-dual to various Wrapped Brane theories which contain OM theory and the ODp theories as special states. 
  We present a systematic study of a new type of consistent ``Brane-world Kaluza-Klein Reduction,'' which describe fully non-linear deformations of co-dimension one objects that arise as solutions of a large class of gauged supergravity theories in diverse dimensions, and whose world-volume theories are described by ungauged supergravities with one half of the original supersymmetry. In addition, we provide oxidations of these Ansatze which are in general related to sphere compactified higher dimensional string theory or M-theory. Within each class we also provide explicit solutions of brane configurations localised on the world-brane. We show that at the Cauchy horizon (in the transverse dimension of the consistently Kaluza-Klein reduced world-brane) there is a curvature singularity for any configuration with a non-null Riemann curvature or a non-vanishing Ricci scalar that lives in the world-brane. Since the massive Kaluza-Klein modes can be consistently decoupled, they cannot participate in regulating these singularities. 
  It has recently been shown that the Randall-Sundrum brane-world may be obtained from an appropriate doubled D3-brane configuration in type IIB theory. This corresponds, in five dimensions, to a sphere compactification of the original IIB theory with a non-trivial breathing mode supporting the brane. In this paper, we shall study the supersymmetry of this reduction to massive five-dimensional supergravity, and derive the effective supersymmetry transformations for the fermionic superpartners to the breathing mode. We also consider the sphere compactifications of eleven-dimensional supergravity to both four and seven dimensions. For the compactifications on S^5 and S^7, we include a squashing mode scalar and discuss the truncation from N=8 to N=2 supersymmetry. 
  We show that two-dimensional (2D) AdS gravity induces on the spacetime boundary a conformally invariant dynamics that can be described in terms of a de Alfaro-Fubini-Furlan model coupled to an external source with conformal dimension two. The external source encodes the information about the gauge symmetries of the 2D gravity system. Alternatively, there exists a description in terms of a mechanical system with anholonomic constraints. The considered systems are invariant under the action of the conformal group generated by a Virasoro algebra, which occurs also as asymptotic symmetry algebra of two-dimensional anti-de Sitter space. We calculate the central charge of the algebra and find perfect agreement between statistical and thermodynamical entropy of AdS_2 black holes. 
  In this letter we analyze the Hamiltonian formulation of the Jackiw-Teitelboim model of 2D gravity and calculate the central charge associated with the asymptotic symmetries, taking care of boundary terms. For black hole solutions, we show that there is no sqrt{2} discrepancy between the thermodynamical entropy and the statistical one obtained via Cardy's formula. 
  The holographic correspondence between 2d, N=2 quantum field theories and classical 4d, N=2 supergravity coupled to hypermultiplet matter is proposed. The geometrical constraints on the target space of the 4d, N=2 non-linear sigma-models in N=2 supergravity background are interpreted as the exact renormalization group flow equations in two dimensions. Our geometrical description of the renormalization flow is manifestly covariant under general reparametrization of the 2d coupling constants. An explicit exact solution to the 2d renormalization flow, based on its dual holographic description in terms of the Zamolodchikov metric, is considered in the particular case of the four-dimensional NLSM target space described by the SU(2)-invariant (Weyl) anti-self-dual Einstein metrics. The exact regular (Tod-Hitchin) solutions to these metrics are governed by the Painlev'e VI equation, and describe domain walls. 
  We explicitly show that the new polynomial invariants for knots, upto nine crossings, agree with the Ooguri-Vafa conjecture relating Chern-Simons gauge theory to topological string theory on the resolution of the conifold. 
  In this paper we would like to show that from N non-BPS D0-branes in type IIB theory we can obtain all BPS and non-BPS D-branes through the tachyon condensation in the large N limit. 
  The Dirac quantization is performed for the constrained system of the open string with different charges located at both ends in the constant background B field. Noncommutativity reveals to commutators [X, X], [P, P] and also [X,P] at both ends of the string. We consider a dependence on the change of the "cyclotron frequency" of the charged string. The g-factor of the charged string is also calculated in the framework of our formulation. 
  It has been shown recently that the background independent open string field theory provides an exact description of the tachyon condensation on unstable D-branes of bosonic string theory. In this analysis the overall normalisation of the action was chosen so that it reproduces the conjectured relations involving tachyon condensation. In this paper we fix this normalisation by comparing the on-shell three tachyon amplitude computed from the background independent open string field theory with the same amplitude computed from the cubic open string field theory, which in turn agrees with the result of the first quantised theory. We find that this normalisation factor is in precise agreement with the one required for verifying the conjectured properties of the tachyon potential. 
  We use the method of Banerjee, Banerjee and Mitra and minimal homotopy paths to compute the consistent gauge anomaly for several superspace models of SSYM coupled to matter. We review the derivation of the anomaly for N=1 in four dimensions and then discuss the anomaly for two-dimensional models with (2,0) supersymmetry. 
  We derive the vertex operators that are expected to govern the emission of the massless d=11 supermultiplet from the supermembrane in the light cone gauge. Our results immediately imply the linear coupling of matrix theory to an arbitrary supergravity background to all orders in anticommuting coordinates. Finally we address the definition of n-point tree level and one-loop scattering amplitudes. The resulting 3-point tree level amplitudes turn out to agree with d=11 supergravity and are completely fixed by supersymmetry and the existence of a normalizable ground state. 
  This work consist of two interrelated parts. First, we derive massive gauge-invariant generalizations of geometric actions on coadjoint orbits of arbitrary (infinite-dimensional) groups $G$ with central extensions, with gauge group $H$ being certain (infinite-dimensional) subgroup of $G$. We show that there exist generalized ``zero-curvature'' representation of the pertinent equations of motion on the coadjoint orbit. Second, in the special case of $G$ being Kac-Moody group the equations of motion of the underlying gauged WZNW geometric action are identified as additional-symmetry flows of generalized Drinfeld-Sokolov integrable hierarchies based on the loop algebra ${\hat \cG}$. For ${\hat \cG} = {\hat {SL}}(M+R)$ the latter hiearchies are equivalent to a class of constrained (reduced) KP hierarchies called ${\sl cKP}_{R,M}$, which contain as special cases a series of well-known integrable systems (mKdV, AKNS, Fordy-Kulish, Yajima-Oikawa etc.). We describe in some detail the loop algebras of additional (non-isospectral) symmetries of ${\sl cKP}_{R,M}$ hierarchies. Apart from gauged WZNW models, certain higher-dimensional nonlinear systems such as Davey-Stewartson and $N$-wave resonant systems are also identified as additional symmetry flows of ${\sl cKP}_{R,M}$ hierarchies. Along the way we exhibit explicitly the interrelation between the Sato pseudo-differential operator formulation and the algebraic (generalized) Drinfeld-Sokolov formulation of ${\sl cKP}_{R,M}$ hierarchies. Also we present the explicit derivation of the general Darboux-B\"acklund solutions of ${\sl cKP}_{R,M}$ preserving their additional (non-isospectral) symmetries, which for R=1 contain among themselves solutions to the gauged $SL(M+1)/U(1)\times SL(M)$ WZNW field equations. 
  We construct monopole solutions in SU(2) Einstein-Yang-Mills-Higgs theory, carrying magnetic charge n. For vanishing and small Higgs selfcoupling, these multimonopole solutions are gravitationally bound. Their mass per unit charge is lower than the mass of the n=1 monopole. For large Higgs selfcoupling only a repulsive phase exists. 
  We consider noncommutative {\cal N}=4 supersymmetric U(N) Yang-Mills theory. Using the {\cal N}=1 superfield formalism and the background field method we compute one-loop four point contributions to the effective action and compare the result with the field theory limit from open string amplitudes in the presence of a constant B-field. 
  In the very early Universe matter can be described as a conformal invariant ultra-relativistic perfect fluid, which does not contribute, on classical level, to the evolution of the isotropic and homogeneous metric. However, in this situation the vacuum effects of quantum matter fields become important. The vacuum effective action depends, essentially, on the particle content of the underlying gauge model. If we suppose that there is some desert in the particle spectrum, just below the Planck mass, then the effect of conformal trace anomaly is dominating at the corresponding energies. With some additional constraints on the gauge model (which favor extended or supersymmetric versions of the Standard Model rather than the minimal one), one arrives at the stable inflation. In this article we report about the calculation of the gravitational waves in this model. The result for the perturbation spectrum is close to the one for the conventional inflaton model, and is in agreement with the existing Cobe data. 
  We discuss various aspect of the holographic correspondence between 5-d gravity and 4-d field theory. First of all, we describe deformations of N=4 Super Yang-Mills (SYM) theories in terms of 5-d gauged supergravity. In particular, we describe N=0 and N=1 deformations of N=4 SYM to confining theories. Secondly, we describe recent proposals for the holographic dual of the renormalization group and for 4-d central charges associated to it. We conclude with a ``holographic'' proof of the Goldstone theorem. 
  We consider an infinite-volume brane world setup where a codimension one brane is coupled to bulk gravity plus a scalar field with vanishing potential. The latter is protected by bulk supersymmetry, which is intact even if brane supersymmetry is completely broken as the volume of the extra dimension is infinite. Within this setup we discuss a flat solution with a ``self-tuning'' property, that is, such a solution exists for a continuous range of values for the brane tension. This infinite-volume solution is free of any singularities, and has the property that the brane cosmological constant is protected by bulk supersymmetry. We, however, also point out that consistency of the coupling between bulk gravity and brane matter generically appears to require that the brane world-volume theory be conformal. 
  From a path integral point of view (e.g. \cite{Q98}) physicists have shown how {\it duality} in antisymmetric quantum field theories on a closed space-time manifold $M$ relies in a fundamental way on Fourier Transformations of formal infinite-dimensional volume measures. We first review these facts from a measure theoretical point of view, setting the importance of the Hodge decomposition theorem in the underlying geometric picture, ignoring the local symmetry which lead to degeneracies of the action. To handle these degeneracies we then apply Schwarz's Ansatz showing how duality leads to a factorization of the analytic torsion of $M$ in terms of the partition functions associated to degenerate "dual" actions, which in the even dimensional case corresponds to the identification of these partition functions. 
  A concise geometrical formulation of N=4 supergravity containing an antisymmetric tensor gauge field is given in central charge superspace: graviphotons are identified in the super-vielbein on the same footing as the vierbein and the Rarita-Schwinger fields. As a consequence of superspace soldering, Chern-Simons terms in the fieldstrength of the antisymmetric tensor arise as an intrinsic property of superspace with central charge coordinates. 
  We review (mainly) quantum effects in the theories where gravity sector is described by metric and dilaton. The one-loop effective action for dilatonic gravity in two and four dimensions is evaluated. Renormalization group equations are constructed. The conformal anomaly and induced effective action for 2d and 4d dilaton coupled theories are found. It is applied to study of quantum aspects of black hole thermodynamics, like calculation of Hawking radiation and quantum corrections to black hole parameters and investigation of quantum instability for such objects with multiple horizons. The use of above effective action in the construction of non-singular cosmological models in Einstein or Brans-Dicke (super)gravity and investigation of induced wormholes in supersymmetric Yang-Mills theory are given.   5d dilatonic gravity (bosonic sector of compactified IIB supergravity) is discussed in connection with bulk/boundary (or AdS/CFT) correspondence. Running gauge coupling and quark-antiquark potential for boundary gauge theory at zero or non-zero temperature are calculated from d=5 dilatonic Anti-de Sitter-like background solution which represents Anti-de Sitter black hole for periodic time. 
  We review the geometric definition of C-function in the context of field theories that admit a holographic gravity dual. 
  The one-loop vacuum polarization tensor is computed in QED with an external constant, homogeneous magnetic field at finite temperature. The Schwinger proper-time formalism is used and the computations are done in Euclidian space. The well-known results are recovered when the temperature and/or the magnetic field are switched off and the effect of the magnetic field on the Debye screening is discussed. 
  Abelian and center gauges are considered in continuum Yang-Mills theory in order to detect the magnetic monopole and center vortex content of gauge field configurations. Specifically we examine the Laplacian Abelian and center gauges, which are free of Gribov copies, as well as the center gauge analog of the (Abelian) Polyakov gauge. In particular, we study meron, instanton and instanton-anti-instanton field configurations in these gauges and determine their monopole and vortex content. While a single instanton does not give rise to a center vortex, we find center vortices for merons. Furthermore we provide evidence, that merons can be interpreted as intersection points of center vortices. For the instanton-anti-instanton pair, we find a center vortex enclosing their centers, which carries two monopole loops. 
  The behaviour of the chiral condensates in the SU(2) gauge theory with broken N=2 supersymmetry is reviewed. The calculation of monopole, dyon, and charge condensates is described. It is shown that the monopole and charge condensates vanish at the Argyres-Douglas point where the monopole and charge vacua collide. This phenomenon is interpreted as a deconfinement of electric and magnetic charges at the Argyres-Douglas point. 
  Fate of branes in external fields is reviewed. Spontaneous creation of the Brane World in $AdS_{5}$ with external field is described. The resulting Brane World consists of a flat 4d spatially finite expanding Universe and curved expanding "regulator" branes. All branes have a positive tension. 
  I reply to the comment by Dr S. Nishigaki (hep-th/0007042) to my papers Phys. Rev. D61 (2000) 056005 and Phys. Rev. D62 (2000) 016005. 
  We discuss aspects of the algebraic geometry of compact non-commutative Calabi-Yau manifolds. In this setting, it is appropriate to consider local holomorphic algebras which can be glued together into a compact Calabi-Yau algebra. We consider two examples: a toroidal orbifold T^6/Z_2 x Z_2, and an orbifold of the quintic in CP_4, each with discrete torsion. The non-commutative geometry tools are enough to describe various properties of the orbifolds. First, one describes correctly the fractionation of branes at singularities. Secondly, for the first example we show that one can recover explicitly a large slice of the moduli space of complex structures which deform the orbifold. For this example we also show that we get the correct counting of complex structure deformations at the orbifold point by using traces of non-commutative differential forms (cyclic homology). 
  A previous calculation of the phase transition in the Wilson loop correlator in the zero temperature AdS/CFT correspondence is extended to the case where the loops are concentric circles of unequal radii. This phase transition occurs due to the instability of the classical string stretched between the loops. We compute the string action and its expansion in the distance h between the loops for small h. We also find that the connected minimal surface is subleading or does not even exist when h=0 and the radii are considerably different. This feature has no analogue in flat space. 
  We consider warped type IIB supergravity solutions with three-form flux and ${\cal N}=1$ supersymmetry, which arise as the supergravity duals of confining gauge theories. We first work in a perturbation expansion around $AdS_5 \times S^5$, as in the work of Polchinski and Strassler, and from the ${\cal N}=1$ conditions and the Bianchi identities recover their first-order solution generalized to an arbitrary ${\cal N}=1$ superpotential. We find the second order dilaton and axion by the same means. We also find a simple family of exact solutions, which can be obtained from solutions found by Becker and Becker, and which includes the recent Klebanov--Strassler solution. 
  A supersymmetric Randall-Sundrum brane-world demands that not merely the graviton but the entire supergravity multiplet be trapped on the brane. To demonstrate this, we present a complete ansatz for the reduction of (D=5,N=4) gauged supergravity to (D=4,N=2) ungauged supergravity in the Randall-Sundrum geometry. We verify that it is consistent to lowest order in fermion terms. In particular, we show how the graviphotons avoid the `no photons on the brane' result because they do not originate from Maxwell's equations in D=5 but rather from odd-dimensional self-duality equations. In the case of the fivebrane, the Randall-Sundrum mechanism also provides a new Kaluza-Klein way of obtaining chiral supergravity starting from non-chiral. 
  It has been conjectured that a pair of D5 - anti D5 branes wrapped on some non-trivial two cycle of a Calabi-Yau manifold becomes a stable BPS D3 brane in the presence of a very large B field and magnetic fluxes on their worldvolumes. We discuss this by considering the non-commutative field theory on the worldvolume of the pair of branes whose field multiplication is made with respect to two different * products due to the presence of different F fields on the two branes. The tachyonic field becomes massless for a specific choice of the magnetic fluxes and it allows a trivial solution. Our discussion generalizes recent results concerning stability of brane-antibrane systems on Calabi-Yau spaces to the case of non-commutative branes. 
  In this paper we investigate the tachyon instability of open bosonic string theory applying methods of boundary conformal field theory. We consider compactifications on maximal tori of various simple Lie algebras with a specific background coupled to the string boundaries. The resulting world-sheet CFT is a free theory perturbed by a boundary term that is marginal but not truly marginal. Assuming that the theory flows to a nontrivial infrared fixed point that is similar to the one in Kondo model, we calculate the new spectrum and some of the Green's functions. We find that in some of the sectors the tachyon mass gets lifted that can be interpreted as a result of switching on appropriate Wilson lines. Various compactifications and patterns of flows are investigated. 
  We consider the boundary states which describe D-branes in a constant B-field background. We show that the two-form field Phi, which interpolates commutative and noncommutative descriptions of D-branes, can be interpreted as the invariant field strength in the T-dual picture. We also show that the extended algebra parametrized by theta and Phi naturally appears as the commutation relations of the original and the T-dual coordinates. 
  We propose the model of $D-$dimensional massless particle whose Lagrangian is given by the $N-$th extrinsic curvature of world-line. The system has $N+1$ gauge degrees of freedom constituting $W-$like algebra; the classical trajectories of the model are space-like curves which obey the conditions $k_{N+a}=k_{N-a}$, $k_{2N}=0$, $a=1,...,N-1$, $N\leq[(D-2)/2]$, while the first $N$ curvatures $k_i$ remain arbitrary. We show that the model admits consistent formulation on the anti-De Sitter space. The solutions of the system are the massless irreducible representations of Poincar\'e group with $N$ nonzero helicities, which are equal to each other. 
  Representations of four-dimensional superconformal groups on harmonic superfields are discussed. It is argued that any representation can be given as a superfield on many superflag manifolds. Representations on analytic superspaces do not require constraints. We discuss short representations and how to obtain them as explicit products of fundamental fields. We also discuss superfields that transform under supergroups. 
  The motion of spinning relativistic particles in external electromagnetic and gravitational fields is considered. Covariant equations for this motion are demonstrated to possess pathological solutions, when treated nonperturbatively in spin. A self-consistent approach to the problem is formulated, based on the noncovariant description of spin and on the usual, ``naive'' definition of the coordinate of a relativistic particle. A simple description of the gravitational interaction of first order in spin, is pointed out for a relativistic particle. The approach developed allows one to consider effects of higher order in spin. Explicit expression for the second-order Hamiltonian is presented. We discuss the gravimagnetic moment, which is a special spin effect in general relativity. 
  The study of boundary conditions in rational conformal field theories is not only physically important. It also reveals a lot on the structure of the theory ``in the bulk''. The same graphs classify both the torus and the cylinder partition functions and provide data on their hidden ``quantum symmetry''. The Ocneanu triangular cells -- the 3j-symbols of these symmetries, admit various interpretations and make a link between different problems. 
  The problem of consistent definition of the quantum corrected gravitational field is considered in the framework of the $S$-matrix method. Gauge dependence of the one-particle-reducible part of the two-scalar-particle scattering amplitude, with the help of which the potential is usually defined, is investigated at the one-loop approximation. The $1/r^2$-terms in the potential, which are of zero order in the Planck constant $\hbar,$ are shown to be independent of the gauge parameter weighting the gauge condition in the action. However, the $1/r^3$-terms, proportional to $\hbar,$ describing the first proper quantum correction, are proved to be gauge-dependent. With the help of the Slavnov identities, their dependence on the weighting parameter is calculated explicitly. The reason the gauge dependence originates from is briefly discussed. 
  Following a suggestion by Vafa, we present a quantum-mechanical model for S-duality symmetries observed in the quantum theories of fields, strings and branes. Our formalism may be understood as the topological limit of Berezin's metric quantisation of the upper half-plane H, in that the metric dependence has been removed. Being metric-free, our prescription makes no use of global quantum numbers. Quantum numbers arise only locally, after the choice of a local vacuum to expand around. Our approach may be regarded as a manifestly non perturbative formulation of quantum mechanics, in that we take no classical phase space and no Poisson brackets as a starting point. The reparametrisation invariance of H under SL(2,R) induces a natural SL(2,R) action on the quantum mechanical operators that implements S-duality. We also link our approach with the equivalence principle of quantum mechanics recently formulated by Faraggi and Matone. 
  We evaluate the gravitational Schwinger terms for the specific two-dimensional model of Weyl fermions in a gravitational background field using a technique introduced by Kallen and find a relation which connects the Schwinger terms with the linearized gravitational anomalies. 
  We apply the modified triplectic formalism for quantizing several popular gauge models - non-abelian antisymmetric tensor field model, W2-gravity and two-dimensional gravity with dynamical torsion. The explicit solutions are obtained for the generating equations of the quantum action and the gauge-fixing functional. Using these solutions we construct the vacuum functional and obtain the corresponding transformations of the extended BRST symmetry. 
  We analyse the spectrum of non-BPS branes in the type I theory on the orbifold $T^4/{\cal I}_4$. We present a detailed analysis of the action of the worldsheet parity $\Omega$ on the different D-brane boundary state sectors of type IIB on $T^4/{\cal I}_4$, using the covariant formulation. Using these results we derive the spectrum of branes in the type I orbifold. We find $\Z_2$- and $\Z$- charged non-BPS branes. A study of the stability of these branes in the type I orbifold is also presented. We find that the type I non-BPS D-particle and D-instanton remain stable in the orbifold. The D-particle carries no charge whereas the non-BPS D-instanton can carry twisted R-R charge. 
  We derive the worldsheet propagator for an open string with different magnetic fields at the two ends, and use it to compute two distinct noncommutativity parameters, one at each end of the string. The usual scaling limit that leads to noncommutative Yang-Mills can be generalized to a scaling limit in which both noncommutativity parameters enter. This corresponds to expanding a theory with U(N) Chan-Paton factors around a background U(1)^N gauge field with different magnetic fields in each U(1). 
  We discuss light-cone gauge description of type IIB Green-Schwarz superstring in AdS_5 x S^5 with a hope to make progress towards understanding spectrum of this theory. As in flat space, fixing light cone gauge consists of two steps: (i) fixing kappa symmetry in such a way that the fermionic part of the action does not depend on x^-; (ii) fixing 2-d reparametrizations by x^+ = tau and a condition on 2-d metric. In curved AdS space the latter cannot be the standard conformal gauge and breaks manifest 2-d Lorentz invariance. It is natural, therefore, to work in phase-space framework, imposing the GGRT gauge conditions x^+= tau, P^+ =const. We obtain the resulting light cone superstring Hamiltonian. This is a review of hep-th/0007036 and hep-th/0009171. 
  We study monotonicity and other properties of the canonical c-function (defined through the correlator of the energy-momentum tensor) in some holographic duals of 4-d quantum field theories. The canonical c-function and its derivatives are related to the 5-d Green's function of the dual supergravity theory. While positivity of the canonical c-function is obvious, we have not found a general proof of its monotonicity, even though c is monotonic in the few explicit examples we examine in this paper. 
  We study new nonperturbative phenomena in N=1 heterotic string vacua corresponding to pointlike bundle singularities in codimension three. These degenerations result in new four-dimensional infrared physics characterized by light solitonic states whose origin is explained in the dual F-theory model. We also show that such phenomena appear generically in $E_7 \to E_6$ Higgsing and describe in detail the corresponding bundle transition. 
  The Z_k-parafermion Hall state is an incompressible fluid of k-electron clusters generalizing the Pfaffian state of paired electrons. Extending our earlier analysis of the Pfaffian, we introduce two ``parent'' abelian Hall states which reduce to the parafermion state by projecting out some neutral degrees of freedom. The first abelian state is a generalized (331) state which describes clustering of k distinguishable electrons and reproduces the parafermion state upon symmetrization over the electron coordinates. This description yields simple expressions for the quasi-particle wave functions of the parafermion state. The second abelian state is realized by a conformal theory with a (2k-1)-dimensional chiral charge lattice and it reduces to the Z_k-parafermion state via the coset construction su(k)_1+su(k)_1/su(k)_2. The detailed study of this construction provides us a complete account of the excitations of the parafermion Hall state, including the field identifications, the Z_k symmetry and the partition function. 
  There is an interesting dichotomy between a space-time metric considered as external field in a flat background and the same considered as an intrinsic part of the geometry of space-time. We shall describe and compare two other external fields which can be absorbed into an appropriate redefinition of the geometry, this time a noncommutative one. We shall also recall some previous incidences of the same phenomena involving bosonic field theories. It is known that some such theories on the commutative geometry of space-time can be re-expressed as abelian-gauge theory in an appropriate noncommutative geometry. The noncommutative structure can be considered as containing extra modes all of whose dynamics are given by the one abelian action. 
  An equation for the quantum average of the gauge invariant Wilson loop in non-commutative Yang-Mills theory with gauge group U(N) is obtained. In the 't Hooft limit, the equation reduces to the loop equation of ordinary Yang-Mills theory. At finite $N$, the equation involves the quantum averages of the additional gauge invariant observables of the non-commutative theory associated with open contours in space-time. We also derive equations for correlators of several gauge invariant (open or closed) Wilson lines. Finally, we discuss a perturbative check of our results. 
  The only calculations performed beyond one-loop level in the light-cone gauge make use of the Mandelstam-Leibbrandt (ML) prescription in order to circumvent the notorious gauge dependent poles. Recently we have shown that in the context of negative dimensional integration method (NDIM) such prescription can be altogether abandoned, at least in one-loop order calculations. We extend our approach, now studying two-loop integrals pertaining to two-point functions. While previous works on the subject present only divergent parts for the integrals, we show that our prescriptionless method gives the same results for them, besides finite parts for arbitrary exponents of propagators. 
  Anomalies arising from nonplanar triangle diagrams of noncommutative gauge theory are studied. Local chiral gauge anomalies for both noncommutative U(1) and U(N) gauge theories with adjoint matter fields are shown to vanish. For noncommutative QED with fundamental matters, due to UV/IR mixing a finite anomaly emerges from the nonplanar contributions. It involves a generalized $\star$-product of gauge fields. 
  We analyze a broad class of stationary solutions with residual N=1 supersymmetry of four-dimensional N=2 supergravity theories with terms quadratic in the Weyl tensor. These terms are encoded in a holomorphic function, which determines the most relevant part of the action and which plays a central role in our analysis. The solutions include extremal black holes and rotating field configurations, and may have multiple centers. We prove that they are expressed in terms of harmonic functions associated with the electric and magnetic charges carried by the solutions by a proper generalization of the so-called stabilization equations. Electric/magnetic duality is manifest throughout the analysis.   We also prove that spacetimes with unbroken supersymmetry are fully determined by electric and magnetic charges. This result establishes the so-called fixed-point behavior according to which the moduli fields must flow towards certain prescribed values on a fully supersymmetric horizon, but now in a more general context with higher-order curvature interactions. We briefly comment on the implications of our results for the metric on the moduli space of extremal black hole solutions. 
  Some of the consequences that follow from the C_2 condition of Zhu are analysed. In particular it is shown that every conformal field theory satisfying the C_2 condition has only finitely many n-point functions, and this result is used to prove a version of a conjecture of Nahm, namely that every representation of such a conformal field theory is quasirational. We also show that every such vertex operator algebra is a finite W-algebra, and we give a direct proof of the convergence of its characters as well as the finiteness of the fusion rules. 
  We extend the Coleman-Hill analysis to non-Abelian Chern-Simons theories containing a tree level topological mass term. We show, in the case of a pure Yang-Mills-Chern-Simons theory, that there are no corrections to the coefficient of the Chern-Simons term beyond one loop in the axial gauge. Our arguments use constraints coming only from small gauge Ward identities as well as the analyticity of the amplitudes, much like the proof in the Abelian case. Some implications of this result are also discussed. 
  The microscopic spectral correlators of the Dirac operator in three-dimensional Yang-Mills theory coupled to fundamental fermions and with three or more colours are derived from the supersymmetric formulation of partially quenched effective Lagrangians. The flavour supersymmetry breaking patterns are appropriately identified and used to calculate the corresponding finite volume partition functions from Itzykson-Zuber type integrals over supersymmetric cosets. New and simple determinant expressions for the spectral correlators in the mesoscopic scaling region are thereby found. The microscopic spectrum derived from the effective finite volume partition function of three-dimensional QCD agrees with earlier results based on the unitary ensemble of random matrix theory and extends the corresponding calculations for QCD in four dimensions. 
  We present a brief introduction to the Berline-Vergne localization formula which expresses the integral of an equivariant cohomology class as a sum over zeros of a vector field to which that class is related. 
  The quantization of the Brink-Schwarz-Casalbuoni superparticle is performed in an explicitly covariant way using the antibracket formalism. Since an infinite number of ghost fields are required, within a suitable off-shell twistor-like formalism, we are able to fix the gauge of each ghost sector without modifying the physical content of the theory. The computation reveals that the antibracket cohomology contains only the physical degrees of freedom. 
  We construct modular invariant partition functions for type II strings on a conifold and a singular Eguchi-Hanson instanton by means of the SL(2,R)/U(1) version of Gepner models. In the conifold case, we find an extra massless hypermultiplet in the IIB spectrum and argue that it may be identified as a soliton. In the Eguchi-Hanson case, our formula is new and different from the earlier result, in particular does not contain graviton. The lightest IIB fields are combined into a six-dimensional (2,0) tensor multiplet with a negative mass square. We give an interpretation to it as a doubleton-like mode. 
  We show that a new expansion which sums seagull and bubble graphs to all orders, can be applied to the O(N) linear sigma model at finite temperature. We prove that this expansion can be renormalised with the usual counterterms in a mass independent scheme and that Goldstone's theorem is satisfied at each order. At the one loop order of this expansion, the Hartree result for the effective potential (daisy and superdaisy graphs) is recovered. We show that at one loop 2PPI order, the self energy of the sigma meson can be calculated exactly and that diagrams are summed beyond the Hartree approximation. 
  Some properties of N=2 superstrings on AdS_3 x N are studied. Their spectrum has the same pattern as 2-d CFTs in the moduli space of symmetric products M^p/S_p, where p is associated with the number of long fundamental strings required to construct the AdS_3 background, and M is the spacetime CFT corresponding to short string excitations of a single long string. Worldsheet operators associated with w long strings correspond to twisted sectors of M^p/S_p. 
  We study a field theory formulation of a fluid mechanical model. We implement the Hamiltonian formalism by using the BFFT conjecture in order to build a gauge invariant fluid field theory. We also generalize previous known classical dynamical field solutions for the fluid model. 
  We use the partition functions on S^1 x S^n of various conformal field theories in four and six dimensions in the limit of vanishing coupling to study the high temperature thermodynamics. Certain modular properties exhibited by the partition functions help to determine the finite volume corrections, which play a role in the discussion of entropy bounds. 
  The reflection amplitudes in non-affine Toda theories which possess extended conformal symmetry are calculated. Considering affine Toda theories as perturbed non-affine Toda theories and using reflection relations which relate different fields with the same conformal dimension, we deduce the vacuum expectation values of local fields for all dual pairs of non-simply laced affine Toda field theories. As an application, we calculate the leading term in the short and long distance predictions of the two-point correlation functions in the massive phase of two coupled minimal models. The central charge of the unperturbed models ranges from $c=1$ to $c=2$, where the perturbed models correspond to two magnetically coupled Ising models and Heisenberg spin ladders, respectively. 
  A recently proposed \ell=\infty field theory model of tachyon dynamics for unstable bosonic D-branes has been shown to arise as the two-derivative truncation of (boundary)-string field theory. Using an \ell\to \infty limit appropriate to stable kinks we obtain a model for the tachyon dynamics on unstable D-branes or D-brane anti-D-brane pairs of superstring theory. The tachyon potential is a positive definite even function of the tachyon, and at the stable global minima there is no on-shell dynamics. The kink solution mimics nicely the properties of stable D-branes: the spectrum of the kink consists of infinite levels starting at zero mass, with spacing double the value of the tachyon mass-squared. It is natural to expect that this model will arise in (boundary) superstring field theory. 
  In these lectures, two different aspects of brane world scenarios in 5d gravity or string theory are discussed. In the first two lectures, work on how warped compactifications of 5d gravity theories can change the guise of the hierarchy problem and the cosmological constant problem is reviewed, and a discussion of several issues which remain unclear in this context is provided. In the next two lectures, microscopic constructions in string theory which involve D-branes wrapped on cycles of Calabi-Yau manifolds are described. The focus is on computing the superpotential in the brane worldvolume field theory. Such calculations may be a necessary step towards understanding e.g. supersymmetry breaking and moduli stabilization in stringy realizations of such scenarios, and are of intrinsic interest as probes of the quantum geometry of the Calabi-Yau space. 
  Derivations of consistent equations of motion for the massive spin two field interacting with gravity is reviewed. From the field theoretical point of view the most general classical action describing consistent causal propagation in vacuum Einstein spacetime is given. It is also shown that the massive spin two field can be consistently described in arbitrary background by means of lagrangian equations representing an infinite series in curvature. Within string theory equations of motion for the massive spin two field coupled to gravity is derived from the requirement of quantum Weyl invariance of the corresponding two dimensional sigma-model. In the lowest order in string length the effective equations of motion are demonstrated to coincide with the general form of consistent equations derived in field theory. 
  The problem of time and the quantization of three dimensional gravity in the strong coupling regime is studied following path integral methods. The time is identified with the volume of spacetime. We show that the effective action describes an infinite set of massless relativistic particles moving in a curved three-dimensional target space, i.e. a tensionless 3-brane on a curved background. If the cosmological constant is zero the target space is flat and there is no ` ` graviton" propagation (i.e., $G[g_{ij} (2), g_{ij} (1)] = 0$). If the cosmological constant is different from zero, 3D gravity is both classical and quantum mechanically soluble. Indeed, we find the following results: i) The general exact solutions of the Einstein equations are singular at $t=0$ showing the existence of a big-bang in this regime and ii) the propagation amplitude between two geometries $<g_{ij} (2), t_2| g_{ij} (1), t_1>$ vanishes as $t \to 0$, suggesting that big-bang is suppressed quantum mechanically. This result is also valid for $D>3$. 
  We define the scaling supersymmetric Yang-Lee model with boundary as the (1,3) perturbation of the superconformal minimal model SM(2/8) (or equivalently, the (1,5) perturbation of the conformal minimal model M(3/8)) with a certain conformal boundary condition. We propose the corresponding boundary S matrix, which is not diagonal for general values of the boundary parameter. We argue that the model has an integral of motion corresponding to an unbroken supersymmetry, and that the proposed S matrix commutes with a similar quantity. We also show by means of a boundary TBA analysis that the proposed boundary S matrix is consistent with massless flow away from the ultraviolet conformal boundary condition. 
  We derive the precise relation between level matching condition and fractional instanton numbers in six dimensional, abelian and supersymmetric orbifolds of E8 x E8 heterotic string theory. The fractional part of the two E8 instanton numbers is explicitly calculated in terms of the gauge twist. This relation is then used to show that the classification of these orbifolds can be given in terms of flat bundles away from the orbifold singularities under the only constraint that the sum of the fractional parts of the gauge instanton numbers match the fractional part of the gravitational instanton number locally at every fixed point. This directly carries over to M-theory on S^1/Z_2 
  Type I string theory provides eight classes of unstable D-brane systems. We determine the gauge group and tachyon spectrum for each one, and thereby describe the gauge symmetry breaking pattern in the low-energy world-volume field theory. The topologies of the resulting coset vacuum manifolds are related to the real K-theory groups KO^{-n}, extending the known relations between the Type II classifying spaces BU and U and the complex K-theory groups K^0 and K^{-1}. We also comment on the role of the background D9-branes. 
  In the noncommutative Dirac-Born-Infeld action with Chern-Simons term, an interpolation field $\Phi$ is used in both DBI action and Chern-Simons term. The Morita equivalence is discussed in both the lagrangian and the Hamiltonian formalisms, which is more transparent in this treatment. 
  We study superstring propagations on the Calabi-Yau manifold which develops an isolated ADE singularity. This theory has been conjectured to have a holographic dual description in terms of N=2 Landau-Ginzburg theory and Liouville theory. If the Landau-Ginzburg description precisely reflects the information of ADE singularity, the Landau-Ginzburg model of $D_4,E_6,E_8$ and Gepner model of $A_2\otimes A_2, A_2\otimes A_3, A_2\otimes A_4$ should give the same result. We compute the elements of $D_4,E_6,E_8$ modular invariants for the singular Calabi-Yau compactification in terms of the spectral flow invariant orbits of the tensor product theories with the theta function which encodes the momentum mode of the Liouville theory. Furthermore we find the interesting identity among characters in minimal models at different levels. We give the complete proof for the identity. 
  We examine the issue of renormalizability of asymptotically free field theories on non-commutative spaces. As an example, we solve the non-commutative O(N) invariant Gross-Neveu model at large N. On commutative space this is a renormalizable model with non-trivial interactions. On the noncommutative space, if we take the translation invariant ground state, we find that the model is non-renormalizable. Removing the ultraviolet cutoff yields a trivial non-interacting theory. 
  The particular model of d5 higher derivative gravity which is dual to ${\cal N}=2$ $Sp(N)$ SCFT is considered. A (perturbative) AdS black hole in such theory is constructed in the next-to-leading order of the AdS/CFT correspondence. The surface counterterms are fixed by the conditions required for a well-defined variational procedure and the finiteness of AdS space (when the brane goes to infinity). A dynamical brane is realized at the boundary of an AdS black hole with a radius that is larger than the horizon radius. The AdS/CFT correspondence dictates the parameters of the gravitational dual in such a way that the dynamical brane (the observable universe) always occurs outside the horizon. 
  We discuss recent progress in the determination of correlators of chiral primary operators in N=4 Super-Yang-Mills theory, based on a combination of superconformal covariance arguments in N=2 harmonic superspace, and Intriligator's insertion formula. Applying this technique to the calculation of the supercurrent four - point function we obtain a compact and explicit result for its three-loop contribution with comparatively little effort. 
  Non-singular instantons are shown to exist on noncommutative R^4 even with a U(1) gauge group. Their existence is primarily due to the noncommutativity of the space. The relation between U(1) instantons on noncommutative R^4 and the projection operators acting on the representation space of the noncommutative coordinates is reviewed. The integer number of instantons on the noncommutative R^4 can be understood as the winding number of the U(1) gauge field as well as the dimension of the projection on the representation space. 
  Based on the DBI action for the four coincident non-BPS D9-branes in the type IIA string theory we demonstrate that the gauge symmetry breaking through the tachyon condensation into the generalized monopole of codimension five produces a pair of two coincident BPS D4-branes. The nontrivial gauge field configuration is studied and shown to yield the non-zero generalized magnetic charge. We discuss how this explicit demonstration is related with the higher K-theory group. 
  It is known that the one-loop effective action of ${QED}_2$ is a quadratic in the field strength when the fermion mass is zero: all potential higher order contributions beyond second order vanish. For nonzero fermion mass it is shown that this behavior persists for a general class of fields for at least one value of the fermion mass when the external field's flux $\Phi$ satisfies $0<|e\Phi|<2\pi$. For ${QED}_4$ the mass-shell renormalized one-loop effective action vanishes for at least one value of the fermion mass for a class of smooth, square integrable background gauge fields provided a plausible zero-mass limit exists. 
  We define a normalized Weyl-type $\star$-product on general K\"{a}hler manifolds. Expanding this product perturbatively we show that the cumbersome term, which appears in a Berezin-type product, does not appear at least in the first order of $\hbar$. This means a normalization factor, which is introduced by Reshetikhin and Takhtajan for a Berezin-type product, is unnecessary for our Weyl-type product at that order. 
  It is shown that the deformed conifold solution with three-form flux, found by Klebanov and Strassler, is supersymmetric, and that it admits a simple F-theory description in terms of a direct product of the deformed conifold and a torus. Some general remarks on Ramond-Ramond backgrounds and warped compactifications are included. 
  The p-hierarchy of Schwarzschild type metrics obtained in a preceing paper is generalised here to a corresponding Reissner-Nordstrom (RN) type hierarchy in the presence of a point charge q in d-dimensions. Certain special features arising, concerning the horizons and the interior region, are discussed. 
  The structure of a new family of factorised $S$-matrix theories with resonance poles is reviewed. They are conjectured to correspond to the Homogeneous sine-Gordon theories associated with simply laced compact Lie groups. Two of their more remarkable properties are, first, that some of the resonance poles can be traced to the presence of unstable particles in the spectrum, and, second, that they involve several independent mass scales. The conjectured relationship with the simply laced HSG theories has been checked by means of the Thermodynamic Bethe ansatz (TBA) and, more recently, through the explicit calculation of the Form Factors. The main results of the TBA analysis are summarized. 
  Quantum dynamical semigroups are applied to the study of the time evolution of harmonic oscillators, both bosonic and fermionic. Explicit expressions for the density matrices describing the states of these systems are derived using the holomorphic representation. Bosonic and fermionic degrees of freedom are then put together to form a supersymmetric oscillator; the conditions that assure supersymmetry invariance of the corresponding dynamical equations are explicitly derived. 
  We study the cosmological evolution of a D3-brane Universe in a type 0 string background. We follow the brane-universe along the radial coordinate of the background and we calculate the energy density which is induced on the brane because of its motion in the bulk. We find that for some typical values of the parameters and for a particular range of values of the scale factor of the brane-universe, the effective energy density is dominated by a term proportional to $\frac{1}{(loga)^{4}}$ indicating a slow varying inflationary phase. For larger values of the scale factor the effective energy density takes a constant value and the brane-universe enters its usual inflationary period. 
  The arcane ADHM construction of Yang-Mills instantons can be very naturally understood in the framework of D-brane dynamics in string theory. In this point-of-view, the mysterious auxiliary symmetry of the ADHM construction arises as a gauge symmetry and the instantons are modified at short distances where string effects become important. By decoupling the stringy effects, one can recover all the instanton formalism, including the all-important volume form on the instanton moduli space. We describe applications of the instanton calculus to the AdS/CFT correspondence and higher derivative terms in the D3-brane effective action. In these applications, there is an interesting relation between instanton partition functions, the Euler characteristic of instanton moduli space and modular symmetry. We also describe how it is now possible to do multi-instanton calculations in gauge theory and we resolve an old puzzle involving the gluino condensate in supersymmetric QCD. 
  We show that the decay of the D2-\bar{D2} system with large worldvolume magnetic fields can be described in noncommutative gauge theory. Tachyon condensation in this system describes the annihilation of D2-\bar{D2} into $D0$-branes. From the 2+1 dimensional point of view, this is the decay of a nonabelian magnetic flux into vortices. The semiclassical approximation is valid over a long period of the decay. Our analysis allows us to clarify earlier results in the literature related to tachyon condensation and noncommutative gauge theory. 
  We find a new gauge in which U(1) noncommutative instantons are explicitly non-singular on the whole noncommutative R^4, thus resolving the previous confusions of the author. We start with the pedagogical introduction to the noncommutative gauge theories. 
  The quantization of the electroweak theory is performed starting from the Lagrangian given in the so-called unitary gauge in which the unphysical Goldstone fields disappear. In such a Lagrangian, the unphysical longitudinal components of the gauge fields and the residual gauge degrees of freedom are naturally eliminated by introducing the Lorentz gauge condition and the ghost equation. In this way, the quantum theory given in $\alpha $-gauge is perfectly established in the Lagangian formalism by the Faddeev-Popov approach or the Lagrange multiplier method in the framework of SU(2)$\times $U(1) gauge symmetry. The theory established is not only simpler than the ordinary R$_\alpha -$gauge theory, but also explicitly renormalizable. The unitarity of the S-matrix is ensured by the $\alpha -$limiting procedure proposed previously. Especially, it is shown that the electroweak theory without involving the Higgs boson can equally be formulated whitin the $SU(2)\times U(1)$ symmetry and exhibits good renormalizability. The unitarity of such a theory may also be guaranteed by the $\alpha $-limiting procedure. 
  In this article the action of T-duality on a mixed brane is studied in the boundary state formalism. We also obtain a two dimensional mixed brane with non-zero electric and magnetic fields, from a D$_1$-brane. 
  We review the relevance to the black hole entropy problem of boundary dynamics in Chern-Simons gravity. We then describe a recent derivation of the action induced on the four dimensional boundary in a five dimensional Chern-Simons gravity theory with gauge invariant, anti-deSitter boundary conditions. 
  We discuss the problem of tachyon condensation in the framework of background independent open string field theory. We show, in particular, that the computation of the string field theory action simplifies considerably if one looks at closed string backgrounds with a large B field, and can be carried out exactly for a generic tachyon profile. We confirm previous results on the form of the exact tachyon potential, and we find, within this framework, solitonic solutions which correspond to lower dimensional unstable branes. 
  The dependence on the torsion H=db of the Wess-Zumino couplings of D-branes that are trivially embedded in space-time is studied. We show that even in this simple set-up some torsion components can be turned on, with a non-trivial effect on the RR couplings. In the special cases in which either the tangent or the normal bundle are trivial, the torsion dependence amounts to substitute the standard curvature with its generalization in the presence of torsion, in the usual couplings involving the roof genus A. 
  Armed with the explicit computation of Schur Multipliers, we offer a classification of SU(n) orbifolds for n = 2,3,4 which permit the turning on of discrete torsion. This is in response to the host of activity lately in vogue on the application of discrete torsion to D-brane orbifold theories. As a by-product, we find a hitherto unknown class of N = 1 orbifolds with non-cyclic discrete torsion group. Furthermore, we supplement the status quo ante by investigating a first example of a non-Abelian orbifold admitting discrete torsion, namely the ordinary dihedral group as a subgroup of SU(3). A comparison of the quiver theory thereof with that of its covering group, the binary dihedral group, without discrete torsion, is also performed. 
  The energy of extended classical objects, such as vortices, depends on their shape. In particular, we show that the curvature energy of a kink in two spatial dimensions, as a prototype of extended classical solutions, is always negative. We obtain a closed form for the curvature energy, assuming small deviations from the straight line. 
  One of the important consequences of the no-force condition for BPS states is the existence of stable static multi-center solutions, at least in ungauged supergravities. This observation has been at the heart of many developments in brane physics, including the construction of intersecting branes and reduced symmetry D-brane configurations corresponding to the Coulomb branch of the gauge theory. However the search for multi-center solutions to gauged supergravities has proven rather elusive. Because of the background curvature, it appears such solutions cannot be static. Nevertheless even allowing for time dependence, general multi-center solutions to gauged supergravity have yet to be constructed. In this letter we investigate the construction of such solutions for the case of D=5, N=2 gauged supergravity coupled to an arbitrary number of vector multiplets. Formally, we find a family of time dependent multi-center black hole solutions which are easily generalized to the case of AdS supergravities in general dimensions. While these are not true solutions, as they have a complex metric and gauge potential, they may be related to a Wick rotated theory or to a theory where the coupling is taken to be imaginary. These solutions thus provide a partial realization of true multi-center black-holes in gauged supergravities. 
  We revisit the non-linear BPS equation: the Dirac monopole of the Born-Infeld theory in the B-field background. The rotation used in our previous papers to discuss the scalar field by transforming the BPS equation into a linear one is extended to the case of gauge field. We also find that this transformation is a symmetry of the action. Moreover using the Legendre-dual formalism we present a simple expression of the BPS equation. 
  The perturbative dynamics of noncommutative field theory (NCFT) is discussed from a point view of string field theory. As in the commutative case it is inevitable to introduce a closed string, which may be described as a bound state of two noncommutative open strings. We point out that the closed string, interacting nontrivially with the open string, plays an essential role in the ultraviolet region. The contribution of the closed string is responsible for the discrepancy between the NCFT and the string field theory. It clarifies the controversial issues associated with the ultraviolet/infrared (UV/IR) behaviour of the perturbative dynamics of the NCFT. 
  We analyze the tachyon field in the bosonic open string theory in a constant B-field background using the background independent open string field theory. We show that in the large noncommutativity limit the action of tachyon field is given exactly by the potential term which has the same form as in the case without B-field but the product of tachyon field is taken to be the star product. 
  The covariant descripion is constructed for the Wick-type symbols on symplectic manifolds by means of the Fedosov procedure. The geometry of the manifolds admitting this symbol is explored. The superextended version of the Wick-type star-product is introduced and a possible application of the construction to the noncommutative field theory is discussed. 
  The correlation functions of two-dimensional anyon fields in a KMS-state are studied. For T=0 the $n$-particle wave functions of noncanonical fermions of level $\alpha$, $\alpha$ odd, are shown to be of Laughlin type of order $\alpha$. For $T>0$ they are given by a simple finite-temperature generalization of Laughlin's wave function. This relates the first and second quantized pictures of the fractional quantum Hall effect. 
  The properties of center vortices are discussed within continuum Yang-Mills theory. By starting from the lattice theory and carefully performing the continuum limit the gauge potential of center vortices is obtained and the continuum analog of the maximal center gauge fixing is extracted. It is shown, that the Pontryagin index of center vortices is given by their self-intersection number, which vanishes unless the center vortices host magnetic monpoles, which make the vortex sheets non-oriented. 
  I give a brief informal introduction to the idea and tests of large extra dimensions, focusing on the case in which the space-time manifold has a direct product structure. I then describe some attractive implementations in which the internal space comprises a compact hyperbolic manifold. This construction yields an exponential hierarchy between the usual Planck scale and the true fundamental scale of physics by tuning only ${\cal O}(1)$ coefficients, since the linear size of the internal space remains small. In addition, this allows an early universe cosmology with normal evolution up to substantial temperatures, and completely evades astrophysical constraints. 
  The quantum-mechanical many-body system with the potential proportional to the pairwise inverse-square distance possesses a strong-weak coupling duality. Based on this duality, particle and/or quasiparticle states are described as SU(1,1) coherent states. The constructed quasiparticle states are of hierarchical nature. 
  In the search for the exact minimum of the tachyon potential in the Witten's cubic string field theory we try to learn as much as possible from the string field theory in the large B-field background. We offer a simple alternative proof of the Witten's factorization, carry out the analysis of string field equations also for the noncommutative torus and find some novel relations to the algebraic K-theory. We note an intriguing relation between Chern-Simons and Chern classes of two noncommutative bundles. Finally we observe a certain pattern which enables us to make a plausible conjecture about the exact form of the minimum. 
  We present the consistent approach to finding the discrete transformations in the representation spaces of the proper Poincar\'e group. To this end we use the possibility to establish a correspondence between involutory automorphisms of the proper Poincar\'e group and the discrete transformations. As a result, we derive rules of the discrete transformations for arbitrary spin-tensor fields without the use of relativistic wave equations. Besides, we construct explicitly fields carrying representations of the extended Poincar\'e group, which includes the discrete transformations as well. 
  This paper has been withdrawn by the authors; its contents are superseded by that of hep-th/0105305 and hep-th/0212275. 
  The space-like asymptotic limit of the bilocal composite field of the state consisting of a nucleus and an electron is studied. It is shown that the resulting local field of an atom satisfies the proper commutation relations in the sub-Fock-space of the atom. 
  We study large N behavior of the IIB matrix model using the equivalence between the IIB matrix model for finite N and a field theory on a non-commutative periodic lattice with N x N sites. We find that the large N dependences of correlation functions can be obtained by naively counting the number of fields in the field theory on the non-commutative periodic lattice. Furthermore the large N scaling behavior of the coupling constant g is determined if we impose that the expectation values of Wilson loops are calculable. 
  In the quantum mechanics of collision problems we must consider scattering states of the system. For these states, the wave functions do not remain in Hilbert space, but they are expressible in terms of generalized functions of a Gel'fand triplet. Supersymmetric quantum mechanics for dealing with the scattering states is here proposed. 
  We give a representation of the volume preserving diffeomorphism of $\bR^p$ in terms of the noncommutative (p-2)-branes whose kinetic term is described by the Hopf term. In the static gauge, the (p-2)-brane can be described by the free fields and it suggests that the quantization of the algebra is possible. 
  Field theories compactified on non-simply connected spaces, which in general allow to impose twisted boundary conditions, are found to unexpectedly have a rich phase structure. One of characteristic features of such theories is the appearance of critical radii, at which some of symmetries are broken/restored. A phase transition can occur at the classical level, or can be caused by quantum effects. The spontaneous breakdown of the translational invariance of compactified spaces is another characteristic feature. As an illustrative example,the O(N) $\phi^4$ model on $M^3\otimes S^1$ is studied and the novel phase structure is revealed. 
  The low-energy effective dynamics of M-theory, eleven-dimensional supergravity, is taken off-shell in a manifestly supersymmetric superspace formulation. We show that a previously proposed relaxation of the torsion constraints can indeed accomodate a current supermultiplet. We comment on the relation and application of this completely general formalism to higher-derivative (R^4) corrections. This talk was presented by Bengt EW Nilsson at the Triangle Meeting 2000 ``Non-perturbative Methods in Field and String Theory'', NORDITA, Copenhagen, June 19-22, 2000, and by Martin Cederwall at the International Conference ``Quantization, Gauge Theory and Strings'' in memory of Efim Fradkin, Moscow, June 5-10, 2000. 
  Starting from a Lie algebroid ${\cal A}$ over a space V we lift its action to the canonical transformations on the principle affine bundle ${\cal R}$ over the cotangent bundle $T^*V$. Such lifts are classified by the first cohomology $H^1({\cal A})$. The resulting object is the Hamiltonian algebroid ${\cal A}^H$ over ${\cal R}$ with the anchor map from $\G({\cal A}^H)$ to Hamiltonians of canonical transformations. Hamiltonian algebroids generalize the Lie algebras of canonical transformations. We prove that the BRST operator for ${\cal A}^H$ is cubic in the ghost fields as in the Lie algebra case. To illustrate this construction we analyze two topological field theories. First, we define a Lie algebroid over the space $V_3$ of $\SL$-opers on a Riemann curve $\Si_{g,n}$ of genus g with n marked points. The sections of this algebroid are the second order differential operators on $\Si_{g,n}$. The algebroid is lifted to the Hamiltonian algebroid over the phase space of $W_3$-gravity. We describe the BRST operator leading to the moduli space of $W_3$-gravity. In accordance with the general construction the BRST operator is cubic in the ghost fields. We present the Chern-Simons explanation of our results. The second example is the Hamiltonian algebroid structure in the Poisson sigma-model invoked by Cattaneo and Felder to describe the Kontsevich deformation quantization formula. The hamiltonian description of the Poisson sigma-model leads to the Lie algebraic form of the BRST operator. 
  We propose a new version of the superfield action for a closed D=10, N=1 superstring where the Lorentz harmonics are used as auxiliary superfields. The incorporation of Lorentz harmonics into the superfield action makes possible to obtain superfield constraints of the induced worldsheet supergravity as equations of motion. Moreover, it becomes evident that a so-called 'Wess-Zumino part' of the superfield action is basically a Lagrangian form of the generalized action principle. We propose to use the second Noether theorem to handle the essential terms in the transformation lows of hidden gauge symmetries, which remove dynamical degrees of freedom from the Lagrange multiplier superfield. 
  We calculate the string loop corrections to the tachyon potential for stable non-BPS Dp-branes on the orbifold T^4/Z_2. We find a non-trivial phase structure and we show that, after tachyon condensation, the non-BPS Dp-branes are attracted to each other for p=0,1,2. We then identify the corresponding closed string boundary states together with the massless long range fields they excite. For p=3,4 the string loop correction diverge. We identify the massless closed string fields responsible for these divergencies and regularise the partition function using a Fischler-Susskind mechanism. 
  Basics of some topics on perturbative and non-perturbative string theory are reviewed. After a mathematical survey of the Standard Model of particle physics and GUTs, the bosonic string kinematics for the free case and with interaction is described. The effective action of the bosonic string and the spectrum is also discussed. Five perturbative superstring theories and their spectra is briefly outlined. Calabi-Yau three-fold compactifications of heterotic strings and their relation to some four-dimensional physics are given. T-duality in closed and open strings are surveyed. D-brane definition is provided and some of their properties and applications to brane boxes configurations, in particular to the cube model are discussed. Finally, non-perturbative issues like S-duality, M-theory, F-theory and basics of their non-perturbative Calabi-Yau compactifications are considered. 
  We introduce a basis for a bi-dimensional finite matrix calculus and a bi-dimensional finite matrix action principle. As an application, we analyze scalar and spinorial fields in $D=4n+2$ in this approach. We verify that to establish a bi-dimensional matrix action principle we have to define a Dirac-algebra-modified Lebniz rule. From the bi-dimensional equations of motion, we obtain a matrix holomorphic feature for massless matrix scalar and spinorial fields. 
  We study correlators of R-symmetry currents in the Coulomb branch of N = 4 supersymmetric gauge theory in the large-N limit, using the AdS/CFT correspondence. In particular, we consider gauge fields in the presence of gravity and scalar fields parameterizing the coset SL(6,R)/SO(6) in the context of five-dimensional gauged supergravity. From a ten-dimensional point of view these backgrounds correspond to continuous D3-brane distributions. We find the surprising result that all 2-point functions of gauge currents fall into the same universality class, irrespectively of whether they correspond to broken or unbroken symmetries. We show that the problem of finding the spectrum can be mapped into an equivalent Schroedinger problem for supersymmetric quantum mechanics. The corresponding potential is the supersymmetric partner of the potential arising in studies of the spectrum for massless scalars and transverse graviton fluctuations in these backgrounds and the associated spectra are also identical. We discuss in detail two examples where these computations can be done explicitly as in the conformal case. 
  These lectures provide an elementary introduction to Chern Simons Gravity and Supergravity in $d=2n+1$ dimensions. 
  This work extends to the D-dimensional space-time the topological mass generation mechanism of the nonabelian BF model in four dimensions. In order to construct the gauge invariant nonabelian kinetic terms for a (D-2)-form B and a 1-form A, we introduce an auxiliary (D-3)-form V. Furthermore, we obtain a complete set of BRST and anti-BRST transformation rules of the fields using the so called horizontality condition, and construct a BRST/anti-BRST invariant quantum action for the model in D-dimensional space-time. 
  We consider a new momentum cut-off scheme for sums over zero-point energies, containing an arbitrary function f(k) which interpolates smoothly between the zero-point energies of the modes around the kink and those in flat space. A term proportional to df(k)/dk modifies the result for the one-loop quantum mass M^(1) as obtained from naive momentum cut-off regularization, which now agrees with previous results, both for the nonsusy and susy case. We also introduce a new regularization scheme for the evaluation of the one-loop correction to the central charge Z^(1), with a cut-off K for the Dirac delta function in the canonical commutation relations and a cut-off \Lambda for the loop momentum. The result for Z^(1) depends only on whether K>\Lambda or K<\Lambda or K=\Lambda. The last case yields the correct result and saturates the BPS bound, M^(1)=Z^(1),in agreement with the fact that multiplet shortening does occur in this N=(1,1) system. We show how to apply mode number regularization by considering first a kink-antikink system, and also obtain the correct result with this method. Finally we discuss the relation of these new schemes to previous approaches based on the Born expansion of phase shifts and higher-derivative regularization. 
  The quantization of the complex linear superfield requires an infinite tower of ghosts. Using the Batalin-Vilkovisky technique, Grisaru, Van Proeyen, and Zanon have been able to define a correct procedure to construct a gauge-fixed action. We generalize their technique by introducing the Lagrange multipliers into the non-minimal sector and we study the characteristic BRST cohomology. We show how the physical subspace is singled out. Finally, we quantize the model in the presence of a background and of a quantum gauge superfield. 
  Einstein gravity in the Palatini first order formalism is shown to possess a vector supersymmetry of the type encountered in the topological gauge theories. A peculiar feature of the gravitationel theory is the link of this vector supersymmetry with the field equation of motion of the Faddeev-Popov ghost associated to diffeomorphism invariance. 
  From the AdS/CFT correspondence, we learn that the classical evolution of supergravity in the bulk can be reduced to a RG-flow equation for the dual low-energy, strongly coupled and large N gauge theory on the boundary. This result has been used to obtain interesting relations between the various terms in the gravitational part of the boundary effective action, in particular the term that affect the cosmological constant. It is found that once the cosmological constant is cancelled in the UV theory, the RG-flow symmetry of the boundary effective action automatically implies the existence of zero cosmological constant solutions that extend all the way into the IR. Given the standard (and well founded) contradiction between the RG-flow idea and the observational evidence of a small cosmological constant, this is considered to be an important progress, albeit incomplete, towards the final solution. Motivated by this success, it would be interesting to see whether this RG-stability extends outside the scope of strong 't Hooft coupling and large N regime that are implicitly assumed in the de Boer-Verlinde-Verlinde Hamilton-Jacobi formulation of the holographic RG-flow equations of the boundary theory. In this paper, we address this question. Taking into account the leading order corrections in the 1/N and $\alpha'/{R^2}$ parameters, we derive new bulk/boundary relations, from which one can read all the local terms in the boundary effective action. Next, we use the resulting constraints, to examine whether the RG-stability of the cosmological constant extends to the new coupling regime. It would be also interesting to use these constraints to study the Randall-Sundrum scenario in this case. 
  We discuss the higher dimensional generalization of gravitational instantons by using volume-preserving vector fields. We give special attention to the case of 8-dimensions and present a new construction of the Ricci flat metric with holonomy in Spin(7). An example of the metric is explicitly given. Further it is shown that our formulation has a natural interpretation in the Chern-Simons theory written by the language of superconnections. 
  We present generalisations of N-extended supersymmetry algebras in four dimensions, using Lorentz covariance and invariance under permutation of the N supercharges as selection criteria. 
  Spacetime is modelled by binary relations - by the classes of the automorphisms $\GL(\C^2)$ of a complex 2-dimensional vector space with respect to the definite unitary subgroup $\U(2)$. In extension of Feynman propagators for particle quantum fields representing only the tangent spacetime structure, global spacetime representations are given, formulated as residues using energy-momentum distributions with the invariants as singularities. The associatated quantum fields are characterized by two invariant masses - for time and position - supplementing the one mass for the definite unitary particle sector with another mass for the indefinite unitary interaction sector without asymptotic particle interpretation. 
  The solution representing a brane-anti-brane system in matrix models breaks the usual matrix spacetime symmetry. We show that the spacetime symmetry on the branes is not breaking, rather appears as a combination of the matrix spacetime transformation and a gauge transformation. As a result, the tachyon field, itself an off-diagonal entry in longitudinal matrices, transforms nontrivially under rotations, decomposing into tensors of different ranks. We also show that the tachyon field can never be gauged away, and conjecture that this field is related to the usual complex scalar tachyon by a field redefinition. We also briefly discuss tachyon condensation. 
  We review the structures imposed on perturbative QFT by the fact that its Feynman diagrams provide Hopf and Lie algebras. We emphasize the role which the Hopf algebra plays in renormalization by providing the forest formulas. We exhibit how the associated Lie algebra originates from an operadic operation of graph insertions. Particular emphasis is given to the connection with the Riemann--Hilbert problem. Finally, we outline how these structures relate to the numbers which we see in Feynman diagrams. 
  We find exact solitons in a large class of noncommutative gauge theories using a simple solution generating technique. The solitons in the effective field theory description of open string field theory are interpreted as D-branes for any value of the noncommutativity. We discuss the vacuum structure of open string field theory in view of our results. 
  We review various aspects of integrable hierarchies appearing in N=2 supersymmetric gauge theories. In particular, we show that the blowup function in Donaldson-Witten theory, up to a redefinition of the fast times, is a tau function for a g-gap solution of the KdV hierarchy. In the case of four-manifolds of simple type, instead, the blowup function becomes a tau function corresponding to a multisoliton solution. We obtain a new expression for the contact terms that links these results to the Whitham hierarchy formulation of Seiberg-Witten theories. 
  This is a short review on the thermal, spectral representation in the real-time version of the finite temperature quantum field theory. After presenting a clear derivation of the spectral representation, we discuss the properties of its spectral function. Two applications of this representation are then considered. One is the solution of the Dyson equation for the thermal propagator. The other is the formulation of the QCD sum rules at finite temperature. 
  The present article analyses the impact on cosmology, in particular on the evolution of cosmological perturbations, of the existence of extra-dimensions. The model considered here is that of a five-dimensional Anti-de Sitter spacetime where ordinary matter is confined to a brane-universe. The homogeneous cosmology is recalled. The equations governing the evolution of cosmological perturbations are presented in the most transparent way: they are rewritten in a form very close to the equations of standard cosmology with two types of corrections: a. corrections due to the unconventional evolution of the homogeneous solution, which change the background-dependent coefficients of the equations; b. corrections due to the curvature along the fifth dimension, which act as source terms in the evolution equations. 
  We consider the effect of the (heavy) fundamental quarks on the low energy effective Lagrangian description of nonabelian gauge theories in 2+1 dimensions. We show that in the presence of the fundamental charges, the magnetic $Z_N$ symmetry becomes local. We construct the effective Lagrangian representing this local symmetry in terms of magnetic vortex fields, and discuss its physical consequences. We show that the finite energy states described by this Lagrangian have distinct bag-like structure. The point-like quarks are confined to the region of space where the value of the vortex field is much smaller than in the surrounding vacuum. 
  We consider Standard Model fields living inside a thick four dimensional flat brane embedded in a (possibly warped) five dimensional space-time and estimate the electromagnetic corrections to the anomalous magnetic moment (g-2) of the electron and muon by including virtual massive fermion and gauge boson states. Constraints on the mass of the ``excited'' states (or thickness of the brane-world) are obtained. 
  The gaussian damping factor (g.d.f.) and the new interaction vertex with the symplectic tensor are the characteristic properties of the N-point scalar-vector scattering amplitudes of the p-p' (p < p') open string system which realizes noncommutative geometry. The g.d.f. is here interpreted as a form factor of the Dp-brane by noncommutative U(1) current. Observing that the g.d.f. is in fact equal to the Fourier transform of the noncommutative projector soliton introduced by Gopakumar, Minwalla and Strominger, we further identify the Dp-brane in the zero slope limit with the noncommutative soliton state. It is shown that the g.d.f. depends only on the total momentum of N-2 incoming/outgoing photons in the zero slope limit. In the description of the low-energy effective action (LEEA) proposed before, this is shown to follow from the delta function propagator and the form of the initial/final wave functions in the soliton sector which resides in x^{m} m= p+1, ...p' dependent part of the scalar field \Phi(x^\mu, x^m). The three and four point amplitudes computed from LEEA agree with string calculation. We discuss related issues which are resummation/lifting of infinite degeneracy and conservation of momentum transverse to the Dp-brane. 
  A formulation of anomalies in terms of star products is suggested which promises insight from an alternative and unifying point of view. 
  We discuss the generic definition of the $\tau$ function for the arbitrary Hamiltonian system. The different approaches concerning the deformations of the curves and surfaces are compared. It is shown that the Baker-Akhiezer function for the secondary integrable system of the Toda lattice type can be identified with the coherent wave function of the initial dynamical system. The $\tau$ function appears to be related to the filling of the interior of the classical trajectory by coherent states. Transition from dispersionless to dispersionful Toda lattice corresponds to the quantization of the initial dynamical system. 
  We study high energy scattering amplitudes in a strongly coupled (confining) gauge theory using the AdS/CFT correspondence. The scattering described by a Wilson line/loop correlation function was shown earlier to correspond to minimal surfaces of the helicoid type, and gave amplitudes with unit intercept and linear trajectory. In this paper we find the correction to the intercept from quadratic fluctuations of the string worldsheet around the helicoid. The relevant term comes from analytical continuation of a L\"uscher like term. It is coupling-constant independent and proportional to the number of effective transverse flat dimensions. The shift of the intercept, under our assumptions, is $n_\perp /96$, and for $n_\perp=7,8$ gives respectively 0.0729, 0.083. Incidentally we note that this is surprisingly close to the observed value of 0.08. 
  I present clear evidences that for $d$=2 a first order transition takes place when the coherence length $\xi$ becomes of the order of the lattice spacing and that this is connected with a sudden proliferation of vortices. Similar results where reported in cond-mat/0010119 although using a different parametrization of the G-L model which obscures the comparison with recent results obtained by means of a variational approximation. 
  We propose a mechanism ensuring (quasi)localization of massless gauge fields on a brane. The mechanism does not rely on BPS properties of the brane and can be realized in any theory where charged particles are confined to the world-volume. The localized matter fluctuations induce a gauge kinetic term on the brane. At short distances the resulting propagator for the gauge field is {\it identical} to the four-dimensional propagator. The gauge theory on the brane is effectively four-dimensional at short distances; it becomes higher-dimensional on very large (cosmic) scales. The brane-bulk system exhibits the phenomenon of ``infrared transparency''. As a result, only very low frequency modes can escape into extra dimensions. In this framework the large wavelength cosmic radiation can dissipate in extra space at a rate that may be observable, in principle. We briefly discuss some astrophysical consequences of this scenario.   The same mechanism of localization of gauge fields may work in Kaplan's framework for domain wall chiral fermions on lattices. 
  We study a deformation of N=4 Super-Yang-Mills theory by a dimension-5 vector operator. There is a simple nonlocal "dipole" field-theory that realizes this deformation. We present evidence that this theory is realized in the setting of "pinned-branes". The dipoles correspond to open strings that arch out of the brane. We find the gravitational dual of the theory at large N. We also discuss the generalization to the (2,0) theory. 
  We present a new estimate of the fine structure constant and the $\beta$-function of QED at an arbitrary scale. Using the non-perturbative but convergent series expression of the one loop effective action of QED that has been available recently we make a non-perturbative estimate of the running coupling and the $\beta$-function, and prove the renormalization group invariance of the effective action. The contrast between our result and the perturbative result is remarkable. 
  The motivations for the construction of an 8-component representation of fermion fields based on a two dimensional representation of time reversal transformation and CPT invariance are discussed. Some of the elementary properties of the quantum field theory in the 8-component representation are studied. It includes the space-time and charge conjugation symmetries, the implementation of a reality condition, the construction of interaction theories, the field theoretical imaginary- and real-time approach to thermodynamics of fermionic systems, the quantization of fermion fields, their particle content and the Feynman rules for perturbation theories. It is shown that in the new presentation, a CPT violation can be formulated in principle. The construction of interaction theories in the 8-component theory for fermions is shown to be constrained by the CPT invariance. The short distance behavior and relativistic covariance are studied. In the path integral representation of the thermodynamical potential, the conventional imaginary-time approach is shown to be smoothly connected to a real-time thermal field theory in the 8-component representation for fermion fields without any additional subtraction of infinities. The metastability at zero density and the nature of the spontaneous CP violation in color superconducting phases of strong interaction ground states are clarified. 
  We study the two-point function for the gauge boson in the axial-type gauges. We use the exact treatment of the axial gauges recently proposed that is intrinsically compatible with the Lorentz type gauges in the path-integral formulation and has been arrived at from this connection and which is a ``one-vector'' treatment. We find that in this treatment, we can evaluate the two-point functions without imposing any additional interpretation on the axial gauge 1/(n.q)^p-type poles. The calculations are as easy as the other treatments based on other known prescriptions. Unlike the ``uniform-prescription'' /L-M prescription, we note, here, the absence of any non-local divergences in the 2-point proper vertex. We correlate our calculation with that for the Cauchy Principal Value prescription and find from this comparison that the 2-point proper vertex differs from the CPV calculation only by finite terms. For simplicity of treatment, the divergences have been calculated here with n^2>0 and these have a smooth light cone limit. 
  We construct maximally supersymmetric gauged N=16 supergravity in three dimensions, thereby obtaining an entirely new class of AdS supergravities. These models are not derivable from any known higher-dimensional theory, indicating the existence of a new type of supergravity beyond D=11. They are expected to be of special importance also for the conjectured AdS/CFT correspondence. One of their noteworthy features is a nonabelian generalization of the duality between scalar and vector fields in three dimensions. Among the possible gauge groups, SO(8)xSO(8) is distinguished as the maximal compact gauge group, but there are also more exotic possibilities such as F_4 x G_2. 
  Sigma model ($\alpha^{\prime}$) corrections to the confining string background are obtained. The main result is that the Poincar\'e invariant ansatz is maintained. Physical conditions for the dissapearance of the naked singularity are discussed. 
  This is a very brief review of relations between Seiberg-Witten theories and integrable systems with emphasis on the perturbative prepotentials presented at the E.S.Fradkin Memorial Conference. 
  The mechanism of color confinement as a consequence of an unbroken non-abelian gauge symmetry and asymptotic freedom is elucidated and compared with that of other models based on an analogy with the type II superconductor. It is demonstrated that a sufficient condition for color confinement is given by $Z_3^{-1}=0$ where $Z_3$ denotes the renormalization constant of the color gauge field. It is shown that this condition is actually satisfied in quantum chromodynamics and that some of the characteristic features of other models follow from it. 
  Using a synthesis of the functional integral and operator approaches we discuss the fermion-boson mapping and the role played by the Bose field algebra in the Hilbert space of two-dimensional gauge and anomalous gauge field theories with massive fermions. In the $QED_2$ with quartic self-interaction among massive fermions, the use of an auxiliary vector field introduces a redundant Bose field algebra that should not be considered as an element of the intrinsic algebraic structure defining the model. In the anomalous chiral $QED_2$ with massive fermions the effect of the chiral anomaly leads to the appearance in the mass operator of a spurious Bose field combination. This phase factor carries no fermion selection rule and the expected absence of $\theta$-vacuum in the anomalous model is displayed from the operator solution. Even in the anomalous model with massive Fermi fields, the introduction of the Wess-Zumino field replicates the theory, changing neither its algebraic content nor its physical content. 
  We consider the minimal chiral Schwinger model, by embedding the gauge noninvariant formulation into a gauge theory following the Batalin-Fradkin-Fradkina-Tyutin point of view. Within the BFFT procedure, the second class constraints are converted into strongly involutive first-class ones, leading to an extended gauge invariant formulation. We also show that, like the standard chiral model, in the minimal chiral model the Wess-Zumino action can be obtained by performing a q-number gauge transformation into the effective gauge noninvariant action. 
  We review recent developments in the theory of renormalisation group flows in minimal models with boundaries. Among these, we discuss in particular the perturbative calculations of Recknagel et al, not only as a tool to predict the IR endpoints of certain flows, but also as a motivation for considering the particular limiting case of c = 1. By treating this limit, we are able to investigate a wide class of perturbations by considering them as deformations away from the c = 1 point. We also present the truncated conformal space approach as a tool for investigating the space of RG flows and checking particular predictions. 
  We construct and discuss solutions of SO(1,2) x SO(1,2) Chern-Simons theory which correspond to multiple BTZ black holes. These solutions typically have additional singularities, the simplest cases being special conical singularities with a 2 pi surplus angle. There are solutions with singularities inside a common outer horizon, and other solutions with naked conical singularities. Previously such singularities have been ruled out on physical grounds, because they do not obey the geodesic equation. We find however that the Chern-Simons gauge symmetry may be used to locate all such singularities to the horizons, where they necessarily follow geodesics. We are therefore led to conclude that these singular solutions correspond to physically sensible geometries.   Boundary charges at infinity are only sensitive to the total mass and spin of the black holes, and not to the distribution among the black holes. We therefore argue that a holographic description in terms of a boundary conformal field theory should represent both single and multiple BTZ solutions with the same asymptotic charges. Then sectors with multiple black holes would contribute to the black hole entropy calculated from a boundary CFT. 
  We argue that apart from the standard closed and open strings one may consider a third possibility that we call monodromic strings. The monodromic string propagating on a target looks like an ordinary open string (a mapping from a segment to the target) but its space of states is isomorphic to that of a closed string.  It is shown that the monodromic strings naturally appear in T-dualizing closed strings moving on simply connected targets. As a nontrivial topology changing example we show that the monodromic strings on a compact Poisson-Lie group are T-dual to the standard closed strings propagating on the noncompact dual PL group. 
  In the context of the pre-Big Bang scenario of string cosmology, we propose a modified equation for the evolution of the tensor perturbations, which includes the full contribution of possible higher-order curvature and coupling corrections required to regularise the background evolution. We then discuss the high-frequency branch of the spectrum of primordial gravitons. 
  We discuss the existence of a non-perturbative gauge sector that can raise the rank of the gauge group of the N_4=2 heterotic string up to 48. These gauge bosons, that don't exist in six dimensions, co-exist with those originating from small instantons shrinking to zero size. 
  We analyze the structure of heterotic M-theory on K3 orbifolds by presenting a comprehensive sequence of M-theoretic models constructed on the basis of local anomaly cancellation. This is facilitated by extending the technology developed in our previous papers to allow one to determine "twisted" sector states in non-prime orbifolds. These methods should naturally generalize to four-dimensional models, which are of potential phenomenological interest. 
  The type IIB supergravity solution describing a collection of regular and fractional D3 branes on the conifold (hep-th/0002159) was recently generalized to the case of the deformed conifold (hep-th/0007191). Here we present another generalization -- when the conifold is replaced by the resolved conifold. This solution can be found in two different ways: (i) by first explicitly constructing the Ricci-flat Kahler metric on resolved conifold and then solving the supergravity equations for the D3-brane ansatz with constant dilaton and (self-dual) 3-form fluxes; (ii) by generalizing the ``conifold'' ansatz of hep-th/0002159 in a natural ``asymmetric'' way so that the 1-d action describing radial evolution still admits a superpotential and then solving the resulting 1-st order system. The superpotentials for the ``resolved'' and ``deformed'' conifold cases turn out to have essentially the same simple structure. The solution in the resolved conifold case has the same asymptotic UV behaviour as in the conifold case, but unlike the deformed conifold case is still singular in the IR. The naked singularity is of repulson type and may have a brane resolution. 
  The connection between spin and statistics implied by the continuous Lorentz group together with strong reflection (TCP) is shown to hold also for the q-Lorentz group. 
  We present a unified treatment of classical solutions of noncommutative gauge theories. We find all solutions of the noncommutative Yang-Mills equations in 2 dimensions; and show that they are labelled by two integers -- the rank of gauge group and the magnetic charge. The magnetic vortex solutions are unstable in 2+1 dimensions, but correspond to the full, stable BPS solutions of N=4 U(1) noncommutative gauge theory in 4 dimensions, that describes N infinite D1 strings that pierce a D3 brane at various points, in the presence of a background B field in the Seiberg-Witten limit. We discuss the behaviour of gauge invariant observables in the background of the solitons. We use these solutions to construct a panoply of BPS and non-BPS solutions of supersymmetric gauge theories that describe various configurations of D-branes. We analyze the instabilities of the non-BPS solutions. We also present an exact analytic solution of noncommutative gauge theory that describes a U(2) monopole. 
  We develop techniques to construct general discrete Wilson lines in four-dimensional N=1 Type IIB orientifolds, their T-dual realization corresponds to branes positioned at the orbifold fixed points. The explicit order two and three Wilson lines along with their tadpole consistency conditions are given for D=4 N=1 Z_6 Type IIB orientifold. The systematic search for all models with general order three Wilson lines leads to a small class of inequivalent models. There are only two inequivalent classes of a potentially phenomenologically interesting model that has a possible SU(3)_{color} x SU(2)_L x SU(2)_R x U(1)_{B-L} gauge structure, arising from a set of branes located at the Z_6 orbifold fixed point. We calculate the spectrum and Yukawa couplings for this model. On the other hand, introduction of anti-branes allows for models with three families and realistic gauge group assignment, arising from branes located at the Z_3 orbifold fixed points. 
  We use wrapped D-brane probes to measure position dependent perturbations of compactification moduli. Due to the backreaction of the D-branes on the local geometry, we suspect that measuring the fluctuations of one modulus to high precision will generically affect the others. These considerations lead us to conjecture a novel uncertainty principle on the Calabi-Yau moduli space. We begin our investigation with a gedanken experiment on a torus. 
  We obtain the full 5D graviton propagator in the Randall-Sundrum model with the Gauss-Bonnet interaction. From the decomposition of the graviton propagator on the brane, we show that localization of gravity arises in the presence of the Gauss-Bonnet term. We also obtain the metric perturbation for observers on the brane with considering the brane bending and compute the amplitude of one massless graviton exchange. For the positive definite amplitude or no ghost states, the sign of the Gauss-Bonnet coefficient should be negative in our convention, which is compatible with string amplitude computations. In that case, the ghost-free condition is sufficient for obtaining the Newtonian gravity. For a vanishing Gauss-Bonnet coefficient, the brane bending allows us to reproduce the correct graviton polarizations for the effective 4D Einstein gravity. 
  The purpose of this talk is to sketch some recent progress which has been made in calculating non-perturbatively the reflection factors for the sinh-Gordon model restricted to a half-line by integrable boundary conditions. The essential idea is to calculate the energy spectrum of boundary breathers in two independent ways; firstly by using the boundary bootstrap and secondly by quantizing the classical solutions corresponding to boundary breathers. Comparing these two calculations provides a way to determine the dependence of the reflection factors on the parameters introduced into the Lagrangian by the boundary conditions. The basic idea is illustrated using a massive free scalar field with a linear boundary condition confining it to a half-line. 
  The determination of the critical exponents by means of the Exact Renormalizion Group approach is still a topic of debate. The general flow equation is by construction scheme independent, but the use of the truncated derivative expansion generates a model dependence in the determination of the universal quantities. We derive new nonperturbative flow equations for the one-component, $Z_2$ symmetric scalar field to the next-to-leading order of the derivative expansion by means of a class of proper time regulators. The critical exponents $\eta$, $\nu$ and $\omega$ for the Wilson-Fisher fixed point are computed by numerical integration of the flow equations, without resorting to polynomial truncations. We show that by reducing the width of the cut-off employed, the critical exponents become rapidly insensitive to the cut-off width and their values are in good agreement with the results of entirely different approaches. 
  We consider a deformation of three dimensional BF theory by means of the antifield BRST formalism. Possible deformations for the action and the gauge symmetries are analyzed. We find a new class of gauge theories which include nonabelian BF theory, higher dimensional nonlinear gauge theory and topological membrane theory. 
  We exactly calculate the thermal distribution and temperature of Hawking radiation for a two-dimensional charged dilatonic black hole after it has settled down to an "equilibrium" state. The calculation is carried out using the Bogoliubov coefficients. The background of the process is furnished by a preexisting black hole and not by collapsing matter as considered by Giddings and Nelson for the case of a Schwarzschild black hole. Furthermore, the vanishing of the temperature and/or the Hawking radiation in the extremal case is obtained as a regular limit of the general case. 
  As opposed to usual Einstein gravity in four dimensions, the Brane-World scenario allows the construction of a local density of gravitational energy (and also of momentum, of angular momentum, etc...). This is a direct consequence of the hypothesis that our universe is located at the boundary of a five-dimensional diffeomorphism invariant manifold. We compute these Brane-World densities of charge using the Lanczos-Israel boundary conditions. To proceed, we implement an explicitely covariant generalization of the Hamiltonian procedure of Regge and Teitelboim given in a previous work. We finally study two simple Brane-World examples. 
  We first recall a covariant formalism used to compute conserved charges in gauge invariant theories. We then study the case of gravity for two different boundary conditions, namely spatial infinity and a Brane-World boundary. The new conclusion of this analysis is that the gravitational energy (and linear and angular momentum) is a local expression if our universe is really a boundary of a five-dimensional spacetime. 
  A self-consistent renormalization group flow equation for the scalar lambda phi^4 theory is analyzed and compared with the local potential approximation. The two prescriptions coincide in the sharp cutoff limit but differ with a smooth cutoff. The dependence of the critical exponent nu on the smoothness parameter and the field of expansion is explored. An optimization scheme based on the minimum sensitivity principle is employed to ensure the most rapid convergence of nu with the level of polynomial truncation. 
  We discuss noncommutative solitons on a noncommutative torus and their application to tachyon condensation. In the large B limit, they can be exactly described by the Powers-Rieffel projection operators known in the mathematical literature. The resulting soliton spectrum is consistent with T-duality and is surprisingly interesting. It is shown that an instability arises for any D-branes, leading to the decay into many smaller D-branes. This phenomenon is the consequence of the fact that K-homology for type II von Neumann factor is labeled by R. 
  We consider Wilson loop observables for Chern-Simons theory at large N and its topological string dual and extend the previous checks for this duality to the case of links. We find an interesting structure involving representation/spin degeneracy of branes ending on branes which features in the large N dual description of Chern-Simons theory. This leads to a refinement of the integer invariants for links and knots. We illustrate our results with explicit computations on the Chern-Simons side. 
  We study topological gauge theories with N=(2,0) supersymmetry based on stable bundles on general Kahler 3-folds. In order to have a theory that is well defined and well behaved, we consider a model based on an extension of the usual holomorphic bundle by including a holomorphic 3-form. The correlation functions of the model describe complex 3-dimensional generalizations of Donaldson-Witten type invariants. We show that the path integral can be written as a sum of contributions from stable bundles and a complex 3-dimensional version of Seiberg-Witten monopoles. We study certain deformations of the theory, which allow us to consider the situation of reducible connections. We shortly discuss situations of reduced holonomy. Afterdimensional reduction to a Kahler 2-fold, the theory reduces to Vafa-Wittentheory. On a Calabi-Yau 3-fold, the supersymmetry is enhanced to N=(2,2). This model may be used to describe classical limits of certain compactifications of(matrix) string theory. 
  We derive the Green-Schwarz action on AdS_5 x S^5 using an alternate version of the coset superspace construction. By Wick rotations and Lie algebra identifications we bring the coset to GL(4|4)/(Sp(4) x GL(1))^2, which allows us to represent the conformal transformations on unconstrained matrices. The derivation is more streamlined even for the bosonic sector, and conformal symmetry is manifest at every step. Kappa-symmetry gauge fixing is more transparent. 
  In this paper we show how two dimensional electron systems can be modeled by strings interacting with D-branes. The dualities of string theory allow several descriptions of the system. These include descriptions in terms of solitons in the near horizon D6-brane theory, non-commutative gauge theory on a D2-brane, the Matrix Theory of D0-branes and finally as a giant graviton in M-theory. The soliton can be described as a D2-brane with an incompressible fluid of D0-branes and charged string-ends moving on it. Including an NS5 brane in the system allows for the existence of an edge with the characteristic massless chiral edge states of the Quantum Hall system. 
  We consider a geometric zero-radius limit for strings on $AdS_5\times S^5$, where the anti-de Sitter hyperboloid becomes the projective lightcone. In this limit the fifth dimension becomes nondynamical, yielding a different "holographic" interpretation than the usual "bulk to boundary" one. When quantized on the random lattice, the fifth coordinate acts as a new kind of Schwinger parameter, producing Feynman rules with normal propagators at the tree level: For example, in the bosonic case ordinary massless $\phi^4$ theory is obtained. In the superstring case we obtain new, manifestly ${\cal N}=4$ supersymmetric rules for ${\cal N}=4$ super Yang-Mills. These gluons are also different from those of the usual AdS/CFT correspondence: They are the "partons" that make up the usual "hadrons" of the open and closed strings in the familiar QCD string picture. Thus, their coupling $g_{YM}$ and rank $N$ of the "color" gauge group are different from those of the "flavor" gauge group of the open string. As a result we obtain different perturbation expansions in radius, coupling, and 1/N. 
  The properties of A-statistics, related to the class of simple Lie algebras sl(n+1) (Palev, T.D.: Preprint JINR E17-10550 (1977); hep-th/9705032), are further investigated. The description of each sl(n+1) is carried out via generators and their relations, first introduced by Jacobson. The related Fock spaces W_p (p=1,2,...) are finite-dimensional irreducible sl(n+1)-modules. The Pauli principle of the underlying statistics is formulated. In addition the paper contains the following new results: (a) The A-statistics are interpreted as exclusion statistics; (b) Within each W_p operators B(p)_1^\pm, ..., B(p)_n^\pm, proportional to the Jacobson generators, are introduced. It is proved that in an appropriate topology the limit of B(p)_i^\pm for p going to infinity is equal to B_i^\pm, where B_i^\pm are Bose creation and annihilation operators; (c) It is shown that the local statistics of the degenerated hard-core Bose models and of the related Heisenberg spin models is p=1 A-statistics. 
  We generalize recent results on tachyon condensation in boundary string field theory to the superstring. 
  I study a class of Wilsonian formulations of non-Abelian gauge theories in algebraic noncovariant gauges where the Wilsonian infrared cutoff $\Lambda$ is inserted as a mass term for the propagating fields. In this way the Ward-Takahashi identities are preserved to all scales. Nevertheless the BRS-invariance in broken and the theory is gauge-dependent and unphysical at $\Lambda\neq0$. Then I discuss the infrared limit $\Lambda\to0$. I show that the singularities of the axial gauge choice are avoided in planar gauge and in light-cone gauge. Finally the rectangular Wilson loop of size $2L\times 2T$ is evaluated at lowest order in perturbation theory and a noncommutativity between the limits $\Lambda\to0$ and $T\to\infty$ is pointed out. 
  Topological integrals appear frequently in Lagrangian field theories. On manifolds without boundary, they can be treated in the framework of (absolute) (co)homology using the formalism of Cheeger--Simons differential characters. String and D--brane theory involve field theoretic models on worldvolumes with boundary. On manifolds with boundary, the proper treatment of topological integrals requires a generalization of the usual differential topological set up and leads naturally to relative (co)homology and relative Cheeger--Simons differential characters. In this paper, we present a construction of relative Cheeger--Simons differential characters which is computable in principle and which contains the ordinary Cheeger--Simons differential characters as a particular case. 
  We consider a family of four-dimensional non-linear sigma models based on an O(5) symmetric group, whose fields take their values on the 4-sphere S4. An SO(4)-subgroup of the model is gauged. The solutions of the model are characterised by two distinct topological charges, the Chern-Pontryagin charge of the gauge field and the degree of the map, i.e. the winding number, of the S4 field. The one dimensional equations arising from the variation of the action density subjected to spherical symmetry are integrated numerically. Several properties of the solutions thus constructed are pointed out. The only solution with unit Chern-Pontryagin charge are the usual BPST instantons with zero S4 winding number, while solutions with unit S4 winding number have zero Chern-Pontryagin charge. 
  We study five-dimensional solutions to Einstein equations coupled to a scalar field. Bounce-type solutions for the scalar field are associated with AdS_5 spaces with smooth warp functions. Gravitons are dynamically localized in this framework in analogy to the Randall-Sundrum solution whereas, a bulk fermion gives rise to a single chiral zero mode localized at the bounce. Additional bulk scalar fields are incorporated in this picture. The dilaton, as a bulk scalar leads, through its coupling, to localized gauge boson fields, something that holds also in the case that the bounce system is replaced by a brane. 
  We derive the noncommutative Chern-Simons action induced by Dirac fermions coupled to a background gauge field, for the fundamental, antifundamental, and the adjoint representation. We discuss properties of the noncommutative Chern-Simons action showing in particular that the Seiberg-Witten formula maps it into the standard commutative Chern-Simons action. 
  The complete exact solution of the Schwinger model with compact gauge group U(1), in the Hamiltonian approach, is presented . The compactification is imposed by demanding that the only surviving true electromagnetic degree of freedom has angular character. Not surprinsingly, this topological condition defines a version of the Schwinger model which is different from the standard one, where $c$ takes values on the line . The main consequences are: the spectra of the zero modes is not degenerated and does not correspond to the equally spaced harmonic oscillator, both the electric charge and a modified gauge invariant chiral charge are conserved (nevertheless, the axial-current anomaly is still present) and, finally, there is no need to introduce a $\theta$-vacuum. A comparison with the results of the standard Schwinger model is pointed out along the text. 
  We study N=4 supersymmetric Yang-Mills (SYM) theory with gauge group SU(2) compactified to three dimensions on a circle of circumference beta. The eight fermion terms in the effective action on the Coulomb branch are determined exactly, for all beta, assuming the existence of an interacting Spin(8) invariant fixed point at the origin. The resulting formulae are manifestly invariant under the SL(2,Z) duality of four dimensional N=4 SYM and lead to interesting quantitative predictions for instanton effects in gauge theory and in Type II string theory. 
  A consistent truncation of IIB on S^5 has been obtained in the sector of the metric and the 4-form potential. The ansatz contains 20 scalars and all 15 gauge fields of ${\cal N}=8$ gauged supergravity in five dimensions. With this fully non-linear ansatz, the calculations for n-point correlators of super Yang-Mills (SYM) theory via AdS/CFT are simpler than those in the literature that use {\em linear} ansatz followed by non-linear field redefinitions. We work out the SYM operators that couple to the scalars by expanding the Dirac-Born-Infeld (DBI) action plus Wess-Zumino (WZ) terms around an $AdS_5\timesS^5$ background with the metric fluctuations. The resulting operators agree with those based on a superconformal symmetry argument. We discuss the significance of our results. 
  We perform a general analysis of representations of the superconformal algebras OSp(8/4,R) and OSp(8*/2N) in harmonic superspace. We present a construction of their highest-weight UIR's by multiplication of the different types of massless conformal superfields ("supersingletons"). In particular, all "short multiplets" are classified. Representations undergoing shortening have "protected dimension" and may correspond to BPS states in the dual supergravity theory in anti-de Sitter space. These results are relevant for the classification of multitrace operators in boundary conformally invariant theories as well as for the classification of AdS black holes preserving different fractions of supersymmetry. 
  We use the recently developed dimensional regularization (DR) scheme for quantum mechanical path integrals in curved space and with a finite time interval to compute the trace anomalies for a scalar field in six dimensions. This application provides a further test of the DR method applied to quantum mechanics. It shows the efficiency in higher loop computations of having to deal with covariant counterterms only, as required by the DR scheme. 
  An anti-self-dual instanton solution in Yang-Mills theory on noncommutative ${\R}^4$ with an anti-self-dual noncommutative parameter is constructed. The solution is constructed by the ADHM construction and it can be treated in the framework of the IIB matrix model. In the IIB matrix model, this solution is interpreted as a system of a Dp-brane and D(p+4)-branes, with the Dp-brane dissolved in the worldvolume of the D(p+4)-branes. The solution has a parameter that characterises the size of the instanton. The zero of this parameter corresponds to the singularity of the moduli space. At this point, the solution is continuously connected to another solution which can be interpreted as a system of a Dp-brane and D(p+4)-branes, with the Dp-brane separated from the D(p+4)-branes. It is shown that even when the parameter of the solution comes to the singularity of the moduli space, the gauge field itself is non-singular. A class of multi-instanton solutions is also constructed. 
  We investigated the bubble collisions during the first order phase transitions. Numerical results indicate that within the certain range of parameters the collision of two bubbles leads to formation of separate relatively long-lived quasilumps - configurations filled with scalar field oscillating around the true vacuum state. Energy is perfectly localized, and density is slightly pulsating around its maximum. This process is accompanied by radiation of scalar waves. 
  We consider (n + m + 1) dimensional bulk spacetime, containing flat (n + m - 1) dimensional parallel branes, with topology R^{n - 1} \times T^m. We assume that the graviton and an m-form field are the only bulk fields and that the m-form field has non vanishing components along the T^m directions only. We then find that the m-form field, with suitable bulk and brane potentials, can stabilise the radion modulus at the required value with no fine tuning. We find self tuning solutions also. 
  By evaluating string scattering amplitudes, we investigate various low energy interactions for the massless scalars on a nonabelian Dirichlet brane. We confirm the existence of couplings of closed string fields to the world-volume scalars, involving commutators of the latter. Our results are consistent with the recently proposed nonabelian world-volume actions for Dp-branes. 
  Starting with its classical parafermion algebra, we consider the quantisation of the SL(2,R)/U(1) WZNW black hole model. 
  We consider supersymmetry algebras in space-times with arbitrary signature and minimal number of spinor generators. The interrelation between super Poincar\'e and super conformal algebras is elucidated. Minimal super conformal algebras are seen to have as bosonic part a classical semimisimple algebra naturally associated to the spin group. This algebra, the Spin$(s,t)$-algebra, depends both on the dimension and on the signature of space time. We also consider maximal super conformal algebras, which are classified by the orthosymplectic algebras. 
  The chiral anomaly can be considered as an object defined either on the space of gauge potentials or on the orbit space. We will discuss the relation between the two descriptions. We will also relate to the cohomology of the group of gauge transformations. 
  We present simple models which exhibit some of the remarkable features expected to hold for the as yet unknown non-perturbative formulation of string theories. Among these are: (a) the absence of a background or embedding space for the full theory; (b) perturbative ground states (local minima of the action) having the characteristics of spaces of different dimension; (c) duality transformations between large- and small-coupling expansions; and (d) perturbative excitations of these ground states which can be interpreted as string worldsheets or p-brane worldvolumes. In this context we formulate gauge theories on arbitrary graphs and speculate concerning actions for graphs which in a continuum and/or thermodynamic limit might be related to the Einstein-Hilbert action. 
  In this note we calculate the Casimir effect of free thermal gravitons in Einstein universe and discuss how it changes the entropy bound condition proposed recently by Verlinde [hep-th/0008140] as a higher dimensional generalization of Cardy's formula for conformal field theories (CFT). We find that the graviton's Casimir effect is necessary in order not to violate Verlinde's bound for weakly coupled CFT. We also comment on the implication of this new Cardy's formula to the thermodynamics of black $p$-brane. 
  We describe a functional method to obtain the exact evolution equation of the effective action with a parameter of the bare theory. When this parameter happens to be the bare mass of the scalar field, we find a functional generalization of the Callan-Symanzik equations. Another possibility is when this parameter is the Planck constant and controls the amplitude of the fluctuations. We show the similarity of these equations with the Wilsonian renormalization group flows and also recover the usual one loop effective action. 
  The Schwinger proper-time method is an effective calculation method, explicitly gauge invariant and nonperturbative. We make use of this method to investigate the radiatively induced Lorentz and CPT-violating effects in quantum electrodynamics when an axial vector interaction term is introduced in the fermionic sector. The induced Lorentz and CPT-violating Chern-Simons term coincides with the one obtained using a covariant derivative expansion but differs from the result usually obtained in other regularization schemes. A possible ambiguity in the approach is also discussed. 
  A new instanton solution is found in the quantum-mechanical double-well potential with a four-fermion term. The solution has finite action and depends on four fermionic collective coordinates. We explain why in general the instanton action can depend on collective coordinates. 
  It is shown that a nontrivial quantum structure of our space at macroscopic scales, which may exist as a relic of quantum gravity processes in the early universe, gives rise to a new phenomenon: spontaneous origin of an interference picture in every physical process. This explains why statistical distributions in radioactivity measurements may be different from the Poisson distribution. 
  Starting with the homogeneous Bethe-Salpeter equation for two fermions, we perform a 3D reduction using a series expansion around an unspecified positive-energy instantaneous approximation of the kernel. A second series expansion is made, at the 3D level, in order to get an "hermitian" potential. The combination of both series gives a very simple result, which does not depend of the initial approximation of the kernel anymore, and could be obtained directly by starting with an approximation of the free propagator. The generalisation of this result to a system of f (=0,...N) fermions and N-f bosons is easy. 
  We study vector perturbations about four-dimensional brane-world cosmologies embedded in a five-dimensional vacuum bulk. Even in the absence of matter perturbations, vector perturbations in the bulk metric can support vector metric perturbations on the brane. We show that during de Sitter inflation on the brane vector perturbations in the bulk obey the same wave equation for a massless five-dimensional field as found for tensor perturbations. However, we present the second-order effective action for vector perturbations and find no normalisable zero-mode in the absence of matter sources. The spectrum of normalisable states is a continuum of massive modes that remain in the vacuum state during inflation. 
  We argue that higher dimensional classical, nonabelian gauge theory may lead to a lower dimensional quantum field theory due to its inherent chaotic dynamics which acts like stohastic quantization. The dimensional reduction is based upon magnetic screening effects analagous to that in nonabelian plasmas. 
  In this paper some properties of the irreducible multiplets of representation for the N = (p, q) - extended supersymmetry in one dimension are discussed. Essentially two results are here presented. At first a peculiar property of the one dimension is exhibited, namely that any multiplet containing 2d (d bosonic and d fermionic) particles in M different spin states, is equivalent to a (d,d) multiplet of just 2 spin states (all bosons and all fermions being grouped in the same spin). Later, it is shown that the classification of all multiplets of this kind carrying an irreducible representation of the N - extended supersymmetry is in one-to-one correspondence with the classification of real-valued Clifford Gamma-matrices of Weyl type. In particular, p+q is mapped into D, the space-time dimensionality, while 2d is determined to be the dimensionality of the corresponding Gamma-matrices. The implications of these results to the theory of spinning particles are analyzed. 
  We study the heat trace asymptotics defined by a time dependent family of operators of Laplace type which naturally appears for time dependent metrics. 
  We study perturbative and instanton corrections to the Operator Product Expansion of the lowest weight Chiral Primary Operators of N=4 SYM_4. We confirm the recently observed non-renormalization of various operators (notably of the double-trace operator with dimension 4 in the 20 irrep of SU(4)), that appear to be unprotected by unitarity restrictions. We demonstrate the splitting of the free-field theory stress tensor and R-symmetry current in supermultiplets acquiring different anomalous dimensions in perturbation theory and argue that certain double-trace operators also undergo a perturbative splitting into operators dual to string and two-particle gravity states respectively. The instanton contributions affect only those double-trace operators that acquire finite anomalous dimensions at strong coupling. For the leading operators of this kind, we show that the ratio of their anomalous dimensions at strong coupling to the anomalous dimensions due to instantons is the same number. 
  We consider asymtotically anti-de Sitter spacetimes in general dimensions. We review the origin of infrared divergences in the on-shell gravitational action, and the construction of the renormalized on-shell action by the addition of boundary counterterms. In odd dimensions, the renormalized on-shell action is not invariant under bulk diffeomorphisms that yield conformal transformations in the boundary (holographic Weyl anomaly). We obtain formulae for the gravitational stress energy tensor, defined as the metric variation of the renormalized on-shell action, in terms of coefficients in the asymptotic expansion of the metric near infinity. The stress energy tensor transforms anomalously under bulk diffeomorphisms broken by infrared divergences. 
  We analyze the superfield constraints of the D=4, N=3 SYM-theory using light-cone gauge conditions. The SU(3)/U(1)xU(1) harmonic variables are interpreted as auxiliary spectral parameters, and the transform to the harmonic-superspace representation is considered. Our nilpotent gauge for the basic harmonic superfield simplifies the SYM-equations of motion. A partial Grassmann decomposition of these equations yields the solvable linear system of iterative equations. 
  In [hep-th/0009173] Milton has presented a brief review of some interesting aspects of the theory of the Casimir effect. This comment is aimed at correcting some imprecise statements in that work with respect to the relevance of the Casimir effect for explaining the phenomenon of sonoluminescence. 
  We reformulate the Randall-Sundrum (RS) model on the compactified AdS by adding a term proportional to the area of the boundary to the usual gravity action with a negative cosmological constant and show that gravity can still be localized on the boundary without introducing singular brane sources. The boundary conditions now follow from the field equations, which are obtained by letting the induced metric vary on the boundary. This approach gives similar modes that are obtained in [1] and clarifies the complementarity of the RS and the AdS/CFT pictures. Normalizability of these modes is checked by an inner-product in the space of linearized perturbations. The same conclusions hold for a massless scalar field in the bulk. 
  In this letter is shown that it is possible to obtain scalar hypersurfaces in 5D N=2 SUGRA where the allowed regions with positive definite scalar metric have a non-trivial topology. This situation may aid in the construction of domain wall solutions which confine gravity to 4 dimensions. 
  We propose a M(atrix) model for N=4 $SU(k)$ Super-Yang-Mills theory compactified on $T^4$. In this model it is possible to make $T^4$ noncommutative and it is easy to turn on all 6 components of the noncommutativity on $T^4$. The action of S-duality on the noncommutativity parameters is also manifest. The M(atrix)-model is given by the large $N$ limit of a $\sigma$-model on $T^2$ whose target space is the moduli space of $k$ SU(N) instantons on $T^3\times R$. We also propose that the $SU(k)$ 2+1D $Spin(8)$ theory (the low-energy description of $k$ M2-branes) on $T^3$ corresponds to the large $N$ limit of an integral over the latter instanton moduli space. The identification is based on the fact that Euclidean wrapped M2-branes in toroidally compactified M-theory correspond to instantons in the M(atrix)-model. In the new M(atrix) models, operators with nonzero momentum along $T^3$ (or $T^4$) correspond to insertions of Wilson lines along a 1-cycle that is determined by the momentum. Momentum is conserved in the large $N$ limit. 
  We review results on several interesting phenomena in warped compactifications of M theory, as presented at Strings 2000. The behavior of gauge fields in dimensional reduction from $d+1$ to $d$ dimensions in various backgrounds is explained from the point of view of the holographic duals (and a point raised in the question session at the conference is addressed). We summarize the role of additional fields (in particular scalar fields) in 5d warped geometries in making it possible for Poincare-invariant domain wall solutions to exist to a nontrivial order in a controlled approximation scheme without fine-tuning of parameters in the $5d$ action (and comment on the status of the singularities arising in the general relativistic description of these solutions). Finally, we discuss briefly the emergence of excitations of wrapped branes in warped geometries whose effective thickness, as measured along the Poincare slices in the geometry, grows as the energy increases. 
  Gravitational holography is argued to render the cosmological constant stable against divergent quantum corrections. This provides a technically natural solution to the cosmological constant problem. Evidence for quantum stability of the cosmological constant is illustrated in a number of examples including, bulk descriptions in terms of delocalized degrees of freedom, boundary screen descriptions on stretched horizons, and non-supersymmetric conformal field theories as dual descriptions of anti-de Sitter space. In an expanding universe, holographic quantum contributions to the stress-energy tensor are argued to be at most of order the energy density of the dominant matter component. 
  Conformal symmetry is investigated within the context of axion-dilaton-modulus theory of gravity of Brans-Dicke-type. A distinction is made between general conformal symmetry and invariance under transformations of the physical units. The conformal degree of symmetry of the theory is studied when quantum fermion (lepton) modes with electromagnetic interaction are considered. Based on the requirement of invariance of the physical laws under general transformations of the units of measure, arguments are given that point at a matter action with non-minimal coupling of the dilaton to the matter fields as the most viable description of the world within the context of the model studied. The geometrical implications of the results obtained are discussed. 
  In order to investigate the composite gauge field, we consider the compositeness condition (i.e. renormalization constant $Z_3=0$) in the general non-abelian gauge field theory. We calculate $Z_3$ at the next-to-leading order in $1/N_f$ expansion ($N_f$ is the number of fermion flavors), and obtain the expression to the gauge coupling constant through the compositeness condition. Then the gauge coupling constant is proportional to $1/\sqrt{4N_f T(R)-11C_2(G)}$ where T(R) is the index for a representation R of gauge group G, and $C_2(G)$ is the quadratic Casimir. It is found that the gauge boson compositeness take place only when $N_f T(R)/C_2(G) > 11/4$, in which the asymptotic freedom in the non-abelian gauge field theory fails. 
  We review a recently-discovered link between the functional relations approach to integrable quantum field theories and the properties of certain ordinary differential equations in the complex domain. 
  We show that the algebra of functions on noncommutative space allows two different representations. One is describing the genuine noncommutative space, while another one can be rewritten in commutative form by a redefinition of generators. 
  The main object of the proposed theory is not a pseudometric, but a symmetric affine connection on the Minkowski space. The coefficients of this connection have one upper and two lower indices. These coefficients are symmetric with respect to the permutation of the lower indices. We identify the convolution of the connection coefficients with the vector - potential of the electromagnetic field. Then the gravity is the Lorentz force of this electromagnetic field. 
  We define an iterative procedure to obtain a non-abelian generalization of the Born-Infeld action. This construction is made possible by the use of the severe restrictions imposed by kappa-symmetry. We have calculated all bosonic terms in the action up to terms quartic in the Yang-Mills field strength and all fermion bilinear terms up to terms cubic in the field strength. Already at this order the fermionic terms do not satisfy the symmetric trace-prescription. 
  Three dimensional SO(3) gauged Skyrme models characterised by specific potentials imposing special asymptotic values on the chiral field are considered. These models are shown to support finite energy solutions with nonvanishing magnetic and electrix flux, whose energies are bounded from below by two distinct charges - the magnetic (monopole) charge and a non-integer version of the Baryon charge. Unit magnetic charge solutions are constructed numerically and their properties characterised by the chosen asymptotics and the Skyrme coupling are studied. For a particular value of the chosen asymptotics, charge-2 axially symmetric solutions are also constructed and the attractive nature of the like-monopoles of this system are exhibited. As an indication towards the possible existence of large clumps of monopoles, some consideration is given to axially symmetric monopoles of charges-2,3,4. 
  We show that an $SL(2,R)_L \times SL(2,R)_R$ Chern-Simons theory coupled to a source on a manifold with the topology of a disk correctly describes the entropy of the AdS$_3$ black hole. The resulting boundary WZNW theory leads to two copies of a twisted Kac-Moody algebra, for which the respective Virasoro algebras have the same central charge $c$ as the corresponding untwisted theory. But the eigenvalues of the respective $L_0$ operators are shifted. We show that the asymptotic density of states for this theory is, up to logarithmic corrections, the same as that obtained by Strominger using the asymptotic symmetry of Brown and Henneaux. 
  In this note we reexamine the possibility of constructing stable non-supersymmetric theories that exhibit an exponential density of states. For weakly coupled closed strings there is a general theorem, according to which stable theories with an exponential density of states must exhibit an almost exact cancellation of spacetime bosons and fermions (not necessarily level by level). We extend this result to open strings by showing that if the above cancellation between bosons and fermions does not occur, the open strings do not decouple from a closed string tachyon even in the NCOS scaling limit. We conclude with a brief comment on the proposed generalization of the AdS/CFT correspondence to non-supersymmetric theories. 
  An overview of new 4d supersymmetric gauge theories with 2-form gauge potentials constructed by various authors during the past five years is given. The key role of three particular types of interaction vertices is emphasized. These vertices are used to develop a connecting perspective on the new models and to distinguish between them. One example is presented in detail to illustrate characteristic features of the models. A new result on couplings of 2-form gauge potentials to Chern-Simons forms is presented. 
  We present solution generating techniques which permit to construct exact inhomogeneous and anisotropic cosmological solutions to a four-dimensional low energy limit of string theory containing non-minimally interacting electromagnetic and dilaton fields. Some explicit homogeneous and inhomogeneous cosmological solutions are constructed. For example, inhomogeneous exact solutions presenting Gowdy - type EMD universe are obtained. The asymptotic behaviour of the solutions is investigated. The asymptotic form of the metric near the initial singularity has a spatially varying Kasner form. The character of the space-time singularities is discussed. The late evolution of the solutions is described by a background homogeneous and anisotropic universe filled with weakly interacting gravitational, dilatonic and electromagnetic waves. 
  We establish that in Quantum Chromodynamics (QCD) at zero temperature, SU_(L+R)(N_F) exhibits the vector mode conjectured by Georgi and SU_(L-R)(N_F)is realized in either the Nambu-Goldstone mode or else the axial-vector charge is also screened from view at infinity. The Wigner-Weyl mode is ruled out unless the beta function in QCD develops an infrared stable zero. 
  The six-dimensional (2,0) field theory admits a generalized ``noncommutative'' deformation associated with turning on a large null 3-form field strength. This theory is studied using its discrete light-cone formulation as quantum mechanics on a blow-up of the ADHM moduli space. We show how to interpret the ADHM manifold as configurations of open membranes, and check our results against basic space-time considerations. 
  In the framework of nonassociative geometry (hep-th/0003238) a unified description of continuum and discrete spacetime is proposed. In our approach at the Planck scales the spacetime is described as a so-called "diodular discrete structure" which at large spacetime scales `looks like' a differentiable manifold. After a brief review of foundations of nonassociative geometry,we discuss the nonassociative smooth and discrete de Sitter spacetimes. 
  We evaluate the one-loop correction to the spectrum of Kaluza-Klein system for the $\phi^3$ model on $R^{1,d}\times (T_\theta^2)^L$, where $1+d$ dimensions are the ordinary flat Minkowski spacetimes and the extra dimensions are the L two-dimensional noncommutative tori with noncommutativity $\theta$. The correction to the Kaluza-Klein mass spectrum is then used to compute the Casimir energy. The results show that when $L>2$ the Casimir energy due to the noncommutativity could give repulsive force to stabilize the extra noncommutative tori in the cases of $d = 4n - 2$, with $n$ a positive integral. 
  We propose a regularization-independent method for studying a renormalizable field theory nonperturbatively through its Dyson-Schwinger equations. Using QED_4 as an example, we show how the coupled equations determining the nonperturbative fermion and photon propagators can be written entirely in terms of renormalized quantities, which renders the equations manifestly finite in a regularization-independent manner. As an illustration of the technique, we apply it to a study of the fermion propagator in quenched QED_4 with the Curtis-Pennington electron-photon vertex. At large momenta the mass function, and hence the anomalous mass dimension gamma_m(alpha), is calculated analytically and we find excellent agreement with previous work. Finally, we show that for the CP vertex the perturbation expansion of gamma_m(alpha) has a finite radius of convergence. 
  In this paper we derive the general expression of a one-loop effective potential of the nonintegrable phases of Wilson lines for an SU(N) gauge theory with a massless adjoint fermion defined on the spactime manifold $R^{1,d-3}\times T^2$ at finite temperature and fermion density. The Phase structure of the vacuum is presented for the case with $d=4$ and N=2 at zero temperature. It is found that gauge symmetry is broken and restored alternately as the fermion density increases, a feature not found in the Higgs mechanism. It is the manifestation of the quantum effects of the nonintegrable phases. 
  We investigate the dynamics of vacuum brane and the bulk in dilatonic brane world. We present exact dynamical solutions which describe the vacuum dilatonic brane world. We find that the solution has initial singularity and singularity at spatial infinity. 
  The scaling form of the free--energy near a critical point allows for the definition of various universal ratios of thermodynamical amplitudes. Together with the critical exponents they characterize the universality classes and may be useful experimental quantities. We show how these universal quantities can be computed for a particular class of universality by using several Quantum Field Theory methods 
  Using a D0-brane as a probe, we study the spacetime geometry in the neighborhood of N D-branes in matrix theory. We find that due to fermionic zero modes, the coordinates of the probe in the transverse directions are noncommutative, and the angular part is a fuzzy sphere. 
  In the world line representation of the fermionic effective action for QCD the interaction between Fermions and the gauge field is contained in the fermionic Wilson loop, namely the Wilson loop for a spin-half particle. It is argued that a string representation of the fermionic Wilson loop can provide a link connecting QCD with a dual description of a meson as a quark and an antiquark connected by a string. This is illustrated by obtaining such a representation in compact U(1) gauge theories. The resulting description contains information about the interaction of the spin of the quark with the world sheet degrees of freedom. Such interactions may be of importance in the realization of chiral symmetry in the string picture of QCD, and for delineating the possible presence of a world sheet supersymmetry in QCD strings. 
  Higher-derivative terms in the string and M-theory effective actions are strongly constrained by supersymmetry. Using a mixture of techniques, involving both string amplitude calculations and an analysis of supersymmetry requirements, we determine the supersymmetric completion of the R^4 action in eleven dimensions to second order in the fermions, in a form compact enough for explicit further calculations. Using these results, we obtain the modifications to the field transformation rules and determine the resulting field-dependent modifications to the coefficients in the supersymmetry algebra. We then make the link to the superspace formulation of the theory and discuss the mechanism by which higher-derivative interactions lead to modifications to the supertorsion constraints. For the particular interactions under discussion we find that no such modifications are induced. 
  We show that the Skyrme theory possesses a submodel with an infinite number of local conserved currents. The constraints leading to the submodel explore a decomposition of SU(2) with a complex field parametrizing the symmetric space SU(2)/U(1) and a real field in the direction of U(1). We demonstrate that the Skyrmions of topological charges $\pm 1$ belong to such integrable sector of the theory. Our results open ways to the development of exact methods, compensating for the non-existence of a BPS type sector in the Skyrme theory. 
  We examine the Hagedorn behavior of little string theory using its conjectured duality with near-horizon NS5-branes. In particular, by studying the string-corrected NS5-brane supergravity solution, it is shown that tree-level corrections to the temperature vanish, while the leading one-loop string correction generates the correct temperature dependence of the entropy near the Hagedorn temperature. Finally, the Hagedorn behavior of ODp-brane theories, which are deformed versions of little string theory, is considered via their supergravity duals. 
  By exploiting the correspondence between the Cardy boundary state in SU(2) group manifold and the BPS D3-brane configuration in the full asymptotically flat geometry of NS5-branes, we show that the Hanany-Witten effect in 10D background is encoded in the Cardy boundary states. The two RR Page D0 charges of the $n$-th spherical D2-brane due to the contraction to $e$ or ($-e$) is interpreted, and attributed to the Hanany-Witten effect. 
  It has been claimed that whereas scalars can be bound to a Randall-Sundrum brane, higher p-form potentials cannot, in contradiction with the Hodge duality between 0-form and 3-form potentials in the five-dimensional bulk. Here we show that a 3-form in the bulk correctly yields a 2-form on the brane, in complete agreement with both bulk and brane duality. We also emphasize that the phenomenon of photon screening in the Randall-Sundrum geometry is ruled out by the bulk Einstein equation. 
  We analyze the nature of space-time nonlocality in string theory. After giving a brief overview on the conjecture of the space-time uncertainty principle, a (semi-classical) reformulation of string quantum mechanics, in which the dynamics is represented by the noncommutativity between temporal and spatial coordinates, is outlined. The formalism is then compared to the space-time noncommutative field theories associated with nonzero electric B-fields. 
  Orientable open string theories containing both bosons and fermions without the GSO projection are expected to have the 10 dimensional N=2(A) space-time supersymmetry in a spontaneously broken phase. We study the low-energy theorem for the nonlinearly realized N=2 supersymmetry using the effective action for an unstable D9-brane. It is explicitly confirmed that the 4-fermion open string amplitudes without the GSO projection obey the low-energy theorem derived from the nonlinear N=2 supersymmetry. An intimate connection between the existence of the hidden supersymmetry and the open-open string (s-t) duality is pointed out. 
  A description of generalized coherent states and geometric phases in the light of the general theory of smooth loops is given. 
  We have calculated the energy levels of the hydrogen atom and as well the Lamb shift within the noncommutative quantum electrodynamics theory. The results show deviations from the usual QED both on the classical and on the quantum levels. On both levels, the deviations depend on the parameter of space/space noncommutativity. 
  Within a self-consistent framework of q-deformed Heisenberg algebra and its equivalent framework of q-deformed boson commutation relations, which relate to the under-cutting phenomenon of Heisenberg's minimal uncertainty relation, special q-deformed squeezed states are constructed. Besides the similar local maximum squeezing as the one in the undeformed case, new strong squeezing appears when the amplitude of the related coherent state increases to large values. A critical phenomenon appears at a large value of the amplitude: the variance of one component of the quadrature of the light field approaches zero, but the variance of the corresponding conjugate quantity remains finite, which is a surprising deviation from Heisenberg's uncertainty relation. The qualitative character exposed by this q-squeezed state may provide some evidence about q-deformed effects in current experiments. 
  We extend the general method of hep-th/0009192 to compute the consistent gauge anomaly for noncommutative 4d SSYM coupled to chiral matter. The choice of the minimal homotopy path allows us to obtain a simple and compact result. We perform the reduction to components in the WZ gauge proving that our result contains, as lowest component, the bosonic chiral anomaly for noncommutative YM theories recently obtained in literature. 
  We focus on the massive Thirring model in 1+1 dimensions at finite temperature and non-zero chemical potential, and comment on some parallels between this model and QCD. In QCD, calculations of physical quantities such as transport coefficients are extremely difficult. In the massive Thirring model, similar calculations are greatly simplified by exploiting the duality which exists with the sine-Gordon model and its relation, at high temperature, to the exactly solvable classical Coulomb gas on the line. 
  We determine all D-branes in the non-tachyonic 0'B orientifold, examine their world-volume anomalies and study orbifold compactifications. We find that the spectrum of the D-branes contains chiral fermions in the symmetric, antisymmetric and fundamental representations of (unitary) gauge groups on the branes. The cancellation of the world-volume anomalies requires Wess-Zumino terms which we determine explicitly. We examine a non-tachyonic compactification to 9D whose closed part interpolates between 0B and IIB and revisit compactifications on orbifolds. The D3-brane allows to conjecture, via the AdS/CFT correspondence, a supergravity dual to a non-supersymmetric and infrared-free gauge theory. The D-string gives hints concerning the S-dual of the 0'B orientifold. 
  We explore the applicability of the exact renormalization group to the study of tunnelling phenomena. We investigate quantum-mechanical systems whose energy eigenstates are affected significantly by tunnelling through a barrier in the potential. Within the approximation of the derivative expansion, we find that the exact renormalization group predicts the correct qualitative behaviour for the lowest energy eigenvalues. However, quantitative accuracy is achieved only for potentials with small barriers. For large barriers, the use of alternative methods, such as saddle-point expansions, can provide quantitative accuracy. 
  The interpretation of closed fundamental strings as solitons in open string field theory is reviewed. Noncommutativity is introduced to facilitate an explicit construction. The tension is computed exactly and the correct spectrum is recovered at long wave length. 
  We present the supersymmetrisation of the anomaly-related R^4 term in eleven dimensions and show that it induces no non-trivial modifications to the on-shell supertranslation algebra and the superspace torsion constraints before inclusion of gauge-field terms. 
  We show that, in the absence of matter in the bulk, the Einstein equations and the Gauss-normal form of the metric place stringent restrictions on the form of the event horizon in a brane world. As a consequence, the off-brane extension of the standard 4-D Schwarzschild horizon in the Randall-Sundrum $AdS_5$ spacetime, as it is viewed from the brane can only be of a tubular shape, instead of a pancake shape. When it is viewed from the $AdS_5$ horizon, such a tubular horizon is absent. 
  We develop algebro-geometrical approach for the open Toda lattice. For a finite Jacobi matrix we introduce a singular reducible Riemann surface and associated Baker-Akhiezer functions. We provide new explicit solution of inverse spectral problem for a finite Jacoby matrix. For the Toda lattice equations we obtain the explicit form of the equations of motion, the symplectic structure and Darboux coordinates. We develop similar approach for 2D open Toda. Explaining some the machinery we also make contact with the periodic case. 
  A recent investigation of SU(2) Yang-Mills theory found several classical solutions with bad behaviour at infinity : one of the potential components oscillated and another tended to infinity. In this paper we apply an idea due to Heisenberg about the quantization of strongly interacting nonlinear fields to these classical singular solutions. We find that this quantization procedure eliminates the bad long distance features while retaining the interesting short distance aspects of these solutions. 
  We discuss the cosmology of a 3-brane embedded in a 5D bulk space-time with a cosmological constant when an intrinsic curvature Ricci scalar is included in the brane action. After deriving the `brane-Friedmann' equations for a Z_2 symmetrical metric, we focus on the case of a Minkowski bulk. We show that there exist two classes of solutions, close to the usual Friedmann-Lemaitre-Robertson-Walker cosmology for small enough Hubble radii. When the Hubble radius gets larger one either has a transition to a fully 5D regime or to a self-inflationary solution which produces a late accelerated expansion. We also compare our results with a perturbative approach and eventually discuss the embedding of the brane into the Minkowski space-time. This latter part of our discussion also applies when no intrinsic curvature term is included. 
  We construct a star product associated with an arbitrary two dimensional Poisson structure using generalized coherent states on the complex plane. From our approach one easily recovers the star product for the fuzzy torus, and also one for the fuzzy sphere. For the latter we need to define the `fuzzy' stereographic projection to the plane and the fuzzy sphere integration measure, which in the commutative limit reduce to the usual formulae for the sphere. 
  The quantization of the SU(2)$\times $U(1) gauge-symmetric electroweak theory is performed in the Hamiltonian path-integral formalism. In this quantization, we start from the Lagrangian given in the unitary gauge in which the unphysical Goldstone fields are absent, but the unphysical longitudinal components of the gauge fields still exist. In order to eliminate the longitudinal components, it is necessary to introduce the Lorentz gauge conditions as constraints. These constraints may be incorporated into the Lagrangian by the Lagrange undetermined multiplier method. In this way, it is found that every component of a four-dimensional vector potential has a conjugate counterpart. Thus, a Lorentz-covariant quantization in the Hamiltonian path-integral formalism can be well accomplished and leads to a result which is the same as given by the Faddeev-Popov approach of quantization. 
  Open superstring theory is formulated in terms of a nondegenerate supertranslation algebra. A supercharge for a tachyonic superstring can be also defined classically by taking into account the leakage of the supercurrent which is compensated by fermionic and bosonic auxiliary fields. The anticommutator of two supercharges of the tachyonic superstring does not contain the zero eigenvalue and so this string is not a BPS state. Brane-antibrane annihilation scenarios are described by these superalgebras defined on the sum of world-volumes of a D-brane, an anti-D-brane and a tachyonic superstring. 
  A discrete symmetry of the string field tachyon condensate noted by Hata and Shinohara is identified as a discrete subgroup of an SU(1,1) symmetry acting on the ghost coordinates. This symmetry, known from early studies of free gauge invariant string field actions, extends to off-shell interactions only for very restricted kinds of string vertices, among them the associative vertex of cubic string field theory. It follows that the string field relevant for tachyon condensation can be trimmed down to SU(1,1) singlets. 
  I consider the giant gravitons in AdS5 x S5. By numerical simulation, I show a strong indication that there is no instanton solution describing the direct tunneling between the giant graviton in the S5 and its dual counterpart in the AdS5. I argue that it supports the supersymmetry breaking scenario suggested in hep-th/0008015 
  The dimensional reduction of eleven dimensional supergravity on a Calabi-Yau manifold gives N=2 supergravity in five dimensions with $h_{1,1}$ vector and $h_{2,1}+1$ hypermultiplets. In this paper instanton solutions are constructed which are responsible for nonperturbtative corrections to the hypermultiplet moduli spaces. These instantons are wrapped Euclidean membranes and fivebranes. For vanishing fivebrane charge the BPS conditions for these solutions define a flow in the hypermultiplet moduli space and are isomorphic to the attractor equations for four dimensional black holes. 
  We show that the Gauss-Bonnet term is the only consistent curvature squared interaction in the Randall-Sundrum model and various static and inflationary solutions can be found. And from metric perturbations around the RS background with a single brane embedded, we also show that for a vanishing Gauss-Bonnet coefficient, the brane bending allows us to reproduce the 4D Einstein gravity at the linearized level. 
  We show how OSp(1|32) gives a unifying framework to describe d=10 type II string theories, d=11 M-theory and d=12 F-theory. The theories are related by different identifications of their symmetry operators as generators of OSp(1|32). T- and S-dualities are recognized as redefinitions of generators. Some (s,t) signatures of spacetime allow reality conditions on the generators. All those that allow a real structure are related again by redefinitions within the algebra, due to the fact that the algebra OSp(1|32) has only one real realization. The redefinitions include space/space, time/time and space/time dualities. A further distinction between the theories is made by the identification of the translation generator. This distinguishes various versions of type II string theories, in particular the so-called *-theories, characterized by the fact that the P_0 generator is not the (unique) positive-definite energy operator in the algebra. 
  The d=10 type II string theories, d=11 M-theory and d=12 F-theory have the same symmetry group. It can be viewed either as a subgroup of a conformal group OSp(1|64) or as a contraction of OSp(1|32). The theories are related by different identifications of their symmetry operators as generators of OSp(1|32). T- and S-dualities are recognized as redefinitions of generators. Some (s,t) signatures of spacetime allow reality conditions on the generators. All those that allow a real structure are related again by redefinitions within the algebra, due to the fact that the algebra OSp(1|32) has only one real realization. The redefinitions include space/space, time/time and space/time dualities. A further distinction between the theories is made by the identification of the translation generator. This distinguishes various versions of type II string theories, in particular the so-called *-theories, characterized by the fact that the P_0 generator is not the (unique) positive-definite energy operator in the algebra. 
  We present a method based on mutations of helices which leads to the construction (in the large volume limit) of exceptional coherent sheaves associated with the $(\sum_al_a=0)$ orbits in Gepner models. This is explicitly verified for a few examples including some cases where the ambient weighted projective space has singularities not inherited by the Calabi-Yau hypersurface. The method is based on two conjectures which lead to the analog,in the general case, of the Beilinson quiver for $\BP^n$. We discuss how one recovers the McKay quiver using the gauged linear sigma model (GLSM) near the orbifold or Gepner point in K\"ahler moduli space. 
  Given a truncated perturbation expansion of a physical quantity, one can, under certain circumstances, obtain lower or upper bounds (or both) to the sum of the full perturbation series by using the Borel transform and a variational conformal map. The method is illustrated by applying it to various mathematical toy-models for which exact results are known. One of these models is used to exemplify how non-perturbative contributions supplement the sum of a Borel-nonsummable series to give the final exact and unambiguous result. Finally, the method is applied to some physical problems. In particular, some speculations are made on the phase of quantum electrodynamics at super-high temperatures from a study of its perturbative free-energy density. 
  We discuss the compactification of type I strings on a torus with additional background gauge flux on the D9-branes. The solutions to the cancellation of the RR tadpoles display various phenomenologically attractive features: supersymmetry breaking, chiral fermions and the opportunity to reduce the rank of the gauge group as desired. We also point out the equivalence of the concept of various different background fields and noncommutative deformations of the geometry on the individual D9-branes by constructing the relevant boundary states to describe such objects. 
  The spectral problem where the field satisfies Dirichlet conditions on one part of the boundary of the relevant domain and Neumann on the remainder is discussed. It is shown that there does not exist a classical asymptotic expansion for short time in terms of fractional powers of $t$ with locally computable coefficients. 
  We obtain charged rotating black hole solutions to the theory of Einstein-Maxwell gravity with cosmological constant in five dimensions. Some of the physical properties of these black holes are discussed. 
  We analyze the finite temperature deconfining phase transition in 2+1 dimensional Georgi-Glashow model. We show explicitly that the transition is due to the restoration of the magnetic $Z_2$ symmetry and that it is in the Ising universality class. We find that neglecting effects of the charged $W$ bosons leads to incorrect predictions for the value of the critical temperature and the universality class of the transition, as well as for various correlation functions in the high temperature phase. We derive the effective action for the Polyakov loop in the high temperature phase and calculate the correlation functions of magnetic vortex operators. 
  We investigate the cosmology of (4+1)-dimensional gravity coupled to a scalar field and a bulk anisotropic fluid within the context of the single-brane Randall-Sundrum scenario. Assuming a separable metric, a static fifth radius and the scalar to depend only on the fifth direction, we find that the warp factor is given as in the papers of Kachru, Schulz and Silverstein [hep-th/0001206, hep-th/0002121] and that the cosmology on a self-tuning brane is standard. In particular, for a radiation-dominated brane the pressure in the fifth direction vanishes. 
  The effective action for quantum fields on a $d$-dimensional spacetime can be computed using a non local expansion in powers of the curvature. We show explicitly that, for conformal fields and up to quadratic order in the curvature, the non local effective action is equivalent to the $d+1$ action for classical gravity in $AdS_{d+1}$ restricted to a $d-1$-brane. This generalizes previous results about quantum corrections to the Newtonian potential and provides an alternative method for making local a non-local effective action. The equivalence can be easily understood by comparing the Kallen-Lehmann decomposition of the classical propagator with the spectral representation of the non local form factors in the quantum effective action. 
  Higher order terms in the effective action of noncommutative gauge theories exhibit generalizations of the *-product (e.g. *' and *-3). These terms do not manifestly respect the noncommutative gauge invariance of the tree level action. In U(1) gauge theories, we note that these generalized *-products occur in the expansion of some quantities that are invariant under noncommutative gauge transformations, but contain an infinite number of powers of the noncommutative gauge field. One example is an open Wilson line. Another is the expression for a commutative field strength tensor in terms of the noncommutative gauge field. Seiberg and Witten derived differential equations that relate commutative and noncommutative gauge transformations, gauge fields and field strengths. In the U(1) case we solve these equations neglecting terms of fourth order in the gauge field but keeping all orders in the noncommutative parameter. 
  We consider the quantum effects of bulk matter (scalars, spinors) in the Randall-Sundrum AdS$_5$ brane-world at nonzero temperature. The thermodynamic energy (modulus potential) is evaluated at low and high temperatures. This potential has an extremum which could be a minimum in some cases (for example, for a single fermion). That suggests a new dynamical mechanism to stabilize the thermal AdS$_5$ brane-world. It is shown that the brane separation required to solve the hierarchy scale problem may occur at a quite low temperature. A natural generalization in terms of the AdS/CFT correspondence (through the supergravity thermal contribution) is also possible. 
  We introduce a class of supersymmetric cycles in spacetimes of the form AdS times a sphere or $T^{1,1}$ which can be considered as generalizations of the giant gravitons. Branes wrapped on these cycles preserve $1\over 2$, $1\over 4$ or $1\over 8$ of the supersymmetry. On the CFT side these configurations correspond to superpositions of the large number of BPS states. 
  We study topology change in M theory compactifications on Calabi-Yau three-folds in the presence of G flux (the four form field strength). In particular, we discuss vacuum solutions in strongly coupled heterotic string theory in which the topology change is inevitable within a single spacetime background. For rather generic choices of initial conditions, the field equations drive the Kahler moduli outside the classical moduli space of a Calabi-Yau manifold. Consistency of the solution suggests that degenerate flop curves - just as wrapped M theory fivebranes - carry magnetic charges under the four form field strength. 
  We prove the no-go theorem that the gravivector $(h_{5\mu})$ and graviscalar $(h_{55})$ cannot have any on-shell propagation on the Randall-Sundrum (RS) brane. For this purpose, we analyze all of their linearized equations with the de Donder gauge (5D transverse-tracefree gauge).   But we do not introduce any matter source. We use the Z$_2$-symmetry argument and their ($h_{5\mu}, h_{55}$) compatibility conditions with the tensor $h_{\mn}$-equation. It turns out that $h_{55}$ does not have any bulk (massive) and brane (massless) propagations. Although $h_{5\mu}$ has a sort of massive propagations, they do not belong to the physical solution. Hence we confirm that the Randall-Sundrum gauge suffices the on-shell brane physics. 
  We consider properties of connected diagrams with fermion-photon interaction and such fermion and photon propagators and vertex function that the values of these diagrams are finite. We establish the properties of these propagators and vertex function that imply that these diagrams are invariant under C, P, T, CP, CT, PT, or CPT transformations up to some phase factor common to each process. We introduce eight new transformations related to Hermiticity and establish the conditions under which they leave the tree transition probabilities invariant. We determine such general Lorentz form-invariant fermion and photon propagators and fermion-photon vertex functions that make diagrams Lorentz-invariant. 
  We derive constraints on the non-perturbative 3-point fermion-boson transverse vertex in massless QED3 from its perturbative calculation to order $\alpha$. We also check the transversality condition to two loops and evaluate the fermion propagator to the same order. We compare a conjecture of the non-perturbative vertex by Burden and Tjiang against our results and comment on its draw backs. Our calculation calls for the need to construct a non-perturbative form for the fermion-boson vertex which agrees with its perturbative limit to ${\cal O}(\alpha)$. 
  We solve the Schwinger-Dyson equations for (2+1)-dimensional QED in the presence of a strong external magnetic field. The calculation is done at finite temperature and the fermionic self energy is not supposed to be momentum-independent, which is the usual simplification in such calculations. The phase diagram in the temperature-magnetic field plane is determined. For intermediate magnetic fields the critical temperature turns out to have a square root dependence on the magnetic field, but for very strong magnetic fields it approaches a B-independent limiting value. 
  The dynamics governing the evolution of a many body system is constrained by a nonabelian local symmetry. We obtain explicit forms of the global macroscopic condition assuring that at the microscopic level the evolution respects the overall symmetry constraint. 
  The paper is concerned with thermostatistics of both $D$-dimensional Bose and Fermi ideal gases in a confining potential of type $Ar^{n}+Br^{-n}$. The investigation is performed in the framework of the semiclassical approximation. Some physical quantities for such systems are derived, like density of states, density profiles and number of particles. Bose-Einstein condensation (BEC) is discussed in the high and low temperature regimes. 
  General formalism of quantum field theory and addition theorem for Bessel functions are applied to derive formula for Casimir-Polder energy of interaction between a polarizable particle and a dilute dielectric ball. The equivalence of dipole-dipole interaction and Casimir energy for dilute homogeneous dielectrics is shown. A novel method is used to derive Casimir energy of a dilute dielectric ball without divergences in calculations. Physically realistic model of a dilute ball is discussed. Different approaches to the calculation of Casimir energy of a dielectric ball are reviewed. 
  We consider cosmological models where the universe, governed by Einstein's equations, is a piece of a five dimensional double-sided anti-de Sitter spacetime (that is, a "$Z_2$-symmetric bulk") with matter confined to its four dimensional Robertson-Walker boundary or "brane". We study the perturbations of such models. We use conformally minkowskian coordinates to disentangle the contributions of the bulk gravitons and of the motion of the brane. We find the restrictions put on the bulk gravitons when matter on the brane is taken to be a scalar field and we solve in that case the brane perturbation equations. 
  An approach is proposed enabling to effectively describe the behaviour of a bosonic system. The approach uses the quantum group $GL_{p,q}(2)$ formalism. In effect, considering a bosonic Hamiltonian in terms of the $GL_{p,q}(2)$ generators, it is shown that its thermodynamic properties are connected to deformation parameters $p$ and $q$. For instance, the average number of particles and the pressure have been computed. If $p$ is fixed to be the same value for $q$, our approach coincides perfectly with some results developed recently in this subject. The ordinary results, of the present system, can be found when we take the limit $p=q=1$. 
  We investigate further on the correspondence between branes on a Calabi-Yau in the large volume limit and in the orbifold limit. We conjecture a new procedure which improves computationally the McKay correspondence and prove it in a non trivial example. We point out the relevance of helices and try to draw some general conclusions about Beilinson theorem and McKay correspondence. 
  We discuss what information can be safely extracted from background independent off-shell string theory. The major obstacle in doing so is that renormalization conditions of the underlying world-sheet theories are not exactly known. To get some insight, we first consider the tachyon and gauge field backgrounds and carry out computations in different renormalization schemes for both, bosonic string and superstring. Next, we use a principle of universality (renormalization scheme independence) to somehow compensate the missing of the renormalization conditions and get information we are looking for. It turns out that some asymptotics which are responsible for the potentials only obey the principle of universality. 
  This review gives results on vertex operators for the Type IIB superstring in an AdS3 x S3 background with Ramond-Ramond flux, which were presented at Strings 2000. Constraint equations for these vertex operators are derived, and their components are shown to satisfy the supergravity linearized equations of motion for the six-dimensional (2,0) theory of a supergravity and tensor multiplet expanded around AdS3 x S3 spacetime. 
  A general non-commutative quantum mechanical system in a central potential $V=V(r)$ in two dimensions is considered. The spectrum is bounded from below and for large values of the anticommutative parameter $\theta $, we find an explicit expression for the eigenvalues. In fact, any quantum mechanical system with these characteristics is equivalent to a commutative one in such a way that the interaction $V(r)$ is replaced by $V = V ({\hat H}_{HO}, {\hat L}_z)$, where ${\hat H}_{HO}$ is the hamiltonian of the two-dimensional harmonic oscillator and ${\hat L}_z$ is z- component of the angular momentum. For other finite values of $\theta$ the model can be solved by using perturbation theory. 
  We construct new exact BPS solitons in various noncommutative gauge theories by the ``gauge'' transformation of known BPS solitons. This ``gauge'' transformation introduced by Harvey, Kraus and Larsen adds localized solitons to the known soliton. These solitons include, for example, the bound state of a noncommutative Abelian monopole and N fluxons at threshold. This corresponds, in superstring theories, to a D-string which attaches to a D3-brane and N D-strings which pierce the D3-brane, where all D-strings are parallel to each other. 
  In this talk, I review how four dimensional stationary supergravity solutions that are more general than spherically symmetric black holes emerge naturally in the low energy description of BPS states in type II Calabi-Yau compactifications. An explicit construction of multicenter solutions using single center attractor flows as building blocks is presented, and some interesting properties of these solutions are examined. We end with a brief remark on non-BPS configurations. 
  We study the topological zero mode sector of type II strings on a K\"ahler manifold $X$ in the presence of boundaries. We construct two finite bases, in a sense bosonic and fermionic, that generate the topological sector of the Hilbert space with boundaries. The fermionic basis localizes on compact submanifolds in $X$. A variation of the FI terms interpolates between the description of these ground states in terms of the ring of chiral fields at the boundary at small volume and helices of exceptional sheaves at large volume, respectively. The identification of the bosonic/fermionic basis with the dual bases for the non-compact/compact K-theory group on $X$ gives a natural explanation of the McKay correspondence in terms of a linear sigma model and suggests a simple generalization of McKay to singular resolutions. The construction provides also a very effective way to describe D-brane states on generic, compact Calabi--Yau manifolds and allows to recover detailed information on the moduli space, such as monodromies and analytic continuation matrices, from the group theoretical data of a simple orbifold. 
  Pseudoaffine theories are characterized by formal replacement of the level to the fractional number: $k\to\frac{k}{q}$, where $q$ is integer strange to $k(g+k)$ ($g$ - dual Coxeter number). An example of "forbidden" $q$ is considered (SU(2), $q=2$). The fusions of obtained theory are similar to the affine ones. Spectra of minimal models are calculated. 
  We review the physics of topological objects in QCD. Topics include: solitons, vortices, magnetic monopoles, instantons, (effective theories of) confinement. 
  We consider the gauge invariance of the standard Yang-Mills model in the framework of the causal approach of Epstein-Glaser and Scharf and determine the generic form of the anomalies. The method used is based Epstein-Glaser approach to renormalization theory. In the case of quantum electrodynamics we obtain quite easily the absence of anomalies in all orders. 
  We show that nodal points of ground states of some quantum systems with magnetic interactions can be identified in simple geometric terms. We analyse in detail two different archetypical systems: i) a planar rotor with a non-trivial magnetic flux $\Phi$, ii) Hall effect on a torus. In the case of the planar rotor we show that the level repulsion generated by any reflection invariant potential $V$ is encoded in the nodal structure of the unique vacuum for $\theta=\pi$. In the second case we prove that the nodes of the first Landau level for unit magnetic charge appear at the crossing of the two non-contractible circles $\alpha_-$, $\beta_-$ with holonomies $h_{\alpha_-}(A)= h_{\beta_-}(A)=-1$ for any reflection invariant potential $V$. This property illustrates the geometric origin of the quantum translation anomaly. 
  We consider the fixed-dimension perturbative expansion. We discuss the nonanalyticity of the renormalization-group functions at the fixed point and its consequences for the numerical determination of critical quantities. 
  We compute the graviton induced corrections to Maxwell's equations in the one-loop and weak field approximations. The corrected equations are analogous to the classical equations in anisotropic and inhomogeneous media. We analyze in particular the corrections to the dispersion relations. When the wavelength of the electromagnetic field is much smaller than a typical length scale of the graviton two-point function, the speed of light depends on the direction of propagation and on the polarisation of the radiation. In the opposite case, the speed of light may also depend on the energy of the electromagnetic radiation. We study in detail wave propagation in two special backgrounds, flat Robertson-Walker and static, spherically symmetric spacetimes. In the case of a flat Robertson-Walker gravitational background we find that the corrected electromagnetic field equations correspond to an isotropic medium with a time-dependent effective refractive index. For a static, spherically symmetric background the graviton fluctuations induce a vacuum structure which causes birefringence in the propagation of light. 
  We re-examine the historically important decay of the neutral pion into two photons. Schwinger's Equivalence Theorem is confirmed. We then consider radiative corrections to the famous Adler-Bell-Jackiw (ABJ) anomaly. The result depends crucially on a physically motivated regularization scheme. Our approach is largely based on Schwinger's source (dispersion) method. 
  The canonical formalism in classical theory of QCD is constructed on a space-like hypersurface. The Poisson bracket on the space-like hypersurface is defined and it plays an important role to describe every algebraic relation in the canonical formalism into Lorentz covariant form. Surface integrals are introduced as alternatives of field equations for quarks, gluons, and Faddeev-Popov ghosts. It is shown that deformations of the space-like hypersurface for surface integrals are generated by the interaction term of QCD Hamiltonian density. By converting the Poisson bracket on the space-like hypersurface to four-dimensional commutator, we can pass over to QCD in the Heisenberg picture without spoiling the explicit Lorentz covariance. 
  This paper investigates Starobinsky's model of inflation driven by the trace anomaly of conformally coupled matter fields. This model does not suffer from the problem of contrived initial conditions that occurs in most models of inflation driven by a scalar field. The universe can be nucleated semi-classically by a cosmological instanton that is much larger than the Planck scale provided there are sufficiently many matter fields. There are two cosmological instantons: the four sphere and a new ``double bubble'' solution. This paper considers a universe nucleated by the four sphere. The AdS/CFT correspondence is used to calculate the correlation function for scalar and tensor metric perturbations during the ensuing de Sitter phase. The analytic structure of the scalar and tensor propagators is discussed in detail. Observational constraints on the model are discussed. Quantum loops of matter fields are shown to strongly suppress short scale metric perturbations, which implies that short distance modifications of gravity would probably not be observable in the cosmic microwave background. This is probably true for any model of inflation provided there are sufficiently many matter fields. This point is illustrated by a comparison of anomaly driven inflation in four dimensions and in a Randall-Sundrum brane-world model. 
  We here give a first indication that there exists a Seiberg-Witten curve for   SU(N) Seiberg-Witten theory with matter transforming in the totally antisymmetric rank three tensor representation. We present a derivation of the leading order hyperelliptic approximation of a curve for this case. Since we are only interested in the asymptotic free theory we are restricted to $N=6,7,8$. The derivation is carried out by reversed engineering starting from the known form of the prepotential at tree level. We also predict the form of the one instanton correction to the prepotential. 
  What do simple clocks, simple computers, black holes, space-time foam, and holographic principle have in common? I will show that the physics behind them is inter-related, linking together our concepts of information, gravity, and quantum uncertainty. Thus, the physics that sets the limits to computation and clock precision also yields Hawking radiation of black holes and the holographic principle. Moreover, the latter two strongly imply that space-time undergoes much larger quantum fluctuations than what the folklore suggests --- large enough to be detected with modern gravitational-wave interferometers through future refinements. 
  The aim of this article is to review Fradkin's contribution in the realm of eikonal physics. In particular, the so-called Fradkin representation is employed to investigate a certain subclass of Feynman diagrams resulting in an expression for the scattering amplitude at Planckian energies. The 't Hooft poles are reproduced. 
  We introduce two simple models which feature an Alice electrodynamics phase. In a well defined sense the Alice flux solutions we obtain in these models obey first order equations similar to those of the Nielsen-Olesen fluxtube in the abelian higgs model in the Bogomol'nyi limit. Some numerical solutions are presented as well. 
  In certain supergravity backgrounds, D0 branes may polarize into higher dimensional Dp branes. We study this phenomenon in some generality from the perspective of a local inertial observer and explore polarization effects resulting from tidal-like forces. We find D2 brane droplets made of D0 branes at an extremum of the Born-Infeld action even in scenarios where the RR fields may be zero. These solutions lead us to a local formulation of the UV-IR correspondence. A holographic Planck scale bound on the number of D0 branes plays an important role in the analysis. We focus on the impact of higher order moments of background fields and work out extensions of the non-commutative algebra beyond the Lie and Heisenberg structures. In this context, it appears that q-deformed algebras come into play. 
  We show the existence of a global anomaly in the one-loop graphs of N=2 string theory, defined by sewing tree amplitudes, unless spacetime supersymmetry is imposed. The anomaly is responsible for the non-vanishing maximally helicity violating amplitudes. The supersymmetric completion of the N=2 string spectrum is formulated by extending the previous cohomological analysis with an external spin factor; the target space-time spin-statistics of these individual fields in a selfdual background are compatible with previous cohomological analysis as fields of arbitrary spin may be bosonized into one another. We further analyze duality relations between the open and closed string amplitudes and demonstrate this in the supersymmetric extension of the target space-time theory through the insertion of zero-momentum operators. 
  The full list of conserved conformal higher spin currents built from massless scalar and spinor fields is presented. It is shown that, analogously to the relationship between usual conformal and AdS symmetries, the set of the conformal higher spin symmetry parameters associated with the conformal conserved currents in d dimensions is in the one-to-one correspondence with the result of the dimensional reduction of the usual (i.e., non-conformal) higher spin symmetry parameters in d+1 dimensions. 
  We show that the world-volume theory on a D-p-brane at the tachyonic vacuum has solitonic string solutions whose dynamics is governed by the Nambu-Goto action of a string moving in (25+1) dimensional space-time. This provides strong evidence for the conjecture that at this vacuum the full (25+1) dimensional Poincare invariance is restored. We also use this result to argue that the open string field theory at the tachyonic vacuum must contain closed string excitations. 
  We study BPS saturated domain walls in supersymmetric SU(N) Yang-Mills theories in the large N limit. We focus on the Seiberg-Witten regime of ${\cal N}=2$ theory perturbed by a small mass ($m<O(\Lambda/N^4)$) and determine the wall profile by numerically minimizing its energy density. Similar to the SU(2) wall studied in a previous work, the SU(N) wall has a five layer structure, with two confinement phases on the outside, a Coulomb phase in the middle and two transition regions. 
  The relation among spacetime supersymmetry algebras and superbrane actions is further explored. It is proved that $SL(2,\bR)$ belongs to the automorphism group of the ${\cal N}=2$ D=10 type IIB SuperPoincar\'e algebra. Its SO(2) subgroup is identified with a non-local SO(2) transformation found in hep-th/9806161. Performing T-duality, new non-local transformations are found in type IIA relating, among others, BIon configurations with two D2-branes intersecting at a point. Its M-theory origin is explained. These results show that part of the SuperPoincar\'e algebra automorphism group might be realized on the field theory as non-local transformations. 
  Flat connections for unitary gauge groups on a 3--torus with twisted boundary conditions as well as recently discovered periodic nontrivial flat connections with ``nondiagonalizable'' triples of holonomies for higher orthogonal and exceptional groups are constructed explicitly in terms of Jacobi theta functions with rational characteristics. The (fractional) Chern-Simons numbers of these vacuum gauge field configurations are verified by direct computation. 
  In the context of the noncommutative QED we consider few phenomena which reflect the noncommutativity. In all of them the new interactions in the Feynmann diagrams that are responsible for the deviation from the standard QED results. These deviations appear as the violations of Lorentz symmetry. We suggest experimental situations where these effects may be observed. The extra phases have far reaching consequences including violation of crossing symmetry. Considering the e-p scattering and Compton scattering the electric dipole moments of the electron and the photon is calculated. 
  We study the supersymmetric kink with higher derivative and momentum cut-off regularization schemes. We introduce the new momentum cut-off regularization scheme which we call ``generalized momentum cut-off''. A new, explicit computation for the central charge anomaly for this scheme is described in detail. The calculation of the first order mass corrections for the bosonic and supersymmetric kink within the momentum cut-off is presented in hep-th/0010051, so that one can compare the one-loop central charge and mass computed independently within the same regularization setup. We confirm that the BPS bound is saturated in one loop level. Thus the Wilsonian momentum cut-off regularization scheme is rehabilitated as a bona fide procedure for computing quantum corrections in the topologically nontrivial backgrounds. We lead the reader to the idea that a consistent regularization in general does not only assume that one regulates loops properly, but also requires caution in defining the total number of modes involved in the quantization of the theory. We also study the higher-derivative regularization scheme of Shifman et al. in great detail. We extend the Noether method for the case when the higher-derivatives are present in the Lagrangian. The extensive discussion of the technical aspects and the consistency of the higher-derivative scheme is given. We show that higher-derivative regularization gives, in general, a nonlocal topological current, which leads to the correct value of the anomaly in the central charge. We also comment on the status of the dimensional regularization approach to the computation of the central charge anomaly and kink mass. 
  We show how particle-vortex duality implies the existence of a large non-abelian discrete symmetry group which relates the electromagnetic response for dual two-dimensional systems in a magnetic field. For conductors with charge carriers satisfying Fermi statistics (or those related to fermions by the action of the group), the resulting group is known to imply many, if not all, of the remarkable features of Quantum Hall systems. For conductors with boson charge carriers (modulo group transformations) a different group is predicted, implying equally striking implications for the conductivities of these systems, including a super-universality of the critical exponents for conductor/insulator and superconductor/insulator transitions in two dimensions and a hierarchical structure, analogous to that of the quantum Hall effect but different in its details. Our derivation shows how this symmetry emerges at low energies, depending only weakly on the details of dynamics of the underlying systems. 
  Renormalization group analysis of boundary conformal field theory on bosonic D25-brane is used to study tachyon condensation. Placing the lump on a finite circle and triggering only the first three tachyon modes, the theory flows to nearby IR fixed point representing lumps that are extended object with definite profile. The boundary entropy corresponding to the D24-brane tension is calculated in the leading order in perturbative analysis which decreases under RG flow and agrees with the expected result to an accuracy of 8%. Multicritical behaviour of the IR theory suggests that the end point of the flow represents a configuration of two D24-branes. Analogy with Kondo physics is discussed. 
  Manifestly supersymmetric formulation of eleven dimensional supergravity in the framework of light-cone approach is discussed. 
  We prove that the Fourier transform of the exponential $e^{-\b V(R)}$ of the {\bf static} interquark potential in QCD is positive. It has been shown by Eliott Lieb some time ago that this property allows in the same limit of static spin independent potential proving certain mass relation between baryons with different quark flavors. 
  In this short note we show that (I) in a QCD-like theory with four (rather than two) degenerate flavors $ud u'd'$, the $\pi\pi'$ scattering length is positive (attractive); and (II) in QCD with only two (u,d) degenerate flavors the I=2 (say, $\pi^+\pi^+$ hadronic) scattering length is, in the large $N_C$ limit, repulsive. $\pi(\pi')$ are the lowest physical states coupling to $J^p = \bar{u}(x)\gamma_5d(x)$ and $J^{p'} = \bar{u}'(x)\gamma_5d'(x),$ respectively. 
  We show that the fluxon solution of the non-commutative gauge theory and its variations are obtained by the soliton generation method recently given by J. A. Harvey, P. Kraus and F. Larsen [hep-th/0010060]. Although this method generally produces non-BPS solutions of equations of motion, the solutions we obtained are BPS. We give the brane interpretation of these BPS solutions and study their counterparts in the ordinary description by the Seiberg-Witten map. 
  We argue that the total observable entropy is bounded by the inverse of the cosmological constant. This holds for all space-times with a positive cosmological constant, including cosmologies dominated by ordinary matter, and recollapsing universes. The argument involves intermediate steps which may be of interest in their own right. We note that entropy cannot be observed unless it lies both in the past and in the future of the observer's history. This truncates space-time to a diamond-shaped subset well-suited to the application of the covariant entropy bound. We further require, and derive, a novel Bekenstein-like bound on matter entropy in asymptotically de Sitter spaces. Our main result lends support to the proposal that universes with positive cosmological constant are described by a fundamental theory with only a finite number of degrees of freedom. 
  Connes' gauge theory on $M_4\times Z_2$ is reformulated in the Lagrangian level. It is pointed out that the field strength in Connes' gauge theory is not unique. We explicitly construct a field strength different from Connes' one and prove that our definition leads to the generation-number independent Higgs potential. It is also shown that the nonuniqueness is related to the assumption that two different extensions of the differential geometry are possible when the extra one-form basis $\chi$ is introduced to define the differential geometry on $M_4\times Z_2$. Our reformulation is applied to the standard model based on Connes' color-flavor algebra. A connection between the unimodularity condition and the electric charge quantization is then discussed in the presence or absence of $\nu_R$. 
  We extend previously proposed generalized gauge theory formulation of Chern-Simons type and topological Yang-Mills type actions into Yang-Mills type actions. We formulate gauge fields and Dirac-K\"ahler matter fermions by all degrees of differential forms. The simplest version of the model which includes only zero and one form gauge fields accommodated with the graded Lie algebra of $SU(2|1)$ supergroup leads Weinberg-Salam model. Thus the Weinberg-Salam model formulated by noncommutative geometry is a particular example of the present formulation. 
  The non-perturbative corrections to the universal hypermultiplet moduli space metric in the type-IIA superstring compactification on a Calabi-Yau threefold are investigated in the presence of 4d, N=2 supergravity. These corrections come from multiple wrapping of the BPS (Euclidean) D2-branes around certain (BPS) Calabi-Yau 3-cycles, and they are known as the D-instantons. The exact universal hypermultiplet metric is governed by a quaternionic potential that satisfies the SU(\infty) Toda equation. The mechanism is proposed, which elevates any four-dimensional hyper-K"ahler metric with a rotational isometry to the quaternionic metric of the same dimension. A generic separable solution to the Toda equation appears to be related to the Eguchi-Hanson metric, whereas another solution originating from the Atiyah-Hitchin metric describes the gravitationally dressed (mixed) D-instantons. 
  We discuss Wilson loop averages in 4-dimensional non-commutative superYang-Mills theory using the dual supergravity description. We postulate that the Wilson loops are located at the mimimum length scale $R$ in the fifth radial coordinate. We find that they exhibit a crossover from Coulomb type of behaviour for large loops, for which non-commutativity is unimportant, to area law for small loops, for which non-commutativity effects are large. The string tension, which can be read off from the area law, is controlled by the non-commutativity scale. The crossover itself, however, appears to involve loops of size of order $R$ which is much larger than the non-commutativity scale. The existence of the area law in non-commutative super Yang-Mills theory which persists up to a large crossover length scale provides further evidence for connection to an underlying string theory. 
  A class of non abelian affine Toda models is constructed in terms of the axial and vector gauged WZW model. It is shown that the multivacua structure of the potential together with non abelian nature of the zero grade subalgebra allows soliton solutions with non trivial electric and topological charges. Their zero curvature representation and the classical $r$-matrix are also constructed in order to prove their classical integrability. 
  Recent work has proposed the principle of `asymptotic past triviality' to characterize the initial state in the pre-big bang scenario of string cosmology, that it is a generic perturbative solution of the low-energy effective action. Among the more generic sets of solutions which is simple enough to investigate thoroughly, yet complex enough to exhibit interesting behavior, is the gravity-dilaton system in spherical symmetry. Since, in the Einstein frame, this system reduces to a massless minimally coupled scalar, which has been target of a large body of previous investigation, we will draw on this and interpret it in the cosmological context. Since this scenario necessarily involves the transition from weak field initial data into the strong field regime, gravitational collapse, we have made numerical computations to answer some of the questions raised on the road to the proposal that `the pre-big bang is as generic as gravitational collapse'. 
  We introduce and solve a generalized model of 1+1D Lorentzian triangulations in which a certain subclass of outgrowths is allowed, the occurrence of these being governed by a coupling constant \beta. Combining transfer matrix-, saddle point- and path integral techniques we show that for \beta<1 it is possible to take a continuum limit in which the model is described by a 1D quantum Calogero Hamiltonian. The coupling constant \beta survives the continuum limit and appears as a parameter of the Calogero potential. 
  Using a one-loop renormalization group improvement for the effective potential in the Higgs model of electrodynamics with electrically and magnetically charged scalar fields, we argue for the existence of a triple (critical) point in the phase diagram ($\lambda_{run}, g_{run}^4$), where $\lambda_{run}$ is the renormalised running selfinteraction constant of the Higgs scalar monopoles and $g_{run}$ is their running magnetic charge. This triple point is a boundary point of three first-order phase transitions in the dual sector of the Higgs scalar electrodynamics: The "Coulomb" and two confinement phases meet together at this critical point. Considering the arguments for the one-loop approximation validity in the region of parameters around the triple point A we have obtained the following triple point values of the running couplings: $(\lambda_{(A)}, g^2_{(A)})\approx(-13.4; 18.6)$, which are independent of the electric charge influence and two--loop corrections to $g^2_{run}$ with high accuracy of deviations. At the triple point the mass of monopoles is equal to zero. The corresponding critical value of the electric fine structure constant turns out to be $\alpha_{crit} = \pi/g^2_{(A)}\approx{0.17}$ by the Dirac relation. This value is close to the $\alpha_{crit}^{lat}\approx{0.20\pm 0.015}$, which in a U(1) lattice gauge theory corresponds to the phase transition between the "Coulomb" and confinement phases. In our theory for $\alpha \ge \alpha_{crit}$ there are two phases for the confinement of the electrically charged particles. The results of the present paper are very encouraging for the Anti--grand unification theory which was developed previously as a realistic alternative to SUSY GUTs. The paper is also devoted to the discussion of this problem. 
  The origin of fermion generations is one of the great mysteries in particle physics. We consider here a possible solution within the Standard Model framework based on a nonabelian generalization of electric-magnetic duality. First, nonabelian duality says that dual to the colour (electric) symmetry SU(3), there is a ``colour magnetic symmetry'' $\widetilde{SU}(3)$, which by a result of 't~Hooft is spontaneously broken and can thus play the role of the "horizontal symmetry" of generations. Second, nonabelian duality suggests the manner this symmetry is broken with frame vectors in internal symmetry space acting as Higgs fields. As a result, mass matrices factorize leading to fermion mass hierarchy. A calculation to first order gives mixing (CKM and MNS) matrices in general agreement with experiment. In particular, quark mixing is seen naturally to be weak compared with leptons, while within the lepton sector, $\mu-\tau$ mixing turns out near maximal but $e-\tau$ mixing small, just as seen in recent $\nu$ oscillation experiments. In addition, the scheme leads to many testable predictions ranging from rare FCNC meson decays and $\mu-e$ conversion in nuclei to cosmic ray air showers above $10^{20}$ eV. 
  We provide a systematic construction for all n-particle form factors of the SU(N)_2/U(1)^{N-1}-homogeneous Sine-Gordon model in terms of general determinant formulae for a huge class of local operators. The ultraviolet limit is carried out and the corresponding Virasoro central charge together with the conformal dimensions of various operators are identified. The renormalization group flow is studied and we find a precise rule, depending on the relative order of magnitude of the resonance parameters, according to which the theory decouples into new cosets along the flow. 
  We use the correspondence between string states and local operators on the world-sheet boundary defined by vertex operators in open string theory to put in correspondence, holographically, the bosonic open string with the large N limit of a mechanical system living on the world-sheet boundary. We give a natural interpretation of this system in terms of a one-dimensional stochastic process and show that the correspondence takes the form of a map between two conformal field theories with central charge c=24 and c=1. 
  It is pointed out that the space noncommutativity parameters $theta^{\mu \nu}$ in noncommutative gauge theory can be considered as a set of superselection parameters, in analogy with the theta-angle in ordinary gauge theories. As such, they do not need to enter explicitly into the action. A simple generic formula is then suggested to reproduce the Chern-Simons action in noncommutative gauge theory, which reduces to the standard action in the commutative limit but in general implies a cascade of lower-dimensional Chern-Simons terms. The presence of these terms in general alters the vacuum structure of the theory and nonstandard gauge theories can emerge around the new vacua. 
  We consider gluodynamics in case when both color and magnetic charges are present. We discuss first short distance physics, where only the fundamental |Q|=1 monopoles introduced via the `t Hooft loop can be considered consistently. We show that at short distances the external monopoles interact as pure Abelian objects. This result can be reproduced by a Zwanziger-type Lagrangian with an Abelian dual gluon. We introduce also an effective dual gluodynamics which might be a valid approximation at distances where the monopoles |Q|=2 can be considered as point-like as well. Assuming the monopole condensation we arrive at a model which is reminiscent in some respect of the Abelian Higgs model but, unlike the latter leaves space for the Casimir scaling. 
  We perform the antifield BRST quantization of duality-symmetric Maxwell theory and show explicitly the quantum equivalence of the different formulations (covariant and non-covariant). The non-covariant gauge-fixed action is used in the computation of propagators for this model. 
  Using the splitting of a $Q$-deformed boson, in the $Q \to q= e^{\frac{\rm 2\pi i}{\rm k}}$ limit, the fractional decomposition of the quantum affine algebra $\hat A(n)$ and the quantum affine superalgebra $\hat A(n,m)$ are found. This decomposition is based on the oscillator representation and can be related to the fractional supersymmetry and k-fermionic spin. We establish also the equivalence between the quantum affine algebra $\hat A(n)$ and the classical one in the fermionic realization. 
  The presence of a constant background antisymmetric tensor for open strings or D-branes forces the space-time coordinates to be noncommutative. An immediate consequence of this is that all fields get complexified. By applying this idea to gravity one discovers that the metric becomes complex. Complex gravity is constructed by gauging the symmetry $U(1,D-1)$. The resulting action gives one specific form of nonsymmetric gravity. In contrast to other theories of nonsymmetric gravity the action is both unique and gauge invariant. It is argued that for this theory to be consistent one must prove the existence of generalized diffeomorphism invariance. The results are easily generalized to noncommutative spaces. 
  I discuss the general formalism of two-dimensional topological field theories defined on open-closed oriented Riemann surfaces, starting from an extension of Segal's geometric axioms. Exploiting the topological sewing constraints allows for the identification of the algebraic structure governing such systems. I give a careful treatment of bulk-boundary and boundary-bulk correspondences, which are responsible for the relation between the closed and open sectors. The fact that these correspondences need not be injective nor surjective has interesting implications for the problem of classifying `boundary conditions'. In particular, I give a clear geometric derivation of the (topological) boundary state formalism and point out some of its limitations. Finally, I formulate the problem of classifying (on-shell) boundary extensions of a given closed topological field theory in purely algebraic terms and discuss their reducibility. 
  It is shown that the entropy of fourdimensional black holes in string theory compactified on weighted Calabi-Yau hypersurfaces shows scaling behavior in a certain limit. This leads to non-monotonic functions on the moduli space. 
  We study supersymmetric domain walls on S_1/Z_2 orbifolds. The supergravity solutions in the bulk are given by the attractor equation associated with Calabi-Yau spaces and have a naked space-time singularity at some |y_s|. We are looking for possibilities to cut off this singularity with the second wall by a stringy mechanism. We use the collapse of the CY cycle at |y_c| which happens before and at a finite distance from the space-time singularity. In our example with three Kahler moduli the second wall is at the end of the moduli space at |y_c| where also the enhancement of SU(2) gauge symmetry takes place so that |y_e|=|y_c|< |y_s|. The physics of the excision of a naked singularity via the enhancon in the context of domain wall has an interpretation on the heterotic side related to R -> 1/R duality. The position of the enhancon is given by the equation R(|y_e|)=1. 
  We investigate non-perturbative structures of the two-dimensional N=2 supersymmetric nonlinear sigma model on the quadric surface Q^{n-2}(C) = SO(n)/SO(n-2)xU(1), which is a Hermitian symmetric space, and therefore Kahler, by using the auxiliary field and large-n methods. This model contains two kinds of non-perturbatively stable vacua; one of them is the same vacuum as that of supersymmetric CP^{n-1} model, and the other is a new kind of vacuum, which has not yet been known to exist in two-dimensional nonlinear sigma models, the Higgs phase. We show that both of these vacua are asymptotically free. Although symmetries are broken in these vacua, there appear no massless Nambu-Goldstone bosons, in agreement with Coleman's theorem, due to the existence of two different mechanisms in these vacua, the Schwinger and the Higgs mechanisms. 
  An explicit Lorentz covariant formulation of the canonical theory for classical fields is established on a space-like hypersurface. Hamilton's equations and a Poisson bracket are defined on the space-like hypersurface. The Poisson bracket relations between total momentum and total angular momentum satisfies the Poincar{\'e} algebra. It is shown that our Poisson bracket has the same symplectic structure that was brought in the covariant symplectic approach. 
  We propose that the expectation value of a circular BPS-Wilson loop in N=4 SUSYM can be calculated exactly, to all orders in a 1/N expansion and to all orders in g^2 N. Using the AdS/CFT duality, this result yields a prediction of the value of the string amplitude with a circular boundary to all orders in alpha' and to all orders in g_s. We then compare this result with string theory. We find that the gauge theory calculation, for large g^2 N and to all orders in the 1/N^2 expansion does agree with the leading string theory calculation, to all orders in g_s and to lowest order in alpha'. We also find a relation between the expectation value of any closed smooth Wilson loop and the loop related to it by an inversion that takes a point along the loop to infinity, and compare this result, again successfully, with string theory. 
  We compute the one-loop four-point function in {\cal N}=4 supersymmetric Yang-Mills theory with gauge group U(N). We perform the calculation in {\cal N}=1 superspace using the background field method and obtain the complete off-shell contributions to the effective action from planar and non planar supergraphs. In the low-energy approximation the result simplifies and we can study its properties under gauge transformations. It appears that the nonplanar contributions do not maintain the gauge invariance of the classical action. 
  We present metric solutions in six and higher dimensions with a bulk cosmological constant, where gravity is localized on a 3-brane. The corrections to four-dimensional gravity from the bulk continuum modes are power-law suppressed. Furthermore, the introduction of a bulk ``hedgehog'' magnetic field leads to a regular geometry, and can localize gravity on the 3-brane with either positive, zero or negative bulk cosmological constant. 
  The quantisation of the reduced first-order dynamics of the nonrelativistic model for Chern-Simons vortices introduced by Manton is studied on a sphere of given radius. We perform geometric quantisation on the moduli space of static solutions, using a Kaehler polarisation, to construct the quantum Hilbert space. Its dimension is related to the volume of the moduli space in the usual classical limit. The angular momenta associated with the rotational SO(3) symmetry of the model are determined for both the classical and the quantum systems. The results obtained are consistent with the interpretation of the solitons in the model as interacting bosonic particles. 
  We discuss the finite-size properties of a simple integrable quantum field theory in 1+1 dimensions with non-trivial boundary conditions. Novel off-critical identities between cylinder partition functions of models with differing boundary conditions are derived. 
  In a classic paper, Fradkin and Tseytlin showed how magnetic deformations can be introduced in open strings. In this contribution we review some recent work on type-I vacua with magnetised branes and describe the role of additional discrete deformations, related to quantised values of the NS-NS antisymmetric tensor B_ab. 
  We present the canonical and quantum cosmological investigation of a spatially flat, four-dimensional Friedmann-Robertson-Walker (FRW) model that is derived from the M-theory effective action obtained originally by Billyard, Coley, Lidsey and Nilsson (BCLN). The analysis makes use of two sets of canonical variables, the Shanmugadhasan gauge invariant canonical variables and the ``hybrid'' variables which diagonalise the Hamiltonian. We find the observables and discuss in detail the phase space of the classical theory. In particular, a region of the phase space exists that describes a four-dimensional FRW spacetime first contracting from a strong coupling regime and then expanding to a weak coupling regime, while the internal space ever contracts. We find the quantum solutions of the model and obtain the positive norm Hilbert space of states. Finally, the correspondence between wave functions and classical solutions is outlined. 
  We explicitly determine the locations of G orbifold conformal field theories, G=Z_M, M=2,3,4,6, G=\hat D_n, n=4,5, or G the binary tetrahedral group \hat T, within the moduli space M^{K3} of N=(4,4) superconformal field theories associated to K3. This is achieved purely from the known description of the moduli space [AM94] and the requirement of a consistent embedding of orbifold conformal field theories within M^{K3}. We calculate the Kummer type lattices for all these orbifold limits. Our method allows an elementary derivation of the B-field values in direction of the exceptional divisors that arise from the orbifold procedure [Asp95,Dou97,BI97], without recourse to D-geometry. We show that our consistency requirement fixes these values uniquely and determine them explicitly. The relation of our results to the classical McKay correspondence is discussed. 
  We consider compactifications of ${\cal M}$-theory to four-dimensional Minkowski space on seven-dimensional non-compact manifolds. These compactifications include a warp factor which is non-constant due to the presence of sources coming from fivebranes wrapping two-dimensional submanifolds of the internal seven-dimensional space. We derive the expression for the field strengths and consider an explicit example of this general class of solutions. 
  We study scalar field theories on M^{D-1} \otimes S^1, which allow to impose twisted boundary conditions for the S^1 direction, in detail and report several interesting properties overlooked so far. One of characteristic features is the appearance of critical radii of the circle S^1. A phase transition can occur at the classical level or can be caused by quantum effects. Radiative corrections can restore broken symmetries or can break symmetries for small radius. A surprising feature is that the translational invariance for the S^1 direction can spontaneously be broken. A particular class of coordinate-dependent vacuum configurations is clarified and the O(N) \phi^4 model on M^{D-1}\otimes S^1 is extensively studied, as an illustrative example. 
  A covariant description of the canonical theory for interacting classical fields is developed on a space-like hypersurface. An identity invariant under the canonical transformations is obtained. The identity follows a canonical equation in which the interaction Hamiltonian density genarates a deformation of the space-like hypersurface. The equation just corresponds to the Yang-Feldman equation in the Heisenberg pictures in quantum field theory. 
  We extend an implicit regularization scheme to be applicable in the $n$-dimensional space-time. Within this scheme divergences involving parity violating objects can be consistently treated without recoursing to dimensional continuation. Special attention is paid to differences between integrals of the same degree of divergence, typical of one loop calculations, which are in principle undetermined. We show how to use symmetries in order to fix these quantities consistently. We illustrate with examples in which regularization plays a delicate role in order to both corroborate and elucidate the results in the literature for the case of CPT violation in extended $QED_4$, topological mass generation in 3-dimensional gauge theories and the Schwinger Model and its chiral version. 
  We consider the effective action of M-theory compactified on a S^1/Z_2 orbifold with R^2 interaction in the Gauss-Bonnet combination. We derive equations of motion with source terms arising from the Gauss-Bonnet terms and find the M2-brane solution up to order \kappa^{2/3}. It receives a correction which depends on the orbifold coordinate in the same form as the gauge 5-brane solution. 
  A new set of exact scattering matrices in 1+1 dimensions is proposed by solving the bootstrap equations. Extending earlier constructions of colour valued scattering matrices this new set has its colour structure associated to non simply-laced Lie algebras. This in particular leads to a coupling of different affine Toda models whose fusing structure has to be matched in a suitable manner. The definition of the new S-matrices is motivated by the semi-classical particle spectrum of the non simply-laced Homogeneous Sine-Gordon (HSG) models, which are integrable perturbations of WZNW cosets. In particular, the S-matrices of the simply-laced HSG models are recovered as a special case. 
  A brief review is presented of motivations and results in one-loop supergravity on manifolds with boundary, whose consideration is suggested by quantum cosmology. 
  We discuss the stability of vacua in two-dimensional gauge theory for any simple, simply connected gauge group. Making use of the representation of a vacuum in terms of a Wilson line at infinity, we determine which vacua are stable against pair production of heavy matter in the adjoint of the gauge group. By calculating correlators of Wilson loops, we reduce the problem to a problem in representation theory of Lie groups, that we solve in full generality. 
  In conformally invariant quantum field theories one encounters besides the standard DHR superselection theory based on spacelike (Einstein-causal) commutation relations and their Haag duality another timelike (''Huygens'') based superselection structure. Whereas the DHR theory based on spacelike causality of observables confirmed the Lagrangian internal symmetry picture on the level of the physical principles of local quantum physics, the attempts to understand the timelike based superselection charges (associated with the center of the conformal covering group) in terms of timelike localized charges lead to a more dynamical role of charges outside the DR theorem and even outside the Coleman-Mandula setting. The ensuing plektonic timelike structure of conformal theories explains the spectrum of the anomalous scale dimensions in terms of phase factors in admissable braid group representations, similar to the explanation of the possible anomalous spin spectrum expected from stringlike d=1+2 fields with braid group statistics. 
  We analyse spin and statistics of quantum dyon fields, i.e. fields carrying both electric and magnetic charge, in 3+1 space-time dimensions. It has been shown long time ago that, at the quantum mechanical level, a composite dyon made out of a magnetic pole of charge g and a particle of electric charge e possesses half-integral spin and fermionic statistics, if the constituents are bosons and the Dirac quantization condition $eg=2\pi n$ holds, with n odd. This phenomenon is called spin-statistics transmutation. We show that the same phenomenon occurs at the quantum field theory level for an elementary dyon. This analysis requires the construction of gauge invariant charged dyon fields. Dirac's proposal for such fields, relying on a Coulomb-like photon cloud, leads to quantum correlators exhibiting an unphysical dependence on the Dirac-string. Recently Froehlich and Marchetti proposed a recipe for charged dyon fields, based on a sum over Mandelstam-strings, which overcomes this problem. Using this recipe we derive explicit expressions for the quantum field theory correlators and we provide a proof of the occurrence of spin-statistics transmutation. The proof reduces to a computation of the self-linking numbers of dyon worldlines and Mandelstam strings, projected on a fixed time three-space. Dyon composites are also analysed. The transmutation discussed in this paper bares some analogy with the appearance of anomalous spin and statistics for particles or vortices in Chern-Simons theories in 2+1 dimensions. However, peculiar features appear in 3+1 dimensions e.g. in the spin addition rule. 
  We discuss toroidal compactifications of maximal supergravity coupled to an extended configuration of BPS states which transform consistently under the U-duality group. Under certain conditions this leads to theories that live in more than eleven spacetime dimensions, with maximal supersymmetry but only partial Lorentz invariance. We demonstrate certain features of this construction for the case of nine-dimensional N=2 supergravity. 
  A vertex algebra is an algebraic counterpart of a two-dimensional conformal field theory. We give a new definition of a vertex algebra which includes chiral algebras as a special case, but allows for fields which are neither meromorphic nor anti-meromorphic. To any complex torus equipped with a flat Kahler metric and a closed 2-form we associate an N=2 superconformal vertex algebra (N=2 SCVA) in the sense of our definition. We find a criterion for two different tori to produce isomorphic N=2 SCVA's. We show that for algebraic tori isomorphism of N=2 SCVA's implies the equivalence of the derived categories of coherent sheaves corresponding to the tori or their noncommutative generalizations (Azumaya algebras over tori). We also find a criterion for two different tori to produce N=2 SCVA's related by a mirror morphism. If the 2-form is of type (1,1), this condition is identical to the one proposed by Golyshev, Lunts, and Orlov, who used an entirely different approach inspired by the Homological Mirror Symmetry Conjecture of Kontsevich. Our results suggest that Kontsevich's conjecture must be modified: coherent sheaves must be replaced with modules over Azumaya algebras, and the Fukaya category must be ``twisted'' by a closed 2-form. We also describe the implications of our results for BPS D-branes on Calabi-Yau manifolds. 
  Using the Vafa-Witten twisted version of N=4 Super Yang-Mills a subset of the supercharges actually relevant for the nonrenormalization properties of the theory is identified. In particular, a relationship between the gauge-fixed action and the chiral primary operator tr(\phi)^2 is worked out. This result can be understood as an off-shell extension of the reduction formula introduced by Intriligator. 
  We analyze how lower-dimensional bosonic D-branes further decay, using the boundary string field theory. Especially we find that the effective tachyon potential of the lower-dimensional D-brane has the same profile as that of D25-brane. 
  After a short introduction to the UV/IR mixing in non-commutative field theories we review the properties of scalar quasi-particles in non-commutative supersymmetric gauge theories at finite temperature. In particular we discuss the appearance of super-luminous wave propagation. 
  We investigate properties of the entropy density related to a generalized extensive statistics and derive the thermodynamic Bethe ansatz equation for a system of relativistic particles obeying such a statistics. We investigate the conformal limit of such a system. We also derive a generalized Y-system. The Gentile intermediate statistics and the statistics of gamma-ons are considered in detail. In particular, we observe that certain thermodynamic quantities for the Gentile statistics majorize those for the Haldane-Wu statistics. Specifically, for the effective central charges related to affine Toda models we obtain nontrivial inequalities in terms of dilogarithms. 
  We present a unified treatment of three cases of quasi-exactly solvable problems, namely, charged particle moving in Coulomb and magnetic fields, for both the Schr\"odinger and the Klein-Gordon case, and the relative motion of two charged particles in an external oscillator potential. We show that all these cases are reducible to the same basic equation, which is quasi-exactly solvable owing to the existence of a hidden $sl_2$ algebraic structure. A systematic and unified algebraic solution to the basic equation using the method of factorization is given. Analytic expressions of the energies and the allowed frequencies for the three cases are given in terms of the roots of one and the same set of Bethe ansatz equations. 
  We discuss the four-dimensional cosmological constant problem in a five-dimensional setting. A scalar field coupled to the SM forms dynamically a smooth brane with four-dimensional Poincare invariance, independently of SM physics. In this respect, our solution may be regarded as a self-tuning solution, free of any singularities and fine-tuning problems. 
  The general method of reduction in the number of coupling parameters is applied in a Chern-Simons-matter model with several independent couplings. We claim that considering the asymptotic region, and expressing all dimensionless coupling parameters as functions of the Chern-Simons coupling, it is possible to show that all $\beta$-functions vanish to any order of perturbative series. Therefore, the model is asymptotically scale invariant. 
  The presentation at Strings 2000 was intended to be in two main parts, but there was only time for part one. However both parts appeared on the online proceedings, and are also included in this document. The first part concerns an exploration of the connection between the physics of the `enhancon' geometry arising from wrapping N D6-branes on the K3 manifold in Type IIA string theory and that of a charge N BPS multi-monopole. This also relates to the physics of 2+1 dimensional SU(N) gauge theory with eight supercharges. The main results uncovered by this exploration are: a) better insight into the non-perturbative geometry of the enhancon; b) the structure of the moduli space geometry, and its characterisation in terms of generalisations of an Atiyah-Hitchin-like manifold; c) the use of Nahm data to describe aspects of the geometry, showing that the enhancon locus itself has a description as a fuzzy sphere. Part two discusses the addition of extra D2-branes into the geometry. Two probe computations show the difference between the geometry as seen by D2-branes and that seen by wrapped D6-branes, and the accompanying gauge theory interpretations are discussed. 
  We propose a refinement of the physical picture describing different vacua in bosonic string theory. The vacua with closed strings and open strings are connected by the string field theory version of the Higgs mechanism, generalizing the Higgs mechanism of an abelian gauge field interacting with a complex scalar. In accordance with Sen's conjecture, the condensation of the tachyon is an essential part of the story. We consider this phenomenon from the point of view of both a world-sheet sigma-model and the target-space theory. In the Appendix the relevant remarks regarding the choice of the coordinates in the background independent open string field theory are given. 
  We derive Brown-Henneaux's commutation relation and central charge in the framework of the path integral. If we use the leading part of the asymptotic symmetry to derive the Ward-Takahashi identity, we can see the central charge arises from the fact that the boundary condition of the path integral is not invariant under the transformation. 
  We establish the equivalence of the Maxwell-Chern-Simons-Proca model to a doublet of Maxwell-Chern-Simons models at the level of polarization vectors of the basic fields using both Lagrangian and Hamiltonian formalisms. The analysis reveals a U(1) invariance of the polarization vectors in the momentum space. Its implications are discussed. We also study the role of Wigner's little group as a generator of gauge transformations in three space-time dimensions. 
  We analyze the superfield equations of the 4-dimensional N=2 and N=4 SYM-theories using light-cone gauge conditions and the harmonic-superspace approach. The harmonic superfield equations of motion are drastically simplified in this gauge, in particular, the basic harmonic-superfield matrices and the corresponding harmonic analytic gauge connections become nilpotent on-shell. 
  SU(N) reduced, quenched, gauge theories have been shown to be related to string theories. We extend this result and show how a 4-dimensional, reduced, quenched, Yang-Mills theory, supplemented by the topological term, can be related through the Wigner-Weyl-Moyal correspondence to an open 3-brane model. The boundary of the 3-brane is described by a Chern-Simons 2-brane. We identify the bulk of the 3-brane with the interior of a hadronic bag and the world-volume of the Chern-Simons 2-brane with the dynamical boundary of the bag. We estimate the value of the induced bag constant to be a little less than 200MeV. 
  After the separation of the center-of-mass motion, a new privileged class of canonical Darboux bases is proposed for the non-relativistic N-body problem by exploiting a geometrical and group theoretical approach to the definition of {\it body frame} for deformable bodies. This basis is adapted to the rotation group SO(3), whose canonical realization is associated with a symmetry Hamiltonian {\it left action}. The analysis of the SO(3) coadjoint orbits contained in the N-body phase space implies the existence of a {\it spin frame} for the N-body system. Then, the existence of appropriate non-symmetry Hamiltonian {\it right actions} for non-rigid systems leads to the construction of a N-dependent discrete number of {\it dynamical body frames} for the N-body system, hence to the associated notions of {\it dynamical} and {\it measurable} orientation and shape variables, angular velocity, rotational and vibrational configurations. For N=3 the dynamical body frame turns out to be unique and our approach reproduces the {\it xxzz gauge} of the gauge theory associated with the {\it orientation-shape} SO(3) principal bundle approach of Littlejohn and Reinsch. For $N \geq 4$ our description is different, since the dynamical body frames turn out to be {\it momentum dependent}. The resulting Darboux bases for $N\geq 4$ are connected to the coupling of the {\it spins} of particle clusters rather than the coupling of the {\it centers of mass} (based on Jacobi relative normal coordinates). One of the advantages of the spin coupling is that, unlike the center-of-mass coupling, it admits a relativistic generalization. 
  The classical Maxwell-Dirac and Maxwell-Klein-Gordon theories admit solutions of the field equations where the corresponding electric current vanishes in the causal complement of some bounded region of Minkowski space. This poses the interesting question of whether states with a similarly well localized charge density also exist in quantum electrodynamics. For a large family of charged states, the dominant quantum corrections at spacelike infinity to the expectation values of local observables are computed. It turns out that certain moments of the charge density decrease no faster than the Coulomb field in spacelike directions. In contrast to the classical theory, it is therefore impossible to define the electric charge support of these states in a meaningful way. 
  We find a general class of rotating charged black hole solutions to N=2, D=5 gauged supergravity coupled to vector supermultiplets. The supersymmetry properties of these solutions are studied, and their mass and angular momenta are obtained. We also compute the stress tensor of the dual D=4, N=4 super Yang-Mills theory in the limit of strong `t Hooft coupling. It is shown that closed timelike curves occur outside the horizon, indicating loss of unitarity in the dual CFT. For imaginary coupling constant of the gravitini to the gauge fields, one can obtain multi-centered rotating charged de Sitter black holes. Some physical properties of these solutions are also discussed. 
  We show that boundary conditions in topological open string theory on Calabi-Yau manifolds are objects in the derived category of coherent sheaves, as foreseen in the homological mirror symmetry proposal of Kontsevich. Together with conformal field theory considerations, this leads to a precise criterion determining the BPS branes at any point in CY moduli space, completing the proposal of Pi-stability. 
  We define an iterative procedure to obtain a non-abelian generalization of the Born-Infeld action. This construction is made possible by the use of the severe restrictions imposed by kappa-symmetry. In this paper we will present all bosonic terms in the action up to terms quartic in the Yang-Mills field strength and all fermion bilinear terms up to terms cubic in the field strength. Already at this order the fermionic terms do not satisfy the symmetric trace-prescription. 
  In the presence of the constant background NS two-form gauge field, we construct the worldsheet partition functions, bulk propagators and boundary propagators for the worldsheets with a handle and a boundary. We analyze the noncommutative $\phi^3$ field theory amplitudes that correspond to the general two-point insertions on the two-loop nonplanar vacuum bubble. By the direct string theory amplitude computations on the worldsheets with a handle, which reduce to the aforementioned field theory amplitudes in the decoupling limit, we find that the stretched string interpretation remains valid for the types of amplitudes in consideration. This completes the demonstration that the stretched string picture holds up in the general multiloop context. 
  The geometrical (superembedding) approach is used as a tool for deriving from the worldvolume dynamics of superbranes field theoretical models exhibiting partial supersymmetry breaking. In this way we obtain nonlinear actions for Goldstone superfields associated with physical degrees of freedom of the superbranes. 
  We consider the set of partition functions that result from the insertion of twist operators compatible with conformal invariance in a given 2D Conformal Field Theory (CFT). A consistency equation, which gives a classification of twists, is written and solved in particular cases. This generalises old results on twisted torus boundary conditions, gives a physical interpretation of Ocneanu's algebraic construction, and might offer a new route to the study of properties of CFT. 
  For massless quenched QED in three dimensions, we evaluate a non-perturbative expression for the fermion propagator which agrees with its two loop perturbative expansion in the weak coupling regime. This calculation is carried out by making use of the Landau-Khalatnikov-Fradkin transformations. Any improved construction of the fermion-boson vertex must make sure that the solution of the Schwinger-Dyson equation for the fermion propagator reproduces this result. For two different gauges, we plot the fermion propagator against momentum. We then make a comparison with a similar plot, using the earlier expression for the fermion propagator, which takes into account only the one loop result. 
  Modifications to the singularity structure of D3-branes that result from turning on a flux for the R-R and NS-NS 3-forms (fractional D3-branes) provide important gravity duals of four-dimensional N=1 super-Yang-Mills theories. We construct generalisations of these modified p-brane solutions in a variety of other cases, including heterotic 5-branes, dyonic strings, M2-branes, D2-branes, D4-branes and type IIA and type IIB strings, by replacing the flat transverse space with a Ricci-flat manifold M_n that admits covariantly constant spinors, and turning on a flux built from a harmonic form in M_n, thus deforming the original solution and introducing fractional branes. The construction makes essential use of the Chern-Simons or ``transgression'' terms in the Bianchi-identity or equation of motion of the field strength that supports the original undeformed solution. If the harmonic form is L^2 normalisable, this can result in a deformation of the brane solution that is free of singularities, thus providing viable gravity duals of field theories in diverse dimensions that have less than maximal supersymmetry. We obtain examples of non-singular heterotic 5-branes, dyonic strings, M2-branes, type IIA strings, and D2-branes. 
  We study the cosmological evolution of the four-dimensional universe on the probe D3-brane in geodesic motion in the curved background of the source Dp-brane with non-zero NS B field. The Friedman equations describing the expansion of the brane universe are obtained and analyzed for various limits. We elaborate on corrections to the cosmological evolution due to nonzero NS B field. 
  We generalize a regularization method of Stumpf 1984 in the case of non-linear spinor field models to fourth order theories and to non-scalar interactions. The involved discrete symmetries can be connected with C,P,T transformations. 
  We give arguments for the necessity to employ Quantum Clifford Hopf Gebras in quantum field theory. The role of the antipode is examined, Feynman diagrams are re-interpreted as tangles of graphical calculus. Regularization due to the design of convolution Hopf gebras is given as a program for further research. 
  We consider multiplet shortening for BPS solitons in N=1 two-dimensional models. Examples of the single-state multiplets were established previously in N=1 Landau-Ginzburg models. The shortening comes at a price of loosing the fermion parity $(-1)^F$ due to boundary effects. This implies the disappearance of the boson-fermion classification resulting in abnormal statistics. We discuss an appropriate index that counts such short multiplets.   A broad class of hybrid models which extend the Landau-Ginzburg models to include a nonflat metric on the target space is considered. Our index turns out to be related to the index of the Dirac operator on the soliton reduced moduli space (the moduli space is reduced by factoring out the translational modulus). The index vanishes in most cases implying the absence of shortening. In particular, it vanishes when there are only two critical points on the compact target space and the reduced moduli space has nonvanishing dimension.   We also generalize the anomaly in the central charge to take into account the target space metric. 
  This is a summary of a series of papers hep-th/9910263,0005283,0010066 written with B. Chen, T. Matsuo and K. Murakami on a p-p', (p<p^{\prime}) open string with B_{ij} field, which has led us to the explicit identification of the Dp-brane with the noncommutative projector soliton via the gaussian damping factor. A lecture given at Summer Institute 2000, FujiYoshida, Yamanashi, Japan, at August 7-14, 2000. 
  In this short note we would like to show the relation between the cubic string field theory for N D-instantons and the string field theory in the presence of the background B field. 
  In the early Universe matter can be described as a conformal invariant ultra-relativistic perfect fluid, which does not contribute, on classical level, to the evolution of the isotropic and homogeneous metric. If we suppose that there is some desert in the particle spectrum just below the Planck mass, then the effect of conformal trace anomaly is dominating at the corresponding energies. With some additional constraints on the particle content of the underlying gauge model (which favor extended or supersymmetric versions of the Standard Model rather than the minimal one), one arrives at the stable inflation. We review the model and report about the calculation of the gravitational waves on the background of the anomaly-induced inflation. The result for the perturbation spectrum is close to the one for the conventional inflaton model, and is in agreement with the existing Cobe data (see also [hep-th/0009197]). 
  It is shown that the twisted sector spectrum, as well as the associated Chern-Simons interactions, can be determined on M-theory orbifold fixed planes that do not admit gravitational anomalies. This is demonstrated for the seven-planes arising within the context of an explicit $R^6 \times S^1/Z_2 \times T^4/Z_2$ orbifold, although the results are completely general. Local anomaly cancellation in this context is shown to require fractional anomaly data that can only arise from a twisted sector on the seven-planes, thus determining the twisted spectrum up to a small ambiguity. These results open the door to the construction of arbitrary M-theory orbifolds, including those containing fixed four-planes which are of phenomenological interest. 
  The dissertation is devoted to the description and further investigation of the properties of null p-branes. 
  Motivated by recent discussions of actions for tachyon and vector fields related to tachyon condensation in open string theory we review and clarify some aspects of their derivation within sigma model approach. In particular, we demonstrate that the renormalized partition function $Z(T,A)$ of boundary sigma model gives the effective action for massless vectors which is consistent with string S-matrix and beta function, resolving an old problem with this suggestion in bosonic string case at the level of the leading $F^2 (dF)^2$ derivative corrections to Born-Infeld action. We give manifestly gauge invariant definition of $Z(T,A)$ in non-abelian NSR open string theory and check that its derivative reproduces the tachyon beta function in a particular scheme. We also discuss derivation of similar actions for tachyon and massless modes in closed bosonic and NSR (type 0) string theories. 
  Spatial noncommutativity is similar and can even be related to the non-Abelian nature of multiple D-branes. But they have so far seemed independent of each other. Reflecting this decoupling, the algebra of matrix valued fields on noncommutative space is thought to be the simple tensor product of constant matrix algebra and the Moyal-Weyl deformation. We propose scenarios in which the two become intertwined and inseparable. Therefore the usual separation of ordinary or noncommutative space from the internal discrete space responsible for non-Abelian symmetry is really the exceptional case of an unified structure. We call it non-Abelian geometry. This general structure emerges when multiple D-branes are configured suitably in a flat but varying B field background, or in the presence of non-Abelian gauge field background. It can also occur in connection with Taub-NUT geometry. We compute the deformed product of matrix valued functions using the lattice string quantum mechanical model developed earlier. The result is a new type of associative algebra defining non-Abelian geometry. Possible supergravity dual is also discussed. 
  A noncommutative and non-anticommutative quantum field theory is formulated in a superspace, in which the superspace coordinates satisfy noncommutative and non-anticommutative relations. A perturbative scalar field theory is investigated in which only the non-anticommutative algebraic structure is kept, and one loop diagrams are calculated and found to be finite due to the damping caused by a Gaussian factor in the propagator. 
  The QED effective action encodes nonlinear interactions due to quantum vacuum polarization effects. While much is known for the special case of electrons in a constant electromagnetic field (the Euler-Heisenberg case), much less is known for inhomogeneous backgrounds. Such backgrounds are more relevant to experimental situations. One way to treat inhomogeneous backgrounds is the "derivative expansion", in which one formally expands around the soluble constant-field case. In this talk I use some recent exactly soluble inhomogeneous backgrounds to perform precision tests on the derivative expansion, to learn in what sense it converges or diverges. A closely related question is to find the exponential correction to Schwinger's pair-production formula for a constant electric field, when the electric background is inhomogeneous. 
  The one-loop correction to the spectrum of Kaluza-Klein system for the $\phi^3$ model on $R^{1,d}\times (T_\theta^2)^L$ is evaluated in the high temperature limit, where the $1+d$ dimensions are the ordinary flat Minkowski spacetimes and the $L$ extra two-dimensional tori are chosen to be the noncommutative torus with noncommutativity $\theta$. The corrections to the Kaluza-Klein mass formula are evaluated and used to compute the Casimir energy with the help of the Schwinger perturbative formula in the zeta-function regularization method. The results show that the one-loop Casimir energy is independent of the radius of torus if L=1. However, when $L>1$ the Casimir energy could give repulsive force to stabilize the extra noncommutative torus if $d-L$ is a non-negative even integral. This therefore suggests a possible stabilization mechanism of extra radius in high temperature, when the extra spaces are noncommutative. 
  It is known that noncommutative Yang-Mills is equivalent to IIB matrix model with a noncommutative background, which is interpreted as a twisted reduced model. In noncommutative Yang-Mills, long range interactions can be seen in nonplanar diagrams after integrating high momentum modes. These interactions can be understood as block-block interactions in the matrix model. Using this relation, we consider long range interactions in noncommutative Yang-Mills associated with fermionic backgrounds. Exchanges of gravitinos, which couple to a supersymmetry current, are examined. 
  We reconsider entropy of black holes which do not have finite area horizon. It is suggested that some of them should have nonzero entropy from both supergravity and string theory point of view. We also refine our arguments in our previous papers for the existence of the microstates of the black hole. 
  Various aspects of the four point function for scalar fields in conformally invariant theories are analysed. This depends on an arbitrary function of two conformal invariants u,v. A recurrence relation for the function corresponding to the contribution of an arbitrary spin field in the operator product expansion to the four point function is derived. This is solved explicitly in two and four dimensions in terms of ordinary hypergeometric functions of variables z,x which are simply related to u,v. The operator product expansion analysis is applied to the explicit expressions for the four point function found for free scalar, fermion and vector field theories in four dimensions. The results for four point functions obtained by using the AdS/CFT correspondence are also analysed in terms of functions related to those appearing in the operator product discussion. 
  We discuss some features of the regular supergravity solution for fractional branes on a deformed conifold, recently found by Klebanov-Strassler, mostly adapting it to a type 0 non-sypersymmetric context. The non-supersymmetric gauge theory is SU(M)*SU(M) with two bi-fundamental Weyl fermions. The tachyon is now stabilized by the RR antisymmetric tensor flux. We briefly discuss the most general non-supersymmetric theory on electric, magnetic and fractional type 0 D3-branes on a conifold. This includes the pure SU(N) theory. 
  These short (personal) notes appeared as the result of my attempt to address the question raised in the title. 
  We discuss the spectra of multi-flavor massless QCD_2. An approximation in which the Hilbert space is truncated to two currents states is used. We write down a 't Hooft like equation for the wave function of the two currents states. We solve this equation for the lowest massive state and find an excellent agreement with the DLCQ results. In addition, the 't Hooft model and the large N_f limit spectra are re-derived by using a description in terms of currents. 
  We consider the first subleading terms in the low-energy cross section for the absorption of dilaton partial waves by D3-branes. We demonstrate that these corrections, computed previously via supergravity, can be reproduced exactly in a worldvolume calculation using a deformation of N=4 SYM theory by a dimension eight chiral operator. The calculation does not depend on how the theory is regularized. This result provides another hint that holographic duality between the D3-brane worldvolume theory and the corresponding supergravity solution may be valid beyond the near horizon limit. 
  We study brane-world solutions of five-dimensional supergravity in singular spaces. We exhibit a self-tuned four-dimensional cosmological constant when five-dimensional supergravity is broken by an arbitrary tension on the brane-world. The brane-world metric is of the FRW type corresponding to a cosmological constant $\Omega_{\Lambda}={5/7}$ and an equation of state $\omega=-{5/7}$ which are consistent with experiment. 
  We continue the study of string theory on AdS_3 x SU(3)/U(1) and AdS_3 x SO(5)/SO(3). We compute the spacetime spectrum of the N=3 supersymmetric dual CFT using worldsheet techniques. The spectrum of chiral primaries coincides for the two models. Unlike N=4 theories, the building block of the symmetric product in spacetime (corresponding to a single long string) is not by itself in the moduli space of a symmetric product. 
  The problem of describing the boundary states of unstable non-BPS D-branes of type-II string theories in light-cone Green-Schwarz (GS) formalism is addressed. Regarding the type II theories in light-cone gauge as different realizations of the $\hat{SO}(8)_{k=1}$ Kac-Moody algebra, the non-BPS D-brane boundary states of these theories are given in terms of the relevant Ishibashi states constructed in this current algebra. Using the expressions for the current modes in terms of the GS variables it is straightforward to reexress the boundary states in the GS formalism. The problem that remains is the lack of manifest SO(8) covariance in these expressions. We also derive the various known expressions for the BPS and non-BPS D-brane boundary states by starting with the current algebra Ishibashi states. 
  We study RR charge cancellation consistency conditions in string compactifications with open string sectors, by introducing D-brane probes in the configuration. We show that uncancelled charges manifest as chiral gauge anomalies in the world-volume of suitable probes. RR tadpole cancellation can therefore be described as the consistency of the effective compactified theory not just in the vacuum, but also in all topological sectors (presence of D-brane probes). The result explains why tadpole cancellation is usually much stronger than anomaly cancellation of the compactified theory (in the vacuum sector). We use the probe criterion to construct consistent six-dimensional orientifolds of curved K3 spaces, where usual CFT techniques to compute tadpoles are not valid. As a last application, we consider compactifications where standard RR charge cancels but full K-theory charge does not. We show the inconsistency of such models manifests as a global gauge anomaly on the world-volume of suitable probes. 
  Scale invariance is considered in the context of gravitational theories where the action, in the first order formalism, is of the form $S = \int L_{1} \Phi d^4x$ + $\int L_{2}\sqrt{-g}d^4x$ where the volume element $\Phi d^4x$ is independent of the metric. For global scale invariance, a "dilaton" $\phi$ has to be introduced, with non-trivial potentials $V(\phi)$ = $f_{1}e^{\alpha\phi}$ in $L_1$ and $U(\phi)$ = $f_{2}e^{2\alpha\phi}$ in $L_2$. This leads to non-trivial mass generation and a potential for $\phi$ which is interesting for inflation. Interpolating models for natural transition from inflation to a slowly accelerated universe at late times appear naturally. This is also achieved for "Quintessential models", which are scale invariant but formulated with the use of volume element $\Phi d^4x$ alone. For closed strings and branes (including the supersymmetric cases), the modified measure formulation is possible and does not require the introduction of a particular scale (the string or brane tension) from the begining but rather these appear as integration constants. 
  Using the path integral method, we calculate the partition function and generating functional (of the field strengths) on the nonlocal generalized 2D Yang - Mills theories ($nlgYM_2$'s), which is nonlocal in auxiliary field [14]. Our calculations is done for general surfaces. We find a general expression for free energy of $W(\phi) = \phi^{2k}$ in $nlgYM_2$ theories at the strong coupling phase (SCP) regime ($A > A_c$) for large groups. In the specific $\phi^4$ model, we show that the theory has a third order phase transition. 
  We study the cosmological evolution of a D3-brane universe in a type 0 string background. We follow the brane-universe along the radial coordinate of the background and we calculate the energy density which is induced on the brane because of its motion in the bulk. We find that for some typical values of the parameters the brane-universe has an inflationary phase. 
  Starting with Lagrangians, which turn out to be degenerate, the Hamiltonian operators for integrable systems can be constructed using Dirac's theory of constraints. We illustrate this by giving a systematic discussion of the first Hamiltonian structure of KdV. The first symplectic 2-form obtained from Dirac's theory is the time component of the covariant Witten-Zuckerman symplectic 2-form current. We derive the flux of the first symplectic 2-form. Then we turn to a new Lagrangian for KdV recently obtained by Pavlov and present the corresponding new covariant symplectic structure for KdV. This shows that the inverse of Magri's second Hamiltonian operator is also a Hamiltonian operator for KdV. 
  We review shortly present status of quantum deformations of Poincar\'{e} and conformal supersymmetries. After recalling the $\kappa$-deformation of $\hbox{D=4}$ Poincar\'{e} supersymmetries we describe the corresponding star product multiplication for chiral superfields. In order to describe the deformation of chiral vertices in momentum space the integration formula over $\kappa$-deformed chiral superspace is proposed. 
  We calculate the amplitude for a non-excited closed string with nonzero winding number to scatter from a D-string with a near critical E field. We go to the NCOS limit and observe that we get the same result if we adopt another approach put forward by Gomis and Ooguri. 
  We give a supersymmetric generalization of the sine algebra and the quantum algebra $U_{t}(sl(2))$. Making use of the $q$-pseudo-differential operators graded with a fermionic algebra, we obtain a supersymmetric extension of sine algebra. With this scheme we also get a quantum superalgebra $U_{t}(sl(2/1)$. 
  We discuss the role of singletons and logarithmic operators in AdS_3 string theory in the context of AdS_3/CFT_2 correspondence. 
  We show how the T-duality is realized for D-branes with noncommutative world-volume coordinates. We discuss D-branes wrapped on tori and the result is that the recently found noncommutative actions form a consistent collection due to the T-duality mapping between noncommutative D-branes and rotated commutative D-branes on deformed tori. 
  We propose a new mechanism of spontaneous supersymmetry breaking. The existence of extra dimensions with nontrivial topology plays an important role. We investigate new features resulting from this mechanism. One noteworthy feature is that there exists a phase in which the translational invariance for the compactified directions is broken spontaneously. The mechanism we propose also yields quite different vacuum structures for models with different global symmetries. 
  Much has been learned about string theory over the last few years by studying properties of cycles and branes in a given background geometry. Here we discuss three situations (quantum volume, attractor flows/D-brane stability, and dynamical topology change) in which cycles in a Calabi-Yau background evolve and/or degenerate in some manner, yielding various insights into aspects of quantum geometry. Based in part on talk given at Strings 2000. 
  Boundary states for D-branes at orbifold fixed points are constructed in close analogy with Cardy's derivation of consistent boundary states in RCFT. Comments are made on the interpretation of the various coefficients in the explicit expressions, and the relation between fractional branes and wrapped branes is investigated for $\mathbb{C}^2/\Gamma$ orbifolds. The boundary states are generalised to theories with discrete torsion and a new check is performed on the relation between discrete torsion phases and projective representations. 
  The vacuum sector of the Brans-Dicke theory is studied from the viewpoint of a non-conformally invariant gravitational model. We show that, this theory can be conformally symmetrized using an appropriate conformal transformation. The resulting theory allows a particle interpretation, and suggests that the quantum aspects of matter may be geometrized. 
  We present a conformally invariant generalized form of the free particle action by connecting the wave and particle aspects through gravity. Conformal invariance breaking is introduced by choosing a particular configurat$ of dynamical variables. This leads to the geometrization of the quantum aspects of matter. 
  We study a model for analyzing the effect of a principal violation of the Lorentz-invariance on the structure of vacuum. The model is based on the divergence theory developed by Salehi (1997). It is shown that the divergence theory can be used to model an ensemble of particles. The ensemble is characterized by the condition that its members are basically at rest in the rest frame of a preferred inertial observer in vacuum. In this way we find a direct dynamical interplay between a particle and its associated ensemble. We show that this effect can be understood in terms of the interaction of a particle with a relativistic pilot wave through an associated quantum potential. 
  We present a model in which the breackdown of conformal symmetry of a quantum stress-tensor due to the trace anomaly is related to a cosmological effect in a gravitational model. This is done by characterizing the traceless part of the quantum stress-tensor in terms of the stress-tensor of a conformal invariant classical scalar field. We introduce a conformal frame in which the anomalous trace is identified with a cosmological constant. In this conformal frame we establish the Einstein field equations by connecting the quantum stress-tensor with the large scale distribution of matter in the universe. 
  What are strings made of? The possibility is discussed that strings are purely mathematical objects, made of logical axioms. More precisely, proofs in simple logical calculi are represented by graphs that can be interpreted as the Feynman diagrams of certain large-N field theories. Each vertex represents an axiom. Strings arise, because these large-N theories are dual to string theories. These ``logical quantum field theories'' map theorems into the space of functions of two parameters: N and the coupling constant. Undecidable theorems might be related to nonperturbative field theory effects. 
  We present an alternative N=2 supergravity multiplet coupled to n copies of vector multiplets and n' copies of hypermultiplets in five dimensions. Our supergravity multiplet contains a single antisymmetric tensor and a dilaton, which are natural Neveu-Schwarz massless fields in superstring theory. The absence of the explicit Chern-Simons terms in our lagrangian deletes the non-trivial constraints on the couplings of vector multiplets in the conventional formulation. The scalars in the vector multiplets form the sigma-model for the coset SO(n,1) / SO(n), like those in the vector multiplets coupled to N=1 supergravity in nine dimensions, while the scalars in the hypermultiplets form that for the quaternionic Kahler manifold Sp(n',1) / Sp(n') X Sp(1). We also perform the gauging of the SO(2) subgroup of the Sp(1) = SL(2,R) automorphism group of N=2 supersymmetry. Our result is also generalized to singular 5D space-time as in the conventional formulation, as a preliminary for supersymmetric Randall-Sundrum brane world scenario. 
  We are dealing with two-dimensional gravitational anomalies, specifically with the Einstein anomaly and the Weyl anomaly, and we show that they are fully determined by dispersion relations independent of any renormalization procedure (or ultraviolet regularization). The origin of the anomalies is the existence of a superconvergence sum rule for the imaginary part of the relevant formfactor. In the zero mass limit the imaginary part of the formfactor approaches a $\delta$-function singularity at zero momentum squared, exhibiting in this way the infrared feature of the gravitational anomalies. We find an equivalence between the dispersive approach and the dimensional regularization procedure. The Schwinger terms appearing in the equal time commutators of the energy momentum tensors can be calculated by the same dispersive method. Although all computations are performed in two dimensions the method is expected to work in higher dimensions too. 
  A prescription for center gauge fixing for pure Yang-Mills theory in the continuum with general gauge groups is presented. The emergence of various types of singularities (magnetic monopoles and center vortices) appearing in the course of the gauge fixing procedure are discussed. 
  A generalization of the Coulomb Gas model with modular SL(2, Z)-symmetry allows for a discrete infinity of phases which are characterized by the condensation of dyonic pseudoparticles and the breaking of parity and time reversal. Here we study the phase diagram of such a model by using renormalization group techniques. Then the symmetry SL(2,Z) acting on the two-dimensional parameter space gives us a nested shape of its global phase diagram and all the infrared stable fixed points. Finally we propose a connection with the 2-dimensional Conformal Field Theory description of the Fractional Quantum Hall Effect. 
  It is shown that the reduced particle dynamics of 2+1 dimensional gravity in the maximally slicing gauge has hamiltonian form. This is proved directly for the two body problem and for the three body problem by using the Garnier equations for isomonodromic transformations. For a number of particles greater than three the existence of the hamiltonian is shown to be a consequence of a conjecture by Polyakov which connects the auxiliary parameters of the fuchsian differential equation which solves the SU(1,1) Riemann-Hilbert problem, to the Liouville action of the conformal factor which describes the space-metric. We give the exact diffeomorphism which transforms the expression of the spinning cone geometry in the Deser, Jackiw, 't Hooft gauge to the maximally slicing gauge. It is explicitly shown that the boundary term in the action, written in hamiltonian form gives the hamiltonian for the reduced particle dynamics. The quantum mechanical translation of the two particle hamiltonian gives rise to the logarithm of the Laplace-Beltrami operator on a cone whose angular deficit is given by the total energy of the system irrespective of the masses of the particles thus proving at the quantum level a conjecture by 't Hooft on the two particle dynamics. The quantum mechanical Green's function for the two body problem is given. 
  We compute off-shell three- and four-tachyon amplitudes at tree level by using a prescription based on the requirement of projective invariance. In particular we show that the off-shell four-tachyon amplitude can be put in the same form as the corresponding on-shell one, exhibiting therefore the same analyticity properties. This is shown both for the bosonic and the fermionic string. The result obtained in the latter case can be extended to the off-shell four-tachyon amplitude in type 0 theory. 
  Using the replica method, we analyze the mass dependence of the QCD3 partition function in a parameter range where the leading contribution is from the zero momentum Goldstone fields. Three complementary approaches are considered in this article. First, we derive exact relations between the QCD3 partition function and the QCD4 partition function continued to half-integer topological charge. The replica limit of these formulas results in exact relations between the corresponding microscopic spectral densities of QCD3 and QCD4. Replica calculations, which are exact for QCD4 at half-integer topological charge, thus result in exact expressions for the microscopic spectral density of the QCD3 Dirac operator. Second, we derive Virasoro constraints for the QCD3 partition function. They uniquely determine the small-mass expansion of the partition function and the corresponding sum rules for inverse Dirac eigenvalues. Due to de Wit-'t Hooft poles, the replica limit only reproduces the small mass expansion of the resolvent up to a finite number of terms. Third, the large mass expansion of the resolvent is obtained from the replica limit of a loop expansion of the QCD3 partition function. Because of Duistermaat-Heckman localization exact results are obtained for the microscopic spectral density in this way. 
  Intersecting Dp-branes often give rise to chiral fermions living on their intersections. We study the construction of four-dimensional chiral gauge theories by considering configurations of type II D(3+n)-branes wrapped on non-trivial n-cycles on T^{2n} x(R^{2(3-n)}/Z_N), for n=1,2,3. The gauge theories on the four non-compact dimensions of the brane world-volume are generically chiral and non-supersymmetric. We analyze consistency conditions (RR tadpole cancellation) for these models, and their relation to four-dimensional anomaly cancellation. Cancellation of U(1) gauge anomalies involves a Green-Schwarz mechanism mediated by RR partners of untwisted and/or twisted moduli. This class of models is of potential phenomenological interest, and we construct explicit examples of SU(3) x SU(2) x U(1) three-generation models. The models are non-supersymmetric, but the string scale may be lowered close to the weak scale so that the standard hierarchy problem is avoided. We also comment on the presence of scalar tachyons and possible ways to avoid the associated instabilities. We discuss the existence of (meta)stable configurations of D-branes on 3-cycles in (T^2)^3, free of tachyons for certain ranges of the six-torus moduli. 
  A universal framework is proposed, where all laws are regularities of relations between things or agents. Parts of the world at one or all times are modeled as networks called SYSTEMS with a minimum of axiomatic properties. A notion of locality is introduced by declaring some relations direct (or links). Dynamics is composed of "atomic" constituents called mechanisms. They are conditional actions of basic local structural transformations (``enzymes''): indirect relations become direct (friend of friend becomes friend), links are removed, objects copied. This defines a kind of universal chemistry.   I show how to model basic life processes in a self contained fashion as a kind of enzymatic computation. The framework also accommodates the gauge theories of fundamental physics. Emergence creates new functionality by cooperation - nonlocal phenomena arise out of local interactions. I explain how this can be understood in a reductionist way by multiscale analysis (e.g. renormalization group). 
  We study type IIA string theory compactified on Calabi-Yau fourfolds and heterotic string theory compactified on Calabi-Yau threefolds times a two-torus. We derive the resulting effective theories which have two space-time dimensions and preserve four supercharges. The duality between such vacua is established at the level of the effective theory. For type IIA vacua with non-trivial Ramond-Ramond background fluxes a superpotential is generated. We show that for a specific choice of background fluxes and a fourfold which has the structure of a threefold fibred over a sphere the superpotential coincides with the superpotential recently proposed by Taylor and Vafa in compactifications of type IIB string theory on a threefold. 
  We discuss the covariant formulation of local field theories described by the Companion Lagrangian associated with p-branes. The covariantisation is shown to be useful for clarifying the geometrical meaning of the field equations and also their relation to the Hamilton-Jacobi formulation of the standard Dirac-Born-Infeld theory. 
  We study the classical geometry associated to fractional D3-branes of type IIB string theory on R^4/Z_2 which provide the gravitational dual for N=2 super Yang-Mills theory in four dimensions. As one can expect from the lack of conformal invariance on the gauge theory side, the gravitational background displays a repulson-like singularity. It turns out however, that such singularity can be excised by an enhancon mechanism. The complete knowledge of the classical supergravity solution allows us to identify the coupling constant of the dual gauge theory in terms of the string parameters and to find a logarithmic running that is governed precisely by the beta-function of the N=2 super Yang-Mills theory. 
  The string theory introduced in early 1971 by Ramond, Neveu, and myself has two-dimensional world-sheet supersymmetry. This theory, developed at about the same time that Golfand and Likhtman constructed the four-dimensional super-Poincar\'e algebra, motivated Wess and Zumino to construct supersymmetric field theories in four dimensions. Gliozzi, Scherk, and Olive conjectured the spacetime supersymmetry of the string theory in 1976, a fact that was proved five years later by Green and myself. 
  We consider a Dp brane within a D9 brane in the presence of a B-field whose polarization is {\em transverse} to the Dp brane. To be definite, we take a D3-D9 system. It is observed that the system has the same pattern of supersymmetry breaking as that of a soliton of the six dimensional non-commutative gauge theory that is obtained by dimensional reduction of an {\cal N}=1, D=10 gauge theory. These results indicate that the soliton solution is the low energy realization of a D3 brane in a D9 brane with a transverse B-field, hence can be viewed as a generalization of the previous results in the literature where similar observations were made for lower codimensional cases. 
  Connes' gauge theory is defined on noncommutative space-times. It is applied to formulate a noncommutative Glashow-Weinberg-Salam (GWS) model in the leptonic sector. It is shown that the model has two Higgs doublets and the gauge bosons sector after the Higgs mechanism contains the massive charged gauge fields, two massless and two massive neutral gauge fields. It is also shown that, in the tree level, the neutrino couples to one of two `photons', the electron interacts with both `photons' and there occurs a nontrivial $\nu_R$-interaction on noncommutative space-times. Our noncommutative GWS model is reduced to the GWS theory in the commutative limit. Thus in the neutral gauge bosons sector there are only one massless photon and only one $Z^0$ in the commutative limit. 
  We study solutions of the Bogomolny equation on R^2\times S^1$ with prescribed singularities. We show that Nahm transform establishes a one-to-one correspondence between such solutions and solutions of the Hitchin equations on a punctured cylinder with the eigenvalues of the Higgs field growing at infinity in a particular manner. The moduli spaces of solutions have natural hyperkahler metrics of a novel kind. We show that these metrics describe the quantum Coulomb branch of certain N=2 d=4 supersymmetric gauge theories on R^3\times S^1. The Coulomb branches of the corresponding uncompactified theories have been previously determined by E. Witten using the M-theory fivebrane. We show that the Seiberg-Witten curves of these theories are identical to the spectral curves associated to solutions of the Bogomolny equation on R^2\times S^1. In particular, this allows us to rederive Witten's results without recourse to the M-theory fivebrane. 
  By applying the covariant Taylor expansion method, the fifth lower coefficients the asymptotic expansion of the heat kernel associated with a fermion of spin 1/2 in Riemann-Cartan space are manifestly given. These coefficients in Riemann-Cartan space is derived from those obtained in Riemannian space by simple replacements. 
  The blocked composite operators are defined in the one-component Euclidean scalar field theory, and shown to generate a linear transformation of the operators, the operator mixing. This transformation allows us to introduce the parallel transport of the operators along the RG trajectory. The connection on this one-dimensional manifold governs the scale evolution of the operator mixing. It is shown that the solution of the eigenvalue problem of the connection gives the various scaling regimes and the relevant operators there. The relation to perturbative renormalization is also discussed in the framework of the $\phi^3$ theory in dimension $d=6$. 
  We study a non-commutative generalization of the standard electroweak model proposed by Balakrishna, Gursey and Wali [ Phys.Lett. B254(1991)430] that is formulated in terms of the derivations Der_2(M_3) of a three-dimensional representation of the su(2) Lie algebra of weak isospin. The linearized Higgs field equations and the scalar boson mass eigenvalues are explicitly given. A light Higgs boson with mass around 130 GeV together with four very heavy scalar bosons are predicted. 
  Conformal invariant new forms of $p$-brane and D$p$-brane actions are proposed. The field content of these actions are: an induced metric, gauge fields, an auxiliary metric and an auxiliary scalar field that implements the Weyl invariance. This scalar field transforms with a conformal weight that depends on the brane dimension. The proposed actions are Weyl invariant in any dimension and the elimination of the auxiliary metric and the scalar field reproduces the Nambu-Goto action for $p$-branes and the Born-Infeld action for D$p$-branes. As a consequence of the fact that in the $p$-brane case, the action is quadratic in $\partial_0 X$, we develop a complete construction for the associated canonical formalism, solving the problem in previous formulations of conformal $p$-brane actions where the Hamiltonian can not be constructed. We obtain the algebra of constraints and identify the generator of the Weyl symmetry. In the construction of the corresponding supersymmetric generalization of this conformal $p$-brane action we show that the associated $\kappa$-symmetry is consistent with the conformal invariance. In the D$p$-brane case, the actions are quadratic in the gauge fields. These actions can be used to construct new conformal couplings in any dimension $p$ to the auxiliary scalar field now promoted as a dynamical variable. 
  A survey of ideas, techniques and results from d=5 supergravity for the conformal and mass-perturbed phases of d=4 ${\cal N}$=4 Super-Yang-Mills theory 
  Based on experimental evidences supporting the hypothesis that neutrinos might be tachyonic fermions, a new Dirac-type equation is proposed and a spin-1/2 tachyonic quantum theory is developed. The new Dirac-type equation provides a solution for the puzzle of negative mass-square of neutrinos. This equation can be written in two spinor equations coupled together via nonzero mass while respecting the maximum parity violation, and it reduces to one Weyl equation in the massless limit. Some peculiar features of tachyonic neutrino are discussed in this theoretical framework. 
  We give an explicit proof that the noncommutative U(N) gauge theories are one-loop renormalizable 
  Pure N=1 super Yang-Mills theory can be realised as a certain low energy limit of M theory near certain singularities in $G_2$-holonomy spaces. For SU(n) and SO(2n) gauge groups these $M$ theory backgrounds can be regarded as strong coupling limits of wrapped D6-brane configurations in Type IIA theory on certain non-compact Calabi-Yau spaces such as the deformed conifold. Various aspects of such realisations are studied including the generation of the superpotential, domain walls, QCD strings and the relation to recent work of Vafa. In the spirit of this recent work we propose a `gravity dual' of M theory near these singularities. 
  We analyse the noncommutative U(1) sigma model, which arises from the vacuum dynamics of the noncommutative charged tachyonic field. The sector of ``spherically symmetric'' excitations of the model is equivalent to a chain of rotators. Classical solutions for this model are found, which are static and ``spherically symmetric'' in noncommutative spatial dimensions. The limit of small noncommutativity reveals the presence of Polyakov vortices in the model. A generalisation of the model to q-deformed space, which may serve as a regularisation of the non-deformed model is also considered. 
  We investigate D-branes on a noncompact singular Calabi-Yau manifold by using the boundary CFT description, and calculate the open string Witten indices between the boundary states. The B-type D-branes turn out to be characterized by the properties of a compact positively curved manifold. We give geometric interpretations to these boundary states in terms of coherent sheaves of the manifold. 
  Using the AdS/CFT correspondence, we show that the Anti-de Sitter (AdS) rotating (Kerr) black holes in five and seven dimensions provide us with examples of non-trivial field theories which are scale, but not conformally invariant. This is demonstrated by our computation of the actions and the stress-energy tensors of the four and six dimensional field theories residing on the boundary of these Kerr-AdS black holes spacetimes. 
  These lectures are devoted to the low energy limit of \N2 SUSY gauge theories, which is described in terms of integrable systems. A special emphasis is on a duality that naturally acts on these integrable systems. The duality turns out to be an effective tool in constructing the double elliptic integrable system which describes the six-dimensional Seiberg-Witten theory. At the same time, it implies a series of relations between other Seiberg-Witten systems. 
  The D2brane-anti-D2brane system is described in the framework of BFSS Matrix model and noncommutative (NC) gauge theory. The physical spectrum of fields is found by appropriate gauge fixing. The exact tachyon potential is computed in terms of these variables and an exact description of tachyon condensation provided. We exhibit multiple vortex production with increasing topological charge and interpret this as gradual conversion of the brane-antibrane system to $D0$ branes. The entire analysis is carried out using the known hamiltonian of the Matrix model, which is equivalent to the hamiltonian of the NC gauge theory. We identify the supersymmetric ground state of this hamiltonian with the tachyonic vacuum; Sen's conjecture about the latter follows simply from this identification. We also find two types of closed string excitations, solitonic (a la Dijkgraaf, Verlinde and Verlinde) as well as perturbative, around the tachyonic vacuum. 
  We present a pedagogical introduction into noncommutative gauge theories, their stringy origin, and non-perturbative effects, including monopole and instanton solutions 
  We obtain three new solvable, real, shape invariant potentials starting from the harmonic oscillator, P\"oschl-Teller I and P\"oschl-Teller II potentials on the half-axis and extending their domain to the full line, while taking special care to regularize the inverse square singularity at the origin. The regularization procedure gives rise to a delta-function behavior at the origin. Our new systems possess underlying non-linear potential algebras, which can also be used to determine their spectra analytically. 
  Asymptotically locally AdS black hole geometries of dimension d > 2 are studied for nontrivial topologies of the transverse section. These geometries are static solutions of a set of theories labeled by an integer 0 < k < [(d-1)/2] which possess a unique globally AdS vacuum. The transverse sections of these solutions are d-2 surfaces of constant curvature, allowing for different topological configurations. The thermodynamic analysis of these solutions reveals that the presence of a negative cosmological constant is essential to ensure the existence of stable equilibrium states. In addition, it is shown that these theories are holographically related to [(d-1)/2] different conformal field theories at the boundary. 
  A possibility that the origin of the CP violation in our world is a domain wall itself, on which we are living, is investigated in the context of the brane world scenario. We estimate the amount of the CP violation on our wall, and show that either of order one or suppressed CP phases can be realized. An interesting case where CP is violated due to the {\it coexistence} of the walls, which conserve CP individually, is also considered. We also propose a useful approximation for the estimation of the CP violation in the double-wall background. 
  We consider the noncommutative Abelian-Higgs theory and investigate general static vortex configurations including recently found exact multi-vortex solutions. In particular, we prove that the self-dual BPS solutions cease to exist once the noncommutativity scale exceeds a critical value. We then study the fluctuation spectra about the static configuration and show that the exact non BPS solutions are unstable below the critical value. We have identified the tachyonic degrees as well as massless moduli degrees. We then discuss the physical meaning of the moduli degrees and construct exact time-dependent vortex configurations where each vortex moves independently. We finally give the moduli description of the vortices and show that the matrix nature of moduli coordinates naturally emerges. 
  In this work we construct an embedding of a nontrivial central extension of the contact superconformal algebra K'(4) into the Lie superalgebra of pseudodifferential symbols on the supercircle S^{1|2}. Associated with this embedding is a one-parameter family of spinor-like tiny irreducible representations of K'(4) realized just on 4 fields instead of the usual 16. 
  We study the Landau-Ginzburg (LG) mirror theory of the non-linear sigma model on the ALE space ${\cal M}$ obtained by resolving the singularity of the orbifold ${\bf C}^2/{\bf Z}_N$. In the LG description, the data of the BPS spectrum and the lines of marginal stability are encoded in the special Lagrangian submanifolds of the mirror manifold $\hat{{\cal M}}$. Our LG description is quite simple as compared with the quiver gauge theory analysis near the orbifold point. Furthermore our mirror analysis can be applied to any point on the moduli space of ${\cal M}$. 
  We elaborate further on a recently proposed scenario for generating a mass hierarchy through quantum fluctuations of a single D3 brane, which represents our world embedded in a bulk five-dimensional space time. In this scenario, the quantum fluctuations of the D3-brane world in the bulk direction, quantified to leading order via a `recoil' world-sheet logarithmic conformal field theory approach, result in the dynamical appearance of a supersymmetry breaking (obstruction) scale alpha. This may be naturally taken to be at the TeV range, in order to provide a solution to the conventional gauge-hierarchy problem. The bulk spatial direction is characterized by the dynamical appearance of an horizon located at +- 1/alpha, inside which the positive energy conditions for the existence of stable matter are satisfied. To ensure the correct value of the four-dimensional Planck mass, the bulk string scale M_s is naturally found to lie at an intermediate energy scale of 10^{14} GeV. As an exclusive feature of the D3-brane quantum fluctuations (`recoil') we find that, for any given M_5, there is a discrete mass spectrum for four-dimensional Kaluza-Klein (KK) modes of bulk graviton and/or scalar fields. KK modes with masses 0 <= m < sqrt{2}alpha << M_s are found to have wavefunctions peaked, and hence localized, on the D3 brane at z=0. 
  The origin of non-renormalization theorems in field theories with global supersymmetry can be traced to the fact that supersymmetric actions can be viewed as the highest components of respective supermultiplets. Supersymmetric interactions in particular can therefore be represented as supersymmetry variations of lower dimensional field polynomials. We investigate here this algebraic structure in the context of the Wess-Zumino model and N=1 and N=2 supersymmetric Yang-Mills theories. 
  The correlation function of two dimensional Ising model with the nearest neighbours interaction on the finite size lattice with the periodical boundary conditions is derived. The expressions similar to the form factor representation are obtained both for the ferromagnetic and paramagnetic regions of coupling constant. The peculiarities caused by the finite size of the system are discussed. The asymptotic behaviour of the corresponding quantities as functions of size is calculated. Some generalization of the obtained result is conjectured. 
  A new family of non critical bosonic string backgrounds in arbitrary space time dimension $D$ and with $ISO(1,D-2)$ Poincar\'e invariance are presented. The metric warping factor and dilaton agree asymptotically with the linear dilaton background. The closed string tachyon equation of motion enjoys, in the linear approximation, an exact solution of ``kink'' type interpolating between different expectation values. A renormalization group flow interpretation ,based on a closed string tachyon potential of type $-T^{2}e^{-T}$, is suggested. 
  Recent string theory developments suggest the necessity to understand supersymmetric gauge theories non-perturbatively, in various dimensions. In this work we show that there is a standard Hamiltonian formulation that generates a finite and supersymmetric result at every order of the DLCQ approximation scheme. We present this DLCQ renormalized Hamiltonian and solve for the bound states and the wave functions to verify that it exactly reproduces the large N SDLCQ results. We find that it has two novel features: it automatically chooses the t'Hooft prescription for renormalizing the singularities and it introduces irrelevant operators that serve to preserve the supersymmetry and improve the convergence. This is a first step in extending the advantages of SDLCQ to non-supersymmetric theories. 
  We investigate boundary states of N=2 coset models based on Grassmannians Gr(n,n+k), and find that the underlying intersection geometry is given by the fusion ring of U(n). This is isomorphic to the quantum cohomology ring of Gr(n,n+k+1), and thus can be encoded in a ``boundary'' superpotential whose critical points correspond to the boundary states. In this way the intersection properties can be represented in terms of a soliton graph that forms a generalized, Z_{n+k+1} symmetric McKay quiver. We investigate the spectrum of bound states and find that the rational boundary CFT produces only a small subset of the possible quiver representations. 
  We calculate the background independent action for bosonic and supersymmetric open string field theory in a constant B-field. We also determine the tachyon effective action in the presence of constant B-field. 
  This is an introduction to two-dimensional conformal field theory and its applications in string theory. Modern concepts of conformal field theory are explained, and it is outlined how they are used in recent studies of D-branes in the strong curvature regime by means of CFT on surfaces with boundary. 
  These notes provide a brief introduction to modern cosmology, focusing primarily on theoretical issues. Some attention is paid to aspects of potential interest to students of string theory, on both sides of the two-way street of cosmological constraints on string theory and stringy contributions to cosmology. Slightly updated version of lectures at the 1999 Theoretical Advanced Study Institute at the University of Colorado, Boulder. 
  We construct a solitonic 3-brane solution in the 5-dimensional Einstein-Hilbert-Gauss-Bonnet theory. This solitonic brane is delta-function like, and has the property that gravity is completely localized on the brane. That is, there are no propagating degrees of freedom in the bulk, while on the brane we have purely 4-dimensional Einstein gravity. Thus, albeit the classical background is 5-dimensional, the quantum theory (perturbatively) is 4-dimensional. Our solution can be embedded in the supergravity context, where we have completely localized supergravity on the corresponding solitonic brane, which is a BPS object preserving 1/2 of the original supersymmetries. By including a scalar field, we also construct a smooth domain wall solution, which in a certain limit reduces to the delta-function-like solitonic brane solution (this is possible for the latter breaks diffeomorphisms only spontaneously). We then show that in the smooth domain wall background the only normalizable mode is the 4-dimensional graviton zero mode, while all the other (including massive Kaluza-Klein) modes are not even plane-wave normalizable. Finally, we observe that in compactifications of Type IIB on 5-dimensional Einstein manifolds other than a 5-sphere the corresponding dual gauge theories on D3-branes are not conformal in the ultra-violet, and at the quantum level we expect the Einstein-Hilbert term to be generated in their world-volumes. We conjecture that in full string theory on Type IIB side this is due to higher curvature terms, which cannot be ignored in such backgrounds. A stronger version of this conjecture also states that (at least in some cases) in such backgrounds D3-branes are solitonic objects with completely localized (super)gravity in their world-volumes. 
  Light-cone approach to field dynamics in AdS space-time is discussed. 
  We provide a unified picture of the domain wall spectrum in supersymmetric QCD with Nc colors and Nf flavors of quarks in the (anti-) fundamental representation. Within the framework of the Veneziano-Yankielowicz-Taylor effective Lagrangian, we consider domain walls connecting chiral symmetry breaking vacua, and we take the quark masses to be degenerate. For Nf/Nc<1/2, there is one BPS saturated domain wall for any value of the quark mass m. For 1/2=<Nf/Nc<1 there are two critical masses, m* and m**, which depend on the number of colors and flavors only through the ratio Nf/Nc; if m<m*, there are two BPS walls, if m*<m<m** there is one non-BPS wall, and if m>m** there is no domain wall. We numerically determine m* and m** as a function of Nf/Nc, and we find that m** approaches a constant value in the limit that this ratio goes to one. 
  In this note we consider compactifications of ${\cal M}$-theory on $Spin(7)$-holonomy manifolds to three-dimensional Minkowski space. In these compactifications a warp factor is included. The conditions for unbroken N=1 supersymmetry give rise to determining equations for the 4-form field strength in terms of the warp factor and the self-dual 4-form of the internal manifold. 
  The role of conformal anomaly in AdS/CFT and related issues is clarified. The comparison of holographic and QFT conformal anomalies (with account of brane quantum gravity contribution) indicates on the possibility for brane quantum gravity to occur within AdS/CFT set-up. 3d quantum induced inflationary (or hyperbolic) brane-world is shown to be realized in frames of AdS3/CFT2 correspondence where the role of 2d brane cosmological constant is played by effective tension due to two-dimensional conformal anomaly. The dynamical equations to describe 4d FRW-Universe with account of quantum effects produced by conformal anomaly are obtained. The quantum corrected energy, pressure and entropy are found. Dynamical evolution of entropy bounds in inflationary Universe is estimated and its comparison with quantum corrected entropy is done. It is demonstrated that entropy bounds for quantum corrected entropy are getting the approximate ones and occur for some limited periods of evolution of inflationary Universe. 
  A Born-Infeld theory describing a D2-brane coupled to a 4-form RR field strength is considered, and the general solutions of the static and Euclidean time equations are derived and discussed. The period of the bounce solutions is shown to allow a consideration of tunneling and quantum-classical transitions in the sphaleron region. The order of such transitions, depending on the strength of the RR field strength, is determined. A criterion is then derived to confirm these findings. 
  It has been conjectured that at the stationary point of the tachyon potential for the non-BPS D-brane or brane-anti-D-brane pair, the negative energy density cancels the brane tension. We study this conjecture using a cubic superstring field theory with insertion of a double-step inverse picture changing operator. We compute the tachyon potential at levels (1/2,1) and (2,6). In the first case we obtain that the value of the potential at the minimum is 97.5% of the non BPS D-brane tension. Using a special gauge in the second case we get 105.8% of the tension. 
  The vanishing cosmological constant in the four dimensional space-time is obtained in a 5D Randall-Sundrum model with a brane (B1) located at $y=0$. The matter fields can be located at the brane. For settling any vacuum energy generated at the brane to zero, we need a three index antisymmetric tensor field $A_{MNP}$ with a specific form for the Lagrangian. For the self-tuning mechanism, the bulk cosmological constant should be negative. 
  The `winding state' behavior appears in the two-loop nonplanar contribution to the partition function in thermal noncommutative field theories. We derive this feature directly from the purely open string theory analysis in the presence of the constant background $B$-field; we compute the two-loop partition function for worldsheets with a handle and a boundary when the time direction of the Euclideanized target space is compactified. In contrast to the closed-string-inspired approach, it is not necessary to add infinite number of extra degrees of freedom. Furthermore, we find a piece of supporting evidence toward the conjecture that, in the UV limit, the noncommutativity parameter plays the role of the effective string scale in noncommutative field theories. 
  Starting from a solution to the classical Batalin-Vilkovisky master equation,an extended solution to an extended master equation is constructed by coupling all the observables, the anomaly candidates and the generators of global symmetries. The construction of the formalism and its applications in the context of the renormalization of generic and potentially anomalous gauge theories are reviewed. 
  By studying various, known extrema of 1) SU(3) sectors, 2) SO(5) sectors and 3) SO(3)xSO(3) sectors of gauged N =8 supergravity in four-dimensions, one finds that the deformation of seven sphere \S^7 gives rise to non-trivial renormalization group(RG) flow in three-dimensional boundary conformal field theory from UV fixed point to IR fixed point. For SU(3) sectors, this leads to four-parameter subspace of the supergravity scalar-gravity action and we identify one of the eigenvalues of A_1 tensor of the theory with a superpotential of scalar potential that governs RG flows on this subspace. We analyze some of the structure of the superpotential and discuss first-order BPS domain-wall solutions, using some algebraic relations between superpotential and derivatives of it with respect to fields, that determine a (super)symmetric kink solution in four-dimensional N =8 supergravity, which generalizes all the previous considerations. The BPS domain-wall solutions are equivalent to vanishing of variation of spin 1/2, 3/2 fields in the supersymmetry preserving bosonic background of gauged N=8 supergravity. For SO(5) sectors, there exist only nontrivial nonsupersymmetric critical points that are unstable and included in SU(3) sectors. For SO(3)xSO(3) sectors, we construct the scalar potential(never been written) explicitly and study explicit construction of first-order domain-wall solutions. 
  We investigate Matrix theory in the large-N limit following the conjectured correspondence between Matrix theory and supergravity on the near-horizon limit of the D0-brane background. We analyze the complete fermionic spectrum of supergravity and obtain two-point functions of the supercurrents in Matrix theory. By examining the large-N scaling properties of the correlators, we discuss the consistency of the 11-dimensional interpretation of the supersymmetry of Matrix theory. 
  Left-right symmetric models are analyzed in the context of noncommutative geometry where we show that spontaneous parity violation is ruled out. 
  Confinement of electrically charged test particles in the dilute plasma of monopoles and pointlike magnetic dipoles is studied. We calculate the tension of the string emerging between the infinitely separated test particles. The string tension is an increasing function of the dipole density provided other parameters of the plasma are fixed. The relevance of our results to confining gauge theories is discussed. 
  Generalizations of the *-product (e.g. n-ary *_n operations) appear in various places in the discussion of noncommutative gauge theories. These include the one-loop effective action of noncommutative gauge theories, the couplings between massless closed and open string modes, and the Seiberg-Witten map between the ordinary and noncommutative Yang-Mills fields. We propose that the natural way to understand the *_n operations is through the expansion of an open Wilson line. We establish the connection between an open Wilson line and the *_n operations and use it to: (I) write down a gauge invariant effective action for the one-loop F^4 terms in the noncommutative N=4 SYM theory; (II) find the gauge invariant couplings between the noncommutative SYM modes and the massless closed string modes in flat space; (III) propose a closed form for the Seiberg-Witten map in the U(1) case. 
  We introduce a generalization of the Heisenberg algebra which is written in terms of a functional of one generator of the algebra, $f(J_0)$, that can be any analytical function. When $f$ is linear with slope $\theta$, we show that the algebra in this case corresponds to $q$-oscillators for $q^2 = \tan \theta$. The case where $f$ is a polynomial of order $n$ in $J_0$ corresponds to a $n$-parameter deformed Heisenberg algebra. The representations of the algebra, when $f$ is any analytical function, are shown to be obtained through the study of the stability of the fixed points of $f$ and their composed functions. The case when $f$ is a quadratic polynomial in $J_0$, the simplest non-linear scheme which is able to create chaotic behavior, is analyzed in detail and special regions in the parameter space give representations that cannot be continuously deformed to representations of Heisenberg algebra. 
  We examine the thermodynamic stability of large black holes in four-dimensional anti-de Sitter space, and we demonstrate numerically that black holes which lack local thermodynamic stability often also lack stability against small perturbations. This shows that no-hair theorems do not apply in anti-de Sitter space. A heuristic argument, based on thermodynamics only, suggests that if there are any violations of Cosmic Censorship in the evolution of unstable black holes in anti-de Sitter space, they are beyond the reach of a perturbative analysis. 
  In this note we study bulk timelike geodesics in the presence of a brane-world black hole, where the brane-world is a two-brane moving in a 3+1-dimensional asymptotically $adS_4$ spacetime. We show that for a certain range of the parameters measuring the black hole mass and bulk cosmological constant, there exist stable timelike geodesics which orbit the black hole and remain bound close to the two-brane. 
  We show that a non-trivial topological effect breaks the conformal invariance of pure Yang-Mills theory. Thus it is possible that classic particle-like solutions exists in pure non-Abelian Yang-Mills theory. We find a static, non-singular solution in source-free SU(2) Yang-Mills theory in four-dimensional Minkowski space. This solution is a stable soliton characterized by non-trivial topology and imaginary $A_0^a$, i.e., $A_0^aA_0^a<0$. It yields hermitian Hamilton, and finite, positive energy. 
  We consider topology-changing transitions between 7-manifolds of holonomy G_2 constructed as a quotient of CY x S^1 by an antiholomorphic involution. We classify involutions for Complete Intersection CY threefolds, focussing primarily on cases without fixed points. The ordinary conifold transition between CY threefolds descends to a transition between G_2 manifolds, corresponding in the N=1 effective theory incorporating the light black hole states either to a change of branch in the scalar potential or to a Higgs mechanism. A simple example of conifold transition with a fixed nodal point is also discussed. As a spin-off, we obtain examples of G_2 manifolds with the same value for the sum of Betti numbers b_2+b_3, and hence potential candidates for mirror manifolds. 
  Noncommutative gauge theories can be constructed from ordinary $U(\infty)$ gauge theories in lower dimensions. Using this construction we identify the operators on noncommutative D-branes which couple to linearized supergravity backgrounds, from a knowledge of such couplings to lower dimensional D-branes with no $B$ field. These operators belong to a class of gauge invariant observables involving open Wilson lines. Assuming a DBI form of the coupling we show, to second order in the gauge potential but to all orders of the noncommutativity parameter, that our proposal agrees with the operator obtained in terms of ordinary gauge fields by considering brane actions in backgrounds and then using the Seiberg-Witten map to rewrite this in terms of noncommutative gauge fields. Our result clarify why a certain {\it commutative} but {\it non-associative} ``generalized star product'' appears both in the expansion of the open Wilson line, as well as in string amplitude computations of open string - closed string couplings. We outline how our procedure can be used to obtain operators in the noncommutative theory which are holographically dual to supergravity modes. 
  An algebraic method is used to work out the mass spectra and symmetry breaking patterns of general vacuum states in N=2 supersymmetric SU(n) Chern-Simons-Higgs systems with the matter fields being in the adjoint representation. The approach provides with us a natural basis for fields, which will be useful for further studies in the self-dual solutions and quantum corrections. As the vacuum states satisfy the SU(2) algebra, it is not surprising to find that their spectra are closely related to that of angular momentum addition in quantum mechanics. The analysis can be easily generalized to other classical Lie groups. 
  "Equivalent unconstrained systems" for QCD obtained by resolving the Gauss law are discussed. We show that the effects of hadronization, confinement, spontaneous chiral symmetry breaking and $\eta_0$-meson mass can be hidden in solutions of the non-Abelian Gauss constraint in the class of functions of topological gauge transformations, in the form of a monopole, a zero mode of the Gauss law, and a rising potential. 
  We study the confining and screening aspects of the Schwinger model on curved static backgrounds. 
  The symplectic leaves of W-algebras are the intersections of the symplectic leaves of the Kac-Moody algebras and the hypersurface of the second class constraints, which define the W-algebra. This viewpoint enables us to classify the symplectic leaves and also to give a representative for each of them. The case of the (W_{2}) (Virasoro) algebra is investigated in detail, where the positivity of the energy functional is also analyzed. 
  We discuss Weyl (conformal) transformations in two-dimensional matterless dilaton gravity. We argue that both classical and quantum dilaton gravity theories are invariant under Weyl transformations. 
  In the context of phase-space quantization, matrix elements and observables result from integration of c-number functions over phase space, with Wigner functions serving as the quasi-probability measure. The complete sets of Wigner functions necessary to expand all phase-space functions include off-diagonal Wigner functions, which may appear technically involved. Nevertheless, it is shown here that suitable generating functions of these complete sets can often be constructed, which are relatively simple, and lead to compact evaluations of matrix elements. New features of such generating functions are detailed and explored for integer-indexed sets, such as for the harmonic oscillator, as well as continuously indexed ones, such as for the linear potential and the Liouville potential. The utility of such generating functions is illustrated in the computation of star functions, spectra, and perturbation theory in phase space. 
  We show that, by considering physics in dS_4 or AdS_4 spacetime, one can circumvent the van Dam - Veltman - Zakharov theorem which requires that the extra polarization states of a massive graviton do not decouple in the massless limit. It is shown that the smoothness of the m->0 limit is ensured if the H (``Hubble'') parameter, associated with the horizon of the dS_4 or AdS_4 space, tends to zero slower than the mass of the graviton m. 
  We study the variational equations for solitons in noncommutative scalar field theories in an even number of spatial dimensions. We prove the existence of spherically symmetric solutions for a sufficiently large noncommutativity parameter $\theta$ and we prove the absence of spherically symmetric solutions for small $\theta$. 
  We study the gauge transformation of the recently computed one-loop four-point function of {\cal N}=4 supersymmetric Yang-Mills theory with gauge group U(N). The contributions from nonplanar diagrams are not gauge invariant. We compute their gauge variation and show that it is cancelled by the variation from corresponding terms of the one-loop five-point function. This mechanism is general: it insures the gauge invariance of the noncommutative one-loop effective action. 
  We propose a new ``bigravity'' model with two positive tension AdS_4 branes in AdS_5 bulk and no negative tension branes. The bounce of the ``warp'' factor mimics the effect of a negative brane and thus gives rise to an anomalously light graviton KK mode. This configuration satisfies the weak energy condition and has no ghost state. In addition, the extra polarization states of the massive graviton practically decouple and thus it does not contradict to Einsteinian gravity. However, the model has certain phenomenological difficulties associated with the presence of a negative cosmological constant on the branes. 
  Bubble nucleation has been studied on lattices using phenomenological Langevin equations. Recently there have been theoretical motivations for using these equations. These studies also conclude that the simple Langevin description requires some modification. We study bubble nucleation on a lattice and determine effects of the modified Langevin equations. 
  We provide the exact time-dependent cosmological solutions in the Randall-Sundrum (RS) setup with bulk matter, which may be smoothly connected to the static RS metric. In the static limit of the extra dimension, the solutions are reduced to the standard Friedmann equations. In view of our solutions, we also propose an explanation for how the extra dimension is stabilized in spite of a flat modulus potential at the classical level. 
  Randall-Sundrum model, which has a scalar field, is used to investigate the domain structure of the extra dimension and to obtain a possible solution of the mass hierarchy problem. It is found that when the domain wall size is comparable to that of domains, domains become unstable. To construct a reliable theory, a region of physical parameter space, where domains are stable, is identified. Analytic forms of field configurations are obtained by perturbative expansions in term of a small parameter that is approximately equal to the relative size of domain wall with respect to domains. By placing a single 3-brane in one of the domain, one can solve the mass hierarchy problem. 
  We study the elliptic C_n and BC_n Ruijsenaars-Schneider models which is elliptic generalization of system given in hep-th/0006004. The Lax pairs for these models are constructed by Hamiltonian reduction technology. We show that the spectral curves can be parameterized by the involutive integrals of motion for these models. Taking nonrelativistic limit and scaling limit, we verify that they lead to the systems corresponding to Calogero-Moser and Toda types. 
  Naked singularities in the gravitational backgrounds dual to gauge theories can be hidden behind the black hole horizon. We present an exact black hole solution in the Klebanov-Tseytlin geometry [hep-th/0002159]. Our solution realizes Maldacena dual of the finite temperature N=1 duality cascade of [hep-th/0007191] above the temperature of the chiral symmetry breaking. We compare the Bekenstein-Hawking entropy of the black hole with the entropy of the SU(N+M)xSU(N) gauge theory. 
  We explicitly evaluate the disk S-matrix elements of one closed string and an arbitrary number of open string states in the presence of a large background B-flux. From this calculation, we show that in the world-volume action of D-branes in terms of non-commutative fields, the closed string fields must be treated as functionals of the non-commutative gauge fields. We also find the generalized multiplication rule *_N between N open string fields on the world-volume of the D-brane. In particular, this result indicates that the difference between the familiar * and the *_N product is just some total derivative terms. We show that the *_N product and the dependence of the closed string fields on the non-commutative gauge fields emerge also from transforming the ordinary Dirac-Born-Infeld action(including the closed string fields) under the Seiberg-Witten map. We then conjecture a non-commutative DBI action for the transformation of the commutative DBI action under the SW map. 
  Making use of the N=2 Liouville theory and world-sheet techniques, we study the properties of D-branes wrapped around vanishing SUSY cycles of singular Calabi-Yau n-folds (n=2,3,4). After constructing boundary states describing the wrapped branes, we evaluate the disc amplitudes corresponding to the periods of SUSY cycles. We use the old technique of KPZ scaling in Liouville theory and derive holomorphicity and scaling behavior of vanishing cycles which are in agreement with geometrical considerations.   We also discuss the open string Witten index using the N=2 Liouville theory and obtain the intersection numbers among SUSY cycles which also agree with geometrical expectation. 
  We find the analogue of the Boillat metric of Born-Infeld theory for the M5-brane. We show that it provides the propagation cone of {\sl all} 5-brane degrees. In an arbitrary background field, this cone never lies outside the Einstein cone. An energy momentum tensor for the three-form is defined and shown to satisfy the Dominant Energy Condition. The theory is shown to be well defined for all values of the magnetic field but there is a limiting electric field strength. We consider the strong coupling limit of the M5-brane and show that the corresponding theory is conformally invariant and admits infinitely many conservation laws. On reduction to the Born-Infeld case this agrees with the work of Bia{\l}nicki-Birula. 
  The quon algebra describes particles, ``quons,'' that are neither fermions nor bosons using a label q that parametrizes a smooth interpolation between bosons (q = +1) and fermions (q = -1). We derive ``conservation of statistics'' relations for quons in relativistic theories, and show that in relativistic theories quons must be either bosons or fermions. 
  The response of vacuum to the presence of external conditions is the subject of this work. We consider a generalization of the Casimir effect in the presence of curved boundaries on which a sharp potential is concentrated. The profile of the potential is a delta function, which has some features in common with a hard boundary and some with a smooth background field. The boundaries investigated are: i) a spherical shell, ii) a cylindrical shell, iii) a magnetic flux tube. The vacuum energy is calculated by means of the Jost function of the scattering problem related to the field equation. The energy is then renormalized by means of a zeta functional approach adopting the heat-kernel expansion. The heat kernel coefficients are calculated and a discussion of the UV-divergences of the model is presented. The renormalized vacuum energy $E^{ren}$ is then numerically studied and plotted. The sign of $E^{ren}$ is found to be always negative in the case of the cylindrical shell, while in the case of the spherical shell and of the magnetic flux tube the sign depends on the value of the radius. 
  We prove that the van Dam-Veltman-Zakharov discontinuity arising in the massless limit of massive gravity theories is peculiar to Minkowski space and it is not present in Anti De Sitter space, where the massless limit is smooth. More generally, the massless limit is smooth whenever the square of the graviton mass vanishes faster than the cosmological constant. 
  Static solutions with a bulk dilaton are derived in the context of six dimensional warped compactification. In the string frame, exponentially decreasing warp factors are identified with critical points of the low energy $\beta$-functions truncated at a given order in the string tension corrections. The stability of the critical points is discussed in the case of the first string tension correction. The singularity properties of the obtained solutions are analyzed and illustrative numerical examples are provided. 
  The coefficient of the Chern-Simons term in the effective action for massive Dirac fermions in three dimensions is computed by using the point-splitting regularization method. We show that in this framework no ambiguities arise. This is related to the fact that the point-splitting regularization does not introduce additional parity breaking effects, implementing one possible physical criterion in order to uniquely characterize the system. 
  In this note we investigate the stability of the classical ground state of the Quantum Hall Soliton proposed recently in hep-th/0010105 . We explore two possible perturbations which are not spherically symmetric and we find that the potential energy decreases in both case. This implies that the system either decays or is dynamically stabilized (because of the presence of magnetic fields). If one makes an extra assumption that in the real quantum treatment of the problem string ends and D0 branes move together (as electrons and vortices in the Quantum Hall effect), a static equilibrium configuration is possible. 
  We study the fluctuation spectrum of linearized gravity around non-fine-tuned branes. We focus on the case of an AdS4 brane in AdS5. In this case, for small cosmological constant, the warp factor near the brane is essentially that of a Minkowski brane. However, far from the brane, the metric differs substantially. The space includes the AdS5 boundary, so it has infinite volume. Nonetheless, for sufficiently small AdS4 cosmological constant, there is a bound state graviton in the theory, and four-dimensional gravity is reproduced. However, it is a massive bound state that plays the role of the four-dimensional graviton. 
  In this paper we study the existence of theta-vacuum states in Yang-Mills theories defined over asymptotically flat space-times taking into account not only the topology but the complicated causal structure of these space-times, too. By a result of Galloway, apparently causality makes all vacuum states, seen by a distant observer, homotopically equivalent making the introduction of theta-terms unnecessary.   But a more careful analysis shows that certain twisted classical vacuum states survive even in this case eventually leading to the conclusion that the concept of ``theta-vacua'' is meaningful in the case of general Yang-Mills theories. We give a classification of these vacuum states based on Isham's results showing that the Yang-Mills vacuum has the same complexity as in the flat Minkowskian case hence the general CP-problem is not more complicated than the well-known flat one. 
  We explore a nonlinear realization of the (2+1)-dimensional Lorentz symmetry with a constant vacuum expectation value of the second rank anti-symmetric tensor field. By means of the nonlinear realization, we obtain the low-energy effective action of the Nambu-Goldstone bosons for the spontaneous Lorentz symmetry breaking. 
  We generalize the results of Randall and Sundrum to a wider class of four-dimensional space-times including the four-dimensional Schwarzschild background and de Sitter universe. We solve the equation for graviton propagation in a general four dimensional background and find an explicit solution for a zero mass bound state of the graviton. We find that this zero mass bound state is normalizable only if the cosmological constant is strictly zero, thereby providing a dynamical reason for the vanishing of cosmological constant within the context of this model. We also show that the results of Randall and Sundrum can be generalized without any modification to the Schwarzschild background. 
  This short note contains a detailed analysis to find the right power law the lowest eigenvalue of a localized massive graviton bound state in a four dimensional AdS background has to satisfy. In contrast to a linear dependence of the cosmological constant we find a quadratic one [hep-th/0011156]. 
  We study the renormalization of massless QED from the point of view of the Hopf algebra discovered by D. Kreimer. For QED, we describe a Hopf algebra of renormalization which is neither commutative nor cocommutative. We obtain explicit renormalization formulas for the electron and photon propagators, for the vacuum polarization and the electron self-energy, which are equivalent to Zimmermann's forest formula for the sum of all Feynman diagrams at a given order of interaction.   Then we extend to QED the Connes-Kreimer map defined by the coupling constant of the theory (i.e. the homomorphism between some formal diffeomorphisms and the Hopf algebra of renormalization) by defining a noncommutative Hopf algebra of diffeomorphisms, and then showing that the renormalization of the electric charge defines a homomorphism between this Hopf algebra and the Hopf algebra of renormalization of QED.   Finally we show that Dyson's formulas for the renormalization of the electron and photon propagators can be given in a noncommutative (e.g. matrix-valued) form. 
  The BRST formulation is used in order to derive the existence criterion for classical bi-Hamiltonian systems, based on non-anomalous deformation of the gauge-fixing structure. The recursion operator is then used to provide the entire hierarchy of integrable models associated to the original BRST and anti-BRST charges. 
  An extension of the non-local regularization scheme is formulated in the Sp(2) symmetric Lagrangian BRST quantization framework. It generates a systematic treatment of the anomalous quantum master equations and allows to substract the divergences as well as to calculate genuine higher loop BRST and anti-BRST anomalies. 
  In this paper we resolve a contradiction posed in a recent paper by Horowitz and Hubeny. The contradiction concerns the way small objects in AdS space are described in the holographic dual CFT description. 
  Although the AdS/CFT correspondence is rigorous only for an infinite $N \to \infty$ stack of D3-branes, it can be fruitfully studied for finite $N$ as a source of gauge structures and choices for chiral fermions and complex scalars which solve the hierarchy problem by a conformal fixed point. We emphasize orbifolds $AdS_5 \times S^5/\Gamma$ where the resulting GFT has ${\cal N} = 0$ supersymmetry. The fact that the complex scalars are prescribed by the construction limits the possible spontaneous symmetry breaking. Both abelian and nonabelian $\Gamma$ are illustrated by simple examples. An accurate $sin^2 \theta$ in electroweak unification can be obtained, suggesting that this approach merits further study. 
  We present the results of probing the ten dimensional type IIB supergravity solution corresponding to a renormalisation group flow of supersymmetric SU(n) Yang-Mills theory from pure gauge N=4 to N=1 with two massless adjoint flavours. The endpoint of the flow is an infrared fixed point theory, and because of this simplicity of the theory, the effective Lagrangian for the probe is very well-behaved, having no zeros or singularities in the tension, and a smooth potential, all of which we exhibit. Specialising to the locus of points where the potential vanishes, we also characterise a part of the Coulomb branch of the N=1 theory. The simplicity of the gauge theory physics allows us to isolate and emphasise a key holographic feature of brane probe physics which has wider applications in the study of geometry/gauge theory duals. 
  We show that the singular sources in the energy-momentum tensor for the Randall-Sundrum brane world, viewed as a solution of type IIB supergravity, are composed of two elements. One of these is a D3-brane source with tension opposite in sign to the RS tension in five dimensions; the other arises from patching two regions of flat ten-dimensional spacetime. This resolves an apparent discrepancy between supersymmetry and the sign and magnitude of the RS tension. 
  We study disk amplitudes whose boundaries have heterogeneous matter states in a system of $(4,5)$ conformal matter coupled to 2-dim gravity. They are analysed by using the 3-matrix chain model in the large $N$ limit. Each of the boundaries is composed of two or three parts with distinct matter states. From the obtained amplitudes, it turns out that each heterogeneous boundary loop splits into several loops and we can observe properties in the splitting phenomena that are common to each of them. We also discuss the relation to boundary operators. 
  We demonstrate by explicit construction that while the untwisted Harrington-Shepard caloron $A_\mu$ is manifestly periodic in Euclidean time, with period $\beta=\frac{1}{T}$, when transformed to the Weyl ($A_0=0$) gauge, the caloron gauge field $A_i$ is periodic only up to a large gauge transformation, with winding number equal to the caloron's topological charge. This helps clarify the tunneling interpretation of these solutions, and their relation to Chern-Simons numbers and winding numbers. 
  Noncommutative solitons are easier to find in a noncommutative field theory. Similarly, the one-loop quantum corrections to the mass of a noncommutative soliton are easier to compute, in a real scalar theory in 2+1 dimensions. We carry out this computation in this paper. We also discuss the model with a double-well potential, and conjecture that there is a partial symmetry restoration in a vacuum state. 
  The proposal that a strong coupling limit of the five-dimensional type II string theory (M-theory compactified on a 6-torus) in which the Planck length becomes infinite could give a six-dimensional superconformal phase of M-theory is reviewed. This limit exists for the free theory, giving a 6-dimensional theory with (4,0) supersymmetry compactified on a circle whose radius gives the 5-dimensional Planck length. The free 6-dimensional theory has a fourth rank tensor gauge field with the symmetries of the Riemann tensor instead of a symmetric tensor gauge field, but its dimensional reduction gives conventional linearised gravity in five dimensions. The possibility of an interacting form of this theory existing and the consequences it would have for the geometric picture of gravity are discussed. 
  We consider the quantum mechanics of a particle on a noncommutative plane. The case of a charged particle in a magnetic field (the Landau problem) with a harmonic oscillator potential is solved. There is a critical point, where the density of states becomes infinite, for the value of the magnetic field equal to the inverse of the noncommutativity parameter. The Landau problem on the noncommutative two-sphere is also solved and compared to the plane problem. 
  We use normal coordinate methods to obtain the expansion with respect to fermionic coordinates of the 11-dimensional supermembrane action in a supergravity background. Likewise, expansions for various branes in other dimensions can be obtained. These methods allow a systematic and unambiguous expansion of the vielbein to any order in the fermionic coordinates and avoid the complications encountered in the gauge completion approach. 
  The Wess-Zumino model is analysed in the framework of the causal approach of Epstein-Glaser. The condition of invariance with respect to supersymmetry transformations is similar to the gauge invariance in the Z\"urich formulation. We prove that this invariance condition can be implemented in all orders of perturbation theory, i.e. the anomalies are absent in all orders. This result is of purely algebraic nature. We work consistently in the quantum framework based on Bogoliubov axioms of perturbation theory so no Grassmann variables are necessary. 
  In this paper, we consider bosonic reduced Yang-Mills integrals by using some approximation schemes, which are a kind of mean field approximation called Gaussian approximation and its improved version. We calculate the free energy and the expectation values of various operators including Polyakov loop and Wilson loop. Our results nicely match to the exact and the numerical results obtained before. Quite good scaling behaviors of the Polyakov loop and of the Wilson loop can be seen under the 't Hooft like large $N$ limit for the case of the loop length smaller. Then, simple analytic expressions for the loops are obtained. Furthermore, we compute the Polyakov loop and the Wilson loop for the case of the loop length sufficiently large, where with respect to the Polyakov loop there seems to be no known results in appropriate literatures even in numerical calculations. The result of the Wilson loop exhibits a strong resemblance to the result simulated for a few smaller values of $N$ in the supersymmetric case. 
  We consider the entropy of a quantum scalar field on a background black hole geometry in asymptotically anti-de Sitter space-time, using the ``brick wall'' approach. In anti-de Sitter space, the theory has no infra-red divergences, and all ultra-violet divergences can be absorbed into a renormalization of the coupling constants in the one-loop effective gravitational Lagrangian. We then calculate the finite renormalized entropy for the Schwarzschild-anti-de Sitter and extremal Reissner-Nordstrom-anti-de Sitter black holes, and show that, at least for large black holes, the entropy is entirely accounted for by the one-loop Lagrangian, apart possibly from terms proportional to the logarithm of the event horizon radius. For small black holes, there are indications that non-perturbative quantum gravity effects become important. 
  We explore physics on the boundary of a Randall-Sundrum type model when the brane tension is slightly sub-critical. We calculate the masses of the Kaluza-Klein decomposition of the graviton and use a toy model to show how localized gravity emerges as the brane tension becomes critical. Finally, we discuss some aspects of the boundary conformal field theory and the AdS/CFT correspondence. 
  We study the way Lorentz covariance can be reconstructed from Matrix Theory as a IMF description of M-theory. The problem is actually related to the interplay between a non abelian Dirac-Born-Infeld action and Super-Yang-Mills as its generalized non-relativistic approximation. All this physics shows up by means of an analysis of the asymptotic expansion of the Bessel functions $K_\nu$ that profusely appear in the computations of amplitudes at finite temperature and solitonic calculations. We hope this might help to better understand the issue of getting a Lorentz covariant formulation in relation with the $N\to +\infty$ limit. There are also some computations that could be of some interest in Relativistic Statistical Mechanics. 
  We compute the two point correlation function of a dimension 4 operator in a nonconformal cascading N=1 SUSY gauge theory using the supergravity dual found by Klebanov and Strassler[hep-th/0007191]. The two point function has a logarithmic correction to conformal behavior which is related to the scale dependence of the effective number of colors. The nonsingular behavior of this correlator suggests that the theory remains a local 4-dimensional quantum field theory at all scales. We also compute the spectrum of low-lying glueball modes corresponding to the above operator. 
  Covariant classical particle dynamics is described, and the associated covariant relativistic particle quantum mechanics is derived. The invariant symmetric bracket is defined on the space of quantum amplitudes, and its relation to a generalized Hamiltonian dynamics and to a covariant Schr\"odinger type equation is shown. Examples for relativistic potential problems are solved. Mathematically and physically acceptable probability densities for the Klein-Gordon equation and for the Dirac equation are derived, and a new continuity equation for each case is given. The quantum distribution for mass is discussed, and unambiguous representations of four-velocity and four-acceleration operators are given. 
  We examine the structure of the Yukawa couplings in the 11 dimensional Horava-Witten M-theory based on non-standard embeddings. We find that the CKM and quark mass hierarchies can be explained in M Theory without introducing undue fine tuning. A phenomenological example is presented satisfying all CKM and quark mass data requiring the 5-branes cluster near the second orbifold plane, and that the instanton charges of the physical orbifold plane vanish. The latter condition is explicitly realized on a Calabi-Yau manifold with del Pezzo base dP_7. 
  Holographic Renormalization Group (RG) in nine dimensions is considered. The d8 holographic conformal anomaly is found. It should correspond to d8 CFT in AdS_9/CFT_8 correspondence. The comparison of holographic and QFT anomalies in d8 de Sitter space is done. It may give the indication for rigorous AdS_9/CFT_8 correspondence proposal. 
  The translation group T(2), contained in Wigner's little group for massless particles, is shown to generate gauge transformations in the Kalb-Ramond theory, exactly as happens in Maxwell case. For the topologically massive ($B\wedge$F) gauge theory, both T(2) and T(3), act as the corresponding generators. 
  We discuss a new feature of the 5d Kaluza-Klein cosmology. For that purpose, we obtain the simplest x^5-dependent solution which, in the reduced description, is associated with a radiation-dominated Robertson-Walker universe, and also can be regarded as an extension of the Schwarzschild solution. This solution enables us to deduce an important result that an evolving universe is related with a static universe by the gauge transformation, i.e., they are gauge equivalent. This means that having a different universe simply corresponds to choosing a different gauge 
  Stable non-BPS states can be constructed and studied in a variety of contexts in string theory. Here we review some interesting constructions that arise from suspended and wrapped branes. We also exhibit some stable non-BPS states that have holographic duals. 
  A systematic analysis is presented of compactifications of the IIB superstring on $AdS_5 \times S^5/\Gamma$ where $\Gamma$ is a non-abelian discrete group. Every possible $\Gamma$ with order $g \leq 31$ is considered. There exist 45 such groups but a majority cannot yield chiral fermions due to a certain theorem that is proved. The lowest order to embrace the nonSUSY standard $SU(3) \times SU(2) \times U(1)$ model with three chiral families is $\Gamma = D_4 \times Z_3$, with $g=24$; this is the only successful model found in the search. The consequent uniqueness of the successful model arises primarily from the scalar sector, prescribed by the construction, being sufficient to allow the correct symmetry breakdown. 
  A class of non abelian affine Toda models arising from the axial gauged two-loop WZW model is presented. Their zero curvature representation is constructed in terms of a graded Kac-Moody algebra. It is shown that the discrete multivacua structure of the potential together with non abelian nature of the zero grade subalgebra allows soliton solutions with non trivial electric and topological charges.  The dressing transformation is employed to explicitly construct one and two soliton solutions and their bound states in terms of the tau functions. A discussion of the classical spectra of such solutions and the time delays are given in detail. 
  We give a review of the works devoted to the treatment of the Kerr super black hole solution as a spinning particle. The real, complex and stringy structures of the Kerr and super-Kerr geometries are discussed, as well as the recent results on the regular matter source for the Kerr spinning particle. It is shown that the source has to represent a rotating bag-like bubble having (A)dS interior and a smooth domain wall boundary. The given by Morris supersymmetric generalization of the U(I) x U'(I) field model (which was used by Witten to describe cosmic superconducting strings) is considered, and it is shown that this model can be adapted for description of superconducting bags having a long range external electromagnetic field and another gauge field confined inside the bag. 
  By making use of current-algebra Ward identities we study renormalization of general anisotropic current-current interactions in 2D. We obtain a set of algebraic conditions that ensure the renormalizability of the theory to all orders. In a certain minimal prescription we compute the beta function to all orders. 
  We construct supergravity solutions corresponding to fivebranes wrapping associative three-cycles of constant curvature in manifolds of G_2-holonomy. The solutions preserve 2 supercharges and are first constructed in D=7 gauged supergravity and then lifted to D=10,11. We show that the low-energy theory of M-fivebranes wrapped on a compact hyperbolic three-space is dual to a superconformal field theory in D=3 by exhibiting a flow to an AdS_4 region. For IIB-fivebranes wrapped on a three-sphere we speculate on a connection with spontaneous supersymmetry breaking of pure N=1 super Yang-Mills theory in D=3. 
  We discuss superparticle and superstring dynamics in AdS_3 x S^3 supported by R-R 3-form background using light-cone gauge approach. Starting with the superalgebra psu(1,1|2) + psu(1,1|2) representing the basic symmetry of this background we find the light-cone superparticle Hamiltonian. We determine the harmonic decomposition of light-cone superfield describing fluctuations of type IIB supergravity fields expanded near AdS_3 x S^3 background and compute the corresponding Kaluza-Klein spectrum. We fix the fermionic and bosonic light-cone gauges in the covariant Green-Schwarz AdS_3 x S^3 superstring action and find the light-cone string Hamiltonian. We also obtain a realization of the generators of psu(1,1|2) + psu(1,1|2) in terms of the superstring 2-d fields in the light-cone gauge. 
  Without recourse to the sophisticated machinery of twisted group algebras, projective character tables and explicit values of 2-cocycles, we here present a simple algorithm to study the gauge theory data of D-brane probes on a generic orbifold G with discrete torsion turned on. We show in particular that the gauge theory can be obtained with the knowledge of no more than the ordinary character tables of G and its covering group G*. Subsequently we present the quiver diagrams of certain illustrative examples of SU(3)-orbifolds which have non-trivial Schur Multipliers. The paper serves as a companion to our earlier work (arXiv:hep-th/0010023) and aims to initiate a systematic and computationally convenient study of discrete torsion. 
  We study the N = 2 supersymmetric analog of the Klebanov-Strassler system. We first review the resolution of singularities by the enhancon mechanism, and the physics of fractional branes on an orbifold. We then describe the exact N = 2 solution. This exhibits a duality cascade as in the N = 1 case, but the singularity resolution is the characteristic N = 2 enhancon. We discuss some related solutions and open issues. 
  Just as for non-abelian gauge theories at strong coupling, discrete lattice methods are a natural tool in the study of non-perturbative quantum gravity. They have to reflect the fact that the geometric degrees of freedom are dynamical, and that therefore also the lattice theory must be formulated in a background-independent way. After summarizing the status quo of discrete covariant lattice models for four-dimensional quantum gravity, I describe a new class of discrete gravity models whose starting point is a path integral over Lorentzian (rather than Euclidean) space-time geometries. A number of interesting and unexpected results that have been obtained for these dynamically triangulated models in two and three dimensions make discrete Lorentzian gravity a promising candidate for a non-trivial theory of quantum gravity. 
  We study the brane world motion in non-static bulk by generalizing the second Randall-Sundrum scenario Explicitly, we take the bulk to be a Vaidya-AdS metric, which describes the gravitational collapse of a spherically symmetric null dust fluid in Anti-de Sitter spacetime. We point out that during an inflationary phase on the brane, black holes will tend to be thermally nucleated in the bulk We analyze the thermodynamical properties of this brane-world.We point out that during an inflationary phase on the brane, black holes will tend to be thermally nucleated in the bulk. Thermal equilibrium of the system is discussed. We calculate the late time behavior of this system, including 1-loop effects. We argue that at late times a sufficiently large black hole will relax to a point of thermal equilibrium with the brane-world environment. This result has interesting implications for early-universe cosmology. 
  Recently we have obtained a non-perturbative but convergent series expression of the one loop effective action of QED, and discussed the renormalization of the effective action. In this paper we establish the electric-magnetic duality in the quantum effective action. 
  The review studies connections between integrable many-body systems and gauge theories. It is shown how the degrees of freedom in integrable systems are related with topological degrees of freedom in gauge theories. The relations between families of integrable systems and N=2 supersymmetric gauge theories are described. It is explained that the degrees of freedom in the many-body systems can be identified with collective coordinates of D-branes, solitons in string theory. 
  The Casimir energy for massless scalar field of two parallel conductor, in two dimensional Schwarzchild black hole background, with Dirichlet boundary conditions is calculated by making use of general properties of renormalized stress tensor. We show that vacuum expectation value of stress tensor can be obtain by Casimir effect, trace anomaly and Hawking radiation. Four-dimensional of this problem, by this method, is under progress by this author. 
  We present the results of a numerical investigation of percolation properties in a version of the classical Heisenberg model. In particular we study the percolation properties of the subsets of the lattice corresponding to equatorial strips of the target manifold ${\cal S}^2$. As shown by us several years ago, this is relevant for the existence of a massless phase of the model. Our investigation yields strong evidence that such a massless phase does indeed exist. It is further shown that this result implies lack of asymptotic freedom in the massive continuum limit. A heuristic estimate of the transition temperature is given which is consistent with the numerical data. 
  The quantum completion of the space of connections in a manifold can be seen as the set of all morphisms from the groupoid of the edges of the manifold to the (compact) gauge group. This algebraic construction generalizes an analogous description of the gauge-invariant quantum configuration space of Ashtekar and Isham, clarifying the relation between the two spaces. We present a description of the groupoid approach which brings the gauge-invariant degrees of freedom to the foreground, thus making the action of the gauge group more transparent. 
  We study in details on how gauge bosons can acquire mass when the chiral symmetry dynamically breaks down for massless gauge theory without scalars. Introducing dynamical scalar fields into the original gauge theory, we show that when the chiral symmetry breaks down, the theory gives gauge boson masses different from what would be obatained if an elemetary Higgs is included. We clarify the reason and propose one method how to calculate gauge boson masses in the case of dynamical gauge symmtry breakdown. We explain the method by using an example in which SU(5) massless gauge theory breaks down to SU(4) with massless fermions in appropriate representations. 
  We present a semiclassical calculation of instanton effects in N=4 supersymmetric Yang-Mills theory formulated on R^{3}XS^{1} and also in the N=1 theory obtained by introducing chiral multiplet masses. In the N=4 case, these instanton effects are related to the bulk contribution to the index which counts BPS dyons in the corresponding four dimensional theory. In both cases, the calculations provide semiclassical tests of recently proposed exact results for the lowest non-trivial terms in the derivative expansion of the Wilsonian effective action. 
  We describe finite temperature N=4 superstrings in D=5 by an effective four-dimensional supergravity of the thermal winding modes that can become tachyonic and trigger the instabilities at the Hagedorn temperature. Using a domain-wall ansatz, exact solutions to special BPS-type first order equations are found. They preserve half of the supersymmetries, contrary to the standard perturbative superstring at finite temperature that breaks all supersymmetries. Our solutions show no indication of any tachyonic instability and provide evidence for a new BPS phase of finite temperature superstrings that is stable for all temperatures. This would have important consequences for a stringy description of the early universe. 
  We obtain the spectrum of glueball masses for the N=1 non-conformal cascade theory whose supergravity dual was recently constructed by Klebanov and Strassler. The glueball masses are calculated by solving the supergravity equations of motion for the dilaton and the two-form in the deformed conifold background. 
  We construct an exact regular vortex solution to the self-dual equations of the Abelian Higgs model in non-commutative space for arbitrary values of $\theta$. To this end, we propose an ansatz which is the analogous, in Fock space, to the one leading to exact solutions for the Nielsen-Olesen vortex in commutative space. We compute the flux and energy of the solution and discuss its relevant properties. 
  We analyze open and mixed sector tree-level amplitudes of N=2 strings in a space-time with (2,2) signature, in the presence of constant B field. The expected topological nature of string amplitudes in the open sector is shown to impose nontrivial constraints on the corresponding noncommutative field theory. In the mixed sector, we first compute a 3-point function and show that the corresponding field theory is written in terms of a generalized *-product. We also analyze a 4-point function (A_{oooc}) of the mixed sector in Theta ---> infinity limit. 
  The holographic description of supersymmetric RG flows in supergravity is considered from both the five-dimensional and ten-dimensional perspectives. An N=1* flow of N=4 super-Yang Mills is considered in detail, and the infra-red limit is studied in terms of IIB supergravity in ten dimensions. Depending on the vevs and the direction of approach to the core, the supergravity solution can be interpreted in terms of either 5-branes or 7-branes. Generally, it is shown that it is essential to use the ten-dimensional description in order to study the infra-red asymptotics in supergravity. 
  The purpose of this note is to show that W3 algebras originate from an unusual interplay between the breakings of the reparametrization invariance under the diffemorphism action on the cotangent bundle of a Riemann surface. It is recalled how a set of smooth changes of local complex coordinates on the base space are collectively related to a background within a symplectic framework. The power of the method allows to calculate explicitly some primary fields whose OPEs generate the algebra as explicit functions in the coordinates: this is achieved only if well defined conditions are satisfied, and new symmetries emerge from the construction. Moreoverer, when primary flelds are introduced outside of a coordinate description the W3 symmetry byproducts acquire a good geometrical definition with respect to holomorphic changes of charts. 
  It is shown that all four superintegrable quantum systems on the Euclidean plane possess the same underlying hidden algebra $sl(3)$. The gauge-rotated Hamiltonians, as well as their integrals of motion, once rewritten in appropriate coordinates, preserve a flag of polynomials. This flag corresponds to highest-weight finite-dimensional representations of the $sl(3)$-algebra, realized by first order differential operators. 
  The dimensional reduction of the three-dimensional fermion-Chern-Simons model (related to Hopf maps) of Adam et el. is shown to be equivalent to (i) either the static, fixed--chirality sector of our non-relativistic spinor-Chern-Simons model in 2+1 dimensions, (ii) or a particular Heisenberg ferromagnet in the plane. 
  We discuss the supersymmetry operator in the cohomological formulation of dimensionally reduced SYM. By establishing the cohomology, a large class of invariants are classified. 
  The model of a classical particle with the weak linear AAD potential is subjected to path integral quantization. The light cone constraints and peculiar properies of its internal variables permit to use in calculations commutative dynamics and apply path integrals for a matrix form of the transition amplitude. Quantization leads to description of a Dirac particle. 
  The classical relativistic linear AAD interaction, introduced by the author, leads in the case of weak coupling to a pointlike particle capable to be sub- mitted to quantization via Feynman's path integrals along the line adequate to the requirements of the Pauli equation. In the discussed nonrelativistic case of the model the concept of spin is considered within early Feynman's ideas. 
  The model of nonrelativistic particles coupled to nonstandard (2+1)-gravity [1] is extended to include Abelian or non-Abelian charges coupled to Chern-Simons gauge fields. Equivalently, the model may be viewed as describing the (Abelian or non-Abelian) anyonic dynamics of Chern-Simons particles coupled, in a reparametrization invariant way, to a translational Chern-Simons action. The quantum two-body problem is described by a nonstandard Schr\"{o}dinger equation with a noninteger angular momentum depending on energy as well as particle charges. Some numerical results describing the modification of the energy levels by these charges in the confined regime are presented. The modification involves a shift as well as splitting of the levels. 
  The free (4,0) superconformal theory in 6 dimensions and its toroidal dimensional reductions are studied. The reduction to four dimensions on a 2-torus has an $SL(2,\Z)$ duality symmetry that acts non-trivially on the linearised gravity sector, interchanging the linearised Einstein equations and Bianchi identities and giving a self-duality between strong and weak coupling regimes. The possibility of this extending to an interacting form of the theory is discussed and implications for the non-geometric picture of gravity that could emerge are considered. 
  Starting from the geometric calculus based on Clifford algebra, the idea that physical quantities are Clifford aggregates ("polyvectors") is explored. A generalized point particle action ("polyvector action") is proposed. It is shown that the polyvector action, because of the presence of a scalar (more precisely a pseudoscalar) variable, can be reduced to the well known, unconstrained, Stueckelberg action which involves an invariant evolution parameter. It is pointed out that, starting from a different direction, DeWitt and Rovelli postulated the existence of a clock variable attached to particles which serve as a reference system for identification of spacetime points. The action they postulated is equivalent to the polyvector action. Relativistic dynamics (with an invariant evolution parameter) is thus shown to be based on even stronger theoretical and conceptual foundations than usually believed. 
  Conical defects, or point particles, in AdS_3 are one of the simplest non-trivial gravitating systems, and are particularly interesting because black holes can form from their collision. We embed the BPS conical defects of three dimensions into the N=4b supergravity in six dimensions, which arises from IIB string theory compactified on K3. The required Kaluza-Klein reduction of the six dimensional theory on a sphere is analyzed in detail, including the relation to the Chern-Simons supergravities in three dimensions. We show that the six dimensional spaces obtained by embedding the 3d conical defects arise in the near-horizon limit of rotating black strings. Various properties of these solutions are analyzed and we propose a representation of our defects in the CFT dual to asymptotically AdS_3 x S^3 spaces. Our work is intended as a first step towards analyzing colliding defects that form black holes. 
  Using background field perturbation theory we study Wilsonian effective actions of noncommutative gauge theories with an arbitrary matter content. We determine the Wilsonian coupling constant and the gauge boson polarization tensor as functions of the momentum scale k at the one-loop level and study their short-distance behaviour as theta k ->0, where theta is the noncommutativity parameter. The mixing between the short-distance and the long-distance degrees of freedom characteristic of noncommutative field theories violates the universality of the Wilsonian action and leads to IR-singularities. We find, in agreement with known results, that the quadratic IR divergencies cancel in supersymmetric gauge theories. The logarithmic divergencies disappear in mass-deformed N=4 theories, but not in other finite N=2 theories. We next concentrate on finite N=2 and mass-deformed N=4 supersymmetric U(1) gauge theories with massive hypermultiplets. The Wilsonian running coupling exhibits a non-trivial threshold behaviour at and well below the noncomutativity scale, eventually becoming flat in the extreme infrared in N=4 theories, but not in N=2 theories. This is interpreted as the (non)-existence of a non-singular commutative limit where the theory is described by a commutative N=2 pure U(1) theory. We expect that our analysis of finite theories is exact to all orders in perturbation theory. 
  The static gauge potential between heavy sources is computed from a recently introduced non-critical bosonic string background. When the sources are located at the infinity of the holographic coordinate, the linear dilaton behavior is recovered, which means that the potential is exactly linear in the separation between the sources. When the sources are moved towards the origin, a competing overconfining cubic branch appears, which is however disfavored energetically. 
  The main new result here is the cancellation of global anomalies in the Type I superstring, with and without D-branes. Our argument here depends on a precise interpretation of the 2-form abelian gauge field using KO-theory; then the anomaly cancellation follows from a geometric form of the full Atiyah-Singer index theorem for families of Dirac operators. This is a refined version of the Green-Schwarz mechanism. It seems that a geometric interpretation of this mechanism-the cancellation of local and global fermion anomalies against local and global anomalies in the electric coupling of an abelian gauge field-always proceeds in a similar manner. For example, a previous paper with M. Hopkins (hep-th/0002027) explains the cancellation of anomalies in Type II with D-branes in these terms. The focal point of this paper is a general discussion about abelian gauge fields and Dirac charge quantization. Namely, we argue that quantization of charge is implemented in the functional integral by interpreting abelian gauge fields as cochains in a generalized differential cohomology theory. Our exposition includes elementary examples as well as examples from superstring theory. The mathematical underpinnings of differential cohomology are currently under development; we only give a sketch here. The anomaly cancellation in Type I depends on properties of a certain quadratic form in KO-theory, which we analyze in an appendix written jointly with M. Hopkins. In particular, the usual equation ``Tr R^2 = Tr F^2'' is refined to an equation in the KO-theory of spacetime. 
  We outline the the geometry of locally anisotropic (la) superspaces and la-supergravity. The approach is backgrounded on the method of anholonomic superframes with associated nonlinear connection structure. Following the formalism of enveloping algebras and star product calculus we propose a model of gauge la-gravity on noncommutative spaces. The corresponding Seiberg-Witten maps are established which allow the definition of dynamics for a finite number of gravitational gauge field components on noncommutative spaces. 
  We consider the Seiberg-Witten Toda chains arising in the context of exact solutions to N=2 SUSY Yang-Mills and their relation to the properties of N=1 SUSY gauge theories. In particular, we discuss their "perturbative" and "solitonic" degenerations and demonstrate some relations of the latter ones to the confining properties of N=1 vacua. 
  We investigate and interpret a large class of soliton solutions found in noncommutative tachyon condensation. These constructions make extensive use of the idea that Dp-branes may be built out of lower dimensional branes. Finally we comment on a recently proposed solution generating technique. 
  The counterterms, which must be included into Light-Front Hamiltonian of $QED_2$ to get the equivalence with conventional Lorentz-covariant formulation, are found. This is done to all orders of perturbation theory in fermion mass, using the bosonization at intermediate steps and comparing Light-Front and Lorentz-covariant perturbation theories for bosonized model. The obtained Light-Front Hamiltonian contains all terms, present in the $QED_2$ theory, canonically (naively) quantized on the Light-Front (in the Light-Front gauge) and an unusual counterterm. This counterterm is proportional to linear combination of fermion zero modes (which are multiplied by some operator factors neutralizing their charge and fermionic number). The coefficients before these zero mode operators are UV finite and depend on condensate parameter in the $\theta$-vacuum. These coefficients are proportional to fermion mass, when this mass goes to zero. 
  A set of consistency conditions is derived from Einstein equations for brane world scenarios with a spatially periodic internal space. In particular, the sum of the total tension of the flat branes and the non-negative integral of the gradient energy of the bulk scalars must vanish. This constraint allows us to make a simple consistency check of several models. We show that the two-brane Randall-Sundrum model satisfies this constraint, but it does not allow a generalization with smooth branes (domain walls), independently of the issue of supersymmetry. The Goldberger-Wise model of brane stabilization has to include the backreaction on the metric and the fine tuning of the cosmological constant to satisfy the constraints. We check that this is achieved in the DeWolfe-Freedman-Gubser-Karch scenario. Our constraints are automatically satisfied in supersymmetric brane world models. 
  In this paper we incorporate gauge fields into the tachyon field theory models for unstable D-branes in bosonic and in Type II string theories. The chosen couplings yield massless gauge fields and an infinite set of equally spaced massive gauge fields on codimension one branes. A lack of a continuum spectrum is taken as evidence that the stable tachyon vacuum does not support conventional gauge excitations. For the bosonic string model we find two possible solvable couplings, one closely related to Born-Infeld forms and the other allowing a detailed comparison to the open string modes on bosonic D-branes. We also show how to include fermions in the type II model. They localize correctly on stable codimension one branes resulting in bose-fermi degeneracy at the massless level. Finally, we establish the solvability of a large class of models that include kinetic terms with more than two derivatives. 
  We investigate some aspects of N=2 twisted theories with matter hypermultiplets in the fundamental representation of the gauge group. A consistent formulation of these theories on a general four-manifold requires turning on a particular magnetic flux, which we write down explicitly in the case of SU(2k). We obtain the blowup formula and show that the blowup function is given by a hyperelliptic sigma-function with singular characteristic. We compute the contact terms and find, as a corollary, interesting identities between hyperelliptic Theta functions. 
  We try to give a qualitative description of the Godbillon-Vey class and its relation on the one hand to the holonomy and on the other hand to the topological entropy of a foliation, using a remarkable theorem proved recently by G. Duminy (which still remains unpublished), relating these three notions in the case of codim-1 foliations. Moreover we shall investigate its possible consequences on string theory. In particular we shall present a conceptual argument according to which the curvature of the B-field (rank two antisymmetric tensor) of open strings might be related to the Godbillon-Vey class using a suitable generalisation of ``Non-Abelian Geometry'' which has just appeared in physics literature. Our starting point again is the Connes-Douglas-Schwarz article on compactifications of matrix models to noncommutative tori. 
  The unitarity condition for scattering amplitudes in a non-anticommutative quantum field theory is investigated. The Cutkosky rules are shown to hold for Feynman diagrams in Euclidean momentum space and unitarity of amplitudes can be satisfied. An analytic continuation of the diagrams to physical Minkowski spacetime can be performed without invoking unphysical singularities in amplitudes. The high energy behavior of amplitudes is found to be regular at infinity provided that only space-space non-anticommutativity is allowed. 
  We study gravitational aspects of Brane-World scenarios. We show that the bulk Einstein equations together with the junction condition imply that the induced metric on the brane satisfies the full non-linear Einstein equations with a specific effective stress energy tensor. This result holds for any value of the bulk cosmological constant. The analysis is done by either placing the brane close to infinity or by considering the local geometry near the brane. In the case that the bulk spacetime is asymptotically AdS, we show that the effective stress energy tensor is equal to the sum of the stress energy tensor of matter localized on the brane and of the holographic stress energy tensor appearing in the AdS/CFT duality. In addition, there are specific higher-curvature corrections to Einstein's equations. We analyze in detail the case of asymptotically flat spacetime. We obtain asymptotic solutions of Einstein's equations and show that the effective Newton's constant on the brane depends on the position of the brane. 
  Dynamical systems whose symplectic structure degenerates, becoming noninvertible at some points along the orbits are analyzed. It is shown that for systems with a finite number of degrees of freedom, like in classical mechanics, the degeneracy occurs on domain walls that divide phase space into nonoverlapping regions each one describing a nondegenerate system, causally disconnected from each other. These surfaces are characterized by the sign of the Liouville's flux density on them, behaving as sources or sinks of orbits. In this latter case, once the system reaches the domain wall, it acquires a new gauge invariance and one degree of freedom is dynamically frozen, while the remaining degrees of freedom evolve regularly thereafter. 
  Using a covariant method to regularize the composite operators, we obtain the bosonized action of the massive Schwinger model on a classical curved background. Using the solution of the bosonic effective action, the energy of two static external charges with finite and large distance separation on a static curved space-time is obtained. The confining behavior of this model is also explicitly discussed. 
  The generalized *-products, or the $*_N$-products, appear both in the one-loop effective action of noncommutative Yang-Mills theories and in the coupling of a closed string to N open strings on a disk when the D-brane world-volume is noncommutative. Factorization of the string amplitudes provides a uniform understanding of the $*_N$-products and a hint to obtain a simple, explicit formula (in the momentum space) for arbitrary N in the non-Abelian case. Possible extension to a more general ${}_M *_{N}$-product in the M-loop context is discussed. 
  We study the supersymmetry of the charged rotating toroidal black hole solutions found by Lemos and Zanchin, and show that the only configurations that are supersymmetric are: (i) the non-rotating electrically charged naked singularities already studied by Caldarelli and Klemm, and (ii) an extreme rotating toroidal black hole with zero magnetic and electric charges. For this latter case, the extreme uncharged black hole, we calculate the Killing spinors and show that the configuration preserves the same supersymmetries as the background spacetime. 
  A plausible physical interpretation of the renormalizability condition is given. It is shown that renormalizable quantum field theories describe such systems wherein the tendency to collapse associated with vacuum fluctuations of attractive forces is suppressed by vacuum fluctuations of kinetic energy. Relaying on the classification of topological types of evolution of point particles and analysing the problem of the fall to the centre, we obtain a general criterion for preventability of collapse which states that the spectrum of the Hamiltonian must be bounded from below. The holographic principle is used to explain the origin of anomalies and make precise the relation between the renormalizability and the reversibility. 
  Starting from the (q,p) 5-brane solution of type IIB string theory, we here construct the low energy configuration corresponding to (NS5,Dp)-brane bound states (for $0\leq p\leq 4$) using the T-duality map between type IIB and type IIA string theories. We use the SL(2,Z) symmetry on the type IIB bound state (NS5,D3) to construct (NS5,D5,D3) bound state. We then apply T-duality transformation again on this state to construct the bound states of the form (NS5,D(p+2),Dp) (for $0\leq p\leq 2$) of both type IIB and type IIA string theories. We give the tension formula for these states and show that they form non-threshold bound states. All these states preserve half of the space-time supersymmetries of string theories. We also briefly discuss the ODp-limits corresponding to (NS5,Dp) bound state solutions. 
  A novel method of transplanting algebras of observables from de Sitter space to a large class of Robertson-Walker space-times is exhibited. It allows one to establish the existence of an abundance of local nets on these spaces which comply with a recently proposed condition of geometric modular action. The corresponding modular symmetry groups appearing in these examples also satisfy a condition of modular stability, which has been suggested as a substitute for the requirement of positivity of the energy in Minkowski space. Moreover, they exemplify the conjecture that the modular symmetry groups are generically larger than the isometry and conformal groups of the underlying space-times. 
  Using anomalous symmetries of the cubic string field theory vertex we derive set of relations between the coefficients of the tachyon condensate. They are in agreement with the results obtained from level truncation approximation. 
  In this review we discuss hidden symmetries of toroidal compactifications of eleven-dimensional supergravity. We recall alternative versions of this theory which exhibit traces of the hidden symmetries when still retaining the massive Kaluza-Klein states. We reconsider them in the broader perspective of M-theory which incorporates a more extended variety of BPS states. We also argue for a new geometry that may underly these theories. All our arguments point towards an extension of the number of space-time coordinates beyond eleven. 
  In this habilitation thesis we provide an introduction to gravitational models in two spacetime dimensions. Focus is put on exactly solvable models. We begin by introducing and motivating different possible gravitational actions, including those of generalized dilaton theories as well as of purely geometrical, higher derivative theories with and without torsion. The relation among them as well as to Poisson sigma models is worked out in some detail. In the exactly solvable cases, such as pure gravity-Yang-Mills systems, the general solution to the field equations on a global level is reviewed. Quantization of such models is performed in the Dirac approach, where, by use of the formulation as Poisson sigma models, all admissible physical quantum states are obtained.   Table of contents: 1. Introduction, 2. 2d geometry and gravitational actions, 3. Generalized dilaton theories and matter actions, 4. 2d gravity-Yang-Mills systems in terms of Poisson sigma models, 5. Classical solutions on a local level, 6. Classical solutions on a global level, 7. Towards quantum gravity.   (In part this work contains/summarizes previous joint work with T. Kloesch and P. Schaller). 
  We briefly discuss some possible cosmological implications of noncommutative geometry. While the noncommutativity we consider does not affect gravity, it can play an important role in the dynamics of other fields that are present in the early universe. We point out the possibility that noncommutativity may cause inflation induced fluctuations to become non-gaussian and anisotropic, and may modify the short distance dispersion relations. 
  A stable non-commutative solution with symmetry breaking is presented for a system of Dp-branes in the presence of a RR (p+5)-form. 
  A submodel of the so-called conformal affine Toda model coupled to the matter field (CATM) is defined such that its real Lagrangian has a positive-definite kinetic term for the Toda field and a usual kinetic term for the (Dirac) spinor field. After spontaneously broken the conformal symmetry by means of BRST analysis, we end up with an effective theory, the off-critical affine Toda model coupled to the matter (ATM). It is shown that the ATM model inherits the remarkable properties of the general CATM model such as the soliton solutions, the particle/soliton correspondence and the equivacence between the Noether and topological currents. The classical solitonic spectrum of the ATM model is also discussed. 
  We study the gauge theories on noncommutative space. We employ the idea of the covariant position to understand the linear and angular momenta, the center of mass position, and to express all gauge invariant observables including the Wilson line. In addition, we utilize the universality of the U(1) gauge theory, which originates from the underlying matrix theory, to analyze various solitons on U(N) theories, like the unstable static vortex solutions in two dimensions and BPS dyonic fluxon solutions. 
  We discuss the gravitational Higgs mechanism in domain wall background solutions that arise in the theory of 5-dimensional Einstein-Hilbert gravity coupled to a scalar field with a non-trivial potential. The scalar fluctuations in such backgrounds can be completely gauged away, and so can be the graviphoton fluctuations. On the other hand, we show that the graviscalar fluctuations do not have normalizable modes. As to the 4-dimensional graviton fluctuations, in the case where the volume of the transverse dimension is finite the massive modes are plane-wave normalizable, while the zero mode is quadratically normalizable. We then discuss the coupling of domain wall gravity to localized 4-dimensional matter. In particular, we point out that this coupling is consistent only if the matter is conformal. This is different from the Randall-Sundrum case as there is a discontinuity in the delta-function-like limit of such a smooth domain wall - the latter breaks diffeomorphisms only spontaneously, while the Randall-Sundrum brane breaks diffeomorphisms explicitly. Finally, at the quantum level both the domain wall as well as the Randall-Sundrum setups suffer from inconsistencies in the coupling between gravity and localized matter, as well as the fact that gravity is generically expected to be delocalized in such backgrounds due to higher curvature terms. 
  We analyze previously proposed order parameters for the confinement - deconfinement transition in lattice SU(2) Yang-Mills theory, defined as vacuum expectation value (v.e.v.) of monopole fields in abelian projection gauges. We show that they exhibit some inconsistency in the treatment of small scales, due to a violation of Dirac quantization condition for fluxes. We propose a new order parameter avoiding this inconsistency. It can be interpreted as v.e.v. of the field of a regular monopole in any abelian projection gauge, but it is independent of the choice of the abelian projection. Furthermore, being constructed in terms of surfaces of center vortices, it has also a natural interpretation in the approach of center dominance. 
  The D-instanton partition function is a fascinating quantity because in the presence of N D3-branes, and in a certain decoupling limit, it reduces to the functional integral of N=4 U(N) supersymmetric gauge theory for multi-instanton solutions. We study this quantity as a function of non-commutativity in the D3-brane theory, VEVs corresponding to separating the D3-branes and alpha'. Explicit calculations are presented in the one-instanton sector with arbitrary N, and in the large-N limit for all instanton charge. We find that for general instanton charge, the matrix theory admits a nilpotent fermionic symmetry and that the action is Q-exact. Consequently the partition function localizes on the minima of the matrix theory action. This allows us to prove some general properties of these integrals. In the non-commutative theory, the contributions come from the ``Higgs Branch'' and are equal to the Gauss-Bonnet-Chern integral of the resolved instanton moduli space. Separating the D3-branes leads to additional localizations on products of abelian instanton moduli spaces. In the commutative theory, there are additional contributions from the ``Coulomb Branch'' associated to the small instanton singularities of the instanton moduli space. We also argue that both non-commutativity and alpha'-corrections play a similar role in suppressing the contributions from these singularities. Finally we elucidate the relation between the partition function and the Euler characteristic of the instanton moduli space. 
  PhD thesis TU-Vienna, May 1994. Table of Contents: 1. Introduction, 2. Poisson Structure Induced Two Dimensional Field Theories, 3. Models of Gravity in 1+1 Dimensions 
  We investigate the relation between instantons and monopoles in the Laplacian Abelian Gauge using analytical methods in the continuum. Our starting point is the fact that the 't Hooft instanton with its high symmetry leads to a pointlike defect with Hopf invariant one. In order to generalise this result we partly break the symmetry by a local perturbation. We find that for generic configurations near the 't Hooft instanton the defects become loops. The analytical results show explicitly that these defects are magnetic monopoles with unit charge. In addition, the monopoles are twisted to account for the instanton number of the background. 
  Haisch and Rueda have recently proposed a model in which the inertia of charged particles is a consequence of their interaction with the electromagnetic zero-point field. This model is based on the observation that in an accelerated frame the momentum distribution of vacuum fluctuations is not isotropic. We analyze this issue through standard techniques of relativistic field theory, first by regarding the field A_mu as a classical random field, and then by making reference to the mass renormalization procedure in Quantum Electrodynamics and scalar-QED. 
  In this note we analyse the dynamical potential of a system of four $Dp$-branes at arbitrary angles. The equilibrium configurations for various values of the relative angles and distances among branes are discussed. The known configurations of parallel branes and brane-antibranes are obtained at extrema of the dynamical potential. 
  The effective potentials for massless scalar and vector quantum field theories on D dimensional manifolds with p compact noncommutative extra dimensions are evaluated by means of dimensional regularization implemented by zeta function techniques. It is found that the zeta function associated with the one-loop operator may not be regular at the origin. Thus, the related heat-kernel trace has a logarithmic term in the short t asymptotics expansion. Consequences of this fact are briefly discussed. 
  Expanded version of the author's contribution to the Concise Encyclopaedia of Supersymmetry, eds. J. Bagger, S. Duplij and W. Siegel 
  In this note we reconsider linearised metric perturbations in the one-brane Randall-Sundrum Model. We present a simple formalism to describe metric perturbations caused by matter perturbations on the brane and remedy some misconceptions concerning the constraints imposed on the metric and matter perturbations by the presence of the brane. 
  I argue that isolated vacua of M-theory, cannot in any conventional way be said to live in the same theory as other disconnected parts of the moduli space. The usual field theoretic mechanisms, which allow an observer in one disconnected component of a moduli space to verify the existence of other components, fail. The failure is a consequence of robust properties of black holes. When barriers between components are much smaller than the Planck scale, the usual field theoretic picture is approximately valid. 
  We show how a recently proposed large $N$ duality in the context of type IIA strings with ${\cal N}=1$ supersymmetry in 4 dimensions can be derived from purely geometric considerations by embedding type IIA strings in M-theory. The phase structure of M-theory on $G_2$ holonomy manifolds and an $S^3$ flop are the key ingredients in this derivation. 
  I use the universal instanton formalism to discuss quantum effects in the open-closed topological string theory of a Calabi-Yau A-model, in the presence of a multiply-wrapped `Floer' D-brane. This gives a precise meaning (up to the issue of compactifying the relevant moduli spaces) to the instanton corrections which affect sigma model and topological string amplitudes. The cohomological formalism I use recovers the homological approach used by Fukaya and collaborators in the singly-wrapped case, even though it is not a naive generalization of the latter. I also prove some non-renormalization theorems for amplitudes with low number of insertions. The non-renormalization argument is purely geometric and based on the universal instanton formulation, and thus it does not assume that the background satisfies the string equations of motion. These results are valid even though the D-brane background typically receives worldsheet instanton corrections. I also point out that the localized form of the boundary BRST operator receives instanton corrections and make a few comments on the consequences of this effect. 
  To obtain the one-loop corrections to the mass of a kink by mode regularization, one may take one-half the result for the mass of a widely separated kink-antikink (or sphaleron) system, where the two bosonic zero modes count as two degrees of freedom, but the two fermionic zero modes as only one degree of freedom in the sums over modes. For a single kink, there is one bosonic zero mode degree of freedom, but it is necessary to average over four sets of fermionic boundary conditions in order (i) to preserve the fermionic Z$_2$ gauge invariance $\psi \to -\psi$, (ii) to satisfy the basic principle of mode regularization that the boundary conditions in the trivial and the kink sector should be the same, (iii) in order that the energy stored at the boundaries cancels and (iv) to avoid obtaining a finite, uniformly distributed energy which would violate cluster decomposition. The average number of fermionic zero-energy degrees of freedom in the presence of the kink is then indeed 1/2. For boundary conditions leading to only one fermionic zero-energy solution, the Z$_2$ gauge invariance identifies two seemingly distinct `vacua' as the same physical ground state, and the single fermionic zero-energy solution does not correspond to a degree of freedom. Other boundary conditions lead to two spatially separated $\omega \sim 0 $ solutions, corresponding to one (spatially delocalized) degree of freedom. This nonlocality is consistent with the principle of cluster decomposition for correlators of observables. 
  A calculation of the one loop gravitational self-energy graph in non-anticommutative quantum gravity reveals that graviton loops are damped by internal momentum dependent factors in the modified propagator and the vertex functions. The non-anticommutative quantum gravity perturbation theory is finite for matter-free gravity and for matter interactions. 
  Following my plenary lecture on ICMP2000 I review my results concerning two closely related topics: topological quantum field theories and the problem of quantization of gauge theories. I start with old results (first examples of topological quantum field theories were constructed in my papers in late seventies) and I come to some new results, that were not published yet. 
  I review my results about noncommutative gauge theories and about the relation of these theories to M(atrix) theory following my lecture on ICMP 2000. 
  We critically review several recent approaches to solving the two cosmological constant problems. The "old" problem is the discrepancy between the observed value of $\Lambda$ and the large values suggested by particle physics models. The second problem is the "time coincidence" between the epoch of galaxy formation $t_G$ and the epoch of $\Lambda$-domination $t_\L$. It is conceivable that the "old" problem can be resolved by fundamental physics alone, but we argue that in order to explain the "time coincidence" we must account for anthropic selection effects. Our main focus here is on the discrete-$\Lambda$ models in which $\Lambda$ can change through nucleation of branes. We consider the cosmology of this type of models in the context of inflation and discuss the observational constraints on the model parameters. The issue of multiple brane nucleation raised by Feng {\it et. al.} is discussed in some detail. We also review continuous-$\L$ models in which the role of the cosmological constant is played by a slowly varying potential of a scalar field. We find that both continuous and discrete models can in principle solve both cosmological constant problems, although the required values of the parameters do not appear very natural. M-theory-motivated brane models, in which the brane tension is determined by the brane coupling to the four-form field, do not seem to be viable, except perhaps in a very tight corner of the parameter space. Finally, we point out that the time coincidence can also be explained in models where $\Lambda$ is fixed, but the primordial density contrast $Q=\delta\rho/\rho$ is treated as a random variable. 
  A general action is proposed for the fields of $q$-dimensional differential form over the compact Riemannian manifold of arbitrary dimensions. Mathematical tools are based on the well-known de Rham-Kodaira decomposing theorem on harmonic integral. A field-theoretic action for strings, $p$-branes and high-spin fields is naturally derived. We also have, naturally, the generalized Maxwell equations with an electromagnetic and monopole current on a curved space-time. A new type of gauge transformations ({\it dual} gauge transformations) plays an essential role for coboundary $q$-forms. 
  We review an iterative construction of the supersymmetric non-abelian Born-Infeld action. We obtain the action through second order in the fieldstrength. Kappa-invariance fixes the ordenings which turn out to deviate from the symmetrized trace proposal. 
  The hypercharge-isospin-color symmetry of the standard model interaction is drastically reduced to a remaining Abelian electromagnetic $\U(1)$-symmetry for the particles. It is shown that such a symmetry reduction comes as a consequence of the central correlation in the internal group as represented by the standard fields where the hypercharge properties are given by the central isospin-color properties. A maximal diagonalizable symmetry subgroup (Cartan torus) of the interaction group for the particles as eigenvectors has to discard either color (confinement) or isospin. An additional diagonalization for the external spin properties which come centrally correlated with the isospin properties enforces the weak isospin breakdown. 
  We discuss the large N limit of Calogero-Moser models for the classical infinite families of simple Lie algebras A_N, B_N, C_N and D_N. We show that the limit defines two different Conformal Field Theories with central charge c>1. The value of c and the dimension of the primary field are dictated by the underlying algebraic symmetries of the model. 
  We use some BRS techniques to construct Chern-Simons forms generalizing the Chern character of K_1 groups in the Cuntz-Quillen description of cyclic homology. 
  Field theories on "quantum" or deformed space-time are considered here. The Moyal-Weyl deformation breaks the Lorentz invariance of the theory, but one can still require invariance under the supertranslation algebra. We investigate some aspects of the Wess-Zumino model, super Yang-Mills theories and analyze the correspondence of the later with the supersymmetric Born-Infeld action. 
  We analyze the backreaction of dilaton tadpoles on the geometry of non-supersymmetric strings. After finding explicit warped solutions for a T-dual version of the Sugimoto model, we examine the possibility of realizing large extra dimension scenarios within the context of non-supersymmetric string models. Our analysis reveals an appealing mechanism to dynamically reduce the number of flat, non-compact directions in non-supersymmetric string theories. 
  We give a definition of mass for spacelike hypersurfaces in space-times with metrics which are asymptotic to the anti-de Sitter one, or to a class of generalizations thereof. We present the results of gr-qc/0110014 which show that our definition provides a geometric invariant for a spacelike hypersurface embedded in a space-time. Some further global invariants are also given. 
  We construct the scalar profile for the non-abelian self dual string connecting two M5-branes compactified on a light-like circle. The construction is based on a conjectured modified version of Nahm's equations describing a D2-brane, with a magnetic field on it, suspended between two D4-branes. Turning on a constant magnetic field on the D2-brane corresponds to a boost in the eleventh direction. In the limit of infinite boost the D4-branes correspond to light-like compactified M5-branes. The solution for the scalar profile of the brane remains finite in this limit and displays all the correct expected features such as smooth interpolation between the unbroken and broken phase with the correct value for the Higgs field at infinity. 
  We explain what is the challenge of light-front quantisation, and how we can now answer it because of recent progress in solving the problem of zero modes in the case of non-Abelian gauge theories. We also give a description of the light-front Hamiltonian for SU(2) finite volume gluodynamics resulting from this recent solution to the problem of light-front zero modes. 
  We review our work on computations of the quantum corrections to the mass and the central charge of the susy kink. For the mass corrections, we find that the widely used momentum cut-off scheme gives an incorrect result, but we deduce through smoothing of the cut-off an extra term in the mass formula, which produces the correct result. We discover the importance of boundary effects for the mode number cut-off regularization scheme. We introduce the notion of delocalized boundary energy. We discuss two discrete $Z_2$ symmetries and their importance to the mode number approach. For the central charge corrections, we use momentum cut-off regularization with two cut-offs, one for propagators and another for Dirac delta functions. We then compute the quantum anomaly in the central charge, and find that it restores the BPS bound at the one-loop level if the two cut-offs are equal. 
  John Bell's emphasis of the essential ambiguities in anomaly calculations is recalled. Some descendants of the anomaly are reviewed. 
  Chern-Simons terms are well-known descendants of chiral anomalies, when the latter are presented as total derivatives. Here I explain that also Chern-Simons terms, when defined on a 3-manifold, may be expressed as total derivatives. 
  We have recently introduced a discrete model of Lorentzian quantum gravity, given as a regularized non-perturbative state sum over simplicial Lorentzian space-times, each possessing a unique Wick rotation to Euclidean signature. We investigate here the phase structure of the Wick-rotated path integral in three dimensions with the aid of computer simulations. After fine-tuning the cosmological constant to its critical value, we find a whole range of the gravitational coupling constant $k_0$ for which the functional integral is dominated by non-degenerate three-dimensional space-times. We therefore have a situation in which a well-defined ground state of extended geometry is generated dynamically from a non-perturbative state sum of fluctuating geometries. Remarkably, its macroscopic scaling properties resemble those of a semi-classical spherical universe. Measurements so far indicate that $k_0$ defines an overall scale in this extended phase, without affecting the physics of the continuum limit. These findings provide further evidence that discrete {\it Lorentzian} gravity is a promising candidate for a non-trivial theory of quantum gravity. 
  The universal method of expansion of integrals is suggested. It allows in particular to derive the threshold expansion of Feynman integrals. 
  Adding explicit mass terms for the spin 2 and spin 3/2 field of N=1 anti-de Sitter supergravity, the limit M --> 0 for these mass terms is smooth: there is no van Dam-Veltman-Zakharov mass discontinuity in the propagators when the cosmological constant is non-vanishing. 
  We construct supergravity solutions for Dp-branes at orbifold points. The solutions are written in terms of a single function, which is the solution to a nonlinear differential equation. The near horizon limits of these solutions are dual, in the AdS/CFT sense, to super-Yang-Mills theories with 8 supercharges in various dimensions. In particular, we present a dual to N=2 SU(N) SYM theory in 3+1 dimensions, and analyse some aspects of the duality. 
  The calculation of bound state properties using renormalization group techniques to compute the corresponding Regge trajectories is presented. In particular, we investigate the bound states in different charge sectors of a scalar theory with interaction (phi^dagger phi chi). The resulting bound state spectrum is surprisingly rich. Where possible we compare and contrast with known results of the Bethe-Salpeter equation in the ladder approximation and, in the non-relativistic limit, with the corresponding Schr"odinger equation. 
  We compute the tree-level four-point scattering amplitudes in string models where matter fields live on D-brane intersections. Extracting the contribution of massless modes, we are left with dimension-six four-fermion operators which in general receive contributions from three different sources: exchange of massive Kaluza--Klein excitations, winding modes and string oscillator states. We compute their coefficients and extract new bounds on the string scale in the brane-world scenario. This is contrasted with the situation where matter fields arise from open strings with both ends confined on the same collection of D-branes, in which case the exchange of massive string modes leads to dimension-eight operators that have been studied in the past. When matter fields live on brane intersections, the presence of dimension-six operators increases the lower bound on the string scale to 2--3 TeV, independently of the number of large extra dimensions. 
  We examine noncommutative Yang-Mills and open string theories using magnetically and electrically deformed supergravity duals. The duals are near horizon regions of Dp-brane bound state solutions which are obtained by using O(p+1,p+1) transformations of Dp-branes. The action of the T-duality group implies that the noncommutativity parameter is constant along holographic RG-flows. The moduli of the noncommutative theory, i.e., the open string metric and coupling constant, as well as the zero-force condition are shown to be invariant under the O(p+1,p+1) transformation, i.e., deformation independent. We find sufficient conditions, including zero force and constant dilaton in the ISO(3,1)-invariant D3 brane solution, for exact S-duality between noncommutative Yang-Mills and open string theories. These results are used to construct noncommutative field and string theories with N=1 supersymmetry from the T^(1,1) and Pilch-Warner solutions. The latter has a non-trivial zero-force condition due to the warping. 
  We construct Spin(p+1,p+1) covariant Dp-brane bound states by using that the potentials in the RR sector of toroidically compactified type II supergravity transform as a chiral spinor of the T-duality group. As an application, we show the invariance of the zero-force condition for a probe D-brane under noncommutative deformations of the background, which gives a holographic proof of the stability of the corresponding field theory ground state under noncommutative deformations. We also identify the Spin(p+1,p+1) transformation laws by examining the covariance of the D-brane Lagrangians. 
  The issue of black hole entropy is reexamined within a finite lattice framework along the lines of Wheeler, 't Hooft and Susskind, with an additional criterion to identify physical horizon states contributing to the entropy. As a consequence, the degeneracy of physical states is lower than that attributed normally to black holes. This results in corrections to the Bekenstein-Hawking area law that are logarithmic in the horizon area. Implications for the holographic entropy bound on bounded spaces are discussed. Theoretical underpinnings of the criterion imposed on the states, based on the `quantum geometry' formulation of quantum gravity, are briefly explained. 
  We discuss various aspects of the description of branes as topological solitons in unstable brane systems of higher dimensions. We first describe a classification of all the possible realisations of branes of M and type II theories as topological solitons of a brane-antibrane system. We then present a description of type IIB NS-NS p-branes in terms of topological solitons in systems of spacetime-filling NS9, anti-NS9 pairs and discuss the implications of these constructions in the description of BPS and non-BPS states in the strongly coupled Heterotic SO(32) theory. We finally present briefly the construction of a conjectured spacetime-filling non-BPS M10-brane, starting point for a brane descent construction of the branes of M-theory. 
  We briefly review the microscopic modeling of black holes as bound states of branes in the context of the soluble D1-D5 system. We present a discussion of the low energy brane dynamics and account for black hole thermodynamics and Hawking radiation rates. These considerations are valid in the regime of supergravity due to the non-renormalization of the low energy dynamics in this model. Using Maldacena duality and standard statistical mechanics methods one can account for black hole thermodynamics and calculate the absorption cross section and the Hawking radiation rates. Hence, at least in the case of this model black hole, since we can account for black hole properties within a unitary theory, there is no information paradox. 
  We study static vortex type solutions of pure gravity for $D \geq 4+1 $. Non-singular vortex solutions can be obtained by considering periodic Kaluza-Klein monopoles. We also show that away from the center of the vortices the space is described by the gravitational instantons derived from minimal surfaces. 
  We construct dual supergravity descriptions of D3-branes wrapping associative 3-cycles $L$. We analyse the conditions for having five-dimensional background solutions of the form $AdS_2 \times L$ and show that they require $L$ to be of constant negative curvature type. This provides $AdS_2$ background solutions when $L$ is the hyperbolic space $H^3$ or its quotients by subgroups of its isometry group. We construct a regular numerical solution interpolating between $AdS_5$ in the UV and $AdS_2 \times H^3$ in the IR. The IR fixed point exists at the ``intersection'' of the Coulomb and Higgs branches. We analyse the singular supergravity solutions which correspond to moving into the Higgs and the Coulomb branches. For negative constant curvature spaces the singularity is of a ``good'' type in the Higgs branch and of a ``bad'' type in the Coulomb branch. For positive constant curvature spaces such as $S^3$ the singularity is of a ``bad'' type in both the Higgs and the Coulomb branches. We discuss the meaning of these results. 
  Type IIB superstring models with the standard model gauge group on D3-branes and with massless matter associated with open strings joining D3-branes to D3-branes or D3-branes to ${\rm D}7_3$-branes are studied. Models with gauge coupling constant unification at an intermediate scale between about $10^{10}$ and $10^{12}$GeV and consistency with the observed value of $\sin^2 \theta_W (M_Z)$ are obtained. Extra vector-like states and extra pairs of Higgs doublets play a crucial role. 
  We study the cosmological evolution of the closed universe on a spherical probe brane moving in the AdS$_m\times S^n$ background and the near-horizon background of the dilatonic D-branes. The Friedmann equations describing the evolution of the brane universe, and the effective energy density and pressure simulated on the probe brane due to its motion in the curved background spacetime are obtained and analyzed. We also comment on the relevance of the spherical probe brane to the giant graviton for the special value of the probe energy. 
  The concept of the random discretization of the space-time is suggested. It is the way to consistent compatible synthesis of quantum and relativistic principles and principle of geometrization. The basic idea of this concept is physical reality of the finite sizes fundamental element of the quantized space-time. The flat space-time with random discretization is described as the probability measure space with the set of all possible discretizations of the flat continual space-time as the set of points. The probability measure can depend on the geometric parameters of discretizations (a number of regions of a discretization, their volumes, areas etc.). In this concept the fundamental length can be defined as the average value of the linear size of a fundamental element. In this concept the "particle" quantum and the space-time quantum are identical. 
  We consider a system of D5/D1 branes in the supergravity background AdS_3xS^3xX, where X is T^4 or K3. By investigating the structure of the missing states in the conformal description, we are able to extend the AdS/CFT correspondence to W algebras. As a test of this new formulation the results are compared to Hilbert schemes and more general supergravity backgrounds as deformations by D3-branes or six-dimensional Calabi-Yau manifolds. 
  In this talk, we present a parity-preserving QED3 model with spontaneous breaking of a local U(1)-symmetry. The breaking is accomplished by a potential of the phi^6-type. It is shown that a net attractive interaction appears in the Moeller scattering (s- and p-wave scatterings between two electrons) as mediated by the gauge field and a Higgs scalar. We show, by solving numerically the Schroedinger equation for both the scattering potentials (s- and p-wave), that in the weak-coupling regime only s-wave bound states appear, whereas in the strong-coupling regime s- and p-wave bound states show up. Also, we discuss possible applications of the model to the phenomenology of high-Tc superconductors and to the re-entrant superconductivity effect. 
  We show that, in perturbative string models where the source of CP violation is a complex vacuum expectation value (v.e.v.) for one or more compactification moduli, CP is conserved if a CP transformation acting on the modulus values is an element of a target-space (self-)duality group. Where the duality group is SL(2,Z) the result confirms a conjecture of Bailin et al. that CP is conserved for v.e.v.'s of the T modulus on the boundary of the fundamental domain, and generalises Giedt's result on the removability of complex Yukawa couplings in such models. Our result applies to any model of spontaneous CP violation where the CP-odd scalar transforms under a symmetry that is not explicitly broken. We consider whether similar results could be obtained in ``brane worlds''. 
  We present an extension of quantum field theory to the case when the spacetime topology fluctuates (spacetime foam). In this extension the number of bosonic fields becomes a variable and the ground state is characterized by a finite particle number density. It is shown that when the number of fields remains a constant, the standard field theory is restored. However, in the complete theory the ground state has a nontrivial properties. In particular, it produces an increase in the level of quantum fluctuations in the field potentials and an additional renormalization of masses of particles. We examine fluctuations of massless fields and show that in the presence of a temperature (thermal state) these fluctuations has 1/f spectrum. Thus, the main prediction of the theory is that our universe should be filled with a random electromagnetic field which should produce an additional 1/f - noise in electric circuits. 
  Harmonic oscillators with a centrifugal spike are analysed, via a non-Hermitian regularization, within a complexified SUSY quantum mechanics. The formalism enables us to construct the factorized creation and annihilation operators. We show how the real though, generically, non-equidistant spectrum complies with the current SUSY-type isospectrality and degeneracy in an unusual way. 
  It is a general belief that the only possible way to consistently deform the Pauli-Fierz action, changing also the gauge algebra, is general relativity. Here we show that a different type of deformation exists in three dimensions if one allows for PT non-invariant terms. The new gauge algebra is different from that of diffeomorphisms. Furthermore, this deformation can be generalized to the case of a collection of massless spin-two fields. In this case it describes a consistent interaction among them. 
  We present a new kind of defect in Abelian Projections, stemming from pointlike zeros of second order. The corresponding topological quantity is the Hopf invariant pi_3(S^2) (rather than the winding number pi_2(S^2) for magnetic monopoles). We give a visualisation of this quantity and discuss the simplest non-trivial example, the Hopf map. Such defects occur in the Laplacian Abelian gauge in a non-trivial instanton sector. For general Abelian projections we show how an ensemble of Hopf defects accounts for the instanton number. 
  The path integral for 3+1 abelian gauge theory is rewritten in terms of a real antisymmetric field allowing a dual action that couples the electric and magnetic currents to the photon and each other in a gauge invariant manner. Standard perturbative abelian quantum electrodynamics reemerges when the monopole current vanishes. For certain simple relationships between the monopole current and the electric current, the altered photon propagator can exhibit abelian charge confinement or develop mass, modeling effects believed to be present in non-abelian theories. 
  The flow of U(1) charge through dense fishnet diagrams, in a non-hermitian matrix scalar field theory g_1Tr(\Sigma^\dagger\Sigma)^2 + 2g_1vTr\Sigma^{\dagger 2}\Sigma^2, is described by a 6-vertex model on a ``diamond'' lattice [1]. We give a direct calculation of the continuum properties of the 6-vertex model on this novel lattice, explicitly confirming the conclusions of [1], that, for 1/2 < v< \infty, they are identical to those of a world-sheet scalar field compactified on a circle S_1. The radius of the circle is related to the ratio v of quartic couplings by R^{-2} = 2T_0 arccos(1-1/2v^2). This direct computational approach may be of value in generalizing the conclusions to the non-Abelian O(n) case. 
  This is a summary of a talk at Strings2000 explaining three ways in which string theory and M-theory are related to the mathematics of K-theory. 
  In the framework of the Sine-Gordon (SG) theory we will present the construction of a dynamical Virasoro symmetry which has nothing to do with the space-time Virasoro symmetry of 2D CFT. Although, it is non-local in the SG field theory, nevertheless it gives rise to a local action on specific N-soliton solution variables. These {\it analytic} variables possess a beautiful geometrical meaning and enter the Form Factor expressions. At the end, we will also give some preliminary hints about the quantisation. 
  Noncommutative ${\cal N}=1$ and ${\cal N}=2$ supersymmetric Yang-Mills theories with gauge group U(N) are studied here using the background field method and superspace background covariant D-algebra in perturbation theory. At one loop divergences arise only in the two-point functions. They are logarithmic UV/IR divergences in the planar/nonplanar sectors that play a role dual to each other. We compute the planar and non planar contributions to the three- and four-point functions with vector external lines. We find that the three-point function vanishes, while the four-point function receives contributions both from vector and from chiral matter loops. 
  We study T-duality in the Green-Schwarz formalism to all orders in superspace coordinates. We find two analogs of Buscher rules for the supervielbein and clarify their meaning from the superstring point of view. The transformation rules for the dilaton, spin 1/2 fermions and Ramond-Ramond superfields are also derived. 
  We discuss the geometry and topology of the complete, non-compact, Ricci-flat Stenzel metric, on the tangent bundle of S^{n+1}. We obtain explicit results for all the metrics, and show how they can be obtained from first-order equations derivable from a superpotential. We then provide an explicit construction for the harmonic self-dual (p,q)-forms in the middle dimension p+q=(n+1) for the Stenzel metrics in 2(n+1) dimensions. Only the (p,p)-forms are L^2-normalisable, while for (p,q)-forms the degree of divergence grows with |p-q|. We also construct a set of Ricci-flat metrics whose level surfaces are U(1) bundles over a product of N Einstein-Kahler manifolds, and we construct examples of harmonic forms there. As an application, we construct new examples of deformed supersymmetric non-singular M2-branes with such 8-dimensional transverse Ricci-flat spaces. We show explicitly that the fractional D3-branes on the 6-dimensional Stenzel metric found by Klebanov and Strassler is supported by a pure (2,1)-form, and thus it is supersymmetric, while the example of Pando Zayas-Tseytlin is supported by a mixture of (1,2) and (2,1) forms. We comment on the implications for the corresponding dual field theories of our resolved brane solutions. 
  Renormalized Hamiltonians for gluons are constructed using a perturbative boost-invariant renormalization group procedure for effective particles in light-front QCD, including terms up to third order. The effective gluons and their Hamiltonians depend on the renormalization group parameter lambda, which defines the width of momentum space form factors that appear in the renormalized Hamiltonian vertices. Third-order corrections to the three-gluon vertex exhibit asymptotic freedom, but the rate of change of the vertex with lambda depends in a finite way on regularization of small-x singularities. This dependence is shown in some examples, and a class of regularizations with two distinct scales in x is found to lead to the Hamiltonian running coupling constant whose dependence on lambda matches the known perturbative result from Lagrangian calculus for the dependence of gluon three-point Green's function on the running momentum scale at large scales. In the Fock space basis of effective gluons with small lambda, the vertex form factors suppress interactions with large kinetic energy changes and thus remove direct couplings of low energy constituents to high energy components in the effective bound state dynamics. This structure is reminiscent of parton and constituent models of hadrons. 
  We study the localization of various bulk fields on an $AdS_4$ brane in $AdS_5$. In this case, for a small brane cosmological constant, all the bulk fields ranging from scalar to graviton are naturally confined to the brane only through the gravitational interaction. In particular, for the vector field we can find an interesting zero mode solution which satisfies the box boundary condition. In the cases of spin 1/2 spinor and 3/2 gravitino fields, the form of zero modes is very similar to as in a flat Minkowski brane, but they are trapped on the $AdS_4$ brane even without introducing a mass term with a 'kink' profile. 
  We extend to its spin~3/2 supersymmetric partner the very recent demonstration that the massless limit of massive spin~2 exchange amplitudes can be made continuous in background AdS spaces, in contrast to the known flat space discontinuities for both systems. In an AdS background, unlike spin~2 where the limit m\to0 is the massless one, spin~3/2 ``masslessness'' requires m\to\sqrt{-\Lambda/3}, the supergravity value tuning the mass and cosmological constant that uniquely provides gauge invariance and two helicities. We find that continuity of the spin~3/2--mediated exchange amplitude can be regained in two ``massless'' limits m\to0 and m\to\sqrt{-\Lambda/3}; only the latter corresponds to cosmological supergravity. 
  We prove that both global monopole and minimally coupled static zero mass scalar field are electrogravity dual of the Schwarzschild solution or flat space and they share the same equation of state, $T^0_0 - T^i_i = 0$. This property was however known for the global monopole spacetime while it is for the first time being established for the scalar field. In particular, it turns out that the Xanthopoulos - Zannias scalar field solution is dual to flat space. 
  The static Yang-Mills-Higgs dyonic instanton is shown to have a non-vanishing, but anti-self-dual, angular momentum 2-form with skew eigenvalues equal to the electric charge; for large charge the angular momentum causes the instanton to expand into a hyper-spherical shell. A class of exact multi dyonic instantons is then found and then generalized to a new class of 1/4 supersymmetric, non-singular, stationary, exact solutions of the ten-dimensional supergravity/Yang-Mills theory. These self-gravitating dyonic instantons yield new heterotic string solitons, to leading order in the inverse string tension. 
  In this paper we investigate the effects of gravitational backreaction for the late time Hawking radiation of evaporating near-extremal black holes. This problem can be studied within the framework of an effective one-loop solvable model on AdS_2. We find that the Hawking flux goes down exponentially and it is proportional to a parameter which depends on details of the collapsing matter. This result seems to suggest that the information of the initial state is not lost and that the boundary of AdS_2 acts, at least at late times, as a sort of stretched horizon in the Reissner-Nordstrom spacetime. 
  We construct an explicit solution of the Seiberg-Witten map for a linear gauge field on the non-commutative plane. We observe that this solution as well as the solution for a constant curvature diverge when the non-commutativity parameter theta reaches certain event horizon in the theta-space. This implies that an ordinary Yang-Mills theory can be continuously deformed by the Seiberg-Witten map into a non-commutative theory only within one connected component of the theta-space. 
  The AdS_2\timesS^2 geometry of near-extremal Reissner-Nordstrom black holes can be described by an effective solvable model which allows to follow analytically the evaporation process including the backreaction. We find that an infinite amount of time is required for the black hole to decay to extremality. 
  The thermodynamics of nearly-extreme charged black holes depends upon the number of ground states at fixed large charge and upon the distribution of excited energy states. Here three possibilities are examined: (1) Ground state highly degenerate (as suggested by the large semiclassical Hawking entropy of an extreme Reissner-Nordstrom black hole), excited states not. (2) All energy levels highly degenerate, with macroscopic energy gaps between them. (3) All states nondegenerate (or with low degeneracy), separated by exponentially tiny energy gaps. I suggest that in our world with broken supersymmetry, this last possibility seems most plausible. An experiment is proposed to distinguish between these possibilities, but it would take a time that is here calculated to be more than about 10^837 years. 
  Higher dimensional conformal QFT possesses an interesting braided structure which, different from the d=1+1 models, is restricted to the timelike region and therefore easily escapes euclidean action methods. It lies behind the spectrum of anomalous dimensions which may be viewed as a kind of substitute for a missing particle interpretation in the presence of interactions. 
  It is shown that the reduced particle dynamics of 2+1 dimensional gravity in the maximally slicing gauge is of hamiltonian nature. We give the exact diffeomorphism which transforms the expression of the spinning cone geometry in the Deser, Jackiw, 't Hooft gauge to the maximally slicing gauge. It is explicitly shown that the boundary term in the action, written in hamiltonian form gives the Hamiltonian for the reduced particle dynamics. The quantum mechanical translation of the two particle Hamiltonian is given and the Green function computed. 
  The nonlinear supersymmetry of one-dimensional systems is investigated in the context of the quantum anomaly problem. Any classical supersymmetric system characterized by the nonlinear in the Hamiltonian superalgebra is symplectomorphic to a supersymmetric canonical system with the holomorphic form of the supercharges. Depending on the behaviour of the superpotential, the canonical supersymmetric systems are separated into the three classes. In one of them the parameter specifying the supersymmetry order is subject to some sort of classical quantization, whereas the supersymmetry of another extreme class has a rather fictive nature since its fermion degrees of freedom are decoupled completely by a canonical transformation. The nonlinear supersymmetry with polynomial in momentum supercharges is analysed, and the most general one-parametric Calogero-like solution with the second order supercharges is found. Quantization of the systems of the canonical form reveals the two anomaly-free classes, one of which gives rise naturally to the quasi-exactly solvable systems. The quantum anomaly problem for the Calogero-like models is ``cured'' by the specific superpotential-dependent term of order $\hbar^2$. The nonlinear supersymmetry admits the generalization to the case of two-dimensional systems. 
  We show that U(N) Yang-Mills theory on noncommutative Minkowski space-time can be renormalized, in a BRS invariant way, at the one-loop level, by multiplicative dimensional renormalization of its coupling constant, its gauge parameter and its fields. It is shown that the Slavnov-Taylor equation, the gauge-fixing equation and the ghost equation hold, up to order $\hbar$, for the MS renormalized noncommutative U(N) Yang-Mills theory. We give the value of the pole part of every 1PI diagram which is UV divergent. 
  We consider certain BPS supergravity solutions of string theory which have singularities and we show that the singularity goes away when we add angular momentum. These smooth solutions enable us to obtain {\it global} $AdS_3$ as the near horizon geometry of a BPS brane system in an asymptotically flat space. 
  We discuss manifestly SL(2,C)x SU(4) and kappa invariant superstring action in AdS(5) x S(5) background in the framework of Green-Schwarz formulation. The action is formulated in terms of 16 Poincare fermionic coordinates which through AdS/CFT correspondence should represent N=4 SYM superspace and 16 superconformal fermionic coordinates. The action is also manifestly invariant with respect to the usual N=4 Poincare superalgebra transformations. kappa-symmetry gauge fixing and the derivation of light-cone gauge action is simplified. 
  We show that in the functional integral formalism of U(1) gauge field theory some formal manipulation such as interchange of order of integration can yield erroneous results. The example studied is analysed by Fubini theorem. 
  Quantum gravity arguments and the entropy bound for effective field theories proposed in PRL 82, 4971 (1999) lead to consider two correlated scales which parametrize departures from relativistic quantum field theory at low and high energies. A simple estimate of their possible phenomenological implications leads to identify a scale of around 100 TeV as an upper limit on the domain of validity of a quantum field theory description of Nature. This fact agrees with recent theoretical developments in large extra dimensions. Phenomenological consequences in the beta-decay spectrum and cosmic ray physics associated to possible Lorentz invariance violations induced by the infrared scale are discussed. It is also suggested that this scale might produce new unexpected effects at the quantum level. 
  We discuss how the fine-tuning of the cosmological constant enters brane world setups. After presenting the Randall Sundrum model as a prototype case, we focus on single brane models with curvature singularities which are separated from the brane in the additional dimension. Finally, the issue of the existence of nearby curved solutions is addressed. 
  We investigate a near-horizon geometry of NS5-branes wrapping on a Riemann surface, which asymptotically approaches to linear dilaton backgrounds. We concretely find a fully localized solution of the near-horizon geometry of intersecting NS5-branes. We also discuss a relation to a description of Landau-Ginzburg theories. 
  We present a numerical study of the fermion-induced effective action in the presence of a static inhomogeneous magnetic field for both 3+1 and 2+1 dimensional QED using a novel approach. This approach is appropriate for cylindrically symmetric magnetic fields with finite magnetic flux $\Phi$. We consider families of magnetic fields, dependent on two parameters, a typical value $B_{m}$ for the field and a typical range d. We investigate the behavior of the effective action for three distinct cases: 1) keeping $\Phi$ (or $B_{m}d^{2}$) constant and varying d, 2) keeping $B_{m}$ constant and varying d and 3) keeping d constant and varying $\Phi$ (or $B_{m}d^{2}$). We note an interesting difference as d tends to infinity (case 2) between smooth and discontinuous magnetic fields. In the strong field limit (case 3) we also derive an explicit asymptotic formula for the 3+1 dimensional action. We study the stability of the magnetic field and we show that magnetic fields of the type we examine remain unstable, even in the presence of the fermions. In the appropriate regions we check our numerical results against the Schwinger formula (constant magnetic field), the derivative expansion and the numerical work of M. Bordag and K. Kirsten. The role of the Landau levels for the effective action, and the appearance of metastable states for large magnetic flux, are discussed in an appendix. 
  The so-called ``massive 11-dimensional supergravity'' theory gives, for one Killing vector, Romans' massive 10-dimensional supergravity in 10 dimensions, for two Killing vectors an Sl(2,Z) multiplet of massive 9-dimensional supergravity theories that can be obtained by standard generalized dimensional reduction type IIB supergravity and has been shown to contain a gauged supergravity. We consider a straightforward generalization of this theory to three Killing vectors and a 3\times 3 symmetric mass matrix and show that it gives an Sl(3,Z) multiplet of 8-dimensional supergravity theories that contain an SO(3) gauged supergravity which is, in some way, the dual to the one found by Salam and Sezgin by standard generalized dimensional reduction. 
  We study the localization of gravity on string-like defects in codimension two. We point out that the gravity-localizing `local cosmic string' spacetime has an orbifold singularity at the horizon. The supergravity embedding and the AdS/CFT correspondence suggest ways to resolve the singularity. We find two resolutions of the singularity that have a semiclassical gravity description and study their effect on the low-energy physics on the defect. The first resolution leads, at long distances, to a codimension one Randall-Sundrum scenario. In the second case, the infrared physics is like that of a conventional finite-size Kaluza-Klein compactification, with no power-law corrections to the gravitational potential. Similar resolutions apply also in higher codimension gravity-localizing backgrounds. 
  We investigate solutions of type II supergravity which have the product R^4 x M^6 structure with non-compact M^6 factor and which preserve at least four supersymmetries. In particular, we consider various conifolds and the N=1 supersymmetric NS5-brane wrapped on 2-sphere solution recently discussed in hep-th/0008001. In all of these cases, we explicitly construct the complex structures, and the Kaehler and parallel (3,0) forms of the corresponding M^6. In addition, we verify that the above solutions preserve, respectively, eight and four supersymmetries of type II theory. We also demonstrate that the ordinary and fractional D3-brane solutions on singular, resolved and deformed conifolds, and the (S-dual of) NS5-brane wrapped on 2-sphere can be obtained as special cases from a universal ansatz for the supergravity fields and a single 1-d action governing their radial evolution. We show that like the 3-branes on conifolds, the NS5-brane on 2-sphere background can be found as a solution of first order system following from a superpotential. 
  By looking at fractional Dp-branes of type IIA on T_4/Z_2 as wrapped branes and by using boundary state techniques we construct the effective low-energy action for the fields generated by fractional branes, build their world-volume action and find the corresponding classical geometry. The explicit form of the classical background is consistent only outside an enhancon sphere of radius r_e, which encloses a naked singularity of repulson-type. The perturbative running of the gauge coupling constant, dictated by the NS-NS twisted field that keeps its one-loop expression at any distance, also fails at r_e. 
  Our proposal here is to set up the conceptual framework for an eventual {Theory of Everything}.   We formulate the arena -language- to build up {\it any} QG. In particular, we show how the objects of fundamental theories, such as p-branes (strings, loops and others) could be posed in this language. 
  We conjecture that there exists a strong coupling limit of bosonic string theory which is related to the 26 dimensional theory in the same way that 11 dimensional M theory is related to superstring theory. More precisely, we believe that bosonic string theory is the compactification on a line interval of a 27 dimensional theory whose low energy limit contains gravity and a three-form potential. The line interval becomes infinite in the strong coupling limit, and this may provide a stable ground state of the theory. We discuss some of the consequences of this conjecture. 
  We review the euclidean path-integral formalism in connection with the one-dimensional non-relativistic particle. The configurations which allow to construct a semiclassical approximation classify themselves into either topological (instantons) and non-topological (bounces) solutions. The quantum amplitudes consist on an exponential associated with the classical contribution multiplied by the fluctuation factor which is given by a functional determinant. The eigenfunctions as well as the energy eigenvalues of the quadratic operators at issue can be written in closed form due to the shape-invariance property. Accordingly we resort to the zeta-function method to compute the functional determinants in a systematic way. The effect of the multi-instantons configurations is also carefully considered. To illustrate the instanton calculus in a relevant model we go to the double-well potential. The second popular case is the periodic-potential where the initial levels split into bands. The quantum decay rate of the metastable states in a cubic model is evaluated by means of the bounce-like solution. 
  In this paper we evaluate the induced Lorentz and CPT violating Chern-Simons term in the QED action at finite temperature. We do this using the method of derivative expansion of fermion determinants. Also, we use the imaginary-time formalism to calculate the temperature dependence of the Chern-Simons term 
  We review some aspects of the use of a technique known as group averaging, which provides a tool for the study of constrained systems. We focus our attention on the case where the gauge group is non-compact, and a `renormalized' group averaging method must be introduced. We discuss the connection between superselection sectors and the rate of divergence of the group averaging integral. 
  We consider a class of special Lagrangian subspaces of Calabi-Yau manifolds and identify their mirrors, using the recent derivation of mirror symmetry, as certain holomorphic varieties of the mirror geometry. This transforms the counting of holomorphic disc instantons ending on the Lagrangian submanifold to the classical Abel-Jacobi map on the mirror. We recover some results already anticipated as well as obtain some highly non-trivial new predictions. 
  We present a detailed analysis of non-supersymmetric spacetime varying string vacua which can lead to an exponential hierarchy between the electroweak and the gravitational scales. In particular, we identify a limit in which these vacua can be interpreted as supersymmetric vacua of F-theory. Furthermore, we study the properties of these solutions as seen by $D7$-brane probes and establish a non-supersymmetric analogue of the enhancon mechanism. 
  We study the regularization and renormalization of the Yang-Mills theory in the framework of the manifestly invariant formalism, which consists of a higher covariant derivative with an infinitely many Pauli-Villars fields. Unphysical logarithmic divergence, which is the problematic point on the Slavnov's method, does not appear in our scheme, and the well-known vale of the renormalization group functions are derived. The cancellation mechanism of the quadratic divergence is also demonstrated by calculating the vacuum polarization tensor of the order of $\Lambda^0$ and $\Lambda^{-4}$. These results are the evidence that our method is valid for intrinsically divergent theories and is expected to be available for the theory which contains the quantity depending on the space-time dimensions, like supersymmetric gauge theories. 
  In brane-world cosmologies of Randall-Sundrum type, we show that evolution of large-scale curvature perturbations may be determined on the brane, without solving the bulk perturbation equations. The influence of the bulk gravitational field on the brane is felt through a projected Weyl tensor which behaves effectively like an imperfect radiation fluid with anisotropic stress. We define curvature perturbations on uniform density surfaces for both the matter and Weyl fluids, and show that their evolution on large scales follows directly from the energy conservation equations for each fluid. The total curvature perturbation is not necessarily constant for adiabatic matter perturbations, but can change due to the Weyl entropy perturbation. To relate this curvature perturbation to the longitudinal gauge metric potentials requires knowledge of the Weyl anisotropic stress which is not determined by the equations on the brane. We discuss the implications for large-angle anisotropies on the cosmic microwave background sky. 
  We consider D-branes in group manifolds, from the point of view of open strings and using the Born-Infeld action on the brane worldvolume. D-branes correspond to certain integral (twined) conjugacy classes. We explain the integrality condition on the conjugacy classes in both approaches. In the Born-Infeld description, the D-brane worldvolume is stabilized against shrinking by a subtle interplay of quantized U(1) fluxes and the non-triviality of the B-field. 
  These are notes on twisted K-homology theory and twisted Ext-theory from the C*-algebra viewpoint, part of a series of talks on ``C*-algebras, noncommutative geometry and K-theory'', primarily for physicists. 
  I briefly describe two motivations, two mechanisms and two possible tests of the hypothesis that the physical parameters of the ground state of a theory can vary in different regions of the universe. 
  A massless scalar field minimally coupled to gravity and propagating in the Schwarzschild spacetime is considered. After dimensional reduction under spherical symmetry the resulting 2D field theory is canonically quantized and the renormalized expectation values $< T_{a b} >$ of the relevant energy-momentum tensor operator are investigated. Asymptotic behaviours and analytical approximations are given for $< T_{a b} >$ in the Boulware, Unruh and Hartle-Hawking states. Special attention is devoted to the black-hole horizon region where the WKB approximation breaks down. 
  Some time ago Dvali, Gabadadze and Senjanovic (hep-ph/9910207) discussed brane world scenarios with time-like extra dimensions. In this paper we construct a solitonic 3-brane solution in the 5-dimensional Einstein-Hilbert-Gauss-Bonnet theory with the space-time signature (-,+,+,+,-). The direction transverse to the brane is the second time-like direction. The solitonic brane is $\delta$-function like, and has the property that gravity is completely localized on the brane. That is, there are no propagating degrees of freedom in the bulk, while on the brane we have purely 4-dimensional Einstein gravity. In particular, there are no propagating tachyonic or negative norm states even though the extra dimension is time-like. 
  In this paper we present simplifying techniques which allow one to compute the quiver diagrams for various D-branes at (non-Abelian) orbifold singularities with and without discrete torsion. The main idea behind the construction is to take the orbifold of an orbifold. Many interesting discrete groups fit into an exact sequence $N\to G\to G/N$. As such, the orbifold $M/G$ is easier to compute as $(M/N)/(G/N)$ and we present graphical rules which allow fast computation given the $M/N$ quiver. 
  We present and study a new chain of 10-d T duality related solutions and their 11-d parents whose existence had been predicted in the literature based in U duality requirements in 4 d. The first link is the S dual of the D7-brane. The next link has 6 spatial worldvolume dimensions, it is charged w.r.t. the RR 7-form but depends only on 2 transverse dimensions since the third has to be compactified in a circle and is isometric and hence is similar in this respect to the KK monopole. The next link has 5 spatial worldvolume dimensions, it is charged w.r.t. the RR 6-form but depends only on 2 transverse dimensions since the 3rd and 4th have to be compactified in circles and are isometric and so on. All these solutions are identical when reduced over the p spatial worldvolume dimensions and preserve a 1/2 on the supersymmetries. Their masses depend on the square of the radii of the isometric directions, just as for the KK monopole. We give a general map of these branes and their duality relations and show how they must appear in the supersymmetry algebra. 
  The D-bound on the entropy of matter systems in de Sitter space is shown to be closely related to the Bekenstein bound, which applies in a flat background. This holds in arbitrary dimensions if the Bekenstein bound is calibrated by a classical Geroch process. We discuss the relation of these bounds to the more general bound on the entropy to area ratio. We find that black holes do not saturate the Bekenstein bound in dimensions greater than four. 
  We investigate how the integrability conditions for conformal anomalies constrain the form of the effective action in even-dimensional quantum geometry. We show that the effective action of four-dimensional quantum geometry (4DQG) satisfying integrability has a manifestly diffeomorphism invariant and regularization scheme-independent form. We then generalize the arguments to six dimensions and propose a model of 6DQG. A hypothesized form of the 6DQG effective action is given. 
  The D0-D6 system, which is not supersymmetric in the absence of a Neveu-Schwarz B-field, becomes supersymmetric if a suitable constant B-field is turned on. On one side of the supersymmetric locus, this system has a BPS bound state, and on the other side it does not. After compactification on T^6, this gives a simple example in which the number of 1/8 BPS states jumps as the moduli of the compactification are changed. The D0-D8 system in a B-field has two different supersymmetric loci, only one of which is continuously connected to the familiar supersymmetric D0-D8 system without a B-field. In a certain range, the D0-D8 system also has a BPS bound state. In the limit in which the B-field goes to infinity, supersymmetric D0-D6 and D0-D8 systems and their bound states can be studied using noncommutative Yang-Mills theory. 
  When it comes to the topological aspects, gravity may have profound effects even at the level of particle physics despite its negligibly small relative strength well below the Planck scale. In spite of this intriguing possibility, relatively little attempt has been made toward the exhibition of this phenomenon in relevant physical systems. In the present work, perhaps the simplest and the most straightforward new algorithm for generating solutions to (anti) self-dual Yang-Mills (YM) equation in the typical gravitational instanton backgrounds is proposed and then applied to find the solutions practically in all the gravitational instantons known. Solutions thus obtained turn out to be some kind of instanton-meron hybrids possessing mixed features of both. Namely, they are rather exotic type of configurations obeying first order (anti) self-dual YM equation which are everywhere non-singular and have finite Euclidean YM actions on one hand while exhibiting meron-like large distance behavior and carrying generally fractional topological charge values on the other. Close inspection, however, reveals that the solutions are more like instantons rather than merons in their generic natures. 
  The star product technique translates the framework of local fields on noncommutative space-time into nonlocal fields on standard space-time. We consider the example of fields on $\kappa$- deformed Minkowski space, transforming under $\kappa$-deformed Poincar\'{e} group with noncommutative parameters. By extending the star product to the tensor product of functions on $\kappa$-deformed Minkowski space and $\kappa$-deformed Poincar\'{e} group we represent the algebra of noncommutative parameters of deformed relativistic symmetries by functions on classical Poincar\'{e} group. 
  We show how the Hopf algebra structure of renormalization discovered by Kreimer can be found in the Epstein-Glaser framework without using an analogue of the forest formula of Zimmermann. 
  We construct and study a class of N particle supersymmetric Hamiltonians with nearest and next-nearest neighbor inverse-square interaction in one dimension. We show that inhomogeneous XY models in an external non-uniform magnetic field can be obtained from these super-Hamiltonians in a particular limit decoupling the fermionic degrees of freedom from the kinematic ones. We further consider a suitable deformation of these super-models such that inhomogeneous XXZ Hamiltonians in an external non-uniform magnetic field are obtained in the same limit. We show that this deformed Hamiltonian with rational potential is, (i) mapped to a set of free super-oscillators through a similarity transformation and (ii) supersymmetric in terms of a new, non-standard realization of the supercharge. We construct many exact eigenstates of this Hamiltonian and discuss about the applicability of this technique to other models. 
  The superstring/M-theory suggests the Born-Infeld type modification of the classical gauge field lagrangian. We discuss how this changes topological issues related to vacuum periodicity in the SU(2) theory in four spacetime dimensions. A new feature, which is due to the breaking of scale invariance by the non-Abelian Born-Infeld (NBI) action, is that the potential barrier between the neighboring vacua is lowered to a finite height. At the top of the barrier one finds an infinite family of sphaleron-like solutions mediating transitions between different topological sectors. We review these solutions for two versions of the NBI action: with the ordinary and symmetrized trace. Then we show the existence of sphaleron excitations of monopoles in the NBI theory with the triplet Higgs. Soliton solutions in the constant external Kalb-Ramond field are also discussed which correspond to monopoles in the gauge theory on non-commutative space. A non-perturbative monopole solution for the non-commutative U(1) theory is presented. 
  Stability and causality are investigated for quantum field theories incorporating Lorentz and CPT violation. Explicit calculations in the quadratic sector of a general renormalizable lagrangian for a massive fermion reveal that no difficulty arises for low energies if the parameters controlling the breaking are small, but for high energies either energy positivity or microcausality is violated in some observer frame. However, this can be avoided if the lagrangian is the sub-Planck limit of a nonlocal theory with spontaneous Lorentz and CPT violation. Our analysis supports the stability and causality of the Lorentz- and CPT-violating standard-model extension that would emerge at low energies from spontaneous breaking in a realistic string theory. 
  We study dynamical effects of introducing noncommutativity on string worldsheets by using a matrix model obtained from the zero-volume limit of four-dimensional SU($N$) Yang-Mills theory. Although the dimensionless noncommutativity parameter is of order 1/N, its effect is found to be non-negligible even in the large $N$ limit due to the existence of higher Fourier modes. We find that the Poisson bracket grows much faster than the Moyal bracket as we increase $N$, which means in particular that the two quantities do not coincide in the large $N$ limit. The well-known instability of bosonic worldsheets due to long spikes is shown to be cured by the noncommutativity. The extrinsic geometry of the worldsheet is described by a crumpled surface with a large Hausdorff dimension. 
  There appears to be three, perhaps related, ways of approaching the nature of vacuum energy . The first is to say that it is just the lowest energy state of a given, usually quantum, system. The second is to equate vacuum energy with the Casimir energy. The third is to note that an energy difference from a complete vacuum might have some long range effect, typically this energy difference is interpreted as the cosmological constant. All three approaches are reviewed, with an emphasis on recent work. It is hoped that this review is comprehensive in scope. There is a discussion on whether there is a relation between vacuum energy and inertia. The solution suggested here to the nature of the vacuum is that Casimir energy can produce short range effects because of boundary conditions, but that at long range there is no overall effect of vacuum energy, unless one considers lagrangians of higher order than Einstein's as vacuum induced. No original calculations are presented in support of this position. This is not a review of the cosmological constant {\it per se}, but rather vacuum energy in general, my approach to the cosmological constant is not standard. 
  A short article describing the derivation of the string deviation equation: $\dot{\Pi}^\nu_\tau+\Pi'^\mu_\sigma=R^\nu_{.\alpha\beta\gamma}r^\gamma (P^\beta_\tau\dot{x}^\alpha+P^\beta_\sigma x'^\alpha)$. 
  A first order formulation for the Maxwell field in five dimensions is dimensionally reduced using the Randall-Sundrum mechanism. We will see that massive photons can not be localized on the brane. 
  Using perturbation theory, we explore the universal high momentum behavior of correlation functions of gauge invariant operators in planar noncommutative gauge theories. We find that the correlation functions are strongly enhanced when pairs of momenta become antiparallel. In particular, there is a transition from the previously noted exponential suppression of correlation functions at high momenta to a more field theoretic behavior when the momenta of pairs of operators antialign within a critical angle. Some of our calculations can be extrapolated to strong coupling, and in particular we are able to reproduce precisely the supergravity prediction for the behavior of two point functions, including the coupling dependence. 
  We consider the physical implications of various choices of the three-momentum basis in the kappa-deformed Poincare algebra. In particular, we find that the energy dependence of the velocity of a kappa-particle leads to unexpected features in kappa-deformed kinematics. We also discuss the notion of kappa-deformed dynamics, and as a tool example we investigate the motion of a kappa-deformed particle under the action of a constant force. 
  The Pasti-Sorokin-Tonin model for describing chiral forms is considered at the quantum level. We study the ultraviolet and infrared behaviour of the model in two, four and six dimensions in the framework of algebraic renormalization. The absence of anomalies, as well as the finiteness, up to non-physical renormalizations, are shown in all dimensions analyzed. 
  We investigate D2-branes and D4-branes parallel to D8-branes. The low energy world volume theory on the branes is non-supersymmetric Chern-Simons theory. We identify the fundamental strings as the anyons of the 2+1 Chern-Simons theory and the D0-branes as solitons. The Chern-Simons theory with a boundary is modeled using NS 5-branes with ending D6-branes. The brane set-up provides for a graphical description of anomaly inflow. We also model the 4+1 Chern-Simons theory using branes and conjecture that D4-branes with a boundary describes a supersymmetric version of Kaplan's theory of chiral fermions. 
  This article contains a brief pedagogical introduction to various renormalization group related aspects of quantum gravity with an emphasis on the scale dependence of Newton's constant and on black hole physics. 
  The A_{N - 1} (2, 0) superconformal theory has an observable associated with every two-cycle in six dimensions. We make a natural guess for the commutation relations of these operators, which reduces to the commutation relations of Wilson and 't Hooft lines in four-dimensional SU(N) N = 4 super Yang-Mills theory upon compactification on a two-torus. We then verify these commutation relations by considering the theory at a generic point of its moduli space and including in the surface operators only contributions from the light degrees of freedom, which amount to N - 1 (2, 0) tensor multiplets. 
  The massless spectrum of the ten dimensional USp(32) type I string has an N=1 supergravity multiplet coupled to non-supersymmetric matter. This raises the question of the consistency of the gravitino interactions. We resolve this apparent puzzle by arguing for the existence of a local supersymmetry which is non-linearly realised in the open sector. We determine the leading order low energy effective Lagrangian. Similar results are expected to apply to lower dimensional type I models where supergravity is coupled to non-supersymmetric branes. 
  We study some aspects of the Kaluza-Klein Melvin solution in M-theory. The associated magnetic field has a maximal critical value $B=\pm 1/R$ where $R$ is the radius of the compactification circle. It is argued that the Melvin background of type IIA with magnetic field $B$ and of type 0A with magnetic field $B'=B-1/R$ are equivalent. Evidence for this conjecture is provided using a further circle compactification and a `9-11' flip. We show that partition functions of nine-dimensional type IIA strings and of a $(-1)^F\sigma_{1/2}$ type IIA orbifold both with NS-NS Melvin fluxtubes are related by such shift of the magnetic field. Then the instabilities of both IIA and 0A Melvin solutions are analyzed. For each theory there is an instanton associated to the decay of spacetime. In the IIA case the decay mode is associated to the nucleation of $D6/D\bar{6}$-brane pairs, while in the 0A case spacetime decays through Witten's bubble production. 
  In this paper, we consider cosmological perturbations on a brane universe embedded in an Anti-de Sitter bulk. We use a novel gauge, in which the full five-dimensional problem is in principle solvable. In this gauge we derive the equations for scalar, vector and tensor perturbations. These equations are necessary in order to calculate microwave background anisotropies in this particular scenario. Throughout the paper, we draw attention to the influence of the bulk gravitons, which act as a source for the perturbations on the brane. In addition, we find that isocurvature modes are generated due to the influence of bulk gravitons. 
  We propose a method of determining masses in brane scenarios which is independent of coordinate transformations. We apply our method to the scenario of Randall and Sundrum (RS) with two branes, which provides a solution to the hierarchy problem. The core of our proposal is the use of covariant equations and expressing all coordinate quantities in terms of invariant distances. In the RS model we find that massive brane fields propagate proper distances inversely proportional to masses that are not exponentially suppressed. The hierarchy between the gravitational and weak interactions is nevertheless preserved on the visible brane due to suppression of gravitational interactions on that brane. The towers of Kaluza-Klein states for bulk fields are observed to have different spacings on different branes when all masses are measured in units of the fundamental scale. Ratios of masses on each brane are the same in our covariant and the standard interpretations. Since masses of brane fields are not exponentiated, the fundamental scale of higher-dimensional gravity must be of the order of the weak scale. 
  The calculation of the partition function for N M5-branes is addressed for the case in which the worldvolume wraps a manifold $T^2\times M_4$, where $M_4$ is simply connected and Kaehler. This is done in a compactification of M-theory which induces the Vafa-Witten theory on $M_4$ in the limit of vanishing torus volume. The results follow from the equivalence of the BPS spectrum counting in the complementary limit of vanishing $M_4$ volumes and from a classification of the the moduli space of quantum vacua of the supersymmetric twisted theory in terms of associated spectral covers. This reduces the problem of the moduli counting to algebraic equations. 
  This paper considers the spacetimes describing pp-waves propagating on extremal non-dilatonic branes. It is shown that an observer moving along a geodesic will experience infinite curvature at the horizon of the brane, which should therefore be regarded as singular. Taking the decoupling limit of these brane-wave spacetimes gives a pp-wave in AdS, the simplest example being the Kaigorodov spacetime. It has been conjectured that gravity in this spacetime is dual to a CFT in the infinite momentum frame with constant momentum density. If correct, this implies that the CFT must resolve the singularity of the bulk spacetime. Evidence in favour of this conjecture is presented. The unbroken conformal symmetries determine the scalar 2-point function up to an arbitrary function of one variable. However, an AdS/CFT calculation shows that this function is constant (to leading order in $1/N^2$) and the result is therefore the same as when the full conformal symmetry is unbroken. This paper also discusses a recently discovered Virasoro symmetry of metrics describing pp-waves in AdS and naked singularities in the Randall-Sundrum scenario. 
  A noncommutative version of the (anti-) self-dual Yang-Mills equations is shown to be related via dimensional reductions to noncommutative formulations of the generalized (SO(3)/SO(2)) nonlinear Schrodinger (NS) equations, of the super-Korteweg- de Vries (super-KdV) as well as of the matrix KdV equations. Noncommutative extensions of their linear systems and bicomplexes associated to conserved quantities are discussed. 
  The construction of brane setups for the exceptional series E6,E7,E8 of SU(2) orbifolds remains an ever-haunting conundrum. Motivated by techniques in some works by Muto on non-Abelian SU(3) orbifolds, we here provide an algorithmic outlook, a method which we call stepwise projection, that may shed some light on this puzzle. We exemplify this method, consisting of transformation rules for obtaining complex quivers and brane setups from more elementary ones, to the cases of the D-series and E6 finite subgroups of SU(2). Furthermore, we demonstrate the generality of the stepwise procedure by appealing to Frobenius' theory of Induced Representations. Our algorithm suggests the existence of generalisations of the orientifold plane in string theory. 
  We apply ideas that have appeared in the study of D-branes on Calabi-Yau compactifications to the derivation of the BPS spectrum of field theories. In particular, we identify an orbifold point whose fractional branes can be thought of as ``partons'' of the BPS spectrum of N=2 pure SU(N) SYM. We derive the BPS spectrum and lines of marginal stability branes near that orbifold, and compare our results with the spectrum of the field theories. 
  We show that a low-energy action for massless fluctuations around a tachyonic soliton background representing a codimension one D-brane coincides with the Dirac-Born-Infeld action. The scalar modes which describe transverse oscillations of the D-brane are translational collective coordinates of the soliton. The appearance of the DBI action is a universal feature independent of details of a tachyon effective action, provided it has the structure implied by the open string sigma model partition function. 
  We study the physics of open strings in bosonic and type II string theories in the presence of unstable D-branes. When the potential energy of the open string tachyon is at its minimum, Sen has argued that only closed strings remain in the perturbative spectrum. We explore the scenario of Yi and of Bergman, Hori and Yi, who argue that the open string degrees of freedom are strongly coupled and disappear through confinement. We discuss arguments using open string field theory and worldsheet boundary RG flows, which seem to indicate otherwise. We then describe a solitonic excitation of the open string tachyon and gauge field with the charge and tension of a fundamental closed string. This requires a double scaling limit where the tachyon is taken to its minimal value and the electric field is taken to its maximum value. The resulting flux tube has an unconstrained spatial profile; and for large fundamental string charge, it appears to have light, weakly coupled open strings living in the core. We argue that the flux tube acquires a size or order $\alpha'$ through sigma model and string coupling effects; and we argue that confinement effects make the light degrees of freedom heavy and strongly interacting. 
  We prove that in anti de Sitter space, there is no van Dam-Veltman-Zakharov discontinuity in the graviton propagator. Here we obtain the mass term of $M^2\propto \Lambda^2$ from the Gauss-Bonnet term, which is a ghost-free one. The condition that the massless limit is smooth is automatically satisfied for this case. 
  This paper is a revised version of our recent publication Faber et al., Phys. Rev. D62 (2000) 025019, hep-th/9907048. The main revision concerns the expansion into group characters that we have used for the evaluation of path integrals over gauge degrees of freedom. In the present paper we apply the expansion recommended by Diakonov and Petrov in hep-lat/0008004. Our former expansion was approximate and in the region of the particular values of parameters violated the completeness condition by 1.4%. We show that by using the expansion into characters recommended by Diakonov and Petrov in hep-lat/0008004 our previous results are retained and the path integral over gauge degrees of freedom for Wilson loops derived by Diakonov and Petrov (Phys. Lett. B224 (1989) 131 and hep-lat/0008004) by using a special regularization is erroneous and predicts zero value for the Wilson loop. We give comments on the paper hep-lat/0008004. 
  A relation between the Schroedinger wave functional and the Clifford-valued wave function which appears in what we call precanonical quantization of fields and fulfills a Dirac-like generalized covariant Schroedinger equation on the space of field and space-time variables is discussed. The Schroedinger wave functional is argued to be the trace of the positive frequency part of the continual product over all spatial points of the values of the aforementioned wave function restricted to a Cauchy surface. The standard functional differential Schroedinger equation is derived as a consequence of the Dirac-like covariant Schroedinger equation. 
  It is shown that two$(1 + 1)$-dimensional (2D) free Abelian- and self-interacting non-Abelian gauge theories (without any interaction with matter fields) belong to a new class of topological field theories. These new theories capture together some of the key features of Witten- and Schwarz type of topological field theories because they are endowed with symmetries that are reminiscent of the Schwarz type theories but their Lagrangian density has the appearance of the Witten type theories. The topological invariants for these theories are computed on a 2D compact manifold and their recursion relations are obtained. These new theories are shown to provide a class of tractable field theoretical models for the Hodge theory in two dimensions of flat (Minkowski) spacetime where there are no propagating degrees of freedom associated with the 2D gauge boson. 
  We study topological symmetry breaking via the behaviour of Wilson and 't Hooft loop operators for the 2+1 dimenesional Abelian-Higgs model with Chern-Simons term. The topological linking of instantons, which are closed vortex loop configurations, give rise to a long-range, logarithmic, confining potential between electric charges and magnetic flux tubes even though all perturbative forces are short range. Gauss' law forces the concomitance of charge and magnetic flux, hence the confinement is actually of anyons. 
  We study the Yang-Mills-Chern-Simons theory systematically in an effort to generalize the Coleman-Hill result to the non-Abelian case. We show that, while the Chern-Simons coefficient is in general gauge dependent in a non-Abelian theory, it takes on a physical meaning in the axial gauge. Using the non-Abelian Ward identities as well as the analyticity of the amplitudes in the momentum variables, we show that, in the axial gauge, the Chern-Simons coefficient does not receive any quantum correction beyond one loop. This allows us to deduce that the ratio ${4\pi m\over g^{2}}$ is unrenormalized, in a non-Abelian theory, beyond one loop in any infrared safe gauge. This is the appropriate generalization of the Coleman-Hill result to non-Abelian theories. Various other interesting properties of the theory are also discussed. 
  This work starts with the observation of a certain "rule" (up to now unexplored) in the fundamental laws of Nature. We show some evidence of this, and formulate it as a fundamental principle which exhibits a number physical consequences. In particular, a new, very simple and extremely aesthetic unified model, which includes supersymmetry and supergravity, naturally arises from this principle, together with some new "physics". Furthermore, the new interpretation of Kaluza-Klein extra dimensions we advocate here provides a natural argument for dimensional reduction, and the agreement with the observed phenomenology is recovered. In the high energy regime, a new physics is expected. Consequences in QFT are shortly commented. Finally, we observe a structure of "levels" and formulate a general conjecture about such a concept. 
  It has been recently proposed that the background independent open superstring field theory action is given by the disc partition function with all possible open string operators inserted at the boundary of the disc. We use this proposal to study tachyon condensation in the D0-D2 system. We evaluate the disc partition function for the D0-D2 system in presence of a large Neveu-Schwarz B-field using perturbation theory. This perturbative expansion of the disc partition function makes sense as the boundary tachyon operator for the large Neveu-Schwarz B-field is almost marginal. We find that the mass defect for the formation of the D0-D2 bound state agrees exactly with the expected result in the large B-field limit. 
  We discuss some algebraic properties of the background field method. We introduce an extra gauge-fixing term for the background gauge field right at the beginning in the action in such a way that BRST invariance is preserved. The background effective action is considered and it is shown to satisfy both the Slavnov-Taylor identities and the Ward identities. This allows to prove the background equivalence theorem by means of the standard techniques. We show that the Legendre transform W_{bg} of the background gauge invariant action gives the same physical amplitudes as the original one we started with. Moreover, we point out that W_{bg} cannot in general be derived from a classical action by the Gell-Mann-Low formula. Finally, we show that the BRST doublet generated from the background field does not modify the anomaly of the original underlying gauge theory. The proof is algebraic and makes no use of arguments based on power-counting. 
  When gravity couples to scalar fields in Anti-de Sitter space, the geometry becomes non-AdS and develops singularities generally. We propose a criterion that the singularity is physically admissible if the integral of the on-shell Lagrangian density over the finite range is finite everywhere. For all classes of the singularities studied here, the criterion suggested in this paper coincides with an independent proposal made by Gubser that the potential should be bounded from above in the solution. This gives a reason why Gubser's conjecture works. 
  A review of various aspects of superstrings in background electromagnetic fields is presented. Topics covered include the Born-Infeld action, spectrum of open strings in background gauge fields, the Schwinger mechanism, finite-temperature formalism and Hagedorn behaviour in external fields, Debye screening, D-brane scattering, thermodynamics of D-branes, and noncommutative field and string theories on D-branes. The electric field instabilities are emphasized throughout and contrasted with the case of magnetic fields. A new derivation of the velocity-dependent potential between moving D-branes is presented, as is a new result for the velocity corrections to the one-loop thermal effective potential. 
  We study various dynamical aspects of solitons in non-commutative gauge theories and find surprising results. Among them is the observation that the solitons can travel faster than the speed of light for arbitrarily long distances. 
  We prove that, given a time-independent Lagrangian defined in the first tangent bundle of configuration space, every infinitesimal Noether symmetry that is defined in the $n$-tangent bundle and is not vanishing on-shell, can be written as a canonical symmetry in an enlarged phase space, up to constraints that vanish on-shell. The proof is performed by the implementation of a change of variables from the the $n$-tangent bundle of the Lagrangian theory to an extension of the Hamiltonian formalism which is particularly suited for the case when the Lagrangian is singular. This result proves the assertion that any Noether symmetry can be canonically realized in an enlarged phase space. Then we work out the regular case as a particular application of this ideas and rederive the Noether identities in this framework. Finally we present an example to illustrate our results. 
  We demonstrate that pure space-like axial gauge quantizations of gauge fields can be constructed in ways which are free from infrared divergences. We begin by constructing an axial gauge formulation in auxiliary coordinates: $x^+=x^0\sin{\theta}+x^1\cos{\theta}, x^-=x^0\cos{\theta}-x^1\sin{\theta}$. For \theta less than \pi\over 4 we can take $x^-$ as the evolution parameter and construct a traditional canonical formulation of the temporal gauge Schwinger model in which residual gauge fields dependent only on $x^+$ are static canonical variables. Then we extrapolate the temporal gauge operator solution into the axial region, \theta > \pi \over 4, where $x^+$ is taken as the evolution parameter. In the axial region we find that we have to change representations of the residual gauge fields from one realizing the PV prescription to one realizing the ML prescription in order for the infrared divergences resulting from $({\partial}_-)^{-1}$ to be canceled by corresponding ones resulting from the inverse of the hyperbolic Laplace operator. Finally, by taking the limit ${\theta}\to\frac{\pi}{2}-0$ we obtain an operator solution and the Hamiltonian of the axial gauge (Coulomb gauge )Schwinger model in ordinary coordinates. That solution includes auxiliary fields and the representation space is of indefinite metric, providing further evidence that ``physical'' gauges are no more physical than ``unphysical'' gauges. 
  We give an off-shell formulation of the N=2 supersymmetric new nonlinear vector-tensor multiplet. Interactions arise in this model as a consequence of gauging the central charge of the supersymmetry algebra, which in contrast to previous models with local central charge is achieved without a coupling to a vector multiplet. Furthermore, we present a new action formula that follows from coupling the N=2 linear multiplet to the vector-tensor multiplet. 
  In the AdS/CFT correspondence motion in the radial direction of the AdS space is identified with renormalization group flow in the field theory. For the N=4 Yang-Mills theory this motion is trivial. More interesting examples of renormalization group flow occur when the N=4 theory is deformed. Aspects of the flows are discussed for the N=4 theory on its moduli space, and deformed to N=2 in the infra-red within the context of 5d SUGRA. 10d lifts and brane probing are crucial tools for linking the spacetime backgrounds to the dual field theory. 
  We investigate the nonlinear realization of partially broken $N$=2 global supersymmetry in the D=2 and 3 anti de-Sitter (AdS) space. We particularly study Nambu-Goldstone degrees of freedom for the $N$=2 AdS supersymmetry partially broken down to $N$=1 AdS supersymmetry, where we observe a NG fermion of the broken supersymmetry and a NG boson of the internal symmetry which form a NG multiplet. Based on the nonlinear realization method, we construct a superspace formalism for 2 and 3 dimensional AdS space and evaluate the covariant derivatives and supervielbeins for the AdS superspace. Finally we obtain the nonlinear transformation laws and the lowest order of effective Lagrangians. 
  Chiral conformal blocks in a rational conformal field theory are a far going extension of Gauss hypergeometric functions. The associated monodromy representations of Artin's braid group capture the essence of the modern view on the subject, which originates in ideas of Riemann and Schwarz. Physically, such monodromy representations correspond to a new type of braid group statistics, which may manifest itself in two-dimensional critical phenomena, e.g. in some exotic quantum Hall states. The associated primary fields satisfy R-matrix exchange relations. The description of the internal symmetry of such fields requires an extension of the concept of a group, thus giving room to quantum groups and their generalizations. We review the appearance of braid group representations in the space of solutions of the Knizhnik - Zamolodchikov equation, with an emphasis on the role of a regular basis of solutions which allows us to treat the case of indecomposable representations as well. 
  The interplay between topological defects (branes) and black holes has been a subject of recent study, motivated in part by interest in brane-world scenarios. In this paper we analyze in detail the description of a black hole bound to a domain wall (a two-brane in four dimensions), for which an exact description in the limit of zero wall thickness has been given recently. We show how to smooth this singular solution with a thick domain wall. We also show that charged extremal black holes of a size (roughly) smaller than the brane thickness expel the wall, thereby extending the phenomenon of flux expulsion. Finally, we analyze the process of black hole nucleation {\it on} a domain wall, and argue that it is preferred over a previously studied mechanism of black hole nucleation {\it away} from the wall. 
  Using the brick wall method we compute the statistical entropy of a scalar field in a nontrivial background, in two different cases. These background are generated by four and five dimensional black holes with four and three U(1) charges respectively. The Bekenstein entropy formula is generally obeyed, but corrections are discussed in the latter case. 
  We study the behavior of gravitational waves and their backreaction on the background in cosmological solutions of the five-dimensional Ho\v{r}ava-Witten theory. As a dynamical background, we consider two cosmological solutions with spatially flat expanding FRW branes, called $(\uparrow)$- and $(\downarrow)$-solutions, in which the orbifold size increases and decreases in time, respectively. % For these background solutions, the wave equation for the tensor perturbation can be solved by the method of separation of variables, and the mode functions are classified by a separation constant which can be regarded as a graviton mass. We show that the spatial behavior of the mode functions are the same for both background solutions, but the temporal behavior is significantly different. We further show that for the $(\uparrow)$ solution, the background bulk geometry is unstable against the backreaction of the perturbation, while for the $(\downarrow)$ solution, the backreaction on the bulk geometry can be neglected. We also show that, in contrast to the effect to the bulk geometry, the backreaction of the perturbation significantly alters the intrinsic geometry of the brane for the $(\downarrow)$ solution. 
  In stochastic quantization, ordinary 4-dimensional Euclidean quantum field theory is expressed as a functional integral over fields in 5 dimensions with a fictitious 5th time. This is advantageous, in particular for gauge theories, because it allows a different type of gauge fixing that avoids the Gribov problem. Traditionally, in this approach, the fictitious 5th time is the analog of computer time in a Monte Carlo simulation of 4-dimensional Euclidean fields. A Euclidean probability distribution which depends on the 5th time relaxes to an equilibrium distribution. However a broader framework, which we call ``bulk quantization", is required for extension to fermions, and for the increased power afforded by the higher symmetry of the 5-dimensional action that is topological when expressed in terms of auxiliary fields. Within the broader framework, we give a direct proof by means of Schwinger-Dyson equations that a time-slice of the 5-dimensional theory is equivalent to the usual 4-dimensional theory. The proof does not rely on the conjecture that the relevant stochastic process relaxes to an equilibrium distribution. Rather, it depends on the higher symmetry of the 5-dimensional action which includes a BRST-type topological invariance, and invariance under translation and inversion in the 5-th time. We express the physical S-matrix directly in terms of the truncated 5-dimensional correlation functions, for which ``going off the mass-shell'' means going from the 3 physical degrees of freedom to 5 independent variables. We derive the Landau-Cutokosky rules of the 5-dimensional theory which include the physical unitarity relation. 
  We study the effective low energy supergravity of the strongly coupled heterotic string compactified on a Calabi-Yau 3-fold with generic E8 x E8 gauge bundle. We focus on the effective potential for the chiral scalars. The effective superpotential can be studied using the dual 11-dimensional M-theory background involving insertions of M5 branes along an interval. In such backgrounds, in some regions of moduli space, the leading nonperturbative contributions are due to open membrane instantons. These instantons lead to both attractive and repulsive forces between the 5-branes and the orientifold ``M9-branes,'' depending on the region of moduli space. The resulting dynamics on moduli space include a strong coupling dual to the Dine-Seiberg instability, in which the interval grows. We discuss conditions under which the 5-branes are attracted to the wall and comment on the relevance of these results to the study of chirality-changing phase transitions in heterotic M-theory. 
  Gravitational properties of domain walls in fundamental theory and their implications for the trapping of gravity are reviewed. In particular, the difficulties to embed gravity trapping configurations within gauged supergravity is reviewed and the status of the domain walls obtained via the breathing mode of sphere reduced Type IIB supergravity is presented. 
  We address the problem of the cosmological constant within the Randall-Sundrum scenario with a brane stabilization mechanism. We consider brane tensions of general form. We examine the conditions under which a small change of the positive tension of the first brane can be absorbed in a small modification of the two-brane configuration, instead of manifesting itself as a cosmological constant. We demonstrate that it is possible to have a cosmological constant in the range predicted by recent observational data, if there is an ultraviolet cutoff of order 10 TeV in the contributions to the brane tension from vacuum fluctuations. 
  Studies of first-order phase transitions through the use of the exact renormalization group are reviewed. In the first part the emphasis is on universal aspects: We discuss the universal critical behaviour near weakly first-order phase transitions for a three-dimensional model of two coupled scalar fields -- the cubic anisotropy model. In the second part we review the application of the exact renormalization group to the calculation of bubble-nucleation rates. More specifically, we concentrate on the pre-exponential factor. We discuss the reliability of homogeneous nucleation theory that employs a saddle-point expansion around the critical bubble for the calculation of the nucleation rate. 
  A Born--Infeld theory describing a $D2$--brane coupled to a 3--form RR potential is reconsidered and a new type of static solution is obtained which is even stable. 
  It is proved that fermions can acquire the mass through the additional non-integrable exponential factor. For this propose the special vector potential associated with the spinor field was introduced. Such a vector potential has close relation with the triality property in Dirac spinors and plays crucial role in the construction of massive term. It is shown that the change in phase of a wavefunction round any closed curve with the possibility of there being singularities in our vector potential will lead to the law of quantization of physical constants including the mass. The triality properties of Dirac's spinors are studied and it leads to a double covering vector representation of Dirac spinor field. It is proved that massive Dirac equation in the bosonic representation is self-dual. 
  We review the construction of actions with supersymmetry on spaces with a domain wall. The latter objects act as sources inducing a jump in the gauge coupling constant. Despite these singularities, supersymmetry can be formulated, maintaining its role as a square root of translations in this singular space. The setup is designed for the application in five dimensions related to the Randall-Sundrum (RS) scenario. The space has two domain walls. We discuss the solutions of the theory with fixed scalars and full preserved supersymmetry, in which case one of the branes can be pushed to infinity, and solutions where half of the supersymmetries are preserved. 
  Exact Renormalization Group techniques are applied to supersymmetric models in order to get some insights into the low energy effective actions of such theories. Starting from the ultra-violet finite mass deformed N=4 supersymmetric Yang-Mills theory, one varies the regularising mass and compensates for it by introducing an effective Wilsonian action. (Polchinski's) Renormalization Group equation is modified in an essential way by the presence of rescaling (a.k.a. Konishi) anomaly, which is responsible for the beta-function. When supersymmetry is broken up to N=1 the form of effective actions in terms of massless fields is quite reasonable, while in the case of the N=2 model we appear to have problems related to instantons. 
  We present the discussion of the energy-momentum tensor of the scalar $\phi^4$- theory on a noncommutative space. The Noether procedure is performed at the operator level. Additionally, the broken dilatation symmetry will be considered in a Moyal-Weyl deformed scalar field theory at the classical level. 
  A superspace formulation is proposed for the osp(1,2)-covariant Lagrangian quantization of general massive gauge theories. Thereby, osp(1,2) is considered as subalgebra of the superalgebra sl(1,2) which is interpreted as conformal algebra acting on two anticommuting coordinates. The mass-dependent (anti)BRST symmetries of the quantum action in the osp(1,2) superfield formalism are realized as translations associated by mass-dependent special conformal transformations. 
  We analyze the superfield constraints of the D=4, N=3 SYM-theory using light-cone gauge conditions. The SU(3)/U(1) x U(1) harmonic variables are interpreted as auxiliary spectral parameters, and the transform to the harmonic-superspace representation is considered. The harmonic superfield equations of motion are drastically simplified in our gauge, in particular, the basic matrix of the harmonic transform and the corresponding harmonic analytic gauge connections become nilpotent on-shell. It is shown that these harmonic SYM-equations are equivalent to the finite set of solvable linear iterative equations. 
  We embed the O(N) nonlinear sigma model in a non-Abelian gauge theory. As a first class system, it is quantized using two different approaches: the functional Schr\"odinger method and the non-local field-antifield procedure. Firstly, the quantization is performed with the functional Schr\"{o}dinger method, for N=2, obtaining the wave functionals for the ground and excited states. In the second place, using the BV formalism we compute the one-loop anomaly. This important result shows that the classical gauge symmetries, appearing due to the conversion via BFFT method, are broken at the quantum level. 
  Using a metric-based formalism to treat cosmological perturbations, we discuss the connection between anisotropic stress on the brane and brane bending. First we discuss gauge-transformations, and draw our attention to gauges, in which the brane-positions remain unperturbed. We provide a unique gauge, where perturbations both on the brane and in the bulk can be treated with generality. For vanishing anisotropic stresses on the brane, this gauge reduces to the generalized longitudinal gauge. We further comment on the gravitational interaction between the branes and the bulk. 
  We consider string theory on a background of the form AdS_3 x N. Our aim is to give a description of the dual CFT in a general set up. With the requirement that we have N=2 supersymmetry in spacetime, we provide evidence that the dual CFT is in the moduli space of a certain symmetric product M^p/S_p. On the way to show this, we reproduce some recent results on string propagation on AdS_3 and extend them to the superstring. 
  We propose CFT descriptions of the D1/D5 system in a class of freely acting Z_2 orbifolds/orientifolds of type IIB theory, with sixteen unbroken supercharges. The CFTs describing D1/D5 systems involve N=(4,4) or N=(4,0) sigma models on $(R^3\times S^1\times T^4\times (T^4)^N/S_N)/Z_2$, where the action of Z_2 is diagonal and its precise nature depends on the model. We also discuss D1(D5)-brane states carrying non-trivial Kaluza-Klein charges. The resulting multiplicities for two-charge bound states are shown to agree with the predictions of U-duality. We raise a puzzle concerning the multiplicities of three-charge systems, which is generically present in all vacuum configurations with sixteen unbroken supercharges studied so far, including the more familiar type IIB on K3 case: the constraints put on BPS counting formulae by U-duality are apparently in contradiction with any CFT interpretation. We argue that the presence of RR backgrounds appearing in the symmetric product CFT may provide a resolution of this puzzle. 
  We consider the existence of a Noether symmetry in the scalar-tensor theory of gravity in flat Friedman Robertson Walker (FRW) cosmology. The forms of coupling function $\omega(\phi)$ and generic potential $V(\phi)$ are obtained by requiring the existence of a Noether symmetry for such theory. We derive exact cosmological solutions of the field equations from a point-like Lagrangian. 
  Three-dimensional spin models of the Ising and XY universality classes are studied by a combination of high-temperature expansions and Monte Carlo simulations applied to improved Hamiltonians. The critical exponents and the critical equation of state are determined to very high precision. 
  We discuss the interactions of Goldstone particles with solitonic states. We observe that, contrary to the familiar situation in the vacuum sector, the Goldstone particles can have non-derivative interactions with the solitons. This result is applied to brane physics and in particular leads to the possibility that neutrinos in brane world scenarios are Goldstone particles for broken supersymmetry. 
  We discuss questions related to renormalization group and to nonperturbative aspects of non-Abelian gauge theories with N=2 and/or N=1 supersymmetry. Results on perturbative and nonperturbative $\beta$ functions of these theories are reviewed, and new mechanisms of confinement and dynamical symmetry breaking recently found in a class of $SU(n_c)$, $USp(2n_c)$ and $SO(n_c)$ theories are discussed. 
  Using noncommutative geometry we do U(1) gauge theory on the permutation group $S_3$. Unlike usual lattice gauge theories the use of a nonAbelian group here as spacetime corresponds to a background Riemannian curvature. In this background we solve spin 0, 1/2 and spin 1 equations of motion, including the spin 1 or `photon' case in the presence of sources, i.e. a theory of classical electromagnetism. Moreover, we solve the U(1) Yang-Mills theory (this differs from the U(1) Maxwell theory in noncommutative geometry), including the moduli spaces of flat connections. We show that the Yang-Mills action has a simple form in terms of Wilson loops in the permutation group, and we discuss aspects of the quantum theory. 
  We present a complete string theory analysis of all mixed gauge, gravitational and target-space anomalies potentially arising in the simplest heterotic Z_N orbifold models, with N odd and standard embedding. These anomalies turn out to be encoded in an elliptic index, which can be easily computed; they are found to cancel through a universal GS mechanism induced by the dilaton multiplet. The target-space symmetry is then shown to have a nice geometric interpretation in terms of torsion, and the target-space dependence of the four-dimensional GS couplings can be alternatively rederived from the implicit torsion dependence of the standard ten-dimensional GS couplings. The result is universal and consists essentially of a Bianchi identity for the NSNS B field depending on all the curvatures, and in particular on the target-space curvature. 
  We review the main algebraic aspects that characterize and determine the domain wall solutions of maximal gauged supergravity in various spacetime dimensions by considering consistent truncations that retain a number of components in the diagonal of the coset space scalar manifold of the theory. Starting from the algebraic classification of domain walls in D=4 gauged supergravity, we also derive the corresponding solutions in D=5 and D=7 dimensions. From a higher dimensional point of view, these solutions describe the gravitatonal field, in the field theory limit, of a large number of M2-, D3- and M5-branes that are distributed on hypersurfaces in the transverse space to the branes. As a new result we employ a smearing procedure as well as various dualities to list the irreducible curves and the symmetry groups of p-brane distributions for all values of p that are of interest in current applications of string theory. Some emphasis is placed on the presentation of new results in the case of NS5-branes. 
  We present a model for the Dirac magnetic monopole, suitable for the strong coupling regime. The magnetic monopole is static, has charge g and mass M, occupying a volume of radius R ~ O (g^2/M). It is shown that inside each n-monopole there exist infinite multipoles. It is given an analytical proof of the existence of monopole-antimonopole bound state. Theses bound-states might give additional strong light to light scattering in the proton-antiproton collision process at FermiLab TEVATRON. 
  We study the eigenvalues E_{n\ell} of the Salpeter Hamiltonian H = \beta\sqrt(m^2 + p^2) + vr^2, v>0, \beta > 0, in three dimensions. By using geometrical arguments we show that, for suitable values of P, here provided, the simple semi-classical formula E = min_{r > 0} {v(P/r)^2 + \beta\sqrt(m^2 + r^2)} provides both upper and lower energy bounds for all the eigenvalues of the problem. 
  We discuss several closely related concepts in the NSR formulation of superstring theory. We demonstrated that recently proposed NSR model for superstrings on $AdS_5 \times S^5$ is described by the world-sheet logarithmic conformal field theory (LCFT). The origin of LCFT on a world-sheet is closely connected to the matter-ghost mixing in the structure of a brane-like vortex operators. We suggest a dynamical origin of M theory as a string theory with an extra dimension given by bosonised superconformal ghosts. 
  We elaborate on the recently proposed static brane world scenario, where the effective 4-D cosmological constant is exponentially small when parallel 3-branes are far apart. We extend this result to a compactified model with two positive tension branes. Besides an exponentially small effective 4-D cosmological constant, this model incorporates a Randall-Sundrum-like solution to the hierarchy problem. Furthermore, the exponential factors for the hierarchy problem and the cosmological constant problem obey an inequality that is satisfied in nature. This inequality implies that the cosmological constant problem can be explained if the hierarchy problem is understood. The basic idea generalizes to the multibrane world scenario. We discuss models with piecewise adjustable bulk cosmological constants (to be determined by the 5-dimensional Einstein equation), a key element of the scenario. We also discuss the global structure of this scenario and clarify the physical properties of the particle (Rindler) horizons that are present. Finally, we derive a 4-D effective theory in which all observers on all branes not separated by particle horizons measure the same Newton's constant and 4-D cosmological constant. 
  The obtention (up to five or six loop orders) of nonasymptotic critical behavior, above and below Tc, from the field theoretical framework is presented and discussed. 
  We study the geometric engineering of supersymmetric quantum field theories (QFT), with non simply laced gauge groups, obtained from superstring and F-theory compactifications on local Calabi-Yau manifolds. First we review the main lines of the toric method for ALE spaces with ADE singularities which we extend to non simply laced ordinary and affine singularities. Then, we develop two classes of solutions depending on the two possible realisations of the outer-automorphism group of the toric graph $\Delta$(ADE). In F-theory on elliptic Calabi-Yau manifolds, we give explicit results for the affine non simply laced toric data and the corresponding BCFG mirror geometries. The latters extend known results obtained in litterature for the affine ADE cases. We also study the geometric engineering of $ N=1$ supersymmetric gauge theory in eight dimensions. In type II superstring compactifications on local Calabi-Yau threefolds, we complete the analysis for ordinary ADE singularities by giving the explict derivation of the lacking non simply ones. Finally we develop the basis of polyvalent toric geometry. The latter extends bivalent and trivalent geometries, considered in the geometric engineering method, and use it to derive a new solution for the affine $\hat D_4$ singularity. Other features are also discussed. 
  In this paper we present a gauge-invariant formalism for perturbations of the brane-world model developed by the author, A. Ishibashi and O. Seto recently, and analyze the behavior of cosmological perturbations in a spatially flat expanding universe realized as a boundary 3-brane in AdS$^5$ in terms of this formalism. For simplicity we restrict arguments to scalar perturbations. We show that the behavior of cosmological perturbations on superhorizon scales in the brane-world model is the same as that in the standard no-extradimension model, irrespective of the initial condition for bulk perturbations, in the late stage when the cosmic expansion rate $H$ is smaller than the inverse of the bulk curvature scale $\ell$. Further, we give rough estimates which indicate that in the early universe when $H$ is much larger than $1/\ell$, perturbations in these two models behave quite differently, and the conservation of the Bardeen parameter does not hold for superhorizon perturbations in the brane-world model. 
  An associative algebra of holomorphic differential forms is constructed associated with pure N=2 Super-Yang-Mills theory for the Lie algebra F4. Existence and associativity of this algebra, combined with the general arguments in the work of Marshakov, Mironov and Morozov, proves that the prepotential of this theory satisfies the generalized WDVV system. 
  In the standard methodology for evaluating the Hawking radiation emanating from a black hole, the background geometry is fixed. Trying to be more realistic we consider a dynamical geometry for a two-dimensional charged black hole and we evaluate the Hawking radiation as tunneling process. This modification to the geometry gives rise to a nonthermal part in the radiation spectrum. We explore the consequences of this new term for the extremal case. 
  A primary goal in holographic theories of gravity is to study the causal structure of spacetime from the field theory point of view. This is a particularly difficult problem when the spacetime has a non-trivial causal structure, such as a black hole. We attempt to study causality through the UV/IR relation between field theory and spacetime quantities, which encodes information about bulk position. We study the UV/IR relations for charged black hole spacetimes in the AdS/CFT correspondence. We find that the UV/IR relations have a number of interesting features, but find little information about the presence of a horizon in the bulk. The scale of Wilson loops is simply related to radial position, whether there is a horizon or not. For time-dependent probes, the part of the history near the horizon only effects the late-time behaviour of field theory observables. Static supergravity probes have a finite scale size related to radial position in generic black holes, but there is an interesting logarithmic divergence as the temperature approaches zero. 
  We study the large N limit of SO(N) and Sp(N) Chern-Simons gauge theory on S^3 and identify its closed string dual as topological strings on an orientifold of the small resolution of the conifold. Applications to large N dualities for N=1 supersymmetric gauge systems in 4 dimensions are also discussed. 
  The analysis of the worldvolume effective actions of the M-theory Kaluza-Klein monopole and 9-brane suggests that it should be possible to describe non-abelian configurations of M2-branes or M5-branes if the M2-branes are transverse to the eleventh direction and the M5-branes are wrapped on it. This is determined by the fact that the Kaluza-Klein monopole and the M9-brane are constrained to move in particular isometric spacetimes. We show that the same kind of situation is implied by the analysis of the brane descent relations in M-theory. We compute some of the non-commutative couplings of the worldvolume effective actions of these non-abelian systems of M2 and M5 branes and show that they indicate the existence of configurations corresponding to N branes expanding into a higher dimensional M-brane. The reduction to Type II brings up new descriptions of coincident D-branes at strong coupling. We show that these systems have the right non-commutative charges to describe certain expanded configurations playing a role in the framework of the AdS/CFT correspondence. Finally, we discuss the realization of non-commutative brane configurations as topological solitons in non-abelian brane-antibrane systems. 
  It is shown that a simple modification of the dimensional regularization allows to compute in a consistent and gauge invariant way any diagram with less than four loops in the SO(10) unified model. The method applies also to the Standard Model generated by the symmetry breaking $SO(10) \to SU(3)\times SU(2)\times U(1)$. A gauge invariant regularization for arbitrary diagram is also described. 
  We examine the boundary behaviour of the gauged N=(2,0) supergravity in D=3 coupled to an arbitrary number of scalar supermultiplets which parametrize a Kahler manifold. In addition to the gravitational coupling constant, the model depends on two parameters, namely the cosmological constant and the size of the Kahler manifold. It is shown that regular and irregular boundary conditions can be imposed on the matter fields depending on the size of the sigma model manifold. It is also shown that the super AdS transformations in the bulk produce the transformations of the N=(2,0) conformal supergravity and scalar multiplets on the boundary, containing fields with nonvanishing Weyl weights determined by the ratio of the sigma model and the gravitational coupling constants. Various types of (2,0) superconformal multiplets are found on the boundary and in one case the superconformal symmetry is shown to be realized in an unconventional way. 
  We compare the canonical quantization and the effective action method to derive expectation values of the stress energy tensor for scalar fields conformally coupled to a 2D Schwarzschild black hole spacetime. Particular attention is devoted to the thermal equilibrium Hartle-Hawking state where the striking disagreement of the results may be reconduced to the incomplete knowledge of the effective action. We show how to reconcile the two procedures and find physically meaningful analytical approximate expressions for the stress tensor in the Hartle-Hawking state. 
  By exploiting the BV quantization of topological bosonic open membrane, we argue that flat 3-form C-field leads to deformations of the algebras of multi-vectors on the Dirichlet-brane world-volume as 2-algebras. This would shed some new light on geometry of M-theory 5-brane and associated decoupled theories. We show that, in general, topological open p-brane theory has a structure of (p+1)-algebra in the bulk, while a structure of p-algebra in the boundary. The bulk/boundary correspondences are exactly as the generalized Deligne conjecture (a theorem of Kontsevich) in the algebraic world of p-algebras. It also imply that the algebras of quantum observables of (p-1)-brane are ``close to'' the algebras of its classical observables as p-algebras. We interpret above as deformation quantization of (p-1)-brane, generalizing the p=1 case. We argue that there is such quantization based on the direct relation between BV master equation and Ward identity of the bulk topological theory. The path integral of the theory will lead to the explicit formula. We also discuss some applications to topological strings and conjecture that the homological mirror symmetry has further generalizations to the categories of p-algebras. 
  This is a short review of the newly discovered ODp-theories that are non-gravitational six-dimensional theories defined as the decoupling limit of NS5-branes in the presence of a near-critical (p+1)-form RR fields. We discuss the motivation for these new theories, their definitions and properties, and their relation to NCOS theory, OM theory and Little String Theory, focusing on the cases p=1,2. 
  We investigate spacetimes in which the speed of light along flat 4D sections varies over the extra dimensions due to different warp factors for the space and the time coordinates (``asymmetrically warped'' spacetimes). The main property of such spaces is that while the induced metric is flat, implying Lorentz invariant particle physics on a brane, bulk gravitational effects will cause apparent violations of Lorentz invariance and of causality from the brane observer's point of view. An important experimentally verifiable consequence of this is that gravitational waves may travel with a speed different from the speed of light on the brane, and possibly even faster. We find the most general spacetimes of this sort, which are given by AdS-Schwarzschild or AdS-Reissner-Nordstrom black holes, assuming the simplest possible sources in the bulk. Due to the gravitational Lorentz violations these models do not have an ordinary Lorentz invariant effective description, and thus provide a possible way around Weinberg's no-go theorem for the adjustment of the cosmological constant. Indeed we show that the cosmological constant may relax in such theories by the adjustment of the mass and the charge of the black hole. The black hole singularity in these solutions can be protected by a horizon, but the existence of a horizon requires some exotic energy densities on the brane. We investigate the cosmological expansion of these models and speculate that it may provide an explanation for the accelerating Universe, provided that the timescale for the adjustment is shorter than the Hubble time. In this case the accelerating Universe would be a manifestation of gravitational Lorentz violations in extra dimensions. 
  We analyze the statistical properties of the spectrum of the QCD Dirac operator at low energy in a finite box of volume $L^4$ by means of partially quenched Chiral Perturbation Theory (pqChPT), a low-energy effective field theory based on the symmetries of QCD. We derive the two-point spectral correlation function from the discontinuity of the chiral susceptibility. For eigenvalues much smaller than $E_c=F^2/\Sigma L^2$, where $F$ is the pion decay constant and $\Sigma$ is the absolute value of the quark condensate, our result for the two-point correlation function coincides with the result previously obtained from chiral Random Matrix Theory (chRMT). The departure from the chRMT result above that scale is described by the contribution of the nonzero momentum modes. In terms of the variance of the number of eigenvalues in an interval containing $n$ eigenvalues on average, it results in a crossover from a $\log n$-behavior to a $n^2 \log n$-behavior. 
  Noncommutative geometry is based on an idea that an associative algebra can be regarded as "an algebra of functions on a noncommutative space". The major contribution to noncommutative geometry was made by A. Connes, who, in particular, analyzed Yang-Mills theories on noncommutative spaces, using important notions that were introduced in his papers (connection, Chern character, etc). It was found recently that Yang-Mills theories on noncommutative spaces appear naturally in string/M-theory; the notions and results of noncommutative geometry were applied very successfully to the problems of physics.   In this paper we give a mostly self-contained review of some aspects of M(atrix) theory, of Connes' noncommutative geometry and of applications of noncommutative geometry to M(atrix) theory. The topics include introduction to BFSS and IKKT matrix models, compactifications on noncommutative tori, a review of basic notions of noncommutative geometry with a detailed discussion of noncommutative tori, Morita equivalence and $SO(d,d|{\mathbb Z})$-duality, an elementary discussion of instantons and noncommutative orbifolds. The review is primarily intended for physicists who would like to learn some basic techniques of noncommutative geometry and how they can be applied in string theory and to mathematicians who would like to learn about some new problems arising in theoretical physics. 
  The Hopf algebra of undecorated rooted trees has tamed the combinatorics of perturbative contributions, to anomalous dimensions in Yukawa theory and scalar $\phi^3$ theory, from all nestings and chainings of a primitive self-energy subdivergence. Here we formulate the nonperturbative problems which these resummations approximate. For Yukawa theory, at spacetime dimension $d=4$, we obtain an integrodifferential Dyson-Schwinger equation and solve it parametrically in terms of the complementary error function. For the scalar theory, at $d=6$, the nonperturbative problem is more severe; we transform it to a nonlinear fourth-order differential equation. After intensive use of symbolic computation we find an algorithm that extends both perturbation series to 500 loops in 7 minutes. Finally, we establish the propagator-coupling duality underlying these achievements making use of the Hopf structure of Feynman diagrams. 
  We evaluate the evolution operator Z_Reg(R_2,R_1) introduced by Diakonov and Petrov for the definition of the Wilson loop in terms of a path integral over gauge degrees of freedom. We use the procedure suggested by Diakonov and Petrov (Phys. Lett. B224 (1989) 131) and show that the evolution operator vanishes. 
  We examine various aspects of the conjectured duality between warped AdS$_5$ geometries with boundary branes and strongly coupled (broken) conformal field theories coupled to dynamical gravity. We also examine compactifications with 5-d gauge fields, in which case the holographic dual is a broken CFT weakly coupled to dynamical gauge fields in addition to gravity. The holographic picture is used to clarify a number of important phenomenological issues in these and related models, including the questions of black hole production, radius stabilization, early universe cosmology, and gauge coupling unification. 
  We propose a general frame work for deriving the OPEs within a logarithmic conformal field theory (LCFT). This naturally leads to the emergence of a logarithmic partner of the energy momentum tensor within an LCFT, and implies that the current algebra associated with an LCFT is expanded. We derive this algebra for a generic LCFT and discuss some of its implications. We observe that two constants arise in the OPE of the energy-momentum tensor with itself. One of these is the usual central charge. 
  We put forward an idea that the boundary entropy associated with integrable massless flow of thermodynamic Bethe ansatz (TBA) is identified with tachyon action of boundary string field theory. We show that the temperature parameterizing a massless flow in the TBA formalism can be identified with tachyon energy for the classical action at least near the ultraviolet fixed point, i.e. the open string vacuum. 
  First order rotational perturbations of the flat Friedmann-Robertson-Walker (FRW) metric are considered in the framework of four dimensional Neveu-Schwarz-Neveu-Schwarz (NS-NS) string cosmological models coupled with dilaton and axion fields. The decay rate of rotation depends mainly upon the dilaton field potential U. The equation for rotation imposes strong limitations upon the functional form of U, restricting the allowed potentials to two: the trivial case U=0 and a generalized exponential type potential. In these two models the metric rotation function can be obtained in an exact analytic form in both Einstein and string frames. In the potential-free case the decay of rotational perturbations is governed by an arbitrary function of time while in the presence of a potential the rotation tends rapidly to zero in both Einstein and string frames. 
  In the framework of heterotic M-theory compactified on a Calabi-Yau threefold 'times' an interval, the relation between geometry and four-flux is derived {\it beyond first order}. Besides the case with general flux which cannot be described by a warped geometry one is naturally led to consider two special types of four-flux in detail. One choice shows how the M-theory relation between warped geometry and flux reproduces the analogous one of the weakly coupled heterotic string with torsion. The other one leads to a {\it quadratic} dependence of the Calabi-Yau volume with respect to the orbifold direction which avoids the problem with negative volume of the first order approximation. As in the first order analysis we still find that Newton's Constant is bounded from below at just the phenomenologically relevant value. However, the bound does not require an {\it ad hoc} truncation of the orbifold-size any longer. Finally we demonstrate explicitly that to leading order in $\kappa^{2/3}$ no Cosmological Constant is induced in the four-dimensional low-energy action. This is in accord with what one can expect from supersymmetry. 
  Operator product expansions are applied to dilaton-axion four-point functions. In the expansions of the bilocal fields $\tilde{\Phi}\tilde{\Phi}$, $\tilde{C}\tilde{C}$ and $\tilde{\Phi}\tilde{C}$, the conformal fields which are symmetric traceless tensors of rank $l$ and have dimensions $\delta=2+l$ or $8+l+\eta(l)$ and $\eta(l)=\mathcal{O}(N^{-2})$ are identified. The unidentified fields have dimension $\delta=\lambda+l+\eta(l)$ with $\lambda\geq 10$. The anomalous dimensions $\eta(l)$ are calculated at order $\mathcal{O}(N^{-2})$ for both $2^{-{1/2}}(-\tilde{\Phi}\tilde{\Phi} + \tilde{C}\tilde{C})$ and $2^{-{1/2}}(\tilde{\Phi}\tilde{C} + \tilde{C}\tilde{\Phi})$ and are found to be the same, proving $U(1)_Y$ symmetry. The relevant coupling constants are given at order $\mathcal{O}(1)$. 
  Explicit formulas for the zeta functions $\zeta_\alpha (s)$ corresponding to bosonic ($\alpha =2$) and to fermionic ($\alpha =3$) quantum fields living on a noncommutative, partially toroidal spacetime are derived. Formulas for the most general case of the zeta function associated to a quadratic+linear+constant form (in {\bf Z}) are obtained. They provide the analytical continuation of the zeta functions in question to the whole complex $s-$plane, in terms of series of Bessel functions (of fast, exponential convergence), thus being extended Chowla-Selberg formulas. As well known, this is the most convenient expression that can be found for the analytical continuation of a zeta function, in particular, the residua of the poles and their finite parts are explicitly given there. An important novelty is the fact that simple poles show up at $s=0$, as well as in other places (simple or double, depending on the number of compactified, noncompactified, and noncommutative dimensions of the spacetime), where they had never appeared before. This poses a challenge to the zeta-function regularization procedure. 
  We study the recently proposed D-brane configuration [hep-th/0010105] modeling the quantum Hall effect, focusing on the nature of the interactions between the charged particles. Our analysis indicates that the interaction is repulsive, which it should be for the ground state of the system to behave as a quantum Hall liquid. The strength of interactions varies inversely with the filling fraction, leading us to conclude that a Wigner crystal is the ground state at small nu. For larger rational nu (still less than unity), it is reasonable to expect a fractional quantum Hall ground state. 
  We present non-supersymmetric toroidal compactifications of type I string theory with both constant background NSNS two-form flux and non-trivial magnetic flux on the various D9-branes. The non-vanishing B-flux admits four-dimensional models with three generations of chiral fermions in standard model like gauge groups. Additionally, we consider the orbifold T^4/Z_2, again with both kinds of background flux present, leading to non-supersymmetric as well as supersymmetric models in six dimensions. All models have T-dual descriptions as intersecting brane worlds. 
  We discuss some issues related to D(2p)-D0 branes with background magnetic fluxes respectively, in a T-dual picture, Dp-Dp branes at angles. In particular, we describe the nature of the supersymmetric bound states appearing after tachyon condensation. We present a very elementary derivation of the conditions to be satisfied by such general supersymmetric gauge configurations, which are simply related by T-duality to the conditions for supersymmetric p-cycles in C^p. 
  We show that exponentially large warp factor hierarchies can be dynamically generated in supersymmetric compactifications. The compactification we consider is the supersymmetric extension of the Randall-Sundrum model. The crucial issue is the stabilization of the radius modulus for large warp factor. The stabilization sector we employ is very simple, consisting of two pure Yang--Mills sectors, one in the bulk and the other localized on a brane. The only fine-tuning required in our model is the cancellation of the cosmological constant, achieved by balancing the stabilization energy against supersymmetry breaking effects. Exponentially large warp factors arise naturally, with no very large or small input parameters. To perform the analysis, we derive the 4-dimensional effective theory for the supersymmetric Randall-Sundrum model, with a careful treatment of the radius modulus. The manifestly (off-shell) supersymmetric form of this effective lagrangian allows a straightforward and systematic treatment of the non-perturbative dynamics of the stabilization sector. 
  In this paper by means of harmonic analysis we derive the complete spectrum of Osp(2|4) x SU(2) x SU(2) x SU(2) multiplets that one obtains compactifying D=11 supergravity on the homogeneous space Q^{111}. In particular we analyze the structure of the short multiplets and compare them with the corresponding composite operators of the N=2 conformal field theory dual to such a compactification, found in a previous publication. We get complete agreement between the quantum numbers of the supergravity multiplets on one side and those of the conformal operators on the other side, confirming the structure of the conjectured SCFT. However the determination of the actual spectrum by harmonic analysis teaches us a lot more: indeed we find out which multiplets are present for each representation of the isometry group, how many they are, the exact values of the hypercharge and of the energy for each multiplet. 
  We discuss the junction conditions in the context of the Randall-Sundrum model with the Gauss-Bonnet interaction. We consider the $Z_2$ symmetric model where the brane is embedded in an $AdS_5$ bulk, as well as a model without $Z_2$ symmetry in which the brane (in this case called by tradition ``shell'') separates two metrically different $AdS_5$ regions. We show that the Israel junction conditions across the membrane (that is either a brane or a shell) have to be modified if more general equations than Einstein's, including higher curvature terms, hold in the bulk, as is likely to be the case in a low energy limit of string theory. We find that the membrane can then no longer be treated in the thin wall approximation. We derive the junction conditions for the Einstein-Gauss-Bonnet theory including second order curvature terms and show that the microphysics of Gauss-Bonnet thick membranes may, in some instances, be simply hidden in a renormalization of Einstein's constant. 
  We analyze the universal transport behavior in 1D and 2D fermionic systems by following the unified framework provided by bosonization. The role played by the adiabatic transition between interacting and noninteracting regions is emphasized. 
  A concise survey is given of the general method of reduction in the number of coupling parameters. Theories with several independent couplings are related to a set of theories with a single coupling. The reduced theories may or may not have particular symmetries. A few have asymptotic power series expansions, others contain non-integer powers and/or logarithmic factors. An example is given with two power series solutions, one with N = 2 Supersymmetry, and one with no known symmetry. In a second example, the reduced Yukawa coupling of the superpotential in a dual magnetic supersymmetric gauge theory is uniquely given by the square of the magnetic gauge coupling with a known factor. 
  We derive the effective action for the radion supermultiplet in the supersymmetric Randall-Sundrum model with two opposite tension branes. 
  In this work we study the dynamics of branes on group manifolds G deep in the stringy regime. After giving a brief overview of the various branes that can be constructed within the boundary conformal field theory approach, we analyze in detail the condensation processes that occur on stacks of such branes. At large volume our discussion is based on certain effective gauge theories on non-commutative `fuzzy' spaces. Using the `absorption of the boundary spin'-principle which was formulated by Affleck and Ludwig in their work on the Kondo model, we extrapolate the brane dynamics into the stringy regime. For supersymmetric theories, the resulting condensation processes turn out to be consistent with the existence of certain conserved charges taking values in some non-trivial discrete abelian groups. We obtain strong constraints on these charge groups for G = SU(N). The results may be compared with a recent proposal of Bouwknegt and Mathai according to which charge groups on curved spaces X (with a non-vanishing NSNS 3-form field strength H) are given by the twisted K-groups K*_H(X). 
  We find a class of extremal black hole-like global p-brane in higher-dimensional gravity with a negative cosmological constant. The region inside the p-brane horizon possesses all essential features required for the Randall-Sundrum-type brane world scenario. The set-up allows to interpret the horizon size as the compactification size in that the Planck scale is determined by the fundamental scale M* and the horizon size r_H via the familiar relation M_Pl^2 ~ M*^{2+n} r_{H}^n, and the gravity behaves as expected in a world with n-extra dimensions compactified with size r_H. Most importantly, a stable mass hierarchy between M_Pl and M* can be generated from topological charge of the p-brane and the horizon size r_H therein. We also offer a new perspective on various issues associated to the brane world scenarios including the cosmological constant problem. 
  Explicit exact formulas are presented, up to fourth order in a strict chiral covariant derivative expansion, for the normal parity component of the Euclidean effective action of even-dimensional Dirac fermions. The bosonic background fields considered are scalar, pseudo-scalar, vector and axial vector. No assumptions are made on the internal symmetry group and, in particular, the scalar and pseudo-scalar fields need not be on the chiral circle. 
  We review the boundary state description of D-branes in type I string theory and show that the only stable non-BPS configurations are the D-particle and the D-instanton. We also compute the gauge and gravitational interactions of the non-BPS D-particles and compare them with the interactions of the dual non-BPS states of the heterotic string, finding complete agreement. 
  We examine the curvature expansion of a the field equations of a four-dimensional higher spin gauge theory extension of anti-de Sitter gravity. The theory contains massless particles of spin 0,2,4,... that arise in the symmetric product of two spin 0 singletons. We cast the curvature expansion into manifestly covariant form and elucidate the structure of the equations and observe a significant simplification. 
  We review how to describe the stable non-BPS D-branes of type II string theory from a classical perspective, and discuss the properties of the space-time geometry associated to these configurations. This is relevant in order to see whether and how the gauge/gravity correspondence can be formulated in non-conformal and non-supersymmetric settings. 
  We investigate the high energy behavior of the correlation functions of the open Wilson lines in noncommutative gauge theory. We obtain a very simple physical picture that they are bound to form a group of closed Wilson loops. We prove our claim in the weak coupling region by perturbative analysis. We emphasize the importance of respecting the cyclic symmetry of the straight Wilson lines to compute the correlation functions. The implications for stringy calculation of the correlators are also discussed. 
  It is argued that the quadratic and linear non-commutative IR divergences that occur in U(1) theory on non-commutative Minkowski spacetime for small non-commutativity matrices $\theta^{\mu\nu}$ are gauge-fixing independent. This implies in particular that the perturbative tachyonic instability produced by the quadratic divergences of this type in the vacuum polarization tensor is not a gauge-fixing artifact. Supersymmetry can be introduced to remove from the renormalized Green functions at one loop, not only the non-logarithmic non-commutative IR divergences, but also all terms proportional to $\theta^{\mu\nu}p_\nu$ 
  It is shown that the never ending oscillatory behaviour of the generic solution, near a cosmological singularity, of the massless bosonic sector of superstring theory can be described as a billiard motion within a simplex in 9-dimensional hyperbolic space. The Coxeter group of reflections of this billiard is discrete and is the Weyl group of the hyperbolic Kac-Moody algebra E$_{10}$ (for type II) or BE$_{10}$ (for type I or heterotic), which are both arithmetic. These results lead to a proof of the chaotic (``Anosov'') nature of the classical cosmological oscillations, and suggest a ``chaotic quantum billiard'' scenario of vacuum selection in string theory. 
  Vector supersymmetry is shown to exist also in light-cone gauge Chern-Simons theory. Using a gauge invariant regularization scheme, we demonstrate explicitly that the finite quantum correction to the coupling constant of Chern-Simons theory is intimately associated with the breaking of vector supersymmetry at the regularization level. The advantage of investigating such a quantum phenomenon in the light-cone gauge is emphasized and the BRST and vector supersymmetry invariance of quantum effective action is discussed. 
  Explicit exact formulas are presented, for the leading order term in a strict chiral covariant derivative expansion, for the abnormal parity component of the effective action of two- and four-dimensional Dirac fermions in presence of scalar, pseudo-scalar, vector and axial vector background fields. The formulas hold for completely general internal symmetry groups and general configurations. In particular the scalar and pseudo-scalar fields need not be on the chiral circle. 
  The Aharonov-Bohm effect on the noncommutative plane is considered. Developing the path integral formulation of quantum mechanics, we find the propagation amplitude for a particle in a noncommutative space. We show that the corresponding shift in the phase of the particle propagator due to the magnetic field of a thin solenoid receives certain gauge invariant corrections because of the noncommutativity. Evaluating the numerical value for this correction, an upper bound for the noncommutativity parameter is obtained. 
  We consider a (3+1)-dimensional local field theory defined on the sphere. The model possesses exact soliton solutions with non trivial Hopf topological charges, and infinite number of local conserved currents. We show that the Poisson bracket algebra of the corresponding charges is isomorphic to that of the area preserving diffeomorphisms of the sphere. We also show that the conserved currents under consideration are the Noether currents associated to the invariance of the Lagrangian under that infinite group of diffeomorphisms. We indicate possible generalizations of the model. 
  In this paper we continue the study of the geometrical features of a functional approach to classical mechanics proposed some time ago. In particular we try to shed some light on a N=2 "universal" supersymmetry which seems to have an interesting interplay with the concept of ergodicity of the system. To study the geometry better we make this susy local and clarify pedagogically several issues present in the literature. Secondly, in order to prepare the ground for a better understanding of its relation to ergodicity, we study the system on constant energy surfaces. We find that the procedure of constraining the system on these surfaces injects in it some local grassmannian invariances and reduces the N=2 global susy to an N=1. 
  Massive maximally-supersymmetric sigma models are shown to exhibit multiple static kink-domain wall solutions that preserve 1/2 of the supersymmetry. The kink moduli space admits a natural Kahler metric. We examine in some detail the case when the target of the sigma model is given by the co-tangent bundle of CP^n equipped with the Calabi metric, and we show that there exist BPS solutions corresponding to n kinks at arbitrary separation. We also describe how 1/4-BPS charged and intersecting domain walls are described in the low-energy dynamics on the kink moduli space. We comment on the similarity of these results to monopole dynamics. 
  We construct a class of supersymmetric boundary interactions in N=2 field theories on the half-space, which depend on parameters that are not at all renormalized or not renormalized in perturbation theory beyond one-loop. This can be used to study D-branes wrapped on a certain class of Lagrangian submanifolds as well as holomorphic cycles. The construction of holomorphic D-branes is in close relationship with the background independent open string field theory approach to brane/anti-brane systems. As an application, mirror pairs of Lagrangian and holomorphic D-branes are identified. The mirror pairs are studied by twisting to open topological field theories. 
  We study the relativistic dynamics of brane probes in singular warped spacetimes and establish limits for such analysis. The behavior of the semiclassical brane probe wave functions implies that unitarity boundary conditions can be imposed at the singularity. 
  We investigate the possibility of Lorentz and CPT violations in the photon sector, of the Chern-Simons form, be induced by radiative corrections arising from the Lorentz and CPT non-invariant fermionic sector of an extended version of QED. By analyzing the modified vacuum polarization tensor, three contributions are considered: two of them can be identified with well known amplitudes; the (identical) QED vacuum polarization tensor and the (closely related) $AVV$ triangular amplitude. These amplitudes are evaluated in their most general form (to include in our discussion automatically the question of ambiguities) on the point of view of a strategy to manipulate and calculate divergent amplitudes that can avoid the explicit calculation of divergent integrals. Rather than this only general properties are used in intermediary steps. With this treatment, the results obtained by others authors can be easily recovered and we show that, if we choose to impose U(1) gauge invariance maintenance in the pure QED calculated amplitudes, to be consistent with the renormalizability, the induced Chern-Simons term assumes a nonvanishing ambiguities free value. However if, in addition, we choose to get an answer consistent with renormalizability by anomaly cancellation of the Standard Model a vanishing value can be obtained, in accordance with what was previously conjectured by other authors. 
  We have obtained an exact solution for the BPS domain wall junction for a N=1 supersymmetric theory in four dimensions and studied its properties. The model is a simplified version of the N=2 SU(2) gauge theory with N_f=1 broken to N=1 by the mass of the adjoint chiral superfield. We define mode equations and demonstrate explicitly that fermion and boson with the same mass have to come in pairs except massless modes. We work out explicitly massless Nambu-Goldstone (NG) modes on the BPS domain wall junction. We find that their wave functions extend along the wall to infinity (not localized) and are not normalizable. It is argued that this feature is a generic phenomenon of NG modes on domain wall junctions in the bulk flat space in any dimensions. NG fermions exhibit a chiral structure in accordance with unitary representations of (1, 0) supersymmetry algebra where fermion and boson with the same mass come in pairs except massless modes which can appear singly. 
  Recently it was shown that NCOS theories are part of a ten-dimensional theory known as Non-relativistic Wound string theory. We clarify the sense in which gravity is present in this theory. We show that Wound string theory contains exceptional unwound strings, including a graviton, which mediate the previously discovered instantaneous long-range interactions, but are negligible as asymptotic states. Unwound strings also provide the expected collective coordinates for the transverse D-branes in the theory. These and other results are shown to follow both from a direct analysis of the effect of the NCOS limit on the parent string theory, and from the worldsheet formalism developed by Gomis and Ooguri, about which we make some additional remarks. We also devote some attention to supergravity duals, and in particular show that the open and closed strings of the theory are respectively described by short and long strings on the supergravity side. 
  In this short paper we would like to present a simple topological matrix model which has close relation with the noncommutative Chern-Simons theory. 
  The vacuum expectation values for the energy-momentum tensor of a massive scalar field with general curvature coupling and obeying the Robin boundary condition on spherically symmetric boundaries in D-dimensional space are investigated. The expressions are derived for the regularized vacuum energy density, radial and azimuthal stress components (i) inside and outside a single spherical surface and (ii) in the intermediate region between two concentric spheres. Regularization procedure is carried out by making use of the generalized Abel-Plana formula for the series over zeros of cylinder functions. Asymptotic behavior of the vacuum densities near the sphere and at large distances is investigated. A decomposition of the Casimir energy into volumic and surface parts is provided for both cases (i) and (ii). We show that the mode sum energy, evaluated as a sum of the zero-point energies for each normal mode of frequency, and the volume integral of the energy density in general are different, and argue that this difference is due to the existence of an additional surface energy contribution. 
  We describe spinors in Minkowskian spaces with arbitrary signature and their role in the classification of space-time superalgebras and their R-symmetries in any dimension. 
  The quantum BRST-anti-BRST operators are explicitely derived and the consequences related to correlation functions are investigated. The connection with the standard formalism and the loopwise expansions for quantum operators and anomalies in Sp(2) approach are analyzed. 
  We study open and unoriented strings in a Topological Membrane (TM) theory through orbifolds of the bulk 3D space. This is achieved by gauging discrete symmetries of the theory. Open and unoriented strings can be obtained from all possible realizations of $C$, $P$ and $T$ symmetries. The important role of $C$ symmetry to distinguish between Dirichlet and Neumman boundary conditions is discussed in detail. 
  For certain dimensionally-regulated one-, two- and three-loop diagrams, problems of constructing the epsilon-expansion and the analytic continuation of the results are studied. In some examples, an arbitrary term of the epsilon-expansion can be calculated. For more complicated cases, only a few higher terms in epsilon are obtained. Apart from the one-loop two- and three-point diagrams, the examples include two-loop (mainly on-shell) propagator-type diagrams and three-loop vacuum diagrams. As a by-product, some new relations involving Clausen function, generalized log-sine integrals and certain Euler--Zagier sums are established, and some useful results for the hypergeometric functions of argument 1/4 are presented. 
  To classify the classical field theories with W-symmetry one has to classify the symplectic leaves of the corresponding W-algebra, which are the intersection of the defining constraint and the coadjoint orbit of the affine Lie algebra if the W-algebra in question is obtained by reducing a WZNW model. The fields that survive the reduction will obey non-linear Poisson bracket (or commutator) relations in general. For example the Toda models are well-known theories which possess such a non-linear W-symmetry and many features of these models can only be understood if one investigates the reduction procedure. In this paper we analyze the SL(n,R) case from which the so-called W_n-algebras can be obtained. One advantage of the reduction viewpoint is that it gives a constructive way to classify the symplectic leaves of the W-algebra which we had done in the n=2 case which will correspond to the coadjoint orbits of the Virasoro algebra and for n=3 which case gives rise to the Zamolodchikov algebra. Our method in principle is capable of constructing explicit representatives on each leaf. Another attractive feature of this approach is the fact that the global nature of the W-transformations can be explicitly described. The reduction method also enables one to determine the ``classical highest weight (h. w.) states'' which are the stable minima of the energy on a W-leaf. These are important as only to those leaves can a highest weight representation space of the W-algebra be associated which contains a ``classical h. w. state''. 
  We discuss a possible explanation of the hierarchy problem within the theories with spacetime dimensions higher than four. We show that the presence of relatively (not hierarchically) large extra dimensions can significantly alter the evolution of the Higgs field VEV, driving it to an infrared stable fixed point $\sim M_{W}$. Such a behaviour results in self-organizing criticality and naturally explains gauge hierarchy without any fine tuning of the parameters. 
  The near-horizon geometry of parallel NS5-branes is described by an exact superconformal 2d field theory based on K^10 = W(k)^4 x M^6, with M^6 a flat 5+1 space time and W(k)^4 =U(1) x SU(2)(k), a four dimensional background with non-trivial dilaton. The ten-dimensional ``BULK'' spectrum of excitations of K^10 can be derived combining unitary representations of the N=4 superconformal theory of W(k)^4 and M^6 in a modular-invariant way. All Bulk states are massive and belong to the long representations of the N=2 six dimensional space-time supersymmetry. The NS5-brane states, propagating on M^6, belong to the short representations of N=2. Both the bulk and the brane spectrum is derived using the powerful worldsheet technics of the N=4 superconformal theories. We claim that both bulk and brane states are necessary to well define the theory. The non abelian U(n) or SO(2n) structure of the 5-brane fields follows from the fusion coefficients appearing in the correlation functions involving SU(2)(k) conformal fields. 
  We study the world volume action of fractional Dp-branes of type IIA string theory compactified on the orbifold T^4/Z_2. The geometric relation between these branes and wrapped branes is investigated using conformal techniques. In particular we examine in detail various scattering amplitudes and find that the leading low-energy interactions are consistent with the boundary action derived geometrically. 
  We describe how to construct solutions to 11-dimensional supergravity corresponding to M5-branes wrapped on holomorphic 2-cycles embedded in C^3. These solutions preserve N=1 supersymmetry in four dimensions. In the near-horizon limit they are expected to be dual to N=1 large N gauge theories in four dimensions by Maldacena's duality. 
  We construct supergravity solutions dual to the twisted field theories arising when M-theory fivebranes wrap general supersymmetric cycles. The solutions are constructed in maximal D=7 gauged supergravity and then uplifted to D=11. Our analysis covers Kahler, special Lagrangian and exceptional calibrated cycles. The metric on the cycles are Einstein, but do not necessarily have constant curvature. We find many new examples of AdS/CFT duality, corresponding to the IR superconformal fixed points of the twisted field theories. 
  Compactified AdS space can not be mapped into just one Poincare coordinate chart. This implies that the bulk field spectrum is discrete despite the infinite range of the coordinates. We discuss here why this discretization of the field spectrum seems to be a necessary ingredient for the holographic mapping. For the Randall Sundrum model we show that this discretization appears even without the second brane. 
  The BRST algebraic proofs of the the nonrenormalization theorems for the beta functions of N=2 and N=4 Super Yang-Mills theories are reviewed. 
  We develop the boundary string field theory approach to tachyon condensation on the DDbar system. Particular attention is paid to the gauge fields, which combine with the tachyons in a natural way. We derive the RR-couplings of the system and express the result in terms of Quillen's superconnection. The result is related to an index theorem, and is thus shown to be exact. 
  We construct low-energy Goldstone superfield actions describing various patterns of the partial spontaneous breakdown of two-dimensional N=(1,1), N=(2,0) and N=(2,2) supersymmetries, with the main focus on the last case. These nonlinear actions admit a representation in the superspace of the unbroken supersymmetry as well as in a superspace of the full supersymmetry. The natural setup for implementing the partial breaking in a self-consistent way is provided by the appropriate central extensions of D=2 supersymmetries, with the central charges generating shift symmetries on the Goldstone superfields. The Goldstone superfield actions can be interpreted as manifestly world-sheet supersymmetric actions in the static gauge of some superstrings and D1-branes in D=3 and D=4 Minkowski spaces. As an essentially new example, we elaborate on the action representing the 1/4 partial breaking pattern N=(2,2) -> N=(1,0). 
  In the presence of D-branes, fermionic N=2 strings in 2+2 dimensions can be coupled to a K"ahler NS-NS two-form B. We present the corresponding action which produces N=2 supersymmetric boundary conditions and discuss the Seiberg-Witten zero-slope limit. After recalling the constraints on the Chan-Paton gauge group, we demonstrate for U(n) groups that the open N=2 string with a nonzero B-field coincides on tree level with noncommutative self-dual Yang-Mills. Several misconceptions of hep-th/0011206 are corrected. 
  We review recent results in the study of regular four dimensional BPS black holes in toroidally compactified type II (or M) theory. We discuss the generating solution for this kind of black holes, its microscopic description(s), and compute the corresponding microscopic entropy. These achievements, which provide a description of the fundamental degrees of freedom accounting for the entropy of any regular BPS black hole in the theory under consideration, are inscribed within a research project aimed to the study of the microscopic properties of this kind of solutions in relation to U--duality invariants computed on the corresponding macroscopic (supergravity) description. 
  We find a static solution to Einstein's field equations on a five-dimensional orbifold with a compact S_1/Z_2 fifth direction and Poincare invariant 3+1 sections. The solution describes a theory with bulk cosmological constant and 3-branes at the orbifold fixed points which carry matter density and pressure in addition to tension. The radius of the fifth dimension is determined by the matter content of the branes. The ratio of the space and time components of the metric depends on the fifth coordinate. Thus, the speed of propagation of massless fields is path dependent. For example, bulk and brane fields propagate with different speeds. 
  The recent construction of the non-semisimple gaugings of maximal D=5 supergravity is reported here. This construction is worked out in the so-called rheonomic approach, based on Free Differential Algebras and the solution of their Bianchi identities. In this approach the dualization mechanism between one-forms and two-forms is more transparent. The lagrangian is unnecessary since the field equations follow from closure of the supersymmetry algebra. These theories contain 12-r self-dual two-forms and 15+r gauge vectors, r of which are abelian and neutral. Such theories, whose existence has been proved and their supersymmetry algebra constructed, have potentially interesting properties in relation with domain wall solutions and the trapping of gravity. 
  Certain linear objects, termed physical lines, are considered, and initial assumptions concerning their properties are introduced. A closed physical line in the form of a circle, termed J-string, is singled out for investigation. It is shown that this curve consists of indivisible line segments of length $\ell_\Delta$. It is assumed that a J-string has an angular momentum whose value is $\hbar$.It is then established that a J-string of radius $R$ possesses a mass $m_J$, equal to $h/2\pi c R$, a corresponding energy, as well as a charge $q_J$, where $q_J = (hc/2\pi)^{1/2}$. It is also established that $\ell_\Delta = 2\pi(hG/c^3)^{1/2}$, where $c$ is the speed of light and $G$ is the gravitational constant. % Based upon investigation of the properties and characteristics of J-strings, a method is developed for the computation of the Planck length and mass $(\ell^*_P, m^*_P)$. The values of $\ell^*_P$ and $m^*_P$ are computed according to the resulting formulae (and given in the paper); these values differ from the currently accepted ones. 
  Superfield expansions over four-dimensional graded spacetime $(x^\mu,\theta^\nu)$, with Minkowski coordinates $x$ extended by vector Grassmann variables $\theta$, are investigated. By appropriate identification of the physical Lorentz algebra in the even and odd parts of the superfield, a typology of `schizofields' containing both integer and half-integer spin fields is established. For two of these types, identified as `gauge potential'-like and `field strength'-like schizofields, an $sl(2/1,{\mathbb C})_{\mathbb R}$ supersymmetry at the component field level is demonstrated. Prospects for a schizofield calculus, and application of these types of fields to the particle spectrum, are adumbrated. 
  The exceptional superalgebra $\D21a$ has been classified as a candidate conformal supersymmetry algera in two dimensions. We propose an alternative interpretation of it as an extended BFV-BRST quantisation superalgebra in 2D ($D(2,1;1) \simeq osp(2,2|2)$). A superfield realization is presented wherein the standard extended phase space coordinates can be identified. The physical states are studied via the cohomology of the BRST operator. Finally we reverse engineer a classical action corresponding to the algebraic model we have constructed, and identify the Lagrangian equations of motion. 
  We give a quantum field theoretical treatment of a one dimensional electron system with a fixed chemical potential $\mu$. The non-perturbative Lindhard response function is found for an electron system in a sinusoidal potential. 
  The RG flow for the sine-Gordon model is determined by means of the method of Wegner and Houghton in next-to-leading order of the derivative expansion. For small values of the fugacity this agrees with the well-known RG flow of the two-dimensional Coulomb-gas found in the dilute gas approximation and a systematic way of obtaining higher-order corrections to this approximation is given. 
  Doplicher, Fredenhagen, and Roberts (1994, 1995) proposed a simple model of a particle in quantum spacetime. We give a new formulation of the model and propose some small changes and additions which improve the physical interpretation. In particular, we show that the internal degrees of freedom e and m of the particle represent external forces acting on the particle. To obtain this result we follow a constructive approach. The model is formulated as a covariance system. It has projective representations in which not only the spacetime coordinates but also the conjugated momenta are two-by-two noncommuting. These momenta are of the form P_mu-(b/c)A_mu, where b is the charge of the particle. The electric and magnetic fields obtained from the vector potential A_mu coincide with the variables e and m postulated by DFR. Similarly, the spacetime position operators are of the form Q_mu-(al^2/hbar c) Omega_mu where a is a generalized charge, l a fundamental length, and with vector potentials Omega_mu which are in some sense dual w.r.t. the A_mu. 
  In this paper we give the boundary string field theory description of brane-antibrane systems. From the world-sheet action of brane-antibrane systems we obtain the tachyon potential and discuss the tachyon condensation exactly. We also find the world-volume action including the gauge fields. Moreover we determine RR-couplings exactly for non-BPS branes and brane-antibranes. These couplings are written by superconnections and correspond to K^1(M) and K^0(M) for the non-BPS branes and brane-antibranes, respectively. We also show that Myers terms appear if we include the transverse scalars in the boundary sigma model action. 
  We obtain the supergravity solution which describes a bound state of D-string/anti-D-string pairs attached to different fixed planes of an orbifold, in type IIB string theory compactified on T^4/Z_2. For parameters at which the conformal field theory point of view predicts stability, the solution displays a repulson-like singularity. However, we observe that a D-string/anti-D-string pair probe in this background becomes tensionless before reaching the singularity, suggesting a resolution by the enhancon mechanism. Moreover, the force feels by this probe is attractive, in contrast to the repulsive behaviour observed in the non-BPS D-brane description. 
  We give in this paper a partial classification of the consistent quadratic gauge actions that can be written in terms of s-form fields. This provides a starting point to study the uniqueness of the Yang-Mills action as a deformation of Maxwell-like theories. We also show that it is impossible to write kinetic 1-form terms that can be consistently added to other 1-form actions such as tetrad gravity in four space-time dimensions even in the presence of a Minkowskian metric background. 
  We discuss the vacuum structure of type IIA/B Calabi-Yau string compactifications to four dimensions in the presence of n-form H-fluxes. These will lift the vacuum degeneracy in the Calabi-Yau moduli space, and for generic points in the moduli space, N=2 supersymmetry will be broken. However, for certain `aligned' choices of the H-flux vector, supersymmetric ground states are possible at the degeneration points of the Calabi-Yau geometry. We will investigate in detail the H-flux induced superpotential and the corresponding scalar potential at several degeneration points, such as the Calabi-Yau large volume limit, the conifold loci, the Seiberg-Witten points, the strong coupling point and the conformal points. Some emphasis is given to the question whether partial supersymmetry breaking can be realized at those points. We also relate the H-flux induced superpotential to the formalism of gauged N=2 supergravity. Finally we point out the analogies between the Calabi-Yau vacuum structure due to H-fluxes and the attractor formalism of N=2 black holes. 
  The issue of microstate counting for general black holes in D=5, N=2 supergravity coupled to vector multiplets is discussed from various viewpoints. The statistical entropy is computed for the near-extremal case by using the central charge appearing in the asymptotic symmetry algebra of AdS_2. Furthermore, we show that the considered supergravity theory enjoys a duality invariance which connects electrically charged black holes and magnetically charged black strings. The near-horizon geometry of the latter turns out to be AdS_3 x S^2, which allows a microscopic calculation of their entropy using the Brown-Henneaux central charges in Cardy's formula. In both approaches we find perfect agreement between statistical and thermodynamical entropy. 
  The Skyrme model is a classical field theory which has topological soliton solutions. These solitons are candidates for describing nuclei, with an identification between the numbers of solitons and nucleons. We have computed numerically, using two different minimization algorithms, minimum energy configurations for up to 22 solitons. We find, remarkably, that the solutions for seven or more solitons have nucleon density isosurfaces in the form of polyhedra made of hexagons and pentagons. Precisely these structures arise, though at the much larger molecular scale, in the chemistry of carbon shells, where they are known as fullerenes. 
  Modelling space-time foam using a non-critical Liouville-string model for the quantum fluctuations of D branes with recoil, we discuss the issues of momentum and energy conservation in particle propagation and interactions. We argue that momentum should be conserved exactly during propagation and on the average during interactions, but that energy is conserved only on the average during propagation and is in general not conserved during particle interactions, because of changes in the background metric. We discuss the possible modification of the GZK cutoff on high-energy cosmic rays, in the light of this energy non-conservation as well as the possible modification of the usual relativistic momentum-energy relation. 
  Projector equivalences used in the definition of the K-theory of operator algebras are shown to lead to generalizations of the solution generating technique for solitons in NC field theories, which has recently been used in the construction of branes from other branes in B-field backgrounds and in the construction of fluxon solutions of gauge theories. The generalizations involve families of static solutions as well as solutions which depend on euclidean time and interpolate between different configurations. We investigate the physics of these generalizations in the brane-construction as well as the fluxon context. These results can be interpreted in the light of recent discussions on the topology of the configuration space of string fields. 
  We study coupling of noncommutative gauge theories on branes to closed string in the bulk. We derive an expression for the gauge theory operator dual to the bulk graviton, both in bosonic string theory and superstring theory. In either case, we find that the coupling is different from what was expected in the literature when the graviton is polarized in the noncommutative directions. In the case of superstring, the expression for the energy-momentum tensor is consistent with the way the bulk metric appears in the action for the noncommutative gauge theory. We also clarify some aspects of the correspondence between operators in the gauge theory and boundary conditions in the dual gravitational description. 
  We provide the geometric actions for most general N=1 supergravity in two spacetime dimensions. Our construction implies an extension to arbitrary N. This provides a supersymmetrization of any generalized dilaton gravity theory or of any theory with an action being an (essentially) arbitrary function of curvature and torsion.   Technically we proceed as follows: The bosonic part of any of these theories may be characterized by a generically nonlinear Poisson bracket on a three-dimensional target space. In analogy to a given ordinary Lie algebra, we derive all possible N=1 extensions of any of the given Poisson (or W-) algebras. Using the concept of graded Poisson Sigma Models, any extension of the algebra yields a possible supergravity extension of the original theory, local Lorentz and super-diffeomorphism invariance follow by construction. Our procedure automatically restricts the fermionic extension to the minimal one; thus local supersymmetry is realized on-shell. By avoiding a superfield approach we are also able to circumvent in this way the introduction of constraints and their solution. For many well-known dilaton theories different supergravity extensions are derived. In generic cases their field equations are solved explicitly. 
  Strong/weak coupling duality in Chern-Simons supergravity is studied. It is argued that this duality can be regarded as an example of superduality. The use of supergroup techniques for the description of Chern-Simons supergravity greatly facilitates the analysis. 
  We consider a brane world residing in the interior region inside the horizon of the D3-brane. The horizon size then can be interpreted as the compactification size. The macroscopically large size of extra dimensions can be derived from the underlying string theory that has only one physical scale, {\it i.e.,} the string scale. Then, the hierarchy between the string scale and the Planck scale is provided by Ramon-Ramon charge of the D3-brane. This picture also offers a new perspective on various issues associated with the brane world scenarios including the cosmological constant. 
  We propose a form of the effective action of the tachyon and gauge fields for brane-antibrane systems and non-BPS Dp-branes, written in terms of the supercurvature. Kink and vortex solutions with constant infinite gauge field strength reproduce the exact tensions of the lower-dimensional D-branes. We discuss the relation to BSFT and other models in the literature. 
  We consider the massive tricritical Ising model M(4,5) perturbed by the thermal operator phi_{1,3} in a cylindrical geometry and apply integrable boundary conditions, labelled by the Kac labels (r,s), that are natural off-critical perturbations of known conformal boundary conditions. We derive massive thermodynamic Bethe ansatz (TBA) equations for all excitations by solving, in the continuum scaling limit, the TBA functional equation satisfied by the double-row transfer matrices of the A_4 lattice model of Andrews, Baxter and Forrester (ABF) in Regime III. The complete classification of excitations, in terms of (m,n) systems, is precisely the same as at the conformal tricritical point. Our methods also apply on a torus but we first consider (r,s) boundaries on the cylinder because the classification of states is simply related to fermionic representations of single Virasoro characters chi_{r,s}(q). We study the TBA equations analytically and numerically to determine the conformal UV and free particle IR spectra and the connecting massive flows. The TBA equations in Regime IV and massless RG flows are studied in Part II. 
  A class of indecomposable representations of U_q(sl_n) is considered for q an even root of unity (q^h = -1) exhibiting a similar structure as (height h) indecomposable lowest weight Kac-Moody modules associated with a chiral conformal field theory. In particular, U_q(sl_n) counterparts of the Bernard-Felder BRS operators are constructed for n=2,3. For n=2 a pair of dual d_2(h) = h dimensional U_q(sl_2) modules gives rise to a 2h-dimensional indecomposable representation including those studied earlier in the context of tensor product expansions of irreducible representations. For n=3 the interplay between the Poincare'-Birkhoff-Witt and (Lusztig) canonical bases is exploited in the study of d_3(h) = h(h+1)(2h+1)/6 dimensional indecomposable modules and of the corresponding intertwiners. 
  Noncommutative gauge fields (similar to the type that arises in string theory with background B-fields) are constructed for arbitrary nonabelian gauge groups with the help of a map that relates ordinary nonabelian and noncommutative gauge theories (Seiberg-Witten map). As in our previous work we employ Kontsevich's formality and the concept of equivalent star products. As a byproduct we obtain a ``Mini Seiberg-Witten map'' that explicitly relates ordinary abelian and nonabelian gauge fields. 
  We consider a variant of the brane-world model in which the universe is the direct product of a Friedmann, Robertson-Walker (FRW) space and a compact hyperbolic manifold of dimension $d\geq2$. Cosmology in this space is particularly interesting. The dynamical evolution of the space-time leads to the injection of a large entropy into the observable (FRW) universe. The exponential dependence of surface area on distance in hyperbolic geometry makes this initial entropy very large, even if the CHM has relatively small diameter (in fundamental units). This provides an attractive reformulation of the cosmological entropy problem, in which the large entropy is a consequence of the topology, though we would argue that a final solution of the entropy problem requires a dynamical explanation of the topology of spacetime. Nevertheless, it is reassuring that this entropy can be achieved within the holographic limit if the ordinary FRW space is also a compact hyperbolic manifold. In addition, the very large statistical averaging inherent in the collapse of the initial entropy onto the brane acts to smooth out initial inhomogeneities. This smoothing is then sufficient to account for the current homogeneity of the universe. With only mild fine-tuning, the current flatness of the universe can also then be understood. Finally, recent brane-world approaches to the hierarchy problem can be readily realized within this framework. 
  In quantum field theory the parameters of the vacuum action are subject to renormalization group running. In particular, the ``cosmological constant'' is not a constant in a quantum field theory context, still less should be zero. In this paper we continue with previous work, and derive the particle contributions to the running of the cosmological and gravitational constants in the framework of the Standard Model in curved space-time. At higher energies the calculation is performed in a sharp cut off approximation. We assess, in two different frameworks, whether the scaling dependences of the cosmological and gravitational constants spoil primordial nucleosynthesis. Finally, the cosmological implications of the running of the cosmological constant are discussed. 
  We study matrix quantum mechanics at a finite temperature equivalent to one dimensional compactified string theory with vortex (winding) excitations. It is explicitly demonstrated that the states transforming under non-trivial U(N) representations describe various configurations vortices and anti-vortices. For example, for the adjoint representation the Feynman graphs (representing discretized world-sheets) contain two faces with the boundaries wrapping around the compactified target space which is equivalent to a vortex-anti-vortex pair. A technique is developed to calculate partition functions in a given representation for the standard matrix oscillator. It enables us to obtain the partition function in the presence of a vortex-anti-vortex pair in the double scaling limit using an analytical continuation to the upside-down oscillator. The Berezinski-Kosterlitz-Thouless phase transition occurs in a similar way and at the same temperature as in the flat 2D space. A possible generalization of our technique to any dimension of the embedding space is discussed. 
  The low-energy electron-electron scattering potential is derived and discussed for the Maxwell-Chern-Simons model coupled to QED_3 with spontaneous symmetry breaking. One shows that the Higgs mechanism might favour electron-electron bound states. 
  We examine the ultraviolet behaviour of supergravity theories as a function of dimension and number of supercharges. We do so by the computation of one and two-loop physical on-shell four point amplitudes. For maximal supergravity, our computations prove the non-renomalisability of supergravity for $D \geq 6$ (including the maximal D=11 case) and give strong evidence for the existance of a five-loop counterterm in D=4. For type I supergravity our results indicate similar patterns. e shall also explore a remarkable relationship between gravity amplitudes and those of Yang-Mills theories. In many ways gravity calculations discover features which relate to the equivalent Yang-Mills features by a squaring proceedure. 
  We find explicitly the induced graviton propagator on de Sitter branes embedded in various five-dimensional spacetimes; de Sitter branes in AdS and Minkowski space are particular cases. By studying the structure of the momentum-space propagator, we are able to extract interesting physics, much of which is qualitatively different from that of flat branes. We find that 1) there can be a set of graviton-like particles which mediate brane gravity at different scales; 2) localized gravity can exist even on de Sitter branes in Minkowski space; 3) Kaluza-Klein modes also contribute to conventional 4-D gravity for de Sitter branes in AdS; and 4) Newton's constant can vary considerably with scale. We comment on the implications for the effective cosmological constant. 
  We discuss explicit examples of BPS solutions in four-dimensional N=2 supergravity with R^2-interactions. We demonstrate how to construct solutions by iteration. Generically, the presence of higher-curvature interactions leads to non-static spacetime line elements. We comment on the existence of horizons for multi-centered solutions. 
  We study Calabi-Yau manifolds defined over finite fields. These manifolds have parameters, which now also take values in the field and we compute the number of rational points of the manifold as a function of the parameters. The intriguing result is that it is possible to give explicit expressions for the number of rational points in terms of the periods of the holomorphic three-form. We show also, for a one parameter family of quintic threefolds, that the number of rational points of the manifold is closely related to as the number of rational points of the mirror manifold. Our interest is primarily with Calabi-Yau threefolds however we consider also the interesting case of elliptic curves and even the case of a quadric in CP_1 which is a zero dimensional Calabi-Yau manifold. This zero dimensional manifold has trivial dependence on the parameter over C but a not trivial arithmetic structure. 
  We study D-branes of the ${SL(2,\bR)}$ WZW model, and of its discrete orbifolds. Gluing the currents by group automorphisms leads to three types of D-branes: two-dimensional hyperbolic planes ($H_2$), de Sitter branes (dS$_2$), or anti-de-Sitter branes (AdS$_2$). We explain that the dS$_2$ branes are unphysical, because the electric field on their worldvolume is supercritical. By combining ${SL(2,\bR)}$ and SU(2), we exhibit a class of supersymmetric AdS$_2\times S^2$ three-brane worldvolumes, and consider their possible embeddings in the near-horizon region of stringy black holes. We point out the intriguing difference between the induced and the effective geometries of these D-branes, and speculate on its possible significance. 
  A new gravitational interaction of spin 3/2 Nambu-Goldstone(N-G) fermion is constructed, which gives a new framework for the consistent gravitational coupling of spin 3/2 massless field. The action is invariant under a new global supersymmetry. 
  Starting from nonlinear realizations of the partially broken central-charge extended N=4 and N=8 Poincar\'e supersymmetries in D=4, we derive the superfield equations of N=2 and N=4 Born-Infeld theories. The basic objects are the bosonic Goldstone N=2 and N=4 superfields associated with the central charge generators. By construction, the equations are manifestly N=2 and N=4 supersymmetric and enjoy covariance under another nonlinearly realized half of the original supersymmetries. They provide a manifestly worldvolume supersymmetric static-gauge description of D3-branes in D=6 and D=10. For the N=2 case we find, to lowest orders, the equivalence transformation to the standard N=2 Maxwell superfield strength and restore, up to the sixth order, the off-shell N=2 Born-Infeld action with the second hidden N=2 supersymmetry. 
  We investigate the deconfining phase transition in QCD_4 and QED_4 at finite temperature using a perturbative deformation of topological quantum field theory (TQFT). A modified maximal abelian gauge (MAG) is utilized in the analysis. In this case, we can derive the linear potential studying the 2D theory through Parisi-Sourlas (PS) dimensional reduction. The mechanism of deconfining phase transition is proposed. It is geometrical to discuss the thermal effect on the linear potential. All we have to do is to investigate the behavior of topological objects as such instantons and vortices on a cylinder. This is the great advantage of our scenario. This mechanism is also applied in the case of QED_4. The phase structure at the high temperature of QED is investigated using the Coulomb potential on a cylinder. It coincides with the result in the lattice compact U(1) gauge theory. Also, QCD with MAG has the property called the abelian dominance, which enables us to discuss the deconfinement of QCD_4. 
  I propose a general class of space-times whose structure is governed by observer-independent scales of both velocity ($c$) and length (Planck length), and I observe that these space-times can naturally host a modification of FitzGerald-Lorentz contraction such that lengths which in their inertial rest frame are bigger than a "minimum length" are also bigger than the minimum length in all other inertial frames. With an analysis in leading order in the minimum length, I show that this is the case in a specific illustrative example of postulates for Relativity with velocity and length observer-independent scales. 
  A V-duality conjecture for noncommutative open string theories (NCOS) that result from decoupling D-branes in Lorentz-boost related backgrounds was put forward recently in hep-th/0006013. The aim of this paper is to test the Galilean nature of this conjecture in the gravity dual setup. We start with an (F, D3) bound state Lorentz-boosted along one D3-brane direction perpendicular to the F-string, and show that insisting a decoupled NCOS allows only infinitesimal Lorentz boosts. In this way, it is shown that the V-duality relates a family of NCOS by Galileo boosts. Starting with a Lorentz-boosted (D1,D3) bound state, we show that a similar V-duality works for noncommutative Yang-Mills (NCYM) theories as well. In addition, we deduce by a holography argument that the running string tension, as a function of the energy scale, for NCOS (or NCYM) remains unchanged under V-duality. 
  We present a menagerie of solutions to the vacuum Einstein equations in six, eight and ten dimensions. These solutions describe spacetimes which are either locally asymptotically adS or locally asymptotically flat, and which have non-trivial topology. We discuss the global structure of these solutions, and their relevance within the context of M-theory. 
  A new expansion is established for the Green's function of the electromagnetic field in a medium with arbitrary $\epsilon$ and $\mu$. The obtained Born series are shown to consist of two types of interactions - the usual terms (denoted $\cal P$) that appear in the Lifshitz theory combined with a new kind of terms (which we denote by $\cal Q$) associated with the changes in the permeability of the medium. Within this framework the case of uniform velocity of light ($\epsilon\mu={\rm const}$) is studied. We obtain expressions for the Casimir energy density and the first non-vanishing contribution is manipulated to a simplified form. For (arbitrary) spherically symmetric $\mu$ we obtain a simple expression for the electromagnetic energy density, and as an example we obtain from it the Casimir energy of a dielectric-diamagnetic ball. It seems that the technique presented can be applied to a variety of problems directly, without expanding the eigenmodes of the problem and using boundary condition considerations. 
  We derive the partition function of N=4 supersymmetric Yang-Mills theory on orbifold-$T^4/{\bf Z}_2$. In classical geometry, K3 surface is constructed from the orbifold-$T^4/{\bf Z}_2$. Along the same way as the orbifold construction, we construct the partition function of K3 surface from orbifold-$T^4/{\bf Z}_2$. The partition function is given by the product of the contribution of the untwisted sector of $T^4/{\bf Z}_2$, and that of the twisted sector of $T^4/{\bf Z}_2$ i.e., ${\cal O}(-2)$ curve blow-up formula. 
  In this paper we calculate the effects produced by temperature in the renormalized vaccum expectation value of the square of the massless scalar field in the pointlike global monopole spacetime. In order to develop this calculation, we had to construct the Euclidean thermal Green function associated with this field in this background. We also calculate the high-temperature limit for the thermal average of the zero-zero component of the energy-momentum tensor. 
  We construct a Heisenberg-like algebra for the one dimensional infinite square-well potential in quantum mechanics. The ladder operators are realized in terms of physical operators of the system as in the harmonic oscillator algebra. These physical operators are obtained with the help of variables used in a recently developed non commutative differential calculus. This \textquotedblleft square-well algebra\textquotedblright is an example of an algebra in a large class of generalized Heisenberg algebras recently constructed. This class of algebras also contains $q$-oscillators as a particular case. We also discuss the physical content of this large class of algebras. 
  The influence of vector fields on the origin of a classical space in quantum cosmologies and on the possible compactification process in multidimensional gravity is investigated. It is shown that all general features of the transition between classical and quantum regimes of the evolution can be obtained within the simplest Bianchi-I model for arbitrary number of dimensions. It is shown that the classical space appears when the horizon size reaches the smallest of characteristic scales (the characteristic scale of inhomogeneity or a scale associated with vector fields). In multidimensional case the presence of vector fields completely removes the initial stage of the compactification process which takes place in the case of vacuum models. 
  We study cosmological solutions in the dilatonic brane world models. The effective four-dimensional equations on the brane are analyzed for the models with one positive tension brane and two branes with tensions of opposite signs. Just as in the non-dilatonic brane case, the conventional Friedmann equations of the four-dimensional universe are reproduced to the leading order in matter energy density for the model with one brane and the introduction of a radion potential is required in order to reproduce the Friedmann equations with the correct sign for the model with two branes. 
  We study the dynamics of Green-Schwarz superstring on the gravitational wave background corresponding to the Matrix string theory and the supersymmetry transformation rules of the superstring. The dynamics is obtained in the light-cone formulation and is shown to agree with that derived from the Matrix string theory. The supersymmetry structure has the corrections due to the effect of the background and is identified with that of the low energy one-loop effective action of Matrix string theory in two superstring background in the weak string coupling limit. 
  We discuss some issues about the holographic interpretation of the compact Randall-Sundrum model, which is conjectured to be dual to a 4d field theory with non-linearly realized conformal symmetry. We make several checks of this conjecture. In particular, we show that the radion couples conformally to a background 4d metric. We also discuss the interpretation of the Goldberger-Wise mechanism for stabilizing the radion. We consider situations where the electroweak breaking stabilizes the radion and we discuss the issue of natural conservation of flavor quantum numbers. 
  After reviewing how M-theory subsumes string theory, I report on some new and interesting developments, focusing on the ``brane-world'': circumventing no-go theorems for supersymmetric brane-worlds, complementarity of the Maldacena and Randall-Sundrum pictures; self-tuning of the cosmological constant. I conclude with the top ten unsolved problems. 
  We study the confinement scenario in N=2 supersymmetric SU(2) gauge theory near the monopole point upon breaking of N=2 supersymmetry by the adjoint matter mass term. We confirm claims made previously that the Abrikosov-Nielsen-Olesen string near the monopole point fails to be a BPS state once next-to-leading corrections in the adjoint mass parameter taken into account. Our results shows that type I superconductivity arises upon monopole condensation. This conclusion allows us to make qualitative predictions on the structure of the hadron mass spectrum near the monopole point. 
  Assuming that around the tachyon vacuum the kinetic term of cubic open string field theory is made purely of ghost operators we are led to gauge invariant actions which manifestly implement the absence of open string dynamics around this vacuum. We test this proposal by showing the existence of lump solutions of arbitrary codimension in this string field theory. The key ingredients in this analysis are certain assumptions about the analyticity properties of tachyon Green's functions. With the help of some further assumptions about the properties of these Green's functions, we also calculate the ratios of tensions of lump solutions of different dimensions. The result is in perfect agreement with the known answers for the ratios of tensions of D-branes of different dimensions. 
  We study a nonminimal massive scalar field in a 2-dimensional black hole spacetime. We consider the black hole which is the solution of the 2d dilaton gravity derived from string-theoretical models. We found an explicit solution in a closed form for all modes and the Green function of the scalar field with an arbitrary mass and a nonminimal coupling to the curvature. Greybody factors, the Hawking radiation, and $ < \phi^2 >^{ren} $ are calculated explicitly for this exactly solvable model. 
  The cosmological constant problem arises at the intersection between general relativity and quantum field theory, and is regarded as a fundamental problem in modern physics. In this paper we describe the historical and conceptual origin of the cosmological constant problem which is intimately connected to the vacuum concept in quantum field theory. We critically discuss how the problem rests on the notion of physical real vacuum energy, and which relations between general relativity and quantum field theory are assumed in order to make the problem well-defined. 
  We show that N=2 supersymmetric Maxwell-Chern-Simons-Higgs systems in three dimension can be realized as gauge theories on a D2-brane in D8-branes background with a non-zero B-field. We reproduce a potential of Coulomb branch of the Chern-Simons theory as a potential of a D2-brane in a classical D8-brane solution and show that each Coulomb vacuum is realized by a D2-brane stabilized in the bulk at a certain distance from D8-branes. 
  A solution of the 6D gravitational model, which has the pole configuration, is found. The vacuum setting is done by the 6D Higgs potential and the solution is for a family of the vacuum and boundary parameters. The boundary condition is solved by the $1/k^2$-expansion (thin pole expansion) where $1/k$ is the {\it thickness} of the pole. The obtained analytic solution is checked by the numerical method. This is a dimensional reduction model from 6D to 4D by use of the soliton solution (brane world). It is regarded as a higher dimensional version of the Randall-Sundrum 5D model. The mass hierarchy is examined. Especially the {\it geometrical see-saw} mass relation is obtained. Some physical quantities in 4D world such as the Planck mass, the cosmological constant, and fermion masses are focussed. Comparison with the 5D model is made. 
  In the framework of a (2+1)-dimensional P-even massive Gross-Neveu model, an external magnetic field is shown to induce a parity breaking first order phase transition. Possibility of applying the results obtained to description of magnetic phase transitions in high-temperature superconductors is discussed. 
  Using two WZNW theories for Lie algebras $g$ and $h, h\subset g,$ we construct the associative quotient algebra which includes a class of $g/h$ coset primary fields and currents. 
  We study the high energy thermodynamics of Little String Theory, using its holographic description. This leads to the entropy-energy relation $S=\beta_H E+\alpha\log E+O(1/E)$. We compute $\alpha$ and show that it is negative; as a consequence, the high energy thermodynamics is unstable. We exhibit a mode localized near the horizon of the black brane, which has winding number one around Euclidean time and a mass that vanishes at large $E$ (or $\beta\to\beta_H$). We argue that the high temperature phase of the theory involves condensation of this mode. 
  We study solutions at the minima of scalar field potentials for Moyal spaces and torii in the large non-commutativity and interprete these solitons in terms of non-BPS D-branes of string theory. We derive a mass spectrum formula linking different D-branes together on quantum torii and suggest that it describes general systems of D-brane bound states extending the D2-D0 one. Then we propose a shape for the effective potential approaching these quasi-stable bound states. We give the gauge symmetries of these systems of branes and show that they depend on the quantum torii representations. 
  We discuss how cosmic strings can be used to mine energy from black holes. A string attached to the black hole gives rise to an additional channel for the energy release. It is demonstrated that when a string crosses the event horizon, its transverse degrees of freedom are thermally excited and thermal string perturbations propagate along the string to infinity. The internal metric induced on the 2D worldsheet of the static string crossing the horizon describes a 2D black hole. For this reason thermal radiation of string excitations propagating along the string can be interpreted as Hawking radiation of the 2D black hole. It is shown that the rate of energy emission through the string channel is of the same order of magnitude as the bulk radiation of the black hole. Thus, for N strings attached to the black hole the efficiency of string channels is increased by factor N. We discuss restrictions on N which exist because of the finite thickness of strings, the gravitational backreaction and quantum fluctuations. Our conclusion is that the energy emission rate by strings can be increased as compared to the standard emission in the bulk by the factor 10^3 for GUT strings and up to the factor 10^{31} for electroweak strings. 
  This study proposes that the longstanding problems of quantum chromodynamics (QCD) as an SU(3)_C gauge theory, the confinement mechanism and \Theta vacuum, can be resolved by dynamical spontaneous symmetry breaking (DSSB) through the condensation of singlet gluons and quantum nucleardynamics (QND) as an SU(2)_N x U(1)_Z gauge theory is produced. The confinement mechanism is the result of massive gluons and the Yukawa potential provides hadron formation. The evidences for the breaking of discrete symmetries (C, P, T, CP) during DSSB appear explicitly: baryons and mesons without their parity partners, the conservation of vector current and the partial conservation of the axial vector current, the baryon asymmetry \delta_B \simeq 10^{-10}, and the neutron electric dipole moment \Theta < 10^{-9}. Hadron mass generation mechanism is suggested in terms of DSSB due to the \Theta vacuum. 
  We show that the p-dimensional noncommutative Yang--Mills model corresponding to a (p-1)-brane allows solutions which correspond to lower branes. This may be interpreted as the Morita equivalence of noncommutative planes of various dimensions. 
  An introduction to supersymmetric (SUSY) solutions defined on the product of Ricci-flat spaces in D= 11 supergravity is presented. The paper contains some background information: (i) decomposition relations for SUSY equations and (ii) 2^{-k}-splitting theorem that explains the appearance of N = 2^{-k} fractional supersymmetries. Examples of M2 and M5 branes on the product of two Ricci-flat spaces are considered and formulae for (fractional) numbers of unbroken SUSY are obtained. 
  Using the zeta function regularization method we calculate the ground state energy of scalar massive field inside a spherical region in the space-time of a point-like global monopole. Two cases are investigated: (i) We calculate the Casimir energy inside a sphere of radius $R$ and make an analytical analysis about it. We observe that this energy may be positive or negative depending on metric coefficient $\alpha$ and non-conformal coupling $\xi$. In the limit $R\to\infty$ we found a zero result. (ii) In the second model we surround the monopole by additional sphere of radius $r_0<R$ and consider scalar field confined in the region between these two spheres. In the latter, the ground state energy presents an additional contribution due to boundary at $r_0$ which is divergent for small radius. Additional comments about renormalization are considered. 
  The smallness of the cosmological constant is one of the basic problems in particle physics and cosmology. Various attempts have been made to explain this mystery, but no satisfactory solution has been found yet. The appearance of extra dimensions in the framework of brane world systems seems to provide some new ideas to address this problem from a different point of view. We shall discuss some of these new approaches and see whether or not they lead to an improvement of the situation. We shall conclude that we are still far from a solution of the problem. 
  We construct exact soliton solutions to the Chern-Simons-Higgs system in noncommutative space, for non-relativistic and relativistic models. In both cases we find regular vortex-like solutions to the BPS equations which approach the ordinary selfdual non-topological and topological solitons when the noncommutative parameter $\theta$ goes to zero. 
  In the thesis we analize different problems related to the supersymmetric extension of the Dirac-Born-Infeld action. In chapter 2 we introduce the DBI action and show how it appears in string theory, we discuss also it's connection with Dp-branes. Chapter 3 is a self contained introduction to supersymmetry, with emphasis on BPS states. In chapter 4 we construct the N=2 supersymmetric extension of the Born-Infeld-Higgs in three space-time dimensions and discuss it's BPS states and Bogomol'nyi bounds. In chapter 5 we construct the N=1 supersymmetric extension of the non-abelian Born-Infeld theory in four space-time dimensions. Chapter 6 deals with the analisis of BPS and non-BPS solutions of the Dirac-Born-Infeld action and their interpretation in superstring theory. Chapter 7 is devoted to the conclusions. Three appendix complete the work. 
  We develop a systematic perturbative expansion and compute the one-loop two-points, three-points and four-points correlation functions in a non-commutative version of the U(N) Wess-Zumino-Witten model in different regimes of the $\theta$-parameter showing in the first case a kind of phase transition around the value $\theta_c = \frac{\sqrt{p^2 + 4 m^2}}{\Lambda^2 p}$, where $\Lambda$ is a ultraviolet cut-off in a Schwinger regularization scheme. As a by-product we obtain the functions of the renormalization group, showing they are essentially the same as in the commutative case but applied to the whole U(N) fields; in particular there exists a critical point where they are null, in agreement with a recent background field computation of the beta-function, and the anomalous dimension of the Lie algebra-valued field operator agrees with the current algebra prediction. The non-renormalization of the level $k$ is explicitly verified from the four-points correlator, where a left-right non-invariant counter-term is needed to render finite the theory, that it is however null on-shell. These results give support to the equivalence of this model with the commutative one. 
  We extend a non local and non covariant version of the Thirring model in order to describe a many-body system with spin-flipping interactions By introducing a model with two fermion species we are able to avoid the use of non abelian bosonization which is needed in a previous approach. We obtain a bosonized expression for the partition function, describing the dynamics of the collective modes of this system. By using the self-consistent harmonic approximation we found a formula for the gap of the spin-charge excitations as functional of arbitrary electron-electron potentials. 
  We study low-energy propagation modes on string network lattice. Specifically, we consider an infinite two-dimensional regular hexagonal string network and analyze the low frequency propagation modes on it. The fluctuation modes tangent to the two-dimensional plane respect the spatial rotational symmetry on the plane, and are described by Maxwell theory. The gauge symmetry comes from the marginal deformation of changing the sizes of the loops of the lattice. The effective Lorentz symmetry respected at low energy will be violated at high energy. 
  We consider the polarization of unstable type IIB D0-branes in the presence of a background five-form field strength. This phenomenon is studied from the point of view of the leading terms in the non-abelian Born Infeld action of the unstable D0-branes. The equations have SO(4) invariant solutions describing a non-commutative 3-sphere, which becomes a classical 3-sphere in the large N limit. We discuss the interpretation of these solutions as spherical D3-branes. The tachyon plays a tantalizingly geometrical role in relating the fuzzy S^3 geometry to that of a fuzzy S^4. 
  We investigate an action which resembles the effective action of brane-antibrane system derived from boundary string field theory. We find that the action has smooth vortex solutions which saturate the Bogomol'nyi bound. 
  AdS/CFT induced quantum dilatonic brane-world where 4d boundary is flat or de Sitter (inflationary) or Anti-de Sitter brane is considered. The classical brane tension is fixed but boundary QFT produces the effective brane tension via the account of corresponding conformal anomaly induced effective action. This results in inducing of brane-worlds in accordance with AdS/CFT set-up as warped compactification. The explicit, independent construction of quantum induced dilatonic brane-worlds in two frames: string and Einstein one is done. It is demonstrated their complete equivalency for all quantum cosmological brane-worlds under discussion, including several examples of classical brane-world black holes. This is different from quantum corrected 4d dilatonic gravity where de Sitter solution exists in Einstein but not in Jordan (string) frame. The role of quantum corrections on massive graviton perturbations around Anti-de Sitter brane is briefly discussed. 
  In $D$-dimensional dilaton gravitational model stationary Schwarzschild type solutions (centrally symmetrical solutions in vacuum) are obtained. They contain the two-parameter set of solutions: the first parameter is proportional to a central mass and the second one describes intensity of the dilaton field. If the latter parameter equals zero we have pure Schwarzschild solution with the black hole and constant dilaton field. But when the mentioned parameter is non-zero and the dilaton is non-constant, the black hole horizon vanishes. 
  A generalization of the two-dimensional Yang-Mills and generalized Yang-Mills theory is introduced in which the building B-F theory is nonlocal in the auxiliary field. The classical and quantum properties of this nonlocal generalization are investigated and it is shown that for large gauge groups, there exist a simple correspondence between the properties a nonlocal theory and its corresponding local theory. 
  We compute the quark-antiquark potential employing the continuum action for QCD-like random lattice strings proposed by Siegel. The model leads to a potential similar to those obtained from Nambu-Goto string theory but has some modifications which we interpret as velocity dependent contributions. We also propose to add extra terms in the action which lead to physically interesting propagators for partons for the infrared region. 
  We provide the explicit gauging of all the SU(2,1) isometries of one N=2 supergravity hypermultiplet, which spans SU(2,1)/U(2) coset space parameterized in terms of two complex projective coordinate fields z_1 and z_2. We derive the full, explicit Killing prepotential that specifies the most general superpotential. As an application we consider the supersymmetric flow (renormalization group) equations for: (i) the flow from a null singularity to the flat, supersymmetric space-time and (ii) the flow that violates c-theorem with the superpotential crossing zero. 
  We study a cosmic string solution of an N=1-supersymmetric version of the Cremmer-Scherk-Kalb-Ramond (CSKR) Lagrangian coupled to a vector superfield by means of a topological mass term. The 2-form gauge potential is proposed to couple non-minimally to matter, here described by a chiral scalar superfield. The important outcome is that supersymmetry is kept exact both in the core and in the exterior region of the string. We contemplate the bosonic configurations and it can be checked that the solutions saturate the Bogomolnyi bound. A glimpse on the fermionic zero modes is also given. 
  A careful ab initio construction of the finite-mass (1/2,1/2) representation space of the Lorentz group reveals it to be a spin-parity multiplet. In general, it does not lend itself to a single-spin interpretation. We find that the (1/2,1/2) representation space for massive particles naturally bifurcates into a triplet and a singlet of opposite relative intrinsic parties. The text-book separation into spin one and spin zero states occurs only for certain limited kinematical settings. We construct a wave equation for the (1/2,1/2) multiplet, and show that the particles and antiparticles in this representation space do not carry a definite spin but only a definite relative intrinsic parity. In general, both spin one and spin zero are covariantly inseparable inhabitants of massive vector fields. This last observation suggests that scalar particles, such as the Higgs, are natural inhabitants of massive (1/2,1/2) representation space. 
  We argue that type II string theories contain unstable NS4 branes, which descend from a conjectured unstable M4 brane of M-theory. Assuming that an M2 brane can arise in M5 brane/anti-brane annihilation, the unstable M4 brane, and also an unstable M3 brane, must exist as sphalerons. We compare the tensions of the unstable NS4 branes, M4 brane, and related type II unstable D-branes, and present 11d supergravity solutions for unstable Mp branes for all p. We study the Z_2 gauge symmetry on the worldvolume of unstable branes, and argue that it can never be unbroken in the presence of lower brane charge. 
  We construct and study a matrix model that describes two dimensional string theory in the Euclidean black hole background. A conjecture of V. Fateev, A. and Al. Zamolodchikov, relating the black hole background to condensation of vortices (winding modes around Euclidean time) plays an important role in the construction. We use the matrix model to study quantum corrections to the thermodynamics of two dimensional black holes. 
  I review the black hole uniqueness theorem and the no hair theorems established for physical black hole stationary states by the early 80'. This review presents the original and decisive work of Carter, Robinson, Mazur and Bunting on the problem of no bifurcation and uniqueness of physical black holes. Its original version was written only few years after my proof of the Kerr-Newman et al. black hole uniqueness theorem has appeared in print. The proof of the black hole uniqueness theorem relies heavily on the positivity properties of nonlinear sigma models on the Riemannian noncompact symmetric spaces with negative sectional curvature. It is hoped that the first hand description of the original developments leading to our current understanding of the black hole uniqueness will be found useful to all interested in the subject. 
  We discuss the field theory interpretation (via holographic duality) of some recently-discovered string theory solutions with varying flux, focusing on four dimensional theories with N=2 supersymmetry and with N=1 supersymmetry which arise as the near-horizon limits of "fractional D3-branes". We argue that in the N=2 case the best interpretation of the varying flux in field theory is via a Higgs mechanism reducing the rank of the gauge group, and that there is no need to invoke a duality to explain the varying flux in this case. We discuss why a similar interpretation does not seem to apply to the N=1 case of Klebanov and Strassler, which was interpreted as a "duality cascade". However, we suggest that it might apply to different vacua of the same theory, such as the one constructed by Pando Zayas and Tseytlin. 
  We test the validity of the Siegel gauge condition for the lump solution of cubic open bosonic string field theory by checking the equations of motion of the string field components outside the Siegel gauge. At level (3,6) approximation, the linear and quadratic terms of the equations of motion of these fields are found to cancel within about 20%. 
  We show, with a 2-dimensional example, that the low energy effective action which describes the physics of a single D-brane is compatible with T-duality whenever the corresponding U(N) non-abelian action is form-invariant under the non-commutative Seiberg-Witten transformations. 
  We discuss the supergravity couplings of noncommutative D-branes by considering the disk amplitudes with one closed string insertion. The result confirms a recent proposal for the general form of the noncommutative Yang-Mills operators coupling to the massless closed string modes. The construction involves smearing Yang-Mills field variables along an open Wilson line. For multiple D-branes interacting with background supergravity fields, this prescription reduces, in the B=0 limit, to the ``symmetrized trace'' prescription, and the supergravity fields are seen to be functionals of the nonabelian scalar fields on the branes. 
  We consider N=1,2 superconformal mechanics in 0+1 dimensions and show that if the Hamiltonian is invertible the superconformal generators can be used to construct half of the super Virasoro algebra. The full algebra can be derived if the special conformal generator is also invertible. The generators are quantized and a general prescription is given for the construction of the N=1 algebra independently of the specific details of the superconformal mechanics provided that in addition its quadratic Casimir operator vanishes. 
  We probe the effective worldvolume theory of a set of coinciding D-branes by switching on constant electro-magnetic fields on them. The comparison of the mass spectrum predicted by string theory with the mass spectrum obtained from the effective action provides insights in the structure of the effective theory. 
  We first obtain the analogue of Vaidya's solution on the brane for studying the collapse of null fluid onto a flat Minkowski cavity on the brane. Since the back-reaction of the bulk onto the brane is supposed to strengthen gravity on the brane, it would favour formation of black hole as against naked singularity. That is the parameter window in the initial data set giving rise to naked singularity in the 4D Vaidya case would now get partially covered. 
  We derive type II supergravity solutions corresponding to space-filling regular and fractional Dp branes on (9-p)-dimensional conical transverse spaces. Fractional Dp-branes are wrapped D(p+2)-branes; therefore, our solutions exist only if the base of the cone has a non-vanishing Betti number b_2. We also consider 11-dimensional SUGRA solutions corresponding to regular and fractional M2 branes on 8-dimensional cones whose base has a non-vanishing b_3. (In this case a fractional M2-brane is an M5-brane wrapped over a 3-cycle.) We discuss the gauge theory intepretation of these solutions, as well as of the solutions constructed by Cvetic et al. in hep-th/0011023 and hep-th/0012011. 
  Superconvergence relations for the transverse gauge field propagator can be used in order to show that the corresponding gauge quanta are not elements of the physical state space, as defined by the BRST algebra. With a given gauge group, these relations are valid for a limited region in the number of matter fields, indicating a phase transition at the boundary. In the case of SUSY gauge theories with matter fields in the fundamental representation, the results predicted by superconvergence can be compared directly with those obtained on the basis of duality and the conformal algebra. There is exact agreement. 
  We propose modified action which is equivalent to N=1 Green-Schwarz superstring and which allows one to realize the supplementation trick [26]. Fermionic first and second class constraints are covariantly separated, the first class constraints (1CC) turn out to be irreducible. We discuss also equations of motion in the covariant gauge for $\kappa$-symmetry. It is shown how the usual Fock space picture can be obtained in this gauge. 
  We show how the level matching condition in six dimensional, abelian and supersymmetric orbifolds of the E_8 x E_8 heterotic string can be given equivalently in terms of fractional gauge and gravitational instanton numbers. This relation is used to restate the classification of the orbifolds in terms of flat bundles away from the orbifold singularities under the constraint of the level matching condition. In an outlook these results are applied to Kaluza-Klein monopoles of the heterotic string on S^1 in Wilson line backgrounds. 
  A massless spinor field is quantized in the background of a singular static magnetic vortex in 2+1-dimensional space-time. The method of self-adjoint extensions is employed to define the most general set of physically acceptable boundary conditions at the location of the vortex. Under these conditions, all effects of polarization of the massless fermionic vacuum in the vortex background are determined. Absence of anomaly is demonstrated, and patterns of both parity and chiral symmetry breaking are discussed. 
  A T-dualized selfdual inspired formulation of massive vector fields coupled to arbitrary matter is generated; subsequently its perturbative series modeling a spontaneously broken gauge theory is analyzed. The new Feynman rules and external line factors are chirally minimized in the sense that only one type of spin index occurs in the rules. Several processes are examined in detail and the cross-sections formulated in this approach. A double line formulation of the Lorentz algebra for Feynman diagrams is produced in this formalism, similar to color ordering, which follows from a spin ordering of the Feynman rules. The new double line formalism leads to further minimization of gauge invariant scattering in perturbation theory. The dualized electroweak model is also generated. 
  In this lecture, we review the derivation of the holographic renormalization group given in hep-th/9912012. Some extra background material is included, and various applications are discussed. 
  We present a self-tuning solution of the cosmological constant problem with one extra dimension which is curved with a warp factor. To separate out the extra dimension and to have a self-tuning solution, a three index antisymmetric tensor field is introduced with the $1/H^2$ term in the Lagrangian. The standard model fields are located at the $y=0$ brane. The existence \cite{kklcc} of the self-tuning solution (which results without any fine tuning among parameters in the Lagrangian) is crucial to obtain a vanishing cosmological constant in a 4D effective theory. The de Sitter and anti de Sitter space solutions are possible. The de Sitter space solutions have horizons. Restricting to the spaces which contain the $y=0$ brane, the vanishing cosmological constant is chosen in the most probable universe. For this interpretaion to be valid, the existence of the self-tuning solution is crucial in view of the phase transitions. In this paper, we show explicitly a solution in case the brane tension shifts from one to another value. We also discuss the case with the $H^2$ term which leads to one-fine-tuning solutions at most. 
  We discuss Holographic Renormalization Group equations in the presence of fermions and form fields in the bulk. The existence of a holographically dual quantum field theory for a given bulk gravity theory imposes consistency conditions on the ranks of the form fields, the fermion - form field couplings, and leads to a novel Ward identity. 
  The first part of this paper is a review of the author's work with S. Bahcall which gave an elementary derivation of the Chern Simons description of the Quantum Hall effect for filling fraction $1/n$. The notation has been modernized to conform with standard gauge theory conventions.   In the second part arguments are given to support the claim that abelian non-commutative Chern Simons theory at level $n$ is exactly equivalent to the Laughlin theory at filling fraction $1/n$. The theory may also be formulated as a matrix theory similar to that describing D0-branes in string theory. Finally it can also be thought of as the quantum theory of mappings between two non-commutative spaces, the first being the target space and the second being the base space. 
  A very simple mechanism is proposed that stabilizes the orbifold geometry within the context of the Randall-Sundrum proposal for solving the hierarchy problem. The electroweak TeV scale is generated from the Planck scale by spontaneous breaking of the orbifold symmetry 
  The classical equations of motion of Maxwell and Born-Infeld theories are known to be invariant under a duality symmetry acting on the field strengths. We implement the SL(2,Z) duality in these theories as linear but non-local transformations on the potentials. We show that the action and the partition function in the Hamiltonian formalism are modular invariant in any gauge. For the Born-Infeld theory we find that the longitudinal part of the fields have to be complexified. 
  A coordinate-free definition for Wick-type symbols is given for symplectic manifolds by means of the Fedosov procedure. The main ingredient of this approach is a bilinear symmetric form defined on the complexified tangent bundle of the symplectic manifold and subject to some set of algebraic and differential conditions. It is precisely the structure which describes a deviation of the Wick-type star-product from the Weyl one in the first order in the deformation parameter. The geometry of the symplectic manifolds equipped by such a bilinear form is explored and a certain analogue of the Newlander-Nirenberg theorem is presented. The 2-form is explicitly identified which cohomological class coincides with the Fedosov class of the Wick-type star-product. For the particular case of K\"ahler manifold this class is shown to be proportional to the Chern class of a complex manifold. We also show that the symbol construction admits canonical superextension, which can be thought of as the Wick-type deformation of the exterior algebra of differential forms on the base (even) manifold. Possible applications of the deformed superalgebra to the noncommutative field theory and strings are discussed. 
  We argue that a system of interacting D-branes, generalizing a recent proposal, can be modelled as a Quantum Hall fluid. We show that tachyon condensation in such a system is equivalent to one particle tunnelling. In a conformal field theory effective description, that induces a transition from a theory with central charge c=2 to a theory with c=3/2, with a corresponding symmetry enhancement. 
  Sigma models arise frequently in particle physics and condensed-matter physics as low-energy effective theories. In this paper I compute the exact free energy at any temperature in two hierarchies of integrable sigma models in two dimensions. These theories, the SU(N)/SO(N) and O(2P)/O(P) x O(P) models, are asymptotically free and exhibit charge fractionalization. When the instanton coupling theta=pi, they flow to the SU(N)_1 and O(2P)_1 conformal field theories, respectively. I also generalize the free energy computation to massive and massless perturbations of the coset conformal field theories SU(N)_k/SO(N)_{2k} and O(2P)_k/O(P)_k x O(P)_k. 
  We derive Virasoro constraints for the zero momentum part of the QCD-like partition functions in the sector of topological charge $\nu$. The constraints depend on the topological charge only through the combination $N_f +\beta\nu/2$ where the value of the Dyson index $\beta$ is determined by the reality type of the fermions. This duality between flavor and topology is inherited by the small-mass expansion of the partition function and {\em all} spectral sum-rules of inverse powers of the eigenvalues of the Dirac operator. For the special case $\beta=2$ but arbitrary topological charge the Virasoro constraints are solved uniquely by a Generalized Kontsevich Model with potential ${\cal V}(X) = 1/X$. 
  We review the status and present range of applications of the ``string-inspired'' approach to perturbative quantum field theory. This formalism offers the possibility of computing effective actions and S-matrix elements in a way which is similar in spirit to string perturbation theory, and bypasses much of the apparatus of standard second-quantized field theory. Its development was initiated by Bern and Kosower, originally with the aim of simplifying the calculation of scattering amplitudes in quantum chromodynamics and quantum gravity. We give a short account of the original derivation of the Bern-Kosower rules from string theory. Strassler's alternative approach in terms of first-quantized particle path integrals is then used to generalize the formalism to more general field theories, and, in the abelian case, also to higher loop orders. A considerable number of sample calculations are presented in detail, with an emphasis on quantum electrodynamics. 
  In relation to the superspace modifications of 11D supergeometry required to describe the M-theory low-energy effective action, we present an analysis of infinitesimal supergravity fluctuations about the flat superspace limit. Our investigation confirms Howe's interpretation of our previous Bianchi identity analysis. However, the analysis also shows that should 11D supergravity obey the rules of other off-shell supergravity theories, the complete M-theory corrections will necessarily excite our previously anticipated spin-1/2 engineering dimension-1/2 spinor auxiliary multiplet superfield. The analysis of fluctuations yields more evidence that Howe's 1997 theorem is specious when applied to Poincar\' e supergravity or 11D supergravity/M-theory. We end by commenting upon recent advances in this area. 
  A simple five-dimensional brane world model is proposed, motivated by M-theory compactified on a six-dimensional manifold of small radius and an $S^1/Z_2$ of large radius. We include a leading-order higher curvature correction to the tree-level bulk action since in brane world scenarios the curvature scale in the bulk may be comparable to the five-dimensional Planck scale and, thus, higher curvature corrections may become important. As a tractable model of the bulk theory we consider pure gravity including a $(Ricci-scalar)^4$-correction to the Einstein-Hilbert action. In this model theory, after a conformal transformation to the Einstein frame, we numerically obtain static solutions, each of which consists of a positive tension brane and a negative tension brane. For these solutions, we obtain two relations between the warp factor and the brane tensions. The existence of these relations implies that, contrary to the original Randall-Sundrum model, the so called radion is no longer a zero mode. We conclude that the tension of our brane should be negative and that fine-tuning of the tension of both branes is necessary for a large warp factor to explain the large hierarchy between the Planck scale and the electroweak scale. 
  We discuss origin of equivalence between noncommutative and ordinary Yang-Mills from point of view of string theory. Working in BRST/Hamiltonian framework first we investigate string model in the decoupling limit and show that change of variables and applying the conversion of constraints of decoupled string theory gives commuting coordinates on the D-brane. Also, we discuss algebra of constraints in general case and show the ways of having commutative coordinates without going to decoupling limit. It could be argued that noncommutative string in B-field is equivalent to the commutative model. We investigate the case of the membrane ending on the M-5-brane in constant C-field and discuss noncommutative/commutative equivalence in this case. 
  The renormalization of effective potentials for the noncommutative scalar field theory at high temperature are investigated to the two-loop approximation. The Feynman diagrams in evaluating the effective potential may be classified into two types: the planar diagrams and nonplanar diagrams. The nonplanar diagrams, which depend on the parameter of noncommutativity, do not appear in the one-loop potential. Despite their appearance in the two-loop level, they do not have an inclination to restore the symmetry breaking in the tree level, in contrast to the planar diagrams. This phenomenon is explained as a consequence of the drastic reduction of the degrees of freedom in the nonplanar diagrams when the thermal wavelength is smaller than the noncommutativity scale. Our results show that the nonplanar two-loop contribution to the effective potential can be neglected in comparsion with that from the planar diagrams. 
  We consider a hybrid of nonlinear sigma models in which two complex projective spaces are coupled with each other under a duality. We study the large N effective action in 1+1 dimensions. We find that some of the dynamically generated gauge bosons acquire radiatively induced masses which, however, vanish along the self-dual points where the two couplings characterizing each complex projective space coincide. These points correspond to the target space of the Grassmann manifold along which the gauge symmetry is enhanced, and the theory favors the non-Abelian ultraviolet fixed point. 
  A solution of Einstein equations is obtained for our four-dimensional world as an intersection of a wall and a string-like defect in seven-dimensional spacetime with a negative cosmological constant. A matter energy-momentum tensor localized on the wall and on the string is needed. A single massless graviton is found and is localized around the intersection. The leading correction to the gravitational Newton potential from massive spin 2 graviton is found to be almost identical to that of a wall in five dimensions, contrary to the case of a string in six dimensions. The generalization to the intersection of a string and $n$ orthogonally intersecting walls is also obtained and a similar result is found for the gravitational potential. 
  We study supergravity solutions representing D3-branes with transverse 6-space having R x S^2 x S^3 topology. We consider regular and fractional D3-branes on a natural one-parameter extensions of the standard Calabi-Yau metrics on the singular and resolved conifolds. After imposing a Z_2 identification on an angular coordinate these generalized "6-d conifolds" are nonsingular spaces. The backreaction of D3-branes creates a curvature singularity that coincides with a horizon. In the presence of fractional D3-branes the solutions are similar to the original ones in hep-th/0002159, hep-th/0010088: the metric has a naked repulson-type singularity located behind the radius where the 5-form flux vanishes. The semiclassical behavior of the Wilson loop suggests that the corresponding gauge theory duals are confining. 
  The relativistic generalization of the Chaplygin gas, put forward by Jackiw and Polychronakos, is derived in Duval's Kaluza-Klein framework, using a universal quadratic Lagrangian. Our framework yields a simplified proof of the field-dependent Poincar\'e symmetry Our action is related to the usual Nambu-Goto action [world volume] of $d$-branes in the same way as the Polyakov and the Nambu action are in strings theory. 
  The recently found non-critical open string theories is reviewed. These open strings, noncommutative open string theories (NCOS), arise as consistent quantum theories describing the low energy theory of D-branes in a background electric B-field in the critical limit. Focusing on the D3-brane case, we construct the most general (3+1) NCOS, which is described by four parameters. We study S and T -dualities of these theories and argue the existence of a U-duality group. 
  We study the general form of the noncommutative associative product (the star-product) on the Grassman algebra; the star-product is treated as a deformation of the usual "pointwise" product. We show that up to a similarity transformation, there is only one such product. The relation of the algebra ${\cal F}$, the algebra of elements of the Grassman algebra with the star-product as a product, to the Clifford algebra is discussed. 
  In a recent paper Shiromizu, Maeda and Sasaki derived the gravitational equations of motion which would hold on a brane which is embedded in a higher dimensional bulk spacetime, showing that even when the Einstein equations are imposed in the bulk, nonetheless the embedding leads to a modification of the Einstein equations on the brane. In this comment on their work we explicitly identify and evaluate a delta function singularity effect at the brane which they do not appear to have discussed in their paper, an effect which, while actually being of interest in and of itself, nonetheless turns out not to modify their reported results. 
  We embed the Wess-Zumino (WZ) model in a wider superspace than the one described by chiral and anti-chiral superfields. 
  A formalism for calculating the open supermembrane contribution to the non-perturbative superpotential of moduli in heterotic M-theory is presented. This is explicitly applied to the Calabi-Yau (1,1)-moduli and the separation modulus of the end-of-the-world BPS three-branes, whose non-perturbative superpotential is computed. The role of gauge bundles on the boundaries of the open supermembranes is discussed in detail, and a topological criterion presented for the associated superpotential to be non-vanishing. 
  We discuss a general feature of Freund Rubin compactifications that was previously overlooked. It consist in a curious pairing, which we call a shadow relation, of completely different (in terms of spin and mass) fields of the dimensionally reduced theory. Particularly interesting is the case where the compactification preserves a certain amount of supersymmetry, giving rise to a shadowing phenomenon between whole supermultiplets of fields. In particular, there are strong suggestions about the consistency of a massive truncation of 11D supergravity to the massless modes of the graviton supermultiplet plus the massive modes of its shadow partner. This fact has important consequences in the ${\cal N}=2$ and ${\cal N}=3$ cases, which seem to realize respectively a Higgs or a superHiggs phenomenon. In other words, we are led to reinterpret the dimensionally reduced theory as a spontaneously broken phase of some higher (super)symmetric theory. 
  QCD(1+1) in the limit of a large number of flavours N_F and a large number of colours N_C is examined in the small N_F/N_C regime. Using perturbation theory in N_F/N_C, stringent results for the leading behaviour of the spectrum departing from N_F/N_C = 0 are obtained. These results provide benchmarks in the light of which previous truncated treatments of QCD(1+1) at large N_F and N_C are critically reconsidered. 
  We show that keeping only the topologically trivial contribution to the average of a class function on U(N) amounts to integrating over its algebra. The goal is reached first by decompactifying an expansion over the instanton basis and then directly, by means of a geometrical procedure. 
  We discuss the appearance of additional, hidden supersymmetries for simple 0+1 $Ad(G)$-invariant supersymmetric models and analyse some geometrical mechanisms that lead to them. It is shown that their existence depends crucially on the availability of odd order invariant skewsymmetric tensors on the (generic) compact Lie algebra $\cal G$, and hence on the cohomology properties of the Lie algebra considered. 
  A uniformly accelerated system will get thermally excited due to interactions with the vacuum fluctuations of the quantum fields. This is the Unruh effect. Also a system accelerated in a circular orbit will be heated, but in this case complications arise relative to the linear case. An interesting question is in what sense the real quantum effects for orbital and spin motion of a circulating electron can be viewed as a demonstration of the Unruh effect. This question has been studied and debated. I review some of the basic points concerning the relation to the Unruh effect, and in particular look at how the electron can be viewed as a thermometer or detector that probes thermal and other properties of the vacuum state in the accelerated frame. 
  These are expanded notes of lectures given at the summer school "Gif 2000" in Paris. They constitute the first part of an "Introduction to supersymmetry and supergravity" with the second part on supergravity by J.-P. Derendinger to appear soon. The present introduction is elementary and pragmatic. I discuss: spinors and the Poincar\'e group, the susy algebra and susy multiplets, superfields and susy lagrangians, susy gauge theories, spontaneously broken susy, the non-linear sigma model, N=2 susy gauge theories, and finally Seiberg-Witten duality. 
  N=1^* gauge theories are believed to have fractional instanton contributions in the confining vacua. D3 brane probe computations in gravitation dual of large-N N=2^* gauge theories point to the absence of such contributions in the low energy gauge dynamics. We study fractional instantons in N=2 SU(2) Yang-Mills theory from the field theoretical perspective. We present new solutions to the Seiberg-Witten SU(2) monodromy problem with the same perturbative asymptotic, a massless monopole and a dyon singularity on the moduli space, and fractional instanton corrections to N=2 prepotential in the semi-classical region of the moduli space. We show that fractional instantons lead to infinite monopole (dyon) condensate in mass deformed N=2 gauge theories. 
  We consider five-dimensional domain-wall solutions which arise from a sphere reduction in M-theory or string theory and have the higher-dimensional interpretation as the near-horizon region of p-branes in constant, background B fields. We analyze the fluctuation spectrum of linearized gravity and find that there is a massive state which is localized and plays the role of the four-dimensional graviton. 
  In this paper we point out that the spacetime uncertainty relation proposed for string theory has strong cosmological implications that can solve the flatness problem and the horizon problem without the need of inflation. We make minimal assumptions about the very early universe. 
  We consider the complex scalar field coupled to background NC U(1) YM and calculate the correlator of two gauge invariant composite operators. We show how the noncommutative gauge invariance is restored for higher correlators (though the Green's function itself is not invariant). It is interesting that the recently discovered noncommutative solitons appear in the calculation. 
  In this work we consider Randall-Sundrum brane-world type scenarios, in which the spacetime is described by a five-dimensional manifold with matter fields confined in a domain wall or three-brane. We present the results of a systematic analysis, using dynamical systems techniques, of the qualitative behaviour of the Friedmann-Lemaitre-Robertson-Walker and the Bianchi I and V cosmological models in these scenarios. We construct the state spaces for these models and discuss how their structure changes with respect to the general-relativistic case, in particular, what new critical points appear and their nature, the occurrence of bifurcations and the dynamics of anisotropy. 
  We present the generic junction conditions obeyed by a co-dimension one brane in an arbitrary background spacetime. As well as the usual Darmois-Israel junction conditions which relate the discontinuity in the extrinsic curvature to the to the energy-momentum tensor of matter which is localized to the brane, we point out that another condition must also be obeyed. This condition, which is the analogous to Newton's second law for a point particle, is trivially satisfied when $Z_2$ symmetry is enforced by hand, but in more general circumstances governs the evolution of the brane world-volume. As an illustration of its effect we compute the force on the brane due to a form field. 
  The Casimir energy of a dilute homogeneous nonmagnetic dielectric ball at zero temperature is derived analytically for the first time for an arbitrary physically possible frequency dispersion of dielectric permittivity $\epsilon(i\omega)$. A microscopic model of dielectrics is considered, divergences are absent in calculations because an average interatomic distance $\lambda$ is a {\it physical} cut-off in the theory. This fact has been overlooked before, which led to divergences in various macroscopic approaches to the Casimir energy of connected dielectrics. 
  A state vector description for relativistic resonances is derived from the first order pole of the $j$-th partial $S$-matrix at the invariant square mass value $\sm_R=(m-i\Gamma/2)^2$ in the second sheet of the Riemann energy surface. To associate a ket, called Gamow vector, to the pole, we use the generalized eigenvectors of the four-velocity operators in place of the customary momentum eigenkets of Wigner, and we replace the conventional Hilbert space assumptions for the in- and out-scattering states with the new hypothesis that in- and out-states are described by two different Hardy spaces with complementary analyticity properties. The Gamow vectors have the following properties:  - They are simultaneous generalized eigenvectors of the four velocity operators with real eigenvalues and of the self-adjoint invariant mass operator $M=(P_\mu P^\mu)^{1/2}$ with complex eigenvalue $\sqrt{\sm_R}$.  - They have a Breit-Wigner distribution in the invariant square mass variable $\sm$ and lead to an exactly exponential law for the decay rates and probabilities. 
  We construct an M-theory dual of a 6 dimensional little string theory with reduced supersymmetry, along the lines of Polchinski and Strassler. We find that upon perturbing the (2,0) theory with an R-current, the M5 branes polarize into a wrapped Kaluza Klein monopole, whose isometry direction is along the R current. We investigate the properties of this theory. 
  The geometric models of N=4 supersymmetric mechanics with $(2d.2d)_{\DC}$-dimensional phase space are proposed, which can be viewed as one-dimensional counterparts of two-dimensional N=2 supersymmetric sigma-models by Alvarez-Gaum\'e and Freedman. The related construction of supersymmetric mechanics whose phase space is a K\"ahler supermanifold is considered. Also, its relation with antisymplectic geometry is discussed. 
  F(4) supergravity, the gauge theory of the exceptional six-dimensional Anti-de Sitter superalgebra, is coupled to an arbitrary number of vector multiplets whose scalar components parametrize the quaternionic manifold SO(4,n)/SO(4)\times SO(n). By gauging the compact subgroup SU(2)_d \otimes G, where SU(2)_d is the diagonal subgroup of SO(4)= SU(2)_L\otimes SU(2)_R (the R-symmetry group of six-dimensional Poincare' supergravity) and G is a compact group such that dim G = n. The potential admits an AdS background for g=3m, as the pure F(4)-supergravity. The boundary F(4) superconformal fields are realized in terms of a singleton superfield (hypermultiplet) in harmonic superspace with flag manifold SU(2)/U(1)=S^2. We analize the spectrum of short representations in terms of superconformal primaries and predict general features of the K-K specrum of massive type IIA supergravity compactified on warped AdS_6\otimes S^4. 
  We investigate the equation where the commutation relation in 2-dimensional zero-curvature equation composed of the algebra-valued potentials is replaced by the Moyal bracket and the algebra-valued potentials are replaced by the non-algebra-valued ones with two more new variables. We call the 4-dimensional equation the noncommutative zero-curvature equation. We show that various soliton equations are derived by the dimensional reduction of the equation. 
  We study the general form of the *-commutator treated as a deformation of the Poisson bracket on the Grassman algebra. We show that, up to a similarity transformation, there are other deformations of the Poisson bracket in addition to the Moyal commutator (one at even and one at odd n, n is the number of the generators of the Grassman algebra) which are not reduced to the Moyal commutator by a similarity transformation. 
  We show that the (M5, M2, M2$'$, MW) bound state solution of eleven dimensional supergravity recently constructed in hep-th/0009147 is related to the (M5, M2) bound state one by a finite Lorentz boost along a M5-brane direction perpendicular to the M2-brane. Given the (M5, M2) bound state as a defining system for OM theory and the above relation between this system and the (M5, M2, M2', MW) bound state, we test the recently proposed V-duality conjecture in OM theory. Insisting to have a decoupled OM theory, we find that the allowed Lorentz boost has to be infinitesimally small, therefore resulting in a family of OM theories related by Galilean boosts. We argue that such related OM theories are equivalent to each other. In other words, V-duality holds for OM theory as well. Upon compactification on either an electric or a `magnetic' circle (plus T-dualities as well), the V-duality for OM theory gives the known one for either noncommutative open string theories or noncommutative Yang-Mills theories. This further implies that V-duality holds in general for the little m-theory without gravity. 
  We consider the reduced, quenched version of a generalized Yang-Mills action in 4k-dimensional spacetime. This is a new kind of matrix theory which is mapped through the Weyl-Wigner-Moyal correspondence into a field theory over a non-commutative phase space. We show that the ``classical'' limit of this field theory is encoded into the effective action of an open, (4k-1)-dimensional, bulk brane enclosed by a dynamical, Chern-Simons type, (4k-2)-dimensional, boundary brane. The bulk action is a pure volume term, while the boundary action carries all the dynamical degrees of freedom. 
  We prove that SU(N) bosonic Yang-Mills matrix integrals are convergent for dimension (number of matrices) $D\ge D_c$. It is already known that $D_c=5$ for N=2; we prove that $D_c=4$ for N=3 and that $D_c=3$ for $N\ge 4$. These results are consistent with the numerical evaluations of the integrals by Krauth and Staudacher. 
  Working in light-cone coordinates, we study the zero-modes and the vacuum in a 2+1 dimensional SU(3) gauge model. Considering the fields as independent of the tranverse variables, we dimensionally reduce this model to 1+1 dimensions. After introducing an appropriate su(3) basis and gauge conditions, we extract an adjoint field from the model. Quantization of this adjoint field and field equations lead to two constrained and two dynamical zero-modes. We link the dynamical zero-modes to the vacuum by writing down a Schrodinger equation and prove the non-degeneracy of the SU(3) vacuum provided that we neglect the contribution of constrained zero-modes. 
  A relation between the Friedmann equation and the Cardy formula has been found for de Sitter closed and Anti de Sitter flat universes. For the remaining (Anti) de Sitter universes the arguments fail, and we speculate whether the general philosophy of holography can be satisfied in such contents. 
  We describe the complete coupling of $(1,0)$ six-dimensional supergravity to tensor, vector and hypermultiplets. The generalized Green-Schwarz mechanism implies that the resulting theory embodies factorized gauge and supersymmetry anomalies, to be disposed of by fermion loops. Consequently, the low-energy theory is determined by the Wess-Zumino consistency conditions, rather than by the requirement of supersymmetry. As already shown for the case without hypermultiplets, this procedure does not fix a quartic coupling for the gauginos. With respect to these previous results, the inclusion of charged hypermultiplets gives additional terms in the supersymmetry anomaly. We also consider the case in which abelian vectors are present. As in the absence of hypermultiplets, abelian vectors allow additional couplings. Finally, we apply the Pasti-Sorokin-Tonin prescription to this model. 
  We report the discovered class of exact static solutions of several 4D Einstein-Maxwell-dilaton systems: string-induced, Liouville, trigonometric, polynomial, etc., for three basic topologies (spherical, hyperbolical and flat) being uniformly treated. In addition to the usual electric-magnetic duality this class obeys a certain extended duality between Maxwell-dilaton coupling and dilaton mass. Though major solutions we obtain are dyonic, the class also comprises interesting neutral models.   As a by-product, we significantly succeded in resolving of the two important problems, one of which has been standing more than a decade (system with the string-inspired exponential Maxwell-dilaton coupling and non-vanishing dilaton mass) and another one - gravity coupled to massive neutral scalar field: generalized Liouville, Sin(h), Cos(h) - is about fifty years old.   Finally, we demonstrate the full separability of the static EMD system and publicize the simple procedure of how to generate new integrability classes. 
  We study the ratio of the entropy to the total energy in conformal field theories at finite temperature. For the free field realizations of {\cal N}=4 super Yang-Mills theory in D=4 and the (2,0) tensor multiplet in D=6, the ratio is bounded from above. The corresponding bounds are less stringent than the recently proposed Verlinde bound. We show that entropy bounds arise generically in CFTs in connection to monotonicity properties with respect to temperature changes of a generalized C-function. For strongly coupled CFTs with AdS duals, we show that the ratio obeys the Verlinde bound even in the presence of rotation. For such CFTs, we point out an intriguing resemblance in their thermodynamic formulas with the corresponding ones of two-dimensional CFTs. We show that simple scaling forms for the free energy and entropy of CFTs with AdS duals reproduce the thermodynamical properties of (D+1)-dimensional AdS black holes. 
  We consider the behaviour of neutral non-BPS branes probed by scalars and gravitons. We show that the naked singularity of the non-BPS branes is a {\it repulson} absorbing no incoming radiation. The naked singularity is surrounded by an infinite potential well breaking the unitarity of the scattering S-matrix. We compute the absorption cross section which is infrared divergent. In particular this confirms that gravity does not decouple for the non-BPS branes. 
  We introduce a novel decomposition of the four dimensional SU(2) gauge field. This decomposition realizes explicitely a symmetry between electric and magnetic variables, suggesting a duality picture between the corresponding phases. It also indicates that at large distances the Yang-Mills theory involves a three component unit vector field, a massive Lorentz vector field, and a neutral scalar field that condenses which yields the mass scale. Our results are consistent with the proposal that the physical spectrum of the theory contains confining strings which are tied into stable knotted solitons. 
  A perturbative formulation of algebraic field theory is presented, both for the classical and for the quantum case, and it is shown that the relation between them may be understood in terms of deformation quantization. 
  At sufficiently low energy the dynamics of a string web is dominated by zero modes involving rigid motion of the internal strings. The dimension of the associated moduli space equals the maximal number of internal faces in the web. The generic web moduli space has boundaries and multiple branches, and for webs with three or more faces the geometry is curved. Webs can also be studied in a lift to M-theory, where a string web is replaced by a membrane wrapped on a holomorphic curve in spacetime. In this case the moduli space is complexified and admits a Kaehler metric. 
  The possibility of detecting noncommutive space relics is analyzed by using the Aharonov-Bohm effect. If space is non-commutative, it turns out that the holonomy receives kinematical corrections that tend to diffuse the fringe pattern. This fringe pattern has a non-trivial energy dependence and, therefore, one could observe noncommutative effects by modifying the energy of the incident electrons beam in the Tonomura experimental arrangement 
  We classify supersymmetric D0-Dp bound states with a non-zero B-field by considering T-dualities of intersecting branes at angles. Especially, we find that the D0-D8 system with the B-field preserves 1/16, 1/8 and 3/16 of supercharges if the B-field satisfies the ``(anti-)self-dual'' condition in dimension eight. The D0-branes in this system are described by eight dimensional instantons on non-commutative R^8. We also discuss the extended ADHM construction of the eight-dimensional instantons and its deformation by the B-field. The modified ADHM equations admit a sort of the `fuzzy sphere' (embeddings of SU(2)) solution. 
  The causal entropy bound (CEB) is confronted with recent explicit entropy calculations in weakly and strongly coupled conformal field theories (CFTs) in arbitrary dimension $D$. For CFT's with a large number of fields, $N$, the CEB is found to be valid for temperatures not exceeding a value of order $M_P/N^{{1\over D-2}}$, in agreement with large $N$ bounds in generic cut-off theories of gravity, and with the generalized second law. It is also shown that for a large class of models including high-temperature weakly coupled CFT's and strongly coupled CFT's with AdS duals, the CEB, despite the fact that it relates extensive quantities, is equivalent to (a generalization of) a purely holographic entropy bound proposed by E. Verlinde. 
  We simulate a supersymmetric matrix model obtained from dimensional reduction of 4d SU(N) super Yang-Mills theory. The model is well defined for finite N and it is found that the large N limit obtained by keeping g^2 N fixed gives rise to well defined operators which represent string amplitudes. The space-time structure which arises dynamically from the eigenvalues of the bosonic matrices is discussed, as well as the effect of supersymmetry on the dynamical properties of the model. Eguchi-Kawai equivalence of this model to ordinary gauge theory does hold within a finite range of scale. We report on new simulations of the bosonic model for N up to 768 that confirm this property, which comes as a surprise since no quenching or twist is introduced. 
  Some examples of Type-I vacua related to non geometric orbifolds are shown. In particular, the open descendants of the diagonal $Z_3$ orbifold are compared with the geometric ones. Although not chiral, these models exhibit some interesting properties, like twisted sectors in the open-string spectra and the presence of ``quantized'' geometric moduli, a key ingredient to ensure their perturbative consistency and to explain the rank reduction of their Chan-Paton groups. 
  Comments on NC Abelian Higgs model withdrawn. An improved treatment of NC U(2) adjoint models to appear elsewhere. 
  We construct an exact soliton, which represents a BPS Dp-brane, in a boundary string field theory action of infinitely many non-BPS D(p-1)-branes. Furthermore, we show that this soliton can be regarded as an exact soliton in the full string field theory. The world-volume theory of the BPS Dp-brane constructed in this way becomes usual gauge theory on commutative R^{p+1}, instead of non-commutative plane. We also construct a D$p$-brane from non-BPS D(p-n)-branes by the ABS like configuration. We confirm that the Dp-brane has correct RR charges and the tension. 
  A flow invariant is a quantity depending only on the UV and IR conformal fixed points and not on the flow connecting them. Typically, its value is related to the central charges a and c. In classically-conformal field theories, scale invariance is broken by quantum effects and the flow invariant a_{UV}-a_{IR} is measured by the area of the graph of the beta function between the fixed points. There exists a theoretical explanation of this fact. On the other hand, when scale invariance is broken at the classical level, it is empirically known that the flow invariant equals c_{UV}-c_{IR} in massive free-field theories, but a theoretical argument explaining why it is so is still missing. A number of related open questions are answered here. A general formula of the flow invariant is found, which holds also when the stress tensor has improvement terms. The conditions under which the flow invariant equals c_{UV}-c_{IR} are identified. Several non-unitary theories are used as a laboratory, but the conclusions are general and an application to the Standard Model is addressed. The analysis of the results suggests some new minimum principles, which might point towards a better understanding of quantum field theory. 
  We propose an explicit construction of the deformation quantization of the general second-class constrained system, which is covariant with respect to local coordinates on the phase space. The approach is based on constructing the effective first-class constraint (gauge) system equivalent to the original second-class one and can also be understood as a far-going generalization of the Fedosov quantization. The effective gauge system is quantized by the BFV-BRST procedure. The star product for the Dirac bracket is explicitly constructed as the quantum multiplication of BRST observables. We introduce and explicitly construct a Dirac bracket counterpart of the symplectic connection, called the Dirac connection. We identify a particular star product associated with the Dirac connection for which the constraints are in the center of the respective star-commutator algebra. It is shown that when reduced to the constraint surface, this star product is a Fedosov star product on the constraint surface considered as a symplectic manifold. 
  We apply the supplementation trick [26] to the Green-Schwarz superstring. For type IIB theory both first and second class constraints are covariantly separated and then arranged into irreducible sets in the initial formulation. For N=1 Green-Schwarz superstring we propose a modified action which is equivalent to the initial one. Fermionic first and second class constraints are covariantly separated, the first class constraints turn out to be irreducible. We discuss also equations of motion in the covariant gauge for $\kappa$-symmetry. For type IIA theory the same modification leads to formulation with irreducible second class constraints. 
  We include the brane curvature scalar to study its cosmological implication in the brane world cosmology. This term is usually introduced to obtain the well-defined stress-energy tensor on the boundary of anti de Sitter-Schwarzschild space. Here we treat this as the perturbed term for cosmological purpose. We find corrections to the well-known equation of the brane cosmology. It contains new interesting terms which may play the important role in the brane cosmology. 
  One of the key ingredients of A. Connes' noncommutative geometry is a generalized Dirac operator which induces a metric(Connes' distance) on the state space. We generalize such a Dirac operator devised by A. Dimakis et al, whose Connes' distance recovers the linear distance on a 1D lattice, into 2D lattice. This Dirac operator being "naturally" defined has the so-called "local eigenvalue property" and induces Euclidean distance on this 2D lattice. This kind of Dirac operator can be generalized into any higher dimensional lattices. 
  This is the translation to appear in the "SUPERSYMMETRY 2000 - Encyclopaedic Dictionary" of the original paper, published in March 1980, (C.R. Acad. Sci. Paris, Ser. A-B, 290, 1980) in which basic notions of noncommutative geometry were introduced and applied to noncommutative tori. These include connections on finite projective modules, their curvature, and the Chern character. Finite projective modules on the noncommutative two-torus $ \Tb^2_{\theta}$ were realized as Schwartz spaces of vector valued functions on $\Rb$. Explicit constant curvature connections were constructed and a basic integrality phenomenon of the total curvature was displayed. The pseudo-differential calculus and the Atiyah-Singer index theorems were extended to Lie group actions on $C^*$ algebras and used to explain the above integrality of the total curvature by an index formula for finite difference-differential operators on the line. Recent interest in the hep-th literature for basic notions of noncommutative geometry in the case of noncommutative tori (cf for instance hep-th/0012145 for an excellent review) prompted us to make the English translation of the original paper available. 
  n-Dimensional pure gravity theory can be obtained as the effective theory of an n+1 model (with non-compact extra dimension) where general n+1 reparametrization invariance is explicitly broken in the extra dimension. As was pointed out in the literature, a necessary consistency condition for having a non-vanishing four dimensional Newton constant is the normalizability in the extra dimension of the zero mass graviton. This, in turn, implies that gravity localization is produced around the local minima of a potential in the extra dimension. We study gravity in the neighborhood of the soft ("thick") local minima. 
  We examine the intersections, fluctuations and deformations of codimension two solitons in field theory on noncommutative $R^4$, in the limit of large noncommutativity. We find that holomorphic deformations are zero modes of flat branes, and we show that there is a zero mode localized at the intersection of two solitons. 
  We obtain regular deformed D2-brane solutions with fractional D2-branes arising as wrapped D4-branes. The space transverse to the D2-brane is a complete Ricci-flat 7-manifold of G_2 holonomy, which is asymptotically conical with principal orbits that are topologically CP^3 or the flag manifold SU(3)/(U(1) x U(1)). We obtain the solution by first constructing an L^2 normalisable harmonic 3-form. We also review a previously-obtained regular deformed D2-brane whose transverse space is a different 7-manifold of G_2 holonomy, with principal orbits that are topologically S^3 x S^3. This describes D2-branes with fractional NS-NS 2-branes coming from the wrapping of 5-branes, which is supported by a non-normalisable harmonic 3-form on the 7-manifold. We prove that both types of solutions are supersymmetric, preserving 1/16 of the maximal supersymmetry and hence that they are dual to {\cal N}=1 three-dimensional gauge theories. In each case, the spectrum for minimally-coupled scalars is discrete, indicating confinement in the infrared region of the dual gauge theories. We examine resolutions of other branes, and obtain necessary conditions for their regularity. The resolution of many of these seems to lie beyond supergravity. In the process of studying these questions, we construct new explicit examples of complete Ricci-flat metrics. 
  Using the Fock-Schwinger proper time method, we calculate the induced Chern-Simons term arising from the Lorentz- and CPT-violating sector of quantum electrodynamics with a $b_\mu \bar{\psi}\gamma^\mu \gamma_5 \psi$ term. Our result to all orders in $b$ coincides with a recent linear-in-$b$ calculation by Chaichian et al. [hep-th/0010129 v2]. The coincidence was pointed out by Chung [Phys. Lett. {\bf B461} (1999) 138] and P\'{e}rez-Victoria [Phys. Rev. Lett. {\bf 83} (1999) 2518] in the standard Feynman diagram calculation with the nonperturbative-in-$b$ propagator. 
  We obtain finite parts (as well as $\epsilon$-pole parts) of massive three-loop vacuum diagrams with three-point and/or four-point interaction vertices by reducing them to tetrahedron diagrams with both massive and massless lines, whose finite parts were given analytically in a recent paper by Broadhurst. In the procedure of reduction, the method of integration-by-parts recurrence relations is employed. We use our result to compute the $\bar{\rm MS}$ effective potential of the massive $\phi^4$ theory. 
  AdS/CFT duality is a conjectured dual correspondence between the large $N$ limit of Conformal Field Theory (CFT) in $d$-dimensions and the supergravity (SUGRA) in $d+1$-dimensional Anti de Sitter (AdS) space. By using this conjecture, we can study various properties of large $N$ CFT by simple calculations in SUGRA. Recently much attention has been paid to the Renormalization Group (RG) flow viewed from the SUGRA side. Such RG flow in CFT is known to be characterized by the c-function which connects CFTs with different central charges. Therefore, we are interested in deriving this c-function from SUGRA with the help of AdS/CFT correspondence. To derive the c-function, we calculate the conformal anomaly (CA) in SUGRA, since it is closely related to the central charge. In this thesis, we discuss the various aspects of CA from AdS/CFT duality, especially for the cases of SUGRA in 3 and 5-dimensions which correspond to 2 and 4-dimensional CFTs, respectively. It is known that the bosonic part of SUGRA with scalar (dilaton) and arbitrary scalar potential describes the special RG flows in dual quantum field theory. So we calculate dilaton-dependent CA from dilatonic gravity with arbitrary potential. After that, we propose candidates of c-functions from such dilatonic gravity and investigate the properties of them. 
  Quantum geometry gives a regularization scheme-independent effective action, whoes equation of motion for the conformal mode has a stable de Sitter solution at the high-energy region where the coupling of the self-interactions of the traceless mode can be neglected because of the asymptotic freedom. However, the dynamics of the traceless mode suggests that inflation ends at the low-energy region. 
  Recently we made a proposal for realization of an effective BRS symmetry along the Wilsonian renormalization group flow. In this paper we show that the idea can be naturally extended for the most general gauge theories. Extensive use of the antifield formalism is made to reveal some remarkable structure of the effective BRS symmetry. The average action defined with a continuum analog of the block spin transformation obeys the quantum master equation (QME), provided that an UV action does so. We show that the RG flow described by the exact flow equations is generated by canonical transformations in the field-antifield space. Using the relation between the average action and the Legendre effective action, we establish the equivalence between the QME for the average action and the modified Ward-Takahashi identity for the Legendre action. The QME remains intact when the regularization is removed. 
  We derive a noncommutative U(1) and U(n) gauge theory on the fuzzy sphere from a three dimensional matrix model by expanding the model around a classical solution of the fuzzy sphere. Chern-Simons term is added in the matrix model to make the fuzzy sphere as a classical solution of the model. Majorana mass term is also added to make it supersymmetric. We consider two large $N$ limits, one corresponding to a gauge theory on a commutative sphere and the other to that on a noncommutative plane. We also investigate stability of the fuzzy sphere by calculating one-loop effective action around classical solutions. In the final part of this paper, we consider another matrix model which gives a supersymmetric gauge theory on the fuzzy sphere. In this matrix model, only Chern-Simons term is added and supersymmetry transformation is modified. 
  A Rayleigh-Schr\"{o}dinger perturbation theory based on the Gaussian wavefunctional is constructed. The method can be used for calculating the energies of both the vacuum and the excited states. A model calculation is carried out for the vacuum state of the $\lambda\phi^4$ field theory. 
  We discuss the interplay between IR and UV divergences in vacuum configurations with open and unoriented strings. We establish a general one-to-one correspondence between anomalies and R-R tadpoles associated to sectors with non-trivial Witten index. The result does not require any supersymmetry to be preserved by the configuration. Under very mild conditions of supersymmetry, a similar correspondence is found between NS-NS tadpoles and RG-flows in gauge theories on D-branes and O-planes. We briefly comment on the AdS/CFT counterpart of the results. 
  We study the spectral geometry of an operator of Laplace type on a manifold with a singular surface. We calculate several first coefficients of the heat kernel expansion. These coefficients are responsible for divergences and conformal anomaly in quantum brane-world scenario. 
  We provide a brief characterization of the main features of the homogeneous sine-Gordon models and discuss a general construction principle for colour valued S-matrices, associated to a pair of simply laced Lie algebras, which contain the homogeneous sine-Gordon models as a subclass. We give a brief introduction to the thermodynamic Bethe ansatz and the form factor approach and discuss explicit solutions for both methods related to the homogeneous Sine-Gordon models and its generalization. 
  We present a simple and new method of constructing superdistributions on superspace over a Grassmann-Banach algebra, which close to the de Rham's ``currents'' defined as dual objects to differential forms. The paper also contains the extension of the H\"ormander's description of the singularity structure (wavefront set) of a distribution to include the supersymmetric case. 
  In this talk the gauge symmetry for Wilsonian flows in pure Yang-Mills theories is discussed. The background field formalism is used for the construction of a gauge invariant effective action. The symmetries of the effective action under gauge transformations for both the gauge field and the auxiliary background field are separately evaluated. Modified Ward-Takahashi and background field identities are used in my study. Finally it is shown how the symmetry properties of the full theory are restored in the limit where the cut-off is removed. 
  This paper investigates the non-commutative version of the Abelian Higgs model at the one loop level. We find that the BRST invariance of the theory is maintained at this order in perturbation theory, rendering the theory one-loop renormalizable. Upon removing the gauge field from the theory we also obtain a consistent continuum renormalization of the broken O(2) linear sigma model, contradicting results found in the literature. The beta functions for the various couplings of the gauged U(1) theory are presented, as are the divergent contributions to every one particle irreducible (1PI) function. We find that all physical couplings and masses are gauge independent. A brief discussion concerning the symmetries $P$, $C$, and $T$ in this theory is also given. 
  After a brief presentation of the exact renormalization group equation, we illustrate how the field theoretical (perturbative) approach to critical phenomena takes place in the more general Wilson (nonperturbative) approach. Notions such as the continuum limit and the renormalizability and the presence of singularities in the perturbative series are discussed. 
  A kind of Dirac-Connes operator defined in the framework of Connes' NCG is introduced on discrete abelian groups; it satisfies a Junk-free condition, and bridges the NCG composed by Dimakis, M\"{u}ller-Hoissen and Sitarz and the NCG of Connes. Then we apply this operator to d-dimensional lattices. 
  A functional partial differential equation is set for the proper graphs generating functional of QED in external electromagnetic fields. This equation leads to the evolution of the proper graphs with the external field amplitude and the external field gauge dependence of the complete fermion propagator and vertex is derived non-perturbativally. 
  We provide a complete algebraic description of BPS states in M-theory in terms of primary constituents that we call BPS preons. We argue that any BPS state preserving $k$ of the 32 supersymmetries is a composite of (32-k) BPS preons. In particular, the BPS states corresponding to the basic M2 and M5 branes are composed of 16 BPS preons. By extending the M-algebra to a generalized D=11 conformal superalgebra $osp(1|64)$ we relate the BPS preons with its fundamental representation, the D=11 supertwistors. 
  The spectrum of stable electrically and magnetically charged supersymmetric particles can change discontinuously as one changes the vacuum on the Coulomb branch of gauge theories with extended supersymmetry in four dimensions. We show that this decay process can be understood and is well described by semiclassical field configurations purely in terms of the low energy effective action on the Coulomb branch even when it occurs at strong coupling. The resulting picture of the stable supersymmetric spectrum is a generalization of the ``string web'' picture of these states found in string constructions for certain theories. 
  We consider the 3+1 visible sector to live on a Hanany-Witten D-brane construction in type IIA string theory. The messenger sector consists of stretched strings from the visible brane to a hidden D6-brane in the extra spatial dimensions. In the open string channel supersymmetry is broken by gauge mediation while in the closed string channel supersymmetry is broken by gravity mediation. Hence, we call this kind of mediation ``string mediation''. We propose an extension of the Dimopoulos-Georgi theorem to brane models: only detached probe branes can break supersymmetry without generating a tachyon. Fermion masses are generated at one loop if the branes break a sufficient amount of the ten dimensional Lorentz group while scalar potentials are generated if there is a force between the visible brane and the hidden brane. Scalars can be lifted at two loops through a combination of brane bending and brane forces. We find a large class of stable non-supersymmetric brane configurations of ten dimensinoal string theory. 
  SW(3/2,2) superconformal algebra is W algebra with two Virasoro operators. The Kac determinant is calculated and the complete list of unitary representations is determined. Two types of extensions of SW(3/2,2) algebra are discussed. A new approach to construction of W algebras from rational conformal field theories is proposed. 
  Symmetry based approaches to the black hole entropy problem have a number of attractive features; in particular they are very general and do not depend on the details of the quantization method. However we point out that, of the two available approaches, one faces conceptual problems (also emphasized by others), while the second contains certain technical flaws. We correct these errors and, within the new, improved scheme, calculate the entropy of 3-dimensional black holes. We find that, while the new symmetry vector fields are well-defined on the ``stretched horizon,'' and lead to well-defined Hamiltonians satisfying the expected Lie algebra, they fail to admit a well-defined limit to the horizon. This suggests that, although the formal calculation can be carried out at the classical level, its real, conceptual origin probably lies in the quantum theory. 
  We consider the CFT that arises from the D1-D5 system in the presence of a constant background gauge potential which couples to the R-charge of the theory: this potential effectively changes the periodicities of the fermions. By the AdS/CFT correspondence, the effect of this connnection should be obtained by finding a smooth solution to the vector field in AdS space which couples to the constant mode of the R-current in the CFT. We investigate such solutions for small values of the connection, and contrast these with spacetimes which have `Wilson lines for the bulk ' -- spacetimes that are locally AdS times a sphere but have a global deformation. The latter class are in general singular spacetimes. We comment on some aspects of the recently found geometries corresponding to the D1-D5 state with angular momentum, observing relations between scales in the microscopic theory and scales in the geometry. 
  Supersymmetric solutions, such as BPS domain walls or black holes, in four- and five-dimensional supergravity theories with eight supercharges can be described by effective quantum mechanics with a potential term. We show how properties of the latter theory can help us to learn about the physics of supersymmetric vacua and BPS solutions in these supergravity theories. The general approach is illustrated in a number of specific examples where scalar fields of matter multiplets take values in symmetric coset spaces. 
  Recent advances in string theory have highlighted the need for reliable numerical methods to calculate correlators at strong coupling in supersymmetric theories. We present a calculation of the correlator <0|T^{++}(r)T^{++}(0)|0> in N=1 SYM theory in 2+1 dimensions. The numerical method we use is supersymmetric discrete light-cone quantization (SDLCQ), which preserves the supersymmetry at every order of the approximation and treats fermions and bosons on the same footing. This calculation is done at large $N_c$. For small and intermediate r the correlator converges rapidly for all couplings. At small r the correlator behaves like 1/r^6, as expected from conformal field theory. At large r the correlator is dominated by the BPS states of the theory. There is, however, a critical value of the coupling where the large-r correlator goes to zero, suggesting that the large-r correlator can only be trusted to some finite coupling which depends on the transverse resolution. We find that this critical coupling grows linearly with the square root of the transverse momentum resolution. 
  The definition of mass and width of relativistic resonances and in particular of the $Z$-boson is discussed. For this we use the theory based on time asymmetric boundary conditions given by Hardy class spaces ${\mathbf \Phi}_-$ and ${\mathbf \Phi}_+$ for prepared in-states and detected out-states respectively, rather than time symmetric Hilbert space theory. This Hardy class boundary condition is a mathematically rigorous form of the singular Lippmann-Schwinger equation. In addition to the rigorous definition of the Lippmann-Schwinger kets $|[j,{\mathsf s}]^{\pm}>$ as functionals on the spaces ${\mathbf \Phi}_{\mp}$, one obtains Gamow kets $|[j,{\mathsf s}_R]^- >$ with complex centre-of-mass energy value ${\mathsf s}_R=(M_R-i\Gamma_R/2)^2$. The Gamow kets have an exponential time evolution given by $\exp{(-iM_Rt-\Gamma_Rt/2)}$ which suggests that $(M_R,\Gamma_R)$ is the right definition of the mass and width of a resonance. This is different from the two definitions of the $Z$-boson mass and width used in the Particle Data Table and leads to a numerical value of $M_R=(91.1626\pm 0.0031) {\rm GeV}$ from the $Z$-boson lineshape data. 
  In this lecture we summarize some recent work on the understanding of instanton effects in string theories with 16 supersymmetries. In particular, we consider F^4 couplings using the duality between the heterotic string on T^4 and type IIA on K_3 at an orbifold point, as well as higher and lower dimensional versions of this string-string duality. At the perturbative level a non-trivial test of the duality, requiring several miraculous identities, is presented by matching a purely one-loop heterotic amplitude to a purely tree-level type II result. A wide variety of non-perturbative effects is shown to occur in this setting, including D-brane instantons for type IIA on K_3 x S^1 and NS5-brane instantons for type IIA on K_3 x T^2. Moreover, the analysis of the three-dimensional case, which possesses a non-perturbative SO(8,24,Z) U-duality, reveals the presence of Kaluza-Klein 5-brane instanton effects, both on the heterotic and the type II side. 
  Superconformal algebras embedding space-time in any dimension and signature are considered. Different real forms of the $R$-symmetries arise both for usual space-time signature (one time) and for Euclidean or exotic signatures (more than one times). Application of these superalgebras are found in the context of supergravities with 32 supersymmetries, in any dimension $D \leq 11$. These theories are related to $D = 11, M, M^*$ and $M^\prime$ theories or $D = 10$, IIB, IIB$^*$ theories when compactified on Lorentzian tori. All dimensionally reduced theories fall in three distinct phases specified by the number of (128 bosonic) positive and negative norm states: $(n^+,n^-) = (128,0), (64,64), (72,56)$. 
  We briefly review the universal supersymmetry present in classical hamiltonian systems and show its applications to field theories. 
  We construct intersecting D-branes as noncommutative solitons in bosonic and type II string theory. ``Defect'' branes which are D-branes containing bubbles of the closed string vacuum play an important role in the construction. 
  A self-contained review is given of the matrix model of M-theory. The introductory part of the review is intended to be accessible to the general reader. M-theory is an eleven-dimensional quantum theory of gravity which is believed to underlie all superstring theories. This is the only candidate at present for a theory of fundamental physics which reconciles gravity and quantum field theory in a potentially realistic fashion. Evidence for the existence of M-theory is still only circumstantial---no complete background-independent formulation of the theory yet exists. Matrix theory was first developed as a regularized theory of a supersymmetric quantum membrane. More recently, the theory appeared in a different guise as the discrete light-cone quantization of M-theory in flat space. These two approaches to matrix theory are described in detail and compared. It is shown that matrix theory is a well-defined quantum theory which reduces to a supersymmetric theory of gravity at low energies. Although the fundamental degrees of freedom of matrix theory are essentially pointlike, it is shown that higher-dimensional fluctuating objects (branes) arise through the nonabelian structure of the matrix degrees of freedom. The problem of formulating matrix theory in a general space-time background is discussed, and the connections between matrix theory and other related models are reviewed. 
  Using the N = 2 off-shell formulation in harmonic superspace for N = 4 super Yang-Mills theory, we present a representation of the one-loop effective action which is free of so-called coinciding harmonic singularities and admits a straightforward evaluation of low-energy quantum corrections in the framework of an N = 2 superfield heat kernel technique. We illustrate our approach by computing the low-energy effective action on the Coulomb branch of SU(2) N = 4 super Yang-Mills. Our work provides the first derivation of the low-energy action of N = 4 super Yang-Mills theory directly in N = 2 superspace without any reduction to N = 1 superfields and for a generic background N = 2 Yang-Mills multiplet. 
  We develop a mathematically precise framework for the Casimir effect. Our working hypothesis, verified in the case of parallel plates, is that only the regularization-independent Ramanujan sum of a given asymptotic series contributes to the Casimir pressure. As an illustration, we treat two cases: parallel plates, identifying a previous cutoff free version (by G. Scharf and W. W.) as a special case, and the sphere.We finally discuss the open problem of the Casimir force for the cube. We propose an Ansatz for the exterior force and argue why it may provide the exact solution, as well as an explanation of the repulsive sign of the force. 
  Fibrewise T-duality (Fourier-Mukai transform) for D-branes on an elliptic Calabi-Yau $X$ is shown to require naturally an appropriate twisting of the operation respectively a twisted charge. The fibrewise T-duality is furthermore expressed through known monodromies in the context of Kontsevich's interpretation of mirror symmetry. 
  Differential structure of a d-dimensional lattice, which is essentially a noncommutative exterior algebra, is defined using reductions in first order and second order of universal differential calculus in the context of noncommutative geometry (NCG) developed by Dimakis et al. This differential structure can be realized adopting a Dirac-Connes operator proposed by us recently within Connes' NCG. With matrix representations being specified, our Dirac-Connes operator corresponds to staggered Dirac operator, in the case that dimension of the lattice equals to 1, 2 and 4. 
  We discuss radiative corrections to the Casimir effect from an effective field theory point of view. It is an improvement and more complete version of a previous discussion by Kong and Ravndal. By writing down the most general effective Lagrangian respecting the symmetries and the boundary conditions, we are able to reproduce earlier results of Bordag, Robaschik and Wieczorek calculated in full QED. They obtained the correction E_0^(1) = \pi^2\alpha/2560mL^4 to the Casimir energy. We find that this leading correction is due to surface terms in the effective theory, which we attribute to having dominant fluctuations localized on the plates. 
  We compute the one-loop contribution to the free energy in eleven-dimensional supergravity, with the eleventh dimension compactified on a circle of radius $R_{11}$. We find a finite result, which, in a small radius expansion, has the form of the type IIA supergravity free energy plus non-perturbative corrections in the string coupling $g_A$, whose coefficients we determine. We then study type IIA superstring theory at finite temperature in the strong coupling regime by considering M-theory on $R^9\times T^2$, one of the sides of the torus being the euclidean time direction, where fermions obey antiperiodic boundary conditions. We find that a certain winding membrane state becomes tachyonic above some critical temperature, which depends on $g_A$. At weak coupling, it coincides with the Hagedorn temperature, at large coupling it becomes $T_{\rm cr} \cong 0.31 l_P^ {-1} $ (so it is very small in string units). 
  We investigated the evolution of a scalar field propagating in small Schwarzschild Anti-de Sitter black holes. The imaginary part of the quasinormal frequency decreases with the horizon size and the real part of the quasinormal frequency keeps nearly as a constant. The object-picture clarified the question on the quasinormal modes for small Anti-de Sitter black holes. Dependence of quasinormal modes on spacetime dimensions and the multipole index for small Anti-de Sitter black holes have also been illustrated. 
  We reexamine the thermodynamics of adS black holes with Ricci flat horizons using the adS soliton as the thermal background. We find that there is a phase transition which is dependent not only on the temperature, but also on the black hole area, which is an independent parameter. As in the spherical adS black hole, this phase transition is related via the adS/CFT correspondence to a confinement-deconfinement transition in the large N gauge theory on the conformal boundary at infinity. 
  We investigate the spectrum of type IIA BPS D-branes on the quintic from a four dimensional supergravity perspective and the associated split attractor flow picture. We obtain some very concrete properties of the (quantum corrected) spectrum, mainly based on an extensive numerical analysis, and to a lesser extent on exact results in the large radius approximation. We predict the presence and absence of some charges in the BPS spectrum in various regions of moduli space, including the precise location of the lines of marginal stability and the corresponding decay products. We explain how the generic appearance of multiple basins of attraction is due to the presence of conifold singularities and give some specific examples of this phenomenon. Some interesting space-time features of these states are also uncovered, such as a nontrivial, moduli independent lower bound on the area of the core of arbitrary BPS solutions, whether they are black holes, empty holes, or more complicated composites. 
  Using the generalised AdS/CFT correspondence, we show that there are certain ten-dimensional differentiable manifolds such that string theory on such a manifold is unstable [to the emission of "large branes"] no matter what the metric may be. The instability is thus due to the [differential] topology of the manifold, not to any particular choice of its geometry. We propose a precise criterion for this "topology selection mechanism", and prove it in many cases. The techniques employed may be useful in more general cases. 
  This paper presents the transition from Classical Electrodynamics (CED) to Extended Electrodynamics (EED) from the electromagnetic duality point of view, and emphasizes the role of the canonical complex structure in ${\cal R}^2$ in, both, nonrelativistic and relativistic formulations of CED and EED. We begin with summarizing the motivations for passing to EED, as well as we motivate and outline the way to be followed in pursuing the right extension of Maxwell equations. Further we give the nonrelativistic and relativistic approaches to the extension and give explicitly the new equations as well as some properties of the nonlinear vacuum solutions. 
  We find exact solutions of the Einstein equations which describe a black hole pierced by infinitely thin cosmic strings. The string segments enter the black hole along the radii and their positions coincide with the symmetry axes of a regular polyhedron. Each string produces an angle deficit proportional to its tension, while the metric outside the strings is locally Schwarzschild one. There are three configurations corresponding to tetrahedra, octahedra and icosahedra where the number of string segments is 14, 26 and 62, respectively. There is also a "double pyramid" configuration where the number of string segments is not fixed. There can be two or three independent types of strings in one configuration. Tensions of strings belonging to the same type are equal. Analogous polyhedral multi-string configurations can be combined with other spherically symmetric solutions of the Einstein equations. 
  The spectrum of stable electrically and magnetically charged supersymmetric particles can change discontinuously due to the decay of these particles as the vacuum on the Coulomb branch is varied. We show that this decay process is well described by semi-classical field configurations purely in terms of the low energy effective action on the Coulomb branch even when it occurs at strong coupling. The resulting picture of the stable supersymmetric spectrum is a generalization of the ``string web'' picture of these states found in string constructions of certain theories. 
  The temperature dependence of commutator anomalies is discussed on the explicit example of particular (anyonic) field operators in two dimensions. The correlation functions obtained show that effects of the non-zero temperature might manifest themselves not only globally but also locally. 
  The quantum completion of the space of connections in a manifold can be seen as the set of all morphisms from the groupoid of the edges of the manifold to the (compact) gauge group. This algebraic construction generalizes the analogous description of the gauge-invariant quantum configuration space of Ashtekar and Isham. We present a description of the groupoid approach which brings the gauge-invariant degrees of freedom to the foreground, thus making the action of the gauge group more transparent. 
  We give graphical rules, based on earlier work for the functional Schrodinger equation, for constructing the density matrix for scalar and gauge fields at finite temperature T. More useful is a dimensionally-reduced effective action (DREA) constructed from the density matrix by further functional integration over the arguments of the density matrix coupled to a source. The DREA is an effective action in one less dimension which may be computed order by order in perturbation theory or by dressed-loop expansions; it encodes all thermal matrix elements. The DREA is useful because it gives a dimensionally-reduced field theory usable at any T including infinity, where it yields the usual dimensionally-reduced field theory (DRFT). However, it cannot and does not have spurious infinities which sometimes occur in the density matrix or the DRFT; these come from ln T factors at infinite temperature. An example of spurious divergences in the DRFT occurs in d 2+1 $\phi^4$ theory dimensionally reduced to d=2. We show that the rules for the DREA replace these "wrong" divergences in physical parameters by calculable powers of ln T; we also compute the phase transition temperature of this theory at one loop order. 
  The consistency of the orbifold action on open strings between D-branes in orbifold theories with and without discrete torsion is analysed carefully. For the example of the C^3/Z_2 x Z_2 theory, it is found that the consistency of the orbifold action requires that the D-brane spectrum contains branes that give rise to a conventional representation of the orbifold group as well as branes for which the representation is projective. It is also shown how the results generalise to the orbifolds C^3/Z_N x Z_N for which a number of novel features arise. In particular, the N>2 theories with minimal discrete torsion have non-BPS branes charged under twisted R-R potentials that couple to none of the (known) BPS branes. 
  I review the relationship between AdS/CFT (anti-de Sitter / conformal field theory) dualities and the general theory of unitary lowest weight (ULWR) (positive energy) representations of non-compact space-time groups and supergroups. The ULWR's have the remarkable property that they can be constructed by tensoring some fundamental ULWR's (singletons or doubletons). Furthermore, one can go from the manifestly unitary compact basis of the ULWR's of the conformal group (Wigner picture) to the manifestly covariant coherent state basis (Dirac picture) labelled by the space-time coordinates. Hence every irreducible ULWR corresponds to a covariant field with a definite conformal dimension. These results extend to higher dimensional generalized spacetimes (superspaces) defined by Jordan (super) algebras and Jordan (super) triple systems. In particular, they extend to the ULWR's of the M-theory symmetry superalgebra OSp(1/32,R). 
  Non-commutative gauge theory on fuzzy sphere was obtained by Alekseev et al. as describing the low energy dynamics of a spherical D2-brane in S^3 with the background b-field. We identify a subset of solutions of this theory which are analogs of ``unstable'' solitons on a non-commutative flat D2-brane found by Gopakumar et al. Analogously to the flat case, these solutions have the interpretation as describing D0-branes ``not yet dissolved'' by the D2-brane. We confirm this interpretation by showing the precise agreement of the binding energy computed in the non-commutative and ordinary Born-Infeld descriptions. We then study stability of the solution describing a single D0-brane off a D2-brane. Similarly to the flat case, we find an instability when the D0-brane is located close to the D2-brane. We furthermore obtain the complete mass spectrum of 0-2 fluctuations, which thus gives a prediction for the low energy spectrum of the 0-2 CFT in S^3. We also discuss in detail how the instability to a formation of the fuzzy sphere modifies the usual Higgs mechanism for small separation between the branes. 
  The phase structure of a non-isotropic non-Abelian SU(3) lattice gauge model at finite temperature is investigated to the third order in the variational-cumulant expansion (VCE) approach. The layer phase exists in this model in the cases of dimensions D=4, D=5 (d=D-1). 
  It is known that the Seiberg-Witten invariants, derived from supersymmetric Yang-Mill theories in four-dimensions, do not distinguish smooth structure of certain non-simply-connected four manifolds. We propose generalizations of Donaldson-Witten and Vafa-Witten theories on a K\"{a}hler manifold based on Higgs Bundles. We showed, in particular, that the partition function of our generalized Vafa-Witten theory can be written as the sum of contributions our generalized Donaldson-Witten invariants and generalized Seiberg-Witten invariants. The resulting generalized Seiberg-Witten invariants might have, conjecturally, information on smooth structure beyond the original Seiberg-Witten invariants for non-simply-connected case. 
  We derive expressions for the phase-space of a particle of momentum $p$ decaying into $N$ particles, that are valid for any number of dimensions. These are the imaginary parts of so-called `sunset' diagrams, which we also obtain. The results are given as a series of hypergeometric functions, which terminate for odd dimensions and are also well-suited for deriving the threshold behaviour. 
  The creation of massless scalar particles from the quantum vacuum by spherical shell with time varying radius is studied. In the general case of motion the equations are derived for the instantaneous basis expansion coefficients. The examples are considered when the mean number of particles can be explicitly evaluated in the adiabatic approximation. 
  Topological Chern-Simons (CS) and BF theories and their holomorphic analogues are discussed in terms of de Rham and Dolbeault cohomologies. We show that Cech cohomology provides another useful description of the above topological and holomorphic field theories. In particular, all hidden (nonlocal) symmetries of non-Abelian CS and BF theories can be most clearly seen in the Cech approach. We consider multidimensional Manin-Ward integrable systems and describe their connections with holomorphic BF theories. Dressing symmetries of these generic integrable systems are briefly discussed. 
  We define generalised chiral vertex operators covariant under the Ocneanu ``double triangle algebra'' {\cal A}, a novel quantum symmetry intrinsic to a given rational 2-d conformal field theory. This provides a chiral approach, which, unlike the conventional one, makes explicit various algebraic structures encountered previously in the study of these theories and of the associated critical lattice models, and thus allows their unified treatment. The triangular Ocneanu cells, the 3j-symbols of the weak Hopf algebra {\cal A}, reappear in several guises. With {\cal A} and its dual algebra {hat A} one associates a pair of graphs, G and {\tilde G}. While G are known to encode complete sets of conformal boundary states, the Ocneanu graphs {\tilde G} classify twisted torus partition functions. The fusion algebra of the twist operators provides the data determining {\hat A}. The study of bulk field correlators in the presence of twists reveals that the Ocneanu graph quantum symmetry gives also an information on the field operator algebra. 
  Liouville field theory is considered with boundary conditions corresponding to a quantization of the classical Lobachevskiy plane (i.e. euclidean version of $AdS_2$). We solve the bootstrap equations for the out-vacuum wave function and find an infinite set of solutions. This solutions are in one to one correspondence with the degenerate representations of the Virasoro algebra. Consistency of these solutions is verified by both boundary and modular bootstrap techniques. Perturbative calculations lead to the conclusion that only the ``basic'' solution corresponding to the identity operator provides a ``natural'' quantization of the Lobachevskiy plane. 
  In the Wess-Zumino gauge, supersymmetry transformations become non-linear and are usually incorporated together with BRS transformations in the form of Slavnov-Taylor identities, such that they appear at first sight to be even non-local. Furthermore, the gauge fixing term breaks supersymmetry. In the present paper, we clarify in which sense supersymmetry is still a symmetry of the system and how it is realized on the level of quantum fields. 
  We study the evaporation of black holes in space-times with extra dimensions of size L. We first obtain a description which interpolates between the expected behaviors of very large and very small black holes and then show that the luminosity is greatly damped when the horizon shrinks towards L from a larger value. Analogously, black holes born with an initial size smaller than L are almost stable. This effect is due to the dependence of both the Hawking temperature and the grey-body factor of a black hole on the dimensionality of space. Although the picture of what happens when the horizon becomes of size L is still incomplete, we argue that there occurs a (first order) phase transition, possibly signaled by an outburst of energy which leaves a quasi-stable remnant. 
  The Casimir effect for general Robin conditions on the surface of a cylinder in $D$-spacetime dimensions is studied for massive scalar field with general curvature coupling. The energy distribution and vacuum stress are investigated. We separate volumic and superficial energy contributions, for both interior and exterior space regions. The possibility that some special conditions may be energetically singled out is indicated. 
  In the framework of causal perturbation theory we consider a massive scalar field coupled to gravity. In the field theoretic approach to quantum gravity (QG) we start with a massless second rank tensor field. This tensor field is then quantized in a covariant way in Minkowski space. This article deals with the adiabatic limit for graviton radiative corrections in a scattering process of two massive scalar particles. We compute the differential cross-section for bremsstrahlung processes in which one of the outgoing particles emites a graviton of low energy, a so called soft graviton. Since the emited graviton will not be detected we have to integrate over all soft gravitons. 
  Using the higher covariant derivatives regularization of gauge theories in the framework of the background field method, supplemented with one-loop Pauli-Villars regulator fields, we obtain a version of the renormalization group equation for the regulator fields, whose vacuum energy depends on the background gauge field. It is evaluated using an anomalous Ward-Takahashi identity, which is related to the rescaling anomaly of the auxiliary fields, obtained by the Fujikawa approach. In this way the anomalous origin of the one-loop beta-function in QCD is clearly shown in terms of scaling of effective Lagrangians without the use of any Feynman diagram. The simplicity of the method is due to the preservation of the background and quantum gauge invariance in any step of the calculation. 
  We construct a new example of the high derivative four-dimensional conformal operator. This operator acts on fermions, and its contribution to the trace anomaly has opposite sign, as compared to conventional scalars, spinors and vectors. Possible generalizations and applications are discussed. 
  The non-commutative Wess-Zumino model is used as a prototype for studying the low energy behaviour of a renormalizable non-commutative field theory. We start by deriving the potential mediating the fermion-fermion and boson-boson interactions in the non-relativistic regime. The quantum counterparts of these potentials are afflicted by irdering ambiguities but we show that there exists an ordering prescription which makes them hermitean. For space/space noncommutativity it turns out that Majorana fermions may be pictured as rods oriented perpendicularly to the direction of motion showing a lack of localituy, while bosons remain insensitive to the effects of noncommutativity. For time/space noncommutativity bosopns and fermions can be regarded as rods oriented along the direction of motion. For both cases of noncommutativity the scattering state described scattered waves, with at least one wave having negative time delay signalizing the underlying nonlocality. The superfield formulation of the model is used to compute the corresponding effective action in the one- and two-loop approximations. In the case of time/space noncommutativity, unitarity is violated in the relativistic regime. However, this does not preclude the existence of the unitary low energy limit. 
  Our starting point is an iterative construction suited to combinatorics in arbitarary dimensions d, of totally anisymmetrised p-Riemann 2p-forms (2p\le d) generalising the (1-)Riemann curvature 2-forms. Superposition of p-Ricci scalars obtained from the p-Riemann forms defines the maximally Gauss--Bonnet extended gravitational Lagrangian. Metrics, spherically symmetric in the (d-1) space dimensions are constructed for the general case. The problem is directly reduced to solving polynomial equations. For some black hole type metrics the horizons are obtained by solving polynomial equations. Corresponding Kruskal type maximal extensions are obtained explicitly in complete generality, as is also the periodicity of time for Euclidean signature. We show how to include a cosmological constant and a point charge. Possible further developments and applications are indicated. 
  We study hyperkahler cones and their corresponding quaternion-Kahler spaces. We present a classification of 4(n-1)-dimensional quaternion-Kahler spaces with n abelian quaternionic isometries, based on dualizing superconformal tensor multiplets. These manifolds characterize the geometry of the hypermultiplet sector of perturbative moduli spaces of type-II strings compactified on a Calabi-Yau manifold. As an example of our construction, we study the universal hypermultiplet in detail, and give three inequivalent tensor multiplet descriptions. We also comment on the construction of quaternion-Kahler manifolds that may describe instanton corrections to the moduli space. 
  It has been conjectured that the bosonic open string theory around the non-perturbative tachyon vacuum has no open string dynamics at all. We explore, in the cubic open string field theory with level truncation approximation, the possibility that this conjecture is realized by the absence of kinetic terms of the string field fluctuations. We study the kinetic terms with two and four derivatives for the lower level scalar modes as well as their BRST transformation properties. The behavior of the coefficients of the kinetic terms in the neighborhood of the non-perturbative vacuum supports our expectation that the BRST invariant scalar component lacks its kinetic term. 
  Fermions in D=4 self-dual Euclidean Taub-NUT space are investigated. Dirac-type operators involving Killing-Yano tensors of the Taub-NUT geometry are explicitly given showing that they anticommute with the standard Dirac operator and commute with the Hamiltonian as it is expected. They are connected with the hidden symmetries of the space allowing the construction of a conserved vector operator analogous to the Runge-Lenz vector of the Kepler problem. This operator is written down pointing out its algebraic properties. 
  We develop a formalism to calculate the effective action of the strongly coupled conformal field theory (CFT) in curved spacetime. The effective action of the CFT is obtained from AdS/CFT correspondence. The anti de-Sitter (AdS) spacetime has various slicing which give various curved spacetime on its boundary. We show the de Sitter spacetime and the Friedmann-Robertson-Walker (FRW) universe can be embedded in the AdS spacetime and derive the scalar two-point function of the conformal fields in those spacetime. In curved spacetime, the two-point function depends on the vacuum state of the CFT. A method to specify the vacuum state in AdS/CFT calculations is shown. Because the classical action in AdS spacetime diverges near the boundary, we need the counter terms to regulate the result. The simple derivation of the counter terms using the Hamilton-Jacobi equation is also presented in the appendix. 
  We study the transformation law of quantum fields in super Yang-Mills theory quantized in the Wess-Zumino gauge. It can be derived from a local version of generalized Slavnov-Taylor identities for general Green functions. Under suitable normalization conditions the transformations are local. Within the vector multiplet anomalous dimensions become equal. The breaking of susy shows up in Fock but not in Hilbert space and is not reflected in the transformation law for the fields. 
  We reconsider the supersymmetric Wess-Zumino-Witten (SWZW) term in four dimensions. It has been known that the manifestly supersymmetric form of the SWZW term includes derivative terms on auxiliary fields, the highest components of chiral superfields, and then we cannot eliminate them by their equations of motion. We discuss a possibility for the elimination of such derivative terms by adding total derivative terms. Although the most of derivative terms can be eliminated as in this way, we find that all the derivative terms can be canceled, if and only if an anomalous term in SWZW term vanishes. As a byproduct, we find the first example of a higher derivative term free from such a problem. 
  We study the quantization of chiral QED by means of an extended Slavnov-Taylor (ST) identity under which an external source and the ghost field form a doublet. This ST identity incorporates the Adler-Bardeen anomaly. We prove that the cohomology classes of the extended classical linearized ST operator ${\cal S}'_0$ are modified with respect to the ones of the classical BRST differential in the FP neutral sector (physical observables). This provides a counterexample showing that the introduction of a doublet can modify the cohomology of the model, as a consequence of the fact that the counting operator for the doublet does not commute with ${\cal S}'_0$. 
  Hamiltonian formulation of the string with dynamical geometry and two-dimensional gravity with torsion is given. Canonical Hamiltonian equals to the linear combination of first class constraints satisfying closed algebra. It is the semidirect sum of the Virasoro algebra and the abelian subalgebra corresponding to the local Lorentz rotation. After making the canonical transformation the theory is quantized. It is proved that there exists Fock space representation of pure two-dimensional gravity with torsion containing no central charge in the Virasoro algebra. Also constructed is the new Fock representation of a standard bosonic string. It is shown that two-dimensional string with dynamical geometry is anomaly free and describes two physical degrees of freedom. 
  We describe a class of asymptotically AdS scalar field spacetimes, and calculate the associated conserved charges for three, four and five spacetime dimensions using the conformal and counter-term prescriptions. The energy associated with the solutions in each case is proportional to $\sqrt{M^2 - k^2}$, where $M$ is a constant and $k$ is a scalar charge. In five spacetime dimensions, the counter-term prescription gives an additional vacuum (Casimir) energy, which agrees with that found in the context of AdS/CFT correspondence. We find a surprising degeneracy: the energy of the ``extremal'' scalar field solution $M=k$ equals the energy of pure AdS. This result is discussed in light of the AdS/CFT conjecture. 
  We present an analysis of the canonical structure of the WZW theory with untwisted conformal boundary conditions. The phase space of the boundary theory on a strip is shown to coincide with the phase space of the Chern-Simons theory on a solid cylinder (a disc times a line) with two Wilson lines. This reveals a new aspect of the relation between two-dimensional boundary conformal field theories and three-dimensional topological theories. A decomposition of the Chern-Simons phase space on a punctured disc in terms of the one on a punctured sphere and of coadjoint orbits of the loop group easily lends itself to quantization, providing at the same time a quantization of the boundary WZW model. 
  We show that correlators of some local operators in gauge theories are sensitive to the presence of the instantons even at high temperature where the latter are bound into instanton-anti-instanton "molecules". We calculate correlation functions of such operators in the deconfined phase of the 2+1 dimensional Georgi-Glashow model and discuss analogous quantities in the chirally symmetric phase of QCD. We clarify the mechanism by which the instanton-anti-instanton molecules contribute to the anomaly of axial U(1) at high temperature. 
  Using parafermionic field theoretical methods, the fundamentals of 2d fractional supersymmetry ${\bf Q}^{K} =P$ are set up. Known difficulties induced by methods based on the $U_{q}(sl(2))$ quantum group representations and non commutative geometry are overpassed in the parafermionic approach. Moreover we find that fractional supersymmetric algebras are naturally realized as matrix models. The K=3 case is studied in details. Links between 2d $({1\over 3},0)$ and $(({1\over 3}^{2}),0)$ fractional supersymmetries and N=2 U(1) and N=4 su(2) standard supersymmetries respectively are exhibited. Field theoretical models describing the self couplings of the matter multiplets $(0^{2},({1\over 3})^{2},({2\over 3})^{2})$ and $(0^{4},({1\over 3})^{4},({2\over 3})^{4})$ are given. 
  An exact integral expression is found for the amplitude of a Bosonic string with ends separated by a fixed distance $R$ evolving over a time $T$ between arbitrary initial and final configurations. It is impossible to make a covariant subtraction of a covariant quantity which would render the amplitude non-zero. It is suggested that this fact (and not the tachyon) is responsible for the lack of a continuum limit of regularized random-surface models with target-space dimension greater than one. It appears consistent, however, to remove this quantity by hand. The static potential of Alvarez and Arvis $V(R)$ is recovered from the resulting finite amplitude for $R>R_{c}$. For $R<R_{c}$, we find $V(R)=-\infty$, instead of the usual tachyonic result. A rotation-invariant expression is proposed for special cases of the off-shell disk amplitude. {\it None} of the finite amplitudes discussed are Nambu or Polyakov functional integrals, except through an unphysical analytic continuation. We argue that the Liouville field does not decouple in off-shell amplitudes, even when the space-time dimension is twenty-six. 
  A brief survey of recent results in the study of boundary integrable quantum field theories, indicating some currently open problems. Based on lectures given at the 2000 Eotvos Summer School in Physics on `Nonperturbative QFT methods and their applications'. 
  I discuss a model for quantized gravitation based on the simplicial lattice discretization. It has been studied in some detail using a comprehensive finite size scaling analysis combined with renormalization group methods. The results are consistent with a value for the universal critical exponent for gravitation $\nu=1/3$, and suggest a simple relationship between Newton's constant, the gravitational correlation length and the observable average space-time curvature. Some perhaps testable phenomenological implications are discussed, such as the scale dependence of Newton's constant and properties of quantum curvature fluctuations. 
  Recently, it has been proposed that the dimension of the Hilbert space of quantum gravity in deSitter space is finite and moreover it is expressed in terms of the coupling constants by using the entropy formula. A weaker conjecture would be that the coupling constant in deSitter space should take only discrete values not necessarily given by the entropy formula. We discuss quantization of the horizon in deSitter space by using Berezin's functorial quantization of Kahler manifolds and argue that the weak conjecture is valid for Euclidean deSitter space. Moreover it can be valid for a class of bounded complex symmetric spaces. 
  We consider an open Wilson line as a momentum representation of a boundary state which describes a D-brane in a constant B-field background. Using this picture, we study the Seiberg-Witten map which relates the commutative and noncommutative gauge fields, and determine the products of fields appearing in the general terms in the expansion of this map. 
  The spacetime symmetries of SGM action proposed as the gravitational coupling of N-G fermions are investigated. The commutators of new nonlinear supersymmetry (NL SUSY) transformations form a closed algebra, which reveals N-G fermion (NL SUSY) nature and a generalized general coordinate transformation. A generalized local Lorentz transformation, which forms a closed algebra, is also introduced. 
  Thus far, there seem to be no complete criteria that can settle the issue as to what the correct generalization of the Dirac-Born-Infeld (DBI) action, describing the low-energy dynamics of the D-branes, to the non-abelian case would be. According to recent suggestions, one might pass the issue of worldvolume solitons from abelian to non-abelian setting by considering the stack of multiple, coincident D-branes and use it as a guideline to construct or censor the relevant non-abelian version of the DBI action. In this spirit, here we are interested in the explicit construction of SU(2) Yang-Mills (YM) instanton solutions in the background geometry of two coincident probe D4-brane worldspaces particularly when the metric of target spacetime in which the probe branes are embedded is given by the Ricci-flat, magnetic extremal 4-brane solution in type IIA supergravity theory with its worldspace metric being given by that of Taub-NUT and Eguchi-Hanson solutions, the two best-known gravitational instantons. And then we demonstrate that with this YM instanton- gravitational instanton configuration on the probe D4-brane worldvolume, the energy of the probe branes attains its minimum value and hence enjoys stable state provided one employs the Tseytlin's non-abelian DBI action for the description of multiple probe D-branes. In this way, we support the arguments in the literature in favor of Tseytlin's proposal for the non-abelian DBI action. 
  We determine the form factor expansion of the one-point functions in integrable quantum field theory at finite temperature and find that it is simpler than previously conjectured. We show that no singularities are left in the final expression provided that the operator is local with respect to the particles and argue that the divergences arising in the non-local case are related to the absence of spontaneous symmetry breaking on the cylinder. As a specific application, we give the first terms of the low temperature expansion of the one-point functions for the Ising model in a magnetic field. 
  $AdS_{5}$ with linear dilaton and non vanishing $B$-field is shown to be a solution of the non critical string beta function equations. A non critical $(D=5)$ solution interpolating between flat space-time and $AdS_{5}$, with asymptotic linear dilaton and non vanishing $B$-field is also presented. This solution is free of space-time singularities and has got the string coupling constant everywhere bounded. Both solutions admit holographic interpretation in terms of ${\cal N}=0$ field theories. Closed string tachyon stability is also discussed. 
  I adapt the Gauge String, representing the strong coupling (SC) expansion in the continuous D>=3 Yang-Mills theory (YM_{D}) with a sufficiently large bare coupling constant \lambda>\lambda_{cr} and a fixed ultraviolet cut off \Lambda, to the analysis of the regularized Wilson's loop-averages. When generalized to describe the fat (rather than infinitely thin) flux-tubes, the pattern of thus modified U(N) Gauge String is proved to be consistent with the chain of the judiciously regularized U(N) Loop equations. In particular, we reveal the dimensional reduction YM_{D}=>YM_{2}, taking place in the extreme SC limit \lambda=>\infty, and compare it with the implications of the AdS/CFT correspondence conjecture. On the other hand, for the loop-averages associated to the sufficiently large minimal areas, the proposed stringy pattern is supposed to be in the one infrared universality class (provided the loops are without zig-zag backtrackings) with the novel implementation of the noncritical D-dimensional Nambu-Goto string. The peculiarity is due to the nonstandard \Lambda^{2}-scaling, \Lambda^{2}=O(\sigma_{ph}), of the physical string tension \sigma_{ph}. Being well-motivated from the viewpoint of the standard YM_{4} theory with \lambda=>0, this scaling is argued to entail that the considered modification of the Nambu-Goto system is in the stringy (rather than in the branched polymer) regime. In sum, the confinement in the continuous D>=3 U(N) (and, similarly, SU(N)) gauge theory is justified, for the first time, at least when both N and \lambda are sufficiently large. As a by-product, when continued to N=1, the Gauge String is shown to describe the continuous U(1) gauge theory with the monopoles. 
  The role of supercharge operators is studied in the case of a Dirac particle moving in a constant chromomagnetic field. The Hamiltonian is factorised and the ground state wave function in the case of unbroken supersymmetry is determined. 
  Differential structure of lattices can be defined if the lattices are treated as models of noncommutative geometry. The detailed construction consists of specifying a generalized Dirac operator and a wedge product. Gauge potential and field strength tensor can be defined based on this differential structure. When an inner product is specified for differential forms, classical action can be deduced for lattice gauge fields. Besides the familiar Wilson action being recovered, an additional term, related to the non-unitarity of link variables and loops spanning no area, emerges. 
  Motivated by a recently proposed "bigravity" model with two positive tension $AdS_4$ branes in $AdS_5$ by Kogan et al.[hep-th/0011141], we study behavior of bulk gauge field in the model. In this case, the zero mode of the gauge field is not constant but depends on the fifth dimensional coordinate transverse to the brane. The zero mode then becomes massive on the brane. From the physical requirement that the mass must be small, a parameter of the model is constrained. We also discuss the Kaluza-Klein modes. 
  Recent understanding of {\cal N}=1* supersymmetric theory (mass deformed {\cal N}=4) has made it possible to find an exact superpotential which encodes the properties of the different phases of the theory. We consider this superpotential as an illustrative example for the source of a nontrivial scalar potential for the string theory dilaton and study its properties. The superpotential is characterized by the rank of the corresponding gauge group (N) and integers p,q,k labelling the different massive phases of the theory. For generic values of these parameters, we find the expected runaway behaviour of the potential to vanishing string coupling. But there are also supersymmetric minima at weak coupling stabilizing the dilaton field. An interesting property of this potential is that there is a proliferation of supersymmetric vacua in the confining phases, with the number of vacua increasing with N and leading to a kind of staircase potential. For a range of parameters, it is possible to obtain realistic values for the gauge coupling. 
  We consider perturbations of unitary minimal models by boundary fields. Initially we consider the models in the limit as c -> 1 and find that the relevant boundary fields all have simple interpretations in this limit. This interpretation allows us to conjecture the IR limits of flows in the unitary minimal models generated by the fields \phi_{rr} of `low' weight. We check this conjecture using the truncated conformal space approach. In the process we find evidence for a new series of integrable boundary flows. 
  Using the Chern-Simons effective model of fractional quantum Hall (FQH) systems, we complete partial results obtained in the literature on FQHE concerning topological orders of FQH states. We show that there exists a class of effective FQH models having the same filling fraction $\nu$, interchanged under $ Gl(n,Z)$ transformations and extends results on Haldane hierarchy. We also show that Haldane states at any generic hierarhical level n may be realised in terms of n Laughlin states composites and rederive results for the n=2,3 levels respectively associated with $\nu= {{2}\over{5}}$ and $\nu= {{3}\over{7}}$ filling fractions. We study symmetries of the filling fractions series $\nu= {{p_2}\over{p_1 p_2 -1}}$ and $\nu= {{p_1 p_2 -1}\over{p_1 p_2 p_3 -p_1 -p_2}}$, with $p_1$ odd and $p_2$ and $p_3$ even integers, and show that, upon imposing the Gl(n,Z) invariance, we get remarkable informations on their stability. Then, we reconsider the Rausch de Traubenberg and Slupinsky (RdTS) algebra recently obtained in [1,2] and analyse its limit on the boundary $\partial({AdS_3})$ of the (1+2) dimensional manifold $AdS_3$. We show that generally one may distinguish bulk highest weight states (BHWS) living in $AdS_3$ and edge highest weight states (EHWS)living on the border $\partial({AdS_3})$. We explore these two kinds of RdTS representations carrying fractional values of the spin and propose them as condidates to describe the FQH states. 
  All fiber bundle with a given set of characteristic classes are viewable as particular projections of a more general bundle called a universal classifying space. This notion of projector valued field, a global definition of connections and gauge fields, can be useful to define vector bundles for non commutative base spaces. In this paper we derive the projector valued field for the fuzzy sphere, defining non-commutative n-monopole configurations, and check that in the classical limit, using the machinery of non-commutative geometry, the corresponding topological charges (Chern class) are integers. 
  We construct the covariant, spinor sets of relativistic wave equations for a massless field on the basis of the two copies of the R-deformed Heisenberg algebra. For the finite-dimensional representations of the algebra they give a universal description of the states with integer and half-integer helicity. The infinite-dimensional representations correspond formally to the massless states with fractional (real) helicity. The solutions of the latter type, however, break down the (3+1)$D$ Poincar\'e invariance to the (2+1)$D$ Poincar\'e invariance, and via a compactification on a circle a consistent theory for massive anyons in $D$=2+1 is produced. A general analysis of the ``helicity equation'' shows that the (3+1)$D$ Poincar\'e group has no massless irreducible representations with the trivial non-compact part of the little group constructed on the basis of the infinite-dimensional representations of $sl(2,\CC)$. This result is in contrast with the massive case where integer and half-integer spin states can be described on the basis of such representations, and means, in particular, that the (3+1)$D$ Dirac positive energy covariant equations have no massless limit. 
  We investigate the Polchinski ERG equation for d-dimensional O(N) scalar field theory. In the context of the non-perturbative derivative expansion we find families of regular solutions and establish their relation with the physical fixed points of the theory. Special emphasis is given to the large N limit for which many properties can be studied analytically. 
  To gain insight into the non-abelian Born-Infeld (NBI) action, we study coinciding D-branes wrapped on tori, and turn on magnetic fields on their worldvolume. We then compare predictions for the spectrum of open strings stretching between these D-branes, from perturbative string theory and from the effective NBI action. Under some plausible assumptions, we find corrections to the Str-prescription for the NBI action at order F^6. In the process we give a way to classify terms in the NBI action that can be written in terms of field strengths only, in terms of permutation group theory. 
  A problem of constructing excited state swave functions of the discrete spectrum of completely integrable quantum systems is considered. Recurrence relations defining wave functions up to the normalizing constant are obtained. 
  We establish an exact connection between the Choptuik scaling parameter for the three-dimensional BTZ black hole, and the imaginary part of the quasinormal frequencies for scalar perturbations. Via the AdS/CFT correspondence, this leads to an interpretation of Choptuik scaling in terms of the timescale for return to equilibrium of the dual conformal field theory. 
  We propose a systematic way of constructing $N=2, d=4$ superfield Born-Infeld action with a second nonlinearly realized N=2 supersymmetry. The latter, together with the manifest N=2 supersymmetry, form a central-charge extended $N=4, d=4$ supersymmetry. We embed the Goldstone-Maxwell N=2 multiplet into an infinite-dimensional off-shell supermultiplet of this N=4 supersymmetry and impose an infinite set of covariant constraints which eliminate all extra N=2 superfields through the Goldstone-Maxwell one. The Born-Infeld superfield Lagrangian density is one of these composite superfields. The constraints can be solved by iterations to any order in the fields. We present the sought N=2 Born-Infeld action up to the 10th order. It encompasses the action found earlier by Kuzenko and Theisen to the 8th order from a self-duality requirement. This is a strong indication that the complete N=2 Born-Infeld action with partially broken N=4 supersymmetry is also self-dual. 
  We show that the massive noncommutative U(1) can be embedded in a gauge theory by using the BFFT Hamiltonian formalism. By virtue of the peculiar non-Abelian algebraic structure of the noncommutative massive U(1) theory, several specific identities involving Moyal commutators had to be used in order to make the embedding possible. This leads to an infinite number of steps in the iterative process of obtaining first-class constraints. We also shown that the involutive Hamiltonian can be constructed. 
  The generalization of the Vafa-Witten theorem ruling out parity violation to QCD at finite temperature is considered. It is shown that this generalization of the theorem rules out Lorentz-invariant parity violating operators from spontaneously acquiring vacuum expectation values. However, it does not rule out Lorentz-noninvariant parity-violating operators from acquiring expectation values. Other situations where the theorem is inapplicable are also discussed. 
  We examine the inclusion of brane self-gravity in brane-world scenarios with three or more compact extra dimensions. If the brane is a thin, localized one, then we find that the geometry in its vicinity is warped in such a way that gravity on the brane can become very weak, independently of the volume of the extra dimensions. As a consequence, self-gravity can make the brane structure enter into the determination of the hierarchy between the Planck scale and a lower fundamental scale. In an extreme case, one can obtain a novel reformulation of the hierarchy problem in brane worlds, without the need for large-size extra dimensions; the hierarchy would be generated when the ratio between the scales of brane tension and brane thickness is large. In a sense, such a scenario is half-way between the one of Arkani-Hamed et al.(ADD) (although with TeV-mass Kaluza-Klein states) and that of Randall and Sundrum (RS1) (but with only a TeV brane, and of positive tension). We discuss in detail the propagation of fields in the background of this geometry, and find that no problems appear even if the brane is taken to be very thin. We also discuss the presence of black branes and black holes in this setting, and the possibility of having a Planck brane. 
  In the noncommutative field theory of open strings in a B-field, D-branes arise as solitons described as projection operators or partial isometries in a $C^*$ algebra. We discuss how D-branes on orbifolds fit naturally into this algebraic framework, through the examples of $R^n/G$, $T^n=R^n/Z^n$, and $T^n/G$. We also propose a framework for formulating D-branes on asymmetric orbifolds. 
  Recently a boundary string field theory that had been proposed some time ago, was used to calculate correctly the ratios of D-brane (both BPS and non-BPS) tensions. We discuss how this work is related to the boundary state formalism and open string closed string duality, and argue that the latter clarifies why the correct tension ratios are obtained in these recent calculations. 
  The lagrangian description of irreducible massless representations of the Poincare group with the corresponding Young tableaux having two rows along with some explicit examples including the notoph and Weyl tensor is given. For this purpose is used the method of the BRST constructions adopted to the systems of second class constraints by the construction of an auxiliary representations of the algebras of constraints in terms of Verma modules. 
  We study essentially non-Abelian backgrounds in the five dimensional N=4 gauged SU(2)$\times$U(1) supergravity. Static configurations that are invariant under either the SO(4) spatial rotations or with respect to the SO(3) rotations and translations along the fourth spatial coordinate are considered. By analyzing consistency conditions for the equations for supersymmetric Killing spinors we derive the Bogomol'nyi equations and obtain their globally regular solutions. The SO(4) symmetric configurations contain the purely magnetic non-Abelian fields together with the purely electric Abelian field and possess two unbroken supersymmetries. The SO(3) configurations have only the non-Abelian fields and preserve four supersymmetries. 
  We give a prescription to add the gravitational field of a global topological defect to a solution of Einstein's equations in an arbitrary number of dimensions. We only demand that the original solution has a O(n) invariance with n greater or equal 3. We will see that the general effect of a global defect is to introduce a deficit solid angle. We also show how the same kind of scalar field configurations can be used for spontaneous compactification of "n" extra dimensions on an n-sphere. 
  Using product integrals we review the unambiguous mathematical representation of Wilson line and Wilson loop operators, including their behavior under gauge transformations and the non-abelian Stokes theorem. Interesting consistency conditions among Wilson lines are also presented. 
  We consider compact four-dimensional ${\bf Z_N}\times {\bf Z_M}$ type IIB orientifolds, for certain values of $N$ and $M$. We allow the additional feature of discrete torsion and discuss the modification of the consistency conditions arising from tadpole cancellation. We point out the differences between the cases with and without discrete torsion. 
  The low energy physics of M theory near certain singularities of $G_2$-holonomy spacetimes can be described by pure N=1 super Yang-Mills theory in four dimensions. In this note we consider the cases when the gauge group is SO(2n), E6, E7 or E8. Confining strings with precisely the expected charges are naturally identified in proposed ``gravity duals'' of these singular M theory spacetimes. 
  We study Witten's background independent open-string field theory in the presence of a constant B-field at one loop level. The Green's function and the partition function with a constant B-field are evaluated for an annulus. 
  Using the method developed by Callan and Thorlacius, we study the low energy effective geometry on a two-dimensional string lattice by examining the energy-momentum relations of the low energy propagation modes on the lattice. We show that the geometry is identical for both the oscillation modes tangent and transverse to the network plane. We determine the relation between the geometry and the lattice variables. The lowest order effective field theory is given by the dimensional reduction of the ten-dimensional N=1 Maxwell theory. The gauge symmetry is related to a property of a three-string junction but not of a higher order junction. A half of the supersymmetries in the effective field theory should be broken at high energy. 
  After discussing the peculiarities of quantum systems on noncommutative (NC) spaces with non-trivial topology and the operator representation of the $\star$-product on them, we consider the Aharonov-Bohm and Casimir effects for such spaces. For the case of the Aharonov-Bohm effect, we have obtained an explicit expression for the shift of the phase, which is gauge invariant in the NC sense. The Casimir energy of a field theory on a NC cylinder is divergent, while it becomes finite on a torus, when the dimensionless parameter of noncommutativity is a rational number. The latter corresponds to a well-defined physical picture. Certain distinctions from other treatments based on a different way of taking the noncommutativity into account are also discussed. 
  We consider supersymmetric models in 5-dimensional space-time compactified on S**1/Z(2) orbifold where N=2 supersymmetry is explicitly broken down to N=1 by the orbifold projection. We find that the residual N=1 supersymmetry is broken spontaneously by a stable classical wall-like field configurations which can appear even in the simple models discussed. We also consider some simple models of bulk fields interacting with those localized on the 4-dimensional boundary wall where N=1 supersymmetry can survive in a rather non-trivial way. 
  We study D-branes in SU(2) WZW model by means of the boundary state techniques. We realize the ``fuzzy sphere'' configuration of multiple D0-branes as the boundary state with the insertion of suitable Wilson line. By making use of the path-integral representation we show that this boundary state preserves the appropriate boundary conditions and leads to the Cardy state describing a spherical D2-brane under the semi-classical approximation. This result directly implies that the spherical D2-brane in SU(2) WZW model can be well described as the bound state of D0-branes.   After presenting the supersymmetric extension, we also investigate the BPS and the non-BPS configurations of D-branes in the NS5 background. We demonstrate that the non-BPS configurations are actually unstable, since they always possess the open string tachyons. We further notice that the stable BPS bound state constructed by the tachyon condensation is naturally interpreted as the brane configuration of fuzzy sphere. 
  Domain wall solutions have attracted much attention due to their relevance for brane world scenarios and the holographic RG flow. In this talk I discuss the following aspects for these applications: (i) derivation of the first order flow equations as Bogomol'nyi bound; (ii) different types of critical points of the superpotential; (iii) the superpotential needed to localize gravity; (iv) the constraints imposed by supersymmetry including an example for an $N$=1 flow and finally (v) sources and exponential trapping of gravity. 
  We study the world-volume effective action of Dp-brane at the tachyonic vacuum which is equivalent to the zero tension limit. Using the Hamiltonian formalism we discuss the algebra of constraints and show that there is a non-trivial ideal of the algebra which corresponds to Virasoro like constraints. The Lagrangian treatment of the model is also considered. For the gauge fixed theory we construct the important subset of classical solutions which is equivalent to the string theory solutions in conformal gauge. We speculate on a possible quantization of the system. At the end a brief discussion of different background fields and fluctuations around the tachyonic vacuum is presented. 
  We summarise the present status of supersymmetric Randall-Sundrum brane-world scenarios and report on their possible realisation within five-dimensional matter coupled N=2 gauged supergravity. 
  We study the effects of interference between the self-dual and anti self-dual massive modes of the linearized Einstein-Chern-Simons topological gravity. The dual models to be used in the interference process are carefully analyzed with special emphasis on their propagating spectrum. We identify the opposite dual aspects, necessary for the application of the interference formalism on this model. The soldered theory so obtained displays explicitly massive modes of the Proca type. It may also be written in a form of Polyakov-Weigman identity to a better appreciation of its physical contents. 
  A review of the relationships between matrix models and noncommutative gauge theory is presented. A lattice version of noncommutative Yang-Mills theory is constructed and used to examine some generic properties of noncommutative quantum field theory, such as UV/IR mixing and the appearence of gauge-invariant open Wilson line operators. Morita equivalence in this class of models is derived and used to establish the generic relation between noncommutative gauge theory and twisted reduced models. Finite dimensional representations of the quotient conditions for toroidal compactification of matrix models are thereby exhibited. The coupling of noncommutative gauge fields to fundamental matter fields is considered and a large mass expansion is used to study properties of gauge-invariant observables. Morita equivalence with fundamental matter is also presented and used to prove the equivalence between the planar loop renormalizations in commutative and noncommutative quantum chromodynamics. 
  We apply quantum group methods for noncommutative geometry to the $Z_2\times Z_2$ lattice to obtain a natural Dirac operator on this discrete space. This then leads to an interpretation of the Higgs fields as the discrete part of spacetime in the Connes-Lott formalism for elementary particle Lagrangians. The model provides a setting where both the quantum groups and the Connes approach to noncommutative geometry can be usefully combined, with some of Connes' axioms, notably the first-order condition, replaced by algebraic methods based on the group structure. The noncommutative geometry has nontrivial cohomology and moduli of flat connections, both of which we compute. 
  We show that in the context of topological string theories N branes and M anti-branes give rise to Chern-Simons gauge theory with the gauge supergroup $U(N|M)$. We also identify a deformation of the theory which corresponds to brane/anti-brane annihilation. Furthermore we show that when $N=M$ all open string states are BRST trivial in the deformed theory. 
  We investigate the deformation of D-brane world-volumes in curved backgrounds. We calculate the leading corrections to the boundary conformal field theory involving the background fields, and in particular we study the correlation functions of the resulting system. This allows us to obtain the world-volume deformation, identifying the open string metric and the noncommutative deformation parameter. The picture that unfolds is the following: when the gauge invariant combination \omega = B + F is constant one obtains the standard Moyal deformation of the brane world-volume. Similarly, when d\omega = 0 one obtains the noncommutative Kontsevich deformation, physically corresponding to a curved brane in a flat background. When the background is curved, H = d\omega \not= 0, we find that the relevant algebraic structure is still based on the Kontsevich expansion, which now defines a nonassociative star product with an A_\infty homotopy associative algebraic structure. We then recover, within this formalism, some known results of Matrix theory in curved backgrounds. In particular, we show how the effective action obtained in this framework describes, as expected, the dielectric effect of D-branes. The polarized branes are interpreted as a soliton, associated to the condensation of the brane gauge field. 
  Using a Hamiltonian approach, we construct the classical and quantum theory of open WZW strings on a strip. (These are the strings which end on WZW branes.) The development involves non-abelian generalized Dirichlet images in an essential way. At the classical level, we find a new non-commutative geometry in which the equal-time coordinate brackets are non-zero at the world-sheet boundary, and the result is an intrinsically non-abelian effect which vanishes in the abelian limit. Using the classical theory as a guide to the quantum theory, we also find the operator algebra and the analogue of the Knizhnik-Zamolodchikov equations for the the conformal field theory of open WZW strings. 
  We present a technique that can be used to generate a static, axisymmetric solution of the Einstein-Maxwell-Dilaton equations from a stationary, axisymmetric solution of the vacuum Einstein equations. Starting from the Kerr solution, Davidson and Gedalin have previously made use of this technique to obtain a pair of oppositely charged, extremal dilatonic black holes, known as a black dihole. In this paper, we shall instead start from the Kerr-NUT solution. It will be shown that the new solution can also be interpreted as a dihole, but with the black holes carrying unbalanced magnetic charges. The effect of the NUT-parameter is to introduce a net magnetic charge into the system. Finally, we uplift our solution to ten dimensions to describe a system consisting of D6 and anti-D6-branes with unbalanced charges. The limit in which they coincide agrees with a solution recently derived by Brax et al.. 
  We study the spontaneous Lorentz symmetry breaking in a field theoretical model in (2+1)-dimension, inspired by string theory. This model is a gauge theory of an anti-symmetric tensor field and a vector field (photon). The Nambu-Goldstone (NG) boson for the spontaneous Lorentz symmetry breaking is identified with the unphysical massless photon in the covariant quantization. We also discuss an analogue of the equivalence theorem between the amplitudes for emission or absorption of the physical massive anti-symmetric tensor field and those of the unphysical massless photon. The low-energy effective action of the NG-boson is also discussed. 
  We study the spectrum of the spinless-Salpeter Hamiltonian H = \beta \sqrt{m^2 + p^2} + V(r), where V(r) is an attractive central potential in three dimensions. If V(r) is a convex transformation of the Coulomb potential -1/r and a concave transformation of the harmonic-oscillator potential r^2, then upper and lower bounds on the discrete eigenvalues of H can be constructed, which may all be expressed in the form E = min_{r>0} [ \beta \sqrt{m^2 + P^2/r^2} + V(r) ] for suitable values of P here provided. At the critical point the relative growth to the Coulomb potential h(r)=-1/r must be bounded by dV/dh < 2\beta/\pi. 
  The elliptic Gaudin model was obtained as the Hitchin system on an elliptic curve with two fixed points. In the present paper the algebraic-geometrical structure of the system with two fixed points is clarified. We identify this system with poles dynamics of the finite gap solutions of Davey-Stewartson equation. The solutions of this system in terms of theta-functions and the action-angle variables are constructed. We also discuss the geometry of its degenerations. 
  Based on the AdS/CFT correspondence, string theory has given exact predictions for circular Wilson loops in U(N) ${\cal N}=4$ supersymmetric Yang-Mills theory to all orders in a 1/N expansion. These Wilson loops can also be derived from Random Matrix Theory. In this paper we show that the result is generically insensitive to details of the Random Matrix Theory potential. We also compute all higher $k$-point correlation functions, which are needed for the evaluation of Wilson loops in arbitrary irreducible representations of U(N). 
  We construct 1/8, 1/4, and 1/2 BPS solutions spanned by diagonal elements of U(N) constant fluxes in the 6+1 dimensional U(N) super Yang-Mills theory on T^6 with topological stability. These solutions represent BPS bound states of D0, D2, D4, and D6 branes. The consistency with the D-brane charge conservation implies that unstable D0-D2-D6-brane systems in a B field decay to the BPS solutions, in which lower dimensional D-branes are dissolved in D6-branes. 
  The geometric action of modular groups for wedge regions (Bisognano-Wichmann property) is derived from the principles of local quantum physics for a large class of Poincare covariant models in d=4. As a consequence, the CPT theorem holds for this class. The models must have a complete interpretation in terms of massive particles. The corresponding charges need not be localizable in compact regions: The most general case is admitted, namely localization in spacelike cones. 
  We study cosmologies in the Randall-Sundrum models, incorporating the possibility of time-varying speed of light and Newton's constant. The cosmologies with varying speed of light (VSL) were proposed by Moffat and by Albrecht and Magueijo as an alternative to inflation for solving the cosmological problems. We consider the case in which the speed of light varies with time after the radion or the scale of the extra dimension has been stabilized. We elaborate on the conditions under which the flatness problem and the cosmological constant problem can be resolved. We find that the RS models are more restrictive about possible desirable VSL cosmological models than the standard general relativity. Particularly, the VSL cosmologies may provide with a possible mechanism for bringing the quantum corrections to the fine-tuned brane tensions after the SUSY breaking under control. 
  We derive a Melvin Universe type solution describing a magnetic field permeating the whole Universe in gravity minimally coupled to any non-linear electromagnetic theory, including Born-Infeld Theory. For a large set of non-linear electrodynamics theories, our solution is complete and non-singular, as long as the magnetic field is sub-critical. We examine some properties of the solution; in particular there is a shift of the symmetry axis and a non-standard period along the orbits of the U(1) symmetry to avoid a conical singularity. We show these are consistent with the usual Dirac quantization condition for the magnetic flux. We find exact solutions describing propagation of waves in the `generalized Melvin Universe' along the principal null directions of the electromagnetic field, where the Boillat and Einstein light-cone touch. By electric-magnetic duality we show that similar Melvin electric and dyonic universes can be obtained. 
  This review is devoted to Anti-Grand Unification and to the Multiple Point Model solution of problems of the unification of gauge interactions. According to this model, near the Planck scale there exists a Multiple Critical Point (MCP), where vacua of all fields existing in nature are degenerate. The MCP rules over the evolution of all inverse finestructure constants in the Standard model and beyond it. 
  Local commuting charges in sigma-models with classical Lie groups as target manifolds are shown to be related to the conserved quantities appearing in the Drinfeld-Sokolov (generalized mKdV) hierarchies. Conversely, the Drinfeld-Sokolov construction can be used to deduce the existence of commuting charges in these and in wider classes of sigma-models, including those whose target manifolds are exceptional groups or symmetric spaces. This establishes a direct link between commuting quantities in integrable sigma-models and in affine Toda field theories. 
  We perform here a critical analysis of some non-supersymmetric gravity solutions in Type 0B string theory. We first consider the most general configuration of parallel N electric and M magnetic D3-branes. The field theory living on their worldvolume is non-supersymmetric and non-conformal (if N is different from M) and has gauge group SU(N)XSU(M). We study the IR regime of the conjectured dual gravity background. A fine tuned solution exists with an asymptotically vanishing tachyon and a running dilaton, which could correspond to a flow towards an IR isolated fixed point at strong coupling. This opens the question upon the possibility of extending AdS/CFT techniques to flows towards IR isolated fixed points. We then use D3-branes as probes of this and other Type 0 backgrounds available in literature which cover the M=0 (or N=0) case. We shaw that for the fine tuned IR conformal solutions the stacks of parallel branes are plagued by instabilities, due to the repulsive force between branes of the same type. Curiously, for a particular solution which could be dual to a confining gauge theory, the stack should instead be stable. 
  We present an explicit formulation of supersymmetric Yang-Mills theories from $\D=$ 5 to 10 dimensions in the familiar $\N=1,\D=4$ superspace. This provides the rules for globally supersymmetric model building with extra dimensions and in particular allows us to simply write down $\N=1$ SUSY preserving interactions between bulk fields and fields localized on branes. We present a few applications of the formalism by way of illustration, including supersymmetric ``shining'' of bulk fields, orbifolds and localization of chiral fermions, anomaly inflow and super-Chern-Simons theories. 
  We consider the homogeneous cosmological radion, which we define as the interbrane distance in a two brane and $Z_2$ symmetrical configuration. In a coordinate system where one of the brane is at rest, the junction conditions for the second (moving) brane give directly the (non-linear) equations of motion for the radion. We analyse the radion fluctuations and solve the non-linear dynamics in some simple cases of interest. 
  In the context of $AdS/CFT$ correspondence the two Wilson loop correlator is examined at both zero and finite temperatures. On the basis of an entirely analytical approach we have found for Nambu-Goto strings the functional relation $d S_c^{(Reg)} / dL = 2 \pi k$ between Euclidean action $S_c$ and loop separation $L$ with integration constant $k$, which corresponds to the analogous formula for point-particles. The physical implications of this relation are explored in particular for the Gross-Ooguri phase transition at finite temperature. 
  It is shown that a double compactified D=11 supermembrane with non trivial wrapping may be formulated as a symplectic non-commutative gauge theory on the world volume. The symplectic non commutative structure is intrinsically obtained from the symplectic 2-form on the world volume defined by the minimal configuration of its hamiltonian. The gauge transformations on the symplectic fibration are generated by the area preserving diffeomorphisms on the world volume. Geometrically, this gauge theory corresponds to a symplectic fibration over a compact Riemman surface with a symplectic connection. 
  We consider the quasilocal thermodynamics of rotating black holes in asymptotically flat and asymptotically anti de Sitter spacetimes. Using the minimal number of intrinsic boundary counterterms inspired by the AdS/CFT correspondence, we find that we are able to carry out an analysis of the thermodynamics of these black holes for virtually all possible values of the rotation parameter and cosmological constant that leave the quasilocal boundary well-defined, going well beyond what is possible with background subtraction methods. Specifically, we compute the quasilocal energy $E$ and angular momentum $J$ for arbitrary values of the rotation, mass and cosmological constant parameters for the 3+1 dimensional Kerr, Kerr-AdS black holes and 2+1 dimensional BTZ black hole. We perform a quasilocal stability analysis and find phase behavior that is commensurate with previous analyses carried out at infinity. 
  We consider the classical equations of the Born-Infeld-Abelian-Higgs model (with and without coupling to gravity) in an axially symmetric ansatz. A numerical analysis of the equations reveals that the (gravitating) Nielsen-Olesen vortices are smoothly deformed by the Born-Infeld interaction, characterized by a coupling constant $\beta^2$, and that these solutions cease to exist at a critical value of $\beta^2$. When the critical value is approached, the length of the magnetic field on the symmetry axis becomes infinite. 
  We study a bosonic string with one end free and the other confined to a D-brane. Only the odd oscillator modes are allowed, which leads to a Virasoro algebra of even Virasoro modes only. The theory is quantized in a gauge where world-sheet time and ordinary time are identified. There are no negative or null norm states, and no tachyon. The Regge slope is twice that of the open string; this can serve as a test of the usefulness of the the model as a semi-quantitative description of mesons with one light and one extremely heavy quark when such higher spin mesons are found. The Virasoro conditions select specific SO(D-1) irreps. The asymptotic density of states can be estimated by adapting the Hardy-Ramanujan analysis to a partition of odd integers; the estimate becomes exact as D goes to infinity. 
  Previous $\lambda$-deformed {\it non-Hermitian} Hamiltonians with respect to the usual scalar product of Hilbert spaces dealing with harmonic oscillator-like developments are (re)considered with respect to a new scalar product in order to take into account their property of self-adjointness. The corresponding deformed $\lambda$-states lead to new families of coherent states according to the DOCS, AOCS and MUCS points of view. 
  We study supersymmetry breaking effects in N=1 SYM from the point of view of quantum effective actions. Restrictions on the geometry of the effective potential from superspace are known to be problematic in quantum effective actions, where explicit supersymmetry breaking can and must be studied. On the other hand the true ground state can be determined from this effective action, only. We study whether some parts of superspace geometry are still relevant for the effective potential and discuss whether the ground states found this way justify a low energy approximation based on this geometry. The answer to both questions is negative: Essentially non-semiclassical effects change the behavior of the auxiliary fields completely and demand for a new interpretation of superspace geometry. These non-semiclassical effects can break supersymmetry. 
  I briefly review how supersymmetry helps in the extraction of exact nonperturbative information from field theories, and then discuss some open problems in strongly coupled gauge theories. (Talk given at ``30 Years of Supersymmetry'' symposium in Minneapolis, Minnesota on October 15, 2000.) 
  We discuss the Seiberg-Witten solution of the non-commutative  N=2 U(N) SYM model. The solution is described in terms of the ordinary Seiberg-Witten curve of the SU(N) theory plus an additional free U(1). Hence, at the two-derivative approximation the theory flows to the ordinary commutative theory in the infra-red (k<1/sqrt(theta)). In particular, the center U(1) is free and it decouples from the other U(1)'s. In addition, no UV/IR mixing is found. 
  It is proven that the Poincare symmetry determines equations of motion, which are for massless particles of any spin in d-dimensional spaces linear in the momentum. The proof is made only for even d and for fields with no gauge symmetry. We comment on a few examples. We pay attention only to spin degrees of freedom. 
  We solve the superspace Bianchi identities for ten-dimensional supersymmetric Yang-Mills theory without imposing any kind of constraints apart from the standard conventional one. In this way we obtain a set of algebraic conditions on certain fields which in the on-shell theory are constructed as composite ones out of the physical fields. These conditions must hence be satisfied by any kind of theory in ten dimensions invariant under supersymmetry and some, abelian or non-abelian, gauge symmetry. Deformations of the ordinary SYM theory (as well as the fields) are identified as elements of a certain spinorial cohomology, giving control over field redefinitions and the distinction between physically relevant higher-order corrections and those removable by field redefinitions. The conditions derived severely constrain theories involving F^2-level terms plus higher-order corrections, as for instance those derived from open strings as effective gauge theories on D-branes. 
  We propose a supersymmetric generalization of Cardy's equation for consistent N=1 superconformal boundary states. We solve this equation for the superconformal minimal models SM(p/p+2) with p odd, and thereby provide a classification of the possible superconformal boundary conditions. In addition to the Neveu-Schwarz (NS) and Ramond (R) boundary states, there are NS~ states. The NS and NS~ boundary states are related by a Z_2 "spin-reversal" transformation. We treat the tricritical Ising model as an example, and in an appendix we discuss the (non-superconformal) case of the Ising model. 
  A gauge invariant regularisation which can be used for non-perturbative treatment of Yang-Mills theories within the exact renormalization group approach is constructed. It consists of a spontaneously broken SU(N|N) super-gauge extension of the initial Yang-Mills action supplied with covariant higher derivatives. We demonstrate that the extended theory in four dimensions is ultra-violet finite perturbatively and argue that it has a sensible limit when the regularisation cutoff is removed. 
  Reminiscences on the String origins of Supersymmetry are followed by a discussion of the importance of confusing bosons with fermions in building superstring theories in 9+1 dimensions. In eleven dimensions, the kinship between bosons and fermions is more subtle, and may involve the exceptional group $F_4$. 
  We study the relationship between three non-Abelian topologically massive gauge theories, viz. the naive non-Abelian generalization of the Abelian model, Freedman-Townsend model and the dynamical 2-form theory, in the canonical framework. Hamiltonian formulation of the naive non-Abelian theory is presented first. The other two non-Abelian models are obtained by deforming the constraints of this model. We study the role of the auxiliary vector field in the dynamical 2-form theory in the canonical framework and show that the dynamical 2-form theory cannot be considered as the embedded version of naive non-Abelian model. The reducibility aspect and gauge algebra of the latter models are also discussed. 
  Within the framework of the q-deformed Heisenberg algebra a dynamical equation of q-deformed quantum mechanics is discussed. The perturbative aspects of the q-deformed Schr\"odinger equation are analyzed. General representations of the additional momentum-dependent interaction originating from the q-deformed effects are presented in two approaches. As examples, such additional interactions related to the harmonic-oscillator potential and the Morse potential are demonstrated. 
  We review the brane world sum rules of Gibbons at al for compact five dimensional warped models with identical four-geometries and bulk dynamics involving scalar fields with generic potential. We show that the absence of dilaton tadpoles in the action functional of the theory is linked to one of these sum rules. Moreover, we calculate the dilaton mass term and derive the condition that is necessary for stabilizing the system. 
  We find soliton solutions in the 2+1 dimensional non-commutative Maxwell Chern-Simons Higgs theories. In the limit of the Chern-Simons coefficient going to zero, these solutions go over to the previously found solutions in the non-commutative Maxwell Higgs theories. The new solutions may have relevance in the theory of the fractional quantum Hall effect and possibly in string vacua corresponding to open strings terminating on D2 branes in the presence of D0 branes. 
  In earlier papers on the loop variable approach to gauge invariant interactions in string theory, a ``wave functional'' with some specific properties was invoked.  It had the purpose of converting the generalized momenta to space time fields. In this paper we describe this object in detail and give some explicit examples. We also work out the interacting equations of the massive mode of the bosonic string, interacting with electromagnetism, and discuss in detail the gauge invariance. This is naturally described in this approach as a massless spin two field interacting with a massless spin one field in a higher dimension. Dimensional reduction gives the massive system. We also show that in addition to describing fields perturbatively, as is required for reproducing the perturbative equations, the wave functional can be chosen to reproduce the Born-Infeld equations, which are non-perturbative in field strengths. This makes contact with the sigma model approach. 
  In a geometrical background, D-brane charge is classified by topological K-theory. The corresponding classification of D-brane charge in an arbitrary, nongeometrical, compactification is still a mystery. We study D-branes on non-simply-connected Calabi-Yau 3-folds, with particular interest in the D-branes whose charges are torsion elements of the K-theory. We argue that we can follow the D-brane charge through the nongeometrical regions of the Kahler moduli space and, as evidence, explicitly construct torsion D-branes at the Gepner point in some examples. In one of our examples, the Gepner theory is a nonabelian orbifold of a tensor product of minimal models, and this somewhat exotic situation seems to be essential to the physics. 
  In this paper we use the analyticity properties of the scattering amplitude in the context of the conformal mapping techniques. The Schwarz-Christoffel and Riemann-Schwarz functions are used to map the upper half -plane onto a triangle. We use the known asymptotic and threshold behaviors of the scattering amplitude to establish a connection between the values of the Regge trajectory functions and the angles of the triangle. This geometrical interpretation allows a link between values of the Regge trajectory functions and the generators of the invariance group of Moebius transformations associated with the underlying automorphic function. The formalism provides useful new relations between analyticity, geometry, Regge trajectory functions, Veneziano model, groups of Moebius transformations and automorphic functions. It is hoped that they will provide avenues for further work. 
  String/Field theory correspondences have been discussed heavily in recent years. Here, we describe a testing scenario involving a non-perturbative field theory calculation using the framework of supersymmetric discrete light-cone quantization (SDLCQ). We consider a Maldacena-type conjecture applied to the near horizon geometry of a D1-brane in the supergravity approximation. Numerical results of a test of this conjecture are presented with orders of magnitude more states than we previously considered. These results support the Maldacena conjecture and are within 10-15% of the predicted results. We present a method for using a ``flavor'' symmetry to greatly reduce the size of the Fock basis and discuss a numerical method that we use which is particularly well suited for this type of matrix element calculation. Our results are still not sufficient to demonstrate convergence, and, therefore, cannot be considered to be a numerical proof of the conjecture. We update our continuous efforts to improve on these results and present some results on the way to higher dimensional scenarios. 
  We discuss the regularization and renormalization of QED with Lorentz and CPT violation, and argue that the coefficient of the Chern-Simons term is an independent parameter not determined by gauge invariance. We also study these issues in a model with spontaneous breaking of Lorentz and CPT symmetries and find an explicit relation with the ABJ anomaly. This explains the observed convergence of the induced Chern-Simons term. 
  We examine noncommutative linear sigma models with U(N) global symmetry groups at the one-loop quantum level, and contrast the results with our previous study of the noncommutative O(N) linear sigma models where we have shown that Nambu-Goldstone symmetry realization is inconsistent with continuum renormalization. Specifically we find no violation of Goldstone's theorem at one-loop for the U(N) models with the quartic term ordering consistent with possible noncommutative gauging of the model. The difference is due to terms involving noncommutative commutator interactions, which vanish in the commutative limit. We also examine the U(2), and O(4) linear sigma models with matter in the adjoint representation, and find that the former is consistent with Goldstone's theorem at one-loop if we include only trace invariants consistent with possible noncommutative gauging of the model, while the latter exhibits violations of Goldstone's theorem of the kind seen in the fundamental of O(N) for N>2. 
  We study the linearized metric perturbation corresponding to the radion for the generalization of the five dimensional two brane setup of Randall and Sundrum to the case when the curvature of each brane is locally constant but non-zero. We find the wave fuction of the radion in a coordinate system where each brane is sitting at a fixed value of the extra coordinate. We find that the radion now has a mass$^2$, which is negative for the case of de Sitter branes but positive for anti de Sitter branes. We also determine the couplings of the radion to matter on the branes, and construct the four dimensional effective theory for the radion valid at low energies. In particular we find that in AdS space the wave function of the radion is always normalizable and hence its effects, though small, remain finite at arbitrarily large brane separations. 
  Vacuum expectation values of local fields for all dual pairs of non-simply laced affine Toda field theories recently proposed are checked against perturbative analysis. The computations based on Feynman diagram expansion are performed upto two-loops. Agreement is obtained. 
  This paper has been withdrawn. 
  The Cho-Faddeev-Niemi-Shabanov decomposition of the SU(2) Yang-Mills field is employed for the calculation of the corresponding Wilsonian effective action to one-loop order with covariant gauge fixing. The generation of a mass scale is observed, and the flow of the marginal couplings is studied. Our results indicate that higher-derivative terms of the color-unit-vector $\mathbf{n}$ field are necessary for the description of topologically stable knotlike solitons which have been conjectured to be the large-distance degrees of freedom. 
  The convergence of the derivative expansion of the exact renormalisation group is investigated via the computation of the beta function of massless scalar lambda phi^4 theory. The derivative expansion of the Polchinski flow equation converges at one loop for certain fast falling smooth cutoffs. Convergence of the derivative expansion of the Legendre flow equation is trivial at one loop, but also can occur at two loops and in particular converges for an exponential cutoff. 
  Contents: 1. Introduction, 2. Chiral gauge theories & the gauge anomaly, 3. The regularization problem, 4. Weyl fermions from 4+1 dimensions, 5. The Ginsparg-Wilson relation, 6. Gauge-invariant lattice regularization of anomaly-free theories. 
  We derive the hard thermal loop action for soft electromagnetic fields in the finite temperature world-line formulation at imaginary time, by first integrating out the hard fermion modes from the microscopic QED action. Further, using the finite T world-line method, we calculate all static higher order terms in the soft electromagnetic field. At high T, the leading non-linear terms are independent of the temperature and, except for a term quartic in the time component of the vector potential, they cancel exactly against the vacuum contribution. The remaining T-dependent non-linear terms become more strongly suppressed by the temperature as the number of soft fields increases, thus making the expansion reliable. Applications of this method to other theories and problems at the soft scale are also briefly discussed. 
  Starting from SO(n,m) groups, we are in search of groups that: (1.) in a simple way, include N supersymmetric generators(see \cite{Nahm}) (2.) contain as subgroup: the de Sitter group SO(4,1) or the Anti-de Sitter group SO(3,2) (3.) permit nontrivial gauge symmetry groups. The smallest groups satisfyng above conditions are the OSp(N|4) groups, which contain Sp(4)xSO(N) ($Sp(4)\sim SO(3,2)$) or OSp(1|4)xSO(N-1). Because of this, it is possible to generate P(3,1)xG using groups contraction mechanism, which may be: $SO(3,2)\to P(3,1)$ o $OSp(N|4)\to ^SP(3,1|N)$ where P(3,1) is the Poincar\'e group and G is a gauge group, say SO(N) or SO(N-1). This group contraction mechanism and its consequences upon different groups representations including SO(3,2) or SO(4,1), is clarified and extended to OSp(N|4) representations (see \cite{Nicolai}), contracted to its N-extensi\'on SuperPoincar\'e group $\ ^SP(3,1|N)$. 
  Relations between the global structure of the gauge group in elliptic F-theory compactifications, fractional null string junctions, and the Mordell-Weil lattice of rational sections are discussed. We extend results in the literature, which pertain primarily to rational elliptic surfaces and obtain pi^1(G) where G is the semi-simple part of the gauge group. We show how to obtain the full global structure of the gauge group, including all U(1) factors. Our methods are not restricted to rational elliptic surfaces. We also consider elliptic K3's and K3-fibered Calabi-Yau three-folds. 
  The quantum-induced dilatonic brane world (New Brane World) is created by brane CFT quantum effects (giving effective brane tension) in accordance with AdS/CFT set-up which also defines surface term. Considering the bosonic sector of 5d gauged supergravity with single scalar and taking the boundary action as predicted by supersymmetry, the possibility to supersymmetrize dilatonic New Brane World is discussed. It is demonstrated that for a number of superpotentials the flat SUSY dilatonic brane-world (with dynamically induced brane dilaton) or quantum-induced de Sitter dilatonic brane-world (not Anti-de Sitter one) where SUSY is broken by the quantum effects occurs. The analysis of graviton perturbations indicates that gravity is localized on such branes. 
  In this talk we construct BPS-saturated domain walls in supersymmetric QCD, for any values of the masses of the chiral matter superfields. We compare our results to those already obtained in the literature and we also discuss their range of applicability, as well as future directions that would be desirable to explore in order to achieve a complete understanding of supersymmetric gluodynamics as a step in improving our knowledge of how QCD works. 
  It is shown that the local coupling of a higher dimensional graviton to a closed degenerate two-form produces dimensional reduction by spontaneous breakdown of extra-dimensional translational symmetry. Four dimensional Poincar\'e invariance emerges as residual symmetry. As a specific example, a six dimensional geometry coupled to a closed rank 2 two-form yields the `ground state' $$ds^2={\rm e}^{-|\xi|^2/4l^2}\eta_{\mu\nu}dx^\mu dx^\nu+\delta_{ij} d\xi^i d\xi^j$$ with $l$ a fundamental length scale. At low energies, space-time reduces to four observable dimensions and general relativistic gravity is reproduced. 
  We study the asymptotic behavior of the bulk spacetimes with the negative cosmological constant in the context of the brane-world scenario. We show that, in Euclidean bulk, or in Lorentzian static bulk, some sequences of hypersurfaces with the positive Ricci scalar evolve to the warped geometries like the anti-de Sitter spacetime. Based on the AdS/CFT correspondence, we discuss that the positivity of the Ricci scalar is related to the stability of CFT on the brane. In addition, the brane-world is described from the holographic point of view. The asymptotic local structure of the conformal infinity is also investigated. 
  The Lax pair of the Ruijsenaars-Schneider model with interaction potential of trigonometric type based on Dn Lie algebra is presented. We give a general form for the Lax pair and prove partial results for small n. Liouville integrability of the corresponding system follows a series of involutive Hamiltonians generated by the characteristic polynomial of the Lax matrix. The rational case appears as a natural degeneration and the nonrelativistic limit exactly leads to the well-known Calogero-Moser system associated with Dn Lie algebra. 
  The path integral formulation of Quantum Field Theory implies an infinite set of local, Schwinger-Dyson-like relations. Exact renormalization group equations can be cast as a particular instance of these relations. Furthermore, exact scheme independence is turned into a vector field transformation of the kernel of the exact renormalization group equation under field redefinitions. 
  We show that the Higgs branch of a four-dimensional Yang-Mills theory, with gauge and matter content summarised by an E_8 quiver diagram, is identical to the generalised Coulomb branch of a four-dimensional superconformal strongly coupled gauge theory with E_8 global symmetry. This is the final step in showing that there is a Higgs-Coulomb identity of this kind for each of the cases {0}, A_1, A_2, D_4, E_6, E_7 and E_8. This series of equivalences suggests the existence of a mirror symmetry between the quiver theories and the strongly coupled theories. We also discuss how to interpret the parameters of the quiver gauge theory in terms of the Hanany-Witten picture. 
  A new family of A_N-type Dunkl operators preserving a polynomial subspace of finite dimension is constructed. Using a general quadratic combination of these operators and the usual Dunkl operators, several new families of exactly and quasi-exactly solvable quantum spin Calogero-Sutherland models are obtained. These include, in particular, three families of quasi-exactly solvable elliptic spin Hamiltonians. 
  The multiplicative anomaly, recently introduced in QFT, plays a fundamental role in solving some mathematical inconsistencies of the widely used zeta-function regularization method. Its physical relevance is still an open question and is here analyzed in the light of a non-perturbative method. Even in this approach the ``different physics'' seems to hold and not to be easily removable by renormalization. 
  We discuss in details a simple, purely bosonic, quantum field theory belonging to larger class of models with the following properties: a) They are asymptotically free, with a dynamically generated mass scale. b) They have a space of parameters which gets quantum corrections drastically modifying the classical singularity structure. The quantum theory can have massless solitons, Argyres-Douglas-like CFTs, exhibit confinement, etc... c) The physics can, to a large extent, be worked out in models with a large number of supersymmetries as well as in purely bosonic ones. In the former case, exact BPS mass formulas can be derived, brane constructions and embedding in M theory do exist. d) The models have an interesting 1/N expansion, and it is possible to define a double scaling limit in the sense of the ``old'' matrix models when approaching the singularities in parameter space. These properties make these theories very good toy models for four dimensional gauge theories with Higgs fields, and provide a framework where the effects of breaking supersymmetry can be explicitly studied. In our model, we work out in details the quantum space of parameters. We obtain the non-local lagrangian description of the Argyres-Douglas-like CFT, and show that it admits a strongly coupled fixed point. We also explicitly demonstrate property d). The possibility of defining such double scaling limits was not anticipated on the gauge theory side, and could be of interest to understand the gauge theory/string theory correspondence. 
  We consider a brane-universe in the background of an Anti-de Sitter/ Schwarschild geometry. We show that the induced geometry of the brane is exactly given by that of a standard radiation dominated FRW-universe. The radiation is represented by a strongly coupled CFT with an AdS-dual description. We show that when the brane crosses the horizon of the AdS-black hole the entropy and temperature are simply expressed in the Hubble constant and its time derivative. We present formulas for the entropy of the CFT which are generally valid, and which at the horizon coincide with the FRW equations. These results shed new light on recently proposed entropy bounds in the context of cosmology. 
  We discuss spectra of $AdS_3$ supergravities, arising in the near horizon geometry of D1/D5 systems in orbifolds/orientifolds of type IIB theory with 16 supercharges. These include models studied in a recent paper (hep-th/0012118), where the group action involves also a shift along a transversal circle, as well as IIB/$\Omega I_4$, which is dual to IIB on $K3$. After appropriate assignements of the orbifold group eigenvalues and degrees to the supergravity single particle spectrum, we compute the supergravity elliptic genus and find agreement, in the expected regime of validity, with the elliptic genus obtained using U-duality map from (4,4) CFTs of U-dual backgrounds. Since this U-duality involves the exchange of KK momentum $P$ and D1 charge $N$, it allows us to test the (4,4) CFTs in the $P < N/4$ and $N < P/4$ regimes by two different supergravity duals. 
  We investigate the quantization of the theta-expanded noncommutative U(1) Yang-Mills action, obtained via the Seiberg-Witten map. As expected we find non-renormalizable terms. The one-loop propagator corrections are gauge independent, and lead us to a unique extention of the noncommutative classical action. We interpret our results as a requirement that also the trace in noncommutative field theory should be deformed. 
  We study the low-energy dynamics of noncommutative $\N=2$ supersymmetric U(N) Yang-Mills theories in the Coulomb phase. Exact results are derived for the leading terms in the derivative expansion of the Wilsonian effective action. We find that in the infrared regime the U(1) subgroup decouples, and the remaining SU(N) is described by the ordinary commutative Seiberg-Witten solution. IR/UV mixing is present in the U(1), but not in SU(N). Our analysis is based on explicit perturbative and multi-instanton calculations. 
  The open supermembrane contribution to the non-perturbative superpotential of bulk space five-branes in heterotic M-theory is presented. We explicitly compute the superpotential for the modulus associated with the separation of a bulk five-brane from an end-of-the-world three-brane. The gauge and kappa-invariant boundary strings of such open supermembranes are given and the role of the holomorphic vector bundle on the orbifold fixed plane boundary is discussed in detail. 
  In a classical, quartic field theory with $SU(N) \times Z_2$ symmetry, a class of kink solutions can be found analytically for one special choice of parameters. We construct these solutions and determine their energies. In the limit $N\to \infty$, the energy of the kink is equal to that of a kink in a $Z_2$ model with the same mass parameter and quartic coupling (coefficient of ${\rm Tr}(\Phi^4)$). We prove the stability of the solutions to small perturbations but global stability remains unproven. We then argue that the continuum of choices for the boundary conditions leads to a whole space of kink solutions. The kinks in this space occur in classes that are determined by the chosen boundary conditions. Each class is described by the coset space $H/I$ where $H$ is the unbroken symmetry group and $I$ is the symmetry group that leaves the kink solution invariant. 
  The noncommutative generalisation of the standard electroweak model due to Balakrishna, Gursey and Wali is formulated in terms of the derivations Der_2(M_3) of a three dimensional representation of the su(2) Lie algebra of weak isospin. A light Higgs boson of mass about 130 GeV, together with four very heavy scalar bosons are predicted. 
  It is shown how the old Cartan's conjecture on the fundamental role of the geometry of simple (or pure) spinors, as bilinearly underlying euclidean geometry, may be extended also to quantum mechanics of fermions (in first quantization), however in compact momentum spaces, bilinearly constructed with spinors, with signatures unambiguously resulting from the construction, up to sixteen component Majorana-Weyl spinors associated with the real Clifford algebra $\Cl(1,9)$, where, because of the known periodicity theorem, the construction naturally ends. $\Cl(1,9)$ may be formulated in terms of the octonion division algebra, at the origin of SU(3) internal symmetry.   In this approach the extra dimensions beyond 4 appear as interaction terms in the equations of motion of the fermion multiplet; more precisely the directions from 5$^{th}$ to 8$^{th}$ correspond to electric, weak and isospin interactions $(SU(2) \otimes U(1))$, while those from 8$^{th}$ to 10$^{th}$ to strong ones SU(3). There seems to be no need of extra dimension in configuration-space. Only four dimensional space-time is needed - for the equations of motion and for the local fields - and also naturally generated by four-momenta as Poincar\'e translations.   This spinor approach could be compatible with string theories and even explain their origin, since also strings may be bilinearly obtained from simple (or pure) spinors through sums; that is integrals of null vectors. 
  One of the simplest example of non-commutative (NC) spaces is the NC plane. In this article we investigate the consequences of the non-commutativity to the quantum mechanics on a plane. We derive corrections to the standard (commutative) Hamiltonian spectrum for hydrogen-like atom and isotropic linear harmonic oscillator (LHO) and formulate the problem of the potential scattering on the NC plane. In the case of LHO we consider the noncommutativity of the momentum operators, too. 
  The near-horizon properties of a black hole are studied within an algebraic framework, using a scalar field as a simple probe to analyze the geometry. The operator H governing the near-horizon dynamics of the scalar field contains an inverse square interaction term. It is shown that the operators appearing in the corresponding algebraic description belong to the representation space of the Virasoro algebra. The operator H is studied using the representation theory of the Virasoro algebra. We observe that the wave functions exhibit scaling behaviour in a band-like region near the horizon of the black hole. 
  We describe graded contractions of Virasoro algebra. The highest weight representations of Virasoro algebra are constructed. The reducibility of representations is analysed. In contrast to standart representations the contracted ones are reducible except some special cases. Moreover we find an exotic module with null-plane on fifth level. 
  A root systems in Carroll spaces with degenerate metric are defined. It is shown that their Cartan matrices and reflection groups are affine. With the help of the geometric consideration the root system structure of affine algebras is determined by a sufficiently simple algorithm. 
  A gauge invariant regularisation for dealing with pure Yang-Mills theories within the exact renormalization group approach is proposed. It is based on the regularisation via covariant higher derivatives and includes auxiliary Pauli-Villars fields which amounts to a spontaneously broken SU(N|N) super-gauge theory. We demonstrate perturbatively that the extended theory is ultra-violet finite in four dimensions and argue that it has a sensible limit when the regularization cutoff is removed. 
  In generalized Yang-Mills theories scalar fields can be gauged just as vector fields in a usual Yang-Mills theory, albeit it is done in the spinorial representation. The presentation of these theories is aesthetic in the following sense: A physical theory using Yang-Mills theories requires several terms and irreducible representations, but with generalized Yang-Mills theories, only two terms and two irreducible representations are required. These theories are constructed based upon the maximal subgroups of the gauge Lie group. The two terms of the lagrangian are the kinetic energy of fermions and of bosons. A brief review of Yang-Mills theories and covariant derivatives is given, then generalized Yang-Mills theories are defined through a generalization of the covariant derivative. Two examples are given, one pertaining the Glashow-Weinberg-Salam model and another SU(5) grand unification. The first is based upon a $U(3)\supset U(1)\times U(1)\times SU(2)$ generalized Yang-Mills theory, and the second upon a $SU(6)\supset U(1)\times SU(5)$ theory. The possibility of expressing generalized Yang-Mills theories using a five-dimensional formalism is also studied. The situation is unclear in this case. At the end a list of comments and criticisms is given. 
  The dynamics of a Dp brane can be described either by an open string ending on this brane or by an open D(p - 2) brane ending on the same Dp brane. The ends of the open string couple to a Dp brane worldvolume gauge field while the boundary of the open D(p - 2) brane couples to a (p - 2)-form worldvolume potential whose field strength is Poincare dual to that of the gauge field on the Dp-brane worldvolume. With this in mind, we find that the Poincare dual of the fixed rank-2 magnetic field used in defining a (1 + p)-dimensional noncommutative Yang-Mills (NCYM) gives precisely a near-critical electric field for the open D(p - 2) brane. We therefore find (1 + p)-dimensional open D(p - 2) brane theories along the same line as for obtaining noncommutative open string theories (NCOS), OM theory and open Dp brane theories (ODp) from NS5 brane. Similarly, the Poincare dual of the near-critical electric field used in defining a (1 + p)-dimensional NCOS gives a fixed magnetic-like field. This field along with the same bulk field scalings defines a (1 + p)-dimensional noncommutative field theory. In the same spirit, we can have various (1 + 5)-dimensional noncommutative field theories resulting from the existence of ODp if the description of open D(4 - p) brane ending on the NS5 brane is insisted. 
  I put forward the stringy representation of the 1/N strong coupling (SC) expansion for the regularized Wilson's loop-averages in the continuous D>=3 Yang-Mills theory (YM_{D}) with a sufficiently large bare coupling constant \lambda>\lambda_{cr} and a fixed ultraviolet cut off \Lambda. The proposed representation is proved to provide with the confining solution of the Dyson-Schwinger chain of the judiciously regularized U(N) Loop equations. Building on the results obtained, we suggest the stringy pattern of the low-energy theory associated to the D=4 U(\infty)=SU(\infty) gauge theory in the standard \lambda=>0 phase with the asymptotic freedom in the UV domain. A nontrivial test, to clarify whether the AdS/CFT correspondence conjecture may be indeed applicable to the large N pure YM_{4} theory in the \lambda=>\infty limit, is also discussed. 
  Supersymmetry transformations may be represented by unitary operators in a formulation of supersymmetry without numbers that anti-commute. The physical relevance of this formulation hinges on whether or not one may add states of even and odd fermion number, a question which soon may be settled by experiment. 
  We consider the derivation of equivalent unconstrained systems for QCD given in the class of functions of nontrivial topological gauge transformations. We show that the unconstrained QCD obtained by resolving the Gauss law constraint contains a monopole, a zero mode of the Gauss law, and a rising potential, which can explain the phenomena of confinment and hadronization as well as spontaneous chiral symmetry breaking and the $\eta$-$\eta'$-mass difference. 
  We explore the embedding of Spin groups of arbitrary dimension and signature into simple superalgebras in the case of extended supersymmetry. The R-symmetry, which generically is not compact, can be chosen compact for all the cases that are congruent mod 8 to the physical conformal algebra so($D-2$,2), $D\geq 3$. An $\rm{so}(1,1)$ grading of the superalgebra is found in all cases. Central extensions of super translation algebras are studied in this framework. 
  We quantize the spherically symmetric sector of generic charged black holes. Thermal properties are encorporated by imposing periodicity in Euclidean time, with period equal to the inverse Hawking temperature of the black hole. This leads to an exact quantization of the area (A) and charge (Q) operators. For the Reissner-Nordstr\"om black hole, $A=4\pi G \hbar (2n+p+1)$ and $Q=me$, for integers $n,p,m$. Consistency requires the fine structure constant to be quantized: $e^2/\hbar=p/m^2$. Remarkably, vacuum fluctuations exclude extremal black holes from the spectrum, while near extremal black holes are highly quantum objects. We also prove that horizon area is an adiabatic invariant. 
  We examine, in the context of certain string compactifications resulting in five dimensional brane worlds the mechanisms of (self) tuning of the cosmological constant and the recovery of standard cosmological evolution. We show that self tuning can occur only as long as supersymmetry is unbroken (unless additional assumptions are made) and that the adjustment of the cosmological constant to zero after supersymmetry breaking and the recovery of standard evolution are the same problem verifying previously made statements in the context of general i.e. not necessarily string theoretic brane worlds. We emphasize, however, that contrary to general brane worlds where the above adjustment requires a fine tuning, stringy brane worlds contain an additional integration constant due to the presence of the compact space thus allowing the adjustment to be done only with integration constants. 
  In this paper we would like to discuss the emergence of D-branes from infinite many D-instantons in bosonic and type IIA string theory in the framework of boundary string field theory. 
  We review recent progress in understanding the anti-de Sitter/conformal field theory correspondence in the context of two-dimensional dilaton gravity theory. 
  We discuss various superstring effective actions and, in particular, their common sector which leads to the so-called pre-big-bang cosmology (cosmology in a weak coupling limit of heterotic superstring). Then, we review the main ideas of the Horava-Witten theory which is a strong coupling limit of heterotic superstring theory. Using the conformal relationship between these two theories we present Kasner asymptotic solutions of Bianchi type IX geometries within these theories and make predictions about possible emergence of chaos. Finally, we present a possible method of generating Horava-Witten cosmological solutions out of the well-known general relativistic pre-big-bang solutions. 
  Chiral gauge anomalies on noncommutative Minkowski Space-time are computed and have their origin elucidated. The consistent form and the covariant form of the anomaly are obtained. Both Fujikawa's method and Feynman diagram techniques are used to carry out the calculations 
  The proof is presented that the Poincare symmetry determines the equations of motion for massless particles of any spin in 2n-dimensional spaces, which are linear in the momentum. 
  We show that in the system of two fermions interacting by scalar exchange, the solutions for J$^{\pi}$=$0^+$ bound states are stable without any cutoff regularization for coupling constant below some critical value. 
  The Inozemtsev limit (IL) or the scaling limit is known to be a procedure applied to the elliptic Calogero Model. It is a combination of the trigonometric limit, infinite shifts of particles coordinates and rescalings of the coupling constants. As a result, one obtains an exponential type of interaction. In the recent paper it is shown that the IL applied to the $sl(N,\bf C)$ elliptic Euler-Calogero Model and the elliptic Gaudin Model produces new Toda-like systems of $N$ interacting particles endowed with additional degrees of freedom corresponding to a coadjoint orbit in $sl(n,\bf C)$. The limits corresponding to the complete degeneration of the orbital degrees provide only ordinary periodic and non periodic Toda systems. We introduce a classification of the systems appearing in the $sl(3,\bf C)$ case via IL. The classification is represented on two-dimensional space of parameters describing the infinite shifts of the coordinates. This space is subdivided into symmetric domains. The mixture of the Toda and the trigonometric Calogero-Sutherland potentials emerges on the low dimensional domain walls of this picture. Due to obvious symmetries this classification can be generalized to the arbitrary number of particles. We also apply IL to $sl(2,\bf C)$ elliptic Gaudin Model with two marked points on the elliptic curve and discuss main features of its possible limits. The limits of Lax matrices are also considered. 
  The non-commutative version of the euclidean $g^2\phi^4$ theory is considered. By using Wilsonian flow equations the ultraviolet renormalizability can be proved to all orders in perturbation theory. On the other hand, the infrared sector cannot be treated perturbatively and requires a resummation of the leading divergencies in the two-point function. This is analogous to what is done in the Hard Thermal Loops resummation of finite temperature field theory. Next-to-leading order corrections to the self-energy are computed, resulting in $O(g^3)$ contributions in the massless case, and $O(g^6\log g^2)$ in the massive one. 
  We study the fluctuation modes for lump solutions of the tachyon effective potential in p-adic open string theory. We find a discrete spectrum with equally spaced mass squared levels. We also find that the interactions derived from this field theory are consistent with p-adic string amplitudes for excited string states. 
  We discuss the $\phi^4$ and $\phi^6$ theory defined in a flat $D$-dimensional space-time. We assume that the system is in equilibrium with a thermal bath at temperature $\beta^{-1}$. To obtain non-perturbative result, the $ 1/N $ expansion is used. The method of the composite operator (CJT) for summing a large set of Feynman graphs, is developed for the finite temperature system. The ressumed effective potential and the analysis of the D=3 and D=4 cases are given. 
  The helicity flip of a spin-1/2 Dirac fermion interacting with a torsion- field endowed with a pseudo-tensorial extension is analysed. Taking the torsion to be represented by a Kalb-Ramond field, we show that there is a finite amplitude for helicity flip for massive fermions. The lowest order contribution is proportional to the pseudo-tensor term. 
  These notes are a short review of the q-deformed fuzzy sphere S^2_{q,N}, which is a ``finite'' noncommutative 2-sphere covariant under the quantum group U_q(su(2)). We discuss its real structure, differential calculus and integration for both real q and q a phase, and show how actions for Yang-Mills and Chern- Simons-like gauge theories arise naturally. It is related to D-branes on the SU(2)_k WZW model for q = exp(\frac{i \pi}{k+2}). 
  The progress brought to the study of chiral fermions and gauge theories by quantization methods with a bulk time suggests their usefulness in supersymmetric theories. Using superspace methods, we show how an explicitly supersymmetric version of such quantization methods may be given. 
  These lectures provide an introduction to noncommutative geometry and its origins in quantum mechanics and to the construction of solitons in noncommutative field theory. These ideas are applied to the construction of D-branes as solitons of the tachyon field in noncommutative open string theory. A brief discussion is given of the K-theory classification of D-brane charge in terms of the K-theory of operator algebras. Based on lectures presented at the Komaba 2000 workshop, Nov. 14-16 2000. 
  We discuss the construction of the analog of an S-matrix for space-times that begin with a Big-Bang and asymptote to an FRW universe with nonnegative cosmological constant. When the cosmological constant is positive there are many such S-matrices, related mathematically by gauge transformations and physically by an analog of the principle of black hole complementarity. In the limit of vanishing $\Lambda$ these become (approximate) Poincare transforms of each other. Considerations of the initial state require a quantum treatment of space-time, and some preliminary steps towards constructing such a theory are proposed. In this context we propose a model for the earliest semiclassical state of the universe, which suggests a solution for the horizon problem different from that provided by inflation. 
  We show that the coset ^sl(2)+^sl(2)/^sl(2) is a quantum Hamiltonian reduction of the exceptional affine Lie superalgebra ^D(2|1;\alpha) and that the corresponding W algebra is the commutant of the U_{q}D(2|1;\alpha) quantum group. 
  We propose explicit expressions for the form factors, including their normalization constants, of topologically charged (or soliton-creating) operators in the sine-Gordon model. The normalization constants, which constitute the main content of our proposal, allow one to find exact relations between the short- and long-distance asymptotics of the correlation functions. We make predictions concerning asymptotics of fermion correlation functions in the massive Thirring model, SU(2)-Thirring model with anisotropy, and in the half-filled Hubbard chain. 
  We study new solutions of the low-energy equations of motion for the non-abelian D-string. We find a "fuzzy funnel" solution consisting of a noncommutative four-sphere geometry which expands along the length of the D-string. We show that this funnel solution has an interpretation as D-strings ending on a set of orthogonal D5-branes. Although not supersymmetric, the system appears to be stable within this framework. We also give a dual description of this configuration as a bion spike in the non-abelian world volume theory of coincident D5-branes. 
  Monopole field configurations have been extensively studied in both Abelian and non-Abelian gauge theories. The question of the quantum corrections to these systems is a difficult one, since the classical monopoles have non-perturbatively large couplings, which makes the standard, perturbative methods for calculating quantum corrections suspect. Here we apply a modified version of Heisenberg's quantization technique for strongly interacting, nonlinear fields to a classical solution of the SU(3) Yang-Mills field equations. This classical solution is not monopole-like and has an energy density which diverges as $r \to \infty$. However, the quantized version of this solution has a monopole-like far field, and a non-divergent energy density as $r \to \infty$. This may point to the conclusion that monopoles may arise not from quantizing classical monopole configurations, but from quantizing field configurations which at the classical level do not appear monopole-like. 
  Low-energy effective M-theory (11d supergravity or 11d SUSY theory) is considered in 11d flat or AdS background. After orbifold compactification of eleventh dimension one gets flat 10d brane at non-zero temperature. Quantum effective potential (free energy at low-temperature approximation) is calculated on such brane with account of only lowest mass from Kaluza-Klein modes. Such effective potential may stabilize the radius of 11th dimension (radion stabilization) as it is demonstrated explicitly. 
  The SO(4,1) gauge-invariant theory of the Dirac fermions in the external field of the Kaluza-Klein monopole is investigated. It is shown that the discrete quantum modes are governed by reducible representations of the o(4) dynamical algebra generated by the components of the angular momentum operator and those of the Runge-Lenz operator of the Dirac theory in Taub-NUT background. The consequence is that there exist central and axial discrete modes whose spinors have no separated variables. 
  The M-9-brane Wess-Zumino action is constructed, and by using it, consistency of the relation of p-branes for $p \ge 8$, suggested on the basis of superalgebra, is discussed. 
  We review the recent studies of tachyon condensation in string field theory. After introducing the open string field theory both for bosonic string and for superstring, we use them to examine the conjecture that the unstable configurations of the D-brane will decay into the `closed string vacuum' through the tachyon condensation. And we describe the attemps to construct a lower dimensional bosonic D-brane as an unstable lump solution of the string field equation. We obtain exact results from another formulation, background independent open string field theory. We also discuss some other topics which are related to tachyon condensation in string theory, such as the construction of a D-brane as a noncommutative soliton and some field theory models. This paper is based on my master's thesis submitted to Department of Physics, Faculty of Science, University of Tokyo on January 2001. 
  Bethe-Salpeter equation in the non-commutative space for a scalar-scalar bound state is considered. It is shown that in the non-relativistic limit, the effect of spatial non-commutativity appears as if there exist a magnetic dipole moment coupled to each particle. 
  In the Wigner-covariant rest-frame instant form of dynamics it is possible to develop a relativistic kinematics for the N-body problem. The Wigner hyperplanes define the intrinsic rest frame and realize the separation of the center-of-mass. Three notions of {\it external} relativistic center of mass can be defined only in terms of the {\it external} Poincar\'e group realization. Inside the Wigner hyperplane, an {\it internal} unfaithful realization of the Poincar\'e group is defined. The three concepts of {\it internal} center of mass weakly {\it coincide} and are eliminated by the rest-frame conditions. An adapted canonical basis of relative variables is found. The invariant mass is the Hamiltonian for the relative motions. In this framework we can introduce the same {\it dynamical body frames}, {\it orientation-shape} variables, {\it spin frame} and {\it canonical spin bases} for the rotational kinematics developed for the non-relativistic N-body problem. 
  We study renormalization effects in the Abelian Chern-Simons (CS) action. These effects can be non-trivial when the gauge field is coupled to dynamical matter, since the regularization of the UV divergences in the model forces the introduction of a parity even piece in the gauge field action. This changes the classical (odd) transformation properties of the pure CS action. This effect, already discussed for the case of a lattice regularization by F. Berruto, M.C. Diamantini and P. Sodano in hep-th/0004203, is also present when the theory is defined in the continuum and, indeed, it is a manifestation of a more general `anomalous' effect, since it happens for every regularization scheme. We explore the physical consequences of this anomaly. We also show that generalized, nonlocal parity transformations can be defined in such a way that the regularized theory is odd, and that those transformations tend to the usual ones when the cutoff is removed. These generalized transformations play a role that is tantamount to the deformed symmetry corresponding to Ginsparg-Wilson fermions [2] (in an even number of spacetime dimensions). 
  We extend the path-integral formulation of Poisson-Lie duality found by Tyurin and von Unge to N=1 supersymmetric sigma-models. Using an explicit representation of the generators of the Drinfel'd double corresponding to GxU(1)^dimG we discuss an application to non-abelian duality. The paper also contains the relevant background and some comments on Poisson-Lie duality. 
  A broad class of higher dimensional instanton solutions are found for a theory which contains gravity, a scalar field and antisymmetric tensor fields of arbitrary rank. The metric used, a warp product of an arbitrary number of any compact Einstein manifolds, includes many of great interest in particle physics and cosmology. For example 4D FRW universes with additional dimensions compactified on a Calabi-Yau three fold, a torus, a compact hyperbolic manifold or a sphere are all included. It is shown that the solution of this form which dominates the Hartle Hawking path integral is always a higher dimensional generalisation of a Hawking Turok instanton when the potential of the scalar field is such that these instantons can exist. On continuation to Lorentzian signature such instantons give rise to a spacetime in which all of the spatial dimensions are of equal size and where the spatial topology is that of a sphere. The extra dimensions are thus not hidden. In the case where the potential for the scalar field is generated solely by a dilatonic coupling to the form fields we find no integrable instantons at all. In particular we find no integrable solutions of the type under consideration for the supergravity theories which are the low energy effective field theories of superstrings. 
  We study the objects (called spectral branes or S-branes) which are obtained by imposing non-local spectral boundary conditions at the boundary of the world sheet of the bosonic string. They possess many nice properties which make them an ideal test ground for the string theory methods. Depending on a particular choice of the boundary operator S-branes may be commutative or non-commutative. We demonstrate that projection of the B-field on the brane directions (i.e. on the components which actually influence the boundary conditions) is done with the help of the chirality operator. We show that the T-duality transformation maps an S-brane to another S-brane. At the expense of introducing non-local interactions in the bulk we construct also a duality transformation between S-branes and D-branes or open strings. 
  The three dimensional Chern-Simons theory on $\rr^2_{\theta}\times \rr$ is studied. Considering the gauge transformations under the group elements which are going to one at infinity, we show that under arbitrary (finite) gauge transformations action changes with an integer multiple of $2\pi$ {\it if}, the level of noncommutaitive Chern-Simons is {\it quantized}. We also briefly discuss the case of the noncommutaitve torus and some other possible extensions. 
  We show that the recently demonstrated absence of the usual discontinuity for massive spin 2 with a Lambda term is an artifact of the tree approximation, and that the discontinuity reappears at one loop. 
  It appears that string-M-theory is the only viable candidate for a complete theory of matter. It must therefore contain both gravity and QCD. What is particularly surprising is the recent conjecture that strongly coupled QCD matrix elements can be evaluated though a duality with weakly coupled gravity. To date there has been no direct verification of this conjecture by Maldacena because of the difficulty of direct strong coupling calculations in gauge theories. We report here on some progress in evaluating a gauge-invariant correlator in the non-perturbative regime in two and three dimensions in SYM theories. The calculations are made using supersymmetric discrete light-cone quantization (SDLCQ). We consider a Maldacena-type conjecture applied to the near horizon geometry of a D1-brane in the supergravity approximation, solve the corresponding N=(8,8) SYM theory in two dimensions, and evaluate the correlator of the stress-energy tensor. Our numerical results support the Maldacena conjecture and are within 10-15% of the predicted results. We also present a calculation of the stress-energy correlator in N=1 SYM theory in 2+1 dimensions. While there is no known duality relatingthis theory to supergravity, the theory does have massless BPS states, and the correlator gives important information about the BPS wave function in the non-perturbative regime. 
  The orbifold lines IIA/I_8 and IIB/$I_8 (-1)^{F_L} possess BPS discrete torsion variants which carry fundamental string (NSNS) charge. We show that these variants are actually classified by an integral electric field F from the twisted RR sector, and compute their tension and NSNS charge as a function of F. The analysis employs equivariant K-theory and the string creation phenomenon. The K-theory results demonstrate the corrections to cohomology in the case of torsion; it is found that 8 units of F are invisible at transverse infinity for IIA, and correspondingly 16 units for IIB. 
  We show how the Implicit Regularization Technique (IRT) can be used for the perturbative renormalization of a simple field theoretical model, generally used as a test theory for new techniques. While IRT has been applied successfully in many problems involving symmetry breaking anomalies and nonabelian gauge groups, all at one loop level, this is the first attempt to a generalization of the technique for perturbative renormalization. We show that the overlapping divergent loops can be given a completely algebraic treatment. We display the connection between renormalization and counterterms in the Lagrangian. The algebraic advantages make IRT worth studying for perturbative renormalization of gauge theories. 
  We rederive the Brown-Henneaux commutation relation and central charge in the framework of the path integral. To obtain the Ward-Takahashi identity, we can use either the asymptotic symmetry or its leading part. If we use the asymptotic symmetry, the central charge arises from the transformation law of the charge itself. Thus, this central charge is clearly different from the quantum anomaly which can be understood as the Jacobian factor of the path integral measure. Alternatively, if we use the leading transformation, the central charge arises from the fact that the boundary condition of the path integral is not invariant under the transformation. This is in contrast to the usual quantum central charge which arises from the fact that the measure of the path integral is not invariant under the relevant transformation. Moreover, we discuss the implications of our analysis in relation to the black hole entropy. 
  It has been observed recently by Giovanni Amelino-Camelia \cite{gac1, gac2} that the hypothesis of existence of a minimal observer-independent (Planck) length scale is hard to reconcile with special relativity. As a remedy he postulated to modify special relativity by introducing an observer-independent length scale. In this letter we set forward a proposal how one should modify the principles of special relativity, so as to assure that the values of mass and length scales are the same for any inertial observer. It turns out that one can achieve this by taking dispersion relations such that the speed of light goes to infinity for finite momentum (but infinite energy), proposed e.g., in the framework of the quantum $\kappa$-Poincar\'e symmetry. It follows that at the Planck scale the world may be non-relativistic. 
  Some instanton corrections to the universal hypermultiplet moduli space metric of the type-IIA string theory compactified on a Calabi-Yau threefold arise due to multiple wrapping of BPS membranes and fivebranes around certain cycles of Calabi-Yau. The classical universal hypermultipet metric is locally equivalent to the Bergmann metric of the symmetric quaternionic space SU(2,1)/U(2), whereas its generic quaternionic deformations are governed by the integrable SU(infinity) Toda equation. We calculate the exact (non-perturbative) UH metrics in the special cases of (i) the D-instantons (the wrapped D2-branes) in the absence of fivebranes, and (ii) the fivebrane instantons with vanishing charges, in the absence of D-instantons. The solutions of the first type preserve the U(1)xU(1) classical symmetry, while they can be interpreted as the gravitational dressing of the hyper-K"ahler D-instanton solutions. The second type solution preserves the non-abelian SU(2) classical symmetry, while it can be interpreted as a gradient flow in the universal hypermultiplet moduli space. 
  The $A^{(1)}_{n-1}$-face model with boundary reflection is considered on the basis of the boundary CTM bootstrap. We construct the fused boundary Boltzmann weights to determine the normalization factor. We derive difference equations of the quantum Knizhnik-Zamolodchikov type for correlation functions of the boundary model. The simplest difference equations are solved the in the case of the free boundary condition. 
  We show that the noncommutativity of space-time destroys the renormalizability of the 1/N expansion of the O(N) Gross-Neveu model. A similar statement holds for the noncommutative nonlinear sigma model. However, we show that, up to the subleading order in 1/N expansion, the noncommutative supersymmetric O(N) nonlinear sigma model becomes renormalizable in D=3. We also show that dynamical mass generation is restored and there is no catastrophic UV/IR mixing. Unlike the commutative case, we find that the Lagrange multiplier fields, which enforce the supersymmetric constraints, are also renormalized. For D=2 the divergence of the four point function of the basic scalar field, which in D=3 is absent, cannot be eliminated by means of a counterterm having the structure of a Moyal product. 
  We consider Dirac's free electron theory on the first quantized level. We decompose its canonical spin current \'a la Gordon and find a conserved ``Gordon spin'' current which turns out to be equivalent to the Hilgevoord-Wouthuysen spin. We can conclude therefrom that the Gordon-type decomposition mentioned above corresponds to a Foldy-Wouthuysen transformation which transforms the Dirac wave function from the conventional Dirac-Pauli to the Newton-Wigner representation. 
  With the help of the Seiberg-Witten map for photons and fermions we define a theta-deformed QED at the classical level. Two possibilities of gauge-fixing are discussed. A possible non-Abelian extension for a pure theta-deformed Yang-Mills theory is also presented. 
  We show that the parent Lagrangian method gives a natural generalization of the dual theories concept for non p-form fields. Using this generalization we construct here a three-parameter family of Lagrangians that are dual to the Fierz-Pauli description of a free massive spin-two system. The dual field is a three-index tensor T, which dinamically belongs to the (2,1) representation of the Lorentz group. As expected, the massless limit of our Lagrangian, which is parameter independent, has two propagating degrees of freedom per space point. 
  The world volume theory on $N$ regular and $M$ fractional D3-branes at the conifold singularity is a non-conformal ${\cal N}=1$ supersymmetric $SU(N+M)\times SU(N)$ gauge theory. In previous work the extremal Type IIB supergravity dual of this theory at zero temperature was constructed. Regularity of the solution requires a deformation of the conifold: this is a reflection of the chiral symmetry breaking. To study the non-zero temperature generalizations non-extremal solutions have to be considered, and in the high temperature phase the chiral symmetry is expected to be restored. Such a solution is expected to have a regular Schwarzschild horizon. We construct an ansatz necessary to study such non-extremal solutions and show that the simplest possible solution has a singular horizon. We derive the system of second order equations in the radial variable whose solutions may have regular horizons. 
  We consider possible superfield representations of extended BRST symmetry for general gauge theories within the principle of gauge-fixing based on a generating equation for the gauge functional. We examine admissible superfield choices for an extended antibracket and delta-operator with given algebraic properties and show that only one of these choices is compatible with the requirement of extended BRST symmetry realized in terms of supertranslations along Grassmann coordinates. We demonstrate that this realization leads to the gauge-independence of the S-matrix. 
  The coefficients in the 1/N expansions of the vacuum expectation values and correlation functions of Wilson loops, in continuum SU(N) gauge theories in 3+1 dimensions, are shown to be determined by a closed and complete set of equations, called the Group-Variation Equations, that exhibit a simple and robust mechanism for the emergence of massive glueballs and the Wilson area law. The equations predict that the cylinder-topology minimal-area spanning surface term in the two-glueball correlation function, when it exists, must be multiplied by a pre-exponential factor, which for large area A of the minimal-area cylinder-topology surface, decreases with increasing A at least as fast as $1/\ln(\sigma A)$. If this factor decreases faster than $1/\ln(\sigma A)$, then the mass $m_{0^{++}}$ of the lightest glueball, and the coefficient $\sigma$ of the area in the Wilson area law, are determined in a precisely parallel manner, and the equations give a zeroth-order estimate of $m_{0^{++}}/\sqrt{\sigma}$ of 2.38, about 33% less than the best lattice value, without the need for a full calculation of any of the terms in the right-hand sides. The large distance behaviour of the vacuum expectation values and correlation functions is completely determined by terms called island diagrams, the dominant contributions to which come from islands of \emph{fixed} size of about $1/\sqrt{\sigma}$. The value of $\sigma$ is determined by the point at which $|\beta(g)/g|$ reaches a critical value, and since the large distance behaviour of all physical quantities is determined by islands of the fixed size $1/\sqrt{\sigma}$, the running coupling $g^2$ never increases beyond the value at which $|\beta(g)/g|$ reaches the critical value. 
  This is the second in a series of two contributions in which we set out to establish a novel momentum space framework to treat field theoretical infinities in perturbative calculations when parity-violating objects occur. Since no analytic continuation on the space-time dimension is effected, this framework can be particularly useful to treat dimension-specific theories. Moreover arbitrary local terms stemming from the underlying infinities of the model can be properly parametrized. We (re)analyse the undeterminacy of the radiatively generated CPT violating Chern-Simons term within an extended version of $QED_4$ and calculate the Adler-Bardeen-Bell-Jackiw triangle anomaly to show that our framework is consistent and general to handle the subtleties involved when a radiative corretion is finite. 
  We consider the T-duality transformations of the low-energy quantum string theory effective action in the presence of classical fundamental string source and demonstrate explicitly that T-duality still holds. 
  In this paper we study K3 compactification of ten-dimensional massive type IIA theory with all possible Ramond-Ramond background fluxes turned on. The resulting six-dimensional theory is a new massive (gauged) supergravity with an action that is manifestly invariant under an O(4,20) / (O(4) times O(20)) duality symmetry. We discover that this six-dimensional theory interpolates between vacua of ten-dimensional massive IIA supergravity and vacua of massless IIA supergravity with appropriate background fluxes turned on. This in turn suggests a new 11-dimensional interpretation for the massive type IIA theory. 
  A physical and geometrical interpretation of previously introduced tensor operator algebras of U(2,2) in terms of algebras of higher-conformal-spin quantum fields on the anti-de Sitter space AdS_5 is provided. These are higher-dimensional W-like algebras and constitute a potential gauge guide principle towards the formulation of induced conformal gravities (Wess-Zumino-Witten-like models) in realistic dimensions. Some remarks on quantum (Moyal) deformations are given and potentially tractable versions of noncommutative AdS spaces are also sketched. The role of conformal symmetry in the microscopic description of Unruh and Hawking's radiation effects is discussed. 
  In a previous paper [hep-th/0012251] we proposed a simple class of actions for string field theory around the tachyon vacuum. In this paper we search for classical solutions describing D-branes of different dimensions using the ansatz that the solutions factorize into the direct product of a matter state and a universal ghost state. We find closed form expressions for the matter state describing D-branes of all dimensions. For the space filling D25-brane the state is the matter part of the zero angle wedge state, the ``sliver'', built in [hep-th/0006240]. For the other D-brane solutions the matter states are constructed using a solution generating technique outlined in [hep-th/0008252]. The ratios of tensions of various D-branes, requiring evaluation of determinants of infinite dimensional matrices, are calculated numerically and are in very good agreement with the known results. 
  In a recent paper hep-th/0008140 by E. Verlinde, an interesting formula has been put forward, which relates the entropy of a conformal formal field in arbitrary dimensions to its total energy and Casimir energy. This formula has been shown to hold for the conformal field theories that have AdS duals in the cases of AdS Schwarzschild black holes and AdS Kerr black holes. In this paper we further check this formula with various black holes with AdS asymptotics. For the hyperbolic AdS black holes, the Cardy-Verlinde formula is found to hold if we choose the ``massless'' black hole as the ground state, but in this case, the Casimir energy is negative. For the AdS Reissner-Nordstr\"om black holes in arbitrary dimensions and charged black holes in D=5, D=4, and D=7 maximally supersymmetric gauged supergravities, the Cardy-Verlinde formula holds as well, but a proper internal energy which corresponds to the mass of supersymmetric backgrounds must be subtracted from the total energy. It is failed to rewrite the entropy of corresponding conformal field theories in terms of the Cardy-Verlinde formula for the AdS black holes in the Lovelock gravity. 
  In this series of lectures I present a review of the geometric structures of supergravity in diverse dimensions mostly relevant to p-brane physics and to pinpoint the correspondence between the macroscopic and microscopic description of branes. In particular I review duality transformations, coset manifold structures and the general steps involved by the process of gauging supergravity lagrangians both with respect to compact, non compact and non semisimple groups. I focus specifically on the issue of the Domain Wall field theory correspondence and its relation with the gaugings of supergravity in p+2 dimensions. A complete review of the geometries involved by D=5, N=2 supergravity and of its most general form is given with emphasis on the problem of finding smooth supersymmetric realizations of the Randall Sundrum scenarios. I also give a general review of the algebraic machinery involved by the Solvable Lie algebra description of the scalar manifolds of supergravity and I emphasize its distinguished role in pinpointing the superstring interpretation of supergravity p brane solutions and the macroscopic/microscopic correspondence. 
  Extended Schwinger's quantization procedure is used for constructing quantum mechanics on a manifold with a group structure. The considered manifold $M$ is a homogeneous Riemannian space with the given action of isometry transformation group. Using the identification of $M$ with the quotient space $G/H$, where $H$ is the isotropy group of an arbitrary fixed point of $M$, we show that quantum mechanics on $G/H$ possesses a gauge structure, described by the gauge potential that is the connection 1-form of the principal fiber bundle $G(G/H, H)$. The coordinate representation of quantum mechanics and the procedure for selecting the physical sector of states are developed. 
  Consistent couplings between an Abelian gauge field and three types of matter fields are investigated by means of the Hamiltonian BRST deformation theory based on cohomological techniques. In this manner, scalar electrodynamics, the Stuckelberg theory for Abelian zero- and one-forms, respectively, spinor electrodynamics, are inferred. 
  Using extended Schwinger's quantization approach quantum mechanics on a Riemannian manifold $M$ with a given action of an intransitive group of isometries is developed. It was shown that quantum mechanics can be determined unequivocally only on submanifolds of $M$ where $G$ acts simply transitively (orbits of $G$-action). The remaining part of degrees of freedom can be described unequivocally after introducing some additional assumptions. Being logically unmotivated, these assumptions are similar to canonical quantization postulates. Besides this ambiguity that has a geometrical nature there is undetermined gauge field of $\hbar$ (or higher) order, vanishing in the classical limit $\hbar\to 0$. 
  An irreducible Hamiltonian BRST-anti-BRST treatment of reducible first-class systems based on homological arguments is proposed. The general formalism is exemplified on the Freedman-Townsend model. 
  Contrary to the conventional belief, it was shown that the Breit equation has the eigenvalues for bound states of two oppositely charged Dirac particles interacting through the (static) Coulomb potential. All eigenvalues reduced to those of the Sch\"odinger case in the non-relativistic limit. 
  A gauge invariant Wilsonian effective action is constructed for pure SU(N) Yang-Mills theory by formulating the corresponding flow equation. Manifestly gauge invariant calculations can be performed i.e. without gauge fixing or ghosts. Regularisation is implemented in a novel way which realises a spontaneously broken SU(N|N) supergauge theory. As an example we sketch the computation of the one-loop beta function, performed for the first time without any gauge fixing. 
  We consider the long range forces between two BPS particles on the Coulomb branch of N=2 and N=4 supersymmetric gauge theories. The 1/r potential is unambiguously fixed, even at strong coupling, by the moduli dependence of central charges supported by the BPS states. The effective Coulombic coupling vanishes on marginal stability curves, while sign changes on crossing these curves explain the restructuring of the spectrum of composite BPS states. This restructuring proceeds via the delocalization of the composite state on approach to the curve of marginal stability. Therefore the spectrum of BPS states can be inferred by analyzing the submanifolds of the moduli space where the long range potential is attractive. This method also allows us to find certain non-BPS bound states and their stability domains. As examples, we consider the dissociation of the W boson and higher charge dyons at strong coupling in N=2 SU(2) SYM, quark-monopole bound states in N=2 SYM with one flavor, and composite dyons in N=2 SU(3) SYM. 
  I discuss the axiomatic framework of (tree-level) associative open string field theory in the presence of D-branes by considering the natural extension of the case of a single boundary sector. This leads to a formulation which is intimately connected with the mathematical theory of differential graded categories. I point out that a generic string field theory as formulated within this framework is not closed under formation of D-brane composites and as such does not allow for a unitary description of D-brane dynamics. This implies that the collection of boundary sectors of a generic string field theory with D-branes must be extended by inclusion of all possible D-brane composites. I give a precise formulation of a weak unitarity constraint and show that a minimal extension which is unitary in this sense can always be obtained by promoting the original D-brane category to an enlarged category constructed by using certain generalized complexes of D-branes. I give a detailed construction of this extension and prove its closure under formation of D-brane composites. These results amount to a completely general description of D-brane composite formation within the framework of associative string field theory. 
  This letter is an attempt to carry out a first-principle computation in M-theory using the point of view that the eleven-dimensional membrane gives the fundamental degrees of freedom of M-theory. Our aim is to derive the exact BPS $R^4$ couplings in M-theory compactified on a torus $T^{d+1}$ from the toroidal BPS membrane, by pursuing the analogy with the one-loop string theory computation. We exhibit an $Sl(3,\Zint)$ modular invariance hidden in the light-cone gauge (but obvious in the Polyakov approach), and recover the correct classical spectrum and membrane instantons; the summation measure however is incorrect. It is argued that the correct membrane amplitude should be given by an exceptional theta correspondence lifting $Sl(3,\Zint)$ modular forms to $\exc(\Zint)$ automorphic forms, generalizing the usual theta lift between $Sl(2,\Zint)$ and $SO(d,d,\Zint)$ in string theory. The exceptional correspondence $Sl(3)\times E_{6(6)}\subset E_{8(8)}$ offers the interesting prospect of solving the membrane small volume divergence and unifying membranes with five-branes. 
  The association of the variational method with supersymmetric quantum mechanics through an ansatz for the superpotential is reviewed and the approximate energy spectra of non-exactly solvable potentials, such like the Hulthen, the Morse and the screened Coulomb potentials, in 3 dimensions, are presented. A sketch of a new exactly shape invariant potential is also presented. 
  We discuss non-conformal gauge theories from type IIB D3-branes embedded in orbifolded space-times. Such theories can be obtained by allowing some non-vanishing logarithmic twisted tadpoles. In certain cases with N=0,1 supersymmetry correlation functions in the planar limit are the same as in the parent N=2 supersymmetric theories. In particular, the effective action in such theories perturbatively is not renormalized beyond one loop in the planar limit. In the N=2 as well as such N=0,1 theories quantum corrections in the D3-brane gauge theories are encoded in the corresponding classical higher dimensional field theories whose actions contain the twisted fields with non-vanishing tadpoles. We argue that this duality can be extended to the non-perturbative level in the N=2 theories. We give some evidence that this might also be the case for N=0,1 theories as well. 
  We consider Randall-Sundrum(RS) model in generalized gravities and see that the localization of gravity happens in generic situations though its effectiveness depends on the details of the configuration. It is shown that RS picture is robust against quantum gravity corrections ($\phi \R$) as long as the correction is reasonably small. We extend our consideration to the model of scalar(dilaton) coupled gravity which leads us to the specific comparison between RS model and inflation models. The exponential and power law hierarchy in RS model are shown to correspond to the exponential and power law inflation respectively. 
  We consider the D-dimensional massive dilaton gravity coupled to Maxwell and antisymmetric tensor fields (EMATD). We derive the full separability of this theory in static case. This discloses the core structure of the theory and yields the simple procedure of how to generate integrability classes.   As an example we take a certain new class, obtain the two-parametric families of dyonic solutions. It turns out that at some conditions they tend to the D-dimensional dyonic Reissner-Nordstr\"om-deSitter solutions but with ``renormalized'' dyonic charge plus a small logarithmic correction. The latter has the significant influence on the global structure of the non-perturbed solution - it may shift and split horizons, break down extremality, and dress the naked singularity.   We speculate on physical importance of the deduced integrability classes, in particular on their possible role in understanding of the problem of unknown dilaton potential in modern cosmological and low-energy string models. 
  We construct four-dimensional domain wall solutions of N=2 gauged supergravity coupled to vector and to hypermultiplets. The gauged supergravity theories that we consider are obtained by performing two types of Abelian gauging. In both cases we find that the behaviour of the scalar fields belonging to the vector multiplets is governed by the so-called attractor equations known from the study of BPS black hole solutions in ungauged N=2 supergravity theories. The scalar fields belonging to the hypermultiplets, on the other hand, are either constant or exhibit a run-away behaviour. These domain wall solutions preserve 1/2 of supersymmetry and they are, in general, curved. We briefly comment on the amount of supersymmetry preserved by domain wall solutions in gauged supergravity theories obtained by more general gaugings. 
  The concept of covariant coordinates on noncommutative spaces leads directly to gauge theories with generalized noncommutative gauge fields of the type that arises in string theory with background B-fields. The theory is naturally expressed in terms of cochains in an appropriate cohomology; we discuss how it fits into the framework of projective modules. The equivalence of star products that arise from the background field with and without fluctuations and Kontsevich's formality theorem allow an explicitly construction of a map that relates ordinary gauge theory and noncommutative gauge theory (Seiberg-Witten map.) As application we show the exact equality of the Dirac-Born-Infeld action with B-field in the commutative setting and its semi-noncommutative cousin in the intermediate picture. Using noncommutative extra dimensions the construction is extended to noncommutative nonabelian gauge theory for arbitrary gauge groups; an explicit map between abelian and nonabelian gauge fields is given. All constructions are also valid for non-constant B-field, Poisson structure and metric. 
  In this letter we study the dilatonic corrections to the static gauge potential between heavy sources. These corrections come from the solutions to the the lowest order beta equations. In the energetically favoured branch, the potential obtained is characterised by having a linear confining term, an $L$ independent term and another 1/L piece. This is indicative of a L\"uscher-type behaviour in the strong-coupling regime of the dual gauge theory. On the other hand, we also explore the singularity as a point where the theory becomes free. 
  We consider a Dirac field in 2+1 dimensions with a domain wall like defect in its mass, minimally coupled to a dynamical Abelian vector field. The mass of the fermionic field is assumed to have just one linear domain wall, which is externally fixed and unaffected by the dynamics. We show that, under some general conditions on the parameters, the localized zero modes predicted by the Callan and Harvey mechanism are stable under the electromagnetic interaction of the fermions. 
  The results conjectured here are based on tachyon condensation in compactified bosonic strings. They can in fact be proven from compactification alone, but this step was not treated consistently here. The proof is given in a new version of the paper: hep-th/0106235 
  We investigate the quantum volume of D-branes wrapped around cycles of various dimension in Calabi-Yau fourfolds and fivefolds. Examining the cases of the sextic and heptic hypersurface Calabi-Yau varieties, as well as one example in weighted projective space, we find expressions for periods which vanish at the singular point analogous to the conifold point. As in the known three-dimensional cases, it is the top dimensional cycle which attains zero quantum volume, even though lower dimensional cycles remain non-degenerate, indicating this phenomena to be a general feature of quantum geometry. 
  The algebra of the generators of translations in superspace is unstable, in the sense that infinitesimal perturbations of its structure constants lead to non-isomorphic algebras. We show how superspace extensions remedy this situation (after arguing that remedy is indeed needed) and review the benefits reaped in the description of branes of all kinds in the presence of the extra dimensions. 
  We consider the collision in 2+1 dimensions of a black hole and a negative tension brane on an orbifold. Because there is no gravitational radiation in 2+1 dimensions, the horizon area shrinks when part of the brane falls through. This provides a potential violation of the generalized second law of thermodynamics. However, tracing the details of the dynamical evolution one finds that it does not proceed from equilibrium configuration to equilibrium configuration. Instead, a catastrophic space-time singularity develops similar to the `big crunch' of $\Omega >1$ FRW space-times. In the context of classical general relativity, our result demonstrates a new instability of constructions with negative tension branes. 
  1) We identify new parameter branches for the ultra-local boundary Poisson bracket in d spatial dimension with a (d-1)-dimensional spatial boundary. There exist 2^{r(r-1)/2} r-dimensional parameter branches for each d-box, r-row Young tableau. The already known branch (hep-th/9912017) corresponds to a vertical 1-column, d-box Young tableau. 2) We consider a local distribution product among the so-called boundary distributions. The product is required to respect the associativity and the Leibnitz rule. We show that the consistency requirements on this product correspond to the Jacobi identity conditions for the boundary Poisson bracket. In other words, the restrictions on forming a boundary Poisson bracket can be related to the more fundamental distribution product construction. 3) The definition of the higher functional derivatives is made independent of the choice of integral kernel representative for a functional. 
  The Chern-Simons theories on a noncommutative plane, which is shown to be describing the quantum Hall liquid, is considered. We introduce matter fields fundamentally coupled to the noncommutative Chern-Simons field. Exploiting BPS equations for the nonrelativistic Chern-Simons theory, we find the exact solutions of multi vortices that are closely packed and exponentially localized. We determine the position, the size and the angular momentum explicitly. We then construct the solutions of two spatially separated vortices and determine the moduli dependence of the size and the angular momentum. We also consider the relativistic Chern-Simons theory and find nontopological solutions whose properties are similar to the nonrelativistic counterpart. However, unlike the nonrelativistic case, there are two branches of solutions for a given magnetic field and they cease to exist below certain noncommutativity scale. 
  Recently, Verlinde noted a surprising similarity between Friedmann equation governing radiation dominated universe and Cardy's entropy formula in conformal field theory. In this note, we study a brane-universe filled with radiation and stiff-matter. We analyze Friedmann equation in this context and compare our results with Cardy's entropy formula. 
  Schwinger's quantization scheme is extended in order to solve the problem of the formulation of quantum mechanics on a space with a group structure. The importance of Killing vectors in a quantization scheme is showed. Usage of these vectors provides algebraic properties of operators to be consistent with the geometrical structure of a manifold. The procedure of the definition of the quantum Lagrangian of a free particle and the norm of velocity (momentum) operators is given. These constructions are invariant under a general coordinate transformation. The unified procedure for constructing the quantum theory on a space with a group structure is developed. Using it quantum mechanics on a Riemannian manifold with a simply transitive group acting on it is investigated. 
  The constraints of the superfield method in two-dimensional supergravity are adapted to allow for nonvanishing bosonic torsion. As the analysis of the Bianchi identities reveals, a new vector superfield is encountered besides the well-known scalar one. The constraints are solved both with superfields using a special decomposition of the supervielbein, and explicitly in terms of component fields in a Wess-Zumino gauge. The graded Poisson Sigma Model (gPSM) is the alternative method used to construct supersymmetric gravity theories. In this context the graded Jacobi identity is solved algebraically for general cases. Some of the Poisson algebras obtained are singular, or several potentials contained in them are restricted. This is discussed for a selection of representative algebras. It is found, that the gPSM is far more flexible and it shows the inherent ambiguity of the supersymmetric extension more clearly than the superfield method. Among the various models spherically reduced Einstein gravity and gravity with torsion are treated. Also the Legendre transformation to eliminate auxiliary fields, superdilaton theories and the explicit solution of the gPSM equations of motion for a typical model are presented. Furthermore, the PSM field equations are analyzed in detail, leading to the so called "symplectic extension". Thereby, the Poisson tensor is extended to become regular by adding new coordinates to the target space. For gravity models this is achieved with one additional coordinate. Finally, the relation of the gPSM to the superfield method is established by extending the base manifold to become a supermanifold. 
  In this paper some properties of the superstring with noncommutative worldsheet are studied. We study the noncommutativity of the spacetime, generalization of the Poincar\'e symmetry of the superstring, the changes of the metric, antisymmetric tensor and dilaton. 
  Anthropic solutions to the cosmological constant problem require seemingly unnatural scalar field potentials with a very small slope or domain walls (branes) with a very small coupling to a four-form field. Here we introduce a class of models in which the smallness of the corresponding parameters can be attributed to a spontaneously broken discrete symmetry. We also demonstrate the equivalence of scalar field and four-form models. Finally, we show how our models can be naturally embedded into a left-right extension of the standard model. 
  In this paper I present a spacetime of two open universes connected by a Lorentzian wormhole. The spacetime has the following features: (1) It can exactly solve the Einstein equations; (2) The weak energy condition is satisfied everywhere; (3) It has a topology of R^2\times T_g (g\ge 2); (4) It has no event horizons. 
  We derive dynamical equations to describe a single 3-brane containing fluid matter and a scalar field coupling to the dilaton and the gravitational field in a five dimensional bulk. First, we show that a scalar field or an arbitrary fluid on the brane cannot evolve to cancel the cosmological constant in the bulk. Then we show that the Randall-Sundrum model is unstable under small deviations from the fine-tuning between the brane tension and the bulk cosmological constant and even under homogeneous gravitational perturbations. Implications for brane world cosmologies are discussed. 
  We give the explicit form of the four dimensional effective supergravity action, which describes low energy physics of the Randall-Sundrum model with moduli fields in the bulk and charged chiral matter living on the branes. The relation between 5d and 4d physics is explicit: the low energy action is derived from the compactification of a locally supersymmetric model in five dimension. The presence of odd $Z_2$ parity scalars in the bulk gives rise to effective potential for the radion in four dimensions. We describe the mechanism of supersymmetry breaking mediation, which relies on non-trivial configuration of these $Z_2$-odd bulk fields. Broken supersymmetry leads to stabilization of the interbrane distance. 
  Fractional branes on the non-compact orbifold $\C^3/\Z_5$ are studied. First, the boundary state description of the fractional branes are obtained. The open-string Witten index calculated using these states reproduces the adjacency matrix of the quiver of $\Z_5$. Then, using the toric crepant resolution of the orbifold $\C^3/\Z_5$ and invoking the local mirror principle, B-type branes wrapped on the holomorphic cycles of the resolution are studied. The boundary states corresponding to the five fractional branes are identified as bound states of BPS D-branes wrapping the 0-, 2- and 4-cycles in the exceptional divisor of the resolution of $\C^3/\Z_5$. 
  We explore the possibility of generalizing the locally localized gravity model in five space-time dimensions to arbitrary higher dimensions. In a space-time with negative cosmological constant, there are essentially two kinds of higher-dimensional cousins which not only take an analytic form but also are free from the naked curvature singularity in a whole bulk space-time. One cousin is a trivial extension of five-dimensional model, while the other one is in essence in higher dimensions. One interesting observation is that in the latter model, only anti-de Sitter ($AdS_p$) brane is physically meaningful whereas de Sitter ($dS_p$) and Minkowski ($M_p$) branes are dismissed. Moreover, for $AdS_p$ brane in the latter model, we study the property of localization of various bulk fields on a single brane. In particular, it is shown that the presence of the brane cosmological constant enables bulk gauge field and massless fermions to confine to the brane only by a gravitational interaction. We find a novel relation between mass of brane gauge field and the brane cosmological constant. 
  The study of Finite Size Effects in Quantum Field Theory allows the extraction of precious perturbative and non-perturbative information. The use of scaling functions can connect the particle content (scattering theory formulation) of a QFT to its ultraviolet Conformal Field Theory content. If the model is integrable, a method of investigation through a nonlinear integral equation equivalent to Bethe Ansatz and deducible from a light-cone lattice regularization is available. It allows to reconstruct the S-matrix and to understand the locality properties in terms of Bethe root configurations, thanks to the link to ultraviolet CFT guaranteed by the exact determination of scaling function. This method is illustrated in practice for Sine-Gordon / massive Thirring models, clarifying their locality structure and the issues of equivalence between the two models. By restriction of the Sine-Gordon model it is also possible to control the scaling functions of minimal models perturbed by Phi_1,3 
  Over the past decade there has been an increasing interest in the study of black holes, and related objects, in higher (and lower) dimensions, motivated to a large extent by developments in string theory. The aim of the present paper is to obtain higher dimensional analogues of some well known results for black holes in 3+1 dimensions. More precisely, we obtain extensions to higher dimensions of Hawking's black hole topology theorem for asymptotically flat ($\Lambda=0$) black hole spacetimes, and Gibbons' and Woolgar's genus dependent, lower entropy bound for topological black holes in asymptotically locally anti-de Sitter ($\Lambda<0$) spacetimes. In higher dimensions the genus is replaced by the so-called $\sigma$-constant, or Yamabe invariant, which is a fundamental topological invariant of smooth compact manifolds. 
  We study quantum corrections on four derivative term of vector multiplets in ${\cal{N}}=2$ supersymmetric Yang-Mills theory. We first show splitting of quantum correction on gauge neutral hypermultiplets from U(1) vector multiplets at four derivative order. We then revisit the non-renormalization theorem given by N. Seiberg and M. Dine and show the non-renormalization theorem in mixed (coulomb plus Higgs) branch even though gauge neutral hypermultiplet develops the vacuum expectation value. 
  Quantum corrections to the energy of D0-branes in a constant RR 4-form background are studied. Using the Born-Oppenheimer approximation, we compute the long-range interaction between two spherical D2-D0 bound states. We extend this calculation to the case where some mass terms are added. For the special value of masses at which the classical energy vanishes, we find complete cancellations of two boson-fermion pairs in the quantum mechanical expression of the zero-point energy, suggesting possible restoration of (partial) supersymmetries. We also briefly discuss the interaction between a dielectric 2-brane and a single D0-brane. 
  Basic concepts and definitions in differential geometry and topology which are important in the theory of solitons and instantons are reviewed. Many examples from soliton theory are discussed briefly, in order to highlight the application of various geometrical concepts and techniques. 
  Classical Calogero-Moser models with rational potential are known to be superintegrable. That is, on top of the r involutive conserved quantities necessary for the integrability of a system with r degrees of freedom, they possess an additional set of r-1 algebraically and functionally independent globally defined conserved quantities. At the quantum level, Kuznetsov uncovered the existence of a quadratic algebra structure as an underlying key for superintegrability for the models based on A type root systems. Here we demonstrate in a universal way the quadratic algebra structure for quantum rational Calogero-Moser models based on any root systems. 
  We study the warped geometry of heterotic M-Theory in five dimensions where five-branes are included in the bulk. Five-branes wrapping holomorphic curves lead to BPS configurations where the junction conditions are automatically satisfied. We consider five-branes wrapped around non-supersymmetric cycles and show that the configuration is unstable. We describe explicitly the resulting time-dependent geometry where the bulk five-branes move towards the Horova-Witten boundary walls. The five-branes collide with the boundary walls in a finite time resulting in the restoration of supersymmetry. 
  In this review, we summarize exact results for the three-dimensional BTZ black hole. We use rigorous mathematical results to clarify the general structure and properties of this black hole spacetime and its microscopic description. In particular, we study the formation of the black hole by point particle collisions, leading to an exact analytic determination of the Choptuik scaling parameter. We also show that a `No Hair Theorem' follows immediately from a mathematical theorem of hyperbolic geometry, due to Sullivan. A microscopic understanding of the Bekenstein-Hawking entropy, and decay rate for massless scalars, is shown to follow from standard results of conformal field theory. 
  We study four point correlation functions of the spin 1 operators in the SU(2)_0 WZNW model. The general solution which is everywhere single-valued has logarithmic terms and thus has a natural interpretation in terms of logarithmic conformal field theory. These are not invariant under all the crossing symmetries but can remain if fields possess additional quantum numbers. 
  We analyze the algebra of observables of a charged particle on a noncommutative torus in a constant magnetic field. We present a set of generators of this algebra which coincide with the generators for a commutative torus but at a different value of the magnetic field, and demonstrate the existence of a critical value of the magnetic field for which the algebra reduces. We then obtain the irreducible representations of the algebra and relate them to noncommutative bundles. Finally we comment on Landau levels, density of states and the critical case. 
  We discuss non-commutative field theories in coordinate space. To do so we introduce pseudo-localized operators that represent interesting position dependent (gauge invariant) observables. The formalism may be applied to arbitrary field theories, with or without supersymmetry.   The formalism has a number of intuitive advantages. First it makes clear the appearance of new degrees of freedom in the infrared. Second, it allows for a study of correlation functions of (composite) operators. Thus we calculate the two point function in position space of the insertion of certain composite operators. We demonstrate that, even at tree level, many of the by now familiar properties of non-commutative field theories are manifest and have simple interpretations. The form of correlation functions are such that certain singularities may be interpreted in terms of dimensional reduction along the non-commutative directions: this comes about because these are theories of fundamental dipoles. 
  In this note we study a massive IIA supergravity theory obtained in hep-th/9707139 by compactification of M-theory. We point out that de Sitter space in arbitrary dimensions arises naturally as the vacuum of this theory. This explicitly shows how de Sitter space can be embedded into eleven-dimensional supergravity. In addition we discuss the novel way in which this theory avoids various `no-go theorems' which assert that de Sitter space is not a consistent vacua of eleven-dimensional supergravity theory. We also point out that the eight-branes of this theory, which couple electrically to the ten-form, can sweep out de Sitter world-volumes. 
  We obtain Q-ball solutions in noncommutative scalar field theory with a global U(1) invariance. The Q-ball solutions are shown to be classically and quantum mechanically stable. We also find that "excited Q-ball" states exist for some class of scalar potentials, which are classically stable in the large noncommutativity limit. 
  We show that a pure gauge theory in higher dimensions may lead to an effective lower-dimensional theory with massive vector field, broken gauge symmetry and no fundamental Higgs boson. The mechanism we propose employs the localization of a vector field on a lower-dimensional defect. No non-zero expectation values of the vector field components along extra dimensions are required. New possibilities for the solution to the gauge hierarchy problem are discussed. 
  We discuss the concept of composite fields in flat CFT as well as in the context of AdS/CFT. Furthermore we show how to represent Green functions using generalized hypergeometric functions and apply these techniques to four-point functions. Finally we prove an identity of $U(1)_Y$ symmetry for four-point functions. 
  We study the reflection amplitudes of affine Toda field theories with boundary, following the ideas developed by Fring and Koberle and focusing our attention on the $E_{n}$ series elements, because of their interesting structure of higher order poles. We also investigate the corresponding minimal reflection matrices, finding, with respect to the bulk case, a more complicated relation between the spectra of bound states associated to the minimal and to the ''dressed'' amplitudes. 
  The multiloop amplitudes for open bosonic string in presence of a constant B-field are derived from first principles. The basic ingredients of the construction are the commutation relations for the string modes and the Reggeon vertex describing the interaction among three generic string states. The modifications due to the presence of the B-field affect non--trivially only the zero modes. This makes it possible to write in a simple and elegant way the general expression for multiloop string amplitudes in presence of a constant B-field. The field theory limit of these string amplitudes is also considered. We show that it reproduces exactly the Feynman diagrams of noncommutative field theories. Issues of UV/IR are briefly discussed. 
  We find a new class of cosmic string solutions with non-vanishing magnetic flux of $\cN=1$, D=4 supergravity with a cosmological constant and coupled to any number of Maxwell and scalar multiplets. We show that these magnetic cosmic string solutions preserve 1/2 of supersymmetry. We give an explicit example of such a solution for which the complex scalars are constant and the spacetime is smooth with topology $R^{1,1}\times S^2$. Two more examples are explored for which a complex scalar field takes values in $\bC P^1$ and in $SL(2,\bR)/U(1)$. 
  The (m^2,\Lambda) plane of spin s>1 massive fields in (A)dS backgrounds is shown to consist of separate phases, divided by lines of novel ``partially massless'' gauge theories that successively remove helicities, starting from the lowest, 0 or +/-(1/2). The norms of the excluded states flip as the gauge lines are crossed and only the region containing the massive Minkowski theory is unitary. The partially massless gauge theories are unitary or not, depending on the ordering of the gauge lines. This ``level splitting'' of massless Minkowski gauge theories is specific to non-zero \Lambda. 
  A two-dimensional integrable system being a deformation of the rational Calogero-Moser system is constructed via the symplectic reduction, performed with respect to the Sklyanin algebra action. We explicitly resolve the respective classical equations of motion via the projection method and quantize the system. 
  We investigate several matrix models based on super Lie algebras, osp(1|32,R), u(1|16,16) and gl(1|32,R). They are natural generalizations of IIB matrix model and were first proposed by Smolin. In particular, we study the supersymmetry structures of these models and discuss possible reductions to IIB matrix model. We also point out that diffeomorphism invariance is hidden in gauge theories on noncommutative space which are derived from matrix models. This symmetry is independent of the global SO(9,1) invariance in IIB matrix model and we report our trial to extend the global Lorentz invariance to local symmetry by introducing u(1|16,16) or gl(1|32,R) super Lie algebras. 
  We derive the central charge and BPS equations from the low-energy effective action for N=2 SU(2) Yang-Mills theory in the Coulomb phase, using a systematic, canonical procedure. We then obtain solutions for monopole and dyon BPS states, whose core structure is described by a dual Lagrangian containing the monopole or dyon as a fundamental field. Spherically symmetric states possess a shell of charge at a characteristic radius. 
  Keeping in mind the several models of M(atrix) theory we attempt to understand the possible structure of the topological M(atrix) theory ``underlying'' these approaches. In particular we are motivated by the issue about the nature of the structure of the vacuum of the topological M(atrix) theory and how this could be related to the vacuum of the electroweak theory. In doing so we are led to a simple topological matrix model. Moreover it is intuitively expected from the current understanding that the noncommutative nature of ``spacetime'' and background independence should lead to a topological Model. The main purpose of this note is to propose a simple topological Matrix Model which bears relation to F and M theories. Suggestions on the origin of the chemical potential term appearing in the matrix models are given. 
  We study the non-local regularization for the case of a spontaneously broken abelian gauge theory in the R_xi gauge with an arbitrary gauge parameter xi. We consider a simple abelian-Higgs model with chiral couplings as an example. We show that if we apply the nonlocal regularization procedure [to construct a nonlocal theory with finite mass parameter ] to the spontaneously broken R_xi gauge Lagrangian, using the quadratic forms as appearing in this Lagrangian, we find that a physical observable in this model, an analogue of the muon anomalous magnetic moment, evaluated to order O[g^{2}] does indeed show xi-dependence.We then apply the modified form of nonlocal regularization that was recently advanced and studied for the unbroken non-abelian gauge theories and discuss the resulting WT identities and xi-independence of the S-matrix elements. 
  The supergravity dual of $N$ regular and $M$ fractional D3-branes on the conifold has a naked singularity in the infrared. Supersymmetric resolution of this singularity requires deforming the conifold: this is the supergravity dual of chiral symmetry breaking. Buchel suggested that at sufficiently high temperature there is no need to deform the conifold: the singularity may be cloaked by a horizon. This would be the supergravity manifestation of chiral symmetry restoration. In previous work [hep-th/0102105] the ansatz and the system of second-order radial differential equations necessary to find such a solution were written down. In this paper we find smooth solutions to this system in a perturbation theory that is valid when the Hawking temperature of the horizon is very high. 
  In a tachyon model proposed by Minahan and Zwiebach and derived in the boundary string field theory, we construct various new solutions which correspond to nontrivial brane configurations in string theory. Our solutions include Dp-D(p-2) bound states, (F, Dp) bound states, string junctions, D(p-2)-branes ending on a Dp-brane, D(p-2)-branes suspended between parallel Dp-branes and their non-commutative generalizations. We find the Bogomol'nyi bounds and the BPS equations for some of our solutions, and check the physical consistency of our solutions with the D-brane picture by looking at the distributions of their energies and RR-charges in space. We also give conjectures for a few other brane configurations. 
  We explored a variety of brane configurations in our previous paper within the two derivative truncation of the unstable D9-brane effective theory. In this paper we extend our previous results with emphasis on the inclusion of the higher derivative corrections for the tachyon and the gauge fields computed in the boundary string field theories. We give the exact solutions to BPS brane configurations studied in our previous paper and find remarkable exact agreements of their energies and RR-charges with the expected results. We further find a few more solutions that we could not construct in the two derivative truncations, such as a (F,D6) bound state ending on a D8-brane whose existence turns out to be due to a higher derivative effect and also the dielectric brane of Emparan and Myers as a nonsupersymmetric example. These are also in exact agreements with the results obtained in the effective theory of supersymmetric D-branes. 
  We show that cohomology of the variational complex in field-antifield BRST theory on an arbitrary manifold is equal to the de Rham cohomology of this manifold. 
  We investigate a noncompact Gepner model, which is composed of a number of SL(2,R)/U(1) Kazama-Suzuki models and N=2 minimal models. The SL(2,R)/U(1) Kazama-Suzuki model contains the discrete series among the SL(2,R) unitary representations as well as the continuous series. We claim that the discrete series contain the vanishing cohomology and the vanishing cycles of the associated noncompact Calabi-Yau manifold. We calculate the Elliptic genus and the open string Witten indices. In the A_{N-1} ALE models, they actually agree with the vanishing cohomology and the intersection form of the vanishing cycles. 
  Condsidering a massive self-interacting phi ^6 scalar field coupled arbitrarily to a (2+1) dimensional Bianchi type-I spacetime, we evaluate the one-loop effective potential. It is found that phi ^6 potential can be regularized in (2+1) dimensional curved spacetime. A finite expression for the energy-momentum tensor is obtained for this model. Evaluating the finite temperature effective potential, the temperature dependence of phase transitions is studied. The crucial dependence of the phase transitions on the spacetime curvature and on the coupling to gravity are also verified. We also discuss the nucleation of bubbles in a phi ^6 model. It is found that there exists an exact solution for the damped motion of the bubble in the thin wall regime. 
  The RR Page charges for the D2-, D4-, D6-brane in B fields are constructed explicitly from the equations of motion and the nonvanishing (modified) Bianchi identities by exploiting their properties --- conserved and localized. It is found that the RR Page charges are independent of the backgound B fields, which provides further evidence that the RR Page charge should be quantized. In our construction, it is highly nontrivial that the terms like B x B x B, B x B x F, B x F x F from different sources are exactly cancelled with each other. 
  We propose a new scenario to implement spontaneous symmetry breaking in the space-time of an arbitrary dimension (D>2) by introducing the non-minimal coupling between the scalar field and the gravity. In this scenario, the usage of the familiar lambda Phi ^4 term, which is non-renormalizable for D >= 5, can be avoided altogether. 
  The paper deals with the analytic theory of the quantum q-deformed Toda chain; the technique used combines the methods of representation theory and the Quantum Inverse Scattering Method. The key phenomenon which is under scrutiny is the role of the modular duality concept (first discovered by L.Faddeev) in the representation theory of noncompact semisimple quantum groups. Explicit formulae for the Whittaker vectors are presented in terms of the double sine functions and the wave functions of the N-particle q-deformed open Toda chain are given as a multiple integral of the Mellin-Barnes type. For the periodic chain the two dual Baxter equations are derived. 
  We show that the coefficient of the three-dimensional Chern-Simons action on the noncommutative plane must be quantized. Similar considerations apply in other dimensions as well. 
  We discuss instantons on noncommutative four-dimensional Euclidean space. In commutative case one can consider instantons directly on Euclidean space, then we should restrict ourselves to the gauge fields that are gauge equivalent to the trivial field at infinity. However, technically it is more convenient to work on four-dimensional sphere.   We will show that the situation in noncommutative case is quite similar. One can analyze instantons taking as a starting point the algebra of smooth functions vanishing at infinity, but it is convenient to add a unit element to this algebra (this corresponds to a transition to a sphere at the level of topology). Our approach is more rigorous than previous considerations ; it seems that it is also simpler and more transparent. In particular, we obtain the ADHM equations in a very simple way. 
  This is a short nontechnical note summarizing the motivation and results of my recent work on D-brane categories. I also give a brief outline of how this framework can be applied to study the dynamics of topological D-branes and why this has a bearing on the homological mirror symmetry conjecture. This note can be read without any knowledge of category theory. 
  We discuss the brane cosmology in the 5D anti de Sitter Schwarzschild (AdSS$_5$) spacetime. A brane with the tension $\sigma$ is defined as the edge of an AdSS$_5$ space. We point out that the location of the horizon is an apparently, singular point at where we may not define an embedding of the AdSS$_5$ spacetime into the moving domain wall (MDW). We resolve this problem by introducing a radially infalling brane (RIB) in AdSS$_5$ space, where an apparent singularity turns out to be a coordinate one. Hence the CFT/FRW-cosmology is well-defined at the horizon. As an example, an universal Cardy formula for the entropy of the CFT can be given by the Friedmann equation at the horizon. 
  We obtain a simple explicit expression for the hyper-Kahler Calabi metric on the co-tangent bundle of CP^{n+1}, for all n, in which it is constructed as a metric of cohomogeneity one with SU(n+2)/U(n) principal orbits. These results enable us to obtain explicit expressions for an L^2-normalisable harmonic 4-form in D=8, and an L^2-normalisable harmonic 6-form in D=12. We use the former in order to obtain an explicit resolved M2-brane solution, and we show that this solution is invariant under all three of the supersymmetries associated with the covariantly-constant spinors in the 8-dimensional Calabi metric. We give some discussion of the corresponding dual N=3 three-dimensional field theory. Various other topics are also addressed, including superpotentials for the Calabi metrics and the metrics of exceptional G_2 and Spin(7) holonomy in D=7 and D=8. We also present complex and quaternionic conifold constructions, associated with the cone metrics whose resolutions are provided by the Stenzel T^*S^{n+1} and Calabi T^*\CP^{n+1} metrics. In the latter case we relate the construction to the hyper-Kahler quotient. We then use the hyper-K\"ahler quotient to give a quaternionic rederivation of the Calabi metrics. 
  We discuss string theory relations between gravity and gauge theory tree amplitudes. Together with $D$-dimensional unitarity, these relations can be used to perturbatively quantize gravity theories, i.e. they contain the necessary information for calculating complete gravity $S$-matrices to any loop orders. This leads to a practical method for computing non-trivial gravity $S$-matrix elements by relating them to much simpler gauge theory ones. We also describe arguments that N=8 D=4 supergravity is less divergent in the ultraviolet than previously thought. 
  Anisotropic cosmological spacetimes are constructed from spherically symmetric solutions to Einstein's equations coupled to nonlinear electrodynamics and a positive cosmological constant. This is accomplished by finding solutions in which the roles of $r$ and $t$ are interchanged for all $r>0$ (i.e. $r$ becomes timelike and $t$ becomes spacelike). Constant time hypersurfaces have topology $R\times S^2$ and in all the spacetimes considered the radius of the two sphere vanishes as $t$ goes to zero. The scale factor of the other dimension diverges as $t$ goes to zero in some solutions and vanishes (or goes to a constant) in other solutions. At late times local observers would see the universe to be homogeneous and isotropic. 
  We investigate U(N) Chern-Simons theories on noncommutative plane. We show that for the theories to be consistent quantum mechanically, the coefficient of the Chern-Simons term should be quantized $\kappa = n/2\pi$ with an integer $n$. This is a surprise for the U(1) gauge theory. When uniform background charge density $\rho_e$ is present, the quantization rule changes to $\kappa +\rho_e\theta = n/2\pi$ with noncommutative parameter $\theta$. With the exact expression for the angular momentum, we argue in the U(1) theory that charged particles in the symmetric phase carry fractional spin $1/2n$ and vortices in the broken phase carry half-integer or integer spin $-n/2$. 
  Abelian and nonabelian gauge invariant states are directly compared to revisit how the unconfined abelian theory is expressed. It is argued that the Yang-Mills equations have no obvious physical content apart from their relation to underlying physical states. The main observation is that the physical states of electrostatics can be regarded as point charges connected by a uniform superposition of all possible Faraday lines. These states are gauge invariant only in the abelian case. 
  A proof that the prepotential for pure N=2 Super-Yang-Mills theory associated with Lie algebras B_r and C_r satisfies the generalized WDVV (Witten-Dijkgraaf-Verlinde-Verlinde) system was given by Marshakov, Mironov and Morozov. Among other things, they use an associative algebra of holomorphic differentials. Later Ito and Yang used a different approach to try to accomplish the same result, but they encountered objects of which it is unclear whether they form structure constants of an associative algebra. We show by explicit calculation that these objects are none other than the structure constants of the algebra of holomorphic differentials. 
  We propose a technique called dimensional descent to show that Wigner's little group for massless particles, which acts as a generator of gauge transformation for usual Maxwell theory, has an identical role even for topologically massive gauge theories. The examples of $B\wedge F$ theory and Maxwell-Chern-Simons theory are analyzed in details. 
  We investigate spectrum of open strings on D-branes after tachyon condensation in bosonic string theory. We calculate 1-loop partition function of the string and show that its limiting forms coincide with partition functions of open strings with different boundary conditions. 
  We calculate the \hkks for the \elm field in the background of a dielectric cylinder with non equal speeds of light inside and outside. The coefficient  $a_{2}$ whose vanishing makes the vacuum energy of a massless field unique, turns out to be zero in dilute order, i.e., in order $(\ep-1)^{2}$, and nonzero beyond. As a consequence, the vanishing of the vacuum energy in the presence of a dielectric cylinder found by Casimir-Polder summation must take place irrespectively of the methods by which it might be calculated. 
  We argue that the four-dimensional universe on the TeV brane of the Randall-Sundrum scenario takes the bimetric structure of Clayton and Moffat, with gravitons traveling faster than photons instead, while the radion varies with time. We show that such brane world bimetric model can thereby solve the flatness and the cosmological constant problems, provided the speed of a graviton decreases to the present day value rapidly enough. The resolution of other cosmological problems such as the horizon problem and the monopole problem requires supplementation by inflation, which may be achieved by the radion field provided the radion potential satisfies the slow-roll approximation. 
  In this work we use the method of consistent deformations of the master equation by Barnich and Henneaux in order to prove that an abelian topological coupling between a zero and a two form fields in D=3 has no nonabelian generalization. We conclude that a topologically massive model involving the Kalb-Ramond two-form field does not admit a nonabelian generalization. The introduction of a connection-type one form field keeps the previous result. 
  The algebraic structure of the neutrino mass Hamiltonian is presented for two neutrino flavors considering both Dirac and Majorana mass terms. It is shown that the algebra is Sp(8) and also discussed how the algebraic structure generalizes for the case of more than two neutrino flavors. 
  In this paper we describe explicitly how the twisted ``bundles'' on a D-brane worldvolume in the presence of a nontrivial B field, can be understood in terms of sheaves on stacks. We also take this opportunity to provide the physics community with a readable introduction to stacks and generalized spaces. 
  We explore some aspects of monodromies of D-branes in the Kahler moduli space of Calabi-Yau compactifications. Here a D-brane is viewed as an object of the derived category of coherent sheaves. We compute all the interesting monodromies in some nontrivial examples and link our work to recent results and conjectures concerning helices and mutations. We note some particular properties of the 0-brane. 
  In this report we introduce the basic techniques (of the Closed Time Path - Coarse Grained Effective Action CTP-CGEA) and ideas (scaling, coarse-graining and backreaction) behind the treatment of quantum processes in dynamical background spacetimes and fields. We show how they are useful for the construction of renormalization group (RG) theories for studying these nonequilibrium processes and discuss the underlying issues. Examples are drawn from quantum field processes in an inflationary universe, semiclassical cosmology and stochastic gravity. In Part I we show how eternal inflation can be treated as static critical phenomena, while a `slow-roll' or power-law inflation can be treated as dynamical critical phenomena. In Part II we introduce the key concepts in open systems and discuss the relation of coarse-graining and backreaction. In Part III we present the CTP - CGEA. We perform perturbative and nonperturbative evaluations, and show how to derive RG equations. In Part IV, we use the RG equations to derive the Einstein-Langevin equation in stochastic semiclassical gravity. We end with a discussion on why a stochastic component of RG equations is expected for nonequilibrium processes 
  Gravitational tidal forces may induce polarization of D0 branes, in analogy to the same effects arising in the context of constant background gauge fields. Such phenomena can teach us about the correspondence between smooth curved spacetime and its underlying non-commutative structure. However, unlike polarization by gauge fields, the gravitational counterpart involves concerns regarding the classical stability of the corresponding polarized states. In this work, we study this issue with respect to the solutions presented in hep-th/0010237 and find that they are classically unstable. The instability however appears with intricate features with all but a few decay channels being lifted. Through a detailed analysis, we then argue that these polarized states may be expected to be long-lived in a regime where the string coupling is small and the number of D0 branes is large. 
  We study deformations of topological closed strings. A well-known example is the perturbation of a topological closed string by itself, where the associative OPE product is deformed, and which is governed by the WDVV equations. Our main interest will be closed strings that arise as the boundary theory for topological open membranes, where the boundary string is deformed by the bulk membrane operators. The main example is the topological open membrane theory with a nonzero 3-form field in the bulk. In this case the Lie bracket of the current algebra is deformed, leading in general to a correction of the Jacobi identity. We identify these deformations in terms of deformation theory. To this end we describe the deformation of the algebraic structure of the closed string, given by the BRST operator, the associative product and the Lie bracket. Quite remarkably, we find that there are three classes of deformations for the closed string, two of which are exemplified by the WDVV theory and the topological open membrane. The third class remains largely mysterious, as we have no explicit example. 
  1. Preliminaries.   2. Heterotic string and motivations for large volume compactifications;     2.1 Gauge coupling unification; 2.2 Supersymmetry breaking by compactification.   3. M-theory on S^1/Z_2 \times Calabi-Yau.   4. Type I/I' string theory and D-branes;     4.1 Low-scale strings and extra-large transverse dimensions; 4.2 Relation type I/I' -- heterotic.   5. Type II theories;     5.1 Low-scale IIA strings and tiny coupling; 5.2 Large dimensions in type IIB; 5.3 Relation type II -- heterotic.   6. Theoretical implications;     6.1 U.V./I.R. correspondence; 6.2 Unification ; 6.3 Supersymmetry breaking and scales hierarchy ; 6.4 Electroweak symmetry breaking in TeV-scale strings.   7. Scenarios for studies of experimental constraints.   8. Extra-dimensions along the world brane: KK excitations of gauge bosons;     8.1 Production at hadron colliders; 8.2 High precision data low-energy bounds; 8.3 One extra dimension for other cases; 8.4 More than one extra dimension.   9. Extra-dimensions transverse to the brane world: KK excitations of gravitons;     9.1 Signals from missing energy experiments; 9.2 Gravity modification and sub-millimeter forces.   10. Dimension-eight operators and limits on the string scale.   11. D-brane Standard Model;     11.1 Hypercharge embedding and the weak angle; 11.2 The fate of U(1)'s and proton stability.   12. Appendix: Supersymmetry breaking in type I strings;     12.1 Scherk-Schwarz deformations; 12.2 Brane supersymmetry breaking. 
  We give an explicit quantum super field construction of the N=2 super Casimir WA(n)-algebras, which is obtained from supersymmetric Miura transformation for the Lie superalgebra A(n,n-1). And also we give an extension of this algebra including a super vertex operator which depends on simple root system of A(n,n-1). 
  Recently, a condition is derived for a nontrivial solution of the Schwinger-Dyson equation to be accompanied by a Goldstone bound state in a special quantum electrodynamics model. This result is extended and a new form of the Goldstone theorem is obtained in a general quantum field theory framework. 
  In three-dimensional QED, which is analyzed in the 1/$N$ expansion, we obtain a sufficient and necessary condition for a nontrivial solution of the Dyson-Schwinger equation to be chiral symmetry breaking solution. In the derivation, a normalization condition of the Goldstone bound state is used. It is showed that the existent analytical solutions satisfy this condition. 
  We derive a condition for a nontrivial solution of the Schwinger-Dyson equation to be accompanied by a Goldstone bound state. It implies that, for quenched planar QED, although chiral symmetry breaking occurs when there is a cutoff, the continuum limit fails to exist. 
  We construct an intersecting brane configuration in six-dimensional space with one extra space-like and one extra time-like dimensions. With a certain additional symmetry imposed on the extra space-time we have found that effective four-dimensional cosmological constant vanishes automatically, providing the static solution with gravity fully localized at the intersection region as there are no propagating massive modes of graviton. In this way, the same symmetry allows us to eliminate tachyonic states of graviton from the spectrum of the effective four-dimensional theory, thus avoiding phenomenological difficulties comming from the matter instability usually induced in theories with extra time-like dimensions. 
  This is a review aimed at a physics audience on the relation between Poisson sigma models on surfaces with boundary and deformation quantization. These models are topological open string theories. In the classical Hamiltonian approach, we describe the reduced phase space and its structures (symplectic groupoid), explaining in particular the classical origin of the non-commutativity of the string end-point coordinates. We also review the perturbative Lagrangian approach and its connection with Kontsevich's star product. Finally we comment on the relation between the two approaches. 
  Using the methods of the 'form factor program' exact expressions of all matrix elements are obtained for several operators of the quantum sine-Gordon model alias the massive Thirring model. A general formula is presented which provides form factors in terms of an integral representation. In particular charge-less operators as for example the current of the topological charge, the energy momentum tensor and all higher currents are considered. In the breather sector it is found the quantum sine-Gordon field equation holds with an exact relation between the 'bare' mass and the normalized mass. Also a relation for the trace of the energy momentum is obtained. All results are compared with Feynman graph expansion and full agreement is found. 
  In phenomenological models with extra dimensions there is a natural symmetry group associated to a brane universe, -- the group of rotations of normal bundle of the brane. We consider Kaluza-Klein gauge fields corresponding to this group and show that they can be localized on the brane in models with warped extra dimensions. These gauge fields are coupled to matter fields which have nonzero rotation moment around the brane. In a particular example of a four-dimensional brane embedded into six-dimensional asymptotically anti-deSitter space, we calculate effective four-dimensional coupling constant between the localized fermion zero modes and the Kaluza-Klein gauge field. 
  In this note we observe that, contrary to the usual lore, string orbifolds do not describe strings on quotient spaces, but rather seem to describe strings on objects called quotient stacks, a result that follows from simply unraveling definitions, and is further justified by a number of results. Quotient stacks are very closely related to quotient spaces; for example, when the orbifold group acts freely, the quotient space and the quotient stack are homeomorphic. We explain how sigma models on quotient stacks naturally have twisted sectors, and why a sigma model on a quotient stack would be a nonsingular CFT even when the associated quotient space is singular. We also show how to understand twist fields in this language, and outline the derivation of the orbifold Euler characteristic purely in terms of stacks. We also outline why there is a sense in which one naturally finds B nonzero on exceptional divisors of resolutions. These insights are not limited to merely understanding existing string orbifolds: we also point out how this technology enables us to understand orbifolds in M-theory, as well as how this means that string orbifolds provide the first example of an entirely new class of string compactifications. As quotient stacks are not a staple of the physics literature, we include a lengthy tutorial on quotient stacks. 
  By considering scalar theories on the fuzzy sphere as matrix models, we construct a renormalization scheme and calculate the one-loop effective action. Because of UV-IR mixing, the two- and the four-point correlators at low energy are not slowly varying functions of external momenta. Interestingly, we also find that field theories on fuzzy RP^2 avoid UV-IR mixing and hence are much more like conventional field theories. We calculate the one-loop beta-function for the O(N) theory on fuzzy RP^2 at large N and show that in addition to the trivial one, it has a nontrivial fixed point that is accessible in perturbation theory. 
  We consider the perturbation of giant gravitons in the background of dilatonic D-branes whose geometry is not of a conventional form of ${\rm AdS}_m \times {\rm S}^n$. We use the quadratic approximation to the brane action to investigate their vibrations around the equilibrium configuration. We found the normal modes of small vibrations of giant gravitons and these vibrations are turned out to be stable. 
  Spontaneous symmetry breaking in (1+1)-dimensional $\phi^{4}$ theory is studied with discretized light-front quantization. Taking effects of non-diagonal interactions into account, the first few terms of the commutation relations $[a_{0},a_{n}]$ are recalculated in the $\hbar$ expansion. Our result of the critical coupling is still consistent with the equal-time result $22\mu^{2}/\hbar \le \lambda_{\rm{cr}} \le 55.5\mu^{2}/\hbar$. We also have examined effects of regarding the ratio of the bare coupling constant to a renormalized mass as an independent parameter in the $\hbar$ expansion. 
  This paper reports the investigation of a matrix model via super Lie algebra, following the proposal of L. Smolin. We consider the osp(1|32,R) nongauged matrix model and gl(1|32,R) gauged matrix model, especially paying attention to the supersymmetry and the relationship with IKKT model. This paper is based on the collaboration with the collaboration with S.Iso, H.Kawai and Y.Ohwashi. 
  We discuss geometrical aspects of different dualities in the integrable systems of the Hitchin type and its various generalizations. It is shown that T duality known in the string theory context is related to the separation of variables procedure in dynamical system. We argue that there are analogues of S duality as well as mirror symmetry in the many-body systems of Hitchin type. The different approaches to the double elliptic systems are unified using the geometry behind the Mukai-Odesskii algebra. 
  We present two examples of SUSY mechanics related with K\"ahler geometry. The first system is the N=4 supersymmetric one-dimensional sigma-model proposed in hep-th/0101065. Another system is the N=2 SUSY mechanics whose phase space is the external algebra of an arbitrary K\"ahler manifold. The relation of these models with antisymplectic geometry is discussed. 
  We study the quantization of chiral fermions coupled to generalized Dirac operators arising in NCG Yang-Mills theory. The cocycles describing chiral symmetry breaking are calculated. In particular, we introduce a generalized locality principle for the cocycles. Local cocycles are by definition expressions which can be written as generalized traces of operator commutators. In the case of pseudodifferential operators, these traces lead in fact to integrals of ordinary local de Rham forms. As an application of the general ideas we discuss the case of noncommutative tori. We also develop a gerbe theoretic approach to the chiral anomaly in hamiltonian quantization of NCG field theory. 
  The question of boundary conditions in conformal field theories is discussed, in the light of recent progress. Two kinds of boundary conditions are examined, along open boundaries of the system, or along closed curves or ``seams''. Solving consistency conditions known as Cardy equation is shown to amount to the algebraic problem of finding integer valued representations of (one or two copies of) the fusion algebra. Graphs encode these boundary conditions in a natural way, but are also relevant in several aspects of physics ``in the bulk''. Quantum algebras attached to these graphs contain information on structure constants of the operator algebra, on the Boltzmann weights of the corresponding integrable lattice models etc. Thus the study of boundary conditions in Conformal Field Theory offers a new perspective on several old physical problems and offers an explicit realisation of recent mathematical concepts. 
  In this short review we outline some recent developments in understanding string orbifolds. In particular, we outline the recent observation that string orbifolds do not precisely describe string propagation on quotient spaces, but rather are literally sigma models on objects called quotient stacks, which are closely related to (but not quite the same as) quotient spaces. We show how this is an immediate consequence of definitions, and also how this explains a number of features of string orbifolds, from the fact that the CFT is well-behaved to orbifold Euler characteristics. Put another way, many features of string orbifolds previously considered ``stringy'' are now understood as coming from the target-space geometry; one merely needs to identify the correct target-space geometry. 
  We study the localization property of antisymmetric tensor fields in the locally localized gravity models. It is shown that all the antisymmetric tensor fields, including the vector field, in a bulk space-time are trapped on an $AdS$ brane by a gravitational interaction where the presence of the brane cosmological constant plays an important role as in the cases of the other bulk fields. The normalized zero-modes spread rather widely in extra space so small extra dimensions might be needed in order not to conflict with experiment. 
  We present a method for computing the one-loop, renormalized quantum energies of symmetrical interfaces of arbitrary dimension and codimension using elementary scattering data. Internal consistency requires finite-energy sum rules relating phase shifts to bound state energies. 
  We study the BPS domain walls of supersymmetric Yang-Mills for arbitrary gauge group. We describe the degeneracies of domain walls interpolating between arbitrary pairs of vacua. A recently proposed large N duality sheds light on various aspects of such domain walls. In particular, for the case of G = SU(N) the domain walls correspond to wrapped D-branes giving rise to a 2+1 dimensional U(k) gauge theory on the domain wall with a Chern-Simons term of level N. This leads to a counting of BPS degeneracies of domain walls in agreement with expected results. 
  We consider the addition of charged matter (``fundametals'') to noncommutative Yang-Mills theory and noncommutative QED, derive Feynman rules and tree-level potentials for them, and study the divergence structure of the theory. These particles behave very much as they do in the commutative theory, except that (1) they occupy bound-state wavefunctions which are essentially those of charged particles in magnetic fields, and (2) there is slight momentum nonconservation at vertices. There is no reduction in the degree of divergence of charged fermion loops like that which affects nonplanar noncommutative Yang-Mills diagrams. 
  We propose a finite Chern-Simons matrix model on the plane as an effective description of fractional quantum Hall fluids of finite extent. The quantization of the inverse filling fraction and of the quasiparticle number is shown to arise quantum mechanically and to agree with Laughlin theory. We also point out the effective equivalence of this model, and therefore of the quantum Hall system, with the Calogero model. 
  We calculate normalization factors and reflection amplitudes in the W-invariant conformal quantum field theories. Using these CFT data we derive vacuum expectation values of exponential fields in affine Toda theories and related perturbed conformal field theories. We apply these results to evaluate explicitly the expectation values of order parameters in the field theories associated with statistical systems, like XY, Z_n-Ising and Ashkin-Teller models. The same results are used for the calculation of the asymptotics of cylindrically symmetric solutions of the classical Toda equations which appear in topological field theories. The integrable boundary Toda theories are considered. We derive boundary reflection amplitudes in non-affine case and boundary one point functions in affine Toda theories. The boundary ground state energies are cojectured. In the last section we describe the duality properties and calculate the reflection amplitudes in integrable deformed Toda theories. 
  Starting from BPS solutions to Yang-Mills which define a stable holomorphic vector bundle, we investigate its deformations. Assuming slowly varying fieldstrengths, we find in the abelian case a unique deformation given by the abelian Born-Infeld action. We obtain the deformed Donaldson-Uhlenbeck-Yau stability condition to all orders in alpha'. This result provides strong evidence supporting the claim that the only supersymmetric deformation of the abelian d=10 supersymmetric Yang-Mills action is the Born-Infeld action. 
  We describe in superspace a classical theory of of two dimensional $(1,1)$ dilaton supergravity coupled to a super-Liouville field, and find exact super black hole solutions to the field equations that have non-constant curvature. We consider the possibility that a gravitini condensate forms and look at the implications for the resultant spacetime structure. We find that all such condensate solutions have a condensate and/or naked curvature singularity. 
  I recall my collaboration with David Gross. A result about descendants of the chiral anomaly is presented: Chern-Simons terms can be written as total derivatives. 
  We explicitly construct noncommutative * products on circularly symmetric two dimensional space by using the technique of Fedosov's deformation quantization. Especially, on constant curvature spaces i.e., S^2 and H^2, we get su(2) and su(1,1) algebra respectively. These are candidates of * products applicable to noncommutative field theories or noncommutative gauge theories on spaces with nontrivial symplectic structure. 
  We point out that scenarios in which the universe is born from the interior of a black hole may not posses many of the problems of the Standard Big-Bang (SBB) model. In particular we demonstrate that the horizon problem, flatness, and the structure formation problem might be solved naturally, not necessarily requiring a long period of cosmological inflation. The black hole information loss problem is also discussed. Our conclusions are completely independent of the details of general models. 
  In this paper we demonstrate that coordinate noncommutativity at short distances can show up in critical phenomena through UV-IR mixing. In the symmetric phase of the Landau-Ginsburg model, noncommutativity is shown to give rise to a non-zero anomalous dimension at one loop, and to cause instability towards a new phase at large noncommutativity. In particular, in less than four dimensions, the one-loop critical exponent $\eta$ is non-vanishing at the Wilson-Fisher fixed point. 
  We study tachyon condensation on brane-antibrane systems in orbifold theories from the viewpoint of boundary string field theory. We show that the condensation of holomorphic tachyon fields generates various fractional D-branes. The boundary N=2 supersymmetry in the world-sheet theory ensures this result exactly. Furthermore, our results are consistent with the twisted RR-charges from detailed calculations of boundary states. We also discuss the generation of RR-charges due to holomorphic tachyon fields on multiple brane-antibrane pairs in flat space. 
  Maldacena and Nunez [hep-th/0008001] identified a gravity solution describing pure N=1 Yang-Mills (YM) in the IR. Their (smooth) supergravity solution exhibits confinement and the U(1)_R chiral symmetry breaking of the dual YM theory, while the singular solution corresponds to the gauge theory phase with unbroken U(1)_R chiral symmetry. In this paper we discuss supersymmetric type IIB compactifications on resolved conifolds with torsion. We rederive singular background of [hep-th/0008001] directly from the supersymmetry conditions. This solution is the relevant starting point to study non-BPS backgrounds dual to the high temperature phase of pure YM. We construct the simplest black hole solution in this background. We argue that it has a regular Schwarzschild horizon and provides a resolution of the IR singularity of the chirally symmetric extremal solution as suggested in [hep-th/0011146]. 
  Regularization of quantum field theories (QFT's) can be achieved by quantizing the underlying manifold (spacetime or spatial slice) thereby replacing it by a non-commutative matrix model or a ``fuzzy manifold''. Such discretization by quantization is remarkably successful in preserving symmetries and topological features, and altogether overcoming the fermion-doubling problem. In this paper, we report on our work on the ``fuzzification'' of the four-dimensional CP2 and its QFT's. CP2 is not spin, but spin${}_c$. Its Dirac operator has many unique features. They are explained and their fuzzy versions are described. 
  Based on results about open string correlation functions, a nonassociative algebra was proposed in a recent paper for D-branes in a background with nonvanishing $H$. We show that our associative algebra obtained by quantizing the endpoints of an open string in an earlier work can also be used to reproduce the same correlation functions. The novelty of this algebra is that functions on the D-brane do not form a closed algebra. This poses a problem to define gauge transformations on such noncommutative spaces. We propose a resolution by generalizing the description of gauge transformations which naturally involves global symmetries. This can be understood in the context of matrix theory. 
  Renormalisation Group (RG) flows in theory space (the space of couplings) are generated by a vector field -- the $\beta$ function. Using a specific metric ansatz in theory space and certain methods employed largely in the context of General Relativity, we examine the nature of the expansion, shear and rotation of geodesic RG flows. The expansion turns out to be a negative quantity inversely related to the norm of the $\beta$ function. This implies the focusing of the flows towards the fixed points of a given field theory. The evolution equation for the expansion along geodesic RG flows is written down and analysed. We illustrate the results for a scalar field theory with a $j\phi$ coupling and pointers to other areas are briefly mentioned. 
  We apply two very different approaches to calculate Skyrmions with baryon number B less than 23. The first employs the rational map ansatz, where approximate charge B Skyrmions are constructed from a degree B rational map between Riemann spheres. We use a simulated annealing algorithm to search for the minimal energy rational map of a given degree B. The second involves the numerical solution of the full non-linear time dependent equations of motion, with initial conditions consisting of a number of well separated Skyrmion clusters. In general, we find a good agreement between the two approaches. For B greater than 6 almost all the solutions are of fullerene type, that is, the baryon density isosurface consists of twelve pentagons and 2B-14 hexagons arranged in a trivalent polyhedron. There are exceptional cases where this structure is modified, which we discuss in detail. We find that for a given value of B there are often many Skyrmions, with different symmetries, whose energies are very close to the minimal value, some of which we discuss. We present rational maps which are good approximations to these Skyrmions and accurately compute their energy by relaxation using the full non-linear dynamics. 
  We study a recently proposed quantum action depending on temperature. We construct a renormalisation group equation describing the flow of action parameters with temperature. At zero temperature the quantum action is obtained analytically and is found free of higher time derivatives. It makes the quantum action an ideal tool to investigate quantum chaos and quantum instantons. 
  An action describing the dynamics of an infinite collection of massless integer spin fields with spin s=0,1,2,3, ...$\infty$ corresponding to totally symmetric Young tableaux representations of Poincare and anti-de Sitter groups is constructed, in any dimension d, in terms of two functions on a 2d-dimensional manifold. The action is represented by an integral localized on a 2d-1-dimensional hypersurface. 
  We argue that, contrary to previous claims, the supersymmetric sine-Gordon model with boundary has a two-parameter family of boundary interactions which preserves both integrability and supersymmetry. We also propose the corresponding boundary S matrix for the first supermultiplet of breathers. 
  It is shown that a IIA superstring carrying D0-brane charge can be `blown-up', in a {\it Minkowski vacuum} background, to a (1/4)-supersymmetric tubular D2-brane, supported against collapse by the angular momentum generated by crossed electric and magnetic Born-Infeld fields. This `supertube' can be viewed as a worldvolume realization of the sigma-model Q-lump. 
  We construct a solitonic 3-brane solution in the 6-dimensional Einstein-Hilbert-Gauss-Bonnet theory with a (negative) cosmological term. This solitonic brane world is delta-function-like. Near the brane the metric is that for a product of the 4-dimensional flat Minkowski space with a 2-dimensional ``wedge'' with a deficit angle (which depends on the solitonic brane tension). Far from the brane the metric approaches that for a product of the 5-dimensional AdS space and a circle. This solitonic solution exists for a special value of the Gauss-Bonnet coupling (for which we also have a delta-function-like codimension-1 solitonic solution), and the solitonic brane tension can take values in a continuous range. We discuss various properties of this solitonic brane world, including coupling between gravity and matter localized on the brane. 
  We present the maximally supersymmetric three-dimensional gauged supergravities. Owing to the special properties of three dimensions -- especially the on-shell duality between vector and scalar fields, and the purely topological character of (super)gravity -- they exhibit an even richer structure than the gauged supergravities in higher dimensions. The allowed gauge groups are subgroups of the global E_8 symmetry of ungauged N=16 supergravity. They include the regular series SO(p,8-p) x SO(p,8-p) for all p=0,1,...,4, the group E_8 itself, as well as various noncompact forms of the exceptional groups E_7, E_6 and F_4 x G_2. We show that all these theories admit maximally supersymmetric ground states, and determine their background isometries, which are superextensions of the anti-de Sitter group SO(2,2). The very existence of these theories is argued to point to a new supergravity beyond the standard D=11 supergravity. 
  We propose a framework where the string scale as well as all compact dimensions are at the electroweak scale $\sim$ TeV$^{-1}$. The weakness of gravity is attributed to the small value of the string coupling $g_s \sim 10^{-16}$, presumably a remnant of the dilaton's runaway behavior, suggesting the possibility of a common solution to the hierarchy and dilaton-runaway problems. In spite of the small $g_s$, in type II string theories with gauge interactions localized in the vicinity of NS5-branes, the standard model gauge couplings are of order one and are associated with the sizes of compact dimensions. At a TeV these theories exhibit higher dimensional and stringy behavior. The models are holographically dual to a higher dimensional non-critical string theory and this can be used to compute the experimentally accessible spectrum and self-couplings of the little strings. In spite of the stringy behavior, gravity remains weak and can be ignored at collider energies. The Damour-Polyakov mechanism is an automatic consequence of our scenario and suggests the presence of a massless conformally-coupled scalar, leading to potentially observable deviations from Einstein's theory, including violations of the equivalence principle. 
  This work deals with the theory of a quantized spin-2 field in the framework of causal perturbation theory. It is divided into two parts. In the first part we analyze the gauge structure of a massless self-interacting quantum tensor field. We look at this theory from a pure field theoretical point of view without assuming any geometrical aspect from general relativity. To first order in the perturbation expansion of the $S$-matrix we derive necessary and sufficient conditions for such a theory to be gauge invariant, by which we mean that the gauge variation of the self-coupling with respect to the gauge charge operator $Q$ is a divergence in the sense of vector analysis. The most general trilinear self-coupling of the graviton field turns out to be the one derived from the Einstein-Hilbert action plus coboundaries and divergences.   In the second part of this work (sect.9) we consider a massive scalar field coupled to gravity. We are interested in the long range behaviour of this theory. Radiative corrections for two particle scattering are investigated in the adiabatic limit, where the cutoff of the interaction at infinity is removed. We compute the differential cross section for graviton bremsstrahlung in which one of the scattered particles emits a graviton of low energy. It is shown that such processes are logarithmically divergent in the adiabatic limit. Furthermore we show that the differential cross section for two particle scattering with a graviton self-energy insertion is finite in the adiabatic limit while for matter self-energy it is logarithmically divergent, too. 
  We consider linearized 5-d gravity in the Randall-Sundrum brane world. The class of static solutions for linearized Einstein equations is found. Also we obtaine wave solutions describing radiation from an imaginary point source located at the Planck distance from the brane. We analyze the fields asymptotic behavior and peculiarities of matter sources. 
  We obtain (3+1) and (3+2) dimensional global flat embeddings of (2+1) uncharged and charged black strings, respectively. In particular, the charged black string, which is the dual solution of the Banados-Teitelboim-Zanelli black holes, is shown to be embedded in the same global embedding Minkowski space structure as that of the (2+1) charged de Sitter black hole solution. 
  We propose the multiple reflection expansion as a tool for the calculation of heat kernel coefficients. As an example, we give the coefficients for a sphere as a finite sum over reflections, obtaining as a byproduct a relation between the coefficients for Dirichlet and Neumann boundary conditions. Further, we calculate the heat kernel coefficients for the most general matching conditions on the surface of a sphere, including those cases corresponding to the presence of delta and delta prime background potentials. In the latter case, the multiple reflection expansion is shown to be non-convergent. 
  The coupling to gravity in D=5 spacetime dimensions is considered for the particle-like and vortex-type solutions obtained by uplifting the D=4 Yang-Mills instantons and D=3 Yang-Mills-Higgs monopoles. It turns out that the particles become completely destroyed by gravity, while the vortices admit a rich spectrum of gravitating generalizations. Such vortex defects may be interesting in view of the AdS/CFT correspondence or in the context of the brane world scenario. 
  We analyze zero energy solutions of the Dirac equation in the background of a string-like configuration in an extension of the standard model which accommodates the most general fermionic mass matrix for neutrinos. If either the left- or the right-handed Majorana mass vanishes, neutrino and electron zero modes are found to exist. If the harmonic mean of the neutrino Majorana masses is large compared to the Dirac mass, we can prove that normalizable neutrino zero modes cease to exist. This leads to an odd number of fermionic zero modes -- namely, only the electron zero mode -- which we argue implies that the bosonic background is in a topologically distinct sector from the vacuum. This is confirmed by noting that the bosonic sector of the model involves the breaking of a global U(1) symmetry and hence possesses topological global U(1) strings. 
  We discuss the origin of the effectively free dynamics of the charge in the magnetic monopole field to apply it for finding the alternative treatment of the charge-monopole as a particle with spin, for tracing out the relation of the charge-monopole to the free relativistic anyon and for clarifying the nature of the non-standard nonlinear supersymmetry of the fermion-monopole system. 
  We investigate bosonic sectors of supersymmetric field theories. We consider superpotentials described by one and by two real scalar fields, and we show how the equations of motion can be factorized into a family of first order Bogomol'nyi equations, so that all the topological defects are of the Bogomol'nyi-Prasad-Sommerfield type. We examine explicit models, that engender the Z_N symmetry, and we identify all the topological sectors, illustrating their integrability. 
  All possible interactions of a point particle with background electromagnetic, gravitational and higher-spin fields is considered in the two-time physics worldline formalism in (d,2) dimensions. This system has a counterpart in a recent formulation of two-time physics in non-commutative field theory with local Sp(2) symmetry. In either the worldline or field theory formulation, a general Sp(2) algebraic constraint governs the interactions, and determines equations that the background fields of any spin must obey. The constraints are solved in the classical worldline formalism (h-bar=0 limit) as well as in the field theory formalism (all powers of h-bar). The solution in both cases coincide for a certain 2T to 1T holographic image which describes a relativistic particle interacting with background fields of any spin in (d-1,1) dimensions. Two disconnected branches of solutions exist, which seem to have a correspondence as massless states in string theory, one containing low spins in the zero Regge slope limit, and the other containing high spins in the infinite Regge slope limit. 
  Interactions and particles in the standard model are characterized by the action of internal and external symmetry groups. The four symmetry regimes involved are related to each other in the context of induced group representations. In addition to Wigner's induced representations of external Poincar\'e group operations, parametrized by energy-momenta, and the induced internal hyperisospin representations, parametrized by the standard model Higgs field, the external operations, including the Lorentz group, can be considered to be induced also by representations of the internal hypercharge-isospin group. In such an interpretation nonlinear spacetime is parametrized by the orbits of the internal action group in the external action group. 
  We study the properties of {\bf exact} (all level $k$) quantum coherent states in the context of string theory on a group manifold (WZWN models). Coherent states of WZWN models may help to solve the unitarity problem: Having positive norm, they consistently describe the very massive string states (otherwise excluded by the spin-level condition). These states can be constructed by (at least) two alternative procedures: (i) as the exponential of the creation operator on the ground state, and (ii) as eigenstates of the annhilation operator. In the $k\to\infty$ limit, all the known properties of ordinary coherent states are recovered. States (i) and (ii) (which are equivalent in the context of ordinary quantum mechanics and string theory in flat spacetime) are not equivalent in the context of WZWN models. The set (i) was constructed by these authors in a previous article. In this paper we provide the construction of states (ii), we compare the two sets and discuss their properties. We analyze the uncertainty relation, and show that states (ii) satisfy automatically the {\it minimal uncertainty} condition for any $k$; they are thus {\it quasiclassical}, in some sense more classical than states (i) which only satisfy it in the $k\to\infty$ limit. Modification to the Heisenberg relation is given by $2 {\cal H}/k$, where ${\cal H}$ is connected to the string energy. 
  A new moving domain wall solution is obtained for a flat 3-universe. This consists of a bulk metric depending on both time and the extra coordinate, plus a dynamically interacting domain wall, admitted by the metric and inhabited by the three-universe. The matter contents are cosmological constants on the domain wall and the bulk. The bulk space is shown to be (A)dS_(5). A remarkable fact concerning the three-universe is that its scale factor never vanishes, even though the corresponding scale factor of the bulk metric vanishes. The inclusion of a bulk scalar field is discussed, neglecting back-reaction. Its normalizability and the existence of a positive frequency or adiabatic bulk vacuum are shown. 
  We study static solutions of the Skyrme model on the two-sphere of radius L, for various choices of potential. The high-density Skyrmion phase corresponds to the ratio beta=L/(size of Skyrmion) being small, whereas the low-density phase corresponds to beta being large. The transition between these two phases, and in particular the behaviour of a relevant order parameter, is examined. 
  We find the classical supersymmetric vacuum states of a class of N = 1* field theories obtained by mass deforming superconformal models with simple gauge groups and N = 4 or N =2 supersymmetry. In particular, new classical vacuum states for mass-deformed N = 4 models with Sp(2N) and SO(N) gauge symmetry are found. We also derive the classical vacua for various mass-deformed N=2 models with Sp(2N) and SU(N) gauge groups and antisymmetric (and symmetric) hypermultiplets. We suggest interpretations of the mass-deformed vacua in terms of three-branes expanded into five-brane configurations. 
  The explicit form of non-Abelian noncommutative supersymmetric (SUSY) chiral anomaly is calculated, the Wess-Zumino consistency condition is verified and the correspondence of the Yang-Mills sector to the previously obtained results is shown. We generalize the Seiberg-Witten map to the case of N=1 SUSY Yang-Mills theory and calculations up to the second order in the noncommutativity parameter are done. 
  We reconsider the issue of embedding space-time fermions into the four-dimensional N=2 world-sheet supersymmetric string. A new heterotic theory is constructed, taking the right-movers from the N=4 topological extension of the conventional N=2 string but a c=0 conformal field theory supporting target-space supersymmetry for the left-moving sector. The global bosonic symmetry of the full formalism proves to be U(1,1), just as in the usual N=2 string. Quantization reveals a spectrum of only two physical states, one boson and one fermion, which fall in a multiplet of (1,0) supersymmetry. 
  We consider a dielectric medium with an ultraviolet behavior as it follows from the Drude model. Compared with dilute models, this has the advantage that, for large frequencies, two different media behave the same way. As a result one expects the Casimir energy to contain less divergencies than for the dilute media approximation. We show that the Casimir energy of a spherical dielectric ball contains just one divergent term, a volume one, which can be renormalized by introducing a contact term analogous to the volume energy counterterm needed in bag models.   PACS: 12.20.Ds, 03.70.+k, 77.22.Ch 
  The one-dimensional Schrodinger equation for the potential $x^6+\alpha x^2 +l(l+1)/x^2$ has many interesting properties. For certain values of the parameters l and alpha the equation is in turn supersymmetric (Witten), quasi-exactly solvable (Turbiner), and it also appears in Lipatov's approach to high energy QCD. In this paper we signal some further curious features of these theories, namely novel spectral equivalences with particular second- and third-order differential equations. These relationships are obtained via a recently-observed connection between the theories of ordinary differential equations and integrable models. Generalised supersymmetry transformations acting at the quasi-exactly solvable points are also pointed out, and an efficient numerical procedure for the study of these and related problems is described. Finally we generalise slightly and then prove a conjecture due to Bessis, Zinn-Justin, Bender and Boettcher, concerning the reality of the spectra of certain PT-symmetric quantum-mechanical systems. 
  We propose a new mechanism for trapping bulk gauge field, giving rise to a massless photon on a flat Minkowski 3-brane in the Randall-Sundrum model in five space-time dimensions. The mechanism we propose employs the topological Higgs mechansim where a topological term and a 3-form gauge potential play an important role. This new mechanism might be considered as a gauge field's analog of the localization of bulk fermions with the mass term of a 'kink' profile. 
  If particle creation is described by a Bogoliubov transformation, then, in the Heisenberg picture, the raising and lowering operators are time dependent. On the other hand, this time dependence is not consistent with field equations and the conservation of the stress-energy tensor. Possible physical interpretations and resolutions of this inconsistency are discussed. 
  Although eigenspectra of one dimensional shape invariant potentials with unbroken supersymmetry are easily obtained, this procedure is not applicable when the parameters in these potentials correspond to broken supersymmetry, since there is no zero energy eigenstate. We describe a novel two-step shape invariance approach as well as a group theoretic potential algebra approach for solving such broken supersymmetry problems. 
  We propose a renormalization group scaling function which is constructed from q-deformed fermionic versions of Virasoro characters. By comparison with alternative methods, which take their starting point in the massive theories, we demonstrate that these new functions contain qualitatively the same information. We show that these functions allow for RG-flows not only amongst members of a particular series of conformal field theories, but also between different series such as N=0,1,2 supersymmetric conformal field theories. We provide a detailed analysis of how Weyl characters may be utilized in order to solve various recurrence relations emerging at the fixed points of these flows. The q-deformed Virasoro characters allow furthermore for the construction of particle spectra, which involve unstable pseudo-particles. 
  The amplitudes in perturbative open string theory are examined as functions of the tachyon condensate parameter. The boundary state formalism demonstrates the decoupling of the open string modes at the non-perturbative minima of the tachyon potential via a degeneration of open world-sheets and identifies an independence of the coupling constants $g_s$ and $g_{YM}$ at general values of the tachyon condensate. The closed sector is generated at the quantum level; it is also generated at the classical level through the condensation of the propagating open string modes on the D-brane degrees of freedom. 
  The exact solution of the Schwinger model with compact gauge group U(1) is presented. The compactification is imposed by demanding that the only surviving true electromagnetic degree of freedom has angular character. Not surprinsingly, this topological condition defines a version of the Schwinger model which is different from the standard one, where $c$ takes values on the line. The main consequences are: the spectra of the zero modes is not degenerated and does not correspond to the equally spaced harmonic oscillator, both the electric charge and a modified gauge invariant chiral charge are conserved (nevertheless, the axial-current anomaly is still present) and, finally, there is no need to introduce a $\theta$-vacuum. A comparison with the results of the standard Schwinger model is pointed out along the text. 
  The AdS/CFT correspondence establishes a string representation for Wilson loops in N=4 SYM theory at large N and large 't Hooft coupling. One of the clearest manifestations of the stringy behaviour in Wilson loop correlators is the string-breaking phase transition. It is shown that resummation of planar diagrams without internal vertices predicts the strong-coupling phase transtion in exactly the same setting in which it arises from the string representation. 
  We apply Kadanoff's theory of marginal deformations of conformal field theories to twistfield deformations of Z_2 orbifold models in K3 moduli space. These deformations lead away from the Z_2 orbifold sub-moduli-space and hence help to explore conformal field theories which have not yet been understood. In particular, we calculate the deformation of the conformal dimensions of vertex operators for p^2<1 in second order perturbation theory. 
  The stability conditions for coordinate gauge independent perturbations of brane-worlds are analyzed. It is shown that, these conditions lead to the Einstein-Hilbert dynamics and to a confined gauge potential, independently of models and metric ansatzes. The size of the extra dimensions are estimated without assuming a fixed topology. The quantum modes corresponding to high frequency gravitational waves are defined through a canonical structure. 
  We analyze the phemomenon of induced fermion number at finite temperature. At finite temperature, the induced fermion number $<N>$ is a thermal expectation value, and we compute the finite temperature fluctuations, $(\Delta N)^2=<N^2>-<N>^2$. While the zero temperature induced fermion number is topological and is a sharp observable, the finite temperature induced fermion number is generically nontopological, and is not a sharp observable. The fluctuations are due to the mixing of states inherent in any finite temperature expectation value. We analyze in detail two different cases in 1+1 dimensional field theory: fermions in a kink background, and fermions in a chiral sigma model background. At zero temperature the induced fermion numbers for these two cases are very similar, but at finite temperature they are very different. The sigma model case is generic and the induced fermion number is nontopological, but the kink case is special and the fermion number is topological, even at finite temperature. There is a simple physical interpretation of all these results in terms of the spectrum of the fermions in the relevant background, and many of the results generalize to higher dimensional models. 
  We show that in 6d models localizing gravity on stringlike defects and satisfying the dominant energy condition, the metric exterior to the string inevitably depends on the string's thickness. As a consequence, in the limit of thin string either the gravity delocalizes, or the six-dimensional Planck scale must be much larger that the four-dimensional one. 
  The Moyal-Lax representation and the Moyal momentum algebra are introduced and systematically investigated. It is shown that the Moyal-Lax equation can be interpreted as a Hamiltonian equation and can be derived from an action. We show that the parameter of non-commutativity, in this case, is related to the central charge of the second Hamiltonian structure of the system. The Moyal-Lax description leads in a natural manner to the dispersionless limit and provides the second Hamiltonian structure of dispersionless integrable models, which has been an open question for sometime. 
  There exist logarithmic CFTs(LCFTs) such as the $c_{p,1}$ models. It is also well known that it generally contains Jordan cell structure. In this paper, we obtain the boundary Ishibashi state for a rank-2 Jordan cell structure and, with these states in $c=-2$ rational LCFT, we derive boundary states in the closed string picture, which correspond to boundary conditions in the open string picture. We also discuss the Verlinde formula for LCFT and possible applications to string theory. 
  One and two loop self-energies are worked out explicitly for a heavy scalar field interacting weakly with a light self-interacting scalar field at finite temperature. The ring/daisy diagrams and a set of necklace diagrams can be summed simultaneously. This simple model serves to illustrate the connection between multi-loop self-energy diagrams and multiple scattering in a medium. 
  Abelian Chern-Simons gauge theory is known to possess a `$S$-self-dual' action where its coupling constant $k$ is inverted {\it i.e.} $k \leftrightarrow {1 \over k}$. Here a vector non-abelian duality is found in the pure non-abelian Chern-Simons action at the classical level. The dimensional reduction of the dual Chern-Simons action to two-dimensions constitutes a dual Wess-Zumino-Witten action already given in the literature. 
  We propose a large N dual of 4d, N=1 supersymmetric, SU(N) Yang-Mills with adjoint field \Phi and arbitrary superpotential W(\Phi). The field theory is geometrically engineered via D-branes partially wrapped over certain cycles of a non-trivial Calabi-Yau geometry. The large N, or low-energy, dual arises from a geometric transition of the Calabi-Yau, where the branes have disappeared and have been replaced by suitable fluxes. This duality yields highly non-trivial exact results for the gauge theory. The predictions indeed agree with expected results in cases where it is possible to use standard techniques for analyzing the strongly coupled, supersymmetric gauge theories. Moreover, the proposed large N dual provides a simpler and more unified approach for obtaining exact results for this class of supersymmetric gauge theories. 
  We perform the Kaluza-Klein reduction of M-theory on warped Calabi-Yau fourfolds with non-trivial four-form flux turned on. The resulting scalar- and superpotential is computed and compared with the superpotential obtained by Gukov, Vafa and Witten using different methods. 
  We propose a physical interpretation of the perturbative breakdown of unitarity in time-like noncommutative field theories in terms of production of tachyonic particles. These particles may be viewed as a remnant of a continuous spectrum of undecoupled closed-string modes. In this way, we give a unified view of the string-theoretical and the field-theoretical no-go arguments against time-like noncommutative theories. We also perform a quantitative study of various locality and causality properties of noncommutative field theories at the quantum level. 
  We analyze the expansion of the fuzzy sphere non-commutative product in powers of the non-commutativity parameter. To analyze this expansion we develop a graphical technique that uses spin networks. This technique is potentially interesting in its own right as introducing spin networks of Penrose into non-commutative geometry. Our analysis leads to a clarification of the link between the fuzzy sphere non-commutative product and the usual deformation quantization of the sphere in terms of the star-product. 
  We address the behaviour of the scalar field with negative mass squared in the five dimensional AdS space where the Randall-Sundrum brane-world is embedded. We point out that the tachyonic scalar allowed in the bulk space destabilizes the embedded brane-world where the cosmological constant is zero. The resolution of this instability is discussed from the viewpoint of AdS/CFT correspondence. 
  We have used a two-dimensional analog of the Hadamard state-condition to study the local constraints on the two-point function of a linear quantum field conformally coupled to a two-dimensional gravitational background. We develop a dynamical model in which the determination of the state of the quantum field is essentially related to the determination of a conformal frame. A particular conformal frame is then introduced in which a two-dimensional gravitational equation is established. 
  We calculate the integrated trace anomaly for a real spin-0 scalar field in six dimensions in a torsionless curved space without a boundary. We use a path integral approach for a corresponding supersymmetric quantum mechanical model. Weyl ordering the corresponding Hamiltonian in phase space, an extra two-loop counterterm ${1/8}\bigg(R + g^{ij} \Gamma^{l}_{k i} \Gamma^{k}_{l j} \bigg)$ is produced in the action. Applying a recursive method we evaluate the components of the metric tensor in Riemann normal coordinates in six dimensions and construct the interaction Langrangian density by employing the background field method.   The calculation of the anomaly is based on the end-point scalar propagator and not on the string inspired center-of-mass propagator which gives incorrect results for the local trace anomaly. The manipulation of the Feynman diagrams is partly relied on the factorization of four dimensional subdiagrams and partly on a brute force computer algebra program developed to serve this specific purpose. The computer program enables one to perform index contractions of twelve quantum fields (10395 in the present case) a task which cannot be accomplished otherwise. We observe that the contribution of the disconnected diagrams is no longer proportional to the two dimensional trace anomaly (which vanishes in four dimensions). The integrated trace anomaly is finally expressed in terms of the 17 linearly independent scalar monomials constructed out of covariant derivatives and Riemann tensors. 
  The noncommutative harmonic oscillator in arbitrary dimension is examined. It is shown that the $\star$-genvalue problem can be decomposed into separate harmonic oscillator equations for each dimension. The noncommutative plane is investigated in greater detail. The constraints for rotationally symmetric solutions and the corresponding two-dimensional harmonic oscillator are solved. The angular momentum operator is derived and its $\star$-genvalue problem is shown to be equivalent to the usual eigenvalue problem. The $\star$-genvalues for the angular momentum are found to depend on the energy difference of the oscillations in each dimension. Furthermore two examples of assymetric noncommutative harmonic oscillator are analysed. The first is the noncommutative two-dimensional Landau problem and the second is the three-dimensional harmonic oscillator with symmetrically noncommuting coordinates and momenta. 
  We discuss the phase structure of the four-dimensional compact U(1) gauge theory at finite temperature using a deformation of the topological model. Its phase structure can be determined by the behavior of the Coulomb gas (CG) system on the cylinder. We utilize the relation between the CG system and the sine-Gordon (SG) model, and investigate the phase structure of the gauge theory in terms of the SG model. Especially, the critical-line equation of the gauge theory in the strong-coupling and high-temperature region is obtained by calculating the one-loop effective potential of the SG model. 
  Semiclassical Hamiltonian field theory is investigated from the axiomatic point of view. A notion of a semiclassical state is introduced. An "elementary" semiclassical state is specified by a set of classical field configuration and quantum state in this external field. "Composed" semiclassical states viewed as formal superpositions of "elementary" states are nontrivial only if the Maslov isotropic condition is satisfied; the inner product of "composed" semiclassical states is degenerate. The mathematical proof of Poincare invariance of semiclassical field theory is obtained for "elementary" and "composed" semiclassical states. The notion of semiclassical field is introduced; its Poincare invariance is also mathematically proved. 
  We examine the lowest order quantum corrections to the effective action arising from a quantized real scalar field in the Randall-Sundrum background spacetime. The leading term is the familiar vacuum, or Casimir, energy density. The next term represents an induced gravity term that can renormalize the 4-dimensional Newtonian gravitational constant. The calculations are performed for an arbitrary spacetime dimension. Two inequivalent boundary conditions, corresponding to twisted and untwisted field configurations, are considered. A careful discussion of the regularization and renormalization of the effective action is given, with the relevant counterterms found. It is shown that the requirement of self-consistency of the Randall-Sundrum solution is not simply a matter of minimizing the Casimir energy density. The massless, conformally coupled scalar field results are obtained as a special limiting case of our results. We clarify a number of differences with previous work. 
  It is shown that quantum-induced (inflationary) brane Universe occurs in the bulk 5d AdS black hole in accordance with AdS/CFT correspondence. Brane stress tensor is induced by quantum effects of dual CFT and brane crosses the horizon of AdS black hole. Quantum-corrected Hubble constant, Hawking temperature and entropy are found on the brane (and at the horizon). The similarity between CFT entropy at the horizon and FRW equations is extended on the quantum level. This suggests the way to understand cosmological entropy bounds in quantum gravity. 
  In this paper we extend our previous work hep-th/0102063 to the case D0-brane+antibrane system. 
  We construct supergravity solutions dual to twisted field theories that are the worldvolume theories of D4-branes wrapped on 2, 3-cycles, and NS-fivebranes on 3-cycles. We first obtain the solutions for the Romans' six-dimensional gauged supergravity theories and then up-lift them to ten dimensions. In particular, we find solutions for field configurations with either non-Abelian fields or B-fields being excited. One of these solutions, in the massless case, is up-lifted to the massless type IIA string theory. This is the first example of such a kind. The cases studied provide new examples of the AdS/CFT duality involving twisted field theories. 
  We present the explicit form of the four-dimensional effective supergravity action which describes low-energy physics of the Randall--Sundrum model with moduli fields in the bulk and charged chiral matter living on the branes. The low-energy action is derived from the compactification of a locally supersymmetric model in five dimension. We describe the mechanism of supersymmetry breaking mediation which relies on the non-trivial configuration of the $Z_2$-odd bulk fields. Broken supersymmetry leads to stabilization of the interbrane distance. 
  The behavior of the divergent part of the bulk AdS/CFT effective action is considered with respect to the special finite diffeomorphism transformations acting on the boundary as a Weyl transformation of the boundary metric. The resulting 1-cocycle of the Weyl group is in full agreement with the 1-cocycle of the Weyl group obtained from the cohomological consideration of the effective action of the corresponding CFT. 
  We consider the Gelfand-Dickey (GD) structure defined by the Moyal $\star$-product with parameter $\ka$, which not only defines the bi-Hamiltonian structure for the generalized Moyal KdV hierarchy but also provides a $W_n^{(\ka)}$ algebra containing the Virasoro algebra as a subalgebra with central charge $\ka^2(n^3-n)/3$.  The free-field realization of the $W_n^{(\ka)}$ algebra is given through the Miura transformation and the cases for $W_3^{(\ka)}$ and $W_4^{(\ka)}$ are worked out in detail. 
  We show that the Higgs branch of a four-dimensional Yang-Mills theory, with gauge and matter content summarised by an ADE quiver diagram, is identical to the generalised Coulomb branch of a four-dimensional superconformal strongly coupled gauge theory with ADE global symmetry. This equivalence suggests the existence of a mirror symmetry between the quiver theories and the strongly coupled theories. 
  Witten's cubic open string field theory is expanded around the perturbatively stable vacuum, including all scalar fields at levels 0, 2, 4 and 6. The (approximate) BRST cohomology of the theory is computed, giving strong evidence for the absence of physical open string states in this vacuum. 
  I review electric-magnetic duality from the perspective of extended supergravity theories in four spacetime dimensions 
  We present an analysis of currents and charges for a system of two mixed fields, both for spinless bosons and for Dirac fermions. This allows us to obtain in a straightforward way the exact field theoretical oscillation formulas exhibiting corrections with respect to the usual ones derived in quantum mechanics. 
  Massive arbitrary spin fields in AdS(3) spacetime are discussed in a framework of light-cone gauge formulation. We also consider compactification of AdS spacetime on manifold which is warped product of AdS space and sphere. Mass spectra of massless and massive fields upon such compactification are found. 
  We propose a Batalin-Vilkovisky (BV) formulation of boundary superstring field theory. The superstring field action is defined in terms of a closed one-form in the space of couplings, and we compute it explicitly for exactly solvable tachyon perturbations. We also argue that the superstring field action defined in this way is the partition function on the disc, in accord with a previous proposal. 
  The gravity duals of nonlocal field theories in the large N limit exhibit a novel behavior near the boundary. To explore this, we present and study the duals of dipole theories - a particular class of nonlocal theories with fundamental dipole fields. The nonlocal interactions are manifest in the metric of the gravity dual and type-0 string theories make a surprising appearance. We compare the situation to that in noncommutative SYM. 
  The notion of the Wick star-product is covariantly introduced for a general symplectic manifold equipped with two transverse polarisations. Along the lines of Fedosov method, the explicit procedure is given to construct the Wick symbols on the manifold. The cohomological obstruction is identified to the equivalence between the Wick star-product and the Fedosov one. In particular in the K\"ahler case, the Wick star-product is shown to be equivalent the Weyl one, iff the manifold is a Calabi-Yau one. 
  Dixon's multipoles for a system of N relativistic positive-energy scalar particles are evaluated in the rest-frame instant form of dynamics. The Wigner hyperplanes (intrinsic rest frame of the isolated system) turn out to be the natural framework for describing multipole kinematics. In particular, concepts like the {\it barycentric tensor of inertia} can be defined in special relativity only by means of the quadrupole moments of the isolated system. 
  We review many quantum aspects of torsion theory and discuss the possibility of the space-time torsion to exist and to be detected. The paper starts, in Chapter 2, with an introduction to the classical gravity with torsion, that includes also interaction of torsion with matter fields. In Chapter 3, the renormalization of quantum theory of matter fields and related topics, like renormalization group, effective potential and anomalies, are considered. Chapter 4 is devoted to the action of particles in a space-time with torsion, and to possible physical effects generated by the background torsion. In particular, we review the upper bounds for the background torsion which are known from the literature. In Chapter 5, the comprehensive study of the possibility of a theory for the propagating completely antisymmetric torsion field is presented. We show, that the propagating torsion may be consistent with the principles of quantum theory only in the case when the torsion mass is much greater than the mass of the heaviest fermion coupled to torsion. Then, universality of the fermion-torsion interaction implies that torsion itself has a huge mass, and can not be observed in realistic experiments. In Chapter 6, we briefly discuss the string-induced torsion and the possibility to induce torsion action and torsion itself through the quantum effects of matter fields. 
  Some time ago, it was found that the never-ending oscillatory chaotic behaviour discovered by Belinsky, Khalatnikov and Lifshitz (BKL) for the generic solution of the vacuum Einstein equations in the vicinity of a spacelike ("cosmological") singularity disappears in spacetime dimensions $D= d+1>10$. Recently, a study of the generalization of the BKL chaotic behaviour to the superstring effective Lagrangians has revealed that this chaos is rooted in the structure of the fundamental Weyl chamber of some underlying hyperbolic Kac-Moody algebra. In this letter, we show that the same connection applies to pure gravity in any spacetime dimension $\geq 4$, where the relevant algebras are $AE_d$. In this way the disappearance of chaos in pure gravity models in $D > 10$ dimensions becomes linked to the fact that the Kac-Moody algebras $AE_d$ are no longer hyperbolic for $d > 9$. 
  We discuss a simple cosmological model derived from M-theory. Three assumptions lead naturally to a pre-big bang scenario: (a) 11-dimensional supergravity describes the low-energy world; (b) non-gravitational fields live on a three-dimensional brane; and (c) asymptotically past triviality. 
  The recently proposed all orders beta function is further investigated. By using a strong-weak coupling duality of the beta function, and some added topology of the space of couplings we are able to extend the flows to arbitrarily large or small scales. Using a non-trivial RG invariant we are able to identify sine-Gordon, sinh-Gordon and Kosterlitz-Thouless phases. We also find an additional phase with cyclic or roaming RG trajectories. 
  In assumption,that string model is the integrable model for particular kind of the background fields, the closed string model in the background gravity field and the antisymmetric B-field is considered as the bihamiltonian system. It is shown, that bihamiltonity is origin of T-duality of the string models. The new Poisson brackets, depending of the background fields and of their derivatives, are obtained. The integrability condition is obtained as the compactability of the bihamoltonity condition and the Jacobi identity of the new Poisson bracket. The B-chiral string model is dual to the chiral string model for the constant background fields. 
  When an NS5-brane crosses a D6-brane, a D4-brane is created stretching between them by the Hanany-Witten effect. However, the T-dual situation involves a Kaluza-Klein monopole crossing a D5-brane and should not result in brane creation. Thus, the newly created D4-brane disappears when T-duality is applied. The T-duality is in a direction transverse to the D4-brane so that one would naively have expected the creation of a D5-brane in the T-dual picture. This situation is investigated via the corresponding supergravity solutions and the tension with the naive result is resolved. Along the way, a differential form version of the supergravity T-duality relations is developed and some comments are made concerning the flux-expulsion properties of D6-branes. 
  String theories with branes can often be generalized by adding brane-antibrane pairs. We explore the cancellation of anomalies in this more general context, extending the familiar anomaly-cancelling mechanisms, both for ten-dimensional string theories with D-branes and for certain supersymmetric compactifications. 
  We show how topological open string theory amplitudes can be computed by using relative stable morphisms in the algebraic category. We achieve our goal by explicitly working through an example which has been previously considered by Ooguri and Vafa from the point of view of physics. By using the method of virtual localization, we successfully reproduce their results for multiple covers of a holomorphic disc, whose boundary lies in a Lagrangian submanifold of a Calabi-Yau 3-fold, by Riemann surfaces with arbitrary genera and number of boundary components. In particular we show that in the case we consider there are no open string instantons with more than one boundary component ending on the Lagrangian submanifold. 
  New string models in D=4 space-time extended by tensor central charge coordinates $z_{mn}$ are constructed. We use the $z_{mn}$ coordinates to generate string tension using a minimally extended string action linear in $dz^{mn}$. It is shown that the presence of $z_{mn}$ lifts the light-like character of the tensionless string worldsheet and the degeneracy of its induced metric. We analyse the equations of motion and find a solution of the string equations in the generalized D=(4+6)-dimensional space $X^{\cal M}=(x^{m},z^{mn})$ with $z_{mn}$ describing a spin wave process. A supersymmetric version of the proposed model is formulated. 
  Using the Batalin-Vilkovisky formalism we provide a detailed analysis of the NS sector of boundary superstring field theory. We construct explicitly the relevant BV structure and derive the master action. Furthermore, we show that this action is exactly equal to the superdisk worldsheet partition function as was recently conjectured. 
  The operators K_n are generators of reparameterization symmetries of Witten's cubic open string field theory. One pertinent question is whether they can be utilised to generate deformations of the tachyon vacuum and thereby violate its uniqueness. We use level truncation to show that these transformations on the vacuum are in fact pure gauge transformations to a very high accuracy, thus giving new evidence for the uniqueness of the perturbatively stable vacuum. Equivalently, this result implies the vanishing of some discrete cohomology classes of the BRST operator in the stable vacuum. 
  In this thesis, we investigate various integral submodels and generalize them. In part I, we study the submodel of the nonlinear $\mathbf{C}P^1$-model and the related submodels in $(1+2)$ dimensions. In part II, we construct integrable submodels of the nonlinear Grassmann models in any dimension. We call them the Grassmann submodels. To show that our submodels are integrable, we construct an infinite number of conserved currents in two ways. One is that we make full use of the Noether currents of the nonlinear Grassmann models. The other is that we use a method of multiplier. Next we investigate symmetries of the Grassmann submodel. By using the symmetries, we can construct a wide class of exact solutions for our submodels. In part III, keeping some properties of our submodels, we generalize our submodels to higher-order equations. First we prepare the Bell polynomials and the generalized Bell polynomials which play the most important roles in our theory of generalized submodels. Next we generalize the $\mathbf{C}P^1$-submodel to higher-order equations. Lastly we generalize the Grassmann submodel to higher-order equations. By using the generalized Bell polynomials, we can show that the generalized Grassmann submodels are also integrable. As a result, we obtain a hierarchy of systems of integrable equations in any dimension which includes Grassmann submodels. These results lead to the conclusion that the integrable structures of our generalized submodels are closely related to some fundamental properties of the Bell polynomials. 
  The purpose of this article is to initiate a study of a class of Lorentz invariant, yet tractable, Lagrangian Field Theories which may be viewed as an extension of the Klein-Gordon Lagrangian to many scalar fields in a novel manner. These Lagrangians are quadratic in the Jacobians of the participating fields with respect to the base space co-ordinates. In the case of two fields, real valued solutions of the equations of motion are found and a phenomenon reminiscent of instanton behaviour is uncovered; an ansatz for a subsidiary equation which implies a solution of the full equations yields real solutions in three-dimensional Euclidean space. Each of these is associated with a spherical harmonic function. 
  We construct the general form of matter coupled N=4 gauged supergravity in five dimensions. Depending on the structure of the gauge group, these theories are found to involve vector and/or tensor multiplets. When self-dual tensor fields are present, they must be charged under a one-dimensional Abelian group and cannot transform non-trivially under any other part of the gauge group. A short analysis of the possible ground states of the different types of theories is given. It is found that AdS ground states are only possible when the gauge group is a direct product of a one-dimensional Abelian group and a semi-simple group. In the purely Abelian, as well as in the purely semi-simple gauging, at most Minkowski ground states are possible. The existence of such Minkowski ground states could be proven in the compact Abelian case. 
  Unlike noncommutative space, when space and time are noncommutative, it seems necessary to modify the usual scheme of quantum mechanics. We propose in this paper a simple generalization of the time evolution equation in quantum mechanics to incorporate the feature of a noncommutative spacetime. This equation is much more constraining than the usual Schr\"odinger equation in that the spatial dimension noncommuting with time is effectively reduced to a point in low energy. We thus call the new evolution equation the spacetime bootstrap equation, the dimensional reduction called for by this evolution seems close to what is required by the holographic principle. We will discuss several examples to demonstrate this point. 
  We consider the conformal field theory which is dual to the Taub-Bolt-AdS spacetime. It is shown that the Cardy-Verlinde formula for the entropy of the conformal field theory agrees precisely with the entropy of the Taub-Bolt-AdS spacetime, at high temperatures. This result may be viewed as providing a conformal field theory interpretation of Taub-Bolt-AdS entropy. 
  In the semiclassical approximation of Grassmann-valued electric charges for regularizing Coulomb self-energies, we extract the unique acceleration-independent interaction hidden in any Lienard-Wiechert solution for the system of N positive-energy spinning particles plus the electromagnetic field in the radiation gauge of the rest-frame instant form. With the help of a semiclassical Foldy-Wouthuysen transformation, this allows us to find the relativistic semiclassical Darwin potential. In the 2-body case the quantization of the lowest order reproduces exactly the results from the reduction of the Bethe-Salpeter equation. 
  An algebraic criterion for the vanishing of the beta function for renormalizable quantum field theories is presented. Use is made of the descent equations following from the Wess-Zumino consistency condition. In some cases, these equations relate the fully quantized action to a local gauge invariant polynomial. The vanishing of the anomalous dimension of this polynomial enables us to establish a nonrenormalization theorem for the beta function $\beta_g$, stating that if the one-loop order contribution vanishes, then $\beta_g$ will vanish to all orders of perturbation theory. As a by-product, the special case in which $\beta_g$ is only of one-loop order, without further corrections, is also covered. The examples of the N=2,4 supersymmetric Yang-Mills theories are worked out in detail. 
  The Weyl anomaly in the Holographic Renormalisation Group as implemented using Hamilton-Jacobi language is studied in detail. We investigate the breakdown of the descent equations in order to isolate the Weyl anomaly of the dual field theory close to the (UV) fixed point. We use the freedom of adding finite terms to the renormalised effective action in order to bring the anomalies in the expected form. We comment on different ways of describing the bare and renormalised schemes, and on possible interpretations of the descent equations as describing the renormalisation group flow non-perturbatively. We find that under suitable assumptions these relations may lead to a class of c-functions. 
  I summarize perturbative and non-perturbative field theory tests of the holographic correspondence between type IIB superstring on AdS_5xS^5 and N=4 SYM theory. The holographic duality between D-instantons and YM instantons is briefly described. Non renormalization of two- and three-point functions of CPO's and their extremal and next-to-extremal correlators are then reviewed. Finally, partial non-renormalization of four-point functions of lowest CPO's is analyzed in view of the interpretation of short distance logarithmic behaviours in terms of anomalous dimensions of unprotected operators. 
  Quantization is still a central problem of modern physics. One example of an unsolved problem is the quantization of Nambu mechanics. After a brief comment on the role of Harrison cohomology, this review concentrates on the central problem of quantization of QCD and, more generally, quark confinement seen as a problem of quantization. Several suggestions are made, some of them rather extravagant. 
  The boundary chiral ring of a 2d gauged linear sigma model on a K\"ahler manifold $X$ classifies the topological D-brane sectors and the massless open strings between them. While it is determined at small volume by simple group theory, its continuation to generic volume provides highly non-trivial information about the $D$-branes on $X$, related to the derived category $D^\flat(X)$. We use this correspondence to elaborate on an extended notion of McKay correspondence that captures more general than orbifold singularities. As an illustration, we work out this new notion of McKay correspondence for a class of non-compact Calabi-Yau singularities related to Grassmannians. 
  We show that M-theory on spaces with irreducible holonomy represent Type IIA backgrounds in which a collection of D6-branes wrap a supersymmetric cycle in a manifold with a holonomy group different from the one appearing in the M-theory description. For example, we show that D6-branes wrapping a supersymmetric four-cycle on a manifold with G_2 holonomy is described in eleven dimensions by M-theory on a space with Spin(7) holonomy. Examples of such Type IIA backgrounds which lift to M-theory on spaces with SU(3), G_2, SU(4) and Spin(7) holonomy are considered. The M-theory geometry can then be used to compute exact quantities of the gauge theory on the corresponding D-brane configuration. 
  As a step toward clarification of the power of supersymmetry (SUSY) in Matrix theory, a complete calculation, including all the spin effects, is performed of the effective action of a probe D-particle, moving along an arbitrary trajectory in interaction with a large number of coincident source D-particles, at one loop at order 4 in the derivative expansion. Furthermore, exploiting the SUSY Ward identity developed previously, the quantum-corrected effective supersymmetry transformation laws are obtained explicitly to the relevant order and are used to verify the SUSY-invariance of the effective action. Assuming that the agreement with 11-dimensional supergravity persists, our result can be regarded as a prediction for supergravity calculation, which, yet unavailable, is known to be highly non-trivial. 
  We study orbifold group actions on locally defined fields upon M-theory branes in a three-form C-fields background. We derive some constraints from the consistency of the orbifold group actions. We show the possibility of the existence of M-theory discrete torsion for the fields on the worldvolume and discuss its features. 
  A dynamical system is canonically associated to every Drinfeld double of any affine Kac-Moody group. The choice of the affine Lu-Weinstein-Soibelman double gives a smooth one-parameter deformation of the standard WZW model. In particular, the worldsheet and the target of the classical version of the deformed theory are the ordinary smooth manifolds. The quasitriangular WZW model is exactly solvable and it admits the chiral decomposition.Its classical action is not invariant with respect to the left and right action of the loop group, however it satisfies the weaker condition of the Poisson-Lie symmetry. The structure of the deformed WZW model is characterized by several ordinary and dynamical r-matrices with spectral parameter. They describe the q-deformed current algebras, they enter the definition of q-primary fields and they characterize the quasitriangular exchange (braiding) relations. Remarkably, the symplectic structure of the deformed chiral WZW theory is cocharacterized by the same elliptic dynamical r-matrix that appears in the Bernard generalization of the Knizhnik-Zamolodchikov equation, with q entering the modular parameter of the Jacobi theta functions. This reveals a tantalizing connection between the classical q-deformed WZW model and the quantum standard WZW theory on elliptic curves and opens the way for the systematic use of the dynamical Hopf algebroids in the rational q-conformal field theory. 
  We discuss the possible relevance of some recent mathematical results and techniques on four-manifolds to physics. We first suggest that the existence of uncountably many R^4's with non-equivalent smooth structures, a mathematical phenomenon unique to four dimensions, may be responsible for the observed four-dimensionality of spacetime. We then point out the remarkable fact that self-dual gauge fields and Weyl spinors can live on a manifold of Euclidean signature without affecting the metric. As a specific example, we consider solutions of the Seiberg-Witten Monopole Equations in which the U(1) fields are covariantly constant, the monopole Weyl spinor has only a single constant component, and the 4-manifold M_4 is a product of two Riemann surfaces Sigma_{p_1} and Sigma_{p_2}. There are p_{1}-1(p_{2}-1) magnetic(electric) vortices on \Sigma_{p_1}(\Sigma_{p_2}), with p_1 + p_2 \geq 2 (p_1=p_2= 1 being excluded). When the two genuses are equal, the electromagnetic fields are self-dual and one obtains the Einstein space \Sigma_p x \Sigma_p, the monopole condensate serving as the cosmological constant. 
  In this paper we present explicit formulas for the *-product on quantum spaces which are of particular importance in physics, i.e., the q-deformed Minkowski space and the q-deformed Euclidean space in 3 and 4 dimensions, respectively. Our formulas are complete and formulated using the deformation parameter q. In addition, we worked out an expansion in powers of h=lnq up to second order, for all considered cases. 
  We derive an effective Lagrangian for meson fields. This is done in the light-cone gauge for two-dimensional large-N_c QCD by using the bilocal auxiliary field method. The auxiliary fields are bilocal on light-cone space and their Fourier transformation determines the parton momentum distribution. As the first test of our method, the 't Hooft equation is derived from the effective Lagrangian. 
  We study a four-dimensional spacetime induced by the recoil of a D(irichlet)-particle, embeded in it, due to scattering by a moving string. The induced spacetime has curvature only up to a radius that depends on the energy of the incident string. Outside that region (`bubble') the spacetime is matched with the Minkowski spacetime. The interior of the bubble is consistent with the effective field theory obtained from strings, with non-trivial tachyon-like and antisymmetric tensor fields (in four dimensions the latter gives rise to an axion pseudoscalar field). The tachyonic mode, however, does not represent the standard flat-spacetime string tachyon, but merely expresses the instability of the distorted spacetime. Due to the non-trivial matter content of the interior of the bubble, there is entropy production, which expresses the fact that information is carried away by the recoil degrees of freedom. We also demonstrate that a particle can be captured by the bubble, depending on the particle's impact parameter. This will result in information loss for an external asymptotic observer, corresponding to production of entropy propotional to the area of the bubble. For the validity of our approach it is essential that the string length is a few orders of magnitude larger than the Planck length, which is a typical situation encountered in many D-brane-world models. A very interesting feature of our model is the emission of high-energy photons from the unstable bubble, which might be related to the observed apparent ``violations'' of the GZK cutoff. 
  A complete analysis of the canonical structure for a gauge fixed PST bosonic five brane action is performed. This canonical formulation is quadratic in the dependence on the antisymmetric field and it has second class constraints. We remove the second class constraints and a master canonical action with only first class constraints is proposed. The nilpotent BRST charge and its BRST invariant effective theory is constructed. The construction does not assume the existence of the inverse of the induced metric. Singular configurations are then physical ones. We obtain the physical Hamiltonian of the theory and analyze its stability properties. Finally, by studying the algebra of diffeomorphisms we find under mild assumptions the general structure for the Hamiltonian constraint for theories invariant under 6 dimensional diffeomorphisms and we give an algebraic characterization of the constraint associated with the bosonic five brane action. We also identify the constraint for the bosonic five brane action upgraded with a cosmological term, it contains a Born-Infeld type term. 
  The energy-momentum tensor of Matrix Theory is derived by computing disk amplitudes with one closed string and an arbitrary number of open strings and by taking the DKPS limit. We clarify its relation to the energy-momentum tensor of the noncommutative gauge theory derived in our previous paper. 
  The restriction of space-time dimensions to "2+1" leads us to a novel quantum field theory which has the Chern-Simons term in its action. This term changes the nature of gauge interaction by giving a so-called topological mass to a gauge field without breaking the gauge symmetry. We investigate how a dynamical mass of fermion is affected by the topological mass in the non-perturbative Schwinger-Dyson method. 
  Following Marleau, we study an extended version of the Skyrme model to which a sixth order term has been added to the Lagrangian and we analyse some of its classical properties. We compute the multi-Skyrmion solutions numerically for up to B=5 and show that they have the same symmetries as the usual Skyrmion solutions. We use the rational map ansatz introduced by Houghton et al. to evaluate the energy and the radius for multi-skyrmion solutions of up to B=6 for both the SU(2) and SU(3) models and compare these results to the ones obtained numerically. We show that the rational map ansatz works as well for the generalised model as for the pure Skyrme model. 
  We describe the extension of the Wigner`s infinite-dimensional unitary representations of Poincar\'{e} group to the case of $\kappa$-deformed Poincar\'{e} group. We show that the corresponding coordinate wave functions on noncommutative space-time are described by free field equations on $\kappa$-deformed Minkowski space. The cases of Klein--Gordon, Dirac, Proca and Maxwell fields are considered. Finally some aspects of second quantization are also discussed. 
  The first quantum correction to the finite temperature partition function for a self-interacting massless scalar field on a $D-$dimensional flat manifold with $p$ non-commutative extra dimensions is evaluated by means of dimensional regularization, suplemented with zeta-function techniques. It is found that the zeta function associated with the effective one-loop operator may be nonregular at the origin. The important issue of the determination of the regularized vacuum energy, namely the first quantum correction to the energy in such case is discussed. 
  We formulate Lorentz group representations in which ordinary complex numbers are replaced by linear functions of real quaternions and introduce dotted and undotted quaternionic one-dimensional spinors. To extend to parity the space-time transformations, we combine these one-dimensional spinors into bi-dimensional column vectors. From the transformation properties of the two-component spinors, we derive a quaternionic chiral representation for the space-time algebra. Finally, we obtain a quaternionic bi-dimensional version of the Dirac equation. 
  We study the spectrum of massive excitations in three-dimensional models belonging to the Ising universality class. By solving the Bethe-Salpeter equation for 3D $\phi^4$ theory in the broken symmetry phase we show that recently found non-perturbative states can be interpreted as bound states of the fundamental excitation. We show that duality predicts an exact correspondence between the spectra of the Ising model in the broken symmetry phase and of the Z(2) gauge theory in the confining phase. The interpretation of the glueball states of the gauge theory as bound states of the dual spin system allows us to explain the qualitative features of the glueball spectrum, in particular, its peculiar angular momentum dependence. 
  We discuss some properties of noncommutative supersymmetric field theories which do not involve gauge fields. We concentrate on the renormalizability issue of these theories. 
  Spontaneously broken gauge theories are described as a perturbation of selfdual gauge theory. Instead of the incorporation of scalar degrees of freedom, the massive component of the gauge field is obtained from an anti-selfdual field strength consisting of three components before gauge fixing. The interactions describe a massive gauge theory that is non-polynomial with an expansion containing an infinite number of terms. The Lagrangian generalizes the form of the axial anomaly in two dimensions. Unitary propagation of the tensor field occurs upon gauge fixing an additional symmetry. 
  We study supersymmetric charged rotating black holes in AdS$_5$, and show that closed timelike curves occur outside the event horizon. Also upon lifting to rotating D3 brane solutions of type IIB supergravity in ten dimensions, closed timelike curves are still present. We believe that these causal anomalies correspond to loss of unitarity in the dual ${\cal N}=4$, D=4 super Yang-Mills theory, i.e. the chronology protection conjecture in the AdS bulk is related to unitarity bounds in the boundary CFT. We show that no charged or uncharged geodesic can penetrate the horizon, so that the exterior region is geodesically complete. These results still hold true in the quantum case, i.~e.~the total absorption cross section for Klein-Gordon scalars propagating in the black hole background is zero. This suggests that the effective temperature is zero instead of assuming the naively found imaginary value. 
  We show that for two non-trivial lambda phi ^4 problems (the anharmonic oscillator and the Landau-Ginzburg hierarchical model), improved perturbative series can be obtained by cutting off the large field contributions. The modified series converge to values exponentially close to the exact ones. For lambda larger than some critical value, the method outperforms Pade's approximants and Borel summations. The method can also be used for series which are not Borel summable such as the double-well potential series. We show that semi-classical methods can be used to calculate the modified Feynman rules, estimate the error and optimize the field cutoff. 
  We discuss the form of the chiral anomaly on an S1/Z2 orbifold with chiral boundary conditions. We find that the 4-divergence of the higher-dimensional current evaluated at a given point in the extra dimension is proportional to the probability of finding the chiral zero mode there. Nevertheless the anomaly, appropriately defined as the five dimensional divergence of the current, lives entirely on the orbifold fixed planes and is independent of the shape of the zero mode. Therefore long distance four dimensional anomaly cancellation ensures the consistency of the higher dimensional orbifold theory. 
  We describe, on a few instructive examples, a systematic way of deducing the superfield equations of motion of superbranes in the approach of partial breaking of global supersymmetry (PBGS) from the nonlinear-realizations formalism. For D-branes these equations simultaneously represent the appropriate supersymmetric Born-Infeld theories. We also discuss how to construct an off-shell superfield action for the $N=2, d=4$ Dirac-Born-Infeld theory corresponding to the partial supersymmetry breaking $N=4 \to N=2$ in $d=4$. 
  A geometric approach to the standard model in terms of the Clifford algebra Cl_7 is advanced. A key feature of the model is its use of an algebraic spinor for one generation of leptons and quarks. Spinor transformations separate into left-sided ("exterior") and right-sided ("interior") types. By definition, Poincare transformations are exterior ones. We consider all rotations in the seven-dimensional space that (1) conserve the spacetime components of the particle and antiparticle currents and (2) do not couple the right-chiral neutrino. These rotations comprise additional exterior transformations that commute with the Poincare group and form the group SU(2)_L, interior ones that constitute SU(3)_C, and a unique group of coupled double-sided rotations with U(1)_Y symmetry. The spinor mediates a physical coupling of Poincare and isotopic symmetries within the restrictions of the Coleman--Mandula theorem. The four extra spacelike dimensions in the model form a basis for the Higgs isodoublet field, whose symmetry requires the chirality of SU(2). The charge assignments of both the fundamental fermions and the Higgs boson are produced exactly. 
  The fermion representations and boundary conditions in five dimensional anti de Sitter space are described in detail. In each case the one loop effective action is calculated for massless fermions. The possibility of topological or Wilson loop symmetry breaking is discussed. 
  We analyze the structure of noncommutative pure Chern-Simons theory systematically in the axial gauge. We show that there is no IR/UV mixing in this theory in this gauge. In fact, we show, using the usual BRST identities as well as the identities following from vector supersymmetry, that this is a free theory. As a result, the tree level Chern-Simons coefficient is not renormalized. It also holds that the Chern-Simons coefficient is not modified at finite temperature. As a byproduct of our analysis, we prove that the ghosts completely decouple in the axial gauge in a noncommutative gauge theory. 
  We sketch briefly the essentials of the quantum groups and their application to the dynamics of a q-deformed simple harmonic oscillator moving on a quantum line, defined in the q-deformed cotangent (momentum phase) space. In this endeavour, the quantum group $GL_{qp} (2)$- and the conventional rotational invariances are respected together. During the course of this discussion, we touch upon Rajaji's personality as a critical physicist and a bold and adventurous man of mathematical physics. 
  We argue that the BRST and the anti-BRST super symmetries in the four-dimensional Yang-Mills theory can be spontaneously broken in a nonlinear partial gauge due to ghost-anti-ghost condensation. However, we show that the spontaneous BRST symmetry breaking can be avoided if we adopt the modified Maximal Abelian gauge which is an orthosymplectic $OSp(4|2)$ invariant renormalizable gauge proposed by the author to derive quark confinement. We compare the Maximal Abelian gauge with the conventional $OSp(4|2)$ invariant gauge proposed by Delbourgo-Jarvis and Baulieu-Thierry-Mieg. Finally, an implication to the Gribov problem is briefly mentioned. 
  BRST construction of $D$-branes in SU(2) WZW model is proposed. 
  The construction of lagrangians describing the various representations of the Poincare group is given in terms of the BRST approach. 
  The free field realization of the eight-vertex model is extended to form factors. It is achieved by constructing off-diagonal with respect to the ground state sectors matrix elements of the $\Lambda$ operator which establishes a relation between corner transfer matrices of the eight-vertex model and of the SOS model. As an example, the two-particle form factor of the $\sigma^z$ operator is evaluated explicitly. 
  Using the nonabelian Dirac-Born-Infeld action with the Wess-Zumino term that is constructed in consistent with T duality we examine the Myers dielectric effect for multiple D0-branes in the near-horizon geometry of D4-branes. The effect in the curved spacetime is also confirmed by the dual formulation based on the abelian Dirac-Born-Infeld action of a D2-brane. Putting a system of muliple D-strings in the external electric RR five-form flux, we construct a noncommutative non-BPS solution where the D-strings expand into a spherical D3-brane. We discuss the external field dependence of the funnel-like or wormhole solution. 
  We find new solutions corresponding to torus-like generalization of dielectric D2-brane from the viewpoint of D2-brane action and N D0-branes one. These are meta-stable and would decay to the spherical dielectric D2-brane. 
  We calculate the entropy of the brane-world black hole in the Randall-Sundrum(RS) model by using the brick-wall method. The modes along the extra dimension are semi-classically quantized on the extra dimension. The number of modes in the extra dimension is given as a simple form with the help of the RS mass relation, and then the entropy for the scalar modes in the five-dimensional spacetime is described by the two-dimensional area of the black brane world. 
  We investigate supersymmetric tubular configurations in the matrix theory. We construct a host of BPS configurations of eight supersymmetries. They can be regarded as cylindrical D2 branes carrying nonvanishing angular momentum. For the simplest tube, the world volume can be described as noncommutative tube and the world volume dynamics can be identified as a noncommutative gauge theory. Among the BPS configurations, some describe excitations on the tube and others describe many parallel tubes of different size and center. 
  We describe a systematic method of studying the action of the T-duality group O(d,d) on space-time fermions and R-R field strengths and potentials in type-II string theories, based on space-time supersymmetry. The formalism is then used to show that the couplings of non-Abelian D-brane charges to R-R potentials can be described by an appropriate Clifford multiplication. 
  We study electric-magnetic duality rotations for noncommutative electromagnetism (NCEM). We express NCEM as a nonlinear commutative U(1) gauge theory and show that it is self-dual when the noncommutativity parameter \theta is light-like (e.g. \theta^{0i}=\theta^{1i}). This implies, in the slowly varying field approximation, self-duality of NCEM to all orders in \theta. 
  Two classical scalar fields are minimally coupled to gravity in the Kachru-Shulz-Silverstein scenario with a rolling fifth radius. A Tolman wormhole solution is found for a R x S^3 brane with Lorentz metric and for a R x AdS_3 brane with positive definite metric. 
  We discuss a four-dimensional Volkov-Akulov supersymmetric theory on a D3-brane with N=2 supersymmetry broken down to N=1. 
  In any low energy effective supergravity theory general formulae exist which allow one to discuss fermion masses, the scalar potential and breaking of symmetries in a model independent set up. A particular role in this discussion is played by Killing vectors and Killing prepotentials. We outline these relations in general and specify then in the context of N=1 and N=2 supergravities in four dimensions. Useful relations of gauged quaternionic geometry underlying hypermultiplets dynamics are discussed. 
  We show how the classical string dynamics in $D$-dimensional gravity background can be reduced to the dynamics of a massless particle constrained on a certain surface whenever there exists at least one Killing vector for the background metric. We obtain a number of sufficient conditions, which ensure the existence of exact solutions to the equations of motion and constraints. These results are extended to include the Kalb-Ramond background. The $D1$-brane dynamics is also analyzed and exact solutions are found. Finally, we illustrate our considerations with several examples in different dimensions. All this also applies to the tensionless strings. 
  We construct new explicit metrics on complete non-compact Riemannian 8-manifolds with holonomy Spin(7). One manifold, which we denote by A_8, is topologically R^8 and another, which we denote by B_8, is the bundle of chiral spinors over $S^4$. Unlike the previously-known complete non-compact metric of Spin(7) holonomy, which was also defined on the bundle of chiral spinors over S^4, our new metrics are asymptotically locally conical (ALC): near infinity they approach a circle bundle with fibres of constant length over a cone whose base is the squashed Einstein metric on CP^3. We construct the covariantly-constant spinor and calibrating 4-form. We also obtain an L^2-normalisable harmonic 4-form for the A_8 manifold, and two such 4-forms (of opposite dualities) for the B_8 manifold. We use the metrics to construct new supersymmetric brane solutions in M-theory and string theory. In particular, we construct resolved fractional M2-branes involving the use of the L^2 harmonic 4-forms, and show that for each manifold there is a supersymmetric example. An intriguing feature of the new A_8 and B_8 Spin(7) metrics is that they are actually the same local solution, with the two different complete manifolds corresponding to taking the radial coordinate to be either positive or negative. We make a comparison with the Taub-NUT and Taub-BOLT metrics, which by contrast do not have special holonomy. In an appendix we construct the general solution of our first-order equations for Spin(7) holonomy, and obtain further regular metrics that are complete on manifolds B^+_8 and B^-_8 similar to B_8. 
  The interplay between string theory and cosmology is very promising. Since string theory will yield a quantum theory of space-time and unify all forces of nature, it has the potential of addressing many of the conceptual problems of the current models of early Universe cosmology. In turn, cosmology is the most obvious testing ground in the effort to construct non-perturbative string theory, and can provide the crucial connection between theory and experiment/observation. 
  We analyze the vacuum structure of spontaneously broken N=2 supersymmetric gauge theory with the Fayet-Iliopoulos term. Our theory is based on the gauge group SU(2) \times U(1) with N_f=1,2 massless quark hypermultiplets having the same U(1) charges. In the classical potential, there are degenerate vacua even in the absence of supersymmetry. It is shown that this vacuum degeneracy is smoothed out, once quantum corrections are taken into account. In N_f=1 case, the effective potential is found to be so-called runaway type, and there is neither well-defined vacuum nor local minimum. On the other hand, in N_f=2 case, while there is also the runaway direction in the effective potential, we find the possibility that there appears the local minimum with broken supersymmetry at the degenerate dyon point. 
  We study the Kaluza-Klein dimensional reduction of zero-modes of bulk antisymmetric tensor fields on a non-compact extra dimension in the Randall-Sundrum model. It is shown that in the Kaluza-Klein reduction on a non-compact extra dimension we have in general a zero-mode depending on a fifth dimension in addition to a conventional constant zero-mode in the Kaluza-Klein reduction on a circle. We examine the localization property of these zero-modes on a flat Minkowski 3-brane. In particular, it is shown that a 2-form and a 3-form on the brane can be respectively obtained from a 3-form and 4-form in the bulk by taking the zero-mode dependent on the fifth dimension. 
  We consider the partition function and correlation functions in the bosonic and supersymmetric Yang-Mills matrix models with compact semi-simple gauge group. In the supersymmetric case, we show that the partition function converges when $D=4,6$ and 10, and that correlation functions of degree $k< k_c=2(D-3)$ are convergent independently of the group. In the bosonic case we show that the partition function is convergent when $D \geq D_c$, and that correlation functions of degree $k < k_c$ are convergent, and calculate $D_c$ and $k_c$ for each group, thus extending our previous results for SU(N). As a special case these results establish that the partition function and a set of correlation functions in the IKKT IIB string matrix model are convergent. 
  We evaluated some particular type of functional integral over the local gauge group C^{\infty}({\bf R}^n, U(1)) by going to a discretized lattice. The results explicitly violates the property of the Haar measure. We also analysed the Faddeev-Popov method through a toy example. The results also violates the property of the Haar measure. 
  The aim of this note is to discuss, in a somewhat informal language, the cancellation of anomalies (in topologically trivial space-time) for 5-branes using as "building blocks": i) a generalization to p-branes of the Dirac strings of monopoles (Dirac branes) and a refinement of this idea involving a geometric regularization of Dirac branes, leading to the formalism of "characteristic currents" ii) the PST formalism . As an example of the potentiality of the developed framework we discuss in some detail the anomaly cancellation in the D=10 effective theory of heterotic string and 5-brane coupled to supergravity, where the anomaly inflow is automatically generated. Some remarks are also made on a similar approach to the problem of anomaly cancellation in the effective theory of M5-brane coupled to D=11 supergravity, developed in collaboration with M.Tonin, where however still as open problem remains a Dirac anomaly. 
  The singularity structure of many IIB supergravity solutions asymptotic to $AdS_5 \times S^5$ becomes clearer when one considers the full ten dimensional solution rather than the dimensionally reduced solution of gauged supergravity. It has been shown that all divergences in the gravitational action of the dimensionally reduced spacetime can be removed by the addition of local counterterms on the boundary. Here we attempt to formulate the counterterm action directly in ten dimensions for a particular class of solutions, the ${\cal N} = 0$ Polchinski-Strassler solutions, which are dual to an ${\cal N} =4$ SYM theory perturbed by mass terms for all scalars and spinors. This involves constructing the solution perturbatively near the boundary. There is a contribution to the Weyl anomaly from the mass terms (which break the classical conformal invariance of the action). The coefficient of this anomaly is reproduced by a free field calculation indicating a non-renormalisation theorem inherited from the ${\cal N} =4$ theory. We comment on the structure of the full solutions and their construction from uplifting particular $ {\cal N} = 0$ flows in five dimensions. 
  We analyze the holographic description of several properties of $\N=1$ confining gauge dynamics. In particular we discuss Wilson loops including the issues of a Luscher term and the broadening of the flux tubes, 't Hooft loops, baryons, instantons, chiral symmetry breaking, the gluino condensate and BPS domain walls. 
  We study the second quantization of field theory on the q-deformed fuzzy sphere for real q. This is performed using a path-integral over the modes, which generate a quasiassociative algebra. The resulting models have a manifest U_q(su(2)) symmetry with a smooth limit q -> 1, and satisfy positivity and twisted bosonic symmetry properties. A systematic way to calculate n-point correlators in perturbation theory is given. As examples, the 4-point correlator for a free scalar field theory and the planar contribution to the tadpole diagram in \phi^4 theory are computed. The case of gauge fields is also discussed, as well as an operator formulation of scalar field theory in 2_q + 1 dimensions. An alternative, essentially equivalent approach using associative techniques only is also presented. The proposed framework is not restricted to 2 dimensions. 
  The low-energy effective action of supersymmetric D-brane systems consists of two terms, one of which is of the Born-Infeld type and one of which is of the Chern-Simons type. I briefly review the status of our understanding of these terms for both the Abelian and non-Abelian cases. 
  We investigate systems of real scalar fields in bidimensional spacetime, dealing with potentials that are small modifications of potentials that admit supersymmetric extensions. The modifications are controlled by a real parameter, which allows implementing a perturbation procedure when such parameter is small. The approach allows obtaining the energy and topological charge in closed forms, up to first order in the parameter. We illustrate the procedure with some examples. In particular, we show how to remove the degeneracy in energy for the one-field and the two-field solutions that appear in a model of two real scalar fields. 
  We study the supergravity duals of supersymmetric theories arising in the world-volume of D6 branes wrapping holomorphic two-cycles and special Lagrangian three-cycles within the framework of eight dimensional gauged supergravity. When uplifted to 11d, our solutions represent M-theory on the background of, respectively, the small resolution of the conifold and a manifold with G_2 holonomy. We further discuss on the flop and other possible geometrical transitions and its implications. 
  We study aspects of gauge theory on tori which are a consequences of Morita equivalence. In particular we study the behavior of gauge theory on noncommutative tori for arbitrarily close rational values of Theta. For such values of Theta, there are Morita equivalent descriptions in terms of Yang-Mills theories on commutative tori with very different magnetic fluxes and rank. In order for the correlators of open Wilson lines to depend smoothly on Theta, the correlators of closed Wilson lines in the commutative Yang-Mills theory must satisfy strong constraints. If exactly satisfied, these constraints give relations between small and large N gauge theories. We verify that these constraints are obeyed at leading order in the 1/N expansion of pure 2-d QCD and of strongly coupled N=4 super Yang-Mills theory. 
  The D1-D5 system is believed to have an `orbifold point' in its moduli space where its low energy theory is a N=4 supersymmetric sigma model with target space M^N/S^N, where M is T^4 or K3. We study correlation functions of chiral operators in CFTs arising from such a theory. We construct a basic class of chiral operators from twist fields of the symmetric group and the generators of the superconformal algebra. We find explicitly the 3-point functions for these chiral fields at large N; these expressions are `universal' in that they are independent of the choice of M. We observe that the result is a significantly simpler expression than the corresponding expression for the bosonic theory based on the same orbifold target space. 
  We study string compactifications with sixteen supersymmetries. The moduli space for these compactifications becomes quite intricate in lower dimensions, partly because there are many different irreducible components. We focus primarily, but not exclusively, on compactifications to seven or more dimensions. These vacua can be realized in a number ways: the perturbative constructions we study include toroidal compactifications of the heterotic/type I strings, asymmetric orbifolds, and orientifolds. In addition, we describe less conventional M and F theory compactifications on smooth spaces. The last class of vacua considered are compactifications on singular spaces with non-trivial discrete fluxes.   We find a number of new components in the string moduli space. Contained in some of these components are M theory compactifications with novel kinds of ``frozen'' singularities. We are naturally led to conjecture the existence of new dualities relating spaces with different singular geometries and fluxes. As our study of these vacua unfolds, we also learn about additional topics including: F theory on spaces without section, automorphisms of del Pezzo surfaces, and novel physics (and puzzles) from equivariant K-theory. Lastly, we comment on how the data we gain about the M theory three-form might be interpreted. 
  Semiclassical gravity predicts that de Sitter space has a finite entropy. We suggest a picture for Euclidean de Sitter space in string theory, and use the AdS/CFT correspondence to argue that de Sitter entropy can be understood as the number of degrees of freedom in a quantum mechanical dual. 
  At present an algebra of strongly interacting fields is unknown. In this paper it is assumed that the operators of strongly nonlinear field can form a non-associative algebra. It is shown that such algebra can be described as an algebra of some pairs. The comparison of presented techniques with the Green's functions method in the superconductivity theory is made. A possible application to the QCD and High-T$_c$ superconductivity theory is discussed. 
  Consistency conditions for the local existence of massless spin 3/2 fields has been explored that the field equations for massless helicity 3/2 are consistent iff the space-time is Ricci-flat and that in Minkowski space-time the space of conserved charges for the fields is its twistor space itself. After considering the twistorial methods to study such massless helicity 3/2 fields, we derive in flat space-time that the charges of spin-3/2 fields defined topologically by the first Chern number of their spin-lowered self-dual Maxwell fields, are given by their twistor space, and in curved space-time that the (anti-)self-duality of the space-time is the necessary condition. Since in N=1 supergravity torsions are the essential ingredients, we generalize our space-time to that with torsion (Einstein-Cartan theory) and have investigated the consistency of existence of spin 3/2 fields in it. A simple solution is found that the space-time has to be conformally (anti-)self-dual, left-(or right-)torsion-free. The integrability condition on $\alpha$-surface shows that the (anti-)self-dual Weyl spinor can be described only by the covariant derivative of the right-(left-)handed-torsion. 
  We analyze the vacuum structure of spontaneously broken N=2 supersymmetric gauge theory with the Fayet-Iliopoulos term. Our theory is based on the gauge group SU(2) \times U(1) with N_f=2 massless quark hypermultiplets having the same U(1) charges. In the classical potential, there are degenerate vacua even in the absence of supersymmetry. It is shown that this vacuum degeneracy is smoothed out, once quantum corrections are taken into account. While there is the runaway direction in the effective potential, we found the promising possibility that there appears the local minimum with broken supersymmetry at the degenerate dyon point. 
  In this work we prove in a precise way that the soldering formalism can be applied to the Srivastava chiral boson (SCB), in contradiction with some results appearing in the literature. We have promoted a canonical transformation that shows directly that the SCB is composed of two Floreanini-Jackiw's particles with the same chirality which spectrum is a vacuum-like one. As another conflictive result we have proved that a Wess-Zumino term used in the literature consists of the scalar field, once again denying the assertion that the WZ term adds a new degree of freedom to the SCB theory in order to modify the physics of the system. 
  We suggest that trialgebraic symmetries migth be a sensible starting point for a notion of integrability for two dimensional spin systems. For a simple trialgebraic symmetry we give an explicit condition in terms of matrices which a Hamiltonian realizing such a symmetry has to satisfy and give an example of such a Hamiltonian which realizes a trialgebra recently given by the authors in another paper. Besides this, we also show that the same trialgebra can be realized on a kind of Fock space of q-oscillators, i.e. the suggested integrability concept gets via this symmetry a close connection to a kind of noncommutative quantum field theory, paralleling the relation between the integrability of spin chains and two dimensional conformal field theory. 
  We establish T-duality between NS5 branes stuck on an orientifold 8-plane in type I' and an orientifold construction in type IIB with D7 branes intersecting at angles. Two applications are discussed. For one we obtain new brane constructions, realizing field theories with gauge group a product of symplectic factors, giving rise to a large new class of conformal N=1 theories embedded in string theory. Second, by studying a D2 brane probe in the type I' background, we get some information on the still elusive (0,4) linear sigma model describing a perturbative heterotic string on an ADE singularity. 
  A new generalization of the Jack polynomials that incorporates fermionic variables is presented. These Jack superpolynomials are constructed as those eigenfunctions of the supersymmetric extension of the trigonometric Calogero-Moser-Sutherland (CMS) model that decomposes triangularly in terms of the symmetric monomial superfunctions. Many explicit examples are displayed. Furthermore, various new results have been obtained for the supersymmetric version of the CMS models: the Lax formulation, the construction of the Dunkl operators and the explicit expressions for the conserved charges. The reformulation of the models in terms of the exchange-operator formalism is a crucial aspect of our analysis. 
  In this note, we study a matrix-regularized version of non-commutative U(1) Chern-Simons theory proposed recently by Polychronakos. We determine a complete minimal basis of exact wavefunctions for the theory at arbitrary level k and rank N and show that these are in one-to-one correspondence with Laughlin-type wavefunctions describing excitations of a quantum Hall droplet composed of N electrons at filling fraction 1/k. The finite matrix Chern-Simons theory is shown to be precisely equivalent to the theory of composite fermions in the lowest Landau level, believed to provide an accurate description of the filling fraction 1/k fractional quantum Hall state. In the large N limit, this implies that level k noncommutative U(1) Chern-Simons theory is equivalent to the Laughlin theory of the filling fraction 1/k quantum Hall fluid, as conjectured recently by Susskind. 
  The response of a gravitational wave detector to scalar waves is analysed in the framework of the debate on the choice of conformal frames for scalar-tensor theories. A correction to the geodesic equation arising in the Einstein conformal frame modifies the geodesic deviation equation. This modification is due to the non-metricity of the theory in the Einstein frame, yielding a longitudinal mode that is absent in the Jordan conformal frame. 
  A one-instanton level test is performed for the proposed reparameterisation scheme matching the conjectured exact low energy results and instanton predictions for N=2 supersymmetric SU(N) gauge theories with 2N massless fundamental matter hypermultiplets across the entire quantum moduli space. The constants within the scheme which ensure agreement between the exact results and the instanton predictions for general N are derived. This constitutes a non-trivial test of the scheme, which eliminates the discrepancies arising when the two sets of results are compared. 
  A relation between dimensional reduction and space-time symmetry gauging is outlined. 
  We analyze the role of RR fluxes in orientifold backgrounds from the point of view of K-theory, and demonstrate some physical implications of describing these fluxes in K-theory rather than cohomology. In particular, we show that certain fractional shifts in RR charge quantization due to discrete RR fluxes are naturally explained in K-theory. We also show that some orientifold backgrounds, which are considered distinct in the cohomology classification, become equivalent in the K-theory description, while others become unphysical. 
  We study bulk fermion fields in various multi-brane models with localized gravity. The chiral zero mode that these models support can be identified as a right-handed sterile neutrino. In this case small neutrino Dirac masses can naturally appear due to a localization of the bulk fermion zero mode wavefunction, in an analogous way to graviton, without invoking the see-saw mechanism. The conditions and the options for localization are discussed in detail. It is shown that, considering a well motivated five dimensional mass term, the localization behaviour of this mode can resemble the graviton's at least in a region of the parameter space. As a result, the ''+-+'', ''++'' models can support, in addition to the ultralight graviton KK state, an ultralight localized and strongly coupled bulk fermion KK mode. We find that there are severe constraints on the parameter space of ''+-+'' and ''++'' models if the neutrino properties resulting from this light fermion state are to be reasonable. Furthermore, in the case that also the Bigravity scenario is realized the above special KK mode can induce too large mixing between the neutrino and the KK tower sterile modes restricting even more the allowed parameter space. 
  Starting from a recently proposed Abelian topological model in (2+1) dimensions, which involve the Kalb-Ramond two form field, we study a non-Abelian generalization of the model. An obstruction for generalization is detected. However we show that the goal is achieved if we introduce a vectorial auxiliary field. Consequently, a model is proposed, exhibiting a non-Abelian topological mass generation mechanism in D=3, that provides mass for the Kalb-Ramond field. The covariant quantization of this model requires ghosts for ghosts. Therefore in order to quantize the theory we construct a complete set of BRST and anti-BRST equations using the horizontality condition. 
  Starting from the space of Lorentzian metrics, we examine the full gravitational path integral in 3 and 4 space-time dimensions. Inspired by recent results obtained in a regularized, dynamically triangulated formulation of Lorentzian gravity, we gauge-fix to proper-time coordinates and perform a non-perturbative ``Wick rotation'' on the physical configuration space. Under certain assumptions about the behaviour of the partition function under renormalization, we find that the divergence due to the conformal modes of the metric is cancelled non-perturbatively by a Faddeev-Popov determinant contributing to the effective measure. We illustrate some of our claims by a 3d perturbative calculation. 
  We investigate the AdS/CFT correspondence for higher-derivative gravity systems, and develop a formalism in which the generating functional of the boundary field theory is given as a functional that depends only on the boundary values of bulk fields. We also derive a Hamilton-Jacobi-like equation that uniquely determines the generating functional, and give an algorithm calculating the Weyl anomaly. Using the expected duality between a higher-derivative gravity system and N=2 superconformal field theory in four dimensions, we demonstrate that the resulting Weyl anomaly is consistent with the field theoretic one. 
  We examine magnetic and electric near horizon regions of maximally supersymmetric D-brane and NS5-brane bound states and find transformations between near horizon regions with worldvolume dual magnetic and electric fluxes. These point to dual formulations of NCYM, NCOS and OD$p$ theories in the limit of weak coupling and large spatial or temporal non-commutativity length scale in terms of weakly coupled theories with fixed worldvolume dual non-commutativity based on open D-branes. We also examine the strong coupling behavior of the open D-brane theories and propose a unified web of dualities involving strong/weak coupling as well as large/small non-commutativity scale. 
  Motivated by the renewed interest in the role of Q-balls in cosmological evolution, we present a discussion of the main properties of Q-balls, including some new results. 
  We construct several new families of exactly and quasi-exactly solvable BC_N-type Calogero-Sutherland models with internal degrees of freedom. Our approach is based on the introduction of two new families of Dunkl operators of B_N type which, together with the original B_N-type Dunkl operators, are shown to preserve certain polynomial subspaces of finite dimension. We prove that a wide class of quadratic combinations involving these three sets of Dunkl operators always yields a spin Calogero-Sutherland model, which is (quasi-)exactly solvable by construction. We show that all the spin Calogero-Sutherland models obtainable within this framework can be expressed in a unified way in terms of a Weierstrass P function with suitable half-periods. This provides a natural spin counterpart of the well-known general formula for a scalar completely integrable potential of BC_N type due to Olshanetsky and Perelomov. As an illustration of our method, we exactly compute several energy levels and their corresponding wavefunctions of an elliptic quasi-exactly solvable potential for two and three particles of spin 1/2. 
  Superembeddings which have bosonic codimension zero are studied in 3,4 and 6 dimensions. The worldvolume multiplets of these branes are off-shell vector multiplets in these dimensions, and their self-interactions include a Born-Infeld term. It is shown how they can be written in terms of standard vector multiplets in flat superspace by working in the static gauge. The action formula is used to determine both Green-Schwarz type actions and superfield actions. 
  We consider a reduced model of four-dimensional Yang-Mills theory with a mass term. This matrix model has two classical solutions, two-dimensional fuzzy sphere and two-dimensional fuzzy torus. These classical solutions are constructed by embedding them into three or four dimensional flat space. They exist for finite size matrices, that is, the number of the quantum on these manifolds is finite. Noncommutative gauge theories on these noncommutative manifolds are derived by expanding the model around these classical solutions and studied by taking two large $N$ limits, a commutative limit and a large radius limit. The behaviors of gauge invariant operators are also discussed. 
  We review the stabilization of the radion in the Randall-Sundrum model through the Casimir energy due to a bulk conformally coupled field. We also show some exact self-consistent solutions taking into account the backreaction that this energy induces on the geometry. 
  The behaviour of the 3D axial next-nearest neighbour Ising (ANNNI) model at the uniaxial Lifshitz point is studied using Monte Carlo techniques. A new variant of the Wolff cluster algorithm permits the analysis of systems far larger than in previous studies. The Lifshitz point critical exponents are $\alpha=0.18(2)$, $\beta=0.238(5)$ and $\gamma=1.36(3)$. Data for the spin-spin correlation function are shown to be consistent with the explicit scaling function derived from the assumption of local scale invariance, which is a generalization of conformal invariance to the anisotropic scaling {\em at} the Lifshitz point. 
  We study the optimisation of exact renormalisation group (ERG) flows. We explain why the convergence of approximate solutions towards the physical theory is optimised by appropriate choices of the regularisation. We consider specific optimised regulators for bosonic and fermionic fields and compare the optimised ERG flows with generic ones. This is done up to second order in the derivative expansion at both vanishing and non-vanishing temperature. We find that optimised flows at finite temperature factorise. This corresponds to the disentangling of thermal and quantum fluctuations. A similar factorisation is found at second order in the derivative expansion. The corresponding optimised flow for a ``proper-time renormalisation group'' is also provided to leading order in the derivative expansion. 
  Coincident D2-branes in open N=2 fermionic string theory with a B-field background yield an integrable modified U(n) sigma model on noncommutative R^{2,1}. This model provides a showcase for an established method (the `dressing approach') to generate solutions for integrable field equations, even in the noncommutative case. We demonstrate the technique by constructing moving U(1) and U(2) solitons and by computing their energies. It is outlined how to derive multi-soliton configurations with arbitrary relative motion; they correspond to D0-branes moving inside the D2-branes. 
  We study the $c=-2$ model of logarithmic conformal field theory in the presence of a boundary using symplectic fermions. We find boundary states with consistent modular properties. A peculiar feature of this model is that the vacuum representation corresponding to the identity operator is a sub-representation of a ``reducible but indecomposable'' larger representation. This leads to unusual properties, such as the failure of the Verlinde formula. Despite such complexities in the structure of modules, our results suggest that logarithmic conformal field theories admit bona fide boundary states. 
  Massive spin s>=3/2 fields can become partially massless in cosmological backgrounds. In the plane spanned by m^2 and \Lambda, there are lines where new gauge invariances permit intermediate sets of higher helicities, rather than the usual flat space extremes of all 2s+1 massive or just 2 massless helicities. These gauge lines divide the (m^2,\Lambda) plane into unitarily allowed or forbidden intermediate regions where all 2s+1 massive helicities propagate but lower helicity states can have negative norms. We derive these consequences for s=3/2,2 by studying both their canonical (anti)commutators and the transmutation of massive constraints to partially massless Bianchi identities. For s=2, a Hamiltonian analysis exhibits the absence of zero helicity modes in the partially massless sector. For s=5/2,3 we derive Bianchi identities and their accompanying gauge invariances for the various partially massless theories with propagating helicities (+/-5/2,+/-3/2) and (+/-3,+/-2), (+/-3,+/-2,+/-1), respectively. Of these, only the s=3 models are unitary. To these ends, we also provide the half integer generalization of the integer spin wave operators of Lichnerowicz. Partial masslessness applies to all higher spins in (A)dS as seen by their degree of freedom counts. Finally a derivation of massive d=4 constraints by dimensional reduction from their d=5 massless Bianchi identity ancestors is given. 
  The non-commutative O(N) Gross-Neveu model is solved in the large N limit in two and three space-time dimensions. The commutative version of the two dimensional model is a renormalizable quantum field theory, both in a coupling constant expansion and an expansion in 1/N. The non-commutative version has a renormalizable coupling constant expansion where ultraviolet divergences can be removed by adjusting counterterms to each order. On the other hand, in a previous work, we showed that the non-commutative theory is not renormalizable in the large N expansion. This is argued to be due to a combined effect of asymptotic freedom and the ultraviolet/infrared mixing that occurs in a non-commutative field theory. In the present paper we will elaborate on this result and extend it to study the large N limit of the three dimensional Gross-Neveu model. We shall see that the large N limit of the three dimensional theory is also trivial when the ultraviolet cutoff is removed. 
  Discrete symmetries are studied in warped space-times with one extra dimension. In particular, we analyze the compatibility of five- and four-dimensional charge conjugation, parity, time reversal and the orbifold symmetry Z_2 with localization of fermions on the four-dimensional brane-world and Lorentz invariance. We then show that, when a suitable topological scalar field (the ``kink'') is included, fermion localization is a consequence of (five-dimensional) CPT invariance. 
  This is a brief overview of my work on the realization of contact superconformal algebras in terms of pseudodifferential symbols on a supercircle. Also the 2-cocycles on K'(4) are given for this realization. 
  We provide the formalism for the quantization of systems of coupled bosonic and fermionic fields in a time dependent classical background. The occupation numbers of the particle eigenstates can be clearly defined and computed, through a generalization of the standard procedure valid for a single field in which Bogolyubov coefficients are employed. We apply our formalism to the problem of nonthermal gravitino production in a two-fields model where supersymmetry is broken gravitationally in the vacuum. Our explicit calculations show that this production is strongly suppressed in the model considered, due to the weak coupling between the sector which drives inflation and the one responsible for supersymmetry breakdown. 
  We analyze certain brane bound states in M-theory and their descendants in type IIA string theory, all involving 3-form or 2-form background fluxes. Among them are configurations which represent NCYM, NCOS and ODp-theories in the scaling limit of OM-theory. In particular, we show how the conditions for the embedding to preserve supersymmetry are modified by the presence of the flux and discuss their relations for the various different bound states. Via the formalism of geometric quantization such a deformation of a supersymmetric cycle is related to a non-commutativity of its coordinates. We also study possible non-commutative deformations of the Seiberg-Witten curve of N=2 supersymmetric gauge theories due to non-trivial H-flux. 
  We review the deformed instanton equations making connection with Hilbert schemes and integrable systems. A single U(1) instanton is shown to be \asd\ with respect to the Burns metric. 
  We consider a class of models with infinite extra dimension, where bulk space does not possess SO(1,3) invariance, but Lorentz invariance emerges as an approximate symmetry of the low-energy effective theory. In these models, the maximum attainable speeds of the graviton, gauge bosons and scalar particles are automatically equal to each other and smaller than the maximum speed in the bulk. Additional fine-tuning is needed in order to assure that the maximum attainable speed of fermions takes the same value. A peculiar feature of our scenario is that there are no truly localized modes. All four-dimensional particles are resonances with finite widths. The latter depends on the energy of the particle and is naturally small at low energies. 
  We construct a Dirac operator on the quantum sphere $S^2_q$ which is covariant under the action of $SU_q(2)$. It reduces to Watamuras' Dirac operator on the fuzzy sphere when $q\to 1$. We argue that our Dirac operator may be useful in constructing $SU_q(2)$ invariant field theories on $S^2_q$ following the Connes-Lott approach to noncommutative geometry. 
  We consider a brane world residing in the interior region inside the horizon of extreme black branes. In this picture, the size of the horizon can be interpreted as the compactification size. The large mass hierarchy is simply translated into the large horizon size, which is provided by the magnitude of charges carried by the black branes. Hence, the macroscopic compactification size is a quantity calculable from the microscopic theory which has only one physical scale, and its stabilization is guaranteed from the charge conservation. 
  A simple realization of the conformal higher spin symmetry on the free $3d$ massless matter fields is given in terms of an auxiliary Fock module both in the flat and $AdS_3$ case. The duality between non-unitary field-theoretical representations of the conformal algebra and the unitary (singleton--type) representations of the $3d$ conformal algebra $sp(4,\R)$ is formulated explicitly in terms of a certain Bogolyubov transform. 
  The extension of the noncommutative u*(N) Lie algebra to noncommutative orthogonal and symplectic Lie algebras is studied. Using an anti-automorphism of the star-matrix algebra, we show that the u*(N) can consistently be restricted to o*(N) and usp*(N) algebras that have new mathematical structures. We give explicit fundamental matrix representations of these algebras, through which the formulation for the corresponding noncommutative gauge field theories are obtained. In addition, we present a D-brane configuration with an orientifold which realizes geometrically our algebraic construction, thus embedding the new noncommutative gauge theories in superstring theory in the presence of a constant background magnetic field. Some algebraic generalizations that may have applications in other areas of physics are also discussed. 
  We review the history of non-renormalisation theorems in global supersymmetry, as well as their importance in all attempts to apply supersymmetry to the real world. 
  A solution with the pole configuration in six dimensions is analysed both analytically and numerically. It is a dimensional reduction model of Randall-Sundrum type. The soliton configuration is induced by the bulk Higgs mechanism. The boundary condition is systematically solved up to the 6th order. The Riemann curvature is finite everywhere. An exact solution for the no potential case is also presented. 
  We review the principal steps leading to drive the wave function $\psi _{\{k_1,k_2,...,k_N \}}(1,2,...,N)$ of a gaz of $N$ identical particle states with exotic statistics. For spins $s=1/M$ $mod(1)$, we show that the quasideterminant conjectured in [19], by using $2d$ conformal field theoretical methods, is indeed related to the quantum determinant of noncommutative geometry. The q-number $[N]!=\prod_{n=1}^N(\sum_{j=0}^{N-1} q^j)$ carrying the effect of the generalized Pauli exclusion principle, according to which no more than $(M-1)$ identical particles of spin $s=1/M$ $mod(1)$ can live altogether on the same quantum state, is rederived in rigourous from the q-antisymmetry. Other features are also given. 
  Topological charges of the $D6$-brane in the presence of a Neveu-Schwarz B-field are computed by methods of twisted $K$-theory. 
  Two different theoretical formulations of the finite temperature effects have been recently proposed for integrable field theories. In order to decide which of them is the correct one, we perform for a particular model an explicit check of their predictions for the one-point function of the trace of the stress-energy tensor, a quantity which can be independently determined by the Thermodynamical Bethe Ansatz. 
  If violation of Lorentz and CPT symmetry is introduced into the fermion sector of conventional quantum electrodynamics, then the Chern-Simons term is radiatively induced with finite nonzero coefficient, as well as the Maxwell term is with logarithmically divergent one. The heat kernel expansion and the proper time methods are used to determine the effective action in the one-loop approximation unambiguously. 
  We discuss massless and massive neutrino zero modes in the background of an electroweak string. We argue that the eventual absence of the neutrino zero mode implies the existence of topologically stable strings where the required non-trivial topology has been induced by the fermionic sector. 
  We determine the stability conditions for a radially symmetric noncommutative scalar soliton at finite noncommutivity parameter $\theta$. We find an intriguing relationship between the stability and existence conditions for all level-1 solutions, in that they all have nearly-vanishing stability eigenvalues at critical $\theta m^2$. The stability or non-stability of the system may then be determined entirely by the $\phi^3$ coefficient in the potential. For higher-level solutions we find an ambiguity in extrapolating solutions to finite $\theta$ which prevents us from making any general statements. For these stability may be determined by comparing the fluctuation eigenvalues to critical values which we calculate. 
  The physical meaning, the properties and the consequences of a discrete scalar field are discussed; limits for the validity of a mathematical description of fundamental physics in terms of continuous fields are a natural outcome of discrete fields with discrete interactions. The discrete scalar field is ultimately the gravitational field of general relativity, necessarily, and there is no place for any other fundamental scalar field, in this context. Part of the paper comprehends a more generic discussion about the nature, if continuous or discrete, of fundamental interactions. There is a critical point defined by the equivalence between the two descriptions. Discrepancies between them can be observed far away from this point as a continuous-interaction is always stronger below it and weaker above it than a discrete one. It is possible that some discrete-field manifestations have already been observed in the flat rotation curves of galaxies and in the apparent anomalous acceleration of the Pioneer spacecrafts. The existence of a critical point is equivalent to the introduction of an effective-acceleration scale which may put Milgrom's MOND on a more solid physical basis. Contact is also made, on passing, with inflation in cosmological theories and with Tsallis generalized one-parameter statistics which is regarded as proper for discrete-interaction systems. The validity of Botzmann statistics is then reduced to idealized asymptotic states which, rigorously, are reachable only after an infinite number of internal interactions . Tsallis parameter is then a measure of how close a system is from its idealized asymptotic state. 
  A quantum-field theoretical interpretation is given to the holographic RG equation by relating it to a field-theoretical local RG equation which determines how Weyl invariance is broken in a quantized field theory. Using this approach we determine the relation between the holographic C theorem and the C theorem in two-dimensional quantum field theory which relies on the Zamolodchikov metric. Similarly we discuss how in four dimensions the holographic C function is related to a conjectured field-theoretical C function. The scheme dependence of the holographic RG due to the possible presence of finite local counterterms is discussed in detail, as well as its implications for the holographic C function. We also discuss issues special to the situation when mass deformations are present. Furthermore we suggest that the holographic RG equation may also be obtained from a bulk diffeomorphism which reduces to a Weyl transformation on the boundary. 
  We find that tachyonic orbifold examples of AdS/CFT have corresponding instabilities at small radius, and can decay to more generic gauge theories. We do this by computing a destabilizing Coleman-Weinberg effective potential for twisted operators of the corresponding quiver gauge theories, generalizing calculations of Tseytlin and Zarembo and interpreting them in terms of the large-N behavior of twisted-sector modes. The dynamically generated potential involves double-trace operators, which affect large-N correlators involving twisted fields but not those involving only untwisted fields, in line with large-N inheritance arguments. We point out a simple reason that no such small radius instability exists in gauge theories arising from freely acting orbifolds, which are tachyon-free at large radius. When an instability is present, twisted gauge theory operators with the quantum numbers of the large-radius tachyons aquire VEVs, leaving a gauge theory with fewer degrees of freedom in the infrared, analogous to but less extreme than ``decays to nothing'' studied in other systems with broken supersymmetry. In some cases one is left with pure glue QCD plus decoupled matter and U(1) factors in the IR, which we thus conjecture is described by the corresponding (possibly strongly coupled) endpoint of tachyon condensation in the M/String-theory dual. 
  We study systematically the higher order corrections to the parity violating part of the effective action for the Abelian Chern-Simons theory in 2+1 dimensions, using the method of derivative expansion. We explicitly calculate the parity violating parts of the quadratic, cubic and the quartic terms (in fields) of the effective action. We show that each of these actions can be summed, in principle, to all orders in the derivatives. However, such a structure is complicated and not very useful. On the other hand, at every order in the powers of the derivatives, we show that the effective action can also be summed to all orders in the fields. The resulting actions can be expressed in terms of the leading order effective action in the static limit. We prove gauge invariance, both large and small of the resulting effective actions. Various other features of the theory are also brought out. 
  We study Abrikosov-Nielsen-Olesen (ANO) flux tubes on the Higgs branch of N=2 QCD with SU(2) gauge group and two flavors of fundamental matter. In particular, we consider this theory near Argyres-Douglas (AD) point where the mass of monopoles connected by these ANO strings becomes small. In this regime the effective QED which describes the theory on the Higgs branch becomes strongly coupled. We argue that the appropriate description of the theory is in terms of long and thin flux tubes (strings) with small tension. We interpret this as another example of duality between field theory in strong coupling and string theory in weak coupling. Then we consider the non-critical string theory for these flux tubes which includes fifth (Liouville) dimension. We identify CFT at the AD point as UV fix point corresponding to AdS metric on the 5d "gravity" side. The perturbation associated with the monopole mass term creates a kink separating UV and IR behavior. We estimate the renormalized string tension and show that it is determined by the small monopole mass. In particular, it goes to zero at the AD point. 
  We demonstrate that M-theory compactifications on 7-manifolds of G_2 holonomy, which yield 4d N=1 supersymmetric systems, often admit at special loci in their moduli space a description as type IIA orientifolds. In this way, we are able to find new dualities of special IIA orientifolds, including dualities which relate orientifolds of IIA strings on manifolds of different topology with different numbers of wrapped D-branes. We also discuss models which incorporate, in a natural way, compact embeddings of gauge theory/gravity dualities similar to those studied in the recent work of Atiyah, Maldacena and Vafa. 
  By analyzing eleven-dimensional superspace fourth-rank superfield strength F-Bianchi identities, we show that M-theory corrections to eleven-dimensional supergravity can not be embedded into the mass dimension zero constraints, such as the (\g^{a b})_{\a\b} X_{a b}{}^c or i (\g^{a_1... a_5})_{\a\b} X_{a_1... a_5}{}^c -terms in the supertorsion constraint T_{\a\b}{}^c. The only possible modification of superspace constraint at dimension zero is found to be the scaling of F_{\a\b c d} like F_{\a\b c d} = (1/2) \big(\g_{c d}\big)_{\a\b} e^\Phi for some real scalar superfield \Phi, which alone is further shown not enough to embed general M-theory corrections. This conclusion is based on the dimension zero F-Bianchi identity under the two assumptions: (i) There are no negative dimensional constraints on the F-superfield strength: F_{\a\b\g\d} = F_{\a\b\g d} =0; (ii) The supertorsion T-Bianchi identities and F-Bianchi identities are not modified by Chern-Simons terms. Our result can serve as a powerful tool for future exploration of M-theory corrections embedded into eleven-dimensional superspace supergravity. 
  We examine the form of the cosmological constant in the loop expansion of broken maximally supersymmetric supergravity theories, and after embedding, within superstring and M-theory. Supersymmetry breaking at the TeV scale generates values of the cosmological constant that are in agreement with current astrophysical data. The form of perturbative quantum effects in the loop expansion is consistent with this parameter regime. 
  We construct a vertex operator which describes an emission of the ground-state tachyon of the closed string out of the noncommutative open string. Such a vertex operator is shown to exist only when the momentum of the closed-string tachyon is subject to some constraints coming from the background $B$ field. The vertex operator has a multiplicative coupling constant $g(\sigma )$ which depends on $\sigma $ as $g(\sigma)=\sin^2 \sigma $ in $0 \le \sigma \le \pi $. This behavior is the same as in the ordinary B=0 case. 
  We consider the process of collision between a hard photon and a soft photon producing an electron-positron pair, under the assumption that the kinematics be described according to the $\kappa$-deformation of the D=4 Poincar\'{e} algebra. We emphasize the relevance of this analysis for the understanding of the puzzling observations of multi-TeV photons from Markarian 501. We find a significant effect of the $\kappa$-deformation for processes above threshold, while, in agreement with a previous study, we find that there is no leading-order deformation of the threshold condition. 
  A reformulation of fermionic QFT in electromagnetic backgrounds is presented which uses methods analogous to those of conventional multiparticle quantum mechanics. Emphasis is placed on the (Schr\"odinger picture) states of the system, described in terms of Slater determinants of Dirac states, and not on the field operator $\hat{\psi}(x)$ (which is superfluous in this approach). The vacuum state `at time $\tau$' is defined as the Slater determinant of a basis for the span of the negative spectrum of the `first quantized' Hamiltonian $\hat{H}(\tau)$, thus providing a concrete realisation of the Dirac Sea. The general S-matrix element of the theory is derived in terms of time-dependent Bogoliubov coefficients, demonstrating that the S-matrix follows directly from the definition of inner product between Slater determinants. The process of `Hermitian extension', inherited directly from conventional multiparticle quantum mechanics, allows second quantized operators to be defined without appealing to a complete set of orthonormal modes, and provides an extremely straightforward derivation of the general expectation value of the theory. The concept of `radar time', advocated by Bondi in his work on k-calculus, is used to generalise the particle interpretation to an arbitrarily moving observer. A definition of particle results, which depends {\it only} on the observer's motion and the background present, not on any choice of coordinates or gauge, or of the particle detector. We relate this approach to conventional methods by comparing and contrasting various derivations. Our particle definition can be viewed as a generalisation to arbitrary observers of Gibbons' approach. 
  We study the dynamics of an open membrane with a cylindrical topology, in the background of a constant three form, whose boundary is attached to p-branes. The boundary closed string is coupled to a two form potential to ensure gauge invariance. We use the action, due to Bergshoeff, London and Townsend, to study the noncommutativity properties of the boundary string coordinates. The constrained Hamiltonian formalism due to Dirac is used to derive the noncommutativity of coordinates. The chain of constraints is found to be finite for a suitable gauge choice, unlike the case of the static gauge, where the chain has an infinite sequence of terms. It is conjectured that the formulation of closed string field theory may necessitate introduction of a star product which is both noncommutative and nonassociative. 
  We show that certain classes of apparently unprotected operators in N=4 SYM_4 do not receive quantum corrections as a consequence of a partial non-renormalization theorem for the 4-point function of chiral primary operators. We develop techniques yielding the asymptotic expansion of the 4-point function of CPOs up to order O(\lambda^2) and we perform a detailed OPE analysis. Our results reveal the existence of new non-renormalized operators of approximate dimension 6. 
  We revisit the problem of precursors in the AdS/CFT correspondence. Identification of the precursors is expected to improve our understanding of the tension between holography and bulk locality and of the resolution of the black hole information paradox. Previous arguments that the precursors are large, undecorated Wilson loops are found to be flawed. We argue that the role of precursors should become evident when one saturates a certain locality bound. The spacetime uncertainty principle is a direct consequence of this bound. 
  We study integrable and conformal boundary conditions for ^sl(2) Z_k parafermions on a cylinder. These conformal field theories are realized as the continuum scaling limit of critical A-D-E lattice models with negative spectral parameter. The conformal boundary conditions labelled by (a,m) in (G, Z_{2k}) are identified with associated integrable lattice boundary conditions labelled by (r,a) in (A_{g-2},G) where g is the Coxeter number of the A-D-E graph G. We obtain analytically the boundary free energies, present general expressions for the parafermion cylinder partition functions and confirm these results by numerical calculations. 
  We discuss a generalized form of IIA/IIB supergravity depending on all R-R potentials C^(p) (p=0,1,...,9) as the effective field theory of Type IIA/IIB superstring theory. For the IIA case we explicitly break this R-R democracy to either p<=3 or p>=5 which allows us to write a new bulk action that can be coupled to N=1 supersymmetric brane actions.   The case of 8-branes is studied in detail using the new bulk & brane action. The supersymmetric negative tension branes without matter excitations can be viewed as orientifolds in the effective action. These D8-branes and O8-planes are fundamental in Type I' string theory. A BPS 8-brane solution is given which satisfies the jump conditions on the wall. It implies a quantization of the mass parameter in string units. Also we find a maximal distance between the two walls, depending on the string coupling and the mass parameter. We derive the same results via supersymmetric flow equations. 
  On the basis of a new quantum field theory without divergence presented by us we identify W-matter with dark matter and guess that the two new stars RXJ1865 and 3C58 and quasi-stellar objects are the compounds of a F-celestial body with a W-celestial body with massive enough mass, and some earthquakes, some tsunamis and some disasters in some areas like Bermuda are caused by W-objects. Keywords: Dark matter; W-matter; Quasi-stellar objects. 
  We study the behaviour of Polyakov confining string in the Georgi-Glashow model in three dimensions near confining-deconfining phase transition described in hep-th/0010201. In the string language, the transition mechanism is the decay of the confining string into D0 branes (charged W bosons of the Georgi-Glashow model). In the world-sheet picture the world-lines of heavy D0 branes at finite temperature are represented as world-sheet vortices of a certain type, and the transition corresponds to the condensation of these vortices. We also show that the ``would be'' Hagedorn transition in the confining string (which is not realized in our model) corresponds to the monopole binding transition in the field theoretical language. The fact that the decay into D0 branes occurs at lower than the Hagedorn temperature is understood as the consequence of the large thickness of the confining string and finite mass of the $D0$ branes. 
  A supersymmetric $D = 1, N =1$ model with a Grassmann-odd Lagrangian is proposed which, in contrast to the model with an even Lagrangian, contains not only a kinetic term but also an interaction term for the coordinates entering into one real scalar Grassmann-even (bosonic) superfield. 
  The general form of the stress-tensor three-point function in four dimensions is obtained by solving the Ward identities for the diffeomorphism and Weyl symmetries. Several properties of this correlator are discussed, such as the renormalization and scheme independence and the analogies with the anomalous chiral triangle. At the critical point, the coefficients a and c of the four-dimensional trace anomaly are related to two finite, scheme-independent amplitudes of the three-point function. Off-criticality, the imaginary parts of these amplitudes satisfy sum rules which express the total renormalization-group flow of a and c between pairs of critical points. Although these sum rules are similar to that satisfied by the two-dimensional central charge, the monotonicity of the flow, i.e. the four-dimensional analogue of the c-theorem, remains to be proven. 
  Spontaneous breaking of global symmetries can produce ``Alice'' strings: line defects which make unbroken symmetries multivalued, induce apparent charge violation via Aharonov-Bohm interactions, and form point defects when twisted into loops. We demonstrate this behavior for both divergent and textured global Alice strings. Both adiabatically scatter charged particles via effective Wilson lines. For textured Alice strings, such Wilson lines occur at all radii, and are multivalued only inside the string. This produces measurable effects, including path-dependent charge violation. 
  We propose a cosmological scenario in which the hot big bang universe is produced by the collision of a brane in the bulk space with a bounding orbifold plane, beginning from an otherwise cold, vacuous, static universe. The model addresses the cosmological horizon, flatness and monopole problems and generates a nearly scale-invariant spectrum of density perturbations without invoking superluminal expansion (inflation). The scenario relies, instead, on physical phenomena that arise naturally in theories based on extra dimensions and branes. As an example, we present our scenario predominantly within the context of heterotic M-theory. A prediction that distinguishes this scenario from standard inflationary cosmology is a strongly blue gravitational wave spectrum, which has consequences for microwave background polarization experiments and gravitational wave detectors. 
  We present exact solutions of the gravitational field equations in the generalized Randall-Sundrum model for an anisotropic brane with Bianchi type I and V geometry, with perfect fluid and scalar fields as matter sources. Under the assumption of a conformally flat bulk (with vanishing Weyl tensor) for a cosmological fluid obeying a linear barotropic equation of state the general solution of the field equations can be expressed in an exact parametric form for both Bianchi type I and V space-times. In the limiting case of a stiff cosmological fluid with pressure equal to the energy density, for a Bianchi type I Universe the solution of the field equations are obtained in an exact analytic form. Several classes of scalar field models evolution on the brane are also considered, corresponding to different choices of the scalar field potential. For all models the behavior of the observationally important parameters like shear, anisotropy and deceleration parameter is considered in detail. 
  We discuss the brane cosmology in the 5D anti de Sitter Reissner-Nordstrom (AdSRN$_5$) spacetime. A brane with the tension $\sigma$ is defined as the edge of an AdSRN$_5$ space with mass $M$ and charge $Q$. In this case we get the CFT-radiation term $(\rho_{CFT})$ from $M$ and the charged dust $(-\rho^2_{cd})$ from $Q^2$ in the Friedmann-like equation.  However, this equation is not justified because it contains $\rho_{cd}^2$-term with the negative sign. This is unconventional in view of the standard cosmology. In order to resolve this problem, we introduce a localized dust matter which satisfies $P_{dm}=0$. If $\rho_{dm}=\fr{\sqrt 3}{2}\rho_{cd}$, the unwanted $-\rho_{cd}^2$ is cancelled against $\rho_{dm}^2$ and thus one recovers a standard Friedmann-Robertson-Walker universe with CFT-radiation and dust matter. For the stiff matter consideration, we can set $\rho_{csti}\sim Q^2$ with the negative sign. Here we introduce a massless scalar which plays the role of a stiff matter with $P_{sca}=\rho_{sca}$ to cancel $-\rho_{csti}$. In this case, however, we find a mixed version of the standard and brane cosmologies. 
  We construct the lattice gauge theory of the group G_N, the semidirect product of the permutation group S_N with U(1)^N, on an arbitrary Riemann surface. This theory describes the branched coverings of a two-dimensional target surface by strings carrying a U(1) gauge field on the world sheet. These are the non-supersymmetric Matrix Strings that arise in the unitary gauge quantization of a generalized two-dimensional Yang-Mills theory. By classifying the irreducible representations of G_N, we give the most general formulation of the lattice gauge theory of G_N, which includes arbitrary branching points on the world sheet and describes the splitting and joining of strings. 
  After a brief introduction into the use of Calabi--Yau varieties in string dualities, and the role of toric geometry in that context, we review the classification of toric Calabi-Yau hypersurfaces and present some results on complete intersections. While no proof of the existence of a finite bound on the Hodge numbers is known, all new data stay inside the familiar range $h_{11}+h_{12}\le 502$. 
  We consider the T-duality relations between Type 0A and 0B theories, and show that this constraints the possible couplings of the tachyon to the RR-fields. Due to the `doubling' of the RR sector in Type 0 theories, we are able to introduce a democratic formulation for the Type 0 effective actions, in which there is no Chern-Simons term in the effective action. Finally we discuss how to embed Type II solutions into Type 0 theories. 
  The non-abelian self-dual action in three dimensions is derived using the self-interaction mechanism. 
  By using the recursion relations found in the framework of N=2 Super Yang-Mills theory with gauge group SU(2), we reconstruct the structure of the instanton moduli space and its volume form for all winding numbers. The construction is reminiscent of the Deligne-Knudsen-Mumford compactification and uses an analogue of the Wolpert restriction phenomenon which arises in the case of moduli spaces of Riemann surfaces. 
  Consequences of a strong version of the AdS/CFT correspondence for extremely stringy physics are examined. In particular, properties of N = 4 supersymmetric Yang-Mills theory are used to extract results about interacting tensionless strings and massless higher spin fields in an AdS_5 x S^5 background. Furthermore, the thermodynamics of this model signals the presence of a Hawking-Page phase transition between AdS_5 space and a "black hole"-like high temperature configuration even in the extreme string limit. 
  We study the averaged action of the Randall-Sundrum model with a time dependent metric ansatz. It can be reformulated in terms of a Brans-Dicke action with time dependent Newton's constant. We show that the physics of early universe, particularly inflation, is governed by the Brans-Dicke theory. The Brans-Dicke scalar, however, quickly settles to its equilibrium value and decouples from the post-inflationary cosmology. The deceleration parameter is negative to start with but changes sign before the Brans-Dicke scalar settles to its equilibrium value. Consequently, the brane metric smoothly exits inflation. We have also studied the slow-roll inflation in our model and investigated the spectra of the density perturbation generated by the radion field and find them consistent with the current observations. 
  We confront Cardy's suggested c-function for four-dimensional field theories with the spontaneous breaking of chiral symmetries in asymptotically free vectorlike gauge theories with fermions transforming according to different representations under the gauge group. Assuming that the infrared limit of the c-function is determined by the dimension of the associated Goldstone manifold, we find that this c-function always decreases between the ultraviolet and infrared fixed points. 
  This is a summary of progress made in understanding the occurrence and properties of local, conserved, commuting charges in non-linear sigma-models, including principal chiral models (PCMs) and WZW models. (Contribution to the NATO-ASI meeting: Integrable Hierarchies and Modern Physical Theories, University of Illinois at Chicago, July 2000.) 
  The theory of quantum fields in classical backgrounds is re-examined for the cases in which the Lagrangian is quadratic in quantum fields. Various methods that describe particle production are discussed. It is found that all methods suffer from certain ambiguities, related to the choice of coordinates, gauge, or counter-terms. They also seem to be inconsistent with the conservation of energy. This suggests that such classical backgrounds may not cause particle production. 
  We propose a general construction principle which allows to include an infinite number of resonance states into a scattering matrix of hyperbolic type. As a concrete realization of this mechanism we provide new S-matrices generalizing a class of hyperbolic ones, which are related to a pair of simple Lie algebras, to the elliptic case. For specific choices of the algebras we propose elliptic generalizations of affine Toda field theories and the homogeneous sine-Gordon models. For the generalization of the sinh-Gordon model we compute explicitly renormalization group scaling functions by means of the c-theorem and the thermodynamic Bethe ansatz. In particular we identify the Virasoro central charges of the corresponding ultraviolet conformal field theories. 
  We investigate the influence of the fermion field boundary conditions on the spectrum and wavefunctions of QED$_{1+1}$ in the Discretized Light-Cone Quantization formalism suggested by Pauli and Brodsky.   The basic lesson is that one Fourier mode (namely the zero mode) can be decisive for even the ground state mass. The choice of boundary conditions is seen to be not important, if and only if all degrees of freedom are treated properly. This is still a problem for light-cone quantization in general. 
  These brief notes record our puzzles and findings surrounding Givental's recent conjecture which expresses higher genus Gromov-Witten invariants in terms of the genus-0 data. We limit our considerations to the case of a projective line, whose Gromov-Witten invariants are well-known and easy to compute. We make some simple checks supporting his conjecture. 
  We analyze the physics of massive spin 2 fields in (A)dS backgrounds and exhibit that: The theory is stable only for masses m^2 >= 2\Lambda/3, where the conserved energy associated with the background timelike Killing vector is positive, while the instability for m^2<2\Lambda/3 is traceable to the helicity 0 energy. The stable, unitary, partially massless theory at m^2=2\Lambda/3 describes 4 propagating degrees of freedom, corresponding to helicities (+/-2,+/-1) but contains no 0 helicity excitation. 
  We find the moduli space of multi-solitons in noncommutative scalar field theories at large theta, in arbitrary dimension. The existence of a non-trivial moduli space at leading order in 1/theta is a consequence of a Bogomolnyi bound obeyed by the kinetic energy of the theta=infinity solitons. In two spatial dimensions, the parameter space for k solitons is a Kahler de-singularization of the symmetric product (R^2)^k/S_k. We exploit the existence of this moduli space to construct solitons on quotient spaces of the plane: R^2/Z_k, cylinder, and T^2. However, we show that tori of area less than or equal to (2 pi theta) do not admit stable solitons. In four dimensions the moduli space provides an explicit Kahler resolution of (R^4)^k/S_k. In general spatial dimension 2d, we show it is isomorphic to the Hilbert scheme of k points in C^d, which for d > 2 (and k > 3) is not smooth and can have multiple branches. 
  We show that as in abelian gauge fields, nonabelian gauge fields are also trapped on a brane in the Randall-Sundrum model by applying a new mechanism based on topological Higgs mechanism. It is pointed out that although almost massless gauge fields are localized on the brane by the new mechanism, exactly massless gauge fields are not localized. This fact does not yield any problem to abelian gauge fields, but may give some problem to nonabelian gauge fields since it is known that there is a discontinuity between massless and massive gauge fields in the case of nonabelian gauge groups. 
  The Spinning Particle Model for anyon is analysed in the Batalin-Tyutin scheme of quantisation in extended phase space. Here additional degrees of freedom are introduced in the phase space such that all the constraints in the theory are rendered First Class that is commuting in the sense of Poisson Brackets. Thus the theory can be studied without introducing the Dirac Brackets which appear in the presence of non-commuting or Second Class constraints. In the present case the Dirac Brackets make the configuration space of the anyon non-canonical and also being dynamical variable dependent, poses problems for the quantisation programme. We show that previously obtained results, (e.g. gyromagnetic ratio of anyon being 2), are recovered in the Batalin-Tyutin variable independent sector in the extended space. The Batalin-Tyutin variable contributions are significant and are computable in a straightforward manner. The latter can beunderstood as manifestations of the non-commutative space-time in the enlarged phase space. 
  We investigate quantum field theory in a bulk space with boundary, which represents a 3-brane. Both flat and anti-de Sitter backgrounds are considered. The basic idea is to keep local commutativity only on the brane, giving up this requirement in the bulk. We explore the consequences of this proposal, constructing a large family of nonlocal bulk fields, whose brane relatives are local. We estimate the ultraviolet behavior of these local brane fields, characterizing a subfamily which generates renormalizable theories on the brane. The issue of brane conformal invariance and the relation between bulk and brane conserved currents are also examined in this framework. 
  We adress ourselves the question of the quantum equivalence of non abelian dualised $\si$-models on the simple example of the T-dualised $SU(2) \si$-model. This theory is classically canonically equivalent to the standard chiral $SU(2) \si$-model. It is known that the equivalence also holds at the first order in perturbations with the same $\be$ functions. However, this model has been claimed to be non-renormalisable at the two-loop order. The aim of the present work is the proof that it is - at least up to this order - still possible to define a correct quantum theory. Its target space metric being only modified in a finite manner, all divergences are reabsorbed into coupling and fields (infinite) renormalisations. 
  A regularized model of the double compactified D=11 supermembrane with nontrivial winding in terms of SU(N) valued maps is obtained. The condition of nontrivial winding is described in terms of a nontrivial line bundle introduced in the formulation of the compactified supermembrane. The multivalued geometrical objects of the model related to the nontrivial wrapping are described in terms of a SU(N) geometrical object which in the $ N\to \infty$ limit, converges to the symplectic connection related to the area preserving diffeomorphisms of the recently obtained non-commutative description of the compactified D=11 supermembrane.(I. Martin, J.Ovalle, A. Restuccia. 2000,2001)   The SU(N) regularized canonical lagrangian is explicitly obtained. In the $ N\to \infty$ limit it converges to the lagrangian in (I.Martin, J.Ovalle, A.Restuccia. 2000,2001) subject to the nontrivial winding condition. The spectrum of the hamiltonian of the double compactified D=11 supermembrane is discussed.  Generically, it contains local string like spikes with zero energy.  However the sector of the theory corresponding to a principle bundle characterized by the winding number $n \neq 0$, described by the SU(N) model we propose, is shown to have no local string-like spikes and hence the spectrum of this sector should be discrete. 
  We consider the matrix quantum mechanics of N D0-branes in the background of the 1-form RR field. It is observed that the transformations of matrix coordinates of D0-branes induce on the Abelian RR field symmetry transformations that are like those of non-Abelian gauge fields. The Lorentz-like equations of motion for matrix coordinates are derived. The field strengths appearing in the Lorentz-like equations transform in the adjoint representation of U(N) under symmetry transformations. A possible relation between D0-brane dynamics in RR background, and the semi-classical dynamics of charged particles in Yang-Mills background is mentioned. 
  We investigate the presence of irrelevant operators in the 2d Ising model perturbed by a magnetic field, by studying the corrections induced by these operators in the spin-spin correlator of the model. To this end we perform a set of high precision simulations for the correlator both along the axes and along the diagonal of the lattice. By comparing the numerical results with the predictions of a perturbative expansion around the critical point we find unambiguous evidences of the presence of such irrelevant operators. It turns out that among the irrelevant operators the one which gives the largest correction is the spin 4 operator T^2 + \bar T^2 which accounts for the breaking of the rotational invariance due to the lattice. This result agrees with what was already known for the correlator evaluated exactly at the critical point and also with recent results obtained in the case of the thermal perturbation of the model. 
  In the warped compactification with a single Randall-Sundrum brane, a puzzling claim has been made that scalar fields can be bound to the brane but their Hodge dual higher-rank anti-symmetric tensors cannot. By explicitly requiring the Hodge duality, a prescription to resolve this puzzle was recently proposed by Duff and Liu. In this note, we implement the Hodge duality via path integral formulation in the presence of the background gravity fields of warped compactifications. It is shown that the prescription of Duff and Liu can be naturally understood within this framework. 
  We solve the Einstein equations in the Randall-Sundrum framework using an isotropic ansatz for the metric and obtain an exact expression to first order in the gravitational coupling. The solution is free from metric singularities away from the source and it satisfies the Israel matching condition on a straight brane. At distances far away from the source and on the physical brane this solution coincides with the 4-D Schwarzschild metric in isotropic coordinates. Furthermore we show that the extension of the standard Schwarzschild horizon in the bulk is tubular for any diagonal form of the metric while there is no restriction for the extension of the Schwarzschild horizon in isotropic coordinates. 
  We consider in detail how the quantum-mechanical tunneling phenomenon occurs in a well-behaved octic potential. Our main tool will be the euclidean propagator just evaluated between two minima of the potential at issue. For such a purpose we resort to the standard semiclassical approximation which takes into account the fluctuations over the instantons, i.e. the finite-action solutions of the euclidean equation of motion. As regards the one-instanton approach, the functional determinant associated with the so-called stability equation is analyzed in terms of the asymptotic behaviour of the zero-mode. The conventional ratio of determinants takes as reference the harmonic oscillator whose frequency is the average of the two different frequencies derived from the minima of the potential involved in the computation. The second instanton of the model is studied in a similar way. The physical effects of the multi-instanton configurations are included in this context by means of the alternate dilute-gas approximation where the two instantons participate to provide us with the final expression of the propagator. 
  A generalization of the Yang-Baxter algebra is found in quantizing the monodromy matrix of two (m)KdV equations discretized on a space lattice. This braided Yang-Baxter equation still ensures that the transfer matrix generates operators in involution which form the Cartan sub-algebra of the braided quantum group. Representations diagonalizing these operators are described through relying on an easy generalization of Algebraic Bethe Ansatz techniques. The conjecture that this monodromy matrix algebra leads, {\it in the cylinder continuum limit}, to a Perturbed Minimal Conformal Field Theory description is analysed and supported. 
  We study classical N=2 string within the framework of the N=4 topological formalism by Berkovits and Vafa. Special emphasis is put on the demonstration of a classical equivalence of the theories and the construction of an action for the N=4 topological string. The SO(2,2) Lorentz invariance missing in the conventional Brink--Schwarz action for the N=2 string is restored in the N=4 topological action. 
  Foliated manifolds are particular examples of noncommutative spaces. In this article we try to give a qualitative description of the Godbillon-Vey class and its relation on the one hand to the holonomy and on the other hand to the topological entropy of a foliation, using a remarkable theorem proved recently by G. Duminy relating these three notions in the case of codim-1 foliations. Moreover we shall investigate its possible relation with the black hole entropy adopting the superstring theory origin of the black hole entropy in the extremal case. This situation we believe has some striking similarities with the explanation due to Bellissard of the integrality of the Hall conductivity in the quantum Hall effect. Our starting point is the Connes-Douglas-Schwarz article on compactifications of matrix models to noncommutative tori.  
  We construct renormalizable, asymptotically free, four dimensional gauge theories that dynamically generate a fifth dimension. 
  When a potential for a scalar field has two local minima, there arises structure of spherical shells due to gravitational interactions. 
  We present a formalism for local composite operators. The corresponding effective potential is unique, multiplicatively renormalizable, it is the sum of 1PI diagrams and can be interpreted as an energy-density. First we apply this method to $\lambda \Phi^4$ theory where we check renormalizability up to three loops and secondly to the Coleman-Weinberg model where the gauge independence of the effective potential for the local composite operator $\phi\phi^*$ is explicitely checked up to two loops. 
  Riemann zeta function is an important object of number theory. It was also used for description of disordered systems in statistical mechanics. We show that Riemann zeta function is also useful for the description of integrable model. We study XXX Heisenberg spin 1/2 anti-ferromagnet. We evaluate a probability of formation of a ferromagnetic string in the anti-ferromagnetic ground state in thermodynamics limit. We prove that for short strings the probability can be expressed in terms of Riemann zeta function with odd arguments. 
  We obtain a solution describing a gravitational shockwave propagating along a Randall-Sundrum brane. The interest of such a solution is twofold: on the one hand, it is the first exact solution for a localized source on a Randall-Sundrum three-brane. On the other hand, one can use it to study forward scattering at Planckian energies, including the effects of the continuum of Kaluza-Klein modes. We map out the different regimes for the scattering obtained by varying the center-of-mass energy and the impact parameter. We also discuss exact shockwaves in ADD scenarios with compact extra dimensions. 
  We engineer a configuration of branes in type IIB string theory whose mechanical structure is that of a DNA molecule. We obtain it by considering a T-dual description of the quantum Hall soliton. Using a probe analysis, we investigate the dynamics of the system and show that it is stable against radial perturbations. We exercise a certain amount of restraint in discussing applications to biophysics. 
  In our previous work (Phys. Rev. D63, 085010, hep-th/0011290), we showed that the brane universe on the giant graviton moving in the near-horizon background of the dilatonic D(6-p)-brane is described by the mirage cosmology. We study thermodynamic properties of the moving giant graviton by applying thermodynamics of cosmology and the recently proposed holographic principles of cosmology. We find that the Fischler-Susskind holographic bound is satisfied by the closed brane universe on the moving giant graviton with p>3. The Bekenstein and the Hubble entropy bounds and the recently proposed Verlinde's holographic principle applied to the brane universe on the giant graviton are also studied. 
  We study a (1+1)-dimensional $\lambda\phi^4$ model with a light-cone zero mode and constant external source to describe spontaneous symmetry breaking. In the broken phase, we find degenerate vacua and discuss their stability based on effective-potential analysis. The vacuum triviality is spurious in the broken phase because these states have lower energy than Fock vacuum. Our results are based on the variational principle. 
  We find an algorithm of numerical renormalization group for spin chain models. The essence of this algorithm is orthogonal transformation of basis states, which is useful for reducing the number of relevant basis states to create effective Hamiltonian. We define two types of rotations and combine them to create appropriate orthogonal transformation. 
  The vacuum diagram is calculated at second order for theories with self-interacting massless fields in the framework of finite causal perturbation theory. It is pointed out that the infrared behaviour of the vacuum diagram leads to unstable Fock vacua for QCD or massless QED, but not for quantum gravity. Therefore a radical rearrangement of the physical system must take place for such theories. Conversely, stability of the Fock vacuum for massless interacting fields is another hint at the possibility that quantum gravity should be treated as an effective theory. 
  A recently proposed world-volume equivalence principal involving the Boillat, as opposed to the Einstein, metric is examined in the context of some colliding wave solutions of the Born-Infeld equations for which two plane polarized pulses pass through one another without distortion. They suffer a delay with respect to the usual Einstein metric but not with respect to the Boillat metric. Both metrics are flat in this case, and the closed string and open string causal structures are interchanged by the Legendre transformation that is used for solving the associated Monge-Amp\`ere equation. In 1+1 dimensions the equations are known to be equivalent to the vorticity free motion of a Chaplygin gas. The latter is shown to be described by the scalar Born-Infeld equation in all dimensions and it is pointed out that the equation of state is Hagedorn-like: there is an upper bound to the pressure and temperature. 
  We study perturbative and non-perturbative properties of the Konishi multiplet in N=4 SYM theory in D=4 dimensions. We compute two-, three- and four-point Green functions with single and multiple insertions of the lowest component of the multiplet, and of the lowest component of the supercurrent multiplet. These computations require a proper definition of the renormalized operator and lead to an independent derivation of its anomalous dimension. The O(g^2) value found in this way is in agreement with previous results. We also find that instanton contributions to the above correlators vanish. From our results we are able to identify some of the lowest dimensional gauge-invariant composite operators contributing to the OPE of the correlation functions we have computed. We thus confirm the existence of an operator belonging to the representation 20', which has vanishing anomalous dimension at order g^2 and g^4 in perturbation theory as well as at the non-perturbative level, despite the fact that it does not obey any of the known shortening conditions. 
  We find the N-soliton solution at infinite theta, as well as the metric on the moduli space corresponding to spatial displacements of the solitons. We use a perturbative expansion to incorporate the leading 1/theta corrections, and find an effective short range attraction between solitons. We study the stability of various solutions. We discuss the finite theta corrections to scattering, and find metastable orbits. Upon quantization of the two-soliton moduli space, for any finite theta, we find an s-wave bound state. 
  Presented are the integral solutions to the quantum Knizhnik-Zamolodchikov equations for the correlation functions of both the bulk and boundary XXZ models in the anti-ferromagnetic regime. The difference equations can be derived from Smirnov-type master equations for correlation functions on the basis of the CTM bootstrap. Our integral solutions with an appropriate choice of the integral kernel reproduce the formulae previously obtained by using the bosonization of the vertex operators of the quantum affine algebra $U_q (\hat{\mathfrak{sl}_2})$. 
  The propagator and complete sets of in- and out-solutions of wave equation, together with Bogoliubov coefficients, relating these solutions, are obtained for vector $W$-boson (with gyromagnetic ratio $g=2$) in a constant electromagnetic field. When only electric field is present the Bogoliubov coefficients are independent of boson polarization and are the same as for scalar boson. When both electric and magnetic fields are present and collinear, the Bogoliubov coefficients for states with boson spin perpendicular to the field are again the same as in scalar case. For $W^-$ spin along (against) the magnetic field the Bogoliubov coefficients and the contributions to the imaginary part of the Lagrange function in one loop approximation are obtained from corresponding expressions for scalar case by substitution $m^2\to m^2+2eH$ $(m^2\to m^2-2eH)$. For gyromagnetic ratio $g=2$ the vector boson interaction with constant electromagnetic field is described by the functions, which can be expected by comparing wave functions for scalar and Dirac particle in constant electromagnetic field. 
  We generalize the Goldberger-Wise mechanism and study the stability of the Crystal Universe models. We show that the model can be stabilized, however for configurations of Crystal Universe in the absence of fine-tuning, brane crystals are not equidistant, i.e. a $"-+"$ pair is far away from adjacent $"-+"$ pair, except for the fixed points of the orbifold, which differs from the assumptions taken in the literature. 
  We investigate Schwinger-Dyson equations for correlators of Wilson line operators in non-commutative gauge theories. We point out that, unlike what happens for closed Wilson loops, the joining term survives in the planar equations. This fact may be used to relate the correlator of an arbitrary number of Wilson lines eventually to a set of {\it closed} Wilson loops, obtained by joining the individual Wilson lines together by a series of well-defined cutting and joining manipulations. For closed loops, we find that the non-planar contributions do not have a smooth limit in the limit of vanishing non-commutativity and hence the equations do not reduce to their commutative counterparts. We use the Schwinger-Dyson equations to derive loop equations for the correlators of Wilson observables. In the planar limit, this gives us a {\it new} loop equation which relates the correlators of Wilson lines to the expectation values of closed Wilson loops. We discuss perturbative verification of the loop equation for the 2-point function in some detail. We also suggest a possible connection between Wilson line based on an arbitrary contour and the string field of closed string. 
  We obtain the induced Lorentz- and CPT-violating term in QED at finite temperature using imaginary-time formalism and dimensional regularization. Its form resembles a Chern-Simons-like structure, but, unexpectedly, it does not depend on the temporal component of the fixed $b_\mu$ constant vector that is coupled to the axial current. Nevertheless Ward identities are respected and its coefficient vanishes at T=0, consistently with previous computations with the same regularization procedure, and it is a non-trivial function of temperature. We argue that at finite $T$ a Chern-Simons-like Lorentz- and CPT-violating term is generically present, the value of its coefficient being unambiguously determined up to a $T-$independent constant, related to the zero-temperature renormalization conditions. 
  We consider the cosmological evolution of a bulk scalar field and ordinary matter living on the brane world in the light of the constraints imposed by the matter dominated cosmological evolution and a small cosmological constant now. We rule out models with a self-tuned minimum of the four dimensional potential as they would lead to rapid oscillations of the Hubble parameter now. A more natural framework is provided by supergravity in singular spaces where the brane coupling and the bulk potential are related by supersymmetry leading to a four dimensional run-away potential. For late times we obtain an accelerating universe due to the breaking of supersymmetry on the brane with an acceleration parameter of q_0=-4/7 and associated equation of state omega=-5/7. 
  We consider the Landau-Ginzburg-Wilson Hamiltonian with O(n)x O(m) symmetry and compute the critical exponents at all fixed points to O(n^{-2}) and to O(\epsilon^3) in a \epsilon=4-d expansion. We also consider the corresponding non-linear sigma model and determine the fixed points and the critical exponents to O(\tilde{\epsilon}^2) in the \tilde{\epsilon}=d-2 expansion. Using these results, we draw quite general conclusions on the fixed-point structure of models with O(n)xO(m) symmetry for n large and all 2 < d < 4. 
  This is an introductory set of lectures on elliptic differential operators and boundary problems, and their associated spectral functions. The role of zeta functions and traces of heat kernels in the regularization of Casimir energies is emphasized, and the renormalization issue is discussed through simple examples. 
  We analyse the Seiberg Witten curve describing the N=2 gauge theory dual to the supergravity solution with fractional branes. Emphasis is given to those aspects that are related to stringy mechanism known as the enhancon. We also compare our results with the features of the supergravity duals, which have been variously interpreted in the literature. Known aspects of the N=2 gauge theories seem to agree with the supergravity solution, whenever the two theories can be faithfully compared. 
  The aim of this paper is to find out a correspondence between one-loop effective action $W_E$ defined by means of path integral in Euclidean gravity and the free energy $F$ obtained by summation over the modes. The analysis is given for quantum fields on stationary space-times of a general form. For such problems a convenient procedure of a "Wick rotation" from Euclidean to Lorentzian theory becomes quite non-trivial implying transition from one real section of a complexified space-time manifold to another. We formulate conditions under which $F$ and $W_E$ can be connected and establish an explicit relation of these functionals. Our results are based on the Kaluza-Klein method which enables one to reduce the problem on a stationary space-time to equivalent problem on a static space-time in the presence of a gauge connection. As a by-product, we discover relation between the asymptotic heat-kernel coefficients of elliptic operators on a $D$ dimensional stationary space-times and the heat-kernel coefficients of a $D-1$ dimensional elliptic operators with an Abelian gauge connection. 
  Using Random Matrix Theory we set out to compute the microscopic correlators of the Euclidean Dirac operator in four dimensions. In particular we consider: the chiral Orthogonal Ensemble (chOE), corresponding to a Yang-Mills theory with two colors and fermions in the fundamental representation, and the chiral Symplectic Ensemble (chSE), corresponding to any number of colors and fermions in the adjoint representation. In both cases we deal with an arbitrary number of massive fermions. We use a recent method proposed by H. Widom for deriving closed formulas for the scalar kernels from which all spectral correlation functions of the chGOE and chGSE can be determined. Moreover, we obtain complete analytic expressions of such correlators in the double microscopic limit, extending previously known results of four-dimensional QCD at beta=1 and beta=4 to the general case with N_f flavors, with arbitrary quark masses and arbitrary topological charge. 
  Confining strings and RG flow at finite temperature are investigated in the (2+1)-dimensional Georgi-Glashow model. This is done in the limit when the electric coupling constant is much larger than the square root of mass of the Higgs field, but much smaller than the vacuum expectation value of this field. The modification of the Debye mass of the dual photon with respect to the case when it is considered to be negligibly small compared to the Higgs mass, is found. Analogous modifications of the potential of monopole densities and string coupling constants are found. At finite temperature, the mass of the Higgs field scales according to a novel RG equation. It is checked that in the limit when the original theory is reduced by the RG flow to the 2D XY model, the so-evolved Higgs mass is still much smaller than the squared electric coupling constant. The SU(3)-theory of confining strings is also discussed within the same approximations. 
  Covariant relativistic quantum theory is used to study the covariant Green's function, which can be used to determine the proper time evolved wave functions that are solutions to the covariant Schr\"odinger type equation for a massive spin zero particle. The concept of covariant action is used to obtain the Green's function for an accelerated relativistic particle. 
  In this work we investigate connections between superalgebras and their realizations in terms of particles, branes and field theory models. We start from Poincar\'e superalgebras with brane charges and study its representations. The existence of new supermultiplets in different dimensions including an ultra short supermultiplet in D=11 different from the supergravity multiplet is shown. Generalizations of superalgebras containing brane charges, including those in D$>$11 are considered. The realization of these algebras at the level of relativistic particle models and, upon quantization, at the level of field theory is presented. Application of Hamiltonian/BRST methods of quantization of systems with mixture of first and second class constraints as well as a conversion method are discussed for the models of interest. Using quantization of particle mechanics we obtain information on the spectrum and linearized equations of motion of the perturbative, linearized M-theory. The generalization of particle models to p-branes is made using a geometrical formulation of superembedding approach to study the example of L-branes which have a linear multiplet on their worldvolume. The p-branes and strings in B-field are considered as well as the origin of noncommutativity and non-associativity in their low-energy limit. It is shown that the application of Hamiltonian/BRST methods for those models leads to stringy version of Seiberg-Witten map and the removal of the non-associativity/noncommutativity. 
  At large N, a field theory and its orbifolds (given by projecting out some of its fields) share the same planar graphs. If the parent-orbifold relation continues even nonperturbatively, then properties such as confinement and chiral symmetry breaking will appear in both parent and orbifold. N=1 supersymmetric Yang-Mills has many nonsupersymmetric orbifolds which resemble QCD. A nonperturbative parent-orbifold relation predicts many surprising effects, exactly valid at large N, and expected to suffer only mild 1/N corrections. These include degeneracies among bosonic hadrons and exact predictions for domain wall tensions. Other predictions are valid even when supersymmetry in the parent is broken. Since these theories are QCD-like, simulation is possible, so these predictions may be numerically tested. The method also relates wide classes of nonsupersymmetric theories. 
  For a limited number of matter fields, the discontinuity of the transverse gauge field propagator can satisfy an exact sum rule. With controlled and limited gauge dependence, this supercconvergence relation is of physical interest. 
  We present a generalization of the infinitesimal gauge transformation for nonabelian fields on the stack of branes up to the third order in $\Phi$. We test the gauge invariance of the action up to the fifth order in $\Phi$ for $D$-instantons. This substantiates the Myers formula for the Chern-Simons term in action, which describes interaction with the RR fields of $N$ coincident $\Dp$ branes. 
  We construct a manifestly gauge invariant Lagrangian in 3+1 dimensions for N Kaluza-Klein modes of an SU(m) gauge theory in the bulk. For example, if the bulk is 4+1, the effective theory is \Pi_{i=1}^{N+1} SU(m)_i with N chiral (\bar{m},m) fields connecting the groups sequentially. This can be viewed as a Wilson action for a transverse lattice in x^5, and is shown explicitly to match the continuum 4+1 compactifed Lagrangian truncated in momentum space. Scale dependence of the gauge couplings is described by the standard renormalization group technique with threshold matching, leading to effective power law running. We also discuss the unitarity constraints, and chiral fermions. 
  We derive an exact expression for the Seiberg-Witten map of noncommutative gauge theory. It is found by studying the coupling of the gauge field to the Ramond-Ramond potentials in string theory. Our result also proves the earlier conjecture by Liu. 
  We extend the large N duality of four dimensional N=1 supersymmetric Yang-Mills theory with additional chiral fields and arbitrary superpotential recently proposed by Cachazo, Intriligator and Vafa to the case of SO/Sp gauge groups. By orientifolding the geometric transition, we investigate a large N duality between N=1, SO/Sp supersymmetric theories with arbitrary superpotential and an Abelian N=2 theory with supersymmetry broken to N=1 by electric and magnetic Fayet-Iliopoulos terms. 
  We interpret the unimodularity condition in almost commutative geometries as central extensions of spin lifts. In Connes' formulation of the standard model this interpretation allows to compute the hypercharges of the fermions. 
  We show that for an eikonal limit of gravity in a space-time of any dimension with a non-vanishing cosmological constant, the Einstein -- Hilbert action reduces to a boundary action. This boundary action describes the interaction of shock-waves up to the point of evolution at which the forward light-cone of a collision meets the boundary of the space-time. The conclusions are quite general and in particular generalize the previous work of E. and H. Verlinde. The role of the off-diagonal Einstein action in removing the bulk part of the action is emphasised. We discuss the sense in which our result is a particular example of holography and also the relation of our solutions in $AdS$ to those of Horowitz and Itzhaki. 
  We compute the spectral index for scalar perturbations generated in a primordial inflationary model. In this model, the transition of the inflationary phase to the radiative era is achieved through the decay of the cosmological term leading a second order phase transition and the characteristics of the model allow to implement a set of initial conditions where the perturbations display a thermal spectrum when they emerge from the horizon. The obtained value for the spectral index is equal to 2, a result that depends very weakly on the various parameters of the model and on the initial conditions used. 
  We explore the effects of non-abelian dynamics of D-branes on their stability and introduce Hitchin-like modifications to previously-known stability conditions. The relation to brane-antibrane systems is used in order to rewrite the equations in terms of superconnections and arrive at deformed vortex equations. 
  We analyze several problems related to off-shell structure of open string sigma model by using a combination of derivative expansion and expansion in powers of the fields. According to the sigma model approach to bosonic open string theory, the tachyon effective action $S(T)$ coincides with the renormalized partition function $Z(T)$ of sigma model on a disk, up to a term vanishing on shell. On the other hand, $Z(T)$ is a generating functional of perturbative open string scattering amplitudes. If $S(T) = Z(T)$, then there should be no contribution of exchange diagrams to string amplitudes computed using $S(T)$. We compute the cubic term in the effective action, and show that it vanishes if some but not all external legs are on shell, and, therefore, any exchange diagram involving the cubic term vanishes too. Then, we discuss a problem of turning on nonrenormalizable boundary interactions, corresponding to massive string modes. We compute the quadratic term for a symmetric tensor field, and show that despite nonrenormalizability of the model one can consistently remove all divergent terms, and obtain a quadratic action reproducing the on-shell condition for the field. We also briefly discuss fermionic (NS) sigma model, compute the tachyon quadratic term, and show that it reproduces the correct tachyon mass. We note that turning on a massive symmetric tensor field leads to the appearance of a term linear in it, which can be removed by adding a higher-derivative term to the boundary of the disc. 
  In this note we discuss bound states of un- or meta-stable brane configurations in various non-trivial (curved) backgrounds. We begin by reviewing some known results concerning brane dynamics on group manifolds. These are then employed to study condensation in cosets of the WZW model. While the basic ideas are more general, our presentation focuses on parafermion theories and, closely related, N=2 superconformal minimal models. We determine the (non-commutative) low energy effective actions for all maximally symmetric branes in a decoupling limit of the two theories. These actions are used to show that the lightest branes can be regarded as elementary constituents for all other maximally symmetric branes. 
  We present supersymmetric solutions for the theory of gauged supergravity in five dimensions obtained by gauging the shift symmetry of the axion of the universal hypermultiplet. This gauged theory can also be obtained by dimensionally reducing M-theory on a Calabi-Yau threefold with background flux. The solution found preserves half of the N=2 supersymmetry, carries electric fields and has nontrivial scalar field representing the CY-volume. We comment on the possible solutions of more general hypermultiplet gauging. 
  We derive the couplings of noncommutative D-branes to spatially varying Ramond-Ramond fields, extending our earlier results in hep-th/0009101. These couplings are expressed in terms of *n products of operators involving open Wilson lines. Equivalence of the noncommutative to the commutative couplings implies interesting identities as well as an expression for the Seiberg-Witten map that was previously conjectured. We generalise our couplings to include transverse scalars, thereby obtaining a Seiberg-Witten map relating commutative and noncommutative descriptions of these scalars. RR couplings for unstable non-BPS branes are also proposed. 
  The worldvolume theory of a D0-brane contains a multiplet of fermions which can couple to background spacetime fields. This coupling implies that a D0-brane may possess multipole moments with respect to the various type IIA supergravity fields. Different such polarization states of the D0-brane will thus generate different long-range supergravity fields, and the corresponding semi-classical supergravity solutions will have different geometries. In this paper, we reconsider such solutions from an eleven-dimensional perspective. We thus begin by deriving the ``superpartners'' of the eleven-dimensional graviton. These superpartners are obtained by acting on the purely bosonic solution with broken supersymmetries and, in theory, one can obtain the full BPS supermultiplet of states. When we dimensionally reduce a polarized supergraviton along its direction of motion, we recover a metric which describes a polarized D0-brane. On the other hand, if we compactify along the retarded null direction we obtain the short distance, or ``near-horizon'', geometry of a polarized D0-brane, which is related to finite $N$ Matrix theory. The various dipole moments in this case can only be defined once the eleven-dimensional metric is ``regularized'' and, even then, they are formally infinite. We argue, however, that this is to be expected in such a non-asymptotically flat spacetime. Moreover, we find that the superpartners of the D0-brane, in this $r \ra 0$ limit, possess neither spin nor D2-brane dipole moments. 
  We analyse the deconfining phase transition in the SU(N) Georgi-Glashow model in 2+1 dimensions. We show that the phase transition is second order for any N, and the universality class is different from the Z(N) invariant Villain model. At large N the conformal theory describing the fixed point is a deformed SU(N)_1 WZNW model which has N-1 massless fields. It is therefore likely that its self-dual infrared fixed point is described by the Fateev-Zamolodchikov theory of Z(N) parafermions. 
  We prove that a sum of free non-covariant duality-symmetric actions does not allow consistent, continuous and local self-interactions that deform the gauge transformations. For instance, non-Abelian deformations are not allowed, even in 4 dimensions where Yang-Mills type interactions of 1-forms are allowed in the non-manifestly duality-symmetric formulation. This suggests that non-Abelian duality should require to leave the standard formalism of perturbative local field theories. The analyticity of self-interactions for a single duality-symmetric gauge field in four dimensions is also analyzed. 
  We discuss the non-linear gravitational interactions in the Randall-Sundrum single brane model. If we naively write down the 4-dimensional effective action integrating over the fifth dimension with the aid of the decomposition with respect to eigen modes of 4-dimensional d'Alembertian, the Kaluza-Klein mode coupling seems to be ill-defined. We carefully analyze second order perturbations of the gravitational field induced on the 3-brane under the assumption of the static and axial-symmetric 5-dimensional metric. It is shown that there remains no pathological feature in the Kaluza-Klein mode coupling after the summation over all different mass modes. Furthermore, the leading Kaluza-Klein corrections are shown to be sufficiently suppressed in comparison with the leading order term which is obtained by the zero mode truncation. We confirm that the 4-dimensional Einstein gravity is approximately recovered on the 3-brane up to second order perturbations. 
  A new matrix model is described, based on the exceptional Jordan algebra. The action is cubic, as in matrix Chern-Simons theory. We describe a compactification that, we argue, reproduces, at the one loop level, an octonionic compactification of the matrix string theory in which SO(8) is broken to G2. There are 27 matrix degrees of freedom, which under Spin(8) transform as the vector, spinor and conjugate spinor, plus three singlets, which represent the two longitudinal coordinates plus an eleventh coordinate. Supersymmetry appears to be related to triality of the representations of Spin(8). 
  It is noted that the internal space-time symmetries of relativistic particles are dictated by Wigner's little groups. The symmetry of massive particles is like the three-dimensional rotation group, while the symmetry of massless particles is locally isomorphic to the two-dimensional Euclidean group. It is noted also that, while the rotational degree of freedom for a massless particle leads to its helicity, the two translational degrees of freedom correspond to its gauge degrees of freedom. It is shown that the E(2)-like symmetry of of massless particles can be obtained as an infinite-momentum and/or zero-mass limit of the O(3)-like symmetry of massive particles. This mechanism is illustrated in terms of a sphere elongating into a cylinder. In this way, the helicity degree of freedom remains invariant under the Lorentz boost, but the transverse rotational degrees of freedom become contracted into the gauge degree of freedom. 
  The local Gauss law of quantum chromodynamics on a finite lattice is investigated. It is shown that it implies a gauge invariant, additive law giving rise to a gauge invariant ${\mathbb Z}_3$-valued global charge in QCD. The total charge contained in a region of the lattice is equal to the flux through its boundary of a certain ${\mathbb Z}_3$-valued, additive quantity. Implications for continuous QCD are discussed. 
  Standard derivations of ``time-independent perturbation theory'' of quantum mechanics cannot be applied to the general case where potentials are energy dependent or where the inverse free Green function is a non-linear function of energy. Such derivations cannot be used, for example, in the context of relativistic quantum field theory. Here we solve this problem by providing a new, general formulation of perturbation theory for calculating the changes in the energy spectrum and wave function of bound states and resonances induced by perturbations to the Hamiltonian. Although our derivation is valid for energy-dependent potentials and is not restricted to inverse free Green functions that are linear in the energy, the expressions obtained for the energy and wave function corrections are compact, practical, and maximally similar to the ones of quantum mechanics. For the case of relativistic quantum field theory, our approach provides a direct covariant way of obtaining corrections to bound and resonance state masses, as well as to wave functions that are not in the centre of mass frame. 
  In this contribution we review some recent work on the non-commutative geometry of branes on group manifolds. In particular, we show how fuzzy spaces arise in this context from an exact world-sheet description and we sketch the construction of a low-energy effective action for massless open string modes. The latter is given by a combination of a Yang-Mills and a Chern-Simons like functional on the fuzzy world-volume. It can be used to study condensation on various brane configurations in curved backgrounds. 
  In this paper we display a direct and physically attractive derivation of the screening contribution to the interaction potential in the Chiral Schwinger model and generalized Maxwell-Chern-Simons gauge theory. It is shown that these results emerge naturally when a correct separation between gauge-invariant and gauge degrees of freedom is made. Explicit expressions for gauge-invariant fields are found. 
  We establish general properties of supersymmetric flow equations and of the superpotential of five-dimensional N = 2 gauged supergravity coupled to vector and hypermultiplets. We provide necessary and sufficient conditions for BPS domain walls and find a set of algebraic attractor equations for N = 2 critical points.  As an example we describe in detail the gauging of the universal hypermultiplet and a vector multiplet. We study a two-parameter family of superpotentials with supersymmetric AdS critical points and we find, in particular, an N = 2 embedding for the UV-IR solution of Freedman, Gubser, Pilch and Warner of the N = 8 theory. We comment on the relevance of these results for brane world constructions. 
  Within a $D$-dimensional superstring spacetime, we construct a non-supersymmetric brane-world with localized gravity and large hierarchy between the scale in the bulk, $M_D$, and the scale on the brane, $M_{D-2}$. The localization of gravity and the large hierarchy are both guaranteed by the presence of non-trivial stringy moduli, such as the axion-dilaton system for the Type-IIB string theory. 
  The effective action with homogeneous valence (off-diagonal) gluons as background fields in the extended SU(2) model of QCD is obtained in one-loop approximation. We keep the manifest gauge and Lorentz invariance during whole calculation by using a special gauge-fixing recipe and taking into account the quartic interaction terms in the original Lagrangian as well. It has been shown that the effective Lagrangian gains a positive imaginary part. 
  Spacetime non-commutativity appears in string theory. In this paper, the non-commutativity in string theory is reviewed. At first we review that a Dp-brane is equivalent to a configuration of infinitely many D($p-2$)-branes. If we consider the worldvolume as that of the Dp-brane, coordinates of the Dp-brane is commutative. On the other hand if we deal with the worldvolume as that of the D($p-2$)-branes, since coordinates of many D-branes are promoted to matrices the worldvolume theory is non-commutative one. Next we see that using a point splitting reguralization gives a non-commutative D-brane, and a non-commutative gauge field can be rewritten in terms of an ordinary gauge field. The transformation is called the Seiberg-Witten map. And we introduce second class constraints as boundary conditions of an open string. Since Neumann and Dirichlet boundary conditions are mixed in the constraints when the open string is coupled to a NS B field, the end points of the open string is non-commutative. 
  We find that with uniform mesh, the numerical schemes derived from finite element method can keep a preserved symplectic structure in one-dimensional case and a preserved multisymplectic structure in two-dimentional case in certain discrete version respectively. These results are in fact the intrinsic reason that the numerical experiments indicate that such finite element algorithms are accurate in practice. 
  We refine the relation between the renormalized partition function of the open bosonic string in background fields and the effective action. In the process, we get some leading derivative corrections to the Born-Infeld action which include all powers of the gauge field. 
  An outline is given of recent work concerning the electromagnetic duality properties of Maxwell theory on curved space-times with or without spin structures. 
  The question as to whether the integrality of the spectrum of observed electric charges is due to a quantum effext has fascinated theoretical physicists throughout the last century. It leads to unanswered questions at the heart of quantum field theory and its role as a framework for particle physics. 
  We determine the vacuum structure and phases of N=1 theories obtained via a mass \mu for the adjoint chiral superfield in N=2, SO(n_c) SQCD. For large number of flavors these theories have two groups of vacua. The first exhibits dynamical breaking of flavor symmetry \USp(2n_f) \to U(n_f) and arises as a relevant deformation of a non-trivial superconformal theory. These are in the confined phase. The second group, in an IR-free phase with unbroken flavor symmetry, is produced from a Coulomb branch singularity with Seiberg's dual gauge symmetry. In the large-\mu regime both groups of vacua are well-described by dual quarks and mesons, and dynamical symmetry breaking in the first group occurs via meson condensation. We follow the description of these vacua from weak to strong coupling and demonstrate a nontrivial agreement between the phases and the number of vacua in the two regimes. We construct the semiclassical monopole flavor multiplets and argue that their multiplicity is consistent with the number of N=1 vacua. 
  We study the absorption of a minimally coupled scalar in the gravitational background created by a stack of near-extremal black three-branes, and more generally by M2, M5 and Dp branes. The absorption probability has the form P(l) = P_0(l) f_l(\lambda), where P_0(l) is the partial wave's absorption probability in the extremal case, and the thermal factor f_l(\lambda) depends on the ratio of the frequency of the incoming wave and the Hawking temperature, \lambda = \omega/\pi T. Using Langer-Olver's method, we obtain a low-temperature (\lambda \gg 1) asymptotic expansion for P(l) with coefficients determined recursively. This expansion, which turns out to be a fairly good approximation even for \lambda \sim 1, accounts for all power-like finite-temperature corrections to P_0(l), and we calculate a few terms explicitly. We also show that at low temperature the absorption probability contains exponentially suppressed terms, and attempt to develop an approximation scheme to calculate those. The high-temperature expansion is also considered. For the s-wave, the low-temperature gravity result is consistent with the free finite-temperature field theory calculation, while for high temperature and higher partial waves we find a disagreement. As a check of the approximation methods used, we apply them to the D1-D5-brane system, and compare results to the known exact solution. 
  Using the anti-de Sitter/conformal field theory correspondence, we relate the shear viscosity \eta of the finite-temperature N=4 supersymmetric Yang-Mills theory in the large N, strong-coupling regime with the absorption cross section of low-energy gravitons by a near-extremal black three-brane. We show that in the limit of zero frequency this cross section coincides with the area of the horizon. From this result we find \eta=\pi/8 N^2T^3. We conjecture that for finite 't Hooft coupling (g_YM)^2N the shear viscosity is \eta=f((g_YM)^2N) N^2T^3, where f(x) is a monotonic function that decreases from O(x^{-2}\ln^{-1}(1/x)) at small x to \pi/8 when x\to\infty. 
  We investigate brane physics in a universe with an extra dimensional global monopole and negative bulk cosmological constant. The graviton zero mode is naturally divergent; we thus invoke a physical cut-off to induce four dimensional gravity on a brane at the monopole core. Independently, the massive Kaluza-Klein modes have naturally compactified extra dimensions, inducing a discrete spectrum. This spectrum remains consistent with four dimensional gravity on the brane, even for small mass gap. Extra dimensional matter fields also induce four dimensional matter fields on the brane, with the same Kaluza-Klein spectrum of excited states. We choose parameters to solve the hierarchy problem; that is, to induce the observed hierarchy between particle and Planck scales in the effective four dimensional universe. 
  Bosonic boundary states at finite temperature are constructed as solutions of boundary conditions at $T\neq 0$ for bosonic open strings with a constant gauge field $F_{ab}$ coupled to the boundary. The construction is done in the framework of thermo field dynamics where a thermal Bogoliubov transformation maps states and operators to finite temperature. Boundary states are given in terms of states from the direct product space between the Fock space of the closed string and another identical copy of it. By analogy with zero temperature, the boundary states heve the interpretation of $Dp$-brane at finite temperature. The boundary conditions admit two different solutions. The entropy of the closed string in a $Dp$-brane state is computed and analysed. It is interpreted as the entropy of the $Dp$-brane at finite temperature. 
  The supergravity dual of $N$ regular and $M$ fractional D1-branes on the cone over the Einstein manifold $Q^{1,1,1}$ has a naked singularity in the infrared. The supergravity dual of $N$ regular and $M$ fractional D3-branes on the conifold also has such a singularity. Buchel suggested and Gubser et al. have shown that in the D3-brane case, the naked singularity is cloaked by a horizon at a sufficiently high temperature. In this paper we derive the system of second-order differential equations necessary to find such a solution for $Q^{1,1,1}$. We also find solutions to this system in perturbation theory that is valid when the Hawking temperature of the horizon is very high. 
  Cosmological inflation is discussed in the realm of Einstein-Cartan-Brans-Dicke (ECBD) gravity by constructing an effective inflaton potential and computing the number of e-folds.It is shown that spin-torsion density contributes to a decrease on the number of e-folds of inflation when the ratio of spin-torsion density to matter density is aprecciable as happens in the Early Universe.Quantum fluctuations of spin and inflatons are also investigated.Our results seems to be in agreement with Palle's proposal that quantum fluctuations of spin may trigger the primordial era of the universe.The main esult of the paper is the appearence of a supersymmetric type inflaton potential from the ECBD gravity where the coefficient of the one-loop type correction is a constant spin-torsion density. 
  It is argued that many non-extremal black branes exhibit a classical Gregory-Laflamme instability if, and only if, they are locally thermodynamically unstable. For some black branes, the Gregory-Laflamme instability must therefore disappear near extremality. For the black $p$-branes of the type II supergravity theories, the Gregory-Laflamme instability disappears near extremality for $p=1,2,4$ but persists all the way down to extremality for $p=5,6$ (the black D3-brane is not covered by the analysis of this paper). This implies that the instability also vanishes for the near-extremal black M2 and M5-brane solutions. 
  We investigate the possibility of constructing a locally supersymmetric extension of NGT (Nonsymmetric Gravitation Theory), based on the graded extension of the Poincare group. In the framework of the simple model that we propose, we end up with a no-go result, namely the impossibility of cancelling some linear contribution in the gravitino field. This drawback seems to seriously undermine the construction of a supergravity based on NGT. 
  One of the central points of the ekpyrotic cosmological scenario based on Horava-Witten theory is that we live on a negative tension brane. However, the tension of the visible brane is positive in the usual HW phenomenology with stronger coupling on the hidden brane, both for standard and non-standard embedding. To make ekpyrotic scenario realistic one must solve the problem of the negative cosmological constant on the visible brane and fine-tune the bulk brane potential with an accuracy of $10^{-50}$. In terms of a canonically normalized scalar field $\phi$ describing the position of the brane, this potential must take a very unusual form $V(\phi)\sim e^{-{5000 \phi\over M_p}}$. We describe the problems which appear when one attempts to obtain this potential in string theory. The mechanism for the generation of density perturbations in this scenario is not brane-specific; it is a particular limiting case of the mechanism of tachyonic preheating. Unlike inflation, this mechanism exponentially amplifies not only quantum fluctuations, but also initial inhomogeneities. As a result, to solve the homogeneity problem in this scenario, one would need the branes to be parallel to each other with an accuracy better than $10^{-60}$ on a scale $10^{30}$ times greater than the distance between the branes. Thus, at present, inflation remains the only robust mechanism that produces density perturbations with a flat spectrum and simultaneously solves all major cosmological problems. 
  We formulate incomplete classical statistics for situations where the knowledge about the probability distribution outside a local region is limited. The information needed to compute expectation values of local observables can be collected in a quantum mechanical state vector, whereas further statistical information about the probability distribution outside the local region becomes irrelevant. The translation of the available information between neighboring local regions is expressed by a Hamilton operator. A quantum mechanical operator can be associated to each local observable, such that expectation values of ``classical'' observables can be computed by the usual quantum mechanical rules. The requirement that correlation functions should respect equivalence relations for local obeservables induces a non-commutative product in classical statistics, in complete correspondence to the quantum mechanical operator product. We also discuss the issue of interference and the complex structure of quantum mechanics within our classical statistical setting. 
  By using anholonomic frames in (pseudo) Riemannian spaces we define anisotropic extensions of Euclidean Taub-NUT spaces. With respect to coordinate frames such spaces are described by off-diagonal metrics which could be diagonalized by corresponding anholonomic transforms. We define the conditions when the 5D vacuum Einstein equations have as solutions anisotropic Taub-NUT spaces. The generalized Killing equations for the configuration space of anisotropically spinning particles (anisotropic spinning space) are analyzed. Simple solutions of the homogeneous part of these equations are expressed in terms of some anisotropically modified Killing-Yano tensors. The general results are applied to the case of the four-dimensional locally anisotropic Taub-NUT manifold with Euclidean signature. We emphasize that all constructions are for(pseudo) Riemannian spaces defined by vacuum soltions, with generic anisotropy, of 5D Einstein equations, the solutions being generated by applying the moving frame method. 
  We consider a pair of noncommutative lumps in the noncommutative Yang--Mills/M(atrix) model. In the case when the lumps are separated by a finite distance their ``polarisations'' do not belong to orthogonal subspaces of the Hilbert space. In this case the interaction between lumps is nontrivial. We analyse the dynamics arisen due to this interaction in both naive approach of rigid lumps and exactly as described by the underlying gauge model. It appears that the exact description is given in terms of finite matrix models/multidimensional mechanics whose dimensionality depends on the initial conditions. 
  We review some aspects of the correspondence between analytic gauge invariants and supersymmetric flat directions for vanishing D-terms and propose a criterion to include the F-term constraints. 
  Starting from the generic harmonic superspace action of the quaternion-K\"ahler sigma models and using the quotient approach we present, in an explicit form, a quaternion-K\"ahler extension of the double Taub-NUT metric. It possesses $U(1)\times U(1)$ isometry and supplies a new example of non-homogeneous Einstein metric with self-dual Weyl tensor. 
  We describe an algebraic framework for studying the symmetry properties of integrable quantum systems on the half line. The approach is based on the introduction of boundary operators. It turns out that these operators both encode the boundary conditions and generate integrals of motion. We use this direct relationship between boundary conditions and symmetry content to establish the spontaneous breakdown of some internal symmetries, due to the boundary. 
  It is shown how to map the quantum states of a system of free Bose particles one-to-one onto the states of a completely deterministic model. It is a classical field theory with a large (global) gauge group. 
  We argue that eleven dimensional supergravity can be described by a non-linear realisation based on the group E_{11}. This requires a formulation of eleven dimensional supergravity in which the gravitational degrees of freedom are described by two fields which are related by duality. We show the existence of such a description of gravity. 
  We study some wrapped configurations of branes in the near-horizon geometry of a stack of other branes. The common feature of all the cases analyzed is a quantization rule and the appearance of a finite number of static configurations in which the branes are partially wrapped on spheres. The energy of these configurations can be given in closed form and the analysis of their small oscillations shows that they are stable. The cases studied include D(8-p)-branes in the type II supergravity background of Dp-branes (for p less or equal than 5), M5-branes in the M5-brane geometry in M-theory and D3-branes in a (p,q) fivebrane background in the type IIB theory. The brane configurations found admit the interpretation of bound states of strings (or M2-branes in M-theory) which extend along the unwrapped directions. We check this fact directly in a particular case by using the Myers polarization mechanism. 
  We consider the case of an integrable quantum spin chain with ``soliton non-preserving'' boundary conditions. This is the first time that such boundary conditions have been considered in the spin chain framework. We construct the transfer matrix of the model, we study its symmetry and we find explicit expressions for its eigenvalues. Moreover, we derive a new set of Bethe ansatz equations by means of the analytical Bethe ansatz method. 
  We study $Q$-ball type solitons in arbitrary spatial dimensions in the setting recently described by Kusenko, where the scalar field potential has a flat direction which rises much slower than $\phi^2$. We find that the general formula for energy as a function of the charge is, $E_d\sim Q_d^{(d/d+1)}$ in spatial dimension $d$. We show that the Hamiltonian governing the stability analysis of certain $Q$-wall configurations, which are one dimensional $Q$-ball solutions extended to planar, wall-like configurations in three dimensions, can be related to supersymmetric quantum mechanics. $Q$-wall and $Q$-string (the corresponding extensions of 2 dimensional $Q$-balls in 3 spatial dimensions) configurations are seen to be unstable, and will tend to bead and form planar or linear arrays of 3 dimensional $Q$-balls. The lifetime of these wall-like and string-like configurations is, however, arbitrarily large and hence they could be of relevance to cosmological density fluctuations and structure formation in the early Universe. 
  In cases of both abelian and nonabelian gauge groups, we consider the Higgs mechanism in topologically massive gauge theories in an arbitrary space-time dimension. It is shown that the presence of a topological term makes it possible to shift mass of gauge fields in a nontrivial way compared to the conventional value at the classical tree level. We correct the previous misleading statement with respect to the counting of physical degrees of freedom, where it is shown that gauge fields become massive by 'eating' the Nambu-Goldstone boson and a higher-rank tensor field, but a new massless scalar appears in the spectrum so the number of the physical degrees of freedom remains unchanged before and after the spontaneous symmetry breakdown. Some related phenomenological implications and applications to superstring theory are briefly commented. 
  We present in this short note an idea about a possible extension of the standard noncommutative algebra to the formal differential operators framework. In this sense, we develop an analysis and derive an extended noncommutative structure given by $[x_{a}, x_{b}]_{\star} = i(\theta + \chi)_{ab}$ where $\theta_{ab}$ is the standard noncommutative parameter and $\chi_{ab}(x)\equiv \chi^{\mu}_{ab}(x)\partial_{\mu} ={1/2}(x_a \theta^{\mu}_{b} - x_b \theta^{\mu}_{a})\partial_\mu$ is an antisymmetric non-constant vector-field shown to play the role of the extended deformation parameter. This idea was motivated by the importance of noncommutative geometry framework, with nonconstant deformation parameter, in the current subject of string theory and D-brane physics. 
  In this paper we study in detail the equivalence of the recently introduced Born-Infeld self dual model to the Abelian Born-Infeld-Chern-Simons model in 2+1 dimensions. We first apply the improved Batalin, Fradkin and Tyutin scheme, to embed the Born-Infeld Self dual model to a gauge system and show that the embedded model is equivalent to Abelian Born-Infeld-Chern-Simons theory. Next, using Buscher's duality procedure, we demonstrate this equivalence in a covariant Lagrangian formulation and also derive the mapping between the n-point correlators of the (dual) field strength in Born-Infeld Chern-Simons theory and of basic field in Born-Infeld Self dual model. Using this equivalence, the bosonization of a massive Dirac theory with a non-polynomial Thirring type current-current coupling, to leading order in (inverse) fermion mass is also discussed. We also re-derive it using a master Lagrangian. Finally, the operator equivalence between the fermionic current and (dual) field strength of Born-Infeld Chern-Simons theory is deduced at the level of correlators and using this the current-current commutators are obtained. 
  This paper aims to present explicit photon-like (3+1) spatially finite soliton solutions of screw type to the vacuum field equations of Extended Electrodynamics (EED) in relativistic formulation. We begin with emphasizing the need for spatially finite soliton modelling of microobjects. Then we briefly comment the properties of solitons and photons and recall some facts from EED. Making use of the localizing functions from differential topology (used in the partition of unity) we explicitly construct spatially finite screw solutions. Further a new description of the spin momentum inside EED, based on the notion for energy-momentum exchange between $F$ and $*F$, isintroduced and used to compute the integral spin momentum of a screw soliton. The consistency between the spatial and time periodicity naturally leads to a particular relation between the longitudinal and transverse sizes of the screw solution, namely, it is equal to $\pi$. The Planck's formula $E=h\nu$ in the form of $ET=h$ arizes as a measure of the integral spin momentum. 
  We study a special combination of the compact directions of the spacetime (compactified on tori) and the worldsheet parity transformed of these directions. The transformations that change the compact part of the spacetime to this combination, are more general than the T-duality transformations. Many properties of this combination and also of the corresponding worldsheet fermions are studied. By using the boundary state formalism, we study the effects of the above transformations to D-branes. For the special cases the resulted branes reduce to the known mixed branes, or reduce to the modified mixed branes. 
  We construct exact solitons on noncommutative tori for the type of actions arising from open string field theory. Given any projector that describes an extremum of the tachyon potential, we interpret the remaining gauge degrees of freedom as a gauge theory on the projective module determined by the tachyon. Whenever this module admits a constant curvature connection, it solves exactly the equations of motion of the effective string field theory. We describe in detail such a construction on the noncommutative tori. Whereas our exact solution relies on the coupling to a gauge theory, we comment on the construction of approximate solutions in the absence of gauge fields. 
  We show that the renormalized U(N) noncommutative Chern-Simons theory can be defined in perturbation theory so that there are no loop corrections to the 1PI functional of the theory in an arbitrary homogeneous axial (time-like, light-like or space-like) gauge. We define the free propagators of the fields of the theory by using the Leibbrandt-Mandelstam prescription --which allows Wick rotation and is consistent with power-counting-- and regularize its Green functions with the help of a family of regulators which explicitly preserve the infinitesimal vector Grassmann symmetry of the theory. We also show that in perturbation theory the nonvanishing Green functions of the elementary fields of the theory are products of the free propagators. 
  We show that reductions of KP hierarchies related to the loop algebra of $SL_n$ with homogeneous gradation give solutions of the Darboux-Egoroff system of PDE's. Using explicit dressing matrices of the Riemann-Hilbert problem generalized to include a set of commuting additional symmetries, we construct solutions of the Witten--Dijkgraaf--E. Verlinde--H. Verlinde equations. 
  We examine the cosmological implications of space-time non-commutativity, discovering yet another realization of the varying speed of light model.   Our starting point is the well-known fact that non-commutativity leads to deformed dispersion relations, relating energy and momentum, implying a frequency dependent speed of light. A Hot Big Bang Universe therefore experiences a higher speed of light as it gets hotter. We study the statistical physics of this "deformed radiation", recovering standard results at low temperatures, but a number of novelties at high temperatures: a deformed Planck's spectrum, a temperature dependent equation of state $w=p/\rho$ (ranging from 1/3 to infinity), a new Stephan-Boltzmann law, and a new entropy relation. These new photon properties closely mimic those of phonons in crystals, hardly a surprising analogy. They combine to solve the horizon and flatness problems, explaining also the large entropy of the Universe. We also show how one would find a direct imprint of non-commutativity in the spectrum of a cosmic graviton background, should it ever be detected. 
  We calculate one- and two-point correlators of winding operators in the matrix model of 2D string theory compactified on a circle, recently proposed for the description of string dynamics on the 2D black hole background. 
  We study the non-commutative instanton solution proposed in hep-th/0009142 and obtain the spectrum of small oscillations. The spectrum thus obtained is in exact agreement with the spectrum of stringy excitations in a configuration of point like D0 branes sitting on top of D4-branes with a uniform magnetic field turned on in the world-volume of the D4-branes in the Seiberg-Witten decoupling limit. This provides further evidence for the solution of hep-th/0009142 and also enables us recover the ADHM data from the 0-4 string spectrum. Generalizations to higher co-dimension solitons are also discussed. 
  Real-time anomalous fermion number violation is investigated for massless chiral fermions in spherically symmetric SU(2) Yang-Mills gauge field backgrounds which can be weakly dissipative or even nondissipative. Restricting consideration to spherically symmetric fermion fields, the zero-eigenvalue equation of the time-dependent effective Dirac Hamiltonian is studied in detail. For generic spherically symmetric SU(2) gauge fields in Minkowski spacetime, a relation is presented between the spectral flow and two characteristics of the background gauge field. These characteristics are the well-known ``winding factor,'' which is defined to be the change of the Chern-Simons number of the associated vacuum sector of the background gauge field, and a new ``twist factor,'' which can be obtained from the zero-eigenvalue equation of the effective Dirac Hamiltonian but is entirely determined by the background gauge field. For a particular class of (weakly dissipative) Luscher-Schechter gauge field solutions, the level crossings are calculated directly and nontrivial contributions to the spectral flow from both the winding factor and the twist factor are observed. The general result for the spectral flow may be relevant to electroweak baryon number violation in the early universe. 
  We show that the photon self-energy in quantum electrodynamics on noncommutative $\mathbb{R}^4$ is renormalizable to all orders (both in $\theta$ and $\hbar$) when using the Seiberg-Witten map. This is due to the enormous freedom in the Seiberg-Witten map which represents field redefinitions and generates all those gauge invariant terms in the $\theta$-deformed classical action which are necessary to compensate the divergences coming from loop integrations. 
  We study the equivalence between a nonlinear self-dual model (NSD) with the Born-Infeld-Chern-Simons (BICS) models using an iterative gauge embedding procedure that produces the duality mapping, including the case where the NSD model is minimally coupled to dynamical, U(1) charged fermionic matter. The duality mapping introduces a current-current interaction term while at the same time the minimal coupling of the original nonlinear self-dual model is replaced by a non-minimal magnetic like coupling in the BICS side. 
  We study quantum corrections at the one loop level in open superstring tachyon condensation using the boundary string field theory (BSFT) method. We find that the tachyon field action has the same form as at the disc level, but with a renormalized effective coupling $\lambda' =g_s^{ren}e^{-T^2/4}$ ($g_s^{ren}$ is the renormalized dimensionless closed string coupling, $e^{-T^2/4}$ is the tachyonic field expectation value) and with an effective string tension. This result is in agreement with that based on general analysis of loop effects. 
  Building on earlier work, we construct linear sigma models for strings on curved spaces in the presence of branes. Our models include an extremely general class of brane-worldvolume gauge field configurations. We explain in an accessible manner the mathematical ideas which suggest appropriate worldsheet interactions for generating a given open string background. This construction provides an explanation for the appearance of the derived category in D-brane physics complementary to that of recent work of Douglas. 
  The stability analysis of an anisotropic inflationary universe of the four dimensional Neveu-Schwarz--Neveu-Schwarz string model with a nonvanishing cosmological constant is discussed in this paper. The accelerating expansion solution found earlier is shown to be stable against the perturbations with respect to the dilaton and axion fields once the dilaton field falls close to the local minimum of the symmetry-breaking potential. This indicates that the Bianchi I space tends to evolve to an isotropic flat Friedmann-Robertson-Walker space. This expanding solution is also shown to be stable against the perturbation with respect to anisotropic spatial directions. 
  We show that a string-inspired Planck scale modification of general relativity can have observable cosmological effects. Specifically, we present a complete analysis of the inflationary perturbation spectrum produced by a phenomenological Lagrangian that has a standard form on large scales but incorporates a string-inspired short distance cutoff, and find a deviation from the standard result. We use the de Sitter calculation as the basis of a qualitative analysis of other inflationary backgrounds, arguing that in these cases the cutoff could have a more pronounced effect, changing the shape of the spectrum. Moreover, the computational approach developed here can be used to provide unambiguous calculations of the perturbation spectrum in other heuristic models that modify trans-Planckian physics and thereby determine their impact on the inflationary perturbation spectrum. Finally, we argue that this model may provide an exception to constraints, recently proposed by Tanaka and Starobinsky, on the ability of Planck-scale physics to modify the cosmological spectrum. 
  We discuss the form of the string-loop-corrected effective action and the loop-corrected solutions of the equations of motion. At the string-tree level, a solution we consider is the extremal magnetic black hole, in which case the tree-level effective gauge couplings decrease at small r, and in this region string-loop corrections to the gauge couplings become important. The effective 4D theory is the N=2 supergravity interacting with matter. Using the N=2 structure of the theory, we calculate the loop corrections to the effective action and solve the loop-corrected equations of motion. In the resulting perturbative solution for the metric, singularity at the origin is smeared by quantum effects. 
  We present a covariant and supersymmetric theory of relativistic hydrodynamics in four-dimensional Minkowski space. 
  We investigate the local geometry on the moduli space of G_2 structures that arises in compactifications of M-theory on holonomy G_2 manifolds. In particular, we determine the homogeneity properties of couplings of the associated N=1, D=4 supergravity under the scaling of moduli space coordinates. We then find some brane solitons of N=1, D=4 supergravity that are associated with wrapping M-branes on cycles of the compact space. These include cosmic strings and domain walls that preserve half of supersymmetry of the four-dimensional theory, and non-supersymmetric electrically and magnetically charged black holes. The geometry of some of the black holes is that of non-extreme M-brane configurations reduced to four-dimensions on a seven torus. 
  We study the noncommutative $\phi^4$ theory with spontaneously broken global O(2) symmetry in 4 dimensions. We demonstrate the renormalizability at one loop. This does not require any choice of ordering of the fields in the interaction terms. It involves regulating the ultraviolet and infrared divergences in a manner consistent with the Ward identities. 
  We consider the Hamiltonian and Lagrangian embedding of a first-order, massive spin-one, gauge non-invariant theory involving anti-symmetric tensor field. We apply the BFV-BRST generalised canonical approach to convert the model to a first class system and construct nil-potent BFV-BRST charge and an unitarising Hamiltonian. The canonical analysis of the St\"uckelberg formulation of this model is presented. We bring out the contrasting feature in the constraint structure, specifically with respect to the reducibility aspect, of the Hamiltonian and the Lagrangian embedded model. We show that to obtain manifestly covariant St\"uckelberg Lagrangian from the BFV embedded Hamiltonian, phase space has to be further enlarged and show how the reducible gauge structure emerges in the embedded model. 
  We compute the metric associated to noncommutative spaces described by a tensor product of spectral triples. Well known results of the two-sheets model (distance on a sheet, distance between the sheets) are extended to any product of two spectral triples. The distance between different points on different fibres is investigated. When one of the triple describes a manifold, one find a Pythagorean theorem as soon as the direct sum of the internal states (viewed as projections) commutes with the internal Dirac operator. Scalar fluctuations yield a discrete Kaluza-Klein model in which the extra metric component is given by the internal part of the geometry. In the standard model, this extra component comes from the Higgs field. 
  We discuss the path integral representation for the fermionic particles and strings and concentrate at the problems arising when some target-space dimensions are compact. An example of partition function for fermionic particle at finite temperature or with one compact target-space dimension is considered in detail. It is demonstrated that the first-quantized path integral requires, in general, presence of nonvanishing "Wilson loops" and modulo some common problems for real fermions in Grassmannian formulation one can try to reinterpret them in terms of condensates of the world-line fermions. The properties of corresponding path integrals in string theory are also discussed. 
  We first derive all world-sheet action functionals for NSR superstring models with (1,1) supersymmetry and any number of abelian gauge fields, for gauge transformations of the standard form. Then we prove for these models that the BRST cohomology groups $H^g(s)$, $g<4$ (with the antifields taken into account) are isomorphic to those of the corresponding bosonic string models, whose cohomology is fully known. This implies that the nontrivial global symmetries, Noether currents, background charges, consistent deformations and candidate gauge anomalies of an NSR (1,1) superstring model are in one-to-one correspondence with their bosonic counterparts. 
  If we assume that the initial conditions for the universe were such that there was no volume-extensive entropy `at the beginning of time' (which is true in Linde's chaotic inflation), we can formulate a covariant holographic bound on the entanglement entropy inside or outside closed space-like surfaces. This bound should hold even for regions where the coarse-grained entropy exceeds the surface area. We find that Bousso's bound gives strong support for this conjecture. We also present a speculative interpretation of the entropy bound, according to which any observer of interest can be surrounded by a holographic screen, providing a non-redundant description of the rest of the universe. 
  We study the discrete light-cone quantization (DLCQ) of closed strings in the background of Minkowski space-time and a constant Neveu-Schwarz $B$-field. For the Bosonic string, we identify the $B$-dependent part of the thermodynamic free energy to all orders in string perturbation theory. For every genus, $B$ appears in a constraint in the path integral which restricts the world-sheet geometries to those which are branched covers of a certain torus. This is the extension of a previous result where the $B$-field was absent \cite{Grignani:2000zm}. We then discuss the coupling of a $B$-field to the Matrix model of M-theory. We show that, when we consider this theory at finite temperature and in a finite $B$-field, the Matrix variables are functions which live on a torus with the same Teichm\"uller parameter as the one that we identified in string theory. We show explicitly that the thermodynamic partition function of the Matrix string model in the limit of free strings reproduces the genus 1 thermodynamic partition function of type IIA string. This is strong evidence that the Matrix model can reproduce perturbative string theory. We also find an interesting behavior of the Hagedorn temperature. 
  We construct the Weyl multiplets of N=2 conformal supergravity in five dimensions. We show that there exist two different versions of the Weyl multiplet, which contain the same gauge fields but differ in the matter field content: the Standard Weyl multiplet and the Dilaton Weyl multiplet. At the linearized level we obtain the transformation rules for the Dilaton Weyl multiplet by coupling it to the multiplet of currents corresponding to an on-shell vector multiplet. We construct the full non-linear transformation rules for both multiplets by gauging the D=5 superconformal algebra F^2(4). We show that the Dilaton Weyl multiplet can also be obtained by solving the equations of motion for an improved vector multiplet coupled to the Standard Weyl multiplet. 
  We analyze the superfield equations of the 4-dimensional N=4 SYM-theory using light-cone gauge conditions and the harmonic-superspace approach. The harmonic superfield equations of motion are drastically simplified in this gauge, in particular, the basic harmonic-superfield matrices and the corresponding harmonic analytic gauge connections become nilpotent on-shell. 
  The non-abelian version of the self-dual model proposed by Townsend, Pilch and van Nieuwenhuizen presents some well known difficulties not found in the abelian case, such as well defined duality operation leading to self-duality and dual equivalence with the Yang-Mills-Chern-Simons theory, for the full range of the coupling constant. These questions are tackled in this work using a distinct gauge lifting technique that is alternative to the master action approach first proposed by Deser and Jackiw. The master action, which has proved useful in exhibiting the dual equivalence between theories in diverse dimensions, runs into trouble when dealing with the non-abelian case apart from the weak coupling regime. This new dualization technique on the other hand, is insensitive of the non-abelian character of the theory and generalize straightforwardly from the abelian case. It also leads, in a simple manner, to the dual equivalence for the case of couplings with dynamical fermionic matter fields. As an application, we discuss the consequences of this dual equivalence in the context of 3D non-abelian bosonization. 
  One discusses the issue of low-energy electron-electron bound states in the Maxwell-Chern-Simons model coupled to QED3 with spontaneous breaking of a local U(1)-symmetry. The scattering potential, in the non-relativistic limit, steaming from the electron-electron Moller scattering, mediated by the Maxwell-Chern-Simons-Proca gauge field and the Higgs scalar, might be attractive by fine-tuning properly the physical parameters of the model. 
  I present a twistor action functional for null 2-surfaces (null strings) in 4D Minkowski spacetime. The proposed formulation is reparametrization invariant and free of algebraic and differential constraints. Proposed approach results in derivation of evolution equations for the null strings. It is shown that non-geodesic null strings are contained in the presented formalism. A discussion of the problem of minimality for 2-surfaces with degenerate induced metric is given. I also speculate on the possible description of strings (time-like 2-surfaces) and conventional (space-like) 2-surfaces. 
  We consider the possibility of localizing gravity on a Nielsen-Olesen vortex in the context of the Abelian Higgs model. The vortex lives in a six-dimensional space-time with negative bulk cosmological constant. In this model we find a region of the parameter space leading, simultaneously, to warped compactification and to regular space-time geometry. A thin defect limit is studied. Regular solutions describing warped compactifications in the case of higher winding number are also presented. 
  The appearances of complex eigenvalues in the spectra of PT-symmetric quantum-mechanical systems are usually associated with a spontaneous breaking of PT. In this letter we discuss a family of models for which this phenomenon is also linked with an explicit breaking of supersymmetry. Exact level-crossings are located, and connections with N-fold supersymmetry and quasi-exact solvability in certain special cases are pointed out. 
  It has been recently discovered in the context of the six vertex or XXZ model in the fundamental representation that new symmetries arise when the anisotropy parameter $(q+q^{-1})/2$ is evaluated at roots of unity $q^{N}=1$. These new symmetries have been linked to an $U(A^{(1)}_1)$ invariance of the transfer matrix and the corresponding spin-chain Hamiltonian.In this paper these results are generalized for odd primitive roots of unity to all vertex models associated with trigonometric solutions of the Yang-Baxter equation by invoking representation independent methods which only take the algebraic structure of the underlying quantum groups $U_q(\hat g)$ into account. Here $\hat g$ is an arbitrary Kac-Moody algebra. Employing the notion of the boost operator it is then found that the Hamiltonian and the transfer matrix of the integrable model are invariant under the action of $U(\hat{g})$. For the simplest case $\hat g=A_1^{(1)}$ the discussion is also extended to even primitive roots of unity. 
  The article is a natural continuation of the papers by Gavrilov and Gitman (Class.Quant.Grav. {\bf 17} (2000) L133; Int. J. Mod. Phys. A15 (2000) 4499) devoted to relativistic particle quantization. Here we generalize the problem, considering the quantization of a spinning particle in arbitrary gravitational background. The nontriviality of such a generalization is related to the neccessity of solving complicated ordering problems. Similar to the flat space-time case, we show in the course of the canonical quantization how a consistent relativistic quantum mechanics of spinning particle in gravitational and electromagnetic backgrounds can be constructed. 
  A five dimensional rotating black string in a Randall-Sundrum brane world is considered. The black string intercepts the three brane in a four dimensional rotating black hole. The geodesic equations and the asymptotics in this background are discussed. 
  The S-matrix is invariant with respect to the variation of any (global) parameter involved in the gauge fixing conditions, if that variation is accompanied by a certain redefinition of the basis of polarization vectors. Renormalizability of the underlying gauge theory is not required. The proof is nonperturbative and, using the `extended' BRS transformation, quite simple. 
  We consider M-theory on compact spaces of G_2 holonomy constructed as orbifolds of the form (CY x S^1)/Z_2 with fixed point set \Sigma on the CY. This describes N=1 SU(2) gauge theories with b_1(\Sigma) chiral multiplets in the adjoint. For b_1=0, it generalizes to compact manifolds the study of the phase transition from the non-Abelian to the confining phase through geometrical S^3 flops. For b_1=1, the non-Abelian and Coulomb phases are realized, where the latter arises by desingularization of the fixed point set, while an S^2 x S^1 flop occurs. In addition, an extremal transition between G_2 spaces can take place at conifold points of the CY moduli space where unoriented membranes wrapped on CP^1 and RP^2 become massless. 
  We argue that the holographic dual to little string theories at finite temperature suffers from a Gregory-Laflamme like instability, providing an alternative explanation to the results of hep-th/0012258. 
  We construct boundary conditions in the gauged linear sigma model for B-type D-branes on Calabi-Yau manifolds that correspond to coherent sheaves given by the cohomology of a monad. This necessarily involves the introduction of boundary fields, and in particular, boundary fermions. The large-volume monodromy for these D-brane configurations is implemented by the introduction of boundary contact terms. We also discuss the construction of D-branes associated to coherent sheaves that are the cohomology of complexes of arbitrary length. We illustrate the construction using examples, specifically those associated with the large-volume analogues of the Recknagel-Schomerus states with no moduli. Using some of these examples we also construct D-brane states that arise as bound states of the above rigid configurations and show how moduli can be counted in these cases. 
  An extension of the fundamental laws of thermodynamics and of the concept of entropy to the ground state fluctuations of the quantum fields is studied and some new results are found. At the end a device to extract energy from the vacuum recently proposed by an author is critically analyzed. It is found that no energy can be extracted cyclically from the vacuum. 
  The vacuum energy density or free energy of a free charged Bose gas at non-zero densities is studied in the context of the debate about Multiplicative Anomalies. Some zeta-function regularised calculations of the free energy in the literature are reexamined, clarified and extended. A range of apparently distinct answers can obtained. Equivalent dimensional regularisation results are also presented for comparison. I conclude that operator ordering and normal ordering are not responsible for these differences. Rather it is an undesirable but unavoidable property of zeta-function regularisation which leads to these different results, making it a bad scheme in general. By comparison I show how dimensional regularisation calculations give a consistent result without any complications, making this a good scheme in this context. 
  A geometrical interpretation of Grassmannian anticommuting coordinates is given. They are taken to represent an indefiniteness inherent in every spacetime point on the level of the spacetime foam. This indeterminacy is connected with the fact that in quantum gravity in some approximation we do not know the following information : are two points connected by a quantum wormhole or not ? It is shown that: (a) such indefiniteness can be represented by Grassmanian numbers, (b) a displacement of the wormhole mouth is connected with a change of the Grassmanian numbers (coordinates). In such an interpretation of supersymmetry the corresponding supersymmetrical fields must be described in an invariant manner on the background of the spacetime foam. 
  We present a full superconformal tensor calculus in five spacetime dimensions in which the Weyl multiplet has 32 Bose plus 32 Fermi degrees of freedom. It is derived by the dimensional reduction from the 6D superconformal tensor calculus. We present two types of 32+32 Weyl multiplets, vector multiplet, linear multiplet, hypermultiplet and nonlinear multiplet. Their superconformal transformation laws and the embedding and invariant action formulas are given. 
  It has recently been argued that non--BPS brane world scenarios can reproduce the small value of the cosmological constant that seems to have been measured. Objections against this proposal are discussed and necessary (but not sufficient) conditions are stated under which it may work. At least n=2 extra dimensions are needed. Also, the mass matrix in the supergravity sector must satisfy $Str M^2=0$. Moreover, the proposal can be ruled out experimentally if Newton's constant remains unchanged down to scales of 10 micrometers. If, on the other hand, such a ``running Newton constant'' is observed, it could provide crucial experimental input for superstring phenomenology. 
  The strongly coupled limit of string scattering and the automorphic construction of the graviton S-matrix is compared with the eleven dimensional formulation of M-theory. In a particular scaling limit at strong string coupling, M-theory is described by eleven-dimensional supergravity which does not possess a dilaton, but rather a perturbative expansion in the gravitational coupling and derivatives. The latter theory provides an off-shell description of the string, upon dimensional reduction. 
  We extend our previous work on the quasi-particle excitations in N=4 non-commutative U(1) Yang-Mills theory at finite temperature. We show that above some critical temperature there is a tachyon in the spectrum of excitations. It is a collective transverse photon mode polarized in the non-commutative plane. Thus the theory seems to undergo a phase transition at high temperature. Furthermore we find that the group velocity of quasi-particles generically exceeds the speed of light at low momentum. 
  We conjecture that the extra dimensions are physical non-compact at high energy scale or high temperature; after the symmetry breaking or cosmological phase transition, the bulk cosmological constant may become negative, and then, the extra dimensions may become physical compact at low energy scale. We show this in a five-dimensional toy brane model with three parallel 3-branes and a real bulk scalar whose potential is temperature dependent. We also point out that after the global or gauge symmetry breaking, or the supersymmetry breaking in supergravity theory, the spontaneous physical compactification of the extra dimensions might be realized. 
  Phase-space and its relativistic extension is a natural space for realizing Sp(2,R) symmetry through canonical transformations. On a Dx2 dimensional covariant phase-space, we formulate noncommutative field theories, where Sp(2,R) plays a role as either a global or a gauge symmetry group. In both cases these field theories have potential applications, including certain aspects of string theories, M-theory, as well as quantum field theories. If interpreted as living in lower dimensions, these theories realize Poincare' symmetry linearly in a way consistent with causality and unitarity. In case Sp(2,R) is a gauge symmetry, we show that the spacetime signature is determined dynamically as (D-2,2). The resulting noncommutative Sp(2,R) gauge theory is proposed as a field theoretical formulation of two-time physics: classical field dynamics contains all known results of `two-time physics', including the reduction of physical spacetime from D to (D-2) dimensions, with the associated `holography' and `duality' properties. In particular, we show that the solution space of classical noncommutative field equations put all massless scalar, gauge, gravitational, and higher-spin fields in (D-2) dimensions on equal-footing, reminiscent of string excitations at zero and infinite tension limits. 
  A flux p-brane in D dimensions has (p+1)-dimensional Poincare invariance and a nonzero rank (D-p-1) field strength tangent to the transverse dimensions. We find a family of such solutions in string theory and M-theory and investigate their properties. 
  We study the properties of a non-abelian gauge theory subjected to a gauge invariant constraint given by the classical equations of motion. The constraint is not imposed by hand, but appears naturally when we study a particular type of local gauge transformations. In this way, all standard techniques to treat gauge theories are available. We will show that this theory lives at one-loop. Also this model retains some quantum characteristic of the usual non-abelian gauge theories as asymptotic freedom. 
  Trying to interpret recent matrix model results (hep-th/0101011) we discuss computation of classical free energy of exact dilatonic 2-d black hole from the effective action of string theory. The euclidean space-time action evaluated on the black hole background is divergent due to linear dilaton vacuum contribution, and its finite part depends on a subtraction procedure. The thermodynamic approach based on subtracting the vacuum contribution for fixed values of temperature and dilaton charge at the "wall" gives (as in the leading-order black hole case) S= M/T for the entropy and zero value for the free energy F. We suggest that in order to establish a correspondence with a non-vanishing matrix model result for F one may need an alternative reparametrization-invariant subtraction procedure using analogy with non-critical string theory (i.e. replacing the spatial coordinate by the dilaton field). The subtraction of the dilaton divergence then produces a finite value for the free energy. We also propose a microscopic estimate for the entropy and energy of the black hole based on the contribution of non-singlet states of the matrix model. 
  We propose the leading couplings, in an alpha' expansion, of noncommutative D-branes to RR potentials in a constant NSNS B-field for an arbitrary choice of noncommutative parameter. The proposal is motivated by some string amplitude computations. The zero momentum couplings are topological in nature and involve Elliott's noncommutative Chern character. The finite momentum couplings are given by smearing the zero momentum operators along an open Wilson line. Comparisons between the RR couplings in different descriptions lead to a better understanding of the field redefinitions between gauge field variables (the Seiberg-Witten map) and help constrain alpha' corrections. In particular we recover the Seiberg-Witten map conjectured by one of the authors in hep-th/0011125. We also discuss the dynamics of the transverse scalar fields and find evidence for a new derivative-driven dielectric effect. 
  We study the Euler-Lagrange cohomology and explore the symplectic or multisymplectic geometry and their preserving properties in classical mechanism and classical field theory in Lagrangian and Hamiltonian formalism in each case respectively. By virtue of the Euler-Lagrange cohomology that is nontrivial in the configuration space, we show that the symplectic or multisymplectic geometry and related preserving property can be established not only in the solution space but also in the function space if and only if the relevant closed Euler-Lagrange cohomological condition is satisfied in each case. We also apply the cohomological approach directly to Hamiltonian-like ODEs and Hamiltonian-like PDEs no matter whether there exist known Lagrangian and/or Hamiltonian associated with them. 
  We argue that the thermodynamics of conformal field theories with AdS duals exhibits a remarkable universality. At strong coupling, a Cardy-Verlinde entropy formula holds even when R-charges or bulk supergravity scalars are turned on. In such a setting, the Casimir entropy can be identified with a generalized C-function that changes monotonically with temperature as well as when non-trivial bulk scalar fields are introduced. We generalize the Cardy-Verlinde formula to cases where no subextensive part of the energy is present and further observe that such a formula is valid for the N=4 super Yang-Mills theory in D=4 even at weak coupling. Finally we show that a generalized Cardy-Verlinde formula holds for asymptotically flat black holes in any dimension. 
  We consider the enhancement of SL(2,R) to Virasoro algebra in a system of N particles on AdS2. We restrict our discussion to the case of non-interacting particles, and argue that they must be treated as fermions. We find operators L_n whose commutators on the ground state, |vac>, satisfy relations that are reminisent of c=1 Virasoro algebra, provided N \geq n \geq -N. Same relations hold also on the states L_{-k}|vac>, if (N-k) \geq n \geq -(N-k). The conditions L_n^\dag = L_{-n}, and L_k|vac> = 0 for k \geq 1 are also satisfied. 
  We construct the exact noncommutative solutions on tori. This gives an exact description of tachyon condensation on bosonic D-branes, non-BPS D-branes and brane-antibrane systems. We obtain various bound states of D-branes after the tachyon condensation. Our results show that these solutions can be generated by applying the gauge Morita equivalence between the constant curvature projective modules. We argue that there is a general framework of the noncommutative geometry based on the notion of Morita equivalence which underlies this specific example. 
  Finite-dimensional representations of Onsager's algebra are characterized by the zeros of truncation polynomials. The Z_N-chiral Potts quantum chain hamiltonians (of which the Ising chain hamiltonian is the N=2 case) are the main known interesting representations of Onsager's algebra and the corresponding polynomials have been found by Baxter and Albertini, McCoy and Perk in 1987-89 considering the Yang-Baxter-integrable 2-dimensional chiral Potts model. We study the mathematical nature of these polynomials. We find that for N>2 and fixed charge Q these don't form classical orthogonal sets because their pure recursion relations have at least N+1-terms. However, several basic properties are very similar to those required for orthogonal polynomials. The N+1-term recursions are of the simplest type: like for the Chebyshev polynomials the coefficients are independent of the degree. We find a remarkable partial orthogonality, for N=3,5 with respect to Jacobi-, and for N=4,6 with respect to Chebyshev weight functions. The separation properties of the zeros known from orthogonal polynomials are violated only by the extreme zero at one end of the interval. 
  We effectively sew two vertices with ghosts in order to obtain a third, composite vertex in the most general case of cycling transformations. In order to do this, we separate the vertices into two parts: a bosonic oscillator part and a ghost oscillator part and write them as canonical forms. 
  In hep-th/9903210 (curvature)$^2$ terms of the effective D-brane action were derived to lowest order in the string coupling. Their results are correct up to ambiguous terms which involve the second fundamental form of the D-brane. We compute five point string amplitudes on the disk. We compare the subleading order in $\alpha'$ of the string amplitudes with the proposed lagrangian of hep-th/9903210 supplemented by the ambiguous terms. The comparison determines the complete form of the gravitational terms in the effective D-brane action to order ${\calO}(\alpha^{' 2})$. Our results are valid for arbitrary ambient geometries and world-volume embeddings. 
  We define a particular class of topological field theories associated to open strings and prove the resulting D-branes and open strings form the bounded derived category of coherent sheaves. This derivation is a variant of some ideas proposed recently by Douglas. We then argue that any 0-brane on any Calabi-Yau threefold must become unstable along some path in the Kahler moduli space. As a byproduct of this analysis we see how the derived category can be invariant under a birational transformation between Calabi-Yaus. 
  A quantitative prediction of Conformal Field Theory (CFT), which relates the second moment of the energy-density correlator away from criticality to the value of the central charge, is verified in the sine-Gordon model. By exploiting the boson-fermion duality of two-dimensional field theories, this result also allows to show the validity of the prediction in the strong coupling regime of the Thirring model. 
  We study the semiclassical partition function in the frame work of the Morse theory, to clarify the phase factor of the partition function and to relate it to the eta invariant of Atiyah. Converting physical system with potential into a curved manifold, we exploit the Jacobi fields and their corresponding eigenvalue equations to be associated with geodesics on the curved manifold and the Hamilton-Jacobi theory. 
  We obtain the localized gravity on the intersection of two orthogonal  non-solitonic or solitonic 4-branes in D=6 in the presence of the Gauss-Bonnet term. The tension of the intersection is allowed to exist unlike the case without the Gauss-Bonnet term. We show that gravity could be confined to the solitonic 4-branes for a particular choice of the Gauss-Bonnet coupling. If the extra dimensions are compactified with the $T^2/(Z_2\times Z_2)$ orbifold symmetry, the mass hierarchy between the Planck scale and the weak scale can be explained by putting our universe at the TeV intersection of positive tension located at the orbifold fixed point. 
  By the simple finite element method, we study the symplectic, multisymplectic structures and relevant preserving properties in some semi-linear elliptic boundary value problem in one-dimensional and two-dimensional spaces respectively. We find that with uniform mesh, the numerical schemes derived from finite element method can keep a preserved symplectic structure in one-dimensional case and a preserved multisymplectic structure in two-dimentional case respectively. These results are in fact the intrinsic reason that the numerical experiments indicate that such finite element schemes are accurate in practice. 
  We study the equivalence between the self-dual and the Maxwell-Chern-Simons (MCS) models coupled to dynamical, U(1) charged matter, both fermionic and bosonic. This is done through an iterative procedure of gauge embedding that produces the dual mapping of the self-dual vector field theory into a Maxwell-Chern-Simons version. In both cases, to establish this equivalence a current-current interaction term is needed to render the matter sector unchanged. Moreover, the minimal coupling of the original self-dual model is replaced by a non-minimal magnetic like coupling in the MCS side. Unlike the fermionic instance however, in the bosonic example the dual mapping proposed here leads to a Maxwell-Chern-Simons theory immersed in a field dependent medium. 
  We present a formalism to explicitly construct non-Abelian gauge theories on noncommutative spaces (induced via a star product with aconstant Poisson tensor) from a consistency relation. This results in an expansion of the gauge parameter, the noncommutative gauge potential and fields in the fundamental representation, in powers of a parameter of the noncommutativity. This allows the explicit construction of actions for these gauge theories. 
  The boundary conditions of the bosonic string theory in non-zero $B$-field background are equivalent to the second class constraints of a discretized version of the theory. By projecting the original canonical coordinates onto the constraint surface we derive a set of coordinates of string that are unconstrained.These coordinates represent a natural framework for the quantization of the theory. 
  We construct the so far unknown Lagrangian of D=6, N=2 F(4) Supergravity coupled to an arbitrary number of vector multiplets whose scalars span the coset manifold SO(4,n)/{SO(4) x SO(n)}.   This is done first in the ungauged case and then extended to the compact gauging of SU(2) x G, where SU(2) is the R-symmetry diagonal subgroup of SU(2)_L x SU(2)_R = SO(4) and G is a compact subgroup of SO(n), n being the number of vector multiplets, and such that dim G = n.   The knowledge of the Lagrangian allows in principle to refine the AdS_6/CFT_5 correspondence already discussed, as far as supersymmetric multiplets are concerned, in a previous related paper. With respect to the latter we also give a more exaustive treatment of the construction of the theory at the level of superspace Bianchi identities and in particular of the scalar potential. 
  We analyze the possibility of constructing supersymmetric curved domain wall solutions in five-dimensional ${\cal N}=2$ gauged supergravity, which are supported by non-constant scalar fields belonging either to vector multiplets only or to vector and hypermultiplets. We show that the BPS equations for the warp factor and for the vector scalars are modified by the presence of a four-dimensional cosmological constant on the domain wall, in agreement with earlier results by DeWolfe, Freedman, Gubser and Karch. We also show that the cosmological constant on the domain wall is anti-de Sitter like and that it constitutes an independent quantity, not related to any of the objects appearing in the context of very special geometry. 
  We investigate the power-law tails in the evolution of a charged massless scalar field around a fixed background of a dilaton black hole. Using both analytical and numerical methods we find the inverse power-law relaxation of charged fields at future timelike infinity, future null infinity, and along the outer horizon of the considered black hole. We invisage that a charged hair decays slower than neutral ones. The oscillatory inverse power law along the outer horizon of the dilaton black hole is of a great importance for a mass inflation scenario along the Cauchy horizon of a dynamically formed dilaton black hole. 
  We try to develop a coherent picture on Liouville theory as a two-dimensional conformal field theory that takes into account the perspectives of path-integral approach, bootstrap, canonical quantization and operator approach. To do this, we need to develop further some of these approaches. This includes in particular a construction of general exponential field operators from a set of covariant chiral operators. The latter are shown to satisfy braid relations that allow one to prove the locality of the former. 
  We study the holographic principle in the brane cosmology. Especially we describe how to accommodate the 5D anti de Sitter Schwarzschild (AdSS$_5$) black hole in the Binetruy-Deffayet-Langlois (BDL) approach of brane cosmology. It is easy to make a connection between a mass $M$ of the AdSS$_5$ black hole and a conformal field theory (CFT)-radiation dominated universe on the brane in the moving domain wall approach. But this is not established in the BDL approach. In this case we use two parameters $C_1, C_2$ in the Friedmann equation. These arise from integration and are really related to the choice of initial bulk matter. If one chooses a bulk energy density $\rho_B$ to account for a mass $M$ of the AdSS$_5$ black hole and the static fifth dimension, a CFT-radiation term with $\rho_{CFT} \sim M/a^{4}$ comes out from the bulk matter without introducing a localized matter distribution on the brane. This means that the holographic principle can be established in the BDL brane cosmology. 
  The Casimir stress on two parallel plates in de Sitter background for massless scalar field satisfying Robin boundary conditions on the plates is calculated. The metric is written in conformally flat form to make maximum use of the Minkowski space calculations. Different cosmological constants are assumed for the space between and outside of the plates to have general results applicable to the case of domain wall formations in the early universe. 
  We consider the Einstein-Yang-Mills-Higgs equations for an SU(3) gauge group in a spherically symmetric ansatz. Several properties of the gravitating monopole solutions are obtained an compared with their SU(2) counterpart. 
  We show that a suitable deformation of the algebra $h_k(1)$ of the creation and annihilation operators for a complex scalar field, initially quantized in Minkowski space--time, induces the canonical quantization of the same field in a generic gravitational background. This discloses the physical meaning of the deformation parameter $q$ which turns out to be related to the gravitational field. The thermal properties are re-obtained in this formalism, and the application to Schwarzschild and Rindler space-times are carried out. 
  We study string-loop corrections to magnetic black hole. Four-dimensional theory is obtained by compactification of the heterotic string theory on the manifold $K3\times T^2$ or on a suitable orbifold yielding N=1 supersymmetry in 6D. The resulting 4D theory has N=2 local supersymmetry. Prepotential of this theory receives only one-string-loop correction. The tree-level gauge couplings are proportional to the inverse effective string coupling and decrease at small distances from the center of magnetic black hole, so that loop corrections to the gauge couplings are important in this region. We solve the system of spinor Killing equations (conditions for the supersymmetry variations of the fermions to vanish) and Maxwell equations. At the string-tree level, we reproduce the magnetic black hole solution which can be also obtained by solving the system of the Einstein-Maxwell equations and the equations of motion for the moduli. String-loop corrections to the tree-level solution are calculated in the first order in string coupling. The resulting corrections to the metric and dilaton are large at small distances from the center of the black hole. Possible smearing of the singularity at the origin by quantum corrections is discussed. 
  We compute one-loop correction to the string field theory action of the tachyon for unstable D-branes in the framework of the boundary superstring field theory. We would expect that the one-loop correction comes from the partition function of the two-dimensional world-sheet theory on the annulus. The annulus correction suggests that the genus expansion is, somehow, governed by the effective string coupling defined in terms of the tachyon \lambda=g_s exp(-T^2/4). 
  A method is described for the development of the one-loop effective action expansion as an asymptotic series in inverse powers of the fermion mass. The method is based on the Schwinger-DeWitt proper-time technique, which allows for loop particles with non-degenerate masses. The case with SU(2)xSU(2) as the symmetry group is considered. The obtained novel series generalizes the well-known Schwinger-De Witt inverse mass expansion for equal masses, and is chiral invariant at each order. We calculate the asymptotic coefficients up to fifth order and clarify their relationship with the standard Seeley-DeWitt coefficients. 
  Stability analysis of the Bianchi type I universe in pure gravity theory is studied in details. We first derive the non-redundant field equation of the system by introducing the generalized Bianchi type I metric. This non-redundant equation reduces to the Friedmann equation in the isotropic limit. It is shown further that any unstable mode of the isotropic perturbation with respect to a de Sitter background is also unstable with respect to anisotropic perturbations. Implications to the choice of physical theories are discussed in details in this paper. 
  We analyze the unitarity of a non-relativistic non-commutative scalar field theory. We show that electric backgrounds spoil unitarity while magnetic ones do not. Furthermore, unlike its relativistic counterparts, unitarity can not be restored (at least at the level of one-to-one scattering amplitude) by adding new states to the theory. This is a signal that the model cannot be embedded in a natural way in string theory. 
  In this paper we make an SL(2,Z)-covariant generalisation of the noncommutative theories, NCYM and NCOS on the D3-brane, and NCOS on the D5-brane in type IIB. Usually, the noncommutative theories are obtained by studying perturbative F-string theory, and the parameters governing the noncommutative theories are given by the open string data. The S-duality of NCYM and NCOS on the D3-brane has been seen by dualising the background, keeping the F-string under study fixed. We give an SL(2,Z)-covariant generalisation of the open string data relevant when one instead studies perturbative (p,q)-string theory. The S-duality of NCYM and NCOS on the D3-brane is reproduced by instead keeping the background fixed and studying different (p,q)-string theories. We also obtain new noncommutative open (p,q)-string theories on the D3-brane and the D5-brane which are S-dual to ordinary NCOS. The theories are studied using the supergravity duals of the D3-brane and the D5-brane, corresponding to a probe brane in the relevant background. 
  We extend the study of the nature of the Hagedorn transition in NCOS systems in various dimensions. The canonical analysis results in a microscopic ionization picture of a bound state system in which the Hagedorn transition is postponed till irrelevancy. A microcanonical analysis leads to a limiting Hagedorn behaviour dominated by highly excited, long open strings. The study of the full phase diagram of the NCOS system using the AdS/CFT correspondence suggests that the microscopic ionization picture is the correct one. We discuss some refinements of the ionization mechanism for $d>2$ NCOS systems, including the formation of a temperature-dependent barrier for the process. Some possible consequences of this behaviour, including a potential puzzle for $d=5$, are discussed. Phase diagrams of a regularized form of NCOS systems are introduced and do accomodate a phase of long open strings which disappears in the strict NCOS limit. 
  Field theories in the presence of branes encounter localized divergences that renormalize brane couplings. The sources of these brane-localized divergences are understood as arising either from broken translation invariance, or from short distance singularities as the brane thickness vanishes. While the former are generated only by quantum corrections, the latter can appear even at the classical level. Using as an example six-dimensional scalar field theory in the background of a 3-brane, we show how to interpret such classical divergences by the usual regularization and renormalization procedure of quantum field theory. In our example, the zero thickness divergences are logarithmic, and lead classically to non-trivial renormalization group flows for the brane couplings. We construct the tree level renormalization group equations for these couplings as well as the one-loop corrections to these flows from bulk-to-brane renormalization effects. 
  Starting from the bosonic part of N=2 Super QCD with a 'Seiberg-Witten' N=2 breaking mass term, we obtain string BPS conditions for arbitrary semi-simple gauge groups. We show that the vacuum structure is compatible with a symmetry breaking scheme which allows the existence of Z_k-strings and which has Spin(10) -> SU(5) x Z_2 as a particular case. We obtain BPS Z_k-string solutions and show that they satisfy the same first order differential equations as the BPS string for the U(1) case. We also show that the string tension is constant, which may cause a confining potential between monopoles increasing linearly with their distance. 
  We construct a BPS-saturated representation of the six-dimensional (2, 0) algebra with a certain non-zero value of the `central' charge. This representation is naturally carried by strings with internal degrees of freedom rather than by point particles. Upon compactification on a circle, it reduces to a massive vector multiplet in five dimensions. We also construct quantum fields out of the creation and annihilation operators of the states of this representation, and show how they give rise to a conserved two-form current that can be coupled to a tensor multiplet. We hope that these results may be relevant for understanding the degrees of freedom associated with strings in interacting (2, 0) theories. 
  We describe the fate of the Type I non-BPS D7-brane, which is tachyonic but carries a non-trivial K-theory $\IZ_2$ charge. It decays to topologically non-trivial gauge field configurations on the background D9-branes. In the uncompactified theory the decay proceeds to infinity, while with a transverse torus the decay reaches a final state, a toron gauge configuration with vanishing Chern classes but non-trivial $\IZ_2$ charge. A similar behaviour is obtained for the type I non-BPS D8-brane, and other related systems. We construct explicit examples of type IIB orientifolds with non-BPS D7-branes, which are hence non-supersymmetric, but for which supersymmetry is restored upon condensation of the tachyon. We also report on the interesting structure of non-BPS states of type IIA theory in the presence of an O6-plane, their M-theory lifts, the relation between string theory K-theory and M-theory cohomology, and its interplay with NS-NS charged objects. We discuss several new effects, including: i) transmutation between NS-NS and RR torsion charges, ii) non-BPS states classified by K-theory but not by cohomology in string theory, but whose lift to M-theory is cohomological. 
  We show that for non-conformally flat bulk spacetime, there exist no bound modes for zero mass graviton on the 3-brane. The brane world model is therefore unstable for the bulk spacetime being different from the conformally flat anti - de Sitter space. 
  We show how the motion of a charged particle near the horizon of an extreme Reissner-Nordstrom black hole can lead to different forms of conformal mechanics, depending on the choice of the time coordinate. 
  In this paper we extend the boundary string field theory action for a non-BPS D-brane to the one including the target space fermions and the nonlinear supersymmetry with 32 supercharges up to some order. This is based on the idea that the vacuum with a non-BPS D-brane belongs to the spontaneously broken phase of the supersymmetry. As a result, we find that the action is almost uniquely determined up to the field redefinition ambiguities. 
  We consider a 6D space-time which is periodic in one of the extra dimensions and compact in the other. The periodic direction is defined by two 4-brane boundaries. Both static and non-static exact solutions, in which the internal spacetime has constant radius of curvature, are derived. In the case of static solutions, the brane tensions must be tuned as in the 5D Randall-Sundrum model, however, no additional fine-tuning is necessary between the brane tensions and the bulk cosmological constant. By further relaxing the sole fine-tuning of the model, we derive non-static solutions, describing de Sitter or Anti de Sitter 4D spacetimes, that allow for the fixing of the inter-brane distance and the accommodation of pairs of positive-negative and positive-positive tension branes. Finally, we consider the stability of the radion field in these configurations by employing small, time-dependent perturbations around the background solutions. In analogy with results drawn in 5 dimensions, the solutions describing a de Sitter 4D spacetime turn out to be unstable while those describing an Anti de Sitter geometry are shown to be stable. 
  We construct the manifestly Lorentz-invariant twistorial formulation of N=1 D=4 superparticle with tensorial central charges which describes massive and massless cases in a uniform manner. The tensorial central charges are realized in terms of even spinor variables and central charge coordinates. The full analysis of the number of conserved supersymmetries has been carried out. In the massive case the superparticle preserves 1/4 or 1/2 of target-space supersymmetries whereas the massless superparticle preserves two or three supersymmetries. 
  We construct the manifestly gauge invariant effective Lagrangian in 3+1 dimensions describing the Standard Model in 4+1 dimensions, following the transverse lattice technique. We incorporate split generation fermions and we explore naturalness for two Higgs configurations: a universal Higgs VEV, common to each transverse brane, and a local Higgs VEV centered on a single brane with discrete exponential attenuation to other branes, emulating the split-generation model. Extra dimensions, with explicit Higgs, do not ameliorate the naturalness problem. 
  We discuss the obstacles for defining a set of observable quantities analogous to an S-matrix which are needed to formulate string theory in an accelerating universe. We show that the quintessence models with the equations of state $-1 < w <-1/3$ have future horizons and may be no better suited to an S-matrix or S-vector description. We also show that in a class of theories with a stable supersymmetric vacuum, a system cannot relax into a zero-energy supersymmetric vacuum while accelerating if the evolution is dominated by a single scalar field with a stable potential. Thus describing an eternally accelerating universe may be a challenge for string theory as presently defined. 
  Recent astronomical observations indicate that the universe is accelerating. We argue that generic quintessence models that accommodate the present day acceleration tend to accelerate eternally. As a consequence the resulting spacetimes exhibit event horizons. Hence, quintessence poses the same problems for string theory as asymptotic de Sitter spaces. 
  The aim of this work is to show, on the example of the behaviour of the spinless charged particle in the homogeneous electric field, that one can quantized the velocity of particle by the special gauge fixation. The work gives also the some information about the theory of second quantisation in the space of Hilbert- Fock and the theory of projectors in the Hilbert space. One consider in Appendix the theory of the spinless charged particle in the homogeneous addiabatical changed electrical field. 
  We formulate the Chern-Simons action for any compact Lie group using Deligne cohomology. This action is defined as a certain function on the space of smooth maps from the underlying 3-manifold to the classifying space for principal bundles. If the 3-manifold is closed, the action is a function with values in complex numbers. If the 3-manifold is not closed, then the action is a section of a Hermitian line bundle associated with the Riemann surface which appears as the boundary. 
  We propose a class of N=2 supersymmetric nonlinear sigma models on the Ricci-flat Kahler manifolds with O(n) symmetry. 
  Cosmological perturbations in the brane-world cosmology with a positive tension brane in the AdS background bulk geometry is analyzed by using the doubly gauge-invariant formalism. We derive four independent equations for scalar perturbations in the plane symmetric (K=0) background. Three of these equations are differential equations written in terms of gauge invariant variables on the brane only, and another of them is an integro-differential equation whose kernel is constructed formally from the Z_2-symmetric retarded Green's function of the bulk gravitational waves. We compare these four equations with the corresponding equations in the standard cosmology. As a by-product, we also obtain a set of equations which may be useful in numerical calculations. 
  The fermionic measure in the functional integral of a gauge theory suffers from an ambiguity in the form of a chiral phase. By fixing it, one is led once again to the conclusion that a chiral phase in the quark mass term of QCD has no effect and cannot cause CP violation. 
  In this report I give the short historical review some of the first steps that were done to the invention of SUSY in Kharkov team headed by D.Volkov.   This paper is dedicated to the memory of Prof. Yu. Gol'fand, whose ideas of SUSY inspired the most active developments in High Energy Physics over thirty years. 
  We study the renormalizability of (massive) topological QCD based on the algebraic BRST technique by adopting a non-covariant Landau type gauge and making use of the full topological superalgebra. The most general local counter terms are determined and it is shown that in the presence of central charges the BRST cohomology remains trivial. By imposing an additional set of stability constraints it is proven that the matter action of topological QCD is perturbatively finite. 
  We show that the specific operators V^a appearing in the triplectic formalism can be viewed as the anti-Hamiltonian vector fields generated by a second rank irreducible Sp(2) tensor. This allows for an explicit realization of the triplectic algebra being constructed from an arbitrary Poisson bracket on the space of the fields only, equipped by the flat Poisson connection. We show that the whole space of fields and antifields can be equipped with an even supersymplectic structure when this Poisson bracket is non-degenerate. This observation opens the possibility to provide the BRST/antiBRST path integral by a well-defined integration measure, as well as to establish a direct link between the Sp(2) symmetric Lagrangian and Hamiltonian BRST quantization schemes. 
  We investigate the most general non(anti)commutative geometry in N=1 four-dimensional superspace, invariant under the classical (i.e., undeformed) supertranslation group. We find that a nontrivial non(anti)commutative superspace geometry compatible with supertranslations exists with non(anti)commutation parameters which may depend on the spinorial coordinates. The algebra is in general nonassociative. Imposing associativity introduces additional constraints which however allow for nontrivial commutation relations involving fermionic coordinates. We obtain explicitly the first three terms of a series expansion in the deformation parameter for a possible associative *-product. We also consider the case of N=2 euclidean superspace where the different conjugation relations among spinorial coordinates allow for a more general supergeometry. 
  The super Moyal-Lax representation and the super Moyal momentum algebra are introduced and the properties of simple and extended supersymmetric integrable models are systematically investigated. It is shown that, much like in the bosonic cases, the super Moyal-Lax equation can be interpreted as a Hamiltonian equation and can be derived from an action. Similarly, we show that the parameter of non-commutativity, in this case, is related to the central charge of the second Hamiltonian structure of the system. The super Moyal-Lax description allows us to go to the dispersionless limit of these models in a singular limit and we discuss some of the properties of such systems. 
  We extend the duality between massive and topologically massive antisymmetric tensor gauge theories in arbitrary space-time dimensions to include topological defects. We show explicitly that the condensation of these defects leads, in 4 dimensions, to confinement of electric strings in the two dual models. The dual phase, in which magnetic strings are confined is absent. The presence of the confinement phase explicitly found in the 4-dimensional case, is generalized, using duality arguments, to arbitrary space-time dimensions. 
  In this talk, I summarize the status of our understanding of the puzzle of large gauge invariance at finite temperature. 
  1-brane nonmaximally supersymmetric solutions of D=10 chiral supergravity are discussed. In the dual frame, their near brane geometry is the product of a 3-dimensional domain wall spacetime and a 7-dimensional homogeneous Einstein space G/H. 
  We study the structure of the tenacious (existing for all values of masses of the matter fields) BPS domain walls interpolating between different chirally asymmetric vacua in supersymmetric QCD in the limit of large masses.  We show that the wall consists in this case of three layers: two outer layers form a ``coat'' with the characteristic size determined by \Lambda_{SYM} and there is also the core with width of order of inverse mass. The core always carries a significant fraction of the total wall energy. This fraction depends on $N_f$ and on the ``windings'' of the matter fields. 
  The coordinate transformations which establish the direct relationship between the actions of linear and nonlinear realizations of supermembranes are proposed. It is shown that the Rocek-Tseytlin constraint known in the framework of the linear realization of the theory is simply equivalent to a limit of a "pure" nonlinear realization in which the field describing the massive mode of the supermembrane puts to zero. 
  We have studied the conformal models WD_{n}^{(p)}, n=3,4,5,..., in the presence of disorder which couples to the energy operator of the model. In the limit of p<<1 where p is the corresponding minimal model index, the problem could be analyzed by means of the perturbative renormalization group, with $epsilon$-expansion in $\epsilon$=1/p. We have found that the disorder makes to flow the model WD_{n}^{(p)} to the model WD_{n}^{(p-1)} without disorder. In the related problem of N coupled regular WD_{n}^{(p)} models (no disorder), coupled by their energy operators, we find a flow to the fixed point of N decoupled WD_{n}^{(p-1)}. But in addition we find in this case two new fixed points which could be reached by a fine tuning of the initial values of the couplings. The corresponding critical theories realize the permutational symmetry in a non-trivial way, like this is known to be the case for coupled Potts models, and they could not be identified with the presently known conformal models. 
  A uniformly acclerated observer in anti-deSitter space-time is known to detect thermal radiation when the acceleration exceeds a critical value. We investigate the holographic interpretation of this phenomenon. For uniformly accelerated trajectories transverse to the boundary of the AdS space, the hologram is a blob which expands along the boundary. Observers on the boundary co-moving with the hologram become observers in cosmological space-times. For supercritical accelerations one gets a Milne universe when the holographic screen is the boundary in Poincare coordinates, while for the boundary in hyperspherical coordinates one gets deSitter spacetimes. The presence or absence of thermality is then interpreted in terms of specific classes of observers in these cosmologies. 
  Given a Riemann surface and a riemannian manifold M with certain restrictions, we construct a cobordism invariant of M. This invariant is a generalization of the elliptic genus and it shares some similar properties. 
  We construct D-brane categories in B-type topological string theory as solutions to string field equations of motion. Using the formalism of superconnections, we show that these solutions form a variant of a construction of Bondal and Kapranov. This analysis is an elaboration on recent work of Lazaroiu. We also comment on the relation between string field theory and the derived category approach of Douglas, and Aspinwall and Lawrence. Non-holomorphic deformations make a somewhat unexpected appearance in this construction. 
  We present a model with an infinite volume bulk in which a braneworld with a cosmological constant evolves to a static, 4-dimensional Minkowski spacetime. This evolution occurs for a generic class of initial conditions with positive energy densities. The metric everywhere outside the brane is that of a 5-dimensional Minkowski spacetime, where the effect of the brane is the creation of a frame with a varying speed of light. This fact is encoded in the structure of the 4-dimensional graviton propagator on the braneworld, which may lead to some interesting Lorentz symmetry violating effects. In our framework the cosmological constant problem takes a different meaning since the flatness of the Universe is guaranteed for an arbitrary negative cosmological constant. Instead constraints on the model come from different concerns which we discuss in detail. 
  We prove the equivalence of the SL(2,R)/U(1) Kazama-Suzuki model, which is a fermionic generalization of the 2d Black Hole, and N=2 Liouville theory. We show that this duality is an example of mirror symmetry. The essential part of the derivation is to realize the fermionic 2d Black Hole as the low energy limit of a gauged linear sigma-model. Liouville theory is obtained by dualizing the charged scalar fields and taking into account the vortex-instanton effects, as proposed recently in non-dilatonic models. The gauged linear sigma-model we study has many useful generalizations which we briefly discuss. In particular, we show how to construct a variety of dilatonic superstring backgrounds which generalize the fermionic 2d Black Hole and admit a mirror description in terms of Toda-like theories. 
  The theory of permutation orbifolds is reviewed and applied to the study of symmetric product orbifolds and the congruence subgroup problem. The issue of discrete torsion, the combinatorics of symmetric products, the Galois action and questions related to the classification of RCFTs are also discussed. 
  We investigate the enlarged CP(N) model in 2+1 dimensions. This is a hybrid of two CP(N) models coupled with each other in a dual symmetric fashion, and it exhibits the gauge symmetry enhancement and radiative induction of the finite off-diagonal gauge boson mass as in the 1+1 dimensional case. We solve the mass gap equations and study the fixed point structure in the large-N limit. We find an interacting ultraviolet fixed point which is in contrast with the 1+1 dimensional case. We also compute the large-N effective gauge action explicitly. 
  For a class of system, the potential of whose Bosonic Hamiltonian has a Fourier representation in the sense of tempered distributions, we calculate the Gaussian effective potential within the framework of functional integral formalism. We show that the Coleman's normal-ordering prescription can be formally generalized to the functional integral formalism. 
  We study configurations consisting of a pair of non-extremal black holes in four dimensions, both with the same mass, and with charges of the same magnitude but opposite sign---diholes, for short. We present such exact solutions for Einstein-Maxwell theory with arbitrary dilaton coupling, and also solutions to the U(1)^4 theories that arise from compactified string/M-theory. Despite the fact that the solutions are very complicated, physical properties of these black holes, such as their area, charge, and interaction energy, admit simple expressions. We also succeed in providing a microscopic description of the entropy of these black holes using the `effective string' model, and taking into account the interaction between the effective string and anti-string. 
  We compute the anomalous dimension of the fermion field with N_f flavours in the fundamental representation of a general Lie colour group in the non-abelian Thirring model at four loops. The implications on the renormalization of the two point Green's function through the loss of multiplicative renormalizability of the model in dimensional regularization due to the appearance of evanescent four fermi operators are considered at length. We observe the appearance of one new colour group Casimir, d_F^{abcd} d_F^{abcd}, in the final four loop result and discuss its consequences for the relation of the Knizhnik-Zamolodchikov critical exponents in the Wess Zumino Witten Novikov model to the non-abelian Thirring model. Renormalization scheme changes are also considered to ensure that the underlying Fierz symmetry broken by dimensional regularization is restored. 
  We study Ricci-flat metrics on non-compact manifolds with the exceptional holonomy $Spin(7), G_2$. We concentrate on the metrics which are defined on ${\bf R} \times G/H$. If the homogeneous coset spaces $G/H$ have weak $G_2$, SU(3) holonomy, the manifold ${\bf R} \times G/H$ may have $Spin(7), G_2$ holonomy metrics. Using the formulation with vector fields, we investigate the metrics with $Spin(7)$ holonomy on ${\bf R}\times Sp(2)/Sp(1), {\bf R}\times SU(3)/U(1)$. We have found the explicit volume-preserving vector fields on these manifold using the elementary coordinate parameterization. This construction is essentially dual to solving the generalized self-duality condition for spin connections. We present most general differential equations for each coset. Then, we develop the similar formulation in order to calculate metrics with $G_2$ holonomy 
  We describe the coupled system of supergravity and a superbrane source by the sum of the group manifold action for D--dimensional supergravity and the action for a super--$p$--brane. We derive the generalized Einstein equation with the source and discuss the local fermionic symmetries of the coupled action. Our scheme could be especially relevant in $D=11, 10$, in which the superfield actions for supergravity are not known. 
  We consider the dynamics of a charged particle in a space whose coordinates are $N\times N$ hermitian matrices. Putting things in the framework of D0-branes of String Theory, we mention that the transformations of the matrix coordinates induce non-Abelian transformations on the gauge potentials. The Lorentz equations of motion for matrix coordinates are derived, and it is observed that the field strengths also transform like their non-Abelian counterparts. The issue of the map between theory on matrix space and ordinary non-Abelian gauge theory is discussed. The phenomenological aspect of "finite-N non-commutativity" for the bound states of D0-branes appears to be very attractive. 
  Based on the recently discovered K-theory description of D-branes in string theory a K-theory interpretation of the topology of gauge conditions for four dimensional gauge theories, in particular 't Hooft abelian projection, is presented. The interpretation of the dyon effect in terms of K^{-1} is also discussed. In the context of type IIA strings, D-p branes are also interpreted as gauge fixing singularities. 
  We investigate integrable boundary conditions (BCs) for the principal chiral model on the half-line, and rational solutions of the boundary Yang-Baxter equation (BYBE). In each case we find a connection with (type I, Riemannian, globally) symmetric spaces G/H: there is a class of integrable BCs in which the boundary field is restricted to lie in a coset of H; these BCs are parametrized by G/H x G/H; there are rational solutions of the BYBE in the defining representations of all classical G parametrized by G/H; and using these we propose boundary S-matrices for the principal chiral model, parametrized by G/H x G/H, which correspond to our boundary conditions. 
  In the five-dimensional Einstein gravity with negative cosmological constant in the presence/absence of a {\it non-fine-tuned} 3-brane, we have investigated the classical stability of black string solutions which are foliations of four-dimensional $AdS/dS$-Schwarzschild black holes. Such black strings are generically unstable as in the well-known Gregory-Laflamme instability. For $AdS$ black strings, however, it turns out that they become stable if the longitudinal size of horizon is larger than the order of the $AdS_4$ radius. Even in the case of unstable black strings, the $AdS$ black strings have a very different feature of string fragmentations from that in the flat brane world. Some implications of our results on the Gubser-Mitra conjecture are also discussed. 
  A high temperature expansion is employed to map some complex anisotropic nonhermitian three and four dimensional Ising models with algebraic long range interactions into a solvable two dimensional variant. We also address the dimensional reductions for anisotropic two dimensional XY and other models. For the latter and related systems it is possible to have an effective reduction in the dimension without the need of compactifying some dimensions. Some solutions are presented. This framework further allows for some very simple general observations. It will be seen that the absence of a ``phase interference'' effect plays an important role in high dimensional problems. A very forbidding purely algebraic recursive series solution to the three dimensional nearest neighbor Ising model will be given. In the aftermath, the full-blown three dimensional nearest neighbor Ising model is exactly mapped onto a single spin 1/2 particle with nontrivial dynamics. All this allows for a formal high dimensional Bosonization. 
  We extend our previous results on the relation between quaternion-Kahler manifolds and hyperkahler cones and we describe how isometries, moment maps and scalar potentials descend from the cone to the quaternion-Kahler space. As an example of the general construction, we discuss the gauging and the corresponding scalar potential of hypermultiplets with the unitary Wolf spaces as target spaces. This class includes the universal hypermultiplet. 
  The Dirac equation for an electron in an external electromagnetic field can be regarded as a singular set of linear equations for the vector potential. Radford's method of algebraically solving for the vector potential is reviewed, with attention to the additional constraints arising from non-maximality of the rank. The extension of the method to general spacetimes is illustrated by examples in diverse dimensions with both $c$- and $a$-number wavefunctions. 
  We study the IR/UV connection of the four-dimensional non-commutative phi^4 theory by using the Wilsonian Renormalization Group equation. Extending the usual formulation to the non-commutative case we are able to prove UV renormalizability to all orders in perturbation theory. The full RG equations are finite in the IR, but perturbative approximations of them are plagued by IR divergences. The latter can be systematically resummed in a way analogous to what is done in finite temperature field theory. As an application, next-to-leading order corrections to the two-point function are explicitly computed. The usual Wilsonian picture, i.e. the insensitivity of the IR regime to the UV, does not hold in the non-commutative case. Nevertheless it can be partially recovered by a matching procedure, in which a high-energy theory, defined in the deep non-commutative regime, is connected at some intermediate scale to a commutative low-energy theory. The latter knows about non-commutativity only through the boundary conditions for two would-be irrelevant couplings. 
  We reconstruct the tachyon effective action for unstable D-branes in superstring theory by examining its behaviour near exactly marginal deformations, where the ambigous higher derivative terms can be eliminated. We then compare this action with that obtained in boundary string field theory and find remarkable agreement. In particular, the tension for lower dimensional branes and the BI-action for the centre of mass motion are reprodued exactly. We also comment on the action for tachyons on the kink in a D-brane/anti-D-brane system and on bosonic string theory. 
  In deterministic theories, one can start from a set of ontological states to formulate the dynamical laws, but these may not be directly observable. Observable are only equivalence classes of states, and these will span a basis of "beables", to be promoted to an orthonormal basis of Hilbert Space. After transforming this basis to a more conventional basis, a theory may result that is fundamentally quantum mechanical. It is conjectured that the quantum laws of the real world may be understood from exactly such a procedure. 
  The exact vacuum expectation values of the second level descendent fields $<(\partial\phi)^2({\overline\partial}\phi)^2e^{a\phi}>$ in the Bullough-Dodd model are calculated. By performing quantum group restrictions, we obtain $<L_{-2}{\overline L}_{-2}{\Phi}_{lk}>$ in the $\Phi_{12}$, $\Phi_{21}$ and $\Phi_{15}$ perturbed minimal CFTs. In particular, the exact expectation value $<T{\overline T}>$ is found to be proportional to the square of the bulk free energy. 
  We discuss an optimisation criterion for the exact renormalisation group based on the inverse effective propagator, which displays a gap. We show that a simple extremisation of the gap stabilises the flow, leading to better convergence of approximate solutions towards the physical theory. This improves the reliability of truncations, most relevant for any high precision computation. These ideas are closely linked to the removal of a spurious scheme dependence and a minimum sensitivity condition. The issue of predictive power and a link to the Polchinski RG are discussed as well. We illustrate our findings by computing critical exponents for the Ising universality class. 
  There are only three stable singularities of a differentiable map between three-dimensional manifolds, namely folds, cusps and swallowtails. A Skyrme configuration is a map from space to SU(2), and its singularities correspond to the points where the baryon density vanishes. In this paper we consider the singularity structure of Skyrme configurations. The Skyrme model can only be solved numerically. However, there are good analytic ansaetze. The simplest of these, the rational map ansatz, has a non-generic singularity structure. This leads us to introduce a non-holomorphic ansatz as a generalization. For baryon number two, three and four, the approximate solutions derived from this ansatz are closer in energy to the true solutions than any other ansatz solution. We find that there is a tiny amount of negative baryon density for baryon number three, but none for two or four. We comment briefly on the relationship to Bogomolny-Prasad-Sommerfield monopoles. 
  We propose the 4D, N=1 supergravitational analogues (avatars) of the 4D, N=1 supersymmetric Born-Infeld action in four dimensions for the first time, by using superspace. In particular, a new Born-Infeld type generalization of the Weyl supergravity action is given. A natural new Born-Infeld type generalization of the Einstein supergravity is found as well. We also brielfy discuss a construction of the four-dimensional Born-Infeld-Einstein supergravity from the AdS supergravity in five dimensions, which seems to be very natural in our approach. 
  The conditions under which noncommutative quantum mechanics and the Landau problem are equivalent theories is explored. If the potential in noncommutative quantum mechanics is chosen as $V= \Omega \aleph$ with $\aleph$ defined in the text, then for the value ${\tilde \theta} = 0.22 \times 10^{-11} cm^2$ (that measures the noncommutative effects of the space), the Landau problem and noncommutative quantum mechanics are equivalent theories in the lowest Landau level. For other systems one can find differents values for ${\tilde \theta}$ and, therefore, the possible bounds for ${\tilde \theta}$ should be searched in a physical independent scenario. This last fact could explain the differents bounds for $\tilde \theta$ found in the literature. 
  It was recently proposed a novel discretization for nonlinear Klein-Gordon field theories in which the resulting lattice preserves the topological (Bogomol'nyi) lower bound on the kink energy and, as a consequence, has no Peierls-Nabarro barrier even for large spatial discretizations (h~1.0). It was then suggested that these ``topological discrete systems'' are a natural choice for the numerical study of continuum kink dynamics. Giving particular emphasis to the phi^4 theory, we numerically investigate kink-antikink scattering and breather formation in these topological lattices. Our results indicate that, even though these systems are quite accurate for studying free kinks in coarse lattices, for legitimate dynamical kink problems the accuracy is rather restricted to fine lattices (h~0.1). We suggest that this fact is related to the breaking of the Bogomol'nyi bound during the kink-antikink interaction, where the field profile loses its static property as required by the Bogomol'nyi argument. We conclude, therefore, that these lattices are not suitable for the study of more general kink dynamics, since a standard discretization is simpler and has effectively the same accuracy for such resolutions. 
  A treatment of linearized gravity with the Einstein-Gauss-Bonnet interaction terms in $D\geq$ 5 is given in the Randall-Sundrum brane background. This Letter has outlined some interesting features of the brane world gravity and Newtonian potential correction with the GB interaction term. We find that the GB coupling $\alpha$ renormalizes the effective four-dimensional Newton constant on the brane, and also additionally contributes to the correction term of the Newtonian potential. Indeed, the GB term does not affect the massless graviton mode and the Einstein gravity on the brane, and quite interestingly, such term in $D\geq 5$ appears to give more information about the necessary boundary condition(s) to be satisfied by the zero mode wavefunction on the brane(s). small (and preferably positive) GB coupling $\alpha$. 
  A general solution to the Einstein field equations with Gauss-Bonnet(GB) term in the $AdS_5$ bulk background implies that the GB coupling $\alpha$ can take either sign (+ or -), though a positive $\alpha$ will be more meaningful. By considering linearized gravity with the GB term in the Randall-Sundrum(RS) a singular 3-brane model, we study the gravitational interactions between matter sources localized on the brane. With a correctly defined boundary condition on the brane, we find a smooth behavior of graviton propagator and hence the zero-mode solution as a 4d massless graviton localized on the brane with correct momentum and tensor structures. The coupling $\alpha$ modifies the graviton propagators both on the brane and in the bulk. The issue on ghost state of the GB term is resolved, and we find that there is no real ghost (negative norm) state of the GB term in the RS single brane picture. The latter condition leads to a consistency in the coupling between the brane matter and the bulk gravity. We also elucidate about the possibilities for behavior of a test particle on the brane and in the bulk. 
  We study the rate of true vacuum bubble nucleation numerically for a phi^4 field system coupled to a source of thermal noise. We compare in detail the cases of additive and multiplicative noise. We pay special attention to the choice of initial field configuration, showing the advantages of a version of the quenching technique. We advocate a new method of extracting the nucleation time scale that employs the full distribution of nucleation times. Large data samples are needed to study the initial state configuration choice and to extract nucleation times to good precision. The 1+1 dimensional models afford large statistics samples in reasonable running times. We find that for both additive and multiplicative models, nucleation time distributions are well fit by a waiting time, or gamma, distribution for all parameters studied. The nucleation rates are a factor three or more slower for the multiplicative compared to the additive models with the same dimensionless parameter choices. Both cases lead to high confidence level linear fits of ln(nucleation time) vs. 1/T plots, in agreement with semiclassical nucleation rate predictions. 
  We study soliton solutions in scalar field theory with a variety of unbounded potentials. A subset of these potentials have Gaussian lump solutions and their fluctuation spectrum is governed by the harmonic oscillator problem. These lumps are unstable with one tachyonic mode. Soliton solutions in several other classes of potentials are stable and are of kink type. The problem of the stability of these solutions is related to a supersymmetric quantum mechanics problem. The fluctuation spectrum is not equispaced and does not contain any tachyonic mode. The lowest energy mode is the massless Goldstone mode which restores broken translation invariance. 
  We construct a kink solution on a non-BPS D-brane using Berkovits' formulation of superstring field theory in the level truncation scheme. The tension of the kink reproduces 95% of the expected BPS D-brane tension. We also find a lump-like solution which is interpreted as a kink--antikink pair, and investigate some of its properties. These results may be considered as successful tests of Berkovits' superstring field theory combined with the modified level truncation scheme. 
  In this short review we compare the rigid Noether charges to topological gauge charges. One important extension is that one should consider each boundary component of spacetime independently. The argument that relates bulk charges to surface terms can be adapted to the perfect fluid situation where one can recognise the helicity and enstrophies as Noether charges. More generally a forcing procedure that increases for instance any Noether charge is demonstrated. In the gauge theory situation, the key idea can be summarized by one sentence: ``go to infinity and stay there''. A new variational formulation of Einstein's gravity is given that allows for local GL(D,R) invariance. The a priori indeterminacy of the Noether charges is emphasized and a covariant ansatz due to S. Silva for the surface charges of gauge theories is analysed, it replaces the (non-covariant) Regge-Teitelboim procedure. 
  Using the AdS/CFT correspondence we study UV behavior of Wilson loops in various noncommutative gauge theories. We get an area law in most cases and try to identify its origin. In D3 case, we may identify the the origin as the D1 dominance over the D3: as we go to the boundary of the AdS space, the effect of the flux of the D3 charge is highly suppressed, while the flux due to the D1 charge is enhenced. So near the boundary the theory is more like a theory on D1 brane than that on D3 brane. This phenomena is closely related to the dimensional reduction due to the strong magnetic field in the charged particle in the magnetic field. The linear potential is not due to the confinement by IR effect but is the analogue of Coulomb's potential in 1+1 dimension. 
  A new renormalization scheme for theories with nontrivial internal symmetry is proposed. The scheme is regularization independent and respects the symmetry requirements. 
  We discuss a simplified method for computing trace anomalies in d=6 and d<6 dimensions. It is known that in the quantum mechanical approach trace anomalies in d dimensions are given by a (1+d/2)-loop computation in an auxiliary 1d sigma model with arbitrary geometry. We show how one can obtain the same information using a simpler d/2-loop calculation on an arbitrary geometry supplemented by a (1+d/2)-loop calculation on the simplified geometry of a maximally symmetric space. 
  The main goal of this paper is the manifestly covariant derivation of the classical relation between the gauge-fixed Grassmann variables of the NSR string and the Type II GS superstring. To this end we analyze the superembedding equation for the Type II superstring to derive the relation between the original variables of the NSR string and the Type II GS superstring and, further, by means of Lorentz harmonic variables we fix $\kappa-$symmetry of the GS superstring in the manifestly Lorentz covariant way. 
  Using superspace techniques, the complete and most general action of D=10 super-Yang--Mills theory is constructed at the alpha'^2 level. No other approximations, e.g., keeping only a subset of the allowed derivative terms, are used. The Lorentz structure of the alpha'^2 corrections is completely determined, while (depending on the gauge group) there is some freedom in the adjoint structure, which is given by a totally symmetric four-index tensor. We examine the second, non-linearly realised supersymmetry that may be present when the gauge group has a U(1) factor, and find that the constraints from linear and non-linear supersymmetry to a large extent coincide. However, the additional restrictions on the adjoint structure of the order alpha'^2 interactions following from the requirement of non-linear supersymmetry do not completely specify the symmetrised trace prescription. 
  A classical dynamical system in a four-dimensional Euclidean space with universal time is considered. The space is hypothesized to be originally occupied by a uniform substance, pictured as a liquid, which at some time became supercooled. Our universe began as a nucleation event initiating a liquid to solid transition. The universe we inhabit and are directly aware of consists of only the three-dimensional expanding phase boundary - a crystalline surface. Random energy transfers to the boundary from thermal fluctuations in the adjacent bulk phases are interpreted by us as quantum fluctuations, and give a physical realization to the stochastic quantization technique. Fermionic matter is modeled as screw dislocations; gauge bosons as surface acoustic waves. Minkowski space emerges dynamically through redefining local time to be proportional to the spatial coordinate perpendicular to the boundary. Lorentz invariance is only approximate, and the photon spectrum (now a phonon spectrum) has a maximum energy. Other features include a geometrical quantum gravitational theory based on elasticity theory, and a simple explanation of the quantum measurement process as a spontaneous symmetry breaking. Present, past and future are physically distinct regions, the present being a unique surface where our universe is being continually constructed. 
  Models of closed superstrings in certain curved NS-NS magnetic flux backgrounds are exactly solvable in terms of free fields. They interpolate between free superstring theories with periodic and antiperiodic boundary conditions for fermions around some compact direction, and, in particular, between type 0 and type II string theories. Using ``9-11 flip'', this interpolation can be extended to M-theory and provides an interesting setting for a study of tachyon problem in closed string theory. Starting with a general 2-parameter family of such Melvin-type models, we present several new magnetic flux backgrounds in 10-d string theory and 11-d M-theory and discuss their tachyonic instabilities. In particular, we suggest a description of type 0B theory in terms of M-theory in curved magnetic flux background, which supports its conjectured SL(2,Z) symmetry, and in which the type 0 tachyon appears to correspond to a state in 11-d supergravity multiplet. In another ``T-dual'' description, the tachyon is related to a winding membrane state. 
  Generic classes of string compactifications include ``brane throats'' emanating from the compact dimensions and separated by effective potential barriers raised by the background gravitational fields. The interaction of observers inside different throats occurs via tunnelling and is consequently weak. This provides a new mechanism for generating small numbers in Nature. We apply it to the hierarchy problem, where supersymmetry breaking near the unification scale causes TeV sparticle masses inside the standard model throat. We also design naturally long-lived cold dark matter which decays within a Hubble time to the approximate conformal matter of a long throat. This may soften structure formation at galactic scales and raises the possibility that much of the dark matter of the universe is conformal matter. Finally, the tunnelling rate shows that the coupling between throats, mediated by bulk modes, is stronger than a naive application of holography suggests. 
  We present the first polytope volume formulas for the multiplicities of affine fusion, the fusion in Wess-Zumino-Witten conformal field theories, for example. Thus, we characterise fusion multiplicities as discretised volumes of certain convex polytopes, and write them explicitly as multiple sums measuring those volumes. We focus on su(2), but discuss higher-point (N>3) and higher-genus fusion in a general way. The method follows that of our previous work on tensor product multiplicities, and so is based on the concepts of generalised Berenstein-Zelevinsky diagrams, and virtual couplings. As a by-product, we also determine necessary and sufficient conditions for non-vanishing higher-point fusion multiplicities. In the limit of large level, these inequalities reduce to very simple non-vanishing conditions for the corresponding tensor product multiplicities. Finally, we find the minimum level at which the higher-point fusion and tensor product multiplicities coincide. 
  Dynamical generation of 4d gauge theories and gravity at low energy from the 3d ones at high energy is studied, based on the fermion condensation mechanism recently proposed by Arkani-Hamed, Cohen and Georgi. For gravity, 4d Einstein gravity is generated from the multiple copy of the 3d ones, by referring to the two form gravity. Since the 3d Einstein action without matter coupling is topological, ultraviolet divergences are less singular in our model. In the gauge model, matter fermions are introduced on the discrete lattice following Wilson. Then, the 4d gauge interactions are correctly generated from the 3d theories even in the left-right asymmetric theories of the standard model. In order for this to occur, the Higgs fields as well as the gauge fields of the extra dimension should be generated by the fermion condensates. Therefore, the generation of the 4d standard model from the multiple copy of the 3d ones is quite promising. To solve the doubling problem in the weak interaction sector, two kinds of discrete lattices have to be introduced separately for L- and R-handed sectors, and the two types of Higgs fields should be generated. 
  Despite its simplicity, the unitary gauge is not a popular choice for practical loop calculations in gauge theories, due to the lack of off-shell renormalizability. We study the renormalization properties of the off-shell Green functions of the elementary electron fields in the massive QED, in order to elucidate the origin and structure of the extra ultraviolet divergences which exist only in the unitary gauge. We find that all these divergences affect the Green functions in a trivial way such that in coordinate space the off-shell Green functions are in fact multiplicatively renormalizable. This result may generalize to the abelian and non-abelian Higgs theories, for which the unitary gauge might bring much simplification to the loop calculations. 
  We study the system of the Dp'-brane with Dp-brane (p<p') inside, in the case where B_{ij} field is a nonvanishing constant. In order to understand how the Dp-brane is viewed from the Dp'-brane worldvolume theory, we investigate the process in which the Dp-brane is probed with p'-p' open string. We calculate the scattering amplitudes among p-p' open strings and p'-p' open strings and show that not only the Weyl transform of the projection operator onto the ground state but also those onto higher excited states emerge as multiplicative factors of the amplitudes. 
  In a non-commutative field theory, the energy-momentum tensor obtained from the Noether method needs not be symmetric; in a massless theory, it needs not be traceless either. In a non-commutative scalar field theory, the method yields a locally conserved yet non-symmetric energy-momentum tensor whose trace does not vanish for massless fields. A non-symmetric tensor also governs the response of the action to a general coordinate transformation. In non-commutative gauge theory, if translations are suitably combined with gauge transformations, the method yields a covariantly constant tensor which is symmetric but only gauge covariant. Using suitable Wilson functionals, this can be improved to yield a locally conserved and gauge invariant, albeit non-symmetric, energy-momentum tensor. 
  A gauge theory equivalent to the hamiltonian reduction scheme for rational Calogero - Moser model is presented. 
  General properties of the theory of higher spin gauge fields in $AdS_4$ are discussed. Some new results on the 4d conformal higher spin gauge theory are announced. Talk given at the International Conference "Quantization, Gauge Theory and Strings" in memory of E.S.Fradkin, Moscow, June 5-10, 2000. 
  Recently, the superstring was covariantly quantized using the BRST-like operator $Q = \oint \lambda^\alpha d_\alpha$ where $\lambda^\alpha$ is a pure spinor and $d_\alpha$ are the fermionic Green-Schwarz constraints. By performing a field redefinition and a similarity transformation, this BRST-like operator is mapped to the sum of the Ramond-Neveu-Schwarz BRST operator and $\eta_0$ ghost. This map is then used to relate physical vertex operators and tree amplitudes in the two formalisms. Furthermore, the map implies the existence of a $b$ ghost in the pure spinor formalism which might be useful for loop amplitude computations. 
  A brief summary is given of the Group-Variation Equations and the island diagram confinement mechanism, with an explanation of the prediction that the cylinder-topology minimal-area spanning surface term in the correlation function of two Wilson loops at large $N_c$, when it exists, must have a pre-exponential factor, which for large area $A$ of the minimal-area cylinder-topology spanning surface, decreases with increasing $A$ at least as fast as $1/\ln(\sigma A)$, where $\sigma$ is the area law parameter. This prediction is expected to be testable in lattice calculations. 
  The method of nonlinear realizations is applied for the conformally invariant description of the spinning particles in terms of geometrical quantities of the parameter spaces of the one dimensional N - extended superconformal groups. We develop the superspace approach to the cases of spin 0, 1/2, 1 particles and describe the alternative component approach in the application to the spin-1/2 particle. 
  We present a first-quantized formulation of the quadratic non-commutative field theory in the background of abelian (gauge) field. Even in this simple case the Hamiltonian of a propagating particle depends non-trivially on the momentum (since external fields depend on location of the Landau orbit) so that one can not integrate out momentum to obtain a local theory in the second order formalism. The cases of scalar and spinning particles are considered. A representation for exact propagators is found and the result is applied to description of the Schwinger-type processes (pair-production in homogenous external field). 
  AdS dynamics for massive scalar field is studied both by solving exactly the equation of motion and by constructing bulk-boundary propagator. A Robertson-Walker-like metric is deduced from the familiar SO(2,n) invariant metric. The metric allows us to present a time-like Killing vector, which is not only invariant under space-like transformations but also invariant under the isometric transformations of SO(2,n) in certain sense. A horizon appears in this coordinate system. Singularities of field variables at boundary are demonstrated explicitly. It is shown that there is a one-to-one correspondence among the exact solutions and the bulk fields obtained by using the bulk-boundary propagator. 
  Paper contains description of the fields nonlinear modes successive quantization scheme. It is shown that the path integrals for absorption part of amplitudes are defined on the Dirac ($\d$-like) functional measure. This permits arbitrary transformation of the functional integral variables. New form of the perturbation theory achieved by mapping the quantum dynamics in the space $W_G$ of the ({\it action, angle})-type collective variables. It is shown that the transformed perturbation theory contributions are accumulated exactly on the boundary $\pa W_G$. Abilities of the developed formalism are illustrated by the Coulomb problem. This model is solved in the $W_C$=({\it angle, angular momentum, Runge-Lentz vector}) space and the reason of its exact integrability is `emptiness' of $\pa W_C$. 
  We formulate the problem of finding self-dual Hamiltonians (associated with integrable systems) as deformations of free systems given on various symplectic manifolds and discuss several known explicit examples, including recently found double elliptic Hamiltonians. We consider as basic the notion of self-duality, while the duality in integrable systems (of the Toda/Calogero/Ruijsenaars type) comes as a derivative notion (degenerations of self-dual systems). This is a talk presented at the Workshop "Classical and Quantum Integrable Systems", Protvino, January, 2001. 
  We study D-branes on smooth noncompact toric Calabi-Yau manifolds that are resolutions of abelian orbifold singularities. Such a space has a distinguished basis {S_i} for the compactly supported K-theory. Using local mirror symmetry we demonstrate that the S_i have simple transformation properties under monodromy; in particular, they are the objects that generate monodromy around the principal component of the discriminant locus. One of our examples, the toric resolution of C^3/(Z_2 x Z_2), is a three parameter model for which we are able to give an explicit solution of the GKZ system. 
  A cutoff regularization for a pure Yang-Mills theory is implemented within the background field method keeping explicit the gauge invariance of the effective action. The method has been applied to compute the beta function at one loop order. 
  Bohr and Rosenfeld carried out an analysis of the consequences of field theory commutation relations. In this note the analysis is sharpened. A conjecture of Heisenberg that volume is quantized is shown to be a consequence of the second quantization of gauge fields. A way to generalize the equations of physics to include the Planck length is indicated. 
  We study the effect of a background flux string on the vacuum energy of massive Dirac fermions in 3+1 dimensions confined to a finite spatial region through MIT boundary conditions. We treat two admissible self-adjoint extensions of the Hamiltonian. The external sector is also studied and unambiguous results for the Casimir energy of massive fermions in the whole space are obtained. 
  The classical theory of non-relativistic charged particle interacting with U(1) gauge field is reformulated as the Schr\"odinger wave equation modified by the de-Broglie-Bohm quantum potential nonlinearity. For, (1 - $\hbar^2$) deformed strength of quantum potential the model is gauge equivalent to the standard Schr\"odinger equation with Planck constant $\hbar$, while for the strength (1 + $\hbar^2$), to the pair of diffusion-anti-diffusion equations. Specifying the gauge field as Abelian Chern-Simons (CS) one in 2+1 dimensions interacting with the Nonlinear Schr\"odinger field (the Jackiw-Pi model), we represent the theory as a planar Madelung fluid, where the Chern-Simons Gauss law has simple physical meaning of creation the local vorticity for the fluid flow. For the static flow, when velocity of the center-of-mass motion (the classical velocity) is equal to the quantum one (generated by quantum potential velocity of the internal motion), the fluid admits N-vortex solution. Applying the Auberson-Sabatier type gauge transform to phase of the vortex wave function we show that deformation parameter $\hbar$, the CS coupling constant and the quantum potential strength are quantized. Reductions of the model to 1+1 dimensions, leading to modified NLS and DNLS equations with resonance soliton interactions are discussed. 
  Harnessing the unimodular degree of freedom in the definition of any toric diagram, we present a method of constructing inequivalent gauge theories which are world-volume theories of D-branes probing the same toric singularity. These theories are various phases in partial resolution of Abelian orbifolds. As examples, two phases are constructed for both the zeroth Hirzebruch and the second del Pezzo surfaces. We show that such a phenomenon is a special case of ``Toric Duality'' proposed in hep-th/0003085. Furthermore, we investigate the general conditions that distinguish these different gauge theories with the same (toric) moduli space. 
  In string or M theories, the spontaneous breaking of 10D or 11D Lorentz symmetry is required to describe our space-time. A direct approach to this issue is provided by the IIB matrix model. We study its 4D version, which corresponds to the zero volume limit of 4D super SU(N) Yang-Mills theory. Based on the moment of inertia as a criterion, spontaneous symmetry breaking (SSB) seems to occur, so that only one extended direction remains, as first observed by Bialas, Burda et al. However, using Wilson loops as probes of space-time we do not observe any sign of SSB in Monte Carlo simulations where N is as large as 48. This agrees with an earlier observation that the phase of the fermionic integral, which is absent in the 4D model, should play a crucial role if SSB of Lorentz symmetry really occurs in the 10D IIB matrix model. 
  The gauge model of nonrelativistic particles on a line interacting with nonstandard gravitational fields [5] is supplemented by the addition of a (non)-Abelian gauge interaction. Solving for the gauge fields we obtain equations, in closed form, for a classical two particle system. The corresponding Schr\"{o}dinger equation, obtained by the Moyal quantization procedure, is solved analytically. Its solutions exhibit two different confinement mechanisms - dependent on the sign of the coupling $\lambda$ to the nonstandard gravitational fields. For $\lambda>0$ confinement is due to a rising potential whereas for $\lambda<0$ it is due to to the dynamical (geometric) bag formation. Numerical results for the corresponding energy spectra are given. For a particular relation between two coupling constants the model fits into the scheme of supersymmetrical quantum mechanics. 
  We find the gravitational field of a `nested' domain wall living entirely within a brane-universe, or, a localised vortex within a wall. For a vortex living on a critical Randall-Sundrum brane universe, we show that the induced gravitational field on the brane is identical to that of an (n-1)-dimensional vacuum domain wall. We also describe how to set-up a nested Randall-Sundrum scenario using a flat critical vortex living on a subcritical (adS) brane universe. 
  We propose an ansatz for the equations of motion of the noncommutative model of a tachyonic scalar field interacting with a gauge field, which allows one to find time-dependent solutions describing decaying solitons. These correspond to the collapse of lower dimensional branes obtained through tachyon condensation of unstable brane systems in string theory. 
  We formulate noncommutative three-dimensional (3d) gravity by making use of its connection with 3d Chern-Simons theory. In the Euclidean sector, we consider the particular example of topology $T^2 \times R$ and show that the 3d black hole solves the noncommutative equations. We then consider the black hole on a constant U(1) background and show that the black hole charges (mass and angular momentum) are modified by the presence of this background. 
  We present a calculation of the correlator <0|T^{++}(r) T^{++}(0) |0> in N=1 SYM theory in 2+1 dimensions. In the calculation, we use supersymmetric discrete light-cone quantization (SDLCQ), which preserves the supersymmetry at every step of the calculation. For small and intermediate r the correlator converges rapidly for all couplings. At small r the correlator behaves like 1/r^6, as expected. At large r the correlator is dominated by the BPS states of the theory. We find a critical coupling where the large-r correlator goes to zero; it grows like the square root of the transverse cutoff. 
  A classical field system of two interacting fields -- a real Higgs field and a complex scalar field -- is considered. It is shown that in such field system a non-trivial solution exists, which is U(1) charged topological kink. Some questions of stability of the obtained solution are discussed. An improved variational procedure for searching of topological U(1) charged solutions is given. 
  A new exact analytically solvable Eckart-type potential is presented, a generalisation of the Hulthen potential. The study through Supersymmetric Quantum Mechanics is presented together with the hierarchy of Hamiltonians and the shape invariance property. 
  We use Grassmann even spinor oscillators to construct a bosonic higher spin extension hs(2,2) of the five-dimensional anti-de Sitter algebra SU(2,2), and show that the gauging of hs(2,2) gives rise to a spectrum S of physical massless fields with spin s=0,2,4,... that is a UIR of hs(2,2). In addition to a master gauge field which contains the massless s=2,4,.. fields, we construct a scalar master field containing the massless s=0 field, the generalized Weyl tensors and their derivatives. We give the appropriate linearized constraint on this master scalar field, which together with a linearized curvature constraint produces the correct linearized field equations. A crucial step in the construction of the theory is the identification of a central generator K which is eliminated by means of a coset construction. Its charge vanishes in the spectrum S, which is the symmetric product of two spin zero doubletons. We expect our results to pave the way for constructing an interacting theory whose curvature expansion is dual to a CFT based on higher spin currents formed out of free doubletons in the large N limit. Thus, extending a recent proposal of Sundborg (hep-th/0103247), we conjecture that the hs(2,2) gauge theory describes a truncation of the bosonic massless sector of tensionless  Type IIB string theory on AdS_5 x S^5 for large N. This implies AdS/CFT correspondence in a parameter regime where both boundary and bulk theories are perturbative. 
  A new approach to constructing the noncommutative scalar field theory is presented. Not only between x_i and p_j, we impose commutation relations between x_is as well as p_js, and give a new representation of x_i,p_js. We carry out both first- and second-quantization explicitly. The second-quantization is performed in both the operator formalism and the functional integral one. 
  The recent discoveries of new forms of quantum statistics require a close look at the under-lying Fock space structure. This exercise becomes all the more important in order to provide a general classification scheme for various forms of statistics, and establish interconnections among them whenever it is possible. We formulate a theory of generalized Fock spaces, which has a three tired structure consisting of Fock space, statistics and algebra. This general formalism unifies various forms of statistics and algebras, which were earlier considered to describe different systems. Besides, the formalism allows us to construct many new kinds of quantum statistics and the associated algebras of creation and destruction operators. Some of these are: orthostatistics, null statistics or statistics of frozen order, quantum group based statistics and its many avatars, and `doubly-infinite' statistics. The emergence of new forms of quantum statistics for particles interacting with singular potential is also highlighted. 
  Orthofermi statistics is characterized by an exclusion principle which is more ``exclusive'' than Pauli's exclusion principle: an orbital state shall not contain more than one particle, no matter what the spin direction is. The wavefunction is antisymmetric in spatial indices alone with arbitrary symmetry in the spin indices. Orthobose statistics is corresponding Bose analog: the wavefunction is symmetric in spatial indices, with arbitrary symmetry in spin indices. We construct the quantum field theory of particles obeying these new kinds of quantum statistics. Non-relativistic as well as relativistic quantum field theories with interactions are considered. 
  The anomalous dimension of single and multi-trace composite operators of scalar fields is shown to vanish at all orders of the perturbative series. The proof hold for theories with N=2 supersymmetry with any number of hypermultiplets in a generic representation of the gauge group. It then applies to the finite N=4 theory as well as to non conformal N=2 models. 
  We construct spherical harmonics for fuzzy spheres of even and odd dimensions, generalizing the correspondence between finite matrix algebras and fuzzy two-spheres.   The finite matrix algebras associated with the various fuzzy spheres have a natural basis which falls in correspondence with tensor constructions of irreducible representations of orthogonal groups SO(n). This basis is useful in describing fluctuations around various D-brane constructions of fuzzy spherical objects. The higher fuzzy spheres are non-associative algebras that appear as projections of associative algebras related to Matrices. The non-associativity (as well as the non-commutativity) disappears in the leading large $N$ limit, ensuring the correct classical limit. Some simple aspects of the combinatorics of the fuzzy four-sphere can be accounted by a heuristic picture of giant fractional instantons. 
  The self-tuning brane scenario is an attempt to solve the cosmological constant problem in the context of extra dimensions. Rather than making the vacuum energy small, this approach proceeds by removing the gravitational effect of vacuum energy on the expansion of the universe. Such behavior is only possible through changing the Friedmann equation of conventional cosmology, and we discuss difficulties in obtaining cosmological evolution compatible with observation in this context. Specific models considered include a bulk scalar field coupling to the brane via a conformal transformation of the brane metric, and via a rescaling of the brane volume element. 
  In a previous paper we showed that the absence of the van Dam-Veltman-Zakharov discontinuity as M^2 -> 0 for massive spin-2 with a Lambda term is an artifact of the tree approximation, and that the discontinuity reappears at one loop, as a result of going from five degrees of freedom to two. In this paper we show that a similar classical continuity but quantum discontinuity arises in the "partially massless" limit M^2 -> 2Lambda/3, as a result of going from five degrees of freedom to four. 
  We extend calculational techniques for static solitons to the case of field configurations with simple time dependence in order to consider quantum effects on the stability of Q-balls. These nontopological solitons exist classically for any fixed value of an unbroken global charge Q. We show that one-loop quantum effects can destabilize very small Q-balls. We show how the properties of the soliton are reflected in the associated scattering problem, and find that a good approximation to the full one-loop quantum energy of a Q-ball is given by $\omega - E_0$, where $\omega$ is the frequency of the classical soliton's time dependence, and $E_0$ is the energy of the lowest bound state in the associated scattering problem. 
  Gauge theories of conformal spacetime symmetries are presented which merge features of Yang-Mills theory and general relativity in a new way. The models are local but nonpolynomial in the gauge fields, with a nonpolynomial structure that can be elegantly written in terms of a metric (or vielbein) composed of the gauge fields. General relativity itself emerges from the construction as a gauge theory of spacetime translations. The role of the models within a general classification of consistent interactions of gauge fields is discussed as well. 
  The simplest possible noncommutative harmonic oscillator in two dimensions is used to quantize the free closed bosonic string in two flat dimensions. The partition function is not deformed by the introduction of noncommutativity, if we rescale the time and change the compactification radius appropriately. The four point function is deformed, preserving, nevertheless, the sl(2,C) invariance. Finally the first Ward identity of the deformed theory is derived. 
  We study scattering of noncommutative solitons in 2+1 dimensional scalar field theory. In particular, we investigate a system of two solitons with level n and n' (the (n,n')-system) in the large noncommutativity limit. We show that the scattering of a general (n,n')-system occurs at right angles in the case of zero impact parameter. We also derive an exact Kahler potential and the metric of the moduli space of the (n,1)-system. We examine numerically the (n,1) scattering and find that the closest distance for a fixed scattering angle is well approximated by a function a+b*sqrt{n} where a and b are some numerical constants. 
  We investigate a quantum system possessing a parasupersymmetry of order 2, an orthosupersymmetry of order $p$, a fractional supersymmetry of order $p+1$, and topological symmetries of type $(1,p)$ and $(1,1,...,1)$. We obtain the corresponding symmetry generators, explore their relationship, and show that they may be expressed in terms of the creation and annihilation operators for an ordinary boson and orthofermions of order $p$. We give a realization of parafermions of order~2 using orthofermions of arbitrary order $p$, discuss a $p=2$ parasupersymmetry between $p=2$ parafermions and parabosons of arbitrary order, and show that every orthosupersymmetric system possesses topological symmetries. We also reveal a correspondence between the orthosupersymmetry of order $p$ and the fractional supersymmetry of order $p+1$. 
  We discuss the recent proposal that BPS D-branes in Calabi-Yau compactification of type II string theory are Pi-stable objects in the derived category of coherent sheaves. 
  A bicomplex structure is associated to the Leznov-Saveliev equation of integrable models. The linear problem associated to the zero curvature condition is derived in terms of the bicomplex linear equation. The explicit example of a Non-Abelian Conformal Affine Toda model is discussed in detail and its conservation laws are derived from the zero curvature representation of its equation of motion. 
  A massive version of T-duality in six dimensions is given, that maps the K3 compactification of Romans' theory onto the K3 compactification of Type IIB theory. This is done by performing a (standard) Kaluza-Klein reduction on six-dimensional massive Type IIA and a Scherk-Schwarz reduction on Type IIB, mapping both theories onto the same five-dimensional theory. We also comment shortly on the difficulties arising if one intends to construct a massive generalisation of the six-dimensional string-string duality. 
  The existence of future horizons in spacetime geometries poses serious problems for string theory and quantum field theories. The observation that the expansion of the universe is accelerating has recently been shown to lead to a crisis for the mathematical formalism of string and M-theories, since the existence of a future horizon for an eternally accelerating universe does not allow the formulation of physical S-matrix observables. Postulating that the speed of light varies in an expanding universe in the future as well as in the past can eliminate future horizons, allowing for a consistent definition of S-matrix observables. 
  We investigate the possibility that the dimension 2 condensate A_mu^2 has a non zero non-perturbative value in Yang-Mills theory. We introduce a multiplicatively renormalisable effective potential for this condensate and show through two loop calculations that a non zero condensate is energetically favoured. 
  We study the SO(4)-symmetric solution of the five-dimensional SU(2) x U(1) gauged N=4 supergravity theory obtained in [hep-th/0101202]. This solution contains purely magnetic non-Abelian and electric Abelian fields. It can be interpreted as a reduction of seven-dimensional gauged supergravity on a torus, which comes from type IIB supergravity on S^3. We also show how to obtain that solution from six-dimensional Romans' theory on a circle. We then up-lift the solution to massless type IIA supergravity. The dual gauge field theory is twisted and is defined on the worldvolume of a NS-fivebrane wrapped on S^3. Two other spatial directions of the NS-fivebrane are on a torus. In the IR limit it corresponds to a three-dimensional gauge field theory with two supercharges. 
  A special class of solutions to the generalised WDVV equations related to a finite set of covectors is considered. We describe the geometric conditions ($\vee$-conditions) on such a set which are necessary and sufficient for the corresponding function to satisfy the generalised WDVV equations. These conditions are satisfied for all Coxeter systems but there are also other examples discovered in the theory of the generalised Calogero-Moser systems. As a result some new solutions for the generalized WDVV equations are found. 
  In the No-boundary Universe with $d=11$ supergravity, under the $S_n \times S_{11-n}$ Kaluza-Klein ansatz, the only seed instanton for the universe creation is a $S_7 \times S_4$ space. It is proven that for the Freund-Rubin, Englert and Awada-Duff-Pope models the macroscopic universe in which we are living must be 4- instead of 7-dimensional without appealing to the anthropic principle. 
  In this work we present a possible way to relate the method of covariantizing the gauge dependent pole and the negative dimensional integration method for computing Feynman integrals pertinent to the light-cone gauge fields. Both techniques are applicable to the algebraic light-cone gauge and dispense with prescriptions to treat the characteristic poles. 
  We consider the generalization to String and M-theory of the Melvin solution. These are flux p-branes which have (p+1)-dimensional Poincare invariance and are associated to an electric (p+1)-form field strength along their worldvolume. When a stack of Dp-branes is placed along the worldvolume of a flux (p+3)-brane it will expand to a spherical D(p+2)-brane due to the dielectric effect. This provides a new setup to consider the gauge theory/gravity duality. Compactifying M-theory on a circle we find the exact gravity solution of the type IIA theory describing the dielectric expansion of N D4-branes into a spherical bound state of D4-D6-branes, due to the presence of a flux 7-brane. In the decoupling limit, the deformation of the dual field theory associated with the presence of the flux brane is irrelevant in the UV. We calculate the gravitational radius and energy of the dielectric brane which give, respectively, a prediction for the VEV of scalars and vacuum energy of the dual field theory. Consideration of a spherical D6-brane probe with n units of D4-brane charge in the dielectric brane geometry suggests that the dual theory arises as the Scherk-Schwarz reduction of the M5-branes (2,0) conformal field theory. The probe potential has one minimum placed at the locus of the bulk dielectric brane and another associated to an inner dielectric brane shell. 
  We show that the triviality of the entire cohomology of the new BRST operator Q around the tachyon vacuum is equivalent to the Q-exactness of the identity I of the star-algebra. We use level truncation to show that as the level is increased, the identity becomes more accurately Q-exact. We carry our computations up to level nine, where an accuracy of 3% is attained. Our work supports, under a new light, Sen's conjecture concerning the absence of open string degrees of freedom around the tachyon vacuum. As a by-product, a new and simple expression for I in terms of Virasoro operators is found. 
  A systematic presentation of spinors in various dimensions is given. 
  The Hamiltonian analysis of Polyakov action is reviewed putting emphasis in two topics: Dirac observables and gauge conditions. In the case of the closed string it is computed the change of its action induced by the gauge transformation coming from the first class constraints. As expected, the Hamiltonian action is not gauge invariant due to the Hamiltonian constraint quadratic in the momenta. However, it is possible to add a boundary term to the original action to build a fully gauge-invariant action at first order. In addition, two relatives of string theory whose actions are fully gauge-invariant under the gauge symmetry involved when the spatial slice is closed are built. The first one is pure diffeomorphism in the sense it has no Hamiltonian constraint and thus bosonic string theory becomes a sub-sector of its space of solutions. The second one is associated with the tensionless bosonic string, its boundary term induces a canonical transformation and the fully gauge-invariant action written in terms of the new canonical variables becomes linear in the momenta. 
  We borrow the minisuperspace approximation from Quantum Cosmology and the quenching approximation from QCD in order to derive a new form of the bosonic p-brane propagator. In this new approximation we obtain an exact description of both the collective mode deformation of the brane and the center of mass dynamics in the target spacetime. The collective mode dynamics is a generalization of string dynamics in terms of area variables. The final result is that the evolution of a p-brane in the quenched-minisuperspace approximation is formally equivalent to the effective motion of a particle in a spacetime where points as well as hypersurfaces are considered on the same footing as fundamental geometrical objects. This geometric equivalence leads us to define a new tension-shell condition that is a direct extension of the Klein-Gordon condition for material particles to the case of a physical p-brane. 
  The non-renormalization theorem of chiral vertices and the generalized non-renormalization theorem of the photon self energy are derived in SQED on the basis of algebraic renormalization. For this purpose the gauge coupling is extended to an external superfield. This extension already provides detailed insight into the divergence structure. Moreover, using the local supercoupling together with an additional external vector multiplet that couples to the axial current, the model becomes complete in the sense of multiplicative renormalization, with two important implications. First, a Slavnov--Taylor identity describing supersymmetry, gauge symmetry, and axial symmetry including the axial anomaly can be established to all orders. Second, from this Slavnov-Taylor identity we can infer a Callan-Symanzik equation expressing all aspects of the non-renormalization theorems. In particular, the gauge $\beta$-function appears explicitely in the closed form. 
  Siegel's action is generalized to the D=2(p+1) (p even) dimensional space-time. The investigation of self-duality of chiral p-forms is extended to the momentum frame, using Siegel's action of chiral bosons in two space-time dimensions and its generalization in higher dimensions as examples. The whole procedure of investigation is realized in the momentum space which relates to the configuration space through the Fourier transformation of fields. These actions correspond to non-local Lagrangians in the momentum frame. The self-duality of them with respect to dualization of chiral fields is uncovered. The relationship between two self-dual tensors in momentum space, whose similar form appears in configuration space, plays an important role in the calculation, that is, its application realizes solving algebraically an integral equation. 
  Feynman's path integral in adelic quantum mechanics is considered. The propagator K(x'',t'';x',t') for one-dimensional adelic systems with quadratic Lagrangians is analytically evaluated. Obtained exact general formula has the form which is invariant under interchange of the number fields R and Q_p. 
  Compactifications on tori may seem to have revealed their beauty long ago but the mystery of 11d Supergravity remains and fresh attempts at a conceptual breakthrough are worth the effort and quite timely. We shall concentrate here on the analogy with Instanton Mathematics and discuss some open questions and work in progress. 
  We demonstrate that the initial conditions for inflation are met when D5 and \bar{D}5 branes annihilate. This scenario uses Sen's conjecture that a co-dimension two vortex forms on the worldvolume of the annihilated 5-brane system. Analogous to a "Big Bang", when the five branes annihilate, a vortex localized on a 3-brane forms and its false vacuum energy generates an inflationy space-time. We also provide a natural mechanism for ending inflation via the motion of the vortex in the bulk due to its extrinsic curvature. We also suggest a consistent way to end inflation and localize matter on our space-time. 
  We consider the critical alternating quantum spin chain with ${q_{+}\over 2}$, ${q_{-} \over2}$ spins. Using the Bethe ansatz technique we find explicit expressions for the $S$-matrix of the model. We show that in the limit that $q_{\pm} \rightarrow \infty$ our results coincide with the ones obtained for the principal chiral model level one, for the LL (RR) LR scattering. We also study the scattering of the bound states of the model and we recover the results of the XXZ (sine -Gordon) model. 
  We recall how the idea of Softly Broken Supersymmetry led to the construction of the Supersymmetric Standard Model in 1981. Its first prediction, the supersymmetric unification of gauge couplings, was conclusively verified by the LEP and SLC experiments 10 years later. Its other predictions include: the existence of superparticles at the electroweak scale; a stable lightest superparticle (LSP) with a mass of $\sim 100$ GeV, anticipated to be a neutral electroweak gaugino; the universality of scalar and gaugino masses at the unification scale. The original motivation for the model, solving the hierarchy problem, indicates that the superparticles should be discovered at the LHC or the TeVatron. 
  We discuss dual formulations of D-brane intersections. The duality is between world volume field theories of different dimensionalities which both describe the same D-brane configuration but are valid in complementary regions of parameter space. We discuss the duality in terms of bion configurations involving D-strings orthogonally intersecting both D3-branes and D5-branes. 
  We show that the modified Cardy-Verlinde formula without the Casimir effect term is satisfied by asymptotically flat charged black holes in arbitrary dimensions. Thermodynamic quantities of the charged black holes are shown to satisfy the energy-temperature relation of a two-dimensional CFT, which supports the claim in our previous work (Phys. Rev. D61, 044013, hep-th/9910244) that thermodynamics of charged black holes in higher dimensions can be effectively described by two-dimensional theories. We also check the Cardy formula for the two-dimensional black hole compactified from a dilatonic charged black hole in higher dimensions. 
  We consider quantum electrodynamics with an additional Lorentz- and CPT-violating axial-vector term in the fermionic sector and discuss the possibility that radiative corrections induce a Lorentz- and CPT-violating Chern-Simons-like term for the gauge field. From the requirement of causality and the assumed validity of perturbation theory, we conclude that the induced Chern-Simons-like term must be absent in the full quantum field theory. 
  We show that one can construct D-branes in parafermionic and WZW theories (and their orbifolds) which have very natural geometrical interpretations, and yet are not automatically included in the standard Cardy construction of D-branes in rational conformal field theory. The relation between these theories and their T-dual description leads to an analogy between these D-branes and the familiar A-branes and B-branes of N=2 theories. 
  The quark-monopole potential is computed at finite temperature in the context of $AdS/CFT$ correspondence. It is found that the potential is invariant under $g \to 1/g$ and $U_T \to U_T / g$. As in the quark-quark case there exists a maximum separation between quark and monopole, and $L$-dependence of the potential exhibits a bifurcation behavior. We find a functional relation $dE_{QM}^{Reg} / dL = [(1/E_{(1,0)}^{Reg}(U_0))^2 + (1/E_{(0,1)}^{Reg}(U_0))^2]^{-1/2}$ which is responsible for the bifurcation. The remarkable property of this relation is that it makes a relation between physical quantities defined at the $AdS$ boundary through a quantity defined at the bulk. The physical implication of this relation for the existence of the extra dimension is speculated. 
  We consider the motion of a particle described by an action that is a functional of the Frenet-Serret [FS] curvatures associated with the embedding of its worldline in Minkowski space. We develop a theory of deformations tailored to the FS frame. Both the Euler-Lagrange equations and the physical invariants of the motion associated with the Poincar\'e symmetry of Minkowski space, the mass and the spin of the particle, are expressed in a simple way in terms of these curvatures. The simplest non-trivial model of this form, with the lagrangian depending on the first FS (or geodesic) curvature, is integrable. We show how this integrability can be deduced from the Poincar\'e invariants of the motion. We go on to explore the structure of these invariants in higher-order models. In particular, the integrability of the model described by a lagrangian that is a function of the second FS curvature (or torsion) is established in a three dimensional ambient spacetime. 
  We use the recursive structure of the compactification of the instanton moduli space of N=2 Super Yang-Mills theory with gauge group SU(2), to construct, by inductive limit, a universal moduli space which includes all the multi-instanton moduli spaces. Furthermore, with the aim of understanding the field theoretic structure of the strong coupling expansion, we perform the Borel sum which acts on the parameter defining such a universal moduli space. 
  We discuss the Coulomb propagator in the formalism developed recently in which we construct the Coulomb gauge path-integral by correlating it with the well-defined Lorentz gauge path-integrals through a finite field-dependent BRS transformation. We discover several features of the Coulomb gauge from it. We find that the singular Coulomb gauge HAS to be treated as the gauge parameter lambda --> 0 limit. We further find that the propagator so obtained has good high energy behavior (k_0^{-2}) for lambda and epsilon nonzero. We further find that the behavior of the propagator so obtained is sensitive to the order of limits k_0 -->infinity, lambda -->0 and epsilon --> 0; so that these have to be handled carefully in a higher loop calculation. We show that we can arrive at the result of Cheng and Tsai for the ambiguous two loop Feynman integrals without the need for an extra ad hoc regularization and within the path integral formulation. 
  We study the dynamical symmetry algebra of the N-body Calogero model describing the structure of degenerate levels and demonstrate that the algebra is intrisically polynomial. We discuss some general properties of an algebra of S_N-symmetric operators acting on the S_N-symmetric subspace of the Fock space for any statistical parameter \nu. In the bosonic case (\nu=0) we find the algebra of generators for every N. For \nu\neq 0, we explicitly reproduce the finite algebra for the 4-particle model, demonstrating some general features of our construction. 
  We study non-supersymmetric SO(3)-invariant deformations of d=3, N=8 super Yang-Mills theory and their type IIA string theory dual. By adding both gaugino mass and scalar mass, dielectric D4-brane potential coincides with D5-brane potential in type IIB theory. We find the region of parameter space where the non-supersymmetric vacuum is described by stable dielectric NS5-branes. By considering the generalized action for NS5-branes in the presence of D4-flux, we also analyze the properties of dielectric NS5-branes. 
  We apply the methods recently developed for computation of type IIA disk instantons using mirror symmetry to a large class of D-branes wrapped over Lagrangian cycles of non-compact Calabi-Yau 3-folds. Along the way we clarify the notion of ``flat coordinates'' for the boundary theory. We also discover an integer IR ambiguity needed to define the quantum theory of D-branes wrapped over non-compact Lagrangian submanifolds. In the large $N$ dual Chern-Simons theory, this ambiguity is mapped to the UV choice of the framing of the knot. In a type IIB dual description involving $(p,q)$ 5-branes, disk instantons of type IIA get mapped to $(p,q)$ string instantons. The M-theory lift of these results lead to computation of superpotential terms generated by M2 brane instantons wrapped over 3-cycles of certain manifolds of $G_2$ holonomy. 
  The fusion products of admissible representations of the su(2) WZW model at the fractional level k=-4/3 are analysed. It is found that some fusion products define representations for which the spectrum of L_0 is not bounded from below. Furthermore, the fusion products generate representations that are not completely reducible and for which the action of L_0 is not diagonalisable. The complete set of representations that is closed under fusion is identified, and the corresponding fusion rules are derived. 
  We study the stability of Freund-Rubin compactifications, AdS_p x M_q, of p+q-dimensional gravity theories with a q-form field strength and no cosmological term. We show that the general AdS_p x S^q vacuum is classically stable against small fluctuations, in the sense that all modes satisfy the Breitenlohner-Freedman bound. In particular, the compactifications used in the recent discussion of the proposed bosonic M-theory are perturbatively stable. Our analysis treats all modes arising from the graviton and the q-form, and is completely independent of supersymmetry. From the masses of the linearized perturbations, we obtain the dimensions of some operators in possible holographic dual CFT's. Solutions with more general compact Einstein spaces need not be stable, and in particular AdS_p x S^n x S^{q-n} is unstable for q < 9 but is stable for q >= 9. We also study the AdS_4 x S^6 compactification of massive type IIA supergravity, which differs from the usual Freund-Rubin compactification in that there is a cosmological term already in ten dimensions. This nonsupersymmetric vacuum is unstable. 
  It has been proposed that Randall-Sundrum models can be holographically described by a regularized (broken) conformal field theory. We analyze the foundations of this duality using a regularized version of the AdS/CFT correspondence. We compare two- and three-point correlation functions and find the same behaviour in both descriptions. In particular, we show that the regularization of the deformed CFT generates kinetic terms for the sources, which hence can be naturally treated as dynamical fields. We also discuss the counterterms required for two- and three-point correlators in the renormalized AdS/CFT correspondence. 
  We study the large $N$ limit of a little string theory that reduces in the IR to U(N) ${\cal N} =1$ supersymmetric Yang-Mills with Chern Simons coupling $k$. Witten has shown that this field theory preserves supersymmetry if $k\geq N/2$ and he conjectured that it breaks supersymmetry if $k< N/2$. We find a non-singular solution that describes the $k=N/2$ case, which is confining. We argue that increasing $k$ corresponds to adding branes to this solution, in a way that preserves supersymmetry, while decreasing $k$ corresponds to adding anti-branes, and therefore breaking supersymmetry. 
  The ten-dimensional superparticle is covariantly quantized by constructing a BRST operator from the fermionic Green-Schwarz constraints and a bosonic pure spinor variable. This same method was recently used for covariantly quantizing the superstring, and it is hoped that the simpler case of the superparticle will be useful for those who want to study this quantization method. It is interesting that quantization of the superparticle action closely resembles quantization of the worldline action for Chern-Simons theory. 
  Matrix models have been shown to be equivalent to noncommutative field theories. In this work we study noncommutative X-Y model and try to understand Kosterlitz Thouless transition in it by analysing the equivalent matrix model. We consider the cases of a finite lattice and infinite lattice separately. We show that the critical value of the matrix model coupling is identical for the finite and infinite lattice cases. However, the critical values of the coupling of the continuum field theory, in the large $ N $ limit, is finite in the infinite lattice case and zero in the case of finite lattice. 
  We analyze different types of quantum corrections to the Cardy-Verlinde entropy formula in a Friedmann-Robertson-Walker universe and in an (anti)-de Sitter space. In all cases we show that quantum corrections can be represented by an effective cosmological constant which is then used to redefine the parameters entering the Cardy-Verlinde formula so that it becomes valid also with quantum corrections, a fact that we interpret as a further indication of its universality. A proposed relation between Cardy-Verlinde formula and the ADM Hamiltonian constraint is given. 
  We study SU(N) supersymmetric Yang-Mills theory with massless adjoint matter defined on $M^3\otimes S^1$. The SU(N) gauge symmetry is broken maximally to $U(1)^{N-1}$, independent of the number of flavor and the boundary conditions of the fields associated with the Scherk-Schwarz mechanism of supersymmetry breaking. The mass of the Higgs scalar is generated through quantum corrections in the extra dimensions. The quantum correction can become manifest by a finite Higgs boson mass at low energies even in the limit of small extra dimensions thanks to the supersymmetry breaking parameter of the Scherk-Schwarz mechanism. 
  We study how the effect of closed-string tachyon condensation can enter into the on-shell effective action of open-string tachyons in the bosonic case. We also consider open-string one-loop quantum corrections to the on-shell action. We use a sigma-model approach with boundary terms, and we utilize some results of boundary string field theory (BSFT) to define the on-shell effective action. We regard D-instanton-like objects with appropriate weight as closed-string tachyon tadpoles, and we insert them into worldsheets to analyze the effect of closed-string tachyons. 
  NSR superstring theory contains a tower of physical vertex operators (brane-like states) which exist at non-zero pictures only, i.e. are essentially mixed with superconformal ghosts. Some of these states are massless, they are responsible for creating D-branes. Other states are tachyonic (called ghost tachyons) creating a problem for the vacuum stability of the NSR model. In this paper we explore the role played by these tachyonic states in string dynamics. We show that the ghost tachyons condense on D-branes, created by massless brane-like states. Thus the vacuum stability is achieved dynamically, as the effective ghost tachyon potential exactly cancels the D-brane tension, in full analogy with Sen's mechanism. As a result, from perturbative NSR model point of view, massless and tachyonic brane-like states appear to live in a parallel world, as the brane is screened by the tachyonic veil. We extend the analysis to the brane-antibrane pair in AdS space and show that in this case due to the effect of the ghost tachyon condensation one can construct extra time-dimensional phenomenological models without tachyons and antibranes. 
  We study T-duality for open strings on tori $\T^d$. The general boundary conditions for the open strings are constructed, and it is shown that T-duality group, which preserves the mass spectrum of closed strings, preserves also the mass spectrum of the open strings. The open strings are transformed to those with different boundary conditions by T-duality. We also discuss the T-duality for D-brane mass spectrum, and show that the D-branes and the open strings with both ends on them are transformed together consistently. 
  We investigate the equivalence between Thirring model and sine-Gordon model in the chirally broken phase of the Thirring model. This is unlike all other available approaches where the fermion fields of the Thirring model were quantized in the chiral symmetric phase. In the path integral approach we show that the bosonized version of the massless Thirring model is described by a quantum field theory of a massless scalar field and exactly solvable, and the massive Thirring model bosonizes to the sine-Gordon model with a new relation between coupling constants. We show that the non-perturbative vacuum of the chirally broken phase in the massless Thirring model can be described in complete analogy with the BCS ground state of superconductivity. The Mermin-Wagner theorem and Coleman's statement concerning the absence of Goldstone bosons in the 1+1-dimensional quantum field theories are discussed. We investigate the current algebra in the massless Thirring model and give a new value of the Schwinger term. We show that the topological current in the sine-Gordon model coincides with the Noether current responsible for the conservation of the fermion number in the Thirring model. This allows to identify the topological charge in the sine-Gordon model with the fermion number. 
  A sliver state is a classical solution of the string field theory of the tachyon vacuum that represents a background with a single D25-brane. We show that the sliver wavefunctional factors into functionals of the left and right halves of the string, and hence can be naturally regarded as a rank-one projector in a space of half-string functionals. By developing an algebraic oscillator approach we are able construct higher rank projectors that describe configurations of multiple D-branes of various dimensionalities and located at arbitrary positions. The results bear remarkable similarities with non-commutative solitons. 
  We describe projection operators in the matter sector of Witten's cubic string field theory using modes on the right and left halves of the string. These projection operators represent a step towards an analytic solution of the equations of motion of the full string field theory, and can be used to construct Dp-brane solutions of the string field theory when the BRST operator Q is taken to be pure ghost, as suggested in the recent conjecture by Rastelli, Sen and Zwiebach. We show that a family of solutions related to the sliver state are rank one projection operators on the appropriate space of half-string functionals, and we construct higher rank projection operators corresponding to configurations of multiple D-branes. 
  A BRST perturbative analysis of SU(N) Yang-Mills theory in a class of maximal Abelian gauges is presented. We point out the existence of a new nonintegrated renormalizable Ward identity which allows to control the dependence of the theory from the diagonal ghosts. This identity, called the diagonal ghost equation, plays a crucial role for the stability of the model under radiative corrections implying, in particular, the vanishing of the anomalous dimension of the diagonal ghosts. Moreover, the Ward identity corresponding to the Abelian Cartan subgroup is easily derived from the diagonal ghost equation. Finally, a simple proof of the fact that the beta function of the gauge coupling can be obtained from the vacuum polarization tensor with diagonal gauge fields as external legs is given. A possible mechanism for the decoupling of the diagonal ghosts at low energy is also suggested. 
  We discuss a generalized form of IIA/IIB supergravity depending on all R-R potentials C^(p) (p=0,1,...,9) as the effective field theory of Type IIA/IIB superstring theory. For the IIA case we explicitly break this R-R democracy to either p<=3 or p>=5 which allows us to write a new bulk action that can be coupled to N=1 supersymmetric brane actions.   The case of 8-branes is studied in detail using the new bulk & brane action. The supersymmetric negative tension branes without matter excitations can be viewed as orientifolds in the effective action. These D8-branes and O8-planes are fundamental in Type I' string theory. A BPS 8-brane solution is given which satisfies the jump conditions on the wall. As an application of our results we derive a quantization of the mass parameter and the cosmological constant in string units. 
  We describe the construction of new configurations of self-gravitating p-branes with worldvolume geometries of the form R^{1,p-s} x S^s, with 1\leq s\leq p, ie, tubular branes. Since such branes are typically unstable against collapse of the sphere, they must be held in equilibrium by a fluxbrane. We present solutions for string loops with non-singular horizons, as well as M5-branes intersecting over such loops. We also construct tubular branes which carry in their worldvolume a dissolved, lower dimensional brane (as in the dielectric effect), or an F-string. However, the connection between our solutions and related configurations that have been studied earlier in the absence of brane self-gravity, is unclear. It is argued that, at least in some instances, the self-gravitating solutions do not appear to be able to reproduce stable configurations of tubular branes. 
  I point out that (BPS saturated) A-type D-branes in superstring compactifications on Calabi-Yau threefolds correspond to {\em graded} special Lagrangian submanifolds, a particular case of the graded Lagrangian submanifolds considered by M. Kontsevich and P. Seidel. Combining this with the categorical formulation of cubic string field theory in the presence of D-branes, I consider a collection of {\em topological} D-branes wrapped over the same Lagrangian cycle and {\em derive} its string field action from first principles. The result is a {\em $\Z$-graded} version of super-Chern-Simons field theory living on the Lagrangian cycle, whose relevant string field is a degree one superconnection in a $\Z$-graded superbundle, in the sense previously considered in mathematical work of J. M. Bismutt and J. Lott. This gives a refined (and modified) version of a proposal previously made by C. Vafa. I analyze the vacuum deformations of this theory and relate them to topological D-brane composite formation, by using the general formalism developed in a previous paper. This allows me to identify a large class of topological D-brane composites (generalized, or `exotic' topological D-branes) which do not admit a traditional description. Among these are objects which correspond to the `covariantly constant sequences of flat bundles' considered by Bismut and Lott, as well as more general structures, which are related to the enhanced triangulated categories of Bondal and Kapranov. I also give a rough sketch of the relation between this construction and the large radius limit of a certain version of the `derived category of Fukaya's category'. 
  The Sp(2)-gauge fixing of N = 1 super-Yang-Mills theory is considered here. We thereby apply the triplectic scheme, where two classes of gauge-fixing bosons are introduced. The first one depends only on the gauge field, whereas the second boson depends on this gauge field and also on a pair of Majorana fermions. In this sense, we build up the BRST extended (BRST plus antiBRST) algebras for the model, for which the nilpotency relations, s^2_1=s^2_2=s_1s_2+s_2s_1=0, hold. 
  We briefly review the results of our paper hep-th/0009013: we study certain perturbative solutions of left-unilateral matrix equations. These are algebraic equations where the coefficients and the unknown are square matrices of the same order, or, more abstractly, elements of an associative, but possibly noncommutative algebra, and all coefficients are on the left. Recently such equations have appeared in a discussion of generalized Born-Infeld theories. In particular, two equations, their perturbative solutions and the relation between them are studied, applying a unified approach based on the generalized Bezout theorem for matrix polynomials. 
  We study Vafa's geometric transition from a brane setup in M-theory. In this transition D5 branes wrapped on P^1 cycles of a resolved conifold disappear and are replaced by fluxes on a deformed conifold. In the limit of small sized P^1, we describe this mechanism as a transition from curved M5 branes to plane M5 branes which replaces SU(N) MQCD by U(1) theories on the bulk. This agrees with the results expected from the geometric transition. We also discuss the reduction to ten dimensions and a brane creation mechanism in the presence of fluxes. 
  A model considered in the paper generalizes membrane theory to the case of delocalized membranes. The model admits covariant formulation, which involves no constraints. It generalizes the notion of membrane to the case of smooth distribution of non-intersecting membranes. A generalization of p-brane solution with delocalized membranes is presented. 
  d5 dilatonic gravity action with surface counterterms motivated by AdS/CFT correspondence and with contributions of brane quantum CFTs is considered around AdS-like bulk. The effective equations of motion are constructed. They admit two (outer and inner) or multi-brane solutions where brane CFTs may be different. The role of quantum brane CFT is in inducing of complicated brane dilatonic gravity. For exponential bulk potentials the number of AdS-like bulk spaces is found in analytical form. The correspondent flat or curved (de Sitter or hyperbolic) dilatonic two branes are created, as a rule, thanks to quantum effects. The observable early Universe may correspond to inflationary brane. The found dilatonic quantum two brane-worlds usually contain the naked singularity but in couple explicit examples the curvature is finite and horizon (corresponding to wormhole-like space) appears. 
  Recently, using a local action satisfying the Wess-Zumino condition as a kinetic term of the conformal mode, we formulated a four-dimensional quantum geometry (4DQG). The conformal mode can be treated exactly, and it was shown that the part of the effective action related to this mode is given by the scale-invariant non-local Riegert action. As for the traceless mode, we introduce dimensionless coupling, which is a unique gravitational coupling of this theory satisfying the conditions of renormalizability and asymptotic freedom. Although this theory is asymptotically free, the physical states are non-trivial, which should be described as composite fields, like the spectrum of 2DQG. The possibility that the physical state conditions representing background-metric independence conceal ghosts is pointed out. The usual graviton state would be realized when the physical state condition breaks down dynamically. 
  We obtain a Bekenstein entropy bound for the charged objects in arbitrary dimensions ($D\ge 4$) using the D-bound recently proposed by Bousso. With the help of thermodynamics of CFTs corresponding to AdS Reissner-Norstr\"om (RN) black holes, we discuss the relation between the Bekenstein and Bekenstein-Verlinde bounds. In particular we propose a Bekenstein-Verlinde-like bound for the charged systems. In the Einstein-Maxwell theory with a negative cosmological constant, we discuss the brane cosmology with positive tension using the Binetruy-Deffayet-Langlois approach. The resulting Friedman-Robertson-Walker equation can be identified with the one obtained by the moving domain wall approach in the AdS RN black hole background. Finally we also address the holographic property of the brane universe. 
  Due to the nonvanishing average photon population of the squeezed vacuum state, finite corrections to the scattering matrix are obtained. The lowest order contribution to the electron mass shift for a one mode squeezed vacuum state is given by $\delta m(\Omega, s)/m=\alpha (2/\pi)(\Omega /m)^2\sinh^2(s)$, where $\Omega$ and $s$ stand for the mode frequency and the squeeze parameter and $\alpha$ for the fine structure constant, respectively. 
  We construct geometrically a gerbe assigned to a connection on a principal SU(2)-bundle over an oriented closed 1-dimensional manifold. If the connection is given by the restriction of a connection on a bundle over a compact 2-manifold bounding the 1-manifold, then we have a natural object in the gerbe. The gerbes and the objects satisfy certain fundamental properties, e.g. gluing law. 
  Tensor reduction of vacuum diagrams uses contraction and decomposition matrices. We present general recurrence relations for the calculation of those matrices and an explicit formula for the 3-loop decomposition matrix and its determinant. 
  It is demonstrated that the breather solutions recently discovered in the weakly coupled topological discrete sine-Gordon system are spectrally unstable. This is in contrast with more conventional spatially discrete systems, whose breathers are always spectrally stable at sufficiently weak coupling. 
  We study the Coulomb-Higgs duality of N=2 supersymmetric Abelian Chern-Simons theories in 2+1 dimensions, by compactifying dual pairs on a circle of radius R and comparing the resulting N=(2,2) theories in 1+1 dimensions. Below the compactification scale, the theory on the Higgs branch reduces to the non-linear sigma model on a toric manifold. In the dual theory on the Coulomb branch, the Kaluza-Klein modes generate an infinite tower of contributions to the superpotential. After resummation, in the limit R->0 the superpotential becomes that of the Landau-Ginzburg model which is the two-dimensional mirror of the toric sigma model. We further examine the conjecture of all-scale three-dimensional mirror symmetry and observe that it is consistent with mirror symmetry in 1+1 dimensions. 
  I review the physical properties of different vacua in the background independent open string field theory. Talk presented at Strings 2001, Mumbai, India, http://theory.theory.tifr.res.in/strings/Proceedings/#sha-s. 
  The enhancon mechanism removes a family of time-like singularities from certain supergravity spacetimes by forming a shell of branes on which the exterior geometry terminates. The problematic interior geometry is replaced by a new spacetime, which in the prototype extremal case is simply flat. We show that this excision process, made inevitable by stringy phenomena such as enhanced gauge symmetry and the vanishing of certain D-branes' tension at the shell, is also consistent at the purely gravitational level. The source introduced at the excision surface between the interior and exterior geometries behaves exactly as a shell of wrapped D6-branes, and in particular, the tension vanishes at precisely the enhancon radius. These observations can be generalised, and we present the case for non-extremal generalisations of the geometry, showing that the procedure allows for the possibility that the interior geometry contains an horizon. Further knowledge of the dynamics of the enhancon shell itself is needed to determine the precise position of the horizon, and to uncover a complete physical interpretation of the solutions. 
  We consider an integrable conformally invariant two dimensional model associated to the affine Kac-Moody algebra SL(3). It possesses four scalar fields and six Dirac spinors. The theory does not possesses a local Lagrangian since the spinor equations of motion present interaction terms which are bilinear in the spinors. There exists a submodel presenting an equivalence between a U(1) vector current and a topological current, which leads to a confinement of the spinors inside the solitons. We calculate the one-soliton and two-soliton solutions using a procedure which is a hybrid of the dressing and Hirota methods. The soliton masses and time delays due to the soliton interactions are also calculated. We give a computer program to calculate the soliton solutions. 
  It has been shown that D0-D$p$ $(p=2, 4, 6, 8)$ systems can be BPS in the presence of $B$-field even if they are not otherwise. We review the number of remaining supersymmetries, the open string ground state spectrum and the construction of the D0-D$p$ systems as solitonic solutions in the noncommutative super Yang-Mills theory. We derive the complete mass spectrum of the fluctuations to discuss the stability of the systems. The results are found to agree with the analysis in the string picture. In particular, we show that supersymmetry is enhanced in D0-D8 depending on the $B$-fields and it is consistent with the degeneracy of mass spectrum. We also derive potentials and discuss their implications for these systems. 
  We discuss extra timelike dimensions and their effects on the gravitational stability of spherical massive bodies. Here we specifically report our results for the case of one extra timelike dimension where we have made analytically rigorous investigations on the tachyonic graviton exchange due to the infinite tower of the Kaluza-Klein mode. With the scale $L$ of the extra timelike dimension we find that some spherical bodies of radius $R$ can be stable at critical radii $R=2\pi Lp$ for some positive integer $p$. We also obtain the generic property of massive bodies that for the short distance range $0<R\leq \pi L$ the gravitational force due to the ordinary massless graviton exchange is screened by the Kaluza-Klein mode exchange of tachyonic gravitons. 
  A proof is given of Polyakov conjecture about the accessory parameters of the SU(1,1) Riemann-Hilbert problem for general elliptic singularities on the Riemann sphere. Its relevance to 2+1 dimensional gravity is stressed. 
  The role of Lorentz symmetry in noncommutative field theory is considered. Any realistic noncommutative theory is found to be physically equivalent to a subset of a general Lorentz-violating standard-model extension involving ordinary fields. Some theoretical consequences are discussed. Existing experiments bound the scale of the noncommutativity parameter to (10 TeV)^{-2}. 
  By using O(7,7) transformations, to deform D6-branes, we obtain half-supersymmetric bound state solutions of type IIA supergravity, containing D6, D4, D2, D0, F1-branes and waves. We lift the solutions to M-theory which gives half-supersymmetric M-theory bound states, e.g. KK6-M5-M5-M5-M2-M2-M2-MW. We also take near horizon limits for the type IIA solutions, which gives supergravity duals of 7-dimensional non-commutative open string theory (with space-time and space-space non-commutativity), non-commutative Yang-Mills theory (with space-space and light-like non-commutativity) and an open D4-brane theory. 
  In this paper we develop the formalism to study superforms in N=2 harmonic superspace. We perform a thorough (if not complete) analysis of the superforms starting from 0-form and moving all the way up to 6-form. Like the N=1 case we find that the lower superforms (0,1,2,3) describe the various important N=2 supermultiplets. Also, the forms form chains, the field strength of 0-form being related to the gauge form of the 1-form, and so on. However, an important difference with the N=1 case is that there is now more than one chain.   Our main aim was to study the higher-forms (4,5,6) to obtain the various N=2 action formulas via the ectoplasmic approach. Indeed, we reproduce the three known action formulas involving a volume integral, a contour integral and no integral over the harmonic subspace from the 6, 5 and 4-form analysis respectively. The next aim is to generalize the analysis to curved space-time (supergravity). 
  We discuss the 't Hoof ansatz for instanton solutions in noncommutative U(2) Yang-Mills theory. We show that the extension of the ansatz leading to singular solutions in the commutative case, yields to non self-dual (or self-antidual) configurations in noncommutative space-time. A proposal leading to selfdual solutions with Q=1 topological charge (the equivalent of the regular BPST ansatz) can be engineered, but in that case the gauge field and the curvature are not Hermitian (although the resulting Lagrangian is real). 
  We have developed a Mathematica package capable of performing gamma-matrix algebra in arbitrary (integer) dimensions. As an application we can compute Fierz transformations. 
  We construct the CP^n model on fuzzy sphere. The Bogomolny bound is saturated by (anti-)self-dual solitons and the general solutions of BPS equation are constructed. The dimension of moduli space describing the BPS solution on fuzzy sphere is exactly the same as that of the commutative sphere or the (noncommutative) plane. We show that in the soliton backgrounds, the number of zero modes of Dirac operator on fuzzy sphere, Atiyah-Singer index, is exactly given by the topological charge of the background solitons. 
  It is shown that an (n-1)-dimensional inflating brane world instantonically created from nothing can exist in the region beyond the Rindler horizon of the Lorentzian spacetime associated with another inflating brane world in n dimensions which is also instantonically created from nothing. Generalizing this construction we obtain an unbounded from above tower of successive brane worlds, each having one more dimension than the one which it nests and one less dimension than the one which nests it. 
  We show that the matrix formulation of non-commutative field theories is equivalent, in the continuum, to a formulation in a mixed configuration-momentum space. In this formulation, the non-locality of the interactions, that leads to the IR/UV mixing, becomes transparent. We clarify the relation between long range effects (and IR divergences) and the non-planarity of the corresponding Feynman diagrams. 
  We review four basic examples where string theory and/or field theory dualities predict the existence of soliton bound-states. These include the existence of threshold bound-states of D0 branes required by IIA/M duality and the closely-related bound-states of instantons in the maximally supersymmetric five dimensional gauge theory. In the IIB theory we discuss (p,q)-strings as bound-states of D and F strings, as well as the corresponding bound-states of monopoles and dyons in N=4 supersymmetric Yang-Mills theory whose existence was predicted by Sen. In particular we consider the L^2-index theory relevant for counting these states. In each case we show that the bulk contribution to the index can be evaluated by relating it to an instanton effect in the corresponding theory with a compact Euclidean time dimension. The boundary contribution to the index can be determined by considering the asymptotic regions of the relevant moduli space. 
  We consider a brane-world of co-dimension one without the reflection symmetry that is commonly imposed between the two sides of the brane. Using the coordinate-free formalism of the Gauss-Codacci equations, we derive the effective Einstein equations by relating the local curvature to the matter on the brane in the case when its bare tension is much larger than the localized matter, and hence show that Einstein gravity is a natural consequence of such models in the weak field limit. We find agreement with the recently derived cosmological case, which can be solved exactly, and point out that such models can be realized naturally in the case where there is a minimally coupled form field in the bulk. 
  We propose an all-orders beta function for current-current interactions in 2d with flavor anisotropy. When the number of left-moving and right-moving flavors are unequal, the beta function has a non-trivial fixed point at finite values of the couplings. We also extend the computation to simple cases with both flavor and color anisotropy. 
  We consider the brane universe in the background of the topological AdS-Schwarzschild black holes. The induced geometry of the brane is that of a flat or an open radiation dominated FRW-universe. Just like the case of a closed radiation dominated FRW-universe, the temperature and entropy are simply expressed in terms of the Hubble parameter and its time derivative when the brane crosses the black hole horizon. We propose the modified Cardy-Verlinde formula which is valid for any values of the curvature parameter k in the Friedmann equations. 
  We study dimensional reductions of noncommutative electrodynamics on flat space which lead to gauge theories of gravitation. For a general class of such reductions, we show that the noncommutative gauge fields naturally yield a Weitzenbock geometry on spacetime and that the induced diffeomorphism invariant field theory can be made equivalent to a teleparallel formulation of gravity which macroscopically describes general relativity. The Planck length is determined in this setting by the Yang-Mills coupling constant and the noncommutativity scale. The effective field theory can also contain higher-curvature and non-local terms which are characteristic of string theory. Some applications to D-brane dynamics and generalizations to include the coupling of ordinary Yang-Mills theory to gravity are also described. 
  We study `Myers effect' for a bunch of $D1$-branes with $IIB$ superstrings moving in one direction along the branes. We show that the `blown-up' configuration is the helical $D1$-brane, which is self-supported from collapse by the axial momentum flow. The tilting angle of the helix is determined by the number of $D1$-branes. The radius of the helix is stabilized to a certain value depending on the number of $D1$-branes and the momentum carried by $IIB$ superstrings. This is actually T-dual version of the supertube recently found as the `blown-up' configuration of a bunch of $IIA$ superstrings carrying $D0$-brane charge. It is found that the helical $D1$ configuration preserves one quarter of the supersymmetry of $IIB$ vacuum. 
  We construct supersymmetric M3-brane solutions in D=11 supergravity. They can be viewed as deformations of backgrounds taking the form of a direct product of four-dimensional Minkowski spacetime and a non-compact Ricci-flat manifold of G_2 holonomy. Although the 4-form field strength is turned on it carries no charge, and the 3-branes are correspondingly massless. We also obtain 3-branes of a different type, arising as M5-branes wrapped over S^2. 
  Warped compactifications with significant warping provide one of the few known mechanisms for naturally generating large hierarchies of physical scales. We demonstrate that this mechanism is realizable in string theory, and give examples involving orientifold compactifications of IIB string theory and F-theory compactifications on Calabi-Yau four-folds. In each case, the hierarchy of scales is fixed by a choice of RR and NS fluxes in the compact manifold. Our solutions involve compactifications of the Klebanov-Strassler gravity dual to a confining N=1 supersymmetric gauge theory,and the hierarchy reflects the small scale of chiral symmetry breaking in the dual gauge theory. 
  We compute the one-loop partition function for quadratic tachyon background in open string theory.   Both closed and open string representations are developed. Using these representations we study the one-loop divergences in the partition function in the presence of the tachyon background. The divergences due to the open and closed string tachyons are treated by analytic continuation in the tachyon mass squared. We pay particular attention to the imaginary part of the analytically continued expressions. The last one gives the decay rate of the unstable vacuum. The dilaton tadpole is also given some partial consideration. The partition function is further used to study corrections to tachyon condensation processes describing brane descent relations. Assuming the boundary string field theory prescription for construction of the string field action via partition function holds at one loop level we study the one-loop corrections to the tachyon potential and to the tensions of lower-dimensional branes. 
  In a classical Hamiltonian theory with second class constraints the phase space functions on the constraint surface are observables. We give general formulas for extended observables, which are expressions representing the observables in the enveloping unconstrained phase space. These expressions satisfy in the unconstrained phase space a Poisson algebra of the same form as the Dirac bracket algebra of the observables on the constraint surface. The general formulas involve new differential operators that differentiate the Dirac bracket. Similar extended observables are also constructed for theories with first class constraints which, however, are gauge dependent. For such theories one may also construct gauge invariant extensions with similar properties. Whenever extended observables exist the theory is expected to allow for a covariant quantization. A mapping procedure is proposed for covariant quantization of theories with second class constraints. 
  In this paper we deal with the cosmological dynamics of Randall-Sundrum brane-world type scenarios in which the five-dimensional Weyl tensor has a non-vanishing projection onto the three-brane where matter fields are confined. Using dynamical systems techniques, we study how the state space of Friedmann-Lemaitre-Robertson-Walker (FLRW) and Bianchi type I cosmological models is affected by the bulk Weyl tensor, focusing in the differences that appear with respect to general relativity and also Randall-Sundrum cosmological scenarios without the Weyl tensor contribution. 
  We study type I compactification on a 4-torus, with a non-trivial discrete background RR 4-form field. By using string dualities and recent insights for gauge theories on tori, we find that a non-trivial background for the RR 4-form is correlated with Spin(32)/Z_2 bundles that are described by a ``non-trivial quadruple'' of holonomies. We also briefly discuss other discrete moduli for the type I string, and variants of orientifold planes. 
  Basics of the geometrical formulation of the dynamics of supersymmetric objects are considered and its relation to conventional formulations of superbranes is discussed. In particular, we demonstrate how the kappa-symmetry of the Green-Schwarz formulation shows up from local worldvolume supersymmetry, and briefly discuss applications of the superembedding approach. 
  We consider adelic approach to strings and spatial noncommutativity. Path integral method to string amplitudes is emphasized. Uncertainties in spatial measurements in quantum gravity are related to noncommutativity between coordinates. p-Adic and adelic Moyal products are introduced. In particular, p-adic and adelic counterparts of some real noncommutative scalar solitons are constructed. 
  We study the ratio of the entropy to the total energy in conformal field theories at finite temperature. For the free field realizations of N=4 super Yang-Mills theory in D=4 and the (2,0) tensor multiplet in D=6, the ratio is bounded from above. The corresponding bounds are less stringent than the recently proposed Verlinde bound. For strongly coupled CFTs with AdS duals, we show that the ratio obeys the Verlinde bound even in the presence of rotation. For such CFTs, we point out an intriguing resemblance in their thermodynamic formulas with the corresponding ones of two-dimensional CFTs. The discussion is based on hep-th/0101076. 
  It is shown how to map the quantum states of a system of free scalar particles one-to-one onto the states of a completely deterministic model. It is a classical field theory with a large (global) gauge group. The mapping is now also applied to free Maxwell fields. Lorentz invariance is demonstrated. 
  The RR Page charges for the D(2p+1)-branes with B-field in type IIB supergravity are constructed consistently from brane source currents. The resulting Page charges are B-independent in the nontrivial and intricate way. It is found that in topologically trivial B-field the Page charge is conserved, but in the topologically nontrivial B-field it is no longer to be conserved, instead there is a jump between two Page charges defined in each patch, and we interpret this jump as Hanany-Witten effect. 
  We call superpartitions the indices of the eigenfunctions of the supersymmetric extension of the trigonometric Calogero-Moser-Sutherland model. We obtain an ordering on superpartitions from the explicit action of the model's Hamiltonian on monomial superfunctions. This allows to define Jack superpolynomials as the unique eigenfunctions of the model that decompose triangularly, with respect to this ordering, on the basis of monomial superfunctions. This further leads to a simple and explicit determinantal expression for the Jack superpolynomials. 
  We propose a string realization of the AdS4 brane in AdS5 that is known to localize gravity. Our theory is M D5 branes in the near horizon geometry of N D3 branes, where M and N are appropriately tuned. 
  In this paper a non-relativistic particle moving on a hypersurface in a curved space and the multidimensional rotator are investigated using the Hamilton-Jacobi formalism. The equivalence with the Dirac Hamiltonian formalism is demonstrated in both Cartesian and curvilinear coordinates. The energy spectrum of the multidimensional rotator is equal to that of a pure Laplace-Beltrami operator with no additional constant arising from the curvature of the sphere. 
  We elucidate the properties of a gas of free closed bosonic strings in thermal equilibrium. Our starting point is the intensive generating functional of connected one-loop closed vacuum string graphs given by the Polyakov path integral. Invariance of the path integral under modular transformations gives a thermal duality invariant expression for the free energy of free closed strings at finite temperature. The free bosonic string gas exhibits a self-dual Kosterlitz-Thouless phase transition. The thermodynamic potentials of the gas of free bosonic closed strings are shown to exhibit an infinite hierarchy of thermal self-duality relations. Note Added (Sep 2005). 
  Gregory and Laflamme showed that certain nonextremal black strings (and p-branes) are unstable to linearized perturbations. It is widely believed that this instability will cause the black string horizon to classically pinch off and then quantum mechanically separate, resulting in higher dimensional black holes. We argue that this cannot happen. Under very mild assumptions, classical event horizons cannot pinch off. Instead, they settle down to new static black string solutions which are not translationally invariant along the string. 
  We study quasinormal modes of massless scalar and fermion fields in the near extremal Reissner-Nordstr{\"o}m black hole, and relate them to Choptuik scaling form following a recently proposed analytic approach. For both massless cases, quasinormal modes are shown to be proportional to the black hole horizon and the Hawking temperature, and the critical exponents are the same, although for the fermionic case there are two possible discrete quasinormal modes. In addition, the critical exponent of the massive boson is also equivalent to that of the massless case. Finally, we discuss quasinormal modes and critical exponents in the other models, and obtain some different critical exponents between massless boson and massive one. 
  We investigate the dynamics of dilatonic D-dimensional 0-branes in the near-horizon regime. The theory is given in a twofold form: two-dimensional dilaton gravity and nonlinear sigma model. Using asymptotic symmetries, duality relations, and sigma model techniques we find that the theory has three conformal points which correspond to (a) the asymptotic (Anti-de Sitter) region of the two-dimensional spacetime, (b) the horizon of the black hole, and (c) the infinite limit of the coupling parameter. We show that the conformal symmetry is perturbatively preserved at one-loop, identify the corresponding conformal field theories, and calculate the associated central charges. Finally, we use the conformal field theories to explain the thermodynamical properties of the two-dimensional black holes. 
  The O(3) non-linear sigma model is investigated using multi Hamilton-Jacobi formalism. The integrability conditions are investigated and the results are in agreement with those obtained by Dirac's method. By choosing an adequate extension of phase space we describe the transformed system by a set of three Hamilton-Jacobi equations and calculate the corresponding action. 
  We discuss the tachyon condensation in a single unstable D-brane in the framework of boundary state formulation. The boundary state in the background of the tachyon condensation and the NS B-field is explicitly constructed. We show in both commutative theory and noncommutative theory that the unstable D-branes behaves like an extended object and eventually reduces to the lower dimensional D-branes as the system approaches the infrared fixed point. We clarify the relationship between the commutative field theoretical description of the tachyon condensation and the noncommutative one. 
  Five dimensional geodesic equation is used to study the gravitational force acted on a test particle in the bulk of the Randall-Sundrum two-brane model.This force could be interpreted as the gravitational attraction from matters on the two branes and may cause the model to be unstable. By analogy with star models in astrophysics, a fluid RS model is proposed in which the bulk is filled with a fluid and this fluid has an anisotropic pressure to balance the gravity from the two branes. Thus a class of exact bulk solutions is obtained which shows that any 4D Einstein solution with a perfect fluid source can be embedded in $y=$ constant hypersurfaces in the bulk to form an equilibrium state of the brane model. By requiring a 4D effective curvature to have a minimum, the compactification size of the extra dimension is discussed. 
  Higher derivative bulk gravity (without Riemann tensor square term) admits AdS-Schwarzschild black hole as exact solution. It is shown that induced brane geometry on such background is open, flat or closed FRW radiation dominated Universe. Higher derivative terms contributions appear in the Hawking temperature, entropy and Hubble parameter via the redefinition of 5-dimensional gravitational constant and AdS scale parameter. These higher derivative terms do not destroy the AdS-dual description of radiation represented by strongly-coupled CFT. Cardy-Verlinde formula which expresses cosmological entropy as square root from other parameters and entropies is derived in $R^2$ gravity. The corresponding cosmological entropy bounds are briefly discussed. 
  In this paper we exhibit a one-parameter family of new Taub-NUT instantons parameterized by a half-line. The endpoint of the half-line will be the reducible Yang-Mills instanton corresponding to the Eguchi-Hanson-Gibbons L^2 harmonic 2-form, while at an inner point we recover the Pope-Yuille instanton constructed as a projection of the Levi-Civita connection onto the positive su(2) subalgebra of the Lie algebra so(4).   Our method imitates the Jackiw-Nohl-Rebbi construction originally designed for flat R^4. That is we find a one-parameter family of harmonic functions on the Taub-NUT space with a point singularity, rescale the metric and project the obtained Levi-Civita connection onto the other negative su(2) subalgebra of so(4). Our solutions will possess the full U(2) symmetry, and thus provide more solutions to the recently proposed U(2) symmetric ansatz of Kim and Yoon. 
  Canonical differential calculus is defined for finitely generated abelian group with an involution existing consistently. Two such canonical calculi are found out. Fermionic representation for canonical calculus is defined based on quantized calculus. Fermionic representations for above-mentioned two canonical calculi are searched. 
  We propose a new approach to field theory on $\kappa$-Minkowski non-commutative space-time, a popular example of Lie-algebra space-time. Our proposal is essentially based on the introduction of a star product, a technique which is proving to be very fruitful in analogous studies of canonical non-commutative space-times, such as the ones recently found to play a role in the description of certain string-theory backgrounds. We find to be incorrect the expectation, previously reported in the literature, that the lack of symmetry of the $\kappa$-Poincare' coproduct should lead to interaction vertices that are not symmetric under exchanges of the momenta of identical particles entering the relevant processes. We show that in $\kappa$-Minkowski the coproduct and the star product must indeed treat momenta in a non-symmetric way, but the overall structure of interaction vertices is symmetric under exchange of identical particles. We also show that in $\kappa$-Minkowski field theories it is convenient to introduce the concepts of "planar" and "non-planar" Feynman loop-diagrams, again in close analogy with the corresponding concepts previously introduced in the study of field theories in canonical non-commutative space-times. 
  In an earlier paper (hep-th/0101127), we developed heat kernel techniques in N = 2 harmonic superspace for the calculation of the low-energy effective action of N = 4 SYM theory, which can be considered as the most symmetric N = 2 SYM theory. Here, the results are extended to generic N = 2 SYM theories. This involves a prescription for computing the variation of the hypermultiplet effective action. Integrability of this variation allows the hypermultiplet effective action to be deduced. This prescription permits a very simple superfield derivation of the perturbative holomorphic prepotential. Explicit calculations of the prepotential and the leading non-holomorphic correction are presented. 
  A variant of an earlier proposal by the author and SenGupta, to describe four dimensional Maxwell electrodynamics in Einstein-Cartan spacetimes through a Kalb-Ramond field as an intermediary, is shown to lead to a new Maxwell-Kalb-Ramond coupling that violates spatial parity, even when the KR gauge field has its standard parity assignment. One consequence of this coupling seems to be a modulation, independent of wavelength but dependent on the KR field strength, of the intensity of synchrotron radiation observed from distant galactic sources. 
  Logarithmic conformal field theory is investigated using the AdS/CFT correspondence and a novel method based on nilpotent weights. Using this device we add ghost fermions and point to a BRST invariance of the theory. 
  The Hamiltonian dynamics of a compressible inviscid fluid is formulated as a gauge theory. The idea of gauge equivalence is exploited to unify the study of apparantly distinct physical problems and solutions of new models can be generated from known fluid velocity profiles. 
  We present here a cohomological analysis of the new spacetime superalgebras that arise in the context of superbrane theory. They lead to enlarged superspaces that allow us to write D-brane actions in terms of fields associated with the additional superspace variables. This suggests that there is an extended superspace/worldvolume fields democracy for superbranes. 
  We discuss the second quantization of scalar field theory on the q-deformed fuzzy sphere S^2_{q,N} for q \in \R, using a path-integral approach. We find quantum field theories which are manifestly covariant under U_q(su(2)), have a smooth limit q -> 1, and satisfy positivity and twisted bosonic symmetry properties. Using a Drinfeld twist, they are equivalent to ordinary but slightly "nonlocal" QFT's on the undeformed fuzzy sphere, which are covariant under SU(2). 
  We revisit the quantum Hall system with no Zeeman splitting energy using the noncommutative field theory. We analyze the BPS condition for the delta-function interaction near the filling factor $\nu=1$. Multi-skyrmions are shown to saturate the BPS bounds. The dimension of the moduli space of $k$ skyrmions is $4k+2$. Advantage of the noncommutative field description is demonstrated through the derivation of the effective nonlinear $\sigma$ model Lagrangian. 
  We find $(N+1)/2$ distinct classes (``generations'') of kink solutions in an $SU(N)\times Z_2$ field theory. The classes are labeled by an integer $q$. The members of one class of kinks will be globally stable while those of the other classes may be locally stable or unstable. The kink solutions in the $q^{th}$ class have a continuous degeneracy given by the manifold $\Sigma_q=H/K_q$, where $H$ is the unbroken symmetry group and $K_q$ is the group under which the kink solution remains invariant. The space $\Sigma_q$ is found to contain incontractable two spheres for some values of $q$, indicating the possible existence of certain incontractable spherical structures in three dimensions. We explicitly construct the three classes of kinks in an SU(5) model with quartic potential and discuss the extension of these ideas to magnetic monopole solutions in the model. 
  We present a new representation of the string vertices of the cubic open string field theory. By using this three-string vertex, we attempt to identify open string fields as huge-sized matrices by following Witten's idea. By using these huge matrices, we obtain some results about the construction of partial isometries in the algebra of open string fields. 
  We construct a Heisenberg-like algebra for the one dimensional quantum free Klein-Gordon equation defined on the interval of the real line of length $L$. Using the realization of the ladder operators of this type Heisenberg algebra in terms of physical operators we build a 3+1 dimensional free quantum field theory based on this algebra. We introduce fields written in terms of the ladder operators of this type Heisenberg algebra and a free quantum Hamiltonian in terms of these fields. The mass spectrum of the physical excitations of this quantum field theory are given by $\sqrt{n^2 \pi^2/L^2+m_q^2}$, where $n= 1,2,...$ denotes the level of the particle with mass $m_q$ in an infinite square-well potential of width $L$. 
  We study the ratio of the entropy to the total energy in conformal field theories at finite temperature. For the free field realizations of N=4 super Yang-Mills theory in D=4 and the (2,0) tensor multiplet in D=6, the ratio is bounded from above. The corresponding bounds are less stringent than the recently proposed Verlinde bound. For strongly coupled CFTs with AdS duals, we show that the ratio obeys the Verlinde bound even in the presence of rotation. For such CFTs, we point out an intriguing resemblance in their thermodynamic formulas with the corresponding ones of two-dimensional CFTs. The discussion is based on hep-th/0101076. 
  We consider certain supersymmetric configurations of intersecting branes and branes ending on branes and analyze the duality between their open and closed string interpretation. The examples we study are chosen such that we have the lower dimensional brane realizing an n+1 dimensional conformal field theory on its worldvolume and the higher dimensional one introducing a conformal boundary. We also consider two CFTs, possibly with different central charges, interacting along a common conformal boundary. We show with a probe calculation that the dual closed string description is in terms of gravity in an AdS_{n+2} bulk with an AdS_{n+1} defect or two different AdS_{n+2} spaces joined along a defect. We also comment briefly on the expected back-reaction. 
  Treating the gravitational field as a dynamical field, we study the spontaneous symmetry breaking induced by a scalar field under its self-interaction and non-minimal interaction with gravity in four dimensional space-time. In particular, we explore the feasibility of inducing spontaneous symmetry breaking after introducing the non-minimal coupling, and discuss briefly the cosmological constant problem corresponding to the phase transition associated with the spontaneous symmetry breaking. 
  We show that the Abelian Higgs field equations in the four dimensional anti de Sitter spacetime have a vortex line solution. This solution, which has cylindrical symmetry in AdS$_4$, is a generalization of the flat spacetime Nielsen-Olesen string. We show that the vortex induces a deficit angle in the AdS$_4$ spacetime that is proportional to its mass density. Using the AdS/CFT correspondence, we show that the mass density of the string is uniform and dual to the discontinuity of a logarithmic derivative of correlation function of the boundary scalar operator. 
  The nonlinear $n$-supersymmetry with holomorphic supercharges is investigated for the 2D system describing the motion of a charged spin-1/2 particle in an external magnetic field. The universal algebraic structure underlying the holomorphic $n$-supersymmetry is found. It is shown that the essential difference of the 2D realization of the holomorphic $n$-supersymmetry from the 1D case recently analysed by us consists in appearance of the central charge entering non-trivially into the superalgebra. The relation of the 2D holomorphic $n$-supersymmetry to the 1D quasi-exactly solvable (QES) problems is demonstrated by means of the reduction of the systems with hyperbolic or trigonometric form of the magnetic field. The reduction of the $n$-supersymmetric system with the polynomial magnetic field results in the family of the one- dimensional QES systems with the sextic potential. Unlike the original 2D holomorphic supersymmetry, the reduced 1D supersymmetry associated with $x^6+...$ family is characterized by the non-holomorphic supercharges of the special form found by Aoyama et al. 
  We consider a string wrapped many times around a compact circle in space, and let this string carry a right moving wave which imparts momentum and angular momentum to the string. The angular momentum causes the strands of the `multiwound' string to separate and cover the surface of a torus. We compute the supergravity solution for this string configuration. We map this solution by dualities to the D1-D5 system with angular momentum that has been recently studied. We discuss how constructing this multiwound string solution may help us to relate the microscopic and macroscopic pictures of black hole absorption. 
  We study supergravity models in four dimensions where the hidden sector is superconformal and strongly-coupled over several decades of energy below the Planck scale, before undergoing spontaneous breakdown of scale invariance and supersymmetry. We show that large anomalous dimensions can suppress Kahler contact terms between the hidden and visible sectors, leading to models in which the hidden sector is "sequestered" and anomaly-mediated supersymmetry breaking can naturally dominate, thus solving the supersymmetric flavor problem. We construct simple, explicit models of the hidden sector based on supersymmetric QCD in the conformal window. The present approach can be usefully interpreted as having an extra dimension responsible for sequestering replaced by the many states of a (spontaneously-broken) strongly-coupled superconformal hidden sector, as dictated by the AdS/CFT correspondence. 
  We present an extended study of our previous work on an alternative five-dimensional N=2 supergravity theory that has a single antisymmetric tensor and a dilaton as a part of supergravity multiplet. The new fields are natural Neveu-Schwarz massless fields in superstring theory. Our total matter multiplets include n copies of vector multiplets forming the sigma-model coset space SO(n,1) / SO(n), and n' copies of hypermultiplets forming the quaternionic K\"ahler manifold Sp(n',1) / Sp(n') X Sp(1). We complete the couplings of matter multiplets to supergravity with the gauged group of the type SO(2) X Sp(n') X Sp(1) X H X [ U(1) ]^{n-p+1} for an arbitrary gauge group H with p = dim H + 1, and the isotropy group Sp(n') X Sp(1) of the coset Sp(n',1) / Sp(n') X Sp(1) formed by the hypermultiplets. We also describe the generalization to singular 5D space-time as in the conventional formulation 
  We write the recently conjectured action for transformation of the ordinary Born-Infeld action under the Seiberg-Witten map with one open Wilson contour in a manifestly non-commutative gauge invariant form. This action contains the non-constant closed string fields, higher order derivatives of the non-commutative gauge fields through the $*_N$-product, and a Wilson operator. We extend this non-commutative $D_9$-brane action to the action for $D_p$-brane by transforming it under T-duality. Using this non-commutative $D_p$-brane action we then evaluate the linear couplings of the graviton and dilaton to the brane for arbitrary non-commutative parameters. By taking the Seiberg-Witten limit we show that they reduce exactly to the known results of the energy-momentum tensor of the non-commutative super Yang-Mills theory. We take this as an evidence that the non-commutative action in the Seiberg-Witten limit includes properly all derivative correction terms. 
  Using the Baxter's T-Q relation derived from the transfer matrix technique, we consider the diagonalization problem of discrete quantum pendulum and discrete quantum sine-Gordon Hamiltonian from the algebraic geometry aspect. For a finite chain system of the size L, when the spectral curve degenerates into rational curves, we have reduced Baxter's T-Q relation into a polynomial equation; the connection of T-Q polynomial equation with the algebraic Bethe Ansatz is clearly established . In particular, for L=4 it is the case of rational spectral curves for the discrete quantum pendulum and discrete sine-Gordon model. To these Baxter's T-Q polynomial equations, we have obtained the complete and explicit solutions with a detailed understanding of their quantitative and qualitative structure. In general the model possesses a spectral curve with a generic parameter. We have conducted certain qualitative study on the algebraic geometry of this high-genus Riemann surface incorporating Baxter's T-Q relation. 
  The four dimensional O(3) non-linear sigma model introduced by Faddeev and Niemi, with a Skyrme-like higher order term to stabilise static knot solutions classified by the Hopf invariant, can be rewritten in terms of the complex two-component CP1 variables. A further rewriting of these variables in terms of SU(2) curvature free gauge fields is performed. This leads us to interpret SU(2) pure gauge vacuum configurations, in a particular maximal abelian gauge, in terms of knots with the Hopf invariant equal to the winding number of the gauge configuration. 
  The strong structural similarity between the deformed conifold of Candelas and de la Ossa (a noncompact Calabi-Yau manifold) and the moduli space of unit charge CP^1 lumps equipped with its L^2 metric is pointed out. This allows one to reinterpret certain recent results on D3 branes in terms of lump dynamics, and to deduce certain curvature properties of the deformed conifold. 
  We consider Euclidean D4 and D6-branes filling the whole ${\bf R}^4$ and ${\bf R}^6$ space, respectively. In both cases, with a constant background B-field turned on for D4-branes, we propose actions which are the same as the DBI actions up to some constant or total derivative terms. These extra terms allow us to write the action as a square of nonlinear instanton equations. As such, the actions can easily be supersymmetrized using the methods of topological field theory. 
  We study XXX Heisenberg spin 1/2 anti-ferromagnet. We evaluate a probability of formation of a ferromagnetic string in the anti-ferromagnetic ground state in thermodynamics limit. We prove that for short strings the probability can be expressed in terms of Riemann zeta function with odd arguments. 
  Formulas for the most general case of the zeta function associated to a quadratic+linear+constant form (in {\bf Z}) are given. As examples, the spectral zeta functions $\zeta_\alpha (s)$ corresponding to bosonic ($\alpha =2$) and to fermionic ($\alpha =3$) quantum fields living on a noncommutative, partially toroidal spacetime are investigated. Simple poles show up at $s=0$, as well as in other places (simple or double, depending on the number of compactified, noncompactified, and noncommutative dimensions of the spacetime). This poses a challenge to the zeta-function regularization procedure. 
  The SU(2) collective coordinates expansion of the Born-Infeld\break Skyrmion Lagrangian is performed. The classical Hamiltonian is computed from this special Lagrangian in approximative way: it is derived from the expansion of this non-polynomial Lagrangian up to second-order variable in the collective coordinates. This second-class constrained model is quantized by Dirac Hamiltonian method and symplectic formalism. Although it is not expected to find symmetries on second-class systems, a hidden symmetry is disclosed by formulating the Born-Infeld Skyrmion %model as a gauge theory. To this end we developed a new constraint conversion technique based on the symplectic formalism. Finally, a discussion on the role played by the hidden symmetry on the computation of the energy spectrum is presented. 
  Here we examine O(n) systems with arbitrary two spin interactions (of unspecified range) within a general framework. We shall focus on translationally invariant interactions. In the this case, we determine the ground states of the $O(n \ge 2)$ systems. We further illustrate how one may establish Peierls bounds for many Ising systems with long range interactions. We study the effect of thermal fluctuations on the ground states and derive the corresponding fluctuation integrals. The study of the thermal fluctuation spectra will lead us to discover a very interesting odd-even $n$ (coupling-decoupling) effect. We will prove a generalized Mermin-Wagner-Coleman theorem for all two dimensional systems (of arbitrary range) with analytic kernels in $k$ space. We will show that many three dimensional systems have smectic like thermodynamics. We will examine the topology of the ground state manifolds for both translationally invariant and spin glass systems. We conclude with a discussion of O(n) spin dynamics in the general case. 
  We find the exact spectrum and degeneracies for the Gross-Neveu model in two dimensions. This model describes N interacting Majorana fermions; it is asymptotically free, and has dynamical mass generation and spontaneous chiral symmetry breaking. We show here that the spectrum contains 2^{N/2} kinks for any $N$. The unusual \sqrt{2} in the number of kinks for odd $N$ comes from restrictions on the allowed multi-kink states. These kinks are the BPS states for a generalized supersymmetry where the conserved current is of dimension N/2; the N=3 case is the {\cal N}=1 supersymmetric sine-Gordon model, for which the spectrum consists of 2\sqrt{2} kinks. We find the exact S matrix for these kinks, and the exact free energy for the model. 
  In a previous paper, the BRST cohomology in the pure spinor formalism of the superstring was shown to coincide with the light-cone Green-Schwarz spectrum by using an SO(8) parameterization of the pure spinor. In this paper, the SO(9,1) Lorentz generators are explicitly constructed using this SO(8) parameterization, proving the Lorentz invariance of the pure spinor BRST cohomology. 
  We give a simple proof that there does not exist a Haar measure on the group $C^{\infty}({\bf R}^n,U(1))$. 
  There are two important examples of physical systems which violate the strong energy condition : Universes (like, it would seem, our own) with a positive cosmological constant, and wormholes. We suggest that a positive cosmological constant can be reconciled with string theory by considering wormholes in string backgrounds. This is argued in two directions : first, we show that brane-worlds with positive cosmological constants give rise to bulk singularities which are best resolved by embedding the brane-world in an AdS/CFT wormhole; and second, for the simplest kind of wormhole in an asymptotically AdS space, we show that the IR stability of the matter needed to keep the wormhole open requires the presence of a brane-world. UV stability conditions then forbid a negative cosmological constant on the brane-world. 
  We introduce so-called chaotic strings (coupled 1-dimensional noise strings underlying the Parisi-Wu approach of stochastic quantization on a small scale) as a possible amendment of ordinary string theories. These strings are strongly self-interacting and exhibit strongest possible chaotic behavior. Constraints on the vacuum energy of the strings fix a certain discrete set of allowed string couplings. We provide extensive numerical evidence that these string couplings numerically coincide with running standard model coupling constants, evaluated at energy scales given by the masses of the known quarks, leptons and gauge bosons. Chaotic strings can thus be used to provide a theoretical argument why certain standard model parameters are realized in nature, others are not, assuming that the a priori free standard model parameters evolve to the minima of the effective potentials. The chaotic string spectrum correctly reproduces the numerical values of the electroweak and strong coupling constants with a precision of 4-5 digits, as well as the (free) masses of the known quarks and leptons with a precision of 3-4 digits. Neutrino mass predictions consistent with present experiments are obtained. The W boson mass also comes out correctly, and a Higgs mass prediction is obtained. 
  In this paper the model considered by Arkani-Hamed, Cohen and Georgi in the context of (de)constructing dimensions has been studied by making use of non-commutative geometry. The non-commutative geometry provides a natural framework to study this model with or without gravity. 
  In this paper we construct the 2+1 effective theory of the light states in D2/D4-brane system in the context of noncommutative Yang-Mills theory. This effective theory is noncommutative and tachyonic, however, it is not taking the form of an Abelian Higgs model as naively expected. We solve the classical solutions of the effective theory which are nicely corresponding to different states during the tachyon condensation process of the dissolution of D2-brane into D4-brane. We also find that if the expected stable self-dual D0/D4 configuration as the unit-winding vortex exists, it would be highly calibrated in the effective theory and be out of the reach of the analytic solutions. 
  We present what we believe are the first specific string (D-brane) constructions whose low-energy limit yields just a three generation $SU(3)\times SU(2)\times U(1)$ standard model with no extra fermions nor U(1)'s (without any further effective field theory assumption). In these constructions the number of generations is given by the number of colours. The Baryon, Lepton and Peccei-Quinn symmetries are necessarily gauged and their anomalies cancelled by a generalized Green-Schwarz mechanism.   The corresponding gauge bosons become massive but their presence guarantees automatically proton stability. There are necessarily three right-handed neutrinos and neutrino masses can only be of Dirac type. They are naturally small as a consequence of a PQ-like symmetry. There is a Higgs sector which is somewhat similar to that of the MSSM and the scalar potential parameters have a geometric interpretation in terms of brane distances and intersection angles. Some other physical implications of these constructions are discussed. 
  The gauge invariance of open string field theory is considered from the point of view of level truncation, and applications to the tachyon condensation problem are discussed. We show that the region of validity of Feynman-Siegel gauge can be accurately determined using the level truncation method. We then show that singularities previously found in the tachyon effective potential are gauge artifacts arising from the boundary of the region of validity of Feynman-Siegel gauge. The problem of finding the stable vacuum and tachyon potential without fixing Feynman-Siegel gauge is addressed. 
  We study the finite temperature phase transition in 2+1 dimensional compact QED and its dual theory: Josephson junction. Duality of these theories at zero temperature was established long time ago by Hosotani. Phase transition in compact QED is well studied and we employ the `duality' to study the superconductivity phase transition in a Josephson junction. For a thick junction we obtain a critical temperature in terms of the geometrical properties of the junction. 
  Theories in 5 dimensions with minimal supersymmetry are studied for domain-wall solutions and in the context of the AdS/CFT correspondence. The scalar manifold is a product of a very special real manifold and a quaternionic-Kaehler manifold. Superconformal methods can clarify the structure of these manifolds, which are part of the family of special manifolds. BPS solutions depending on the scalars and a warp factor of the 5-dimensional metric with a flat 4-dimensional metric can interpolate between critical points determined by algebraic attractor equations. The mixing of vector and hypermultiplets is essential to obtain UV and IR critical points. 
  We revisit the physics of five-dimensional black holes constructed from D5- and D1-branes and momentum modes in type IIB string theory compactified on K3. Since these black holes incorporate D5-branes wrapped on K3, an enhancon locus appears in the spacetime geometry. With a `small' number of D1-branes, the entropy of a black hole is maximised by including precisely half as many D5-branes as there are D1-branes in the black hole. Any attempts to introduce more D5-branes, and so reduce the entropy, are thwarted by the appearance of the enhancon locus above the horizon, which then prevents their approach. The enhancon mechanism thereby acts to uphold the Second Law of Thermodynamics. This result generalises: For each type of bound state object which can be made of both types of brane, we show that a new type of enhancon exists at successively smaller radii in the geometry, again acting to prevent any reduction of the entropy just when needed. We briefly explore the appearance of the enhancon in the black hole interior. 
  In this report we consider brane-world universe (New Brane World) where an arbitrary large $N$ quantum CFT exists on the domain wall. This corresponds to implementing of Randall-Sundrum compactification within the context of AdS/CFT correspondence. Using anomaly induced effective action for domain wall CFT, the possibility of self-consistent quantum creation of 4d de Sitter wall universe (inflation) is demonstrated. In case of maximally SUSY Yang-Mills theory the exact correspondence with radius and effective tension found by Hawking-Hertog-Reall is obtained.   We also discuss the bosonic sector of 5d gauged supergravity with single scalar and taking the boundary action as predicted by supersymmetry and discuss the possibility to supersymmetrize dilatonic New Brane World. It is demonstrated that for a number of superpotentials the flat SUSY dilatonic brane-world (with dynamically induced brane dilaton) or quantum-induced de Sitter dilatonic brane-world (not Anti-de Sitter one) where SUSY is broken by the quantum effects occurs. The analysis of graviton perturbations indicates that gravity is localized on such branes.   New Brane World is useful in the study of FRW dynamics and cosmological entropy bounds. Brane stress tensor is induced by quantum effects of dual CFT and brane crosses the horizon of AdS black hole. The similarity between CFT entropy at the horizon and FRW equations is extended on the quantum level. This suggests the way to understand cosmological entropy bounds in quantum gravity. 
  We describe actions that correspond to the interaction of the Super Virasoro algebra with supergravitons. These new field theories introduce a superfield that corresponds to dual elements of the super Virasoro algebra. We are also able to extend the definition of these field theories to higher dimensions. We explicitly exhibit the 2, 3 and 4 dimensional cases. Remarkably, the fundamental prepotentials describing these dual elements of the super Virasoro algebra, in each model, agrees with the known prepotentials of the corresponding supergravity theory. These theories might be important in the quantization of the super Virasoro group, supergravity and in AdS and super AdS gravity. 
  The commutation relations of the composite fields are studied in the 3, 2 and 1 space dimensions. It is shown that the field of an atom consisting of a nucleus and an electron fields satisfies, in the space-like asymptotic limit, the canonical commutation relations within the sub-Fock-space of the atom. The composite anyon fields are shown to satisfy the proper anyonic commutation relations with the additive phase exponents. Then, (quasi)particle pictures of the anyons are clarified. The hierarchy of the fractional quantum Hall effect is rather simply nderstood by utilizing the (quasi)particle charactors of the anyons. The commutation relations of the scalar object in the Schwinger(Thirring) model are mentioned briefly. 
  We discuss the physical impacts of the ``Cho decomposition'' (or the ``Cho-Faddeev-Niemi-Shabanov decomposition'') of the non-Abelian gauge potential on QCD. We show how the decomposition makes a subtle but important modification on the non-Abelian dynamics, and present three physically equivalent quantization schemes of QCD which are consistent with the decomposition. In particular, we show that the decomposition enlarges the dynamical degrees of QCD by making the topological degrees of the non-Abelian gauge symmetry dynamical. Furthermore with the decomposition we show that the Skyrme-Faddeev theory of non-linear sigma model and QCD have almost identical topological structures. In specific we show that an essential ingredient in both theories is the Wu-Yang type non-Abelian monopole, and that the Faddeev-Niemi knots of the Skyrme-Faddeev theory can actually be interpreted to describe the multiple vacua of the SU(2) QCD. Finally we argue that the Skyrme-Faddeev theory is, just like QCD, a theory of confinement which confines the magnetic flux of the monopoles. 
  For any root system $\Delta$ and an irreducible representation ${\cal R}$ of the reflection (Weyl) group $G_\Delta$ generated by $\Delta$, a {\em spin Calogero-Moser model} can be defined for each of the potentials: rational, hyperbolic, trigonometric and elliptic. For each member $\mu$ of ${\cal R}$, to be called a "site", we associate a vector space ${\bf V}_{\mu}$ whose element is called a "spin". Its dynamical variables are the canonical coordinates $\{q_j,p_j\}$ of a particle in ${\bf R}^r$, ($r=$ rank of $\Delta$), and spin exchange operators $\{\hat{\cal P}_\rho\}$ ($\rho\in\Delta$) which exchange the spins at the sites $\mu$ and $s_{\rho}(\mu)$. Here $s_\rho$ is the reflection generated by $\rho$. For each $\Delta$ and ${\cal R}$ a {\em spin exchange model} can be defined. The Hamiltonian of a spin exchange model is a linear combination of the spin exchange operators only. It is obtained by "freezing" the canonical variables at the equilibrium point of the corresponding classical Calogero-Moser model. For $\Delta=A_r$ and ${\cal R}=$ vector representation it reduces to the well-known Haldane-Shastry model. Universal Lax pair operators for both spin Calogero-Moser models and spin exchange models are presented which enable us to construct as many conserved quantities as the number of sites for {\em degenerate} potentials. 
  We numerically construct static and spherically symmetric electrically charged black hole solutions in Einstein-Born-Infeld gravity with massive dilaton. The numerical solutions show that the dilaton potential allows many more black hole causal structures than the massless dilaton. We find that depending on the black hole mass and charge and the dilaton mass the black holes can have either one, two, or three horizons. The extremal solutions are also found out. As an interesting peculiarity we note that there are extremal black holes with an inner horizon and with triply degenerated horizon. 
  We propose a way to identify the gauge invariant operator in noncommutative gauge theory on a D-brane with nonzero B field which couples to a specific supergravity mode in the bulk. This uses the description of noncommutative gauge theories in terms of ordinary $U(\infty)$ gauge theories in lower dimensions. The proposal is verified in the DBI approximation. Other authors have shown that the proposal is also consistent with explicit string worldsheet calculations. We comment on implications to holography. 
  A five-dimensional scenario with a non compact extra dimension of infinite extent is studied, in which a single three-brane is affected by small Gaussian fluctuations in the extra dimension. The average magnitude of the fluctuations is of order of the electro-weak length scale ($\sigma\sim m_{EW}^{-1}$). The model provides an stochastic approach to gravity that accounts for an alternative resolution of the mass hierarchy problem. The cosmological constant problem can be suitably treated as well. Surprisingly the Mach's principle finds a place in the model. It is argued that the Mach's principle, the mass hierarchy and the cosmological constant problem, are different aspects of a same property of gravity in this model: its stochastic character. Thin-brane scenarios are recovered in the "no-fluctuations" limit ($\sigma\to 0$). 
  In previous papers we built (multiple) D-branes in flat space-time as classical solutions of the string field theory based on the tachyon vacuum. In this paper we construct classical solutions describing all D-branes in any fixed space-time background defined by a two-dimensional CFT of central charge 26. A key role is played by the geometrical definition of the sliver state in general boundary CFT's. The correct values for ratios of D-brane tensions arise because the norm of the sliver solution is naturally related to the disk partition function of the appropriate boundary CFT. We also explore the possibility of reproducing the known spectrum of physical states on a D-brane as deformations of the sliver. 
  Two consistency conditions for partition functions established by Akemann and Dam-gaard in their studies of the fermionic mass dependence of the QCD partition function at low energy ({\it a la} Leutwiller-Smilga-Verbaarschot) are interpreted in terms of integrable hierarchies. Their algebraic relation is shown to be a consequence of Wick's theorem for 2d fermionic correlators (Hirota identities) in the special case of the 2-reductions of the KP hierarchy (that is KdV/mKdV). The consistency condition involving derivatives is an incarnation of the string equation associated with the particular matrix model (the particular kind of the Kac-Schwarz operator). 
  We find solutions of the six-dimensional maximal supergravity by adding a perturbation of vector fields to the solution AdS${}_3$ $\times$ S${}^3$. For certain perturbations the solution represents a dual description of an ${\cal N}=(4,0)$ field theory in two dimensions by the AdS/CFT correspondence. 
  We present the details of a mean-field approximation scheme for the quantum mechanics of N D0-branes at finite temperature. The approximation can be applied at strong 't Hooft coupling. We find that the resulting entropy is in good agreement with the Bekenstein-Hawking entropy of a ten-dimensional non-extremal black hole with 0-brane charge. This result is in accord with the duality conjectured by Itzhaki, Maldacena, Sonnenschein and Yankielowicz. We study the spectrum of single-string excitations within the quantum mechanics, and find evidence for a clear separation between light and heavy degrees of freedom. We also present a way of identifying the black hole horizon. 
  We consider a generic supersymmetric matter theory coupled to linearized supergravity, and analyze scenarios for spontaneous symmetry breaking in terms of vacuum expectation values of components of the current supermultiplet. When the vacuum expectation of the energy momentum tensor is zero, but the scalar current or pseudoscalar current gets an expectation, evaluation of the gravitino self energy using the supersymmetry current algebra shows that there is an induced gravitino mass term. The structure of this term generalizes the supergravity action with cosmological constant to theories with CP violation. When the vacuum expectation of the energy momentum tensor is nonzero, supersymmetry is broken; requiring cancellation of the cosmological constant gives the corresponding generalized gravitino mass formula. 
  We study the features of the vacuum of the harmonic oscillator in the Moyal quantization. The vacuums with and without using the normal ordering look different. The vacuum without the normal ordering is shown to be expressed using the Weyl ordering. The Weyl ordered vacuum is then compared with the normal ordered vacuum, and the implication of the difference between them is discussed. 
  We propose a new prescription of how to represent D-branes in Gepner models in more general homology classes than those in the previous constructions. The central role is played by a certain projection acting on the Recknagel-Schomerus boundary states. Consequently, the boundary states are in most cases no longer a sum of products of N=2 Ishibashi states, but nevertheless preserve spacetime supersymmetry and satisfy the Cardy condition. We demonstrate these in the (k=1)^3 Gepner model in detail, and construct boundary states for D-branes wound around arbitrary rigid 1-cycles on the corresponding 2-torus. We also emphasize the necessity of some angle-dependent transformations in identifying a proper free-field realization for each brane tilted at an angle. In particular, this is essential for the Witten index to give the correct intersection numbers between the different D-branes. 
  We give a comment on the possible role of the sliver state in the generic boundary conformal field theory. We argue that for each Cardy state, there exists at least one projector in the string field theory. 
  In these two lectures, delivered at the XXXVII Karpacz Winter School, February 2001, I review some applications of superspace in various topics related to string theory and M-theory. The first lecture is mainly devoted to descriptions of brane dynamics formulated in supergravity backgrounds. The second lecture concerns the use of superspace techniques for determining consistent interactions in supersymmetric gauge theory and supergravity, e.g. alpha'-corrections from string/M-theory. 
  In this paper we study the Yang-Baxter integrable structure of Conformal Field Theories with extended conformal symmetry generated by the W_3 algebra. We explicitly construct various T- and Q-operators which act in the irreducible highest weight modules of the W_3 algebra. These operators can be viewed as continuous field theory analogues of the commuting transfer matrices and Q-matrices of the integrable lattice systems associated with the quantum algebra U_q(\hat{sl}(3)). We formulate several conjectures detailing certain analytic characteristics of the Q-operators and propose exact asymptotic expansions of the T- and Q-operators at large values of the spectral parameter. We show, in particular, that the asymptotic expansion of the T-operators generates an infinite set of local integrals of motion of the W_3 CFT which in the classical limit reproduces an infinite set of conserved Hamiltonians associated with the classical Boussinesq equation. We further study the vacuum eigenvalues of the Q-operators (corresponding to the highest weight vector of the W_3 module) and show that they are simply related to the expectation values of the boundary exponential fields in the non-equilibrium boundary affine Toda field theory with zero bulk mass. 
  A general model independent approach using the `off-shell Bethe Ansatz' is presented to obtain an integral representation of generalized form factors. The general techniques are applied to the quantum sine-Gordon model alias the massive Thirring model. Exact expressions of all matrix elements are obtained for several local operators. In particular soliton form factors of charge-less operators as for example all higher currents are investigated. It turns out that the various local operators correspond to specific scalar functions called p-functions. The identification of the local operators is performed. In particular the exact results are checked with Feynman graph expansion and full agreement is found. Furthermore all eigenvalues of the infinitely many conserved charges are calculated and the results agree with what is expected from the classical case. Within the frame work of integrable quantum field theories a general model independent `crossing' formula is derived. Furthermore the `bound state intertwiners' are introduced and the bound state form factors are investigated. The general results are again applied to the sine-Gordon model. The integrations are performed and in particular for the lowest breathers a simple formula for generalized form factors is obtained. 
  There is a very natural map from the configuration space of n distinct points in Euclidean 3-space into the flag manifold U(n)/U(1)^n, which is compatible with the action of the symmetric group. The map is well-defined for all configurations of points provided a certain conjecture holds, for which we provide numerical evidence. We propose some additional conjectures, which imply the first, and test these numerically. Motivated by the above map, we define a geometrical multi-particle energy function and compute the energy minimizing configurations for up to 32 particles. These configurations comprise the vertices of polyhedral structures which are dual to those found in a number of complicated physical theories, such as Skyrmions and fullerenes. Comparisons with 2-particle and 3-particle energy functions are made. The planar restriction and the generalization to hyperbolic 3-space are also investigated. 
  We study behavior of bulk gauge field in the bigravity model in which two positive tension $AdS_4$ branes in $AdS_5$ bulk are included. We solve the equations of motions for Kaluza-Klein modes and determine the mass spectrum. It is shown that unlike the case of graviton, we find no ultralight Kaluza-Klein modes in the spectrum. 
  We show explicitly that all partially and strictly massless fields with spins s<=3 in (A)dS have null propagation. Assuming that this property holds also for s>3, we derive the mass-cosmological constant tunings required to yield all higher spin partially massless theories. As s increases, the unitarily allowed region for massive spins is squeezed around \Lambda=0, so that an infinite tower of massive particles forces vanishing \Lambda. We also speculate on the relevance of this result to string theory and supergravity in (A)dS backgrounds. 
  We construct a broad family of exact solutions to the five-dimensional Einstein equations coupled to a scalar field with an exponential potential. Embedding a three-brane in these bulk space-times in a particular way we obtain a class of self-tuned curved brane worlds in which the vacuum energy on the brane is gravitationally idle, the four-dimensional geometry being insensitive to the value of the brane tension. This self-tuning arises from cancellations, enforced by the junction conditions, between the scalar field potential, the brane vacuum energy and the matter on the brane. Finally, we study some physically relevant examples and their dynamics. 
  We obtain static and rotating electrically charged black holes of a Einstein-Maxwell-Dilaton theory of the Brans-Dicke type in (2+1)-dimensions. The theory is specified by three fields, the dilaton, the graviton and the electromagnetic field, and two parameters, the cosmological constant and the Brans-Dicke parameter. It contains eight different cases, of which one distinguishes as special cases, string theory, general relativity and a theory equivalent to four dimensional general relativity with one Killing vector. We find the ADM mass, angular momentum, electric charge and dilaton charge and compute the Hawking temperature of the solutions. Causal structure and geodesic motion of null and timelike particles in the black hole geometries are studied in detail. 
  We study ghost number one excitations on the sliver to investigate the solution of string field actions around the tachyon vacuum. The generalized gluing and resmoothing theorem is used to develop a method for evaluating the effective action for excitations on both the wedge states and the sliver state. We analyze the discrete symmetries of the resulting effective action for excitations on the sliver. The gauge unfixed effective action till level two excitations on the sliver is evaluated. This is done for the case with the BRST operator $c_0$ and $c_0 + (c_2 + c_{-2})/2$ with excitations purely in the ghost sector. We find that the values of the effective potential at the local maximum lie close by for the zeroth and the second level of approximation. This indicates that level truncation in string field theory around the tachyon vacuum using excitations on the sliver converges for both choices of the BRST operator. It also provides evidence for the conjectured string field theory actions around the tachyon vacuum. 
  We derive the Kadomtsev-Petviashvili (KP) equation defined over a general associative algebra and construct its N-soliton solution. For the example of the Moyal algebra, we find multi-soliton solutions for arbitrary space-space noncommutativity. The noncommutativity of coordinates is shown to obstruct the general construction of a tau function for these solitons. We investigate the two-soliton solution in detail and show that asymptotic observers of soliton scattering are unable to detect a finite spatial noncommutativity. An explicit example shows that a pair of solitons in a noncommutative background can be interpreted as several pairs of image solitons. Finally, a dimensional reduction gives the general N-soliton solution for the previously discussed noncommutative KdV equation. 
  We solve the Einstein equations in higher dimensions with warped geometry where an extra dimension is assumed to have orbifold symmetry $S^{1}/Z_{2}$. The setup considered here is an extension of the five-dimensional Randall-Sundrum model to $5+D$ dimensions, and hidden and observable branes are fixed on the orbifold. It is assumed that the brane tension (self-energy) of each brane with $(4+D)$-dimensional spacetime is anisotropic and that the warped metric function of the four dimensions is generally different from that of the extra $D$ dimensions. We point out that the forms of the warped metric functions and the relations between the tensions of two branes depend on the integration constant appearing in the Einstein equations as well as on the sign of the bulk cosmological constant. 
  We propose a new mechanism of spontaneous supersymmetry breaking in noncommutative gauge theories. We find that in N=1 noncommutative gauge theories both supersymmetry and gauge invariance are dynamically broken. Supersymmetry is broken spontaneously by a Fayet-Iliopoulos D-term which naturally arises in a noncommutative U(n) theory. For a non-chiral matter content the Fayet-Iliopoulos term is not renormalized and its tree-level value can be chosen to be much smaller than the relevant string/noncommutativity scale. In the low energy theory, the noncommutative U(n) gauge symmetry is broken down to a commutative U(1) x SU(n). This breaking is triggered by the IR/UV mixing and manifests itself at and below the noncommutativity mass scale M_{NC}\sim \theta^{-1/2}. In particular, the U(1) degrees of freedom decouple from the SU(n) in the infrared and become arbitrarily weakly coupled, thus playing the role of the hidden sector for supersymmetry breaking. 
  We study tachyon condensation on noncommutative toric orbifolds with a $Z_2$ discrete group and explore the various kins of brane bound states arising in the case of irrational values of the B-field. We show that $Z_$ symmetry of the orbifolds incorporates naturally anti-branes in the spectrum and leads to equivalent results as those obtained by starting from an original pair of D-barD system on quantum torii. A specific analysis is deserved to the irrational representation of NC orbifolds and to the unstable bound states generated by the condensation. 
  In $D$-dimensional gauge theory with a kinetic term based on the p-form tensor gauge field, we introduce a gauge invariant operator associated with the composite formed from a electric $(p-1)$-brane and a magnetic $(q-1)$-brane in $D=p+q+1$ spacetime dimensions. By evaluating the partition function for this operator, we show that the expectation value of this operator gives rise to the topological contributions identical to those in gauge theory with a topological Chern-Simons BF term. 
  A general master action in terms of superfields is given which generates generalized Poisson sigma models by means of a natural ghost number prescription. The simplest representation is the sigma model considered by Cattaneo and Felder. For Dirac brackets considerably more general models are generated. 
  In this paper we try to construct noncommutative Yang-Mills theory for generic Poisson manifolds. It turns out that the noncommutative differential calculus defined in an old work is exactly what we need. Using this calculus, we generalize results about the Seiberg-Witten map, the Dirac-Born-Infeld action, the matrix model and the open string quantization for constant B field to non-constant background with H=0. 
  We present a cohomological method for obtaining the non-Abelian Seiberg-Witten map for any gauge group and to any order in theta. By introducing a ghost field, we are able to express the equations defining the Seiberg-Witten map through a coboundary operator, so that they can be solved by constructing a corresponding homotopy operator. 
  We review different approaches to the graphical generation of the tadpole-free Feynman diagrams of the self-energy and the one-particle irreducible four-point function. These are needed for calculating the critical exponents of the euclidean multicomponent scalar phi^4-theory with renormalization techniques in d=4-epsilon dimensions. 
  We consider the compactification of the IIA string to (1+1) dimensions on non-compact 4-folds that are ALE fibrations. Supersymmetry requires that the compactification include 4-form fluxes, and a particular class of these models has been argued by Gukov, Vafa and Witten to give rise to a set of perturbed superconformal coset models that also have a Landau-Ginzburg description. We examine all these ADE models in detail, including the exceptional cosets. We identify which perturbations are induced by the deformation of the singularity, and compute the Landau-Ginzburg potentials exactly. We also show how the the Landau-Ginzburg fields and their superpotentials arise from the geometric data of the singularity, and we find that this is most naturally described in terms of non-compact, holomorphic 4-cycles. 
  We review recent results in the matrix model approach to the 2-d non-critical string theory compactified in time, in the phase of condensation of the world sheet vortices (above the Berezinski-Kosterlitz-Thouless phase transition). This phase is known to describe strings on the 2-d black hole background, due to the conjecture of V.Fateev, A. and Al.Zamolodchikov. The corresponding matrix model has an integrable Toda structure which allows to compute many interesting physical quantities, such as string partition functions of various genera and the 1- and 2-point correlators of vorticities of arbitrary charges. 
  We propose a field theoretical model that exhibits spontaneous breaking of the rotational symmetry. The model has a two-dimensional sphere as extra dimensions of the space-time and consists of a complex scalar field and a background gauge field. The Dirac monopole, which is invariant under the rotations of the sphere, is taken as the background field. We show that when the radius of the sphere is larger than a certain critical radius, the vacuum expectation value of the scalar field develops vortices, which pin down the rotational symmetry to lower symmetries. We evaluate the critical radius and calculate configurations of the vortices by the lowest approximation. The original model has a $U(1) \times SU(2)$ symmetry and it is broken to U(1), U(1), D_3 for each case of the monopole number q = 1/2, 1, 3/2, respectively, where D_3 is the symmetry group of a regular triangle. Moreover, we show that the vortex configurations are stable against higher corrections of the perturbative approximation. 
  The universal formulation of spin exchange models related to Calogero-Moser models implies the existence of integrable hierarchies, which have not been explored. We show the general structures and features of the spin exchange model hierarchies by taking as examples the well-known Heisenberg spin chain with the nearest neighbour interactions. The energy spectra of the second member of the hierarchy belonging to the models based on the $A_r$ root systems $(r=3,4,5)$ are explicitly and {\em exactly} evaluated. They show many many interesting features and in particular, much higher degree of degeneracy than the original Heisenberg model, as expected from the integrability. 
  We calculate a one loop effective action of SU(2) QCD in the presence of the monopole background, and find a possible connection between the resulting QCD effective action and a generalized Skyrme-Faddeev action of the non-linear sigma model. The result is obtained using the gauge-independent decomposotion of the gauge potential into the topological degrees which describes the non-Abelian monopoles and the local dynamical degrees of the potential, and integrating out all the dynamical degrees of QCD. 
  It is shown that there exist large classes of BPS vacua in heterotic M-theory which have negative tension on the visible orbifold plane, positive tension on the hidden plane and positive tension, physical five-branes in the bulk space. Explicit examples of such vacua are presented. Furthermore, it is demonstrated that the ratio, beta/|alpha|, of the bulk five-brane tension to the visible plane tension can, for several large classes of such vacua, be made arbitrarily small. Hence, it is straightforward to find vacua with the properties required in the examples of the Ekpyrotic theory of cosmology - a visible brane with negative tension and beta/|alpha| small. This contradicts recent claims in the literature. 
  The computation of two and three point functions in the Coulomb gas free field approach to string theory in the SL(2,R)/U(1) black hole background is reviewed. An interesting relation arises when comparing the results obtained using two different screening operators. The formalism is then modified to study string theory propagating in AdS$_3$ which is considered as the direct product of the SL(2)/U(1) coset times a timelike free boson. This representation allows to naturally include the spectral flow symmetry and winding number in vertex operators and correlation functions. Two and three point tachyon amplitudes are computed in this new scenario and the results coincide with previous reports in the literature. Novel expressions are found for processes violating winding number conservation. 
  In a previous article it was proved that the extensions of the Taub-NUT geometry do not admit Killing-Yano tensors, even if they possess St\" {a}ckel-Killing tensors. Here the analysis is taken further, and it is shown that, in general, this class of metrics does not even admit f-symbols. The only exception is the original Taub-NUT metric which possesses four Killing-Yano tensors of valence two. 
  Supersymmetry (SUSY) breaking without messenger fields is proposed. We assume that our world is on a wall and SUSY is broken only by the coexistence of another wall with some distance from our wall. The Nambu-Goldstone (NG) fermion is localized on the distant wall. Its overlap with the wave functions of physical fields on our wall gives the mass splitting of physical fields on our wall thanks to a low-energy theorem. We propose that this overlap provides a practical method to evaluate mass splitting in models with SUSY breaking due to the coexistence of walls. 
  We discuss a calculable version of brane inflation, in which a set of parallel D-brane and anti-D-brane worlds, initially displaced in extra dimension, slowly attract each other. In the effective four-dimensional theory this slow motion of branes translates into a slow-roll of a scalar field (proportional to their separation) with a flat potential that drives inflation. The number of possible e-foldings is severely constrained. The scalar spectral index is found to be 0.97, while the effective compactification scale is of order $10^{12}$ GeV. Reheating of the Universe is provided by collision and subsequent annihilation of branes. 
  We show how the motion through the extra dimensions of a gas of branes and antibranes can, under certain circumstances, produce an era of inflation as seen by observers trapped on a 3-brane, with the inflaton being the inter-brane separation. Although most of our discussion refers to arbitrary p-branes, when we need to be specific we assume that they are D-branes of Type II or Type I string theory. For realistic brane couplings, such as those arising in string theory, the inter-brane potentials are too steep to inflate the universe for acceptably long times. However, for special regions of the parameter space of brane-antibrane positions the brane motion is slow enough for there to be sufficient inflation. Inflation would be more generic in models where the inter-brane interactions are much weaker. The spectrum of primordial density fluctuations predicted has index n slightly less than 1, and an acceptable amplitude, provided that the extra dimensions have linear size 1/r ~ 10^{12} GeV. Reheating occurs as in hybrid inflation, with the tachyonic instability of the brane-antibrane system taking over for small separations. The tachyon field can induce a cascade mechanism within which higher-dimension branes annihilate into lower-dimension ones. We argue that such a cascade naturally stops with the production of 3-branes in 10-dimensional string theory. 
  We present a candidate supergravity solution for a stacked configuration of stable non-BPS D-branes in Type II string theory compactified on T^4/Z_2. This gives a supergravity description of nonabelian tachyon condensation on the brane worldvolume. 
  An accelerating Universe can be accommodated naturally within non-critical string theory, in which scattering is described by a superscattering matrix \$ that does not factorize as a product of $S$- and $S^\dagger$-matrix elements and time evolution is described by a modified Liouville equation characteristic of open quantum-mechanical systems. We describe briefly alternative representations in terms of the stochastic Ito and Fokker-Planck equations. The link between the vacuum energy and the departure from criticality is stressed. We give an explicit example in which non-marginal \$tring couplings cause a departure from criticality, and the corresponding cosmological vacuum energy relaxes to zero \`a la quintessence. 
  We study curved domain wall solutions for gauged supergravity theories obtained by gauging some of the isometries of the manifold spanned by the scalars of vector and hypermultiplets. We first consider the case obtained by compactifying M-theory on a Calabi-Yau threefold in the presence of G-fluxes. It is found that supersymmetry allows for the construction of domain wall configurations with curved worldvolume and a cosmological constant. However it turns out that the equations of motion, if one insists on the supersymmetric ansatz for the scalars and warp factor, rule out solutions with a cosmological constant and allows only for Ricci-flat worldvolumes. Moreover, in the absence of flux, there are non-supersymmetric solutions with worldvolumes given by Einstein manifolds. We also generalize our results to all five dimensional gauged supergravity. 
  We present supersymmetry breaking four dimensional orientifolds of type IIA strings. The compact space is a torus times a four dimensional orbifold. The orientifold group reflects one direction in each torus. RR tadpoles are cancelled by D6-branes intersecting at angles in the torus and in the orbifold. The angles are chosen such that supersymmetry is broken. The resulting four dimensional theories contain chiral fermions. The tadpole cancellation conditions imply that there are no non-abelian gauge anomalies. The models also contain anomaly-free U(1) factors. 
  We examine Kaluza-Klein branes in detail. Specifically, we show that codimension four submanifolds that are stationary under a semi-free circle action may be interpreted as branes or antibranes in the Kaluza-Klein reduced space that are magnetically charged under the Kaluza-Klein field strength. We derive the equation in cohomology that is satisfied by such a brane using an explicit construction of the Thom class of the normal bundle of the brane worldvolume in the reduced space. This may be applied to both the D6-brane of Type IIA String Theory, and also to various recent constructions of magnetic branes immersed in fluxbrane backgrounds. We then go on to study the special case of monopole-antimonopole production in a five-dimensional Kaluza-Klein theory, illustrating our arguments with various concrete examples. 
  We outline, on a few instructive examples, the characteristic features of the approach to superbranes and super Born-Infeld theories based on the concept of partial spontaneous breaking of global supersymmetry (PBGS). The examples include the N=1, D=4 supermembrane and the ``space-filling'' D2- and D3-branes. Besides giving a short account of the available results for these systems, we present some new developments. For the supermembrane we prove the equivalence of the equation of motion following from the off-shell Goldstone superfield action and the one derived directly from the nonlinear realizations formalism. We give a new derivation of the off-shell Goldstone superfield actions for the considered systems, using a universal procedure inspired by the relationship between linear and nonlinear realizations of PBGS. 
  We use the moduli space approximation to study the time evolution of magnetically charged configurations in a theory with an SU(N+2) gauge symmetry spontaneously broken to U(1) x SU(N) x U(1). We focus on configurations containing two massive and N-1 massless monopoles. The latter do not appear as distinct objects, but instead coalesce into a cloud of non-Abelian field. We find that at large times the cloud and the massless particles are decoupled, with separately conserved energies. The interaction between them occurs through a scattering process in which the cloud, acting very much like a thin shell, contracts and eventually bounces off the cores of the massive monopoles. The strength of the interaction, as measured, e.g., by the amount of energy transfer, tends to be greatest if the shell is small at the time that it overlaps the massive cores. We also discuss the corresponding behavior for the case of the SU(3) multimonopole solutions studied by Dancer. 
  We respond to the criticisms by Kallosh, Kofman and Linde concerning our proposal of the ekpyrotic universe scenario. We point out a number of errors in their considerations and argue that, at this stage, the ekpyrotic model is a possible alternative to inflationary cosmology as a description of the very early universe. 
  Solitons of a nonlinear field interacting with fermions often acquire a fermionic number or an electric charge if fermions carry a charge. We show how the same mechanism (chiral anomaly) gives solitons statistical and rotational properties of fermions. These properties are encoded in a geometrical phase, i.e., an imaginary part of a Euclidian action for a nonlinear sigma-model. In the most interesting cases the geometrical phase is non-perturbative and has a form of an integer-valued theta-term. 
  We consider a class of domain-wall black hole solutions in the dilaton gravity with a Liouville-type dilaton potential. Using the surface counterterm approach we calculate the stress-energy tensor of quantum field theory (QFT) corresponding to the domain-wall black hole in the domain-wall/QFT correspondence. A brane universe is investigated in the domain-wall black hole background. When the tension term of the brane is equal to the surface counterterm, we find that the equation of motion of the brane can be mapped to the standard form of FRW equations, but with a varying gravitational constant on the brane. A Cardy-Verlinde-like formula is found, which relates the entropy density of the QFT to its energy density. At the moment when the brane crosses the black hole horizon of the background, the Cardy-Verlinde-like formula coincides with the Friedmann equation of the brane universe, and the Hubble entropy bound is saturated by the entropy of domain-wall black holes. 
  We propose an extended BRST invariant Lagrangian quantization scheme of general gauge theories based on an explicit realization of the modified triplectic algebra that was announced in our previous investigation (hep-th/0104189). The algebra includes, besides the odd operators $V^a$ appearing in the triplectic formalism, also the odd operators $U^a$ introduced within modified triplectic quantization, both of which being anti-Hamiltonian vector fields. We show that some even supersymplectic structure defined on the space of fields and antifields provides the extended BRST path integral with a well-defined integration measure. All the known Lagrangian quantization schemes based on the extended BRST symmetry are obtained by specifying the (free) parameters of that method. 
  We review some recent results on D-branes on Calabi-Yau (CY) manifolds. We show the existence of structures (helices and quivers) which enable one to make statements about large families of D-branes in various phases of the Gauged Linear Sigma Model (GLSM) associated with the CY manifold. A comparison of the quivers of two phases leads to the prediction that certain D-brane configurations will decay as one moves across phases. We discuss how boundary fermions can be used to realise various D-brane configurations associated with coherent sheaves in the GLSM with boundary.   This is based on the talk presented by S.G. at Strings 2001, Mumbai. 
  We calculate the three-point functions in the sine-Liouville theory explicitly. The same calculation was done in the (unpublished) work of Fateev, Zamolodchikov and Zamolodchikov to check the conjectured duality between the sine-Liouville and the SL(2,R)/U(1) coset CFTs. The evaluation of correlators boils down to that of a free-field theory with a certain number of insertion of screening operators. We prove that the winding number conservation is violated up to (+-)1 in three-point functions, which is in agreement with the result of FZZ that in generic N-point correlators the winding number conservation is violated up to N-2 units. A new integral formula of Dotsenko-Fateev type is derived, using which we write down the generic three-point functions of tachyons explicitly. When the winding number is conserved, the resultant expression is shown to reproduce the correlators in the coset model correctly, including the group-theoretical factor. As an application, we also study the superstring theory on linear dilaton background which is described by super-Liouville theory. We obtain the three-point amplitude of tachyons in which the winding number conservation is violated. 
  A discussion of the AdS/CFT correspondence in IIB is given in a superspace context. The main emphasis is on the properties of SCFT correlators on the boundary which are studied using harmonic superspace techniques. These techniques provide the easiest way of implementing the superconformal Ward identities. The Ward identities, together with analyticity, can be used to give a compelling argument in support of the non-renormalisation theorems for two- and three-point functions, and to establish the triviality of extremal and next-to-extremal correlation functions. The OPE in is also briefly discussed. 
  The encoding of an inflating patch of space-time in terms of a dual theory is discussed. Following Bousso's interpretation of the holographic principle, we find that those are generically described not by states in the dual theory but by density matrices. We try to implement this idea on simple deformations of the AdS/CFT examples, and an argument is given as to why inflation is so elusive to string theory. 
  It is shown how magnetic fluxbrane, Fp-brane, solutions are related to electric black p-branes by analytic continuation. Viewing the transverse space of branes as a warped cone, one finds that the cone base of the p-brane becomes the world-volume of the F(D-p-3)-brane and the world volume of the p-brane becomes the cone base of the F(D-p-3)-brane. An explicit example of the correspondence is given for a 2-brane and F6-brane of 11D supergravity. 
  We establish the ultra-violet finiteness of various classes of noncommutative gauge theories. 
  The problem of identifying the dynamical degrees of freedom for SU(2) gauge theories is discussed. After studying SU(N) theories, it is shown that classical pure SU(2) gauge theory is equivalent to an abelian theory. Finally, we prove that the Maximal Abelian Gauge correctly identifies the abelian degrees of freedom of the SU(2) theory. 
  The Dirac equation for an electron in two spatial dimensions in the Coulomb and homogeneous magnetic fields is an example of the so-called quasi-exactly solvable models. The solvable parts of its spectrum was previously solved from the recursion relations. In this work we present a purely algebraic solution based on the Bethe ansatz equations. It is realised that, unlike the corresponding problems in the Schr\"odinger and the Klein-Gordon case, here the unknown parameters to be solved for in the Bethe ansatz equations include not only the roots of wave function assumed, but also a parameter from the relevant operator. We also show that the quasi-exactly solvable differential equation does not belong to the classes based on the algebra $sl_2$. 
  An infra-red fixed point of ${\mathcal N}=1$ super-Yang-Mills theory is believed to be dual to a solution of five-dimensional gauged ${\mathcal N}=8$ supergravity. We test this conjecture at next to leading order in the large $N$ expansion by computing bulk one-loop corrections to the anomaly coefficient $a-c$. The one-loop corrections are non-zero for all values of the bulk mass, and not just special ones as claimed in previous work. 
  By applying mirror symmetry to D-branes in a Calabi-Yau geometry we shed light on a $G_2$ flop in M-theory relevant for large $N$ dualities in ${\cal N}=1$ supersymmetric gauge theories. Furthermore, we derive superpotential for M-theory on corresponding $G_2$ manifolds for all A-D-E cases. This provides an effective method for geometric engineering of ${\cal N}=1$ gauge theories for which mirror symmetry gives exact information about vacuum geometry. We also find a number of interesting dual descriptions. 
  A concise derivation of all uncertainty relations is given entirely within the context of phase-space quantization, without recourse to operator methods, to the direct use of Weyl's correspondence, or to marginal distributions of x and p. 
  We study loop corrections in boundary string field theory (BSFT). After commenting on problems with quantizing the tree level BSFT as an ordinary field theory, we discuss the tree level coupling to closed strings and define the loop corrections via factorization in the closed string channel. This description is weakly coupled in the vicinity of the closed string vacuum. Our proposal for the one-loop effective action differs in general from computing the annulus or cylinder partition functions. We also compute the decay rates and the loop corrections to the tensions of unstable branes in perturbative string theory. 
  We consider the couplings of RR fields with open string sector for $Dp$-${\overline{Dp}}$ backgrounds of various $p$. Proposed approach, based on the approximation of the open string algebra by the algebra of differential operators, provides the unified description of these couplings and their interrelations. 
  In this paper we propose a unified approach to (topological) string theory on certain singular spaces in their large volume limit. The approach exploits the non-commutative structure of D-branes, so the space is described by an algebraic geometry of non-commutative rings. The paper is devoted to the study of examples of these algebras. In our study there is an auxiliary commutative algebraic geometry of the center of the (local) algebras which plays an important role as the target space geometry where closed strings propagate. The singularities that are resolved will be the singularities of this auxiliary geometry. The singularities are resolved by the non-commutative algebra if the local non-commutative rings are regular. This definition guarantees that D-branes have a well defined K-theory class. Homological functors also play an important role. They describe the intersection theory of D-branes and lead to a formal definition of local quivers at singularities, which can be computed explicitly for many types of singularities. These results can be interpreted in terms of the derived category of coherent sheaves over the non-commutative rings, giving a non-commutative version of recent work by M. Douglas. We also describe global features like the Betti numbers of compact singular Calabi-Yau threefolds via global holomorphic sections of cyclic homology classes. 
  I review the construction of an action for open superstring field theory which does not suffer from the contact term problems of other approaches. This action resembles a Wess-Zumino-Witten action and can be constructed in a manifestly D=4 super-Poincar\'e covariant manner. This review is based on lectures given at the ICTP Latin-American String School in Mexico City and the Komaba 2000 Workshop in Tokyo. 
  Withdrawn by the authors 
  One of the most intriguing aspects of Chern-Simons-type topological models is the fractional statistics of point particles which has been shown essential for our understanding of the fractional quantum Hall effects. Furthermore these ideas are applied to the study of high Tc superconductivity. We present here an recently proposed model for fractional spin with the Pauli term. On the other hand, in D=4 space-time, a Schwarz-type topological gauge theory with antisymmetric tensor gauge field, namely BF model, is reviewed. Antisymmetric tensor fields are conjectured as mediator of string interaction. A dimensional reduction of the previous model provides a (2+1) dimensional topological theory, which involves a 2-form B and a 0-form $\phi$. Some recent results on this model are reported. Recently, there have been thoughts of generalizing unusual statistics to extended objects in others space-time dimensions, and in particular to the case of strings in four dimensions. In this context, discussions about fractional spin and antisymmetric tensor field are presented. 
  We study the conformal spectra of the critical square lattice Ising model on the Klein bottle and M\"obius strip using Yang-Baxter techniques and the solution of functional equations. In particular, we obtain expressions for the finitized conformal partition functions in terms of finitized Virasoro characters. This demonstrates that Yang-Baxter techniques and functional equations can be used to study the conformal spectra of more general exactly solvable lattice models in these topologies. The results rely on certain properties of the eigenvalues which are confirmed numerically. 
  The Lagrangian Sp(3) BRST symmetry for irreducible gauge theories is constructed in the framework of homological perturbation theory. The canonical generator of this extended symmetry is shown to exist. A gauge-fixing procedure specific to the standard antibracket-antifield formalism, that leads to an effective action, which is invariant under all the three differentials of the Sp(3) algebra, is given. 
  The metric of the gravity dual of a field theory should contain precisely the same information as the field theory. We discuss this connection in the N=4 theory where a scalar vev may be introducedat the level of 5d supergravity and the solutions lifted to 10d. We stress the role of brane probing in finding the coordinates appropriate to the field theory. In these coordinates the metric parametrizes the gauge invariant operators of the field theory and either side of the duality is uniquely determined by the other. We follow this same chain of computations for the 10d lift of the N=2* geometry of Pilch and Warner. The brane probe of the metric reveals the 2d moduli space and the functional form of the gauge coupling. In the coordinates appropriate to the field theory the metric on moduli space takes a very simple form and one can read off the gravity predictions for operators in the field theory. Surprisingly there is logarithmic renormalization even in the far UV where the field theory reverts to N=4 super Yang-Mills. We extend the analysis of Buchel et al to find the D3 brane source distribution that generates the supergravity prediction for the gauge coupling for the whole class of solutions corresponding to different points on moduli space. This distribution does not account for the logarithmic behaviour in the rest of the metric though. We discuss possible resolutions of the discrepancy. 
  The Casimir stress on two concentric spherical shell in de Sitter background for massless scalar field is calculated. The scalar field satisfies Dirichlet boundary conditions on the spheres. The metric is written in conformally flat form to make maximum use of Minkowski space calculations. Then the Casimir stress is calculated for inside and outside of the shell with different backgrounds. This model may be used to study the effect of the Casimir stress on the dynamics of the domain wall formation in inflationary models of early universe. 
  We investigate conformal mechanics associated with the rotating Bertotti-Robinson (RBR) geometry found recently as the near-horizon limit of the extremal rotating Einstein-Maxwell-dilaton-axion black holes. The solution breaks the $SL(2,R)\times SO(3)$ symmetry of Bertotti-Robinson (BR) spacetime to $SL(2,R)\times U(1)$ and breaks supersymmetry in the sense of $N=4, d=4$ supergravity as well. However, it shares with BR such properties as confinement of timelike geodesics and discreteness of the energy of test fields on the geodesically complete manifold. Conformal mechanics governing the radial geodesic motion coincides with that for a charged particle in the BR background (a relativistic version of the De Alfaro-Fubini-Furlan model), with the azimuthal momentum playing the role of a charge. Similarly to the BR case, the transition from Poincar\'e to global coordinates leads to a redefinition of the Hamiltonian making the energy spectrum discrete. Although the metric does not split into a product space even asymptotically, it still admits an infinite-dimensional extension of $SL(2,R)$ as asymptotic symmetry. The latter is shown to be given by the product of one copy of the Virasoro algebra and U(1), the same being valid for the extremal Kerr throat. 
  We compute the Boundary Superstring Field Theory partition function on the annulus in the presence of independent linear tachyon profiles on the two boundaries. The R-R sector is found to contribute non-trivially to the derivative terms of the space-time effective action. In the process we construct a boundary state description of D-branes in the presence of a linear tachyon. We quantize the open string in a tachyonic background and address the question of open/closed string duality. 
  Representation theory, for the classical binary polyhedral groups is encoded by the affine Dynkin diagrams E6^{(1)}, E7^{(1)} and E8^{(1)} (McKay correspondance). The quantum versions of these classical geometries are associated with representation theories described by the usual Dynkin diagrams E6, E7 and E8. The purpose of these notes is to compare several chosen aspects of the classical and quantum geometries by using the study of spaces of paths and spaces of essential paths (Ocneanu theory) on these diagrams. To keep the size of this contribution small enough, most of our discussion will be limited to the cases of diagrams E6 and E6^{(1)}, i.e. to the quantum and classical tetrahedra. We shall in particular interpret the A11 labelling of the vertices of E6 diagram as a quantum analogue of the usual decomposition of spaces of sections for vector bundles above homogeneous spaces. We also show how to recover Klein invariants of polyhedra by paths algebra techniques and discuss their quantum generalizations. 
  We show the instability of two self-tuning brane world models with gauge invariant linear perturbation theory. This general method confirms a known instability of the original model with vanishing bulk potential. We also show the dynamical instability of a recently proposed self-tuning "smooth" brane model and its limit, the Randall-Sundrum model. Astonishingly, we also find instability under purely gravitational perturbations. 
  We present integral equations for the scattering amplitudes of three scalar particles, using the Faddeev channel decomposition, which can be readily extended to any finite number of particles of any helicity. The solution of these equations, which have been demonstrated to be calculable, provide a non-perturbative way of obtaining relativistic scattering amplitudes for any finite number of particles that are Lorentz invariant, unitary, cluster decomposable and reduce unambiguously in the non-relativistic limit to the non-relativistic Faddeev equations. The aim of this program is to develop equations which explicitly depend upon physically observable input variables, and do not require renormalization or dressing of these parameters to connect them to the boundary states. 
  I argue that the gauge group of noncommutative gauge theory consists of maps into unitary operators on Hilbert space of the form $u=1+K$ with $K$ compact. Some implications of this proposal are outlined. 
  We argue that any viable mechanism of gauge field localization should automatically imply charge universality on the brane. We study whether this condition is satisfied in the two known proposals aimed to localize vector field in flat bulk space. We construct a simple calculable model with confinement in the bulk and deconfinement on the brane, as in the Shifman--Dvali set up. We find that in our model the 4-dimensional Coulomb law is indeed reproduced on the brane due to the massless localized photon mode. The charge universality is enforced by the presence of ``confining strings''. On the other hand, charge universality condition is not satisfied in another, brane-induced localization mechanism when the number of extra dimensions d is larger than two. We demonstrate that in the non-Abelian case the gauge fields inside the brane are never four-dimensional and their self-interaction is strong at all distances of interest. Hence this mechanism does not work for d>2. At d=2 the charge universality is still a problem, but it holds automatically at d=1. At d=1, however, the bulk gauge fields are strongly coupled in the non-Abelian case. 
  We find that the gas of IIA strings undergoes a phase transition into a gas of IIB strings at the self-dual temperature. A gas of free heterotic strings undergoes a Kosterlitz-Thouless duality transition with positive free energy and positive specific heat but vanishing internal energy at criticality. We examine the consequences of requiring a tachyon-free thermal string spectrum. We show that in the absence of Ramond-Ramond fluxes the IIA and IIB string ensembles are thermodynamically ill-defined. The 10D heterotic superstrings have nonabelian gauge fields and in the presence of a temperature dependent Wilson line background are found to share a stable and tachyon-free ground state at all temperatures starting from zero with gauge group SO(16)xSO(16). The internal energy of the heterotic string is a monotonically increasing function of temperature with a stable and supersymmetric zero temperature limit. Our results point to the necessity of gauge fields in a viable weakly coupled superstring theory. Note Added (Sep 2005). 
  Multi-brane backgrounds are studied in the framework of the background independent open string field theory. A simple description of the non-abelian degrees of freedom is given. Algebra of the differential operators acting on the space of functions on the space-time provides a natural tool for the discussion of this phenomena. 
  We construct tachyon lump solutions of bosonic D2-branes on SU(2) group manifolds by level truncation approximation in cubic string field theory, which are regarded as D0-branes. The energies for these solutions show good agreement with the expected values. 
  Quadratic tachyon profile has been discussed in the boundary string field theory. We here compute the g-function by factorizing the cylinder amplitude. The answer is compared with the disc partition function. The boundary state is constructed. We extend these computations to those of the boundary sine-Gordon model at the free fermion point. 
  The free field description of logarithmic and prelogarithmic operators in non compact Wess-Zumino-Witten model is analysed. We study the structure of the Jordan blocks of the SL(2)_k affine algebra and the role of the puncture operator in the theory in relation with the unitarity bound. 
  We consider the brane universe in the bulk background of the charged topological AdS black holes. The evolution of the brane universe is described by the Friedmann equations for a flat or an open FRW-universe containing radiation and stiff matter. We find that the temperature and entropy of the dual CFT are simply expressed in terms of the Hubble parameter and its time derivative, and the Friedmann equations coincide with thermodynamic formulas of the dual CFT at the moment when the brane crosses the black hole horizon. We obtain the generalized Cardy-Verlinde formula for the CFT with an R-charge, for any values of the curvature parameter k in the Friedmann equations. 
  We construct supergravity solutions dual to the twisted field theories arising when M-theory membranes wrap holomorphic curves in Calabi-Yau n-folds. The solutions are constructed in an Abelian truncation of maximal D=4 gauged supergravity and then uplifted to D=11. For four-folds and five-folds we find new smooth AdS/CFT examples and for all cases we analyse the nature of the singularities that arise. Our results provide an interpretation of certain charged topological AdS black holes. We also present the generalised calibration two-forms for the solutions. 
  We report on joint work, past and in progress, with K.Fredenhagen and with J.E,Roberts, on the quantum structure of spacetime in the small which is dictated by the principles of Quantum Mechanics and of General Relativity; we comment on how these principles point to a deep link between coordinates and fields. This is an expanded version of a lecture delivered at the 37th Karpacz School in Theoretical Physics, February 2001. 
  We present some preliminary investigations about the AdS2*S2 D3-branes in AdS3*S3. We analyse the quadratic fluctuations of Dirac-Born-Infeld action around a given semi-classical D-brane configuration and compare them with results obtained by using conformal field theory techniques. We finally study classical motions of open strings attached to those D-branes and analyse the role of the spectral flow in this context. 
  In this short note we provide a proof that the Chern-Simons part of the action for N D-branes is invariant under gauge transformations of the RR fields of the type C_p ->C_p + d\Lambda_{p-1}, and rewrite the action in a form that makes this symmetry manifest. 
  The OPE of two N=2 R-symmetry current (short) multiplets is determined by the possible superspace three-point functions that two such multiplets can form with a third, a priori long multiplet. We show that the shortness conditions on the former put strong restrictions on the quantum numbers of the latter. In particular, no anomalous dimension is allowed unless the third supermultiplet is an R-symmetry singlet. This phenomenon should explain many known non-renormalization properties of correlation functions, including the one of four stress-tensor multiplets in N=4 SYM_4. 
  The radion dynamics related to the presence of moving branes with both positive or negative tensions is studied in the linearized approximation. The radion effective Lagrangian is computed for a compact system with three branes and in particular we examine the decompactification limit when one brane is sent to infinity. In the non-compact case we calculate the coupling of the gravitational modes (graviton, dilaton and radion) to matter on the branes. The character of gravity on the two branes for all possible combinations of brane tensions is also discussed. It turns out that one can have a normalizable dilaton mode even in the non-compact case. Finally, we speculate on the role of moving branes as a possible source of radion emission. 
  We consider the brane-world in the holographic point of view. Bearing the realistic models in mind, the bulk massless scalar field is introduced. First of all, we find the constraint on the coupling of the scalar fields with the matter(not holographic CFT) on the brane. We show that the traceless part of the energy-momentum tensor of holographic CFT is a part of the bulk Weyl tensor. The trace part which comes from the trace-anomaly is corresponding to the $\rho^2$-term appeared in the generalized FRW equation in the brane-world. 
  We study the perturbative unitarity of non-commutative quantum Yang-Mills theories, extending previous investigations on scalar field theories to the gauge case where non-locality mingles with the presence of unphysical states. We concentrate our efforts on two different aspects of the problem. We start by discussing the analytical structure of the vacuum polarization tensor, showing how Cutkoski's rules and positivity of the spectral function are realized when non-commutativity does not affect the temporal coordinate. When instead non-commutativity involves time, we find the presence of extra troublesome singularities on the $p_0^2$-plane that seem to invalidate the perturbative unitarity of the theory. The existence of new tachyonic poles, with respect to the scalar case, is also uncovered. Then we turn our attention to a different unitarity check in the ordinary theories, namely time exponentiation of a Wilson loop. We perform a $O(g^4)$ generalization to the (spatial) non-commutative case of the familiar results in the usual Yang-Mills theory. We show that exponentiation persists at $O(g^4)$ in spite of the presence of Moyal phases reflecting non-commutativity and of the singular infrared behaviour induced by UV/IR mixing. 
  We investigate the classical geometry corresponding to a collection of fractional D3 branes in the orbifold limit of an ALE space. We discuss its interpretation in terms of the world-volume gauge theory on the branes, which is in general a non conformal N=2 Yang-Mills theory with matter. The twisted fields reproduce the perturbative behaviour of the gauge theory. We regulate the IR singularities for both twisted and untwisted fields by means of an enhancon mechanism qualitatively consistent with the gauge theory expectations. The five-form flux decreases logarithmically towards the IR with a coefficient dictated by the gauge theory beta-functions. 
  We introduce normal coordinates on the infinite dimensional group $G$ introduced by Connes and Kreimer in their analysis of the Hopf algebra of rooted trees. We study the primitive elements of the algebra and show that they are generated by a simple application of the inverse Poincar\'e lemma, given a closed left invariant 1-form on $G$. For the special case of the ladder primitives, we find a second description that relates them to the Hopf algebra of functionals on power series with the usual product. Either approach shows that the ladder primitives are given by the Schur polynomials. The relevance of the lower central series of the dual Lie algebra in the process of renormalization is also discussed, leading to a natural concept of $k$-primitiveness, which is shown to be equivalent to the one already in the literature. 
  We discuss the origin of the associativity (WDVV) equations in the context of quasiclassical or Whitham hierarchies. The associativity equations are shown to be encoded in the dispersionless limit of the Hirota equations for KP and Toda hierarchies. We show, therefore, that any tau-function of dispersionless KP or Toda hierarchy provides a solution to associativity equations. In general, they depend on infinitely many variables. We also discuss the particular solution to the dispersionless Toda hierarchy that describes conformal mappings and construct a family of new solutions to the WDVV equations depending on finite number of variables. 
  We show how the presence of a very light scalar with a cubic self-interaction in six dimensions can stabilize the extra dimensions at radii which are naturally exponentially large, $r \sim \ell \exp [(4\pi)^3/g^2]$, where $\ell$ is a microscopic physics scale and $g$ is the (dimensionless) cubic coupling constant. The resulting radion mode of the metric becomes a very light degree of freedom whose mass, $m \sim 1/(M_p r^2)$ is stable under radiative corrections. For $1/r \sim 10^{-3}$ eV the radion is extremely light, $m \sim 10^{-33}$ eV. Its couplings cause important deviations from General Relativity in the very early universe, but naturally evolve to phenomenologically acceptable values at present. 
  In the previous paper\cite{KS00b}, we derived the Abelian projected effective gauge theory as a low energy effective theory of the SU(N) Yang-Mills theory by adopting the maximal Abelian gauge. At that time, we have demonstrated the multiplicative renormalizability of the propagators for the diagonal gluon and the dual Abelian anti-symmetric tensor field. In this paper, we show the multiplicative renormalizability of the Green's functions also for the off-diagonal gluon. Moreover we complement the previous results by calculating the anomalous dimension and the renormalization group functions which are undetermined in the previous paper. 
  We study four kinds of 1/4 BPS solutions in massive IIA supergravity corresponding to D8-D0-F1, D8-D2, D8-D4 and D8-D6-NS5 systems. We show that these solutions are reproduced without making nontrivial assumptions by using supersymmetry conditions. D8-D2 and D8-D4 solutions are represented by harmonic functions, as usual, while the other two are represented by solutions of non-linear differential equations. Because these four solutions can be treated in almost identical ways, we mainly focus on the D8-D6-NS5 systems. We first discuss D6-NS5 solutions with uniform mass parameters. Then, we introduce D8-branes as domain walls by connecting two solutions with different values of the mass parameter. We also discuss boundary conditions and supersymmetry on domain walls. 
  We perform canonical quantization of the open string on a unstable D-brane in the background of the tachyon condensation. Evaluating the Polyakov path-integral on a stripe, we obtain the field theoretical propagator in the open string theory. As the condensation occurs the string field theory is continuously deformed. At the infrared fixed point of the condensation, the open string field on the unstable D-brane transmutes to that on the lower dimensional D-brane with the correct D-brane tension. 
  We consider deformations of a toroidal orbifold $T^4/Z_2$ and an orbifold of quartic in $CP^3$. In the $T^4/Z_2$ case, we construct a family of noncommutative K3 surfaces obtained via both complex and noncommutative deformations. We do this following the line of algebraic deformation done by Berenstein and Leigh for the Calabi-Yau threefold. We obtain 18 as the dimension of the moduli space both in the noncommutative deformation as well as in the complex deformation, matching the expectation from classical consideration. In the quartic case, we construct a $4 \times 4$ matrix representation of noncommutative K3 surface in terms of quartic variables in $CP^3$ with a fourth root of unity. In this case, the fractionation of branes occurs at codimension two singularities due to the presence of discrete torsion. 
  We investigate the mathematical structure of the world sheet in two-dimensional conformal field theories. 
  Fruitful ideas on how to quantize gravity are few and far between. In this paper, we give a complete description of a recently introduced non-perturbative gravitational path integral whose continuum limit has already been investigated extensively in d less than 4, with promising results. It is based on a simplicial regularization of Lorentzian space-times and, most importantly, possesses a well-defined, non-perturbative Wick rotation. We present a detailed analysis of the geometric and mathematical properties of the discretized model in d=3,4. This includes a derivation of Lorentzian simplicial manifold constraints, the gravitational actions and their Wick rotation. We define a transfer matrix for the system and show that it leads to a well-defined self-adjoint Hamiltonian. In view of numerical simulations, we also suggest sets of Lorentzian Monte Carlo moves. We demonstrate that certain pathological phases found previously in Euclidean models of dynamical triangulations cannot be realized in the Lorentzian case. 
  We construct the most general gauge fixing and the associated Faddeev-Popov ghost term for the SU(2) Yang-Mills theory, which leaves the global U(1) gauge symmetry intact (i.e., the most general Maximal Abelian gauge). We show that the most general form involves eleven independent gauge parameters. Then we require various symmetries which help to reduce the number of independent parameters for obtaining the simpler form. In the simplest case, the off-diagonal part of the gauge fixing term obtained in this way is identical to the modified maximal Abelian gauge term with two gauge parameters which was proposed in the previous paper from the viewpoint of renormalizability.   In this case, moreover, we calculate the beta function, anomalous dimensions of all fields and renormalization group functions of all gauge parameters in perturbation theory to one-loop order. We also discuss the implication of these results to obtain information on low-energy physics of QCD. 
  Brane-world singularities are analysed, emphasizing the case of supergravity in singular spaces where the singularity puzzle is naturally resolved. These naked singularities are either time-like or null, corresponding to the finite or infinite amount of conformal time that massless particles take in order to reach them. Quantum mechanically we show that the brane-world naked singularities are inconsistent. Indeed we find that time-like singularities are not wave-regular, so the time-evolution of wave packets is not uniquely defined in their vicinity, while null singularities absorb incoming radiation. Finally we stress that for supergravity in singular spaces there is a topological obstruction, whereby naked singularities are necessarily screened off by the second boundary brane. 
  We consider the holographic duality for a generic bulk theory of scalars coupled to gravity. By studying the fluctuations around Poincare invariant backgrounds with non-vanishing scalars, with the scalar and metric boundary conditions considered as being independent, we obtain all one- and two-point functions in the dual renormalization group flows of the boundary field theory. Operator and vev flows are explicitly distinguished by means of the physical condensates. The method is applied to the GPPZ and Coulomb branch flows, and field theoretical expectations are confirmed. 
  We discuss deformation quantization of the Kaehler coset space by using the Fedosov formalism. We show that the Killing potentials of the Kaehler coset space satisfy the fuzzy algebrae, when the coset space is irreducible. 
  We investigate classical bosonic open-string field theory from the perspective of the Wilson renormalization group of world-sheet theory. The microscopic action is identified with Witten's covariant cubic action and the short-distance cut-off scale is introduced by length of open-string strip which appears in the Schwinger representation of open-string propagator. {\it Classical open-string field theory} in the title means open-string field theory governed by a classical part of the low energy action. It is obtained by integrating out suitable tree interactions of open-strings and is of non-polynomial type. We study this theory by using the BV formalism. It turns out to be deeply related with deformation theory of $A_{\infty}$-algebra. We introduce renormalization group equation of this theory and discuss it from several aspects. It is also discussed that this theory is interpreted as a boundary open-string field theory. Closed-string BRST charge and boundary states of closed-string field theory in the presence of open-string field play important roles. 
  The equation of motion of test particles in the geometry of a black string embedded in a 5-dimensional AdS spacetime is studied. 
  We determine the non-abelian Born-Infeld action, including fermions, as it results from the four-point tree-level open superstring scattering amplitudes at order alpha'^2. We find that, after an appropriate field redefinition all terms at this order can be written as a symmetrised trace. We confront this action with the results that follow from kappa-symmetry and conclude that the recently proposed non-abelian kappa-symmetry cannot be extended to cubic orders in the Born-Infeld curvature. 
  Using a 2-loop approximation for $\beta$-functions, we have considered the corresponding renormalization group improved effective potential in the Dual Abelian Higgs Model (DAHM) of scalar monopoles and calculated the phase transition (critical) couplings in U(1) and SU(N) regularized gauge theories. In contrast to our previous result $\alpha_{crit} \approx 0.17$, obtained in the one-loop approximation with the DAHM effective potential (see Ref.[20]), the critical value of the electric fine structure constant in the 2-loop approximation, calculated in the present paper, is equal to $\alpha_{crit}\approx 0.208$ and coincides with the lattice result for compact QED [10]: $\alpha_{crit}^{lat} \approx 0.20\pm 0.015$. Following the 't Hooft's idea of the "abelization" of monopole vacuum in the Yang--Mills theories, we have obtained an estimation of the SU(N) triple point coupling constants, which is $\alpha_{N,crit}^{-1}=\frac{N}{2}\sqrt{\frac{N+1}{N-1}}  \alpha_{U(1),crit}^{-1}$. This relation was used for the description of the Planck scale values of the inverse running constants $\alpha_i^{-1}(\mu)$ (i=1,2,3 correspond to U(1), SU(2) and SU(3) groups), according to the ideas of the Multiple Point Model [16]. 
  We apply the formalism of holographic renormalization to domain wall solutions of 5-dimensional supergravity which are dual to deformed conformal field theories in 4 dimensions. We carefully compute one- and two-point functions of the energy-momentum tensor and the scalar operator mixing with it in two specific holographic flows, resolving previous difficulties with these correlation functions. As expected, two-point functions have a 0-mass dilaton pole for the Coulomb branch flow in which conformal symmetry is broken spontaneously but not for the flow dual to a mass deformation in which it is broken explicitly. A previous puzzle of the energy scale in the Coulomb branch flow is explained. 
  We consider the N=4 supersymmetric Yang-Mills theory in four dimensions. We compute the one-loop contributions to the effective action with five external vector fields and compare them with corresponding results in open superstring theory. Our calculation determines the structure of the F^5 terms that appear in the nonabelian generalization of the Born Infeld action. The trace operation on the gauge group indices receives contributions from the symmetric as well as the antisymmetric part. We find that in order to study corrections to the symmetrized trace prescription one has to consistently take into account derivative contributions not only with antisymmetrized products \nabla_{[\mu}\nabla_{\nu]} but also with symmetrized ones \nabla_{(\mu}\nabla_{\nu)}. 
  The dynamics of a particle moving in background electromagnetic and gravitational fields is revisited from a Lie group cohomological perspective. Physical constants characterising the particle appear as central extension parameters of a group which is obtained from a centrally extended kinematical group (Poincare or Galilei) by making local some subgroup. The corresponding dynamics is generated by a vector field inside the kernel of a presymplectic form which is derived from the canonical left-invariant one-form on the extended group. A non-relativistic limit is derived from the geodesic motion via an Inonu-Wigner contraction. A deeper analysis of the cohomological structure reveals the possibility of a new force associated with a non-trivial mixing of gravity and electromagnetism leading to in principle testable predictions. 
  We argue that the quantized non-Abelian gauge theory can be obtained as the infrared limit of the corresponding classical gauge theory in a higher dimension. We show how the transformation from classical to quantum field theory emerges and calculate Planck's constant from quantities defined in the underlying classical gauge theory. 
  We consider Dirac-supersymmetric interactions, which produce CP-conserving separation of positive and negative energy solutions in the Dirac equation in order to investigate an alternative to the Kaluza-Klein mechanism. We review conditions under which separation is possible into free particle and compactified behaviors in different dimensions, with attention to spin degrees of freedom. We show a U(1) constant magnetic field produces such kind of behavior; an explicit treatment is given to the 6-d to 4-d and 4-d to 2-d breaking cases and the spectrum is obtained. A dynamical mass-creation mechanism is suggested from the procedure. 
  A connection between solutions of the relativistic d-brane system in (d+1) dimensions with the solutions of a Galileo invariant fluid in d-dimensions is by now well established. However, the physical nature of the light-cone gauge description of a relativistic membrane changes after the reduction to the fluid dynamical model since the gauge symmetry is lost. In this work we argue that the original gauge symmetry present in a relativistic d-brane system can be recovered after the reduction process to a d-dimensional fluid model. To this end we propose, without introducing Wess-Zumino fields, a gauge invariant theory of isentropic fluid dynamics and show that this symmetry corresponds to the invariance under local translation of the velocity potential in the fluid dynamics picture. We show that different but equivalent choices of the sympletic sector lead to distinct representations of the embedded gauge algebra. 
  We study the creation of massless scalar particles from the quantum vacuum due to the dynamical Casimir effect by spherical shell with oscillating radius. In the case of a small amplitude of the oscillation, to solve the infinite set of coupled differential equations for the instantaneous basis expansion coefficients we use the method based on the time-dependent perturbation theory of the quantum mechanics. To the first order of the amplitude we derive the expressions for the number of the created particles for both parametric resonance and non-resonance cases. 
  In the conformal field theories having affine SL(2) symmetry, we study the operator product expansion (OPE) involving primary fields in highest weight representations. For this purpose, we analyze properties of primary fields with definite SL(2) weights, and calculate their two- and three-point functions. Using these correlators, we show that the correct OPE is obtained when one of the primary fields belongs to the degenerate highest weight representation. We briefly comment on the OPE in the SL(2,R) WZNW model. 
  In this paper, we consider a reduced supersymmetric Yang-Mills integral with four supercharges by using a Gaussian approximation scheme and its improved version. We calculate the free energy and the expectation values of Polyakov loop and Wilson loop operators by extending the method employed in the bosonic case in the previous paper. Our results nicely match to the exact and the numerical results obtained before. The loop amplitudes exhibit good scaling behaviors similarly as in the bosonic case. The 't Hooft like large $N$ limit leads simple formulas for the case of the loop length smaller. Also, the Polyakov loop and the Wilson loop are computed for the case of the loop length sufficiently large, where we see that the behavior of the Wilson loop reproduces the result simulated for a few smaller values of $N$ at least qualitatively. 
  We develop two numerical methods to solve the differential equations with deviating arguments for the motion of two charges in the action-at-a-distance electrodynamics. Our first method uses St\"urmer's extrapolation formula and assumes that a step of integration can be taken as a step of light ladder, which limits its use to shallow energies. The second method is an improvement of pre-existing iterative schemes, designed for stronger convergence and can be used at high-energies. 
  We consider a deformation of the BF theory in any dimension by means of the antifield BRST formalism. Possible consistent interaction terms for the action and the gauge symmetries are analyzed and we find a new class of topological gauge theories. Deformations of the world volume BF theory are considered as possible deformations of the topological open membrane. Therefore if we consider these theories on open membranes, we obtain noncommutative structures of the boundaries of open membranes, and we propose a generalization of the path integral representation of the star deformation. 
  We present a method for classifying conformal field theories based on Coulomb gases (bosonic free-field construction). Given a particular geometric configuration of the screening charges, we give necessary conditions for the existence of degenerate representations and for the closure of the vertex-operator algebra. The resulting classification contains, but is more general than, the standard one based on classical Lie algebras. We then apply the method to the Coulomb gas theory for the two-flavoured loop model of Jacobsen and Kondev. The purpose of the study is to clarify the relation between Coulomb gas models and conformal field theories with extended symmetries. 
  We describe properties of the M-theory five-brane containing $Q$ coincident self-dual strings on its worldvolume. This is the five-brane description of Q membranes ending on the five-brane. In particular, we consider a Maldacena-like low energy limit in the six-dimensional worldvolume which yields a near `horizon' description of the self-dual string using light open membranes, i.e. OM theory, in an AdS_3 x S^3 geometry. 
  We consider the associativity (or WDVV) equations in the form they appear in Seiberg-Witten theory and prove that they are covariant under generic electric-magnetic duality transformations. We discuss the consequences of this covariance from various perspectives. 
  The space of local integrals of motion for the Sine-Gordon theory (the free fermion point) and the theory of free fermions in the light cone coordinates is investigated. Some important differences between the spaces of local integrals of motion of these theories are obtained. The equivalence is broken on the level of the integrals of motion between bosonic and fermionic theories (in the free fermion point). The integrals of motion are constracted without Quantum Inverse Scattering Method (QISM)and the additional quantum integrals of motion are obtaned. So the QISM is not absolutely complete. 
  We study curves of marginal stability (CMS) in five-dimensional N=1 E_N theories compactified on a circle using the D3-brane probe realization. In this realization, BPS states correspond to string webs in the affine E_N 7-brane background and junction positions of the webs deteremine CMS. We find that there exist string webs involving infinitely many junctions. Consequently the E_N theories have infinitely many CMS. We also find that there exists a transition from open strings to string webs involving loops. The transition describes a new phenomenon occurring on CMS. 
  We generalize recent study of the stability of isotropic (spherical) rotating membranes to the anisotropic ellipsoidal membrane. We find that while the stability persists for deformations of spin $l=1$, the quadrupole and higher spin deformations ($l\geq 2$) lead to instabilities. We find the relevant instability modes and the corresponding eigenvalues. These indicate that the ellipsoidal rotating membranes generically decay into finger-like configurations. 
  Recently a brane world perspective on the cosmological constant and the hierarchy problems was presented. Here, we elaborate on some aspects of that particular scenario and discuss the stability of the stationary brane solution and the dynamics of a probe brane. Even though the brane is unstable under a small perturbation from its stationary position, such instability is harmless when the 4-D cosmological constant is very small, as is the case of our universe. One may also introduce radion stabilizing potentials in a more realistic scenario. 
  Three-point functions of Wess-Zumino-Witten models are investigated. In particular, we study the level-dependence of three-point functions in the models based on algebras $su(3)$ and $su(4)$. We find a correspondence with Berenstein-Zelevinsky triangles. Using previous work connecting those triangles to the fusion multiplicities, and the Gepner-Witten depth rule, we explain how to construct the full three-point functions. We show how their level-dependence is similar to that of the related fusion multiplicity. For example, the concept of threshold level plays a prominent role, as it does for fusion. 
  A semiclassical gravitation model is outlined which makes use of the Casimir energy density of vacuum fluctuations in extra compactified dimensions to produce the present-day cosmological constant as rho_LAMBDA ~ M^8/M_P^4, where M_P is the Planck scale and M is the weak interaction scale. The model is based on (4+D)-dimensional gravity, with D = 2 extra dimensions with radius b(t) curled up at the ADD length scale b_0 = M_P/M^2 ~ 0.1 mm. Vacuum fluctuations in the compactified space perturb b_0 very slightly, generating a small present-day cosmological constant.   The radius of the compactified dimensions is predicted to be b_0 = k^{1/4} 0.09 mm (or equivalently M = 2.4 TeV/k^{1/8}), where the Casimir energy density is k/b^4.   Primordial inflation of our three-dimensional space occurs as in the cosmology of the ADD model as the inflaton b(t), which initially is on the order of 1/M ~ 10^{-17} cm, rolls down its potential to b_0. 
  We explore the idea of a network of domain walls to appear at the surface of a soliton star. We show that for a suitable fine tuning among the parameters of the model we can find localized fermion zero modes only on the network of domain walls. In this scenario the soliton star gets unstable and decays into free particles before the cold matter upper mass limit is achieved. However, if fermions do not bind to the network of domain walls, the network becomes neutral, imposing a new lower bound on the charge of the soliton star, slightly raising its critical mass. 
  We give an explicit parameterization of the general 8k -3 instanton. 
  The presence of a distant D4-brane is used to further investigate the duality between M-theory and D0-brane quantum mechanics. Although the D4-brane background fields are not strong enough to induce a classical dielectric effect in the D0 system, a polarization of the quantum mechanical ground state does result. A similar deformation arises for the bubble of normal space found near D0-branes in classical supergravity solutions. These deformations are compared and are shown to have the same structure in each case. Brief comments are included on the relation of D0-branes in this background to D0-branes as instantons in the D4-brane field theory and an appendix addresses certain infrared issues associated with 't Hooft scaling in 0+1 dimensions. 
  We propose a vacuum condensate of mass dimension 2 consisting of gluons and ghosts in the framework of the manifestly covariant gauge fixing of the SU(N) Yang-Mills theory. This quantity is both BRST and anti-BRST invariant for any gauge. It includes the ghost condensation $C^a \bar{C}^a$ proposed first in the modified Maximal Abelian gauge and reduces to the gluon condensates $(\mathscr{A}_\mu)^2$ of mass dimension 2 proposed recently in the Landau gauge of the Lorentz gauge fixing. The vacuum condensate of dimensions 2 can provide the effective mass for gluons and ghosts. The possible existence of such condensations is demonstrated by calculating the operator product expansion of the gluon and ghost propagators in both gauges. Its implications to quark confinement are also discussed in consistent with the previous works. 
  The estimation of the cosmological constant in inflationary Brane New World models is done. It is shown that basically it is quite large, of the same order as in anomaly-driven inflation. However, for some fine-tuning of bulk gravitational constant and AdS scale parameter $l^2$ it may be reduced to sufficiently small value. Bulk higher derivative AdS gravity with quantum brane matter may also serve as the model where small positive cosmological constant occurs. 
  As an alternative to the covariant Ostrogradski method, we show that higher-derivative relativistic Lagrangian field theories can be reduced to second differential-order by writing them directly as covariant two-derivative theories involving Lagrange multipliers and new fields. Despite the intrinsic non-covariance of the Dirac's procedure used to deal with the constraints, the explicit Lorentz invariance is recovered at the end. We develop this new setting on the grounds of a simple scalar model and then its applications to generalized electrodynamics and higher-derivative gravity are worked out. For a wide class of field theories this method is better suited than Ostrogradski's for a generalization to 2n-derivative theories 
  Linear recursion relations for the instanton corrections to the effective prepotential are derived for N=2 supersymmetric gauge theories with one hypermultiplet in the adjoint representation of SU(N) using the Calogero-Moser parameterization of the Seiberg-Witten spectral curves. S-duality properties of the Calogero-Moser parameterization and conjectures on the Seiberg-Witten spectral curves generalized to arbitrary simply laced classical gauge groups are also discussed. 
  We study a system of spinless electrons moving in a two dimensional noncommutative space subject to a perpendicular magnetic field $\vec B$ and confined by a harmonic potential type ${1\over 2}mw_{0}r^2$. We look for the orbital magnetism of the electrons in different regimes of temperature $T$, magnetic field $\vec B$ and noncommutative parameter $\te$. We prove that the degeneracy of Landau levels can be lifted by the $\te$-term appearing in the electron energy spectrum at weak magnetic field. Using the {\it Berezin-Lieb} inequalities for thermodynamical potential, it is shown that in the high temperature limit, the system exibits a magnetic $\te$-dependent behaviour, which is missing in the commutative case. Moreover, a correction to susceptibility at low $T$ is observed. Using the {\it Fermi-Dirac} trace formulas, a generalization of the thermodynamical potential, the average number of electrons and the magnetization is obtained. There is a critical point where the thermodynamical potential becomes infinite in both of two methods above. So at this point we deal with the partition function by adopting another approach. The standard results in the commutative case for this model can be recovered by switching off the $\te$-parameter. 
  We construct the second variation Lagrangian for Randall-Sundrum model with two branes, study its gauge invariance, find the corresponding equations of motion and decouple them. We also derive an effective Lagrangian for this model in the unitary gauge, which describes a massless graviton, massive gravitons and a massless scalar radion. It is shown that the mass spectra of these fields and their couplings to the matter are different on different branes. 
  We obtain the operator algebra of each twisted sector of all WZW orbifolds, including the general twisted current algebra and the algebra of the twisted currents with the twisted affine primary fields. Surprisingly, the twisted right and left mover current algebras are not a priori copies of each other. Using the operator algebra we also derive world-sheet differential equations for the twisted affine primary fields of all WZW orbifolds. Finally we include ground state properties to obtain the twisted Knizhnik-Zamolodchikov equations of the WZW permutation orbifolds and the inner-automorphic WZW orbifolds. 
  The symplectic and Poisson structures of the Liouville theory are derived from the symplectic form of the SL(2,R) WZNW theory by gauge invariant Hamiltonian reduction. Causal non-equal time Poisson brackets for a Liouville field are presented. Using the symmetries of the Liouville theory, symbols of chiral fields are constructed and their *-products calculated. Quantum deformations consistent with the canonical quantisation result, and a non-equal time commutator is given. 
  We discuss the problem of noncommutative SO(N) gauge field theories from the string one-loop point of view. To this end we propose an expression for the string propagator on the boundary of the Moebius strip in the presence of a constant B field. We discuss in detail the problems related to its derivation. Then we use it to compute the one-loop corrections to two-, three- and four-gluon amplitudes in an open string theory with orthogonal Chan-Paton factors. We show that these corrections in the field theory limit in 4D are compatible with the one-loop corrections of a renormalizable noncommutative SO(N) gauge field theory. 
  We find M-theory solutions with homogeneous fluxes for which the spacetime is a lorentzian symmetric space. We show that generic solutions preserve sixteen supersymmetries and that there are two special points in their moduli space of parameters which preserve all thirty-two supersymmetries. We calculate the symmetry superalgebra of all these solutions. We then construct various M-theory and string theory branes with homogeneous fluxes and we also find new homogeneous flux-brane solutions. 
  We propose that a novel deformation of string perturbation theory, involving non-local interactions between strings, is required to describe the gravity duals of field theories deformed by multiple-trace operators. The new perturbative expansion involves a new parameter, which is neither the string coupling nor the coefficient of a vertex operator on the worldsheet. We explore some of the properties of this deformation, focusing on a special case where the deformation in the field theory is exactly marginal. 
  We consider the CPT anomaly of two-dimensional chiral U(1) gauge theory on a torus with topologically nontrivial zweibeins corresponding to the presence of spacetime torsion. The resulting chiral determinant can be expressed in terms of the standard chiral determinant without torsion, but with modified spinor boundary conditions. This implies that the two-dimensional CPT anomaly can be moved from one spin structure to another by choosing appropriate zweibeins. Similar results apply to higher-dimensional chiral gauge theories. 
  We evaluate the Seiberg-Witten map for solitons and instantons in noncommutative gauge theories in various dimensions. We show that solitons constructed using the projection operators have delta-function supports when expressed in the commutative variables. This gives a precise identification of the moduli of these solutions as locations of branes. On the other hand, an instanton solution in four dimensions allows deformation away from the projection operator construction. We evaluate the Seiberg-Witten transform of the U(2) instanton and show that it has a finite size determined by the noncommutative scale and by the deformation parameter \rho. For large \rho, the profile of the D0-brane density of the instanton agrees surprisingly well with that of the BPST instanton on commutative space. 
  We compute loop corrections in p-adic open string field theory. We argue that quantum effects induce a pole with m^2 ~ - ln g for the open string field at the locally stable vacuum. We also compute the one loop effective potential and show that the potential develops an imaginary piece when the field becomes tachyonic. 
  A supersymmetric affinization of the algebra of octonions is introduced. It satisfies a super-Malcev property and is N=8 supersymmetric. Its Sugawara construction recovers, in a special limit, the non-associative N=8 superalgebra of Englert et al. This paper extends to supersymmetry the results obtained by Osipov in the bosonic case. 
  The type 0 theories have twice as many stable D-branes as the type II theories. In light of this added complication, we find the descent relations for D-branes in the type 0A and 0B theories. In addition, we work out how the two types of D-branes differ in their couplings to NS-NS fields. 
  Renormalization is cast in the form of a Lie algebra of infinite triangular matrices. By exponentiation, these matrices generate counterterms for Feynman diagrams with subdivergences. As representations of an insertion operator, the matrices are related to the Connes-Kreimer Lie algebra. In fact, the right-symmetric nonassociative algebra of the Connes-Kreimer insertion product is equivalent to an "Ihara bracket" in the matrix Lie algebra. We check our results in a three-loop example in scalar field theory. Apart from possible applications in high-precision phenomenology, we give a few ideas about possible applications in noncommutative geometry and functional integration. 
  We apply a new general method of anholonomic frames with associated nonlinear connection structure to construct new classes of exact solutions of Einstein-Dirac equations in five dimensional (5D)gravity. Such solutions are parametrized by off-diagonal metrics in coordinate (holonomic) bases, or, equivalently, by diagonal metrics given with respect to some anholonomic frames (pentads, or funfbiends, satisfing corresponding constraint relations). We consider two possibilities of generalization of the Taub NUT metric in order to obtain vacuum solutions of 5D Einsitein equations with effective renormalization of constants having distinguished anisotropies on an angular parameter or on extra dimension coordinate. The constructions are extended to solutions describing self-consistent propagations of 3D Dirac wave packets in 5D anisotropic Taub NUT spacetimes. We show that such anisotropic configurations of spinor matter can induce gravitational 3D solitons being solutions of Kadomtsev-Petviashvili or of sine-Gordon equations. 
  Some simple (namely, single-channel) correlation functions involving an arbitrary number of fields are computed by means of a direct application of the residue calculus, through partial fraction expansions. Examples are presented in minimal models and parafermionic conformal theories. A generic factorized expression is deduced for the corresponding single-channel structure constants. 
  Trace anomaly for conformally invariant higher derivative 4D scalar-dilaton theory is obtained by means of calculating divergent part of one-loop effective action for such system. Its applications are briefly mentioned 
  In this short note we would like to discuss general solutions of the Berkovits superstring field theory, in particular the string field action for fluctuation around such a solution. We will find that fluctuations obey the same equation of motion as the original field with the new BRST operator. Then we will argue that the superstring field theory action for fluctuation field has the same form as the original one. 
  This is an introductory review of gravity on branes with an emphasis on codimension 1 models. However, for a new result it is also pointed out that the cosmological evolution of the 3-brane in the model of Dvali, Gabadadze and Porrati may follow the standard Friedmann equation.  Contents:   1. Introduction   2. Conventions   3. The Lanczos-Israel matching conditions   4. The action principle with codimension 1 hypersurfaces: Need for the Gibbons-Hawking term   5. The Newtonian limit on thin branes   6. A remark on black holes in the model of Dvali, Gabadadze and Porrati   7. The cosmology of codimension 1 brane worlds 
  We calculate fusion rules for the admissible representations of the affine superalgebra sl(2|1;C) at fractional level k=-1/2 in the Ramond sector. By representing 3-point correlation functions involving a singular vector as the action of differential operators on the sl(2|1;C) invariant 3-point function, we obtain conditions on permitted quantum numbers involved. We find that in this case the primary fields close under fusion. 
  We address the question whether a graviton could have a small nonzero mass. The issue is subtle for two reasons: there is a discontinuity in the mass in the lowest tree-level approximation, and, moreover, the nonlinear four-dimensional theory of a massive graviton is not defined unambiguously. First, we reiterate the old argument that for the vanishing graviton mass the lowest tree-level approximation breaks down since the higher order corrections are singular in the graviton mass. However, there exist nonperturbative solutions which correspond to the summation of the singular terms and these solutions are continuous in the graviton mass. Furthermore, we study a completely nonlinear and generally covariant five-dimensional model which mimics the properties of the four-dimensional theory of massive gravity. We show that the exact solutions of the model are continuous in the mass, yet the perturbative expansion exhibits the discontinuity in the leading order and the singularities in higher orders as in the four-dimensional case. Based on exact cosmological solutions of the model we argue that the helicity-zero graviton state which is responsible for the perturbative discontinuity decouples from the matter in the limit of vanishing graviton mass in the full classical theory. 
  The boundary string field theory approach is used to evaluate the one-loop tachyon potential. We first discuss the boundary condition at the two boundaries on annulus diagram and then the exact form of corrected potentials at zero and high temperature are obtained. The profile of the tachyon potential is found to be temperature independent and tachyon will condense at high temperature. Our investigations also provide an easy way to prove the universality of the tachyon potential, even if the string is in the thermal background. 
  The multi-field generalisation of the Bateman equation arises from considerations of the continuation of String and Brane equations to the case where the base space is of higher dimension than the target space. The complex extension of this equation possesses a remarkably large invariance group, and admits a very simple implicit form for its general solution, in addition to the special case of holomorphic and anti-holomorphic explicit solutions. A class of inequivalent Lagrangians for this equation is discovered. 
  We use the conjectured strong-weak coupling worldsheet duality between the SL(2)/U(1) and Sine-Liouville conformal field theories to study some properties of degenerate operators and to compute correlation functions in CFT on AdS_3. The same quantities have been computed in the past by other means. The agreement between the different approaches provides new evidence for the duality. We also discuss the supersymmetric analog of this duality, the correspondence between SCFT on the cigar and N=2 Liouville. We show that in the spacetime CFT dual to string theory on AdS_3 via the AdS/CFT correspondence, the central term in the Virasoro algebra takes different values in different sectors of the theory. In a companion paper we use the results described here to study D-branes in AdS_3. 
  We study D-branes that preserve a diagonal SL(2) affine Lie algebra in string theory on AdS_3. We find three classes of solutions, corresponding to the following representations of SL(2): (1) degenerate, finite dimensional representations with half integer spin, (2) principal continuous series, (3) principal discrete series. We solve the bootstrap equations for the vacuum wave functions and discuss the corresponding open string spectrum. We argue that from the point of view of the AdS/CFT correspondence, the above D-branes introduce boundaries with conformal boundary conditions into the two dimensional spacetime. Open string vertex operators correspond to boundary perturbations. We also comment on the geometric interpretation of the branes. 
  We study radial waves in (2+1)-dimensional noncommutative scalar field theory, using operatorial methods. The waves propagate along a discrete radial coordinate and are described by finite series deformations of Bessel-type functions. At radius much larger than the noncommutativity scale $\sqrt{\theta}$, one recovers the usual commutative behaviour. At small distances, classical divergences are smoothed out by noncommutativity. 
  In this work we consider the action for a set of complex scalar supermultiplets interacting with the scale factor in the supersymmetric cosmological models. We show that the local conformal supersymmetry leads to a scalar field potential defined in terms of the K\"ahler potential and superpotential. Using supersymmetry breaking we are able to obtain a normalizable wave function for the FRW cosmological model. 
  We study the two-centred AdS_7 x S^4 solution of eleven-dimensional supergravity using the Euclidean path-integral approach, and find that it can be interpreted as an instanton, signalling the splitting of the throat of the M5 brane. The instanton is interpreted as indicating a coherent superposition of the quantum states corresponding to classically distinct solutions. This is a surprising result since it leads, through the AdS/CFT correspondence, to contradictory implications for the dual (2,0) superconformal field theory on the M5 brane. We also argue that similar instantons should exist for other branes in ten- and eleven-dimensional supergravity. The counterterm subtraction technique for gravitational instantons, which arose from the AdS/CFT correspondence, is examined in terms of its applicability to our results. Connections are also made to the work of Maldacena et al on anti-de Sitter fragmentation. 
  We propose that at low energy four dimensional bosonic strings may form bound states where they become bundled together much like the filaments in a cable. We inspect the properties of these bundles in terms of their extrinsic geometry. This involves both torsion and curvature contributions, and leads to four dimensional field theories where the bundles appear as closed knotted solitons. Field theories that allow for this interpretation include the Faddeev model, the Skyrme model, the Yang-Mills theory, and a Hartree-type field theory model of a two-component plasma. 
  This is a brief review of vacuum string field theory, a new approach to open string field theory based on the stable vacuum of the tachyon. We discuss the sliver state explaining its role as a projector in the space of half-string functionals. We review the construction of D-brane solutions in vacuum string field theory, both in the algebraic approach and in the more general geometrical approach that emphasizes the role of boundary CFT. -- To appear in the Proceedings of Strings 2001, Mumbai, India. 
  We propose a unitary matrix Chern-Simons model representing fractional quantum Hall fluids of finite extent on the cylinder. A mapping between the states of the two systems is established. Standard properties of Laughlin theory, such as the quantization of the inverse filling fraction and of the quasiparticle number, are reproduced by the quantum mechanics of the matrix model. We also point out that this system is holographically described in terms of the one-dimensional Sutherland integrable particle system. 
  We find the supergravity solution sourced by a supertube: a (1/4)-supersymmetric D0-charged IIA superstring that has been blown up to a cylindrical D2-brane by angular momentum. The supergravity solution captures all essential features of the supertube, including the D2-dipole moment and an upper bound on the angular momentum: violation of this bound implies the existence of closed timelike curves, with a consequent ghost-induced instability of supertube probes. 
  A very simple field theory in noncommutative phase space X^{M},P^{M} in d+2 dimensions, with a gauge symmetry based on noncommutative u*(1,1), furnishes the foundation for the field theoretic formulation of Two-Time Physics. This leads to a remarkable unification of several gauge principles in d dimensions, including Maxwell, Einstein and high spin gauge principles, packaged together into one of the simplest fundamental gauge symmetries in noncommutative quantum phase space in d+2 dimensions. A gauge invariant action is constructed and its nonlinear equations of motion are analyzed. Besides elegantly reproducing the first quantized worldline theory with all background fields, the field theory prescribes unique interactions among the gauge fields. A matrix version of the theory, with a large N limit, is also outlined 
  We consider configurations of D7-branes and whole and fractional D3-branes with N=2 supersymmetry. On the supergravity side these have a warp factor, three-form flux and a nonconstant dilaton. We discuss general IIB solutions of this type and then obtain the specific solutions for the D7/D3 system. On the gauge side the D7-branes add matter in the fundamental representation of the D3-brane gauge theory. We find that the gauge and supergravity metrics on moduli space agree. However, in many cases the supergravity curvature is large even when the gauge theory is strongly coupled. In these cases we argue that the useful supergravity dual must be a IIA configuration. 
  We compute the coefficients of an infinite family of chiral primary operators in the local operator expansion of a circular Wilson loop in N=4 supersymmetric Yang-Mills theory. The computation sums all planar rainbow Feynman graphs. We argue that radiative corrections from planar graphs with internal vertices cancel in leading orders and we conjecture that they cancel to all orders in perturbation theory. The coefficients are non-trivial functions of the 'tHooft coupling and their strong coupling limits are in exact agreement with those previously computed using the AdS/CFT correspondence. They predict the subleading orders in strong coupling and could in principle be compared with string theory calculations. 
  Using a coherent state representation we derive many-body probability distributions and wavefunctions for the Chern-Simons matrix model proposed by Polychronakos and compare them to the Laughlin ones. We analyze two different coherent state representations, corresponding to different choices for electron coordinate bases. In both cases we find that the resulting probability distributions do not quite agree with the Laughlin ones. There is agreement on the long distance behavior, but the short distance behavior is different. 
  The measure for the one loop scattering of one and $N$ bosonic strings is calculated using the Group Theoretic approach to String Theory. The calculation is done for the case when the projective transformation associated with the loop can be parametrized in terms of two finite fixed points and the multiplier. 
  We give simple representations for quantum theories in which the position commutators are non vanishing constants. A particular representation reproduces results found using the Moyal star product. The notion of exact localization being meaningless in these theories, we adapt the notion of ``maximally localized states'' developed in another context . We find that gaussian functions play this role in a 2+1 dimensional model in which the non commutation relations concern positions only. An interpretation of the wave function in this non commutative geometry is suggested. We also analyze higher dimensional cases. A possible incidence on the causality issue for a Q.F.T with a non commuting time is sketched. 
  We derive and solve the full set of scalar perturbation equations for a class of $Z_2$-symmetric five-dimensional geometries generated by a bulk cosmological constant and by a 3-brane non-minimally coupled to a bulk dilaton field. The massless scalar modes, like their tensor analogues, are localized on the brane, and provide long-range four-dimensional dilatonic interactions, which are generically present even when matter on the brane carries no dilatonic charge. The shorter-range corrections induced by the continuum of massive scalar modes are always present: they persist even in the case of a trivial dilaton background (the standard Randall--Sundrum configuration) and vanishing dilatonic charges. 
  We show that for a large class of d=4 N=2 conformal field theories the 1/N correction to the chiral anomaly of the U(1) R-current can be shown to arise from the D7-branes present in the dual orientifolded orbifold string theories, generalizing a result in the literature for the simplest case. We also study the U(1)_R anomaly for d=4 N=1 conformal field theories that arise from orientifolds of the conifold. We find agreement between the field- and string-theoretic calculations, confirming a prediction of the AdS/CFT correspondence at order 1/N for string theories on AdS_5 x T^{11}/Z_2. 
  The physics that is traditionally formulated in one--time-physics (1T-physics) can also be formulated in two-time-physics (2T-physics). The physical phenomena in 1T or 2T physics are not different, but the spacetime formalism used to describe them is. The 2T description involves two extra dimensions (one time and one space), is more symmetric, and makes manifest many hidden features of 1T-physics. One such hidden feature is that families of apparently different 1T-dynamical systems in d dimensions holographically describe the same 2T system in d+2 dimensions. In 2T-physics there are two timelike dimensions, but there is also a crucial gauge symmetry that thins out spacetime, thus making 2T-physics effectively equivalent to 1T-physics. The gauge symmetry is also responsible for ensuring causality and unitarity in a spacetime with two timelike dimensions. What is gained through 2T-physics is a unification of diverse 1T dynamics by making manifest hidden symmetries and relationships among them. Such symmetries and relationships is the evidence for the presence of the underlying higher dimensional spacetime structure. 2T-physics could be viewed as a device for gaining a better understanding of 1T-physics, but beyond this, 2T-physics offers new vistas in the search of the unified theory while raising deep questions about the meaning of spacetime. In these lectures, the recent developments in the powerful gauge field theory formulation of 2T-physics will be described after a brief review of the results obtained so far in the more intuitive worldline approach. 
  This paper is an investigation of the effects of a thermal bulk fluid in brane world models compactified on AdS_5. Our primary purpose is to study how such a fluid changes the bulk dynamics and to compare these effects with those generated by matter localized to the branes. We find an exact cosmological solution for a thermally excited massless bulk field, as well as perturbative solutions with matter on the brane and in the bulk. We then perturb around these solutions to find solutions for a massive bulk mode in the limit where the bulk mass (m_B) is small compared to the AdS curvature scale and T< m_B. We find that without a stabilizing potential there are no physical solutions for a thermal bulk fluid. We then include a stabilizing potential and calculate the shift in the radion as well as the time dependence of the weak scale as a function of the bulk mass. It is shown that, as opposed to a brane fluid, the bulk fluid contribution to the bulk dynamics is controlled by the bulk mass. 
  We write an action, in four dimensional N=1 curved superspace, which contains a pure R^4 term with a coupling constant. Starting from the off-shell solution of the Bianchi identities, we compute the on-shell torsions and curvatures with this term. We show that their complete solution includes, for some of them, an infinite series in the R^4 coupling constant, which can only be computed iteratively. We explicitly compute the superspace torsions and curvatures up to second order in this coupling constant. Finally, we comment on the lifting of this result to higher dimensions. 
  We show that an infinite subset of the higher-derivative alpha' corrections to the DBI and Chern-Simons actions of ordinary commutative open-string theory can be determined using noncommutativity. Our predictions are compared to some lowest order alpha' corrections that have been computed explicitly by Wyllard (hep-th/0008125), and shown to agree. 
  We demonstrate how nonlocal regularization is applied to gauge invariant models with spontaneous symmetry breaking. Motivated by the ability to find a nonlocal BRST invariance that leads to the decoupling of longitudinal gauge bosons from physical amplitudes, we show that the original formulation of the method leads to a nontrivial relationship between the nonlocal form factors that can appear in the model. 
  We obtain a generalisation of the original complete Ricci-flat metric of G_2 holonomy on R^4\times S^3 to a family with a non-trivial parameter \lambda. For generic \lambda the solution is singular, but it is regular when \lambda={-1,0,+1}. The case \lambda=0 corresponds to the original G_2 metric, and \lambda ={-1,1} are related to this by an S_3 automorphism of the SU(2)^3 isometry group that acts on the S^3\times S^3 principal orbits. We then construct explicit supersymmetric M3-brane solutions in D=11 supergravity, where the transverse space is a deformation of this class of G_2 metrics. These are solutions of a system of first-order differential equations coming from a superpotential. We also find M3-branes in the deformed backgrounds of new G_2-holonomy metrics that include one found by A. Brandhuber, J. Gomis, S. Gubser and S. Gukov, and show that they also are supersymmetric. 
  We study the identity projectors of the string field theory in the generic BCFT background. We explain how it can be identified as the projector in the linking algebra of the noncommutative geometry. We show that their (regularized) trace is exactly given by the boundary entropy which is proportional to the D-brane tension. 
  In this paper we study spatially quenched, SU(N) Yang-Mills theory in the large-N limit. The resulting reduced action shows the same formal look as the Banks-Fischler-Shenker-Susskind M-theory action. The Weyl-Wigner-Moyal symbol of this matrix model is the Moyal deformation of a p(=2)-brane action. Thus, the large-N limit of the spatially quenched SU(N) Yang-Mills is seen to describe a dynamical membrane. By assuming spherical symmetry we compute the mass spectrum of this object in the WKB approximation. 
  Lattice Gauge Theory in 4-dimensional Euclidean space-time is generalized to ribbon categories which replace the category of representations of the gauge group. This provides a framework in which the gauge group becomes a quantum group while space-time is still given by the `classical' lattice. At the technical level, this construction generalizes the Spin Foam Model dual to Lattice Gauge Theory and defines the partition function for a given triangulation of a closed and oriented piecewise-linear 4-manifold. This definition encompasses both the standard formulation of d=4 pure Yang-Mills theory on a lattice and the Crane-Yetter invariant of 4-manifolds. The construction also implies that a certain class of Spin Foam Models formulated using ribbon categories are well-defined even if they do not correspond to a Topological Quantum Field Theory. 
  We give a simple proof of why there is a Matrix theory approximation for a membrane shaped like an arbitrary Riemann surface. As corollaries, we show that noncompact membranes cannot be approximated by matrices and that the Poisson algebra on any compact phase space is U(infinity). The matrix approximation does not appear to work properly in theories such as IIB string theory or bosonic membrane theory where there is no conserved 3-form charge to which the membranes couple. 
  We construct an eleven-dimensional superspace with superspace coordinates and formulate a finite M-theory using non-anticommutative geometry. The conjectured M-theory has the correct eleven-dimensional supergravity low energy limit. We consider the problem of finding a stable finite M-theory which has de Sitter space as a natural ground state, and the problem of eliminating possible future horizons. 
  We find a three-parameter family of solutions to IIB supergravity that corresponds to N=1 supersymmetric holographic RG flows of N=4 supersymmetric Yang Mills theory. This family of solutions allows one to give a mass to a single chiral superfield, and to probe a two-dimensional subspace of the Coulomb branch. In particular, we examine part of the Coulomb branch of the Leigh-Strassler fixed point. We look at the infra-red asymptotics of these flows from the ten-dimensional perspective. We also make general conjectures for the lifting Ansatz of five-dimensional scalar configurations to ten-dimensional tensor gauge fields. Our solution provides a highly non-trivial test of these conjectures. 
  We examine the application of boundary states in computing amplitudes in off-shell open string theory. We find a straightforward generalization of boundary state which produces the correct matrix elements with on-shell closed string states. 
  We find a one-parameter family of new G_2 holonomy metrics and demonstrate that it can be extended to a two-parameter family. These metrics play an important role as the supergravity dual of the large $N$ limit of four dimensional supersymmetric Yang-Mills. We show that these G_2 holonomy metrics describe the M-theory lift of the supergravity solution describing a collection of D6-branes wrapping the supersymmetric three-cycle of the deformed conifold geometry for any value of the string coupling constant. 
  Various ambiguous results on radiatively induced Lorentz and CPT violation in quantum electrodynamics with a modified fermionic sector are reviewed and possible explanations for this ambiguity appearing in the literature are commentated. Furthermore, joint between stringent limit from astrophysical observation and theoretical prediction on Lorenz and CPT violation is discussed. 
  We describe the ghost sector of cubic string field theory in terms of degrees of freedom on the two halves of a split string. In particular, we represent a class of pure ghost BRST operators as operators on the space of half-string functionals. These BRST operators were postulated by Rastelli, Sen, and Zwiebach to give a description of cubic string field theory in the closed string vacuum arising from condensation of a D25-brane in the original tachyonic theory. We find a class of solutions for the ghost equations of motion using the pure ghost BRST operators. We find a vanishing action for these solutions, and discuss possible interpretations of this result. The form of the solutions we find in the pure ghost theory suggests an analogous class of solutions in the original theory on the D25-brane with BRST operator Q_B coupling the matter and ghost sectors. 
  In contrast with pseudo-gravitational effects that are mathematically analogous but physically quite distinct from gravity, this presentation deals with a kind of quasi-gravitational effect that can act in an asymmetrically moving brane worldsheet in a manner that approximates (and in a crude analysis might be physically indistinguishable from) the effect that would arise from genuine gravitation, of ordinary Newtonian type in non-relativistic applications, and of scalar - tensor (Jordan - Brans - Dicke rather than pure Einstein) type in relativistic applications. 
  We consider the physics of enhancons as applied to four dimensional black holes which are constructed by wrapping both D-branes and NS-branes on K3. As was recently shown for the five dimensional black holes, the enhancon is crucial in maintaining consistency with the second law of thermodynamics. This is true for both the D-brane and NS-brane sectors of these black holes. In particular NS5-branes in both type IIA and IIB string theory are found to exhibit enhancon physics when wrapped on a K3 manifold. 
  A physical system should be in a local equilibrium if it cannot be distinguished from a global equilibrium by ``infinitesimally localized measurements''. This seems to be a natural characterization of local equilibrium, however the problem is to give a precise meaning to the qualitative phrase ``infinitesimally localized measurements''.   A solution is suggested in form of a {\em Local Equilibrium Condition} (LEC) which can be applied to non-interacting quanta.   The Unruh temperature of massless quanta is derived by applying LEC to an arbitrary point inside the Rindler Wedge.   Massless quanta outside a hot sphere are analyzed. A stationary spherically symmetric local equilibrium does only exist according to LEC if the temperature is globally constant.   Using LEC a non-trivial stationary local equilibrium is found for rotating massless quanta between two concentric cylinders of different temperatures. This shows that quanta may behave like a fluid with a B\'enard instability. 
  We propose a general method to study open/closed string dualities from transitions in M theory which is valid for a large class of geometrical configurations. By T-duality we can transform geometrically engineered configurations into N = 1 brane configurations and study the transitions of the corresponding branes by lifting the configurations to M-theory. We describe the transformed degenerated M5 branes and extract the field theory information on gluino condensation by factorization of the Seiberg-Witten curve. We also include massive flavors and orientifolds and discuss Seiberg duality which appears in this case as a birational flop. After the transition, the Seiberg duality becomes an abelian electric-magnetic duality. 
  A generalization of the Bardeen formalism to the case of warped geometries is presented. The system determining the gauge-invariant fluctuations of the metric induced by the scalar fluctuations of the brane is reduced to a set of Schr\"odinger-like equations for the Bardeen potentials and for the canonical normal modes of the scalar-tensor action. Scalar, vector and tensor modes of the geometry are classified according to four-dimensional Lorentz transformations. While the tensor modes of the geometry live on the brane determining the corrections to Newton law, the scalar and and vector fluctuations exhibit non normalizable zero modes and are, consequently, not localized on the brane. The spectrum of the massive modes of the fluctuations is analyzed using supersymmetric quantum mechanics. 
  We study realization of the target space diffeomorphisms in the type $C$ topological string. We found that the charges, which generate transformations of the boundary observables, form an algebra, which differs from that of bulk charges by the contribution of the bubbled disks. We discuss applications to noncommutative field theories. 
  We report on recent progress in understanding mirror symmetry. Some of more recent generalizations and applications are also presented. --- A contribution to the Proceedings of ``Strings 2001'' at Mumbai, India. 
  The effect of non-commutativity on electromagnetic waves violates Lorentz invariance: in the presence of a background magnetic induction field b, the velocity for propagation transverse to b differs from c, while propagation along b is unchanged. In principle, this allows a test by the Michelson-Morley interference method. We also study non-commutativity in another context, by constructing the theory describing a charged fluid in a strong magnetic field, which forces the fluid particles into their lowest Landau level and renders the fluid dynamics non-commutative, with a Moyal product determined by the background magnetic field. 
  Linear recursion relations for the instanton corrections to the effective prepotential are derived for two cases of N=2 supersymmetric gauge theories; the first case with an arbitrary number of hypermultiplets in the fundamental representation of an arbitrary classical gauge group, and the second case with one hypermultiplet in the adjoint representation of SU(N). The construction for both cases proceed from the Seiberg-Witten solutions and the renormalization group type equations for the prepotential. Successive iterations of these recursion relations allow us to simply obtain instanton corrections to an arbitrarily high order. Checks with previous results in the literature were performed. Other theoretical properties and generalizations are also discussed. 
  We discuss the reduction of the open membrane metric and determine the (previously unknown) conformal factor. We also construct SL(2,R) invariant open string metrics and complex open string coupling constants by reducing the open membrane metric on a 2-torus. In doing so we also clarify some issues on manifest SL(2,R) symmetry of the D3-brane. We remark on the consequences of our results for the recently conjectured existence of decoupled (p,q) non-commutative open string theories in type IIB string theory. 
  With the help of the Seiberg-Witten map, one can obtain an effective action of a deformed QED from a noncommutative QED. Starting from the deformed QED, we investigate the propagation of photons in the background of electromagnetic field, up to the leading order of the noncommutativity parameter. In our setting (both the electric and magnetic fields are parallel to the coordinate axis $x^1$ and the nonvanishing component of the noncommutativity parameter is $\theta^{23}$), we find that the electric field has no effect on the propagation of photons, but the velocity of photons can be larger than the speed of light ($c=1$) when the propagating direction of photons is perpendicular to the direction of background magnetic field, while the light-cone condition does not change when the propagating direction is parallel to the background magnetic field. The causality associated with the superluminal photons is discussed briefly. 
  We review the generalization of field theory to space-time with noncommuting coordinates, starting with the basics and covering most of the active directions of research. Such theories are now known to emerge from limits of M theory and string theory, and to describe quantum Hall states. In the last few years they have been studied intensively, and many qualitatively new phenomena have been discovered, both on the classical and quantum level.   To appear in Reviews of Modern Physics. 
  We consider diquark condensation in external chromomagnetic fields at non--zero temperature. The general features of this process are investigated for various field configurations in relation to their symmetry properties and the form of the quark spectrum. According to the fields, there arises dimensional reduction by one or two units. In all cases there exists diquark condensation even at arbitrary weak quark attraction, confirming the idea about universality of this mechanism in a chromomagnetic field. Possible influence of a nonzero chemical potential on the results obtained is also discussed. 
  The induced 4d gravity on a brane world is analyzed. The case of a thick brane is considered and the unexpected appearance of a new threshold scale, much larger than the thickness scale is found. In cases of phenomenological interest, this new length scale turns out to be in the submillimeter range. The effect of $R^2$ corrections, both in the bulk and on the brane, is also studied. It is shown that they introduce new threshold scales and may induce drastic modifications to the leading behavior. A concrete string/D-brane realization of the induced gravity scenario is also presented. 
  An off-shell formulation is given for the ``supersymmetry in singular spaces'' which has recently been developed in an on-shell formalism by Bergshoeff, Kallosh and Van Proeyen using supersymmetry singlet `coupling constant' field and 4-form multiplier field in five-dimensional space-time. We present this formulation for a general supergravity-Yang-Mills-hypermultiplet coupled system compactified on an orbifold $S^1/Z_2$. Relations between the bulk cosmological constant and brane tensions of the boundary planes are discussed. 
  It is proved that general consistency requirements of stability under complex analytic change of charts show that primary currents in finite chiral W-algebras are described in terms of pure gravitational variables. 
  In the presence of a non-vanishing chemical potential the eigenvalues of the Dirac operator become complex. We calculate spectral correlation functions of complex eigenvalues using a random matrix model approach. Our results apply to non-Hermitian Dirac operators in three-dimensional QCD with broken flavor symmetry and in four-dimensional QCD in the bulk of the spectrum. The derivation follows earlier results of Fyodorov, Khoruzhenko and Sommers for complex spectra exploiting the existence of orthogonal polynomials in the complex plane. Explicit analytic expressions are given for all microscopic k-point correlation functions in the presence of an arbitrary even number of massive quarks, both in the limit of strong and weak non-Hermiticity. In the latter case the parameter governing the non-Hermiticity of the Dirac matrices is identified with the influence of the chemical potential. 
  Recently quintessence is proposed to explain the observation data of supernova indicating a time-varying cosmological constant and accelerating universe. Inspired by this and its mysterious origin, we look for the possibility of quintessence as the holographic dark matters dominated in the late time in the brane world scenarios. We consider both the cases of static and moving brane in a dilaton gravity background. For the static brane we use the Hamilton-Jacobi method motivated by holographic renormalization group to study the intrinsic FRW cosmology on the brane and find out the constraint on the bulk potential for the quintessence. This constraint requires a negative slowly varying bulk potential which implies an anti-de Sitter-like bulk geometry and could be possibly realized from the higher dimensional supergravities or string theory. We find the similar constraint for the moving brane cases and that the quintessence on it has the effect as a mildly time-varying Newton constant. 
  We obtain a supergravity solution arising when D6-branes are wrapped on coassociative four-cycles of constant curvature in seven manifolds of G_2 holonomy. The solutions preserve two supercharges and thus represent supergravity duals of three dimensional Yang-Mills with N=1 supersymmetry. When uplifted to eleven dimensions our solution describes M-theory on the background of an eight manifold with Spin(7) holonomy. 
  We recast the Podle\`s spheres in the noncommutative physics context by showing that they can be regarded as slices along the time coordinate of the different regions of the quantum Minkowski space-time. The investigation of the transformations of the quantum sphere states under the left coaction of the ${\cal SO}_{q}(3)$ group leads to a decomposition of the transformed Hilbert space states in terms of orthogonal subspaces exhibiting the periodicity of the quantum sphere states. 
  We show that ultraviolet divergences found in fermionic Green's functions of massless $QED_2$ have an essentially non-perturbative nature. We investigate their origin both in gauge invariant formalism (the one where we introduce Wess-Zumino fields to restore quantum gauge invariance) and in gauge non-invariant formalism, mapping two different but equivalent mechanisms responsible for their appearance. We find the same results in both approaches, what contradicts a previous work of Jian-Ge, Qing-Hai and Yao-Yang, that found no divergences in the chiral Schwinger model considered in the gauge invariant formalism. 
  We propose a framework in which the quantum gravity scale can be as low as $10^{-3}$ eV. The key assumption is that the Standard Model ultraviolet cutoff is much higher than the quantum gravity scale. This ensures that we observe conventional weak gravity. We construct an explicit brane-world model in which the brane-localized Standard Model is coupled to strong 5D gravity of infinite-volume flat extra space. Due to the high ultraviolet scale, the Standard Model fields generate a large graviton kinetic term on the brane. This kinetic term ``shields'' the Standard Model from the strong bulk gravity. As a result, an observer on the brane sees weak 4D gravity up to astronomically large distances beyond which gravity becomes five-dimensional. Modeling quantum gravity above its scale by the closed string spectrum we show that the shielding phenomenon protects the Standard Model from an apparent phenomenological catastrophe due to the exponentially large number of light string states. The collider experiments, astrophysics, cosmology and gravity measurements {\it independently} point to the same lower bound on the quantum gravity scale, $10^{-3}$ eV. For this value the model has experimental signatures both for colliders and for sub-millimeter gravity measurements. Black holes reveal certain interesting properties in this framework. 
  These notes provide a hopefully pedagogic introduction to Born-Infeld theory and some of its uses in String/M-Theory from the perspective of some-one interested in spacetime geometry. Causality and the consequences of having both the open string and closed string metric are stressed. This leads to some new insighst into old problems, such as strings at finite temperature. Various strong coupling limits of both Born-Infeld and the M5-brane theories are described and their relevance for tachyon condensations are mentioned. 
  From the basic chiral and anti-chiral Poisson bracket algebra of the SL(2,R) WZNW model, non-equal time Poisson brackets are derived. Through Hamiltonian reduction we deduce the corresponding brackets for its coset theories. 
  D-instanton calculus has proved to be able to probe the AdS near horizon geometry for $N$ D-branes systems which, when decoupled from gravity, yield four dimensional superconformal gauge theories with various matter content. In this work we extend previous analysis to encompass fractional brane models which give rise to non conformal N=2 Super Yang-Mills theories. Via D-instanton calculus we study the geometry of such models for finite $N$ and recover the $\beta$ function of the gauge coupling constants which is expected in non conformal gauge theories. We also give a topological matrix theory formulation for the D-instanton action of these theories. Finally, we revisit the related system where the D3-branes wrap a ${\real}^4/{\zet}_p$ orbifold singularity and the D(-1) branes are associated to instanton solutions of four-dimensional gauge theories in the blown down ALE space. 
  We determine higher-derivative terms in the open superstring effective action with U(N) gauge group up to and including order alpha'^4 as can be extracted from 4 boson, 2 boson - 2 fermion and 4 fermion string scattering amplitudes. This yields corrections to the non-abelian Born-Infeld action involving higher derivatives as is relevant for studying D-branes beyond the slowly varying field approximation. While at order alpha'^2 the action has recently been shown to be a symmetrised trace, this no longer is true at order alpha'^3 or alpha'^4. We argue that these terms including higher derivatives are as important in a low-energy expansion as e.g. the much-discussed alpha'^4 F^6 terms. In particular a computation of the fluctuation spectra at order alpha'^4 has to take into account these non-symmetrised trace higher-derivative terms computed here. 
  We analyze the operator product expansion T_{\mu \nu}(z) W[C] in N=4 4-dimensional Super-Yang-Mills (SYM) theory with U(N) gauge group, and clarify that the closed Wilson loop does not possess an anomalous dimension and that only the shape of the loop is changed by the conformal transformation. 
  The well-known fact that classical automorphisms of (compactified) Minkowski spacetime (Poincare or conformal trandsformations) also allow a natural derivation/interpretation in the modular setting (in the operator-algebraic sense of Tomita and Takesaki) of the algebraic formulation of QFT has an interesting nontrivial chiral generalization to the diffeomorphisms of the circle. Combined with recent ideas on algebraic (d-1)-dimensional lightfront holography, these diffeomorphisms turn out to be images of ``fuzzy'' acting groups in the original d-dimensional (massive) QFT. These actions do not require any spacetime noncommutativity and are in complete harmony with causality and localization principles. Their use tightens the relation with kinematic chiral structures on the causal horizon and makes recent attempts to explain the required universal structure of a possible future quantum Bekenstein law in terms of Virasoro algebra structures more palatable. 
  In this paper, two things are done. (i) Using cohomological techniques, we explore the consistent deformations of linearized conformal gravity in 4 dimensions. We show that the only possibility involving no more than 4 derivatives of the metric (i.e., terms of the form $\partial^4 g_{\mu \nu}$, $\partial^3 g_{\mu \nu} \partial g_{\alpha \beta}$, $\partial^2 g_{\mu \nu} \partial^2g_{\alpha \beta}$, $\partial^2 g_{\mu \nu} \partial g_{\alpha \beta} \partial g_{\rho \sigma}$ or $\partial g_{\mu \nu} \partial g_{\alpha \beta} \partial g_{\rho \sigma} \partial g_{\gamma \delta}$ with coefficients that involve undifferentiated metric components - or terms with less derivatives) is given by the Weyl action $\int d^4x \sqrt{-g} W_{\a\b\g\d} W^{\a\b\g\d}$, in much the same way as the Einstein-Hilbert action describes the only consistent manner to make a Pauli-Fierz massless spin-2 field self-interact with no more than 2 derivatives. No a priori requirement of invariance under diffeomorphisms is imposed: this follows automatically from consistency. (ii) We then turn to "multi-Weyl graviton" theories. We show the impossibility to introduce cross-interactions between the different types of Weyl gravitons if one requests that the action reduces, in the free limit, to a sum of linearized Weyl actions. However, if different free limits are authorized, cross-couplings become possible. An explicit example is given. We discuss also how the results extend to other spacetime dimensions. 
  It is shown that the operator algebraic setting of local quantum physics leads to a uniqueness proof for the inverse scattering problem. The important mathematical tool is the thermal KMS aspect of wedge-localized operator algebras and its strengthening by the requirement of crossing symmetry for generalized formfactors. The theorem extends properties which were previously seen in d=1+1 factorizing models. 
  We find new closed string couplings on Dp-branes for the bosonic string. These couplings are quadratic in derivatives and therefore take the form of induced kinetic terms on the brane. For the graviton in particular we find the induced Einstein-Hilbert term as well as terms quadratic in the second fundamental tensor. We comment on tachyon dependences of these brane-localized couplings. 
  We study some properties of the tachyonic lumps in the level truncation scheme of bosonic cubic string field theory. We find that several gauges work well and that the size of the lump as well as its tension is approximately independent of these gauge choices at level (2,4). We also examine the fluctuation spectrum around the lump solution, and find that a tachyon with m^2=-0.96 and some massive scalars appear on the lump world-volume. This result strongly supports the conjecture that a codimension 1 lump solution is identified with a D-brane of one lower dimension within the framework of bosonic cubic string field theory. 
  The sine-Gordon model with Neumann boundary condition is investigated. Using the bootstrap principle the spectrum of boundary bound states is established. Somewhat surprisingly it is found that Coleman-Thun diagrams and bound state creation may coexist. A framework to describe finite size effects in boundary integrable theories is developed and used together with the truncated conformal space approach to confirm the bound states and reflection factors derived by bootstrap. 
  The bound state spectrum and the associated reflection factors are determined for the sine-Gordon model with arbitrary integrable boundary condition by closing the bootstrap. Comparing the symmetries of the bound state spectrum with that of the Lagrangian it is shown how one can "derive" the relationship between the UV and IR parameters conjectured earlier. 
  In the framework of the Euclidean path integral approach we derive the exact formula for the general N-point chiral densities correlator in the Schwinger model on a torus 
  We present a finite dimensional matrix model associated to the noncommutative Chern-Simons theory, obtained by inserting a Wilson line. For a specific choice of the representation of the Wilson line the model is equivalent to the minimal modification of the matrix model which is compatible with finite dimensional matrices, and was introduced previously to study droplets of quantum Hall fluid. For other representations we obtain generalizations corresponding to regularized U(n) Chern-Simons theoris, representing multilayered quantum Hall fluids. 
  We analyse the implications of the fact that there are two claims for a dual to \N=4 superconformal SU(N) Yang-Mills theory (SCYM), the Maldacena conjecture and the theorem of Rehren. While the Maldacena dual is conjectured to be a non-perturbative string theory for large string coupling $g_s$ and small $N$, the Rehren dual is an ordinary quantum field theory on $AdS_5$ for all values of the parameters. We argue that as a result, if we accept the Maldacena conjecture, one of the following statements must be true: 1) SCYM does not satisfy the axioms of algebraic quantum field theory for finite $N$ because its observables do not obey the causal structure of conformal Minkowski space;. 2) String theory on $AdS_5 \times S^5$ is not a quantum theory of gravity in 10 dimensions because it is dual to an ordinary quantum field theory on $AdS_5$ whose causal structure remains fixed for all values of its couplings; or 3) there is no consistent quantisation of string theory on $AdS_5 \times S^5$ for finite string scale $l_s$ and $g_s$.   In evaluating the evidence for each of these conclusions we point out that many of the tests of the Maldacena conjecture can be explained by a weaker form of an $AdS/CFT$ correspondence. 
  There have been comments on this paper which point out unclear motivation and definitions on noncommutative momentum introduced. Therefore, this paper is withdrawn by the author for more clear presentation. 
  There have been comments on the starting paper, hep-th/0106074, which point out unclear motivation and definitions on noncommutative momentum introduced. Thus, this paper is withdrawn by the author for more clear presentation. 
  The lowest order quantum corrections to the effective action arising from quantized massive fermion fields in the Randall-Sundrum background spacetime are computed. The boundary conditions and their relation with gauge invariance are examined in detail. The possibility of Wilson loop symmetry breaking in brane models is also analysed. The self-consistency requirements, previously considered in the case of a quantized bulk scalar field, are extended to include the contribution from massive fermions. It is shown that in this case it is possible to stabilize the radius of the extra dimensions but it is not possible to simultaneously solve the hierarchy problem, unless the brane tensions are dramatically fine tuned, supporting previous claims. 
  Effects of the configuration of an external static magnetic field in the form of a singular vortex on the vacuum of a quantized massless spinor field are studied. The most general boundary conditions at the punctured singular point which make the twodimensional Dirac Hamiltonian to be self-adjoint are employed. 
  We consider a large coupling limit of a Born-Infeld action in a curved background of an arbitrary metric and a constant two form field. Following hep-th/0009061, we go to the Hamiltonian description. The Hamiltonian can be dualized and the dual action admits a string-like configuration as its solution. We interpret it as a closed string configuration. The procedure can be viewed as a novel way of bringing out the appropriate degrees of freedom, a closed string, for a open string under the strong coupling limit. We argue that this interpretation implies a large number of dual pairs of gauge and gravity theories whose particular examples are AdS/CFT and matrix theory conjectures. 
  We calculate partition functions for supersymmetric heterotic theories on Melvin background with Wilson line. These functions coincide with partition functions for some of non-supersymmetric heterotic theories in appropriate limits. This suggests that non-supersymmetric heterotic theories are equivalent to supersymmetric theories on supersymmetry-breaking backgrounds, just as in the case of recently conjectured IIA-0A duality. 
  We study examples where conformal invariance implies rational critical indices, triviality of the underlying quantum field theory and emergence of hypergeometric functions as solutions of the field equations. 
  Curved, intersecting brane configurations satisfying the type IIA supergravity equations of motion are found. In eleven dimensions, the models are interpreted in terms of orthogonally intersecting M5--branes, where the world--volumes are curved due to the effects of one or more massless scalar fields. Duality symmetries are employed to generate further type II and heterotic solutions. Some cosmological implications are discussed. 
  We uncover a surprising correspondence between a non-perturbative formulation of three-dimensional Lorentzian quantum gravity and a hermitian two-matrix model with ABAB-interaction. The gravitational transfer matrix can be expressed as the logarithm of a two-matrix integral, and we deduce from the known structure of the latter that the model has two phases. In the phase of weak gravity, well-defined two-dimensional universes propagate in proper time, whereas in the strong-coupling phase the spatial hypersurfaces disintegrate into many components connected by wormholes. 
  There are now two cosmological constant problems: (i) why the vacuum energy is so small and (ii) why it comes to dominate at about the epoch of galaxy formation. Anthropic selection appears to be the only approach that can naturally resolve both problems. The challenge presented by this approach is that it requires scalar fields with extremely flat potentials or four-form fields coupled to branes with an extremely small charge. Some recent suggestions are reviewed on how such features can arise in particle physics models. 
  The possibility of mass in the context of scale-invariant, generally covariant theories, is discussed. Scale invariance is considered in the context of a gravitational theory where the action, in the first order formalism, is of the form $S = \int L_{1} \Phi d^4x$ + $\int L_{2}\sqrt{-g}d^4x$ where $\Phi$ is a density built out of degrees of freedom independent of the metric. For global scale invariance, a "dilaton" $\phi$ has to be introduced, with non-trivial potentials $V(\phi)$ = $f_{1}e^{\alpha\phi}$ in $L_1$ and $U(\phi)$ = $f_{2}e^{2\alpha\phi}$ in $L_2$. This leads to non-trivial mass generation and a potential for $\phi$ which is interesting for new inflation. Scale invariant mass terms for fermions lead to a possible explanation of the present day accelerated universe and of cosmic coincidences. 
  The cosmological constant problem and the possibility of obtaining a see saw cosmological effect, where the effective vacuum energy is highly suppressed by the existence of a large scale is investigated in the context of scale-invariant, generally covariant theory. Scale invariance is considered in the context of a gravitational theory where the action, in the first order formalism, is of the form $S = \int L_{1} \Phi d^4x$ + $\int L_{2}\sqrt{-g}d^4x$ where $\Phi$ is a density built out of degrees of freedom independent of the metric. 
  We study the brane with arbitrary tension $\sigma$ on the edge of various black holes with AdS asymptotics. We investigate Friedmann equations governing the motion of the brane universes and match the Friedmann equation to Cardy entropy formula. 
  We show that any theory with second class constraints may be cast into a gauge theory if one makes use of solutions of the constraints expressed in terms of the coordinates of the original phase space. We perform a Lagrangian path integral quantization of the resulting gauge theory and show that the natural measure follows from a superfield formulation. 
  There have been comments on the starting paper, hep-th/0106074, which point out unclear motivation and definitions on noncommutative momentum introduced. Therefore, this paper is withdrawn by the author for more clear presentation. 
  Quantum Mechanics in the non-commutative plane is shown to admit the ``exotic'' symmetry of the doubly-centrally-extended Galilei group. When coupled to a planar magnetic field whose strength is the inverse of the non-commutative parameter, the system becomes singular, and ``Faddeev-Jackiw'' reduction yields the ``Chern-Simons'' mechanics of Dunne, Jackiw, and Trugenberger. The reduced system moves according to the Hall law. 
  We construct quantized free superfields and represent them as operator-valued distributions in Fock space starting with Majorana fields. The perturbative construction of the S-matrix for interacting theories is carried through by extending the causal method of Epstein and Glaser to superspace. We propose a scaling and singular order of distributions in superspace by a procedure which scales both commutative and non-commutative variables. Using this singular order the chiral (Wess-Zumino) model appears to be super-renormalizable. 
  We analyse the algebras generated by free component quantum fields together with the susy generators $Q,\bar Q$. Restricting to hermitian fields we first construct the scalar field algebra from which various scalar superfields can be obtained by exponentiation. Then we study the vector algebra and use it to construct the vector superfield. Surprisingly enough, the result is totally different from the vector multiplet in the literature. It contains two hermitian four-vector components instead of one and a spin-3/2 field similar to the gravitino in supergravity. 
  We study, in the context of five dimensional N=2 gauged supergravity with vector and hypermultiplets, curved domain wall solutions with worldvolumes given by four dimensional Einstein manifolds. For a choice of the projection condition on the Killing spinors of the BPS solutions, first order differential equations governing the flow of the scalars are derived. With these equations, we analyze the equations of motion and determine conditions under which gauged supergravity theories may admit Einstein domain wall solutions. 
  We describe orientifold operation defining O3 plane in the conifold background by deriving it from that of O4 plane in the Type IIA brane construction by T-duality. We find that both $O3^+$ and $O3^-$ are at the tip of the cone so that there is no net untwisted RR charge. RG analysis shows that we need two `fractional' branes for the conformal invariance in orientifolded conifold. We argue that the gravity solution is the same as Klebanov and Tsyetlin since SUGRA cannot distinguish the orientifolds and D branes in this case. We describe the duality cascade as well as the quantum deformation of the moduli space of the field theory in the presence of the orientifold.  The finitely resolved conifold does not allow the orientifold, while deformed conifold leaves us an unresolved issue on supersymmetry. 
  Superfield approach in supersymmetric quantum field theory is described. Many examples of its applications to different superfield models are considered. 
  We consider a time-dependent Schr\"odinger equation for the Friedmann-Robertson-Walker (FRW) model. We show that for this purpose it is possible to include an additional action invariant under reparametrization of time. The last one does not change the equations of motion for the minisuperspace model, but changes only the constraint. The same procedure is applied to the supersymmetric case. 
  When eight-dimensional instantons, satisfying F \wedge F = \pm \star_8 (F\wedge F), shrink to zero size, we find stringy objects in higher order ten-dimensional Yang-Mills (viewed as a low-energy limit of open string theory). The associated F^4 action is a combination of two independent parts having a single-trace and a double-trace structure. As a result we get a D-string from the single-trace term and a fundamental string from the double-trace. The latter has (8,0) supersymmetry on the worldsheet and couplings to the background gauge fields of a heterotic string. A correlation between the conformal factor of the instanton and the tachyon field is conjectured. 
  We describe a novel dynamical mechanism to radiate away a positive four dimensional cosmological constant, in the Randall-Sundrum cosmological scenario. We show that there are modes of the bulk gravitational field for which the brane is effectively a mirror. This will generally give rise to an emission of thermal radiation from the brane into the bulk. The temperature turns out to be nonvanishing only if the effective four dimensional cosmological constant is positive. In any theory where the four dimensional vacuum energy is a function of physical degrees of freedom, there is then a mechanism that radiates away any positive four dimensional cosmological constant. 
  The elementary particles are modeled as harmonic oscillator excitations of transverse U(1) gauge fields propagating at v = c, with open and closed string-like propagation paths. One, two and three node states represent the leptons, bosons, and quarks. We incorporate a twist theta for the gauge field components which rotate counterclockwise (L) for the electron, yielding a chiral model. Theta increases by pi from node to node, making the lepton models SU(2) representations. At nodes the twist may reverse, creating new particle states. For three nodes, twist combinations map the SU(3) color states of the quarks. Generations are modeled topologically by the winding number of the strings. Mapping model E fields to distant observers makes understandable how fractional charges arise for the quarks. These models are 3D slices of spacetime, allowing us to make drawings of particle field conformations. From model particle quantum numbers, new mass relationships are derived. 
  Bound states of BPS particles in five-dimensional N=2 supergravity are counted by a topological index. We compute this bound state index exactly for two and three black holes as a function of the SU(2)_L angular momentum. The required regulator for the infrared continuum of near-coincident black holes is chosen in accord with the enhanced superconformal symmetry. 
  In the intersecting braneworld models, higher curvature corrections to the Einstein action are necessary to provide a non-trivial geometry (brane tension) at the brane junctions. By introducing such terms in a Gauss-Bonnet form, we give an effective description of localized gravity on the singular delta-function branes. There exists a non-vanishing brane tension at the four-dimensional brane intersection of two 4-branes. Importantly, we give explicit expressions of the graviton propagator and show that the Randall-Sundrum single-brane model with a Gauss-Bonnet term in the bulk correctly gives a massless graviton on the brane as for the RS model. We explore some crucial features of completely localized gravity in the solitonic braneworld solutions obtained with a choice (\xi=1) of solutions. The no-go theorem known for Einstein's theory may not apply to the \xi=1 solution. As complementary discussions, we provide an effective description of the power-law corrections to Newtonian gravity on the branes or at the common intersection thereof. 
  Warped compactification of the six-dimensional bulk with a negative cosmological constant is realized with a 4-brane along with an abelian gauge theory. No fine tuning of couplings are needed to obtain the vanishing cosmological constant in four dimensions, as is the case for the bulk with a positive cosmological constant. 
  We inspect a particular gauge field theory model that describes the properties of a variety of physical systems, including a charge neutral two-component plasma, a Gross-Pitaevskii functional of two charged Cooper pair condensates, and a limiting case of the bosonic sector in the Salam-Weinberg model. It has been argued that this field theory model also admits stable knot-like solitons. Here we produce numerical evidence in support for the existence of these solitons, by considering stable axis-symmetric solutions that can be thought of as straight twisted vortex lines clamped at the two ends. We compute the energy of these solutions as a function of the amount of twist per unit length. The result can be described in terms of a energy spectral function. We find that this spectral function acquires a minimum which corresponds to a nontrivial twist per unit length, strongly suggesting that the model indeed supports stable toroidal solitons. 
  We propose gauge theories in which the unstable branes and the fundamental string are realized as classical solutions. While the former are represented by domain wall like configurations of a scalar field coupled to the gauge field, the latter is by a confined flux tube in the bulk. It is shown that the confined flux tube is really a source of the bulk B-field. Our model also provides a natural scenario of the confinement on the brane in the context of the open string tachyon condensation. It is also argued that the fundamental string can be realized as a classical solution in a certain IIB matrix model as in our model. 
  We investigate both geometric and conformal field theoretic aspects of mirror symmetry on N=(4,4) superconformal field theories with central charge c=6. Our approach enables us to determine the action of mirror symmetry on (non-stable) singular fibers in elliptic fibrations of Z_N orbifold limits of K3. The resulting map gives an automorphism of order 4,8, or 12, respectively, on the smooth universal cover of the moduli space. We explicitly derive the geometric counterparts of the twist fields in our orbifold conformal field theories. The classical McKay correspondence allows for a natural interpretation of our results. 
  To an RCFT corresponds two combinatorial structures: the amplitude of a torus (the 1-loop partition function of a closed string, sometimes called a modular invariant), and a representation of the fusion ring (called a NIM-rep or equivalently a fusion graph, and closely related to the 1-loop partition function of an open string). In this paper we develop some basic theory of NIM-reps, obtain several new NIM-rep classifications, and compare them with the corresponding modular invariant classifications. Among other things, we make the following fairly disturbing observation: there are infinitely many (WZW) modular invariants which do not correspond to any NIM-rep. The resolution could be that those modular invariants are physically sick. Is classifying modular invariants really the right thing to do? For current algebras, the answer seems to be: Usually but not always. For finite groups a la Dijkgraaf-Vafa-Verlinde-Verlinde, the answer seems to be: Rarely. 
  A previously developed lepton mass equation is extended to include the massive bosons and quarks of all three generations. The particles are modeled as closed, string-like, light front solitons whose key quantum numbers are their node number n and their winding number omega. The simplicity and form of the mass equation suggests that the dominant mass effect comes from how the particles embed themselves in spacetime. 
  In this paper we would like to propose the background independent formulation of Berkovits' superstring field theory. Then we will show that the solution of equation of motion of this theory leads to the Berkovits' superstring field theory formulated around particular CFT background. 
  An explicit holographic correspondence between $AdS$ bulk and boundary quantum states is found in the form of a one to one mapping between scalar field creation/annihilation operators. The mapping requires the introduction of arbitrary energy scales and exhibits an ultraviolet-infrared duality: a small regulating mass in the boundary theory corresponds to a large momentum cutoff in the bulk. In the massless (conformal) limit of the boundary theory the mapping covers the whole field spectrum of both theories. The mapping strongly depends on the discretization of the field spectrum of compactified $AdS$ space in Poincare coordinates. 
  We discuss some general properties of quantum gravity in De Sitter space. It has been argued that the Hilbert space is of finite dimension. This suggests a macroscopic argument that General Relativity cannot be quantized -- unless it is embedded in a more precise theory that determines the value of the cosmological constant. We give a definition of the quantum Hilbert space using the asymptotic behavior in the past and future, without requiring detailed microscopic knowledge. We discuss the difficulties in defining any precisely calculable or measurable observables in an asymptotically de Sitter spacetime, and explore some meta-observables that appear to make mathematical sense but cannot be measured by an observer who lives in the spacetime. This article is an expanded version of a lecture at Strings 2001 in Mumbai. 
  Global properties of abelian noncommutative gauge theories based on $\star$-products which are deformation quantizations of arbitrary Poisson structures are studied. The consistency condition for finite noncommutative gauge transformations and its explicit solution in the abelian case are given. It is shown that the local existence of invertible covariantizing maps (which are closely related to the Seiberg-Witten map) leads naturally to the notion of a noncommutative line bundle with noncommutative transition functions. We introduce the space of sections of such a line bundle and explicitly show that it is a projective module. The local covariantizing maps define a new star product $\star'$ which is shown to be Morita equivalent to $\star$. 
  There are numerous derivations of the Hawking effect available in the literature. They emphasise different features of the process, and sometimes make markedly different physical assumptions. This article presents a ``minimalist'' argument, and strips the derivation of as much excess baggage as possible. All that is really necessary is quantum physics plus a slowly evolving future apparent horizon (*not* an event horizon). In particular, neither the Einstein equations nor Bekenstein entropy are necessary (nor even useful) in deriving Hawking radiation. 
  We propose a dual non-perturbative description for maximally extended Schwarzschild Anti-de-Sitter spacetimes. The description involves two copies of the conformal field theory associated to the AdS spacetime and an initial entangled state. In this context we also discuss a version of the information loss paradox and its resolution. 
  A holographic duality is proposed relating quantum gravity on dS_D (D-dimensional de Sitter space) to conformal field theory on a single S^{D-1} ((D-1)-sphere), in which bulk de Sitter correlators with points on the boundary are related to CFT correlators on the sphere, and points on I^+ (the future boundary of dS_D) are mapped to the antipodal points on S^{D-1} relative to those on I^-. For the case of dS_3, which is analyzed in some detail, the central charge of the CFT_2 is computed in an analysis of the asymptotic symmetry group at I^\pm. This dS/CFT proposal is supported by the computation of correlation functions of a massive scalar field. In general the dual CFT may be non-unitary and (if for example there are sufficently massive stable scalars) contain complex conformal weights. We also consider the physical region O^- of dS_3 corresponding to the causal past of a timelike observer, whose holographic dual lives on a plane rather than a sphere. O^- can be foliated by asymptotically flat spacelike slices. Time evolution along these slices is generated by L_0+\bar L_0, and is dual to scale transformations in the boundary CFT_2. 
  We examine a Wess-Zumino term, written in bilinear of superinvariant currents, for a superstring in anti-de Sitter (AdS) space. The standard Inonu-Wigner contraction does not give the correct flat limit but gives zero. This originates from the fact that the fermionic metric of the super-Poincare group is degenerate. We propose a generalization of the Inonu-Wigner contraction which reduces the super-AdS group to the "nondegenerate" super-Poincare group, therefore it gives a correct flat limit of this Wess-Zumino term. We also discuss the M-algebra obtained by this generalized Inonu-Wigner contraction from osp(1|32). 
  The relation that exists in quantum mechanics among action variables, angle variables and the phases of quantum states is clarified, by referring to the system of a generalized oscillator. As a by-product, quantum-mechanical meaning of the classical Hamilton-Jacobi equation and related matters is clarified, where a new picture of quantum mechanics is introduced, to be called the Hamilton-Jacobi picture. 
  A sigma model action is constructed for the type II string in the $Ads_5\times S_5$ back grounds with Ramond-Ramond flux. 
  We construct D=10 supergravity solutions corresponding to type IIB fivebranes wrapping a two-sphere in a Calabi-Yau two-fold. These are related in the IR to the large N limit of pure N=2 SU(N) super Yang-Mills theory. We show that the singularities in the IR correspond to the wrapped branes being distributed on a ring. We analyse the dynamics of a probe fivebrane and show that it incorporates the full perturbative structure of the gauge theory. For a class of solutions the two-dimensional moduli space is non-singular and we match the result for the corresponding slice of the Coulomb branch of the gauge theory. 
  The Sp(3) BRST symmetry for Yang-Mills theory is derived in the framework of the antibracket-antifield formalism. 
  We suggest the Hamiltonian approach for fluid mechanics based on the dynamics, formulated in terms of Lagrangian variables. The construction of the canonical variables of the fluid sheds a light of the origin of Clebsh variables, introduced in the previous century. The developed formalism permits to relate the circulation conservation (Tompson theorem) with the invariance of the theory with respect to special diffiomorphisms and establish also the new conservation laws. We discuss also the difference of the Eulerian and Lagrangian description, pointing out the incompleteness of the first. The constructed formalism is also applicable for ideal plasma. We conclude with several remarks on the quantization of the fluid. 
  We construct an n+q+2 dimensional background that has dilatonic q-brane singularities and that is charged under an antisymmetric tensor field, the background spacetime being maximally symmetric in n-dimensions with constant curvature k=0,+1,-1. For k=1 the bulk solutions correspond to black q-branes. For k=0,-1 the geometry resembles the `white hole' region of the Reissner-N"ordstrom solution with a past Cauchy horizon. The metric between the (timelike) singularity and the horizon is static whereas beyond the horizon it is cosmological. In the particular case of q=0, we study the motion of a codimension one n-brane in these charged dilatonic backgrounds that interpolate between the original scalar self-tuning and the black hole geometry and provide a way to avoid the naked singularity problem and/or the need of having exotic matter on the brane. These backgrounds are asymmetrically warped and so break 4D Lorentz symmetry in a way that is safe for particle physics but may lead to faster than light propagation in the gravitational sector. 
  Open Wilson line operators and generalized star product have been studied extensively in noncommutative gauge theories. We show that they also show up in noncommutative scalar field theories as universal structures. We first point out that dipole picture of noncommutative geometry provides an intuitive argument for robustness of the open Wilson lines and generalized star products therein. We calculate one-loop effective action of noncommutative scalar field theory with cubic self-interaction and show explicitly that the generalized star products arise in the nonplanar part. It is shown that, at low-energy, large noncommutativity limit, the nonplanar part is expressible solely in terms of the {\sl scalar} open Wilson line operator and descendants. 
  We examine the notion of symmetry in quantum field theory from a fundamental representation theoretic point of view. This leads us to a generalization expressed in terms of quantum groups and braided categories. It also unifies the conventional concept of symmetry with that of exchange statistics and the spin-statistics relation. We show how this quantum group symmetry is reconstructed from the traditional (super) group symmetry, statistics and spin-statistics relation.   The old question of extending the Poincare group to unify external and internal symmetries (solved by supersymmetry) is reexamined in the new framework. The reason why we should allow supergroups in this case becomes completely transparent. However, the true symmetries are not expressed by groups or supergroups here but by ordinary (not super) quantum groups. We show in this generalized framework that supersymmetry remains the most general unification of internal and space-time symmetries provided that all particles are either bosons or fermions.   Finally, we demonstrate with some examples how quantum geometry provides a natural setting for the construction of super-extensions, super-spaces, super-derivatives etc. 
  Weyl groups are ubiquitous, and efficient algorithms for them -- especially for the exceptional algebras -- are clearly desirable. In this paper we provide several of these, addressing practical concerns arising naturally for instance in computational aspects of the study of affine algebras or Wess-Zumino-Witten conformal field theories. We also discuss the efficiency and numerical accuracy of these algorithms. 
  We study two-dimensional nonlinear sigma models in which the target spaces are the coset supermanifolds U(n+m|n)/[U(1)\times U(n+m-1|n)] \cong CP^{n+m-1|n} (projective superspaces) and OSp(2n+m|2n)/OSp(2n+m-1|2n) \cong S^{2n+m-1|2n} (superspheres), n, m integers, -2\leq m\leq 2; these quantum field theories live in Hilbert spaces with indefinite inner products. These theories possess non-trivial conformally-invariant renormalization-group fixed points, or in some cases, lines of fixed points. Some of the conformal fixed-point theories can also be obtained within Landau-Ginzburg theories. We obtain the complete spectra (with multiplicities) of exact conformal weights of states (or corresponding local operators) in the isolated fixed-point conformal field theories, and at one special point on each of the lines of fixed points. Although the conformal weights are rational, the conformal field theories are not, and (with one exception) do not contain the affine versions of their superalgebras in their chiral algebras. The method involves lattice models that represent the strong-coupling region, which can be mapped to loop models, and then to a Coulomb gas with modified boundary conditions. The results apply to percolation, dilute and dense polymers, and other statistical mechanics models, and also to the spin quantum Hall transition in noninteracting fermions with quenched disorder. 
  Quantum mechanics in a noncommutative plane is considered. For a general two dimensional central field, we find that the theory can be perturbatively solved for large values of the noncommutative parameter ($\theta$) and explicit expressions for the eigenstates and eigenvalues are given. The Green function is explicitly obtained and we show that it can be expressed as an infinite series. For polynomial type potentials, we found a smooth limit for small values of $\theta$ and for non-polynomial ones this limit is necessarily abrupt. The Landau problem, as a limit case of a noncommutative system, is also considered. 
  We present four-dimensional gauge theories in Minkowski spacetime which effectively generate in certain energy regimes five-dimensional warped geometries whereas, in general, the fifth dimension is latticized. After discussing in detail several general aspects in such theories we present a number of exactly solvable examples. We also point out how a particular case, defined in an N-sided polygon and having a Z_N symmetry, has a similar realization in an appropriate supersymmetric setting with D3-branes. 
  Using the thin-shell formalism we discuss the motion of domain walls in de Sitter and anti-de Sitter (AdS) time-dependent bulks. This motion results in a dynamics for the brane scale factor. We show that in the case of a clean brane the scale factor describes both singular and non-singular universes, with phases of contraction and expansion. These phases can be understood as quotients of AdS spacetime by a discrete symmetry group. We discuss this effect in some detail, and suggest how the AdS/CFT correspondence could be applied to obtain a non perturbative description of brane-world string cosmology. 
  A generic F-theory compactification containing many D3 branes develops multiple brane throats. The interaction of observers residing inside different throats involves tunneling suppression and, as a result, is very weak. This suggests a new mechanism for generating small numbers in Nature. One application is to the hierarchy problem: large supersymmetry breaking near the unification scale inside a shallow throat causes TeV-scale SUSY-breaking inside the standard-model throat. Another application, inspired by nuclear-decay, is in designing naturally long-lived particles: a cold dark matter particle residing near the standard model brane decays to an approximate CFT-state of a longer throat within a Hubble time. This suggests that most of the mass of the universe today could consist of CFT-matter and may soften structure formation at sub-galactic scales. The tunneling calculation demonstrates that the coupling between two throats is dominated by higher dimensional modes and consequently is much larger than a naive application of holography might suggest. 
  We study the spectrum of open strings on AdS_2 branes in AdS_3 in an NS-NS background, using the SL(2,R) WZW model. When the brane carries no fundamental string charge, the open string spectrum is the holomorphic square root of the spectrum of closed strings in AdS_3. It contains short and long strings, and is invariant under spectral flow. When the brane carries fundamental string charge, the open string spectrum again contains short and long strings in all winding sectors. However, branes with fundamental string charge break half the spectral flow symmetry. This has different implications for short and long strings. As the fundamental string charge increases, the brane approaches the boundary of AdS_3. In this limit, the induced electric field on the worldvolume reaches its critical value, producing noncommutative open string theory on AdS_2. 
  Examples of scalar conserved currents are presented for trigonometric, hyperbolic and elliptic versions of the Hubbard model with non-nearest neighbour variable range hopping. They support for the first time the hypothesis about the integrability of the elliptic version. The two- electron wave functions are constructed in an explicit form. 
  The localization of metric fluctuations on scalar brane configurations breaking spontaneously five-dimensional Poincar\'e invariance is discussed. Assuming that the four-dimensional Planck mass is finite and that the geometry is regular, it is demonstrated that the vector and scalar fluctuations of the metric are not localized on the brane. 
  Superconformal ghost current generators of conformal dimension 3/2 are constructed using the conformal ghosts and anticommuting infinite dimensional gamma matrices of the Clifford algebra. The super-Virasoro algebra for the ghosts in both the N.S. and R sectors are presented. The anomaly terms in both cases are deduced using Jacobi identity. 
  Homogeneous and isotropic cosmologies of the Planck era before the classical Einstein equations become valid are studied taking quantum gravitational effects into account. The cosmological evolution equations are renormalization group improved by including the scale dependence of Newton's constant and of the cosmological constant as it is given by the flow equation of the effective average action for gravity. It is argued that the Planck regime can be treated reliably in this framework because gravity is found to become asymptotically free at short distances. The epoch immediately after the initial singularity of the Universe is described by an attractor solution of the improved equations which is a direct manifestation of an ultraviolet attractive renormalization group fixed point. It is shown that quantum gravity effects in the very early Universe might provide a resolution to the horizon and flatness problem of standard cosmology, and could generate a scale-free spectrum of primordial density fluctuations. 
  We deform gravity with higher curvature terms in four dimensions and argue that the non-relativistic limit is of the same form as the non-relativistic limit of the theories with large extra dimensions. Therefore the experiments that perform sub-millimeter tests of inverse-square law cannot distinguish the effects of large extra dimensions from the effects of higher dimensional operators. In other words instead of detecting the presence of sub-millimeter dimensions; the experiments could be detecting the existence of massive modes of gravity with large masses ($\ge 10^{-3}$ eV) 
  We study an N-body Calogero model in the S_N-symmetric subspace of the positive definite Fock space. We construct a new algebra of S_N-symmetric operators represented on the symmetric Fock space, and find a natural orthogonal basis by mapping the algebra onto the Heisenberg algebra. Our main result is the bosonic realization of nonlinear symmetry algebra describing the structure of degenerate levels of Calogero model. 
  We study some aspects of recent proposals to use the noncommutative Chern-Simons theory as an effective description of some planar condensed matter models in strong magnetic fields, such as the Quantum Hall Effect. We present an alternative justification for such a description, which may be extended to other planar systems where a uniform magnetic field is present. 
  The field-dependent invariant representation (the "dynamical" representation) of the Poincar\'e algebra is considered as a dynamical principle in order to get the corresponding "dynamical" electromagnetic coupling for higher spins ($s\geq 1$). If in lower-spin ($s=0,1/2$) cases the "dynamical" coupling is taken to coincide with the minimal electromagnetic coupling the higher-spin coupling is inevitably non-minimal, containing a term linear in the field strength tensor $F_{\mu\nu}$. This term leads to $g=2$. 
  We consider quantum mechanics on the noncommutative plane in the presence of magnetic field $B$. We show, that the model has two essentially different phases separated by the point $B\theta=c\hbar^2/e$, where $\theta$ is a parameter of noncommutativity. In this point the system reduces to exactly-solvable one-dimensional system. When $\kappa=1-eB\theta/c\hbar^2<0$ there is a finite number of states corresponding to the given value of the angular momentum. In another phase, i.e. when $\kappa>0$ the number of states is infinite. The perturbative spectrum near the critical point $\kappa=0$ is computed. 
  We develop a technique of construction of integrable models with a Z_2 grading of both the auxiliary (chain) and quantum (time) spaces. These models have a staggered disposition of the anisotropy parameter. The corresponding Yang-Baxter Equations are written down and their solution for the gl(N) case are found. We analyze in details the N=2 case and find the corresponding quantum group behind this solution. It can be regarded as quantum U_{q,B}(gl(2)) group with a matrix deformation parameter qB with (qB)^2=q^2. The symmetry behind these models can also be interpreted as the tensor product of the (-1)-Weyl algebra by an extension of U_q(gl(N)) with a Cartan generator related to deformation parameter -1. 
  We consider ``brane world sum rules'' for compactifications involving an arbitrary number of spacetime dimensions. One of the most striking results derived from such consistency conditions is the necessity for negative tension branes to appear in five--dimensional scenarios. We show how this result is easily evaded for brane world models with more than five dimensions. As an example, we consider a novel realization of the Randall--Sundrum scenario in six dimensions involving only positive tension branes. 
  We discuss a version of Hamiltonian (2+1)-dimensional dynamics, in which one allows nonvanishing Poisson brackets also between the coordinates, and between the momenta. The resulting equations of motion are not any more derivable from a Lagrangian. However, taking a specific limit, in which the symplectic form becomes singular, one can recover a first-order Lagrangian description. This signals the dimensional reduction of the phase-space to half its initial number of degrees of freedom. We reach the same limit from another point of view, studying a particular form of the Poisson brackets, which is singled out geometrically and easy to handle algebraically. For comparison, a discussion of quantum mechanics with extended Heisenberg algebra is included. The quantum theory constrains the antisymmetric matrix providing the algebra to the above mentioned classically singular limit. 
  In this short note we present a mathematical interpretation of tachyon condensation on (three dimensional) orbifolds within the framework of boundary string field theory (BSFT). We explicitly show that important parts of decay modes in brane-antibrane systems with N=2 boundary supersymmetry can be interpreted as the McKay correspondence described as complexes. This will give an example of the recent interpretation of D-branes as derived category. We also discuss the N=4 boundary supersymmetry as a more refined structure. 
  We consider a model in which the universe is the direct product of a (3+1)-dimensional Friedmann, Robertson-Walker (FRW) space and a compact hyperbolic manifold (CHM). Standard Model fields are confined to a point in the CHM (i.e. to a brane). In such a space, the decay of massive Kaluza-Klein modes leads to the injection of any initial bulk entropy into the observable (FRW) universe. Both Kolmogoro-Sinai mixing due to the non-integrability of flows on CHMs and the large statistical averaging inherent in the collapse of the initial entropy onto the brane smooth out any initial inhomogeneities in the distribution of matter and of 3-curvature on any slice of constant 3-position. If, as we assume, the initial densities and curvatures in each fundamental correlation volume are drawn from some universal underlying distributions independent of location within the space, then these smoothing mechanisms effectively reduce the density and curvature inhomogeneities projected onto the FRW. This smoothing is sufficient to account for the current homogeneity and flatness of the universe. The fundamental scale of physics can be $\gsim 1$TeV. All relevant mass and length scales can have natural values in fundamental units. All large dimensionless numbers, such as the entropy of the universe, are understood as consequences of the topology of spacetime which is not explained. No model for the origin of structure is proffered. 
  The interaction of an electron with a polar molecule is shown to be the simplest realization of a quantum anomaly in a physical system. The existence of a critical dipole moment for electron capture and formation of anions, which has been confirmed experimentally and numerically, is derived. This phenomenon is a manifestation of the anomaly associated with quantum symmetry breaking of the classical scale invariance exhibited by the point-dipole interaction. Finally, analysis of symmetry breaking for this system is implemented within two different models: point dipole subject to an anomaly and finite dipole subject to explicit symmetry breaking. 
  We propose a mechanism to trap massive vector fields as a photon on the Randall-Sundrum brane embedded in the five dimensional AdS space. This localization-mechanism of the photon is realized by considering a brane action, to which a quadratic potential of the bulk-vector fields is added. We also point out that this potential gives several constraints on the fluctuations of the vector fields in the bulk space. 
  We demonstrate that an impulse action (`recoil') on a D-particle embedded in a (four-dimensional) cosmological Robertson-Walker (RW) spacetime is described, in a $\sigma$-model framework, by a suitably extended higher-order logarithmic world-sheet algebra of relevant deformations. We study in some detail the algebra of the appropriate two-point correlators, and give a careful discussion as to how one can approach the world-sheet renormalization group infrared fixed point, in the neighborhood of which the logarithmic algebra is valid. It is found that, if the initial RW spacetime does not have cosmological horizons, then there is no problem in approaching the fixed point. However, in the presence of horizons, there are world-sheet divergences which imply the need for Liouville dressing in order to approach the fixed point in the correct way. A detailed analysis on the subtle subtraction of these divergences in the latter case is given. In both cases, at the fixed point, the recoil-induced spacetime is nothing other than a coordinate transformation of the initial spacetime into the rest frame of the recoiling D-particle. However, in the horizon case, if one identifies the Liouville mode with the target time, which expresses physically the back reaction of the recoiling D-particle onto the spacetime structure, it is found that the induced spacetime distortion results in the removal of the initial cosmological horizon and the eventual stopping of the acceleration of the Universe. In this latter sense, our model may be thought of as a conformal field theory description of a (toy) Universe characterized by a sort of `phase transition' at the moment of impulse, implying a time-varying speed of light. 
  Primordial black hole formation has been studied using an inflaton field with a variable cosmological term as the potential. 
  String theoretic resolution of classical spacetime singularities is discussed. Particular emphasis is on the use of brane probes, and the connection [1] of the enhancon phenomenon [2] to the n=2* Pilch-Warner flow spacetime [3]. Some comments and details on the singularity of the PW spacetime are added. For the proceedings of Strings 2001, Mumbai, India. 
  Realization of the conformal higher spin symmetry on the 4d massless field supermultiplets is given. The self-conjugated supermultiplets, including the linearized ${\cal N}=4$ SYM theory, are considered in some detail. Duality between non-unitary field-theoretical representations and the unitary doubleton--type representations of the 4d conformal algebra $su(2,2)$ is formulated in terms of a Bogolyubov transform. The set of 4d massless fields of all spins is shown to form a representation of $sp(8)$.   The obtained results are extended to the generalized superspace invariant under $osp(L, 2M)$ supersymmetries. World line particle interpretation of the free higher spin theories in the $osp(2\N, 2M)$ invariant (super)space is given. Compatible with unitarity free equations of motion in the $osp(L,2M)$ invariant (super)space are formulated. A conjecture on the chain of $AdS_{d+1}/CFT_d \to AdS_{d}/CFT_{d-1} \to ...$ dualities in the higher spin gauge theories is proposed. 
  We establish a theorem about non-trivial 11D supergravity fluctuations that are conformally related to flat superspace geometry. Under the assumption that a theory of conformal 11D supergravity exists, similar in form to that of previously constructed theories in lower dimensions, this theorem demands the appearance of non-vanishing dimension 1/2 torsion tensors in order to accommodate a non-trivial 11D conformal compensator and thus M-theory corrections that break super-conformal symmetry. At the complete non-linear level, a presentation of a conventional minimal superspace realization of Weyl symmetry in eleven dimensional superspace is also described. All of our results taken together imply that there exists some realization of conformal symmetry relevant for the M-theory effective action. We thus led to conjecture this is also true for the full and complete M-theory. 
  We present a general method of deriving the effective action for conformal anomalies in any even dimension, which satisfies the Wess-Zumino consistency condition by construction. The method relies on defining the coboundary operator of the local Weyl group, and giving a cohomological interpretation to counterterms in the effective action in dimensional regularization with respect to this group. Non-trivial cocycles of the Weyl group arise from local functionals that are Weyl invariant in and only in the physical even integer dimension. In the physical dimension the non-trivial cocycles generate covariant non-local action functionals characterized by sensitivity to global Weyl rescalings. The non-local action so obtained is unique up to the addition of trivial cocycles and Weyl invariant terms, both of which are insensitive to global Weyl rescalings. These distinct behaviors under rigid dilations can be used to distinguish between infrared relevant and irrelevant operators in a generally covariant manner. Variation of the $d=4$ non-local effective action yields two new conserved geometric stress tensors with local traces. The method may be extended to any even dimension by making use of the general construction of conformal invariants given by Fefferman and Graham. As a corollary, conformal field theory behavior of correlators at the asymptotic infinity of either anti-de Sitter or de Sitter spacetimes follows, i.e. AdS$_{d+1}$ or deS$_{d+1}$/CFT$_d$ correspondence. The same construction naturally selects all infrared relevant terms (and only those terms) in the low energy effective action of gravity in any even integer dimension. The infrared relevant terms arising from the known anomalies in d=4 imply that the classical Einstein theory is modified at large distances. 
  We study a scale invariant two measures theory where a dilaton field \phi has no explicit potentials. The scale transformations include a shift \phi\to\phi+const. The theory demonstrates a new mechanism for generation of the exponential potential: in the conformal Einstein frame (CEF), after SSB of scale invariance, the theory develops the exponential potential and, in general, non-linear kinetic term is generated as well. The possibility of quintessence and of halo dark matter solutions are shown. The regime where the fermionic matter dominates (as compared to the dilatonic contribution) is analyzed. There it is found that starting from a single fermionic field we obtain exactly three different types of spin 1/2 particles in CEF that appears to suggest a new approach to the family problem of particle physics. It is automatically achieved that for two of them, fermion masses are constants, gravitational equations are canonical and the "fifth force" is absent. For the third type of particles, four fermionic interaction appears from SSB of scale invariance. 
  Following the recent construction of maximal (N=16) gauged supergravity in three dimensions, we derive gauged D=3, N=8 supergravities in three dimensions as deformations of the corresponding ungauged theories with scalar manifolds SO(8,n)/(SO(8)x SO(n)). As a special case, we recover the N=(4,4) theories with local SO(4) = SO(3)_L x SO(3)_R, which reproduce the symmetries and massless spectrum of D=6, N=(2,0) supergravity compactified on AdS_3 x S^3. 
  We show that melting the quark mass, the scalar $\sigma $ mass and the quark condensate leads uniquely to the quark-level SU(2) linear $\sigma $ model field theory. Upon thermalization, the chiral phase transition curve requires $T_c =2f^{CL}_{\pi}\approx 180 MeV$ when $\mu = 0$, while the critical chemical potential is $\mu_c =m_q\approx 325 MeV$. Transition to the superconductive phase occurs at $T^{(SC)}_c=\Delta /\pi e^{-\gamma_E}$. Coloured diquarks suggest $T_c^{(SC)}<180 MeV$. 
  In this paper we consider heterotic compactifications on K3 x T2 as well as type II compactifications on K3-fibred Calabi-Yau spaces with certain fluxes for the gauge and RR field strengths F and H turned on. By providing an identification of corresponding fluxes we show that the well-known N=2 heterotic/type II string-string duality still holds for a subset of all possible fluxes, namely those which arise from six-dimensional gauge fields with internal magnetic flux on the common two-sphere P1, which is the base space of the type II K3-fibration. On the other hand, F- and H-fluxes without P1-support, such as heterotic F-fluxes on the torus T2 or type II H-fluxes on cycles of the K3-fibre cannot be matched in any simple way, which is a challenge for heterotic/type II string-string duality. Our analysis is based on the comparison of terms in the effective low-energy heterotic and type II actions which are induced by the fluxes, such as the Green-Schwarz couplings related to flux-induced U(1) anomalies, the effective superpotential and the Fayet-Iliopoulos scalar potential. 
  We adress the problem of Fock space representations of (free) multiplet component fiels encountered in supersymmetric quantum field theory insisting on positivity and causality. We look in detail on the scalar and Majorana components of the chiral supersymmetric multiplet. Several Fock space representations are introduced. The last section contains a short application to the supersymmetric Epstein-Glaser method. The present paper is written in the vane of axiomatic quantum field theory with applications to the causal approach to supersymmetry. 
  It is shown that Witten's star product in string field theory, defined as the overlap of half strings, is equivalent to the Moyal star product involving the relativistic phase space of even string modes. The string field A(x[\sigma]) can be rewritten as a phase space field of the even modes $x_{2n},x_{0}, p_{2n}$ where $x_{2n}$ are the positions of the even string modes, and $p_{2n}$ are related to the Fourier space of the odd modes $x_{2n+1}$ up to a linear transformation. The $p_{2n}$ play the role of conjugate momenta for the even modes $x_{2n}$ under the string star product. The split string formalism is used in the intermediate steps to establish the map from Witten's star-product to Moyal's star-product. An ambiguity related to the midpoint in the split string formalism is clarified by considering odd or even modding for the split string modes, and its effect in the Moyal star product formalism is discussed. The noncommutative geometry defined in this way is technically similar to the one that occurs in noncommutative field theory, but it includes the timelike components of the string modes, and is Lorentz invariant. This map could be useful to extend the computational methods and concepts from noncommutative field theory to string field theory and vice-versa. 
  We present a minimal model for the Universe evolution fully extracted from effective String Theory. This model is by its construction close to the standard cosmological evolution, and it is driven selfconsistently by the evolution of the string equation of state itself. The inflationary String Driven stage is able to reach enough inflation, describing a Big Bang like evolution for the metric. By linking this model to a minimal but well established observational information, (the transition times of the different cosmological epochs), we prove that it gives realistic predictions on early and current energy density and its results are compatible with General Relativity. Interestingly enough, the predicted current energy density is found Omega = 1 and a lower limit Omega \geq 4/9 is also found. The energy density at the exit of the inflationary stage also gives | Omega |_{inf}=1. This result shows an agreement with General Relativity (spatially flat metric gives critical energy density) within an inequivalent Non-Einstenian context (string low energy effective equations). The order of magnitude of the energy density-dilaton coupled term at the beginning of the radiation dominated stage agrees with the GUT scale. The predicted graviton spectrum is computed and analyzed without any free parameters. Peaks and asymptotic behaviours of the spectrum are a direct consequence of the dilaton involved and not only of the scale factor evolution. Drastic changes are found at high frequencies: the dilaton produces an increasing spectrum (in no string cosmologies the spectrum is decreasing). Without solving the known problems about higher order corrections and graceful exit of inflation, we find this model closer to the observational Universe than the current available string cosmology scenarii. 
  We define a non-commutative product for arbitrary gauge and B-field backgrounds in terms of correlation functions of open strings. While off-shell correlations are, of course, not conformally invariant, it turns out that, at least to first derivative order, our product has the trace property and is associative up to surface terms if the background fields are put on-shell. No on-shell conditions for the inserted functions are needed, but it is essential to include the full contribution of the Born-Infeld measure. We work with a derivative expansion and avoid any topological limit, which would effectively constrain $H$. 
  We present string duals of four dimensional N=2 pure SU(N) SYM theory. The theory is obtained as the low energy limit of D5-branes wrapped on non-trivial two-cycles. Using seven dimensional gauged supergravity and uplifting the result to ten dimensions, we obtain solutions corresponding to various points of the N=2 moduli space. The more symmetric solution may correspond to a point with rotationally invariant classical vevs. By turning on seven dimensional scalar fields, we find a solution corresponding to a linear distribution of vevs. Both solutions are conveniently studied with a D5-probe, which also confirms many of the standard expectations for N=2 solutions. 
  We study the positive energy unitary representations of 2N extended superconformal algebras OSp(8*|2N) in six dimensions. These representations can be formulated in a particle basis or a supercoherent state basis, which are labeled by the superspace coordinates in d=6. We show that the supercoherent states that form the bases of positive energy representations of OSp(8*|2N) can be identified with conformal superfields in six dimensions. The massless conformal superfields correspond precisely to the ultra short doubleton supermultiplets of OSp(8*|2N). The other positive energy unitary representations correspond to massive conformal superfields in six dimensions and they can be obtained by tensoring an arbitrary number of doubleton supermultiplets with each other. The supermultiplets obtained by tensoring two copies of the doubletons correspond to massless anti-de Sitter supermultiplets in d = 7. 
  Recently, a new example of a complete non-compact Ricci-flat metric of G_2 holonomy was constructed, which has an asymptotically locally conical structure at infinity with a circular direction whose radius stabilises. In this paper we find a regular harmonic 3-form in this metric, which we then use in order to obtain an explicit solution for a fractional D2-brane configuration. By performing a T-duality transformation on the stabilised circle, we obtain the type IIB description of the fractional brane, which now corresponds to D3-brane with one of its world-volume directions wrapped around the circle. 
  We consider five dimensional non-factorizable geometries where the transverse dimension is bounded and the remaining (parallel) dimensions are not. We study the construction of effective theories at distances much longer than the transverse size. An observer unable to resolve the transverse direction can only measure distances along the parallel dimensions, but the non-factorizable geometry makes the length of a curve along the parallel dimension sensitive to where on the transverse direction the curve lies. We show that long geodesics that differ in their endpoints only by shifts along the transverse direction all have the same length to within the observer's resolution. We argue that this is the correct notion of distance in the effective theory for a bulk observer. This allows us to present a consistent interpretation of what is measured by observers that live either on a brane or in the bulk. 
  The motion of a string in curved spacetime is discussed in detail. The basic formalism for string motion in Minkowski spacetime and in curved spacetime is presented. The description applies to cosmic strings as well as to fundamental strings. Major ansatze for solving the string equations of motion are reviewed. In particular, the ``null string ansatz'', which is relevant to strings in strong gravitational fields, is emphasised. The formalism is applied to the motion of strings in 5-dimensional Kaluza-Klein black hole backgrounds with electric and magnetic charge. Such background spacetimes have been of interest lately, particularly from the point of view of fundamental string theory. It is shown that interesting results, relating to the extra dimension as seen by the string, are obtained even at the classical level. 
  We compute gravitational and axial anomaly for D-type (2,0) theories realized on N pairs of coincident M5-branes at R^5/Z_2 orbifold fixed point. We first summarize work by Harvey, Minasian, and Moore on A-type (2,0) theories, and then extend it to include the effect of orbifold fixed point. The net anomaly inflow follows when we further take into account the consistency of T^5/Z_2 M-theory orbifold. We deduce that the world-volume anomaly is given by N{\cal J}_8 + N(2N-1)(2N-2) p_2/24 where {\cal J}_8 is the anomaly polynomial of a free tensor multiplet and p_2 is the second Pontryagin class associated with the normal bundle. This result is in accord with Intriligator's conjecture. 
  3+1-dimensional free inviscid fluid dynamics is shown to satisfy the criteria for exact integrability, i.e. having an infinite set of independent, conserved quantities in involution, with the Hamiltonian being one of them. With (density dependent) interaction present, distinct infinite serieses of conserved quantities in involution are discovered. Clebsch parametrization of the velocity field is used in the the latter analysis. Relativistic generalization of the free system is also shown to be integrable. 
  We develop a systematic DLCQ perturbation theory and show that DLCQ S-matrix does not have a covariant continuum limit for processes with $p^+=0$ exchange. This implies that the role of the zero mode is more subtle than ever considered in DLCQ and hence must be treated with great care also in non-perturbative approach. We also make a brief comment on DLCQ in string theory. 
  Attention has been recently called upon the fact that the weak and null energy conditions and the second law of thermodynamics are violated in wormhole solutions of Einstein's theory with classical, nonminimally coupled, scalar fields as material source. It is shown that the discussion is only meaningful when ambiguities in the definitions of stress-energy tensor and energy density of a nonminimally coupled scalar are resolved. The three possible approaches are discussed with emphasis on the positivity of the respective energy densities and covariant conservation laws. The root of the ambiguities is traced to the energy localization problem for the gravitational field. 
  A new quantization scheme (WL-scheme), using world lines as objects of quantization, is proposed. Applying to nonlinear scalar field, the WL -scheme is investigated and compared with the conventional PA-scheme of quantization. In the PA-scheme objects of quantization are particles and antiparticles, which are fragments of the total physical object -- world line (WL). Applying to the nonlinear field, the PA-scheme of quantization leads to such difficulties as nonstationary vacuum, obligatory use of perturbation theory technique, normal ordering and cut-off in the scattering problem. These difficulties are corollaries of inconsistency of PA-scheme. The WL-scheme is free of these difficulties. These difficulties are connected with the reconstruction problem of the total world lines from their fragments (particles and antiparticles). In the case, when these fragments interact between themselves, such a reconstruction is very complicated problem. The new WL-scheme of quantization is free of all these problems, because it does not cut the total world line into fragments (particles and antiparticles). Formally appearance of fragments in the conventional quantization PA-scheme is a corollary of identification of the energy with the Hamiltonian. In fact such an identification is not necessary. It leads only to difficulties. The new WL-scheme of quantization does not use this identification and enables to go around all these problems. The WL-scheme enables not to use additional (to nonrelativistic QM) quantization rules, used in the relativistic QFT (normal ordering, perturbation technique, renormalization). 
  There have been comments on the starting paper, hep-th/0106074, which point out unclear motivation and definitions on noncommutative momentum introduced. Therefore, to give more clear presentation, this paper is withdrawn. 
  We consider worldsheet string theory on $Z_N$ orbifolds of $AdS_3$ associated with conical singularities. If the orbifold action includes a similar twist of $S^3$, supersymmetry is preserved, and there is a moduli space of vacua arising from blowup modes of the orbifold singularity. We exhibit the spectrum, including the properties of twisted sectors and states obtained by fractional spectral flow. A subalgebra of the spacetime superconformal symmetry remains intact after the $Z_N$ quotient, and serves as the spacetime symmetry algebra of the orbifold. 
  A new T-duality transformation is found in two-dimensional non-linear sigma models. This is a straightforward generalisation of Abelian and non-Abelian T-dualities. 
  We have developed dualization of ordinary and ``Stueckelberg compensated'' massive phase for the Kalb-Ramond theory. The compensated phase allows to study the interplay between spin jumping and duality. We show that spin jumping is caused by mass, while gauge symmetry is not necessary for this effect to take place. 
  The region of moduli space of string theories which is most likely to describe the "real world" is where the string coupling is about unity and the volume of extra compact dimensions is about the same size as the string volume. Here we map the landscape of this "central" region in a model-independent way, assuming only that the string coupling and compact volume moduli are chiral superfields of N=1 supergravity in 4 dimensions, and requiring only widely accepted conditions: that the supersymmetry (SUSY) breaking scale is about the weak scale and that the cosmological constant be of acceptably small magnitude. We find that the superpotential has (in the supersymmetric limit) a fourth order zero in the SUSY breaking direction. The potential near the minimum is very steep in the SUSY preserving directions, and very flat in the SUSY breaking direction, consequently the SUSY breaking field has a weak scale mass, while other moduli are heavy. We also argue that there will be additional near by minima with a large negative cosmological constant. 
  We discuss general properties and possible types of magnetic vortices in non-Abelian gauge theories (we consider here $G= SU(N), SO(N), USp(2N)$) in the Higgs phase. The sources of such vortices carry "fractional" quantum numbers such as $Z_n$ charge (for SU(N)), but also full non-Abelian charges of the dual gauge group. If such a model emerges as an effective dual magnetic theory of the fundamental (electric) theory, the non-Abelian vortices can provide for the mechanism of quark-confinement in the latter. 
  We consider chiral condensates in SU(2) gauge theory with broken N=2 supersymmetry and one fundamental flavor in the matter sector. Matter and gaugino condensates are determined by integrating out the adjoint field. The only nonperturbative input is the Affleck-Dine-Seiberg one-instanton superpotential. The results are consistent with those obtained by the `integrating in' procedure. We then calculate monopole, dyon, and charge condensates using the Seiberg-Witten approach. The key observation is that the monopole and charge condensates vanish at the Argyres-Douglas point where the monopole and charge vacua collide. We interpret this phenomenon as a deconfinement of electric and magnetic charges at the Argyres-Douglas point. 
  We review the construction of regular p-brane solutions of M-theory and string theory with less than maximal supersymmetry whose transverse spaces have metrics with special holonomy, and where additional fluxes allow brane resolutions via transgression terms. We focus on the properties of resolved M2-branes and fractional D2-branes, whose transverse spaces are Ricci flat eight-dimensional and seven-dimensional spaces of special holonomy. We also review fractional M2-branes with transverse spaces corresponding to the new two-parameter Spin(7) holonomy metrics, and their connection to fractional D2-branes with transverse spaces of G_2 holonomy. 
  We discuss the nonabelian world-volume action which governs the dynamics of N coincident Dp-branes. In this theory, the branes' transverse displacements are described by matrix-valued scalar fields, and so this is a natural physical framework for the appearance of noncommutative geometry. One example is the dielectric effect by which Dp-branes may be polarized into a noncommutative geometry by external fields. Another example is the appearance of noncommutative geometries in the description of intersecting D-branes of differing dimensions, such as D-strings ending on a D3- or D5-brane. We also describe the related physics of giant gravitons. 
  Spontaneous symmetry breaking usually occurs due to the tachyonic (spinodal) instability of a scalar field near the top of its effective potential at $\phi = 0$. Naively, one might expect the field $\phi$ to fall from the top of the effective potential and then experience a long stage of oscillations with amplitude O(v) near the minimum of the effective potential at $\phi = v$ until it gives its energy to particles produced during these oscillations. However, it was recently found that the tachyonic instability rapidly converts most of the potential energy V(0) into the energy of colliding classical waves of the scalar field. This conversion, which was called "tachyonic preheating," is so efficient that symmetry breaking typically completes within a single oscillation of the field distribution as it rolls towards the minimum of its effective potential. In this paper we give a detailed description of tachyonic preheating and show that the dynamics of this process crucially depend on the shape of the effective potential near its maximum. In the simplest models where $V(\phi) \sim -m^2\phi^2$ near the maximum, the process occurs solely due to the tachyonic instability, whereas in the theories $-\lambda\phi^n$ with n > 2 one encounters a combination of the effects of tunneling, tachyonic instability and bubble wall collisions. 
  We construct a new class of scalar noncommutative multi-solitons on an arbitrary Kahler manifold by using Berezin's geometric approach to quantization and its generalization to deformation quantization. We analyze the stability condition which arises from the leading 1/hbar correction to the soliton energy and for homogeneous Kahler manifolds obtain that the stable solitons are given in terms of generalized coherent states. We apply this general formalism to a number of examples, which include the sphere, hyperbolic plane, torus and general symmetric bounded domains. As a general feature we notice that on homogeneous manifolds of positive curvature, solitons tend to attract each other, while if the curvature is negative they will repel each other. Applications of these results are discussed. 
  We study the effect of noncommutativity of space on the physics of a quantum interferometer located in a rotating disk in a gauge field background. To this end, we develop a path-integral approach which allows defining an effective action from which relevant physical quantities can be computed as in the usual commutative case. For the specific case of a constant magnetic field, we are able to compute, exactly, the noncommutative Lagrangian and the associated shift on the interference pattern for any value of $\theta$. 
  We construct integrable realizations of conformal twisted boundary conditions for ^sl(2) unitary minimal models on a torus. These conformal field theories are realized as the continuum scaling limit of critical A-D-E lattice models with positive spectral parameter. The integrable seam boundary conditions are labelled by (r,s,\zeta) in (A_{g-2},A_{g-1},\Gamma) where \Gamma is the group of automorphisms of G and g is the Coxeter number of G. Taking symmetries into account, these are identified with conformal twisted boundary conditions of Petkova and Zuber labelled by (a,b,\gamma) in (A_{g-2}xG, A_{g-2}xG,Z_2) and associated with nodes of the minimal analog of the Ocneanu quantum graph. Our results are illustrated using the Ising (A_2,A_3) and 3-state Potts (A_4,D_4) models. 
  The field theory dual to the Freedman-Townsend model of a non-Abelian anti-symmetric tensor field is a nonlinear sigma model on the group manifold G. This can be extended to the duality between the Freedman-Townsend model coupled to Yang-Mills fields and a nonlinear sigma model on a coset space G/H. We present the supersymmetric extension of this duality, and find that the target space of this nonlinear sigma model is a complex coset space, GC/HC. 
  Based on a heuristic boost argument, we propose that the 4 dimensional de Sitter space can be described by a spherical Chern-Simons matrix model near the cosmological horizon, or models generalizing this simple choice. The dimension of the Hilbert space is naturally finite. We also make some comments on possible realization of holography in this approach, and on possible relation to the conformal field theory approach. 
  In the context of four-dimensional SU(N) gauge theories, we study the spectrum of the confining strings. We compute, for the SU(6) gauge theory formulated on a lattice, the three independent string tensions sigma_k related to sources with Z_N charge k=1,2,3, using Monte Carlo simulations. Our results, whose uncertainty is approximately 2% for k=2 and 4% for k=3, are consistent with the sine formula sigma_k/sigma = sin(k pi/N) / \sin(pi/N) for the ratio between sigma_k and the standard string tension sigma, and show deviations from the Casimir scaling.   The sine formula is known to emerge in supersymmetric SU(N) gauge theories and in M-theory. We comment on an analogous behavior exhibited by two-dimensional SU(N)xSU(N) chiral models. 
  We construct a new family of classical BPS solutions of type IIB supergravity describing 3-branes transverse to a 6-dimensional space with topology R^2*ALE. They are characterized by a non-trivial flux of the supergravity 2-forms through the homology 2-cycles of a generic smooth ALE manifold. Our solutions have two Killing spinors and thus preserve N=2 supersymmetry. They are expressed in terms of a quasi harmonic function H (the ``warp factor''), whose properties we study in the case of the simplest ALE, namely the Eguchi-Hanson manifold. The equation for H is identified as an instance of the confluent Heun equation. We write explicit power series solutions and solve the recurrence relation for the coefficients, discussing also the relevant asymptotic expansions. While, as in all such N=2 solutions, supergravity breaks down near the brane, the smoothing out of the vacuum geometry has the effect that the warp factor is regular in a region near the cycle. We interpret the behavior of the warp factor as describing a three-brane charge ``smeared'' over the cycle and consider the asymptotic form of the geometry in that region, showing that conformal invariance is broken even when the complex type IIB 3-form field strength is assumed to vanish. We conclude with a discussion of the basic features of the gauge theory dual. 
  We use boundary string field theory to study open string tachyon condensation on a three-sphere closed string background. We consider the closed string background described by $SU(2)_k$ WZW model in the limit of large $k$. We compute the exact tachyon potential and analyse the decay modes. 
  Noncommutative versions of theories with a gauge freedom define (when they exist) consistent deformations of their commutative counterparts. General aspects of Seiberg-Witten maps are discussed from this point of view. In particular, the existence of the Seiberg-Witten maps for various noncommutative theories is related to known cohomological theorems on the rigidity of the gauge symmetries of the commutative versions. In technical terms, the Seiberg-Witten maps define canonical transformations in the antibracket that make the solutions of the master equation for the commutative and noncommutative versions coincide in their antifield-dependent terms. As an illustration, the on-shell reducible noncommutative Freedman-Townsend theory is considered. 
  We discuss the issue of the cosmological constant in non-commutative non-supersymmetric gauge theories. In particular, in orbifold field theories non-commutativity acts as a UV cut-off. We suggest that in these theories quantum corrections give rise to a vacuum energy \rho, that is controlled by the non-commutativity parameter \theta, \rho ~ 1/theta^2 (only a soft logarithmic dependence on the Planck scale survives). We demonstrate our claim in a two-loop computation in field theory and by certain higher loop examples. Based on general expressions from string theory, we suggest that the vacuum energy is controlled by non-commutativity to all orders in perturbation theory. 
  In the context of a five-dimensional brane-world model motivated from heterotic M-theory, we develop a framework for potential-driven brane-world inflation. Specifically this involves a classification of the various background solutions of (A)dS_5 type, an analysis of five-dimensional slow-roll conditions and a study of how a transition to the flat vacuum state can be realized. It is shown that solutions with bulk potential and both bane potentials positive exist but are always non-separating and have a non-static orbifold. It turns out that, for this class of backgrounds, a transition to the flat vacuum state during inflation is effectively prevented by the rapidly expanding orbifold. We demonstrate that such a transition can be realized for solutions where one boundary potential is negative. For this case, we present two concrete inflationary models which exhibit the transition explicitly. 
  In frames of dS/CFT correspondence suggested by Strominger we calculate holographic conformal anomaly for dual euclidean CFT. The holographic renormalization group method is used for this purpose. It is explicitly demonstrated that two-dimensional and four-dimensional conformal anomalies (or corresponding central charges) have the same form as those obtained in AdS/CFT duality. 
  We investigate the large Nc limit of pure N=2 supersymmetric gauge theory with gauge group SU(Nc) by using the exact low energy effective action. Typical one-complex dimensional sections of the moduli space parametrized by a global complex mass scale v display three qualitatively different regions depending on the ratio between |v| and the dynamically generated scale Lambda. At large |v|/Lambda, instantons are exponentially suppressed as N goes to infinity. When |v| is of order Lambda, singularities due to massless dyons occur. They are densely distributed in rings of calculable thicknesses in the v-plane. At small |v|/Lambda, instantons disintegrate into fractional instantons of charge 1/(2N). These fractional instantons give non-trivial contributions to all orders of 1/N, unlike a planar diagrams expansion which generates a series in 1/N^2, implying the presence of open strings. We have explicitly calculated the fractional instantons series in two representative examples, including the 1/N and 1/N^2 corrections. Our most interesting finding is that the 1/N expansion breaks down at singularities on the moduli space due to severe infrared divergencies, a fact that has remarkable consequences. 
  We apply supersymmetric discrete light-cone quantization (SDLCQ) to the study of supersymmetric Yang-Mills theory on R x S^1 x S^1. One of the compact directions is chosen to be light-like and the other to be space-like. Since the SDLCQ regularization explicitly preserves supersymmetry, this theory is totally finite, and thus we can solve for bound-state wave functions and masses numerically without renormalizing. We present an overview of all the massive states of this theory, and we see that the spectrum divides into two distinct and disjoint sectors. In one sector the SDLCQ approximation is only valid up to intermediate coupling. There we find a well defined and well behaved set of states, and we present a detailed analysis of these states and their properties. In the other sector, which contains a completely different set of states, we present a much more limited analysis for strong coupling only. We find that, while these state have a well defined spectrum, their masses grow with the transverse momentum cutoff. We present an overview of these states and their properties. 
  In this note we introduce the notion of bundle gerbe K-theory and investigate the relation to twisted K-theory. We provide some examples. Possible applications of bundle gerbe K-theory to the classification of D-brane charges in non-trivial backgrounds are discussed. 
  We review some aspects of string tachyon dynamics with special emphasis on effective actions 
  We analyse several open and mixed sector tree-level amplitudes in N=2 p-p' systems with a constant magnetic B turned on. The 3-point function vanishes on-shell. The 4-point function, in the Seiberg-Witten (SW) low energy limit\cite{SW}, is local, {\it indicating the possible topological nature of the theory (in the SW low energy limit)} and the {\it possible relation between noncommutative N=2 p-p' system in two complex dimensions and in the SW limit, and (non)commutative N=2 p'-p' system in two real dimensions.} We discuss three extreme noncommutativity limits (after having taken the Seiberg-Witten low energy limit) of the mixed 3-point function, and get two kinds of commutative non-associaitive generalized star products. We make some speculative remarks related to reproducing the above four-point tree level amplitude in the open sector, from a field theory. 
  A Siegel-type chiral p-form action is proposed in D=2(p+1) spacetime dimensions. The approach we adopt is to realize the symmetric second-rank Lagrange-multiplier field, introduced in Siegel's action, in terms of a normalized multiplication of two (q+1)-form fields with q indices of each field contracted in the even p case, or of two pairs of (q+1)-form fields with q indices of each pair of fields contracted in the odd p case, where the (q+1)-form fields are of external derivatives of one auxiliary q-form field for the former, or a pair of auxiliary q-form fields for the latter. Using this action, it is straightforward to deduce the recently constructed PST action for q equal to zero. It is found that the Siegel-type chiral p-form action with a fixed p (even or odd) is doubly self-dual in D=2(p+1) spacetime dimensions when the auxiliary field(s) is/are also chosen to be of p-form. This result includes PST's as a special case where only the chiral 0-form action is doubly self-dual in D=2 dimensions. 
  We perform the group analysis of Witten's equations of motion for a particle moving in the presence of a magnetic monopole, and also when constrained to move on the surface of a sphere, which is the classical Wess-Zumino-Witten model. We also consider variations of this model. Our analysis gives the generators of the corresponding Lie point symmetries. The Lie symmetry corresponding to Kepler's third law is obtained in two related examples. 
  In this report we study the dual equivalence between the generalized self-dual (SD) and topologically massive (TM) models. To this end we linearize the model using an auxiliary field and apply a gauge embedding procedure to construct a gauge equivalent model. We clearly show that, under the above conditions, a nonlinear SD model always has a duality equivalent TM action.The general result obtained is then particularized for a number of examples, including the Born-Infeld-Chern-Simons (BICS) model recently discussed in the literature. 
  The dynamics of totally symmetric free massless higher spin fields in $AdS_d$ is reformulated in terms of the compensator formalism for AdS gravity. The $AdS_5$ higher spin algebra is identified with the star product algebra with the $su(2,2)$ vector (i.e., $o(4,2)$ spinor) generating elements. Cubic interactions of the totally symmetric bosonic higher spin gauge fields in $AdS_5$, including the interaction with gravity, are formulated at the action level. 
  We consider D-brane/anti-D-brane systems at T>0. Starting at the closed string vacuum, we argue that a finite temperature leads to the reappearance of open string degrees of freedom. We also show that, at a sufficiently large temperature, the open string vacuum becomes stable. Building upon this observation and previous work by Horowitz, Maldacena and Strominger, we formulate a microscopic brane-antibrane model for the non-extremal black three-brane in ten dimensions (as well as for the black two- and five-branes in eleven dimensions). Under reasonable assumptions, and using known results from the AdS/CFT correspondence, the microscopic entropy agrees with the supergravity result up to a factor of 2^(p/p+1), with p the dimension of the brane. The negative specific heat and pressure of the black brane have a simple interpretation in terms of brane-antibrane annihilation. We also find in the model states resembling black holes and other lower-dimensional black branes. 
  We analyze the anti-de Sitter (AdS) superparticle and superstring systems described in terms of supermatrix valued coordinates proposed by Roiban and Siegel. This approach gives simple symmetry transformations and equations of motion. We examine their kappa-transformations, infinite reducibility and kappa-gauge fixing conditions. A closed first class constraint set for the AdS superparticle is GL(4|4) covariant and keeping superconformal symmetry manifestly. For the AdS superstring $\sigma$-dependence breaks the GL(4|4) covariance, where supercovariant derivatives and currents satisfy the inhomogeneous GL(4|4). A closed first class constraint set for the AdS superstring turns out to be the same as the one for a superstring in flat space, namely ABCD constraints. 
  We consider a class of higher order corrections in the form of Euler densities of arbitrary rank $n$ to the standard gravity action in $D$ dimensions. We have previously shown that this class of corrections allows for domain wall solutions despite the presence of higher powers of the curvature. In the present paper we explicitly solve the linearized equation of motion for gravity fluctuations around the domain wall background and show that there always exist one massless state (graviton) propagating on the wall and a continuous tower of massive states propagating in the bulk. 
  We study the UV properties of the three-dimensional ${\cal N}=4$ SUSY nonlinear sigma model whose target space is $T^*(CP^{N-1})$ (the cotangent bundle of $CP^{N-1}$) to higher orders in the 1/N expansion. We calculate the $\beta$-function to next-to-leading order and verify that it has no quantum corrections at leading and next-to-leading orders. 
  We study the one loop dynamics of QFT on the fuzzy sphere and calculate the planar and nonplanar contributions to the two point function at one loop. We show that there is no UV/IR mixing on the fuzzy sphere. The fuzzy sphere is characterized by two moduli: a dimensionless parameter N and a dimensionful radius R. Different geometrical phases can obtained at different corners of the moduli space. In the limit of the commutative sphere, we find that the two point function is regular without UV/IR mixing; however quantization does not commute with the commutative limit, and a finite ``noncommutative anomaly'' survives in the commutative limit. In a different limit, the noncommutative plane R^2_theta is obtained, and the UV/IR mixing reappears. This provides an explanation of the UV/IR mixing as an infinite variant of the ``noncommutative anomaly''. 
  We comment on the recent papers by Costa et al and Emparan, which show how one might generate supergravity solutions describing certain dielectric branes in ten dimensions. The ``basic'' such solutions describe either N fundamental strings or N D4-branes expanding into a D6-brane, with topology M^2 x S^5 or M^5 x S^2 respectively. Treating these solutions in a unified way, we note that they allow for precisely two values of the radius of the relevant sphere, and that the solution with the smaller value of the radius has the lower energy. Moreover, the possible radii in both cases agree up to numerical factors with the corresponding solutions of the D6-brane worldvolume theory. We thus argue that these supergravity solutions are the correct gravitational description of the dielectric branes of Emparan and Myers. 
  We construct an action for the N=2 supersymmetric sine-Gordon model on the half-line, which we argue is both supersymmetric and integrable. The boundary interaction depends on three continuous boundary parameters, as well as the bulk mass parameter. 
  Apart from the flat space with an angular deficit, Einstein general relativity possesses another cylindrically symmetric solution. Because this configuration displays circles whose "circumferences" tend to zero when their "radius" go to infinity, it has not received much attention in the past. We propose a geometric interpretation of this feature and find that it implies field boundary conditions different from the ones found in the literature if one considers a source consisting of the scalar and the vector fields of a U(1) system . To obtain a non increasing energy density the gauge symmetry must be unbroken . For the Higgs potential this is achieved only with a vanishing vacuum expectation value but then the solution has a null scalar field. A non trivial scalar behaviour is exhibited for a potential of sixth order. The trajectories of test particles in this geometry are studied, its causal structure discussed. We find that this bosonic background can support a normalizable fermionic condensate but not such a current. 
  I present asymmetric orientifold models which, with the addition of RR fluxes, fix all the NS NS moduli including the dilaton. In critical string theory, this gives new AdS backgrounds with (discretely tunably) weak string coupling. Extrapolating to super-critical string theory, this construction leads to a promising candidate for a metastable de Sitter background with string coupling of order 1/10 and dS radius of order 100 times the string scale. Extrapolating further to larger and larger super-critical dimension suggests the possibility of finding de Sitter backgrounds with weaker and weaker string coupling. This note is an updated version of the last part of my Strings 2001 talk. 
  In cases of both abelian and nonabelian gauge groups, we study the Higgs mechanism in the topologically massive gauge theories in an arbitrary space-time dimension. We show that when the conventional Higgs potential coexists with a topological term, gauge fields become massive by 'eating' simultaneously both the Nambu-Goldstone boson and a higher-rank tensor field, and instead a new massless scalar field is 'vomitted' in the physical spectrum. Because of the appearance of this new massless field, the number of the physical degrees of freedom remains unchanged before and after the spontaneous symmetry breakdown. Moreover, the fact that the new field is a physical and positive norm state is rigorously proved by performing the manifestly covariant quantization of the model in three and four dimensions. In the mechanism at hand, the presence of a topological term makes it possible to shift mass of gauge fields in a nontrivial manner compared to the conventional value. 
  The symmetry properties of the bosonic string effective action under Poisson-Lie duality transformations are investigated. A convenient and simple formulation of these duality transformations is found, that allows the reduction of the string effective action in a Kaluza-Klein framework. It is shown that the action is invariant provided that the two Lie algebras, forming the Drinfeld double, have traceless structure constants. Finally, a functional relation is found between the Weyl anomaly coefficients of the original and dual non-linear sigma models. 
  We consider properties of a covariant worldvolume action for a system of N coincident Dp-branes in D=(p+2) dimensional space-time (so called codimension one branes). In the case of N coincident D0-branes in D=2 we then find a generalization of this action to a model which includes fermionic degrees of freedom and is invariant under target-space supersymmetry and worldline kappa-symmetry. We find that the type IIA D=2 superalgebra generating the supersymmetry transformations of the ND0-brane system acquires a non-trivial "central extension" due to a nonlinear contribution of U(N) adjoint scalar fields. Peculiarities of space-time symmetries of coincident Dp-branes are discussed. 
  The study of noncommutative solitons is greatly facilitated if the field equations are integrable, i.e. result from a linear system. For the example of a modified but integrable U(n) sigma model in 2+1 dimensions we employ the dressing method to construct explicit multi-soliton configurations on noncommutative R^{2,1}. These solutions, abelian and nonabelian, feature exact time-dependence for any value of the noncommutativity parameter theta and describe various lumps of finite energy in relative motion. We discuss their scattering properties and prove asymptotic factorization for large times. 
  There have been comments on the starting paper, hep-th/0106074, which point out unclear motivation and definition of noncommutative momentum introduced. This paper is withdrawn by the author for more clear presentation. 
  In the framework of superfield formalism, we discuss some aspects of the cohomological features of a two (1+1)-dimensional free Abelian gauge theory described by a Becchi-Rouet-Stora-Tyutin (BRST) invariant Lagrangian density. We demonstrate that the conserved and nilpotent (anti-)BRST- and (anti-)co-BRST charges are the generators of translations along the Grassmannian directions of the four (2+2)-dimensional supermanifold. A bosonic symmetry is shown to be generated by a Noether conserved charge that generates a translation along a bosonic direction of the supermanifold which turns out to be equivalent to a couple of successive translations along the two different and independent Grassmannian directions of the same supermanifold. Algebraically, these charges are found to be analogous to the de Rham cohomology operators of differential geometry. 
  We apply 't Hooft's deterministic quantum mechanics approach to free vector bosons in three dimensions and check Lorentz invariance. This approach does not work for the conformal group, for free bosons in two dimensions. This presents a technical difficulty for constructing a ``deterministic string theory''. 
  A recently proposed connection between closed string field and an open Wilson line defined on an arbitrary contour is further explored here. We suggest that reparametrization invariance of a Wilson line is the principle which determines the coupling of non-commutative gauge theory/matrix model to the modes of the closed string. An analogue of the level matching condition on the gauge theory/matrix model operators emerges quite naturally from the cyclic symmetry of the straight Wilson line. We show that the generating functional of correlation functions of these operators has the space-time gauge symmetry that one expects to find in closed string field theory. We also identify an infinite number of conserved operators in gauge theory/matrix model, the first of which is known to be the conserved stress tensor. 
  In a previous work, we computed the fully off-shell effective action $\Gamma$ and the corresponding quantum-corrected supersymmetry (SUSY) transformation operator $\delta_\epsilon$ for the so-called source-probe configuration in Matrix theory at one loop at order 4 in the derivative expansion, and showed that they satisfy the SUSY Ward identity $\delta_\epsilon \Gamma=0$. In this article, starting from the most general form of $\Gamma$, we demonstrate that, conversely, given such $\delta_\epsilon$ the SUSY Ward identity determines $\Gamma$ uniquely to the order specified above. Our demonstration does not require the explicit knowledge of the quantum-corrected supersymmetry transformation and hence strongly suggests that the uniqueness property would persist to all orders in perturbation theory. 
  We present a relativistic three-body equation to study correlations in a medium of finite temperatures and densities. This equation is derived within a systematic Dyson equation approach and includes the dominant medium effects due to Pauli blocking and self energy corrections. Relativity is implemented utilizing the light front form. The equation is solved for a zero-range force for parameters close to the confinement-deconfinement transition of QCD. We present correlations between two- and three-particle binding energies and calculate the three-body Mott transition. 
  It has been proposed that translationally-invariant black branes are classically stable if and only if they are locally thermodynamically stable. Reall has outlined a general argument to demonstrate this, and studied in detail the case of charged black p-branes in type II supergravity. We consider the application of his argument in the simplest non-trivial case, an uncharged asymptotically flat brane enclosed in a finite cylindrical cavity. In this simple context, it is possible to give a more complete argument than in the cases considered earlier, and it is therefore a particularly attractive example of the general approach. 
  String theory properly describes black-hole evaporation. The quantum string emission by Black Holes is computed. The black-hole temperature is the Hawking temperature in the semiclassical quantum field theory (QFT) regime and becomes the intrinsic string temperature, T_s, in the quantum (last stage) string regime. The QFT-Hawking temperature T_H is upper bounded by the string temperature T_S. The black hole emission spectrum is an incomplete gamma function of (T_H - T_S). For T_H << T_S, it yields the QFT-Hawking emission. For T_H \to T_S, it shows highly massive string states dominate the emission and undergo a typical string phase transition to a microscopic `minimal' black hole of mass M_{\min} or radius r_{\min} (inversely proportional to T_S) and string temperature T_S. The string back reaction effect (selfconsistent black hole solution of the semiclassical Einstein equations) is computed. Both, the QFT and string black hole regimes are well defined and bounded.The string `minimal' black hole has a life time tau_{min} simeq (k_B c)/(G hbar [T_S]^3). The semiclassical QFT black hole (of mass M and temperature T_H) and the string black hole (of mass M_{min} and temperature T_S) are mapped one into another by a `Dual' transform which links classical/QFT and quantum string regimes. 
  Accurate and powerful analytic and computational methods developped by the author allow to obtain the highly non trivial total absorption spectrum of the Black Hole, as well as phase shifts and cross sections (elastic and inelastic), the angular distribution of absorbed and scattered waves, and the Hawking emission rates. The exact total absorption spectrum of waves by the Black Hole presents as a function of frequency a remarkable oscillatory behaviour characteristic of a diffraction pattern. It oscillates around its optical geometric limit (27/4) pi (r_s)^2 with decreasing amplitude and almost constant period. This is an unique distinctive feature of the black hole absorption, and due to its r=0 singularity. Ordinary absorptive bodies and optical models do not present these features. The Hamiltonian describing the wave-black hole interaction is non hermitian (despite being real) due to its singularity at the origin (r=0). The unitarity optical theorem of scattering theory is generalized to the black hole case explicitely showing that absorption takes place only at the origin (r = 0). All these results allow to understand and reproduce the Black Hole absorption spectrum in terms of Fresnel-Kirchoff diffraction theory. These fundamental features will be present for generic higher dimensional Black Hole backgrounds, and whatever the low energy effective theory they arise from. In recent and increasing litterature on absorption cross sections (`grey body factors') of black holes (whatever ordinary, stringy, D-braned), the fundamental remarkable features of the Black Hole Absorption spectrum are overlooked. 
  We propose the exact boundary S matrix for breathers of the N=2 supersymmetric sine-Gordon model. We argue that this S matrix has three independent parameters, in agreement with a recently-proposed action. We also show, contrary to a previous claim, that the ``universal'' supersymmetric boundary S matrix commutes with two supersymmetry charges. General N=2 supersymmetric boundary integrable models are expected to have boundary S matrices with a similar structure. 
  First the "frame problem" is sketched: the motion of an isolated particle obeys a simple law in galilean frames, but how does the galilean character of the frame manifest itself at the place of the particle? A description of vacuum as a system of virtual particles will help to answer this question. For future application to such a description, the notion of global particle is defined and studied. To this end, a systematic use of the Fourier transformation on the Poincare group is needed. The state of a system of n free particles is represented by a statistical operator W, which defines an operator-valued measure on the n-th power of the dual of the Poincare group. The inverse Fourier-Stieltjes transform of that measure is called the characteristic function of the system; it is a function on the n-th power of the Poincare group. The main notion is that of global characteristic function: it is the restriction of the characteristic function to the diagonal subgroup ; it represents the state of the system, considered as a single particle. The main properties of characteristic functions, and particularly of global characteristic functions, are studied. A mathematical Appendix defines two functional spaces involved. 
  Quantum effects of bulk matter, in the form of massive fermions, are considered in the Randall-Sundrum $AdS_5$ brane world at finite temperatures. The thermodynamic energy (modulus potential) is calculated in the limiting case when the temperature is low, and is shown to possess a minimum, thus suggesting a new dynamical mechanism for stabilizing the brane world. Moreover, these quantum effects may solve the hierarchy scale problem, at quite low temperatures. The present note reviews essentially the fermion-related part of the recent article by I. Brevik, K. A. Milton, S. Nojiri, and S. D. Odintsov, hep-th/0010205 . 
  We study the large mass asymptotics of the Dirac operator with a nondegenerate mass matrix m={diag}(m_1,m_2,m_3) in the presence of scalar and pseudoscalar background fields taking values in the Lie algebra of the U(3) group. The corresponding one-loop effective action is regularized by the Schwinger's proper-time technique. Using a well-known operator identity, we obtain a series representation for the heat kernel which differs from the standard proper-time expansion, if m_1\ne m_2\ne m_3. After integrating over the proper-time we use a new algorithm to resum the series. The invariant coefficients which define the asymptotics of the effective action are calculated up to the fourth order and compared with the related Seeley-DeWitt coefficients for the particular case of a degenerate mass matrix with m_1=m_2=m_3. 
  We study gravitating monopoles and non-abelian black holes of SU(2) Einstein-Yang-Mills-Higgs theory coupled to a massless dilaton. The domain of existence of these solutions decreases with increasing dilaton coupling constant. The critical solutions of this system are Einstein-Maxwell-dilaton solutions. 
  The effective action of the radion in the Randall-Sundrum model is analysed. Fine tunings are needed to obtain the observed mass hierarchy and an invisible radion. since the kinetic terms are important for determining the radion mass, the finite quantum corrections from massless conformally coupled fermions are analysed and found to vanish at one loop order. 
  We describe vacuum as a system of virtual particles, some of which have negative energies. Any system of vacuum particles is a part of a keneme, i.e. of a system of n particles which can, without violating the conservation laws, annihilate in the strict sense of the word (transform into nothing). A keneme is a homogeneous system, i.e. its state is invariant by all transformations of the invariance group. But a homogeneous system is not necessarily a keneme. In the simple case of a spin system, where the invariance group is SU(2), a homogeneous system is a system whose total spin is unpolarized; a keneme is a system whose total spin is zero. The state of a homogeneous system is described by a statistical operator with infinite trace (von Neumann), to which corresponds a characteristic distribution. The characteristic distributions of the homogeneous systems of vacuum are defined and studied. Finally it is shown how this description of vacuum can be used to solve the frame problem posed in (I). 
  We consider some applications of the Renormalization Group flow equations obtained by resorting to a specific class of proper time regulators. Within this class a particular limit that corresponds to a sharpening of the effective width of the regulator is investigated and a procedure to analytically implement this limit on the flow equations is shown. We focus on the critical exponents determination for the O(N) symmetric scalar theory in three dimensions. The large N limit and some perturbative features in four dimensions are also analysed. In all problems examined the results are optimized when the mentioned limit of the proper time regulator is taken. 
  The normalization of the boundary superstring field theory action is determined by computing the near on-shell amplitude involving three gauge fields. 
  We consider quantization of open string theories in linear dilaton and constant antisymmetric tensor backgrounds and discuss the noncommutativity of space-time coordinates arising in such theories, including their relationship with light-like noncommutativity as well as backgrounds with null isometries. It is argued that the results can also be understood using space-time equations of motion of the string modes. We then present N= 2 supersymmetric generalization of these theories and the associated noncommutativity structure. 
  In modern fundamental theories there is consideration of higher dimensions, often in the context of what can be written as a Schr\"odinger equation. Thus, the energetics of bound states in different dimensions is of interest. By considering the quantum square well in continuous $D$ dimensions, it is shown that there is always a bound state for $0<D \le 2$. This binding is complete for D \to 0 and exponentially small for D \to 2_-. For D>2, a finite-sized well is always needed for there to be a bound state. This size grows like D^2 as D gets large. By adding the proper angular momentum tail a volcano, zero-energy, bound state can be obtained. 
  I review some properties of D-branes in the SU(2) and SL(2,R) WZW models. I comment on a potential difficulty for the realization of `warped brane worlds' in string theory. This short note is based on a talk given at the Strings'01 conference in Mumbay. 
  We review the emergence of the ten-dimensional fermionic closed string theories from subspaces of the Hilbert space of the 26-dimensional bosonic closed string theory compactified on an $E_8\times SO(16)$ lattice. They arise from a consistent truncation procedure which generates space-time fermions out of bosons. This procedure is extended to open string sectors. We prove that truncation of the unique tadpole-free $SO(2^{13})$ bosonic string theory compactified on the above lattice determines the anomaly free Chan-Paton group of the Type I theory and the consistent Chan-Paton groups of Type O theories. It also predicts the tension of space-filling D-branes in these fermionic theories. The derivation of these fermionic string properties from bosonic considerations alone points towards a dynamical origin of the truncation process. Space-time fermions and supersymmetries would then arise from bosonic degrees of freedom and no fermionic degrees of freedom would be needed in a fundamental theory of quantum gravity. 
  The possibility of parity violation through spacetime torsion has been explored in a scenario containing fields with different spins. Taking the Kalb-Ramond field as the source of torsion, an explicitly parity violating $U(1)_{EM}$ gauge invariant theory has been constructed by extending the KR field with a Chern-Simons term. 
  We consider a static brane in the background of a topological black hole, in arbitrary dimensions. For hyperbolic horizons, we find a solution only when the black hole mass assumes its minimum negative value. In this case, the tension of the brane vanishes, and the brane position coincides with the location of the horizon. For an elliptic horizon, we show that the massless mode of Randall-Sundrum is recovered in the limit of large black hole mass. 
  We discuss composite operators in N=4 super Yang-Mills theory and their realisations as superfields on different superspaces. The superfields that realise various operators on analytic superspace may be different in the free, interacting and quantum theories. In particular, in the quantum theory, there is a restricted class of operators that can be written as analytic tensor superfields. This class includes all series B and C operators in the theory as well as some series A operators which saturate the unitarity bounds. Operators of this type are expected to be protected from renormalisation. 
  We compute the effective potential of SU(2) Yang-Mills theory using the background field method and the Faddeev-Niemi decomposition of the gauge fields. In particular, we find that the potential will depend on the values of two scalar fields in the decomposition and that its structure will give rise to a symmetry breaking. 
  We study the localization properties of bulk form potentials on dilatonic domain walls. We find that bulk form potentials of any ranks can be localized as form potentials of the same ranks or one lower ranks, for any values of the dilaton coupling parameter. For large enough values of the dilaton coupling parameter, bulk form potentials of any ranks can be localized as form potentials of both the same ranks and one lower ranks. 
  The possibility to study the M/string theory cosmology via 5d bulk & brane action is investigated. The role of the 4-form field in the theory of BPS branes in 5d is clarified. We describe arguments suggesting that the effective 4d description of the universe in the ekpyrotic scenario (hep-th/0103239) should lead to contraction rather than expansion of the universe. To verify these arguments, we study the full 5d action prior to its integration over the 5th dimension. We show that if one adds the potential V(Y) to the action of the bulk brane, then the metric ansatz used in the ekpyrotic scenario does not solve the dilaton and gravitational equations. To find a consistent cosmological solution one must use a more general metric ansatz and a complete 5d description of the brane interaction instead of simply adding an effective 4d bulk brane potential V(Y). 
  Motivated by noncommutative Chern-Simons theory, we construct an infinite class of field theories that satisfy the axioms of Witten's string field theory. These constructions have no propagating open string degrees of freedom. We demonstrate the existence of non-trivial classical solutions. We find Wilson loop-like observables in these examples. 
  We study domain walls in two different extensions of super Yang--Mills characterized by the absence of a logarithmic term in their effective superpotential. The models, defined by the usual gaugino condensate and an extra field Y, give different patterns of domain walls despite both leading to the same effective limit for heavy Y, i.e. the Veneziano--Yankielowicz effective Lagrangian of super Yang--Mills. We explain the origin of those differences and also give a physical motivation for introducing the field Y. 
  We consider a sigma model formulation of open string theory with boundary fermions carrying Chan-Paton charges at the string ends. This formalism is particularly suitable for studying world-volume potentials on D-branes. We perform explicit two-loop sigma model computations of the potential T-dual to the non-abelian Born-Infeld action. We also discuss the world-volume couplings of NS fluxes which are responsible for Myers' dielectric effect. 
  We consider ``cosmologically symmetric'' (i.e. solutions with homogeneity and isotropy along three spatial dimensions) five-dimensional spacetimes with a scalar field and a three-brane representing our universe. We write Einstein's equations in a conformal gauge, using light-cone coordinates. We obtain explicit solutions: a. assuming proportionality between the scalar field and the logarithm of the (bulk) scale factor; b. assuming separable solutions. We then discuss the cosmology in the brane nduced by these solutions. 
  After an introduction to N=2 susy Yang-Mills theories, I review in some detail, for the SU(2) gauge group, how the low-energy effective action is obtained using duality and the constraints arising from the supersymmetry. Then I discuss how knowledge of this action, duality and certain discrete symmetries allow us to determine the spectra of stable BPS states at any point in moduli space. This is done for gauge group SU(2), without and with fundamental matter hypermultiplets which may have non-vanishing bare masses. In the latter case non-trivial four-dimensional CFTs arise at Argyres-Douglas type points. 
  We discuss several aspects of the proposed correspondence between quantum gravity on de Sitter spaces and Euclidean conformal field theories. The central charge appearing in the asymptotic symmetry algebra of three-dimensional de Sitter space is derived both from the conformal anomaly and the transformation law of the CFT stress tensor when going from dS_3 in planar coordinates to dS_3 with cosmological horizon. The two-point correlator for CFT operators coupling to bulk scalars is obtained in static coordinates, corresponding to a CFT on a cylinder. Correlation functions are also computed for CFTs on two-dimensional hyperbolic space. We furthermore determine the energy momentum tensor and the Casimir energy of the conformal field theory dual to the Schwarzschild-de Sitter solution in five dimensions. Requiring the pressure to be positive yields an upper bound for the black hole mass, given by the mass of the Nariai solution. Beyond that bound, which is similar to the one found by Strominger requiring the conformal weights of CFT operators to be real, one encounters naked singularities. 
  We study the S-matrix of two-dimensional \lambda\phi^4 theory in Discretized Light Cone Quantization and show how the correct continuum limit is reached for various processes in lowest order perturbation theory. 
  The content of the OPE of two 1/2 BPS operators in N=4 SCFT$_4$ is given by their superspace three-point functions with a third, a priori long operator. For certain 1/2 BPS short superfields these three-point functions are uniquely determined by superconformal invariance. We focus on the cases where the leading ($\theta=0$) components lie in the tensor products $[0,m,0]\otimes[0,n,0]$ and $[m,0,0]\otimes[0,0,n]$ of SU(4).   We show that the shortness conditions at the first two points imply selection rules for the supermultiplet at the third point. Our main result is the identification of all possible protected operators in such OPE's. Among them we find not only BPS short multiplets, but also series of special long multiplets which satisfy current-like conservation conditions in superspace. 
  The correspondence between BRST-BFV, Dirac and projection operator approaches to quantize constrained systems is analyzed. It is shown that the component of the BFV wave function with maximal number of ghosts and antighosts in the Schrodinger representation may be viewed as a wave function in the projection operator approach. It is shown by using the relationship between different quantization techniques that the Marnelius inner product for BRST-BFV systems should be in general modified in order to take into account the topology of the group; the Giulini-Marolf group averaging prescription for the inner product is obtained from the BRST-BFV method. The relationship between observables in different approaches is also found. 
  This paper has been withdrawn by the authors. 
  Using a Kaluza-Klein framework, we consider a relativistic fluid whose projection yields the supersymmetric non-relativistic Chaplygin gas introduced by Bergner-Jackiw-Polychronakos and by Hoppe. The conserved (super)charges of the Chaplygin gas are obtained as the projection of those arising in the extended model. 
  We study Myers world-volume effective action of coincident D-branes. We investigate a system of N_0 D0-branes in the geometry of Dp-branes with p=2 or p=4. The choice of coordinates can make the action simplified and tractable. For p=4, we show that a certain point-like D0-brane configuration solving equations of motion of the action can expand to form a fuzzy two-sphere without changing quantum numbers. We compare non-commutative D0-brane configurations with dual spherical D(6-p)-brane systems. We also discuss the relation between these configurations and giant gravitons in 11-dimensions. 
  Using the embedded defect method, we classify the possible embeddings of a 't Hooft-Polyakov monopole in a general gauge theory. We then discuss some similarities with embedded vortices and relate our results to fundamental monopoles. 
  We consider the supergravity dual descriptions of non-conformal super Yang-Mills theories realized on the world-volume of Dp-branes. We use the dual description to compute stress-energy tensor and current correlators. We apply the results to the study of dilatonic brane-worlds described by non-conformal field theories coupled to gravity. We find that brane-worlds based on D4 and D5 branes exhibit a localization of gauge and gravitational fields. We calculate the corrections to the Newton and Coulomb laws in these theories. 
  We present an N=1 superfield formulation of supersymmetric gauge theories with a compact extra dimension. The formulation incorporates the radion superfield and allows to write supersymmetric theories on warped gravitational backgrounds. We apply it to study the breaking of supersymmetry by the F-term of the radion, and show that, for flat extra dimensions, this leads to the same mass spectrum as in Scherk-Schwarz models of supersymmetry breaking. We also consider scenarios where supersymmetry is broken on a boundary of a warped extra dimension and compare them with anomaly mediated models. 
  In this work we have constructed the n=4 extended local conformal time supersymmetry for the Friedmann-Robertson-Walker (FRW) cosmological models. This is based on the superfield construction of the action, which is invariant under wordline local n=4 supersymmetry with $SU(2)_{local} \otimes SU(2)_{global}$ internal symmetry. It is shown that the supersymmetric action has the form of the localized (or superconformal) version of the action for n=4 supersymmetric quantum mechanics. This superfield procedure provides a well defined scheme for including supermatter. 
  We construct a gauge invariant regularisation scheme for pure SU(N) Yang-Mills theory in fixed dimension four or less (for N = infinity in all dimensions), with a physical cutoff scale Lambda, by using covariant higher derivatives and spontaneously broken SU(N|N) supergauge invariance. Providing their powers are within certain ranges, the covariant higher derivatives cure the superficial divergence of all but a set of one-loop graphs. The finiteness of these latter graphs is ensured by properties of the supergroup and gauge invariance. In the limit Lambda tends to infinity, all the regulator fields decouple and unitarity is recovered in the renormalized pure SU(N) Yang-Mills theory. By demonstrating these properties, we prove that the regularisation works to all orders in perturbation theory. 
  We construct the consistent supersymmetric extensions of the operators describing the recoil of a D-brane and show that they realize an N=1 logarithmic superconformal algebra. The corresponding supersymmetric vertex operator is related to the action of a twisted superparticle with twist field determined by the angular momentum of the recoiling D-brane and with explicitly broken kappa-symmetry. We show that the superconformal completion removes the logarithmic modular divergences that are present in the bosonic string loop scattering amplitudes. These features are all consequences of the relationship that exists in these models between worldsheet rescaling and the time evolution of the D-brane in target space. 
  It is shown that the large $N$ limit of SU(N) YM in $curved$ $m$-dim backgrounds can be subsumed by a higher $m+n$ dimensional gravitational theory which can be identified to an $m$-dim generally invariant gauge theory of diffs $N$, where $N$ is an $n$-dim internal space (Cho, Sho, Park, Yoon). Based on these findings, a very plausible geometrical interpretation of the $AdS/CFT$ correspondence could be given. Conformally invariant sigma models in $D=2n$ dimensions with target non-compact SO(2n,1) groups are reviewed. Despite the non-compact nature of the SO(2n,1), the classical action and Hamiltonian are positive definite. Instanton field configurations are found to correspond geometrically to conformal ``stereographic'' mappings of $R^{2n}$ into the Euclidean signature $AdS_{2n}$ spaces. The relation between Self Dual branes and Chern-Simons branes, High Dimensional Knots, follows. A detailed discussion on $W_\infty $ symmetry is given and we outline the Vasiliev procedure to construct an action involving higher spin massless fields in $AdS_4$. This $AdS_4$ spacetime higher spin theory should have a one-to-one correspondence to noncritical $W_\infty$ strings propagating on $AdS_4 \times S^7$. 
  We clarify the role of gauge invariance for the theory of an AdS4 brane embedded in AdS5. The presence of a nonvanishing mass parameter even for the lightest KK mode of the graviton indicates that all of the spin-2 modes propagate five polarization states. Despite this fact, it was shown earlier that the classical theory has a smooth limit as the mass parameter is taken to zero. We argue that locality in the fifth dimension ensures that this property survives at the quantum level. 
  We study D-branes on Calabi-Yau manifolds, carrying charges which are torsion elements of the K-theory. Interesting physics ensues when we follow these branes into nongeometrical phases of the compactification. On the level of K-theory, we determine the monodromies of the group of charges as we circle singular loci in the closed string moduli space. Going beyond K-theory, we discuss the stability of torsion D-branes as a function of the K\"ahler moduli. When the fundamental group of the Calabi-Yau is nonabelian, we find evidence for new threshold bound states of BPS branes. In a two-parameter example, we compare our results with computations in the Gepner model. Our study of the torsion D-branes in the compactification of [FHSV] sheds light on the physics of that model. In particular, we develop a proposal for the group of allowed D-brane charges in the presence of discrete RR fluxes. 
  We consider the dynamics of a viscous cosmological fluid in the generalized Randall-Sundrum model for an isotropic brane. To describe the dissipative effects we use the Israel-Hiscock-Stewart full causal thermodynamic theory. In the limiting case of a stiff cosmological fluid with pressure equal to the energy density, the general solution of the field equations can be obtained in an exact parametric form for a cosmological fluid with constant bulk viscosity and with a bulk viscosity coefficient proportional to the square root of the energy density, respectively. The obtained solutions describe generally non-inflationary brane worlds, starting from a singular state. During this phase of evolution the comoving entropy of the Universe is an increasing function of time, and thus a large amount of entropy is created in the brane world due to viscous dissipative processes. 
  We consider the use of interpolating gauges (with a gauge function (F[A;alpha ]) in gauge theories to connect the results in a set of different gauges in the path-integral formulation. We point out that the results for physical observables are very sensitive to the epsilon term that we have to add to deal with singularities and thus cannot be left out of a discussion of gauge-independence generally. We further point out, with reasons, that the fact that we can ignore this term in the discussion of gauge independence while varying of the gauge parameter in Lorentz-type covariant gauges is an exception rather than a rule . We show that generally gauge-independence requires that the epsilon-term has to be varied with alpha. We further show that if we make a naive use of the epsilon term -i\int d^{4}x[{1/2}A^{2}-\bar{c}c]) (that is appropriate for the Feynman gauge) for general interpolating gauges with arbitrary parameter values [i.e.alpha], we cannot preserve gauge independence [except when we happen to be in the infinitesimal neighborhood of the Lorentz-type gauges]. We show with an explicit example that for such a naive use of an epsilon-term, we develop serious pathology in the path-integral as alpha is/are varied. We point out that correct way to fix the epsilon-term in a path-integral in a non-Lorentz gauge is by connecting the path-integral to the Lorentz-gauge path-integral with correct epsilon-term as has been done using the finite field-dependent BRS transformations in recent years. 
  We examine the stability of ${\rm AdS}_p \times {\rm S}^n \times {\rm S}^{q-n}$. The initial data constructed by De Wolfe et al \cite{Gary} has been carefully analyised and we have confirmed that there is no lower bound for the total mass for $q< 9$. The effective action on ${\rm AdS}_p$ has been derived for dilatonic compactification of the system to describe the non-linear fluctuation of the background space-time. The stability is discussed applying the positive energy theorem to the effective theory on AdS, which again shows the stability for $q \geq 9$. 
  We use a series of reductions, T-dualities and liftings to construct connections between fractional brane solutions in IIA, IIB and M-theory. We find a number of phantom branes that are not supported by the geometry, however materialize upon untwisting and/or Hopf-reduction. 
  The discrete tensorial charges carried by orientifold planes define n-gerbes in space-time. The simplest way to ensure a consistent string compactification is to require these gerbes to be flat. This results in expressions for the local gerbe-holonomies around each orientifold plane, describing its charges. Inverting the procedure and considering all flat gerbes leads to a classification of orientifold configurations. Requiring that the tadpole is cancelled by adding D-branes, we classify all supersymmetric orientifolds on T^k/Z_2 with 2^k O(9-k) planes at the fixed points, for k less or equal to 6. For k=6 these theories organize in orbits of the SL(2,Z) S-duality symmetry of N=4 supersymmetric gauge theories. 
  Critical systems are described by conformal field theories, whose dynamics can be exactly solved in two dimensions. In the presence of a boundary, with the so-called method of images it is possible to study the surface critical behaviour of these systems, and the conformal boundary conditions can be related to the bulk operator content of the theory. After an overview of the basic concepts of bulk and boundary conformal field theory, we present an explicit calculation of some two-point correlation functions in Virasoro minimal models with boundary. In the second part of the thesis, we summarize the known results about perturbed conformal field theories, which describe the dynamics of systems away from the critical point. We concentrate on cases in which the off-critical massive field theory is integrable, with a corresponding purely elastic and factorized scattering theory, focusing our attention on the bootstrap approach. We also present the basic properties of affine Toda field theories, whose S-matrices are closely related to the ones of some perturbed minimal models. In the presence of a boundary, assuming that the boundary conditions are compatible with integrability, the scattering theory is still elastic and factorized. The analytic structure of the reflection matrix encodes the boundary spectrum of the theory, in the light of a bootstrap approach analogous to the bulk one. Starting from two kinds of known reflection amplitudes, we have performed a detailed study of the boundary bound states structure for the three E_{n} affine Toda field theories and for the corresponding perturbed minimal models. 
  We study the topological properties of fuzzy sphere. We show that the topological charge is only defined modulo N+1, that is finite integer quotient Z_{N+1}, where N is a cut-off spin of fuzzy sphere. This periodic structure on topological charges is shown based on the boson realizations of SU(2) algebra, Schwinger vs. Holstein-Primakoff. We argue that this result can have a natural K-theory interpretation and the topological charges on fuzzy sphere can be classified by the twisted K-theory. We also outline how solitons on fuzzy sphere can realize D-brane solitons in the presence of Neveu-Schwarz fivebranes proposed by Harvey and Moore. 
  The generalization of the form factor representation of the 2D Ising model correlation function to the case of the general disposition of correlating spins on a cylinder is given. The magnetic susceptibility on the lattice where one of the dimensions $N$ is finite is calculated in both the para- and ferromagnetic regions of the Ising coupling parameter. The singularity structure in the complex temperature plane and the thermodynamic limit $N\to\infty$ are discussed. 
  Cosmological brane world solutions are found for five-dimensional bulk spacetimes with a scalar field. A supergravity inspired method for obtaining static solutions is combined with a method for finding brane cosmologies with constant bulk energies. This provides a way to generate full (bulk and brane) cosmological solutions to brane worlds with bulk scalar fields. Examples of these solutions, and their cosmological evolution, are discussed. 
  We study the Toda field theory with finite Lie algebras using an extension of the Goulian-Li technique. In this way, we show that, after integrating over the zero mode in the correlation functions of the exponential fields, the resulting correlation function resembles that of a free theory. Furthermore, it is shown that for some ratios of the charges of the exponential fields the four-point correlation functions which contain a degenerate field satisfy the Riemann ordinary differential equation. Using this fact and the crossing symmetry, we derive a set of functional equations for the structure constants of the A_2 Toda field theory. 
  The Dirac equation is not semisimple. We therefore regard it as a contraction of a simpler decontracted theory. The decontracted theory is necessarily purely algebraic and non-local. In one simple model the algebra is a Clifford algebra with 6N generators. The quantum imaginary $\hbar i$ is the contraction of a dynamical variable whose back-reaction provides the Dirac mass. The simplified Dirac equation is exactly Lorentz invariant but its symmetry group is SO(3,3), a decontraction of the Poincare group, and it has a slight but fundamental non-locality beyond that of the usual Dirac equation. On operational grounds the non-locality is ~10^{-25} sec in size and the associated mass is about the Higgs mass.   There is a non-standard small but unique spin-orbit coupling ~1/N, whose observation would be some evidence for the simpler theory. All the fields of the Standard Model call for similar non-local simplification. 
  The inflaton potential in four-dimensional theory is rather arbitrary, and fine-tuning is required generically. By contrast, inflation in the brane world scenario has the interesting feature that the inflaton potential is motivated from higher dimensional gravity, or more generally, from bulk modes or string theory. We emphasize this feature with examples. We also consider the impact on the spectrum of density perturbation from a velocity-dependent potential between branes in the brane inflationary scenario. It is likely that such a potential can have an observable effect on the ratio of tensor to scalar perturbations. 
  We present an integrable Hamiltonian which describes the sinh-Gordon model on the half line coupled to a non-linear oscillator at the boundary. We explain how we apply Sklyanin's formalism to a dynamical reflection matrix to obtain this model. This method can be applied to couple other integrable field theories to dynamical systems at the boundary. We also show how to find the dynamical solution of the quantum reflection equation corresponding to our particular example. 
  The apparent observation of dark energy poses problems for string theory. In de Sitter space, or in quintessence models, one cannot define a gauge-invariant S-matrix. We argue that eternal quintessence does not arise in weakly coupled string theory, but point out that it is difficult to define an $S$-matrix even in the presence of perturbative potentials for the moduli. The solutions of the Fischler-Susskind equations all have Big Bang or Big Crunch Singularities. We believe that an S-matrix (or S-vector) exists in this context but cannot be calculated by purely perturbative methods. We study the possibility of metastable de Sitter vacua in such weakly coupled scenarios, and conclude that the S-matrix of the extreme weak coupling region cannot probe de Sitter physics.  We also consider proposed explanations of the dark energy from the perspective of string theory, and find that most are implausible. We note that it is possible that the axion constitutes both the dark matter and the dark energy. 
  The idea of gauging (i.e. making local) symmetries of a physical system is a central feature of many modern field theories. Usually, one starts with a Lagrangian for some scalar or spinor matter fields, with the Lagrangian being invariant under a global phase symmetry transformation of the matter fields. Making this global phase symmetry local results in the introduction of vector fields. The vector fields can be said to arise as a result of the gauge principle. Here we show that this chain of reasoning can be reversed: by gauging the electric-magnetic dual symmetry of a Lagrangian which originally contains only the vector gauge fields we find that it is necessary to introduce matter fields (scalar fields in our example). In this gauging of the electric-magnetic dual symmetry the traditional roles of the vector fields and the matter fields are interchanged. 
  We construct U(2) BPS monopole superpartner solutions in N=2 non-commutative super Yang-Mills theory. Calculation to the second order in the noncommutative parameter $\theta$ shows that there is no electric quadrupole moment that is expected from the magnetic dipole structure of noncommtative U(2) monopole. This might give an example of the nature of how supersymmetry works not changing between the commutative and noncommutative theories. 
  There have been comments on the starting paper, hep-th/0106074, which point out unclear motivation and definitions on noncommutative momentum introduced. This paper is withdrawn by the author for more clear presentation. 
  Noncommutative algebra in planar quantum mechanics is shown to follow from 't Hooft's recent analysis on dissipation and quantization. The noncommutativity in the coordinates or in the momenta of a charged particle in a magnetic field with an oscillator potential are shown as dual descriptions of the same phenomenon. Finally, noncommutativity in a fluid dynamical model, analogous to the lowest Landau level problem, is discussed. 
  We show that noncommutative electromagnetism and Dirac-Born-Infeld (DBI) theory with scalar fields are SL(2,R) self-dual when noncommutativity is light-like and we are in the slowly varying field approximation. This follows from SL(2,R) self-duality of the commutative DBI Lagrangian and of its zero slope limit that we study in detail.   We study a symmetry of noncommutative static configurations that maps space-noncommutativity into space-time (and light-like) noncommutativity. SL(2,R) duality is thus extended to space-noncommutativity. Via Seiberg-Witten map we study the nontrivial action of this symmetry on commutative DBI theory. In particular space-time noncommutative BPS magnetic monopoles corresponds to commutative BPS type magnetic monopoles with both electric and magnetic B-field background. Energy, charge and tension of these configurations are computed and found in agreement with that of a D1-string D3-brane system. We discuss the dual string-brane configuration. 
  We study brane matter in the ekpyrotic scenario and observe that in order to obtain standard gravity on the visible brane, the tension of the visible brane should be positive. If the sizes of both the fifth dimension and the Calabi-Yau threefold are fixed, the Israel junction conditions do not allow time-dependent brane matter. Relaxing this constraint, it is possible to obtain approximately standard cosmology on the visible brane, with small corrections due to possible time-dependence of the Calabi-Yau threefold. 
  We have examined the deformation of a generic non-Abelian gauge theory obtained by replacing its Lie group by the corresponding quantum group. This deformed gauge theory has more degrees of freedom than the theory from which it is derived. By going over from point particles in the standard theory to solitonic particles in the deformed theory, it is proposed to interpret the new degrees of fredom as descriptive of a non-locality of the deformed theory. It also turns out that the original Lie algebra gets replaced by two dual algebras, one of which lies close to and approaches the original Lie algebra in a correspondence limit, while the second algebra is new and disappears in this same correspondence limit. The exotic field particles associated with the second algebra can be interpreted as quark-like constituents of the solitons, which are themselves described as point particles in the first algebra. These ideas are explored for q-deformed SU(2) and $GL_q(3)$. 
  Recent developments in local quantum physics have led to revolutionary conceptual changes in the thinking about a more intrinsic formulation and in particular about unexpected aspects of localized degrees of freedom. This paradigmatic change is most spectacular in a new rigorous form of ``holography'' and ``transplantation'' as generic properties in QFT beyond the rather special geometric black hole setting in which the geometric manifestations of these properties were first noted. This new setting is also the natural arena for understanding the rich world of ``black hole analogs'' (``dumb holes'' for phonons). The mathematical basis for all this is the extremely powerful Tomita-Takesaki modular theory in operator algebras. The rich consequences of the impressive blend of this theory with physical localization entails among other things the presence of ``fuzzy'' acting infinite dimensional symmetry groups, a spacetime interpretation and derivation of the d=1+1 Zamolodchikov-Faddeev algebra (i.e. a better understanding of the bootstrap-formfactor approach) and the noncommutative multiparticle structure of ``free'' anyons based on the use of Wigner representation theory. 
  We construct cosmological solutions of four-dimensional effective heterotic M-theory with a moving five-brane and evolving dilaton and T modulus. It is shown that the five-brane generates a transition between two asymptotic rolling-radii solutions. Moreover, the five-brane motion always drives the solutions towards strong coupling asymptotically. We present an explicit example of a negative-time branch solution which ends in a brane collision accompanied by a small-instanton transition. The five-dimensional origin of some of our solutions is also discussed. 
  We work on the relation between the local thermodynamic instability and the dynamical instability of large black holes in four-dimensional anti-de Sitter space proposed by Gubser and Mitra. We find that all perturbations suppressing the metric fluctuations at linear order become dynamically unstable when black holes lose the local thermodynamic stability. We discuss how dynamical instabilities can be explained by the Second Law of Thermodynamics. 
  Affine su(3) and su(4) fusion multiplicities are characterised as discretised volumes of certain convex polytopes. The volumes are measured explicitly, resulting in multiple sum formulas. These are the first polytope-volume formulas for higher-rank fusion multiplicities. The associated threshold levels are also discussed. For any simple Lie algebra we derive an upper bound on the threshold levels using a refined version of the Gepner-Witten depth rule. 
  For every ADE Dynkin diagram, we give a realization, in terms of usual fusion algebras (graph algebras), of the algebra of quantum symmetries described by the associated Ocneanu graph. We give explicitly, in each case, the list of the corresponding twisted partition functions 
  In this short note we construct the DLCQ description of the flux seven-branes in type IIA string theory and discuss its basic properties. The matrix model involves dipole fields. We explain the relation of this nonlocal matrix model to various orbifolds. We also give a spacetime interpretation of the Seiberg-Witten-like map, proposed in a different context first by Bergman and Ganor, that converts this matrix model to a local, highly nonlinear theory. 
  The similarity of the commutation relations for bosons and quasibosons (fermion pairs) suggests the possibility that all integral spin particles presently considered to be bosons could be quasibosons. The boson commutation relations for integral spin particles could be just an approximation to the quasiboson commutation relations that contain an extra term. Although the commutation relation for quasibosons are slightly more complex, it is simpler picture of matter in that only fermions and composite particles formed of fermions exist. Mesons are usually referred to as bosons, but they must be quasibosons since their internal structure is fermion (quark) pairs. The photon is usually considered to be an elementary boson, but as shown here, existing experiments do not rule out the possibility that it is also a quasiboson. We consider how the quasiboson, composite nature of such a photon might manifest itself. 
  The form of the most general orbifold breaking of gauge, global and supersymmetries with a single extra dimension is given. In certain theories the Higgs boson mass is ultraviolet finite due to an unbroken local supersymmetry, which is explicitly exhibited. We construct: a 1 parameter SU(3) \times SU(2) \times U(1) theory with 1 bulk Higgs hypermultiplet, a 2 parameter SU(3) \times SU(2) \times U(1) theory with 2 bulk Higgs hypermultiplets, and a 2 parameter SU(5) \to SU(3) \times SU(2) \times U(1) theory with 2 bulk Higgs hypermultiplets, and demonstrate that these theories are unique. We compute the Higgs mass and compactification scale in the SU(3) \times SU(2) \times U(1) theory with 1 bulk Higgs hypermultiplet. 
  The high temperature asymptotics of the Helmholtz free energy of electromagnetic field subjected to boundary conditions with spherical and cylindrical symmetries are constructed by making use of a general expansion in terms of heat kernel coefficients and the related determinant. For this, some new heat kernel coefficients and determinants had to be calculated for the boundary conditions under consideration. The obtained results reproduce all the asymptotics derived by other methods in the problems at hand and involve a few new terms in the high temperature expansions. An obvious merit of this approach is its universality and applicability to any boundary value problem correctly formulated. 
  We present here a manifestly gauge invariant calculation of vacuum polarization to fermions in the presence of a constant Maxwell and a constant Kalb-Ramond field in four dimensions. The formalism is a generalisation of the one used by Schwinger in his famous paper on gauge invariance and vacuum polarization. We get an explicit expression for the vacuum polarization induced effective Lagrangian for a constant Maxwell field interacting with a constant Kalb-Ramond field. In the weak field limit we get the coupling between the Maxwell field and the Kalb-Ramond field to be $(\tilde{H}.\tilde{F})^2$, where ${\tilde H}_{\mu}= {1\over {3!}}\epsilon_{\mu\alpha\beta\lambda}H^{\alpha\beta\lambda}$ and $\tilde F$ is the dual of $F_{\mu\nu}$. 
  We study a matrix version of the purely cubic open string field theory as describing the expansion around the closed string vacuum. Any D-branes in the given closed string background can appear as classical solutions by using the identity projectors. Expansion around this solution gives the correct kinetic term for the open strings on the created D-branes while there are some subtleties in the unwanted degree of freedom. 
  We consider an involutive automorphism of the conformal algebra and the resulting symmetric space. We display a new action of the conformal group which gives rise to this space. The space has an intrinsic symplectic structure, a group-invariant metric and connection, and serves as the model space for a new conformal gauge theory. 
  The quest for unification of particles and fields and for reconciliation of Quantum Mechanics and General Relativity has led us to gauge theories, string theories, supersymmetry and higher-extended objects: membranes... Our spacetime is quantum mechanical but it admits semiclassical descriptions of various ``complementary'' kinds that could be valid approximations in various circumstances. One of them might be supergravity in 11 dimensions the largest known interacting theory of a finite number of fields with gauged Poincar\' e supersymmetry. Its solitons and their dual membranes would be states in its quantum version called M-theory. We shall review the construction of its classical action by deformation of a globally supersymmetric free theory and its on-shell superspace formulation. Then we shall focus on the bosonic matter equations of the dimensional reductions on tori of dimensions 1 to 8 to exhibit their common self-duality nature. In the concluding section we shall discuss possible remnants at the quantum level and beyond the massless sector of generalised discrete U-dualities. We shall also comment on the variable dimension of spacetime descriptions and on the possibility of extending the self dual description to spacetime itself and its metric. 
  There have been comments on the starting paper, hep-th/0106074, which point out unclear motivation and definitions on noncommutative momentum introduced. This paper is withdrawn by the author for more clear presentation. 
  We show that in a Randall-Sundrum II type braneworld, the vacuum exterior of a spherical star is not in general a Schwarzschild spacetime, but has radiative-type stresses induced by 5-dimensional graviton effects. Standard matching conditions do not lead to a unique exterior on the brane because of these 5-dimensional graviton effects. We find an exact uniform-density stellar solution on the brane, and show that the general relativity upper bound $GM/R<{4\over9}$ is reduced by 5-dimensional high-energy effects. The existence of neutron stars leads to a constraint on the brane tension that is stronger than the big bang nucleosynthesis constraint, but weaker than the Newton-law experimental constraint. We present two different non-Schwarzschild exteriors that match the uniform-density star on the brane, and we give a uniqueness conjecture for the full 5-dimensional problem. 
  In harmonic superspace, the classical equations of motion of $D=4, N=2$ supersymmetric Yang-Mills theory for Minkowski and Euclidean spaces are analyzed. We study dual superfield representations of equations and subsidiary conditions corresponding to classical SYM-solutions with different symmetries. In particular, alternative superfield constructions of self-dual and static solutions are described in the framework of the harmonic approach. 
  The loop representation formulation of non-relativistic particles coupled with abelian gauge fields is studied. Both Maxwell and Chern-Simons interactions are separately considered. It is found that the loop-space formulations of these models share significant similarities, although in the Chern-Simons case there exists an unitary transformation that allows to remove the degrees of freedom associated with the paths. The existence of this transformation, which allows to make contact with the anyonic interpretation of the model, is subjected to the fact that the charge of the particles be quantized. On the other hand, in the Maxwell case, we find that charge quantization is necessary in order to the geometric representation be consistent. 
  We perform the consistent quantization of open string D-brane in non-constant  NS-NS closed string B field background by directly imposing the worldsheet conformal symmetry. In addition to the previous noncommutative D-brane coordinates quantization with constant noncommutative parameter $\theta $, we obtain a set of constraints on the B field which can be interpreted as a quantum consistent deformation of a curved noncommutative D-brane. 
  By representing the field content as well as the particle creation operators in terms of fermionic Fock operators, we compute the corresponding matrix elements of the Federbush model. Only when these matrix elements satisfy the form factor consistency equations involving anyonic factors of local commutativity, the corresponding operators are local. We carry out the ultraviolet limit, analyze the momentum space cluster properties and demonstrate how the Federbush model can be obtained from the $SU(3)_3$-homogeneous sine-Gordon model. We propose a new Lagrangian which on one hand constitutes a generalization of the Federbush model in a Lie algebraic fashion and on the other a certain limit of the homogeneous sine-Gordon models. 
  We consider the two complex dimensional moduli space of supersymmetric vacua for low energy effective N=2 SYM with gauge group SU(3). We describe, at the topological level, a consistent model of how the relevant curves of marginal stability (CMS) intertwine with the branch cuts to partition the moduli space into pieces carrying different BPS spectra. At strong coupling we find connected cores which carry a smaller BPS spectrum than that at weak coupling. At the strongest coupling we find double cores which carry a finite BPS spectrum. These include not only states one can deduce from the monodromy group, but three states, bounded away from weak coupling, each of which we interpret as a bound state of two BPS gauge bosons. We find new BPS states at weak coupling corresponding to a excitations of a state with magnetic charge a simple co-root, with respect to the other simple root direction. 
  The free energy of the Ginzburg-Landau theory satisfies a nonlinear functional differential equation which is turned into a recursion relation. The latter is solved graphically order by order in the loop expansion to find all connected vacuum diagrams, and their corresponding weights. In this way we determine the connected vacuum diagrams and their weights up to four loops. 
  We study localization properties of various bulk fields on a dilatonic p-brane which is delocalized along its transverse directions except one. We find that all the bosonic and fermionic bulk fields can be localized on the delocalized dilatonic p-brane in a strict sense, namely the Kaluza-Klein zero modes of the bulk fields are normalizable and are localized around the brane, for any values of the dilaton coupling parameter. 
  The mathematical formalism necessary for the diagramatic evaluation of quantum corrections to a conformally invariant field theory for a self-interacting scalar field on a curved manifold with boundary is considered. The evaluation of quantum corrections to the effective action past one-loop necessitates diagramatic techniques. Diagramatic evaluations and higher loop-order renormalisation can be best accomplished on a Riemannian manifold of constant curvature accommodating a boundary of constant extrinsic curvature. In such a context the stated evaluations can be accomplished through a consistent interpretation of the Feynman rules within the spherical formulation of the theory for which the method of images allows. To this effect, the mathematical consequences of such an interpretation are analyzed and the spherical formulation of the Feynman rules on the bounded manifold is, as a result, developed. 
  We study a proper-time renormalisation group, which is based on an operator cut-off regularisation of the one-loop effective action. The predictive power of this approach is constrained because the flow is not an exact one. We compare it to the Exact Renormalisation Group, which is based on a momentum regulator in the Wilsonian sense. In contrast to the former, the latter provides an exact flow. To leading order in a derivative expansion, an explicit map from the exact to the proper-time renormalisation group is established. The opposite map does not exist in general. We discuss various implications of these findings, in particular in view of the predictive power of the proper-time renormalisation group. As an application, we compute critical exponents for O(N)-symmetric scalar theories at the Wilson-Fisher fixed point in 3d from both formalisms. 
  The character of quantum corrections to the gravitational action of a conformally invariant field theory for a self-interacting scalar field on a manifold with boundary is considered at third loop-order in the perturbative expansion of the zero-point function. Diagramatic evaluations and higher loop-order renormalisation can be best accomplished on a Riemannian manifold of constant curvature accommodating a boundary of constant extrinsic curvature. The associated spherical formulation for diagramatic evaluations reveals a non-trivial effect which the topology of the manifold has on the vacuum processes and which ultimately dissociates the dynamical behaviour of the quantised field from its behaviour in the absence of a boundary. The first surface divergence is evaluated and the necessity for simultaneous renormalisation of volume and surface divergences is shown. 
  We introduce the concept of superfield effective action in noncommutative N=1 supersymmetric field theories containing chiral superfields. One and two loops low-energy contributions to the effective action are found for the noncommutative Wess-Zumino model. The one loop Kahlerian effective potential coincides with its commutative counterpart. We show that the two loops nonplanar contributions to the Kahlerian effective potential are leading in the case of small noncommutativity. The structure of the leading chiral corrections to the effective action and the behaviour of the chiral effective potential in the limit of large noncommutativity are also investigated. 
  We consider the vacuum energy of the electromagnetic field in the background of spherically symmetric dielectrics, subject to a cut-off frequency in the dispersion relations. The effect of this frequency dependent boundary condition between media is described in terms of the {\it incomplete} $\zeta$-functions of the problem. The use of the Debye asymptotic expansion for Bessel functions allows to determine the dominant (volume, area, ...) terms in the Casimir energy. The application of these expressions to the case of a gas bubble immersed in water is discussed, and results consistent with Schwinger's proposal about the role the Casimir energy plays in sonoluminescence are found.   PACS: 03.70.+k,12.20.Ds,78.60.Mq 
  The high temperature asymptotics of thermodynamic functions of electromagnetic field subjected to boundary conditions with spherical and cylindrical symmetries are constructed by making use of a general expansion in terms of heat kernel coefficients and the related determinant. For this, some new heat kernel coefficients and determinants had to be calculated for the boundary conditions under consideration. The obtained results reproduce all the asymptotics derived by other methods in the problems at hand and involve a few new terms in the high temperature expansions. An obvious merit of this approach is its universality and applicability to any boundary value problem correctly formulated. 
  Three types of thick branes, i.e., Poincar\'{e}, de Sitter and Anti-de Sitter brane are considered. They are realized as the non-singular solutions of the Einstein equations with the non-trivial dilatons and the potentials. The scalar perturbations of these systems are also investigated. We find that the effective potentials of the master equations of the scalar perturbations are positive definite and consequently these systems are stable under the small perturbations. 
  We calculate the self-energy and self-force for an electrically charged particle at rest in the background of Gott-Hiscock cosmic string space-time. We found the general expression for the self-energy which is expressed in terms of the $S$ matrix of the scattering problem. The self-energy continuously falls down outward from the string's center with maximum at the origin of the string. The self-force is repulsive for an arbitrary position of the particle. It tends to zero in the string's center and also far from the string and it has a maximum value at the string's surface. The plots of the numerical calculations of the self-energy and self-force are shown. 
  The temperature phase transition in the N-component scalar field theory with spontaneous symmetry breaking is investigated in the perturbative approach. The second Legendre transform is used together with the consideration of the gap equations in the extrema of the free energy. Resummations are performed on the super daisy level and beyond. The phase transition turns out to be weakly of first order. The diagrams beyond the super daisy ones which are calculated correspond to next-to-next-to-leading order in 1/N. It is shown that these diagrams do not alter the phase transition qualitatively. In the limit N goes to infinity the phase transition becomes second order. A comparison with other approaches is done. 
  The one-instanton contribution to a circular BPS Wilson loop in N=4 SU(2) Yang--Mills theory is evaluated in semiclassical approximation. This article amplifies part of a talk given by MBG at the Strings 2001 conference, Mumbai, India (January 5-10, 2001). The results are preliminary and a more complete exposition will be contained in a forthcoming paper. 
  The dual theory describing the 4D Coulomb gas of point-like magnetically charged objects, which confines closed electric strings, is considered. The respective generalization of the theory of confining strings to confining membranes is further constructed. The same is done for the analogous SU(3)-inspired model. We then consider a combined model which confines both electric charges and closed strings. Such a model is nothing, but the mixture of the above-mentioned Coulomb gas with the condensate of the dual Higgs field, described by the dual Abelian Higgs model. It is demonstrated that in a certain limit of this dual Abelian Higgs model, the system under study undergoes naively the dimensional reduction and becomes described by the (completely integrable) 2D sine-Gordon theory. In particular, owing to this fact, the phase transition in such a model must be of the Berezinskii-Kosterlitz-Thouless type, and the respective critical temperature is expressed in terms of the parameters of the dual Abelian Higgs model. However, it is finally discussed that the dimensional reduction is rigorously valid only in the strong coupling limit of the original 4D Coulomb gas. In such a limit, this reduction transforms the combined model to the 2D free bosonic theory. 
  We obtain the binding energy of an infinitely heavy quark-antiquark pair from Dirac brackets by computing the expectation value of the pure QCD Hamiltonian. This procedure exploits the rich structure of the dressing around static fermions. Some subtle points related to exhibing explicitly the interquark energy are considered. 
  We analyse some dynamical issues in a modified type IIA supergravity, recently proposed as an extension of M-theory that admits de Sitter space. In particular we find that this theory has multiple zero-brane solutions. This suggests a microscopic quantum mechanical matrix description which yields a massive deformation of the usual M(atrix) formulation of M-theory and type IIA string theory. 
  In this thesis we study two different approaches to holography, and comment on the possible relation between them. The first approach is an analysis of the high-energy regime of quantum gravity in the eikonal approximation, where the theory reduces to a topological field theory. This is the regime where particles interact at high energies but with small momentum transfer. We do this for the cases of asymptotically dS and AdS geometries and find that in both cases the theory is topological. We discuss the relation of our solutions in AdS to those of Horowitz and Itzhaki. We also consider quantum gravity away from the extreme eikonal limit and explain the sense in which the covariance of the theory is equivalent to taking into account transfer of momentum. The second approach we pursue is the AdS/CFT correspondence. We provide a holographic reconstruction of the bulk space-time metric and of bulk fields on this space-time, out of conformal field theory data. Knowing which sources are turned on is sufficient in order to obtain an asymptotic expansion of the bulk metric and of bulk fields near the boundary to high enough order so that all infrared divergences of the on-shell action are obtained. We provide explicit formulae for the holographic stress-energy tensors associated with an arbitrary asymptotically AdS geometry. We also study warped compactifications, where our d-dimensional world is regarded as a slice of a (d+1)-dimensional space-time, and analyse in detail the question as to where the d-dimensional observer can find the information about the extra dimension. 
  We show that it is possible to construct a quantum field theory that is invariant under the translation of the noncommutative parameter $\theta_{\mu\nu}$. This is realized in a noncommutative cohomological field theory. As an example, a noncommutative cohomological scalar field theory is constructed, and its partition function is calculated. The partition function is the Euler number of Gopakumar, Minwalla and Strominger (GMS) soliton space. 
  We discuss brane world sum rules for compactifications involving an arbitrary number of spacetime dimensions. One of the most striking results derived from such consistency conditions is the necessity for negative tension branes to appear in five-dimensional scenarios. We show how this result can be evaded for brane world models with more than five dimensions. As an example, we consider a novel realization of the Randall-Sundrum scenario in six dimensions involving only positive tension branes. A complete account of our results appeared in hep-th/0106140. 
  Boundary conditions and gluing conditions for open strings and D-branes in the $SL(2,R)$ WZWN model, corresponding to $AdS_3$, are discussed. Some boundary conditions and gluing conditions previously considered in the literature are shown to be incompatible with the variation principle.   We then consider open string boundary conditions corresponding to a certain {\it field-dependent} gluing condition. This allows us to consider open strings with constant energy and angular momentum. Classically, these open strings naturally generalize the open strings in flat Minkowski space. For rigidly rotating open strings, we show that the torsion leads to a bending and an unfolding. We also derive the $SL(2,R)$ Regge relation, which generalizes the linear Minkowski Regge relation. For "high" mass, it takes the form $L\approx \pm M/H$, where $H$ is the scale of the $SL(2,R)$ group manifold. 
  We elucidate some properties of the relation between two T-dual systems in tori, branes at angles and branes wrapping the whole torus carrying fluxes. We analyze different features of these systems: charges, low energy spectrum, tadpole cancellation, symmetry groups, ... and the correspondence between the two viewpoints. Particular attention is paid to supersymmetry and stability conditions. While on the branes at angles side stability and supersymmetry can be expressed as conditions on the angles between the two branes at the intersection, on the dual side supersymmetry has to do with a correction to Hermite Yang-Mills and a modified notion of stability should be considered. 
  Studying the general structure of the noncommutative (NC) local groups, we prove a no-go theorem for NC gauge theories. According to this theorem, the closure condition of the gauge algebra implies that: 1) the local NC $u(n)$ {\it algebra} only admits the irreducible n by n matrix-representation. Hence the gauge fields are in n by n matrix form, while the matter fields {\it can only be} in fundamental, adjoint or singlet states; 2) for any gauge group consisting of several simple-group factors, the matter fields can transform nontrivially under {\it at most two} NC group factors. In other words, the matter fields cannot carry more than two NC gauge group charges. This no-go theorem imposes strong restrictions on the NC version of the Standard Model and in resolving the standing problem of charge quantization in noncommutative QED. 
  We investigate the analogue of S-duality for 2-d gravity. Our analysis is based in a partition function associated to the Katanaev-Volovich type action for 2-d gravity. We find a S-dual 2-d gravitational action which is related to the original 2-d gravitational action by strong-weak duality transformation. 
  We derive finite boost transformations based on the Lorentz sector of the bicross-product-basis $\kappa$-Poincare' Hopf albegra. We emphasize the role of these boost transformations in a recently-proposed new relativistic theory. We find that when the (dimensionful) deformation parameter is identified with the Planck length, which together with the speed-of-light constant has the status of observer-independent scale in the new relativistic theory, the deformed boosts saturate at the value of momentum that corresponds to the inverse of the Planck length. 
  Totally symmetric massless fermionic fields of arbitrary spins in AdS(5) are described as su(2,2) multispinors. The approach is based on the well-known isomorfism o(4,2)=su(2,2). Explicitly gauge invariant higher spin free actions are constructed and free field equations are analyzed. 
  The problem of the M5 brane anomaly cancellation is addressed. We reformulate FHMM construction making explicit the relation with the M5 brane SUGRA solution. We suggest another solution to the magnetic coupling equation which doesn't need anomalous SO(5) variation of the 3-form potential and coincides with the SUGRA solution outside smoothed out core of the magnetic source. Chern-Simons term evaluated on this solution generates the same anomaly inflow as achieved by FHMM. 
  The presence of a cosmological constant, Lambda, in an action with higher powers of the curvature can produce rapidly oscillating metrics. We develop a perturbative approach for generating periodic solutions to the non-linear field equations for such actions based on a small amplitude expansion. We find that these oscillations have an amplitude proportional to \sqrt{\Lambda} and a frequency of order the Planck mass. In a 4+1 dimensional scenario, a family of metrics exists that are periodic in the extra dimension and are parameterized by an effective four-dimensional cosmological constant which drives a rapid oscillation. 
  In this paper we review how to reconstruct scalar field theories in two dimensional spacetime starting from solvable Schrodinger equations. Three different Schrodinger potentials are analyzed. We obtained two new models starting from the Morse and Scarf II hyperbolic potentials, {\it i.e}, the $U(\phi)=\phi^2\ln^2(\phi^2)$ model and $U(\phi)=\phi^2\cos^2(\ln(\phi^2))$ model respectively. 
  We consider warped compactifications of ${\cal M}$-theory to three-dimensional Minkowski space on compact eight-manifolds. Taking all the leading quantum gravity corrections of eleven-dimensional supergravity into account we obtain the solution to the equations of motion and Bianchi identities. Generically these vacua are not supersymmetric and yet have a vanishing three-dimensional cosmological constant. 
  The adiabatic invariant nature of black hole horizon area in classical gravity suggests that in quantum theory the corresponding operator has a discrete spectrum. I here develop further an algebraic approach to black hole quantization which starts from very elementary assumptions, and proceeds by exploiting symmetry. It predicts a uniformly spaced area spectrum for all charges and angular momenta. Area eigenvalues are degenerate; correspondence with black hole entropy then dictates a precise value for the interval between eigenvalues. 
  We construct exact classical solutions in cubic open string field theory. By the redefinition of the string field, we find that the solutions correspond to finite deformations of the Wilson lines. The solutions have well-defined Fock space expressions, and they have no branch cut singularity of marginal parameters which was found in the analysis using level truncation approximation in Feynman-Siegel gauge. We also discuss marginal tachyon lump solutions at critical radius. 
  We study the diagonalization problem of certain discrete quantum integrable models by the method of Baxter's T-Q relation from the algebraic geometry aspect. Among those the Hofstadter type model (with the rational magnetic flux), discrete quantum pendulum and discrete sine-Gordon model are our main concern in this report. By the quantum inverse scattering method, the Baxter's T-Q relation is formulated on the associated spectral curve, a high genus Riemann surface in general, arisen from the study of the spectrum problem of the system. In the case of degenerated spectral curve where the spectral variables lie on rational curves, we obtain the complete and explicit solution of the T-Q polynomial equation associated to the model, and the intimate relation between the Baxter's T-Q relation and algebraic Bethe Ansatz is clearly revealed. The algebraic geometry of a general spectral curve attached to the model and certain qualitative properties of solutions of the Baxter's T-Q relation are discussed incorporating the physical consideration. 
  N-fold supersymmetry is an extension of the ordinary supersymmetry in one-dimensional quantum mechanics. One of its major property is quasi-solvability, which means that energy eigenvalues can be obtained for a portion of the spectra. We show that recently found Type A N-fold supersymmetry can be constructed by using sl(2) algebra, which provides a basis for the quasi-solvability. By this construction we find a condition for the Type A N-fold supersymmetry, which is less restrictive than the condition known previously. Several explicitly known models are also examined in the light of this construction. 
  We consider the two-dimensional "Schwarzschild" and "Reissner-Nordstrom" stringy black holes as systems of Casimir type. We explicitly calculate the energy-momentum tensor of a massless scalar field satisfying Dirichlet boundary conditions on two one-dimensional "walls". These results are obtained using the Wald's axioms. Thermodynamical quantities such as pressure, specific heat, isothermal compressibility and entropy of the two-dimensional stringy black holes are calculated. A comparison is made between the obtained results and the laws of thermodynamics. The results obtained for the extremal (Q=M) stringy two-dimensional charged black hole are identical in all three different vacua used; a fact that indicates its quantum stability. 
  Quantum monodromy matrices coming from a theory of two coupled (m)KdV equations are modified in order to satisfy the usual Yang-Baxter relation. As a consequence, a general connection between braided and {\it unbraided} (usual) Yang-Baxter algebras is derived and also analysed. 
  We propose a six-dimensional framework to calculate the supersymmetric index of M-theory 5-branes wrapped on a six-manifold with product topology $M_4\times T^2$, where $M_4$ is a holomorphic 4-cycle in a Calabi-Yau three-fold. This is obtained by zero-modes counting of the self-dual tensor contribution plus ``little'' string states and correctly reproduces the known results which can be obtained by shrinking or blowing the $T^2$ volume parameter. We also extract the geometric moduli space of the multi M5-brane system and infer the generic structure of the supersymmetric index for more general geometries. 
  I discuss the recent work done in collaboration with Chris Hill, Jing Wang and Hsin-Chia Cheng. We construct four-dimensional renormalizable gauge theories which, in their infrared limit, generate the dynamics of gauge interactions in flat extra dimensions. In this construction, the Kaluza-Klein states of extra dimensions arise from having many gauge symmetries in four dimensions. The IR limit is determined by the dynamics of spontaneous breakdown of those symmetries. 
  We discuss the behavior of the Calogero model and the related model of deformed oscillators with the S_N extended Heisenberg algebra for a special value of the constant of interaction/statistical parameter nu. The problem with finite number of deformed oscillators is analyzed in the algebraic approach, while collective-field theory has been used to investigate the large-N limit. In this limit, system reduces to a large number of collapsing (free) particles, for nu=-1/N. 
  Extending the commutator algebra of quantum $\kappa$-Poincar\'e symmetry to the whole of the phase space, and assuming that this algebra is to be covariant under action of deformed Lorentz generators, we derive the transformation properties of positions under the action of deformed boosts. It turns out that these transformations leave invariant the quadratic form in the position space, which is the Minkowski metric and that the boosts saturate. The issues of massless and massive particles motion, as well as time dilatation and length contraction in this new framework are also studied. 
  A noncommutative version of the usual electro-weak theory is constructed. We discuss how to overcome the two major problems: 1) although we can have noncommutative U(n) (which we denote by $U_{\star}(n)$) gauge theory we cannot have noncommutative SU(n) and 2) the charges in noncommutative QED are quantized to just $0, \pm 1$. We show how the problem with charge quantization, as well as with the gauge group, can be resolved by taking $U_{\star}(3)\times U_{\star}(2)\times U_{\star}(1)$ gauge group and reducing the extra U(1) factors in an appropriate way. Then we proceed with building the noncommutative version of the standard model by specifying the proper representations for the entire particle content of the theory, the gauge bosons, the fermions and Higgs. We also present the full action for the noncommutative Standard Model (NCSM). In addition, among several peculiar features of our model, we address the {\it inherent} CP violation and new neutrino interactions. 
  The negative symmetry flows are incorporated into the Riemann-Hilbert problem for the homogeneous $A_m$-hierarchy and its $\hat{gl} (m+1, C)$ extension. A loop group automorphism of order two is used to define a sub-hierarchy of $\hat{gl} (m+1, C)$ hierarchy containing only the odd symmetry flows. The positive and negative flows of the $\pm 1$ grade coincide with equations of the multidimensional Toda model and of topological-anti-topological fusion. 
  We study a bound state of fractional D3/D7-branes in the ten-dimensional space R^{1,5}*R^{4}/Z_2 using the boundary state formalism. We construct the boundary actions for this system and show that higher order terms in the twisted fields are needed in order to satisfy the zero-force condition. We then find the classical background associated to the bound state and show that the gauge theory living on a probe fractional D3-brane correctly reproduces the perturbative behavior of a four-dimensional N=2 supersymmetric gauge theory with fundamental matter. 
  The quon algebra describes particles, ``quons,'' that are neither fermions nor bosons, using a label $q$ that parametrizes a smooth interpolation between bosons ($q = 1$) and fermions ($q = -1$). Understanding the relation of quons on the one side and bosons or fermions on the other can shed light on the different properties of these two kinds of operators and the statistics which they carry. In particular, local bilinear observables can be constructed from bosons and fermions, but not from quons. In this paper we construct bosons and fermions from quon operators. For bosons, our construction works for $-1 \leq q \leq 1$. The case $q=-1$ is paradoxical, since that case makes a boson out of fermions, which would seem to be impossible. None the less, when the limit $q \to -1$ is taken from above, the construction works. For fermions, the analogous construction works for $-1 \leq q \leq 1$, which includes the paradoxical case $q=1$. 
  The mechanism by which gauge and gravitational anomalies cancel in certain string theories is reviewed. The presentation is aimed at theorists who do not necessarily specialize in string theory. 
  The stability of the shell of wrapped D6-branes on K3 is investigated from the point of view of supergravity. We first construct an effective energy-momentum tensor for the shell under the reasonable conditions and show that supersymmetric solutions satisfy Israel's junction conditions at arbitrary radius of the shell. Next we study the perturbation of the whole system including the self-gravity of the shell. It is found that in spite of the existence of wrapped D6-branes with negative tension, there is no eigenmode whose frequencies of the shell and the fields are imaginary numbers, at any radius of the shell. Furthermore, when the radius of the shell is less than the enhan\c{c}on radius, resonances are produced, and this indicates a kind of ``instability'' of the system. This can even classically explain why the shell is constructed at the enhan\c{c}on radius. 
  We construct an effective action describing an elementary M5-brane interacting with dynamical eleven-dimensional supergravity, which is free from gravitational anomalies. The current associated to the elementary brane is taken as a distribution valued delta-function on the support of the 5-brane itself. Crucial ingredients of the construction are the consistent inclusion of the dynamics of the chiral two-form on the 5-brane, and the use of an invariant Chern-kernel allowing to introduce a D=11 three-form potential which is well defined on the worldvolume of the 5-brane. 
  There have been comments on the starting paper, hep-th/0106074, which point out unclear motivation and definitions on noncommutative momentum introduced. This paper is withdrawn by the author for more clear presentation. 
  We discuss graded D-brane systems of the topological A model on a Calabi-Yau threefold, by means of their string field theory. We give a detailed analysis of the extended string field action, showing that it satisfies the classical master equation, and construct the associated BV system. The analysis is entirely general and it applies to any collection of D-branes (of distinct grades) wrapping the same special Lagrangian cycle, being valid in arbitrary topology. Our discussion employs a $\Z$-graded version of the covariant BV formalism, whose formulation involves the concept of {\em graded supermanifolds}. We discuss this formalism in detail and explain why $\Z$-graded supermanifolds are necessary for a correct geometric understanding of BV systems. For the particular case of graded D-brane pairs, we also give a direct construction of the master action, finding complete agreement with the abstract formalism. We analyze formation of acyclic composites and show that, under certain topological assumptions,all states resulting from the condensation process of a pair of branes with grades differing by one unit are BRST trivial and thus the composite can be viewed as a closed string vacuum. We prove that there are {\em six} types of pairs which must be viewed as generally inequivalent. This contradicts the assumption that `brane-antibrane' systems exhaust the nontrivial dynamics of topological A-branes with the same geometric support. 
  The method of refined algebraic quantization of constrained systems which is based on modification of the inner product of the theory rather than on imposing constraints on the physical states is generalized to the case of constrained systems with structure functions and open gauge algebras. A new prescription for inner product for the open-algebra systems is suggested. It is illustrated on a simple example. The correspondence between refined algebraic and BRST-BFV quantizations is investigated for the case of nontrivial structure functions. 
  We discuss the possibility of quintessence in the dilatonic domain walls including the Randall-Sundrum brane world. We obtain the zero mode effective action for gravitating objects in the dilatonic domain wall. First we consider the four dimensional (4D) gravity and the Brans-Dicke graviscalar with a potential. This can be further rewritten as a minimally coupled scalar with the Liouville-type potential in the Einstein frame. However this model fails to induce the quintessence on the dilatonic domain wall because the potential is negative. Second we consider the 4D gravity with the dilaton. In this case we find also a negative potential. Any negative potential gives us negative energy density and positive pressure, which does not lead to an accelerating universe. Consequently it turns out that the zero mode approach of the dilatonic domain wall cannot accommodate the quintessence in cosmology. 
  For the so-called source-probe configuration in Matrix theory, we prove the following theorem concerning the power of supersymmetry (SUSY): Let $\delta$ be a quantum-corrected effective SUSY transformation operator expandable in powers of the coupling constant $g$ as $\delta = \sum_{n\ge 0} g^{2n} \delta^{(n)}$, where $\delta^{(0)}$ is of the tree-level form. Then, apart from an overall constant, the SUSY Ward identity $\delta \Gamma=0$ determines the off-shell effective action $\Gamma$ uniquely to arbitrary order of perturbation theory, provided that the $ SO(9)$ symmetry is preserved. Our proof depends only on the properties of the tree-level SUSY transformation laws and does not require the detailed knowledge of quantum corrections. 
  Using simple algebraic methods along with an analogy to the BFSS model, we explore the possible (target) spacetime symmetries that may appear in a matrix description of de Sitter gravity. Such symmetry groups could arise in two ways, one from an ``IMF'' like construction and the other from a ``DLCQ'' like construction. In contrast to the flat space case, we show that the two constructions will lead to different groups, i.e. the Newton-Hooke group and the inhomogeneous Euclidean group (or its algebraic cousins). It is argued that matrix quantum mechanics based on the former symmetries look more plausible. Then, after giving a detailed description of the relevant one particle dynamics, a concrete Newton-Hooke matrix model is proposed. The model naturally incorporates issues such as holography, UV-IR relations, and fuzziness, for gravity in $dS_{4}$. We also provide evidence to support a possible phase transition. The lower temperature phase, which corresponds to gravity in the perturbative regime, has a Hilbert space of infinite dimension. In the higher temperature phase where the perturbation theory breaks down, the dimension of the Hilbert space may become finite. 
  We present a series of instanton-like solutions to a matrix model which satisfy a self-duality condition and possess an action whose value is, to within a fixed constant factor, an integer l^2. For small values of the dimension n^2 of the matrix algebra the integer resembles the result of a quantization condition but as n -> \infty the ratio l/n can tend to an arbitrary real number between zero and one. 
  Yang--Mills theory in four dimensions is studied by using the Coulomb gauge. The Coulomb gauge Hamiltonian involves integration of matrix elements of an operator P built from the Laplacian and from a first-order differential operator. The operator P is studied from the point of view of spectral theory of pseudo-differential operators on compact Riemannian manifolds, both when self-adjointness holds and when it is not fulfilled. In both cases, well-defined matrix elements of P are evaluated as a first step towards the more difficult problems of quantized Yang--Mills theory. 
  We study tachyon condensation in a baby version of Witten's open string field theory. For some special values of one of the parameters of the model, we are able to obtain closed form expressions for the stable vacuum state and for the value of the potential at the minimum. We study the convergence rate of the level truncation method and compare our exact results with the numerical results found in the full string field theory. 
  In a {\cal N}=1 superspace setup and using dimensional regularization, we give a general and simple prescription to compute anomalous dimensions of composite operators in {\cal N}=4, SU(N) supersymmetric Yang-Mills theory, perturbatively in the coupling constant g. We show in general that anomalous dimensions are responsible for the appearance of higher order poles in the perturbative expansion of the two-point function and that their lowest contribution can be read directly from the coefficient of the 1/\epsilon^2 pole. As a check of our procedure we rederive the anomalous dimension of the Konishi superfield at order g^2. We then apply this procedure to the case of the double trace, dimension 4, superfield in the 20 of SU(4) recently considered in the literature. We find that its anomalous dimension vanishes for all N in agreement with previous results. 
  We introduce a realization of Seiberg duality in MQCD that does not involve moving branes. Using related arguments in M Theory and IIA, we investigate the flavor symmetry breaking via flavored magnetic monopole condensation of hep-th/0005076. We verify our results by considering visualizations of M5 branes corresponding to $\N=2$ vacua which survive soft breaking by an adjoint mass term. 
  A modification of perturbation theory, known as delta-expansion (variationally improved perturbation), gave rigorously convergent series in some D=1 models (oscillator energy levels) with factorially divergent ordinary perturbative expansions. In a generalization of variationally improved perturbation appropriate to renormalizable asymptotically free theories, we show that the large expansion orders of certain physical quantities are similarly improved, and prove the Borel convergence of the corresponding series for $m_v \lsim 0$, with $m_v$ the new (arbitrary) mass perturbation parameter. We argue that non-ambiguous estimates of quantities relevant to dynamical (chiral) symmetry breaking in QCD, are possible in this resummation framework. 
  We deepen the understanding of the quantization of the Yang-Mills field by showing that the concept of gauge fixing in 4 dimensions is replaced in the 5-dimensional formulation by a procedure that amounts to an $A$-dependent gauge transformation. The 5-dimensional formulation implements the restriction of the physical 4-dimensional gluon field to the Gribov region, while being a local description that is under control of BRST symmetries both of topological and gauge type. The ghosts decouple so the Euclidean probability density is everywhere positive, in contradistinction to the Faddeev-Popov method for which the determinant changes sign outside the Gribov region. We include in our discussion the coupling of the gauge theory to a Higgs field, including the case of spontaneously symmetry breaking. We introduce a minimizing functional on the gauge orbit that could be of interest for numerical gauge fixing in the simulations of spontaneously broken lattice gauge theories. Other new results are displayed, such as the identification of the Schwinger-Dyson equation of the five dimensional formulation in the (singular) Landau gauge with that of the ordinary Faddeev-Popov formulation, order by order in perturbation theory. 
  We study soliton solutions in supersymmetric scalar field theory with a class of potentials. We study both bosonic and fermionic zero-modes around the soliton solution. We study two possible couplings of gauge fields to these models. While the Born-Infeld like coupling has one normalizable mode (the zero mode), the other kind of coupling has no normalizable modes. We show that quantum mechanical problem which determines the spectrum of fluctuation modes of the scalar, fermion and the gauge field is identical. We also show that only the lowest lying mode, i.e., the zero mode, is normalizable and the rest of the spectrum is continuous. 
  We present a coherent proof of the spin-statistics theorem in path integral formulation. The local path integral measure and Lorentz invariant local Lagrangian, when combined with Green's functions defined in terms of time ordered products, ensure causality regardless of statistics. The Feynman's $m-i\epsilon$ prescription ensures the positive energy condition regardless of statistics, and the abnormal spin-statistics relation for both of spin-0 scalar particles and spin-1/2 Dirac particles is excluded if one imposes the positive norm condition in conjunction with Schwinger's action principle. The minus commutation relation between one Bose and one Fermi field arises naturally in path integral. The Feynman's $m-i\epsilon$ prescription also ensures a smooth continuation to Euclidean theory, for which the use of the Weyl anomaly is illustrated to exclude the abnormal statistics for the scalar and Dirac particles not only in 4-dimensional theory but also in 2-dimensional theory. 
  Sometime ago it was shown that the operatorial approach to classical mechanics, pioneered in the 30's by Koopman and von Neumann, can have a functional version. In this talk we will extend this functional approach to the case of classical field theories and in particular to the Yang-Mills ones. We shall show that the issues of gauge-fixing and Faddeev-Popov determinant arise also in this classical formalism. 
  We propose a phase-space path integral formulation of noncommutative quantum mechanics, and prove its equivalence to the operatorial formulation. As an illustration, the partition function of a noncommutative two-dimensional harmonic oscillator is calculated. 
  I present a brief review on some of the recent developments in topological quantum field theory. These include topological string theory, topological Yang-Mills theory and Chern-Simons gauge theory. It is emphasized how the application of different field and string theory methods has led to important progress, opening entirely new points of view in the context of Gromov-Witten invariants, Donaldson invariants, and quantum-group invariants for knots and links. 
  The problem of consistent formulation of the correspondence principle in quantum gravity is considered. The usual approach based on the use of the two-particle scattering amplitudes is shown to be in disagreement with the classical result of General Relativity given by the Schwarzschild solution. It is shown also that this approach fails to describe whatever non-Newtonian interactions of macroscopic bodies. An alternative interpretation of the correspondence principle is given directly in terms of the effective action. Gauge independence of the \hbar^0 part of the one-loop radiative corrections to the gravitational form factors of the scalar particle is proved, justifying the interpretation proposed. Application to the black holes is discussed. 
  We describe how the D-brane spectra of the various ten-dimensional string theories can be related to general properties of the open-closed duality, encoded in the $S$ and $P$ matrices of the conformal field theory. We also complete the classification and the description of non-BPS branes in these string theories, elucidating their non-Abelian structures and the nature of the corresponding super-Higgs mechanisms. We find that the type 0 theories and their orientifolds have two types of uncharged branes, distinguished by their couplings to the closed string tachyon. We also find that the 0A orientifold has the unusual feature of having charged and uncharged branes with identical world-volume dimensions. We conclude with some comments on fractional branes, elucidating their role in connection with the boundary states of $D_{odd}$ SU(2) WZW models. 
  We extend a recent work by Mussardo and Penati on integrable quantum field theories with a single stable particle and an infinite number of unstable resonance states, including the presence of a boundary. The corresponding scattering and reflection amplitudes are expressed in terms of Jacobian elliptic functions, and generalize the ones of the massive thermal Ising model and of the Sinh-Gordon model. In the case of the generalized Ising model we explicitly study the ground state energy and the one-point function of the thermal operator in the short-distance limit, finding an oscillating behaviour related to the fact that the infinite series of boundary resonances does not decouple from the theory even at very short-distance scales. The analysis of the generalized Sinh-Gordon model with boundary reveals an interesting constraint on the analytic structure of the reflection amplitude. The roaming limit procedure which leads to the Ising model, in fact, can be consistently performed only if we admit that the nature of the bulk spectrum uniquely fixes the one of resonance states on the boundary. 
  I give a short guide into applications of the heat kernel technique to string/brane physics with an emphasis on the emerging boundary value problems. 
  We analyse the OPE of any two 1/2 BPS operators of (2,0) SCFT$_6$ by constructing all possible three-point functions that they can form with another, in general long operator. Such three-point functions are uniquely determined by superconformal symmetry. Selection rules are derived, which allow us to infer ``non-renormalization theorems'' for an abstract superconformal field theory. The latter is supposedly related to the strong-coupling dynamics of $N_c$ coincident M5 branes, dual, in the large-$N_c$ limit, to the bulk M-theory compactified on AdS$_7 \times$S$_4$. An interpretation of extremal and next-to-extremal correlators in terms of exchange of operators with protected conformal dimension is given. 
  We compute the quantum fluctuations of a 3-brane with tension, energy density and stiffness. As a result of the fluctuations there are induced forces between massive objects living on the brane. We study various limiting cases of the induced potential between 2 and 3 massive objects. One quite interesting finding is that for tensionless brane world there are universal (mass independent) $1/r^3$ forces between the objects on the brane. These forces are in principle measurable. 
  We present a generalization of the five dimensional multigravity models to six dimensions. The key characteristic of these constructions is that that we obtain solutions which do not have any negative tension branes while at the same time the branes are kept flat. This is due to the fact that in six dimensions the internal space is not trivial and its curvature allows bounce configurations with the above feature. These constructions give for the first time a theoretically and phenomenologically viable realization of multigravity. 
  We obtain the couplings of noncommutative branes of type II string theories to constant Ramond-Ramond backgrounds, for BPS as well as non-BPS branes, in the background-independent description. For the BPS branes, we also generalize these couplings to other descriptions, and thereby argue their equivalence to the known couplings in the commutative description. The first part is a review of earlier work while the second part contains some additional observations. 
  Perturbation theory for gravity in dimensions greater than two requires higher derivatives in the free action. Higher derivatives seem to lead to ghosts, states with negative norm. We consider a fourth order scalar field theory and show that the problem with ghosts arises because in the canonical treatment, $\phi$ and $\Box \phi $ are regarded as two independent variables. Instead, we base quantum theory on a path integral, evaluated in Euclidean space and then Wick rotated to Lorentzian space. The path integral requires that quantum states be specified by the values of $\phi$ and $\phi_{,\tau}$. To calculate probabilities for observations, one has to trace out over $\phi_{,\tau}$ on the final surface. Hence one loses unitarity, but one can never produce a negative norm state or get a negative probability. It is shown that transition probabilities tend toward those of the second order theory, as the coefficient of the fourth order term in the action tends to zero. Hence unitarity is restored at the low energies that now occur in the universe. 
  A review of computations of free energy for Gibbs states on stationary but not static gravitational and gauge backgrounds is given. On these backgrounds wave equations for free fields are reduced to eigen-value problems which depend non-linearly on the spectral parameter. We present a method to deal with such problems. In particular, we demonstrate how some results of the spectral theory of second order elliptic operators, such as heat kernel asymptotics, can be extended to a class of non-linear spectral problems. The method is used to trace down the relation between the canonical definition of the free energy based on summation over the modes and the covariant definition given in Euclidean quantum gravity. As an application, high-temperature asymptotics of the free energy and of the thermal part of the stress-energy tensor in the presence of rotation are derived. We also discuss statistical mechanics in the presence of Killing horizons where canonical and Euclidean theories are related in a non-trivial way. 
  Orientifold vacua allow the simultaneous presence of supersymmetric bulks, with one or more gravitinos, and non-supersymmetric combinations of BPS branes. This ``brane supersymmetry breaking'' raises the issue of consistency for the resulting gravitino couplings, and Dudas and Mourad recently provided convincing arguments to this effect for the ten-dimensional $USp(32)$ model. These rely on a non-linear realization of local supersymmetry {\it \`a la} Volkov-Akulov, although no gravitino mass term is present, and the couplings have a nice geometrical interpretation in terms of ``dressed'' bulk fields, aside from a Wess-Zumino-like term, resulting from the supersymmetrization of the Chern-Simons couplings. Here we show that {\it all} couplings can be given a geometrical interpretation, albeit in the dual 6-form model, whose bulk includes a Wess-Zumino term, so that the non-geometric ones are in fact demanded by the geometrization of their duals. We also determine the low-energy couplings for six-dimensional (1,0) models with brane supersymmetry breaking. Since these include both Chern-Simons and Wess-Zumino terms, only the resulting field equations are geometrical, aside from contributions due to vectors of supersymmetric sectors. 
  We show how to construct a set of Euclidean conformal correlation functions on the boundary of a de Sitter space from an interacting bulk quantum field theory with a certain asymptotic behaviour. We discuss the status of the boundary theory w.r.t. the reflection positivity and conclude that no obvious physical holographic interpretation is available. 
  The Bekenstein-Hawking black hole area entropy law suggests that the quantum degrees of freedom of black holes may be realized as projections of quantum states unto the event horizon of the black hole. In this paper, we provide further evidence for this interpretation in the context of string theory. In particular, we argue that increase in the quantum entropy due to the capture of infalling fundamental strings appears in the form of horizon degrees of freedom. 
  In this letter we consider models with N U(1) gauge fields together with N Kalb-Ramond fields in the large N limit. These models can be solved explicitely and exhibit confinement for a large class of bare actions. The confining phase is characterized by an approximate "low energy" vector gauge symmetry under which the Kalb-Ramond fields transform. A duality transformation shows that confinement is associated with magnetic monopoles condensation. 
  The 1988 book, now free, with corrections and bookmarks (for pdf). 
  We derive a generalized Nielsen identity for the case of Yang-Mills theories that include some classical fields. We discuss under which circumstances the effective action of the classical fields (i.e., after integration of quantum fields) becomes gauge fixing independent. We conclude that classical test fields provide a physical insight into the problem of the gauge-fixing dependence of the quantum effective action. 
  The large Nc expansion of N=2 supersymmetric Yang-Mills theory with gauge group SU(Nc) has recently been shown to break down at singularities on the moduli space. We conjecture that by taking Nc to infinity and approaching the singularities in a correlated way, all the observables of the theory have a finite universal limit yielding amplitudes in string theories dual to field theories describing the light degrees of freedom. We explicitly calculate the amplitudes corresponding to the Seiberg-Witten period integrals for an A_{n-1} series of multicritical points as well as for other critical points exhibiting a scaling reminiscent of the c=1 matrix model. Our results extend the matrix model approach to non-critical strings in less than one dimension to non-critical strings in four dimensions. 
  We have constructed a bulk & brane action of IIA theory which describes a pair of BPS domain walls on S_1/Z_2, with strings attached. The walls are given by two orientifold O8-planes with coincident D8-branes and `F1-D0'-strings are stretched between the walls. This static configuration satisfies all matching conditions for the string and domain wall sources and has 1/4 of unbroken supersymmetry. 
  We use the BSFT method to study the unoriented open string field theory (type I). The partition function on the Mobius strip is calculated. We find that, at the one-loop level, the divergence coming from planar graph and unoriented graph cancel each other as expected. 
  We derive an explicit expression for an associative *-product on fuzzy complex projective spaces. This generalises previous results for the fuzzy 2-sphere and gives a discrete non-commutative algebra of functions on fuzzy complex projective spaces, represented by matrix multiplication. The matrices are restricted to ones whose dimension is that of the totally symmetric representations of SU(N). In the limit of infinite dimensional matrices we recover the commutative algebra of functions on ordinary projective space. Derivatives on the fuzzy projective space are also expressed as matrix commutators. 
  We present the Ricci-flat metric and its Kahler potential on the conifold with the O(N) isometry, whose conical singularity is repaired by the complex quadric surface Q^{N-2} = SO(N)/SO(N-2)xU(1). 
  We study the matter part of the algebra of open string fields using the 3-string vertex over the sliver state, which we call ``comma vertex''. By generalizing this comma vertex to the $N$-string overlap, we obtain a closed form of the Neumann coefficients in the $N$-string vertex and discuss its relation to the oscillator representation of wedge states. 
  The high energy thermodynamics of Little String Theory (LST) is known to be unstable. An unresolved question is whether the corresponding instability in LST holographic dual is of stringy or supergravity origin. We study UV thermodynamics of a large metric deformation of the LST dual realized (in the extremal case) by type IIB fivebranes wrapping a two-sphere of a resolved conifold, and demonstrate that the resulting black hole has negative specific heat. This explicitly shows that the high energy thermodynamic instability of the LST holographic dual is of the supergravity origin. 
  We study the world-volume theory of a bosonic membrane perturbatively and discuss if one can obtain any conditions on the number of space-time dimensions from the consistency of the theory. We construct an action which is suitable for such a study. In order to study the theory perturbatively we should specify a classical background around which perturbative expansion is defined. We will discuss the conditions which such a background should satisfy to deduce the critical dimension. Unfortunately we do not know any background satisfying such conditions. In order to get indirect evidences for the critical dimension of the membrane, we next consider two string models obtained via double dimensional reduction of the membrane. The first one reduces to the Polyakov string theory in the conformal gauge. The second one is described by the Schild action. We show that the critical dimension is 26 for these string theories, which implies that the critical dimension is 27 for the membrane theory. 
  A novel inhomogeneous gauge transformation law is proposed for a non-Abelian adjoint two-form in four dimensions. Rules for constructing actions invariant under this are given. The auxiliary vector field which appears in some of these models transforms like a second connection in the theory. Another local symmetry leaves the compensated three-form field strength invariant, but does not seem to be generated by any combination of local constraints. Both types of symmetries change the action by total divergences, suggesting that boundary degrees of freedom have to be taken into account for local quantization. 
  We show that U(\infty) symmetry transformations of the noncommutative field theory in the Moyal space are generated by a combination of two W_{1+\infty} algebras in the Landau problem. Geometrical meaning of this infinite symmetry is illustrated by examining the transformations of an invariant subgroup on the noncommutative solitons, which generate deformations and boosts of solitons. In particular, we find that boosts of solitons, which was first introduced in Ref[22], are valid only in the infinite "theta" limit. 
  One-loop effective action of noncommutative scalar field theory with cubic self-interaction is studied. Utilizing worldline formulation, both planar and nonplanar part of the effective action are computed explicitly. We find complete agreement of the result with Seiberg-Witten limit of string worldsheet computation and standard Feynman diagrammatics. We prove that, at low-energy and large noncommutativity limit, nonplanar part of the effective action is simplified enormously and is resummable into a quadratic action of scalar open Wilson line operators. 
  Quaternion quantum mechanics is examined at the level of unbroken SU(2) gauge symmetry. A general quaternionic phase expression is derived from formal properties of the quaternion algebra. 
  We derive the general equations of motion for a braneworld containing a (localized) domain wall. Euclideanization gives the geometry for braneworld false vacuum decay. We explicitly derive these instantons and compute the amplitude for false vacuum decay. We also show how to construct a toy ekpyrotic instanton. Finally, we comment on braneworlds with a compact spatial direction and the adS soliton. 
  We investigate a sequence of quadratic topological terms of the Chern-Simons type in different spacetime dimensions, related by dimensional compactification and sharing the properties of topological mass generation and statistical transmutation. The implications for bosonization in several dimensions are also analyzed. 
  We show that noncommutative U(r) gauge theories with a chiral fermion in the adjoint representation can be constructed on the lattice with manifest star-gauge invariance in arbitrary even dimensions. Chiral fermions are implemented using a Dirac operator which satisfies the Ginsparg-Wilson relation. A gauge-invariant integration measure for the fermion fields can be given explicitly, which simplifies the construction as compared with lattice chiral gauge theories in ordinary (commutative) space-time. Our construction includes the cases where continuum calculations yield a gauge anomaly. This reveals a certain regularization dependence, which is reminiscent of parity anomaly in commutative space-time with odd dimensions. We speculate that the gauge anomaly obtained in the continuum calculations in the present cases can be cancelled by an appropriate counterterm. 
  We study the propagation of a massless minimally coupled scalar in the near horizon geometry of non-extremal NS5-branes. Using the holographic principle for dilatonic backgrounds we compute the two-point function of an operator in Little String Theory at the Hagedorn temperature. We then comment on relations with correlation functions in two dimensional string theory. 
  We consider the path-sum of Ponzano-Regge with additional boundary contributions in the context of the holographic principle of Quantum Gravity. We calculate an holographic projection in which the bulk partition function goes to a semi-classical limit while the boundary state functional remains quantum-mechanical. The properties of the resulting boundary theory are discussed. 
  The geometry of the D1-D5 system with a small angular momentum j has a long throat ending in a conical defect. We solve the scalar wave equation for low energy quanta in this geometry. The quantum is found to reflect off the end of the throat, and stay trapped in the throat for a long time. The length of the throat for j=1/2 equals n_1n_5 R, the length of the effective string in the CFT; we also find that at this distance the incident wave becomes nonlinear. Filling the throat with several quanta gives a `hot tube' which has emission properties similar to those of the near extremal black hole. 
  Variation of coupling constants of integrable system can be considered as canonical transformation or, infinitesimally, a Hamiltonian flow in the space of such systems. Any function $T(\vec p, \vec q)$ generates a one-parametric family of integrable systems in vicinity of a single system: this gives an idea of how many integrable systems there are in the space of coupling constants. Inverse flow is generated by a dual "Hamiltonian", $\widetilde T(\vec p, \vec q)$ associated with the dual integrable system. In vicinity of a self-dual point the duality transformation just interchanges momenta and coordinates in such a "Hamiltonian": $\widetilde T(\vec p, \vec q) = T(\vec q, \vec p)$. For integrable system with several coupling constants the corresponding "Hamiltonians" $T_i(\vec p, \vec q)$ satisfy Whitham equations and after quantization (of the original system) become operators satisfying the zero-curvature condition in the space of coupling constants: [ d/dg_a - T_a(p,q), d/dg_b - T_b(p,q) ] = 0. Some explicit formulas are given for harmonic oscillator and for Calogero-Ruijsenaars-Dell system. 
  A detailed description of the infinite-dimensional Lie algebra of star-gauge transformations in noncommutative Yang-Mills theory is presented. Various descriptions of this algebra are given in terms of inner automorphisms of the underlying deformed algebra of functions on spacetime, of deformed symplectic diffeomorphisms, of the infinite unitary Lie algebra, and of the algebra of compact operators on a quantum mechanical Hilbert space. The spacetime and string interpretations are also elucidated. 
  We give an interpretation of the non-supersymmetric heterotic theories as the supersymmetric heterotic theories on Melvin background with Wilson lines. The coincidence of the partition functions are shown for most of the non-supersymmetric theories. We also discuss the tachyonic instability in terms of the non-trivial background fields. The behavior in the strong coupling region is investigated by using the duality relations of the supersymmetric theories. 
  The correlation function of the two dimensional Ising model with the nearest neighbours interaction on the finite size lattice with the periodical boundary conditions is derived. The expressions similar to the form factor expansion are obtained both for the paramagnetic and ferromagnetic regions of coupling parameter. The peculiarities caused by finite size are analyzed. The scaling limit of the lattice form factor expansion is evaluated. 
  We investigate the limit of minimal model conformal field theories where the central charge approaches one. We conjecture that this limit is described by a non-rational CFT of central charge one. The limiting theory is different from the free boson but bears some resemblance to Liouville theory. Explicit expressions for the three point functions of bulk fields are presented, as well as a set of conformal boundary states. We provide analytic and numerical arguments in support of the claim that this data forms a consistent CFT. 
  Giant gravitons in AdS_5 x S^5, and its orbifolds, have a dual field theory representation as states created by chiral primary operators. We argue that these operators are not single-trace operators in the conformal field theory, but rather are determinants and subdeterminants of scalar fields; the stringy exclusion principle applies to these operators. Evidence for this identification comes from three sources: (a) topological considerations in orbifolds, (b) computation of protected correlators using free field theory and (c) a Matrix model argument. The last argument applies to AdS_7 x S^4 and the dual (2,0) theory, where we use algebraic aspects of the fuzzy 4-sphere to compute the expectation value of a giant graviton operator along the Coulomb branch of the theory. 
  We study the parity breaking effective action in 2+1 dimensions, generated, at finite temperature, by massive fermions interacting with a non-Abelian gauge background. We explicitly calculate, in the static limit, parity violating amplitudes up to the seven point function, which allows us to determine the corresponding effective actions. We derive the exact parity violating effective action when $\vec{E}=0$. When $\vec{E}\neq 0$, there are families of terms that can be determined order by order in perturbation theory. We attempt to generalize our results to non-static backgrounds through the use of time ordered exponentials and prove gauge invariance, both {\it small} and {\it large}, of the resulting effective action. We also point out some open questions that need to be further understood. 
  We establish existence and stabilty results for solitons in noncommutative scalar field theories in even space dimension $2d$. In particular, for any finite rank spectral projection $P$ of the number operator ${\mathcal N}$ of the $d$-dimensional harmonic oscillator and sufficiently large noncommutativity parameter $\theta$ we prove the existence of a rotationally invariant soliton which depends smoothly on $\theta$ and converges to a multiple of $P$ as $\theta\to\infty$.   In the two-dimensional case we prove that these solitons are stable at large $\theta$, if $P=P_N$, where $P_N$ projects onto the space spanned by the $N+1$ lowest eigenstates of ${\mathcal N}$, and otherwise they are unstable. We also discuss the generalisation of the stability results to higher dimensions. In particular, we prove stability of the soliton corresponding to $P=P_0$ for all $\theta$ in its domain of existence.   Finally, for arbitrary $d$ and small values of $\theta$, we prove without assuming rotational invariance that there do not exist any solitons depending smoothly on $\theta$. 
  In a variety of systems which exhibit aging, the two-time response function scales as $R(t,s)\approx s^{-1-a} f(t/s)$. We argue that dynamical scaling can be extended towards conformal invariance, obtaining thus the explicit form of the scaling function $f$. This quantitative prediction is confirmed in several spin systems, both for $T<T_c$ (phase ordering) and $T=T_c$ (non-equilibrium critical dynamics). The 2D and 3D Ising models with Glauber dynamics are studied numerically, while exact results are available for the spherical model with a non-conserved order parameter, both for short-ranged and long-ranged interactions, as well as for the mean-field spherical spin glass. 
  It is often claimed [PST1] that the (Hodge type) duality operation is defined only in even dimensional spacetimes and that self-duality is further restricted to twice-odd dimensional spacetime theories.   The purpose of this paper is to extend the notion of both duality symmetry as well as self-duality.   By considering tensorial doublets, we introduce a novel well-defined notion of self-duality based on a duality Hodge-type operation in arbitrary dimension and for any rank of these tensors. Thus, a generalized Self-Dual Action is defined such that equations of motion are the claimed generalized self-duality relations. We observe in addition, that taking the proper limit on the parameters of this action, it always provides us with a master-action, which interpolates models well-studied in physics; by considering a particular limit, we find an action which describes an interesting type of relation, referred to as semi-self-duality, which results to be the parent action between Maxwell- type actions.   Finally, we apply these ideas to construct manifest Hodge-type self-dual solutions in a (2+1)-dimensional version of the Maxwell's theory. 
  We study the cosmological evolution of a type-0 string theory by employing non-criticality, which may be induced by fluctuations of the D3 brane worlds. We check the consistency of the approach to O(alpha ') in the corresponding sigma-model. The ten-dimensional theory is reduced to an effective four-dimensional model, with only time dependent fields. We show that the four-dimensional universe has an inflationary phase and graceful exit from it, while the other extra dimensions are stabilized to a constant value, with the fifth dimension much larger than the others. We pay particular attention to demonstrating the role of tachyonic matter in inducing these features. The Universe asymptotes, for large times, to a non-accelerating linearly-expanding Universe with a time-dependent dilaton and a relaxing to zero vacuum energy a la quintessence. 
  We report on recent results showing that a rich non-perturbative vacuum structure is associated with flavor mixing in Quantum Field Theory.   We treat explicitly the case of mixing among three generations of Dirac fermions.   Exact oscillation formulas are presented exhibiting new features with respect to the usual ones. CP and T violation are also discussed. 
  We find and study solutions to the Einstein equations in D dimensions coupled to a scalar field source with a Liouville potential under the assumption of D-2 planar symmetry. The general static or time-dependent solutions are found yielding three classes of SO(D-2) symmetric spacetimes. In D=4 homogeneous and isotropic subsets of these solutions yield planar scalar field cosmologies. In D=5 they represent the general static or time-dependent backgrounds for a dilatonic wall-type brane Universe of planar cosmological symmetry. Here we apply these solutions as SO(8) symmetric backgrounds to non-supersymmetric 10 dimensional string theories, the open USp(32) type I string and the heterotic string SO(16)XSO(16). We obtain the general SO(9) solutions as a particular case. All static solutions are found to be singular with the singularity sometimes hidden by a horizon. The solutions are not asymptotically flat or of constant curvature. The singular behavior is no longer true once we permit space and time dependence of the spacetime metric much like thick domain wall or global vortex spacetimes. We analyze the general time and space dependent solutions giving implicitly a class of time and space dependent solutions and describe the breakdown of an extension to Birkhoff's theorem in the presence of scalar matter. We argue that the solutions described constitute the general solution to the field configuration under D-2 planar symmetry. 
  We generalize the non-threshold bound state in type IIB supergravity of the form (NS5, D5, D3) constructed by the present authors (in hep-th/0011236) to non-zero asymptotic value of the axion $(\chi_0$). We identify the decoupling limits corresponding to both the open D3-brane theory and open D5-brane theory for this supergravity solution as expected. However, we do not find any non-commutative Yang-Mills theory (NCYM) limit for this solution in the presence of NS5 branes. We then study the $SL(2, Z)$ duality symmetry of type IIB theory for both OD3-limit and OD5-limit. We find that for OD3 theory, a generic $SL(2, Z)$ duality always gives another OD3-theory irrespective of the value of $\chi_0$ being rational or not. This indicates that OD3-theory is self-dual. But, under a special set of $SL(2, Z)$ transformations for which $\chi_0$ is rational OD3-theory goes over to a 5+1 dimensional NCYM theory and these two theories in this case are related to each other by strong-weak duality symmetry. On the other hand, for OD5-theory, a generic $SL(2, Z)$ duality gives another OD5-theory if $\chi_0$ is irrational, but when $\chi_0$ is rational it gives the little string theory limit indicating that OD5-theory is S-dual to the type IIB little string theory. 
  We reconsider the phenomenon of mass generation via coordinate-dependent compatifications of higher-dimensional theories on orbifolds. For definiteness, we study a generic five-dimensional (5D) theory compactified on S^1/Z_2. We show that the presence of fixed points, where the fields or their derivatives may be discontinuous, permits new realizations of the Scherk-Schwarz mechanism where, for example, the mass terms are localized at the orbifold fixed points. Our technique can be used to describe the explicit breaking of global flavor symmetries and supersymmetries by brane-localized mass terms. It can also be applied to the spontaneous breaking of local symmetries, such as gauge symmetries or supergravities. 
  Due to unsuitable attempt to connect the base theory in hep-th/0106074 to noncommutative field theory, this paper has been withdrawn by the author. 
  A theory of gravity in $d+1$ dimensions is dynamically generated from a theory in $d$ dimensions. As an application we show how $N$ dynamically coupled gravity theories can reduce the effective Planck mass. 
  We study string theory on singular Z_N quotients of AdS_3, corresponding to spaces with conical defects. The spectrum is computed using the orbifold procedure. It is shown that spectral flow may be used to generate the twisted sectors. We further compute the thermal partition function and show that it correctly reproduces the spectrum. 
  The asymptotic symmetries of the near-horizon geometry of a lifted (near-extremal) Reissner-Nordstrom black hole, obtained by inverting the Kaluza-Klein reduction, explain the deviation of the Bekenstein-Hawking entropy from extremality. We point out the fact that the extra dimension allows us to justify the use of a Virasoro mode decomposition along the time-like boundary of the near-horizon geometry, AdS$_2\times$S$^n$, of the lower-dimensional (Reissner-Nordstrom) spacetime. 
  Following Osipov and Hiller, a generalized heat kernel expansion is considered for the effective action of bosonic operators. In this generalization, the standard heat kernel expansion, which counts inverse powers of a c-number mass parameter, is extended by allowing the mass to be a matrix in flavor space. We show that the generalized heat kernel coefficients can be related to the standard ones in a simple way. This holds with or without trace and integration over spacetime, to all orders and for general flavor spaces. Gauge invariance is manifest. 
  The creation of 4d de Sitter (inflationary) boundary gluing two d5 de Sitter bulks on the classical as well as on quantum level (with account of brane QFT via corresponding trace anomaly induced effective action) is discussed. Quantum effects decrease the classical de Sitter brane radius or create new de Sitter brane with even smaller radius. It is important that brane CFT may be chosen to be dual to one of 5d de Sitter bulks, making the explicit relation of de Sitter brane-world with dS/CFT correspondence. In this way, the localization of gravity on the brane is shown. Moving (time-dependent) de Sitter brane in d5 SdS BH is considered. In the special coordinate system where brane equations look like quantum-corrected FRW equations the comparison with similar brane equations in SAdS BH bulk is done. 
  The derivation of a convergent series representation for the quantum electrodynamic effective action obtained by two of us (S.R.V. and D.R.L.) in [Can. J. Phys. vol. 71, p. 389 (1993)] is reexamined. We present more details of our original derivation. Moreover, we discuss the relation of the electric-magnetic duality to the integral representation for the effective action, and we consider the application of nonlinear convergence acceleration techniques which permit the efficient and reliable numerical evaluation of the quantum correction to the Maxwell Lagrangian. 
  We construct two SU(5) models on the space-time $M^4 \times T^2/(Z_2 \times Z_2^{\prime})$ where the gauge and Higgs fields are in the bulk and the Standard Model fermions are on the brane at the fixed point or line. For the zero modes, the SU(5) gauge symmetry is broken down to $SU(3) \times SU(2) \times U(1) $ due to non-trivil orbifold projection. In particular, if we put the Standard Model fermions on the 3-brane at the fixed point in Model II, we only have the zero modes and KK modes of the Standard Model gauge fields and two Higgs doublets on the observable 3-brane.   So, we can have the low energy unification, and solve the triplet-doublet splitting problem, the gauge hierarchy problem, and the proton decay problem. 
  We study the decay of a near-extremal black hole in AdS$_2$, related to the near-horizon region of 3+1-dimensional Reissner-Nordstr\"om spacetime, following Fabbri, Navarro, and Navarro-Salas. Back-reaction is included in a semiclassical approximation. Calculations of the stress-energy tensor of matter coupled to the physical spacetime for an affine null observer demonstrate that the black hole evaporation proceeds smoothly and the near-extremal black hole evolves back to an extremal ground state, until this approximation breaks down. 
  We analyze the perturbative stability of non-supersymmetric intersecting brane world models on tori. Besides the dilaton tadpole, a dynamical instability in the complex structure moduli space occurs at string disc level, which drives the background geometry to a degenerate limit. We show that in certain orbifold models this latter instability is absent as the relevant moduli are frozen. We construct explicit examples of such orbifold intersecting brane world models and discuss the phenomenological implications of a three generation Standard Model which descends naturally from an SU(5) GUT theory. It turns out that various phenomenological issues require the string scale to be at least of the order of the GUT scale. As a major difference compared to the Standard Model, some of the Yukawa couplings are excluded so that the standard electroweak Higgs mechanism with a fundamental Higgs scalar is not realized in this set-up. 
  We discuss the dynamical situation which arises in a local quantum field theory after renormalization. By using the example of the three-dimensional theory of a neutral scalar field interacting through the quartic coupling, we show that after renormalization the dynamics of a theory is governed by a generalized dynamical equation with a nonlocal interaction operator. It is shown that the generalized dynamical equation allows one to formulate this theory in an ultraviolet-finite way. 
  The hybrid formalism is used to quantize the superstring compactified to two-dimensional target-space in a manifestly spacetime supersymmetric manner. A quantizable sigma model action is then constructed for the Type II superstring in curved two-dimensional supergravity backgrounds which can include Ramond-Ramond flux. Such curved backgrounds include Calabi-Yau four-fold compactifications with Ramond-Ramond flux, and new extremal black hole solutions in two-dimensional dilaton supergravity theory. These black hole solutions are a natural generalization of the CGHS model and might be possible to describe using a supergroup version of the SL(2,R)/U(1) WZW model. We also study some dynamical aspects of the new black holes, such as formation and evaporation. 
  We study through holography the compact Randall-Sundrum (RS) model at finite temperature. In the presence of radius stabilization, the system is described at low enough temperature by the RS solution. At high temperature it is described by the AdS-Schwarzschild solution with an event horizon replacing the TeV brane. We calculate the transition temperature T_c between the two phases and we find it to be somewhat smaller than the TeV scale. Assuming that the Universe starts out at T >> T_c and cools down by expansion, we study the rate of the transition to the RS phase. We find that the transition is very slow so that an inflationary phase at the weak scale begins. The subsequent evolution depends on the stabilization mechanism: in the simplest Goldberger-Wise case inflation goes on forever unless tight bounds are satisfied by the model parameters; in slightly less-minimal cases these bounds may be relaxed. 
  We study the motion of a D(8-p)-brane probe in the background created by a stack of non-threshold (D(p-2), Dp) bound states for $2\le p\le 6$. The brane probe and the branes of the background have two common directions. We show that for a particular value of the worldvolume gauge field there exist configurations of the probe brane which behave as massless particles and can be interpreted as gravitons blown up into a fuzzy sphere and a noncommutative plane. We check this behaviour by studying the motion and energy of the brane and by determining how supersymmetry is broken by the probe as it moves under the action of the background. 
  We construct the first three family N=1 supersymmetric string model with Standard Model gauge group SU(3)_C x SU(2)_L x U(1)_Y from an orientifold of type IIA theory on T^6/(Z_2 x Z_2) and D6-branes intersecting at angles. In addition to the minimal supersymmetric Standard Model particles, the model contains right-handed neutrinos, a chiral (but anomaly-free) set of exotic multiplets, and extra vector-like multiplets. We discuss some phenomenological features of this model. 
  The gauge field term in the Standard Model Lagrangian is slightly rewritten, suggesting that the three gauge couplings have absorbed factors which depend on the dimensions of the corresponding gauge groups. The ratios of the physical couplings may turn out to be dominated by these factors, with deviations due to quantum corrections. 
  We analyse supermembrane instantons (fully wrapped supermembranes) by computing the partition function of the three-dimensional supersymmetrical U(N) matrix model under periodic boundary conditions. By mapping the model to a cohomological field theory and considering a mass-deformation of the model, we show that the partition function exactly leads to the U-duality invariant measure factor entering supermembrane instanton sums. On the other hand, a computation based on the quasi-classical assumption gives the non U-duality invariant result of the zero-mode analysis by Pioline et al. This is suggestive of the importance of supermembrane self-interactions and shows a crucial difference from the matrix string case. 
  The analog of the principal SO(3) subalgebra of a finite dimensional simple Lie algebra can be defined for any hyperbolic Kac Moody algebra g(A) associated with a symmetrizable Cartan matrix A, and coincides with the non-compact group SO(1,2). We exhibit the decomposition of g(A) into representations of SO(1,2); with the exception of the adjoint SO(1,2) algebra itself, all of these representations are unitary. We compute the Casimir eigenvalues; the associated ``exponents'' are complex and non-integer. 
  We perform a perturbative ${\cal O}(g^4)$ Wilson loop calculation for the U(N) Yang-Mills theory defined on non-commutative one space - one time dimensions. We choose the light-cone gauge and compare the results obtained when using the Wu-Mandelstam-Leibbrandt ($WML$) and the Cauchy principal value ($PV$) prescription for the vector propagator. In the $WML$ case the $\theta$-dependent term is well-defined and regular in the limit $\theta \to 0$, where the commutative theory is recovered; it provides a non-trivial example of a consistent calculation when non-commutativity involves the time variable. In the $PV$ case, unexpectedly, the result differs from the $WML$ one only by the addition of two singular terms with a trivial $\theta$-dependence. We find this feature intriguing, when remembering that, in ordinary theories on compact manifolds, the difference between the two cases can be traced back to the contribution of topological excitations. 
  Much work has been devoted to the phenomenology and cosmology of the so-called braneworld universe, where our (3+1)-dimensional universe lies on a brane surrounded by a (4+1)-dimensional bulk spacetime that is essentially empty except for a negative cosmological constant and the various modes associated with gravity. For such a braneworld cosmology, the difficulty of justifying some preferred initial conditions inevitably arises. The various proposals for inflation restricted to the brane only partially explain the homogeneity and isotropy of the resulting braneworld universe because the homogeneity and isotropy of the bulk must be assumed. We propose a mechanism by which a brane surrounded by AdS space arises naturally so that the homogeneity and isotropy of both the brane and the bulk are guaranteed. We postulate an initial false vacuum phase of (4+1)-dimensional Minkowski or de Sitter space subsequently decaying to a true vacuum of anti-de Sitter space, assumed discretely degenerate. This decay takes place through bubble nucleation. When two bubbles of the true AdS vacuum collide, a brane (or domain wall) inevitably forms between the two AdS phases. We live on this brane. The SO(3,1) symmetry of the collision geometry ensures the three-dimensional spatial homogeneity and isotropy of the universe on the brane as well as of the bulk. In the semi-classical limit, this symmetry is exact. We sketch how the leading quantum corrections translate into cosmological perturbations. 
  Dual field theory realisations are given for linearised gravity in terms of gauge fields in exotic representations of the Lorentz group. The field equations and dual representations are discussed for a wide class of higher spin gauge fields. For non-linear Einstein gravity, such transformations can be implemented locally in light-cone gauge, or partially implemented in the presence of a Killing vector. Sources and the relation to Kaluza-Klein monopoles are discussed. 
  We consider the isotropic two-dimensional abelian sandpile model from a perspective based on two-dimensional (conformal) field theory. We compute lattice correlation functions for various cluster variables (at and off criticality), from which we infer the field-theoretic description in the scaling limit. We find a perfect agreement with the predictions of a c=-2 conformal field theory and its massive perturbation, thereby providing direct evidence for conformal invariance and more generally for a description in terms of a local field theory. The question of the height 2 variable is also addressed, with however no definite conclusion yet. 
  We generate by computer a basis of invariants for the fundamental representations of the exceptional Lie groups E(6) and E(7), up to degree 18. We discuss the relevance of this calculation for the study of supersymmetric gauge theories, and revisit the self-dual exceptional models. We study the chiral ring of G(2) to degree 13, as well as a few classical groups. The homological dimension of a ring is a natural estimator of its complexity and provides a guideline for identifying theories that have a good chance to be amenable to a solution. 
  We explain microscopically why split attractor flows, known to underlie certain stationary BPS solutions of four dimensional N=2 supergravity, are the relevant data to describe wrapped D-branes in Calabi-Yau compactifications of type II string theory. We work entirely in the context of the classical geometry of A-branes, i.e. special Lagrangian submanifolds, avoiding both the use of homological algebra and explicit constructions of special Lagrangians. Our results provide a way to disassemble and assemble arbitrary special Lagrangians to and from more simple building blocks, giving a concrete way to determine for example marginal stability walls and deformation moduli spaces. 
  We analyze string theory backgrounds that include different kinds of orientifold planes and map out a natural correspondence to (twisted) affine Kac-Moody algebras. The low-energy description of specific BPS states in these backgrounds leads to a construction of explicit twisted magnetic monopole solutions on R^3 x S^1. These backgrounds yield new low-energy field theories with twisted boundary conditions and the link with affine algebras yields a natural guess for the superpotentials of the corresponding pure N=1, and N=1* gauge theories. 
  We call attention to a series of mistakes in a paper by S. Nam [JHEP 10 (2000) 044, hep-th/0008083]. 
  We present a superspace formulation of N=1 eleven-dimensional supergravity with no manifest local Lorentz covariance, which we call teleparallel superspace. This formulation will be of great importance, when we deal with other supergravity theories in dimensions higher than eleven dimensions, or a possible formulation of noncommutative supergravity. As an illustrative example, we apply our teleparallel superspace formulation to the case of N=1 supergravity in twelve-dimensions. We also show the advantage of teleparallel superspace as backgrounds for supermembrane action. 
  The effect of temperature is investigated in the Randall-Sundrum brane-world scenario. It is shown that for a spacetime ansatz motivated by similarity with AdS/CFT correspondence several features of the model, such as its $Z_2$ symmetry, are not maintained at nonzero temperatures. 
  We study a large N_{c} limit of a two-dimensional Yang-Mills theory coupled to bosons and fermions in the fundamental representation. Extending an approach due to Rajeev we show that the limiting theory can be described as a classical Hamiltonian system whose phase space is an infinite-dimensional supergrassmannian. The linear approximation to the equations of motion and the constraint yields the 't Hooft equations for the mesonic spectrum. Two other approximation schemes to the exact equations are discussed. 
  The Cartan's equations definig simple spinors (renamed pure by C. Chevalley) are interpreted as equations of motion in momentum spaces, in a constructive approach in which at each step the dimesions of spinor space are doubled while those momentum space increased by two. The construction is possible only in the frame of geometry of simple or pure spinors, which imposes contraint equations on spinors with more than four components, and the momentum spaces result compact, isomorphic toinvariant-mass-spheres imbedded in each other, since the signatures appear to be unambiguously defined and result steadily lorentzian; up to dimension ten with Clifford algebra Cl(1,9), where the construction naturally ends. The equations of motion met in the construction are most of those traditionally postulated ad hoc for multicomponent fermions. The 3 division algebras: complex numbers, quaternions and octonions appear to be strictly correlated with this spinor geometry, from which they appear to gradually emerge in the construction, where they play a basic role for the physical interpretation. In fact they seem then to be at the origin of electroweak and strong charges, of the 3 families and of the groups of the standard model. In this approach there seems to be no need of higher dimensional (>4) space-time, here generated merely by Poincare translations, and dimensional reduction from Cl(1,9) to Cl(1,3) is equivalent to decoupling of the equations of motion. 
  We construct a general class of chiral four-dimensional string models with Scherk--Schwarz supersymmetry breaking, involving freely acting orbifolds. The basic ingredient is to combine an ordinary supersymmetry-preserving Z_N projection with a supersymmetry-breaking projection Z_M' acting freely on a subspace of the internal manifold. A crucial condition is that any generator of the full orbifold group Z_N x Z_M' must either preserve some supersymmetry or act freely in order to become irrelevant in some large volume limit. Tachyons are found to be absent or limited to a given region of the tree-level moduli space. We find several new models with orthogonal supersymmetries preserved at distinct fixed-points. Particular attention is devoted to an interesting Z_3 x Z_3' heterotic example. 
  We show that the SU(2)_0 WZNW model has a hidden OSp(2|2)_{-2} symmetry. Both these theories are known to have logarithms in their correlation functions. We also show that, like OSp(2|2)_{-2}, the logarithmic structure present in the SU(2)_0 model is due to the underlying c=-2 sector. We also demonstrate that the quantum Hamiltonian reduction of SU(2)_0 leads very directly to the correlation functions of the c=-2 model. We also discuss some of the novel boundary effects which can take place in this model. 
  We give a Lagrangian and display all local symmetries for N=4 topological string by Berkovits, Vafa and Siegel, the latter previously known in the superconformal gauge. Leading to a small N=4 superconformal algebra and exhibiting the manifest Lorentz invariance the model is proposed to be a framework for restoring the manifest Lorentz invariance in N=2 string scattering amplitudes. 
  I discuss tree-level amplitudes in cubic topological string field theory, showing that a certain family of gauge conditions leads to an A-infty algebra of tree-level string products which define a potential describing the dynamics of physical states. Upon using results of modern deformation theory, I show that the string moduli space admits two equivalent descriptions, one given in standard Maurer-Cartan fashion and another given in terms of a `homotopy Maurer-Cartan problem', which describes the critical set of the potential. By applying this construction to the topological A and B models, I obtain an intrinsic formulation of `D-brane superpotentials' in terms of string field theory data. This gives a prescription for computing such quantities to all orders, and proves the equivalence of this formulation with the fundamental description in terms of string field moduli. In particular, it clarifies the relation between the Chern-Simons/holomorphic Chern-Simons actions and the superpotential for A/B-type branes. 
  We add a simple boundary term to the Polyakov action and construct a new class of D-branes with a single null direction. On the string world-sheet the system is described by a single quantized left-mode sector of a conformal field theory. By a Wick rotation of spacetime, we map open strings attached to these branes into chiral closed strings. We suggest that these so-called H-branes describe quantum horizons - black hole, cosmological (de-Sitter), etc. We show how one can get a space/phase space transmutation near the horizon and discuss the new features of boundary states which become squeezed states. 
  In the limit of small velocities, the dynamics of half-BPS Yang-Mills-Higgs solitons can be described by the geodesic approximation. Recently, it has been shown that quarter-BPS states require the addition of a potential term to this approximation. We explain the logic behind this modification for a larger class of models and then analyse in detail the dynamics of two five-dimensional dyonic instantons, using both analytical and numerical techniques. Nonzero-modes are shown to play a crucial role in the analysis of these systems, and we explain how these modes lead to qualitatively new types of dynamics. 
  We generalize the Jackiw-Rebbi-Hasenfratz-'t Hooft construction of fermions from bosons to demonstrate the fermionic nature of certain bound states involving SU(N) instantons in even spatial dimensions and SO(N) instantons in $8k+1$ spatial dimensions. We use this result to identify several fermionic excitations in various perturbatively bosonic string theories. In some examples we are able to identify these fermions as excitations in known conformal field theories and independently confirm their fermionic nature. Examples of the fermions we find include certain 3-string junctions in type 0B theory, excitations of the 0-p system in type 0A theory, excitations of the stable D-particle of type O theory, and a rich spectrum of fermions in the bosonic string compactified on the SO(32) group lattice. 
  We construct N=1 supersymmetric four-dimensional orientifolds of type IIA on T^6/(Z_2 x Z_2) with D6-branes intersecting at angles. The use of D6-branes not fully aligned with the O6-planes in the model allows for a construction of many supersymmetric models with chiral matter, including those with the Standard Model and grand unified gauge groups. We perform a search for realistic gauge sectors, and construct the first example of a supersymmetric type II orientifold with SU(3)_C x SU(2)_L x U(1)_Y gauge group and three quark-lepton families. In addition to the supersymmetric Standard Model content, the model contains right-handed neutrinos, a (chiral but anomaly-free) set of exotic multiplets, and diverse vector-like multiplets. The general class of these constructions are related to familiar type II orientifolds by small instanton transitions, which in some cases change the number of generations, as discussed in specific models. These constructions are supersymmetric only for special choices of untwisted moduli. We briefly discuss the supersymmetry breaking effects away from that point. The M-theory lift of this general class of supersymmetric orientifold models should correspond to purely geometrical backgrounds admitting a singular G_2 holonomy metric and leading to four-dimensional M-theory vacua with chiral fermions. 
  We discuss non-conformal non-supersymmetric large N gauge theories with vanishing vacuum energy density to all orders in perturbation theory. These gauge theories can be obtained via a field theory limit of Type IIB D3-branes embedded in orbifolded space-times. We also discuss gravity in this setup. 
  We study the spectrum of the Chern-Simons matrix model and identify an orthogonal set of states. The connection to the spectrum of the Calogero model is discussed. 
  We show that the entropy of de Sitter space in any dimension can be understood as the entropy of a highly excited string located near the horizon. The string tension is renormalized to $T \sim \Lambda$ due to the large gravitational redshift near the horizon. The de Sitter temperature is given by the Hagedorn temperature of the string. 
  The two-component approach to the one-dimensional Dirac equation is applied to the Woods-Saxon potential. The scattering and bound state solutions are derived and the conditions for a transmission resonance (when the transmission coefficient is unity) and supercriticality (when the particle bound state is at E=-m) are then derived. The square potential limit is discussed. The recent result that a finite-range symmetric potential barrier will have a transmission resonance of zero-momentum when the corresponding well supports a half-bound state at E=-m is demonstrated. 
  1.An expression for the smoothed counting function in terms of the fractional derivatives of the delta-function is presented. 2. The Neumann-Dirichlet (ND) boundary problem is introduced via some elementary examples based on a functorial relation between the ND, DD, NN and periodic boundary conditions. 
  We give a detailed discussion of the disk amplitudes with one closed string insertion, which we used to construct the supergravity couplings of noncommutative D-branes to the RR potentials, given in hep-th/0104139. We prove the inclusion of Elliott's formula, the integer-valued modification of the noncommutative Chern character, to all orders in the gauge field. We also give a detailed comparison between the form of the result in which Elliott's formula is manifest, and the form expressed in Matrix model variables. 
  A holographic correspondence between data on horizon and space-time physics is investigated. We find similarities with the AdS/CFT correspondence, based on the observation that the optical metric near the horizon describes a Euclidean asymptotically anti-de Sitter space. This picture emerges for a wide class of static space-times with a non-degenerate horizon, including Schwarzschild black holes as well as de Sitter space-time. We reveal a asymptotic conformal symmetry at the horizon. We compute the conformal weights and 2-point functions for a scalar perturbation and discuss possible connections with a (non-unitary) conformal field theory located on the horizon. We then reconstruct the scalar field and the metric from the data given on the horizon. We show that the solution for the metric in the bulk is completely determined in terms of a specified metric on the horizon. From the General Relativity point of view our solutions present a new class of space-time metrics with non-spherical horizons. The horizon entropy associated with these solutions is also discussed. 
  We study static brane configurations in the bulk background of the topological black holes in asymptotically flat spacetime. We find that such configurations are possible even for flat black hole horizon, unlike the AdS black hole case. We construct the brane world model with an orbifold structure S^1/Z_2 in such bulk background. We also study massless bulk scalar field. 
  We discuss the brane world model of Dvali, Gabadadze and Porrati in which branes evolve in an infinite bulk and the brane curvature term is added to the action. If Z_2 symmetry between the two sides of the brane is not imposed, we show that the model admits the existence of "stealth branes" which follow the standard 4D internal evolution and have no gravitational effect on the bulk space. Stealth branes can nucleate spontaneosly in a Minkowski bulk. This process is described by the standard 4D quantum cosmology formalism with tunneling boundary conditions for the brane world wave function. The notorious ambiguity in the choice of boundary conditions is fixed in this case due to the presence of the embedding spacetime. We also point to some problematic aspects of models admitting stealth brane solutions. 
  Inflationary theory is already 20 years old, and it is impossible to describe all of its versions and implications in a short talk. I will concentrate on several subjects which I believe to be most important. First of all, I will give a brief review of the first versions of inflationary theory, from Starobinsky model to new inflation. Then I will describe chaotic inflation, the theory of quantum fluctuations and density perturbations, the theory of eternal inflation, and recent observational data. In the second part of the talk I will discuss the recently proposed ekpyrotic scenario and argue that in its present form it does not provide a viable alternative to inflation. 
  We analyze the dynamics of M-theory on a manifold of G_2 holonomy that is developing a conical singularity. The known cases involve a cone on CP^3, where we argue that the dynamics involves restoration of a global symmetry, SU(3)/U(1)^2, where we argue that there are phase transitions among three possible branches corresponding to three classical spacetimes, and S^3 x S^3 and its quotients, where we recover and extend previous results about smooth continuations between different spacetimes and relations to four-dimensional gauge theory. 
  Using branes in massive Type IIA string theory, and a novel decoupling limit, we provide an explicit correspondence between non-commutative Chern-Simons theory and the fractional quantum Hall fluid. The role of the electrons is played by D-particles, the background magnetic field corresponds to a RR 2-form flux, and the two-dimensional fluid is described by non-commutative D2-branes. The filling fraction is given by the ratio of the number of D2-branes and the number of D8-branes, and therefore by the ratio rank/level of the Chern-Simons gauge theory. Quasiparticles and quasiholes are realized as endpoints of fundamental strings on the D2-branes, and are found to possess fractional D-particle charges and fractional statistics. 
  We examine noncommutative Chern-Simons theory on a bounded spatial domain. We argue that upon `turning on' the noncommutativity, the edge observables, which characterized the commutative theory, move into the bulk. We show this to lowest order in the noncommutativity parameter appearing in the Moyal star product.   If one includes all orders, the Hamiltonian formulation of the gauge theory ceases to exist, indicating that the Moyal star product must be modified in the presence of a boundary. Alternative descriptions are matrix models. We examine one such model, obtained by a simple truncation of Chern-Simons theory on the noncommutative plane, and express its observables in terms of Wilson lines. 
  Two dimensional Non-Commutative Open String (NCOS) theory, well-defined perturbatively, may also be studied at strong coupling and for large D-string charge by making use of the Holographic duality. We analyze the zero mode dynamics of a closed string in the appropriate background geometry and map the results onto a sector of strongly coupled NCOS dynamics. We find an elaborate classical picture that shares qualitative similarities with the SL(2,R) WZW model. In the quantum problem, we compute propagators and part of the energy spectrum of the theory; the latter involves interesting variations in the density of states as a function of the level number, and energies scaling inversely with the coupling. Finally, the geometry exhibits a near horizon throat, associated with NCOS dynamics, yet it is found that the whole space is available for Holography. This provides a setting to extend the Maldacena duality beyond the near horizon limit. 
  We formulate the bosonic sector of IIB supergravity as a non-linear realisation. We show that this non-linear realisation contains the Borel subalgebras of SL(11) and $E_7$ and argue that it can be enlarged so as to be based on the rank eleven Kac-Moody algebra $E_{11}$ 
  We propose a superfield description of osp(1,2) covariant quantization by extending the set of admissibility conditions for the quantum action. We realize a superfield form of the generating equations, specify the vacuum functional and obtain the corresponding transformations of extended BRST symmetry. 
  An M-theory constructed in an eleven-dimensional supermanifold with a <>-product of field operators is shown to have a de Sitter space solution. Possible implications of this result for cosmology are mentioned. 
  In this article we study the noncommutative description of the DBI Lagrangian and its T-dual counterpart. We restrict the freedoms of the noncommutativity parameters of these Lagrangians. Therefore the noncommutativity parameter, the effective metric, the effective coupling constant of the string and the extra modulus ${\tilde \Phi}$ of the effective T-dual theory, can be expressed in terms of the closed string variables $g$, $B$, $g_s$ and the noncommutativity parameter of the effective theory of open string. 
  We dimensionally reduce the four-derivative corrections to the parity-conserving part of the D9-brane effective action involving all orders of the gauge field, to obtain corrections to the actions for the lower-dimensional Dp-branes. These corrections involve the second fundamental form and correspond to a non-geodesic embedding of the Dp-brane into (flat) ten-dimensional space. In addition, we study the transformation of the corrections under the Seiberg-Witten map relating the ordinary and non-commutative theories. A speculative discussion about the higher-order terms in the derivative expansion is also included. 
  The AdS_5 superalgebra PSU(2,2|4) has an infinite dimensional extension, which we denote by hs(2,2|4). We show that the gauging of hs(2,2|4) gives rise to a spectrum of physical massless fields which coincides with the symmetric tensor product of two AdS_5 spin-1 doubletons (i.e. the N=4 SYM multiplets living on the boundary of AdS_5). This product decomposes into levels \ell=0,1,2,..,\infty of massless supermultiplets of PSU(2,2|4). In particular, the D=5, N=8 supergravity multiplet arises at level \ell=0. In addition to a master gauge field, we construct a master scalar field containing the s=0,1/2 fields, the anti-symmetric tensor field of the gauged supergravity and its higher spin analogs. We define the linearized constraints and obtain the linearized field equations of the full spectrum, including those of D=5,N=8 gauged supergravity and in particular the self-duality equations for the 2-form potentials of the gauged supergravity (forming a 6-plet of SU(4)), and their higher spin cousins with s=2,3,...,\infty. 
  The Faddeev-Skyrme model, a modified O(3) nonlinear sigma model in three space dimensions, is known to admit topological solitons that are stabilized by the Hopf charge. The Faddeev-Skyrme model is also related to the low-energy limits of SU(2) Yang-Mills theory. Here, the model is reformulated into its gauge-equivalent expression, which turns out to be Skyrme-like. The solitonic solutions of this Skyrme-like model are analyzed by the rational map ansatz. The energy function and the Bogomolny-type lower bound of the energy are established. The generalized Faddeev-Skyrme model that originates from the infrared limits of SU(N) Yang-Mills theory is presented. 
  We study the meson sector of 2+1 dimensional light-front QCD using a Bloch effective Hamiltonian in the first non-trivial order. The resulting two dimensional integral equation is converted into a matrix equation and solved numerically. We investigate the efficiency of Gaussian quadrature in achieving the cancellation of linear and logarithmic light-front infrared divergences. The vanishing energy denominator problem which leads to severe infrared divergences in 2+1 dimensions is investigated in detail. Our study indicates that in the context of Fock space based effective Hamiltonian methods to tackle gauge theories in 2+1 dimensions, approaches like similarity renormalization method may be mandatory due to uncanceled infrared divergences caused by the vanishing energy denominator problem. We define and study numerically a reduced model which is relativistic, free from infrared divergences, and exhibits logarithmic confinement. The manifestation and violation of rotational symmetry as a function of the coupling are studied quantitatively. 
  We study D-branes in the Lorentzian AdS_3 background from the viewpoint of boundary states, emphasizing the role of open-closed duality in string theory. Employing the world sheet with Lorentzian signature, we construct the Cardy states with the discrete series. We show that they are compatible with (1) unitarity and normalizability, and (2) the spectral flow symmetry, in the open string spectrum. We also discuss their brane interpretation. We further show that in the case of superstrings on AdS_3 x S^3 x T^4, our Cardy states yield an infinite number of physical BPS states in the open string channel, on which the spectral flows act consistently. 
  A consistent framework has been put forward to quantize the isentropic, compressible and inviscid fluid model in the Hamiltonian framework, using the Clebsch parameterization. The naive quantization is hampered by the non-canonical (in particular field dependent) Poisson Bracket algebra. To overcome this problem, the Batalin-Tyutin \cite{12} quantization formalism is adopted in which the original system is converted to a local gauge theory and is embedded in a {\it canonical} extended phase space. In a different reduced phase space scheme \cite{vy} also the original model is converted to a gauge theory and subsequently the two distinct gauge invariant formulations of the fluid model are related explicitly. This strengthens the equivalence between the relativistic membrane (where a gauge invariance is manifest) and the fluid (where the gauge symmetry is hidden). Relativistic generalizations of the extended model is also touched upon. 
  Old fashioned duality used to derive the closed string field theory for magnetic vortex from the gauge theory with Higgs scalar, is applied to the string theories. The bosonic sring theory coupled to the Kalb-Ramond 2-form field is dually transformed to the 6-brane theory coupled to the 7-form field. The old fashioned dual transformation is also examined for the Type IIA and IIB superstrings. For this study, the string field theoretical treatment of the bosonic and fermionic strings is developped based on the reparametrization invariant formulation of strings by Marshall and Ramond. In order for the self-duality of the Type IIB superstring to appear, the following dual correspondence may happen: the dual transformation of the NS-NS field functional is the bosonization of the R-R one. 
  An exact time-dependent solution of field equations for the 3-d gauge field model with a Chern-Simons (CS) topological mass is found. Limiting cases of constant solution and solution with vanishing topological mass are considered. After Lorentz boost, the found solution describes a massive nonlinear non-abelian plane wave. For the more complicate case of gauge fields with CS mass interacting with a Higgs field, the stochastic character of motion is demonstrated. 
  We analyze a completely integrable two-dimensional quantum-mechanical model that emerged in the recent studies of the compound gluonic states in multi-color QCD at high energy. The model represents a generalization of the well-known homogenous Heisenberg spin magnet to infinite-dimensional representations of the SL(2,C) group and can be reformulated within the Quantum Inverse Scattering Method. Solving the Yang-Baxter equation, we obtain the R-matrix for the SL(2,C) representations of the principal series and discuss its properties. We explicitly construct the Baxter Q-operator for this model and show how it can be used to determine the energy spectrum. We apply Sklyanin's method of the Separated Variables to obtain an integral representation for the eigenfunctions of the Hamiltonian. We demonstrate that the language of Feynman diagrams supplemented with the method of uniqueness provide a powerful technique for analyzing the properties of the model. 
  I derive a procedure to generate sum rules for the trace anomalies a and a'. Linear combinations of Delta a = a_UV-a_IR and Delta a' = a'_UV-a'_IR are expressed as multiple flow integrals of the two-, three- and four-point functions of the trace of the stress tensor. Eliminating Delta a', universal flow invariants are obtained, in particular sum rules for Delta a. The formulas hold in the most general renormalizable quantum field theory (unitary or not), interpolating between UV and IR conformal fixed points. I discuss the relevance of these sum rules for the issue of the irreversibility of the RG flow. The procedure can be generalized to derive sum rules for the trace anomaly c. 
  A-infinity algebras and categories are known to be the algebraic structures behind open string field theories. In this note we comment on the relevance of the homology construction of A-infinity categories to superpotentials. 
  We consider models with N U(1) gauge fields A_{\mu}^n, N Kalb-Ramond fields B_{\mu \nu}^n, an arbitrary bare action and a fixed UV cutoff \Lambda. Under mild assumptions these can be obtained as effective low energy theories of SU(N+1) Yang Mills theories in the maximal abelian gauge. For a large class of bare actions they can be solved in the large N limit and exhibit confinement. The confining phase is characterized by an approximate ``low energy'' vector gauge symmetry under which the Kalb-Ramond fields B_{\mu\nu}^n transform. The same symmetry allows for a duality transformation showing that magnetic monopoles have condensed. The models allow for various mechanisms of confinement, depending on which sources for A_{\mu}^n or B_{\mu \nu}^n are switched on, but the area law for the Wilson loop is obtained in any case. 
  The gauge invariance of cubic open superstring field theory is considered in a framework of level truncation, and applications to the tachyon condensation problem are discussed. As it is known, in the bosonic case the Feynman-Siegel gauge is not universal within the level truncation method. We explore another gauge that is more suitable for calculation of the tachyon potential for fermionic string at level (2,6). We show that this new gauge has no restrictions on the region of its validity at least at this level. 
  We derive a simple no-go theorem relating to self-tuning solutions to the cosmological constant for observers on a brane, which rely on a singularity in an extra dimension. The theorem shows that it is impossible to shield the singularity from the brane by a horizon, unless the positive energy condition (rho+p >= 0) is violated in the bulk or on the brane. The result holds regardless of the kinds of fields which are introduced in the bulk or on the brane, whether Z_2 symmetry is imposed at the brane, or whether higher derivative terms of the Gauss-Bonnet form are added to the gravitational part of the action. However, the no-go theorem can be evaded if the three-brane has spatial curvature. We discuss explicit realizations of such solutions which have both self-tuning and a horizon shielding the singularity. 
  As pointed out, chiral non-commutative theories exist, and examples can be constructed via string theory. Gauge anomalies require the matter content of individual gauge group factors, including U(1) factors, to be non-chiral. All ``bad'' mixed gauge anomalies, and also all ``good'' (e.g. for $\pi ^0\to \gamma \gamma$) ABJ type flavor anomalies, automatically vanish in non-commutative gauge theories. We interpret this as being analogous to string theory, and an example of UV/IR mixing: non-commutative gauge theories automatically contain ``closed string,'' Green-Schwarz fields, which cancel these anomalies. 
  In a recent paper Bernevig, Brodie, Susskind and Toumbas constructed a brane realization of the Quantum Hall fluid. Since then it has been realized that the Quantum Hall system is very closely related to non--commutative Chern Simons theory and this suggests alternative brane constructions which we believe are more reliable and clear. In this paper a brane construction is given for the non--commutative Chern Simons Matrix formulation of the Quantum Hall system as described by in recent papers by Susskind, Polychronakos and by Hellerman and Van Raamsdonk. The system is a generalized version of Berkooz's ``Rigid Light Cone Membrane which occurs as an excition of the DLCQ description of the M5--brane in a background 3--form field. The original construction of Berkooz corresponds to the fully filled $\nu =1$ state of the QH system. To change the filling fraction to $\nu = 1/(k+1)$ a system of $k$ background D8-branes is required. Quasi--hole excitations can be generated by passing a D6-brane though the Rigid Membrane. 
  In this paper we discuss a physical observable which is drastically different in a brane-world scenario. To date, the Randall-Sundrum model seems to be consistent with all experimental tests of general relativity. Specifically, we examine the so-called gravitomagnetic effect in the context of the Randall-Sundrum (RS) model. This treatment, of course, assumes the recovery of the Kerr metric in brane-worlds which we have found to the first order in the ratio of the brane separation to the radius of the AdS$_5$, $(\ell/r)$. We first show that the second Randall-Sundrum model of one brane leaves the gravitomagnetic effect unchanged. Then, we consider the two-brane scenario of the original Randall-Sundrum proposal and show that the magnitude of the gravitomagnetic effect depends heavily on the ratio of $(\ell/r)$. Such dependence is a result of the geometrodynamic spacetime and does not appear in static scenarios. We hope that we will be able to test this proposal experimentally with data from NASA's Gravity Probe B (GP-B)and possibly disprove either the Randall-Sundrum two-brane scenario or standard general relativity. 
  We present the locally supersymmetric formulation of unimodular gravity theory in D (1\le D \le 11) dimensions, namely supergravity theory with the metric tensor whose determinant is constrained to be unity. In such a formulation, the usual fine-tuning of cosmological constant is no longer needed, but its value is understood as an initial condition. Moreover, the zero-ness of the cosmological constant is concluded as the most probable configuration, based on the effective vacuum functional. We also show that the closure of supersymmetry gauge algebra is consistent with the unimodular condition on the metric. 
  It was suggested by Sugimoto that there is a new supersymmetry breaking mechanism by an orientifold plane which is oppositely charged as the usual one. Here we prove the trace formula for this system to show that the supersymmetry is broken not explicitly but spontaneously. We also discuss the possibility of interpreting the orientifold plane as an intrinsic object of the superstring theory. 
  A SUSY breaking mechanism with no messenger fields is proposed. We assume that our world is on a domain wall and SUSY is broken only by the coexistence of another wall with some distance from our wall. We find an ${\cal N}=1$ model in four dimensions which admits an exact solution of a stable non-BPS configuration of two walls and studied its properties explicitly. We work out how various soft SUSY breaking terms can arise in our framework. Phenomenological implications are briefly discussed. We also find that effective SUSY breaking scale becomes exponentially small as the distance between two walls grows. 
  The algebra of conserved observables of the SO(4,1) gauge-invariant theory of the Dirac fermions in the external field of the Kaluza-Klein monopole is investigated. It is shown that the Dirac conserved operators have physical parts associated with Pauli operators that are also conserved in the sense of the Klein-Gordon theory. In this way one gets simpler methods of analyzing the properties of the conserved Dirac operators and their main algebraic structures including the representations of dynamical algebras governing the Dirac quantum modes. 
  A model considered in the paper generalizes supergravity type model to the case of delocalized membrane sources. A generalization of intersecting p-brane solution with delocalized membranes is presented. 
  We clarify the relation between the Dixmier-Douady class on the space of self adjoint Fredholm operators (`universal B-field') and the curvature of determinant bundles over infinite-dimensional Grassmannians. In particular, in the case of Dirac type operators on a three dimensional compact manifold we obtain a simple and explicit expression for both forms. 
  The multiplicative anomaly related to the functional regularized determinants involving products of elliptic operators is introduced and some of its properties discussed. Its relevance concerning the mathematical consistency is stressed. With regard to its possible physical relevance, some examples are illustrated. 
  We discuss the dimensional reduction of various effective actions, particularly that of the closed Bosonic string and pure gravity, to two and three dimensions. The result for the closed Bosonic string leads to coset symmetries which are in agreement with those recently predicted and argued to be present in a new unreduced formulation of this theory. We also show that part of the Geroch group appears in the unreduced duality symmetric formulation of gravity recently proposed. We conjecture that this formulation can be extended to a non-linear realisation based on a Kac-Moody algebra which we identify. We also briefly discuss the proposed action of Bosonic M-theory. 
  For redundant second-class constraints the Dirac brackets cannot be defined and new brackets must be introduced. We prove here that the Jacobi identity for the new brackets must hold on the surface of the second-class constraints. In order to illustrate our proof we work out explicitly the cases of a fractional spin particle in 2+1 dimensions and the original Brink-Schwarz massless superparticle in D=10 dimensions in a Lorentz covariant constraints separation. 
  Deformation quantization (the Moyal deformation) of SDYM equation for the algebra of the area preserving diffeomorphisms of a 2-surface $\Sigma_{2}$, sdiff($\Sigma_{2}$), is studied. Deformed equation we call the master equation (ME) as it can be reduced to many integrable nonlinear equations in mathematical physics. Two sets of concerved charges for ME are found. Then the linear systems for ME (the Lax pairs) associated with the conserved charges are given. We obtain the dressing operators and the infinite algebra of hidden symmetries of ME. Twistor construction is also done. 
  Two- and three-point correlation functions of arbitrary protected operators are constructed in N=4 SYM using analytic superspace methods. The OPEs of two chiral primary multiplets are given. It is shown that the $n$-point functions of protected operators for $n\leq4$ are invariant under $U(1)_Y$ and it is argued that this implies that the two- and three-point functions are not renormalised. It is shown explicitly how unprotected operators can be accommodated in the analytic superspace formalism in a way which is fully compatible with analyticity. Some new extremal correlators are exhibited. 
  We construct an explicit class of solutions of type IIB supergravity that is a smooth deformation of the 3-brane class of solutions. The solution is nonsupersymmetric and involves nontrivial dilaton and axion fields as well as the standard 5-form field strength. One of the main features of the solution is that for large values of the radius the deformation is small and it asymptotically approaches the undeformed 3-brane solution, signaling a restoration of conformal invariance in the UV for the dual gauge theory. We suggest that the supergravity deformation corresponds to a massive deformation on the dual gauge theory and consequently the deformed theory has the undeformed one as an ultraviolet fixed point. In cases where the original 3-brane solution preserves some amount of supersymmetry we suggest that the gauge theory interpretation is that of soft supersymmetry breaking. We discuss the deformation for D3-branes on the conifold and the generalized conifold explicitly. We show that the semiclassical behavior of the Wilson loop suggests that the corresponding gauge theory duals are confining. 
  We study linearized perturbations of eleven-dimensional $OSp(32|1)$ Chern-Simons supergravity. The action contains a term that changes the value of the cosmological constant, as considered by Horava. It is shown that the spectrum contains a 3-form and a 6-form whose field strengths are dual to each other, thus providing a link with the eleven-dimensional supergravity of Cremmer, Julia and Scherk. The linearized equations for the graviton and Rarita-Schwinger field are shown to be the standard ones as well. 
  We investigate the complete phase diagram of the decoupled world-sheet theory of (P,Q) strings. These theories include 1+1 dimensional super Yang-Mills theory and non-commutative open string theory. We find that the system exhibits a rich fractal phase structure, including a cascade of alternating supergravity, gauge theory, and matrix string theory phases. The cascade proceeds via a series of SL(2,Z) S-duality transformations, and depends sensitively on P and Q. In particular, we find that the system may undergo multiple Hagedorn-type transitions as the temperature is varied. 
  We present a teleparallel complex gravity as the foundation for the formulation of noncommutative gravity theory. The negative energy ghosts in the conventional formulation with U(1,3) local Lorentz connection no longer exists, since the local Lorentz invariance is broken down to U(1,3) global Lorentz symmetry. As desired, our teleparallel complex gravity theory also passes the key classical test of perihelion advance of Mercury. Based on this result, we present a lagrangian for the noncommutative teleparallel gravity theory. 
  We study phase structure of the moduli space of a D0-brane on the orbifold C^3/Z_2 \times Z_2 based on stability of quiver representations. It is known from an analysis using toric geometry that this model has multiple phases connected by flop transitions. By comparing the results of the two methods, we obtain a correspondence between quiver representations and geometry of toric resolutions of the orbifold. It is shown that a redundancy of coordinates arising in the toric description of the D-brane moduli space, which is a key ingredient of disappearance of non-geometric phases, is understood from the monodromy around the orbifold point. We also discuss why only geometric phases appear from the viewpoint of stability of D0-branes. 
  The motion of a test Dq-brane in a Dp-brane background is studied. The induced metric on the test brane is then interpreted as the cosmology of the test brane universe. One is then able to resolve the resulting cosmological singularities. In particular, for a D3-brane in a D5-brane background, one finds a 3+1 dimensional FRW universe with equation of state $p = \ed$. It has been argued that this may have been the dominant form of matter at very early times. 
  The Gauge Technique has been applied to QED$_{2+1}$ in the quenched case with infrared subtraction. The behaviour of the fermion propagator near the threshold is then found to be \[ S(p)\to \frac{(\gamma \cdot p+m)}{(p^{2}-m^{2})}(\frac{p^{2}-m^{2}}{% 2m^{2}})^{\zeta}\exp (-\frac{\eta \varsigma}{2}), \] where $\varsigma =e^{2}/(4\pi m)$ and this is gauge invariant except the exponential factor. We also find a spectral function in the Landau and Yennie like gauge. The propagators $S(p)$ are expressed in terms of $\Phi (z,1,\varsigma)$ explicitly .The vacuum expectation value $< \overline{\psi}\psi >$ is gauge dependent . Thus dynamical mass generation does not occur. 
  We find M-theory solutions that are holographic duals of flows of the maximally supersymmetric N=8 scalar-fermion theory in (2+1) dimensions. In particular, we construct the M-theory solution dual to a flow in which a single chiral multiplet is given a mass, and the theory goes to a new infra-red fixed point. We also examine this new solution using M2-brane probes. The (2+1)-dimensional field theory fixed-point is closely related to that of Leigh and Strassler, while the M-theory solution is closely related to the corresponding IIB flow solution. We recast the IIB flow solution in a more geometric manner and use this to obtain an Ansatz for the M-theory flow. We are able to generalize our solution further to obtain flows with del Pezzo sub-manifolds, and we give an explicit solution with a conifold singularity. 
  A renormalization group flow equation with a scale-dependent transformation of field variables gives a unified description of fundamental and composite degrees of freedom. In the context of the effective average action, we study the renormalization flow of scalar bound states which are formed out of fundamental fermions. We use the gauged Nambu--Jona-Lasinio model at weak gauge coupling as an example. Thereby, the notions of bound state or fundamental particle become scale dependent, being classified by the fixed-point structure of the flow of effective couplings. 
  Theta series for exceptional groups have been suggested as a possible description of the eleven-dimensional quantum supermembrane. We present explicit formulae for these automorphic forms whenever the underlying Lie group $G$ is split (or complex) and simply laced. Specifically, we review and construct explicitly the minimal representation of $G$, generalizing the Schr\"odinger representation of symplectic groups. We compute the spherical vector in this representation, i.e. the wave function invariant under the maximal compact subgroup, which plays the role of the summand in the automorphic theta series. We also determine the spherical vector over the complex field. We outline how the spherical vector over the $p$-adic number fields provides the summation measure in the theta series, postponing its determination to a sequel of this work. The simplicity of our result is suggestive of a new Born-Infeld-like description of the membrane where U-duality is realized non-linearly. Our results may also be used in constructing quantum mechanical systems with spectrum generating symmetries. 
  We propose a dynamical mechanism of localization of gauge fields on a brane in which gauge bosons are excitations of the brane itself or composites made out of matter fields localized on the brane. The mechanism is operative for both Abelian and non-Abelian gauge fields. Several scalar and scalar-fermion composite models of gauge fields are considered. The models exhibit exact gauge invariance and therefore charge universality of gauge interactions is automatically preserved. The mechanism is shown to be equivalent to a modification of the Dvali, Gabadadze and Shifman scenario in which gauge bosons have no bulk kinetic terms and only possess induced kinetic terms on the brane. 
  We show that the Abelian Higgs field equations in the background of the four dimensional AdS-Schwarzschild black hole have a vortex line solution. This solution, which has axial symmetry, is a generalization of the AdS spacetime Nielsen-Olesen string. By a numerical study of the field equations, we show that black hole could support the Abelian Higgs field as its Abelian hair. Also, we conside the self gravity of the Abelian Higgs field both in the pure AdS spacetime and AdS-Schwarzschild black hole background and show that the effect of string as a black hole hair is to induce a deficit angle in the AdS-Schwarzschild black hole. 
  In this paper, we describe a method for obtaining the nonabelian Seiberg-Witten map for any gauge group and to any order in theta. The equations defining the Seiberg-Witten map are expressed using a coboundary operator, so that they can be solved by constructing a corresponding homotopy operator. The ambiguities, of both the gauge and covariant type, which arise in this map are manifest in our formalism. 
  Using recently proposed non-linearly realized supersymmetry in non-Abelian gauge theory corrected to the order (alpha')^2, we derive the non-linear BPS equations in the background B-field for the U(2) monopoles and instantons. We show that these non-Abelian non-linear BPS equations coincide with the non-commutative anti-self-dual equations via the Seiberg-Witten map. 
  The stability of 5-D brane-world models under conformal perturbations is investigated. The analysis is carried out in the general case and then it is applied to particular solutions. It is shown that models with the Poincare and the de Sitter branes are unstable because they have negative mass squared of gravexcitons whereas models with the Anti de Sitter branes have positive gravexciton mass squared and are stable. It is also shown that 4-D effective cosmological and gravitational constants on branes as well as gravexciton masses undergo hierarchy: they have different values on different branes. 
  We obtain ADM and Komar surface integrals for the energy density, tension and angular momentum density of stationary $p$-brane solutions of Einstein's equations. We use them to derive a Smarr-type formula for the energy density and thence a first law of black brane mechanics. The intensive variable conjugate to the worldspace p-volume is an `effective' tension that equals the ADM tension for uncharged branes, but vanishes for isotropic boost-invariant charged branes. 
  The collision of plane waves corresponding to massless states of closed string is considered in $D$-dimensional space-time. The reduced tree level effective action is known to be manifestly $O(d,d)$ invariant, $d$ being the number of transverse spatial dimensions in the collision process. We adopt a coset space reformulation of the effective two dimensional theory and discuss the relation of this process with classical integrable systems in two dimensions in the presence of gravity. We show how it is possible to generate new backgrounds for the scattering process, from known background solutions to the equations of motion, in the coset reformulation. We present explicit calculations for the case of four space-time dimensions as an illustrative example. 
  We consider a Dirac field in 2+1 Euclidean dimensions, in the presence of a linear domain wall defect in its mass, and a constant electromagnetic field. We evaluate the exact fermionic determinant for the situation where the defect is assumed to be rectilinear, static, and the gauge field is minimally coupled to the fermions. We discuss the dependence of the result on the (unique) independent geometrical parameter of this system, namely, the relative orientation of the wall and the direction of the external field. We apply the result for the determinant to the evaluation of the vacuum energy. 
  A cubic algebraic equation for the effective parametrizations of the standard gravitational Lagrangian has been obtained without applying any variational principle.It was suggested that such an equation may find application in gravity theory, brane, string and Rundall-Sundrum theories. The obtained algebraic equation was brought by means of a linear-fractional transformation to a parametrizable form, expressed through the elliptic Weierstrass function, which was proved to satisfy the standard parametrizable form, but with $g_{2}$ and $g_{3}$ functions of a complex variable instead of the definite complex numbers (known from the usual arithmetic theory of elliptic functions and curves). The generally divergent (two) infinite sums of the inverse first and second powers of the poles in the complex plane were shown to be convergent in the investigated particular case, and the case of the infinite point of the linear-fractional transformation was investigated. Some relations were found, which ensure the parametrization of the cubic equation in its general form with the Weierstrass function. 
  Inflation is now a basic ingredient of the modern cosmology. After reviewing the basic characteristics of inflation, we briefly discuss inflation in string and brane theories, focusing on the problem of the exit from inflation in these theories. We present a model, based on type-0 string theory, in which our universe after the inflationary phase, passes smoothly to a flat Robertson-Walker universe. 
  Thick branes obtained from a bulk action containing Gauss-Bonnet self-interactions are analyzed in light of the localization properties of the various modes of the geometry. The entangled system describing the localization of the tensor, vector and scalar fluctuation is decoupled in terms of variables invariant for infinitesimal coordinate transformations. The dynamics of the various zero modes is discussed and solved in general terms. Provided the four-dimensional Planck mass is finite and provided the geometry is everywhere regular, it is shown that the vector and scalar zero modes are not localized. The tensor zero mode is localized leading to four-dimensional gravity. The general formalism is illustrated through specific analytical examples 
  We investigate two-dimensional dilaton gravity models with a power-law dilaton potential, whose black hole solutions contain, among others, the dimensional reduction of the Schwarzschild black hole, the Anti-de Sitter black hole and Rindler spacetime. We show that the ground state of these models satisfies simple transformation laws under the SL(2,R) conformal group. We use these transformation laws to explain the scaling behavior of the thermodynamical parameters of the black hole and the nonextensivity of the thermodynamical system. The black hole thermodynamical behavior, in particular its entropy, is reproduced by a mapping into a two-dimensional conformal field theory on the cylinder. 
  The bound state solutions of two fermions interacting by a scalar exchange are obtained in the framework of the explicitly covariant light-front dynamics. The stability with respect to cutoff of the J$^{\pi}$=$0^+$ and J$^{\pi}$=$1^+$ states is studied. The solutions for J$^{\pi}$=$0^+$ are found to be stable for coupling constants $\alpha={g^2\over4\pi}$ below the critical value $\alpha_c\approx 3.72$ and unstable above it. The asymptotic behavior of the wave functions is found to follow a ${1\over k^{2+\beta}}$ law. The coefficient $\beta$ and the critical coupling constant $\alpha_c$ are calculated from an eigenvalue equation. The binding energies for the J$^{\pi}$=$1^+$ solutions diverge logarithmically with the cutoff for any value of the coupling constant. For a wide range of cutoff, the states with different angular momentum projections are weakly split. 
  According to the Second Law of Thermodynamics, cycles applied to thermodynamic equilibrium states cannot perform work (passivity property of thermodyamic equilibrium states). In the presence of matter this can hold only in the rest frame of the matter, as moving matter drives, e.g., windmills and turbines. If, however, a homogeneous and stationary state has the property that no cycle can perform more work than an ideal windmill, then it can be shown that there is some inertial frame where the state is a thermodynamic equilibrium state. This provides a covariant characterization of thermodynamic equilibrium states.   In the absence of matter, cycles should perform work only when driven by nonstationary inertial forces caused by the observer's motion. If a pure state of a relativistic quantum field theory behaves this way, it satisfies the spectrum condition and exhibits the Unruh effect. 
  We show that in the system of two fermions interacting by scalar exchange, the solutions for J$^{\pi}$=$0^+$ bound states are stable without any cutoff regularization, for values of the coupling constant $\alpha$ below a critical value $\alpha_c$. This latter is calculated from an eigenvalue equation. 
  We study the thermodynamic relations of conformal field theories (CFTs), which are holographically dual to anti-de Sitter-Schwarzschild bulk space-times. A Cardy-Verlinde formula is derived thermodynamically for CFTs living on S^n x R with S^n having an arbitrary radius. The Hawking-Page phase transition of the CFT is described using Landau's theory of phase transitions, and an alternative derivation of the Cardy-Verlinde formula is presented. The condensate in the high temperature phase is identified as being composed of radiational matter. 
  We extend the gauge coupling of Super-Yang-Mills theories to an external superfield composed out of a chiral and an antichiral superfield and perform renormalization in the extended model. In one-loop order we find an anomalous breaking of supersymmetry which vanishes in the limit to constant coupling. The anomaly arises form non-local contributions and its coefficient is gauge and scheme independent and strictly of one-loop order. In the perturbative framework of the construction the anomaly cannot be absorbed as a counterterm to the classical action. With local gauge coupling the symmetric counterterms with chiral integration are holomorphic functions and this property is independent from the specific regularization scheme. Thus, the symmetric counterterm to the Yang-Mills part is of one-loop order only - and it is the anomaly which gives rise to the two-loop order of the gauge $\beta$-function in pure Super-Yang-Mills theories and to its closed all-order expression. 
  A new presentation of the Borchers-Buchholz result of the Lorentz-invariance of the energy-momentum spectrum in theories with broken Lorentz symmetry is given in terms of properties of the Green's functions of microcausal Bose and Fermi-fields. Strong constraints based on complex geometry phenomenons are shown to result from the interplay of the basic principles of causality and stability in Quantum Field Theory: if microcausality and energy-positivity in all Lorentz frames are satisfied, then it is unavoidable that all stable particles of the theory be governed by Lorentz-invariant dispersion laws: in all the field sectors, discrete parts outside the continuum as well as the thresholds of the continuous parts of the energy-momentum spectrum, with possible holes inside it, are necessarily represented by mass-shell hyperboloids (or the light-cone). No violation of this geometrical fact can be produced by spontaneous breaking of the Lorentz symmetry. 
  If quantum fields exist in extra compact dimensions, they will give rise to a quantum vacuum or Casimir energy. That vacuum energy will manifest itself as a cosmological constant. The fact that supernova and cosmic microwave background data indicate that the cosmological constant is of the same order as the critical mass density to close the universe supplies a lower bound on the size of the extra dimensions. Recent laboratory constraints on deviations from Newton's law place an upper limit. The allowed region is so small as to suggest that either extra compact dimensions do not exist, or their properties are about to be tightly constrained by experimental data. 
  In logarithmic conformal field theory, primary fields come together with logarithmic partner fields on which the stress-energy tensor acts non-diagonally. Exploiting this fact and global conformal invariance of two- and three-point functions, operator product expansions of logarithmic operators in arbitrary rank logarithmic conformal field theory are investigated. Since the precise relationship between logarithmic operators and their primary partners is not yet sufficiently understood in all cases, the derivation of operator product expansion formulae is only possible under certain assumptions. The easiest cases are studied in this paper: firstly, where operator product expansions of two primaries only contain primary fields, secondly, where the primary fields are pre-logarithmic operators. Some comments on generalization towards more relaxed assumptions are made, in particular towards the case where logarithmic fields are not quasi-primary. We identify an algebraic structure generated by the zero modes of the fields, which proves useful in determining settings in which our approach can be successfully applied. 
  We consider the evolution of a scalar field coupled to curvature in topological black hole spacetimes. We solve numerically the scalar wave equation with different curvature-coupling constant $\xi$ and show that a rich spectrum of wave propagation is revealed when $\xi$ is introduced. Relations between quasinormal modes and the size of different topological black holes have also been investigated. 
  We construct braneworld effective action in two brane Randall-Sundrum model and show that the radion mode plays the role of a scalar field localizing essentially nonlocal part of this action. Non-minimal curvature coupling of this field reflects the violation of AdS/CFT-correspondence for finite values of brane separation. Under small detuning of the brane tension from the Randall-Sundrum flat brane value, the radion mode can play the role of inflaton. Inflationary dynamics corresponds to branes moving apart in the field of repelling interbrane inflaton-radion potential and implies the existence acceleration stage caused by remnant cosmological constant at late (large brane separation) stages of evolution. We discuss the possibility of fixing initial conditions in this model within the concept of braneworld creation from the tunneling or no-boundary cosmological state, which formally replaces the conventional moduli stabilization mechanism. 
  Gauge symmetry breaking by boundary conditions on a manifold is known to be equivalent to Wilson-line breaking through a background gauge field, and is therefore spontaneous. These equivalent pictures are related by a non-periodic gauge transformation. However, we find that boundary condition gauge symmetry breaking on orbifolds is explicit; there is no gauge where all the breaking can be attributed to a background gauge field. In the case of a five-dimensional SU(5) grand unified theory on S^1/Z_2, the vacuum with gauge symmetry broken to SU(3) x SU(2) x U(1) and that with SU(5) preserved are completely disconnected: there is no physical process which causes tunneling between the two. This allows a certain localized explicit breaking of SU(5) on one of the orbifold fixed points in the theory with SU(5) breaking. Split multiplets on this fixed point are shown not to induce violations of unitarity in scattering amplitudes. 
  We present a new class of quantum field theories which are exactly solvable. The theories are generated by introducing Pauli-Villars fermionic and bosonic fields with masses degenerate with the physical positive metric fields. An algorithm is given to compute the spectrum and corresponding eigensolutions. We also give the operator solution for a particular case and use it to illustrate some of the tenets of light-cone quantization. Since the solutions of the solvable theory contain ghost quanta, these theories are unphysical. However, we also discuss how perturbation theory in the difference between the masses of the physical and Pauli-Villars particles could be developed, thus generating physical theories. The existence of explicit solutions of the solvable theory also allows one to study the relationship between the equal-time and light-cone vacua and eigensolutions. 
  We derive a constraint (string equation) which together with the Toda Lattice hierarchy determines completely the integrable structure of the compactified 2D string theory. The form of the constraint depends on a continuous parameter, the compactification radius R. We show how to use the string equation to calculate the free energy and the correlation functions in the dispersionless limit. We sketch the phase diagram and the flow structure and point out that there are two UV critical points, one of which (the sine-Liouville string theory) describes infinitely strong vortex or tachyon perturbation. 
  We consider a real Lagrangian off-critical submodel describing the soliton sector of the so-called conformal affine $sl(3)^{(1)}$ Toda model coupled to matter fields (CATM). The theory is treated as a constrained system in the context of Faddeev-Jackiw and the symplectic schemes. We exhibit the parent Lagrangian nature of the model from which generalizations of the sine-Gordon (GSG) or the massive Thirring (GMT) models are derivable. The dual description of the model is further emphasized by providing the relationships between bilinears of GMT spinors and relevant expressions of the GSG fields. In this way we exhibit the strong/weak coupling phases and the (generalized) soliton/particle correspondences of the model. The $sl(n)^{(1)}$ case is also outlined. 
  Feynman's functional formulation of statistical mechanics is used to study the renormalizability of the well known Linear Chiral Sigma Model in the presence of fermionic fields at finite temperature in an alternative way. It is shown that the renormalization conditions coincide with those of the zero temperature model. 
  We propose a model for noncommutative quantum cosmology by means of a deformation of minisuperspace. For the Kantowski-Sachs metric we are able to find the exact wave function. We construct wave packets and show that new quantum states that ``compete'' to be the most probable state appear, in clear contrast with the commutative case. A tunneling process could be possible among these states. 
  This review paper is a continuation of hep-th/0012145 and it deals primarily with noncommutative ${\mathbb R}^{d}$ spaces. We start with a discussion of various algebras of smooth functions on noncommutative ${\mathbb R}^{d}$ that have different asymptotic behavior at infinity. We pay particular attention to the differences arising when working with nonunital algebras and the unitized ones obtained by adjoining the unit element. After introducing main objects of noncommutative geometry over those algebras such as inner products, modules, connections, etc., we continue with a study of soliton and instanton solutions in field theories defined on these spaces. The discussion of solitons includes the basic facts regarding the exact soliton solutions in the Yang-Mills-Higgs systems as well as an elementary discussion of approximate solitons in scalar theories in the $\theta \to \infty$ limit. We concentrate on the module structure and topological numbers characterizing the solitons. The section on instantons contains a thorough description of noncommutative ADHM construction, a discussion of gauge triviality conditions at infinity and the structure of a module underlying the ADHM instanton solution. Although some familiarity with general ideas of noncommutative geometry reviewed in the first part is expected from the reader, this part is largely independent from the first one. 
  We investigate the dynamics of a pair of (4+1)-dimensional black holes in the moduli approximation and with fixed angular momentum. We find that spinning black holes at small separations are described by the de Alfaro, Fubini and Furlan model. For more than two black holes, we find an explicit expression for the three-body interactions in the moduli metric by associating them with the one-loop three-point amplitude of a four-dimensional $\phi^3$ theory. We also investigate the dynamics of a three black hole system in various approximations. 
  The consequences of certain simple assumptions like smoothness of ground state properties and vanishing of the vacuum energy (at least perturbatively) are explored. It would be interesting from the point of view of building realistic theories to obtain these properties without supersymmetry. Here we show, however, at least in some quantum mechanical models, that these simple assumptions lead to supersymmetric theories. 
  Properties of the domain wall (kink) solution in the 5 dimensional Randall-Sundrum model are examined both {\it analytically} and {\it numerically}. The configuration is derived by the bulk Higgs mechanism. We focus on 1) the convergence property of the solution, 2) the stableness of the solution, 3) the non-singular property of the Riemann curvature, 4) the behaviours of the warp factor and the Higgs field. It is found that the bulk curvature changes the sign around the surface of the wall. We also present some {\it exact} solutions for two simple cases: a) the no potential case, b) the cosmological term dominated case. Both solutions have the (naked) curvature singularity. We can regard the domain wall solution as a singularity resolution of the exact solutions. 
  We study the dynamical consequences of Maggiore's unique generalised uncertainty principle (GUP). We find that it leads naturally, and generically, to novel consequences. In the high temperature limit, there is a drastic reduction in the degrees of freedom, of the type found, for example, in strings far above the Hagedorn temperature. In view of this, the present GUP may perhaps be taken as the new version of the Heisenberg uncertainty principle, conjectured by Atick and Witten to be responsible for such reduction. Also, the present GUP leads naturally to varying speed of light and modified dispersion relations. They are likely to have novel implications for cosmology and black hole physics, a few of which we discuss qualitatively. 
  An informal introduction to our recent work on the principal chiral model with boundary. We found that both classically integrable boundary conditions and quantum boundary S-matrices were classified by the symmetric spaces G/H. The connection is explained by the presence of a twisted Yangian algebra of non-local charges. 
  We consider five-dimensional brane worlds with N=2 gauged supergravity in the bulk coupled supersymmetrically to two boundary branes at the fixed points of a Z_2 symmetry. We analyse two mechanisms that break supersymmetry either by choosing flipped fermionic boundary conditions on the boundary branes or by modifying the gravitino variation to include both Z_2-odd and Z_2-even operators. In all cases we find the corresponding background. Including an even part in the gravitino variation leads to tilted branes. Choosing the flipped boundary conditions leads to AdS_4 branes and stabilized radion in the detuned case, when the expectation value of the even variation is nonzero. Another solution has the interpretation of moving AdS_4 branes separated by a horizon. The solution with moving branes separated by a horizon can be extended to the tuned case. In the presence of a horizon, temperature mediation communicates supersymmetry breakdown to the branes. 
  There is a Nahm transform for two-dimensional gauge fields which establishes a one-to-one correspondence between the orbit space of U(N) gauge fields with topological charge k defined on a torus and that of U(k) gauge fields with charge N on the dual torus. The main result of this paper is to show that a similar duality transform cannot exist for CPn instantons. This fact establishes a significative difference between 4-D gauge theories and CPn models. The result follows from the global analysis of the moduli space of instantons based on a complete and explicit parametrization of all self-dual solutions on the two-dimensional torus. The boundary of the space of regular instantons is shown to coincide with the space of singular instantons. This identification provides a new approach to analyzing the role of overlapping instantons in the infrared sector of CPn sigma models. 
  For physicists, recent developments in astrophysics and cosmology present exciting challenges. We are conducting "experiments" in energy regimes some of which will be probed by accelerators in the near future, and others which are inevitably the subject of more speculative theoretical investigations. Dark matter is an area where we have hope of making discoveries both with accelerator experiments and dedicated searches. Inflation and dark energy lie in regimes where presently our only hope for a fundamental understanding lies in string theory. 
  In this paper we consider superstring propagation in a nontrivial NS-NS background. We deform the world sheet stress tensor and supercurrent with an infinitesimal B_{\mu\nu} field. We construct the gauge-covariant super-Poincare generators in this background and show that the B_{\mu\nu} field spontaneously breaks spacetime supersymmetry. We find that the gauge-covariant spacetime momenta cease to commute with each other and with the spacetime supercharges. We construct a set of "magnetic" super-Poincare generators that are conserved for constant field strength H_{\mu\nu\lambda}, and show that these generators obey a "magnetic" extension of the ordinary supersymmetry algebra. 
  We study the metrics on the families of moduli spaces arising from probing with a brane the ten and eleven dimensional supergravity solutions corresponding to renormalisation group flows of supersymmetric large n gauge theory. In comparing the geometry to the physics of the dual gauge theory, it is important to identify appropriate coordinates, and starting with the case of SU(n) gauge theories flowing from N=4 to N=1 via a mass term, we demonstrate that the metric is Kahler, and solve for the Kahler potential everywhere along the flow. We show that the asymptotic form of the Kahler potential, and hence the peculiar conical form of the metric, follows from special properties of the gauge theory. Furthermore, we find the analogous Kahler structure for the N=4 preserving Coulomb branch flows, and for an N=2 flow. In addition, we establish similar properties for two eleven dimensional flow geometries recently presented in the literature, one of which has a deformation of the conifold as its moduli space. In all of these cases, we notice that the Kahler potential appears to satisfy a simple universal differential equation. We prove that this equation arises for all purely Coulomb branch flows dual to both ten and eleven dimensional geometries, and conjecture that the equation holds much more generally. 
  Some issues related to quantum anomaly induced effects due to matter are considered. Explicit examples corresponding to quantum creation of d4 dilatonic AdS Universe and of d2 dilatonic AdS Black Hole (BH) are discussed. Motivated by holographic RG, in a similar approach, it is shown that, starting from a 5 dimensional AdS Universe, 4-dimensional de Sitter or AdS world is generated on the boundary of such Universe as a result of quantum effects. A 5-dimensional brane-world cosmological scenario is also considered, where the brane tension is not longer a free parameter, but its role is taken by quantum effects induced by the 4-dimensional conformal anomaly associated with conformal coupled matter. As a result, consistent quantum creations of De Sitter or AdS curved branes are possible. 
  With our current level of understanding, the problem of making string theory predictions is not one of "solving" the theory, but rather of trying to determine whether there are any generic expectations. Within this context, we discuss what it would mean to predict low energy supersymmetry, and consider questions like: what is the form of low energy CP violation, is unification a string prediction, and others. 
  We discuss the compactification of type IIB supergravity on a Calabi-Yau manifold in the presence of both RR and NS fluxes for the three-form fields. We obtain the classical potential both by direct compactification and by using the techniques of N=2 gauged supergravity in four-dimensions. We briefly discuss the properties of such potential and compare the result with previous derivations. 
  We discuss a matrix model for D0-branes on S^3 \times M^7 based on quantum group symmetries. For finite radius of S^3 (i.e. for finite k), it gives results beyond the reach of the ordinary matrix model. For large k all known static properties of branes on S^3 are reproduced. 
  We present, using spectral analysis, a possible way to prove the Riemann's hypothesis (RH) that the only zeroes of the Riemann zeta-function are of the form s=1/2+i\lambda_n. A supersymmetric quantum mechanical model is proposed as an alternative way to prove the Riemann's conjecture, inspired in the Hilbert-Polya proposal; it uses an inverse eigenvalue approach associated with a system of p-adic harmonic oscillators. An interpretation of the Riemann's fundamental relation Z(s) = Z(1-s) as a duality relation, from one fractal string L to another dual fractal string L' is proposed. 
  We formulate Poisson Chern-Simons gauge theories on compact group manifolds. These describe a sector of the large representation limit of noncommutative Chern-Simons in the same way as the light-cone formulation of the membrane action describe a sector of the large N Matrix model. While the formulation we give is on a group manifold, only excitations that are invariant under the left action of the stability group of a weight are allowed. 
  We construct boundary states representing D-strings in $AdS_3$. These wrap twisted conjugacy classes of SL(2,R), and the boundary states are therefore based on continuous representations only. We check Cardy's condition and find a consistent open string spectrum. The open string spectrum on all the D-branes is the same. 
  We have calculated interactions between two fuzzy spheres in 3 dimension. It depends on the distance r between the spheres and the radii rho_1, rho_2. There is no force between the spheres when they are far from each other (long distance case). We have also studied the interaction for r=0 case. We find that an attractive force exists between two fuzzy sphere surfaces. 
  We investigate the effect of the bulk gravitational field on the cosmological perturbations on a brane embedded in the 5D Anti-de Sitter (AdS) spacetime. The effective 4D Einstein equations for the scalar cosmological perturbations on the brane are obtained by solving the perturbations in the bulk. Then the behaviour of the corrections induced by the bulk gravitational field to the conventional 4D Einstein equation are determined. Two types of the corrections are found. First we investigate the corrections which become significant at scales below the AdS curvature scales and in the high energy universe with the energy density larger than the tension of the brane. The evolution equation for the perturbations on the brane is found and solved. Another type of the corrections is induced on the brane if we consider the bulk perturbations which do not contribute to the metric perturbations but do contribute to the matter perturbations. At low energies, they have imaginary mass $m^2=-(2/3) \k^2$ in the bulk where $\k$ is the 3D comoving wave number of the perturbations. They diverge at the horizon of the AdS spacetime. The induced density perturbations behave as sound waves with sound velocity $1/\sqrt{3}$ in the low energy universe. At large scales, they are homogeneous perturbations that depend only on time and decay like radiation. They can be identified as the perturbations of the dark radiation. They produce isocurvature perturbations in the matter dominated era. Their effects can be observed as the shifts of the location and the height of the acoustic peak in the CMB spectrum. 
  Special case calculations are presented, which can be used to put restrictions on the general form of heat kernel coefficients for transmittal boundary conditions and for generalized bag boundary conditions. 
  We identify a class of chiral models where the one-loop effective potential for Higgs scalar fields is finite without any requirement of supersymmetry. It corresponds to the case where the Higgs fields are identified with the components of a gauge field along compactified extra dimensions. We present a six dimensional model with gauge group U(3)xU(3) and quarks and leptons accomodated in fundamental and bi-fundamental representations. The model can be embedded in a D-brane configuration of type I string theory and, upon compactification on a T^2/Z_2 orbifold, it gives rise to the standard model with two Higgs doublets. 
  We use a 0-brane to probe a ten-dimensional near-extremal black hole with N units of 0-brane charge. We work directly in the dual strongly-coupled quantum mechanics, using mean-field methods to describe the black hole background non-perturbatively. We obtain the distribution of W boson masses, and find a clear separation between light and heavy degrees of freedom. To localize the probe we introduce a resolving time and integrate out the heavy modes. After a non-trivial change of coordinates, the effective potential for the probe agrees with supergravity expectations. We compute the entropy of the probe, and find that the stretched horizon of the black hole arises dynamically in the quantum mechanics, as thermal restoration of unbroken U(N+1) gauge symmetry. Our analysis of the quantum mechanics predicts a correct relation between the horizon radius and entropy of a black hole. 
  We study in detail the ADHM construction of U(N) instantons on noncommutative Euclidean space-time R_{NC}^4 and noncommutative space R_{NC}^2 x R^2. We point out that the completeness condition in the ADHM construction could be invalidated in certain circumstances. When this happens, regular instanton configuration may not exist even if the ADHM constraints are satisfied. Some of the existing solutions in the literature indeed violate the completeness condition and hence are not correct. We present alternative solutions for these cases. In particular, we show for the first time how to construct explicitly regular U(N) instanton solutions on R_{NC}^4 and on R_{NC}^2 x R^2. We also give a simple general argument based on the Corrigan's identity that the topological charge of noncommutative regular instantons is always an integer. 
  Critical String Theory is by definition an $S$-matrix theory. In this sense, (quantum) gravity situations where a unitary $S$-matrix may not be a well-defined concept, as a consequence of the existence of macroscopic (global) or microscopic (local) gravitational fluctuations with event horizons, present a challenge to string theory. In this article, we take some modest steps in suggesting alternative treatments of such cases via non-critical (Liouville) strings,which do not have a well-defined S matrix, but for which a superscattering \$ matrix is mathematically consistent. After a brief review of the underlying mathematical formalism, we consider a specific stringy model of induced non-criticality, with dynamical formation of horizons, associated with the recoil of a D-particle defect, embedded in our four-dimensional space time, during its scattering with a (macroscopic) number of closed string states. We study in detail the associated spacetime distortion in the neighbourhood of the defect, which has the form of a finite-radius curved `bubble', matched with a Minkowskian space-time in the exterior. As a consequence of the non-criticality of the underlying sigma-model, the space time is unstable, and has non-trivial stochastic properties: thermal properties due to its ``Rindler accelerating nature'', and entropy growth for an asymptotic observer, associated with information being carried away by the `recoil' degrees of freedom. We also discuss phenomenological (and cosmological) constraints on the model. 
  We study a geometry of the partition function which is defined in terms of a solution of the five-term relation. It is shown that the 3-dimensional hyperbolic structure or Euclidean AdS_3 naturally arises in the classical limit of this invariant. We discuss that the oriented ideal tetrahedron can be assigned to the partition function of string. 
  We study spontaneous supersymmetry breaking induced by brane-localized dynamics in five-dimensional supergravity compactified on S^1/Z_2. We consider a model with gravity in the bulk and matter localized on tensionless branes at the orbifold fixed points. We assume that the brane dynamics give rise to effective brane superpotentials that trigger the supersymmetry breaking. We analyze in detail the super-Higgs effect. We compute the full spectrum and show that the symmetry breaking is spontaneous but nonlocal in the fifth dimension. We demonstrate that the model can be interpreted as a new, non-trivial implementation of a coordinate-dependent Scherk-Schwarz compactification. 
  In this talk I review some aspects of the idea that there is an infra-red conformal fixed-point at the TeV scale. In particular, it is shown how gauge coupling unification can be achieved by TeV unification in a semi-simple gauge group. 
  In this paper, we survey the nature of spinors and supersymmetry (SUSY) in various types of spaces. We treat two distinct types of spaces: flat spaces and spaces of constant (non-zero) curvature. The flat spaces we consider are either three or four dimensional of signatures 3 + 1, 4 + 0, 2 + 2 and 3 + 0. In each of these cases, SUSY generators anti-commute to yield the generators of translations in the non-compact flat spaces. The spaces of constant curvature we consider are two-dimensional: the surface of the sphere S_2 and the Anti-deSitter space AdS_2. S_2 is embedded in a 3 + 0 Euclidean space while AdS_2 is embedded in 2 + 1 Minkowski space. The SUSY generators in these cases anti-commute to yield the generators of the isometry groups (SO(3) or SO(2,1)) of the space involved. We also report on some recent developments in looking for superspace realizations of these SUSY algebras. We can report good progress in the 3 + 0 Euclidean and in the AdS_2 case, somewhat less in the S_2 case. In each of the compact cases, we can construct field multiplet models carrying invariance under the full SUSY algebra. 
  A general analysis of the induced brane dynamics is performed when the intrinsic curvature term is included in the action. Such a term is known to cause dramatic changes and is generically induced by quantum corrections coming from the bulk gravity and its coupling with matter living on the brane. The induced brane dynamics is shown to be the usual Einstein dynamics coupled to a well defined modified energy-momentum tensor. In cosmology, conventional general relativity revives for an initial era whose duration depends on the value of the five-dimensional Planck mass. Violations of energy conditions may be possible, as well as matter inhomogeneities on the brane in (A)dS_{5} or Minkowski backgrounds. A new anisotropic cosmological solution is given in the above context. This solution, for a fine-tuned five-dimensional cosmological constant, exhibits an intermediate accelerating phase which is followed by an era corresponding to a 4D perfect fluid solution with no future horizons. 
  By constructing close one cochain density ${\Omega^1}_{2n}$ in the gauge group space we get WZW effective Lagrangian on high dimensional non-commutative space.Especially consistent anomalies derived from this WZW effective action in non-commutative four-dimensional space coincides with those by L.Bonora etc. 
  We derive determinant representations and nonlinear differential equations for the scaled 2-point functions of the 2D Ising model on the cylinder. These equations generalize well-known results for the infinite lattice (Painlev\'e III equation and the equation for the $\tau$-function of Painlev\'e V). 
  We study the enhancon mechanism for fractional D-branes in conifold and orbifold backgrounds and show how it can resolve the repulson singularity of these geometries. In particular we show that the consistency of the excision process requires that the interior space be not empty. In the orbifold case, we exploit the boundary state formalism to obtain an explicit conformal description and emphasize the non trivial role of the volume of the internal manifold. 
  We study tensor perturbations in a model with inflating Randall--Sundrum-type brane embedded in five-dimensional anti-de Sitter (adS$_5$) bulk. In this model, a natural {\it in}-vacuum of gravitons is the vacuum defined in static adS$_5$ frame. We show that this vacuum is, in fact, the same as the {\it in}-vacuum defined in the frame with de Sitter (dS$_4$) slicing, in which the brane is at rest. Thus, during inflation, gravitons on and off the brane remain in their vacuum state. We then study graviton creation by the brane on which inflation terminates at some moment of time. We mostly consider gravitons whose wavelengths at the end of inflation exceed both the horizon size and the adS$_5$ radius. Creation of these gravitons is dominated by (zero mode)--(zero mode) Bogoliubov coefficients and, apart from an overall factor, the spectrum of produced gravitons is the same as in four-dimensional theory. ``Kaluza--Klein'' gravitons are also produced, but this effect is subdominant. Graviton spectra at somewhat higher momenta are also presented for completeness. 
  We introduce Wilson's, or Polchinski's, exact renormalization group, and review the Local Potential Approximation as applied to scalar field theory. Focusing on the Polchinski flow equation, standard methods are investigated, and by choosing restrictions to some sub-manifold of coupling constant space we arrive at a very promising variational approximation method. Within the Local Potential Approximation, we construct a function, C, of the coupling constants; it has the property that (for unitary theories) it decreases monotonically along flows and is stationary only at fixed points - where it `counts degrees of freedom', i.e. is extensive, counting one for each Gaussian scalar. In the latter part of the thesis, the Local Potential Approximation is used to derive a non-trivial Polchinski flow equation to include Fermi fields. Our flow equation does not support chirally invariant solutions and does not reproduce the features associated with the corresponding invariant theories. We solve both for a finite number of components, N, and within the large N limit. The Legendre flow equation provides a comparison with exact results in the large N limit. In this limit, it is solved to yield both chirally invariant and non-invariant solutions. 
  We consider the effects of homogeneous Dirichlet's boundary conditions on two infinite parallel plane surfaces separated by a small distance {\it a}. We find that although spontaneous symmetry breaking does not occur for the theory of a massless, quartically self-interacting real scalar field, the theory becomes a theory of a massive scalar field. 
  We calculate the volumes of a large class of Einstein manifolds, namely Sasaki-Einstein manifolds which are the bases of Ricci-flat affine cones described by polynomial embedding relations in C^n. These volumes are important because they allow us to extend and test the AdS/CFT correspondence. We use these volumes to extend the central charge calculation of Gubser (1998) to the generalized conifolds of Gubser, Shatashvili, and Nekrasov (1999). These volumes also allow one to quantize precisely the D-brane flux of the AdS supergravity solution. We end by demonstrating a relationship between the volumes of these Einstein spaces and the number of holomorphic polynomials (which correspond to chiral primary operators in the field theory dual) on the corresponding affine cone. 
  It is shown that the $F_4$ rational and trigonometric integrable systems are exactly-solvable for {\it arbitrary} values of the coupling constants. Their spectra are found explicitly while eigenfunctions by pure algebraic means. For both systems new variables are introduced in which the Hamiltonian has an algebraic form being also (block)-triangular. These variables are invariant with respect to the Weyl group of $F_4$ root system and can be obtained by averaging over an orbit of the Weyl group. Alternative way of finding these variables exploiting a property of duality of the $F_4$ model is presented. It is demonstrated that in these variables the Hamiltonian of each model can be expressed as a quadratic polynomial in the generators of some infinite-dimensional Lie algebra of differential operators in a finite-dimensional representation. Both Hamiltonians preserve the same flag of polynomials and each subspace of the flag coincides with the finite-dimensional representation space of this algebra. Quasi-exactly-solvable generalization of the rational $F_4$ model depending on two continuous and one discrete parameters is found. 
  The statistical mechanics of nonrelativistic fermions in a constant magnetic field is considered from the quantum field theory point of view. The fermionic determinant is computed using a general procedure that contains all possible regularizations. The nonrelativistic grand-potential can be expressed in terms polylogarithm functions, whereas the partition function in 2+1 dimensions and vanishing chemical potential can be compactly written in terms of the Dedekind eta function. The strong and weak magnetic fields limits are easily studied in the latter case by using the duality properties of the Dedekind function. 
  The role of associativity or WDVV equations in effective supersymmetric quantum theories is discussed and it is demonstrated that for wide class of their solutions when residue formulas are valid the proof of associativity equations can be reduced to solving the system of ordinary linear equations and depends only upon corresponding matching and nondegeneracy conditions. The covariance of WDVV equations upon generic duality transformations and the role of associativity equations in general context of quasiclassical integrable systems is also discussed. 
  We study the Schwinger model on a half-line in this paper. In particular, we investigate the behavior of the chiral condensate near the edge of the line. The effect of the chosen boundary condition is emphasized. The extension to the finite temperature case is straightforward in our approach. 
  We demonstrate the classical stability of the weak/Planck hierarchy within the Randall-Sundrum scenario, incorporating the Goldberger-Wise mechanism and higher-derivative interactions in a systematic perturbative expansion. Such higher-derivative interactions are expected if the RS model is the low-energy description of some more fundamental theory. Generically, higher derivatives lead to ill-defined singularities in the vicinity of effective field theory branes. These are carefully treated by the methods of classical renormalization. 
  Classical version of Born-Infeld electrodynamics is recalled and its most important properties discussed. Then we analyze possible abelian and non-abelian generalizations of this theory, and show how certain soliton-like configurations can be obtained. The relationship with the Standard Model of electroweak interactions is also mentioned. 
  The fractional supersymmetry in the case of the non-relativistic motion of one anyon with fractional spin is realized. Thus the associated Hamiltonian is discussed. 
  We show that the mass of the Fayet hypermultiplet, which represents the matter sector of N=2 supersymmetric Yang-Mills theory, may be induced through a generalization of the central charge constraint usually proposed in the literature. This mass showing up as a parameter of the supersymmetry transformations, we conclude that it will stay unrenormalized at the quantum level. 
  We investigate the applicability of the moduli space approximation in theories with unbroken non-Abelian gauge symmetries. Such theories have massless magnetic monopoles that are manifested at the classical level as clouds of non-Abelian field surrounding one or more massive monopoles. Using an SO(5) example with one massive and one massless monopole, we compare the predictions of the moduli space approximation with the results of a numerical solution of the full field equations. We find that the two diverge when the cloud velocity becomes of order unity. After this time the cloud profile approximates a spherical wavefront moving at the speed of light. In the region well behind this wavefront the moduli space approximation continues to give a good approximation to the fields. We therefore expect it to provide a good description of the motion of the massive monopoles and of the transfer of energy between the massive and massless monopoles. 
  After a brief review of the celebrated 1941 paper of Rarita and Schwinger on the theory of particles with half-integral spins, we present an {\em ab initio} construct of the representation space relevant for the description of spin-3/2 particles. The chosen example case of spin-3/2 shows that covariance of a wave equation, and that of the imposed supplementary conditions, alone is not a sufficient criterion to guarantee the compatibility of a framework with relativity -- a lesson already arrived by Velo and Zwanziger. Here this same lesson is shown to be true at the level of the representation space without invoking any interactions. The presented detailed analysis forces us to abandon the single-spin interpretation of the Rarita and Schwinger framework, and suggests a new interpretation that fully respects the relativity theory. 
  We formulate a method of performing non-perturbative calculations in quantum field theory, based upon a derivative expansion of the exact renormalization group. We then proceed to apply this method to the calculation of critical exponents for three dimensional O(N) symmetric theory. Finally we discuss how the approximation scheme manages to reproduce some exactly known solutions in critical phenomena. 
  The asymptotic expansion of the heat kernel associated with Laplace operators is considered for general irreducible rank one locally symmetric spaces. Invariants of the Chern-Simons theory of irreducible U(n)- flat connections on real compact hyperbolic 3-manifolds are derived 
  It is shown that Hawking radiation of Dirac particles does not exist for $P_1, Q_2$ components but for $P_2, Q_1$ components in a charged Vaidya - de Sitter black hole. Both the location and the temperature of the event horizon change with time. The thermal radiation spectrum of Dirac particles is the same as that of Klein-Gordon particles. Our result demonstrates that there is no new quantum effect in the thermal radiation of Dirac particles in any spherically symmetry black holes. 
  The local action of an SU(2) gauge theory in general covariant Abelian gauges and the associated equivariant BRST symmetry that guarantees the perturbative renormalizability of the model are given. A global SL(2,R) symmetry of the model is spontaneously broken by ghost-antighost condensation at arbitrarily small coupling. This leads to propagators that are finite at Euclidean momenta for all elementary fields except the Abelian ``photon''. Ward Identities show that the symmetry breaking gives rise to massless BRST-quartets with ghost numbers (1,2,-2,-1) and (0,1,-1,0). The latter quartet is interpreted as due to an Abelian Higgs mechanism in the dual description of the model. 
  Assuming the completeness condition for boundaries we derive trace formulas for the annulus coefficients in 2-dimensional conformal field theory. We also derive polynomial equations that relate the annulus, Moebius and Klein bottle coefficients, and conjecture an annulus trace formula that is sensitive to the orientation of the boundaries. 
  The one-loop quantum corrections for BTZ black hole are considered using the dimensionally reduced 2D model. Cases of 3D minimal and conformal coupling are analyzed. Two cases are considered: minimally coupled and conformally coupled 3D scalar matter. In the minimal case, the Hartle-Hawking and Unruh vacuum states are defined and the corresponding semiclassical corrections of the geometry are found. The calculations are done for the conformal case too, in order to make the comparison with the exact results obtained previously in the special case of spinless black hole. Beside that we find exact corrections for AdS black hole for 2D minimally coupled scalar field in the Hatrle-Hawking and Boulware state 
  We derive the fusion hierarchy of functional equations for critical A-D-E lattice models related to, the sl(2) unitary minimal models, the parafermionic models and the supersymmetric models of conformal field theory, and deduce the related TBA functional equations. The derivation uses fusion projectors and applies in the presence of all known integrable boundary conditions on the torus and cylinder. The resulting TBA functional equations are_universal_ in the sense that they depend only on the Coxeter number of the A-D-E graph and are independent of the particular integrable boundary conditions. We conjecture generally that TBA functional equations are universal for all integrable lattice models associated with rational CFTs and their integrable perturbations. 
  We show that in a certain region of a negative coordinate axis, the U(1amaharia) sector of the static dyonic solutions to the noncommutative U(4) N=4 Super Yang-Mills (SYM) can be consistently decoupled from the SU(4) to {\it all orders in the noncommutativity parameter}. We show the above decoupling in two ways. First, we show the noncommutative dyon being the same as the commutative dyon, is a consistent solution to noncommutative equations of motion in that region of noncommutative space. Second, as an example of decoupling of a non-null U(1) sector, we also obtain a family of solutions with nontrivial U(1) components for {\it all} components of the gauge field in the same region of noncommutative space. In both cases, the SU(4) and U(1) components separately satisfy the equations of motion. 
  We find new integrable boundary conditions, depending on a free parameter $g$, for the O(N) nonlinear $\sigma$ model, which are of nondiagonal type, that is, particles can change their ``flavor'' through scattering off the boundary. These boundary conditions are derived from a microscopic boundary lagrangian, which is used to establish their integrability, and exhibit integrable flows between diagonal boundary conditions investigated earlier. We solve the boundary Yang-Baxter equation, connect these solutions to the boundary conditions, and examine the corresponding integrable flows. 
  A new exact renormalization group equation for the effective average action of Euclidean quantum gravity is constructed. It is formulated in terms of the component fields appearing in the transverse-traceless decomposition of the metric. It facilitates both the construction of an appropriate infrared cutoff and the projection of the renormalization group flow onto a large class of truncated parameter spaces. The Einstein-Hilbert truncation is investigated in detail and the fixed point structure of the resulting flow is analyzed. Both a Gaussian and a non-Gaussian fixed point are found. If the non-Gaussian fixed point is present in the exact theory, quantum Einstein gravity is likely to be renormalizable at the nonperturbative level. In order to assess the reliability of the truncation a comprehensive analysis of the scheme dependence of universal quantities is performed. We find strong evidence supporting the hypothesis that 4-dimensional Einstein gravity is asymptotically safe, i.e. nonperturbatively renormalizable. The renormalization group improvement of the graviton propagator suggests a kind of dimensional reduction from 4 to 2 dimensions when spacetime is probed at sub-Planckian length scales. 
  Monte Carlo simulations of a system whose action has an imaginary part are considered to be extremely difficult. We propose a new approach to this `complex-action problem', which utilizes a factorization property of distribution functions. The basic idea is quite general, and it removes the so-called overlap problem completely. Here we apply the method to a nonperturbative study of superstring theory using its matrix formulation. In this particular example, the distribution function turns out to be positive definite, which allows us to reduce the problem even further. Our numerical results suggest an intuitive explanation for the dynamical generation of 4d space-time. 
  The Blau-Thompson N_T=2, D=3 nonequivariant topological model is extended to a N_T=4, D=3 topological theory. The latter, formally, may be regarded as a topological deformation of the N_T=2, D=4 Yamrom-Vafa-Witten theory after dimensional reduction to D=3. For completeness, also the dimensional reduction of the half-twisted N_T=2, D=4 Yamron model is explicitly constructed. 
  The properties of brane-antibrane systems and systems of unstable D-branes in Type II superstring theory are investigated using the formalism of superconnections. The low-energy open string dynamics is shown to be probed by generalized Dirac operators. The corresponding index theorems are used to compute the chiral gauge anomalies in these systems, and hence their gravitational and Ramond-Ramond couplings. A spectral action for the generalized Dirac operators is also computed and shown to exhibit precisely the expected processes of tachyon condensation on the brane worldvolumes. The Chern-Simons couplings are thereby shown to be naturally related to Fredholm modules and bivariant K-theory, confirming the expectations that D-brane charge is properly classified by K-homology. 
  We extend the analysis of the canonical structure of the Wess-Zumino-Witten theory to the bulk and boundary coset G/H models. The phase spaces of the coset theories in the closed and in the open geometry appear to coincide with those of a double Chern-Simons theory on two different 3-manifolds. In particular, we obtain an explicit description of the canonical structure of the boundary G/G coset theory. The latter may be easily quantized leading to an example of a two-dimensional topological boundary field theory. 
  We show that the noncommutative Yang-Mills field forms an irreducible representation of the (undeformed) Lie algebra of rigid translations, rotations and dilatations. The noncommutative Yang-Mills action is invariant under combined conformal transformations of the Yang-Mills field and of the noncommutativity parameter \theta. The Seiberg-Witten differential equation results from a covariant splitting of the combined conformal transformations and can be computed as the missing piece to complete a covariant conformal transformation to an invariance of the action. 
  In built noncommutativity of supermembranes with central charges in eleven dimensions is disclosed. This result is used to construct an action for a noncommutative supermembrane where interesting topological terms appear. In order to do so, we first set up a global formulation for noncommutative Yang Mills theory over general symplectic manifolds. We make the above constructions following a pure geometrical procedure using the concept of connections over Weyl algebra bundles on symplectic manifolds. The relation between noncommutative and ordinary supermembrane actions is discussed. 
  The gauge equivalent formulation of the Faddeev-Skyrme model is used for the study of the quantum theory. The rotational quantum excitations around the soliton solution of Hopf number unity are investigated by the method of collective coordinates. The quantum Hamiltonian of the system is found to coincide with the Hamiltonian of a symmetrical top rotating in SU(2). Thus, the irreducible representations of physical observables can be constructed. 
  This paper has been withdrawn by the author(s), see hep-th/0205152. 
  A quantum Dirac field theory with no divergences of vacuum energy is presented. The vacuum energy divergence is eliminated by removing a extra degree of freedom of the Dirac fields. The conditions for removing the extra degree of freedom, expressed in the form of a conservation law and an orthogonality relation, define another spin 1/2 field with the same rest mass that is just the antifermion field. The anticommutation relations for fermion and antifermion fields are imposed by this conservation law. Both fermion and antifermion fields have only states with positive energies due to the orthogonality relation. The expressions of the charge density and the current density are obtained. The charge conservation law is established. A form for canical quantization applicable to both Boson and Dirac fields is introduced. 
  We calculate perturbative quantum gravity corrections to eternal two-dimensional black holes. We estimate the leading corrections to the AdS_2 black hole entropy and determine the quantum modification of N-dimensional Schwarzschild spacetime. 
  We investigate the convergence of the derivative expansion of the exact renormalisation group, by using it to compute the beta function of scalar theory. We demonstrate that the derivative expansion of the Polchinski flow equation converges at one loop for certain fast falling smooth cutoffs. The derivative expansion of the Legendre flow equation trivially converges at one loop, but also at two loops: slowly with sharp cutoff (as a momentum-scale expansion), and rapidly in the case of a smooth exponential cutoff. We also show that the two loop contributions to certain higher derivative operators (not involved in beta) have divergent momentum-scale expansions for sharp cutoff, but the smooth exponential cutoff gives convergent derivative expansions for all such operators with any number of derivatives. In the latter part of the thesis, we address the problems of applying the exact renormalisation group to gauge theories. A regularisation scheme utilising higher covariant derivatives and the spontaneous symmetry breaking of the gauge supergroup SU(N|N) is introduced and it is demonstrated to be finite to all orders of perturbation theory. 
  A chemical potential may be introduced into the AdS/CFT correspondence by setting the D3 branes of the construction spinning. In the field theory the fermionic modes are expected to condense as Cooper pairs, although at zero temperature the chemical potential destabilizes the scalar sector of the N=4 theory obscuring this phenomena. We show, in the case where a chemical potential is introduced for a small number of the gauge colours, that there is a metastable vacuum for the scalar fields where fermionic Cooper pairing is apparently manifest. In this vacuum the D3 branes expand non-commutatively (to balance the centrifugal force) into a D5 brane, in a mechanism analogous to Harmark and Savvidy's (M)atrix theory construction of a spinning D2 brane. We show that the D5 brane acts as a source for the RR 3-form whose UV scaling and symmetries are those of a fermion bilinear. The D5 brane rotates within the S^5 and so decays by the emission of RR fields which we interpret as the metastable vacuum decaying via higher dimension operators. 
  We consider the Casimir energy of a thick dielectric-diamagnetic shell under a uniform velocity light condition, as a function of the radii and the permeabilities. We show that there is a range of parameters in which the stress on the outer shell is inward, and a range where the stress on the outer shell is outward. We examine the possibility of obtaining an energetically stable configuration of a thick shell made of a material with a fixed volume. 
  A new N=4 superconformal algebra (SCA) is presented. Its internal affine Lie algebra is based on the seven-dimensional Lie algebra su(2)\oplus g, where g should be identified with a four-dimensional non-reductive Lie algebra. Thus, it is the first known example of what we choose to call a non-reductive SCA. It contains a total of 16 generators and is obtained by a non-trivial In\"on\"u-Wigner contraction of the well-known large N=4 SCA. The recently discovered asymmetric N=4 SCA is a subalgebra of this new SCA. Finally, the possible affine extensions of the non-reductive Lie algebra g are classified. The two-form governing the extension appearing in the SCA differs from the ordinary Cartan-Killing form. 
  We study gravity in codimension-2 brane world scenarios with infinite volume extra dimensions. In particular, we consider the case where the brane has non-zero tension. The extra space then is a two-dimensional ``wedge'' with a deficit angle. In such backgrounds we can effectively have the Einstein-Hilbert term on the brane at the classical level if we include higher curvature (Gauss-Bonnet) terms in the bulk. Alternatively, such a term would be generated at the quantum level if the brane matter is not conformal. We study (linearized) gravity in the presence of the Einstein-Hilbert term on the brane in such backgrounds. We find that, just as in the original codimension-2 Dvali-Gabadadze model with a tensionless brane, gravity is almost completely localized on the brane with ultra-light modes penetrating into the bulk. 
  We discuss the semiclassical instability of the Randall-Sundrum brane-world model against a creation of a kind of Kaluza-Klein bubble. An example describing such a bubble space-time is constructed from the five-dimensional AdS-Schwarzschild metric. The induced geometry of the brane looks like the Einstein-Rosen bridge, which connects the positive and the negative tension branes. The bubble rapidly expands and there also form a trapped region around it. 
  The spectrum of the Hermitian Hamiltonian ${1\over2}p^2+{1\over2}m^2x^2+gx^4$ ($g>0$), which describes the quantum anharmonic oscillator, is real and positive. The non-Hermitian quantum-mechanical Hamiltonian $H={1\over2}p^2+{1 \over2}m^2x^2-gx^4$, where the coupling constant $g$ is real and positive, is ${\cal PT}$-symmetric. As a consequence, the spectrum of $H$ is known to be real and positive as well. Here, it is shown that there is a significant difference between these two theories: When $g$ is sufficiently small, the latter Hamiltonian exhibits a two-particle bound state while the former does not. The bound state persists in the corresponding non-Hermitian ${\cal PT}$-symmetric $-g\phi^4$ quantum field theory for all dimensions $0\leq D<3$ but is not present in the conventional Hermitian $g\phi^4$ field theory. 
  It is shown that the dimensional reduction of the N_T=2, D=3 Blau-Thompson model to D=2, i.e., the novel topological twist of N=8, D=2 super Yang-Mills theory, provides an example of a Hodge-type cohomological theory. In that theory the generators of the topological shift, co-shift and gauge symmetry, together with a discrete duality operation, are completely analogous to the de Rham cohomology operators and the Hodge *-operation. 
  The integrability of the generalized Benney hierarchy with three primary fields is investigated from the point of view of two-dimensional topological field theories coupled to gravity. The associated primary free energy and correlation functions at genus zero are obtained via Landau-Ginzburg formulation and the string equation is derived using the twistor construction for the Orlov operators. By adopting the approach of Dubrovin and Zhang we obtain the genus-one corrections of the Poisson brackets of the generalized Benney hierarchy. 
  We use an extended quantum field theory (EQFT) hep-th/9911168 to explore possible observational effects of the spacetime. It is shown that as it was expected the spacetime foam can provide quantum bose fields with a cutoff at very small scales, if the energy of zero - point fluctuations of fields is taken into account. It is also shown that EQFT changes the behaviour of massless fields at very large scales (in the classical region). We show that as $r\gg 1/\mu $ the Coulomb and Newton forces acquire the behaviour $\sim 1/r$ (instead of $1/r^{2}$). 
  Within loop quantum gravity we construct a coarse-grained approximation for the Einstein-Maxwell theory that yields effective Maxwell equations in flat spacetime comprising Planck scale corrections.   The corresponding Hamiltonian is defined as the expectation value of the electromagnetic term in the Einstein-Maxwell Hamiltonian constraint, regularized a la Thiemann, with respect to a would-be semiclassical state. The resulting energy dispersion relations entail Planck scale corrections to those in flat spacetime. Both the helicity dependent contribution of Gambini and Pullin [GP] and, for a value of a parameter of our approximation, that of Ellis et. al. [ELLISETAL] are recovered. The electric/magnetic asymmetry in the regularization procedure yields nonlinearities only in the magnetic sector which are briefly discussed. Observations of cosmological Gamma Ray Bursts might eventually lead to the needed accuracy to study some of these quantum gravity effects. 
  The relation between BRST cohomology and the N=1 supersymmetric Yang-Mills action in 4 dimensions is discussed. In particular, it is shown that both off and on shell N=1 SYM actions are related to a lower dimensional field polynomial by solving the descent equations, which is obtained from the cohomological analysis of linearized Slavnov-Taylor operator $\B$, in the framework of Algebraic Renormalization. Furthermore we show that off and on shell solutions differ only by a $\B$- exact term, which is a consequence of the fact that the cohomology of both cases are same. 
  A new basis of states for highest-weight modules in $\ZZ_k$ parafermionic conformal theories is displayed. It is formulated in terms of an effective exclusion principle constraining strings of $k$ fundamental parafermionic modes. The states of a module are then built by a simple filling process, with no singular-vector subtractions. That results in fermionic-sum representations of the characters, which are exactly the Lepowsky-Primc expressions. We also stress that the underlying combinatorics -- which is the one pertaining to the Andrews-Gordon identities -- has a remarkably natural parafermionic interpretation. 
  We study the framing dependence of the Wilson loop observable of U(N) Chern-Simons gauge theory at large N. Using proposed geometrical large N dual, this leads to a direct computation of certain topological string amplitudes in a closed form. This yields new formulae for intersection numbers of cohomology classes on moduli of Riemann surfaces with punctures (including all the amplitudes of pure topological gravity in two dimensions). The reinterpretation of these computations in terms of BPS degeneracies of domain walls leads to novel integrality predictions for these amplitudes. Moreover we find evidence that large N dualities are more naturally formulated in the context of U(N) gauge theories rather than SU(N). 
  We consider exact solutions of Einstein equations defining static black holes parametrized by off-diagonal metrics which by anholonomic mappings can be equivalently transformed into some diagonal metrics with coefficients being very similar to those from the Schwarzschild and/or Reissner-N\"ordstrom solutions with anisotropic renormalizations of constants. We emphasize that such classes of solutions, for instance, with ellipsoidal symmetry of horizons, can be constructed even in general relativity theory if off-diagonal metrics and anholonomic frames are introduced into considerations. Such solutions do not violate the Israel's uniqueness theorems on static black hole configurations because at long radial distances one holds the usual Schwarzschild limit. We show that anisotropic deformations of the Reissner-N\"ordstrom metric can be an exact solution on the brane, re-interpreted as a black hole with an effective electromagnetic like charge anisotropically induced and polarized by higher dimension gravitational interactions. 
  An alternative approach to introducing gravitational dynamics on a brane embedded in a higher dimensional spacetime is presented. The brane is treated as a boundary of a higher dimensional manifold in which the bulk action is described by a metric independent topological quantum field theory. The example of a five dimensional non-Abelian BF theory with a boundary brane is considered. A natural boundary condition is adopted chosen for consistency of the topological action despite the presence of a boundary. The resulting effective action on the brane is the action of general relativity in first order form plus terms involving the extrinsic curvature of the brane. 
  Why does the physical 4-dimensional space have a 3 + 1 signature rather than a 4 + 0 or a 2 + 2 for its metric? We give a simple explanation based largely on a group-theoretic argument a la Wigner. Applied to flat spaces of higher dimensions the same approach indicates that metrics with more than one time dimension are physically unacceptable because the corresponding irreducible unitary representations are infinite dimensional (besides the trivial representation). 
  A new mechanism which leads to a linearized massless graviton localized on the brane is found in the $AdS$/CFT setting, {\it i.e.} in a single copy of $AdS_5$ spacetime with a singular brane on the boundary, within the Randall-Sundrum brane-world scenario. With an help of a recent development in path-integral techniques, a one-parameter family of propagators for linearized gravity is obtained analytically, in which a parameter $\xi$ reflects various kinds of boundary conditions that arise as a result of the half-line constraint. In the case of a Dirichlet boundary condition ($\xi = 0$) the graviton localized on the brane can be massless {\it via} coupling constant renormalization. Our result supports a conjecture that the usual Randall-Sundrum scenario is a regularized version of a certain underlying theory. 
  The amplitudes for emission and scattering of N=2 strings off D-branes are calculated. We consider in detail the amplitudes <cc> and <occ> for the different types of D-branes. For some D-branes we find massive poles in the scattering spectrum that are absent in the ordinary N=2 spectrum. 
  We present a class of solvable SO(D) symmetric matrix models with D bosonic matrices coupled to chiral fermions. The SO(D) symmetry is spontaneously broken due to the phase of the fermion integral. This demonstrates the conjectured mechanism for the dynamical generation of four-dimensional space-time in the IIB matrix model, which was proposed as a nonperturbative definition of type IIB superstring theory in ten dimensions. 
  We study the free energy of the 1+1 dimensional O(N) nonlinear sigma-models for even N using the TBA equations proposed recently. We give explicit formulae for the constant solution of the TBA equations (Y-system) and calculate the first two virial coefficients. The free energy is also compared to the leading large N result. 
  A boundary-state computation is performed to obtain derivative corrections to the Chern-Simons coupling between a p-brane and the RR gauge potential C(p-3). We work to quadratic order in the gauge field strength F, but all orders in derivatives. In a certain limit, which requires the presence of a constant B-field background, it is found that these corrections neatly sum up into the *2 product of (commutative) gauge fields. The result is in agreement with a recent prediction using noncommutativity. 
  We study a scalar-tensor bimetric cosmology in the Randall-Sundrum model with one positive tension brane, where the biscalar field is assumed to be confined on the brane. The effective Friedmann equations on the brane are obtained and analyzed. We comment on resolution of cosmological problems in this bimetric model. 
  The simplest supersymmetry algebra and superspace in three dimensional Euclidean (3dE) space is examined. Representations of the algebra are considered and the implications of restricting the space of states to states with positive definite norm are determined.   A superspace is defined and superfields are introduced. Supersymmetric field theory models in 3dE are described both in superfield and component field forms. The relationship between these models, and similar models in four dimensional Minkowski space is described. 
  We consider closed string tachyons localized at the fixed points of noncompact nonsupersymmetric orbifolds. We argue that tachyon condensation drives these orbifolds to flat space or supersymmetric ALE spaces. The decay proceeds via an expanding shell of dilaton gradients and curvature which interpolates between two regions of distinct angular geometry. The string coupling remains weak throughout. For small tachyon VEVs, evidence comes from quiver theories on D-branes probes, in which deformations by twisted couplings smoothly connect non-supersymmetric orbifolds to supersymmetric orbifolds of reduced order. For large tachyon VEVs, evidence comes from worldsheet RG flow and spacetime gravity. For $\IC^2/\IZ_n$, we exhibit infinite sequences of transitions producing SUSY ALE spaces via twisted closed string condensation from non-supersymmetric ALE spaces. In a $T$-dual description this provides a mechanism for creating NS5-branes via {\it closed} string tachyon condensation similar to the creation of D-branes via {\it open} string tachyon condensation. We also apply our results to recent duality conjectures involving fluxbranes and the type 0 string. 
  We show that the Quantum Hall Soliton constructed in \cite{giantbob} is stable under small perturbations. We find that creating quasiparticles actually lowers the energy of the system, and discuss whether this indicates an instability on the time scales relevant to the problem. 
  The sinh-Gordon model on a half-line with integrable boundary conditions is considered in low order perturbation theory developed in affine Toda field theory. The quantum corrections to the classical reflection factor of the model are studied up to the second order in the difference of the two boundary parameters and to one loop order in the bulk coupling. It is noticed that the general form of the second order quantum corrections are consistent with Ghoshal's formula. 
  Gravitational and gauge anomalies provide stringent constraints on which subset of chiral models can effectively describe M-theory at low energy. In this paper, we explicitly construct an abelian orbifold of M-theory to obtain an N=1, chiral super Yang-Mills theory in four dimensions, using anomaly matching to determine the entire gauge and representation structure. The model described in this paper is the simplest four dimensional model which one can construct from M-theory compactified on an abelian orbifold without freely-acting involutions. The gauge group is SO(12) x SU(8) x SU(2) x SU(2) x U(1). 
  In this talk we will report on few results of discrete physics on the fuzzy sphere . In particular non-trivial field configurations such as monopoles and solitons are constructed on fuzzy ${\bf S}^2$ using the language of K-theory, i.e projectors . As we will show, these configurations are intrinsically finite dimensional matrix models . The corresponding monopole charges and soliton winding numbers are also found using the formalism of noncommutative geometry and cyclic cohomology . 
  We construct supergravity duals of D6-branes wrapping Kahler four-cycles inside a Calabi-Yau threefold, CY_3. We obtain the purely gravitational M-theory description, which turns out to be a Calabi-Yau fourfold, CY_4. We also analyze the dynamics of a probe D6 in this background. 
  I briefly review the cosmological constant problem and attempts toward its solution, and present the first nontrivial example for the self-tuning mechanism with a $1/H^2$ term with the antisymmetric field strength $H_{MNPQ}$ in a 5D RS-II setup. 
  The holonomy of an SU(2) N-instanton in the x^4-direction gives a map from R^3 into SU(2), which provides a good model of an N-Skyrmion. Combining this map with the standard Hopf map from SU(2)=S^3 to S^2 gives a configuration for a Hopf soliton of charge N. In this way, one may define a collective-coordinate manifold for Hopf solitons. This paper deals with instanton approximations to Hopf solitons in the Skyrme-Faddeev model, focussing in particular on the N=1 and N=2 sectors, and the two-soliton interaction. 
  We introduce the concept of general gauge theory which includes Yang-Mills models. In the framework of the causal approach and show that the anomalies can appear only in the vacuum sector of the identities obtained from the gauge invariance condition by applying derivatives with respect to the basic fields. Then we provide a general result about the absence of anomalies in higher orders of perturbation theory. This result reduces the renormalizability proof to the study of lower orders of perturbation theory. For the Yang-Mills model one can perform this computation explicitly and obtains its renormalizability in all orders. 
  Using techniques of supersymmetric gauge theories, we present the Ricci-flat metrics on non-compact Kahler manifolds whose conical singularity is repaired by the Hermitian symmetric space. These manifolds can be identified as the complex line bundles over the Hermitian symmetric spaces. Each of the metrics contains a resolution parameter which controls the size of these base manifolds, and the conical singularity appears when the parameter vanishes. 
  In this paper, we study a new matrix theory based on non-BPS D-instantons in type IIA string theory and D-instanton - anti D-instanton system in type IIB string theory, which we call K-matrix theory. The theory correctly incorporates the creation and annihilation processes of D-branes. The configurations of the theory are identified with spectral triples, which are the noncommutative generalization of Riemannian geometry a la Connes, and they represent the geometry on the world-volume of higher dimensional D-branes. Remarkably, the configurations of D-branes in the K-matrix theory are naturally classified by a K-theoretical version of homology group, called K-homology. Furthermore, we argue that the K-homology correctly classifies the D-brane configurations from a geometrical point of view. We also construct the boundary states corresponding to the configurations of the K-matrix theory, and explicitly show that they represent the higher dimensional D-branes. 
  An embedding superspace, whose Bosonic part is the flat 2 + 1 dimensional embedding space for AdS2, is introduced. Superfields and several supersymmetric models are examined in the embedded AdS2 superspace. 
  We propose a new approach for coupling the type II superstring to the Ramond-Ramond background of D-branes in the RNS formalism, alternative to introducing RR vertex operators. It is based on the mixing between Ramond-Ramond p-form excitations in the closed string spectrum and transversally polarized excitations in the open string spectrum. 
  The calculation of the symmetry factor corresponding to a given Feynman diagram is well known to be a tedious problem. We have derived a simple formula for these symmetry factors. Our formula works for any diagram in scalar theory ($\phi^3$ and $\phi^4$ interactions), spinor QED, scalar QED, or QCD. 
  We construct four dimensional gauge theories in which the successful supersymmetric unification of gauge couplings is preserved but accelerated by N-fold replication of the MSSM gauge and Higgs structure. This results in a low unification scale of $10^{13/N}$ TeV. 
  The 1+1 dimensional massive Dirac equation is solved exactly in light-cone coordinates for $x^+ > 0$ and $x^- > -L$, in the presence of an arbitrary $x^+$ dependent electric field. Our solution resolves the ambiguity at $p^+ = 0$. We also obtain the one loop rate for pair production for a positive electric field, compute the expectation values of the vector and axial vector currents, and recover the well known anomaly $e^2 E/\pi$ in the divergence of the latter. A final intriguing result is that the theory seems to exhibit a phase transition in the limit of infinite $L$. 
  We construct modular invariant partition functions for strings propagating on non-compact manifolds of G_2 holonomy. Our amplitudes involve a pair of N=1 minimal models M_m, M_{m+2} (m=3,4,...) and are identified as describing strings on manifolds of G_2 holonomy associated with A_{m-2} type singularity. It turns out that due to theta function identities our amplitudes may be cast into a form which contain tricritical Ising model for any m. This is in accord with the results of Shatashvili and Vafa. We also construct a candidate partition function for string compactified on a non-compact Spin(7) manifold. 
  We study issues of the quintessence in the brane cosmology. The initial bulk spacetime consists of two 5D topological anti de Sitter black hole joined by the brane (moving domain wall). Here we do not introduce any conventional radiation and matter. Instead we include a localized scalar on the brane as a stress-energy tensor, and thus we find the quintessence which gives an accelerating universe. Importantly, we obtain a $\rho^2$-term as well as a holographic matter term of $\alpha/a^4$ from the masses of the topological black holes. We discuss a possibility that in the early universe, $\rho^2$-term makes a large kinetic term which induces a decelerating universe. This may provide a hint of avoiding from the perpetually accelerating universe of the present-day quintessence. If a holographic matter term exists, it will plays the role of a CFT-radiation in the early universe. 
  We study Liouville theory on worldsheets with boundary using the solutions of Knizhnik-Zamolodchikov equation involving a degenerate representation of the Virasoro algebra. The expression for bulk-boundary propagator on a disc is proposed. 
  Motivated by the possibility that physics may be effectively five-dimensional over some range of distance scales, we study the possible gaugings of five-dimensional N=2 supergravity. Using a constructive approach, we derive the conditions that must be satisfied by the scalar fields in the vector, tensor and hypermultiplets if a given global symmetry is to be gaugeable. We classify all those theories that admit the gauging of a compact group that is either Abelian or semi-simple, or a direct product of a semi-simple and an Abelian group. In the absence of tensor multiplets, either the gauge group must be semi-simple or the Abelian part has to be U(1)_R and/or an Abelian isometry of the hyperscalar manifold. On the other hand, in the presence of tensor multiplets the gauge group cannot be semi-simple. As an illustrative exercise, we show how the Standard Model SU(3) X SU(2) X U(1) group may be gauged in five-dimensional N=2 supergravity. We also show how previous special results may be recovered within our general formalism. 
  Dynamics of D-branes on $SU(2)/U(1)$ gauged WZW model are investigated. We find the effective action for infinite $k$, where $k$ is the level of WZW model. We also consider finite $k$ correction to the effective action which is compatible with Fedosov's deformation quantization of the background. 
  We develop the non commutative Chern-Simons gauge model analysis modeling the description of the hierarchical states of fractional quantum Hall fluids. For a generic level $n$ of the hierarchy, we show that the order parameter matrix $K_{ab}$ is given by $ \theta^{-2}Tr(\tau_a \tau_b)$, where $\{\tau_a, 1\leq a \leq n \}$ define a specific set of $n\times n$ matrices depending on the parameter $\theta$ and the levels $l_a$ of the CS effective field theory. Our analysis predicts the existence of a third order tensor of order parameters $C_{abc}$ induced by the external magnetic field. It is shown that the $C_{abc}$'s are not new order parameters and are given by $\theta d_{abc}$, where $d_{abc}$ are numbers depending on the $l_a$'s. We also give the generalized quantum Hall soliton extending the one obtained in the case of the Laughlin state. 
  Quantum Yang-Mills theory can be rewritten in terms of gauge-invariant variables: it has the form of the so-called BF gravity, with an additional `aether' term. The BF gravity based on the gauge group SU(N) is actually a theory of high spin fields (up to J=N) with high local symmetry mixing up fields with different spins -- like in supergravity but without fermions. At N going to infinity one gets a theory with an infinite tower of spins related by local symmetry, similar to what one has in string theory. We, thus, outline a way to derive string theory from a local Yang-Mills theory in the large N limit. 
  The conditions for validity of the Causal Entropy Bound (CEB) are verified in the context of non-singular cosmologies with classical sources. It is shown that they are the same conditions that were previously found to guarantee validity of the CEB: the energy density of each dynamical component of the cosmic fluid needs to be sub-Planckian and not too negative, and its equation of state needs to be causal. In the examples we consider, the CEB is able to discriminate cosmologies which suffer from potential physical problems more reliably than the energy conditions appearing in singularity theorems. 
  We construct a class of spin foam models describing matter coupled to gravity, such that the gravitational sector is described by the unitary irreducible representations of the appropriate symmetry group, while the matter sector is described by the finite-dimensional irreducible representations of that group. The corresponding spin foam amplitudes in the four-dimensional gravity case are expressed in terms of the spin network amplitudes for pentagrams with additional external and internal matter edges. We also give a quantum field theory formulation of the model, where the matter degrees of freedom are described by spin network fields carrying the indices from the appropriate group representation. In the non-topological Lorentzian gravity case, we argue that the matter representations should be appropriate SO(3) or SO(2) representations contained in a given Lorentz matter representation, depending on whether one wants to describe a massive or a massless matter field. The corresponding spin network amplitudes are given as multiple integrals of propagators which are matrix spherical functions. 
  We discuss some physical issues related to the K-theoretic classification of D-brane charges, putting an emphasis on the role of D-brane instantons. The relation to D-instantons provides a physical interpretation to the mathematical algorithm for computing K-theory known as the ``Atiyah-Hirzebruch spectral sequence.'' Conjecturally, a formulation in terms of D-instantons leads to a computationally useful formulation of K-homology in general. As an application and illustration of this viewpoint we discuss some issues connected with D-brane charges associated with branes in WZW models. We discuss the case of SU(3) in detail, and comment on the general picture of branes in SU(N), based on a recent result of M. Hopkins. 
  We assemble a few remarks on the supergravity solution of hep-th/0007191, whose UV asymptotic form was previously found in hep-th/0002159. First, by normalizing the R-R fluxes, we compare the logarithmic flow of couplings in supergravity with that in field theory, and find exact agreement. We also write the 3-form field strength $G_3 = F_3 - \tau H_3$ present in the solution in a manifestly SO(4) invariant (2,1) form. In addition, we discuss various issues related to the chiral symmetry breaking and wrapped branes. 
  For the case of the SU(2) WZW model at level one, the boundary states that only preserve the conformal symmetry are analysed. Under the assumption that the usual Cardy boundary states as well as their marginal deformations are consistent, the most general conformal boundary states are determined. They are found to be parametrised by group elements in SL(2,C). 
  The noncommutative string theory is described by embedding open string theory in a constant second rank antisymmetric $B_{\mu\nu}$ field and the noncommutative gauge theory is defined by a deformed $\star$ product. As a check, study of various scattering amplitudes in both noncommutative string and noncommutative gauge theory confirm that in the $\alpha^{'}\to 0$ limit, the noncommutative string theoretic amplitude goes over to the noncommutative gauge theoretic amplitude, and the couplings are related as $g_{YM}=G_0\sqrt{\frac{1}{2\alpha^{'}}}$. Furthermore we show that in this limit there will not be any correction to the gauge theoretic action because of absence of massive modes. We get sin/cos factors in the scattering amplitudes depending on the odd/even number of external photons. 
  We show that the higher order derivative $\alpha^{'}$ corrections to the DBI and Chern-Simon action is derived from non-commutativity in the Seiberg-Witten limit, and is shown to agree with Wyllard's (hep-th/0008125) result, as conjectured by Das et al., (hep-th/0106024). In calculating the corrections, we have expressed $\hat F$ in terms F, $\hat A$ in terms of A up to order ${\cal O}(A^3)$, and made use of it. 
  We show the existence of some new local, covariant and continuous symmetries for the BRST invariant Lagrangian density of a free two ($1 + 1$)-dimensional (2D) Abelian U(1) gauge theory in the framework of superfield formalism. The Noether conserved charges corresponding to the above local continuous symmetries find their geometrical origin as the translation generators along the odd (Grassmannian)- and even (bosonic) directions of the four ($2 + 2)$-dimensional compact supermanifold. Some new discrete symmetries are shown to exist in the superfield formulation. The logical origin for the existence of BRST- and co-BRST symmetries is shown to be encoded in the Hodge decomposed versions (of the 2D fermionic vector fields) that are consistent with the discrete symmetries of the theory. 
  We study leading string corrections to the type IIB supergravity solution dual to the ${\cal N}=1$ supersymmetric $SU(N+M)\times SU(N)$ gauge theory coupled to bifundamental chiral superfields $A_i, B_j$, $i,j=1,2$. This solution was found in hep-th/0007191, and its asymptotic form describing logarithmic RG flow was constructed in hep-th/0002159. The leading tree-level string correction to the type IIB string effective action is represented by the invariant of the form $\a'{}^3 (R^4 + ...)$. Since the background contains 3-form field strengths, we need to know parts of this invariant that depend on them. By analyzing the 5-point superstring scattering amplitudes we show that only a few specific $R^3 (H_3)^2$ and $R^3 (F_3)^2$ terms are present in the effective action.   Their contribution to the holographic RG flow turns out to be of the same order as of the $R^4$ terms. Using this fact we show that it is possible to have agreement between the $\alpha'$-corrected radial dependence of the supergravity fields and the RG flow dictated by the NSVZ beta functions in field theory. The agreement with field theory requires that the anomalous dimension of the operators $\Tr (A_i B_j)$ is corrected by a term of order $(M/N)^4 \lambda^{-1/2}$ from its value $-{1 \over 2} $ found for M=0 ($\lambda$ is the appropriate 't Hooft coupling which is assumed to be strong). 
  From spinor and scalar 2+1 dimensional QED effective actions at finite temperature and density in a constant magnetic field background, we calculate the corresponding virial coefficients for particles in the lowest Landau level. These coefficients depend on a parameter theta related to the time-component of the gauge field, which plays an essential role for large gauge invariance. The variation of the parameter theta might lead to an interpolation between fermionic and bosonic virial coefficients, although these coefficients are singular for theta=pi/2. 
  Two forms of anomalies for chiral spinors living on submanifolds of the spacetime are obtained from the integrality theorem for immersions. The first form of the chiral anomaly is the usual for chiral spinors living on D-brane and O-plane intersections, the second form is exotic. 
  We present a simple brane-world model in five dimensions. In this model we do not need any fine-tuning between the five dimensional cosmological constant and the brane tension to obtain four dimensional flat Minkowski space. The space-time of our solution has no naked singularities. Further the compactification scale of the fifth direction is automatically determined. 
  The Hamiltonian theory of zero-curvature equations with spectral parameter on an arbitrary compact Riemann surface is constructed. It is shown that the equations can be seen as commuting flows of an infinite-dimensional field generalization of the Hitchin system. The field analog of the elliptic Calogero-Moser system is proposed. An explicit parameterization of Hitchin system based on the Tyurin parameters for stable holomorphic vector bundles on algebraic curves is obtained. 
  Corresponding to two ways of realizing the q-deformed Heisenberg algebra by the undeformed variables there are two q-perturbative Hamiltonians with the additional momentum-dependent interactions, one originates from the perturbative expansion of the potential, the other originates from that of the kinetic energy term. At the level of operators, these two q-perturbative Hamiltonians are different. In order to establish a reliable foundation of the perturbative calculations in q-deformed dynamics, except examples of the harmonic-oscillator and the Morse potential demonstrated before, the general q-perturbative equivalent theorem is demonstrated, which states that for any regular potential which is singularity free the expectation values of two q-perturbative Hamiltonians in the eigenstates of the undeformed Hamiltonian are equivalent. For the q-deformed ``free'' particle case, the perturbative Hamiltonian originated from the kinetic energy term still keeps its general expression, but it does not lead to energy shift. 
  We discuss the gauge natural formulation of supersymmetric theories and supergravity, with the aim to show that the standard and the supersymmetric frameworks admit in fact a unifying mathematical language. 
  In this paper we show the equivalence of various (non-threshold) bound state solutions of branes, or equivalently branes in background potentials, in ten- and eleven-dimensional supergravity. We compare solutions obtained in two very different ways. One method uses a zero mode analysis to make an Ansatz which makes it possible to solve the full non-linear supergravity equations. The other method utilises T-duality techniques to turn on the fields on the brane. To be specific, in eleven dimensions we show the equivalence for the (M2,M5) bound state, or equivalently an M5-brane in a C_3 field, where we also consider the (MW,M2,M2',M5) solution, which can be obtained from the (M2,M5) bound state by a boost. In ten dimensions we show the equivalence for the ((F,D1),D3) bound state as well as the bound states of (p,q) 5-branes with lower dimensional branes in type IIB, corresponding to D3-branes in B_2 and C_2 fields and (p,q) 5-branes in B_2, C_2 and C_4 fields. We also comment on the recently proposed V-duality related to infinitesimally boosted solutions. 
  We consider gauge bosons in the bulk of AdS5 in a two-brane theory that addresses the hierarchy problem. We show such a theory can be consistent with gauge coupling unification at a high scale. We discuss subtleties in this calculation and show how to regulate consistently in a bounded AdS5 background. Our regularization is guided by the holographic dual of the calculation. 
  In AdS5, the coupling for bulk gauge bosons runs logarithmically, not as a power law. For this reason, one can preserve perturbative unification of couplings. Depending on the cutoff, this can occur at a high scale. We discuss subtleties in the calculation and present a regularization scheme motivated by the holographic correspondence. We find that generically, as in the standard model, the couplings almost unify. For specific choices of the cutoff and number of scalar multiplets, there is good agreement between the measured couplings and the assumption of high scale unification. 
  The Casimir energy for the transverse oscillations of a piecewise uniform closed string is calculated. The great adaptability of this string model with respect to various regularization methods is pointed out. We survey several regularization methods: the cutoff method, the complex contour integration method, and the zeta-function method. The most powerful method in the present case is the contour integration method. The Casimir energy turns out to be negative, and more so the larger is the number of pieces in the string. The thermodynamic free energy and the critical Hagedorn temperature are calculated for a two-piece string. Mass and decay spectra are calculated for quantum massive excitations and the physical meaning of the critical temperatures characterizing the radiation in the decay of a massive microstate is discussed. 
  We calculate the 1-loop effective potential in a supersymmetric model in 5D with $S^1/(Z_2\times Z_2)$ orbifold compactification. The procedure of calculation consists of evaluating first the integrals over four-momenta using the dimensional regularization and then the sum over Kaluza-Klein modes using the zeta-regularization. We show that both fermionic and bosonic contributions are separately finite and argue that, supersymmetry is not necessary for the finiteness of the theory at 1-loop. Also, some general arguments on the finiteness of the theory with arbitrary number of extra dimensions are presented. 
  Interactions of noncommutative solitons in a modified U(n) sigma model in 2+1 dimensions can be analyzed exactly. Using an extension of the dressing method, we construct explicit time-dependent solutions of its noncommutative field equation by iteratively solving linear equations. The approach is illustrated by presenting bound states and right-angle scattering configurations for two noncommutative solitons. 
  We consider the quantization of the massless minimally coupled scalar field in de Sitter spacetime. The no-boundary Euclidean prescription naturally picks out the de Sitter invariant vacuum state of Kirsten and Garriga. We extend Strominger's dS/CFT correspondence to this case which allows us to interpret the massless field in terms of a Euclidean CFT. The extension is non-trivial and requires careful treatment of the zero mode. Since the graviton is massless, this work may also be considered a step towards a theory of gravity in de Sitter space. 
  We construct N=1 supersymmetric theories on worldvolumes of D5 branes wrapped around 2-cycles of threefolds which are A-D-E fibrations over a plane. We propose large N duals as geometric transitions involving blowdowns of two cycles and blowups of three-cycles. This yields exact predictions for a large class of N=1 supersymmetric gauge systems including U(N) gauge theories with two adjoint matter fields deformed by superpotential terms, which arise in A-D-E fibered geometries with non-trivial monodromies. 
  We show that crossing symmetry of four point functions in the $H_3^+$ WZNW model follows from similar properties of certain five point correlation functions in Liouville theory that have already been proven previously. 
  We describe some remarkable properties of the so-called Information Metric on instanton moduli space. This Metric is manifestly gauge and conformally invariant and coincides with the Euclidean AdS_5 metric on the one-instanton SU(2) moduli space for the standard metric on R^4. We propose that for an arbitrary boundary metric the AdS/CFT bulk space-time is the instanton moduli space equipped with the Information Metric.   To test this proposal, we examine the variation of the instanton moduli and the Information Metric for first-order perturbations of the boundary metric and obtain three non-trivial and somewhat surprising results: (1) The perturbed Information Metric is Einstein. (2) The perturbed instanton density is the corresponding massless boundary-to-bulk scalar propagator. (3) The regularized boundary-to-bulk geodesic distance is proportional to the logarithm of the perturbed instanton density. The Hamilton-Jacobi equation implied by (3) equips the moduli space with a rich geometrical structure which we explore.   These results tentatively suggest a picture in which the one-instanton sector of SU(2) Yang-Mills theory (rather than some large-N limit) is in some sense holographically dual to bulk gravity. 
  We construct a generalized BF theory in superspace that can embed eleven-dimensional supergravity theory. Our topological BF theory can accommodate all the necessary Bianchi identities for teleparallel superspace supergravity in eleven-dimensions, as the simplest but nontrivial solutions to superfield equations for our superspace action. This indicates that our theory may have solutions other than eleven-dimensional supergravity, accommodating generalized theories of eleven-dimensional supergravity. Therefore our topological theory can be a good candidate for the low energy limit of M-theory, as an underlying fundamental theory providing a `missing link' between eleven-dimensional supergravity and M-theory. 
  Type A N-fold supersymmetry of one-dimensional quantum mechanics can be constructed by using sl(2) generators represented on a finite dimensional functional space. Using this sl(2) formalism we show a general method of constructing Type A N-fold supersymmetric models. We also present systematic generation of known models and several new models using this method. 
  Classical nonlinear canonical (Poisson) maps have a distinguished role in quantum mechanics. They act unitarily on the quantum phase space and generate $\hbar$-independent quantum canonical maps. It is shown that such maps act in the noncommutative phase space as dictated by the classical covariance. A crucial observation made is that under the classical covariance the local quantum mechanical picture can become nonlocal in the Hilbert space. This nonlocal picture is made equivalent by the Weyl map to a noncommutative picture in the phase space formulation of the theory. The connection between the entanglement and nonlocality of the representation is explored and specific examples of the generation of entanglement are provided by using such concepts as the generalized Bell states. That the results have direct application in generating vacuum soliton configurations in the recently popular scalar field theories of noncommutative coordinates is also demonstrated. 
  We study branes and open strings in a large class of orbifolds of a curved background using microscopic techniques of boundary conformal field theory. In particular, we obtain factorizing operator product expansions of open string vertex operators for such branes. Applications include branes in Z2 orbifolds of the SU(2) WZW model and in the D-series of unitary minimal models considered previously by Runkel. 
  We briefly review attempts to construct string theories that yield the standard model, concentrating on models with a geometric interpretation. Calabi-Yau compactifications are discussed in the context of both the weakly coupled heterotic string and the strongly coupled version, heterotic M-theory. Similarly, we consider orbifold compactifications of weakly coupled heterotic theory and of Type II theories in the presence of D-branes. The latter allows a "bottom-up" construction that so far seems the most natural and direct route to the standard model. 
  We discuss bosonic and supersymmetric Yang-Mills matrix models with compact semi-simple gauge group. We begin by finding convergence conditions for the partition and correlation functions. Moving on, we specialise to the SU(N) models with large N. In both the Yang-Mills and cohomological formulations, we find all quantities which are invariant under the supercharges. Finally, we apply the deformation method of Moore, Nekrasov and Shatashvili directly to the Yang-Mills model. We find a deformation of the action which generates mass terms for all the matrix fields whilst preserving some supersymmetry. This allows us to rigorously integrate over a BRST quartet and arrive at the well known formula of MNS. 
  We investigate static axially symmetric monopole and black hole solutions with magnetic charge n > 1 in Einstein-Yang-Mills-Higgs theory. For vanishing and small Higgs selfcoupling, multimonopole solutions are gravitationally bound. Their mass per unit charge is lower than the mass of the n=1 monopole. For large Higgs selfcoupling only a repulsive phase exists. The static axially symmetric hairy black hole solutions possess a deformed horizon with constant surface gravity. We consider their properties in the isolated horizon framework, interpreting them as bound states of monopoles and black holes. Representing counterexamples to the ``no-hair'' conjecture, these black holes are neither uniquely characterized by their horizon area and horizon charge. 
  Formulas for the expansion of arbitrary invariant group functions in terms of the characters for the Sp(2N), SO(2N+1), and SO(2N) groups are derived using a combinatorial method. The method is similar to one used by Balantekin to expand group functions over the characters of the U(N) group. All three expansions have been checked for all N by using them to calculate the known expansions of the generating function of the homogeneous symmetric functions. An expansion of the exponential of the traces of group elements, appearing in the finite-volume gauge field partition functions, is worked out for the orthogonal and symplectic groups. 
  We construct a one-parameter set of intersecting D4-brane models, with six stacks, that yield the (non-supersymmetric) standard model plus extra vector-like matter. Twisted tadpoles and gauge anomalies are cancelled, and the model contains all of the Yukawa couplings to the tachyonic Higgs doublets that are needed to generate mass terms for the fermions.   A string scale in the range 1-10 TeV and a Higgs mass not much greater than the current bound is obtained for certain values of the parameters, consistently with the observed values of the gauge coupling constants. 
  We discuss the propagation of gravity in five-dimensional Minkowski space in the presence of a four-dimensional brane. We show that there exists a solution to the wave equation that leads to a propagator exhibiting four-dimensional behavior at low energies (long distances) with five-dimensional effects showing up as corrections at high energies (short distances). We compare our results with propagators derived in previous analyses exhibiting five-dimensional behavior at low energies. We show that different solutions correspond to different physical systems. 
  BPS lumps in ${\cal N}=2$ SUSY nonlinear sigma models on hyper-\kahler manifolds in four dimensions are studied. We present new lump solutions with various kinds of topological charges. New BPS equations and a new BPS bound, expressed by the three complex structures on hyper-\kahler manifolds, are found. We show that any states satisfying these BPS equations preserve 1/8 (1/4) SUSY of ${\cal N}=2$ SUSY nonlinear sigma models with (without) a potential term. These BPS states include non-parallel multi-(Q-)lumps. 
  This is a short guide to some uses of the zeta-function regularization procedure as a a basic mathematical tool for quantum field theory in curved space-time (as is the case of Nambu-Jona-Lasinio models), in quantum gravity models (in different dimensions), and also in cosmology, where it appears e.g. in the calculation of possible `contributions' to the cosmological constant coming through manifestations of the vacuum energy density. Part of this research was carried out in fruitful and enjoyable collaboration with people from Tomsk State Pedagogical University. 
  Horava--Witten M theory -- heterotic string duality poses special problems for the twisted sectors of heterotic orbifolds. In [1] we explained how in M theory the twisted states couple to gauge fields apparently living on M9 branes at both ends of the eleventh dimension at the same time. The resolution involves 7D gauge fields which live on fixed planes of the (T^4/Z_N) x (S^1/Z_2) x R^{5,1} orbifold and lock onto the 10D gauge fields along the intersection planes. The physics of such intersection planes does not follow directly from the M theory but there are stringent kinematic constraints due to duality and local consistency, which allowed us to deduce the local fields and the boundary conditions at each intersection. In this paper we explain various phenomena at the intersection planes in terms of duality between HW and type I' superstring theories. The orbifold fixed planes are dual to stacks of D6 branes, the M9 planes are dual to O8 orientifold planes accompanied by D8 branes, and the intersections are dual to brane junctions. We engineer several junction types which lead to distinct patterns of 7D/10D gauge field locking, 7D symmetry breaking and/or local 6D fields. Another aspect of brane engineering is putting the junctions together; sometimes, the combined effect is rather spectacular from the HW point of view and the quantum numbers of some twisted states have to `bounce' off both ends of the eleventh dimension before their heterotic identity becomes clear. Some models involve D6/O8 junctions where the string coupling diverges towards the orientifold plane. We use the heterotic-HW-I' duality to predict what should happen at such junctions. 
  It is reviewed how Renormalization Group, species doubling and CKM mixing are known to appear in numerical analysis: Butcher group, parasitic solutions and higher order methods 
  We study N=1 four dimensional quiver theories arising on the worldvolume of D3-branes at del Pezzo singularities of Calabi-Yau threefolds. We argue that under local mirror symmetry D3-branes become D6-branes wrapped on a three torus in the mirror manifold. The type IIB (p,q) 5-brane web description of the local del Pezzo, being closely related to the geometry of its mirror manifold, encodes the geometry of 3-cycles and is used to obtain gauge groups, quiver diagrams and the charges of the fractional branes. 
  We perform a path-integral analysis of the string representation of the dual Abelian Higgs (DAH) model beyond the London limit, where the string describing the vortex of a flux tube has a finite thickness. We show that besides an additional vortex core contribution to the string tension, a modified Yukawa interaction appears as a boundary contribution in the type-II dual superconducting vacuum. In the London limit, the modified Yukawa interaction is reduced to the Yukawa one. 
  Two simple proofs are presented for the first order virial expansion of the self-energy of a particle moving through a medium, characterised by temperature and/or chemical potential(s). One is based on the virial expansion of the self-energy operator itself, while the other is on the analysis of its Feynman diagrams in configuration space. 
  We discuss some aspects of closed superstrings in Melvin-type magnetic backgrounds. A 2-parameter family of such NS-NS backgrounds are exactly solvable as weakly coupled string models with the spectrum containing tachyonic modes. Magnetic field allows one to interpolate between free superstring theories with periodic and antiperiodic boundary conditions for the space-time fermions around some compact direction, and, in particular, between type 0 and type II string theories. Using ``9-11'' flip, this interpolation can be extended to M-theory and may be used to study the issue of tachyon in type 0 string theory. We review, following hep-th/0104238, related duality proposals, and, in particular, consider a description of type 0 theory in terms of M-theory in a curved magnetic flux background in which the type 0 tachyon appears to correspond to a state in $d=11$ supergravity fluctuation spectrum. 
  We establish a relationship between the zeros of the partition function in the complex mass plane and the spectral properties of the Dirac operator in QCD. This relation is derived within the context of chiral Random Matrix Theory and applies to QCD when chiral symmetry is spontaneously broken. Further, we introduce and examine the concept of normal modes in chiral spectra. Using this formalism we study the consequences of a finite Thouless energy for the zeros of the partition function. This leads to the demonstration that certain features of the QCD partition function are universal. 
  The algebraic classification of Cardy for boundary states on a $G/H$ coset CFT of a compact group G, is geometrically realized on the corresponding manifold resulting from gauging the WZW model. The branes consist of H orbits of quantized G conjugacy classes shifted by quantized H conjugacy classes. 
  Using the algebraic geometry method of Berenstein et al (hep-th/0005087), we reconsider the derivation of the non commutative quintic algebra ${\mathcal{A}}_{nc}(5)$ and derive new representations by choosing different sets of Calabi-Yau charges ${C_{i}^{a}}$. Next we extend these results to higher $d$ complex dimension non commutative Calabi-Yau hypersurface algebras ${\mathcal{A}}_{nc}(d+2)$. We derive and solve the set of constraint eqs carrying the non commutative structure in terms of Calabi-Yau charges and discrete torsion. Finally we construct the representations of ${\mathcal{A}}_{nc}(d+2) $ preserving manifestly the Calabi-Yau condition $ \sum_{i}C_{i}^{a}=0$ and give comments on the non commutative subalgebras. 
  We discuss the absorption cross section for the minimally-coupled massless scalar field into a stationary and circularly symmetric black hole with nonzero angular velocity in four or higher dimensions. In particular, we show that it equals the horizon area in the zero-frequency limit provided that the solution of the scalar field equation with an incident monochromatic plane wave converges pointwise to a smooth time-independent solution outside the black hole and on the future horizon, with the error term being at most linear in the frequency. We also show that this equality holds for static black holes which are not necessarily spherically symmetric. The zero-frequency scattering cross section is found to vanish in both cases. It is shown in an Addendum that the equality holds for any stationary black hole with vanishing expansion if the limit solution is known to be a constant. 
  We generalize the dimensionally reduced Yang-Mills matrix model by adding d=1 Chern-Simons term and terms for a bosonic vector. The coefficient, \kappa of the Chern-Simons term must be integer, and hence the level structure. We show at the bottom of the Yang-Mills potential, the low energy limit, only the linear motion is allowed for D0 particles. Namely all the particles align themselves on a single straight line subject to \kappa^2/r^2 repulsive potential from each other. We argue the relevant brane configuration to be D0-branes in a D4 after \kappa of D8's pass the system. 
  We give an explicit proof of equivalence of the two-point function to one-loop order in the two formalisms of thermal $\lambda \phi^3$ theory based on the expressions in the real-time formalism. It is indicated that the key-point of completing the proof is to separate carefully the imaginary part of the zero-temperature loop integral from relevant expressions and this fact will certainly be very useful for examination of the equivalence problem of the two formalisms of thermal field theory in other theories, including the one of the propagators for scalar bound states in a NJL model. 
  Hawking radiation emanating from two-dimensional charged and uncharged dilatonic black holes - dimensionally reduced from (2+1) spinning and spinless, respectively, BTZ black holes - is viewed as a tunnelling process. Two dimensional dilatonic black holes (AdS(2) included) are treated as dynamical backgrounds in contrast to the standard methodology where the background geometry is fixed when evaluating Hawking radiation. This modification to the geometry gives rise to a nonthermal part in the radiation spectrum. Nonzero temperature of the extremal two-dimensional charged black hole is found.  The Bekenstein-Hawking area formula is easily derived for these dynamical geometries. 
  We give the bare-bone description of the quasitriangular chiral WZW model for the particular choice of the Lu-Weinstein-Soibelman Drinfeld double of the affine Kac-Moody group. The symplectic structure of the model and its Poisson-Lie symmetry are completely characterized by two $r$-matrices with spectral parameter. One of them is ordinary and trigonometric and characterizes the $q$-current algebra. The other is dynamical and elliptic (in fact Felder's one) and characterizes the braiding of $q$-primary fields. 
  Motivated by the formalism of string bit models, or quantum matrix models, we study a class of simple Hamiltonian models of quantum gravity type in two space-time dimensions. These string bit models are special cases of a more abstract class of models defined in terms of the sl(2) subalgebra of the Virasoro algebra. They turn out to be solvable and their scaling limit coincides in special cases with known transfer matrix models of two-dimensional quantum gravity. 
  We construct a classical solution of vacuum string field theory (VSFT) and study whether it represents the perturbative open string vacuum. Our solution is given as a squeezed state in the Siegel gauge, and it fixes the arbitrary coefficients in the BRST operator in VSFT. We identify the tachyon and massless vector states as fluctuation modes around the classical solution. The tachyon mass squared \alpha m_t^2 is given in a closed form using the Neumann coefficients defining the three-string vertex, and it reproduces numerically the expected value of -1 to high precision. The ratio of the potential height of the solution to the D25-brane tension is also given in terms of the Neumann coefficients. However, the behavior of the potential height in level truncation does not match our expectation, though there are subtle points in the analysis. 
  We discuss an N=2 quantum mechanics with or without a central charge. A representation is constructed with the number of bosonic degrees of freedom less that one half of the fermionic degrees of freedom. We suggest a systematic method of reducing the bosonic degrees of freedom called ``dynamical reduction."  Our consideration opens a problem of a general classification of nonstandard representations of N=2$ superalgebra. 
  We discuss the discrete Z_k D-brane charges (twisted K-theory charges) in five-brane backgrounds from several different points of view. In particular, we interpret it as a result of a standard Higgs mechanism. We show that certain degrees of freedom (singletons) on the boundary of space can extend the corresponding Z_k symmetry to U(1). Related ideas clarify the role of AdS singletons in the AdS/CFT correspondence. 
  We consider multiplet shortening for BPS solitons in N=1 two-dimensional models. Examples of the single-state multiplets were established previously in N=1 Landau-Ginzburg models. The shortening comes at a price of loosing the fermion parity $(-1)^F$ due to boundary effects. This implies the disappearance of the boson-fermion classification resulting in abnormal statistics. To count such short multiplets we introduce a new index. We consider the phenomenon of shortening in a broad class of hybrid models which extend the Landau-Ginzburg models to include a nonflat metric on the target space. Our index turns out to be related to the index of the Dirac operator on the soliton moduli space. The latter vanishes in most cases implying the absence of shortening. We also generalize the anomaly in the central charge to take into account the target space metric. 
  We present an action for noncommutative supersymmetric Yang-Mills theory in ten-dimensions, and confirm its invariance under supersymmetry. We next add higher-order derivative terms to such a noncommutative supersymmetric action. These terms contain fields as high as the quartic order. This resulting action can be regarded as supersymmetric generalization of noncommutative non-Abelian Dirac-Born-Infeld action. Some ambiguities related to field redefinitions are also clarified. 
  The problem of finding most general form of the classical integrable relativistic models of many-body interaction of the $BC_{n}$ type is considered. In the simplest nontrivial case of $n=2$,the extra integral of motion is presented in explicit form within the ansatz similar to the nonrelativistic Calogero-Moser models. The resulting Hamiltonian has been found by solving the set of two functional equations. 
  Six-dimensional Nielsen-Olesen vortices are analyzed in the context of a quadratic gravity theory containing Euler-Gauss-Bonnet self-interactions. The relations among the string tensions can be tuned in such a way that the obtained solutions lead to warped compactification on the vortex. New regular solutions are possible in comparison with the case where the gravity action only consists of the Einstein-Hilbert term. The parameter space of the model is discussed 
  We examine the spectrum and boundary energy in boundary sine-Gordon theory, based on our recent results on the complete spectrum predicted by closing the boundary bootstrap. We check the spectrum and the reflection factors against truncated conformal space, together with a (still unpublished) prediction by Al.B. Zamolodchikov for the boundary energy and the relation between the parameters of the scattering amplitudes and of the perturbed CFT Hamiltonian. In addition, we give a derivation of Zamolodchikov's formulae. We find an entirely consistent picture and strong evidence for the validity of the conjectured spectrum and scattering amplitudes, which together give a complete description of the boundary sine-Gordon theory on mass shell. 
  We calculate conformal anomalies in noncommutative gauge theories by using the path integral method (Fujikawa's method). Along with the axial anomalies and chiral gauge anomalies, conformal anomalies take the form of the straightforward Moyal deformation in the corresponding conformal anomalies in ordinary gauge theories. However, the Moyal star product leads to the difference in the coefficient of the conformal anomalies between noncommutative gauge theories and ordinary gauge theories. The $\beta$ (Callan-Symanzik) functions which are evaluated from the coefficient of the conformal anomalies coincide with the result of perturbative analysis. 
  The so-called renormalization group (RG) method is applied to derive kinetic and transport equations from the respective microscopic equations. The derived equations include Boltzmann equation in classical mechanics, Fokker-Planck equation, a rate equation in a quantum field theoretical model. Utilizing the formulation of the RG method which elucidates the important role played by the choice of the initial conditions, the general structure and the underlying assumptions in the derivation of kinetic equations in the RG method is clarified. It is shown that the present formulation naturally leads to the choice for the initial value of the microscopic distribution function at arbitrary time $t_0$ to be on the averaged distribution function to be determined. The averaged distribution function may be thought as an integral constant of the solution of microscopic evolution equation; the RG equation gives the slow dynamics of the would-be initial constant, which is actually the kinetic equation governing the averaged distribution function. It is further shown that the averaging as given above gives rise to a coarse-graining of the time-derivative which is expressed with the initial time $t_0$, thereby leads to time-irreversible equations even from a time-reversible equation. It is shown that a further reduction of Boltzmann equation to fluid dynamical equations and the adiabatic elimination of fast variables in Fokker-Planck equation are also performed in a unified way in the present method. 
  We show that the existence of algebraic forms of exactly-solvable $A-B-C-D$ and $G_2, F_4$ Olshanetsky-Perelomov Hamiltonians allow to develop the {\it algebraic} perturbation theory, where corrections are computed by pure algebraic means. A classification of perturbations leading to such a perturbation theory based on representation theory of Lie algebras is given. In particular, this scheme admits an explicit study of anharmonic many-body problems. Some examples are presented. 
  In this paper we study the action for N D0-branes in a curved background. In particular, we focus on the meaning of space-time diffeomorphism invariance. For a single D-brane, diffeomorphism invariance acts in a naive way on the world-volume fields, but for multiple D-branes, the meaning of diffeomorphism invariance is much more obscure. The problem goes beyond the determination of an ordering of the U(N)-valued fields, because one can show that there is no lift of ordinary diffeomorphisms to matrix-valued diffeomorphisms. On the other hand, the action can presumably be constructed from perturbative string theory calculations. Based on the general characteristics of such calculations we determine a set of constraints on the action for N D0-branes, that ensure space-time covariance. These constraints can be solved order by order, but they are insufficient to determine the action completely. All solutions to the constraints obey the axioms of D-geometry. Moreover the action must contain new terms. This exhibits clearly that the answer is more than a suitable ordering of the action of a single D0 brane. 
  We consider the classical null p-brane dynamics in D-dimensional curved backgrounds and apply the Batalin-Fradkin-Vilkovisky approach for BRST quantization of general gauge theories. Then we develop a method for solving the tensionless $p$-brane equations of motion and constraints. This is possible whenever there exists at least one Killing vector for the background metric. It is shown that the same method can be also applied for the tensile 1-branes. Finally, we give two examples of explicit exact solutions in four dimensions. 
  We calculate the finite temperature effective potential of $\lambda\phi^4$ at the two loop order of the 2PPI expansion. This expansion contains all diagrams which remain connected when two lines meeting at the same point are cut and therefore sums systematically the bubble graphs. At one loop in the 2PPI expansion, the symmetry restoring phase transition is first order. At two loops, we find a second order phase transition with mean field critical exponents. 
  We discuss the relation between two different models which are recently proposed as the model of localizing bulk gauge fields on a brane. In the former model, the localization of gauge field is achieved by adding both bulk and boundary mass terms while in the latter, it is done by taking into consideration the coupling between the gauge field and the dilaton field (this model is also regarded as the gauge theory with nontrivial dielectric ``constant''). We make a certain transformation for the gauge field in the latter Lagrangian. As the result, we find those two models are closely related to each other. 
  Smooth manifolds of G_2 holonomy, used to compactify M-theory to four dimensions, give only abelian gauge groups without charged matter multiplets. But singular G_2-manifolds can give abelian or nonabelian gauge groups with chiral fermions. We describe the mechanism of anomaly cancellation in these models, which depends upon anomaly inflow from the bulk. We also compare the anomaly predictions to what has been learned by more explicit arguments in some special cases. 
  The partition function of QCD is analyzed for an arbitrary number of flavors, N_f, and arbitrary quark masses including the contributions from all topological sectors in the Leutwyler--Smilga regime. For given N_f and arbitrary vacuum angle, \theta, the partition function can be reduced to N_f-2 angular integrations of single Bessel functions. For two and three flavors, the \theta dependence of the QCD vacuum is studied in detail. For N_f= 2 and 3, the chiral condensate decreases monotonically as \theta increases from zero to \pi and the chiral condensate develops a cusp at \theta=\pi for degenerate quark masses in the macroscopic limit. We find a discontinuity at \theta=\pi in the first derivative of the energy density with respect to \theta for degenerate quark masses. This corresponds to the first--order phase transition in which CP is spontaneously broken, known as Dashen's phenomena. 
  We show that the entropy of any black object in any dimension can be understood as the entropy of a highly excited string on the stretched horizon. The string has a gravitationally renormalized tension due to the large redshift near the horizon. The Hawking temperature is given by the Hagedorn temperature of the string. As examples, we consider black holes with one (black p-branes) or two charges, Reissner-Nordstrom black holes and the BTZ black hole in addition to Schwarzschild black holes. We show that the vanishing and nonvanishing extremal entropies can be obtained as smooth limits of the near-extreme cases. 
  We discuss non-renormalization properties of some composite operators in N=4 supersymmetric Yang-Mills theory. 
  Using the method developed in {\tt hep-th/0103015}, we determine the non-abelian Born-Infeld action through ${\cal O}(\alpha'{}^3)$. We start from solutions to a Yang-Mills theory which define a stable holomorphic vector bundle. Subsequently we investigate its deformation away from this limit. Through $ {\cal O}(\alpha'{}^2)$, a unique, modulo field redefinitions, solution emerges. At $ {\cal O}(\alpha'{}^3)$ we find a one-parameter family of allowed deformations. The presence of derivative terms turns out to be essential. Finally, we present a detailed comparison of our results to existing, partial results. 
  The stability of physical systems depends on the existence of a state of least energy. In gravity, this is guaranteed by the positive energy theorem. For topological reasons this fails for nonsupersymmetric Kaluza-Klein compactifications, which can decay to arbitrarily negative energy. For related reasons, this also fails for the AdS soliton, a globally static, asymptotically toroidal $\Lambda<0$ spacetime with negative mass. Nonetheless, arguing from the AdS/CFT correspondence, Horowitz and Myers (hep-th/9808079) proposed a new positive energy conjecture, which asserts that the AdS soliton is the unique state of least energy in its asymptotic class. We give a new structure theorem for static $\Lambda<0$ spacetimes and use it to prove uniqueness of the AdS soliton. Our results offer significant support for the new positive energy conjecture and add to the body of rigorous results inspired by the AdS/CFT correspondence. 
  We study the gauge invariance of the massive modes in the compactification of gauge theories from D=5 to D=4. We deal with Abelian gauge theories of rank one and two, and with non-Abelian ones of rank one. We show that St\"uckelberg fields naturally appear in the compactification mechanism, contrarily to what usually occurs in literature where they are introduced by hand, as a trick, to render gauge invariance for massive theories. We also show that in the non-Abelian case they appear in a very different way when compared with their usual implementation in the non-Abelian Proca model. 
  General model of multidimensional $R^2$-gravity including Riemann tensor square term (non-zero $c$ case) is considered. The number of brane-worlds in such model is constructed (mainly in five dimensions) and their properties are discussed. Thermodynamics of S-AdS BH (with boundary) is presented when perturbation on $c$ is used. The entropy, free energy and energy are calculated. For non-zero $c$ the entropy (energy) is not proportional to the area (mass). The equation of motion of brane in BH background is presented as FRW equation. Using dual CFT description it is shown that dual field theory is not conformal one when $c$ is not zero. In this case the holographic entropy does not coincide with BH entropy (they coincide for Einstein gravity or $c=0$ HD gravity where AdS/CFT description is well applied).   Asymmetrically warped background (analog of charged AdS BH) where Lorentz invariance violation occurs is found. The cosmological 4d dS brane connecting two dS bulk spaces is formulated in terms of parameters of $R^2$-gravity. Within proposed dS/CFT correspondence the holographic conformal anomaly from five-dimensional higher derivative gravity in de Sitter background is evaluated. 
  We review a recent development in the theoretical understanding of the nu=5/2 quantum Hall plateau and propose a new conformal field theory, slightly different from the Moore-Read one, to describe another universality class relevant for this plateau. The ground state is still given by the Pfaffian and is completely polarized, however, the elementary quasiholes are charge 1/2 anyons with abelian statistics theta=pi/2, which obey complete spin-charge separation. The physical hole is represented by two such quasiholes plus a free neutral Majorana fermion. We also compute the periods and amplitudes of the chiral persistent currents in both states and show that they have different temperature dependence. Finally, we find indications of a classical two-step phase transition between the new and the Moore-Read states, through a compressible state, which is characterized by the spontaneous breaking of a hidden Z_2 symmetry corresponding to the conservation of the chiral fermion parity. We believe that this transition could explain the "kink" observed in the activation experiment for nu=5/2. 
  We show the existence of non-threshold bound states of (p, q) string networks and D3-branes, preserving 1/4 of the full type IIB supersymmetry, interpreted as string networks dissolved in D3-branes. We also write down the expression for the mass density of the system and discuss the extension of the construction to other Dp-branes. Differences in our construction of string networks with the ones interpreted as dyons in N=4 gauge theories are also pointed out. 
  We obtain the renormalized equations of motion for matter and semi-classical gravity in an inhomogeneous space-time. We use the functional Schrodinger picture and a simple Gaussian approximation to analyze the time evolution of the $\lambda\phi^4$ model, and we establish the renormalizability of this non-perturbative approximation. We also show that the energy-momentum tensor in this approximation is finite once we consider the usual mass and coupling constant renormalizations, without the need of further geometrical counter-terms. 
  We develop a systematic method of directly embedding supermembrane wrapped around a circle into matrix string theory. Our purpose is to study connection between matrix string and membrane from an entirely 11 dimensional point of view. The method does neither rely upon the DLCQ limit nor upon string dualities. In principle, this enables us to construct matrix string theory with arbitrary backgrounds from the corresponding supermembrane theory. As a simplest application of the formalism, the matrix-string action with a 7 brane background (Kaluza-Klein Melvin solution) with nontrivial RR vector field is given. 
  We study topological defects as inhomogeneous (localized) condensates of particles in Quantum Field Theory. In the framework of the Closed-Time-Path formalism, we consider explicitly a $(1+1)$ dimensional $\la \psi^4$ model and construct the Heisenberg picture field operator $\psi$ in the presence of kinks. We show how the classical kink solutions emerge from the vacuum expectation value of such an operator in the Born approximation and/or $\la \to 0$ limit. The presented method is general in the sense that applies also to the case of finite temperature and to non-equilibrium; it also allows for the determination of Green's functions in the presence of topological defects. We discuss the classical kink solutions at $T\neq 0$ in the high temperature limit. We conclude with some speculations on the possible relevance of our method for the description of the defect formation during symmetry-breaking phase transitions. 
  As a step toward constructing realistic brane world models in string theory, we consider the interactions of a pair of non-BPS branes. We construct a dyonic generalization of the non-BPS branes first constructed by Bergman, Gaberdiel and Sen as orbifolds of D-branes on $T^4/\BZ_2$. The force between a dyonic brane and an electric brane is computed and is found to vanish at a nontrivial critical separation. This equilibrium point is unstable. For smaller separations the branes coalesce to form a composite dyonic state, while for larger separations the branes run off to infinity. We suggest generalizations that will lead to potentials with stable local minima. 
  BPS walls and junctions are studied in ${\cal N}=1$ SUSY nonlinear sigma models in four spacetime dimensions. New BPS junction solutions connecting N discrete vacua are found for nonlinear sigma models with several chiral scalar superfields. A nonlinear sigma model with a single chiral scalar superfield is also found which has a moduli space of the topology of $S^1$ and admits BPS walls and junctions connecting arbitrary points in moduli space. SUSY condition in nonlinear sigma models are classified either as stationary points of superpotential or singularities of the K\"ahler metric in field space. The total number of SUSY vacua is invariant under holomorphic field redefinitions if we count ``runaway vacua'' also. 
  For renormalizable models a method is presented to unambiguously compute the energy that is carried by localized field configurations (solitons). A variational approach for the total energy is utilized to search for soliton configurations. As an example a 1+1 dimensional model is considered. The quantum energy of configurations that are translationally invariant for a subset of coordinates is discussed. 
  We evaluate $N$ dependences of correlation functions in the bosonic part of the IIB matrix model by the Monte Carlo method. We also evaluate those in two sorts of regularized Schild models and find that the $N$ dependences are different from those in the matrix model. In particular, the distribution of the eigenvalues are logarithmically divergent in the regularized Schild model when $g^2N$ is fixed. 
  We present a full two-loop O(g^6) perturbative field theoretic calculation of the expectation value of two circular Maldacena-Wilson loops in D=4 N=4 supersymmetric U(N) gauge theory. It is demonstrated that, after taking into account very subtle cancellations of bulk and boundary divergences, the result is completely finite without any renormalization. As opposed to previous lower order calculations existing in the literature, internal vertex diagrams no longer cancel identically and lead to subleading corrections to the dominant ladder diagrams. Taking limits, we proceed to extract the two-loop static potential corresponding to two infinite anti-parallel lines. Our result gives some evidence that the existing strong-coupling calculations using the AdS/CFT conjecture might sum up the full set of large N planar Feynman diagrams. 
  Effective field theories with (large) extra dimensions are studied within a physical regularization scheme provided by string theory. Explicit string calculations then allow us to consistently analyze the ultraviolet sensitivity of Kaluza--Klein theories in the presence or absence of low energy supersymmetry. 
  In the context of a five dimensional N=1 Kaluza Klein model compactified on S_1/Z_2 x Z_2' we compute the one-loop gauge corrections to the self energy of the (zero-mode) scalar field. The result is quadratically divergent due to the appearance of a Fayet-Iliopoulos term. 
  An extension of dimensional regularization to the case of compact dimensions is presented. The procedure preserves the Kaluza-Klein tower structure, but has a regulator specific to the compact dimension. Possible 5 and 4 dimensional divergent as well as manifest finite contributions of (one-loop) Feynman graphs can easily be identified in this scheme. 
  Free noncommutative fields constitute a natural and interesting example of constrained theories with higher derivatives. The quantization methods involving constraints in the higher derivative formalism can be nicely applied to these systems. We study real and complex free noncommutative scalar fields where momenta have an infinite number of terms. We show that these expressions can be summed in a closed way and lead to a set of Dirac brackets which matches the usual corresponding brackets of the commutative case. 
  We consider conditions under which a universe contracting towards a big crunch can make a transition to an expanding big bang universe. A promising example is 11-dimensional M-theory in which the eleventh dimension collapses, bounces, and re-expands. At the bounce, the model can reduce to a weakly coupled heterotic string theory and, we conjecture, it may be possible to follow the transition from contraction to expansion. The possibility opens the door to new classes of cosmological models. For example, we discuss how it suggests a major simplification and modification of the recently proposed ekpyrotic scenario. 
  In the preceding paper (Phys. Lett. B463 (1999) 257), the authors presented a q-analog of the ADHMN construction and obtained a family of anti-selfdual configurations with a parameter q for classical SU(2) Yang-Mills theory in four-dimensional Euclidean space. The family of solutions can be seen as a q-analog of the single BPS monopole preserving (anti-)selfduality. Further discussion is made on the relation to axisymmetric ansatz on anti-selfdual equation given by Witten in the late seventies. It is found that the q-exponential functions familiar in q-analysis appear as analytic functions categorizing the anti-selfdual configurations yielded by axisymmetric ansatz. 
  Born-Infeld theory is the non-linear generalization of Maxwell electrodynamics. It naturally arises as the low-energy effective action of open strings, and it is also part of the world-volume effective action of D-branes. The N=1 and N=2 supersymmetric generalizations of the Born-Infeld action are closely related to partial spontaneous breaking of rigid extended supersymmetry. We review some remarkable features of the Born-Infeld action and outline its supersymmetric generalizations in four dimensions. The non-abelian N=1 supersymmetric extension of the Born-Infeld theory and its N=1 supergravitational avatars are given in superspace. 
  We show how a radiation dominated universe subject to space-time quantization may give rise to inflation as the radiation temperature exceeds the Planck temperature. We consider dispersion relations with a maximal momentum (i.e. a mimimum Compton wavelength, or quantum of space), noting that some of these lead to a trans-Planckian branch where energy increases with decreasing momenta. This feature translates into negative radiation pressure and, in well-defined circumstances, into an inflationary equation of state. We thus realize the inflationary scenario without the aid of an inflaton field. As the radiation cools down below the Planck temperature, inflation gracefully exits into a standard Big Bang universe, dispensing with a period of reheating. Thermal fluctuations in the radiation bath will in this case generate curvature fluctuations on cosmological scales whose amplitude and spectrum can be tuned to agree with observations. 
  Supersymmetric domain-wall spacetimes that lift to Ricci-flat solutions of M-theory admit generalized Heisenberg (2-step nilpotent) isometry groups. These metrics may be obtained from known cohomogeneity one metrics of special holonomy by taking a "Heisenberg limit", based on an In\"on\"u-Wigner contraction of the isometry group. Associated with each such metric is an Einstein metric with negative cosmological constant on a solvable group manifold. We discuss the relevance of our metrics to the resolution of singularities in domain-wall spacetimes and some applications to holography. The extremely simple forms of the explicit metrics suggest that they will be useful for many other applications. We also give new but incomplete inhomogeneous metrics of holonomy SU(3), $G_2$ and Spin(7), which are $T_1$, $T_2$ and $T_3$ bundles respectively over hyper-K\"ahler four-manifolds. 
  In this Letter we consider the problem of partial masslessness and unitarity in (A)dS using gauge invariant description of massive high spin particles. We show that for S = 2 and S = 3 cases such formalism allows one correctly reproduce all known results. Then we construct a gauge invariant formulation for massive particles of arbitrary integer spin s in arbitrary space-time dimension d. For d = 4 our results confirm the conjecture made recently by Deser and Waldron. 
  We clarify a mechanism to obtain a massless gauge boson from the Kaluza-Klein approach of the Randall-Sundrum(RS) brane world. This corresponds exactly to the same mechanism of achieving a localization of the gauge boson by adding both the bulk and brane mass terms. Accordingly this work puts another example for a localization-mechanism of the gauge boson on the brane. 
  We argue that the van Dam-Veltman-Zakharov discontinuity arising in the $M^2 \to 0$ limit of the massive graviton through an explicit Pauli-Fierz mass term could be absent in anti de Sitter space. This is possible if the graviton can acquire mass spontaneously from the higher curvature terms or/and the massless limit $M^2\to 0$ is attained faster than the cosmological constant $\Lambda \to 0$. We discuss the effects of higher-curvature couplings and of an explicit cosmological term ($\Lambda$) on stability of such continuity and of massive excitations. 
  We describe low energy physics in the CFL and LOFF phases by means of effective lagrangians. In the CFL case we present also how to derive expressions for the parameters appearing in the lagrangian via weak coupling calculations taking advantage of the dimensional reduction of fermion physics around the Fermi surface. The Goldstone boson of the LOFF phase turns out to be a phonon satisfying an anisotropic dispersion relation. 
  We consider the construction of fluxbranes in certain curved geometries, generalizing the familiar construction of the Melvin fluxtube as a quotient of flat space. The resulting configurations correspond to fluxbranes wrapped on cycles in curved spaces. The non-trivial transverse geometry leads in some instances to solutions with asymptotically constant dilaton profiles. We describe explicitly several supersymmetric solutions of this kind. The solutions inherit some properties from their flat space cousins, like flux periodicity. Interestingly type IIA/0A fluxbrane duality holds near the core of these fluxbranes, but does not persist in the asymptotic region, precisely where it would contradict perturbative inequivalence of IIA/0A theories. 
  The basic arguments underlying the symplectic projector method are presented. By this method, local free coordinates on the constrait surface can be obtained for a broader class of constrained systems. Some interesting examples are analyzed. 
  We consider the problem of the interaction between D0-brane bound state and 1-form RR photons by the world-line theory. Based on the fact that in the world-line theory the RR gauge fields depend on the matrix coordinates of D0-branes, the gauge fields also appear as matrices in the formulation. At the classical level, we derive the Lorentz-like equations of motion for D0-branes, and it is observed that the center-of-mass is colourless with respect to the SU(N) sector of the background. Using the path integral method, the perturbation theory for the interaction between the bound state and the RR background is developed. We discuss what kind of field theory may be corresponded to the amplitudes which are calculated by the perturbation expansion in world-line theory. Qualitative considerations show that the possibility of existence of a map between the world-line theory and the non-Abelian gauge theory is very considerable. 
  We give some remarks on twisted determinant line bundles and Chern-Simons topological invariants associated with real hyperbolic manifolds. Index of a twisted Dirac operator is derived. We discuss briefly application of obtained results in topological quantum field theory. 
  The 1983 book, free at last, with corrections and bookmarks. From the original troff, but now with CM (TeX) fonts. 
  We consider the Born-Infeld action for symmetry-preserving, orientable D-branes in compact group manifolds. We find classical solutions that obey the flux quantization condition. They correspond to conformally invariant boundary conditions on the world sheet. We compute the spectrum of quadratic fluctuations and find agreement with the predictions of conformal field theory, up to a missing level-dependent truncation. Our results extend to D-branes with the geometry of twined conjugacy classes; they illustrate the mechanism of flux stabilization of D-branes. 
  A dual pair of supersymmetric string theories that involves an asymmetric orbifold and an orientifold of Type II is considered. The D-branes of the orbifold theory (that were recently determined by Gutperle) are all non-BPS and do not carry any conserved gauge charges. It is shown that they carry non-trivial K-theory charges, and that they can be understood in terms of branes wrapping certain homology classes of the M-theory compactification. Using the adiabatic argument, dual partners of some of these non-BPS D-branes are proposed. The relations between these dual states are found to be in agreement with the M-theory description of the D-branes. 
  The limitations of the approach based on using fields restricted to the lightfront (Lightfront Quantization or p$\to \infty $ Frame Approach) which drive quantum fields towards canonical and ultimately free fields are well known. Here we propose a new concept which does not suffer from this limitation. It is based on a procedure which cannot be directly formulated in terms of pointlike fields but requires ``holographic'' manipulations of the algebras generated by those fields. We illustrate the new concepts in the setting of factorizable d=1+1 models and show that the known fact of absence of ultraviolet problems in those models (in the presence of higher than canonical dimensions) also passes to their holographic images. In higher spacetime dimensions d\TEXTsymbol{>}1+1 the holographic image lacks the transversal localizability; however this can be remedied by doing holography on d-2 additional lightfronts which share one lightray (Scanning by d-1 chiral conformal theories). 
  We construct and check by explicit Feynman diagram calculations the BRST Ward identities for N=2 rigid super Yang-Mills theory and N=2 extended supergravity in four-dimensional Euclidean space without auxiliary fields. We use the Batalin-Vilkovisky formalism. In the supergravity case we need one new contractible pair of complex spinor fields to obtain the usual gauge-fixing term and corresponding Nielsen-Kallosh ghosts. 
  We study extremal and non-extremal generalizations of the regular non-abelian monopole solution of hep-th/9707176, interpreted in hep-th/0007018 as 5-branes wrapped on a shrinking S^2. Naively, the low energy dynamics is pure N=1 supersymmetric Yang-Mills. However, our results suggest that the scale of confinement and chiral symmetry breaking in the Yang-Mills theory actually coincides with the Hagedorn temperature of the little string theory. We find solutions with regular horizons and arbitrarily high Hawking temperature. Chiral symmetry is restored at high energy density, corresponding to large black holes. But the entropy of the black hole solutions decreases as one proceeds to higher temperatures, indicating that there is a thermodynamic instability and that the canonical ensemble is ill-defined. For certain limits of the black hole solutions, we exhibit explicit non-linear sigma models involving a linear dilaton. In other limits we find extremal non-BPS solutions which may have some relevance to string cosmology. 
  We study a boundary version of the gauged WZW model with a Poisson-Lie group G as the target. The Poisson-Lie structure of G is used to define the Wess-Zumino term of the action on surfaces with boundary. We clarify the relation of the model to the topological Poisson sigma model with the dual Poisson-Lie group G^* as the target and show that the phase space of the theory on a strip is essentially the Heisenberg double of G introduced by Semenov-Tian-Shansky 
  We begin by reviewing the noncommutative supersymmetric tubular configurations in the matrix theory. We identify the worldvolume gauge fields, the charges and the moment of R-R charges carried by the tube. We also study the fluctuations around many tubes and tube-D0 systems. Based on the supersymmetric tubes, we have constructed more general configurations that approach supersymmetric tubes asymptotically. These include a bend with angle and a junction that connects two tubes to one. The junction may be interpreted as a finite-energy domain wall that interpolates U(1) and U(2) worldvolume gauge theories. We also construct a tube along which the noncommutativity scale changes. Relying upon these basic units of operations, one may build physical configurations corresponding to any shape of Riemann surfaces of arbitrary topology. Variations of the noncommutativity scale are allowed over the Riemann surfaces. Particularly simple such configurations are Y-shaped junctions. 
  We formulate models of complex scalar fields in the space-time that has a two-dimensional sphere as extra dimensions. The Dirac-Wu-Yang monopole is set in two-sphere S^2 as a background gauge field. The nontrivial topology of the monopole induces topological defects, i.e. vortices. When the radius of S^2 is larger than a critical radius, the scalar field develops a vacuum expectation value and creates vortices in S^2. Then the vortices break the rotational symmetry of S^2. We exactly evaluate the critical radius as r_q = \sqrt{|q|}/\mu, where q is the monopole number and \mu is the imaginary mass of the scalar. We show that the vortices repel each other. We analyze the vacua of the models with one scalar field in each case of q=1/2, 1, 3/2 and find that: when q=1/2, a single vortex exists; when q=1, two vortices sit at diametrical points on S^2; when q=3/2, three vortices sit at the vertices of the largest triangle on S^2. The symmetry of the model G = U(1) x SU(2) x CP is broken to H_{1/2} = U(1)', H_{1} = U(1)'' x CP, H_{3/2} = D_{3h}, respectively. Here D_{3h} is the symmetry group of a regular triangle. We extend our analysis to the doublet scalar fields and show that the symmetry is broken from G_{doublet} = U(1) x SU(2) x SU(2)_f x P to H_{doublet} = SU(2)' x P. Finally we obtain the exact vacuum of the model with the multiplet (q_1, q_2,..., q_{2j+1}) = (j, j,..., j) and show that the symmetry is broken from G_{multiplet} = U(1) x SU(2) x SU(2j+1)_f x CP to H_{multiplet} = SU(2)' x CP'. Our results caution that a careful analysis of dynamics of the topological defects is required for construction of a reliable model that possesses such a defect structure. 
  We discuss some properties of higher derivative (HD) bulk gravity without Riemann tensor square term. Such a kind of gravity admits Schwarzschild Anti de Sitter (SAdS) black hole as exact solution. It is shown that induced brane geometry on such background is Friedmann-Robertson-Walker (FRW) radiation dominated Universe. We show that HD terms contributions appear in the Hawking temperature, entropy and Hubble parameter via the redefinition of 5-dimensional gravitational constant and AdS scale parameter. These HD terms do not destroy the AdS-dual description of radiation represented by strongly-coupled CFT. So-called Cardy-Verlinde formula which expresses cosmological entropy as square root from other parameters and entropies is also derived in R^2-gravity. This talk is based on works with Shin'ichi Nojiri and Sergei D. Odintsov. 
  D=2,N=2 generalized Wess-Zumino theory is investigated by the dimensional reduction from D=4,N=1 theory. For each solitonic configuration (i,j) the classical static solution is solved by the Hamilton-Jacobi method of equivalent one-dimensional classical mechanics. It is easily shown that the Bogomol'nyi mass bound is saturated by these solutions and triangular mass inequality M_{ij}<M_{ik}+M_{kj} is automatically satisfied. 
  We review our recent results on the on-shell description of sine-Gordon model with integrable boundary conditions. We determined the spectrum of boundary states by closing the boundary bootstrap and gave a derivation of Al.B. Zamolodchikov's (unpublished) formulae for the boundary energy and the relation between the Lagrangian (ultraviolet) and bootstrap (infrared) parameters. These results have been checked against numerical finite volume spectra coming from the truncated conformal space approach. We find an entirely consistent picture and strong evidence for the validity of the conjectured spectrum and scattering amplitudes, which together give a complete description of the boundary sine-Gordon theory on mass shell. 
  The Polyakov bosonic string is quantised in Euclidean Anti-de Sitter space-time using functional methods. Regularisation of the functional determinants using both heat-kernel and zeta function techniques shows that the AdS_{D+1} string reduces to the Liouville field theory, as in the flat space-time case. A Feynman graph expansion is used to evaluate (approximately) the coefficient multiplying the Liouville action; this then gives the critical dimension for this space-time as 22 (that is, 21 flat directions plus one radial direction). 
  A new way of constructing fusion bases (i.e., the set of inequalities governing fusion rules) out of fusion elementary couplings is presented. It relies on a polytope reinterpretation of the problem: the elementary couplings are associated to the vertices of the polytope while the inequalities defining the fusion basis are the facets. The symmetry group of the polytope associated to the lowest rank affine Lie algebras is found; it has order 24 for $\su(2)$, 432 for $\su(3)$ and quite surprisingly, it reduces to 36 for $\su(4)$, while it is only of order 4 for $\sp(4)$. This drastic reduction in the order of the symmetry group as the algebra gets more complicated is rooted in the presence of many linear relations between the elementary couplings that break most of the potential symmetries. For $\su(2)$ and $\su(3)$, it is shown that the fusion-basis defining inequalities can be generated from few (1 and 2 respectively) elementary ones. For $\su(3)$, new symmetries of the fusion coefficients are found. 
  We propose a general scheme to construct multiple Lagrangians for completely integrable non-linear evolution equations that admit multi- Hamiltonian structure. The recursion operator plays a fundamental role in this construction. We use a conserved quantity higher/lower than the Hamiltonian in the potential part of the new Lagrangian and determine the corresponding kinetic terms by generating the appropriate momentum map. This leads to some remarkable new developments. We show that nonlinear evolutionary systems that admit $N$-fold first order local Hamiltonian structure can be cast into variational form with $2N-1$ Lagrangians which will be local functionals of Clebsch potentials. This number increases to $3N-2$ when the Miura transformation is invertible. Furthermore we construct a new Lagrangian for polytropic gas dynamics in $1+1$ dimensions which is a {\it local} functional of the physical field variables, namely density and velocity, thus dispensing with the necessity of introducing Clebsch potentials entirely. This is a consequence of bi-Hamiltonian structure with a compatible pair of first and third order Hamiltonian operators derived from Sheftel's recursion operator. 
  We investigate aspects of cosmology in brane world theories with a bulk scalar field. We concentrate on a recent model motivated from supergravity in singular spaces. After discussing the background evolution of such a brane-world, we present the evolution of the density contrast. We compare our results to those obtained in the (second) Randall-Sundrum scenario and usual 4D scalar-tensor theories. 
  We show that 2D noncommutative harmonic oscillator has an isotropic representation in terms of commutative coordinates. The noncommutativity in the new mode, induces energy level splitting, and is equivalent to an external magnetic field effect. The equivalence of the spectra of the isotropic and anisotropic representation is traced back to the existence of SU(2) invariance of the noncommutative model. 
  We discuss the off-shell renormalization properties of the abelian Higgs model in the unitary gauge. The model is not renormalizable according to the usual power counting rules. In this paper, however, we show that with a proper choice of interpolating fields for the massive photon and the Higgs particle, their off-shell Green functions can be renormalized. An analysis of the nature of the extra singularities in the unitary gauge is given, and a recipe for the off-shell renormalization is provided. 
  We consider properties of the CKM phase in the heterotic orbifold models. We find that at the renormalizable level the CKM phase vanishes identically for the prime orbifolds, whereas it can be non-zero for some non-prime orbifolds. In particular, we study in detail the Z_6-I orbifold which allows for a non-trivial CKM phase and analyze the modular properties of the corresponding Jarlskog invariant. The CKM phase is shown to vanish if the moduli fields are stabilized at Im T_i=\pm 1/2. 
  We investigate hetrotic string theory on special holonomy manifolds including exceptional holonomy G_2 and Spin(7) manifolds. The gauge symmetry is F_4 in a G_2 manifold compactification, and so(9) in a Spin(7) manifold compactification. We also study the cascade of the holonomies: so(8) > Spin(7) > G_2 > su(3) > su(2). The differences of adjoining groups are described by Ising, tricritical Ising, 3-state Potts and u(1) models. These theories are essential for spacetime supersymmetries and gauge group enhancements. As concrete examples, we construct modular invariant partition functions and analyze their massless spectra for G_2 and Spin(7) orbifolds. We obtain the relation between topological numbers of the manifolds and multiplicities of matters in specific representations. 
  We examine heterotic M-theory compactified on a Calabi-Yau manifold with an additional parallel M5 brane. The dominant non-perturbative effect stems from open membrane instantons connecting the M5 with the boundaries. We derive the four-dimensional low-energy supergravity potential for this situation including subleading contributions as it turns out that the leading term vanishes after minimisation. At the minimum of the potential the M5 gets stabilised at the middle of the orbifold interval while the vacuum energy is shown to be manifestly positive. Moreover, induced by the non-trivial running of the Calabi-Yau volume along the orbifold which is driven by the G-fluxes, we find that the orbifold-length and the Calabi-Yau volume modulus are stabilised at values which are related by the G-flux of the visible boundary. Finally we determine the supersymmetry-breaking scale and the gravitino mass for this open membrane vacuum. 
  We study relevant deformations of the N=2 superconformal theory on the world-volume of N D3 branes at an A_{k-1} singularity. In particular, we determine the vacuum structure of the mass-deformed theory with N=1 supersymmetry and show how the different vacua are permuted by an extended duality symmetry. We then obtain exact, modular covariant formulae (for all k, N and arbitrary gauge couplings) for the holomorphic observables in the massive vacua in two different ways: by lifting to M-theory, and by compactification to three dimensions and subsequent use of mirror symmetry. In the latter case, we find an exact superpotential for the model which coincides with a certain combination of the quadratic Hamiltonians of the spin generalization of the elliptic Calogero-Moser integrable system. 
  The calculation of the standard model Lagrangian of classical field theory within the framework of noncommutative geometry is sketched using a variant with 18 parameters. Improvements compared with the traditional formulation are contrasted with remaining deviations from the requirements of physics. 
  The spectrum of the fermionic operators depending on external fields is an important object in Quantum Field Theory. In this paper we prove, using transition to the alternative basis for the $\gamma$-matrices, that this spectrum does not depend on the sign of the fermion mass, up to a constant factor. This assumption has been extensively used, but usually without proof. As an illustration, we calculated the coincidence limit of the coefficient $a_2(x,x^\prime)$ on the general metric background, vector and axial vector fields. 
  We propose tubular field theory, which is a continuum analogue of lattice field theory. One-dimensional links (and zero-dimensional sites) in lattice field theory are replaced by two-dimensional tubes to result in two-dimensional spacetime microscopically. As an example, scalar and gauge fields are considered in `three-dimensional' tubular spacetime. 
  In order to estimate the effect of dynamical gluons to chiral condensate, the gap equation of SU(2) gauged Nambu--Jona-Lasinio model, under a constant background magnetic field, is investigated up to the two-loop order in 2+1 and 3+1 dimensions. We set up a general formulation allowing both cases of electric as well as magnetic background field. We rely on the proper time method to maintain gauge invariance. In 3+1 dimensions chiral symmetry breaking ($\chi$SB) is enhanced by gluons even in zero background magnetic field and becomes much striking as the background field grows larger. In 2+1 dimensions gluons also enhance $\chi$SB but whose dependence on the background field is not simple: dynamical mass is not a monotone function of background field for a fixed four-fermi coupling. 
  We investigate the $Spin(7)$ holonomy metric of cohomogeneity one with the principal orbit $SU(3)/U(1)$. A choice of U(1) in the two dimensional Cartan subalgebra is left as free and this allows manifest $\Sigma_3=W(SU(3))$ (= the Weyl group) symmetric formulation. We find asymptotically locally conical (ALC) metrics as octonionic gravitational instantons. These ALC metrics have orbifold singularities in general, but a particular choice of the U(1) subgroup gives a new regular metric of $Spin(7)$ holonomy. Complex projective space ${\bf CP}(2)$ that is a supersymmetric four-cycle appears as a singular orbit. A perturbative analysis of the solution near the singular orbit shows an evidence of a more general family of ALC solutions. The global topology of the manifold depends on a choice of the U(1) subgroup. We also obtain an $L^2$-normalisable harmonic 4-form in the background of the ALC metric. 
  A general procedure is outlined which allows one to construct superintegrable models of Winternitz type. Some examples are presented. 
  It is shown that deformations of twistor space compatible with the Moyal deformation of Minkowski space-time must take the form recently suggested by Kapustin, Kuznetsov and Orlov. 
  We show that the exact N=1 superpotential of a class of 4d string compactifications is computed by the closed topological string compactified to two dimensions. A relation to the open topological string is used to define a special geometry for N=1 mirror symmetry. Flat coordinates, an N=1 mirror map for chiral multiplets and the exact instanton corrected superpotential are obtained from the periods of a system of differential equations. The result points to a new class of open/closed string dualities which map individual string world-sheets with boundary to ones without. It predicts an mathematically unexpected coincidence of the closed string Gromov-Witten invariants of one Calabi-Yau geometry with the open string invariants of the dual Calabi-Yau. 
  We study gauge theories which are associated with classical vacua of perturbative Type II string theory that allows for a conformal field theory description. We show that even if we compactify seven spatial dimensions (allowing for one macroscopic dimension to arise non-perturbatively) the Standard Model cannot be obtained from the perturbative sector of Type II superstrings. Therefore, the construction of the Standard Model (or extensions thereof) from M theory must involve fields that are non-perturbative from the Type II perspective. We also address the case of eight compact dimensions. 
  We review a recent progress in constructing low-energy effective action in N=4 super Yang-Mills theories. Using harmonic superspace approach we consider N=4 SYM in terms of unconstrained N=2 superfield and apply N=2 background field method to finding effective action for N=4 SU(n) SYM broken down to U(1)^(n-1). General structure of leading low-energy corrections to effective action is discussed. 
  We calculate the tension of the D3-brane in the fivebrane background which is described by the exactly solvable SU(2)_k x U(1) world-sheet conformal field theory with large Kac-Moody level k. The D3-brane tension is extracted from the amplitude of one closed string exchange between two parallel D3-branes, and the amplitude is calculated by utilizing the open-closed string duality. The tension of the D3-brane in the background does not coincide with the one in the flat space-time even in the flat space-time limit: k -> infinity. The finite curvature effect should vanish in the flat space-time limit and only the topological effect can remain. Therefore, the deviation indicates the condensation of gravitino and/or dilatino which has been expected in the fivebrane background as a gravitational instanton. 
  In my talk I shall consider the mechanism of self-expansion of a system of N D0-branes into high-dimensional non-commutative world-volume investigated by Harmark and Savvidy. Here D2-brane is formed due to the internal angular momentum of D0-brane system. The idea is that attractive force of tension should be cancelled by the centrifugal motion preventing a D-brane system from collapse to a lower-dimensional one. I shall also present a new extended solution where a total of 9 space dimensions is used to embed a D0-brane system. In the last section, by performing linear analysis, the stability of the system is demonstrated. 
  We construct a linear sigma model for open-strings ending on special Lagrangian cycles of a Calabi-Yau manifold. We illustrate the construction for the cases considered by Aganagic and Vafa in hep-th/0012041. This leads naturally to concrete models for the moduli space of open-string instantons. These instanton moduli spaces can be seen to be intimately related to certain auxiliary boundary toric varieties. By considering the relevant Gelfand-Kapranov-Zelevinsky (GKZ) differential equations of the boundary toric variety, we obtain the contributions to the worldvolume superpotential on the A-branes from open-string instantons. By using an ansatz due to Aganagic, Klemm and Vafa in hep-th/0105045, we obtain the relevant change of variables from the linear sigma model to the non-linear sigma model variables - the open-string mirror map. Using this mirror map, we obtain results in agreement with those of AV and AKV for the counting of holomorphic disc instantons. 
  Representations of four dimensional superconformal groups are constructed as fields on many different superspaces, including super Minkowski space, chiral superspace, harmonic superspace and analytic superspace. Any unitary irreducible representation can be given as a field on any one of these spaces if we include fields which transform under supergroups. In particular, on analytic superspaces, the fields are unconstrained. One can obtain all representations of the N=4 complex superconformal group $PSL(4|4)$ with integer dilation weight from copies of the Maxwell multiplet on $(4,2,2)$ analytic superspace. This construction is compared with the oscillator construction and it is shown that there is a natural correspondence between the oscillator construction of superconformal representations and those carried by superfields on analytic superspace. 
  This article reviews some recent progress in our understanding of the structure of Rational Conformal Field Theories, based on ideas that originate for a large part in the work of A. Ocneanu. The consistency conditions that generalize modular invariance for a given RCFT in the presence of various types of boundary conditions --open, twisted-- are encoded in a system of integer multiplicities that form matrix representations of fusion-like algebras. These multiplicities are also the combinatorial data that enable one to construct an abstract ``quantum'' algebra, whose $6j$- and $3j$-symbols contain essential information on the Operator Product Algebra of the RCFT and are part of a cell system, subject to pentagonal identities. It looks quite plausible that the classification of a wide class of RCFT amounts to a classification of ``Weak $C^*$- Hopf algebras''. 
  We study the brane world cosmology in the RS2 model where the electric charge varies with time in the manner described by the varying fine-structure constant theory of Bekenstein. We map such varying electric charge cosmology to the dual variable-speed-of-light cosmology by changing system of units. We comment on cosmological implications for such cosmological models. 
  A family of conformal boundary states for a free boson on a circle is constructed. The family contains superpositions of conventional U(1)-preserving Neumann and Dirichlet branes, but for general parameter values the boundary states are fundamental and preserve only the conformal symmetry. The relative overlaps satisfy Cardy's condition, and each boundary state obeys the factorisation constraint. It is also argued that, together with the conventional Neumann and Dirichlet branes, these boundary states already account for all fundamental conformal D-branes of the free boson theory. The results can be generalised to the situation with N=1 world-sheet supersymmetry, for which the family of boundary states interpolates between superpositions of non-BPS branes and combinations of conventional brane anti-brane pairs. 
  We demonstrate that AdS_5 x T^{pq} is unstable, in the sense of Breitenlohner and Freedman, for unequal p and q. This settles, negatively, the long-standing question of whether the T^{pq} manifolds for unequal p and q might correspond to non-supersymmetric fixed points of the renormalization group. We also show that the AdS_3 x S^7 vacuum of Sugimoto's USp(32) open string theory is unstable. This explains, at a heuristic level, the apparent absence of a heterotic string dual. 
  We compute the leading radiative correction to the Casimir force between two parallel plates in the $\lambda\Phi^4$ theory. Dirichlet and periodic boundary conditions are considered. A heuristic approach, in which the Casimir energy is computed as the sum of one-loop corrected zero-point energies, is shown to yield incorrect results, but we show how to amend it. The technique is then used in the case of periodic boundary conditions to construct a perturbative expansion which is free of infrared singularities in the massless limit. In this case we also compute the next-to-leading order radiative correction, which turns out to be proportional to $\lambda^{3/2}$. 
  We adopt a physically motivated empirical approach to the characterisation of the distributions of twin and triplet primes within the set of primes, rather than in the set of all natural numbers. Remarkably, the occurrences of twins or triplets in any finite sequence of primes are like fixed-probability random events. The respective probabilities are not constant, but instead depend on the length of the sequence in ways that we have been able to parameterise. For twins the ``decay constant'' decreases as the reciprocal of the logarithm of the length of the sequence, whereas for triplets the falloff is faster: decreasing as the square of the reciprocal of the logarithm of the number of primes. The manner of the decrease is consistent with the Hardy--Littlewood Conjectures, developed using purely number theoretic tools of analysis. 
  We show that the topological massive BF theories can be written as a pure BF term through field redefinitions. The fields are rewritten as power expansion series in the inverse of the mass parameter $m$. We also give a cohomological justification of this expansion through BRST framework. In this approach the BF term can be seen as a topological generator for massive BF theories. 
  The superspace Lagrangian formulation of N=1 supersymmetric quantum mechanics is presented. The general Lagrangian constructed out of chiral and antichiral supercoordinates containing up to two derivatives and with a canonically normalized kinetic energy term describes the motion of a nonrelativistic spin 1/2 particle with Land\'e g-factor 2 moving in two spatial dimensions under the influence of a static but spatially dependent magnetic field. Noether's theorem is derived for the general case and is used to construct superspace dependent charges whose lowest components give the superconformal generators. The supercoordinate of charges containing an R symmetry charge, the supersymmetry charges and the Hamiltonian are combined to form a supercharge supercoordinate. Superconformal Ward identities for the quantum effective action are derived from the conservation equations and the source of potential symmetry breaking terms are identified. 
  General features of the spectra of matter states in all 175 models found in a previous work by the author are discussed. Only twenty patterns of representations are found to occur. Accomodation of the Minimal Supersymmetric Standard Model (MSSM) spectrum is addressed. States beyond those contained in the MSSM and nonstandard hypercharge normalization are shown to be generic, though some models do allow for the usual hypercharge normalization found in SU(5) embeddings of the Standard Model gauge group. The minimum value of the hypercharge normalization consistent with accomodation of the MSSM is determined for each model. In some cases, the normalization can be smaller than that corresponding to an SU(5) embedding of the Standard Model gauge group, similar to what has been found in free fermionic models. Bizzare hypercharges typically occur for exotic states, allowing for matter which does not occur in the decomposition of SU(5) representations---a result which has been noted many times before in four-dimensional string models. Only one of the twenty patterns of representations, comprising seven of the 175 models, is found to be without an anomalous U(1). The sizes of nonvanishing vacuum expectation values induced by the anomalous U(1) are studied. It is found that large radius moduli stabilization may lead to the breakdown of sigma-model perturbativity. Various quantities of interest in effective supergravity model building are tabulated for the set of 175 models. In particular, it is found that string moduli masses appear to be generically quite near the gravitino mass. String scale gauge coupling unification is shown to be possible, albeit contrived, in an example model. The intermediate scales of exotic particles are estimated and the degree of fine-tuning is studied. 
  In this paper, we look for metrics of cohomogeneity one in D=8 and D=7 dimensions with Spin(7) and G_2 holonomy respectively. In D=8, we first consider the case of principal orbits that are S^7, viewed as an S^3 bundle over S^4 with triaxial squashing of the S^3 fibres. This gives a more general system of first-order equations for Spin(7) holonomy than has been solved previously. Using numerical methods, we establish the existence of new non-singular asymptotically locally conical (ALC) Spin(7) metrics on line bundles over \CP^3, with a non-trivial parameter that characterises the homogeneous squashing of CP^3. We then consider the case where the principal orbits are the Aloff-Wallach spaces N(k,\ell)=SU(3)/U(1), where the integers k and \ell characterise the embedding of U(1). We find new ALC and AC metrics of Spin(7) holonomy, as solutions of the first-order equations that we obtained previously in hep-th/0102185. These include certain explicit ALC metrics for all N(k,\ell), and numerical and perturbative results for ALC families with AC limits. We then study D=7 metrics of $G_2$ holonomy, and find new explicit examples, which, however, are singular, where the principal orbits are the flag manifold SU(3)/(U(1)\times U(1)). We also obtain numerical results for new non-singular metrics with principal orbits that are S^3\times S^3. Additional topics include a detailed and explicit discussion of the Einstein metrics on N(k,\ell), and an explicit parameterisation of SU(3). 
  Abelian deformations of ordinary algebras of functions are studied. The role of Harrison cohomology in classifying such deformations is illustrated in the context of simple examples chosen for their relevance to physics. It is well known that Harrison cohomology is trivial on smooth manifolds and that, consequently, abelian *-products on such manifolds are trivial to first order in the deformation parameter. The subject is nevertheless interesting; first because varieties with singularities appear in the physical context and secondly, because deformations that are trivial to first order are not always (indeed not usually) trivial as exact deformations. We investigate cones, to illustrate the situation on algebraic varieties, and we point out that the coordinate algebra on (anti-) de Sitter space is a nontrivial deformation of the coordinate algebra on Minkowski space -- although both spaces are smooth manifolds. 
  We use the general $N = 1$ supersymmetric formulation of one dimensional sigma models on non trivial manifolds and its subsequent quantization to formulate the classical and quantum dynamics of the $ N= 2 $ supersymmetric charged particle moving on a sphere in the field of a monopole. The factorization method accommodated with the general covariance and it is used to integrate the corresponding system. 
  A path-integral approach for the computation of quantum-mechanical propagators and energy Green's functions is presented. Its effectiveness is demonstrated through its application to singular interactions, with particular emphasis on the inverse square potential--possibly combined with a delta-function interaction. The emergence of these singular potentials as low-energy nonrelativistic limits of quantum field theory is highlighted. Not surprisingly, the analogue of ultraviolet regularization is required for the interpretation of these singular problems. 
  We consider the four-dimensional effective field theory which has been used in previous studies of perturbations in the Ekpyrotic Universe, and discuss the spectrum of cosmological fluctuations induced on large scales by quantum fluctuations of the bulk brane. By matching cosmological fluctuations on a constant energy density hypersurface we show that the growing mode during the very slow collapsing pre-impact phase couples only to the decaying mode in the expanding post-impact phase, and that hence no scale-invariant spectrum of adiabatic fluctuations is generated. Note that our conclusions may not apply to improved toy models for the Ekpyrotic scenario. 
  We give a new construction of the minimal unitary representation of the exceptional group E_8(8) on a Hilbert space of complex functions in 29 variables. Due to their manifest covariance with respect to the E_7(7) subgroup of E_8(8) our formulas are simpler than previous realizations, and thus well suited for applications in superstring and M theory. 
  Head-on collisions between two solitons in the pure $CP^1$ model on a flat torus are investigated via numerical simulations. The charge-two lumps, written out in terms of Weierstrass' elliptic $\wp$-function, are found to scatter at 90$^{\circ}$. The phenomenon of singularity formation is also seen. 
  We present our results of a numerical investigation of the behaviour of a system of two solitons in the (2+1) dimensional $CP^1$ model on a torus. Defined by the elliptic function of Weierstrass, and working in the Skyrme version of the model, the soliton lumps exhibit splitting, scattering at right angles and motion reversal in the various configurations considered. The work is restricted to systems with no initial velocity. 
  Inozemtsev models are classically integrable multi-particle dynamical systems related to Calogero-Moser models. Because of the additional q^6 (rational models) or sin^2(2q) (trigonometric models) potentials, their quantum versions are not exactly solvable in contrast with Calogero-Moser models. We show that quantum Inozemtsev models can be deformed to be a widest class of partly solvable (or quasi-exactly solvable) multi-particle dynamical systems. They posses N-fold supersymmetry which is equivalent to quasi-exact solvability. A new method for identifying and solving quasi-exactly solvable systems, the method of pre-superpotential, is presented. 
  Motivated by applications of twisted current algebras in description of the entropy of $Ads_3$ black hole, we investigate the simplest twisted current algebra $sl(3,{\bf C})^{(2)}_k$. Free field representation of the twisted algebra and the corresponding twisted Sugawara energy-momentum tensor are obtained by using three $(\beta,\gamma)$ pairs and two scalar fields. Primary fields and two screening currents of the first kind are presented. 
  By studying already known extrema of non-semi-simple Inonu-Wigner contraction CSO(p, q)^{+} and non-compact SO(p, q)^{+}(p+q=8) gauged N=8 supergravity in 4-dimensions developed by Hull sometime ago, one expects there exists nontrivial flow in the 3-dimensional boundary field theory. We find that these gaugings provide first-order domain-wall solutions from direct extremization of energy-density. We also consider the most general CSO(p, q, r)^{+} with p+q+r=8 gauging of N=8 supergravity by two successive SL(8,R) transformations of the de Wit-Nicolai theory, that is, compact SO(8) gauged supergravity. The theory found earlier has local SU(8)x CSO(p, q, r)^{+} gauge symmetry as well as local N=8 supersymmetry. The gauge group CSO(p, q, r)^{+} is spontaneously reduced to its maximal compact subgroup SO(p)^{+} x SO(q)^{+} x U(1)^{+r(r-1)/2}. The T-tensor we obtain describes a two-parameter family of gauged N=8 supergravity from which one can construct A_1 and A_2 tensors. The effective nontrivial scalar potential can be written as the difference of positive definite terms. We examine the scalar potential for critical points at which the expectation value of the scalar field is SO(p)^{+} x SO(q)^{+} x SO(r)^{+} invariant. It turns out that there is no new extra critical point. However, we do have flow equations and domain-wall solutions for the scalar fields are the gradient flow equations of the superpotential that is one of the eigenvalues of A_1 tensor. 
  We use the generalised rational map ansatz introduced by Ioannidou et al. to construct analytically some topologically non-trivial solutions of the generalised SU(3) Skyrme model defined by adding a sixth order term to the usual Lagrangian. These solutions are radially symmetric and some of them can be interpreted as bound states of Skyrmions. The same ansatz is used to construct low-energy configuration of the SU(N) Skyrme model. 
  A particular initial state for the construction of a perturbative QCD expansion is investigated. It is formed as a coherent superposition of zero momentum gluon pairs and shows Lorentz as well as global $SU(3)$ symmetries. The general form of the Wick theorem is discussed, and it follows that the gluon and ghost propagators determined by the proposed vacuum state, coincides with the ones used in an alternative of the usual perturbation theory proposed in a previous work, and reviewed here. Therefore, the ability of such a procedure of producing a finite gluon condensation parameter already in the first orders of perturbation theory is naturally explained. It also follows that this state satisfies the physicality condition of the BRST procedure in its Kugo and Ojima formulation. A brief review of the canonical quantization for gauge fields, developed by Kugo and Ojima, is done and the value of the gauge parameter $\alpha$ is fixed to $\alpha=1$ where the procedure is greatly simplified. Therefore, after assuming that the adiabatic connection of the interaction does not take out the state from the interacting physical space, the predictions of the perturbation expansion for the physical quantities, at the value $\alpha=1$, should have meaning. The validity of this conclusion solves the gauge dependence indeterminacy remained in the proposed perturbation expansion. 
  We show why and when D0-branes at the Gepner point of Calabi-Yau manifolds given as Fermat hypersurfaces exist. 
  Using functional derivatives with respect to free propagators and interactions we derive a closed set of Schwinger-Dyson equations in quantum electrodynamics. Its conversion to graphical recursion relations allows us to systematically generate all connected and one-particle irreducible Feynman diagrams for the $n$-point functions and the vacuum energy together with their correct weights. 
  We discuss the ultra-violet properties of bosonic and supersymmetric noncommutative non-linear sigma-models in two dimensions, both with and without a Wess-Zumino-Witten term. 
  A four-fermion model in 2+1 dimensions describing N Dirac fermions interacting via SU(N) invariant N^2-1 four-fermion interactions is solved in the leading order of the 1/N expansion. The 1/N expansion corresponds to 't Hoofts topological 1/N expansion in which planar Feynman diagrams prevail. For the symmetric phase of this model, it is argued that the planar expansion corresponds to the ladder approximation. A truncated set of Schwinger-Dyson equations for the fermion propagator and composite boson propagator representing the relevant planar diagrams is solved analytically. The critical four-fermion coupling and various critical exponents are determined as functions of N. The universality class of this model turns out to be quite distinct from the Gross-Neveu model in the large N limit. 
  We argue that the Karch-Randall compactification is holographically dual to a 4-d conformal field theory coupled to gravity on Anti de Sitter space. Using this interpretation we recover the mass spectrum of the model. In particular, we find no massless spin-2 states. By giving a purely 4-d interpretation to the compactification we make clear that it represents the first example of a local 4-d field theory in which general covariance does not imply the existence of a massless graviton. We also discuss some variations of the Karch-Randall model discussed in the literature, and we examine whether its properties are generic to all conformal field theory. 
  In this paper a formulation of U(1) gauge theory on a fuzzy torus is discussed. The theory is regulated in both the infrared and ultraviolet. It can be thought of as a non-commutative version of lattice gauge theory on a periodic lattice. The construction of Wilson loops is particularly transparent in this formulation. Following Ishibashi, Iso, Kawai and Kitazawa, we show that certain Fourier modes of open Wilson lines are gauge invariant. We also introduce charged matter fields which can be thought of as fundamentals of the gauge group. These particles behave like charges in a strong magnetic field and are frozen into the lowest Landau levels. The resulting system is a simple matrix quantum mechanics which should reflect much of the physics of charged particles in strong magnetic fields.   The present results were first presented as a talk at the Institute for Mathematical Science, Chennai, India; the author wishes to thank Prof. T. R. Govindarajan and the IMS for hospitality and financial support, and the audience for pointing me out the work of Ambjorn, Makeenko, Nishimura and Szabo. The author is also grateful to the organizers of the Summer School in Modern Mathematical Physics, Sokobanja, Serbia, Yugoslavia, for financial support and hospitality during presentation of the present work at international conference FILOMAT 2001, Nis, Yugoslavia. 
  An F-theory dual of a nonsupersymmetric orientifold is considered. It is argued that the condensation of both open and closed string tachyons in the orientifold corresponds to the annihilation of branes and anti-branes in the F-theory dual. The end-point of tachyon condensation is thus expected to be the vacuum of Type-IIB superstring. Some speculations are presented about the F-theory dual of the bosonic string and tachyon condensation thereof. 
  The Fedosov deformation quantization of the symplectic manifold is determined by a 1-form differential r. We identify a class of r for which the $\star$ product becomes the Moyal product by taking appropriate Darboux coordinates, but invariant by canonically transforming the coordinates. This respect of the $\star$ product is explained by studying the fuzzy algebrae of the Kaehler coset space. 
  We consider the CFT of a free boson compactified on a circle, such that the compactification radius $R$ is an irrational multiple of $R_{selfdual}$. Apart from the standard Dirichlet and Neumann boundary states, Friedan suggested [1] that an additional 1-parameter family of boundary states exists. These states break U(1) symmetry of the theory, but still preserve conformal invariance. In this paper we give an explicit construction of these states, show that they are uniquely determined by the Cardy-Lewellen sewing constraints, and we study the spectrum in the `open string channel', which is given here by a continous integral with a nonnegative measure on the space of conformal weights. 
  We investigate the existence of bound states in a one-dimensional quantum system of $N$ identical particles interacting with each other through an inverse square potential. This system is equivalent to the Calogero model without the confining term. The effective Hamiltonian of this system in the radial direction admits a one-parameter family of self-adjoint extensions and the negative energy bound states occur when most general boundary conditions are considered. We find that these bound states exist only when $N=3,4$ and for certain values of the system parameters. The effective Hamiltonian for the system is related to the Virasoro algebra and the bound state wavefunctions exhibit a scaling behaviour in the limit of small inter-particle separation. 
  In the present paper the phase transition in the regularized U(1) gauge theory is investigated using the dual Abelian Higgs model of scalar monopoles. The corresponding renormalization group improved effective potential, analogous to the Coleman-Weinberg's one, was considered in the two-loop approximation for $\beta$ functions, and the phase transition (critical) dual and non-dual couplings were calculated in the U(1) gauge theory. It was shown that the critical value of the renormalized electric fine structure constant $\alpha_{\text{crit}}\approx 0.208$ obtained in this paper coincides with the lattice result for compact QED: $\alpha_{\text{crit}}^{\text{lat}}\approx 0.20\pm 0.015$. This result and the behavior of $\alpha$ in the vicinity of the phase transition point were compared with the Multiple Point Model prediction for the values of $\alpha$ near the Planck scale. Such a comparison is very encouraging for the Multiple Point Model assuming the existence of the multiple critical point at the Planck scale. 
  We consider brane cosmology studying the shortest null path on the brane for photons, and in the bulk for gravitons. We derive the differential equation for the shortest path in the bulk for a 1+4 cosmological metric. The time cost and the redshifts for photons and gravitons after traveling their respective path are compared. We consider some numerical solutions of the shortest path equation, and show that there is no shortest path in the bulk for the Randall-Sundrum vacuum brane solution, the linear cosmological solution of Bin\'etruy, et al for $\omega = -1, -{2/3}$, and for some expanding brane universes. 
  We study M-theory on two classes of manifolds of Spin(7) holonomy that are developing an isolated conical singularity. We construct explicitly a new class of Spin(7) manifolds and analyse in detail the topology of the corresponding classical spacetimes. We discover also an intricate interplay between various anomalies in M-theory, string theory, and gauge theory within these models, and in particular find a connection between half-integral G-fluxes in M-theory and Chern-Simons terms of the N=1, D=3 effective theory. 
  In Quantum Mechanics operators must be hermitian and, in a direct product space, symmetric. These properties are saved by Lie algebra operators but not by those of quantum algebras. A possible correspondence between observables and quantum algebra operators is suggested by extending the definition of matrix elements of a physical observable, including the eventual projection on the appropriate symmetric space. This allows to build in the Lie space of representations one-parameter families of operators belonging to the enveloping Lie algebra that satisfy an approximate symmetry and have the properties required by physics. 
  We extend the Vecchia-Veneziano-Witten (VVW) model of QCD in the chiral limit and for large colour number $N_c$, by introducing an effective dilaton-gluon coupling from which we derive both the axion and dilaton potentials. Furthermore, using a string inspired model, we determine a new interquark potential as a perturbative series in terms of the interquark distance $r$. Our potential goes beyond Dick one obtained in [8] and shares the same features as the Bian-Huang-Shen potential $V_{BHS}$ which depends only on odd powers of $r$ [22]. 
  We explicitly construct a Hamiltonian whose exact eigenfunctions are the generalized Laguerre functions. Moreover, we present the related raising and lowering operators. We investigate the corresponding coherent states by adopting the Gazeau-Klauder approach, where resolution of unity and overlapping properties are examined. Coherent states are found to be similar to those found for a particle trapped in a P\"oschl-Teller potential of the trigonometric type. Some comparisons with Barut-Girardello and Klauder-Perelomov methods are noticed. 
  This is a pedagogical introduction to D-branes, addressed to graduate students in field theory and particle physics and to other beginners in string theory. I am not going to review the most recent results since there are already many good papers on web devoted to that. Instead, I will present some old techniques in some detail in order to show how some basic properties of strinfs and branes as the massless spectrum of string, the effective action of D-branes and their tension can be computed using QFT techniques. Also, I will present shortly the boundary state description of D-branes. The details are exposed for bosonic branes since I do not assume any previous knowledge of supersymmetry which is not a requirement for this school. However, for completeness and to provide basic notions for other lectures, I will discuss some properties of supersymemtric branes. The present lectures were delivered at Jorge Andr\'{e} Swieca School on Particle and Fields, 2001, Campos do Jord\~{a}o, Brazil. 
  Recently, a proposal for the full non-abelian open superstring effective action through ${\cal O}(\alpha'{}^3)$ has been formulated in {\tt hep-th/0108169}. We test this result by calculating the spectrum in the presence of constant magnetic background fields and by comparing the result to string theoretic predictions. The agreement is perfect. Other proposals for the superstring effective action through this order do not reproduce the spectrum correctly. 
  A family of solutions with mass and electric charge of five-dimensional Chern-Simons supergravity is displayed. The action contains an extra term that changes the value of the cosmological constant, as considered by Horava. It is shown that the solutions approach asymptotically the Reissner-Nordstrom spacetime. The role of the torsion tensor in providing charged solutions is stressed. 
  We discuss the open string one-loop partition function in tachyon condensation background of a unstable D-brane system. We evaluate the partition function by using the boundary state formulation and find that it is in complete agreement with the result obtained in the boundary string field theory. It suggests that the open string higher loop diagrams may be produced consistently by a closed string field theory, where the D-brane plays a role of source for the closed string field. 
  To help focus ideas regarding possible routes to the breakdown of Lorentz invariance, it is extremely useful to explore concrete physical models that exhibit similar phenomena. In particular, acoustics in Bose--Einstein condensates has the interesting property that at low-momentum the phonon dispersion relation can be written in a ``relativistic'' form exhibiting an approximate ``Lorentz invariance''. Indeed all of low-momentum phonon physics in this system can be reformulated in terms of relativistic curved-space quantum field theory. In contrast, high-momentum phonon physics probes regions where the dispersion relation departs from the relativistic form and thus violates Lorentz invariance. This model provides a road-map of at least one route to broken Lorentz invariance. Since the underlying theory is manifestly physical this type of breaking automatically avoids unphysical features such as causality violations. This model hints at the type of dispersion relation that might be expected at ultra-high energies, close to the Planck scale, where quantum gravity effects are suspected to possibly break ordinary Lorentz invariance. 
  The solitons of the SO(3) gauged Skyrme model with no pion-mass potential were studied in Refs. {nl,jmp}. Here, the effects of the inclusion of this potential are studied. In contrast with the (ungauged) Skyrme model, where the effect of this potential on the solitons is marginal, here it turns out to be decisive, resulting in very different dependence of the energy as a function of the Skyrme coupling constant. 
  We show that the imaginary part of the $\star$-genvalue equation in the Moyal quantization reveals the symmetries of the Hamiltonian by which we obtain the conserved quantities. Applying to the Toda lattice equation, we derive conserved quantities which are used as the independent variables of Wigner function. 
  We present a twistor description for null two-surfaces (null strings) in 4D Minkowski space-time. The Lagrangian density for a variational principle is taken as a surface-forming null bivector. The proposed formulation is reparametrization invariant and free of any algebraic and differential constraints. The spinor formalism of Cartan-Penrose allows us to derive a non-linear evolution equation for the world-sheet coordinate. An example of null two-surface given by the two-dimensional self-intersection (caustic) of a null hypersurface is studied. 
  A new renormalization group treatment is proposed for the critical exponents of an m-fold Lifshitz point. The anisotropic cases (m not equal 8) are described by two independent fixed points associated to two independent momentum flow along the quadratic and quartic directions, respectively. The isotropic case is described separately. In that case, the fixed point is due to renormalization group transformations along the quartic directions. The new scaling laws are derived for both cases and generalize the ones previously reported. 
  We study a class of braneworlds where the cosmological evolution arises as the result of the movement of a three-brane in a five-dimensional static dilatonic bulk, with and without reflection symmetry. The resulting four-dimensional Friedmann equation includes a term which, for a certain range of the parameters, effectively works as a quintessence component, producing an acceleration of the universe at late times. Using current observations and bounds derived from big-bang nucleosynthesis we estimate the parameters that characterize the model. 
  We construct D=11 supergravity solutions dual to the twisted field theories arising when M-theory fivebranes wrap supersymmetric cycles. The cases considered are M-fivebranes wrapped on (i) a complex Lagrangian four-cycle in a D=8 hyper-Kaehler manifold corresponding to a D=2 field theory with (2,1) supersymmetry (ii) a product of two holomorphic two-cycles in a product of two Calabi-Yau two-folds corresponding to a D=2 field theory with (2,2) supersymmetry and (iii) a product of a holomorphic two-cycle and a SLAG three-cycle in a product of a Calabi-Yau two-fold and a Calabi-Yau three-fold corresponding to a quantum mechanics with two supercharges. In each case we construct BPS equations and find IR superconformal fixed points corresponding to new examples of AdS/CFT duality arising from the twisted field theories. 
  We explore the diversity of warped metric function in five-dimensional gravity including a scalar field and a 3-brane.   We point out that the form of the function is determined by a parameter introduced here.   For a particular value of the parameter, the warped metric function is smooth without having a singularity, and we show that the bulk cosmological constant have a upper bound and must be positive and that the lower bound of five-dimensional fundamental scale is controlled by both the brane tension and four-dimensional effective Planck scale.   The general warp factor obtained here may relate to models inspired by SUGRA or M-theory. 
  We show that the velocity-dependent forces between parallel fundamental strings moving apart in a D-dimensional spacetime imply an expanding universe in (D-1)-dimensional spacetime. 
  The fundamental action of superon-graviton model(SGM) for space-time and matter is written down explicitly in terms of the fields of the graviton and superons by using the affine and the spin connection formalisms, alternatively. Some characteristic structures including some hidden symmetries of the gravitational coupling of superons are manifested (in two dimensional space-time) with some details of the calculations. SGM cosmology is discussed briefly. 
  We show that the spectral theory of the Dirac operator $D = i\delsl-\sigma(x) -i\pi(x)\gam_5$ in a static background $(\sigma(x),\pi(x))$ in 1+1 space-time dimensions, is underlined by a certain generalization of supersymmetric quantum mechanics, and explore its consequences. 
  Special kind of closed strings is considered. It is shown that these closed strings behave as two (an even number of) open strings at the classical level and almost one open string at the quantum level. They contain photons in their spectrum and can lie on D branes. Some properties of closed string field effective action are declared. 
  We study static fermion bags in the 1+1 dimensional Gross-Neveu and Nambu-Jona-Lasinio models. It has been known, from the work of Dashen, Hasslacher and Neveu (DHN), followed by Shei's work, in the 1970's, that the self-consistent static fermion bags in these models are reflectionless. The works of DHN and of Shei were based on inverse scattering theory. Several years ago, we offered an alternative argument to establish the reflectionless nature of these fermion bags, which was based on analysis of the spatial asymptotic behavior of the resolvent of the Dirac operator in the background of a static bag, subjected to the appropriate boundary conditions. We also calculated the masses of fermion bags based on the resolvent and the Gelfand-Dikii identity. Based on arguments taken from a certain generalized one dimensional supersymmetric quantum mechanics, which underlies the spectral theory of these Dirac operators, we now realize that our analysis of the asymptotic behavior of the resolvent was incomplete. We offer here a critique of our asymptotic argument. 
  The derivative expansion of the effective action is a perturbative development in derivatives of the fields. The expansion breaks down when some of the derivatives are too large. We show how to sum exactly the first and second derivatives and treat perturbatively derivatives higher than second. 
  Talk given at NATO ARW in Kiev (September 2000) "Non-commutative Structures in Mathematics and Physics". 
  We study how the solution of the two-dimensional Dirichlet boundary problem for smooth simply connected domains depends upon variations of the data of the problem. We show that the Hadamard formula for the variation of the Dirichlet Green function under deformations of the domain reveals an integrable structure. The independent variables corresponding to the infinite set of commuting flows are identified with harmonic moments of the domain. The solution to the Dirichlet boundary problem is expressed through the tau-function of the dispersionless Toda hierarchy. We also discuss a degenerate case of the Dirichlet problem on the plane with a gap. In this case the tau-function is identical to the partition function of the planar large $N$ limit of the Hermitean one-matrix model. 
  In this paper we would like to suggest matrix form of the string field for any configuration of N D-instantons in bosonic string field theory. 
  We study the generation of density perturbations in the ekpyrotic scenario for the early universe, including gravitational backreaction. We expose interesting subtleties that apply to both inflationary and ekpyrotic models. Our analysis includes a detailed proposal of how the perturbations generated in a contracting phase may be matched across a `bounce' to those in an expanding hot big bang phase. For the physical conditions relevant to the ekpyrotic scenario, we re-obtain our earlier result of a nearly scale-invariant spectrum of energy density perturbations. We find that the perturbation amplitude is typically small, as desired to match observation. 
  We study the covariant quantization of the Green-Schwarz (GS) superstrings proposed recently by Berkovits. In particular, we reformulate the Berkovits approach in a way that clarifies its relation with the GS approach and allows to derive in a straightforward way its extension to curved spacetime background. We explain the procedure working explicitly in the case of the heterotic string. 
  The most general N=1 Lagrangian for the spinning particle with local supersymmetry is found and the constraints of the system are analysed. The Dirac quantisation of the model is also investigated. 
  We study four N=1 SU(N)^6 gauge theories, with bi-fundamental chiral matter and a superpotential. In the infrared, these gauge theories all realize the low-energy world-volume description of N coincident D3-branes transverse to the complex cone over a del Pezzo surface dP_3 which is the blowup of P^2 at three generic points. Therefore, the four gauge theories are expected to fall into the same universality class--an example of a phenomenon that has been termed "toric duality." However, little independent evidence has been given that such theories are infrared-equivalent.   In fact, we show that the four gauge theories are related by the N=1 duality of Seiberg, vindicating this expectation. We also study holographic aspects of these gauge theories. In particular we relate the spectrum of chiral operators in the gauge theories to wrapped D3-brane states in the AdS dual description. We finally demonstrate that the other known examples of toric duality are related by N=1 duality, a fact which we conjecture holds generally. 
  We propose a consistent string theory framework for embedding brane world scenarios with infinite-volume extra dimensions. In this framework the Standard Model fields are localized on D3-branes sitting on top of an orientifold 3-plane. The transverse 6-dimensional space is a non-compact orbifold or a more general conifold. The 4-dimensional gravity on D3-branes is reproduced due to the 4-dimensional Einstein-Hilbert term induced at the quantum level. The orientifold 3-plane plays a crucial role, in particular, without it the D3-brane world-volume theories would be conformal due to the tadpole cancellation. We point out that in some cases the 4-dimensional Planck scale is controlled by the size of certain relevant (as opposed to marginal) orbifold blow-ups. We can then have a scenario with the desirable 4-dimensional Planck scale, the string scale of order TeV, and the cross-over to 10-dimensional gravity around the present Hubble size. We discuss some general features as well as concrete models in this ``Orientiworld'' framework, including those with D7-branes. We point out that the D7-brane gauge symmetry at the quantum level becomes part of the 4-dimensional gauge symmetry. We present an N=1 supersymmetric model with 3 chiral generations of quarks and leptons, where the original gauge group (which contains an SU(6) subgroup) can be Higgsed to obtain a Pati-Salam model with 3 chiral generations, the Pati-Salam Higgs fields required to break the gauge group further to that of the Standard Model, as well as the desired electroweak Higgs doublets. The superpotential in this model is such that we have precisely one heavy (top-like) generation. 
  Compactifications of M-theory on singular manifolds contain additional charged massless states descending from M-branes wrapped on vanishing cycles. We construct the first explicit example of a complete supergravity Lagrangian that includes such extra states. This is done for a compactification on a Calabi-Yau threefold that develops a genus zero curve of A_1 singularities at the boundary of the K\"ahler cone with a resulting SU(2) gauge symmetry enhancement. The corresponding SU(2) gauged supergravity Lagrangian includes two charged and two neutral vector multiplets, and turns out to be uniquely fixed by the Calabi-Yau geometry and by the effective ungauged Lagrangian describing the Coulomb branch. One can see explicitly how resolving the singularity corresponds to a supersymmetric Higgs effect in the gauged supergravity description. The elementary transformation relating the two families of smooth Calabi-Yau resolutions of the singularity acts as the SU(2) Weyl twist. The resulting structure appears to be very rigid and is likely to apply to other types of singularities and manifolds as well. 
  The holographic principle is often (and hastily) attributed to quantum gravity and domains of the Planck size. Meanwhile it can be usefully applied to problems where gravitation effects are negligible and domains of less exotic size. The essence of this principle is that any physical system can be taken to be either classical, placed in a D+1-dimensional spacetime, or quantum-mechanical, located in its D-dimensional boundary. For example, one believes that a hydrogen atom is a typical quantum system living in a four-dimensional spacetime, but it can also be conceived as a classical system living in a five-dimensional embracing spacetime. The subnuclear realm is more intricate since the gluon vacuum reveals two phases, the hadronic and plasma phases. They differ in energetics and symmetry. Moreover, the classical four-dimensional picture is pertinent to the behavior of constituent quarks while the plasma phase is expected to be grasped by standard four-dimensional QCD. The relation between the holographic standpoint and the symmetry treatment of these two phases is outlined. Exact retarded solutions to the classical SU(N) four-dimensional Yang-Mills equations with the source composed of several point-like colored particles is considered. Features of these solutions in the large-N limit provide insight into the gauge symmetries of two gluon vacua. 
  The gravitational energy is examined in asymptotically de Sitter space-times. The positivity will be shown for certain cases. The de Sitter/CFT(dS/CFT) correspondence recently proposed and cosmic no-hair conjecture are testified in the aspect of the gravitational energy. From the holographic renormalization group point of view, the two conjectures are deeply connected with each other. 
  It is reviewed how space-time supersymmetry is realized nonlinearly in open superstring theory without making the GSO projection. We show that the world-sheet string dualities, viz. dualities of open-closed strings and of open-open strings, play crucial roles for the existence of 10 dimensional N=2 supersymmetry in a spontaneously broken phase. We also speculate on a possible mechanism of the restoration of supersymmetry from the viewpoint of world-sheet dynamics. 
  In the SL(2) conformal field theory, we write down and analyze the analytic expression of the three-point functions of generic primary fields with definite SL(2) weights. Using these results, we discuss the operator product expansion in the SL(2,R) WZW model. We propose a prescription of the OPE, the classical limit of which is in precise agreement with the tensor products of the representations of SL(2,R). 
  Nonlinear sigma models compatible with the aratyn-Ferreira-Zimerman ansatz are discussed, the latter ansatz automatically leading to configurations with definite values of the Hopf index. These models are allowed to involve a weight factor which is a function of one of the toroidal coordinates. Depending on the choice of the weight factor, the field equation takes various forms. In one model with a special weight factor, the field equation turns out to be the fifth Painleve equation. This model suggests the existence of a knot soliton strictly confined in a finite spatial volume. Some other interesting cases are also discussed. 
  In this paper, we argue that the Landau singularity of the coupling constant in QED reflects the instability of vacuum state, in other words the perturbative ground state in QED.We do not have a rigorous mathematical proof. We rather provide strong arguments derived from the theory of stability of matter developed by Lieb et al on the hand and the theory of weakly interacting fermion systems developed by Feldman et al on the other hand. 
  We develop the basis of the two dimensional generalized quantum statistical systems by using results on $r$-generalized Fibonacci sequences. According to the spin value $s$ of the 2d-quasiparticles, we distinguish four classes of quantum statistical systems indexed by $ s=0,1/2:mod(1)$, $s=1/M:mod(1)$, $s=n/M:mod(1)$ and $0\leq s\leq 1:mod(1)$. For quantum gases of quasiparticles with $s=1/M:mod(1)$, $M\geq 2,$, we show that the statistical weights densities $\rho_M$ are given by the integer hierarchies of Fibonacci sequences. This is a remarkable result which envelopes naturally the Fermi and Bose statistics and may be thought of as an alternative way to the Haldane interpolating statistical method. 
  We use field theory and brane diamond techniques to demonstrate that Toric Duality is Seiberg duality for N=1 theories with toric moduli spaces. This resolves the puzzle concerning the physical meaning of Toric Duality as proposed in our earlier work. Furthermore, using this strong connection we arrive at three new phases which can not be thus far obtained by the so-called ``Inverse Algorithm'' applied to partial resolution of C^3/Z_3 x Z_3. The standing proposals of Seiberg duality as diamond duality in the work by Aganagic-Karch-L\"ust-Miemiec are strongly supported and new diamond configurations for these singularities are obtained as a byproduct. We also make some remarks about the relationships between Seiberg duality and Picard-Lefschetz monodromy. 
  Chiral primary operators annihilated by a quarter of the supercharges are constructed in the four dimensional N=4 Super-Yang-Mills theory with gauge group SU(N). These quarter-BPS operators share many non-renormalization properties with the previously studied half-BPS operators. However, they are much more involved, which renders their construction nontrivial in the fully interacting theory. In this paper we calculate order g^2 two-point functions of local, polynomial, scalar composite operators within a given representation of the SU(4) R-symmetry group. By studying these two-point functions, we identify the eigenstates of the dilatation operator, which turn out to be complicated mixtures of single and multiple trace operators.   Given the elaborate combinatorics of this problem, we concentrate on two special cases. First, we present explicit computations for quarter-BPS operators with scaling dimension Delta < 8. In this case, the discussion applies to arbitrary N of the gauge group. Second, we carry out a leading plus subleading large N analysis for the particular class of operators built out of double and single trace operators only. The large N construction addresses quarter-BPS operators of general dimension. 
  In a recent paper hep-th/0109064, quarter-BPS chiral primaries were constructed in the fully interacting four dimensional N=4 Super-Yang-Mills theory with gauge group SU(N). These operators are annihilated by four supercharges, and at order g^2 have protected scaling dimension and normalization. Here, we compute three-point functions involving these quarter-BPS operators along with half-BPS operators. The combinatorics of the problem is rather involved, and we consider the following special cases: (1) correlators < half half BPS > of two half-BPS primaries with an arbitrary chiral primary; (2) certain classes of < half quarter quarter > and < quarter quarter quarter > three-point functions; (3) three-point functions involving the Delta < 8 operators found in hep-th/0109064; (4) < half quarter quarter> correlators with the special quarter-BPS operator made of single and double trace operators only. The analysis in cases (1)-(3) is valid for general N, while (4) is a large N approximation. Order g^2 corrections to all three-point functions considered in this paper are found to vanish.   In the AdS/CFT correspondence, quarter-BPS chiral primaries are dual to threshold bound states of elementary supergravity excitations. We present a supergravity discussion of two- and three-point correlators involving these bound states. 
  We construct various kinds of gauged noncommutative WZW models. In particular, axial gauged noncommutative U(2)/U(1) WZW model is studied and by integrating out the gauge fields, we obtain a noncommutative non-linear $\sigma$-model. 
  We construct the Lagrangian description of arbitrary integer higher spin massless fields on the background of the D-dimensional anti-de Sitter space. The operator constraints in auxiliary Fock space corresponding to subsidiary conditions for irreducible unitary massless representations of the D-dimensional anti-de Sitter group are formulated. Unlike flat space, the algebra of the constraints turns out to be nonlinear and analogous to the $W_3$ algebra. We construct the nilpotent BRST charge for this nonlinear algebra and derive on its basis the correct field content and gauge invariant action describing the consistent arbitrary integer spin field dynamics in $AdS$ space. 
  A topological monopole-like field configuration exists for Yang-Mills gauge fields in a 4+1 dimensions. When the extra dimension is compactified to 3+1 dimensions with periodic lattice boundary conditions, these objects reappear in the low energy effective theory as a novel solution, a gauged-bosonic Skyrmion. When the low energy theory spontaneously breaks, the Nambu-Goldstone mode develops a VEV, and the gauged-bosonic Skyrmion morphs into a `t Hooft--Polyakov monopole. 
  We formulate and study a class of massive N=2 supersymmetric gauge field theories coupled to boundary degrees of freedom on the strip. For some values of the parameters, the infrared limits of these theories can be interpreted as open string sigma models describing D-branes in large-radius Calabi-Yau compactifications. For other values of the parameters, these theories flow to CFTs describing branes in more exotic, non-geometric phases of the Calabi-Yau moduli space such as the Landau-Ginzburg orbifold phase. Some simple properties of the branes (like large radius monodromies and spectra of worldvolume excitations) can be computed in our model. We also provide simple worldsheet models of the transitions which occur at loci of marginal stability, and of Higgs-Coulomb transitions. 
  We study the relationship between ADHM/Nahm construction and ``solution generating technique'' of BPS solitons in noncommutative gauge theories. ADHM/Nahm construction and ``solution generating technique'' are the most strong ways to construct exact BPS solitons. Localized solitons are the solitons which are generated by the ``solution generating technique.'' The shift operators which play crucial roles in ``solution generating technique'' naturally appear in ADHM/Nahm construction and we can construct various exact localized solitons including new solitons: localized periodic instantons (=localized calorons) and localized doubly-periodic instantons. Nahm construction also gives rise to BPS fluxons straightforwardly from the appropriate input Nahm data which is expected from the D-brane picture of BPS fluxons. We also show that the Fourier-transformed soliton of the localized caloron in the zero-period limit exactly coincides with the BPS fluxon. 
  Confining strings are investigated in the (2+1)D Georgi-Glashow model. This is done in the limit when the electric coupling constant is much larger than the square root of the mass of the Higgs field, but much smaller than the vacuum expectation value of this field. The modification of the Debye mass of the dual photon with respect to the case when it is considered to be negligibly small compared to the Higgs mass, is found. Analogous modifications of the potential of monopole densities and string coupling constants are found as well. 
  Self-duality equations for Yang-Mills fields in d-dimensional Euclidean spaces consist of linear algebraic relations amongst the components of the curvature tensor which imply the Yang-Mills equations. For the extension to superspace gauge fields, the super self-duality equations are investigated, namely, systems of linear algebraic relations on the components of the supercurvature, which imply the self-duality equations on the even part of superspace. A group theory based algorithm for finding such systems is developed. Representative examples in various dimensions are provided, including the Spin(7) and G(2) invariant systems in d=8 and 7, respectively. 
  I present here some new results which make explicit the role of the division algebras R,C,H,O in the construction and classification of, respectively, N=1,2,4,8 global supersymmetric quantum mechanical and classical dynamical systems. In particular an N=8 Malcev superaffine algebra is introduced and its relation to the non-associative N=8 SCA is discussed. A list of present and possible future applications is given. 
  We discuss physical implications of the four-dimensional effective supergravity, that describes low-energy physics of the Randall--Sundrum model with moduli fields in the bulk and charged chiral matter living on the branes. Cosmological constant can be cancelled through the introduction of a brane Polonyi field and a brane superpotential for the 4d dilaton. We deduce a generalization of the effective 4d action to the case of a general, not necessarily exponential, warp factor. We note, that breakdown of supersymmetry in generic warped models may lead to the stabilization of the interbrane distance. 
  We propose localization techniques for computing Gromov-Witten invariants of maps from Riemann surfaces with boundaries into a Calabi-Yau, with the boundaries mapped to a Lagrangian submanifold. The computations can be expressed in terms of Gromov-Witten invariants of one-pointed maps. In genus zero, an equivariant version of the mirror theorem allows us to write down a hypergeometric series, which together with a mirror map allows one to compute the invariants to all orders, similar to the closed string model or the physics approach via mirror symmetry. In the noncompact example where the Calabi-Yau is $K_{\PP^2},$ our results agree with physics predictions at genus zero obtained using mirror symmetry for open strings. At higher genera, our results satisfy strong integrality checks conjectured from physics. 
  We show that M-theory compactified on a compact Joyce 8-manifold of $Spin(7)$-holonomy, which yields an effective theory in $D = 3$ with $\N$ = 1 supersymmetry, admits at some special points in it moduli space a description in terms of type IIA theory on an orientifold of compact Joyce 7-manifold of $G_2$-holonomy. We find the evidence in favour of this duality by computing the massless spectra on both M-thory side and type IIA side. For the latter, we compute the massless spectra by going to the orbifold limit of the Joyce 7-manifold. 
  We give an introductory survey to the use of Hopf algebras in several problems of noncommutative geometry. The main example, the Hopf algebra of rooted trees, is a graded, connected Hopf algebra arising from a universal construction. We show its relation to the algebra of transverse differential operators introduced by Connes and Moscovici in order to compute a local index formula in cyclic cohomology, and to the several Hopf algebras defined by Connes and Kreimer to simplify the combinatorics of perturbative renormalization. We explain how characteristic classes for a Hopf module algebra can be obtained from the cyclic cohomology of the Hopf algebra which acts on it. Finally, we discuss the theory of noncommutative spherical manifolds and show how they arise as homogeneous spaces of certain compact quantum groups. 
  Some general properties of perturbed (rational) CFT in the background metric of symmetric 2D sphere of radius $R$ are discussed, including conformal perturbation theory for the partition function and the large $R$ asymptotic. The truncated conformal space scheme is adopted to treat numerically perturbed rational CFT's in the spherical background. Numerical results obtained for the scaling Lee-Yang model lead to the conclusion that the partition function is an entire function of the coupling constant. Exploiting this analytic structure we are able to describe rather precisely the ``experimental'' truncated space data, including even the large $R$ behavior, starting only with the CFT information and few first terms of conformal perturbation theory. 
  We study strong coupling phenomena in U(1) gauge theory on the non-commutative plane. To do so, we make use of a T-dual description in terms of an $N\to\infty$ limit of U(N) gauge theory on a commutative torus. The magnetic flux on this torus is taken to be $m=N-1$, while the area scales like 1/N, keeping $\Lambda_{QCD}$ fixed. With a few assumptions, we argue that the speed of high frequency light in pure non-commutative QED is modified in the non-commutative directions by the factor $1 + \Lambda_{QCD}^4 \theta^2$, where $\theta$ is the non-commutative parameter. If charged flavours are included, there is an upper bound on the momentum of a photon propagating in the non-commutative directions, beyond which it is unstable against production of charged pairs. We also discuss a particular $\theta\to\infty$ limit of pure non-commutative QED which is T-dual to a more conventional $N\to\infty$ limit with $m/N$ fixed. In the non-commutative description, this limit gives rise to an exotic theory of open strings. 
  We use the approach to generate spin foam models by an auxiliary field theory defined on a group manifold (as recently developed in quantum gravity and quantization of BF-theories) in the context of topological quantum field theories with a 3-form field strength. Topological field theories of this kind in seven dimensions are related to the superconformal field theories which live on the worldvolumes of fivebranes in M-theory. The approach through an auxiliary field theory for spinfoams gives a topology independent formulation of such theories. 
  We search for a possible bosonic (i.e. non-supersymmetric) string/gauge theory correspondence by using IIB and 0B strings as a guide. Our construction is based on the low-energy bosonic string effective action modified by an extra form flux. The closed string tachyon can be stabilyzed if the AdS scale L does not exceed certain critical value, L<L_c. We argue that the extra form may be generated as a soliton from 3-string junctions, similarly to the known non-perturbative (Jackiw-Rebbi-'t Hooft-Hasenfratz) mechanism in gauge theories. The stable AdS_{13} x S^{13} solution is found, which apparently implies the existence of a 12-dimensional AdS-boundary conformal field theory with the SO(14) global symmetry in the large N 't Hooft limit. We also generalize the conjectured AdS/CFT duality to finite temperature, and calculate the `glueball' masses from the dilaton wave equation in the AdS black hole background, in various spacetime dimensions. 
  We present examples of four dimensional, non-supersymmetric field theories in which ultraviolet supersymmetry breaking effects, such as bose-fermi splittings and the vacuum energy, are suppressed by $(\alpha/4 \pi)^{N}$, where $\alpha$ is a weak coupling factor and $N$ can be made arbitrarily large. The particle content and interactions of these models are conveniently represented by a graph with sites and links, describing the gauge theory space structure. While the theories are supersymmetric ``locally'' in theory space, supersymmetry can be explicitly broken by topological obstructions. 
  We construct integrable boundary conditions for sl(2) coset models with central charges c=3/2-12/(m(m+2)) and m=3,4,... The associated cylinder partition functions are generating functions for the branching functions but these boundary conditions manifestly break the superconformal symmetry. We show that there are additional integrable boundary conditions, satisfying the boundary Yang-Baxter equation, which respect the superconformal symmetry and lead to generating functions for the superconformal characters in both Ramond and Neveu-Schwarz sectors. We also present general formulas for the cylinder partition functions. This involves an alternative derivation of the superconformal Verlinde formula recently proposed by Nepomechie. 
  We demonstrate that the UV/IR mixing problems found recently for a scalar $\phi^4$ theory on the fuzzy sphere are localized to tadpole diagrams and can be overcome by a suitable modification of the action. This modification is equivalent to normal ordering the $\phi^4$ vertex. In the limit of the commutative sphere, the perturbation theory of this modified action matches that of the commutative theory. 
  The division algebras R, C, H, O are used to construct and analyze the N=1,2,4,8 supersymmetric extensions of the KdV hamiltonian equation. In particular a global N=8 super-KdV system is introduced and shown to admit a Poisson bracket structure given by the "Non-Associative N=8 Superconformal Algebra". 
  We study the analogue of the Oppenheimer-Snyder model of a collapsing sphere of homogeneous dust on the Randall-Sundrum type brane. We show that the collapsing sphere has the Vaidya radiation envelope which is followed by the brane analogue of the Schwarzschild solution described by the Reissner-Nordstrom metric. The collapsing solution is matched to the brane generalized Vaidya solution and which in turn is matched to the Reissner-Nordstrom metric. The mediation by the Vaidya radiation zone is the new feature introduced by the brane. Since the radiating mediation is essential, we are led to the remarkable conclusion that a collapsing sphere on the brane does indeed, in contrast to general relativity, radiate null radiation. 
  A new simple mechanism for SUSY breaking is proposed due to the coexistence of BPS domain walls. It is assumed that our world is on a BPS domain wall and that the other BPS wall breaks the SUSY preserved by our wall. This mechanism requires no messenger fields nor complicated SUSY breaking sector on any of the walls. We obtain an ${\cal N}=1$ model in four dimensions which has an exact solution of a stable non-BPS configuration of two walls. We propose that the overlap of the wave functions of the N-G fermion and those of physical fields provides a practical method to evaluate SUSY breaking mass splitting on our wall thanks to a low-energy theorem. This is based on our recent works hep-th/0009023 and hep-th/0107204. 
  In this paper, the analog of Maxwell electromagnetism for hydrodynamic turbulence, the metafluid dynamics, is extended in order to reformulate the metafluid dynamics as a gauge field theory. That analogy opens up the possibility to investigate this theory as a constrained system. Having this possibility in mind, we propose a Lagrangian to describe this new theory of turbulence and, subsequently, analyze it from the symplectic point of view.   From this analysis, a hidden gauge symmetry is revealed, providing a clear interpretation and meaning of the physics behind the metafluid theory. Further, the geometrical interpretation to the gauge symmetries is discussed and the spectrum for 3D turbulence computed. 
  In this paper we reformulate Abelian and non-Abelian noninvariant systems as gauge invariant theories using a new constraint conversion scheme, developed on the symplectic framework. This conversion method is not plagued by the ambiguity problem that torments the BFFT and iterative methods and also it seems more powerful since it does not require special modifications to handle with non-Abelian systems. 
  From the requirement of continuous matching of bulk metric around the point of brane collision we derive a conservation law for collisions of p-branes in (p+2)-dimensional space-time. This conservation law relates energy densities on the branes before and after the collision. Using this conservation law we are able to calculate the amount of matter produced in the collision of orbifold-fixed brane with a bulk brane in the ``ekpyrotic/pyrotechnic type'' models of brane cosmologies. 
  In this paper, we set up the bi-module of the algebra ${\cal A}$ on fuzzy sphere. Based on the differential operators in moving frame, we generalize the ABS construction into fuzzy sphere case. The applications of ABS construction are investigated in various physical systems. 
  This dissertation is devoted to deriving the bosonic sectors of certain gauged supergravities in various dimensions from reducing eleven-dimensional supergravity, type IIA and type IIB supergravities in ten dimensions on certain spherical spaces. Explicit non-linear Kaluza-Klein ans\"atze for reductions of eleven-dimensional supergravity and of type IIA and type IIB supergravities on $S^n$ and $S^n\times T^m$ are presented. Knowing explicit non-linear ans\"atze is proven to be very useful in finding super Yang-Mills operators of gauge theories via AdS/CFT correspondence. We present a sample calculation which allows us to find a super Yang-Mills operator using a non-linear ansatz. Knowing non-linear ans\"atze is also useful for finding supergravity duals to certain twisted supersymmetric gauge theories. These supergravity solutions are branes wrapped on certain supersymmetric cycles. Some solutions, which are dual to gauge theories in three and five dimensions, are presented. 
  By means of a simple model we investigate the possibility that spacetime is a membrane embedded in higher dimensions. We present cosmological solutions of d-dimensional Einstein-Maxwell theory which compactify to two dimensions. These solutions are analytically continued to obtain dual solutions in which a (d-2)-dimensional Einstein spacetime "membrane" is embedded in d-dimensions. The membrane solutions generalise Melvin's 4-dimensional flux tube solution. The flat membrane is shown to be classically stable. It is shown that there are zero mode solutions of the d-dimensional Dirac equation which are confined to a neighbourhood of the membrane and move within it like massless chiral (d-2)-dimensional fermions. An investigation of the spectrum of scalar perturbations shows that a well-defined mass gap between the zero modes and massive modes can be obtained if there is a positive cosmological term in (d-2) dimensions or a negative cosmological term in d dimensions. 
  BPS solutions of 5-dimensional supergravity correspond to certain gradient flows on the product M x N of a quaternionic-Kaehler manifold M of negative scalar curvature and a very special real manifold N of dimension n >=0. Such gradient flows are generated by the `energy function' f = P^2, where P is a (bundle-valued) moment map associated to n+1 Killing vector fields on M. We calculate the Hessian of f at critical points and derive some properties of its spectrum for general quaternionic-Kaehler manifolds. For the homogeneous quaternionic-Kaehler manifolds we prove more specific results depending on the structure of the isotropy group. For example, we show that there always exists a Killing vector field vanishing at a point p in M such that the Hessian of f at p has split signature. This generalizes results obtained recently for the complex hyperbolic plane (universal hypermultiplet) in the context of 5-dimensional supergravity. For symmetric quaternionic-Kaehler manifolds we show the existence of non-degenerate local extrema of f, for appropriate Killing vector fields. On the other hand, for the non-symmetric homogeneous quaternionic-Kaehler manifolds we find degenerate local minima. 
  The muon anomalous $g$ value, $a_\mu=(g-2)/2$, is calculated up to one-loop level in noncommutative QED. We argue that relativistic muon in E821 experiment nearly always stays at the lowest Landau level. So that spatial coordinates of muon do not commute each other. Using parameters of E821 experiment, $B=14.5$KG and muon energy 3.09GeV/c, we obtain the noncommutativity correction to $a_\mu$ is about $1.57\times 10^{-9}$, which significantly makes standard model prediction close to experiment. 
  The AdS/CFT correspondence is explored for ``partially massless'' fields in AdS space (which have fewer helicity states than a massive field but more than a conventional massless field). Such fields correspond in the boundary conformal field theory to fields obeying a certain conformally-invariant differential equation that has been described by Eastwood et al. The first descendant of such a field is a conformal field of negative norm. Hence, partially massless fields may make more physical sense in de Sitter as opposed to Anti de Sitter space. 
  In theories with flat directions containing vortices, such as supersymmetric QED, there is a vacuum selection effect in the allowed asymptotic configurations. We explain the role played by gauge fields in this effect and give a simple criterion for determining what vacua will be chosen, namely those that minimise the vector mass. We then consider the effect of vacuum selection on stable (BPS) non-topological vortices in a simple Abelian model with N=2 supersymmetry which occurs as a low energy limit of Calabi-Yau compactifications of type II superstrings. In this case the magnetic flux spreads over an arbitrarily large area. We discuss the implications for cosmology and for superstring inspired magnetic confinement scenarios. 
  We discuss D-branes of the topological A-model (A-branes), which are believed to be closely related to the Fukaya category. We give string theory arguments which show that A-branes are not necessarily Lagrangian submanifolds in the Calabi-Yau: more general coisotropic branes are also allowed, if the line bundle on the brane is not flat. We show that a coisotropic A-brane has a natural structure of a foliated manifold with a transverse holomorphic structure. We argue that the Fukaya category must be enlarged with such objects for the Homological Mirror Symmetry conjecture to be true. 
  We construct a family of (p+3)-dimensional brane worlds in which the brane has one compact extra dimension, the bulk has two extra dimensions, and the bulk closes regularly at codimension two submanifolds known as bolts. The (p+1)-dimensional low energy spacetime M_{low} may be any Einstein space with an arbitrary cosmological constant, the value of the bulk cosmological constant is arbitrary, and the only fields are the metric and a bulk Maxwell field. The brane can be chosen to have positive tension, and the closure of the bulk provides a singularity-free boundary condition for solutions that contain black holes or gravitational waves in M_{low}. The spacetimes admit a nonlinear gravitational wave whose properties suggest that the Newtonian gravitational potential on a flat M_{low} will behave essentially as the static potential of a massless minimally coupled scalar field with Neumann boundary conditions. When M_{low} is (p+1)-dimensional Minkowski with p\ge3 and the bulk cosmological constant vanishes, this static scalar potential is shown to have the long distance behaviour characteristic of p spatial dimensions. 
  Although the equations of motion for the Neveu-Schwarz (NS) and Ramond (R) sectors of open superstring field theory can be covariantly expressed in terms of one NS and one R string field, picture-changing problems prevent the construction of an action involving these two string fields. However, a consistent action can be constructed by dividing the NS and R states into three string fields which are real, chiral and antichiral.   The open superstring field theory action includes a WZW-like term for the real field and holomorphic Chern-Simons-like terms for the chiral and antichiral fields. Different versions of the action can be constructed with either manifest d=8 Lorentz covariance or manifest N=1 d=4 super-Poincar\'e covariance. The lack of a manifestly d=10 Lorentz covariant action is related to the self-dual five-form in the Type IIB R-R sector. 
  We investigate how the mass correction appears in 5-D with Scherk-Schwarz compactification and clarify whether the KK regularization is reliable method or not. In the extremely sharp cutoff limit of the 5-D regulator which preserves 5-D Lorentz invariance, we prove that the one loop correction to the mass does not depend on the ultraviolet physics for the Scherk-Schwarz breaking of supersymmetry. This is a unique property of Scherk-Schwarz breaking which is given by the boundary condition. 
  We study solitons in scalar theories with polynomial interactions on the fuzzy sphere. Such solitons are described by projection operators of rank k, and hence the moduli space for the solitons is the Grassmannian Gr(k,2j+1). The gradient term of the action provides a non-trivial potential on Gr(k,2j+1), thus reducing the moduli space. We construct configurations corresponding to well-separated solitons, and show that although the solitons attract each other, the attraction vanishes in the limit of large j. In this limit, it is argued that the moduli space is CP^k. For the k-soliton bound state, the moduli space is simply CP^1, all other moduli being lifted. We find that the moduli space of multi-solitons is smooth and that there are no singularities as several solitons coalesce. When the fuzzy S^2 is flattened to a noncommutative plane, we find agreement with the known results, modulo some operator-ordering ambiguities. This suggests that the fuzzy sphere is a natural way to regulate the noncommutative plane both in the ultraviolet and infrared. 
  In this paper we analyze two local extensions of a model introduced some time ago to obtain a path integral formalism for Classical Mechanics. In particular, we show that these extensions exhibit a nonrelativistic local symmetry which is very similar to the well known kappa-symmetry introduced in the literature almost 20 years ago. Differently from the latter, this nonrelativistic local symmetry gives no problem in separating 1st from 2nd-class constraints. 
  The problem of the electromagnetic coupling for spin 3/2 fields is discussed. Following supergravity and some recent works in the field of classical supersymmetric particles, we find that the electromagnetic coupling must not obey a minimal coupling in the sense that one needs to consider not only the electromagnetic potential but also the coupling of the electromagnetic field strenght. This coupling coincides with the one found by Ferrara {\it et al} by requiring that the gyromagnetic ratio be 2. Coupling with non-Abelian Yang-Mills fields is also discussed. 
  We rediscuss the recent controversy on a possible Chern-Simons like term generated through radiative corrections in QED with a CPT violating term. We emphasize the fact that any absence of an {\sl a priori} divergence should be explained by some symmetry or some non-renormalisation theorem : otherwise, no prediction can be made on the corresponding quantity. 
  In an earlier work extensions of supersymmetry and super Lie algebras were constructed consistently starting from any representation $\D$ of any Lie algebra $\g$. Here it is shown how infinite dimensional Lie algebras appear naturally within the framework of fractional supersymmetry. Using a differential realization of $\g$ this infinite dimensional Lie algebra, containing the Lie algebra $\g$ as a sub-algebra, is explicitly constructed. 
  We investigate the consequences of generalizing certain well established properties of the open string metric to the conjectured open membrane and open Dp-brane metrics. By imposing deformation independence on these metrics their functional dependence on the background fields can be determined including the notorious conformal factor. In analogy with the non-commutativity parameter $\Theta^{\mu\nu}$ in the string case, we also obtain `generalized' theta parameters which are rank q+1 antisymmetric tensors (polyvectors) for open Dq-branes and rank 3 for the open membrane case. The expressions we obtain for the open membrane quantities are expected to be valid for general background field configurations, while the open D-brane quantities are only valid for one parameter deformations. By reducing the open membrane data to five dimensions, we show that they, modulo a subtlety with implications for the relation between OM-theory and NCYM, correctly generate the open string and open D2-data. 
  Hawking radiation is viewed as a tunnelling process. In this way the effect of self-gravitation gives rise to semiclassical corrections to the entropy of the (2+1) BTZ black hole. The modified entropy, due to specific modelling of the self-gravitation effect, of the (2+1) BTZ black hole is evaluated. To first order in $\omega$ which is a shell of energy radiated outwards the event horizon of the BTZ black hole, modified entropy is proportional to the horizon. In this semiclassical analysis, corrections to the Bekenstein-Hawking formula $S_{BH}=\mathcal{A}_{H} / 4l_{P}^{2}$ are found to be negative and the proportionality factor connecting the modified entropy, $S_{bh}$, of the (2+1) BTZ black hole to the Bekenstein-Hawking entropy, $S_{BH}$, is evaluated to first order in $\omega$. 
  In the following, we will review the fundamental problem that prevents a complete understanding of a theory of supersymmetrical field representations and describe its possible relation to a similar problem facing superstring/M-theory. 
  Direct calculation of the one-loop contributions to the energy density of bosonic and supersymmetric phi-to-the-fourth kinks exhibits: (1) Local mode regularization. Requiring the mode density in the kink and the trivial sectors to be equal at each point in space yields the anomalous part of the energy density. (2) Phase space factorization. A striking position-momentum factorization for reflectionless potentials gives the non-anomalous energy density a simple relation to that for the bound state. For the supersymmetric kink, our expression for the energy density (both the anomalous and non-anomalous parts) agrees with the published central charge density, whose anomalous part we also compute directly by point-splitting regularization. Finally we show that, for a scalar field with arbitrary scalar background potential in one space dimension, point-splitting regularization implies local mode regularization of the Casimir energy density. 
  We found an exact solution of elongated U(1) instanton on noncommutative R^4 for general instanton number k. The deformed ADHM equation was solved with general k and the gauge connection and the curvature were given explicitly. We also checked our solutions and evaluated the instanton charge by a numerical calculation. 
  The presence of a distant D4-brane is used to further investigate the duality between M-theory and D0-brane quantum mechanics. A polarization of the quantum mechanical ground state is found. A similar deformation arises for the bubble of normal space found near D0-branes in classical supergravity solutions. These deformations are compared and are shown to have the same structure in each case. 
  We analyze superfield representations of BPS-conditions for the self-dual static solutions of D=4, N=2 supersymmetric Yang-Mills theory. 
  We study anisotropic reheating (entropy production) on 3D brane from a decaying bulk scalar field in the brane-world picture of the Universe by considering off-diagonal metrics and anholonomic frames. We show that a significant amount of, in general, anisotropic dark radiation is produced in this process unless only the modes which satisfy a specific relation are excited. We conclude that subsequent entropy production within the brane is required in general before primordial nucleosynthesis and that the presence of off-diagonal components (like in the Salam, Strathee and Petracci works) of the bulk metric modifies the processes of entropy production which could substantially change the brane-world picture of the Universe. 
  For the classical principal chiral model with boundary, we give the subset of the Yangian charges which remains conserved under certain integrable boundary conditions, and extract them from the monodromy matrix. Quantized versions of these charges are used to deduce the structure of rational solutions of the reflection equation, analogous to the 'tensor product graph' for solutions of the Yang-Baxter equation. We give a variety of such solutions, including some for reflection from non-trivial boundary states, for the SU(N) case, and confirm these by constructing them by fusion from the basic solutions. 
  A recently derived basic theorem on the decomposition of SO(2N) vertices is used to obtain a complete analytic determination of all SO(10) invariant cubic superpotential couplings involving $16_{\pm}$ semispinors of SO(10) chirality $\pm$ and tensor representations. In addition to the superpotential couplings computed previously using the basic theorem involving the 10, 120 and $\bar{126}$ tensor representations we compute here couplings involving the 1, 45 and 210 dimensional tensor representations, i.e., we compute the $\bar{16}_{\mp}16_{\pm}1$,$\bar{16}_{\mp}16_{\pm}45$ and $\bar{16}_{\mp}16_{\pm}210$ Higgs couplings in the superpotential. A complete determination of dimension five operators in the superpotential arising from the mediation of the 1, 45 and 210 dimensional representations is also given. The vector couplings $\bar{16}_{\pm}16_{\pm}1$, $\bar{16}_{\pm}16_{\pm}45$ and $\bar{16}_{\pm}16_{\pm}210$ are also analyzed. The role of large tensor representations and the possible application of results derived here in model building are discussed. 
  A general construction of affine Non Abelian Toda models in terms of gauged two loop WZNW model is discussed. Its connection to non relativistic models corresponding to constrained KP hierarchies is established in terms of time evolution associated to positive and negative grading of the Lie algebra. 
  We present an alternative way to determine the unknown parameter associated to a gaussian approximation in a generic two-dimensional model. Instead of the standard variational approach, we propose a procedure based on a quantitative prediction of conformal invariance, valid for systems in the scaling regime, away from criticality. We illustrate our idea by considering, as an example, the sine-Gordon model. Our method gives a good approximation for the soliton mass as function of $\beta$. 
  We investigate the issue of regularization/renormalization in the presence of a nontrivial background in the case of 1+1-(supersymmetric) solitons. In particular we study and compare the commonly employed regularization methods (mode- energy/momentum- cutoff and derivative regularization). We show that the main point for a consistent regularization/renormalization is to find a relation between the ``cutoffs'' in the vacuum and the nontrivial sector so that they can be related in a consistent manner. For each scheme we give a principle to derive this relation and to perform calculations in a consistent way. These principles are simple and not restricted to two dimensions or supersymmetric theories. 
  The existence of light (a massless U(1) gauge boson) is one of unresolved mysteries in nature. In this paper, we would like to propose that light is originated from certain quantum orders in our vacuum. We will construct quantum models on lattice to demonstrate that some quantum orders can give rise to light without breaking any symmetries and without any fine tuning. Through our models, we show that the existence of light can simply be a phenomenon of quantum coherence in a system with many degrees of freedom. Massless gauge fluctuations appears commonly and naturally in strongly correlated systems. 
  We study U(1) and U(2) noncommutative instantons on R^2_{NC} x R^2_C based on the ADHM construction. It is shown that a mild singularity in the instanton solutions for both self-dual and anti-self-dual gauge fields always disappears in gauge invariant quantities and thus physically regular solutions can be constructed even though any projected states are not involved in the ADHM construction. Furthermore the instanton number is also an integer. 
  The thermodynamics of d5 AdS BHs with positive, negative or zero curvature spatial section in higher derivative (HD) gravity is described. HD contribution to free energy may change its sign which leads to more complicated regime for Hawking-Page phase transitions. Some variant of d5 HD gravity is dual to ${\cal N}=2$ $Sp(N)$ SCFT up to the next-to-leading order in large $N$. Then, according to Witten interpretation the stable AdS BH phase corresponds to deconfinement while global AdS phase corresponds to confinement. Unlike to Einstein gravity in HD theory the critical $N$ appears. It may influence the phase transition structure. In particulary, what was confining phase above the critical value becomes the deconfining phase below it and vice-versa. 
  We examine the low-energy dynamics of CP^1 lumps coupled to gravity, taking into account the gravitational back-reaction of the spacetime geometry. We show that the single lump moduli space is equipped with a three-dimensional metric, and we derive stability bounds on the scalar coupling constant. We also derive an expression for the multi-lump moduli space metric. 
  We rehabilitate the M_1(C)+ M_2(C)+ M_3(C) model of electro-magnetic, weak and strong forces as an almost commutative geometry in the setting of the spectral action. 
  We point out that when a D-brane is placed in an NS-NS B field background with non-vanishing field strength (H=dB) along the D-brane worldvolume, the coordinate of one end of the open string does not commute with that of the other in the low energy limit. The degrees of the freedom associated with both ends are not decoupled and accordingly, the effective action must be quite different from that of the ordinary noncommutative gauge theory for a constant B background. We construct an associative and noncommutative product which operates on the coordinates of both ends of the string and propose a new type of noncommutative gauge action for the low energy effective theory of a Dp-brane. This effective theory is bi-local and lives in twice as large dimensions (2D=2(p+1)) as in the H=0 case. When viewed as a theory in the D-dimensional space, this theory is non-local and we must force the two ends of the string to coincide. We will then propose a prescription for reducing this bi-local effective action to that in D dimensions and obtaining a local effective action. 
  We examine domain wall solutions of N=1, D=4 supergravity which preserve half of the supersymmetry and arise from Euclidean M2-brane instantons on M5-branes wrapping associative 3-cycles of G_2 holonomy manifolds. We also investigate composite solutions which break an additional half of the supersymmetry. 
  We examine a family of BPS solutions of ten-dimensional type IIb supergravity. These solutions asymptotically approach AdS_5 X S^5 and carry internal `angular' momentum on the five-sphere. While a naked singularity appears at the center of the anti-de Sitter space, we show that it has a natural physical interpretation in terms of a collection of giant gravitons. We calculate the distribution of giant gravitons from the dipole field induced in the Ramond-Ramond five-form, and show that these sources account for the entire internal momentum carried by the BPS solutions. 
  This article presents the derivation of the stress-energy tensor of a free scalar field with a general non-linear dispersion relation in curved spacetime. This dispersion relation is used as a phenomelogical description of the short distance structure of spacetime following the conventional approach of trans-Planckian modes in black hole physics and in cosmology. This stress-energy tensor is then used to discuss both the equation of state of trans-Planckian modes in cosmology and the magnitude of their backreaction during inflation. It is shown that gravitational waves of trans-Planckian momenta but subhorizon frequencies cannot account for the form of cosmic vacuum energy density observed at present, contrary to a recent claim. The backreaction effects during inflation are confirmed to be important and generic for those dispersion relations that are liable to induce changes in the power spectrum of metric fluctuations. Finally, it is shown that in pure de Sitter inflation there is no modification of the power spectrum except for a possible magnification of its overall amplitude independently of the dispersion relation. 
  Some thermodynamical properties of Lovelock gravity are discussed in several space-time dimensions, the holographic principle being one of the ingredients of the discussion. As it turns out, the area law and the brickwall method, though correct for the Einstein-Hilbert theory, may fail to work in general. 
  We discuss a formulation of harmonic superspace approach for noncommuative N=2 supersymmetric field theories paying main attention on new features arising because of noncommutativity. We begin with the known notions of the harmonic superfield models and results obtained and consider how these notions and results are modified in the noncommutative case. The actions of basic N=2 models, like hypermultiplet and N=2 vector multiplet, are given in terms of unconstrained off-shell superfields on noncommutative harmonic superspace. We calculate the low-energy contributions to the effective actions of these models. The background field method in noncommutative harmonic superspace is developed and it is applied to study the 1-loop effective action in general noncommutative N=2 model and N=4 SYM theory. 
  We propose the relationships between the noncommutative solitons and the (fractional) D-branes on the C^2/Z_n orbifold and extend the solution generating technique for the orbifold. As applications, we determine how tachyon condensations occur in various D-Dbar systems on the orbifolds. The calculations give results consistent with BSFT. The extended solution generating technique enables us to calculate more general decay modes of D-Dbar systems. 
  ``Extremely'' localized wavefunctions in noncommutative geometry have disturbances that are localized to distances smaller than $\sqrt{\theta}$, where $\theta$ is the ``area'' parameter that measures noncommutativity. In particular, distributions such as the sign function or the Dirac delta function are limiting cases of extremely localized wavefunctions. It is shown that Moyal star products of extremely localized wavefunctions cannot be correctly computed perturbatively in powers of $\theta$. Nonperturbative effects as a function of $\theta$ are explicitly displayed through exact computations in several examples. In particular, for distributions, star products end up being functions of $\theta ^{-1}$ and have no expansion in positive powers of $\theta$. This result provides a warning for computations in noncommutative space that often are performed with perturbative methods. Furthermore, the result may have interesting applications that could help elucidate the role of noncommutative geometry in several areas of physics. 
  We study thermodynamic properties and phase structures of topological black holes in Einstein theory with a Gauss-Bonnet term and a negative cosmological constant. The event horizon of these topological black holes can be a hypersurface with positive, zero or negative constant curvature. When the horizon is a zero curvature hypersurface, the thermodynamic properties of black holes are completely the same as those of black holes without the Gauss-Bonnet term, although the two black hole solutions are quite different. When the horizon is a negative constant curvature hypersurface, the thermodynamic properties of the Gauss-Bonnet black holes are qualitatively similar to those of black holes without the Gauss-Bonnet term. When the event horizon is a hypersurface with positive constant curvature, we find that the thermodynamic properties and phase structures of black holes drastically depend on the spacetime dimension $d$ and the coefficient of the Gauss-Bonnet term: when $d\ge 6$, the properties of black hole are also qualitatively similar to the case without the Gauss-Bonnet term, but when $d=5$, a new phase of locally stable small black hole occurs under a critical value of the Gauss-Bonnet coefficient, and beyond the critical value, the black holes are always thermodynamically stable. However, the locally stable small black hole is not globally preferred, instead a thermal anti-de Sitter space is globally preferred. We find that there is a minimal horizon radius, below which the Hawking-Page phase transition will not occur since for these black holes the thermal anti de Sitter space is always globally preferred. 
  We perform the Batalin-Vilkovisky quantization of Yang-Mills theory on a 2-point space, discussing the formulation of Connes-Lott as well as Connes' real spectral triple approach. Despite of the model's apparent simplicity the gauge structure reveals infinite reducibility and the gauge fixing is afflicted with the Gribov problem. 
  The effects of introducing a harmonic spatial inhomogeneity into the Kalb-Ramond field, interacting with the Maxwell field according to a `string-inspired' proposal made in earlier work are investigated. We examine in particular the effects on the polarization of synchrotron radiation from cosmologically distant (i.e. of redshift greater than 2) galaxies, as well as the relation between the electric and magnetic components of the radiation field. The rotation of the polarization plane of linearly polarized radiation is seen to acquire an additional contribution proportional to the square of the frequency of the dual Kalb-Ramond axion wave, assuming that it is far smaller compared to the frequency of the radiation field. 
  A new formula relating the analytic continuation of the Hurwitz zeta function to the Euler gamma function and a polylogarithmic function is presented. In particular, the values of the first derivative of the real part of the analytic continuation of the Hurwitz zeta function for even negative integers and the imaginary one for odd negative integers are explicitly given. The result can be of interest both on mathematical and physical side, because we are able to apply our new formulas in the context of the Spectral Zeta Function regularization, computing the exact pair production rate per space-time unit of massive Dirac particles interacting with a purely electric background field. 
  We discuss the possibility of a smooth transition from the pre- to the post-big bang regime, in the context of the lowest-order string effective action (without higher-derivative corrections), taking into account with a phenomenological model of source the repulsive gravitational effects due to the back-reaction of the quantum fluctuations outside the horizon. We determine a set of necessary conditions for a successful and realistic transition, and we find that such conditions can be satisfied (by an appropriate model of source), provided the background is higher-dimensional and anisotropic. 
  We introduce three non-local observables for the two-dimensional Ising model. At criticality, conformal field theory may be used to obtain theoretical predictions for their behavior. These formulae are explicit enough to show that their asymptotics are described by highest weights $h_{pq}$ from the Kac table for c=1/2 distinct from those of the three unitary representations (0, 1/16 and 1/2). 
  Using two new well defined 4-dimensional potential vectors, we formulate the classical Maxwell's field theory in a form which has manifest Lorentz covariance and SO(2) duality symmetry in the presence of magnetic sources. We set up a consistent Lagrangian for the theory. Then from the action principle we get both Maxwell's equation and the equation of motion of a dyon moving in the electro-magnetic field. 
  Based on the assumption that the warp factor of four-dimensional spacetime and the one of fifth dimension are tied through a parameter $\alpha$, we consider five-dimensional gravity with a 3-brane coupled to a bulk scalar field. For arbitrary value of $\alpha$, the form of the warp factor is implicitly determined by hypergeometric function. Concretely, we show that the warp factor becomes explicit form for appropriate value of $\alpha$, and study the relation between four-dimensional effective Planck scale and the brane tension. This setup allows the possibility of extending the diversity of brane world. 
  Relative moduli spaces of periodic monopoles provide novel examples of Asymptotically Locally Flat hyperkahler manifolds. By considering the interactions between well-separated periodic monopoles, we infer the asymptotic behavior of their metrics. When the monopole moduli space is four-dimensional, this construction yields interesting examples of metrics with self-dual curvature (gravitational instantons). We discuss their topology and complex geometry. An alternative construction of these gravitational instantons using moduli spaces of Hitchin equations is also described. 
  Semiclassical Hamiltonian field theory is investigated from the axiomatic point of view. A notion of a semiclassical state is introduced. An "elementary" semiclassical state is specified by a set of classical field configuration and quantum state in this external field. "Composed" semiclassical states viewed as formal superpositions of "elementary" states are nontrivial only if the Maslov isotropic condition is satisfied; the inner product of "composed" semiclassical states is degenerate. The mathematical proof of Poincare invariance of semiclassical field theory is obtained for "elementary" and "composed" semiclassical states. The notion of semiclassical field is introduced; its Poincare invariance is also mathematically proved. 
  It is shown that the alternative Klein-Gordon equation with positive definite probability density proposed in a letter by M.D. Kostin does not meet the requirements of relativistic (quantum) field theory and therefore does not allow for a meaningful physical interpretation. 
  We will show that 2-dimensional N=2-extended supersymmetric theory can have solitonic solution using the Hamilton-Jacobi method of classical mechanics. Then it is shown that the Bogomol'nyi mass bound is saturated by these solutions and triangular mass inequality is satisfied. At the end, we will mention domain-wall structure in 3-dimensional spacetime. 
  The quantum gravity is formulated based on principle of local gauge invariance. The model discussed in this paper has local gravitational gauge symmetry and gravitational field appears as gauge field. The problems on quantization and renormalization of the theory are also discussed in this paper. In leading order approximation, the gravitational gauge field theory gives out classical Newton's theory of gravity. In first order approximation and for vacuum, the gravitational gauge field theory gives out Einstein's general theory of relativity. This quantum gauge theory of gravity is a renormalizable quantum theory. 
  We implement the Hamiltonian treatment of a nonAbelian noncommutative gauge theory, considering with some detail the algebraic structure of the noncommutative symmetry group. The first class constraints and Hamiltonian are obtained and their algebra derived, as well as the form of the gauge invariance they impose on the first order action. 
  We study $T^2$ compactification of massive type IIA supergravity in presence of possible Ramond-Ramond (RR) background fluxes. The resulting theory in D=8 is shown to possess full $SL(2,R)\times SL(2,R)$ T-duality symmetry similar to the massless case. It is shown that elements of duality symmetry interpolate between massive type IIA compactified on $T^2$ and ordinary type IIA compactified on $T^2$ with RR 2-form flux. We also discuss relationship between M-theory vacua and massive type IIA vacua. The D8-brane is found to correspond to M-theory `pure gravity' solution which is a direct product of 7-dimensional Minkowski space and a 4-dimensional instanton. We also construct D6-D8 bound state which preserves 1/2 supersymmetries. We then discuss massive IIA compactification on $T^4$ and point out that when all possible RR fluxes on $T^4$ are turned on the six-dimensional theory appears to assume a nice SO(4,4) invariant form. 
  This talk discusses two topological features in non-abelian gauge theories, related by the notion of abelian projection and the Hopf invariant. Minimising the energy of the non-linear sigma model with a Skyrme-like term (the Faddeev-Niemi model), can be identified with a non-linear maximal abelian gauge fixing of the SU(2) gauge vacua with a winding number equal to the Hopf invariant. In the context of abelian projection the Hopf invariant can also be associated to a monopole world line, through the Taubes winding, measuring its contribution to topological charge. Calorons with non-trivial holonomy provide an explicit realisation. We discuss the identification of its constituent monopoles through degenerate eigenvalues of the Polyakov loop (the singularities or defects of the abelian projection). It allows us to study the correlation between the defect locations and the explicit constituent monopole structure, through a specific SU(3) example. 
  After a short introduction to Matrix theory, we explain how can one generalize matrix models to describe toroidal compactifications of M-theory and the heterotic vacua with 16 supercharges. This allows us, for the first time in history, to derive the conventional perturbative type IIA string theory known in the 80s within a complete and consistent nonperturbative framework, using the language of orbifold conformal field theory and conformal perturbation methods. A separate chapter is dedicated to the vacua with Horava-Witten domain walls that carry E8 gauge supermultiplets. Those reduce the gauge symmetry of the matrix model from U(N) to O(N). We also explain why these models contain open membranes. The compactification of M-theory on T4 involves the so-called (2,0) superconformal field theory in six dimensions, compactified on T5. A separate chapter describes an interesting topological contribution to the low energy equations of motion on the Coulomb branch of the (2,0) theory that admits a skyrmionic solution that we call ``knitting fivebranes''. Then we return to the orbifolds of Matrix theory and construct a formal classical matrix model of the Scherk-Schwarz compactification of M-theory and type IIA string theory as well as type 0 theories. We show some disastrous consequences of the broken supersymmetry. Last two chapters describe a hyperbolic structure of the moduli spaces of one-dimensional M-theory. 
  We investigate the relation between the algebraic construction of boundary states in AdS_3 and the target space analysis of D-branes and show the consistency of the two descriptions. We compute, in the semiclassical regime, the overlap of a localized closed string state with boundary states and identify the latter with D-branes wrapping conjugacy classes in AdS_3. The string partition function on the disk is shown to reproduce the spacetime DBI action. Other consistency checks are performed. We also comment on the role of the spectral flow symmetry of the underlying SL(2,R)/U(1) coset model in constructing D-branes that correspond to degenerate representations of SL(2,R). 
  We investigate non-local dualities between suitably compactified higher-dimensional Einstein gravity and supergravities which can be revealed if one reinterprets the dualised Kaluza-Klein two-forms in $D>4$ as antisymmetric forms belonging to supergravities. We find several examples of such a correspondence including one between the six-dimensional Einstein gravity and the four-dimensional Einstein-Maxwell-dilaton-axion theory (truncated N=4 supergravity), and others between the compactified eleven and ten-dimensional supergravities and the eight or ten-dimensional pure gravity. The Killing spinor equation of the D=11 supergravity is shown to be equivalent to the geometric Killing spinor equation in the dual gravity. We give several examples of using new dualities for solution generation and demonstrate how $p$-branes can be interpreted as non-local duals of pure gravity solutions. New supersymmetric solutions are presented including $M2\subset 5$-brane with two rotation parameters. 
  M-theory compactification on a manifold X of $G_2$ holonomy can give chiral fermions in four dimensions only if X is singular. A number of examples of conical singularities that give chiral fermions are known; the present paper is devoted to describing some additional examples. In some of them, the physics can be determined but the metric is not known explicitly, while in others the metric can be described explicitly but the physics is more challenging to understand. 
  The spectrum of the Hamiltonian of the double compactified D=11 supermembrane with non-trivial central charge or equivalently the non-commutative symplectic super Maxwell theory is analyzed. In distinction to what occurs for the D=11 supermembrane in Minkowski target space where the bosonic potential presents string-like spikes which render the spectrum of the supersymmetric model continuous, we prove that the potential of the bosonic compactified membrane with non-trivial central charge is strictly positive definite and becomes infinity in all directions when the norm of the configuration space goes to infinity. This ensures that the resolvent of the bosonic Hamiltonian is compact. We find an upper bound for the asymptotic distribution of the eigenvalues. 
  Near-extremal black holes are obtained by exciting the Ramond sector of the D1-D5 CFT, where the ground state is highly degenerate. We find that the dual geometries for these ground states have throats that end in a way that is characterized by the CFT state. Below the black hole threshold we find a detailed agreement between propagation in the throat and excitations of the CFT. We study the breakdown of the semiclassical approximation and relate the results to the proposal of gr-qc/0007011 for resolving the information paradox: semiclassical evolution breaks down if hypersurfaces stretch too much during an evolution. We find that a volume V stretches to a maximum throat depth of V/2G. 
  We propose a way to examine N=1 and N=2 string dualities on Calabi-Yau three-folds or extensions. Our way is to find out or to construct two types of toric representations of a Calabi-Yau three-fold, which contain phases topologically equivalent or phases connected by flops. We discuss how to find relations among Calabi-Yau three-folds realized in different toric representations. We examine several examples of Calabi-Yau three-folds which have the Hodge numbers, $(h^{1,1},h^{2,1})=(5,185)$ and the various numbers of K3 fibers. We observe that each phase of our examples contains Del Pezzo 4-cycles, $B_8$ in six ways. 
  A lattice regularization of the supersymmetric Wess-Zumino model is studied by using Ginsparg-Wilson operators. We recognize a certain conflict between the lattice chiral symmetry and the Majorana condition for Yukawa couplings, or in Weyl representation a conflict between the lattice chiral symmetry and Yukawa couplings. This conflict is also related, though not directly, to the fact that the kinetic (K\"{a}hler) term and the superpotential term are clearly distinguished in the continuum Wess-Zumino model, whereas these two terms are mixed in the Ginsparg-Wilson operators. We illustrate a case where lattice chiral symmetry together with naive Bose-Fermi symmetry is imposed by preserving a SUSY-like symmetry in the free part of the Lagrangian; one-loop level non-renormalization of the superpotential is then maintained for finite lattice spacing, though the finite parts of wave function renormalization deviate from the supersymmetric value. All these properties hold for the general Ginsparg-Wilson algebra independently of the detailed construction of lattice Dirac operators. 
  We consider the dynamics of two interacting lumps/solitons in a noncommutative gauge model. We show that equations of motion describing this dynamics can be reduced to ones of a two-dimensional mechanical system which is well studied and was shown to exhibit stochastic behaviour. 
  We consider interaction of two lumps corresponding to 0-branes in noncommutative gauge theory 
  We derive the partition function of {\cal N}=4 supersymmetric Yang-Mills theory on orbifold-T^4/{\bf Z}_2 for gauge group SU(N). We generalize the method of our previous work for the SU(2) case to the SU(N) case. The resulting partition function is represented as the sum of the product of G\"ottche formula of singular quotient space $T^4/{\bf Z}_2 $ and of blow-up formulas including A_{N-1} theta series with level N. 
  Two results are presented for reduced Yang-Mills integrals with different symmetry groups and dimensions: the first is a compact integral representation in terms of the relevant variables of the integral, the second is a method to analytically evaluate the integrals in cases of low order. This is exhibited by evaluating a Yang-Mills integral over real symmetric matrices of order 3. 
  We analyse in detail the quantization of a simple noncommutative model of spontaneous symmetry breaking in zero dimensions taking into account the noncommutative setting seriously. The connection to the counting argument of Feyman diagrams of the corresponding theory in four dimensions is worked out explicitly. Special emphasis is put on the motivation as well as the presentation of some well-known basic notions of quantum field theory which in the zero-dimensional theory can be studied without being spoiled by technical complications due to the absence of divergencies. 
  A pedagogical and self-contained introduction to noncommutative quantum field theory is presented, with emphasis on those properties that are intimately tied to string theory and gravity. Topics covered include the Weyl-Wigner correspondence, noncommutative Feynman diagrams, UV/IR mixing, noncommutative Yang-Mills theory on infinite space and on the torus, Morita equivalences of noncommutative gauge theories, twisted reduced models, and an in-depth study of the gauge group of noncommutative Yang-Mills theory. Some of the more mathematical ideas and techniques of noncommutative geometry are also briefly explained. 
  Discrete versions of the Yang-Mills and Einstein actions are proposed for any finite group. These actions are invariant respectively under local gauge transformations, and under the analogues of Lorentz and general coordinate transformations. The case Z_n \times Z_n \times...\times Z_n is treated in some detail, recovering the Wilson action for Yang-Mills theories, and a new discretized action for gravity. 
  We analyze D-brane states and their central charges on the resolution of C^2/Z_n by using local mirror symmetry. There is a point in the moduli space where all n(n-1)/2 branches of the principal component of the discriminant locus coincide. We argue that this is the point where compactifications of Type IIA theory on a K3 manifold containing such a local geometry acquire a non-perturbative gauge symmetry of the type A_{n-1}. This analysis, which involves an explicit solution of the GKZ system of the local geometry, explains how the quantum geometry exhibits all positive roots of A_{n-1} and not just the simple roots that manifest themselves as the exceptional curves of the classical geometry. We also make some remarks related to McKay correspondence. 
  Brane Gas Cosmology (BGC) is an approach to M-theory cosmology in which the initial state of the Universe is taken to be small, dense and hot, with all fundamental degrees of freedom near thermal equilibrium. Such a starting point is in close analogy with the Standard Big Bang (SBB) model. The topology of the Universe is assumed to be toroidal in all nine spatial dimensions and is filled with a gas of p-branes. The dynamics of winding modes allow, at most, three spatial dimensions to become large, thus explaining the origin of our macroscopic 3+1-dimensional Universe. Here we conduct a detailed analysis of the loitering phase of BGC. We do so by including into the equations of motion that describe the dilaton gravity background some new equations which determine the annihilation of string winding modes into string loops. Specific solutions are found within the model that exhibit loitering, i.e. the Universe experiences a short phase of slow contraction during which the Hubble radius grows larger than the physical extent of the Universe. As a result the brane problem (generalized domain wall problem) in BGC is solved. The initial singularity and horizon problems of the SBB scenario are solved without relying on an inflationary phase. 
  An analytical approximation of $<T_{\mu\nu}>$ for a quantized scalar field in a static spherically symmetric spacetime with a topology $S^2 \times R^2$ is obtained. The gravitational background is assumed slowly varying. The scalar field is assumed to be both massive and massless, with an arbitrary coupling $\xi$ to the scalar curvature and in a zero temperature vacuum state. It is demonstrated that for some values of curvature coupling the stress-energy has the properties needed to support the wormhole geometry. 
  We study the partition function of both Close-Packed Dimers and the Critical Ising Model on a square lattice embedded on a genus two surface. Using numerical and analytical methods we show that the determinants of the Kasteleyn adjacency matrices have a dependence on the boundary conditions that, for large lattice size, can be expressed in terms of genus two theta functions. The period matrix characterizing the continuum limit of the lattice is computed using a discrete holomorphic structure. These results relate in a direct way the lattice combinatorics with conformal field theory, providing new insight to the lattice regularization of conformal field theories on higher genus Riemann Surfaces. 
  We present a (unique?) possibility of de Sitter solution in the framework of N=2 supersymmetry (hypersymmetry). We show that a model with a vector and a charged hypermultiplet has a hybrid inflation type potential. It leads to a slow-roll regime in de Sitter type background with all supersymmetries broken spontaneously. Beyond the bifurcation point the waterfall stage abruptly brings the system into a ground state with 2 unbroken supersymmetries. De Sitter stage exists under condition that the superconformal SU(2,2|2) symmetry of the model is softly broken down to N=2 supersymmetry by the vev of the Killing prepotential triplet P^r. This hybrid hypersymmetry model may describe P-term inflation and/or acceleration of the universe at the present epoch. 
  This paper was withdrawn. It has been superseded by the latest version of hep-th/0107069. 
  We study IIB string theory on the orbifold R^8/Gamma with discrete torsion where Gamma is an arbitrary subgroup of U(4). We extend some previously known identities for discrete torsion in abelian groups to nonabelian groups. We construct explicit formulas for a large class of fractional D-brane states and prove that the physical states are classified by projective representations of the orbifold group as predicted originally by Douglas. The boundary states are found to be linear combinations of Ishibashi states with the coefficients being characters of the projective representations. 
  We show how to derive systematically new forms of the BRST transformations for a generic gauge fixed action. They arise after a symmetry of the gauge fixed action is found in the sector involving the Lagrange multiplier and its canonical momentum. 
  This is the written version of the supersymmetry lectures delivered at the 30th and 31st British Universities Summer Schools in Theoretical Elementary Particle Physics (BUSSTEPP) held in Oxford in September 2000 and in Manchester in August-September 2001. 
  The effective action for five-dimensional heterotic M-theory in the presence of five-branes is systematically derived from Horava-Witten theory coupled to an M5-brane world-volume theory. This leads to a five-dimensional N=1 gauged supergravity theory on S^1/Z_2 coupled to four-dimensional N=1 theories residing on the two orbifold fixed planes and an additional bulk three-brane. We analyse the properties of this action, particularly the four-dimensional effective theory associated with the domain-wall vacuum state. The moduli Kahler potential and the gauge-kinetic functions are determined along with the explicit relations between four-dimensional superfields and five-dimensional component fields. 
  We consider high-energy fixed-angle scattering of glueballs in confining gauge theories that have supergravity duals. Although the effective description is in terms of the scattering of strings, we find that the amplitudes are hard (power law). This is a consequence of the warped geometry of the dual theory, which has the effect that in an inertial frame the string process is never in the soft regime. At small angle we find hard and Regge behaviors in different kinematic regions. 
  We show that there exists a correspondence between four dimensional gauge theories with simple groups and higher dimensional gauge theories at large N. As an example, we show that a four dimensional {N}=2 supersymmetric SU(N) gauge theory, on the Higgs branch, has the same correlators as a five dimensional SU(N) gauge theory in the limit of large N provided the couplings are appropriately rescaled. We show that our results can be applied to the AdS/CFT correspondence to derive correlators of five or more dimensional gauge theories from solutions of five dimensional supergravity in the large t'Hooft coupling limit. 
  We study some dynamical aspects of gauge theories on noncommutative tori. We show that Morita duality, combined with the hypothesis of analyticity as a function of the noncommutativity parameter Theta, gives information about singular large-N limits of ordinary U(N) gauge theories, where the large-rank limit is correlated with the shrinking of a two-torus to zero size. We study some non-perturbative tests of the smoothness hypothesis with respect to Theta in theories with and without supersymmetry. In the supersymmetric case this is done by adapting Witten's index to the present situation, and in the nonsupersymmetric case by studying the dependence of energy levels on the instanton angle. We find that regularizations which restore supersymmetry at high energies seem to preserve Theta-smoothness whereas nonsupersymmetric asymptotically free theories seem to violate it. As a final application we use Morita duality to study a recent proposal of Susskind to use a noncommutative Chern-Simons gauge theory as an effective description of the Fractional Hall Effect. In particular we obtain an elegant derivation of Wen's topological order. 
  We find a new solution of Type IIB supergravity which represents a collection of D5 branes wrapped on the topologically non-trivial S^3 of the deformed conifold geometry T^*S^3. The Type IIB solution is obtained by lifting a new solution of D=7 SU(2)_L x SU(2)_R gauged supergravity to ten dimensions in which SU(2)_D gauge fields in the diagonal subgroup are turned on. The supergravity solution describes a slice of the Coulomb branch in the large N limit of N=2 SYM in three dimensions. 
  A discrete total variation calculus with variable time steps is presented in this letter. Using this discrete variation calculus, we generalize Lee's discrete mechanics and derive variational symplectic-energy-momentum integrators by Kane, Marsden and Ortiz. The relationship among discrete total variation, Lee's discrete mechanics and Kane-Marsden-Ortiz's integrators is explored. 
  We investigate particular models which can be N-fold supersymmetric at specific values of a parameter in the Hamiltonians. The models to be investigated are a periodic potential and a parity-symmetric sextic triple-well potential. Through the quantitative analyses on the non-perturbative contributions to the spectra by the use of the valley method, we show how the characteristic features of N-fold supersymmetry which have been previously reported by the authors can be observed. We also clarify the difference between quasi-exactly solvable and quasi-perturbatively solvable case in view of the dynamical property, that is, dynamical N-fold supersymmetry breaking. 
  We consider the dynamical stability of a static brane model that incorporates a three-index antisymmetric tensor field and has recently been proposed as a possible solution to the cosmological constant problem. Ultimately, we are able to establish the existence of time-dependent, purely gravitational perturbations. As a consequence, the static solution of interest is ``dangerously'' located at an unstable saddle point. This outcome is suggestive of a hidden fine tuning in what is an otherwise self-tuning model. 
  We initiate a systematic study of natural differential operators in Riemannian geometry whose leading symbols are not of Laplace type. In particular, we define a discrete leading symbol for such operators which may be computed pointwise, or from spectral asymptotics. We indicate how this can be applied to the computation of another kind of spectral asymptotics, namely asymptotic expansions of fundamental solutions, and to the computation of conformally covariant operators. 
  In this Ph.D. thesis, we study tachyon condensation in string field theories.   In chapter 2, we review Witten's bosonic string field theory and calculate the tachyon potential.   In chapter 3, we calculate the tachyon potential in Berkovits' superstring field theory.   In chapter 4, we look for exact solutions in a toy model.   Unpublished result: we use conservation laws to calculate the level (4,8) approximation of the tachyon potential in Berkovits' superstring field theory. We verify Sen's conjecture up to 94.4%. 
  The bosonic parts of D3-brane actions in AdS(5) backgrounds are known to have symmetries which are field-dependent extensions of conformal transformations of the worldvolume coordinates. Using the coset space SU(2,2|1)/SO(4,1), we apply the method of nonlinear realizations to construct a four-dimensional N = 1 off-shell supersymmetric action which has a generalized field-dependent superconformal invariance. The Goldstone fields for broken scale, chiral and S-supersymmetry transformations form a chiral supermultiplet. 
  We write down one-to-one mappings between the singular vectors of the Neveu-Schwarz N=2 superconformal algebra and $16 + 16$ types of singular vectors of the Topological and of the Ramond N=2 superconformal algebras. As a result one obtains construction formulae for the latter using the construction formulae for the Neveu-Schwarz singular vectors due to D\"orrzapf. The indecomposable singular vectors of the Topological and of the Ramond N=2 algebras (`no-label' and `no-helicity' singular vectors) cannot be mapped to singular vectors of the Neveu-Schwarz N=2 algebra, but to {\it subsingular} vectors, for which no construction formulae exist. 
  A recent analytic test of the instanton method performed by comparing the exact spectrum of the Lam${\acute e}$ potential (derived from representations of a finite dimensional matrix expressed in terms of $su(2)$ generators) with the results of the tight--binding and instanton approximations as well as the standard WKB approximation is commented upon. It is pointed out that in the case of the Lam${\acute e}$ potential as well as others the WKB--related method of matched asymptotic expansions yields the exact instanton result as a result of boundary conditions imposed on wave functions which are matched in domains of overlap. 
  We study the problem of a charged particle in the presence of a uniform magnetic field plus a vortex in noncommutative planar space considering the two possible non-commutative extensions of the corresponding Hamiltonian, namely the ``fundamental'' and the ``antifundamental'' representations. Using a Fock space formalism we construct eigenfunctions and eigenvalues finding in each case half of the states existing in the ordinary space case. In the limit of $\theta \to 0$ we recover the two classes of states found in ordinary space, relevant for the study of anyon physics. 
  We compute the one-loop corrections to the tachyon potentials in both, bosonic and supersymmetric, cases by worldsheet methods. In the process, we also get some insight on tachyon condensation from the viewpoint of closed string modes and show that there is an instability due to the closed string tachyon in the bosonic case. 
  We compute the number of linearly independent ways in which a tensor of Weyl type may act upon a given irreducible tensor-spinor bundle V over a Riemannian manifold. Together with the analogous but easier problem involving actions of tensors of Einstein type, this enumerates the possible curvature actions on V. 
  The (2+1)D Georgi-Glashow (or Polyakov) model with the additional fundamental massless quarks is explored at finite temperature. In the case of vanishing Yukawa coupling, it is demonstrated that the interaction of a monopole and an antimonopole in the molecule via quark zero modes leads to the decrease of the Berezinsky-Kosterlitz-Thouless critical temperature when the number of quark flavors is equal to one. If the number of flavors becomes larger, monopoles are shown to exist only in the molecular phase at any temperatures exceeding a certain exponentially small one. This means that for such a number of flavors and at such temperatures, no fundamental matter can be confined by means of the monopole mechanism. 
  We discuss the quantization of a self-interacting string consisting of maximally charged matter. We construct the Hamiltonian in the non-relativistic limit by expanding around a static solution of the Einstein-Maxwell field equations. Conformal symmetry is broken on the worldsheet, but a subgroup of the conformal group acts as the gauge group of the theory. Thus, the Faddeev-Popov quantization procedure of fixing the gauge is applicable. We calculate the Hamiltonian and show that, if properly quantized, the system possesses a well-defined ground state and the spacing of its energy levels is of order the Planck mass. This generalizes earlier results on a system of maximally charged black holes to the case of continuous matter distributions. 
  We outline a phenomenological programme for the search of effects induced by (string-motivated) canonical noncommutative spacetime. The tests we propose are based, in analogy with a corresponding programme developed over the last few years for the study of Lie-algebra noncommutative spacetimes, on the role of the noncommutativity parameters in the $E(p)$ dispersion relation. We focus on the role of deformed dispersion relations in particle-production collision processes, where the noncommutativity parameters would affect the threshold equation, and in the dispersion of gamma rays observed from distant astrophysical sources. We emphasize that the studies here proposed have the advantage of involving particles of relatively high energies, and may therefore be less sensitive to "contamination" (through IR/UV mixing) from the UV sector of the theory. We also explore the possibility that the relevant deformation of the dispersion relations could be responsible for the experimentally-observed violations of the GZK cutoff for cosmic rays and could have a role in the observation of hard photons from distant astrophysical sources. 
  We study the stability of the giant gravitons in the string theory background with NSNS B field. We consider the perturbation of giant gravitons formed by a probe D$(8-p)$ brane in the background generated by D$(p-2)$-D$(p)$ branes for $2 \le p \le 5$. We use the quadratic approximation to the brane action to find the equations of motion. For $p=5$, giant graviton configurations are stable independent of the size of the brane. For $p \ne 5$, we calculated the range of the size of the brane where they are stable. We also present the mode frequencies explicitly for some special cases. 
  We examine whether any type II asymmetric orbifolds have the same massless spectrum as the dimensional reduction of D=5 simple supergravity, which, besides the eleven-dimensional supergravity, is the only known supergravity above four dimensions with no moduli. We attempt to construct such models by further twisting the orbifolds which yield D=4, N=4 pure supergravity to find that, unfortunately, none of the models have that spectrum. We provide supergravity arguments explaining why this is so. As a by-product, we list all possible momentum-winding lattices that give D=4, N=4 pure supergravity. 
  We construct an explicit model to describe fermions confined on a four dimensional brane embedded in a five dimensional anti-de Sitter spacetime. We extend previous works to accommodate massive bound states on the brane and exhibit the transverse structure of the fermionic fields. We estimate analytically and calculate numerically the fermion mass spectrum on the brane, which we show to be discrete. The confinement life-time of the bound states is evaluated, and it is shown that existing constraints can be made compatible with the existence of massive fermions trapped on the brane for durations much longer than the age of the Universe. 
  We consider the ultra-violet divergence structure of general noncommutative supersymmetric $U(N_c)$ gauge theories, and seek theories which are all-orders finite. 
  We formulate a linear difference equation which yields averaged semi-inclusive decay rates for arbitrary, not necessarily large, values of the masses. We show that the rates for decays $M \to m+\M'$ of typical heavy open strings are independent of the masses $M$ and $m$, and compute the ``mass deffect''$M-m-M'$. For closed strings we find decay rates proportional to $M m_{R}^{(1-D)/2}$, where $m_{R}$ is the reduced mass of the decy products. Our method yields exact interaction rates valid for all mass ranges and may provide a fully microscopic basis, not limited to the long string approximation, for the interactions in the Boltzmann equation approach to hot string gases. 
  We argue that there are two distinct classes of type I compactification to four dimensions on any space. These two classes are distinguished in a mysterious way by the presence (or absence) of a discrete 6-form potential. In simple examples, duality suggests that the new class of compactifications have reduced numbers of moduli. We also point out analogous discrete choices in M, F and type II compactifications, including some with $G_2$ holonomy. These choices often result in spaces with frozen singularities. 
  This paper studies aspects of ``holography'' for Euclidean signature pure gravity on asymptotically AdS 3-manifolds. This theory can be described as SL(2,C) CS theory. However, not all configurations of CS theory correspond to asymptotically AdS 3-manifolds. We show that configurations that do have the metric interpretation are parameterized by the so-called projective structures on the boundary. The corresponding asymptotic phase space is shown to be the cotangent bundle over the Schottky space of the boundary. This singles out a ``gravitational'' sector of the SL(2,C) CS theory. It is over this sector that the path integral has to be taken to obtain the gravity partition function. We sketch an argument for holomorphic factorization of this partition function. 
  The tetrad gauge invariant theory of the free Dirac field in two special moving charts of the de Sitter spacetime is investigated pointing out the operators that commute with the Dirac one. These are the generators of the symmetry transformations corresponding to isometries that give rise to conserved quantities according to the Noether theorem. With their help the plane wave spinor solutions of the Dirac equation with given momentum and helicity are derived and the final form of the quantum Dirac field is established. It is shown that the canonical quantization leads to a correct physical interpretation of the massive or massless fermion quantum fields. 
  We examine D-branes on $AdS_3$, and find a three-brane wrapping the entire $AdS_3$, in addition to 1-branes and instantonic 2-branes previously discussed in the literature. The three-brane is found using a construction of Maldacena, Moore, and Seiberg. We show that all these branes satisfy Cardy's condition and extract the open string spectrum on them. 
  We present a derivation of Hawking radiation based on canonical quantization of a massless scalar field in the background of a Schwarzschild black hole using Lemaitre coordinates and show that in these coordinates the Hamiltonian of the massless field is time-dependent. This result exhibits the non-static nature of the problem and shows it is better to talk about the time dependence of physical quantities rather than the existence of a time-independent vacuum state for the massless field. We then demonstrate the existence of Hawking radiation and show that despite the fact that the flux looks thermal to an outside observer, the time evolution of the massless field is unitary. 
  Using the Pauli-Villars regularization, we make a perturbative analysis of the quantum master equation (QME), $\Sigma =0$, for the Wilsonian effective action. It is found that the QME for the UV action determines whether exact gauge symmetry is realized along the renormalization group (RG) flow. The basic task of solving the QME can be reduced to compute the Troost-van Niuwenhuizen-Van Proyen jacobian factor for the classical UV action. When the QME cannot be satisfied, the non-vanishing $\Sigma$ is proportional to a BRS anomaly, which is shown to be preserved along the RG flow. To see how the UV action fulfills the QME in anomaly free theory, we calculate the jacobian factor for a pure Yang-Mills theory in four dimensions. 
  It is shown that two definitions for the exterior differential in superspace, giving the same exterior calculus, when applied to the Poisson bracket lead to the different results. Examples of the even and odd linear brackets, corresponding to semi-simple Lie groups, are given and their natural connection with BRST and anti-BRST charges is indicated. 
  Using cohomological methods we discuss several issues related to chiral anomalies in noncommutative U(N) YM theories in any even dimension. We show that for each dimension there is only one solution of the WZ consistency condition and that there cannot be any reducible anomaly, nor any mixed anomaly when the gauge group is a product group. We also clarify some puzzling aspects of the issue of the anomaly when chiral fermions are in the adjoint representation. 
  In the usual and current understanding of planar gauge choices for Abelian and non Abelian gauge fields, the external defining vector $n_\mu$ can either be space-like ($n^2<0$) or time-like ($n^2>0$) but not light-like ($n^2=0$). In this work we propose a light-like planar gauge that consists in defining a modified gauge-fixing term, $\cal{L}_{GF}$, whose main characteristic is a two-degree violation of Lorentz covariance arising from the fact that four-dimensional space-time spanned entirely by null vectors as basis necessitates two light-like vectors, namely $n_\mu$ and its dual $m_\mu$, with $n^2=m^2=0, n\cdot m\neq 0$, say, e.g. normalized to $n\cdot m=2$. 
  In this paper the linear representations of analytic Moufang loops are investigated. We prove that every representation of semisimple analytic Moufang loop is completely reducible and find all nonassociative irreducible representations. We show that such representations are closely associated with the (anti-)self-dual Yang-Mills equations in ${\bf R}^8$ 
  A short review of some of the aspects of quintessence model building is presented. We emphasize the role of tracking models and their possible supersymmetric origin. 
  Within the covariant formulation of Light-Front Dynamics, in a scalar model with the interaction Hamiltonian $H=-g\psi^{2}(x)\phi(x)$, we calculate nonperturbatively the renormalized state vector of a scalar "nucleon" in a truncated Fock space containing the $N$, $N\pi$ and $N\pi\pi$ sectors. The model gives a simple example of non-perturbative renormalization which is carried out numerically. Though the mass renormalization $\delta m^2$ diverges logarithmically with the cutoff $L$, the Fock components of the "physical" nucleon are stable when $L\to\infty$. 
  We employ the twistor approach to the construction of U(2) multi-instantons `a la 't Hooft on noncommutative R^4. The noncommutative deformation of the Corrigan-Fairlie-'t Hooft-Wilczek ansatz is derived. However, naively substituting into it the 't Hooft-type solution is unsatisfactory because the resulting gauge field fails to be self-dual on a finite-dimensional subspace of the Fock space. We repair this deficiency by a suitable Murray-von Neumann transformation after a specific projection of the gauge potential. The proper noncommutative 't Hooft multi-instanton field strength is given explicitly, in a singular as well as in a regular gauge. 
  We derive a tensorial formula for a fourth-order conformally invariant differential operator on conformal 4-manifolds. This operator is applied to algebraic Weyl tensor densities of a certain conformal weight, and takes its values in algebraic Weyl tensor densities of another weight. For oriented manifolds, this operator reverses duality: For example in the Riemannian case, it takes self-dual to anti-self-dual tensors and vice versa. We also examine the place that this operator occupies in known results on the classification of conformally invariant operators, and we examine some related operators. 
  We consider the problem of supersymmetry breaking in 5 dimensional N=1 supersymmetric models with $S^1/Z_2$ compactification. 
  The bootstrap program for 1+1-dimensional integrable Quantum Field Theories (QFT's) is developed to a large extent for the Homogeneous sine-Gordon (HSG) models. This program can be divided into various steps, which include the computation of the exact S-matrix, Form Factors of local operators and correlation functions, as well as the identification of the operator content of the QFT and the development of various consistency checks. Taking as an input the S-matrix proposal for the HSG-models, we confirm its consistency by carrying out both a Thermodynamic Bethe Ansatz (TBA) and a Form Factor analysis. In contrast to many other 1+1-dimensional integrable models studied in the literature, the HSG-models break parity, both at the level of the Lagrangian and S-matrix, and their spectrum includes unstable particles. These features have specific consequences in our analysis which are given a physical interpretation. By exploiting the Form Factor approach, we develop further the QFT advocated to the HSG-models. We evaluate correlation functions of various local operators of the model as well as Zamolodchikov's c-function and $\Delta$-sum rules. For the $SU(3)_2$-HSG model we show how the form factors of different local operators are interrelated by means of the momentum space cluster property. We find closed formulae for all $n$-particle form factors of a large class of operators of the $SU(N)_2$-HSG models. These formulae are expressed in terms of universal building blocks which allow both a determinant and an integral representation. 
  Two ways in which de Sitter space can arise in supergravity theories are discussed. In the first, it arises as a solution of a conventional supergravity, in which case it necessarily has no Killing spinors. For example, de Sitter space can arise as a solution of N=8 gauged supergravities in four or five dimensions. These lift to solutions of 11-dimensional supergravity or D=10 IIB supergravity which are warped products of de Sitter space and non-compact spaces of negative curvature. In the second way, de Sitter space can arise as a supersymmetric solution of an unconventional supergravity theory, which typically has some kinetic terms with the `wrong' sign; such solutions are invariant under a de Sitter supergroup. Such solutions lift to supersymmetric solutions of unconventional supergravities in D=10 or D=11, which nonetheless arise as field theory limits of theories that can be obtained from M-theory by timelike T-dualities and related dualities. Brane solutions interpolate between these solutions and flat space and lead to a holographic duality between theories in de Sitter vacua and Euclidean conformal field theories. Previous results are reviewed and generalised, and discussion is included of Kaluza-Klein theory with non-compact internal spaces, brane and cosmological solutions, and holography on de Sitter spaces and product spaces. 
  The integrality of Ooguri-Vafa disk invariants is verified using discrete symmetries of the superpotential of the mirror Landau-Ginzburg theory to calculate quantum corrections to the boundary variables. We show that these quantum corrections are completely determined if we assume that the discrete symmetry of the superpotential also holds in terms of the quantum corrected variables. We discuss the case of local P^2 blown up at three points and local F_2 blown up at two points in detail. 
  This is a continuation of the preceding paper (hep-ph/0108219).   First of all we make a brief review of generalized coherent states based on Lie algebra su(1,1) and prove that the resolution of unity can be obtained by the curvature form of some bundle. Next for a set of generalized coherent states we define a universal bundle over the infinite-dimensional Grassmann manifold and construct the pull-back bundle making use of a projector from the parameter space to this Grassmann one. We mainly study Chern characters of these bundles. Although the Chern characters in the infinite-dimensional case are in general not easy to calculate, we can perform them for the special cases. In this paper we report our calculations and propose some problems. 
  We present a general setup for junctions of semi-infinite 4-branes in $AdS_6$ with the Gauss-Bonnet term. The 3-brane tension at the junction of 4-branes can be nonzero. Using the brane junctions as the origin of the $Z_N$ discrete rotation symmetry, we identify 3-brane tensions at three fixed points of the orbifold $T^2/Z_3$ in terms of the 4-brane tensions. As a result, the three 3-brane tensions can be simultaneously positive, which enables us to explain the mass hierarchy by taking one of two branes apart from the hidden brane as the visible brane, and hence does not introduce a severe cosmological problem. 
  We find one explicit L^2 harmonic form for every Calabi manifold. Calabi manifolds are known to arise in low energy dynamics of solitons in Yang-Mills theories, and the L^2 harmonic form corresponds to the supersymmetric ground state. As the normalizable ground state of a single U(N) instanton, it is related to the bound state of a single D0 to multiple coincident D4's in the non-commutative setting, or equivalently a unit Kaluza-Klein mode in DLCQ of fivebrane worldvolume theory. As the ground state of nonabelian massless monopoles realized around a monopole-``anti''-monopole pair, it shows how the long range force between the pair is screened in a manner reminiscent of large N behavior of quark-anti-quark potential found in AdS/CFT correspondence. 
  The review is devoted to a relativistic formulation of the first Dirac quantization of QED (1927) and its generalization to the non-Abelian theories (Yang-Mills and QCD) with the topological degeneration of initial data. Using the Dirac variables we give a systematic description of relativistic nonlocal bound states in QED with a choice of the time axis of quantization along the eigenvectors of their total momentum operator.   We show that the Dirac variables of the non-Abelian fields are topologically degenerated, and there is a pure gauge Higgs effect in the sector of the zero winding number that leads to a nonperturbative physical vacuum in the form of the Wu-Yang monopole. Phases of the topological degeneration in the new perturbation theory are determined by an equation of the Gribov ambiguity of the constraint-shell gauge defined as an integral of the Gauss equation with zero initial data. The constraint-shell non-Abelian dynamics includes zero mode of the Gauss law differential operator, and a rising potential of the instantaneous interaction, that rearranges the perturbation series and changes the assymptotic freedom formula.   The Dirac variables in QCD with the topological degeneration of initial data describe constituent gluon, and quark masses, the spontaneous chiral symmetry breaking, color confinement in the form of quark-hadron duality as a consequence of summing over the Gribov copies. A solution of U(1)-problem is given by mixing the zero mode with $\eta_0$ - meson. We discuss reasons why all these physical effects disappear for arbitrary gauges of physical sources in the standard Faddeev-Popov integral. 
  We present some new results on the rational solutions of the Knizhnik-Zamolodchikov equation for the four-point conformal blocks of isospin I primary fields in the SU(2)_k Wess-Zumino-Novikov-Witten model. The rational solutions corresponding to integrable representations of the affine algebra su(2)_k have been classified by Michel, Stanev and Todorov; provided that the conformal dimension is an integer, they are in one-to-one correspondence with the local extensions of the chiral algebra. Here we give another description of these solutions as specific braid-invariant combinations of the so called regular basis and display a new series of rational solutions for isospins I = k+1 corresponding to non-integrable representations of the affine algebra. 
  It is proposed that a non-Abelian adjoint two-form in BF type theories transform inhomogeneously under the gauge group. The resulting restrictions on invariant actions are discussed. The auxiliary one-form which is required for maintaining vector gauge symmetry transforms like a second gauge field, and hence cannot be fully absorbed in the two-form. But it can be replaced, via a vector gauge transformation, by the usual gauge field, leading to gauge equivalences between different types of theories. A new type of symmetry also appears, one which depends on local functions but cannot be generated by constraints. It is connected to the identity in the limit of a vanishing global parameter, so it should be called a semiglobal symmetry. The corresponding conserved currents and BRST charges are parametrized by the space of flat connections. 
  We construct the explicit Euclidean scalar Green function associated with a massless field in a higher dimensional global monopole spacetime, i.e., a $(1+d)$-spacetime with $d\geq3$ which presents a solid angle deficit. Our result is expressed in terms of a infinite sum of products of Legendre functions with Gegenbauer polynomials. Although this Green function cannot be expressed in a closed form, for the specific case where the solid angle deficit is very small, it is possible to develop the sum and obtain the Green function in a more workable expression. Having this expression it is possible to calculate the vacuum expectation value of some relevant operators. As an application of this formalism, we calculate the renormalized vacuum expectation value of the square of the scalar field, $<\Phi^2(x)>_{Ren.}$, and the energy-momentum tensor, $<T_{\mu\nu}(x)>_{Ren.}$, for the global monopole spacetime with spatial dimensions $d=4$ and $d=5$. 
  We study the superspace formulation of the noncommutative nonlinear supersymmetric O(N) invariant sigma-model in 2+1 dimensions. We prove that the model is renormalizable to all orders of 1/N and explicitly verify that the model is asymptotically free. 
  We construct the N=1 supersymmetric extension of the polytropic gas dynamics. We give both the Lagrangian as well as the Hamiltonian description of this system. We construct the infinite set of "Eulerian'' conserved charges associated with this system and show that they are in involution, thereby proving complete integrability of this system. We construct the SUSY -B extension of this system as well and comment on the similarities and differences between this system and an earlier construction of the supersymmetric Chaplygin gas. We also derive the N=1 supersymmetric extension of the elastic medium equations, which, however, do not appear to be integrable. 
  We discuss Faddeev-Popov quantization at the non-perturbative level and show that Gribov's prescription of cutting off the functional integral at the Gribov horizon does not change the Schwinger-Dyson equations, but rather resolves an ambiguity in the solution of these equations. We note that Gribov's prescription is not exact, and we therefore turn to the method of stochastic quantization in its time-independent formulation, and recall the proof that it is correct at the non-perturbative level. The non-perturbative Landau gauge is derived as a limiting case, and it is found that it yields the Faddeev-Popov method in Landau gauge with a cut-off at the Gribov horizon, plus a novel term that corrects for over-counting of Gribov copies inside the Gribov horizon. Non-perturbative but truncated coupled Schwinger-Dyson equations for the gluon and ghost propagators $D(k)$ and $G(k)$ in Landau gauge are solved asymptotically in the infrared region. The infrared critical exponents or anomalous dimensions, defined by $D(k) \sim 1/(k^2)^{1 + a_D}$ and $G(k) \sim 1/(k^2)^{1 + a_G}$ are obtained in space-time dimensions $d = 2, 3, 4$. Two possible solutions are obtained with the values, in $d = 4$ dimensions, $a_G = 1, a_D = -2$, or $ a_G = [93 - (1201)^{1/2}]/98 \approx 0.595353, a_D = - 2a_G$. 
  We study the problem of vortex solutions in the background of rotating black holes in both asymptotically flat and asymptoticlly anti de Sitter spacetimes. We demonstrate the Abelian Higgs field equations in the background of four dimensional Kerr, Kerr-AdS and Reissner-Nordstrom-AdS black holes have vortex line solutions. These solutions, which have axial symmetry, are generalization of the Nielsen-Olesen string. By numerically solving the field equations in each case, we find that these black holes can support an Abelian Higgs field as hair. This situation holds even in the extremal case, and no flux-expulsion occurs. We also compute the effect of the self gravity of the Abelian Higgs field show that the the vortex induces a deficit angle in the corresponding black hole metrics. 
  In this talk we show that the tachyon annihilation combined with an approximation, in which string theory non-commutativity structure is captured by the algebra of differential operators on space-time, gives a unified point of view on: non-Abelian structures on $D$-branes; all lowest energy excitations on $D$-branes; all RR couplings in type II string theory. 
  We consider a conformal unified theory as the basis of conformal-invariant cosmological model where the permanent rigid state of the universe is compatible with the primordial element abundance and supernova data. We show that the cosmological creation of vector Z and W bosons, in this case, is sufficient to explain the CMB temperature (2.7 K). The primordial bosons violate the baryon number in the standard model as a result of anomalous nonconservation of left-handed currents and a nonzero squeezed vacuum expectation value of the topological Chern-Simons functional. 
  We report an attempt of deriving a string representation of QCD based on a novel vacuum condensate of mass dimension 2. 
  Based on an argument for the noncommutativity of momenta in noncommutative directions, we arrive at a generalization of the ${\cal N}=1$ super $E^2$ algebra associated to the deformation of translations in a noncommutative Euclidean plane. The algebra is obtained using appropriate representaions of its generators on the space of superfields in a $D=2, {\cal N}=1$ ``noncommutative superspace.'' We find that the (anti)commutators between several (super)translation generators are no longer vanishing, but involve a new set of generators which together with the (super)translation and rotation generators form a consistent closed algebra. We then analyze the spectrum of this algebra in order to obtain its fundamental and adjoint representations. 
  Regularization of quantum field theories (QFT's) can be achieved by quantizing the underlying manifold (spacetime or spatial slice) thereby replacing it by a non-commutative matrix model or a ``fuzzy manifold'' . Such discretization by quantization is remarkably successful in preserving symmetries and topological features, and altogether overcoming the fermion-doubling problem . In this thesis, the fuzzification of coadjoint orbits and their QFT's are put forward. 
  These lectures present an elementary discussion of some background material relevant to the problem of de Sitter quantum gravity. The first two lectures discuss the classical geometry of de Sitter space and properties of quantum field theory on de Sitter space, especially the temperature and entropy of de Sitter space. The final lecture contains a pedagogical discussion of the appearance of the conformal group as an asymptotic symmetry group, which is central to the dS/CFT correspondence. A (previously lacking) derivation of asymptotically de Sitter boundary conditions is also given. 
  For the D-branes on the SL(2,R) WZW model we present a particular choice of outer automorphism for the gluing condition of currents that leads to a special AdS_2 brane configuration. This configuration is shown to be a static solution in the cylindrical coordinates, and a nonstatic solution in the Poincar\'e coordinates to the nonlinear equation of motion for the Dirac-Born-Infeld action of a D-string. The generalization of it gives a family of nonstatic AdS_2 brane solutions. They are deomonstrated to transform to each other under the isometry group of AdS_3 spacetime. 
  Representation of the algebra of FP (anti)ghosts in string theory is studied by generalizing the recursive fermion system in the Cuntz algebra constructed previously. For that purpose, the pseudo Cuntz algebra, which is a $\ast$-algebra generalizing the Cuntz algebra and acting on indefinite-metric vector spaces, is introduced. The algebra of FP (anti)ghosts in string theory is embedded into the pseudo Cuntz algebra recursively in two different ways. Restricting a certain permutation representation of the pseudo Cuntz algebra, representations of these two recursive FP ghost systems are obtained. With respect to the zero-mode operators of FP (anti)ghosts, it is shown that one corresponds to the four-dimensional representation found recently by one of the present authors (M.A.) and Nakanishi, while the other corresponds to the two-dimensional one by Kato and Ogawa. 
  Using the path-integral formalism, we generalize the 't Hooft-Veltman method of unitary regulators to put forward a framework for finite, alternative quantum theories to a given quantum field theory. Feynman-like rules of such a finite, alternative quantum theory lead to alternative, perturbative Green functions. Which are acceptably regularized perturbative expansions of the original Green functions, causal, and imply no unphysical free particles. To demonstrate that the proposed framework is feasible, we take the quantum field theory of a single, self-interacting real scalar field and show how we can alter, covariantly and locally, its free-field Lagrangian to obtain finite, alternative perturbative Green functions. 
  We consider particlelike solutions to supergravity based on the Kerr-Newman BH solution. The BH singularity is regularized by means of a phase transition to a new superconducting vacuum state near the core region. We show that this phase transition can be controlled by gravity in spite of the extreme smallness of the local gravitational field.   Supersymmetric BPS domain wall model is suggested which can provide this phase transition and formation the stable charged (dual) superconducting core. 
  We argue that the temperature inversion symmetry present in the original Casimir setup and also in other Casimir systems for which symmetrical boundary conditions are imposed is not related to the duality transformations that in the context defined in Ref. [1] are transformations relating spatial extension and temperature, and pressure and energy density. We provide an example of a Casimir system for which in principle there is no temperature inversion symmetry but nevertheless these duality transformations can be found. 
  We study null bulk geodesic motion in the brane world cosmology in the RS2 scenario and in the static universe in the bulk of the charged topological AdS black hole. We obtain equations describing the null bulk geodesic motion as observed in one lower dimensions. We find that the null geodesic motion in the bulk of the brane world cosmology in the RS2 scenario is observed to be under the additional influence of extra non-gravitational force by the observer on the three-brane, if the brane universe does not possess the Z_2 symmetry. As for the null geodesic motion in the static universe in the bulk of the charged AdS black hole, the extra force is realized even when the brane universe has the Z_2 symmetry. 
  In this paper, we establish basic material for future investigations of the analysis and geometry of the twistor bundle, and of differential operators with the twistor bundle as source and/or target, especially the Rarita-Schwinger operator, a first order differential operator taking twistors to twistors. 
  We study a quantum system in a Riemannian manifold M on which a Lie group G acts isometrically. The path integral on M is decomposed into a family of path integrals on a quotient space Q=M/G and the reduced path integrals are completely classified by irreducible unitary representations of G. It is not necessary to assume that the action of G on M is either free or transitive. Hence the quotient space M/G may have orbifold singularities. Stratification geometry, which is a generalization of the concept of principal fiber bundle, is necessarily introduced to describe the path integral on M/G. Using it we show that the reduced path integral is expressed as a product of three factors; the rotational energy amplitude, the vibrational energy amplitude, and the holonomy factor. 
  The Hamiltonian reduction of classical SU(2) Yang-Mills field theory to the equivalent unconstrained theory of gauge invariant local dynamical variables is generalized to the case of nonvanishing theta angle. It is shown that for any theta angle the elimination of the pure gauge degrees of freedom leads to a corresponding unconstrained nonlocal theory of self-interacting second rank symmetric tensor fields, and that the obtained classical unconstrained gluodynamics with different theta angles are canonically equivalent as on the original constrained level. 
  We construct the fusion ring of a quasi-rational $\hat{sl}(4)_k$ WZNW theory at generic level $k \not\in Q$. It is generated by commutative elements in the group ring $Z[\tilde{W}]$ of the affine Weyl group $\tilde{W}$ which extend polynomially the formal characters of finite dimensional representations of $sl(4)$. 
  We study a black hole domain wall system in dilatonic gravity being the low-energy limit of the string theory. Using the C-metric construction we derive the metric for an infinitisemally thin domain wall intersecting dilaton black hole. The behavior of the domain wall in the spacetime of dilaton black hole was analyzed and it was revealed that the extereme dilaton black hole always expelled the domain wall. We elaborated the back reaction problem and concluded that topological kink solution smoothes out singularity of the considered topological defect. Finally we gave some comments concerning nucleation of dilaton black holes in the presence of the domain wall. We found that domain walls would rather prefer to nucleate small black holes on them, than large ones inside them. 
  In this paper we study the noncommutativity of a moving membrane with background fields. The open string variables are analyzed. Some scaling limits are studied. The equivalence of the magnetic and electric noncommutativities is investigated. The conditions for equivalence of noncommutativity of the T-dual theory in the rest frame and noncommutativity of the original theory in the moving frame are obtained. 
  We give infinite dimensional and finite dimensional examples of $F-$fold Lie superalgebras. The finite dimensional examples are obtained by an inductive procedure from Lie algebras and Lie superalgebras. 
  We find considerable evidence supporting the conjecture that four-dimensional Quantum Einstein Gravity is ``asymptotically safe'' in Weinberg's sense. This would mean that the theory is likely to be nonperturbatively renormalizable and thus could be considered a fundamental (rather than merely effective) theory which is mathematically consistent and predictive down to arbitrarily small length scales. For a truncated version of the exact flow equation of the effective average action we establish the existence of a non-Gaussian renormalization group fixed point which is suitable for the construction of a nonperturbative infinite cutoff-limit. The truncation ansatz includes the Einstein-Hilbert action and a higher derivative term. 
  We study the transmission of supersymmetry breaking via gravitational interactions in a five-dimensional brane-world compactified on $S^1/Z_2$. We assume that chiral matter and gauge fields are confined at the orbifold fixed points, where supersymmetry is spontaneously broken by effective brane superpotentials. Using an off-shell supergravity multiplet we integrate out the auxiliary fields and examine the couplings between the bulk supergravity fields and boundary matter fields. The corresponding tree-level shift in the bulk gravitino mass spectrum induces one-loop radiative masses for the boundary fields. We calculate the boundary gaugino and scalar masses for arbitrary values of the brane superpotentials, and show that the mass spectrum reduces to the Scherk-Schwarz limit for arbitrarily large values of the brane superpotentials. 
  We show that in modified Faddeev-Jackiw formalism, first and second class constraints appear at each level, and the whole constraint structure is in exact correspondence with level by level method of Dirac formalism. 
  Using the AdS/CFT correspondence in a confining backgroundand the worldline formalism of gauge field theories,we compute scattering amplitudes with an exchange of quark andantiquark in the $t$-channel corresponding to Reggeon exchange. Itrequires going beyond the eikonal approximation, which was used when studying Pomeron exchange. The wordline path integral is evaluated through the determination of minimal surfaces and their boundaries by the saddle-point method at large gauge coupling g^2N_c. We find a Regge behaviour with linear Regge trajectories. The slope is related to the $q\bar q$ static potential and is four times the Pomeronslope obtained in the same framework. A contribution to the intercept, related to the L\"uscher term, comes from the fluctuations around the minimal surface. 
  We discuss why the N_6=1 heterotic string has to be viewed as something similar to a ``non-compact orbifold''. Only the perturbative spectrum is forced to satisfy the constraints imposed by the vanishing of six-dimensional anomalies. These do not apply to the states of the non-perturbative spectrum, such as those appearing when small instantons shrink to zero size. 
  These introductory notes are about functional renormalization group equations and some of their applications. It is emphasised that the applicability of this method extends well beyond critical systems, it actually provides us a general purpose algorithm to solve strongly coupled quantum field theories. The renormalization group equation of F. Wegner and A. Houghton is shown to resum the loop-expansion. Another version, due to J. Polchinski, is obtained by the method of collective coordinates and can be used for the resummation of the perturbation series. The genuinely non-perturbative evolution equation is obtained in a manner reminiscent of the Schwinger-Dyson equations. Two variants of this scheme are presented where the scale which determines the order of the successive elimination of the modes is extracted from external and internal spaces. The renormalization of composite operators is discussed briefly as an alternative way to arrive at the renormalization group equation. The scaling laws and fixed points are considered from local and global points of view. Instability induced renormalization and new scaling laws are shown to occur in the symmetry broken phase of the scalar theory. The flattening of the effective potential of a compact variable is demonstrated in case of the sine-Gordon model. Finally, a manifestly gauge invariant evolution equation is given for QED. 
  This lecture consists of two parts. The first is a (totally unsystematic) survey of some of the high points in the evolution of gravity and its successors, primarily in the course of the past century. The second summarizes some new work on surprising properties of higher $(> 1)$ spin fields in cosmological backgrounds: the presence of $\L$ gives rise to discrete sets of massive models endowed with gauge invariances, that divide the ($m^2, \L$) plane into unitary and non-unitary phases. The unitary region common to fermions and bosons shrinks to flat space ($ \L \to 0 $) as their spins increase. 
  We study the dynamics of a large class of N=1 quiver theories, geometrically realized by type IIB D-brane probes wrapping cycles of local Calabi-Yau threefolds. These include N=2 (affine) A-D-E quiver theories deformed by superpotential terms, as well as chiral N=1 quiver theories obtained in the presence of vanishing 4-cycles inside a Calabi-Yau. We consider the various possible geometric transitions of the 3-fold and show that they correspond to Seiberg-like dualities (represented by Weyl reflections in the A-D-E case or `mutations' of bundles in the case of vanishing 4-cycles) or large N dualities involving gaugino condensates (generalized conifold transitions). Also duality cascades are naturally realized in these classes of theories, and are related to the affine Weyl group symmetry in the A-D-E case. 
  Positive frequency Wightman function and vacuum expectation values of the energy-momentum tensor are computed for a massive scalar field with general curvature coupling parameter and satisfying Robin boundary condition on a uniformly accelerated infinite plate. The both regions of the right Rindler wedge, (i) on the right (RR region) and (ii) on the left (RL region) of the plate are investigated. For the case (ii) the electromagnetic field is considered as well. The mode summation method is used with combination of a variant of the generalized Abel-Plana formula. This allows to present the expectation values in the form of a sum of the purely Rindler and boundary parts. Near the plate surface the vacuum energy-momentum tensor is dominated by the boundary term. At large distances from the plate and near the Rindler horizon the main contribution comes from the purely Rindler part. In the RL region the vacuum energy density of the electromagnetic field is negative near the horizon and is positive near the plate. 
  In this ultra short note, gauge field propagation in D-brane configuration of M theory in the BFSS matrix formulation is considered. Noncommutativity of the space plays a key role for appearance of gauge fields as physical degrees of freedom. 
  We show that the asymptotic dynamics of three-dimensional gravity with positive cosmological constant is described by Euclidean Liouville theory. This provides an explicit example of a correspondence between de Sitter gravity and conformal field theories. In the case at hand, this correspondence is established by formulating Einstein gravity with positive cosmological constant in three dimensions as an SL(2,C) Chern-Simons theory. The de Sitter boundary conditions on the connection are divided into two parts. The first part reduces the CS action to a nonchiral SL(2,C) WZNW model, whereas the second provides the constraints for a further reduction to Liouville theory, which lives on the past boundary of dS_3. 
  The formulation of 2d-dilaton theories, like spherically reduced Einstein gravity, is greatly facilitated in a formulation as a first order theory with nonvanishing bosonic torsion. This is especially also true at the quantum level. The interpretation of superextensions as graded Poisson sigma models is found to cover generically all possible 2d supergravities. Superfields and thus superfluous auxiliary fields are avoided altogether. The procedure shows that generalizations of bosonic 2d models are highly ambiguous. 
  It is sometimes claimed that one cannot describe charged particles in gauge theories. We identify the root of the problem and present an explicit construction of charged particles. This is shown to have good perturbative properties and, asymptotically before and after scattering, to recover particle modes. 
  We construct ten-dimensional supergravity solutions corresponding to the near horizon limit of IIB fivebranes wrapping special Lagrangian three-cycles of constant curvature. The case of branes wrapping a three-sphere provides a gravity dual of pure N=2 super-Yang-Mills theory in D=3. The non-trivial part of the solutions are seven manifolds that admit two G_2 structures each of which is covariantly constant with respect to a different connection with torsion. We derive a formula for the generalised calibration for this general class of solutions. We discuss analogous aspects of the geometry that arises when fivebranes wrap other supersymmetric cycles which lead to Spin(7) and SU(N) structures. In some cases there are two covariantly constant structures and in others one. 
  We present several results concerning non-commutative instantons and the Seiberg-Witten map. Using a simple ansatz we find a large new class of instanton solutions in arbitrary even dimensional non-commutative Yang-Mills theory. These include the two dimensional ``shift operator'' solutions and the four dimensional Nekrasov-Schwarz instantons as special cases. We also study how the Seiberg-Witten map acts on these instanton solutions. The infinitesimal Seiberg-Witten map is shown to take a very simple form in operator language, and this result is used to give a commutative description of non-commutative instantons. The instanton is found to be singular in commutative variables. 
  In previous work we considered M-theory five branes wrapped on elliptic Calabi-Yau threefold near the smooth part of the discriminant curve. In this paper, we extend that work to compute the light states on the worldvolume of five-branes wrapped on fibers near certain singular loci of the discriminant. We regulate the singular behavior near these loci by deforming the discriminant curve and expressing the singularity in terms of knots and their associated braids. There braids allow us to compute the appropriate string junction lattice for the singularity and,hence to determine the spectrum of light BPS states. We find that these techniques are valid near singular points with N=2 supersymmetry. 
  We demonstrate how one can describe explicitly the present arbitrariness in solutions of relativistic wave equations in external electromagnetic fields of special form. This arbitrariness is connected to the existence of a transformation, which reduces effectively the number of variables in the initial equations. Then we use the corresponding representations to construct new sets of exact solutions, which may have a physical interest. Namely, we present new sets of stationary and nonstationary solutions in magnetic field and in some superpositions of electric and magnetic fields. 
  We consider Yang-Mills theory in Euclidean space-time $(R^4)$ and construct its configuration space. The orbits are first shown to form a congruence set. Then we discuss the orthogonal gauge condition in Abelian theory and show that Coulomb-like surfaces foliate the entire configuration space. In the non-Abelian case, where these exists no global orthogonal gauge, we derive the non-linear gauge proposed previously by the author by modifying the orthogonality condition. However, unlike the Abelian case, the entire configuration space cannot be foliated by submanifolds defined by the non-linear gauge. The foliation is only limited to the non-perturbative regime of Yang-Mills theory. 
  We find a class of flat supersymmetric brane-antibrane configurations. They follow from ordinary brane-antibrane systems by turning on a specific worldvolume background electric field, which corresponds to dissolved fundamental strings. We have clarified in detail how they arise and identified their constituent charges as well as the corresponding supergravity solutions. Adopting the matrix theory description, we construct the worldvolume gauge theories and prove the absence of any tachyonic degrees. We also study supersymmetric solitons of the worldvolume theories. 
  We study a scale invariant two measures theory where a dilaton field \phi has no explicit potentials. The scale transformations include a shift \phi\to\phi+const. The theory demonstrates a new mechanism for gene- ration of the exponential potential: in the conformal Einstein frame (CEF), after SSB of scale invariance, the theory develops the exponential potential and, in general, non-linear kinetic term is generated as well. The possibility of quintessence solutions are shown. As an example, for one choice of the parameters we obtain standard scaling solutions usually used in the context of the quintessential scenario. For other choice of the parameters, the theory allows a scaling solution with equation of state p_{\phi}=w\rho_{\phi} where w is restricted by -1<w<-0.82. The regime where the fermionic matter dominates (as compared to the dilatonic contribution) is analyzed. There it is found that starting from a single fermionic field we obtain exactly three different types of spin 1/2 particles in CEF that appears to suggest a new approach to the family problem of particle physics. It is automatically achieved that for two of them, fermion masses are constants, gravitational equations are canonical and the "fifth force" is absent. For the third type of particles, a fermionic self-interaction appears as a result of SSB of scale invariance. 
  We consider the classification of BPS and non-BPS D-branes in orientifold models. In particular we construct all stable BPS and non-BPS D-branes in the Gimon-Polchinski (GP) and Dabholkar-Park-Blum-Zaffaroni (DPBZ) orientifolds and determine their stability regions in moduli space as well as decay products. We find several kinds of integrally and torsion charged non-BPS D-branes. Certain of these are found to have projective representations of the orientifold $\times$ GSO group on the Chan-Paton factors. It is found that the GP orientifold is not described by equivariant orthogonal K-theory as may have been at first expected. Instead a twisted version of this K-theory is expected to be relevant. 
  We show that relativistic strings of open and closed types in Minkowski space-time of dimension 3 and 4 have topologically stable singular points. This paper describes the structure of singularities, derives their normal forms, and introduces a local characteristic of singularity (topological charge), possessing a global law of conservation. Two other types of solutions (breaking and exotic strings) are also considered, which have singularities at arbitrary value of dimension. 
  Regularization modifies the (odd) behaviour of the Abelian Chern-Simons action under parity. This effect happens for any sensible regularization; in particular, on the lattice. However, as in the chiral symmetry case, there exist generalized parity transformations such that the regularized theory is odd, and the corresponding operator verifies a Ginsparg-Wilson like relation. We present a derivation of such a relation and of the corresponding symmetry transformations. 
  The local Casimir energy is investigated for a wedge with and without a circular outer boundary due to the confinement of a massless scalar field with general curvature coupling parameter and satisfying the Dirichlet boundary conditions. Regularization procedure is carried out making use of a variant of the generalized Abel-Plana formula, previously established by one of the authors. The surface divergences in the vacuum expectation values of the energy density near the boundaries are considered. The corresponding results can be applied to the cosmic strings. 
  A solution to Slavnov-Taylor identities in a general four dimensional N=1 supersymmetric Yang-Mills theory containing arbitrary matter superfields is proposed. The solution proposed appears just a simple generalization of an analogous solution in the pure supersymmetric Yang-Mills theory. 
  The covariant form of the non-Abelian gauge anomaly on noncommutative R2n is computed for U(N) groups. Its origin and properties are analyzed. Its connection with the consistent form of the gauge anomaly is established. We show along the way that bi-fundamental $U(N)\times U(M)$ chiral matter carries no mixed anomalies, and interpret this result as a consequence of the half-dipole structure which characterizes the charged non-commutative degrees of freedom. 
  Negative dimensional integration method (NDIM) is a technique which can be applied, with success, in usual covariant gauge calculations. We consider three two-loop diagrams: the scalar massless non-planar double-box with six propagators and the scalar pentabox in two cases, where six virtual particles have the same mass and in the case where all of them are massless. Our results are given in terms hypergeometric functions of Mandelstam variables and for arbitrary exponents of propagators and dimension $D$ as well. 
  BPS domain wall solutions of gauged supergravities are found, including those theories which have non-compact gauge groups. These include models that have both an unstable de Sitter solution and stable domain wall solutions. 
  We show that generically the initial singularity is isotropic in spatially homogeneous cosmological models in the brane-world scenario. We then argue that it is plausible that the initial singularity is isotropic in typical brane world cosmological models. Therefore, brane cosmology naturally gives rise to a set of initial data that provide the conditions for inflation to subsequently take place, thereby solving the initial conditions problem and leading to a self--consistent and viable cosmology. 
  We study Vafa's geometric transition and Klebanov - Strassler solution from various points of view in M-theory. In terms of brane configurations, we show the detailed equivalences between the two models. In some limits, both models have an alternative realization as fourfolds in M-theory with appropriate G-fluxes turned on. We discuss some aspects of the fourfolds including how to see the transition and a possible extension to the non-supersymmetric case. 
  A simplified higher dimensional Randall-Sundrum-like model in 6 dimensions is considered. It has been observed previously by Goldberger and Wise that in such a self-interacting scalar theory on the bulk with a conical singularity there is mixing of renormalization of 4d brane couplings with that of the bulk couplings. We study the influence of the running bulk couplings on the running of the 4d brane couplings. We find that bulk quantum effects may completely alter the running of brane couplings. In particular, the structure of the Landau pole may be drastically altered and non-asymptotically free running may turn into asymptotically safe (or free) behavior. 
  Soliton models are used in elementary particle physics and nuclear physics to model extended objects such as nucleons, using effective field theories derived from more fundamental theories such as QCD. Computer simulation requires some sort of discretisation procedure, a notable example being lattice gauge theory applied in the imaginary time formulation.   In this lecture we focus on simulation in real time and the problems which arise when invariants of the motion are monitored in a simulation.   Ad hoc discretisations invariably introduce drift in numerically computed invariants, due not only to numerical accuracy (which can be negligible), but with the fact that the discretisation may not necessarily imply conservation. We discuss an approach developed which gives principles for the discretisation of such systems and for the construction of exact discrete invariants of the motion. This approach is applied to the Skyrme model.   Talk at the International Workshop on Nuclear Theory, Rila, June 2001, to be published in the Proceedings. 
  We study the SU(2) electroweak model in which the standard Yang-Mills coupling is supplemented by a Born-Infeld term. The deformation of the sphaleron and bisphaleron solutions due to the Born-Infeld term is investigated and new branches of solutions are exhibited. Especially, we find a new branch of solutions connecting the Born-Infeld sphaleron to the first solution of the Kerner-Gal'tsov series. 
  The exact renormalization group equation for pure quantum gravity is used to derive the non-perturbative $\Fbeta$-functions for the dimensionless Newton constant and cosmological constant on the theory space spanned by the Einstein-Hilbert truncation. The resulting coupled differential equations are evaluated for a sharp cutoff function. The features of these flow equations are compared to those found when using a smooth cutoff. The system of equations with sharp cutoff is then solved numerically, deriving the complete renormalization group flow of the Einstein-Hilbert truncation in $d=4$. The resulting renormalization group trajectories are classified and their physical relevance is discussed. The non-trivial fixed point which, if present in the exact theory, might render Quantum Einstein Gravity nonperturbatively renormalizable is investigated for various spacetime dimensionalities. 
  This review is devoted to strings and branes. Firstly, perturbative string theory is introduced. The appearance of various types of branes is discussed. These include orbifold fixed planes, D-branes and orientifold planes. The connection to BPS vacua of supergravity is presented afterwards. As applications, we outline the role of branes in string dualities, field theory dualities, the AdS/CFT correspondence and scenarios where the string scale is at a TeV. Some issues of warped compactifications are also addressed. These comprise corrections to gravitational interactions as well as the cosmological constant problem. 
  Point particle may interact to traceless symmetric tensors of arbitrary rank. Free gauge theories of traceless symmetric tensors are constructed, that provides a possibility for a new type of interactions, when particles exchange by those gauge fields. The gauge theories are parameterized by the particle's mass m and otherwise are unique for each rank s. For m=0, they are local gauge models with actions of 2s-th order in derivatives, known in d=4 as "pure spin", or "conformal higher spin" actions by Fradkin and Tseytlin. For nonzero m, each rank-s model undergoes a unique nonlocal deformation which entangles fields of all ranks, starting from s. There exists a nonlocal transform which maps m > 0 theories onto m=0 ones, however, this map degenerates at some m > 0 fields whose polarizations are determined by zeros of Bessel functions. Conformal covariance properties of the m=0 models are analyzed, the space of gauge fields is shown to admit an action of an infinite-dimensional "conformal higher spin" Lie algebra which leaves gauge transformations intact. 
  Noncommuting spatial coordinates and fields can be realized in actual physical situations. Plane wave solutions to noncommuting photodynamics exhibit violaton of Lorentz invariance (special relativity). 
  Extending previous work, we calculate in this note the fermionic spectrum of two-dimensional QCD (QCD_2) in the formulation with SU(N_c) currents. Together with the results in the bosonic sector this allows to address the as yet unresolved task of finding the single-particle states of this theory as a function of the ratio of the numbers of flavors and colors, \lambda=N_f/N_c, anew. We construct the Hamiltonian matrix in DLCQ formulation as an algebraic function of the harmonic resolution K and the continuous parameter \lambda. Amongst the more surprising findings in the fermionic sector chiefly considered here is that the fermion momentum is a function of \lambda. This dependence is necessary in order to reproduce the well-known 't Hooft and large N_f spectra. Remarkably, those spectra have the same single-particle content as the ones in the bosonic sectors. The twist here is the dramatically different sizes of the Fock bases in the two sectors, which makes it possible to interpret in principle all states of the discrete approach. The hope is that some of this insight carries over into the continuum. We also present some new findings concerning the single-particle spectrum of the adjoint theory. 
  The Maxwell theory on non-commutative spaces has been considered. The non-linear equations of electromagnetic fields on non-commutative spaces were obtained in the compact spin-tensor (quaternion) form. It was shown that the plane electromagnetic wave is the solution of the system of non-linear wave equations of the second order for the electric and magnetic induction fields. We have found the canonical and symmetrical energy-momentum tensors and their non-zero traces. So, the trace anomaly of the energy-momentum tensor was obtained in electrodynamics on non-commutative spaces. It was noted that the dual transformations of electromagnetic fields on non-commutative spaces are broken. 
  Tensor and matrix formulations of Dirac-K\"ahler equation for massive and massless fields are considered. The equation matrices obtained are simple linear combinations of matrix elements in the 16-dimensional space. The projection matrix-dyads defining all the 16 independent equation solutions are found. A method of computing the traces of 16-dimensional Petiau-Duffin-Kemmer matrix product is considered. It is shown that the symmetry group of the Dirac-K\"ahler tensor fields is SO(4,2). The conservation currents corresponding this symmetry are constructed. Supersymmetry of the Dirac-K\"ahler fields with tensor and spinor parameters is analyzed. We show the possibility of constructing a gauge model of interacting Dirac-K\"ahler fields where the gauge group is the noncompact group under consideration. 
  The exact solutions of the wave equation for arbitrary spin particles with electric dipole and magnetic moments in the constant and uniform electromagnetic field were found. The differential probability of pair production of particles by an external electromagnetic field has been calculated on the basis of the exact solutions. We have also estimated the imaginary part of the constant and uniform electromagnetic field. The nonlinear corrections to the Maxwell Lagrangian have been calculated taking into account the vacuum polarization of arbitrary spin particles. The role of electric dipole and magnetic moments of arbitrary spin particles in instability of the vacuum is discussed. 
  The dS/CFT correspondence differs from its AdS/CFT counterpart in some ways, yet is strikingly similar to it in many others. For example, both involve CFTs defined on connected spaces (despite the fact that the conformal boundary of deSitter space is not connected), and both impose constraints on scalar masses (Strominger's bound for deSitter, and the Breitenlohner-Freedman bound for Anti-deSitter). We argue that these similarities can be explored and exploited using a slight extension of the Euclidean approach to AdS/CFT. The methods are particularly compatible with Hull's embedding of deSitter Space in a timelike T-dual version of M-theory. 
  We study the possibility of spontaneous CP violation in string models with the dilaton field stabilized at a phenomenologically acceptable value. We consider three mechanisms to stabilize the dilaton: multiple gaugino condensates, a nonperturbative Kahler potential, and a superpotential based on S-duality, and analyze consequent CP phases in the soft SUSY breaking terms. Due to non-universality forced upon the theory by requiring a non-trivial CKM phase, the EDM problem becomes more severe. Even if there are no complex phases in the VEVs of the SUSY breaking fields, the electric dipole moments are overproduced by orders of magnitude. We also address the question of modular invariance of the physical CP phases. 
  d5 higher derivative gravity on the Schwarzschild-de Sitter (SdS) black hole background is considered. Two horizons SdS BHs are not in thermal equilibrium and Hawking-Page phase transitions are not expected there, unlike to the case of AdS BHs. It is demonstrated that there exists the regime of d5 theory where Nariai BH which is extremal limit of SdS BH is stable. It is in the contrast with Einstein gravity on such backgroundwhere only pure de Sitter space is always stable. Speculating on the applications in proposed dS/CFT correspondence, these two (de Sitter and Nariai) stable spaces may correspond to confining-deconfining phases in dual CFT. 
  We study the supersymmetric GUT models where the supersymmetry and GUT gauge symmetry can be broken by the discrete symmetry. First, with the ansatz that there exist discrete symmetries in the branes' neighborhoods, we discuss the general reflection $Z_2$ symmetries and GUT breaking on $M^4\times M^1$ and $M^4\times M^1\times M^1$. In those models, the extra dimensions can be large and the KK states can be set arbitrarily heavy. Second, considering the extra space manifold is the annulus $A^2$ or disc $D^2$, we can define any $Z_n$ symmetry and break any 6-dimensional N=2 supersymmetric SU(M) models down to the 4-dimensional N=1 supersymmetric $SU(3)\times SU(2)\times U(1)^{M-4}$ models for the zero modes. In particular, there might exist the interesting scenario on $M^4\times A^2$ where just a few KK states are light, while the others are relatively heavy. Third, we discuss the complete global discrete symmetries on $M^4\times T^2$ and study the GUT breaking. 
  We compute, at two-loop order, one-particle-irreducible Green functions and effective action in noncommutative $\lambda[\Phi^3]_\star$-theory for both planar (g=0, h=3) and nonplanar (g=1, h=1) contributions. We adopt worldline formulation of the Feynman diagrammatics so that relation to string theory diagrammatics is made transparent in the Seiberg-Witten limit. We argue that the resulting two-loop effective action is expressible via open Wilson lines: one-particle-irreducible effective action is generating functional of connected diagrams for interacting open Wilson lines. 
  In this work are consider several topics in the Topological Membrane (TM) approach to string theory. The string dynamics is generated from the bulk physics, namely from the 3D Topologically Massive Gauge Theory (TMGT) and Topologically Massive Gravity (TMG). Both (equivalent) path integral and canonical methods of quantizing TMGT are studied. It is shown that Narain constraints on toroidal compactification (integer, even, self-dual momentum lattice) have a natural interpretation in purely three dimensional terms. The Heterotic string and the block structure of c=1 RCFT are derived from the point of view of three dimensional field theory. Open and unoriented strings in TM(GT) theory are also studied through orbifolds of the bulk 3D space. This is achieved by gauging discrete symmetries of the theory. Open and unoriented strings can be obtained from all possible realizations of C, P and T symmetries. The important role of C symmetry to distinguish between Dirichlet and Neumann boundary conditions is discussed in detail. Future directions of research in this field are also suggested and discussed. 
  Einstein derived general relativity from Riemannian geometry. Connes extends this derivation to noncommutative geometry and obtains electro-magnetic, weak and strong forces. These are pseudo forces, that accompany the gravitational force just as in Minkowskian geometry the magnetic force accompanies the electric force. The main physical input of Connes' derivation is parity violation. His main output is the Higgs boson which breaks the gauge symmetry spontaneously and gives masses to gauge and Higgs bosons. 
  Fields in supersymmetric gauge theories may be seen as elements in a spinorial cohomology. We elaborate on this subject, specialising to maximally supersymmetric theories, where the superspace Bianchi identities, after suitable conventional constraints are imposed, put the theories on shell. In these cases, the spinorial cohomologies describe in a unified manner gauge transformations, fields and possible deformations of the models, e.g. string-related corrections in an alpha' expansion. Explicit cohomologies are calculated for super-Yang-Mills theory in D=10, for the N=(2,0) tensor multiplet in D=6 and for supergravity in D=11, in the latter case from the point of view of both the super-vielbein and the super-3-form potential. The techniques may shed light on some questions concerning the alpha'-corrected effective theories, and result in better understanding of the role of the 3-form in D=11 supergravity. 
  We examine several different types of five dimensional stationary spacetimes with bulk scalar fields and parallel 3-branes. We study different methods for avoiding the appearance of spacetime singularities in the bulk for models with and without cosmological expansion. For non-expanding models, we demonstrate that in general the Randall-Sundrum warp factor is recovered in the asymptotic bulk region, although elsewhere the warping may be steeper than exponential. We show that nonsingular expanding models can be constructed as long as the gradient of the bulk scalar field vanishes at zeros of the warp factor, which are then analogous to the particle horizons found in expanding models with a pure AdS bulk. Since the branes in these models are stabilized by bulk scalar fields, we expect there to be no linearly unstable radion modes. As an application, we find a specific class of expanding, stationary solutions with no singularities in the bulk in which the four dimensional cosmological constant and mass hierarchy are naturally very small. 
  We review the idea of Pi-stability for B-type D-branes on a Calabi-Yau manifold. It is shown that the octahedral axiom from the theory of derived categories is an essential ingredient in the study of stability. Various examples in the context of the quintic Calabi-Yau threefold are studied and we plot the lines of marginal stability in several cases. We derive the conjecture of Kontsevich, Horja and Morrison for the derived category version of monodromy around a "conifold" point. Finally, we propose an application of these ideas to the study of supersymmetry breaking. 
  The status of exponential scalar potentials in supergravity is reviewed, and updated. One version of N=8 D=4 supergravity with a positive exponential potential, obtainable from a `non-compactification' of M-theory, is shown to have an accelerating cosmological solution that realizes `eternal quintessence'. Some implications for a de Sitter version of the domain wall/QFT correspondence are discussed. 
  We study the constraints on models with extra dimensions arising from local anomaly cancellation. We consider a five-dimensional field theory with a U(1) gauge field and a charged fermion, compactified on the orbifold S^1/(Z_2 x Z_2'). We show that, even if the orbifold projections remove both fermionic zero modes, there are gauge anomalies localized at the fixed points. Anomalies naively cancel after integration over the fifth dimension, but gauge invariance is broken, spoiling the consistency of the theory. We discuss the implications for realistic supersymmetric models with a single Higgs hypermultiplet in the bulk, and possible cancellation mechanisms in non-minimal models. 
  We construct an off-shell N=3 supersymmetric extension of the abelian D=4 Born-Infeld action starting from the action of supersymmetric Maxwell theory in N=3 harmonic superspace. A crucial new feature of the N=3 super BI action is that its interaction part contains only terms of the order 4k in the N=3 superfield strengths. The correct component bosonic BI action arises as the result of elimination of auxiliary tensor field which is present in the off-shell N=3 vector multiplet in parallel with the gauge field strength. In this new Legendre-type representation, the bosonic BI action is fully specified by a real function of the single variable quartic in the auxiliary tensor field. The generic choice of this function amounts to a wide set of self-dual nonlinear extensions of the Maxwell action. All of them admit an off-shell N=3 supersymmetrization. 
  The q-electroweak theory obtained by replacing SU(2) by $SU_q(2)$ in the Weinberg-Salam model is experimentally not distinguishable from the standard model at the level of the doublet representation. However, differences between the two theories should be observable when higher dimensional representations are taken into account. In addition the possibility of probing non-local structure may be offered by the q-theory. 
  We demonstrate the existence of stable knot solitons in the standard electroweak theory whose topological quantum number $\pi_3(S^2)$ is fixed by the Chern-Simon index of the $Z$ boson. The electroweak knots are made of the helical magnetic flux tube of $Z$ boson which has a non-trivial dressing of the Higgs field, which could also be viewed as two quantized flux rings linked together whose linking number becomes the knot quantum number. We estimate the mass of the lightest knot to be around $21 TeV$. 
  We investigate some properties of string theory on Kaluza-Klein Melvin background. We discuss what happens when closed string tachyons in the theory condense, by using a linear sigma model of the Kaluza-Klein Melvin background. 
  The quantization rules recently proposed by M. Navarro (and independently I.V. Kanatchikov) for a finite-dimensional formulation of quantum field theory are applied to the Klein-Gordon and the Dirac fields to obtain the quantum equations of motion of both fields. In doing so several problems arise.   Solving these difficulties leads us to propose a new classical canonical formalism, which, in turn, leads us to new, improved rules of quantization. We show that the new classical equations of motion and rules of quantization overcome several known unsatisfactory features of the previous formalism. We argue that the new formalism is a general improvement with respect to the previous one.   Further we show that the quantum field theory of the Dirac and Klein-Gordon field describes particles with extra, harmonic-oscillator-like degrees of freedom. We argue that these degrees of freedom should give rise to a multi-particle interpretation of the formalism. 
  We start from a new theory (discussed earlier) in which the arena for physics is not spacetime, but its straightforward extension-the so called Clifford space ($C$-space), a manifold of points, lines, areas, etc..; physical quantities are Clifford algebra valued objects, called polyvectors. This provides a natural framework for description of supersymmetry, since spinors are just left or right minimal ideals of Clifford algebra. The geometry of curved $C$-space is investigated. It is shown that the curvature in $C$-space contains higher orders of the curvature in the underlying ordinary space. A $C$-space is parametrized not only by 1-vector coordinates $x^\mu$ but also by the 2-vector coordinates $\sigma^{\mu \nu}$, 3-vector coordinates $\sigma^{\mu \nu \rho}$, etc., called also {\it holographic coordinates}, since they describe the holographic projections of 1-lines, 2-loops, 3-loops, etc., onto the coordinate planes. A remarkable relation between the "area" derivative $\p/ \p \sigma^{\mu \nu}$ and the curvature and torsion is found: if a scalar valued quantity depends on the coordinates $\sigma^{\mu \nu}$ this indicates the presence of torsion, and if a vector valued quantity depends so, this implies non vanishing curvature. We argue that such a deeper understanding of the $C$-space geometry is a prerequisite for a further development of this new theory which in our opinion will lead us towards a natural and elegant formulation of $M$-theory. 
  We study axially symmetric monopoles of both the SU(2) Yang-Mills-Higgs-Dilaton (YMHD) as well as of the SU(2) Einstein-Yang-Mills-Higgs-Dilaton (EYMHD) system. We find that equally to gravity, the presence of the dilaton field can render an attractive phase. We also study the influence of a massive dilaton on the attractive phase in the YMHD system. 
  We consider the integrable open XX quantum spin chain with nondiagonal boundary terms. We derive an exact inversion identity, using which we obtain the eigenvalues of the transfer matrix and the Bethe Ansatz equations. For generic values of the boundary parameters, the Bethe Ansatz solution is formulated in terms of Jacobian elliptic functions. 
  In this paper we present the effects produced by the temperature in the renormalized vacuum expectation value of the zero-zero component of the energy-momentum tensor associated with massless left-handed spinor field in the pointlike global monopole spacetime. In order to develop this calculation we had to obtain the Euclidean thermal Green function in this background. Because the expression obtained for the thermal energy density cannot be expressed in a closed form, its explicit dependence on the temperature is not completely evident. So, in order to obtain concrete information about its thermal behavior, we develop a numerical analysis of our result in the high-temperature limit for specific values of the parameter $\alpha$ which codify the presence of the monopole. 
  We consider the bosonic fields which describe a particle which may exist in states with spins one and zero with different masses. All the linearly independent solutions of the equation for a free particle are obtained in the form of the projection matrix-dyads (density-matrices). The interaction with anomalous magnetic dipole and quadruple electric moments of a particle is studied. The Hamiltonian form of the first order equation has been investigated. 
  A new class of warped Anti-de Sitter solutions is found, arising as the near-horizon region of various semi-localized brane intersections. The dual gauge theories of AdS in warped spacetimes have reduced supersymmetry, which is pertinent to the study of viable supersymmetric extensions of the Standard Model. In addition, various supergravity solutions are probed via the absorption of various fields. In particular, we calculate exact absorption probabilities which provide finite-energy probes of supergravity solutions away from the conformal limit. Lastly, we discuss how brane world scenarios may arise from the near-horizon region of various p-brane configurations. Motivated by the dual non-commutative gauge theory, it is shown how a background B field mimics a negative four-dimensional cosmological constant, such that in both cases there is a massive four-dimensional graviton. 
  We investigate the transition between singular and non-singular geometries from the vantage point of perturbative field dynamics. In particular, we obtain the closed-form absorption probability for minimally-coupled massless scalars propagating in the background of a heterotic 5-brane on a Taub-NUT instanton. This is an exact calculation for arbitrary incident frequencies. For the singular geometry, the absorption probability vanishes when the frequency is below a certain threshold, and for the non-singular case it vanishes for all frequencies. We discuss the connection between this phenomenon and the behavior of geodesics in this background. We also obtain exact quasinormal modes. 
  We explain how perturbative string theory can be viewed as an exactly renormalizable Weyl invariant quantum mechanics in the worldsheet representation clarifying why string scattering amplitudes are both finite and unambiguously normalized and explaining the origin of UV-IR relations in spacetime. As applications we examine the worldsheet representation of nonperturbative type IB states and of string solitons. We conclude with an analysis of the thermodynamics of a free closed string gas establishing the absence of the Hagedorn phase transition. We show that the 10D heterotic strings share a stable finite temperature ground state with gauge group SO(16)xSO(16). The free energy at the self-dual Kosterlitz-Thouless phase transition is minimized with finite entropy and positive specific heat. The open and closed string gas transitions to a confining long string phase at a temperature at or below the string scale in the presence of an external electric field. 
  It is speculated that the observed universe has a dual representation as renormalization group flow between two conformal fixed points of a three-dimensional Euclidean field theory. The infrared fixed point corresponds to the inflationary phase in the far past. The ultraviolet fixed point corresponds to a de Sitter phase dominated by the cosmological constant indicated in recent astronomical data. The monotonic decrease of the Hubble parameter corresponds to the irreversibility of renormalization group flow. 
  The most general geometrical scenario in which the brane-world program can be implemented is investigated. The basic requirement is that it should be consistent with the confinement of gauge interaction, the existence of quantum states and the embedding in a bulk with arbitrary dimensions, signature and topology.   It is found that the embedding equations are compatible with a wide class of Lagrangians, starting with a modified Einstein-Hilbert Lagrangian as the simplest one, provided minimal boundaries are added to the bulk.  A non-trivial canonical structure is derived, suggesting a canonical quantization of the brane-world geometry relative to the extra dimensions, where the quantum states are set in correspondence with high frequency gravitational waves. It is shown that in the cases of at least six dimensions, there exists a confined gauge field included in the embedding structure. The size of extra dimensions compatible with the embedding is calculated and found to be different from the one derived with product topology. 
  We have studied scalar potentials V of gauged N=8,4,2 supergravities in d=4. Extrema of these potentials may correspond to de Sitter, anti de Sitter and Minkowski vacua. All de Sitter extrema that we have studied correspond to unstable maximum/saddle points with negative curvature |V''|=2V for the fields canonically normalized at the extremum. This is equivalent to the relation |m^2| =|R|/2 = 6H^2 for the tachyonic mass m, the curvature scalar R, and the Hubble constant H. This prevents the use of de Sitter extrema for slow-roll inflation in the early universe, which would require |V''| \ll V . Moreover, in all models that we were able to analyse the potential is unbounded from below. On the other hand, barring the question how realistic such models could be, one can use them for the description of the accelerated expansion of the universe at the present epoch. This is related to a novel possibility of the fast-roll inflation with |V''| > V . We also display some potentials that have flat directions with vanishing cosmological constant, and discuss their possible cosmological implications. 
  We test the consistency of the use of a noncommutative theory description for charged particles in a strong magnetic field, by deriving the induced Chern-Simons (CS) term for an external Abelian gauge field in 2+1 dimensions. In this description, the system is modeled by a noncommutative matter field coupled to a U(1) noncommutative gauge field, related to the original, commutative one, by a Seiberg-Witten transformation. We show that an Abelian CS term for the commutative gauge field is indeed induced, and moreover that it matches the result of previous commutative field theory calculations. 
  We calculate the gravitational potential energy between infinitely long parallel strings with tensions \tau_1 and \tau_2. Classically, it vanishes, but at one loop, we find that the long range gravitational potential energy per unit length is U/L = 24G_N^2\tau_1\tau_2/(5 \pi a^2) + ..., where a is the separation between the strings, G_N is Newton's constant, and we set \hbar = c =1. The ellipses represent terms suppressed by more powers of G_N \tau_i. Typically, massless bulk fields give rise at one loop to a long range potential between p-branes in space-times of dimension p+2+1. The contribution to this potential from bulk scalars is computed for arbitrary p (strings correspond to p=1) and in the case of three-branes its possible relevance for cosmological quintessence is commented on. 
  Connection between the partition function for the 2D sigma model with boundary pertubations and the low energy effective action for massless fields from in the open string theory is discussed. In the non-abelian case with a stack of $N$ D-branes, the terms up to the order of $\alpha'^3$ are found 
  We point out that the leading infrared singular terms in the effective actions of noncommutative gauge theories arising from nonplanar loop diagrams have a natural interpretation in terms of the matrix model (operator) formulation of these theories. In this formulation (for maximal spatial noncommutativity), noncommutative space arises as a configuration of an infinite number of D-particles. We show that the IR singular terms correspond to instantaneous linear potentials between these D-particles resulting from the zero point energies of fluctuations about this background. For theories with fewer fermionic than bosonic degrees of freedom, such as pure noncommutative gauge theory, the potential is attractive and renders the theory unstable. With more fermionic than bosonic degrees of freedom, the potential is repulsive and we argue that the theory is stable, though oddly behaved. 
  We consider interactions of a NS open string with first important states of a NS closed string, i.e., a closed-string tachyon and a graviton, where both ends of the NS open string are attached on a D-brane, and a constant background B field is lying along directions parallel to the D-brane world volume. Contrary to general expectations, there are no constraints on these vertex operators coming from the B field. However, we point out that these vertex operators have singularities at both ends of the NS open string when external momenta take some values. These kinds of singularities essentially come from the Dirichlet boundary conditions along directions transverse to the D-brane world volume. 
  In usual dimensional counting, momentum has dimension one. But a function f(x), when differentiated n times, does not always behave like one with its power smaller by n. This inevitable uncertainty may be essential in general theory of renormalization, including quantum gravity. As an example, we classify possible singularities of a potential for the Schr\"{o}dinger equation, assuming that the potential V has at least one $C^2$ class eigen function. The result crucially depends on the analytic property of the eigen function near its 0 point. 
  We find giant graviton configurations of an M5-brane probe in the D=11 supergravity background generated by a stack of non-threshold (M2,M5) bound states. The M5-brane probe shares three directions with the background and wraps a two-sphere transverse to the bound states. For a particular value of the worldvolume gauge field of the PST formalism, there exist solutions of the equations of motion for which the M5-brane probe behaves as a wave propagating in the (M2,M5) background. We have checked that the probe breaks the supersymmetry of the background exactly as a massless particle moving along the trajectory of its center of mass. 
  A standard Lax representation for the polytropic gas dynamics is derived by exploiting various properties of the Lucas and Fibonacci polynomials. The two infinite sets of conserved charges are derived from this representation and shown to coincide with the ones derived from the known non-standard representation. The same Lax function is shown to also give the standard Lax description for the elastic medium equations. In addition, some results on possible dispersive extensions of such models are presented. 
  The integrable model corresponding to the ${\cal N}=2$ supersymmetric SU(N) gauge theory with matter in the symmetric representation is constructed. It is a spin chain model, whose key feature is a new twisted monodromy condition. 
  In this paper we explicitly show that the various noncompact abelian orbifolds are realized as special limits of parameters in type II (NSNS) Melvin background and its higher dimensional generalizations. As a result the supersymmetric ALE spaces (A-type C^2/Z_N) and nonsupersymmetric orbifolds in type II and type 0 theory are all connected with each other by the exactly marginal deformation. Our results provide new examples of the duality between type II and type 0 string theory. We also discuss the decay of unstable backgrounds in this model which include closed string tachyons. 
  We study the dynamics of an open membrane with a cylindrical topology, in the background of a constant three form. We use the action, due to Bergshoeff, London and Townsend, to study the noncommutativity properties of the boundary string coordinates. The constrained Hamiltonian formalism due to Dirac is used to derive the noncommutativity of coordinates. The chain of constraints is found to be finite for a suitable gauge choice. 
  A cohomological BRST characterization of the Seiberg-Witten (SW) map is given. We prove that the coefficients of the SW map can be identified with elements of the cohomology of the BRST operator modulo a total derivative. As an example, it will be illustrated how the first coefficients of the SW map can be written in terms of the Chern-Simons three form. This suggests a deep topological and geometrical origin of the SW map. The existence of the map for both Abelian and non-Abelian case is discussed. By using a recursive argument and the associativity of the $\star$-product, we shall be able to prove that the Wess-Zumino consistency condition for non-commutative BRST transformations is fulfilled. The recipe of obtaining an explicit solution by use of the homotopy operator is briefly reviewed in the Abelian case. 
  The N=1 Volkov-Akulov model of nonlinear supersymmetry is explicitly related to a vector supermultiplet model with a Fayet-Iliopoulos D term of linear supersymmetry. The physical significance of the results is discussed briefly. 
  We consider flat solutions in the brane background with a massless scalar field appearing in 5D $H^2_{MNPQ}$. Since there exist bulk singularities or arises the divergent 4D Planck mass, we should introduce a compact extra dimension, the size of which is then fixed by brane tension(s) and a bulk cosmological constant. Inspecting scalar perturbations around the flat solutions, we find that the flat solutions are stable vacua from the positive mass spectrum of radion. We show that the massless radion mode is projected out by the boundary condition arising in cutting off the extra dimension. Thus, the fixed extra dimension is not alterable, which is not useful toward a self-tuning of the cosmological constant. 
  Exact renormalization group techniques are applied to mass deformed N=4 supersymmetric Yang-Mills theory, viewed as a regularised N=2 model. The solution of the flow equation, in the local potential approximation, reproduces the one-loop (perturbatively exact) expression for the effective action of N=2 supersymmetric Yang-Mills theory, when the regularising mass, M, reaches the value of the dynamical cutoff. One speculates about the way in which further non-perturbative contributions (instanton effects) may be accounted for. 
  We start with some methodic remarks referring to purely bosonic quantum systems and then explain how corrections to the leading--order quasiclassical result for the fermion--graded partition function Tr{(-1)^F exp(-\beta H)} can be calculated at small \beta. We perform such calculation for certain supersymmetric quantum mechanical systems where such corrections are expected to appear. We consider in particular supersymmetric Yang-Mills theory reduced to (0+1) dimensions and were surprised to find that the correction of order \beta^2 vanishes in this case.   We discuss also a nonstandard N =2 supersymmetric sigma model defined on S^3 and other 3--dimensional conformally flat manifolds and show that the quasiclassical expansion breaks down for this system. 
  We consider a new matrix model based on the simply connected compact exceptional Lie group E6. A matrix Chern-Simons theory is directly derived from the invariant on E6. It is stated that the similar argument as Smolin which derives an effective action of the matrix string type can also be held in our model. An important difference is that our model has twice as many degrees of freedom as Smolin's model has. One way to introduce the cosmological term is the compactification on directions. It is of great interest that the properties of the product space $\Vec{\mathfrak{J}^c} \times \Vec{\mathcal{G}}$, in which the degrees of freedom of our model live, are very similar to those of the physical Hilbert space. 
  We consider NS-NS superstring model with several ``magnetic'' parameters $b_s$ (s=1, ...,N) associated with twists mixing a compact $S^1$ direction with angles in $N$ spatial 2-planes of flat 10-dimensional space. It generalizes the Kaluza-Klein Melvin model which has single parameter $b$. The corresponding U-dual background is a R-R type IIA solution describing an orthogonal intersection of $N$ flux 7-branes. Like the Melvin model, the NS-NS string model with $N$ continuous parameters is explicitly solvable; we present its perturbative spectrum and torus partition function explicitly for the N=2 case. For generic $b_s$ (above some critical values) there are tachyons in the $S^1$ winding sector. A remarkable feature of this model is that while in the Melvin N=1 case all supersymmetry is broken, a fraction of it may be preserved for $N >1$ by making a special choice of the parameters $b_s$. Such solvable NS-NS models may be viewed as continuous-parameter analogs of non-compact orbifold models. They and their U-dual R-R fluxbrane counterparts may have some ``phenomenological'' applications. In particular, in N=3 case one finds a special 1/4 supersymmetric R-R 3-brane background. Putting Dp-branes in flat twisted NS-NS backgrounds leads to world-volume gauge theories with reduced amount of supersymmetry. We also discuss possible ways of evolution of unstable backgrounds towards stable ones. 
  We propose a novel prescription for computing the boundary stress tensor and charges of asymptotically de Sitter (dS) spacetimes from data at early or late time infinity. If there is a holographic dual to dS spaces, defined analogously to the AdS/CFT correspondence, our methods compute the (Euclidean) stress tensor of the dual. We compute the masses of Schwarzschild-de Sitter black holes in four and five dimensions, and the masses and angular momenta of Kerr-de Sitter spaces in three dimensions. All these spaces are less massive than de Sitter, a fact which we use to qualitatively and quantitatively relate de Sitter entropy to the degeneracy of possible dual field theories. Our results in general dimension lead to a conjecture: Any asymptotically de Sitter spacetime with mass greater than de Sitter has a cosmological singularity. Finally, if a dual to de Sitter exists, the trace of our stress tensor computes the RG equation of the dual field theory. Cosmological time evolution corresponds to RG evolution in the dual. The RG evolution of the c function is then related to changes in accessible degrees of freedom in an expanding universe. 
  In this paper we study wrapped brane configurations that give rise to three dimensional pure Yang-Mills theory with eight supercharges. The corresponding supergravity solution is singular and it was conjectured that the singularity is removed by an enhancon mechanism. Instead, we incorporate non-perturbative gauge fields into supergravity and find a smooth solution for these configurations. Along the way we derive a non-abelian supergravity Lagrangian for type IIA on K3 and explicit formulae for the toroidal reduction of the heterotic string with non-abelian gauge fields. We proceed to analyse the duality with Yang-Mills theory and find that the dual background is a monopole configuration in little string theory. 
  We consider the stabilization of string moduli and resulting soft supersymmetry-breaking terms in heterotic string orbifolds. Among the results obtained are: formulae for the scalar interaction soft terms without integrating out the hidden sector gaugino condensate, which reduce to standard expressions in the usual "truncated" limit; an expression for the modular transformation of A-terms; a study of the minima of the scalar potential in the Kaehler modulus direction; and a discussion of the implications for CP violation phenomenology.   Some closely related results have appeared in a recent paper of Khalil, Lebedev and Morris, namely the exact modular invariance of A-terms up to unitary mixing, and the existence of certain complex minima for the moduli. 
  We prove that the equations of motion describing domain walls in a Wess-Zumino theory involving only one chiral matter multiplet can be factorized into first order Bogomol'nyi equations, so that all the topological defects are of the Bogomol'nyi-Prasad-Sommerfield type. 
  We examine the renormalizability problem of spontaneously broken non-Abelian gauge theory on noncommutative spacetime. We show by an explicit analysis of the U(2) case that ultraviolet divergences can be removed at one loop level with the same limited number of renormalization constants as required on commutative spacetime. We thus push forward the efforts towards constructing realistic models of gauge interactions on noncommutative spacetime. 
  We discuss UV/IR mixing effects in non-supersymmetric non-commutative U(N) gauge theories. We show that the singular (non-planar) terms in the 2- and 3-point functions, namely the poles and the logarithms, can be obtained from a manifestly gauge invariant effective action. The action, which involves open Wilson line operators, can be derived from closed strings exchange between two stacks of D-branes. Our concrete example is type 0B string theory and the field theory that lives on a collection of N electric D3-branes. We show that one of the closed string modes that couple to the field theory operator which is responsible for the infrared poles, is the type 0 tachyon. 
  Wigner's quasi-probability distribution function in phase-space is a special (Weyl) representation of the density matrix. It has been useful in describing quantum transport in quantum optics; nuclear physics; decoherence (eg, quantum computing); quantum chaos; "Welcher Weg" discussions; semiclassical limits. It is also of importance in signal processing.   Nevertheless, a remarkable aspect of its internal logic, pioneered by Moyal, has only emerged in the last quarter-century: It furnishes a third, alternative, formulation of Quantum Mechanics, independent of the conventional Hilbert Space, or Path Integral formulations. In this logically complete and self-standing formulation, one need not choose sides--coordinate or momentum space. It works in full phase-space, accommodating the uncertainty principle. This is an introductory overview of the formulation with simple illustrations. 
  We consider the interplay between brane constructions and type IIA, IIB or M-theory geometries on Calabi-Yau (CY) and G_2 holonomy manifolds. This is related to N=1 (and N=2) gauge theories in four dimensions. We first discuss simple geometric transitions corresponding to brane set ups involving orthogonal (or parallel) Neveu-Schwarz branes that approach each other. This is related to confinement and Seiberg duality in SQCD. In particular, we argue that in type IIA, a CP^1 of singularities and one unit of Ramond-Ramond (RR) flux is dual to a D6 brane wrapped on a Lens space, describing the UV and IR of N=1 Super-Yang-Mills (SYM), respectively. Also, in the large N_c duality that relates D6 branes on S^3 to an S^2 with RR flux, we implement the presence of $N_f$ flavors of quarks. We then compactify M-theory on T^*(S^3) X S^1)/Z_2 and observe that one phase describes SO(4) SYM in the UV and two others describe confinement. Moreover, we consider compact 7-spaces (CY X S^1)/Z_2. We describe transitions where disconnected S^3's approach and connect each other before they vanish. These effects correspond to non-Abelian Higgs mechanism and confinement. The similar transitions involving S^2 X S^1's are also considered. Finally, we present transitions at finite distance in moduli space, where the first Betti number b_1 of 3-cycles of singularities changes. 
  We consider the open XXZ quantum spin chain with nondiagonal boundary terms. For bulk anisotropy value \eta = i \pi/(p+1), p= 1, 2, ..., we propose an exact (p+1)-order functional relation for the transfer matrix, which implies Bethe-Ansatz-like equations for the corresponding eigenvalues. The key observation is that the fused spin-(p+1)/2 transfer matrix can be expressed in terms of a lower-spin transfer matrix, resulting in the truncation of the fusion hierarchy. 
  We discuss the asymptotic dynamical evolution of spatially homogeneous brane-world cosmological models close to the initial singularity. We find that generically the cosmological singularity is isotropic in Bianchi type IX brane-world models and consequently these models do not exhibit Mixmaster or chaotic-like behaviour close to the initial singularity. We argue that this is typical of more general cosmological models in the brane-world scenario. In particular, we show that an isotropic singularity is a past-attractor in all orthogonal Bianchi models and is a local past-attractor in a class of inhomogeneous brane-world models. 
  The so-called ``brick-wall model'' is a semi-classical approach that has been used to explain black hole entropy in terms of thermal matter fields. Here, we apply the brick-wall formalism to thermal bulk fields in a Randall-Sundrum brane world scenario. In this case, the black hole entity is really a string-like object in the anti-de Sitter bulk, while appearing as a Schwarzchild black hole to observers living on the brane. In spite of these exotic circumstances, we establish that the Bekenstein-Hawking entropy law is preserved. Although a similar calculation was recently considered in the literature, this prior work invoked a simplifying assumption (which we avoid) that can not be adequately justified. 
  In the first part of this work we consider an unstable non-BPS Dp-\bar{Dp}-brane pair in Type II superstring theory. Turning on a background NS-NS B-field (constant and nonzero along two spatial directions), we show that the tachyon responsible for the unstability has a complex GMS solitonic solution, which is interpreted as the low energy remnant of the resulting D(p-2)-brane. In the second part, we apply these results to construct the noncommutative soliton analogous of Witten's superconducting string. This is done by considering the complex GMS soliton arising from the D3-\bar{D3}-brane annihilation in Type IIB superstring theory. In the presence of left-handed fermions, we apply the Weyl-Wigner-Moyal correspondence and the bosonization technique to show that this object behaves like a superconducting wire. 
  We analyse the large momentum behaviour of 4-dimensional massive euclidean Phi-4-theory using the flow equations of Wilson's renormalization group. The flow equations give access to a simple inductive proof of perturbative renormalizability. By sharpening the induction hypothesis we prove new and, as it seems, close to optimal bounds on the large momentum behaviour of the correlation functions. The bounds are related to what is generally called Weinberg's theorem. 
  We investigate instanton expansions of partition functions of several toric E-string models using local mirror symmetry and elliptic modular forms. We also develop a method to obtain the Seiberg--Witten curve of E-string with arbitrary Wilson lines with the help of elliptic functions. 
  For the noncommutative torus ${\cal T}$, in case of the N.C. parameter$\theta = \frac{Z}{n}$ and the area of ${\cal T}$ is an integer, we construct the basis of Hilbert space ${\cal H}_n$ in terms of $\theta$functions of the positions $z_i$ of $n$ solitons. The loop wrapping a$the torus generates the algebra ${\cal A}_n$. We show that ${\cal A}_$isomorphic to the $Z_n \times Z_n$ Heisenberg group on $\theta$ funct$We find the explicit form for the local operators, which is the gener$$g$ of an elliptic $su(n)$, and transforms covariantly by the global gauge transformation of the Wilson loop in ${\cal A}_n$. By acting on ${\cal H}_n$ we establish the isomorphism of ${\cal A}_n$ and $g$. Th$is easy to give the projection operators corresponding to the soliton$the ABS construction for generating solitons. We embed this $g$ into $$L$-matrix of the elliptic Gaudin and C.M. models to give the dynamic$For $\theta$ generic case, we introduce the crossing parameter $\eta$ related with $\theta$ and the modulus of ${\cal T}$. The dynamics of solitons is determined by the transfer matrix $T$ of the elliptic qua$group ${\cal A}_{\tau, \eta}$, equivalently by the elliptic Ruijsenaa$operators $M$. The eigenfunctions of $T$ found by Bethe ansatz appear$be twisted by $\eta$. 
  We study the three dimensional Schwarzschild-de Sitter (SdS$_3$) black hole which corresponds essentially to a conical defect. We compute the mass of the SdS$_3$ black hole from the correct definition of the mass in asymptotically de Sitter space. Then we clarify the relation between the mass, entropy and temperature for this black hole without any ambiguity. Also we establish the SdS$_3$/CFT$_2$-correspondence for the entropy by applying the Cardy formula to a CFT with a central charge $c=3\ell/2G_3$. Finally we discuss the entropy bounds for the SdS$_3$ black hole. 
  We examine string field algebra which is generated by star product in Witten's string field theory including ghost part. We perform calculations using oscillator representation consistently. We construct wedge like states in ghost part and investigate algebras among them. As a by-product we have obtained some solutions of vacuum string field theory. We also discuss some problems about identity state. We hope these calculations will be useful for further investigation of Witten type string field theory. 
  In these talks, I discuss a few selected topics in integrable models that are of interest from various points of view. Some open questions are also described. 
  We consider irreducible cyclic representations of the algebra of monodromy matrices corresponding to the R-matrix of the six-vertex model. In roots of unity the Baxter Q-operator can be represented as a trace of a tensor product of L-operators corresponding to one of these cyclic representations and satisfies the TQ-equation. We find a new algebraic structure generated by these L-operators and, as a consequence, by the Q-operators. 
  The present paper is based on the modified part of the review "Random Dynamics and Multiple Point Model" by L.V.Laperashvili, H.B.Nielsen, D.A.Ryzhikh and N.Stillits, in preparation for publication in Russian, which contains the results of our joint activity with H.B.Nielsen concerning the investigations of phase transitions in gauge theories. In this review we have presented the main ideas of the Nielsen's Random Dynamics (RD) and his achievements (with co-authors) in the Anti-Grand Unification Theory (AGUT) and Multiple Point Model (MPM). We have considered also the theory of Scale Relativity (SR) by L.Nottale, which has a lot in common with RD: both theories lead to the discreteness of our space-time, giving rise to the new description of physics at very small distances. In this paper we have demonstrated the possibility of [SU(5)]$^3$ SUSY unification with superparticles of masses $M\approx 10^{18.3}$ GeV and calculated its critical point -- critical value of the inverse finestucture constant -- at $\alpha_{5,crit}^{-1} = \alpha_5^{-1}(\mu_{Pl})\approx 34.0$ (close to $\alpha_{GUT}^{-1}\approx 34.4$) with a hope that such an unified theory approaches the (multi)critical point at the Planck scale. 
  Since the particles such as molecules, atoms and nuclei are composite particles, it is important to recognize that physics must be invariant for the composite particles and their constituent particles, this requirement is called particle invariance in this paper. But difficulties arise immediately because for fermion we use Dirac equation, for meson we use Klein-Gordon equation and for classical particle we use Newtonian mechanics, while the connections between these equations are quite indirect. Thus if the particle invariance is held in physics, i.e., only one physical formalism exists for any particle, we can expect to find out the differences between these equations by employing the particle invariance. As the results, several new relationships between them are found, the most important result is that the obstacles that cluttered the path from classical mechanics to quantum mechanics are found, it becomes possible to derive the quantum wave equations from relativistic mechanics after the obstacles are removed. An improved model is proposed to gain a better understanding on elementary particle interactions. This approach offers enormous advantages, not only for giving the first physically reasonable interpretation of quantum mechanics, but also for improving quark model. 
  In a toy model of gauge and gravitational interactions in $D \ge 4$ dimensions, endowed with an invariant UV cut-off $\Lambda$, and containing a large number $N$ of non-self-interacting matter species, the physical gauge and gravitational couplings at the cut-off, $\alpha_g \equiv g^2 \Lambda^{D-4}$ and $\alpha_G \equiv G_N \Lambda^{D-2}$, are shown to be bounded by appropriate powers of ${1\over N}$. This implies that the infinite-bare-coupling (so-called compositeness) limit of these theories is smooth, and can even resemble our world. We argue that such a result, when extended to more realistic situations, can help avoid large-N violations of entropy bounds, solve the dilaton stabilization and GUT-scale problems in superstring theory, and provide a new possible candidate for quintessence. 
  We investigate the Casimir effect at finite temperature for a charged scalar field in the presence of an external uniform and constant magnetic field, perpendicular to the Casimir plates. We have used a boundary condition characterized by a deformation parameter $\theta$; for $\theta=0$ we have a periodic condition and for $\theta=\pi$, an antiperiodic one, for intermediate values, we have a deformation. The temperature was introduced using the imaginary time formalism and both the lagrangian and free energy were obtained from Schwinger proper time method for computing the effective action. We also computed the permeability and its asymptotic expressions for low and high temperatures. 
  We develop a superspace Noether procedure for supersymmetric field theories in 4-dimensions for which an off-shell formulation in ordinary superspace exists. In this way we obtain an elegant and compact derivation of the various supercurrents in these theories. We then apply this formalism to compute the central charges for a variety of effective actions. As a by-product we also obtain a simple derivation of the anomalous superconformal Ward-identity in N=2 Yang-Mills theory. The connection with linearized supergravity is also discussed. 
  We consider five dimensional theories compactified on the orbifold S^1/Z_2 and prove that spontaneous local supersymmetry breaking by Wilson lines and by the Scherk-Schwarz mechanism are equivalent. Wilson breaking is triggered by the SU(2)_R symmetry which is gauged in off-shell N=2 supergravity by auxiliary fields. The super-Higgs mechanism disposes of the would-be Goldstinos which are absorbed by the gravitinos to become massive. The breaking survives in the flat limit, where we decouple all gravitational interactions, and the theory becomes softly broken global supersymmetry. 
  We formulate two-dimensional rational conformal field theory as a natural generalization of two-dimensional lattice topological field theory. To this end we lift various structures from complex vector spaces to modular tensor categories. The central ingredient is a special Frobenius algebra object A in the modular category that encodes the Moore-Seiberg data of the underlying chiral CFT. Just like for lattice TFTs, this algebra is itself not an observable quantity. Rather, Morita equivalent algebras give rise to equivalent theories. Morita equivalence also allows for a simple understanding of T-duality.   We present a construction of correlators, based on a triangulation of the world sheet, that generalizes the one in lattice TFTs. These correlators are modular invariant and satisfy factorization rules. The construction works for arbitrary orientable world sheets, in particular for surfaces with boundary. Boundary conditions correspond to representations of the algebra A. The partition functions on the torus and on the annulus provide modular invariants and NIM-reps of the fusion rules, respectively. 
  A one-loop renormalization group (RG) analysis is performed for noncommutative Landau-Ginsburg theory in an arbitrary dimension. We adopt a modern version of the Wilsonian RG approach, in which a shell integration in momentum space bypasses the potential IR singularities due to UV-IR mixing. The momentum-dependent trigonometric factors in interaction vertices, characteristic of noncommutative geometry, are marginal under RG transformations, and their marginality is preserved at one loop. A negative $\Theta$-dependent anomalous dimension is discovered as a novel effect of the UV-IR mixing. We also found a noncommutative Wilson-Fisher (NCWF) fixed point in less than four dimensions. At large noncommutativity, a momentum space instability is induced by quantum fluctuations, and a consequential first-order phase transition is identified together with a Lifshitz point in the phase diagram. In the vicinity of the Lifshitz point, we introduce two critical exponents $\nu_m$ and $\beta_k$, whose values are determined to be 1/4 and 1/2, respectively, at mean-field level. 
  A new mechanism for T and CPT violation is reviewed, which relies on chiral fermions, gauge interactions and nontrivial spacetime topology. Also discussed are the possible effects on the propagation of electromagnetic waves in vacuo, in particular for the cosmic microwave background radiation. 
  Starting from the boundary CFT definition for the D-branes in vacuum string field theory (VSFT) given in hep-th/0105168, we derive the oscillator expression for the D24-brane solution in the VSFT on D25-brane. We show that the state takes the form of a squeezed state, similar to the one found directly in terms of the oscillators and reported in hep-th/0102112. Both the solutions are actually one parameter families of solutions. We also find numerical evidence that at least for moderately large values of the parameter $(b)$ in the oscillator construction the two families of solutions are same under a suitable redefinition of the parameter. Finally we generalize the method to computing the oscillator expression for a D-brane solution with constant gauge field strength turned on along the world volume. 
  Under the hypothesis that the cosmological constant vanishes in the true ground state with lowest possible energy density, we argue that the observed small but finite vacuum-like energy density can be explained if we consider a theory with two or more degenerate perturbative vacua, which are unstable due to quantum tunneling, and if we still live in one of such states. An example is given making use of the topological vacua in non-Abelian gauge theories. 
  The necessary and sufficient conditions are established for the second-class constraint surface to be (an almost) K\"ahler manifold. The deformation quantisation for such systems is scetched resulting in the Wick-type symbols for the respective Dirac brackets. 
  Four-dimensional Manin triples and Drinfeld doubles are classified and corresponding two-dimensional Poisson-Lie T-dual sigma models on them are constructed. The simplest example of a Drinfeld double allowing decomposition into two nontrivially different Manin triples is presented. 
  A superfield algorithm for master actions of a class of gauge field theories including topological ones in arbitrary dimensions is presented generalizing a previous treatment in two dimensions. General forms for master actions in superspace are given, and possible theories are determined by means of a ghost number prescription and the master equations. The resulting master actions determine the original actions together with their gauge invariances. Generalized Poisson sigma models in arbitrary dimensions are constructed by means of this algorithm, and other applications in low dimensions are given including the Chern-Simon model. 
  We clarify the relation between orbifold and interval pictures in 5d brane worlds. We establish this correspondence for Z_2-even and Z_2-odd orbifold fields. In the interval picture Gibbons-Hawking terms are necessary to fulfill consistency conditions. We show how the brane world consistency conditions arise in the interval picture. We apply the procedure to the situation where the transverse dimension is terminated by naked singularities. In particular, we find the boundary terms needed when the naive vacuum action is infinite. 
  It is by now well established that divergences of the on-shell action for asymptotically AdS solutions can be cancelled by adding covariant local boundary counterterms to the action. Here we show that although one can still renormalise the action for asymptotically $AdS_p \times S^q$ solutions using local boundary counterterms the counterterm action is not covariant since the conformal boundary is degenerate. Any given counterterm action is defined with respect to specific coordinate frame and gauge choices. 
  We study a Randall-Sundrum model modified by a Gauss-Bonnet interaction term. We consider, in particular, a Friedmann-Robertson-Walker metric on the brane and analyse the resulting cosmological scenario. It is shown that the usual Friedmann equations are recovered on the brane. The equation of state relating the enery density and the pressure is uniquely determined by the matching conditions. A cosmological solution with negative pressure is found. 
  We introduce, by means of the Brans-Dicke scalar field, space-time fluctuations at scale comparable to Planck length near the event horizon of a black hole and examine their dramatic effects. 
  We show that the correctly evaluated effective Lagrangian should include short-distance interaction terms which have been avoided under the protection of usual regularization and must be properly identified and reinstated if regularization is to be removed. They have special physical and mathematical significance as well as restoring gauge invariance and suppressing divergence in the effective Lagrangian. The rich structure of the short-distance interaction terms can open up challenging opportunities where the conventional regularization with rigid structure is unavailable and inappropriate. It becomes clear that gauge invariance is preserved with or without regularization and therefore there is no Lorentz-Violating Chern-Simons term in QED. 
  We argue that two four-dimensional strongly coupled superconformal field theories, on the Higgs branch in certain large N limits, become respectively (2,0) theory and (1,1) little string theory in six dimensions. We identify the spectrum of states responsible for the generation of the two extra dimensions and string winding modes. We establish the equivalence using orbifold realizations of the field theories and exploiting string dualities. We also speculate on deconstructions of M-theory. 
  We propose a constraint on the noncommutative gauge theory with U(N) gauge group which gives rise to a noncommutative version of the SU(N) gauge group. The baryon operator is also constructed. 
  We describe the construction of the quantum deformed affine Lie algebras using the vertex operators in the free field theory. We prove the Serre relations for the quantum deformed Borel subalgebras of affine algebras, namely the case of $ \hat{\it sl}_{2}$ is considered in details. We provide the formulas for generators of affine algebra. 
  If spacetime possesses extra dimensions of size and curvature radii much larger than the Planck or string scales, the dynamics of these extra dimensions should be governed by classical general relativity. We argue that in general relativity, it is highly nontrivial to obtain solutions where the extra dimensions are static and are dynamically stable to small perturbations. We also illustrate that intuition on equilibrium and stability built up from non-gravitational physics can be highly misleading. For all static, homogeneous solutions satisfying the null energy condition, we show that the Ricci curvature of space must be nonnegative in all directions. Much of our analysis focuses on a class of spacetime models where space consists of a product of homogeneous and isotropic geometries. A dimensional reduction of these models is performed, and their stability to perturbations that preserve the spatial symmetries is analyzed. We conclude that the only physically realistic examples of classically stabilized large extra dimensions are those in which the extra-dimensional manifold is positively curved. 
  Nodland Ralston (PRL,1997) investigated the cosmological anisotropy of electromagnetic fields.In this paper we show that it is possible obtain a torsion correction to Nodland-Ralston action starting from the massive Proca electrodynamics in Riemannian spacetime and performing the minimal coupling with torsion.We end up with an action which contains the Nodland Ralston action without breaking the gauge invariance.This mechanism however gives a photon a mass generated by the nonlinear torsion terms.The torsion vector is along the cosmic rotation axis and interacts with the massive photon.This method which breaks conformal invariance allow us to determine a primordial torsion of the order $10^{-29}eV$ from the well-known photon mass limits. 
  A recent notion in theoretical physics is that not all quantum theories arise from quantising a classical system. Also, a given quantum model may possess more than just one classical limit. These facts find strong evidence in string duality and M-theory, and it has been suggested that they should also have a counterpart in quantum mechanics. In view of these developments we propose "dequantisation", a mechanism to render a quantum theory classical. Specifically, we present a geometric procedure to "dequantise" a given quantum mechanics (regardless of its classical origin, if any) to possibly different classical limits, whose quantisation gives back the original quantum theory. The standard classical limit $\hbar\to 0$ arises as a particular case of our approach. 
  The effect of certain simple backgrounds on the Hagedorn temperature in theories of closed strings is examined. The background of interest are constant Neveu-Schwarz $B$-fields, a constant offset of the space-time metric and a compactified spatial dimension. We find that the Hagedorn temperature of string theory depends on the parameters of the background. We comment on an interesting non-extensive feature of the Hagedorn transition, including a subtlety with decoupling of closed strings in the NCOS limit of open string theory and on the large radius limit of discrete light-cone quantized closed strings. 
  We consider the phase structure of a pure compact U(1) gauge theory in four dimensions at finite temperature by treating this system as a perturbative deformation of the topological model. Phases of a gauge theory can be investigated from the phase structure of the topological model. The thermal pressure of the topological model has been calculated, from which its phase structure can be derived. We have obtained phases of a compact U(1) gauge theory. Moreover, the critical-line equation has been explicitly evaluated. 
  The field-theoretic one-loop effective action in a static scalar background depending nontrivially on a single spatial coordinate is related, in the proper-time formalism, to the trace of the evolution kernel (or heat kernel) for an appropriate, one dimensional, quantum-mechanical Hamiltonian. We describe a recursive procedure applicable to these traces for shape-invariant Hamiltonians, resolving subtleties from the continuum mode contributions by utilizing the expression for the regularized Witten index. For some cases which include those of domain-wall-type scalar backgrounds, our recursive procedure yields the full expression for the scalar or fermion one-loop effective action in both (1+1) and (3+1)-dimensions. 
  We draw an analogy between the chiral extrapolation of lattice QCD calculations from large to small quark masses and the interpolation between the large mass (weak field) and small mass (strong field) limits of the Euler--Heisenberg QED effective action. In the latter case, where the exact answer is known, a simple extrapolation of a form analogous to those proposed for the QCD applications is shown to be surprisingly accurate over the entire parameter range. 
  Recently we have studied the Bloch effective Hamiltonian approach to bound states in 2+1 dimensional gauge theories. Numerical calculations were carried out to investigate the vanishing energy denominator problem. In this work we study similarity renormalization approach to the same problem. By performing analytical calculations with a step function form for the similarity factor, we show that in addition to curing the vanishing energy denominator problem, similarity approach generates linear confining interaction for large transverse separations. However, for large longitudinal separations, the generated interaction grows only as the square root of the longitudinal separation and hence produces violations of rotational symmetry in the spectrum. We carry out numerical studies in the G{\l}azek-Wilson and Wegner formalisms and present low lying eigenvalues and wavefunctions. We investigate the sensitivity of the spectra to various parameterizations of the similarity factor and other parameters of the effective Hamiltonian, especially the scale $\sigma$. Our results illustrate the need for higher order calculations of the effective Hamiltonian in the similarity renormalization scheme. 
  We examine Dashen's phenomenon in the Leutwyler--Smilga regime of QCD with any number of colors and quarks in either the fundamental or adjoint representations of the gauge group. In this limit, the theories only depend on simple combinations of quark masses, volume, chiral condensate and vacuum angle. Based upon this observation, we derive simple expressions for the chiral condensate and the topological density and show that they are in fact related. By examining the zeros of the various partition functions, we elucidate the mechanism leading to Dashen's phenomena in QCD. 
  Topological field theory in three dimensions provides a powerful tool to construct correlation functions and to describe boundary conditions in two-dimensional conformal field theories. 
  In this Letter we establish a relationship between symmetric SU(2) Yang--Mills instantons and metrics with Spin(7)-holonomy. Our method is based on a slight extension of that of Bryant and Salamon developed to construct explicit manifolds with special holonomies in 1989.    More precisely, we prove that making use of symmetric SU(2) Yang--Mills instantons on Riemannian spin-manifolds, we can construct metrics on the chiral spinor bundle whose holonomies are within Spin(7). Moreover if the resulting space is connected, simply connected and complete, the holonomy coincides with Spin(7).    The basic example is the metric constructed on the chiral spinor bundle of the round four-sphere by using a generic SU(2)-instanton of unit action; hence it is a five-parameter deformation of the Bryant--Salamon example, also found by Gibbons, Page and Pope. 
  The finite temperature parity-violating contributions to the polarization tensor are computed at one loop in a system without fermions. The system studied is a Maxwell-Chern-Simons-Higgs system in the broken phase, for which the parity-violating terms are well known at zero temperature. At nonzero temperature the static and long-wavelength limits of the parity violating terms have very different structure, and involve non-analytic log terms depending on the various mass scales. At high temperature the boson loop contribution to the Chern-Simons term goes like T in the static limit and like T log T in the long-wavelength limit, in contrast to the fermion loop contribution which behaves like 1/T in the static limit and like log T/T in the long wavelength limit. 
  We consider the Maldacena conjecture applied to the near horizon geometry of a D1-brane in the supergravity approximation and present numerical results of a test of the conjecture against the boundary field theory calculation using supersymmetric discrete light-cone quantization (SDLCQ). We present numerical results with approximately 1000 times as many states as we previously considered. These results support the Maldacena conjecture and are within 10-15% of the predicted numerical results in some regions. Our results are still not sufficient to demonstrate convergence, and, therefore, cannot be considered to a numerical proof of the conjecture. We present a method for using a ``flavor'' symmetry to greatly reduce the size of the basis and discuss a numerical method that we use which is particularly well suited for this type of matrix element calculation. 
  Large extra dimensions provide interesting extensions of our parameter space for gravitational theories. There exist now brane models which can perfectly reproduce standard four-dimensional Friedmann cosmology. These models are not motivated by observations, but they can be helpful in developing new approaches to the dimensionality problem in string theory.   I describe the embedding of standard Friedmann cosmology in the DGP model, and in particular the realization of our current (dust+Lambda)-dominated universe in this model. 
  Certain linear objects, termed physical lines, are considered, and initial assumptions concerning their properties are introduced. A physical line in the form of a circle is called a \emph{$J$-string}. It is assumed that a $J$-string has an angular momentum whose value is $\hbar$. It is then established that a $J$-string of radius $R$ possesses a mass $m_J$, equal to $h/2\pi c R$, a corresponding energy, as well as a charge $q_J$, where $q_J = (hc/2\pi)^{1/2}$. It is shown that this physical curve consists of indivisible line segments of length $\ell_\Delta = 2\pi(hG/c^3)^{1/2}$, where $c$ is the speed of light and $G$ is the gravitational constant. Quantum features of $J$-strings are studied. Based upon investigation of the properties and characteristics of $J$-strings, a method is developed for the computation of the Planck length and mass $(\ell^*_P, m^*_P)$. The values of $\ell^*_P$ and $m^*_P$ are computed according to the resulting formulae (and given in the paper); these values differ from the currently accepted ones. 
  We investigate fluxbrane solutions to the Einstein-antisymmetric form-dilaton theory in arbitrary space-time dimensions for a transverse space of cylindrical topology $S^k\times R^n$, corresponding to smeared and unsmeared solutions. A master equation for a single metric function is derived. This is a non-linear second-order ordinary differential equation admitting an analytic solution, singular at the origin, which serves as an attractor for globally regular solutions, whose existence is demonstrated numerically. For all fluxbranes of different levels of smearing the metric function diverges at infinity as the same power of the radial coordinate except for the maximally smeared case, where a global solution is known in closed form and can be obtained algebraically using U-duality. The particular cases of F6 and F3 fluxbranes in D=11 supergravity and fluxbranes in IIA, IIB supergravities are discussed. 
  A new type of nonlocal currents (quasi-particles), which we call twisted parafermions, and its corresponding twisted $Z$-algebra are found. The system consists of one spin-1 bosonic field and six nonlocal fields of fractional spins. Jacobi-type identities for the twisted parafermions are derived, and a new conformal field theory is constructed from these currents. As an application, a parafermionic representation of the twisted affine current algebra $A^{(2)}_2$ is given. 
  The metric determines the casual structure of spacetime, but in quantum gravity it is also a dynamical field which must be quantized using this causal structure; this is the famous problem of time. A radical resolution of this paradox is proposed: remove the concept of space-like separation entirely. This can be done by describing all fields in terms on p-jets, living on the observer's trajectory; all points on the trajectory have time-like separations. Such a description is necessary to construct well-defined representations the N-dimensional generalization of the Virasoro algebra Vir(N); this is the natural quantum extension of vect(N), which is the correct symmetry algebra of general relativity in N dimensions. The limit p -> oo, necessary for a field theory interpretation, only exists if N = 4 and there are three fermions for every two bosons, a relation that is satisfied in the standard model coupled to gravity. 
  A long-standing problem of theoretical physics is the exceptionally small value of the cosmological constant $\Lambda \sim 10^{-120}$ measured in natural Planckian units. Here we derive this tiny number from a toroidal string cosmology based on closed strings. In this picture the dark energy arises from the correlation between momentum and winding modes that for short distances has an exponential fall-off with increasing values of the momenta.The freeze-out by the expansion of the background universe for these transplanckian modes may be interpreted as a frozen condensate of the closed-string modes in the three non-compactified spatial dimensions. 
  The hybrid formalism for the superstring is used to compute one-loop amplitudes with an arbitrary number of external d=4 supergravity states. These one-loop N-point amplitudes are expressed as Koba-Nielsen-like formulas with manifest d=4 supersymmetry. 
  Certain aspects of three dimensional asymptotically de Sitter spaces are studied, with emphasis on the mapping between gravity observables and the representation of the conformal symmetry of the boundary. In particular, we show that non-real conformal weights for the boundary theory correspond to space-times that have non-zero angular momentum. Some miscellaneous results on the role of the holonomies and isometry groups are also presented. 
  We classify generalised supersymmetric fluxbranes in type II string theory obtained as Kaluza-Klein reductions of the Minkowski space vacuum of eleven-dimensional supergravity. We obtain two families of smooth solutions which contains all the known solutions, new solutions called nullbranes, and solutions interpolating between them. We explicitly construct all the solutions and we study the U-duality orbits of some of these backgrounds. 
  We construct Calabi-Yau geometries with wrapped D6 branes which realize ${\cal N}=1$ supersymmetric $A_r$ quiver theories, and study the corresponding geometric transitions. This also yields new large $N$ dualities for topological strings generalizing topological strings/large $N$ Chern-Simons duality. Lifting up to M-theory yields smooth quantum geometric transitions without branes or fluxes, in the context of $G_2$ holonomy manifolds. In addition we construct a linear sigma model realization which is relevant for the worldsheet theory of superstrings propagating in local manifolds with $G_2$ holonomy, and obtain mirror geometries for this class of supersymmetric sigma models. 
  We study the dynamics of fuzzy two-spheres in a matrix model which represents string theory in the presence of RR flux. We analyze the stability of known static solutions of such a theory which contain commuting matrices and SU(2) representations. We find that irreducible as well as reducible representations are stable. Since the latter are of higher energy, this stability poses a puzzle. We resolve this puzzle by noting that reducible representations have marginal directions corresponding to non-spherical deformations. We obtain new static solutions by turning on these marginal deformations. These solutions now have instability or tachyonic directions. We discuss condensation of these tachyons which correspond to classical trajectories interpolating from multiple, small fuzzy spheres to a single, large sphere. We briefly discuss spatially independent configurations of a D3/D5 system described by the same matrix model which now possesses a supergravity dual. 
  We compute the leading low-energy term in the planar part of the 2-loop contribution to the effective action of $\N=4$ SYM theory in 4 dimensions, assuming that the gauge group $SU(N+1)$ is broken to $SU(N) x U(1)$ by a constant scalar background $X$. While the leading 1-loop correction is the familiar $c_1 F^4/|X|^4$ term, the 2-loop expression starts with $c_2 F^6/|X|^8$. The 1-loop constant $c_1$ is known to be equal to the coefficient of the $F^4$ term in the Born-Infeld action for a probe D3-brane separated by distance $|X|$ from a large number $N$ of coincident D3-branes. We show that the same is true also for the 2-loop constant $c_2$: it matches the coefficient of the $F^6$ term in the D3-brane probe action. In the context of the AdS/CFT correspondence, this agreement suggests a non-renormalization of the coefficient of the $F^6$ term beyond two loops. Thus the result of hep-th/9706072 about the agreement between the $v^6$ term in the D0-brane supergravity interaction potential and the corresponding 2-loop term in the 1+0 dimensional reduction of $\N=4$ SYM theory has indeed a direct generalization to 1+3 dimensions, as conjectured earlier in hep-th/9709087. We also discuss the issue of gauge theory -- supergravity correspondence for higher order ($F^8$, etc.) terms. 
  We give a general analysis of OPEs of 1/2 BPS superfield operators for the $D=3,4,5,6$ superconformal algebras OSp(8/4,R), PSU(2,2), F${}_4$ and OSp($8^*/4$) which underlie maximal AdS supergravity in $4\leq D+1\leq 7$. \\ The corresponding three-point functions can be formally factorized in a way similar to the decomposition of a generic superconformal UIR into a product of supersingletons. This allows for a simple derivation of branching rules for primary superfields. The operators of protected conformal dimension which may appear in the OPE are classified and are shown to be either 1/2 or 1/4 BPS, or semishort. As an application, we discuss the "non-renormalization" of extremal $n$-point correlators. 
  We examine in details Friedmann-Robertson-Walker models in 2+1 dimensions in order to investigate the cosmic holographic principle suggested by Fischler and Susskind. Our results are rigorously derived differing from the previous one found by Wang and Abdalla. We discuss the erroneous assumptions done in this work. The matter content of the models is composed of a perfect fluid, with a $\gamma$-law equation of state. We found that closed universes satisfy the holographic principle only for exotic matter with a negative pressure. We also analyze the case of a collapsing flat universe. 
  A path-integral approach for delta-function potentials is presented. Particular attention is paid to the two-dimensional case, which illustrates the realization of a quantum anomaly for a scale invariant problem in quantum mechanics. Our treatment is based on an infinite summation of perturbation theory that captures the nonperturbative nature of the delta-function bound state. The well-known singular character of the two-dimensional delta-function potential is dealt with by considering the renormalized path integral resulting from a variety of schemes: dimensional, momentum-cutoff, and real-space regularization. Moreover, compatibility of the bound-state and scattering sectors is shown. 
  Using the point-splitting regularisation, we calculate the axial anomaly in an arbitrary even dimensional Non-Commutative (NC) field theory. Our result is (star) gauge invariant in its {\it unintegrated} form, to the leading order in the NC parameter.  Exploiting the Seiberg Witten map, this result gets transformed to the familiar Adler-Bell-Jackiw anomaly in ordinary space-time. Furthermore, using this map, we derive an expression for the unintegrated axial anomaly for constant fields in NC space-time, that is valid to all finite orders of the NC parameter. 
  We consider Schwinger type processes involving the creation of the charge and monopole pairs in the external fields and propose interpretation of these processes via corresponding brane configurations in Type IIB string theory. We suggest simple description of some new interesting nonperturbative processes like monopole/dyon transitions in the electric field and W-boson decay in the magnetic field using the brane language. Nonperturbative pair production in the strong coupling regime using the AdS/CFT correspondence is studied. The treatment of the similar processes in the noncommutative theories when noncommutativity is traded for the background fields is presented and the possible role of the critical magnetic field which is S-dual to the critical electric field is discussed. 
  Plan of this report is given below: 1. Motivation from Physical and Mathematical Point of View; 2. Differential Calculi on Finite Groups; 3. Metrics; 4. Lagrangian Field Theory and Symplectic Structure; 5. Scalar Field Theory and Spectral of Finite Groups. 
  We compute the effective action of QED at one loop order for an electric field which points in the $\hat{z}$ direction and depends arbitrarily upon the light cone time coordinate, $x^+ = (x^0 + x^3)/\sqrt{2}$. This calculation generalizes Schwinger's formula for the vacuum persistence probability in the presence of a constant electric field. 
  In contrast to what happens for ferromagnets, the lattice structure participates in a crucial way to determine existence and type of critical behaviour in antiferromagnetic systems. It is an interesting question to investigate how the memory of the lattice survives in the field theory describing a scaling antiferromagnet. We discuss this issue for the square lattice three-state Potts model, whose scaling limit as T->0 is argued to be described exactly by the sine-Gordon field theory at a specific value of the coupling. The solution of the scaling ferromagnetic case is recalled for comparison. The field theory describing the crossover from antiferromagnetic to ferromagnetic behaviour is also introduced. 
  The formalism of Quantum Mechanics is based by definition on conserving probabilities and thus there is no room for the description of dissipative systems in Quantum Mechanics. The treatment of time-irreversible evolution (the arrow of time) is therefore ruled out by definition in Quantum Mechanics. In Quantum Field Theory it is, however, possible to describe time-irreversible evolution by resorting to the existence of infinitely many unitarily inequivalent representations of the canonical commutation relations (ccr). In this paper I review such a result by discussing the canonical quantization of the damped harmonic oscillator (dho), a prototype of dissipative systems. The irreversibility of time evolution is expressed as tunneling among the unitarily inequivalent representations. The exact action for the dho is derived in the path integral formalism of the quantum Brownian motion developed by Schwinger and by Feynman and Vernon. The doubling of the phase-space degrees of freedom for dissipative systems is related to quantum noise effects. Finally, the role of dissipation in the quantum model of the brain and the occurrence that the cosmological arrow of time, the thermodynamical one and the biological one point into the same direction are shortly mentioned. 
  To study noncommutativity properties of the open string with constant B-field we construct a mechanical action which reproduces classical dynamics of the string sector under consideration. It allows one to apply the Dirac quantization procedure for constrained systems in a direct and unambiguous way. The mechanical action turns out to be the first order system without taking the strong field limit $B\longrightarrow\infty$. In particular, it is true for zero mode of the string coordinate which means that the noncommutativity is intrinsic property of this mechanical system. We describe the arbitrariness in the relation existent between the mechanical and the string variables and show that noncommutativity of the string variables on the boundary can be removed. It is in correspondence with the result of Seiberg and Witten on relation among noncommutative and ordinary Yang-Mills theories. 
  The possibility of having discrete degrees of freedom at singularities associated to `conifolds with discrete torsion' is studied. We find that the field theory of D-brane probes near these singularities is identical to ordinary conifolds, so that there are no additional discrete degrees of freedom located at the singularity. We shed light on how the obstructions to resolving the singularity are global topological issues rather that local obstrucions at the singularity itself. We also analyze the geometric transitions and duality cascades when one has fractional branes at the singularity and compute the moduli space of an arbitrary number of probes in the geometry. We provide some evidence for a conjecture that there are no discrete degrees of freedom localized at any Calabi-Yau singularity that can not be guessed from topological data away from the singularity. 
  Using an expansion in powers of an infinitesimally small coupling constant $g$, all generators of the Poincar\'e group in local scalar quantum field theory with interaction term $g \phi^3$ are expressed in terms of annihilation and creation operators $a_\lambda$ and $a^\dagger_\lambda$ that result from a boost-invariant renormalization group procedure for effective particles. The group parameter $\lambda$ is equal to the momentum-space width of form factors that appear in vertices of the effective-particle Hamiltonians, $H_\lambda$. It is verified for terms order 1, $g$, and $g^2$, that the calculated generators satisfy required commutation relations for arbitrary values of $\lambda$. One-particle eigenstates of $H_\lambda$ are shown to properly transform under all Poincar\'e transformations. The transformations are obtained by exponentiating the calculated algebra. From a phenomenological point of view, this study is a prerequisite to construction of observables such as spin and angular momentum of hadrons in quantum chromodynamics. 
  We determine the consistent D-brane configurations in type II nonsupersymmetric Melvin Background. The D-branes are analysed from three complementary points of view: the effective Born-Infeld action, the open string partition function and the boundary state approach. We show the agreement of the results obtained by the three different approaches. Among the surprising features is the existence of supersymmetric branes, some of them having a quasi-periodic direction. We also discuss the generalisation to backgrounds with several magnetic fields, some of them preserving in the closed and the open spectra some amount of supersymmetry. The case of rational magnetic flux, equivalent to freely-acting noncompact orbifolds, is also studied. It allows more possibilities of consistent D-brane configurations. 
  We discuss compactifications of the heterotic string in the presence of background fluxes. Specifically we consider compactifications on T^6, T^5, K3 x T^2 and K3 x S^1 for which we derive the bosonic sector of the low energy effective action. The consistency with the corresponding gauged supergravities is demonstrated. 
  We consider non-perturbative effects in theories with extra dimensions and the deconstructed versions of these theories. We establish the rules for instanton calculations in 5D theories on the circle, and use them for an explicit one-instanton calculation in a supersymmetric gauge theory. The results are then compared to the known exact Seiberg-Witten type solution for this theory, confirming the validity both of the exact results and of the rules for instanton calculus for extra dimensions introduced here. Next we consider the non-perturbative results from the perspective of deconstructed extra dimensions. We show that the non-perturbative results of the deconstructed theory do indeed reproduce the known results for the continuum extra dimensional theory, thus providing the first non-perturbative evidence in favor of deconstruction. This way deconstruction also allows us to make exact predictions in higher dimensional theories which agree with earlier results, and helps to clarify the interpretation of 5D instantons. 
  We consider a scalar field theory on AdS in both minimally and non-minimally coupled cases. We show that there exist constraints which arise in the quantization of the scalar field theory on AdS which cannot be reproduced through the usual AdS/CFT prescription. We argue that the usual energy, defined through the stress-energy tensor, is not the natural one to be considered in the context of the AdS/CFT correspondence. We analyze a new definition of the energy which makes use of the Noether current corresponding to time displacements in global coordinates. We compute the new energy for Dirichlet, Neumann and mixed boundary conditions on the scalar field and for both the minimally and non-minimally coupled cases. Then, we perform the quantization of the scalar field theory on AdS showing that, for `regular' and `irregular' modes, the new energy is conserved, positive and finite. We show that the quantization gives rise, in a natural way, to a generalized AdS/CFT prescription which maps to the boundary all the information contained in the bulk. In particular, we show that the divergent local terms of the on-shell action contain information about the Legendre transformed generating functional, and that the new constraints for which the irregular modes propagate in the bulk are the same constraints for which such divergent local terms cancel out. In this situation, the addition of counterterms is not required. We also show that there exist particular cases for which the unitarity bound is reached, and the conformal dimension becomes independent of the effective mass. This phenomenon has no bulk counterpart. 
  In this paper we investigate whether spacetime torsion induced by a Kalb-Ramond field in a string inspired background can generate a mass for the left-handed neutrino. We consider an Einstein-Dirac-Kalb-Ramond lagrangian in higher dimensional spacetime with torsion generated by the Kalb-Ramond antisymmetric field in the presence of a bulk fermion. We show that such a coupling can generate a mass term for the four dimensional neutrino after a suitable large radius compactification of the extra dimensions. 
  We derive noncommutative multi-particle quantum mechanics from noncommutative quantum field theory in the nonrelativistic limit. Paricles of opposite charges are found to have opposite noncommutativity. As a result, there is no noncommutative correction to the hydrogen atom spectrum at the tree level. We also comment on the obstacles to take noncommutative phenomenology seriously, and propose a way to construct noncommutative SU(5) grand unified theory. 
  The chiral primary operators of the D=6 superconformal (2,0) theory corresponding to 14 scalars of N=4 D=7 supergravity are obtained by expanding the world volume action for the M5-brane around an AdS_7 x S^4 background. In the leading order, the operators take their values in the symmetric traceless representation of the SO(5) R-symmetry group in consistency with the early conjecture on their structure based on the superconformal symmetry and Matrix-like model arguments. 
  Certain black branes are unstable toward fluctuations that lead to non-uniform mass distributions. We study static, non-uniform solutions that differ only perturbatively from uniform ones. For uncharged black strings in five dimensions, we find evidence of a first order transition from uniform to non-uniform solutions. 
  Frames normal for linear connections in vector bundles are defined and studied. In particular, such frames exist at every fixed point and/or along injective path. Inertial frames for gauge fields are introduced and on this ground the principle of equivalence for (system of) gauge fields is formulated. 
  We show that in the simplest theories of spontaneous symmetry breaking one can have a stage of a fast-roll inflation. In this regime the standard slow-roll condition |m^2| << H^2 is violated. Nevertheless, this stage can be rather long if |m| is sufficiently small. Fast-roll inflation can be useful for generating proper initial conditions for the subsequent stage of slow-roll inflation in the very early universe. It may also be responsible for the present stage of accelerated expansion of the universe. We also make two observations of a more general nature. First of all, the universe after a long stage of inflation (either slow-roll or fast-roll) cannot reach anti-de Sitter regime even if the cosmological constant is negative. Secondly, the theories with the potentials with a "stable" minimum at V(\phi)<0 in the cosmological background exhibit the same instability as the theories with potentials unbounded from below. This instability leads to the development of singularity with the properties practically independent of V(\phi). However, the development of the instability in some cases may be so slow that the theories with the potentials unbounded from below can describe the present stage of cosmic acceleration even if this acceleration occurs due to the fast-roll inflation. 
  In this article I attempt to collect some ideas,opinions and formulae which may be useful in solving the problem of gauge/ string / space-time correspondence This includes the validity of D-brane representation, counting of gauge-invariant words, relations between the null states and the Yang-Mills equations and the discussion of the strong coupling limit of the string sigma model. The article is based on the talk given at the "Odyssey 2001" conference. 
  The commutation relations of the composite fields are studied in the 3, 2 and 1 space dimensions. It is shown that the field of an atom consisting of a nucleus and an electron fields satisfies, in the space-like asymptotic limit, the canonical commutation relations within the sub-Fock-space of the atom. The field-particle duality in the bound state is discussed from the statistics point of view. Then, the commutation relations of the scalar object in the Schwinger(Thirring) model are mentioned briefly and are shown consistent with its interpretation as the Nambu-Goldstone boson.   The composite anyon fields are shown to satisfy the proper anyonic commutation relations with the additive phase exponents. Then, quasiparticle picture of the anyons is clarified under the restriction of this additibity. The difference between field and particle aspects becomes more prominent in the 2 space dimension. It is argued that the hierarchy of the fractional quantum Hall effect is rather simply understood by utilizing the quasiparticle charactors of the anyons when the background-boson gauge is assumed. In contrast to it, the coposite fermion theories are critically reviewed 
  An elementary introduction is given to the problem of black hole entropy as formulated by Bekenstein and Hawking, based on the so-called Laws of Black Hole Mechanics. Wheeler's `It from Bit' picture is presented as an explanation of plausibility of the Bekenstein-Hawking Area Law. A variant of this picture that takes better account of the symmetries of general relativity is shown to yield corrections to the Area Law that are logarithmic in the horizon area, with a finite, fixed coefficient. The Holographic hypothesis, tacitly assumed in the above considerations, is briefly described and the beginnings of a general proof of the hypothesis is sketched, within an approach to quantum gravitation which is non-perturbative in nature, namely Non-perturbative Quantum General Relativity (also known as Quantum Geometry). The holographic entropy bound is shown to be somewhat tightened due to the corrections obtained earlier. A brief summary of Quantum Geometry approach is included, with a sketch of a demonstration that precisely the log area corrections obtained from the variant of the It from Bit picture adopted earlier emerges for the entropy of generic black holes within this formalism. 
  I present an exact solution for the Heisenberg picture, Dirac electron in the presence of an electric field which depends arbitrarily upon the light cone time parameter $x^+ = (t+x)/\sqrt{2}$. This is the largest class of background fields for which the mode functions have ever been obtained. The solution applies to electrons of any mass and in any spacetime dimension. The traditional ampiguity at $p^+ = 0$ is explicitly resolved. It turns out that the initial value operators include not only $(I + \gamma^0 \gamma^1) \psi$ at $x^+ = 0$ but also $(I - \gamma^0 \gamma^1) \psi$ at $x^- = -L$. Pair creation is a discrete and instantaneous event on the light cone, so one can compute the particle production rate in real time. In $D=1+1$ dimensions one can also see the anomaly. Another novel feature of the solution is that the expectation value of the currents operators depends non-analytically upon the background field. This seems to suggest a new, strong phase of QED. 
  In this paper we discuss D-branes in the Melvin background and its supersymmetric generalizations. In particular we determine the D-brane spectra in these backgrounds by constructing their boundary states explicitly, where some of the D-branes are supersymmetric. The results sensitively depend on whether the value of magnetic flux in the Melvin background is rational or irrational. For the rational case the D-branes are regarded as the generalizations of fractional D-branes in abelian orbifolds ${\bf C^n/Z_N}$ of type II or type 0 string theory. For the irrational case we found a very limited spectrum. Since the background includes the nontrivial H-flux, the D-branes will provide interesting examples from the viewpoint of the noncommutative geometry. 
  We investigate the perturbative and non-perturbative correspondence of a class of four dimensional dual string constructions with N=4 and N=2 supersymmetry, obtained as Z_2 or Z_2 x Z_2 orbifolds of the type II, heterotic and type I string. In particular, we discuss the heterotic and type I dual of all the symmetric Z_2 x Z_2 orbifolds of the type II string, classified in hep-th/9901123. . 
  We obtain spacetimes generated by static and spinning magnetic string sources in Einstein Relativity with negative cosmological constant. Since the spacetime is asymptotically a cylindrical anti-de Sitter spacetime, we will be able to calculate the mass, momentum, and electric charge of the solutions. We find two families of solutions, one with longitudinal magnetic field and the other with angular magnetic field. The source for the longitudinal magnetic field can be interpreted as composed by a system of two symmetric and superposed electrically charged lines with one of the electrically charged lines being at rest and the other spinning. The angular magnetic field solution can be similarly interpreted as composed by charged lines but now one is at rest and the other has a velocity along the axis. This solution cannot be extended down to the origin. 
  We consider the noncommutative hypermultiplet model within harmonic superspace approach. The 1-loop four-point contributions to the effective action of selfinteracting q-hypermultiplet are computed. This model has two coupling constants instead of a single one in commutative case. It is shown that both these coupling constants are generated by 1-loop quantum corrections in the model of q-hypermultiplet interacting with vector multiplet. The holomorphic effective action of q-hypermultiplet in external gauge superfield is calculated. For the fundamental representation there is no UV/IR-mixing and the holomorphic potential is a star-product generalization of a standard commutative one. For the adjoint representation of U(N) gauge group the leading contributions to the holomorphic effective action are given by the terms respecting for the UV/IR-mixing which are related to U(1) phase of U(N) group. 
  We show that the D25 sliver wavefunction, just as the D-instanton sliver, factorizes when expressed in terms of half-string coordinates. We also calculate analytically the star-product of two zero-momentum eigenstates of $\hat{x}$ using the vertex in the oscillator basis, thereby showing that the star-product in the matter sector can indeed be seen as multiplication of matrices acting on the space of functionals of half strings. We then use the above results to establish that the matrices $\rho_{1,2}$, conjectured by Rastelli, Sen and Zwiebach to be left and right projectors on the sliver, are indeed so. 
  We show that the nondemolition measurement of a spacelike Wilson loop operator W(C) is impossible in a relativistic non-Abelian gauge theory. In particular, if two spacelike-separated magnetic flux tubes both link with the loop C, then a nondemolition measurement of W(C) would cause electric charge to be transferred from one flux tube to the other, a violation of relativistic causality. A destructive measurement of W(C) is possible in a non-Abelian gauge theory with suitable matter content. In an Abelian gauge theory, many cooperating parties distributed along the loop C can perform a nondemolition measurement of the Wilson loop operator if they are equipped with a shared entangled ancilla that has been prepared in advance. We also note that Abelian electric charge (but not non-Abelian charge) can be transported superluminally, without any accompanying transmission of information. 
  We develop a new formalism for the bosonic sector of low-energy heterotic string theory toroidally compactified to three dimensions. This formalism is based on the use of some single non-quadratic real matrix potential which transforms linearly under the action of subgroup of the three-dimensional charging symmetries. We formulate a new charging symmetry invariant approach for the symmetry generation and straightforward construction of asymptotically flat solutions. Finally, using the developed approach and the established formal analogy between the heterotic and Einstein-Maxwell theories, we construct a general class of the heterotic string theory extremal solutions of the Israel-Wilson-Perjes type. This class is asymptotically flat and charging symmetry complete; it includes the extremal solutions constructed before and possesses the non-trivial bosonic string theory limit. 
  We establish a symmetry map which relates two low-energy heterotic string theories with different numbers of the Abelian gauge fields compactified from the diverse to three dimensions on a torus. We discuss two applications of the established symmetry: a generation of the heterotic string theory solutions from the stationary Einstein-Maxwell fields and one non-trivial submersion of the heterotic string theory into the bosonic one. 
  We discuss the problems of dark matter, quantum gravity, and vacuum energy within the context of a theory for which Lorentz invariance is not postulated, but instead emerges as a natural consequence in the physical regimes where it has been tested. 
  We study the O3-plane in the conifold. On the D3-brane world-volume we obtain SO x USp gauge theory that exhibits a duality cascade phenomenon. The orientifold projection is determined on the type IIB string side, and corresponds to that of O4-plane on the dual type IIA side. We show that SUGRA solutions of Klebanov-Tseytlin and Klebanov-Strassler survive under the projection. We also investigate the orientifold projection in the generalized conifolds, and verify desired features of the O4-projection in the type IIA picture. 
  $N$ conformal theory models $WD^{(p)}_{3}$ coupled locally by their energy operators are analyzed by means of a perturbative renormalization group. New non-trivial fixed points are found. 
  It has been argued by some authors that the parent action approach cannot be used in order to establish the duality between the 2+1 Abelian and non-Abelian Self-Dual (SD) and Yang-Mills-Chern-Simons (YMCS) models for all the coupling regimes. We propose here an alternative (perturbative) point of view, and show that this equivalence can be achieved with the parent action approach. 
  Our formalism described recently in (Dolby et al, hep-th/0103228) is applied to the study of particle creation in spatially uniform electric fields, concentrating on the cases of a time-invariant electric field and a so-called `adiabatic' electric field. Several problems are resolved by incorporating the `Bogoliubov coefficient' approach and the `tunnelling' approaches into a single consistent, gauge invariant formulation. The value of a time-dependent particle interpretation is demonstrated by presenting a coherent account of the time-development of the particle creation process, in which the particles are created with small momentum (in the frame of the electric field) and are then accelerated by the electric field to make up the `bulge' of created particles predicted by asymptotic calculations. An initial state comprising one particle is also considered, and its evolution is described as being the sum of two contributions: the `sea of current' produced by the evolved vacuum, and the extra current arising from the initial particle state. 
  We study the properties of an exact multi-membrane solution in seven-dimensional maximal SO(5)-gauged supergravity. Unlike previously known multi-centered solutions, the present one is asymptotically anti-de Sitter. We show that this multi-membrane configuration preserves only a quarter of the supersymmetries. When lifted to eleven dimensions, this solution is interpreted as a set of open membranes ending on self-dual strings on a stack of M5-branes, in the near M5 limit. 
  We search for Ricci flat, K\"{a}hler geometries which are asymptotic to the cone whose base is the space T^{11} by working out covariantly constant spinor equations. The metrics we find are singular in the interior and introducing parallel D3-branes does not form regular event horizons cloaking the naked singularities. We also work out a supersymmetric ansatz involving only the metric and the 5-form field corresponding to D3-branes wrapping over the non-trivial 2-cycle of T^{11}. We find a system of first-order equations and argue that the solution has an event horizon and the ADM mass per unit volume diverges logarithmically. 
  In noncommutative field theories, it was known that one-loop effective action describes propagation of non-interacting open Wilson lines, obeying the flying dipole's relation. We show that two-loop effective action describes cubic interaction among `closed string' states created by open Wilson lines. Taking d-dimensional noncommutative [\Phi^3] theory as the simplest setup, we compute nonplanar contribution at low-energy and large noncommutativity limit. We find that the contribution is expressible in a remarkably simple cubic interaction involving scalar open Wilson lines only and nothing else. We show that the interaction is purely geometrical and noncommutative in nature, depending only on sizes of each open Wilson line. 
  We construct the non-compact Calabi-Yau manifolds interpreted as the complex line bundles over the Hermitian symmetric spaces. These manifolds are the various generalizations of the complex line bundle over CP^{N-1}. Imposing an F-term constraint on the line bundle over CP^{N-1}, we obtain the line bundle over the complex quadric surface Q^{N-2}. On the other hand, when we promote the U(1) gauge symmetry in CP^{N-1} to the non-abelian gauge group U(M), the line bundle over the Grassmann manifold is obtained. We construct the non-compact Calabi-Yau manifolds with isometries of exceptional groups, which we have not discussed in the previous papers. Each of these manifolds contains the resolution parameter which controls the size of the base manifold, and the conical singularity appears when the parameter vanishes. 
  The first heat kernel coefficients are calculated for a dispersive ball whose permittivity at high frequency differs from unity by inverse powers of the frequency. The corresponding divergent part of the vacuum energy of the electromagnetic field is given and ultraviolet divergencies are seen to be present. Also in a model where the number of atoms is fixed the pressure exhibits infinities. As a consequence, the ground-state energy for a dispersive dielectric ball cannot be interpreted easily. 
  We study the ultraviolet asymptotics in $A_n$ affine Toda theories with integrable boundary actions. The reflection amplitudes of non-affine Toda theories in the presence of conformal boundary actions have been obtained from the quantum mechanical reflections of the wave functional in the Weyl chamber and used for the quantization conditions and ground-state energies. We compare these results with the thermodynamic Bethe ansatz derived from both the bulk and (conjectured) boundary scattering amplitudes. The two independent approaches match very well and provide the non-perturbative checks of the boundary scattering amplitudes for Neumann and $(+)$ boundary conditions. Our results also confirm the conjectured boundary vacuum energies and the duality conjecture between the two boundary conditions. 
  We investigate D-branes in orientifolds of WZW models. A connection between the conformal field theory approach to orientifolds and the target space motivated analysis is established. In particular, we associate previously constructed crosscap states to involutions of the group manifold and their fixed point sets. Whereas our analysis of D-branes in orientifolds of general WZW models is restricted to special D0-branes, we investigate all symmetry preserving branes of SU(2)-orientifolds in detail. For that case, the location of the orientifold fixed point set is independently determined by scattering localized graviton wave packets. 
  We analyze the (discrete) spectrum of the semirelativistic ``spinless-Salpeter'' Hamiltonian H = \beta \sqrt{m^2 + p^2} + V(r), beta > 0, where V(r) represents an attractive, spherically symmetric potential in three dimensions. In order to locate the eigenvalues of H, we extend the ``envelope theory,'' originally formulated only for nonrelativistic Schroedinger operators, to the case of Hamiltonians H involving the relativistic kinetic-energy operator. If V(r) is a convex transformation of the Coulomb potential -1/r and a concave transformation of the harmonic-oscillator potential r^2, both upper and lower bounds on the discrete eigenvalues of H can be constructed, which may all be expressed in the form E = min_{r>0} [ \beta \sqrt{m^2 + P^2/r^2} + V(r) ] for suitable values of the numbers P here provided. At the critical point, the relative growth to the Coulomb potential h(r) = -1/r must be bounded by dV/dh < 2 \beta/\pi. 
  A spontaneously broken SU(2)xU(1) gauge theory with just one "primordial" generation of fermions is formulated in the context of generally covariant theory which contains two measures of integration in the action: the standard \sqrt{-g}d^{4}x and a new \Phi d^{4}x, where \Phi is a density built out of degrees of freedom independent of the metric. Such type of models are known to produce a satisfactory answer to the cosmological constant problem. Global scale invariance is implemented. After SSB of scale invariance and gauge symmetry it is found that with the conditions appropriate to laboratory particle physics experiments, to each primordial fermion field corresponds three physical fermionic states. Two of them correspond to particles with constant masses and they are identified with the first two generations of the electro-weak theory. The third fermionic states at the classical level get non-polynomial interactions which indicate the existence of fermionic condensate and fermionic mass generation. 
  Stability of massive antisymmetric tensor fields with the Chern-Simons type action in anti de Sitter spacetime is studied. It is found that there exists a complete set of solutions whose energy is conserved and positive definite if the mass is positive. Scalar products of the solutions are shown to be well-defined and conserved. In contrast to the previously studied scalar field case there is no other set of stable solutions with a different kind of boundary condition. 
  We consider three-point couplings in simple Lie algebras -- singlets in triple tensor products of their integrable highest weight representations. A coupling can be expressed as a linear combination of products of finitely many elementary couplings. This carries over to affine fusion, the fusion of Wess-Zumino-Witten conformal field theories, where the expressions are in terms of elementary fusions. In the case of su(4) it has been observed that there is a purely affine elementary fusion, i.e., an elementary fusion that is not an elementary coupling. In this note we show by construction that there is at least one purely affine elementary fusion associated to every su(N>3). 
  We develop a formalism to carry out coarse-grainings in quantum field theoretical systems by using a time-dependent projection operator in the Heisenberg picture. A systematic perturbative expansion with respect to the interaction part of the Hamiltonian is given, and a Langevin-type equation without a time-convolution integral term is obtained. This method is applied to a quantum field theoretical model, and coupled transport equations are derived. 
  We initiate the study of Brane Gas Cosmology (BGC) on manifolds with non-trivial holonomy. Such compactifications are required within the context of superstring theory in order to make connections with realistic particle physics. We study the dynamics of brane gases constructed from various string theories on background spaces having a K3 submanifold. The K3 compactifications provide a stepping stone for generalising the model to the case of a full Calabi-Yau three-fold. Duality symmetries are discussed within a cosmological context. Using a duality, we arrive at an N=2 theory in four-dimensions compactified on a Calabi-Yau manifold with SU(3) holonomy. We argue that the Brane Gas model compactified on such spaces maintains the successes of the trivial toroidal compactification while greatly enhancing its connection to particle physics. The initial state of the universe is taken to be a small, hot and dense gas of p-branes near thermal equilibrium. The universe has no initial singularity and the dynamics of string winding modes allow three spatial dimensions to grow large, providing a possible solution to the dimensionality problem of string theory. 
  We analyze the impact of certain modifications to short distance physics on the inflationary perturbation spectrum. For the specific case of power-law inflation, we find distinctive -- and possibly observable -- effects on the spectrum of density perturbations. 
  A mechanism to find different Weyl invariant p-branes and Dp-branes actions is explained. Our procedure clarifies the Weyl invariance for such systems. Besides, by considering gravity-dilaton effective action in higher dimensions we also derive a Weyl invariant action for p-branes. We argue that this derivation provides a geometrical scenario for the Weyl invariance of p-branes. Our considerations can be extended to the case of super-p-branes. 
  We aimed to obtain the energy levels of spin-1 particles moving in a constant magnetic field. The method used here is completely algebraic. In the process to obtain the energy levels the wave function is choosen in terms of Laguerre Polynomials. 
  By rigorous reanalysis of the results, we have proven that the propagators at finite temperature for scalar bound states in one-generation fermion condensate scheme of electroweak symmetry breaking are in fact identical in the imaginary-time and the real-time formalism. This dismisses the doubt about possible discrepancy between the two formalisms in this problem. Identity of the derived thermal transformation matrices of the real-time matrix propagators for scalar bound states without and with chemical potential and the ones for corresponding elementary scalar particles shows similarity of thermodynamic property between the two types of particles. Only one former inference is modified, i.e. when the two flavors of fermions have unequal nonzero masses, the amplitude of the composite Higgs particle will decay instead grow in time. 
  Recently established rationality of correlation functions in a globally conformal invariant quantum field theory satisfying Wightman axioms is used to construct a family of soluble models in 4-dimensional Minkowski space-time. We consider in detail a model of a neutral scalar field $\phi$ of dimension 2. It depends on a positive real parameter c, an analogue of the Virasoro central charge, and admits for all (finite) c an infinite number of conserved symmetric tensor currents. The operator product algebra of $\phi$ is shown to coincide with a simpler one, generated by a bilocal scalar field $V(x_1,x_2)$ of dimension (1,1). The modes of V together with the unit operator span an infinite dimensional Lie algebra $L_V$ whose vacuum (i.e. zero energy lowest weight) representations only depend on the central charge c. Wightman positivity (i.e. unitarity of the representations of $L_V$) is proven to be equivalent to $c \in N$. 
  In this letter we apply the methods of our previous paper hep-th/0108045 to noncommutative fermions. We show that the fermions form a spin-1/2 representation of the Lorentz algebra. The covariant splitting of the conformal transformations into a field-dependent part and a \theta-part implies the Seiberg-Witten differential equations for the fermions. 
  We present the construction of exactly solvable superconformal field theories describing Type II string models compactified on compact G_2 manifolds. These models are defined by anti-holomorphic quotients of the form (CY*S^1)/Z_2, where we realize the Calabi-Yau as a Gepner model. In the superconformal field theory the Z_2 acts as charge conjugation implying that the representation theory of a W(2,4,6,8,10) algebra plays an important role in the construction of these models. Intriguingly, in all three examples we study, including the quintic, the massless spectrum in the Z_2 twisted sector of the superconformal field theory differs from what one expects from the supergravity computation. This discrepancy is explained by the presence of a discrete NS-NS background two-form flux in the Gepner model. 
  Various forms of derivative dispersion relations, in which the dispersion integral is replaced by a series of derivatives of the imaginary part of a scattering amplitude, are reviewed. Conditions of their validity and practical applicability as well as their relevance to high-energy small-angle hadron-hadron scattering are discussed. 
  Recently an interesting conjecture was put forward which states that any asymptotically de Sitter space whose mass exceeds that of exact de Sitter space contains a cosmological singularity. In order to test this mass bound conjecture, we present two solutions. One is the topological de Sitter solution and the other is its dilatonic deformation. Although the latter is not asymptotically de Sitter space, the two solutions have a cosmological horizon and a cosmological singularity. Using surface counterterm method we compute the quasilocal stress-energy tensor of gravitational field and the mass of the two solutions. It turns out that this conjecture holds within the two examples. Also we show that the thermodynamic quantities associated with the cosmological horizon of the two solutions obey the first law of thermodynamics. Furthermore, the nonconformal extension of dS/CFT correspondence is discussed. 
  We examine the connection between three dimensional gravity with negative cosmological constant and two-dimensional CFT via the Chern-Simons formulation. A set of generalized spectral flow transformations are shown to yield new sectors of solutions. One implication is that the microscopic calculation of the entropy of the Banados-Teitelboim-Zanelli (BTZ) black hole is corrected by a multiplicative factor with the result that it saturates the Bekenstein-Hawking expression. 
  We show that the correspondence between SU(5) monopoles and the elementary particles, which underlies the construction of a dual standard model, has some simpler analogues associated with the strong, weak and hypercharge interactions. We then discuss how these analogues relate to Bais' generalization of the Montenon-Olive conjecture and find the representations of the monopoles under the dual gauge group; these representations agree with those of the elementary particles. 
  Real-time anomalous fermion number violation has been investigated for massless chiral fermions in spherically symmetric SU(2) Yang-Mills gauge field backgrounds which can be weakly dissipative or even nondissipative. Restricting consideration to spherically symmetric fermion fields, a relation has been found between the spectral flow of the Dirac Hamiltonian and two characteristics of the background gauge field. This new result may be relevant to electroweak baryon number violation in the early universe. 
  We obtain new results for consistent braneworld Kaluza-Klein reductions, showing how we can derive four-dimensional N=2 gauged supergravity ``localised on the AdS_4 brane'' as an exact embedding in five-dimensional N=4 gauged supergravity. Similarly, we obtain five-dimensional N=2 gauged supergravity localised on an AdS_5 brane as a consistent Kaluza-Klein reduction from six-dimensional N=4 gauged supergravity. These embeddings can be lifted to type IIB and massive type IIA supergravity respectively. The new AdS braneworld Kaluza-Klein reductions are generalisations of earlier results on braneworld reductions to ungauged supergravities. The lower-dimensional cosmological constant in our AdS braneworld reductions is an adjustable parameter, and so it can be chosen to be small enough to be phenomenologically realistic, even if the higher-dimensional one is of Planck scale. We also discuss analytic continuations to give a de Sitter gauged supergravity in four dimensions as a braneworld Kaluza-Klein reduction. We find that there are two distinct routes that lead to the same four-dimensional theory. In one, we start from a five-dimensional de Sitter supergravity, which itself arises from a Kaluza-Klein reduction of type IIB^* supergravity on the hyperbolic 5-sphere. In the other, we start from AdS gauged supergravity in five dimensions, with an analytic continuation of the two 2-form potentials, and embed the four-dimensional de Sitter supergravity in that. The five-dimensional theory itself comes from an O(4,3)/O(3,2) reduction of Hull's type IIB_{7+3} supergravity in ten dimensions. 
  We test a candidate for a four-dimensional C-function. This is done by considering all asymptotically free, vectorlike gauge theories with N_f flavors and fermions in arbitrary representations of any simple Lie group. Assuming spontaneous breaking of chiral symmetry in the infrared limit and that the value of the C-function in this limit is determined by the number of Goldstone bosons, we find that only in the case of a theory with two colors and fermions in one single pseudo-real representation of SU(2) the C-theorem seems to be violated. Conversely, this might also be a sign of new constraints, restricting the number of flavors consistent with spontaneous chiral symmetry breaking. For all other groups and representations we find that this candidate C-function decreases along the renormalization group flow. 
  The $n$-instanton contribution to the Seiberg-Witten prepotential of ${\bf N}=2$ supersymmetric $d=4$ Yang Mills theory is represented as the integral of the exponential of an equivariantly exact form. Integrating out an overall scale and a U(1) angle the integral is rewritten as $(4n-3)$ fold product of a closed two form. This two form is, formally, a representative of the Euler class of the Instanton moduli space viewed as a principal U(1) bundle, because its pullback under bundel projection is the exterior derivative of an angular one-form. We comment on a recent speculation of Matone concerning an analogy linking the instanton problem and classical Liouville theory of punctured Riemann spheres. 
  The dual Meissner effect scenario of confinement is discussed by studying the low energy regime of SU(2) Yang-Mills in a maximal Abelian gauge. The Abelian projected effective action is computed perturbatively. This serves as an input for a study of the non-perturbative regime, which is undertaken using exact renormalisation group methods. It is argued that the effective action derived here contains the relevant degrees of freedom for confinement if ultraviolet irrelevant vertices are retained. 
  We present a maximally supersymmetric IIB string background. The geometry is that of a conformally flat lorentzian symmetric space G/K with solvable G, with a homogeneous five-form flux. We give the explicit supergravity solution, compute the isometries, the 32 Killing spinors, and the symmetry superalgebra, and then discuss T-duality and the relation to M-theory. 
  We study the two-dimensional gauge theory of the symmetric group S_n describing the statistics of branched n-coverings of Riemann surfaces. We consider the theory defined on the disk and on the sphere in the large-n limit. A non trivial phase structure emerges, with various phases corresponding to different connectivity properties of the covering surface. We show that any gauge theory on a two-dimensional surface of genus zero is equivalent to a random walk on the gauge group manifold: in the case of S_n, one of the phase transitions we find can be interpreted as a cutoff phenomenon in the corresponding random walk. A connection with the theory of phase transitions in random graphs is also pointed out. Finally we discuss how our results may be related to the known phase transitions in Yang-Mills theory. We discover that a cutoff transition occurs also in two dimensional Yang-Mills theory on a sphere, in a large N limit where the coupling constant is scaled with N with an extra log N compared to the standard 't Hooft scaling. 
  Liouville field theory is considered on domains with conformally invariant boundary conditions. We present an explicit expression for the three point function of boundary fields in terms of the fusion coefficients which determine the monodromy properties of the conformal blocks. 
  We analyze a nonabelian extension of Born--Infeld action for the SU(2) group. In the class of spherically symmetric solutions we find that, besides the Gal'tsov--Kerner glueballs, only the analytic dyons have finite energy. The presented analytic and numerical investigation excludes the existence of pure magnetic monopoles of 't Hooft--Polyakov type. 
  It is shown that the fermionic Heisenberg-Weyl algebra with 2N=D fermionic generators is equivalent to the generalized Grassmann algebra with two fractional generators. The 2,3 and 4 dimensional Heisenberg - Weyl algebra is explicitly given in terms of the fractional generators. These algebras are used for the formulation of the N=2,3,4 extended supersymmetry. As an example we reformulate the Lax approach of the supersymmetric Korteweg - de Vries equation in terms of the generators of the generalized Grassmann algebra. 
  An unambiguous and slice-independent formula for the two-loop superstring measure on moduli space for even spin structure is constructed from first principles. The construction uses the super-period matrix as moduli invariant under worldsheet supersymmetry. This produces new subtle contributions to the gauge-fixing process, which eliminate all the ambiguities plaguing earlier gauge-fixed formulas.   The superstring measure can be computed explicitly and a simple expression in terms of modular forms is obtained. For fixed spin structure, the measure exhibits the expected behavior under degenerations of the surface. The measure allows for a unique modular covariant GSO projection. Under this GSO projection, the cosmological constant, the 1-, 2- and 3- point functions of massless supergravitons all vanish pointwise on moduli space without the appearance of boundary terms. A certain disconnected part of the 4-point function is shown to be given by a convergent, finite integral on moduli space. A general slice-independent formula is given for the two-loop cosmological constant in compactifications with central charge c=15 and N=1 worldsheet supersymmetry in terms of the data of the compactification conformal field theory.   In this paper, a summary of the above results is presented with detailed constructions, derivations and proofs to be provided in a series of subsequent publications. 
  We show that the central charges that group theory allows in the (2,0) supersymmetry translations algebra arise from a string and a 3-brane by commuting two supercharges. We show that the net force between two such parallel strings vanishes. We show that all the coupling constants are fixed numbers, due to supersymmetry, and self-duality of the three-form field strength. We obtain a charge quantization for the self-dual field strength, and show that when compactifying on a two-torus, it reduces to the usual quantization condition of N=4 SYM with gauge group SU(2), and with coupling constant and theta angle given by the tau-parameter of the two-torus, provided that we pick that chiral theory which corresponds to a theta function with zero characteristics, as expected on manifolds of this form. 
  We present a simple field transformation which changes the field arguments from the ordinary position-space coordinates to the oblique phase-space coordinates that are linear in position and momentum variables. This is useful in studying quantum field dynamics in the presence of external uniform magnetic field: here, the field transformation serves to separate the dynamics within the given Landau level from that between different Landau levels. We apply this formalism to both nonrelativistic and relativistic field theories. In the large external magnetic field our formalism provides an efficient method for constructing the relevant lower-dimensional effective field theories with the field degrees defined only on the lowest Landau level. 
  It is formulated a new 'anholonomic frame' method of constructing exact solutions of Einstein equations with off--diagonal metrics in 4D and 5D gravity. The previous approaches and results are summarized and generalized as three theorems which state the conditions when two types of ansatz result in integrable gravitational field equations. There are constructed and analyzed different classes of anisotropic and/or warped vacuum 5D and 4D metrics describing ellipsoidal black holes with static anisotropic horizons and possible anisotropic gravitational polarizations and/or running constants. We conclude that warped metrics can be defined in 5D vacuum gravity without postulating any brane configurations with specific energy momentum tensors. Finally, the 5D and 4D anisotropic Einstein spaces with cosmological constant are investigated. 
  Tensor, matrix and quaternion formulations of Dirac-K\"ahler equation for massive and massless fields are considered. The equation matrices obtained are simple linear combinations of matrix elements in the 16-dimensional space. The projection matrix-dyads defining all the 16 independent equation solutions are found. A method of computing the traces of 16-dimensional Petiau-Duffin-Kemmer matrix product is considered. We show that the symmetry group of the Dirac-K\"ahler tensor fields for charged particles is SO(4,2). The conservation currents corresponding this symmetry are constructed. We analyze transformations of the Lorentz group and quaternion fields. Supersymmetry of the Dirac-K\"ahler fields with tensor and spinor parameters is investigated. We show the possibility of constructing a gauge model of interacting Dirac-K\"ahler fields where the gauge group is the noncompact group under consideration. 
  We study the spectrum of bosonic string theory on rotating BTZ black holes, using a SL(2,R) WZW model. Previously, Natsuume and Satoh have analyzed strings on BTZ black holes using orbifold techniques. We show how an appropriate spectral flow in the WZW model can be used to generate the twisted sectors, emphasizing how the spectral flow works in the hyperbolic basis natural for the BTZ black hole. We discuss the projection condition which leads to the quantization condition for the allowed quantum numbers for the string excitations, and its connection to the anomaly in the corresponding conserved Noether current. 
  Probability theory can be modified in essentially one way while maintaining consistency with the basic Bayesian framework. This modification results in copies of standard probability theory for real, complex or quaternion probabilities. These copies, in turn, allow one to derive quantum theory while restoring standard probability theory in the classical limit. The argument leading to these three copies constrain physical theories in the same sense that Cox's original arguments constrain alternatives to standard probability theory. This sequence is presented in some detail with emphasis on questions beyond basic quantum theory where new insights are needed. 
  The Schwinger-Dyson equation of fermion self-energy in the linearization approximation is solved exactly in a theory with gauge and effective four-fermion interactions. Different expressions for the indepedent solutions which respectively submit to irregular and regular ultraviolet boundary condition are derived and expounded. 
  We study the conditions for the existence of black holes that can be produced in colliders at TeV-scale if the space-time is higher dimensional. On employing the microcanonical picture, we find that their life-times strongly depend on the details of the model. If the extra dimensions are compact (ADD model), microcanonical deviations from thermality are in general significant near the fundamental TeV mass and tiny black holes decay more slowly than predicted by the canonical expression, but still fast enough to disappear almost instantaneously. However, with one warped extra dimension (RS model), microcanonical corrections are much larger and tiny black holes appear to be (meta)stable. Further, if the total charge is not zero, we argue that naked singularities do not occur provided the electromagnetic field is strictly confined on an infinitely thin brane. However, they might be produced in colliders if the effective thickness of the brane is of the order of the fundamental length scale (~1/TeV). 
  In these lectures I shall explain how a new-found nonabelian duality can be used to solve some outstanding questions in particle physics. The first lecture introduces the concept of electromagnetic duality and goes on to present its nonabelian generalization in terms of loop space variables. The second lecture discusses certain puzzles that remain with the Standard Model of particle physics, particularly aimed at nonexperts. The third lecture presents a solution to these problems in the form of the Dualized Standard Model, first proposed by Chan and the author, using nonabelian dual symmetry. The fundamental particles exist in three generations, and if this is a manifestation of dual colour symmetry, which by 't Hooft's theorem is necessarily broken, then we have a natural explanation of the generation puzzle, together with tested and testable consequences not only in particle physics, but also in astrophysics, nuclear and atomic physics.   Reported is mainly work done in collaboration with Chan Hong-Mo, and also various parts with Peter Scharbach, Jacqueline Faridani, Jos\'e Bordes, Jakov Pfaudler, Ricardo Gallego severally. 
  We define a class of orthosymplectic superalgebras $osp(m;j|2n;\omega)$ which may be obtained from $osp(m|2n)$ by contractions and analytic continuations in a similar way as the orthogonal and the symplectic Cayley-Klein algebras are obtained from the corresponding classical ones. Contractions of $osp(1|2)$ and $osp(3|2)$ are regarded as an examples. 
  It was shown by Weyl that the general static axisymmetric solution of the vacuum Einstein equations in four dimensions is given in terms of a single axisymmetric solution of the Laplace equation in three-dimensional flat space. Weyl's construction is generalized here to arbitrary dimension $D\ge 4$. The general solution of the D-dimensional vacuum Einstein equations that admits D-2 orthogonal commuting non-null Killing vector fields is given either in terms of D-3 independent axisymmetric solutions of Laplace's equation in three-dimensional flat space or by D-4 independent solutions of Laplace's equation in two-dimensional flat space. Explicit examples of new solutions are given. These include a five-dimensional asymptotically flat ``black ring'' with an event horizon of topology S^1 x S^2 held in equilibrium by a conical singularity in the form of a disc. 
  We construct a generalization of pure lattice gauge theory (LGT) where the role of the gauge group is played by a tensor category. The type of tensor category admissible (spherical, ribbon, symmetric) depends on the dimension of the underlying manifold (<=3, <=4, any). Ordinary LGT is recovered if the category is the (symmetric) category of representations of a compact Lie group. In the weak coupling limit we recover discretized BF-theory in terms of a coordinate free version of the spin foam formulation. We work on general cellular decompositions of the underlying manifold.   In particular, we are able to formulate LGT as well as spin foam models of BF-type with quantum gauge group (in dimension <=4) and with supersymmetric gauge group (in any dimension).   Technically, we express the partition function as a sum over diagrams denoting morphisms in the underlying category. On the LGT side this enables us to introduce a generalized notion of gauge fixing corresponding to a topological move between cellular decompositions of the underlying manifold. On the BF-theory side this allows a rather geometric understanding of the state sum invariants of Turaev/Viro, Barrett/Westbury and Crane/Yetter which we recover.   The construction is extended to include Wilson loop and spin network type observables as well as manifolds with boundaries. In the topological (weak coupling) case this leads to TQFTs with or without embedded spin networks. 
  The vacuum Einstein equations in five dimensions are shown to admit a solution describing an asymptotically flat spacetime regular on and outside an event horizon of topology S^1 x S^2. It describes a rotating ``black ring''. This is the first example of an asymptotically flat vacuum solution with an event horizon of non-spherical topology. There is a range of values for the mass and angular momentum for which there exist two black ring solutions as well as a black hole solution. Therefore the uniqueness theorems valid in four dimensions do not have simple higher dimensional generalizations. It is suggested that increasing the spin of a five dimensional black hole beyond a critical value results in a transition to a black ring, which can have an arbitrarily large angular momentum for a given mass. 
  The purpose of the "bootstrap program" is to construct integrable quantum field theories in 1+1 dimensions in terms of their Wightman functions explicitly. As an input the integrability and general assumptions of local quantum field theories are used. The object is to be achieved in tree steps: 1) The S-matrix is obtained using a qualitative knowledge of the particle spectrum and the Yang-Baxter equations. 2) Matrix elements of local operators are calculated by means of the "form factor program" using the S-matrix as an input. 3) The Wightman functions are calculated by taking sums over intermediate states. The first step has been performed for a large number of models and also the second one for several models. The third step is unsolved up to now. Here the program is illustrated in terms of the sine-Gordon model alias the massive Thirring model. Exploiting the "off-shell" Bethe Ansatz we propose general formulae for form factors. For example the n-particle matrix element for all higher currents are given and in particular all eigenvalues of the higher conserved charges are calculated. Furthermore quantum operator equations are obtained in terms of their matrix elements, in particular the quantum sine-Gordon field equation. Exact expressions for the finite wave function and mass renormalization constants are calculated. 
  Non-singular two and three dimensional string cosmologies are constructed using the exact conformal field theories corresponding to SO(2,1)/SO(1,1) and SO(2,2)/SO(2,1). {\it All} semi-classical curvature singularities are canceled in the exact theories for both of these cosets, but some new quantum curvature singularities emerge. However, considering different patches of the global manifolds, allows the construction of non-singular spacetimes with cosmological interpretation. In both two and three dimensions, we construct non-singular oscillating cosmologies, non-singular expanding and inflationary cosmologies including a de Sitter (exponential) stage with positive scalar curvature as well as non-singular contracting and deflationary cosmologies. Similarities between the two and three dimensional cases suggest a general picture for higher dimensional coset cosmologies: Anisotropy seems to be a generic unavoidable feature, cosmological singularities are generically avoided and it is possible to construct non-singular cosmologies where some spatial dimensions are experiencing inflation while the others experience deflation. 
  Special geometry is most known from 4-dimensional N=2 supergravity, though it contains also quaternionic and real geometries. In this review, we first repeat the connections between the various special geometries. Then the constructions are given starting from the superconformal approach. Without going in detail, we give the main underlying principles. We devote special attention to the quaternionic manifolds, introducing the notion of hypercomplex geometry, being manifolds close to hyperkaehler manifolds but without a metric. These are related to supersymmetry models without an action. 
  We thereby prove that a large class of topologically massive theories of the Cremmer-Scherk-Kalb-Ramond-type in any $d$ dimensions corresponds to gauge non-invariant first-order theories that can be interpreted as self-dual models. 
  In this paper we show that the entropy of de Sitter space with a black hole in arbitrary dimension can be understood using a modified Cardy-Verlinde entropy formula. We also comment on the observer dependence of the de Sitter entropy. 
  We study the relationship between the holomorphic unitary connection of Chern-Simons theory with temporal Wilson lines and the Richardson's exact solution of the reduced BCS Hamiltonian. We derive the integrals of motion of the BCS model, their eigenvalues and eigenvectors as a limiting case of the Chern-Simons theory. 
  We analyze unoriented Wess-Zumino-Witten models from a geometrical point of view. We show that the geometric interpretation of simple current crosscap states is as centre orientifold planes localized on conjugacy classes of the group manifold. We determine the locations and dimensions of these planes for arbitrary simply-connected groups and orbifolds thereof. The dimensions of the O-planes turn out to be given by the dimensions of symmetric coset manifolds based on regular embeddings. Furthermore, we give a geometrical interpretation of boundary conjugation in open unoriented WZW models; it yields D-branes together with their images under the orientifold projection. To find the agreement between O-planes and crosscap states, we find explicit answers for lattice extensions of Gaussian sums. These results allow us to express the modular P-matrix, which is directly related to the crosscap coefficient, in terms of characters of the horizontal subgroup of the affine Lie algebra. A corollary of this relation is that there exists a formal linear relation between the modular P- and the modular S-matrix. 
  We study the scalar quantum field theory on a generic noncommutative two-sphere as a special case of noncommutative curved space, which is described by the deformation quantization algebra obtained from symplectic reduction and parametrized by $H^2(S^2, \QR)$. The fuzzy sphere is included as a special case parametrized by the integer two-cohomology class $H^2(S^2, \QZ)$, which has finite number of degrees of freedom and the field theory has a well defined Hilbert space. When the two-cohomology class is not integer valued, the scalar quantum field theory based on the deformation algebra is not unitary: the signature of the inner product on the space of functions is indefinite. Hence the existence of deformation quantization does not guarantee a physically acceptable deformed geometric background. For the deformation quantization on a general curved space, this obstruction of unitarity can be given by an explicit topological formula. 
  We investigate spacetimes in which the speed of light along flat 4D sections varies over the extra dimensions due to different warp factors for the space and the time coordinates (``asymmetrically warped'' spacetimes). The main property of such spaces is that while the induced metric is flat, implying Lorentz invariant particle physics on a brane, bulk gravitational effects will cause apparent violations of Lorentz invariance and of causality from the brane observer's point of view. An important experimentally verifiable consequence of this is that gravitational waves may travel with a speed different from the speed of light on the brane, and possibly even faster. We find the most general spacetimes of this sort, which are given by certain types of black hole spacetimes characterized by the mass and the charge of the black hole. We show how to satisfy the junction conditions and analyze the properties of these space-times. 
  We investigate D-branes with maximal symmetry on general group manifolds in terms of boundary states and effective actions. We show that in large $k$ limit boundary states with an suitable Wilson line form boundary states of the other types of D-branes, extending the known fact in SU(2) case. We also show that fluctuation mass spectrum around D-brane solutions of the effective action agrees with that of boundary CFT in large $k$ limit. 
  We find a description of hybrid inflation in (3+1)-dimensions using brane dynamics of Hanany-Witten type. P-term inflation/acceleration of the universe with the hybrid potential has a slow-roll de Sitter stage and a waterfall stage which leads towards an N=2 supersymmetric ground state. We identify the slow-roll stage of inflation with a non-supersymmetric `Coulomb phase' with Fayet-Iliopoulos term. This stage ends when the mass squared of one of the scalars in the hypermultiplet becomes negative. At that moment the brane system starts undergoing a phase transition via tachyon condensation to a fully Higgsed supersymmetric vacuum which is the absolute ground state of P-term inflation. A string theory/cosmology dictionary is provided, which leads to constraints on parameters of the brane construction from cosmological experiments. We display a splitting of mass levels reminiscent of the Zeeman effect due to spontaneous supersymmetry breaking. 
  In this paper we construct and analyze new classes of wormhole and flux tube-like solutions for the 5D vacuum Einstein equations. These 5D solutions possess generic local anisotropy which gives rise to a gravitational running or scaling of the Kaluza-Klein ``electric'' and ``magnetic'' charges of these solutions. It is also shown that it is possible to self-consistently construct these anisotropic solutions with various rotational 3D hypersurface geometries (i.e. ellipsoidal, cylindrical, bipolar and toroidal). The local anisotropy of these solutions is handled using the technique of anholonomic frames with their associated nonlinear connection structures [vst]. Through the use of the anholonomic frames the metrics are diagonalized, in contrast to holonomic coordinate frames where the metrics would have off-diagonal components. In the local isotropic limit these solutions are shown to be equivalent to spherically symmetric 5D wormhole and flux tube solutions. 
  A (1 + d)-dimensional thick "brane world" model with varying Lambda-term is considered. The model is generalized to the case of a chain of Ricci-flat internal spaces when the matter source is an anisotropic perfect fluid. The "horizontal" part of potential is obtained in the Newtonian approximation. In the multitemporal case (with a Lambda-term) a set of equations for potentials is presented. 
  This topical review deals with a multidimensional gravitational model containing dilatonic scalar fields and antisymmetric forms. The manifold is chosen in the form M = M_0 x M_1 x ...x M_n, where M_i are Einstein spaces (i >0). The sigma-model approach and exact solutions in the model are reviewed and the solutions with p-branes (e.g. Majumdar-Papapetrou-type, cosmological, spherically symmetric, black-brane and Freund-Rubin-type ones) are considered. 
  Multiple scale techniques are well-known in classical mechanics to give perturbation series free from resonant terms. When applied to the quantum anharmonic oscillator, these techniques lead to interesting features concerning the solution of the Heisenberg equations of motion and the Hamiltonian spectrum. 
  A-statistics is defined in the context of the Lie algebra sl(n+1). Some thermal properties of A-statistics are investigated under the assumption that the particles interact only via statistical interaction imposed by the Pauli principle of A-statistics. Apart from the general case, three particular examples are studied in more detail: (a) the particles have one and the same energy and chemical potential; (b) equidistant energy spectrum; (c) two species of particles with one and the same energy and chemical potential within each class. The grand partition functions and the average number of particles are among the thermodynamical quantities written down explicitly. 
  We consider the possible consistent truncation of N-extended supergravities to lower N' theories. The truncation, unlike the case of N-extended rigid theories, is non trivial and only in some cases it is sufficient just to delete the extra N-N' gravitino multiplets. We explore different cases (starting with N=8 down to N'\geq 2) where the reduction implies restrictions on the matter sector. We perform a detailed analysis of the interesting case N=2 \to N=1. This analysis finds applications in different contexts of superstring and M-theory dynamics. 
  A Green's function approach is presented for the D-dimensional inverse square potential in quantum mechanics. This approach is implemented by the introduction of hyperspherical coordinates and the use of a real-space regulator in the regularized version of the model. The application of Sturm-Liouville theory yields a closed expression for the radial energy Green's function. Finally, the equivalence with a recent path-integral treatment of the same problem is explicitly shown. 
  The possibility of a small modification of spinor Quantum Electro-Dynamics is reconsidered, in which Lorentz and CPT non-covariant kinetic terms for photons and fermions are present. The corresponding free field theory is carefully discussed. The finite one-loop parity-odd induced effective action is unambiguously calculated using the physical cutoff method, which manifestly encodes the maximal residual symmetry group allowed by the presence of the Lorentz and CPT breaking axial-vector. This very same induced effective action, which is different from those ones so far quoted in the Literature, is also re-derived by means of the dimensional regularization, provided the maximal residual symmetry is maintained in the enlarged $D$-dimensional space-time. As a consequence, it turns out that the requirement of keeping the maximal residual symmetry at the quantum level just corresponds to the physical renormalization prescription which naturally fixes the one-loop parity-odd induced effective action. 
  We construct, using harmonic superspace and the quaternionic quotient approach, a quaternionic-K\"ahler extension of the most general two centres hyper-K\"ahler metric. It possesses $U(1)\times U(1)$ isometry, contains as special cases the quaternionic-K\"ahler extensions of the Taub-NUT and Eguchi-Hanson metrics and exhibits an extra one-parameter freedom which disappears in the hyper-K\"ahler limit. Some emphasis is put on the relation between this class of quaternionic-K\"ahler metrics and self-dual Weyl solutions of the coupled Einstein-Maxwell equations. The relation between our explicit results and the recent general ansatz of Calderbank and Pedersen for quaternionic-K\"ahler metrics with $U(1)\times U(1)$ isometries is traced in detail. 
  In this talk I review the recent construction of a new family of classical BPS solutions of type IIB supergravity describing 3-branes transverse to a 6-dimensional space with topology $\mathbb{R}^{2}\times$ALE. They are characterized by a non-trivial flux of the supergravity 2-forms through the homology 2-cycles of a generic smooth ALE manifold. These solutions have two Killing spinors and thus preserve $\mathcal{N}=2$ supersymmetry. They are expressed in terms of a quasi harmonic function $H$ (the ``warp factor''), whose properties was studied in detail in the case of the simplest ALE, namely the Eguchi-Hanson manifold. The equation for $H$ was identified as an instance of the confluent Heun equation. 
  The universal amplitude ratio $\tilde{R}_{\xi}$ for percolation in two dimensions is determined exactly using results for the dilute A model in regime 1, by way of a relationship with the q-state Potts model for q<4. 
  A detailed derivation from first principles is given for the unambiguous and slice-independent formula for the two-loop superstring chiral measure which was announced in the first paper of this series. Supergeometries are projected onto their super period matrices, and the integration over odd supermoduli is performed by integrating over the fibers of this projection. The subtleties associated with this procedure are identified. They require the inclusion of some new finite-dimensional Jacobian superdeterminants, a deformation of the worldsheet correlation functions using the stress tensor, and perhaps paradoxically, another additional gauge choice, ``slice \hat\mu choice'', whose independence also has to be established. This is done using an important correspondence between superholomorphic notions with respect to a supergeometry and holomorphic notions with respect to its super period matrix. Altogether, the subtleties produce precisely the corrective terms which restore the independence of the resulting gauge-fixed formula under infinitesimal changes of gauge-slice. This independence is a key criterion for any gauge-fixed formula and hence is verified in detail. 
  The 'anholonomic frame' method (see gr-qc/0005025, gr-qc/0001060 and hep-th/0110250) is applied for constructing new classes of exact solutions of vacuum Einstein equations with off-diagonal metrics in 4D and 5D gravity. We examine several black tori solutions generated by anholonomic transforms with non-trivial topology of the Schwarzshild metric, which have a static toroidal horizon. We define ansatz and parametrizations which contain warping factors, running constants (in time and extra dimension coordinates) and effective nonlinear gravitational polarizations. Such anisotropic vacuum toroidal metrics, the first example was given in gr-qc/0005025, differ substantially from the well known toroidal black holes (see hep-th/9511188 and gr-qc/9709013) which were constructed as non-vacuum solutions of the Einstein-Maxwell gravity with cosmological constant. Finally, we analyze two anisotropic 5D and 4D black tori solutions with cosmological constant. 
  Goldstone's theorem states that there is a massless mode for each broken symmetry generator. It has been known for a long time that the naive generalization of this counting fails to give the correct number of massless modes for spontaneously broken spacetime symmetries. We explain how to get the right count of massless modes in the general case, and discuss examples involving spontaneously broken Poincare and conformal invariance. 
  We propose an alternative inflationary universe scenario in the context of Randall-Sundrum braneworld cosmology. In this new scenario the existence of extra-dimension(s) plays an essential role. First, the brane universe is initially in the inflationary phase driven by the effective cosmological constant induced by small mismatch between the vacuum energy in the 5-dimensional bulk and the brane tension. This mismatch arises since the bulk is initially in a false vacuum. Then, the false vacuum decay occurs, nucleating a true vacuum bubble with negative energy inside the bulk. The nucleated bubble expands in the bulk and consequently hits the brane, bringing a hot big-bang brane universe of the Randall-Sundrum type. Here, the termination of the inflationary phase is due to the change of the bulk vacuum energy. The bubble kinetic energy heats up the universe. As a simple realization, we propose a model, in which we assume an interaction between the brane and the bubble. We derive the constraints on the model parameters taking into account the following requirements: solving the flatness problem, no force which prohibits the bubble from colliding with the brane, sufficiently high reheating temperature for the standard nucleosynthesis to work, and the recovery of Newton's law up to 1mm. We find that a fine tuning is needed in order to satisfy the first and the second requirements simultaneously, although, the other constraints are satisfied in a wide range of the model parameters. 
  We compare N=2 string and N=4 topological string within the framework of the sigma model approach. Being classically equivalent on a flat background, the theories are shown to lead to different geometries when put in a curved space. In contrast to the well studied Kaehler geometry characterising the former case, in the latter case a manifold has to admit a covariantly constant holomorphic two-form in order to support an N=4 twisted supersymmetry. This restricts the holonomy group to be a subgroup of SU(1,1) and leads to a Ricci--flat manifold. We speculate that, the N=4 topological formalism is an appropriate framework to smooth down ultraviolet divergences intrinsic to the N=2 theory. 
  We discuss gauge-fixing, propagators and effective potentials for topological A-brane composites in Calabi-Yau compactifications. This allows for the construction of a holomorphic potential describing the low-energy dynamics of such systems, which generalizes the superpotentials known from the ungraded case. Upon using results of homotopy algebra, we show that the string field and low energy descriptions of the moduli space agree, and that the deformations of such backgrounds are described by a certain extended version of `off-shell Massey products' associated with flat graded superbundles. As examples, we consider a class of graded D-brane pairs of unit relative grade. Upon computing the holomorphic potential, we study their moduli space of composites. In particular, we give a general proof that such pairs can form acyclic condensates, and, for a particular case, show that another branch of their moduli space describes condensation of a two-form. 
  It is often convenient to view Hawking (black hole) radiation as a process of quantum tunneling. Within this framework, Kraus and Wilczek (KW) have initiated an analytical treatment of black hole emission. Notably, their methodology incorporates the effects of a dynamical black hole geometry. In the current paper, the KW formalism is applied to the case of a charged BTZ black hole. In the context of this interesting model, we are able to demonstrate a non-thermal spectrum, with the usual Hawking result being reproduced at zeroth order in frequency. Considerable attention is then given to the examination of near-extremal thermodynamics. 
  Using the heat kernel method and the analytic continuation of the zeta function, we calculate the canonical and improved vacuum stress tensors, ${T_{\mu \nu}(\vec{x})}$ and ${\Theta_{\mu \nu}(\vec{x})}$, associated with a massless scalar field confined in the interior of an infinitely long rectangular waveguide. The local depence of the renormalized energy for two special configurations when the total energy is positive and negative are presented using ${T_{00}(\vec{x})}$ and ${\Theta_{00}(\vec{x})}$. From the stress tensors we obtain the local Casimir forces in all walls by introducing a particular external configuration. It is shown that this external configuration cannot give account of the edge divergences of the local forces. The local form of the forces is obtained for three special configurations. 
  Using the Hopf fibration and starting from a four dimensional noncommutative Moyal plane, $R^2_{\theta}\times R^2_{\theta}$, we obtain a star-product for the noncommutative (fuzzy) $R^3_{\lambda}$ defined by $[x^i,x^j]=i\lambda\epsilon_{ijk}x^k$. Furthermore, we show that there is a projection function which allows us to reduce the functions on $R^3_{\lambda}$ to that of the fuzzy sphere, and hence we introduce a new star-product on the fuzzy sphere. We will then briefly discuss how using our method one can extract information about the field theory on fuzzy sphere and $\rrlam$ from the corresponding field theories on $R_{theta}\times R_{\theta}$ space. 
  I study various properties of the critical limits of correlators containing insertions of conserved and anomalous currents. In particular, I show that the improvement term of the stress tensor can be fixed unambiguously, studying the RG interpolation between the UV and IR limits. The removal of the improvement ambiguity is encoded in a variational principle, which makes use of sum rules for the trace anomalies a and a'. Compatible results follow from the analysis of the RG equations. I perform a number of self-consistency checks and discuss the issues in a large set of theories. 
  In this paper we shall investigate the possibility of solving U(1) theories on the non-commutative (NC) plane for arbitrary values of $\theta$ by exploiting Morita equivalence. This duality maps the NC U(1) on the two-torus with a rational parameter $\theta$ to the standard U(N) theory in the presence of a 't Hooft flux, whose solution is completely known. Thus, assuming a smooth dependence on $\theta$, we are able to construct a series rational approximants of the original theory, which is finally reached by taking the large $N-$limit at fixed 't Hooft flux. As we shall see, this procedure hides some subletities since the approach of $N$ to infinity is linked to the shrinking of the commutative two-torus to zero-size. The volume of NC torus instead diverges and it provides a natural cut-off for some intermediate steps of our computation. In this limit, we shall compute both the partition function and the correlator of two Wilson lines. A remarkable fact is that the configurations, providing a finite action in this limit, are in correspondence with the non-commutative solitons (fluxons) found independently by Polychronakos and by Gross and Nekrasov, through a direct computation on the plane. 
  We investigate the decay of a spherically symmetric near-extremal charged black hole, including back-reaction effects, in the near-horizon region. The non-locality of the effective action controlling this process allows and also forces us to introduce a complementary set of boundary conditions which permit to determine the asymptotic late time Hawking flux. The evaporation rate goes down exponentially and admits an infinite series expansion in Planck's constant. At leading order it is proportional to the total mass and the higher order terms involve higher order momenta of the classical stress-tensor. Moreover we use this late time behaviour to go beyond the near-horizon approximation and comment on the implications for the information loss paradox. 
  In this letter we show that in the extra dimension model, contrary to the widely accepted conception, the simply truncated $\phi^4$ and non-Abelian SU(N) Kaluza-Klein theories are not renormalizable, i.e. the tree level relations of the effective theories can not sustain the quantum corrections. The breaking down of the tree level relations of the effective theories can be traced back to several factors: the breaking of the higher dimension Lorentz symmetry and higher dimension gauge symmetry, interactions assumed in the underlying Lagrangians, and the dimension reduction and rescaling procedure. 
  The Standard Model plus gravitation is derived from a Kaluza-Klein theory of pure gravitation in six dimensions with two higher dimensions of time. 3DT is different than other Kaluza-Klein theories because it allows dependence upon the higher-dimensional coordinates. It shows how predictions at the Planck mass can be tested at low energies. 3DT explains the origins of the elementary particles, Maxwell's equations, the Dirac equation, the weak interactions and the strong interactions. Quark confinement and aymptotic freedom are produced. 3DT provides an explanation for the mass of the electron, the value 1/137 of the fine structure constant, the masses of the muon and tau, the masses of the electron's, muon's and tau's neutrinos, the masses of the W, Z and the photon. The calculation of these parameters is made possible by a better way of doing quantum field theory. 3DT is anomaly-free. The relationship between quantum mechanics and general relativity is demonstrated. 3DT predicts that there is no Higgs particle and no supersymmetric particles. Instead, it predicts that there are seven new, superweak vectors with masses of 4.56 TeV, 7.32 TeV, 27.36 TeV, 29.43 TeV, 31.22 TeV, 33.04 TeV and 38.79 TeV. 
  We show how to translate boundary conditions into constraints in the symplectic quantization method by an appropriate choice of generalized variables. This way the symplectic quantization of an open string attached to a brane in the presence of an antisymmetric background field reproduces the non commutativity of the brane coordinates. 
  We investigate the possibility of obtaining localized black hole solutions in brane worlds by introducing a dependence of the four-dimensional line--element on the extra dimension. An analysis, performed for the cases of an empty bulk and of a bulk containing either a scalar or a gauge field, reveals that no conventional type of matter can support such a dependence. Considering a particular ansatz for the five-dimensional line--element that corresponds to a black hole solution with a ``decaying'' horizon, we determine the bulk energy--momentum tensor capable of sustaining such a behaviour. It turns out that an exotic, shell-like distribution of matter is required. For such solutions, the black hole singularity is indeed localized near the brane and the spacetime is well defined near the AdS horizon, in contrast to the behaviour found in black string type solutions. 
  We consider a brane world scenario which arises as the near-horizon region of a non-extremal D5-brane. There is a quasi-localized massive graviton mode, as well as harmonic modes of higher mass which are bound to the brane to a lesser degree. Lorentz invariance is slightly broken, which may have observable effects due to the leakage of the metastable graviton states into the bulk. Unlike a brane world arising from an extremal D5-brane, there is no mass gap. We also find that a brane world arising from a non-extremal M5/M5-brane intersection has the same graviton dynamics as that of a non-extremal D5-brane. This is evidence that a previously conjectured duality relation between the dual quantum field theories of each p-brane background may hold away from extremality. 
  We analyze the algebra of observables and the physical Fock space of the finite Chern-Simons matrix model. We observe that the minimal algebra of observables acting on that Fock space is identical to that of the Calogero model. Our main result is the identification of the states in the l-th tower of the Chern-Simons matrix model Fock space and the states of the Calogero model with the interaction parameter nu=l+1. We describe quasiparticle and quasihole states in the both models in terms of Schur functions, and discuss some nontrivial consequences of our algebraic approach. 
  The N_c to infinity limit of a matrix quantum field theory is equivalent to summing only planar Feynman diagrams. The possibility of interpreting this sum as some kind world-sheet theory has been in the air ever since 't Hooft's original paper. We establish here just such a world sheet description for a scalar quantum field with interaction term g\Tr\phi^3, and we indicate how the approach might be extended to more general field theories. 
  We study conformal field theories for strings propagating on compact, seven-dimensional manifolds with G_2 holonomy. In particular, we describe the construction of rational examples of such models. We argue that analogues of Gepner models are to be constructed based not on N=1 minimal models, but on Z_2 orbifolds of N=2 models. In Z_2 orbifolds of Gepner models times a circle, it turns out that unless all levels are even, there are no new Ramond ground states from twisted sectors. In examples such as the quintic Calabi-Yau, this reflects the fact that the classical geometric orbifold singularity can not be resolved without violating G_2 holonomy. We also comment on supersymmetric boundary states in such theories, which correspond to D-branes wrapping supersymmetric cycles in the geometry. 
  We compute leading order corrections to the entropy of any thermodynamic system due to small statistical fluctuations around equilibrium. When applied to black holes, these corrections are shown to be of the form $-k\ln(Area)$. For BTZ black holes, $k=3/2$, as found earlier. We extend the result to anti-de Sitter Schwarzschild and Reissner-Nordstrom black holes in arbitrary dimensions. Finally we examine the role of conformal field theory in black hole entropy and its corrections. 
  We study the geometry of orientifolds in the SU(2) WZW model. They correspond to the two inequivalent, orientation-reversing involutions of $S^3$, whose fixed-point sets are: the north and south poles (O0), or the equator two-sphere (O2). We show how the geometric action of these involutions leads unambiguously to the previously obtained algebraic results for the Klein bottle and Moebius amplitudes. We give a semiclassical derivation of the selection rules and signs in the crosscap couplings, paying particular attention to discrete B-fluxes.  A novel observation, which does not follow from consistency of the one-loop vacuum diagrams, is that in the case of the O0 orientifolds only integer- or only half-integer-spin Cardy states may coexist. 
  On Riemannian signature conformal 4-manifolds we give a conformally invariant extension of the Maxwell operator on 1-forms. We show the extension is in an appropriate sense injectively elliptic, and recovers the invariant gauge operator of Eastwood and Singer. The extension has a natural compatibility with the de Rham complex and we prove that, given a certain restriction, its conformally invariant null space is isomorphic to the first de Rham cohomology. General machinery for extending this construction is developed and as a second application we describe an elliptic extension of a natural operator on perturbations of conformal structure. This operator is closely linked to a natural sequence of invariant operators that we construct explictly. In the conformally flat setting this yields a complex known as the conformal deformation complex and for this we describe a conformally invariant Hodge theory which parallels the de Rham result. 
  String propagation on a cone with deficit angle $2\pi(1-{1\over N})$ is considered for the purpose of computing the entropy of a large mass black hole. The entropy computed using the recent results on condensation of twisted-sector tachyons in this theory is found to be in precise agreement with the Bekenstein-Hawking entropy. 
  A new approach for treating boundary Poisson structures based on causality and locality analysis is proposed for a single scalar field with boundary interaction. For the case of linear boundary condition, it is shown that the usual canonical quantization can be applied systematically. 
  The ambiguities of the Seiberg-Witten map for gauge field coupled with fermionic matter are discussed. We find that only part of the ambiguities can be absorbed by gauge transformation and/or field redefinition and thus are negligible. The existence of matter field makes some other part of the ambiguities difficult to be absorbed by gauge transformation or field redefinition. 
  We show that flavor 't Hooft anomalies automatically vanish in noncommutative field theories which are obtained from string theory in the decoupling limit. We claim that this is because the flavor symmetries are secretly local, because of coupling to closed string bulk modes. An example is the SU(4) R-symmetry of N=4 D=4 NCSYM. The gauge fields, along with all closed string bulk modes, are not on-shell external states but do appear as off-shell intermediate states in non-planar processes; these closed string modes are thereby holographically encoded in the NCFT. 
  The relationship between the entropy of de Sitter (dS) Schwarzschild space and that of the CFT, which lives on the brane, is discussed by using Friedmann-Robertson-Walker (FRW) equations and Cardy-Verlinde formula. The cosmological constant appears on the brane with time-like metric in dS Schwarzschild background. On the other hand, in case of the brane with space-like metric in dS Schwarzschild background, the cosmological constant of the brane does not appear because we can choose brane tension to cancel it. We show that when the brane crosses the horizon of dS Schwarzschild black hole, both for time-like and space-like cases, the entropy of the CFT exactly agrees with the black hole entropy of 5-dimensional AdS Schwarzschild background as it happens in the AdS/CFT correspondence. 
  The problem of the restoring of the equivalence between Light-Front (LF) Hamiltonian and conventional Lorentz-covariant formulations of gauge theory is solved for QED(1+1) and (perturbatively to all orders) for QCD(3+1). For QED(1+1) the LF Hamiltonian is constructed which reproduces the results of Lorentz-covariant theory. This is achieved by bosonization of the model and by analysing the resulting bosonic theory to all orders in the fermion mass. For QCD(3+1) we describe nonstandard regularization that allows to restore mentioned equivalence with finite number of counterterms in LF Hamiltonian. 
  We find a relationship between the partition function mass zeros and the spectral properties of the QCD Dirac operator in the context of chiral Random Matrix Theory. Introducing the concept of normal modes we see that certain features of the QCD partition function are universal. 
  We present three groups of examples of Wigner Quantum Systems related to the Lie superalgebras $osp(1/6n)$, $sl(1/3n)$ and $sl(n/3)$ and discuss shortly their physical features. In the case of $sl(1/3n)$ we indicate that the underlying geometry is noncommutative. 
  We study the type II string theories compactified on manifolds of $G_2$ holonomy of the type $({Calabi-Yau 3-fold} \times S^1)/\bz_2$ where $CY_3$ sectors realized by the Gepner models. We construct modular invariant partition functions for $G_2$ manifold for arbitrary Gepner models of the Calabi-Yau sector. We note that the conformal blocks contain the tricritical Ising model and find extra massless states in the twisted sectors of the theory when all the levels $k_i$ of minimal models in Gepner constructions are even. 
  We discuss different dualities of QHE in the framework of the noncommutative Chern-Simons theory. First, we consider the Morita or T-duality transformation on the torus which maps the abelian noncommutative CS description of QHE on the torus into the nonabelian commutative description on the dual torus. It is argued that the Ruijsenaars integrable many-body system provides the description of the QHE with finite amount of electrons on the torus. The new IIB brane picture for the QHE is suggested and applied to Jain and generalized hierarchies. This picture naturally links 2d $\sigma$-model and 3d CS description of the QHE. All duality transformations are identified in the brane setup and can be related with the mirror symmetry and S duality. We suggest a brane interpretation of the plateu transition in IQHE in which a critical point is naturally described by $SL(2,R)$ WZW model. 
  The Hamiltonian formulation of the dynamics of a relativistic particle described by a higher-derivative action that depends both on the first and the second Frenet-Serret curvatures is considered from a geometrical perspective. We demonstrate how reparametrization covariant dynamical variables and their projections onto the Frenet-Serret frame can be exploited to provide not only a significant simplification of but also novel insights into the canonical analysis. The constraint algebra and the Hamiltonian equations of motion are written down and a geometrical interpretation is provided for the canonical variables. 
  We investigate several models described by real scalar fields, searching for topological defects, and investigating their linear stability. We also find bosonic zero modes and examine the thermal corrections at the one-loop level. The classical investigations are of interest to high energy physics and applications in condensed matter, in particular to spatially extended systems where fronts and interfaces separating different phase states may appear. The thermal investigations show that the finite temperature corrections that appear in a specific model induce a second-order phase transition in the system, although the thermal effects do not suffice to fully restore the symmetry at high temperature. 
  The chiral superstring measure constructed in the earlier papers of this series for general gravitino slices is examined in detail for slices supported at two points x_\alpha. In this case, the invariance of the measure under infinitesimal changes of gravitino slices established previously is strengthened to its most powerful form: the measure is shown, point by point on moduli space, to be locally and globally independent from the points x_\alpha, as well as from the superghost insertion points p_a, q_\alpha introduced earlier as computational devices. In particular, the measure is completely unambiguous. The limit x_\alpha = q_\alpha is then well defined. It is of special interest, since it elucidates some subtle issues in the construction of the picture-changing operator Y(z) central to the BRST formalism. The formula for the chiral superstring measure in this limit is derived explicitly. 
  We construct a supergravity dual to the cascading $SU(N+M) x SU(N)$ supersymmetric gauge theory (related to fractional D3-branes on conifold according to Klebanov et al) in the case when the 3-space is compactified on $S^3$ and in the phase with unbroken chiral symmetry. The size of $S^3$ serves as an infrared cutoff on the gauge theory dynamics. For a sufficiently large $S^3$ the dual supergravity background is expected to be nonsingular. We demonstrate that this is indeed the case: we find a smooth type IIB supergravity solution using a perturbation theory that is valid when the radius of $S^3$ is large. We consider also the case with the euclidean world-volume being $S^4$ instead of $R x S^3$, where the supergravity solution is again found to be regular. This ``curved space'' resolution of the singularity of the fractional D3-branes on conifold solution is analogous to the one in the non-extremal (finite temperature) case discussed in our previous work. 
  A definition of non-abelian genus zero open Wilson surfaces is proposed. The ambiguity in surface-ordering is compensated by the gauge transformations. 
  Quantum field theory in 4+1 dimensional bulk space with boundary, representing a 3-brane, is considered. We study the impact of the initial conditions in the bulk on the field dynamics on the brane. We demonstrate that these conditions determine the Kaluza-Klein measure. We also establish the existence of a rich family of quantum fields on the brane, generated by the same bulk action, but corresponding to different initial conditions. A simple classification of these fields is proposed and it is shown that some of them lead to ultraviolet finite theories, which have some common features with strings. 
  We derive an explicit expression for an associative star product on non-commutative versions of complex Grassmannian spaces, in particular for the case of complex 2-planes. Our expression is in terms of a finite sum of derivatives. This generalises previous results for complex projective spaces and gives a discrete approximation for the Grassmannians in terms of a non-commutative algebra, represented by matrix multiplication in a finite-dimensional matrix algebra. The matrices are restricted to have a dimension which is precisely determined by the harmonic expansion of functions on the commutative Grassmannian, truncated at a finite level. In the limit of infinite-dimensional matrices we recover the commutative algebra of functions on the complex Grassmannians. 
  Principles of a new approach (binary geometrophysics) are presented to construct the unified theory of spacetime and the familiar kinds of physical interactions. Physically, the approach is a modified S-matrix theory involving ideas of the multidimensional geometric models of physical interactions of Kaluza-Klein's type as well as Fokker-Feynman's action-at-a-distance theory. Mathematically, this is a peculiar binary geometry being described in algebraic terms. In the present approach the binary geometry volume is a prototype of three related notions: the S-matrix, the physical action (Lagrangian) of both strong and electroweak interactions, and the multidimensional metric. A transition from microworld geometrophysics to the conventional physical theory in classical spacetime are characterized. 
  We perform the exact renormalization of two-dimensional massless gauge theories. Using these exact results we discuss the cluster property and confinement in both the anomalous and chiral Schwinger models. 
  The Variational Method is applied within the context of Supersymmetric Quantum Mechanics to provide information about the energy states of a hydrogenic donor in a quantum dot. 
  We investigate the modular transformation properties of observable (light) fields in heterotic orbifolds, in the light of recent calculations of CP-violating quantities. Measurable quantities must be modular invariant functions of string moduli, even if the light fields are noninvariant. We show that physical invariance may arise by patching smooth functions that are separately noninvariant. CP violation for <T> on the unit circle, which requires light and heavy states to mix under transformation, is allowed in principle, although the Jarlskog parameter J_CP(T) must be amended relative to previous results. However, a toy model of modular invariant mass terms indicates that the assumption underlying these results is unrealistic. In general the mass eigenstate basis is manifestly modular invariant and coupling constants are smooth invariant functions of T, thus CP is unbroken on the unit circle. We also discuss the status of CP-odd quantities when CP is a discrete gauge symmetry, and point out a link with baryogenesis. 
  We analyse the cosmological implications of brane-antibrane systems in string-theoretic orbifold and orientifold models. In a class of realistic models, consistency conditions require branes and antibranes to be stuck at different fixed points, and so their mutual attraction generates a potential for one of the radii of the underlying torus or the 4D string dilaton. Assuming that all other moduli have been fixed by string effects, we find that this potential leads naturally to a period of cosmic inflation with the radion or dilaton field as the inflaton. The slow-roll conditions are satisfied more generically than if the branes were free to move within the space. The appearance of tachyon fields at certain points in moduli space indicates the onset of phase transitions to different non-BPS brane systems, providing ways of ending inflation and reheating the corresponding observable brane universe. In each case we find relations between the inflationary parameters and the string scale to get the correct spectrum of density perturbations. In some examples the small numbers required as inputs are no smaller than 0.01, and are the same small quantities which are required to explain the gauge hierarchy. 
  D2-branes are studied in the context of Born-Infeld theory as a source of the 3-form RR gauge potential. Considering the static case with only a radial magnetic field it is shown that a locally stable hemispherical deformation of the brane exists which minimises the energy locally. Since the D2-brane carries also the charge of D0-branes, and the RR spacetime potential is unbounded from below, these can tunnel to condense on the D2-brane. The corresponding instanton-like configuration and the tunneling rate are derived and discussed. 
  It is well-known that ($N$, $M$) 5-branes of type IIB supergravity form a non-threshold bound state with ($N'$, $M'$) strings called the (F, D1, NS5, D5) bound state where the strings lie along one of the spatial directions of the 5-branes (hep-th/9905056). By taking low energy limits in appropriate ways on this supergravity configuration, we obtain the supergravity duals of various decoupled theories in (5+1) dimensions corresponding to noncommutative open string (NCOS) theory, open D-string (OD1) theory and open D5-brane (OD5) theory. We then study the $SL(2, Z)$ transformation properties of these theories. We show that when the asymptotic value of the axion ($\chi_0$) is rational (for which $\chi_0$ can be put to zero), NCOS theory is always related to OD1 theory by strong-weak or S-duality symmetry. We also discuss the self-duality conjecture (hep-th/0006062) of both NCOS and OD1 theories. On the other hand, when $\chi_0$ is irrational, we find that $SL(2, Z)$ duality on NCOS theory gives another NCOS theory with different values of the parameters, but for OD1 theory $SL(2,Z)$ duality always gives an NCOS theory. $SL(2, Z)$ transformation on OD5 theory reveals that it gives rise to Little String Theory (LST) when $\chi_0$ = rational, but it gives another OD5 theory with different values of the parameters when $\chi_0$ is irrational. 
  The globalization of Kontsevich's local formula (resp., the perturbative expansion of the Poisson sigma model) is described in down-to-earth terms. 
  Based on a general analysis of Green functions in the real-time thermal field theory, we have proven that the four-point amputated functions in a NJL model in the fermion bubble diagram approximation behave like usual two-point functions. We expound the thermal transformations of the matrix propagator for a scalar bound state in the $F\bar{F}$ basis and in the $RA$ basis. The resulting physical causal, advanced and retarded propagator are respectively identical to corresponding ones derived in the imaginary-time formalism and this shows once again complete equivalence of the two formalisms of thermal field theory on the discussed problem in the NJL model. 
  We propose a cosmological model in which the universe undergoes an endless sequence of cosmic epochs each beginning with a `bang' and ending in a `crunch.' The temperature and density are finite at each transition from crunch to bang. Instead of having an inflationary epoch, each cycle includes a period of slow accelerated expansion (as recently observed) followed by slow contraction. The combination produces the homogeneity, flatness, density fluctuations and energy needed to begin the next cycle. 
  We discuss Kowalski-Glikman type pp-wave solutions with unbroken supersymmetry in 6 and 5 dimensional supergravity theories. 
  It is known that there is a quantum mechanical tunneling process which, through nucleation of a membrane, induces a transition between two de Sitter spaces, lowering the cosmological constant. It is shown in this paper that a different, new, membrane nucleation process exists, which, in addition to lowering the cosmological constant, leaves a black hole behind. Once a black hole is present, the relaxation of the cosmological constant may proceed via an analog of the old process, which decreases the black hole horizon area, or via the new process, which increases it. 
  The ``Laughlin'' picture of the Fractional Quantum Hall effect can be derived using the ``exotic'' model based on the two-fold centrally-extended planar Galilei group. When coupled to a planar magnetic field of critical strength determined by the extension parameters, the system becomes singular, and ``Faddeev-Jackiw'' reduction yields the ``Chern-Simons'' mechanics of Dunne, Jackiw, and Trugenberger. The reduced system moves according to the Hall law. 
  We reveal a novel mathematical structure in physical observables, the mass of tachyon fluctuation mode and the energy density, associated with a classical solution of vacuum string field theory constructed previously [hep-th/0108150]. We find that they are expressed in terms of quantities which apparently vanish identically due to twist even-odd degeneracy of eigenvalues of a Neumann coefficient matrix defining the three-string interactions. However, they can give non-vanishing values because of the breakdown of the degeneracy at the edge of the eigenvalue distribution. We also present a general prescription of correctly simplifying the expressions of these observables. Numerical calculation of the energy density following our prescription indicates that the present classical solution represents the configuration of two D25-branes. 
  New exact solutions of Einstein equations which describe black hole with radial cosmic strings are constructed in the paper. The case of infinitely thin strings and the case of delocalized strings are considered. The case of delocalized strings allows generalization to dimensions greater than 4. 
  Using the superembedding formalism we construct supermembrane actions with higher derivative terms which can be viewed as possible higher order terms in effective actions. In particular, we provide the first example of an action for an extended supersymmetric object with fully $\kappa$-symmetric extrinsic curvature terms. 
  A role of the renormalization group invariance in calculations of the ground state energy for models with confined fermion fields is discussed. The case of the (1+1)D MIT bag with massive fermions is studied in detail. 
  We present a brief introduction to the construction of gauge theories on noncommutative spaces with star products. Particular emphasis is given to issues related to non-Abelian gauge groups and charge quantization. This talk is based on joined work with B. Jurco, J. Madore, L. Moeller, S. Schraml and J. Wess. 
  Many years ago Friedman and Sorkin [1] established the existence of certain topological solitonic excitations in quantum gravity called (topological) geons. Geons can have quantum numbers like charge and can be tensorial or spinorial having integer or half-odd integer spin. A striking result is that geons can violate the canonical spin-statistics connection [2,3]. Such violation induces novel physical effects at low energies. The latter will be small since the geon mass is expected to be of the order of Plank mass. Nevertheless, these effects are very striking and include CPT and causality violations and distortion of the cosmic microwave spectrum. Interesting relations of geon dynamics to supersymmetry are also discussed. 
  The slice-independent gauge-fixed superstring chiral measure in genus 2 derived in the earlier papers of this series for each spin structure is evaluated explicitly in terms of theta-constants. The slice-independence allows an arbitrary choice of superghost insertion points q_1, q_2 in the explicit evaluation, and the most effective one turns out to be the split gauge defined by S_{\delta}(q_1,q_2)=0. This results in expressions involving bilinear theta-constants M. The final formula in terms of only theta-constants follows from new identities between M and theta-constants which may be interesting in their own right. The action of the modular group Sp(4,Z) is worked out explicitly for the contribution of each spin structure to the superstring chiral measure. It is found that there is a unique choice of relative phases which insures the modular invariance of the full chiral superstring measure, and hence a unique way of implementing the GSO projection for even spin structure. The resulting cosmological constant vanishes, not by a Riemann identity, but rather by the genus 2 identity expressing any modular form of weight 8 as the square of a modular form of weight 4. The degeneration limits for the contribution of each spin structure are determined, and the divergences, before the GSO projection, are found to be the ones expected on physical grounds. 
  Consistent boundary Poisson structures for open string theory coupled to background $B$-field are considered using the new approach proposed in hep-th/0111005. It is found that there are infinitely many consistent Poisson structures, each leads to a consistent canonical quantization of open string in the presence of background $B$-field. Consequently, whether the $D$-branes to which the open string end points are attached is noncommutative or not depends on the choice of a particular Poisson structure. 
  We consider the fermionic bound states associated with a soliton-antisoliton pair in 1+1 dimensions which have zero energy when the solitons are infinitely far apart. We calculate the energies of these states when the solitons are separated by a finite distance. The energies decay exponentially with the distance between the soliton and antisoliton. When the fermion mass is much larger than the boson mass, the energy simplifies substantially. These energies may be interpreted as a contribution to the effective potential between the soliton and antisoliton. The character of this contribution depends upon which fermionic states are occupied. Performing the analogous calculation for the simplest (3+1)-dimensional soliton system, we find no fermionic energy shift. 
  We describe mirror symmetry on higher dimensional tori, paying special attention to the behaviour of D-branes under mirror symmetry. To find the mirror D-branes the description of mirror symmetry on D-branes due to Ooguri, Oz en Yin is used. This method allows us to deal with the coisotropic D-branes recently introduced by Kapustin and Orlov. We compare this to the description of mirror symmetry on D-branes using the Fourier-Mukai transform of charges. 
  The enhancon mechanism is studied in the heterotic string theory. We consider the N_L=0 winding strings with momentum (NS1-W*) and the Kaluza-Klein dyons (KK5-NS5*). The NS1-W* and KK5-NS5* systems are dualized to the D4-D0* and D6-D2* systems, respectively, under the d=6 heterotic/IIA S-duality. The heterotic form has a number of advantages over the type IIA form. We study these backgrounds and obtain the enhancon radii by brane probe analysis. The results are consistent with S-duality. 
  In this paper we apply the anholonomic frames method developed in refs. [1-4] to construct and study anisotropic vacuum field configurations in 5D gravity. Starting with an off--diagonal 5D metric, parameterized in terms of several ansatz functions, we show that using anholonomic frames greatly simplifies the resulting Einstein field equations. These simplified equations contain an interesting freedom in that one can chose one of the ansatz functions and then determine the remaining ansatz functions in terms of this choice. As examples we take one of the ansatz functions to be a solitonic solution of either the Kadomtsev-Petviashvili equation or the sine-Gordon equation. There are several interesting physical consequences of these solutions. First, a certain subclass of the solutions discussed in this paper have an exponential warp factor similar to that of the Randall-Sundrum model. However, the warp factor depends on more than just the 5$^{th}$ coordinate. In addition the warp factor arises from anisotropic vacuum solution rather than from any explicit matter. Second, the solitonic character of these solutions might allow them to be interpreted either as gravitational models for particles (i.e. analogous to the ' t Hooft-Polyakov monopole, but in the context of gravity), or as nonlinear, anisotropic gravitational waves. 
  We consider a five-dimensional supergravity model with SU(5) gauge symmetry and the minimal field content. Studying the arising scalar potential we find that the gauging of the $U(1)_R$ symmetry of the five-dimensional supergravity causes instabilities. Lifting the instabilities the vacua are of Anti-de-Sitter type and SU(5) is broken along with supersymmetry. Keeping the $U(1)_R$ ungauged the potential has flat directions along which supersymmetry is unbroken. 
  Hawking radiation is computed in different coordinate systems using the method of complex paths. In this procedure the event horizon of the 2D Schwarzschild stringy black hole is treated as a singularity for the semiclassical action functional. After the regularization of the event horizon's singularity the emission/absorption probabilities and the Hawking temperature in the different coordinate representations are derived. The identical results obtained indicate the covariance of the Hawking radiation. 
  We present a construction of superconformal field theories for manifolds with Spin(7) holonomy. Geometrically these models correspond to the realization of Spin(7) manifolds as anti-holomorphic quotients of Calabi-Yau fourfolds. Describing the fourfolds as Gepner models and requiring anomaly cancellation we determine the resulting Betti numbers of the Spin(7) superconformal field theory. As in the G_2 case, we find that the Gepner model and the geometric result disagree. 
  We study the space-time symmetries and transformation properties of the non-commutative U(1) gauge theory, by using Noether charges. We carry out our analysis by keeping an open view on the possible ways $\theta^{\mu \nu}$ could transform. We conclude that $\theta^{\mu \nu}$ cannot transform under any space-time transformation since the theory is not invariant under the conformal transformations, with the only exception of space-time translations. The same analysis applies to other gauge groups. 
  We consider type IIB string theory on a seven dimensional orbifold with holonomy in G2. The motivation is to use D1-branes as probes of the geometry. The low energy theory on the D1-brane is a sigma-model with two real supercharges (N = (1,1) in two dimensional language). We study in detail the closed and open string sectors and propose a coupling of the twisted fields to the brane that modifies the vacuum moduli space so that the singularity at the origin is removed. Instead of coming from D-terms, which are not present here, the modification comes from a ``twisted'' mass term for the seven scalar multiplets on the brane. The proposed mechanism involves a generalization of the moment map. 
  We study closed string tachyon condensation using the RG flow of the worldsheet theory. In many cases the worldsheet theory enjoys N=2 supersymmetry, which provides analytic control over the flow, due to non-renormalization theorems. Moreover, Mirror symmetry sheds light on the RG flow in such cases.   We discuss the relevant tachyon condensation in the context of both compact and non-compact situations which lead to very different conclusions. Furthermore, the tachyon condensation leads to non-trivial dualities for non-supersymmetric probe theories. 
  We study the cosmological evolution of a D3-brane Universe in a type 0 string background. We follow the brane universe along the radial coordinate of the background and we calculate the energy density which is induced on the brane because of its motion in the bulk. For constant values of tachyon and dilaton an inflationary phase is appearing. For non constant values of tachyon and dilaton and for a particular range of values of the scale factor of the brane-universe, the effective energy density is dominated by a term proportional to $\frac{1}{(\log\alpha)^{4}}$ indicating a slowly varying inflationary phase. 
  We prove, for an arbitrary finite root system, the periodicity conjecture of Al.B.Zamolodchikov concerning Y-systems, a particular class of functional relations arising in the theory of thermodynamic Bethe ansatz. Algebraically, Y-systems can be viewed as families of rational functions defined by certain birational recurrences formulated in terms of the underlying root system. In the course of proving periodicity, we obtain explicit formulas for all these rational functions, which turn out to always be Laurent polynomials.   In a closely related development, we introduce and study a family of simplicial complexes that can be associated to arbitrary root systems. In type A, our construction produces Stasheff's associahedron, whereas in type B, it gives the Bott-Taubes polytope, or cyclohedron. We enumerate the faces of these complexes, prove that their geometric realization is always a sphere, and describe them in concrete combinatorial terms for the classical types ABCD. 
  We study the Morita equivalence for fermion theories on noncommutative two-tori. For rational values of the $\theta$ parameter (in appropriate units) we show the equivalence between an abelian noncommutative fermion theory and a nonabelian theory of twisted fermions on ordinary space. We study the chiral anomaly and compute the determinant of the Dirac operator in the dual theories showing that the Morita equivalence also holds at this level. 
  In Brane Gas Cosmology (BGC) the initial state of the universe is taken to be small, dense and hot, with all fundamental degrees of freedom near thermal equilibrium. This starting point is in close analogy with the Standard Big Bang (SBB) model. In the simplest example, the topology of the universe is assumed to be toroidal in all nine spatial dimensions and is filled with a gas of $p$-branes. The dynamics of winding modes allow, at most, three spatial dimensions to become large, providing a possible explanation to the origin of our macroscopic (3+1)-dimensional universe. Specific solutions are found within the model that exhibit loitering, i.e. the universe experiences a short phase of slow contraction during which the Hubble radius grows larger than the physical extent of the universe. This phase is studied by combining the dilaton gravity background equations of motion with equations that determine the annihilation of string winding modes into string loops. Loitering provides a solution to the brane problem (generalised domain wall problem) in BGC and the horizon problem of the SBB scenario. In BGC the initial singularity problem of the SBB scenario is solved, without relying on an inflationary phase due to the presence of the T-duality symmetry in the theory. 
  We discuss the possible existence of new generic vacuum solutions of Robertson-Walker form in higher derivative gravity theories in four dimensions. These solutions illustrate how a dynamical coupling between very low and very high frequency modes can occur when the cosmological constant is small. 
  The non-linear behaviour of Randall-Sundrum gravity with one brane is examined. Due to the non-compact extra dimension, the perturbation spectrum has no mass gap, and the long wavelength effective theory is only understood perturbatively. The full 5-dimensional Einstein equations are solved numerically for static, spherically symmetric matter localized on the brane, yielding regular geometries in the bulk with axial symmetry. An elliptic relaxation method is used, allowing both the brane and asymptotic radiation boundary conditions to be simultaneously imposed. The same data that specifies stars in 4-dimensional gravity, uniquely constructs a 5-dimensional solution. The algorithm performs best for small stars (radius less than the AdS length) yielding highly non-linear solutions. An upper mass limit is observed for these small stars, and the geometry shows no global pathologies. The geometric perturbation is shown to remain localized near the brane at high densities, the confinement interestingly increasing for both small and large stars as the upper mass limit is approached. Furthermore, the static spatial sections are found to be approximately conformal to those of AdS. We show that the intrinsic geometry of large stars, with radius several times the AdS length, is described by 4-dimensional General Relativity far past the perturbative regime. This indicates that the non-linear long wavelength effective action remains local, even though the perturbation spectrum has no mass gap. The implication is that Randall-Sundrum gravity, with localized brane matter, reproduces relativistic astrophysical solutions, such as neutron stars and massive black holes, consistent with observation. 
  A brief overview is given of recent progress in understanding the dynamics of hot gauge theories. 
  We investigate quantum aspects of Gopakumar-Minwalla-Strominger (GMS) solutions of noncommutative field theory (NCFT) at large noncommutativity limit, $\theta \to \infty$. Building upon a quantitative map between operator formulation of 2-(respectively, (2+1))-dimensional NCFTs and large $N$ matrix models of $c=0$ (respectively, $c=1$) noncritical strings, we show that GMS solutions are quantum mechanically sensible only if we make appropriate joint scaling of $\theta$ and $N$. For 't Hooft's planar scaling, GMS solutions are replaced by large $N$ saddle-point solutions. GMS solutions are recovered from saddle-point solutions at small 't Hooft coupling regime, but are destabilized at large 'tHooft coupling regime by quantum effects. We make comparisons between these large $N$ effects and recently studied infrared effects in NCFTs. We estimate U(N) symmetry breaking gradient effects and argue that they are suppressed only at small 't Hooft coupling regime. 
  We find regular string duals of three dimensional N=1 SYM with a Chern-Simons interaction at level k for SO and Sp gauge groups. Using the string dual we exactly reproduce the conjectured pattern of supersymmetry breaking proposed by Witten by showing that there is dynamical supersymmetry breaking for k<h/2 while supersymmetry remains unbroken for k>= h/2, where h is the dual Coxeter number of the gauge group. We also find regular string duals of four dimensional N=1 SYM for SO and Sp gauge groups and exactly reproduce the expected pattern of chiral symmetry breaking Z_{2h}--> Z_2 by analyzing the symmetries of the string solution. 
  We perform a general reduction of the open membrane metric in a worldvolume direction of the M5-brane. Using reduction rules analogous to the bulk, we show that the open membrane metric leads to the standard open string metric and open string coupling constant on the D4-brane only for an ``electric'' reduction in which case the open membrane metric has no off-diagonal components and the Born-Infeld curvature tensor is a matrix of rank 2. Instead, if we perform a general reduction, with nonzero off-diagonal components of the open membrane metric, we obtain a rank 4 Born-Infeld tensor corresponding to a bound state of an open string with an open D2--brane. Next, we identify and reduce a 3-form open membrane ``noncommutativity'' tensor on the M5-brane. This open membrane parameter only reduces to the open string noncommutativity tensor on the D4-brane provided we constrain ourselves to an ``electric'' or a ``magnetic'' reduction. 
  Using the ${\cal N}=2$ superfield approach, we construct full ${\cal N}=4$ supersymmetric low-energy effective actions for ${\cal N}=4$ SYM models, with both ${\cal N}=2$ gauge superfield strengths and hypermultiplet superfields included. The basic idea is to complete the known non-holomorphic effective potentials which depend only on ${\cal N}=2$ superfield strengths $W$ and ${\bar W}$ to the full on-shell ${\cal N}=4$ invariants by adding the appropriate superfield hypermultiplet terms. We prove that the effective potentials of the form ${ln} W {ln} \bar W$ can be ${\cal N} = 4$ completed in this way and present the precise structure of the corresponding completions. However, the effective potentials of the non-logarithmic form suggested in hep-th/9811017 and hep-th/9909020 do not admit the ${\cal N}=4$ completion. Therefore, such potentials cannot come out as (perturbative or non-perturbative) quantum corrections in ${\cal N}=4$ SYM models. 
  In a generalized Heisenberg/Schroedinger picture we use an invariant space-time transformation to describe the motion of a relativistic particle. We discuss the relation with the relativistic mechanics and find that the propagation of the particle may be defined as space-time transition between states with equal eigenvalues of the first and second Casimir operators of the Lorentz algebra. In addition we use a vector on the light-cone. A massive relativistic particle with spin 0 is considered. We also consider the nonrelativistic limit. 
  We study field theories defined in regions of the spatial non-commutative (NC) plane with a boundary present delimiting them, concentrating in particular on the U(1) NC Chern-Simons theory on the upper half plane. We find that classical consistency and gauge invariance lead necessary to the introduction of $K_0$-space of square integrable functions null together with all their derivatives at the origin. Furthermore the requirement of closure of $K_0$ under the *-product leads to the introduction of a novel notion of the *-product itself in regions where a boundary is present, that in turn yields the complexification of the gauge group and to consider chiral waves in one sense or other. The canonical quantization of the theory is sketched identifying the physical states and the physical operators. These last ones include ordinary NC Wilson lines starting and ending on the boundary that yield correlation functions depending on points on the one-dimensional boundary. We finally extend the definition of the *-product to a strip and comment on possible relevance of these results to finite Quantum Hall systems. 
  We study and explore the symmetry properties of fermions coupled to dynamical torsion and electromagnetic fields. The stability of the theory upon radiative corrections as well as the presence of anomalies are investigated. 
  We consider the double-elliptic generalisation of dynamical systems of Calogero-Toda-Ruijsenaars type using finite-dimensional Mukai-Sklyanin algebras. The two-body system, which involves an elliptic dependence both on coordinates and momenta, is investigated in detail and the relation with Nambu dynamics is mentioned. We identify the 2D complex manifold associated with the double elliptic system as an elliptically fibered rational ("1/2K3 ") surface. Some generalisations are suggested which provide the ground for a description of the N-body systems. Possible applications to SUSY gauge theories with adjoint matter in $d=6$ with two compact dimensions are discussed. 
  We use inverse scattering methods, generalized for a specific class of complex potentials, to construct a one parameter family of complex potentials V(s, r) which have the property that the zero energy s-wave Jost function, as a function of s alone, is identical to Riemann's $\xi$ function whose zeros are the non-trivial zeros of the zeta function. These potentials have an asymptotic expansion in inverse powers of s(s-1) with real coefficients V_n(r) which are explicitly calculated. We show that the validity of the Riemann hypothesis depends essentially on simple integrability properties of the first order coefficient, V_1(r). In the case studied in this paper, this coefficient does not satisfy these conditions, but proof of that fact does indicate several possibilities for proceeding further. 
  We establish a correspondence between toroidal compactifications of M-theory and del Pezzo surfaces. M-theory on T^k corresponds to P^2 blown up at k generic points; Type IIB corresponds to P^1\times P^1. The moduli of compactifications of M-theory on rectangular tori are mapped to Kahler moduli of del Pezzo surfaces.The U-duality group of M-theory corresponds to a group of classical symmetries of the del Pezzo represented by global diffeomorphisms. The half-BPS brane charges of M-theory correspond to spheres in the del Pezzo, and their tension to the exponentiated volume of the corresponding spheres. The electric/magnetic pairing of branes is determined by the condition that the union of the corresponding spheres represent the anticanonical class of the del Pezzo. The condition that a pair of half-BPS states form a bound state is mapped to a condition on the intersection of the corresponding spheres. We present some speculations about the meaning of this duality. 
  We consider "sliver" states which act as projection operators in the matter star product of Witten's cubic string field theory. These sliver states, which might be associated with Dirichlet p-branes, are not finite norm states in the matter string Hilbert space. We describe the singularities of these states, and demonstrate that the sliver states are composed of strings having singular geometric features. These singularities take a particularly simple form in the zero slope limit alpha' -> 0, where the star algebra factorizes into a product of the algebra of functions on space-time and the noncommutative star product of fields associated with higher string modes. An analogy to the sliver geometry suggests a natural mechanism for describing closed string states in open string field theory. 
  We investigate the semiclassical instability of the Randall-Sundrum brane world. We carefully analyze the bubble solution with the Randall-Sundrum background, which expresses the decay of the brane world. We evaluate the decay probability following the Euclidean path integral approach to quantum gravity. Since a bubble rapidly expands after the nucleation, the entire spacetime will be occupied by such bubbles. 
  This article describes concepts and mechanisms used in porting of EPICS (Experimental Physical and Industrial Control System) codes to platform of operating system UNIX. Without destruction of EPICS architecture, new features of EPICS provides the support for real time operating system LynxOS/x86 and equipment produced by INP (Budker Institute of Nuclear Physics). Application of ported EPICS reduces the cost of software and hardware is used for automation of FEL (Free Electron Laser) complex. 
  The solutions of 10 and 11 dimensional supergravity that are warped products of de Sitter space with a non-compact `internal' space are investigated. A convenient form of the metric is found and it is shown that in each case the internal space is asymptotic to a cone over a product of spheres. A consistent truncation gives gauged supergravities with non-compact gauge groups. The BPS domain wall solutions of the non-compact gauged supergravities are lifted to warped solutions in 10 or 11 dimensions. 
  In the presence of a boundary interaction, Neumann boundary conditions should be modified to contain a function S of the boundary fields: (\nabla_N +S)\phi =0. Information on quantum boundary dynamics is then encoded in the $S$-dependent part of the effective action. In the present paper we extend the multiple reflection expansion method to the Robin boundary conditions mentioned above, and calculate the heat kernel and the effective action (i) for constant S, (ii) to the order S^2 with an arbitrary number of tangential derivatives. Some applications to symmetry breaking effects, tachyon condensation and brane world are briefly discussed. 
  The problem of interpretation of the \hbar^0-order part of radiative corrections to the effective gravitational field is considered. It is shown that variations of the Feynman parameter in gauge conditions fixing the general covariance are equivalent to spacetime diffeomorphisms. This result is proved for arbitrary gauge conditions at the one-loop order. It implies that the gravitational radiative corrections of the order \hbar^0 to the spacetime metric can be physically interpreted in a purely classical manner. As an example, the effective gravitational field of a black hole is calculated in the first post-Newtonian approximation, and the secular precession of a test particle orbit in this field is determined. 
  Two-body and three-body systems of scalar bosons are considered in the framework of covariant constraint dynamics. The reduced equation obtained after eliminating redundant degrees of freedom can be viewed as an eigenvalue equation for an observable which is intimately related with the relative motion. We display the connection of this observable with binding energy. 
  The noncommutative star product of phase space functions is, by construction, associative for both non-degenerate and degenerate case (involving only second class constraints) as has been shown by Berezin, Batalin and Tyutin. However, for the latter case, the manifest associativity is lost if an arbitrary coordinate system is used but can be restored by using an unconstrained canonical set.  The existence of such a canonical transformation is guaranteed by a theorem due to Maskawa and Nakajima. In terms of these new variables, the Kontsevich series for the star product reduces to an exponential series which is manifestly associative. We also show, using the star product formalism, that the angular momentum of a particle moving on a circle is quantized. 
  We investigate field theories on the non-commutative torus upon varying theta, the parameter of non-commutativity. We argue that one should think of Morita equivalence as a symmetry of algebras describing the same space rather than of theories living on different spaces (as is T-duality). Then we give arguments why physical observables depend on theta non-continuously. 
  In some models of N=1 supersymmetric SU(M) gauge dynamics (hep-th/9503163 and hep-th/9707244), the tension of a string ending on q external quarks is proportional to sin(pi q/M), q=1,..., M-1. In this paper we calculate the ratios of the q-string tensions using the recently derived type IIB gravity duals of N=1 SUSY gauge theories. Far in the IR these gravity duals contain a three-sphere with M units of R-R 3-form flux which, upon S-duality, turns into NS-NS 3-form flux. The confining q-string is described by a D3-brane wrapping a two-sphere inside the three-sphere with q units of world volume flux. For one of the gravity dual backgrounds (Maldacena-Nunez) a D3-brane probe calculation exactly reproduces the sin(pi q/M) dependence, while for another (Klebanov-Strassler) we find approximate agreement. We speculate on the connection of the q-string tensions with D-brane tensions in the SU(2) WZW model. 
  We argue that the worldvolume theories of D-branes probing orbifolds with discrete torsion develop, in the large quiver limit, new non-commutative directions. This provides an explicit `deconstruction' of a wide class of noncommutative theories. This also provides insight into the physical meaning of discrete torsion and its relation to the T-dual B field. We demonstrate that the strict large quiver limit reproduces the matrix theory construction of higher-dimensional D-branes, and argue that finite `fuzzy moose' theories provide novel regularizations of non-commutative theories and explicit string theory realizations of gauge theories on fuzzy tori. We also comment briefly on the relation to NCOS, (2,0) and little string theories. 
  The universal hypermultiplet moduli space metric in the type-IIA superstring theory compactified on a Calabi-Yau threefold is related to integrable systems. The instanton corrections in four dimensions arise due to multiple wrapping of BPS membranes and fivebranes around certain (supersymmetric) cycles of Calabi-Yau. The exact (non-perturbative) metrics can be calculated in the special cases of (i) the D-instantons (or the wrapped D2-branes) in the absence of fivebranes, and (ii) the fivebrane instantons with vanishing charges, in the absence of D-instantons. The solutions of the first type are governed by the three-dimensional Toda equation, whereas the solutions of the second type are governed by the particular Painleve VI equation. 
  The finite temperature effect is examined in Randall-Sundrum brane-world scenario with inclusion of the matter fields on the brane. At zero temperature it is found that the theory on the brane is conformally invariant, which guarantees $AdS$/CFT. At 4d effective action we derived a temperature-dependent nonvanishing cosmological constant at the flat spacetime limit of brane worldvolume. At the cosmological temperature $3 {\bf K}$ the cosmological constant is roughly $(0.0004 eV)^4$ which is within the upper bound of the recent experimental value $(0.01 eV)^4$ 
  We investigate supergravity solutions describing D5 branes wrapped on a two cycle which are dual to N=2 super Yang Mills theory. Brane probing these solutions allows the moduli space of the field theory to be identified. There are a unique set of coordinates in which the field theory on the probe takes an N=2 form and in these coordinates the running coupling of the gauge theory may be identified. We show that the geometry, when restricted to the moduli space, takes a very simple form involving only two functions. One is the running gauge coupling whilst the other parametrizes the scalar operators of the field theory. The D5 brane distributions, for the full set of solutions in the literature,can be determined by assuming the field theory's form for the running coupling as a function of scalar vevs. We show that the resulting distributions also correctly reproduce the scalar operators parametrized elsewhere providing the first non-trivial consistency check on the duality. 
  The finite-size scaling spectra of the spin-1/2 XXZ Heisenberg chain with toroidal boundary conditions and an even number of sites provide a projection mechanism yielding the spectra of models with a central charge c<1 including the unitary and non-unitary minimal series. Taking into account the half-integer angular momentum sectors - which correspond to chains with an odd number of sites - in many cases leads to new spinor operators appearing in the projected systems. These new sectors in the XXZ chain correspond to a new type of frustration lines in the projected minimal models. The corresponding new boundary conditions in the Hamiltonian limit are investigated for the Ising model and the 3-state Potts model and are shown to be related to duality transformations which are an additional symmetry at their self-dual critical point. By different ways of projecting systems we find models with the same central charge sharing the same operator content and modular invariant partition function which however differ in the distribution of operators into sectors and hence in the physical meaning of the operators involved. Related to the projection mechanism in the continuum there are remarkable symmetry properties of the finite XXZ chain. The observed degeneracies in the energy and momentum spectra are shown to be the consequence of intertwining relations involving U_q[sl(2)] quantum algebra transformations. 
  We solve for the spectrum and eigenfunctions of Dirac operator on the sphere. The eigenvalues are nonzero whole numbers. The eigenfunctions are two-component spinors which may be classified by representations of the SU(2) group with half-integer angular momenta. They are not the conventional spherical spinors but special linear combinations of those. 
  We study branes moving in an AdS Schwarzschild black hole background. When the brane tension exceeds a critical value, the induced metric on the brane is of FRW type and asymptotically de Sitter. We discuss the relevance of such configurations to dS/CFT correspondence. When the black hole mass reaches a critical value that depends on the brane tension, the brane interpolates in the infinite past and future between a dS space and a finite space of zero Hubble constant. This corresponds to a cosmological evolution without a Big Bang or a Big Crunch. Moreover, the central charge of the CFT dual to the dS brane enters the Cardy-Verlinde formula that gives the entropy of the thermal CFT dual to the bulk AdS black hole. 
  We use the Witten index in the open string sector to determine tadpole charges of orientifold planes and D-branes. As specific examples we consider type I compactifications on Calabi Yau manifolds and noncompact orbifolds. The tadpole constraints suggest that the standard embedding is not a natural choice for the gauge bundle. Rather there should be a close connection of the gauge bundle and the spin bundle. In the case of a four fold, the standard embedding does not in general fulfill the tadpole conditions. We show that this agrees with the Green-Schwarz mechanism. In the case of noncompact orbifolds we are able to solve the tadpole constraints with a gauge bundle, which is related to the spin bundle. We compare these results to anomaly cancellation on the fixed plane of the orbifold. In the case of branes wrapping noncompact cycles, there are fractional intersection numbers and anomaly coefficients, which we explain in geometric terms. 
  We study the star algebra of ghost sector in vacuum string field theory (VSFT). We show that the star product of two states in the Siegel gauge is BRST exact if we take the BRST charge to be the one found in hep-th/0108150, and the BRST exact states are nil factors in the star algebra. By introducing a new star product defined on the states in the Siegel gauge, the equation of motion of VSFT is characterized as the projection condition with respect to this new product. We also comment on the comma form of string vertex in the ghost sector. 
  We present a short review of adelic quantum mechanics pointing out its non-Archimedean and noncommutative aspects. In particular, $p$-adic path integral and adelic quantum cosmology are considered. Some similarities between $p$-adic analysis and q-analysis are noted. The $p$-adic Moyal product is introduced. 
  We compute the cosmological perturbations generated in the colliding bubble braneworld universe in which bubbles filled with five-dimensional anti-de Sitter space (AdS5)expanding within a five dimensional de Sitter space (dS5) or Minkowski space (M5) collide to form a (3+1) dimensional local brane on which the cosmology is virtually identical to that of the Randall-Sundrum model. The perturbation calculation presented here is valid to linear order but treats the fluctuations of the expanding bubbles as (3+1) dimensional fields localized on the bubble wall. We find that for bubbles expanding in dS5 the dominant contribution to the power spectrum is `red' but very small except in certain cases where the fifth dimension is not large or the bubbles have expanded to far beyond the dS5 apparent horizon length. This paper supersedes a previous version titled "Exactly Scale-Invariant Cosmological Perturbations From a Colliding Bubble Braneworld Universe" in which we erroneously claimed that a scale-invariant spectrum results for the case of bubbles expanding in M5. This present paper corrects the errors of the previous version and extends the analysis to the more interesting and general case of bubbles expanding in dS5. 
  We study the spectrum of the confining strings in four-dimensional SU(N) gauge theories. We compute, for the SU(4) and SU(6) gauge theories formulated on a lattice, the string tensions sigma_k related to sources with Z_N charge k, using Monte Carlo simulations. Our results are consistent with the sine formula sigma_k/sigma = sin k pi/N / sin pi/N for the ratio between sigma_k and the standard string tension sigma.   For the SU(4) and SU(6) cases the accuracy is approximately 1% and 2%, respectively. The sine formula is known to emerge in various realizations of supersymmetric SU(N) gauge theories. On the other hand, our results show deviations from Casimir scaling. We also discuss an analogous behavior exhibited by two-dimensional SU(N) x SU(N) chiral models. 
  In this paper, we investigate the near-extremal thermodynamics of the Reissner-Nordstrom (RN) black hole. Our methodology is based on a duality that exists between the near-horizon geometry of the near-extremal RN sector and Jackiw-Teitelboim (JT) theory. First, the described correspondence is reviewed at the classical level. Next, we consider first-order perturbations in the dual JT geometry by incorporating a quantum scalar field into the formalism. The novelty of our approach is that the matter field is endowed with a 4-dimensional pedigree. We ultimately find that back-reaction effects prohibit the JT black hole from losing all of its mass. This outcome directly implies that an RN black hole can not reach extremality and, moreover, can not even come arbitrarily close to an extremal state. 
  We study gauge invariant operators of open string field theory and find a precise correspondence with on-shell closed strings. We provide a detailed proof of the gauge invariance of the operators and a heuristic interpretation of their correlation functions in terms of on-shell scattering amplitudes of closed strings. We also comment on the implications of these operators to vacuum string field theory. 
  In this paper we discuss the question of whether the entropy of cosmological horizon in some asymptotically de Sitter spaces can be described by the Cardy-Verlinde formula, which is supposed to be an entropy formula of conformal field theory in any dimension. For the Schwarzschild-de Sitter solution, although the gravitational mass is always negative (in the sense of the prescription in hep-th/0110108 to calculate the conserved charges of asymptotically de Sitter spaces), we find that indeed the entropy of cosmological horizon can be given by using naively the Cardy-Verlinde formula. The entropy of pure de Sitter spaces can also be expressed by the Cardy-Verlinde formula. For the topological de Sitter solutions, which have a cosmological horizon and a naked singularity, the Cardy-Verlinde formula also works well. Our result is in favour of the dS/CFT correspondence. 
  We study the algebra ${\cal A}_n$ and the basis of the Hilbert space ${\cal H}_n$ in terms of the $\theta$ functions of the positions of $n$ solitons. Then we embed the Heisenberg group as the quantum operator factors in the representation of the transfer matrice of various integrable models. Finally we generalize our result to the generic $\theta$ case. 
  Several models in NCG with mild changes to the standard model(SM)are introduced to discuss the neutrino mass problem. We use two constraints, Poincar$\acute{e}$ duality and gauge anomaly free, to discuss the possibility of containing right-handed neutrinos in them. Our work shows that no model in this paper, with each generation containing a right-handed neutrino, can satisfy these two constraints in the same time. So, to consist with neutrino oscillation experiment results, maybe fundamental changes to the present version of NCG are usually needed to include Dirac massive neutrinos. 
  We discuss some new metrics of special holonomy, and their roles in string theory and M-theory. First we consider Spin(7) metrics denoted by C_8, which are complete on a complex line bundle over CP^3. The principal orbits are S^7, described as a triaxially squashed S^3 bundle over S^4. The behaviour in the S^3 directions is similar to that in the Atiyah-Hitchin metric, and we show how this leads to an M-theory interpretation with orientifold D6-branes wrapped over S^4. We then consider new G_2 metrics which we denote by C_7, which are complete on an R^2 bundle over T^{1,1}, with principal orbits that are S^3\times S^3. We study the C_7 metrics using numerical methods, and we find that they have the remarkable property of admitting a U(1) Killing vector whose length is nowhere zero or infinite. This allows one to make an everywhere non-singular reduction of an M-theory solution to give a solution of the type IIA theory. The solution has two non-trivial S^2 cycles, and both carry magnetic charge with respect to the R-R vector field. We also discuss some four-dimensional hyper-Kahler metrics described recently by Cherkis and Kapustin, following earlier work by Kronheimer. We show that in certain cases these metrics, whose explicit form is known only asymptotically, can be related to metrics characterised by solutions of the su(\infty) Toda equation, which can provide a way of studying their interior structure. 
  Explicit solutions to the conifold equations with complex dimension $n=3,4$ in terms of {\it{complex coordinates (fields)}} are employed to construct the Ricci-flat K\"{a}hler metrics on these manifolds. The K\"{a}hler 2-forms are found to be closed. The complex realization of these conifold metrics are used in the construction of 2-dimensional non-linear sigma model with the conifolds as target spaces. The action for the sigma model is shown to be bounded from below. By a suitable choice of the 'integration constants', arising in the solution of Ricci flatness requirement, the metric and the equations of motion are found to be {\it{non-singular}}. As the target space is Ricci flat, the perturbative 1-loop counter terms being absent, the model becomes topological. The inherent U(1) fibre over the base of the conifolds is shown to correspond to a gauge connection in the sigma model.   The same procedure is employed to construct the metric for the resolved conifold, in terms of complex coordinates and the action for a non-linear sigma model with resolved conifold as target space, is found to have a minimum value, which is topological. The metric is expressed in terms of the six real coordinates and compared with earlier works. The harmonic function, which is the warp factor in Type II-B string theory, is obtained and the ten-dimensional warped metric has the $AdS_{5}\times X_{5}$ geometry. 
  Based on concepts drawn from the ekpyrotic scenario and M-theory, we elaborate our recent proposal of a cyclic model of the Universe. In this model, the Universe undergoes an endless sequence of cosmic epochs which begin with the Universe expanding from a `big bang' and end with the Universe contracting to a `big crunch.'  Matching from `big crunch' to `big bang' is performed according to the prescription recently proposed with Khoury, Ovrut and Seiberg. The expansion part of the cycle includes a period of radiation and matter domination followed by an extended period of cosmic acceleration at low energies. The cosmic acceleration is crucial in establishing the flat and vacuous initial conditions required for ekpyrosis and for removing the entropy, black holes, and other debris produced in the preceding cycle. By restoring the Universe to the same vacuum state before each big crunch, the acceleration insures that the cycle can repeat and that the cyclic solution is an attractor. 
  We investigate homogeneous and isotropic cosmological solutions supported by the SU(2) gauge field governed by the Born-Infeld lagrangian. In the framework of the Friedmann-Robertson-Walker cosmology, with or without cosmological constant $\lambda$, we derive dynamical systems that give rather complete description of the space of solutions. For $\lambda=0$ the effective equation of state $\ve(p)$ is shown to interpolate between $p=-\ve/3$ in the regime of the strong field and $p=\ve/3$ for the weak field. Correspondingly, the Universe starts with zero acceleration and gradually enters the decelerating regime, asymptotically approaching the Tolman solution. 
  In the present talk we briefly demonstrate an elegant and effective technique for calculation of the trace expansion in the derivatives of background fields. One of main advantages of the technique is manifestly (super)symmetrical and gauge invariant form of expressions on all stages of calculations. Other advantage is the universality of the calculation method. Some particular examples and results are presented. 
  In the framework of perturbative quantum field theory (QFT) we propose a new, universal (re)normalization condition (called 'master Ward identity') which expresses the symmetries of the underlying classical theory. It implies for example the field equations, energy-momentum, charge- and ghost-number conservation, renormalized equal-time commutation relations and BRST-symmetry.   It seems that the master Ward identity can nearly always be satisfied, the only exceptions we know are the usual anomalies. We prove the compatibility of the master Ward identity with the other (re)normalization conditions of causal perturbation theory, and for pure massive theories we show that the 'central solution' of Epstein and Glaser fulfills the master Ward identity, if the UV-scaling behavior of its individual terms is not relatively lowered.   Application of the master Ward identity to the BRST-current of non-Abelian gauge theories generates an identity (called 'master BRST-identity') which contains the information which is needed for a local construction of the algebra of observables, i.e. the elimination of the unphysical fields and the construction of physical states in the presence of an adiabatically switched off interaction. 
  We study the spontaneous breakdown of SO(10) symmetry in the IIB matrix model, a conjectured nonperturbative definition of type IIB superstring theory in ten dimensions. Our analysis is based on a Gaussian expansion technique, which was originally proposed by Kabat-Lifschytz and applied successfully to the strong coupling dynamics of the Matrix Theory. We propose a prescription for including higher order corrections, which yields a rapid convergence in a simple example. This prescription is then applied to the IIB matrix model up to the third order. We find that the `self-consistency equations' allow various symmetry breaking solutions. Among them, the solution preserving SO(4) symmetry is found to have the smallest free energy. The value of the free energy comes closer to the analytic formula obtained by Krauth-Nicolai-Staudacher as we increase the order. The extent of the space-time in the 4 directions is larger than the remaining 6 directions, and the ratio increases with the order. These results provide the first analytical evidence that four-dimensional space-time is generated dynamically in the IIB matrix model. 
  New exact string solutions in non-constant background fields are found and it is shown that some of them are compatible with the boundary conditions for the open string - D-brane system. Extension of the constraint algebra is proposed and discussed. 
  In this talk we describe the application of the AdS/CFT correspondence for a confining background to the study of high energy scattering amplitudes in gauge theory. We relate the energy behaviour of scattering amplitudes to properties of minimal surfaces of the helicoidal type. We describe the results of hep-th/0003059 and hep-th/0010069 for amplitudes with vacuum quantum number exchange and, very briefly, hep-th/0110024 on the extension of this formalism to Reggeon exchange. 
  We consider supersymmetry algebras in arbitrary spacetime dimension and signature. Minimal and maximal superalgebras are given for single and extended supersymmetry. It is seen that the supersymmetric extensions are uniquely determined by the properties of the spinor representation, which depend on the dimension $D$ mod 8 and the signature $|\rho|$ mod 8 of spacetime. 
  We consider partial spontaneous breaking of N=1 AdS4 supersymmetry OSp(1|4) down to N=1, d=3 Poincare supersymmetry in the nonlinear realizations framework. We construct the corresponding worldvolume Goldstone superfield action and show that it describes the N=1 AdS4 supermembrane. It enjoys OSp(1|4) supersymmetry realized as a field-dependent modification of N=1, d=3 superconformal symmetry and goes into the superfield action of ordinary N=1, D=4 supermembrane in the flat limit. Its bosonic core is the Maldacena-type conformally invariant action of the AdS4 membrane. We show how to reproduce the latter action within a nonlinear realization of the AdS4 group SO(2,3). The same universal nonlinear realizations techniques can be used to construct conformally-invariant worldvolume actions for (d-2)-branes in generic AdSd spaces. 
  We study the holographic dual of asymmetrically warped space-times, which are asymptotically AdS. The self-tuning of the cosmological constant is reinterpreted as a cancellation of the visible sector stress-energy tensor by the contribution of a hidden CFT, charged under a spontaneously broken global symmetry. The apparent violation of 4D causality due to bulk geodesics is justified by considering that the CFT feels the background metric as smeared out over a length of the order of the AdS radius. 
  Recently we constructed two new $(1+1)$-dimensional scalar field theory models that posses solitonic solutions. They are the $U(\phi)=\phi^2\ln^2(\phi^2)$ and the $U(\phi)=\phi^2\cos^2(\phi^2)$ models . The first quantum corrections for these models are given by exactly solvable Schrodinger equations. In this paper we first examine the quantum meaning of the solitonic solutions and study the scattering of the mesons by the quantum soliton at order $\hbar$. Finally we give a finite expression for the soliton masses of both models and evaluate such expression approximately in the case of the second model. 
  Motivated by possible applications to condensed matter systems, in this paper we construct U(N) noncommutative Chern-Simons (NCCS) action for a disc and for a double-layer geometry, respectively. In both cases, gauge invariance severely constrains the form of the NCCS action. In the first case, it is necessary to introduce a group-valued boson field with a non-local chiral boundary action, whose gauge variation cancels that of the bulk action. In the second case, the coefficient matrix $K$ in the double U(N) NCCS action is restricted to be of the form with all the matrix elements being the same integer $k$. We suggest that this double NCCS theory with U(1) gauge group describes the so-called Halperin $(kkk)$ state in a double-layer quantum Hall system. Possible physical consequences are addressed. 
  Recently Amelino--Camelia proposed a ``Doubly Special Relativity'' theory with two observer independent scales (of speed and mass) that could replace the standard Special Relativity at energies close to the Planck scale. Such a theory might be a starting point in construction of quantum theory of space-time. In this paper we investigate the quantum and statistical mechanical consequences of such a proposal. We construct the generalized Newton--Wigner operator and find relations between energy/momentum and frequency/wavevector for position eigenstates of this operator. These relations indicate the existence of a minimum length scale. Next we analyze the statistical mechanics of the corresponding systems. We find that depending on the value of a parameter defining the canonical commutational algebra one has to do either with system with maximal possible temperature or with the one, which in the high temperature limit becomes discrete. 
  The purpose of this talk is to review a few issues concerning noncommutativity arising from String Theory. In particular, it is shown how in Type IIB Theory, the annihilation of a D3-\bar{D3} brane pair yields a D1-string. This object, in the presence of a large B-field and fermions, happens to be a complex noncommutative soliton endowed with superconductivity. 
  We present a method to solve the master equation for the Wilsonian action in the antifield formalism. This is based on a representation theory for cutoff dependent global symmetries along the Wilsonian renormalization group (RG) flow. For the chiral symmetry, the master equation for the free theory yields a continuum version of the Ginsparg-Wilson relation. We construct chiral invariant operators describing fermionic self-interactions. The use of canonically transformed variables is shown to simplify the underlying algebraic structure of the symmetry. We also give another non-trivial example, a realization of SU(2) vector symmetry. Our formalism may be used for a non-perturbative truncation of the Wilsonian action preserving global symmetries. 
  We study the open string extension of the mirror map for N=1 supersymmetric type II vacua with D-branes on non-compact Calabi-Yau manifolds. Its definition is given in terms of a system of differential equations that annihilate certain period and chain integrals. The solutions describe the flat coordinates on the N=1 parameter space, and the exact disc instanton corrected superpotential on the D-brane world-volume. A gauged linear sigma model for the combined open-closed string system is also given. It allows to use methods of toric geometry to describe D-brane phase transitions and the N=1 K\"ahler cone. Applications to a variety of D-brane geometries are described in some detail. 
  The exactly solvable BCS Hamiltonian of superconductivity is considered from several viewpoints: Richardson's ansatz, conformal field theory, integrable inhomogenous vertex models and Chern-Simons theory. 
  Kaluza-Klein theory in which the geometry of an additional dimension is fractal has been considered. In such a theory the mass of an elementary electric charge appears to be many orders of magnitude smaller than the Planck mass, and the "tower" of masses which correspond to higher integer charges becomes aperiodic. 
  Huyghens principle and Planck law are studied in Maxwell and Maxwell-Chern-Simons frameworks in (2+1) dimensions. Contrary to (3+1) dimensions, massless photons are shown to violate Huyghens principle in planar world. In addition, we obtain that Planck law is no longer proportional to $\nu^3$, but to the squared frequency, $\nu^2$, of the planar photons. We also briefly discuss possible physical consequences of these results. 
  The dependence on the topological theta angle term in quantum field theory is usually discussed in the context of instanton calculus. There the observables are 2 pi periodic, analytic functions of theta. However, in strongly coupled theories, the semi-classical instanton approximation can break down due to infrared divergences. Instances are indeed known where analyticity in theta can be lost, while the 2 pi periodicity is preserved. In this short note we exhibit a simple two dimensional example where the 2 pi periodicity is lost. The observables remain periodic under the transformation theta -> theta + 2 k pi for some k >= 2. We also briefly discuss the case of four dimensional N=2 supersymmetric gauge theories. 
  We study a varying electric charge brane world cosmology in the RS2 model obtained from a varying-speed-of-light brane world cosmology by redefining the system of units. We elaborate conditions under which the flatness problem and the cosmological constant problem can be resolved by such cosmological model. 
  We analyze properties of the Sp(2M) conformally invariant field equations in the recently proposed generalized $\half M(M+1)$-dimensional space-time $\M_M$ with matrix coordinates. It is shown that classical solutions of these field equations define a causal structure in $\M_M$ and admit a well-defined decomposition into positive and negative frequency solutions that allows consistent quantization in a positive definite Hilbert space. The effect of constraints on the localizability of fields in the generalized space-time is analyzed. Usual d-dimensional Minkowski space-time is identified with the subspace of the matrix space $\M_M$ that allows true localization of the dynamical fields. Minkowski coordinates are argued to be associated with some Clifford algebra in the matrix space $\M_M$. The dynamics of a conformal scalar and spinor in $\M_2$ and $\M_4$ is shown to be equivalent, respectively, to the usual conformal field dynamics of a scalar and spinor in the 3d Minkowski space-time and the dynamics of massless fields of all spins in the 4d Minkowski space-time. An extension of the electro-magnetic duality transformations to all spins is identified with a particular generalized Lorentz transformation in $\M_4$. The M=8 case is shown to correspond to a 6d chiral higher spin theory. The cases of M=16 (d=10) and M=32 (d=11) are discussed briefly. 
  The exact mass gap of the O(N) Gross-Neveu model is known, for arbitrary $N$, from non-perturbative methods. However, a "naive" perturbative expansion of the pole mass exhibits an infinite set of infrared renormalons at order 1/N, formally similar to the QCD heavy quark pole mass renormalons, potentially leading to large ${\cal O}(\Lambda)$ perturbative ambiguities. We examine the precise vanishing mechanism of such infrared renormalons, which avoids this (only apparent)contradiction, and operates without need of (Borel) summation contour prescription, usually preventing unambiguous separation of perturbative contributions. As a consequence we stress the direct Borel summability of the (1/N) perturbative expansion of the mass gap. We briefly speculate on a possible similar behaviour of analogous non-perturbative QCD quantities. 
  A new configuration of non-abelian D1-branes growing into D5-branes is found. This time the effect is triggered by a non-trivial electric field on the world-volume of the D1-branes and a constant RR 4-form potential. Based on the these configurations and other observations regarding non-abilean effective actions, a new action for matrix string theory in non-trivial backgrounds is conjectured. As an application we found that fundamental strings can grow into Dp-branes, in particular by placing the strings in the background of a group of near horizon D3-branes we found D5-branes. These types of configurations were found from the supergravity point of view in previous works. 
  We show how to define gauge-covariant coordinate transformations on a noncommuting space. The construction uses the Seiberg-Witten equation and generalizes similar results for commuting coordinates. 
  We show that the general Lorentz- and CPT-violating extension of quantum electrodynamics is one-loop renormalizable. The one-loop Lorentz-violating beta functions are obtained, and the running of the coefficients for Lorentz and CPT violation is determined. Some implications for theory and experiment are discussed. 
  A review is given on the recently proposed two dimensional axion model (O(3) sigma-model with a dynamical Hopf-term) and the T-duality relating it to the SU(2)xU(1) symmetric anisotropic sigma-model. Strong evidence is presented for the correctness of the proposed S-matrix for both models comparing perturbative and Thermodynamical Bethe Ansatz calculations for different types of free energies. This also provides a very stringent test of the validity of T-duality transformation at the quantum level. The quantum non-integrability of the O(3) sigma-model with a non-dynamical Hopf-term, in contradistinction to the axion model, is illustrated by calculating the 2-->3 particle production amplitude to lowest order. 
  We discuss some subtleties in connection with the new attempts to provide a firm basis for ths Witten-Veneziano formula. 
  We study the BPS spectrum of Little String Theory for bound states of M5-branes wrapped on six manifold of product topology $M_4\times\Sigma_2$ and the apparence of multi-loop $\theta$-functions in a supersymmetric index calculation. We find a total reconstruction of the g-loop heterotic contribution in the case of a double K3 M-theory compactification. Moreover, we consider total wrapping of M5-branes on del Pezzo surfaces $B_k$ and, by studying the relevant amplitude, we notice the arising of $\theta$-functions relative to BPS strings on $T^{k-1}$, i.e. membranes on $T^k$. This happens because of beautiful relations between four dimensional SYM theories and CFTs in two dimensions and seems to be linked to a duality recently observed by A.Iqbal, A.Neitzke and C.Vafa in. 
  We solve the Einstein equations in the Dvali-Gabadadze-Porrati model with a static, spherically symmetric matter distribution on the physical brane and obtain an exact expression for the gravitational field outside the source to the first order in the gravitational coupling. Although when confined on the physical brane this expression reproduces the correct 4D Newtonian potential for distances r_s << r << {\lambda}, where {\lambda} is a characteristic length scale of the model, it does not coincide with the standard linearized form of the 4D Schwarzschild metric. The solution reproduces the 5D Schwarzschild metric in the linearized approximation for r >> {\lambda}. 
  Motivated by examples that appeared in the context of string theory - gauge theory duality, we consider corrections to supergravity backgrounds induced by higher derivative R^4+... terms in superstring effective action. We argue that supersymmetric solutions that solve BPS conditions at the leading (supergravity) order continue to satisfy a 1-st order ``RG-type'' system of equations with extra source terms encoding string (or M-theory) corrections. We illustrate this explicitly on the examples of R^4 corrections to the generalized resolved and deformed 6-d conifolds and to a class of non-compact 7-d spaces with G_2 holonomy. Both types of backgrounds get non-trivial modifications which we study in detail, stressing analogies between the two cases. 
  We complete the construction of vacuum string field theory by proposing a canonical choice of ghost kinetic term -- a local insertion of the ghost field at the string midpoint with an infinite normalization. This choice, supported by level expansion studies in the Siegel gauge, allows a simple analytic treatment of the ghost sector of the string field equations. As a result, solutions are just projectors, such as the sliver, of an auxiliary CFT built by combining the matter part with a twisted version of the ghost conformal theory. Level expansion experiments lead to surprising new projectors -- butterfly surface states, whose analytical expressions are obtained. With the help of a suitable open-closed string vertex we define open-string gauge invariant operators parametrized by on-shell closed string states. We use regulated vacuum string field theory to sketch how pure closed string amplitudes on surfaces without boundaries arise as correlators of such gauge invariant operators. 
  By applying generalized dimensional reduction on the type IIB supersymmetry variations, we derive the supersymmetry variations for the massive 9-dimensional supergravity. We use these variations and the ones for massive type IIA to derive the supersymmetry transformation of the gravitino for the proposed massive 11-dimensional supergravity. 
  An analysis of the one-loop vacuum fluctuations associated with a scalar field confined in the interior of a infinite waveguide of rectangular cross section is presented. We first consider the massless scalar field defined in a four-dimensional Euclidean space. To identify the infinities of the vacuum fluctuations we use a combination of dimensinal and zeta function analytic regularization procedures. The infinities which occur in the one-loop vacuum fluctuations fall into two distinct classes: ultraviolet divergences that are renormalized by the introduction of bulk counterterms and also surface and edges divergences that demand countertems concentrated on the boundaries. We present the detailed form of the surface and edge divergences. Finally we discuss how to generalize our calculations for a confined massive scalar field defined in a higher dimensional Euclidean space. 
  Universe is considered as a brane in infinite (2+4)-space.It is shown that zero modes of all kinds of matter fields and 4-gravity are localized on the brane by increasing transversal gravitational potential. 
  In this note, we extend the noncommutative bion core solution of Constable, Myers and Tafjord (hep-th/9911136) to include the effects of a nonzero NS-NS two-form B. The result is a `tilted bion' in which the core expands out to a single D3-brane at an angle to the D1-brane core. Its properties agree perfectly with an analysis of the dual situation, that of a magnetic charge on an abelian D3-brane in a background worldvolume magnetic field. We also demonstrate that this agreement extends beyond geometry to include the field strength on the D3-brane. We make a proposal for including possible worldvolume gauge fields when mapping a noncommutative geometrical brane solution onto a corresponding commutative brane description. 
  We show that the two-loop Euler-Heisenberg effective Lagrangian for scalar QED in a constant Euclidean self-dual background has a simple explicit closed form expression in terms of the digamma function. This result leads to a simple analysis of the weak- and strong-field expansions, the two-loop scalar QED beta function, and the analytic continuation properties of the effective Lagrangian and its imaginary part. 
  We develop both the gravity and field theory sides of the Karch-Randall conjecture that the near-horizon description of a certain D5-D3 brane configuration in string theory, realized as AdS_5 x S^5 bisected by an AdS_4 x S^2 "brane", is dual to N=4 Super Yang-Mills theory in R^4 coupled to an R^3 defect. We propose a complete Lagrangian for the field theory dual, a novel "defect superconformal field theory" wherein a subset of the fields of N=4 SYM interacts with a d=3 SU(N) fundamental hypermultiplet on the defect preserving conformal invariance and 8 supercharges. The Kaluza-Klein reduction of wrapped D5 modes on AdS_4 x S^2 leads to towers of short representations of OSp(4|4), and we construct the map to a set of dual gauge-invariant defect operators O_3 possessing integer conformal dimensions. Gravity calculations of <O_4> and <O_4O_3> are given. Spacetime and N-dependence matches expectations from dCFT, while the behavior as functions of lambda = g^2 N at strong and weak coupling is generically different. We comment on a class of correlators for which a non-renormalization theorem may still exist. Partial evidence for the conformality of the quantum theory is given, including a complete argument for the special case of a U(1) gauge group. Some weak coupling arguments which illuminate the duality are presented. 
  We review the relation between entropy bounds to rewrite Friedmann equation on the brane in terms of three entropy bounds: Bekenstein-Verlinde ($S_{BV}$); Bekenstein-Hawking bound ($S_{BH}$); Hubble bound ($S_H$). For a strongly coupled conformal field theory (CFT) with a dual 5-dimensional anti de Sitter Schwarzschild (AdSS$_5$) black hole, we can easily establish the connection between the Cardy-Verlinde formula on the CFT side and the entropy representation of Friedmann equation in cosmology. In this case its cosmological evolution for entropy is given by the semi-circle. However, for the matter-dominated case, we find that the cosmological evolution diagram takes a different form of the cycloid. Here we propose two different entropy relations for matter-dominated case. It turns out that the Verlinde's entropy relation so restricted that it may not be valid for the matter-dominated universe. 
  The problem of the scalar pair production by a one-dimensional vector- potential $A_{\mu}(x_3)$ is reduced to the $S-$ matrix formalism of the theory with an unstable vacuum. Our choice of in- and out-states does not coincide with that of other authors and we argue extensively in favor of our choice. In terms of our classification the states that can be created by the field enter into the field operator in the same way as do the states that cannot be created by the field, i.e. the field operator has the usual form. We show that the norm of a solution of the wave equation is determined by one of the amplitude of its asymptotic form for $x_3\to \pm\infty$. For the step potential and for the constant field potential we get the explicit expressions for the complete in- and out-sets of orthonormalized wave functions. For the constant electric field we obtain the scalar particle propagator in terms of the stationary states and show that with our choice of in- and out-states it has the form dictated by the general theory. 
  As shown recently 2d quantum gravity theories -- including spherically reduced Einstein-gravity -- after an exact path integral of its geometric part can be treated perturbatively in the loops of (scalar) matter. Obviously the classical mechanism of black hole formation should be contained in the tree approximation of the theory. This is shown to be the case for the scattering of two scalars through an intermediate state which by its effective black hole mass is identified as a "virtual black hole". We discuss the lowest order tree vertex for minimally and non-minimally coupled scalars and find a non-trivial finite S-matrix for gravitational s-wave scattering in the latter case. 
  An application of the Gordan-Hilbert finite algebraic basis theorem is suggested. 
  We begin a study of possibilities of describing hadrons in terms of monolocal fields which transform as proper Lorentz group representations decomposable into an infinite direct sum of finite-dimensional irreducible representations. The additional requirement that the free-field Lagrangians be invariant under the secondary symmetry transformations generated by the polar or the axial four-vector representation of the orthochronous Lorentz group provides an effective mechanism for selecting the class representations considered and eliminating an infinite number of arbitrary constants allowed by the relativistic invariance of the Lagrangians. 
  I investigate the Poisson-sigma model on the classical and quantum level. First I show how the interaction can be obtained by a deformation of the classical master equation of an Abelian BF theory in two dimensions. On the classical level this model includes various known two-dimensional field theories, in particular the Yang-Mills theory. On the quantum level the perturbation expansion of the path integral in the covariant gauge yields the Kontsevich deformation formula. Finally I perform the calculation of the path integral in a general gauge, and demonstrate how the derived partition function reduces in the special case of a linear Poisson structure to the familiar form of 2d Yang-Mills theory. 
  We present a new cosmological model, based on the holographic principle, which shares many of the virtues of inflation. The very earliest semiclassical era of the universe is dominated by a dense gas of black holes, with equation of state $p=\rho$. Fluctuations lead to an instability to a phase with a dilute gas of black holes, which later decays via Hawking radiation to a radiation dominated universe. The quantum fluctuations of the initial state give rise to a scale invariant spectrum of density perturbations, for a range of scales. We point out a problem, that appears to prevent the range of scales predicted by the model from coinciding with the range where such a spectrum has been observed. We speculate that this may be related to our field theoretic treatment of fluctuations in the highly holographic $p=\rho$ background. The monopole problem is solved in a manner completely different from inflationary models, and a relic density of highly charged extremal black monopoles is predicted. We discuss the nature of the entropy and flatness problems in our model. 
  We discuss the Schrodinger equation in presence of quaternionic potentials. The study is performed analytically as long as it proves possible, when not, we resort to numerical calculations. The results obtained could be useful to investigate an underlying quaternionic quantum dynamics in particle physics. Experimental tests and proposals to observe quaternionic quantum effects by neutron interferometry are briefly reviewed. 
  The universal amplitude ratio $R_{\xi}$ for the ($q\leqslant 4$)-state Potts model in two dimensions is determined by using results for the dilute A model in regime 1. The nature of the relationship between the Potts model and the dilute A model, both related to $\phi_{2,1}$ perturbed conformal field theory, is discussed. 
  We study the IR behavior of noncommutative gauge theory in the matrix formulation. We find that in this approach, the nature of the UV/IR mixing is easily understood, which allows us to perform a reliable calculation of the quantum effective action for the long wavelength modes of the noncommutative gauge field. At one loop, we find that our description is weakly coupled only in the supersymmetric theory. At two loops, we find non-trivial interaction terms suggestive of dipole degrees of freedom. These dipoles exhibit a channel duality reminiscent of string theory. 
  We discuss several different constructions of non-Abelian large gauge transformations at finite temperature. Pisarski's ansatz with even winding number is related to Hopf mappings, and we present a simple new ansatz that has any integer winding number at finite temperature. 
  All leptons, quarks, and gauge bosons can be placed in the periodic table of elementary particles. As the periodic table of elements derived from atomic orbital, the periodic table of elementary particles is derived from the two sets of seven orbitals: principal dimensional orbital and auxiliary dimensional orbital. (Seven orbitals come indirectly from the seven extra dimensions in eleven-dimensional space-time.) Principal dimensional orbital derived from varying space-time dimension, varying speed of light, and varying supersymmetry explains gauge bosons and low-mass leptons. Auxiliary dimensional orbital derived from principal dimensional orbital accounts for high-mass leptons and individual quarks. For hadrons as the composites of individual quarks, hadronic dimensional orbital derived from auxiliary dimensional orbital is responsible. These three sets of seven orbitals explain all elementary particles and hadrons. QCD, essentially, describes the different occupations of quarks in the three sets of seven orbitals at different temperatures. The periodic table of elementary particles and the compositions of hadrons relate to the Barut lepton mass formula, the Polazzi mass formula for stable hadrons, and the MacGregor-Akers constituent quark model. The calculated masses for elementary particles and hadrons are in good agreement with the observed masses. For examples, the calculated masses for the top quark, neutron, and pion are 176.5 GeV, 939.54MeV, and 135.01MeV in excellent agreement with the observed masses, 174.3 GeV, 939.57 MeV, and 134.98 MeV, respectively. 
  It is shown that the BRS-formulated two-dimensional BF theory in the light-cone gauge (coupled with chiral Dirac fields) is solved very easily in the Heisenberg picture. The structure of the exact solution is very similar to that of the BRS-formulated two-dimensional quantum gravity in the conformal gauge. In particular, the BRS Noether charge has anomaly. Based on this fact, a criticism is made on the reasoning of Kato and Ogawa, who derived the critical dimension D=26 of string theory on the basis of the anomaly of the BRS Noether charge. By adding the $\widetilde{B}^2$ term to the BF-theory Lagrangian density, the exact solution to the two-dimensional Yang-Mills theory is also obtained. 
  The formalism of spacetime dependent lagrangians developed in Ref.1 is applied to the Sine Gordon and massive Thirring models.It is shown that the well-known equivalence of these models (in the context of weak-strong duality) can be understood in this approach from the same considerations as described in [1] for electromagnetic duality. A further new result is that all these can be naturally linked to the fact that the holographic principle has analogues at length scales much larger than quantum gravity. There is also the possibility of {\it noncommuting coodinates} residing on the boundaries. PACS: 11.15.-q: 11.10/Ef 
  We consider the quantum theory of a two-form gauge field on a space-time which is a direct product of time and a spatial manifold, taken to be a compact five-manifold with no torsion in its cohomology. We show that the Hilbert space of this non-chiral theory is a certain subspace of a tensor product of two spaces, that are naturally interpreted as the Hilbert spaces of a chiral and anti-chiral two-form theory respectively. We also study the observable operators in the non-chiral theory that correspond to the electric and magnetic field strengths, the Hamiltonian, and the exponentiated holonomy of the gauge-field around a spatial two-cycle. All these operators can be decomposed into contributions pertaining to the chiral and anti-chiral sectors of the theory. 
  We study the wrapping of N type IIB Dp-branes on a compact Riemann surface $\Sigma$ in genus $g>1$ by means of the Sen-Witten construction, as a superposition of N' type IIB Dp'-brane/antibrane pairs, with $p'>p$. A background Neveu-Schwarz field B deforms the commutative $C^{\star}$-algebra of functions on $\Sigma$ to a noncommutative $C^{\star}$-algebra. Our construction provides an explicit example of the $N'\to\infty$ limit advocated by Bouwknegt-Mathai and Witten in order to deal with twisted K-theory. We provide the necessary elements to formulate M(atrix) theory on this new $C^{\star}$-algebra, by explicitly constructing a family of projective $C^{\star}$-modules admitting constant-curvature connections. This allows us to define the $g>1$ analogue of the BPS spectrum of states in $g=1$, by means of Donaldson's formulation of the Narasimhan-Seshadri theorem. 
  An exact evolution equation, the functional generalization of the Callan-Symanzik method, is given for the effective action of QED where the electron mass is used to turn the quantum fluctuations on gradually. The usual renormalization group equations are recovered in the leading order but no Landau pole appears. 
  We discuss the proposal of Hata and Kawano for the tachyon fluctuation around a solution of vacuum string field theory representing a D25 brane. We give a conformal field theory construction of their state -- a local insertion of a tachyon vertex operator on the sliver surface state, and explain why the on-shell momentum condition emerges correctly. We also show that a naive computation of the D25-brane tension using data for the three point coupling of this state gives an answer that is $(\pi^2/3)(16/27\ln 2)^3 \simeq 2.0558$ times the expected answer. We demonstrate that this problem arises because the HK state does not satisfy the equations of motion in a strong sense required for the computation of D-brane tension from the on-shell 3-tachyon coupling. 
  We study condensation of closed string tachyons living on defects, such as orbifold fixed planes and Neveu-Schwarz fivebranes. We argue that the high energy density of localized states decreases in the process of condensation of such tachyons. In some cases this means that $c_{eff}$ decreases along the flow; in others, $c_{eff}$ remains constant and the decreasing quantity is a closed string analog, $g_{cl}$, of the ``boundary entropy'' of D-branes. We discuss the non-supersymmetric orbifolds $C/Z_n$ and $C^2/Z_n$. In the first case tachyon condensation decreases $n$ and in some cases connects type II and type 0 vacua. In the second case non-singular orbifolds are related by tachyon condensation to both singular and non-singular ones. We verify that $g_{cl}$ decreases in flows between non-singular orbifolds. The main tools in the analysis are the structure of the chiral ring of the perturbed theory, the geometry of the resolved orbifold singularities, and the throat description of singular conformal field theories. 
  We propose a closed string tachyon action including kinetic and potential terms for non-supersymmetric orbifolds. The action is given in terms of solutions to $tt^*$ equations which captures the geometry of vacua of the corresponding N=2 worldsheet theory. In certain cases the solutions are well studied. In case of tachyons of ${\bf C}/Z_n$, solutions to affine toda equations determine the action. We study the particular case of ${\bf C}/Z_3\to {\bf C}$ in detail and find that the Tachyon action is determined in terms of a solution to Painleve III equation. 
  We give matrix and supergravity descriptions of type IIA F-strings polarizing into cylindrical D2 branes. When a RR four-form field strength F_4 is turned on in a supersymmetric fashion (with 4 supercharges), a complete analysis of the solutions reveals the existence of a moduli space of F1 -> D2 polarizations (Caesar) for some fractional strengths of the perturbation, and of no polarization whatsoever (nihil) for all other strengths of the perturbation. This is a very intriguing phenomenon, whose physical implications we can only speculate about. In the matrix description of the polarization we use the Non-Abelian Born-Infeld action in an extreme regime, where the commutators of the fields are much larger than 1. The validity of the results we obtain, provides a direct confirmation of this action, although is does not confirm or disprove the symmetrized trace prescription. 
  The Ising quantum chain with a peculiar twisted boundary condition is considered. This boundary condition, first introduced in the framework of the spin-1/2 XXZ Heisenberg quantum chain, is related to the duality transformation, which becomes a symmetry of the model at the critical point. Thus, at the critical point, the Ising quantum chain with the duality-twisted boundary is translationally invariant, similar as in the case of the usual periodic or antiperiodic boundary conditions. The complete energy spectrum of the Ising quantum chain is calculated analytically for finite systems, and the conformal properties of the scaling limit are investigated. This provides an explicit example of a conformal twisted boundary condition and a corresponding generalised twisted partition function. 
  We present a prescription for constructing the monodromy matrix, $\hat{\cal M}(\omega)$, for $O(d,d)$ invariant string effective actions and derive its transformation properties under the $T$-duality group. This allows us to construct $\hat{\cal M}(\omega)$ for new backgrounds, starting from known ones, which are related by $T$-duality. As an application, we derive the monodromy matrix for the exactly solvable Nappi-Witten model, both when B=0 and $B\neq 0$. 
  We study the convergence of the derivative expansion for flow equations. The convergence strongly depends on the choice for the infrared regularisation. Based on the structure of the flow, we explain why optimised regulators lead to better physical predictions. This is applied to O(N)-symmetric real scalar field theories in 3d, where critical exponents are computed for all N. In comparison to the sharp cut-off regulator, an optimised flow improves the leading order result up to 10%. An analogous reasoning is employed for a proper time renormalisation group. We compare our results with those obtained by other methods. 
  We review a perturbative approach to deal with Lagrangians with higher or infinite order time derivatives. It enables us to construct a consistent Poisson structure and Hamiltonian with only first time derivatives order by order in coupling. To the lowest order, the Hamiltonian is bounded from below whenever the potential is. We consider spacetime noncommutative field theory as an example. 
  We study an N=1 two-dimensional non-linear sigma model with boundaries representing, e.g., a gauge fixed open string. We describe the full set of boundary conditions compatible with N=1 superconformal symmetry. The problem is analyzed in two different ways: by studying requirements for invariance of the action, and by studying the conserved supercurrent. We present the target space interpretation of these results, and identify the appearance of partially integrable almost product structures. 
  Starting from the known representation of the Kac-Moody algebra in terms of the coordinates and momenta, we extend it to the representation of the super Kac-Moody and super Virasoro algebras. Then we use general canonical method to construct an action invariant under local gauge symmetries, where components of the super energy-momentum tensor $L_\pm$ and $G_\pm$ play the role of the diffeomorphisms and supersymmetries generators respectively. We obtain covariant extension of WZNW theory with respect to local supersymmetry as well as explicit expressions for gauge transformations. 
  We propose a correspondence between the physics of certain small charge black holes in AdS_k x S^l and large charge black holes in AdS_l x S^k. The curvature singularities of these solutions arise, following Myers and Tafjord, from a condensate of giant gravitons. When the number of condensed giants N_g is much greater than the number of background branes N, we propose that the system has an equivalent description in terms of N giant gravitons condensed in a background created by N_g branes. Our primary evidence is an exact correspondence between gravitational entropy formulae of small and large charge solutions in different dimensions. 
  The canonical treatment of dynamic systems with manifest Lagrangian constraints proposed by Berezin is applied to concrete examples: a special Lagrangian linear in velocities, relativistic particles in proper time gauge, a relativistic string in orthonormal gauge, and the Maxwell field in the Lorentz gauge, 
  We discuss fluxes of RR and NSNS background fields in type II string compactifications on non-compact Calabi-Yau threefolds together with their dual brane description which involves bound states of branes. Simultaneously turning on RR and NSNS 2-form fluxes in an 1/2 supersymmetric way can be geometrically described in M-theory by a SL(2,Z) family of metrics of G(2) holonomy. On the other hand, if the flux configuration only preserves 1/4 of supersymmetries, we postulate the existence of a new eight-dimensional manifold with spin(7) holonomy, which does not seem to fit into the classes of known examples. The latter situation is dual to a 1/4 supersymmetric web of branes on the deformed conifold. In addition to the 2-form fluxes, we also present some considerations on type IIA NSNS 4-form and 6-form fluxes. 
  We construct new classes of exact solutions of the 4D vacuum Einstein equations which describe ellipsoidal black holes, black tori and combined black hole -- black tori configurations. The solutions can be static or with anisotropic polarizations and running constants. They are defined by off--diagonal metric ansatz which may be diagonalized with respect to anholonomic moving frames. We examine physical properties of such anholonomic gravitational configurations and discuss why the anholonomy may remove the restriction that horizons must be with spherical topology. 
  We obtain the solution that corresponds to a screwed superconducting cosmic string (SSCS) in the framework of a general scalar-tensor theory including torsion. We investigate the metric of the SSCS in Brans-Dicke theory with torsion and analyze the case without torsion. We show that in the case with torsion the space-time background presents other properties different from that in which torsion is absent. When the spin vanish, this torsion is a $\phi$-gradient and then it propagates outside of the string. We investigate the effect of torsion on the gravitational force and on the geodesics of a test-particle moving around the SSCS. The accretion of matter by wakes formation when a SSCS moves with speed $v $ is investigated. We compare our results with those obtained for cosmic strings in the framework of scalar-tensor theory. 
  If the graviton possesses an arbitrarily small (but nonvanishing) mass, perturbation theory implies that cosmic strings have a nonzero Newtonian potential. Nevertheless in Einstein gravity, where the graviton is strictly massless, the Newtonian potential of a cosmic string vanishes. This discrepancy is an example of the van Dam--Veltman--Zakharov (VDVZ) discontinuity. We present a solution for the metric around a cosmic string in a braneworld theory with a graviton metastable on the brane. This theory possesses those features that yield a VDVZ discontinuity in massive gravity, but nevertheless is generally covariant and classically self-consistent. Although the cosmic string in this theory supports a nontrivial Newtonian potential far from the source, one can recover the Einstein solution in a region near the cosmic string. That latter region grows as the graviton's effective linewidth vanishes (analogous to a vanishing graviton mass), suggesting the lack of a VDVZ discontinuity in this theory. Moreover, the presence of scale dependent structure in the metric may have consequences for the search for cosmic strings through gravitational lensing techniques. 
  We study the dyonic charge of type IIA D6-branes interpreted as M-theory Kaluza-Klein monopoles. The connection between the dyonic properties of type IIA D6-branes and the existence of eleven dimensions is discussed. We also study dyonic properties of non-commutative monopoles. Variations of the external antisymmetric B-field generate the electric charge of non-commutative dyons. 
  This paper has been withdrawn. 
  We probe the U(N) Gross-Neveu model with a source-term $J\bar{\Psi}\Psi$. We find an expression for the renormalization scheme and scale invariant source $\hat{J}$, as a function of the generated mass gap. The expansion of this function is organized in such a way that all scheme and scale dependence is reduced to one single parameter d. We get a non-perturbative mass gap as the solution of $\hat{J}=0$. In one loop we find that any physical choice for d gives good results for high values of N. In two loops we can determine d self-consistently by the principle of minimal sensitivity and find remarkably accurate results for N>2. 
  In a representation theoretic approach a free q-relativistic wave equation must be such, that the space of solutions is an irreducible representation of the q-Poincare algebra. It is shown how this requirement uniquely determines the q-wave equations. As examples, the q-Dirac equation (including q-gamma matrices which satisfy a q-Clifford algebra), the q-Weyl equations, and the q-Maxwell equations are computed explicitly. 
  Systems built out of N-body interactions, beyond 2-body interactions, are formulated on the plane, and investigated classically and quantum mechanically (in phase space). Their Wigner Functions--the density matrices in phase-space quantization--are given and analyzed. 
  We consider gauge theories on noncommutative euclidean space . In particular, we discuss the structure of gauge group following standard mathematical definitions and using the ideas of hep-th/0102182. 
  Following a previous work on Abelian (2,0)-gauge theories, one reassesses here the task of coupling (2,0) relaxed Yang-Mills superpotentials to a (2,0)- nonlinear $\sigma$-model, by gauging the isotropy or the isometry group of the latter. One pays special attention to the extra ``chiral-like'' component-field gauge potential that comes out from the relaxation of superspace constraints. 
  Loop quantum gravity effective theories are reviewed in the context of the observed GZK limit anomaly and related processes. This is accomplished through a kinematical analysis of the modified threshold conditions for the involved decay reactions, arising from the theory. Specially interesting is the possibility of an helicity dependant violation of the limit, whose primary effect would be the observation of favoured helicity states for highly energetic particles. 
  In this note we study aspects of the interplay between fluxbranes and p-branes. We describe how a fluxbrane can be physically realized as a limit of a brane-antibrane configuration, in a manner similar to the way a uniform electric field appears in between the plates of a capacitor. We also study the evolution of a fluxbrane after nucleation of p-branes. We find that Kaluza-Klein fluxbranes do relax by forming brane-antibrane pairs or spherical branes, but we also find that for fluxtubes with dilaton coupling in a different range, the field strength does not relax, instead it becomes stronger after each nucleation bounce. We speculate on a possible runaway instability of such fluxtubes an an eventual breakdown of their classical description. 
  Following hep-th/0109127, we show that a certain class of BPS naked singularities (superstars) found in compactifications of M-theory can be interpreted as being composed of giant gravitons. More specifically, we study superstars which are asymptotically AdS_7 x S^4 and AdS_4 x S^7 and show that these field configurations can be interpreted as being sourced by continuous distributions of spherical M2- and M5-branes, respectively, which carry internal momenta and have expanded on the spherical component of the space-time. 
  We summarize some recent progress in constructing four-dimensional supersymmetric chiral models from Type II orientifolds. We present the construction a supersymmetric Standard-like Model and a supersymmetric GUT model to illustrate the new features of this approach and its connection to M theory on compact, singular G_2 holonomy spaces. The Standard-like model presented is the first example of a three-family supersymmetic orientifold model with the Standard Model as part of the gauge structure. We also discuss the connection of how chiral fermions arise in this class of models with recent results of M theory compactified on G_2 holonomy spaces. 
  We consider correlation functions for string theory on AdS_3. We analyze their singularities and we provide a physical interpretation for them. We explain which worldsheet correlation functions have a sensible physical interpretation in terms of the boundary theory. We consider the operator product expansion of the four point function and we find that it factorizes only if a certain condition is obeyed. We explain that this is the correct physical result. We compute correlation functions involving spectral flowed operators and we derive a constraint on the amount of winding violation. 
  We determine the energy eigenvalues and eigenfunctions of the harmonic oscillator where the coordinates and momenta are assumed to obey the modified commutation relations [x_i,p_j]=i hbar[(1+ beta p^2) delta_{ij} + beta' p_i p_j]. These commutation relations are motivated by the fact they lead to the minimal length uncertainty relations which appear in perturbative string theory. Our solutions illustrate how certain features of string theory may manifest themselves in simple quantum mechanical systems through the modification of the canonical commutation relations. We discuss whether such effects are observable in precision measurements on electrons trapped in strong magnetic fields. 
  We consider a brane universe in a Reissner-Nordstrom-de Sitter background spacetime of arbitrary dimensionality. It is shown that the brane evolution is described by generalized Friedmann equations for radiative matter along with a stiff-matter contribution. On the basis of the (conjectured) dS/CFT correspondence, we identify various thermodynamic properties of the brane. It is then demonstrated that, when the brane crosses the de Sitter cosmological horizon, the CFT thermodynamics and Friedmann-like equations coincide. Moreover, the CFT entropy is shown to be expressible in a generalized Cardy-Verlinde form. Finally, we consider the holographic entropy bounds in this scenario. 
  It is pointed out that phase structures of gauge theories compactified on non-simply connected spaces are not trivial. As a demonstration, an SU(2) gauge model on $M^3\otimes S^1$ is studied and is shown to possess three phases: Hosotani, Higgs and coexisting phases. The critical radius and the order of the phase transitions are explicitly determined. A general discussion about phase structures for small and large scales of compactified spaces is given. The appearance of phase transitions suggests a GUT scenario in which the gauge hierarchy problem is replaced by a dynamical problem of how to stabilize a radius of a compactified space in close vicinity to a critical radius. 
  In this paper we investigated the problem of the existence of invariant meaures on the local gauge group. We prove that it is impossible to define a {\it finite} translationally invariant measure on the local gauge group $C^{\infty}({\bf R}^n,G)$(where $G$ is an arbitrary matrix Lie group). 
  We present a discrete total variation calculus in Hamiltonian formalism in this paper. Using this discrete variation calculus and generating functions for the flows of Hamiltonian systems, we derive two-step symplectic-energy integrators of any finite order for Hamiltonian systems from a variational perspective. The relationship between symplectic integrators derived directly from the Hamiltonian systems and the variationally derived symplectic-energy integrators is explored. 
  Critical behaviour of the 2D scalar field theory in the LC framework is reviewed. The notion of dynamical zero modes is introduced and shown to lead to a non trivial covariant dispersion relation only for Continuous LC Quantization (CLCQ). The critical exponent $\eta$ is found to be governed by the behaviour of the infinite volume limit under conformal transformations properties preserving the local LC structure. The $\beta$-function is calculated exactly and found non-analytic, with a critical exponent $\omega=2$, in agreement with the conformal field theory analysis of Calabrese et al. 
  Unabridged version of the Thesis presented to the University of Rome ``Tor Vergata'', in partial fulfillment of the requirements for the ``Laurea'' degree in Physics, May 2001. 
  The previously developed algebraic lightfront holography is used in conjunction with the tensor splitting of the chiral theory on the causal horizon. In this way a universal area law for the entanglement entropy of the vacuum relative to the split (tensor factorized) vacuum is obtained. The universality of the area law is a result of the kinematical structure of the properly defined lightfront degrees of freedom. We consider this entropy associated with causal horizon of the wedge algebra in Minkowski spacetime as an analog of the quantum Bekenstein black hole entropy similar to the way in which the Unruh temperature for the wedge algebra may be viewed as an analog in Minkowski spacetime of the Hawking thermal behavior. My more recent preprint hep-th/20202085 presents other aspects of the same problem. 
  The analysis of D-branes in coset models G/H provides a natural extension of recent studies on branes in WZW-theory and it has various interesting applications to physically relevant models. In this work we develop a reduction procedure that allows to construct the non-commutative gauge theories which govern the dynamics of branes in G/H. We obtain a large class of solutions and interprete the associated condensation processes geometrically. The latter are used to propose conservation laws for the dynamics of branes in coset models at large level k. In super-symmetric theories, conserved charges are argued to take their values in the representation ring of the denominator theory. Finally, we apply the general results to study boundary fixed points in two examples, namely for parafermions and minimal models. 
  A new topological conformal field theory in four Euclidean dimensions is constructed from N=4 super Yang-Mills theory by twisting the whole of the conformal group with the whole of the R-symmetry group, resulting in a theory that is conformally invariant and has two conformally invariant BRST operators. A curved space generalisation is found on any Riemannian 4-fold. This formulation has local Weyl invariance and two Weyl-invariant BRST symmetries, with an action and energy-momentum tensor that are BRST-exact. This theory is expected to have a holographic dual in 5-dimensional de Sitter space. 
  We discuss the perturbative expansion of several one-loop improved renormalisation group equations. It is shown that in general the integrated renormalisation group flows fail to reproduce perturbation theory beyond one loop. 
  We use simple iterated one-loop graphs in massless Yukawa theory and QED to pose the following question: what are the symmetries of the residues of a graph under a permutation of places to insert subdivergences. The investigation confirms partial invariance of the residue under such permutations: the highest weight transcendental is invariant under such a permutation. For QED this result is gauge invariant, ie the permutation invariance holds for any gauge. Computations are done making use of the Hopf algebra structure of graphs and employing GiNaC to automate the calculations. 
  Topological properties of Fortuin-Kasteleyn clusters are studied on the torus. Namely, the probability that their topology yields a given subgroup of the first homology group of the torus is computed for Q=1, 2, 3 and 4. The expressions generalize those obtained by Pinson for percolation (Q=1). Numerical results are also presented for three tori of different moduli. They agree with the theoretical predictions for Q=1, 2 and 3. For Q=4 agreement is not ruled out but what seems logarithmic corrections makes it harder to decide. 
  In this paper we use Modified Self-Consistent Resummation (MSCR) in order to obtain the scalar dressed mass by evaluating the self-energy up to two loops in the neutral scalar $\lambda \phi^4$ model at finite temperature. With this laboratory model we show that, if a theory is renormalizable at zero temperature, using the MSCR it is always possible to obtain a finite corrected mass at finite temperature. This feature of the MSCR is not observed in some other approximation techniques usually found in the literature. 
  We discuss the new integrable boundary conditions for the O(N) nonlinear $\sigma$ model and related solutions of the boundary Yang-Baxter equation, which were presented in our previous paper hep-th/0108039. 
  The old idea that the photon is a Goldstone boson emergent from a spontaneously broken theory of interacting fermions is revisited. It is conjectured that the gauge-potential condensate has a vacuum expectation value which is very large, perhaps the GUT/Planck momentum scale, but that the magnitude of the effective potential which generates it is very small, so small that in the limit of vanishing cosmological constant it would vanish. In this way, the threat of unacceptably large observable, noncovariant residual effects is mitigated. The linkage of these ideas to other speculative ideas involving black holes and parametrizations of Standard-Model coupling constants is also described. 
  I argue that the correlations that are predicted by Quantum Field Theory should not be interpreted as a real sign of non locality. 
  We continue the investigation of Spin(7) holonomy metric of cohomogeneity one with the principal orbit SU(3)/U(1). A special choice of U(1) embedding in SU(3) allows more general metric ansatz with five metric functions. There are two possible singular orbits in the first order system of Spin(7) instanton equation. One is the flag manifold $SU(3)/T^2$ also known as the twister space of CP(2) and the other is CP(2) itself. Imposing a set of algebraic constraints, we find a two-parameter family of exact solutions which have SU(4) holonomy and are asymptotically conical. There are two types of asymptotically locally conical (ALC) metrics in our model, which are distingushed by the choice of $S^1$ circle whose radius stabilizes at infinity. We show that this choice of M theory circle selects one of possible singular orbits mentioned above. Numerical analyses of solutions near the singular orbit and in the asymptotic region support the existence of two families of ALC Spin(7) metrics: one family consists of deformations of the Calabi hyperKahler metric, the other is a new family of metrics on a line bundle over the twister space of CP(2). 
  We consider a D-brane probe in unstable string background associated with flux branes. The twist in spacetime metric reponsible for the supersymmetry breaking is shown to manifest itself in mixing of open Wilson lines with the phases of some adjoint matter fields, resulting in a nonlocal and nonsupersymmetric form of Yang-Mills theory as the probe dynamics. This provides a setup where one can study fate of a large class of unstable closed string theories that include as a limit type 0 theories and various orbifolds of type II and type 0 theories. We discuss the limit of ${\bf C}/Z_n$ orbifold in some detail and speculate on couplings with closed string tachyons. 
  We formulate the Landau problem in the context of the noncommutative analog of a surface of constant negative curvature, that is $AdS_2$ surface, and obtain the spectrum and contrast the same with the Landau levels one finds in the case of the commutative $AdS_2$ space. 
  We propose a new holographic C-function for the accelerating universe defined in the stringy frame motivated mainly by the fact that the number of degrees of freedom should be infinite for a physical system of infinite size. This is the generalization of Strominger's recent proposal of the holographic C-function to the asymptotically non-de Sitter universe. We find that the corresponding C-theorem holds true if the universe accelerates toward the weak coupling regime driven by the exponential dilaton potential. It also holds in other simple cases. 
  A proof is given of Polyakov conjecture about the auxiliary parameters of the SU(1,1) Riemann-Hilbert problem for general elliptic singularities. Such a result is related to the uniformization of the the sphere punctured by n conical defects. Its relevance to the hamiltonian structure of 2+1 dimensional gravity in the maximally slicing gauge is stressed. 
  We study quantum corrections in the earlier proposed string theory, which is based on Weyl invariant purely extrinsic curvature action. At one-loop level it remains Weyl invariant irrespective of the dimension D of the embedding spacetime. To some extent the counterterms are reminiscent of the ones in pure quantum gravity. At classical level the string tension is equal to zero and quarks viewed as open ends of the surface are propagating freely without interaction. We demonstrate that quantum fluctuations generate nonzero area term (string tension). 
  A (M4 x M4) x Z4 model, describing an extended particle composed of two local modes and represented by a field psi(x, xi ;z), is formulated in its most general form (x, xi ; z) belong to (M4 x M4) x Z4. The 'z' argument specifies whether the particle is observable, unobservable, or partially observable (the latter case appears in two forms). In this four-sheeted structure, each sheet posses its own symmetry localized with respect to both space-times M4 inducing thereby connections in the continuous directions. Connections in the discrete direction describe transitions between observable, unobservable, and partially observable states. Curvatures and propagators are determined. 
  In this note we consider boundary perturbations in the A-Series unitary minimal models by phi_{r,r+2} fields on superpositions of boundaries. In particular, we consider perturbations by boundary condition changing operators. Within conformal perturbation theory we explicitly map out the space of perturbative renormalisation group flows for the example phi_{1,3} and find that this sheds light on more general phi_{r,r+2} perturbations. Finally, we find a simple diagrammatic representation for the space of flows from a single Cardy boundary condition. 
  There are $N-1$ classes of kink solutions in $SU(N)\times Z_2$. We show how interactions between various kinks depend on the classes of individual kinks as well as on their orientations with respect to each other in the internal space. In particular, we find that the attractive or repulsive nature of the interaction depends on the trace of the product of charges of the two kinks. We calculate the interaction potential for all combinations of kinks and antikinks in $SU(5)\times Z_2$ and study their collisions. The outcome of kink-antikink collisions, as expected from previous studies, is sensitive to their initial relative velocity. We find that heavier kinks tend to break up into lighter ones, while interactions between the lightest kinks and antikinks in this model can be repulsive as well as attractive. 
  We have investigated the effects of a generic bulk first-order phase transition on thick Minkowski branes in warped geometries. As occurs in Euclidean space, when the system is brought near the phase transition an interface separating two ordered phases splits into two interfaces with a disordered phase in between. A remarkable and distinctive feature is that the critical temperature of the phase transition is lowered due to pure geometrical effects. We have studied a variety of critical exponents and the evolution of the transverse-traceless sector of the metric fluctuations. 
  In this lecture notes we explain and discuss some ideas concerning noncommutative geometry in general, as well as noncommutative field theories and string field theories. We consider noncommutative quantum field theories emphasizing an issue of their renormalizability and the UV/IR mixing. Sen's conjectures on open string tachyon condensation and their application to the D-brane physics have led to wide investigations of the covariant string field theory proposed by Witten about 15 years ago. We review main ingredients of cubic (super)string field theories using various formulations: functional, operator, conformal and the half string formalisms. The main technical tools that are used to study conjectured D-brane decay into closed string vacuum through the tachyon condensation are presented. We describe also methods which are used to study the cubic open string field theory around the tachyon vacuum: construction of the sliver state, ``comma'' and matrix representations of vertices. 
  We study a class of type I string models with supersymmetry broken on the world-volume of some D-branes and vanishing tree-level potential. Despite the non-supersymmetric spectrum, supersymmetry is non-linearly realized on these D-branes, while it is spontaneously broken in the bulk by Scherk-Schwarz boundary conditions. These models can easily accommodate 3-branes with interesting gauge groups and chiral fermions. We also study the effective field theory and in particular we compute the four-fermion couplings of the localized Goldstino with the matter fermions on the brane. 
  We study conformal field theories in two dimensions separated by domain walls, which preserve at least one Virasoro algebra. We develop tools to study such domain walls, extending and clarifying the concept of `folding' discussed in the condensed-matter literature. We analyze the conditions for unbroken supersymmetry, and discuss the holographic duals in AdS3 when they exist. One of the interesting observables is the Casimir energy between a wall and an anti-wall. When these separate free scalar field theories with different target-space radii, the Casimir energy is given by the dilogarithm function of the reflection probability. The walls with holographic duals in AdS3 separate two sigma models, whose target spaces are moduli spaces of Yang-Mills instantons on T4 or K3. In the supergravity limit, the Casimir energy is computable as classical energy of a brane that connects the walls through AdS3. We compare this result with expectations from the sigma-model point of view. 
  We reconsider the $CP^{1}$ model with the Hopf term by using the Batalin-Fradkin-Tyutin (BFT) scheme, which is an improved version of the Dirac quantization method. We also perform a semi-classical quantization of the topological charge Q sector by exploiting the collective coordinates to explicitly show the fractional spin statistics. 
  We study IIA/B string theory compactified on twisted circles. These models possess closed string tachyons and reduce to type 0B/A theory in a special limit. Using methods of gauged linear sigma models and mirror symmetry we construct a conformal field theory which interpolates between these models and flat space via an auxiliary Liouville direction. Interpreting motion in the Liouville direction as renormalization group flow, we argue that the end point of tachyon condensation in all these models (including 0B/A theory) is supersymmetric type II theory. We also find a zero-slope limit of these models which is best described in a T-dual picture as a type II NS-NS fluxbrane. In this limit tachyon condensation is an interesting and well posed problem in supergravity. We explicitly determine the tachyon as a fluctuation of supergravity fields, and perform a rudimentary numerical analysis of the relevant flows. 
  In many PT symmetric models with real spectra, apparently, energy levels "merge and disappear" at a point of the spontaneous PT-symmetry breaking. We argue that such an oversimplified and discontinuous physical interpretation of this mechanism as proposed, e.g., by one of us in Phys. Lett. A 285 (2001), p. 7 would be inappropriate. Using the elementary square-well model of the above reference in the strongly non-Hermitian regime we exemplify how the doublets of states with broken PT symmetry continue to exist at complex conjugate energies. In contrast to many other exactly solvable examples of such a mechanism (we listed some of them in quant-ph/0110064), our present model of symmetry breaking does not complexify all the spectrum at once. "Realistically", it rather proceeds step by step, starting from the low-lying part of the spectrum and involving more and more excited states with an increasing degree of non-Hermiticity of the underlying interaction. 
  The paper has been withdrawn 
  We briefly review the basics of electric-magnetic duality symmetry and their geometric interpretation in the M-theory context. Then we recall some no-go theorems that prevent a simple extension of the duality symmetry to non-Abelian gauge theories. 
  We report yes-go and no-go results on consistent cross-couplings for a collection of gravitons. Motivated by the search of theories where multiplets of massless spin-two fields cross-interact, we look for all the consistent deformations of a positive sum of Pauli-Fierz actions. We also investigate the problem of deforming a (positive and negative) sum of linearized Weyl gravity actions and show explicitly that there exists multi-Weyl-graviton theories. As the single-graviton Weyl theory, these theories do not have an energy bounded from below. 
  We investigate a recent proposal for defining a conserved mass in asymptotically de Sitter spacetimes that is based on a conjectured holographic duality between such spacetimes and Euclidean conformal field theory. We show that an algorithm for deriving such terms in asymptotically anti de Sitter spacetimes has an asymptotically de Sitter counterpart, and derive the explicit form for such terms up to 9 dimensions. We show that divergences of the on-shell action for de Sitter spacetime are removed in any dimension in inflationary coordinates, but in covering coordinates a linear divergence remains in odd dimensions that cannot be cancelled by local terms that are polynomial in boundary curvature invariants. We show that the class of Schwarzschild-de Sitter black holes up to 9 dimensions has finite action and conserved mass, and construct a definition of entropy outside the cosmological horizon by generalizing the Gibbs-Duhem relation in asymptotically dS spacetimes. The entropy is agreement with that obtained from CFT methods in $d=2$. In general our results provide further supporting evidence for a dS/CFT correspondence, although some important interpretive problems remain. 
  A manifestly gauge-invariant theory of gravitational fluctuations of brane-world scenarios is discussed. Without resorting to any specific gauge choice, a general method is presented in order to disentangle the fluctuations of the brane energy-momentum from the fluctuations of the metric. As an application of the formalism, the localization of metric fluctuations on scalar branes breaking spontaneously five-dimensional Poincar\'e invariance is addressed. Only assuming that the four-dimensional Planck mass is finite and that the geometry is regular, it is demonstrated that the vector and scalar fluctuations of the metric are not localized on the brane. 
  Black p-brane solutions for a wide class of intersection rules and Ricci-flat ``internal'' spaces are considered. They are defined up to moduli functions H_s obeying non-linear differential equations with certain boundary conditions imposed. A new solution with intersections corresponding to the Lie algebra C_2 is obtained. The functions H_1 and H_2 for this solution are polynomials of degree 3 and 4. 
  We compactify M-theory on seven-manifolds with a warp-factor and G-fluxes on the internal space. Because of non-zero G-fluxes, we are forced to adopt a Majorana supersymmetry spinor ansatz which does not have the usual direct product structure of two lower dimensional Majorana spinors. For the spinor ansatz that we choose, we find that supersymmetry puts strong constraints on the internal space namely that it must be conformal to a Ricci-flat seven-manifold of the form $X^7= X^6 \times X^1$. The holonomy of $X^6$ must be larger than 1 if the warp-factor is to be non-trivial. The warp-factor depends only on the $X^1$ direction and is singular. We argue that to avoid this singularity one has to embed this solution in a Horava-Witten setup and thus has natural links to much studied brane-world scenarios. 
  We give a systematic account of symmetric D-branes in the Lie group SU(3). We determine both the classical and quantum moduli space of (twisted) conjugacy classes in terms of the (twisted) Stiefel diagram of the Lie group. We show that the allowed (twisted) conjugacy classes are in one-to-one correspondence with the integrable highest weight representations of the (twisted) affine Lie algebra. In particular, we show how the charges of these D-branes fit in the twisted K-theory groups. 
  A class of correlation functions of half-BPS composite operators are computed exactly (at finite $N$) in the zero coupling limit of N=4 SYM theory. These have a simple dependence on the four-dimensional spacetime coordinates and are related to correlators in a one-dimensional Matrix Model with complex Matrices obtained by dimensional reduction of N=4 SYM on a three-sphere. A key technical tool is Frobenius-Schur duality between symmetric and Unitary groups and the results are expressed simply in terms of U(N) group integrals or equivalently in terms of Littlewood-Richardson coefficients. These correlation functions are used to understand the existence/properties of giant gravitons and related solutions in the string theory dual on $ AdS_5 \times S^5$. Some of their properties hint at integrability in N=4 SYM. 
  The renormalization-group (RG) flow in the finite-temperature (2+1)-dimensional Georgi-Glashow model is explored. This is done in the limit when the squared electric coupling constant is much larger than the mass of the Higgs field. The novel equation describing the evolution of the Higgs mass is derived and integrated along the separatrices of the RG flow in the limit when the original theory reduces to the 2D XY model. In particular, it is checked that in the vicinity of the phase-transition point, there exists a range of parameters allowing to the Higgs mass evolved along some of the separatrices to remain much smaller than the squared electric coupling constant. 
  From first principles, I present a concrete realization of Carlip's idea on the black hole entropy from the conformal field theory on the horizon in any dimension. New formulation is free of inconsistencies encountered in Carlip's. By considering a correct gravity action, whose variational principle is well defined at the horizon, I $derive$ a correct $classical$ Virasoro generator for the surface deformations at the horizon through the canonical method. The existence of the classical Virasoro algebra is crucial in obtaining an operator Virasoro algebra, through canonical quantization, which produce the right central charge and conformal weight $\sim A_+/\hbar G$ for the semiclassical black hole entropy. The coefficient of proportionality depends on the choice of ground state, which has to be put in by hand to obtain the correct numerical factor 1/4 of the Bekenstein-Hawking (BH) entropy. The appropriate ground state is different for the rotating and the non-rotating black holes but otherwise it has a $universality$ for a wide variety of black holes. As a byproduct of my results, I am led to conjecture that {\it non-commutativity of taking the limit to go to the horizon and computing variation is proportional to the Hamiltonian and momentum constraints}. It is shown that almost all the known uncharged black hole solutions satisfy the conditions for the universal entropy formula. 
  A conjectured finite M-theory based on eleven-dimensional supergravity formulated in a superspace with a non-anticommutative <>-product of field operators is proposed. Supermembranes are incorporated in the superspace <>-product formalism. When the deformed supersymmetry invariant action of eleven-dimensional supergravity theory is expanded about the standard supersymmetry invariant action, the spontaneously compactified M-theory can yield a four-dimensional de Sitter spacetime inflationary solution with dark energy described by the four-form F-fields. A fit to the present cosmological data for an accelerating universe is obtained from matter fields describing the dominant dark matter and the four-form F-field dark energy. Chiral fermions are obtained from the M-theory by allowing singularities in the compact internal seven-dimensional space. The possibility of obtaining a realistic M-theory containing the standard model is discussed. 
  It is proposed that certain techniques from arithmetic algebraic geometry provide a framework which is useful to formulate a direct and intrinsic link between the geometry of Calabi-Yau manifolds and the underlying conformal field theory. Specifically it is pointed out how the algebraic number field determined by the fusion rules of the conformal field theory can be derived from the number theoretic structure of the cohomological Hasse-Weil L-function determined by Artin's congruent zeta function of the algebraic variety. In this context a natural number theoretic characterization arises for the quantum dimensions in this geometrically determined algebraic number field. 
  Brane-like states are defined by physical vertex operators in NSR superstring theory, existing at nonzero pictures only. These states exist both in open and closed string theories, in the NS and NS-NS sectors respectively. In this paper we present a detailed analysis of their BRST properties, giving a proof that these vertex operators are physical, i.e. BRST invariant and BRST non-trivial. 
  These are notes of my lectures held at the first School & Workshop on Logarithmic Conformal Field Theory and its Applications, September 2001 in Tehran, Iran.   These notes cover only selected parts of the by now quite extensive knowledge on logarithmic conformal field theories. In particular, I discuss the proper generalization of null vectors towards the logarithmic case, and how these can be used to compute correlation functions. My other main topic is modular invariance, where I discuss the problem of the generalization of characters in the case of indecomposable representations, a proposal for a Verlinde formula for fusion rules and identities relating the partition functions of logarithmic conformal field theories to such of well known ordinary conformal field theories.   These two main topics are complemented by some remarks on ghost systems, the Haldane-Rezayi fractional quantum Hall state, and the relation of these two to the logarithmic c=-2 theory. 
  All previous Randall-Sundrum type models have required a Z_2 identification source which does not have a known string theoretic origin. We show that the near-horizon of various resolved branes on an Eguchi-Hanson instanton dimensionally reduce to a five-dimensional domain wall that traps gravity, without an additional delta-function source. This brings us substantially closer to embedding infinite extra dimensions in M-theory. Also, this provides us with a brane world model for a strongly-coupled Yang-Mills field theory with quark-antiquark charge screening at finite separation distance. 
  We construct various boundary states in the coset conformal field theory G/H. The G/H theory admits the twisted boundary condition if the G theory has an outer automorphism of the horizontal subalgebra that induces an automorphism of the H theory. By introducing the notion of the brane identification and the brane selection rule, we show that the twisted boundary states of the G/H theory can be constructed from those of the G and the H theories. We apply our construction to the su(n) diagonal cosets and the su(2)/u(1) parafermion theory to obtain the twisted boundary states of these theories. 
  We present a simple four-dimensional model in which anomaly mediated supersymmetry breaking naturally dominates. The central ingredient is that the hidden sector is near a strongly-coupled infrared fixed-point for several decades of energy below the Planck scale. Strong renormalization effects then sequester the hidden sector from the visible sector. Supersymmetry is broken dynamically and requires no small input parameters. The model provides a natural and economical explanation of the hierarchy between the supersymmetry-breaking scale and the Planck scale, while allowing anomaly mediation to address the phenomenological challenges posed by weak scale supersymmetry. In particular, flavor-changing neutral currents are naturally near their experimental limits. 
  The worldline representation of the one loop fermionic effective action is used to obtain the vertex operator for the pion and the sigma in QCD strings. The vertex operator of the scalar sigma is distinguished from that of the pseudo-scalar pion by the presence of an additional operator m V_{0} where m is the current quark mass and V_{0} is a vertex operator that would describe a tachyon in the open bosonic string theory. This leads to a relation between the sigma propagator and the pion propagator, when chiral symmetry is spontaneously broken this relation implies that the propagator constructed from V_{0} must behave like a massless ghost state. The presence of this state ensures that the sigma is massive and no longer degenerate with the pion. The expectation value of V_{0} in a string description is related to the vacuum expectation value of the chiral condensate in QCD. Our analysis emphasizes the need for boundary fermions in any string representation of mesons and is suggestive of world sheet supersymmetry. 
  We apply to the XYZ model the technique of construction of integrable models with staggered parameters, presented recently for the XXZ case. The solution of modified Yang-Baxter equations is found and the corresponding integrable zig-zag ladder Hamiltonian is calculated. The result is coinciding with the XXZ case in the appropriate limit. 
  Our 1992 remarks about Alain Connes' interpretation of the standard model within his theory of non-commutative riemannian spin manifolds. 
  In brane world models of nature, supersymmetry breaking is often isolated on a distant brane in a higher dimensional space. The form of the Kahler potential in generic string and M-theory brane world backgrounds is shown to give rise to tree-level non-universal squark and slepton masses. This results from the exchange of bulk supergravity fields and warping of the internal geometry. This is contrary to the notion that bulk locality gives rise to a sequestered no-scale form of the Kahler potential with vanishing tree-level masses and solves the supersymmetric flavor problem. As a result, a radiatively generated anomaly mediated superpartner spectrum is not a generic outcome of these theories. 
  We try to give a pedagogical introduction to Connes' derivation of the standard model of electro-magnetic, weak and strong forces from gravity. 
  Current attempts to find a unified theory that would reconcile Einstein's General Relativity and Quantum Mechanics, and explain all known physical phenomena, invoke the Kaluza-Klein idea of extra spacetime dimensions. The best candidate is M-theory, which lives in eleven dimensions, the maximum allowed by supersymmetry of the elementary particles. We give a non-technical account.   An Appendix provides an updated version of Edwin A. Abbott's 1884 satire {\it Flatland: A Romance of Many Dimensions}. Entitled {\it Flatland, Modulo 8}, it describes the adventures of a superstring theorist, A. Square, who inhabits a ten-dimensional world and is initially reluctant to accept the existence of an eleventh dimension. 
  We consider a brane universe in an asymptotically de Sitter background spacetime of arbitrary dimensionality. In particular, the bulk spacetime is described by a ``topological de Sitter'' solution, which has recently been investigated by Cai, Myung and Zhang. In the current study, we begin by showing that the brane evolution is described by Friedmann-like equations for radiative matter. Next, on the basis of the dS/CFT correspondence, we identify the thermodynamic properties of the brane universe. We then demonstrate that many (if not all) of the holographic aspects of analogous AdS-bulk scenarios persist. These include a (generalized) Cardy-Verlinde form for the CFT entropy and various coincidences when the brane crosses the cosmological horizon. 
  I consider the semiclassical approximation of the graded Chern-Simons field theories describing certain systems of topological A type branes in the large radius limit of Calabi-Yau compactifications. I show that the semiclassical partition function can be expressed in terms of a certain (differential) numerical invariant which is a version of the analytic torsion of Ray and Singer, but associated with flat graded superbundles. I also discuss a `twisted' version of the Ray-Singer norm, and show its independence of metric data. As illustration, I consider graded D-brane pairs of unit relative grade with a scalar condensate in the boundary condition changing sector. For the particularly simple case when the reference flat connections are trivial, I show that the generalized torsion reduces to a power of the classical Ray-Singer invariant of the base 3-manifold. 
  We propose a new way of second quantizing string theory. The method is based on considering the Fock space of strings described by constituents which make up the $X^\mu_R$ and the $X^\mu_L$ i.e. the right and left mover modes separately. A state with any number of strings get represented by the Cartesian product of two free particle Fock spaces, one for right mover degrees of freedom, and one for left. The resulting string field theory is a free theory. 
  Heisenberg's matrix formulation of quantum mechanics can be generalized to relativistic systems by evolving in light-front time tau = t+z/c. The spectrum and wavefunctions of bound states, such as hadrons in quantum chromodynamics, can be obtained from matrix diagonalization of the light-front Hamiltonian on a finite dimensional light-front Fock basis defined using periodic boundary conditions in the light-front space coordinates. This method, discretized light-cone quantization (DLCQ), preserves the frame-independence of the front form even at finite resolution and particle number. Light-front quantization can also be used in the Hamiltonian form to construct an event generator for high energy physics reactions at the amplitude level. The light-front partition function, summed over exponentially-weighted light-front energies, has simple boost properties which may be useful for studies in heavy ion collisions. I also review recent work which shows that the structure functions measured in deep inelastic lepton scattering are affected by final-state rescattering, thus modifying their connection to light-front probability distributions. In particular, the shadowing of nuclear structure functions is due to destructive interference effects from leading-twist diffraction of the virtual photon, physics not included in the nuclear light-front wavefunctions. 
  The basic results in calculations of the thermodynamic functions of electromagnetic field in the background of a dilute dielectric ball at zero and finite temperature are presented. Summation over the angular momentum values is accomplished in a closed form by making use of the addition theorem for the relevant Bessel functions. The behavior of the thermodynamic characteristics in the low and high temperature limits is investigated. The $T^3$-term in the low temperature expansion of the free energy is recovered (this term has been lost in our previous calculations). 
  An integrable structure behind Witten--Dijkgraaf--Verlinde--Verlinde (WDVV) equations is identified with reduction of a Riemann-Hilbert problem for a homogeneous GL(N, C) loop group. Reduction requires the dressing matrices to be fixed points of a loop group automorphism of order two resulting in a sub-hierarchy of gl(N,C) hierarchy containing only odd symmetry flows. The model possesses Virasoro symmetry and imposing Virasoro constraints ensures homogeneity property of the Darboux-Egoroff structure. Dressing matrices of the reduced model provide solutions of the WDVV equations. 
  The current status of bounds on and limits of fermion determinants in two, three and four dimensions in QED and QCD is reviewed. A new lower bound on the two-dimensional QED determinant is derived. An outline of the demonstration of the continuity of this determinant at zero mass when the background magnetic field flux is zero is also given. 
  We compute RR charges of D2-branes on a background with H-field which belongs to a nontrivial cohomology class. We discover that the RR charge depends on the configuration of the background `electric' RR field. This result explains the ambiguity in the definition of the RR charge previously observed in the SU(2) WZW model. 
  Under suitable assumptions on the boundary conditions, it is shown that there is a bijective correspondence between equivalence classes of asymptotic reducibility parameters and asymptotically conserved n-2 forms in the context of Lagrangian gauge theories. The asymptotic reducibility parameters can be interpreted as asymptotic Killing vector fields of the background, with asymptotic behaviour determined by a new dynamical condition. A universal formula for asymptotically conserved n-2 forms in terms of the reducibility parameters is derived. Sufficient conditions for finiteness of the charges built out of the asymptotically conserved n-2 forms and for the existence of a Lie algebra g among equivalence classes of asymptotic reducibility parameters are given. The representation of g in terms of the charges may be centrally extended. An explicit and covariant formula for the central charges is constructed. They are shown to be 2-cocycles on the Lie algebra g. The general considerations and formulas are applied to electrodynamics, Yang-Mills theory and Einstein gravity. 
  We examine the cosmological evolution equations of de Sitter, flat and anti-de Sitter braneworlds sandwiched in between two n dimensional AdS-Schwarzschild spacetimes. We are careful to use the correct form for the induced Newton's constant on the brane, and show that it would be naive to assume the energy of the bulk spacetime is just given by the sum of the black hole masses. By carefully calculating the energy of the bulk for large mass we show that the induced geometry of the braneworld is just a radiation dominated FRW universe with the radiation coming from a CFT that is dual to the AdS bulk. 
  The evaluation of field theoretic correlators at strong couplings is especially interesting in the light of recently discovered string/field theory correspondences. We present a calculation of the stress-tensor correlator in N=1 SYM theory in 2+1 dimensions. We calculate this object numerically with the method of supersymmetric discrete light-cone quantization (SDLCQ) at large N_c. For small distances we reproduce the conformal field theory result with the correlator behaving like 1/r^6. In the large r limit the correlator is determined by the (massless) BPS states of the theory. We find a critical value of the coupling where the correlator goes to zero in this limit. This critical coupling is shown to grow linearly with the square root of the transverse momentum resolution. 
  The energy spectrum of a nonrelativistic particle on a noncommutative sphere in the presence of a magnetic monopole field is calculated. The system is treated in the field theory language, in which the one-particle sector of a charged Schroedinger field coupled to a noncommutative U(1) gauge field is identified. It is shown that the Hamiltonian is essentially the angular momentum squared of the particle, but with a nontrivial scaling factor appearing, in agreement with the first-quantized canonical treatment of the problem. Monopole quantization is recovered and identified as the quantization of a commutative Seiberg-Witten mapped monopole field. 
  A 6-component "wave function" (not field, but S-matrix interpretable) for a massive spin-1 particle parallels the Dirac "chirality-doubled" 4-component wave function for a spin-1/2 particle, by pairing two wave functions for same spin but opposite "handedness". The correlated "opposite-parity" pair of complex 3-vectors defines a fluctuating spin-correlated lightlike ``internal velocity" as well as an independent "external rapidity". Extension from fermions to vector bosons of the velocity-fluctuation ("zitterbewegung") interpretation of rest mass weakens theoretical motivation for elementary scalar bosons. 
  Cosmological limits on Lorentz invariance breaking in Chern-Simons $(3+1)-dimensional$ electrodynamics are used to place limits on torsion. Birefrigence phenomena is discussed by using extending the propagation equation to Riemann-Cartan spacetimes instead of treating it in purely Riemannian spaces. The parameter of Lorentz violation is shown to be proportional to the axial torsion vector which allows us to place a limit on cosmological background torsion from the Lorentz violation constraint which is given by $ 10^{-33} eV <|S^{\mu}| < 10^{-32} eV$ where $|S^{\mu}|$ is the axial torsion vector. 
  In this note we complement recent results on the exchange $r$-matrices appearing in the chiral WZNW model by providing a direct, purely finite-dimensional description of the relationship between the monodromy dependent 2-form that enters the chiral WZNW symplectic form and the exchange $r$-matrix that governs the corresponding Poisson brackets. We also develop the special case in which the exchange $r$-matrix becomes the `canonical' solution of the classical dynamical Yang-Baxter equation on an arbitrary self-dual Lie algebra. 
  In the framework of superfield formalism, we demonstrate the existence of a new local, covariant, continuous and nilpotent (dual-BRST) symmetry for the BRST invariant Lagrangian density of a self-interacting two ($1 + 1$)-dimensional (2D) non-Abelian gauge theory (having no interaction with matter fields). The local and nilpotent Noether conserved charges corresponding to the above continuous symmetries find their geometrical interpretation as the translation generators along the odd (Grassmannian) directions of the four ($2 + 2)$-dimensional supermanifold. 
  We discuss the form of the string-loop-corrected effective action obtained by compactification of the heterotic string theory on the manifold $K3\times T^2$ or on its orbifold limit and the loop-corrected magnetic black hole solutions of the equations of motion. Effective 4D theory has N=2 local supersymmetry. Using the string-loop-corrected prepotential of the N=2 supersymmetric theory, which receives corrections only from the string world sheets of torus topology, we calculate the loop corrections to the tree-level gauge couplings and solve the loop-corrected equations of motion. At the string-tree level, the effective gauge couplings decrease at small distances from the origin, and in this region string-loop corrections to the gauge couplings become important. A possibility of smearing the singularity of the tree-level supersymmetric solution with partially broken supersymmetry by quantum corrections is discussed. 
  We propose a general framework for studying quantum field theory on the anti-de-Sitter space-time, based on the assumption of positivity of the spectrum of the possible energy operators. In this framework we show that the n-point functions are analytic in suitable domains of the complex AdS manifold, that it is possible to Wick rotate to the Euclidean manifold and come back, and that it is meaningful to restrict AdS quantum fields to Poincare' branes. We give also a complete characterization of two-point functions which are the simplest example of our theory. Finally we prove the existence of the AdS-Unruh effect for uniformly accelerated observers on trajectories crossing the boundary of AdS at infinity, while that effect does not exist for all the other uniformly accelerated trajectories. 
  We discuss the renormalization of a BRST and anti-BRST invariant composite operator of mass dimension 2 in Yang-Mills theory with the general BRST and anti-BRST invariant gauge fixing term of the Lorentz type. The interest of this study stems from a recent claim that the non-vanishing vacuum condensate of the composite operator in question can be an origin of mass gap and quark confinement in any manifestly covariant gauge, as proposed by one of the authors. First, we obtain the renormalization group flow of the Yang-Mills theory. Next, we show the multiplicative renormalizability of the composite operator and that the BRST and anti-BRST invariance of the bare composite operator is preserved under the renormalization. Third, we perform the operator product expansion of the gluon and ghost propagators and obtain the Wilson coefficient corresponding to the vacuum condensate of mass dimension 2. Finally, we discuss the connection of this work with the previous works and argue the physical implications of the obtained results. 
  We present a method to find solutions of the Weyl or the Dirac equation without specifying a representation choice for $\gamma^a$'s. By taking $\gamma^a$'s formally as independent variables, we construct solutions out of $2^d$ orthonormal polynomials of $\gamma^a$'s (in $d$-dimensional space) operating on a ``vacuum state''. Polynomials reduce into $2^{d/2}$ or $2^{(d+1)/2}$ repetitions of the Dirac spinors for $d$ even or odd, respectively. We further propose the corresponding graphic presentation of basic states, which offers an easy way to see all the quantum numbers of states with respect to the generators of the Lorentz group, as well as transformation properties of the states under any operator. 
  Topological field theories of Schwarz-type generally admit symmetries whose algebra does not close off-shell, e.g. the basic symmetries of BF models or vector supersymmetry of the gauge-fixed action for Chern-Simons theory (this symmetry being at the origin of the perturbative finiteness of the theory). We present a detailed discussion of all these symmetries within the algebraic approach to the Batalin-Vilkovisky formalism. Moreover, we discuss the general algebraic construction of topological models of both Schwarz- and Witten-type. 
  In previous publications (J.Geom.Phys.38 (2001) 81-139 and references therein) the partition function for 2+1 gravity was constructed for the fixed genus Riemann surface. With help of this function the dynamical transition from pseudo-Anosov to periodic (Seifert-fibered) regime was studied.In this paper the periodic regime is studied in some detail in order to recover major results of Kontsevich (Comm.Math.Phys. 147 (1992) 1-23) ispired by earlier work by Witten on topological two dimensional quantum gravity.To achieve this goal some results from enumerative combinatorics have been used. The logical developments are extensively illustrated using geometrically convincing figures. This feature is helpful for development of some non traditional applications (mentioned through the entire text) of obtained results to fields other than theoretical particle physics 
  A comprehensive introduction to logarithmic conformal field theory, using an algebraic point of view, is given. A number of examples are explained in detail, including the c=-2 triplet theory and the k=-4/3 affine su(2) theory. We also give some brief introduction to the work of Zhu. 
  This thesis considers massive field theories in 1+1 dimensions known as affine Toda quantum field theories. We first consider the boundary sine-Gordon model, deriving a complete picture of the boundary bound state structure for general integrable boundary conditions, and then more general ATFTs in the bulk, discovering a "generalised bootstrap" equation which explicitly encodes the Lie algebra into the S-matrix. This last is related to a number of S-matrix identities, as well as a generalisation of the idea that the conserved charges of the theory form an eigenvector of the Cartan matrix. 
  We examine the structure of the Clifford algebra associated with a Hermitian bilinear form and apply the result to a dynamical model of the relativistic point particle. The dynamics of the particle is described by a Dirac spinor with components in a Clifford algebra. This spinor determines, through the Clifford algebra, both the space-time coordinates and their conjugate momenta and satisfies a first order equation of motion which leads to the usual space-time canonical equations of motion. The constraints appear as the equations of motion for the einbein and spin connection which are needed to ensure the reparametrization invariance and local Lorentz invariance of the action. 
  We show that, in 't Hooft's large N limit, matrix models can be formulated as a classical theory whose equations of motion are the factorized Schwinger--Dyson equations. We discover an action principle for this classical theory. This action contains a universal term describing the entropy of the non-commutative probability distributions. We show that this entropy is a nontrivial 1-cocycle of the non-commutative analogue of the diffeomorphism group and derive an explicit formula for it. The action principle allows us to solve matrix models using novel variational approximation methods; in the simple cases where comparisons with other methods are possible, we get reasonable agreement. 
  By studying the effects of the shape moduli associated with toroidal compactifications, we demonstrate that Planck-sized extra dimensions can cast significant ``shadows'' over low-energy physics. These shadows can greatly distort our perceptions of the compactification geometry associated with large extra dimensions, and place a fundamental limit on our ability to probe the geometry of compactification simply by measuring Kaluza-Klein states. We also discuss the interpretation of compactification radii and hierarchies in the context of geometries with non-trivial shape moduli. One of the main results of this paper is that compactification geometry is effectively renormalized as a function of energy scale, with ``renormalization group equations'' describing the ``flow'' of geometric parameters such as compactification radii and shape angles as functions of energy. 
  Semiclassical mechanics of systems with first-class constraints is developed. Starting from the quantum theory, one investigates such objects as semiclassical states and observables, semiclassical inner product, semiclassical gauge transformations and evolution. Quantum mechanical semiclassical substitutions (not only the WKB-ansatz) can be viewed as "composed semiclassical states" being infinite superpositions of wave packets with minimal uncertainties of coordinates and momenta ("elementary semiclassical states"). Each elementary semiclassical state is specified by a set (X,f) of classical variables X (phase, coordinates, momenta) and quantum function f ("shape of the wave packet" or "quantum state in the background X"). A notion of an elemantary semiclassical state can be generalized to the constrained systems, provided that one uses the refined algebraic quantization approach based on modifying the inner product rather than on imposing the constrained conditions on physical states. The inner product of physical states is evaluated. It is obtained that classical part of X the semiclassical state should belong to the constrained surface; otherwise, the semiclassical state (X,f) will have zero norm for all f. Even under classical constraint conditions, the semiclassical inner product is degenerate. One should factorize then the space of semiclassical states. Semiclassical gauge transformations and evolution of semiclassical states are studied. The correspondence with semiclassical Dirac approach is discussed. 
  We discuss a model in which the fundamental scale of gravity is restricted to 10^{-3} eV. An observable modification of gravity occurs simultaneously at the Hubble distance and at around 0.1 mm. These predictions can be tested both by the table-top experiments and by cosmological measurements. The model is formulated as a brane-world theory embedded in a space with two or more infinite-volume extra dimensions. Gravity on the brane reproduces the four-dimensional laws at observable distances but turns to the high-dimensional behavior at larger scales. To determine the crossover distance we smooth out the singularities in the Green's functions by taking into account softening of the graviton propagator due to the high-dimensional operators that are suppressed by the fundamental scale. We find that irrespective of the precise nature of microscopic gravity the ultraviolet and infrared scales of gravity-modification are rigidly correlated. This fixes the fundamental scale of gravity at 10^{-3} eV. The result persists for nonzero thickness branes. 
  We consider electrons in uniform external magnetic and electric fields which move on a plane whose coordinates are noncommuting. Spectrum and eigenfunctions of the related Hamiltonian are obtained. We derive the electric current whose expectation value gives the Hall effect in terms of an effective magnetic field. We present a receipt to find the action which can be utilized in path integrals for noncommuting coordinates. In terms of this action we calculate the related Aharonov--Bohm phase and show that it also yields the same effective magnetic field. When magnetic field is strong enough this phase becomes independent of magnetic field. Measurement of it may give some hints on spatial noncommutativity. The noncommutativity parameter \theta can be tuned such that electrons moving in noncommutative coordinates are interpreted as either leading to the fractional quantum Hall effect or composite fermions in the usual coordinates. 
  We derive and solve the full set of scalar perturbation equations for a class of five-dimensional brane--world solutions, with a dilaton scalar field coupled to the bulk cosmological constant and to a 3-brane. The spectrum contains one localized massless scalar mode, to be interpreted as an effective dilaton on the brane, inducing long--range scalar interactions. Two massive scalar modes yield corrections to Newton's law at short distances, which persist even in the limit of vanishing dilaton (namely, in the standard Randall--Sundrum configuration). 
  The minimal embedding of the Standard Model in type I string theory is described. The SU(3) color and SU(2) weak interactions arise from two different collections of branes. The correct prediction of the weak angle is obtained for a string scale of 6-8 TeV. Two Higgs doublets are necessary and proton stability is guaranteed. It predicts two massive vector bosons with masses at the TeV scale, as well as a new superweak interaction. 
  Correspondence between BRST-BFV, Dirac and refined algebraic (group averaging, projection operator) approaches to quantize constrained systems is analyzed. For the closed-algebra case, it is shown that the component of the BFV wave function with maximal (minimal) number of ghosts and antighosts in the Schrodinger representation may be viewed as a wave function in the refined algebraic (Dirac) quantization approach. The Giulini-Marolf group averaging formula for the inner product in the refined algebraic quantization approach is obtained from the Batalin-Marnelius prescription for the BRST-BFV inner product which should be generally modified due to topological problems. The considered prescription for the correspondence of states is observed to be applicable to the open-algebra case. Refined algebraic quantization approach is generalized then to the case of nontrivial structure functions. A simple example is discussed. Correspondence of observables in different quantization methods is also investigated. 
  In this report I wish to recall how the basic concepts and ingredients of supergravity were formulated by Dmitrij V. Volkov and the present author in 1973-74 under the first investigation of the super-Higgs effect . 
  In the present work the gauge invariance of causal Yang-Mills theory will be proven with the aid of the gauge-factor group. For that purpose it must be shown, that the operator valued distributions T_n and D_n(ret) occurring in the causal S-matrix construction can be written, after applying the gauge variation d_Q, as a divergence. Since merely local terms lead to gauge destroying expressions, one has to focus on them exclusively. In the first part of the work the local gauge-factor group will be defined in the style of the concept of gauge cohomology theory. It will be shown, that every element out of the so defined factor group under the transformation d_Q leads to a divergence of the entire operator valued distribution d_Q(T_n). In the second part all local terms arising in causal Yang-Mills theory are systematically investigated. Without further restrictions there can be proven, that every local operator valued distribution is an element of the gauge factor group or equal to zero. This concludes the demonstration of gauge invariance of causal Yang-Mills theory. 
  The perturbative finiteness of various topological models (e.g. BF models) has its origin in an extra symmetry of the gauge-fixed action, the so-called vector supersymmetry. Since an invariance of this type also exists for gravity and since gravity is closely related to certain BF models, vector supersymmetry should also be useful for tackling various aspects of quantum gravity. With this motivation and goal in mind, we first extend vector supersymmetry of BF models to generic manifolds by incorporating it into the BRST symmetry within the Batalin-Vilkovisky framework. Thereafter, we address the relationship between gravity and BF models, in particular for three-dimensional space-time. 
  The aim of this paper is two-fold. First, we provide a simple and pedagogical discussion of how compactifications of M-theory or supergravity preserving some four-dimensional supersymmetry naturally lead to reduced holonomy or its generalization, reduced weak holonomy. We relate the existence of a (conformal) Killing spinor to the existence of certain closed and co-closed p-forms, and to the metric being Ricci flat or Einstein. Then, for seven-dimensional manifolds, we show that octonionic self-duality conditions on the spin connection are equivalent to G_2 holonomy and certain generalized self-duality conditions to weak G_2 holonomy. The latter lift to self-duality conditions for cohomogeneity-one spin(7) metrics. To illustrate the power of this approach, we present several examples where the self-duality condition largely simplifies the derivation of a G_2 or weak G_2 metric. 
  `Old' String Theory is a theory of one-dimensional extended objects, whose vibrations correspond to excitations of various target-space field modes including gravity. In these lectures I will give an introduction to low-energy Effective Target-Space Actions derived from conformal invariance conditions of the underlying sigma models in string theory, discuss cosmology, emphasizing the role of the dilaton field in inducing inflationary scenaria and in general expanding string universes. Specifically, I shall analyse some exact solutions of string theory with a linear dilaton, and discuss their role in inducing expanding Robertson-Walker Universes. I will mention briefly pre-Big-Bang scenaria of String Cosmology, in which the dilaton plays a crucial role. In view of recent claims on experimental evidence on the existence of cosmic acceleration in the universe today, with a positive non-zero cosmological constant (de Sitter type), I shall also discuss difficulties of incorporating such Universes with eternal acceleration in the context of critical string theory, and present scenaria for a graceful exit from such a phase. 
  We consider the brane universe in the bulk background of the topological AdS-Schwarzschild black holes, where the brane tension takes larger value than the fine-tunned value. The resulting universe is radiation dominated and has positive cosmological constant. We obtain the associated cosmological Cardy formula and the Cardy-Verlinde formula. We also derive the Hubble and the Bekenstein entropy bounds from the conjectured holography bound on the Casimir entropy. 
  We consider a class of 5-D brane-world solutions with a power-law warp factor $a(y)\propto y^{q}$, and bulk dilaton with profile $\phi \propto \ln y$, where $y$ is the proper distance in the extra dimension. This class includes the Heterotic M-theory brane-world of Refs. \cite{ovrut,ekpy} and the Randall-Sundrum (RS) model as a limiting case. In general, there are two moduli fields $y_{\pm}$, corresponding to the "positions" of two branes (which live at the fixed points of an orbifold compactification). Classically, the moduli are massless, due to a scaling symmetry of the action. However, in the absence of supersymmetry, they develop an effective potential at one loop. Local terms proportional to $K_\pm^4$, where $K_\pm=q/y_\pm$ is the local curvature scale at the location of the corresponding brane, are needed in order to remove the divergences in the effective potential. Such terms break the scaling symmetry and hence they may act as stabilizers for the moduli. When the branes are very close to each other, the effective potential induced by massless bulk fields behaves like $V\sim d^{-4}$, where $d$ is the separation between branes. When the branes are widely separated, the potentials for each one of the moduli generically develop a "Coleman-Weinberg"-type behaviour of the form $a^4(y_\pm)K_\pm^4 \ln(K_\pm/\mu_\pm)$, where $\mu_\pm$ are renormalization scales. In the RS case, the bulk geometry is $AdS$ and $K_\pm$ are equal to a constant, independent of the position of the branes, so these terms do not contribute to the mass of the moduli. However, for generic warp factor, they provide a simple stabilization mechanism. For $q\gtrsim 10$, the observed hierarchy can be naturally generated by this potential, giving the lightest modulus a mass of order $m_- \lesssim TeV$. 
  Matrix descriptions of even dimensional fuzzy spherical branes $S^{2k} $ in Matrix Theory and other contexts in Type II superstring theory reveal, in the large $N$ limit, higher dimensional geometries $SO(2k+1)/U(k)$, which have an interesting spectrum of $SO(2k+1)$ harmonics and can be up to 20 dimensional, while the spheres are restricted to be of dimension less than 10. In the case $k=2$, the matrix description has two dual field theory formulations. One involves a field theory living on the non-commutative coset $SO(5)/U(2)$ which is a fuzzy $S^2$ fibre bundle over a fuzzy $S^4$. In the other, there is a U(n) gauge theory on a fuzzy $S^4$ with $ {\cal O}(n^3)$ instantons. The two descriptions can be related by exploiting the usual relation between the fuzzy two-sphere and U(n) Lie algebra. We discuss the analogous phenomena in the higher dimensional cases, developing a relation between fuzzy $SO(2k)/U(k)$ cosets and unitary Lie algebras. 
  We derive the five-dimensional metrics which describe a non-singular boundary brane collision in the ekpyrotic scenario in the context of general relativity, taking into account brane tension. We show that the metrics constrain matter created in the collision to have negative energy density or pressure. In particular, the minimal field content of heterotic M-theory leads to negative energy density. We also consider bulk brane-boundary brane collisions and show that the collapse of the fifth dimension is an artifact of the four-dimensional effective theory. 
  We report a simplification in the large N matrix mechanics of light-cone matrix field theories. The absence of pure creation or pure annihilation terms in the Hamiltonian formulation of these theories allows us to find their reduced large N Hamiltonians as explicit functions of the generators of the Cuntz algebra. This opens up a free-algebraic playground of new reduced models -- all of which exhibit new hidden conserved quantities at large N and all of whose eigenvalue problems are surprisingly simple. The basic tool we develop for the study of these models is the infinite dimensional algebra of all normal-ordered products of Cuntz operators, and this algebra also leads us to a special number-conserving subset of these models, each of which exhibits an infinite number of new hidden conserved quantities at large N. 
  The spectrum of the infinite dimensional Neumann matrices M^{11}, M^{12} and M^{21} in the oscillator construction of the three-string vertex determines key properties of the star product and of wedge and sliver states. We study the spectrum of eigenvalues and eigenvectors of these matrices using the derivation K_1 = L_1 + L_{-1} of the star algebra, which defines a simple infinite matrix commuting with the Neumann matrices. By an exact calculation of the spectrum of K_1, and by consideration of an operator generating wedge states, we are able to find analytic expressions for the eigenvalues and eigenvectors of the Neumann matrices and for the spectral density. The spectrum of M^{11} is continuous in the range [-1/3, 0) with degenerate twist even and twist odd eigenvectors for every eigenvalue except for -1/3. 
  I present an exact solution for the Heisenberg picture, Dirac electron in the presence of an electric field which depends arbitrarily upon the light cone time parameter $x^+ = (t+x)/\sqrt{2}$. This is the largest class of background fields for which the mode functions have ever been obtained. The solution applies to electrons of any mass and in any spacetime dimension. The traditional ambiguity at $p^+ = 0$ is explicitly resolved. It turns out that the initial value operators include not only $(I + \gamma^0 \gamma^1) \psi$ at $x^+ = 0$ but also $(I - \gamma^0 \gamma^1) \psi$ at $x^- = -L$. Pair creation is a discrete and instantaneous event on the light cone, so one can compute the particle production rate in real time. In $D=1+1$ dimensions one can also see the anomaly. Another novel feature of the solution is that the expectation value of the current operators depends nonanalytically upon the background field. This seems to suggest a new, strong phase of QED. 
  The fundamental action of superon-graviton model(SGM) for space-time and matter is written down explicitly in terms of the fields of the graviton and superons by using the affine and the spin connection formalisms, alternatively. Some characteristic structures including some hidden symmetries of the gravitational coupling of superons are manifested (in two dimensional space-time) with some details of the calculations. SGM cosmology is discussed briefly. 
  Unabridged version of the Thesis presented to the University of L' Aquila, in partial fulfillment of the requirements for the ``Laurea'' degree in Physics, October 1998. Work carried out at the University of L'Aquila and at the University of Rome ``Tor Vergata''. 
  Twisted Eguchi-Kawai reduced chiral models are shown to be formally equivalent to a U(1) non-commutative parent theory. The non-commutative theory describes the vacuum dynamics of the non-commutative charged tachyonic field of a brane system. To make contact with the continuum non-commutative theory, a double scaling large N limit for the reduced model is required. We show a possible limiting procedure, which we propose to investigate numerically. Our numerical results show substantial consistency with the outlined procedure 
  We consider SU(2) gauge potentials over a space with a compactified dimension. A non-Abelian Fourier transform of the gauge potential in the compactified dimension is defined in such a way that the Fourier coefficients are (almost) gauge invariant. The functional measure and the gauge field strengths are expressed in terms of these Fourier coefficients. The emerging formulation of the non-Abelian gauge theory turns out to be an Abelian gauge theory of a set of fields defined over the initial space with the compactified dimension excluded. The Abelian theory contains an Abelian gauge field, a scalar field, and an infinite tower of vector matter fields, some of which carry Abelian charges. Possible applications of this formalism are discussed briefly. 
  We analyze the interplay between dispersion relations for the spectrum of Hawking quanta and the statistical mechanics of such a radiation. We first find the general relation between the occupation number density and the energy spectrum of Hawking quanta and then study several cases in details. We show that both the canonical and the microcanonical picture of the evaporation lead to the same linear dispersion relation for relatively large black holes. We also compute the occupation number obtained by instead assuming that the spectrum levels out (and eventually falls to zero) for very large momenta and show that the luminosity of black holes is not appreciably affected by the modified statistics. 
  The ultra-violet behavior of Kaluza-Klein theories on a one dimensional orbifold is discussed. An extension of dimensional regularization that can be applied to a compact dimension is presented. Using this, the FI-tadpole is calculated in the effective KK theory resulting from compactifying supersymmetric theories in 5 dimensions. 
  As a prototype of powerful non-abelian symmetry in an Integrable System, I will show the appearance of a Witt algebra of vector fields in the SG theory. This symmetry does not share anything with the well-known Virasoro algebra of the conformal $c=1$ unperturbed limit. Although it is quasi-local in the SG field theory, nevertheless it gives rise to a local action on $N$-soliton solution variables. I will explicitly write the action on special variables, which possess a beautiful geometrical meaning and enter the Form Factor expressions of quantum theory. At the end, I will also give some preliminary hints about the quantisation. 
  We further analyze a recent perturbative two-loop calculation of the expectation value of two axi-symmetric circular Maldacena-Wilson loops in N=4 gauge theory. Firstly, it is demonstrated how to adapt the previous calculation of anti-symmetrically oriented circles to the symmetric case. By shrinking one of the circles to zero size we then explicitly work out the first few terms of the local operator expansion of the loop. Our calculations explicitly demonstrate that circular Maldacena-Wilson loops are non-BPS observables precisely due to the appearance of unprotected local operators. The latter receive anomalous scaling dimensions from non-ladder diagrams. Finally, we present new insights into a recent conjecture claiming that coincident circular Maldacena-Wilson loops are described by a Gaussian matrix model. We report on a novel, supporting two-loop test, but also explain and illustrate why the existing arguments in favor of the conjecture are flawed. 
  By considering 5--dimensional cosmological models with a bulk filled with a perfect fluid and a cosmological constant, we have found regular instantonic solution which is free from any singularity at the origin of the extra--coordinate and describe 5--dimensional asymptotically anti de Sitter wormhole, when the bulk has a topology $ R \times S^4 $ and is filled with dust and a negative cosmological constant. Compactified brane-world instantons which are built up from such instantonic solution describe either a single brane or a string of branes. Their analytical continuation to the pseudo--Riemannian metric can give rise to either 4-dimensional inflating branes or solutions with the same dynamical behaviour for extra--dimension and branes, in addition to multitemporal solutions. Dust brane-world models with other spatial topologies are also considered. 
  Finite groups are of the greatest importance in science. Loops are a simple generalization of finite groups: they share all the group axioms except for the requirement that the binary operation be associative. The least loops that are not themselves groups are those of order five. We offer a brief discussion of these loops and challenge the reader (especially Holger) to find useful applications for them in physics. 
  We derive an elegant analytic formula for the energy spectrum of the relativistic Dirac-Morse problem, which has been solved recently. The new formula displays the properties of the spectrum more vividly. 
  We consider the Kerr-de Sitter (Kerr-dS) black hole in various dimensions. Introducing a counterterm, we show that the total action of these spacetimes are finite. We compute the masses and the angular momenta of Kerr-dS spaces with one rotational parameter in four, five and seven dimensions. These conserved charges are also computed for the case of Kerr-dS space with two rotational parameters in five dimensions. Although the angular momentum density due to the counterterm is nonzero, it gives a vanishing contribution to the total angular momentum. We also find that the total angular momentum of the spacetime is independent of the radius of the boundary for all cases, a fact that is not true for the total mass of the system. 
  Dirac equation for a charged particle in static electromagnetic field is written for special cases of spherically symmetric potentials. Besides the well known Dirac-Coulomb and Dirac-Oscillator potentials, we obtain a relativistic version of the S-wave Morse potential. This is accomplished by adding a simple exponential potential term to the Dirac operator, which in the nonrelativistic limit reproduces the usual Morse potential. The relativistic bound states spectrum and spinor wavefunctions are obtained. 
  We apply the coset character identities (generalization of Jacobi's abstruse identity) to compact and noncompact Gepner models. In the both cases, we prove that the partition function actually vanishes due to the spacetime supersymmetry. In the case of the compact models and discrete parts of the noncompact models, the partition function includes the expected vanishing factor. But the character identities used to the continuous part of the noncompact models suggest that these models have twice as many supersymmetry as expected. This fact is an evidence for the conjecture that the holographically dual of the string theory on an actually singular Calabi-Yau manifold is a super CONFORMAL field theory. The extra SUSY charges are interpreted as the superconformal S generators. 
  We study Landau Ginzburg (LG) theories mirror to 2D N=2 gauged linear sigma models on toric Calabi-Yau manifolds. We derive and solve new constraint equations for Landau Ginzburg elliptic Calabi-Yau superpotentials, depending on the physical data of dual linear sigma models. In Calabi-Yau threefolds case, we consider two examples. First, we give the mirror symmetry of the canonical line bundle over the Hirzebruch surfaces $\bf F_n$. Second, we find a special geometry with the affine so(8) Lie algebra toric data extending the geometry of elliptically fibered K3. This geometry leads to a pure N=1 six dimensional SO(8) gauge model from the F-theory compactification. For Calabi-Yau fourfolds, we give a new algebraic realization for ADE hypersurfaces. 
  We discuss geometric aspects of orbifold conformal field theories in the moduli space of N=(4,4) superconformal field theories with central charge c=6. Part of this note consists of a summary of our earlier results on the location of these theories within the moduli space [NW01,Wen] and the action of a specific version of mirror symmetry on them [NW]. We argue that these results allow for a direct translation from geometric to conformal field theoretic data. Additionally, this work contains a detailed discussion of an example which allows the application of various versions of mirror symmetry on K3. We show that all of them agree in that point of the moduli space. 
  Dirac equation for a charged spinor in electromagnetic field is written for special cases of spherically symmetric potentials. This facilitates the introduction of relativistic extensions of shape invariant potential classes. We obtain the relativistic spectra and spinor wavefunctions for all potentials in one of these classes. The nonrelativistic limit reproduces the usual Rosen-Morse I & II, Eckart, Poschl-Teller, and Scarf potentials. 
  We discuss the partners of the stress energy tensor and their structure in Logarithmic conformal field theories. In particular we draw attention to the fundamental differences between theories with zero and non-zero central charge. However they are both characterised by at least two independent parameters. We show how, by using a generalised Sugawara construction, one can calculate the logarithmic partner of T. We show that such a construction works in the c=-2 theory using the conformal dimension one primary currents which generate a logarithmic extension of the Kac-Moody algebra. 
  Some recent literature has examined the holographic-induced cosmology of a brane universe in the background of an anti-de Sitter-black hole geometry. In this regard, curved-brane scenarios have begun to receive considerable attention. Our current interest is in a formal discrepancy that exists between two such works by Padilla (hep-th/0111247) and Youm (hep-th/0111276). In particular, these authors have incorporated different values for a conformal factor that is used to relate the thermodynamics of the relevant (AdS bulk and CFT brane) spacetimes. After a more general review, we clarify this issue and discuss the implications on the prior results. 
  It was found that deformation of S^7 gives rise to renormalization group(RG) flow from N=8, SO(8)-invariant UV fixed point to N=1, G_2-invariant IR fixed point in four-dimensional gauged N=8 supergravity. Also BPS supersymmetric domain wall configuration interpolated between these two critical points. In this paper, we use the G_2-invariant RG flow equations for both scalar fields and domain-wall amplitude and apply them to the nonlinear metric ansatz developed by de Wit, Nicolai and Warner some time ago. We carry out the M-theory lift of the G_2-invariant RG flow through a combinatoric use of the four-dimensional RG flow equations and eleven-dimensional Einstein-Maxwell equations. The nontrivial r(that is the coordinate transverse to the domain wall)-dependence of vacuum expectation values becomes consistent with not only at the critical points but also along the supersymmetric RG flow connecting two critical points. By applying an ansatz for an eleven-dimensional three-form gauge field with varying scalars, we discover an exact solution to the eleven-dimensional Einstein-Maxwell equations corresponding to the M-theory lift of the G_2-invariant RG flow. 
  A harmonic oscillator is an indefinite-frequency one if the parameter $\omega$ is replaced by an operator. An ensemble of $N$ such oscillators may be regarded as a toy model of a bosonic quantum field. All the possible frequencies associated with a given problem are present already in a single oscillator and $N$ can be finite. Due to the operator character of $\omega$ the resulting algebra of creation-annihilation operators is non-canonical. In the limit of large $N$ one recovers perturbation theory formulas of the canonical quantum field theory but with form factors automatically built in. Vacuum energy of the ensemble is finite, a fact discussed in the context of the cosmological constant problem. Space of states is given by a vector bundle with Fock-type fibers. Interactions of the field with 2-level systems, including Rabi oscillations and spontaneous emission, are discussed in detail. 
  Quantum non-perturbative geometry of the universal hypermultiplet is investigated. We consider the simple case when the D-instantons, originating from the Calabi-Yau wrapped D2-branes, preserve a U(1)xU(1) symmetry of the universal hypermultiplet moduli space. The cluster decomposition of D-instantons is proved to be valid in a curved spacetime. We find an SL(2,Z) duality-invariant quaternionic solution to the effective NLSM metric of the universal hypermultiplet, which is governed by a modular-invariant function. This function appears to be the same function found by Green and Gutperle, and describing the modular invariant completion of the R^4 term by the D-instanton effects in the type-II superstring/M-theory. We argue that our solution interpolates between the perturbative (large CY volume) region and the superconformal (Landau-Ginzburg) region in the universal hypermultiplet moduli space. We also calculate a non-perturbative scalar potential in the hyper-K\"ahler limit, when an abelian isometry of the universal hypermultiplet moduli space is gauged in the presence of D-instantons. 
  The nonlinear gravitational interaction is investigated in the Randall-Sundrum two branes model with the radius stabilization mechanism. As the stabilization model, we assume a single scalar field that has a potential in the bulk and a potential on each brane. We develop a formulation of the second order gravitational perturbations under the assumption of a static and axial-symmetric five-dimensional metric that is spherically symmetric in the four-dimensional sense. After deriving the formal solutions for the perturbations, we discuss the gravity on each brane induced by the matter on its own side, taking the limit of large coupling of the scalar field interaction term on the branes. We show using the Goldberger-Wise stabilization model that four-dimensional Einstein gravity is approximately recovered in the second order perturbations. 
  We renormalize the Curci-Ferrari model at two loops in the MSbar scheme in an arbitrary covariant gauge. 
  We review the construction and phenomenology of intersecting brane worlds. The breaking of chiral and supersymmetry together with a reduction of gauge symmetry by introducing D-branes at generic angles into type I or type II string compactifications is taken as a starting point for the engineering of phenomenologically appealing four-dimensional vacua. Examples that come very close to Standard Model and GUT physics are considered and issues like the cancellation of anomalies and the emergence of global symmetries studied in some detail. Concerning the gravitational backreaction, the perturbative potential generated at the disc level is shown to lead to a run-away instability of geometric moduli within the purely toroidal compactifications. It can at least partly be avoided in certain orbifold vacua, where the relevant moduli are frozen. These more restricted models still possess sufficient freedom to find semi-realistic brane world orbifolds. 
  We discuss, using the imaginary time method, some aspects of the connection between the Ward identity, the non-analyticity of amplitudes and the causality relation in QED at finite temperature. 
  The covariant light-front equations have been solved exactly for a two fermion system with different boson exchange ladder kernels. We present a method to study the cutoff dependence of these equations and to determine whether they need to be regularized or not. Results are presented for scalar and pseudo-scalar exchange. This latter furthermore exhibits some strange particularities which will be discussed. 
  This contribution presents the running triple-gluon-vertex coupling constant, g_lambda, in Hamiltonians for the gluons that are characterized by the size 1/lambda. The coupling constant is obtained from renormalization group equations for effective particles in canonical light-front QCD in third order perturbation theory. lambda plays the role of a finite cutoff parameter and is varied from 100 Gev down to 100 MeV. 
  An analysis of a spherically symmetric braneworld configuration is performed when the intrinsic curvature scalar is included in the bulk action; the vanishing of the electric part of the Weyl tensor is used as the boundary condition for the embedding of the brane in the bulk. All the solutions outside a static localized matter distribution are found; some of them are of the Schwarzschild-(A)dS_{4} form. Two modified Oppenheimer-Volkoff interior solutions are also found; one is matched to a Schwarzschild-(A)dS_{4} exterior, while the other does not. A non-universal gravitational constant arises, depending on the density of the considered object; however, the conventional limits of the Newton's constant are recovered. An upper bound of the order of TeV for the energy string scale is extracted from the known solar system measurements (experiments). On the contrary, in usual brane dynamics, this string scale is calculated to be larger than TeV. 
  The brane-world scenario offers the possibility for signals to travel outside our visible universe and reenter it. We find the condition for a signal emitted from the brane to return to the brane. We study the propagation of such signals and show that, as seen by a 4D observer, these signals arrive earlier than light traveling along the brane. We also study the horizon problem and find that, while the bulk signals can travel far enough to homogenize the visible universe, it is unlikely that they have a significant effect since they are redshifted in the gravitational field of the bulk black hole. 
  We stress that in contradiction with what happens in space dimensions $n \geq 3$, there is no strict bound on the number of bound states with the same structure as the semi-classical estimate for large coupling constant and give, in two dimensions, examples of weak potentials with one or infinitely many bound states. We derive bounds for one and two dimensions which have the "right" coupling constant behaviour for large coupling. 
  The multi-loop amplitudes for the closed, oriented superstring are represented by finite dimensional integrals of explicit functions calculated through the super-Schottky group parameters and interaction vertex coordinates on the supermanifold. The integration region is proposed to be consistent with the group of the local symmetries of the amplitude and with the unitarity equations. It is shown that, besides the SL(2) group, super-Schottky group and modular one, the total group of the local symmetries includes an isomorphism between sets of the forming group transformations, the period matrix to be the same. The singular integration configurations are studied. The calculation of the integrals over the above configurations is developed preserving all the local symmetries of the amplitude, the amplitudes being free from divergences. The nullification of the 0-, 1-, 2- and 3-point amplitudes of massless states is verified. Vanishing the amplitudes for a longitudinal gauge boson is argued. 
  We consider the sine-Gordon and affine Toda field theories on the half-line with classically integrable boundary conditions, and show that in the quantum theory a remnant survives of the bulk quantized affine algebra symmetry generated by non-local charges. The paper also develops a general framework for obtaining solutions of the reflection equation by solving an intertwining property for representations of certain coideal subalgebras of quantized affine algebras. 
  We consider quantum N=2 string embedded into the N=4 topological framework from the perspective of the old covariant quantisation. Making use of the causality and cyclic symmetry of tree amplitudes we argue that no Lorentz covariant boson emission vertex can be constructed within the N=4 topological formalism. 
  We deform the interaction between nonrelativistic point particles on a plane and a Chern-Simons field to obtain an action invariant with respect to time-dependent area-preserving diffeomorphisms. The deformed and undeformed Lagrangians are connected by a point transformation leading to a classical Seiberg-Witten map between the corresponding gauge fields. The Schroedinger equation derived by means of Moyal-Weyl quantization from the effective two-particle interaction exhibits - a singular metric, leading to a splitting of the plane into an interior (bag-) and an exterior region, - a singular potential (quantum correction) with singularities located at the origin and at the edge of the bag. We list some properties of the solutions of the radial Schroedinger equation. 
  In this note I try to clarify the problem of perturbations in the ekpyrotic universe. I write down the most general matching conditions and specify the choices taken by the two debating sides. I also bring up the problem of surface stresses which always have to be present when a transition from a collapsing to an expanding phase is made. 
  The Seiberg-Witten map links noncommutative gauge theories to ordinary gauge theories, and allows to express the noncommutative variables in terms of the commutative ones. Its explicit form can be found order by order in the noncommutative parameter theta and the gauge potential A by the requirement that gauge orbits are mapped on gauge orbits. This of course leaves ambiguities, corresponding to gauge transformations, and there is an infinity of solutions. Is there one better, clearer than the others ? In the abelian case, we were able to find a solution, linked by a gauge transformation to already known formulas, which has the property of admitting a recursive formulation, uncovering some pattern in the map. In the special case of a pure gauge, both abelian and non-abelian, these expressions can be summed up, and the transformation is expressed using the parametrisation in terms of the gauge group. 
  Bucher [Bucher2001] has recently proposed an interesting brane-world cosmological scenario where the ``Big Bang'' hypersurface is the locus of collision of two vacuum bubbles which nucleate in a five dimensional flat space. This gives rise to an open universe, where the curvature can be very small provided that $d/R_0$ is sufficiently large. Here, d is the distance between bubbles and $R_0$ is their size at the time of nucleation. Quantum fluctuations develop on the bubbles as they expand towards each other, and these in turn imprint cosmological perturbations on the initial hypersurface. We present a simple formalism for calculating the spectrum of such perturbations and their subsequent evolution. We conclude that, unfortunately, the spectrum is very tilted, with spectral index $n_s=3$. The amplitude of fluctuations at horizon crossing is given by $<(\delta \rho/\rho)^2> \sim (R_0/d)^2 S_E^{-1} k^2$, where $S_E\gg 1$ is the Euclidean action of the instanton describing the nucleation of a bubble and k is the wavenumber in units of the curvature scale. The spectrum peaks on the smallest possible relevant scale, whose wave-number is given by $k\sim d/R_0$. We comment on the possible extension of our formalism to more general situations where a Big Bang is ignited through the collision of 4D extended objects. 
  We perform the Batalin-Vilkovisky analysis of gauge-fixing for graded Chern-Simons theories. Upon constructing an appropriate gauge-fixing fermion, we implement a Landau-type constraint, finding a simple form of the gauge-fixed action. This allows us to extract the associated Feynman rules taking into account the role of ghosts and antighosts. Our gauge-fixing procedure allows for zero-modes, hence is not limited to the acyclic case. We also discuss the semiclassical approximation and the effective potential for massless modes, thereby justifying some of our previous constructions in the Batalin-Vilkovisky approach. 
  The induced fractional fermion number at zero temperature is topological (in the sense that it is only sensitive to global asymptotic properties of the background field), and is a sharp observable (in the sense that it has vanishing rms fluctuations). In contrast, at finite temperature, it is shown to be generically nontopological, and not a sharp observable. 
  The relativistic J-matrix is investigated in the case of Coulomb-free scattering for a general short-range spin-dependent perturbing potential and in two different L2 bases. The resulting recursion relation of the reference problem, in this case, has an analytic solution. The non-relativistic limit is obtained and shown to be identical to the familiar non-relativistic J-matrix. Scattering examples are given to verify the non-relativistic limit and calculate the relativistic effects in the phase shift. 
  Unrenormalizable theories contain infinitely many free parameters. Considering these theories in terms of the Wilsonian renormalization group (RG), we suggest a method for removing this large ambiguity. Our basic assumption is the existence of the maximal ultraviolet cutoff in a cutoff theory, and we require that the theory be so fine-tuned as to reach the maximal cutoff. The theory so obtained behaves as a local continuum theory to the shortest distance. In concrete examples of the scalar theory we find that at least in a certain approximation to the Wilsonian RG, this requirement enable us to make unique predictions in the infrared regime in terms of a finite number of independent parameters. Therefore, the method might provide a way for calculating quantum corrections in a low-energy effective theory of quantum gravity. 
  Based on the gauge potential decomposition theory and the $\phi $-mapping theory, the topological inner structure of the Chern-Simons-Higgs vortex has been showed in detail. The evolution of CSH vortices is studied from the topological properties of the Higgs scalar field. The vortices are found generating or annihilating at the limit points and encountering, splitting or merging at the bifurcation points of the scalar field $\phi .$ 
  We consider the flat supersymmetric D2 and anti-D2 system, which follows from ordinary noncommutative D2 anti-D2 branes by turning on an appropriate worldvolume electric field describing dissolved fundamental strings. We study the strings stretched between D2 and anti-D2 branes and show explicitly that the would-be tachyonic states become massless. We compute the string spectrum and clarify the induced noncommutativity on the worldvolume. The results are compared with the matrix theory description of the worldvolume gauge theories. 
  We study supersymmetric vortex solutions in three-dimensional abelian gauged supergravity. First, we construct the general U(1)-gauged D=3, N=2 supergravity whose scalar sector is an arbitrary Kahler manifold with U(1) isometry. This construction clarifies the connection between local supersymmetry and the specific forms of some scalar potentials previously found in the literature -- in particular, it provides the locally supersymmetric embedding of the abelian Chern-Simons Higgs model. We show that the Killing spinor equations admit rotationally symmetric vortex solutions with asymptotically conical geometry which preserve half of the supersymmetry. 
  In this paper, the evolution of galaxies is by the incompatibility between dark matter and baryonic matter. Due to the structural difference, baryonic matter and dark matter are incompatible to each other as oil droplet and water in emulsion. In the interfacial zone between dark matter and baryonic matter, this incompatibility generates the modification of Newtonian dynamics to keep dark matter and baryonic matter apart. The five periods of baryonic structure development in the order of increasing incompatibility are the free baryonic matter, the baryonic droplet, the galaxy, the cluster, and the supercluster periods. The transition to the baryonic droplet generates density perturbation in the CMB. In the galaxy period, the first-generation galaxies include elliptical, normal spiral, barred spiral, irregular, and dwarf spheroidal galaxies. In the cluster period, the second-generation galaxies include modified giant ellipticals, cD, evolved S0, dwarf elliptical, BCD, and tidal dwarf galaxies. The whole observable expanding universe behaves as one unit of emulsion with increasing incompatibility between dark matter and baryonic matter. 
  Classical higher-derivative gravity is investigated in the context of the holographic renormalization group (RG). We parametrize the Euclidean time such that one step of time evolution in (d+1)-dimensional bulk gravity can be directly interpreted as that of block spin transformation of the d-dimensional boundary field theory. This parametrization simplifies the analysis of the holographic RG structure in gravity systems, and conformal fixed points are always described by AdS geometry. We find that higher-derivative gravity generically induces extra degrees of freedom which acquire huge mass around stable fixed points and thus are coupled to highly irrelevant operators at the boundary. In the particular case of pure R^2-gravity, we show that some region of the coefficients of curvature-squared terms allows us to have two fixed points (one is multicritical) which are connected by a kink solution. We further extend our analysis to Minkowski time to investigate a model of expanding universe described by the action with curvature-squared terms and positive cosmological constant, and show that, in any dimensionality but four, one can have a classical solution which describes time evolution from a de Sitter geometry to another de Sitter geometry, along which the Hubble parameter changes drastically. 
  Recent progress in the study of solitons and black holes in non-Abelian field theories coupled to gravity is reviewed. New topics include gravitational binding of monopoles, black holes with non-trivial topology, Lue-Weinberg bifurcation, asymptotically AdS lumps, solutions to the Freedman-Schwarz model with applications to holography, non-Abelian Born-Infeld solutions 
  Nonperturbative corrections in type II string theory corresponding to Riemann surfaces with one boundary are calculated in several noncompact geometries of desingularized orbifolds. One of these models has a complicated phase structure which is explored. A general condition for integrality of the numerical invariants is discussed. 
  We study large N dualities for a general class of N=1 theories realized on type IIB D5 branes wrapping 2-cycles of local Calabi-Yau threefolds or as effective field theories on D4 branes in type IIA brane configurations. We completely solve the issue of the classical moduli space for N=2, U(N_1)x ... x U(N_n) theories deformed by a general superpotential for the adjoint and bifundamental fields. The N=1 geometries in type IIB and its T-dual brane configurations are presented and they agree with the field theory analysis. We investigate the geometric transitions in the ten dimensional theories as well as in M-theory. Strong coupling effects in field theory are analyzed in the deformed geometry with fluxes. Gluino condensations are identified the normalizable deformation parameters while the vacuum expectation values of the bifundamental fields are with the non-normalizable ones. By lifting to M theory, we get a transition from finite coverings of non-hyperelliptic curves to non-hyperelliptic curves. We also discuss orientifold theories, Seiberg dualities and mirror symmetries. 
  We study the excitations of a massive Schwarzschild black hole of mass M resulting from the capture of infalling matter described by a massless scalar field. The near-horizon dynamics of this system is governed by a Hamiltonian which is related to the Virasoro algebra and admits a one-parameter family of self-adjoint extensions described by a parameter z \in R . The density of states of the black hole can be expressed equivalently in terms of z or M, leading to a consistent relation between these two parameters. The corresponding black hole entropy is obtained as S = S(0) - 3/2 log S(0) + C, where S(0) is the Bekenstein-Hawking entropy, C is a constant with other subleading corrections exponentially suppressed. The appearance of this precise form of the black hole entropy within our formalism, which is expected on general grounds in any conformal field theoretic description, provides strong evidence for the near-horizon conformal structure in this system. 
  An on-shell model for the supersymmetric index counting multiplicities of BPS states of the M5-brane theory is reviewed. In particular we explicitly study the tensionless Little String states appearing at intersections in a bound state of $N$ 5-branes wrapped on a six-manifold with product topology $M_4\times T^2$. 
  In this lecture I will review some results about the discrete light-cone quantization (DLCQ) of strings and some connections of the results with matrix string theory. I will review arguments which show that, in the path integral representation of the thermal free energy of a string, the compactifications which are necessary to obtain discrete light-cone quantization constrains the integral over all Riemann surfaces of a given genus to the set of those Riemann surfaces which are branched covers of a particular torus. I then review an explicit check of this result at genus 1. I discuss the intriguing suggestion that these branched covers of a torus are related to those which are found in a certain limit of the matrix string model 
  We construct the covariant $\kappa$-symmetric superstring action for type $IIB$ superstring on plane wave space supported by Ramond-Ramond background. The action is defined as a 2d sigma-model on the coset superspace. We fix the fermionic and bosonic light-cone gauges in the covariant Green-Schwarz superstring action and find the light-cone string Lagrangian and the Hamiltonian. The resulting light-cone gauge action is quadratic in both the bosonic and fermionic superstring 2d fields, and therefore, this model can be explicitly quantized. We also obtain a realization of the generators of the basic superalgebra in terms of the superstring 2d fields in the light-cone gauge. 
  We investigate the charged Schwarzschild-Anti-deSitter (SAdS) BH thermodynamics in 5d Einstein-Gauss-Bonnet gravity with electromagnetic field. The Hawking-Page phase transitions between SAdS BH and pure AdS space are studied. The corresponding phase diagrams (with critical line defined by GB term coefficient and electric charge) are drawn. The possibility to account for higher derivative Maxwell terms is mentioned.  In frames of proposed dS/CFT correspondence it is demonstrated that brane gravity maybe localized similarly to AdS/CFT.  SdS BH thermodynamics in 5d Einstein and Einstein-Gauss-Bonnet gravity is considered. The corresponding (complicated) surface counterterms are found and used to get the conserved BH mass, free energy and entropy. The interesting feature of of higher derivative gravity is the possibility for negative (or zero) SdS (or SAdS) BH entropy which depends on the parameters of higher derivative terms. We speculate that the appearence of negative entropy may indicate a new type instability where a transition between SdS (SAdS) BH with negative entropy to SAdS (SdS) BH with positive entropy would occur. 
  We study the $N=2$ supersymmetric $E_6$ models on the 6-dimensional space-time where the supersymmetry and gauge symmetry can be broken by the discrete symmetry. On the space-time $M^4\times S^1/(Z_2\times Z_2') \times S^1/(Z_2\times Z_2')$, for the zero modes, we obtain the 4-dimensional $N=1$ supersymmetric models with gauge groups $SU(3)\times SU(2) \times SU(2) \times U(1)^2$, $SU(4)\times SU(2) \times SU(2) \times U(1)$, and $SU(3)\times SU(2) \times U(1)^3$ with one extra pair of Higgs doublets from the vector multiplet. In addition, considering that the extra space manifold is the annulus $A^2$ and disc $D^2$, we list all the constraints on constructing the 4-dimensional $N=1$ supersymmetric $SU(3)\times SU(2) \times U(1)^3$ models for the zero modes, and give the simplest model with $Z_9$ symmetry. We also comment on the extra gauge symmetry breaking and its generalization. 
  We study a generalization of the Randall-Sundrum mechanism for generating the weak/Planck hierarchy, which uses two rather than one warped extra dimension, and which requires no negative tension branes. A 4-brane with one exponentially large compact dimension plays the role of the Planck brane. We investigate the dynamical stability with respect to graviton, graviphoton and radion modes. The radion is shown to have a tachyonic instability for certain models of the 4-brane stress-energy, while it is stable in others, and massless in a special case. If stable, its mass is in the milli-eV range, for parameters of the model which solve the hierarchy problem. The radion is shown to couple to matter with gravitational strength, so that it is potentially detectable by submillimeter-range gravity experiments. The radion mass can be increased using a bulk scalar field in the manner of Goldberger and Wise, but only to order MeV, due to the effect of the large extra dimension. The model predicts a natural scale of 10^{13} GeV on the 4-brane, making it a natural setting for inflation from the ultraviolet brane. 
  We discuss the relation between the superembedding method for deriving worldvolume actions for D-branes and the method of Partially Broken Global Supersymmetry based upon linear and non-linear realisations of SUSY. We give the explicit relation for the cases of space filling branes in 3 and 4 dimensions and show that the standard F-constraint of the superembedding method is the source of the required covariant non-linear constraints for the PBGS method. 
  Second quantization of a classical nonrelativistic one-particle system as a deformation quantization of the Schrodinger spinless field is considered. Under the assumption that the phase space of the Schrodinger field is $C^{\infty}$, both, the Weyl-Wigner-Moyal and Berezin deformation quantizations are discussed and compared. Then the geometric quantum mechanics is also quantized using the Berezin method under the assumption that the phase space is $CP^{\infty}$ endowed with the Fubini-Study Kahlerian metric. Finally, the Wigner function for an arbitrary particle state and its evolution equation are obtained. As is shown this new "second quantization" leads to essentially different results than the former one. For instance, each state is an eigenstate of the total number particle operator and the corresponding eigenvalue is always ${1 \over \hbar}$. 
  We study the moduli space of the D0-brane system on Dp-branes realized in the noncommutative super Yang-Mills theory. By examining the fluctuations around the solitonic solutions generated by solution generating technique, we confirm the interpretation of the moduli as the positions of D0-branes on Dp-branes. Low-energy scattering process is also examined for two D0-branes. We find that the D0-branes scatter at right angle for head-on collision in the D0-D4 system. For D0-D6 and D0-D8 systems we find special solutions which reduce to the D0-D4 case, giving the same behavior. This suggests that the scattering at right angle for head-on collision is a universal behavior of this kind of soliton scatterings. 
  A possible mechanism accounting for monopole configurations in continuum Yang-Mills theories is discussed. The presence of the gauge fixing term is taken into account. 
  The fundamental action of superon-graviton model(SGM) of Einstein-Hilbert type for space-time and matter is written down explicitly in terms of the fields of the graviton and superons by using the affine connection formalism and the spin connection formalism. Some characteristic structures including some hidden symmetries of the gravitational coupling of superons are manifested (in two dimensional space-time) with some details of the calculations. SGM cosmology is discussed briefly. 
  It is shown that quantum mechanics on noncommutative spaces (NQM) can be obtained by the canonical quantization of some underlying second class constrained system formulated in extended configuration space. It leads, in particular, to an intriguing possibility of quantization in terms of the initial (noncommutative) variables. Two different formulations are discissed. The first one is appropriate for at most quadratic potential. The noncommutativity parameter and rank of matrix of the constraint brackets depend on the potential. It explains appearance of two phases of the resulting NQM. The second formulation is appropriate for an arbitrary potential. In both cases the corresponding Lagrangian action is presented and quantized, which leads to quantum mechanics with ordinary product replaced by the Moyal product. 
  We find supertubes with arbitrary (and not necessarily planar) cross section; the stability against the D2-brane tension is due to a compensation by the local momentum generated by Born-Infeld fields. Stability against long-range supergravity forces is also established. We find the corresponding solutions of the infinite-N M(atrix) model. The supersymmetric D2/anti-D2 system is a special case of the general supertube, and we show that there are no open-string tachyons in this system via a computation of the open-string one-loop vacuum energy. 
  We obtain exact expressions for the quasi-normal modes of various spin for the BTZ black hole. These modes determine the relaxation time of black hole perturbations. Exact agreement is found between the quasi-normal frequencies and the location of the poles of the retarded correlation function of the corresponding perturbations in the dual conformal field theory. This then provides a new quantitative test of the AdS/CFT correspondence. 
  In a recent paper (hep-th/0111091), the near-extremal thermodynamics of a 4-dimensional Reissner-Nordstrom black hole had been considered. In the current letter, we extend this prior treatment to the more general case of a spherically symmetric, charged black hole of arbitrary dimensionality. After summarizing the earlier work, we demonstrate a duality that exists between the near-extremal sector of spherically symmetric black holes and Jackiw-Teitelboim theory. On the basis of this correspondence, we argue that back-reaction effects prohibit any of these ``RN-like'' black holes from reaching extremality and, moreover, from coming arbitrarily close to an extremal state. 
  We extend a non local and non covariant version of the Thirring model in order to describe a many-body system with backward and umklapp scattering processes. We express the vacuum to vacuum functional in terms of a non trivial fermionic determinant. Using path-integral methods we find a bosonic representation for this determinant which allows us to obtain an effective action for the collective excitations of the system. By introducing a non local version of the self-consistent harmonic approximation, we get an expression for the gap of the charge-density excitations as functional of arbitrary electron-electron potentials. As an example we also consider the case of a non contact umklapp interaction. 
  We study the addition of an irrelevant operator to the N=4 supersymmetric large n SU(n) gauge theory, in the presence of finite temperature, T. In the supergravity dual, the effect of the operator is known to correspond to a deformation of the AdS_5 x S^5 ``throat'' which restores the asymptotic ten dimensional Minkowski region of spacetime, completing the full D3-brane solution. The system at non-zero T is interesting, since at the extremes of some of the geometrical parameters the geometry interpolates between a seven dimensional spherical Minkowskian Schwarzschild black hole (times R^3) and a five dimensional flat AdS Schwarzschild black hole (times S^5). We observe that when the coupling of the operator reaches a critical value, the deconfined phase, which is represented by the geometry with horizon, disappears for all temperatures, returning the system to a confined phase which is represented by the thermalised extremal geometry. 
  We construct supergravity solutions describing the near horizon limit of D1D5 systems with non-trivial boundary conditions. Upon reduction to five dimensions they define Melvin universes with NS--NS/RR fluxes, that smoothly interpolate between two different AdS_3 geometries which define fixed points for the RG--flow of the dual field theory. We discuss the decoupling limits at the two ends of the flow. We also present a systematic study of the global properties of our solution. In particular we show how, although the AdS_3 x S^3 global isometry group is broken down to SU(2)_R x U(1)^3 by global identifications, a full two-dimensional conformal group of isometries, with the expected central charge, is restored at infinity. 
  The vacuum expectation value of the Wilson loop in the dual representation is calculated in the dual Higgs model with dual Dirac strings. It is shown that the averaged value of the Wilson loop in the dual representation obeys the area-law falloff. Quantum fluctuations of the dual-vector and the Higgs field around Abrikosov flux lines induced by dual Dirac strings in a dual superconducting vacuum and string shape fluctuations are taken into account. 
  Geodetic evolution of a de-Sitter brane is exclusively driven by a Higgs potential, rather than by a plain cosmological constant. The deviation from Einstein gravity, parameterized by the conserved bulk energy, is characterized by a hairy horizon which serves as the locus of unbroken symmetry. The quartic structure of the potential, singled out on finiteness grounds of the total (including the dark component) energy density, chooses the no-boundary proposal. 
  The quantum gravity is formulated based on principle of local gauge invariance. The model discussed in this paper has local gravitational gauge symmetry and gravitational field is represented by gauge field. In leading order approximation, it gives out classical Newton's theory of gravity. It can also give out Einstein's field equation with cosmological constant. For classical tests, it gives out the same theoretical predictions as those of general relativity. This quantum gauge theory of gravity is a renormalizable quantum theory. 
  We write the type IIB worldsheet action in classes of bosonic curved backgrounds threaded with Ramond-Ramond fluxes. The fixing of the kappa symmetry in the light-cone gauge and the use of the Bianchi identities of the supergravity theory lead to an expression of a relatively simple form, yet rich with new physical information about how fundamental strings react to the presence of RR fields. The results are useful in particular to the study of vacuum structure and dynamics in the context of the Holographic duality; and to possibly formulate an open-closed string duality at the level of the worldsheet. 
  Assigning an intrinsic constant dipole moment to any field, we present a new kind of associative star product, the dipole star product, which was first introduced in [hep-th/0008030]. We develop the mathematics necessary to study the corresponding noncommutative dipole field theories. These theories are sensible non-local field theories with no IR/UV mixing. In addition we discuss that the Lorentz symmetry in these theories is ``softly'' broken and in some particular cases the CP (and even CPT) violation in these theories may become observable. We show that a non-trivial dipole extension of N=4, D=4 gauge theories can only be obtained if we break the SU(4) R (and hence super)-symmetry. Such noncommutative dipole extensions, which in the maximal supersymmetric cases are N=2 gauge theories with matter, can be embedded in string theory as the theories on D3-branes probing a smooth Taub-NUT space with three form fluxes turned on or alternatively by probing a space with R-symmetry twists. We show the equivalences between the two approaches and also discuss the M-theory realization. 
  We reanalyse the gravitational couplings of the perturbative orientifold planes $Op^-$, $Op^+$ (and D-branes). We first compute their $D_{-1}$ instantonic corrections for $p=3$. Then, by using U-dualities, we obtain the Wess-Zumino terms of orientifolds with RR flux for $p \leq 5$. The expressions for the effective actions can be partially checked via M-theory. We point out a previous oversimplification and we show in fact that the difficulty still stands in the way of the full computation of 7 Brane instanton corrections. 
  Recent observations suggest that the cosmological equation-of-state parameter w is close to -1. To say this is to imply that w could be slightly less than -1, which leads to R.Caldwell's "Phantom cosmologies". These often have the property that they end in a "Big Smash", a final singularity in which the Universe is destroyed in a finite proper time by excessive *expansion*. We show that, classically, this fate is not inevitable: there exist Smash-free Phantom cosmologies, obtained by a suitable perturbation of the deSitter equation of state, in which the spacetime is in fact asymptotically deSitter. [Contrary to popular belief, such cosmologies, which violate the Dominant Energy Condition, do not necessarily violate causality.] We also argue, however, that the physical interpretation of these classically acceptable spacetimes is radically altered by ``holography'', as manifested in the dS/CFT correspondence. It is shown that, if the boundary CFTs have conventional properties, then recent ideas on "time as an inverse renormalization group flow" can be used to rule out these cosmologies. Very recently, however, it has been argued that the CFTs in dS/CFT are of a radically unconventional form, and this opens up the possibility that Smash-free Phantom spacetimes offer a simple model of a "bouncing" cosmology in which the quantum-mechanical entanglement of the field theories in the infinite past and future plays an essential role. 
  Following Dahl's method an exact Runge-Lenz vector M with two components M and M is obtained as a constant of motion for a two particle-system with charges e and e whose electromagnetic interaction is based on Chern-Simons electrodynamics. The Poisson bracket {M, M} = L but is modified by the appearance of the product e e as central charges. 
  The thermodynamic free energy F(\beta) is calculated for a gas consisting of the transverse oscillations of a piecewise uniform bosonic string. The string consists of 2N parts of equal length, of alternating type I and type II material, and is relativistic in the sense that the velocity of sound everywhere equals the velocity of light. The present paper is a continuation of two earlier papers, one dealing with the Casimir energy of a 2N--piece string [I. Brevik and R. Sollie (1997)], and another dealing with the thermodynamic properties of a string divided into two (unequal) parts [I. Brevik, A. A. Bytsenko and H. B. Nielsen (1998)]. Making use of the Meinardus theorem we calculate the asymptotics of the level state density, and show that the critical temperatures in the individual parts are equal, for arbitrary spacetime dimension D. If D=26, we find \beta= (2/N)\sqrt{2\pi /T_{II}}, T_{II} being the tension in part II. Thermodynamic interactions of parts related to high genus g is also considered. 
  We construct a realization of the algebra of the Z_3-graded topological symmetry of type (1,1,1) in terms of a pair of operators D_1: H_1 -> H_2, and D_2: H_2 -> H_3 satisfying [D_1D_1^\dagger,D_2^\dagger D_2]=0. We show that the sequence of the restriction of these operators to the zero-energy subspace forms a complex and establish the equality of the corresponding topological invariants with the analytic indices of these operators. 
  We show that the system where $CP^1$ model coupled to Hopf term can reveal fractional spin in a collective coordinate quantization scheme, provided one makes a transition to physically inequivalent sector within a same solitonic sector characterized by a nonvanishing topological number 
  We study compactifications of string theory and M-theory to six dimensions with background fluxes. The nonzero fluxes lead to additional mass parameters. We derive the S- and T-duality rules for the corresponding (massive) supergravity theories. Specifically, we investigate the massive T-duality between Type IIA superstring theory compactified on K3 with background fluxes and Type IIB superstring theory compactified on K3. Furthermore, we generalise to the massive case the 6D 'string-string' S-duality between M-theory on K3 x S^1 and the Heterotic String on T^4. Whereas in the case of massive T--duality the mass parameters are in the fundamental representation of the U-duality group O(4,20) we find that in the case of massive S-duality they are in the 3-index antisymmetric representation. In the latter case the mass parameters involved extend those of Kaloper and Myers. We apply our duality rules to massive brane solutions, like the domain wall solutions corresponding to the mass parameters and find new massive brane solutions. Finally, we discuss the higher-dimensional interpretation of the dualities and brane solutions. 
  We attempt to deal with the orbifold singularities in the moduli space of flat connections for supersymmetric gauge theories on the torus. At these singularities the energy gap in the transverse fluctuations vanishes and the resulting breakdown of the adiabatic approximation is resolved by considering the full set of zero-momentum fields. These can not be defined globally, due to the problem of Gribov copies. For this reason we restrict the fields to the fundamental domain, containing no gauge copies, but requiring a boundary condition in field space. 
  Classical configurations of a M2-brane, a D2-brane and D0-branes are investigated in the background of an infinite array of M5-branes or NS5-branes. On the M2-brane, we discuss three kinds of configurations, such as a sphere, a cylinder and a torus-like one. These are stabilized by virtue of the background fluxes of M5-branes. The torus-like M2-brane configuration has winding and momentum numbers of 11th direction, and in terms of the type IIA superstring theory, this corresponds to a torus-like D2-brane with electric and magnetic fluxes on it. We also reproduce the same configuration from a non-abelian Born-Infeld action for D0-branes. It will be a construction of closed strings from D0-branes. An electric flux quantization condition on the D2-brane is also discussed in terms of D0-branes. 
  Constrained Hamiltonian systems fall into the realm of presymplectic geometry. We show, however, that also Poisson geometry is of use in this context.   For the case that the constraints form a closed algebra, there are two natural Poisson manifolds associated to the system, forming a symplectic dual pair with respect to the original, unconstrained phase space. We provide sufficient conditions so that the reduced phase space of the constrained system may be identified with a symplectic leaf in one of those. In the second class case the original constrained system may be reformulated equivalently as an abelian first class system in an extended phase space by these methods.   Inspired by the relation of the Dirac bracket of a general second class constrained system to the original unconstrained phase space, we address the question of whether a regular Poisson manifold permits a leafwise symplectic embedding into a symplectic manifold. Necessary and sufficient for this is the vanishing of the characteristic form-class of the Poisson tensor, a certain element of the third relative cohomology. 
  We study quantum mechanics of a massive superparticle in d=4 which preserves 1/4 of the target space supersymmetry with eight supercharges, and so corresponds to the partial breaking N=8 down to N=2. Its worldline action contains a Wess-Zumino term, explicitly breaks d=4 Lorentz symmetry and exhibits one complex fermionic kappa-symmetry. We perform the Hamiltonian analysis of the model and quantize it in two different ways, with gauge-fixed kappa-symmetry and in the Gupta-Bleuler formalism. Both approaches give rise to the same supermultiplet structure of the space of states. It contains three irreducible N=2 multiplets with the total number of (4+4) complex on-shell components. These states prove to be in one-to-one correspondence with the de Rham complex of p-forms on a three-dimensional subspace of the target x-manifold. We analyze the vacuum structure of the model and find that the non-trivial vacua are given by the exact harmonic one- and two-forms. Despite the explicit breaking of the d=4 Lorentz symmetry in the fermionic sector, the d=4 mass-shell condition is still valid in the model. 
  We study the role of context, complex of physical conditions, in quantum as well as classical experiments. It is shown that by taking into account contextual dependence of experimental probabilities we can derive the quantum rule for the addition of probabilities of alternatives. Thus we obtain quantum interference without applying to wave or Hilbert space approach. The Hilbert space representation of contextual probabilities is obtained as a consequence of the elementary geometric fact: $\cos$-theorem. By using another fact from elementary algebra we obtain complex-amplitude representation of probabilities. Finally, we found contextual origin of noncommutativity of incompatible observables. 
  The spectrum of supersymmetric domain wall solitons of the Wess-Zumino model is known to be discontinuous across a curve (of marginal stability) in the moduli space of quartic superpotentials. Here we show how this phenomenon can be understood from the behavior of the long-range inter-soliton force, which we compute by a method due to Manton. 
  Gluing conditions are proposed to characterize the D-branes in gauged WZW models. From them the boundary conditions for the group-valued and the subgroup-valued fields are determined. We construct a gauged WZW action for open strings that coincides classically with those written previously, when the gluing conditions are imposed. 
  We continue our study of codimension two solutions of warped space-time varying compactifications of string theory. In this letter we discuss a non-supersymmetric solution of the classical type IIB string theory with de Sitter gravity on a codimension two uncompactified part of spacetime. A non-zero positive value of the cosmological constant is induced by the presence of non-trivial stringy moduli, such as the axion-dilaton system for the type IIB string theory. Furthermore, the naked singularity of the codimension two solution is resolved by the presence of a small but non-zero cosmological constant. 
  An AdS_4 brane embedded in AdS_5 exhibits the novel feature that a four-dimensional graviton is localized near the brane, but the majority of the infinite bulk away from the brane where the warp factor diverges does not see four-dimensional gravity. A naive application of the holographic principle from the point of view of the four-dimensional observer would lead to a paradox; a global holographic mapping would require infinite entropy density. In this paper, we show that this paradox is resolved by the proper covariant formulation of the holographic principle. This is the first explicit example of a time-independent metric for which the spacelike formulation of the holographic principle is manifestly inadequate. Further confirmation of the correctness of this approach is that light-rays leaving the brane intersect at the location where we expect four-dimensional gravity to no longer dominate. We also present a simple method of locating CFT excitations dual to a particle in the bulk. We find that the holographic image on the brane moves off to infinity precisely when the particle exits the brane's holographic domain. Our analysis yields an improved understanding of the physics of the AdS_4/AdS_5 model. 
  We derive leading terms in the effective actions describing the coupling of bulk supergravity fields to systems of arbitrary numbers of Dp-branes and D(p+4)-branes in type IIA/IIB string theory. We use these actions to investigate the physics of Dp-D(p+4) systems in the presence of weak background fields. In particular, we construct various solutions describing collections of Dp-branes blown up into open D(p+2)-branes ending on D(p+4)-branes. The configurations are stabilized by the presence of background fields and represent an open-brane analogue of the Myers dielectric effect. To deduce the D-brane actions, we use supersymmetry to derive operators corresponding to moments of various conserved currents in the Berkooz-Douglas matrix model of M-theory in the presence of longitudinal M5-branes and then use dualities to relate these operators to the worldvolume operators appearing in the Dp-D(p+4)-brane effective actions. 
  Kutasov--type duals of supersymmetric gauge theories had been studied only in the dual regime and the s-confining case. Here we extend the discussion to the case of less flavor, analogous to the case of quantum-modified moduli space in Seiberg duality. Unlike the Seiberg duality, however, we find that parts of the moduli space become superconformal, generalizing the so far isolated example of SU(2) theory with two doublets and a triplet. We also point out that the magnetic superpotential needs to be augmented by an additional instanton-generated piece when the magnetic group is SU(2). 
  We discuss the gravity dual description for a non-commutative Yang-Mills theory, which reduces to that on AdS_{5} x S_{5} in the commutative limit. It is found that doubletons do not decouple in this dual gravity description unless one takes the commutative limit. The decoupling of the doubletons in AdS_{5} space implies that the dual gauge theory has SU(N) gauge symmetry. Our result implies that this gravity description is dual to non-commutative U(N) gauge theory. It is compatible with the claim that U(1) and SU(N) gauge symmetries can not separate in non-commutative U(N) gauge theory. 
  Recently Maldacena, Moore, and Seiberg (MMS) have proposed a physical interpretation of the Atiyah-Hirzebruch spectral sequence, which roughly computes the K-homology groups that classify D-branes. We note that in IIB string theory, this approach can be generalized to include NS charged objects and conjecture an S-duality covariant, nonlinear extension of the spectral sequence. We then compute the contribution of the MMS double-instanton configuration to the derivation d_5. We conclude with an M-theoretic generalization reminiscent of 11-dimensional E_8 gauge theory. 
  We study global defects coupled to higher-dimensional gravity with a negative cosmological constant. This paper is mainly devoted to studying global black brane solutions which are extended global defects surrounded by horizons. We find series solutions in a few separated regions and confirm numerically that they can be mutually connected. When the world volume of the brane is Ricci-flat, the brane is surrounded by a degenerated horizon, while it is surrounded by two horizons when the world volume has a positive constant curvature. Each solution corresponds to an extremal and a non-extremal state, respectively. Their causal structures resemble those of the Reissner-Nordstr\"{o}m black holes in anti-de Sitter spacetime. However, the non-extremal black brane is not a static object, but an inflating brane. In addition, we briefly discuss a brane world model in the context of the global black branes. We comment on a few thermodynamic properties of the global black branes, and discuss a decrease of the cosmological constant on the brane world through the thermodynamic instability of the non-extremal global black brane. 
  We perform the calculation of the partition function of the Poisson-sigma model on the world sheet with the topology of a two-dimensional disc. Considering the special case of a linear Poisson structure we recover the partition function of the Yang-Mills theory. Using a glueing procedure we are able to calculate the partition function for arbitrary base manifolds. 
  Within a self-consistent proper-time Renormalization Group (RG) approach we investigate an effective QCD trace anomaly realization with dilatons and determine the finite-temperature behavior of the gluon condensate. Fixing the effective model at vanishing temperature to the glueball mass and the bag constant a possible gluonic phase transition is explored in detail. Within the RG framework the full non-truncated dilaton potential analysis is compared with a truncated potential version. 
  We investigate the possibility that stringy nonperturbative instabilities are described by worldsheet methods. We focus on the case of open bosonic string theory, where the D-instanton plays a role of the bounce, i.e. it describes barrier penetration. In the process, we compute the exponential factor in a decay probability. 
  We summarize recent evidence supporting the conjecture that four-dimensional Quantum Einstein Gravity (QEG) is nonperturbatively renormalizable along the lines of Weinberg's asymptotic safety scenario. This would mean that QEG is mathematically consistent and predictive even at arbitrarily small length scales below the Planck length. For a truncated version of the exact flow equation of the effective average action we establish the existence of a non-Gaussian renormalization group fixed point which is suitable for the construction of a nonperturbative infinite cutoff-limit. The cosmological implications of this fixed point are discussed, and it is argued that QEG might solve the horizon and flatness problem of standard cosmology without an inflationary period. 
  We propose a modification of special relativity in which a physical energy, which may be the Planck energy, joins the speed of light as an invariant, in spite of a complete relativity of inertial frames and agreement with Einstein's theory at low energies. This is accomplished by a non-linear modification of the action of the Lorentz group on momentum space, generated by adding a dilatation to each boost in such a way that the Planck energy remains invariant. The associated algebra has unmodified structure constants, and we highlight the similarities between the group action found and a transformation previously proposed by Fock. We also discuss the resulting modifications of field theory and suggest a modification of the equivalence principle which determines how the new theory is embedded in general relativity. 
  We look at the supersymmetric generalization of harmonic maps into Lie groups, known to physicists as the chiral model. Explicit solutions to the equations are found and examined using Backlund transformations. 
  While there has been growing interest for noncommutative spaces in recent times, most examples have been based on the simplest noncommutative algebra: [x_i,x_j]=i theta_{ij}. Here we present new classes of (non-formal) deformed products associated to linear Lie algebras of the kind [x_i,x_j]=ic_{ij}^k x_k. For all possible three-dimensional cases, we define a new star product and discuss its properties. To complete the analysis of these novel noncommutative spaces, we introduce noncompact spectral triples, and the concept of star triple, a specialization of the spectral triple to deformations of the algebra of functions on a noncompact manifold. We examine the generalization to the noncompact case of Connes' conditions for noncommutative spin geometries, and, in the framework of the new star products, we exhibit some candidates for a Dirac operator. On the technical level, properties of the Moyal multiplier algebra M(R_\theta^{2n) are elucidated. 
  We derive the Cardy--Verlinde entropy formula for the field theory that lives on the boundary of an asymptotically de Sitter space with a black hole. The boundary theory which is not conformal has a monotonic $C$--function defined by the Casimir energy. The instability of the space due to Hawking radiation from the black hole corresponds to an RG flow from the IR to the UV during which $C$ increases. The endpoint of black hole evaporation is de Sitter space which is described by a conformal theory at the UV fixed point of the RG flow. 
  We investigate the W-algebras generated by the integer dimension chiral primary operators of the SU(2)_0 WZNW model. These have a form almost identical to that found in the c=-2 model but have, in addition, an extended Kac-Moody structure. Moreover on Hamiltonian reduction these SU(2)_0 W-algebras exactly reduce to those found in c=-2. We explicitly find the free field representations for the chiral j=2 and j=3 operators which have respectively a fermionic doublet and bosonic triplet nature. The correlation functions of these operators accounts for the rational solutions of the Knizhnik-Zamolodchikov equation that we find. We explicitly compute the full algebra of the j=2 operators and find that the associativity of the algebra is only guaranteed if certain null vectors decouple from the theory. We conjecture that these algebras may produce a quasi-rational conformal field theory. 
  We conjecture the factorized scattering description for OSP(m/2n)/OSP(m-1/2n) supersphere sigma models and OSP(m/2n) Gross Neveu models. The non-unitarity of these field theories translates into a lack of `physical unitarity' of the S matrices, which are instead unitary with respect to the non-positive scalar product inherited from the orthosymplectic structure. Nevertheless, we find that formal thermodynamic Bethe ansatz calculations appear meaningful, reproduce the correct central charges, and agree with perturbative calculations. This paves the way to a more thorough study of these and other models with supergroup symmetries using the S matrix approach. 
  The Hamiltonian theory of isomonodromy equations for meromorphic connections with irregular singularities on algebraic curves is constructed. An explicit formula for the symplectic structure on the space of monodromy and Stokes matrices is obtained. The Whitham equations for the isomonodromy equations are derived. It is shown that they provide a flat connection on the space of the spectral curves of the Hitchin systems. 
  We compare two different methods of computing form factors. One is the well established procedure of solving the form factor consistency equations and the other is to represent the field content as well as the particle creation operators in terms of fermionic Fock operators. We compute the corresponding matrix elements for the complex free fermion and the Federbush model. The matrix elements only satisfy the form factor consistency equations involving anyonic factors of local commutativity when the corresponding operators are local. We carry out the ultraviolet limit, analyze the momentum space cluster properties and demonstrate how the Federbush model can be obtained from the $SU(3)_3$-homogeneous sine-Gordon model. We propose a new class of Lagrangians which constitute a generalization of the Federbush model in a Lie algebraic fashion. For these models we evaluate the associated scattering matrices from first principles, which can alternatively also be obtained in a certain limit of the homogeneous sine-Gordon models. 
  Seven-manifolds of G_2 holonomy provide a bridge between M-theory and string theory, via Kaluza-Klein reduction to Calabi-Yau six-manifolds. We find first-order equations for a new family of G_2 metrics D_7, with S^3\times S^3 principal orbits. These are related at weak string coupling to the resolved conifold, paralleling earlier examples B_7 that are related to the deformed conifold, allowing a deeper study of topology change and mirror symmetry in M-theory. The D_7 metrics' non-trivial parameter characterises the squashing of an S^3 bolt, which limits to S^2 at weak coupling. In general the D_7 metrics are asymptotically locally conical, with a nowhere-singular circle action. 
  We provide examples of gravitational collapse and black hole formation in AdS, either from collapsing matter shells or in analogy to the Oppenheimer-Sneider solution. We then investigate boundary properties of the corresponding states. In particular, we describe the boundary two-point function corresponding to a shell outside its horizon; if the shell is quasistatically lowered into the horizon, the resulting state is the Boulware state. We also describe the more physical Hartle-Hawking state, and discuss its connection to the quasistatic shell states and to thermalization on the boundary. 
  We construct the minimal bosonic higher spin extension of the 7D AdS algebra SO(6,2), which we call hs(8*). The generators, which have spin s=1,3,5,..., are realized as monomials in Grassmann even spinor oscillators. Irreducibility, in the form of tracelessness, is achieved by modding out an infinite dimensional ideal containing the traces. In this a key role is played by the tree bilinear traces which form an SU(2)_K algebra. We show that gauging of hs(8*) yields a spectrum of physical fields with spin s=0,2,4,...which make up a UIR of hs(8*) isomorphic to the symmetric tensor product of two 6D scalar doubletons. The scalar doubleton is the unique SU(2)_K invariant 6D doubleton. The spin s\geq 2 sector comes from an hs(8*)-valued one-form which also contains the auxiliary gauge fields required for writing the curvature constraints in covariant form. The physical spin s=0 field arises in a separate zero-form in a `quasi-adjoint' representation of hs(8*). This zero-form also contains the spin s\geq 2 Weyl tensors, i.e. the curvatures which are non-vanishing on-shell. We suggest that the hs(8*) gauge theory describes the minimal bosonic, massless truncation of M theory on AdS_7\times S^4 in an unbroken phase where the holographic dual is given by N free (2,0) tensor multiplets for large N. 
  We discuss the condensation of charged tachyons in the heterotic theory on the Kaluza-Klein Melvin background. The arguments are based on duality relations which are expected to hold from the adiabatic argument. It is argued that in many cases the rank of the gauge group is not changed by the condensation, as opposed to naive expectations. 
  After briefly reviewing the hamiltonian approach to 2+1 dimensional gravity in absence of matter on closed universes, we consider 2+1 dimensional gravity coupled to point particles in an open universe. We show that the hamiltonian structure of the theory is the result of a conjecture put forward by Polyakov in a different context. A proof is given of such a conjecture. Finally we give the exact quantization of the two particle problem in open space. 
  The way of finding all the constraints in the Hamiltonian formulation of singular (in particular, gauge) theories is called the Dirac procedure. The constraints are naturally classified according to the correspondig stages of this procedure. On the other hand, it is convenient to reorganize the constraints such that they are explicitly decomposed into the first-class and second-class constraints. We discuss the reorganization of the constraints into the first- and second-class constraints that is consistent with the Dirac procedure, i.e., that does not violate the decomposition of the constraints according to the stages of the Dirac procedure. The possibility of such a reorganization is important for the study of gauge symmetries in the Lagrangian and Hamiltonian formulations. 
  We continue the development of the topological membrane approach to open and unoriented string theories. We study orbifolds of topologically massive gauge theory defined on the geometry $[0,1]\times\Sigma$, where $\Sigma$ is a generic compact Riemann surface. The orbifold operations are constructed by gauging the discrete symmetries of the bulk three-dimensional field theory. Multi-loop bosonic string vacuum amplitudes are thereby computed as bulk correlation functions of the gauge theory. It is shown that the three-dimensional correlators naturally reproduce twisted and untwisted sectors in the case of closed worldsheet orbifolds, and Neumann and Dirichlet boundary conditions in the case of open ones. The bulk wavefunctions are used to explicitly construct the characters of the underlying extended Kac-Moody group for arbitrary genus. The correlators for both the original theory and its orbifolds give the expected modular invariant statistical sums over the characters. 
  Within a variational calculation we investigate the role of baryons for the structure of dense matter in the Gross-Neveu model. We construct a trial ground state at finite baryon density which breaks translational invariance. Its scalar potential interpolates between widely spaced kinks and antikinks at low density and the value zero at infinite density. Its energy is lower than the one of the standard Fermi gas at all densities considered. This suggests that the discrete gamma_5 symmetry of the Gross-Neveu model does not get restored in a first order phase transition at finite density, at variance with common wisdom. 
  An exact solution, in which a D2-brane and an anti-D2-brane are connected by an elliptically tubular D2-brane, is obtained without any junction condition. The solution is shown to preserve one quarter of the supersymmetries of the type-IIA Minkowski vacuum. We show that the configuration cannot be obtained by "blowing-up" from some inhomogeneously D0-charged superstrings. The BPS bound tells us that it is rather composed of D0-charged D2-brane-anti-D2-brane pair and a strip of superstrings connecting them. We obtain the correction to the charges of the string end points in the constant magnetic background. 
  I study the enhancon mechanism for fractional D-branes in conifold and orbifold backgrounds and show how it can resolve the ``repulson'' singularity of these geometries. In particular I show that the consistency of the excision process requires that the interior space be not empty. 
  The Complex Angular Momentum (CAM) representation of (scalar) four-point functions has been previously established starting from the general principles of local relativistic Quantum Field Theory (QFT). Here, we carry out the diagonalization of the general $t$-channel Bethe--Salpeter (BS) structure of four point functions in the corresponding CAM variable $\lambda_{t},$ for all negative values of the squared-energy variable $t$. This diagonalization is closely related to the existence of BS-equations for the absorptive parts in the crossed channels, interpreted as convolution equations with spectral properties. The production of Regge poles equipped with factorized residues involving Euclidean three-point functions appears as conceptually built-in in the analytic axiomatic framework of QFT. The existence of leading Reggeon terms governing the asymptotic behaviour of the four-point function at fixed $t$ is strictly conditioned by the asymptotic behaviour of a global Bethe-Salpeter kernel of the theory. 
  We show how the elliptic Calogero-Moser integrable systems arise from a symplectic quotient construction, generalising the construction for A_{N-1} by Gorsky and Nekrasov to other algebras. This clarifies the role of (twisted) affine Kac-Moody algebras in elliptic Calogero-Moser systems and allows for a natural geometric construction of Lax operators for these systems. We elaborate on the connection of the associated Hamiltonians to superpotentials for N=1* deformations of N=4 supersymmetric gauge theory, and argue how non-perturbative physics generates the elliptic superpotentials. We also discuss the relevance of these systems and the associated quotient construction to open problems in string theory. In an appendix, we use the theory of orbit algebras to show the systematics behind the folding procedures for these integrable models. 
  We discuss the radiation reaction problem for an electric charge moving in flat space-time of arbitrary dimensions. It is shown that four is the unique dimension where a local differential equation exists accounting for the radiation reaction and admitting a consistent mass-renormalization (the Dirac-Lorentz equation). In odd dimensions the Huygens principle does not hold; as a result, the radiation reaction force depends on the whole past history of a charge (radiative tail). We show that the divergence in the tail integral can be removed by the mass renormalization only in the 2+1 theory. In even dimensions higher than four, divergences can not be removed by a renormalization. 
  We extend the previous series of articles [HPA] devoted to finding mappings between the Weinberg-Tucker-Hammer formalism and antisymmetric tensor fields. Now we take into account solutions of different parities of the Weinberg-like equations. Thus, the Proca, Duffin-Kemmer and Bargmann-Wigner formalisms are generalized. 
  In this paper we use the anholonomic frames method to construct exact solutions for vacuum 5D gravity with metrics having off-diagonal components. The solutions are in general anisotropic and possess interesting features such as an anisotropic warp factor with respect to the extra dimension, or a gravitational scaling/running of some of the physical parameters associated with the solutions. A certain class of solutions are found to describe Schwarzschild black holes which ``solitonically'' propagate in spacetime. The solitonic character of these black hole solutions arises from the embedding of a 3D soliton configuration (e.g. the soliton solutions to the Kadomtsev-Petviashvily or sine-Gordon equations) into certain ansatz functions of the 5D metric. These solitonic solutions may either violate or preserve local Lorentz invariance. In addition there is a connection between these solutions and noncommutative field theory. 
  We construct several new G(2) holonomy metrics that play an important role in recent studies of geometrical transitions in compactifications of M-theory to four dimensions. In type IIA string theory these metrics correspond to D6 branes wrapped on the three-cycle of the deformed conifold and the resolved conifold with two-form RR flux on the blown-up two-sphere, which are related by a conifold transition. We also study a G(2) metric that is related in type IIA to the line bundle over S^2 x S^2 with RR two-form flux. Our approach exploits systematically the definition of torsion-free G(2) structures in terms of three-forms which are closed and co-closed. Besides being an elegant formalism this turns out to be a practical tool to construct G(2) holonomy metrics. 
  Noncommutative three-dimensional gravity can be described in terms of a noncommutative Chern-Simons theory. We extend this structure and also propose an action for gravitational fields on an even dimensional noncommutative space. The action is worked out in some detail for fields on a noncommutative ${\bf CP}^2$ and on $S^4$. 
  Four dimensional gravity in the low energy limit of a higher dimensional theory has been expected to be a (generalized) Brans-Dicke theory. A subtle point in brane world scenarios is that the system of four dimensional effective gravitational equations is not closed due to bulk gravitational waves and bulk scalars. Nonetheless, weak gravity on the brane can be analyzed completely. We revisit the theory of weak brane gravity using gauge-invariant gravitational and scalar perturbations around a background warped geometry with a bulk scalar between two flat branes. We obtain a simple condition for the radion stabilization in terms of the scalar field potentials. We show that for general potentials of the scalar field which provides radion stabilization and a general conformal transformation to a frame in which matter on the branes are minimally coupled to the metric, 4-dimensional Einstein gravity, not BD gravity, is restored at low energies on either brane. In contrast, in RS brane world scenario without a bulk scalar, low energy gravity is BD one. We conjecture that in general brane world scenarios with more than one scalar field, one will again encounter the situation that low energy gravity is not described by the Einstein theory. Equipped with the weak gravity results, we discuss the properties of 4d brane gravitational equations, in particular, the value and sign of 4d Newton's gravitational coupling. 
  The string theory on symmetric product describes the second-quantized string theory. The development for the bosonic open string was discussed in the previous work. In this paper, we consider the open superstring theory on the symmetric product and examine the nature of the second quantization. The fermionic partition functions are obtained from the consistent fermionic extension of the twisted boundary conditions for the non-abelian orbifold, and they can be interpreted in terms of the long string language naturally. In the closed string sector, the boundary/cross-cap states are also constructed. These boundary states are classified into three types in terms of the long string language, and explain the change of the topology of the world-sheet. To obtain the anomaly-free theory, the dilaton tadpole must be cancelled. This condition gives SO(32) Chan-Paton group as ordinary superstring theory. 
  There is remarkable relation between self-dual Yang-Mills and self-dual Einstein gravity in four Euclidean dimensions. Motivated by this we investigate the Spin(7) and G_2 invariant self-dual Yang-Mills equations in eight and seven Euclidean dimensions and search for their possible analogs in gravitational theories. The reduction of the self-dual Yang-Mills equations to one dimension results into systems of first order differential equations. In particular, the Spin(7)-invariant case gives rise to a 7-dimensional system which is completely integrable. The different solutions are classified in terms of algebraic curves and are characterized by the genus of the associated Riemann surfaces. Remarkably, this system arises also in the construction of solutions in gauged supergravities that have an interpretation as continuous distributions of branes in string and M-theory. For the G_2 invariant case we perform two distinct reductions, both giving rise to 6-dimensional systems. The first reduction, which is a complex generalization of the 3-dimensional Euler spinning top system, preserves an SU(2) X SU(2) X Z_2 symmetry and is fully integrable in the particular case where an extra U(1) symmetry exists. The second reduction we employ, generalizes the Halphen system familiar from the dynamics of monopoles. Finally, we analyze massive generalizations and present solitonic solutions interpolating between different degenerate vacua. 
  We discuss the reduction of N=2 supergravity to N=1, by a consistent truncation of the second gravitino multiplet. 
  We systematically develop the procedure of holographic renormalization for RG flows dual to asymptotically AdS domain walls. All divergences of the on-shell bulk action can be cancelled by adding covariant local boundary counterterms determined by the near-boundary behavior of bulk fields. This procedure defines a renormalized action from which correlation functions are obtained by functional differentiation. The correlators are finite and well behaved at coincident points. Ward identities, corrected for anomalies, are satisfied. The correlators depend on parts of the solution of the bulk field equations which are not determined by near-boundary analysis. In principle a full nonlinear solution is required, but one can solve linearized fluctuation equations to define a bulk-to-boundary propagator from which 2-point correlation functions are easily obtained. We carry out the procedure explicitly for two known RG flows obtained from the maximal gauged D=5 supergravity theory, obtaining new results on correlators of vector currents and related scalar operators and giving further details on a recent analysis of the stress tensor sector. 
  We present the N=2 supersymmetric formulation for the classical and quantum dynamics of a nonrelativistic charged particle on a curved surface in the presence of a perpendicular magnetic field. For a particle moving on a constant-curvature surface in a constant magnetic field, our Hamiltonian possesses the shape-invariance property in addition. On the surface of a sphere and also on the hyperbolic plane, we exploit the supersymmetry and shape-invariance properties to obtain complete solutions to the corresponding quantum mechanical problems. 
  We consider models involving the higher (third) derivative extension of the abelian Chern-Simons (CS) topological term in D=2+1 dimensions. The polarisation vectors in these models reveal an identical structure with the corresponding expressions for usual models which contain, at most, quadratic structures. We also investigate the Hamiltonian structure of these models and show how Wigner's little group acts as gauge generator. 
  In this paper, we calculate the spectrum of scalar field fluctuations in a bouncing, asymptotically flat Universe, and investigate the dependence of the result on changes in the physics on length scales shorter than the Planck length which are introduced via modifications of the dispersion relation. In this model, there are no ambiguities concerning the choice of the initial vacuum state. We study an example in which the final spectrum of fluctuations depends sensitively on the modifications of the dispersion relation without needing to invoke complex frequencies. Changes in the amplitude and in the spectral index are possible, in addition to modulations of the spectrum. This strengthens the conclusions of previous work in which the spectrum of cosmological perturbations in expanding inflationary cosmologies was studied, and it was found that, for dispersion relations for which the evolution is not adiabatic, the spectrum changes from the standard prediction of scale-invariance. 
  We argue that a spontaneous breakdown of local Weyl invariance offers a mechanism in which gravitational interactions contribute to the generation of particle masses and their electric charge. The theory is formulated in terms of a spacetime geometry whose natural connection has both dynamic torsion and non-metricity. Its structure illuminates the role of dynamic scales used to determine measurable aspects of particle interactions and it predicts an additional neutral vector boson with electroweak properties. 
  This paper has been withdrawn by the authors, due a crucial error in Sec. 3. 
  After a short discussion of the intimate relation between the generalized statistics and supersymmetry, we review the recent results on the nonlinear supersymmetry obtained in the context of the quantum anomaly problem and of the universal algebraic construction associated with the holomorphic nonlinear supersymmetry. 
  We compare two applications of the gauge/gravity correspondence to a non conformal gauge theory, based respectively on the study of D-branes wrapped on supersymmetric cycles and of fractional D-branes on orbifolds. We study two brane systems whose geometry is dual to N=4, D=2+1 super Yang-Mills theory, the first one describing D4-branes wrapped on a two-sphere inside a Calabi-Yau two-fold and the second one corresponding to a system of fractional D2/D6-branes on the orbifold R^4/Z_2. By probing both geometries we recover the exact perturbative running coupling constant and metric on the moduli space of the gauge theory. We also find a general expression for the running coupling constant of the gauge theory in terms of the "stringy volume" of the two-cycle which is involved in both types of brane systems. 
  We study the behavior of the photon two point function, in non-commutative QED, in a general covariant gauge and in arbitrary space-time dimensions. We show, to all orders, that the photon self-energy is transverse. Using an appropriate extension of the dimensional regularization method, we evaluate the one-loop corrections, which show that the theory is renormalizable. We also prove, to all orders, that the poles of the photon propagator are gauge independent and briefly discuss some other related aspects. 
  By considering a simplified but exact model for realizing the ekpyrotic scenario, we clarify various assumptions that have been used in the literature. In particular, we discuss the new ekpyrotic prescription for passing the perturbations through the singularity which we show to provide a spectrum depending on a non physical normalization function. We also show that this prescription does not reproduce the exact result for a sharp transition. Then, more generally, we demonstrate that, in the only case where a bounce can be obtained in Einstein General Relativity without facing singularities and/or violation of the standard energy conditions, the bounce cannot be made arbitrarily short. This contrasts with the standard (inflationary) situation where the transition between two eras with different values of the equation of state can be considered as instantaneous. We then argue that the usually conserved quantities are not constant on a typical bounce time scale. Finally, we also examine the case of a test scalar field (or gravitational waves) where similar results are obtained. We conclude that the full dynamical equations of the underlying theory should be solved in a non singular case before any conclusion can be drawn. 
  We consider thorny spheres, that is 2-dimensional compact surfaces which are everywhere locally isometric to a round sphere $S^2$ except for a finite number of isolated points where they have conical singularities. We use thorny spheres to generate, from a spherically symmetric solution of the Einstein equations, new solutions which describe spacetimes pierced by an arbitrary number of infinitely thin cosmic strings radially directed. Each string produces an angle deficit proportional to its tension, while the metric outside the strings is a locally spherically symmetric solution. We prove that there can be arbitrary configurations of strings provided that the directions of the strings obey a certain equilibrium condition. In general this equilibrium condition can be written as a force-balance equation for string forces defined in a flat 3-space in which the thorny sphere is isometrically embedded, or as a constraint on the product of holonomies around strings in an alternative 3-space that is flat except for the strings. In the case of small string tensions, the constraint equation has the form of a linear relation between unit vectors directed along the string axes. 
  This is a brief review of some recent results on the geometric approach to symmetric D-branes in group manifolds, both twisted and untwisted. We describe the geometry of the gluing conditions and the quantisation condition in the boundary WZW model, and we illustrate this by determining the consistent twisted and untwisted D-branes in the Lie group SU_3. 
  It is well known that a weakly coupled U(N) gauge theory on a torus with sides of length L has extra light states with energies of order 1/NL. We show that a similar result holds for gauge theories on M/G where M is any compact Riemannian manifold and G is any freely acting discrete isometry group. As in the toroidal case, this is achieved by adding a suitable nontrivial flat connection. As one application, we consider the AdS/CFT correspondence on spacetimes asymptotic to AdS_5/G. By considering finite size effects at nonzero temperature, we show that consistency requires these extra light states of the gauge theory on S^3/G. 
  We present some results of studying certain axially symmetric supergravity geometries corresponding to a distribution of BPS D6-branes wrapped on K3, obtained as extremal limits of a rotating solution. The geometry's unphysical regions resulting from the wrapping can be repaired by the enhancon mechanism, with the result that there are two nested enhancon shells. For a range of parameters, the two shells merge into a single toroidal surface. Given the quite intricate nature of the geometry, it is an interesting system in which to test previous techniques that have been brought to bear in spherically symmetric situations. We are able to check the consistency of the construction using supergravity surgery techniques, and probe brane results. Implications for the Coulomb branch of (2+1)-dimensional pure SU(N) gauge theory are extracted from the geometry. Related results for wrapped D4- and D5-brane distributions are also discussed. 
  We generalize the basic enhancon solution of Johnson, Peet and Polchinski by constructing solutions without spherical symmetry. A careful consideration of boundary conditions at the enhancon surface indicates that the interior of the supergravity solution is still flat space in the general case. We provide some explicit analytic solutions where the enhancon locus is a prolate or oblate sphere. 
  Quantum mechanics in the presence of $\delta$-function potentials is known to be plagued by UV divergencies which result from the singular nature of the potentials in question. The standard method for dealing with these divergencies is by constructing self-adjoint extensions of the corresponding Hamiltonians. Two particularly interesting examples of this kind are nonrelativistic spin zero particles in $\delta$-function potential and Dirac particles in Aharonov-Bohm magnetic background. In this paper we show that by extending the corresponding Schr\"odinger and Dirac equations onto the flat noncommutative space a well-defined quantum theory can be obtained. Using a star product and Fock space formalisms we construct the complete sets of eigenfunctions and eigenvalues in both cases which turn out to be finite. 
  We propose a pregeometrical formulation of Berkovits' open Ramond-Neveu-Schwarz (RNS) superstring field theories. We show that Berkovits' open RNS superstring field theories arise by expanding around particular solutions of the classical equations of motion for this theory. Our action contains pure ghost operators only and so is formally background independent. 
  We discuss a class of 4-dimensional non-homogeneous quaternionic spaces, which become the two known homogeneous spaces (EAdS_4$ and SU(2,1)/U(2)) in certain limits. These moduli spaces have two regions where the metric is positive definite, separated by a non-physical region where the metric has timelike directions and which contains a curvature singularity. They admit four isometries and we consider their general Abelian gauging. The critical points of the resulting superpotential and hence the possible domain wall solutions differ significantly in the two regions. On one side one can construct only singular walls, whereas in the other we found a smooth domain wall interpolating between two infra-red critical points located exactly on the boundary of the physical allowed parameter region. 
  The fact that both the D6-brane and the orientifold 6-plane have smooth, horizon-free descriptions in M-theory makes them especially useful in understanding certain aspects of brane physics. We briefly review how this connection has been used to understand a number of effects, several of which are associated with the Hanany-Witten transition. One particular outcome is a "confinement mod 2" effect for zero-branes in the background of a single D8-brane. We also discuss an interesting puzzle associated with flux-expulsion from D6-branes in this context. Finally, we discuss the promise of using a similar M-theoretic description of the orientifold 6-plane to understand the consistency of stringy negative energy objects with the 2nd law of black hole thermodynamics. 
  We find general first-order equations for G_2 metrics of cohomogeneity one with S^3\times S^3 principal orbits. These reduce in two special cases to previously-known systems of first-order equations that describe regular asymptotically locally conical (ALC) metrics \bB_7 and \bD_7, which have weak-coupling limits that are S^1 times the deformed conifold and the resolved conifold respectively. Our more general first-order equations provide a supersymmetric unification of the two Calabi-Yau manifolds, since the metrics \bB_7 and \bD_7 arise as solutions of the {\it same} system of first-order equations, with different values of certain integration constants. Additionally, we find a new class of ALC G_2 solutions to these first-order equations, which we denote by \wtd\bC_7, whose topology is an \R^2 bundle over T^{1,1}. There are two non-trivial parameters characterising the homogeneous squashing of the T^{1,1} bolt. Like the previous examples of the \bB_7 and \bD_7 ALC metrics, here too there is a U(1) isometry for which the circle has everywhere finite and non-zero length. The weak-coupling limit of the \wtd\bC_7 metrics gives S^1 times a family of Calabi-Yau metrics on a complex line bundle over S^2\times S^2, with an adjustable parameter characterising the relative sizes of the two S^2 factors. 
  We propose a way to introduce matter fields transforming in arbitrary representations of the gauge group in noncommutative U(N) gauge theories. We then argue that in the presence of supersymmetry, an ordinary commutative SU(N) gauge theory with a general matter content can always be embedded into a noncommutative U(N) theory at energies above the noncommutativity mass scale M_{NC} ~ \theta^{-1/2}. At energies below M_{NC}, the U(1) degrees of freedom decouple due to the IR/UV mixing, and the noncommutative theory reduces to its commutative counterpart. Supersymmetry can be spontaneously broken by a Fayet-Iliopoulos D-term introduced in the noncommutative U(N) theory. U(1) degrees of freedom become arbitrarily weakly coupled in the infrared and naturally play the role of the hidden sector for supersymmetry breaking. To illustrate these ideas we construct a noncommutative U(5) GUT model with Fayet-Iliopoulos supersymmetry breaking, which reduces to a realistic commutative theory in the infrared. 
  We explore the dynamic dS/CFT correspondence using the moving domain wall(brane) approach in the brane cosmology. The bulk spacetimes are given by the Schwarzschild-de Sitter (SdS) black hole and the topological-de Sitter (TdS) solutions. We consider the embeddings of (Euclidean) moving domain walls into the (Euclidean) de Sitter spaces. The TdS solution is better to describe the static dS/CFT correspondence than the SdS black hole, while in the dynamic dS/CFT correspondence the SdS solution provides situation better than that of the TdS solution. However, we do not find a desirable cosmological scenario from the SdS black hole space. 
  We construct integrable generalizations of the elliptic Calogero-Sutherland-Moser model of particles with spin, involving noncommutative spin interactions. The spin coupling potential is a modular function and, generically, breaks the global spin symmetry of the model down to a product of U(1) phase symmetries. Previously known models are recovered as special cases. 
  In the present paper we demonstrate the possibility of the [SUSY(5)]^3 SUSY unification at the energy scale $\mu_{GUT}\approx 10^{18.3} GeV$ with the value of GUT inverse finestructure constant $\alpha_{GUT}^{-1}\approx 34.4$, which is very close to its critical value $\alpha_{5, crit}^{-1}\approx 34.0$, existing at the Planck scale. 
  The unitarity equations for the boson interaction amplitudes in the superstring theory are used to calculate the interaction amplitudes including the Ramond states, which are 10-fermion and Ramond bosons. The n-loop, 4-point amplitude with two massless Neveu-Schwarz bosons and two massless Ramond states is obtained explicitly. It is shown that, in addition, the unitarity equations require some integral relations for local functions determining the amplitude. For the tree amplitude the validness of the above integral relations is verified. 
  In this paper we extend the idea of integration to generic algebras. In particular we concentrate over a class of algebras, that we will call self-conjugated, having the property of possessing equivalent right and left multiplication algebras. In this case it is always possible to define an integral sharing many of the properties of the usual integral. For instance, if the algebra has a continuous group of automorphisms, the corresponding derivations are such that the usual formula of integration by parts holds. We discuss also how to integrate over subalgebras. Many examples are discussed, starting with Grassmann algebras, where we recover the usual Berezin's rule. The paraGrassmann algebras are also considered, as well as the algebra of matrices. Since Grassmann and paraGrassmann algebras can be represented by matrices we show also that their integrals can be seen in terms of traces over the corresponding matrices. An interesting application is to the case of group algebras where we show that our definition of integral is equivalent to a sum over the unitary irreducuble representations of the group. We show also some example of integration over non self-conjugated algebras (the bosonic and the $q$-bosonic oscillators), and over non-associative algebras (the octonions). 
  In the first part of this article, we find fractional D3-branes supergravity solutions on orientifolded C^2/Z_2 orbifolds of type IIB string theory. The one-loop corrected gauge couplings for the symplectic or orthogonal groups living on the D-branes are reproduced on the world-volume of probes. In the second part of the paper, we construct a D3-brane solution on the two-centers Taub-NUT manifold which interpolates between fractional D3-branes in the ALE space limit and a T-dual smeared type IIA configuration. Then, we lift this configuration to M-theory and comment on the connections with wrapped M5-branes solutions. 
  Connes and Kreimer have discovered a Hopf algebra structure behind renormalization of Feynman integrals. We generalize the Hopf algebra to the case of ribbon graphs, i.e. to the case of theories with matrix fields. The Hopf algebra is naturally defined in terms of surfaces corresponding to ribbon graphs. As an example, we discuss renormalization of $\Phi^4$ theory and the 1/N expansion. 
  We describe a simple mechanism that can lead to inflation within string-based brane-world scenarios. The idea is to start from a supersymmetric configuration with two parallel static Dp-branes, and slightly break the supersymmetry conditions to produce a very flat potential for the field that parametrises the distance between the branes, i.e. the inflaton field. This breaking can be achieved in various ways: by slight relative rotations of the branes with small angles, by considering small relative velocities between the branes, etc. If the breaking parameter is sufficiently small, a large number of e-folds can be produced within the D-brane, for small changes of the configuration in the compactified directions. Such a process is local, i.e. it does not depend very strongly on the compactification space nor on the initial conditions. Moreover, the breaking induces a very small velocity and acceleration, which ensures very small slow-roll parameters and thus an almost scale invariant spectrum of metric fluctuations, responsible for the observed temperature anisotropies in the microwave background. Inflation ends as in hybrid inflation, triggered by the negative curvature of the string tachyon potential. In this paper we elaborate on one of the simplest examples: two almost parallel D4-branes in a flat compactified space. 
  I show how to compute the exact one-loop thermal correction to the free energy of a soliton. The method uses the effective potential as an auxiliary step to ensure that the soliton is quantized around the appropriate vacuum. The exact result is then computed using scattering theory techniques, and includes all orders in the derivative expansion. It can be efficiently combined with a calculation of the exact quantum correction to yield the full free energy to one loop. I demonstrate this technique with explicit computations in $\phi^4$ models, obtaining the free energy for a kink in 1+1 dimensions and a domain wall in 2+1 dimensions. 
  One shows here the existence of solutions to the Callan-Symanzik equation for the non-Abelian SU(2) Chern-Simons-matter model which exhibits asymptotic conformal invariance to every order in perturbative theory. The conformal symmetry in the classical domain is shown to hold by means of a local criteria based on the trace of the energy-momentum tensor. By using the recently exhibited regimes for the dependence between the several couplings in which the set of $\beta$-functions vanish, the asymptotic conformal invariance of the model appears to be valid in the quantum domain. By considering the SU(n) case the possible non validity of the proof for a particular n would be merely accidental. 
  We consider holographic RG flows which interpolate between non-trivial ultra-violet (UV) and infra-red (IR) conformal fixed points. We study the ``superpotentials'' which characterize different flows and discuss their expansions near the fixed points. Then we focus on the holographic $c$-function as defined from the two-point function of the stress-energy tensor. We point out that the equation for the metric fluctuations in the background flow is equivalent to a scattering problem and we use this to obtain an expression for the $c$-function in terms of the associated Jost functions. We propose two explicit models that realize UV-IR flows. In the first example we consider a thin wall separating two AdS spaces with different radii, while in the second one the UV region is replaced with an asymptotically AdS space. We find that the holographic $c$-function obeys the expected properties. In particular it reduces to the correct -- nonzero -- central charge in the IR limit. 
  A supersymmetric formulation of a three-dimensional SYM-Chern-Simons theory using light-cone quantization is presented, and the supercharges are calculated in light-cone gauge. The theory is dimensionally reduced by requiring all fields to be independent of the transverse dimension. The result is a non-trivial two-dimensional supersymmetric theory with an adjoint scalar and an adjoint fermion. We perform a numerical simulation of this SYM-Chern-Simons theory in 1+1 dimensions using SDLCQ (Supersymmetric Discrete Light-Cone Quantization). We find that the character of the bound states of this theory is very different from previously considered two-dimensional supersymmetric gauge theories. The low-energy bound states of this theory are very ``QCD-like.'' The wave functions of some of the low mass states have a striking valence structure. We present the valence and sea parton structure functions of these states. In addition, we identify BPS-like states which are almost independent of the coupling. Their masses are proportional to their parton number in the large-coupling limit. 
  We calculate Wilson loop (quark anti-quark potential) in dS/CFT correspondence. The brane-world model is considered where bulk is two 5d Euclidean de Sitter spaces and boundary (brane) is 4d de Sitter one. Starting from the Nambu-Goto action, the calculation of the effective tension (average energy) is presented. Unlike to the case of supergravity calculation of Wilson loop in AdS/CFT set-up, there is no need to regularize the Nambu-Goto action (the volume of de Sitter space is finite). It turns out that at sufficiently small curvature of 5d background the energy (potential) is positive and linear on the distance between quark and anti-quark what indicates to the possibility of confinement. 
  We consider open strings attached to a moving D$_p$-brane with motion along itself, in the presence of the backgrounds $B_{\mu \nu}$-field and a U(1) gauge field $A_{\alpha}$. The effects of the motion of the brane on the open string propagator and on the open string variables are studied. We observe that some free parameters appear in the open string variables and in the propagator of it. 
  We describe a supersymmetric RG flow between conformal fixed points of a two-dimensional quantum field theory as an analytic domain wall solution of the three-dimensional SO(4) x SO(4) gauged supergravity. Its ultraviolet fixed point is an N=(4,4) superconformal field theory related, through the double D1-D5 system, to theories modeling the statistical mechanics of black holes. The flow is driven by a relevant operator of conformal dimension \Delta=3/2 which breaks conformal symmetry and breaks supersymmetry down to N=(1,1), and sends the theory to an infrared conformal fixed point with half the central charge.   Using the supergravity description, we compute counterterms, one-point functions and fluctuation equations for inert scalars and vector fields, providing the complete framework to compute two-point correlation functions of the corresponding operators throughout the flow in the two-dimensional quantum field theory. This produces a toy model for flows of N=4 super Yang-Mills theory in 3+1 dimensions, where conformal-to-conformal flows have resisted analytical solution. 
  Possible ways of generalization of the superembedding approach for the supersurfaces with the number of Grassmann directions being less than the half of that for the target superspace are considered on example of Type II superstrings. Focus is on n=(1,1) superworldsheet embedded into D=10 Type II superspace that is of the interest for establishing a relation with the NSR string. 
  It is shown that the amplitude for reflection of a Dirac particle with arbitrarily low momentum incident on a potential of finite range is -1 and hence the transmission coefficient T=0 in general. If however the potential supports a half-bound state at k=0 this result does not hold. In the case of an asymmetric potential the transmission coefficient T will be non-zero whilst for a symmetric potential T=1. 
  We report on progress made in the construction of higher-derivative superinvariants for type-II theories in ten dimensions. The string amplitude calculations required for this analysis exhibit interesting features which have received little attention in the literature so far. We discuss two examples from a forthcoming publication: the construction of the (H_{NS})^2 R^3 terms and the fermionic completion of the \epsilon\epsilon R^4 terms. We show that a correct answer requires very careful treatment of the chiral splitting theorem, implies unexpected new relations between fermionic correlators, and most interestingly, necessitates the use of worldsheet gravitino zero modes in the string vertex operators. In addition, we discuss the relation of our results to the predictions of the linear scalar superfield of the type-IIB theory and find (and explain) an interesting discrepancy. 
  Recently, it was noticed by us that the nonlinear holomorphic supersymmetry of order $n\in\N, n>1$, ($n$-HSUSY) has an algebraic origin. We show that the Onsager algebra underlies $n$-HSUSY and investigate the structure of the former in the context of the latter. A new infinite set of mutually commuting charges is found which, unlike those from the Dolan-Grady set, include the terms quadratic in the Onsager algebra generators. This allows us to find the general form of the superalgebra of $n$-HSUSY and fix it explicitly for the cases of $n=2,3,4,5,6$. The similar results are obtained for a new, contracted form of the Onsager algebra generated via the contracted Dolan-Grady relations. As an application, the algebraic structure of the known 1D and 2D systems with $n$-HSUSY is clarified and a generalization of the construction to the case of nonlinear pseudo-supersymmetry is proposed. Such a generalization is discussed in application to some integrable spin models and with its help we obtain a family of quasi-exactly solvable systems appearing in the $PT$-symmetric quantum mechanics. 
  We propose a new procedure to embed second class systems by introducing Wess-Zumino (WZ) fields in order to unveil hidden symmetries existent in the models. This formalism is based on the direct imposition that the new  Hamiltonian must be invariant by gauge-symmetry transformations. An interesting feature in this approach is the possibility to find a representation for the WZ fields in a convenient way, which leads to preserve the gauge symmetry in the original phase space. Consequently, the gauge-invariant Hamiltonian can be written only in terms of the original phase-space variables. In this situation, the WZ variables are only auxiliary tools that permit to reveal the hidden symmetries present in the original second class model. We apply this formalism to important physical models: the reduced-SU(2) Skyrme model, the Chern-Simons-Proca quantum mechanics and the chiral bosons field theory. In all these systems, the gauge-invariant Hamiltonians are derived in a very simple way. 
  It has recently been shown that the ten-dimensional superstring can be quantized using the BRST operator $Q=\oint\lambda^\alpha d_\alpha$ where $\lambda^\alpha$ is a pure spinor satisfying $\lambda \gamma^m \lambda=0$ and $d_\alpha$ is the fermionic supersymmetric derivative. In this paper, the pure spinor version of superstring theory is formulated in a curved supergravity background and it is shown that nilpotency and holomorphicity of the pure spinor BRST operator imply the on-shell superspace constraints of the supergravity background. This is shown to lowest order in $\alpha'$ for the heterotic and Type II superstrings, thus providing a compact pure spinor version of the ten-dimensional superspace constraints for N=1, Type IIA and Type IIB supergravities. Since quantization is straightforward using the pure spinor version of the superstring, it is expected that these methods can also be used to compute higher-order $\alpha'$ corrections to the ten-dimensional superspace constraints. 
  We discuss a systematic way to dimensionally regularize divergent sums arising in field theories with an arbitrary number of physical compact dimensions or finite temperature. The method preserves the same symmetries of the action as the conventional dimensional regularization and allows an easy separation of the regulated divergence from the finite term that depends on the compactification radius (temperature). 
  We construct a covariant quantum superstring, extending Berkovits' approach by introducing new ghosts to relax the pure spinor constraints. The central charge of the underlying Kac-Moody algebra, which would lead to an anomaly in the BRST charge, is treated as a new generator with a new b-c system. We construct a nilpotent BRST current, an anomalous ghost current and an anomaly-free energy-momentum tensor. For open superstrings, we find the correct massless spectrum. In addition, we construct a Lorentz invariant B-field to be used for the computation of the integrated vertex operators and amplitudes. 
  Integrable Hamiltonians for higher spin periodic XXZ chains are constructed in terms of the spin generators; explicit examples for spins up to 3/2 are given. Relations between Hamiltonians for some U_q(sl_2)-symmetric and U(1)-symmetric universal r-matrices are studied; their properties are investigated. A certain modification of the higher spin periodic chain Hamiltonian is shown to be an integrable U_q(sl_2)-symmetric Hamiltonian for an open chain. 
  We consider the theory of pure gravity in 2+1 dimensions, with negative cosmological constant. The theory contains simple matter in the form of point particles; the later are classically described as lines of conical singularities. We propose a formalism in which quantum amplitudes for process involving black holes and point particles are obtained as Liouville field theory (LFT) correlation functions on Riemann surfaces X. Point particles are described by LFT vertex operators, black holes (asymptotic regions) are in correspondence with boundaries of X. We analyze two examples: the amplitude for emission of a particle by the BTZ black hole, and the amplitude of black hole creation by two point particles. We then define an inner product between quantum states. The value of this inner product can be interpreted as the amplitude for one set of point particles to go into another set producing black holes. The full particle S-matrix is then given by the sum of all such amplitudes. This S-matrix is that of a non-critical string theory, with the world-sheet CFT being essentially the Liouville theory. Lambda<0 quantum gravity in 2+1 dimensions is thus a string theory. 
  We study the dynamical equations for extra-dimensional dependence of a warp factor and a bulk scalar in 5d brane world scenarios with induced brane metric of constant curvature. These equations are similar to those for the time dependence of the scale factor and a scalar field in 4d cosmology, but with the sign of the scalar field potential reversed. Based on this analogy, we introduce novel methods for studying the warped geometry. We construct the full phase portraits of the warp factor/scalar system for several examples of the bulk potential. This allows us to view the global properties of the warped geometry. For flat branes, the phase portrait is two dimensional. Moving along typical phase trajectories, the warp factor is initially increasing and finally decreasing. All trajectories have timelike gradient-dominated singularities at one or both of their ends, which are reachable in a finite distance and must be screened by the branes. For curved branes, the phase portrait is three dimensional. However, as the warp factor increases the phase trajectories tend towards the two dimensional surface corresponding to flat branes. We discuss this property as a mechanism that may stretch the curved brane to be almost flat, with a small cosmological constant. Finally, we describe the embedding of branes in the 5d bulk using the phase space geometric methods developed here. In this language the boundary conditions at the branes can be described as a 1d curve in the phase space. We discuss the naturalness of tuning the brane potential to stabilize the brane world system. 
  We show that standard Einstein gravity coupled to a free conformal field theory (CFT) in Anti de Sitter space can undergo a Higgs phenomenon whereby the graviton acquires a nonzero mass (and three extra polarizations). We show that the essential ingredients of this mechanism are the discreteness of the energy spectrum in AdS space, and unusual boundary conditions on the elementary fields of the CFT. These boundary conditions can be interpreted as implying the existence of a 3-d defect CFT living at the boundary of the AdS space. Our free-field computation sheds light on the essential, model-independent features of AdS that give rise to massive gravity. 
  We study the analytic properties of the scaling function associated with the 2D Ising model free energy in the critical domain $T \to T_c$, $H \to 0$. The analysis is based on numerical data obtained through the Truncated Free Fermion Space Approach. We determine the discontinuities across the Yang-Lee and Langer branch cuts. We confirm the standard analyticity assumptions and propose "extended analyticity"; roughly speaking, the latter states that the Yang-Lee branching point is the nearest singularity under Langer's branch cut. We support the extended analyticity by evaluating numerically the associated "extended dispersion relation". 
  Wigner's irreducible positive energy representations of the Poincare group are often used to give additional justifications for the Lagrangian quantization formalism of standard QFT. Here we study another more recent aspect. We explain in this paper modular concepts by which we are able to construct the local operator algebras for all standard positive energy representations directly i.e. without going through field coordinatizations. In this way the artificial emphasis on Lagrangian field coordinates is avoided from the very beginning. These new concepts allow to treat also those cases of ``exceptional'' Wigner representations associated with anyons and the famous Wigner ``spin tower''which have remained inaccessible to Lagrangian quantization. Together with the d=1+1 factorizing models (whose modular construction has been studied previously), they form an interesting family of theories with a rich vacuum-polarization structure (but no on shell real particle creation) to which the modular methods can be applied for their explicit construction. We explain and illustrate the algebraic strategy of this construction. We also comment on possibilities of formulating the Wigner theory in a setting of a noncommutative spacetime substrate. This is potentially interesting in connection with recent unitarity- and Lorentz invariance- preserving results of the special nonlocality caused by this kind of noncommutativity. 
  We concretely define the identity string field as a surface state and deal with it consistently in terms of conformal field theory language, never using its formal properties nor oscillator representation of it. The generalized gluing and resmoothing theorem provides us with a powerful computational tool which fits into our framework. Among others, we can prove that in some situations the identity state defined this way actually behaves itself like an identity element under the *-product. We use these CFT techniques to give an explicit expression of the classical solution in the ordinary cubic string field theory having the property that the conjectured vacuum string field theory action arises when the cubic action is expanded around it. 
  Following the symplectic approach we show how to embed the Abelian Proca model into a first-class system by extending the configuration space to include an additional pair of scalar fields, and compare it with the improved Dirac scheme. We obtain in this way the desired Wess-Zumino and gauge fixing terms of BRST invariant Lagrangian. Furthermore, the integrability properties of the second-class system described by the Abelian Proca model are investigated using the Hamilton-Jacobi formalism, where we construct the closed Lie algebra by introducing operators associated with the generalized Poisson brackets. 
  Symmetry invariant local interaction of a many body system leads to global constraints. We obtain explicit forms of the global macroscopic condition assuring that at the microscopic level the evolution respects the overall symmetry. 
  It is substantiated that spin is a notion associated with the group of internal symmetry that is tightly connected with the geometrical structure of spacetime. The wave equation for the description of the particles with spin one half is proposed. On this ground it is shown that the spin of electron is exhibited through the quantum number and accordingly the Dirac equation describes properties of particles with the projection of spin equal plus or minus one half. On contrary, we put forward the conjecture that the spin of the quark cannot be considered as a quantum number, but only as an origin of a non-abelian gauge field. It is a deep reason for understanding quark-lepton symmetry and such important phenomena as quark confinement. 
  We discuss some aspects of the behaviour of a string gas at the Hagedorn temperature from a Euclidean point of view. Using AdS space as an infrared regulator, the Hagedorn tachyon can be effectively quasi-localized and its dynamics controled by a finite energetic balance. We propose that the off-shell RG flow matches to an Euclidean AdS black hole geometry in a generalization of the string/black-hole correspondence principle. The final stage of the RG flow can be interpreted semiclassically as the growth of a cool black hole in a hotter radiation bath. The end-point of the condensation is the large Euclidean AdS black hole, and the part of spacetime behind the horizon has been removed. In the flat-space limit, holography is manifest by the system creating its own transverse screen at infinity. This leads to an argument, based on the energetics of the system, explaining why the non-supersymmetric type 0A string theory decays into the supersymmetric type IIB vacuum. We also suggest a notion of `boundary entropy', the value of which decreases along the line of flow. 
  We present special classes of orientifold models involving supersymmetry breaking via branes at angles. Type II superstring theories are compactified on a two torus times a four-dimensional orbifold. Combining worldsheet parity with a reflection of half of the compact coordinates leads to D6-branes at angles which are mapped onto each other by the orbifold group, while applying the geometric action only along one coordinate leads to intersecting D8-branes with non-trivial transformation properties under the orbifold group. The models differ in the gauge groups and matter content. 
  We propose a new BRST-like quantization procedure which is applicable to dynamical systems containing both first and second class constraints. It requires no explicit separation into first and second class constraints and therefore no conversion of second class constraints is needed. The basic ingredient is instead an invariant projection operator which projects out the maximal subset of constraints in involution. The hope is that the method will enable a covariant quantization of models for which there is no covariant separation into first and second class constraints. An example of this type is given. 
  We develop a method to compute the one-loop effective action of noncommutative U(1) gauge theory based on the bosonic worldline formalism, and derive compact expressions for N-point 1PI amplitudes. The method, resembling perturbative string computations, shows that open Wilson lines emerge as a gauge invariant completion of certain terms in the effective action. The terms involving open Wilson lines are of the form reminiscent of closed string exchanges between the states living on the two boundaries of a cylinder. They are also consistent with recent matrix theory analysis and the results from noncommutative scalar field theories with cubic interactions. 
  We consider simple superalgebras which are a supersymmetric extension of $\fspin(s,t)$ in the cases where the number of odd generators does not exceed 64. All of them contain a super Poincar\'e algebra as a contraction and another as a subalgebra. Because of the contraction property, some of these algebras can be interpreted as de Sitter or anti de Sitter superalgebras. However, the number of odd generators present in the contraction is not always minimal due to the different splitting properties of the spinor representations under a subalgebra. We consider the general case, with arbitrary dimension and signature, and examine in detail particular examples with physical implications in dimensions $d=10$ and $d=4$. 
  We exhibit a simple class of exactly marginal "double-trace" deformations of two dimensional CFTs which have AdS_3 duals, in which the deformation is given by a product of left and right-moving U(1) currents. In this special case the deformation on AdS_3 is generated by a local boundary term in three dimensions, which changes the physics also in the bulk via bulk-boundary propagators. However, the deformation is non-local in six dimensions and on the string worldsheet, like generic non-local string theories (NLSTs). Due to the simplicity of the deformation we can explicitly make computations in the non-local string theory and compare them to CFT computations, and we obtain precise agreement. We discuss the effect of the deformation on closed strings and on D-branes. The examples we analyze include a supersymmetry-breaking but exactly marginal "double-trace" deformation, which is dual to a string theory in which no destabilizing tadpoles are generated for moduli nonperturbatively in all couplings, despite the absence of supersymmetry. We explain how this cancellation works on the gravity side in string perturbation theory, and also non-perturbatively at leading order in the deformation parameter. We also discuss possible flat space limits of our construction. 
  Dirac-like monopoles are studied in three-dimensional Abelian Maxwell and Maxwell-Chern-Simons models. Their scalar nature is highlighted and discussed through a dimensional reduction of four-dimensional electrodynamics with electric and magnetic sources. Some general properties and similarities of them when are considered in Minkowski or Euclidian space are mentioned. However, by virtue of the structure of the space-time in which they are considered a number of differences among them take place. Furthermore, we pay attention to some consequences of these objects when acting upon usual particles. Among other subjects, special attention is given to the study of a Lorentz-violating non-minimal coupling between neutral fermions and the field generated by a monopole alone. In addition, an analogue of the Aharonov-Casher effect is discussed in this framework. 
  Matrix string theory is equivalent to type IIA superstring theory in the light-cone gauge, together with extra degrees of freedom representing D-brane states. It is therefore the appropriate framework in which to study systems of multiple fundamental strings expanding into higher-dimensional D-branes. Starting from Matrix theory in a weakly curved background, we construct the linear couplings of closed string fields to type IIA Matrix strings. As a check, we show that at weak coupling the resulting action reproduces light-cone gauge string theory in a weakly curved background. Further dualities give a type IIB Matrix string theory and a type IIA theory of Matrix strings with winding. We comment on the dielectric effect in each of these theories, giving some explicit solutions describing fundamental strings expanding into various Dp-branes. 
  The projection of a two dimensional planar system on the higher Landau levels of an external magnetic field is formulated in the language of the non commutative plane and leads to a new class of star products. 
  We examine the causality and degrees of freedom (DoF) problems encountered by charged, gravitating, massive higher spin fields. For spin s=3/2, making the metric dynamical yields improved causality bounds. These involve only the mass, the product eM_P of the charge and Planck mass and the cosmological constant \Lambda. The bounds are themselves related to a gauge invariance of the timelike component of the field equation at the onset of acausality. While propagation is causal in arbitrary E/M backgrounds, the allowed mass ranges of parameters are of Planck order. Generically, interacting spins s>3/2 are subject to DoF violations as well as to acausality; the former must be overcome before analysis of the latter can even begin. Here we review both difficulties for charged s=2 and show that while a g-factor of 1/2 solves the DoF problem, acausality persists for any g. Separately we establish that no s=2 theory --DoF preserving or otherwise -- can be tree unitary. 
  Correlation functions of fermionic fields described by the massless Thirring model are analysed within the operator formalism developed by Klaiber and the path-integral approach with massless fermions quantized in the chiral symmetric phase. We notice that Klaiber's composite fermion operators possess non-standard properties under parity transformations and construct operators with standard parity properties. We find that Klaiber's parameterization of a one-parameter family of solutions of the massless Thirring model is not well defined, since it is not consistent with the requirement of chiral symmetry. We show that the dynamical dimensions of correlation functions depend on an arbitrary parameter induced by ambiguities of the evaluation of the chiral Jacobian. A non-perturbative renormalization of the massless Thirring model is discussed. We demonstrate that the infrared divergences of Klaiber's correlation functions can be transferred into ultra-violet divergences by renormalization of the wave function of fermionic fields. This makes Klaiber's correlation functions non-singular in the infrared limit. We show that the requirement of non-perturbative renormalizability of the massless Thirring model fixes a free parameter of the path-integral approach. In turn, the operator formalism is inconsistent with non-perturbative renormalizability of the massless Thirring model. We carry out a non-perturbative renormalization of the sine-Gordon model and show that it is not an asymptotically free theory as well as the massless Thirring model. We calculate the fermion condensate by using the Fourier transform of the two-point Green function of massless Thirring fermion fields quantized in the chiral symmetric phase. 
  We construct a consistent quantum field theory of a free massless (pseudo)scalar field in 1+1-dimensional space-times free of infrared divergences. We show that a continuous symmetry of (pseudo)scalar field translations is spontaneously broken. Goldstone bosons appear as quanta of a free massless (pseudo)scalar field. We show that the main inequality between vacuum expectation values of certain operators which has been used for the derivation of the Mermin-Wagner-Hohenberg theorem is fulfilled in the quantum field theory of a free massless (pseudo)scalar field free of infrared divergences. 
  We study the spectrum of density fluctuations of Fractional Hall Fluids in the context of the noncommutative hidrodynamical model of Susskind. We show that, within the weak-field expansion, the leading correction to the noncommutative Chern--Simons Lagrangian (a Maxwell term in the effective action,) destroys the incompressibility of the Hall fluid due to strong UV/IR effects at one loop. We speculate on possible relations of this instability with the transition to the Wigner crystal, and conclude that calculations within the weak-field expansion must be carried out with an explicit ultraviolet cutoff at the noncommutativity scale. We point out that the noncommutative dipoles exactly match the spatial structure of the Halperin--Kallin quasiexcitons. Therefore, we propose that the noncommutative formalism must describe accurately the spectrum at very large momenta, provided no weak-field approximations are made. We further conjecture that the noncommutative open Wilson lines are `vertex operators' for the quasiexcitons. 
  We transform static solutions of space-noncommutative Dirac-Born-Infeld theory (DBI) into static solutions of space-time noncommutative DBI. Via Seiberg-Witten map we match this symmetry transformation with a corresponding symmetry of commutative DBI. This allows to: 1) study new BPS type magnetic monopoles, with constant electric and magnetic background and describe them both in the commutative and in the noncommutative setting; 2) relate by S-duality space-noncommutative magnetic monopoles to space-noncommutative electric monopoles 
  We discuss fractional D3-branes on the orbifold C^3/Z_2*Z_2. We study the open and the closed string spectrum on this orbifold. The corresponding N=1 theory on the brane has, generically, a U(N_1)*U(N_2)*U(N_3)*U(N_4) gauge group with matter in the bifundamental. In particular, when only one type of brane is present, one obtains pure N=1 Yang-Mills. We study the coupling of the branes to the bulk fields and present the corresponding supergravity solution, valid at large distances. By using a probe analysis, we are able to obtain the Wilsonian beta-function for those gauge theories that possess some chiral multiplet. Although, due to the lack of moduli, the probe technique is not directly applicable to the case of pure N=1 Yang-Mills, we point out that the same formula gives the correct result also for this case. 
  We construct boundary states for the AdS_2 D-branes in AdS_3. We show that, in the semi-classical limit, the boundary states correctly reproduce geometric configurations of these branes. We use the boundary states to compute the one loop free energy of open string stretched between the branes. The result agrees precisely with the open string computation in hep-th/0106129. 
  We describe a class of exact solutions of super Yang-Mills theory on even-dimensional noncommutative tori. These solutions generalize the solitons on a noncommutative plane introduced in hep-th/0009142 that are conjectured to describe unstable D2p-D0 systems. We show that the spectrum of quadratic fluctuations around our solutions correctly reproduces the string spectrum of the D2p-D0 system in the Seiberg-Witten decoupling limit. In particular the fluctuations correctly reproduce the 0-0 string winding modes. For p=1 and p=2 we match the differences between the soliton energy and the energy of an appropriate SYM BPS state with the binding energies of D2-D0 and D4-D0 systems. We also give an example of a soliton that we conjecture describes branes of intermediate dimension on a torus such as a D2-D4 system on a four-torus. 
  We derive the classical type IIB supergravity solution describing fractional D3-branes transverse to a C^2/Gamma orbifold singularity, for Gamma any Kleinian ADE subgroup. This solution fully describes the N=2 gauge theory with appropriate gauge groups and matter living on the branes, up to non-perturbative instanton contributions. 
  We develop a recursive algorithm for the investigation of infinite sequences of quasi-primary fields obtained from chiral primary operators (CPOs) $O^I_k(x)$ and eventually their derivatives by applying operator product expansions and singling out SO(6) representations. We show that normal products of $O_2$ operators can be expressed in terms of projection operators on representations of SO(20) and discuss intertwining operators for SO(6) representations. Furthermore we derive $\mathcal{O}(\frac{1}{N^2})$ corrections to AdS/CFT 4-point functions by graphical combinatorics and finally extract anomalous dimensions by applying the method of conformal partial wave analysis. We find infinite sequences of quasi-primary fields with vanishing anomalous dimensions and interpret them as 1/2-BPS or 1/4-BPS fields. 
  We analyze the constraints which follow both on the geometry and on the gauge sector for a consistent supergravity reduction of a general matter-coupled N=2 supergravity theory in four dimensions. These constraints can be derived in an elegant way by looking at the fermionic sector of the theory. 
  We discuss D-branes on a line of conformal field theories connected by an exact marginal deformation. The line contains an SU(2) WZW model and two mutually T-dual SU(2)/U(1) cosets times a free boson. We find the D-branes preserving a U(1) isometry, an F-flux quantization condition and conformal invariance. Away from the SU(2) point a U(1) times U(1) symmetry is broken to U(1) times Z_k, i.e. continuous rotations of branes are accompanied by rotations along the branes. Requiring decoupling of the cosets from the free boson at the endpoints of the deformation breaks the continuous rotation of branes to Z_k. At the SU(2) point the full U(1) times U(1) symmetry is restored. This suggests the occurrence of phase transitions for branes at angles in the coset model, at a semiclassical level. We also discuss briefly the orientifold planes along the deformation line. 
  Analyzing the constraint structure of electrodynamics, massive vector bosons, Dirac fermions and electrodynamics coupled to fermions, we show that Dirac quantization method leads to appropriate creation-annihilation algebra among the Forier coefficients of the fields. 
  This is a pedagogical and extended version of the results published in Refs. [1,2] and presented by the authors in various talks during the last year. We discuss the type II D-branes (both regular and fractional) of the orbifold R^{1,5}*R^4/Z_2, we determine their corresponding supergravity solution and show how this can be used to study the properties of N=2 super Yang-Mills. Supergravity is able to reproduce the perturbative moduli space of the gauge theory, while it does not encode the non-perturbative corrections. The short distance region of space-time, which corresponds to the infrared region of the gauge theory, is excised by an enhancon mechanism, and more states should be included in the low energy effective action in order to enter inside the enhancon and recover the instanton corrections. (To be published on a Memorial Volume commemorating Michael Marinov) 
  The AdS/CFT correspondence can be realized in spaces that are globally different but share the same asymptotic behavior. Two known cases are: a compact AdS space and the space generated by a large number of coincident branes. We discuss the physical consistency, in the sense of the Cauchy problem, of these two formulations. We show that the role of the boundary in the compact AdS space is equivalent to that of the flat asymptotic region in the brane space. We also show, by introducing a second coordinate chart for the pure AdS space, that a point at its spatial infinity corresponds to a horizon in the brane system. 
  We consider the dynamics of p anti-D3 branes inside the Klebanov-Strassler geometry, the deformed conifold with M units of RR 3-form flux around the S^3. We find that for p<<M the system relaxes to a nonsupersymmetric NS 5-brane ``giant graviton'' configuration, which is classically stable, but quantum mechanically can tunnel to a nearby supersymmetric vacuum with M-p D3 branes. This decay mode is exponentially suppressed and proceeds via the nucleation of an NS 5-brane bubble wall. We propose a dual field theory interpretation of the decay as the transition between a nonsupersymmetric ``baryonic'' branch and a supersymmetric ``mesonic'' branch of the corresponding SU(2M-p)x SU(M-p) low energy gauge theory. The NS 5-brane tunneling process also provides a simple explanation of the geometric transition by which D3-branes can dissolve into 3-form flux. 
  In this work we propose an exact microscopic description of maximally symmetric branes in a Euclidean $AdS_3$ background. As shown by Bachas and Petropoulos, the most important such branes are localized along a Euclidean $AdS_2 \subset AdS_3$. We provide explicit formulas for the coupling of closed strings to such branes (boundary states) and for the spectral density of open strings. The latter is computed in two different ways first in terms of the open string reflection amplitude and then also from the boundary states by world-sheet duality. This gives rise to an important Cardy type consistency check. All the results are compared in detail with the geometrical picture. We also discuss a second class of branes with spherical symmetry and finally comment on some implications for D-branes in a 2D back hole geometry. 
  In this paper we reexamine the D-brane spectrum in the Melvin background with nonconstant NSNS B-field from the viewpoint of its world-volume and string world-sheet theory. We find that the stable D2-D0 bound state exists even though it does not wrap any nontrivial cycles. We show that this system is stabilized by the presence of the NSNS B-field and the magnetic flux F. Moreover from the non-abelian world-volume theory of D0-branes the bound state is regarded as a system of D0-branes expanding into a fuzzy torus. 
  We study Little String Theories (LST) with ${\cal N}=(1,0)$ supersymmetry arising, in a suitable double scaling limit, from 5-branes in heterotic string theory or in the heterotic-like type II/$(-)^{F_L}\times {\rm shift}$. The limit in question, previously studied in the type II case, is such that the resulting holographically dual pairs, i.e. bulk string theory and LST are at a finite effective coupling. In particular, the internal $(2,2)$ SCFT on the string theory side is non-singular and given by $SL(2)/U(1)\times SU(2)/U(1)$ coset. In the type II orbifold case, we determine the orbifold action on the internal SCFT and construct the boundary states describing the non-BPS massive states of a completely broken $SO$ gauge theory, in agreement with the dual picture of D5-branes in type II/$\Omega\times {\rm shift}$. We also describe a different orbifold action which gives rise to a $Sp$ gauge theory with $(1,1)$ supersymmetry. In both the heterotic SO(32) and $E_8\times E_8$ cases, we determine the gauge bundles which correspond to the above SCFT and break down the gauge groups to $SU(2)\times SO(28)$ and $E_7\times E_8$ respectively. The double scaling limit in this case involves taking small instanton together with small string coupling constant limit. We determine the spectrum of chiral gauge invariant operators with the corresponding global symmetry charges on the LST side and compare with the massless excitations on the string theory side, finding agreement for multiplicities and global charges. 
  Abelian Lagrangians containing $\lambda\phi^{4}$-type vertices are regularized by means of a suitable point-splitting scheme combined with generalized gauge transformations. The calculation is developped in details for a general Lagrangian whose fields (gauge and matter ones) satisfy usual conditions. We illustrate our results by considering some special cases, such as the $(\bar{\psi}\psi)^{2}$ and the Avdeev-Chizhov models. Possible application of our results to the Abelian Higgs model, whenever spontaneous symmetry breaking is considered, is also discussed. We also pay attention to a number of features of the point-split action such as the regularity and non-locality of its new ``interacting terms''. 
  We re-examine the recent proposal of Rastelli, Sen and Zwiebach on the tachyon fluctuation of the vacuum string field theory representing a D25 brane, originally considered by Hata and Kawano. We show that the tachyon state satisfies the linearized equations of motion on-shell in the strong sense thereby allowing us to calculate the ratio of energy density to the tension of the D-brane to be $E_c/T_{25}\simeq \pi^2/3[1/16(ln2)^3]\simeq 0.62$. Our proof relies on a careful handling of the limits ($n\to\infty$) involved in the conformal theory description of the sliver and tachyon states. We conjecture that the sliver state represents a single D25 brane. 
  We present a generalization of the boundary state formalism for the bosonic string that allows us to calculate the overlap of the boundary state with arbitrary closed string states. We show that this generalization exactly reproduces world-sheet sigma model calculations, thus giving the correct overlap with both on- and off-shell string states, and that this new boundary state automatically satisfies the requirement for integrated vertex operators in the case of non-conformally invariant boundary interactions. 
  We begin by outlining the ancient puzzle of off shell currents and infinite size particles in a string theory of hadrons. We then consider the problem from the modern AdS/CFT perspective. We argue that although hadrons should be thought of as ideal thin strings from the 5-dimensional bulk point of view, the 4-dimensional strings are a superposition of "fat" strings of different thickness.   We also find that the warped nature of the target geometry provides a mechanism for taming the infinite zero point fluctuations which apparently produce a divergent result for hadronic radii.   Finally a calculation of the large momentum behavior of the form factor is given in the limit of strong 't Hooft parameter where the classical gravity limit is appropriate. We find agreement with parton model expectations. 
  In brane world scenarios with a bulk scalar field between two branes it is known that 4-dimensional Einstein gravity is restored at low energies on either brane. By using a gauge-invariant gravitational and scalar perturbation formalism we extend the theory of weak gravity in the brane world scenarios to higher energies, or shorter distances. We argue that weak gravity on either brane is indistinguishable from 4-dimensional higher derivative gravity, provided that the inter-brane distance (radion) is stabilized, that the background bulk scalar field is changing near the branes and that the background bulk geometry near the branes is warped. This argument holds for a general conformal transformation to a frame in which matter on the branes is minimally coupled to the metric. In particular, Newton's constant and the coefficients of curvature-squared terms in the 4-dimensional effective action are determined up to an ambiguity of adding a Gauss-Bonnet topological term. In other words, we provide the brane-world realization of the so called $R^2$-model without utilizing a quantum theory. We discuss the appearance of composite spin-2 and spin-0 fields in addition to the graviton on the brane and point out a possibility that the spin-0 field may play the role of an effective inflaton to drive brane-world inflation. Finally, we conjecture that the sequence of higher derivative terms is an infinite series and, thus, indicates non-locality in the brane world scenarios. 
  A new class of twistor-like string models in four-dimensional space-time extended by the addition of six tensorial central charge (TCC) coordinates $z_{mn}$ is studied. The Hamiltonian of tensionless string in the extended space-time is derived and its symmetries are investigated. We establish that the string constraints reduce the number of independent TCC coordinates $z_{mn}$ to one real effective coordinate which composes an effective 5-dimensional target space together with the $x^{m}$ coordinates. We construct the P.B. algebra of the first class constraints and discover that it coincides with the P.B. algebra of tensionless strings. The Lorentz covariant antisymmetric Dirac $\hat{\mathbf C}$-matrix of the P.B. of the second class constraints is constructed and its algebraic structure is further presented. 
  We study the coupling of supergravity with a purely bosonic brane source (bosonic p-brane). The interaction, described by the sum of their respective actions, is self-consistent if the bosonic p-brane is the pure bosonic limit of a super-p-brane. In that case the dynamical system preserves 1/2 of the local supersymmetry characteristic of the `free' supergravity. 
  Stable states (particles), ghosts and unstables states (particles) are discussed with respect to the time representations involved, their unitary groups and the induced Hilbert spaces. Unstable particles with their decay channels are treated as higher dimensional probability collectives with nonabelian probability groups U(n) generalizing the individual abelian U(1)-normalization for stable states (particles). 
  We extent the previous determinations of nonasymptotic critical behavior of Phys. Rev B32, 7209 (1985) and B35, 3585 (1987) to accurate expressions of the complete classical-to-critical crossover (in the 3-d field theory) in terms of the temperature-like scaling field (i.e., along the critical isochore) for : 1) the correlation length, the susceptibility and the specific heat in the homogeneous phase for the n-vector model (n=1 to 3) and 2) for the spontaneous magnetization (coexistence curve), the susceptibility and the specific heat in the inhomogeneous phase for the Ising model (n=1). The present calculations include the seventh loop order of Murray and Nickel (1991) and closely account for the up-to-date estimates of universal asymptotic critical quantities (exponents and amplitude combinations) provided by Guida and Zinn-Justin [J. Phys. A31, 8103 (1998)]. 
  We review a method to find the non-abelian open superstring effective action, thereby settling the issue of the ordering ambiguities. We start from solutions to Yang-Mills which, in D-brane context, define certain BPS configurations. Studying their deformations in the abelian case shows that the Born-Infeld action is the unique deformation which admits solutions of this type. By applying the method to the non-abelian case we calculated the full effective action through O(\alpha'^3). The presence of derivative terms turns out to be essential. Testing the result by comparing the spectrum in the presence of a constant magnetic background field with the string theory prediction, we obtain perfect agreement. 
  The n-instanton contribution to the Seiberg-Witten prepotential of N=2 supersymmetric d=4 Yang Mills theory is represented as the integral of the exponential of an equivariantly exact form. Integrating out an overall scale and a U(1) angle the integral is rewritten as (4n-3) fold product of a closed two form. This two form is, formally, a representative of the Euler class of the Instanton moduli space viewed as a principal U(1) bundle, because its pullback under bundel projection is the exterior derivative of an angular one-form. 
  We generalize the worldline variational approach to field theory by introducing a trial action which allows for anisotropic terms to be induced by external 4-momenta of Green's functions. By solving the ensuing variational equations numerically we demonstrate that within the (quenched) scalar Wick-Cutkosky model considerable improvement can be achieved over results obtained previously with isotropic actions. In particular, the critical coupling associated with the instability of the model is lowered, in accordance with expectations from Baym's proof of the instability in the unquenched theory. The physical picture associated with a different quantum mechanical motion of the dressed particle along and perpendicular to its classical momentum is discussed. Indeed, we find that for large couplings the dressed particle is strongly distorted in the direction of its four-momentum. In addition, we obtain an exact relation between the renormalized coupling of the theory and the propagator. Along the way we introduce new and efficient methods to evaluate the averages needed in the variational approach and apply them to the calculation of the 2-point function. 
  The theory of scale relativity provides a new insight into the origin of fundamental laws in physics. Its application to microphysics allows to recover quantum mechanics as mechanics on a non-differentiable (fractal) space-time. The Schr\"odinger and Klein-Gordon equations have already been demonstrated as geodesic equations in this framework. We propose here a new development of the intrinsic properties of this theory to obtain, using the mathematical tool of Hamilton's bi-quaternions, a derivation of the Dirac equation, which, in standard physics, is merely postulated. The bi-quaternionic nature of the Dirac spinor is obtained by adding to the differential (proper) time symmetry breaking, which yields the complex form of the wave-function in the Schr\"odinger and Klein-Gordon equations, the breaking of further symmetries, namely, the differential coordinate symmetry ($dx^{\mu} \leftrightarrow - dx^{\mu}$) and the parity and time reversal symmetries. 
  Using algebraic methods the Neveu-Schwarz fermionic matter sliver is constructed. Inspirited by the wedge algebra two equations for the sliver, linear and quadratic, are considered. It is shown that both equations give the same nontrivial answer. The sliver is considered also using CFT methods where it is defined as the limit of the wedge states in the NS sector of the superstring. 
  The topology of center vortices is studied. For this purpose it is sufficient to consider mathematically idealised vortices, defined in a gauge invariant way as closed (infinitely thin) flux surfaces (in D=4 dimensions) which contribute the n'th power of a non-trivial center element to Wilson loops when they are n-foldly linked to the latter. In ordinary 3-space generic center vortices represent closed magnetic flux loops which evolve in time. I show that the topological charge of such a time-dependent vortex loop can be entirely expressed by the temporal changes of its writhing number. 
  The off-shell and the on-shell Sudakov form factors in theories with broken gauge symmetry are calculated in the double-logarithmic approximation. We have used different infrared cut-offs, i.e. different mass scales, for virtual photons and weak gauge bosons. 
  We search for static solitons stabilized by heavy fermions in a 3+1 dimensional Yukawa model. We compute the renormalized energy functional, including the exact one-loop quantum corrections, and perform a variational search for configurations that minimize the energy for a fixed fermion number. We compute the quantum corrections using a phase shift parameterization, in which we renormalize by identifying orders of the Born series with corresponding Feynman diagrams. For higher-order terms in the Born series, we develop a simplified calculational method. When applicable, we use the derivative expansion to check our results. We observe marginally bound configurations at large Yukawa coupling, and discuss their interpretation as soliton solutions subject to general limitations of the model. 
  The dS/CFT correspondence is illuminated through an analysis of massive scalar field theory in d-dimensional de Sitter space. We consider a one-parameter family of dS-invariant vacua related by Bogolyubov transformations and compute the corresponding Green functions. It is shown that none of these Green functions correspond to the one obtained by analytic continuation from AdS. Among this family of vacua are in (out) vacua which have no incoming (outgoing) particles on past (future) infinity. Surprisingly, it is shown that in odd spacetime dimensions the in and out vacua are the same, implying the absence of particle production for this state. The correlators of the boundary CFT, as defined by the dS/CFT correspondence, are shown to depend on the choice of vacuum state - the correlators with all points on past infinity vanish in the in vacuum. For dS_3 we argue that this bulk vacuum dependence of the correlators is dual to a deformation of the boundary CFT_2 by a specific marginal operator. It is also shown that Witten's non-standard de Sitter inner product (slightly modified) reduces to the standard inner product of the boundary field theory. Next we consider a scalar field in the Kerr-dS_3 Euclidean vacuum. A density matrix is constructed by tracing out over modes which are causally inaccessible to a single geodesic observer. This is shown to be a thermal state at the Kerr-dS_3 temperature and angular potential. It is further shown that, assuming Cardy's formula, the microscopic entropy of such a thermal state in the boundary CFT precisely equals the Bekenstein-Hawking value of one quarter the area of the Kerr-dS_3 horizon. 
  We construct new D-brane bound states using charged macroscopic type IIB string solutions.A generic bound state solution, when dimensionally reduced, carries multiple gauge charges. Starting with D=9 charged macroscopic strings, we obtain solutions in D=10, which are interpreted as carrying (F, D0, D2) charges as well as nonzero momenta. The masses and charges are also explicitly shown to satisfy the non-threshold bound of 1/2 BPS objects. Our solutions reduce to the known D-brane bound state solutions with appropriate restrictions in the parameter space. We further generalize the results to (Dp- D(p+2)) bound state in IIA/B theories, giving an explicit example with p=1. 
  We find an infinite number of noncommutative geometries which posses a differential structure. They generalize the two dimensional noncommutative plane, and have infinite dimensional representations. Upon applying generalized coherent states we are able to take the continuum limit, where we recover the punctured plane with non constant Poisson structures. 
  We study D-branes in Kazama-Suzuki models by means of the boundary state description. We can identify the boundary states of Kazama-Suzuki models with the solitons in N=2 Landau-Ginzburg theories. We also propose a geometrical interpretation of the boundary states in Kazama-Suzuki models. 
  Introduction to two dimensional conformal field theory on open and unoriented surfaces. The construction is illustrated in detail on the example of SU(2) WZW models. 
  We propose a definition of dS/CFT correlation functions by equating them to S-matrix elements for scattering particles from I^- to I^+. In planar coordinates, which cover half of de Sitter space, we consider instead the S-vector obtained by specifying a fixed state on the horizon. We construct the one-parameter family of de Sitter invariant vacuum states for a massive scalar field in these coordinates, and show that the vacuum obtained by analytic continuation from the sphere has no particles on the past horizon. We use this formalism to provide evidence that the one-parameter family of vacua corresponds to marginal deformations of the CFT by computing a three-point function. 
  We study the behavior of the four$-$dimensional Newton's law in warped braneworlds. The setup considered here is a $(3+n)$-brane embedded in $(5+n)$ dimensions, where $n$ extra dimensions are compactified and a dimension is infinite. We show that the wave function of gravity is described in terms of the Bessel functions of $(2+n/2)$-order and that estimate the correction to Newton's law. In particular, the Newton's law for $n=1$ can be exactly obtained. 
  We suggest a simple modification of the usual procedures of analysis for the high-temperature (strong-coupling or hopping-parameter) expansions of the renormalized four-point coupling constant in the fourdimensional phi^4 lattice scalar field theory. As a result we can more convincingly validate numerically the triviality of the continuum limit taken from the high temperature phase. 
  One of the remarkable outcomes of the AdS/CFT correspondence has been the generalization of Cardy's entropy formula for arbitrary dimensionality, as well as a variety of anti-de Sitter scenarios. More recently, related work has been done in the realm of asymptotically de Sitter backgrounds. Such studies presume a well-defined dS/CFT duality, which has not yet attained the credibility of its AdS analogue. In this paper, we derive and interpret generalized forms of the Cardy entropy for a selection of asymptotically de Sitter spacetimes. These include the Schwarzschild-de Sitter black hole (as a review of hep-th/0112093), the Reissner-Nordstrom-de Sitter black hole and a special class of topological de Sitter solutions. Each of these cases is found to have interesting implications in the context of the proposed correspondence. 
  In the Randall-Sundrum (RS) brane-world model a singular delta-function source is matched by the second derivative of the warp factor. So one should take possible curvature corrections in the effective action of the RS models in a Gauss-Bonnet (GB) form. We present a linearized treatment of gravity in the RS brane-world with the Gauss-Bonnet modification to Einstein gravity. We give explicit expressions for the Neumann propagator in arbitrary D dimensions and show that a bulk GB term gives, along with a tower of Kaluza-Klein modes in the bulk, a massless graviton on the brane, as in the standard RS model. Moreover, a non-trivial GB coupling can allow a new branch of solutions with finite Planck scale and no naked bulk singularity, which might be useful to avoid some of the previously known ``no--go theorems'' for RS brane-world compactifications. 
  We discuss general properties of classical string field theories with symmetric vertices in the context of deformation theory. For a given conformal background there are many string field theories corresponding to different decomposition of moduli space of Riemann surfaces. It is shown that any classical open string field theories on a fixed conformal background are $A_\infty$-quasi-isomorphic to each other. This indicates that they have isomorphic moduli space of classical solutions. The minimal model theorem in $A_\infty$-algebras plays a key role in these results. Its natural and geometric realization on formal supermanifolds is also given. The same results hold for classical closed string field theories, whose algebraic structure is governed by $L_\infty$-algebras. 
  Gravitational time delay in asymptotically Anti de Sitter spaces has consequences for holographic duality. We argue that the requirement of bulk causality implies that it is not possible for a collection of boundary observers, performing local measurements, to extract information from precursors. Using similar arguments, we derive an integrated weak energy constraint on spacetimes which can admit a holographic dual. 
  In this paper we further explore the question of topological charge in the center vortex-nexus picture of gauge theories. Generally, this charge is locally fractionalized in units of 1/N for gauge group SU(N), but globally quantized in integral units. We show explicitly that in d=4 global topological charge is a linkage number of the closed two-surface of a center vortex with a nexus world line, and relate this linkage to the Hopf fibration, with homotopy $\Pi_3(S^3)\simeq Z$; this homotopy insures integrality of the global topological charge. We show that a standard nexus form used earlier, when linked to a center vortex, gives rise naturally to a homotopy $\Pi_2(S^2)\simeq Z$, a homotopy usually associated with 't Hooft-Polyakov monopoles and similar objects which exist by virtue of the presence of an adjoint scalar field which gives rise to spontaneous symmetry breaking. We show that certain integrals related to monopole or topological charge in gauge theories with adjoint scalars also appear in the center vortex-nexus picture, but with a different physical interpretation. We find a new type of nexus which can carry topological charge by linking to vortices or carry d=3 Chern-Simons number without center vortices present; the Chern-Simons number is connected with twisting and writhing of field lines, as the author had suggested earlier. In general, no topological charge in d=4 arises from these specific static configurations, since the charge is the difference of two (equal) Chern-Simons number, but it can arise through dynamic reconnection processes. We complete earlier vortex-nexus work to show explicitly how to express globally-integral topological charge as composed of essentially independent units of charge 1/N. 
  We extend some aspects of vacuum string field theory to superstring field theory in Berkovits' formulation, and we study the star algebra in the fermionic matter sector. After clarifying the structure of the interaction vertex in the operator formalism of Gross and Jevicki, we provide an algebraic construction of the supersliver state in terms of infinite-dimensional matrices. This state is an idempotent string field and solves the matter part of the equation of motion of superstring field theory with a pure ghost BRST operator. We determine the spectrum of eigenvalues and eigenvectors of the infinite-dimensional matrices of Neumann coefficients in the fermionic matter sector. We then analyze coherent states based on the supersliver and use them in order to construct higher-rank projector solutions, as well as to construct closed subalgebras of the star algebra in the fermionic matter sector. Finally, we show that the geometric supersliver is a solution to the superstring field theory equations of motion, including the (super)ghost sector, with the canonical choice of vacuum BRST operator recently proposed by Gaiotto, Rastelli, Sen and Zwiebach. 
  Bosonization of the extended Thirring model with SU(2) symmetry in the Minkowski path integral method is discussed. We argue that it is not an easy task to bosonize such a model if we derive correctly the fermion determinant which is induced with the decoupling transformation because it seems that there arise ghost fields. This is contrary to what is shown in a textbook and some people believe. 
  Free field equations, with various spins, for space-time algebras with second-rank tensor (instead of usual vector) momentum are constructed. Similar algebras are appearing in superstring/M theories. The most attention is payed to the gauge invariance properties, particularly the spin two equations with gauge invariance are constructed for dimensions 2+2 and 2+4 and connection to Einstein equation and diffeomorphism invariance is established. 
  The closed string model in the background gravity field and the antisymmetric B-field is considered as the bihamiltonian system in assumption,that string model is the integrable model for particular kind of the background fields. It is shown, that bihamiltonity is origin of two types of the T-duality of the closed string models. The dual nonlocal Poisson brackets, depending of the background fields and of their derivatives, are obtained. The integrability condition is formulated as the compatibility of the bihamoltonity condition and the Jacobi identity of the dual Poisson bracket. It is shown, that the dual brackets and dual hamiltonians can be obtained from the canonical (PB) and from the initial hamiltonian by imposing of the second kind constraints on the initial dynamical system, on the closed string model in the constant background fields, as example. The closed string model in the constant background fields is considered without constraints, with the second kind constraints and with first kind constraints as the B-chiral string. The two particles discrete closed string model is considered as two relativistic particle system to show the difference between the Gupta-Bleuler method of the quantization with the first kind constraints and the quantization of the Dirac bracket with the second kind constraints. 
  Exact $SU(2)\times U(1)$ self-gravitating BPS global monopoles in four dimensions are constructed by dimensional reduction of eight dimensional metrics with $G_2$ holonomy asymptotic to cones over $S^3\times S^3$. The solutions carry two topological charges in an interesting way. They are generically axially but not spherically symmetric. This last fact is related to the isometries and asymptotic topology of the $G_2$ metrics. It is further shown that some $G_2$ metrics known numerically reduce to supersymmetric cosmic strings. 
  We discuss various aspects of gauge theories realized on the world-volume of wrapped branes. In particular we analyze the coupling of SYM operators to space-time fields both in N=1 and N=2 models and give a description of the gluino condensate in the Maldacena-Nunez N=1 solution. We also explore the seven dimensional BPS equations relevant for these solutions and their generalizations. 
  We use effective field theory techniques to examine the quantum corrections to the gravitational metrics of charged particles, with and without spin. In momentum space the masslessness of the photon implies the presence of nonanalytic pieces $\sim \sqrt{-q^2},q^2\log -q^2$ etc. in the form factors of the energy-momentum tensor. We show how the former reproduces the classical non-linear terms of the Reissner-Nordstr\"{o}m and Kerr-Newman metrics while the latter can be interpreted as quantum corrections to these metrics, of order $G\alpha\hbar/mr^3$ 
  We use the nonabelian action of N coincident D(-1) branes in constant background fields, in the N $\to \infty$ limit, to construct noncommutative D-brane actions in an arbitrary noncommutative description and comment on tachyon condensation from this perspective. 
  Some problems on variations are raised for classical discrete mechanics and field theory and the difference variational approach with variable step-length is proposed motivated by Lee's approach to discrete mechanics and the difference discrete variational principle for difference discrete mechanics and field theory on regular lattice. Based upon Hamilton's principle for the vertical variations and double operation of vertical exterior differential on action, it is shown that for both continuous and variable step-length difference cases there exists the nontrivial Euler-Lagrange cohomology as well as the necessary and sufficient condition for symplectic/multi-symplectic structure preserving properties is the relevant Euler-Lagrange 1-form is closed in both continuous and difference classical mechanics and field theory. While the horizontal variations give rise to the relevant identities or relations of the Euler-Lagrange equation and conservation law of the energy/energy-momentum tensor for continuous or discrete systems. The total variations are also discussed. Especially, for those discrete cases the variable step-length of the difference is determined by the relation between the Euler-Lagrange equation and conservation law of the energy/energy-momentum tensor. In addition, this approach together with difference version of the Euler-Lagrange cohomology can be applied not only to discrete Lagrangian formalism but also to the Hamiltonian formalism for difference mechanics and field theory. 
  Dynamics of the multi-component, multi-field quintessence and gravity is formulated as relativistic N-particle dynamics, embedded in a static viscus flat space and under the forces given by an interacting Lorentz scalar potential via exchange of field bosons. The Ratra-Peebles power-law potential of effective single-field quintessence can be derived from this microscopic perspective. In certain situations, the effective dynamics can be made identical to that of the single complex quintessence, except for that the overall U(1) symmetry is not manifestly broken. The present formulation provides a convenient gauge for analyzing the superhorizon perturbations and possibly for quantization of superhorizon fields and gravity. 
  A gauge-invariant color-charge operator is defined and related to an integral of the gauge-invariant chromoelectric field over a closed surface. We discuss the case of a surface all of whose points are a macroscopic distance from a system of quarks and gluons which it entirely surrounds. When this system of quarks and gluons forms a hadron or an object composed of hadrons, such as a nucleus, it is argued that the gauge-invariant color charge enclosed within this surface must vanish and the system of hadrons in the interior of the surface must be a color singlet. 
  We review the implementation, in a temporal-gauge formulation of QCD, of the non-Abelian Gauss's law and the construction of gauge-invariant gauge and matter fields. We then express the QCD Hamiltonian in terms of these gauge-invariant operator-valued fields, and discuss the relation of this Hamiltonian and the gauge-invariant fields to the corresponding quantities in a Coulomb gauge formulation of QCD. We argue that a representation of QCD in terms of gauge-invariant quantities could be particularly useful for understanding low-energy phenomenology. We present the results of an investigation into the topological properties of the gauge-invariant fields, and show that there are Gribov copies of these gauge-invariant gauge fields, which are constructed in the temporal gauge, even though the conditions that give rise to Gribov copies do not obtain for the gauge-dependent temporal-gauge fields. 
  Supersymmetric field theories can be constructed that violate Lorentz and CPT symmetry. We illustrate this with some simple examples related to the original Wess-Zumino model. 
  A simple mechanism for SUSY breaking is proposed due to the coexistence of BPS domain walls. It requires no messenger fields nor complicated SUSY breaking sector on any of the walls. We assumed that our world is on a BPS domain wall and that the other BPS wall breaks the SUSY preserved by our wall. We obtain an ${\cal N}=1$ model in four dimensions which admits an exact solution of a stable non-BPS configuration of two walls. The stability is assured by a topological quantum number associated with the winding on the field space of the topology of $S^1$. We propose that the overlap of the wave functions of the Nambu-Goldstone fermion and those of physical fields provides a practical method to evaluate SUSY breaking mass splitting on our wall thanks to a low-energy theorem. This is based on our recent works hep-th/0009023 and hep-th/0107204. 
  The character of the principal series of representations of SL(n,R) is evaluated by using Gel'fand and Naimark's definition of character. This representation is realized in the space of functions defined on the right coset space of SL(n,R) with respect to the subgroup of real triangular matrices. This form of the representations considerably simplifies the problem of determination of the integral kernel of the group ring which is fundamental in the Gel'fand-Naimark theory of character. An important feature of the principal series of representations is that the `elliptic' elements of SL(n,R) do not contribute to its character. 
  Using the bicomplex approach we discuss a noncommutative system in two--dimensional Euclidean space. It is described by an equation of motion which reduces to the ordinary sine--Gordon equation when the noncommutation parameter is removed, plus a constraint equation which is nontrivial only in the noncommutative case. We show that the system has an infinite number of conserved currents and we give the general recursive relation for constructing them. For the particular cases of lower spin nontrivial currents we work out the explicit expressions and perform a direct check of their conservation. These currents reduce to the usual sine-Gordon currents in the commutative limit. We find classical ``localized'' solutions to first order in the noncommutativity parameter and describe the Backlund transformations for our system. Finally, we comment on the relation of our noncommutative system to the commutative sine-Gordon system. 
  We clarify RR tadpole cancellation conditions for intersecting D4-/D5-branes. We find all of the D4-brane models which have D=4 three-generation chiral fermions with the SU(3)XSU(2)XU(1)^n symmetries. For the D5-brane case, we present a solution to the conditions which gives exactly the matter contents of standard model with U(1) anomalies. 
  Computing all divergent one-loop Green's functions of \theta-expanded noncommutative quantum electrodynamics up to first order in \theta, we show that this model is not renormalizable. The reason is a divergence in the electron four-point function which cannot be removed by field redefinitions. Ignoring this problem, we find however clear hints for new symmetries in massless \theta-expanded noncommutative QED: Four additional divergences which would be compatible with gauge and Lorentz symmetries and which are not reachable by field redefinitions are absent. 
  In Pre-Big-Bang and in Ekpyrotic Cosmology, perturbations on cosmological scales today are generated from quantum vacuum fluctuations during a phase when the Universe is contracting (viewed in the Einstein frame). The backgrounds studied to date do not yield a scale invariant spectrum of adiabatic fluctuations. Here, we present a new contracting background model (neither of Pre-Big-Bang nor of the Ekpyrotic form) involving a single scalar field coupled to gravity in which a scale-invariant spectrum of curvature fluctuations and gravitational waves results. The equation of state of this scalar field corresponds to cold matter. We demonstrate that if this contracting phase can be matched via a nonsingular bounce to an expanding Friedmann cosmology, the scale-invariance of the curvature fluctuations is maintained. We also find new background solutions for Pre-Big-Bang and for Ekpyrotic cosmology, which involve two scalar fields with exponential potentials with background values which are evolving in time. We comment on the difficulty of obtaining a scale-invariant spectrum of adiabatic fluctuations with background solutions which have been studied in the past. 
  We survey some algebraic geometric aspects of mirror symmetry and duality in string theory. Some applications of computer algebra to algebraic geometry and string theory are shortly reviewed. 
  Superconformal transformations are derived for the $\N=2,4 supermultiplets corresponding to the simplest chiral primary operators. These are applied to two, three and four point correlation functions. When $\N=4$, results are obtained for the three point function of various descendant operators, including the energy momentum tensor and SU(4) current. For both $\N=2$ or 4 superconformal identities are derived for the functions of the two conformal invariants appearing in the four point function for the chiral primary operator. These are solved in terms of a single arbitrary function of the two conformal invariants and one or three single variable functions. The results are applied to the operator product expansion using the exact formula for the contribution of an operator in the operator product expansion in four dimensions to a scalar four point function. Explicit expressions representing exactly the contribution of both long and possible short supermultiplets to the chiral primary four point function are obtained. These are applied to give the leading perturbative and large N corrections to the scale dimensions of long supermultiplets. 
  We consider the simplest class of Lie-algebraic deformations of space-time algebra, with the selection of $\kappa$-deformations as providing quantum deformation of relativistic framework. We recall that the $\kappa$-deformation along any direction in Minkowski space can be obtained. Some problems of the formalism of $\kappa$-deformations will be considered. We shall comment on the conformal extension of light-like $\kappa$-deformation as well as on the applications to astrophysical problems. 
  We continue the study of thermodynamics of black holes in de Sitter spaces. In a previous paper (hep-th/0111093), we have shown that the entropy of cosmological horizon in the Schwarzschild-de Sitter solution and topological de Sitter solution can be expressed in a form of the Cardy-Verlinde formula, if one adopts the prescription to compute the gravitational mass from data at early or late time infinity of de Sitter space. However, this definition of gravitational mass cannot give a similar expression like the Cardy-Verlinde formula for the entropy associated with the horizon of black holes in de Sitter spaces. In this paper, we first generalize the previous discussion to the case of Reissner-Nordstr\"om-de Sitter and Kerr-de Sitter solutions. Furthermore, we find that the entropy of black hole horizon can also be rewritten in terms of the Cardy-Verlinde formula for these black holes in de Sitter spaces, if we use the definition due to Abbott and Deser for conserved charges in asymptotically de Sitter spaces. We discuss the implication of our result. In addition, we give the first law of de Sitter black hole mechanics. 
  We study the collision between a BTZ black hole and a test particle coupled to a scalar field. We compute the power spectrum, the energy radiated and the plunging waveforms for this process. We show that for late times the signal is dominated by the quasinormal ringing. In terms of the AdS/CFT correspondence the bulk gravity process maps into a thermal state, an expanding bubble and gauge particles decaying into bosons of the associated operator. These latter thermalize in a timescale predicted by the bulk theory. 
  We study the principles of the gauge symmetry and supersymmetry breaking due to the local or global discrete symmetries on the extra space manifold. We show that the gauge symmetry breaking by Wilson line is the special case of the discrete symmetry approach where all the discrete symmetries are global and act freely on the extra space manifold. As applications, we discuss the N=2 supersymmetric SO(10) and $E_8$ breaking on the space-time $M^4\times A^2$ and $M^4\times D^2$, and point out that similarly one can study any N=2 supersymmetric $SO(M)$ breaking. We also comment on the one-loop effective potential, the possible questions and generalization. 
  We study the features of the vacuum of the harmonic oscillator in the Moyal quantization. Two vacua are defined, one with the normal ordering and the other with the Weyl ordering. Their equivalence is shown by using a differential equation satisfied by the normal ordered vacuum. 
  The Ryder relation between left- and right- spinors has been generalized in my previous works. On this basis Ahluwalia presented a physical content following from this generalization. It is related to non-locality. A similar conclusion can be drawn on the basis of a generalization of the Sakurai-Gersten consideration. I correct several calculating and conceptual misunderstandings of the previous works. 
  We argue that multi-trace interactions in quantum field theory on the boundary of AdS space can be incorporated in the AdS/CFT correspondence by using a more general boundary condition for the bulk fields than has been considered hitherto. We illustrate the procedure for a renormalizable four-dimensional field theory with a $(\Tr \Phi^2)^2$ interaction. In this example, we show how the AdS fields with the appropriate boundary condition reproduce the renormalization group effects found in the boundary field theory. We also construct in related examples a line of fixed points with a nonperturbative duality, and a flow between two methods of quantization. 
  Based on the twisted R-R tadpole cancellation conditions at the singularities of D=4 Type IIB orbifold $T^6/ Z_3$, we propose a new bottom-up approach to embed standard model with three generations into string theory. 
  We discuss some aspects of the topological features of a non-interacting two (1+1)-dimensional Abelian gauge theory in the framework of superfield formalism. This theory is described by a BRST invariant Lagrangian density in the Feynman gauge. We express the local and continuous symmetries, Lagrangian density, topological invariants and symmetric energy momentum tensor of this theory in the language of superfields by exploiting the nilpotent (anti-)BRST- and (anti-)co-BRST symmetries. In particular, the Lagrangian density and symmetric energy momentum tensor of this topological theory turn out to be the sum of terms that geometrically correspond to the translations of some local superfields along the Grassmannian directions of the four (2+2)-dimensional supermanifold. In this interpretation, the (anti-)BRST- and (anti-)co-BRST symmetries, that emerge after the imposition of the (dual) horizontality conditions, play a very important role. 
  Nature's attraction to unique mathematical structures provides powerful hints for unraveling her mysteries. None is at present as intriguing as eleven-dimensional M-theory. The search for exceptional structures specific to eleven dimensions leads us to exceptional groups in the description of space-time. One specific connection, through the coset $F_4/SO(9)$, may provide a generalization of eleven-dimensional supergravity. Since this coset happens to be the projective space of the Exceptional Jordan Algebra, its charge space may be linked to the fundamental degrees of freedom underlying M-theory. 
  A covariant approach to the conformal property associated with Moyal-Lax operators is given. By identifying the conformal covariance with the second Gelfand-Dickey flow, we covariantize Moyal-Lax operators to construct the primary fields of one-parameter deformation of classical $W$-algebras. 
  The prequantization map for a Poisson-Gerstenhaber algebra of dynamical variables represented by differential forms within the polysymplectic formulation of the De Donder--Weyl covariant Hamiltonian field theory is presented and the corresponding prequantum Schroedinger equation for a non-homogeneous form valued wave function is derived. This is the first step toward understanding the procedures of covariant precanonical field quantization from the point of view of geometric quantization. 
  Double-trace deformations of the AdS/CFT duality result in a new perturbation expansion for string theory, based on a non-local worldsheet. We discuss some aspects of the deformation in the low energy gravity approximation, where it appears as a change in the boundary condition of fields. We relate unique features of the boundary of AdS to the worldsheet becoming non-local, and conjecture that non-local worldsheet actions may be generic in other classes of backgrounds. 
  The review of recent results in the s=1/2 quantum spin chains with $1/\sinh^2(\kappa r$ exchange is presented. Related problems in the theory of classical and quantum Calogero-Sutherland-Moser systems with inverse square hyperbolic and elliptic potentials are discussed. The attention is paid to finding the explicit form of corresponding Bethe-Ansatz equations and to connection with generalized Hubbard chains in one dimension. 
  It is shown that elementary black hole can not be distinguished from an elementary particle in the non-commutative space-times (space/space and space/time) at the Planck scale. But, the non-commutative space-times can not be ``directly'' measured in the elementary black hole system. A time-varying non-commutative parameter $\theta(t)$ is suggested in accordance with the time-varying-G scenario. By identifying the elementary black hole with an elementary particle, the large hierarchy between the weak scale and Planck scale is naturally understood. For large black hole, the discrete spectrum of the horizon area is derived by identifying the black hole horizon with a non-commutative sphere. 
  The non-abelian Chern-Simons field interacting with $N$ component complex field is treated as a constrained system using the Hamilton-Jacobi approach. The reduced phase space Hamiltonian density is obtained without introducing Lagrange multipliers and with out any additional gauge fixing condition. The quantization of this system is also discussed. 
  We investigate the slicing dependence of the relationship between conserved quantities in the (A)dS/CFT correspondence. Specifically, we show that the Casimir energy depends upon the topology and geometry of spacetime foliations of the bulk near the conformal boundary. We point out that the determination of the brane location in brane-world scenarios exhibits a similar slicing dependence, and we comment on this in the context of the AdS/CFT correspondence conjecture. 
  Maxwell-Chern-Simons models in the presence of an instanton anti-instanton background are studied. The saddle-point configuration corresponds to the creation and annihilation of a vortex localized around the Dirac string needed to support the nontrivial background. This configuration is generalized to the case in which a nonlocal Maxwell term is allowed in order to fulfill the finite action requirement. Following 't Hooft procedure, we compute the vortex correlation functions and we study the possibility of obtaining spin 1/2 excitations. A possible connection with the bosonization of interacting three-dimensional massive fermionic systems is also discussed. 
  A 3-dimensional model dual to the Rozansky-Witten topological sigma-model with a hyper-Kaehler target space is considered. It is demonstrated that a Feynman diagram calculation of the classical part of its partition function yields the Milnor linking number. 
  The Landau problem is discussed in two similar but still different non-commutative frameworks. The ``standard'' one, where the coupling to the gauge field is achieved using Poisson brackets, yields all Landau levels. The ``exotic'' approach, where the coupling to the gauge field is achieved using the symplectic structure, only yields lowest-Landau level states, as advocated by Peierls and used in the description of the ground states of the Fractional Quantum Hall Effect. The same reduced model also describes vortex dynamics in a superfluid ${}^4$He film. Remarkably, the spectrum depends crucially on the quantization scheme. The system is symmetric w. r. t. area-preserving diffeomorphisms. 
  The integrable XXX spin s quantum chain and the alternating $s^{1}$, $s^{2}$ ($s^{1}-s^{2}={1\over 2}$) chain with boundaries are considered. The scattering of their excitations with the boundaries via the Bethe ansatz method is studied, and the exact boundary S matrices are computed in the limit $s, s^{1, 2} \to \infty$. Moreover, the connection of these models with the SU(2) Principal Chiral, WZW and the RSOS models is discussed. 
  We study the Hawking-Turok (HT) instanton solutions which have been employed to describe the creation of an open inflationary universe, in the context of higher derivative theories. We consider the effects of adding quadratic and cubic terms of the forms $\alpha R^{2}$ and $\beta R^{3}$ to the gravitational action. Using a conformal transformation to convert the higher derivative theories into theories of self interacting scalar fields minimally coupled to Einstein gravity, we argue that the cubic term represents a generic perturbation of the polynomial type to the action and obtain the conditions on the parameters of these theories for the existence of singular and non-singular instanton solutions. We find that, relative to the quadratic case, there are significant changes in the nature of the constraints on the parameters for the existence of these instantons, once cubic (and higher order perturbations) are added to the action. 
  It has been argued by Witten and others that in the presence of a nontrivial B-field, D-brane charges in type IIB string theories are measured by twisted K-theory. In joint work with Bouwknegt, Carey and Murray it was proved that twisted K-theory is canonically isomorphic to bundle gerbe K-theory, whose elements are ordinary vector bundles on a principal projective unitary bundle, with an action of the bundle gerbe determined by the principal projective unitary bundle. The principal projective unitary bundle is in turn determined by the twist. In this paper, we study in more detail the Chern-Weil representative of the Chern character of bundle gerbe K-theory that was introduced previously, and we also extend it to the equivariant and holomorphic cases. Included is a discussion of interesting examples. 
  Evaluation of variation of a Green's function in a gauge field theory with a gauge parameter theta involves field transformations that are (close to) singular. Recently, we had demonstrated {hep-th/0106264}some unusual results that follow from this fact for an interpolating gauge interpolating between the Feynman and the Coulomb gauge (formulated by Doust). We carry out further studies of this model. We study properties of simple loop integrals involved in an interpolating gauge. We find that the proof of continuation of a Green's function from the Feynman gauge to the Coulomb gauge via such a gauge in a gauge-invariant manner seems obstructed by the lack of differentiability of the path-integral with respect to theta (at least at discrete values for a specific Green's function considered) and/or by additional contributions to the WT-identities. We show this by the consideration of simple loop diagrams for a simple scattering process. The lack of differentiability, alternately, produces a large change in the path-integral for a small enough change in theta near some values. We find several applications of these observations in a gauge field theory. We show that the usual procedure followed in the derivation of the WT-identity that leads to the evaluation of a gauge variation of a Green's function involves steps that are not always valid in the context of such interpolating gauges. We further find new results related to the need for keeping the epsilon-term in the in the derivation of the WT-identity and and a nontrivial contribution to gauge variation from it. We also demonstrate how arguments using Wick rotation cannot rid us of these problems. This work brings out the pitfalls in the use of interpolating gauges in a clearer focus. 
  First order rotational perturbations of the Friedmann-Robertson-Walker metric are considered in the framework of the brane-world cosmological models. A rotation equation, relating the perturbations of the metric tensor to the angular velocity of the matter on the brane is derived under the assumption of slow rotation. The mathematical structure of the rotation equation imposes strong restrictions on the temporal and spatial dependence of the brane matter angular velocity. The study of the integrable cases of the rotation equation leads to three distinct models, which are considered in detail. As a general result we find that, similarly to the general relativistic case, the rotational perturbations decay due to the expansion of the matter on the brane. One of the obtained consistency conditions leads to a particular, purely inflationary brane-world cosmological model, with the cosmological fluid obeying a non-linear barotropic equation of state. 
  (2+1)-dimensional Georgi-Glashow model is explored in the regime when the Higgs boson is not infinitely heavy, but its mass is rather of the same order of magnitude as the mass of the W boson. In the weak-coupling limit, the Debye mass of the dual photon and the expression for the monopole potential are found. The cumulant expansion applied to the average over the Higgs field is checked to be convergent for the known data on the monopole fugacity. These results are further generalized to the SU(N)-case. In particular, it is found that the requirement of convergence of the cumulant expansion establishes a certain upper bound on the number of colours. This bound, expressed in terms of the parameter of the weak-coupling approximation, allows the number of colours to be large enough. Finally, the string tension and the coupling constant of the so-called rigidity term of the confining string are found at arbitrary number of colours. 
  The kinematical part of general theory of deformational structures on smooth manifolds is developed. We introduce general concept of d-objects deformation, then within the set of all such deformations we develop some special algebra and investigate group and homotopical properties of the set. In case of proper deformations some propositions, generalizing isometry theory on Riemannian manifolds are formulated. 
  Using the data of eigenvalues and eigenvectors of Neumann matrices in the 3-string vertex, we prove analytically that the ghost kinetic operator of vacuum string field theory obtained by Hata and Kawano is equal to the ghost operator inserted at the open string midpoint. We also comment on the values of determinants appearing in the norm of sliver state. 
  We construct a string theory realization of the 4+1d quantum Hall effect recently discovered by Zhang and Hu. The string theory picture contains coincident D4-branes forming an S^4 and having D0-branes (i.e. instantons) in their world-volume. The charged particles are modelled as string ends. Their configuration space approaches in the large n limit a CP^3, which is an S^2 fibration over S^4, the extra S^2 being made out of the Chan-Paton degrees of freedom. An alternative matrix theory description involves the fuzzy four-sphere. We also find that there is a hierarchy of quantum Hall effects in odd-dimensional spacetimes, generalizing the known cases in 2+1d and 4+1d. 
  We investigate the effect of the minimal length uncertainty relation, motivated by perturbative string theory, on the density of states in momentum space. The relation is implemented through the modified commutation relation [x_i,p_j]=i hbar[(1 + beta p^2) delta_{ij} + beta' p_i p_j]. We point out that this relation, which is an example of an UV/IR relation, implies the finiteness of the cosmological constant. While our result does not solve the cosmological constant problem, it does shed new light on the relation between this outstanding problem and UV/IR correspondence. We also point out that the blackbody radiation spectrum will be modified at higher frequencies, but the effect is too small to be observed in the cosmic microwave background spectrum. 
  The spectrum of quenched Yang-Mills theory in the large-N limit displays strings and higher dimensional extended objects. The effective dynamics of string-like excitations is encoded into area preserving Schild action. In this letter, we bridge the gap between SU(N) gauge models and fully reparametrization invariant Nambu-Goto string models by introducing an extra matrix degree of freedom in the Yang-Mills action. In the large-N limit this matrix variable becomes the world-sheet auxiliary field allowing a smooth transition between the Schild and Nambu-Goto strings. The new improved matrix model we propose here can be extended to p-branes provided we enlarge the dimensionality of the target spacetime. 
  The strong form of the AdS/CFT correspondence implies that the leading $N$ expressions for the connected correlation functions of the gauge invariant operators in the free ${\cal N}=4$ supersymmetric Yang-Mills theory with the gauge group SU(N) correspond to the boundary S matrix of the classical interacting theory in the Anti de Sitter space. It was conjectured recently that the theory in the bulk should be a local theory of infinitely many higher spin fields. In this paper we study the free higher spin fields ($N=\infty$) corresponding to the free scalar fields on the boundary. We explicitly construct the boundary to bulk propagator for the higher spin fields and show that the classical solutions in the bulk are in one to one correspondence with the deformations of the free action on the boundary by the bilinear operators. We also discuss the constraints on the correlation functions following from the higher spin symmetry. We show that the higher spin symmetries fix the correlation functions up to the finite number of parameters. We formulate sufficient conditions for the bulk theory to reproduce the free field correlation functions on the boundary. 
  A new concept for the geometrisation of electromagnetic interaction is proposed. Instead of the concept "extended field--point sources", interacting Maxwell's and Dirac's fields are considered as a unified closed noneuclidean and nonriemannean space--time 4-manifold. This manifold can be considered as geometrical realisation of the "dressed electron" idea. Within this approach, the Dirac equation proves to be a relation that accounts for topological and metric characteristics of this manifold. Dirac's spinors serve as basis vectors of its fundamental group representation, while the electromagnetic field components prove to be components of a curvature tensor of the manifold covering space. Energy, momentum components, mass, charge, spin and particle--antiparticle states appear to be geometrical characteristics of the above manifold. 
  Explicit solutions for one completely-integrable system of Calogero-Moser type in external fields are found in case of three and four interacting particles. Relation between coupling constant, initial values of coordinates and time of falling into the singularity of potential is derived. 
  We discuss a number of exact results in N=1 supersymmetric field theories. We review the results obtained by Seiberg in Super-Yang-Mills (SYM) theories with matter in fundamental representation. We then consider Kutasov-type SYM theories, which also contain matter in the adjoint representation and an appropriate tree--level superpotential. We finally focus on one particular case in the latter theories, a generalization of the theories with equal number of flavors and colors studied by Seiberg, in which non--trivial superconformal theories appear at certain sections of the quantum--modified moduli space. Throughout the paper we stress the role played by duality in the search for exact results. 
  We construct consistent interacting gauge theories for M conformal massless spin-2 fields ("Weyl gravitons") with the following properties: (i) in the free limit, each field fulfills the equation ${\cal B}^{\mu \nu} = 0$, where ${\cal B}^{\mu \nu}$ is the linearized Bach tensor, (ii) the interactions contain no more than four derivatives, just as the free action and (iii) the internal metric for the Weyl gravitons is not positive definite. The interacting theories are obtained by gauging appropriate non-semi-simple extensions of the conformal algebra $so(4,2)$ with commutative, associative algebras of dimension M. By writing the action in terms of squares of supercurvatures, supersymmetrization is immediate and leads to consistent conformal supergravities with M interacting gravitons. 
  The statics and dynamics of a surface separating two phases of a relativistic quantum field theory at or near the critical temperature typically make use of a free energy as a functional of an order parameter. This free energy functional also affords an economical description of states away from equilibrium. The similarities and differences between using a scalar field as the order parameter versus the energy density are examined, and a peculiarity is noted. We also point out several conceptual errors in the literature dealing with the dynamical prefactor in the nucleation rate. 
  We consider noncommutative gauge theories which have zero mass states propagating along both commutative and noncommutative dimensions. Solitons in these theories generically carry U(m) gauge group on their world-volume. From the point of view of string theory, these solitons correspond to  ``branes within branes''. We show that once the world-volume U(m) gauge theory is in the Higgs phase, light states become quasi-localized, rather than strictly localized on the soliton, i.e. they mix with light bulk modes and have finite widths to escape into the noncommutative dimensions. At small values of U(m) symmetry breaking parameters, these widths are small compared to the corresponding masses. Explicit examples considered are adjoint scalar field in the background of a noncommutative vortex in U(1)-Higgs theory, and gauge fields in instanton backgrounds in pure gauge noncommutative theories. 
  We investigate the ABJ anomaly in the framework of an effective field theory for a 3-brane scenario and show that the contribution from induced gravity on the brane depends on both the topological structure of the bulk space-time and the embedding of the brane in the bulk. This fact implies the existence of a non-trivial vacuum structure of bulk quantum gravity. Furthermore, we argue that this axial gravitational anomaly may not necessarily be cancelled by choosing the matter content on the brane since it could be considered as a possible effect from bulk quantum gravity. 
  We compute the three-graviton tree amplitude in Type IIB superstring theory compactified to six dimensions using the manifestly (6d) supersymmetric Berkovits-Vafa-Witten worldsheet variables. We consider two cases of background geometry: the flat space example R6xK3, and the curved example AdS3xS3xK3 with Ramond flux, and compute the correlation functions in the bulk. 
  We study novel type IIB compactifications on the T^6/Z_2 orientifold. This geometry arises in the T-dual description of Type I theory on T^6, and one normally introduces 16 space-filling D3-branes to cancel the RR tadpoles. Here, we cancel the RR tadpoles either partially or fully by turning on three-form flux in the compact geometry. The resulting (super)potential for moduli is calculable. We demonstrate that one can find many examples of N=1 supersymmetric vacua with greatly reduced numbers of moduli in this system. A few examples with N>1 supersymmetry or complete supersymmetry breaking are also discussed. 
  Orientifolds with three-form flux provide some of the simplest string examples of warped compactification. In this paper we show that some models of this type have the unusual feature of D=4, N=3 spacetime supersymmetry. We discuss their construction and low energy physics. Although the local form of the moduli space is fully determined by supersymmetry, to find its global form requires a careful study of the BPS spectrum. 
  Five nontrivial stationary points are found for maximal gauged N=16 supergravity in three dimensions with gauge group $SO(8)\times SO(8)$ by restricting the potential to a submanifold of the space of $SU(3)\subset(SO(8)\times SO(8))_{\rm diag}$ singlets. The construction presented here uses the embedding of $E_{7(+7)}\subset E_{8(+8)}$ to lift the analysis of N=8, D=4 supergravity performed by N. Warner to N=16, D=3, and hence, these stationary points correspond to some of the known extrema of gauged N=8, D=4 supergravity. 
  It is shown that the matrix models which give non-perturbative definitions of string and M theory may be interpreted as non-local hidden variables theories in which the quantum observables are the eigenvalues of the matrices while their entries are the non-local hidden variables. This is shown by studying the bosonic matrix model at finite temperature, with T taken to scale as 1/N. For large N the eigenvalues of the matrices undergo Brownian motion due to the interaction of the diagonal elements with the off diagonal elements, giving rise to a diffusion constant that remains finite as N goes to infinity. The resulting probability density and current for the eigenvalues are then found to evolve in agreement with the Schroedinger equation, to leading order in 1/N. The quantum fluctuations and uncertainties in the eigenvalues are then consequences of ordinary statistical fluctuations in the values of the off-diagonal matrix elements. This formulation of the quantum theory is background independent, as the definition of the thermal ensemble makes no use of a particular classical solution. The derivation relies on Nelson's stochastic formulation of quantum theory, which is expressed in terms of a variational principle. 
  We present three lectures on heterotic M-theory and a fourth lecture extending this theory to more general orbifolds. In Lecture 1, Horava-Witten theory is briefly discussed. We then compactify this theory on Calabi-Yau threefolds, choosing the "standard" embedding of the spin connection in the gauge connection. We derive, in detail, both the five-dimensional effective action and the associated actions of the four-dimensional "end-of-the-world" branes. Lecture 2 is devoted to showing that this theory naturally admits static, N=1 supersymmetry preserving BPS three-branes, the minimal vacuum having two such branes. One of these, the "visible" brane, is shown to support a three-generation E_6 grand unified theory, whereas the other emerges as the "hidden" brane with unbroken E_8 gauge group. Thus heterotic M-theory emerges as a fundamental paradigm for so-called "brane world" scenarios of particle physics. In Lecture 3, we introduce the concept of "non-standard" embeddings. These are shown to permit a vast generalization of allowed vacua, leading on the visible brane to new grand unified theories, such as SO(10) and SU(5), and to the standard model SU(3)_C X SU(2)_L X U(1)_Y. It is demonstrated that non-standard embeddings generically imply the existence of five-branes in the bulk space. The physical properties of these bulk branes is discussed in detail. Finally, in Lecture 4 we move beyond Horava-Witten theory and consider orbifolds larger than S^1/Z_2. For explicitness, we consider M-theory orbifolds on S^1/Z_2 X T^4/Z_2, discussing their anomaly structure in detail and completely determining both the untwisted and twisted sector spectra. 
  The model of the bosonic string with the noncommutative world-sheet geometry is proposed in the framework of Fedosov's deformation quantization. The re-interpretation of the model in terms of bosonic string coupled to infinite multiplet of background fields is given and the link to the W-symmetry is discussed. For the case of d=4 Euclidean target space, the stringy counterparts of the Yang-Mills instantons are constructed in both commutative and noncommutative regimes. 
  It is shown that quantum mechanics on noncommutative (NC) spaces can be obtained by canonical quantization of some underlying constrained systems. Noncommutative geometry arises after taking into account the second class constraints presented in the models. It leads, in particular, to a possibility of quantization in terms of the initial NC variables. For a two-dimensional plane we present two Lagrangian actions, one of which admits addition of an arbitrary potential. Quantization leads to quantum mechanics with ordinary product replaced by the Moyal product. For a three-dimensional case we present Lagrangian formulations for a particle on NC sphere as well as for a particle on commutative sphere with a magnetic monopole at the center, the latter is shown to be equivalent to the model of usual rotor. There are several natural possibilities to choose physical variables, which lead either to commutative or to NC brackets for space variables. In the NC representation all information on the space variable dynamics is encoded in the NC geometry. Potential of special form can be added, which leads to an example of quantum mechanics on the NC sphere. 
  We present a construction of open-closed string field theory based on disc and RP2 geometries. Finding an appropriate BRS operator in the case of the RP2 geometry, we generalize the background independent open string field theory (or boundary string field theory) of Witten on a unit disc. The coupling constant flow at the closed string side is driven by the scalar operator inserted at the nontrivial loop of RP2. We discuss the off-shell extension of the boundary/crosscap states. Our construction provides an interpolation of orientifold planes of various dimensions as well as that of D-branes. 
  According to the holography principle (due to G.`t Hooft, L. Susskind, J. Maldacena, et al.), quantum gravity and string theory on certain manifolds with boundary can be studied in terms of a conformal field theory on the boundary. Only a few mathematically exact results corroborating this exciting program are known. In this paper we interpret from this perspective several constructions which arose initially in the arithmetic geometry of algebraic curves. We show that the relation between hyperbolic geometry and Arakelov geometry at arithmetic infinity involves exactly the same geometric data as the Euclidean AdS_3 holography of black holes. Moreover, in the case of Euclidean AdS_2 holography, we present some results on bulk/boundary correspondence where the boundary is a non-commutative space. 
  We present orientifold models of type IIA string theory with D8-branes compactified on a two torus times a four dimensional orbifold. The orientifold group is chosen such that one coordinate of the two torus is reversed when applying worldsheet parity. RR tadpole cancellation requires D8-branes which wrap 1-cycles on the two torus and transform non-trivially under the orbifold group. These models are T-dual to orientifolds with D4-branes only which admit large volume compactifications. The intersections of the D8-branes are chosen in such a way that supersymmetry is broken in the open string sector and chiral fermions arise. Stability of the models is discussed in the context of NSNS tadpoles. Two examples with the SM gauge group and two left-right symmetric models are given. 
  I present a summary of the recent progress made in field and string theory which has led to a reformulation of quantum-group polynomial invariants for knots and links into new polynomial invariants whose coefficients can be described in topological terms. The approach opens a new point of view in the theory of knot and link invariants. 
  We discuss the possibility of a transition from a contracting flat space - big crunch - to an expanding flat space - big bang. 
  We present a new scenario for baryogenesis in the context of heterotic brane-world models. The baryon asymmetry of the universe is generated at a small-instanton phase transition which is initiated by a moving brane colliding with the observable boundary. We demonstrate, in the context of a simple model, that reasonable values for the baryon asymmetry can be obtained. As a byproduct we find a new class of moving-brane cosmological solutions in the presence of a perfect fluid. 
  We derive the corrections to the Friedmann equation of order rho^2 in the Randall-Sundrum (RS) model, where two 3-branes bound a slice of five-dimensional Anti-deSitter space. The effects of radion stabilization by the Goldberger-Wise mechanism are taken into account. Surprisingly, we find that an inflaton on either brane will experience no order rho^2 corrections in the Hubble rate H due to its own energy density, although an observer on the opposite brane does see such a correction. Thus there is no enhancement of the slow-roll condition unless inflation is simultaneously driven by inflatons on both branes. Similarly, during radiation domination, the rho^2 correction to H on a given brane vanish unless there is nonvanishing energy density on the opposite brane. During the electroweak phase transition the correction can be large, but is has the wrong sign for causing sphalerons to go out of thermal equilibrium, so it cannot help electroweak baryogenesis. We discuss the differences between our results and exact solutions in RS-II cosmology. 
  We propose that higher-dimensional extended objects (p-branes) are created by super-Planckian scattering processes in theories with TeV scale gravity. As an example, we compute the cross section for p-brane creation in a (n+4)-dimensional spacetime with asymmetric compactification. We find that the cross section for the formation of a brane which is wounded on a compact submanifold of size of the fundamental gravitational scale is larger than the cross section for the creation of a spherically symmetric black hole. Therefore, we predict that branes are more likely to be created than black holes in super-Planckian scattering processes in these manifolds. The higher rate of p-brane production has important phenomenological consequences, as it significantly enhances possible detection of non-perturbative gravitational events in future hadron colliders and cosmic rays detectors. 
  We show that any generally covariant coupling of matter fields to gravity gives rise to a conserved, on-shell symmetric energy-momentum tensor equivalent to the canonical energy-momentum tensor of the flat-space theory. For matter fields minimally coupled to gravity our algorithm gives the conventional Belinfante tensor. We establish that different matter-gravity couplings give metric energy-momentum tensors differing by identically conserved tensors. We prove that the metric energy-momentum tensor obtained from an arbitrary gravity theory is on-shell equivalent to the canonical energy-momentum tensor of the flat-space theory. 
  The problem of dynamical chiral symmetry breaking (DCSB) in multidimensional quantum electrodynamics (QED) is considered. It is shown that for six-dimensional QED the phenomenon of DSCB exists in ladder model for any coupling. 
  Using algebraic Bethe ansatz and the solution of the quantum inverse scattering problem, we compute compact representations of the spin-spin correlation functions of the XXZ-1/2 Heisenberg chain in a magnetic field. At lattice distance m, they are typically given as the sum of m terms. Each term n of this sum, n = 1,...,m is represented in the thermodynamic limit as a multiple integral of order 2n+1; the integrand depends on the distance as the power m of some simple function. The root of these results is the derivation of a compact formula for the multiple action on a general quantum state of the chain of transfer matrix operators for arbitrary values of their spectral parameters. 
  The effective equations of motion for a point charged particle taking account of radiation reaction are considered in various space-time dimensions. The divergencies steaming from the pointness of the particle are studied and the effective renormalization procedure is proposed encompassing uniformly the cases of all even dimensions. It is shown that in any dimension the classical electrodynamics is a renormalizable theory if not multiplicatively beyond d=4. For the cases of three and six dimensions the covariant analogs of the Lorentz-Dirac equation are explicitly derived. 
  We introduce and study new integrable models of A_n^{(1)}-Non-Abelian Toda type which admit U(1)\otimes U(1) charged topological solitons. They correspond to the symmetry breaking SU(n+1) \to SU(2)\otimes SU(2)\otimes U(1)^{n-2} and are conjectured to describe charged dyonic domain walls of N=1 SU(n+1) SUSY gauge theory in large n limit.   It is shown that this family of relativistic IMs corresponds to the first negative grade q={-1} member of a dyonic hierarchy of generalized cKP type. The explicit relation between the 1-soliton solutions (and the conserved charges as well) of the IMs of grades q=-1 and q=2 is found. The properties of the IMs corresponding to more general symmetry breaking SU(n+1) \to SU(2)^{\otimes p}\otimes U(1)^{n-p} as well as IM with global SU(2) symmetries are discussed. 
  We calculate the subleading terms in the Born--Oppenheimer expansion for the effective zero-mode Hamiltonian of N = 1, d=4 supersymmetric Yang--Mills theory with any gauge group. The Hamiltonian depends on 3r abelian gauge potentials A_i, lying in the Cartan subalgebra, and their superpartners (r being the rank of the group). The Hamiltonian belongs to the class of N = 2 supersymmetric QM Hamiltonia constructed earlier by Ivanov and I. Its bosonic part describes the motion over the 3r--dimensional manifold with a special metric. The corrections explode when the root forms \alpha_j(A_i) vanish and the Born--Oppenheimer approximation breaks down. 
  It is shown that the N-th power of the light-cone evolution operator of 2N-periodic quantum discrete Liouville model can be identified with the Dehn twist operator in quantum Teichmuller theory. 
  We show that the apparently periodic Charap-Duff Yang-Mills `instantons' in time-compactified Euclidean Schwarzschild space are actually time independent. For these solutions, the Yang-Mills potential is constant along the time direction (no barrier) and therefore, there is no tunneling. We also demonstrate that the solutions found to date are three dimensional monopoles and dyons. We conjecture that there are no time-dependent solutions in the Euclidean Schwarzschild background. 
  The embedding of a thick de Sitter 3-brane into a five-dimensional bulk is studied, assuming a scalar field with potential is present in the bulk. A class of solutions is found in closed form that can represent a thick de Sitter 3-brane interpolating either between two dynamical black holes with a $R \times S_{4}$ topology or between two Rindler-like spacetimes with a $R_{2}\times S_{3}$ topology. The gravitational field is localized in a small region near the center of the 3-brane. The analysis of graviton fluctuations shows that a zero mode exists and separates itself from a set of continuous modes by a mass gap. The existence of such a mass gap is shown to be universal. The scalar perturbations are also studied and shown to be stable. 
  Using the recently discovered Clifford statistics we propose a simple model for the grand canonical ensemble of the carriers in the theory of fractional quantum Hall effect. The model leads to a temperature limit associated with the permutational degrees of freedom of such an ensemble.  We also relate Schur's theory of projective representations of the permutation groups to physics, and remark on possible extensions of the second quantization procedure. 
  This short note was born out of discussions on anyons in the FQHE at the YITP workshop "Fundamental problems of quantum field theories" (December 19-21, 2001, YITP, Kyoto). At that time, I felt that there might not be a sound consensus of opinion on the subject. Now, I would like to show my understanding here, the essential part of which is based on hep-th/0110197.   The first problem discussed is a notion of ``bosonized electrons (bosonization)", which are unphysical objects from the standpoint of the Chern-Simons gauge invariance. Therefore, their condensate is merely of mathematical concept and the true physical state realized is a liquid-like one made of degeneration of anyons. Based on this recognition, I argue about a mechanism of the degeneration that results from the genuine CS gauge field theory. It is noted that the Ginzburg-Landau effective theory is not necessary. As the last problem, the gauge invariance in the "composite-fermion theories" is discussed. 
  In this note we construct exact solution of Berkovits'superstring field theory. 
  We present a short review of the recent 5D self-tuning solution of the cosmological constant problem with $1/H^2$ term, and present the dual description of the solution. In the dual description, we show that the presence of the coupling of the dual field($\sigma$) to the brane(which is a bit different from the original theory) maintains the self-tuning property. 
  SUSY partnership between singular potentials often breaks down. Via regularization it can be restored on certain ad hoc subspaces of Hilbert space [Das and Pernice, Nucl. Phys. B 561 (1999) 357]. Within the naturally complexified (so called PT symmetric) quantum mechanics we show how SUSY between strongly singular harmonic oscillators can completely be re-established. Our recipe leads to a new form of the bosonic creation and annihilation operators and proves continuous near the usual regular (i.e., linear harmonic) limit. 
  It is argued that the derivative expansion is a suitable method to deal with finite temperature field theory, if it is restricted to spatial derivatives only. Using this method, a simple and direct calculation is presented for the radiatively induced Chern-Simons--like piece of the effective action of (2+1)-dimensional fermions at finite temperature coupled to external gauge fields. The gauge fields are not assumed to be subjected to special constraints, and in particular, they are not required to be stationary nor Abelian. 
  We obtain the magnetic counterpart of the BTZ solution, i.e., the rotating spacetime of a point source generating a magnetic field in three dimensional Einstein gravity with a negative cosmological constant. The static (non-rotating) magnetic solution was found by Clement, by Hirschmann and Welch and by Cataldo and Salgado. This paper is an extension of their work in order to include (i) angular momentum, (ii) the definition of conserved quantities (this is possible since spacetime is asymptotically anti-de Sitter), (iii) upper bounds for the conserved quantities themselves, and (iv) a new interpretation for the magnetic field source. We show that both the static and rotating magnetic solutions have negative mass and that there is an upper bound for the intensity of the magnetic field source and for the value of the angular momentum. The magnetic field source can be interpreted not as a vortex but as being composed by a system of two symmetric and superposed electric charges, one of the electric charges is at rest and the other is spinning. The rotating magnetic solution reduces to the rotating uncharged BTZ solution when the magnetic field source vanishes. 
  A two-dimensional nonlinear gauge theory that can be proposed for generalization to higher dimensions is derived by means of cohomological arguments. 
  We show that squeezed state solutions for solitonic lumps in Vacuum String Field Theory still exist in the presence of a constant B field. We show in particular that, just as in the B=0 case, we can write down a compact explicit form for such solutions. 
  The existence of supersymmetric D0-branes at an arbitrary distance from a flux 5-brane is proven. The physical picture in type IIA is consistent with the Kaluza-Klein reduction origin of the flux 5-brane. An analysis of the fluctuations around these vacua is performed and the backreaction of the D0-branes computed. Non-threshold D0-D2 bound states and similar stabilization mechanisms for D2-branes in such backgrounds are also briefly discussed. 
  In 1981, covariantly constant spinors were introduced into Kaluza-Klein theory as a way of counting the number of supersymmetries surviving compactification. These are related to the {\it holonomy} group of the compactifying manifold. The first non-trivial example was provided in 1982 by D=11 supergravity on the squashed $S^{7}$, whose $G_{2}$ holonomy yields N=1 in D=4. In 1983, another example was provided by D=11 supergravity on $K3$, whose SU(2) holonomy yields half the maximum supersymmetry. In 2002, $G_{2}$ and $K3$ manifolds continue to feature prominently in the full D=11 M-theory and its dualities. In particular, singular $G_{2}$ compactifications can yield chiral $(N=1,D=4)$ models with realistic gauge groups. The notion of generalized holonomy is also discussed. 
  We consider radion stabilization in hyperbolic brane-world scenarios. We demonstrate that in the context of Einstein gravity, matter fields which stabilize the extra dimensions must violate the null energy condition. This result is shown to hold even allowing for FRW-like expansion on the brane. In particular, we explicitly demonstrate how one putative source of stabilizing matter fails to work, and how others violate the above condition. We speculate on a number of ways in which we may bypass this result, including the effect of Casimir energy in these spaces. A brief discussion of supersymmetry in these backgrounds is also given. 
  We apply a self-consistent relativistic mean-field variational ``Gaussian functional'' (or Hartree) approximation to the linear $\sigma$ model with spontaneously and explicitly broken chiral O(4) symmetry. We set up the self-consistency, or ``gap'' and the Bethe-Salpeter equations. We check and confirm the chiral Ward-Takahashi identities, among them the Nambu-Goldstone theorem and the (partial) axial current conservation [CAC], both in and away from the chiral limit. With explicit chiral symmetry breaking we confirm the Dashen relation for the pion mass and partial CAC. We solve numerically the gap and Bethe-Salpeter equations, discuss the solutions' properties and the particle content of the theory. 
  A new approach to the quenched propagator in QED beyond the IR limit is proposed. The method is based on evolution equations in the proper time. 
  We apply methods of dynamical systems to study the behaviour of the Randall-Sundrum models. We determine evolutionary paths for all possible initial conditions in a 2-dimensional phase space and we investigate the set of accelerated models. The simplicity of our formulation in comparison to some earlier studies is expressed in the following: our dynamical system is a 2-dimensional Hamiltonian system, and what is more advantageous, it is free from the degeneracy of critical points so that the system is structurally stable. The phase plane analysis of Randall-Sundrum models with isotropic Friedmann geometry clearly shows that qualitatively we deal with the same types of evolution as in general relativity, although quantitatively there are important differences. 
  We present here a detailed analysis of the local symmetries of supergravity in an arbitrary dimension D, both in the component and superfield approaches, using a field-space democracy point of view. As an application, we discuss briefly how a complete description of the local gauge symmetries clarifies the properties of the supergravity-superbrane coupled system in the standard background superfield approximation for supergravity. 
  We present new exact solutions (in 3+1 and 2+1 dimensions) of relativistic wave equations (Klein-Gordon and Dirac) in external electromagnetic fields of special form. These fields are combinations of Aharonov-Bohm solenoid field and some additional electric and magnetic fields. In particular, as such additional fields, we consider longitudinal electric and magnetic fields, some crossed fields, and some special non-uniform fields. The solutions obtained can be useful to study Aharonov-Bohm effect in the corresponding electromagnetic fields. 
  We review the procedure by which we implemented the non-Abelian Gauss's law and constructed gauge-invariant fields for QCD in the temporal (Weyl) gauge. We point out that the operator-valued transformation that transforms gauge-dependent temporal-gauge fields into gauge-invariant ones has the formal structure of a gauge transformation. We express the ``standard'' Hamiltonian for temporal-gauge QCD entirely in terms of gauge-invariant fields, calculate the commutation rules for these fields, and compare them to earlier work on Coulomb-gauge QCD. We also discuss multiplicities of gauge-invariant temporal-gauge fields that belong to different topological sectors and that, in previous work, were shown to be based on the same underlying gauge-dependent temporal-gauge fields. We relate these multiplicities of gauge-invariant fields to Gribov copies. We argue that Gribov copies appear in the temporal gauge, but not when the theory is represented in terms of gauge-dependent fields and Gauss's law is left unimplemented. There are Gribov copies of the gauge-invariant gauge field, which can be constructed when Gauss's law is implemented. 
  We study the quantum mechanics of a charged particle on a constant curvature noncommutative Riemann surface in the presence of a constant magnetic field. We formulate the problem by considering quantum mechanics on the noncommutative AdS_2 covering space and gauging a discrete symmetry group which defines a genus-g surface. Although there is no magnetic field quantization on the covering space, a quantization condition is required in order to have single-valued states on the Riemann surface. For noncommutative AdS_2 and subcritical values of the magnetic field the spectrum has a discrete Landau level part as well as a continuum, while for overcritical values we obtain a purely noncommutative phase consisting entirely of Landau levels. 
  It is shown that the notion of W_\infty-algebra originally carried out over a (compact) Riemann surface can be extended to n complex dimensional (compact) manifolds within a symplectic geometrical setup. The relationships with the Kodaira-Spencer deformation theory of complex structures are discussed. Subsequently, some field theoretical aspects at the classical level are briefly underlined. 
  We discuss some aspects of heterotic-Type I duality. We focus on toroidal compactification, with special attention for the topology of the gauge group, and the topology of the bundle. We review the arguments leading to a classification of Spin(32)/Z_2-bundles over tori, suitable for string compactifications. A central role is played by n-gerbes with connection, a generalization of bundles with connection. 
  Consistent couplings between a set of vector fields and a system of matter fields are analysed in the framework of Lagrangian BRST cohomology. 
  In this paper we extend Schwinger's quantization approach to the case of a supermanifold considered as a coset space of the Poincare group by the Lorentz group. In terms of coordinates parametrizing a supermanifold, quantum mechanics for a superparticle is constructed. As in models related to the usual Riemannian manifold, the key role in analyzes is played by Killing vectors. The main feature of quantum theory on the supermanifold consists of the fact that the spatial coordinates are not commute with each other and therefore are represented on wave functions by integral operators. 
  We describe a new technique for calculating instanton effects in supersymmetric gauge theories applicable on the Higgs or Coulomb branches. In these situations the instantons are constrained and a potential is generated on the instanton moduli space. Due to existence of a nilpotent fermionic symmetry the resulting integral over the instanton moduli space localizes on the critical points of the potential. Using this technology we calculate the one- and two-instanton contributions to the prepotential of SU(N) gauge theory with N=2 supersymmetry and show how the localization approach yields the prediction extracted from the Seiberg-Witten curve. The technique appears to extend to arbitrary instanton number in a tractable way. 
  We give a constructive classification of the positive energy (lowest weight) unitary irreducible representations of the D=6 superconformal algebras osp(8*/2N). Our results confirm all but one of the conjectures of Minwalla (for N=1,2) on this classification. Our main tool is the explicit construction of the norms of the states that has to be checked for positivity. We give also the reduction of the exceptional UIRs. 
  Based on experimental discovery that the mass-square of neutrino is negative, a quantum equation for superluminal neutrino is proposed in comparison with Dirac equation and Dirac equation with imaginary mass. A virtual particle may also be viewed as superluminal one. Both the basic symmetry of space-time inversion and the maximum violation of space-inversion symmetry are emphasized. 
  We study the spectrum of stable static fermion bags in the 1+1 dimensional Gross-Neveu model with $N$ flavors of Dirac fermions, in the large $N$ limit. In the process, we discover a new kink, heavier than the Callan-Coleman-Gross-Zee (CCGZ) kink, which is marginally stable (at least in the large $N$ limit). The connection of this new kink and the conjectured $S$ matrix of the Gross-Neveu model is obscured at this point. After identifying all stable static fermion bags, we arrange them into a periodic table, according to their $O (2N)$ and topological quantum numbers. 
  Various studies have explored the possibility of explaining the Bekenstein-Hawking (black hole) entropy by way of some suitable state-counting procedure. Notably, many of these treatments have used the well-known Cardy formula as an intermediate step. Our current interest is a recent calculation in which Carlip has deduced the leading-order quantum correction to the (otherwise) classical Cardy formula. In this paper, we apply Carlip's formulation to the case of a generic model of two-dimensional gravity with coupling to a dilaton field. We find that the corrected Cardy entropy includes the anticipated logarithmic ``area'' term. Such a term is also evident when the entropic correction is derived independently by thermodynamic means. However, there is an apparent discrepancy between the two calculations with regard to the factor in front of the logarithm. In fact, the two values of this prefactor can only agree for very specific two-dimensional models, such as that describing Jackiw-Teitelboim theory. 
  Two new methods for investigation of two-dimensional quantum systems, whose Hamiltonians are not amenable to separation of variables, are proposed. 1)The first one - $SUSY-$ separation of variables - is based on the intertwining relations of Higher order SUSY Quantum Mechanics (HSUSY QM) with supercharges allowing for separation of variables. 2)The second one is a generalization of shape invariance. While in one dimension shape invariance allows to solve algebraically a class of (exactly solvable) quantum problems, its generalization to higher dimensions has not been yet explored. Here we provide a formal framework in HSUSY QM for two-dimensional quantum mechanical systems for which shape invariance holds. Given the knowledge of one eigenvalue and eigenfunction, shape invariance allows to construct a chain of new eigenfunctions and eigenvalues. These methods are applied to a two-dimensional quantum system, and partial explicit solvability is achieved in the sense that only part of the spectrum is found analytically and a limited set of eigenfunctions is constructed explicitly. 
  We show that the maximally supersymmetric pp-waves of IIB superstring and M-theories can be obtained as a Penrose limit of the supersymmetric AdS x S solutions. In addition we find that in a certain large tension limit, the geometry seen by a brane probe in an AdS x S background is either Minkowski space or a maximally supersymmetric pp-wave. 
  Gravitational stability of torsion and inflaton potential in a four-dimensional spacetime de Sitter solution in scalar-tensor cosmology where Cartan torsion propagates is investigated in detail. Inflaton and torsion evolution equations are derived by making use of a Lagrangean method. Stable and unstable modes for torsion and inflatons are found. Present astrophysical observations favour a stable mode for torsion since this would explain why no relic torsion imprint has been found on the Cosmic Background Radiation in the universe. 
  In general relativity, only relative acceleration has an observer-independend meaning: curvature and non-gravitational forces determine the rate at which world lines of test bodies diverge or converge. We derive the equations governing both in the conventional geometric formalism as well as using the background field method. This allows us to generalize the results to test bodies with charge and/or spin. The application of the equations to the motion of particles in a central field results in an elegant, fully relativistic version of the Ptolemaean epicycle scheme. 
  A method for describing the quantum kink states in the semi-classical limit of several (1+1)-dimensional field theoretical models is developed. We use the generalized zeta function regularization method to compute the one-loop quantum correction to the masses of the kink in the sine-Gordon and cubic sinh-Gordon models and another two ${\rm P}(\phi)_2$ systems with polynomial self-interactions. 
  The projection formalism for calculating effective Hamiltonians and resonances is generalized to the nonlocal and/or nonhermitian case, so that it is applicable to the reduction of relativistic systems (Bethe-Salpeter equations), and to dissipative systems modeled by an optical potential.   It is also shown how to recover all solutions of the time-independent Schroedinger equation in terms of solutions of the effective Schroedinger equation in the reduced state space and a Schroedinger equation in a reference state space.   For practical calculations, it is important that the resulting formulas can be used without computing any projection operators. This leads to a modified coupled reaction channel/resonating group method framework for the calculation of multichannel scattering information. 
  The Chern-Simons coupling shift is calculated within the framework of the hybrid regularization based on a local higher covariant derivative regulator. When the Yang-Mills term is present in the theory the well-know integer-shift is obtained, but is absent, the shift value is non-integer. These results show a possibility that a non-integer-shift can be derived using a local higher covariant derivative and also suggest that the Yang-Mills term plays an important role in the integer-shift of the Chern-Simons coupling. 
  We show by some counterexamples that Lagrangian sysytems with nonlocality of finite extent are not necessarily unstable. 
  I extend upon the paper by Batalin and Marnelius, in which they show how to construct and quantize a gauge theory from a Hamiltonian system with second class constraints. Among the avenues explored, their technique is analyzed in relation to other well-known methods of quantization and a bracket is defined, such that the operator formalism can be fully developed. I also extend to systems with mixed class constraints and look at some simple examples. 
  We study branes and orientifolds on the group manifold of SO(3). We consider particularly the case of the equatorial branes, which are projective planes. We show that a Dirac-Born-Infeld action can be defined on them, although they are not orientable. We find that there are two orientifold projections with the same spacetime action, which differ by their action on equatorial branes. 
  Non-canonical quantization is based on certain reducible representations of canonical commutation relations. Relativistic formalism for electromagnetic non-canonical quantum fields is introduced. Unitary representations of the Poincar\'e group at the level of fields and states are explicitly given. Multi-photon and coherent states are introduced. Statistics of photons in a coherent state is Poissonian if an appropriately defined thermodynamic limit is performed. Radiation fields having a correct $S$ matrix are constructed. The $S$ matrix is given by a non-canonical coherent-state displacement operator, a fact automatically eliminating the infrared catastrophe. This, together with earlier results on elimination of vacuum and ultraviolet infinities, suggests that non-canonical quantization leads to finite field theories. Renormalization constant $Z_3$ is found as a parameter related to wave functions of non-canonical vacua. 
  We construct logarithmic conformal field theories starting from an ordinary conformal field theory -- with a chiral algebra C and the corresponding space of states V -- via a two-step construction: i) deforming the chiral algebra representation on V\tensor End K[[z,1/z]], where K is an auxiliary finite-dimensional vector space, and ii) extending C by operators corresponding to the endomorphisms End K. For K=C^2, with End K being the two-dimensional Clifford algebra, our construction results in extending C by an operator that can be thought of as \partial^{-1}E, where \oint E is a fermionic screening. This covers the (2,p) Virasoro minimal models as well as the sl(2) WZW theory. 
  We show the Standard Model and SuperString Theories can be naturally based on a Quantum Computer foundation. The Standard Model of elementary particles can be viewed as defining a Quantum Computer Grammar and language. A Quantum Computer in a certain limit naturally forms a Superspace upon which Supersymmetry rotations can be defined - a Continuum Quantum Computer. Quantum high-level computer languages such as Quantum C and Quantum Assembly language are also discussed. In these new linguistic representations, particles become literally symbols or letters, and particle interactions become grammar rules. This view is NOT the same as the often-expressed view that Mathematics is the language of Physics. Some new developments relating to Quantum Computers and Quantum Turing Machines are also described. 
  One can geometrically engineer supersymmetric field theories theories by placing D-branes at or near singularities. The opposite process is described, where one can reconstruct the singularities from quiver theories. The description is in terms of a noncommutative quiver algebra which is constructed from the quiver diagram and the superpotential. The center of this noncommutative algebra is a commutative algebra, which is the ring of holomorphic functions on a variety V. If certain algebraic conditions are met, then the reverse geometric engineering produces V as the geometry that D-branes probe. It is also argued that the identification of V is invariant under Seiberg dualities. 
  We continue the development of the actions, S_{AFF}, by examining the cases where there are N fermionic degrees of freedom associated with a 0-brane. These actions correspond to the interaction of the N-extended super Virasoro algebra with the supergraviton and the associated SO(N) gauge field that accompanies the supermultiplet. The superfield formalism is used throughout so that supersymmetry is explicit. 
  The wedge states form an important subalgebra in the string field theory. We review and further investigate their various properties. We find in particular a novel expression for the wedge states, which allows to understand their star products purely algebraically. The method allows also for treating the matter and ghost sectors separately. It turns out, that wedge states with different matter and ghost parts violate the associativity of the algebra. We introduce and study also wedge states with insertions of local operators and show how they are useful for obtaining exact results about convergence of level truncation calculations. These results help to clarify the issue of anomalies related to the identity and some exterior derivations in the string field algebra. 
  We present a Lagrangian formulation for the free superspin-3/2 massive 4D, N=1 superfield. The model is described by a dynamical real vector superfield and an auxiliary real scalar superfield. The corresponding multiplet contains spin-1, spin-2 and two spin-3/2 propagating component fields on-shell. The auxillary superfield is needed to ensure that the on-shell vector superfield carries only the irreducible representation of the Poincare supergroup with a given mass and the fixed superspin of 3/2. The bosonic sector of the component level of the model is also presented. 
  The proof of one-loop renormalizability of the general Lorentz- and CPT-violating extension of quantum electrodynamics is described. Application of the renormalization-group method is discussed and implications for theory and experiment are considered. 
  I consider a class of Grand Unified models, in which E8 is broken to SU(3)xSU(2)xSU(2)xSU(2)xU(1)_Y, then to SU(3)xSU(2)_{diag}xU(1)_Y. The breaking of (SU(2))^3 to SU(2)_{diag} reduces the SU(2) coupling constant, at unification, by a factor of 1/\sqrt{3}, so that the ratio of the SU(3) and SU(2)_{diag} coupling constants, at unification, is equal to the ratio observed at about 1 TeV. By choosing a suitable alignment of U(1)_Y, and introducing a generalization of the CKM matrix, the U(1)_Y coupling constants of the observed fermions, at unification, can also be arranged to have the ratios, to the SU(3) coupling constant, that are observed at about 1 TeV. This suggests a model of Heterotic M-Theory, with a standard embedding of the spin connection in one of the E8s, but with the visible sector now having the E8 that is unbroken at unification. The universe is pictured as a thick pipe, where the long direction of the pipe represents the four extended dimensions, the circumference represents a compact six-manifold, and the radial direction represents the eleventh dimension. The inner radius of the pipe is about 10^{-19} metres, and the outer radius of the pipe is about 10^{-14} metres. We live on the inner surface of the pipe. The low-energy generation structure and the high mass of the top quark follow from the breaking pattern, and gravity and the Yang-Mills interactions are unified at about 1 TeV. Parity breaking must occur spontaneously in four dimensions, rather than being inherited from ten dimensions. The stability of the proton might be correlated with the entries in the CKM matrix. 
  We show that logarithmic conformal field theories may be derived using nilpotent scale transformation. Using such nilpotent weights we derive properties of LCFT's, such as two and three point correlation functions solely from symmetry arguments. Singular vectors and the Kac determinant may also be obtained using these nilpotent variables, hence the structure of the four point functions can also be derived. This leads to non homogeneous hypergeometric functions. Also we consider LCFT's near a boundary. Constructing "superfields" using a nilpotent variable, we show that the superfield of conformal weight zero, composed of the identity and the pseudo identity is related to a superfield of conformal dimension two, which comprises of energy momentum tensor and its logarithmic partner. This device also allows us to derive the operator product expansion for logarithmic operators. Finally we discuss the AdS/LCFT correspondence and derive some correlation functions and a BRST symmetry. 
  An improved correspondence formula is proposed for the calculation of correlation functions of a conformal field theory perturbed by multi-trace operators from the analysis of the dynamics of the dual field theory in Anti-de Sitter space. The formula reduces to the usual AdS/CFT correspondence formula in the case of single-trace perturbations. 
  We consider the equilibria of point particles under the action of two body central forces in which there are both repulsive and attractive interactions, often known as central configurations, with diverse applications in physics, in particular as homothetic time-dependent solutions to Newton's equations of motion and as stationary states in the One Component Plasma model. Concentrating mainly on the case of an inverse square law balanced by a linear force, we compute numerically equilibria and their statistical properties. When all the masses (or charges) of the particles are equal, for small numbers of points they are regular convex deltahedra, which on increasing the number of points give way to a multi-shell structure. In the limit of a large number of points we argue using an analytic model that they form a homogeneous spherical distribution of points, whose spatial distribution appears, from our preliminary investigation, to be similar to that of a Bernal hard-sphere liquid. 
  D-instantons are used to probe the near-horizon geometry of D3-branes systems on orbifold spaces. For fractional D3-branes, D-instanton calculus correctly reproduces the gauge beta-function and U(1)_R anomaly of the corresponding N=2 non-conformal Super Yang-Mills theories. For D3-branes wrapping the orbifold singularity, D-instantons can be identified with gauge instantons on ALE space, providing evidence of AdS/CFT duality for gauge theories on curved spaces. 
  We present a Lorentzian version of three-dimensional noncommutative Einstein-AdS gravity by making use of the Chern-Simons formulation of pure gravity in 2+1 dimensions. The deformed action contains a real, symmetric metric and a real, antisymmetric tensor that vanishes in the commutative limit. These fields are coupled to two abelian gauge fields. We find that this theory of gravity is invariant under a class of transformations that reduce to standard diffeomorphisms once the noncommutativity parameter is set to zero. 
  There is strong evidence coming from Lorentzian dynamical triangulations that the unboundedness of the gravitational action is no obstacle to the construction of a well-defined non-perturbative path integral. In a continuum approach, a similar suppression of the conformal divergence comes about as the result of a non-trivial path-integral measure. 
  In this paper we have considered the renormalized one-loop effective action for massless self-interacting scalar field in the 3-dimensional ball. The scalar field satisfies Dirichlet boundary condition on the ball. Using heat kernel expansion method we calculate the divergent part of effective action, then by bag model renormalization procedure we obtain the renormalized one-loop effective action. 
  We discuss recent results on one-loop contributions to the effective action in {\cal N}=4 supersymmetric Yang-Mills theory in four dimensions. Contributions with five external vector fields are compared with corresponding ones in open superstring theory in order to understand the relation with the F^5 terms that appear in the nonabelian generalization of the Born-Infeld action. 
  We compute the non-abelian couplings in the Chern-Simons action for a set of coinciding fundamental strings in both the type IIA and type IIB Matrix string theories. Starting from Matrix theory in a weakly curved background, we construct the linear couplings of closed string fields to type IIA Matrix strings. Further dualities give a type IIB Matrix string theory and a type IIA theory of Matrix strings with winding. 
  A connection between integrable quantum field theory and the spectral theory of ordinary differential equations is reviewed, with particular emphasis being given to its relevance to certain problems in PT-symmetric quantum mechanics. 
  We construct a topological theory for euclidean gravity in four dimensions, by enforcing self-duality conditions on the spin connection. The corresponding topological symmetry is associated to the SU(2) X diffeomorphism X U(1) invariance. The action of this theory is that of d=4, N=2 supergravity, up to a twist. The topological field theory is SU(2) invariant, but the full SO(4) invariance is recovered after untwist. This suggest that the topological gravity is relevant for manifolds with special holonomy. The situation is comparable to that of the topological Yang-Mills theory in eight dimensions, for which the SO(8) invariance is broken down to Spin(7), but is recovered after untwisting the topological theory. 
  We explain that a bulk with arbitrary dimensions can be added to the space over which a quantum field theory is defined. This gives a TQFT such that its correlation functions in a slice are the same as those of the original quantum field theory. This generalizes the stochastic quantization scheme, where the bulk is one dimensional. 
  Some selected applications of KT and HKT geometries in string theory, supergravity, black hole moduli spaces and hermitian geometry are reviewed. It is shown that the moduli spaces of a large class of five-dimensional supersymmetric black holes are HKT spaces. In hermitian geometry, it is shown that a compact, conformally balanced, strong KT manifold whose associated KT connection has holonomy contained in SU(n) is Calabi-Yau. The implication of this result in the context of some string compactifications is explained. 
  The superstring action in AdS_5 x S^5 depends on two parameters: the inverse string tension a' and the radius R. The standard AdS/CFT correspondence requires that the string coordinates are rescaled so that the action depends only on one combination of the two: (\lambda)^{1/2} = R^2/a'. Then \lambda \to 0 limit is equivalent to R \to 0 for fixed $a'$ or to the zero-tension limit in AdS_5 x S^5: a' \to \infty for fixed R. After reviewing previous work hep-th/0009171 on the light cone superstring we explicitly obtain the \lambda= 0 form of its action. Its zero-mode part is the same as the superparticle action in AdS_5 x S^5, and thus the \lambda=0 string spectrum must include, as expected, the ``protected'' type IIB supergravity states. Following recent suggestions, it is conjectured that the spectrum of this tensionless string should as well contain higher spin massless states in AdS_5. We also discuss the case of another parametrization of the string action which has straightforward R\to\infty flat space limit but where R \to 0 and a' \to \infty limits are not equivalent. There R \to 0 corresponds to shrinking S^5 to zero the size and ``freezing'' the fluctuations of the radial coordinate of AdS_5. This case is the basis of the ``non-standard'' AdS/CFT correspondence suggested in hep-th/0010106. Parts of this work were presented in the talk at ``Supergravity at 25'' conference, Stony Brook, December 1-2, 2001. 
  An introduction to the construction of D-branes using conformal field theory methods is given. A number of examples are discussed in detail, in particular the construction of all conformal D-branes for the theory of a single free boson on a circle. 
  A brief, example-oriented introduction is given to special holonomy and its uses in string theory and M-theory. We discuss A_k singularities and their resolution; the construction of a K3 surface by resolving T^4/Z_2; holomorphic cycles, calibrations, and worldsheet instantons; aspects of the low-energy effective action for string compactifications; the significance of the standard embedding of the spin connection in the gauge group for heterotic string compactifications; G_2 holonomy and its relation to N=1 supersymmetric compactifications of M-theory; certain isolated G_2 singularities and their resolution; the Joyce construction of compact manifolds of G_2 holonomy; the relation of D6-branes to M-theory on special holonomy manifolds; gauge symmetry enhancement from light wrapped M2-branes; and chiral fermions from intersecting branes. These notes are based on lectures given at TASI '01. 
  This paper explains the origins of the inflation, the four normal force fields, and the extreme force fields. It proposes that the background of the multiverse is the homogeneous static universe, consisting of 11D (space-time dimensional) positive energy membrane and negative energy anti-membrane with the Planck energy as the vacuum energy. The only force in the multiverse background is the attractive pre-strong force, the predecessor of the strong force. Vacuum energy decreases with decreasing space-time number based on quantized varying speed of light. (The vacuum energy of 4D space-time is zero.) With such vacuum energy differences, the local dimensional oscillation between high and space-time dimensions results in eternal local inflation-deflation. (For our observable universe, such vacuum energy differences become the mass-energy differences for 4D elementary particles, including quarks, leptons, and gauge bosons.) Each region of the universe follows a particular path of the dimensional oscillation, leading to a particular set of force fields. For our universe, gravity appears in the first dimensional oscillation between the 11 D membrane and the 10 D string. The asymmetrical weak force appears in the asymmetrical second dimensional oscillation between the 10D particle and the 4D particle. Electromagnetism appears as the force in the transition between the first and the second dimensional oscillations. The cosmology explains the origins of the four forces. Under extreme conditions such as zero absolute temperature and extremely high pressure, the extreme force fields form for superconductor, the fractional quantum Hall effect, gravastar, supernova, neutron star, and gamma ray burst. 
  Within the closed time path formalism a general nonperturbative expression is derived which resums through the Bethe-Salpter equation all leading order contributions to the shear viscosity in hot scalar field theory. Using a previously derived generalized fluctuation-dissipation theorem for nonlinear response functions in the real-time formalism, it is shown that the Bethe-Salpeter equation decouples in the so-called (r,a) basis. The general result is applied to scalar field theory with pure lambda*phi**4 and mixed g*phi**3+lambda*phi**4 interactions. In both cases our calculation confirms the leading order expression for the shear viscosity previously obtained in the imaginary time formalism. 
  We study thermodynamic evaporation of Schwarzschild-de Sitter black holes in terms of a low energy perturbation theory. A small black hole which is far from the cosmological horizon and observers at the spacelike hypersurface where black hole attraction and expansion of cosmological horizon balance exactly are considered. In the low energy perturbation, scalar field equations are solved in both regions of the hypersurface and scattering amplitudes are derived. And then the desired thermal temperatures from the two horizons are obtained as a ``minimal'' value of the statistical thermal temperature, and the fine-tuning between amplitudes gives a relation of the two temperatures. 
  In string theory, there are various physical situations where the world-sheet fields have a shifted moding. For instance, this is the case for the twisted closed string in Z_N orbifold or for the charged open string in a constant electro-magnetic field. Because of this feature, it is quite challenging to give explicit formulae describing the string interaction, even for the bosonic case. In this note, we focus on the case of the charged open bosonic string and construct the 1-loop tadpole which is an object generating all 1-point functions from the annulus in the presence of an external field. In the operator formalism, this represents one of the basic building blocks for the construction of a general loop amplitude. 
  The calculation of the minimum of the effective potential using the zeta function method is extremely advantagous, because the zeta function is regular at $s=0$ and we gain immediately a finite result for the effective potential without the necessity of subtratction of any pole or the addition of infinite counter-terms. The purpose of this paper is to explicitly point out how the cancellation of the divergences occurs and that the zeta function method implicitly uses the same procedure used by Bollini-Giambiagi and Salam-Strathdee in order to gain finite part of functions with a simple pole. 
  In this work we derive the Hamiltonian formalism of the O(N) non-linear sigma model in its original version as a second-class constrained field theory and then as a first-class constrained field theory. We treat the model as a second-class constrained field theory by two different methods: the unconstrained and the Dirac second-class formalisms. We show that the Hamiltonians for all these versions of the model are equivalent. Then, for a particular factor-ordering choice, we write the functional Schrodinger equation for each derived Hamiltonian. We show that they are all identical which justifies our factor-ordering choice and opens the way for a future quantization of the model via the functional Schrodinger representation. 
  We summarise the approach to brane cosmology known as ``mirage cosmology'' and use it to determine the Friedmann equation on a 3-brane embedded in different bulk spacetimes all with one or more extra dimensions. Usually, when there is more than one extra dimension the junction conditions, central to the usual brane world scenarios, are difficult to apply. This problem does not arise in mirage cosmology because the brane is treated as a ``test particle'' in the background spacetime. We discuss in detail the dynamics of a brane embedded in two specific 10D bulk spacetimes, namely Sch-AdS$_5 \times$S$_5$ and a rotating black hole, and from the dynamics--which are now rather more complicated since the brane can move in all the extra dimensions--determine the new ``dark fluid'' terms in the brane Friedmann equation. Some of these, such as the cosmological constant term, are seen to be bulk dependent. However, for both bulks we show that there exists a critical brane angular momentum, $\ell_c$, and discuss its significance. We then show explicitly how this mirage cosmology approach matches with the familiar junction condition approach when there is just one extra dimension. The issue of a varying speed of light in mirage cosmology is reviewed and we find a scenario in which $c_{\bf eff}$ always increases, tending asymptotically to a constant $c_0$ as the universe expands. Finally some comments are made regarding brane inflation and limitations of the mirage cosmology approach are also discussed. 
  We characterize the dependence on doublets of the cohomology of an arbitrary nilpotent differential s (including BRST differentials and classical linearized Slavnov-Taylor (ST) operators) in terms of the cohomology of the doublets-independent component of s. All cohomologies are computed in the space of local integrated formal power series. We drop the usual assumption that the counting operator for the doublets commutes with s (decoupled doublets) and discuss the general case where the counting operator does not commute with s (coupled doublets). The results are purely algebraic and do not rely on power-counting arguments. 
  An analysis of CPN models is given in terms of general coordinates or arbitrary interpolating fields.Only closed expressions made from simple functions are involved.Special attention is given to CP2 and CP4. In the first of these the retrieval of stereographic coordinates reveals the hermitian form of the metric. A similar analysis for the latter case allows comparison with the Fubini-Study metric. 
  BRST-methods provide elegant and powerful tools for the construction and analysis of constrained systems, including models of particles, strings and fields. These lectures provide an elementary introduction to the ideas, illustrated with some important physical applications. 
  In this work we discuss the effect of the quartic fermion self-interaction of Thirring type in QED in D=2 and D=3 dimensions. This is done through the computation of the effective action up to quadratic terms in the photon field. We analyze the corresponding nonlocal photon propagators nonperturbatively in % \frac{k}{m}, where k is the photon momentum and m the fermion mass. The poles of the propagators were determined numerically by using the Mathematica software. In D=2 there is always a massless pole whereas for strong enough Thirring coupling a massive pole may appear . For D=3 there are three regions in parameters space. We may have one or two massive poles or even no pole at all. The inter-quark static potential is computed analytically in D=2. We notice that the Thirring interaction contributes with a screening term to the confining linear potential of massive QED_{2}. In D=3 the static potential must be calculated numerically. The screening nature of the massive QED$_{3}$ prevails at any distance, indicating that this is a universal feature of % D=3 electromagnetic interaction. Our results become exact for an infinite number of fermion flavors. 
  We present a formulation of the stationary bosonic string sector of the whole toroidally compactified effective field theory of the heterotic string as a double Ernst system which, in the framework of General Relativity describes, in particular, a pair of interacting spinning black holes; however, in the framework of low--energy string theory the double Ernst system can be particularly interpreted as the rotating field configuration of two interacting sources of black hole type coupled to dilaton and Kalb--Ramond fields. We clarify the rotating character of the $B_{t\phi}$--component of the antisymmetric tensor field of Kalb--Ramond and discuss on its possible torsion nature. We also recall the fact that the double Ernst system possesses a discrete symmetry which is used to relate physically different string vacua. Therefore we apply the normalized Harrison transformation (a charging symmetry which acts on the target space of the low--energy heterotic string theory preserving the asymptotics of the transformed fields and endowing them with multiple electromagnetic charges) on a generic solution of the double Ernst system and compute the generated field configurations for the 4D effective field theory of the heterotic string. This transformation generates the $U(1)^n$ vector field content of the whole low--energy heterotic string spectrum and gives rise to a pair of interacting rotating black holes endowed with dilaton, Kalb--Ramond and multiple electromagnetic fields where the charge vectors are orthogonal to each other. 
  For the 2-brane Randall-Sundrum model, we calculate the bulk geometry for strong gravity, in the low matter density regime, for slowly varying matter sources. This is relevant for astrophysical or cosmological applications. The warped compactification means the radion can not be written as a homogeneous mode in the orbifold coordinate, and we introduce it by extending the coordinate patch approach of the linear theory to the non-linear case. The negative tension brane is taken to be in vacuum. For conformally invariant matter on the positive tension brane, we solve the bulk geometry as a derivative expansion, formally summing the `Kaluza-Klein' contributions to all orders. For general matter we compute the Einstein equations to leading order, finding a scalar-tensor theory with $\omega(\Psi) \propto \Psi / (1 - \Psi)$, and geometrically interpret the radion. We comment that this radion scalar may become large in the context of strong gravity with low density matter. Equations of state allowing $(\rho - 3 P)$ to be negative, can exhibit behavior where the matter decreases the distance between the 2 branes, which we illustrate numerically for static star solutions using an incompressible fluid. For increasing stellar density, the branes become close before the upper mass limit, but after violation of the dominant energy condition. This raises the interesting question of whether astrophysically reasonable matter, and initial data, could cause branes to collide at low energy, such as in dynamical collapse. 
  We consider the quasilocal thermodynamics of rotating black holes in asymptotic de Sitter spacetimes. Using the minimal number of intrinsic boundary counterterms, we carry out an analysis of the quasilocal thermodynamics of Kerr-de Sitter black holes for virtually all possible values of the mass, rotation parameter and cosmological constant that leave the quasilocal boundary inside the cosmological event horizon. Specifically, we compute the quasilocal energy, the conserved charges, the temperature and the heat capacity for the $(3+1)$-dimensional Kerr-dS black holes. We perform a quasilocal stability analysis and find phase behavior that is commensurate with previous analysis carried out through the use of Arnowitt-Deser-Misner (ADM) parameters. Finally, we investigate the non-rotating case analytically. 
  We propose a nonperturbative framework for the O(32) type I open and closed string theory. The short distance degrees of freedom are bosonic and fermionic hermitian matrices belonging respectively to the adjoint and fundamental representations of the special unitary group SU(N). We identify a closed matrix algebra at finite N which corresponds to the Lorentz, gauge, and supersymmetry algebras of the large N continuum limit. The planar reduction of our matrix theory coincides with the low energy spacetime effective action of the d=10 type I O(32) unoriented open and closed string theory. We show that matrix T-duality transformations can yield a nonperturbative framework for the T-dual type I' closed string theory with 32 D8branes. We show further that under a strong-weak coupling duality transformation the large N reduced action coincides with the low energy spacetime effective action of the d=10 heterotic string, an equivalence at leading order in the inverse string tension and with either gauge group Spin(32)/Z2 or E8xE8. Our matrix formalism has the potential of providing a nonperturbative framework encapsulating all of the weak coupling limits of M theory. 
  We obtain and study an analytical solution of a de-Sitter thick domain wall in five-dimensional Einstein gravity interacting with a scalar field. The scalar field potential is axion-like, V(phi)=a+b cos(phi) with constants a,b satisfying -3b<5a<3b, and the solution is expressed in terms of elliptic functions. 
  Differential geometric formulation of quantum gauge theory of gravity is studied in this paper. The quantum gauge theory of gravity which is proposed in the references hep-th/0109145 and hep-th/0112062 is formulated completely in the framework of traditional quantum field theory. In order to study the relationship between quantum gauge theory of gravity and traditional quantum gravity which is formulated in curved space, it is important to find the differential geometric formulation of quantum gauge theory of gravity. We first give out the correspondence between quantum gauge theory of gravity and differential geometry. Then we give out differential geometric formulation of quantum gauge theory of gravity. 
  We consider condensation of localized closed string tachyons by examining the recent proposal of Harvey, Kutasov, Martinec, and Moore. We first observe that the $g_{\rm cl}$ defined by HKMM does not reflect the space-time supersymmetry when the model has the SUSY. Especially for ${\bf C}^2/{\bf Z}_N$ models, $g_{\rm cl}$ defined by them is highly peaked along the supersymmetric points in the space of orbifolds, which is unsatisfactory property of the "potential" of the RG-flow. We give the modified definition of the $g_{\rm cl}$ in type II cases such that it has a valley along the supersymmetric points in the orbifold moduli space. New definition predicts that the processes suggested by Adams, Polchinski and Silverstein and was argued to be forbidden by HKMM are in fact allowed. 
  We numerically integrate the semiclassical equations of motion for spherically symmetric Einstein-Maxwell theory with a dilaton coupled scalar field and look for zero temperature configurations. The solution we find is studied in detail close to the horizon and comparison is made with the corresponding one in the minimally coupled case. 
  Using a multiple integral representation for the correlation functions, we compute the emptiness formation probability of the XXZ spin-1/2 Heisenberg chain at anisotropy Delta=1/2. We prove it is expressed in term of the number of alternating sign matrices. 
  A generalization of the standard electroweak model to noncommutative spacetime would involve a product gauge group which is spontaneously broken. Gauge interactions in terms of physical gauge bosons are canonical with respect to massless gauge bosons as required by the exact gauge symmetry, but not so with respect to massive ones; and furthermore they are generally asymmetric in the two sets of gauge bosons. On noncommutative spacetime this already occurs for the simplest model of U(1) x U(1). We examine whether the above feature in gauge interactions can be perturbatively maintained in this model. We show by a complete one loop analysis that all ultraviolet divergences are removable with a few renormalization constants in a way consistent with the above structure. 
  We show analytically that the ratio of the norm of sliver states agrees with the ratio of D-brane tensions. We find that the correct ratio appears as a twist anomaly. 
  It has been proposed some time ago that the large $N-$limit can be understood as a ``classical limit'', where commutators in some sense approach the corresponding Poisson brackets. We discuss this in the light of some recent numerical results for an SU(N) gauge model, which do not agree with this ``classicality'' of the large N-limit. The world sheet becomes very crumpled. We speculate that this effect would disappear in supersymmetric models. 
  We study some configurations of brane probes which are partially wrapped on spheres transverse to a stack of non-threshold bound states. The latter are represented by the corresponding supergravity background. Two cases are studied: D(10-p)-branes in the background of (D(p-2), Dp) bound states and D(8-p)-branes in the (NS5, Dp) geometry. By using suitable flux quantization rules of the worldvolume gauge field, we determine the stable configurations of the probe. The analysis of the energy and supersymmetry of these configurations reveals that they can be interpreted as bound states of lower dimensional objects polarized into a D-brane. 
  The formulation of Seiberg-Witten maps from the point of view of consistent deformations of gauge theories in the context of the Batalin-Vilkovisky antifield formalism is reviewed. Some additional remarks on noncommutative Yang-Mills theory are made. 
  Through direct examination of the effect of the OM limit on the M2-brane worldvolume action, we derive a membrane action for OM theory, and more generally, for the eleven-dimensional M-theoretic construct known as Galilean or Wrapped M2-brane (WM2) theory, which contains OM theory as a special class of states. In the static gauge, the action in question implies a discrete spectrum for the closed membrane of WM2 theory, which under double dimensional reduction is shown to reproduce the known NCOS/Wound closed string spectrum. We examine as well open membranes ending on each of the three types of M5-branes in WM2 theory (OM theory arising from the 'longitudinal' type), and show that the 'fully transverse' fivebrane is tensionless. As a prelude to the membrane, we also study the case of the string, where we likewise obtain a reparametrization-invariant action, and make contact with previous work. 
  In the SU(2)_{L} x U(1)_{Y} standard electroweak theory coupled with the Einstein gravity, new topological configurations naturally emerge, if the spatial section of the universe is globally a three-sphere(S^3) with a small radius. The SU(2)_L gauge fields and Higgs fields wrap the space nontrivially, residing at or near a local minimum of the potential. As the universe expands, however, the shape of the potential rapidly changes and the local minimum eventually disappears. The fields then start to roll down towards the absolute minimum. In the absence of the U(1)_Y gauge interaction the resulting space is a homogeneous and isotropic S^3, but the U(1)_Y gauge interaction necessarily induces anisotropy while preserving the homogeneity of the space. Large magnetic fields are generically produced over a substantial period of the rolling-over transition. The magnetic field configuration is characterized by the Hopf map. 
  We establish that the Yang-Baxter equations in the presence of an impurity can in general only admit solutions of simultaneous tranmission and reflection when the transmission and reflection amplitudes commute in the defect degrees of freedom with an additional exchange of the corresponding rapidities. In the absence of defect degrees of freedom we show in complete generality, that the only exceptions to this are theories which possess rapidity independent bulk scattering matrices. In particular bulk theories with diagonal scattering matrices, can only be the free Boson and Fermion, the Federbush model and their generalizations. These anyonic solutions do not admit the possibility of excited impurity states. 
  We consider a class of Lagrangian theories where part of the coordinates does not have any time derivatives in the Lagrange function (we call such coordinates degenerate). We advocate that it is reasonable to reconsider the conventional definition of singularity based on the usual Hessian and, moreover, to simplify the conventional Hamiltonization procedure. In particular, in such a procedure, it is not necessary to complete the degenerate coordinates with the corresponding conjugate momenta. 
  The IR/UV mixing and the violation of unitarity are two of the most intriguing aspects of noncommutative quantum field theories. In this paper the relation between these two phenomena is explained and established. We start out by showing that the S-matrix of noncommutative field theories is hermitian analytic. As a consequence, a noncommutative field theory is unitary if the discontinuities of its Feynman diagram amplitudes agree with the expressions calculated using the Cutkosky formulae. These unitarity constraints relate the discontinuities of amplitudes with physical intermediate states; and allow us to see how the IR/UV mixing may lead to a breakdown of unitarity. Specifically, we show that the IR/UV singularity does not lead to the violation of unitarity in the space-space noncommutative case, but it does lead to its violation in a space-time noncommutative field theory. As a corollary, noncommutative field theory without IR/UV mixing will be unitary in both the space-space and space-time noncommutative case. To illustrate this, we introduce and analyse the noncommutative Lee model--an exactly solvable quantum field theory. We show that the model is free from the IR/UV mixing in both the space-space and space-time noncommutative cases. Our analysis is exact. Due to absence of the IR/UV mixing one can expect that the theory is unitary. We present some checks supporting this claim. Our analysis provides a counter example to the generally held beliefs that field theories with space-time noncommutativity are non-unitary. 
  We study the structure of the four-point correlation function of the lowest-dimension 1/2 BPS operators (stress-tensor multiplets) in the (2,0) six-dimensional theory. We first discuss the superconformal Ward identities and the group-theoretical restrictions on the corresponding OPE. We show that the general solution of the Ward identities is expressed in terms of a single function of the two conformal cross-ratios ("prepotential"). Using the maximally extended gauged seven-dimensional supergravity, we then compute the four-point amplitude in the supergravity approximation and identify the corresponding prepotential. We analyze the leading terms in the OPE by performing a conformal partial wave expansion and show that they are in agreement with the non-renormalization theorems following from representation theory. The investigation of the (2,0) theory is carried out in close parallel with the familiar four-dimensional N=4 super-Yang-Mills theory. 
  We present a construction of boundary states based on the Coulomb-gas formalism of Dotsenko and Fateev. It is shown that Neumann-like coherent states on the charged bosonic Fock space provide a set of boundary states with consistent modular properties. Such coherent states are characterised by the boundary charges, which are related to the number of bulk screening operators through the charge neutrality condition. We illustrate this using the Ising model as an example, and show that all of its known consistent boundary states are reproduced in our formalism. This method applies to $c<1$ minimal conformal theories and provides an unified computational tool for studying boundary states of such theories. 
  The notion of Poincare gauge manifold ($G$), proposed in the context of an (1+1) gravitational theory by Cangemi and Jackiw (D. Cangemi and R. Jackiw, Ann. Phys. (N.Y.) 225 (1993) 229), is explored from a geometrical point of view. First $G$ is defined for arbitrary dimensions, and in the sequence a symplectic structure is attached to $T*G$. Treating the case of five dimensions, a (4,1)-de Sitter space, aplications are presented studing representations of the Poincare group in association with kinetic theory and the Weyl operators in phase space. The central extension in the Aghassi-Roman-Santilli group (J. J. Aghassi, P. Roman and R. M. Santilli, Phys. Rev. D 1(1970) 2573) is derived as a subgroup of linear transformations in $G$ with six dimensions. 
  We review the framework we and our collaborators have developed for the study of one-loop quantum corrections to extended field configurations in renormalizable quantum field theories. We work in the continuum, transforming the standard Casimir sum over modes into a sum over bound states and an integral over scattering states weighted by the density of states. We express the density of states in terms of phase shifts, allowing us to extract divergences by identifying Born approximations to the phase shifts with low order Feynman diagrams. Once isolated in Feynman diagrams, the divergences are canceled against standard counterterms. Thus regulated, the Casimir sum is highly convergent and amenable to numerical computation. Our methods have numerous applications to the theory of solitons, membranes, and quantum field theories in strong external fields or subject to boundary conditions. 
  We analytically prove that the matter solution of vacuum string field theory constructed by Kostelecky and Potting is the matter sliver state. We also give an analytical proof that the ghost solution by Hata and Kawano is the sliver state in the twisted ghost CFT. It is also proved that the candidate state for the tachyon proposed by Hata and Kawano can be identified with the state constructed by Rastelli, Sen and Zwiebach using CFT. Our proofs are based on the techniques recently developed by Okuyama. 
  We compute the Casimir energy of a real scalar field in the presence of a pair of partially transparent plane mirrors, modeled by Dirac delta potentials. 
  By replacing ten-dimensional pure spinors with eleven-dimensional pure spinors, the formalism recently developed for covariantly quantizing the d=10 superparticle and superstring is extended to the d=11 superparticle and supermembrane. In this formalism, kappa symmetry is replaced by a BRST-like invariance using the nilpotent operator Q=int (lambda^alpha d_alpha) where d_alpha is the worldvolume variable corresponding to the d=11 spacetime supersymmetric derivative and lambda^alpha is an SO(10,1) pure spinor variable satisfying (lambda Gamma^c lambda)=0 for c=1 to 11.   Super-Poincare covariant unintegrated and integrated supermembrane vertex operators are explicitly constructed which are in the cohomology of Q. After double-dimensional reduction of the eleventh dimension, these vertex operators are related to Type IIA superstring vertex operators where Q=Q_L+Q_R is the sum of the left and right-moving Type IIA BRST operators and the eleventh component of the pure spinor constraint, (lambda Gamma^{11} lambda)=0, replaces the (b_L^0 - b_R^0) constraint of the closed superstring. A conjecture is made for the computation of M-theory scattering amplitudes using these supermembrane vertex operators. 
  Explicit expressions for the expectation values and the variances of some observables, which are bilinear quantities in the quantum fields on a D-dimensional manifold, are derived making use of zeta function regularization. It is found that the variance, related to the second functional variation of the effective action, requires a further regularization and that the relative regularized variance turns out to be 2/N, where N is the number of the fields, thus being independent on the dimension D. Some illustrating examples are worked through. 
  We calculate the ring of differential operators on some singular affine varieties (intersecting stacks, a point on a singular curve or an orbifold). Our results support the proposed connection of the ring of differential operators with geometry of D-branes in (bosonic) string theory. In particular, the answer does know about the resolution of singularities in accordance with the string theory predictions. 
  Consistent Hamiltonian couplings between a set of vector fields and a system of matter fields are derived by means of BRST cohomological techniques. 
  Using two dimensional (2D) N=4 sigma model, with $U(1)^r$ gauge symmetry, and introducing the ADE Cartan matrices as gauge matrix charges, we build " toric" hyper-Kahler eight real dimensional manifolds X_8. Dividing by one toric geometry circle action of X_8 manifolds, we present examples describing quotients $X_7={X_8\over U(1)}$ of G_2 holonomy. In particular, for the A_r Cartan matrix, the quotient space is a cone on a $ {S^2}$ bundle over r intersecting $\bf WCP^2_{(1,2,1)}$ projective spaces according to the A_r Dynkin diagram. 
  Two bases of states are presented for modules of the graded parafermionic conformal field theory associated to the coset $\osp(1,2)_k/\uh(1)$. The first one is formulated in terms of the two fundamental (i.e., lowest dimensional) parafermionic modes. In that basis, one can identify the completely reducible representations, i.e., those whose modules contain an infinite number of singular vectors; the explicit form of these vectors is also given.  The second basis is a quasi-particle basis, determined in terms of a modified version of the $\ZZ_{2k}$ exclusion principle. A novel feature of this model is that none of its bases are fully ordered and this reflects a hidden structural $\Z_3$ exclusion principle. 
  The Lie algebra of Feynman graphs gives rise to two natural representations, acting as derivations on the commutative Hopf algebra of Feynman graphs, by creating or eliminating subgraphs. Insertions and eliminations do not commute, but rather establish a larger Lie algebra of derivations which we here determine. 
  We systematically investigate the effect of short distance physics on the spectrum of temperature anistropies in the Cosmic Microwave Background produced during inflation. We present a general argument-assuming only low energy locality-that the size of such effects are of order H^2/M^2, where H is the Hubble parameter during inflation, and M is the scale of the high energy physics.   We evaluate the strength of such effects in a number of specific string and M theory models. In weakly coupled field theory and string theory models, the effects are far too small to be observed. In phenomenologically attractive Horava-Witten compactifications, the effects are much larger but still unobservable. In certain M theory models, for which the fundamental Planck scale is several orders of magnitude below the conventional scale of grand unification, the effects may be on the threshold of detectability.   However, observations of both the scalar and tensor fluctuation contributions to the Cosmic Microwave Background power spectrum-with a precision near the cosmic variance limit-are necessary in order to unambiguously demonstrate the existence of these signatures of high energy physics. This is a formidable experimental challenge. 
  In open string field theory the kinetic operator mixes matter and ghost sectors, and thus the ghost structure of classical solutions is not universal. Nevertheless, we have found from numerical analysis that certain ratios of expectation values for states involving pure ghost excitations appear to be universal. We give an analytic expression for these ratios and find good evidence that they are common to all known solutions of open string field theory, including the tachyon vacuum solution, lump solutions and string fields representing marginal deformations. We also draw attention to a close correspondence between the expectation values for the pure matter components in the tachyon vacuum solution and those in the solution of a simpler equation for a ghost number zero string field. Finally we observe that the action of L_0 on the tachyon condensate gives a state that is approximately factorized into a matter and a ghost part. 
  In order to extend the limits of classical theory application in the microworld some weak generalization of Maxwell electrodynamics is suggested. It is shown that slightly generalized classical Maxwell electrodynamics can describe the intraatomic phenomena with the same success as relativistic quantum mechanics can do. Group-theoretical grounds for the description of fermionic states by bosonic system are presented briefly. The advantages of generalized electrodynamics in intraatomic region in comparison with standard Maxwell electrodynamics are demonstrated on testing example of hydrogen atom. We are able to obtain some results which are impossible in the framework of standard Maxwell electrodynamics. The Sommerfeld - Dirac formula for the fine structure of the hydrogen atom spectrum is obtained on the basis of such Maxwell equations without appealing to the Dirac equation. The Bohr postulates and the Lamb shift are proved to be the consequences of the equations under consideration. The relationship of the new model with the Dirac theory is investigated. Possible directions of unification of such electrodynamics with gravity are mentioned. 
  We review some selected aspects of the construction of gauge invariant operators in field theories on non-commutative spaces and their relation to the energy momentum tensor as well as to the non-commutative loop equations. 
  We present a complete relativistic analysis for the scalar radiation emitted by a particle in circular orbit around a Schwarzschild-anti-de Sitter black hole. If the black hole is large, then the radiation is concentrated in narrow angles- high multipolar distribution- i.e., the radiation is synchrotronic. However, small black holes exhibit a totally different behavior: in the small black hole regime, the radiation is concentrated in low multipoles. There is a transition mass at $M=0.427 R$, where $R$ is the AdS radius. This behavior is new, it is not present in asymptotically flat spacetimes. 
  Gravitational S-duality is defined by the contraction of two indices of the Riemann tensor with the epsilon tensor. We review its realization in linearized gravity, and study its generalization to full non-linear gravity by means of explicit examples: Up to a rescaling of the coordinates, it relates two Taub-NUT-Schwarzschild metrics by interchanging m with l, provided both parameters are non-zero. In the presence of a cosmological constant gravitational S-duality can be implemented at the expense of the introduction of a three-form field whose value turns out to be dual to the cosmological constant. 
  The Lie algebroids are generalization of the Lie algebras. They arise, in particular, as a mathematical tool in investigations of dynamical systems with the first class constraints. Here we consider canonical symmetries of Hamiltonian systems generated by a special class of Lie algebroids. The ``coordinate part'' of the Hamiltonian phase space is the Poisson manifold $M$ and the Lie algebroid brackets are defined by means of the Poisson bivector. The Lie algebroid action defined on $M$ can be lifted to the phase space. The main observation is that the classical BRST operator has the same form as in the case of the Lie groups action. Two examples are analyzed. In the first, $M$ is the space of $SL(3,C)$-opers on Riemann curves with the Adler-Gelfand-Dikii brackets. The corresponding Hamiltonian system is the $W_3$-gravity. Its phase space is the base of the algebroid bundle. The sections of the bundle are the second order differential operators on Riemann curves. They are the gauge symmetries of the theory. The moduli space of $W_3$ geometry of Riemann curves is the symplectic quotient with respect to their action. It is demonstrated that the nonlinear brackets and the second order differential operators arise from the canonical brackets and the standard gauge transformations in the Chern-Simons field theory, as a result of the partial gauge fixing. The second example is $M=C^4$ endowed with the Sklyanin brackets. The symplectic reduction with respect to the algebroid action leads to a generalization of the rational Calogero-Moser model. As in the previous example the Sklyanin brackets can be derived from a ``free theory.'' In this case it is a ``relativistic deformation'' of the $SL(2,C)$ Higgs bundle over an elliptic curve. 
  We study a class of one-matrix models with an action containing nonpolynomial terms. By tuning the coupling constants in the action to criticality we obtain that the eigenvalue density vanishes as an arbitrary real power at the origin, thus defining a new class of multicritical matrix models. The corresponding microscopic scaling law is given and possible applications to the chiral phase transition in QCD are discussed. For generic coupling constants off-criticality we prove that all microscopic correlation functions at the origin of the spectrum remain in the known Bessel universality class. An arbitrary number of Dirac mass terms can be included and the corresponding massive universality is maintained as well. We also investigate the critical behavior at the edge of the spectrum: there, in contrast to the behavior at the origin, we find the same critical exponents as derived from matrix models with a polynomial action. 
  We investigate the possibility of mixing between open and closed string excitations in D-brane models with the fundamental string scale at the TeV. The open string modes describe the Standard Model Higgs, while closed strings describe graviscalars living in the bulk. This provides a string setup for computing the Higgs-graviscalar mixing, that leads to a phenomenologically interesting invisible width of the Higgs in low scale quantum gravity models, as suggested previously by Giudice, Rattazzi and Wells. 
  In this paper we construct a class of random matrix ensembles labelled by a real parameter $\alpha \in (0,1)$, whose eigenvalue density near zero behaves like $|x|^\alpha$. The eigenvalue spacing near zero scales like $1/N^{1/(1+\alpha)}$ and thus these ensembles are representatives of a {\em continous} series of new universality classes. We study these ensembles both in the bulk and on the scale of eigenvalue spacing. In the former case we obtain formulas for the eigenvalue density, while in the latter case we obtain approximate expressions for the scaling functions in the microscopic limit using a very simple approximate method based on the location of zeroes of orthogonal polynomials. 
  It is shown that permanent confinement in three-dimensional compact U(1) gauge theory can be destroyed by matter fields in a deconfinement transition. This is a consequence of a non-trivial infrared fixed point caused by matter, and an anomalous scaling dimension of the gauge field. This leads to a logarithmic interaction between the defects of the gauge-fields, which form a gas of magnetic monopoles. In the presence of logarithmic interactions, the original electric charges are unconfined. The confined phase which is permanent in the absence of matter fields is reached at a critical electric charge, where the interaction between magnetic charges is screened by a pair unbinding transition in a Kosterlitz-Thouless type of phase-transition. 
  We begin with an effective string theory for long distance QCD, and evaluate the semiclassical expansion of this theory about a classical rotating string solution, taking into account the the dynamics of the boundary of the string. We show that, after renormalization, the zero point energy of the string fluctuations remains finite when the masses of the quarks on the ends of the string approach zero. The theory is then conformally invariant in any spacetime dimension D. For D=26 the energy spectrum of the rotating string formally coincides with that of the open string in classical Bosonic string theory. However, its physical origin is different. It is a semiclassical spectrum of an effective string theory valid only for large values of the angular momentum. For D=4, the first semiclassical correction adds the constant 1/12 to the classical Regge formula. 
  We consider 2+1 gravity minimally coupled to a self-interacting scalar field. The case in which the fall-off of the fields at infinity is slower than that of a localized distribution of matter is analyzed. It is found that the asymptotic symmetry group remains the same as in pure gravity (i.e., the conformal group). The generators of the asymptotic symmetries, however, acquire a contribution from the scalar field, but the algebra of the canonical generators possesses the standard central extension. In this context, new massive black hole solutions with a regular scalar field are found for a one-parameter family of potentials. These black holes are continuously connected to the standard zero mass black hole. 
  We construct Poisson brackets at boundaries of open strings and membranes with constant background fields which are compatible with their boundary conditions. The boundary conditions are treated as primary constraints which give infinitely many secondary constraints. We show explicitly that we need only two (the primary one and one of the secondary ones) constraints to determine Poisson brackets of strings. We apply this to membranes by using canonical transformations. 
  We show that multi-trace interactions can be consistently incorporated into an extended AdS/CFT prescription involving the inclusion of generalized boundary conditions and a modified Legendre transform prescription. We find new and consistent results by considering a self-contained formulation which relates the quantization of the bulk theory to the AdS/CFT correspondence and the perturbation at the boundary by double-trace interactions. We show that there exist particular double-trace perturbations for which irregular modes are allowed to propagate as well as the regular ones. We perform a detailed analysis of many different possible situations, for both minimally and non-minimally coupled cases. In all situations, we make use of a new constraint which is found by requiring consistence. In the particular non-minimally coupled case, the natural extension of the Gibbons-Hawking surface term is generated. 
  We study the boundary states for the rational points in the moduli spaces of c=1 conformal and c=3/2 superconformal field theories, including the isolated Ginsparg points. We use the orbifold and simple-current techniques to relate the boundary states of different theories and to obtain symmetry-breaking, non-Cardy boundary states. We show some interesting examples of fractional and twisted branes on orbifold spaces. 
  Asymptotically de Sitter spaces can be described by Euclidean boundary theories with entropies given by the modified Cardy--Verlinde formula. We show that the Cardy--Verlinde formula describes a string with a rescaled tension which in fact is a string at the stretched cosmological horizon as seen from the boundary. The temperature of the boundary theory is the rescaled Hagedorn temperature of the string. Our results agree with an alternative description of asymptotically de Sitter spaces in terms of strings on the stretched horizon. The relation between the two descriptions is given by the large gravitational redshift between the boundary and the stretched horizon and a shift in energy. 
  The orientifolds of SU(2)/U(1) gauged WZW models are investigated. In particular, we construct the new type orientifolds and identify their geometries. We closely follow the analysis of D-branes in the SU(2)/U(1) WZW models, which was given by Maldacena, Moore and Seiberg. 
  We study the wave equation for a minimally coupled massive scalar in three-dimensional de Sitter space. We compute the absorption cross section to investigate its cosmological horizon in the southern diamond. Although the absorption cross section is not defined exactly, we can be determined it from the fact that the low-energy $s(j=0)$-wave absorption cross section for a massless scalar is given by the area of the cosmological horizon. On the other hand, the low-temperature limit of $j\not=0$-mode absorption cross section is useful for extracting information surrounding the cosmological horizon. Finally we mention a computation of the absorption cross section on the CFT-side using the dS/CFT correspondence. 
  In vacuum string field theory, the sliver state solution has been proposed as a candidate of a D-brane configuration. Physical observables associated with this solution, such as its energy density and the tachyon mass, are written in terms of the Neumann coefficients. These observables, though vanish naively due to twist symmetry, acquire non-vanishing values arising from their singular behavior. Therefore, this phenomenon is called twist anomaly. In this paper we present an analytical derivation of these physical observables with the help of the star algebra spectroscopy. We also identify in our derivation the origin of the twist anomaly in the finite-size matrix regularization. 
  In this paper we perform some non-trivial tests for the recently obtained open membrane/D-brane metrics and `generalized' noncommutativity parameters using Dp/NS5/M5-branes which have been deformed by light-like fields. The results obtained give further evidence that these open membrane/D-brane metrics and `generalized' noncommutativity parameters are correct. Further, we use the open brane data and supergravity duals to obtain more information about non-gravitational theories with light-like noncommutativity, or `generalized' light-like noncommutativity. In particular, we investigate various duality relations (strong coupling limits). In the light-like case we also comment on the relation between open membrane data (open membrane metric etc.) in six dimensions and open string data in five dimensions. Finally, we investigate the strong coupling limit (high energy limit) of five dimensional NCYM with \Theta^{12}=\Theta^{34}. In particular, we find that this NCYM theory can be UV completed by a DLCQ compactification of M-theory. 
  Based on the gauge independent decomposition of the non-Abelian gauge field into the dual potential and the valence potential, we calculate the one loop effective action of SU(2) QCD in an arbitrary constant monopole background, using the background field method. Our result provides a strong evidence for a dynamical symmetry breaking through the monopole condensation, which can induce the dual Meissner effect and establish the confinement of color, in the non-Abelian gauge theory. The result is obtained by separating the topological degrees which describe the non-Abelian monopoles from the dynamical degrees of the gauge potential, and integrating out all the dynamical degrees of QCD. 
  We discuss the problem of the motion of classical strings in some black hole and cosmological spacetimes. In particular, the null string limit (zero tension) of tensile strings is considered. We present some new exact string solutions in Reissner-Nordstr\"om black hole background as well as in the Einstein Static Universe and in the Einstein-Schwarzschild (a black hole in the Einstein Static Universe) spacetime. These solutions can give some insight into a general nature of propagation of strings (cosmic and fundamental) in curved backgrounds. 
  Branched n-coverings of Riemann surfaces are described by a 2d lattice gauge theory of the symmetric group S(n) defined on a cell discretization of the surface. We study the theory in the large-n limit, and we find a rich phase diagram with first and second order transition lines. The various phases are characterized by different connectivity properties of the covering surface. We point out some interesting connections with the theory of random walks on group manifolds and with random graph theory. 
  We review aspects of quantisation of the 11-dimensional supermembrane world volume theory. We explicitly construct vertex operators for the massless states and study interactions of supermembranes. The open supermembrane and its vertex operators are discussed. We show how our results have direct applications to Matrix theory by appropriate regularisation of the supermembrane. 
  Taking seriously the hypothesis that the full symmetry algebra of M-theory is osp(1|32,R), we derive the supersymmetry transformations for all fields that appear in 11- and 12-dimensional realizations and give the associated SUSY algebras. We study the background-independent osp(1|32,R) cubic matrix model action expressed in terms of representations of the Lorentz groups SO(10,2) and SO(10,1). We explore further the 11-dimensional case and compute an effective action for the BFSS-like degrees of freedom. We find the usual BFSS action with additional terms incorporating couplings to transverse 5-branes, as well as a mass-term and an infinite tower of higher-order interactions. 
  In this paper we introduce the concept of Deligne cohomology of an orbifold. We prove that the third Deligne cohomology group of a smooth \'{e}tale groupoid classify gerbes with connection over the groupoid. We argue that the $B$-field and the discrete torsion in type II superstring theories are special kinds of gerbes with connection, and finally, for each one of them, using Deligne cohomology we construct a flat line bundle over the inertia groupoid, namely a Ruan inner local in the case of an orbifold. 
  We conjecture that the end point of bulk closed string tachyon decay at any non-zero coupling, is the annihilation of space time by Witten's bubble of nothing, resulting in a topological phase of the theory. In support of this we present a variety of situations in which there is a correspondence between the existence of perturbative tachyons in one regime and the semi-classical annihilation of space-time. Our discussion will include many recently investigated scenarios in string theory including Scherk-Schwarz compactifications, Melvin magnetic backgrounds, and noncompact orbifolds. We use this conjecture to investigate a possible web of dualities relating the eleven-dimensional Fabinger-Horava background with nonsupersymmetric string theories. Along the way we point out where our conjecture resolves some of the puzzles associated with bulk closed string tachyon condensation. 
  Regular classical solutions of pure SU(3) gauge theories, in Minkowsky spacetime, are computed in the Landau gauge. The classical fields have an intrinsic energy scale and produce quark confinement if interpreted in the sense of a nonrelativistic potential. Moreover, the quark propagator in the background of these fields vanishes at large positive and negative time and space separations. 
  It has been suggested that the observed value of the cosmological constant is related to the supersymmetry breaking scale M_{susy} through the formula Lambda \sim M_p^4 (M_{susy}/M_p)^8. We point out that a similar relation naturally arises in the codimension two solutions of warped space-time varying compactifications of string theory in which non-isotropic stringy moduli induce a small but positive cosmological constant. 
  The stability of the Bianchi type I anisotropic brane cosmology is analyzed in this paper. We also study the effect of the brane solution by comparing the models on the 3-brane and the models in the conventional Einstein's space. Analysis is presented for two different models: one with a perfect fluid and the other one with a dilaton field. It is shown that the anisotropic expansion is smeared out dynamically for both theories in the large time limit independent of the models with different types of matter. The initial states are, however, dramatically different. A primordial anisotropic expansion will grow for the conventional Einstein's theory. On the other hand, it is shown that the initial state is highly isotropic for the brane universe except for a very particular case. Moreover, it is also shown that the Bianchi type I anisotropic cosmology is stable against any anisotropic perturbation for both theories in the large time limit. 
  In this Letter we study the dependence of the spectrum of fluctuations in inflationary cosmology on possible effects of trans-Planckian physics, using the Corley/Jacobson dispersion relations as an example. We compare the methods used in previous work [1] with the WKB approximation, give a new exact analytical result, and study the dependence of the spectrum obtained using the approximate method of Ref. [1] on the choice of the matching time between different time intervals. We also comment on recent work subsequent to Ref. [1] on the trans-Planckian problem for inflationary cosmology. 
  We consider the su(2) and su(3) affine theories on a cylinder, from the point of view of their discrete internal symmetries. To this end, we adapt the usual treatment of boundary conditions leading to the Cardy equation to take the symmetry group into account. In this context, the role of the Ishibashi states from all (non periodic) bulk sectors is emphasized. This formalism is then applied to the su(2) and su(3) models, for which we determine the action of the symmetry group on the boundary conditions, and we compute the twisted partition functions. Most if not all data relevant to the symmetry properties of a specific model are hidden in the graphs associated with its partition function, and their subgraphs. A synoptic table is provided that summarizes the many connections between the graphs and the symmetry data that are to be expected in general. 
  The behaviour of the distance between two branes (the `radion') in a braneworld model with a bulk scalar field is investigated. We show that the BPS conditions of supergravity ensure that the dynamics of the scalar field and the radion are not independent; we derive the four-dimensional effective action, showing that the effective theory is of scalar--tensor nature, coupling the radion to four-dimensional gravity. 
  In this paper we study the large N_c limit of SO(N_c) gauge theory coupled to a Majorana field and a real scalar field in 1+1 dimensions extending ideas of Rajeev. We show that the phase space of the resulting classical theory of bilinears, which are the mesonic operators of this theory, is OSp_1(H|H )/U(H_+|H_+), where H|H refers to the underlying complex graded space of combined one-particle states of fermions and bosons and H_+|H_+ corresponds to the positive frequency subspace. In the begining to simplify our presentation we discuss in detail the case with Majorana fermions only (the purely bosonic case is treated in our earlier work). In the Majorana fermion case the phase space is given by O_1(H)/U(H_+), where H refers to the complex one-particle states and H_+ to its positive frequency subspace. The meson spectrum in the linear approximation again obeys a variant of the 't Hooft equation. The linear approximation to the boson/fermion coupled case brings an additonal bound state equation for mesons, which consists of one fermion and one boson, again of the same form as the well-known 't Hooft equation. 
  In this paper we study the large $N_c$ limit of SO(N_c) gauge theory coupled to a real scalar field following ideas of Rajeev. We see that the phase space of this resulting classical theory is Sp_1(H)/U(H_+) which is the analog of the Siegel disc in infinite dimensions. The linearized equations of motion give us a version of the well-known 't Hooft equation of two dimensional QCD. 
  In recent work, it was shown that velocity-dependent forces between parallel fundamental strings moving apart in a D-dimensional spacetime implied an expanding universe in D-1-dimensional spacetime. Here we expand on this work to obtain exact solutions for various string/brane cosmological toy models. 
  The energy eigenvalues of the superintegrable chiral Potts model are determined by the zeros of special polynomials which define finite representations of Onsager's algebra. The polynomials determining the low-sector eigenvalues have been given by Baxter in 1988. In the Z_3-case they satisfy 4-term recursion relations and so cannot form orthogonal sequences. However, we show that they are closely related to Jacobi polynomials and satisfy a special "partial orthogonality" with respect to a Jacobi weight function. 
  In noncommutative spaces, it is unknown whether the Pontrjagin class gives integer, as well as, the relation between the instanton number and Pontrjagin class is not clear. Here we define ``Instanton number'' by the size of $B_{\alpha}$ in the ADHM construction. We show the analytical derivation of the noncommuatative U(1) instanton number as an integral of Pontrjagin class (instanton charge) with the Fock space representation. Our approach is for the arbitrary converge noncommutative U(1) instanton solution, and is based on the anti-self-dual (ASD) equation itself. We give the Stokes' theorem for the number operator representation. The Stokes' theorem on the noncommutative space shows that instanton charge is given by some boundary sum. Using the ASD conditions, we conclude that the instanton charge is equivalent to the instanton number. 
  In the framework of the Sen conjectures a construction of vacuum superstring field theory on a non-BPS brane is discussed. A distinguished feature of this theory is a presence of a ghost kinetic operator mixing GSO+/- sectors. A candidate for such kinetic operator with zero cohomology is discussed. 
  The SW(3/2,3/2,2) superconformal algebra is a W algebra with two free parameters. It consists of 3 superconformal currents of spins 3/2, 3/2 and 2. The algebra is proved to be the symmetry algebra of the coset (su(2)+su(2)+su(2))/su(2). At the central charge c=21/2 the algebra coincides with the superconformal algebra associated to manifolds of G_2 holonomy. The unitary minimal models of the SW(3/2,3/2,2) algebra and their fusion structure are found. The spectrum of unitary representations of the G_2 holonomy algebra is obtained. 
  This is a new version of the paper, which uses the same methods as in the previous version, but the model is now different. We study two complex scalar fields coupled through a quadratic interaction in 2+1 dimensions. We use the method of bilinears as suggested by Rajeev. The resulting classical theory is studied within the linear approximation and we show that there is a possible bound state for the composite type particles for a range of coupling constant strengths. 
  In this paper we describe the moduli space of kinks in a class of systems of two coupled real scalar fields in (1+1) Minkowskian space-time. The main feature of the class is the spontaneous breaking of a discrete symmetry of (real) Ginzburg-Landau type that guarantees the existence of kink topological defects. 
  We discuss some issues arising in studying (linearized) gravity on non-BPS higher codimension branes in an infinite-volume bulk. In particular, such backgrounds are badly singular for codimension-3 and higher delta-function-like branes with non-zero tension. As we discuss in this note, non-trivial issues arise in smoothing out such singularities. Thus, adding higher curvature terms might be necessary in this context. 
  In this article, we study how the Grothendieck group of coherent sheaves can be used to describe D-branes. We show how global bound state construction in topological $K$-theory can be adapted to our context, showing that D-branes wrapping a subvariety are holomorphically classified by a relative $K$-group. By taking the duality between the relative $K$-groups and the $K$-homologies, we show that D-brane charge of type IIB superstrings is properly classified by the $K$-homology. 
  The second virial coefficient $B_{2}^{nc}(T)$ for non-interacting particles moving in a two-dimensional noncommutative space and in the presence of a uniform magnetic field $\vec B$ is presented. The noncommutativity parameter $\te$ can be chosen such that the $B_{2}^{nc}(T)$ can be interpreted as the second virial coefficient for anyons of statistics $\al$ in the presence of $\vec B$ and living on the commuting plane. In particular in the high temperature limit $\be\lga 0$, we establish a relation between the parameter $\te$ and the statistics $\al$. Moreover, $B_{2}^{nc}(T)$ can also be interpreted in terms of composite fermions. 
  The influence of an external electromagnetic field on the vacuum structure of a quantized Dirac field is investigated by considering the quantum corrections to classical Maxwell's lagrangian density induced by fluctuations of the non-perturbative vacuum. Effective Lagrangian densities for Maxwell's theory in (3+1) and (2+1) dimensions are derived from the vacuum zero-point energy of the fermion field in the context of a consistent Pauli-Villars-Rayski subtraction scheme, recovering Euler-Kockel-Heisenberg and Maxwell-Chern-Simons effective theories. Effective scalar quantum electrodymanics as well as low temperature effects in both spinor and scalar theories are also discussed. 
  We present a class of chiral non-supersymmetric D=4 field theories in which quadratic divergences appear only at two loops. They may be depicted as ``SUSY quivers'' in which the nodes represent a gauge group with extended e.g., N=4 SUSY whereas links represent bifundamental matter fields which transform as chiral multiplets with respect to different N=1 subgroups. One can obtain this type of field theories from simple D6-brane configurations on Type IIA string theory compactified on a six-torus. We discuss the conditions under which this kind of structure is obtained from D6-brane intersections. We also discuss some aspects of the effective low-energy field theory. In particular we compute gauge couplings and Fayet-Iliopoulos terms from the Born-Infeld action and show how they match the field theory results. This class of theories may be of phenomenological interest in order to understand the modest hierarchy problem i.e., the stability of the hierarchy between the weak scale and a fundamental scale of order 10-100 TeV which appears e.g. in low string scale models. Specific D-brane models with the spectrum of the SUSY Standard Model and three generations are presented. 
  In this paper we study $T^2$ compactification of 6-dimensional massive type IIA supergravity in presence of  Ramond-Ramond background fluxes. The resulting theory in four dimensions is shown to possess $SL(2,R)\times SL(2,R)\times O(4,20)$ duality symmetry. It is shown that specific elements of this symmetry relate massive type IIA compactified on $K3\times T^2$ (with fluxes along $K3$) to the ordinary type IIA compactified on $K3\times T^2$ (with fluxes along $T^2$). In turn, this relationship is exploited to relate Romans theory to heterotic strings. The D8-brane (domain-wall) wrapped on $K3\times T^2$ is related to {\it pure gravity} heterotic solution which is a direct product of 6-dimensional flat spacetime and a 4-dimensional Taub-NUT instanton. 
  In this paper, we systematically study the question of screening length in Abelian Chern-Simons theories. In the Abelian Higgs theory, where there are two massive poles in the gauge propagator at the tree level, we show that the coefficient of one of them becomes negligible at high temperature and that the screening length is dominantly determined by the parity violating part of the self-energy. In this theory, static magnetic fields are screened. In the fermion theory, on the other hand, the parity conserving part of the self-energy determines the screening length and static magnetic fields are not screened. Several other interesting features are also discussed. 
  Supersymmetry (SUSY) in non-relativistic quantum mechanics (QM) is applied to a 2-dimensional physical system: a neutron in an external magnetic field. The superpotential and the two-component wave functions of the ground state are found out. 
  These lectures provide an introduction to perturbative string theory and its construction on spaces with background Ramond flux. Traditional covariant quantization of the string and its connection with vertex operators and conformal invariance of the worldsheet theory are reviewed. A supersymmetric covariant quantization of the superstring in six and ten spacetime dimensions is discussed. Correlation functions are computed with these variables. Applications to strings in anti-de Sitter backgrounds with Ramond flux are analyzed. Based on lectures presented at the Theoretical Advanced Study Institute TASI 2001, June 3-29, 2001. 
  The number of asymptotically de Sitter (non-singular) solutions of 5d dilatonic gravity with positive cosmological constant is found. These solutions are similar to the previously known asymptotically AdS spaces where dilaton may generate the singularity. Using these solutions the consistent $c$-function is proposed in the same way as in AdS/CFT. The consistency of RG flow gives further support for dS/CFT correspondence. From holographic RG flow equations we calculate the holographic 4d conformal anomaly with dilatonic contributions. This conformal anomaly turns out to be the same as in AdS/CFT correspondence. 
  The relationship between Elliptic Ruijsenaars-Schneider (RS) and Calogero-Moser (CM) models with Sklyanin algebra is presented. Lax pair representations of the Elliptic RS and CM are reviewed. For n=2 case, the eigenvalue and eigenfunction for Lame equation are found by using the result of the Bethe ansatz method. 
  A model of spherically symmetric SU(2) gauge theory is considered. The self-duality equations are written and it is shown that they are compatible with the Einstein-Yang-Mills equations. It is proven that this property is true for any gauge theory with curved base space-time and having a compact Lie group as structural group. 
  In this paper we calculate the Casimir energy for spherical shell with massless self-interacting scalar filed which satisfying Dirichlet boundary conditions on the shell. Using zeta function regularization and heat kernel coefficients we obtain the divergent contributions inside and outside of Casimir energy. The effect of self-interacting term is similar with existing of mass for filed. In this case some divergent part arises. Using the renormalization procedure of bag model we can cancel these divergent parts. 
  We give a path-integral proof of level-rank duality in Kazama-Suzuki models for world-sheets of spherical topology. 
  The microscopic origin of black hole entropy remains one of the more intriguing open questions in theoretical physics. A subplot in this drama is the renowned Cardy-Verlinde formula, which uses two-dimensional conformal formalism to explain the entropy of an arbitrary-dimensional black hole. In this paper, by exploiting the AdS/CFT and black hole-string dualities, we are able to provide a physical picture for this paradoxical behavior. Following a recent study by Haylo (in a dS context), we show that the dual CFT for an asymptotically AdS spacetime actually conforms to a string-like description. Moreover, we demonstrate that this stringy CFT is directly related to a string that lives on the stretched horizon of an AdS-Schwarzschild-like black hole. In fact, after an appropriate renormalization, these two boundary theories are shown to be thermodynamically equivalent. 
  We give a simplified and more complete description of the loop variable approach for writing down gauge invariant equations of motion for the fields of the open string. A simple proof of gauge invariance to all orders is given. In terms of loop variables, the interacting equations look exactly like the free equations, but with a loop variable depending on an extra parameter, thus making it a band of finite width. The arguments for gauge invariance work exactly as in the free case. We show that these equations are Wilsonian RG equations with a finite world-sheet cutoff and that in the infrared limit, equivalence with the Callan-Symanzik $\beta$-functions should ensure that they reproduce the on-shell scattering amplitudes in string theory. It is applied to the tachyon-photon system and the general arguments for gauge invariance can be easily checked to the order calculated. One can see that when there is a finite world sheet cutoff in place, even the U(1) invariance of the equations for the photon, involves massive mode contributions. A field redefinition involving the tachyon is required to get the gauge transformations of the photon into standard form. 
  We study the possible quantum anomaly for the transverse Ward-Takahashi relations in four dimensional gauge theories based on the method of computing the axial-vector and the vector current operator equations. In addition to the well-known anomalous axial-vector divergence equation (the Adler-Bell-Jackiw anomaly), we find the anomalous axial-vector curl equation, which leads to the quantum anomaly of the transverse Ward-Takahashi relation for the axial-vector vertex. The computation shows that there is no anomaly for the transverse Ward-Takahashi relation for the vector vertex. 
  The vertex operators for the supergravity multiplet can be constructed through the Wilson lines in IIB matrix model. We investigate the structure of the vertex operators and the symmetry of their correlation functions. For this purpose, we perturb the theory by the Wilson lines dual to the supergravity multiplet. The structure of the Wilson lines can be determined by requiring $\cal{N}$=2 SUSY under the low energy approximation. We argue that the generating functional of the correlators is invariant under local SUSY transformations of IIB supergravity. 
  We study asymptotic distribution of eigen-values $\omega$ of a quadratic operator polynomial of the following form $(\omega^2-L(\omega))\phi_\omega=0$, where $L(\omega)$ is a second order differential positive elliptic operator with quadratic dependence on the spectral parameter $\omega$. We derive asymptotics of the spectral density in this problem and show how to compute coefficients of its asymptotic expansion from coefficients of the asymptotic expansion of the trace of the heat kernel of $L(\omega)$. The leading term in the spectral asymptotics is the same as for a Laplacian in a cavity. The results have a number of physical applications. We illustrate them by examples of field equations in external stationary gravitational and gauge backgrounds. 
  When a quantum field theory has a symmetry, global or local like in gauge theories, in the tree or classical approximation formal manipulations lead to believe that the symmetry can also be implemented in the full quantum theory, provided one uses the proper quantization rules. While this is often true, it is not a general property and therefore requires a proof because simple formal manipulations ignore the unavoidable divergences of perturbation theory. The existence of invariant regularizations allows solving the problem in most cases but the combination of gauge symmetry and chiral fermions leads to subtle issues. Depending on the specific group and field content, anomalies are found: obstructions to the quantization of chiral gauge symmetries.Because anomalies take the form of local polynomials in the fields, are linked to local group transformations, but vanish for global (rigid) transformations they have a topological character.In these notes we review various perturbative and non-perturbative regularization techniques, and show that they leave room for possible anomalies when both gauge fields and chiral fermions are present. We determine the form of anomalies in simple examples. We relate anomalies to the index of the Dirac operator in a gauge background. We exhibit gauge instantons that contribute to the anomaly in the example of the CP(N-1) models and SU(2) gauge theories. We briefly mentioned a few physical consequences. For many years the problem of anomalies had been discussed only within the framework of perturbation theory. New non-perturbative solutions based on lattice regularization have recently been proposed. We describe the so-called overlap and domain wall fermion formulations. 
  We describe a topological effect on configurations of D-branes in the presence of NS-NS and RR field strength fluxes. The fluxes induce the appearance of chiral anomalies on lower dimensional submanifolds of the D-brane worldvolume. This anomaly is not associated to a dynamical chiral fermion degree of freedom, but rather should be regarded as an explicit flux-induced anomalous term (Wess-Zumino term) in the action. The anomaly is cancelled by an inflow mechanism, which exploits the fact that fluxes can act as sources of RR fields. We discuss several applications of this flux-induced anomaly; among others, its role in understanding anomaly cancellation in compactifications with D-branes and fluxes, and the possibility of phase transitions where a chiral fermion disappears from the D-brane world-volume spectrum, being replaced by an explicit Wess-Zumino term. We comment on the relation among different mechanisms to obtain four-dimensional chirality in string theory. 
  It is shown that the violation of unitarity observed in space/time noncommutative field theories is due to an improper definition of quantum field theory on noncommutative spacetime. 
  Discretized nonabelian gauge theories living on finite group spaces G are defined by means of a geometric action \int Tr F \wedge *F. This technique is extended to obtain discrete versions of the Born-Infeld action. The discretizations are in 1-1 correspondence with differential calculi on finite groups.   A consistency condition for duality invariance of the discretized field equations is derived for discretized U(1) actions S[F] living on a 4-dimensional abelian G. Discretized electromagnetism satisfies this condition and therefore admits duality rotations.   Yang-Mills and Born-Infeld theories are also considered on product spaces M^D x G, and we find the corresponding field theories on M^D after Kaluza-Klein reduction on the G discrete internal spaces. We examine in some detail the case G=Z_N, and discuss the limit N -> \infty.   A self-contained review on the noncommutative differential geometry of finite groups is included. 
  We study a deformation of the type IIB Maldacena-Nunez background which arises as the near-horizon limit of NS5 branes wrapped on a two-cycle. This background is dual to a "little string theory" compactified on a two-sphere, a theory which at low energies includes four-dimensional N = 1 super Yang-Mills theory. The deformation we study corresponds to a mass term for some of the scalar fields in this theory, and it breaks supersymmetry completely. In the language of seven-dimensional SO(4) gauged supergravity the deformation involves (at leading order) giving a VEV, depending only on the radial coordinate, to a particular scalar field. We explicitly construct the corresponding solution at leading order in the deformation, both in seven-dimensional and in ten-dimensional supergravity, and we verify that it completely breaks supersymmetry. Since the original background had a mass gap and we are performing a small deformation, the deformed background is guaranteed to be stable even though it is not supersymmetric. 
  It is shown that an action inspired from a BF and Chern-Simons model, based on the $AdS_4$ isometry group SO(3, 2), with the inclusion of a Higgs potential term, furnishes the MacDowell-Mansouri-Chamseddine-West action for gravity, with a Gauss-Bonnet and cosmological constant term. The $AdS_4$ space is a natural vacuum of the theory. Using Vasiliev's procedure to construct higher spin massless fields in AdS spaces and a suitable star product, we discuss the preliminary steps to construct the corresponding higher-spin action in $AdS_4$ space representing the higher spin extension of this model. Brief remarks on Noncommutative Gravity are made. 
  Free arbitrary spin massless and self-dual massive fields propagating in $AdS_5$ are investigated. We study totally symmetric and mixed symmetry fields on an equal footing. Light-cone gauge action for such fields is constructed. As an example of application of light-cone formalism we discuss $AdS/CFT$ correspondence for massless arbitrary spin $AdS$ fields and corresponding boundary operators at the level of two point function. 
  In this note we study warped compactifications of M-theory on manifolds of Spin(7) holonomy in the presence of background 4-form flux. The explicit form of the superpotential can be given in terms of the self -dual Cayley calibration on the Spin(7) manifold, in agreement with the general formula propsed in hep-th/9911011. 
  We show that the Lukierski et al. model, invariant with respect to the two-fold centrally extended Galilei group, can be decomposed into an infinite number of independent copies (differing in their spin) of the ``exotic'' particle of Duval et al. The difference between the two models is found to be sensitive to electromagnetic coupling. The nature of the noncommutative plane coordinates is discussed in the light of the exotic Galilean symmetry. We prove that the first model, interpreted as describing a non-relativistic anyon, is the non-relativistic limit of a particle with torsion related to relativistic anyons. 
  A previous calculation on the tachyon state arising as fluctuations of a $D$ brane in vacuum string field theory is extended to include the vector state. We use the boundary conformal field theory approach of Rastelli, Sen and Zwiebach to construct a vector state. It is shown that the vector field satisfies the linearized equations of motion provided the two conditions $k^2=0$ and $k^\mu A_\mu=0$ are satisfied. Earlier calculations using Fock space techniques by Hata and Kawano have found massless vector states that are not necessarily transverse. 
  The semi-classical quantisation of the two lowest energy static solutions of boundary sine-Gordon model is considered. A relation between the Lagrangian and bootstrap parameters is established by comparing their quantum corrected energy difference and the exact one. This relation is also confirmed by studying the semi-classical limit of soliton reflections on the boundary. 
  Two dimensional conformal field theories with central charge one are discussed. After a short review of theories based on one free boson, a different CFT is described, which is obtained as a limit of minimal models. 
  We consider a minimally coupled massless scalar propagating in the background of NS5-branes in the presence of a 2-form RR electric field. The supergravity solution also called the (NS5,D1) bound state in the appropriate decoupling limit is the holographic dual of Open D-string (OD1) theory. Using this information, we compute the two-point function of the operators in OD1 theory which couples to the massless string states such as the dilaton. We will indicate how to obtain the absorption cross-section of a scalar on NS5-branes (i.e. Little String Theory (LST)) obtained earlier, from our results in the low-energy limit. 
  We describe generalized D=11 Poincar\'{e} and conformal supersymmetries. The corresponding generalization of twistor and supertwistor framework is outlined with $OSp(1|64)$ superspinors describing BPS preons. The $\frac{k}{32}$ BPS states as composed out of $n=32 - k$ preons are introduced, and basic ideas concerning BPS preon dynamics is presented. The lecture is based on results obtained by J.A. de Azcarraga, I. Bandos, J.M. Izquierdo and the author$^1$. 
  We consider the geometrical formulation in central charge superspace of the N=4 supergravity containing an antisymmetric tensor gauge field. The theory is on-shell, so clearly, the constraints used for the identification of the multiplet together with the superspace Bianchi identities imply equations of motion for the component fields. We deduce these equations of motion in terms of supercovariant quantities and then, we give them in terms of component fields. These equations of motion, deduced from the geometry, without supposing the existence of a Lagrangian, are found to be the same as those derived from the Lagrangian given in the component formulation of this N=4 supergravity multiplet by Nicolai and Townsend. 
  Chiral condensates in the trivial light-cone vacuum emerge if defined as short-time limits of fermion propagators. In gauge theories, the necessary inclusion of a gauge string in combination with the characteristic light-cone infrared singularities contain the relevant non-perturbative ingredients responsible for formation of the condensate, as demonstrated for the 't Hooft model. 
  We study the N=2 super Yang Mills theory living in the world volume of a bound state made of fractional D3/D7 branes at the orbifold R^{1,5}*R^4/ Z_2, by using the probe technique. We also discuss the boundary action for the system. 
  The requirement that the quantum partition function be invariant under a renormalization group transformation results in a wide class of exact renormalization group equations, differing in the form of the kernel. Physical quantities should not be sensitive to the particular choice of the kernel. We demonstrate this scheme independence in four dimensional scalar field theory by showing that, even with a general kernel, the one-loop beta function may be expressed only in terms of the effective action vertices, and thus, under very general conditions, the universal result is recovered. 
  We investigate the compatibility of Lorentz-violating quantum field theories with the requirements of causality and stability. A general renormalizable model for free massive fermions indicates that these requirements are satisfied at low energies provided the couplings controlling the breaking are small. However, for high energies either microcausality or energy positivity or both are violated in some observer frame. We find evidence that this difficulty can be avoided if the model is interpreted as a sub-Planckian approximation originating from a nonlocal theory with spontaneous Lorentz violation. The present study thereby supports the validity of the standard-model extension as the low-energy limit of any realistic string theory that exhibits spontaneous Lorentz breaking. 
  We compute the representations (``nimreps'') of the fusion algebra of affine sl(N), which determine the boundary conditions of sl(N) WZW theories twisted by the charge conjugation. This is done following two procedures, one of general validity, the other specific to the problem at hand. The problem is related to the classical problem of decomposition of the fundamental representations of sl(N) onto representations of $B_l=so(2l+1)$ or $C_l =sp(2l)$ algebras. The relevant nimreps and their diagonalisation matrix are thus expressed in terms of modular data of the affine B or C algebras 
  Recent papers by Finkelstein, Galiautdinov, and coworkers {[J. Math. Phys. 42, 1489, 3299 (2001)]} discuss a suggestion by Wilczek that nonabelian projective representations of the permutation group can be used as a new type of particle statistics, valid in any dimension. Wilczek's suggestion was based in part on an analysis by Nayak and Wilczek (NW) of the nonabelian representation of the braid group in a quantum Hall system. We point out that projective permutation statistics is not possible in a local quantum field theory as it violates locality, and show that the NW braid group representation is not equivalent to a projective representation of the permutation group. The structure of the finite image of the braid group in a 2^{n/2-1}-dimensional representation is obtained. 
  It has recently been shown that there exist stable inhomogeneous neutral black strings in higher dimensional gravity. These solutions were motivated by the fact that the corresponding homogeneous solutions are unstable. We show that there exist new inhomogeneous black string and black p-brane solutions even when the corresponding translationally invariant solutions are stable. In particular, we show there exist inhomogeneous near-extremal black strings and p-branes. Some of these solutions remain inhomogeneous even when the size of the compact direction (at infinity) is very small. 
  We present a review work on Supersymmetric Classical Mechanics in the context of a Lagrangian formalism, with $N=1-$supersymmetry. We show that the N=1 supersymmetry does not allow the introduction of a potential energy term depending on a single commuting supercoordinate, $\phi (t;\Theta)$. 
  The entropy of the states associated to the solutions of the equations of motion of the bosonic open string with combinations of Neumann and Dirichlet boundary conditions is given. Also, the entropy of the string in the states $| A^i > = \alpha^{i}_{-1} |0>$ and $| \phi^a > = \alpha^{a}_{-1} |0>$ that describe the massless fields on the world-volume of the Dp-brane is computed. 
  A family of SO(10) symmetric instanton solutions in Type IIB supergravity is developed. The instanton of least action is a candidate for the low-energy, semiclassical approximation to the {D=--1} brane. Unlike a previously published solution,[GGP] this admits an interpretation as a tunneling amplitude between perturbatively degenerate asymptotic states, but with action twice that found previously. A number of associated issues are discussed such as the relation between the magnetic and electric pictures, an inversion symmetry of the dilaton and the metric, the $R\times S^9$ topology of the background, and some properties of the solution in an "instanton frame" corresponding to a Lagrangian in which the dilaton's kinetic energy vanishes. 
  We consider the two most studied proposals of relativity theories with observer-independent scales of both velocity and length/mass: the one discussed by Amelino-Camelia as illustrative example for the original proposal (gr-qc/0012051) of theories with two relativistic invariants, and an alternative more recently proposed by Magueijo and Smolin (hep-th/0112090). We show that these two relativistic theories are much more closely connected than it would appear on the basis of a naive analysis of their original formulations. In particular, in spite of adopting a rather different formal description of the deformed boost generators, they end up assigning the same dependence of momentum on rapidity, which can be described as the core feature of these relativistic theories. We show that this observation can be used to clarify the concepts of particle mass, particle velocity, and energy-momentum-conservation rules in these theories with two relativistic invariants. 
  The question of general covariance in quantum gravity is considered in the first post-Newtonian approximation. Transformation properties of observable quantities under deformations of a reference frame, induced by variations of the gauge conditions fixing general invariance, are determined. It is found that the one-loop contributions violate the principle of general covariance, in the sense that the quantities which are classically invariant under such deformations take generally different values in different reference frames. The relative value of this violation is of the order 1/N, where N is the number of particles in a gravitating body. 
  The Cardy-Verlinde formula is further verified by using the Kerr-Newman-AdS$_4$ and Kerr-Newman-dS$_4$ black holes. In the Kerr-Newman-AdS$_4$ spacetime, we find that, for strongly coupled CFTs with AdS duals, to cast the entropy of the CFT into the Cardy-Verlinde formula the Casimir energy must contains the terms $ -n ({\mathcal{J}} \Omega_H+ \frac{Q\Phi}{2}+ \frac{Q\Phi_0}{2})$, which associate with rotational and electric potential energies, and the extensive energy includes the term $-Q \Phi_0$. For the Kerr-Newman-dS$_4$ black hole, we note that the Casimir energy is negative but the extensive energy is positive on the cosmological horizon; while the Casimir energy is positive but the extensive energy is negative on the event horizon (the definitions for the two energies possess the same forms as the corresponding quantities of the Kerr-Newman-AdS$_4$ black hole). Thus we have to take the absolute value of the Casimir (extensive) energy in the Cardy-Verlinde formula for the cosmological (event) horizon. The result for the Kerr-Newman-dS$_4$ spacetime provides support of the dS/CFT correspondence. Furthermore, we also obtain the Bekenstein-Verlinde-like entropy bound for the Kerr-Newman-AdS$_4$ black hole and the D-bound on the entropy of matter system in Kerr-Newman-dS$_4$ spacetime. We find that both the bounds are tightened by the electric charge. 
  We study one-loop effective action of Berkooz-Douglas Matrix theory and obtain non-abelian action of D0-branes in the background field produced by longitudinal 5-branes. Since these 5-branes do not have D0-brane charge and are not present in BFSS Matrix theory, our analysis can be regarded as an independent test for the coupling of D-branes to general weak backgrounds proposed by Taylor and Van Raamsdonk, and also as a check of consistency between the two versions of Matrix theory. The coupling to the 5-branes which we obtain is basically consistent with the previous proposal, but we point out subtleties in the ordering of matrices for the multipole moments. 
  Using the recent proposal for the observables in open string field theory, we explicitly compute the coupling of closed string tachyon and massless states with the open string states up to level two. Using these couplings, we then calculate the tree level S-matrix elements of two closed string tachyons or two massless states in the open string field theory. Up to some contact terms, the results reproduce exactly the corresponding amplitudes in the bosonic string theory. 
  We compute the Schwinger term in the gravitational constraints in two dimensions, starting from the path integral in Hamiltonian form and the Einstein anomaly. 
  We show that the Seiberg-Witten map for a noncommutative gauge theory involves a noncommutative 1-cocycle. The cocycle condition enforces a consistency requirement, which has been previously derived. 
  We report on a test of the Maldacena conjecture. This string/field theory correspondence has interesting applications. When combined with Rehren's theorem, it has implications for issues concerning space-time structure and Lorentz symmetry. Our results indicate that the conjecture is correct. We are within 10-15% of the expected results, although the numerical evidence is not yet decisive. 
  In these lecture notes we first assemble the basic ingredients of supersymmetric gauge theories (particularly N=4 super-Yang-Mills theory), supergravity, and superstring theory. Brane solutions are surveyed. The geometry and symmetries of anti-de Sitter space are discussed. The AdS/CFT correspondence of Maldacena and its application to correlation functions in the the conformal phase of N=4 SYM are explained in considerable detail. A pedagogical treatment of holographic RG flows is given including a review of the conformal anomaly in four-dimensional quantum field theory and its calculation from five-dimensional gravity. Problem sets and exercises await the reader. 
  We comment on the conformal boundary states of the c=1 free boson theory on a circle which do not preserve the U(1) symmetry. We construct these Virasoro boundary states at a generic radius by a simple asymmetric shift orbifold acting on the fundamental boundary states at the self-dual radius. We further calculate the boundary entropy and find that the Virasoro boundary states at irrational radius have infinite boundary entropy. The corresponding open string description of the asymmetric orbifold is given using the quotient algebra construction. Moreover, we find that the quotient algebra associated with a non-fundamental boundary state contains the noncommutative Weyl algebra. 
  We consider the question raised by Unruh and Wald of whether mirrored boxes can self-accelerate in flat spacetime (the ``self-accelerating box paradox'').   From the point of view of the box, which perceives the acceleration as an impressed gravitational field, this is equivalent to asking whether the box can be supported by the buoyant force arising from its immersion in a perceived bath of thermal (Unruh) radiation. The perfect mirrors we study are of the type that rely on light internal degrees of freedom which adjust to and reflect impinging radiation. We suggest that a minimum of one internal mirror degree of freedom is required for each bulk field degree of freedom reflected. A short calculation then shows that such mirrors necessarily absorb enough heat from the thermal bath that their increased mass prevents them from floating on the thermal radiation. For this type of mirror the paradox is therefore resolved. We also observe that this failure of boxes to ``float'' invalidates one of the assumptions going into the Unruh-Wald analysis of entropy balances involving boxes lowered adiabatically toward black holes. Nevertheless, their broad argument can be maintained until the box reaches a new regime in which box-antibox pairs dominate over massless fields as contributions to thermal radiation. 
  String and M-theory realizations of brane world supersymmetry breaking scenarios are considered in which visible sector Standard Model fields are confined on a brane, with hidden sector supersymmetry breaking isolated on a distant brane. In calculable examples with an internal manifold of any volume the Kahler potential generically contains brane--brane non-derivative contact interactions coupling the visible and hidden sectors and is not of the no-scale sequestered form. This leads to non-universal scalar masses and without additional assumptions about flavor symmetries may in general induce dangerous sflavor violation even though the Standard Model and supersymmetry branes are physically separated. Deviations from the sequestered form are dictated by bulk supersymmetry and can in most cases be understood as arising from exchange of bulk supergravity fields between branes or warping of the internal geometry. Unacceptable visible sector tree-level tachyons arise in many models but may be avoided in certain classes of compactifications. Anomaly mediated and gaugino mediated contributions to scalar masses are sub-dominant except in special circumstances such as a flat or AdS pure five--dimensional bulk geometry without bulk vector multiplets. 
  Just as non-commutative gauge theories arise from quantising open strings in a large magnetic field, non-Abelian two-form gauge theories may conceivably be constructed by quantising open membranes in a large three-form magnetic background. We make some observations that arise in following this strategy, with an emphasis on the relation to the quantisation of volume-preserving diffeomorphisms (vpd). In particular, we construct consistent non-Abelian interactions of a two-form in 3+1 dimensions, based on gauge invariance under vpd. 
  We argue that multi-trace deformations of the boundary CFT in AdS/CFT correspondence can arise through the OPE of single-trace operators. We work out the example of a scalar field in AdS_5 with cubic self interaction. By an appropriate reparametrization of the boundary data we are able to deform the boundary CFT by a marginal operator that couples to the conformal anomaly. Our method can be used in the analysis of multi-trace deformations in N=4 SYM where the OPEs of various single-trace operators are known. 
  We study the extent to which the gauge symmetry of abelian Yang-Mills can be deformed under two conditions: first, that the deformation depend on a two-form scale. Second, that the deformation preserve supersymmetry. We show that (up to a single parameter) the only allowed deformation is the one determined by the star product. We then consider the supersymmetry algebra satisfied by NCYM expressed in commutative variables. The algebra is peculiar since the supercharges are not gauge-invariant. However, the action, expressed in commutative variables, appears to be quadratic in fermions to all orders in theta. 
  A proposal towards a microscopic understanding of the Bekenstein-Hawking entropy for D=4 spacetimes with event horizon is made. Since we will not rely on supersymmetry these spacetimes need not be supersymmetric. Euclidean D-branes which wrap the event horizon's boundary will play an important role. After arguing for a discretization of the Euclidean D-brane worldvolume based on the worldvolume uncertainty relation, we count chainlike excitations on the worldvolume of specific dual Euclidean brane pairs. Without the need for supersymmetry it is shown that one can thus reproduce the D=4 Bekenstein-Hawking entropy and its logarithmic correction. 
  We consider Einstein gravity coupled to a CFT made of a single free conformal scalar in 4-d Anti de Sitter space. This simple case is rich enough to explain an unexpected gravitational Higgs phenomenon that has no flat-space counterpart, yet simple enough that many calculations can be carried on exactly. Specifically, in this paper we compute the graviton self energy due to matter, and we exhibit its spectral representation. This enables us to find the spin-2 bound-state content of the system. 
  There are gauge-transformation operators applicable to massless spin-1/2 particles within the little-group framework of internal space-time symmetries of massive and massless particles. It is shown that two of the $SL(2,c)$ spinors are invariant under gauge transformations while the remaining two are not. The Dirac equation contains only the gauge-invariant spinors leading to polarized neutrinos. It is shown that the gauge-dependent $SL(2,c)$ spinor is the origin of the gauge dependence of electromagnetic four-potentials. It is noted also that, for spin-1/2 particles, the symmetry group for massless particles is an infinite-momentum/zero-mass limit of the symmetry group for massive particles. 
  With two typical parent actions we have two kinds of dual worlds: i) one of which contains an electric as well as magnetic current, and ii) the other contains (generalized) Chern-Simons terms. All these fields are defined on a curved spacetime of arbitrary (odd) dimensions. A new form of gauge transformations is introduced and plays an essential role in defining the interaction with a magnetic monopole or in defining the generalized Chern-Simons terms. 
  4-dimensional optics is here introduced axiomatically as the space that supports a Universal wave equation which is applied to the postulated Higgs field. Self-guiding of this field is shown to produce all the modes necessary to provide explanations for the known elementary particles. Forces are shown to appear as evanescent fields due to waveguiding of the Higgs field, which provide coupling between waveguides corresponding to different particles. Carrier particles are also discussed and shown to correspond to waveguided modes existing in 3-dimensional space. 
  We study the spectra of BPS excitations of D1D5 bound states in a class of free orbifolds/orientifolds of type IIB theory and its dual descriptions in terms of chiral primaries of the corresponding $AdS_3$ supergravities. 
  It is well known that the classical equations of motion of Maxwell and Born-Infeld theories are invariant under a duality symmetry acting on the field strengths. We review the implementation of the SL(2,Z) duality in these theories as linear but non-local transformations of the potentials. 
  We consider the associativity or Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations and discuss one of the most relevant for non-perturbative physics class of their solutions based on existence of the residue formulas. It is demonstrated for this case that the proof of associativity equations is reduced to the problem of solving system of algebraic linear equations. The particular examples of solutions related to Landau-Ginzburg topological theories, Seiberg-Witten theories and tau-functions of quasiclassical hierarchies are discussed in detail. We also discuss related questions including covariance of associativity equations, their relation to dispersionless Hirota relations and auxiliary linear problem for the WDVV equations. 
  We generalize the results of hep-th/0008140 to the case of the (n+1)-dimensional closed FRW universe satisfying a general equation of state of the form p=w\rho. We find that the entropy of the universe can no longer be expressed in a form similar to the Cardy formula, when w\neq 1/n. As a result, in general the entropy formula does not coincide with the Friedmann equation when the conjectured bound on the Casimir energy is saturated. Furthermore, the conjectured bound on the Casimir energy generally does not lead to the Hubble and the Bekenstein entropy bounds. 
  We show that the presence of finite-size monopoles can lead to a number of interesting physical processes involving quantum entanglement of charges. Taking as a model the classical solution of the N=2 SU(2) Yang-Mills theory, we study interaction between dyons and scalar particles in the adjoint and fundamental representation. We find that there are bound states of scalars and dyons, which, remarkably, are always an entangled configuration of the form |\psi > =|dyon+> |scalar-> +/- |dyon->|scalar+>. We determine the energy levels and the wave functions and also discuss their stability. 
  We determine, in the context of five-dimensional ${\cal N}=2$ gauged supergravity with vector and hypermultiplets, the conditions under which curved (non Ricci flat) supersymmetric domain wall solutions may exist. These curved BPS domain wall solutions may, in general, be supported by non-constant vector and hyper scalar fields. We establish our results by a careful analysis of the BPS equations as well as of the associated integrability conditions and the equations of motion. We construct an example of a curved BPS solution in a gauged supergravity model with one hypermultiplet. We also discuss the dual description of curved BPS domain walls in terms of RG flows. 
  We find a large family of solutions to the Dirac equation on a manifold of $G_2$ holonomy asymptotic to a cone over $S^3 \times S^3$, including all radial solutions. The behaviour of these solutions is studied as the manifold developes a conical singularity. None of the solutions found are both localised and square integrable at the origin. This result is consistent with the absence of chiral fermions in M-theory on the conifold over $S^3\times S^3$. The approach here is complementary to previous analyses using dualities and anomaly cancellation. 
  Within D=5 N=2 gauged supergravity coupled to hypermultiplets we derive consistency conditions for BPS domain walls with constant negative curvature on the wall. For such wall solutions to exist, the covariant derivative of the projector, governing the constraint on the Killing spinor, has to be non-zero and proportional to the cosmological constant on the domain walls. We also prove that in this case solutions of the Killing spinor equations are solutions of the equations of motion. We present explicit, analytically solved examples of such domain walls, employing the universal hypermultiplet fields. These examples involve the running of two scalar fields and the space-time in the transverse direction that is cut off at a critical distance, governed by the magnitude of the negative cosmological constant on the wall. 
  We describe in detail a general scheme for embedding several BPS monopoles into a theory with a larger gauge group, which generalizes embeddings of SU(2) monopoles discussed by several authors. This construction is applied to obtain explicit fields for monopoles with several charge configurations in the SU(5)\to [SU(3)\times SU(2)\times U(1)]/Z_6 model. 
  Recently, the author has constructed a series of four dimensional non-critical string theories with eight supercharges, dual to theories of light electric and magnetic charges, for which exact formulas for the central charge of the space-time supersymmetry algebra as a function of the world-sheet couplings were obtained. The basic idea was to generalize the old matrix model approach, replacing the simple matrix integrals by the four dimensional matrix path integrals of N=2 supersymmetric Yang-Mills theory, and the Kazakov critical points by the Argyres-Douglas critical points. In the present paper, we study qualitatively similar toy path integrals corresponding to the two dimensional N=2 supersymmetric non-linear sigma model with target space CP^n and twisted mass terms. This theory has some very strong similarities with N=2 super Yang-Mills, including the presence of critical points in the vicinity of which the large n expansion is IR divergent. The model being exactly solvable at large n, we can study non-BPS observables and give full proofs that double scaling limits exist and correspond to universal continuum limits. A complete characterization of the double scaled theories is given. We find evidence for dimensional transmutation of the string coupling in some non-critical string theories. We also identify en passant some non-BPS particles that become massless at the singularities in addition to the usual BPS states. 
  It is argued that whereas supersymmetry requires the instanton contribution to the expectation value of a straight Wilson line in the N=4 supersymmetric SU(2) Yang-Mills theory to vanish, it is not required to vanish in the case of a circular Wilson loop. A non-vanishing value can arise from a subtle interplay between a divergent integral over bosonic moduli and a vanishing integral over fermionic moduli. The one-instanton contribution to such Wilson loops is explicitly evaluated in semi-classical approximation. The method utilizes the symmetries of the problem to perform the integration over the bosonic and fermionic collective coordinates of the instanton. The integral is singular for small instantons touching the loop and is regularized by introducing a cutoff at the boundary of the (euclidean) AdS_5 moduli space. In the case of a circular loop a nonzero finite result is obtained when the cutoff is removed and a perimeter divergence subtracted. This is contrasted with the case of the straight line where the result is zero after subtraction of an identical divergence per unit length. The linear divergence is an artifact of our non supersymmetric regulator that deserves further consideration. The generalization to gauge group SU(N) with arbitrary N is straightforward in the limit of small 't Hooft coupling. The extension to strong 't Hooft coupling is more challenging and only a qualitative discussion is given of the AdS/CFT correspondence 
  We discuss an environmentally friendly renormalization group approach to analyze phase transitions. We intend to apply this method to the Electroweak Phase Transition. This work is in progress. We present some previously obtained results concerning a scalar theory, where the main features of this algorithm are introduced. 
  We study black hole evaporation of near-extremal black holes in spherically reduced models with asymptotically Minkowskian spacetime, with the effects of the back-reaction on the geometry included semi-classically. The stress-energy tensor is calculated for null in-falling observers. It is shown that the evaporation proceeds smoothly and there are no instabilities of the outer or inner apparent horizon before the endpoint of evaporation. 
  We present a pair of symmetric formulations of the matter sector of the stationary effective action of heterotic string theory that arises after the toroidal compactification of d dimensions. The first formulation is written in terms of a pair of matrix potentials Z_1 and Z_2 which exhibits a clear symmetry between them and can be used to generate new families of solutions on the basis of either Z_1 or Z_2; the second one is an O(d+1,d+n+1)-invariant formulation which is written in terms of a matrix vector W endowed with an O(d+1,d+n+1)-invariant scalar product which linearizes the action of the O(d+1,d+n+1) symmetry group on the coset space O(d+1,d+n+1)/[O(d+1)XO(d+n+1)]; this fact opens as well a simple solution--generating technique which can be applied on the basis of known solutions. A special class of extremal solutions is indicated by asuming a simple ansatz for the matrix vector W that reduces the equation of motion to the Laplace equation for a real scalar function. 
  It is known that electric-magnetic duality transformations are symmetries of the generalized Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations. In Seiberg-Witten theory the solutions to these equations come in certain sets according to the gauge group. We show that the duality transformations transform solutions within a set to another solution within the same set, by using a subset of the Picard-Fuchs equations on the Seiberg-Witten family of Riemann surfaces. The electric-magnetic duality transformations can be thought of as changes of a canonical homology basis on the surfaces which in our derivation is clearly responsible for the covariance of the generalized WDVV system. 
  Assuming the existence of a $dS/CFT$ correspondence we study the holograms of sources moving along geodesics in the bulk by calculating the one point functions they induce in the boundary theory. In analogy with a similar study of uniformly accelerated sources in $AdS$ spacetime, we argue that comoving geodesic observers correspond to a coordinate system on the boundary in which the one point function is {\it constant}. For $dS_3$ we show that the conformal transformations on the boundary which achieve this - when continued suitably to Lorentzian signature - induce nontrivial Bogoliubov transformations between modes, leading to a thermal spectrum. This may be regarded as a holographic signature of thermality detected by bulk geodesic observers. 
  We derive the second variation Lagrangian of the Randall-Sundrum model with two branes, study its gauge invariance and diagonalize it in the unitary gauge. We also show that the effective four-dimensional theory looks different on different branes and calculate the observable mass spectra and the couplings of the physical degrees of freedom of 5-dimensional gravity to matter. 
  A model of matter-coupled gravity in two dimensions is quantized. The crucial requirement for performing the quantization is the vanishing of the conformal anomaly, which is achieved by tuning a parameter in the interaction potential. The spectrum of the theory is determined by mapping the model first onto a field theory with a Liouville interaction, then onto a linear dilaton conformal field theory. In absence of matter fields a pure gauge theory with massless ground state is found; otherwise it is possible to minimally couple up to 11 matter scalar fields: in this case the ground state is tachyonic and the matter sector decouples, like the transverse oscillators in the critical bosonic string. 
  Noncommutative IR singularities and UV/IR mixing in relation with the Goldstone theorem for complex scalar field theory are investigated. The classical model has two coupling constants, $\lambda_1$ and $\lambda_2$, associated to the two noncommutative extensions $\phi^*\star\phi\star\phi^*\star\phi$ and $\phi^*\star\phi^*\star\phi\star\phi$ of the interaction term $|\phi|^4$ on commutative spacetime. It is shown that the symmetric phase is one-loop renormalizable for all $\lambda_1$ and $\lambda_2$ compatible with perturbation theory, whereas the broken phase is proved to exist at one loop only if $\lambda_2=0$, a condition required by the Ward identities for global U(1) invariance. Explicit expressions for the noncommutative IR singularities in the 1PI Green functions of both phases are given. They show that UV/IR duality does not hold for any of the phases and that the broken phase is free of quadratic noncommutative IR singularities. More remarkably, the pion selfenergy does not have noncommutative IR singularities at all, which proves essential to formulate the Goldstone theorem at one loop for all values of the spacetime noncommutativity parameter $\theta$. 
  N=2 SQED with several flavors admits multiple, static BPS domain wall solutions. We determine the explicit two-kink metric and examine the dynamics of colliding domain walls. The multi-kink metric has a toric Kahler structure and we reduce the Kahler potential to quadrature. In the second part of this paper, we consider semi-local vortices compactified on circle. We argue that, in the presence of a suitable Wilson line, the vortices separate into domain wall constituents. These play the role of fractional instantons in two-dimensional gauge theories and sigma-models. 
  The nonperturbative fermion-boson vertex function in four-dimensional Abelian gauge theories is self-consistently and exactly derived in terms of a complete set of normal (longitudinal) and transverse Ward-Takahashi relations for the The nonperturbative fermion-boson vertex function in four-dimensional Abelian gauge theories is self-consistently and exactly derived in terms of a complete set of normal(longitudinal) and transverse Ward-Takahashi relations for the fermion-boson and the axial-vector vertices in the case of massless fermion, in which the possible quantum anomalies and perturbative corrections are taken into account simultaneously. We find that this nonperturbative fermion-boson vertex function is expressed nonperturbatively in terms of the full fermion propagator and contains the contributions of the transverse axial anomaly and perturbative corrections. The result that the transverse axial anomaly contributes to the nonperturbative fermion-boson vertex arises from the coupling between the fermion-boson and the axial-vector vertices through the transverse Ward-Takahashi relations for them and is a consequence of gauge invariance. 
  The role of the gauge invariance in noncommutative field theory is discussed. A basic introduction to noncommutative geometry and noncommutative field theory is given. Background invariant formulation of Wilson lines is proposed. Duality symmetries relating various noncommutative gauge models are being discussed. 
  We survey geometric quantization of finite dimensional affine Kahler manifolds. Its corresponding prequantization and the Berezin's deformation quantization formulation, as proposed by Cahen et al., is used to quantize their corresponding Kahler quotients. Equivariant formalism greatly facilitates the description. 
  It is important to reveal the brane-bulk correspondence for understanding the brane world cosmology. When gravitational waves exist in the bulk, however, it is difficult to make the analysis of the interrelationship between the brane and the bulk. Hence, the minimal model which allows gravitational waves in the bulk would be useful. As for such models, we adopt the Bianchi type midi-superspace models. In particular, the effects of gravitational waves in the bulk on the brane cosmology is examined using the midi-superspace approach. 
  We investigate cosmological evolution in models where the effective potential V(\phi) may become negative for some values of the field \phi. Phase portraits of such theories in space of variables (\phi,\dot\phi,H) have several qualitatively new features as compared with phase portraits in the theories with V(\phi) > 0. Cosmological evolution in models with potentials with a "stable" minimum at V(\phi)<0 is similar in some respects to the evolution in models with potentials unbounded from below. Instead of reaching an AdS regime dominated by the negative vacuum energy, the universe reaches a turning point where its energy density vanishes, and then it contracts to a singularity with properties that are practically independent of V(\phi). We apply our methods to investigation of the recently proposed cyclic universe scenario. We show that in addition to the singularity problem there are other problems that need to be resolved in order to realize a cyclic regime in this scenario. We propose several modifications of this scenario and conclude that the best way to improve it is to add a usual stage of inflation after the singularity and use that inflationary stage to generate perturbations in the standard way. 
  We discuss Wick rotations in the context of gravity, with emphasis on a non-perturbative Wick rotation proposed in hep-th/0103186 mapping real Lorentzian metrics to real Euclidean metrics in proper-time coordinates. As an application, we demonstrate how this Wick rotation leads to a correct answer for a two dimensional non-perturbative path-integral. 
  Dirac's method for variations of a brane embedded in co-dimension one is demonstrated. The variation in the location of the brane invokes a rest frame formulation of the 'sandwiched' brane action. We first demonstrate the necessity of this method by re-deriving Snell's law. Second, we apply the method to a general $N$-dimensional brane embedded in co-dimension one bulk in the presence of gravity. We re-derive the brane equations: (i) Israel junction condition, (ii) Energy/momentum conservation on the brane, and (iii) Geodetic-type equation for the brane. 
  We examine the Dyson-Schwinger equation for the fermion propagator in quenched QED in three and four dimension based on spectral representation with vertex ansatz which preserves Ward-Takahashi Identity.An appropriate renormalization within dispersion integral smoothes the threshold behaviour of the fermion self energy in three dimension.Thus we avoid the infrared singurality in three dimension.The behaviour of the fermion propagator in three dimension near the threshold is then found to be similar to the four dimensional one.There exisit analytic solutions for arbitrary gauges and the full propagators are expressed in terms of hypergeometric function in four dimension.There is a possibility of dynamical chiral symmetry breaking in four dimension with vanishing bare mass. 
  We explain how the string spectrum in flat space and pp-waves arises from the large $N$ limit, at fixed $g^2_{YM}$, of U(N) ${\cal N} =4$ super Yang Mills. We reproduce the spectrum by summing a subset of the planar Feynman diagrams. We give a heuristic argument for why we can neglect other diagrams. We also discuss some other aspects of pp-waves and we present a matrix model associated to the DLCQ description of the maximally supersymmetric eleven dimensional pp-waves. 
  Generalized composite fluxbrane solutions for a wide class of intersection rules are obtained. The solutions are defined on a manifold which contains a product of n Ricci-flat spaces M_1 x ... x M_n with 1-dimensional M_1. They are defined up to a set of functions H_s obeying non-linear differential equations equivalent to Toda-type equations with certain boundary conditions imposed. A conjecture on polynomial structure of governing functions H_s for intersections related to semisimple Lie algebras is suggested. This conjecture is valid for Lie algebras: A_m, C_{m+1}, m > 0. For simple Lie algebras the powers of polynomials coincide with the components of the dual Weyl vector in the basis of simple roots. Explicit formulas for A_1 + ... + A_1 (orthogonal), "block-ortogonal" and A_2 solutions are obtained. Certain examples of solutions in D = 11 and D =10 (II A) supergravities (e.g. with A_2 intersection rules) and Kaluza-Klein dyonic A_2 flux tube, are considered. 
  In a series of papers, we investigate the reformulation of Epstein-Glaser renormalization in coordinate space, both in analytic and Hopf algebraic terms. This first article deals with analytical aspects. Some of the historically good reasons for the divorces of the Epstein-Glaser method, both from mainstream quantum field theory and the mathematical literature on distributions, are made plain; and overcome. 
  Starting from the non-supersymmetric, tachyon-free orientifold of type 0 string theory, we construct four-dimensional brane world models with D6-branes intersecting at angles on internal tori. They support phenomenologically interesting gauge theories with chiral fermions. Despite the theory being non-supersymmetric the perturbative scalar potential induced at leading order is shown to stabilize geometric moduli, leaving only the dilaton tadpole uncanceled. As an example we present a three generation model with gauge group and fermion spectrum close to a left-right symmetric extension of the Standard Model. 
  By using the so-called Information Metric on the moduli space of an anti-selfdual (ASD) Instanton in a Self-Dual (SD) Non-Commutative background, we investigate the geometry of moduli space. The metric is evaluated perturbatively in non-commutativity parameter and we show that by putting a cut-off in the location of instanton in the definition of Information Metric we can recover the five dimensional space time in the presence of a B-field. This result shows that the non-commutative YM-Instanton Moduli corresponds to D-Instanton Moduli in the gravity side where the radial and transverse location of D-Instanton are corresponding to YM-Instanton size and location, respectively. The match is shown in the first order of non-commutativity parameter. 
  The interaction kernel in the Bethe-Salpeter equation for quark-antiquark bound states is derived from the Bethe-Salpeter equations satisfied by the quark-antiquark four-point Green's function. The latter equations are established based on the equations of motion obeyed by the quark and antiquark propagators, the four-point Green's function and some other kinds of Green's functions which follow directly from the QCD generating functional. The B-S kernel derived is given an exact and explicit expression which contains only a few types of Green's functions. This expression is not only convenient for perturbative calculations, but also suitable for nonperturbative investigations. 
  We make two remarks: (i) Renormalization of the effective charge in a 4--dimensional (supersymmetric) gauge theory is determined by the same graphs and is rigidly connected to the renormalization of the metric on the moduli space of the classical vacua of the corresponding reduced quantum mechanical system. Supersymmetry provides constraints for possible modifications of the metric, and this gives us a simple proof of nonrenormalization theorems for the original 4-dimensional theory. (ii) We establish a nontrivial relationship between the effective (0+1)-dimensional and (1+1)-dimensional Lagrangia (the latter represent conventional  Kahlerian sigma models). 
  We present a brief review of recent advances in string cosmology. Starting with the Dilaton-Moduli Cosmology (known also as the Pre Big Bang), we go on to include the effects of axion fields and address the thorny issue of the Graceful Exit in String Cosmology. This is followed by a review of density perturbations arising in string cosmology and we finish with a brief introduction to the impact moving five branes can have on the Dilaton-Moduli cosmological solutions. 
  For a brane world embedded in various ten or eleven-dimensional geometries, we calculate the corrections to the four-dimensional gravitational potential due to graviton modes propagating in the extra dimensions, including those rotating around compact directions. Due to additional "warp" factors, these rotation modes may have as significant an effect as the s-wave modes which propagate in the large or infinite extra dimension. 
  We give a detailed study of the associativity anomaly in open string field theory from the viewpoint of the split string and Moyal formalisms. The origin of the anomaly is reduced to the properties of the special infinite size matrices which relate the conventional open string to the split string variables, and is intimately related to midpoint issues. We discuss two steps to cope with the anomaly. We identify the field subspace that causes the anomaly which is related to the existence of closed string configurations, and indicate a decomposition of open/closed string sectors. We then propose a consistent cut off method with a finite number of string modes that guarantees associativity at every step of any computation. 
  A new parafermionic algebra associated with the homogeneous space $A^{(2)}_2/U(1)$ and its corresponding $Z$-algebra have been recently proposed. In this paper, we give a free boson representation of the $A^{(2)}_2$ parafermion algebra in terms of seven free fields. Free field realizations of the parafermionic energy-momentum tensor and screening currents are also obtained. A new algebraic structure is discovered, which contains a $W$-algebra type primary field with spin two. 
  We study N=1 super Liouville theory on worldsheets with and without boundary. Some basic correlation functions on a sphere or a disc are obtained using the properties of degenerate representations of superconformal algebra. Boundary states are classified by using the modular transformation property of annulus partition functions, but there are some of those whose wave functions cannot be obtained from the analysis of modular property. There are two ways of putting boundary condition on supercurrent, and it turns out that the two choices lead to different boundary states in quality. Some properties of boundary vertex operators are also presented. The boundary degenerate operators are shown to connect two boundary states in a way slightly complicated than the bosonic case. 
  It is built a map between an Abelian Topological Quantum Field Theory, $2+1D$ compact U(1) gauge Maxwell Chern-Simons Theory and the nonrelativistic quantum mechanics Azbel-Hofstadter model of Bloch electrons. The $U_q(sl_2)$ quantum group and the magnetic translations group of the Azbel-Hofstadter model correspond to discretized subgroups of U(1) with linear gauge parameters. The magnetic monopole confining and condensate phases in the Topological Quantum Field Theory are identified with the extended (energy bands) and localized (gaps) phases of the Bloch electron. The magnetic monopole condensate is associated, at the nonrelativistic level, with gravitational white holes due to deformed classical gauge fields. These gravitational solutions render the existence of finite energy pure magnetic monopoles possible. This mechanism constitutes a dynamical symmetry breaking which regularizes the solutions on those localized phases allowing physical solutions of the Shr\"odinger equation which are chains of electron filaments connecting several monopole-white holes.To test these results would be necessary a strong external magnetic field $B\sim 5 T$ at low temperature $T<1 K$. To be accomplished, it would test the existence of magnetic monopoles and classical gravity to a scale of $\sim 10^{-15}$ \textit{meters}, the dimension of the monopole-white hole. A proper discussion of such experiment is out of the scope of this theoretical work. 
  It is shown that the N=4 superalgebra of the Dirac theory in Taub-NUT space has different unitary representations related among themselves through unitary U(2) transformations. In particular the SU(2) transformations are generated by the spin-like operators constructed with the help of the same covariantly constant Killing-Yano tensors which generate Dirac-type operators. A parity operator is defined and some explicit transformations which connect the Dirac-type operators among themselves are given. These transformations form a discrete group which is a realization of the quaternion discrete group. The fifth Dirac operator constructed using the non-covariant Killing-Yano tensor of the Taub-NUT space is quite special. This non-standard Dirac operator is connected with the hidden symmetry and is not equivalent to the Dirac-type operators of the standard N=4 supersymmetry. 
  We find that the quantum monodromy matrix associated with a derivative nonlinear Schrodinger (DNLS) model exhibits U(2) or U(1,1) symmetry depending on the sign of the related coupling constant. By using a variant of quantum inverse scattering method which is directly applicable to field theoretical models, we derive all possible commutation relations among the operator valued elements of such monodromy matrix. Thus, we obtain the commutation relation between creation and annihilation operators of quasi-particles associated with DNLS model and find out the $S$-matrix for two-body scattering. We also observe that, for some special values of the coupling constant, there exists an upper bound on the number of quasi-particles which can form a soliton state for the quantum DNLS model. 
  We obtain an analytic approximation for the effective action of a quantum scalar field in a general static two-dimensional spacetime. We apply this to the dilaton gravity model resulting from the spherical reduction of a massive, non-minimally coupled scalar field in the four-dimensional Schwarzschild geometry. Careful analysis near the event horizon shows the resulting two-dimensional system to be regular in the Hartle-Hawking state for general values of the field mass, coupling, and angular momentum, while at spatial infinity it reduces to a thermal gas at the black-hole temperature. 
  It is shown that a localized four-dimensional Einstein term, induced by quantum corrections, modifies significantly the law of gravity in a Randall-Sundrum brane world. In particular, the short-distance behavior of gravity changes from five- to four-dimensional, while, depending on the values of parameters, there can be an intermediate range where gravity behaves as in five dimensions. The spectrum of graviton fluctuations around the brane, their relative importance for the gravitational force, and the relevance of their emission in the bulk for the brane cosmology are analysed. Finally, constraints on parameters are derived from energy loss in astrophysical and particle physics processes. 
  Non-Fock representations of the canonical commutation relations modeled over an infinite-dimensional nuclear space are constructed in an explicit form. The example of the nuclear space of smooth real functions of rapid decrease results in nonequivalent quantizations of scalar fields. 
  We describe a novel duality symmetry of Phi(4)-theory defined on noncommutative Euclidean space and with noncommuting momentum coordinates. This duality acts on the fields by Fourier transformation and scaling. It is an extension, to interactions defined with a star-product, of that which arises in quantum field theories of non-interacting scalar particles coupled to a constant background electromagnetic field. The dual models are in general of the same original form but with transformed coupling parameters, while in certain special cases all parameters are essentially unchanged. Using a particular regualarization we show, to all orders of perturbation theory, that that this duality also persists at the quantum level. We also point out various other properties of this class of noncommutative field theories. 
  We study chiral symmetries of fermionic non commutative dipole theories. By using Fujikawa's approach we obtain explicit expressions of the anomalies for Dirac and chiral fermions in 2 and 4 dimensions. 
  We show how a lattice approach can be used to derive Thermodynamic Bethe Ansatz (TBA) equations describing all excitations for boundary flows. The method is illustrated for a prototypical flow of the tricritical Ising model by considering the continuum scaling limit of the A4 lattice model with integrable boundaries. Fixing the bulk weights to their critical values, the integrable boundary weights admit two boundary fields $\xi$ and $\eta$ which play the role of the perturbing boundary fields $\phi_{1,3}$ and $\phi_{1,2}$ inducing the renormalization group flow between boundary fixed points. The excitations are completely classified in terms of (m,n) systems and quantum numbers but the string content changes by certain mechanisms along the flow. For our prototypical example, we identify these mechanisms and the induced map between the relevant finitized Virasoro characters. We also solve the boundary TBA equations numerically to determine the flows for the leading excitations. 
  If our 3+1-dimensional universe is a brane or domain wall embedded in a higher dimensional space, then a phenomenon we term the ``clash of symmetries'' provides a new method of breaking some continuous symmetries. A global $G_{\text{cts}} \otimes G_{\text{discrete}}$ symmetry is spontaneously broken to $H_{\text{cts}} \otimes H_{\text{discrete}}$, where the continuous subgroup $H_{\text{cts}}$ can be embedded in several different ways in the parent group $G_{\text{cts}}$, and $H_{\text{discrete}} < G_{\text{discrete}}$. A certain class of topological domain wall solutions connect two vacua that are invariant under {\it differently embedded} $H_{\text{cts}}$ subgroups. There is then enhanced symmetry breakdown to the intersection of these two subgroups on the domain wall. This is the ``clash''. In the brane limit, we obtain a configuration with $H_{\text{cts}}$ symmetries in the bulk but the smaller intersection symmetry on the brane itself. We illustrate this idea using a permutation symmetric three-Higgs-triplet toy model exploiting the distinct $I-$, $U-$ and $V-$spin U(2) subgroups of U(3). The three disconnected portions of the vacuum manifold can be treated symmetrically through the construction of a three-fold planar domain wall junction configuration, with our universe at the nexus. A possible connection with $E_6$ is discussed. 
  In this paper, exact one-point functions of N=1 super-Liouville field theory in two-dimensional space-time with appropriate boundary conditions are presented. Exact results are derived both for the theory defined on a pseudosphere with discrete (NS) boundary conditions and for the theory with explicit boundary actions which preserves super conformal symmetries. We provide various consistency checks. We also show that these one-point functions can be related to a generalized Cardy conditions along with corresponding modular $S$-matrices. Using this result, we conjecture the dependence of the boundary two-point functions of the (NS) boundary operators on the boundary parameter. 
  The aim of these lectures is to give a brief introduction to brane cosmology. After introducing some basic geometrical notions, we discuss the cosmology of a brane universe with matter localized on the brane. Then we introduce an intrinsic curvature scalar term in the bulk action, and analyze the cosmology of this induced gravity. Finally we present the cosmology of a moving brane in the background of other branes, and as a particular example, we discuss the cosmological evolution of a test brane moving in a background of a Type-0 string theory. 
  In this paper we present two solutions of open bosonic string field theory defined on D0-brane background that correspond to the change of the D0-brane position in the transverse space. 
  In this work we develop an approach to obtain analytical expressions for potentials in an impenetrable box. It is illustrated through the particular cases of the harmonic oscillator and the Coulomb potential. In this kind of system the energy expression respect the correct quantum limits, which is a very important quality. The similarity of this kind of problem with the quasi exactly solvable potentials is explored in order to accomplish our goals. 
  Starting from a N=1 scalar supermultiplet in 2+1 dimensions, we demonstrate explicitly the appearance of induced N=1 vector and scalar supermultiplets of composite operators made out of the fundamental supersymmetric constituents. We discuss an extension to a N=2 superalgebra with central extension, due to the existence of topological currents in 2+1 dimensions. As a specific model we consider a supersymmetric $CP^1$ $\sigma$-model as the constituent theory, and discuss the relevance of these results for an effective description of the infrared dynamics of planar high-temperature superconducting condensed matter models with quasiparticle excitations near nodal points of their Fermi surface. 
  Exact calculations are given for the Casimir energy for various fields in $R\times S^3$ geometry. The Green's function method naturally gives a result in a form convenient in the high-temperature limit, while the statistical mechanical approach gives a form convenient for low temperatures. The equivalence of these two representations is demonstrated. Some discrepancies with previous work are noted. In no case, even for ${\cal N}=4$ SUSY, is the ratio of entropy to energy found to be bounded. This deviation, however, occurs for low temperature, where the equilibrium approach may not be relevant. The same methods are used to calculate the energy and free energy for the TE modes in a half-Einstein universe bounded by a perfectly conducting 2-sphere. 
  In this letter we address the problem of inducing boundary degrees of freedom from a bulk theory whose action contains higher-derivative corrections. As a model example we consider a topological theory with an action that has only a ``higher-derivative'' term. By choosing specific coupling of the brane to the bulk we show that the boundary action contains gravity action along with some higher-derivative corrections. The co-dimension of the brane is more than one. In this sense the boundary is singular. 
  The quantum Hall effect is studied by introducing two different matrix variables for electrons and holes, having Chern-Simons type interactions. By generalizing the constraint condition proposed by Susskind to realize the Pauli's exclusion principle in this two component matrix model, the classical exciton solution having excitation of both quasi-electron and quasi-hole is obtained. The constraint condition is also solved quantum mechanically in the infinite-sized matrix case, giving the examples of the physical states. Using these quantum states, the corrections of the exciton energy, coming from the noncommutativity of space (Pauli principle) and from the quantum effects of the background state, are estimated in the lowest order perturbation expansion. As a result, the dispersion relation of exciton is obtained. 
  The joining--splitting interaction of non-critical bosonic string is analyzed in the light-cone formulation. The Mandelstam method of constructing tree string amplitudes is extended to the bosonic massive string models of the discrete series. The general properties of the Liouville longitudinal excitations which are necessary and sufficient for the Lorentz covariance of the light-cone amplitudes are derived. The results suggest that the covariant and the light-cone approach are equivalent also in the non-critical dimensions. Some aspects of unitarity of interacting non-critical massive string theory are discussed. 
  The Cardy-Verlinde formula is generalized to the asymptotically flat rotating charged black holes in the Einstein-Maxwell theory and low-energy effective field theory describing string by using some typical spacetimes, such as the Kerr-Newman, Einstein-Maxwell-dilaton-axion, Kaluza-Klein, and Sen black holes. For the Kerr-Newman black hole, the definition of the Casimir energy takes the same form as that of the Kerr-Newman-AdS$_4$ and Kerr-Newman-dS$_4$ black holes, while the Cardy-Verlinde formula possesses different from since the Casimir energy does not appear in the extensive energy. The Einstein-Maxwell-dilaton-axion, Kaluza-Klein, and Sen black holes have special property: The definition of the Casimir energy for these black holes is similar to that of the Kerr-Newman black hole, but the Cardy-Verlinde formula takes the same form as that of the Kerr black hole. Furthermore, we also study the entropy bounds for the systems in which the matters surrounds these black holes. We find that the bound for the case of the Kerr-Newman black hole is related to its charge, and the bound for the cases of the EMDA, Kaluza-Klein, and Sen black holes can be expressed as a unified form. A surprising result is that the entropy bounds for the Kaluza-Klein and Sen black holes are tighter than the Bekenstein one. 
  A new type of topological matter interactions involving second-rank antisymmetric tensor matter fields with an underlying $N_T \geq 1$ topological supersymmetry are proposed. The construction of the 4-dimensional, $N_T = 1$ Donaldson-Witten theory, the $N_T = 1$ super-BF model and the $N_T = 2$ topological B-model with tensor matter are explicitly worked out. 
  We study spherically and axially symmetric monopoles of the SU(2) Einstein-Yang-Mills-Higgs-dilaton (EYMHD) system with a new coupling between the dilaton field and the covariant derivative of the Higgs field. This coupling arises in the study of (4+1)-dimensional vortices in the Einstein-Yang-Mills (EYM) system. 
  It is shown that the Holographic Renormalization Group can be formulated universally within Quantum Field Theory as (the quantization of) the Hamiltonian flow on the cotangent bundle to the space of gauge-invariant single-trace operators supplied with the canonical symplectic structure. The classical Hamiltonian dynamics is recovered in the large $N$ limit. 
  We study effects associated with the chiral anomaly for a cascading $SU(N+M)\times SU(N)$ gauge theory using gauge/gravity duality. In the gravity dual the anomaly is a classical feature of the supergravity solution, and the breaking of the U(1) R-symmetry down to ${\bf Z}_{2M}$ proceeds via the Higgs mechanism. 
  We study compact non-supersymmetric Z_N orbifolds in various dimensions. We compute the spectrum of several tachyonic type II and heterotic examples and partially classify tachyon-free heterotic models. We also discuss the relation to compactification on K3 and Calabi-Yau manifolds. 
  We use a previously derived integral representation for the four graviton amplitude at two loops in Superstring theory, whose leading term for vanishing momenta gives the two-loop contribution to the R^4 term in the Effective Action. We find by an explicit computation that this contribution is zero, in agreement with a general argument implying the vanishing of the R^4 term beyond one loop. 
  The principle of the fermionic projector is studied for a fermionic projector which in the vacuum is the direct sum of seven identical massive sectors and one massless left-handed sector, each of which is composed of three Dirac seas. It is shown under general assumptions and for an interaction via general chiral and (pseudo)scalar potentials that the sectors spontaneously form pairs, which are referred to as blocks. The resulting so-called effective interaction can be described by chiral potentials corresponding to the effective gauge group $SU(2) \otimes SU(3) \otimes U(1)^3$. The properties of the corresponding gauge fields are analyzed.   The similarity of this model to the standard model gives a strong indication that the principle of the fermionic projector is of physical significance. 
  We consider braneworld scenarios including the leading correction to the Einstein-Hilbert action suggested by superstring theory, the Gauss-Bonnet term. We obtain and study the complete set of equations governing the cosmological dynamics. We find they have the same form as those in Randall-Sundrum scenarios but with time-varying four-dimensional gravitational and cosmological constants. Studying the bulk geometry we show that this variation is produced by bulk curvature terms parameterized by the mass of a black hole. Finally, we show there is a coupling between these curvature terms and matter that can be relevant for early universe cosmology. 
  We introduce the technique of inverse Mellin transform in a problem of strong-field QED. We show that the {\it moments} of pair production width in a uniform background magnetic field are proportional to the derivatives of photon polarization function at the zero momentum. Hence, the pair-production width or the absorptive part of the photon polarization function is calculable from the latter by the inverse Mellin transform. Using the {\it Kramers-Kronig} relation, the dispersive part of photon polarization function can be computed as well. Therefore the analytic property of the photon polarization function in all energy range is obtained. We also discuss briefly the possible extensions of this technique to other problems. 
  An electron moving on plane in a uniform magnetic field orthogonal to plane is known as the Landau problem. Wigner functions for the Landau problem when the plane is noncommutative are found employing solutions of the Schroedinger equation as well as solving the ordinary *-genvalue equation in terms of an effective Hamiltonian. Then, we let momenta and coordinates of the phase space be noncommutative and introduce a generalized *-genvalue equation. We solve this equation to find the related Wigner functions and show that under an appropriate choice of noncommutativity relations they are independent of noncommutativity parameter. 
  A long-standing problem of theoretical physics is the exceptionally small value of the cosmological constant $\Lambda \sim 10^{-120}$ measured in natural Planckian units. Here we derive this tiny number from a toroidal string cosmology based on closed strings. In this picture the dark energy arises from the correlation between momentum and winding modes that for short distances has an exponential fall-off with increasing values of the momenta. The freeze-out by the expansion of the background universe for these transplanckian modes may be interpreted as a frozen condensate of the closed-string modes in the three non-compactified spatial dimensions. 
  We present a simple derivation of the Ricci-flat Kahler metric and its Kahler potential on the canonical line bundle over arbitrary Kahler coset space equipped with the Kahler-Einstein metric. 
  We study the equivalence/duality between various non-commutative gauge models at the classical and quantum level. The duality is realised by a linear Seiberg-Witten-like map. The infinitesimal form of this map is analysed in more details. 
  We show that the Bronnikov theorem on the nonexistence of regular electrically charged black holes can be circumvented. In the frame of nonlinear electrodynamics, we present the exact regular black hole solutions of a hybrid type. They are electrically charged, but contain a `dual' core confining a polarization of magnetic charges.   The considered example is based on a modification of the Ay\'on-Beato & Garc\'\i a solution. It represents a very specific realization of the idea on confinement based on dual electrodynamics and dual superconductivity. 
  The boundary states for a certain class of WZW models are determined. The models include all modular invariants that are associated to a symmetry of the unextended Dynkin diagram. Explicit formulae for the boundary state coefficients are given in each case, and a number of properties of the corresponding NIM-reps are derived. 
  The ground state energy of the massive scalar field with non-conformal coupling $\xi$ on the short-throat flat-space wormhole background is calculated by using zeta renormalization approach. We discuss the renormalization and relevant heat kernel coefficients in detail. We show that the stable configuration of wormholes can exist for $\xi > 0.123$. In particular case of massive conformal scalar field with $\xi=1/6$, the radius of throat of stable wormhole $a\approx 0.16/m$. The self-consistent wormhole has radius of throat $a\approx 0.0141 l_p $ and mass of scalar boson $m\approx 11.35 m_p$ ($l_p$ and $m_p$ are the Planck length and mass, respectively). 
  We consider the N=1 supersymmetric two-dimensional non-linear sigma model with boundaries and nonzero B-field. By analysing the appropriate currents we describe the full set of boundary conditions compatible with N=1 superconformal symmetry. Using this result the problem of finding a correct action is discussed. We interpret the supersymmetric boundary conditions as a maximal integral submanifold of the target space manifold, and speculate about a new geometrical structure, the deformation of an almost product structure. 
  Linear free field theories are one of the few Quantum Field Theories that are exactly soluble. There are, however, (at least) two very different languages to describe them, Fock space methods and the Schroedinger functional description. In this paper, the precise sense in which the two representations are related is reviewed. Several properties of these representations are studied, among them the well known fact that the Schroedinger counterpart of the usual Fock representation is described by a Gaussian measure. A real scalar field theory is considered, both on Minkowski spacetime for arbitrary, non-inertial embeddings of the Cauchy surface, and for arbitrary (globally hyperbolic) curved spacetimes. As a concrete example, the Schroedinger representation on stationary and homogeneous cosmological spacetimes is constructed. 
  We consider the $N=(1,1)$ SYM theory that is obtained by dimensionally reducing SYM theory in 2+1 dimensions to 1+1 dimensions and discuss soft supersymmetry breaking. We discuss the numerical simulation of this theory using SDLCQ when either the boson or the fermion has a large mass. We compare our result to the pure adjoint fermion theory and pure adjoint boson DLCQ calculations of Klebanov, Demeterfi, and Bhanot and of Kutasov. With a large boson mass we find that it is necessary to add additional operators to the theory to obtain sensible results. When a large fermion mass is added to the theory we find that it is not necessary to add operators to obtain a sensible theory. The theory of the adjoint boson is a theory that has stringy bound states similar to the full SYM theory. We also discuss another theory of adjoint bosons with a spectrum similar to that obtained by Klebanov, Demeterfi, and Bhanot. 
  For the 2-charge extremal holes in string theory we show that the Bekenstein entropy obtained from the area of the stretched horizon has a statistical interpretation as a `coarse graining entropy': different microstates give geometries that differ near r=0 and the stretched horizon cuts off the metric at r=b where these geometries start to differ. 
  We show that the asymptotic dynamics of two-dimensional de Sitter or anti-de Sitter Jackiw-Teitelboim (JT) gravity is described by a generalized two-particle Calogero-Sutherland model. This correspondence is established by formulating the JT model of (A)dS gravity in two dimensions as a topological gauge theory, which reduces to a nonlinear 0+1-dimensional sigma model on the boundary of (A)dS space. The appearance of cyclic coordinates allows then a further reduction to the Calogero-Sutherland quantum mechanical model. 
  We study D-branes in the bosonic closed string theory whose automorphism group is the Bimonster group (the wreath product of the Monster simple group with Z_2). We give a complete classification of D-branes preserving the chiral subalgebra of Monster invariants and show that they transform in a representation of the Bimonster. Our results apply more generally to self-dual conformal field theories which admit the action of a compact Lie group on both the left- and right-moving sectors. 
  We obtain singularity resolutions for various overlapping brane configurations, including those of two heterotic 5-branes, type II 5-branes or D4-branes. In these solutions, the ``harmonic'' function H for each brane component depends only on the associated four-dimensional relative transverse space. The resolution is achieved by replacing these transverse spaces with Eguchi-Hanson or Taub-NUT spaces, both of which admit a normalisable self-dual (or anti-self-dual) harmonic 2-form. Due to the manner in which the interaction terms for the form fields modify their Bianchi identities or equations of motion, these normalisable harmonic 2-forms provide regular sources for the branes. We also obtain resolved 5-branes and D4-branes wrapped on S^1, which is fibred over the transverse Eguchi-Hanson or Taub-NUT spaces. The T-duality invariance of the NS-NS 5-brane is retained after the resolution. The resolved 5-branes and D4-branes provide regular supergravity duals of certain supersymmetric Yang-Mills theories in five and four dimensions. 
  It has been argued by several authors that the quantum mechanical spectrum of black hole horizon area must be discrete. This has been confirmed in different formalisms, using different approaches. Here we concentrate on two approaches, the one involving quantization on a reduced phase space of collective coordinates of a Black Hole and the algebraic approach of Bekenstein. We show that for non-rotating, neutral black holes in any spacetime dimension, the approaches are equivalent. We introduce a primary set of operators sufficient for expressing the dynamical variables of both, thus mapping the observables in the two formalisms onto each other. The mapping predicts a Planck size remnant for the black hole. 
  We investigate the nonlinear holomorphic supersymmetry for quantum-mechanical systems on Riemann surfaces subjected to an external magnetic field. The realization is shown to be possible only for Riemann surfaces with constant curvature metrics. The cases of the sphere and Lobachevski plane are elaborated in detail. The partial algebraization of the spectrum of the corresponding Hamiltonians is proved by the reduction to one-dimensional quasi-exactly solvable sl(2,R) families. It is found that these families possess the "duality" transformations, which form a discrete group of symmetries of the corresponding 1D potentials and partially relate the spectra of different 2D systems. The algebraic structure of the systems on the sphere and hyperbolic plane is explored in the context of the Onsager algebra associated with the nonlinear holomorphic supersymmetry. Inspired by this analysis, a general algebraic method for obtaining the covariant form of integrals of motion of the quantum systems in external fields is proposed. 
  In this paper we apply the Symplectic Projectors Method to gauge theories with second class constraints in various space-time dimensions. The conclusion is that this method is equivalent to the standard quantization methods. Although it suffers from some limitations since its applicability is restricted to some simpler situation when one is not interested in full information provided by the BRST, our analysis shows that it is worthwhile to investigate this method further. Relevant reference for this thesis are hep-th/0005214, hep-th/0104154 and hep-th/0108197. 
  We examine Sen's descent relations among (non-)BPS D-branes by using low energy effective field theories of DpDpbar system. We find that the fluctuation around the kink solution reproduces the low energy matter content on a non-BPS D(p-1)-brane. The effective action for these fluctuation modes turns out to be a generalization of Minahan-Zwiebach model. In addition, it is shown that the fluctuations around the vortex solution consist of massless fields on a BPS D(p-2)-brane and they are subject to Dirac-Born-Infeld action. We find the universality that the above results do not refer to particular forms of the effective action. 
  The B_N hyperbolic Sutherland spin model is expressed in terms of a suitable set of commuting Dunkl operators. This fact is exploited to derive a complete family of commuting integrals of motion of the model, thus establishing its integrability. The Dunkl operators are shown to possess a common flag of invariant finite-dimensional linear spaces of smooth scalar functions. This implies that the Hamiltonian of the model preserves a corresponding flag of smooth spin functions. The discrete spectrum of the restriction of the Hamiltonian to this spin flag is explicitly computed by triangularization. The integrability of the hyperbolic Sutherland spin chain of B_N type associated with the dynamical model is proved using Polychronakos's "freezing trick". 
  The dynamics of the phase transition of the continuum $\Phi ^{4}_{1+1}$-theory in Light Cone Quantization is reexamined taking into account fluctuations of the order parameter $< \Phi >$ in the form of dynamical zero mode operators (DZMO) which appear in a natural way via the Haag expansion of the field $\Phi (x)$ of the interacting theory. The inclusion of the DZM-sector changes significantly the value of the critical coupling, bringing it in agreement within 2% with the most recent Monte-Carlo and high temperature/strong coupling estimates. The critical slowing down of the DZMO governs the low momentum behavior of the dispersion relation through invariance of this DZMO under conformal transformations preserving the local light cone structure. The critical exponent $\eta$ characterising the scaling behaviour at $k^2 \to 0$ comes out in agreement with the known value 0.25 of the Ising universality class. $\eta$ is made of two contributions: one, analytic $(75 %$) and another (25%) which can be evaluated only numerically with an estimated error of 3%. The $\beta$-function is then found from the non-perturbative expression of the physical mass. It is non-analytic in the coupling constant with a critical exponent $\omega=2$. However, at D=2, $\omega$ is not parametrisation independent with respect to the space of coupling constants due to this strong non-analytic behaviour. 
  Using differential renormalization, we calculate the complete two-point function of the background gauge superfield in pure N=1 Supersymmetric Yang-Mills theory to two loops. Ultraviolet and (off-shell) infrared divergences are renormalized in position and momentum space respectively. This allows us to reobtain the beta function from the dependence on the ultraviolet renormalization scale in an infrared-safe way. The two-loop coefficient of the beta function is generated by the one-loop ultraviolet renormalization of the quantum gauge field via nonlocal terms which are infrared divergent on shell. We also discuss the connection of the beta function to the flow of the Wilsonian coupling. 
  Recently, Klemm and others [hep-th/0104141] have successfully generalized the Cardy-Verlinde formula for an asymptotically flat spacetime of arbitrary dimensionality. And yet, from a holographic perspective, the interpretation of this formula remains somewhat unclear. Nevertheless, in this paper, we incorporate the implied flat space/CFT duality into a study on boundary descriptions of a $d$-dimensional Schwarzschild-black hole spacetime. In particular, we demonstrate that the (presumably) dual CFT adopts a string-like description and, moreover, is thermodynamically equivalent to a string that lives on the stretched horizon of the bulk black hole. Significantly, a similar equivalence has recently been established in both an asymptotically dS and AdS context. On this basis, we argue that the asymptotically flat Cardy-Verlinde formula does, indeed, have a holographic pedigree. 
  We give the general presciption for calculating the moduli of irreducible, stable SU(n) holomorphic vector bundles with positive spectral covers over elliptically fibered Calabi-Yau threefolds. Explicit results are presented for Hirzebruch base surfaces B=F_r. The transition moduli that are produced by chirality changing small instanton phase transitions are defined and specifically enumerated. The origin of these moduli, as the deformations of the spectral cover restricted to the ``lift'' of the horizontal curve of the M5-brane, is discussed. We present an alternative description of the transition moduli as the sections of rank n holomorphic vector bundles over the M5-brane curve and give explicit examples. Vector bundle moduli appear as gauge singlet scalar fields in the effective low-energy actions of heterotic superstrings and heterotic M-theory. 
  It is shown that an algebraically defined holographic projection of a QFT onto the lightfront changes the local quantum properties in a very drastic way. The expected ubiquitous vacuum polarization characteristic of QFT is confined to the lightray (longitudinal) direction, whereas operators whose localization is transversely separated are completely free of vacuum correlations. This unexpected ''transverse return to QM'' combined with the rather universal nature of the strongly longitudinal correlated vacuum correlations (which turn out to be described by rather kinematical chiral theories) leads to a d-2 dimensional area structure of the d-1 dimensional lightfront theory. An additive transcription in terms of an appropriately defined entropy related to the vacuum restricted to the horizon is proposed and its model independent universality aspects which permit its interpretation as a quantum candidate for Bekenstein's area law are discussed. The transverse tensor product foliation structure of lightfront degrees of freedom is essential for the simplifying aspects of the algebraic lightcone holography. Key-words: Quantum field theory; Mathematical physics, Quantum gravity 
  The purpose is to construct the quantum field theory including gravity, based on physical assumptions as few as possible. Up to now, the work by Prof. Steven Weinberg probably suits this purpose the most. Though the purpose is difficult to reach, my recent preprint was interested in an exceptional case caused by singularity. Therefore, I'd like to explain the motivations and possible applications of the preprint. 
  We establish that the open string star product in the zero momentum sector can be described as a continuous tensor product of mutually commuting two dimensional Moyal star products. Let the continuous variable $\kappa \in [~0,\infty)$ parametrize the eigenvalues of the Neumann matrices; then the noncommutativity parameter is given by $\theta(\kappa) =2\tanh(\pi\kappa/4)$. For each $\kappa$, the Moyal coordinates are a linear combination of even position modes, and the Fourier transform of a linear combination of odd position modes. The commuting coordinate at $\kappa=0$ is identified as the momentum carried by half the string. We discuss the relation to Bars' work, and attempt to write the string field action as a noncommutative field theory. 
  In hep-th/9803002, Maldacena argued that in the light of the AdS/CFT correspondence as formulated by Witten and Gubser, Klebanov and Polyakov as a relation between partition functions, the expectation value of the Wilson loop in $\cal N$=4 SU(N) SYM is given by the worldsheet partition function with the action formulated on an $AdS_5\times S^5$ background and the world sheet ending on the loop on the boundary of $AdS_5$. What we propose to do in this paper is to provide some alternative arguments as to why it should be so. 
  The vacuum energy density of electromagnetic field inside a perfectly conducting wedge is calculated by making use of the local zeta function technique. This regularization completely eliminates divergent expressions in the course of calculations and gives rise to a finite expression for the energy density in question without any subtractions. Employment of the Hertz potentials for constructing the general solution to the Maxwell equations results in a considerable simplification of the calculations. Transition to the global zeta function is carried out by introducing a cutoff nearby the cusp at the origin. Proceeding from this the heat kernel coefficients are calculated and the high temperature asymptotics of the Helmholtz free energy and of the torque of the Casimir forces are found. The wedge singularity gives rise to a specific high temperature behaviour $\sim T^2$ of the quantities under consideration. The obtained results are directly applicable to the free energy of a scalar massless field and electromagnetic field on the background of a cosmic string. 
  Continuing the study in hep-th/0004092 and hep-th/0004092, we investigate a non-trivial string dynamical process related to orientifold planes, i.e., the splitting of physical NS-branes and D(p+2)-branes on orientifold Op-planes. Creation or annihilation of physical Dp-branes usually accompanies the splitting process. In the particular case p=4, we use Seiberg-Witten curves as an independent method to check the results. 
  Negative energy objects generally lead to instabilities and a number of other disturbing behaviors. In particular, negative energy fluxes lead to a breakdown of the classical area theorem for black hole horizons, which can lead to violations of the second law of thermodynamics. The negative energy objects that arise in string theory involve special boundary conditions which remove the perturbative instabilities. We show that they have additional special features which allow them to evade contradiction with the second law. We identify one mechanism which applies for most orientifold planes in string theory, and distinct mechanisms for the O8-plane and the AdS soliton. 
  We discuss $\theta$-deformed Maxwell theory at first order in $\theta$ with the help of the Seiberg-Witten (SW) map. With an appropriate field redefinition consistent with the SW-map we analyse the one-loop corrections of the vacuum polarization of photons. We show that the radiative corrections obtained in a previous work may be described by the Ward-identity of the BRST-shift symmetry corresponding to a field redefinition. 
  We analyse the IR-singularities that appear in a noncommutative scalar quantum field theory on $\mathcal{E}_4$. We demonstrate with the help of the quadratic one-loop effective action and an appropriate field redefinition that no IR-singularities exist. No new degrees of freedom are needed to describe the UV/IR-mixing. 
  We study solutions of Einstein gravity coupled to a positive cosmological constant and matter, which are asymptotically de Sitter and homogeneous. Regarded as perturbations of de Sitter space, a theorem of Gao and Wald implies that generically these solutions are `tall,' meaning that the perturbed universe lives through enough conformal time for an entire spherical Cauchy surface to enter any observer's past light cone. Such observers will realize that their universe is spatially compact. An interesting fact, which we demonstrate with an explicit example, is that this Cauchy surface can have arbitrarily large volume for fixed asymptotically de Sitter behavior. Our main focus is on the implications of tall universes for the proposed dS/CFT correspondence. Particular attention is given to the associated renormalization group flows, leading to a more general de Sitter `c-theorem.' We find, as expected, that a contracting phase always represents a flow towards the infrared, while an expanding phase represents a `reverse' flow towards the ultraviolet. We also discuss the conformal diagrams for various classes of homogeneous flows. 
  The equations of motion of quantum Yang - Mills theory (in the planar `large N' limit), when formulated in Loop-space are shown to have an anomalous term, which makes them analogous to the equations of motion of WZW models. The anomaly is the Jacobian of the change of variables from the usual ones i.e. the connection one form $A $, to the holonomy $U$. An infinite dimensional Lie algebra related to this change of variables (the Lie algebra of loop substitutions) is developed, and the anomaly is interpreted as an element of the first cohomology of this Lie algebra. The Migdal-Makeenko equations are shown to be the condition for the invariance of the Yang-Mills generating functional $Z$ under the action of the generators of this Lie algebra. Connections of this formalism to the collective field approach of Jevicki and Sakita are also discussed. 
  We discuss the quantization of a scalar particle moving in two-dimensional de Sitter space. We construct the conformal quantum mechanical model on the asymptotic boundary of de Sitter space in the infinite past. We obtain explicit expressions for the generators of the conformal group and calculate the eigenvalues of the Hamiltonian. We also show that two-point correlators are in agreement with the Green function one obtains from the wave equation in the bulk de Sitter space. 
  We discuss the localized tachyon condensation in the non-supersymmetric orbifold theories by taking the cosmological constant as the measure of degrees of freedom (d.o.f). We first show asymptotic density of state is not a proper quantity to count the 'localized' d.o.f. We then show that localized d.o.f lead us a c-function given by the lightest tachyon mass, which turns out to be the same as the tachyon potential recently suggested by Dabholkar and Vafa. We also argue that delocalized d.o.f also encode information on the process of localized tachyon condensation in the g-function, based on the fact that the global geometry of the orbifolds is completely determined by the local geometry around the fixed points. For type II, both c- and g-function respect the stability of the supersymmetric models and both allow all the process suggested by Adams, Polchinski and Silverstein. 
  We consider de Sitter brane-world motivated by dS/CFT correspondence where both bulk and brane are de Sitter spaces. The brane tension is fixed by holographic RG. The 4d effective action for metric perturbations and 4d graviton correlator are explicitly found. The induced values of cosmological and Newton constants are calculated. The short distance behaviour of the graviton correlator (when no brane matter presents) turns out to be significally stronger than in the case of General Relativity. It is shown that quantum brane CFT gives the dominant contribution to graviton correlator on small scales like in Brane New World scenario. 
  Noncommutative gravity in three dimensions with no cosmological constant is reviewed. We find a solution which describes the presence of a torsional source. 
  We review the doubly gauge invariant formalism of cosmological perturbations in the Randall-Sundrum brane world. This formalism leads to four independent equations describing the evolution of scalar perturbations. Three of these equations are differential equations written in terms of gauge invariant variables on the brane only, and the other is an integro-differential equation describing non-locality due to bulk gravitational waves. At low energy the evolution of the scalar-type cosmological perturbations in the brane-world cosmology differs from that in the standard cosmology only by non-local effects due to bulk gravitational waves. 
  We construct a family of quasi-solvable quantum many-body systems by an algebraic method. The models contain up to two-body interactions and have permutation symmetry. We classify these models under the consideration of invariance property. It turns out that this family includes the rational, hyperbolic (trigonometric) and elliptic Inozemtsev models as the particular cases. 
  The bulk quantization method is used for regularizing a conventional four dimensional theory of massless fermions coupled to an external non-Abelian gauge field and for subsequently evaluating the associated Ward identity. As a result one obtains the well-known chiral anomaly. 
  We propose a new algebraic deformation of ${\cal N}=4$ SYM via decomposition of spinor and scalar fields in vector supermultiplet. This decomposition generates degrees of freedom of usual quarks and leptons and the deformation model is a low energy effective model. We show that supersymmetry is broken in certain limit and the deformation model reduces to a SM-like model, or a QCD-like model. Meanwhile, gauge symmetry can be spontaneously broken by nontrivial supersymmetry vacuum. 
  Considering the complex n-dimension Calabi-Yau homogeneous hyper-surfaces ${\cal H}_{n}$ and using algebraic geometry methods, we develop the crossed product algebra method, introduced by Berenstein et Leigh in hep-th/0105229, and build the non commutative (NC) geometries for orbifolds ${\cal O}={\cal H}_{n}/{\bf Z}_{n+2}^{n}$ with a discrete torsion matrix $t_{ab}=exp[{\frac{i2\pi}{n+2}}{(\eta_{ab}-\eta_{ba})}]$, $\eta_{ab} \in SL(n,{\bf Z})$. We show that the NC manifolds ${\cal O}^{(nc)}$ are given by the algebra of functions on the real $(2n+4)$ Fuzzy torus ${\cal T}^{2(n+2)}_{\beta_{ij}}$ with deformation parameters $\beta_{ij}=exp{\frac{i2\pi}{n+2}}{[(\eta^{-1}_{ab}-\eta^{-1}_{ba})} q_{i}^{a} q_{j}^{b}]$, $q_{i}^{a}$'s being Calabi-Yau charges of ${\bf Z}_{n+2}^{n}$. We develop graph rules to represent ${\cal O}^{(nc)}$ by quiver diagrams which become completely reducible at singularities. Generic points in these NC geometries are be represented by polygons with $(n+2)$ vertices linked by $(n+2)$ edges while singular ones are given by $(n+2)$ non connected loops. We study the various singular spaces of quintic orbifolds and analyze the varieties of fractional $D$ branes at singularities as well as the spectrum of massless fields. Explicit solutions for the NC quintic ${\cal Q}^{(nc)}$ are derived with details and general results for complex $n$ dimension orbifolds with discrete torsion are presented. 
  We show that it is possible to build up a consistent model describing a superconducting cosmic string (SCS) endowed with torsion. A full string solution is obtained by matching the internal and the external solutions. We derive the deficit angle, the geodesics of and the gravitational force on a test-particle moving under the action of this screwed SCS. A couple of potentially observable astrophysical phenomena are highlighted: the dynamics of compact objects orbiting torsioned SCS and the accretion of matter onto it. 
  We propose an exactly solvable Grassmannian sigma-model coupled to the Chern-Simons theory. In the presence of a novel topological term our model admits exact self-dual vortex solutions which are identical to those of pure Grassmannian model, but the topological charge has a physical meaning as a magnetic flux since the gauge field is no longer auxiliary. We also extend the theory to a noncommutative plane and analyze the BPS solutions. 
  We consider 5-dimensional spacetimes of constant 3-dimensional spatial curvature in the presence of a bulk cosmological constant. We find the general solution of such a configuration in the presence of a Gauss-Bonnet term. Two classes of non-trivial bulk solutions are found. The first class is valid only under a fine tuning relation between the Gauss-Bonnet coupling constant and the cosmological constant of the bulk spacetime. The second class of solutions are static and are the extensions of the AdS-Schwarzchild black holes. Hence in the absence of a cosmological constant or if the fine tuning relation is not true, the generalised Birkhoff's staticity theorem holds even in the presence of Gauss-Bonnet curvature terms. We examine the consequences in brane world cosmology obtaining the generalised Friedmann equations for a perfect fluid 3-brane and discuss how this modifies the usual scenario. 
  We study a string theory which is exclusively based on extrinsic curvature action. It is a tensionless string theory because the action reduces to perimeter for the flat Wilson loop. We are able to solve and quantize this high-derivative nonlinear two-dimensional conformal field theory. The absence of conformal anomaly in quantum theory requires that the space-time should be 13-dimensional. We have found that all particles, with arbitrary large spin, are massless. This pure massless spectrum is consistent with the tensionless character of the theory and we speculate that it may describe unbroken phase of standard string theory. 
  We describe in detail the solution of type IIB superstring theory in the maximally supersymmetric plane-wave background with constant null Ramond-Ramond 5-form field strength. The corresponding light-cone Green-Schwarz action found in hep-th/0112044 is quadratic in both bosonic and fermionic coordinates. We find the spectrum of the light-cone Hamiltonian and the string representation of the supersymmetry algebra. The superstring Hamiltonian has a ``harmonic-oscillator'' form in both the string-oscillator and the zero-mode parts and thus has discrete spectrum in all 8 transverse directions. We analyze the structure of the zero-mode sector of the theory, establishing the precise correspondence between the lowest-lying ``massless'' string states and the type IIB supergravity fluctuation modes in the plane-wave background. The zero-mode spectrum has certain similarity to the supergravity spectrum in AdS_5 x S^5 of which the plane-wave background is a special limit. We also compare the plane-wave string spectrum with expected form of the light-cone gauge spectrum of superstring in AdS_5 x S^5. 
  We review the combinatorial structure of perturbative quantum field theory with emphasis given to the decomposition of graphs into primitive ones. The consequences in terms of unique factorization of Dyson--Schwinger equations into Euler products are discussed. 
  We investigate the Penrose limits of classical string and M-theory backgrounds. We prove that the number of (super)symmetries of a supergravity background never decreases in the limit. We classify all the possible Penrose limits of AdS x S spacetimes and of supergravity brane solutions. We also present the Penrose limits of various other solutions: intersecting branes, supersymmetric black holes and strings in diverse dimensions, and cosmological models. We explore the Penrose limit of an isometrically embedded spacetime and find a generalisation to spaces with more than one time. Finally, we show that the Penrose limit is a large tension limit for all branes including those with fields of Born--Infeld type. 
  We discuss the problem of introducing background spacetimes of the de Sitter type (quintessential backgrounds) in the context of fundamental theories involving supersymmetry. The role of a model presented in 1984, showing that these backgrounds can occur as spontaneously broken phases of locally supersymmetric 4D, N-extended theories, is highlighted. The present twin challenges of the presumed presence of supersymmetry in particle physics and the emergence of experimental evidence for a positive cosmological constant from type-II supernovae data are noted for the continued investigation of superstring/M-theory. Finally we note how the 1984-model may have a role to play in future investigations. 
  The renormalization group (RG) flow for the two-dimensional sine-Gordon model is determined by means of Polchinski's RG equation at next-to-leading order in the derivative expansion. In this work we have two different goals, (i) to consider the renormalization scheme-dependence of Polchinski's method by matching Polchinski's equation with the Wegner-Houghton equation and with the real space RG equations for the two-dimensional dilute Coulomb-gas, (ii) to go beyond the local potential approximation in the gradient expansion in order to clarify the supposed role of the field-dependent wave-function renormalization. The well-known Coleman fixed point of the sine-Gordon model is recovered after linearization, whereas the flow exhibits strong dependence on the choice of the renormalization scheme when non-linear terms are kept. The RG flow is compared to those obtained in the Wegner-Houghton approach and in the dilute gas approximation for the two-dimensional Coulomb-gas. 
  The supersymmetric version of the descent equations following from the Wess-Zumino consistency condition is discussed. A systematic framework in order to solve them is proposed. 
  We introduce and completely describe the analogues of the Riemann curvature tensor for the curved supergrassmannian of the passing through the origin (0|2)-dimensional subsupermanifolds in the (0|4)-dimensional supermanifold with the preserved volume form. The underlying manifold of this supergrassmannian is the conventional Penrose's complexified and compactified version of the Minkowski space, i.e., the Grassmannian of 2-dimensional subspaces in the 4-dimensional space. The result provides with yet another counterexample to Coleman-Mandula's theorem. 
  We consider the reduction of supersymmetry in N-extended four dimensional supergravity via the super Higgs mechanism in theories without cosmological constant. We provide an analysis largely based on the properties of long and short multiplets of Poincare' supersymmetry. Examples of the super Higgs phenomenon are realized in spontaneously broken N=8 supergravity through the Scherk-Schwarz mechanism and in superstring compactification in presence of brane fluxes. In many models the massive vectors count the difference in number of the translation isometries of the scalar sigma-model geometries in the broken and unbroken phase. 
  Using the recently proposed formalism for Lambda<0 quantum gravity in 2+1 dimensions we study the process of black hole production in a collision of two point particles. The creation probability for a BH with a simplest topology inside the horizon is given by the Liouville theory 4-point function projected on an intermediate state. We analyze in detail the semi-classical limit of small AdS curvatures, in which the probability is dominated by the exponential of the classical Liouville action. The probability is found to be exponentially small. We then argue that the total probability of creating a horizon given by the sum of probabilities of all possible internal topologies is of order unity, so that there is no exponential suppression of the total production rate. 
  Using the kappa-symmetric action for a D3-brane, we study the interaction between its world-volume fermions and a bosonic type IIB supergravity background preserving 4-dimensional Lorentz invariance. We find that the renormalizable terms in the action include only coupling between the fermions and the 3-form flux in the combination *G_3-iG_3, which is zero for a class of supersymmetric and nonsupersymmetric solutions. We also find the magnetic and electric dipole moments for the fermions, which are proportional to the derivative of the dilaton-axion. We show that different gauges to fix the kappa-symmetry give the same interaction terms, and prove that these terms are also SL(2,R) self-dual. We interpret our results in terms of N=1 supersymmetric gauge theory on the D-brane. 
  A cosmological model based on Kaluza-Klein theory is studied. A metric, in which the scale factor of the compact space evolves as an inverse power of the radius of the observable universe, is constructed. The Freedmann-Robertson-Walker equations of standard four-dimensional cosmology are obtained precisely. The pressure in our universe is an effective pressure expressed in terms of the components of the higher dimensional energy-momentum tensor. In particular, this effective pressure could be negative and might therefore explain the acceleration of our present universe. A special feature of this model is that, for a suitable choice of the parameters of the metric, the higher dimensional gravitational coupling constant could be negative. 
  We propose a scenario for particle-mass generation, assuming the existence of a physical regime where, firstly, physical particles can be considered as point-like objects moving in a background space-time and, secondly, their mere presence spoils the invariance under the local diffeomorphism group, resulting in an anomalous realization of the latter. Under these hypotheses, we describe mass generation starting from the massless free theory. The mechanism is not sensitive to the detailed description of the underlying theory at higher energies, leaning only on general structural features of it, specifically diffeomorphism invariance. 
  We analyze the hamiltonian quantization of Chern-Simons theory associated to the universal covering of the Lorentz group SO(3,1). The algebra of observables is generated by finite dimensional spin networks drawn on a punctured topological surface. Our main result is a construction of a unitary representation of this algebra. For this purpose, we use the formalism of combinatorial quantization of Chern-Simons theory, i.e we quantize the algebra of polynomial functions on the space of flat SL(2,C)-connections on a topological surface with punctures. This algebra admits a unitary representation acting on an Hilbert space which consists in wave packets of spin-networks associated to principal unitary representations of the quantum Lorentz group. This representation is constructed using only Clebsch-Gordan decomposition of a tensor product of a finite dimensional representation with a principal unitary representation. The proof of unitarity of this representation is non trivial and is a consequence of properties of intertwiners which are studied in depth. We analyze the relationship between the insertion of a puncture colored with a principal representation and the presence of a world-line of a massive spinning particle in de Sitter space. 
  A system of coupled scalar fields is introduced which possesses a spectrum of massive single-soliton solutions. Some of these solutions are unstable and decay into lower mass stable solitons. Some properties of the solutions are obtained using general principles including conservations of energy and topological charges. Rest energies are calculated via a variational scheme, and the dynamics of the coupled fields are obtained by solving the field equations numerically. 
  We obtain the correct cohomology at any ghost number for the open and closed covariant superstring, quantized by an approach which we recently developed. We define physical states by the usual condition of BRST invariance and a new condition involving a new current which is related to a grading of the underlying affine Lie algebra. 
  Non-supersymmetric brane world scenarios in string theory display perturbative instabilities that usually involve run-away potentials for scalar moduli fields. We investigate in the framework of intersecting brane worlds whether the leading order scalar potential for the closed string moduli allows to satisfy the slow-rolling conditions required for applications in inflationary cosmology. Adopting a particular choice of basis in field space and assuming mechanisms to stabilize some of the scalars, we find that slow-rolling conditions can be met very generically. In intersecting brane worlds inflation can end nearly instantaneously like in the hybrid inflation scenario due to the appearance of open string tachyons localized at the intersection of two branes, which signal a corresponding phase transition in the gauge theory via the condensation of a Higgs field. 
  We describe how mirror symmetry of three-dimensional N=1 supersymmetric gauge theories can be used to determine the theory on the world-volume of a D2-brane probe of manifolds with G_2 holonomy. This is a much shortened companion paper to hep-th/0202126. 
  Using D2-brane probes, we study various properties of M-theory on singular, non-compact manifolds of G_2 and Spin(7) holonomy. We derive mirror pairs of N=1 supersymmetric three-dimensional gauge theories, and apply this technique to realize exceptional holonomy manifolds as both Coulomb and Higgs branches of the D2-brane world-volume theory. We derive a ``G_2 quotient construction'' of non-compact manifolds which admit a metric of G_2 holonomy. We further discuss the moduli space of such manifolds, including the structure of geometrical transitions in each case. For completeness, we also include familiar examples of manifolds with SU(3) and Sp(2) holonomy, where some of the new ideas are clarified and tested. 
  We interpret cosmological evolution holographically as a renormalisation group flow in a dual Euclidean field theory, as suggested by the conjectured dS/CFT correspondence. Inflation is described by perturbing around the infra-red fixed point of the dual field theory. The spectrum of the cosmic microwave background radiation is determined in terms of scaling violations in the field theory. The dark energy allows similar, albeit less predictive, considerations. We discuss the cosmological fine-tuning problems from the holographic perspective. 
  The paper is devoted to the comparison of the Fukaya category (it is responcible for the A-side of mirror symmetry) with the category of holonomic modules over the quantized algebra of functions on the same symplectic manifold. We conjecture that these categories become $A_{\infty}$-equivalent after a twist by a kind of integral transformation. 
  We compute the partition function of the conformal field theory on the two-dimensional euclidean black hole background using path-integral techniques. We show that the resulting spectrum is consistent with the algebraic expectations for the SL(2,R)/U(1) coset conformal field theory construction. In particular, we find confirmation for the bound on the spin of the discrete representations and we determine the density of the continuous representations. We point out the relevance of the partition function to all string theory backgrounds that include an SL(2,R)/U(1) coset factor. 
  We study a non-Riemannian modification of 5-dimensional Kaluza-Klein theory. In our proposal the Riemannian structure of the five-dimensional manifold is replaced by a Weyl-integrable one. In this context a 4-dimensional Poincar$\grave{e}$ invariant solution is studied. Spacetime structures with thick smooth branes separated in the extra dimension arise. We focus our attention, mainly, in a case where the massless graviton is located in one of the thick branes at the origin, meanwhile the matter degrees of freedom are confined to the other brane. Due to the small overlap of the graviton's wave-function with the second thick brane, the model accounts for a resolution of the mass hierarchy problem a la Randall-Sundrum. Although, initially, no assumptions are made about the topology of the extra dimension, the solution found yields an extra space that is effectivelly compact and respects $Z_2$ symmetry. Unlike some other models with branes, a preliminary study (not including the effect of scalar perturbations) shows that the spectrum of massive Kaluza-Klein states is quantized and free of tachyonic modes. 
  We consider system of Dp-branes in the presence of a nonzero B field with one leg along brane worldvolume and the other transverse to it. We study the corresponding supergravity solutions and show that the worldvolume theories decouple from gravity for $p\leq 5$. Therefore these solutions provide dual description of large N noncommutative dipole field theories. We shall only consider those systems which preserve 8 supercharges in the branes worldvolume. We analyze the system of M5-branes and NS5-branes in the presence of nonzero C field and RR field with one leg along the transverse direction and the others along the worldvolume of the brane, respectively. This could provide a new deformation of (2,0) and little string field theories. Finally, we study the Wilson loops using the dual gravity descriptions. 
  A sigma model formulation for static dilaton black holes with N Abelian gauge fields was presented. We proved uniqueness of static asymptotically flat spacetimes with non-degenerate black holes for the considered theory. 
  We find marginal and scalar solutions in cubic open string field theory by using left-right splitting properties of a delta function. The marginal solution represents a marginal deformation generated by a U(1) current, and it is a generalized solution of the Wilson lines one given by the present authors. The scalar solution has a well-defined universal Fock space expression, and it is expressed as a singular gauge transform of the trivial vacuum. The expanded theory around it is unable to be connected with the original theory by the string field redefinition. Errors in hep-th/0112124 are corrected in this paper. 
  Coupling any interacting quantum mechanical system to gravity in one dimension requires the cosmological constant to belong to the matter energy spectrum and thus to be quantized, even though the gravity sector is free of any quantum dynamics, while physical states are also confined to the subspace of matter quantum states whose energy coincides with the cosmological constant value. These general facts are illustrated through some simple examples. The physical projector quantization approach readily leads to the correct representation of such systems, whereas other approaches relying on gauge fixing methods are often plagued by Gribov problems in which case the quantization rule is not properly recovered. Whether such a quantization of the cosmological constant as well as the other ensuing consequences in terms of physical states extend to higher dimensional matter-gravity coupled quantum systems is clearly a fascinating open issue. 
  We develop a systematic approach to G_2 holonomy manifolds with an SU(2)xSU(2) isometry using maximal eight-dimensional gauged supergravity to describe D6-branes wrapped on deformed three-spheres. A quite general metric ansatz that generalizes the celebrated Bryant-Salamon metric involves nine functions. We show that only six of them are the independent ones and derive the general first order system of differential equations that they obey. As a byproduct of our analysis, we generalize the notion of the twist that relates the spin and gauge connections in a way that involves non-trivially the scalar fields. 
  The modification to four--dimensional Einstein gravity at low energy in two brane models is investigated within supergravity in singular spaces. Using perturbation theory around a static BPS background, we study the effective four--dimensional gravitational theory, a scalar--tensor theory, and derive the Brans--Dicke parameter when matter is present on the positive tension brane only. We show there is an attractor mechanism towards general relativity in the matter dominated era. The dynamics of the interbrane distance are discussed. Finally, when matter lives on both branes, we find that there is a violation of the equivalence principle whose magnitude is governed by the warping of the extra dimension. 
  Two main problems face the construction of noncommutative actions for gravity with star products: the complex metric and finding an invariant measure. The only gauge groups that could be used with star products are the unitary groups. I propose an invariant gravitational action in D=4 dimensions based on the constrained gauge group U(2,2) broken to $U(1,1)\times U(1,1).$ No metric is used, thus giving a naturally invariant measure. This action is generalized to the noncommutative case by replacing ordinary products with star products. The four dimensional noncommutative action is studied and the deformed action to first order in deformation parameter is computed. 
  We present a matrix action based on the unitary group U(N) whose large N ground states are conjectured to be in precise correspondence with the weak-strong dual effective field theory limits of M theory preserving sixteen supersymmetries. We identify a finite N matrix algebra that corresponds to the spacetime and internal symmetry algebra of the Lorentz invariant field theories obtained in the different large N limits. The manifest diffeomorphism invariance of matrix theory is spontaneously broken upon specification of the large N ground state. We verify that there exist planar limits which yield the low energy spacetime effective actions of all six supersymmetric string theories in nine spacetime dimensions and with sixteen supercharges. 
  The reparametrization symmetries of Witten's vertex in ordinary or vacuum string field theories can be used to extract useful information about classical solutions of the equations of motion corresponding to D-branes. It follows, that the vacuum string field theory in general has to be regularized. For the regularization recently considered by Gaiotto et al., we show that the identities we derive, are so constraining, that among all surface states they uniquely select the simplest butterfly projector discovered numerically by those authors. The reparametrization symmetries are also used to give a simple proof that the butterfly states and their generalizations are indeed projectors. 
  We study anti-de Sitter black holes in the Einstein-Gauss-Bonnet and the generic R^2 gravity theories, evaluate different thermodynamic quantities, and also examine the possibilities of Hawking-Page type thermal phase transitions in these theories. In the Einstein theory, with a possible cosmological term, one observes a Hawking-Page phase transition only if the event horizon is a hypersurface of positive constant curvature (k=1). But, with the Gauss-Bonnet or/and the (Riemann)^2 interaction terms, there may occur a similar phase transition for a horizon of negative constant curvature (k=-1). We examine the finite coupling effects, and find that N>5 could trigger a Hawking-Page phase transition in the latter theory. For the Gauss-Bonnet black holes, one relates the entropy of the black hole to a variation of the geometric property of the horizon based on first law and Noether charge. With (Riemann)^2 term, however, we can do this only approximately, and the two results agree when, r_H>>L, the size of the horizon is much bigger than the AdS curvature scale. We establish some relations between bulk data associated with the AdS black hole and boundary data defined on the horizon of the AdS geometry. Following a heuristic approach, we estimate the difference between Hubble entropy {\cal S}_H and Bekenstein-Hawking entropy {\cal S}_{BH} with (Riemann)^2 term, which, for k=0 and k=-1, would imply {\cal S}_{BH}\leq {\cal S}_H. 
  We consider the superposition of the first two members of the gravitational hierarchy (Einstein plus first Gauss-Bonnet(GB)) interacting with the superposition of the first two members of the $SO_{(\pm)}(d)$ Yang--Mills hierarchy, in $d$ dimensions. Such systems can occur in the low energy effective action of string theory. Particle-like solutions %for the systems with only an Einstein term, and with only a GB term, in dimensions $d=6,8$ are constructed respectively. Our results reveal qualitatively new properties featuring double-valued solutions with critical behaviour. In this preliminary study, we have restricted ourselves to one-node solutions. 
  Following a self-contained review of the basics of the theory of cosmological perturbations, we discuss why the conclusions reached in the recent paper by Kaloper et al are too pessimistic estimates of the amplitude of possible imprints of trans-Planckian (string) physics on the spectrum of cosmic microwave anisotropies in an inflationary Universe. It is shown that the likely origin of large trans-Planckian effects on late time cosmological fluctuations comes from nonadiabatic evolution of the state of fluctuations while the wavelength is smaller than the Planck (string) scale, resulting in an excited state at the time that the wavelength crosses the Hubble radius during inflation. 
  Bogolubov's chain of equations method for temperature Wightman functions is suggested for investigation of relativistic phase transition. The chain equations for the Wightman functions forming momentum--energy tensor are obtained. It is clarified that structure of the chain equations determines the basis approximation (the Hartree - Fock approximation) and corrections calculation algorithm. The basis approximation is investigated in details: renormalized equations for effective masses, order parameter and generating functional which reproduce those equations are obtained. Being considered on the solution of the gap equations for the effective masses the generating functional turns to nonequilibrium functional of free energy density, which allows to obtain phases stability conditions. Thermodynamic observables like heat capacity and sonic speed are calculated. The correction to the Hartree-Fock approximations is ascertained to be small for all temperatures excluding vicinity of the phases equilibrium point. 
  We study the issue of gauge invariance in five-dimensional theories compactified on an orbifold $S^1/(\mathbb{Z}_2\times \mathbb{Z}^\prime_2)$ in the presence of an external U(1) gauge field. From the four-dimensional point We study the issue of gauge invariance in five-dimensional theories compactified on an orbifold $S^1/(\mathbb{Z}_2\times \mathbb{Z}^\prime_2)$ in the presence of an external U(1) gauge field. From the four-dimensional point of view the theory contains a tower of Kaluza-Klein Dirac fermions with chiral couplings and it looks anomalous at the quantum level. We show that this ``anomaly'' is cancelled by a topological Chern-Simons term which is generated in the effective action when the gauge theory is regularized introducing an Pauli-Villars fermion with an odd mass term. In the presence of a classical background gauge field, the fermionic current acquires a vacuum expectation value, thus generating the suitable Chern-Simons term and a gauge invariant theory. 
  The Hamiltonian reduction of SU(2) Yang-Mills theory for an arbitrary \theta angle to an unconstrained nonlocal theory of a self-interacting positive definite symmetric 3 \times 3 matrix field S(x) is performed. It is shown that, after exact projection to a reduced phase space, the density of the Pontryagin index remains a pure divergence, proving the \theta independence of the unconstrained theory obtained. An expansion of the nonlocal kinetic part of the Hamiltonian in powers of the inverse coupling constant and truncation to lowest order, however, lead to violation of the \theta independence of the theory. In order to maintain this property on the level of the local approximate theory, a modified expansion in the inverse coupling constant is suggested, which for a vanishing \theta angle coincides with the original expansion. The corresponding approximate Lagrangian up to second order in derivatives is obtained, and the explicit form of the unconstrained analogue of the Chern-Simons current linear in derivatives is given. Finally, for the case of degenerate field configurations S(x) with rank|S| = 1, a nonlinear \sigma-type model is obtained, with the Pontryagin topological term reducing to the Hopf invariant of the mapping from the three-sphere S^3 to the unit two-sphere S^2 in the Whitehead form. 
  A low energy bound in a class of chiral solitonic field theories related the infrared physics of the SU(N) Yang-Mills theory is established. 
  Some issues on the stability of black string or brane solutions are summarized briefly. The stability of dS/AdS-Schwarzschild black strings has been investigated. Interestingly, the AdS-Schwarzschild black strings turn out to be stable as the horizon size becomes larger than the AdS scale. It is also shown that BTZ black strings in four dimensions are stable regardless of the horizon size. Such stable feature seems to be common for several known black strings in dimensions lower than five. Some implications of our results on the role of non-uniformity in stable black string configurations are also discussed. 
  We argue that the early universe may be described by an initial state of space-filling branes and anti-branes. At high temperature this system is stable. At low temperature tachyons appear and lead to a phase transition, dynamics, and the creation of D-branes. These branes are cosmologically produced in a generic fashion by the Kibble mechanism. From an entropic point of view, the formation of lower dimensional branes is preferred and $D3$ brane-worlds are exponentially more likely to form than higher dimensional branes. Virtually any brane configuration can be created from such phase transitions by adjusting the tachyon profile. A lower bound on the number defects produced is: one D-brane per Hubble volume. 
  We construct the hyperkahler cones corresponding to the Quaternion-Kahler orthogonal Wolf spaces SO(n+4)/(SO(n)xSO(4)) and their non-compact versions, which appear in hypermultiplet couplings to N=2 supergravity. The geometry is completely encoded by a single function, the hyperkahler potential, which we compute from an SU(2) hyperkahler quotient of flat space. We derive the Killing vectors and moment maps for the SO(n+4) isometry group on the hyperkahler cone. For the non-compact case, the isometry group SO(n,4) contains n+2 abelian isometries which can be used to find a dual description in terms of n tensor multiplets and one double-tensor multiplet. Finally, using a representation of the hyperkahler quotient via quiver diagrams, we deduce the existence of a new eight dimensional ALE space. 
  We provide a detailed map between wrapped D3-branes in Anti-de Sitter (AdS) backgrounds and dibaryon operators in the corresponding conformal field theory (CFT). The effective five dimensional action governing the dynamics of AdS space contains a $U(1)_R$ gauge field that mediates interactions between objects possessing R-charge. We show that the $U(1)_R$ charge of these wrapped D3-branes as measured by the gauge field matches the R-charge of the dibaryons expected from field theory considerations. We are able, through a careful probe brane calculation in an $AdS_5\times T^{1,1}$ background, to understand the exact relation between the mass of the wrapped D3-brane and the dimension of the corresponding dibaryon. We also make some steps toward matching the counting of dibaryon operators in the CFT with the ground states of a supersymmetric quantum mechanical system whose target space is the moduli space of D-branes. Finally, we discuss BPS excitations of the D3-brane and compare them with higher dimension operators in the CFT. 
  Surface states are open string field configurations which arise from Riemann surfaces with a boundary and form a subalgebra of the star algebra. We find that a general class of star algebra projectors arise from surface states where the open string midpoint reaches the boundary of the surface. The projector property of the state and the split nature of its wave-functional arise because of a nontrivial feature of conformal maps of nearly degenerate surfaces. Moreover, all such projectors are invariant under constant and opposite translations of their half-strings. We show that the half-string states associated to these projectors are themselves surface states. In addition to the sliver, we identify other interesting projectors. These include a butterfly state, which is the tensor product of half-string vacua, and a nothing state, where the Riemann surface collapses. 
  The Weyl group symmetry W(E_k) is studied from the points of view of the E-strings, Painleve equations and U-duality. We give a simple reformulation of the elliptic Painleve equation in such a way that the hidden symmetry W(E_10) is manifestly realized. This reformulation is based on the birational geometry of the del Pezzo surface and closely related to Seiberg-Witten curves describing the E-strings. The relation of the W(E_k) symmetry to the duality of M-theory on a torus is discussed on the level of string equations of motion. 
  We observe that the pp wave limit of $AdS_5\times M^5$ compactifications of type IIB string theory is universal, and maximally supersymmetric, as long as $M^5$ is smooth and preserves some supersymmetry. We investigate a specific case, $M^5=T^{1,1}$. The dual ${\cal N}=1$ SCFT, describing D3-branes at a conifold singularity, has operators that we identify with the oscillators of the light-cone string in the universal pp-wave background. The correspondence is remarkable in that it relies on the exact spectrum of anomalous dimensions in this CFT, along with the existence of certain exceptional series of operators whose dimensions are protected only in the limit of large `t Hooft coupling. We also briefly examine the singular case $M^5=S^5/Z_2$, for which the pp wave background becomes a $Z_2$ orbifold of the maximally supersymmetric background by reflection of 4 transverse coordinates. We find operators in the corresponding ${\cal N}=2$ SCFT with the right properties to describe both the untwisted and the twisted sectors of the closed string. 
  We present an alternative quantization for irreducible open gauge theories. The method relies on the possibility of modifying the classical BRST operator and the gauge-fixing action written as in Yang-Mills type theories, in order to obtain an on-shell invariant quantum action by using equations characterizing the full gauge algebra. From this follows then the construction of an off-shell version of the theory. We show how it is possible to build off-shell BRST algebra together with an invariant extension of the classical action. This is realized via a systematic prescription for the introduction of auxiliary fields. 
  We derive the Wilson-Polchinski RG equation in the planar limit. We explain that the equation necessarily involves also non-planar amplitudes with sphere topology, which represent multi-trace contributions to the effective action. The resulting RG equation turns out to be of the Hamilton-Jacobi type since loop effects manifest themselves through terms which are linear in first order derivatives of the effective action with respect to the sources. We briefly outline applications to renormalization of non-commutative field theories, matrix models with external sources and holography. 
  We review Wilson loops in N=4 supersymmetric Yang-Mills theory with emphasis on the exact results. The implications are discussed in the context of the AdS/CFT correspondence. 
  We find a Penrose limit of AdS_5 x T^{1,1} which gives the pp-wave geometry identical to the one that appears in the Penrose limit of AdS_5 x S^5. This leads us to conjecture that there is a subsector of the corresponding N=1 gauge theory which has enhanced N=4 supersymmetry. We identify operators in the N=1 gauge theory with stringy excitations in the pp-wave geometry and discuss how the gauge theory operators fall into N=4 supersymmetry multiplets. We find similar enhancement of symmetry in some other models, but there are also examples in which there is no supersymmetry enhancement in the Penrose limit. 
  It was recently shown that velocity-dependent forces between parallel fundamental strings moving apart in a $D-$dimensional spacetime implied an accelerating expanding universe in $D-1$-dimensional space-time. Exact solutions were obtained for the early time expansion in $D=5,6$. Here we show that this result also holds for fundamental strings in the background of a fivebrane, and argue that the feature of an accelerating universe would hold for more general $p$-brane-seeded models. 
  The equation describing the stochastic motion of a classical particle in 1+1-dimensional space-time is connected to the Dirac equation with external gauge fields. The effects of assigning different turning probabilities to the forward and the backward moving particles in time are discussed. 
  In this talk we consider the issue of stabilization of compact hyperbolic brane-world scenarios from the point of view of 4-dimensional effective theories. The idea is to clarify the status of stabilization for these models. Possible ways to overcome a no-go theorem that appeared in a recent paper are shown invoking the holographic framework and type IIA*/IIB* theories. A brief discussion on supersymmetry is also given. 
  Motivated by recent proposals for a de Sitter version of the AdS/CFT correspondence, we give some topological restrictions on spacetimes of de Sitter type, i.e., spacetimes with $\Lambda>0$, which admit a regular past and/or future conformal boundary. For example we show that if $M^{n+1}$, $n \ge 2$, is a globally hyperbolic spacetime obeying suitable energy conditions, which is of de Sitter type, with a conformal boundary to both the past and future, then if one of these boundaries is compact, it must have finite fundamental group and its conformal class must contain a metric of positive scalar curvature. Our results are closely related to theorems of Witten and Yau hep-th/9910245 pertaining to the Euclidean formulation of the AdS/CFT correspondence. 
  There exist both continuum and lattice regularizations of gauge theories with fermions which preserve chiral U(1) invariance ("fermion number"). Such regularizations necessarily break gauge invariance but, in a covariant gauge, one recovers gauge invariance to all orders in perturbation theory by including suitable counterterms.   At the non-perturbative level, an apparent conflict then arises between the chiral U(1) symmetry of the regularized theory and the existence of 't Hooft vertices in the renormalized theory. The only possible resolution of the paradox is that the chiral U(1) symmetry is broken spontaneously in the enlarged Hilbert space of the covariantly gauge-fixed theory. The corresponding Goldstone boson is unphysical. The theory must therefore be defined by introducing a small fermion-mass term that breaks explicitly the chiral U(1) invariance, and is sent to zero after the infinite-volume limit has been taken. Using this careful definition (and a lattice regularization) for the calculation of correlation functions in the one-instanton sector, we show that the 't Hooft vertices are recovered as expected. 
  In this paper a de Sitter Space version of Black Hole Complementarity is formulated which states that an observer in de Sitter Space describes the surrounding space as a sealed finite temperature cavity bounded by a horizon which allows no loss of information. We then discuss the implications of this for the existence of boundary correlators in the hypothesized dS/cft correspondence. We find that dS complementarity precludes the existence of the appropriate limits. We find that the limits exist only in approximations in which the entropy of the de Sitter Space is infinite. The reason that the correlators exist in quantum field theory in the de Sitter Space background is traced to the fact that horizon entropy is infinite in QFT. 
  We present reasons as to why an ab initio analysis of the spacetime structure of massive gravitino is necessary. Afterwards, we construct the relevant representation space, and finally, give a new physical interpretation of massive gravitino. 
  We analyse unstable D-brane systems in type I string theory. Generalizing the proposal in hep-th/0108085, we give a physical interpretation for real KK-theory and claim that the D-branes embedded in a product space X x Y which are made from the unstable Dp-brane system wrapped on Y are classified by a real KK-theory group KKO^{p-1}(X,Y). The field contents of the unstable D-brane systems are systematically described by a hidden Clifford algebra structure.   We also investigate the matrix theory based on non-BPS D-instantons and show that the spectrum of D-branes in the theory is exactly what we expect in type I string theory, including stable non-BPS D-branes with Z_2 charge. We explicitly construct the D-brane solutions in the framework of BSFT and analyse the physical property making use of the Clifford algebra. 
  We study the four-dimensional gravitational fluctuation on anisotropic brane tension embedded in braneworlds with vanishing bulk cosmological constant. In this setup, warp factors have two types (A and B) and we point out that the two types correspond to positive and negative tension brane, respectively. We show that volcano potential in the model of type A has singularity and the usual Newton's law is reproduced by the existence of normalizable zero mode. While, in the case of type B, the effective Planck scale is infinite so that there is no normalizable zero mode. 
  The renormalization group flow equation obtained by means of a proper time regulator is used to calculate the two loop beta function and anomalous dimension eta of the field for the O(N) symmetric scalar theory. The standard perturbative analysis of the flow equation does not yield the correct results for both beta and eta. We also show that it is still possible to extract the correct beta and eta from the flow equation in a particular limit of the infrared scale. A modification of the derivation of the Exact Renormalization Group flow, which involves a more general class of regulators, to recover the proper time renormalization group flow is analyzed. 
  Compactifications of type II theories on Calabi-Yau threefolds including electric and magnetic background fluxes are discussed. We derive the bosonic part of the four-dimensional low energy effective action and show that it is a non-canonical N=2 supergravity which includes a massive two-form. The symplectic invariance of the theory is maintained as long as the flux parameters transform as a symplectic vector and a massive two-form which couples to both electric and magnetic field strengths is present. The mirror symmetry between type IIA and type IIB compactified on mirror manifolds is shown to hold for R-R fluxes at the level of the effective action. We also compactify type IIA in the presence of NS three-form flux but the mirror symmetry in this case remains unclear. 
  We use a model where the cosmological term can be related to the chiral gauge anomaly of a possible quantum scenario of the initial evolution of the universe. We show that this term is compatible with the Friedmann behavior of the present universe. 
  We study the phase structure of nonlocal two dimensional generalized Yang - Mills theories (nlgYM$_2$) and it is shown that all order of $\phi^{2k}$ model of these theories has phase transition only on compact manifold with $g = 0$(on sphere), and the order of phase transition is 3. Also it is shown that the $\phi^2 + \frac{2\alpha}{3}\phi^3$ model of nlgYM$_2$ has third order phase transition on any compact manifold with $1 < g < 1+ \frac{\hat{A}}{|\eta_c|}$, and has no phase transition on sphere. 
  UV/IR mixing is one of the most important features of noncommutative field theories. As a consequence of this coupling of the UV and IR sectors, the configuration of fields at the zero momentum limit in these theories is a very singular configuration. We show that the renormalization conditions set at a particular momentum configuration with a fixed number of zero momenta, renormalizes the Green's functions for any general momenta only when this configuration has same set of zero momenta. Therefore only when renormalization conditions are set at a point where all the external momenta are nonzero, the quantum theory is renormalizable for all values of nonzero momentum. This arises as a result of different scaling behaviors of Green's functions with respect to the UV cutoff ($\Lambda$) for configurations containing different set of zero momenta. We study this in the noncommutative $\phi^4$ theory and analyse similar results for the Gross-Neveu model at one loop level. We next show this general feature using Wilsonian RG of Polchinski in the globally O(N) symmetric scalar theory and prove the renormalizability of the theory to all orders with an infrared cutoff. In the context of spontaneous symmetry breaking (SSB) in noncommutative scalar theory, it is essential to note the different scaling behaviors of Green's functions with respect to $\Lambda$ for different set of zero momenta configurations. We show that in the broken phase of the theory the Ward identities are satisfied to all orders only when one keeps an infrared regulator by shifting to a nonconstant vacuum. 
  We study confinement in softly broken N=2 SUSY QCD with gauge group SU(N_c) and N_f hypermultiplets of fundamental matter (quarks) when the Coulomb branch is lifted by small mass of adjoint matter. Concentrating mostly on the theory with SU(3) gauge group we discuss the N=1 vacua which arise in the weak coupling at large values of quark masses and study flux tubes and monopole confinement in these vacua. In particular we find the BPS strings in SU(3) gauge theory formed by two interacting U(1) gauge fields and two scalar fields generalizing ordinary Abrikosov-Nielsen-Olesen vortices. Then we focus on the SU(3) gauge theories with N_f=4 and N_f=5 flavors with equal masses. In these theories there are N=1 vacua with restored SU(2) gauge subgroup in quantum theory since SU(2) subsectors are not asymptotically free. We show that although the confinement in these theories is due to Abelian flux tubes the multiplicity of meson spectrum is the same as expected in a theory with non-Abelian confinement. 
  We propose a generalization of the standard geometric formulation of quantum mechanics, based on the classical Nambu dynamics of free Euler tops. This extended quantum mechanics has in lieu of the standard exponential time evolution, a nonlinear temporal evolution given by Jacobi elliptic functions. In the limit where latter's moduli parameters are set to zero, the usual geometric formulation of quantum mechanics, based on the Kahler structure of a complex projective Hilbert space, is recovered. We point out various novel features of this extended quantum mechanics, including its geometric aspects. Our approach sheds a new light on the problem of quantization of Nambu dynamics. Finally, we argue that the structure of this nonlinear quantum mechanics is natural from the point of view of string theory. 
  We argue that the cosmological constant problem can be solved in a braneworld model with infinite-volume extra dimensions, avoiding no-go arguments applicable to theories that are four-dimensional in the infrared. Gravity on the brane becomes higher-dimensional at super-Hubble distances, which entails that the relation between the acceleration rate and vacuum energy density flips upside down compared to the conventional one. The acceleration rate decreases with increasing the energy density. The experimentally acceptable rate is obtained for the energy density larger than (1 TeV)$^4$. The results are stable under quantum corrections because supersymmetry is broken only on the brane and stays exact in the bulk of infinite volume extra space. Consistency of 4D gravity and cosmology on the brane requires the quantum gravity scale to be around $10^{-3}$ eV. Testable predictions emerging within this approach are: (i) simultaneous modifications of gravity at sub-millimeter and the Hubble scales; (ii) Hagedorn-type saturation in TeV energy collisions due to the Regge spectrum with the spacing equal to $10^{-3}$ eV. 
  We construct a D3-brane wrapped on S^1, which is fibred over the resolved conifold as its transverse space. Whereas a fractional D3-brane on the resolved conifold is not supersymmetric and has a naked singularity, our solution is supersymmetric and regular everywhere. We also consider an $S^1$-wrapped D3-brane on the resolved cone over T^{1,1}/Z_2, as well as on the deformed conifold. In the former case, we obtain a regular supergravity dual to a certain four-dimensional field theory whose Lorentz and conformal symmetries are broken in the IR region and restored in the UV limit. 
  We calculate the spectrum of the matrix M' of Neumann coefficients of the Witten vertex, expressed in the oscillator basis including the zero-mode a_0. We find that in addition to the known continuous spectrum inside [-1/3,0) of the matrix M without the zero-modes, there is also an additional eigenvalue inside (0,1). For every eigenvalue, there is a pair of eigenvectors, a twist-even and a twist-odd. We give analytically these eigenvectors as well as the generating function for their components. Also, we have found an interesting critical parameter b_0 = 8 ln 2 on which the forms of the eigenvectors depend. 
  Regular magnetic monopoles in the non-Abelian Born-Infeld-Higgs theory are known to exist in the region of the field strength parameter $\beta>\beta_{{\rm cr}}$, bounded from below. Beyond this region, only pointlike (embedded abelian) monopoles exist, and we show that the transition from the regular to singular structure is reminiscent of gravitational collapse. Near the threshold behavior is characterized by the rapidly increasing negative pressure, which typically arises in the high density NBI matter. Another feature, shared both the NBI and gravitating monopoles, is the existence of excited states, which can be thought of as bound states of monopoles and sphalerons. These are labeled by the number $N$ of nodes of the Yang-Mills function. Their masses are greater than the mass of the ground state monopole, and they are expected to be unstable. The sequence of masses $M_N$ rapidly converges to the mass of the embedded Abelian solution with constant Higgs. The ratio of the sphaleron size to that of the monopole grows with decreasing $\beta$, and, at the same time, both fall down until the solutions cease to exist, again exhibiting collapse to the pointlike monopole. The results are presented and compared both for the ordinary and the symmetrized trace NBI actions. 
  We show that, contrary to previous string models, the high-temperature behaviour of the recently proposed confining strings reproduces exactly the correct large-N QCD result, a {\it necessary} condition for any string model of confinement. 
  We consider type IIB string in the two plane-wave backgrounds which may be interpreted as special limits of the AdS_3 x S^3 metric supported by either the NS-NS or R-R 3-form field. The NS-NS plane-wave string model is equivalent to a direct generalization of the Nappi-Witten model, with its spectrum being similar to that of strings in constant magnetic field. The R-R model can be solved in the light-cone gauge, where the Green-Schwarz action describes 4 massive and 4 massless copies of free bosons and fermions. We find the spectra of the two string models and study the asymptotic density of states. We also discuss a more general class of exactly solvable plane-wave models with reduced supersymmetry which is obtained by adding twists in two spatial 2-planes. 
  We discuss the different bounds on entropy in the context of two-dimensional cosmology. We show that in order to obtain well definite bounds one has to use the scale symmetry of the gravitational theory. We then extend the recently found relation between the Friedmann equation and the Cardy formula to the case of two dimensions. In particular, we find that in two dimensions this relation requires that the central charge c of the conformal field theory is given in terms of the Newton constant G of the gravitational theory by c=6/G. 
  The quantum equivalence between sigma-models and their non-abelian T-dualised partners is examined for a large class of four dimensional non-homogeneous and quasi-Einstein metrics with an isometry group SU(2) times U(1). We prove that the one-loop renormalisability of the initial torsionless sigma-models is equivalent to the one-loop renormalisability of the T-dualised torsionful model. For a subclass of Kahler original metrics, the dual partners are still Kahler (with torsion). 
  We present an exact normalisable zero-energy chiral fermion solution for abelian BPS dipoles. For a single dipole, this solution is contained within the high temperature limit of the SU(2) caloron with non-trivial holonomy. 
  We study cosmological backgrounds from the point of view of the dS/CFT correspondence and its renormalization group flow extension. We focus on the case where gravity is coupled to a single scalar with a potential. Depending on the latter, the scalar can drive both inflation and the accelerated expansion (dS) phase in the far future. We also comment on quintessence scenarios, and flows familiar from the AdS/CFT correspondence. We finally make a tentative embedding of this discussion in string theory where the scalar is the dilaton and the potential is generated at the perturbative level. 
  Solutions of D=7 maximal gauged supergravity are constructed with metrics that are a product of a n-dimensional anti-de Sitter (AdS) space, with n=2,3,4,5, and certain Einstein manifolds. The gauge fields have the same form as in the recently constructed solutions describing the near-horizon limits of M5-branes wrapping supersymmetric cycles. The new solutions do not preserve any supersymmetry and can be uplifted to obtain new solutions of D=11 supergravity, which are warped and twisted products of the D=7 metric with a squashed four-sphere. Some aspects of the stability of the solutions are discussed. 
  The methodology based on the association of the Variational Method with Supersymmetric Quantum Mechanics is used to evaluate the energy states of the confined hydrogen atom. 
  We discuss various Penrose limits of conformal and nonconformal backgrounds. In AdS_5 x T^{1,1}, for a particular choice of the angular coordinate in T^{1,1} the resulting Penrose limit coincides with the similar limit for AdS_5 x S^5. In this case an identification of a subset of field theory operators with the string zero modes creation operators is possible. For another limit we obtain a light-cone string action that resembles a particle in a magnetic field. We also consider three different types of backgrounds that are dual to nonconformal field theories: The Schwarzschild black hole in AdS_5, D3-branes on the small resolution of the conifold and the Klebanov-Tseytlin background. We find that in all three cases the introduction of nonconformality renders a theory that is no longer exactly solvable and that the form of the deformation is universal. The corresponding world sheet theory in the light-cone gauge has a \tau=x^+ dependent mass term. 
  We explore bosonic strings and Type II superstrings in the simplest time dependent backgrounds, namely orbifolds of Minkowski space by time reversal and some spatial reflections. We show that there are no negative norm physical excitations. However, the contributions of negative norm virtual states to quantum loops do not cancel, showing that a ghost-free gauge cannot be chosen. The spectrum includes a twisted sector, with strings confined to a ``conical'' singularity which is localized in time. Since these localized strings are not visible to asymptotic observers, interesting issues arise regarding unitarity of the S-matrix for scattering of propagating states. The partition function of our model is modular invariant, and for the superstring, the zero momentum dilaton tadpole vanishes. Many of the issues we study will be generic to time-dependent cosmological backgrounds with singularities localized in time, and we derive some general lessons about quantizing strings on such spaces. 
  We study different renormalisation group flows for scale dependent effective actions, including exact and proper-time renormalisation group flows. These flows have a simple one loop structure. They differ in their dependence on the full field-dependent propagator, which is linear for exact flows. We investigate the inherent approximations of flows with a non-linear dependence on the propagator. We check explicitly that standard perturbation theory is not reproduced. We explain the origin of the discrepancy by providing links to exact flows both in closed expressions and in given approximations. We show that proper-time flows are approximations to Callan-Symanzik flows. Within a background field formalism, we provide a generalised proper-time flow, which is exact. Implications of these findings are discussed. 
  We investigate some of the issues relating to the dynamical instability of general static spacetimes with horizons. Our paper will be partially pedagogical and partially exploratory in nature. After discussing the current understanding, we generalize the proposal of Gubser and Mitra, which identifies dynamical instability of black branes with local thermodynamic instability, to include all product spacetimes with the horizon uniformly smeared over an internal space. We support our conjecture by explicitly exhibiting a threshold unstable mode for Schwarzschild-AdS_5 x S^5 black hole. Using the AdS/CFT correspondence, this simultaneously yields a prediction for a phase transition in the dual gauge theory. We also discuss implications for spacetimes with cosmological horizons. 
  The isometry algebras of the maximally supersymmetric solutions of IIB supergravity are derived by the Inonu-Wigner contractions of the super-AdS_5xS^5 algebra. The super-AdS_5xS^5 algebra allows introducing two contraction parameters; the one for the Penrose limit to the maximally supersymmetric pp-wave algebra and the AdS_5xS^5 radius for the flat limit. The fact that the Jacobi identity of three supercharges holds irrespectively of these parameters reflects the fact that the number of supersymmetry is not affected under both contractions. 
  It has been shown that (2+1)-dimensional N=8 super Yang-Mills (SYM) theory with electric flux is related to (2+1)- dimensional noncommutative open string (NCOS) theory by `2-11' flip. This implies that the instanton process in SYM theory, which corresponds to D0-brane exchange (M-momentum transfer) between D2-branes, is dual to the KK momentum exchange in NCOS theory, which is perturbative process in nature. In order to confirm this, we obtain the effective action of probe M2-brane on the background of tilted M2-branes, which would correspond to the one-loop effective action of SYM theory with non-perturbative instanton corrections. Then we consider the dual process in NCOS theory, which is the scattering amplitude of the wound graviton off the D2-F1 bound state involving KK-momentum transfer in x^2-direction. Both of them give the same interaction terms. Remarkably they also have the same behavior on the nontrivial velocity dependence. All these strongly support the duality between those two theories with completely different nature. 
  We consider an extension of the gauge-fixing procedure in the framework of the Lagrangian superfield BRST and BRST-antiBRST quantization schemes for arbitrary gauge theories, taking into account the possible ambiguity in the choice of the superfield antibracket. We show that this ambiguity is fixed by the algebraic properties of the antibracket and the form of the BRST and antiBRST transformations, realized in terms of superspace translations. The Ward identities related to the generalized gauge-fixing procedure are obtained. 
  Recently, the holographic aspects of asymptotically de Sitter spacetimes have generated substantial literary interest. The plot continues in this paper, as we investigate a certain class of dilatonically deformed ``topological'' de Sitter solutions (which were introduced in hep-th/0110234). Although such solutions possess a detrimental cosmological singularity, their interpretation from a holographic perspective remains somewhat unclear. The current focus is on the associated generalized $C$-functions, which are shown to maintain their usual monotonicity properties in spite of this exotic framework. These findings suggest that such topological solutions may still play a role in our understanding of quantum gravity with a positive cosmological constant. 
  We probe the U(N) chiral Gross-Neveu model with a source-term $J\l{\Psi}\Psi$. We find an expression for the renormalization scheme and scale invariant source $\hat{J}$, as a function of the generated mass gap. The expansion of this function is organized in such a way that all scheme and scale dependence is reduced to one single parameter $d$. We obtain a non-perturbative mass gap as the solution of $\hat{J}=0$. A physical choice for $d$ gives good results for $N>2$. The self-consistent minimal sensitivity condition gives a slight improvement. 
  In this note we show how the anomalies of both pure and matter coupled N=1,2 supersymmetric gauge theories describing the low energy dynamics of fractional branes on orbifolds can be derived from supergravity. 
  The Hamilton-Jacobi formalism was applied to quantize the front-form Schwinger model. The importance of the surface term is discussed in detail. The BRST-anti-BRST symmetry was analyzed within Hamilton-Jacobi formalism. 
  In the context of softly-broken N=4 to N=2 supersymmetric SU(N) gauge theory, we calculate using semi-classical instanton methods, the lowest order non-trivial terms in the mass expansion of the prepotential for all instanton number. We find exact agreement with Seiberg-Witten theory and thereby achieve the most powerful test yet of this theory. We also calculate the one- and two-instanton contributions completely and also find consistency with Seiberg-Witten theory. Our approach relies on the fact that the instanton calculus admits a nilpotent fermionic symmetry, or BRST operator, whose existence implies that the integrals over the instanton moduli space, which give the coefficients of the prepotential, localize on the space of resolved point-like instantons or what we call ``topicons''. 
  Braneworld cosmology for a domain wall embedded in the charged (Anti)-de Sitter-Schwarzschildblack hole of the five--dimensional Einstein-Gauss-Bonnet-Maxwell theory is considered. The effective Friedmann equation for the brane is derived by introducing the necessary surface counterterms required for a well-defined variational principlein the Gauss--Bonnet theory and for the finiteness of the bulk space. The asymptotic dynamics of the brane cosmology is determined and it is found that solutions with vanishingly small spatial volume are unphysical. The finiteness of the bulk action is related to the vanishing of the effective cosmological constant on the brane. An analogy between the Friedmann equation and a generalized Cardy--Verlinde formula is drawn. 
  Solutions of the Polchinski exact renormalization group equation in the scalar O(N) theory are studied. Families of regular solutions are found and their relation with fixed points of the theory is established. Special attention is devoted to the limit $N=\infty$, where many properties can be analyzed analytically. 
  There are several equivalent descriptions for constant B-field background of open string. The background can be interpreted as constant B-field as well as constant gauge field strength or infinitely many D-branes with non-commuting Chan-Paton matrices. In this article, the equivalence of these open string theories is studied in Witten's cubic open string field theory. Through the map between these equivalent descriptions, both algebra of non-commutative coordinates as well as Chan-Paton matrix algebra are identified with subalgebras of open string field algebra. 
  We give a brief account of supersymmetric Born-Infeld theories with extended supersymmetry, including those with partially broken supersymmetry. Some latest developments in this area are presented. One of them is N=3 supersymmetric Born-Infeld theory which admits a natural off-shell formulation in N=3 harmonic superspace. 
  We study the behaviour of the two- and three-point thermal Green functions, to one loop order in noncommutative U(N) Yang-Mills theory, at temperatures $T$ much higher than the external momenta $p$. We evaluate the amplitudes for small and large values of the variable $\theta p T$ ($\theta$ is the noncommutative parameter) and exactly compute the static gluon self-energy for all values of $\theta p T$. We show that these gluon functions, which have a leading $T^2$ behaviour, are gauge independent and obey simple Ward identities. We argue that these properties, together with the results for the lowest order amplitudes, may be sufficient to fix uniquely the hard thermal loop effective action of the noncommutative theory. 
  We elaborate on a new representation of Lagrangians of 4D nonlinear electrodynamics including the Born-Infeld theory as a particular case. In this new formulation, in parallel with the standard Maxwell field strength $F_{\alpha\beta}, \bar{F}_{\dot\alpha\dot\beta}$, an auxiliary bispinor field $V_{\alpha\beta}, \bar{V}_{\dot\alpha\dot\beta}$ is introduced. The gauge field strength appears only in bilinear terms of the full Lagrangian, while the interaction Lagrangian $E$ depends on the auxiliary fields, $E = E(V^2, \bar V^2)$. The generic nonlinear Lagrangian depending on $F,\bar{F}$ emerges as a result of eliminating the auxiliary fields. Two types of self-duality inherent in the nonlinear electrodynamics models admit a simple characterization in terms of the function $E$. The continuous SO(2) duality symmetry between nonlinear equations of motion and Bianchi identities amounts to requiring $E$ to be a function of the SO(2) invariant quartic combination $V^2\bar V^2$, which explicitly solves the well-known self-duality condition for nonlinear Lagrangians. The discrete self-duality (or self-duality under Legendre transformation) amounts to a weaker condition $E(V^2, \bar{V}^2) = E(-V^2, -\bar{V}^2)$. We show how to generalize this approach to a system of $n$ Abelian gauge fields exhibiting U(n) duality. The corresponding interaction Lagrangian should be U(n) invariant function of $n$ bispinor auxiliary fields. 
  A new approach to the concept of particles and their production in quantum field theory is developed. A local operator describing the current of particle density is constructed for scalar and spinor fields in arbitrary gravitational and electromagnetic backgrounds. This enables one to describe particles in a local, general-covariant and gauge-invariant way. However, the current depends on the choice of a 2-point function. There is a choice that leads to the local non-conservation of the current in a gravitational or an electromagnetic background, which describes local particle production consistent with the usual global description based on the Bogoliubov transformation. The most natural choice based on the Green function calculated using the Schwinger-DeWitt method leads to the local conservation of the current, provided that interactions with quantum fields are absent. Interactions with quantum fields lead to the local non-conservation of the current which describes local particle production consistent with the usual global description based on the interaction picture. 
  Recently, the author has proposed a generalization of the matrix and vector models approach to the theory of random surfaces and polymers. The idea is to replace the simple matrix or vector (path) integrals by gauge theory or non-linear sigma model (path) integrals. We explain how this solves one of the most fundamental limitation of the classic approach: we automatically obtain non-perturbative definitions in non-Borel summable cases. This is exemplified on the simplest possible examples involving O(N) symmetric non-linear sigma models with N-dimensional target spaces, for which we construct (multi)critical metrics. The non-perturbative definitions of the double scaled, manifestly positive, partition functions rely on remarkable identities involving (path) integrals. 
  We study the strong coupling dynamics of the heterotic E_8 x E_8 string theory on the orbifolds T^6/Z_3 and C^3/Z_3 using the duality with the Horava-Witten M-theory picture.  This leads us to a conjecture about the low energy description of the five dimensional E_0-theory (the CFT that describes the the singularity region of M-theory on C^3/Z_3) compactified on S^1/Z_2. 
  The effective average action of Yang-Mills theory is analyzed in the framework of exact renormalization group flow equations. Employing the background-field method and using a cutoff that is adjusted to the spectral flow, the running of the gauge coupling is obtained on all scales. In four dimensions and for the gauge groups SU(2) and SU(3), the coupling approaches a fixed point in the infrared. 
  We consider orientifolds of Calabi-Yau 3-folds in the context of Type IIA and Type IIB superstrings. We show how mirror symmetry can be used to sum up worldsheet instanton contributions to the superpotential for Type IIA superstrings. The relevant worldsheets have the topology of the disc and ${\bf RP^2}$. 
  By considering the B-field dynamical and studying its interaction with Ramond-Ramond (RR) background we observe the breaking of the B-field gauge symmetry in the effective action. This effect takes place due to non-perturbative coupling of the B-field to membrane topological charge. As a result, the B-field is renormalized in the RR backgrounds, making it impossible to obtain consistent non-commutative models with constant B-field.  We argue that the gauge invariance is restored by introducing appropriate external D-brane configuration. 
  Scalar field theories with appropriate potentials in Minkowski space can have time-dependent classical solutions containing topological defects which correspond to S-branes - i.e. branes all of whose tangential dimensions are spacelike. It is argued that such S-branes arise in string theory as time-dependent solutions of the worldvolume tachyon field of an unstable D-brane or D-brane-anti-D-brane pair. Using the known coupling of the spacetime RR fields to the worldvolume tachyon it is shown that these S-branes carry a charge, defined as the integral of a RR field strength over a sphere (containing a time as well as spatial dimensions) surrounding the S-brane. This same charge is carried by SD-branes, i.e. Dirichlet branes arising from open string worldsheet conformal field theories with a Dirichlet boundary condition on the timelike dimension. The corresponding SD-brane boundary state is constructed. Supergravity solutions carrying the same charges are also found for a few cases. 
  We present a new approach for generating solutions in heterotic string theory compactified down to three dimensions on a torus with d+n>2, where d and n stand for the number of compactified space--time dimensions and Abelian gauge fields, respectively. It is shown that in the case when d=2k+1 and n is arbitrary, one can apply a solution--generating procedure starting from solutions of the stationary Einstein theory with k Maxwell fields; our approach leads to classes of solutions which are invariant with respect to the total group of three-dimensional charging symmetries. We consider a particular extension of the stationary Einstein--multi-Maxwell theory obtained on the basis of the Kerr--multi-Newman--NUT special class of solutions and establish the conditions under which the resulting multi--dimensional metric is free of Dirac string peculiarities. 
  We present a new approach for generation of solutions in the four-dimensional heterotic string theory with one vector field and in the five-dimensional bosonic string theory starting from the static Einstein-Maxwell fields. Our approach allows one to construct the solution classes invariant with respect to the total subgroup of the three-dimensional charging symmetries of these string theories. The new generation procedure leads to the extremal Israel-Wilson-Perjes subclass of string theory solutions in a special case and provides its natural continuous extension to the realm of non-extremal solutions. We explicitly calculate all string theory solutions related to three-dimensional gravity coupled to an effective dilaton field which arises after an appropriate charging symmetry invariant reduction of the static Einstein-Maxwell system. 
  We investigate vortex solutions to the Abelian Higgs field equations in a four dimensional de Sitter spacetime background. We obtain both static and dynamic solutions with axial symmetry that are generalizations of the Nielsen-Olesen gauge vortices in flat spacetime. The static solution is located in the static patch of de Sitter space. We numerically solve the field equations in an inflationary (big bang) patch and find a time dependent vortex soution, whose effect to create a deficit angle in the spacetime. We show that this solution can be interpreted in terms of a renormalization group flow in accord with a generalized {$c$}-theorem, providing evidence in favour of a dS/CFT correspondence. 
  High-energy scattering in non-conformal gauge theories is investigated using the AdS/CFT dual string/gravity theory. It is argued that strong-gravity processes, such as black hole formation, play an important role in the dual dynamics. Further information about this dynamics is found by performing a linearized analysis of gravity for a mass near an infrared brane; this gives the far field approximation to black hole or other strong-gravity effects, and in particular allows us to estimate their shape. From this shape, one can infer a total scattering cross-section that grows with center of mass energy as ln^2 E, saturating the Froissart bound. 
  Gauge theories in axial gauges are studied using Exact Renormalisation Group flows. We introduce a background field in the infrared regulator, but not in the gauge fixing, in contrast to the usual background field gauge. It is shown how heat-kernel methods can be used to obtain approximate solutions to the flow and the corresponding Ward identities. Expansion schemes are discussed, which are not applicable in covariant gauges. As an application, we derive the one-loop effective action for covariantly constant field strength, and the one-loop beta-function for arbitrary regulator. 
  Within the exact renormalisation group, the scaling solutions for O(N) symmetric scalar field theories are studied to leading order in the derivative expansion. The Gaussian fixed point is examined for d>2 dimensions and arbitrary infrared regularisation. The Wilson-Fisher fixed point in d=3 is studied using an optimised flow. We compute critical exponents and subleading corrections-to-scaling to high accuracy from the eigenvalues of the stability matrix at criticality for all N. We establish that the optimisation is responsible for the rapid convergence of the flow and polynomial truncations thereof. The scheme dependence of the leading critical exponent is analysed. For all N > 0, it is found that the leading exponent is bounded. The upper boundary is achieved for a Callan-Symanzik flow and corresponds, for all N, to the large-N limit. The lower boundary is achieved by the optimised flow and is closest to the physical value. We show the reliability of polynomial approximations, even to low orders, if they are accompanied by an appropriate choice for the regulator. Possible applications to other theories are outlined. 
  We propose a generalization of Heisenbergs' matrix mechanics based on many-index objects. It is shown that there exists a solution describing a harmonic oscillator and many-index objects lead to a generalization of spin algebra. 
  In a preceding report (hep-th/0112261), Pierre Ramond outlined a programme we have followed recently. An important ingredient in a future unique theory for basic physics must be a unique mathematical structure. It is then interesting to investigate the underlying mathematical structure of 11-dimensional space-time. A search for exceptional group structures specific to eleven dimensions leads us then to a study of the coset F_4/SO(9),which may provide a generalization of eleven-dimensional supergravity. This scheme is more general though and in this report we will show the corresponding structure related to the coset SU(3)/SU(2) x U(1), where we interpret the U(1) as the helicity group in four dimensions. 
  We construct boundary state and crosscap state in D=4,N=1 type-IIB Z_N orientifold and investigate properties of amplitude. We find that the boundary state of a cylinder is different from the boundary state of a M\"{o}bius strip. Using these states, we find that amplitudes do not factorize in Z_N(N=even) orientifold. Tadpole divergence remain in Z_4, Z_8, Z'_8 and Z'_{12} model due to volume dependence of boundary and crosscap state. On the other hand the amplitude of Z_3 and Z_7 orientifolds factorize so that we obtain the gauge groups of the model by employing the massless tadpole cancellation condition. 
  We discuss the Hawking radiation in the Higgs-Yukawa system and we show dynamical formation of a spherical domain wall around the black hole. The formation of the spherical wall is a general property of the black hole whose Hawking temperature is equal to or greater than the energy scale of the system. The formation of the electroweak wall and that of the GUT wall are shown as realistic cases. We also discuss a phenomenon of the spontaneous charging-up of the black hole. The Hawking radiation can charge-up the black hole itself by the charge-transportation due to the spherical wall when C- and CP-violation in the wall are assumed. The black hole with the electroweak wall can obtain a large amount of the hyper charge. 
  We combine linear response theory and dimensional regularization in order to derive the dynamical Casimir force in the low frequency regime. We consider two parallel plates moving along the normal direction in $D-$dimensional space. We assume the free-space values for the mass of each plate to be known, and obtain finite, separation-dependent mass corrections resulting from the combined effect of the two plates. The global mass correction is proportional to the static Casimir energy, in agreement with Einstein's law of equivalence between mass and energy for stressed rigid bodies. 
  The zero modes of the Dirac operator in the background of center vortex gauge field configurations in $\R^2$ and $\R^4$ are examined. If the net flux in D=2 is larger than 1 we obtain normalizable zero modes which are mainly localized at the vortices. In D=4 quasi-normalizable zero modes exist for intersecting flat vortex sheets with the Pontryagin index equal to 2. These zero modes are mainly localized at the vortex intersection points, which carry a topological charge of $\pm 1/2$. To circumvent the problem of normalizability the space-time manifold is chosen to be the (compact) torus $\T^2$ and $\T^4$, respectively. According to the index theorem there are normalizable zero modes on $\T^2$ if the net flux is non-zero. These zero modes are localized at the vortices. On $\T^4$ zero modes exist for a non-vanishing Pontryagin index. As in $\R^4$ these zero modes are localized at the vortex intersection points. 
  We construct a simple non singular cosmological model in which the currently observed expansion phase was preceded by a contraction. This is achieved, in the framework of pure general relativity, by means of a radiation fluid and a free scalar field having negative energy. We calculate the power spectrum of the scalar perturbations that are produced in such a bouncing model and find that, under the assumption of initial vacuum state for the quantum field associated with the hydrodynamical perturbation, this leads to a spectral index n=-1. The matching conditions applying to this bouncing model are derived and shown to be different from those in the case of a sharp transition. We find that if our bounce transition can be smoothly connected to a slowly contracting phase, then the resulting power spectrum will be scale invariant. 
  In both old and recent literature, it has been argued that the celebrated van Dam-Veltman-Zakharov (vDVZ) discontinuity of massive gravity is an artifact due to linearization of the true equations of motion. In this letter, we investigate that claim. First, we exhibit an explicit -albeit somewhat arbitrary- fully covariant set of equations of motion that, upon linearization, reduce to the standard Pauli-Fierz equations. We show that the vDVZ discontinuity still persists in that non-linear, covariant theory. Then, we restrict our attention to a particular system that consistently incorporates massive gravity: the Dvali-Gabadadze-Porrati (DGP) model. DGP is fully covariant and does not share the arbitrariness and imperfections of our previous covariantization, and its linearization exhibits a vDVZ discontinuity. Nevertheless, we explicitly show that the discontinuity does disappear in the fully covariant theory, and we explain the reason for this phenomenon. 
  Standard superspace Feynman diagram rules give one estimate of the onset of ultraviolet divergences in supergravity and super Yang-Mills theories. Newer techniques motivated by string theory but which also make essential use of unitarity cutting rules give another in certain cases. We trace the difference to the treatment of higher-dimensional gauge invariance in supersymmetric theories that can be dimensionally oxidized to pure supersymmetric gauge theories. 
  In this note we discuss the possibility to get a time rather than space in the scenario of (de)construction of new dimension. 
  We elaborate the idea that the matrix models equipped with the gauge symmetry provide a natural framework to describe identical particles. After demonstrating the general prescription, we study an exactly solvable harmonic oscillator type gauged matrix model. The model gives a generalization of the Calogero-Sutherland system where the strength of the inverse square potential is not fixed but dynamical bounded by below. 
  Using the supergravity solution of $N_1$ D3-branes probing $A_{N_2-1}$ singularities we study the pp-wave limit of $AdS_5\times S^5/Z_{N_2}$. We show that there are two different pp-wave limits. One is the orbifold of the pp-wave limit of $AdS_5\times S^5$. In this case there is no symmetry enhancement. The other case is the same as the pp-wave limit of $AdS_5\times S^5$ and therefore we again see the maximal supersymmetry. We will also identify operators in the four dimensional ${\cal N}=2$ $SU(N_1)^{N_2}$ gauge theory which correspond to stringy excitations in the orbifolded pp-wave geometry. The existence of the maximal pp-wave geometry indicates that there is a subsector of the corresponding ${\cal N}=2$ gauge theories which has enhanced ${\cal N}=4$ supersymmetry. We also study the pp-wave limits of $AdS_{7,4}\times S^{4,7}/Z_{N}$. 
  A proposal is made for a cosmological D3/D7 model with a constant magnetic flux along the D7 world-volume. It describes an N=2 gauge model with Fayet-Iliopoulos terms and the potential of the hybrid P-term inflation. The motion of the D3-brane towards D7 in a phase with spontaneously broken supersymmetry provides a period of slow-roll inflation in the de Sitter valley, the role of the inflaton being played by the distance between D3 and D7-branes. After tachyon condensation a supersymmetric ground state is formed: a D3/D7 bound state corresponding to an Abelian non-linear (non-commutative) instanton. In this model the existence of a non-vanishing cosmological constant is associated with the resolution of the instanton singularity. We discuss a possible embedding of this model into a compactified M-theory setup. 
  We study four-dimensional superconformal field theories coupled to three-dimensional superconformal boundary or defect degrees of freedom. Starting with bulk N=2, d=4 theories, we construct abelian models preserving N=2, d=3 supersymmetry and the conformal symmetries under which the boundary/defect is invariant. We write the action, including the bulk terms, in N=2, d=3 superspace. Moreover we derive Callan-Symanzik equations for these models using their superconformal transformation properties and show that the beta functions vanish to all orders in perturbation theory, such that the models remain superconformal upon quantization. Furthermore we study a model with N=4 SU(N) Yang-Mills theory in the bulk coupled to a N=4, d=3 hypermultiplet on a defect. This model was constructed by DeWolfe, Freedman and Ooguri, and conjectured to be conformal based on its relation to an AdS configuration studied by Karch and Randall. We write this model in N=2, d=3 superspace, which has the distinct advantage that non-renormalization theorems become transparent. Using N=4, d=3 supersymmetry, we argue that the model is conformal. 
  In two dimensional conformal field theory the generating functional for correlators of the stress-energy tensor is given by the non-local Polyakov action associated with the background geometry. We study this functional holographically by calculating the regularized on-shell action of asymptotically AdS gravity in three dimensions, associated with a specified (but arbitrary) boundary metric. This procedure is simplified by making use of the Chern-Simons formulation, and a corresponding first-order expansion of the bulk dreibein, rather than the metric expansion of Fefferman and Graham. The dependence of the resulting functional on local moduli of the boundary metric agrees precisely with the Polyakov action, in accord with the AdS/CFT correspondence. We also verify the consistency of this result with regard to the nontrivial transformation properties of bulk solutions under Brown-Henneaux diffeomorphisms. 
  Discrete Mechanics based on finite element methods is presented in this paper. We also explore the relationship between this discrete mechanics and Veselov discrete mechanics. High order discretizations are constructed in terms of high order interpolations. 
  In the Orientiworld framework the Standard Model fields are localized on D3-branes sitting on top of an orientifold 3-plane. The transverse 6-dimensional space is a non-compact orbifold (or a more general conifold). The 4-dimensional gravity on D3-branes is reproduced due to the 4-dimensional Einstein-Hilbert term induced at the quantum level. The orientifold 3-plane plays a crucial role, in particular, without it the D3-brane world-volume theories would be conformal due to the tadpole cancellation. We study non-perturbative gauge dynamics in various N=1 supersymmetric orientiworld models based on the Z_3 as well as Z_5 and Z_7 orbifold groups. Our discussions illustrate that there is a rich variety of supersymmetry preserving dynamics in some of these models. On the other hand, we also find some models with dynamical supersymmetry breaking. 
  We study in some detail the properties of a previously proposed new class of string and brane models whose world-sheet (world-volume) actions are built with a modified reparametrization-invariant measure of integration and which do not contain any ad hoc dimensionfull parameters. The ratio of the new and the standard Riemannian integration measure densities plays the role of a dynamically generated string/brane tension. The latter is identified as (the magnitude of) an effective (non-Abelian) electric field-strength on the world-sheet/world-volume obeying the standard Gauss-law constraint. As a result a simple classical mechanism for confinement via strings is proposed. 
  We construct the Seiberg-Witten curve for the E-string theory in six-dimensions. The curve is expressed in terms of affine E_8 characters up to level 6 and is determined by using the mirror-type transformation so that it reproduces the number of holomorphic curves in the Calabi-Yau manifold and the amplitudes of N=4 U(n) Yang-Mills theory on 1/2 K3. We also show that our curve flows to known five- and four-dimensional Seiberg-Witten curves in suitable limits. 
  Some recent finite temperature calculations arising in the investigation of the Verlinde-Cardy relation are re-analysed. Some remarks are also made about temperature inversion symmetry. 
  First, we review the basic mathematical structures and results concerning the gauge orbit space stratification. This includes general properties of the gauge group action, fibre bundle structures induced by this action, basic properties of the stratification and the natural Riemannian structures of the strata. In the second part, we study the stratification for theories with gauge group $\rmSU(n)$ in space time dimension 4. We develop a general method for determining the orbit types and their partial ordering, based on the 1-1 correspondence between orbit types and holonomy-induced Howe subbundles of the underlying principal $\rmSU(n)$-bundle. We show that the orbit types are classified by certain cohomology elements of space time satisfying two relations and that the partial ordering is characterized by a system of algebraic equations. Moreover, operations for generating direct successors and direct predecessors are formulated, which allow one to construct the set of orbit types, starting from the principal one. Finally, we discuss an application to nodal configurations in topological Chern-Simons theory. 
  We construct the boundary states preserving half the global supersymmetries in string theory propagating on a Hpp background. 
  The Casimir stress on cylinderical shell in de Sitter background for massless scalar field satisfying Dirichlet boundary conditions on the cylinder is calculated. The metric is written in conformally flat form to make maximum use of the Minkowski space calculations. Different cosmological constants are assumed for the space inside and outside the cylinder to have general results applicable to the case of cylindrical domain wall formations in the early universe. 
  Crosscap states for orientifolds of Euclidean AdS_3 are constructed. We show that our crosscap states describe the same orientifolds which were obtained by the classical analysis. The spectral density of open strings in the system with orientifold can be read from the M"obius strip amplitudes and it is compared to that of the open strings stretched between branes and their mirrors. We also compute the Klein bottle amplitudes. 
  We consider new cosmological solutions with a collapsing, an intermediate and an expanding phase. The boundary between the expanding (collapsing) phase and the intermediate phase is seen by comoving observers as a cosmological past (future) horizon. The solutions are naturally embedded in string and M-theory. In the particular case of a two-dimensional cosmology, space-time is flat with an identification under boost and translation transformations. We consider the corresponding string theory orbifold and calculate the modular invariant one-loop partition function. In this case there is a strong parallel with the BTZ black hole. The higher dimensional cosmologies have a time-like curvature singularity in the intermediate region. In some cases the string coupling can be made small throughout all of space-time but string corrections become important at the singularity. This happens where string winding modes become light which could resolve the singularity. The new proposed space-time casual structure could have implications for cosmology, independently of string theory. 
  We perform some analytic continuations of the de Sitter thick domain wall solutions obtained in our previous paper hep-th/0201130 in the system of gravity and a scalar field with an axion-like potential. The obtained new solutions represent anti-de Sitter thick domain walls and cosmology. The anti-de Sitter domain wall solutions are periodic, and correspondingly the cosmological solutions represent cyclic universes. We parameterize the axion-like scalar field potential and determine the parameter regions of each type of solutions. 
  In a series of three projects a new technique which allows for higher-loop renormalisation on a manifold with boundary has been developed and used in order to assess the effects of the boundary on the dynamical behaviour of the theory. Commencing with a conceptual approach to the theoretical underpinnings of the, underlying, spherical formulation of Euclidean Quantum Field Theory this overview presents an outline of the stated technique's conceptual development, mathematical formalism and physical significance. 
  In this paper we extend our previous result on the description of the partcle motion in a generalized Heisenberg picture to a relativistic fermion. The operators of the Lorentz algebra in this picture may be regarded as field operators. In this approach the transition amplitudes for the particle are constructed in terms of two-component functions in the unitary representations of the Lorentz group. 
  An iterative construction of higher order Einstein tensors for a maximally Gauss-Bonnet extended gravitational Lagrangian was introduced in a previous paper. Here the formalism is extended to non-factorisable metrics in arbitrary ($d+1$) dimensions in the presence of superposed Gauss-Bonnet terms. Such a generalisation turns out to be remarkably convenient and elegant. Having thus obtained the variational equations we first construct bulk solutions, with nonzero and zero cosmological constant. It is also pointed out that in the absence of Gauss-Bonnet terms a Schwarzschild type solution can be obtained in the non-factorisable case. Two positive tension branes are then inserted and their tensions are obtained in terms of parameters in the warp factor. Relations to recent studies of several authors are pointed out. 
  We consider cosmological two-brane models with AdS bulk, for which the radion, i.e. the separation between the two branes, is time dependent. In the case of two de Sitter branes (including Minkowski branes as a limiting case), we compute explicitly, without any approximation, the effective four-dimensional action for the radion. With the scale factor on-shell, this provides the non-perturbative dynamics for the radion. We discuss the differences between the dynamics derived from the four-dimensional action with the scale factor off-shell and the true five-dimensional dynamics. 
  Stability and instability bands in classical mechanics are well-studied in connection with systems such as described by the Mathieu equation. We examine whether such band structure can arise in classical field theory in the context of an embedded kink in 1+1 dimensions. The static embedded kink is unstable to perturbations but we show that if the kink is dynamic it can exhibit stability in certain parameter bands. Our results are relevant for estimating the lifetimes of various embedded defects and, in particular, loops of electroweak Z-string. 
  We deform two-dimensional topological gravity by making use of its gauge theory formulation. The obtained noncommutative gravity model is shown to be invariant under a class of transformations that reduce to standard diffeomorphisms once the noncommutativity parameter is set to zero. Some solutions of the deformed model, like fuzzy AdS_2, are obtained. Furthermore, the transformation properties of the model under the Seiberg-Witten map are studied. 
  We study the consistency of orbifold field theories and clarify to what extent the condition of having an anomaly-free spectrum of zero-modes is sufficient to guarantee it. Preservation of gauge invariance at the quantum level is possible, although at the price, in general, of introducing operators that break the 5d local parity. These operators are, however, perfectly consistent with the orbifold projection. We also clarify the relation between localized Fayet-Iliopoulos (FI) terms and anomalies. These terms can be consistently added, breaking neither local supersymmetry nor the gauge symmetry. In the framework of supergravity the localized FI term arises as the boundary completion of a bulk interaction term: given the bulk Lagrangian the FI is fixed by gauge invariance. 
  Doubly Special Relativity (DSR) theory is a theory with two observer-independent scales, of velocity and mass (or length). Such a theory has been proposed by Amelino--Camelia as a kinematic structure which may underline quantum theory of relativity. Recently another theory of this kind has been proposed by Magueijo and Smolin. In this paper we show that both these theories can be understood as particular bases of the $\kappa$--Poincar\'e theory based on quantum (Hopf) algebra. This observation makes it possible to construct the space-time sector of Magueijo and Smolin DSR. We also show how this construction can be extended to the whole class of DSRs. It turns out that for all such theories the structure of space-time commutators is the same. This results lead us to the claim that physical predictions of properly defined DSR theory should be independent of the choice of basis. 
  We discuss time-dependent backgrounds of type IIB supergravity realizing gravitation duals of gauge theories formulated in de Sitter space-time as a tool of embedding de Sitter in a supergravity. We show that only the gravitational duals to non-conformal gauge theories are sensitive to a specific value of a Hubble parameter. We consider two nontrivial solutions of this type: a gravity dual to six-dimensional (1,1) little string theory, and to a four-dimensional cascading SU(N+M)xSU(N) supersymmetric gauge theory (related to fractional D3-branes on a singular conifold according to Klebanov et al), in accelerating universe. In both cases we argue that the IR singularity of the geometry is regulated by the expansion of the gauge theory background space-time. 
  A bound state problem in a topologically massive quantum electrodynamics is investigated by using a non-perturbative method. We formulate the Bethe- Salpeter equation for scalar bound states composed of massive fermion and anti-fermion pair under the lowest ladder approximation. In a large mass expansion for the (anti-) fermion, we derive the Schr{\"o}dinger equation and solve it by a numerical method. The energy eigenvalues of bound states are evaluated for various values of a topological mass and also a fermion mass. Then we find a novel logarithmic scaling behaviour of the binding energy in varying the topological mass, fermion mass and also a quantum number. There exists a critical value of the topological mass, beyond which the bound states disappear. As the topological mass decreases, the energy eigenvalues of the bound states, which are negative, also decrease with a logarithmic dependence on the topological mass. A Chern-Simons term gives the bound system a repulsive effect. 
  We couple three-dimensional Chern-Simons gauge theory with BF theory and study deformations of the theory by means of the antifield BRST formalism. We analyze all possible consistent interaction terms for the action under physical requirements and find a new topological field theory in three dimensions with new nontrivial terms and a nontrivial gauge symmetry. We analyze the gauge symmetry of the theory and point out the theory has the gauge symmetry based on the Courant algebroid. 
  The deformation star product of smooth functions on the momentum phase space of covariant (polysymplectic) Hamiltonian field theory is introduced. 
  Two forms are available for the fermion propagator at finite temperature and density. It is shown that, when one deals with a diquark-condensation-operator inserted Green function in hot and dense QCD, the standard form of the quark propagator does not work. On the other hand, another form of the quark propagator does work. 
  In heterotic string theory compactified to four dimensions with N=2 supersymmetry, string-loop corrections to the universal sector of the low-energy effective action are studied. Within the framework of N=2 supersymmetric formulation of the theory, in the first order in string coupling constant, we solve the system of the loop-corrected Maxwell and Killing spinor equations. Taking as the in-put the tree-level dyonic black hole solution, we calculate string-loop corrections to the string tree-level metric and moduli of dyonic black hole. 
  We construct some classes of instanton solutions of eight dimensional noncommutative ADHM equations generalizing the solutions of eight dimensional commutative ADHM equations found by Papadopoulos and Teschendorff, and interpret them as supersymmetric $D0$-$D8$ bound states in a NS $B$-field. Especially, we consider the $D0$-$D8$ system with anti-self-dual $B$-field preserving 3/16 of supercharges. This system and self-duality conditions are related with the group $Sp(2)$ which is a subgroup of the eight dimensional rotation group SO(8). 
  In this Report we review the microscopic formulation of the five dimensional black hole of type IIB string theory in terms of the D1-D5 brane system. The emphasis here is more on the brane dynamics than on supergravity solutions. We show how the low energy brane dynamics, combined with crucial inputs from AdS/CFT correspondence, leads to a derivation of black hole thermodynamics and the rate of Hawking radiation. Our approach requires a detailed exposition of the gauge theory and conformal field theory of the D1-D5 system. We also discuss some applications of the AdS/CFT correspondence in the context of black hole formation in three dimensions by thermal transition and by collision of point particles. 
  We consider rotating topological black branes with one rotational parameter in various dimensions. Also a general five-dimensional higher genus solution of the Einstein equation with a negative cosmological constant which represents a topological black brane with two rotational parameters is introduced. We find out that the counterterms inspired by conformal field theory introduced by Kraus, Larsen and Sieblink cannot remove the divergences in $r$ of the action in more than five dimensions. We modify the counterterms by adding a curvature invariant term to it. Using the modified counterterms we show that the $r$ divergences of the action, the mass, and the angular momentum densities of these spacetimes are removed. We also find out in the limit of $m=0$ the mass density of these spacetimes in odd dimensions is not zero. 
  In this work we review some features of topological defects in field theory models for real scalar fields. We investigate topological defects in models involving one and two or more real scalar fields. In models involving a single field we examine two different subclasses of models, which support one or more topological defects. In models involving two or more real scalar fields, we explore the presence of defects that live inside topological defects, and junctions and networks of defects. In the case of junctions of defects we investigte structures that simulate nanotubes and fulerenes. Our investigations may also be used to describe nonlinear properties of polymers, Langmuir films and optical solitons in fibers. 
  The effective potential of quantized scalar field on fuzzy sphere is evaluated to the two-loop level. We see that one-loop potential behaves like that in the commutative sphere and the Coleman-Weinberg mechanism of the radiatively symmetry breaking could be also shown in the fuzzy sphere system. In the two-loop level, we use the heavy-mass approximation and the high-temperature approximation to perform the evaluations. The results show that both of the planar and nonplanar Feynman diagrams have inclinations to restore the symmetry breaking in the tree level. However, the contributions from planar diagrams will dominate over those from nonplanar diagrams by a factor N^2. Thus, at heavy-mass limit or high-temperature system the quantum field on the fuzzy sphere will behave like those on the commutative sphere. We also see that there is a drastic reduction of the degrees of freedom in the nonplanar diagrams when the particle wavelength is smaller than the noncommutativity scale. 
  We study the dynamics of M-theory on G2 holonomy manifolds, and consider in detail the manifolds realized as the quotient of the spin bundle over S^3 by discrete groups. We analyse, in particular, the class of quotients where the triality symmetry is broken. We study the structure of the moduli space, construct its defining equations and show that three different types of classical geometries are interpolated smoothly. We derive the N=1 superpotentials of M-theory on the quotients and comment on the membrane instanton physics. Finally, we turn on Wilson lines that break gauge symmetry and discuss some of the implications. 
  We show that the high-temperature behaviour of the recently proposed confining strings reproduces exactly the correct large-N QCD result, for a large class of truncations of the long-range interaction between surface elements. 
  We analyze some consequences of two possible interpretations of the action of the ladder operators emerging from generalized Heisenberg algebras in the framework of the second quantized formalism. Within the first interpretation we construct a quantum field theory that creates at any space-time point particles described by a q-deformed Heisenberg algebra and we compute the propagator and a specific first order scattering process. Concerning the second one, we draw attention to the possibility of constructing this theory where each state of a generalized Heisenberg algebra is interpreted as a particle with different mass. 
  We study models compactified on S^1/Z_2 with bulk and brane matter fields charged under U(1) gauge symmetry. We calculate the FI-terms and show by minimizing the resulting potential that supersymmetry or gauge symmetry is spontaneously broken if the sum of the charges does not vanish. Even if this sum vanishes, there could be an instability as a consequence of localized FI-terms. This leads to a spontaneous localization of charged bulk fields on respective branes. 
  Recently the operator algebra and twisted vertex operator equations were given for each sector of all WZW orbifolds, and a set of twisted KZ equations for the WZW permutation orbifolds were worked out as a large example. In this companion paper we report two further large examples of this development. In the first example we solve the twisted vertex operator equations in an abelian limit to obtain the twisted vertex operators and correlators of a large class of abelian orbifolds. In the second example, the twisted vertex operator equations are applied to obtain a set of twisted KZ equations for the (outer-automorphic) charge conjugation orbifold on su(n \geq 3). 
  We examine orbifolds of the IIB string via gauged supergravity. For the gravity duals of the A_{n-1} quiver gauge theories, we extract the massless degrees of freedom and assemble them into multiplets of N=4 gauged supergravity in five dimensions. We examine the embedding of the gauge group into the isometry group of the scalar manifold, as well as the symmetries of the scalar potential. From this we find that there is a large SU(1,n) symmetry group which relates different RG flows in the dual quiver gauge theory. We find that this symmetry implies an extension of the usual duality between ten-dimensional IIB solutions which involves exchanging geometric moduli with background fluxes. 
  An ensemble of short open strings in equilibrium with the heat bath provided by the Euclidean worldvolume of a stack of Dbranes undergoes a thermal phase transition to a long string phase. The transition temperature is just below the string scale. We point out that this phenomenon provides a simple mechanism within open and closed string theories for altering the strong-electro-weak coupling unification scale relative to the fundamental closed string mass scale in spacetimes with external electromagnetic background. 
  I put forward the low-energy confining asymptote of the solution $<W_{C}>$ (valid for large macroscopic contours C of the size $>>1/\Lambda_{QCD}$) to the large N Loop equation in the D=4 U(N) Yang-Mills theory with the asymptotic freedom in the ultraviolet domain. Adapting the multiscale decomposition characteristic of the Wilsonean renormgroup, the proposed Ansatz for the loop-average is composed in order to sew, along the lines of the bootstrap approach, the large N weak-coupling series for high-momentum modes with the $N\to{\infty}$ limit of the recently suggested stringy representation of the 1/N strong-coupling expansion Dub4 applied to low-momentum excitations. The resulting low-energy stringy theory can be described through such superrenormalizable deformation of the noncritical Liouville string that, being devoid of ultraviolet divergences, does not possess propagating degrees of freedom at short-distance scales $<<1/{\sqrt{\sigma_{ph}}}$, where $\sigma_{ph}\sim{(\Lambda_{QCD})^{2}}$ is the physical string tension. 
  We consider spaces M_7 and M_8 of G_2 holonomy and Spin(7) holonomy in seven and eight dimensions, with a U(1) isometry. For metrics where the length of the associated circle is everywhere finite and non-zero, one can perform a Kaluza-Klein reduction of supersymmetric M-theory solutions (Minkowksi)_4\times M_7 or (Minkowksi)_3\times M_8, to give supersymmetric solutions (Minkowksi)_4\times Y_6 or (Minkowksi)_3\times Y_7 in type IIA string theory with a non-singular dilaton. We study the associated six-dimensional and seven-dimensional spaces Y_6 and Y_7 perturbatively in the regime where the string coupling is weak but still non-zero, for which the metrics remain Ricci-flat but that they no longer have special holonomy, at the linearised level. In fact they have ``almost special holonomy,'' which for the case of Y_6 means almost Kahler, together with a further condition. For Y_7 we are led to introduce the notion of an ``almost G_2 manifold,'' for which the associative 3-form is closed but not co-closed. We obtain explicit classes of non-singular metrics of almost special holonomy, associated with the near Gromov-Hausdorff limits of families of complete non-singular G_2 and Spin(7) metrics. 
  We consider the breaking of N=1 supersymmetry by non-zero G-flux when M-theory is compactified on a smooth manifold X of G_2 holonomy. Gukov has proposed a superpotential W to describe this breaking in the low-energy effective theory. We check this proposal by comparing the bosonic potential implied by W with the corresponding potential deduced from the eleven-dimensional supergravity action. One interesting aspect of this check is that, though W depends explicitly only on G-flux supported on X, W also describes the breaking of supersymmetry by G-flux transverse to X. 
  The path integral for ghost fermions, which is heuristically made use of in the Batalin- Fradkin-Vilkovisky approach to quantization of constrained systems, is derived from first principles. The derivation turns out to be rather different from that of physical fermions since the definition of Dirac states for ghost fermions is subtle. With these results at hand, it is then shown that the nonminimal extension of the Becchi-Rouet-Stora-Tyutin operator must be chosen differently from the notorious choice made in the literature in order to avoid the boundary terms that have always plagued earlier treatments. Furthermore it is pointed out that the elimination of states with nonzero ghost number requires the introduction of a thermodynamic potential for ghosts; the reason is that Schwarz's Lefschetz formula for the partition function of the time- evolution operator is not capable, despite claims to the contrary, to get rid of nonzero ghost number states on its own. Finally, we comment on the problems of global topological nature that one faces in the attempt to obtain the solutions of the Dirac condition for physical states in a configuration space of nontrivial geometry; such complications give rise to anomalies that do not obey the Wess-Zumino consistency conditions. 
  When a quantum field theory possesses topological excitations in a phase with spontaneously broken symmetry, these are created by operators which are non-local with respect to the order parameter. Due to non-locality, such disorder operators have non-trivial correlation functions even in free massive theories. In two dimensions, these correlators can be expressed exactly in terms of solutions of non-linear differential equations. The correlation functions of the one-parameter family of non-local operators in the free charged bosonic and fermionic models are the inverse of each other. We point out a simple derivation of this correspondence within the form factor approach 
  We construct non-extremal fractional D-brane solutions of type-II string theory at the Z_2 orbifold point of K3. These solutions generalize known extremal fractional-brane solutions and provide further insights into N=2 supersymmetric gauge theories and dual descriptions thereof. In particular, we find that for these solutions the horizon radius cannot exceed the non-extremal enhancon radius. As a consequence, we conclude that a system of non-extremal fractional branes cannot develop into a black brane. This conclusion is in agreement with known dual descriptions of the system. 
  We argue that recently proposed by Amelino-Camelia et all [1,2] so-called doubly special relativity (DSR), with deformed boost transformations identical with the formulae for $\kappa$-deformed kinematics in bicrossproduct basis is a classical special relativity in nonlinear disguise. The choice of symmetric composition law for deformed fourmomenta as advocated in [1, 2] implies that DSR is obtained by considering nonlinear fourmomenta basis of classical Poincar\'{e} algebra and it does not lead to noncommutative space-time. We also show how to construct large two classes of doubly special relativity theories - generalizing the choice in [1,2] and the one presented by Magueijo and Smolin [3]. The older version of deformed relativistic kinematics, differing essentially from classical theory in the coalgebra sector and leading to noncommutative $\kappa$-deformed Minkowski space is provided by quantum $\kappa$-deformation of Poincar\'e symmetries. 
  We apply zeta-function regularization to the kink and susy kink and compute its quantum mass. We fix ambiguities by the renormalization condition that the quantum mass vanishes as one lets the mass gap tend to infinity while keeping scattering data fixed. As an alternative we write the regulated sum over zero point energies in terms of the heat kernel and apply standard heat kernel subtractions. Finally we discuss to what extent these procedures are equivalent to the usual renormalization conditions that tadpoles vanish. 
  We do not think that the relativistic Morse potential problem has been correctly formulated and solved by Alhaidari (Phys. Rev. Lett. 87, 210405 (2001)). 
  We obtain by superfield methods the exceptional representations of the OSp(2N/4,R) and SU(2,2/1) superalgebras which extend to supersingletons of SU(2,2/2N) and F(4), respectively. These representations describe superconformally coupled multiplets and appear in three- and four-dimensional superconformal field theories which are holographic descriptions of certain anti-de Sitter supergravities. 
  The loop expansion of the effective action is used to evaluate quantum corrections to a scalar field theory (massive phi^4 model) on AdS. We evaluate one loop corrections and show that they preserve conformal invariance of the boundary theory as conjectured by AdS/CFT correspondence. 
  It is well known but rather mysterious that root spaces of the $E_k$ Lie groups appear in the second integral cohomology of regular, complex, compact, del Pezzo surfaces. The corresponding groups act on the scalar fields (0-forms) of toroidal compactifications of M theory. Their Borel subgroups are actually subgroups of supergroups of finite dimension over the Grassmann algebra of differential forms on spacetime that have been shown to preserve the self-duality equation obeyed by all bosonic form-fields of the theory. We show here that the corresponding duality superalgebras are nothing but Borcherds superalgebras truncated by the above choice of Grassmann coefficients. The full Borcherds' root lattices are the second integral cohomology of the del Pezzo surfaces. Our choice of simple roots uses the anti-canonical form and its known orthogonal complement. Another result is the determination of del Pezzo surfaces associated to other string and field theory models. Dimensional reduction on $T^k$ corresponds to blow-up of $k$ points in general position with respect to each other. All theories of the Magic triangle that reduce to the $E_n$ sigma model in three dimensions correspond to singular del Pezzo surfaces with $A_{8-n}$ (normal) singularity at a point. The case of type I and heterotic theories if one drops their gauge sector corresponds to non-normal (singular along a curve) del Pezzo's. We comment on previous encounters with Borcherds algebras at the end of the paper. 
  We consider tachyonic string-field fluctuations about a Dp-brane background in the geometrical (CFT) formulation of Vacuum String Field Theory. We then extend this analysis to the case of a background B-field. We find that the standard results for D-brane tension are reproduced in both cases. 
  The possibility that extremely long-lived, time-dependent, and localized field configurations (``oscillons'') arise during the collapse of asymmetrical bubbles in 2+1 dimensional phi^4 models is investigated. It is found that oscillons can develop from a large spectrum of elliptically deformed bubbles. Moreover, we provide numerical evidence that such oscillons are: a) circularly symmetric; and b) linearly stable against small arbitrary radial and angular perturbations. The latter is based on a dynamical approach designed to investigate the stability of nonintegrable time-dependent configurations that is capable of probing slowly-growing instabilities not seen through the usual ``spectral'' method. 
  We investigate the issues of unitarity and reality of the spectrum for the imaginary coupled affine Toda field theories based on a_1(1) and a_2(2) and the perturbed minimal models that arise from their various RSOS restrictions. We show that while all theories based on a_1(1) have real spectra in finite volume, the spectra of a_2(2) models is in general complex, with some exceptions. We also correct the S matrices conjectured earlier for the phi(1,5) perturbations of minimal models and give evidence for a conjecture that the RSOS spectra can be obtained as suitable projections of the folded ATFTs in finite volume. 
  We show that there exists a supersymmetric de Sitter background for the D=6, N=2, F(4) supergravity preserving the compact R-symmetry and gauging with respect to the conventional Anti de Sitter version of the theory. We construct the gauged matter coupled F(4) de Sitter supergravity explicitly and show that it contains ghosts in the vector sector. 
  We consider the XXZ model for a chain of particles whose spins are arbitrary with the anisotropy parameter equal to the root of minus one and generalized periodic boundary conditions. The conditions for the truncation of the functional fusion relations of the transfer matrices are obtained. The truncation results in a closed system of equations whose solution allows obtaining the energy spectrum. 
  We review some results on the connection among supergravity central charges, BPS states and Bekenstein-Hawking entropy. In particular, N=2 supergravity in four dimensions is studied in detail. For higher N supergravities we just give an account of the general theory specializing the discussion to the N=8 case when one half of supersymmetry is preserved. We stress the fact that for extremal supergravity black holes the entropy formula is topological, that is the entropy turns out to be a moduli independent quantity and can be written in terms of invariants of the duality group of the supergravity theory. 
  We analyse open string correlators in non-constant background fields, including the metric $g$, the antisymmetric $B$-field, and the gauge field $A$. Working with a derivative expansion for the background fields, but exact in their constant parts, we obtain a tachyonic on-shell condition for the inserted functions and extract the kinetic term for the tachyon action. The 3-point correlator yields a non-commutative tachyon potential. We also find a remarkable feature of the differential structure on the D-brane: Although the boundary metric $G$ plays an essential role in the action, the natural connection on the D-brane is the same as in closed string theory, i.e. it is compatible with the bulk metric and has torsion $H=dB$. This means, in particular, that the parallel transport on the brane is independent of the gauge field $A$. 
  We derive a noncommutative theory description for vortex configurations in a complex field in 2+1 dimensions. We interpret the Magnus force in terms of the noncommutativity, and obtain some results for the quantum dynamics of the system of vortices in that context. 
  We encode dynamical symmetries of Born-Infeld theory in a geometry on the tangent bundle of generally curved spacetime manifolds. The resulting covariant formulation of a maximal acceleration extension of special and general relativity is put to use in the discussion of particular point particle dynamics and the transition to a first quantized theory. 
  We extend the proposal of Berenstein, Maldacena and Nastase to Type IIB superstring propagating on pp-wave over ${\bf R}^4/{\bf Z}_k$ orbifold. We show that first-quantized free string theory is described correctly by the large $N$, fixed gauge coupling limit of ${\cal N}=2$ $[U(N)]^k$ quiver gauge theory. We propose a precise map between gauge theory operators and string states for both untwisted and twisted sectors. We also compute leading-order perturbative correction to the anomalous dimensions of these operators. The result is in agreement with the value deduced from string energy spectrum, thus substantiating our proposed operator-state map. 
  Riemann normal coordinates (RNC) are unsuitable for \kahler manifolds since they are not holomorphic. Instead, \kahler normal coordinates (KNC) can be defined as holomorphic coordinates. We prove that KNC transform as a holomorphic tangent vector under holomorphic coordinate transformations, and therefore that they are natural extensions of RNC to the case of \kahler manifolds. The KNC expansion provides a manifestly covariant background field method preserving the complex structure in supersymmetric nonlinear sigma models. 
  Motivated by the recent discussions of the Penrose limit of AdS_5\times S^5, we examine a more general class of supersymmetric pp-wave solutions of the type IIB theory, with a larger number of non-vanishing structures in the self-dual 5-form. One of the pp-wave solutions can be obtained as a Penrose limit of a D3/D3 intersection. In addition to 16 standard supersymmetries these backgrounds always allow for supernumerary supersymmetries. The latter are in one-to-one correspondence with the linearly-realised world-sheet supersymmetries of the corresponding exactly-solvable type IIB string action. The pp-waves provide new examples where supersymmetries will survive in a T-duality transformation on the x^+ coordinate. The T-dual solutions can be lifted to give supersymmetric deformed M2-branes in D=11. The deformed M2-brane is dual to a three-dimensional field theory whose renormalisation group flow runs from the conformal fixed point in the infra-red regime to a non-conformal theory as the energy increases. At a certain intermediate energy scale there is a phase transition associated with a naked singularity of the M2-brane. In the ultra-violet limit the theory is related by T-duality to an exactly-solvable massive IIB string theory. 
  We propose that the entropy of de Sitter space can be identified with the mutual entropy of a dual conformal field theory. We argue that unitary time evolution in de Sitter space restricts the total number of excited degrees of freedom to be bounded by the de Sitter entropy, and we give a CFT interpretation of this restriction. We also clarify issues arising from the fact that both de Sitter and anti de Sitter have dual descriptions in terms of conformal field theory. 
  We extend the idea of mirage cosmology to M-theory. Considering the motion of a probe brane in the M-theory background generated by a stack of non-threshold (M2,M5) bound states, we study the cosmological evolution of the brane universe in this background. We estimate the range of $r$ where the formalism is valid. Effective energy density on the probe brane is obtained in terms of the scale factor. Comparing the limiting case of the result with that from type IIB background, we confirm that the cosmological evolution by mirage matter is a possible scenario in the M-theory context. 
  Two different kinds of interactions between a ${Z}_{n}$-parafermionic and a Liouville field theory are considered. For generic values of $n$, the effective central charges describing the UV behavior of both models are calculated in the Neveu-Schwarz sector. For $n=2$ exact vacuum expectation values of primary fields of the Liouville field theory, as well as the first descendent fields are proposed. For $n=1$, known results for Sinh-Gordon and Bullough-Dodd models are recovered whereas for $n=2$, exact results for these two integrable coupled Ising-Liouville models are shown to exchange under a weak-strong coupling duality relation. In particular, exact relations between the parameters in the actions and the mass of the particles are obtained. At specific imaginary values of the coupling and $n=2$, we use previous results to obtain exact information about: (a) Integrable coupled models like Ising-${\cal M}_{p/p'}$, homogeneous sine-Gordon model $SU(3)_2$ or the Ising-XY model; (b) Neveu-Schwarz sector of the $\Phi_{13}$ integrable perturbation of N=1 supersymmetric minimal models. Several non-perturbative checks are done, which support the exact results. 
  Beginning with the most general fractal strings/sprays construction recently expounded in the book by Lapidus and Frankenhuysen, it is shown how the complexified extension of El Naschie's Cantorian-Fractal spacetime model belongs to a very special class of families of fractal strings/sprays whose scaling ratios are given by suitable pinary (pinary, $p$ prime) powers of the Golden Mean. We then proceed to show why the logarithmic periodicity laws in Nature are direct physical consequences of the complex dimensions associated with these fractal strings/sprays. We proceed with a discussion on quasi-crystals with p-adic internal symmetries, von Neumann's Continuous Geometry, the role of wild topology in fractal strings/sprays, the Banach-Tarski paradox, tesselations of the hyperbolic plane, quark confinement and the Mersenne-prime hierarchy of bit-string physics in determining the fundamental physical constants in Nature. 
  We use higher derivative classical gravity to study the nonlinear coupling between the cosmological expansion of the universe and metric oscillations of Planck frequency and very small amplitude. We derive field equations at high orders in the derivative expansion and find that the nature of the new dynamics is extremely restricted. For the equation of state parameter $w>0$ the relative importance of the oscillations grows logarithmically. Their effect on the cosmological expansion resembles that of dark energy. 
  Renormalization group limit cycles may be a commonplace for quantum Hamiltonians requiring renormalization, in contrast to experience to date with classical models of critical points, where fixed points are far more common. We discuss the simplest model Hamiltonian identified to date that exhibits a renormalization group limit cycle. The model is a discrete Hamiltonian with two coupling constants and a non-perturbative renormalization group that involves changes in only one of these couplings and is soluble analytically. The Hamiltonian is the discrete analog to a continuum Hamiltonian previously proposed by us. 
  In this short note we give an example of the exact solution of the open bosonic string field theory defined on the background of $N$ coincided D0-branes. This solution leads to the change of the original background to the background where D0-branes are localised in general positions. 
  It is shown that Einstein's equations on the brane can be received from the multi-dimensional vector field equations in pseudo-Euclidean space. The idea is based on the observation that the brane geometry can be equivalently described by the intrinsic metric or by the derivatives of its normal. From the other hand the normal to the brane can be constructed with the components of some multi-dimensional vector fields. For the both cases 4-dimensional effective action for gravity appears to be the same. 
  We consider the low-energy effective action of the 5D Einstein-Maxwell-Kalb-Ramond theory. After compactifying this truncated model on a two-torus and switching off the U(1) vector fields of this theory, we recall a formulation of the resulting three-dimensional action as a double Ernst system coupled to gravity. Further, by applying the so-called normalized Harrison transformation on a generic solution of this double Ernst system we recover the U(1) vector field sector of the theory. Afterward, we compute the field content of the generated charged configuration for the special case when the starting Ernst potentials correspond to a pair of interacting Kerr black holes, obtaining in this way an exact field configuration of the 5D Einstein-Maxwell-Kalb-Ramond theory endowed with effective Coulomb and dipole terms with momenta. Some physical properties of this object are analyzed as well as the effect of the normalized Harrison transformation on the double Kerr seed solution. 
  We study the flow from the theory of D2-branes in a G2 holonomy background to M2-branes in a Spin(7) holonomy background. We consider in detail the UV and IR regimes, and the effect of topology change of the background on the field theory. We conjecture a non-Abelian N=1 mirror symmetry. 
  We show that the string spectrum in the pp-wave limit of AdS_5\times S^5/Z_M (orbifolded pp-wave) is reproduced from the N=2 quiver gauge theory by quantizing the Green-Schwarz string theory on the orbifolded pp-wave in light cone gauge. We find that the twisted boundary condition on the world-sheet is naturally interpreted from the viewpoint of the quiver gauge theory. The correction of order g_{YM}^2 to the gauge theory operators agrees with the result in its dual string theory. We also discuss strings on some other orbifolded pp-waves. 
  We show that a formalism for analyzing the near-horizon conformal symmetry of Schwarzschild black holes using a scalar field probe is capable of describing black hole decay. The equation governing black hole decay can be identified as the geodesic equation in the space of black hole masses. This provides a novel geometric interpretation for the decay of black holes. Moreover, this approach predicts a precise correction term to the usual expression for the decay rate of black holes. 
  We develop the description of non-commutative geometry of the 4-dimensional quantum Hall fluid's theory proposed recently by Zhang and Hu. The non-commutative structure of fuzzy $S^{4}$ appears naturally in this theory. The fuzzy monopole harmonics, which are the essential elements in this non-commutative geometry, are explicitly constructed and their obeying the matrix algebra is obtained. This matrix algebra is associative. We also propose a fusion scheme of the fuzzy monopole harmonics of the coupling system from those of the subsystems, and determine the fusion rule in such fusion scheme. By products, we provide some essential ingredients of the theory of SO(5) angular momentum. In particular, the explicit expression of the coupling coefficients, in the theory of SO(5) angular momentum, are given. It is discussed that some possible applications of our results to the 4-dimensional quantum Hall system and the matrix brane construction in M-theory. 
  The physical world is marked by the phenomenon of spontaneous broken symmetry (SBS) i.e. where the state of a system is assymmetric with respect to the symmetry principles that govern its dynamics. For material systems this is not surprising since more often than not energetic considerations dictate that the ground state or low lying excited states of many body system become ordered i.e. a collective variable, such as magnetization or the Fourier transform of the density of a solid, picks up expectation values which otherwise would vanish by virtue of the dynamical symmetry(isotropy or translational symmetry in the aforementioned examples). More surprising was the discovery of the role of SBS in describing the vacuum or low lyng excitations of a quantum field theory. First came spontaneously broken chiral symmetry which was then applied to soft pion physics. When combined with current algebra, this field dominated particle physics in the 60's. Then came the application of the notion of SBS to situations where the symmetry is locally implemented by gauge fields. In that case the concept of order becomes more subtle. This development lead the way to electroweak unification and it remains one of the principal tools of the theorist in the quest for physics beyond the standard model. This brief review is intended to span the history of SBS with emphasis on conceptual rather than quantitative content. It is a written version of lectures of R.Brout on the ``Paleolithic Age'' and on ``Modern Times'' by F.Englert, i.e. respectively without and with gauge fields. 
  The theory of symmetry breaking in presence of gauge fields is presented, following the historical track. Particular emphasis is placed upon the underlying concepts. 
  We review the emergence of the ten-dimensional fermionic closed string theories from subspaces of the Hilbert space of the 26-dimensional bosonic closed string theory compactified on an $E_8\times SO(16)$ lattice. They arise from a consistent truncation procedure which generates space-time fermions out of bosons. This procedure is extended to open string sectors. We prove, from bosonic considerations alone, that truncation of the unique tadpole-free $SO(2^{13})$ bosonic string theory compactified on the above lattice determines the anomaly free Chan-Paton group of the Type I theory. It also yields the Chan-Paton groups making Type O theories tadpole-free. These results establish a link between all M-theory strings and the bosonic string within the framework of conformal field theory. Its significance is discussed. 
  We investigate the possible effects on the evolution of perturbations in the inflationary epoch due to short distance physics. We introduce a suitable non local action for the inflaton field, suggested by Noncommutative Geometry, and obtained by adopting a generalized star product on a Friedmann-Robertson-Walker background. In particular, we study how the presence of a length scale where spacetime becomes noncommutative affects the gaussianity and isotropy properties of fluctuations, and the corresponding effects on the Cosmic Microwave Background spectrum. 
  On a historical note, we first describe the early superspace construction of counterterms in supergravity and then move on to a brief discussion of selected areas in string theory where higher order supergravity invariants enter the effective theories. Motivated by this description we argue that it is important to understand $p$-brane actions with $\k$-invariant higher order terms, thus re-opening the question of $\k$-invariant ``rigidity'' terms. Finally we describe a recent construction of such an action using the superembedding formalism. 
  There is strong evidence that the area of any surface limits the information content of adjacent spacetime regions, at 10^(69) bits per square meter. We review the developments that have led to the recognition of this entropy bound, placing special emphasis on the quantum properties of black holes. The construction of light-sheets, which associate relevant spacetime regions to any given surface, is discussed in detail. We explain how the bound is tested and demonstrate its validity in a wide range of examples.   A universal relation between geometry and information is thus uncovered. It has yet to be explained. The holographic principle asserts that its origin must lie in the number of fundamental degrees of freedom involved in a unified description of spacetime and matter. It must be manifest in an underlying quantum theory of gravity. We survey some successes and challenges in implementing the holographic principle. 
  A three-dimensional simple N=1 supergravity theory with a supersymmetric sigma-model on the coset E_{8(+8)} / SO(16) is constructed. Both bosons and fermions in the matter multiplets are in the spinorial 128-representation of SO(16) with the same chirality. Due to their common chirality, this model can not be obtained from the maximal N=16 supergravity. By introducing an independent vector multiplet, we can also gauge an arbitrary subgroup of SO(16) together with a Chern-Simons term. Similar N=1 supersymmetric sigma-models coupled to supergravity are also constructed for the cosets F_{4(-20)} / SO(9) and SO(8,n) / SO(8) X SO(n). 
  We prove a relation between the asymptotic behavior of the conformal factor and the accessory parameters of the SU(1,1) Riemann- Hilbert problem. Such a relation shows the hamiltonian nature of the dynamics of N particles coupled to 2+1 dimensional gravity. A generalization of such a result is used to prove a connection between the regularized Liouville action and the accessory parameters in presence of general elliptic singularities. This relation had been conjectured by Polyakov in connection with 2-dimensional quantum gravity. An alternative proof, which works also in presence of parabolic singularities, is given by rewriting the regularized Liouville action in term of a background field. 
  In this letter we investigate the ultra-violet behaviour of four-point one-loop gluon amplitudes in dimensions greater than four coupled to various particles types. We discuss the structure of the counterterms and their inherent symmetries. 
  We consider the unoriented two-dimensional Abelian sandpile model on the half-plane with open and closed boundary conditions. We show that the operator effecting the change from closed to open, or from open to closed, is a boundary primary field of weight -1/8, belonging to a c=-2 logarithmic conformal field theory. 
  We consider the Abelian Higgs model as well as the SU(2) Georgi-Glashow model in which the gauge field action is replaced by a non linear Born-Infeld action. We study soliton solutions arising in these models, namely the vortex and monopole solutions, respectively. We construct formulas which provide good approximations for the mass of the Born-Infeld deformed solitons using only the data of the undeformed solutions. The results obtained indicate that in the self-dual limit, the Born-Infeld interaction leads to bound vortices, while for monopoles it gives rise to repulsion. 
  I discuss cosmological models either derived from, or inspired by, string theory or M-theory. In particular I discuss solutions in the low-energy effective theory and the role of the dilaton, moduli and antisymmetric form fields in the dimensionally reduced effective action. The pre big bang model is an attempt to use cosmological solutions to make observational predictions. I then discuss the effective theory of gravity found in recent brane-world models where we live on a 3-brane embedded in a five-dimensional spacetime and how the study of cosmological perturbations may enable us to test these ideas. 
  In this note, we consider the two derivative truncation of boundary string field theory for unstable D9 branes in Type IIA string theory. We construct multiples of the stable codimension 1 solitons that correspond to stacks of D8 branes. We find the fluctuation modes that correspond to open strings stretching between the branes, and find that their masses are consistent with the string tension. We show that these modes are localized halfway between the branes and that their width is independent of the brane separation. 
  The background independent formulation of the gauge theories on D-branes in flat space-time is considered, some examples of the solutions of their equations of motion are presented, the solutions of Dirac equation in these backgrounds are analyzed, and the generalizations to the curved spaces, like orbifolds, conifolds, and K3 surfaces, are discussed. 
  We propose an algebraic description of (untwisted) D-branes on compact group manifolds $G$ using quantum algebras related to $U_q(\mg)$. It reproduces the known characteristics of stable branes in the WZW models, in particular their configurations in $G$, energies as well as the set of harmonics. Both generic and degenerate branes are covered. 
  The notion of a "point" is essential to describe the topology of spacetime. Despite this, a point probably does not play a particularly distinguished role in any intrinsic formulation of string theory. We discuss one way to try to determine the notion of a point from a worldsheet point of view. The derived category description of D-branes is the key tool. The case of a flop is analyzed and Pi-stability in this context is tied in to some ideas of Bridgeland. Monodromy associated to the flop is also computed via Pi-stability and shown to be consistent with previous conjectures. 
  We analyze the spectrum of the bosonic and superstring on the orbifold of the space-time by a boost, leading to the cosmological singularity. We show that the modular invariance leads to the spectrum where the twisted sector tachyon, together with all other twisted sector fields, present in the Euclidean version of the orbifold, is absent. This makes impossible to resolve the singularity by a marginal deformation of the worldsheet CFT. We also establish a relation between the resolution of rotational orbifolds in Euclidean and Lorentzian setups, and quantum groups. The analysis confirms the impossibility of resolving the cosmological orbifold singularity. 
  We discuss the signature of the scale of short distance physics in the Cosmic Microwave Background. In addition to effects which depend on the ratio of Hubble scale H during inflation to the energy scale M of the short distance physics, there can be effects which depend on $\dot{\phi}^2/M^4$ where $\phi$ is the {\it classical background} of the inflaton field. Therefore, the imprints of short distance physics on the spectrum of Cosmic Microwave Background anisotropies generically involve a {\it double expansion}. We present some examples of a single scalar field with higher order kinetic terms coupled to Einstein gravity, and illustrate that the effects of short distance physics on the Cosmic Microwave Background can be substantial even for H << M, and generically involve corrections that are not simply powers of H/M. The size of such effects can depend on the short distance scale non-analytically even though the action is local. 
  Unification ideas suggest an integral treatment of fermion and boson spin and gauge-group degrees of freedom. Hence, a generalized quantum field equation, based on Dirac's, is proposed and investigated which contains gauge and flavor symmetries, determines vector gauge field and fermion solution representations, and fixes their mode of interaction. The simplest extension of the theory with a 6-dimensional Clifford algebra predicts an SU(2)_L X U(1) symmetry, which is associated with the isospin and the hypercharge, their vector carriers, two-flavor charged and chargeless leptons, and scalar particles. A mass term produces breaking of the symmetry to an electromagnetic U(1), and a Weinberg's angle theta_W with sin^2(theta_W)=.25 . A more realistic 8-d extension gives coupling constants of the respective groups g=1/sqrt 2~.707 and g'=1/sqrt 6~.408, with the same theta_W. 
  We propose a new form of dimensional reduction that constrains dilatation instead of a component of momentum. It corresponds to replacing toroidal compactification in a Cartesian coordinate with that in the logarithm of the radius. Massive theories in de Sitter or anti de Sitter space are thus produced from massless (scale invariant) theories in one higher space or time dimension. As an example, we derive free massive actions for arbitrary representations of the (anti) de Sitter group in arbitrary dimensions. (Previous general results were restricted to symmetric tensors.) We also discuss generalizations to interacting theories. 
  The new method of solving quantum mechanical problems is proposed. The finite, i.e. cut off, Hilbert space is algebraically implemented in the computer code with states represented by lists of variable length. Complete numerical solution of a given system is then automatically obtained. The technique is applied to Wess-Zumino quantum mechanics and D=2 and D=4 supersymmetric Yang-Mills quantum mechanics with SU(2) gauge group. Convergence with increasing cut-off was observed in many cases, well within the reach of present machines. Many old results were confirmed and some new ones, especially for the D=4 system, are derived. Extension to D=10 is possible but computationally demanding for higher gauge groups. 
  It is possible to introduce external time dependent back ground fields in the formulation of a system as fields whose dynamics can not be deduced from Euler Lagrange equations of motion. This method leads to singular Lagrangians for real systems. We discuss quantization of constraint systems in these cases and introduce generalized Gupta-Bleuler quantization. In two examples we show explicitly that this method of quantization leads to true Schr\"{o}dinger equations. 
  Two-loop $\beta$-function and anomalous dimension are calculated for N=1 supersymmetric quantum electrodynamics, regularized by higher derivatives in the minimal subtraction scheme. The result for two-loop contribution to the $\beta$-function appears to be equal to 0, does not depend on the form of regularizing term and does not lead to anomaly puzzle. Two-loop anomalous dimension can be also made independent on parameters of higher derivative regularization by a special choice of subtraction scheme. 
  We propose a variation of spacetime noncommutative field theory to realize the stringy spacetime uncertainty relation without breaking any of the global symmetries of the homogeneous isotropic universe. We study the spectrum of metric perturbations in this model for a wide class of accelerating background cosmologies. Spacetime noncommutativity leads to a coupling between the fluctuation modes and the background cosmology which is nonlocal in time. For each mode, there is a critical time at which the spacetime uncertainty relation is saturated. This is the time when the mode is generated. These effects lead to a spectrum of fluctuations whose spectral index is different from what is obtained for commutative spacetime in the infrared region, but is unchanged in the ultraviolet region. In the special case of an exponentially expanding background, we find a scale-invariant spectrum. but with a different magnitude than in the context of commutative spacetime if the Hubble constant is above the string scale. 
  We consider supersymmetric theories with a warped extra dimension where supersymmetry is broken by boundary conditions that preserve an R-symmetry. It is shown that this supersymmetry breaking mechanism naturally invokes the Stuckelberg formalism for the gravitino in order to give a four-dimensional theory with a smooth massless limit. 
  In this talk we give a brief description of the formulation of chiral and gauge symmetries on the fuzzy sphere . In particular fermion doublers are shown to be absent and the correct anomaly equation in two dimensions is obtained in the corresponding continuum limit . 
  We consider a Dirac equation set on an extended spin space that contains fermion and boson solutions. At given dimension, it determines the scalar symmetries. The standard field equations can be equivalently written in terms of such degrees of freedom, and are similarly constrained. At 9+1 dimensions, the SU(3) X SU(2)_L X U(1) gauge groups emerge, as well as solution representations with quantum numbers of related gauge bosons, leptons, quarks, Higgs-like particles and others as lepto-quarks. Information on the coupling constants is also provided; e. g., for the hypercharge g'=(1/2) sqrt(3/5) ~ >.387, at tree level. 
  We argue that it is possible to maintain both supersymmetry and integrability in the boundary tricritical Ising field theory. Indeed, we find two sets of boundary conditions and corresponding boundary perturbations which are both supersymmetric and integrable. The first set corresponds to a ``direct sum'' of two non-supersymmetric theories studied earlier by Chim. The second set corresponds to a one-parameter deformation of another theory studied by Chim. For both cases, the conserved supersymmetry charges are linear combinations of Q, \bar Q and the spin-reversal operator \Gamma. 
  We obtain explicit realizations of holographic renormalization group (RG) flows from M-theory, from E^{2,1} \times Spin(7) at UV to AdS_4 \times \tilde{S^7} (squashed S^7) at IR, from E^{2,1} \times CY4 at UV to AdS_4 \times Q^{1,1,1} at IR, and from E^{2,1} \times HK (hyperKahler) at UV to AdS_4 \times N^{0,1,0} at IR. The dual type IIA string theory configurations correspond to D2-D6 brane systems where D6 branes wrap supersymmetric four-cycles. We also study the Penrose limits and obtain the pp-wave backgrounds for the above configurations. Besides, we study some examples of non-supersymmetric and supersymmetric flows in five-dimensional gauge theories. 
  We investigate the low-energy dynamics of the BPS solitons of the noncommutative CP^1 model in 2+1 dimensions using the moduli space metric of the BPS solitons. We show that the dynamics of a single soliton coincides with that in the commutative model. We find that the singularity in the two-soliton moduli space, which exists in the commutative CP^1 model, disappears in the noncommutative model.We also show that the two-soliton metric has the smooth commutative limit. 
  In this paper we have considered the particle creation in the spatially closed Robertson-Walker space-time. We considered a real massive scalar field which conformally coupled to the Robertson-Walker background. With the dependence of the scale factor on time, the case under consideration is a dynamical Casimir effect with moving boundaries. 
  The massless bosonic field compactified on the circle of rational $R^2$ is reexamined in the presense of boundaries. A particular class of models corresponding to $R^2=\frac{1}{2k}$ is distinguished by demanding the existence of a consistent set of Newmann boundary states. The boundary states are constructed explicitly for these models and the fusion rules are derived from them. These are the ones prescribed by the Verlinde formula from the S-matrix of the theory. In addition, the extended symmetry algebra of these theories is constructed which is responsible for the rationality of these theories. Finally, the chiral space of these models is shown to split into a direct sum of irreducible modules of the extended symmetry algebra. 
  The $Z_2$ bosonic orbifold models with compactification radius $R^2=1/2k$ are examined in the presence of boundaries.   Demanding the extended algebra characters to have definite conformal dimension and to consist of an integer sum of Virasoro characters, we arrive at the right splitting of the partition function. This is used to derive a free field representation of a complete, consistent set of boundary states, without resorting to a basis of the extended algebra Ishibashi states. Finally the modules of the extended symmetry algebra that correspond to the finitely many characters are identified inside the direct sum of Fock modules that constitute the space of states of the theory. 
  In this note we explore the possibility of obtaining gauge bosons and fermionic spectrum as close as possible to the Standard Model content, by placing D3-branes at a ZN orbifold-like singularity in the presence of D7-branes. Indeed, we find that this is plausible provided a sufficiently high N is allowed for and the singular point is also fixed by an orientifold action. If extra charged matter is not permitted then the singularity should necessarily be non-supersymmetric. Correct hypercharge assignments require a dependence on some Abelian gauge D7-groups. In achieving such a construction we follow a recent observation made in Ref. [hep-th/0105155] about the possibility that, the three left handed quarks, would present different U(2) transformation properties. 
  We investigate proposals of how the form factor approach to compute correlation functions at zero temperature can be extended to finite temperature. For the two-point correlation function we conclude that the suggestion to use the usual form factor expansion with the modification of introducing dressing functions of various kinds is only suitable for free theories. Dynamically interacting theories require a more severe change of the form factor program. 
  Integrable boundary Toda theories are considered. We use boundary one-point functions and boundary scattering theory to construct the explicit solutions corresponding to classical vacuum configurations. The boundary ground state energies are conjectured. 
  We suggest a new method for the calculation of the nonlocal part of the effective action. It is based on resummation of perturbation series for the heat kernel and its functional trace at large values of the proper time parameter. We derive a new, essentially nonperturbative, nonlocal contribution to the effective action in spacetimes with dimensions $d>2$. 
  We calculate the Green function for the Dirac equation describing a spin 1/2 particle in the presence of a potential which is a sum of the Coulomb potential V_C=-A_1/r and a Lorentz scalar potential V_S=-A_2/r. The bound state spectrum is obtained. 
  We study the Penrose limit of various AdS_p X S^q orbifolds. The limiting spaces are waves with parallel rays and singular wave fronts. In particular, we consider the orbifolds AdS_3 X S^3/\Gamma, AdS_5 X S^5/\Gamma and AdS_{4,7} X S^{7,4}/\Gamma where \Gamma acts on the sphere and/or the AdS factor. In the pp-wave limit, the wave fronts are the orbifolds C^2/\Gamma, C^4/\Gamma and R XC^4/\Gamma, respectively. When desingularization is possible, we get asymptotically locally pp-wave backgrounds (ALpp). The Penrose limit of orientifolds are also discussed. In the AdS_5 X RP^5 case, the limiting singularity can be resolved by an Eguchi-Hanson gravitational instanton. The pp-wave limit of D3-branes near singularities in F-theory is also presented. Finally, we give the embedding of D-dimensional pp-waves in flat M^{2,D} space. 
  We study the Euclidean two-point function of Fermi fields in the SU(2)-Thirring model on the whole distance (energy) scale. We perform perturbative and renormalization group analyses to obtain the short-distance asymptotics, and numerically evaluate the long-distance behavior by using the form factor expansion. Our results illustrate the use of bosonization and conformal perturbation theory in the renormalization group analysis of a fermionic theory, and numerically confirm the validity of the form factor expansion in the case of the SU(2)-Thirring model. 
  We study how the DGP (Dvali-Gabadadze-Porrati) brane affects particle dynamics in linearized approximation. We find that once the particle is removed from the brane it is repelled to the bulk. Assuming that the cutoff for gravitational interaction is $M_*\sim 1/\epsilon$, we calculate the classical self energy of a particle as the function of its position. Since the particle wants to go to the region where its self energy is lower, it is repelled from the brane to the bulk where it gains its 5D self energy. Cases when mass of the particle $m<8\pi^2M_*$ and $m>8\pi^2M_*$ are qualitatively different, and in later case one has to take into account effects of strong gravity. In both cases the particle is repelled from the brane. For $m<8\pi^2M_*$ we obtain the same result from the 'electrostatic' analog of the theory. In that language mass (charge) in the bulk induces charge distribution on the brane which shields the other side of the brane and provides repulsive force. The DGP brane acts as a conducting plane in electrostatics (keeping in mind that in gravity different charges repel). The repulsive nature of the brane requires a certain localization mechanism. When the particle overcomes the localizing potential it rapidly moves to the bulk. Particles of mass $m>8\pi^2M_*$ form a black hole within $1/M_*$ distance from the brane. 
  We consider domain walls obtained by embedding the 1+1-dimensional $\phi^4$-kink in higher dimensions. We show that a suitably adapted dimensional regularization method avoids the intricacies found in other regularization schemes in both supersymmetric and non-supersymmetric theories. This method allows us to calculate the one-loop quantum mass of kinks and surface tensions of kink domain walls in a very simple manner, yielding a compact d-dimensional formula which reproduces many of the previous results in the literature. Among the new results is the nontrivial one-loop correction to the surface tension of a 2+1 dimensional N=1 supersymmetric kink domain wall with chiral domain-wall fermions. 
  We discuss some issues related to spontaneous N=2-> N=1 supersymmetry breaking. In particular, we state a set of geometrical conditions which are necessary that such a breaking occurs. Furthermore, we discuss the low energy N=1 effective Lagrangian and show that it satisfies non-trivial consistency conditions which can also be viewed as conditions on the geometry of the scalar manifold. 
  In this paper, we examine the complex sine-Gordon model in the presence of a boundary, and derive boundary conditions that preserve integrability. We present soliton and breather solutions, investigate the scattering of particles and solitons off the boundary and examine the existence of classical solutions corresponding to boundary bound states. 
  The maximally supersymmetric type IIB pp-wave is compactified on spatial circles, with and without an auxiliary rotational twist. All spatial circles of constant radius are identified. Without the twist, an S$^1$ compactification can preserve 24, 20 or 16 supercharges. $T^2$ compactifications can preserve 20, 18 or 16 supercharges; $T^3$ compactifications can preserve 18 or 16 supercharges and higher compactifications preserve 16 supercharges. The worldsheet theory of this background is discussed. The T-dual and decompactified type IIA and M-theoretic solutions which preserve 24 supercharges are given. Some comments are made regarding the AdS parent and the CFT description. 
  The quantization of the non-commutative N=1, U(1) super-Yang-Mills action is performed in the superfield formalism. We calculate the one-loop corrections to the self-energy of the vector superfield. Although the power-counting theorem predicts quadratic ultraviolet and infrared divergences, there are actually only logarithmic UV and IR divergences, which is a crucial feature of non-commutative supersymmetric field theories. 
  Non-supersymmetric multi-wall configurations are generically unstable. It is proposed that the stabilization in compact space can be achieved by introducing a winding number into the model. A BPS-like bound is studied for the energy of configuration with nonvanishing winding number. Winding number is implemented in an ${\cal N}=1$ supersymmetric nonlinear sigma model with two chiral scalar fields and a bound states of BPS and anti-BPS walls is found to exist in noncompact spaces. Even in compactified space $S^1$, this nontrivial bound state persists above a critical radius of the compact dimension. 
  In an alternative interpretation, the Seiberg-Witten map is shown to be induced by a field dependent co-ordinate transformation connecting noncommutative and ordinary space-times. Furthermore, following our previous ideas, it has been demonstrated here that the above (field dependent co-ordinate) transformation can occur naturally in the Batalin-Tyutin extended space version of the relativistic spinning particle model, (in a particular gauge). There is no need to postulate the space-time non-commutativity in an {\it ad hoc} way: It emerges from the spin degrees of freedom. 
  The monodromy matrix, ${\hat{\cal M}}$, is constructed for two dimensional tree level string effective action. The pole structure of ${\hat{\cal M}}$ is derived using its factorizability property. It is found that the monodromy matrix transforms non-trivially under the non-compact T-duality group, which leaves the effective action invariant and this can be used to construct the monodromy matrix for more complicated backgrounds starting from simpler ones. We construct, explicitly, ${\hat{\cal M}}$ for the exactly solvable Nappi-Witten model, both when B=0 and $B\neq 0$, where these ideas can be directly checked. We consider well known charged black hole solutions in the heterotic string theory which can be generated by T-duality transformations from a spherically symmetric `seed' Schwarzschild solution. We construct the monodromy matrix for the Schwarzschild black hole background of the heterotic string theory. 
  We examine whether the cosmologies with varying speed of light (VSL) are compatible with the second law of thermodynamics. We find that the VSL cosmology with varying fundamental constant is severely constrained by the second law of thermodynamics, whereas the bimetric cosmological models are less constrained. 
  Generalised Scherk-Schwarz reductions in which compactification on a circle is accompanied by a twist with an element of a global symmetry G typically lead to gauged supergravities and are classified by the monodromy matrices, up to conjugation by the global symmetry. For compactifications of IIB supergravity on a circle, G=SL(2,R) and there are three distinct gauged supergravities that result, corresponding to monodromies in the three conjugacy classes of SL(2,R). There is one gauging of the compact SO(2) subgroup of the SL(2,R) and two distinct gaugings of non-compact SO(1,1) subgroups, embedded differently in SL(2,R). The non-compact gaugings can be obtained from the compact one via an analytic continuation of the kind used in D=4 gauged supergravities. For the superstring, the monodromy must be in SL(2,Z), and the distinct theories correspond to SL(2,Z) conjugacy classes. The theories consist of two infinite classes with quantised mass parameter m=1,2,3,..., three exceptional theories corresponding to elliptic conjugacy classes, and a set of sporadic theories corresponding to hyperbolic conjugacy classes. 
  We provide a world-sheet description of Neveu-Schwarz five-branes wrapped on a complex projective space. It is an orbifold of the product of an N=2 minimal model and the IR fixed point of a certain linear sigma model. We show how the naked singularity in the supergravity description is resolved by the world-sheet CFT. Applying mirror symmetry, we show that the low-energy theory of NS5-branes wrapped on CP^1 in Eguchi-Hanson space is described by the Seiberg-Witten prepotential for N=2 super-Yang-Mills, with the gauge group given by the ADE-type of the five-brane. The world-sheet CFT is generically regular, but singularities develop precisely at the Argyres-Douglas points and massless monopole points of the space-time theory. We also study the low-energy theory of NS5-branes wrapped on CP^2 in a Calabi-Yau 3-fold and its relation to (2,2) super-Yang-Mills theory in two dimensions. 
  We study gauge fixing in the generalized Gupta-Bleuler quantization. In this method physical states are defined to be simultaneous null eigenstates of a set of quantum invariants. We apply the method to a solvable model proposed by Friedberg, Lee, Pang and Ren and show that no Gribov-type copies appears by construction. 
  We describe the set of generalized Poincare and conformal superalgebras in D=4,5 and 7 dimensions as two sequences of superalgebraic structures, taking values in the division algebras R, C and H. The generalized conformal superalgebras are described for D=4 by OSp(1;8| R), for D=5 by SU(4,4;1) and for D=7 by U_\alpha U(8;1|H). The relation with other schemes, in particular the framework of conformal spin (super)algebras and Jordan (super)algebras is discussed. By extending the division-algebra-valued superalgebras to octonions we get in D=11 an octonionic generalized Poincare superalgebra, which we call octonionic M-algebra, describing the octonionic M-theory. It contains 32 real supercharges but, due to the octonionic structure, only 52 real bosonic generators remain independent in place of the 528 bosonic charges of standard M-algebra. In octonionic M-theory there is a sort of equivalence between the octonionic M2 (supermembrane) and the octonionic M5 (super-5-brane) sectors. We also define the octonionic generalized conformal M-superalgebra, with 239 bosonic generators. 
  It is pointed out that a collider experiment involves a local contribution to the energy-momentum tensor, a circumstance which not a common feature of the current state of the Universe at large characterized by the cosmological constant $\Lambda_0$. This contribution may be viewed as a change in the structure of space-time from its large scale form governed by $\Lambda_0$ to one governed by a $\Lambda$ peculiar to the scale of the experiment. Possible consequences of this effect are explored by exploiting the asymptotic symmetry of space-time for non-vanishing $\Lambda$ and its relation to vacuum energy. 
  We perform a non-perturbative study of pure gauge theory in a two dimensional non-commutative (NC) space. On the lattice, it is equivalent to the twisted Eguchi-Kawai model, which we simulated at N ranging from 25 to 515. We observe a clear large-N scaling for the 1- and 2-point function of Wilson loops, as well as the 2-point function of Polyakov lines. The 2-point functions agree with a universal wave function renormalization. Based on a Morita equivalence, the large-N double scaling limit corresponds to the continuum limit of NC gauge theory, so the observed large-N scaling demonstrates the non-perturbative renormalizability of this NC field theory. The area law for the Wilson loops holds at small physical area as in commutative 2d planar gauge theory, but at large areas we find an oscillating behavior instead. In that regime the phase of the Wilson loop grows linearly with the area. This agrees with the Aharonov-Bohm effect in the presence of a constant magnetic field, identified with the inverse non-commutativity parameter. 
  We discuss quantum mechanics in the moduli space consisting of two maximally charged dilaton black holes. The quantum mechanics of the two black hole system is similar to the one of DFF model, and this system has the $SL(2,R)$ conformal symmetry. Also, we discuss the bound states in this system. 
  A generalization of the limiting procedure of Penrose, which allows non-zero cosmological constants and takes into account metrics that contain homogeneous functions of degree zero, is presented. It is shown that any spacetime which admits a spacelike conformal Killing vector has a limit which is conformal to plane waves. If the spacetime is an Einstein space, its limit exists only if the cosmological constant is negative or zero. When the conformal Killing vector is hypersurface orthogonal, the limits of Einstein spacetimes are certain AdS plane waves. In this case the nonlinear version of the Randall-Sundrum zero mode is obtained as the limit of the brane world scenarios. 
  We study a five-dimensional field theory which contains a monopole (string) solution with chiral fermion zero modes. This monostring solution is a close analog of the fivebrane solution of M-theory. The cancellation of normal bundle anomalies parallels that for the M-theory fivebrane, in particular, the presence of a Chern-Simons term in the low-energy effective U(1) gauge theory plays a central role. We comment on the relationship between the the microscopic analysis of the world-volume theory and the low-energy analysis and draw some cautionary lessons for M-theory. 
  We study the twisted Eguchi-Kawai (TEK) reduction procedure for large-N unitary matrix lattice models. In particular, we consider the case of two-dimensional principal chiral models, and use numerical Monte Carlo (MC) simulations to check the conjectured equivalence of TEK reduced model and standard lattice model in the large-N limit. The MC results are compared with the large-N limit of lattice principal chiral models to verify the supposed equivalence. The consistency of the TEK reduction procedure is verified in the strong-coupling region, i.e. for $\beta<\beta_c$ where $\beta_c$ is the location of the large-N phase transition. On the other hand, in the weak-coupling regime $\beta>\beta_c$, relevant for the continuum limit, our MC results do not support the equivalence of the large-N limits of the lattice chiral model and the corresponding TEK reduction. The implications for the correspondence between TEK model and noncommutative field theory are also discussed. 
  We study the Lagrangian for a sigma model based on the non-compact Heisenberg group. A unique feature of this model -- unlike the case for compact Lie groups -- is that the definition of the Lagrangian has to be regulated since the trace over the Heisenberg group is otherwise divergent. The resulting theory is a real Lagrangian with a quartic interaction term. After a few non-trivial transformations, the Lagrangian is shown to be equivalent -- at the classical level -- to a complex cubic Lagrangian. A one loop computation shows that the quartic and cubic Lagrangians are equivalent at the quantum level as well. The complex Lagrangian is known to classically equivalent to the SU(2) sigma model, with the equivalence breaking down at the quantum level. An explanation of this well known results emerges from the properties of the Heisenberg sigma model. 
  This dissertation reviews some properties of the low-energy effective actions for six dimensional open-string models. The first chapter is a pedagogical introduction about supergravity theories. In the second chapter closed strings are analyzed, with particular emphasis on type IIB, whose orientifold projection, in order to build type-I models, is the subject of the third chapter. Original results are reported in chapters 4 and 5. In chapter 4 we describe the complete coupling of (1,0) six-dimensional supergravity to tensor, vector and hypermultiplets. The generalized Green-Schwarz mechanism implies that the resulting theory embodies factorized gauge and supersymmetry anomalies, to be disposed of by fermion loops. Consequently, the low-energy theory is determined by the Wess-Zumino consistency conditions, rather than by the requirement of supersymmetry, and this procedure does not fix a quartic coupling for the gauginos. In chapter 5 we describe the low-energy effective actions for type-I models with brane supersymmetry breaking, resulting form the simultaneous presence of supersymmetric bulks, with one or more gravitinos, and non-supersymmetric combinations of BPS branes.The consistency of the resulting gravitino couplings implies that local supersymmetry is non-linearly realized on some branes. We analyze in detail the ten-dimensional $USp(32)$ model and the six-dimensional (1,0) models. 
  The mechanism of dynamical mass generation for the gauge field is studied through 1-loop. We find out that torsion is an obstruction to the appearance of a 1-loop mass correction. Contrary, if torsion is not present, a mass gap is generated for the 2-form field. 
  We consider classical and quantum dynamics on potentials that are asymptotically unbounded from below. By explicit construction we find that quantum bound states can exist in certain bottomless potentials. The classical dynamics in these potentials is novel. Only a set of zero measure of classical trajectories can escape to infinity. All other trajectories get trapped as they get further out into the asymptotic region. 
  We analyze a recently constructed class of D-brane theories with the fermion spectrum of the SM at the intersection of D6-branes wrapping a compact toroidal space. We show how the SM Higgs mechanism appears as a brane recombination effect in which the branes giving rise to U(2)_L \times U(1) recombine into a single brane related to U(1)_{em}. We also show how one can construct D6-brane models which respect some supersymmetry at every intersection. These are quasi-supersymmetric models of the type introduced in hep-th/0201205 which may be depicted in terms of SUSY-quivers and may stabilize the hierarchy between the weak scale and a fundamental scale of order 10-100 TeV present in low string scale models. Several explicit D6-brane models with three generations of quarks and leptons and different SUSY-quiver structure are presented. One can prove on general grounds that if one wants to build a (factorizable) D6-brane configuration with the SM gauge group and N = 1 SUSY (or quasi-SUSY), also a massless (B-L) generator must be initially present in any model. If in addition we insist on lef- and right-handed fermions respecting the same N=1 SUSY, the brane configurations are forced to have intersections giving rise to Higgs multiplets, providing for a rationale for the very existence of the SM Higgs sector. 
  We present a large and universal class of new boundary states which break part of the chiral symmetry in the underlying bulk theory. Our formulas are based on coset constructions and they can be regarded as a non-abelian generalization of the ideas that were used by Maldacena, Moore and Seiberg to build new boundary states for SU(N). We apply our expressions to construct defect lines joining two conformal field theories with possibly different central charge. Such defects can occur e.g. in the AdS/CFT correspondence when branes extend to the boundary of the AdS-space. 
  We consider dimensionally reduced three-dimensional supersymmetric Yang-Mills-Chern-Simons theory. Although the N=1 supersymmetry of this theory does not allow true massive Bogomol'nyi-Prasad-Sommerfield (BPS) states, we find approximate BPS states which have non-zero masses that are almost independent of the Yang-Mills coupling constant and which are a reflection of the massless BPS states of the underlying N=1 super Yang-Mills theory. The masses of these states at large Yang-Mills coupling are exactly at the n-particle continuum thresholds. This leads to a relation between their masses at zero and large Yang-Mills coupling. 
  We reanalyze brane inflation with brane-brane interactions at an angle, which include the special case of brane-anti-brane interaction. If nature is described by a stringy realization of the brane world scenario today (with arbitrary compactification), and if some additional branes were present in the early universe, we find that an inflationary epoch is generically quite natural, ending with a big bang when the last branes collide. In an interesting brane inflationary scenario suggested by generic string model-building, we use the density perturbation observed in the cosmic microwave background and the coupling unification to find that the string scale is comparable to the GUT scale. 
  We argue that the gauge theory dual to the Type IIB string theory in ten-dimensional pp-wave background can be thought to `live' on an {\it Euclidean} subspace spanning four of the eight transverse coordinates. We then show that light-cone time evolution of the string is identifiable as the RG flow of the gauge theory -- a relation facilitating `holography' of the pp-wave background. The `holography' reorganizes the dual gauge theory into theories defined over Hilbert subspaces of fixed R-charge. The reorganization breaks the SO(4,2)$\times$SO(6) symmetry to a maximal subgroup SO(4)$\times$ SO(4) spontaneously. We argue that the low-energy string modes may be regarded as Goldstone modes resulting from such symmetry breaking pattern. 
  We discuss nonplanar anomalies in noncommutative gauge theories. In particular we show that a nonplanar anomaly exists when the external noncommutative momentum is zero and that it leads to a non-conservation of the associated axial charge. In the case of nonplanar local anomalies, a cancellation of the anomaly can be achieved by a Green-Schwarz mechanism. In an example of D3 branes placed on orbifold singularity that leads to a chiral theory, the mechanism involves twisted RR fields which propagate with zero noncommutative momentum. Global anomalies are not cancelled and, in particular, the decay pi0 --> 2 gamma is allowed. 
  We show that noncommutative gauge theory in two dimensions is an exactly solvable model. A cohomological formulation of gauge theory defined on the noncommutative torus is used to show that its quantum partition function can be written as a sum over contributions from classical solutions. We derive an explicit formula for the partition function of Yang-Mills theory defined on a projective module for arbitrary noncommutativity parameter \theta which is manifestly invariant under gauge Morita equivalence. The energy observables are shown to be smooth functions of \theta. The construction of noncommutative instanton contributions to the path integral is described in some detail. In general, there are infinitely many gauge inequivalent contributions of fixed topological charge, along with a finite number of quantum fluctuations about each instanton. The associated moduli spaces are combinations of symmetric products of an ordinary two-torus whose orbifold singularities are not resolved by noncommutativity. In particular, the weak coupling limit of the gauge theory is independent of \theta and computes the symplectic volume of the moduli space of constant curvature connections on the noncommutative torus. 
  We extend previous work by developing a worldsheet description of non-abelian gauge theory (Yang-Mills). This task requires the introduction of Grassmann variables on the world sheet analogous to those of the Neveu-Schwarz-Ramond formulation of string theory. A highlight of our construction is that once the three gluon vertices of Yang-Mills Feynman diagrams are given a worldsheet description, the worldsheet formalism automatically produces all of the quartic vertices. 
  We derive the general formula, at a finite cutoff, for the change in the boundary condition of a scalar field in AdS under a Multiple-trace deformation of the dual CFT. Our analysis suggests that fluctuations around the classical solution in AdS should not be constrained by boundary conditions. 
  Using multiple integral representations, we derive exact expressions for the correlation functions of the spin-1/2 Heisenberg chain at the free fermion point. 
  At a formal level, there appears to be no difficulty involved in introducing a chemical potential for a globally conserved quantum number into the partition function for space-time including a black hole. Were this possible, however, it would provide a form of black hole hair, and contradict the idea that global quantum numbers are violated in black hole evaporation. We demonstrate dynamical mechanisms that negate the formal procedure, both for topological charge (Skyrmions) and complex scalar-field charge. Skyrmions collapse to the horizon; scalar-field charge fluctuates uncontrollably. 
  Recollections on how the basic concepts and ingredients of supergravity were formulated by Dmitrij V. Volkov and the present author in 1973-74. 
  We consider the process of black hole formation in particle collisions in the exactly solvable framework of 2+1 dimensional Anti de Sitter gravity. An effective Hamiltonian describing the near horizon dynamics of a head on collision is given. The Hamiltonian exhibits a universal structure, with a formation of a horizon at a critical distance. Based on it we evaluate the action for the process and discuss the semiclassical amplitude for black hole formation. The derived amplitude is seen to contain no exponential suppression or enhancement. Coments on the CFT description of the process are made. 
  We show how to compute terms in an expansion of the world-volume superpotential for fairly general D-branes on the quintic Calabi-Yau using linear sigma model techniques, and show in examples that this superpotential captures the geometry and obstruction theory of bundles and sheaves on this Calabi-Yau. 
  We obtain the existence of a cohomological obstruction to expressing N=2 line bundles as tensor products of N=1 bundles. The motivation behind this paper is an attempt at understanding the N=2 super KP equation via Baker functions, which are special sections of line bundles on supercurves. 
  We solve the problem of finding all eigenvalues and eigenvectors of the Neumann matrix of the matter sector of open bosonic string field theory, including the zero modes, and switching on a background B-field. We give the discrete eigenvalues as roots of transcendental equations, and we give analytical expressions for all the eigenvectors. 
  We investigate the quantized scalar field on the Kaluza-Klein spacetimes of $M^D\times T^d \times S_{FZ}$, where $M^D$ is the ordinary $D$ dimensional flat Minkowski spacetimes, $T^d $ is the $d$ dimensional commutative torus, and $S_{FZ}$ is a noncommutative fuzzy two sphere with a fixed quantized radius. After evaluating the one-loop correction to the spectrum we use the mass-corrected term to compute the Casimir energy of the scalar field on the model spacetime. It is seen that, for some values of $D$ and $d$, the Casimir energy due to vacuum fluctuation in the model spacetimes could give rise a repulsive force to stabilize the commutative torus. 
  We review here the work done on the occurence of chaotic configurations in systems derived from Gauge theories. These include Yang-Mills and associated field theories with modifications including Chern Simons and Higgs fields. 
  Using the new scalar and vector degrees of freedom derived from the non-linear gauge condition (grad-dot-D)(grad-dot-A)=0, we show that the effective dynamics of the vector fields (identified as ``gluons'') in the stochastic vacuum defined by the scalars result in the vector fields acquiring a range of possible masses and losing their self-interactions. From this range of masses, we derive the mass gap in pure Yang-Mills theory. Finally, we comment on the gauge-invariance of the result. 
  We propose a recipe for determination of the partition function of ${\cal N}=4$ $ADE$ gauge theory on $K3$ by generalizing our previous results of the SU(N) case. The resulting partition function satisfies Montonen-Olive duality for $ADE $ gauge group. 
  We calculate the effective action in Yang-Mills and scalar \phi^4 quantum field theory with quantized scale invariant metric treated non-perturbatively in d=4 dimensions. There is no charge renormalization in the one-loop order for matter fields. We show that the electromagnetic energy of point charges can be finite. The temperature dependence of the effective action in inflationary models is changed substantially as a result of an interaction with quantum gravity. 
  A first-order `BPS' equation is obtained for 1/8 supersymmetric intersections of soliton-membranes (lumps) of supersymmetric (4+1)-dimensional massless sigma models, and a special non-singular solution is found that preserves 1/4 supersymmetry. For 4-dimensional hyper-K\"ahler target spaces ($HK_4$) the BPS equation is shown to be the low-energy limit of the equation for a Cayley-calibrated 4-surface in $\bE^4\times HK_4$. Similar first-order equations are found for stationary intersections of Q-lump-membranes of the massive sigma model, but now generic solutions preserve either 1/8 supersymmetry or no supersymmetry, depending on the time orientation. 
  We provide first explicite examples of quantum deformations of D=4 conformal algebra with mass-like deformation parameters, in applications to quantum gravity effects related with Planck mass. It is shown that one of the classical $r$-matrices defined on the Borel subalgebra of $sl(4)$ with $o(4,2)$ reality conditions describes the light-cone $\kappa$-deformation of D=4 Poincar\'{e} algebra. We embed this deformation into the three-parameter family of generalized $\kappa$-deformations, with $r$-matrices depending additionally on the dilatation generator. Using the extended Jordanian twists framework we describe these deformations in the form of noncocommutative Hopf algebra. We describe also another four-parameter class of generalized $\kappa$-deformations, which is obtained by continuous deformation of distinguished $\kappa$-deformation of D=4 Weyl algebra, called here the standard $\kappa$-deformation of Weyl algebra. 
  In this paper, we arrive from different starting points at the conclusion that the symmetry given by an action of the Grothendieck-Teichmueller group GT on the so called extended moduli space of string theory can not be physical - in the sense that it does not survive the inclusion of general nonperturbative vacua given by boundary conditions on the level of two dimensional conformal field theory - but has to be extended to a quantum symmetry given by a self-dual, noncommutative, and noncocommutative Hopf algebra. First, we show that a class of two dimensional boundary conformal field theories always uniquely defines a trialgebra and find the above mentioned Hopf algebra as the universal symmetry of such trialgebras (in analogy to the definition of GT as the universal symmetry of quasi-triangular quasi-Hopf algebras). Second, we argue in a more heuristic approach that this Hopf algebra symmetry can also be found in a more geometric picture using the language of gerbes. 
  The ADHM constraints which implicitly specify instanton gauge field configurations are solved for the explicit general form of instantons with topological charge two and gauge group U(N). 
  We address the conjecture that at the tachyonic vacuum open strings get transformed into closed strings. We show that it is possible in the context of boundary string field theory to interpolate between the conventional open string theory, characterized by having the D25 brane as the boundary state, and an off-shell (open) string theory where the boundary state is identified with the closed string vacuum, where holomorphic and antiholomorphic modes decouple and where bulk vertex operator correlation functions are identical to those of the closed string. 
  The open string ending on a D-brane with a constant B-field in a pp-wave Ramond-Ramond background is exactly solvable. The theory is controlled by three dimensionful parameters: alpha', the mass parameter (RR background times the lightcone momentum) and the B-field. We quantize the open string theory and determine the full noncommutative structure. In particular, we find a fully noncommutative phase space whose noncommutativity depends on all these parameters. The lightcone Hamiltionian is obtained, and as a consequence of the nontrivial commutation relations of the theory, new features of the spectrum are noted. Various scaling limits of the string results are considered. Physical implications are discussed. 
  By employing D6-branes intersecting at angles in $D = 4$ type I strings, we construct the first examples of three generation string GUT models (PS-A class), that contain at low energy exactly the standard model spectrum with no extra matter and/or extra gauge group factors. They are based on the group $SU(4)_C \times SU(2)_L \times SU(2)_R$. The models are non-supersymmetric, even though SUSY is unbroken in the bulk. Baryon number is gauged and its anomalies are cancelled through a generalized Green-Schwarz mechanism. We also discuss models (PS-B class) which at low energy have the standard model augmented by an anomaly free U(1) symmetry and show that multibrane wrappings correspond to a trivial redefinition of the surviving global U(1) at low energies. There are no colour triplet couplings to mediate proton decay and proton is stable. The models are compatible with a low string scale of energy less that 650 GeV and are directly testable at present or future accelerators as they predict the existence of light left handed weak fermion doublets at energies between 90 and 246 GeV. The neutrinos get a mass through an unconventional see-saw mechanism. The mass relation $m_e = m_d$ at the GUT scale is recovered. Imposing supersymmetry at particular intersections generates non-zero Majorana masses for right handed neutrinos as well providing the necessary singlets needed to break the surviving anomaly free U(1), thus suggesting a gauge symmetry breaking method that can be applied in general left-right symmetric models. 
  We continue the analysis of Vacuum String Field Theory in the presence of a constant B field. In particular we give a proof of the ratio of brane tensions is the expected one. On the wake of the recent literature we introduce wedge-like states and orthogonal projections. Finally we show a few examples of the smoothing out effects of the B field on some of the singularities that appear in VSFT. 
  This paper has been withdrawn by the authors, due to an accuracy error in the Maple and Fortran calculations which completely changed the results. 
  We study examples of chiral four-dimensional IIB orientifolds with Scherk--Schwarz supersymmetry breaking, based on freely acting orbifolds. We construct a new Z3xZ3' model, containing only D9-branes, and rederive from a more geometric perspective the known Z6'xZ2' model, containing D9, D5 and \bar D 5 branes. The cancellation of anomalies in these models is then studied locally in the internal space. These are found to cancel through an interesting generalization of the Green--Schwarz mechanism involving twisted Ramond--Ramond axions and 4-forms. The effect of the latter amounts to local counterterms from a low-energy effective field theory point of view. We also point out that the number of spontaneously broken U(1) gauge fields is in general greater than what expected from a four-dimensional analysis of anomalies. 
  Observational evidence suggests that our universe is currently evolving towards an asymptotically de Sitter future. Unfortunately and in spite of much recent attention, various quantum, holographic and cosmological aspects of de Sitter space remain quite enigmatic. With such intrigue in mind, this paper considers the ``construction'' of a toy model that describes an asymptotically de Sitter universe. More specifically, we add fluid-like matter to an otherwise purely de Sitter spacetime, formulate the relevant solutions and then discuss the cosmological and holographic implications. If the objective is to construct an asymptotically de Sitter universe that is free of singularities and has a straightforward holographic interpretation, then the results of this analysis are decidedly negative. Nonetheless, this toy model nicely illustrates the pitfalls that might be encountered in a more realistic type of construction. 
  We present the Lax pair for the field elliptic Calogero-Moser system and establish a connection between this system and the Kadomtsev-Petviashvili equation. Namely, we consider elliptic families of solutions of the KP equation, such that their poles satisfy a constraint of being balanced. We show that the dynamics of these poles is described by a reduction of the field elliptic CM system.   We construct a wide class of solutions to the field elliptic CM system by showing that any N-fold branched cover of an elliptic curve gives rise to an elliptic family of solutions of the KP equation with balanced poles. 
  If the linear perturbation theory is valid through the bounce, the surviving fluctuations from the ekpyrotic scenario (cyclic one as well) should have very blue spectra with suppressed amplitude for the scalar-type structure. We derive the same (and consistent) result using the curvature perturbation in the uniform-field (comoving) gauge and in the zero-shear gauge. Previously, Khoury et al. interpreted results from the latter gauge condition incorrectly and claimed the scale-invariant spectrum, thus generating controversy in the literature. We also correct similar errors in the literature based on wrong mode identification and joining condition. No joining condition is needed for the derivation. 
  We demonstrate the emergence of the conformal group SO(4,2) from the Clifford algebra of spacetime. The latter algebra is a manifold, called Clifford space, which is assumed to be the arena in which physics takes place. A Clifford space does not contain only points (events), but also lines, surfaces, volumes, etc..., and thus provides a framework for description of extended objects. A subspace of the Clifford space is the space whose metric is invariant with respect to the conformal group SO(4,2) which can be given either passive or active interpretation. As advocated long ago by one of us, active conformal transformations, including dilatations, imply that sizes of physical objects can change as a result of free motion, without the presence of forces. This theory is conceptually and technically very different from Weyl's theory and provides when extended to a curved conformal space a resolution of the long standing problem of realistic masses in Kaluza-Klein theories. 
  A generalisation of existing SU(2) results is obtained. In particular, the source-free Gauss law for SU(3)-valued gauge fields is solved using a non-Abelian analogue of the Poincare lemma. When sources are present, the colour-electric field is divided into two parts in a way similar to the Hodge decomposition. Singularities due to coinciding eigenvalues of the colour-magnetic field are also analysed. 
  We discuss D-dimensional scalar and electromagnetic fields interacting with a quantized metric. The gravitons depend solely on twodimensional coordinates. We consider a conformal field theory model for the metric tensor. We show that an interaction with gravity improves the short distance behaviour. As a result there is no charge renormalization in the fourdimensional Higgs model. 
  We carry out the extension of the covariant Ostrogradski method to fermionic field theories. Higher-derivative Lagrangians reduce to second order differential ones with one explicit independent field for each degree of freedom. 
  In this paper we consider the influence of transplanckian physics on the CMBR anisotropies produced by inflation. We consider a simple toy model that allows for analytic calculations and argue on general grounds, based on ambiguities in the choice of vacuum, that effects are expected with a magnitude of the order of $H/\Lambda$, where $H$ is the Hubble constant during inflation and $\Lambda$ the scale for new physics, e.g. the Planck scale. 
  Noncommutativity in an open string moving in a background Neveu-Schwarz field is investigated in a gauge independent Hamiltonian approach, leading to new results. The noncommutativity is shown to be a direct consequence of the non-trivial boundary conditions, which, contrary to several approaches, are not treated as constraints. We find that the noncommutativity persists for all string points. In the conformal gauge our results reduce to the usual noncommutativity at the boundaries only. 
  We discuss the Equivalence Theorem (ET) in the BRST formalism. The existence of a local inverse of the field transformation (at least as a formal power expansion) suggests a formulation of the ET, which allows a nilpotent BRST symmetry. This strategy cannot be implemented at the quantum level if the inverse is non-local. In this case we propose an alternative formulation of the ET, where, by using Faddeev-Popov fields, this difficulty is circumvented. We study the quantum deformation of the associated ST identity, which turns out to be anomaly free, and show that a selected set of Green functions, which in some cases can be identified with the physical observables of the model, does not depend on the choice of the transformation of the fields. In general the transformation of the fields yields a non-renormalizable theory. When the equivalence is established between a renormalizable and a non-renormalizable theory, the ET provides a way to give a meaning to the last one by using the resulting ST identity. In this case the Quantum Action Principle cannot be of any help in the discussion of the ET. We assume and discuss the validity of a Quasi Classical Action Principle, which turns out to be sufficient for the present work. As an example we study the renormalizability and unitarity of massive QED in Proca's gauge by starting from a linear Lorentz-covariant gauge. 
  The geometry of flat spacetime modded out by a null rotation (boost+rotation) is analysed. When embedding this quotient spacetime in String/M-theory, it still preserves one half of the original supersymmetries. Its connection with the BTZ black hole, supersymmetric dilatonic waves and one possible resolution of its singularity in terms of nullbranes are also discussed. 
  We investigate half-supersymmetric domain wall solutions of four maximally supersymmetric D=9 massive supergravity theories obtained by Scherk-Schwarz reduction of D=10 IIA and IIB supergravity. One of the theories does not have a superpotential and does not allow domain wall solutions preserving any supersymmetry. The other three theories have superpotentials leading to half-supersymmetric domain wall solutions, one of which has zero potential but non-zero superpotential.   The uplifting of these domain wall solutions to ten dimensions leads to three classes of half-supersymmetric Type IIB 7-brane solutions. All solutions within each class are related by SL(2,R) transformations. The three classes together contain solutions carrying all possible (quantised) 7-brane charges. One class contains the well-known D7-brane solution and its dual partners and we provide the explicit solutions for the other two classes. The domain wall solution with zero potential lifts up to a half-supersymmetric conical space-time. 
  Starting from the Maldacena-Nunez supergravity dual of N=1 super Yang-Mills theory we study the inclusion of a supersymmetry breaking gaugino mass term. We consider a class of non supersymmetric deformations of the MN solutions which have been recently proposed in the literature. We show that they can be interpreted as corresponding to the inclusion of both a mass and a condensate. We calculate the vacuum energy of the supergravity solutions showing that the N-fold vacuum degeneracy of the N=1 theory is lifted by the inclusion of a mass term. 
  In this paper we calculate the spectrum of Neumann matrix with zero modes in the presence of the constant B field in Witten's cubic string field theory. We find both the continuous spectrum inside $[{-1\over3}, 0)$ and the constraint on the existence of the discrete spectrum. For generic $\theta$, -1/3 is not in the discrete spectrum but in the continuous spectrum. For each eigenvalue in the continuous spectrum there are four twist-definite degenerate eigenvector except for -1/3 at which the degeneracy is two. However, for each twist-definite eigenvector the twist parity is opposite among the two spacetime components. Based upon the result at -1/3 we prove that the ratio of brane tension to be one as expected. Furthermore, we discuss the factorization of star algebra in the presence of B field under zero-slope limit and comment on the implications of our results to the recent proposed map of Witten's star to Moyal's star. 
  We systematically study the question of identification and consistent inclusion of the radion, within the Lagrangian approach, in a two brane Randall-Sundrum model. Exploiting the symmetry properties of the theory, we show how the radion can be identified unambiguously and give the action to all orders in the radion field and the metric. Using the background field method, we expand the theory to quadratic orders in the fields. We show that the most general classical solutions, for the induced metric on the branes in the case of a constant radion and a factorizable 4D metric, correspond to Einstein spaces. We discuss extensively the diagonalization of the quadratic action. Furthermore, we obtain the 4-dimensional effective theory from this and study the question of the spectrum as well as the couplings in these theories. 
  We show that N=8 spontaneously broken supergravity in four dimensions obtained by Scherk-Schwarz generalized dimensional reduction can be obtained from a pure four dimensional perspective by gauging a suitable electric subgroup of E_{7,7}. Owing to the fact that there are non isomorphic choices of maximal electric subgroups of the U-duality group their gaugings give rise to inequivalent theories. This in particular shows that the Scherk-Schwarz gaugings do not fall in previous classifications of possible gauged N=8 supergravities. Gauging of flat groups appear in many examples of string compactifications in presence of brane fluxes. 
  We discuss the partners of the stress energy tensor and their structure in Logarithmic conformal field theories. In particular we draw attention to the fundamental differences between theories with zero and non-zero central charge. We analyze the OPE for T, \bar{T} and the logarithmic partners t and \bar{t} for c=0 theories. 
  The motion of a scalar particle in (d+1)-dimensional AdS space may be described in terms of the Cartesian coordinates that span the (d+2)-dimensional space in which the AdS space is embedded. Upon quantization, the mass hyperboloid defined in terms of the conjugate momenta turns into the wave equation in AdS space. By interchanging the roles of coordinates and conjugate momenta in the (d+2)-dimensional space we arrive at a dual description. For massive modes, the dual description is equivalent to the conventional formulation, as required by holography. For tachyonic modes, this interchange of coordinates and momenta establishes a duality between Euclidean AdS and dS spaces. We discuss its implications on Green functions for the various vacua. 
  We study a disk amplitude which has a complicated heterogeneous matter configuration on the boundary in a system of the (3,4) conformal matter coupled to two-dimensional gravity. It is analyzed using the two-matrix chain model in the large N limit. We show that the disk amplitude calculated by Schwinger-Dyson equations can completely be reproduced through purely geometrical consideration. From this result, we speculate that all heterogeneous loop amplitudes can be derived from the geometrical consideration and the consistency among relevant amplitudes. 
  After reviewing the main characteristics of the spacetime of accelerating universes driven by a quintessence scalar field with constant equation of state $\omega$, we investigate in this paper the classical stability of such spaces to cosmological perturbations, particularizing in the case of a closed geometry and equation of state $\omega=-2/3$. We conclude that this space is classically stable and conjecture that accelerating universes driven by quintessential fields have "no-hair". 
  We discuss construction of classical time dependent solutions in open string (field) theory, describing the motion of the tachyon on unstable D-branes. Despite the fact that the string field theory action contains infinite number of time derivatives, and hence it is not a priori clear how to set up the initial value problem, the theory contains a family of time dependent solutions characterized by the initial position and velocity of the tachyon field. We write down the world-sheet action of the boundary conformal field theories associated with these solutions and study the corresponding boundary states. For D-branes in bosonic string theory, the energy momentum tensor of the system evolves asymptotically towards a finite limit if we push the tachyon in the direction in which the potential has a local minimum, but hits a singularity if we push it in the direction where the potential is unbounded from below. 
  Some problems related to the structure of higher terms of the epsilon-expansion of Feynman diagrams are discussed. 
  We study the geometric interpretation of two dimensional rational conformal field theories, corresponding to sigma models on Calabi-Yau manifolds. We perform a detailed study of RCFT's corresponding to T^2 target and identify the Cardy branes with geometric branes. The T^2's leading to RCFT's admit ``complex multiplication'' which characterizes Cardy branes as specific D0-branes. We propose a condition for the conformal sigma model to be RCFT for arbitrary Calabi-Yau n-folds, which agrees with the known cases. Together with recent conjectures by mathematicians it appears that rational conformal theories are not dense in the space of all conformal theories, and sometimes appear to be finite in number for Calabi-Yau n-folds for n>2. RCFT's on K3 may be dense. We speculate about the meaning of these special points in the moduli spaces of Calabi-Yau n-folds in connection with freezing geometric moduli. 
  The lattice model of scalar quantum electrodynamics (Maxwell field coupled to a complex scalar field) in the Hamiltonian framework is discussed.  It is shown that the algebra of observables ${\cal O}({\Lambda})$ of this model is a $C^*$-algebra, generated by a set of gauge-invariant elements satisfying the Gauss law and some additional relations. Next, the faithful, irreducible and non-degenerate representations of ${\cal O}({\Lambda})$ are found. They are labeled by the value of the total electric charge, leading to a decomposition of the physical Hilbert space into charge superselection sectors. In the Appendices we give a unified description of spinorial and scalar quantum electrodynamics and, as a byproduct, we present an interesting example of weakly commuting operators, which do not commute strongly. 
  While it is clear that in some kinematic regime QCD can be described by an effective (as opposed to fundamental) string theory, it is not at all clear how this string theory should be. The `natural' candidate, the bosonic string, leads to amplitudes with the usual problems related to the existence of the tachyon, absence of the adequate Adler zero, and massless vector particles, not to mention the conformal anomaly. The supersymmetric version does not really solve most of these problems. For a long time it has been believed that the solution of at least some of these difficulties is associated to a proper identification of the vacuum, but this program has remained elusive. We show in this work how the first three problems can be avoided, by using a sigma model approach where excitations above the correct (chirally non-invariant) QCD vacuum are identified. At the leading order in a derivative expansion we recover the non-linear sigma model of pion interactions. At the next-to-leading order the O(p^4) Lagrangian of Gasser and Leutwyler is obtained, with values for the coefficients that match the observed values. We also discuss some issues related to the conformal anomaly. 
  We find examples in string theory of locally wrapped D-branes. These excitations mimic skyrmions in that they correspond to topological excitations of the scalar fields parametrizing the brane motion in the space transverse to its world-volume. While these brane excitations appear to be point-like, evidence is provided that curvature corrections to the probe action might allow for a delocalization of the wrapping on a scale of the order of the string length, therefore rendering the phenomena non-singular. 
  We argue that G_2 manifolds for M-theory admitting string theory Calabi-Yau duals are fibered by coassociative submanifolds. Dual theories are constructed using the moduli space of M5-brane fibers as target space. Mirror symmetry and various string and M-theory dualities involving G_2 manifolds may be incorporated into this framework. To give some examples, we construct two non-compact manifolds with G_2 structures: one with a K3 fibration, and one with a torus fibration and a metric of G_2 holonomy. Kaluza-Klein reduction of the latter solution gives abelian BPS monopoles in 3+1 dimensions. 
  We examine the conjecture that an 11d E_8 bundle, appearing in the calculation of phases in the M-Theory partition function, plays a physical role in M-Theory, focusing on consequences for the classification of string theory solitons. This leads for example to a classification of IIA solitons in terms of that of LE_8 bundles in 10d. Since K(Z,2) approximates LE_8 up to \pi_{14}, this reproduces the K-Theoretic classification of IIA D-branes while treating NSNS and RR solitons more symmetrically and providing a natural interpretation of G_0 as the central extension of LE_8. 
  We study quantum gravity on $dS_{3}$ using the Chern-Simons formulation of three -dimensional gravity. We derive an exact expression for the partition function for quantum gravity on $dS_{3}$ in a Euclidean path integral approach. We show that the topology of the space relevant for studying de Sitter entropy is a solid torus. The quantum fluctuations of de Sitter space are sectors of configurations of point masses taking a {\em discrete} set of values. The partition function gives the correct semi-classical entropy. The sub-leading correction to the entropy is logarithmic in horizon area, with a coefficient -1. We discuss this correction in detail, and show that the sub-leading correction to the entropy from the dS/CFT correspondence agrees with our result. A comparison with the corresponding results for the $AdS_{3}$ BTZ black hole is also presented. 
  On the basis a new conjecture, we present a new Lagrangian density and a new quantization method for QED, construct coupling operators and mass operators, derive scattering operators S_{f} and S_{w} which are dependent on each other and supplement new Feynman rules. S_{f} and S_{w} together determine a Fenman integral. Hence all Feynman integrals are convergent and it is unnecessary to introduce regularization and counterterms. That the energy of the vacuum state is equal to zero is naturally obtained. From this we can easily determine the cosmological constant according to data of astronomical observation, and it is possible to correct nonperturbational methods which depend on the energy of the ground state in quantum field theory. On the same basis as the new QED, we obtain naturally a new SU(2)XU(1) electroweak unified model whose L=L_{F}+L_{W} , here L is left-right symmetric. Thus the world is left-right symmetric in principle, but the part observed by us is asymmetric because L_{W} and L_{F} are all asymmetic. This model do not contain any unknown particle with a massive mass. A conjecture that there is repulsion or gravitation between the W-particles and the F-particles is presented. If the new interaction is gravitation, W-matter is the candidate for dark matter. If the new interaction is repulsion, W-matter is the origin of universe expansion. 
  We reexamine a scenario in which photons and gravitons arise as Goldstone bosons associated with the spontaneous breaking of Lorentz invariance. We study the emergence of Lorentz invariant low energy physics in an effective field theory framework, with non-Lorentz invariant effects arising from radiative corrections and higher order interactions. Spontaneous breaking of the Lorentz group also leads to additional exotic but weakly coupled Goldstone bosons, whose dispersion relations we compute. The usual cosmological constant problem is absent in this context: being a Goldstone boson, the graviton can never develop a potential, and the existence of a flat spacetime solution to the field equations is guaranteed. 
  All possible Bogoliubov operators that generate the thermal transformations in the Thermo Field Dynamics (TFD) form a SU(1,1) group. We discuss this contruction in the bosonic string theory. In particular, the transformation of the Fock space and string operators generated by the most general SU(1,1) unitary Bogoliubov transformation and the entropy of the corresponding thermal string are computed. Also, we construct the thermal $D$-brane solution generated by the SU(1,1) transformation in a constant Kalb-Ramond field and compute its entropy. 
  Six dimensional compactification of the type IIA matrix model on the ${\bf Z}$-orbifold is studied. Introducing a ${\bf Z}_{3}$ symmetry properly on the three mirror images of fields in the $N$-body system of the supersymmetric D0 particles, the action of the Matrix model compactified on the ${\bf Z}$-orbifold is obtained. In this matrix model ${\cal N}=1$ supersymmetry is explicitly demonstrated. 
  This paper presents some cosmological consequences of the five dimensional, two brane Randall-Sundrum brane scenario. The radius of the compact extra dimension is taken to be time dependent. It is shown that the cosmology consistent with the two brane Randall-Sundrum model is a power law expansion of the universe, with scale factor growing as $t^{1/2}$. The two branes tend to move towards each other with time. Some comments are made on the contribution of surface terms in deriving the four dimensional effective action. 
  It is shown that in a quantized space determined by the $B_2\quad (O(5)=Sp(4))$ algebra with three dimensional parameters of the length $L^2$, momentum $(Mc)^2$, and action $S$, the spectrum of the Coulomb problem with conserving Runge-Lenz vector coincides with the spectrum found by Schr\"odinger for the space of constant curvature but with the values of the principal quantum number limited from the side of higher values. The same problem is solved for the spectrum of a harmonic oscillator. 
  It is shown that in the case of multicomponent integrable systems connected with algebras $A_n$, the discrete transformation $T$ possesses the fine structure and can be represented in the form $T=\prod T_i^{l_i}$, where $T_i$ are n commuting basis discrete transformations, and $l_i$ are arbitrary natural numbers. All the calculations are conducted in detail for the case of a 3-wave interacting system. 
  Neveu-Schwarz ghost slivers in pictures zero and minus one are constructed. In particular, using algebraic methods $\beta$, $\gamma$ ghost sliver in the -1 picture is obtained. The algebraic method consists in solving a projector equation in an algebra, where the multiplication is defined by a pure 3-string vertex without any insertions at the string midpoint. We show that this projector is a sliver in a twisted version of $\beta$, $\gamma$ conformal theory. We also show that the product of the twisted $b$, $c$ and $\beta$, $\gamma$ ghost slivers solves an equation that appears after a special rescaling of super VSFT. 
  We present a new class of hermitian one-matrix models originated in the W-infinity algebra: more precisely, the polynomials defining the W-infinity generators in their fermionic bilinear form are shown to expand the orthogonal basis of a class of random hermitian matrix models. The corresponding potentials are given, and the thermodynamic limit interpreted in terms of a simple plasma picture. The new matrix models can be successfully applied to the full bosonization of interesting one-dimensional systems, including all the perturbative orders in the inverse size of the system. As a simple application, we present the all-order bosonization of the free fermionic field on the one-dimensional lattice. 
  We study supersymmetric pp-waves in M-theory, their dimensional reduction to D0-branes or pp-waves in type IIA, and their T-dualisation to solutions in the type IIB theory. The general class of pp-waves that we consider encompass the Penrose limits of AdS_p\times S^q with (p,q)=(4,7), (7,4), (3,3), (3,2), (2,3), (2,2), but includes also many other examples that can again lead to exactly-solvable massive strings, but which do not arise from Penrose limits. All the pp-waves in D=11 have 16 ``standard'' Killing spinors, but in certain cases one finds additional, or ``supernumerary,'' Killing spinors too. These give rise to linearly-realised supersymmetries in the string or matrix models. A focus of our investigation is on the circumstances when the Killing spinors are independent of particular coordinates (x^+ or transverse-space coordinates), since these will survive at the field-theory level in dimensional reduction or T-dualisation. 
  We suppose that there are both particles with negative energies described by L_{W} and particles with positive energies described by L_{F}, L_{W} and L_{F} are independent of each other before quantization, dependent on each other after quantization and symmetric, and L=L_{W} + L_{F}. From this we present a new quantization method for QED. That the energy of the vacuum state is equal to zero is naturally obtained. Thus we can easily determine the cosmological constant according to data of astronomical observation, and it is possible to correct nonperturbational methods which depend on the energy of the ground state in quantum field theory. 
  Dirichlet p-branes in the background of pp waves are constructed using the massive Green-Schwarz worldsheet action for open strings. These branes are localized at the origin and only for $p=7, 5, 3$ preserve half the supersymmetries. The spectrum of the brane theory is analyzed and is found to be in agreement with the spectrum of the small fluctuations of the world-volume super Yang-Mills theory in this background. These branes are expected to correspond to objects that are nonperturbative in $N$ in the dual gauge theory. 
  In earlier work, the planar diagrams of $SU(N_c)$ gauge theory have been regulated on the light-cone by a scheme involving both discrete $p^+$ and $\tau=ix^+$. The transverse coordinates remain continuous, but even so all diagrams are rendered finite by this procedure. In this scheme quartic interactions are represented as two cubics mediated by short lived fictitious particles whose detailed behavior could be adjusted to retain properties of the continuum theory, at least at one loop. Here we use this setup to calculate the one loop three gauge boson triangle diagram, and so complete the calculation of diagrams renormalizing the coupling to one loop. In particular, we find that the cubic vertex is correctly renormalized once the couplings to the fictitious particles are chosen to keep the gauge bosons massless. 
  We study the IR dynamics of the type IIB supergravity solution describing N D3-branes and M fractional D3-branes on the resolved conifold. The baryon mass and the tension of domain wall in the dual gauge theory are evaluated and compared with those for the deformed conifold. The IR behavior of the solution for the general conifold is also discussed. We show that the area law behavior of the Wilson loop is attributed to the existence of the locus in the IR where the D3-brane charge vanishes. 
  We have devoted an effort to study some nonlinear actions, characteristics of the ${\cal W}$-theories, in the framework of the soldering formalism. We have disclosed interesting new results concerning the embedding of the original chiral ${\cal W}$-particles in different metrical spaces in the final soldered action, i.e., the metric gets modified by the soldering interference process. The results are presented in a weak field approximation for the ${\cal W}_N$ case when N is greater than 3 and also in an exact way for ${\cal W}_2$. We have promoted a generalization of the interference phenomena to ${\cal W}_N$-theories of different chiralities and shown that the geometrical features introduced can yield a new understanding about the interference formalism in quantum field theories. 
  Using the de Sitter/CFT correspondence we describe a scenario of holographic inflation which is driven by a three dimensional boundary field theory. We find that inflationary constraints severely restrict the $\beta$--function, the anomalous dimensions and the value of the $C$--function of the boundary theory. The scenario has model independent predictions such as $\epsilon<< \eta$, $n_T<0.04$, $P_{tensor}/P_{scalar}<0.08$ and $H<10^{14} GeV$. We consider some simple boundary theories and find that they do not lead to inflation. Thus, building an acceptable holographic inflation model remains a challenge. We also describe holographic quintessence and find that it closely resembles a cosmological constant. 
  In gauge theories, not all rigid symmetries of the classical action can be maintained manifestly in the quantization procedure, even in the absence of anomalies. If this occurs for an anomaly-free symmetry, the effective action is invariant under a transformation that differs from its classical counterpart by quantum corrections. As shown by Fradkin and Palchik years ago, such a phenomenon occurs for conformal symmetry in quantum Yang-Mills theories with vanishing beta function, such as the N = 4 super Yang-Mills theory. More recently, Jevicki et al demonstrated that the quantum metamorphosis of conformal symmetry sheds light on the nature of the AdS/CFT correspondence. In this paper, we derive the conformal Ward identity for the bosonic sector of the N = 4 super Yang-Mills theory using the background field method. We then compute the leading quantum modification of the conformal transformation for a specific Abelian background which is of interest in the context of the AdS/CFT correspondence. In the case of scalar fields, our final result agrees with that of Jevicki et al. The resulting vector and scalar transformations coincide with those which are characteristic of a D3-brane embedded in AdS5 x S5. 
  We compute the instanton expansions of the holomorphic couplings in the effective action of certain $\cx N=1$ supersymmetric four-dimensional open string vacua. These include the superpotential $W(\phi)$, the gauge kinetic function $f(\phi)$ and a series of other holomorphic couplings which are known to be related to amplitudes of topological open strings at higher world-sheet topologies. The results are in full agreement with the interpretation of the holomorphic couplings as counting functions of BPS domain walls. Similar techniques are used to compute genus one partition function for the closed topological string on Calabi--Yau 4-fold which gives rise to a theory with the same number of supercharges in two dimensions. 
  We construct a gravitational-anomaly-free effective action for the coupled system of IIA D=10 dynamical supergravity interacting with an NS5-brane. The NS5-brane is considered as elementary in that the associated current is a delta-function supported on its worldvolume. Our approach is based on a Chern-kernel which encodes the singularities of the three-form field strength near the brane in an SO(4)-invariant way and provides a solution for its Bianchi identity in terms of a two-form potential. A dimensional reduction of the recently constructed anomaly-free effective action for an elementary M5-brane in D=11 is seen to reproduce our ten-dimensional action. The Chern-kernel approach provides in particular a concrete realization of the anomaly cancellation mechanism envisaged by Witten. 
  We consider world-volume description of M2-branes ending on an M5-brane. The system can be described either as a solitonic solution of the M5-brane field equations or in terms of an effective string propagating in 6-dimensions. We show that the zeroth order scalar scattering amplitudes behave similarly in both pictures. The soliton solution appears to have a horizon-like throat region. Due to the underlying geometric structure of the M5-brane theory, modes propagating near the horizon are subject to a large red-shift. This allows one to define a decoupling limit and implies a holographic duality between two theories which do not contain dynamical gravity. 
  An appropriate generalization of the unitary parasupersymmetry algebra of Beckers-Debergh to arbitrary order is presented in this paper. A special representation for realizing of the even arbitrary order unitary parasupersymmetry algebra of Beckers-Debergh is analyzed by one dimensional shape invariance solvable models, 2D and 3D quantum solvable models obtained from the shape invariance theory as well. In particular in the special representation, it is shown that the isospectrum Hamiltonians consist of the two partner Hamiltonians of the shape invariance theory. 
  Presently there is preliminary observational evidence that the cosmological constant might be non zero, and hence that our Universe is eternally accelerating (de Sitter). This poses fundamental problems for string theory, since a scattering matrix is not well defined in such Universes. In a previous paper we have presented a model, based on (non-equilibrium) non-critical strings, which is characterized by eventual ``graceful'' exit from a de Sitter phase. The model is based on a type-0 string theory, involving D3 brane worlds, whose initial quantum fluctuations induce the non criticality. We argue in this article that this model is compatible with the current observations. A crucial r\^ole for the correct ``phenomenology'' of the model is played by the relative magnitude of the flux of the five form of the type 0 string to the size of five of the extra dimensions, transverse to the direction of the flux-field. We do not claim, at this stage at least, that this model is a realistic physical model for the Universe, but we find it interesting that the model cannot be ruled out immediately, at least on phenomenological grounds. 
  We show that the Ocneanu algebra of quantum symmetries, for an ADE diagram (or for higher Coxeter-Dynkin systems, like the Di Francesco - Zuber system) is, in most cases, deduced from the structure of the modular T matrix in the A series. We recover in this way the (known) quantum symmetries of su(2) diagrams and illustrate our method by studying those associated with the three genuine exceptional diagrams of type su(3), namely E5, E9 and E21. This also provides the shortest way to the determination of twisted partition functions in boundary conformal field theory with defect lines. 
  We discuss the lift of certain D6-antiD6-brane systems to M-theory. These are purely gravitational configurations with a bolt singularity. When reduced along a trivial circle, and for large bolt radius, the bolt is related to a non-supersymmetric orbifold type of singularity where some closed string tachyons are expected in the twisted sectors. This is a kind of open-closed string duality that relates open string tachyons on one side and localised tachyons in the other. We consider the evolution of the system of branes from M-theory point of view. This evolution gives rise to a brane-antibrane annihilation on the brane side. On the gravity side, the evolution is related to a reduction of the order of the orbifold and to a contraction of the bolt to a nut or flat space if the system has non-vanishing or vanishing charge, respectively. We also consider the inverse process of reducing a non-supersymmetric orbifold to a D6-brane system. For $C^2/Z_N\times Z_M$, the reduced system is a fractional D6-brane at an orbifold singularity $C/Z_M$. 
  We examine the Cardy-Verlinde formula for finite temperature N=4 Super Yang-Mills theory on $R\times S^3$, and its AdS dual. We find that curvature effects introduce non-trivial corrections to thermodynamic quantities computed on both sides. We find a modified version of the Cardy-Verlinde formula for the SYM theory, incorporating these. On the gravity side, these corrections imply that the Cardy-Verlinde formula is exact. 
  The algebra of observables of an N-body Calogero model is represented on the S_N-symmetric subspace of the positive definite Fock space. We discuss some general properties of the algebra and construct four different realizations of the dynamical symmetry algebra of the Calogero model. Using the fact that the minimal algebra of observables is common to the Calogero model and the finite Chern-Simons (CS) matrix model, we extend our analysis to the CS matrix model. We point out the algebraic similarities and distinctions of these models. 
  We find that dilaton dominated supersymmetry breaking and spontaneous CP violation can be achieved in heterotic string models with superpotentials singular at the fixed points of the modular group. A semi--realistic picture of CP violation emerges in such models: the CKM phase appears due to a complex VEV of the T-modulus, while the soft supersymmetric CP phases are absent due to an axionic--type symmetry. 
  In this paper we present a classification of possible dynamics of closed string moduli within specific toroidal compactifications of Type II string theories due to the NS-NS tadpole terms in the reduced action. They appear as potential terms for the moduli when supersymmetry is broken due to the presence of D-branes. We particularise to specific constructions with two, four and six-dimensional tori, and study the stabilisation of the complex structure moduli at the disk level. We find that, depending on the cycle on the compact space where the brane is wrapped, there are three possible cases: i) there is a solution inside the complex structure moduli space, and the configuration is stable at the critical point, ii) the moduli fields are driven towards the boundary of the moduli space, iii) there is no stable solution at the minimum of the potential and the system decays into a set of branes. 
  A quantizable worldsheet action is constructed for the superstring in a supersymmetric plane wave background with Ramond-Ramond flux. The action is manifestly invariant under all isometries of the background and is an exact worldsheet conformal field theory. 
  We study the plane wave limit of $AdS_5\times S^5/Z_2$ which arises as the near horizon geometry of D3-branes at an orientifold 7-plane in type I' theory. We analyze string theory in the resulting plane wave background which contains open strings. We identify gauge invariant operators in the dual $Sp(N)$ gauge theory with unoriented closed and open string states. 
  Higher order renormalization in 4D quantum gravity is carried out using dimensional regularization with great care concerning the conformal-mode dependence. In this regularization, resummation can be automatically carried out without making an assumption like that of David, Distler and Kawai. In this paper we consider a model of 4D quantum gravity coupled to QED. Resummation inevitably implies a four-derivative quantum gravity. The renormalizability is directly checked up to $O(e_r^6)$ and $O(t_r^2)$, where $e_r$ and $t_r$ are the running coupling constants of QED and the traceless gravitational mode. There is no other running coupling constant in our model. The conformal mode is treated exactly, which means it is unrenormalized. It is found that Hathrell's results are included in our results. As a by-product, it is found that a higher-order gravitational correction to the beta function of QED is negative. An advantage of our model is that in the very high-energy regime, it closely resembles exactly solvable 2D quantum gravity. Thus, we can study physical states of 4D quantum gravity in this regime in parallel to those of 2D quantum gravity, which can be described with diffeomorphism invariant composite fields. 
  Covariant quantization of the Nambu-Goto spinning particle in 2+1-dimensions is studied. The model is relevant in the context of recent activities in non-commutative space-time. From a technical point of view also covariant quantization of the model poses an interesting problem: the set of second class constraints (in the Dirac classification scheme) is {\it reducible}. The reducibility problem is analyzed from two contrasting approaches: (i) the auxiliary variable method [bn] and (ii) the projection operator method [blm]. Finally in the former scheme, a Batalin-Tyutin quantization has been done. This induces a mapping between the non-commutative and the ordinary space-time. BRST quantization programme in the latter scheme has also been discussed. 
  Using a PT symmetric regularization technique we reminad the reader that and how (a) the SUSY is re-established between the two shifted harmonic oscillator potentials $ V(q)=q^2+{G}/{q^2}+ const$ and (b) many non-equivalent Hermitian and non-Hermitian (PT symmetric) systems may often be studied in parallel (with a schematic solvable illustrative example added). 
  In present paper we show that many properties of the baby skyrmions, which have been determined numerically, can be understood in terms of an analytic approximation. In particular, we show that this approximation captures properties of the multiskyrmion solutions (derived numerically) such as their stability towards decay into various channels, and that it is more accurate for the "new baby Skyrme model" which describes anisotropic physical systems in terms of multiskyrmion fields with axial symmetry. Some universal characteristics of configurations of this kind are demonstrated, which do not depend on their topological number. 
  Recently by us was proposed the model where Einstein's equation on the brane was connected with Maxwell's multi-dimensional equations in pseudo-Euclidean space. Based on this idea unification of 4-dimensional gravity and electromagnetism in (2+4)-space is found. In this picture photon is massless in four dimensions and obtains large mass in extra (1+1)-space normal to the brane. 
  The Killing spinor equations for pp-wave solutions of eleven dimensional supergravity are analysed and it is shown that there are solutions that preserve 18,20,22 and 24 supersymmetries, in addition to the generic solution preserving 16 supersymmetries and the Kowalski-Glikman solution preserving 32 supersymmetries. 
  We study the moduli space of M-theories compactified on G_2 manifolds which are asymptotic to a cone over quotients of S^3 x S^3. We show that the moduli space is composed of several components, each of which interpolates smoothly among various classical limits corresponding to low energy gauge theories with a given number of massless U(1) factors. Each component smoothly interpolates among supersymmetric gauge theories with different gauge groups. 
  We consider open strings ending on a D5-brane in the pp-wave background, which is realized in the Penrose limit of $AdS_5 \times S^5$ with an $AdS_4\times S^2$ brane. A complete list of gauge invariant operators in the defect conformal field theory is constructed which is dual to the open string states. 
  The Euclidean Schwarzschild-de Sitter geometry may be considered as an extremum of two different action principles. If the thermodynamical parameters are held fixed at the cosmological horizon, one deals with the gravitational thermodynamical effects of the black hole but ignores those of the cosmological horizon. Conversely, if the macroscopical variables are held fixed at the black hole horizon, it is only the cosmological horizon thermodynamics which is dealt with. Both cases are analyzed. In particular, the internal energy U is calculated in the semiclassical approximation as a function of the mass parameter m of Schwarzschild de Sitter space. In the first case one finds U=+m, while in the second one gets U=-m. This suggests that de Sitter space is thermodynamically unstable under black hole formation. 
  We review certain emergent notions on the nature of spacetime from noncommutative geometry and their radical implications. These ideas of spacetime are suggested from developments in fuzzy physics, string theory, and deformation quantisation. The review focuses on the ideas coming from fuzzy physics. We find models of quantum spacetime like fuzzy $S^4$ on which states cannot be localised, but which fluctuate into other manifolds like $ CP^3$ . New uncertainty principles concerning such lack of localisability on quantum spacetimes are formulated.Such investigations show the possibility of formulating and answering questions like the probabilty of finding a point of a quantum manifold in a state localised on another one. Additional striking possibilities indicated by these developments is the (generic) failure of $CPT$ theorem and the conventional spin-statistics connection. They even suggest that Planck's `` constant '' may not be a constant, but an operator which does not commute with all observables. All these novel possibilities arise within the rules of conventional quantum physics,and with no serious input from gravity physics. 
  The possibility of detecting noncommutative space relics is analyzed using the Aharonov-Bohm effect. We show that, if space is noncommutative, the holonomy receives non-trivial kinematical corrections that will produce a diffraction pattern even when the magnetic flux is quantized. The scattering problem is also formulated, and the differential cross section is calculated. Our results can be extrapolated to high energy physics and the bound $\theta \sim [ 10 {TeV}]^{-2}$ is found. If this bound holds, then noncommutative effects could be explored in scattering experiments measuring differential cross sections for small angles. The bound state Aharonov- Bohm effect is also discussed. 
  We compare a momentum space implicit regularisation (IR) framework with other renormalisation methods which may be applied to dimension specific theories, namely Differential Renormalisation (DfR) and the BPHZ formalism. In particular, we define what is meant by minimal subtraction in IR in connection with DfR and dimensional renormalisation (DR) .We illustrate with the calculation of the gluon self energy a procedure by which a constrained version of IR automatically ensures gauge invariance at one loop level and handles infrared divergences in a straightforward fashion. Moreover, using the $\phi^4_4$ theory setting sun diagram as an example and comparing explicitly with the BPHZ framework, we show that IR directly displays the finite part of the amplitudes. We then construct a parametrization for the ambiguity in separating the infinite and finite parts whose parameter serves as renormalisation group scale for the Callan-Symanzik equation. Finally we argue that constrained IR, constrained DfR and dimensional reduction are equivalent within one loop order. 
  We here use our non-perturbative, cluster decomposable relativistic scattering formalism to calculate photon-spinor scattering, including the related particle-antiparticle annihilation amplitude. We start from a three-body system in which the unitary pair interactions contain the kinematic possibility of single quantum exchange and the symmetry properties needed to identify and substitute antiparticles for particles. We extract from it unitary two-particle amplitude for quantum-particle scattering. We verify that we have done this correctly by showing that our calculated photon-spinor amplitude reduces in the weak coupling limit to the usual lowest order, manifestly covariant (QED) result with the correct normalization. That we are able to successfully do this directly demonstrates that renormalizability need not be a fundamental requirement for all physically viable models. 
  In modern physics, one of the greatest divides is that between space-time and quantum fields, as the fiber bundle of the Standard Model indicates. However on the operational grounds the fields and space-time are not very different. To describe a field in an experimental region we have to assign coordinates to the points of that region in order to speak of "when" and "where" of the field itself. But to operationally study the topology and to coordinatize the region of space-time, the use of radars (to send and receive electromagnetic signals) is required. Thus the description of fields (or, rather, processes) and the description of space-time are indistinguishable at the fundamental level. Moreover, classical general relativity already says -- albeit preserving the fiber bundle structure -- that space-time and matter are intimately related. All this indicates that a new theory of elementary processes (out of which all the usual processes of creation, annihilation and propagation, and consequently the topology of space-time itself would be constructed) has to be devised.   In this thesis I present the foundations of such a finite, discrete, algebraic, quantum theory and apply it to the description of spin-1/2 quanta of the Standard Model. 
  Following recent work on the quantum Hall effect on $S^4$, we solve the Landau problem on the complex projective spaces ${\bf C}P^k$ and discuss quantum Hall states for such spaces. Unlike the case of $S^4$, a finite spatial density can be obtained with a finite number of internal states for each particle. We treat the case of ${\bf C}P^2$ in some detail considering both Abelian and nonabelian background fields. The wavefunctions are obtained and incompressibility of the Hall states is shown. The case of ${\bf C}P^3$ is related to the case of $S^4$. 
  It is shown that classical decay of unstable D-branes in bosonic and superstring theories produces pressureless gas with non-zero energy density. The energy density is stored in the open string fields, even though around the minimum of the tachyon potential there are no open string degrees of freedom. We also give a description of this phenomenon in an effective field theory. 
  Starting from the geometrical construction of special Lagrangian submanifolds of a toric variety, we identify a certain subclass of A-type D-branes in the linear sigma model for a Calabi-Yau manifold and its mirror with the A- and B-type Recknagel-Schomerus boundary states of the Gepner model, by reproducing topological properties such as their labeling, intersection, and the relationships that exist in the homology lattice of the D-branes. In the non-linear sigma model phase these special Lagrangians reproduce an old construction of 3-cycles relevant for computing periods of the Calabi-Yau, and provide insight into other results in the literature on special Lagrangian submanifolds on compact Calabi-Yau manifolds. The geometrical construction of rational boundary states suggests several ways in which new Gepner model boundary states may be constructed. 
  Looking for string vacua with fixed moduli, we study compactifications of type IIA string theory on Calabi-Yau fourfolds in the presence of generic Ramond-Ramond fields. We explicitly derive the (super)potential induced by Ramond-Ramond fluxes performing a Kaluza-Klein reduction of the ten-dimensional effective action. This can be conveniently achieved in a formulation of the massive type IIA supergravity where all Ramond-Ramond fields appear in a democratic way. The result agrees with the general formula for the superpotential written in terms of calibrations. We further notice that for generic Ramond-Ramond fluxes all geometric moduli are stabilized and one finds non-supersymmetric vacua at positive values of the scalar potential. 
  We show that the spectrum of conical defects in three-dimensional de Sitter space is in one-to-one correspondence with the spectrum of vertex operators in Liouville conformal field theory. The classical conformal dimensions of vertex operators are equal to the masses of the classical point particles in dS_3 that cause the conical defect. The quantum dimensions instead are shown to coincide with the mass of the Kerr-dS_3 solution computed with the Brown-York stress tensor. Therefore classical de Sitter gravity encodes the quantum properties of Liouville theory. The equality of the gravitational and the Liouville stress tensor provides a further check of this correspondence. The Seiberg bound for vertex operators translates on the bulk side into an upper mass bound for classical point particles. Bulk solutions with cosmological event horizons correspond to microscopic Liouville states, whereas those without horizons correspond to macroscopic (normalizable) states. We also comment on recent criticism by Dyson, Lindesay and Susskind, and point out that the contradictions found by these authors may be resolved if the dual CFT is not able to capture the thermal nature of de Sitter space. Indeed we find that on the CFT side, de Sitter entropy is merely Liouville momentum, and thus has no statistical interpretation in this approach. 
  In this paper the interaction of extended waves in a noncommutative modified 2+1 dimensional U(2) sigma model are studied. Using the dressing method, we construct an explicit two-wave solution of the noncommutative field equation. The scattering of these waves and large time factorization are discussed. 
  We study the Seiberg-Witten curves for N=2 SUSY gauge theories arising from type IIA string configurations with two orientifold sixplanes. Such theories lift to elliptic models in M-theory. We express the M-theory background for these models as a nontrivial elliptic fibration over C. We discuss singularities of this surface, and write the Seiberg-Witten curve for several theories as a subvariety of this surface. 
  We discuss the impact of a bulk photon mass in a Dvali-Gabadadze-Porrati type brane model with Maxwell terms both on the brane and in the bulk, as proposed by Dvali, Gabadadze and Shifman. The motivation to include the bulk photon mass is to suppress radiation loss into the bulk. We point out that this modifies the photon propagator in such a way that it generates a small photon mass on the brane. Compatibility with present bounds on a photon mass imply that the transition to five-dimensional distance laws for the electromagnetic potentials would appear only at super-horizon length scales, thus excluding any direct detection possibility of a transition from four-dimensional to five-dimensional distance laws in electromagnetic interactions. We also include results on fermion propagators with Dirac terms on the brane and in the bulk. 
  We investigate strings at singularities of G_2-holonomy manifolds which arise in Z_2 orbifolds of Calabi-Yau spaces times a circle. The singularities locally look like R^4/Z_2 fibered over a SLAG, and can globally be embedded in CICYs in weighted projective spaces. The local model depends on the choice of a discrete torsion in the fibration, and the global model on an anti-holomorphic involution of the Calabi-Yau hypersurface. We determine how these choices are related to each other by computing a Wilson surface detecting discrete torsion. We then follow the same orbifolds to the non-geometric Landau-Ginzburg region of moduli space. We argue that the symmetry-breaking twisted sectors are effectively captured by real Landau-Ginzburg potentials. In particular, we find agreement in the low-energy spectra of strings computed from geometry and Gepner-model CFT. Along the way, we construct the full modular data of orbifolds of N=2 minimal models by the mirror automorphism, and give a real-LG interpretation of their modular invariants. Some of the models provide examples of the mirror-symmetry phenomenon for G_2 holonomy. 
  In three dimensions, there are two distinct mass-generating mechanisms for gauge fields: adding the usual Proca/Pauli-Fierz, or the more esoteric Chern-Simons (CS), terms. Here we analyze the three-term models where both types are present, and their various limits. Surprisingly, in the tensor case, these seemingly innocuous systems are physically unacceptable. If the sign of the Einstein term is ``wrong'' as is in fact required in the CS case, then the excitation masses are always complex; with the usual sign, there is a (known) region of the two mass parameters where reality is restored, but instead we show that a ghost problem arises, while, for the ``pure mass'' two-term system without an Einstein action, complex masses are unavoidable. This contrasts with the smooth behavior of the corresponding vector models. Separately, we show that the ``partial masslessness'' exhibited by (plain) massive spin-2 models in de Sitter backgrounds is formally shared by the three-term system: it also enjoys a reduced local gauge invariance when this mass parameter is tuned to the cosmological constant. 
  The Friedmann law on the brane generically depends quadratically on the brane energy density and involves a ``dark radiation'' term due to the bulk Weyl tensor. Despite its unfamiliar form, we show how it can be derived from a standard four-dimensional Brans-Dicke theory at low energy. In particular, the dark radiation term is found to depend linearly on the brane energy densities. For any equation of state on the branes, the radion evolves such as to generate radiation-dominated cosmology. The radiation-dominated era is conventional and consistent with nucleosynthesis. 
  If a Lagrangian of gauge theory of internal symmetries is not gauge-invariant, the energy-momentum fails to be conserved in general. 
  Superconformal tensor calculus on an orbifold S^1/Z_2 is given in five-dimensional (5D) spacetime. The four-dimensional superconformal Weyl multiplet and various matter multiplets are induced on the boundary planes from the 5D supermultiplets in the bulk. We identify those induced 4D supermultiplets and clarify a general method for coupling the bulk fields to the matter fields on the boundaries in a superconformal invariant manner. 
  We study a matrix model describing type IIB superstring in orbifold backgrounds. We particularly consider a {\bf C}^3/{\bf Z}_3 orbifold model whose six dimensional transverse space is orbifolded by {\bf Z}_3 discrete symmetry. This model is chiral and has d=4 {\cal N}=1 supersymmetry of Yang-Mills type as well as an inhomogeneous supersymmetry specific to matrix models. We calculate one-loop effective action around some backgrounds, and the result can be interpreted as interactions mediated by massless particles in IIB supergravity in orbifold background, if the background is in the Higgs branch. If the background is in the Coulomb branch, the dynamics is governed by the reduced model of d=4 super Yang-Mills theory, which might be interpreted as exchange of massless particles in the twisted sector. But the perturbative calculation cannot reproduce the supergravity result. We also show that this model with a large Higgs vacuum expectation value becomes IIB (IKKT) matrix model. 
  The non-standard intersection of two 5-branes and a string can give rise to AdS_3\times S^3\times S^3\times S^1. We consider the Penrose limit of this geometry and study the supersymmetry of the resulting pp-wave solution. There is a one-parameter family of Penrose limits associated with the orthogonal rotation of the two foliating circles within the two 3-spheres. Supernumerary Killing spinors arise only when the rotation angle is 45 degrees, for which case we obtain the corresponding light-cone string action that has linearly-realised supersymmetry. We also obtain Penrose limits of other non-standard intersections that give rise to the product of AdS_3 or AdS_2 and two spheres. The resulting pp-waves are supported by multiple constant field strengths. 
  Maximally supersymmetric spacetime algebras in eleven-dimensions, which are the isometry superalgebras of Minkowski space, AdS_7 x S^4, AdS_4 x S^7 and pp-wave background, are related by Inonu-Wigner contractions. The super-AdS_{4(7)} x S^{7(4)} algebras allow to introduce two contraction parameters, the one for the flat limit to the super-Poincare algebra and the other for a Penrose limit to the super-pp-wave algebra. Under these contractions supersymmetries are maintained because the Jacobi identity of three supercharges holds for any values of contraction parameters. 
  An analysis of the Schwinger's action principle in Lagrangian quantum field theory is presented. A solution of a problem contained in it is proposed via a suitable definition of a derivative with respect to operator variables. This results in a preservation of Euler-Lagrange equations and a change in the operator structure of conserved quantities. Besides, it entails certain relation between the field operators and their variations (which is identically valid for some fields, e.g. for the free ones). The general theory is illustrated on a number of particular examples 
  Homogeneous gravitational wave backgrounds arise as infinite momentum limits of many geometries with a well-understood holographic description. General global aspects of these geometries are discussed. Using exact CFT techniques, strings in pp-wave backgrounds supported by a Neveu-Schwarz flux are quantized. As in Euclidean $AdS_3$, spectral flow and associated long strings are shown to be crucial in obtaining a complete spectrum. Holography is investigated using conformally flat coordinates analogous to those of the Poincar\'e patch in AdS. It is argued that the holographic direction is the light-cone coordinate $u$, and that the holographic degrees of freedom live on a codimension-one screen at fixed $u$. The usual conformal symmetry on the boundary is replaced by a representation of a Heisenberg-type algebra $H_D\times H_D$, hinting at a new class of field theories realizing this symmetry. A sample holographic computation of 2 and 3-point functions is provided and Ward identities are derived. A complementary screen at fixed $v$ is argued to be necessary in order to encode the vacuum structure. 
  A very simple criterion to ascertain if (D-2)-surfaces are trapped in arbitrary D-dimensional Lorentzian manifolds is given. The result is purely geometric, independent of the particular gravitational theory, of any field equations or of any other conditions. Many physical applications arise, a few shown here: a definition of general horizon, which reduces to the standard one in black holes/rings and other known cases; the classification of solutions with a (D-2)-dimensional abelian group of motions and the invariance of the trapping under simple dimensional reductions of the Kaluza-Klein/string/M-theory type. Finally, a stronger result involving closed trapped surfaces is presented. It provides in particular a simple sufficient condition for their absence. 
  This comment directs attention to some fails of the Alhaidari approach to solve relativistic problems. It is shown that his gauge considerations are way off the mark and that the class of exactly solvable relativistic problems is not so enlarged as Alhaidari thinks it is. 
  New representation for the generating function of correlators of third components of spins in the XX Heisenberg spin chain is considered in the form given by the fermionic Gaussian path integrals. A part of the discrete anti-commuting integration variables is subjected to ``automorphic'' boundary conditions in respect of imaginary time. The situation when only a part of the integration variables is subjected to the unusual boundary conditions generalizes more conventional ones when ``automorphic'' boundary conditions appear for all sites in the lattice spin models. The results of the functional integration are expressed as determinants of the matrix operators. The generating function, as well as the partition function of the model, are calculated by means of zeta-regularization. Certain correlation functions at nonzero temperature are obtained explicitly. 
  The cosmological effects of the tachyon rolling down to its ground state are discussed by coupling a simple effective field theory for the tachyon field to Einstein gravity. As the tachyon rolls down to the minimum of its potential the universe expands. Depending upon initial conditions, the scale factor may or may not start off accelerating, but ultimately it ceases to do so and the final flat spacetime is either static in the rest frame of the tachyon (if $k=0$) or (if $k=-1$) given by the Milne model. 
  Two-dimensional hybrid superstring on singular Calabi-Yau manifolds is studied by the field redefinition of the NSR formalism. The compactification on singular Calabi-Yau fourfold is described by N=2 super-Liouville theory and N=2 Landau-Ginzburg models. We examine the world-sheet topological N=4 superconformal algebra, which is useful to identify physical states of the theory. It is shown that the space-time superconformal symmetry is not compatible with this original topological algebra. A new model is proposed by modifying the topological superconformal generators, which is consistent with space-time N=2 superconformal symmetry. 
  It is shown that the evaluation of the expectation value (EV) of topological charge density over $\theta$-vacuum is reduced to investigation of the Chern-Simons term EV. An equation for this quantity is established and solved. EV of the topological charge density at an arbitrary $\theta$ occurs equal to zero and, as a consequence, topological susceptibility of both QCD and pure Yang-Mills vacua defined in a Wick sense is equal to zero, whereas when defined in a Dyson sense it differs from zero by the quantity proportional to the respective condensate of the chromomagnetic field. Thus, the usual Witten-Veneziano formula for the $\eta^{'}$ meson mass is modified. 
  We revisit classical "on shell" duality, i.e., pseudoduality, in two dimensional conformally invariant classical sigma models and find some new interesting results. We show that any two sigma models that are "on shell" duals have opposite 1-loop renormalization group beta functions because of the integrability conditions for the pseudoduality transformation. A new result states for any two compact Lie groups of the same dimension there is a natural pseudoduality transformation that maps classical solutions of the WZW model on the first group into solutions of the WZW model on the second group. This transformation preserves the stress-energy tensor. The two groups can be non-isomorphic such as B_n and C_n in the Cartan notation. This transformation can be used for a new construction of non-local conserved currents. The new non-local currents on G depend on the choice of dual group G-tilde. 
  We propose a description of open string fields on a D25-brane in vacuum string field theory. We show that the tachyon mass is correctly reproduced from our proposal and further argue that the mass spectrum of all other open string states is correctly obtained as well. We identify the string coupling constant from the three-tachyon coupling and show that the tension of a D25-brane is correctly expressed in terms of the coupling constant, which resolves the controversy in the literature. We also discuss a reformulation of our description which is rather similar to boundary string field theory. 
  We consider N-point deformation of algebraic K3 surfaces. First, we construct two-point deformation of algebraic K3 surfaces by considering algebraic deformation of a pair of commutative algebraic K3 surfaces. In this case, the moduli space of the noncommutative deformations is of dimension 19, the same as the moduli dimension of the complex deformations of commutative algebraic K3 surfaces. Then, we extend this method to the N-point case. In the N-point case, the dimension of deformation moduli space becomes 19N(N-1)/2. 
  Global geometric properties of dS space are presented explicitly in various coordinates. A Robertson-Walker like metric is deduced, which is convenient to be used in study of dynamics in dS space. Singularities of wavefunctions of massive scalar fields at boundary are demonstrated. A bulk-boundary propagator is constructed by making use of the solutions of equations of motion. The dS/CFT correspondence and the Strominger's mass bound is shown. 
  We discuss some of the key topological aspects of a two $(1+1)$-dimensional (2D) self-interacting non-Abelian gauge theory (having no interaction with matter fields) in the framework of {\it chiral} superfield formalism. We provide the geometrical interpretation for the Lagrangian density, symmetric energy momentum tensor, topological invariants, etc., by exploiting the {\it on-shell} nilpotent BRST- and co-BRST symmetries that emerge after the application of (dual) horizontality conditions. We show that the above physically interesting quantities geometrically correspond to the translation of some local (but composite) {\it chiral} superfields along one of the two independent Grassmannian directions of a four ($2+2)$-dimensional supermanifold. This translation is generated by the conserved and on-shell nilpotent (co-)BRST charges that are present in the theory. 
  This set of lectures contain a brief review of some basic supersymmetry and its representations, with emphasis on superspace and superfields. Starting from the Poincar\'e group, the supersymmetric extensions allowed by the Coleman-Mandula theorem and its generalisation to superalgebras, the Haag, Lopuszanski and Sohnius theorem, are discussed. Minkowski space is introduced as a quotient space and Superspace is presented as a direct generalization of this. The focus is then shifted from a general presentation to the relation between supersymmetry and complex geometry as manifested in the possible target space geometries for N=1 and N=2 supersymmetric nonlinear sigma models in four dimensions. Gauging of isometries in nonlinear sigma models is discussed for these cases, and the quotient construction is described. 
  We briefly review three aspects of string cosmology: (1) the ``stochastic'' approach to the pre-big bang scenario, (2) the presence of chaos in the generic cosmological solutions of the tree-level low-energy effective actions coming out of string theory, and (3) the remarkable link between the latter chaos and the Weyl groups of some hyperbolic Kac-Moody algebras. Talk given at the Francqui Colloquium ``Strings and Gravity: Tying the Forces Together'' (Brussels, October 2001). 
  Some applications of the odd Poisson bracket developed by Kharkov's theorists are represented, including the reformulation of classical Hamiltonian dynamics, the description of hydrodynamics as a Hamilton system by means of the odd bracket and the dynamics formulation with the Grassmann-odd Lagrangian. Quantum representations of the odd bracket are also constructed and applied for the quantization of classical systems based on the odd bracket and for the realization of the idea of a composite spinor structure of space-time. At last, the linear odd bracket, corresponding to a semi-simple Lie group, is introduced on the Grassmann algebra. 
  It is shown that a Dirac bracket algebra is isomorphic to the original Poisson bracket algebra of first class functions subject to first class constraints. The isomorphic image of the Dirac bracket algebra in the star-product commutator algebra is found. 
  A computation of the Casimir effect for a real scalar field in four situations: on a segment of a line, on a circle and on both standard commutative and noncommutative two-spheres is given in this paper. The main aim of this paper is to discuss the Casimir energy on the noncommutative sphere within the theory with commutative time. The comparison with the noncommutative cylinder is also done. 
  The effect of the spontaneous nucleation of black holes in de Sitter space is reviewed, and the main steps of the calculation in Nucl.Phys. B 582, 313, 2000 of the one-loop amplitude of this process are summarized. The existence of such an effect suggests that de Sitter space is not a ground state of quantum gravity with a positive cosmological constant. 
  In this comment we point out numerous errors in the paper of Alhaidari cited in the title. 
  We study the N=1 Sp(2N+2M)xSp(2N) cascading gauge theory on a stack of N physical and M fractional (half) D3-branes at the singularity of an orientifolded conifold. In addition to the D3-branes and an O7-plane, the background contains eight D7-branes, which give rise to matter in the fundamental representation of the gauge group. The moduli space of the gauge theory is analyzed and its structure is related to the brane configurations in the dual type IIB theory and in type IIA/M-theory. 
  By considering the Higgs mechanism in the framework of a parity-preserving Planar Quantum Electrodynamics, one shows that an attractive electron-electron interaction may come out. The e-e interaction potential emerges as the non-relativistic limit of the Moller scattering amplitude and it may result attractive with a suitable choice of parameters. Numerical values of the e-e binding energy are obtained by solving the two-dimensional Schrodinger equation. The existence of bound states is to be viewed as an indicative that this model may be adopted to address the pairing mechanism in some systems endowed with parity-preservation. 
  We present classical solutions for a D5 and NS5-branes in a pp-wave background. The worldvolume coordinates for these branes lie along a six dimensional pp-wave configuration obtained from $AdS_3 \times S_3$ in a Penrose limit. One in addition has nontrivial R-R 3-form field strength as well as a dilaton background. Classical solutions for Dp-branes (p> 5), as well as their bound states, are then generated in both IIA and IIB string theories by using T-duality symmetries. We also briefly discuss the construction of Dp-branes in pp-wave backgrounds, obtained from string theories on $AdS_m\times S^m$. Finally, the construction of the above D5-brane solution is presented from the point of view of the Green-Schwarz worldsheet string theory and its low lying spectrum is discussed. 
  We consider aspects of holography in the $pp$-wave limit of $AdS_5\times S5$. This geometry contains two $\RR4$'s, one obtained from $AdS_5$ directions, and the other from the $S 5$. We argue that the holographic direction in the $pp$-wave background can be taken to be $r$, the radial direction in the first $\RR4 $. Normalizable modes correspond to states, and non-normalizable modes correspond to deformations of the boundary theory. In the strict $pp$-wave limit, there are additional non-normalizable modes in the second $\RR 4$, which have no apparent super-Yang-Mills interpretation. We outline the procedure for calculating correlation functions holographically. 
  This is my contribution to Stephen Hawking's 60th birthday party. Happy Birthday Stephen! 
  This paper describes a light-cone quantization of a two-dimensional massive scalar field without periodic boundary conditions in order to make the quantization manifestly consistent to causality. For this purpose, the field is decomposed by the Legendre polynomials. Creation-annihilation operators for this field are defined and the Fock space was constructed. 
  A class of black objects which are solutions of pure gravity with negative cosmological constant are classified through the mapping between the Killing spinors of the ground state and those of the transverse section. It is shown that these geometries must have transverse sections of constant curvature for spacetime dimensions d below seven. For d > 6, the transverse sections can also be Euclidean Einstein manifolds. In even dimensions, spacetimes with transverse section of nonconstant curvature exist only in d = 8 and 10. This classification goes beyond standard supergravity and the eleven dimensional case is analyzed. It is shown that if the transverse section has negative scalar curvature, only extended objects can have a supersymmetric ground state. In that case, some solutions are explicitly found whose ground state resembles a wormhole. 
  We discuss the relation between bulk de Sitter three-dimensional spacetime and the corresponding conformal field theory at the boundary, in the framework of the exact quasinormal mode spectrum. We show that the quasinormal mode spectrum corresponds exactly to the spectrum of thermal excitations of Conformal Field Theory at the past boundary I^-, together with the spectrum of the Conformal Field Theory at the future boundary I^+. 
  We study gauge transformations of the Hata-Kawano vector state on a D25-brane in the framework of vacuum string field theory. We show that among the infinite number of components of the polarization vector, all the components except one spacetime vector degree of freedom are gauge freedom, and give string field gauge transformations reproducing gauge transformations of the constant modes of the U(1) gauge field. These gauge transformations can be used to fix the normalization of the vector field. We also discuss a difficulty in obtaining a gauge invariant action of the vector field. Our arguments rely on the factorization ansatz of gauge transformations. 
  The scalar theory is ultraviolet (UV) quadratically divergent on ordinary spacetime. On noncommutative (NC) spacetime, this divergence will generally induce pole-like infrared (IR) singularities in external momenta through the UV/IR mixing. In spontaneous symmetry breaking theory this would invalidate the Goldstone theorem which is the basis for mass generation when symmetry is gauged. We examine this issue at two loop level in the U(N) linear sigma model which is known to be free of such IR singularities in the Goldstone self-energies at one loop. We analyze the structures in the NC parameter (\theta_{\mu\nu}) dependence in two loop integrands of Goldstone self-energies. We find that their coefficients are effectively once subtracted at the external momentum p=0 due to symmetry relations between 1PI and tadpole contributions, leaving a final result proportional to a quadratic form in p. We then compute the leading IR terms induced by NC to be of order p^2\ln(\theta_{\mu\nu})^2 and p^2\ln\tilde{p}^2 (\tilde{p}_{\mu}=\theta_{\mu\nu}p^{\nu}) which are much milder than naively expected without considering the above cancellation. The Goldstone bosons thus keep massless and the theorem holds true at this level. However, the limit of \theta\to 0 cannot be smooth any longer as it is in the one loop Goldstone self-energies, and this nonsmooth behaviour is not necessarily associated with the IR limit of the external momentum as we see in the term of p^2\ln(\theta_{\mu\nu})^2. 
  We consider the matrix model associated with pp-wave background and construct supersymmetric branes. In addition to the spherical membrane preserving 16 supersymmetries, one may construct rotating elliptic membranes preserving 8 supersymmetries. The other branch describes rotating 1/8 BPS hyperbolic branes in general. When the angular momentum vanishes in this branch, the hyperbolic brane becomes 1/4 BPS preserving 8 real supersymmetries. It may have the shape of hyperboloid of one or two sheets embedded in the flat three space. We study the spectrum of the worldvolume fields on the hyperbolic branes and show that there are no massless degrees. We also compute the spectrum of the 0-2 strings. 
  By fully exploiting the existence of the unitarily inequivalent representations of quantum fields, we exhibit the entanglement between inner and outer particles, with respect to the event horizon of a black hole. We compute the entanglement entropy and we find that the nonunitarity of the mapping, between the vacua in the flat and the curved frames, makes the entanglement very robust. 
  The text is an essentially self-contained introduction to four-dimensional N=1 supergravity, including its couplings to super Yang-Mills and chiral matter multiplets, for readers with basic knowledge of standard gauge theories and general relativity. Emphasis is put on showing how supergravity fits in the general framework of gauge theories and how it can be derived from a tensor calculus for gauge theories of a standard form. Contents: 1. Introduction, 2. Gauge symmetries in the jet space approach, 3. D=4, N=1 pure supergravity, 4. Tensor calculus for standard gauge theories, 5. Off-shell formulations of D=4, N=1 supergravity with matter, A. Lorentz algebra, spinors, Grassmann parity, B. Explicit verification of local supersymmetry. 
  In the light-cone gauge choice for Abelian and non-Abelian gauge fields, the vector boson propagator carries in it an additional ``spurious'' or ``unphysical'' pole intrinsic to the choice requiring a careful mathematical treatment. Research in this field over the years has shown us that mathematical consistency only is not enough to guarantee physically meaningful results. Whatever the prescription invoked to handle such an object, it has to preserve causality in the process. On the other hand the covariantization technique is a well suited one to tackle gauge dependent poles in the Feynman integrals, dispensing the use of {\em ad hoc} prescriptions. In this work we show that the covariantization technique in the light-cone gauge is a direct consequence of the canonical quantization of the theory. 
  We study one-loop corrections in scalar and gauge field theories on the non-commutative torus. For rational theta, Morita equivalence allows these theories to be reformulated in terms of ordinary theories on a commutative torus with twisted boundary conditions. UV/IR mixing does not lead to singularities, however there can be large corrections. In particular, gauge theories show tachyonic instabilities for some of the modes. We discuss their relevance to spontaneous Z_N x Z_N symmetry breaking in the Morita dual SU(N) theory due to electric flux condensation. 
  We consider the contribution to the entropy from fields in the background of a curved time-independent metric. To account for the curvature of space, we postulate a position-dependent UV cutoff. We argue that a UV cutoff on energy naturally implies an IR cutoff on distance. With this procedure, we calculate the scalar contribution in a background anti-de Sitter space, the exterior of a black hole, and de Sitter space. In all cases, we find results that can be simply interpreted in terms of local energy and proper volume, yielding insight into the apparent reduced dimensionality of systems with gravity. 
  Calogero-Moser systems are classical and quantum integrable multi-particle dynamics defined for any root system $\Delta$. The {\em quantum} Calogero systems having $1/q^2$ potential and a confining $q^2$ potential and the Sutherland systems with $1/\sin^2q$ potentials have "integer" energy spectra characterised by the root system $\Delta$. Various quantities of the corresponding {\em classical} systems, {\em e.g.} minimum energy, frequencies of small oscillations, the eigenvalues of the classical Lax pair matrices, etc. at the equilibrium point of the potential are investigated analytically as well as numerically for all root systems. To our surprise, most of these classical data are also "integers", or they appear to be "quantised". To be more precise, these quantities are polynomials of the coupling constant(s) with integer coefficients. The close relationship between quantum and classical integrability in Calogero-Moser systems deserves fuller analytical treatment, which would lead to better understanding of these systems and of integrable systems in general. 
  In the usual d dimensional SO(d) gauged Higgs models with $d$-component Higgs fields, the 'energies' of the topologically stable solitons are bounded from below by the Chern-Pontryagin charges. A new class of Higgs models is proposed here, whose 'energies' are stabilised instead by the winding number of the Higgs field itself, with no reference to the gauge group. Consequently, such Higgs models can be gauged by SO(N), with 2 \le N \le d. 
  We analyze the Scherk-Schwarz (SS) supersymmetry breaking in brane-world five dimensional theories compactified on the orbifold $S^1/\mathbb{Z}_2$. The SS breaking parameter is undetermined at the tree-level (no-scale supergravity) and can be interpreted as the Hosotani vacuum expectation value corresponding to the $U(1)_R$ group in five dimensional N=2 (ungauged) supergravity. We show that the SS breaking parameter is fixed at the loop level to either 0 or 1/2 depending on the matter content propagating in the bulk but in a rather model-independent way. Supersymmetry breaking is therefore fixed through a radiative Scherk-Schwarz mechanism. We also show that the two discrete values of the SS parameter, as well as the supersymmetry breaking shift in the spectrum of the bulk fields, are altered in the presence of a brane-localized supersymmetry breaking arising from some hidden sector dynamics. The interplay between the SS and the brane localized breaking is studied in detail. 
  The computation of long range linear self-interaction forces in string and higher dimensional brane models requires the evaluation of the gradients of regularised values of divergent self-interaction potentials. It is shown that the appropriately regularised gradient in directions orthogonal to the brane surface will always be obtainable simply by multiplying the regularised potential components by just half the trace of the second fundamental tensor, except in the hypermembrane case for which the method fails. Whatever the dimension of the background this result is valid provided the codimension is two (the hyperstring case) or more, so it can be used for investigating brane-world scenarios with more than one extra space dimension. 
  In this thesis exactly marginal deformations of field theories living on D3-branes at low energies are studied. These theories include   N=4 supersymmetric Yang-Mills theory and theories obtained from it via the orbifolding procedure. 
  The S-matrix for a spin 1/2 particle in the presence of a potential which is the sum of the Coulomb potential V_c=-A_1/r and a Lorentz scalar potential V_s= -A_2/r is calculated 
  In this paper we study exactly marginal deformations of field theories living on D3-branes at low energies. These theories include N=4 supersymmetric Yang-Mills theory and theories obtained from it via the orbifolding procedure. We restrict ourselves only to orbifolds and deformations which leave some supersymmetry unbroken. A number of new families of N=1 superconformal field theories are found. We analyze the deformations perturbatively, and also by using general arguments for the dimension of the space of exactly marginal deformations. We find some cases where the space of perturbative exactly marginal deformations is smaller than the prediction of the general analysis at least up to three-loop order), and other cases where the perturbative result (at low orders) has a non-generic form. 
  An optimized Rayleigh-Schr\"{o}dinger expansion scheme of solving the functional Schr\"odinger equation with an external source is proposed to calculate the effective potential beyond the Gaussian approximation. For a scalar field theory whose potential function has a Fourier representation in a sense of tempered distributions, we obtain the effective potential up to the second order, and show that the first-order result is just the Gaussian effective potential. Its application to the $\lambda\phi^4$ field theory yields the same post-Gaussian effective potential as obtained in the functional integral formalism. 
  We take steps toward constructing explicit solutions that describe non-extremal charged dilatonic branes of string/M-theory with a transverse circle. Using a new coordinate system we find an ansatz for the solutions with only one unknown function. We show that this function is independent of the charge and our ansatz can therefore also be used to construct neutral black holes on cylinders and near-extremal charged dilatonic branes with a transverse circle. For sufficiently large mass $M > M_c$ these solutions have a horizon that connects across the cylinder but they are not translationally invariant along the circle direction. We argue that the neutral solution has larger entropy than the neutral black string for any given mass. This means that for $M > M_c$ the neutral black string can gain entropy by redistributing its mass to a solution that breaks translational invariance along the circle, despite the fact that it is classically stable. We furthermore explain how our construction can be used to study the thermodynamics of Little String Theory. 
  It is argued that the zero-point energies of free quantum fields diverge at most quadratically and not quartically, as is generally believed. This is a consequence of the relativistic invariance which requires that the energy density of the vacuum $\rho$ and its pressure $p$ satisfy $\rho=-p$. The usually obtained quartic divergence is an artifact of the use of a noninvariant regularization which violates this relation. One consequence of our results is that the zero-point energies of free massless fields vanish. Implications for the cosmological constant problem are briefly discussed. 
  We continue our investigation of the phenomenological implications of the "deformed" commutation relations [x_i,p_j]=i hbar[(1 + beta p^2) delta_{ij} + beta' p_i p_j]. These commutation relations are motivated by the fact that they lead to the minimal length uncertainty relation which appears in perturbative string theory. In this paper, we consider the effects of the deformation on the classical orbits of particles in a central force potential. Comparison with observation places severe constraints on the value of the minimum length. 
  We compute the moduli space metric of SU(N) Yang-Mills theory with N=2 supersymmetry in the vicinity of the point where the classical moduli vanish. This gauge theory may be realized as a set of N D7-branes wrapping a K3 surface, near the enhancon locus. The moduli space metric determines the low-energy worldvolume dynamics of the D7 branes near this point, including stringy corrections. Non-abelian gauge symmetry is not restored on the worldvolume at the enhancon point, but rather the gauge group remains U(1)^{N-1} and light electric and magnetically charged particles coexist. We also study the moduli space metric for a single probe brane in the background of N-1 branes near the enhancon point. We find quantum corrections to the supergravity probe metric that are not suppressed at large separations, but are down by 1/N factors, due to the response of the N-1 enhancon branes to the probe. A singularity appears before the probe reaches the enhancon point where a dyon becomes massless. We compute the masses of W-bosons and monopoles in a large N limit near this critical point. 
  A world-sheet sigma model approach is applied to string theories dual to four-dimensional gauge theories, and semi-classical soliton solutions representing highly excited string states are identified which correspond to gauge theory operators with relatively small anomalous dimensions. The simplest class of such states are strings on the leading Regge trajectory, with large spin in AdS_5. These correspond to operators with many covariant derivatives, whose anomalous dimension grows logarithmically with the space-time spin. In the gauge theory, the logarithmic scaling violations are similar to those found in perturbation theory. Other examples of highly excited string states are also considered. 
  We construct the Hamiltonian of the super five brane in terms of its physical degrees of freedom. It does not depend on the inverse of the induced metric. Consequently, some singular configurations are physically admissible, implying an interpretation of the theory as a multiparticle one. The symmetries of the theory are analyzed from the canonical point of view in terms of the first and second class constraints. In particular it is shown how the chiral sector may be canonically reduced to its physical degrees of freedom. 
  Confinement in non-Abelian gauge theories, such as QCD, is often explained using an analogy to type II superconductivity. In this analogy the existence of the ``Meissner'' effect for quarks with respect to the QCD vacuum is an important element. Here we show that using the ideas of Abelian projection it is possible to arrive at an effective London equation from a non-Abelian gauge theory. (London's equation gave a phenomenological description of the Meissner effect prior to the Ginzburg-Landau or BCS theory of superconductors). The Abelian projected gauge field acts as the E&M field in normal superconductivity, while the remaining non-Abelian components form a gluon condensate which is described via an effective scalar field. This effective scalar field plays a role similar to the scalar field in Ginzburg-Landau theory. 
  We find half supersymmetric AdS-embeddings in AdS_5 x S^5 corresponding to all quarter BPS orthogonal intersections of D3-branes with Dp-branes. A particular case is the Karch-Randall embedding AdS_4 x S^2. We explicitly prove that these embeddings are supersymmetric by showing that the kappa symmetry projections are compatible with half of the target space Killing spinors and argue that all these cases lead to AdS/dCFT dualities involving a CFT with a defect. We also find an asymptotically AdS_4 x S^2 embedding that corresponds to a holographic RG-flow on the defect. We then consider the pp-wave limit of the supersymmetric AdS-embeddings and show how it leads to half supersymmetric D-brane embeddings in the pp-wave background. We systematically analyze D-brane embeddings in the pp-wave background along with their supersymmetry. We construct all supersymmetric D-branes wrapped along the light-cone using operators in the dual gauge theory: the open string states are constructed using defect fields. We also find supersymmetric D1 (monopoles) and D3 (giant gravitons) branes that wrap only one of the light-cone directions. These correspond to non-perturbative states in the dual gauge theory. 
  We revisit the decoupling phenomenon of massless modes in the noncommutative open string (NCOS) theories. We check the decoupling by explicit computation in (2+1) or higher dimensional NCOS theories and recapitulate the validity of the decoupling to all orders in perturbation theory. 
  We study constant dilaton supersymmetric solutions of type IIB Supergravity with 5-form and 3-form flux with isometry group U(1) $\times$$Z_3$. Some of these solutions correspond to marginal perturbations of N=4 Yang-Mills. We find one line of solutions in particular of AdS$_5$ fibred over an $S^5$. This line is described by a single complex parameter. AdS$_5\times$S$^5$ is obtained when this partameter is tuned to zero. 
  A generic massive Thirring Model in three space-time dimensions exhibits a correspondence with a topologically massive bosonized gauge action associated to a self-duality constraint, and we write down a general expression for this relationship.   We also generalize this structure to $d$ dimensions, by adopting the so-called doublet approach, recently introduced. In particular, a non- conventional formulation of the bosonization technique in higher dimensions (in the spirit of $d=3$), is proposed and, as an application, we show how fermionic (Thirring-like) representations for bosonic topologically massive models in four dimensions may be built up. 
  We study cosmological solutions of Einstein gravity with a positive cosmological constant in diverse dimensions. These include big-bang models that re-collapse, big-bang models that approach de Sitter acceleration at late times, and bounce models that are both past and future asymptotically de Sitter. The re-collapsing and the bounce geometries are all tall in the sense that entire spatial slices become visible to a comoving observer before the end of conformal time, while the accelerating big-bang geometries can be either short or tall. We consider the interpretation of these cosmological solutions as renormalization group flows in a dual field theory and give a geometric interpretation of the associated c-function as the area of the apparent cosmological horizon in Planck units. The covariant entropy bound requires quantum effects to modify the early causal structure of some of our big-bang solutions. 
  A direct relation between two types of topological field theories, Chern-Simons theory and BF theory, is presented by using ``Generalized Differential Calculus'', which extends an ordinary p-form to an ordered pair of p and (p+1)-form. We first establish the generalized Chern-Weil homomormism for generalized curvature invariant polynomials in general even dimensional manifolds, and then show that BF gauge theory can be obtained from the action which is the generalized second Chern class with gauge group G. Particularly when G is taken as SL(2,C) in four dimensions, general relativity with cosmological constant can be derived by constraining the topological BF theory. 
  In the context of quantum field theories in curved spacetime, we compute the effective action of the transition amplitude from vacuum to vacuum in the presence of an external gravitational field. The imaginary part of resulted effective action determines the probability of vacuum decay via quantum tunneling process, giving the rate and spectrum of particle creations. We show that gravitational field polarizes vacuum and discretizes its spectrum for such a polarization gains gravitational energy. On the basis of gravitational vacuum polarization, we discuss the quantum origin of vacuum decay in curved spacetime as pair-creations of particles and anti-particles. The thermal spectrum of particle creations is attributed to (i) the CPT invariance of pair-creations(annihilations) from(into) vacuum and (ii) vacuum acts as a reserve with the temperature determined by gravitational energy-gain. 
  A 7-manifold with G_2 holonomy can be constructed as a R^3 bundle over a quaternionic space. We consider a quaternionic base space which is singular and its metric depends on three parameters, where one of them corresponds to an interpolation between S^4 and CP^2 or its non-compact analogs. This 4-d Einstein space has four isometries and the fixed point set of a generic Killing vector is discussed. When embedded into M-theory the compactification over a given Killing vector gives intersecting 6-branes as IIA configuration and we argue that membrane instantons may resolve the curvature singularity. 
  The TST-dual of the general 1/4-supersymmetric D2-brane supertube is identified as a 1/4-supersymmetric IIA `supercurve': a string with arbitrary transverse displacement travelling at the speed of light. A simple proof is given of the classical upper bound on the angular momentum, which is also recovered as the semi-classical limit of a quantum bound. The classical bound is saturated by a `superhelix', while the quantum bound is saturated by a bosonic oscillator state in a unique SO(8) representation. 
  The analysis of (2+1)-dimensional Yang-Mills ($YM_{2+1})$ theory via the use of gauge-invariant matrix variables is reviewed. The vacuum wavefunction, string tension, the propagator mass for gluons, its relation to the magnetic mass for $YM_{3+1}$ at nonzero temperature and the extension of our analysis to the Yang-Mills-Chern-Simons theory are discussed. 
  We obtain first-order equations for G_2 holonomy of a wide class of metrics with S^3\times S^3 principal orbits and SU(2)\times SU(2) isometry, using a method recently introduced by Hitchin. The new construction extends previous results, and encompasses all previously-obtained first-order systems for such metrics. We also study various group contractions of the principal orbits, focusing on cases where one of the S^3 factors is subjected to an Abelian, Heisenberg or Euclidean-group contraction. In the Abelian contraction, we recover some recently-constructed G_2 metrics with S^3\times T^3 principal orbits. We obtain explicit solutions of these contracted equations in cases where there is an additional U(1) isometry. We also demonstrate that the only solutions of the full system with S^3\times T^3 principal orbits that are complete and non-singular are either flat R^4 times T^3, or else the direct product of Eguchi-Hanson and T^3, which is asymptotic to R^4/Z_2\times T^3. These examples are in accord with a general discussion of isometric fibrations by tori which, as we show, in general split off as direct products. We also give some (incomplete) examples of fibrations of G_2 manifolds by associative 3-tori with either T^4 or K3 as base. 
  We consider the stabilization of de Sitter brane-world. The scalar field bulk-brane theory produces the non-trivial minimum of modulus potential where temporal radion is realized. The hierarchy problem (between Planck and electroweak scales) may be solved. However, the interpretation of radion is not so clear as in AdS brane-world. In particulary, the introduction of two times physics or pair-creation of bulk spaces or identification of one of spatial coordinates with imaginary time (non-zero temperature) may be required. 
  We address the localization of gravity on the Friedmann-Robertson-Walker type brane embedded in either $AdS_{5}$ or $dS_{5}$ bulk space,and derive two definite limits between which the value of the bulk cosmological constant has to lie in order to localize the graviton on the brane.The lower limit implies that the brane should be either $dS_{4}$ or 4d Minkowski in the $AdS_{5}$ bulk.The positive upper limit indicates that the gravity can be trapped also on curved brane in the $dS_{5}$ bulk space.Some implications to recent cosmological scenarios are also discussed. 
  This paper has been withdrawn by the authors because the results obtained here had been corrected and appeared in hep-th/0306008. 
  A chiral random matrix model with complex eigenvalues is solved as an effective model for QCD with non-vanishing chemical potential. We derive new matrix model correlation functions which predict the local fluctuations of complex Dirac operator eigenvalues at zero virtuality. The parameter which measures the non-Hermiticity of the Dirac matrix is identified with the chemical potential. In the phase with broken chiral symmetry all spectral correlations of the Dirac eigenvalues are calculated as functions of quark flavors and chemical potential. The derivation uses the orthogonality of the Laguerre polynomials in the complex plane. Explicit results are given for any finite matrix size N as well in the large-N limit for weak and strong non-Hermiticity. 
  In the first part of this lecture, the 1/N expansion technique is illustrated for the case of the large-N sigma model. In large-N gauge theories, the 1/N expansion is tantamount to sorting the Feynman diagrams according to their degree of planarity, that is, the minimal genus of the plane onto which the diagram can be mapped without any crossings. This holds both for the usual perturbative expansion with respect to powers of {tilde g}^2=g^2 N, as well as for the expansion of lattice theories in positive powers of 1/{tilde g}^2. If there were no renormalization effects, the tilde g expansion would have a finite radius of convergence. The zero-dimensional theory can be used for counting planar diagrams. It can be solved explicitly, so that the generating function for the number of diagrams with given 3-vertices and 4-vertices, can be derived exactly. This can be done for various kinds of Feynman diagrams. We end with some remarks about planar renormalization. 
  Motivated by recent works of Sen and Gibbons, we study the evolution of a flat and homogeneous universe dominated by tachyon matter. In particular, we analyse the necessary conditions for inflation in the early roll of a single tachyon field. 
  In this paper time dependent solutions of supergravities with dilaton and arbitrary rank antisymmetric tensor field are found. Although the solutions are nonsupersymmetric the equations of motions can be integrated in a simple form. Such supergravity solutions are related to Euclidean or spacelike branes (S-branes). 
  We investigate the SU(N) Principal Chiral Model on a half-line with a particular set of boundary conditions (BCs). In previous work these BCs have been shown to correspond to boundary scattering matrices (K-matrices) which are representation conjugating and whose matrix structure corresponds to one of the symmetric spaces SU(N)/SO(N) or SU(N)/Sp(N). Starting from the bulk particle spectrum and the K-matrix for a particle in the vector representation we construct K-matrices for particles in higher rank representations scattering off the boundary. We then perform an analysis of the physical strip pole structure and provide a complete set of boundary Coleman-Thun mechanisms for those poles which do not correspond to particles coupling to the boundary. We find that the model has no non-trivial boundary states. 
  Covariant quantization of self-dual strings in 2+2 flat dimensions reduces them to their zero modes, a consequence of extended world-sheet supersymmetry. We demonstrate how to arrive at the same result more directly by employing a `double' light-cone gauge. An unconventional feature of this gauge is the removal of anticommuting degrees of freedom by commuting symmetries and vice versa. The reducibility of the N=4 string and its equivalence with the N=2 string become apparent. 
  Towards the end of the brane inflationary epoch in the brane world, cosmic strings (but not texture, domain walls or monopoles) are copiously produced during brane collision. These cosmic strings are D$p$-branes with $(p-1)$ dimensions compactified. We elaborate on this prediction of the superstring theory description of the brane world. Using the data on the temperature anisotropy in the cosmic microwave background, we estimate the cosmic string tension $\mu$ to be around $G \mu \simeq 10^{-7}$. This in turn implies that the anisotropy in the cosmic microwave background comes mostly from inflation, but with a component (of order 10%) from cosmic strings. This cosmic string effect should also be observable in gravitational wave detectors and maybe even pulsar timing measurements.   Keywords : Inflation, Brane World, Superstring Theory, Cosmic String, Cosmology 
  We study instanton effects in theories with compact extra dimensions. We perform an instanton calculation in a 5d theory on a circle of radius R, with gauge, scalar, and fermion fields in the bulk of the extra dimension. We show that, depending on the matter content, instantons of size rho << R can dominate the amplitude. Using deconstruction as an ultraviolet definition of the theory allows us to show, in a controlled approximation, that for a small number of bulk fermions, the amplitude for small size instantons exponentially grows as e^{O(1)R/rho}. 
  We use boundary quantum group symmetry to obtain recursion formulas which determine nondiagonal solutions of the boundary Yang-Baxter equation (reflection equation) of the XXZ type for any spin j. 
  The hypothesis of brane-embedded matter appears to gain increasing credibility in astrophysics. However, it can only be truly successful if its implications on particle interaction are consistent with existing knowledge. This letter focuses on the issue of optical interference, and shows that at least one brane-world model can offer plausible interpretations for both Young's double-slit experiment, and the experiments that fit less neatly with it. The basic assumption is that particles can interact at a distance through the vibrations induced by their motion on the brane. The qualitative analysis of this mechanism suggests that fringe visibility in single photon interference depends on the energy levels and the interval between interacting particles. A double-slit experiment, performed with coherent single red photons should reveal the disappearance of interference when the time delay between individual particles is increased over 2.18 seconds. In the case of infrared photons with the frequency of $9\cdot 10^{13}$ Hz, interference must vanish already at the interval of one second. 
  We present a formulation of a matrix model which manifestly possesses the general coordinate invariance when we identify the large $N$ matrices with differential operators. In order to build a matrix model which has the local Lorentz invariance, we investigate how the $so(9,1)$ Lorentz symmetry and the $u(N)$ gauge symmetry are mixed together. We first analyze the bosonic part of the model, and we find that the Einstein gravity is reproduced in the classical low-energy limit. And we present a proposal to build a matrix model which has ${\cal N}=2$ SUSY and reduces to the type IIB supergravity in the classical low-energy limit. 
  We consider the M-theory lifts of configurations of type IIA D6-branes intersecting at angles. In supersymmetry preserving cases, the lifts correspond to special holonomy geometries, like conifolds and $G_2$ holonomy singularities. Transitions in which D6-branes approach and recombine lift to topology changing transition in these geometries. In some instances non-supersymmetric configurations can be reliably lifted, leading to the same topological manifolds, but endowed with non-supersymmetric metrics. In these cases the phase transitions are driven dynamically, due to instabilities localized at the singularities. Even though in non-compact setups the instabilities relax to infinity, in compact situations there exist nearby minima where the instabilities dissappear and the decay reaches a well-defined (in general supersymmetric) endpoint. 
  The evidence for neutrino masses in atmospheric and solar neutrino experiments provides further support for the embedding of the Standard Model fermions in the chiral 16 SO(10) representation. Such an embedding is afforded by the realistic free fermionic heterotic-string models. In this paper we advance the study of these string models toward a non-perturbative analysis by generalizing the work of Donagi, Pantev, Ovrut and Waldram from the case of G=SU(2n+1) to G=SU(2n) stable holomorphic vector bundles on elliptically fibered Calabi-Yau manifolds with fundamental group Z2. We demonstrate existence of G=SU(4) solutions with three generations and SO(10) observable gauge group over Hirzebruch base surface, whereas we show that certain classes of del Pezzo base surface do not admit such solutions. The SO(10) symmetry is broken to SU(5)XU(1) by Wilson line. The overlap with the realistic free fermionic heterotic-string models is discussed. 
  We prove two theorems, announced in hep-th/0108170, for static spacetimes that solve Einstein's equation with negative cosmological constant. The first is a general structure theorem for spacetimes obeying a certain convexity condition near infinity, analogous to the structure theorems of Cheeger and Gromoll for manifolds of non-negative Ricci curvature. For spacetimes with Ricci-flat conformal boundary, the convexity condition is associated with negative mass. The second theorem is a uniqueness theorem for the negative mass AdS soliton spacetime. This result lends support to the new positive mass conjecture due to Horowitz and Myers which states that the unique lowest mass solution which asymptotes to the AdS soliton is the soliton itself. This conjecture was motivated by a nonsupersymmetric version of the AdS/CFT correspondence. Our results add to the growing body of rigorous mathematical results inspired by the AdS/CFT correspondence conjecture. Our techniques exploit a special geometric feature which the universal cover of the soliton spacetime shares with familiar ``ground state'' spacetimes such as Minkowski spacetime, namely, the presence of a null line, or complete achronal null geodesic, and the totally geodesic null hypersurface that it determines. En route, we provide an analysis of the boundary data at conformal infinity for the Lorentzian signature static Einstein equations, in the spirit of the Fefferman-Graham analysis for the Riemannian signature case. This leads us to generalize to arbitrary dimension a mass definition for static asymptotically AdS spacetimes given by Chru\'sciel and Simon. We prove equivalence of this mass definition with those of Ashtekar-Magnon and Hawking-Horowitz. 
  We compute the finite temperature induced fermion number for fermions coupled to a static nonlinear sigma model background in (2+1) dimensions, in the derivative expansion limit. While the zero temperature induced fermion number is well known to be topological (it is the winding number of the background), at finite temperature there is a temperature dependent correction that is nontopological -- this finite T correction is sensitive to the detailed shape of the background. At low temperature we resum the derivative expansion to all orders, and we consider explicit forms of the background as a CP^1 instanton or as a baby skyrmion. 
  We study the dynamical behavior of the dilaton in the background of three-dimensional Kerr-de Sitter space which is inspired from the low-energy string effective action. The perturbation analysis around the cosmological horizon of Kerr-de Sitter space reveals a mixing between the dilaton and other fields. Introducing a gauge (dilaton gauge), we can disentangle this mixing completely and obtain one decoupled dilaton equation. However it turns out that this belongs to the tachyon. The stability of de Sitter solution with J=0 is discussed. Finally we compute the dilaton absorption cross section to extract information on the cosmological horizon of de Sitter space. 
  Brane cosmology driven by the tachyon rolling down to its ground state is investigated. We adopt an effective field theoretical description for the tachyon and Randall-Sundrum type brane world scenario. After formulating basic equations, we show that the standard cosmology with a usual scalar field can mimic the low energy behavior of the system near the tachyon ground state. We also investigate qualitative behavior of the system beyond the low energy regime for positive, negative and vanishing 4-dimensional effective cosmological constant $\Lambda_4=\kappa_5^4V(T_0)^2/12-|\Lambda_5|/2$, where $\kappa_5$ and $\Lambda_5$ are 5-dimensional gravitational coupling constant and (negative) cosmological constant, respectively, and $V(T_0)$ is the (positive) tension of the brane in the tachyon ground state. In particular, for $\Lambda_4<0$ the tachyon never settles down to its potential minimum and the universe eventually hits a big-crunch singularity. 
  Three explicit and equivalent representations for the monodromy of the conformal blocks in the SL(2,C)/SU(2) WZNW model are proposed in terms of the same quantity computed in Liouville field theory. We show that there are two possible fusion matrices in this model. This is due to the fact that the conformal blocks, being solutions to the Knizhnik-Zamolodchikov equation, have a singularity when the SL(2,C) isospin coordinate x equals the worldsheet variable z. We study the asymptotic behaviour of the conformal block when x goes to z. The obtained relation inserted into a four point correlation function in the SL(2,C)/SU(2) WZNW model gives some expression in terms of two correlation functions in Liouville field theory. 
  We discuss a phase structure of compact QED in four dimensions by considering the theory as a perturbed topological model. In this scenario we use the singular configuration with an appropriate regularization, and so obtain the results similar to the lattice gauge theory due to the effect of topological objects. In this paper we calculate the thermal pressure of the topological model by the use of a one-dimensional Coulomb gas approximation, which leads to a phase structure of full compact QED. Furthermore the critical-line equation is explicitly evaluated. We also discuss relations between the monopole condensation in compact QED in four dimensions and the chiral symmetry restoration in the massive Thirring model in two dimensions. 
  We discuss a quantum-theoretical aspect of the massive Abelian antisymmetric tensor gauge theory with antisymmetric tensor current. To this end, an Abelian rank-2 antisymmetric tensor field is quantized both in the covariant gauge with an arbitrary gauge parameter and in the axial gauge of the Landau type. The covariant quantization yields the generating functional written in terms of an antisymmetric tensor current and its divergence. Origins of the terms in the generating functional are clearly understood in comparison with the quantization in the unitary gauge. The quantization in the axial gauge with a suitable axis directly yields the generating functional which is same as that obtained by using Zwanziger's formulation for electric and magnetic charges. It is shown that the generating functionals lead to a composite of the Yukawa and the linear potentials. 
  We study the finite size corrections for the magnetization and the internal energy of the 2d Ising model in a magnetic field by using transfer matrix techniques. We compare these corrections with the functional form recently proposed by Delfino and LeClair-Mussardo for the finite temperature behaviour of one-point functions in integrable 2d quantum field theories. We find a perfect agreement between theoretical expectations and numerical results. Assuming the proposed functional form as an input in our analysis we obtain a relevant improvement in the precision of the continuum limit estimates of both quantities. 
  This review is devoted to open strings, and in particular to the often surprising features of their spectra. It follows and summarizes developments that took place mainly at the University of Rome ``Tor Vergata'' over the last decade, and centred on world-sheet aspects of the constructions now commonly referred to as ``orientifolds''. Our presentation aims to bridge the gap between the world-sheet analysis, that first exhibited many of the novel features of these systems, and their geometric description in terms of extended objects, D-branes and O-planes, contributed by many other colleagues, and most notably by J. Polchinski. We therefore proceed through a number of prototype examples, starting from the bosonic string and moving on to ten-dimensional fermionic strings and their toroidal and orbifold compactifications, in an attempt to guide the reader in a self-contained journey to the more recent developments related to the breaking of supersymmetry. 
  We study the renormalisation group flows between minimal W models by means of a new set of nonlinear integral equations which provide access to the effective central charge of both unitary and nonunitary models. We show that the scaling function associated to the nonunitary models is a nonmonotonic function of the system size. 
  We consider quantum field theory on a spacetime representing the Big Crunch/Big Bang transition postulated in the ekpyrotic or cyclic cosmologies. We show via several independent methods that an essentially unique matching rule holds connecting the incoming state, in which a single extra dimension shrinks to zero, to the outgoing state in which it re-expands at the same rate. For free fields in our construction there is no particle production from the incoming adiabatic vacuum. When interactions are included the total particle production for fixed external momentum is finite at tree level. We discuss a formal correspondence between our construction and quantum field theory on de Sitter spacetime. 
  We study the physical content of quadratic diff-invariant Lagrangians in arbitrary dimensions by using covariant symplectic techniques. This paper extends previous results in dimension four. We discuss the difference between the even and odd dimensional cases. 
  Dyonic black holes with string-loop corrections are studied in the near-horizon region. In perturbative heterotic string theory compactified to four dimensions with N=2 supersymmetry, in the first order in string-loop expansion parameter, we solve the system of Maxwell and Killing spinor equations for dyonic black hole. At the horizon, the string-loop-corrected solution displays restoration of spontaneously broken supersymmetry. 
  A field theory approach to summing threshold effects to the gauge couplings in a two-torus compactification is presented and the link with the (heterotic) string calculation is carefully investigated. We analyse whether the complete UV behaviour of the theory may be described on pure field theory grounds, as due to momentum modes only, and address the role of the winding modes, not included by the field theory approach. ``Non-decoupling'' effects in the low energy limit, due to a small dimension are discussed. The role of modular invariance in ensuring a finite (heterotic) string result is addressed. 
  The emitted power of the radiation from a charged particle moving uniformly on a circle inside a cylindrical waveguide is considered. The expressions for the energy flux of the radiation passing through the waveguide cross-section are derived for both TE and TM waves. The results of the numerical evaluation are presented for the number of emitted quanta depending on the waveguide radius, the radius of the charge rotation orbit and dielectric permittivity of the filling medium. These results are compared with the corresponding quantities for the synchrotron radiation in a homogeneous medium. 
  We prove that there do not exist multisoliton solutions of noncommutative scalar field theory in the Moyal plane which interpolate smoothly between $n$ overlapping solitons and $n$ solitons with an infinite separation. 
  Using the results of previous investigations on sine-Gordon form factors exact expressions of all breather matrix elements are obtained for several operators: all powers of the fundamental bose field, general exponentials of it, the energy momentum tensor and all higher currents. Formulae for the asymptotic behavior of bosonic form factors are presented which are motivated by Weinberg's power counting theorem in perturbation theory. It is found that the quantum sine-Gordon field equation holds and an exact relation between the ``bare'' mass and the renormalized mass is obtained. Also a quantum version of a classical relation for the trace of the energy momentum is proven. The eigenvalue problem for all higher conserved charges is solved. All results are compared with perturbative Feynman graph expansions and full agreement is found. 
  This combines a reply to the Comment [hep-th/0203067 v1] by A. N. Vaidya and R. de L. Rodrigues with an erratum to our Letter [Phys. Rev. Lett. 87, 210405 (2001)] 
  It has long been known that strings wound around incontractible cycles can play a vital role in cosmology. In particular, in a spacetime with toroidal spatial hypersurfaces, the dynamics of the winding modes may help yield three large spatial dimensions. However, toroidal compactifications are phenomenologically unrealistic. In this paper we therefore take a first step toward extending these cosmological considerations to $D$-dimensional toroidal orbifolds. We use numerical simulation to study the timescales over which "pseudo-wound" strings unwind on these orbifolds with trivial fundamental group. We show that pseudo-wound strings can persist for many ``Hubble times'' in some of these spaces, suggesting that they may affect the dynamics in the same way as genuinely wound strings. We also outline some possible extensions that include higher-dimensional wrapped branes. 
  We describe the effective supergravity theory present below the scale of spontaneous gauge symmetry breaking due to an anomalous U(1), obtained by integrating out tree-level interactions of massive modes. A simple case is examined in some detail. We find that the effective theory can be expressed in the linear multiplet formulation, with some interesting consequences. Among them, the modified linearity conditions lead to new interactions not present in the theory without an anomalous U(1). These additional interactions are compactly expressed through a superfield functional. 
  By studying singularities in stationary axisymmetric Kerr and Tomimatsu-Sato solutions with distortion parameter \d = 2, 3, ... in general relativity, we conclude that these singularities can be regarded as nothing other than closed string-like circular mass distributions. We use two different regularizations to identify \d-function type singularities in the energy-momentum tensor for these solutions, realizing a regulator independent result. This result gives supporting evidence that these axisymmetric exact solutions may well be the classical solutions around closed string-like mass distributions, just like Schwarzschild solution corresponding to a point mass distribution. In other words, these axisymmetric exact solutions may well provide the classical backgrounds around closed strings. 
  Following the construction of the projection operators on $T^2$ presented by Gopakumar, Headrick and Spradlin, we construct a set of projection operators on the integral noncommutative orbifold $T^2/G (G=Z_N, N=2, 3, 4, 6)$ which correspond to a set of solitons on $T^2/Z_N$ in noncommutative field theory. In this way, we derive an explicit form of projector on $T^2/Z_6$ as an example. We also construct a complete set of projectors on $T^2/Z_N$ by series expansions for integral case. 
  We find supersymmetric solutions of the D4 brane Born-Infeld action describing D2 supertubes ending on an arbitrary curve inside a D4 brane. From the D4 brane point of view, these are dyonic strings. We also consider various higher dimensional extensions of the usual supertubes, involving expanded D4 and D3 brane configurations. Finally, considering the worldsheet theory for open strings on a supertube, we show that this configuration is an exact solution to all orders in alpha'. Further the causal structure of the open-string metric provides new insight into the arbitrary cross-section of the supertube solutions. From this point of view, it is similar to the arbitrary profile that appears for certain null plane waves. 
  The (2+1) dimensional non-linear electrodynamics, the so called Pagels--Tomboulis electrodynamics, with the Chern--Simons term is considered. We obtain "generalized self--dual equation" and find the corresponding generalized massive Chern--Simons Lagrangian. Similar results for (2+1) massive dilaton electrodynamics have been obtained. 
  We consider a supersymmetric extension of quantum gauge theory based on a vector multiplet containing supersymmetric partners of spin 3/2 for the vector fields. The constructions of the model follows closely the usual construction of gauge models in the Epstein-Glaser framework for perturbative field theory. Accordingly, all the arguments are completely of quantum nature without reference to a classical supersymmetric theory. As an application we consider the supersymmetric electroweak theory. The resulting self-couplings of the gauge bosons agree with the standard model up to a divergence. 
  We investigate all spherically symmetric fundamental monopole solutions with fixed topological charge in the SU(5) --> SU(3)xSU(2)xU(1)/Z_3xZ_2 symmetry breaking. We find that there is three-fold replication of the monopoles. The three copies correspond to bound states with ``monopole clouds'' arising from the non-Abelian nature of the SU(3) factor. The triplication of monopoles could help us understand the observed family structure of standard model particles. 
  We introduce a simple coordinate system covering half of de Sitter space. The new coordinates have several attractive properties: the time direction is a Killing vector, the metric is smooth at the horizon, and constant-time slices are just flat Euclidean space. We demonstrate the usefulness of the coordinates by calculating the rate at which particles tunnel across the horizon. When self-gravitation is taken into account, the resulting tunneling rate is only approximately thermal. The effective temperature decreases through the emission of radiation. 
  In this paper a new supersymmetric extension of conformal mechanics is put forward. The beauty of this extension is that all variables have a clear geometrical meaning and the super-Hamiltonian turns out to be the Lie-derivative of the Hamiltonian flow of standard conformal mechanics. In this paper we also provide a supersymmetric extension of the other conformal generators of the theory and find their "square-roots". The whole superalgebra of these charges is then analyzed in details. We conclude the paper by showing that, using superfields, a constraint can be built which provides the exact solution of the system. 
  We investigate a class of matrix model which describes the dynamics of identical particles in even dimentional space. We show that the degrees of freedom after some constraints are implimented is proportional to particle number and consist of those for positions and internal degrees. The particle dynamics is given by the metric on the smooth moduli space. The moduli space metric for two particles is found. The size of tightly packed $N$ particles grows like $\sqrt{N}$. Our matrix model is related to the matrix model for fractional quantum Hall effect, the ADHM formalism of U(1) instantons on noncommutative space, and supersymmetric D0 branes on D6 branes with nonzero B-field in type IIA theory. 
  There are good indications that fundamental physics gives rise to a modified space-momentum uncertainty relation that implies the existence of a minimum length scale. We implement this idea in the scalar field theory that describes density perturbations in flat Robertson-Walker space-time. This leads to a non-linear time-dependent dispersion relation that encodes the effects of Planck scale physics in the inflationary epoch. Unruh type dispersion relations naturally emerge in this approach, while unbounded ones are excluded by the minimum length principle. We also find red-shift induced modifications of the field theory, due to the reduction of degrees of freedom at high energies, that tend to dampen the fluctuations at trans-Planckian momenta. In the specific example considered, this feature helps determine the initial state of the fluctuations, leading to a flat power spectrum. 
  The low energy dynamics of vortices in selfdual Abelian Higgs theory is of second order in vortex velocity and characterized by the moduli space metric. When Chern-Simons term with small coefficient is added to the theory, we show that a term linear in vortex velocity appears and can be consistently added to the second order expression. We provides an additional check of the first and second order terms by studying the angular momentum in the field theory. We briefly explore other first order term due to small background electric charge density and also the harmonic potential well for vortices given by the moment of inertia. 
  Thermodynamics of five-dimensional Schwarzschild Anti-de Sitter (SAdS) and SdS black holes in the particular model of higher derivative gravity is considered. The free energy, mass (thermodynamical energy) and entropy are evaluated. There exists the parameters region where BH entropy is zero or negative. The arguments are given that corresponding BH solutions are not stable. We also present the FRW-equations of motion of time (space)-like branes in SAdS or SdS BH background. The properties of dual CFT are discussed and it is shown that it has zero Casimir energy when BH entropy (effective gravitational constant) is zero. The Cardy-Verlinde formula for CFT dual to SAdS or SdS BH is found in the universal form. 
  We consider a single brane embedded in five dimensions with vanishing and positive bulk cosmological constant. In this setup, the existence of $dS_{4}$ brane is allowed. We explore the gravitational fluctuations on the brane, and we point out that the usual four-dimensional gravity can be reproduced by a normalizable zero mode and that the continuous massive modes are separated by a mass gap from zero mode. We derive the relation among the cosmological constant in the brane, and the fundamental scale of five dimensions and Planck scale. Finally we show that not zero but tiny observed cosmological constant cannot be realized in this setup. 
  The (2+1)D Georgi-Glashow model is explored at finite temperature in the regime when the Higgs boson is not infinitely heavy. The resulting Higgs-mediated interaction of monopoles leads to the appearance of a certain upper bound for the parameter of the weak-coupling approximation. Namely, when this bound is exceeded, the cumulant expansion used for the average over the Higgs field breaks down. The finite-temperature deconfining phase transition with the account for the same Higgs-mediated interaction of monopoles is further analysed. It is demonstrated that in the general case, accounting for this interaction leads to the existence of two distinct phase transitions separated by the temperature region where W-bosons exist in both, molecular and plasma, phases. The dependence of possible ranges of the critical temperatures corresponding to these phase transitions on the parameters of the Georgi-Glashow model is discussed. The difference in the RG behaviour of the fugacity of W-bosons from the respective behaviour of this quantity in the compact-QED limit of the model is finally pointed out. 
  I review cosmology within the framework of type-0 non-critical strings, proposed in collaboration with G. Diamandis, B. Georgalas, E. Papantonopoulos and I. Pappa. The instabilities of the tachyonic backgrounds, due to the absence of space-time supersymmetry, are treated in this framework as a necessary ingredient to ensure cosmological flow. The model involves D3 brane worlds, whose initial quantum fluctuations induce the non criticality. I argue that this model is compatible with the current astrophysical observations pointing towards acceleration of the Universe. A crucial r\^ole for the correct ``phenomenology'' of the model, in particular the order-one value of the deceleration parameter, is played by the relative magnitude of the flux of the five form of the type-0 string as compared to the size of the volume of five of the extra dimensions, transverse to the direction of the flux-field. 
  We use the Liouville-von Neumann (LvN) approach to study the dynamics and the adiabaticity of a time-dependent driven anharmonic oscillator as an eample of non-equilibrium quantum dynamics. We show that the adiabaticity is minimally broken in the sense that a gaussian wave-packet at the past infinity evolves to coherent states, however slowly the potential changes, its coherence factor is order of the coupling. We also show that the dynamics are governed by an equation of motion similar to the Kepler motion which satisfies the angular momentum conservation. 
  The effective action for the low energy scattering of two gravitons with a D-brane in the presence of a constant antisytmetric $B$ field in bosonic string theory is calculated and the modification to the standard D-brane action to first order in $\alpha'$ is obtained. 
  A new presentation of the Borchers-Buchholz result of the Lorentz-invariance of the energy-momentum spectrum in theories with broken Lorentz symmetry is given in terms of properties of the Green's functions of microcausal Bose and Fermi-fields. Strong constraints based on complex geometry phenomenons are shown to result from the interplay of the basic principles of causality and stability in Quantum Field Theory: if microcausality and energy-positivity in all Lorentz frames are satisfied, then it is unavoidable that all stable particles of the theory be governed by Lorentz-invariant dispersion laws; in all the field sectors, discrete parts outside the continuum as well as the thresholds of the continuous parts of the energy-momentum spectrum, with possible holes inside it, are necessarily represented by mass-shell hyperboloids (or the light-cone). No violation of this geometrical fact can be produced by spontaneous breaking of the Lorentz symmetry, even if the field-theoretical framework is enlarged so as to include short-distance singularities wilder than distributions. 
  We argue that in the presence of instanton-like singularities, the existence of cosmological horizons can become frame-dependent, ie. a horizon which appears in Einstein frame may not appear in string frame. We speculate on the relation between instanton-like singularities and the formulation of quantum gravity in de Sitter space. 
  We study cosmology on a BPS D3-brane evolving in the 10D SUGRA background describing a non-BPS brane. Initially the BPS brane is taken to be a probe whose dynamics we determine in the non-compact non-BPS background. The cosmology observed on the brane is of the FRW type with a scale factor $S(\tau)$. In this mirage cosmology approach, there is no self-gravity on the brane which cannot inflate. Self-gravity is then included by compactifying the background space-time. The low energy effective theory below the compactification scale is shown to be bi-metric, with matter coupling to a different metric than the geometrically induced metric on the brane. The geometrical scale factor on the brane is now $S(\tau) a(\tau)$ where $a(\tau)$ arises from brane self-gravity. In this non-BPS scenario the brane generically inflates. We study the resulting inflationary scenario taking into account the fact that the non-BPS brane eventually decays on a time-scale much larger than the typical inflationary time-scale. After the decay, the theory ceases to be bi-metric and COBE normalization is used to estimate the string scale which is found to be of order $10^{14}$ GeV. 
  Using the pure spinor formalism for the superstring, the vertex operator for the first massive states of the open superstring is constructed in a manifestly super-Poincar\'e covariant manner. This vertex operator describes a massive spin-two multiplet in terms of ten-dimensional superfields. 
  The graviton localized on the 3-brane is examined in Randall-Sundrum brane-world scenario from the viewpoint of one-dimensional singular quantum mechanics. For the Randall-Sundrum single brane scenario the one-parameter family of the fixed-energy amplitude is explicitly computed where the free parameter $\xi$ parametrizes the various boundary conditions at the brane. The general criterion for the localized graviton to be massless is derived when $\xi$ is arbitrary but non-zero. When $\xi=0$, the massless graviton is obtained via a coupling constant renormalization. For the two branes picture the fixed-energy amplitude is in general dependent on the two free parameters. The numerical test indicates that there is no massless graviton in this picture. For the positive-tension brane, however, the localized graviton becomes massless when the distance between branes are infinitely large, which is essentially identical to the single brane picture. For the negative-tension brane there is no massless graviton regardless of the distance between branes and choice of boundary conditions. 
  In the perturbative AdS-CFT correspondence, the dual field whose source are the prescribed boundary values of a bulk field in the functional integral, and the boundary limit of the quantized bulk field are the same thing. This statement is due to the fact that Witten graphs are boundary limits of the corresponding Feynman graphs for the bulk fields, and hence the dual conformal correlation functions are limits of bulk correlation functions. This manifestation of duality is analyzed in terms of the underlying functional integrals of different structure. 
  We present a complete description of the spectrum of compound states of reggeized gluons in QCD in multi-colour limit. The analysis is based on the identification of these states as ground states of noncompact Heisenberg SL(2,C) spin magnet. A unique feature of the magnet, leading to many unusual properties of its spectrum, is that the quantum space is infinite-dimensional and conventional methods, like the Algebraic Bethe Ansatz, are not applicable. Our solution relies on the method of the Baxter Q-operator. Solving the Baxter equations, we obtained the explicit expressions for the eigenvalues of the Q-operator. They allowed us to establish the quantization conditions for the integrals of motion and, finally, reconstruct the spectrum of the model. We found that intercept of the states built from even (odd) number of reggeized gluons, N, is bigger (smaller) than one and it decreases (increases) with N approaching the same unit value for infinitely large N. 
  We study the $\theta$ dependence of four-dimensional SU($N$) gauge theories, for $N\geq 3$ and in the large-N limit. We use numerical simulations of the Wilson lattice formulation of gauge theories to compute the first few terms of the expansion of the ground-state energy $F(\theta)$ around $\theta=0$, $F(\theta)-F(0) = A_2 \theta^2 (1 + b_2 \theta^2 + ...)$. Our results support Witten's conjecture: $F(\theta)-F(0) = {\cal A} \theta^2 + O(1/N)$ for sufficiently small values of $\theta$, $\theta < \pi$.   Indeed we verify that the topological susceptibility has a nonzero large-N limit $\chi_\infty=2 {\cal A}$ with corrections of $O(1/N^2)$, in substantial agreement with the Witten-Veneziano formula which relates $\chi_\infty$ to the $\eta^\prime$ mass. Furthermore, higher order terms in $\theta$ are suppressed; in particular, the $O(\theta^4)$ term $b_2$ (related to the $\eta^\prime - \eta^\prime$ elastic scattering amplitude) turns out to be quite small: $b_2=-0.023(7)$ for N=3, and its absolute value decreases with increasing $N$, consistently with the expectation $b_2=O(1/N^2)$. 
  The two parameter model is reduced to a one parameter model by using simple transformations. Because the separation between different phase regions for a one parameter model is just a point, the equivalence between the two models leads to the exact equation of the line that separates the broken and un-broken phases in the $(\lambda ,\sigma)$ plane. Also, we obtain nontrivial estimates on the stability region for this model. 
  We solve for the effective actions on the Coulomb branches of a class of N=2 supersymmetric theories by finding the complex structure of an M5 brane in an appropriate background hyperkahler geometry corresponding to the lift of two O6^- orientifolds and four D6 branes to M theory. The resulting Seiberg-Witten curves are of finite genus, unlike other solutions proposed in the literature. The simplest theories in this class are the scale invariant Sp(k) theory with one antisymmetric and four fundamental hypermultiplets and the SU(k) theory with two antisymmetric and four fundamental hypermultiplets. Infinite classes of related theories are obtained by adding extra SU(k) factors with bifundamental matter and by turning on masses to flow down to various asymptotically free theories. The N=4 supersymmetric SU(k) theory can be embedded in these asymptotically free theories, allowing a derivation of a subgroup of its S duality group as an exact equivalence of quantum field theories. 
  We deconstruct the non-supersymmetric SU(5) breaking by discrete symmetry on the space-time $M^4\times S^1$ and $M^4\times S^1/(Z_2\times Z_2')$ in the Higgs mechanism deconstruction scenario. And we explain the subtle point on how to exactly match the continuum results with the latticized results on the quotient space $S^1/Z_2$ and $S^1/(Z_2\times Z_2')$. We also propose an effective deconstruction scenario and discuss the gauge symmetry breaking by the discrete symmetry on theory space in this approach. As an application, we suggest the $G^N$ unification where $G^N$ is broken down to $SU(3)\times SU(2)\times U(1)^{n-3}$ by the bifundamental link fields and the doublet-triplet splitting can be achieved. 
  We derive a general expression for the power spectra of scalar and tensor fluctuations generated during inflation given an arbitrary choice of boundary condition for the mode function at a short distance. We assume that the boundary condition is specified at a short-distance cutoff at a scale $M$ which is independent of time. Using a particular prescription for the boundary condition at momentum $p = M$, we find that the modulation to the power spectra of density and gravitational wave fluctuations is of order $(H/M)$, where $H$ is the Hubble parameter during inflation, and we argue that this behavior is generic, although by no means inevitable. With fixed boundary condition, we find that the shape of the modulation to the power spectra is determined entirely by the deviation of the background spacetime from the de Sitter limit. 
  We show the absence of Lee-Yang singularities in the partition function of QCD close to the CP symmetric point theta=0 in the complex theta plane. Analyticity of the theta vacuum energy density at theta=0 provides a key missing link in the Vafa-Witten proof of absence of spontaneous breaking of parity symmetry in vector-like gauge theories. The result follows from renormalizability, unitarity, positivity and existence of BPS bounds. Generalisations of this theorem to other physical systems are also discussed with particular interest focused on the non-linear CP^N sigma model. 
  A theoretical mechanism is devised to determine the large distance physics of spacetime. It is a two dimensional nonlinear model, the lambda model, set to govern the string worldsurface to remedy the failure of string theory. The lambda model is formulated to cancel the infrared divergent effects of handles at short distance on the worldsurface. The target manifold is the manifold of background spacetimes. The coupling strength is the spacetime coupling constant. The lambda model operates at 2d distance $\Lambda^{-1}$, very much shorter than the 2d distance $\mu^{-1}$ where the worldsurface is seen. A large characteristic spacetime distance $L$ is given by $L^2=\ln(\Lambda/\mu)$. Spacetime fields of wave number up to 1/L are the local coordinates for the manifold of spacetimes. The distribution of fluctuations at 2d distances shorter than $\Lambda^{-1}$ gives the {\it a priori} measure on the target manifold, the manifold of spacetimes. If this measure concentrates at a macroscopic spacetime, then, nearby, it is a measure on the spacetime fields. The lambda model thereby constructs a spacetime quantum field theory, cutoff at ultraviolet distance $L$, describing physics at distances larger than $L$. The lambda model also constructs an effective string theory with infrared cutoff $L$, describing physics at distances smaller than $L$. The lambda model evolves outward from zero 2d distance, $\Lambda^{-1} = 0$, building spacetime physics starting from $L=\infty$ and proceeding downward in $L$. $L$ can be taken smaller than any distance practical for experiments, so the lambda model, if right, gives all actually observable physics. The harmonic surfaces in the manifold of spacetimes are expected to have novel nonperturbative effects at large distances. 
  A possibility of 5D gauge unification of $SU(2)_L \times U(1)_Y$ in $SU(3)_W$ is examined. The orbifold compactification allows fixed points where $SU(2)_L\times U(1)_Y$ representations can be assigned. We present a few possibilities which give long proton lifetime, top-bottom mass hierarchy from geometry, and reasonable neutrino masses. In general, these {\it chiral models} can lead to fixed point anomalies. We can show easily, due to the simplicity of the model, that these anomalies are cancelled by the relevant Chern-Simons terms for all the models we consider. It is also shown that the fixed point U(1)--graviton--graviton anomaly cancels without the help from the Chern-Simons term. Hence, we conjecture that the fixed point anomalies can be cancelled if the effective 4D theory is made anomaly free by locating chiral fermions at the fixed points. 
  Using a fully covariant formalism given by Carter for the deformation dynamics of p-branes governed by the Dirac-Nambu-Goto action in a curved background, it is proved that the corresponding Witten's phase space is endowed with a covariant symplectic structure, which can serve as a starting point for a phase space quantization of such objects. Some open questions for further research are outlined. 
  On the basis of the covariant description of the canonical formalism for quantization, we present the basic elements of the symplectic geometry for a restricted class of topological defects propagating on a curved background spacetime. We discuss the future extensions of the present results. 
  It is proved that the projections of the deformation vector field, normal and tangential to the worldsheet manifold swept out by Dirac-Nambu-Goto bosonic extendons propagating in a curved background, play the role of {\it global} symplectic potentials on the corresponding Witten covariant phase space. It is also proved that the {\it presymplectic} structure obtained from such potentials by direct exterior derivation, has not components tangent to the action of the relevant diffeomorphisms group of the theory. 
  We consider self tuning solutions for a brane embedded in an anti de Sitter spacetime. We include the higher derivative Gauss-Bonnet terms in the action and study singularity free solutions with finite effective Newton's constant. Using the methods of Csaki et al, we prove that such solutions, when exist, always require a fine tuning among the brane parameters. We then present a new method of analysis in which the qualitative features of the solutions can be seen easily without obtaining the solutions explicitly. Also, the origin of the fine tuning is transparent in this method. 
  Near-horizon conformal structure of a massive Schwarzschild black hole of mass M is analyzed using a scalar field as a simple probe of the background geometry. The near-horizon dynamics is governed by an operator which is related to the Virasoro algebra and admits a one-parameter family of self-adjoint extensions described by a real parameter z. When z satisfies a suitable contraint, the corresponding wavefunctions exhibit scaling behaviour in a band-like region near the horizon of the black hole. This formalism is consistent with the Bekenstein-Hawking entropy formula and naturally produces the -3/2 log M^2 correction term to the black hole entropy with other subleading corrections exponentially suppressed. This precise form for the black hole entropy is expected on general grounds in any conformal field theoretic description of the problem. The presence of the Virasoro algebra and the scaling properties of the associated wavefunctions in the near-horizon region together with the appearance of the logarithmic correction to the Bekenstein-Hawking entropy provide strong evidence for the near-horizon conformal structure in this system. 
  We study classical solutions of vacuum version of Berkovits' superstring field theory, focusing on the (super)ghost sector. We first argue that the supersliver state which is annihilated by eta_0, though it has the correct quantum numbers and solves the equation of motion, is actually non-perturbatively pure-gauge, and hence it fails to describe a D-brane. As a step toward the construction of non-trivial solutions, we calculate e^{-Phi}Qe^{Phi} for twisted superslivers. As a by-product, we find that the bc-twisted sliver solution in bosonic VSFT can, at least formally, also be written as a pure-gauge configuration. 
  Taking as starting point a perturbative study of the classical equations of motion of the non-Abelian Chern-Simons Theory with non-dynamical sources, we search for analytical expressions for link invarians. In order to present these expressions in a manifestly diffeomorphism-invariant form, we introduce a set of differential forms associated with submanifolds in Euclidean three-space that allow us to write the link invariants as a kind of surface-dependent diffeomorphism-invariants that present certain Abelian gauge symmetry. 
  Modeling the potential by an inverse square law in terms of the tachyon field ($V(T)=\beta T^{-2}$) we find exact solution for spatially flat isotropic universe.We show that for $\beta>2\sqrt{3}/3$ the model undergoes power-law inflation. A way to construct other exact solutions is specified and exemplified. 
  We discuss some controverted aspects of the evaluation of the thermal energy of a scalar field in a one-dimensional compact space. The calculations are carried out using a generalised zeta function approach. 
  A comparison is made between proposals for the exact three point function in Liouville quantum field theory and the nonperturbative weak coupling expansion developed long ago by Braaten, Curtright, Ghandour, and Thorn. Exact agreement to the order calculated (i.e. up to and including corrections of order O(g^{10})) is found. 
  We propose a field theory for describing the tachyon on a brane-antibrane system near the minimum of the potential. This field theory realizes two known properties of the tachyon effective action: 1) absence of plane-wave solutions around the minimum, and 2) exponential fall off of the pressure at late time as the tachyon field evolves from any spatially homogeneous initial configuration towards the minimum of the potential. Classical solutions in this field theory include non-relativistic matter with arbitrary spatial distribution of energy. 
  We construct supergravity solutions corresponding to space-like branes in string theory. Our approach is to apply the usual solution generating techniques to an appropriate time-dependent solution of the eleven dimensional vacuum Einstein equations. In this way all SDp-brane solutions are obtained, as well as the NS and M-theory space branes. Bound states of SDp/SD(p-2)- and SDp/SD(p-4)-branes are also constructed. Finally, we begin an investigation of the near-brane regions and singularity structure of these solutions. 
  It is shown that the super Higgs mechanism that occurs in a wide class of models with vanishing cosmological constant (at the classical level) is obtained by the gauging of a flat group which must be an electric subgroup of the duality group. If the residual massive gravitinos which occur in the partial supersymmetry breaking are BPS saturated, then the flat group is non abelian. This is so for all the models obtained by a Scherk-Schwarz supersymmetry breaking mechanism. If gravitinos occur in long multiplets, then the flat groups may be abelian. This is the case of supersymmetry breaking by string compactifications on an orientifold T^6/Z_2 with non trivial brane fluxes. 
  We construct light-cone gauge superstring field theory in a pp-wave background with Ramond-Ramond flux. The leading term in the interaction Hamiltonian is determined up to an overall function of $p^+$ by requiring closure of the pp-wave superalgebra. The bosonic and fermionic Neumann matrices for this cubic vertex are derived, as is the interaction point operator. We comment on the development of a $1/\mu p^+$ expansion for these results. 
  We consider N=2 moose/quiver gauge theories corresponding to N_1 D3-branes at a C^2/Z_{N_2} singularity in the ``large moose'' limit where N_1 and N_2 are scaled to infinity together. In the dual holographic description, this scaling gives rise to a maximally supersymmetric pp-wave background with a compact light-cone direction. We identify the gauge theory operators that describe the Discrete Light-Cone Quantization (DLCQ) of the string in this background. For each discrete light-cone momentum and winding sector there is a separate ground state and Fock space. The large moose/quiver diagram provides a useful graphical representation of the string and its excitations. This representation has a natural explanation in a T-dual language. The dual theory is a non-relativistic type IIA string wound around the T-dual direction, and bound by a quadratic Newtonian potential. We end with some comments on D-string/D-particle states, a possible lift to M-theory and the relation to deconstruction. 
  We formulate rational conformal field theory in terms of a symmetric special Frobenius algebra A and its representations. A is an algebra in the modular tensor category of Moore-Seiberg data of the underlying chiral CFT. The multiplication on A corresponds to the OPE of boundary fields for a single boundary condition. General boundary conditions are A-modules, and (generalised) defect lines are A-A-bimodules.   The relation with three-dimensional TFT is used to express CFT data, like structure constants or torus and annulus coefficients, as invariants of links in three-manifolds. We compute explicitly the ordinary and twisted partition functions on the torus and the annulus partition functions. We prove that they satisfy consistency conditions, like modular invariance and NIM-rep properties.   We suggest that our results can be interpreted in terms of non-commutative geometry over the modular tensor category of Moore-Seiberg data. 
  It is shown in first order perturbation theory that anharmonic oscillators in non-commutative space behave smoothly in the commutative limit just as harmonic oscillators do. The non-commutativity provides a method for converting a problem in degenerate perturbation theory to a non-degenerate problem. 
  It is an accepted practice in cosmology to invoke a scalar field with potential $V(\phi)$ when observed evolution of the universe cannot be reconciled with theoretical prejudices. Since one function-degree-of-freedom in the expansion factor $a(t)$ can be traded off for the function $V(\phi)$, it is {\it always} possible to find a scalar field potential which will reproduce a given evolution. I provide a recipe for determining $V(\phi)$ from $a(t)$ in two cases:(i) Normal scalar field with Lagrangian ${\cal L} = (1/2)\partial_a\phi \partial^a\phi - V(\phi)$ used in quintessence/dark energy models. (ii) A tachyonic field with Lagrangian ${\cal L} = -V(\phi) [ 1- \partial_a\phi \partial^a\phi]^{1/2} $, motivated by recent string theoretic results. In the latter case, it is possible to have accelerated expansion of the universe during the late phase in certain cases. This suggests a string theory based interpretation of the current phase of the universe with tachyonic condensate acting as effective cosmological constant. 
  The Berezinsky-Kosterlitz-Thouless (BKT) RG flow in the ensemble of monopoles existing in the finite-temperature (2+1)D Georgi-Glashow model is explored in the regime when the Higgs field is not infinitely heavy, but its mass is rather of the same order of magnitude as the mass of the W-boson. The corrections to the standard RG flow are derived to the leading order in the inverse mass of the Higgs boson. According to the obtained RG equations, the scaling of the free-energy density in the critical region and the value of the critical temperature of the phase transition are found to be unaffected by the finiteness of the Higgs-boson mass. The evolution of the Higgs mass itself is also investigated and shown to be rather weak, that enables one to treat this parameter as a constant. The same analysis is further performed in the SU(N)-case at N>2, where the RG invariance is demonstrated to hold only approximately, in a certain sense. Modulo this approximation, the critical behaviour of the SU(N)-model turns out to be identical to that of the SU(2)-one. 
  In this paper we propose a generalization of N=4 three dimensional AdS supergravity to the noncommutative case. This is a supersymmetric version of the results presented in hep-th/0201103. We show that it continues to admit an N=4 supersymmetric solution which is the noncommutative couterpart of AdS_3 space. Some other solutions are also discussed. 
  We perform a systematic string computation of the masses of anomalous U(1) gauge bosons in four-dimensional orientifold vacua, and we study their localization properties in the internal (compactified) space. We find that N=1 supersymmetric sectors yield four-dimensional contributions, localized in the whole six-dimensional internal space, while N=2 sectors give contributions localized in four internal dimensions. As a result, the U(1) gauge fields can be much lighter than the string scale, so that when the latter is at the TeV, they can mediate new non-universal repulsive forces at submillimeter distances much stronger than gravity. We also point out that even U(1)s which are free of four-dimensional anomalies may acquire non-zero masses as a consequence of six-dimensional anomalies. 
  Starting from first principles, a constructive method is presented to obtain boundary states in conformal field theory. It is demonstrated that this method is well suited to compute the boundary states of logarithmic conformal field theories. By studying the logarithmic conformal field theory with central charge c=-2 in detail, we show that our method leads to consistent results. In particular, it allows to define boundary states corresponding to both, indecomposable representations as well as their irreducible subrepresentations. 
  The equation of motion for Berkovits' WZW-like open (super)string field theory is shown to be integrable in the sense that it can be written as the compatibility condition ("zero-curvature condition") of some linear equations. Employing a generalization of solution-generating techniques (the splitting and the dressing methods), we demonstrate how to construct nonperturbative classical configurations of both N=1 superstring and N=2 fermionic string field theories. With and without u(n) Chan-Paton factors, various solutions of the string field equation are presented explicitly. 
  New gaugings of four dimensional N=8 supergravity are constructed, including one which has a Minkowski space vacuum that preserves N=2 supersymmetry and in which the gauge group is broken to $SU(3)xU(1)^2$. Previous gaugings used the form of the ungauged action which is invariant under a rigid $SL(8,R)$ symmetry and promoted a 28-dimensional subgroup ($SO(8),SO(p,8-p)$ or the non-semi-simple contraction $CSO(p,q,8-p-q)$) to a local gauge group. Here, a dual form of the ungauged action is used which is invariant under $SU^*(8)$ instead of $SL(8,R)$ and new theories are obtained by gauging 28-dimensional subgroups of $SU^*(8)$. The gauge groups are non-semi-simple and are different real forms of the $CSO(2p,8-2p)$ groups, denoted $CSO^*(2p,8-2p)$, and the new theories have a rigid SU(2) symmetry. The five dimensional gauged N=8 supergravities are dimensionally reduced to D=4. The $D=5,SO(p,6-p)$ gauge theories reduce, after a duality transformation, to the $D=4,CSO(p,6-p,2)$ gauging while the $SO^*(6)$ gauge theory reduces to the $D=4, CSO^*(6,2)$ gauge theory. The new theories are related to the old ones via an analytic continuation. The non-semi-simple gaugings can be dualised to forms with different gauge groups. 
  We study the coupling of the closed string to the open string in the topological B-model. These couplings can be viewed as gauge invariant observables in the open string field theory, or as deformations of the differential graded algebra describing the OSFT. This is interpreted as an intertwining map from the closed string sector to the deformation (Hochschild) complex of the open string algebra. By an explicit calculation we show that this map induces an isomorphism of Gerstenhaber algebras on the level of cohomology. Reversely, this can be used to derive the closed string from the open string. We shortly comment on generalizations to other models, such as the A-model. 
  We consider the set of controlled time-dependent backgrounds of general relativity and string theory describing ``bubbles of nothing'', obtained via double analytic continuation of black hole solutions. We analyze their quantum stability, uncover some novel features of their dynamics, identify their causal structure and observables, and compute their particle production spectrum. We present a general relation between squeezed states, such as those arising in cosmological particle creation, and nonlocal theories on the string worldsheet. The bubble backgrounds have various aspects in common with de Sitter space, Rindler space, and moving mirror systems, but constitute controlled solutions of general relativity and string theory with no external forces. They provide a useful theoretical laboratory for studying issues of observables in systems with cosmological horizons, particle creation, and time-dependent string perturbation theory. 
  We examine noncommutative Chern Simons theory using operator regularization. Both the zeta-function and the eta-function are needed to determine one loop effects. The contributions to these functions coming from the two point function is evaluated. The U(N) noncommutative model smoothly reduces to the SU(N) commutative model as the noncommutative parameter theta_{mu nu} vanishes. 
  In this letter we show that, in five-dimensional anti-deSitter space (AdS) truncated by boundary branes, effective field theory techniques are reliable at high energy (much higher than the scale suggested by the Kaluza-Klein mass gap), provided one computes suitable observables. We argue that in the model of Randall and Sundrum for generating the weak scale from the AdS warp factor, the high energy behavior of gauge fields can be calculated in a {\em cutoff independent manner}, provided one restricts Green's functions to external points on the Planck brane. Using the AdS/CFT correspondence, we calculate the one-loop correction to the Planck brane gauge propagator due to charged bulk fields. These effects give rise to non-universal logarithmic energy dependence for a range of scales above the Kaluza-Klein gap. 
  We construct 1/4 BPS configurations, `M-ribbons', in M-theory on T^2, which give the supertubes and supercurves in type IIA theory upon dimensional reduction. These M-ribbons are generalized so as to be consistent with the SL(2,Z) modular transformation on T^2. In terms of the type IIB theory, the generalized M-ribbons are interpreted as an SL(2,Z) duality family of super D-helix. It is also shown that the BPS M-ribbons must be straight in one direction. 
  We propose a Lie-algebra model for noncommutative coordinate and momentum space . Based on a rigid commutation relation for the commutators of space time operators the model is quite constrained if one tries to keep Lorentz invariance as much as possible. We discuss the question of invariants esp. the definition of a mass. 
  For the noncommutative torus ${\cal T}$, in case of the N.C. parameter $\theta = \frac{Z}{n}$, we construct the basis of Hilbert space ${\ca$H}_n$ in terms of $\theta$ functions of the positions $z_i$ of $n$ solitons. The wrapping around the torus generates the algebra ${\cal A}_n$, which is the $Z_n \times Z_n$ Heisenberg group on $\theta$ functions. We find the generators $g$ of an local elliptic $su(n)$, w$transform covariantly by the global gauge transformation of ${\cal A}$By acting on ${\cal H}_n$ we establish the isomorphism of ${\cal A}_n$$g$. We embed this $g$ into the $L$-matrix of the elliptic Gaudin and$models to give the dynamics. The moment map of this twisted cotangent $su_n({\cal T})$ bundle is matched to the $D$-equation with Fayet-Illiopoulos source term, so the dynamics of the N.C. solitons becomes that of the brane. The geometric configuration $(k, u)$ of th$spectral curve ${\rm det}|L(u) - k| = 0$ describes the brane configuration, with the dynamical variables $z_i$ of N.C. solitons as$moduli $T^{\otimes n} / S_n$. Furthermore, in the N.C. Chern-Simons theory for the quantum Hall effect, the constrain equation with quasiparticle source is identified also with the moment map eqaution $the N.C. $su_n({\cal T})$ cotangent bundle with marked points. The eigenfunction of the Gaudin differential $L$-operators as the Laughli$wavefunction is solved by Bethe ansatz. 
  We explicitly find the spectrum of the operators $M^{rs}$ and $\widetilde{M}^{rs}$, which specify the star-product in the matter and ghost sectors correspondingly. Further we derive the diagonal representation for the 3-string vertices. Using this representation we identify the appearing Moyal structures in the matter sector. In addition to the continuous non-commutativity parameter $\theta(\kappa)$ found in hep-th/0202087 we find the discrete non-commutativity parametrized by $\theta_{\xi}$. 
  In the Seiberg-Witten limit, the low-energy dynamics of N weakly coupled identical open strings on a D3-brane can behave as two-dimensional incompressible hydrodynamics. Classical vortices are frozen in the fluid and described by an action expressed in terms of two canonical conjugate fields, which can be taken as the new space coordinate. The noncommutative space naturally arises when this pair of conjugate fields are quantized. To the lowest order of $\hbar$, the vorticity can replace the background $B$-field on the D3 brane, thereby yielding a spatially and temporally varying noncommutative parameter $\theta^{ij}$. Demanding a quantum area-preserving transformation between two classical inertial-frame coordinates, we identify the classical solitons that survive in the noncommutative space, and they turn out to be the "electric field" solutions of the Dirac-Born-Infeld Lagrangian created by a $\delta$-function source. The strongly magnetized electron gas in a semiconductor of finite thickness is taken as a case study, where similar quantum column vortices as those in a D3 brane can be present. The electric charges contained in these electron-gas column vortices are quantized, but in a way different from those in the sheet vortices that produce the fractional quantum Hall effect. 
  The conductivity of a finite temperature 1+1 dimensional fermion gas described by the massive Thirring model is shown to be related to the retarded propagator of the dual boson sine-Gordon model. Duality provides a natural resummation which resolves infra-red problems, and the boson propagator can be related to the fermion gas at non-zero temperature and chemical potential or density. In addition, at high temperatures, we can apply a dimensional reduction technique to find resummed closed expressions for the boson self-energy and relate them to the fermion conductivity. Particular attention is paid to the discussion of analytic continuation. The resummation implicit in duality provides a powerful alternative to the standard diagrammatic evaluation of transport coefficients at finite temperature. 
  Infinite set of higher spin conserved charges is found for the $sp(2M)$ symmetric dynamical systems in $\f{1}{2} M (M+1)$-dimensional generalized spacetime $\M_M$. Since the dynamics in $\M_M$ is equivalent to the conformal dynamics of infinite towers of fields in $d$-dimensional Minkowski spacetime with $d=3,4,6,10,...$ for $M=2,4,8,16, ...$, respectively, the constructed currents in $\M_M$ generate infinite towers of (mostly new) higher spin conformal currents in Minkowski spacetime. The charges have a form of integrals of $M$-forms which are bilinear in the field variables and are closed as a consequence of the field equations. Conservation implies independence of a value of charge of a local variation of a $M$-dimensional integration surface $\Sigma \subset \M_M$ analogous to Cauchy surface in the usual spacetime. The scalar conserved charge provides an invariant bilinear form on the space of solutions of the field equations that gives rise to a positive definite norm on the space of quantum states. 
  We consider string theory in a time dependent orbifold with a null singularity. The singularity separates a contracting universe from an expanding universe, thus constituting a big crunch followed by a big bang. We quantize the theory both in light-cone gauge and covariantly. We also compute some tree and one loop amplitudes which exhibit interesting behavior near the singularity. Our results are compatible with the possibility that strings can pass through the singularity from the contracting to the expanding universe, but they also indicate the need for further study of certain divergent scattering amplitudes. 
  This lecture note covers topics on boundary conformal field theory, modular transformations and the Verlinde formula, and boundary logarithmic CFT. An introductory review on CFT with boundary and a discussion of its applications to logarithmic cases are given. LCFT at $c=-2$ is mainly discussed. 
  The fuzzy supersphere $S_F^{(2,2)}$ is a finite-dimensional matrix approximation to the supersphere $S^{(2,2)}$ incorporating supersymmetry exactly. Here the star-product of functions on $S_F^{(2,2)}$ is obtained by utilizing the OSp(2,1) coherent states. We check its graded commutative limit to $S^{(2,2)}$ and extend it to fuzzy versions of sections of bundles using the methods of [1]. A brief discussion of the geometric structure of our star-product completes our work. 
  Contributions to the bound-state dynamics of fermions in local quantum field theory from the region of large relative momenta of the constituent particles, are studied and compared in two different approaches. The first approach is conventionally developed in terms of bare fermions, a Tamm-Dancoff truncation on the particle number, and a momentum-space cutoff that requires counterterms in the Fock-space Hamiltonian. The second approach to the same theory deals with bound states of effective fermions, the latter being derived from a suitable renormalization group procedure. An example of two-fermion bound states in Yukawa theory, quantized in the light-front form of dynamics, is discussed in detail. The large-momentum region leads to a buildup of overlapping divergences in the bare Tamm-Dancoff approach, while the effective two-fermion dynamics is little influenced by the large-momentum region. This is illustrated by numerical estimates of the large-momentum contributions for coupling constants on the order of between 0.01 and 1, which is relevant for quarks. 
  We study the ghost sector of vacuum string field theory where the BRST operator Q is given by the midpoint insertion proposed by Gaiotto, Rastelli, Sen and Zwiebach. We introduce a convenient basis of half-string modes in terms of which Q takes a particularly simple form. We show that there exists a field redefinition which reduces the ghost sector field equation to a pure projection equation for string fields satisfying the constraint that the ghost number is equally divided over the left- and right halves of the string. When this constraint is imposed, vacuum string field theory can be reformulated as a $U(\infty)$ cubic matrix model. Ghost sector solutions can be constructed from projection operators on half-string Hilbert space just as in the matter sector. We construct the ghost sector equivalent of various well-known matter sector projectors such as the sliver, butterfly and nothing states. 
  Noncommutative gravity in three dimensions with vanishing cosmological constant is examined. We find a solution which describes a spacetime in the presence of a torsional source. We estimate the phase shift for each partial wave of a scalar field in the spacetime by the Born approximation. 
  Based on the observation that $AdS_5\times S^5/Z_k$ orbifolds have a maximal supersymmetric PP-wave limit, the description of strings in PP-waves in terms of ${\cal N}=2$ quiver gauge theories is presented. We consider two different, small and large $k$, cases and show that the operators in the gauge theory which correspond to stringy excitations are labelled by two integers, the excitation and winding or momentum numbers. For the large $k$ case, the relation between the space-time and worldsheet deconstructions is discussed. We also comment on the possible duality between these two cases. 
  Static field classical configurations in (1+1)-dimensions for new non-linear potential models are investigated from an isospectral potential class and the concept of bosonic zero- mode solution. One of the models here considered has a static non-topological configuration with a single vacuum state that breaks supersymmetry. 
  Discrete symmetry breaking and possible restoration at finite temperature $T$ are analysed in 2D Gross-Neveu model by the real-time thermal field theory in the fermion bubble approximation. The dynamical fermion mass $m$ is proven to be scale-independent and this fact indicates the equivalence between the fermion bubble diagram approximation and the mean field approximation used in the auxialiary scalar field approach. Reproducing of the non-zero critical temperature $T_c=0.567 m(0)$, ($m(0)$ is the dynamical fermion mass at T=0), shows the equivalence between the real-time and the imaginary-time thermal field theory in this problem. However, in the real-time formalism, more results including absence of scalar bound state, the equation of criticality curve of chemical potential-temperature and the $\ln(T_c/T)$ behavior of $m^2$ at $T\stackrel{<}{\sim} T_c$ can be easily obtained. The last one indicates the second-order phase transition feature of the symmetry restoration. 
  We consider an extended linear $\sigma $ model in which the fermions are quarks and are coupled to gluons. Equivalently, this is QCD extended by coupling the quarks to a colour singlet chiral multiplet of ($ \sigma, \vec \pi $) fields. This theory has a phase governed by a UV fixed point where all couplings are AF (asymptotically free). This implies that the scalars are elementary at high energies (UV) and, as they are colour singlets, they are not confined at low energy (IR). Thus, the scalar particles are elementary at all scales. 
  Looking for a quantum-mechanical implementation of duality, we formulate a relation between coherent states and complex-differentiable structures on classical phase space ${\cal C}$. A necessary and sufficient condition for the existence of locally-defined coherent states is the existence of an almost complex structure on ${\cal C}$. A necessary and sufficient condition for globally-defined coherent states is a complex structure on ${\cal C}$.   The picture of quantum mechanics that emerges is conceptually close to that of a geometric manifold covered by local coordinate charts. Instead of the latter, quantum mechanics has local coherent states. A change of coordinates on ${\cal C}$ may or may not be holomorphic. Correspondingly, a transformation between quantum-mechanical states may or may not preserve coherence. Those that do not preserve coherence are duality transformations. A duality appears as the possibility of giving two or more, apparently different, descriptions of the same quantum-mechanical phenomenon. Coherence becomes a local property on classical phase space. Observers on ${\cal C}$ not connected by means of a holomorphic change of coordinates need not, and in general will not, agree on what is a semiclassical effect vs. what is a strong quantum effect 
  Given a boundary of spacetime preserved by a Diff(S^{1}) sub-algebra, we propose a systematic method to compute the zero mode and the central extension of the associated Virasoro algebra of charges. Using these values in the Cardy formula, we may derive an associated statistical entropy to be compared with the Bekenstein-Hawking result.   To illustrate our method, we study in detail the BTZ and the rotating Kerr-adS_{4} black holes (at spatial infinity and on the horizon). In both cases, we are able to reproduce the area law with the correct factor of 1/4 for the entropy. We also recover within our framework the first law of black hole thermodynamics.   We compare our results with the analogous derivations proposed by Carlip and others. Although similar, our method differs in the computation of the zero mode. In particular, the normalization of the ground state is automatically fixed by our construction. 
  We give observations about dualities where one of the dual theories is geometric. These are illustrated with a duality between the simple harmonic oscillator and a topological field theory. We then discuss the Wilson loop in the context of the AdS/CFT duality. We show that the Wilson loop calculation for certain asymptotically AdS scalar field spacetimes with naked singularities gives results qualitatively similar to that for the AdS black hole. In particular, it is apparent that (dimensional) metric parameters in the singular spacetimes permit a ``thermal screening'' interpretation for the quark potential in the boundary theory, just like black hole mass. This suggests that the Wilson loop calculation merely captures metric parameter information rather than true horizon information. 
  We construct boundary quantum group generators which, through linear intertwining relations, determine nondiagonal solutions of the boundary Yang-Baxter equation for the cases A^1_{n-1} and A^2_2. 
  A novel approach that does not rely on supersymmetry, nor in the AdS/CFT correspondence, to evaluate the Wilson loops asociated with a gauge theory of area-preserving diffeomorphisms in terms of pure string degrees of freedoms is presented. It is based on the Guendelman-Nissimov-Pacheva formulation of composite antisymmetric tensor field theories of volume-preserving diffeomorphisms. Such theories admit p-brane solutions. The quantum effects are discussed and we evaluate exactly the vev of the Wilson loops, in the large N limit of quenched-reduced SU(N) QCD, in terms of a path integral involving pure string degrees of freedom. It is consistent with the recent results based on the AdS/CFT correspondence and dual QCD models (dual Higgs model with dual Dirac strings). More general Loop wave equations in C-spaces (Clifford manifolds) are proposed in terms of holographic variables that contain the dynamics of an aggregate of closed branes (p-loops) of various dimensionalities. 
  We discuss the stringy properties of high-energy QCD using its hidden integrability in the Regge limit and on the light-cone. It is shown that multi-colour QCD in the Regge limit belongs to the same universality class as superconformal $\cal{N}$=2 SUSY YM with $N_f=2N_c$ at the strong coupling orbifold point. The analogy with integrable structure governing the low energy sector of $\cal{N}$=2 SUSY gauge theories is used to develop the brane picture for the Regge limit. In this picture the scattering process is described by a single M2 brane wrapped around the spectral curve of the integrable spin chain and unifying hadrons and reggeized gluons involved in the process. New quasiclassical quantization conditions for the complex higher integrals of motion are suggested which are consistent with the $S-$duality of the multi-reggeon spectrum. The derivation of the anomalous dimensions of the lowest twist operators is formulated in terms of the Riemann surfaces 
  We study in detail the Quantum Field Theory of mixing among three generations of Dirac fermions (neutrinos). We construct the Hilbert space for the flavor fields and determine the generators of the mixing transformations. By use of these generators, we recover all the known parameterizations of the three-flavor mixing matrix and we find a number of new ones. The algebra of the currents associated with the mixing transformations is shown to be a deformed $su(3)$ algebra, when CP violating phases are present. We then derive the exact oscillation formulas, exhibiting new features with respect to the usual ones. CP and T violation are also discussed. 
  In this paper we study the noncommutative extension of a modified U(n) sigma model in 2+1 dimensions. Using the method of dressing transformations, an iterative approach for the construction of solutions from a given seed solution, we demonstrate the construction of multi-soliton and soliton-antisoliton configurations for general n. As illustrative examples we discuss U(3) solitons and consider the head-on collision of a U(2) soliton and an antisoliton explicitly, which will result in a 90^{\text{o}} angle scattering. Further we discuss the head-on collision of one U(2) soliton with two antisolitons. This results in a 60^{\text{o}} angle scattering. 
  The equivalence of NS5-branes and ALF spaces under T-duality is well known. However, a naive application of T-duality transforms the ALF space into a smeared NS5-brane, de-localized on the dual, transverse, circle. In this paper we re-examine this duality, starting from a two-dimensional N=(4,4) gauged linear sigma model describing Taub-NUT space. After dualizing the circle fiber, we find that the smeared NS5-brane target space metric receives corrections from multi-worldsheet instantons. These instantons are identified as Nielsen-Olesen vortices. We show that their effect is to break the isometry of the target space, localizing the NS5-brane at a point. The contribution from the k-instanton sector is shown to be proportional to the weighted integral of the Euler form over the k-vortex moduli space. The duality also predicts the, previously unknown, asymptotic exponential decay coefficient of the BPS vortex solution. 
  We consider the evolution of FRW cosmological models and linear perturbations of tachyon matter rolling towards a minimum of its potential. The tachyon coupled to gravity is described by an effective 4d field theory of string theory tachyon. In the model where a tachyon potential $V(T)$ has a quadratic minimum at finite value of the tachyon field $T_0$ and $V(T_0)=0$, the tachyon condensate oscillates around its minimum with a decreasing amplitude. It is shown that its effective equation of state is $p=-\epsilon/3$. However, linear inhomogeneous tachyon fluctuations coupled to the oscillating background condensate are exponentially unstable due to the effect of parametric resonance. In another interesting model, where tachyon potential exponentially approaches zero at infinity of $T$, rolling tachyon condensate in an expanding universe behaves as pressureless fluid. Its linear fluctuations coupled with small metric perturbations evolve similar to these in the pressureless fluid. However, this linear stage changes to a strongly non-linear one very early, so that the usual quasi-linear stage observed at sufficiently large scales in the present Universe may not be realized in the absence of the usual particle-like cold dark matter. 
  We consider the master Lagrangian of Deser and Jackiw, interpolating between the self-dual and the Maxwell-Chern-Simons Lagrangian, and quantize it following the symplectic approach, as well as the traditional Dirac scheme. We demonstrate the equivalence of these procedures in the subspace of the second-class constraints. We then proceed to embed this mixed first- and second-class system into an extended first-class system within the framework of both approaches, and construct the corresponding generator for this extended gauge symmetry in both formulations. 
  We study a quotient Conformal Field Theory, which describes a 3+1 dimensional cosmological spacetime. Part of this spacetime is the Nappi-Witten (NW) universe, which starts at a ``big bang'' singularity, expands and then contracts to a ``big crunch'' singularity at a finite time. The gauged WZW model contains a number of copies of the NW spacetime, with each copy connected to the preceeding one and to the next one at the respective big bang/big crunch singularities. The sequence of NW spacetimes is further connected at the singularities to a series of non-compact static regions with closed timelike curves. These regions contain boundaries, on which the observables of the theory live. This suggests a holographic interpretation of the physics. 
  The most remarkable and interesting feature of brane world scenario is the use of bulk's curvature to localize gravity on the brane, albeit with fine tuning of the brane and bulk parameters. For FRW expanding universe on the brane, it is a moving hypersurface in Schwarzschild anti de Sitter bulk spacetime. We show that zero mass gravitons have bound state on the brane for suitable values of brane and bulk parameters. There occur various cases giving rise to different cosmological models, in particular we discuss a model with positive cosmological constant on the brane. 
  Using an exact supergravity solution representing the Dp-\bar{Dp} system, it is demonstrated that one can construct a supergravity analogue of the tachyon potential. Remarkably, the (regularized) minimum value of the potential turns out to be V(T_{0})=-2m with m denoting the ADM mass of a single Dp-brane. This result, in a sense, appears to confirm that Sen's conjecture for the tachyon condensation on unstable D-branes is indeed correct although the analysis used here is semi-classical in nature and hence should be taken with some care. Also shown is the fact that the tachyon mass squared m^2_{T} (which has started out as being negative) can actually become positive definite and large as the tachyon rolls down toward the minimum of its potential. It indeed signals the possibility of successful condensation of the tachyon since it shows that near the minimum of its potential, tachyon can become heavy enough to disappear from the massless spectrum. Some cosmological implications of this tachyon potential in the context of ``rolling tachyons'' is also discussed. 
  We show that the high temperature limit of the noncommutative thermal Yang-Mills theory can be directly obtained from the Boltzmann transport equation of classical particles. As an illustration of the simplicity of the Boltzmann method, we evaluate the two and the three-point gluon functions in the noncommutative U(N) theory at high temperatures T. These amplitudes are gauge invariant and satisfy simple Ward identities. Using the constraint satisfied at order T^2 by the covariantly conserved current, we construct the hard thermal loop effective action of the noncommutative theory. 
  Using the AdS/CFT correspondence we study vacua of N=4 SYM for which part of the gauge symmetry is broken by expectation values of scalar fields. A specific subclass of such vacua can be analyzed with gauged supergravity and the corresponding domain wall solutions lift to continuous distributions of D3-branes in type IIB string theory. Due to the non-trivial expectation value of the scalars, the SO(6) R-symmetry is spontaneously broken and field theory predicts the existence of Goldstone bosons. We explicitly show that, in the dual supergravity description, these emerge as massless poles in the current two-point functions, while the bulk gauge fields which are dual to the broken currents become massive via the Higgs mechanism. We find agreement with field theory expectations and, hence, provide a non-trivial test of the AdS/CFT correspondence far away from the conformal point. 
  In this talk I study the topology of mathematically idealised center vortices, defined in a gauge invariant way as closed (infinitely thin) flux surfaces (in D=4 dimensions) which contribute the $n^{th}$ power of a non-trivial center element to Wilson loops when they are n-foldly linked to the latter. In ordinary 3-space generic center vortices represent closed magnetic flux loops which evolve in time. I show that the topological charge of such a time-dependent vortex loop can be entirely expressed by the temporal changes of its writhing number. 
  We consider non-supersymmetric quiver theories obtained by orbifolding the N=4 supersymmetric U(K) gauge theory by a discrete Z_\Gamma group embedded in the SU(4) R-symmetry group. We explicitly find that in such theories there are no one-loop quadratic divergences in the effective potential.  Moreover, when the gauge group U(n)^\Gamma of the quiver theory is spontaneously broken down to the diagonal U(n), we identify a custodial supersymmetry which is responsible for the fermion-boson degeneracy of the mass spectrum. 
  Exploiting insights on strings moving in pp-wave backgrounds, we show how open strings emerge from N = 4 SU(M) Yang-Mills theory as fluctuations around certain states carrying R-charge of order M. These states are dual to spherical D3-branes of AdS_5 x S^5 and we reproduce the spectrum of small fluctuations of these states from Yang Mills theory. When G such D3-branes coincide, the expected G^2 light degrees of freedom emerge. The open strings running between the branes can be quantized easily in a Penrose limit of the spacetime. Taking the corresponding large charge limit of the Yang-Mills theory, we reproduce the open string worldsheets and their spectra from field theory degrees of freedom. 
  A simple method to canonically quantize noncommutative field theories is proposed. As a result, the elementary excitations of a (2n+1)-dimensional scalar field theory are shown to be bilocal objects living in an (n+1)-dimensional space-time. Feynman rules for their scattering are derived canonically. They agree, upon suitable redefinitions, with the rules obtained via star-product methods. The IR/UV connection is interpreted within this framework. 
  For n+1 dimensional asymptotically AdS spacetimes which have holographic duals on their n dimensional conformal boundaries, we show that the imposition of causality on the boundary theory is sufficient to prove positivity of mass for the spacetime when n > 2, without the assumption of any local energy condition. We make crucial use of a generalization of the time-delay formula calculated in gr-qc/9404019, which relates the time delay of a bulk null curve with respect to a boundary null geodesic to the Ashtekar-Magnon mass of the spacetime. We also discuss holographic causality for the negative mass AdS soliton and its implications for the positive energy conjecture of Horowitz and Myers. 
  This paper shows how to construct anomaly free world sheet actions in string theory with $D$-branes. Our method is to use Deligne cohomology and bundle gerbe theory to define geometric objects which are naturally associated to $D$-branes and connections on them. The holonomy of these connections can be used to cancel global anomalies in the world sheet action. 
  We investigate colliding processes of closed strings with large angular momenta with D-branes. We give explicit CFT calculations for closed string states with an arbitrary number of bosonic excitations and no or one fermion excitation. The results reproduce the correspondence between closed string states and single trace operators in the boundary gauge theory recently suggested by Berenstein, Maldacena and Nastase. 
  Five-brane distributions with no strong coupling problems and high symmetry are studied. The simplest configuration corresponds to a spherical shell of branes with S^3 geometry and symmetry. The equations of motions with delta-function sources are carefully solved in such backgrounds. Various other brane distributions with sixteen unbroken supercharges are described. They are associated to exact world-sheet superconformal field theories with domain-walls in space-time. We study the equations of gravitational fluctuations, find normalizable modes of bulk 6-d gravitons and confirm the existence of a mass gap. We also study the moduli of the configurations and derive their (normalizable) wave-functions. We use our results to calculate in a controllable fashion using holography, the two-point function of the stress tensor of little string theory in these vacua. 
  We find out that some unitary minimal models of the N=1 ${\cal SW}(3/2,2)$ superconformal algebra can be realized as the level one coset models based on the Wolf spaces $SU(n)/(SU(n-2)\times SU(2))$. We obtain the expression of the fermionic current with the conformal weight 5/2 in the algebra. Then, these models are twisted to give the topological conformal field theories. 
  Using D-brane effective field theories, we study dynamical decay of unstable brane systems : (i) a parallel brane-antibrane pair with separation l and (ii) a dielectric brane. In particular we give explicitly the decay width of these unstable systems, and describe how the decay proceeds after the tunnel effect. The decay (i) is analysed by the use of a tachyon effective action on the Dp-Dpbar. A pair annihilation starts by nucleation of a bubble of a tachyon domain wall which represents a throat connecting these branes, and the tunneling decay width is found to be proportional to exp(-l^{p+1} T_{Dp}). We study also the decay leaving topological defects corresponding to lower-dimensional branes, which may be relevant for recent inflationary braneworld scenario. As for the decay (ii), first we observe that Dp-branes generically ``curl up'' in a nontrivial RR field strength. Using this viewpoint, we compute the decay width of the dielectric D2-branes by constructing relevant Euclidean bounce solutions in the shape of a funnel. We also give new solutions in doughnut shape which are involved with nucleation of dielectric branes from nothing. 
  We analyse of the effective action of the tachyon field on a D-brane, of both bosonic as well as superstring theory. We find that the non-standard kinetic term of the tachyon field requires a correction to the Born-Infeld type Lagrangian. The cosmological significance of the resulting dynamics is explored. We also examine if the rolling tachyon can provide an effective cosmological constant and contrast its behaviour with quintessence. 
  In some models with infinite extra dimensions, gauge fields are localized on a brane by gravity. A generic property of these models is the existence of arbitrarily light bulk modes of charged fields. This property may lead to interesting low-energy effects such as electric charge non-conservation on the brane (decay of electron to nothing). One may worry that light charged Kaluza--Klein modes would lead to unacceptable phenomenology due to their copious production in, e.g., electron-positron annihilation and/or their contribution to QED observables like anomalous magnetic moments. We argue, however, that both loop effects and production of light charged Kaluza--Klein modes are suppressed due to screening effect of gapless spectrum of bulk photons. 
  We show that D=4 Schwarzschild black holes can arise from a doublet of Euclidean D3-antiD3 pairs embedded in D=10 Lorentzian spacetime. By starting from a D=10 type IIB supergravity description for the D3-antiD3 pairs and wrapping one of them over an external 2-sphere, we derive all vacuum solutions compatible with the symmetry of the problem. Analysing under what condition a Euclidean brane configuration embedded in a Lorentzian spacetime can lead to a time-independent spacetime, enables us to single out the embedded D=4 Schwarzschild spacetime as the unique solution generated by the D3-antiD3 pairs. In particular we argue on account of energy-conservation that time-independent solutions arising from isolated Euclidean branes require those branes to sit at event horizons. In combination with previous work this self-dual brane-antibrane origin of the black hole allows for a microscopic counting of its Bekenstein-Hawking entropy. Finally we indicate how Hawking-radiation can be understood from the associated tachyon condensation process. 
  A description of the bosonic sector of massive IIA supergravity as a non-linear realisation is given. An essential feature of this construction is that the momentum generators have non-trivial commutation relations with the generators associated with the gauge fields. 
  We outline the proofs of several principal statements in conventional renormalization theory. This may be of some use in the light of new trends and new techniques (Hopf algebras, etc.) recently introduced in the field. 
  We consider giant gravitons as probes of a class of ten-dimensional solutions of type IIB supergravity which arise as lifts of solutions of U(1)^3 gauged N=2 supergravity in five-dimensions. Surprisingly it is possible to solve exactly for minimum energy configurations of these spherical D3-brane probes in the compact directions, even in backgrounds which preserve no supersymmetry. The branes behave as massive charged particles in the five non-compact dimensions. As an example we probe geometries which are believed to represent the supergravity background of coherent states of giant gravitons. We comment on the apparently repulsive nature of the naked singularities in these geometries. 
  We formulate a dynamically triangulated model of three-dimensional Lorentzian quantum gravity whose spatial sections are flat two-tori. It is shown that the combinatorics involved in evaluating the one-step propagator (the transfer matrix) is that of a set of vicious walkers on a two-dimensional lattice with periodic boundary conditions and that the entropy of the model scales exponentially with the volume. We also give explicit expressions for the Teichm\"uller parameters of the spatial slices in terms of the discrete parameters of the 3d triangulations, and reexpress the discretized action in terms of them. The relative simplicity and explicitness of this model make it ideally suited for an analytic study of the conformal-factor cancellation observed previously in Lorentzian dynamical triangulations and of its relation to alternative, reduced phase space quantizations of 3d gravity. 
  Divergent part of the one-loop effective action for the Yang-Mills theory in a special gauge containing forth degrees of ghost fields and allowing addition of BRST-invariant mass term is calculated by the generalized t'Hooft-Veltman technique. The result is BRST-invariant and defines running mass, coupling constant and parameter of the gauge. 
  We show why the universe started in an unstable de Sitter state. The quantum origin of our universe implies one must take a `top down' approach to the problem of initial conditions in cosmology, in which the histories that contribute to the path integral, depend on the observable being measured. Using the no boundary proposal to specify the class of histories, we study the quantum cosmological origin of an inflationary universe in theories like trace anomaly driven inflation in which the effective potential has a local maximum. We find that an expanding universe is most likely to emerge in an unstable de Sitter state, by semiclassical tunneling via a Hawking-Moss instanton. Since the top down view is forced upon us by the quantum nature of the universe, we argue that the approach developed here should still apply when the framework of quantum cosmology will be based on M-Theory. 
  We provide a new class of exactly solvable superconformal field theories that corresponds to type II compactification on manifolds with exceptional holonomies. We combine N=1 Liouville field and N=1 coset models and construct modular invariant partition functions of strings moving on these manifolds. The resulting theories preserve spacetime supersymmetry. Also we explicitly construct chiral currents in these models to realize consistent string theories. 
  Low energy theorems of Nambu-Goldstone fermion associated with spontaneously broken supersymmetry are studied for gauge supermultiplets. Two possible terms in the effective Lagrangian are needed to deal with massless gaugino and/or massless gauge boson. As an illustrative example, a concrete model is worked out which can interpolate massless as well as massive gaugino and/or gauge boson to examine the low energy effective interaction of NG-fermion. 
  We study the dynamical features of Maggiore's generalised commutation relations. We focus on their generality and, in particular, their dependence on the Hamiltonian H. We derive the generalisation of the Planck's law for black body spectrum, study the statistical mechanics of free particles, and study the early universe evolution which now exhibits non trivial features. We find that the dynamical features, found here and in our earlier work, are all generic and vary systematically with respect to the asymptotic growth of the Hamiltonian H. 
  We introduce some techniques to investigate dynamical mass generation. The Gross-Neveu model (GN) is used as a toy model, because the GN mass gap is exactly known, making it possible to check reliability of the various methods. Very accurate results are obtained. Also application to SU(N) Yang-Mills (YM) is discussed. 
  In this paper, in the frame of Extended Electrodynamics (EED), we study some of the consequences that can be obtained from the introduced and used by Maxwell equations complex structure \mathcal{J} in the space of 2-forms on \mathbb{R}^4, and also used in EED. First we give the vacuum EED equations with some comments. Then we recall some facts about the invariance group $G$ (with Lie algebra \mathcal{G}) of the standard complex structure $J$ in \mathbb{R}^2. After defining and briefly studying a representation of $G$ in the space of 2-forms on \mathbb{R}^4 and the joint action of $G$ in the space of \mathcal{G}-valued 2-forms on \mathbb{R}^4 we consider its connection with the vacuum solutions of EED. Finally we consider the case with point dependent group parameters and show that the set of the nonlinear vacuum EED solutions is a disjoint union of orbits of the $G$-action, noting some similarities with the quantim mechanical eigen picture and with the QFT creation and anihilation operators. 
  We use a new, exact approach in calculating the energy density measured by an observer living on a brane embedded in a charged black hole spacetime. We find that the bulk Weyl tensor gives rise to non-linear terms in the energy density and pressure in the FRW equations for the brane. Remarkably, these take exactly the same form as the ``unconventional'' terms found in the cosmology of branes embedded in pure AdS, with extra matter living on the brane. Black hole driven cosmologies have the benefit that there is no ambiguity in splitting the braneword energy momentum into tension and additional matter. We propose a new, enlarged relationship between the two descriptions of braneworld cosmology. We also study the exact thermodynamics of the field theory and present a generalised Cardy-Verlinde formula in this set up. 
  Field theory on a fuzzy noncommutative sphere can be considered as a particular matrix approximation of field theory on the standard commutative sphere. We investigate from this point of view the scalar $\phi^4$ theory. We demonstrate that the UV/IR mixing problems of this theory are localized to the tadpole diagrams and can be removed by an appropiate (fuzzy) normal ordering of the $\phi^4$ vertex. The perturbative expansion of this theory reduces in the commutative limit to that on the commutative sphere. 
  One considers the quantum dynamics of a charged spin-1/2 particle in an extended external eletromagnetic field that arises from the reduction of a 5-dimensional Abelian gauge theory. The non-relativistic regime of the reduced 4D-dynamics is worked out and one identifies the system as a sector of an N=2-supersymmetric quantum-mechanical dynamics. The full supersymmetric model is studied and one checks the algebra of the fermionic charges; the conclusion is that no central charge drops out. The possible r\^ole of the extra external fields, a scalar and a magnetic-like field, is discussed. 
  We consider the maximal supersymmetric pure Yang-Mills theories on six and eight dimensional space. We determine, in a systematic way, all the possible fractions of supersymmetry preserved by the BPS states and present the corresponding `self-dual' BPS equations. In six dimensions the intrinsic one has 1/4 supersymmetry, while in eight dimensions, 1/16, 2/16, ..., 6/16. We apply our results to some explicit BPS configurations of finite or infinite energy on commutative or noncommutative spaces. 
  In the paper ``Density perturbations in the ekpyrotic scenario'', it is argued that the expected spectrum of primordial perturbations should be scale invariant in this scenario. Here we show that, contrary to what is claimed in that paper, the expected spectrum depends on an arbitrary choice of matching variable. As no underlying (microphysical) principle exists at the present time that could lift the arbitrariness, we conclude that the ekpyrotic scenario is not yet a predictive model. 
  We have computed one-loop bulk and brane mass renormalization effects in a five-dimensional gauge theory compactified on the M_4 \times S^1/Z_2 orbifold, where an arbitrary gauge group G is broken by the orbifold action to its subgroup H. The space-time components of the gauge boson zero modes along the H generators span the gauge theory on the orbifold fixed point branes while the zero modes of the higher-dimensional components of the gauge bosons along the G/H generators play the role of Higgs fields with respect to the gauge group H. No quadratic divergences in the mass renormalization of the gauge and Higgs fields are found either in the bulk or on the branes. All brane effects for the Higgs field masses vanish (only wave function renormalization effects survive) while bulk effects are finite and can trigger, depending on the fermionic content of the theory, spontaneous Hosotani breaking of the brane gauge group H. For the gauge fields we do find logarithmic divergences corresponding to mass renormalization of their heavy Kaluza-Klein modes. Two-loop brane effects for Higgs field masses are expected from wave function renormalization brane effects inserted into finite bulk mass corrections. 
  A class of spectral problems with a hidden Lie-algebraic structure is considered. We define a duality transformation which maps the spectrum of one quasi-exactly solvable (QES) periodic potential to that of another QES periodic potential. The self-dual point of this transformation corresponds to the energy-reflection symmetry found previously for certain QES systems. The duality transformation interchanges bands at the bottom (top) of the spectrum of one potential with gaps at the top (bottom) of the spectrum of the other, dual, potential. Thus, the duality transformation provides an exact mapping between the weak coupling (perturbative) and semiclassical (nonperturbative) sectors. 
  Using the superembedding approach, the full supersymmetric effective field theory of the D9-brane, super Born-Infeld theory, is fixed by the so called $\mathcal{F}$-constraint. The odd-odd components of the theory's super field strength, $f_{\alpha \beta}$, are implied by this constraint. Given $f_{\alpha \beta}$, the super Bianchi identities imply the theory's equations of motion. We calculate $f_{\alpha \beta}$ up to order 5 in fields, corresponding to order 6 in fields in the Lagrangian. 
  Motivated by recent proposals in hep-th/0202021 and hep-th/0204051 we develop semiclassical quantization of superstring in $AdS_5 x S^5$. We start with a classical solution describing string rotating in $AdS_5$ and boosted along large circle of $S^5$. The energy of the classical solution $E$ is a function of the spin $S$ and the momentum $J$ (R-charge) which interpolates between the limiting cases S=0 and J=0 considered previously. We derive the corresponding quadratic fluctuation action for bosonic and fermionic fields from the GS string action and compute the string 1-loop (large $\lambda= {R^4\over \a'^2}$) correction to the classical energy spectrum in the $(S,J)$ sector. We find that the 1-loop correction to the ground-state energy does not cancel for non-zero $S$. For large $S$ it scales as $\ln S$, i.e. as the classical term, with no higher powers of $\ln S$ appearing. This supports the conjecture made in hep-th/0204051 that the classical $E-S = a \ln S$ scaling can be interpolated to weak coupling to reproduce the corresponding operator anomalous dimension behaviour in gauge theory. 
  With the new cosmological data gathered over the last few years, the inflationary paradigm has seen its predictions largely unchallenged. A recent proposal, called the ekpyrotic scenario, was argued to be a viable competitor as it was claimed that the spectrum of primordial perturbations it produces is scale invariant. By investigating closely this scenario, we show that the corresponding spectrum depends explicitly on an arbitrary function of wavenumber and is therefore itself arbitrary. It can at will be set scale invariant. We conclude that the scenario is not predictive at this stage. 
  In this note, we use results of Aspinwall and Morrison to discuss the F-theory duals of certain $T^4/\bbZ_N$ orbifold compactifications of Ho\v{r}ava--Witten theory. In the M-theory limit an interesting set of rules, based on anomaly cancellation, has been developed for what gauge and matter multiplets must be present on the various orbifold fixed planes. Here we show how several aspects of these rules can be understood directly from F-theory. 
  We prove that the helicity is preserved in the scattering of photons in the Born-Infeld theory (in 4d) on the tree level. 
  The assignment of local observables in the vacuum sector, fulfilling the standard axioms of local quantum theory, is known to determine uniquely a compact group G of gauge transformations of the first kind together with a central involutive element k of G, and a complete normal algebra of fields carrying the localizable charges, on which k defines the Bose/Fermi grading.  We show here that any such pair {G,k}, where G is compact metrizable, does actually appear. The corresponding model can be chosen to fulfill also the split property.  This is not a dynamical phenomenon: a given {G,k} arises as the gauge group of a model where the local algebras of observables are a suitable subnet of local algebras of a possibly infinite product of free field theories. 
  We construct a new (singular) cohomogeneity-three metric of G_2 holonomy. The solution can be viewed as a triple intersection of smeared Taub-NUTs. The metric comprises three non-compact radial-type coordinates, with the principal orbits being a T^3 bundle over S^1. We consider an M-theory vacuum (Minkowski)_4\times M_7 where M_7 is the G_2 manifold. Upon reduction on a circle in the T^3, we obtain the intersection of a D6-brane, a Taub-NUT and a 6-brane with R-R 2-form flux. Reducing the solution instead on the base space S^1, we obtain three intersecting 6-branes all carrying R-R 2-form flux. These two configurations can be viewed as a classical flop in the type IIA string theory. After reducing on the full principal orbits and the spatial world-volume, we obtain a four-dimensional metric describing a lattice universe, in which the three non-compact coordinates of the G_2 manifold are identified with the spatial coordinates of our universe. 
  The gauge-invariant Chern-Simons-type Lorentz- and CPT-breaking term is here reassessed and a spin-projector method is adopted to account for the breaking (vector) parameter. Issues like causality, unitarity, spontaneous gauge-symmetry breaking and vortex formation are investigated, and consistency conditions on the external vector are identified. 
  In this paper we map the D-brane projector states in the vacuum string field theory to the noncommutative GMS solitons based on the recently proposed map of Witten's star to Moyal's star. We find that the singular geometry conditions of Moore and Taylor are associated with the commutative modes of these projector states in our framework. The properties of the candidate closed string state and the wedge state are also discussed, and the possibility of the non-GMS soliton in VSFT is commented. 
  In this paper we study the superstring version of the exactly solvable string model constructed by Russo and Tseytlin. This model represents superstring theory in a curved spacetime and can be seen as a generalization of the Melvin background. We investigate D-branes in this model as probes of the background geometry by constructing the boundary states. We find that spacetime singularities in the model become smooth at high energy from the viewpoint of open string. We show that there always exist bulk (movable) D-branes by the effect of electric flux. The model also includes Nappi-Witten model as the Penrose limit and supersymmetry is enhanced in the limit. We examine this phenomenon in the open string spectrum. We also find the similar enhancement of supersymmetry can be occurred in several coset models. 
  It is shown that negative tension branes in higher dimensions may lead to an effective lower dimensional theory where the gauge-invariant vector fields associated with the fluctuations of the metric are always massless and localized on the brane. Explicit five-dimensional examples of this phenomenon are provided. Furthermore, it is shown that higher dimensional gauge fields can also be localized on these configurations with the zero mode separated from the massive tower by a gap. 
  Whether a string has rotation and shear can be investigated by an anology with the point particle. Rotation and shear involve first covariant spacetime derivatives of a vector field and, because the metric stress tensor for both the point particle and the string have no such derivatives, the best vector fields can be identified by requiring the conservation of the metric stress. It is found that the best vector field is a non-unit accelerating field in x, rather than a unit non-accelerating vector involving the momenta; it is also found that there is an equation obeyed by the spacetime derivative of the Lagrangian. The relationship between membranes and fluids is looked at. 
  We analyse Coleman's theorem asserting the absence Goldstone bosons and spontaneously broken continuous symmetry in the quantum field theory of a free massless (pseudo)scalar field in 1+1-dimensional space-time (Comm. Math. Phys. 31, 259 (1973)). We confirm that Coleman's theorem reproduces well-known Wightman's statement about the non-existence of a quantum field theory of a free massless (pseudo)scalar field in 1+1-dimensional space-time in terms of Wightman's observables defined on the test functions from S(R^2). Referring to our results (Eur. Phys. J. C 24, 653 (2002)) we argue that a formulation of a quantum field theory of a free massless (pseudo)scalar field in terms of Wightman's observables defined on the test functions from S_0(R^2) is motivated well by the possibility to remove a collective zero-mode motion of the ``center of mass'' of a free massless (pseudo)scalar field (Eur. Phys. J. C 24, 653 (2002)) responsible for infrared divergences of the Wightman functions. We show that in the quantum field theory of a free massless (pseudo)scalar field with Wightman's observables defined on the test functions from S_0(R^2) a continuous symmetry is spontaneously broken. Coleman's theorem reformulated for the test functions from S_0(R^2) does not refute this result. We construct a most general version of a quantum field theory of a self-coupled massless (pseudo)scalar field with a conserved current. We show that this theory satisfies Wightman's axioms and Wightman's positive definiteness condition with Wightman's observables defined on the test functions from S(R^2) and possesses spontaneously broken continuous symmetry. Nevertheless, in this theory the generating functional of Green functions exists only when the collective zero-mode is not excited by the external source. 
  We walk out the landscape of K-theoretic Poincare Duality for finite algebras. It paves the way to get continuum Dirac operators from discrete noncommutative manifolds. 
  By diagonalizing the three-string vertex and using a special coordinate representation the matter part of the open superstring star is identified with the continuous Moyal product of functions of anti-commuting variables. We show that in this representation the identity and sliver have simple expressions. The relation with the half-string fermionic variables in continuous basis is given. 
  For the purpose of analyzing non-perturbative dynamics of string theory, Nishimura and Sugino have applied an improved mean field approximation (IMFA) to IIB matrix model. We have extracted the essence of the IMFA and obtained a general scheme, an improved Taylor expansion, that can be applied to a wide class of series which is not necessarily convergent. This approximation scheme with the help of the 2PI free energy enables us to perform higher order calculations. We have shown that the value of the free energy is stable at higher orders, which supports the validity of the approximation. Moreover, the ratio between the extent of ``our'' space-time and that of the internal space is found to increase rapidly as we take the higher orders into account. Our results suggest that the four dimensional space-time emerges spontaneously in IIB matrix model. 
  The relation between defects of Abelian gauges and instantons is discussed for explicit examples in the Laplacian Abelian gauge. The defect coming from an instanton is pointlike and becomes a monopole loop with twist upon perturbation. The interplay between magnetic charge, twist and instanton number - encoded as a Hopf invariant - is investigated with the help of a new method, an auxiliary Abelian fibre bundle. 
  We consider spacetime with torsion in a Randall-Sundrum (RS) scenario where torsion, identified with the rank-2 Kalb-Ramond field, exists in the bulk together with gravity. While the interactions of both graviton and torsion in the bulk are controlled by the Planck mass, an additional exponential suppression comes for the torsion zero-mode on the visible brane. This may serve as a natural explanation of why the effect of torsion is so much weaker than that of curvature on the brane. The massive torsion modes, on the other hand, are correlated with the corresponding gravitonic modes and may be detectable in TeV-scale experiments. 
  The new approach to quantum mechanical problems is proposed. Quantum states are represented in an algebraic program, by lists of variable length, while operators are well defined functions on these lists. Complete numerical solution of a given system can then be automatically obtained. The method is applied to Wess-Zumino quantum mechanics and D=2 and D=4 supersymmetric Yang-Mills quantum mechanics with the SU(2) gauge group. Convergence with increasing size of the basis was observed in various cases. Many old results were confirmed and some new ones, especially for the D=4 system, are derived. Preliminary results in higher dimensions are also presented. In particular the spectrum of the zero-volume glueballs in $ 4 < D < 11 $ is obtained for the first time. 
  We construct modular spaces of all 6-dimensional real semisimple Drinfeld doubles, i.e. the sets of all possible decompositions of the Lie algebra of the Drinfeld double into Manin triples. These modular spaces are significantly different from the known one for Abelian Drinfeld double, since some of these Drinfeld doubles allow decomposition into several non-isomorphic Manin triples and their modular spaces are therefore written as unions of homogeneous spaces of different dimension. Implications for Poisson-Lie T-duality and especially Poisson-Lie T-plurality are mentioned. 
  Doubly Special Relativity (DSR) theory is a recently proposed theory with two observer-independent scales (of velocity and mass), which is to describe a kinematic structure underlining the theory of Quantum Gravity. We observe that there is infinitely many DSR constructions of the energy-momentum sector, each of whose can be promoted to the $\kappa$-Poincar\'e quantum (Hopf) algebra. Then we use the co-product of this algebra and the known construction of $\kappa$-deformed phase space via Heisenberg double in order to derive the non-commutative space-time structure and description of the whole of the DSR phase space. Next we show that contrary to the ambiguous structure of the energy momentum sector, the space-time of the DSR theory is unique and equivalent to the theory with non-commutative space-time proposed long ago by Snyder. This theory provides non-commutative version of Minkowski space-time enjoying ordinary Lorentz symmetry. It turns out that when one builds a natural phase space on this space-time, its intrinsic length parameter $\ell$ becomes observer-independent. 
  We describe in detail the solution of the extension of the chiral Gaussian Unitary Ensemble (chGUE) into the complex plane. The correlation functions of the model are first calculated for a finite number of N complex eigenvalues, where we exploit the existence of orthogonal Laguerre polynomials in the complex plane. When taking the large-N limit we derive new correlation functions in the case of weak and strong non-Hermiticity, thus describing the transition from the chGUE to a generalized Ginibre ensemble. Applications to the Dirac operator eigenvalue spectrum in QCD with non-vanishing chemical potential are briefly discussed. This is an extended version of arXiv:hep-th/0204068. 
  We consider C/Z_N and C^2/Z_N orbifolds of heterotic string theories and Z_N orbifolds of AdS_3. We study theories with N=2 worldsheet superconformal invariance and construct RG flows. Following Harvey, Kutasov, Martinec and Moore, we compute g_cl and show that it decreases monotonically along RG flows- as conjectured by them. For the heterotic string theories, the gauge degrees of freedom do not contribute to the computation of g_cl. 
  We propose a matrix model to describe a class of fractional quantum Hall (FQH) states for a system of (N_1+N_2) electrons with filling factor more general than in the Laughlin case. Our model, which is developed for FQH states with filling factor of the form \nu_{k_1k_2}=\frac{k_1+k_2}{k_1k_2} (k_1 and k_2 odd integers), has a U(N_1)\times U(N_2) gauge invariance, assumes that FQH fluids are composed of coupled branches of the Laughlin type, and uses ideas borrowed from hierarchy scenarios. Interactions are carried, amongst others, by fields in the bi-fundamentals of the gauge group. They simultaneously play the role of a regulator, exactly as does the Polychronakos field. We build the vacuum configurations for FQH states with filling factors given by the series \nu_{p_1p_2}=\frac{p_2}{p_1p_2-1}, p_1 and p_2 integers. Electrons are interpreted as a condensate of fractional D0-branes and the usual degeneracy of the fundamental state is shown to be lifted by the non-commutative geometry behaviour of the plane. The formalism is illustrated for the state at \nu={2/5}. 
  We develop methods to study the singularities of certain $G_2$ cones related to toric hyperkahler spaces and Einstein selfdual orbifolds. This allows us to determine the low energy gauge groups of chiral N=1 compactifications of M-theory on a large family of such backgrounds, which includes the models recently studied by Acharya and Witten. All M-theory compactifications belonging to our family admit a $T^2$ of isometries, and therefore T-dual IIA and IIB descriptions. We argue that reduction through such an isometry leads generically to systems of weakly and strongly coupled IIA 6-branes, T-dual to delocalized type IIB 5-branes. We find a simple criterion for the existence of a `good' isometry which leads to IIA models containing only weakly-coupled D6-branes, and construct examples of such backgrounds. Some of the methods we develop may also apply to different situations, such as the study of certain singularities in the hypermultiplet moduli space of N=2 supergravity in four dimensions. 
  We study path-integrals over reparametrizations of the world-sheet boundary. Such integrals arise when string propagates between fixed space-time contours. In gauge/string duality they are needed to describe gauge theory Wilson loops. We show that (1) in AdS/CFT, the integral is well defined and gives a finite 1-loop correction to the Wilson loop; (2) in critical string theory, the integral is UV divergent, and fixed contour amplitudes are off shell. In the second case, we show that the divergences can be removed by renormalizing the contour. We calculate the 2-loop contour beta-function and explain how it is related to the D0-brane effective action. We also apply this method to compute the first alpha' correction to the effective action of higher dimensional branes. 
  We study the equivalence between the $B\wedge F$ self-dual ($SD_{B\wedge F}$) and the $B\wedge F$ topologically massive ($TM_{B\wedge F}$) models including the coupling to dynamical, U(1) charged fermionic matter. This is done through an iterative procedure of gauge embedding that produces the dual mapping. In the interactive cases, the minimal coupling adopted for both vector and tensor fields in the self-dual representation is transformed into a non minimal magnetic like coupling in the topologically massive representation but with the currents swapped. It is known that to establish this equivalence a current-current interaction term is needed to render the matter sector unchanged. We show that both terms arise naturally from the embedding procedure. 
  In light of the recent work by Sen and Gibbons, we present a phase-plane analysis on the cosmology containing a rolling tachyon field in a potential resulted from string theory. We show that there is no stable point on the phase-plane, which indicated that there is a coincidence problem if one consider tachyon as a candidate of quintessence. Furthermore, we also analyze the phase-plane of the cosmology containing a rolling tachyon field for an exactly solvable toy potential in which the critical point is stable. Therefore, it is possible for rolling tachyon to be quintessence if one give up the strict constraint on the potential or find a more appropriate effective potential for the tachyon from M/string theory. 
  The study of general two dimensional models of gravity allows to tackle basic questions of quantum gravity, bypassing important technical complications which make the treatment in higher dimensions difficult. As the physically important examples of spherically symmetric Black Holes, together with string inspired models, belong to this class, valuable knowledge can also be gained for these systems in the quantum case. In the last decade new insights regarding the exact quantization of the geometric part of such theories have been obtained. They allow a systematic quantum field theoretical treatment, also in interactions with matter, without explicit introduction of a specific classical background geometry. The present review tries to assemble these results in a coherent manner, putting them at the same time into the perspective of the quite large literature on this subject. 
  We obtain the vacuum solutions for M-theory compactified on eight-manifolds with non-vanishing four-form flux by analyzing the scalar potential appearing in the three-dimensional theory. Many of these vacua are not supersymmetric and yet have a vanishing three-dimensional cosmological constant. We show that in the context of Type IIB compactifications on Calabi-Yau threefolds with fluxes and external brane sources alpha'-corrections generate a correction to the supergravity potential proportional to the Euler number of the internal manifold which spoils the no-scale structure appearing in the classical potential. This indicates that alpha'-corrections may indeed lead to a stabilization of the radial modulus appearing in these compactifications. 
  Internal microscopic symmetry of a many body system leads to global constraints. We obtain explicit forms of the global macroscopic condition assuring that at the microscopic level the evolution respects the overall symmetry. 
  We study a noncommutative gauge theory on a fuzzy four-sphere. The idea is to use a matrix model with a fifth-rank Chern-Simons term and to expand matrices around the fuzzy four-sphere which corresponds to a classical solution of this model. We need extra degrees of freedom since algebra of coordinates does not close on the fuzzy four-sphere. In such a construction, a fuzzy two sphere is added at each point on the fuzzy four-sphere as extra degrees of freedom. It is interesting that fields on the fuzzy four-sphere have higher spins due to the extra degrees of freedom. We also consider a theory around the north pole and take a flat space limit. A noncommutative gauge theory on four-dimensional plane, which has Heisenberg type noncommutativity, is considered. 
  We carry out field redefinitions in ten-dimensional Type IIB supergravity and show that they do not give rise to any physical corrections to the holographic renormalization group structure in the AdS/CFT correspondence. We in particular show that the holographic Weyl anomaly of the N=4 SU(N) super Yang-Mills theory does not change under the field redefinition of the ten-dimensional metric of the form G_{MN} -> G_{MN}+\alpha R G_{MN}+\beta R_{MN}. These results are consistent with the fact that classical supergravity represents the on-shell structure of massless modes of superstrings, which should not change under redefinitions of fields. 
  A systematic approach to the description of gauge invariant charges is presented and applied to the construction of both the static colour charge configuration in QCD and the monopole solution in pure SU(2). The gauge invariant non-abelian monopole offers a new style of order parameter for monopole condensation. 
  We show that, at one loop, the magnetic mass vanishes at finite temperature in QED in any dimension. In QED$_{3}$, even the zero temperature part can be regularized to zero. We calculate the two loop contributions to the magnetic mass in QED$_{3}$ with a Chern-Simons term and show that it vanishes. We give a simple proof which shows that the magnetic mass vanishes to all orders at finite temperature in this theory. This proof also holds for QED in any dimension. 
  Using the Moyal star product, we define open bosonic string field theory carefully, with a cutoff, for any number of string oscillators and any oscillator frequencies. Through detailed computations, such as Neumann coefficients for all string vertices, we show that the Moyal star product is all that is needed to give a precise definition of string field theory. The formulation of the theory as well as the computation techniques are considerably simpler in the Moyal formulation. After identifying a monoid algebra as a fundamental mathematical structure in string field theory, we use it as a tool to compute with ease the field configurations for wedge, sliver, and generalized projectors, as well as all the string interaction vertices for perturbative as well as monoid-type nonperturbative states. Finally, in the context of VSFT we analyze the small fluctuations around any D-brane vacuum. We show quite generally that to obtain nontrivial mass and coupling, as well as a closed strings, there must be an associativity anomaly. We identify the detailed source of the anomaly, but leave its study for future work. 
  We propose a new cosmological scenario which resolves the conventional initial singularity problem. The space-time geometry has an unconventional time-like singularity on a lower dimensional hypersurface, with localized energy density. The natural interpretation of this singularity in string theory is that of negative tension branes, for example the orientifolds of type II string theory. Space-time ends at the orientifolds, and it is divided in three regions: a contracting region with a future cosmological horizon; an intermediate region which ends at the orientifols; and an expanding region separated from the intermediate region by a past cosmological horizon. We study the geometry near the singularity of the proposed cosmological scenario in a specific string model. Using D-brane probes we confirm the interpretation of the brane singularity as an orientifold. The boundary conditions on the orientifolds and the past/future transition amplitudes are well defined. Assuming the trivial vacuum in the past, we derive a thermal spectrum in the future. 
  The vacuum of a large-N gauge field on a p-torus has a spatial stress tensor with tension along the direction of smallest periodicity and equal pressures (but p times smaller in magnitude) along the other directions, assuming an AdS/CFT correspondence and a refined form of the Horowitz-Myers positive-energy conjecture. For infinite N, the vacuum exhibits a phase transition when the lengths of the two shortest periodicities cross. A comparison is made with the Surya-Schleich-Witt phase transition at finite temperature. A zero-loop approximation is also given for large but finite N. 
  We describe a global approach to the study of duality transformations between antisymmetric fields with transitions and argue that the natural geometrical setting for the approach is that of gerbes, these objects are mathematical constructions generalizing U(1) bundles and are similarly classified by quantized charges. We address the duality maps in terms of the potentials rather than on their field strengths and show the quantum equivalence between dual theories which in turn allows a rigorous proof of a generalized Dirac quantization condition on the couplings. Our approach needs the introduction of an auxiliary form satisfying a global constraint which in the case of 1-form potentials coincides with the quantization of the magnetic flux. We apply our global approach to refine the proof of the duality equivalence between d=11 supermembrane and d=10 IIA Dirichlet supermembrane. 
  We examine whether tachyon matter is a viable candidate for the cosmological dark matter. First, we demonstrate that in order for the density of tachyon matter to have an acceptable value today, the magnitude of the tachyon potential energy at the onset of rolling must be finely tuned. For a tachyon potential $V(T)\sim M_{Pl}^4\exp(-T/\tau)$, the tachyon must start rolling at $T\simeq 60\tau$ in order for the density of tachyon matter today to satisfy $\Omega_{T,0}\sim 1$, provided that standard big bang cosmology begins at the same time as the tachyon begins to roll. In this case, the value of $\Omega_{T,0}$ is exponentially sensitive to $T/\tau$ at the onset of rolling, so smaller $T/\tau$ is unacceptable, and larger $T/\tau$ implies a tachyon density that is too small to have interesting cosmological effects. If instead the universe undergoes a second inflationary epoch after the tachyon has already rolled considerably, then the tachyon can begin with $T$ near zero, but the increase of the scale factor during inflation must still be finely tuned in order for $\Omega_{T,0} \sim 1$. Second, we show that tachyon matter, unlike quintessence, can cluster gravitationally on very small scales. If the starting value of $T/\tau$ is tuned finely enough that $\Omega_{T,0}\sim 1$, then tachyon matter clusters more or less identically to pressureless dust. Thus, if the fine-tuning problem can be explained, tachyon matter is a viable candidate for cosmological dark matter. 
  We show that, for both scalar and spinor QED, the two-loop Euler-Heisenberg effective Lagrangian for a constant Euclidean self-dual background has an extremely simple closed-form expression in terms of the digamma function. Moreover, the scalar and spinor QED effective Lagrangians are very similar to one another. These results are dramatic simplifications compared to the results for other backgrounds. We apply them to a calculation of the low energy limits of the two-loop massive N-photon `all +' helicity amplitudes. The simplicity of our results can be related to the connection between self-duality, helicity and supersymmetry. 
  We analyze the structure of the imaginary part of the two-loop Euler-Heisenberg QED effective Lagrangian for a constant self-dual background. The novel feature of the two-loop result, compared to one-loop, is that the prefactor of each exponential (instanton) term in the imaginary part has itself an asymptotic expansion. We also perform a high-precision test of Borel summation techniques applied to the weak-field expansion, and find that the Borel dispersion relations reproduce the full prefactor of the leading imaginary contribution. 
  We provide a world-sheet interpretation to the plane wave limit of a large class of exact supergravity backgrounds in terms of logarithmic conformal field theories. As an illustrative example, we consider the two-dimensional conformal field theory of the coset model SU(2)_N/U(1) times a free time-like boson U(1)_{-N}, which admits a space-time interpretation as a three-dimensional plane wave solution by taking a correlated limit \`a la Penrose. We show that upon a contraction of Saletan type, in which the parafermions of the compact coset model are combined with the free time-like boson, one obtains a novel logarithmic conformal field theory with central charge c=3. Our results are motivated at the classical level using Poisson brackets of the fields, but they are also explicitly demonstrated at the quantum level using exact operator product expansions. We perform several computations in this theory including the evaluation of the four-point functions involving primary fields and their logarithmic partners, which are identified. We also employ the extended conformal symmetries of the model to construct an infinite number of logarithmic operators. This analysis can be easily generalized to other exact conformal field theory backgrounds with a plane wave limit in the target space. 
  We consider static solutions of the sine-Gordon theory defined on a cylinder, which can be either periodic or quasi-periodic in space. They are described by the different modes of a simple pendulum moving in an inverted effective potential and correspond to its libration or rotation. We review the decomposition of the solutions into an oscillatory sum of alternating kinks and anti-kinks or into a monotonic train of kinks, respectively, using properties of elliptic functions. The two sectors are naturally related to each other by a modular transformation, whereas the underlying spectral curve of the model can be used to express the energy of the static configurations in terms of contour integrals \`a la Seiberg-Witten in either case. The stability properties are also examined by means of supersymmetric quantum mechanics, where we find that the unstable configurations are associated to singular superpotentials, thus allowing for negative modes in the spectrum of small fluctuations. 
  We consider a Penrose limit of AdS_4 x Q^{1,1,1} that provides the pp-wave geometry equal to the one in the Penrose limit of AdS_4 x S^7. We expect that there exists a subsector of three dimensional N=2 dual gauge theory which has enhanced N=8 maximal supersymmetry. We identify operators in the N=2 gauge theory with 11-dimensional supergravity KK excitations in the pp-wave geometry and describe how both the chiral multiplets and semi-conserved multiplets fall into N=8 supermultiplets. 
  We consider an dS_4 brane embedded in a five-dimensional bulk with a positive, vanishing or negative bulk cosmological constant and derive the localized graviton spectrum that consists of a normalizable zero-mode separated by a gap from a continuum of massive states. We estimate the massive sector contribution to the static potential at short distances and find that only in the case of a negative bulk cosmological constant there is a range, determined by the effective four-dimensional and the bulk cosmological constants, where the conventional Newton law is valid. 
  We study the bosonic and fermionic zero modes in noncommutative instanton backgrounds based on the ADHM construction. In k instanton background in U(N) gauge theory, we show how to explicitly construct 4Nk (2Nk) bosonic (fermionic) zero modes in the adjoint representation and 2k (k) bosonic (fermionic) zero modes in the fundamental representation from the ADHM construction. The number of fermionic zero modes is also shown to be exactly equal to the Atiyah-Singer index of the Dirac operator in the noncommutative instanton background. We point out that (super)conformal zero modes in non-BPS instantons are affected by the noncommutativity. The role of Lorentz symmetry breaking by the noncommutativity is also briefly discussed to figure out the structure of U(1) instantons. 
  We propose a new rule for boundary renormalization group flows in fixed-point free coset models. Our proposal generalizes the 'absorption of boundary spin'-principle formulated by Affleck and Ludwig to a large class of perturbations in boundary conformal field theories. We illustrate the rule in the case of unitary minimal models. 
  Bulk matter modes of higher dimensional models generically become unstable in the presence of additional matter multiplets at the branes. This quantum instability is driven by localized Fayet-Iliopoulos terms that attract the bulk zero modes towards the boundary branes. We study this mechanism in the framework of a 5 dimensional S^1/Z_2 orbifold and give conditions for the various possibilities of localization of (chiral) zero modes. This mechanism is quite relevant for realistic model building, as the standard model contains U(1) hypercharge with potentially localized FI-terms. The analysis is closely related to localized anomalies in higher dimensional gauge theories. Five dimensional gauge invariance of the effective action determines the anomaly constraints and fixes the normalization of Chern-Simons terms. The localization of the bulk modes does not effect the anomaly cancellation globally, but the additional heavy Kaluza-Klein modes of the bulk fields may cancel the Chern-Simons terms. We discuss also the potential appearance of the parity anomaly that might render the construction of some orbifold models inconsistent. 
  We study the Wess-Zumino model with the coupling extended to a chiral superfield. In order to incorporate the renormalization effects a further external real field has to be introduced. It is then possible to derive a Callan-Symanzik equation and to prove renormalizability. By constructing the supercurrent in this context the whole machinery for describing the superconformal symmetries becomes available. The presence of the external fields allows also to define multiple insertions of all relevant composite operators. Interesting relations to the curved superspace treatment show up. 
  The dependence of the free energy of string theory on the temperature at T much larger than the Hagedorn temperature was found long ago by Atick and Witten and is $F(T)\sim \Lambda T^2$, where $\Lambda$ diverges because of a tachyonic instability. We show that this result can be understood assuming that, above the Hagedorn transition, Poincare' symmetry is deformed into a quantum algebra. Physically this quantum algebra describes a non-commutative spatial geometry and a discrete euclidean time. We then show that in string theory this deformed Poincare' symmetry indeed emerges above the Hagedorn temperature from the condensation of vortices on the world-sheet. This result indicates that the endpoint of the condensation of closed string tachyons with non-zero winding is an infinite stack of spacelike branes with a given non-commutative world-volume geometry. On a more technical side, we also point out that $T$-duality along a circle with antiperiodic boundary conditions for spacetime fermions is broken by world-sheet vortices, and the would-be T-dual variable becomes non-compact. 
  We consider corrections to the Penrose limit of AdS_3 \times S^3 with NS-NS flux which are due to the terms next to leading order in inverse radius expansion. The worldsheet theory of a lightcone string is interacting due to the presence of quartic terms in the action. Perturbative corrections to the spectrum are shown to agree with the results from the exact quantization in AdS_3 \times S^3. 
  We use the correspondence between scalar field theory on AdS and induced conformal field theory on its boundary to calculate correlation functions of logarithmic conformal field theory in arbitrary dimensions.Our calculations utilize the newly proposed method of nilpotent weights.We derive expressions for the four point function assuming a generic interaction term 
  Starting with the Lagrangian formalism with N=2 supersymmetry in terms of two Grassmann variables in Classical Mechanics, the Dirac canonical quantization method is implemented. The N=2 supersymmetry algebra is associated to one-component and two-component eigenfunctions considered in the Schr\"odinger picture of Nonrelativistic Quantum Mechanics. Applications are contemplated. 
  We show that the translational subgroup of Wigner's little group for massless particles in 3+1 dimensions generate gauge transformation in linearized Einstein gravity. Similarly a suitable representation of the 1-dimensional translational group T(1) is shown to generate gauge transformation in the linearized Einstein-Chern-Simons theory in 2+1 dimensions. These representations are derived systematically from appropriate representations of translational groups which generate gauge transformations in gauge theories living in spacetime of one higher dimension by the technique of dimensional descent. The unified picture thus obtained is compared with a similar picture available for vector gauge theories in 3+1 and 2+1 dimensions. Finally, the polarization tensor of Einstein-Pauli-Fierz theory in 2+1 dimensions is shown to split into the polarization tensors of a pair of Einstein-Chern-Simons theories with opposite helicities suggesting a doublet structure for Einstein-Pauli-Fierz theory. 
  The Ekpyrotic scenario assumes that our visible Universe is a boundary brane in a five-dimensional bulk and that the hot Big Bang occurs when a nearly supersymmetric five-brane travelling along the fifth dimension collides with our visible brane. We show that the generation of isocurvature perturbations is a generic prediction of the Ekpyrotic Universe. This is due to the interactions in the kinetic terms between the brane modulus parametrizing the position of the five-brane in the bulk and the dilaton and volume moduli. We show how to separate explicitly the adiabatic and isorcuvature modes by performing a rotation in field space. Our results indicate that adiabatic and isocurvature pertubations might be cross-correlated and that curvature perturbations might be entirely seeded by isocurvature perturbations. 
  We construct M5-branes in pp-wave background and find that they preserve exactly half of the background pp-wave supersymmetries. We explicitly write down the standard as well as supernumerary Killing spinors and find that their respective numbers are also half of those for the pp-wave background. This is in line with the recent work of Dabholkar et.al. which shows half-supersymmetric D-branes can be constructed in Hpp-wave background. 
  Starting with the Chern-Simons formulation of (2+1)-dimensional gravity we show that the gravitational interactions deform the Poincare symmetry of flat space-time to a quantum group symmetry. The relevant quantum group is the quantum double of the universal cover of the (2+1)-dimensional Lorentz group, or Lorentz double for short. We construct the Hilbert space of two gravitating particles and use the universal R-matrix of the Lorentz double to derive a general expression for the scattering cross section of gravitating particles with spin. In appropriate limits our formula reproduces the semi-classical scattering formulae found by 't Hooft, Deser, Jackiw and de Sousa Gerbert. 
  A nonlocal method of extracting the positive (or the negative) frequency part of a field, based on knowledge of a 2-point function, leads to certain natural generalizations of the normal ordering of quantum fields in classical gravitational and electromagnetic backgrounds and illuminates the origin of the recently discovered nonlocalities related to a local description of particles. A local description of particle creation by gravitational backgrounds is given, with emphasis on the case of black-hole evaporation. The formalism reveals a previously hidden relation between various definitions of the particle current and those of the energy-momentum tensor. The implications to particle creation by classical backgrounds, as well as to the relation between vacuum energy, dark matter, and cosmological constant, are discussed. 
  A general method to easily build global and relative operators for any number n of elementary systems if they are defined for 2 is presented. It is based on properties of the morphisms valued in the tensor products of algebras of the kinematics and it allows also the generalization to any n of relations demon- strated for two. The coalgebra structures play a peculiar role in the explicit constructions. Three examples are presented concerning the Galilei, Poincare' and deformed Galilei algebras. 
  We compare the non-commutative quantum mechanics (NCQM) on sphere and the discrete part of the spectrum of NCQM on pseudosphere (Lobachevsky plane, or $AdS_2$) in the presence of a constant magnetic field $B$ with planar NCQM. We show, that (pseudo)spherical NCQM has a ``critical point'', where the system becomes effectively one-dimensional, and two different ``phases'', which the phases of the planar system originate from, specified by the sign of the parameter $\kappa=1-B\theta$. The ``critical point'' of (pseudo)spherical NCQM corresponds to the $\kappa\to\infty $ point of conventional planar NCQM, and to the ``critical point'' $\kappa=0$ of the so-called ``exotic'' planar NCQM, with a symplectic coupling of the (commutative) magnetic field. 
  We study the fate of U(1) strings embedded in a non-Abelian gauge theory with a hierarchical pattern of the symmetry breaking: G->U(1) at V->nothing at v, V>>v. While in the low-energy limit the Abrikosov-Nielsen-Olesen string (flux tube) is perfectly stable, being considered in the full theory it is metastable. We consider the simplest example: the magnetic flux tubes in the SU(2) gauge theory with adjoint and fundamental scalars. First, the adjoint scalar develops a vacuum expectation value V breaking SU(2) down to U(1). Then, at a much lower scale, the fundamental scalar (quark) develops a vacuum expectation value v creating the Abrikosov-Nielsen-Olesen string. (We also consider an alternative scenario in which the second breaking, U(1)->nothing, is due to an adjoint field.) We suggest an illustrative ansatz describing an "unwinding" in SU(2) of the winding inherent to the Abrikosov-Nielsen-Olesen strings in U(1). This ansatz determines an effective 2D theory for the unstable mode on the string world-sheet. We calculate the decay rate (per unit length of the string) in this ansatz and then derive a general formula. The decay rate is exponentially suppressed. The suppressing exponent is proportional to the ratio of the monopole mass squared to the string tension, which is quite natural in view of the string breaking through the monopole-antimonopole pair production. We compare our result with the one given by Schwinger's formula dualized for describing the monopole-antimonopole pair production in the magnetic field. 
  We consider, from several complementary perspectives, the physics of confinement and deconfinement in the 2+1 dimensional Georgi-Glashow model. Polyakov's monopole plasma and 't Hooft's vortex condensation are discussed first. We then discuss the physics of confining strings at zero temperature. We review the Hamiltonian variational approach and show how the linear confining potential arises in this framework. The second part of this review is devoted to study of the deconfining phase transition. We show that the mechanism of the transition is the restoration of 't Hooft's magnetic symmetry in the deconfined phase. The heavy charged $W$ bosons play a crucial role in the dynamics of the transition, and we discuss the interplay between the charged $W$ plasma and the binding of monopoles at high temperature. Finally we discuss the phase transition from the point of view of confining strings. We show that from this point of view the transition is not driven by the Hagedorn mechanism (proliferation of arbitrarily long strings), but rather by the "disintegration" of the string due to the proliferation of 0 branes. 
  This is my contribution to the proceedings of Stephen Hawking's 60th birthday celebration. If the ideas of TeV scale gravity are correct, then black holes should be produced at accelerators that probe the TeV scale, and their decays should evidence one of Stephen's greatest discoveries, the phenomenon of black hole radiance. 
  The explicit form of the quantum propagator of a bosonic p-brane, previously obtained by the authors in the quenched-minisuperspace approximation, suggests the possibility of a novel, unified, description of p-branes with different dimensionality. The background metric that emerges in this framework is a quadratic form on a Clifford manifold. Substitution of the Lorentzian metric with the Clifford line element has two far reaching consequences. On the one hand, it changes the very structure of the spacetime fabric since the new metric is built out of a Minimum Length below which it is impossible to resolve the distance between two points; on the other hand, the introduction of the Clifford line element extends the usual relativity of motion to the case of Relative Dimensionalism of all p-branes that make up the spacetime manifold near the Planck scale. 
  Finite temperature boson and fermion field theories on ultrastatic space-times with a d-sphere spatial section are discussed with one eye on the questions of temperature inversion symmetry and modular invariance. For conformally invariant theories it is shown that the total energy at any temperature for any dimension, d, is given as a power series in the d=3 and d=5 energies, for scalars, and the d=1 and d=3 energies for spinors. Further, these energies can be given in finite terms at specific temperatures associated with singular moduli of elliptic function theory. Some examples are listed and numbers given. 
  We describe two extensions of the notion of a self-dual connection in a vector bundle over a manifold M from dim M=4 to higher dimensions. The first extension, Omega-self-duality, is based on the existence of an appropriate 4-form Omega on the Riemannian manifold M and yields solutions of the Yang-Mills equations. The second is the notion of half-flatness, which is defined for manifolds with certain Grassmann structure T^C M \cong E \otimes H. In some cases, for example for hyper-Kaehler manifolds M, half-flatness implies Omega-self-duality. A construction of half-flat connections inspired by the harmonic space approach is described. Locally, any such connection can be obtained from a free prepotential by solving a system of linear first order ODEs. 
  We present a supersymmetric D-brane model that has CP spontaneously broken by discrete torsion. The low energy physics is largely independent of the compactification scheme and the kahler metric has `texture zeros' dictated by the choice of discrete torsion. This motivates a simple ansatz for the kahler metric which results in a CKM matrix given in terms of two free parameters, hence we predict a single mixing angle and the CKM phase. The CKM phase is predicted to be close to Pi/3. 
  The dual Meissner effect scenario of confinement is analysed using exact renormalisation group (ERG) equations. In particular, the low energy regime of SU(2) Yang-Mills is studied in a maximal Abelian gauge. It is shown that under general conditions the effective action derived when integrated using ERG methods contains the relevant degrees of freedom for confinement. In addition, the physics in the confining regime is dual to that of the broken phase of an Abelian Higgs model. 
  The metric of a spacetime with a parallel plane (pp)-wave can be obtained in a certain limit of the space AdS^5xS^5. According to the AdS/CFT correspondence, the holographic dual of superstring theory on that background should be the analogous limit of N=4 supersymmetric Yang-Mills theory. In this paper we shall show that, contrary to widespread expectation, non-planar diagrams survive this limiting procedure in the gauge theory. Using matrix model techniques as well as combinatorial reasoning it is demonstrated that a subset of diagrams of arbitrary genus survives and that a non-trivial double scaling limit may be defined. We exactly compute two- and three-point functions of chiral primaries in this limit. We also carefully study certain operators conjectured to correspond to string excitations on the pp-wave background. We find non-planar linear mixing of these proposed operators, requiring their redefinition. Finally, we show that the redefined operators receive non-planar corrections to the planar one-loop anomalous dimension. 
  We compute the non-holomorphic corrections to low-energy effective action (higher derivative terms) in N=2, SU(2) SYM theory coupled to hypermultiplets on a non-abelian background for a class of gauge fixing conditions. A general procedure for calculating the gauge parameters depending contributions to one-loop superfield effective action is developed. The one-loop non-holomorphic effective potential is exactly found in terms of Euler dilogarithm function for specific choice of gauge parameters. 
  In this paper, following a stream of investigation on supersymmetric gauge theories with cosmic string solutions, we contemplate the possibility of building up a D-and-F term cosmic string by means of a gauge-field mixing in connection with a U(1) x U(1)'-symmetry. The spontaneous break of both gauge symmetry and supersymmetry are thoroughly analysed and the fermion zero-modes are worked out. The role of the gauge-field mixing parameter is elucidated in connection with the string configuration that comes out. As an application of the model presented here, we propose the possibility that the supersimetric cosmic string yield production of fermionic charge carriers that may eject, at their late stages, particles that subsequently decay to produce cosmic rays of ultra-high energy. In our work, it turns out that massive supersymmetric fermionic partners may be produced for a susy breaking scale in the range 10^{11} to 10^{13} GeV, which is compatible with the phenomenology of a gravitino mass at the TeV scale. We also determine the range of the gauge-field mixing parameter, \alpha, in connection with the mass scales of the present model. 
  We derive the Wilsonian renormalization group equation in two dimensional ${\cal N}=2$ supersymmetric nonlinear sigma models. This equation shows that the sigma models on compact Einstein K\"{a}hler manifolds are aymptotically free. This result is gerenal and does not depend on the specific forms of the K\"{a}hler potentials. We also examine the renormalization group flow in a new model which connects two manifolds with different global symmetries. 
  In view of our accelerating universe, one of the outstanding theoretical issues is the absence of a quantum-gravitational description of de Sitter space. Although speculative, an intriguing circumvention may be found in the realm of brane-world scenarios; where the physical universe can be interpreted as a non-critical 3-brane moving in a higher-dimensional, static bulk. In this paper, we focus on the cosmological implications of a positively curved brane world evolving in the background of a ``topological'' anti-de Sitter black hole (i.e, Schwarzschild-like but with an arbitrary horizon topology). We show that the bulk black hole will typically induce either an asymptotically de Sitter ``bounce'' universe or a big bang/big crunch FRW universe, depending on a critical value of mass. Interestingly, the critical mass is only non-vanishing in the case of a spherical horizon geometry. We go on to provide a holographic interpretation of this curiosity. 
  The existence of a CPT anomaly is established for a particular four-dimensional Abelian lattice gauge theory with Ginsparg-Wilson fermions. 
  I review my explanation of the irreversibility of the renormalization-group flow in even dimensions greater than two and address new investigations and tests. 
  PP-waves have recently been of interest to string theorists. This is the original, hard to find, original article on plane polarized gravitational waves. 
  We discuss gravitational perturbations in the Randall-Sundrum two branes model with radius stabilization. Following the idea by Goldberger and Wise for the radius stabilization, we introduce a scalar field which has potentials localized on the branes in addition to a bulk potential. In our previous paper we discussed gravitational perturbations induced by static, spherically symmetric and nonrelativistic matter distribution on the branes under the condition that the values of the scalar field on the respective branes cannot fluctuate due to its extremely narrow brane potentials. We call this case the strong coupling limit. Our concern in this paper is to generalize our previous analysis relaxing the limitation of taking the strong coupling limit. We find that new corrections in metric perturbations due to relaxing the strong coupling limit enhance the deviation from the 4D Einstein gravity only in some exceptional cases. In the case that matter fields reside on the negative tension brane, the stabilized radion mass becomes very small when the new correction becomes large. 
  The "central" region of moduli space of M- and string theories is where the string coupling is about unity and the volume of compact dimensions is about the string volume. Here we argue that in this region the non-perturbative potential which is suggested by membrane instanton effects has the correct scaling and shape to allow for enough slow-roll inflation, and to produce the correct amplitude of CMB anisotropies. Thus, the well known theoretical obstacles for achieving viable slow-roll inflation in the framework of perturbative string theory are overcome. Limited knowledge of some generic properties of the induced potential is sufficient to determine the simplest type of consistent inflationary model and its predictions about the spectrum of cosmic microwave background anisotropies: a red spectrum of scalar perturbations, and negligible amount of tensor perturbations. 
  We discuss the correspondence point between a string state and a black hole, in a pp-wave background, and find that the answer is considerably different from that in a flat spacetime background. 
  We perform a two-loop calculation of the effective Lagrangian for the low--energy modes of the quantum mechanical system obtained by dimensional reduction from 4D, N = 1 supersymmetric QED. The bosonic part of the Lagrangian describes the motion over moduli space of vector potentials A_i endowed with a nontrivial conformally flat metric. We determined the coefficient of the two-loop correction to the metric, which is proportional to 1/A^6. For the matrix model obtained from Abelian 4D, N = 2 theory, this correction vanishes, as it should. 
  We consider the Lagrangian path-integrals in Minkowski space for gauges with a residual gauge-invariance. From rather elementary considerations, we demonstrate the necessity of inclusion of an epsilon-term (even) in the formal treatments, without which one may reach incorrect conclusions. We show, further, that the epsilon-term can contribute to the BRST WT-identities in a nontrivial way (even as epsilon-->0). We also show that the (expectation value of the) correct epsilon-term satisfies an algebraic condition. We show by considering (a commonly used) example of a simple local quadratic epsilon -term, that they lead to additional constraints on Green's function that are not normally taken into account in the BRST formalism that ignores the epsilon-term, and that they are characteristic of the way the singularities in propagators are handled. We argue that for a subclass of these gauges, the Minkowski path-integral could not be obtained by a Wick rotation from a Euclidean path-integral. 
  In this paper we study the regular self-gravitating 't Hooft-Polyakov magnetic monopole in a global monopole spacetime. We show that for the large distance, the structure of the manifold corresponds to the Reissner-Nordstr\"{o}m spacetime with a solid angle deficit factor. Although we analyze static and spherically symmetric solutions, it is not possible to solve analytically the system of coupled differential equations and only numerical evaluations can provide detailed information about the behavior of this system at the neighborhood of the defect's core. So, for this reason we solve numerically the set of differential equations for the metric tensor and for the matter fields for different values of the Higgs field vacuum expectation value, $\eta$, and the self-coupling constant, $\lambda$. 
  Field theories on canonical noncommutative spacetimes, which are being studied also in connection with string theory, and on $\kappa$-Minkowski spacetime, which is a popular example of Lie-algebra noncommutative spacetime, can be naturally constructed by introducing a suitable generating functional for Green functions in energy-momentum space. Direct reference to a star product is not necessary. It is sufficient to make use of the simple properties that the Fourier transform preserves in these spacetimes and establish the rules for products of wave exponentials that are dictated by the non-commutativity of the coordinates. The approach also provides an elementary description of "planar" and "non-planar" Feynman diagrams. We also comment on the rich phenomenology emerging from the analysis of these theories. 
  We investigate the issues of holography and string interactions in the duality between SYM and the pp wave background. We argue that the Penrose diagram of the maximally supersymmetric pp-wave has a one dimensional boundary. This fact suggests that the holographic dual of the pp-wave can be described by a quantum mechanical system. We believe this quantum mechanical system should be formulated as a matrix model. From the SYM point of view this matrix model is built out of the lowest lying KK modes of the SYM theory on an $S^3$ compactification, and it relates to a wave which has been compactified along one of the null directions. String interactions are defined by finite time amplitudes on this matrix model. For closed strings they arise as in AdS-CFT, by free SYM diagrams. For open strings, they arise from the diagonalization of the hamiltonian to first order in perturbation theory. Estimates of the leading behaviour of amplitudes in SYM and string theory agree, although they are performed in very different regimes. Corrections are organized in powers of $1/(\mu \alpha ' p^+)^2$ and $g^2(\mu \alpha ' p^+)^4$. 
  Paper withdrawn due to a crucial algebraic error in section 3. 
  We analyse the geometrical structure of supersymmetric solutions of type II supergravity of the form R^{1,9-n} x M_n with non-trivial NS flux and dilaton. Solutions of this type arise naturally as the near-horizon limits of wrapped NS fivebrane geometries. We concentrate on the case d=7, preserving two or four supersymmetries, corresponding to branes wrapped on associative or SLAG three-cycles. Given the existence of Killing spinors, we show that M_7 admits a G_2-structure or an SU(3)-structure, respectively, of specific type. We also prove the converse result. We use the existence of these geometric structures as a new technique to derive some known and new explicit solutions, as well as a simple theorem implying that we have vanishing NS three-form and constant dilaton whenever M_7 is compact with no boundary. The analysis extends simply to other type II examples and also to type I supergravity. 
  We formulate a prescription for computing Minkowski-space correlators from AdS/CFT correspondence. This prescription is shown to give the correct retarded propagators at zero temperature in four dimensions, as well as at finite temperature in the two-dimensional conformal field theory dual to the BTZ black hole. Using the prescription, we calculate the Chern-Simons diffusion constant of the finite-temperature N=4 supersymmetric Yang-Mills theory in the strong coupling limit. We explain why the quasinormal frequencies of the asymptotically AdS background correspond to the poles of the retarded Green's function of the boundary conformal field theory. 
  We compute the correlation functions of R-charge currents and components of the stress-energy tensor in the strongly coupled large-N finite-temperature N=4 supersymmetric Yang-Mills theory, following a recently formulated Minkowskian AdS/CFT prescription. We observe that in the long-distance, low-frequency limit, such correlators have the form dictated by hydrodynamics. We deduce from the calculations the R-charge diffusion constant and the shear viscosity. The value for the latter is in agreement with an earlier calculation based on the Kubo formula and absorption by black branes. 
  A charged particle in a uniform magnetic field in a two-dimensional torus has a discrete noncommutative translation symmetry instead of a continuous commutative translation symmetry. We study topology and symmetry of a particle in a magnetic field in a torus of arbitrary dimensions. The magnetic translation group (MTG) is defined as a group of translations that leave the gauge field invariant. We show that the MTG on an n-dimensional torus is isomorphic to a central extension of a cyclic group Z_{nu_1} x ... x Z_{nu_{2l}} x T^m by U(1) with 2l+m=n. We construct and classify irreducible unitary representations of the MTG on a three-torus and apply the representation theory to three examples. We shortly describe a representation theory for a general n-torus. The MTG on an n-torus can be regarded as a generalization of the so-called noncommutative torus. 
  Supersymmetric field theories on noncommutative spaces are constructed. We present two different representations of noncommutative space, but we can obtain supersymmetry algebla and supersymmetric Yang-Mills action independent of its representation. As a result, we will see that the action has a close relationship with IIB matrix model. 
  Cosmological observations suggest the existence of two different kinds of energy densities dominating at small ($ \lesssim 500$ Mpc) and large ($\gtrsim 1000 $ Mpc) scales. The dark matter component, which dominates at small scales, contributes $\Omega_m \approx 0.35$ and has an equation of state $p=0$ while the dark energy component, which dominates at large scales, contributes $\Omega_V \approx 0.65$ and has an equation of state $p\simeq -\rho$. It is usual to postulate wimps for the first component and some form of scalar field or cosmological constant for the second component. We explore the possibility of a scalar field with a Lagrangian $L =- V(\phi) \sqrt{1 - \del^i \phi \del_i \phi}$ acting as {\it both} clustered dark matter and smoother dark energy and having a scale dependent equation of state. This model predicts a relation between the ratio $ r = \rho_V/\rho_{\rm DM}$ of the energy densities of the two dark components and expansion rate $n$ of the universe (with $a(t) \propto t^n$) in the form $n = (2/3) (1+r) $. For $r \approx 2$, we get $n \approx 2$ which is consistent with observations. 
  A general framework for studying compactifications in supergravity and string theories was introduced by Candelas, Horowitz, Strominger and Witten. This was further generalised to take into account the warp factor by de Wit, Smit and Hari Dass. Though the prime focus of the latter was to find solutions with nontrivial warp factors (shown not to exist under a variety of circumstances), it was shown there that de Sitter compactifications are generically disfavoured. In this note we place these results in the context of a revived interest in de Sitter spacetimes . 
  We prove the perturbative renormalisability of pure SU(2) Yang-Mills theory in the abelian gauge supplemented with mass terms. Whereas mass terms for the gauge fields charged under the diagonal U(1) allow to preserve the standard form of the Slavnov-Taylor identities (but with modified BRST variations), mass terms for the diagonal gauge fields require the study of modified Slavnov-Taylor identities. We comment on the renormalization group equations, which describe the variation of the effective action with the different masses. Finite renormalized masses for the charged gauge fields, in the limit of vanishing bare mass terms, are possible provided a certain combination of wave function renormalization constants vanishes sufficiently rapidly in the infrared limit. 
  Using path integral method (Fujikawa's method) we calculate anomalies in noncommutative gauge theories with fermions in the bi-fundamental and adjoint representations. We find that axial and chiral gauge anomalies coming from non-planar contributions are derived in the low noncommutative momentum limit $\widetilde{p}^{\mu}(\equiv \theta^{\mu\nu}p_{\nu}) \to 0$. The adjoint chiral fermion carries no anomaly in the non-planar sector in $D=4k (k=1,2,...,)$ dimensions. It is naturally shown from the path integral method that anomalies in non-planar sector originate in UV/IR mixing. 
  Unstable particles, together with their stable decay products, constitute probability collectives which are defined as Hilbert spaces with dimension higher than one, nondecomposable in a particle basis. Their structure is considered in the framework of Birkhoff-von Neumann's Hilbert subspace lattices. Bases with particle states are related to bases with a diagonal scalar product by a Hilbert-bein involving the characteristic decay parameters (in some analogy to the $n$-bein structures of metrical manifolds). Probability predictions as expectation values, involving unstable particles, have to take into account all members of the higher dimensional collective. E.g., the unitarity structure of the $S$-matrix for an unstable particle collective can be established by a transformation with its Hilbert-bein. 
  This is a PhD thesis submitted to the University of the witwatersrand. The PhD focusses on the computation of non-holomorphic corrections and the study of monopole solutions in N=2 SUSY Yang Mills theories from the perspective of threebranes in F-theory. 
  A systematic procedure for performing holographic renormalization, which makes use of the Hamilton-Jacobi method, is proposed and applied to a bulk theory of gravity interacting with a scalar field and a U(1) gauge field in the Stueckelberg formalism. We describe how the power divergences are obtained as solutions of a set of "descent equations" stemming from the radial Hamiltonian constraint of the theory. In addition, we isolate the logarithmic divergences, which are closely related to anomalies. The method allows to determine also the exact one-point functions of the dual field theory. Using the other Hamiltonian constraints of the bulk theory, we derive the Ward identities for diffeomorphisms and gauge invariance. In particular, we demonstrate the breaking of U(1)_R current conservation, recovering the holographic chiral anomaly recently discussed in hep-th/0112119 and hep-th/0202056. 
  Motivated by recent evidence indicating that Quantum Einstein Gravity (QEG) might be nonperturbatively renormalizable, the exact renormalization group equation of QEG is evaluated in a truncation of theory space which generalizes the Einstein-Hilbert truncation by the inclusion of a higher-derivative term $(R^2)$. The beta-functions describing the renormalization group flow of the cosmological constant, Newton's constant, and the $R^2$-coupling are computed explicitly. The fixed point (FP) properties of the 3-dimensional flow are investigated, and they are confronted with those of the 2-dimensional Einstein-Hilbert flow. The non-Gaussian FP predicted by the latter is found to generalize to a FP on the enlarged theory space. In order to test the reliability of the $R^2$-truncation near this FP we analyze the residual scheme dependence of various universal quantities; it turns out to be very weak. The two truncations are compared in detail, and their numerical predictions are found to agree with a suprisingly high precision. Due to the consistency of the results it appears increasingly unlikely that the non-Gaussian FP is an artifact of the truncation. If it is present in the exact theory QEG is probably nonperturbatively renormalizable and ``asymptotically safe''. We discuss how the conformal factor problem of Euclidean gravity manifests itself in the exact renormalization group approach and show that, in the $R^2$-truncation, the investigation of the FP is not afflicted with this problem. Also the Gaussian FP of the Einstein-Hilbert truncation is analyzed; it turns out that it does not generalize to a corresponding FP on the enlarged theory space. 
  Phase Space is the framework best suited for quantizing superintegrable systems, naturally preserving the symmetry algebras of the respective hamiltonian invariants. The power and simplicity of the method is fully illustrated through new applications to nonlinear sigma models, specifically for de Sitter N-spheres and Chiral Models, where the symmetric quantum hamiltonians amount to compact and elegant expressions. Additional power and elegance is provided by the use of Nambu Brackets to incorporate the extra invariants of superintegrable models. Some new classical results are given for these brackets, and their quantization is successfully compared to that of Moyal, validating Nambu's original proposal. 
  In the context of gauge/gravity duality, we try to understand better the proposed duality between the fractional D2-brane supergravity solutions of (Nucl. Phys. B 606 (2001) 18, hep-th/0101096) and a confining 2+1 dimensional gauge theory. Based on the similarities between this fractional D2-brane solution and D3-brane supergravity solutions with more firmly established gauge theory duals, we conjecture that a confining q-string in the 2+1 dimensional gauge theory is dual to a wrapped D4-brane. In particular, the D4-brane looks like a string in the gauge theory directions but wraps a S**3 in S**4 in the transverse geometry. For one of the supergravity solutions, we find a near quadratic scaling law for the tension: $T \sim q (N-q)$. Based on the tension, we conjecture that the gauge theory dual is SU(N) far in the infrared. We also conjecture that a quadratic or near quadratic scaling is a generic feature of confining 2+1 dimensional SU(N) gauge theories. 
  An outline is presented of the Extended Scale Relativity (ESR) in C-spaces (Clifford manifolds), where the speed of light and the minimum Planck scale are the two universal invariants. This represents in a sense an extension of the theory developed by L. Nottale long ago. It is shown how all the dimensions of a C-space can be treated on equal footing by implementing the holographic principle associated with a nested family of p-loops of various dimensionalities. This is achieved by using poly-vector valued coordinates in C-spaces that encode in one stroke points, lines, areas, volumes,... In addition, we review the derivation of the minimal-length string uncertainty relations; the logarithmic corrections (valid in any dimension) to the black hole area-entropy relation. We also show how the higher derivative gravity with torsion and the recent results of kappa-deformed Poincare theories of gravity follow naturally from the geometry of C-spaces. In conclusion some comments are made on the cosmological implications of this theory with respect to the cosmological constant problem, the two modes of time, the expansion of the universe, number four as the average dimension of our world and a variable fine structure constant. 
  It has been recently proposed that string theory in the background of a plane wave corresponds to a certain subsector of the N=4 supersymmetric Yang-Mills theory. This correspondence follows as a limit of the AdS/CFT duality. As a particular case of the AdS/CFT correspondence, it is a priori a strong/weak coupling duality. However, the predictions for the anomalous dimensions which follow from this particular limit are analytic functions of the 't Hooft coupling constant $\lambda$ and have a well defined expansion in the weak coupling regime. This allows one to conjecture that the correspondence between the strings on the plane wave background and the Yang-Mills theory works at the level of perturbative expansions.   In our paper we perform perturbative computations in the Yang-Mills theory that confirm this conjecture. We calculate the anomalous dimension of the operator corresponding to the elementary string excitation. We verify at the two loop level that the anomalous dimension has a finite limit when the R charge $J\to \infty$ keeping $\lambda/J^2$ finite. We conjecture that this is true at higher orders of perturbation theory. We show, by summing an infinite subset of Feynman diagrams, under the above assumption, that the anomalous dimensions arising from the Yang-Mills perturbation theory are in agreement with the anomalous dimensions following from the string worldsheet sigma-model. 
  We consider supersymmetric PP-wave limits for different N=1 orbifold geometries of the five sphere S^5 and the five dimensional Einstein manifold T^{1,1}. As there are several interesting ways to take the Penrose limits, the PP-wave geometry can be either maximal supersymmetric N=4 or half-maximal supersymmetric N=2. We discuss in detail the cases AdS_5 x S^5/Z_3, AdS_5 x S^5/(Z_m x Z_n) and AdS_5 x T^{1,1}/(\Z_m \times \Z_n) and we identify the gauge invariant operators which correspond to stringy excitations for the different limits. 
  We consider a natural generalisation of the class of hyperbolic Kac-Moody algebras. We describe in detail the conditions under which these algebras are Lorentzian. We also construct their fundamental weights, and analyse whether they possess a real principal so(1,2) subalgebra. Our class of algebras include the Lorentzian Kac-Moody algebras that have recently been proposed as symmetries of M-theory and the closed bosonic string. 
  We review recent work showing that there exists a large class of new stable black strings which are not translationally invariant. Both neutral and charged black strings are considered. The discussion includes known properties of these new solutions, attempts to find them explicitly, and a list of open questions. 
  We analyze a class of conical G_2 metrics admitting two commuting isometries, together with a certain one-parameter family of G_2 deformations which preserves these symmetries. Upon using recent results of Calderbank and Pedersen, we write down the explicit G_2 metric for the most general member of this family and extract the IIA reduction of M-theory on such backgrounds, as well as its type IIB dual. By studying the asymptotics of type II fields around the relevant loci, we confirm the interpretation of such backgrounds in terms of localized IIA 6-branes and delocalized IIB 5-branes. In particular, we find explicit, general expressions for the string coupling and R-R/NS-NS forms in the vicinity of these objects. Our solutions contain and generalize the field configurations relevant for certain models considered in recent work of Acharya and Witten. 
  A systematic method for constructing trigonometric R-matrices corresponding to the (multiplicity-free) tensor product of any two affinizable representations of a quantum algebra or superalgebra has been developed by the Brisbane group and its collaborators. This method has been referred to as the Tensor Product Graph Method. Here we describe applications of this method to untwisted and twisted quantum affine superalgebras. 
  It is well known that the Lagrangian and the Hamiltonian formalisms can be combined and lead to "covariant symplectic" methods. For that purpose a "pre-symplectic form" has been constructed from the Lagrangian using the so-called Noether form. However, analogously to the standard Noether currents, this symplectic form is only determined up to total divergences which are however essential ingredients in gauge theories.   We propose a new definition of the symplectic form which is covariant and free of ambiguities in a general first order formulation. Indeed, our construction depends on the equations of motion but not on the Lagrangian. We then define a generalized Hamiltonian which generates the equations of motions in a covariant way. Applications to Yang-Mills, general relativity, Chern-Simons and supergravity theories are given. We also consider nice sets of possible boundary conditions that imply the closure and conservation of the total symplectic form.   We finally revisit the construction of conserved charges associated with gauge symmetries, from both the "covariant symplectic" and the "covariantized Regge-Teitelboim" points of view. We find that both constructions coincide when the ambiguity in the Noetherian pre-symplectic form is fixed using our new prescription. We also present a condition of integrability of the equations that lead to these quantities. 
  In this letter, we investigate corrections to quartic gauge couplings in compactifications of string theories on a 2-torus without vector structure. First, we calculate the threshold corrections to F^4 terms in the heterotic CHL string. Then, using the CHL string/type I duality dictionary, we map these corrections to perturbative and non-perturbative effects in type I string compactified on 2-torus without vector structure. Comparing the perturbative terms provides a quantitative test of the S-duality conjecture. The non-perturbative contributions are due to D-strings wrapping the torus. A T-duality along one of the compactified directions allows us to compare these instanton corrections to the ones obtained in a standard type I compactifications. The most striking feature is that these non-perturbative couplings turn out to be identical for both models. 
  Recently a class of Type IIA orientifold models was constructed yielding just the fermions of the SM at the intersections of D6-branes wrapping a 6-torus. We generalize that construction to the case of Type IIB compactified on an orientifold of T^4 \times (C/Z_N) with D5-branes intersecting at angles on T^4. We construct explicit models in which the massless fermion spectrum is just the one of a three-generation Standard Model.  One of the motivations for these new constructions is that in this case there are 2 dimensions which are transverse to the SM D5-brane configuration. By making those two dimensions large enough one can have a low string scale M_s of order 1-10 TeV and still have a large M_{Planck} in agreement with observations. From this point of view, these are the first explicit D-brane string constructions where one can achieve having just the fermionic spectrum and gauge group of the SM embedded in a Low String Scale scenario. The cancellation of U(1) anomalies turns out to be quite analogous to the toroidal D6-brane case and the proton is automatically stable due to the gauging of baryon number. Unlike the D6-brane case, the present class of models has N = 0 SUSY both in the bulk and on the branes and hence the spectrum is simpler. 
  An elementary introduction to Maldacena's AdS/CFT correspondence is given, with some emphasis in the Fefferman-Graham construction. This is based on lectures given by one of us (E.A.) at the Universidad Autonoma de Madrid. 
  We compute the DC conductance with two different methods, which both exploit the integrability of the theories under consideration. On one hand we determine the conductance through a defect by means of the thermodynamic Bethe ansatz and standard relativistic potential scattering theory based on a Landauer transport theory picture. On the other hand, we propose a Kubo formula for a defect system and evaluate the current-current two-point correlation function it involves with the help of a form factor expansion. For a variety of defects in a fermionic system we find excellent agreement between the two different theoretical descriptions. 
  It is generally believed that weak scale supersymmetry implies weak scale supergravity, in the sense that the masses of the gravitino and gravitationally coupled moduli have masses below 100 TeV. This paper presents a realistic framework for supersymmetry breaking in the hidden sector in which the masses of the gravitino and gravitational moduli can be much larger. This cleanly eliminates the cosmological problems of hidden sector models. Supersymmetry breaking is communicated to the visible sector by anomaly-mediated supersymmetry breaking. The framework is compatible with perturbative gauge coupling unification, and can be realized either in models of "warped" extra dimensions, or in strongly-coupled four-dimensional conformal field theories. 
  We investigate some of the intricate features in a gravity decoupling limit of a open bosonic string theory, in a constant electromagnetic (EM-) field. We explain the subtle nature of space-time at short distances, due to its entanglement with the gauge field windings in the theory. Incorporating the mass-shell condition in the theory, we show that the time coordinate is small, of the order of EM-string scale, and the space coordinates are large. We perform a careful analysis in the critical regime to describe the decoupling of a series of gauge-string windings in successions, just below the Hagedorn temperature. We argue for the condensation of gauge-string at the Hagedorn temperature, which is followed by the decoupling of tachyonic particles. We demonstrate the phenomena by revoking the effective noncommutative dynamics for the D(3)-brane and obtain nonlinear corrections to U(1) gauge theory. We discuss the spontaneous breaking of noncommutative U(1) symmetry and show that the Hagedorn phase is described by the noninteracting gauge particles. The notion of time reappears in the phase at the expense of temperature. It suggests a complementarity between two distinct notions, time and temperature, at short distances. 
  We study possible backgrounds of 2D string theory using its equivalence with a system of fermions in upside-down harmonic potential. Each background corresponds to a certain profile of the Fermi sea, which can be considered as a deformation of the hyperbolic profile characterizing the linear dilaton background. Such a perturbation is generated by a set of commuting flows, which form a Toda Lattice integrable structure. The flows are associated with all possible left and right moving tachyon states, which in the compactified theory have discrete spectrum. The simplest nontrivial background describes the Sine-Liouville string theory. Our methods can be also applied to the study of 2D droplets of electrons in a strong magnetic field. 
  In theories of gravity with a positive cosmological constant, we consider product solutions with flux, of the form (A)dS_p x S^q. Most solutions are shown to be perturbatively unstable, including all uncharged dS_p x S^q spacetimes. For dimensions greater than four, the stable class includes universes whose entropy exceeds that of de Sitter space, in violation of the conjectured "N-bound". Hence, if quantum gravity theories with finite-dimensional Hilbert space exist, the specification of a positive cosmological constant will not suffice to characterize the class of spacetimes they describe. 
  The Casimir energy for massless scalar field which satisfies priodic boundary conditions in two-dimensional domain wall background is calculated by making use of general properties of renormalized stress-tensor. The line element of domain wall is time dependent, the trace anomaly which is the nonvanishing $T^{\mu}_{\mu}$ for a conformally invariant field after renormalization, represent the back reaction of the dynamical Casimir effect. 
  We study N=4 super Yang Mills theory at finite U(1)_R charge density (and temperature) using the AdS/CFT Correspondence. The ten dimensional backgrounds around spinning D3 brane configurations split into two classes of solution. One class describe spinning black holes and have previously been extensively studied, and interpreted, in a thermodynamic context, as the deconfined high density phase of the dual field theory. The other class have naked singularities and in the supersymmetric limit are known to correspond to multi-centre solutions describing the field theory in the Coulomb phase. We provide evidence that the non-supersymmetric members of this class represent naked, spinning D-brane distributions describing the Coulomb branch at finite density. At a critical density a phase transition occurs to a spinning black hole representing the deconfined phase where the higgs vevs have evaporated. We perform a free energy calculation to determine the phase diagram of the Coulomb branch at finite temperature and density. 
  We study the multiplicity of BPS domain walls in N=1 super Yang-Mills theory, by passing to a weakly coupled Higgs phase through the addition of fundamental matter. The number of domain walls connecting two specified vacuum states is then determined via the Witten index of the induced worldvolume theory, which is invariant under the deformation to the Higgs phase. The worldvolume theory is a sigma model with a Grassmanian target space which arises as the coset associated with the global symmetries broken by the wall solution. Imposing a suitable infrared regulator, the result is found to agree with recent work of Acharya and Vafa in which the walls were realized as wrapped D4-branes in IIA string theory. 
  We discuss the scalar cosmological perturbations in a 3-brane world with a 5D bulk. We first show explicitly how the effective perturbed Einstein's equations on the brane (involving the Weyl fluid) are encoded into Mukohyama's master equation. We give the relation between Mukohyama's master variable and the perturbations of the Weyl fluid, we also discuss the relation between the former and the perturbations of matter and induced metric on the brane. We show that one can obtain a boundary condition on the brane for the master equation solely expressible in term of the master variable, in the case of a perfect fluid with adiabatic perturbations on a Randall-Sundrum (RS) or Dvali-Gabadadze-Porrati (DGP) brane. This provides an easy way to solve numerically for the evolution of the perturbations as well as should shed light on the various approximations done in the literature to deal with the Weyl degrees of freedom. 
  We analyse the classical decay process of unstable D-branes in superstring theory using the boundary string field theory (BSFT) action. We show that the solutions of the equations of motion for the tachyon field asymptotically approach to T=x^0 and the pressure rapidly falls off at late time producing the tachyon matter irrespective of the initial condition. We also consider the cosmological evolution driven by the rolling tachyon using the BSFT action as an effective action. 
  In the fall of 1924, Enrico Fermi visited Paul Ehrenfest at Leyden on a 3-month fellowship from the International Education Board (IEB). Fermi was 23 years old. In his trip report to the IEB, Fermi says he learned a lot about cryogenics and worked on two scientific papers, including the following one. It was submitted in German to Zeitschrift fur Physik. The German version was known to Weizsacker and Williams and cited in the papers (10 years) later in which they extended Fermi's method to the Ultra-Relativistic case. The German version was subsequently translated into a Russian version and perhaps other languages. Fermi's Italian version (printed in Nuovo Cimento) is less widely known and does not appear in the ``Collected Works''. Nevertheless, Persico remarks that this was one of Fermi's favorite ideas and that he often used it in later life. So, we would like to think of this as a late 100th birthday present to the Italian Navigator. We would like to thank Professor T.D. Lee for his encouragement of this project and for interesting discussions about Fermi. Also Tom Rosenblum at the Rockefeller Archives for bringing Fermi's correspondence to our attention and Bonnie Sherwood for typing the original manuscript. 
  A black string generaliztion of the Myers-Perry N dimensional rotating black hole is considered in an (N+1) dimensional Randall-Sundrum brane world. The black string intercepts the (N-1) brane in a N dimensional rotating black hole. We examine the diverse cases arising for various non-zero rotation components and obtain the geodesic equations for these space-time. The asymptotics of theresulting brane world geometries and their implications are discussed. 
  We investigate a stability equation involving two-component eigenfunctions which is associated with a potential model in terms of two coupled real scalar fields, which presents non BPS topological defect. 
  Recently, Berenstein et al. have proposed a duality between a sector of N=4 super-Yang-Mills theory with large R-charge J, and string theory in a pp-wave background. In the limit considered, the effective 't Hooft coupling has been argued to be lambda'=g_{YM}^2 N/J^2=1/(mu p^+ alpha')^2. We study Yang-Mills theory at small lambda' (large mu) with a view to reproducing string interactions. We demonstrate that the effective genus counting parameter of the Yang-Mills theory is g_2^2=J^4/N^2=(4 pi g_s)^2 (mu p^+ alpha')^4, the effective two-dimensional Newton constant for strings propagating on the pp-wave background. We identify g_2 sqrt{lambda'} as the effective coupling between a wide class of excited string states on the pp-wave background. We compute the anomalous dimensions of BMN operators at first order in g_2^2 and lambda' and interpret our result as the genus one mass renormalization of the corresponding string state. We postulate a relation between the three-string vertex function and the gauge theory three-point function and compare our proposal to string field theory. We utilize this proposal, together with quantum mechanical perturbation theory, to recompute the genus one energy shift of string states, and find precise agreement with our earlier computation. 
  N=4 supersymmetric Yang-Mills theory with gauge group SU(n) (n>=3) is believed to have two exactly marginal deformations which break the supersymmetry to N=1. We discuss the construction of the string theory dual to these deformations, in the supergravity approximation, in a perturbation series around the AdS_5 * S^5 solution. We construct explicitly the deformed solution at second order in the deformation. We show that deformations which are marginal but not exactly marginal lead to a non-conformal solution with a logarithmically running coupling constant. Surprisingly, at third order in the deformation we find the same beta functions for the couplings in field theory and in supergravity, suggesting that the leading order beta functions (or anomalous dimensions) do not depend on the gauge coupling (the coefficient is not renormalized). 
  It is known that the chiral part of any 2d conformal field theory defines a 3d topological quantum field theory: quantum states of this TQFT are the CFT conformal blocks. The main aim of this paper is to show that a similar CFT/TQFT relation exists also for the full CFT. The 3d topological theory that arises is a certain ``square'' of the chiral TQFT. Such topological theories were studied by Turaev and Viro; they are related to 3d gravity. We establish an operator/state correspondence in which operators in the chiral TQFT correspond to states in the Turaev-Viro theory. We use this correspondence to interpret CFT correlation functions as particular quantum states of the Turaev-Viro theory. We compute the components of these states in the basis in the Turaev-Viro Hilbert space given by colored 3-valent graphs. The formula we obtain is a generalization of the Verlinde formula. The later is obtained from our expression for a zero colored graph. Our results give an interesting ``holographic'' perspective on conformal field theories in 2 dimensions. 
  We examine a bilinear form Wess-Zumino term for a superstring in anti-de Sitter (AdS) spaces. This is composed of two parts; a bilinear term in superinvariant currents and a total derivative bilinear term which is required for the pseudo-superinvariance of the Wess-Zumino term. The covariant supercharge commutator containing a string charge is also obtained. 
  We study the radiative corrections of the noncommutative QED at the one-loop level. A correction of the magnetic dipole moment due to the noncommutativity are evaluated. As in the ordinary QED, IR divergence is shown to vanish when we combine both the tree level Bremsstrahlung diagram and the one-loop electron vertex function. 
  In this letter we derive the Seiberg-Witten map for noncommutative super Yang-Mills theory in Wess-Zumino gauge. Following (and using results of) hep-th/0108045 we split the observer Lorentz transformations into a covariant particle Lorentz transformation and a remainder which gives directly the Seiberg-Witten differential equations. These differential equations lead to a theta-expansion of the noncommutative super Yang-Mills action which is invariant under commutative gauge transformations and commutative observer Lorentz transformation, but not invariant under commutative supersymmetry transformations: The theta-expansion of noncommutative supersymmetry leads to a theta-dependent symmetry transformation. For this reason the Seiberg-Witten map of super Yang-Mills theory cannot be expressed in terms of superfields. 
  The major obstacle to a supersymmetric theory on the lattice is the failure of the Leibniz rule. We analyze this issue by using the Wess-Zumino model and a general Ginsparg-Wilson operator, which is local and free of species doublers. We point out that the Leibniz rule could be maintained on the lattice if the generic momentum $k_{\mu}$ carried by any field variable satisfies $|ak_{\mu}|<\delta$ in the limit $a\to 0$ for arbitrarily small but finite $\delta$. This condition is expected to be satisfied generally if the theory is finite perturbatively, provided that discretization does not induce further symmetry breaking. We thus first render the continuum Wess-Zumino model finite by applying the higher derivative regularization which preserves supersymmetry. We then put this theory on the lattice, which preserves supersymmetry except for a breaking in interaction terms by the failure of the Leibniz rule. By this way, we define a lattice Wess-Zumino model which maintains the basic properties such as $U(1)\times U(1)_{R}$ symmetry and holomorphicity. We show that this model reproduces continuum theory in the limit $a\to 0$ up to any finite order in perturbation theory; in this sense all the supersymmetry breaking terms induced by the failure of the Leibniz rule are irrelevant. We then suggest that this discretization may work to define a low energy effective theory in a non-perturbative way. 
  We study the dynamics of type I strings on Melvin backgrounds, with a single or multiple twisted two-planes. We construct two inequivalent types of orientifold models that correspond to (non-compact) irrational versions of Scherk-Schwarz type breaking of supersymmetry. In the first class of vacua, D-branes and O-planes are no longer localized in space-time but are smeared along the compact Melvin coordinate with a characteristic profile. On the other hand, the second class of orientifolds involves O-planes and D-branes that are both rotated by an angle proportional to the twist. In case of ``multiple Melvin spaces'', some amount of supersymmetry is recovered if the planes are twisted appropriately and part of the original O-planes are transmuted into new ones. The corresponding boundary and crosscap states are determined. 
  Recently, Maldacena, Moore and Seiberg introduced non-maximally symmetric boundary states on group manifold using T-duality. In the work presented here we suggest simple description of these branes in terms of group elements. We show that T-dualization actually reduces to multiplication of conjugacy classes by the corresponding U(1) subgroups. Using this description we find the two-form trivializing the WZW three-form on the branes. SU(2) and $SL(2,R)$ examples are considered in details. 
  We investigate the rolling of the tachyon on the unstable D9 brane in Type IIA string theory by studying the BSFT action. The action is known for linear profiles of the tachyon, which is the expected asymptotic behavior of the tachyon as it approaches the closed string vacuum, as recently described by Sen. We find that the action does indeed seem consistent with the general Sen description, in that it implies a constant energy density with diminishing pressure. However, the details are somewhat different from an effective field theory of Born-Infeld type. For instance, the BSFT action implies there are poles for certain rolling velocities, while a Born-Infeld action would have a cut. We also find that solutions with pressure diminishing from either the positive or negative side are possible. 
  Following an eight-dimensional gauged supergravity approach we construct the most general solution describing D6-branes wrapped on a Kahler four-cycle taken to be the product of two spheres of different radii. Our solution interpolates between a Calabi-Yau four-fold and the spaces S^2xS^2xS^2xR^2 or S^2xS^2xR^4, depending on generic choices for the parameters. Then we turn on a background four-form field strength, corresponding to D2-branes, and show explicitly how our solution is deformed. For a particular choice of parameters it represents a flow from a Calabi-Yau four-fold times the three-dimensional Minkowski space-time in the ultraviolet, to the space-time AdS_4xQ^{1,1,1} in the infrared. In general, the solution in the infrared has a singularity which within type-IIA supergravity corresponds to the near horizon geometry of the solution for the D2-D6 system. Finally, we uncover the relation with work done in the eighties on Freund-Rubin type compactifications. 
  We review extensions of the AdS/CFT correspondence to gauge/ gravity dualities with N=1 supersymmetry. In particular, we describe the gauge/gravity dualities that emerge from placing D3-branes at the apex of the conifold. We consider first the conformal case, with discussions of chiral primary operators and wrapped D-branes. Next, we break the conformal symmetry by adding a stack of partially wrapped D5-branes to the system, changing the gauge group and introducing a logarithmic renormalization group flow. In the gravity dual, the effect of these wrapped D5-branes is to turn on the flux of 3-form field strengths. The associated RR 2-form potential breaks the U(1) R-symmetry to $Z_{2M}$ and we study this phenomenon in detail. This extra flux also leads to deformation of the cone near the apex, which describes the chiral symmetry breaking and confinement in the dual gauge theory. 
  We study string propagation in a spacetime with positive cosmological constant, which includes a circle whose radius approaches a finite value as |t|\to\infty, and goes to zero at t=0. Near this cosmological singularity, the spacetime looks like R^{1,1}/Z. In string theory, this spacetime must be extended by including four additional regions, two of which are compact. The other two introduce new asymptotic regions, corresponding to early and late times, respectively. States of quantum fields in this spacetime are defined in the tensor product of the two Hilbert spaces corresponding to the early time asymptotic regions, and the S-matrix describes the evolution of such states to states in the tensor product of the two late time asymptotic regions. We show that string theory provides a unique continuation of wavefunctions past the cosmological singularities, and allows one to compute the S-matrix. The incoming vacuum evolves into an outgoing state with particles. We also discuss instabilities of asymptotically timelike linear dilaton spacetimes, and the question of holography in such spaces. Finally, we briefly comment on the relation of our results to recent discussions of de Sitter space. 
  We investigate the Kalb-Ramond antisymmetric tensor field as solution to the muon $g-2$ problem. In particular we calculate the lowest-order Kalb-Ramond contribution to the muon anomalous magnetic moment and find that we can fit the new experimental value for the anomaly by adjusting the coupling without affecting the electron anomalous magnetic moment results. 
  Seiberg and Witten have discussed a specifically "stringy" kind of instability which arises in connection with "large" branes in asymptotically AdS spacetimes. It is easy to see that this instability actually arises in most five-dimensional asymptotically AdS black hole string spacetimes with non-trivial horizon topologies. We point out that this is a more serious problem than it may at first seem, for it cannot be resolved even by taking into account the effect of the branes on the geometry of spacetime. [It is ultimately due to the {\em topology} of spacetime, not its geometry.] Next, assuming the validity of some kind of dS/CFT correspondence, we argue that asymptotically deSitter versions of the Hull-Strominger-Gutperle S-brane spacetimes are also unstable in this "topological" sense, at least in the case where the R-symmetries are preserved. We conjecture that this is due to the unrestrained creation of "late" branes, the spacelike analogue of large branes, at very late cosmological times. 
  The superconformal structure of coset superspaces with AdS_m x S^n geometry of bosonic subspaces is studied. It is shown, in particular, that the conventional superspace extensions of the coset manifolds AdS_2 x S^2, AdS_3 x S^3 and AdS_5 x S^5, which arise as solutions of corresponding D=4,6, 10 supergravities and have been extensively studied in connection with AdS/CFT correspondence, are not superconformally flat, though their bosonic submanifolds are conformally flat. We give a group-theoretical reasoning for this fact. We find that in the AdS_2 x S^2 and AdS_3 x S^3 cases there exist different supercosets based on the supergroup OSp(4^*|2) which are superconformally flat. We also argue that in D=2,3,4 and 5 there exist superconformally flat `pure' AdS_D supercosets. Two methods of checking the superconformal flatness are proposed. One of them consists in solving the Maurer-Cartan structure equations and the other is based on embedding the isometry supergroup of the AdS_m x S^n superspace into a superconformal group in (m+n)-dimensional Minkowski space. Finally, we discuss some applications of the above results to the description of supersymmetric dynamical systems. 
  It is currently believed that the local causality of Quantum Field Theory (QFT) is destroyed by the measurement process. This belief is also based on the Einstein-Podolsky-Rosen (EPR) paradox and on the so-called Bell's theorem, that are thought to prove the existence of a mysterious, instantaneous action between distant measurements. However, I have shown recently that the EPR argument is removed, in an interpretation-independent way, by taking into account the fact that the Standard Model of Particle Physics prevents the production of entangled states with a definite number of particles. This result is used here to argue in favor of a statistical interpretation of QFT and to show that it allows for a full reconciliation with locality and causality. Within such an interpretation, as Ballentine and Jarret pointed out long ago, Bell's theorem does not demonstrate any nonlocality. 
  We derive two families of supergravity solutions describing D-branes in the maximally supersymmetric Hpp-wave background. The first family of solutions corresponds to quarter-BPS D-branes. These solutions are delocalised along certain directions transverse to the pp-wave. The second family corresponds to the non-supersymmetric D-branes. These solutions are fully localised. A peculiar feature of the nonsupersymmetric solutions is that gravity becomes repulsive close to the core of the D-brane. Both families preserve the amount of supersymmetry predicted by the D-brane probe/CFT analysis. All solutions are written in Brinkman coordinates. To construct these kind of solutions it is crucial to identify the coordinates in which the ansatz looks the simplest. We argue that the natural coordinates to get the supergravity description of the half-BPS branes are the Rosen coordinates. 
  In this paper, we recast the fermionic ghost sector of Witten's open bosonic string field theory in the language of noncommutative field theory. In particular, following the methods of hep-th/0202087, we find that in Siegel gauge Witten's star product roughly corresponds to a continuous tensor product of Clifford Algebras, and we formulate important operators of the theory in this language, notably the kinetic operator of vacuum string field theory and the BRST operator describing the vacuum of the unstable D-25 brane. We find that the BRST operator is singular in this formulation. We explore alternative operator/Moyal representations of the star product analogous to the split string description and the discrete Moyal basis developed extensively in recent work by Bars and Matsuo (hep-th/0204260). Finally, we discuss some interesting singularities in the formalism and how they may be regulated. 
  We give examples of string compactifications to 4d Minkowski space with different amounts of supersymmetry that can be connected by spherical domain walls. The tension of these domain walls is tunably lower than the 4d Planck scale. The ``stringy'' description of these walls is known in terms of certain configurations of wrapped Dirichlet and NS branes. This construction allows us to connect a variety of vacua with 4d N=4,3,2,1 supersymmetry. 
  We construct a Penrose limit of AdS_4 x M^{1,1,1} where M^{1,1,1}= SU(3) x SU(2) x U(1)/(SU(2) x U(1) x U(1)) that provides the pp-wave geometry equal to the one in the Penrose limit of AdS_4 x S^7. There exists a subsector of three dimensional N=2 dual gauge theory which has enhanced N=8 maximal supersymmetry. We identify operators in the N=2 gauge theory with supergravity KK excitations in the pp-wave geometry and describe how the gauge theory operators made out of two kinds of chiral fields of conformal dimension 4/9, 1/3 fall into N=8 supermultiplets. 
  We study the U(1) and U(2) instanton solutions of gauge theory on general noncommutative $\bf{R}^4$. In all cases considered we obtain explicit results for the projection operators. In some cases we computed numerically the instanton charge and found that it is an integer, independent of the noncommutative parameters $\theta_{1,2}$. 
  We explore a version of the cosmological dilaton-fixing and decoupling mechanism in which the dilaton-dependence of the low-energy effective action is extremized for infinitely large values of the bare string coupling $g_s^2 = e^{\phi}$. We study the efficiency with which the dilaton $\phi$ runs away towards its ``fixed point'' at infinity during a primordial inflationary stage, and thereby approximately decouples from matter. The residual dilaton couplings are found to be related to the amplitude of the density fluctuations generated during inflation. For the simplest inflationary potential, $V (\chi) = {1/2} m_{\chi}^2 (\phi) \chi^2$, the residual dilaton couplings are shown to predict violations of the universality of gravitational acceleration near the $\Delta a / a \sim 10^{-12}$ level. This suggests that a modest improvement in the precision of equivalence principle tests might be able to detect the effect of such a runaway dilaton. Under some assumptions about the coupling of the dilaton to dark matter and/or dark energy, the expected time-variation of natural ``constants'' (in particular of the fine-structure constant) might also be large enough to be within reach of improved experimental or observational data. 
  Nonconfining Schwinger Model [AR] is studied with a one parameter class of kinetic energy like regularization. It may be thought of as a generalization over the regularization considered in [AR]. Phasespace structure has been determined in this new situation. The mass of the gauge boson acquires a generalized expression with the bare coupling constant and the parameters involved in the regularization. Deconfinement scenario has become transparent at the quark-antiquark potential level. 
  $F-$Lie algebras are natural generalisations of Lie algebras (F=1) and Lie superalgebras (F=2). When $F>2$ not many finite-dimensional examples are known. In this paper we construct finite-dimensional $F-$Lie algebras $F>2$ by an inductive process starting from Lie algebras and Lie superalgebras. Matrix realisations of $F-$Lie algebras constructed in this way from $\mathfrak{su}(n), \mathfrak{sp}(2n)$ $\mathfrak{so}(n)$ and $\mathfrak{sl}(n|m)$, $\mathfrak{osp}(2|m)$ are given. We obtain non-trivial extensions of the Poincar\'e algebra by In\"on\"u-Wigner contraction of certain $F-$Lie algebras with $F>2$. 
  Gauge theories in 2+1 dimensions whose gauge symmetry is spontaneously broken to a finite group enjoy a quantum group symmetry which includes the residual gauge symmetry. This symmetry provides a framework in which fundamental excitations (electric charges) and topological excitations (magnetic fluxes) can be treated on equal footing. In order to study symmetry breaking by both electric and magnetic condensates we develop a theory of symmetry breaking which is applicable to models whose symmetry is described by a quantum group (quasitriangular Hopf algebra). Using this general framework we investigate the symmetry breaking and confinement phenomena which occur in (2+1)-dimensional gauge theories. Confinement of particles is linked to the formation of string-like defects. Symmetry breaking by an electric condensate leads to magnetic confinement and vice-versa. We illustrate the general formalism with examples where the symmetry is broken by electric, magnetic and dyonic condensates. 
  I analyze the asymptotic symmetries of a theory of gravity in a background consisting of two patches of ${\rm AdS}_3$ spacetime glued together along an ${\rm AdS}_2$ brane. These are generated by a single Virasoro algebra, as expected from the conjectured dual description in terms of a scale-invariant interface separating two conformal field theories. Contributed to the proceedings of the Francqui Colloquium 2001: `Strings and Gravity: Tying the Forces Together .' 
  Using the Nahm transform we investigate doubly periodic charge one SU(2) instantons with radial symmetry. Two special points where the Nahm zero modes have softer singularities are identified as constituent locations. To support this picture, the action density is computed analytically and numerically within a two dimensional slice containing the two constituents. For particular values of the parameters the torus can be cut in half yielding two copies of a twisted charge 1/2 instanton. Such objects comprise a single constituent. 
  Many two-dimensional physical systems have symmetries which are mathematically described by quantum groups (quasi-triangular Hopf algebras). In this letter we introduce the concept of a spontaneously broken Hopf symmetry and show that it provides an effective tool for analysing a wide variety of phases exhibiting many distinct confinement phenomena. 
  We concur with de Castro's observation that the gauge considerations of our approach are not valid. Nevertheless, except for an error that will be corrected, all of our findings are accurate independent of those considerations. 
  We find extrema of the potential of matter couplings to N=2 supergravity that define de Sitter vacua and no tachyonic modes. There are three essential ingredients in our construction: namely non-abelian non-compact gaugings, de Roo-Wagemans rotation angles, and Fayet-Iliopoulos terms. 
  A new concept of geometrization of electromagnetic field is proposed. Instead of the concept of extended field and its point sources, the interacting Maxwellian and Dirac electron--positron fields are considered as a microscopic unified closed connected nonmetrized space--time 4-manifold. Within this approach, the Dirac equation proves to be a group-theoretic relation that accounts for the topological and metric properties of this manifold. The Dirac spinors serve as basis functions of its fundamental group representation, while the tensor components of electromagnetic field prove to be the components of a curvature tensor of the relevant covering space. A basic distinction of the suggested approach from the geometrization of gravitational field in general relativity is that, first, not only the field is geometrized but also are its microscopic sources and, second, the field and its sources are treated not as a metrized Riemannian space--time but as a nonmetrized space-- time manifold. A possibility to geometrize weak interaction is also discussed. 
  We consider cosmological consequences of string theory tachyon condensation. We show that it is very difficult to obtain inflation in the simplest versions of this theory. Typically, inflation in these theories could occur only at super-Planckian densities, where the effective 4D field theory is inapplicable. Reheating and creation of matter in models where the tachyon potential V(T) has a minimum at infinitely large T is problematic because the tachyon field in such theories does not oscillate. If the universe after inflation is dominated by the energy density of the tachyon condensate, it will always remain dominated by the tachyons. It might happen that string condensation is responsible for a short stage of inflation at a nearly Planckian density, but one would need to have a second stage of inflation after that. This would imply that the tachyon played no role in the post-inflationary universe until the very late stages of its evolution. These problems do not appear in the recently proposed models of hybrid inflation where the complex tachyon field has a minimum at T << M_p. 
  We discuss magnetic monopoles in gauge theories with Wilson loops on orbifolds. We present a simple example in 5 dimensions with the fifth dimension compactified on an S^1/Z_2 orbifold. The Wilson loop in this SO(3) example replaces the adjoint Higgs scalar (needed to break SO(3) to U(1)) in the well-known 't Hooft - Polyakov construction. Our solution is a magnetic monopole string with finite energy, and length equal to the size of the extra dimension. 
  In this work we propose new non-commutative gauge theories that describe the dynamics of branes localized along twisted conjugacy classes on group manifolds. Our proposal is based on a careful analysis of the exact microscopic solution and it generalizes the matrix models (`fuzzy gauge theories') that are used to study e.g. the bound state formation of point-like branes in a curved background. We also construct a large number of classical solutions and interpret them in terms of condensation processes on branes localized along twisted conjugacy classes. 
  Characteristics of Abelian dyon - fermion bound system, parity - violating effects, a new series of energy spectra, effects related to the non - vanishing electric dipole moment, feature of spin orientation etc, are analyzed and compared with hydrogen - like atom. These analyses explore possibility of a new approach of searching for dyons under bound condition. 
  In the framework of the q-deformed Heisenberg algebra the investigation of $q$-deformation of Virial theorem explores that q-deformed quantum mechanics possesses better dynamical property. It is clarified that in the case of the zero potential the theoretical framework for the q-deformed Virial theorem is self-consistent. In the selfadjoint states the q-deformed uncertainty relation essentially deviates from the Heisenberg one. 
  We study the geometry of determinant line bundles associated to Dirac operators on compact odd dimensional manifolds. Physically, these arise as (local) vacuum line bundles in quantum gauge theory. We give a simplified derivation of the commutator anomaly formula using a construction based on noncyclic trace extensions and associated multiplicative renormalized determinants. 
  Casimir energy for solid conducting ball is considered on the base of some finite models. One model is physical and built of a battery of parallel metallic plates. Two finite models are based on the Higgs model of superconductivity. One of them is supersymmetric and based on the Witten field model for superconducting strings. Treatment shows that contribution of Casimir energy can be very essential for superdence state in the neutron stars and nuclear matter. 
  We study the standard angular momentum algebra $[x_i,x_j]=i\lambda \epsilon_{ijk}x_k$ as a noncommutative manifold $R^3_\lambda$. We show that there is a natural 4D differential calculus and obtain its cohomology and Hodge * operator. We solve the spin 0 wave equation and some aspects of the Maxwell or electromagnetic theory including solutions for a uniform electric current density, and we find a natural Dirac operator. We embed $R^3_\lambda$ inside a 4D noncommutative spacetime which is the limit $q\to 1$ of q-Minkowski space and show that $R^3_\lambda$ has a natural quantum isometry group given by the quantum double $D(U(su(2)))$ as a singular limit of the $q$-Lorentz group. We view $\R^3_\lambda$ as a collection of all fuzzy spheres taken together. We also analyse the semiclassical limit via minimum uncertainty states $|j,\theta,\phi>$ approximating classical positions in polar coordinates. 
  We use the anti-de Sitter (AdS) conformal field theory correspondence to calculate the conserved charges and the Euclidean actions of the charged rotating black string in four dimensions both in the canonical and the grand-canonical ensemble. The four-dimensional solution of the Einstein-Maxwell equations with a positive cosmological constant is introduced, and its conserved quantities and actions for fixed charge and fixed electric potential are calculated. We also study the thermodynamics of the asymptotically AdS black strings and perform a stability analysis both in the canonical and the grand-canonical ensembles. We find that the asymptotically AdS black string in the canonical ensemble is locally stable, while in the grand-canonical ensemble it is stable only for a part of the phase space. 
  In the Brans-Dicke model we treat the scalar field exactly and expand the gravitational field in a power series. A comparison with 2D sigma models and \phi^{4} perturbation theory in four dimensions suggests that the perturbation series in 4D Brans-Dicke model is renormalizable. 
  We treat free large N superconformal field theories as holographic duals of higher spin (HS) gauge theories expanded around AdS spacetime with radius R. The HS gauge theories contain massless and light massive AdS fields. The HS current correlators are written in a crossing symmetric form including only exchange of other HS currents. This and other arguments point to the existence of a consistent truncation to massless HS fields. A survey of massless HS theories with 32 supersymmetries in D=4,5,7 (where the 7D results are new) is given and the corresponding composite operators are discussed. In the case of AdS_4, the cubic couplings of a minimal bosonic massless HS gauge theory are described. We examine high energy/small tension limits giving rise to massless HS fields in the Type IIB string on AdS_5 x S^5 and M theory on AdS_{4/7} x S^{7/4}. We discuss breaking of HS symmetries to the symmetries of ordinary supergravity, and a particularly natural Higgs mechanism in AdS_5 x S^5 and AdS_4 x S^7 where the HS symmetry is broken by finite g_{YM}. In AdS_5 x S^5 it is shown that the supermultiplets of the leading Regge trajectory cross over into the massless HS spectrum. We propose that g_{YM}=0 corresponds to a critical string tension of order 1/R^2 and a finite string coupling of order 1/N. In AdS_7 x S^4 we give a rotating membrane solution coupling to the massless HS currents, and describe these as limits of Wilson surfaces in the A_{N-1}(2,0) SCFT, expandable in terms of operators with anomalous dimensions that are asymptotically small for large spin. The minimal energy configurations have semi-classical energy E=s for all s and the geometry of infinitely stretched strings with energy and spin density concentrated at the endpoints. 
  The minimal bosonic higher spin gauge theory in four dimensions contains massless particles of spin s=0,2,4,.. that arise in the symmetric product of two spin 0 singletons. It is based on an infinite dimensional extension of the AdS_4 algebra a la Vasiliev. We derive an expansion scheme in which the gravitational gauge fields are treated exactly and the gravitational curvatures and the higher spin gauge fields as weak perturbations. We also give the details of an explicit iteration procedure for obtaining the field equations to arbitrary order in curvatures. In particular, we highlight the structure of all the quadratic terms in the field equations. 
  We reconsider in some detail a construction allowing (Borel) convergence of an alternative perturbative expansion, for specific physical quantities of asymptotically free models. The usual perturbative expansions (with an explicit mass dependence) are transmuted into expansions in 1/F, where $F \sim 1/g(m)$ for $m \gg \Lambda$ while $F \sim (m/\Lambda)^\alpha$ for $m \lsim \Lambda$, $\Lambda$ being the basic scale and $\alpha$ given by renormalization group coefficients. (Borel) convergence holds in a range of $F$ which corresponds to reach unambiguously the strong coupling infrared regime near $m\to 0$, which can define certain "non-perturbative" quantities, such as the mass gap, from a resummation of this alternative expansion. Convergence properties can be further improved, when combined with $\delta$ expansion (variationally improved perturbation) methods. We illustrate these results by re-evaluating, from purely perturbative informations, the O(N) Gross-Neveu model mass gap, known for arbitrary $N$ from exact S matrix results. Comparing different levels of approximations that can be defined within our framework, we find reasonable agreement with the exact result. 
  We use recently proposed translations isometries of pp-waves to construct D4 and D3 brane solutions, using T-duality transformations, in exactly solvable pp-wave background originating from $AdS_3\times S^3$ geometry. A unique property of the new brane solutions is the breaking of SO(10-p-1) symmetry in the transverse direction of the branes due to the presence of constant NS-NS and R-R background fluxes. We verify that the our `localized' solutions satisfy the field equations and explicitly present the corresponding Killing spinors. We also show the connection of our results to certain M5-branes in pp-wave geometry. 
  Some of the key cohomological features of the two $(1+1)$-dimensional (2D) free Abelian- and self-interacting non-Abelian gauge theories (having no interaction with matter fields) are briefly discussed first in the language of symmetry properties of the Lagrangian densities and the same issues are subsequently addressed in the framework of superfield formulation on the four ($2 + 2)$-dimensional supermanifold. Special emphasis is laid on the on-shell- and off-shell nilpotent (co-)BRST symmetries that emerge after the application of (dual) horizontality conditions on the supermanifold. The (anti-)chiral superfields play a very decisive role in the derivation of the on-shell nilpotent symmetries. The study of the present superfield formulation leads to the derivation of some new symmetries for the Lagrangian density and the symmetric energy-momentum tensor. The topological nature of the above theories is captured in the framework of superfield formulation and the geometrical interpretations are provided for some of the topologically interesting quantities. 
  We compute the propagator for massless and massive scalar fields in the metric of the pp-wave. The retarded propagator for the massless field is found to stay confined to the surface formed by null geodesics. The algebraic form of the massive propagator is found to be related in a simple way to the form of the propagator in flat spacetime. 
  We use the generalized zeta function regularization method to compute the one-loop quantum correction to the masses of the TK1 and TK2 kinks in a deformation of the O(N) linear sigma model on the real line. 
  We present a class of orthogonal membrane configurations which preserve 1/4 of the full type IIA supersymmetry. These membrane configurations carry additional F-string charges. We further analyze the D1-D3 configuration after applying T- duality along the world volume directions of the above orthogonal membranes. 
  The vector and tensor fluctuations of vortices localizing gravity in the context of the six-dimensional Abelian Higgs model are studied. These string-like solutions break spontaneously six-dimensional Poincar\'e invariance leading to a finite four-dimensional Planck mass and to a regular geometry both in the bulk and on the core of the vortex. While the tensor modes of the metric are decoupled and exhibit a normalizable zero mode, the vector fluctuations, present in the gauge sector of the theory, are naturally coupled to the graviphoton fields associated with the vector perturbations of the warped geometry. Using the invariance under infinitesimal diffeomorphisms, it is found that the zero modes of the graviphoton fields are never localized. On the contrary,the fluctuations of the Abelian gauge field itself admit a normalizable zero mode. 
  A cosmological model with an exotic fluid is investigated. We show that the equation of state of this ``modified Chaplygin'' gas can describe the current accelerated expansion of the universe. We then reexpress it as FRW cosmological model containing a scalar field $\phi$ and find its self--interacting potential. Moreover motivated by recent works of Sen [sen1, sen2] and Padmanbhan [pad] on tachyon field theory, a map for this exotic fluid as a normal scalar field $\phi$ with Lagrangian ${\cal L}_{\phi} = \frac{\dot{\phi}^2}{2} - U (\phi)$ to the tachyonic field $T$ with Lagrangian ${\cal L}_{T} = - V(T) \sqrt{1 - \dot{T}^2}$ is obtained. 
  For a conformal theory it is natural to seek the conformal moduli space, M_c to which it belongs, generated by the exactly marginal deformations. By now we should have the tools to determine M_c in the presence of enough supersymmetry. Here it is shown that its dimension is determined in terms of a certain index. Moreover, the D-term of the global group is an obstruction for deformation, in presence of a certain amount of preserved supersymmetry. As an example we find that the deformations of the membrane (3d) field theory, under certain conditions, are in 35/SL(4,C). Other properties including the local geometry of M_c are discussed. 
  We compute the quasinormal modes associated with decay of the massless scalar filed around a small Schwarzschild-Anti-de-Sitter black hole. The computations shows that when the horizon radius is much less than the anti-de-Sitter radius, the imaginary part of the frequency goes to zero as $r_+^{d-2}$ while the real part of $\omega$ decreases to its minimum and then goes to $d-1$. Thus the quasinormal modes approach the usual AdS modes in the limit $r_+ -> 0$. This agrees with suggestions of Horowitz et al (Phys.Rev. D62 024027 (2000)). 
  Recently an action formulation, called the general WZW orbifold action, was given for each sector of every WZW orbifold. In this paper we gauge this action by general twisted gauge groups to find the action formulation of each sector of every coset orbifold. Connection with the known current-algebraic formulation of coset orbifolds is discussed as needed, and some large examples are worked out in further detail. 
  This paper serves to elucidate the nature of toric duality dubbed in hep-th/0003085 in the construction for world volume theories of D-branes probing arbitrary toric singularities. This duality will be seen to be due to certain permutation symmetries of multiplicities in the gauged linear sigma model fields. To this symmetry we shall refer as ``multiplicity symmetry.'' We present beautiful combinatorial properties of these multiplicities and rederive all known cases of torically dual theories under this new light. We also initiate an understanding of why such multiplicity symmetry naturally leads to monodromy and Seiberg duality. Furthermore we discuss certain ``flavor'' and ``node'' symmetries of the quiver and superpotential and how they are intimately related to the isometry of the background geometry, as well as how in certain cases complicated superpotentials can be derived by observations of the symmetries alone. 
  We construct Wigner's continuous spin representations of the Poincar\'e algebra for massless particles in higher dimensions. The states are labeled both by the length of a space-like translation vector and the Dynkin indices of the {\it short little group} $SO(d-3)$, where $d$ is the space-time dimension. Continuous spin representations are in one-to-one correspondence with representations of the short little group. We also demonstrate how combinations of the bosonic and fermionic representations form supermultiplets of the super-Poincar\'e algebra. If the light-cone translations are nilpotent, these representations become finite dimensional, but contain zero or negative norm states, and their supersymmetry algebra contains a central charge in four dimensions. 
  We study a minimally coupled tachyon field rolling down to its ground state on the FRW brane. We construct tacyonic potential which can implements power law inflation in the brane world cosmology. The potential turns out to be ${V_0 \phi^{-1}}$ on the brane and reduces to inverse square potential at late times when brane corrections to the Friedmann equation become negligible. We also do similar exercise with a normal scalar field and discover that the inverse square potential on the brane leads to power law inflation. 
  We construct new D6-brane model vacua (non-supersymmetric) that have at low energy exactly the standard model spectrum (with right handed neutrinos). The minimal version of these models requires five stacks of branes. and the construction is based on D6-branes intersecting at angles in $D = 4$ type toroidal orientifolds of type I strings. Three U(1)'s become massive through their couplings to RR couplings and from the two surviving anomaly free U(1)'s, one is the standard model hypercharge generator while the extra anomaly free U(1) could be broken from its non-zero couplings to RR fields and also by triggering a vev to previously massive particles. We suggest that extra massless U(1)'s should be broken by requiring some intersection to respect N=1 supersymmetry thus supporting the appearance of massless charged singlets at the supersymmetric intersection. Proton is stable as baryon number is gauged and its anomalies are cancelled through a generalized Green-Schwarz mechanism. Neutrinos are of Dirac type with small masses, as in the four stack standard models of hep-th/0105155, as a result of the existence of a similar PQ like-symmetry. The models are unique in the sense that they predict the existence of only one supersymmetric particle, the superpartner of $\nu_R$. 
  We consider non-linear gravitational models with a multidimensional warped product geometry. Particular attention is payed to models with quadratic scalar curvature terms. It is shown that for certain parameter ranges, the extra dimensions are stabilized if the internal spaces have negative constant curvature. In this case, the 4-dimensional effective cosmological constant as well as the bulk cosmological constant become negative. As a consequence, the homogeneous and isotropic external space is asymptotically AdS. The connection between the D-dimensional and the 4-dimensional fundamental mass scales sets a restriction on the parameters of the considered non-linear models. 
  Orbifold compactifications of 10D heterotic strings do allow different sets of chiral fermions at different fixed points. Even if the effective 4D theory is anomaly free by including the bulk fermions, there arise abelian and nonabelian fixed point anomalies from these chiral fermions. As the underlying string theory is well defined these localized fixed point anomalies of the chiral fermions are to be cancelled by a variant of the Green-Schwarz mechanism. 
  We construct the order alpha'^3 terms in the supersymmetric Yang-Mills action in ten dimensions for an arbitrary gauge group. The result can be expressed in terms of the structure constants of the Yang-Mills group, and is therefore independent of abelian factors. The alpha'^3 invariant obtained here is independent of the alpha'^2 invariant, and we argue that additional superinvariants will occur at all odd orders of alpha'. 
  We calculate the exact eigenvalues of the adjoint scalar fields in the massive vacua of N=1* SUSY Yang-Mills with gauge group SU(N). This provides a field theory prediction for the distribution of D3 brane charge in the AdS dual. We verify the proposal of Polchinski and Strassler that the D3-brane's lie on a fuzzy sphere in the supergravity limit and determine the corrections to this distribution due to worldsheet and quantum effects. The calculation also provides several new results concerning the equilibrium configurations of the N-body Calogero-Moser Hamiltonian. 
  We discuss a core instability of 't Hooft Polyakov monopoles in Alice electrodynamics type of models in which charge conjugation symmetry is gauged. The monopole may deform into a toroidal defect which carries an Alice flux and a (non-localizable) magnetic Cheshire charge. 
  We analyse two new versions of \theta-expanded non-commutative quantum electrodynamics up to first order in \theta and first loop order. In the first version we expand the bosonic sector using the Seiberg-Witten map, leaving the fermions unexpanded. In the second version we leave both bosons and fermions unexpanded. The analysis shows that the Seiberg-Witten map is a field redefinition at first order in \theta. However, at higher order in \theta the Seiberg-Witten map cannot be regarded as a field redefinition. We find that the initial action of any \theta-expanded massless non-commutative QED must include one extra term proportional to \theta which we identify by loop calculations. 
  Classical BRST invariance in the pure spinor formalism for the open superstring is shown to imply the supersymmetric Born-Infeld equations of motion for the background fields. These equations are obtained by requiring that the left and right-moving BRST currents are equal on the worldsheet boundary in the presence of the background. The Born-Infeld equations are expressed in N=1 D=10 superspace and include all abelian contributions to the low-energy equations of motion, as well as the leading non-abelian contributions. 
  We recast the phenomenon of duality cascades in terms of the Cartan matrix associated to the quiver gauge theories appearing in the cascade. In this language, Seiberg dualities for the different gauge factors correspond to Weyl reflections. We argue that the UV behavior of different duality cascades depends markedly on whether the Cartan matrix is affine ADE or not. In particular, we find examples of duality cascades that can't be continued after a finite energy scale, reaching a "duality wall", in terminology due to M. Strassler. For these duality cascades, we suggest the existence of a UV completion in terms of a little string theory. 
  The standard demand for the quantum partition function to be invariant under the renormalization group transformation results in a general class of exact renormalization group equations, different in the form of the kernel. Physical quantities should not be sensitive to the particular choice of the kernel. Such scheme independence is elegantly illustrated in the scalar case by showing that, even with a general kernel, the one-loop beta function may be expressed only in terms of the effective action vertices, and in this way the universal result is recovered. 
  We present numerical evidence for the existence of spinning generalizations for non-topological Q-ball solitons in the theory of a complex scalar field with a non-renormalizable self-interaction. To the best of our knowledge, this provides the first explicit example of spinning solitons in 3+1 dimensional Minkowski space. In addition, we find an infinite discrete family of radial excitations of non-rotating Q-balls, and construct also spinning Q-balls in 2+1 dimensions. 
  In this article we present the complete massless and massive one-loop triangle diagram results using the negative dimensional integration method (NDIM). We consider the following cases: massless internal fields; one massive, two massive with the same mass m and three equal masses for the virtual particles. Our results are given in terms of hypergeometric and hypergeometric-type functions of external momenta (and masses for the massive cases) where the propagators in the Feynman integrals are raised to arbitrary exponents and the dimension of the space-time D. Our approach reproduces the known results as well as other solutions as yet unknown in the literature. These new solutions occur naturally in the context of NDIM revealing a promising technique to solve Feynman integrals in quantum field theories. 
  We consider the massive vector $N$-component $(\lambda\phi^{4})_{D}$ theory in Euclidian space and, using an extended Matsubara formalism we perform a compactification on a $d$-dimensional subspace, $d\leq D$. This allows us to treat jointly the effect of temperature and spatial confinement in the effective potential of the model, setting forth grounds for an analysis of phase transitions driven by temperature and spatial boundary. For $d=2$, which corresponds to the heated system confined between two parallel planes (separation $L$), we obtain, in the large $N$ limit at one-loop order, formulas for temperature- and boundary-dependent mass and coupling constant. The equation for the critical curve in the $\beta \times L$ plane is also derived. 
  I construct 1/16, 1/8 and 1/4 BPS Wilson loops in N=4 supersymmetric Yang-Mills theory and argue that expectation values of 1/4 BPS loops do not receive quantum corrections. At strong coupling, non-renormalization of supersymmetric Wilson loops implies subtle cancellations in the partition function of the AdS string with special boundary conditions. The cancellations are shown to occur in the semiclassical approximation. 
  We construct D-branes in a left-right asymmetrically gauged WZW model, with the gauge subgroup embedded differently on the left and the right of the group element. The symmetry-preserving boundary conditions for the group-valued field $g$ are described, and the corresponding action is found. When the subgroup $H=U(1)$, we can implement T-duality on the axially gauged WZW action; an orbifold of the vectorially gauged theory is produced. For the parafermion $SU(2)/U(1)$ coset model, a $\sigma$-model is obtained with vanishing gauge field on D-branes. We show that a boundary condition surviving from the SU(2) parent theory characterizes D-branes in the parafermion theory, determining the shape of A-branes. The gauge field on B-branes is obtained from the boundary condition for A-branes, by the orbifold construction and T-duality. These gauge fields stabilize the B-branes. 
  We consider the tensor theory on coincident E8 5-branes compactified on a three torus. Using string theory, we predict that there must be distinct components in the moduli space of this theory.We argue that new superconformal field theories are to be found in these sectors with, for example, global G2 and F4 symmetries. In some cases, twisted E8 5-branes can be identified with small instantons in non-simply-laced gauge groups. This allows us to determine the Higgs branch for the fixed point theory.   We determine the Coulomb branch by using an M theory dual description involving partially frozen singularities. Along the way, we show that a D0-brane binds to two D4-branes, but not to an Sp-type O4-plane (despite the existence of a Higgs branch). These results are used to check various string/string dualities for which, in one case (quadruple versus NVS), we present a new argument. Finally, we describe the construction of new non-BPS branes as domain walls in various heterotic/type I string theories. 
  We show that a non-trivial dilaton condensation alters the dimensions of orientifold planes. An off-shell crosscap state which naturally interpolates between the usual on-shell crosscap states and their T-duals plays an important role in the analysis. We present an explicit representation of the off-shell crosscap state on an RP2 worldsheet in the gauge in which the worldsheet curvature in the bulk of the fundamental region of the RP2 vanishes. We show that the non-trivial dilaton condensation reproduces the correct descent relation among orientifold plane tensions. 
  We calculate the correction to the Bekenstein-Hawking entropy formula for five dimensional AdS-Schwarzschild black holes due to thermodynamic fluctuations. The result is then compared with the boundary gauge theory entropy corrections via AdS/CFT correspondence. We then further generalise our analysis for the rotating black hole in five dimensional AdS space. 
  We compute the spinorial cohomology of ten-dimensional abelian SYM at order alpha'^3 and we find that it is trivial. Consequently, linear supersymmetry alone excludes the presence of alpha'^3-order corrections. Our result lends support to the conjecture that there may be a unique supersymmetric deformation of ordinary ten-dimensional abelian SYM. 
  We investigate the phenomenon of brane induced supersymmetry breakdown on orbifolds in the presence of a Scherk-Schwarz mechanism. General consistency conditions are derived for arbitrary dimensions and the results are illustrated in the specific example of a 5-dimensional theory compactified on $S^1/Z_2$. This includes a discussion of the Kaluza-Klein spectrum and the possibility of a brane induced supersymmetry restoration. 
  Using the Wigner-Heisenberg algebra for bosonic systems in connection with oscillators we find a new representation for the Virasoro algebra. 
  We amalgamate three seemingly quite different fields of concepts and phenomena and argue that they actually represent closely related aspects of a more primordial space-time structure called by us wormhole spaces. Connes' framework of non-commutative topological spaces and ``points, speaking to each other'', a translocal web of (cor)relations, being hidden in the depth-structure of our macroscopic space-time and made visible by the application of a new geometric renormalisation process, and the apparent but difficult to understand translocal features of quantum theory. We argue that the conception of our space-time continuum as being basically an aggregate of structureless points is almost surely to poor and has to be extended and that the conceptual structure of quantum theory, in particular its translocal features like e.g. entanglement and complex superposition, are exactly a mesoscopic consequence of this microscopic wormhole structure. We emphasize the close connections with the ``small world phenomenon'' and rigorously show that the micro state of our space-time, viewed as a dynamical system, has to be critical in a scale free way as recently observed in other fields of network science. We then briefly indicate the mechanisms by which this non-local structure manages to appear in a seemingly local disguise on the surface level, thus invoking a certain Machian spirit. 
  We propose a dynamical matrix product ansatz describing the stochastic dynamics of two species of particles with excluded-volume interaction and the quantum mechanics of the associated quantum spin chains respectively. Analyzing consistency of the time-dependent algebra which is obtained from the action of the corresponding Markov generator, we obtain sufficient conditions on the hopping rates for identifing the integrable models. From the dynamical algebra we construct the quadratic algebra of Zamolodchikov type, associativity of which is a Yang Baxter equation. The Bethe ansatz equations for the spectra are obtained directly from the dynamical matrix product ansatz. 
  We use the process of quantum hamiltonian reduction of SU(2)_k, at rational level k, to study explicitly the correlators of the h_{1,s} fields in the c_{p,q} models. We find from direct calculation of the correlators that we have the possibility of extra, chiral and non-chiral, multiplet structure in the h_{1,s} operators beyond the `minimal' sector. At the level of the vacuum null vector h_{1,2p-1}=(p-1)(q-1) we find that there can be two extra non-chiral fermionic fields. The extra indicial structure present here permeates throughout the entire theory. In particular we find we have a chiral triplet of fields at h_{1,4p-1}=(2p-1)(2q-1). We conjecture that this triplet algebra may produce a rational extended c_{p,q} model. We also find a doublet of fields at h_{1,3p-1}=(\f{3p}{2}-1)(\f{3q}{2}-1). These are chiral fermionic operators if p and q are not both odd and otherwise parafermionic. 
  This is a set of lectures on the gauge/string duality and non-critical strings, with a particular emphasis on the discretized, or matrix model, approach. After a general discussion of various points of view, I describe the recent generalization to four dimensional non-critical (or five dimensional critical) string theories of the matrix model approach. This yields a fully non-perturbative and explicit definition of string theories with eight (or more) supercharges that are related to four dimensional CFTs and their relevant deformations. The space-time as well as world-sheet dimensions of the supersymmetry preserving world-sheet couplings are obtained. Exact formulas for the central charge of the space-time supersymmetry algebra as a function of these couplings are calculated. They include infinite series of string perturbative contributions as well as all the non-perturbative effects. An important insight on the gauge theory side is that instantons yield a non-trivial 1/N expansion at strong coupling, and generate open string contributions, in addition to the familiar closed strings from Feynman diagrams. We indicate various open problems and future directions of research. 
  Calculation by Douglas and Shenker of the tension ratios for vortices of different N-alities in the softly broken N=2 supersymmetric SU(N) Yang-Mills theory, is carried to the second order in the adjoint multiplet mass m. Corrections to the ratios violating the well-known sine formula are found, showing that it is not a universal quantity. 
  Gauge symmetry breaking through the Hosotani mechanism (the dynamics of nonintegrable phases) in softly broken supersymmetric QCD with $N_F^{fd}$ flavors is studied. For $N=$ even, there is a single SU(N) symmetric vacuum state, while for $N=$ odd, there is a doubly degenerate SU(N) symmetric vacuum state in the model. We also study generalized supersymmetric QCD by adding $N_F^{adj}$ numbers of massless adjoint matter. The gauge symmetry breaking pattern such as $SU(3)\to SU(2)\times U(1)$ is possible for appropriate choices of the matter content and values of the supersymmetry breaking parameter. The massless state of the adjoint Higgs scalar is also discussed in the models. 
  We argue that string interactions in a PP-wave spacetime are governed by an effective coupling $g_{eff}=g_s(\mu p^+\apm)f(\mu p^+ \apm)$ where $f(\mu p^+ \apm)$ is proportional to the light cone energy of the string states involved in the interaction. This simply follows from generalities of a Matrix String description of this background. $g_{eff}$ nicely interpolates between the expected result ($g_s$) for flat space (small $\mu p^+\apm$) and a recently conjectured expression from the perturbative gauge theory side (large $\mu p^+\apm$). 
  The main characteristics of the quantum oscillator coherent states including the two-particle Calogero interaction are investigated. We show that these Calogero coherent states are the eigenstates of the second-order differential annihilation operator which is deduced via R-deformed Heisenberg algebra or Wigner-Heisenberg algebraic technique and correspond exactly to the pure uncharged-bosonic states. They possess the important properties of non-orthogonality and completeness. The minimum uncertainty relation for the Calogero interaction coherent states is investigated. New sets of even and odd Wigner oscillator coherent states are pointed out. 
  A geometric relationship between loop quantum gravity and partitioned (triangulated) string theory is discussed. Combinatorial analysis reveals that three spatial and three curvature dimensions, intrinsic to the partitioned string, are necessary to replicate Standard Model particles and interactions. This analysis has established that particulate mass is determined by a functional relationship involving these six extra dimensions. The combinatorial analysis involves non-commutative 3D-matrix algebra which forms the mathematical underpinnings of Dirac notation. The functional relationship (symbolized by Beth) requires exponential, Randall-Sundrum, scaling to compute mass. Through the proper interpretation of complex gravity a cyclic cosmological model is developed. This formulation of cyclic cosmology inherently involves observed dark energy. Thus, a comprehensive theory is constructed from geometric fundamentals which models both massive, oscillating neutrinos and the current epoch of mini-inflation. 
  This is my contribution to the Festschrift honoring Stephen Hawking on his 60th birthday. Twenty-five years ago, Gibbons and Hawking laid out the semi-classical properties of de Sitter space. After a summary of their main results, I discuss some further quantum aspects that have since been understood. The largest de Sitter black hole displays an intriguing pattern of instabilities, which can render the boundary structure arbitrarily complicated. I review entropy bounds specific to de Sitter space and outline a few of the strategies and problems in the search for a full quantum theory of the spacetime. 
  We investigate for the N = 2 supersymmetry (SUSY) a relation between a vector supermultiplet of the linear SUSY and the Volkov-Akulov model of the nonlinear SUSY. We express component fields of the vector supermultiplet in terms of Nambu-Goldstone fermion fields at the leading orders in a SUSY invariant way, and show the vector nature of the U(1) gauge field explicitly. A relation of the actions for the two models is also discussed briefly. 
  We consider issues related to tachyonic inflation with exponential potential. We find exact solution of evolution equations in the slow roll limit in FRW cosmology. We also carry out similar analysis in case of Brane assisted tachyonic inflation. We investigate the phase space behavior of the system and show that the dust like solution is a late time attractor. The difficulties associated with reheating in the tachyonic model are also indicated. 
  We consider a D3-brane as boundary of a five dimensional charged anti de Sitter black hole. We show that the charge of the black hole induces a regular cosmological evolution for the scale factor of the brane, with a smooth transition between a contracting and an eventual expanding phase. Simple analytical solutions can be obtained in the case of a vanishing effective cosmological constant on the brane. A nonvanishing cosmological constant, or the inclusion of radiation on the brane, does not spoil the regularity of these solutions at small radii, and observational constraints such as the ones from primordial nucleosynthesis can be easily met. Fluctuations of brane fields remain in the linear regime provided the minimal size of the scale factor is sufficiently large. We conclude with an analysis of the Cardy-Verlinde formula in this set up. 
  For the PT symmetric potential of Dorey, Dunning and Tateo we show that in the large angular momentum (i.e., strongly spiked) limit the low-lying eigenstates of this popular non-Hermitian problem coincide with the shifted Hermitian harmonic oscillators calculated at the zero angular momentum. This type of an approximate Hermitization is valid in all the domain where the spectrum of energies remains real. It proves very efficient numerically. The construction is asymmetric with respect to the sign of the subdominant square-root spike, and exhibits a discontinuity at the point where the PT symmetric regularization vanishes. 
  We analyze the worldline formalism in the presence of a gravitational background. In the worldline formalism a path integral is used to quantize the worldline coordinates of the particles. Contrary to the simpler cases of scalar and vector backgrounds, external gravity requires a precise definition of the ultraviolet regularization of the path integral. Taking into account the UV regularization, we describe the first quantized representation of the one-loop effective action for a scalar particle. We compute explicitly the contribution to the graviton tadpole and self-energy to test the validity of the method. The results obtained by usual field theoretical Feynman diagrams are reproduced in an efficient way. Finally, we comment on the technical problems related to the factorization of the zero mode from the path integral on the circle. 
  The cylinder diagrams that determine the static interactions between pairs of Dp-branes in the type IIB plane wave background are evaluated. The resulting expressions are elegant generalizations of the flat-space formulae that depend on the value of the Ramond-Ramond flux of the background in a non-trivial manner. The closed-string and open-string descriptions consistently transform into each other under a modular transformation only when each of the interacting D-branes separately preserves half the supersymmetries. These results are derived for configurations of euclidean signature  D(p+1)-instantons but also generalize to lorentzian signature Dp-branes. 
  We describe how chiral matter charged under SU(N) and SO(2N) gauge groups arises from codimension seven singularities in compactifications of M-theory on manifolds with G(2) holonomy. The geometry of these spaces is that of a cone over a six-dimensional Einstein space which can be constructed by (multiple) unfolding of hyper-Kahler quotient spaces. In type IIA the corresponding picture is given by stacks of intersecting D6-branes and chiral matter arises from open strings stretching between them. Usually one obtains (bi)fundamental representations but by including orientifold six-planes in the type IIA picture we find more exotic representations like the anti-symmetric, which is important for the study of SU(5) grand unification, and trifundamental representations. We also exhibit many cases where the G(2) metrics can be described explicitly, although in general the metrics on the spaces constructed via unfolding are not known. 
  In this paper, we study the matrix model proposed by Berenstein, Maldacena, and Nastase to describe M-theory on the maximally supersymmetric pp-wave. We show that the model may be derived directly as a discretized theory of supermembranes in the pp-wave background, or alternatively, from the dynamics of D0-branes in type IIA string theory. We consider expanding the model about each of its classical supersymmetric vacua and note that for large values of the mass parameter \mu, interaction terms are suppressed by powers of 1/mu, so that the model may be studied in perturbation theory. We compute the exact spectrum about each of the vacua in the large \mu limit and find the complete (infinite) set of BPS states, which includes states preserving 2, 4, 6, 8, or 16 supercharges. Through explicit perturbative calculations, we then determine the effective coupling that controls the perturbation expansion for large \mu and estimate the range of parameters and energies for which perturbation theory is valid. 
  We apply the method of algebraic deformation to N-tuple of algebraic K3 surfaces. When N=3, we show that the deformed triplet of algebraic K3 surfaces exhibits a deformed hyperk\"{a}hler structure. The deformation moduli space of this family of noncommutatively deformed K3 surfaces turns out to be of dimension 57, which is three times of that of complex deformations of algebraic K3 surfaces. 
  In the first part of this work we review the equations of motion for the brane presented in Friedmann-Robertson-Walker (FRW) form, when bulk is five-dimensional (A)dS Black Hole. The spacelike (timelike) FRW brane equations are considered from the point of view of their representation in the form similar to two-dimensional CFT entropy, so-called Cardy-Verlinde (CV) formula. The following five-dimensional gravities are reviewed: Einstein, Einstein-Maxwell and Einstein with brane quantum corrections. The second part of the work is devoted to study FRW brane equations and their representation in CV form, brane induced matter and brane cosmology in Einstein-Gauss-Bonnet (GB) gravity. In particular, we focus on the inflationary brane cosmology. The energy conditions for brane matter are also analyzed. We show that for some values of GB coupling constant (bulk is AdS BH) the brane matter is not CFT. Its energy density and pressure are not always positive. 
  A low energy iteration scheme to study nonlinear gravity in the brane world is developed. As a result, we obtain the brane world effective action at low energies. The relation between the geometrical approach and the approach using the AdS/CFT correspondence is also clarified. In particular, we find generalized dark radiation as homogeneous solutions in our iteration scheme. Moreover, the precise correspondence between the bulk geometry and the brane effective action is established, which gives a holographic view of the brane world. 
  In this paper we study U(1) gauge transformations on the space-time coordinates and on the background fields $g_{\mu\nu}$ and $\phi$. For some special gauge functions, gauged coordinates and gauged U(1) field are equivalent to the rotated coordinates and rotated gauge field. We find gauge transformations that are symmetries of the string action. Also we obtain general $T$-duality transformations for the background fields. For special background fields this duality is equivalent to a gauge transformation. 
  We present a method for explicitly computing the non-perturbative superpotentials associated with the vector bundle moduli in heterotic superstrings and M-theory. This method is applicable to any stable, holomorphic vector bundle over an elliptically fibered Calabi-Yau threefold. For specificity, the vector bundle moduli superpotential, for a vector bundle with structure group G=SU(3), generated by a heterotic superstring wrapped once over an isolated curve in a Calabi-Yau threefold with base B=F1, is explicitly calculated. Its locus of critical points is discussed. Superpotentials of vector bundle moduli potentially have important implications for small instanton phase transitions and the vacuum stability and cosmology of superstrings and M-theory. 
  The structure of the commutator algebra for conformal quantum mechanics is considered. Specifically, it is shown that the emergence of a dimensional scale by renormalization implies the existence of an anomaly or quantum-mechanical symmetry breaking, which is explicitly displayed at the level of the generators of the SO(2,1) conformal group. Correspondingly, the associated breakdown of the conservation of the dilation and special conformal charges is derived. 
  The entropy and information puzzles arising from black holes cannot be resolved if quantum gravity effects remain confined to a microscopic scale. We use concrete computations in nonperturbative string theory to argue for three kinds of nonlocal effects that operate over macroscopic distances. These effects arise when we make a bound state of a large number of branes, and occur at the correct scale to resolve the paradoxes associated with black holes. 
  In non-commutative field theories conventional wisdom is that the unitarity is non-compatible with the perturbation analysis when time is involved in the non-commutative coordinates. However, as suggested by Bahns et.al. recently, the root of the problem lies in the improper definition of the time-ordered product. In this article, functional formalism of S-matrix is explicitly constructed for the non-commutative $\phi^p$ scalar field theory using the field equation in the Heisenberg picture and proper definition of time-ordering. This S-matrix is manifestly unitary. Using the free spectral (Wightmann) function as the free field propagator, we demonstrate the perturbation obeys the unitarity, and present the exact two particle scattering amplitude for 1+1 dimensional non-commutative nonlinear Schr\"odinger model. 
  It is shown that all the states in AdS(5) x S(5) supergravity have zero eigenvalue for the all Casimir operators of its symmetry group SU(2,2|4). To compute this universal zero in supergravity we refine the oscillator methods for studying the lowest weight unitary representations of SU(N,M|R,S). We solve the reduction problem when one multiplies an arbitrary number of super doubletons. This enters in the computation of the quadratic Casimir eigenvalues of the lowest weight representations. We apply the results to SU(2,2|4) that classifies the Kaluza-Klein towers of ten dimensional type IIB supergravity compactified on AdS(5) x S(5). We show that the vanishing of the SU(2,2|4) Casimir eigenvalues for all the states is indeed a group theoretical fact in AdS(5) x S(5) supergravity. By the AdS-CFT correspondence, it is also a fact for gauge invariant states of super Yang-Mills theory with four supersymmetries in four dimensions. This non-trivial and mysterious zero is very interesting because it is predicted as a straightforward consequence of the fundamental local Sp(2) symmetry in 2T-physics. Via the 2T-physics explanation of this zero we find a global indication that this special supergravity hides a twelve dimensional structure with (10,2) signature. 
  The connection between the supersymmetric quantum mechanics involving two-component eigenfunctions and the stability equation associated with two classical configurations is investigated, and a matrix superpotential is deduced. The issue of stability is ensured for the Bogomol'nyi-Prasad-Sommerfield (BPS) states on two domain walls in a scalar potential model containing up to fourth-order powers in the fields, which is explicit demonstrated using the intertwining operators in terms of 2x2-matrix superpotential in the algebraic framework of supersymmetry in quantum mechanics. Also, a non-BPS state is found to be non-stable via fluctuation hessian matrix. 
  Based on the model of a "soft" cellular space and deterministic quantum mechanics developed previously, the scattering of a free moving particle by structural units of the space -- superparticles -- is studied herein. The process of energy and inert mass transmission from the moving particle to superparticles and hence the creation of elementary excitations of the space -- inertons -- are analyzed in detail. The space crystallite made up around the particle in the degenerate space is shown to play the key role in those processes. A comprehensive analysis of the nature of the origin of gravitation, the particle's gravitational potential 1/r, and the gravitational interaction between material objects is performed. It seems reasonably to say that the main idea of the work may briefly be stated in the words: No motion, no gravity. 
  We present the formalism of q-stars with local or global U(1) symmetry. The equations we formulate are solved numerically and provide the main features of the soliton star. We study its behavior when the symmetry is local in contrast to the global case. A general result is that the soliton remains stable and does not decay into free particles and the electrostatic repulsion preserves it from gravitational collapse. We also investigate the case of a q-star with non-minimal energy-momentum tensor and find that the soliton is stable even in some cases of collapse when the coupling to gravity is absent. 
  Recently the time dependent solutions of type II supergravities in $d = 10$, with the metric having the symmetry $ISO(p+1) \times SO(8-p, 1)$ have been given by two groups (Chen-Gal'tsov-Gutperle (CGG), [hep-th/0204071] and Kruczenski-Myers-Peet (KMP), [hep-th/0204144]). The supergravity solutions correspond to space-like D$p$-branes in type II string theory. While the CGG solution is a four parameter solution, the KMP solution is a three parameter solution and so in general they are different. This difference can be attributed to the fact that unlike the CGG solution, KMP uses a specific boundary condition for the metric and the dilaton field. It is shown that when we impose the boundary conditions used in the KMP solution to the CGG solution then both become three parameter solutions and they map to each other under a coordinate transformation along with a Hodge duality of the field strength. We also give the relations between the parameters characterizing the two solutions. 
  We give an expression, in terms of boundary spectral functions, for the spectral asymmetry of the Euclidean Dirac operator in two dimensions, when its domain is determined by local boundary conditions, and the manifold is of product type. As an application, we explicitly evaluate the asymmetry in the case of a finite-length cylinder, and check that the outcome is consistent with our general result. Finally, we study the asymmetry in a disk, which is a non-product case, and propose an interpretation. 
  We study the superstring theory on pp-wave background with NSNS-flux that is realized as the Penrose limit of AdS_3 x S^3 x M^4, where M^4 is T^4 or T^4/Z_2(~ K3). Quantizing this system in the covariant gauge, we explicitly construct the space-time supersymmetry algebra and the complete set of DDF operators. We analyse the spectrum of physical states by using the spectrally flowed representations of current algebra. This spectrum is classified by the ``short string sectors'' and the ``long string sectors'' as in AdS_3 string theory. The states of the latter propagate freely along the transverse plane of pp-wave background, but the states of the former do not. We compare the short string spectrum with the BPS and almost BPS states which have large R-charges in the symmetric orbifold conformal theory, which is known as the candidate of dual theory of superstrings on AdS_3 x S^3 x M^4. We show that every short string states can be embedded successfully in the single particle Hilbert space of symmetric orbifold conformal theory. 
  This is a slightly expanded version of my talk at Future Perspectives in Theoretical Physics and Cosmology, Stephen Hawking's 60th Birthday Worshop. I describe some of the issues that were important in gauged supergravity in the 1980's and how these, and related issues have once again become important in the study of holographic field theories. 
  Integrability in string/field theories is known to emerge when considering dynamics in the moduli space of physical theories. This implies that one has to look at the dynamics with respect to unusual time variables like coupling constants or other quantities parameterizing configuration space of physical theories. The dynamics given by variations of coupling constants can be considered as a canonical transformation or, infinitesimally, a Hamiltonian flow in the space of physical systems. We briefly consider here an example of mechanical integrable systems. Then, any function $T(\vec p, \vec q)$ generates a one-parametric family of integrable systems in vicinity of a single system. For integrable system with several coupling constants the corresponding "Hamiltonians" $T_i(\vec p, \vec q)$ satisfy Whitham equations and after quantization (of the original system) become operators satisfying the zero-curvature condition in the space of coupling constants: $$ [\frac{\partial}{\partial g_a} - \hat T_a(\hat{\vec p},\hat{\vec q}),\ \frac{\partial}{\partial g_b} - \hat T_b(\hat{\vec p},\hat{\vec q})] = 0 $$ 
  We consider a six dimensional brane world model with asymmetric warp factors for time and both extra spatial coordinates, $y$ and $z$. We derive the set of differential equations governing the shortest graviton path and numerically solve it for AdS-Schwarzschild and AdS-Reissner-Nordstr\"om bulks. In both cases we derive a set of conditions for the existence of shortcuts in bulks with shielded singularities and show some examples of shortcuts obtained under these conditions. Consequences are discussed. 
  We consider supergravity solutions of D5 branes wrapped on supersymmetric 2-cycles and use them to discuss relevant features of four-dimensional N=1 and N=2 super Yang-Mills theories with gauge group SU(n). In particular in the N=1 case, using a gravitational dual of the gaugino condensate, we obtain the complete NSVZ beta-function. We also find non-perturbative corrections associated to fractional instantons with charge 2/n. These non-perturbative effects modify the running of the coupling constant which remains finite even at small scales in a way that resembles to the soft confinement scenario of QCD. 
  If TeV-scale gravity describes nature, black holes will be produced in particle accelerators, perhaps even with impressive rates at the Large Hadron Collider. Their decays, largely via the Hawking process, will be spectacular. Black holes also would be produced in cosmic ray collisions with our atmosphere, and their showers may be observable. Such a scenario means the end of our quest to understand the world at shorter distances, but may represent the beginning of the exploration of extra dimensions. 
  We compute the component field 4-dimensional N=1 supergravity Lagrangian that is obtained from a superfield Lagrangian in the U(1)_K formalism with a linear dilaton multiplet. All fermionic terms are presented. In a variety of important ways, our results generalize those that have been reported previously, and are flexible enough to accomodate many situations of phenomenological interest in string-inspired effective supergravity, especially models based on orbifold compactifications of the weakly-coupled heterotic string. We provide for an effective theory of hidden gaugino and matter condensation. We include supersymmetric Green-Schwarz counterterms associated with the cancellation of U(1) and modular duality anomalies; the modular duality counterterm is of a rather general form. Our assumed form for the dilaton Kahler potential is quite general and can accomodate Kahler stabilization methods. We note possible applications of our results. We also discuss the usefulness of the linear dilaton formulation as a complement to the chiral dilaton approach. 
  One of the most intriguing unsolved problems in modern cosmology involves the nature of the observed dark matter in the universe. Various hypothetical particles have been postulated as the source of the dark matter, but as of yet none of these particles have been observed. In this paper I present an alternative view. The observations indicate the presence of a new force that is much weaker than gravity on small scales and stronger than gravity on large scales. This new force is described by a Liouville field that consists of a scalar field with an exponential potential. The solutions have a Newtonian behavior on small scales and produce forces that are weak compared to gravity. On large scales the field has a logarithmic behavior and will therefore produce flat rotation curves. The Liouville field may therefore be able to account for the observations that are usually taken to imply the existence of dark matter. 
  First, we will study the cosmological perturbations from the brane point of view. It turns out that two types of the extra data are necessary to know the evolution of the system. To fix these data, the analysis of the bulk is needed. So, we have solved equations of motion for the bulk gravity and determined the extra data. We would like to stress that, both analysis take complementary roles to achieve this goal. 
  We study the cosmology induced on a brane probing a warped throat region in a Calabi-Yau compactification of type IIB string theory. For the case of a BPS D3-brane probing the Klebanov-Strassler warped deformed conifold, the cosmology described by a suitable brane observer is a bouncing, spatially flat Friedmann-Robertson-Walker universe with time-varying Newton's constant, which passes smoothly from a contracting to an expanding phase. In the Klebanov-Tseytlin approximation to the Klebanov-Strassler solution the cosmology would end with a big crunch singularity. In this sense, the warped deformed conifold provides a string theory resolution of a spacelike singularity in the brane cosmology. The four-dimensional effective action appropriate for a brane observer is a simple scalar-tensor theory of gravity. In this description of the physics, a bounce is possible because the relevant energy-momentum tensor can classically violate the null energy condition. 
  Commutative Yang-Mills theories in 1+1 dimensions exhibit an interesting interplay between geometrical properties and U(N) gauge structures: in the exact expression of a Wilson loop with $n$ windings a non trivial scaling intertwines $n$ and $N$. In the non-commutative case the interplay becomes tighter owing to the merging of space-time and ``internal'' symmetries in a larger gauge group $U(\infty)$. We perform an explicit perturbative calculation of such a loop up to ${\cal O}(g^6)$; rather surprisingly, we find that in the contribution from the crossed graphs (the genuine non-commutative terms) the scaling we mentioned occurs for large $n$ and $N$ in the limit of maximal non-commutativity $\theta=\infty$. We present arguments in favour of the persistence of such a scaling at any perturbative order and succeed in summing the related perturbative series. 
  We investigate de Sitter/conformal field theory (dS/CFT) correspondence in two dimensions. We define the conserved mass of de Sitter spacetime and formulate the correspondence along the lines of anti-de Sitter/conformal field theory duality. Asymptotic symmetry group, mass, and central charge of de Sitter spacetime are equal to those of anti-de Sitter spacetime. The entropy of two-dimensional de Sitter spacetime is evaluated by applying Cardy formula. We calculate the boundary correlators induced by the propagation of the dilaton in two-dimensional de Sitter space. Although the dilaton is a tachyonic perturbation in the bulk, boundary conformal correlators have positive dimension. 
  In this paper we propose that the cosmological constant scale and neutrino masses have a common origin: a new spontaneously broken scalar field. This hypothesis is implemented in a 6-space dimensional model that restores the parity symmetry. The 6-dimensional parity symmetry gives a natural mechanism for the smallness of the vacuum energy density. 
  We formulate matrix models for strings in ten dimensional pp-wave backgrounds and for particles in eleven dimensional ones. This is done by first characterizing the deformations of ten dimensional {\cal N}=1 SYM which are induced by a constant bispinorial coupling $\int \bar\Psi H\Psi$ plus a minimal purely bosonic completition and then by the appropriate dimensional reduction. We find a whole class of new models for the matrix strings and a generalization of the supernumerary supersymmetric models as far as the matrix theory for particles is concerned. A companion deformation of the IKKT matrix model is also discussed. 
  Noncommutative Yang-Mills theories are sensitive to the choice of the representation that enters in the gauge kinetic term. We constrain this ambiguity by considering grand unified theories. We find that at first order in the noncommutativity parameter \theta, SU(5) is not truly a unified theory, while SO(10) has a unique noncommutative generalization. In view of these results we discuss the noncommutative SM theory that is compatible with SO(10) GUT and find that there are no modifications to the SM gauge kinetic term at lowest order in \theta.   We study in detail the reality, hermiticity and C,P,T properties of the Seiberg-Witten map and of the resulting effective actions expanded in ordinary fields. We find that in models of GUTs (or compatible with GUTs) right-handed fermions and left-handed ones appear with opposite Seiberg-Witten map. 
  We calculate a general spectral correlation function of products and ratios of characteristic polynomials for a $N\times N$ random matrix taken from the chiral Gaussian Unitary Ensemble (chGUE). Our derivation is based upon finding an Itzykson-Zuber type integral for matrices from the non-compact manifold ${\sf{Gl(n,{\mathcal{C}})/U(1)\times ...\times U(1)}}$ (matrix Macdonald function). The correlation function is shown to be always represented in a determinant form generalising the known expressions for only positive moments. Finally, we present the asymptotic formula for the correlation function in the large matrix size limit. 
  We obtain the correct all-loop beta-function of pure N=1 super Yang-Mills theory from the supergravity solution of the warped deformed conifold, including also some nonperturbative corrections. The crucial ingredient is the gauge-gravity relation that can be inferred by taking into account the phenomenon of gaugino condensation. 
  We use Berkovits' pure spinor quantization to compute various three-point tree correlation functions in position-space for the Type IIB superstring. We solve the constraint equations for the vertex operators and obtain explicit expressions for the graviton and axion components of the vertex operators. Using these operators we compute tree level correlation functions in flat space and discuss their extension to the AdS5 X S5 background. 
  We study Witten open string field theory in the pp-wave background in the tensionless limit, and construct the N-string vertex in the basis which diagonalizes the string perturbative spectrum. We found that the Witten *-product can be viewed as infinite copies of the Moyal product with the same noncommutativity parameter $\theta=2$. Moreover, we show that this Moyal structure is universal in the sense that, written in the string bit basis, Witten's *-product for any background can always be given in terms of the above-mentioned Moyal structure. We identify some projective operators in this algebra that we argue to correspond to D-branes of the theory. 
  In this paper we discuss deformations of the BRST operator of the fermionic string. These deformations preserve nilpotency of the BRST operator and correspond to turning on infinitesimal Gravitino and Ramond-Ramond spacetime fields. 
  We derive the gauge invariant perturbation equations for a 5-dimensional bulk spacetime in the presence of a brane. The equations are derived in full generality, without specifying a particular energy content of the bulk or the brane. We do not assume Z_2 symmetry, and show that the degree of freedom associated with brane motion plays a crucial role. Our formalism may also be used in the Z_2 symmetric case where it simplifies considerably. 
  A class of exact non-renormalized extremal correlators of half-BPS operators in N=4 SYM, with U(N) gauge group, is shown to satisfy finite factorization equations reminiscent of topological gauge theories. The finite factorization equations can be generalized, beyond the extremal case, to a class of correlators involving observables with a simple pattern of SO(6) charges. The simple group theoretic form of the correlators allows equalities between ratios of correlators in N=4 SYM and Wilson loops in Chern-Simons theories at k=\infty, correlators of appropriate observables in topological G/G models and Wilson loops in two-dimensional Yang-Mills theories. The correlators also obey sum rules which can be generalized to off-extremal correlators. The simplest sum rules can be viewed as large k limits of the Verlinde formula using the Chern-Simons correspondence. For special classes of correlators, the saturation of the factorization equations by a small subset of the operators in the large N theory is related to the emergence of semiclassical objects like KK modes and giant gravitons in the dual ADS \times S background. We comment on an intriguing symmetry between KK modes and giant gravitons. 
  We analyze the fluctuations of Nielsen-Olesen vortices arising in the six-dimensional Abelian-Higgs model. The regular geometry generated by the defect breaks spontaneously six-dimensional Poincar\'e symmetry leading to a warped space-time with finite four-dimensional Planck mass. As a consequence, the zero mode of the spin two fluctuations of the geometry is always localized but the graviphoton fields, corresponding to spin one metric fluctuations, give rise to zero modes which are not localized either because of their behaviour at infinity or because of their behaviour near the core of the vortex. A similar situation occurs for spin zero fluctuations. Gauge field fluctuations exhibit a localized zero mode. 
  We develop a Hamiltonian formalism which can be used to discuss the physics of a massless scalar field in a gravitational background of a Schwarzschild black hole. Using this formalism we show that the time evolution of the system is unitary and yet all known results such as the existence of Hawking radiation can be readily understood. We then point out that the Hamiltonian formalism leads to interesting observations about black hole entropy and the information paradox. 
  It is shown that Electromagnetism creates geometry different from Riemannian geometry. General geometry including Riemannian geometry as a special case is constructed. It is proven that the most simplest special case of General Geometry is geometry underlying Electromagnetism. Action for electromagnetic field and Maxwell equations are derived from curvature function of geometry underlying Electromagnetism. And it is shown that equation of motion for a particle interacting with electromagnetic field coincides exactly with equation for geodesics of geometry underlying Electromagnetism. 
  The Casimir effect giving rise to an attractive force between the configuration boundaries that confine the massless scalar field is rigorously proven for odd dimensional hypercube with the Dirichlet boundary conditions and different spacetime dimensions D by the Epstein zeta function regularization. 
  We present in this paper quantum real lines as quantum defomations of the real numbers $\R$.Upon deforming the Heisenberg algebra $\cL$ generated by $(a, a^\dagger)$ in terms of the Moyal $\ast$-product,we first construct q-deformed algebras of q-differentiable functions in two cases where q is generic (not a root of unity) and q is the N-th root of unity. We then investigate these algebras and finally propose two quantum real lines as the base spaces of these algebras. It is turned out that both quantum lines are discrete spaces and have noncommutative structures.We further find, minimal length, fuzzy structure and infinitesimal structure. 
  A family of de Sitter vacua introduced in hep-th/0203198 as plausible initial conditions for inflation, are discussed from the point of view of de Sitter holography. The vacua are argued to be physically acceptable and the inflationary picture provides a physical interpretation of a subfamily of de Sitter invariant vacua. Some speculations on the issue of vacuum choice and the connection between the CMBR and holography are also provided. 
  The soliton spectrum (massive and massless) of a family of integrable models with local U(1) and U(1)\otimes U(1) symmetries is studied. These models represent relevant integrable deformations of SL(2,R) \otimes U(1)^{n-1} - WZW and SL(2,R) \otimes SL(2,R)\otimes U(1)^{n-2} - WZW models. Their massless solitons appears as specific topological solutions of the U(1) (or U(1)\otimes U(1)) - CFTs. The nonconformal analog of the GKO-coset formula is derived and used in the construction of the composite massive solitons of the ungauged integrable models. 
  We study the one dimensional potentials in q space and the new features that arise. In particular we show that the probability of tunneling of a particle through a barrier or potential step is less than the one of the same particle with the same energy in ordinary space which is somehow unexpected. We also show that the tunneling time for a particle in q space is less than the one of the same particle in ordinary space. 
  We investigate N=2, D=5 supersymmetry and matter-coupled supergravity theories in a superconformal context. In a first stage we do not require the existence of a Lagrangian. Under this assumption, we already find at the level of rigid supersymmetry, i.e. before coupling to conformal supergravity, more general matter couplings than have been considered in the literature. For instance, we construct new vector-tensor multiplet couplings, theories with an odd number of tensor multiplets, and hypermultiplets whose scalar manifold geometry is not hyperkaehler.   Next, we construct rigid superconformal Lagrangians. This requires some extra ingredients that are not available for all dynamical systems. However, for the generalizations with tensor multiplets mentioned above, we find corresponding new actions and scalar potentials. Finally, we extend the supersymmetry to local superconformal symmetry, making use of the Weyl multiplet. Throughout the paper, we will indicate the various geometrical concepts that arise, and as an application we compute the non-vanishing components of the Ricci tensor of hypercomplex group manifolds. Our results can be used as a starting point to obtain more general matter-couplings to Poincare supergravity. 
  Weak gravity in the brane world scenario with a scalar field in the bulk between two end-of-the-world branes is investigated by using the doubly gauge invariant formalism. We review the result that Einstein gravity is restored in the low energy limit and that high-energy corrections to Einstein gravity can be mimicked by higher derivative terms. After that, it is argued that non-locality is one of essential features of brane worlds. 
  After setting up a general model for supersymmetric classical mechanics in more than one dimension we describe systems with centrally symmetric potentials and their Poisson algebra. We then apply this information to the investigation and solution of the supersymmetric Coulomb problem, specified by an 1/|x| repulsive bosonic potential. 
  We reconsider the role that bundle gerbes play in the formulation of the WZW model on closed and open surfaces. In particular, we show how an analysis of bundle gerbes on groups covered by SU(N) permits to determine the spectrum of symmetric branes in the boundary version of the WZW model with such groups as the target. We also describe a simple relation between the open string amplitudes in the WZW models based on simply connected groups and in their simple-current orbifolds. 
  We investigate the physical and mathematical structure of a new class of geometric transitions proposed by Aganagic and Vafa. The distinctive aspect of these transitions is the presence of open string instanton corrections to Chern-Simons theory. We find a precise match between open and closed string topological amplitudes applying a beautiful idea proposed by Witten some time ago. The closed string amplitudes are reproduced from an open string perspective as a result of a fascinating interplay of enumerative techniques and Chern-Simons computations. 
  We present a novel global E_7(7) symmetry in five-dimensional maximal supergravity as well as an E_8(8) symmetry in d=4. These symmetry groups which are known to be present after reduction to d=4 and d=3, respectively, appear as conformal extensions of the respective well-known hidden-symmetry groups. A global scaling symmetry of the Lagrangian is the key to enhancement of E_6(6) to E_7(7) in d=5 and E_7(7) to E_8(8) in d=4. The group action on the physical fields is induced by conformal transformations in auxiliary spaces of dimensions 27 and 56, respectively. The construction is analogous to the one where the conformal group of Minkowski space acts on the boundary of AdS_5 space. A geometrical picture underlying the action of these ``conformal duality groups'' is given. 
  Coupling fundamental quarks to QCD in the dual string representation corresponds to adding the open string sector. Flavors therefore should be represented by space-time filling D-branes in the dual 5d closed string background. This requires several interesting properties of D-branes in AdS. D-branes have to be able to end in thin air in order to account for massive quarks, which only live in the UV region. They must come in distinct sets, representing the chiral global symmetry, with a bifundamental field playing the role of the chiral condensate. We show that these expectations are born out in several supersymmetric examples. To analyze most of these properties it is not necessary to go beyond the probe limit in which one neglects the backreaction of the flavor D-branes. 
  Chern-Simons type gauge field is generated by the means of the singular area preserving transformations in the lowest Landau level of electrons forming fractional quantum Hall state. Dynamics is governed by the system of constraints which correspond to the Gauss law in the non-commutative Chern-Simons gauge theory and to the lowest Landau level condition in the picture of composite fermions. Physically reasonable solution to this constraints corresponds to the Laughlin state. It is argued that the model leads to the non-commutative Chern-Simons theory of the QHE and composite fermions. 
  We examine some excited state energies in the non-unitary integrable quantum field theory obtained from the perturbation of the minimal conformal field theory model $M_{3,5}$ by its operator $\phi_{2,1}$. Using the correspondence of this IQFT to the scaling limit of the dilute $A_2$ lattice model (in a particular regime) we derive the functional equations for the QFT commuting transfer matrices. These functional equations can be transformed to a closed set of TBA-like integral equations which determine the excited state energies in the finite-size system. In particular, we explicitly construct these equations for the ground state and two lowest excited states. Numerical results for the associated energy gaps are compared with those obtained by the truncated conformal space approach (TCSA). 
  A general path integral analysis of the separable Hamiltonian of Liouville-type is reviewed. The basic dynamical principle used is the Jacobi's principle of least action for given energy which is reparametrization invariant, and thus the gauge freedom naturally appears. The choice of gauge in path integral corresponds to the separation of variables in operator formalism. The gauge independence and the operator ordering are closely related. The path integral in this formulation sums over orbits in space instead of space-time. An exact path integral of the Green's function for the hydrogen atom in parabolic coordinates is ilustrated as an example, which is also interpreted as one-dimensional quantum gravity with a quantized cosmological constant. 
  We introduce a new class of effective actions describing dynamically broken supersymmetric theories in an essentially non-perturbative region. Our approach is a generalization of the known supersymmetric non-linear sigma models, but allows in contrast to the latter the description of dynamical supersymmetry breaking by non-perturbative non-semiclassical effects. This non-perturbative breaking mechanism takes place in confined theories, where the effective fields are composite operators. It is necessary within the context of quantum effective actions and the associated concept of symmetry breaking as a hysteresis effect. In this paper we provide a mathematical definition and description of the actions, its application to specific supersymmetic gauge theories is presented elsewhere. 
  We investigate O(4) textures in a background with a positive cosmological constant. We find static solutions which co-move with the expanding background. There exists a solution in which the scalar field is regular at the horizon. This solution has a noninteger winding number smaller than one. There also exist solutions in which scalar-field derivatives are singular at the horizon. Such solutions can complete one winding within the horizon. If the winding number is larger than some critical value, static solutions including the regular one are unstable under perturbations. 
  We find transformation matrices allowing to express non-commutative three dimensional harmonic oscillator in terms of an isotropic commutative oscillator, following ``philosophy of simplicity'' approach. Non-commutative parameters have physical interpretation in terms of an external magnetic field. Furthermore, we show that for a particular choice of noncommutative parameters there is an equivalent anisotropic representation, whose transformation matrices are far more complicated. We indicate a way to obtain the more complex solutions from the simple ones. 
  Using well known Lagrangean techniques for uncovering the gauge symmetries of a Lagrangean, we derive the transformation laws for the phase space variables corresponding to local symmetries of the Hamilton equations of motion. These transformation laws are shown to coincide with those derived by Hamiltonian methods based on the Dirac conjecture. The connection between the Lagrangean and Hamiltonian approach is illustrated for first class systems involving one primary constraint. 
  Semi-classical soliton solutions for superstrings in $AdS_5\times S^5$ are used to predict the dimension of gauge theory operators in $\N=4 $ SU(N) SYM theory. We discuss the possible origin of scaling violations on the gauge theory side. 
  We extend the path-integral formalism for Poisson-Lie T-duality to include the case of Drinfeld doubles which can be decomposed into bi-algebras in more than one way. We give the correct shift of the dilaton, correcting a mistake in the litterature. We then use the fact that the six dimensional Drinfeld doubles have been classified to write down all possible conformal Poisson-Lie T-duals of three dimensional space times and we explicitly work out two duals to the constant dilaton and zero anti-symmetric tensor Bianchi type V space time and show that they satisfy the string equations of motion. This space-time was previously thought to have no duals because of the tracefulness of the structure constants. 
  We explore the bubble spacetimes which can be obtained from double analytic continuations of static and rotating black holes in anti-de Sitter space. In particular, we find that rotating black holes with elliptic horizon lead to bubble spacetimes only in dimension greater than five. For dimension greater than seven, the topology of the bubble can be non-spherical. However, a bubble spacetime is shown to arise from a rotating de Sitter black hole in four dimensions. In all cases, the evolution of the bubble is of de Sitter type. Double analytic continuations of hyperbolic black holes and branes are also discussed. 
  We discuss the low energy effective action for the Bosonic and Fermionic zero-modes of a smooth BPS Randall-Sundrum domain wall, including the induced supergravity on the wall. The result is a pure supergravity in one lower dimension. In particular, and in contrast to non-gravitational domain walls or domain walls in a compact space, the zero-modes representing transverse fluctuations of domain wall have vanishing action. 
  The symmetry algebra of a QFT in the presence of an external EM background (named "residual symmetry") is investigated within a Lie-algebraic, model independent scheme. Some results previously encountered in the literature are here extended. In particular we compute the symmetry algebra for a constant EM background in D=3 and D=4 dimensions. In D= 3 dimensions the residual symmetry algebra is isomorphic to $u(1)\oplus {\cal P}_c(2)$, with ${\cal P}_c(2)$ the centrally extended 2-dimensional Poincar\'e algebra. In D=4 dimension the generic residual symmetry algebra is given by a seven-dimensional solvable Lie algebra which is explicitly computed. Residual symmetry algebras are also computed for specific non-constant EM backgrounds. 
  We discuss Coleman's theorem concerning the energy density of the ground state of the sine-Gordon model proved in Phys. Rev. D 11, 2088 (1975). According to this theorem the energy density of the ground state of the sine-Gordon model should be unbounded from below for coupling constants beta^2 > 8 pi. The consequence of this theorem would be the non-existence of the quantum ground state of the sine-Gordon model for beta^2 > 8 pi. We show that the energy density of the ground state in the sine-Gordon model is bounded from below even for beta^2 > 8 pi. This result is discussed in relation to Coleman's theorem (Comm. Math. Phys. 31, 259 (1973)), particle mass spectra and soliton-soliton scattering in the sine-Gordon model. 
  In a traditional gauge theory, the matter fields \phi^a and the gauge fields A^c_\mu are fundamental objects of the theory. The traditional gauge field is similar to the connection coefficient in the Riemannian geometry covariant derivative, and the field-strength tensor is similar to the curvature tensor. In contrast, the connection in Riemannian geometry is derived from the metric or an embedding space. Guided by the physical principal of increasing symmetry among the four forces, we propose a different construction. Instead of defining the transformation properties of a fundamental gauge field, we derive the gauge theory from an embedding of a gauge fiber F=R^n or F=C^n into a trivial, embedding vector bundle F=R^N or F=C^N where N>n. Our new action is symmetric between the gauge theory and the Riemannian geometry. By expressing gauge-covariant fields in terms of the orthonormal gauge basis vectors, we recover a traditional, SO(n) or U(n) gauge theory. In contrast, the new theory has all matter fields on a particular fiber couple with the same coupling constant. Even the matter fields on a C^1 fiber, which have a U(1) symmetry group, couple with the same charge of +/- q. The physical origin of this unique coupling constant is a generalization of the general relativity equivalence principle. Because our action is independent of the choice of basis, its natural invariance group is GL(n,R) or GL(n,C). Last, the new action also requires a small correction to the general-relativity action proportional to the square of the curvature tensor. 
  There has been substantial interest in obtaining a quantum-gravitational description of de Sitter space. However, any such attempts have encountered formidable obstacles, and new philosophical directions may be in order. One possibility, although somewhat speculative, would be to view the physical universe as a timelike hypersurface evolving in a higher-dimensional bulk spacetime; that is, the renowned brane-world scenario. In this paper, we extend some recent studies along this line, and consider a non-critical 3-brane moving in the background of an anti-de Sitter Reissner-Nordstrom-like black hole. Interestingly, even an arbitrarily small electrostatic charge in the bulk can induce a singularity-free ``bounce'' universe on the brane, whereas a vanishing charge typically implies a singular cosmology. However, under closer examination, from a holographic (dS/CFT) perspective, we demonstrate that the charge-induced bounce cosmologies are not physically viable. This implies the necessity for censoring against charge in a bulk black hole. 
  The well-known Bogomol'nyi-Prasad-Sommerfield (BPS) monopole is considered in the limit of the infinite mass of the Higgs field as a basis of the Yang-Mills field vacuum with the finite energy density. In this limit, the Higgs field disappears, but it leaves its traces as the BPS monopole transforms into the Wu-Yang monopole obtained in the pure Yang-Mills theory by a spontaneous scale symmetry breaking in the class of functions with the zero value of a topological charge. The topological degeneration of the BPS monopole manifests itself as the Gribov copies of the covariant Coulomb gauge in the form of the time integral of the Gauss constraint. We also show that in the theory there is a zero mode of the Gauss constraint leading to an electric monopole and an additional mass of eta_0-meson in QCD. The consequences of the monopole vacuum in the form of a rising potential and topological confinement are studied in the framework of the perturbation theory. An estimation of the vacuum expectation value of the square of the magnetic tension is given through the eta_0-meson mass, and arguments in favour of the stability of the monopole vacuum are considered. We also discuss why all these "smiles" of the Cheshire cat are kept by the Dirac fundamental quantization, but not the conventional Faddeev-Popov integral. 
  We advocate a method to improve systematically the self-consistent harmonic approximation (or the Gaussian approximation), which has been employed extensively in condensed matter physics and statistical mechanics. We demonstrate the {\em convergence} of the method in a model obtained from dimensional reduction of SU($N$) Yang-Mills theory in $D$ dimensions. Explicit calculations have been carried out up to the 7th order in the large-N limit, and we do observe a clear convergence to Monte Carlo results. For $D \gtrsim 10$ the convergence is already achieved at the 3rd order, which suggests that the method is particularly useful for studying the IIB matrix model, a conjectured nonperturbative definition of type IIB superstring theory. 
  We argue that the dielectric effect, mostly studied for systems of coincident D-branes, is also extendible to configurations of multiple gravitational waves. We provide some evidence that Matrix string theory has an alternative interpretation as describing also, in the static gauge, multiple Type IIA gravitational waves. Starting with the linearised action of Matrix string theory in a weakly curved background, we identify the non-Abelian couplings of multiple coinciding gravitons to weak background fields, both in Type IIA and in Type IIB, and we use them to study various dielectric configurations from the point of view of the expanding gravitons. We also identify, in the Abelian limit, some non-perturbative correction terms to the Abelian gravitational wave action. 
  Two approaches to the study of cosmological perturbations in the brane-world scenario are compared: the first uses the 5D equations directly whereas the second approach projects them onto the 4D brane and then uses the effective 4D equations. 
  Statistical systems displaying a strongly anisotropic or dynamical scaling behaviour are characterized by an anisotropy exponent theta or a dynamical exponent z. For a given value of theta, we construct local scale transformations which can be viewed as scale transformations with a space-time-dependent dilatation factor. Two distinct types of local scale transformations are found. The first type may describe strongly anisotropic scaling of static systems with a given value of theta, whereas the second type may describe dynamical scaling with a dynamical exponent z. Local scale transformations act as a dynamical symmetry group of certain non-local free-field theories. Known special cases of local scale invariance are conformal invariance for theta=1 and Schrodinger invariance for theta=2. The hypothesis of local scale invariance implies that two-point functions of quasiprimary operators satisfy certain linear fractional differential equations, which are constructed from commuting fractional derivatives. The explicit solution of these yields exact expressions for two-point correlators at equilibrium and for two-point response functions out of equilibrium. Aparticularly simple and general form is found for the two-time autoresponse function. These predictions are explicitly confirmed at the uniaxial Lifshitz points in the ANNNI and ANNNS models and in the aging behaviour of simple ferromagnets such as the kinetic Glauber-Ising model and the kinetic spherical model with a non-conserved order parameter undergoing either phase-ordering kinetics or non-equilibrium critical dynamics. 
  Euclidean d=3 SU(2) Yang-Mills-Chern-Simons (YMCS) theory, including Georgi-Glashow (GGCS) theory, may have solitons in the presence of appropriate mass terms. For integral CS level k and for solitons carrying integral CS number, YMCS is gauge-invariant and consistent. However, individual solitons such as sphalerons and linked center vortices with CS number of 1/2 and writhing center vortices with arbitrary CS number are non-compact; a condensate of them threatens compactness of the theory. We study various forms of the non-compact theory in the dilute-gas approximation, treating the parameters of non-compact large gauge transformations as collective coordinates. We conclude: 1) YMCS theory dynamically compactifies; non-compact YMCS have infinitely higher vacuum energy than compact YMCS. 2) An odd number of sphalerons is associated with a domain- wall sphaleron, a pure-gauge configuration on a closed surface enclosing them and with a half-integral CS number. 3) We interpret the domain-wall sphaleron in terms of fictitious closed Abelian magnetic field lines that express the links of the Hopf fibration. Sphalerons are over- and under-crossings of knots in the field lines; the domain-wall sphaleron is a superconducting wall confining these knots to a compact domain. 4) Analogous results hold for center vortices and nexuses. 5) For a CS term induced with an odd number of fermion doublets, domain-wall sphalerons are related to non-normalizable fermion modes. 6) GGCS with monopoles is compactified with center-vortex-like strings. 
  We investigate Penrose limits of two classes of non-local theories, little string theories and non-commutative gauge theories. Penrose limits of the near-horizon geometry of NS5-branes help to shed some light on the high energy spectrum of little string theories. We attempt to understand renormalization group flow in these theories by considering Penrose limits wherein the null geodesic also has a radial component. In particular, we demonstrate that it is possible to construct a pp-wave spacetime which interpolates between the linear dilaton and the AdS regions for the Type IIA NS5-brane. Similar analysis is considered for the holographic dual geometry to non-commutative field theories. 
  I will discuss the development of inflationary theory and its present status, as well as some recent attempts to suggest an alternative to inflation. In particular, I will argue that the ekpyrotic scenario in its original form does not solve any of the major cosmological problems. Meanwhile, the cyclic scenario is not an alternative to inflation but rather a complicated version of inflationary theory. This scenario does not solve the flatness and entropy problems, and it suffers from the singularity problem. We describe many other problems that need to be resolved in order to realize a cyclic regime in this scenario, produce density perturbations of a desirable magnitude, and preserve them after the singularity. We propose several modifications of this scenario and conclude that the best way to improve it is to add a usual stage of inflation after the singularity and use that inflationary stage to generate perturbations in the standard way. This modification significantly simplifies the cyclic scenario, eliminates all of its numerous problems, and makes it equivalent to the usual chaotic inflation scenario. 
  A covariant quantization method for physical systems with reducible constraints is presented. 
  We present a conjecture that the universal enveloping algebra of differential operators $\frac{\p}{\p t_k}$ over $\mathbb{C}$ coincides in the origin with the universal enveloping algebra of the (Borel subalgebra of) Virasoro generators from the Kontsevich model. Thus, we can decompose any (pseudo)differential operator to a combination of the Virasoro operators. Using this decomposition we present the r.h.s. of the Givental formula math.AG/0008067 as a constant part of the differential operator we introduce. In the case of $\mathbb{CP}^1$ studied in hep-th/0103254, the l.h.s. of the Givental formula is a unit, which imposes certain constraints on this differential operator. We explicitly check that these constraints are correct up to $O(q^4)$. We also propose a conjecture of factorization modulo Hirota equation of the differential operator introduced and check this conjecture with the same accuracy. 
  The effective action for the Brink-Schwarz Superparticle is constructed in an infinite dimensional phase space using a gauge invariant formulation. 
  Chaos criterion for quantum field theory is proposed. Its correspondence with classical chaos criterion in semi-classical regime is shown. It is demonstrated for real scalar field that proposed chaos criterion can be used to investigate stability of classical solutions of field equations. 
  The space of couplings of a given theory is the arena of interest in this article. Equipped with a metric ansatz akin to the Fisher information matrix in the space of parameters in statistics (similar metrics in physics are the Zamolodchikov metric or the O'Connor--Stephens metric) we investigate the geometry of theory space through a study of specific examples. We then look into renormalisation group flows in theory space and make an attempt to characterise such flows via its isotropic expansion, rotation and shear. Consequences arising from the evolution equation for the isotropic expansion are discussed. We conclude by pointing out generalisations and pose some open questions. 
  After reviewing the supertubes and super brane-antibrane systems in the context of matrix model, we look for more general higher-dimensional configurations. For D3-bar{D3}, we find a non-trivial configuration with E cdot B not equal to 0 and describe the worldvolume gauge theory. We present the string probe of D3-bar{D3} system and study the decoupling limits leading to either noncommutative Super-Yang-Mills or NCOS theories with eight supercharges. 
  We consider the Penrose limit of the solution of D5-brane given in the Anti-de Sitter space and analyse the shape of the D5-brane in the pp-wave background. We find that the D5-brane leads to the branes and the throats connecting the branes. The branes spread on R^4 with periodic values of the light-cone time x^+ and the throats lie along x^+. We also give some comments on holography. 
  We studied the massive scalar wave propagation in the background of Reissner-Nordstr\"{o}m black hole by using numerical simulations. We learned that the value $Mm$ plays an important role in determining the properties of the relaxation of the perturbation. For $Mm << 1$ the relaxation process depends only on the field parameter and does not depend on the spacetime parameters. For $Mm >> 1$, the dependence of the relaxation on the black hole parameters appears. The bigger mass of the black hole, the faster the perturbation decays. The difference of the relaxation process caused by the black hole charge $Q$ has also been exhibited. 
  Spin networks are natural generalization of Wilson loops functionals. They have been extensively studied in the case where the gauge group is compact and it has been shown that they naturally form a basis of gauge invariant observables. Physically the restriction to compact gauge group is enough for the study of Yang-mills theories, however it is well known that non-compact groups naturally arise as internal gauge groups for Lorentzian gravity models. In this context a proper construction of gauge invariant observables is needed. The purpose of this work is to define the notion of spin network states for non-compact groups. We first built, by a careful gauge fixing procedure, a natural measure and a Hilbert space structure on the space of gauge invariant graph connection. Spin networks are then defined as generalized eigenvectors of a complete set of hermitic commuting operators. We show how the delicate issue of taking the quotient of a space by non compact groups can be address in term of algebraic geometry. We finally construct the full Hilbert space containing all spin network states. Having in mind application to gravity we illustrate our results for the groups SL(2,R), SL(2,C). 
  Assuming the S-matrix on noncommutative (NC) spacetime can still be developped perturbatively in terms of the time-ordered exponential of the interaction Lagrangian, we investigate the perturbation theory of NC field theory. We first work out with care some typical Green functions starting from the usual concepts of time-ordering and commutation relations for free fields. The results are found to be very different from those in the naive approach pursued in the literature. A simple framework then appears naturally which can incorporate the new features of our results and which turns out to be the usual time-ordered perturbation theory extended to the NC context. We provide the prescriptions for computing S-matrix elements and Green functions in this framework. We also emphasize that the naive seemingly covariant approach cannot be reproduced from the current one, in contrast to the field theory on ordinary spacetime. We attribute this to the phase-like nonlocal interaction intrinsic in NC field theory which modifies the analytic properties of Green functions significantly. 
  The pp-wave (Penrose limit) in conformal field theory can be viewed as a special contraction of the unitary representations of the conformal group. We study the kinematics of conformal fields in this limit in a geometric approach where the effect of the contraction can be visualized as an expansion of space-time. We discuss the two common models of space-time as carrier spaces for conformal fields: One is the usual Minkowski space and the other is the coset of the conformal group over its maximal compact subgroup. We show that only the latter manifold and the corresponding conformal representation theory admit a non-singular contraction limit. We also address the issue of correlation functions of conformal fields in the pp-wave limit. We show that they have a well-defined contraction limit if their space-time dependence merges with the dependence on the coordinates of the R symmetry group. This is a manifestation of the fact that in the limit the space-time and R symmetries become indistinguishable. Our results might find applications in actual calculations of correlation functions of composite operators in N=4 super Yang-Mills theory. 
  Observational evidence suggests that our universe is presently dominated by a dark energy component and undergoing accelerated expansion. We recently introduced a model, motivated by string theory for short-distance physics, for explaining dark energy without appealing to any fine-tuning. The idea of the transplanckian dark energy (TDE) was based on the freeze-out mechanism of the ultralow frequency modes, $\omega(k)$ of very short distances, by the expansion of the background universe, $\omega(k) \leq H$. In this paper we address the issue of the stress-energy tensor for the nonlinear short-distance physics and explain the need to modify Einstein equations in this regime. From the modified Einstein equations we then derive the equation of state for the TDE model, which has the distinctive feature of being continually time-dependent. The explanation of the coincidence puzzle relies entirely on the intrinsic time-evolution of the TDE equation of state. 
  The Polyakov world-line path integral describing the propagation of gluon field quanta is constructed by employing the background gauge fixing method and is subsequently applied to analytically compute the divergent terms of the one (gluonic) loop effective action to fourth order in perturbation theory. The merits of the proposed approach is that, to a given order, it reduces to performing two integrations, one over a set of Grassmann and one over a set of Feynman-type parameters through which one manages to accomodate all Feynman diagrams entering the computation at once. 
  Recently, Batalin and Marnelius proposed a superfield algorithm for master actions in the BV-formulation for a class of first order gauge field theories. Possible theories are determined by a ghost number prescription and a simple local master equation. We investigate consistent solutions of these local master equations with emphasis on four and six dimensional theories. 
  We explore the 1/4 BPS configurations of the supersymmetric gauge theories on $R^{1+3} \times S^1$. The BPS bound for energy and the BPS equations are obtained and the characteristics of the BPS solutions are studied. These BPS configurations describe electrically charged calorons, which are constituted of dyons and carry linear momentum along the compact direction. We carry out various approaches to the single caloron case in the theory of SU(2) gauge group. 
  We study the physical spectrum of cubic open string field theory around universal solutions, which are constructed using the matter Virasoro operators and the ghost and anti-ghost fields. We find the cohomology of the new BRS charge around the solutions, which appear with a ghost number that differs from that of the original theory. Considering the gauge-unfixed string field theory, we conclude that open string excitations perturbatively disappear after the condensation of the string field to the solutions. 
  In three dimensions, a `master theory' for all Thurston geometries requires imaginary flux. However, these geometries can be obtained from physical three-dimensional theories with various additional scalar fields, which can be interpreted as moduli in various compactifications of a higher-dimensional `master theory'. Three Thurston geometries are of the form N_2 x S^1, where N_2 denotes a two-dimensional Riemannian space of constant curvature. This enables us to twist these spaces, via T-duality, into other Thurston geometries as a U(1) bundle over N_2. In this way, Hopf T-duality relates all but one of the geometries in the higher-dimensional M-theoretic framework. The exception is the `Sol geometry,' which results from the dimensional reduction of the decoupling limit of the D3-brane in a background B-field. 
  The vacuum energy for a massless conformally coupled scalar field in a brane world set up, corresponding to de Sitter branes in a bulk anti-de Sitter spacetime, is calculated. We use the Euclidean version of the metric which can be conformally related to a metric similar in form to the Einstein universe (S^4 times R). Employing zeta-function regularisation we evaluate the one-loop effective potential and show that the vacuum energy is zero for the one brane and non-zero for the two brane configuration. We comment on the back-reaction of this Casimir energy and on the inclusion of a mass term or non-conformal coupling. 
  The two surprising features of gravity are (a) the principle of equivalence and (b) the connection between gravity and thermodynamics. Using principle of equivalence and special relativity in the {\it local inertial frame}, one could obtain the insight that gravity must possess a geometrical description. I show that, using the same principle of equivalence, special relativity and quantum theory in the {\it local Rindler frame} one can obtain the Einstein-Hilbert action functional for gravity and thus the dynamics of the spacetime. This approach, which essentially involves postulating that the horizon area must be proportional to the entropy, uses the local Rindler frame as a natural extension of the local inertial frame and leads to the interpretation that the gravitational action represents the free energy of the spacetime geometry. As an aside, one also obtains a natural explanation as to: (i) why the covariant action for gravity contains second derivatives of the metric tensor and (ii) why the gravitational coupling constant is positive. The analysis suggests that gravity is intrinsically holographic and even intrinsically quantum mechanical. 
  We present an explicit evidence that shows the correspondence between the type IIB supergravity in the pp-wave background and its dual supersymmetric Yang-Mills theory at the interaction level. We first construct the cubic term of the light-cone interaction Hamiltonian for the dilaton-axion sector of the supergravity. Our result agrees with the corresponding part of the light-cone string field theory (SFT) and furthermore fixes its previously undetermined $p^+$-dependent normalization. Adopting thus fixed light-cone SFT, we compute the matrix elements of light-cone Hamiltonian involving three chiral primary states and find an agreement with a prediction from the dual Yang-Mills theory. 
  Following Gubser, Klebanov and Polyakov [hep-th/0204051], we study strings in AdS black hole backgrounds. With respect to the pure AdS case, rotating strings are replaced by orbiting strings. We interpret these orbiting strings as CFT states of large spin similar to glueballs propagating through a gluon plasma. The energy and the spin of the orbiting string configurations are associated with the energy and the spin of states in the dual finite temperature N=4 SYM theory. We analyse in particular the limiting cases of short and long strings. Moreover, we perform a thermodynamic study of the angular momentum transfer from the glueball to the plasma by considering string orbits around rotating AdS black holes. We find that standard expectations, such as the complete thermal dissociation of the glueball, are borne out after subtle properties of rotating AdS black holes are taken into account. 
  We study the Type IIA NS five-brane wrapped on a Calabi-Yau manifold X in a double-scaled decoupling limit. We calculate the euclidean partition function in the presence of a flat RR 3-form field. The classical contribution is given by a sum over fluxes of the self-dual tensor field which reduces to a theta-function. The quantum contributions are computed using a T-dual IIB background where the five-branes are replaced by an ALE singularity. Using the supergravity effective action we find that the loop corrections to the free energy are given by B-model topological string amplitudes. This seems to provide a direct link between the double-scaled little strings on the five-brane worldvolume and topological strings. Both the classical and quantum contributions to the partition function satisfy (conjugate) holomorphic anomaly equations, which explains an observation of Witten relating topological string theory to the quantization of three-form fields. 
  Using gauge formulation of gravity the three-dimensional SU(2) YM theory equations of motion are presented in equivalent form as FRW cosmological equations. With the radiation, the particular (periodic, big bang-big crunch) three-dimensional universe is constructed. Cosmological entropy bounds (so-called Cardy-Verlinde formula) have the standard form in such universe. Mapping such universe back to YM formulation we got the thermal solution of YM theory. The corresponding holographic entropy bounds (Cardy-Verlinde formula) in YM theory are constructed. This indicates to universal character of holographic relations. 
  We study the 1/4 BPS equations in the eight dimensional noncommutative Yang-Mills theory found by Bak, Lee and Park. We explicitly construct some solutions of the 1/4 BPS equations using the noncommutative version of the ADHM-like construction in eight dimensions. From the calculation of topological charges, we show that our solutions can be interpreted as the bound states of the $D0$-$D4$-$D8$ with a $B$-field. We also discuss the structure of the moduli space of the 1/4 BPS solutions and determine the metric of the moduli space of the U(2) one-instanton in four and eight dimensions. 
  The simplest (2+1)-dimensional mechanical systems associated with light-like curves, already studied by Nersessian and Ramos, are reconsidered. The action is linear in the curvature of the particle path and the moduli spaces of solutions are completely exhibited in 3-dimensional Minkowski background, even when the action is not proportional to the pseudo-arc length of the trajectory. 
  We define an operator which for odd-dimensional compact gauge group furnishes unitary equivalence of the bosonic and fermionic sector in the supersymmetric quantum-mechanical matrix model obtained by dimensional reduction from 3-dimensional supersymmetric Yang-Mills theory. 
  We study for subgroups $G\subseteq U(N)$ partial summations of the $\theta$-expanded perturbation theory. On diagrammatic level a summation procedure is established, which in the U(N) case delivers the full star-product induced rules. Thereby we uncover a cancellation mechanism between certain diagrams, which is crucial in the U(N) case, but set out of work for $G\subset U(N)$. In addition, an explicit proof is given that for $G\subset U(N), G\neq U(M), M<N$ there is no partial summation of the $\theta $-expanded rules resulting in new Feynman rules using the U(N) star-product vertices and besides suitable modified propagators at most a $finite$ number of additional building blocks. Finally, we show that certain SO(N) Feynman rules conjectured in the literature cannot be derived from the enveloping algebra approach. 
  I present a novel class of exactly solvable quantum field theories. They describe non-relativistic fermions on even dimensional flat space, coupled to a constant external magnetic field and a four point interaction defined with the Groenewold-Moyal star product. Using Hamiltonian quantization and a suitable regularization, I show that these models have a dynamical symmetry corresponding to $\gl_\infty\oplus \gl_\infty$ at the special points where the magnetic field $B$ is related to the matrix $\theta$ defining the star product as $B\theta=\pm I$. I construct all eigenvalues and eigenstates of the many-body Hamiltonian at these special points. I argue that this solution cannot be obtained by any mean-field theory, i.e. the models describe correlated fermions. I also mention other possible interpretations of these models in solid state physics. 
  String propagation on a three-dimensional Lorentzian string orbifold with a null singularity has been studied by Horowitz and Steif, and more recently by Liu, Moore and Seiberg. We analyze the target space as a classical gravitational background. The singularity becomes spacelike when an arbitrarily small amount of matter is thrown at the singularity. This can be seen directly by studying the null singularity as a limit of the M=0, J=0 BTZ black hole metric. 
  A bosonic string coupled to the generalized Chern-Simons theory in 3+1D acquires a magnetic field along itself, when it is closed, and a topological charge at its extremity, when it is open. We construct the creation operators for the full quantum field states associated to these strings and determine the dual algebra satisfied by them. We show that the creation operator fo the composite state of a quantum closed bosonic string, bearing a magnetic flux, and a topologically charged open bosonic string, possesses generalized statistics. The relation of our results with previous approaches to the problem is also established. 
  We consider ``bubbles of nothing'' constructed by analytically continuing black hole solutions in Anti-de Sitter space. These provide interesting examples of smooth time-dependent backgrounds which can be studied through the AdS/CFT correspondence. Our examples include bubbles constructed from Schwarzschild-AdS, Kerr-AdS and Reissner-Nordstrom AdS. The Schwarzschild bubble is dual to Yang-Mills theory on three dimensional de Sitter space times a circle. We construct the boundary stress tensor of the bubble spacetime and relate it to the properties of field theory on de Sitter. 
  The Penrose limits of a D5-brane wrapped on the sphere of AdS_5 x S^5 and connected to the boundary by M fundamental strings, which is dual to the baryon vertex of the N=4 SU(M) super Yang-Mills theory, are investigated. It is shown that, for null geodesics that lead to the maximally supersymmetric Hpp-wave background, the resulting D5-brane is a 1/2-supersymmetric null brane. For an appropriate choice of radial geodesic, however, the limiting configuration is 1/4-supersymmetric and closely related to the Penrose limit of a flat space BIon. 
  In this note we investigate a new type of non-commutative field theory based on a constant skew-symmetric three-form parameter. In 3+1 dimensions such a three-form parameter can be viewed as a short-distance regulator which nevertheless preserves spatial-rotation and at long range preserves Lorentz invariance approximately. For a scalar field theory with quartic self-interaction we obtain drastically improved ultra-violet behavior of the diagrams, due to the oscillatory dependence of the interaction vertex on the momenta. The radiative corrections to the coupling are rendered finite already at the one-loop level. The key finding of this paper is that what appears as the reemergence of UV divergences as IR singularity in $p \to 0$ limit, must be interpreted simply as the logarithmic running of the coupling. Thus at low energies the theory is virtually indistinguishable from the standard theory. Conversely at high energies the diagram converges exponentially fast, the running of the coupling stops and the theory avoids developing the Landau pole. Bare coupling defined at high energy can be kept small, and in this sense the theory is similar to asymptotically free theories. 
  Following the work of Lima et al on the exact evaluation of the nonperturbative contribution to the superpotential from open-membrane instanton in Heterotic M-Theory, we evaluate systematically the contribution to the superpotential of a membrane instanton obtained by wrapping of a single M2 brane, once, on an isolated supersymmetric 3-cycle in a G_2-holonomy manifold. We then try to relate the results obtained to those sketched out in Harvey and Moore. We also do a heat-kernel asymptotics analysis to see whether one gets similar UV-divergent terms for (one or both of) the bosonic and fermionic determinants indicative of (partial) cancelation among them. The answer is in the affirmative, as expected by the supersymmetry of the starting membrane action. This work is a first step, both, in extending the work of Harvey and Moore to the evaluation of nonperturbative superpotentials for non-rigid supersymmetric 3-cycle wrappings, and in understanding the large N Chern-Simons/closed type-A topological string theory duality of Gopakumar and Vafa from M theory point of view. 
  We present example of exact solution to Witten's open bosonic string field theory. We will analyse the new BRST operator and we will argue that the new solution describes the flow from zero slope limit to the tensionless limit in the string world-sheet action. 
  We study warped compactifications to three dimensions, realized as an orientifold of type IIA string theory on T^7. By turning on 3- and 4-form fluxes on the torus in a supersymmetric way, we generate a potential for the moduli fields. We present various flux configurations with N=1,2,3,4,5,6 supersymmetries and count the number of moduli in each case. In particular, we show that there are N=1 configurations where all but one of the moduli are frozen. 
  We extend the pp-wave correspondence to a non supersymmetric example. The model is the type 0B string theory on the pp-wave R-R background. We explicitly solve the model and give the spectrum of physical states. The field theory counterpart is given by a sector of the large N SU(N) x SU(N) CFT living on a stack of N electric and N magnetic D3-branes. The relevant effective coupling constant is g_{eff}=g_sN/J^2. The string theory has a tachyon in the spectrum, whose light-cone energy can be exactly computed as a function of g_{eff}. We argue that the perturbative analysis in g_{eff} in the dual gauge theory is reliable, with corrections of non perturbative type. We find a precise state/operator map, showing that the first perturbative corrections to the anomalous dimensions of the operators have the behavior expected from the string analysis. 
  We give a worldsheet proof of the equivalence between the U(N) Chern-Simons gauge theory on S^3 and the topological closed string theory on the resolved conifold geometry. When the `t Hooft coupling of the gauge theory is small, the dual closed string worldsheet develops a new branch. We show that the fluctuations of the worldsheet into this branch effectively correspond to ``holes'' on the worldsheet, generating an open string sector. This leads to a microscopic description of how the `t Hooft expansion of gauge theory amplitudes is reproduced in the closed string computation. We find that the closed string amplitudes also contain terms which are not captured in the `t Hooft expansion but are present in the exact computation in the gauge theory amplitudes. These arise when the whole Riemann surface is in the new branch. We also discuss the cases with SO and Sp gauge groups. 
  In the context of a toy model we discuss the phenomenon of colliding five-branes, with two of the extra space dimensions compactified on tori. In one of the branes (hidden world) the torus is magnetised. Assuming opposite-tension branes, we argue that the collision results eventually in a (time-dependent) cosmological vacuum energy, whose value today is tiny, lying comfortably within the standard bounds by setting the breaking of the four-dimensional supersymmetry at a TeV scale. The small value of the vacuum energy as compared with the supersymmetry-breaking scale is attributed to transient phenomena with relaxation times of order of the Age of the Universe. An interesting feature of the approach is the absence of a cosmic horizon, thereby allowing for a proper definition of an S-matrix.As a result of the string non-criticality induced at the collision,our model does not provide an alternative to inflation, given that arguments can be given supporting the occurence of an inflationary phase early after the collision. The physics before the collision is not relevant to our arguments on the cosmological constant hierarchy, which are valid for asymptotically long times after it. 
  In this article we extend previous work by the authors, and elaborate further on the structure of the general solution to the graviton and dilaton equations of motion in brane world scenaria, in the context of five-dimensional effective actions with O(\alpha ')$ higher-curvature corrections, compatible with bulk string-amplitude calculations. We consider (multi)brane scenaria, dividing the bulk space into regions, in which one matches two classes of general solutions, a linear Randall-Sundrum solution and a (logarithmic) dilatonic domain wall (bulk naked singularity).We pay particular attention to examining the possibility of resolving the mass hierarchy problem together with the vanishing of the vacuum energy on the observable world, which is taken to be a positive tension brane. The appearance of naked dilatonic domain walls provides a dynamical restriction of the bulk space time. Of particular interest is a dilatonic-wall solution, which after appropriate coordinate transformation, results in a linear dilaton conformal field theory. The latter may provide a holographic resolution of the naked singularity problem. All the string-inspired models involved have the generic feature that the brane tensions are proportional to the string coupling $g_s$; it remains a challenge for string theory, therefore, to show whether microscopic models respecting this feature can be constructed. 
  We analyze theories in which a supersymmetric sector is coupled to a supersymmetry-breaking sector described by a non-linear realization. We show how to consistently couple N=1 supersymmetric matter to non-supersymmetric matter in such a way that all interactions are invariant under non-linear supersymmetry transformations. We extend this formalism to couple N=2 supersymmetric matter to N=1 superfields that lack N=2 partners but transform in a non-linear representation of the N=2 algebra. In particular, we show how to couple an N=2 vector to N=1 chiral fields in a consistent way. This has important applications to effective field theories describing the interactions of D-brane world-volume fields with bulk fields. We apply our method to study systems where different sectors break different halves of supersymmetry, which appear naturally in models of intersecting branes. 
  We examine a large N duality via geometric transition for N=1 SO/Sp gauge theories with superpotential for adjoint chiral superfield. In this paper, we find that the large N gauge theories are exactly analyzed for the classical quartic superpotentials by the finite rank SO/Sp gauge theories. With this classical superpotentials, we evaluate the confining phase superpotentials using the Seiberg-Witten theory. In the dual theory, we calculate the superpotential generated by the R-R and NS-NS 3-form fluxes. As the non-trivial examples, we discuss for SO(6), SO(8) and Sp(4) gauge theories. In these cases we have the perfect agreement of the confining phase superpotentials up to the 4th order of the glueball superfields. 
  We discuss the Penrose limit of pure AdS space, which is flat Minkowski space. Even though there is no holographic principle, we construct a ``holographic screen'' on which information on the corresponding CFT is encoded. The screen is obtained as a gauge-fixing condition upon restricting the Hilbert space to the states that are annihilated by the generator of scale transformations. This constraint leads to Dirac brackets which turn the Poincare algebra into the algebra of the conformal group on the ``holographic screen.'' 
  We find nonsupersymmetric and supersymmetric solutions of D3 brane configuration in the background of pp wave obtained as a Penrose limit of $AdS_5\times S^5$. 
  The quantum action generated by fermions which are minimally coupled to abelian vortex background fields is studied in D=2+1 and D=3+1 Euclidean dimensions. We present a detailed analysis of single- and binary-vortex configurations using the recently developed method of worldline numerics. The dependence of the fermion-induced quantum action on the fermion mass and the magnetic fluxes carried by the vortices is studied, and the binary-vortex interaction is computed. Additionally, we discuss the chiral condensate generated from a dilute gas of vortices in the intermediate fermion mass range for the case D=3+1. As a byproduct, our findings provide insight into the validity limits of the derivative expansion, which is the standard analytical approach to inhomogeneous backgrounds. 
  We deal with the presence of topological defects in models for two real scalar fields. We comment on defects hosting topological defects, and we search for explicit defect solutions using the trial orbit method. As we know, under certain circumstances the second order equations of motion can be solved by first order differential equations. In this case we show that the trial orbit method can be used very efficiently to obtain explicit solutions. 
  We develop new tools for an in-depth study of our recent proposal for Matrix Theory. We construct the anomaly-free and finite planar continuum limit of the ground state with SO(2^{13}) symmetry matching with the tadpole and tachyon free IR stable high temperature ground state of the open and closed bosonic string. The correspondence between large N limits and spacetime effective actions is demonstrated more generally for an arbitrary D25brane ground state which might include brane-antibrane pairs or NS-branes and which need not have an action formulation. Closure of the finite N matrix Lorentz algebra nevertheless requires that such a ground state is simultaneously charged under all even rank antisymmetric matrix potentials. Additional invariance under the gauge symmetry mediated by the one-form matrix potential requires a ground state charged under the full spectrum of antisymmetric (p+1)-form matrix potentials with p taking any integer value less than 26. Matrix Dbrane democracy has a beautiful large N remnant in the form of mixed Chern-Simons couplings in the effective Lagrangian whenever the one-form gauge symmetry is nonabelian. 
  At critical coupling, the interactions of Ginzburg-Landau vortices are determined by the metric on the moduli space of static solutions. The asymptotic form of the metric for two well separated vortices is shown here to be expressible in terms of a Bessel function. A straightforward extension gives the metric for N vortices. The asymptotic metric is also shown to follow from a physical model, where each vortex is treated as a point-like particle carrying a scalar charge and a magnetic dipole moment of the same magnitude. The geodesic motion of two well separated vortices is investigated, and the asymptotic dependence of the scattering angle on the impact parameter is determined. Formulae for the asymptotic Ricci and scalar curvatures of the N-vortex moduli space are also obtained. 
  We discuss the associativity or WDVV equations and demonstrate that they can be rewritten as certain functional relations between the {\it second} derivatives of a single function, similar to the dispersionless Hirota equations. The properties of these functional relations are further discussed. 
  The origins of the 11-dimensional supermembrane are recalled, and a curious property is discussed: the field theory limit of a supermembrane in a hyper-K\"ahler background is a 3-dimensional sigma-model with N=4 supersymmetry, but the higher-order fermion interactions of the supermembrane generically break this to N=3. 
  We study the thermodynamic consequences of a recently proposed description for a Schwarzschild black hole based on Euclidean (D3,D3)+(\bar{D3},\bar{D3}) brane pairs described in terms of chain-like excitations. A discrete mass-spectrum of Bekenstein-type is inferred and upon identification of the black hole mass with the chain's energy the leading corrections to both Hawking-temperature and specific heat of the black hole are obtained. The results indicate that for small black holes the evaporation process will be considerably altered. 
  In this paper we calculate three-string interaction from light cone string field theory in pp-wave. We find exact agreements with the free planar three point functions of non-chiral BMN operators of ${\cal N} = 4$ super Yang Mills. The three string interaction vertex involving the Neumann matrices was derived in a recent paper hep-th/0204146. We explicitly calculate the bosonic Neumann matrices in the limit of large $\mu p^{+} \alpha^{'}$ . Using the Neumann matrices we are able to compute the cubic interactions of three string modes in a pp-wave background. 
  We study the Euclidean effective action and the full fermion propagator for a Dirac field in the presence of a scalar field with a domain wall defect, in 2+1 dimensions. We include quantum effects due to both fermion and scalar field fluctuations, in a one-loop approximation. The results are interpreted in terms of the quantum stability of the zero mode solution. We also study, for this system, the induced `inertial' electric field for the fermions on the defect, due to the quantum fluctuations of the scalar field. 
  Under certain conditions the number of photons radiated classically by a charged particle following a prescribed trajectory can be finite. An interesting formula for this number is presented and discussed. 
  We consider a family of pp-wave solutions of IIB supergravity. This family has a non-trivial, constant 5-form flux, and non-trivial, (light-cone) time-dependent RR and NS-NS 3-form fluxes. The solutions have either 16 or 20 supersymmetries depending upon the time dependence. One member of this family of solutions is the Penrose limit of the solution obtained by Pilch and Warner as the dual of a Leigh-Strassler fixed point. The family of solutions also provides indirect evidence in support of a recent conjecture concerning a large N duality group that acts on RG flows of N=2 supersymmetric, quiver gauge theories. 
  We discuss some recent attempts to reconcile cosmology with supergravity and M/string theory. First of all, we point out that in extended supergravities the scalar masses are quantized in terms of the cosmological constant in de Sitter vacua: the eigenvalues of the Casimir operator 3 m^2/\Lambda take integer values. For the current value of the cosmological constant extended supergravities predict ultra light scalars with the mass of the order of Hubble constant, 10^{-33} eV. This may have interesting consequences for cosmology. Turning our attention to cosmological implications of M/string theory, we present a possibility to use string theory D-brane constructions to reproduce the main features of hybrid inflation. We stress an important role played by Fayet-Iliopoulos terms responsible for the positive contribution to the potentials and stabilization of moduli. 
  Supercritical string theories in D>10 dimensions with no moduli are described, generalizing the asymmetric orientifold construction of one of the authors. By taking the number of dimensions to be large and turning on fluxes, dilaton potentials are generated with nontrivial minima at arbitrarily small cosmological constant and D-dimensional string coupling, separated by a barrier from a flat-space linear dilaton region, but possibly suffering from strong coupling problems. The general issue of the decay of a de Sitter vacuum to flat space is discussed. For relatively small barriers, such decays are described by gravitational instantons. It is shown that for a sufficiently large potential barrier, the bubble wall crosses the horizon. At the same time the instanton decay time exceeds the Poincare recurrence time. It is argued that the inclusion of such instantons is neither physically meaningful nor consistent with basic principles such as causality. This raises the possibility that such de Sitter vacua are effectively stable. In the case of the supercritical flux models, decays to the linear dilaton region can be forbidden by such large barriers, but decays to lower flux vacua including AdS minima nevertheless proceed consistently with this criterion. These models provide concrete examples in which cosmological constant reduction by flux relaxation can be explored. 
  We demonstrate an inflationary solution to the cosmological horizon problem during the Hagedorn regime in the early universe. Here the observable universe is confined to three spatial dimensions (a three-brane) embedded in higher dimensions. The only ingredients required are open strings on D-branes at temperatures close to the string scale. No potential is required. Winding modes of the strings provide a negative pressure that can drive inflation of our observable universe. Hence the mere existence of open strings on branes in the early hot phase of the universe drives Hagedorn inflation, which can be either power law or exponential. We note the amusing fact that, in the case of stationary extra dimensions, inflationary expansion takes place only for branes of three or less dimensions. 
  We define the notion of energy, and compute its values, for gravitational systems involving terms quadratic in curvature. While our construction parallels that of ordinary Einstein gravity, there are significant differences both conceptually and concretely. In particular, for D=4, all purely quadratic models admit vacua of arbitrary constant curvature. Their energies, including that of conformal (Weyl) gravity, necessarily vanish in asymptotically flat spaces. Instead, they are proportional to that of the Abbott-Deser (AD) energy expression in constant curvature backgrounds and therefore also proportional to the mass parameter in the corresponding Schwarzschild-(Anti) de Sitter geometries. Combined Einstein-quadratic curvature systems reflect the above results: Absent a cosmological constant term, the only vacuum is flat space, with the usual (ADM) energy and no explicit contributions from the quadratic parts. If there is a Lambda term, then the vacuum is also unique with that Lambda value, and the energy is just the sum of the separate contributions from Einstein and quadratic parts to the AD expression . Finally, we discuss the effects on energy definition of both higher curvature terms and higher dimension. 
  A four-dimensional black hole solution of the Einstein equations with a positive cosmological constant, coupled to a conformal scalar field, is given. There is a curvature singularity at the origin, and scalar field diverges inside the event horizon. The electrically charged solution, which has a fixed charge-to-mass ratio is also found. The quartic self-interacting coupling becomes bounded in terms of Newton's and the cosmological constants. 
  We present regular solutions for a brane world scenario in the form of a 't Hooft-Polyakov monopole living in the three-dimensional spherical symmetric transverse space of a seven-dimensional spacetime. In contrast to the cases of a domain-wall in five dimensions and a string in six dimensions, there exist gravity-localizing solutions for both signs of the bulk cosmological constant. A detailed discussion of the parameter space that leads to localization of gravity is given. A point-like monopole limit is discussed. 
  We resolve the mixing of the scalar operators of naive dimension 4 belonging to the representation 20' of the SU(4) R-symmetry in N=4 SYM. We compute the order g^2 corrections to their anomalous dimensions and show the absence of instantonic contributions thereof. Ratios of the resulting expressions are irrational numbers, even in the large N limit where, however, we observe the expected decoupling of double-trace operators from single-trace ones. We briefly comment on the generalizations of our results required in order to make contact with the double scaling limit of the theory conjectured to be holographically dual to type IIB superstring on a pp-wave. 
  We study affine osp(1|2) fusion, the fusion in osp(1|2) conformal field theory, for example. Higher-point and higher-genus fusion is discussed. The fusion multiplicities are characterized as discretized volumes of certain convex polytopes, and are written explicitly as multiple sums measuring those volumes. We extend recent methods developed to treat affine su(2) fusion. They are based on the concept of generalized Berenstein-Zelevinsky triangles and virtual couplings. Higher-point tensor products of finite-dimensional irreducible osp(1|2) representations are also considered. The associated multiplicities are computed and written as multiple sums. 
  We explicitly construct Green functions for a field in an arbitrary representation of gauge group propagating in noncommutative instanton backgrounds based on the ADHM construction. The propagators for spinor and vector fields can be constructed in terms of those for the scalar field in noncommutative instanton background. We show that the propagators in the adjoint representation are deformed by noncommutativity while those in the fundamental representation have exactly the same form as the commutative case. 
  The single 3-brane brane world at six dimension is examined when the extra dimensions are not compact. Although the warp factor diverges at the asymptotic region of the extra dimension, the normalizable zero mode and higher KK spectrum exist in the gravitational fluctuation. We compute the zero mode analytically and KK spectrum numerically. It is explicitly proven that our solution does not obey `brane world sum rule'. 
  Seiberg-Witten maps and a recently proposed construction of noncommutative Yang-Mills theories (with matter fields) for arbitrary gauge groups are reformulated so that their existence to all orders is manifest. The ambiguities of the construction which originate from the freedom in the Seiberg-Witten map are discussed with regard to the question whether they can lead to inequivalent models, i.e., models not related by field redefinitions. 
  We present a numerical study of critical phenomena (including the Lue-Weinberg phenomenon) arising for gravitating monopoles in a global monopole spacetime. The equations of this model have been recently studied by Spinelly et al. in a domain of parameter space away from the critical points. 
  Recently it has been proposed that the coefficient of the three-point function of the BMN operators in N=4 supersymmetric Yang-Mills theory is related to the three-string interactions in the pp-wave background. We calculate three-point functions of these operators to the first order in the effective Yang-Mills coupling lambda' = g_{YM}^2 N/J^2 in planar perturbation theory. On the string theory side, we derive the explicit expressions of the Neumann matrices to all orders in 1/(\mu p^+ \alpha')^2. This allows us to compute the corresponding three-string scattering amplitudes. This provides an all orders prediction for the field theory three-point functions. We compare our field theory results with the string theory results to the subleading order in 1/(\mu p^+ \alpha')^2 and find perfect agreement. 
  This is the full text of a survey talk for nonspecialists, delivered at the 66th Annual Meeting of the German Physical Society in Leipzig, March 2002. We have not taken pains to suppress the colloquial style. References are given only insofar as they help to underline the points made; this is not a full-blooded survey. The connection between noncommutative field theory and string theory is mentioned, but deemphasized. Contributions to noncommutative geometry made in Germany are emphasized. 
  The progress of noncommutative geometry has been crucially influenced, from the beginning, by quantum physics: we review this development in recent years. The Standard Model, with its central role for the Dirac operator, led to several formulations culminating in the concept of a real spectral triple. String theory then came into contact with NCG, leading to an emphasis on Moyal-like algebras and formulations of quantum field theory on noncommutative spaces. Hopf algebras have yielded an unexpected link between the noncommutative geometry of foliations and perturbative quantum field theory. The quest for a suitable foundation of quantum gravity continues to promote fruitful ideas, among them the spectral action principle and the search for a better understanding of "noncommutative spaces". 
  The interrelations between the two definitions of momentum operator, via the canonical energy-momentum tensorial operator and as translation operator (on the operator space), are studied in quantum field theory. These definitions give rise to similar but, generally, different momentum operators, each of them having its own place in the theory. Some speculations on the relations between quantum field theory and quantum mechanics are presented. 
  The noncommutative dipole QED is studied in detail for the matter fields in the adjoint representation. The axial anomaly of this theory is calculated in two and four dimensions using various regularization methods. The Ward-Takahashi identity is proved by making use of a non-perturbative path integral method. The one-loop $\beta$-function of the theory is calculated explicitly. It turns out that the value of the $\beta$-function depends on the direction of the dipole length $\vec{L}$, which defines the noncommutativity. Finally using a semi-classical approximation a non-perturbative definition of the form factors is presented and the anomalous magnetic moment of this theory at one-loop order is computed. 
  We study the modular invariance of strings on pp-waves with RR-flux. We explicitly show that the one-loop partition functions of the maximally supersymmetric pp-waves and their orbifolds can be modular invariant in spite of the mass terms in the light-cone gauge. From this viewpoint, we also determine the spectrum of type 0B theory on pp-wave and discuss its gauge theory dual. Furthermore, we investigate the spectrum of a non-supersymmetric orbifold and point out its supersymmetry enhancement in the Penrose limit. 
  We examine the unitarity issue in the recently proposed time-ordered perturbation theory on noncommutative (NC) spacetime. We show that unitarity is preserved as long as the interaction Lagrangian is explicitly Hermitian. We explain why it makes sense to distinguish the Hermiticity of the Lagrangian from that of the action in perturbative NC field theory and how this requirement fits in the framework. 
  We study D-branes on abelian orbifolds C^d/Z_N for d=2, 3. The toric data describing the D-brane vacuum moduli space, which represents the geometry probed by D-branes, has certain redundancy compared with the classical geometric description of the orbifolds. We show that the redundancy has a simple combinatorial structure and find analytic expressions for degrees of the redundancy. For d=2 the structure of the redundancy has a connection with representations of SU(N) Lie algebra, which provides a new correspondence between geometry and representation theory. We also prove that non-geometric phases do not appear in the Kahler moduli space for d=2. 
  The mapping of topologically nontrivial gauge transformations in noncommutative gauge theory to corresponding commutative ones is investigated via the operator form of the Seiberg-Witten map. The role of the gauge transformation part of the map is analyzed. Chern-Simons actions are examined and the correspondence to their commutative counterparts is clarified. 
  The Lagrange description of an ideal fluid gives rise in a natural way to a gauge potential and a Poisson structure that are classical precursors of analogous noncommuting entities. With this observation we are led to construct gauge-covariant coordinate transformations on a noncommuting space. Also we recognize the Seiberg-Witten map from noncommuting to commuting variables as the quantum correspondent of the Lagrange to Euler map in fluid mechanics. 
  The bound state spectrum of the massive Thirring model is studied in the framework of the canonical quantization in the rest frame. First, we quantize the field with the massless free fermion basis states. Then, we make a Bogoliubov transformation. This leads to the natural mass renormalization. The bound state spectrum is analytically solved by the $q\bar{q}$ Fock space. It is found that the spectrum has the right behaviors both for the weak and for the strong coupling limits after the appropriate wave function regularization. This regularization is quite clear and the treatment is self-consistent for the bound state problem compared to other regularizations. Further, we show that the interaction between $q\bar{q}$ bosons is always repulsive and therefore there is no bound state in the four fermion ($qq \bar q \bar q$) Fock space. This confirms that there is only one bound state in the massive Thirring model. 
  In this paper we develop a formalism to analyze the spectrum of small perturbations about arbitrary solutions of Einstein, Yang-Mills and scalar systems. We consider a general system of gravitational, gauge and scalar fields in a $D-$dimensional space-time and give the bilinear action for the fluctuations of the fields in the system around an arbitrary solution of the classical field equations. We then consider warped geometries, popular in brane world scenarios, and use the light cone gauge to separate the bilinear action into a totally decoupled spin-two, -one and -zero fluctuations. We apply our general scheme to several examples and discuss in particular localization of abelian and non-abelian gauge fields of the standard model to branes generated by scalar fields. We show in particular that the Nielsen-Olsen string solution gives rise to a normalizable localized spin-1 field in any number of dimensions. 
  We provide a proof of the equivalence of N=1 dynamics obtained by deforming N=2 supersymmetric gauge theories by addition of certain superpotential terms, with that of type IIB superstring on Calabi-Yau threefold geometries with fluxes. In particular we show that minimization of the superpotential involving gaugino fields is equivalent to finding loci where Seiberg-Witten curve has certain factorization property. Moreover, by considering the limit of turning off of the superpotential we obtain the full low energy dynamics of N=2 gauge systems from Calabi-Yau geometries with fluxes. 
  I recall the main motivation to study quantum field theories on noncommutative spaces and comment on the most-studied example, the noncommutative R^4. That algebra is given by the *-product which can be written in (at least) two ways: in an integral form or an exponential form. These two forms of the *-product are adapted to different classes of functions, which, when using them to formulate field theory, lead to two versions of quantum field theories on noncommutative R^4. The integral form requires functions of rapid decay and a (preferably smooth) cut-off in the path integral, which therefore should be evaluated by exact renormalisation group methods. The exponential form is adapted to analytic functions with arbitrary behaviour at infinity, so that Feynman graphs can be used to compute the path integral (without cut-off) perturbatively. 
  The Bethe-Salpeter equation in non-commutative QED (NCQED) is considered for three-body bound state. We study the non-relativistic limit of this equation in the instantaneous approximation and derive the corresponding Schr\"{o}dinger equation in non-commutative space. It is shown that the experimental data for Helium atom puts an upper bound on the magnitude of the parameter of non-commutativity, $\theta\sim10^{-9}\lambda_e^2$. 
  We study a class of non-protected local composite operators which occur in the R symmetry singlet channel of the OPE of two stress-tensor multiplets in {\cal N}=4 SYM. At tree level these are quadrilinear scalar dimension four operators, two single-traces and two double-traces. In the presence of interaction, due to a non-trivial mixing under renormalization, they split into linear combinations of conformally covariant operators. We resolve the mixing by computing the one-loop two-point functions of all the operators in an {\cal N}=1 setup, then diagonalizing the anomalous dimension matrix and identifying the quasiprimary operators. We find one operator whose anomalous dimension is negative and suppressed by a factor of 1/N^2 with respect to the anomalous dimensions of the Konishi-like operators. We reveal the mechanism responsible for this suppression and argue that it works at every order in perturbation theory. In the context of the AdS/CFT correspondence such an operator should be dual to a multiparticle supergravity state whose energy is less than the sum of the corresponding individual single-particle states. 
  A gauged $SU_q(2)$ theory is characterized by two dual algebras, the first lying close to the Lie algebra of SU(2) while the second introduces new degrees of freedom that may be associated with non-locality or solitonic structure. The first and second algebras, here called the external and internal algebras respectively, define two sets of fields, also called external and internal. The gauged external fields agree with the Weinberg-Salam model at the level of the doublet representation but differ at the level of the adjoint representation. For example the g-factor of the charged W-boson differs in the two models. The gauged internal fields remain speculative but are analogous to color fields. 
  The nonequilibrium dynamics of coupled quantum oscillators subject to different time dependent quenches are analyzed in the context of the Liouville-von Neumann approach. We consider models of quantum oscillators in interaction that are exactly soluble in the cases of both sudden and smooth quenches. The time evolution of number densities and the final equilibration distribution for the problem of a quantum oscillator coupled to an infinity set of other oscillators (a bath) are explicitly worked out. 
  The canonical and symmetrical energy-momentum tensors and their non-zero traces in Maxwell's theory on non-commutative spaces have been found. Dirac's quantization of the theory under consideration has been performed. I have found the extended Hamiltonian and equations of motion in the general gauge covariant form. 
  We present a supersymmetric version of the two-brane Randall-Sundrum scenario, with arbitrary brane tensions T_1 and T_2, subject to the bound |T_{1,2}| \leq \sqrt{-6\Lambda_5}, where \Lambda_5 < 0 is the bulk cosmological constant. Dimensional reduction gives N=1, D=4 supergravity, with cosmological constant \Lambda_4 in the range \half\Lambda_5 \leq \Lambda_4 \leq 0. The case with \Lambda_4 = 0 requires T_1 = -T_2 = \sqrt{-6\Lambda_5}. This work unifies and generalizes previous approaches to the supersymmetric Randall-Sundrum scenario. It also shows that the Randall-Sundrum fine-tuning is not a consequence of supersymmetry. 
  We consider the problem of condensation of open string tachyon fields which have an O(D) symmetric profile. This problem is described by a boundary conformal field theory with D scalar fields on a disc perturbed by relevant boundary operators with O(D) symmetry. The model is exactly solvable in the large D limit and we analyze its 1/D expansion. We find that this expansion is only consistent for tachyon fields which are polynomials. In that case, we show that the theory is renormalized by normal ordering the interaction. The beta-function for the tachyon field is the linear wave operator. We derive an expression for the tachyon potential and compare with other known expressions. In particular, our technique gives the exact potential for the quadratic tachyon profile. It can be used to correct the action which has been derived in that case iteratively in derivatives of the tachyon field. 
  The detailed description of the method of the construction of the nilpotent BRST charges for nonlinear algebras of constraints appearing in the description of the massless higher spin fields on the $AdS_D$ background is presented. It is shown that the corresponding BRST charge is not uniquely defined, but this ambiguity has no impact on the physical content of the theory. 
  The method of construction of Fock space realizations of Lie algebras is generalized for nonlinear algebras. We consider as an example the nonlinear algebra of constraints which describe the totally symmetric fields with higher spins in the AdS space-time. 
  The superdiffeomorphisms invariant description of $N$ - extended spinning particle is constructed in the framework of nonlinear realizations approach. The action is universal for all values of $N$ and describes the time evolution of $D+2$ different group elements of the superdiffeomorphisms group of the $(1,N)$ superspace. The form of this action coincides with the one-dimensional version of the gravity action, analogous to Trautman's one. 
  We consider M-theory on AdS_4 x V_{5,2} where V_{5,2}= SO(5)/SO(3) is a Stiefel manifold. We construct a Penrose limit of AdS_4 x V_{5,2} that provides the pp-wave geometry. There exists a subsector of three dimensional N=2 dual gauge theory, by taking both the conformal dimension and R charge large with the finiteness of their difference, which has enhanced N=8 maximal supersymmetry. We identify operators in the N=2 gauge theory with supergravity KK excitations in the pp-wave geometry and describe how the gauge theory operators made out of chiral field of conformal dimension 1/3 fall into N=8 supermultiplets. 
  We discuss D-dimensional scalar field interacting with a scale invariant random metric which is either a Gaussian field or a square of a Gaussian field. The metric depends on d-dimensional coordinates (where d is less than D). By a projection to a lower dimensional subspace we obtain a scale invariant non-Gaussian model of Euclidean quantum field theory in D-d or d dimensions. 
  We investigate the presence of domain walls in models described by three real scalar fields. We search for stable defect structures which minimize the energy of the static field configurations. We work out explict orbits in field space and find several analytical solutions in all BPS sectors, some of them presenting internal structure such as the appearence of defects inside defects. We point out explicit applications in high energy physics, and in other branches of nonlinear science. 
  We present a class of static membrane solutions, with non-flat worldvolume geometry, in the eleven dimensional supergravity with source terms. This class of solutions contains supersymmetric as well as a large class of non-supersymmetric configurations. We comment about near horizon limit and stability of these solutions and point out an interesting relation with certain two dimensional dilaton gravity system. 
  The Penrose-Gueven limit simplifies a given supergravity solution into a pp-wave background. Aiming at clarifying its relation to renormalization group flow we study the Penrose-Guven limit of supergravity backgrounds that are dual to non-conformal gauge theories. The resulting backgrounds fall in a class simple enough that the quantum particle is exactly solvable. We propose a map between the effective time-dependent quantum mechanical problem and the RG flow in the gauge theory. As a testing ground we consider explicitly two Penrose limits of the infrared fixed point of the Pilch-Warner solution. We analyze the corresponding gauge theory picture and write down the operators which are the duals of the low lying string states. We also address RG flows of a different nature by considering the Penrose-Gueven limit of a stack of N D_p branes. We note that in the far IR (for p<3)the limit generically has negative mass-squared. This phenomenon signals, in the world sheet picture, the necessity to transform to another description. In this regard, we consider explicitly the cases of M2 from D2 and F1 from D1 . 
  We show that the boson field representation of the massless fermion fields, suggested by Morchio, Pierotti and Strocchi in J. Math. Phys. 33, 777 (1992) for the operator solution of the massless Thirring model, agrees completely with the existence of the chirally broken phase in the massless Thirring model revealed in EPJC 20, 723 (2001) and hep-th/0112183, when the free massless boson fields are described by the quantum field theory, free of infrared divergences in 1+1-dimensional space-time, formulated in hep-th/0112184 and hep-th/0204237. 
  The most popular noncommutative field theories are characterized by a matrix parameter theta^(mu,nu) that violates Lorentz invariance. We consider the simplest algebra in which the theta-parameter is promoted to an operator and Lorentz invariance is preserved. This algebra arises through the contraction of a larger one for which explicit representations are already known. We formulate a star product and construct the gauge-invariant Lagrangian for Lorentz-conserving noncommutative QED. Three-photon vertices are absent in the theory, while a four-photon coupling exists and leads to a distinctive phenomenology. 
  We formulate a theory of quantum processes, extend it to a generic quantum cosmology, formulate a reversible quantum logic for the Quantum Universe As Computer, or Qunivac. Qunivac has an orthogonal group of cosmic dimensionality. It has a Clifford algebra of ``cosmonions,'' extending the quaternions to a cosmological number of anticommuting units. Its qubits obey Clifford-Wilczek statistics and are associated with unit cosmonions. This makes it relatively easy to program the Dirac equation on Qunivac in a Lorentz-invariant way. Qunivac accommodates a field theory and a gauge theory. Its gauge group is necessarily a quantum group. 
  We investigate the group contraction method for various space-time groups, including SO(3)->E_2, SO(3,1)->G_3, SO(5-h,h)->P(3,1) (h=1 or 2), and its consequences for representations of these groups. Following strictly quantum mechanical procedures we specifically pay attention in the asymptotic limiting procedure employed in the contraction G -> G', not only to the respective algebras but to their representations spaces spanned by the eigenvectors of the Cartan subalgebra and the eigenvalues labelling these representation spaces. Where appropriate a physical interpretation is given to the contraction prodecure. 
  We investigate orientifolds of type II string theory on K3 and Calabi-Yau 3-folds with intersecting D-branes wrapping special Lagrangian cycles. We determine quite generically the chiral massless spectrum in terms of topological invariants and discuss both orbifold examples and algebraic realizations in detail. Intriguingly, the developed techniques provide an elegant way to figure out the chiral sector of orientifold models without computing any explicit string partition function. As a new example we derive a non-supersymmetric Standard-like Model from an orientifold of type IIA on the quintic Calabi-Yau 3-fold with wrapped D6-branes. In the case of supersymmetric intersecting brane models on Calabi-Yau manifolds we discuss the D-term and F-term potentials, the effective gauge couplings and the Green-Schwarz mechanism. The mirror symmetric formulation of this construction is provided within type IIB theory. We finally include a short discussion about the lift of these models from type IIB on K3 to F-theory and from type IIA on Calabi-Yau 3-folds to M-theory on G_2 manifolds. 
  A simple iterative procedure is suggested for the deformation quantization of (irregular) Poisson brackets associated to the classical Yang-Baxter equation. The construction is shown to admit a pure algebraic reformulation giving the Universal Deformation Formula (UDF) for any triangular Lie bialgebra. A simple proof of classification theorem for inequivalent UDF's is given. As an example the explicit quantization formula is presented for the quasi-homogeneous Poisson brackets on two-plane. 
  The Euclidean $(\phi^{4})_{3,\epsilon$ model in $R^3$ corresponds to a perturbation by a $\phi^4$ interaction of a Gaussian measure on scalar fields with a covariance depending on a real parameter $\epsilon$ in the range $0\le \epsilon \le 1$. For $\epsilon =1$ one recovers the covariance of a massless scalar field in $R^3$. For $\epsilon =0$ $\phi^{4}$ is a marginal interaction. For $0\le \epsilon < 1$ the covariance continues to be Osterwalder-Schrader and pointwise positive. After introducing cutoffs we prove that for $\epsilon > 0$, sufficiently small, there exists a non-gaussian fixed point (with one unstable direction) of the Renormalization Group iterations. These iterations converge to the fixed point on its stable (critical) manifold which is constructed. 
  We analyze in the spirit of hep-th/0110039 the possible existence of supersymmetric $D p - \bar D p$ brane systems in flat ten dimensional Minkowski space. For $p=3,4$ we show that besides the solutions related by T-duality to the $D2 - \bar D 2$ systems found by Bak and Karch there exist other ansatz whose compatibility is shown from general arguments and that preserve also eight supercharges, in particular a $D4 - \bar D4$ system with D2-branes dissolved on it and Taub-NUT charge. We carry out the explicit construction in Weyl basis of the corresponding Killing spinors and conjecture the existence of new solutions for higher dimensional branes with some compact directions analogous to the supertube recently found. 
  We discuss the fully non-linear formulation of multigravity. The concept of universality classes of effective Lagrangians describing bigravity, which is the simplest form of multigravity, is introduced. We show that non-linear multigravity theories can naturally arise in several different physical contexts: brane configurations, certain Kaluza-Klein reductions and some non-commutative geometry models. The formal and phenomenological aspects of multigravity (including the problems linked to the linearized theory of massive gravitons) are briefly discussed. 
  We give a simple - straightforward and rigorous - derivation that when the eigenvalues of one of the $d=9 (5,3,2)$ matrices in the SU(N) invariant supersymmetric matrix model become large (and well separated from each other) the ground-state wavefunction (resp. asymptotic zero-energy solution of the corresponding differential equation) factorizes, for all $N>1$, into a product of supersymmetric harmonic oscillator wavefunctions (involving the `off-diagonal' degrees of freedom) and a wavefunction $\psi$ that is annihilated by the free supercharge formed out of all `diagonal' (Cartan sub-algebra) degrees of freedom. 
  We explore the cosmological solutions of classes of non-linear bigravity theories. These theories are defined by effective four-dimensional Lagrangians describing the coupled dynamics of two metric tensors, and containing, in the linearized limit, both a massless graviton and an ultralight one. We focus on two paradigmatic cases: the case where the coupling between the two metrics is given by a Pauli-Fierz-type mass potential, and the case where this coupling derives from five-dimensional brane constructions. We find that cosmological evolutions in bigravity theories can be described in terms of the dynamics of two ``relativistic particles'', moving in a curved Lorenzian space, and connected by some type of nonlinear ``spring''. Classes of bigravity cosmological evolutions exhibit a ``locking'' mechanism under which the two metrics ultimately stabilize in a bi-de-Sitter configuration, with relative (constant) expansion rates. In the absence of matter, we find that a generic feature of bigravity cosmologies is to exhibit a period of cosmic acceleration. This leads us to propose bigravity as a source of a new type of dark energy (``tensor quintessence''), exhibiting specific anisotropic features. Bigravity could also have been the source of primordial inflation. 
  A certain conformally invariant N=1 supersymmetric SU(n) gauge theory has a description as an infra-red fixed point obtained by deforming the N=4 supersymmetric Yang-Mills theory by giving a mass to one of its N=1 chiral multiplets. We study the Penrose limit of the supergravity dual of the large n limit of this N=1 gauge theory. The limit gives a pp-wave with R-R five-form flux and both R-R and NS-NS three-form flux. We discover that this new solution preserves twenty supercharges and that, in the light-cone gauge, string theory on this background is exactly solvable. Correspondingly, this latter is the stringy dual of a particular large charge limit of the large n gauge theory. We are able to identify which operators in the field theory survive the limit to form the string's ground state and some of the spacetime excitations. The full string model, which we exhibit, contains a family of non-trivial predictions for the properties of the gauge theory operators which survive the limit. 
  A black hole attached to a brane in a higher dimensional space emitting quanta into the bulk may leave the brane as a result of a recoil. We study this effect. We consider black holes which have a size much smaller than the characteristic size of extra dimensions. Such a black hole can be effectively described as a massive particle with internal degrees of freedom. We consider an interaction of such particles with a scalar massless field and prove that for a special choice of the coupling constant describing the transition of the particle to a state with smaller mass the probability of massless quanta emission takes the form identical to the probability of the black hole emission. Using this model we calculate the probability for a black hole to leave the brane and study its properties. The discussed recoil effect implies that black holes which might be created by interaction of high energy particles in colliders the thermal emission of the formed black hole could be terminated and the energy non-conservation can be observed in the brane experiments. 
  The bare one loop soliton quantum mass corrections can be expressed in two ways: as a sum over the zero-point energies of small oscillations around the classical configuration, or equivalently as the (Euclidean) effective action per unit time. In order to regularize the bare one loop quantum corrections (expressed as the sum over the zero-point energies) we subtract and add from it the tadpole graph that appear in the expansion of the effective action per unit time. The subtraction renders the one loop quantum corrections finite. Next, we use the renormalization prescription that the added tadpole graph cancels with adequate counterterms, obtaining in this way a finite unambiguous expression for the one loop soliton quantum mass corrections. When we apply the method to the solitons of the sine-Gordon and phi^4 kink models we obtain results that agree with known results. Finally we apply the method to compute the soliton quantum mass corrections in the recently introduced phi^2 cos^2 ln(phi^2) model. 
  We review the Batalin-Vilkovisky quantization procedure for Yang--Mills theory on a 2-point space. 
  We prove a uniqueness theorem for asymptotically flat static charged dilaton black hole solutions in higher dimensional space-times. We also construct infinitely many non-asymptotically flat regular static black holes on the same space-time manifold with the same spherical topology. An application to the uniqueness of a certain class of flat $p$-branes is also given. 
  In an attempt to bridge the gap between M-theory and braneworld phenomenology, we present various gravitational Lorentz-violating braneworlds which arise from p-brane systems. Lorentz invariance is still preserved locally on the braneworld. For certain p-brane intersections, the massless graviton is quasi-localized. This also results from an M5-brane in a C-field. In the case of a p-brane perturbed from extremality, the quasi-localized graviton is massive. For a braneworld arising from global AdS_5, gravitons travel faster when further in the bulk, thereby apparently traversing distances faster than light. 
  In this note we explore the stringy interpretation of non-perturbative effects in N=1^* deformations of the A_{k-1} quiver models. For certain types of deformations we argue that the massive vacua are described by Nk fractional D3-branes at the orbifold polarizing into k concentric 5-brane spheres each carrying fractional brane charge. The polarization of the D3-branes induces a polarization of D-instantons into string world-sheets wrapped on the Myers spheres. We show that the superpotentials in these models are indeed generated by these world-sheet instantons. We point out that for certain parameter values the condensates yield the exact superpotential for a relevant deformation of the Klebanov-Witten conifold theory. 
  We show how the Szekeres form of the line element is naturally adapted to study Penrose limits in classical string backgrounds. Relating the "old" colliding wave problem to the Penrose limiting procedure as employed in string theory we discuss how two orthogonal Penrose limits uniquely determine the underlying target space when certain symmetry is imposed. We construct a conformally deformed background with two distinct, yet exactly solvable in terms of the string theory on R-R backgrounds, Penrose limits. Exploiting further the similarities between the two problems we find that the Penrose limit of the gauged WZW Nappi-Witten universe is itself a gauged WZW plane wave solution of Sfetsos and Tseytlin. Finally, we discuss some issues related to singularity, show the existence of a large class of non-Hausdorff solutions with Killing Cauchy Horizons and indicate a possible resolution of the problem of the definition of quantum vacuum in string theory on these time-dependent backgrounds. 
  We derive the equations of time-independent stochastic quantization, without reference to an unphysical 5th time, from the principle of gauge equivalence. It asserts that probability distributions $P$ that give the same expectation values for gauge-invariant observables $<W > = \int dA W P$ are physically indistiguishable. This method escapes the Gribov critique. We derive an exact system of equations that closely resembles the Dyson-Schwinger equations of Faddeev-Popov theory, which we then solve non-perturbatively for the critical exponents that characterize the asymptotic form at $k \approx 0$ of the tranverse and longitudinal parts of the gluon propagator in Landau gauge, $D^T \sim (k^2)^{-1-\a_T}$ and $D^L \sim a (k^2)^{-1-\a_L}$, and obtain $\a_T = - 2\a_L \approx - 1.043$ (short range), and $\a_L \approx 0.521$, (long range). Although the longitudinal part vanishes with the gauge parameter $a$ in the Landau gauge limit, $a \to 0$, there are vertices of order $a^{-1}$, so the longitudinal part of the gluon propagator contributes in internal lines, replacing the ghost that occurs in Faddeev-Popov theory. We compare our results with the corresponding results in Faddeev-Popov theory. 
  Many abelian gauge theories in three dimensions flow to interacting conformal field theories in the infrared. We define a new class of local operators in these conformal field theories which are not polynomial in the fundamental fields and create topological disorder. They can be regarded as higher-dimensional analogues of twist and winding-state operators in free 2d CFTs. We call them monopole operators for reasons explained in the text. The importance of monopole operators is that in the Higgs phase, they create Abrikosov-Nielsen-Olesen vortices. We study properties of these operators in three-dimensional QED using large N_f expansion. In particular, we show that monopole operators belong to representations of the conformal group whose primaries have dimension of order N_f. We also show that monopole operators transform non-trivially under the flavor symmetry group, with the precise representation depending on the value of the Chern-Simons coupling. 
  We study the general form of the possible kinetic terms for 2-form fields in four dimensions, under the restriction that they have a semibounded energy density. This is done by using covariant symplectic techniques and generalizes previous partial results in this direction. 
  Small fluctuations around a constant electric or constant magnetic field F are analyzed in a theory with pseudo scalar [phi] with a coupling g[phi]F~F. It is found that a magnetic external field leads to mass generation for the small perturbations, while an electric field suffers from a tachyonic mass generation in the case in which the field strength is higher than a critical value (related to the pseudo scalar mass). The vacuum energy can be exactly evaluated and it is found that an imaginary part is present when the external electric field exceeds its critical value. 
  We construct all possible orthogonally intersecting S-brane solutions in 11-dimensions corresponding to standard supersymmetric M-brane intersections. It is found that the solutions can be obtained by multiplying the brane and the transverse directions with appropriate powers of two hyperbolic functions of time. This is the S-brane analog of the ``harmonic function rule''. The transverse directions can be hyperbolic, flat or spherical. We also discuss some properties of these solutions. 
  We study the IR/UV connection in non-commutative $\phi^{3}$ theory as well as in non-commutative QED from the point of view of the dispersion relation for the self-energy. We show that, although the imaginary part of the self-energy is well behaved as the parameter of non-commutativity vanishes, the real part becomes divergent as a consequence of the high energy behavior of the dispersion integral. Some other interesting features that arise from this analysis are also briefly discussed. 
  We propose a simple string bit formalism for interacting strings in a plane wave background, in terms of supersymmetric quantum mechanics with a symmetric product target-space. We construct the light-cone supersymmetry generators and Hamiltonian at finite string coupling. We find a precise match between string amplitudes and the non-planar corrections to the correlation functions of BMN operators computed from gauge theory, and conjecture that this correspondence extends to all orders in perturbation theory. We also give a simple RG explanation for why the effective string coupling is g_2 = J^2/N instead of g_s = g_{ym}^2. 
  We study scalar solitons on the fuzzy sphere at arbitrary radius and noncommutativity. We prove that no solitons exist if the radius is below a certain value. Solitons do exist for radii above a critical value which depends on the noncommutativity parameter. We construct a family of soliton solutions which are stable and which converge to solitons on the Moyal plane in an appropriate limit. These solutions are rotationally symmetric about an axis and have no allowed deformations. Solitons that describe multiple lumps on the fuzzy sphere can also be constructed but they are not stable. 
  We examine and implement the concept of non-additive composition laws in the quantum theory of closed bosonic strings moving in (3+1)-dimensional Minkowski space. Such laws supply exact selection rules for the merging and splitting of closed strings. 
  We elaborate on the symmetry breaking pattern involved in the Penrose limit of $AdS_{d+1} \times S^{d+1}$ spacetimes and the corresponding limit of the CFT dual. For d=2 we examine in detail how the symmetries contract to products of rotation and Heisenberg algebras, both from the bulk and CFT points of view. Using a free field realization of these algebras acting on products of elementary fields of the CFT with SO(2) R charge +1, we show that this process of contraction restricts all the fields to a few low angular momentum modes and ensures that the field with R charge -1 does not appear. This provides an understanding of several important aspects of the proposal of Berenstein, Maldacena and Nastase. We also indicate how the contraction can be performed on correlation functions. 
  We describe the modern formalism, ideas and applications of the instanton calculus for gauge theories with, and without, supersymmetry. Particular emphasis is put on developing a formalism that can deal with any number of instantons. This necessitates a thorough review of the ADHM construction of instantons with arbitrary charge and an in-depth analysis of the resulting moduli space of solutions. We review the construction of the ADHM moduli space as a hyper-Kahler quotient. We show how the functional integral in the semi-classical approximation reduces to an integral over the instanton moduli space in each instanton sector and how the resulting matrix partition function involves various geometrical quantities on the instanton moduli space: volume form, connection, curvature, isometries, etc. One important conclusion is that this partition function is the dimensional reduction of a higher-dimensional gauged linear sigma model which naturally leads us to describe the relation of the instanton calculus to D-branes in string theory. Along the way we describe powerful applications of the calculus of many instantons to supersymmetric gauge theories including (i) the gluino condensate puzzle in N=1 theories (ii) Seiberg-Witten theory in N=2 theories; and (iii) the AdS/CFT correspondence in N=2 and N=4 theories. Finally, we briefly review the modifications of the instanton calculus for a gauge theory defined on a non-commutative spacetime and we also describe a new method for calculating instanton processes using a form of localization on the instanton moduli space. 
  We study BPS states in type IIA string compactification on a local Calabi-Yau 3-fold which are related to the BPS states of the E-string. Using Picard-Lefshetz transformations of the 3-cycles on the mirror manifold we determine automorphisms of the K-theory of the compact divisor of the Calabi-Yau which maps certain D-brane configurations to a bound state of single D4-brane with multiple D0-branes. This map allows us to write down the generating functions for the multiplicity of these BPS states. 
  We use the supergravity modes to clarify the role of the prefactor in the light-cone superstring field theory on PP-wave background. We verify some of the proposals of the recent paper hep-th/0205089 and give further evidence for the correspondence between N=4 SYM gauge theory and string theory on PP-wave. We also consider energy-preserving processes and find that they give vanishing cubic interaction Hamiltonian matrix. 
  We reconsider the generalization of standard quantum mechanics in which the position operators do not commute. We argue that the standard formalism found in the literature leads to theories that do not share the symmetries present in the corresponding commutative system. We propose a general prescription to specify a Hamiltonian in the noncommutative theory that preserves the existing symmetries. We show that it is always possible to choose this Hamiltonian in such a way that the energy spectrum of the standard and non-commuting theories are identical, so that experimental differences between the predictions of both theories are to be found only at the level of the detailed structure of the energy eigenstates. 
  We study the replacement of SU(3) by SU_q(3) in standard gauge theories. At the level of a global theory there is a physically sensible SU_q(3) formalism with measurable differences from the SU(3) theory. In contrast to the SU_q(2) case, where it is possible to construct a q-electroweak theory, there is no local (Yang-Mills) formalism for SU_q(3). 
  We formulate gauge invariant interactions of totally symmetric tensor and tensor-spinor higher spin gauge fields in AdS(5) that properly account for higher-spin-gravitational interactions at the action level in the first nontrivial order. 
  In this paper the partition function of N=4 D=0 super Yang-Mills matrix theory with arbitrary simple gauge group is discussed. We explicitly computed its value for all classical groups of rank up to 11 and for the exceptional groups G_2, F_4 and E_6. In the case of classical groups of arbitrary rank we conjecture general formulas for the B_r, C_r and D_r series in addition to the known result for the A_r series. Also, the relevant boundary term contributing to the Witten index of the corresponding supersymmetric quantum mechanics has been explicitly computed as a simple function of rank for the orthogonal and symplectic groups SO(2N+1), Sp(2N), SO(2N). 
  We study the closed and open supermembranes on the maximally supersymmetric pp-wave background. In the framework of the membrane theory, the superalgebra is calculated by using the Dirac bracket and we obtain its central extension by surface terms. The result supports the existence of the extended objects in the membrane theory in the pp-wave limit. When the central terms are discarded, the associated algebra completely agrees with that of Berenstein-Maldacena-Nastase matrix model. We also discuss the open supermembranes on the pp-wave and elaborate the possible boundary conditions. 
  We study the N=1 Sp(2N+2M)xSp(2N) gauge theory on a stack of N physical and M fractional D3-branes in the background of an orientifolded conifold. The gravity dual is a type IIB orientifold of adS_5xT^11 (with certain background fluxes turned on) containing an O7-plane and 8 D7-branes. In the conformal case (M=0), we argue that the alpha'^2-corrections localized on the 8 D7-branes and the O7-plane should give vanishing contributions to the supergravity equations of motion for the bulk fields. In the cascading case (M not equal to 0), we argue that the alpha'^2-terms give rise to corrections which in the dual Sp(2N+2M)xSp(2N) gauge theory can be interpreted as corrections to the anomalous dimensions of the matter fields. 
  Two pictures of BPS bound states in Calabi-Yau compactifications of type II string theory exist, one as a set of particles at equilibrium separations from each other, the other as a fusion of D-branes at a single point of space. We show how quiver quantum mechanics smoothly interpolates between the two, and use this, together with recent mathematical results on the cohomology of quiver varieties, to solve some nontrivial ground state counting problems in multi-particle quantum mechanics, including one arising in the setup of the spherical quantum Hall effect, and to count ground state degeneracies of certain dyons in supersymmetric Yang-Mills theories. A crucial ingredient is a non-renormalization theorem in N=4 quantum mechanics for the first order part of the Lagrangian in an expansion in powers of velocity. 
  In type IIB light-cone superstring field theory, the cubic interaction has two pieces: a delta-functional overlap and an operator inserted at the interaction point. In this paper we extend our earlier work hep-th/0204146 by computing the matrix elements of this operator in the oscillator basis of pp-wave string theory for all mu p^+ alpha'. By evaluating these matrix elements for large mu p^+ alpha', we check a recent conjecture relating matrix elements of the light-cone string field theory Hamiltonian (with prefactor) to certain three-point functions of BMN operators in the gauge theory. We also make several explicit predictions for gauge theory. 
  The effective string action of the color-electric flux tube in the U(1) x U(1) dual Ginzburg-Landau (DGL) theory is studied by performing a path-integral analysis by taking into accountthe finite thickness of the flux tube. The DGL theory, corresponding to the low-energy effective theory of Abelian-projected SU(3) gluodynamics, can be expressed as s [U(1)]^{3} dual Abelian Higgs (DAH) model with a certain constraint in the Weyl symmetric formulation. This formulation allows us to adopt quite similar path-integral techniques as in the U(1) DAH model, and therefore, the resulting effective string action in the U(1) x U(1) DGL theory has also quite a similar structure except the number of color degrees of freedom. A modified Yukawa interaction appears as a boundary contribution, which is completely due to the finite thickness of the flux tube, and is reduced into the ordinary Yukawa interaction in the deep type-II (London) Limit. 
  We consider a matrix model depending on a parameter $\lambda$ which permits the fuzzy sphere as a classical background.By expanding the bosonic matrices around this background ones recovers a U(1) (U(n)) noncommutative gauge theory on the fuzzy sphere. To check classical stability of this background, we look for new classical solutions of this model and find them for $\lambda < 1$, that make the fuzzy sphere solution unstable for $\lambda < \half$ and stable otherwise. \\ 
  We use the structural similarity of certain Coxeter Artin Systems to the Yang--Baxter and Reflection Equations to convert representations of these systems into new solutions of the Reflection Equation. We construct certain Bethe ansatz states for these solutions, using a parameterisation suggested by abstract representation theory. 
  We extend the perturbative approach developed in an earlier work to deal with Lagrangians which have arbitrary higher order time derivative terms for both bosons and fermions. This approach enables us to find an effective Lagrangian with only first time derivatives order by order in the coupling constant. As in the pure bosonic case, to the first order, the quantized Hamiltonian is bounded from below whenever the potential is. We show in the example of a single complex fermion that higher derivative interactions result in an effective mass and change of vacuum for the low energy modes. The supersymmetric noncommutative Wess-Zumino model is considered as another example. We also comment on the higher derivative terms in Witten's string field theory and the effectiveness of level truncation. 
  We consider in detail an approach (proposed by the author earlier) where quantum states are described by elements of a linear space over a Galois field, and operators of physical quantities - by linear operators in this space. The notion of Galois fields (which is extremely simple and elegant) is discussed in detail and we also discuss the conditions when our description gives the same predictions as the conventional one. In quantum theory based on a Galois field, all operators are well defined and divergencies cannot not exist in principle. A particle and its antiparticle are described by the same modular irreducible representation of the symmetry algebra. This automatically explains the existence of antiparticles and shows that a particle and its antiparticle are the different states of the same object. As a consequence, a new symmetry arises, and the structure of the theory is considerably simplified. In particular, one can work with only creation operators or only annihilation ones since they are no more independent. In our approach the problem arises whether the existence of neutral elementary particles (e.g. the photon) is compatible with the usual relation between spin and statistics, or in other words whether neutral particles can be elementary or only composite. 
  In a {\cal N}=1 superspace formulation of {\cal N}=4 Yang-Mills theory we obtain the anomalous dimensions of chiral operators with large R charge J \to \infty keeping g^2 N/J^2 finite, to all orders of perturbation theory in the planar limit. Our result proves the conjecture that the anomalous dimensions are indeed finite in the above limit. This amounts to an exact check of the proposed duality between a sector of {\cal N}=4 Yang-Mills theory with large R charge J and string theory in a pp-wave background. 
  We study some of the novel properties of conformal field theories with noncompact target spaces as applied to string theory. Standard CFT results get corrected by boundary terms in the target space in a way consistent with the expected spacetime physics. For instance, one-point functions of general operators on the sphere and boundary operators on the disk need not vanish; we show that they are instead equal to boundary terms in spacetime. By applying this result to vertex operators for spacetime gauge transformations with support at infinity, we derive formulas for conserved gauge charges in string theory. This approach provides a direct CFT definition of ADM energy-momentum in string theory. 
  We study the behavior of the Wilson loop in the (5+1)-dimensional supersymmetric Yang-Mills theory with the presence of the solitonic object. Using the dual string description of the Yang-Mills theory that is given by the D1/D5 system, we estimate the Wilson loops both in the temporal and spatial cases. For the case of the temporal loop, we obtain the velocity dependent potential. For the spatial loop, we find that the area law is emerged due to the effect of the D1-branes. Further, we consider D1/D5 system in the presence of the constant B field. It is found that the Wilson loop obeys the area law for the effect of the noncommutativity. 
  Massive theories of abelian p-forms are quantized in a generalized path-representation that leads to a description of the phase space in terms of a pair of dual non-local operators analogous to the Wilson Loop and the 't Hooft disorder operators. Special atention is devoted to the study of the duality between the Topologically Massive and the Self-Dual models in 2+1 dimensions. It is shown that these models share a geometric representation in which just one non local operator suffices to describe the observables. 
  In this article we present formulae for q-integration on quantum spaces which could be of particular importance in physics, i.e. q-deformed Minkowski space and q-deformed Euclidean space in 3 or 4 dimensions. Furthermore, our formulae can be regarded as a generalization of Jackson's q-integral to 3 and 4 dimensions and provide a new possibility for an integration over the whole space being invariant under translations and rotations. 
  In this work we compute the spectra, waveforms and total scalar energy radiated during the radial infall of a small test particle coupled to a scalar field into a $d$-dimensional Schwarzschild-anti-de Sitter black hole. We focus on $d=4, 5$ and 7, extending the analysis we have done for $d=3$. For small black holes, the spectra peaks strongly at a frequency $\omega \sim d-1$, which is the lowest pure anti-de Sitter (AdS) mode. The waveform vanishes exponentially as $t \to \infty$, and this exponential decay is governed entirely by the lowest quasinormal frequency. This collision process is interesting from the point of view of the dynamics itself in relation to the possibility of manufacturing black holes at LHC within the brane world scenario, and from the point of view of the AdS/CFT conjecture, since the scalar field can represent the string theory dilaton, and 4, 5, 7 are dimensions of interest for the AdS/CFT correspondence. 
  We obtain geodesically complete spacetimes generated by static and rotating magnetic point sources in an Einstein-Maxwell-Dilaton theory of the Brans-Dicke type in three dimensions (3D). The theory is specified by three fields, the dilaton, the graviton and the electromagnetic field, and two parameters, the cosmological constant and the Brans-Dicke parameter, w. When the Brans-Dicke parameter is infinity, our solution reduces to the magnetic counterpart of the BTZ solution, while the w=0 case is equivalent to 4D general relativity with one Killing vector. The source for the magnetic field can be interpreted as composed by a system of two symmetric and superposed electric charges. One of the electric charges is at rest and the other is spinning. 
  The microscopic correlation functions of non-chiral random matrix models with complex eigenvalues are analyzed for a wide class of non-Gaussian measures. In the large-N limit of weak non-Hermiticity, where N is the size of the complex matrices, we can prove that all k-point correlation functions including an arbitrary number of Dirac mass terms are universal close to the origin. To this aim we establish the universality of the asymptotics of orthogonal polynomials in the complex plane. The universality of the correlation functions then follows from that of the kernel of orthogonal polynomials and a mapping of massive to massless correlators. 
  Within a toroidal orbifold framework, we exhibit intersecting brane-world constructions of flipped SU(5) \times U(1) GUT models with various numbers of generations, other chiral matter representations and Higgs representations. We exhibit orientifold constructions with integer winding numbers that yield 8 or more conventional SU(5) generations, and orbifold constructions with fractional winding numbers that yield flipped SU(5) \times U(1) models with just 3 conventional generations. Some of these models have candidates for the 5 and {\bar 5} Higgs representations needed for electroweak symmetry breaking, but not for the 10 and {\bar 10} representations needed for GUT symmetry breaking. We have also derived models with complete GUT and electroweak Higgs sectors, but these have undesirable extra chiral matter. 
  Recently Hollands and Wald argued that inflation does not solve any of the major cosmological problems. We explain why we disagree with their arguments. They also proposed a new speculative mechanism of generation of density perturbations. We show that in their scenario the inhomogeneities responsible for the large scale structure observed today were generated at an epoch when the energy density of the hot universe was 10^{95} times greater than the Planck density. The only way to avoid this problem is to assume that there was a stage of inflation in the early universe. 
  We consider, in 5 dimensions, the low energy effective action induced by heterotic string theory including the leading stringy correction of order alpha'. In the presence of a single positive tension flat brane, and an infinite extra dimension, we present a particular class of solutions with finite 4-dimensional Planck scale and no naked singularity. A ``self-tuning'' mechanism for relaxing the cosmological constant on the brane, without a drastic fine tuning of parameters, is discussed in this context. Our solutions are distinct from the standard self-tuning solutions discussed in the context of vanishing quantum corrections in alpha', and become singular in this limit. 
  Based on the recently proposed action for Matrix theory describing the DLCQ M theory in the maximally supersymmetric pp-wave background, we obtain the supersymmetry algebra of supercharge density. Using supersymmetry transformation rules for fermions, we identify BPS states with the central charges in the supersymmetry algebra, which can be activated only in the large N limit. They preserve some fraction of supersymmetries and correspond to rotating transverse membranes and longitudinal five branes. 
  We describe a large class of exact string backgrounds with a null Killing vector arising, via a limiting \`a la Penrose procedure, from string backgrounds corresponding to coset conformal field theories for compact groups G_N/H_N times a free time-like boson U(1)_{-N}. In this way a class of novel logarithmic conformal field theories (LCFT) emerges, that includes the one constructed recently as an N\to \infty limit of the SU(2)_N/U(1) X U(1)_{-N} theory. We explicitly give the exact operator algebra for the basic chiral fields as well as their representation in terms of free bosons, even though these are not known in general at finite N. We also compute four-point functions of various operators in the theory. For the cases of the four- and five-dimensional models, corresponding to a limit of the theory SO(D+1)_N/SO(D) X U(1)_{-N} for D=3 and 4, we also present the explicit expressions for the background fields. 
  We study the partition function of type IIA string theory on 10-manifolds of the form T^2 x X where X is 8-dimensional, compact, and spin. We pay particular attention to the effects of the topological phases in the supergravity action implied by the K-theoretic formulation of RR fields, and we use these to check the T-duality invariance of the partition function. We find that the partition function is only T-duality invariant when we take into account the T-duality anomalies in the RR sector, the fermionic path integral (including 4-fermi interaction terms), and 1-loop corrections including worldsheet instantons. We comment on applications of our computation to speculations about the role of the Romans mass in M-theory. We also discuss some issues which arise when one attempts to extend these considerations to checking the full U-duality invariance of the theory. 
  We study asymptotic expansions of spin-spin correlation functions for the XXZ Heisenberg chain in the critical regime. We use the fact that the long-distance effects can be described by the Gaussian conformal field theory. Comparing exact results for form factors in the XYZ and sine-Gordon models, we determine correlation amplitudes for the leading and main sub-leading terms in the asymptotic expansions of spin-spin correlation functions. We also study the isotropic (XXX) limit of these expansions. 
  We study the pp-wave limits of various elliptic models with orientifold planes and D7-branes, as well as the pp-wave limit of an orientifold of adS_5 x T^{11}. Many of the limits contain both open and closed strings. We also present pp-wave limits of theories which give rise to a compact null direction and contain open strings. Maps between the string theory states and gauge theory operators are proposed. 
  We examine Chern-Simons theory written on a noncommutative plane with a `hole', and show that the algebra of observables is a nonlinear deformation of the $w_\infty$ algebra. The deformation depends on the level (the coefficient in the Chern-Simons action), and the noncommutativity parameter, which were identified, respectively, with the inverse filling fraction and the inverse density in a recent description of the fractional quantum Hall effect. We remark on the quantization of our algebra. The results are sensitive to the choice of ordering in the Gauss law. 
  We analyze the properties of stars whose interior is described by the stiffest equation of state consistent with causality. We note the remarkable fact that the entropy of such stars scales like the area. 
  The issue of non-perturbative background independent quantization of matrix models is addressed. The analysis is carried out by considering a simple matrix model which is a matrix extension of ordinary mechanics reduced to 0 dimension. It is shown that this model has an ordinary mechanical system evolving in time as a classical solution. But in this treatment the action principle admits a natural modification which results in algebraic relations describing quantum theory. The origin of quantization is similar to that in Adler's generalized quantum dynamics. The problem with extension of this formalism to many degrees of freedom is solved by packing all the degrees of freedom into a single matrix. The possibility to apply this scheme to field theory and to various matrix models is discussed. 
  We introduce a computational technique for studying non-supersymmetric deformations of domain wall solutions of interest in AdS/CFT. We focus on the Klebanov-Strassler solution, which is dual to a confining gauge theory. From an analysis of asymptotics we find that there are three deformations that leave the ten-dimensional supergravity solution regular and preserve the global bosonic symmetries of the supersymmetric solution. Also, we show that there are no regular near-extremal deformations preserving the global symmetries, as one might expect from the existence of a gap in the gauge theory. 
  A Z_2 orbifold compactification of the heterotic string is considered. The resulting 6D GUT groups can be SO(16) or E_7 times SU(2) plus some hidden sector groups. The N=4 supersymmetry is reduced to N=2. In particular, the SO(16) 6D model with one spinor representation 128 can reduce to the previous 5D SO(16) or SO(14) family unification models after compactifying the sixth dimension. To obtain one spinor, we have to take into account the left-over center of SO(16). We also comment on the E_7 times SU(2) model. 
  We present unified ways of handling the cosmological perturbations in a class of gravity theory covered by a general action in eq. (1). This gravity includes our previous generalized $f(\phi,R)$ gravity and the gravity theory motivated by the tachyonic condensation. We present general prescription to derive the power spectra generated from vacuum quantum fluctuations in the slow-roll inflation era. An application is made to a slow-roll inflation based on the tachyonic condensation with an exponential potential. 
  We define noncommutative gerbes using the language of star products. Quantized twisted Poisson structures are discussed as an explicit realization in the sense of deformation quantization. Our motivation is the noncommutative description of D-branes in the presence of topologically non-trivial background fields. 
  We study the rolling tachyon including the gauge fields in boundary string field theory. We show that there are no plane wave solutions for the gauge fields for large time. The disappearance of the plane wave solutions indicates that there are no excitations of the gauge fields on the tachyon matter, which is consistent with the Sen's conjecture. 
  We find an infinite number of conserved currents and charges in the semiclassical limit $\lambda \to \infty$ of string theory in AdS$_5 \times S^5$ and remark on their relevance to conserved charges in the dual gauge theory. We establish a general procedure of exploring the semiclassical limit by viewing the classical motion as collective motion in the relevant part of the configuration space. We illustrate the procedure for semiclassical expansion around solutions of string theory on AdS$_5 \times (S^5/Z_M)$. 
  We show that the pure spinor formalism proposed by Berkovits to covariantly quantize superstrings is a gauge fixed, twisted version of the complexified n=2 superembedding formulation of the superstring. This provides the Berkovits approach with a geometrical superdiffeomorphism invariant ground. As a consequence, the absence of the worldsheet (super)diffeomorphism ghosts in the pure spinor quantization prescription and the nature of the Berkovits BRST charge and antighost are clarified. Since superembedding is classically equivalent to the Green-Schwarz formulation, we thus also relate the latter to the pure spinor construction. 
  We construct a set of conserved charges for asymptotically deSitter spacetimes that correspond to asymptotic conformal isometries. The charges are given by boundary integrals at spatial infinity in the flat cosmological slicing of deSitter. Using a spinor construction, we show that the charge associated with conformal time translations is necessarilly positive and hence may provide a useful definition of energy for these spacetimes. A similar spinor construction shows that the charge associated with the time translation Killing vector of deSitter in static coordinates has both positive and negative definite contributions. For Schwarzshild-deSitter the conformal energy we define is given by the mass parameter times the cosmological scale factor. The time dependence of the charge is a consequence of a non-zero flux of the corresponding conserved current at spatial infinity. For small perturbations of deSitter, the charge is given by the total comoving mass density. 
  We give a Petrov classification for five-dimensional metrics. We classify Ricci-flat metrics that are static, have an SO(3) isometry group and have Petrov type 22. We use this classification to look for the metric of a black hole on a cylinder, i.e. a black hole with asymptotic geometry four-dimensional Minkowski space times a circle. Although a black string wrapped around the circle and the five-dimensional black hole are both algebraically special, it turns out that the black hole on a cylinder is not. 
  We construct an exact metric which at short distances is the metric of massless particles in 5+1 spacetime (moving along a diameter of the sphere) and is AdS_3\times S^3 at infinity. We also consider a set of a conical defect spacetimes which are locally AdS_3\times S^3 and have the masses and charges of a special set of chiral primaries of the dual orbifold CFT. We find that excitation energies for a scalar field in the latter geometries agree exactly with the excitations in the corresponding CFT state created by twist operators: redshift in the geometry reproduces `long circle' physics in the CFT. We propose a map of string states in AdS_3\times S^3\times T^4 to states in the orbifold CFT, analogous to the recently discovered map for AdS_5\times S^5. The vibrations of the string can be pictured as oscillations of a Fermi sea in the CFT. 
  We construct six stack D6-brane vacua (non-supersymmetric) that have at low energy exactly the standard model (with right handed neutrinos). The construction is based on D6-branes intersecting at angles in $D = 4$ type toroidal orientifolds of type I strings. Three U(1)'s become massive through their couplings to RR fields and from the three surviving massless U(1)'s at low energies, one is the standard model hypercharge generator. The two extra massless U(1)'s get broken, as suggested recently (hep-th/0205147), by requiring some intersections to respect N=1 supersymmetry thus supporting the appearance of massless charged singlets. Proton and lepton number are gauged symmetries and their anomalies are cancelled through a generalized Green-Schwarz mechanism that gives masses to the corresponding gauge bosons through couplings to RR fields. Thus proton is stable and neutrinos are of Dirac type with small masses as a result of a PQ like-symmetry. The models predict the existence of only two supersymmetric particles, superpartners of $\nu_R$'s. 
  We study non-supersymmetric orbifold singularities from the point of view of D-brane probes. We present a description of the decay of such singularities from considerations of the toric geometry of the probe branes. 
  We perform a canonical and BRST analysis of a seven-dimensional Chern-Simons theory on a manifold with boundary. The main result is that the 7D theory induces for consistency a chiral two-form on the 6D boundary. We also comment on similar behaviour in a five-dimensional Chern-Simons theory relevant for $\N=4$ supersymmetric Yang-Mills theory in four dimensions. 
  Massive quarks are included in the Curci-Ferrari model and the theory is renormalized at two loops in the MSbar scheme in an arbitrary covariant gauge. 
  We consider the dynamics of a contracting universe ruled by two minimally coupled scalar fields with general exponential potentials. This model describes string-inspired scenarios in the Einstein frame. Both background and perturbations can be solved analytically in this model. Curvature perturbations are generated with a scale invariant spectrum only for a dust-like collapse, as happens for a single field model with an exponential potential. We find the conditions for which a scale invariant spectrum for isocurvature perturbation is generated. 
  We describe three-dimensional Kerr-de Sitter space using similar methods as recently applied to the BTZ black hole. A rigorous form of the classical connection between gravity in three dimensions and two-dimensional conformal field theory is employed, where the fundamental degrees of freedom are described in terms of two dependent SL(2,C) currents. In contrast to the BTZ case, however, quantization does not give the Bekenstein-Hawking entropy connected to the cosmological horizon of Kerr-de Sitter space. 
  We evaluate the fermionic Casimir effect associated with a massive fermion confined within a planar (d+1) dimensional slab-bag, on which MIT bag model boundary conditions of standard type, along a single spatial direction, are imposed. A simple and effective method for adding up the zero-point energy eigenvalues, corresponding to a quantum field under the influence of arbitrary boundary conditions, imposed on the field on flat surfaces perpendicular to a chosen spatial direction, is proposed. Using this procedure, an analytic result is obtained, from which small and large fermion mass limits, valid for an arbitrary number of dimensions, are derived. They match some known results in particular cases. The method can be easily extended to other configurations. 
  We discuss the hierarchy of Yukawa couplings in a supersymmetric three family Standard-like string Model. The model is constructed by compactifying Type IIA string theory on a Z_2 x Z_2 orientifold in which the Standard Model matter fields arise from intersecting D6-branes. When lifted to M theory, the model amounts to compactification of M-theory on a G_2 manifold. While the actual fermion masses depend on the vacuum expectation values of the multiple Higgs fields in the model, we calculate the leading worldsheet instanton contributions to the Yukawa couplings and examine the implications of the Yukawa hierarchy. 
  In the lightcone frame, where the supermembrane theory and the Matrix model are strikingly similar, the equations of motion admit an elegant complexification in even dimensional spaces. Although the explicit rotational symmetry of the target space is lost, the remaining unitary symmetries apart from providing a simple and unifying description of all known solutions suggest new ones for rotating spherical and toroidal membranes. In this framework the angular momentum is represented by U(1) charges which balance the nonlinear attractive forces of the membrane. We examine in detail a six dimensional rotating toroidal membrane solution which lives in a 3-torus, $T^3$ and admits stable radial modes. In Matrix Theory it corresponds to a toroidal N-$D_{0}$ brane bound state. We demonstrate its existence and discuss its radial stability. 
  I present a heuristic calculation of the critical exponent relating the gravitino mass to the cosmological constant in a de Sitter universe. The ingredients for the calculation are the area law for entropy, an R symmetry of the low energy effective Lagrangian, and a crude picture of the degenerate levels of the cosmological horizon. 
  We start a systematic study of string theory in AdS_3 black hole backgrounds. Firstly, we analyse in detail the geodesic structure of the BTZ black hole, including spacelike geodesics. Secondly, we study the spectrum for massive and massless scalar fields, paying particular attention to the connection between Sl(2,R) subgroups, the theory of special functions and global properties of the BTZ black holes. We construct classical strings that wind the black holes. Finally, we apply the general formalism to the vacuum black hole background, and formulate the boundary spacetime Virasoro algebra in terms of worldsheet operators. We moreover establish the link between a proposal for a ghost free spectrum for Sl(2,R) string propagation and the massless black hole background, thereby claryfing aspects of the AdS3/CFT correspondence. 
  We fully compute the N=1 supersymmetrization of the fourth power of the Weyl tensor in d=4 x-space with the auxiliary fields. In a previous paper, we showed that their elimination requires an infinite number of terms; we explicitely compute those terms to order \kappa^4 (three loop). We also write, in superspace notation, all the possible N=1 actions, in four dimensions, that contain pure R^4 terms (with coupling constants). We explicitely write these actions in terms of the \theta components of the chiral density \epsilon and the supergravity superfields R, G_m, W_{ABC}. Using the method of gauge completion, we compute the necessary \theta components which allow us to write these actions in x-space. We discuss under which circumstances can these extra R^4 correction terms be reabsorbed in the pure supergravity action, and their relevance to the quantum supergravity/string theory effective actions. 
  We consider the classical dynamics of bosonic and fermionic matrix variables in complex Hilbert space, defined by a trace action, assuming cyclic invariance under the trace and the presence of a global unitary invariance. With plausible and explicitly stated assumptions, including the existence of a large hierarchy of scale between the underlying dynamics and observed physics, we show that (1) the equilibrium statistical mechanics of this matrix dynamics, in the canonical ensemble, gives rise to an emergent quantum mechanics for many degrees of freedom, including the standard canonical commutation/anticommutation relations and the usual unitary Heisenberg and Schr\"odinger picture time evolutions of operators and states, and (2) the fluctuation or Brownian motion corrections to this thermodyamics lead to an energy-driven stochastic modification of the Schr\"odinger equation, which is known to imply state vector reduction with Born rule probabilities. Thus, quantum mechanics and its probabilistic interpretation both arise as emergent phenomena in the underlying trace dynamics. 
  We have extended the variational perturbative theory based on the back ground field method to include the optimized expansion of Okopinska and the post Gaussian effective potential of Stansu and Stevenson. This new method provides much simpler way to compute the correction terms to the Gausssian effective action (or potential). We have also renormalized the effective potential in 3+1 dimensions by introducing appropriate counter terms in the lagrangian 
  We show explicitly by the heuristic and practical arguments that for $N = 2$ supersymmetry (SUSY) a SUSY invariant relation between component fields of a vector supermultiplet of linear SUSY and Nambu-Goldstone fermions of the Volkov-Akulov model of nonlinear SUSY is written by using only three arbitrary dimensionless parameters, which can be recasted as the vacuum expectation values of auxiliary fields in the vector supermultiplet. 
  We consider the model in two dimensions with boundary quadratic deformation (BQD), which has been discussed in tachyon condensation. The partition function of this model (BQD) on a cylinder is determined, using the method of zeta function regularization. We show that, for closed channel partition function, a subtraction procedure must be introduced in order to reproduce the correct results at conformal points. The boundary entropy (g-function) is determined from the partition function and the off-shell boundary state. We propose and consider a supersymmetric generalization of BQD model, which includes a boundary fermion mass term, and check the validity of the subtraction procedure. 
  We study a generalization of the group of loops based on sets of signed points, instead of paths or loops. This geometrical setting incorporates the kinematical constraints of the Sigma Model, inasmuch as the the group of loops does with the Bianchi identities of Yang-Mills theories. We employ an Abelian version of this construction to quantize the Self-Dual Model, which allows us to relate this theory with that of a massless scalar field obeying non-trivial boundary conditions. 
  In recent papers, it has been shown that (i) the dynamics of theories involving gravity can be described, in the vicinity of a spacelike singularity, as a billiard motion in a region of hyperbolic space bounded by hyperplanes; and (ii) that the relevant billiard has remarkable symmetry properties in the case of pure gravity in $d+1$ spacetime dimensions, or supergravity theories in 10 or 11 spacetime dimensions, for which it turns out to be the fundamental Weyl chamber of the Kac-Moody algebras $AE_d$, $E_{10}$, $BE_{10}$ or $DE_{10}$ (depending on the model). We analyse in this paper the billiards associated to other theories containing gravity, whose toroidal reduction to three dimensions involves coset models $G/H$ (with $G$ maximally non compact). We show that in each case, the billiard is the fundamental Weyl chamber of the (indefinite) Kac-Moody ``overextension'' (or ``canonical Lorentzian extension'') of the finite-dimensional Lie algebra that appears in the toroidal compactification to 3 spacetime dimensions. A remarkable feature of the billiard properties, however, is that they do not depend on the spacetime dimension in which the theory is analyzed and hence are rather robust, while the symmetry algebra that emerges in the toroidal dimensional reduction is dimension-dependent. 
  We show that any conformal field theory in d-dimensional Minkowski space, in a phase with spontaneously broken conformal symmetry and with the dilaton among its fields, can be rewritten in terms of the static gauge (d-1)-brane on AdS_(d+1) by means of an invertible change of variables. This nonlinear holographic transformation maps the Minkowski space coordinates onto the brane worldvolume ones and the dilaton onto the transverse AdS brane coordinate. One of the consequences of the existence of this map is that any (d-1)-brane worldvolume action on AdS_(d+1)\times X^m (with X^m standing for the sphere S^m or more complicated curved manifold) admits an equivalent description in Minkowski space as a nonlinear and higher-derivative extension of some conventional conformal field theory action, with the conformal group being realized in a standard way. The holographic transformation explicitly relates the standard realization of the conformal group to its field-dependent nonlinear realization as the isometry group of the brane AdS_(d+1) background. Some possible implications of this transformation, in particular, for the study of the quantum effective action of N=4 super Yang-Mills theory in the context of AdS/CFT correspondence, are briefly discussed. 
  We propose to resolve the controversy regarding the stability of the monopole condensation in QCD by calculating the imaginary part of the one-loop effective action perturbatively. We calculate the imaginary part perturbatively to the order $g^2$ with two different methods, with Fyenman diagram and with Schwinger's method. Our result shows that with the magnetic background the effective action has no imaginary part, but with the electric background it acquires a negative imaginary part. This strongly indicates a stable monopole condensation in QCD. 
  We address the relation among the parameters of accelerating brane-universe embedded in five dimensional bulk space. It is pointed out that the tiny cosmological constant of our world can be obtained as quantum corrections around a given brane-solution in the bulk theory or in the field theory on the boundary from the holographic viewpoint.Some implications to the cosmology and constarints on the parameters are also given. 
  A generalization of duality transformations for arbitrary Lorentz tensors is presented, and a systematic scheme for constructing the dual descriptions is developed. The method, a purely Lagrangian approach, is based on a first order parent Lagrangian, from which the dual partners are generated. In particular, a family of theories which are dual to the massive spin two Fierz-Pauli field, both free and coupled to an external source, is constructed in terms of a third rank tensor which is taken as the basic configuration variable. 
  Electromagnetism can be generalized to Yang-Mills theory by replacing the group U(1)$ by a nonabelian Lie group. This raises the question of whether one can similarly generalize 2-form electromagnetism to a kind of "higher-dimensional Yang-Mills theory". It turns out that to do this, one should replace the Lie group by a "Lie 2-group", which is a category C where the set of objects and the set of morphisms are Lie groups, and the source, target, identity and composition maps are homomorphisms. We show that this is the same as a "Lie crossed module": a pair of Lie groups G,H with a homomorphism t: H -> G and an action of G on H satisfying two compatibility conditions. Following Breen and Messing's ideas on the geometry of nonabelian gerbes, one can define "principal 2-bundles" for any Lie 2-group C and do gauge theory in this new context. Here we only consider trivial 2-bundles, where a connection consists of a Lie(G)-valued 1-form together with an Lie(H)-valued 2-form, and its curvature consists of a Lie(G)-valued 2-form together with a Lie(H)-valued 3-form. We generalize the Yang-Mills action for this sort of connection, and use this to derive "higher Yang-Mills equations". Finally, we show that in certain cases these equations admit self-dual solutions in five dimensions. 
  We investigate boundary conditions for open strings in NSNS and RR pp-wave backgrounds constructed by Russo and Tseytlin, which are S-dual to each other. We show that if we do not turn on any boundary term (i.e. gauge field), D-branes in the RR background cannot move away from the origin in most cases, while those in the NSNS background can move anywhere. We construct RR counterparts of D3-branes in the NSNS background as D3-branes with gauge fluxes and show that indeed they can move anywhere, in accord with S-duality. 
  We study the BPS conditions in the closed supermembranes on the maximally supersymmetric pp-wave background. In particular, the 1/2 and 1/4 BPS states are discussed in detail. Moreover, we comment on the zero-modes in the invariant mass formulae of the theory. 
  Following a recent study by Das and Pernice [Nucl. Phys. B561, (1999) 357], we have carefully analyzed the half-harmonic oscillator. In contrast to their observations, our analysis reveals that the spectrum does not allow for a zero energy ground state and hence the supersymmetry is broken when the domain is restricted to the positive half of the real axis. 
  The thermodynamic Bethe ansatz method is employed for the study of the integrable critical $RSOS(q_{1}, q_{2};q)$ model. The high and low temperature behavior are investigated, and the central charge of the effective conformal field theory is derived. The obtained central charge is expressed as the sum of the central charges of two generalized coset models. 
  A sketch is given of a circle of ideas relating quantum field theories with representation theory. The main mathematical ingredients are spinor geometry and the gauge group equivariant K-theory of the space of connections. 
  We prove the uniqueness of higher dimensional (dilatonic) charged black holes in static and asymptotically flat spacetimes for arbitrary vector-dilaton coupling constant. An application to the uniqueness of a wide class of black p-branes is also given. 
  In the first part of this work, a perturbative analysis up to one-loop order is carried out to determine the one-loop $\beta$-function of noncommutative U(1) gauge theory with matter fields in the adjoint representation. In the second part, the conformal anomaly of the same theory is calculated using the Fujikawa's path integral method. The value of the one-loop $\beta$-function calculated in both methods coincides. As it turns out, noncommutative QED with matter fields in the adjoint representation is asymptotically free for the number of flavor degrees of freedom $N_{f}<3$. 
  The ekpyrotic and cyclic universe scenarios have revived the idea that the density perturbations apparent in today's universe could have been generated in a `pre-singularity' epoch before the big bang. These scenarios provide explicit mechanisms whereby a scale invariant spectrum of adiabatic perturbations may be generated without the need for cosmic inflation, albeit in a phase preceding the hot big bang singularity. A key question they face is whether there exists a unique prescription for following perturbations through the bounce, an issue which is not yet definitively settled. This goal of this paper is more modest, namely to study a bouncing Universe model in which neither General Relativity nor the Weak Energy Condition is violated. We show that a perturbation which is pure growing mode before the bounce does not match to a pure decaying mode perturbation after the bounce. Analytical estimates of when the comoving curvature perturbation varies around the bounce are given. It is found that in general it is necessary to evaluate the evolution of the perturbation through the bounce in detail rather than using matching conditions. 
  The aim of this talk is to explain how symmetry breaking in a quantum field theory problem leads to a study of projective bundles, Dixmier-Douady classes, and associated gerbes. A gerbe manifests itself in different equivalent ways. Besides the cohomological description as a DD class, it can be defined in terms of a family of local line bundles or as a prolongation problem for an (infinite-dimensional) principal bundle, with the fiber consisting of (a subgroup of) projective unitaries in a Hilbert space. The prolongation aspect is directly related to the appearance of central extensions of (broken) symmetry groups. We also discuss the construction of twisted K-theory classes by families of supercharges for the supersymmetric Wess-Zumino-Witten model. 
  This paper has been superseded by hep-th/0303164, "The Dirichlet Obstruction in AdS/CFT" 
  We study wrapped brane configurations via possible maximally supersymmetric gauged supergravities. First, we construct various supersymmetric wrapped D3 brane configurations from D=5 N=8 SO(6) gauged supergravity. This procedure provides certain new examples of wrapped D3 branes around supersymmetric cycles inside non-compact special holonomy manifolds. We analyze their behaviors numerically in order to discuss a correspondence to Higgs and Coulomb branches of sigma models on wrapped D3 branes. We also realize supersymmetric wrapped M2 branes from D=4 N=8 SO(8) gauged supergravity. Then, we study supersymmetric wrapped type IIB NS5 branes by D=7 N=4 SO(4) gauged supergravity. We show a method to derive them by using supersymmetric wrapped M5 branes in D=7 N=4 SO(5) gauged supergravity. This method is based on a domain wall like reduction. Solutions include NS5 branes wrapped around holomorphic $CP^2$ inside non-compact Calabi-Yau threefold. Their behavior shows a similar feature to that for NS5 branes wrapped around holomorphic $CP^1$ inside non-compact $K3$ surface. This construction also provides a check of preserved supersymmetry for a solution interpreted within a string world-sheet theory introduced by Hori and Kapustin. Finally, we find new non-supersymmetric solutions including AdS space-times in D=6 N=2 $SU(2)\times U(1)$ massive gauged supergravity. These solutions can be interpreted as non-supersymmetric wrapped D4-D8 configurations which are dual to non-supersymmetric conformal field theories realized on wrapped D4 branes. 
  Spectral triples describe and generalize Riemannian spin geometries by converting the geometrical information into algebraic data, which consist of an algebra $A$, a Hilbert space $H$ carrying a representation of $A$ and the Dirac operator $D$ (a selfadjoint operator acting on $H$). The gravitational action is described by the trace of a suitable function of $D$. In this paper we examine the (path-integral-) quantization of such a system given by a finite dimensional commutative algebra. It is then (in concrete examples) possible to construct the moduli space of the theory, i.e. to divide the space of all Dirac operators by the action of the diffeomorphism group, and to construct an invariant measure on this space. We discuss expectation values of various observables and demonstrate some interesting effects such as the effect of coupling the system to Fermions (which renders finite quantities in cases, where the Bosons alone would give infinite quantities) or the striking effect of spontaneous breaking of spectral invariance. 
  We investigate the influence of a uniform magnetic field on the zero-point energy of charged fields of two types, namely, a massive charged scalar field under Dirichlet boundary conditions and a massive fermion field under MIT boundary conditions. For the first, exact results are obtained, in terms of exponentially convergent functions, and for the second, the limits for small and for large mass are analytically obtained too. Coincidence with previously known, partial result serves as a check of the procedure. For the general case in the second situation --a rather involved one-- a precise numerical analysis is performed. 
  We introduce a new first order formulation of world-volume actions for p-branes with k-supersymmetry. In this language, which involves more auxiliary fields compensated by more local symmetries, the action is provided by a very compact, simple and elegant formula applicable to any supergravity background. The k-supersymmetry variation against which it is invariant is obtained from the bulk supersymmetries by means of a projector that has a simple expression in terms of the auxiliary fields. The distinctive feature of our formalism is that all fermion fields are hidden into the definition of the curvatures and the action is formally the same, in terms of these differential forms as it would be in a purely bosonic theory. Typically our new formulation enables one to discuss the correct boundary actions for non trivial supergravity Dp-brane bulk solutions like the D3-brane solution on smooth ALE manifolds with flux recently constructed in the literature 
  Motivated by the conjecture that the cosmological constant problem could be solved by strong quantum effects in the infrared we use the exact flow equation of Quantum Einstein Gravity to determine the renormalization group behavior of a class of nonlocal effective actions. They consist of the Einstein-Hilbert term and a general nonlinear function F(k, V) of the Euclidean space-time volume V. A partial differential equation governing its dependence on the scale k is derived and its fixed point is analyzed. For the more restrictive truncation of theory space where F(k, V) is of the form V+V ln V, V+V^2, and V+\sqrt{V}, respectively, the renormalization group equations for the running couplings are solved numerically. The results are used in order to determine the k-dependent curvature of the S^4-type Euclidean space-times which are solutions to the effective Einstein equations, i.e. stationary points of the scale dependent effective action. For the V+V ln V-invariant (discussed earlier by Taylor and Veneziano) we find that the renormalization group running enormously suppresses the value of the renormalized curvature which results from Planck-size bare parameters specified at the Planck scale. Hence one can obtain very large, almost flat universes without fine-tuning the cosmological constant. 
  We study in a self-consistent way the impact of the emission of bulk gravitons on the (homogeneous) cosmology of a three-brane embedded in a five-dimensional spacetime. In the low energy regime, we recover the well known result that the bulk affects the Friedmann equation only via a radiation-like term $\C/a^4$, called dark or Weyl radiation. By contrast, in the high energy regime, we find that the Weyl parameter $\C$ is no longer constant but instead grows very rapidly as $\C\propto a^4$. As a consequence, the value of $\C$ today is not a free parameter as usually considered but is a fixed number, which, generically, depends only on the number of relativistic degrees of freedom at the high/low energy transition. Our estimated amount of Weyl radiation satisfies the present nucleosynthesis bounds. 
  We study the evolution of perturbations on a moving probe D3-brane coupled to a 4-form field in an AdS$_5$-Schwarzschild bulk. The unperturbed dynamics are parametrised by a conserved energy $E$ and lead to Friedmann-Robertson-Walker `mirage' cosmology on the brane with scale factor $a(\tau)$. The fluctuations about the unperturbed worldsheet are then described by a scalar field $\phi(\tau,\vec{x})$. We derive an equation of motion for $\phi$, and find that in certain regimes of $a$ the effective mass squared is negative. On an expanding BPS brane with E=0 superhorizon modes grow as $a^4$ whilst subhorizon modes are stable. When the brane contracts, all modes grow. We also briefly discuss the case when $E>0$, BPS anti-branes as well as non-BPS branes. Finally, the perturbed brane embedding gives rise to scalar perturbations in the FRW universe. We show that $\phi$ is proportional to the gauge invariant Bardeen potentials on the brane. 
  A superembedding construction of general non-abelian Born-Infeld actions in three dimensions is described. These actions have rigid target space and local worldvolume supersymmetry(i.e. kappa symmetry). The standard abelian Born-Infeld gauge multiplet is augmented with an additional worldvolume SU(N) gauge supermultiplet. It is shown how to construct single-trace actions and in particular a kappa-supersymmetric extension of the symmetrised trace action. 
  As a simple model for unknown Planck scale physics, we assume that the quantum modes responsible for producing primordial curvature perturbations during inflation are placed in their instantaneous adiabatic vacuum when their proper momentum reaches a fixed high energy scale M. The resulting power spectrum is derived and presented in a form that exhibits the amplitude and frequency of the superimposed oscillations in terms of H/M and the slow roll parameter epsilon. The amplitude of the oscillations is proportional to the third power of H/M. We argue that these small oscillations give the lower bound of the modifications of the power spectrum if the notion of free mode propagation ceases to exist above the critical energy scale M. 
  We give some supersymmetric wave solutions, both chiral (selfdual) and nonchiral, to interacting supersymmetric theories in four dimensions. 
  We construct explicit cohomogeneity two metrics of G_2 holonomy, which are foliated by twistor spaces. The twistor spaces are S^2 bundles over four-dimensional Bianchi IX Einstein metrics with self-dual (or anti-self-dual) Weyl tensor. Generically the 4-metric is of triaxial Bianchi IX type, with SU(2) isometry. We derive the first-order differential equations for the metric coefficients, and obtain the corresponding superpotential governing the equations of motion, in the general triaxial Bianchi IX case. In general our metrics have singularities, which are of orbifold or cosmic-string type. For the special case of biaxial Bianchi IX metrics, we give a complete analysis their local and global properties, and the singularities. In the triaxial case we find that a system of equations written down by Tod and Hitchin satisfies our first-order equations. The converse is not always true. A discussion is given of the possible implications of the singularity structure of these spaces for M-theory dynamics. 
  Quiver theories arising on D3-branes at orbifold and del Pezzo singularities are studied using mirror symmetry. We show that the quivers for the orbifold theories are given by the soliton spectrum of massive 2d N=2 theory with weighted projective spaces as target. For the theories obtained from the del Pezzo singularities we show that the geometry of the mirror manifold gives quiver theories related to each other by Picard-Lefschetz transformations, a subset of which are simple Seiberg duals. We also address how one indeed derives Seiberg duality on the matter content from such geometrical transitions and how one could go beyond and obtain certain ``fractional Seiberg duals.'' Moreover, from the mirror geometry for the del Pezzos arise certain Diophantine equations which classify all quivers related by Picard-Lefschetz. Some of these Diophantine equations can also be obtained from the classification results of Cecotti-Vafa for the 2d N=2 theories. 
  Instanton and wormhole solutions are constructed in a d-dimensional gravity theory with an axion-dilaton pair of scalar fields. We discuss the cases of vanishing, positive and negative cosmological constant. 
  We review the construction of regular p-brane solutions of M-theory and string theory with less than maximal supersymmetry whose transverse spaces have metrics with special holonomy, and where additional fluxes allow for brane resolutions via transgression terms. We summarize properties of resolved M2-branes and fractional D2-branes, whose transverse spaces are Ricci flat eight-dimensional and seven-dimensional spaces of special holonomy. Recent developments in the construction of new G_2 holonomy spaces are also reviewed. 
  We propose a new approach for using the AdS/CFT correspondence to study quantum black hole physics. The black holes on a brane in an AdS$_{D+1}$ braneworld that solve the classical bulk equations are interpreted as duals of {\it quantum-corrected} $D$-dimensional black holes, rather than classical ones, of a conformal field theory coupled to gravity. We check this explicitly in D=3 and D=4. In D=3 we reinterpret the existing exact solutions on a flat membrane as states of the dual 2+1 CFT. We show that states with a sufficiently large mass really are 2+1 black holes where the quantum corrections dress the classical conical singularity with a horizon and censor it from the outside. On a negatively curved membrane, we reinterpret the classical bulk solutions as quantum-corrected BTZ black holes. In D=4 we argue that the bulk solution for the brane black hole should include a radiation component in order to describe a quantum-corrected black hole in the 3+1 dual. Hawking radiation of the conformal field is then dual to classical gravitational bremsstrahlung in the AdS$_5$ bulk. 
  Unification ideas motivate the formulation of field equations on an extended spin space. Demanding that the Poincare symmetry be maintained, one derives scalar symmetries that are associated with flavor and gauge groups. Boson and fermion solutions are obtained with a fixed representation. A field theory can be equivalently written and interpreted in terms of elements of such space and is similarly constrained. At 5+1 dimensions, one obtains isospin and hypercharge SU(2)_L X U(1) symmetries, their vector carriers, two-flavor charged and chargeless leptons, and scalar particles. Mass terms produce breaking of the symmetry to an electromagnetic U(1), a Weinberg's angle with sin^2(theta_W)=.25, and additional information on the respective coupling constants. Their underlying spin symmetry gives information on the particles' masses; one reproduces the standard-model ratio M_Z/M_W, and predicts a Higgs mass of M_H ~114 GeV, at tree level. 
  We investigate how the matrix representation of SU(N) algebra approaches that of the Poisson algebra in the large N limit. In the adjoint representation, the (N^2-1) times (N^2-1) matrices of the SU(N) generators go to those of the Poisson algebra in the large N limit. However, it is not the case for the N times N matrices in the fundamental representation. 
  Recently it has been shown that D-branes in orientifolds are not always described by equivariant Real K-theory. In this paper we define a previously unstudied twisted version of equivariant Real K-theory which gives the D-brane spectrum for such orientifolds. We find that equivariant Real K-theory can be twisted by elements of a generalised group cohomology. This cohomology classifies all orientifolds just as group cohomology classifies all orbifolds. As an example we consider the $\Omega\times\I_4$ orientifolds. We completely determine the equivariant orthogonal K-theory $KO_{\Zop_2}(\R^{p,q})$ and analyse the twisted versions. Agreement is found between K-theory and Boundary Confromal Field Theory (BCFT) results for both integrally- and torsion-charged D-branes. 
  The definition of ``Lie derivative'' of spinors with respect to Killing vectors is extended to all kinds of Lorentz tensors. This Lie-Lorentz derivative appears naturally in the commutator of two supersymmetry transformations generated by Killing spinors and vanishes for Vielbeins. It can be identified as the generator of the action of isometries on supergravity fields and its use for the calculation of supersymmetry algebras is revised and extended. 
  We investigate multi-monopole solutions of a modified version of the BPS Yang-Mills-Higgs model in which a term quartic in the covariant derivatives of the Higgs field (a Skyrme term) is included in the Lagrangian. Using numerical methods we find that this modification leads to multi-monopole bound states. We compute axially symmetric monopoles up to charge five and also monopoles with Platonic symmetry for charges three, four and five. The numerical evidence suggests that, in contrast to Skyrmions, the minimal energy Skyrmed monopoles are axially symmetric. 
  Direct evaluation of the Seiberg-Witten prepotential is accomplished following the localization programme suggested some time ago. Our results agree with all low-instanton calculations available in the literature. We present a two-parameter generalization of the Seiberg-Witten prepotential, which is rather natural from the M-theory/five dimensional perspective, and conjecture its relation to the tau-functions of KP/Toda hierarchy. 
  We study a connection between duality and topological field theories. First, 2d Kramers-Wannier duality is formulated as a simple 3d topological claim (more or less Poincare duality), and a similar formulation is given for higher-dimensional cases. In this form they lead to simple TFTs with boundary coloured in two colours. The statistical models live on the boundary of these TFTs, as in the CS/WZW or AdS/CFT correspondence. Classical models (Poisson-Lie T-duality) suggest a non-abelian generalization in the 2dcase, with abelian groups replaced by quantum groups. Amazingly, the TFT formulation solves the problem without computation: quantum groups appear in pictures, independently of the classical motivation. Connection with Chern-Simons theory appears at the symplectic level, and also in the pictures of the Drinfeld double: Reshetikhin-Turaev invariants of links in 3-manifolds, computed from the double, are included in these TFTs. All this suggests nice phenomena in higher dimensions. 
  We continue the study of a class of geometric transitions proposed by Aganagic and Vafa which exhibit open string instanton corrections to Chern-Simons theory. In this paper we consider an extremal transition for a local del Pezzo model which predicts a highly nontrivial relation between topological open and closed string amplitudes. We show that the open string amplitudes can be computed exactly using a combination of enumerative techniques and Chern-Simons theory proposed by Witten some time ago. This yields a striking conjecture relating the topological amplitudes of all genus of the local del Pezzo model to a system of coupled Chern-Simons theories. 
  We demonstrate the equivalence of all loop closed topological string amplitudes on toric local Calabi-Yau threefolds with computations of certain knot invariants for Chern-Simons theory. We use this equivalence to compute the topological string amplitudes in certain cases to very high degree and to all genera. In particular we explicitly compute the topological string amplitudes for P2 up to degree 12 and P1 x P1 up to total degree 10 to all genera. This also leads to certain novel large N dualities in the context of ordinary superstrings, involving duals of type II superstrings on local Calabi-Yau three-folds without any fluxes. 
  The heterotic string free fermionic formulation produced a large class of three generation models, with an underlying SO(10) GUT symmetry which is broken directly at the string level by Wilson lines. A common subset of boundary condition basis vectors in these models is the NAHE set, which corresponds to Z2 X Z2 orbifold of an SO(12) Narain lattice, with (h11,h21)=(27,3). Alternatively, a manifold with the same data is obtained by starting with a Z2 X Z2 orbifold at a generic point on the lattice, with (h11,h21)=(51,3), and adding a freely acting Z2 involution. The equivalence of the two constructions is proven by examining the relevant partition functions. The explicit realization of the shift that reproduces the compactification at the free fermionic point is found. It is shown that other closely related shifts reproduce the same massless spectrum, but different massive spectrum, thus demonstrating the utility of extracting information from the full partition function. A freely acting involution of the type discussed here, enables the use of Wilson lines to break the GUT symmetry and can be utilized in non-perturbative studies of the free fermionic models. 
  We investigate the pp-wave limit of the AdS_3\times S^3\times K3 compactification of Type IIB string theory from the point of view of the dual Sym_N(K3) CFT. It is proposed that a fundamental string in this pp-wave geometry is dual to the c=6 effective string of the Sym_N(K3) CFT, with the string bits of the latter being composed of twist operators. The massive fundamental string oscillators correspond to certain twisted Virasoro generators in the effective string. It is shown that both the ground states and the genus expansion parameter (at least in the orbifold limit of the CFT) coincide. Surprisingly the latter scales like J^2/N rather than the J^4/N^2 which might have been expected. We demonstrate a leading-order agreement between the pp-wave and CFT particle spectra. For a degenerate special case (one NS 5-brane) an intriguing complete agreement is found. 
  BMN operators are characterized by the fact that they have infinite R-charge and finite anomalous dimension in the BMN double scaling limit. Using this fact, we show that the BMN operators close under operator product expansion and form a sector in the N=4 supersymmetric Yang-Mills theory. We then identify short-distance limits of general BMN n-point correlators, and show how they correspond to the pp-wave string interactions. We also discuss instantons in the light of the pp-wave/SYM correspondence. 
  We derive a general expression for the gauge invariant mass (m_G) for an Abelian gauge field, as induced by vacuum polarization, in 1+1 dimensions. From its relation to the chiral anomaly, we show that m_G has to satisfy a certain quantization condition. This quantization can be, on the other hand, explicitly verified by using the exact general expression for the gauge invariant mass in terms of the fermion propagator. This result is applied to some explicit examples, exploring the possibility of having interesting physical situations where the value of $m_G$ departs from its canonical value. We also study the possibility of generalizing the results to the 2+1 dimensional case at finite temperature, showing that there are indeed situations where a finite and non-vanishing gauge invariant mass is induced. 
  These lectures are intended as a broad introduction to Chern Simons gravity and supergravity. The motivation for these theories lies in the desire to have a gauge invariant action -in the sense of fiber bundles- in more than three dimensions, which could provide a firm ground for constructing a quantum theory of the gravitational field. The case of Chern-Simons gravity and its supersymmetric extension for all odd D is presented. No analogous construction is available in even dimensions. 
  Using the notion of a gauge connection on a flat superspace, we construct a general class of noncommutative ($D=2,$ $\mathcal{N}=1$) supertranslation algebras generalizing the ordinary algebra by inclusion of some new bosonic and fermionic operators. We interpret the new operators entering into the algebra as the generators of a U(1) (super) gauge symmery of the underlying theory on superspace. These superalgebras are gauge invariant, though not closed in general. We then show that these type of superalgebras are naturally realized in a supersymmetric field theory possessing a super U(1) gauge symmetry. As the non-linearly realized symmetries of this theory, the generalized noncommutative (super)translations and super gauge transformations are found to form a closed algebra. 
  We calculate mass spectrum of CHS model deformed by an exactly marginal operator, and find that there are tachyons which are not localized in the target space. Similar deformation is discussed in another CFT which corresponds to separated NS5-branes. A condensation of the tachyons is briefly argued. 
  In a previous paper (Corrigan-Sasaki), many remarkable properties of classical Calogero and Sutherland systems at equilibrium are reported. For example, the minimum energies, frequencies of small oscillations and the eigenvalues of Lax pair matrices at equilibrium are all "integer valued". The equilibrium positions of Calogero and Sutherland systems for the classical root systems (A_r, B_r, C_r and D_r) correspond to the zeros of Hermite, Laguerre, Jacobi and Chebyshev polynomials. Here we define and derive the corresponding polynomials for the exceptional (E_6, E_7, E_8, F_4 and G_2) and non-crystallographic (I_2(m), H_3 and H_4) root systems. They do not have orthogonality but share many other properties with the above mentioned classical polynomials. 
  We discuss within the weak-field approximation and the derivative expansion how the area law of the Wilson loop follows directly from the vacuum condensate of mass dimension 2, i.e., simultaneous Bose-Einstein condensation of gluon pair and ghost-antighost pair. Such a novel vacuum condensate was recently claimed to exist as the non-vanishing vacuum expectation value of a BRST-invariant composite operator of mass dimension 2. First of all, we use a version of the non-Abelian Stokes theorem to rewrite the Wilson loop line integral to a surface integral. Then we convert the Yang-Mills theory with an insertion of the Wilson loop operator into a bosonic string theory with a rigidity term by way of an equivalent antisymmetric tensor gauge theory which couples to the surface spanned by the Wilson loop. This result suggests an intimate relationship between quark confinement and mass gap in Yang-Mills theory. In fact, the dual Ginzburg-Landau theory describing the dual superconductivity is also derivable by making use of duality transformations without using the naive Abelian projection and without breaking the global color invariance of the original Yang-Mills theory. This feature is desirable from the viewpoint of color confinement preserving color symmetry. 
  The complete D-brane spectrum in $\Zop_2$ orientifolds is computed. Stable non-BPS D-branes with both integral and torsion charges are found. The relation to K-theory is discussed and a new K-theory relevant to orientifolds is suggested. 
  The supersymmetric $(AdS_3\times S^3)/Z_N$ orbifold constructed by the authors in hep-th/0106171 is shown to describe AdS fragmentation, where fivebranes are emerging from the F1-NS5 background. The twisted sector moduli of the orbifold are the collective coordinates of groups of $n_5/N$ fivebranes. We discuss the relation between the descriptions of this background as a perturbative string orbifold and as a BPS state in the dual spacetime CFT. Finally, we attempt to apply the lessons learned to the description of BTZ black holes as $AdS_3$ orbifolds and to related big crunch/big bang cosmological scenarios. 
  We consider M-theory on AdS_4 x N^{0,1,0} where N^{0,1,0}= (SU(3) x SU(2))/(SU(2) x U(1)). We review a Penrose limit of AdS_4 x N^{0,1,0} that provides the pp-wave geometry of AdS_4 x S^7. There exists a subsector of three dimensional N=3 dual gauge theory, by taking both the conformal dimension and R-charge large with the finiteness of their difference, which has enhanced N=8 maximal supersymmetry. We identify operators in the N=3 gauge theory with supergravity KK excitations in the pp-wave geometry and describe how the N=2 gauge theory operators originating from both N=3 short vector multiplet and N=3 long gravitino multiplet fall into N=8 supermultiplets. 
  An analytic ghost-free model for the QCD running coupling $\alpha(Q^2)$ is proposed. It is constructed from a more general approach we developed particularly for investigating physical observables of the type $F(Q^2)$ in regions that are inaccessible to perturbative methods of quantum field theory. This approach directly links the infrared (IR) and the ultraviolet(UV) regions together under the causal analyticity requirement in the complex $Q^2-$plane. Due to the inclusion of crucial non-perturbative effects, the running coupling in our model not only excludes unphysical singularities but also freezes to a finite value at the IR limit $Q^2=0$. This makes it consistent with a popular phenomenological hypothesis, namely the IR freezing phenomenon. Applying this model to compute the Gluon condensate, we obtain a result that is in good agreement with the most recent phenomenological estimate. Having calculated the $\beta-$function corresponding to our QCD coupling constant, we find that it behaves qualitatively like its perturbative counterpart, when calculated beyond the leading order and with a number of quark flavours allowing for the occurrence of IR fixed points. 
  A twistorial formulation of the N=1 D=4 superparticle with tensorial central charges describing massive and massless cases in uniform manner is given. The twistors resolve energy-momentum vector whereas the tensorial central charges are written in term of spinor Lorentz harmonics. The model makes possible to describe states preserving all allowed fractions of target-space supersymmetry. The full analysis of the number of conserved supersymmetries in models with N=1 D=4 superalgebra with tensorial central charges has been carried out. 
  The conditions under which a general two-dimensional non-linear sigma model is classically integrable are given. These requirements are found by demanding that the equations of motion of the theory are expressible as a zero curvature relation. Some new integrable two-dimensional sigma models are then presented. 
  We show that gravitational Chern-Simons corrections, associated with the sigma-model anomaly on the M5-brane world-volume, can resolve the singularity of the M2-brane solution with Ricci-flat, special holonomy transverse space. We explicitly find smooth solutions in the cases when the transverse space is a manifold of Spin(7) holonomy and SU(4) holonomy. We comment on the consequences of these results for the holographically related three-dimensional theories living on the world volume of a stack of such resolved M2-branes. 
  Four years ago the Extended Scale Relativity (ESR) theory in C-spaces (Clifford manifolds) was proposed as the plausible physical foundations of string theory. In such theory the speed of light and the minimum Planck scale are the two universal invariants. All the dimensions of a C-space can be treated on equal footing by implementing the holographic principle associated with a nested family of p-loops of various dimensionalities. This is achieved by using polyvector valued coordinates in C-spaces that encode in one stroke points, lines, areas, volumes,.....We review the derivation of the minimal length/time string/brane uncertainty relations and the maximum Planck temperature thermodynamical uncertainty relation. The Weyl-Heisenberg algebra in C-spaces is constructed which $induces$ a Noncommutative Geometric structure in the $ X^A $ coordinates. Hence quantization in C-spaces involves in a natural fashion a Noncommutative Quantum Mechanics and Field Theory, rather than being postulated ad-hoc. A QFT in C-spaces may very likely involve (Braided Hopf) Quantum Clifford algebras and generalized Moyal-like star products associated with multisymplectic geometry. 
  We continue and extend our earlier investigation ``Strings in a Time-Dependent Orbifold'' (hep-th/0204168). We formulate conditions for an orbifold to be amenable to perturbative string analysis and classify the low dimensional orbifolds satisfying these conditions. We analyze the tree and torus amplitudes of some of these orbifolds. The tree amplitudes exhibit a new kind of infrared divergences which are a result of some ultraviolet effects. These UV enhanced IR divergences can be interpreted as due to back reaction of the geometry. We argue that for this reason the three dimensional parabolic orbifold is not amenable to perturbation theory. Similarly, the smooth four dimensional null-brane tensored with sufficiently few noncompact dimensions also appears problematic. However, when the number of noncompact dimensions is sufficiently large perturbation theory in these time dependent backgrounds seems consistent. 
  RR fields in string backgrounds including orientifold planes and branes on top of them are classified by K-theory. Following the idea introduced in hep-th/0103183, we also classify such fluxes by cohomology. Both of them are compared through the Atiyah-Hirzebruch Spectral Sequence. Some new correlations between branes on orientifold planes $Op^\pm$ and obstructions to the existence of some branes are found. Finally, we find a topological condition that avoid the presence of global gauge anomalies in lower dimensional systems. 
  We study the structure of cubic matrix mechanics based on three-index objects. It is shown that there exists a counterpart of canonical structure in classical mechanics. 
  We find and explore a class of dyonic instanton solutions which can be identified as the supertubes connecting two D4 branes. They correspond to a single monopole string and a pair of monopole antimonopole strings from the worldvolume view point of D4 branes. 
  Consistent couplings among a set of scalar fields, two types of one-forms and a system of two-forms are investigated in the light of the Hamiltonian BRST cohomology, giving a four-dimensional nonlinear gauge theory. The emerging interactions deform the first-class constraints, the Hamiltonian gauge algebra, as well as the reducibility relations. 
  The Kaluza-Klein theory and Randall-Sundrum theory are examined comparatively, with focus on the behavior of the five dimensional (Dirac) fermion in the dimensional reduction to four dimensions. They are properly treated using the Cartan formalism. In the KK case, the dual property between the electric and magnetic dipole moments is revealed in relation to the ratio of two massive parameters: the inverse of the radius of the extra-space circle and the 5D fermion mass. The order estimation of the couplings is done. In the RS case, we consider the interaction with the 5D(bulk) Higgs field and the gauge field. The chiral property, localization, anomaly phenomena are examined. We evaluate the bulk quantum effect using the method of the induced effective action. The electric dipole moment term naturally appears. This is a new origin of the CP-violation. In the 4D limit, the dual relation between KK model and RS model appears. 
  We construct the nonlocal braneworld action in the two-brane Randall-Sundrum model in a holographic setup alternative to Kaluza-Klein description: the action is written as a functional of the two metric and radion fields on the branes. This action effectively describes the dynamics of the gravitational field both on the branes and in the bulk in terms of the brane geometries directly accessible for observations. Its nonlocal form factors incorporate the cumulative effect of the bulk Kaluza-Klein modes. We also consider the reduced version of this action obtained by integrating out the fields on the negative-tension brane invisible from the viewpoint of the Planckian brane observer. This effective action features a nontrivial transition (AdS flow) between the local and nonlocal phases of the theory associated with the limits of small and large interbrane separation. Our results confirm a recently proposed braneworld scenario with diverging (repulsive) branes and suggest possible new implications of this phase transition in brane cosmology. 
  We revisit the non-rotating massive BTZ black hole within a pseudo-Riemannian symmetric space context. Using classical symmetric space techniques we find that every such space intrinsically carries a regular Poisson structure whose symplectic leaves are para-hermitian symmetric surfaces. We also obtain a global expression of the metric yielding a dynamical description of the black hole from its initial to its final singularity. 
  We study the Moyal quantization for the constrained system. One of the purposes is to give a proper definition of the Wigner-Weyl(WW) correspondence, which connects the Weyl symbols with the corresponding quantum operators. A Hamiltonian in terms of the Weyl symbols becomes different from the classical Hamiltonian for the constrained system, which is related to the fact that the naively constructed WW correspondence is not one-to-one any more. In the Moyal quantization a geometrical meaning of the constraints is clear. In our proposal, the 2nd class constraints are incorporated into the definition of the WW correspondence by limiting the phasespace to the hypersurface. Even though we assume the canonical commutation relations in the formulation, the Moyal brackets between the Weyl symbols yield the same results as those for the constrained system derived by using the Dirac bracket formulation. 
  We derive the couplings of D-branes to the RR fields from the first principles, i.e. from the nonlinear sigma-model. We suggest a procedure to extract poles from string amplitudes, before the eventual formula for the couplings is obtained. Using the fact that the parts of amplitudes with poles are irrelevant for the effective action this procedure ultimately simplify the derivation since it allows us to omit these poles already from the very beginning. We also use the vertex for the RR fields in the (-3/2,-1/2) ghost picture as the useful tool for the calculation of such string amplitudes. We carry out the calculations in all orders of brane massless excitations and obtain the Myers-Chern-Simons action. Our goal is to present the calculations in full technical details and specify the approximation in which one obtains the action in question. 
  Section I contains introductory remarks about surface motions. Section II gives a detailed derivation of $H=-\Delta-Tr\sum_{i<j}[X_i,X_j]^2$ as describing a quantized discrete analogue of relativistically invariant membrane dynamics. Section III concerns the question of zero-energy bound-states in SU(N)-invariant supersymmetric matrix models. Section IV discusses the space of solutions of some differential matrix equations on $(-\infty,+\infty)$, interpolating between different representations of $su(2)$. Some exercises are added, and one remark/conjecture concerning 5-commutators. 
  It is well known that modifications to the Friedmann equation on a warped brane in an anti de Sitter bulk do not provide any low energy distinguishing feature from standard cosmology. However, addition of a brane curvature scalar in the action produces effects which can serve as a distinctive feature of brane world scenarios and can be tested with observations. The fitting of such a model with supernovae Ia data (including SN 1997ff at $z\approx1.7$) comes out very well and predicts an accelerating universe. 
  It is noted that a finite Penrose limit for brane probes with non-zero worldvolume fluxes does not generically exist; this is closely related to the observation by Blau et al that for a brane probe the Penrose limit is equivalent to an infinite-tension limit. It is shown that when the limit exists, however, the number of supersymmetries preserved by the probe does not decrease. 
  We conduct an exhaustive search for solutions of IIA and IIB supergravity with augmented supersymmetry. We find a two-parameter family of IIB solutions preserving 28 supercharges, as well as several other IIA and IIB families of solutions with 24 supercharges. Given the simplicity of the pp-wave solution, the algorithm described here represents a systematic way of classifying all such solutions with augmented supersymmetry. By T-dualizing some of these solutions we obtain exact non-pp wave supergravity solutions (with 8 or 16 supercharges), which can be interpreted as perturbations of the AdS-CFT correspondence with irrelevant operators. 
  We study string theory on a non-singular time-dependent orbifold of flat space, known as the `null-brane'. The orbifold group, which involves only space-like identifications, is obtained by a combined action of a null Lorentz transformation and a constant shift in an extra direction. In the limit where the shift goes to zero, the geometry of this orbifold reproduces an orbifold with a light-like singularity, which was recently studied by Liu, Moore and Seiberg (hep-th/0204168). We find that the backreaction on the geometry due to a test particle can be made arbitrarily small, and that there are scattering processes which can be studied in the approximation of a constant background. We quantize strings on this orbifold and calculate the torus partition function. We construct a basis of states on the smooth orbifold whose tree level string interactions are nonsingular. We discuss the existence of physical modes in the singular orbifold which resolve the singularity. We also describe another way of making the singular orbifold smooth which involves a sandwich pp-wave. 
  Inspired by our previous finding that supersymmetric Yang-Mills-Chern-Simons (SYM-CS) theory dimensionally reduced to 1+1 dimensions possesses approximate Bogomol'nyi-Prasad-Sommerfield (BPS) states, we study the analogous phenomenon in the three-dimensional theory. Approximate BPS states in two dimensions have masses which are nearly independent of the Yang-Mills coupling and proportional to their average number of partons. These states are a reflection of the exactly massless BPS states of the underlying pure SYM theory. In three dimensions we find that this mechanism leads to anomalously light bound states. While the mass scale is still proportional to the average number of partons times the square of the CS coupling, the average number of partons in these bound states changes with the Yang-Mills coupling. Therefore, the masses of these states are not independent of the coupling. Our numerical calculations are done using supersymmetric discrete light-cone quantization (SDLCQ). 
  Exact global solutions of five-dimensional cosmological models compactified on a $S_1/Z_2$ orbifold with two 3-branes are presented, and evolution of a simple model is studied. It is found that on all 4D spacetime hypersurfaces, except a singular one, the expanding universe was started not from a big bang but from a big bounce, and before the bounce the universe was in a deflationary contracting phase. It is also found that whether the energy density on the second brane is positive, zero, or negative depends on the size of the fifth dimension. 
  We consider the question which potentials in the action of a (1+1) dimensional scalar field theory allowing for spontaneous symmetry breaking have quantum fluctuations corresponding to reflectionless scattering data. The general problem of restoration from known scattering data is formulated and a number of explicit examples is given. Only certain sets of reflectionless scattering data correspond to symmetry breaking and all restored potentials are similar either to the Phi**4-model or to the sine-Gordon model. 
  We show how all known N=2, d=4,5,6 maximally supersymmetric vacua (Hpp-waves and aDSxS solutions) are related through dimensional reduction/oxidation preserving all the unbroken supersymmetries. In particular we show how the N=2, d=5 family of vacua (which are the near-horizon geometry of supersymmetric rotating black holes) interpolates between aDS_2xS^3 and aDS_3xS^2 in parameter space and how it can be dimensionally reduced to an N=2, d=4 dyonic Robinson-Bertotti solution with geometry aDS_2xS^2 and oxidized to an N=2, d=6 solution with aDS_3xS^3 geometry (which is the near-horizon of the self-dual string). 
  We show how VEV's and condensates can be read off from the Schr\"odinger wave-functional without further calculation. This allows us to study non-perturbative physics by solving the Schr\"odinger equation. To illustrate the method we calculate fermion condensates from the exact solution of the Schwinger model, and other (1+1) dimensional models. The chiral condensate is seen to be a large-distance effect due to propagators reflecting off the space-time boundary. 
  We explore a classical instability of spacetimes of dimension $D>4$. Firstly, we consider static solutions: generalised black holes and brane world metrics. The dangerous mode is a tensor mode on an Einstein base manifold of dimension $D-2$. A criterion for instability is found for the generalised Schwarzschild, AdS-Schwarzschild and topological black hole spacetimes in terms of the Lichnerowicz spectrum on the base manifold. Secondly, we consider perturbations in time-dependent solutions: Generalised dS and AdS. Thirdly we show that, subject to the usual limitations of a linear analysis, any Ricci flat spacetime may be stabilised by embedding into a higher dimensional spacetime with cosmological constant. We apply our results to pure AdS black strings. Finally, we study the stability of higher dimensional ``bubbles of nothing''. 
  The non-perturbative superpotential generated by a heterotic superstring wrapped once around a genus-zero holomorphic curve is proportional to the Pfaffian involving the determinant of a Dirac operator on this curve. We show that the space of zero modes of this Dirac operator is the kernel of a linear mapping that is dependent on the associated vector bundle moduli. By explicitly computing the determinant of this map, one can deduce whether or not the dimension of the space of zero modes vanishes. It is shown that this information is sufficient to completely determine the Pfaffian and, hence, the non-perturbative superpotential as explicit holomorphic functions of the vector bundle moduli. This method is illustrated by a number of non-trivial examples. 
  A pp-wave solution to 11-dimensional supergravity is given with precisely 26 supercharges. Its uniqueness and the absence of 11-dimensional pp-waves which preserve (precisely) 28 or 30 supercharges is discussed. Compactification on a spacelike circle gives a IIA configuration with all 26 of the supercharges. For this compactification, D0 brane charge does not appear in the supersymmetry algebra. Indeed, the 26 supercharge IIA background does not admit any supersymmetric D-branes. In an appendix, a 28 supercharge IIB pp-wave is presented along with its supersymmetry algebra. 
  We develop an approximation scheme for our worldsheet model of the sum of planar diagrams based on mean field theory. At finite coupling the mean field equations show a weak coupling solution that resembles the perturbative diagrams and a strong coupling solution that seems to represent a tensionless soup of field quanta. With a certain amount of fine-tuning, we find a solution of the mean field equations that seems to support string formation. 
  We study Gromov-Witten invariants of a rational elliptic surface using holomorphic anomaly equation in [HST1](hep-th/9901151). Formulating invariance under the affine $E_8$ Weyl group symmetry, we determine conjectured invariants, the number of BPS states, from Gromov-Witten invariants. We also connect our holomorphic anomaly equation to that found by Bershadsky,Cecotti,Ooguri and Vafa [BCOV1](hep-th/9302103). 
  We consider gauge theories with multitrace deformations in the context of certain AdS/CFT models with explicit breaking of conformal symmetry and supersymmetry. In particular, we study the standard four-dimensional confining model based on the D4-brane metric at finite temperature. We work in the self-consistent Hartree approximation, which becomes exact in the large-N limit and is equivalent to the AdS/CFT multitrace prescription that has been proposed in the literature. We show that generic multitrace perturbations have important effects on the phase structure of these models. Most notably they can induce new types of large-N first-order phase transitions. 
  We reexamine the oscillator construction of the D25-brane solution and the tachyon fluctuation mode of vacuum string field theory given previously. Both the classical solution and the tachyon mode are found to violate infinitesimally their determining equations in the level cut-off regularization. We study the effects of these violations on physical quantities such as the tachyon mass and the ratio of the energy density of the solution relative to the D25-brane tension. We discuss a possible way to resolve the problem of reproducing the expected value of one for the ratio. 
  In this paper we construct explicit Lagrangian formulation for the massive spin-2 supermultiplets with N = k supersymmetries k = 1,2,3,4. Such multiplets contain 2k particles with spin-3/2, so there must exist N = 2k local supersymmetries in the full nonlinear theories spontaneously broken so that only N = k global supersymmetries remain unbroken. In this paper we unhide these hidden supersymmetries by using gauge invariant formulation for massive high spin particles. Such formulation, operating with the right set of physical degrees of freedom from the very beginning and having non-singular massless limit, turns out to be very well suited for construction of massive supermultiplets from the well known massless ones. For all four cases considered we have managed to show that the massless limit of the supertransformations for N = k massive supermultiplet could be uplifted to N = 2k supersymmetry. This, in turn, allows one to investigate which extended supergravity models such massive multiplets could arise from. Our results show a clear connection of possible models with the five-dimensional extended supergravities. 
  We study an integrable deformation of the super-Liouville theory which generates a RG flows to the critical Ising model as the IR fixed point. This model turns out to be a supersymmetric sinh-Gordon model with spontaneously broken N=1 supersymmetry. The resulting massless Goldstino is the only stable on-shell particle which controls the IR behaviours. We propose the exact $S$-matrix of the Goldstino and compare associated thermodynamic Bethe ansatz equations with the quantization conditions derived from the reflection amplitudes of the the super-Liouville theory to provide nonperturbative checks for both the (NS) and the (R) sectors. 
  By exploiting the relation between static zero modes of massless Dirac operator and Kustaanheimo-Stiefel (Hopf) bundle sections, a general zero modes Ansatz which depends on an arbitrary real vector-function on $R^3$ is constructed. 
  We investigate the dynamics of gauge and scalar fields on unstable D-branes with rolling tachyons. Assuming an FRW metric on the brane, we find a solution of the tachyon equation of motion which is valid for arbitrary tachyon potentials and scale factors. The equations of motion for a U(1) gauge field and a scalar field in this background are derived. These fields see an effective metric which differs from the original FRW metric. The field equations receive large corrections due to the curvature of the effective metric as well as the time variation of the gauge coupling. The equations of state for these fields resemble those of nonrelativistic matter rather than those of massless particles. 
  We show how in the presence of RR two-form field strength the conditions for preserving supersymmetry on six- and seven-dimensional manifolds lead to certain generalizations of monopole equations. For six dimensions the string frame metric is Kaehler with the complex structure that descends from the octonions if in addition we assume F^{(1,1)}=0. The susy generator is a gauge covariantly constant spinor. For seven dimensions the string frame metric is conformal to a G_2 metric if in addition we assume the field strength to obey a selfduality constraint. Solutions to these equations lift to geometries of G_2 and Spin(7) holonomy respectively. 
  A connection between weak and strong tension limits and their perturbative corrections is discussed. New twistor-like models based on D=4, N=1 tensionless superstring and superbrane with tensor central charges are studied. The presence of three, two or less preserved fractions of $\kappa-$symmetry in the actions free of the Wess-Zumino terms is shown. A correlation of extra $\kappa-$symmetry with the R-symmetry is established. The equations of the superstring and superbrane models preserving 3/4 supersymmetry are exactly solved. The general solution for the Goldstone fermion is pure static, but for the Goldstone bosons it also includes a term describing string/brane motions along the fixed directions given by the initial data. These solutions correspond to the partial spontaneous breaking of the D=4, N=1 global supersymmetry and can be associated with a static closed magnetic Nielsen-Olesen vortex or a p-dimensional vortex. 
  This paper investigates an integrable system which is related to hyperbolic monopoles; ie the Bogomolny Yang-Mills-Higgs equations in (2+1) anti-de Sitter space which are integrable and whose solutions can be obtained using analytical methods. In particular, families of soliton solutions have been constructed explicitly and their dynamics has been investigated in some detail. 
  We replace our earlier condition that physical states of the superstring have non-negative grading by the requirement that they are analytic in a new real commuting constant t which we associate with the central charge of the underlying Kac-Moody superalgebra. The analogy with the twisted N=2 SYM theory suggests that our covariant superstring is a twisted version of another formulation with an equivariant cohomology. We prove that our vertex operators correspond in one-to-one fashion to the vertex operators in Berkovits' approach based on pure spinors. Also the zero-momentum cohomology is equal in both cases. Finally, we apply the methods of equivariant cohomology to the superstring, and obtain the same BRST charge as obtained earlier by relaxing the pure spinor constraints. 
  We study Randall-Sundrum brane models from the viewpoint of condensed matter/quantum optics. Following the idea of analog gravity we obtain effective metrics in fluid and Bose-Einstein-condensate systems mimicking those in the brane framework. We find that the effect of warp factors in the bulk geometry translates into finiteness of the analog systems. As an illustration of this identification we give a new interpretation of the peculiar behaviour of the critical temperature for the splitting of thick branes in warped spacetimes. 
  We discuss the Penrose limit of the classical string geometry obtained from a truly marginal deformation of $SL(2)\otimes SU(2)$ WZNW model. 
  We study static spherically symmetric solutions of high derivative gravity theories, with 4, 6, 8 and even 10 derivatives. Except for isolated points in the space of theories with more than 4 derivatives, only solutions that are nonsingular near the origin are found. But these solutions cannot smooth out the Schwarzschild singularity without the appearance of a second horizon. This conundrum, and the possibility of singularities at finite r, leads us to study numerical solutions of theories truncated at four derivatives. Rather than two horizons we are led to the suggestion that the original horizon is replaced by a rapid nonsingular transition from weak to strong gravity. We also consider this possibility for the de Sitter horizon. 
  In the presence of compact dimensions massive solutions of General Relativity may take one of several forms including the black-hole and the black-string, the simplest relevant background being R^{3+1} * S^1. It is shown how Morse theory places constraints on the qualitative features of the phase diagram, and a minimalistic diagram is suggested which describes a first order transition whose only stable phases are the uniform string and the black-hole. The diagram calls for a topology changing ``merger'' transition in which the black-hole evolves continuously into an unstable black-string phase. As evidence a local model for the transition is presented in which the cone over S^2 * S^2 plays a central role. Horizon cusps do not appear as precursors to black hole merger. A generalization to higher dimensions finds that whereas the cone has a tachyon function for d=5, its stability depends interestingly on the dimension - it is unstable for d<10, and stable for d>10. 
  It has recently been observed that IIB string theory in the pp-wave background can be used to calculate certain quantities, such as the dimensions of BMN operators, as exact functions of the effective coupling lambda' = lambda/J^2. These functions interpolate smoothly between the weak and strong effective coupling regimes of N=4 SYM theory at large R charge J. In this paper we use the pp-wave superstring field theory of hep-th/0204146 to study more complicated observables. The expansion of the three-string interaction vertex suggests more complicated interpolating functions which in general give rise to fractional powers of lambda' in physical observables at weak effective coupling. 
  We argue that states with nontrivial horizontal charges of BTZ black hole can be excited by ordinary falling matter including Hawking radiation. The matter effect does not break the integrability condition of the charges on the horizon. Thus we are able to trace the proccesses in which the matter imprints the information on the horizon by use of the charged states. It is naturally expected that in the thermal equilibrium with the Hawking radiation the black hole wanders ergodically through different horizontal states due to thermal fluctuation of incoming matter. This fact strengthens plausibility of the basic part of Carlip's idea. We also discuss some aspects of the quantum horizontal symmetry and conjecture how the precise black hole entropy will be given from our point of view. 
  We consider a five-dimensional constant curvature black hole, which is constructed by identifying some points along a Killing vector in a five-dimensional AdS space. The black hole has the topology M_4 times S^1, its exterior is time-dependent and its boundary metric is of the form of a three-dimensional de Sitter space times a circle, which means that the dual conformal field theory resides on a dynamical spacetime. We calculate the quasilocal stress-energy tensor of the gravitational background and then the stress-energy tenor of the dual conformal field theory. It is found that the trace of the tensor does indeed vanish, as expected. Further we find that the constant curvature black hole spacetime is just the "bubble of nothing" resulting from Schwarzschild-AdS black holes when the mass parameter of the latter vanishes. 
  Presented are two kinds of integral solutions to the quantum Knizhnik-Zamolodchikov equations for the 2n-point correlation functions of the Heisenberg XYZ antiferromagnet. Our first integral solution can be obtained from those for the cyclic SOS model by using the vertex-face correspondence. By the construction, the sum with respect to the local height variables k_0, k_1, >..., k_{2n} of the cyclic SOS model remains other than n-fold integral in the first solution. In order to perform those summations, we improve that to find the second integral solution of (r+1)n-fold integral for r in Z_{>1}, where r is a parameter of the XYZ model. Furthermore, we discuss the relations among our formula, Lashkevich-Pugai's formula and Shiraishi's one. 
  In the 6-dimensional model of warped compactification based on the euclidean AdS Reissner-Nordstrom metric it is possible to escape artificial 4-brane's stress-energy tensor anistropy necessary for fulfillment of Israel junction conditions by introducing in 3+1 spacetime of the cosmological constant which turns out to be ${}\sim G^2$ ($G$ - Newton's constant) and acquires the value compatible with observations. 
  We consider a class of multi-matrix models with an action which is O(D) invariant, where D is the number of NxN Hermitian matrices X_\mu, \mu=1,...,D. The action is a function of all the elementary symmetric functions of the matrix $T_{\mu\nu}=Tr(X_\mu X_\nu)/N$. We address the issue whether the O(D) symmetry is spontaneously broken when the size N of the matrices goes to infinity. The phase diagram in the space of the parameters of the model reveals the existence of a critical boundary where the O(D) symmetry is maximally broken. 
  We suggest that the extrinsic curvature and torsion of a bosonic string can be employed as variables in a two dimensional Landau-Ginzburg gauge field theory. Their interpretation in terms of the abelian Higgs multiplet leads to two different phases. In the phase with unbroken gauge symmetry the ground state describes open strings while in the phase with broken gauge symmetry the ground state involves closed strings. When we allow for an additional abelian gauge structure along the string, we arrive at an interpretation in terms of the two dimensional SU(2) Yang-Mills theory. 
  Time dependent orbifolds with spacelike or null singularities have recently been studied as simple models of cosmological singularities. We show that their apparent simplicity is an illusion: the introduction of a single particle causes the spacetime to collapse to a strong curvature singularity (a Big Crunch), even in regions arbitrarily far from the particle. 
  We consider the N=1 supersymmetric kink on a circle, i.e., on a finite interval with boundary or transition conditions which are locally invisible. For Majorana fermions, the single-particle Dirac Hamiltonian as a differential operator obeys simultaneously the three discrete symmetries of charge conjugation, parity, and time reversal. However, no single locally invisible transition condition can satisfy all three. When calculating sums over zero-point energies by mode number regularization, this gives a new rationale for a previous suggestion that one has to average over different choices of boundary conditions, such that for the combined set all three symmetries are obeyed. In particular it is shown that for twisted periodic or twisted antiperiodic boundary conditions separately both parity and time reversal are violated in the kink sector, as manifested by a delocalized momentum that cancels only in the average. 
  The equations for the solitons arbitrarily rotating in the ordinary and isotopic space are obtained. The wave functions of the corresponding dynamic states in the quantum case are found. The generalized matrix of the moments of inertia is degenerate for the O(2)-invariant configurations characteristic for the nucleon and delta-isobar. The equation for such configurations is established. It is shown that the spin-isospin rotation prevents the collapse of the soliton states in the SU(2) sigma-model. The entire consideration is based on the variational approach to the method of collective variables. 
  The Seiberg-Witten map for noncommutative Yang-Mills theories is studied and methods for its explicit construction are discussed which are valid for any gauge group. In particular the use of the evolution equation is described in some detail and its relation to the cohomological approach is elucidated. Cohomological methods which are applicable to gauge theories requiring the Batalin-Vilkoviskii antifield formalism are briefly mentioned. Also, the analogy of the Weyl-Moyal star product with the star product of open bosonic string field theory and possible ramifications of this analogy are briefly mentioned. 
  Quantum $A_N$-Toda field theory in two dimensions is investigated based on the method of quantizing canonical free field. Toda exponential operator associated with the fundamental weight $\lambda^1$ is constructed. 
  Using the AdS/CFT correspondence, we identify the symmetry algebra of the Laplacian on Euclidean space as an explicit quotient of the universal enveloping algebra of the Lie algebra of conformal motions. We construct analogues of these symmetries on a general conformal manifold. 
  The effective action in gauge theories is known to depend on a choice of gauge fixing conditions. This dependence is such that any change of gauge conditions is equivalent to a field redefinition in the effective action. In this sense, the quantum deformation of conformal symmetry in the N = 4 super Yang-Mills theory, which was computed in 't Hooft gauge in hep-th/9808039 and hep-th/0203236, is gauge dependent. The deformation is an intrinsic property of the theory in that it cannot be eliminated by a local choice of gauge (although we sketch a field redefinition induced by a nonlocal gauge which, on the Coulomb branch of the theory, converts the one-loop quantum-corrected conformal transformations to the classical ones). We explicitly compute the deformed conformal symmetry in R_\xi gauge. The conformal transformation law of the gauge field turns out to be \xi-independent. We construct the scalar field redefinition which relates the 't Hooft and R_\xi gauge results. A unique feature of 't Hooft gauge is that it makes it possible to consistently truncate the one-loop conformal deformation to the terms of first order in derivatives of the fields such that the corresponding transformations form a field realization of the conformal algebra. 
  (A) The momentum density conjugate to a bosonic quantum field splits naturally into the sum of a classical component and a nonclassical component. It is shown that the field and the nonclassical component of the momentum density satisfy an_exact_ uncertainty relation, i.e., an equality, which underlies the Heisenberg-type uncertainty relation for fields.   (B) The above motivates a new approach to deriving and interpreting bosonic quantum fields, based on an exact uncertainty principle. In particular, the postulate that an ensemble of classical fields is subject to nonclassical momentum fluctuations, of a strength determined by the field uncertainty, leads from the classical to the quantum field equations. Examples include scalar, electromagnetic and gravitational fields. For the latter case the exact uncertainty principle specifies a unique (non-Laplacian) operator ordering for the Wheeler-deWitt equation. 
  We consider the leading post-Newtonian and quantum corrections to the non-relativistic scattering amplitude of charged scalars in the combined theory of general relativity and scalar QED. The combined theory is treated as an effective field theory. This allows for a consistent quantization of the gravitational field. The appropriate vertex rules are extracted from the action, and the non-analytic contributions to the 1-loop scattering matrix are calculated in the non-relativistic limit. The non-analytical parts of the scattering amplitude, which are known to give the long range, low energy, leading quantum corrections, are used to construct the leading post-Newtonian and quantum corrections to the two-particle non-relativistic scattering matrix potential for two charged scalars. The result is discussed in relation to experimental verifications. 
  We study classical solutions describing rotating and boosted membranes on AdS_7 x S^4 background in M-theory. We find the dependence of the energy on the spin and R-charge of these solutions. In the flat space limit we get E ~ S^{2/3}, while for AdS at leading order E-S grows as S^{1/3}. The membranes on AdS_4 x S^7 background have briefly been studied as well. 
  After reviewing D-branes as conjugacy classes and various charge quantizations (modulo $k$) in WZW model, we develop the classification and systematic construction of all possible untwisted D-branes in Lie groups of A-D-E series. D-branes are classified according to their positions in the maximal torus. The moduli space of D-branes is naturally identified with a unit cell in the weight space which is exponentiated to be the maximal torus. However, for the D-brane classification, one may consider only the fundamental Weyl domain that is surrounded by the hyperplanes defined by Weyl reflections. We construct all the D-branes by the method of iterative deletion in the Dynkin diagram. The dimension of a D-brane always becomes an even number and it reduces as we go from a generic point of the fundamental domain to its higher co-dimensional boundaries. Quantum mechanical stability requires that only D-branes at discrete positions are allowed. 
  We discuss the continuum limits of Berenstein-Maldacena-Nastase matrix model. They give rise to Poisson bracket gauge field theories on the ordinary two sphere or on a set of two spheres with a gauge groups U(n) depending on the degeneracy of the classical solution about which the model is considered. We show that these models fail to be equivalent among each other in the continuum limit. 
  We suggest supersymmetric extension of conformally invariant string theory which is exclusively based on extrinsic curvature action. At the classical level this is a tension-less string theory. The absence of conformal anomaly in quantum theory requires that the space-time should be 6-dimensional. 
  We analyze the properties of a spontaneously broken D=4, N=4 supergravity without cosmological constant, obtained by gauging translational isometries of its classical scalar manifold. This theory offers a suitable low energy description of the super-Higgs phases of certain Type-IIB orientifold compactifications with 3-form fluxes turned on. We study its N=3,2,1,0 phases and their classical moduli spaces and we show that this theory is an example of no-scale extended supergravity. 
  An important subclass of D-branes on a Calabi-Yau manifold, X, are in 1-1 correspondence with objects in D(X), the derived category of coherent sheaves on X. We study the action of the monodromies in Kaehler moduli space on these D-branes. We refine and extend a conjecture of Kontsevich about the form of one of the generators of these monodromies (the monodromy about the "conifold" locus) and show that one can do quite explicit calculations of the monodromy action in many examples. As one application, we verify a prediction of Mayr about the action of the monodromy about the Landau-Ginsburg locus of the quintic. Prompted by the result of this calculation, we propose a modification of the derived category which implements the physical requirement that the shift-by-6 functor should be the identity. Boundary Linear sigma-Models prove to be a very nice physical model of many of these derived category ideas, and we explain the correspondence between these two approaches 
  We present new non-Abelian solitonic configurations in the low energy effective theory describing a collection of N parallel D1--branes. These configurations preserve 1/4, 1/8, 1/16 and 1/32 of the spacetime supersymmetry. They are solutions to a set of generalised Nahm's equations which are related to self-duality equations in eight dimensions. Our solutions represent D1--branes which expand into fuzzy funnel configurations ending on collections of intersecting D3--branes. Supersymmetry dictates that such intersecting D3--branes must lie on a calibrated three-surface of spacetime and we argue that the generalised Nahm's equations encode the data for the construction of magnetic monopoles on the relevant three-surfaces. 
  We analyse different approaches to the description of the quantum field theory of a free massless (pseudo)scalar field defined in 1+1-dimensional space-time which describes the bosonized version of the massless Thirring model. These are (i) axiomatic quantum field theory, (ii) current algebra and (iii) path-integral. We show that the quantum field theory of a free massless (pseudo)scalar field defined on the class of Schwartz's test functions S_0(R^2) connects all these approaches. This quantum field theory is well-defined within the framework of Wightman's axioms and Wightman's positive definiteness condition. The physical meaning of the definition of Wightman's observables on the class of test functions from S_0(R^2) instead of S(R^2), as required by Wightman's axioms, is the irrelevance of the collective zero-mode related to the collective motion of the ``center of mass'' of the free massless (pseudo)scalar field, which can be deleted from the intermediate states of correlation functions (Eur. Phys. J. C 24, 653 (2002)). In such a theory the continuous symmetry, induced by shifts of the massless (pseudo)scalar field, is spontaneously broken and there is a non-vanishing spontaneous magnetization. The obtained results are discussed in connection with Coleman's theorem asserting the absence of Goldstone bosons and spontaneously broken continuous symmetry in quantum field theories defined in 1+1-dimensional space-time. 
  The simple consequences of the renormalization group invariance in calculations of the ground state energy for models of confined quantum fields are discussed. The case of (1+1)D MIT quark bag model is considered in detail. 
  We introduce the hypothesis that the matter content of the universe can be a product of the decay of primordial vector bosons.  The effect of the intensive cosmological creation of these primordial vector $W, ~Z $ bosons from the vacuum is studied in the framework of General Relativity and the Standard Model where the relative standard of measurement identifying conformal quantities with the measurable ones is accepted.  The relative standard leads to the conformal cosmology with the z-history of masses with the constant temperature, instead of the conventional z-history of the temperature with constant masses in inflationary cosmology.  In conformal cosmology both the latest supernova data and primordial nucleosynthesis are compatible with a stiff equation of state associated with one of the possible states of the infrared gravitation field.  The distribution function of the created bosons in the lowest order of perturbation theory exposes a cosmological singularity as a consequence of the theorem about the absence of the massless limit of massive vector fields in quantum theory. This singularity can be removed by taking into account the collision processes leading to a thermalization of the created particles. The cosmic microwave background (CMB) temperature T=(M_W^2H_0)^{1/3} ~ 2.7 K occurs as an integral of motion for the universe in the stiff state. We show that this temperature can be attained by the CMB radiation being the final product of the decay of primordial bosons.  The effect of anomalous nonconservation of baryon number due to the polarization of the Dirac sea vacuum by these primordial bosons is considered. 
  We study the holographic duals of type II and heterotic matrix string theories described by warped AdS_3 supergravities. By explicitly solving the linearized equations of motion around near horizon D-string geometries, we determine the spectrum of Kaluza-Klein primaries for type I, II supergravities on warped AdS_3 x S^7. The results match those coming from the dual two-dimensional gauge theories living on the D-string worldvolumes. We briefly discuss the connections with the N=(8,8), N=(8,0) orbifold superconformal field theories to which type IIB / heterotic matrix strings flow in the infrared. In particular, we associate the dimension (3/2, 3/2) twisted operator which brings the matrix string theories out from the conformal point (R^8)^N / S_N with the dilaton profile in the supergravity background.   The familiar dictionary between masses and scaling dimensions of fields and operators are modified by the presence of non-trivial warp factors and running dilatons. These modifications are worked out for the general case of domain wall / QFT correspondences between supergravities on warped AdS_{d+1} x S^q geometries and super Yang-Mills theories with 16 supercharges. 
  We consider non-planar contributions to the correlation functions of BMN operators in free N = 4 super Yang Mills theory. We recalculate these non-planar contributions from a different kind of diagram and find some exact agreements. The vertices of these diagrams are represented by free planar three point functions, thus our calculations provide some interesting identities for correlation functions of BMN operators in N = 4 super Yang Mills theory. These diagrams look very much like loop diagrams in a second quantized string field theory, thus these identities could possibly be interpreted as natural consequences of the pp-wave/CFT correspondence. 
  We consider the case of several scalar fields, charged under a number of U(1) factors, acquiring vacuum expectation values due to an anomalous U(1). We demonstrate how to make redefinitions at the superfield level in order to account for tree-level exchange of vector supermultiplets in the effective supergravity theory of the light fields in the supersymmetric vacuum phase. Our approach builds upon previous results that we obtained in a more elementary case. We find that the modular weights of light fields are typically shifted from their original values, allowing an interpretation in terms of the preservation of modular invariance in the effective theory. We address various subtleties in defining unitary gauge that are associated with the noncanonical Kahler potential of modular invariant supergravity, the vacuum degeneracy, and the role of the dilaton field. We discuss the effective superpotential for the light fields and note how proton decay operators may be obtained when the heavy fields are integrated out of the theory at the tree-level. We also address how our formalism may be extended to describe the generalized Green-Schwarz mechanism for multiple anomalous U(1)'s that occur in four-dimensional Type I and Type IIB string constructions. 
  In searching for the essence of special relativity, we have been gradually accumulating ten arguments focusing on one fundamental postulate based on quantum mechanics.A particle is always not pure. It always contain two contradictory fields, $\phi(\vec{x},t)$ and $\chi(\vec{x},t)$,which are coupled together with the symmetry $\phi(-\vec{x},-t)\longrightarrow\chi(\vec{x},t)$ and $\chi(-\vec{x},-t)\longrightarrow\phi(\vec{x},t)$. 
  Ghost condensates of dimension two in SU(N) Yang-Mills theory quantized in the Maximal Abelian Gauge are discussed. These condensates turn out to be related to the dynamical breaking of the SL(2,R) symmetry present in this gauge 
  We consider axially symmetric SU(2) Yang-Mills-Higgs (YMH) multimonopoles in Brans-Dicke theory for winding number n > 1. In analogy to the spherically symmetric n=1 solutions, we find that the axially symmetric solutions exist for higher values of the gravitational coupling than in the pure Einstein gravity case. For large values of the gravitational coupling, the solutions collapse to form a black hole which outside the horizon can be described by an extremal Reissner-Nordstrom solution. Similarly as in the pure Einstein gravity case, like-charged monopoles reside in an attractive phase in a limited domain of parameter space. However, we find that the strength of attraction is decreasing for decreasing Brans-Dicke parameter. 
  We consider probe p-branes dynamics in string theory backgrounds of general type. We use an action, which interpolates between Nambu-Goto and Polyakov type actions. This allows us to give a unified description for the tensile and tensionless branes. Firstly, we perform our analysis in the frequently used static gauge. Then, we obtain exact brane solutions in more general gauges. The same approach is used to study the Dirichlet p-brane dynamics and here exact solutions are also found. As an illustration, we apply our results to the brane world scenario in the framework of the mirage cosmology approach. 
  We consider the higher order gravity with dilaton and with the leading string theory corrections taken into account. The domain wall type solutions are investigated for arbitrary number of space-time dimensions. The explicit formulae for the fixed points and asymptotic behavior of generic solutions are given. We analyze and classify solutions with finite effective gravitational constant. There is a class of such solutions which have no singularities. We discuss in detail the relation between fine tuning and self tuning and clarify in which sense our solutions are fine-tuning free. The stability of such solutions is also discussed. 
  We show that B-model topological strings on local Calabi-Yau threefolds are large N duals of matrix models, which in the planar limit naturally give rise to special geometry. These matrix models directly compute F-terms in an associated N=1 supersymmetric gauge theory, obtained by deforming N=2 theories by a superpotential term that can be directly identified with the potential of the matrix model. Moreover by tuning some of the parameters of the geometry in a double scaling limit we recover (p,q) conformal minimal models coupled to 2d gravity, thereby relating non-critical string theories to type II superstrings on Calabi-Yau backgrounds. 
  We find a new subalgebra of the star product in the matter sector. Its elements are squeezed states whose matrices commute with (K_1)^2. This subalgebra contains a large set of projectors. The states are represented by their eigenvalues and we find a mapping between the eigenvalues representation and other known representations. The sliver is naturally in this subalgebra. Surprisingly, all the generalized butterfly states are also in this subalgebra, enabling us to analyze their spectrum, and to show the orthogonality of different butterfly states. This means that multi D-brane states can be built of butterfly states. 
  We show how allowing non-local terms in the field equations of symmetric tensors uncovers a neat geometry that naturally generalizes the Maxwell and Einstein cases. The end results can be related to multiple traces of the generalized Riemann curvatures R_{alpha_1 ... alpha_s; beta_1 > ... beta_s} introduced by de Wit and Freedman, divided by suitable powers of the D'Alembertian operator \Box. The conventional local equations can be recovered by a partial gauge fixing involving the trace of the gauge parameters Lambda_{alpha_1 ... alpha_{s-1}}, absent in the Fronsdal formulation. The same geometry underlies the fermionic equations, that, for all spins s+(1/2), can be linked via the operator (not hskip 1pt pr)/(\Box) to those of the spin-s bosons. 
  We perform a one-loop test of the holographic interpretation of the Karch-Randall model, whereby a massive graviton appears on an AdS_4 brane in an AdS_5 bulk. Within the AdS/CFT framework, we examine the quantum corrections to the graviton propagator on the brane, and demonstrate that they induce a graviton mass in exact agreement with the Karch-Randall result. Interestingly enough, at one loop order, the spin 0, spin 1/2 and spin 1 loops contribute to the dynamically generated (mass)^2 in the same 1: 3: 12 ratio as enters the Weyl anomaly and the 1/r^3 corrections to the Newtonian gravitational potential. 
  We found a gravity solution of p+1 dimensional extended object with SO(p)xSO(9-p) symmetry which has zero pressure and zero dilaton charge. We expect that this object is a residual tachyon dust after tachyon condensation of brane and anti-brane system discussed by Sen, recently. We also discuss the Hawking temperature and some properties of this object. 
  We construct in this paper a multi-brne model with Minkowski and AdS spacetime alternated in the fifth dimension and study the gravity on the brane of our universe. We will show that in this model there are both the graviton-like resonant states which generates the "quasi-localized gravity" and the resonant states of the massive KK mode. This model gives rise to gravity which deviates from the Newton gravitational potential at small and large scale which, however can be shown to be consistent with the observations in the absence of fine-tuning of the model parameters. 
  We study four-dimensional N=1 gauge theories arising on D3-branes probing toric singularities. Toric dualities and flows between theories corresponding to different singularities are analyzed by encoding the geometric information into (p,q) webs. A new method for identifying global symmetries of the four-dimensional theories using the brane webs is developed. Five-dimensional theories are associated to the theories on the D3-branes by using (p,q) webs. This leads to a novel interpretation of Seiberg duality, as crossing curves of marginal stability in five dimensions. 
  The Atiyah-Singer index theorem is investigated on various compact manifolds which admit finite matrix approximations (``fuzzy spaces'') with a view to applications in a modified Kaluza-Klein type approach in which the internal space consists of a finite number of points. Motivated by the chiral nature of the standard model spectrum we investigate manifolds that do not admit spinors but do admit $Spin^c$ structures. It is shown that, by twisting with appropriate bundles, one generation of the electroweak sector of the standard model, including a right-handed neutrino, can be obtained in this way from the complex projective space $\CP^2$. The unitary Grassmannian $U(5)/(U(3)\times U(2))$ yields a spectrum that contains the correct charges for the Fermions of the standard model, with varying multiplicities for the different particle states. 
  Effective string field equations with zero-constrained torsion have been studied extensively in the literature. But, one may think that the effects of vanishing of the non-metricity have not been explained in detail and also in according to some recent literature [4],[5], the action density in my previous paper [3] is not complete. For these reasons, in this erratum, in according to the effects of vanishing of the non-metricity to the field equations, the action density of my previous paper [3] will be completed in a variational setting. Furthermore, up to now, an unambiguous derivation of the dilaton potential has not been given. If one thinks that vanishing of the non-metricity gives a spontaneous breakdown of local Weyl invariance then a dilaton potential is obtained unambiguously in this framework. 
  The AdS/CFT correspondence is employed to derive logarithmic corrections to the Cardy-Verlinde formula when thermal fluctuations in the Anti-de Sitter black hole are accounted for. The qualitative effect of these corrections on the braneworld cosmology is investigated. The role of such terms in enabling a contracting universe to undergo a bounce is demonstrated. Their influence on the stability of black holes in AdS space and the Hawking-Page-Witten phase transitions is also discussed. 
  The nullification of threshold amplitudes is considered within the conventional framework of quantum field theory. The relevant Ward identities for the reduced theory are derived both on path-integral and diagrammatic levels. They are then used to prove the vanishing of tree-graph threshold amplitudes. 
  Large order asymptotic behaviour of renormalization constants in the minimal subtraction scheme for the $\phi ^4$ $(4-\epsilon)$ theory is discussed. Well-known results of the asymptotic $4-\epsilon $ expansion of critical indices are shown to be far from the large order asymptotic value. A {\em convergent} series for the model $\phi ^4$ $(4-\epsilon)$ is then considered. Radius of convergence of the series for Green functions and for renormalisation group functions is studied. The results of the convergent expansion of critical indices in the $4-\epsilon $ scheme are revalued using the knowledge of large order asymptotics. Specific features of this procedure are discussed. 
  Boundary S matrices for the boundary tricritical Ising field theory (TIM), both with and without supersymmetry, have previously been proposed. Here we provide support for these S matrices by showing that the corresponding boundary entropies are consistent with the expected boundary flows. We develop the fusion procedure for boundary RSOS models, with which we derive exact inversion identities for the TIM. We confirm the TBA description of nonsupersymmetric boundary flows of Lesage et al., and we obtain corresponding descriptions of supersymmetric boundary flows. 
  We discuss the generalization to global gauge anomalies of the familiar procedure for the cancellation of local gauge anomalies in effective theories of spontaneously broken symmetries. We illustrate this mechanism in a recently proposed six-dimensional extension of the standard model. 
  We discuss a general class of boundary conditions for bosons living in an extra spatial dimension compactified on S^1/Z_2. Discontinuities for both fields and their first derivatives are allowed at the orbifold fixed points. We analyze examples with free scalar fields and interacting gauge vector bosons, deriving the mass spectrum, that depends on a combination of the twist and the jumps. We discuss how the same physical system can be characterized by different boundary conditions, related by local field redefinitions that turn a twist into a jump or vice-versa. When the description is in term of discontinuous fields, appropriate lagrangian terms should be localized at the orbifold fixed points. 
  Recently the construction of the non-abelian effective D-brane action was performed through order $\alpha'{}^3$ including the terms quadratic in the gauginos. This result can be tested by calculating the spectrum in the presence of constant magnetic background fields and comparing it to the string theoretic predictions. This test was already performed for the purely bosonic terms. In this note we extend the test to the fermionic terms. We obtain perfect agreement. 
  The local neighborhood of a triple intersection of fivebranes in type IIA string theory is shown to be equivalent to type IIB string theory on a noncompact Calabi-Yau fourfold. The phases and the effective theory of the intersection are analyzed in detail. 
  We study gauge theories based on nonabelian 2-forms. Certain connections on loop space give rise to generalized covariant derivatives that include a nonabelian 2-form. This can be used to find rather straightforward expressions for the field strength and gauge transformations. As a special case we recover formulas for connections on nonabelian gerbe, as recently constructed in mathematics. The general construction gives rise to connections on algebra bundles, which might be relevant for D-branes in the presence of torsion. We construct BV sigma model actions for these connections and discuss their gauge fixing. We find both Yang-Mills type and topological BF type actions. 
  Given a relativistic two-point Green's function for a spinor system with spherical symmetry we show how to obtain another in the same class by extended point canonical transformations (XPCT). 
  zeta-function methods are used to study the properties of the non-relativistic interacting Bose gas at finite temperature and density. Results for the ground state energy and pressure are obtained at both zero and finite temperature. The method used does not restrict the form of the interaction, which can be completely general. A similar procedure is then applied to evaluate the ground state energy of a binary mixture consisting of different bosons. Analytical results are obtained in the case where the two species have the same mass. 
  A topological theory for euclidean gravity in eight dimensions is built by enforcing octonionic self-duality conditions on the spin connection. The eight-dimensional manifold must be of a special type, with G_2 or Spin(7) holonomy. The resulting theory is related to a twisted version of N=1, D=8 supergravity. The situation is comparable to that of the topological Yang--Mills theory in eight dimensions, for which the SO(8) invariance is broken down to Spin(7), but is recovered after untwisting the topological theory. 
  In field theory on a fibre bundle Y->X, an energy-momentum current is associated to a lift onto Y of a vector field on X. Such a lift by no means is unique, and contains a vertical part. It follows that: (i) there are a set of different energy-momentum currents, (ii) the Noether part of an energy-momentum current can not be taken away, (iii) if a Lagrangian is not gauge-invariant, the energy-momentum fails to be conserved. 
  In kappa-deformed relativistic framework we consider three different definitions of kappa-deformed velocities and introduce corresponding addition laws. We show that one of the velocities has classical relativistic addition law. The relation of velocity formulae with the coproduct for fourmomenta and noncommutative space-time structure is exhibited. 
  We compute the one-loop renormalization of the Planck mass for type II string theories compactified to four dimensions on symmetric orbifolds that preserve ${\cal N}=2$ supersymmetry. Depending on the orbifold, the effect can be as large as to compete with the standard tree-level value. 
  As shown by Freedman, Gubser, Pilch and Warner, the RG flow in ${\cal N}=4$ super-Yang-Mills theory broken to an ${\cal N}=1$ theory by the addition of a mass term can be described in terms of a supersymmetric domain wall solution in five-dimensional ${\cal N}=8$ gauged supergravity. The FGPW flow is an example of a holographic RG flow in a field theory on a flat background. Here we put the field theory studied by Freedman, Gubser, Pilch and Warner on a curved $AdS_4$ background, and we construct the supersymmetric domain wall solution which describes the RG flow in this field theory. This solution is a curved (non Ricci flat) domain wall solution. This example demonstrates that holographic RG flows in supersymmetric field theories on a curved $AdS_4$ background can be described in terms of curved supersymmetric domain wall solutions. 
  In the paper "Constraint Quantization of Open String in Background $B$ field and Noncommutative D-brane", it is claimed that the boundary conditions lead to an infinite set of secondary constraints and Dirac brackets result in a non-commutative Poisson structure for D-brain. Here we show that contrary to the arguments in that paper, the set of secondary constraints on the boundary is finite and the non-commutativity algebra can not be obtained by evaluating the Dirac brackets. 
  We study the two-loop F^6 interactions in SO(32) heterotic superstring theory in D=10. By using the generalized Riemann identity we are able to determine the single-trace part of the effective action up to a few constants which are related to certain scattering amplitudes. This two-loop heterotic result is related by duality to Type I interactions at the tree level. However, it turns out to be completely different from any sort of non-Abelian generalization of Born-Infeld theory. We offer an explanation of this discrepancy. 
  A popular way to study N=1 supersymmetric gauge theories is to realize them geometrically in string theory, as suspended brane constructions, D-branes wrapping cycles in Calabi-Yau manifolds, orbifolds, and otherwise. Among the applications of this idea are simple derivations and generalizations of Seiberg duality for the theories which can be so realized.   We abstract from these arguments the idea that Seiberg duality arises because a configuration of gauge theory can be realized as a bound state of a collection of branes in more than one way, and we show that different brane world-volume theories obtained this way have matching moduli spaces, the primary test of Seiberg duality.   Furthermore, we do this by defining ``brane'' and all the other ingredients of such arguments purely algebraically, for a very large class of N=1 quiver supersymmetric gauge theories, making physical intuitions about brane-antibrane systems and tachyon condensation precise using the tools of homological algebra.   These techniques allow us to compute the spectrum and superpotential of the dual theory from first principles, and to make contact with geometry and topological string theory when this is appropriate, but in general provide a more abstract notion of ``noncommutative geometry'' which is better suited to these problems. This makes contact with mathematical results in the representation theory of algebras; in this language, Seiberg duality is a tilting equivalence between the derived categories of the quiver algebras of the dual theories. 
  A perturbative quantum theory of the two Killing vector reduction of Einstein gravity is constructed. Although the reduced theory inherits from the full one the lack of standard perturbative renormalizability, we show that strict cutoff independence can be regained to all loop orders in a space of Lagrangians differing only by a field dependent conformal factor. A closed formula is obtained for the beta functional governing the flow of this conformal factor. The flow possesses a unique fixed point at which the trace anomaly is shown to vanish. The approach to the fixed point is compatible with Weinberg's ``asymptotic safety'' scenario. 
  The low energy effective theory for the Randall-Sundrum two brane system is investigated with an emphasis on the role of the non-linear radion in the brane world. The equations of motion in the bulk is solved using a low energy expansion method. This allows us, through the junction conditions, to deduce the effective equations of motion for the gravity on the brane. It is shown that the gravity on the brane world is described by a quasi-scalar-tensor theory with a specific coupling function omega(Psi) = 3 Psi / 2(1-Psi) on the positive tension brane and omega(Phi) = -3 Phi / 2(1+Phi) on the negative tension brane, where Psi and Phi are non-linear realizations of the radion on the positive and negative tension branes, respectively. In contrast to the usual scalar-tensor gravity, the quasi-scalar-tensor gravity couples with two kinds of matter, namely, the matters on both positive and negative tension branes, with different effective gravitational coupling constants. In particular, the radion disguised as the scalar fields Psi and Phi couples with the sum of the traces of the energy momentum tensor on both branes. In the course of the derivation, it has been revealed that the radion plays an essential role to convert the non-local Einstein gravity with the generalized dark radiation to the local quasi-scalar-tensor gravity. For completeness, we also derive the effective action for our theory by substituting the bulk solution into the original action. It is also shown that the quasi-scalar-tensor gravity works as holograms at the low energy in the sense that the bulk geometry can be reconstructed from the solution of the quasi-scalar-tensor gravity. 
  We show that it is possible to obtain the Gross-Neveu model in 1+1 dimensions from gauge fields only. This is reminiscent of the fact that in 1+1 dimensions the gauge field tensor is essentially a pseudo-scalar. We also show that it is possible in this context to combine the Gross-Neveu model with the massive Schwinger model in the limit where the fermion mass is larger than the electric charge. 
  A kind of doubly special relativity theory proposed by J. Magueijo and L. Smolin [Phys. Rev. Lett. 88, 190403 (2002)] is analysed. It is shown that this theory leads to serious physical difficulties in interpretation of kinematical quantities. Moreover, it is argued that statistical mechanics and thermodynamics cannot be resonably formulate within the model proposed in the mentioned paper. 
  The conjecture that M-theory has the rank eleven Kac-Moody symmetry e11 implies that Type IIA and Type IIB string theories in ten dimensions possess certain infinite dimensional perturbative symmetry algebras that we determine. This prediction is compared with the symmetry algebras that can be constructed in perturbative string theory, using the closed string analogues of the DDF operators. Within the limitations of this construction close agreement is found. We also perform the analogous analysis for the case of the closed bosonic string. 
  Recent studies of the ultraviolet behaviour of pure gravity suggest that it admits a non-Gaussian attractive fixed point, and therefore that the theory is asymptotically safe. We consider the effect on this fixed point of massless minimally coupled matter fields. The existence of a UV attractive fixed point puts bounds on the type and number of such fields. 
  We study the spectrum of the recently proposed matrix model of DLCQ M-theory in a parallel plane (pp)-wave background. In contrast to matrix theory in a flat background this model contains mass terms, which lift the flat directions of the potential and renders its spectrum discrete. The supersymmetry algebra of the model groups the energy eigenstates into supermultiplets, whose members differ by fixed amounts of energy in great similarity to the representation of supersymmetry in AdS spaces. There is a unique and exact zero-energy groundstate along with a multitude of long and short multiplets of excited states. For large masses the quantum mechanical model may be treated perturbatively and we study the leading order energy shifts of the first excited states up to level two. Most interestingly we uncover a protected short multiplet at level two, whose energies do not receive perturbative corrections. Moreover, we conjecture the existence of an infinite series of similar protected multiplets in the pp-wave matrix model. 
  We compute the radiation pressure force on a moving mirror, in the nonrelativistic approximation, assuming the field to be at temperature $T.$ At high temperature, the force has a dissipative component proportional to the mirror velocity, which results from Doppler shift of the reflected thermal photons. In the case of a scalar field, the force has also a dispersive component associated to a mass correction. In the electromagnetic case, the separate contributions to the mass correction from the two polarizations cancel. We also derive explicit results in the low temperature regime, and present numerical results for the general case. As an application, we compute the dissipation and decoherence rates for a mirror in a harmonic potential well. 
  One of the few schemes for obtaining an integrable nonultralocal quantum model is its possible generation from an ultralocal model by a suitable gauge transformation. Applying this scheme we discover two new nonultralocal models, which fit well into the braided Yang-Baxter relations ensuring their quantum integrability. Our first model is generated from a lattice Liouville-like system, while the second one which is an exact lattice version of the light-cone sine-Gordon is gauge transformed from a model, which gives also the quantum mKdV for a different gauge choice. 
  We study type IIA superstring theory on a PP-wave background with 24 supercharges. This model can exactly be solved and then quantized. The open string in this PP-wave background is also studied. We observe that the theory has supersymmetric Dp-branes for p=2,4,6,8. 
  The contributions to the heat kernel coefficients generated by the corners of the boundary are studied. For this purpose the internal and external sectors of a wedge and a cone are considered. These sectors are obtained by introducing, inside the wedge, a cylindrical boundary. Transition to a cone is accomplished by identification of the wedge sides. The basic result of the paper is the calculation of the individual contributions to the heat kernel coefficients generated by the boundary singularities. In the course of this analysis certain patterns, that are followed by these contributions, are revealed. The implications of the obtained results in calculations of the vacuum energy for regions with nonsmooth boundary are discussed. The rules for obtaining all the heat kernel coefficients for the minus Laplace operator defined on a polygon or in its cylindrical generalization are formulated. 
  We consider the Maldacena-Nunez supergravity solution corresponding to N=1 super Yang-Mills within the approach by Di Vecchia, Lerda and Merlatti. We show that if one uses the radial distance as a field theory scale, the corresponding beta function has an infrared fixed point. Assuming this to be a physical property for all four dimensional non-singular renormalization schemes, we use the relation between the gaugino condensate and its dual to investigate the connection between the IR and UV behaviors. Imposing the ``field theory boundary condition'' that the first two terms in the perturbative UV beta function are universal, the fixed point is found to be of first order, and the slope of the IR beta function is also fixed. 
  We show that the N-particle A_{N-1} and B_N rational Calogero models without the harmonic interaction admit a new class of bound and scattering states. These states owe their existence to the self-adjoint extensions of the corresponding Hamiltonians, labelled by a real parameter z. It is shown that the new states appear for all values of N and for specific ranges of the coupling constants. Moreover, they are shown to exist even in the excited sectors of the Calogero models. The self-adjoint extension generically breaks the classical scaling symmetry, leading to quantum mechanical scaling anomaly. The scaling symmetry can however be restored for certain values of the parameter z. We also generalize these results for many particle systems with classically scale invariant long range interactions in arbitrary dimensions. 
  We quantise canonical free-field zero modes $p$, $q$ on a half-plane $p>0$ both, for the Liouville field theory and its reduced Liouville particle dynamics. We describe the particle dynamics in detail, calculate one-point functions of particle vertex operators, deduce their zero mode realisation on the half-plane, and prove that the particle vertex operators act self-adjointly on a Hilbert space $L^2(\rr_+)$ on account of symmetries generated by the $S$-matrix. Similarly, self-adjointness of the corresponding vertex operator of Liouville field theory in the zero mode sector is obtained by applying the Liouville reflection amplitude, which is derived by the operator method. 
  In the framework of brane models the postulates of special relativity theory is revised. It is assumed that there exists preferred frame and relativity principle is violated on the brane. Because of trapping any moving object on the brane is really accelerated and the formulas for gravitational contraction of the intervals (containing the escape speed) appears to be equivalent to ordinary Lorentz ones. 
  We have studied a modified Yang-Mills-Higgs system coupled to Einstein gravity. The modification of the Einstein-Hilbert action involves a direct coupling of the Higgs field to the scalar curvature. In this modified system we are able to write a Bogomol'nyi type condition in curved space and demonstrate that the positive static energy functional is bounded from below. We then investigate non-Abelian sperically symmetric static solutions in a similar fashion to the `t Hooft-Polyakov monopole. After reviewing previously studied monopole solutions of this type, we extend the formalism to included electric charge and we present dyon solutions. 
  We define a sequence of VSFT D-branes whose low energy limit leads exactly to a corresponding sequence of GMS solitons. The D-branes are defined by acting on a fixed VSFT lump with operators defined by means of Laguerre polynomials whose argument is quadratic in the string creation operators. The states obtained in this way form an algebra under the SFT star product, which is isomorphic to a corresponding algebra of GMS solitons under the Moyal product. In order to obtain a regularized field theory limit we embed the theory in a constant background B field. 
  In this paper we consider the presence of the Wu-Yang magnetic monopole in the global monopole spacetime and their influence on the vacuum polarization effects around these two monopoles placed together. According to Wu-Yang [Nucl. Phys. {\bf B107}, 365 (1976)] the solution of the Klein-Gordon equation in such an external field will not be an ordinary function but, instead, {\it section}. Because of the peculiar radial symmetry of the global monopole spacetime, it is possible to cover its space section by two overlapping regions, needed to define the singularity free vector potential, and to study the quantum effects due to a charged scalar field in this system. In order to develop this analysis we construct the explicit Euclidean scalar Green {\it section} associated with a charged massless field in a global monopole spacetime in the presence of the Abelian Wu-Yang magnetic monopole. Having this Green section it is possible to study the vacuum polarization effects. We explicitly calculate the renormalized vacuum expectation value $<\Phi(x)^*\Phi(x) >_{Ren.}$, associated with a charged scalar field operator and the respective energy-momentum tensor,   $<T_{\mu\nu}(x)>_{Ren.}$, which are expressed in terms of the parameter which codify the presence of the global and magnetic monopoles. 
  On the occasion of this ArkadyFest, celebrating Arkady Vainshtein's 60th birthday, I review some selected aspects of the connection between perturbative and nonperturbative physics, a subject to which Arkady has made many important contributions. I first review this connection in quantum mechanics, which was the subject of Arkady's very first paper. Then I discuss this issue in relation to effective actions in field theory, which also touches on Arkady's work on operator product expansions. Finally, I conclude with a discussion of a special quantum mechanical system, a quasi-exactly solvable model with energy-reflection duality, which exhibits an explicit duality between the perturbative and nonperturbative sectors, without invoking supersymmetry. 
  Dynamical symmetries of Born-Infeld theory associated with its maximal field strength are encoded in a geometry on the tangent bundle of spacetime manifolds. The resulting extension of general relativity respecting a finite upper bound on accelerations is put to use in the discussion of particle dynamics, first quantization, and the derivation of a correction to the Thomas precession. 
  The coproduct of a Feynman diagram is set up through identifying the perturbative unitarity of the S-matrix with the cutting equation from the cutting rules. On the one hand, it includes all partitions of the vertex set of the Feynman diagram and leads to the circling rules for the largest time equation. Its antipode is the conjugation of the Feynman diagram. On the other hand, it is regarded as the integration of incoming and outgoing particles over the on-shell momentum space. This causes the cutting rules for the cutting equation. Its antipode is an advanced function vanishing in retarded regions. Both types of coproduct are well-defined for a renormalized Feynman diagram since they are compatible with the Connes--Kreimer Hopf algebra. 
  We analyze the effect of heavy fundamentally charged particles on the finite temperature deconfining phase transition in the 2+1 dimensional Georgi-Glashow model. We show that in the presence of fundamental matter the transition turns into a crossover. The near critical theory is mapped onto the 2 dimensional Ising model in an external magnetic field. Using this mapping we determine the width of the crossover region as well as the specific heat as a function of the fundamental mass. 
  We show that the symmetry algebra governing the interacting part of the matrix model for M-theory on the maximally supersymmetric pp-wave is the basic classical Lie superalgebra SU(4|2). We determine the SU(4|2) multiplets present in the exact spectrum in the limit where \mu (the mass parameter) becomes infinite, and find that these include infinitely many BPS multiplets. Using the representation theory of SU(4|2), we demonstrate that some of these BPS multiplets, including all of the vacuum states of the matrix model plus certain infinite towers of excited states, have energies which are exactly protected non-perturbatively for any value of \mu > 0. In the large N limit, these lead to exact quantum states of M-theory on the pp-wave. We also show explicitly that there are certain BPS multiplets which do receive energy corrections by combining with other BPS multiplets to form ordinary multiplets. 
  We show that the anomalous contribution to the central charge of the 1+1-dimensional N=1 supersymmetric kink that is required for BPS saturation at the quantum level can be linked to an analogous term in the extra momentum operator of a 2+1-dimensional kink domain wall with spontaneous parity violation and chiral domain wall fermions. In the quantization of the domain wall, BPS saturation is preserved by nonvanishing quantum corrections to the momentum density in the extra space dimension. Dimensional reduction from 2+1 to 1+1 dimensions preserves the unbroken N=1/2 supersymmetry and turns these parity-violating contributions into the anomaly of the central charge of the supersymmetric kink. On the other hand, standard dimensional regularization by dimensional reduction from 1 to (1-epsilon) spatial dimensions, which also preserves supersymmetry, obtains the anomaly from an evanescent counterterm. 
  A study of the relation between topology change, energy and Lie algebra representations for fuzzy geometry in connection to $M$-theory is presented. We encounter two different types of topology change, related to the different features of the Lie algebra representations appearing in the matrix models of $M$-theory. From these studies, we propose a new method of obtaining non-commutative solutions for the non-Abelian $D$-brane action found by Myers. This mechanism excludes one of the two topology changing processes previously found in other non-commutative solutions of many matrix-based models in $M$-theory i.e. in M(atrix) theory, Matrix string theory and non-Abelian $D$-brane physics. 
  We construct a positive constant curvature space by identifying some points along a Killing vector in a de Sitter Space. This space is the counterpart of the three-dimensional Schwarzschild-de Sitter solution in higher dimensions. This space has a cosmological event horizon, and is of the topology ${\cal M}_{D-1}\times S^1$, where ${\cal M}_{D-1}$ denotes a $(D-1)$-dimensional conformal Minkowski spacetime. 
  We propose a generalization of cubic matrix mechanics by introducing a canonical triplet and study its relation to Nambu mechanics. The generalized cubic matrix mechanics we consider can be interpreted as a 'quantum' generalization of Nambu mechanics. 
  We study the action of the $SL(2;R)$ group on the noncommutative DBI Lagrangian. The symmetry conditions of this theory under the above group will be obtained. These conditions determine the extra U(1) gauge field. By introducing some consistent relations we observe that the noncommutative (or ordinary) DBI Lagrangian and its $SL(2;R)$ dual theory are dual of each other. Therefore, we find some $SL(2;R)$ invariant equations. In this case the noncommutativity parameter, its $T$-dual and its $SL(2;R)$ dual versions are expressed in terms of each other. Furthermore, we show that on the effective variables, $T$-duality and $SL(2;R)$ duality do not commute. We also study the effects of the $SL(2;R)$ group on the noncommutative Chern-Simons action. 
  We look at vacuum solutions for fields confined in cavities where the boundary conditions can rule out constant field configurations, other than the zero field. If the zero field is unstable, symmetry breaking can occur to a field configuration of lower energy which is not constant. The stability of the zero field is determined by the size of the length scales which characterize the cavity and parameters that enter the scalar field potential. There can be a critical length at which an instability of the zero field sets in. In addition to looking at the rectangular and spherical cavity in detail, we describe a general method which can be used to find approximate analytical solutions when the length scales of the cavity are close to the critical value. 
  We propose a canonical description of the dynamics of quantum systems on a class of Robertson-Walker space-times. We show that the worldline of an observer in such space-times determines a unique orbit in the local conformal group SO(4,1) of the space-time and that this orbit determines a unique transport on the space-time. For a quantum system on the space-time modeled by a net of local algebras, the associated dynamics is expressed via a suitable family of ``propagators''. In the best of situations, this dynamics is covariant, but more typically the dynamics will be ``quasi-covariant'' in a sense we make precise. We then show by using our technique of ``transplanting'' states and nets of local algebras from de Sitter space to Robertson-Walker space that there exist quantum systems on Robertson-Walker spaces with quasi-covariant dynamics. The transplanted state is locally passive, in an appropriate sense, with respect to this dynamics. 
  New features are described for models with multi-particle area-dependent potentials, in any number of dimensions. The corresponding many-body field theories are investigated for classical configurations. Some explicit solutions are given, and some conjectures are made about chaos in such field theories. 
  The symmetry structure of axion-dilaton quantum string cosmology is investigated. The invariance of the string effective action under S-duality group, SU(1,1), facilitates solution of Wheeler-De Witt equation from group theoretic considerations; revealing existence of a new class of wave functions. We discover the an underlying ${W}$-infinity algebra in this formulation. 
  The consequences for the brane cosmological evolution of energy exchange between the brane and the bulk are analysed in detail, in the context of a non-factorizable background geometry with vanishing effective cosmological constant on the brane. A rich variety of brane cosmologies is obtained, depending on the precise mechanism of energy transfer, the equation of state of brane-matter and the spatial topology. An accelerating era is generically a feature of our solutions. In the case of low-density flat universe more dark matter than in the conventional FRW picture is predicted. Spatially compact solutions are found to delay their recollapse. 
  We study the superalgebra of the M-theory on a fully supersymmetric pp-wave. We identify the algebra as the special unitary Lie superalgebra, su(2|4;2,0) or su(2|4;2,4), and analyze its root structure. We discuss the typical and atypical representations deriving the typicality condition explicitly in terms of the energy and other four quantum numbers. We classify the BPS multiplets preserving 4,8,12,16 real supercharges and obtain the corresponding spectrum. We show that in the BPS multiplet either the lowest energy floor is an su(2) singlet or the highest energy floor is an su(4) singlet. 
  A very general calculational strategy is applied to the evaluation of the divergent physical amplitudes which are typical of perturbative calculations. With this approach in the final results all the intrinsic arbitrariness of the calculations due to the divergent character is still present. We show that by using the symmetry properties as a guide to search for the (compulsory) choices in such a way as to avoid ambiguities, a deep and clear understanding of the role of regularization methods emerges. Requiring then an universal point of view for the problem, as allowed by our approach, very interesting conclusions can be stated about the possible justifications of most intriguing aspect of the perturbative calculations in quantum field theory: the triangle anomalies. 
  In this paper, we discuss the self-consistency condition for the spherical symmetric Klein-Gordon equation, and then discuss a natural possibility that gravity and weak coupling constants g_G and g_W may be defined after g_{EM}. In this point of view, gravity and the weak force are subsidiary derived from electricity. Particularly, SU(2)_L * U(1) unification is derived without assuming a phase transition.  A possible origin of the Higgs mechanism is proposed. Each particle pair of the standard model is associated with the corresponding asymptotic expansion of an eigen function. 
  The low energy dynamics of degenerated BPS domain walls arising in a generalized Wess-Zumino model is described as geodesic motion in the space of these topological walls. 
  In this work we show that the Universe evolving in a spacetime with torsion (originated from a second rank antisymmetric Kalb-Ramond field) and dilaton is free from any big bang singularity and can have acceleration during the evolution. Both the matter and radiation dominated era have been considered and the role of the dilaton to explain the decelerating phase in the earlier epoch has also been discussed. 
  Fractional charges, and in particular the spectral asymmetry eta of certain Dirac operators, can appear in the central charge of supersymmetric field theories. This yields unexpected analyticity constraints on eta from which classic results can be recovered in an elegant way. The method could also be applied in the context of string theory. 
  We consider the radial quantization of N=4 super Yang-Mills (SYM) in 4 dimensions, i.e., N=4 SYM on a cylinder R times S^3. We construct the generators of superconformal symmetry in the case of U(N) gauge group, generalizing the earlier work by Nicolai et al. for U(1) gauge group. We study how these generators contract to the symmetry of pp-wave when they act on a state with large R-charge. 
  Recently, mirror symmetry is derived as T-duality applied to gauge systems that flow to non-linear sigma models. We present some of its applications to study quantum geometry involving D-branes. In particular, we show that one can employ D-branes wrapped on torus fibers to reproduce the mirror duality itself, realizing the program of Strominger-Yau-Zaslow in a slightly different context. Floer theory of intersecting Lagrangians plays an essential role. 
  A technique for avoiding infinite integrals in the calculation of the one-loop diagram contribution to the vacuum polarization component of an atomic energy level is presented. This makes renormalization unnecessary. Infinite integrals do not occur because, as it is shown, no delta functions are required for the Green's functions. Thus there are none to overlap. This procedure is shown to produce the same formula as the one obtained by dimensional renormalization. 
  The supersymmetry constraints on the $\hat{G}^{4}\lamda^{16}$ term in the effective action of type IIB superstring theory are studied in order to determine the dependence of its coefficient on the complex scalar field, $\tau$. The resulting expression is consistent with the $SL(2,{\Bbb Z})$ invariant conjectures in the literature. 
  We find a one to one mapping between low energy string dilaton states in AdS bulk and high energy glueball states on the corresponding boundary. This holographic mapping leads to a relation between bulk and boundary scattering amplitudes. From this relation and the dilaton action we find the appropriate momentum scaling for high energy QCD amplitudes at fixed angles. 
  We show that the Bekenstein-Hawking entropy associated with any black hole undergoes logarithmic corrections when small thermodynamic fluctuations around equilibrium are taken into account. Thus, the corrected expression for black hole entropy is given by $S= A/4 - k \ln(A)$, where $A$ is the horizon area and $k$ is a constant which depends on the specific black hole. We apply our result to BTZ black hole, for which $k=3/2$, as found earlier, as well as to anti-de Sitter-Schwarzschild and Reissner-Nordstrom black hole in arbitrary spacetime dimensions. Finally, we examine the role of conformal field theory in black hole entropy and its corrections. 
  The vacuum expectation values of the energy--momentum tensor are investigated for massless scalar fields satisfying Dicichlet or Neumann boundary conditions, and for the electromagnetic field with perfect conductor boundary conditions on two infinite parallel plates moving by uniform proper acceleration through the Fulling--Rindler vacuum. The scalar case is considered for general values of the curvature coupling parameter and in an arbitrary number of spacetime dimension. The mode--summation method is used with combination of a variant of the generalized Abel--Plana formula. This allows to extract manifestly the contributions to the expectation values due to a single boundary. The vacuum forces acting on the boundaries are presented as a sum of the self--action and interaction terms. The first one contains well known surface divergences and needs a further regularization. The interaction forces between the plates are always attractive for both scalar and electromagnetic cases. An application to the 'Rindler wall' is discussed. 
  We study vortex-creating, or monopole, operators in 3d CFTs which are the infrared limit of N=2 and N=4 supersymmetric QEDs in three dimensions. Using large-Nf expansion, we construct monopole operators which are primaries of short representations of the superconformal algebra. Mirror symmetry in three dimensions makes a number of predictions about such operators, and our results confirm these predictions. Furthermore, we argue that some of our large-Nf results are exact. This implies, in particular, that certain monopole operators in N=4 d=3 SQED with Nf=1 are free fields. This amounts to a proof of 3d mirror symmetry in a special case. 
  Vortex dynamics in a thin superfluid ${}^4$He film as well as in a type II superconductor is described by the classical counterpart of the model advocated by Peierls, and used for deriving the ground states of the Fractional Quantum Hall Effect. The model has non-commuting coordinates, and is obtained by reduction from a particle associated with the ``exotic'' extension of the planar Galilei group. 
  We deform the AdS/CFT Correspondence by the inclusion of a non-supersymmetric scalar mass operator. We discuss the behaviour of the dual 5 dimensional supergravity field then lift the full solution to 10 dimensions. Brane probing the resulting background reveals a potential consistent with the operator we wished to insert. 
  We study the classification of D-branes in all compact Lie groups including non-simply-laced ones. We also discuss the global structure of the quantum moduli space of the D-branes. D-branes are classified according to their positions in the maximal torus. We describe rank 2 cases, namely $B_2$, $C_2$, $G_2$, explicitly and construct all the D-branes in $B_r$, $C_r$, $F_4$ by the method of iterative deletion in the Dynkin diagram. The discussion of moduli space involves global issues that can be treated in terms of the exact homotopy sequence and various lattices. We also show that singular D-branes can exist at quantum mechanical level. 
  It is shown that the quarks and leptons of the standard model, including a right-handed neutrino, can be obtained by gauging the holonomy groups of complex projective spaces of complex dimensions two and three. The spectrum emerges as chiral zero modes of the Dirac operator coupled to gauge fields and the demonstration involves an index theorem analysis on a general complex projective space in the presence of topologically non-trivial SU(n)xU(1) gauge fields. The construction may have applications in type IIA string theory and non-commutative geometry. 
  An asymptotic theory is developed for general non-integrable boundary quantum field theory in 1+1 dimensions based on the Langrangean description. Reflection matrices are defined to connect asymptotic states and are shown to be related to the Green functions via the boundary reduction formula derived. The definition of the $R$-matrix for integrable theories due to Ghoshal and Zamolodchikov and the one used in the perturbative approaches are shown to be related. 
  We construct with a geometric procedure the supersymmetry transformation laws and Lagrangian for all the ``variant'' D=11 and D=10 Type IIA supergravities. We identify into our classification the D=11 and D=10 Type IIA ``variant'' theories first introduced by Hull performing T-duality transformation on both spacelike and timelike circles. We find in addition a set of D=10 Type IIA ``variant'' supergravities that can not be obtained trivially from eleven dimensions compactifying on a circle. 
  We consider deterministic chaotic models of vacuum fluctuations on a small (quantum gravity) scale. As a suitable small-scale dynamics, nonlinear versions of strings, so-called `chaotic strings' are introduced. These can be used to provide the `noise' for second quantization of ordinary strings via the Parisi- Wu approach of stochastic quantization. Extensive numerical evidence is presented that the vacuum energy of chaotic strings is minimized for the numerical values of the observed standard model parameters, i.e. in this extended approach to second quantization concrete predictions for vacuum expectations of dilaton-like fields and hence on masses and coupling constants can be given. Low-energy fermion and boson masses are correctly obtained with a precision of 3-4 digits, the electroweak and strong coupling strengths with a precision of 4-5 digits. In particular, the minima of the vacuum energy yield high-precision predictions of the Higgs mass (154 GeV), of the neutrino masses (1.45E-5 eV, 2.57E-3 eV, 4.92E-2 eV) and of the GUT scale (1.73E16 GeV). 
  A deformation of the canonical algebra for kinematical observables of the quantum field theory in Minkowski space-time has been considered under the condition of Lorentz invariance. A relativistic invariant algebra obtained depends on additional fundamental constants M, L and H with the dimensions of mass, length and action, respectively. In some limiting cases the algebra goes over into the well-known Snyder or Yang algebras. In general case the algebra represents a class of Lie algebras, that consists of simple algebras and semidirect sums of simple algebras and integrable ones. Some algebras belonging to this class are noninvariant under T and C transformations. Possible applications of obtained algebras for descriptions of states of matter under extreme conditions are briefly discussed. 
  To better understand the possible breakdown of locality in quantum gravitational systems, we pursue the identity of precursors in the context of AdS/CFT. Holography implies a breakdown of standard bulk locality which we expect to occur only at extremely high energy. We consider precursors that encode bulk information causally disconnected from the boundary and whose measurement involves nonlocal bulk processes. We construct a toy model of holography which encapsulates the expected properties of precursors and compare it with previous such discussions. If these precursors can be identified in the gauge theory, they are almost certainly Wilson loops, perhaps with decorations, but the relevant information is encoded in the high-energy sector of the theory and should not be observable by low energy measurements. This would be in accord with the locality bound, which serves as a criterion for situations where breakdown of bulk locality is expected. 
  In a Randall-Sundrum theory (RS1) 3+1 dimensional black holes and higher dimensional black holes are not the natural continuations of each other. 3+1 dimensional black holes decay into a large number of 4+1 dimensional black holes at a critical mass, $M_{\rm crit}\sim 10^{32}$ TeV. Those black holes themselves may become unstable above another, albeit much smaller critical mass, $M_0\sim 10^3$TeV. 
  Even if the electromagnetic field does not create pairs, virtual pairs lead to the appearance of a phase in vacuum-vacuum amplitude. This makes it necessary to distinguish the in- and out-solutions even when it is commonly assumed that there is only one complete set of solutions as, for example, in the case of a constant magnetic field. Then in- and out-solutions differ only by a phase factor which is in essence the Bogoliubov coefficient. The propagator in terms of in- and out-states takes the same form as the one for pair creating fields. The transition amplitude for an electron to go from an initial in-state to out-state is equal to unity (in diagonal representation). This is in agreement with Pauli principal: if in the field there is an electron with given (conserved) set of quantum numbers, virtual pair cannot appear in this state. So even the phase of transition amplitude remains unaffected by the field. We show how one may redefine the phases of Bogoliubov coefficients in order to express the vacuum-vacuum amplitude through them. 
  We study the theory of noncommutative U(N) Yang-Mills field interacting with scalar and spinor fields in the fundamental and the adjoint representations. We include in the action both the terms describing interaction between the gauge and the matter fields and the terms which describe interaction among the matter fields only. Some of these interaction terms have not been considered previously in the context of noncommutative field theory. We find all counterterms for the theory to be finite in the one-loop approximation. It is shown that these counterterms allow to absorb all the divergencies by renormalization of the fields and the coupling constants, so the theory turns out to be multiplicatively renormalizable. In case of 1PI gauge field functions the result may easily be generalized on an arbitrary number of the matter fields. To generalize the results for the other 1PI functions it is necessary for the matter coupling constants to be adapted in the proper way. In some simple cases this generalization for a part of these 1PI functions is considered. 
  This article illustrates the bound states of Kemmer equation for spin-1 particles. The asymptotic, exact and Coulomb field solutions are obtained by using action principle. In the conclusion the energy spectrum of spin-1 particles moving in a Coulomb potential compared with the energy spectrum of spin-0 and spin-1/2 particles. 
  This article illustrates a completely algebraic method to obtain the energy levels of a massive spin-1 particle moving in a constant magnetic field. In the process to obtain the energy levels the wave function was written by harmonic oscillator solutions. 
  Spacelike branes are new time-dependent systems to explore and it has been observed that related supergravity solutions can be obtained by analytically continuing known D-brane solutions. Here we show that analytic continuation of known solutions of the Dirac-Born-Infeld equations also lead to interesting analogs of time dependent gravity solutions. Properties of these new solutions, which are similar to the Witten bubble of nothing and S-branes, are discussed. We comment on how these new bubble solutions seem relevant to the tachyon condensation process of non-BPS branes, and remark on their application to cosmological scenarios. Unstable brane configurations which resemble S-brane type solutions are also discussed. 
  We propose a resolution to the black-hole information-loss paradox: in one formulation of physical theory, information is preserved and macroscopic causality is violated; in another, causality is preserved and pure states evolve to mixed states. However, no experiments can be performed that would distinguish these two descriptions. We explain how this could work in practice; a key ingredient is the suggested quantum-chaotic nature of black holes. 
  We construct the supergravity solution in 11 dimensions describing D6-branes wrapped around a Kahler four-cycle with a B-field along the flat directions of the brane. The configuration is dual to an N=2 noncommutative gauge theory in 2+1 dimensions. We also construct the four associated independent Killing spinors. The phenomenon of supersymmetry without supersymmetry appears naturally when compactifying to type IIA or 8d gauged supergravity. Therefore, this solution also provides an 11d background with four supercharges and four-form flux, which is not obtainable from 8d gauged supergravity. 
  String representation of the $[U(1)]^{N-1}$ gauge-invariant dual Abelian-Higgs-type theory, which is relevant to the SU(N)-QCD with the $\Theta$-term and provides confinement of quarks, is derived. The N-dependence of the Higgs vacuum expectation value is found, at which the tension of the string joining quarks becomes N-independent, similarly to the real QCD. Contrary to that, the inverse coupling constant of the rigidity term of this string always behaves approximately as 1/N. A long-range Aharonov-Bohm-type interaction of a dyon (i.e., a quark which acquired a magnetic charge due to the $\Theta$-term) with a closed electric string becomes nontrivial at $\Theta$ not equal to $N\pi$ times an integer. On the contrary, at these critical values of $\Theta$, the scattering of dyons over strings is absent. 
  We show that a partition function on the not-flat D1-brane can be written in the same form as that on the flat one in $\alpha^\prime$-order. In this case the information of the curvature of the brane configuration is included in tachyon beta function. 
  I review some recent progress in understanding the properties of AdS2 branes in AdS3. Different methods - classical string motion, Born-Infeld dynamics, boundary states - are evocated and compared. 
  Applied to the electroweak interactions, the theory of Lie algebra extensions suggests a mechanism by which the boson masses are generated without resource to spontaneous symmetry breaking. It starts from a gauge theory without any additional scalar field. All the couplings predicted by the Weinberg-Salam theory are present, and a few others which are nevertheless consistent within the model. 
  The universal perturbative invariants of rational homology spheres can be extracted from the Chern-Simons partition function by combining perturbative and nonperturbative results. We spell out the general procedure to compute these invariants, and we work out in detail the case of Seifert spaces. By extending some previous results of Lawrence and Rozansky, the Chern-Simons partition function with arbitrary simply-laced group for these spaces is written in terms of matrix integrals. The analysis of the perturbative expansion amounts to the evaluation of averages in a Gaussian ensemble of random matrices. As a result, explicit expressions for the universal perturbative invariants of Seifert homology spheres up to order five are presented. 
  It is known that the physics of open strings on a D2-brane on a two-torus is realized from the viewpoint of deformation quantization in the Seiberg-Witten limit. We study its T-dual theory, i.e. D1-brane physics on two-tori. Such theory is described by Kronecker foliation. The algebra of open strings on the D1-brane is then identified with the crossed product representation of a noncommutative two-torus. The Morita equivalence of noncommutative two-tori is also realized geometrically along this line. As an application, Heisenberg modules and the tensor product between them are discussed from these geometric viewpoints. We show they are related to the homological mirror symmetry of two-tori. 
  The loop variable approach is a proposal for a gauge invariant generalization of the sigma-model renormalization group method of obtaining equations of motion in string theory. The basic guiding principle is space-time gauge invariance rather than world-sheet properties. In essence it is a version of Wilson's exact renormalization group equation for the world sheet theory. It involves all the massive modes and is defined with a finite world-sheet cutoff, which allows one to go off the mass-shell. On shell the tree amplitudes of string theory are reproduced. The equations are gauge invariant off shell also. This article is a self-contained discussion of the loop variable approach as well as its connection with the renormalization group. 
  We consider N=1 supersymmetric sine-Gordon theory (SSG) with supersymmetric integrable boundary conditions (boundary SSG = BSSG). We find two possible ways to close the boundary bootstrap for this model, corresponding to two different choices for the boundary supercharge. We argue that these two bootstrap solutions should correspond to the two integrable Lagrangian boundary theories considered recently by Nepomechie. 
  The best-known way of stabilizing textures is by Skyrme-like terms, but another possibility is to use gauge fields. The semilocal vortex may be viewed as an example of this, in two spatial dimensions. In three dimensions, however, the idea (in its simplest form) does not work -- the link between the gauge field and the scalar field is not strong enough to prevent the texture from collapsing. Modifying the |D Phi|^2 term in the Lagrangian (essentially by changing the metric on the Phi-space) can strengthen this link, and lead to stability. Furthermore, there is a limit in which the gauge field is entirely determined in terms of the scalar field, and the system reduces to a pure Skyrme-like one. This is described for gauge group U(1), in dimensions two and three. The non-abelian version is discussed briefly, but as yet no examples of texture stabilization are known in this case. 
  We study N=1,2,4 higher spin superalgebras in four dimensions and higher spin gauge theories based on them. We extend the existing minimal N=2,4 theories and find a minimal N=1 theory. Utilizing the basic structure of the minimal N=8 theory, we express the full field equations for the N=1,2,4 theories in a universal form without introducing Kleinian operators. We also use a non-minimal N=4 higher spin algebra tensored with U(3) to describe a higher spin extension of N=4 supergravity coupled to the massless vector multiplets arising in the KK spectrum of 11D supergravity on the N=3 supersymmetric AdS_4 x N^{010} background. The higher spin theory also contains a triplet of vector multiplets which may play a role in the super-Higgs effect in which N=4 is broken down to N=3. 
  We emphasize the necessity of a delicate interplay between the gauge and gravitational sectors of five-dimensional brane worlds in creating phenomenologically relevant vacua. We discuss locally supersymmetric brane worlds with unflipped and flipped fermionic boundary conditions and with matter on the branes. We point out that a natural separation between the gauge and gravity sectors, very difficult in models with true extra dimensions, may be achieved in 4d models with deconstructed dimensions. 
  The equations of motion and junction conditions for a gravi-dilaton brane world scenario are studied in the string frame. It is shown that they allow Kasner-like solutions on the brane, which makes the dynamics of the brane very similar to the low curvature phase of pre-big bang cosmology. Analogies and differences of this scenario with the Randall-Sundrum one and with the standard bulk pre-big bang dynamics are also discussed. 
  We identify a time-dependent class of metrics with potential applications to cosmology, which emerge from negative-tension branes. The cosmology is based on a general class of solutions to Einstein-dilaton-Maxwell theory, presented in {hep-th/0106120}. We argue that solutions with hyperbolic or planar symmetry describe the gravitational interactions of a pair of negative-tension $q$-branes. These spacetimes are static near each brane, but become time-dependent and expanding at late epoch -- in some cases asymptotically approaching flat space. We interpret this expansion as being the spacetime's response to the branes' presence. The time-dependent regions provide explicit examples of cosmological spacetimes with past horizons and no past naked singularities. The past horizons can be interpreted as S-branes. We prove that the singularities in the static regions are repulsive to time-like geodesics, extract a cosmological `bounce' interpretation, compute the explicit charge and tension of the branes, analyse the classical stability of the solution (in particular of the horizons) and study particle production, deriving a general expression for Hawking's temperature as well as the associated entropy. 
  We discuss a general iterative procedure for constructing time dependent solutions in open string theory describing rolling of a generic tachyon field away from its maximum. These solutions are characterized by two parameters labelling the initial position and velocity of the tachyon field, but one of these parameters can be removed by using time translation symmetry. The Wick rotated version of the resulting one parameter family of inequivalent solutions describes a one parameter family of boundary conformal field theories, each member of which is related to the boundary conformal field theory describing the original D-brane system by a nearly marginal deformation. We apply this technique to construct a time dependent solution on a D-brane in bosonic string theory which can be interpreted as the creation of a lower dimensional brane during the decay of an unstable D-brane. 
  We point out two extensions of the relation between matrix models, topological strings and N=1 supersymmetric gauge theories. First, we note that by considering double scaling limits of unitary matrix models one can obtain large N duals of the local Calabi-Yau geometries that engineer N=2 gauge theories. In particular, a double scaling limit of the Gross-Witten one-plaquette lattice model gives the SU(2) Seiberg-Witten solution, including its induced gravitational corrections. Secondly, we point out that the effective superpotential terms for N=1 ADE quiver gauge theories is similarly computed by large multi-matrix models, that have been considered in the context of ADE minimal models on random surfaces. The associated spectral curves are multiple branched covers obtained as Virasoro and W-constraints of the partition function. 
  Both in string field theory and in p-adic string theory the equations of motion involve infinite number of time derivatives. We argue that the initial value problem is qualitatively different from that obtained in the limit of many time derivatives in that the space of initial conditions becomes strongly constrained. We calculate the energy-momentum tensor and study in detail time dependent solutions representing tachyons rolling on the p-adic string theory potentials. For even potentials we find surprising small oscillations at the tachyon vacuum. These are not conventional physical states but rather anharmonic oscillations with a nontrivial frequency--amplitude relation. When the potentials are not even, small oscillatory solutions around the bottom must grow in amplitude without a bound. Open string field theory resembles this latter case, the tachyon rolls to the bottom and ever growing oscillations ensue. We discuss the significance of these results for the issues of emerging closed strings and tachyon matter. 
  In this lecture, I put forward conjectures asserting that, in all noncommutative field theories, (1) open Wilson lines and their descendants constitute a complete set of interpolating operators of `noncommutative dipoles', obeying dipole relation, (2) infrared dynamics of the noncommutative dipoles is dual to ultraviolet dynamics of the elementary noncommutative fields, and (3) open string field theory is a sort of noncommutative field theory, whose open Wilson lines are interpolating operators for closed strings. I substantiate these conjectures by various intuitive arguments and explicit computations of one- and two-loop Feynman diagrammatics. 
  Local boundary conditions for spinor fields are expressed in terms of a 1-parameter family of boundary operators, and find applications ranging from (supersymmetric) quantum cosmology to the bag model in quantum chromodynamics. The present paper proves that, for massless spinor fields on the Euclidean ball in dimensions d=2,4,6, the resulting zeta(0) value is independent of such a theta parameter, while the various heat-kernel coefficients exhibit a theta-dependence which is eventually expressed in a simple way through hyperbolic functions and their integer powers. 
  We study the the properties of a BRST ghost degree of freedom complementary to a two-state spinor. We show that the ghost may be regarded as a unit carrier of negative entropy. We construct an irreducible representation of the su(2) Lie algebra with negative spin, equal to -1/2, on the ghost state space and discuss the representation of finite SU(2) group elements. The Casimir operator J^2 of the combined spinor-ghost system is nilpotent and coincides with the BRST operator Q. Using this, we discuss the sense in which the positive and negative spin representations cancel in the product to give an effectively trivial representation. We compute an effective dimension, equal to 1/2, and character for the ghost representation and argue that these are consistent with this cancellation. 
  We study $SO(m)$ covariant Matrix realizations of $ \sum_{i=1}^{m} X_i^2 = 1 $ for even $m$ as candidate fuzzy odd spheres following hep-th/0101001. As for the fuzzy four sphere, these Matrix algebras contain more degrees of freedom than the sphere itself and the full set of variables has a geometrical description in terms of a higher dimensional coset. The fuzzy $S^{2k-1} $ is related to a higher dimensional coset $ {SO(2k) \over U(1) \times U(k-1)}$. These cosets are bundles where base and fibre are hermitian symmetric spaces.   The detailed form of the generators and relations for the Matrix algebras related to the fuzzy three-spheres suggests Matrix actions which admit the fuzzy spheres as solutions. These Matrix actions are compared with the BFSS, IKKT and BMN Matrix models as well as some others. The geometry and combinatorics of fuzzy odd spheres lead to some remarks on the transverse five-brane problem of Matrix theories and the exotic scaling of the entropy of 5-branes with the brane number. 
  We point out the flaw in the analysis of Gangopadhyaya and Mallow, hep-th/0206133, where it is claimed that supersymmetry is broken in the SUSY half-oscillator, even with a regularization respecting supersymmetry. 
  We consider an alternative approach to non-linear special-relativistic constructions. Our point of departure is not $\kappa$-deformed algebra (or even group-theoretical considerations) but rather 3 physical postulates defining particle's velocity, mass, and the upper bound on its energy in terms of the respective classical quantities. For a specific definition of particle's velocity we obtain Magueijo-Smolin (MS) version of the dual special-relativistic construction. It is shown that this version follows from the $\kappa$-Poincare algebra by the appropriate choice of on the shell mass. The $\kappa$-deformed Hamiltonian is found which invalidates the some arguments about un-physical predictions of the MS transformation. 
  A conjecture for computing all genus topological closed string amplitudes on toric local Calabi-Yau threefolds, by interpreting the associated 5-brane web as a Feynman diagram, is given. A propagator and a three point vertex is defined which allows us to write down the amplitude associated with 5-brane web. We verify the conjecture that this amplitude is equal to the closed string partition function by computing integer invariants for resolved conifold and certain curves of low degree in local del Pezzo surfaces, local Hirzebruch surfaces and their various blowups. 
  We investigate non-commutative gauge theories in homogeneous spaces G/H. We construct such theories by adding cubic terms to IIB matrix model which contain the structure constants of G. The isometry of a homogeneous space, G must be a subgroup of SO(10) in our construction. We investigate CP^2=SU(3)/U(2) case in detail which gives rise to 4 dimensional non-commutative gauge theory. We show that non-commutative gauge theory on R^4 can be realized in the large N limit by letting the action approach IIB matrix model in a definite way. We discuss possible relevances of these theories to the large N limit of IIB matrix model. 
  We use the group-theoretic interpretation of the AdS/CFT correspondence which we proposed earlier in order to lift intertwining operators acting between boundary conformal representations to intertwining operators acting between bulk conformal representations. Further, we present the classification of the positive energy (lowest weight) unitary irreducible representations of the D=6 superconformal algebras $osp(8^*/2N)$. 
  We discuss a new family of metrics of 7-manifolds with G_2 holonomy, which are R^3 bundles over a quaternionic space. The metrics depend on five parameters and have two Abelian isometries. Certain singularities of the G_2 manifolds are related to fixed points of these isometries; there are two combinations of Killing vectors that possess co-dimension four fixed points which yield upon compactification only intersecting D6-branes if one also identifies two parameters. Two of the remaining parameters are quantized and we argue that they are related to the number of D6-branes, which appear in three stacks. We perform explicitly the reduction to the type IIA model. 
  The non-relativistic version of the non commutative Field Theory, recently introduced by Lozano, Moreno and Schaposnik [1], is shown to admit the ``exotic'' Galilean symmetry found before for point particles. 
  The pressureless tachyonic matter recently found in superstring field theory has an over-abundance problem in cosmology. We argue that this problem is naturally solved in the brane inflationary scenario if almost all of the tachyon energy is drained (via its coupling to the inflaton and matter fields) to heating the universe, while the rest of the tachyon energy goes to a network of cosmic strings (lower-dimensional BPS D-branes) produced during the tachyon rolling at the end of inflation. 
  The Casimir problem is usually posed as the response of a fluctuating quantum field to externally imposed boundary conditions. In reality, however, no interaction is strong enough to enforce a boundary condition on all frequencies of a fluctuating field. We construct a more physical model of the situation by coupling the fluctuating field to a smooth background potential that implements the boundary condition in a certain limit. To study this problem, we develop general new methods to compute renormalized one--loop quantum energies and energy densities. We use analytic properties of scattering data to compute Green's functions in time--independent background fields at imaginary momenta. Our calculational method is particularly useful for numerical studies of singular limits because it avoids terms that oscillate or require cancellation of exponentially growing and decaying factors. To renormalize, we identify potentially divergent contributions to the Casimir energy with low orders in the Born series to the Green's function. We subtract these contributions and add back the corresponding Feynman diagrams, which we combine with counterterms fixed by imposing standard renormalization conditions on low--order Green's functions. The resulting Casimir energy and energy density are finite functionals for smooth background potentials. In general, however, the Casimir energy diverges in the boundary condition limit. This divergence is real and reflects the infinite energy needed to constrain a fluctuating field on all energy scales; renormalizable quantum field theories have no place for ad hoc surface counterterms. We apply our methods to simple examples to illustrate cases where these subtleties invalidate the conclusions of the boundary condition approach. 
  We construct a class of negative spin irreducible representations of the su(2) Lie algebra. These representations are infinite-dimensional and have an indefinite inner product. We analyze the decomposition of arbitrary products of positive and negative representations with the help of generalized characters and write down explicit reduction formulae for the products. From the characters, we define effective dimensions for the negative spin representations, find that they are fractional, and point out that the dimensions behave consistently under multiplication and decomposition of representations. 
  Recent work on the numerical solution of supersymmetric gauge theories is described. The method used is SDLCQ (supersymmetric discrete light-cone quantization). An application to N=1 supersymmetric Yang-Mills theory in 2+1 dimensions at large N_c is summarized. The addition of a Chern-Simons term is also discussed. 
  We investigate the various time-dependent bubble spacetimes that can be obtained from double analytic continuation of asymptotically locally flat/AdS spacetimes with NUT charge. We find different time-dependent explicit solutions of general relativity from double analytic continuations of Taub-Nut(-AdS) and Kerr-Nut(-AdS) spacetimes. One solution in particular has Milne-like evolution throughout, and another is a NUT-charged generalization of the AdS soliton. These solutions are all four dimensional. In certain situations the NUT charge induces an ergoregion into the bubble spacetime and in other situations it quantitatively modifies the evolution of the bubble, as when rotation is present. In dimensions greater than four, no consistent bubble solutions are found that have only one timelike direction. 
  We study the superstring vacua constructed from the conformal field theories of the type H_4 x M, where H_4 denotes the super Nappi-Witten model (super WZW model on the 4-dimensional Heisenberg group H_4) and M denotes an arbitrary N=2 rational superconformal field theory with c=9. We define (type II) superstring vacua with 8 supercharges, which are twice as many as those on the backgrounds of H_4 x CY_3. We explicitly construct as physical vertices the space-time SUSY algebra that is a natural extension of H_4 Lie algebra. The spectrum of physical states is classified into two sectors: (1) strings freely propagating along the transverse plane of pp-wave geometry and possessing the integral U(1)_R-charges in M sector, and (2) strings that do not freely propagate along the transverse plane and possess the fractional U(1)_R-charges in M. The former behaves like the string excitations in the usual Calabi-Yau compactification, but the latter defines new sectors without changing the physics in ``bulk'' space. We also analyze the thermal partition functions of these systems, emphasizing the similarity to the DLCQ string theory. As a byproduct we prove the supersymmetric cancellation of conformal blocks in an arbitrary unitary N=2 SCFT of c=12 with the suitable GSO projection. 
  We elucidate the relationship between supersymmetric D3-branes and chiral baryonic operators in the AdS/CFT correspondence. For supersymmetric backgrounds of the form AdS_5 x H, we characterize via holomorphy a large family of supersymmetric D3-brane probes wrapped on H. We then quantize this classical family of probe solutions to obtain a BPS spectrum which describes D3-brane configurations on H. For the particular examples H = T^{1,1} and H = S^5, we match the BPS spectrum to the spectrum of chiral baryonic operators in the dual gauge theory. 
  Because of the problems arising from the fermion unification in the traditional Grand Unified Theory and the mass hierarchy between the 4-dimensional Planck scale and weak scale, we suggest the low energy gauge unification theory with low high-dimensional Planck scale. We discuss the non-supersymmetric SU(5) model on $M^4\times S^1/Z_2 \times S^1/Z_2$ and the supersymmetric SU(5) model on $M^4\times S^1/(Z_2\times Z_2') \times S^1/(Z_2\times Z_2')$. The SU(5) gauge symmetry is broken by the orbifold projection for the zero modes, and the gauge unification is accelerated due to the SU(5) asymmetric light KK states. In our models, we forbid the proton decay, still keep the charge quantization, and automatically solve the fermion mass problem. We also comment on the anomaly cancellation and other possible scenarios for low energy gauge unification. 
  We study the gravity dual of several wrapped D-brane configurations in presence of 4-form RR fluxes partially piercing the unwrapped directions. We present a systematic approach to obtain these solutions from those without fluxes. We use D=8 gauged supergravity as a starting point to build up these solutions. The configurations include (smeared) M2-branes at the tip of a G_2 cone on S^3 x S^3, D2-D6 branes with the latter wrapping a special Lagrangian 3-cycle of the complex deformed conifold and an holomorphic sphere in its cotangent bundle T^*S^2, D3-branes at the tip of the generalized resolved conifold, and others obtained by means of T duality and KK reduction. We elaborate on the corresponding N=1 and N=2 field theories in 2+1 dimensions. 
  We present a class of magnetically charged brane solutions for the theory of Einstein-Maxwell gravity with a negative cosmological constant in an arbitrary number of spacetime dimensions. 
  We study webs of D3- and D5-branes in type IIB supergravity. These webs preserve at least 8 supercharges. By solving the Killing spinor equations we determine the form of supergravity solutions for the system. We then turn to the sub-class of the intersecting D3/D5 brane system and elucidate some of its features. 
  We review physical motivations, phenomenological consequences, and open problems of the so-called pre-big bang scenario in superstring cosmology. 
  In this short note we compare the endpoint of tachyon condensation of twisted circles with the endpoint of nonpertubative brane nucleation in the Kaluza-Klein Melvin spacetime. 
  A general algorithm is presented which gives a closed-form expression for an arbitrary perturbative diagram of cubic string field theory at any loop order. For any diagram, the resulting expression is given by an integral of a function of several infinite matrices, each built from a finite number of blocks containing the Neumann coefficients of Witten's 3-string vertex. The closed-form expression for any diagram can be approximated by level truncation on oscillator level, giving a computation involving finite size matrices. Some simple tree and loop diagrams are worked out as examples of this approach. 
  We find quasinormal modes of near extremal black branes by solving a singular boundary value problem for the Heun equation. The corresponding eigenvalues determine the dispersion law for the collective excitations in the dual strongly coupled N=4 SYM at finite temperature. 
  This paper has been withdrawn by the authors. 
  We describe the low-energy effective theory of N=1 spontaneously broken supegravity obtained by flux-induced breaking in the presence of n D3 branes. This theory can be obtained by integrating out three massive gravitino multiplets in the hierarchical breaking N=4 to N=3 to N=2 to N=1. This integration also eliminates the IIB complex dilaton. The resulting theory is a no-scale supergravity model, whose moduli are the three chiral multiplets that correspond to the three radii of the three 2-tori in 6-torus, together with the 6n brane coordinates. The U(n) gauge interactions on the branes respect the no-scale structure,and the N=1 goldstino is the fermionic partner of the 6-torus volume. 
  The AdS/dCFT correspondence is used to show that a planar q-dimensional superconformal CFT defect expands, under the addition of electric charge and angular momentum, to a supersymmetric higher-dimensional defect of geometry R^q x C, where C is an arbitrary curve. The dual string theory process is the expansion of D-branes and fundamental strings into a supertube in an AdS background. 
  A quantum treatment of spontaneous symmetry breaking in scalar light front field theories is formulated. It is based on a finite-volume quantization with antiperiodic boundary conditions for scalar fields. This avoids a necessity to solve the zero mode constraint and simultaneously enables one to define a unitary operator which shifts the scalar field by a constant. The operator transforms the light-front vacuum into a coherent-state vacuum that minimizes the light front energy and leads to a spontaneous breakdown of the discrete or continuous symmetry of the Hamiltonian. Simple theories of a self-interacting real scalar field in two dimensions and of a complex scalar field in three dimensions are analyzed in this way including a discussion of the Goldstone theorem. 
  The Casimir effect due to conformally coupled bulk scalar fields on background of conformally flat brane-world geometries is investigated. In the general case of mixed boundary conditions formulae are derived for the vacuum expectation values of the energy-momentum tensor and vacuum forces acting on boundaries. The special case of the AdS bulk is considered and the application to the Randall-Sundrum scenario is discussed. The possibility for the radion stabilization by the vacuum forces is demonstrated. 
  We consider the interacting system of D=4 N=1 supergravity and the Brink-Schwarz massless superparticle as described by the sum of their superfield actions, and derive the complete set of superfield equations of motion for the coupled dynamical system. These include source terms given by derivatives of a vector superfield current density with support on the worldline. This current density is constructed from the spin 3/2 and spin 2 current density `prepotentials'. We analyze the gauge symmetry of the coupled action and show that it is possible to fix the gauge in such a way that the equations of motion reduce to those of the supergravity-bosonic particle coupled system. 
  Recently, we proposed a new braneworld model [GIT] in which the collision of a brane universe and a vacuum bubble coming from the extra-dimension is utilized as a trigger of brane big-bang. In this article, mainly reviewing this model, we briefly summarize cosmological braneworld scenarios in which collision of branes plays an important role. 
  Certain spontaneously broken gauge theories contain massless magnetic monopoles. These are realized classically as clouds of non-Abelian fields surrounding one or more massive monopoles. In order to gain a better understanding of these clouds, we study BPS solutions with four massive and six massless monopoles in an SU(6) gauge theory. We develop an algebraic procedure, based on the Nahm construction, that relates these solutions to previously known examples. Explicit implementation of this procedure for a number of limiting cases reveals that the six massless monopoles condense into four distinct clouds, of two different types. By analyzing these limiting solutions, we clarify the correspondence between clouds and massless monopoles, and infer a set of rules that describe the conditions under which a finite size cloud can be formed. Finally, we identify the parameters entering the general solution and describe their physical significance. 
  This paper comprises the written version of the lectures on string theory delivered at the 31st British Universities Summer School on Theoretical Elementary Particle Physics which was held in Manchester, England, August 28 - September 12 2001. 
  A perturbative quantum theory of the 2-Killing vector reduction of general relativity is constructed. Although non-renormalizable in the standard sense, we show that to all orders of the loop expansion strict cut-off independence can be achieved in a space of Lagrangians differing only by a field dependent conformal factor. In particular the Noether currents and the quantum constraints can be defined as finite composite operators. The form of the field dependence in the conformal factor changes with the renormalization scale and a closed formula is obtained for the beta functional governing its flow. The flow possesses a unique fixed point at which the trace anomaly is shown to vanish. The approach to the fixed point adheres to Weinberg's ``asymptotic safety'' scenario, both in the gravitational wave/cosmological sector and in the stationary sector. 
  Conventional superstring amplitudes in flat space exhibits exponential fall off at wide angle in contrast to the power law behavior found in QCD. It has recently been argued by Polchinski and Strassler that this conflict can be resolved via String/Gauge duality. They carried out their analysis in terms of strings in a deformed $AdS^5$ background. On the other hand, an equally valid approach to the String/Gauge duality for 4-d QCD is based on M-theory in a specific Black Hole deformation of $AdS^7 \times S^4$. We show that a very natural extension to this phenomenologically interesting M-theory background also gives the correct hard scattering power laws. In the Regge limit we extend the analysis to show the co-existence of both the hard BFKL-like Pomeron and the soft Pomeron Regge pole. 
  We consider supergravity configuration of D5 branes wrapped on supersymmetric 2-cycles and use it to calculate one-point and two-point Green functions of some special operators in N=2 super Yang-Mills theory. We show that Green functions obtained from supergravity include two very different parts. One of them corresponds to perturbative results of quantum field theory, and another is a non-perturbative effect which corresponds to contribution from instantons with fractional charge. Comparing Green functions obtained from supergravity and gauge theory, we obtain radial/energy-scale relation for this gauge/gravity correspondence with N=2 supersymmetry. This relation leads right beta-function of N=2 SYM from supergravity configuration. 
  Although the formulas of tachyon inflation correspond to those of the inflation driven by the ordinary scalar field, there is obvious difference between them, which can not be neglected. We calculate the scalar and tensor perturbation of the string theory inspired tachyon inflation, which has been widely studied recently. We also show, through the Hamilton-Jacobi approach, that the the rolling tachyon can essentially produce enough inflation. An exact solution with the inverse squared potential of tachyon field has been proposed and its power spectra has been analyzed. 
  We derive general formulae for the first order variation of the ADM mass, angular momentum for linear perturbations of a stationary background in Einstein-Maxwell axion-dilaton gravity being the low-energy limit of the heterotic string theory. All these variations were expressed in terms of the perturbed matter energy momentum tensor and the perturbed charge current density. Combining these expressions we reached to the form of the {\it physical version} of the first law of black hole dynamics for the stationary black holes in the considered theory being the strong support for the cosmic censorship. 
  We discuss the breakdown of perturbative unitarity of noncommutative quantum field theories in electric-type background in the light of string theory. We consider the analytic structure of string loop two-point functions using a suitable off-shell continuation and then study the zero slope limit of Seiberg and Witten. In this way we pick up how the unphysical tachyonic branch cut appears in the noncommutative field theory. We briefly discuss discontinuities and cutting rules for the full string theory amplitude and relate them to the noncommutative field theoretical results, and also discuss the insight one gains into the magnetic case too. 
  We consider two ways of introducing minimal Abelian gauge interactions into the model presented in [1]. They are different only if the second central charge of the planar Galilei group is nonzero. One way leads to standard gauge transformations and second to a generalized gauge theory with gauge transformations accompanied by time-dependent area-preserving coordinate transformations. Both approaches, however, are related to each other by a classical Seiberg-Witten map supplemented by the noncanonical transformation of the phase space variables for planar particles. We also formulate the two-body problem in the model with a generalized gauge symmetry and consider the case with both CS and background electromagnetic fields, as it is used in the description of fractional quantum Hall effect. 
  We propose a five-dimensional standard model which regards the Higgs field as a weak boson associated with the fifth dimension. The kinetic term of the Higgs field is obtained from the fifth components of field strengths defined in five dimension. The coupling constant of the fermion fields and the Higgs field is only the weak coupling constant. However, since the vacuum expectation value depends on the fifth coordinate, we can explain the various mass spectrum of elementary particles. 
  A stronger version of an anomaly matching theorem (AMT) is proven that allows to anticipate the matching of continuous as well as discrete global anomalies. The AMT shows a connection between anomaly matching and the geometry of the null cone of SYM theories. Discrete symmetries are shown to be broken at the origin of the moduli space in Seiberg-Witten theories. 
  The parametric families of integrable boundary affine Toda theories are considered. We calculate boundary one-point functions and propose boundary S-matrices in these theories. We use boundary one-point functions and S-matrix amplitudes to derive boundary ground state energies and exact solutions describing classical vacuum configurations. 
  A manifestly gauge invariant exact renormalization group for pure SU(N) Yang-Mills theory is proposed, allowing gauge invariant calculations, without any gauge fixing or ghosts. The necessary gauge invariant regularisation which implements the effective cutoff, is naturally incorporated by embedding the theory into a spontaneously broken SU(N|N) super-gauge theory. This guarantees finiteness to all orders in perturbation theory. 
  Using a gauge invariant exact renormalization group, we show how to compute the effective action, and extract the physics, whilst manifestly preserving gauge invariance at each and every step. As an example we give an elegant computation of the one-loop SU(N) Yang-Mills beta function, for the first time at finite N without any gauge fixing or ghosts. It is also completely independent of the details put in by hand, e.g. the choice of covariantisation and the cutoff profile, and, therefore, guides us to a procedure for streamlined calculations. 
  It has been proposed that the geometry of an extra dimension could automatically adjust itself to compensate for an arbitrary energy density on the 3-D brane which we are presumed to inhabit, such that a static solution to Einstein's equation results. This would solve the long-standing cosmological constant problem, of why our universe is not overwhelmed by the enormous energy of the quantum vacuum fluctuations predicted by quantum field theory. I will review some of the attempts along these lines, and present a no-go theorem showing that these attempts are doomed, at least within one of the most promising classes of models. 
  We argue that it may be possible to reheat the universe after inflation driven by D-brane annihilation, due to the coupling of massless fields to the time-dependent tachyon condensate which describes the annihilation process. This mechanism can work if the original branes annihilate to a stable brane containing the standard model. Given reasonable assumptions about the shape of the tachyon background configuration and the size of the relevant extra dimension, the reheating can be efficient enough to overcome the problem of the universe being perpetually dominated by cold dark tachyon matter. In particular, reheating is most efficient when the final brane codimension is large, and when the derivatives of the tachyon background are large. 
  We present a gauge invariant action for a superstring in the plane wave background with Ramond-Ramond (RR) five-form flux. The Wess-Zumino term is given explicitly in a bilinear form of the left invariant currents by introducing a fermionic center to define the nondegenerate group metric. The reparametrization invariance generators, whose combinations are conformal generators, and fermionic constraints, half of which generate kappa-symmetry, are obtained. Equations of motion are obtained in conformal invariant and background covariant manners. 
  The connection between Lorentz invariance violation and noncommutativity of fields in a quantum field theory is investigated. A new dispersion relation for a free field theory with just one additional noncommutative parameter is obtained. While values for the noncommutative scale much larger than 10^{-20} eV^{-1} are ruled out by the present experimental status, cosmic ray physics would be compatible with and sensible to a noncommutativity arising from quantum gravity effects. We explore the matter-antimatter asymmetry which is naturally present in this framework. 
  We derive the low-energy effective theory on the BPS domain wall in 4D N=1 global SUSY theories in terms of the 3D superfields. Our derivation makes the preserved SUSY by the wall manifest and the procedure for integrating out the massive modes easier. Our procedure clarifies how the 3D superfields are embedded into the 4D chiral and vector superfields. We also point out a shortcoming of the conventional procedure for deriving the effective theory on the wall. 
  Using the algebraic geometry method of Berenstein and Leigh for the construction of the toroidal orbifold (T^2 x T^2 x T^2) / (Z_2 x Z_2) with discrete torsion and considering local K3 surfaces, we present non-commutative aspects of the orbifolds of product of K3 surfaces. In this way, the ordinary complex deformation of K3 can be identified with the resolution of stringy singularities by non-commutative algebras using crossed products. We give representations and make some comments regarding the fractionation of branes. Illustrating examples are presented. 
  Magnetic monopoles having non-Abelian charges have been found recently to play a crucial role in the infrared in a class of supersymmetric gauge theories. We argue that these "dual quarks" can naturally be identified with the non-Abelian magnetic monopoles of the type first discussed by Goddard, Nuyts and Olive. Our argument is based on a few simple observations as regards to their charge structure, flavor quantum numbers, and some general properties of electromagnetic duality. 
  We explicitly derive a complementary pair of four-dimensional M-theory brane-world models, linked by a five-dimensional bulk, each of which has a unique anomaly-free chiral spectrum. This is done via resolution of local consistency requirements, in the context of the simplest global quotient T7/G with ten-dimensional fixed-planes, for which a chiral four-dimensional spectrum could arise. 
  We investigate N=1 super Yang-Mills theory using fractional branes. We first define the beta-function with respect to a supergravity coordinate. To provide the relation between the supergravity parameter and the renormalization group scale we use the UV known gauge theory beta-function as a type of boundary condition. We show that there are no privileged renormalization schemes connected to a given supergravity solution while we investigate in some detail two schemes. The Wilsonian one where just one loop is manifest and the one containing multi-loops. A new functional relation between the gaugino condensate and the supergravity coordinates is finally determined. 
  The origin of the anomalies is analyzed. It is shown that they are due to the fact that the generators of the symmetry do not leave invariant the domain of definition of the Hamiltonian and then a term, normally forgotten in the Heisenberg equation, gives an extra contribution responsible for the non conservation of the charges. This explanation is equivalent to that of the Fujikawa in the path integral formalism. Finally, this approach is applied to the conformal symmetry breaking in two-dimensional quantum mechanics. 
  The possibility of parity violation in a gravitational theory with torsion is extensively explored in four and higher dimensions. In the former case,we have listed our conclusions on when and whether parity ceases to be conserved, with both two-and three-index antisymmetry of the torsion field. In the latter, the bulk spacetime is assumed to have torsion, and the survival of parity-violating terms in the four dimensional effective action is studied, using the compactification schemes proposed by Arkani-Hamed-Dimopoulos-Dvali and Randall-Sundrum. An interesting conclusion is that the torsion-axion duality arising in a stringy scenario via the second rank antisymmetric Kalb-Ramond field leads to conservation of parity in the gravity sector in any dimension. However, parity-violating interactions do appear for spin 1/2 fermions in such theories, which can have crucial phenomenological implications. 
  We find non-BPS solutions of the noncommutative CP^1 model in 2+1 dimensions. These solutions correspond to soliton anti-soliton configurations. We show that the one-soliton one-anti-soliton solution is unstable when the distance between the soliton and the anti-soliton is small. We also construct time-dependent solutions and other types of solutions. 
  We consider a model of topological solitons where charged particles have finite mass and the electric charge is quantised already at the classical level. In the electrodynamic limit, which physically corresponds to electrodynamics of solitons of zero size, the Lagrangian of this model has two degrees of freedom only and reduces to the Lagrangian of the Maxwell field in dual representation. We derive the equations of motion and discuss their relations with Maxwell's equations. It is shown that Coulomb and Lorentz forces are a consequence of topology. Further, we relate the U(1) gauge invariance of electrodynamics to the geometry of the soliton field, give a general relation for the derivation of the soliton field from the field strength tensor in electrodynamics and use this relation to express homogeneous electric fields in terms of the soliton field. 
  Brane Gas Cosmology (BGC) is an approach to unifying string theory and cosmology in which matter is described by a gas of strings and branes in a dilaton gravity background. The Universe is assumed to start out with all spatial dimensions compact and small. It has previously been shown that in this context, in the approximation of neglecting inhomogeneities and anisotropies, there is a dynamical mechanism which allows only three spatial dimensions to become large. However, previous studies do not lead to any conclusions concerning the isotropy or anisotropy of these three large spatial dimensions. Here, we generalize the equations of BGC to the anisotropic case, and find that isotropization is a natural consequence of the dynamics. 
  It has been argued by several authors, using different formalisms, that the quantum mechanical spectrum of black hole horizon area is discrete and uniformly spaced. Recently it was shown that two such approaches, namely the one involving quantisation on a reduced phase space, and the algebraic approach of Bekenstein and Gour are equivalent for spherically symmetric, neutral black holes (hep-th/0202076). The observables of one can be mapped to those of the other. Here we extend that analysis to include charged black holes. Once again, we find that the ground state of the black hole is a Planck size remnant. 
  5-dimensional homogeneous and isotropic models with a bulk cosmological constant and a minimally coupled scalar field are considered. We have found that in special cases the scalar field can mimic a frustrated (i.e. disordered) networks of topological defects: cosmic strings, domain walls and hyperdomain walls. This equivalence enabled us to obtain 5-dimensional instantonic solutions which can be used to construct brane-world models. In some cases, their analytic continuation to a Lorentzian metric signature give rise to either 4-dimensional flat or inflating branes. Models with arbitrary dimensions (D $>$ 5) are also briefly discussed. 
  We investigate relevant deformation and the renormalization group flow in a defect conformal field theory from the point of view of the holography. We propose a candidate of g-function in the context of the holography, and prove the g-theorem: the g-function is monotonically non-increasing along the RG flow. We apply this g-theorem to the D5-brane solution which is an asymptotically AdS_4 x S^2 brane in AdS_5 x S^5. This solution corresponds to the mass deformation of the defect CFT. We checked that the g-function is monotonically non-increasing in this solution. 
  The enhancement in the decay rate of the Kaluza-Klein vacuum due to the presence of a brane is studied, both in the test brane approximation and beyond it. Spontaneous brane materialization in the Kaluza-Klein vacuum is also described. 
  We investigate the dynamics of a spherically symmetric vaccum on a Randall and Sundrum 3-brane world. Under certain natural conditions, the effective Einstein equations on the brane form a closed system for spherically symmetric dark radiation. We determine exact dynamical and inhomogeneous solutions, which are shown to depend on the brane cosmological constant, on the dark radiation tidal charge and on its initial energy configuration. We identify the conditions defining these solutions as singular or as globally regular. Finally, we discuss the confinement of gravity to the vicinity of the brane and show that a phase transition to a regime where gravity is not bound to the brane may occur at short distances during the collapse of positive dark energy density on a realistic de Sitter brane. 
  We consider a diagonalization of Witten's star product for a ghost system of arbitrary background charge and Grassmann parity. To this end we use a bosonized formulation of such systems and a spectral analysis of Neumann matrices. We further identify a continuous Moyal product structure for a combined ghosts+matter system. The normalization of multiplication kernel is discussed. 
  In this paper we show how one can obtain very simply the spectra of the PP-wave limits of M-theory over AdS_7(4) x S^4(7) spaces and IIB superstring theory over AdS_5 x S^5 from the oscillator construction of the Kaluza-Klein spectra of these theories over the corresponding spaces. The PP-wave symmetry superalgebras are obtained by taking the number P of ``colors'' of oscillators to be large (infinite). In this large P limit, the symmetry superalgebra osp(8*|4) of AdS_7 x S^4 and the symmetry superalgebra osp(8|4,R) of AdS_4 x S^7 lead to isomorphic PP-wave algebras, which is the semi-direct sum of su(4|2) with H^(18,16), while the symmetry superalgebra su(2,2|4) of AdS_5 x S^5 leads to the semi-direct sum of [psu(2|2) + psu(2|2) + u(1)] with H^(16,16) as its PP-wave algebra [H^(m,n) denoting a super-Heisenberg algebra with m bosonic and n fermionic generators]. The zero mode spectra of M-theory or IIB superstring theory in the PP-wave limit corresponds simply to the unitary positive energy representations of these algebras whose lowest weight vector is the Fock vacuum of all the oscillators. General positive energy supermultiplets including those corresponding to higher modes can similarly be constructed by the oscillator method. 
  We study the algebraic formulation of exact factorizable S-matrices for integrable two-dimensional field theories. We show that different formulations of the S-matrices for the Potts field theory are essentially equivalent, in the sense that they can be expressed in the same way as elements of the Temperley-Lieb algebra, in various representations. This enables us to construct the S-matrices for certain nonlinear sigma models that are invariant under the Lie ``supersymmetry'' algebras sl(m+n|n) (m=1,2; n>0), both for the bulk and for the boundary, simply by using another representation of the same algebra. These S-matrices represent the perturbation of the conformal theory at theta=pi by a small change in the topological angle theta. The m=1, n=1 theory has applications to the spin quantum Hall transition in disordered fermion systems. We also find S-matrices describing the flow from weak to strong coupling, both for theta=0 and theta=pi, in certain other supersymmetric sigma models. 
  We develop a systematic method to solve the Cardy condition for the coset conformal field theory G/H. The problem is equivalent to finding a non-negative integer valued matrix representation (NIM-rep) of the fusion algebra. Based on the relation of the G/H theory with the tensor product theory G x H, we give a map from NIM-reps of G x H to those of G/H. Our map provides a large class of NIM-reps in coset theories. In particular, we give some examples of NIM-reps not factorizable into the G and the H sectors. The action of the simple currents on NIM-reps plays an essential role in our construction. As an illustration of our procedure, we consider the diagonal coset SU(2)_5 x SU(2)_3 /SU(2)_8 to obtain a new NIM-rep based on the conformal embedding su(2)_3 \oplus su(2)_8 \subset sp(6)_1. 
  We analyse some physical consequences when supersymmetry is broken by a set of D-branes and/or orientifold planes in Type II string theories. Generically, there are global dilaton tadpoles at the disk level when the transverse space is compact. By taking the toy model of a set of electric charges in a compact space, we discuss two different effects appearing when global tadpoles are not cancelled. On the compact directions a constant term appears that allows to solve the equations of motion. On the non-compact directions Poincar\'e invariance is broken. We analyse some examples where the Poincar\'e invariance is broken along the time direction (cosmological models).After that, we discuss how to obtain a finite interaction between D-branes and orientifold planes in the compact space at the supergravity level. 
  A Procedure is outlined that may be used as a starting point for a perturbative treatment of theories with permanent confinement. By using a counter term in the Lagrangian that renormalizes the infrared divergence in the Coulomb potential, it is achieved that the perturbation expansion at a finite value of the strong coupling constant may yield reasonably accurate properties of hadrons, and an expression for the string constant as a function of the QCD Lambda parameter. 
  We construct U(2) noncommutative multi-instanton solutions by extending Witten's ansatz [1] which reduces the problem of cylindrical symmetry in four dimensions to that of a set of Bogomol'nyi equations for an Abelian Higgsmodel in two dimensional curved space. Using the Fock space approach, we give explicit vortex solutions to the Bogomol'nyi equations and, from them, we present multi-instanton solutions. 
  We investigate the set of boundary states in the symplectic fermion description of the logarithmic conformal field theory with central charge c=-2. We show that the thus constructed states correspond exactly to those derived under the restrictions of the maximal chiral symmetry algebra for this model, W(2,3,3,3). This connects our previous work to the coherent state approach of Kawai and Wheater. 
  Recently proposed procedure of constructing maximally superintegrable systems of Winternitz type is further developed and illustrated by an example of system admitting an explicit construction of angle variables and additional integrals of motion. A possible application of the method to Liouville system is briefly presented. 
  We consider the stability of the two branches of non-extremal enhancon solutions. We argue that one would expect a transition between the two branches at some value of the non-extremality, which should manifest itself in some instability. We study small perturbations of these solutions, constructing a sufficiently general ansatz for linearised perturbations of the non-extremal solutions, and show that the linearised equations are consistent. We show that the simplest kind of perturbation does not lead to any instability. We reduce the problem of studying the more general spherically symmetric perturbation to solving a set of three coupled second-order differential equations. 
  Group theory indicates the existence of a $SO(8) X SO(7) \subset SO(16)$ invariant self-duality equation for a 3-form in 16 dimensions. It is a signal for interesting topological field theories, especially on 8-dimensional manifolds with holonomy group smaller than or equal to Spin(7), with a gauge symmetry that is SO(8) or SO(7). Dimensional reduction also provides new supersymmetric theories in 4 and lower dimensions, as well as a model with gravitational interactions in 8 dimensions, which relies on the octonionic gravitational self-duality equation. 
  We present the results of work in which we derived the order rho^2 corrections to the Friedmann equations in the Randall-Sundrum I model. The effects of Golberger-Wise stabilization are taken into account. We surprisingly find that in the cases of inflation and radiation domination, the leading corrections on a given brane come exclusively from the effects of energy density located on the opposite brane. 
  Casimir interactions (due to the massless scalar field fluctuations) of two surfaces which are close to each other are studied.   After a brief general presentation, explicit calculations for co-axial cylinders, co-centric spheres and co-axial cones are performed. 
  We prove the uniqueness theorem for self-gravitating non-linear sigma-models in higher dimensional spacetime. Applying the positive mass theorem we show that Schwarzschild-Tagherlini spacetime is the only maximally extended, static asymptotically flat solution with non-rotating regular event horizon with a constant mapping. 
  We present a universal normal algebra suitable for constructing and classifying Calabi-Yau spaces in arbitrary dimensions. This algebraic approach includes natural extensions of reflexive weight vectors to higher dimensions, related to Batyrev's reflexive polyhedra, and their n-ary combinations. It also includes a `dual' construction based on the Diophantine decomposition of invariant monomials, which provides explicit recurrence formulae for the numbers of Calabi-Yau spaces in arbitrary dimensions with Weierstrass, K3, etc., fibrations. Our approach also yields simple algebraic relations between chains of Calabi-Yau spaces in different dimensions, and concrete visualizations of their singularities related to Cartan-Lie algebras. This Universal Calabi-Yau Algebra is a powerful tool for decyphering the Calabi-Yau genome in all dimensions. 
  Recent studies of homogeneous anisotropic universe models in the brane world scenario show that the cosmological singularity in this context is isotropic. It has therefore been suggested that this may be a generic feature of singularities on the brane, even in the inhomogeneous case. Using a perturbative approach, we show that this is not the case. As in the GR case, the presence of decaying modes in the perturbations signal the instability (in the past) of the isotropic singularity. The brane universe is therefore not born with isotropy built in: as in standard cosmology, the observed large-scale isotropy and homogeneity remains to be explained. 
  We study classical solutions in Berenstein-Maldacena-Nastase (BMN) matrix model. A supersymmetric (1/2 BPS) fuzzy sphere is one of the classical solutions and corresponds to a giant graviton. We also consider other classical solutions, such as non-supersymmetric fuzzy sphere and harmonic oscillating gravitons. Some properties of oscillating gravitons are discussed. In particular, oscillating gravitons turn into usual supergravitons in the limit mu -> 0. Moreover, we calculate the one-loop effective action around the supersymmetric fuzzy sphere by the use of the background field method and show the quantum stability of the giant graviton. Also, the instability of the non-supersymmetric fuzzy sphere is proven. 
  I reply to the objections recently raised by J. Llosa to my constructive proof that Lagrangians with nonlocality of finite extent inherit the full Ostrogradskian instability. 
  We consider massless elementary particles in a quantum theory based on a Galois field (GFQT). We previously showed that the theory has a new symmetry between particles and antiparticles, which has no analogue in the standard approach. We now prove that the symmetry is compatible with all operators describing massless particles. Consequently, massless elementary particles can have only the half-integer spin (in conventional units), and the existence of massless neutral elementary particles is incompatible with the spin-statistics theorem. In particular, this implies that the photon and the graviton in the GFQT can only be composite particles. 
  We consider a 5-D gravity plus a bulk scalar field, and with a 3-brane. The Darboux transformation is used to construct some exact solutions. To do this we reduce the system of equations, which describes the 5-D gravity and bulk scalar field to the Schr\"odinger equation. The jump conditions at the branes lead to the jump potential in the Schr\"odinger equation. Using the Darboux transformation with these jump conditions, we offer a new exact solution of the brane equations, which represents a generalization of the Rundall-Sundrum solution. For simplicity, the main attention is focused on the case when Hubble root on the visible brane is zero. However, the argument is given that our method is valid in more realistic models with cosmological expansion. 
  We research the entropy of a black hole in curved space-times by 't Hooft`s approach, so-called the brick wall method. One of these space-time, a asymptotically dS space-time has two physical horizons; one is a black hole horizon and the other is a cosmological horizon. The others have only one horizon, a black hole horizon. Using this model, we calculate all thermodynamic quantities containing the background geometric effect and show that entropy is proportional to area of each boundary. Furthermore, we show that the Cardy-Verlinde formula can be rederived from the physical quantities of the brick wall method and this fact becomes a evidence of the dS/CFT correspondence. 
  We investigate the consequences of two assumptions for String (or M) Theory, namely that: 1) all coordinates are compact and bound by the horizon of observation, 2) the ``dynamics'' of compactification is determined by the ``second law of thermodynamics'', i.e. the principle of entropy. We discuss how this leads to a phenomenologically consistent scenario for our world, both at the elementary particle's and at the cosmological level, without any fine tuning or further ``ad hoc'' constraint. 
  In this paper we explicitly calculate the analogue of the 't Hooft SU(2) Yang--Mills instantons on Gibbons--Hawking multi-centered gravitational instantons which come in two parallel families: the multi-Eguchi--Hanson, or A_k ALE gravitational instantons and the multi-Taub--NUT, or A_k ALF gravitational instantons. We calculate their action and find the reducible ones. Following Kronheimer we also exploit the U(1) invariance of our solutions and study the corresponding explicit singular SU(2) magnetic monopole solutions on Euclidean three-space. 
  We study one-loop effective action of Berkooz-Douglas Matrix theory and obtain non-abelian action of D0-branes in the longitudinal 5-brane background. In this paper, we extend the analysis of hep-th/0201248 and calculate the part of the effective action containing fermions. We show that the effective action is manifestly invariant under the loop-corrected SUSY transformation, and give the explicit transformation laws. The effective action consists of blocks which are closed under the SUSY, and includes the supersymmetric completion of the couplings to the longitudinal 5-branes proposed by Taylor and Van Raamsdonk as a subset. 
  A nonlocal quantum gravity theory is presented which is finite and unitary to all orders of perturbation theory. Vertex form factors in Feynman diagrams involving gravitons suppress graviton and matter vacuum fluctuation loops by introducing a low-energy gravitational scale, Lambda_{Gvac} < 2.4 X 10^{-3} eV. Gravitons coupled to non-vacuum matter loops and matter tree graphs are controlled by a vertex form factor with the energy scale, Lambda_{GM} < 1-10 TeV. A satellite Eotvos experiment is proposed to test a violation of the equivalence principle for coupling of gravitons to pure vacuum energy compared to matter. 
  We construct a non-Abelian world volume effective action for a system of multiple M-theory gravitons. This action contains multipole moment couplings to the eleven-dimensional background potentials. We use these couplings to study, from the microscopical point of view, giant graviton configurations where the gravitons expand into an M2-brane, with the topology of a fuzzy 2-sphere, that lives in the spherical part of the AdS_7 x S^4 background or in the AdS part of AdS_4 x S^7. When the number of gravitons is large we find perfect agreement with the Abelian, macroscopical description of giant gravitons given in the literature. 
  We formulate the planar `large N limit' of matrix models with a continuously infinite number of matrices directly in terms of U(N) invariant variables. Non-commutative probability theory, is found to be a good language to describe this formulation. The change of variables from matrix elements to invariants induces an extra term in the hamiltonian,which is crucual in determining the ground state. We find that this collective potential has a natural meaning in terms of non-commutative probability theory:it is the `free Fisher information' discovered by Voiculescu. This formulation allows us to find a variational principle for the classical theory described by such large N limits. We then use the variational principle to study models more complex than the one describing the quantum mechanics of a single hermitian matrix (i.e., go beyond the so called D=1 barrier). We carry out approximate variational calculations for a few models and find excellent agreement with known results where such comparisons are possible. We also discover a lower bound for the ground state by using the non-commutative analogue of the Cramer-Rao inequality. 
  The bosonic beta-gamma ghost system has long been used in formal constructions of conformal field theory. It has become important in its own right in the last few years, as a building block of field theory approaches to disordered systems, and as a simple representative -- due in part to its underlying su(2)_{-1/2} structure -- of non-unitary conformal field theories. We provide in this paper the first complete, physical, analysis of this beta-gamma system, and uncover a number of striking features. We show in particular that the spectrum involves an infinite number of fields with arbitrarily large negative dimensions. These fields have their origin in a twisted sector of the theory, and have a direct relationship with spectrally flowed representations in the underlying su(2)_{-1/2} theory. We discuss the spectral flow in the context of the operator algebra and fusion rules, and provide a re-interpretation of the modular invariant consistent with the spectrum. 
  We discuss the approach of effective field theory on a d-dimensional Euclidean space in a scalar theory with two different mass scales in the presence of flat surfaces. Then considering Dirichlet and Neumann boundary conditions, we implement the renormalization program in the lambda phi^{4} theory in a region bounded by two parallel hyperplanes in the one-loop approximation. 
  In this talk I first briefly review how far we have come in answering old questions about the most fundamental building blocks of matter. I begin with things we know, which is the Standard Model, and then talk about things we can guess, which is superstring theory. After this review I discuss a key point at which our understanding of superstring theory presently stops: the problem of supersymmetry breaking and the cosmological constant. I explain in which direction I imagine a way out. This way out predicts gravitinos and dilatons with masses of order milli-eV. 
  We examine in the framework of 5d Kaluza-Klein theory the gauge equivalence of $x^5$-dependent cosmological solutions each of which describes in the 4d sector an arbitrarily evolving isotropic, homogeneous universe with some pure gauge. We find that (1)within a certain time scale $\tau_c$   (which is characterized by the compactification radius $R_c$) any arbitrarily evolving 4d universe is allowed to exist by field equations, and these 4d universes with appropriate pure gauges are all gauge equivalent as long as they are of the same topology. (2)Outside $\tau_c$ the gauge equivalence disappears and the evolution of the universe is fixed by field equations. 
  We study the Casimir problem as the limit of a conventional quantum field theory coupled to a smooth background. The Casimir energy diverges in the limit that the background forces the field to vanish on a surface. We show that this divergence cannot be absorbed into a renormalization of the parameters of the theory. As a result, the Casimir energy of the surface and other quantities like the surface tension, which are obtained by deforming the surface, cannot be defined independently of the details of the coupling between the field and the matter on the surface. In contrast, the energy density away from the surface and the force between rigid surfaces are finite and independent of these complications. 
  We analyze the scalar potentials of maximal gauged three-dimensional supergravities which reveal a surprisingly rich structure. In contrast to maximal supergravities in dimensions D>3, all these theories admit a maximally supersymmetric (N=16) ground state with negative cosmological constant Lambda<0, except for the gauge group SO(4,4)^2, for which Lambda=0. We compute the mass spectra of bosonic and fermionic fluctuations around these vacua and identify the unitary irreducible representations of the relevant background (super)isometry groups to which they belong. In addition, we find several stationary points which are not maximally supersymmetric, and determine their complete mass spectra as well. In particular, we show that there are analogs of all stationary points found in higher dimensions, among them de Sitter vacua in the theories with noncompact gauge groups SO(5,3)^2 and SO(4,4)^2, as well as anti-de Sitter vacua in the compact gauged theory preserving 1/4 and 1/8 of the supersymmetries. All the dS vacua have tachyonic instabilities, whereas there do exist non-supersymmetric AdS vacua which are stable, again in contrast to the D>3 theories. 
  We consider a supersymmetric model with a single matter supermultiplet in a five-dimensional space-time with orbifold compactification along the fifth dimension. The boundary conditions on the two orbifold planes are chosen in such a way that supersymmetry remains unbroken on the boundaries. We calculate the vacuum energy-momentum tensor in a configuration in which the boundary branes are moving with constant velocity. The results show that the contribution from fermions cancels that of bosons only in the static limit, but in general a velocity-dependent Casimir energy arises between the branes. We relate this effect to the particle production due to the branes motion and finally we discuss some cosmological consequences. 
  Using mirror pairs (M_3, W_3) in type II superstring compactifications on Calabi-Yau threefolds, we study, geometrically, F-theory duals of M-theory on seven manifolds with G_2 holonomy. We first develop a way for getting Landau Ginzburg (LG) Calabi-Yau threefols W_3, embedded in four complex dimensional toric varieties, mirror to sigma model on toric Calabi-Yau threefolds M_3. This method gives directly the right dimension without introducing non dynamical variables. Then, using toric geometry tools, we discuss the duality between M-theory on (S^1 x M_3)/Z_2 with G_2 holonomy and F-theory on elliptically fibered Calabi-Yau fourfolds with SU(4) holonomy, containing W_3 mirror manifolds. Illustrating examples are presented. 
  We study the behavior of matrix string theory in the strong coupling region, where matrix strings reduce to discrete light-cone type IIA superstrings except at the usual string-interaction points. In the large N limit, this reduction corresponds to the double-dimensional reduction from wrapped supermembranes on R^{10} x S^1 to type IIA superstrings on R^{10} in the light-cone gauge. Such reductions were shown classically, while they are not obvious quantum mechanically. Recently, Sekino and Yoneya analyzed the double-dimensional reduction of the wrapped supermembrane quantum mechanically to one-loop order in the strong coupling expansion. We analyze the problem in matrix string theory by using the same expansion. At the one-loop level, the quantum corrections cancel out as was presented by them. However, at the two-loop level we find that the quantum corrections cancel out only for the leading terms in the large N. 
  A field model of two-component fermions is described, the consequences of which coincide in the main with primary postulates of the standard model. Such a model can be constructed for 4 generations at the minimum. Peculiarities of the relative coordinate space, determining in general an internal symmetry group, are considered. Analogues of the Higgs fields appear in the model naturally after transition to the Grassmannian extra coordinates. 
  The boundary state formalism is used to confirm predictions from non-commutativity for the derivative corrections to the Dirac--Born--Infeld and Chern--Simons actions, at all orders in derivatives. As anticipated by S. Mukhi, the method applies by induction to every coupling in the Chern--Simons action. It is also used to derive the corrections to the Dirac--Born--Infeld action at quadratic order in the field strength. 
  We construct gauge theory of interacting symmetric traceless tensor fields of all ranks s=0,1,2,3, ... which generalizes Weyl-invariant dilaton gravity to the higher spin case, in any dimension d>2. The action is given by the trace of the projector to the subspace with positive eigenvalues of an arbitrary hermitian differential operator H, the symmetric tensor fields emerge after expansion of the latter in power series in derivatives. After decomposition in perturbative series around a conformally flat point H=\Box, the quadratic part of the action breaks up as a sum of free gauge theories of symmetric traceless tensors of rank s with actions of d-4+2s order in derivatives introduced in 4d case by Fradkin and Tseytlin and studied at the cubic order level by Fradkin and Linetsky. Higher orders in interaction are well-defined. We discuss in detail global symmetries of the model which give rise to infinite dimensional conformal higher spin algebras for any d. We stress geometric origin of conformal higher spin fields as background fields of a quantum point particle, and make the conjecture generalizing this geometry to the system "tensionless d-1 brane + Fronsdal higher spin massless fields in d+1 dimensions". We propose a candidate on the role of Higgs-like higher spin compensator able to spontaneously break higher spin symmetries. At last, we make the conjecture that, in even dimensions d, the action of conformal higher spin theory equals the logarithmically divergent term of the action of massless higher spin fields on AdS_{d+1} evaluated on the solutions of Dirichlet-like problem, where conformal higher spin fields are boundary values of massless higher spin fields on AdS_{d+1}, the latter conjecture provides information on the full higher spin action in AdS_{d+1}. 
  Exact cosmological solutions for effective actions in D dimensions inspired by the tree-level superstring action are studied. For a certain range of free parameters existing in the model, nonsingular bouncing solutions are found. Among them, of particular interest can be open hyperbolic models, in which, without any fine tuning, the internal scale factor and the dilaton field (connected with string coupling in string theories) tend to constant values at late times. A cosmological singularity is avoided due to nonminimal dilaton-gravity coupling and, for D > 11, due to pure imaginary nature of the dilaton, which conforms to currently discussed unification models. The existence of such and similar solutions supports the opinion that the Universe had never undergone a stage driven by full-scale quantum gravity. 
  Time-dependent solutions of supergravity and string theory are studied. The examples are obtained from de Sitter deformation of gauge/gravity dualities, analytical continuation of static solutions, and ``exactly solvable'' worldsheet models. Among other things, it is shown that turning on a Hubble parameter in the background of a confining gauge theory in four dimensions can restore chiral symmetry. Some of the solutions obtained from analytical continuation have the interpretation of a universe with a bounce separating a big bang from a big crunch singularity. In the worldsheet context, it is argued why string propagation close to a Milne-type cosmological singularity might be physically non-singular. 
  In many gauge theories at different values of parameters entering Lagrangian, the vacuum is dominated by coherent condensates of different mutually non-local fields (for instance, by condensates of electric or magnetic charges, or by various dyons). It is argued that the transition between these "dual to each other" phases proceeds through the intermediate "mixed phase", having qualitatively different features. The examples considered include: ordinary YM, N=1 SYM, N=1 SQCD, and broken N=2 SYM and SQCD. 
  The equations defining pure spinors are interpreted as equations of motion formulated on the lightcone of a ten-dimensional, lorentzian, momentum space. Most of the equations for fermion multiplets, usually adopted by particle physics, are then naturally obtained and their properties like internal symmetries, charges, families appear to be due to the correlation of the associated Clifford algebras, with the 3 complex division algebras: complex numbers at the origin of U(1) and charges; quaternions at the origin of SU(2); families and octonions at the origin of SU(3). Pure spinors instead could be relevant not only because the underlying momentum space results compact, but also because it may throw light on some aspects of particle physics, like: masses, charges, constaint relations, supersymmetry and epistemology. 
  Exact solutions of classical gauge theories in even-dimensional (D=2n) spacetimes are discussed. Common and specific properties of these solutions are analyzed for the particular dimensions D=2, D=4, and D=6. A consistent formulation of classical gauge field theories with pointlike charged or colored particles is proposed for D=6. The particle Lagrangian must then depend on the acceleration. The self-interaction of a point particle is considered for D=2 and D=6. In D=2, radiation is absent and all processes are reversible. In D=6, the expression for the radiation rate and the equation of motion of a self-interacting particle are derived; from which follows that the Zitterbewegung always leads to radiation. It is shown that non-Abelian solutions are absent for any D not equal to 4; only Coulomb-like solutions, which correspond to the Abelian limit of the D-dimensional Yang--Mills--Wong theory, are admitted. 
  The question of Bohr correspondence in quantum field theory is considered from a dynamical point of view. It is shown that the classical description of particle interactions is inapplicable even in the limit of large particles' masses because of finite quantum fluctuations of the fields produced. In particular, it is found that the relative value of the root mean square fluctuation of the Coulomb and Newton potentials of a massive particle is equal to 1/sqrt{2}. It is shown also that in the case of a macroscopic body, the quantum fluctuations are suppressed by a factor 1/sqrt{N}, where N is the number of particles in the body. An adequate macroscopic interpretation of the correspondence principle is given. 
  Strings propagating in AdS_{D+1} are made Weyl invariant to leading order in large D by a ghost-matter coupling that preserves the Poincare symmetry of the boundary. 
  The spontaneous symmetry breaking of rotational O(N) symmetry in noncommutative field theory is investigated in a 2+1 dimensional model of scalar fields coupled through a combination of quartic and sextuple self-interactions. There are five possible orderings of the fields in the sextuple interaction and two for the quartic interaction. At one loop, we prove that for some choices of these orderings there is the absence of IR/UV mixing and the appearance of massless excitations. A supersymmetric extension of the model is also studied. Supersymmetry puts additional constraints on the couplings but for any given N there is a Moyal ordering of the superfields for which the requirement for the existence of Goldstone bosons is satisfied. For some ordering and when N goes to infinity we find evidence that the model is renormalizable to all orders in perturbation theory. We also consider a generic chiral model in 3+1 dimensions whose superpotential is invariant under local gauge transformations. We find that for any value of N there is no one loop correction to the pion mass and that, at two loops, there are no pion mass corrections for slowly varying superfields so that Goldstone theorem holds true. We also find a new purely noncommutative coupling which gives contributions starting at order N-2 loops. 
  We extend the BMN duality between IIB superstring theory on a pp-wave background and a sector of N=4 super Yang-Mills theory to the non-supersymmetric and unstable background built by Romans as a compactification on a U(1) bundle over CP2 with 3-form and 5-form field strength fluxes. We obtain a stable theory with the fewest number of supercharges (e.g. 16) allowed by this kind of solutions and make conjectures on the dual gauge theory. 
  We study the correlator of two parallel Wilson lines in two-dimensional noncommutative Yang-Mills theory, following two different approaches. We first consider a perturbative expansion in the large-N limit and resum all planar diagrams. The second approach is non-perturbative: we exploit the Morita equivalence, mapping the two open lines on the noncommutative torus (which eventually gets decompacted) in two closed Wilson loops winding around the dual commutative torus. Planarity allows us to single out a suitable region of the variables involved, where a saddle-point approximation of the general Morita expression for the correlator can be performed. In this region the correlator nicely compares with the perturbative result, exhibiting an exponential increase with respect to the momentum p. 
  We study the Penrose limit of the (p,q) fivebranes supergravity background. We consider the different phases of the worldvolume field theory and their weakly coupled descriptions. In the Penrose limit we get a solvable string theory and compute the spectrum. It corresponds to states of the six-dimensional worldvolume theory with large energy and large U(1) charge. We comment on the RG behavior of the gauge theory. 
  (2+1)-dimensional Georgi-Glashow model and its SU(N)-generalization are explored at nonzero temperatures and in the regime when the Higgs boson is not infinitely heavy. The finiteness of the Higgs-boson mass leads to various novel effects. Those include the appearance of two separate phase transitions and of the upper bound on the parameter of the weak-coupling approximation, necessary to maintain the stochasticity of the Higgs vacuum. The modification of the finite-temperature behavior of the model emerging due to the introduction of massless quarks is also discussed. 
  We present a noncommutative version of a plane-wave solution to the gravitational field equations. We start with a given classical solution, admittedly rather simple, and construct an algebra and a differential calculus which supports the metric. In the particular solution presented as an example the 1-forms do not anticommute, to a degree which depends on the amplitude of the deviation of the metric from the standard Minkowski metric. 
  In this paper we examine a certain threebrane solution of type IIB string theory whose long-wavelength dynamics are those of a supersymmetric gauge theory in 2+1 continuous and 1 discrete dimension, all of infinite extent. Low-energy processes in this background are controlled by dimensional deconstruction, a strict limit in which gravity decouples but the lattice spacing stays finite. Relating this limit to the near-horizon limit of our solution we obtain an exact, continuum gravitational dual of a lattice gauge theory with nonzero lattice spacing. H-flux in the translationally invariant solution encodes the spatial discreteness of the gauge theory, and we relate the cutoff on allowed momenta to a giant graviton effect in the bulk. 
  Given the path of a point particle, one can relate its acceleration and, in general, its kinematics to the curvature scalars of its trajectory. Using this, a general Ansatz is made for the Yang Mills connection corresponding to a non-Abelian point source. The Yang Mills field equations are then solved outside the position of the point source under physically reasonable constraints such as finite total energy flux and finite total color charge. The solutions contain the Trautman solution; moreover two of them are exact whereas one of them is found using a series expansion in 1/R, where R is the retarded distance. These solutions are new and, in their most general form, are not gauge equivalent to the original Trautman solution. 
  We consider the construction of nonsingular Pre-Big-Bang and Ekpyrotic type cosmological models realized by the addition to the action of specific higher-order terms stemming from quantum corrections. We study models involving general relativity coupled to a single scalar field with a potential motivated by the Ekpyrotic scenario. We find that the inclusion of the string loop and quantum correction terms in the string frame makes it possible to obtain solutions of the variational equations which are nonsingular and bouncing in the Einstein frame, even when a negative exponential potential is present, as is the case in the Ekpyrotic scenario. We analyze the spectra of perturbations produced during the bouncing phase and find that the spectrum of curvature fluctuations in the model proposed originally to implement the Ekpyrotic scenario has a large blue tilt ($n_R= 3$). Except for instabilities introduced on small scales, the result agrees with what is obtained by imposing continuity of the induced metric and of the extrinsic curvature across a constant scalar field (up to $k^2$ corrections equal to the constant energy density) matching surface between the contracting and the expanding Einstein Universes. We also discuss nonsingular cosmological solutions obtained when a Gauss-Bonnet term with a coefficient suitably dependent on the scalar matter field is added to the action in the Einstein frame with a potential for the scalar field present. 
  We argue that Quantum Gravitation forces us to sum over metrics of all signatures. 
  We present a non-perturbative regularization scheme for Quantum Field Theories which amounts to an embedding of the originally unregularized theory into a spacetime with an extra compactified dimensions of length L ~ Lambda^{-1} (with Lambda an ultraviolet cutoff), plus a doubling in the number of fields, which satisfy different periodicity conditions and have opposite Grassmann parity. The resulting regularized action may be interpreted, for the fermionic case, as corresponding to a finite-temperature theory with a supersymmetry, which is broken because of the boundary conditions. We test our proposal in a perturbative calculation (the vacuum polarization graph for a D-dimensional fermionic theory) and in a non-perturbative one (the chiral anomaly). 
  Based on the old results of Cho, Soh, Park and Yoon, it is shown how higher m + n dimensional pure gravitational actions restricted to AdS_m times S^n backgrounds admit a holographic reduction to a lower m-dimensional Yang-Mills-like gauge theory of diffeomorphisms of S^n interacting with a charged non-linear sigma model plus boundary terms by a simple tuning of the AdS_m-throat and S^n-radius sizes. After performing a harmonic expansion of the fields with respect to the internal coordinates and a subsequent integration one obtains an m-dimensional effective action involving an infinite-component field theory. The supersymmetrization program can be carried out in a straighforward fashion. 
  Starting from N=1 scalar supermultiplets in 2+1 dimensions, we build explicitly the composite superpartners which define a N=2 superalgebra induced by the initial N=1 supersymmetry. The occurrence of this extension is linked to the topologically conserved current out of which the composite superpartners are constructed. 
  A new twisted deformation, U_z(so(4,2)), of the conformal algebra of the (3+1)-dimensional Minkowskian spacetime is presented. This construction is provided by a classical r-matrix spanned by ten Weyl-Poincare generators, which generalizes non-standard quantum deformations previously obtained for so(2,2) and so(3,2). However, by introducing a conformal null-plane basis it is found that the twist can indeed be supported by an eight-dimensional carrier subalgebra. By construction the Weyl-Poincare subalgebra remains as a Hopf subalgebra after deformation. Non-relativistic limits of U_z(so(4,2)) are shown to be well defined and they give rise to new twisted conformal algebras of Galilean and Carroll spacetimes. Furthermore a difference-differential massless Klein-Gordon (or wave) equation with twisted conformal symmetry is constructed through deformed momenta and position operators. The deformation parameter is interpreted as the lattice step on a uniform Minkowskian spacetime lattice discretized along two basic null-plane directions. 
  We discuss the appearance of non-supersymmetric compactifications with exactly the Standard Model (SM) at low energies, in the context of IIB orientifold constructions with  D5 branes intersecting at angles on the $T^4$ tori, of the orientifold of $T^4 \times (\C /Z_N)$.  We discuss constructions where the Standard Model embedding is considering within four, five and six stacks of D5 branes. The appearance of the three generation observable Standard Model at low energies is accompanied by a gauged baryon number, thus ensuring automatic proton stability. Also, a compatibility with a low scale of order TeV is ensured by having a two dimensional space transverse to all branes. The present models complete the discussion of some recently constructed four stack models of D5 branes with the SM at low energy. By embedding the four, five and six stack Standard Model configurations into quiver diagrams, deforming them around the QCD intersection numbers, we find a rich variety of vacua that may have exactly the Standard Model at low energy. Also by using brane recombination on the U(1)'s, we show that the five and six vacua flow into their four stack counterparts. Thus string vacua with five and six stack deformations are continuously connected to the four stack vacua. 
  We study time-dependent solutions of Einstein-Maxwell gravity in four dimensions coupled to tachyon matter--the Dirac-Born-Infeld Lagrangian that provides an effective description of a decaying tachyon on an unstable D-brane in string theory. Asymptotically, the solutions are similar to the recently studied space-like brane solutions and carry S-brane charge. They do not break the Lorentzian R-symmetry. We study the tachyon matter as a probe in such a background and analyze its backreaction. For early/late times, the tachyon field has a constant energy density and vanishing pressure as in flat space. On the other hand, at intermediate times, the energy density of the tachyon diverges and produces a space-like curvature singularity. 
  We start by discussing the classical noncommutative (NC) Q-ball solutions near the commutative limit, then generalize the virial relation. Next we quantize the NC Q-ball canonically. At very small theta quantum correction to the energy of the Q-balls is calculated through summation of the phase shift. UV/IR mixing terms are found in the quantum corrections which cannot be renormalized away. The same method is generalized to the NC GMS soliton for the smooth enough solution. UV/IR mixing is also found in the energy correction and UV divergence is shown to be absent. In this paper only (2+1) dimensional scalar field theory is discussed. 
  In the asymptotically flat two-dimensional dilaton gravity, we present an N-body particle action which has a dilaton coupled mass term for the exact solubility. This gives nonperturbative exact solutions for the N-body self-gravitating system, so the infalling particles form a black hole and their trajectories are exactly described. In our two-dimensional case, the critical mass for the formation of black holes does not exist, so even a single particle forms a black hole, which means that we can treat many black holes. The infalling particles give additional time-like singularities in addition to the space-like black hole singularity. However, the latter singularities can be properly cloaked by the future horizons within some conditions. 
  We extend the search for fermionic subspaces of the bosonic string compactified on E8 X SO(16) lattices to include all fermionic D-branes. This extension constraints the truncation procedure previously proposed and relates the fermionic strings, supersymmetric or not, to the global structure of the SO(16) group. The specific properties of all the fermionic D-branes are found to be encoded in its universal covering, whose maximal toroid defines the configuration space torus of their mother bosonic theory. 
  We study the obtainment of a time-dependent cosmological constant at D=2 in a model based on the Jackiw-Teitelboim cosmology. We show that the cosmological term goes to zero asymptotically and can induce a nonsingular behavior at the origin. 
  We construct the one-loop effective action in Yang-Mills and Pure Quantum Gravity theories with heat kernel(or proper time method), which maintains manifest covariance during and after quantization (gauge and diffeomorphism invariance are always preserved). In this talk, we will basically focus on "What, How, and Why" we prefer heat kernel than the standard Feynman diagram calculation in momentum space at the one loop correction. The beta function of Yang-Mills field in the fixed gravitational background can be more simply obtained. The non-local term which cannot be easily obtained in the expansion method are exactly computed in Yang-Mills in the case of covariantly constant background field. The local term is consistent with asymptotic expansion method or any most standard method. The non-local terms give some physical implication concerning non-perturbative problems such as confinement and instabilities. The modification of this technique to quantum gravity is discussed. 
  We study Wilson loops in N=4 SYM theory which are non-constant in the scalar (S5) directions and open string solutions associated with them in the context of AdS/CFT correspondence. An interplay between Minkowskian and Euclidean pictures turns out to be non-trivial for time-dependent Wilson loops. We find that in the S5-rotating case there appears to be no direct open-string duals for the Minkowskian Wilson loops, and their expectation values should be obtained by analytic continuation from the Euclidean-space results. In the Euclidean case, we determine the dependence of the ``quark - anti-quark'' potential on the rotation parameter, both at weak coupling (i.e. in the 1-loop perturbative SYM theory) and at strong coupling (i.e. in the classical string theory in AdS5 x S5). 
  We introduce in this short note some aspects of the Moyal momentum algebra that we call the Das-Popowicz Mm algebra. Our interest on this algebra is motivated by the central role that it can play in the formulation of integrable models and in higher conformal spin theories. 
  We construct an N=1 supersymmetric Lagrangian model for the massive superspin-1 superfield. The model is described by a dynamical spinor superfield and an auxiliary chiral scalar superfield. On-shell this model leads to a spin-1/2, spin-3/2 and two spin-1 propagating component fields. The superfield action is given and its structure in the fermionic component sector is presented. We prove that the most general theory is characterized by a one parameter family of actions. The massless limit is shown to correspond to the dynamics of both the gravitino and superhelicity-1/2 multiplets. 
  We conjecture a topology changing transition in M-theory on a non-compact asymptotically conical Spin(7) manifold, where a 5-sphere collapses and a CP(2) bolt grows. We argue that the transition may be understood as the condensation of M5-branes wrapping the 5-sphere. Upon reduction to ten dimensions, it has a physical interpretation as a transition of D6-branes lying on calibrated submanifolds of flat space. In yet another guise, it may be seen as a geometric transition between two phases of type IIA string theory on a G_2 holonomy manifold with either wrapped D6-branes, or background Ramond-Ramond flux. This is the first non-trivial example of a topology changing transition with only 1/16 supersymmetry. 
  We explore aspects of the physics of de Sitter (dS) space that are relevant to holography with a positive cosmological constant. First we display a nonlocal map that commutes with the de Sitter isometries, transforms the bulk-boundary propagator and solutions of free wave equations in de Sitter onto the same quantities in Euclidean anti-de Sitter (EAdS), and takes the two boundaries of dS to the single EAdS boundary via an antipodal identification. Second we compute the action of scalar fields on dS as a functional of boundary data. Third, we display a family of solutions to 3d gravity with a positive cosmological constant in which the equal time sections are arbitrary genus Riemann surfaces, and compute the action of these spaces as a functional of boundary data from the Einstein gravity and Chern-Simons gravity points of view. These studies suggest that if de Sitter space is dual to a Euclidean conformal field theory (CFT), this theory should involve two disjoint, but possibly entangled factors. We argue that these CFTs would be of a novel form, with unusual hermiticity conditions relating left movers and right movers. After exploring these conditions in a toy model, we combine our observations to propose that a holographic dual description of de Sitter space would involve a pure entangled state in a product of two of our unconventional CFTs associated with the de Sitter boundaries. This state can be constructed to preserve the de Sitter symmetries and and its decomposition in a basis appropriate to antipodal inertial observers would lead to the thermal properties of static patch. 
  We present massive N=2 supergravity with SO(2)-gauging in nine-dimensions by direct construction. A full lagrangian and transformation rules are fixed, respectively up to quartic and quadratic fermion terms. Corresponding to the generalized Scherk-Schwarz dimensional reduction utilizing SL(2,R) symmetry, this theory allows three arbitrary mass parameters m_0, m_1and m_2 in addition to the minimal gauge coupling g, so that our system has the most general form compared with other results in the past. Unlike ordinary gauged maximal supergravity theories in other dimensions, the scalar potential is positive definite for arbitrary values of the mass parameters. As an application, we also analyze the stability and supersymmetry for 7-brane domain wall solutions for this gauged maximal supergravity, keeping the three mass parameters. 
  Hawking radiation can usefully be viewed as a semi-classical tunneling process that originates at the black hole horizon. The same basic premise should apply to de Sitter background radiation, with the cosmological horizon of de Sitter space now playing the featured role. In fact, a recent work [hep-th/0204107] has gone a long way to verifying the validity of this de Sitter-tunneling picture. In the current paper, we extend these prior considerations to arbitrary-dimensional de Sitter space, as well as Schwarzschild-de Sitter spacetimes. It is shown that the tunneling formalism naturally censors against any black hole with a mass in excess of the Nariai value; thus enforcing a ``third law'' of Schwarzschild-de Sitter thermodynamics. We also provide commentary on the dS/CFT correspondence in the context of this tunneling framework. 
  Langevin equation describing soft modes in the quark-gluon plasma is reformulated on the loop space. The Cauchy problem for the resulting loop equation is solved for the case when the nonvanishing components of the gauge potential correspond to the Cartan generators of the SU(N)-group and are proportional to a constant unit vector in the Cartan subalgebra. The regularized form of the loop equation with an arbitrary gauge potential is found, and perturbation theory in powers of the 't Hooft coupling is discussed. 
  We give a pedagogical introduction to string theory, D-branes and p-brane solutions. 
  In the usual Clifford algebra formulation of electrodynamics the Faraday bivector field F is decomposed into the observer dependent sum of a relative vector E and a relative bivector e_5 B by making a space-time split, which depends on the observer velocity. (E corresponds to the three-dimensional electric field vector, B corresponds to the three-dimensional magnetic field vector and e_5 is the (grade-4) pseudoscalar.) In this paper it is proved that the space-time split and the relative vectors are not relativistically correct, which means that the ordinary Maxwell equations with E and B and the field equations (FE) with F are not physically equivalent. Therefore we present the observer independent decomposition of F by using the 1-vectors of electric E and magnetic B fields. The equivalent, invariant, formulations of relativistic electrodynamics (independent of the reference frame and of the chosen coordinatization for that frame) which use F, E and B, the real multivector Psi = E - e_5 cB and the complex 1-vector Psi = E - icB are developed and presented here. The new observer independent FE are presented in formulations with E and B, with real and complex Psi. When the sources are absent the FE with real and complex Psi become Dirac like relativistic wave equations for the free photon. The expressions for the observer independent stress-energy vector T(v) (1-vector), energy density U (scalar), the Poynting vector S and the momentum density g (1-vectors), the angular momentum density M (bivector) and the Lorentz force K (1-vector) are directly derived from the FE. The local conservation laws are also directly derived from the FE and written in an invariant way. 
  It is shown that the vacuum state of weakly interacting quantum field theories can be described, in the Heisenberg picture, as a linear combination of randomly distributed incoherent paths that obey classical equations of motion with constrained initial conditions. We call such paths "pseudoclassical" paths and use them to define the dynamics of quantum fluctuations. Every physical observable is assigned a time-dependent value on each path in a way that respects the uncertainty principle, but in consequence, some of the standard algebraic relations between quantum observables are not necessarily fulfilled by their time-dependent values on paths. We define "collective observables" which depend on a large number of independent degrees of freedom, and show that the dynamics of their quantum fluctuations can be described in terms of unconstrained classical stochastic processes without reference to any additional external system or to an environment. Our analysis can be generalized to states other than the vacuum. Finally, we compare our formalism to the formalism of coherent states, and highlight their differences. 
  We investigate various aspects of the plane wave geometries obtained from D1/D5-brane system. We study the effect of Hopf-duality on the supersymmetries preserved by the Penrose limit of $AdS_3\times S^3\times T^4$ geometry. In type-IIB case, we first show that the Penrose limit makes the size of the `would-be' internal torus comparable to that of the other directions. Based on this observation, we consider, in taking the Penrose limit, the generalization of the null geodesic to incorporate the tilted direction between the equator of $S^3$ and one of the torus directions. For generic values of the tilting angle, supersymmetries are not preserved. When the limit is taken along the torus direction, 16 supersymmetries are preserved. For the ordinary Penrose limit, 16 generic and 8 `supernumerary' supersymmetries are observed. In the Penrose limit of Hopf-dualized type-IIA geometry, only 4 supersymmetries are preserved. We classify all the Killing spinors according to their periodic properties along some relevant coordinates. 
  Some massless supermultiplets appear as the trivial solution of Kostant's equation, a Dirac-like equation over special cosets. We study two examples; one over the coset SU(3)/SU(2) times U(1) contains the N=2 hypermultiplet in (3+1) dimensions with U(1) as helicity; the other over the coset F_4/SO(9) describes the N=1 supermultiplet in eleven dimensions, where SO(9) is the light-cone little group. We present the general solutions to Kostant's equation for both cases; they describe massless physical states of arbitrary spins which display the same relations as the fields in the supermultiplets. They come in sets of three representations called Euler triplets, but do not display supersymmetry although the number of bosons and fermions is the same when spin-statistics is satisfied. We build the free light-cone Lagrangian for both cases. 
  The quantum gravity is formulated based on gauge principle. The model discussed in this paper has local gravitational gauge symmetry and gravitational field is represented by gauge potential. A preliminary study on gravitational gauge group is presented. Path integral quantization of the theory is discussed in the paper. A strict proof on the renormalizability of the theory is also given. In leading order approximation, the gravitational gauge field theory gives out classical Newton's theory of gravity. It can also give out an Einstein-like field equation with cosmological term. The prediction for cosmological constant given by this model is well consistent with experimental results. For classical tests, it gives out the same theoretical predictions as those of general relativity. Combining cosmological principle with the field equation of gravitational gauge field, we can also set up a cosmological model which is consistent with recent observations. 
  We argue that extremal black holes of N=4 Poincare supergravity coupled to conformal matter have a quantum-exact dual 5-d description in the maximally supersymmetric extension of the Randall-Sundrum theory. This dual is a the classical, static supergravity solution describing a string with both Neveu-Schwarz and Ramond charge with respect to different antisymmetric tensor doublets. We also discuss the issue of the singularity present in a class of such supergravity solutions found by Cvetic, Lu and Pope. 
  By studying cohomological quantum mechanics on the punctured plane,we were led to identify (reduced) Bessel functions with homotopic loops living on the plane.This identification led us to correspondence rules between exponentials and Bessel functions.The use of these rules makes us retrieve known but also new formulas in Bessel functions theory. 
  We study the Penrose limit of ODp theory. There are two different PP-wave limits of the theory. One of them is a ten dimensional PP-wave and the other a four dimensional one. We observe the later one leads to an exactly solvable background for type II string theories where we have both NS and RR fields. The Penrose limit of different branes of string (M-theory) in a nonzero B/E field (C field) is also studied. These backgrounds are conjectured to provide dual description of NCSYM, NCOS and OM theory. We see that under S-duality the subsector of NCSYM$_4$ and NCOS$_4$ which are dual to the corresponding string theory on PP-wave coming from NCYM$_4$ and NCOS$_4$ map to each other for given null geodesic. 
  We present numerical evidence that a domain wall in a background with varying vacuum energy density acquires velocity in the direction of decreasing Hubble parameter. This should lead to at least a partial relaxation of the cosmological constant on the wall. 
  We outline a general geometric structure that underlies the N=1 superpotentials of a certain class of flux and brane configurations in type II string compactifications on Calabi-Yau threefolds. This ``holomorphic N=1 special geometry'' is in many respects comparable to, and in a sense an extension of, the familiar special geometry in N=2 supersymmetric type II string compactifications. It puts the computation of the instanton-corrected superpotential W of the four-dimensional N=1 string effective action on a very similar footing as the familiar computation of the N=2 prepotential F via mirror symmetry. In this note we present some of the main ideas and results, while more details as well as some explicit computations will appear in a companion paper 
  We find more self-tuning solutions by introducing a general form for Lagrangian of a 3-index antisymmetric tensor field $A_{MNP}$ in the RS II model. In particular, for the logarithmic Lagrangian, $\propto\log(-H^2)$, we obtained a closed form weak self-tuning solution. 
  Combining an optimized expansion scheme in the spirit of the background field method with the Coleman's normal-ordering renormalization prescription, we calculate the effective potential of sine-Gordon field theory beyond the Gaussian approximation. The first-order result is just the sine-Gordon Gaussian effective potential (GEP). For the range of the coupling beta^2 <= 3.4 pi (an approximate value), a calculation with Mathematica indicates that the result up to the second order is finite without any further renormalization procedure and tends to improve the GEP more substantially while beta^2 increases from zero. 
  Local decompositions of a Dirac spinor into `charged' and `real' pieces psi(x) = M(x) chi(x) are considered. chi(x) is a Majorana spinor, and M(x) a suitable Dirac-algebra valued field. Specific examples of the decomposition in 2+1 dimensions are developed, along with kinematical implications, and constraints on the component fields within M(x) sufficient to encompass the correct degree of freedom count. Overall local reparametrisation and electromagnetic phase invariances are identified, and a dynamical framework of nonabelian gauge theories of noncompact groups is proposed. Connections with supersymmetric composite models are noted (including, for 2+1 dimensions, infrared effective theories of spin-charge separation in models of high-Tc superconductivity). 
  We consider six dimensional brane world models with a compact and a warped extra dimension with five dimensional branes. We find that such scenarios have many interesting features arising from both ADD and Randall-Sundrum -models. In particular we study a class of models with a single 5D brane and a finite warped extra dimension, where one of the brane dimensions is compact. In these models the hierarchy problem can be solved on a single positive tension brane. 
  M(atrix) theory is known to be mass-deformed in the pp-wave background and still retains all 16 dynamical supersymmetries. We consider generalization of such deformations on super Yang-Mills quantum mechanics (SYQM) with less supersymmetry. In particular this includes N=8 U(N) SYQM with a single adjoint and any number of fundamental hypermultiplets, which is a pp-wave deformation of DLCQ matrix theory of fivebranes. With k >= 1 fivebranes, we show that a rich vacuum structure exists, with many continuous family of solutions that preserve all dynamical supersymmetries. The vacuum moduli space contains copies of CP^{k-1} of various sizes. 
  We treat spherically symmetric black holes in Gauss-Bonnet gravity by imposing boundary conditions on fluctuating metric on the horizon. Obtained effective two-dimensional theory admits Virasoro algebra near the horizon. This enables, with the help of Cardy formula, evaluation of the number of states. Obtained results coincide with the known macroscopic expression for the entropy of black holes in Gauss-Bonnet gravity. 
  Closed string diagrams are derived from cubic open string field theory using a gauge fixed kinetic operator. The basic idea is to use a string propagator that does not generate a boundary to the world sheet. Using this propagator and the closed string vertex, the moduli space of closed string surfaces is covered, so closed string scattering amplitudes should be reproduced. This kinetic operator could be a gauge fixed form of the string field theory action around the closed string vacuum. 
  A formal ``small tension'' expansion of D=11 supergravity near a spacelike singularity is shown to be equivalent, at least up to 30th order in height, to a null geodesic motion in the infinite dimensional coset space E10/K(E10) where K(E10) is the maximal compact subgroup of the hyperbolic Kac-Moody group E10(R). For the proof we make use of a novel decomposition of E10 into irreducible representations of its SL(10,R) subgroup. We explicitly show how to identify the first four rungs of the E10 coset fields with the values of geometric quantities constructed from D=11 supergravity fields and their spatial gradients taken at some comoving spatial point. 
  Quantum field theoretic treatments of fermion oscillations are typically restricted to calculations in Fock space. In this letter we extend the oscillation formulae to include more general quasi-free states, and also consider the case when the mixing is not unitary. 
  By considering N_e-electrons and N_h-holes together in uniform external magnetic and electric fields, we end up with a total Hall conductivity \sigma_{H}^{tot}, which is depending to the difference between N_e and N_h and becomes null when N_e=N_h. Dealing with the same system but requiring that the coordinates of plane are noncommuting, we obtain a new Hall conductivity \sigma_{H}^{(tot,nc)}. In the limit N_e=N_h, we find that \sigma_{H}^{(tot,nc)} is only noncommutativity parameters \theta_i-dependent, which means that theoretically it is possible to have Hall effect without B. Moreover, at the critical points \theta_e=l^2 and \theta_h=-l^2, we find that \sigma_{H}^{(tot,nc)} becomes two times the usual Hall conductivity for an noninteracting mixing system. 
  We construct an exact solution describing the collision of two Kaluza-Klein "bubbles of nothing" in 3+1 dimensions. When the bubbles collide, a curvature singularity forms which is hidden inside an event horizon. However, unlike the formation of ordinary black holes, in this case the spacetime resembles the entire maximally extended Schwarzschild solution. We also point out that there are inequivalent bubbles that can be constructed from Kerr black holes. 
  In 1983, Donaldson shocked the topology world by using instantons from physics to prove new theorems about four-dimensional manifolds, and he developed new topological invariants. In 1988, Witten showed how these invariants could be obtained by correlation functions for a twisted N=2 SUSY gauge theory. In 1994, Seiberg and Witten discovered dualities for such theories, and in particular, developed a new way of looking at four-dimensional manifolds that turns out to be easier, and is conjectured to be equivalent to, Donaldson theory.   This review describes the development of this mathematical subject, and shows how the physics played a pivotal role in the current understanding of this area of topology. 
  We derive various solutions for the gauge field and the gaugino when there are both 5D bulk kinetic terms and 4D brane kinetic terms. Below the compactification scale $1/y_c$, gauge interaction by the massless mode is universal, independent of the locations of sources, but above $1/y_c$, the interaction distinguishes their locations. We consider the $S^1/(Z_2\times Z_2')$ orbifold compactification, in which N=2 SUSY and SU(2) gauge symmetry break down to N=1 and U(1), respectively. While the odd parity gauge fields under $Z_2$, $A^1_\mu$ and $A^2_\mu$, interact with the bulk gauge coupling $g^2$, $A^3_\mu$ at low energy interacts with $e^2g^2/(e^2+g^2)$ due to the brane kinetic term with the coupling $e^2$. Even if $g^2$, which is asymptotically free, blew up at the scale $1/y_c<\mu < \Lambda_{\rm cutoff}$, $e^2g^2/(e^2+g^2)$ could remain small at low energy by brane matter fields' contribution. The condensations of the gauginos $\Psi^1$ and $\Psi^2$ generate soft mass term of $\Psi^3$ by gravity mediation. 
  We describe the general features of the Neveu-Schwarz and Ramond sectors of logarithmic conformal field theories with N=1 supersymmetry. Three particular systems are examined in some detail -- D-brane recoil, a superconformal extension of the c=-2 model, and the supersymmetric SU(2)_2 WZW model. 
  Noncommutative version of D-dimensional relativistic particle is proposed. We consider the particle interacting with the configuration space variable $\theta^{\mu\nu}(\tau)$ instead of the numerical matrix. The corresponding Poincare invariant action has a local symmetry, which allows one to impose the gauge $\theta^{0i}=0, ~ \theta^{ij}=const$. The matrix $\theta^{ij}$ turns out to be the noncommutativity parameter of the gauge fixed formulation. Poincare transformations of the gauge fixed formulation are presented in the manifest form. 
  Class of exact solutions of the Skyrme and the Faddeev model are presented. In contrast to previously found solutions, they are produced by the interplay of the two terms in the Lagrangians of the models. They are not solitonic but of wave character. With an appropriate choice of field variables, the field equations of the two models are written in exactly the same form. The solutions supply us with examples of the superposition of two plane waves in nonlinear field theories. 
  With the two most profound conceptual revolutions of XXth century physics, quantum mechanics and relativity, which have culminated into relativistic spacetime geometry and quantum gauge field theory as the principles for gravity and the three other known fundamental interactions, the physicist of the XXIst century has inherited an unfinished symphony: the unification of the quantum and the continuum. As an invitation to tomorrow's quantum geometers who must design the new rulers by which to size up the Universe at those scales where the smallest meets the largest, these lectures review the basic principles of today's conceptual framework, and highlight by way of simple examples the interplay that presently exists between the quantum world of particle interactions and the classical world of geometry and topology. 
  Based on the physical projector approach, a nonpertubative quantization of the massless Schwinger model is considered which does not require any gauge fixing. The spectrum of physical states, readily identified following a diagonalization of the operator algebra, is that of a massive pseudoscalar field, namely the electric field having acquired a mass proportional to the gauge coupling constant. The physical spectrum need not be identified with confined bound fermion-antifermion pairs, an interpretation which one is otherwise led to given whatever gauge fixing procedure but which is not void of gauge fixing artefacts. 
  We consider the interactions between Dp-branes intersecting at an arbitrary number of angles in the context of type II string theory. For cosmology purposes we concentrate in the theory on R^{3,1} x T^6. Interpreting the distance between the branes as the inflaton field, the branes can intersect at most at two angles on the compact space. If the configuration is non-supersymmetric we will have an interbrane potential that provides an effective cosmological inflationary epoch at the four dimensional intersection between the branes. The end of inflation occurs when the interbrane distance becomes small compared with the string scale, where a tachyon develops triggering the recombination of the branes. We study this recombination due to tachyon instabilities and we find the possibility for the final configuration to be again branes intersecting at two angles. This preserves the interesting features that are present in the intersecting brane models from the string model building point of view also after the end of inflation. This fact was not present in the models of branes intersecting at just one angle. This kind of recombination can be also important in other string contexts. 
  We show that Doubly Special Relativity (DSR) can be viewed as a theory with energy-momentum space being the four dimensional de Sitter space. Different formulations (bases) of the DSR theory considered so far can be therefore understood as different coordinate systems on this space. The emerging geometrical picture makes it possible to understand the universality of the non-commutative structure of space-time of Doubly Special Relativity. Moreover, it suggests how to construct the most natural DSR bases and makes it possible to address the long standing problem of total momentum of many particle systems from a different perspective. 
  We argue that bouncing branes occur naturally when there is more than one extra-dimension. We consider three-branes embedded in space-times with a horizon and an isometry group SO(6). As soon as the brane angular momentum is large enough, a repulsive barrier prevents the branes from reaching the horizon. We illustrate this phenomenon with the case of D3-branes in an AdS_5-Schwarzschild-S_5 background and asymptotically flat space-time. 
  Under certain conditions some solutions to five-dimensional heterotic M-theory can be accurately described by the four-dimensional action of the theory - they have a four-dimensional limit. We consider the connection between solutions of four and five-dimensional heterotic M-theory when moving five-branes are present in the bulk. We begin by describing how to raise the known four-dimensional moving brane solutions to obtain approximate solutions to the five-dimensional theory, presenting for the first time the metric template necessary for this procedure. We then present the first solutions to the five-dimensional theory containing moving five-branes. We can then discuss the connection between our new exact five-dimensional solution and the four-dimensional ones. It is shown that our new solution corresponds to a solution with a static brane in four-dimensions. In other words our new solution could not have been identified as containing a moving brane from a purely four-dimensional viewpoint. 
  We show that the exact RG-flow equation introduced recently in hep-th/0207134 can be obtained in the sharp cut-off limit of the well-known ERGE. This can be expected from the fact that in this limit the new scale-dependent effective action coincides with the one which is usually considered. 
  We consider branes embedded in spacetimes of codimension one and two, with a warped metric tensor for the subspace parallel to the brane. We study a variety of brane-world solutions arising by introducing a Schwarzschild-like black hole metric on the brane and we investigate the properties of the corresponding higher-dimensional spacetime. We demonstrate that normalizable bulk modes lead to a vanishing flow of energy through the naked singularities. From this point of view, these singularities are harmless. 
  We find a general class of pp-wave solutions of type IIB string theory such that the light cone gauge worldsheet lagrangian is that of an interacting massive field theory. When the light cone Lagrangian has (2,2) supersymmetry we can find backgrounds that lead to arbitrary superpotentials on the worldsheet. We consider situations with both flat and curved transverse spaces. We describe in some detail the background giving rise to the N=2 sine Gordon theory on the worldsheet. Massive mirror symmetry relates it to the deformed $CP^1$ model (or sausage model) which seems to elude a purely supergravity target space interpretation. 
  The explicit solutions of the Bogomolny equations for N vortices on a sphere of radius R^2 > N are not known. In particular, this has prevented the use of the geodesic approximation to describe the low energy vortex dynamics. In this paper we introduce an approximate general solution of the equations, valid for R^2 close to N, which has many properties of the true solutions, including the same moduli space CP^N. Within the framework of the geodesic approximation, the metric on the moduli space is then computed to be proportional to the Fubini- Study metric, which leads to a complete description of the particle dynamics. 
  We study the gauge coupling evolution of a unified theory in the compact Randall-Sundrum model with gauge bosons propagating in the bulk. One-loop corrections in AdS are interpreted in the 4d dual theory as the sum of two contributions: CFT insertions subleading in a 1/N expansion and loops of the additional particles coupled to the CFT. We have calculated the scalar loop correction to the low energy gauge couplings both in scenarios where the GUT symmetry is broken by boundary conditions and with the Higgs mechanism. In each case our results are what expected from the holographic dual theory. 
  We study string theory on the extended spacetime of the BTZ black hole, as described by an orbifold of the SL(2,R) WZW model. The full spacetime has an infinite number of disconnected boundary components, each corresponding to a dual CFT. We discuss the computation of bulk and boundary correlation functions for operators inserted on different components. String theory correlation functions are obtained by analytic continuation from an orbifold of the SL(2,C)/SU(2) coset model. This yields two-point functions for general operators, including those describing strings that wind around the horizon of the black hole. 
  We analyse the quantum corrected geometry and radiation in the scattering of extremal black holes by low-energy neutral matter. We point out the fact that the correlators of local observables inside the horizon are the same as those of the vacuum. Outside the horizon the correlators at late times are much bigger than those of the (thermal) case obtained neglecting the backreaction. This suggests that the corrected Hawking radiation could be compatible with unitarity. 
  The sine-Gordon model on the half-line with a dynamical boundary introduced by Delius and one of the authors is considered at quantum level. Classical boundary conditions associated with classical integrability are shown to be preserved at quantum level too. Non-local conserved charges are constructed explicitly in terms of the field and boundary operators. We solve the intertwining equation associated with a certain coideal subalgebra of $U_q(\hat{sl_2})$ generated by these non-local charges. The corresponding solution is shown to satisfy quantum boundary Yang-Baxter equations. Up to an exact relation between the quantization length of the boundary quantum mechanical system and the sine-Gordon coupling constant, we conjecture the soliton/antisoliton reflection matrix and boundstates reflection matrices. The structure of the boundary state is then considered, and shown to be divided in two sectors. Also, depending on the sine-Gordon coupling constant a finite set of boundary bound states are identified. Taking the analytic continuation of the coupling, the corresponding boundary sinh-Gordon model is briefly discussed. In particular, the particle reflection factor enjoys weak-strong coupling duality. 
  Manifestly invariant renormalization scheme for supersymmetric gauge theories is proposed. This scheme is applied to supersymmetric quantum electrodynamics. 
  We study the Heisenberg quantization for the systems of identical particles in noncommtative spaces. We get fermions and bosons as a special cases of our argument, in the same way as commutative case and therefore we conclude that the Pauli exclusion principle is also valid in noncommutative spaces. 
  We consider the dynamics of a FRW brane in a purely AdS background, from the point of view of the bulk, and explicitly construct the geodesical behaviour of gravitational signs leaving and subsequently returning to the brane. In comparison with photons following a geodesic inside the brane, we verify that shortcuts exist, though they are extremely small for today's Universe. However, we show that at times just before nucleosynthesis, if high redshifts were available, the effect could be sufficiently large to solve the horizon problem. Assuming an inflationary epoch in the brane evolution, we argue that the influence of those signs trought the extra dimension in the causal structure cannot be neglected. This effect could be relevant for probing the extra dimension in inflationary scenarios. 
  After discussing the general form of the kinetic operator around the tachyon vacuum, we determine the specific form of the pure-ghost kinetic operator Q^ by requiring the twist invariance of the action. We obtain a novel result that the Grassmann-even piece Q_even of Q^ must act differently on GSO(+) sector and on GSO(-) sector to preserve the twist invariance, and show that this structure is crucial for gauge invariance of the action. With this choice of Q^, we construct a solution in an approximation scheme which is conjectured to correspond to a non-BPS D9-brane. We consider both 0-picture cubic and Berkovits' non-polynomial superstring field theories for the NS sector. 
  We study the AdS/CFT correspondence for string states which flow into plane wave states in the Penrose limit. Leading finite radius corrections to the string spectrum are compared with scaling dimensions of finite R-charge BMN-like operators. We find agreement between string and gauge theory results. 
  We study exact string backgrounds representing a constant magnetic field background in heterotic string theory. These backgrounds are obtained by Kaluza-Klein reduction of a special class of plane wave solutions. For small values of the magnetic field they possess localized closed string tachyons analogous to the Nielsen-Olesen instability of a constant magnetic field in SO(3) Yang-Mills theory. When the magnetic field is embedded in the SO(32) gauge group of the heterotic string it is possible to study the lowest level tachyon in supergravity. We identify the closed string tachyons as fluctuations of the supergravity fields about the background. We argue that the tachyon signals the decay of the background to flat space. Our evidence rests on the study of the closed string tachyon potential, world sheet renormalization group equations in the supergravity approximation and the S-dual of this system in type I theory. In the S-dual description, the closed string tachyons in heterotic string theory correspond to open string tachyons in type I theory. Finally we analyze the non-perturbative stability of these models representing constant magnetic field backgrounds and show that it is stable to pair production of particles. 
  It is shown that AdS(5)xS(5) supergravity has hitherto unnoticed supersymmetric properties that are related to a hidden 12-dimensional structure. The totality of the AdS(5)xS(5) supergravity Kaluza-Klein towers is given by a single superfield that describes the quantum states of a 12-dimensional supersymmetric particle. The particle has super phase space (X,P,Theta) with (10,2) signature and 32 fermions. The worldline action is constructed as a generalization of the supersymmetric particle action in Two-Time Physics. SU(2,2|4) is a linearly realized global supersymmetry of the 2T action. The action is invariant under the gauge symmetries Sp(2,R), SO(4,2),SO(6), and fermionic kappa. These gauge symmetries insure unitarity and causality while allowing the reduction of the 12-dimensional super phase space to the correct super phase space for AdS(5)xS(5) or M(4)xR(6) with 16 fermions and one time, or other dually related one time spaces. One of the predictions of this formulation is that all of the SU(2,2|4) representations that describe Kaluza-Klein towers in AdS(5)xS(5) or M(4)xR(6) supergravity universally have vanishing eigenvalues for all the Casimir operators. This prediction has been verified directly in AdS(5)xS(5) supergravity. This suggests that the supergravity spectrum supports a hidden (10,2) structure. A possible duality between AdS(5)xS(5) and M(4)xR(6) supergravities is also indicated. Generalizations of the approach applicable 10-dimensional super Yang Mills theory and 11-dimensional M-theory are briefly discussed. 
  In this paper we consider the implications of a cosmological constant for the evolution of the universe, under a set of assumptions motivated by the holographic and horizon complementarity principles. We discuss the ``causal patch" description of spacetime required by this framework, and present some simple examples of cosmologies described this way. We argue that these assumptions inevitably lead to very deep paradoxes, which seem to require major revisions of our usual assumptions. 
  We describe the construction of configurations of D6-branes wrapped on compact 3-cycles intersecting at points in non-compact Calabi-Yau threefolds. Such constructions provide local models of intersecting brane worlds, and describe sectors of four-dimensional gauge theories with chiral fermions. We present several classes of non-compact manifolds with compact 3-cycles intersecting at points, and discuss the rules required for model building with wrapped D6-branes. The rules to build 3-cycles are simple, and allow easy computation of chiral spectra, RR tadpoles and the amount of preserved supersymmetry. We present several explicit examples of these constructions, some of which have Standard Model like gauge group and three quark-lepton generations. In some cases, mirror symmetry relates the models to other constructions used in phenomenological D-brane model building, like D-branes at singularities. Some simple N=1 supersymmetric configurations may lead to relatively tractable G_2 manifolds upon lift to M-theory, which would be non-compact but nevertheless yield four-dimensional chiral gauge field theories. 
  We consider giant graviton probes of 11-dimensional supergravity solutions which are lifts of arbitrary solutions of 4-dimensional U(1)^4 and 7-dimensional U(1)^2 gauged supergravities. We show that the structure of the lift ansatze is sufficient to explicitly find a solution for the embedding and motion of the M5- or M2-brane in the internal space. The brane probe action then reduces to that of a massive charged particle in the gauged supergravity, demonstrating that probing the gauged supergravity with such particles is equivalent to the more involved 11-dimensional brane probe calculation. As an application of this we use these particles to probe superstar geometries which are conjectured to be sourced by giant gravitons. 
  We study the thermodynamical observables of the 2d Ising model in the neighborhood of the magnetic axis by means of numerical diagonalization of the transfer matrix. In particular, we estimate the leading order corrections to the Zamolodchikov mass spectrum and find evidence of non-vanishing contributions due to the stress-energy tensor. 
  A method of calculation of the scattering amplitude for fermions and scalar bosons with exchanging of a scalar particle in ladder approximation is suggested. The Bethe-Salpeter ladder integral equations system for the imaginary part of the amplitude is costructed and solution in the Regge asymptotical form is found. The corrections to the amplitude due to the exit from mass shell are calculated and the real part of the amplitude is found. 
  A careful reduction of the three-dimensional gravity to the Liouville description is performed, where all gauge fixing and on-shell conditions come from the definition of asymptotic de Sitter spaces. The roles of both past and future infinities are discussed and the conditions space-time evolution imposes on both Liouville fields are explicited. Space-times which correspond to non-equivalent profiles of the Liouville field at past and future infinities are shown to exist. The qualitative implications of this for any tentative dual theory are presented. 
  We consider scalar Born-Infeld type theories with arbitrary potentials V(T) of a scalar field T. We find that for models with runaway potentials V(T) the generic inhomogeneous solutions after a short transient stage can be very well approximated by the solutions of a Hamilton-Jacobi equation that describes free streaming wave front propagation. The analytic solution for this wave propagation shows the formation of caustics with multi-valued regions beyond them. We verified that these caustics appear in numerical solutions of the original scalar BI non-linear equations. Our results include the scalar BI model with an exponential potential, which was recently proposed as an effective action for the string theory tachyon in the approximation where high-order spacetime derivatives of T are truncated. Since the actual string tachyon dynamics contain derivatives of all orders, the tachyon BI model with an exponential potential becomes inadequate when the caustics develop because high order spatial derivatives of T become divergent. BI type tachyon theory with a potential decreasing at large T could have interesting cosmological applications because the tachyon field rolling towards its ground state at infinity acts as pressureless dark matter. We find that inhomogeneous cosmological tachyon fluctuations rapidly grow and develop multiple caustics. Any considerations of the role of the tachyon field in cosmology will have to involve finding a way to predict the behavior of the field at and beyond these caustics. 
  Much of the recent progress in String Theory can be traced to a precise strategy: a careful study of the few models known since the beginnings of the subject, and the abstraction from them of basic properties that one would like to demand from other models. This could be termed a set of "model-building rules". The approach corresponds to the fact, often a source of embarrassment to specialists, that String Theory, born as a set of rules rather than as a set of principles, has long resisted attempts to reduce it to a logically satisfying structure.   Talk presented at the Cargese Summer Institute on Non-Perturbative Methods in Field Theory, Cargese, France, July 16-30, 1987. 
  We show that QCD Dirac spectra well below Lambda_{QCD}, both at zero and at nonzero chemical potential, can be obtained from a chiral Lagrangian. At nonzero chemical potential Goldstone bosons with nonzero baryon number condense beyond a critical value. Such superfluid phase transition is likely to occur in any system with a chemical potential with the quantum numbers of the Goldstone bosons. We discuss the phase diagram for one such system, QCD with two colors, and show the existence of a tricritical point in an effective potential approach. 
  We construct an effective superpotential that describes dynamical flavor symmetry breaking in supersymmetric N=1 SO(N_c) theories with N_f-flavor quarks for N_f >= N_c. Our superpotential induces spontaneous flavor symmetry breaking of SU(N_f) down to SO(N_c)xSU(N_f-N_c) as nonabelian residual groups and respects the anomaly-matching property owing to the appearance of massless composite Nambu-Goldstone superfields. In massive SO(N_c) theories, our superpotential provides holomorphic decoupling property and consistent vacuum structure with instanton effect if N_c-2 quarks remain massless. This superpotential may reflect the remnant of physics corresponding to N=2 SO(N_c) theories near the Chebyshev point, which also exhibits dynamical flavor symmetry breaking. 
  We consider issues related to tachyonic inflation with inverse-power-law potential. We find the solution of the evolution equations in the slow roll limit in FRW as well as in the brane cosmology. Using the holographic entropy bound, we estimate the quantum-gravitational discreteness of tachyonic inflation perturbations. 
  The possibility that non-supersymmetric quiver theories may have a renormalization-group fixed point at which there is conformal invariance requires non-perturbative information. 
  We solve exactly the equations of motion for linearized gravity in the Randall-Sundrum model with matter on the branes and calculate the Newtonian limit in it. The result contains contributions of the radion and of the massive modes, which change considerably Newton's law at small distances. The effects of "shadow" matter, which lives on the other brane, are considered and compared with those of ordinary matter for both positive and negative tension branes. We also calculate light deflection and Newton's law in the zero mode approximation and explicitly distinguish the contribution of the radion field. 
  We consider the toroidal black holes that arise as a generalization of the AdS_5 times S^5 solution of type IIB supergravity. The symmetries of the horizon space allow T-duality transformations that can be exploited to generate new inequivalent black hole solutions of both type IIB and type IIA supergravity, with non-trivial dilaton, B-field, and RR forms. We examine the asymptotic structure and thermodynamical properties of these solutions. 
  Topological objects resulting from symmetry breakdown may be either stable or metastable depending on the pattern of symmetry breaking. However, if they trap zero-energy modes of fermions, and in the process acquire non-integer fermionic charge, the metastable configurations also get stabilized. In the case of Dirac fermions the spectrum of the number operator shifts by 1/2. In the case of majorana fermions it becomes useful to assign negative values of fermion number to a finite number of states occupying the zero-energy level, constituting a \textit{majorana pond}. We determine the parities of these states and prove a superselection rule. Thus decay of objects with half-integer fermion number is not possible in isolation or by scattering with ordinary particles. The result has important bearing on cosmology as well as condensed matter physics. 
  In this paper we present time dependent solution of the open bosonic string field theory describing the motion of the tachyon on unstable D-brane. 
  We study type IIA string theories on the pp-waves with 24 supercharges. The type IIA pp-wave backgrounds are derived from the maximally supersymmetric pp-wave solution in eleven dimensions through the toroidal compactification on the spatial isometry directions. The associated actions of type IIA strings are obtained by using these metrics and other background fields of the type IIA supergravities on the one hand. On the other hand, we derive these theories from D=11 supermembrane on the pp-wave via double dimensional reduction for the spatial isometry directions. The resulting actions agree with those of type IIA strings obtained in the study of the supergravities. Also, the action of the matrix string is written down. Moreover, the quantization of closed and open strings is discussed. In particular, we study Dp-branes allowed in one of the type IIA theories. 
  We develop a systematic method for classifying supersymmetric orbifold compactifications of M-theory. By restricting our attention to abelian orbifolds with low order, in the special cases where elements do not include coordinate shifts, we construct a "periodic table" of such compactifications, organized according to the orbifolding group (order up to 12) and dimension (up to 7). An intriguing connection between supersymmetric orbifolds and G2-structures is explored. 
  We study physical applications of the Bohm metrics, which are infinite sequences of inhomogeneous Einstein metrics on spheres and products of spheres of dimension 5 <= d <= 9. We prove that all the Bohm metrics on S^3 x S^2 and S^3 x S^3 have negative eigenvalue modes of the Lichnerowicz operator and by numerical methods we establish that Bohm metrics on S^5 have negative eigenvalues too. We argue that all the Bohm metrics will have negative modes. These results imply that higher-dimensional black-hole spacetimes where the Bohm metric replaces the usual round sphere metric are classically unstable. We also show that the stability criterion for Freund-Rubin solutions is the same as for black-hole stability, and hence such solutions using Bohm metrics will also be unstable. We consider possible endpoints of the instabilities, and show that all Einstein-Sasaki manifolds give stable solutions. We show how Wick rotation of Bohm metrics gives spacetimes that provide counterexamples to a strict form of the Cosmic Baldness conjecture, but they are still consistent with the intuition behind the cosmic No-Hair conjectures. We show how the Lorentzian metrics may be created ``from nothing'' in a no-boundary setting. We argue that Lorentzian Bohm metrics are unstable to decay to de Sitter spacetime. We also argue that noncompact versions of the Bohm metrics have infinitely many negative Lichernowicz modes, and we conjecture a general relation between Lichnerowicz eigenvalues and non-uniqueness of the Dirichlet problem for Einstein's equations. 
  We count the BPS states of strings in uniform 3-form fluxes, using supersymmetric quantum mechanics derived from the kappa-symmetric action for D-branes. This problem is relevant to the stringy physics of warped compactifications. We work on a type IIB T^6/Z_2 orientifold with imaginary self-dual, quantized, 3-form flux. Ignoring the orientifold projection, the number of short multiplets living on a single string is the square of the units of 3-form flux present on the torus; the orientifold removes roughly half of the multiplets. We review the well-known case of a superparticle on T^2 as an pedagogical example. 
  We consider the evolution of small rotational perturbations, with azimuthal symmetry, of the brane-world cosmological models. The equations describing the temporal, radial, and angular dependence of the perturbations are derived by taking into account the effects of both scalar and tensor parts of the dark energy term on the brane. The time decay of the initial rotation is investigated for several types of equation of state of the ultra-high density cosmological matter. For an expanding Universe, rotation always decays in the case of the perfect dragging, for which the angular velocity of the matter on the brane equals the rotation metric tensor. For non-perfect dragging, the behavior of the rotation is strongly equation of state dependent. For some classes of dense matter, like the stiff causal or the Chaplygin gas, the angular velocity of the matter on the brane is an increasing function of time. For other types of the ultra-dense matter, like the Hagedorn fluid, rotation is smoothed out by the expansion of the Universe. Therefore the study of dynamics of rotational perturbations of brane world models, as well as in general relativity, could provide some insights on the physical properties and equation of state of the cosmological fluid filling the very early Universe. 
  In these notes we review the method to construct integrable deformations of the compactified c=1 bosonic string theory by primary fields (momentum or winding modes), developed recently in collaboration with S. Alexandrov and V. Kazakov. The method is based on the formulation of the string theory as a matrix model. The flows generated by either momentum or winding modes (but not both) are integrable and satisfy the Toda lattice hierarchy. 
  This is a critical review of inert properties of classical relativistic point objects. The objects are classified as Galilean and non-Galilean. Three types of non-Galilean objects are considered: spinning, rigid, and dressed particles. In the absence of external forces, such particles are capable of executing not only uniform motions along straight lines but also Zitterbewegungs, self-accelerations, self-decelerations, and uniformly accelerated motions. A free non-Galilean object possesses the four-velocity and the four-momentum which are in general not collinear, therefore, its inert properties are specified by two, rather than one, invariant quantities. It is shown that a spinning particle need not be a non-Galilean object. The necessity of a rigid mechanics for the construction of a consistent classical electrodynamics in spacetimes of dimension D+1 is justified for D+1>4. The problem of how much the form of fundamental laws of physics orders four dimensions of our world is revised together with its solution suggested by Ehrenfest. The present analysis made it apparent that the notion of the ``back-reaction'' does not reflect the heart of the matter in classical field theories with point-like sources, the notion of ``dressed'' particles proves more appropriate. 
  Brane-like vertex operators play an important role in a worldsheet formulation of D-branes and M-theory. In this paper we derive the DBI D-brane action from NSR closed string sigma-model with brane-like states. We also show that these operators carry RR charges and define D-brane wavefunctions in a second quantized formalism. 
  We analyse the spacetime structure of the global vortex and its maximal analytic extension in an arbitrary number of spacetime dimensions. We find that the vortex compactifies space on the scale of the Hubble expansion of its worldvolume, in a manner reminiscent of that of the domain wall. We calculate the effective volume of this compactification and remark on its relevance to hierarchy resolution with extra dimensions. We also consider strongly gravitating vortices and derive bounds on the existence of a global vortex solution. 
  We derive boundary conditions for the covariant open string corresponding to D-branes in an Hpp-wave, by requiring kappa symmetry of its bulk action. Both half-supersymmetric and quarter-supersymmetric branes are seen to arise in this way, and the analysis furthermore agrees fully with the existing probe brane and supergravity computations. We elaborate on the origin of dynamical and kinematical supersymmetries from the covariant point of view. In particular we focus on the D-string which only preserves half of the dynamical supersymmetries and none of the kinematical ones. We discuss its origin in AdS_5 x S^5 and its world-volume spectrum. 
  We study the superpotential of a certain class of N=1 supersymmetric type II compactifications with fluxes and D-branes. We show that it has an important two-dimensional meaning in terms of a chiral ring of the topologically twisted theory on the world-sheet. In the open-closed string B-model, this chiral ring is isomorphic to a certain relative cohomology group V, which is the appropriate mathematical concept to deal with both the open and closed string sectors. The family of mixed Hodge structures on V then implies for the superpotential to have a certain geometric structure. This structure represents a holomorphic, N=1 supersymmetric generalization of the well-known N=2 special geometry. It defines an integrable connection on the topological family of open-closed B-models, and a set of special coordinates on the space \cal M of vev's in N=1 chiral multiplets. We show that it can be given a very concrete and simple realization for linear sigma models, which leads to a powerful and systematic method for computing the exact non-perturbative N=1 superpotentials for a broad class of toric D-brane geometries. 
  When we have noncommutativity among coordinates (or conjugate momenta), we consider Wigner's formulation of quantum mechanics, including a new derivation of path integral formula. We also propose the Moyal star product based on the Dirac bracket in constrained systems. 
  We consider a BMN operator with one scalar, phi, and one vector, D_{m}Z, impurity field and compute the anomalous dimension both at planar and torus levels. This "mixed" operator corresponds to a string state with two creation operators which belong to different SO(4) sectors of the background. The anomalous dimension at both levels is found to be the same as the scalar impurity BMN operator. At planar level this constitutes a consistency check of BMN conjecture. Agreement at the torus level can be explained by an argument using supersymmetry and supression in the BMN limit. The same argument implies that a class of fermionic BMN operators also have the same planar and torus level anomalous dimensions. Implications of the results for the map from N=4 SYM theory to string theory in the pp-wave background are discussed. 
  We present several supergravity solutions corresponding to both Dp, as well as Dp-Dp' systems, in NS-NS and R-R PP-wave background originating from AdS_3 times S^3 times R^4. The Dp brane solutions, p=1,..,5 are fully localized, whereas Dp-Dp' are localized along common transverse directions. We also discuss the supersymmetry properties of these solutions and the worldsheet construction for the p-p' system. 
  Lagrangians of the Abelian Gauge Theory and its dual are related in terms of a shifted action. We show that in d=4 constrained Hamiltonian formulation of the shifted action yields Hamiltonian description of the dual theory, without referring to its Lagrangian. We apply this method, at the first order in the noncommutativity parameter theta, to the noncommutative U(1) gauge theory possessing spatial noncommutativity. Its dual theory is effectively a space--time noncommutative U(1) gauge theory. However, we obtain a Hamiltonian formulation where time is commuting. Space-time noncommutative D3--brane worldvolume Hamiltonian is derived as the dual of space noncommutative U(1) gauge theory. We show that a BPS like bound can be obtained and it is saturated for configurations which are the same with the ordinary D3-brane BIon and dyon solutions. 
  Requiring the existence of certain BPS solutions to the equations of motion, we determine the bosonic part of the non-abelian D-brane effective action through order $\alpha'{}^4$. We also propose an economic organizational principle for the effective action. 
  We show that the Woronowicz prescription using a bimodule constructed out of a tensorial product of a bimodule and its conjugate and a bi-coinvariant singlet leads to a trivial differential calculus. 
  We present a consistent framework that enables us to understand the big bang singularity of our universe. 
  We argue that, given the nonlocal nature of precursors in AdS/CFT correspondence, the boundary field theory contains information about events inside a black hole horizon. The essence of our proposal is sketched in figure 1, and relies on the global nature of event horizons. 
  We argue that for a large class of N=1 supersymmetric gauge theories the effective superpotential as a function of the glueball chiral superfield is exactly given by a summation of planar diagrams of the same gauge theory. This perturbative computation reduces to a matrix model whose action is the tree-level superpotential. For all models that can be embedded in string theory we give a proof of this result, and we sketch an argument how to derive this more generally directly in field theory. These results are obtained without assuming any conjectured dualities and can be used as a systematic method to compute instanton effects: the perturbative corrections up to n-th loop can be used to compute up to n-instanton corrections. These techniques allow us to see many non-perturbative effects, such as the Seiberg-Witten solutions of N=2 theories, the consequences of Montonen-Olive S-duality in N=1* and Seiberg-like dualities for N=1 theories from a completely perturbative planar point of view in the same gauge theory, without invoking a dual description. 
  We study the superconformal and super-BRS invariance of the supersymmetric Wess-Zumino-Witten model based on Lie superalgebra. The computation of the critical super-dimension of this model is done using the Fujikawa regularization. Finally, we recover the well-known result which fixes the relative coupling constant a2 = 1 in a rigorous way. 
  Current theoretical investigations seem to indicate the possibility of observing signatures of short distance physics in the Cosmic Microwave Background spectrum. We try to gain a deeper understanding on why all information about this regime is lost in the case of Black Hole radiation but not necessarily so in a cosmological setting by using the moving mirror as a toy model for both backgrounds. The different responses of the Hawking and Cosmic Microwave Background spectra to short distance physics are derived in the appropriate limit when the moving mirror mimics a Black Hole background or an expanding universe. The different sensitivities to new physics, displayed by both backgrounds, are clarified through an averaging prescription that accounts for the intrinsic uncertainty in their quantum fluctuations. We then proceed to interpret the physical significance of our findings for time-dependent backgrounds in the light of nonlocal string theory. 
  We investigate the four-dimensional supergravity theory obtained from the compactification of eleven-dimensional supergravity on a smooth manifold of G_2 holonomy. We give a new derivation for the Kaehler potential associated with the scalar kinetic term of the N=1 four-dimensional theory. We then examine some solutions of the four-dimensional theory which arise from wrapped M-branes. 
  The virtual black hole phenomenon, which has been observed previously in specific models, is established for generic 2D dilaton gravity theories with scalar matter. The ensuing effective line element can become asymptotically flat only for two classes of models, among them spherically reduced theories and the string inspired dilaton black hole (CGHS model).   We present simple expressions for the lowest order scalar field vertices of the effective theory which one obtains after integrating out geometry exactly. Treating the boundary in a natural and simple way asymptotic states, tree-level vertices and tree-level S-matrix are conformally invariant.   Examples are provided pinpointing the physical consequences of virtual black holes on the (CPT-invariant) S-matrix for gravitational scattering of scalar particles. For minimally coupled scalars the evaluation of the S-matrix in closed form is straightforward.   For a class of theories including the CGHS model all tree-graph vertices vanish, which explains the particular simplicity of that model and at the same time shows yet another essential difference to the Schwarzschild case. 
  We present new, non trivial generalizations of the recent Tomaras, Tsamis and Woodard extension of the original Schwinger formula for charged pair production in a constant field. 
  Path integrals can be rigorously defined only in low dimensional systems where the small distance limit can be taken. Particularly non-trivial models in more than four dimensions can only be handled with considerable amount of speculation. In this lecture we try to put these various aspects in perspective. 
  We review the relations between a family of domain-wall solutions to M-theory and gravitational instantons with special holonomy. When oxidized into the maximal-dimension parent supergravity, the transverse spaces of these domain walls become cohomogeneity-one spaces with generalized Heisenberg symmetries and a homothetic conformal symmetry. These metrics may also be obtained as scaling limits of generalized Eguchi-Hanson metrics, or, with appropriate discrete identifications, from generalized Atiyah-Hitchin metrics, thus providing field-theoretic realizations of string-theory orientifolds. 
  A straightforward generalization of the celebrated uniqueness theorem to dimensions greater than four was recently found to fail in two pure gravity cases - the 5d rotating black ring and the black string on R^{3,1} * S^1. Two amendments are suggested here (without proof) in order to rectify the situation. The first is that in addition to specifying the mass and angular momentum (and gauge charges) one needs to specify the horizon topology as well. Secondly, the theorem may survive if applied exclusively to stable solutions. Note that the latter is at odds with the proposed stable but non-uniform string. 
  A physically reasonable model is introduced in order to estimate, in a functional way, the vast number of distinct graphs which are conventionally neglected in eikonal scattering models that lead to total cross sections increasing with energy in the form of the Froissart bound. 
  Using a mathematical framework which provides a generalization of the de Rham complex (well-designed for p-form gauge fields), we study the gauge structure and duality properties of theories for free gauge fields transforming in arbitrary irreducible representations of GL(D,R). We prove a generalization of the Poincare lemma which enables us to solve the above-mentioned problems in a systematic and unified way. 
  We discuss gauge fields on tori in diverse dimensions, mainly in two and four dimensions. We construct various explicit gauge fields which have some topological charges and find the Dirac zero modes in the background of the gauge fields. By using the zero mode, we give new gauge fields on the dual torus, which is a gauge theoretical description of T-duality transformation of the corresponding D-brane systems including DDbar systems. From the transformation, we can easily see the duality expected from the index theorem. It is also mentioned that, for each topological charges, the corresponding constant curvature bundle can be constructed and their duality transformation can be performed in terms of Heisenberg modules. 
  This work is an extension of our previous work, hep-th/0204160, which showed how to systematically calculate the high energy evolution of gauge couplings in compact AdS_5 backgrounds. We first directly compute the one-loop effects of massive charged scalar fields on the low energy couplings of a gauge theory propagating in the AdS background. It is found that scalar bulk mass scales (which generically are of order the Planck scale) enter only logarithmically in the corrections to the tree-level gauge couplings. As we pointed out previously, we show that the large logarithms that appear in the AdS one-loop calculation can be obtained within the confines of an effective field theory, by running the Planck brane correlator from a high UV matching scale down to the TeV scale. This result exactly reproduces our previous calculation, which was based on AdS/CFT duality. We also calculate the effects of scalar fields satisfying non-trivial boundary conditions (relevant for orbifold breaking of bulk symmetries) on the running of gauge couplings. 
  We propose a scalar background in Minkowski spacetime imparting constant proper acceleration to a classical particle. In contrast to the case of a constant electric field the proposed scalar potential does not create particle-antiparticle pairs. Therefore an elementary particle accelerated by such field is a more appropriate candidate for an "Unruh-detector" than a particle moving in a constant electric field. We show that the proposed detector does not reveal the universal thermal response of the Unruh type. 
  We investigate the gauging of a two-dimensional deformation of the Poincare algebra, which accounts for the existence of an invariant energy scale. The model describes 2D dilaton gravity with torsion. We obtain explicit solutions of the field equations and discuss their physical properties. 
  We investigate a method to evaluate quasi-normal modes of D3-brane black holes by wave interpretation of fields on D3-brane based on the Feynman's space-time approach. We perturbatively solve the wave equation which describes propagation of a dilaton wave in a bulk space and its interaction with the D3-brane. The obtained condition for the quasi-normal modes are qualitatively equivalent to that evaluated in the usual scattering of the dilaton in the black 3-brane spacetime in the corresponding supergravity description. 
  Lorentz symmetry violation (LSV) can be generated at the Planck scale, or at some other fundamental length scale, and naturally preserve Lorentz symmetry as a low-energy limit (deformed Lorentz symmetry, DLS). DLS can have important implications for ultra-high energy cosmic-ray physics (see papers physics/0003080, astro-ph/0011181 and astro-ph/0011182, and references quoted in these papers). A crucial question is how DLS can be extended to a deformed Poincare symmetry (DPS), and what can be the dynamical origin of this phenomenon. We discuss recent proposals to identify DPS with a symmetry incorporating the Planck scale (doubly special relativity, DSR) and suggest new ways in this direction. Implications for models of quadratically deformed relativistic kinematics (QDRK) and linearly deformed relativistic kinematics (LDRK) are also discussed. 
  We analyse the spectrum of perturbations of the de Sitter space on the one hand, while on the other hand we compute the location of the poles in the Conformal Field Theory (CFT) propagator at the border. The coincidence is striking, supporting a dS/CFT correspondence. We show that the spectrum of thermal excitations of the CFT at the past boundary $I^{-}$ together with that spectrum at the future boundary $I^{+}$ is contained in the quasi-normal mode spectrum of the de Sitter space in the bulk. 
  We calculate the thermal partition function in the canonical ensemble for type IIB superstrings in the plane wave background with constant null R-R 5-form. The Hagedorn temperature is found to be higher than the corresponding value for strings in flat space. In the limit corresponding to the weakly coupled field theory we find that the Hagedorn temperature is pushed to infinity. The key property of strings in the plane wave background under investigation on which our result relies is that the effective mass of the bosonic and fermionic coordinates in the light-cone gauge is proportional to the momentum $p^+$. The free energy is finite as the Hagedorn temperature is approached from below, suggesting a possible phase transition. 
  We construct massive open string states around a classical solution in the oscillator formulation of Vacuum String Field Theory. In order for the correct mass spectrum to be reproduced, the projection operators onto the modes of the left- and right-half of the string must have an anomalous eigenvalue 1/2, and the massive states are constructed using the corresponding eigenvector. We analyze numerically the projection operators by regularizing them to finite size matrices and confirm that they indeed have eigenvalue 1/2. Beside the desired massive states, we have spurious massive as well as massless states, which are infinitely degenerate. We show that these unwanted states can be gauged away. 
  By intersecting the RR charged D_p - bar{D}_p pair (p=6, 4, 2, 0) with the RR F7-brane and by intersecting the NSNS charged F1-bar{F1} and NS5-bar{NS5} pairs with the NSNS F6-branes, the possibility of stabilizing the brane-antibrane systems is considered. The behavior of the corresponding supergravity solutions indicates that the RR F7-brane content of the solution plays the role of keeping the brane and the antibrane from annihilating each other completely since the two-brane configuration structure still persists in the vanishing inter-brane distance limit of the supergravity solution. In terms of the stringy description, we interpret this as representing that the RR F7-brane ``delays'' the brane-antibrane annihilation process but only until this non-supersymmetric and hence unstable F7-brane itself decays. Then next, the behavior of the supergravity solutions representing F1-bar{F1} and NS5-bar{NS5} again for vanishing inter-brane separation reveals that as they approach, these ``NS''-charged brane and antibrane always collide and annihilate irrespective of the presence or the absence of the NSNS F6-brane. And we have essentially attributed this to the absence of (open) stringy description of the instability in the ``NS''-charged case. This interpretation may provide a resolution to the contrasting features between the instability of ``R''-charged brane-antibrane systems and that of `NS''-charged ones. Certainly, however, it poses another puzzle that in the ``NS''-charged case, the quantum entity, that should take over the semi-classical instability as the inter-brane distance gets smaller, is missing. This is rather an embarrassing state of affair that needs to be treated with great care. 
  We investigate the explicit form of the fermionic zeromodes in heterotic fivebrane backgrounds. By explicitly solving the fermionic field equations in fivebrane backgrounds, two normalizable and physical fermionic zeromodes are obtained. Each of these zeromodes has a non-vanishing gravitino component. We suggest a possible scenario of the gravitino pair condensation. 
  We analyze the spectrum of the N=(2,2) supersymmetric Landau-Ginzburg theory in two dimensions with superpotential W=X^{n+2}-lambda X^2. We find the full BPS spectrum of this theory by exploiting the direct connection between the UV and IR limits of the theory. The computation utilizes results from the Picard-Lefschetz theory of singularities and its extension to boundary singularities. The additional fact that this theory is integrable requires that the BPS states do not close under scattering. This observation fixes the masses of non-BPS states as well. 
  We calculate the full 1-loop corrections to the low energy coupling of bulk gauge boson in a slice of AdS5 which are induced by generic 5-dimensional scalar, Dirac fermion, and vector fields with arbitrary Z_2 times Z_2' orbifold boundary conditions. In supersymmetric limit, our results correctly reproduce the results obtained by an independent method based on 4-dimensional effective supergravity. This provides a nontrivial check of our results and assures the regularization scheme-independence of the results. 
  A procedure to obtain noncommutative version for any nondegenerated dynamical system is proposed and discussed. The procedure is as follow. Let $S=\int dt L(q^A, ~ \dot q^A)$ is action of some nondegenerated system, and $L_1(q^A, ~ \dot q^A, ~ v_A)$ is the corresponding first order Lagrangian. Then the corresponding noncommutative version is $S_N=\int dt[ L_1(q^A, ~ \dot q^A, \~ v_A)+ \dot v_A\theta^{AB}v_B]$. Namely, the system $S_N$ has the following properties: 1) It has the same number of physical degrees of freedom as the initial system $S$. 2) Equations of motion of the system are the same as for the initial system $S$, modulo the term which is proportional to the parameter $\theta^{AB}$. 3) Configuration space variables have the noncommutative brackets: $\{q^A, ~ q^B\}=-2\theta^{AB}$. It is pointed also that quantization of the system $S_N$ leads to quantum mechanics with ordinary product replaced by the Moyal product. 
  We discuss Strassler's hypothesis of matching nonperturbative effects in orbifold pairs of gauge theories which are perturbatively planar equivalent. One of the examples considered is the parent N=1 SU(N) supersymmetric Yang-Mills theory and its nonsupersymmetric orbifold daughter. We apply two strategies allowing us to study nonperturbative effects: (i) low-energy theorems; (ii) putting the theory on small-size T^4 or R^3 x S^1 the parent and daughter theories are weakly coupled and amenable to quasiclassical treatment. In all cases our consideration yields a mismatch between the parent and daughter theories. Thus, regretfully, we present evidence against Strassler's hypothesis. 
  We construct IIA GS superstring action on the ten-dimensional pp-wave background, which arises as the compactification of eleven-dimensional pp-wave geometry along the isometry direction. The background geometry has 24 Killing spinors and among them, 16 components correspond to the non-linearly realized kinematical supersymmetry in the string action. The remaining eight components are linearly realized and shown to be independent of x^+ coordinate, which is identified with the world-sheet time coordinate of the string action in the light-cone gauge. The resultant dynamical N=(4,4) supersymmetry is investigated, which is shown to be consistent with the field contents of the action containing two free massive supermultiplets. 
  Using Darboux transformation one can construct infinite family of potentials which lead to the flat spectrum of scalar field fluctuations with arbitrary multiple precision, and, at the same time, with "essentially blue" spectrum of perturbations of metric. Besides, we describe reconstruction problem: find classical potential V(phi) starting from the known "one-loop potential" u(t) = d^2V(phi(t))/d phi(t)^2. 
  We construct actions for (p,0)- and (p,1)- supersymmetric, 1 <= p <= 4, two-dimensional gauge theories coupled to non-linear sigma model matter with a Wess-Zumino term. We derive the scalar potential for a large class of these models. We then show that the Euclidean actions of the (2,0) and (4,0)-supersymmetric models without Wess-Zumino terms are bounded by topological charges which involve the equivariant extensions of the Kahler forms of the sigma model target spaces evaluated on the two-dimensional spacetime. We give similar bounds for Euclidean actions of appropriate gauge theories coupled to non-linear sigma model matter in higher spacetime dimensions which now involve the equivariant extensions of the Kahler forms of the sigma model target spaces and the second Chern character of gauge fields. The BPS configurations are generalisations of abelian and non-abelian vortices. 
  We consider the plane wave limit of the nonspherical giant gravitons. We compute the Poisson brackets of the coordinate functions and find a nonlinear algebra. We show that this algebra solves the supersymmetry conditions of the matrix model. This is the generalization of the algebraic realization of the spherical membrane as the ``fuzzy sphere''. We describe finite dimensional representations of the algebra corresponding to the fuzzy torus. 
  We consider four-dimensional N=2 superconformal field theories based on ADE quiver diagrams. We use the procedure of hep-th/0206079 and compute the exact anomalous dimensions of operators with large U(1)_R charge to all orders in perturbation in the planar limit. The results are in agreement with the string computation in the dual pp-wave backgrounds. 
  We reproduce the correspondence between open string states |n> and BMN operators O_n in N=2 supersymmetric gauge theory using a worldsheet computation with the relation <B|n>=O_n, where <B| is a boundary state of D3-branes. We regard the NS ground states and their excitations with respect to a bosonic oscillator as the states |n> and obtain a chiral operator and BMN operators, respectively, in correspondence to them. Because we take the coupling constant to be weak and the background spacetime to be flat Minkowski, not AdS, the string states |n> are circular waves around the D3-branes, instead of KK modes on S^5. We also discuss the 1/J correction to the correspondence. 
  We study the system of Schwarzschild anti de Sitter (S-AdS) bulk and FRW brane for localization of gravity; i.e. zero mass gravitons having ground state on the brane, and thereby recovering the Einstein gravity with high energy correction. It has been known that gravity is not localized on AdS brane with AdS bulk. We prove the general result that gravity is not localized for dynamic branes whenever Lambda_4 < 0, and is localized for the curvature index k = 1 only when Lambda_4 > 0 and black hole mass M not equal 0, else it is localized for all other FRW models. If the localization is taken as the brane world compatibility criterion for cosmological models, then it would predict that negative cosmological constant on the brane is not sustainable. 
  We study the duality between IIB string theory on a pp-wave background, arising as a Penrose limit of the AdS_3 times S^3 times M, where M is T^4 (or K3), and the 2D CFT which is given by the N=(4,4) orbifold (M)^N/S_N, resolved by a blowing-up mode. After analizying the action of the supercharges on both sides, we establish a correspondence between the states of the two theories. In particular and for the T^4 case, we identify both massive and massless oscillators on the pp-wave, with certain classes of excited states in the resolved CFT carrying large R-charge n. For the former, the excited states involve fractional modes of the generators of the N=4 chiral algebra acting on the Z_n ground states. For the latter, they involve, fractional modes of the U(1)^4_L times U(1)^4_R super-current algebra acting on the Z_n ground states. By using conformal perturbation theory we compute the leading order correction to the conformal dimensions of the first class of states, due to the presence of the blowing up mode. We find agreement, to this order, with the corresponding spectrum of massive oscillators on the pp-wave. We also discuss the issue of higher order corrections. 
  An off-shell formulation of two distinct tensor multiplets,a massive tensor multiplet and a tensor gauge multiplet, is presented in superconformal tensor calculus in five-dimensional space-time. Both contain a rank 2 antisymmetric tensor field, but there is no gauge symmetry in the former, while it is a gauge field in the latter. Both multiplets commonly have 4 bosonic and 4 fermionic {\em on-shell} modes, but the former consists of 16(boson)+16(fermion) component fields while the latter consists of 8(boson)+8(fermion) component fields. 
  Black p-brane solutions for a wide class of intersection rules and Ricci-flat ``internal'' spaces are considered. They are defined up to moduli functions H_s obeying non-linear differential equations with certain boundary conditions imposed. A new solution with intersections corresponding to the Lie algebra A_3 is obtained. The functions H_1, H_2 and H_3 for this solution are polynomials of degree 3, 4 and 3, correspondingly. An example of A_3-solution with three 3-branes in 12-dimensional model (suggested by N. Khviengia et al) is presented. 
  The origin of the radiatively induced Lorentz and CPT violations, in perturbative evaluations, of an extended version of QED, is investigated. Using a very general calculational method, concerning the manipulations and calculations involving divergent amplitudes, we clearly identify the possible sources of contributions for the violating terms. We show that consistency in the perturbative calculations, in a broader sense, leaves no room for the existence of radiatively induced contributions which is in accordance with what was previously conjectured and recently advocated by some authors supported on general arguments. 
  We construct supergravity solutions for D-branes in nontrivial flux backgrounds. We revisit the issue of charge quantization in this framework, and show that in these backgrounds, charge need not be quantized. We also show in a particular example that the semiclassical description of branes produces integral charges. 
  We study bouncing and cyclic universes from an (n+1)-dimensional brane in the (n+2)-dimensional charged AdS bulk background. In the moving domain wall (MDW) approach this picture is clearly realized with a specified bulk configuration, the 5D charged topological AdS (CTAdS_5) black hole with mass M and charge Q. The bulk gravitational dynamics induces the 4D Friedmann equations with CFT-radiation and exotic stiff matters for a dynamic brane. This provides bouncing universes for k=0, -1 and cyclic universe for k=1, even though it has an exotic stiff matter from the charge Q. In this work we use the other of the Binetruy-Deffayet-Langlos (BDL) approach with the bulk Maxwell field. In this case we are free to determine the corresponding mass M-tilde and charge Q-tilde because the mass term is usually included as an initial condition and the charge is given by an unspecified solution to the Maxwell equation under the BDL metric. Here we obtain only bouncing universes if one does not choose two CTAdS_5 black holes as the bulk spacetime. We provide a way of avoiding the exotic matter on the brane by introducing an appropriate local matter. Finally we discuss an important relation between the exotic holographic matter and Lorentz invariance violation. 
  Recently the operator algebra, including the twisted affine primary fields, and a set of twisted KZ equations were given for the WZW permutation orbifolds. In the first part of this paper we extend this operator algebra to include the so-called orbifold Virasoro algebra of each WZW permutation orbifold. These algebras generalize the orbifold Virasoro algebras (twisted Virasoro operators) found some years ago in the cyclic permutation orbifolds. In the second part, we discuss the reducibility of the twisted affine primary fields of the WZW permutation orbifolds, obtaining a simpler set of single-cycle twisted KZ equations. Finally we combine the orbifold Virasoro algebra and the single-cycle twisted KZ equations to investigate the spectrum of each orbifold, identifying the analogues of the principal primary states and fields also seen earlier in cyclic permutation orbifolds. Some remarks about general WZW orbifolds are also included. 
  A nonperturbative approach for spontaneous symmetry breaking is proposed. It is based on some properties of interacting field operators. As the consequences an additional terms like to m^2 A^2 appears in the initial Lagrangian. 
  In this paper, imposing hermitian conjugate relations on the two-parameter deformed quantum group GL_{p,q}(2) is studied. This results in a non-commutative phase associated with the unitarization of the quantum group. After the achievement of the quantum group U_{p,q}(2) with pq real via a non-commutative phase, the representation of the algebra is built by means of the action of the operators constituting the U_{p,q}(2) matrix on states. 
  We study complex bosons and fermions coupled through a generalized Yukawa type coupling in the large-N_c limit following ideas of Rajeev [Int. Jour. Mod. Phys. A 9 (1994) 5583]. We study a linear approximation to this model. We show that in this approximation we do not have boson-antiboson and fermion-antifermion bound states occuring together. There is a possibility of having only fermion-antifermion bound states. We support this claim by finding distributional solutions with energies lower than the two mass treshold in the fermion sector. This also has implications from the point of view of scattering theory to this model. We discuss some aspects of the scattering above the two mass treshold of boson pairs and fermion pairs. We also briefly present a gauged version of the same model and write down the linearized equations of motion. 
  By using the two 4-dimensional potential formulation of electromagnetic (EM) field theory introduced in [1], we found that the SO(2) duality symmetric EM field theory can be reduced to the magnetic source free case by a special choice of SO(2) parameter,this special case we called nature picture of the EM field theory, the reduction condition led to a result, i.e. the electric charge and magnetic charge are no more independent. Some comments to paper [SD] are also mentioned. 
  Using the U(4) formalism developed ten years ago, the worldsheet action for the superstring in Ramond-Ramond plane wave backgrounds is expressed in a manifestly N=(2,2) superconformally invariant manner. This simplifies the construction of consistent Ramond-Ramond plane wave backgrounds and eliminates the problems associated with light-cone interaction point operators. 
  The possible time variation of dimensionless fundamental constants of nature, such as the fine-structure constant $\alpha$, is a legitimate subject of physical enquiry. By contrast, the time variation of dimensional constants, such as $\hbar$, $c$, $G$, $e$, $k$..., which are merely human constructs whose number and values differ from one choice of units to the next, has no operational meaning. To illustrate this, we refute a recent claim of Davies et al that black holes can discriminate between two contending theories of varying $\alpha$, one with varying $c$ and the other with varying $e$. In Appendix A we respond to criticisms by P. Davies and two {\it Nature} referees. In Appendix B we respond to remarks by Magueijo and by T. Davis. In Appendix C we critique recent claims by Copi, A. Davis and Krauss to have placed constraints on $\Delta G/G$. 
  We investigate light-cone structure on the world-volume of an unstable D-brane with a tachyon decaying inhomogeneously by using a field theoretical description. It is shown that (i) light-cones governing open strings are narrower than those governing closed strings and will eventually collapse inward in all directions except at kinks, where the tachyon remains at the top of its potential; and that (ii) light-cones governing open strings at a kink will be narrowed only in the direction perpendicular to the kink surface. It is also shown that (iii) future-directed light-cones governing open strings near a kink are tilted towards the kink, compared with those governing closed strings. The result (i) implies that open strings except at kinks are redshifted, compared with closed strings, and will eventually cease to be dynamical. On the other hand, the result (ii) shows that open strings on a kink surface can move freely along the kink surface and are dynamical but do not feel the existence of the spatial dimension perpendicular to the kink surface. The result (iii) indicates that open strings near a kink have tendency to move towards the kink. Hence, the light-cone structure vividly illustrates how open strings behave during the dynamical formation of a kink. We also discuss about a possibility that the early universe has a network of various dimensional D-branes, black-branes and tachyon matter. A problem associated with the network and a possible solution to the problem are discussed. 
  It has recently been suggested that, in a large N limit, a particular four dimensional gauge theory is indistinguishable from the six dimensional CFT with (0,2) supersymmetry compactified on a torus. We give further evidence for this correspondence by studying the Seiberg-Witten curve for the "deconstructed" theory and demonstrating that along the reduced Coulomb branch of moduli space (on the intersection of the Higgs and Coulomb branches) it describes the low energy physics on a stack of M5-branes on a torus, which is the (0,2) theory on a torus as claimed. The M-theory construction helps to clarify the enhancement of supersymmetry in the deconstructed theory at low energies, and demonstrates its stability to radiative and instanton corrections. We demonstrate the role of the theta vacuum in the deconstructed theory. We point out that by varying the theta parameters and gauge couplings in the deconstructed theory, the complex structure of the torus can be chosen arbitrarily, and the torus is not metrically S^1 x S^1 in general. 
  We review a solution of the cosmological constant problem in a brane-world model with infinite-volume extra dimensions. The solution is based on a nonlinear generally covariant theory of a metastable graviton that leads to a large-distance modification of gravity.     From the extra-dimensional standpoint the problem is solved due to the fact that the four-dimensional vacuum energy curves mostly the extra space. The four-dimensional curvature is small, being inversely proportional to a positive power of the vacuum energy. The effects of infinite-volume extra dimensions are seen by a brane-world observer as nonlocal operators.     From the four-dimensional perspective the problem is solved because the zero-mode graviton is extremely weakly coupled to localized four-dimensional sources. The observable gravity is mediated not by zero mode but, instead, by a metastable graviton with a lifetime of the order of the present-day Hubble scale. Therefore, laws of gravity are modified in the infrared above the Hubble scale. Large wave-length sources, such as the vacuum energy, feel only the zero-mode interaction and, as a result, curve space very mildly. Shorter wave-length sources interact predominantly via exchange of the metastable graviton. Because of this, all standard properties of early cosmology, including inflation, are intact. 
  We apply the spacetime dependent lagrangian formalism [1] to the action in general relativity. We obtain Barriola-Vilenkin [B-V] [2] type of topological solution by exploiting the electro-gravity duality [4] of the vacuum Einstein equations and using Dadhich's definition of empty space. The monopole mass M is shown to be of order a/G with a/G < M < 2a/G, a a small positive constant and G Newton's gravitational constant. The lower of these bounds can be written as M >= (a/G)beta, where 1< beta(=2k) < 2, 2k is the global monopole charge. This reminds us of the Bogomolny bound for usual monopoles. Comparing with the Barriola-Vilenkin scenario of a spontaneously broken scalar field theory, this can also provide an experimental check on the Grand Unification Scale (presently 10^{16}Gev) if the B-V monopole is ever discovered. PACS:11.15.-q, 11.27.+d, 14.80.Hv, 04. 
  We introduce the perturbative aspects of noncommutative quantum mechanics. Then we study the Berry's phase in the framework of noncommutative quantum mechanics. The results show deviations from the usual quantum mechanics which depend on the parameter of space/space noncommtativity. 
  Chiral p-forms are, in fact, present in many supersymmetric and supergravity models in two, six and ten dimensions. In this work, the dual projection procedure, which is essentially equivalent to a canonical transformation, is used to diagonalize some theories in D=2 (0-forms). The dual projection performed here provides an alternative way of gauging the chiral components without the necessity of constraints. It is shown, through the dual projection, that the nonmover field (the noton) initially introduced by Hull to cancel out the Siegel anomaly, has non-Abelian, PST and supersymmetric formulations. 
  We consider classical solutions for strings ending on magnetically charged black holes in four-dimensional Kaluza-Klein theory. We examine the classical superstring and the global vortex, which can be viewed as a nonsingular model for the superstring. We show how both of these can end on a Kaluza-Klein monopole in the absence of self-gravity. Including gravitational back-reaction gives rise to a confinement mechanism of the magnetic flux of the black hole along the direction of the string. We discuss the relation of this work to localized solutions in ten dimensional supergravity. 
  Composite S-brane solutions in multidimensional gravity with scalar fields and fields of forms related to Toda-like systems are presented. These solutions are defined on a product manifold R_{*} x M_1 x ... x M_n, where R_{*} is a time manifold, M_1 is an Einstein manifold and M_i (i > 1) are Ricci-flat manifolds. Certain examples of S-brane solutions related to A_1 + ... + A_1, A_m Toda chains and those with "block-orthogonal" intersections (e.g. SM-brane solutions) are singled out. Under certain restrictions imposed a Kasner-like asymptotical behaviour of the solutions is shown. 
  A small black hole attached to a brane in a higher dimensional space emitting quanta into the bulk may leave the brane as a result of a recoil. We construct a field theory model in which such a black hole is described as a massive scalar particle with internal degrees of freedom. In this model, the probability of transition between the different internal levels followed by emission of massless quanta is identical to the probability of thermal emission calculated for the Schwarzschild black hole. The discussed recoil effect implies that the thermal emission of the black holes, which might be created by interaction of high energy particles in colliders, could be terminated and the energy non-conservation can be observed in the brane experiments. 
  We construct a new set of intersecting D4-brane models that yield the (non-supersymmetric) standard model up to vector-like matter and, in some cases, extra U(1) factors in the gauge group.   The models are constrained by the requirement that twisted tadpoles cancel, and that the gauge boson coupled to the weak hypercharge U(1)_Y does not get a string-scale mass via a generalised Green-Schwarz mechanism. We find six-stack models that contain all of the Yukawa couplings to the tachyonic Higgs doublets that are needed to generate mass terms for the fermions at renormalisable level, but which have charged-singlet scalar tachyons and an unwanted extra U(1) gauge symmetry after spontaneous symmetry breaking. There is also a six-stack model without any unwanted gauged U(1) symmetries, but which only has the Yukawa couplings to generate masses for the u quarks and charged leptons. A particular eight-stack model is free of charged-singlet tachyons and has gauge coupling strengths whose ratios at the string scale are close to those measured at the electroweak scale, consistent with the string scale being at most a few TeV. 
  In this paper we explicitly work out the precise relationship between Ext groups and massless modes of D-branes wrapped on complex submanifolds of Calabi-Yau manifolds. Specifically, we explicitly compute the boundary vertex operators for massless Ramond sector states, in open string B models describing Calabi-Yau manifolds at large radius, directly in BCFT using standard methods. Naively these vertex operators are in one-to-one correspondence with certain sheaf cohomology groups (as is typical for such vertex operator calculations), which are related to the desired Ext groups via spectral sequences. However, a subtlety in the physics of the open string B model has the effect of physically realizing those spectral sequences in BRST cohomology, so that the vertex operators are actually in one-to-one correspondence with Ext group elements. This gives an extremely concrete physical test of recent proposals regarding the relationship between derived categories and D-branes. We check these results extensively in numerous examples, and comment on several related issues. 
  We investigate a minimal superspace description for 5D superconformal Killing vectors. The vielbein appropriate for AdS symmetry is discussed within the confines of this minimal supergeometry. 
  The systematization of the purely Lagrangean approach to constrained systems in the form of an algorithm involves the iterative construction of a generalized Hessian matrix W taking a rectangular form. This Hessian will exhibit as many left zero-modes as there are Lagrangean constraints in the theory. We apply this approach to a general Lagrangean in the first order formulation and show how the seemingly overdetermined set of equations is solved for the velocities by suitably extending W to a rectangular matrix. As a byproduct we thereby demonstrate the equivalence of the Lagrangean approach to the traditional Dirac-approach. By making use of this equivalence we show that a recently proposed symplectic algorithm does not necessarily reproduce the full constraint structure of the traditional Dirac algorithm. 
  We classify and construct all the smooth Kaluza-Klein reductions to ten dimensions of the M2- and M5-brane configurations which preserve some of the supersymmetry. In this way we obtain a wealth of new supersymmetric IIA backgrounds describing composite configurations of D-branes, NS-branes and flux/nullbranes; bound states of D2-branes and strings, D4-branes and NS5-branes, as well as some novel configurations in which the quotient involves nowhere-vanishing transverse rotations to the brane twisted by a timelike or lightlike translation. From these results there also follow novel M-theory backgrounds locally isometric to the M-branes, some of which are time-dependent and all of which are asymptotic to discrete quotients of eleven-dimensional Minkowski spacetime. We emphasise the universality of the formalism by briefly discussing analogous analyses in type IIA/IIB dual to the ones mentioned above. Some comments on the dual gauge theory description of some of our configurations are also included. 
  We investigate the Kaluza-Klein reductions to ten dimensions of the purely gravitational half-BPS M-theory backgrounds: the M-wave and the Kaluza-Klein monopole. We determine the moduli space of smooth (supersymmetric) Kaluza-Klein reductions by classifying the freely-acting spacelike Killing vectors which preserve some Killing spinor. As a consequence we find a wealth of new supersymmetric IIA configurations involving composite and/or bound-state configurations of waves, D0 and D6-branes, Kaluza-Klein monopoles in type IIA and flux/nullbranes, and some other new configurations. Some new features raised by the geometry of the Taub-NUT space are discussed, namely the existence of reductions with no continuous moduli. We also propose an interpretation of the flux 5-brane in terms of the local description (close to the branes) of a bound state of D6-branes and ten-dimensional Kaluza-Klein monopoles. 
  It is argued by Duff that only the time variation of dimensionless constants of nature is a legitimate subject of enquiry, and that dimensional constants such as c, h-bar and G... are merely human constructs whose value has no operational meaning. We refute this claim and point out that such varying dimensional "constants" can have significant physical consequences for the universe that can be directly measured in experiments. Postulating that dimensional constants vary in time can significantly change the laws of physics. 
  Tensionless super-p-branes in a generalized superspace with additional tensorial central charge coordinates may provide an extended object model for BPS preons, i.e. for the hypothetical constituents of M-theory preserving 31 of 32 supersymmetries [hep-th/0101113]. 
  We discuss the q-state Potts models for q<=4, in the scaling regimes close to their critical or tricritical points. Starting from the kink S-matrix elements proposed by Chim and Zamolodchikov, the bootstrap is closed for the scaling regions of all critical points, and for the tricritical points when 4>q>=2. We also note a curious appearance of the extended last line of Freudenthal's magic square in connection with the Potts models. 
  We show that in the absence of a Ramond-Ramond sector both the type IIA and type IIB free string gases have a thermal instability due to low temperature tachyon modes. The gas of free IIA strings undergoes a thermal duality transition into a gas of free IIB strings at the self-dual temperature. The free heterotic string gas is a tachyon-free ensemble with gauge symmetry SO(16)$\times$SO(16) in the presence of a timelike Wilson line background. It exhibits a holographic duality relation undergoing a self-dual phase transition with positive free energy and positive specific heat. The type IB open and closed string ensemble is related by thermal duality to the type I' string ensemble. We identify the order parameter for the Kosterlitz-Thouless phase transition from a low temperature gas of short open strings to a high temperature long string phase at or below T_C. Note Added (Sep 2005). 
  It is shown that a suitably formulated algebraic lightfront holography, in which the lightfront is viewed as the linear extension of the upper causal horizon of a wedge region, is capable of overcoming the shortcomings of the old lightfront quantization. The absence of transverse vacuum fluctuations which this formalism reveals, is responsible for an area (edge of the wedge) -rearrangement of degrees of freedom which in turn leads to the notion of area density of entropy for a ``split localization''. This area proportionality of horizon associated entropy has to be compared to the volume dependence of ordinary heat bath entropy. The desired limit, in which the split distance vanishes and the localization on the horizon becomes sharp, can at most yield a relative area density which measures the ratio of area densities for different quantum matter. In order to obtain a normalized area density one needs the unknown analog of a second fundamental law of thermodynamics for thermalization caused by vacuum fluctuation through localization on causal horizons. This is similar to the role of the classical Gibbs form of that law which relates Bekenstein's classical area formula with the Hawking quantum mechanism for thermalization from black holes. PACS: 11.10.-z, 11.30.-j, 11.55.-m 
  We find a general class of pp-wave string solutions with NS-NS $H_3$ or R-R $F_3$ field strengths, which are analogous to solutions with non-constant $F_5$ recently considered by Maldacena and Maoz (hep-th/0207284). We show that: (i) all pp-wave solutions supported by non-constant $H_3$ or $F_p$ fields are exact type II superstring solutions to all orders in $\a'$; (ii) the corresponding light-cone gauge Green-Schwarz actions are non-linear in bosons but always quadratic in fermions, and describe UV finite 2-d theories; (iii) the pp-wave backgrounds supported by non-constant $F_3$ field do not have, in contrast to their $F_5$-field counterparts, ``supernumerary'' supersymmetries and thus the associated light-cone GS actions do not possess 2-d supersymmetry. We consider a specific example where the pp-wave $F_3$ background is parametrized by an arbitrary holomorphic function of one complex bosonic coordinate. The corresponding GS action has the same bosonic part, similar Yukawa terms but twice as many interacting world-sheet fermions as the (2,2) supersymmetric model originating from the analogous $F_5$ background. We also discuss the structure of massless scalar vertex operators in the models related to N=2 super sine-Gordon and N=2 super Liouville theories. 
  We show that the high temperature behavior of non-commutative QED may be simply obtained from Boltzmann transport equations for classical particles. The transport equation for the charge neutral particle is shown to be characteristically different from that for the charged particle. These equations correctly generate, for arbitrary values of the non-commutative parameter theta, the leading, gauge independent hard thermal loops, arising from the fermion and the gauge sectors. We briefly discuss the generating functional of hard thermal amplitudes. 
  The topological charge of center vortices is discussed in terms of the self-intersection number of the closed vortex surfaces in 4-dimensional Euclidian space-time and in terms of the temporal changes of the writhing number of the time-dependent vortex loops in 3-dimensional space. 
  Within the exact renormalisation group approach, it is shown that stability properties of the flow are controlled by the choice for the regulator. Equally, the convergence of the flow is enhanced for specific optimised choices for the regularisation. As an illustration, we exemplify our reasoning for 3d scalar theories at criticality. Implications for other theories are discussed. 
  The quantum field algebra of real scalar fields is shown to be an example of infinite dimensional quantum group. The underlying Hopf algebra is the symmetric algebra S(V) and the product is Wick's normal product. Two coquasitriangular structures can be built from the two-point function and the Feynman propagator of scalar fields to reproduce the operator product and the time-ordered product as twist deformations of the normal product. A correspondence is established between the quantum group and the quantum field concepts. On the mathematical side the underlying structures come out of Hopf algebra cohomology. 
  N-fold tensor products of a rational CFT carry an action of the permutation group S_N. These automorphisms can be used as gluing conditions in the study of boundary conditions for tensor product theories. We present an ansatz for such permutation boundary states and check that it satisfies the cluster condition and Cardy's constraints. For a particularly simple case, we also investigate associativity of the boundary OPE, and find an intriguing connection with the bulk OPE. In the second part of the paper, the constructions are slightly extended for application to Gepner models. We give permutation branes for the quintic, together with some formulae for their intersections. 
  On a slice of AdS_5, despite having a dimensionful coupling, gauge theories can exhibit logarithmic dependence on scale. In this paper, we utilize deconstruction to analyze the scaling behavior of the theory, both above and below the AdS curvature scale, and shed light on position-dependent regularizations of the theory. We comment on applications to geometries other than AdS. 
  We derive the complete five-gluon scattering amplitude at tree level, within the context of Open Superstring theory. We find the general expression in terms of kinematic factors, and also find its complete expansion up to ${\cal O}({\alpha'}^3)$ terms. We use our scattering amplitude to test three non-equivalent ${\cal O}({\alpha'}^3)$ effective lagrangians that have recently been matter of some controversy. 
  It is generally believed that inflationary cosmology explains the isotropy, large scale homogeneity and flatness as well as predicting the deviations from homogeneity of our universe. We show that this is not the only cosmology which can explain successfully these features of the universe. We consider anew and modify a model in which local Lorentz invariance is spontaneously broken in the very early universe, and in this epoch the speed of light undergoes a first or second order phase transition to a value ~ 30 orders of magnitude smaller, corresponding to the presently measured speed of light. Before the phase transition at a time t ~ t_c, the entropy of the universe is reduced by many orders of magnitude, allowing for a semiclassical quantum field theory calculation of a scale invariant fluctuation spectrum. After the phase transition has occurred, the radiation density and the entropy of the universe increase hugely and the increase in the entropy follows the arrow of time determined by the spontaneously broken direction of the vev <\phi^a>_0. This solves the enigma of the arrow of time and the second law of thermodynamics. A new calculation of the primordial Gaussian and adiabatic fluctuation spectrum is carried out, leading to a scale invariant scalar component of the power spectrum. We argue that there are several attractive features of VSL theory compared to standard inflationary theory, and that it provides an alternative cosmology with potentially different predictions. 
  Warped compactifications with branes provide a new approach to the hierarchy problem and generate a diversity of four-dimensional thresholds. We investigate the relationships between these scales, which fall into two classes. Geometrical scales, such as thresholds for Kaluza-Klein, excited string, and black hole production, are generically determined soley by the spacetime geometry. Dynamical scales, notably the scale of supersymmetry breaking and moduli masses, depend on other details of the model. We illustrate these relationships in a class of solutions of type IIB string theory with imaginary self-dual fluxes. After identifying the geometrical scales and the resulting hierarchy, we determine the gravitino and moduli masses through explicit dimensional reduction, and estimate their value to be near the four-dimensional Planck scale. In the process we obtain expressions for the superpotential and Kahler potential, including the effects of warping. We identify matter living on certain branes to be effectively sequestered from the supersymmetry breaking fluxes: specifically, such "visible sector" fields receive no tree-level masses from the supersymmetry breaking. However, loop corrections are expected to generate masses, at the phenomenologically viable TeV scale. 
  We consider cosmological inflation driven by the rolling tachyon in the context of the braneworld scenario. We show that sufficient inflation consistent with the observational constraints can be achieved for well defined upper limits on the five-dimensional mass scale, string mass scale and the string coupling for the bosonic string. 
  The Standard Model data suggests the realization of grand unification structures in nature, in particular that of SO(10). A class of string vacua that preserve the SO(10) embedding, are the three generation free fermion heterotic-string models, that are related to Z2xZ2 orbifold compactification. Attempts to use the M-theory framework to explore further this class of models are discussed. Wilson line breaking of the SO(10) GUT symmetry results in super-heavy meta-stable states, which produce several exotic dark matter and UHECR candidates, with differing phenomenological characteristics. Attempts to develop the tools to study the properties of these states in forthcoming UHECR experiments are discussed. It is proposed that quantum mechanics arises from an equivalence postulate, that may lay the foundations for the rigorous formulation of quantum gravity. 
  We study the dynamics of D-branes in a smooth time-dependent background. The theory on the branes is a time-dependent non-commutative field theory. We find the metric and fluxes that determine the dual holographic closed string theory. This provides a concrete example of holography in a cosmological setting. 
  A BPS exact wall solution is found in N=1 supergravity in four dimensions. The model uses chiral scalar field with a periodic superpotential admitting winding numbers. Maintaining the periodicity in supergravity requires a gravitational correction to superpotential which allows the exact solution. By introducing boundary cosmological constants, we construct non-BPS multi-wall solutions for which a systematic analytic approximation is worked out for small gravitational coupling. 
  A quantum theory of the region of pure gravitation was given earlier in two papers [gr-qc/9908036 (Phys. Lett. A {\bf {265}}, 1 (2000)); gr-qc/0101056]. In this paper I provide further insight into the physics of this region. 
  We investigate cosmology with complex scalar field. It is shown that such models can describe two stages of inflation and the oscillatory regime. Thus we don't need both quintessence and $\Lambda$-term. It is enough to have single complex field to obtain both inflation and present accelerated universe. If the present accelerated expansion was created by the complex field then universe must escape eternal acceleration. Besides, we will show that cosmological equations with complex field admit stationary solution without any cosmological constant. 
  We compute the first 5 terms in the short-time heat trace asymptotics expansion for an operator of Laplace type with transfer boundary conditions using the functorial properties of these invariants. 
  If C is a cocommutative coalgebra, a bialgebra structure can be given to the symmetric algebra S(C). The symmetric product is twisted by a Laplace pairing and the twisted product of any number of elements of S(C) is calculated explicitly. This is used to recover important identities in the quantum field theory of interacting scalar bosons. 
  We show that the rational Calogero model with suitable boundary condition admits quantum states with non-equispaced energy levels. Such a spectrum generically consists of infinitely many positive energy states and a single negative energy state. The new states appear for arbitrary number of particles and for specific ranges of the coupling constant. These states owe their existence to the self-adjoint extensions of the corresponding Hamiltonian, which are labelled by a real parameter z. Each value of z corresponds to a particular spectrum, leading to inequivalent quantizations of the model. 
  We consider the radial Schroedinger equation with an attractive potential singular in the origin. The additional continuum of states caused by the singularity, that usually remain nontreatable, are shown to correspond to particles, asymptotically free near the singularity (in the inner channel). Depending on kinematics, they are either confined by the centre or may escape to infinity (to the outer channel).     The orthonormality within the continuum of confined states is established and the scattering phase of the particle emitted by the centre and then reflected back to it is found.     For the deconfinement case a unitary 2x2 S-matrix is found in terms of the Jost functions, and describes transitions within and between the two channels. The volume elements in the two channels are different.      The two-channel situation is analogous to the known behaviour of radiation in the black hole metrics. We discuss the black hole essence of singularly attracting centre for classical motion and the relativity of time inherent to this problem. 
  The Thermo Field Dynamics formalism is presented. In particular, it is applied to the two-dimensional field theory that describes a open bosonic string. The value of entropy operator is computed in various Dirichlet and Neumann boundary states. The Thermal Bogoliubov transformations, the thermal vacuum and the thermal Fock space are computed explicitely for the case of SU(1,1) thermal algebra. 
  We compute the complete bulk one-loop contribution to the Weyl anomaly of the boundary theory for IIB Supergravity compactified on $ AdS_5\times S^5$. The result, that $\delta {\cal A}=(E+I)/(2\pi^2)$, reproduces the subleading term in the exact expression ${\cal A}=-(N^2-1)(E+I)/(2\pi^2)$ for the Weyl anomaly of ${\cal N}=4$ Super-Yang-Mills theory, confirming the Maldacena conjecture. The anomaly receives contributions from all multiplets casting doubt on the possibility of describing the boundary theory beyond leading order in $N$ by a consistent truncation to the `massless' multiplet of IIB Supergravity. 
  The Hamiltonian $H={1\over2} p^2+{1\over2}m^2x^2+gx^2(ix)^\delta$ with $\delta,g\geq0$ is non-Hermitian, but the energy levels are real and positive as a consequence of ${\cal PT}$ symmetry. The quantum mechanical theory described by $H$ is treated as a one-dimensional Euclidean quantum field theory. The two-point Green's function for this theory is investigated using perturbative and numerical techniques. The K\"allen-Lehmann representation for the Green's function is constructed, and it is shown that by virtue of ${\cal PT}$ symmetry the Green's function is entirely real. While the wave-function renormalization constant $Z$ cannot be interpreted as a conventional probability, it still obeys a normalization determined by the commutation relations of the field. This provides strong evidence that the eigenfunctions of the Hamiltonian are complete. 
  The compact 7-manifold arising in the compactification of 11-dimensional supergravity is described by the metric encoded in the vacuum expectation values(vevs) in d=4, N=8 gauged supergravity. Especially, the space of SU(3)-singlet vevs contains various critical points and RG flows(domain walls) developing along AdS_4 radial coordinate. Based on the nonlinear metric ansatz of de Wit-Nicolai-Warner, we show the geometric construction of the compact 7-manifold metric and find the local frames(siebenbeins) by decoding the SU(3)-singlet vevs into squashing and stretching parameters of the 7-manifold. Then the 11-dimensional metric for the whole SU(3)-invariant sector is obtained as a warped product of an asymptotically AdS_4 space with a squashed and stretched 7-sphere. We also discuss the difference in the 7-manifold between two sectors, namely SU(3)xU(1)-invariant sector and G_2-invariant sector. In spite of the difference in base 6-sphere, both sectors share the 4-sphere of CP^2 associated with the common SU(3)-invariance of various 7-manifolds. 
  We present a new physical model that links the maximum speed of light with the minimal Planck scale into a maximal-acceleration Relativity principle in the spacetime tangent bundle and in phase spaces (cotangent bundle). Crucial in order to establish this link is the use of Clifford algebras in phase spaces. The maximal proper-acceleration bound is a = c^2/ \Lambda in full agreement with the old predictions of Caianiello, the Finslerian geometry point of view of Brandt and more recent results in the literature. We present the reasons why an Extended Scale Relativity based on Clifford spaces is physically more appealing than those based on kappa-deformed Poincare algebras and the inhomogeneous quantum groups operating in quantum Minkowski spacetimes. The main reason being that the Planck scale should not be taken as a deformation parameter to construct quantum algebras but should exist already as the minimum scale in Clifford spaces. 
  Asymptotically locally conical (ALC) metric of exceptional holonomy has an asymptotic circle bundle structure that accommodates the M theory circle in type IIA reduction. Taking Spin(7) metrics of cohomogeneity one as explicit examples, we investigate deformations of ALC metrics, in particular that change the asymptotic S^1 radius related to the type IIA string coupling constant. When the canonical four form of Spin(7) holonomy is taken to be anti-self-dual, the deformations of Spin(7) metric are related to the harmonic self-dual four forms, which are given by solutions to a system of first order differential equations, due to the metric ansatz of cohomogeneity one. We identify the L^2-normalizable solution that deforms the asymptotic radius of the M theory circle. 
  Vacuum structure and global cosmic strings are analyzed in the effective theory of self-interacting O(2) scalar fields on (3+1)-manifolds with conical singularities. In the context of one-loop effective action computed by heat-kernel methods with $\zeta$-function regularization, we find an inhomogeneous vacuum of minimum energy and suggest some reason why low-energy global strings are likely to be generated at the conical singularities. 
  We review and develop the construction of crosscap states associated with parity symmetries in rational conformal field theories. A general method to construct crosscap states in abelian orbifold models is presented. It is then applied to rational U(1) and parafermion systems, where in addition we study the geometrical interpretation of the corresponding parities. 
  Using the techniques of two dimensional conformal field theory we construct time dependent classical solutions in open string theory describing the decay of an unstable D-brane in the presence of background electric field, and explicitly evaluate the time dependence of the energy momentum tensor and the fundamental string charge density associated with this solution. The final decay product can be interpreted as a combination of stretched fundamental strings and tachyon matter. 
  The explicit form of the fermionic zero-modes in the fivebrane backgrounds of type IIA and IIB supergravity theories is investigated. In type IIA fivebrane background there are four zero-modes of gravitinos and dilatinos. In type IIB fivebrane background four zero-modes of dilatinos and no zero-modes of gravitinos are found. These zero-modes indicate the four-fermion condensates which have been suggested in a calculation of the tension of the D-brane in fivebrane backgrounds. 
  We develop an effective field model for describing FQH states with rational filling factors that are not of Laughlin type. These kinds of systems, which concern single layer hierarchical states and multilayer ones, were observed experimentally; but have not yet a satisfactory non commutative effective field description like in the case of Susskind model. Using $D$ brane analysis and fiber bundle techniques, we first classify such states in terms of representations characterized, amongst others, by the filling factor of the layers; but also by proper subgroups of the underlying $U(n) $ gauge symmetry. Multilayer states in the lowest Landau level are interpreted in terms of systems of $D2$ branes; but hierarchical ones are realized as Fiber bundles on $D2$ which we construct explicitly. In this picture, Jain and Haldane series are recovered as special cases and have a remarkable interpretation in terms of Fiber bundles with specific intersection matrices. We also derive the general NC commutative effective field and matrix models for FQH states, extending Susskind theory, and give the general expression of the rational filling factors as well as their non abelian gauge symmetries. 
  The double-tensor multiplet naturally appears in type IIB superstring compactifications on Calabi-Yau threefolds, and is dual to the universal hypermultiplet. We revisit the calculation of instanton corrections to the low-energy effective action, in the supergravity approximation. We derive a Bogomolny'i bound for the double-tensor multiplet and find new instanton solutions saturating the bound. They are characterized by the topological charges and the asymptotic values of the scalar fields in the double-tensor multiplet. 
  Recently spatially localized anomalies have been considered in higher dimensional field theories. The question of the quantum consistency and stability of these theories needs further discussion. Here we would like to investigate what string theory might teach us about theories with localized anomalies. We consider the Z_3 orbifold of the heterotic E_8 x E_8 theory, and compute the anomaly of the gaugino in the presence of Wilson lines. We find an anomaly localized at the fixed points, which depends crucially on the local untwisted spectra at those points. We show that non-Abelian anomalies cancel locally at the fixed points for all Z_3 models with or without additional Wilson lines. At various fixed points different anomalous U(1)s may be present, but at most one at a given fixed point. It is in general not possible to construct one generator which is the sole source of the anomalous U(1)s at the various fixed points. 
  A challenge in the theory of integrable systems is to show for every nonultralocal quantum integrable model, a possible connection to an ultralocal model. Some of such gauge connections were discovered earlier. We complete the task by identifying the same for the remaining ones along with two new models. We also unveil the underlying algebraic structure for these nonultralocal models. 
  We consider type IIB string interaction on the maximally supersymmetric pp-wave background and discuss how the bosonic symmetries of the background are realized. This analysis shows that there are some interesting differences with respect to the flat-space case and suggests modifications to the existing form of the string vertex. We focus on the zero-mode part which is responsible for some puzzling string predictions about the N=4 SYM side. We show that these puzzles disappear when a symmetry preserving string interaction is used. 
  Tachyon condensation in the open bosonic string is analyzed using a perturbative expansion of the tachyon potential around the unstable D25-brane vacuum. Using the leading terms in the tachyon potential, Pad\'e approximants can apparently give the energy of the stable vacuum to arbitrarily good accuracy. Level-truncation approximations up to level 10 for the coefficients in the tachyon potential are extrapolated to higher levels and used to find approximants for the full potential. At level 14 and above, the resulting approximants give an energy less than -1 in units of the D25-brane tension, in agreement with recent level-truncation results by Gaiotto and Rastelli. The extrapolated energy continues to decrease below -1 until reaching a minimum near level 26, after which the energy turns around and begins to approach -1 from below. Within the accuracy of this method, these results are completely consistent with an energy which approaches -1 as the level of truncation is taken to be arbitrarily large. 
  Following recent advances in large N matrix mechanics, I discuss here the free (Cuntz) algebraic formulation of the large N limit of two-dimensional conformal field theories of chiral adjoint fermions and bosons. One of the central results is a new {\it affine free algebra} which describes a large N limit of su(N) affine Lie algebra. Other results include the associated {\it free-algebraic partition functions and characters}, a free-algebraic coset construction, free- algebraic construction of osp(1|2), {\it free-algebraic vertex operator constructions} in the large N Bose systems and a provocative new free-algebraic factorization of the ordinary Koba-Nielsen factor. 
  We introduce a particular embedding of seven dimensional self-duality membrane equations in C^3\times R which breaks G_2 invariance down to SU(3). The world-volume membrane instantons define SU(3) special lagrangian submanifolds of C^3. We discuss in detail solutions for spherical and toroidal topologies assuming factorization of time. We show that the extra dimensions manifest themselves in the solutions through the appearance of a non-zero conserved charge which prevents the collapse of the membrane. We find non-collapsing rotating membrane instantons which contract from infinite size to a finite one and then they bounce to infinity in finite time. Their motion is periodic. These generalized complex Nahm equations, in the axially symmetric case, lead to extensions of the continuous Toda equation to complex space. 
  We consider a 3-parametric linear deformation of the Poisson brackets in classical mechanics. This deformation can be thought of as the classical limit of dynamics in so-called "quantized spaces". Our main result is a description of the motion of a particle in the corresponding Kepler-Coulomb problem. 
  We study a bound state of fractional D3-branes localized inside the world-volume of fractional D7-branes on the orbifold C^3/Z_2 x Z_2. We determine the open string spectrum that leads to N=1 U(N1)xU(N2)xU(N3)xU(N4) gauge theory with matter having the number of D7-branes as a flavor index. We derive the linearized boundary action of the D7-brane on this orbifold using the boundary state formalism and we discuss the tadpole cancellation. After computing the asymptotic expression of the supergravity solution the anomalies of the gauge theory are reproduced. 
  We study domain wall solutions in d=5, N=2 supergravity coupled to a single hypermultiplet whose moduli space is described by certain inhomogeneous, toric ESD manifolds constructed recently by Calderbank and Singer. Upon gauging a generic U(1) isometry of these spaces, we obtain an infinite family of models whose "superpotential" admits an arbitrary number of isolated critical points. By investigating the associated supersymmetric flows, we prove the existence of domain walls of Randall-Sundrum type for each member of our family, and find chains of domain walls interpolating between various AdS_5 backgrounds. Our models are described by a discrete infinity of smooth and complete one-hypermultiplet moduli spaces, which live on an open subset of the minimal resolution of certain cyclic quotient singularities. These spaces generalize the Pedersen metrics considered recently by Behrndt and Dall' Agata. 
  Gauge fields in exotic representations of the Lorentz group in D dimensions - i.e. ones which are tensors of mixed symmetry corresponding to Young tableaux with arbitrary numbers of rows and columns - naturally arise through massive string modes and in dualising gravity and other theories in higher dimensions. We generalise the formalism of differential forms to allow the discussion of arbitrary gauge fields. We present the gauge symmetries, field strengths, field equations and actions for the free theory, and construct the various dual theories. In particular, we discuss linearised gravity in arbitrary dimensions, and its two dual forms. 
  We propose a description of dark energy and acceleration of the universe in extended supergravities with de Sitter (dS) solutions. Some of them are related to M-theory with non-compact internal spaces. Masses of ultra-light scalars in these models are quantized in units of the Hubble constant: m^2 = n H^2. If dS solution corresponds to a minimum of the effective potential, the universe eventually becomes dS space. If dS solution corresponds to a maximum or a saddle point, which is the case in all known models based on N=8 supergravity, the flat universe eventually stops accelerating and collapses to a singularity. We show that in these models, as well as in the simplest models of dark energy based on N=1 supergravity, the typical time remaining before the global collapse is comparable to the present age of the universe, t = O(10^{10}) years. We discuss the possibility of distinguishing between various models and finding our destiny using cosmological observations. 
  We discuss the theory of dark energy based on maximally extended supergravity and suggest a possible anthropic explanation of the present value of the cosmological constant and of the observed ratio between dark energy and energy of matter. 
  We show how the Killing spinors of some maximally supersymmetric supergravity solutions whose metrics describe symmetric spacetimes (including $AdS,AdS\times S$ and H$pp$-waves) can be easily constructed using purely geometrical and group-theoretical methods. The calculation of the supersymmetry algebras is extremely simple in this formalism. 
  We present various generalizations of the Dirac formalism. The different-parity solutions of the Weinberg's 2(2J+1)-component equations are found. On this basis, generalizations of the Bargmann-Wigner (BW) formalism are proposed. Relations with modern physics constructs are discussed. 
  The correlation between the neutral electromagnetic pion decay, the Sutherland-Veltman paradox and the $AVV$ triangle anomaly phenomenon is discussed within the framework of an alternative strategy to handle the divergences involved in the perturbative evaluation of the associated physical amplitudes. We show that the general characteristic of the adopted strategy allows us to recover the traditional treatment for the problem as well as allows us to construct an alternative way to look at the problem where the ambiguities play no relevant role. 
  We obtain, in a systematic way, all the classical BPS equations which correspond to the quantum BPS states in the M-theory on a fully supersymmetric pp-wave. The superalgebra of the M-theory matrix model shows that the BPS states always preserve pairs of supersymmetry, implying the possible fractions of the unbroken supersymmetry as 2/16, 4/16, 6/16,.... We study their classical counterparts, and find there are essentially one unique set of 2/16 BPS equations, three inequivalent types of 4/16 BPS equations, and three inequivalent types of 8/16 BPS equations only, in addition to the 16/16 static fuzzy sphere. We discuss various supersymmetric objects as solutions. In particular, when the fuzzy sphere rotates, the supersymmetry is further broken as 16/16 -> 8/16 -> 4/16. 
  We discuss the uncertainty relations in quantum mechanics on noncommutative plane. In particular, we show that, for a given state at most one out of three basic nontrivial uncertainty relations can be saturated. We consider also in some detail the case of angular momentum eigenstates. 
  We discuss the high temperature behaviour of IIB strings in the maximally symmetric plane wave background, and show that there is a Hagedorn temperature. We discuss the map between strings in the pp-wave background and the dual superconformal field theory in the thermal domain. The Hagedorn bound describes a curve in the R-charge chemical potential versus temperature phase diagram of the dual Yang-Mills theory and the theory manifestly exists on both sides. Using a recent observation of Brower, Lowe, and Tan, we update our earlier calculation to reflect that the pp-wave string exists on both sides of the Hagedorn bound as well. 
  We present the Generalized Borel Transform (GBT). This new approach allows one to obtain approximate solutions of Laplace/Mellin transform valid in both, perturbative and non perturbative regimes. We compare the results provided by the GBT for a solvable model of quantum mechanics with those provided by standard techniques, as the conventional Borel sum, or its modified versions. We found that our approach is very efficient for obtaining both the low and the high energy behavior of the model. 
  We study M/D-branes in a null-brane background. By taking a near horizon limit, one is left with cosmological models in the corresponding Poincar\'e patches. To deal with their usual horizons, we either extend these models to global AdS or remain in the Poincar\'e patch and apply a T-duality transformation whenever the effective radius of the compact dimension associated with the null-brane probes distances smaller than the string scale. The first scenario gives rise to null orbifolds in AdS spaces, which are described in detail. Their conformal boundaries are singular. The second has a dual gauge theory description in terms of Super Yang-Mills in the null-brane background. The latter is a good candidate for a non-perturbative definition of string theory in a time-dependent background. 
  We apply supersymmetric discrete light-cone quantization (SDLCQ) to the study of supersymmetric Yang-Mills-Chern-Simons (SYM-CS) theory on R x S^1 x S^1. One of the compact directions is chosen to be light-like and the other to be space-like. Since the SDLCQ regularization explicitly preserves supersymmetry, this theory is totally finite, and thus we can solve for bound-state wave functions and masses numerically without renormalizing. The Chern-Simons term is introduced here to provide masses for the particles while remaining totally within a supersymmetric context. We examine the free, weak and strong-coupling spectrum. The transverse direction is discussed as a model for universal extra dimensions in the gauge sector. The wave functions are used to calculate the structure functions of the lowest mass states. We discuss the properties of Kaluza-Klein states and focus on how they appear at strong coupling. We also discuss a set of anomalously light states which are reflections of the exact Bogomol'nyi-Prasad-Sommerfield states of the underlying SYM theory. 
  There exist a one complex parameter family of de Sitter invariant vacua, known as alpha vacua. In the context of slow roll inflation, we show that all but the Bunch-Davies vacuum generates unacceptable production of high energy particles at the end of inflation. As a simple model for the effects of trans-planckian physics, we go on to consider non-de Sitter invariant vacua obtained by patching modes in the Bunch-Davies vacuum above some momentum scale M_c, with modes in an alpha vacuum below M_c. Choosing M_c near the Planck scale M_pl, we find acceptable levels of hard particle production, and corrections to the cosmic microwave perturbations at the level of H M_pl/M_c^2, where H is the Hubble parameter during inflation. More general initial states of this type with H<< M_c << M_pl can give corrections to the spectrum of cosmic microwave background perturbations at order 1. The parameter characterizing the alpha-vacuum during inflation is a new cosmological observable. 
  Derrick's theorem on the nonexistence of stable time-independent scalar field configurations [G. H. Derrick, J. Math. Phys. 5, 1252 (1964)] is generalized to finite systems of arbitrary dimension. It is shown that the "dilation" argument underlying the theorem hinges upon the fulfillment of specific Neumann boundary conditions, providing thus new means of evading it without resorting to time-dependence or additional fields of higher spin. The theorem in its original form is only recovered when the boundary conditions are such that both the gradient and potential energies vanish at the boundaries, in which case it establishes the nonexistence of stable time-independent solutions in finite systems of more than two spatial dimensions. 
  We detail the global structure of the five-dimensional bulk for the cosmological evolution of Dvali-Gabadadze-Porrati braneworlds. The picture articulated here provides a framework and intuition for understanding how metric perturbations leave (and possibly reenter) the brane universe. A bulk observer sees the braneworld as a relativistically expanding bubble, viewed either from the interior (in the case of the Friedmann-Lemaitre-Robertson-Walker phase) or the exterior (the self-accelerating phase). Shortcuts through the bulk in the first phase can lead to an apparent brane causality violation and provide an opportunity for the evasion of the horizon problem found in conventional four-dimensional cosmologies. Features of the global geometry in the latter phase anticipate a depletion of power for linear metric perturbations on large scales. 
  Given the conventional Maxwell-Lorentz formulation of classical electrodynamics in a flat spacetime of arbitrary odd dimension, the retarded vector potential $A^\mu$ generated by a point-like charge is found to be pure gauge, $A^\mu=\partial^\mu\chi$. By the Gauss law, the charge vanishes. Therefore, classical electromagnetism is missing from odd-dimensional worlds. By contrast, a particle interacting with a scalar field generates nonzero field strength. If the electromagnetic action is augmented by the addition of the Chern-Simons term, the interaction picture in the three-dimensional world becomes nontrivial. 
  We investigate the quantum aspects of three-dimensional gravity with a positive cosmological constant. The reduced phase space of the three-dimensional de Sitter gravity is obtained as the space which consists of the Kerr-de Sitter space-times and their Virasoro deformations. A quantization of the phase space is carried out by the geometric quantization of the coadjoint orbits of the asymptotic Virasoro symmetries. The Virasoro algebras with real central charges are obtained as the quantum asymptotic symmetries. The states of globally de Sitter and point particle solutions become the primary states of the unitary irreducible representations of the Virasoro algebras. It is shown that those states are perturbatively stable at the quantum level. The Virasoro deformations of these solutions correspond to the excited states in the unitary irreducible representations. In view of the dS/CFT correspondence, we also study the relationship between the Liouville field theory obtained by a reduction of the SL(2;$\mathbb{C}$) Chern-Simons theory and the three-dimensional gravity both classically and quantum mechanically. In the analyses of the both theories, the Kerr-de Sitter geometries with nonzero angular momenta do not give the unitary representations of the Virasoro algebras. 
  We propose a Lax equation for the non-linear sigma model which leads directly to the conserved local charges of the system. We show that the system has two infinite sets of such conserved charges following from the Lax equation, much like dispersionless systems. We show that the system has two Hamiltonian structures which are compatible so that it is truly a bi-Hamiltonian system. However, the two Hamiltonian structures act on the two distinct sets of charges to give the dynamical equations, which is quite distinct from the behavior in conventional integrable systems. We construct two recursion operators which connect the conserved charges within a given set as well as between the two sets. We show explicitly that the conserved charges are in involution with respect to either of the Hamiltonian structures thereby proving complete integrability of the system. Various other interesting features are also discussed. 
  Gauge theories on q-deformed spaces are constructed using covariant derivatives. For this purpose a ``vielbein'' is introduced, which transforms under gauge transformations. The non-Abelian case is treated by establishing a connection to gauge theories on commutative spaces, i.e. by a Seiberg-Witten map. As an example we consider the Manin plane. Remarks are made concerning the relation between covariant coordinates and covariant derivatives. 
  We advocate a framework for constructing perturbative closed string compactifications which do not have large-radius limits. The idea is to augment the class of vacua which can be described as fibrations by enlarging the monodromy group around the singular fibers to include perturbative stringy duality symmetries. As a controlled laboratory for testing this program, we study in detail six-dimensional (1,0) supersymmetric vacua arising from two-torus fibrations over a two-dimensional base. We also construct some examples of two-torus fibrations over four-dimensional bases, and comment on the extension to other fibrations. 
  We give the exact construction of Riemannian (or stringy) instantons, which are classical solutions of 2d Yang-Mills theories that interpolate between initial and final string configurations. They satisfy the Hitchin equations with special boundary conditions. For the case of U(2) gauge group those equations can be written as the sinh-Gordon equation with a delta function source. Using techniques of integrable theories based on the zero curvature conditions, we show that the solution is a condensate of an infinite number of one-solitons with the same topological charge and with all possible rapidities. 
  A procedure, allowing to calculate the coefficients of the SW prepotential in the framework of the instanton calculus is presented. As a demonstration explicit calculations for 2, 3 and 4- instanton contributions are carried out. 
  We review our recent work on M-theory compactifications on `toric' G_2 cones, a class of models which generalize those recently considered by Acharya and Witten and lead to chiral matter in four dimensions. We explain our criteria for identifying the gauge group content of such theories and briefly discuss the associated metrics. 
  Correlation functions in perturbative N=4 supersymmetric Yang-Mills theory are examined in the Berenstein-Maldacena-Nastase (BMN) limit. We demonstrate that non-extremal four-point functions of chiral primary fields are ill-defined in that limit. This lends support to the assertion that only gauge theoretic two-point functions should be compared to pp-wave strings. We further refine the analysis of the recently discovered non-planar corrections to the planar BMN limit. In particular, a full resolution to the genus one operator mixing problem is presented, leading to modifications in the map between BMN operators and string states. We give a perturbative construction of the correct operators and we identify their anomalous dimensions. We also distinguish symmetric, antisymmetric and singlet operators and find, interestingly, the same torus anomalous dimension for all three. Finally, it is discussed how operator mixing effects modify three point functions at the classical level and, at one loop, allow us to recover conformal invariance. 
  The three string vertex for Type IIB superstrings in a maximally supersymmetric plane-wave background is investigated. Specifically, we derive a factorization theorem for the Neumann coefficients that generalizes a flat-space result that was obtained some 20 years ago. The resulting formula is used to explore the leading large mu asymptotic behavior, which is relevant for comparison with dual gauge theory results. 
  We present some clues to the study of the renormalization group, at graduate level, as well as some bibliographical pointers to classical resources. Just the kind of things one had liked to hear when starting to study the subject. 
  We show with two numerical examples that the conventional expansion in powers of the field for the critical potential of 3-dimensional O(N) models in the large-N limit, does not converge for values of phi^2 larger than some critical value. This can be explained by the existence of conjugated branch points in the complex phi^2 plane. Pade approximants [L+3/L] for the critical potential apparently converge at large phi^2. This allows high-precision calculation of the fixed point in a more suitable set of coordinates. We argue that the singularities are generic and not an artifact of the large-N limit. We show that ignoring these singularities may lead to inaccurate approximations. 
  In this paper we show that the worldline reparametrization for particles with higher derivative interactions appears as a higher dimensional symmetry, which is generated by the truncated Virasoro algebra. We also argue that for generic nonlocal particle theories the fields on the worldline may be promoted to those living on a two dimensional worldsheet, and the reparametrization symmetry becomes locally the same as the conformal symmetry. 
  The Bogoliubov transformation in thermofield dynamics, an operator formalism for the finite-temperature quantum-field theory, is generalized to describe a field in arbitrary confined regions of space and time. Starting with the scalar field, the approach is extended to the electromagnetic field and the energy-momentum tensor is written via the Bogoliubov transformation. In this context, the Casimir effect is calculated for zero and non-zero temperature, and therefore it can be considered as a vacuum condensation effect of the electromagnetic field. This aspect opens an interesting perspective for using this procedure as an effective scheme for calculations in the studies of confined fields, including the interacting fields. 
  The action for a non-BPS p=2 brane embedded in a flat N=1, D=4 target superspace is obtained through the method of nonlinear realizations of the associated super-Poincare symmetries. The brane excitation modes correspond to the Nambu-Goldstone degrees of freedom resulting from the broken space translational symmetry and the target space supersymmetries. The action for this p=2 brane is found to be an invariant synthesis of the Akulov-Volkov and Nambu-Goto actions. The dual D2-brane Born-Infeld action is derived. The invariant coupling of matter fields localized on the brane to the Nambu-Goldstone modes is also obtained. 
  Hence it excludes proton decay and supersymmetry. The main predictions of a gauge model based on the exceptional simple Lie superalgebra mb(3|8) (a localized version of su(3)+su(2)+u(1)) are reviewed. 
  The semiclassical quantization conditions for all partial waves are derived for bound states of two interacting anyons in the presence of a uniform background magnetic field. Singular Aharonov-Bohm-type interactions between the anyons are dealt with by the modified WKB method of Friedrich and Trost. For s-wave bound state problems in which the choice of the boundary condition at short distance gives rise to an additional ambiguity, a suitable generalization of the latter method is required to develop a consistent WKB approach. We here show how the related self-adjoint extension parameter affects the semiclassical quantization condition for energy levels. For some simple cases admitting exact answers, we verify that our semiclassical formulas in fact provide highly accurate results over a broad quantum number range. 
  We study supergravity solutions of Dp-branes in the time-dependent orbifold background. We show that worldvolume theories decouple from the bulk gravity for p less than six. Along AdS/CFT correspondence, these solutions could provide the gravity description of noncommutative field theory with time-dependent noncommutative parameter. Type II NS5-brane (M5-brane) in the presence of RR n-form for n=0,..., 4 (C field) in this time-dependent background have also been studied. 
  In the context of homogeneous and isotropic superstring cosmology, the T-duality symmetry of string theory has been used to argue that for a background space-time described by dilaton gravity with strings as matter sources, the cosmological evolution of the Universe will be nonsingular. In this Letter we discuss how T-duality extends to brane gas cosmology, an approximation in which the background space-time is again described by dilaton gravity with a gas of branes as a matter source. We conclude that the arguments for nonsingular cosmological evolution remain valid. 
  In this paper we consider the general setting for constructing Action Principles for three-dimensional first order autonomous equations. We present the results for some integrable and non-integrable cases of the Lotka-Volterra equation, and we show Lagrangian descriptions which are valid for systems satisfying Shil'nikov criteria on the existence of strange attractors, though chaotic behavior or homoclinic orbits have not been verified up to now. The Euler-Lagrange equations we get for these systems usually present "time reparameterization" symmetry, though other kinds of invariance may be found according to the kernel of the associated symplectic 2-form. The formulation of a Hamiltonian structure (Poisson brackets and Hamiltonians) for these systems from the Lagrangian viewpoint leads to a method of finding new constants of the motion starting from known ones, which is applied to some systems found in the literature known to possess a constant of the motion, to find the other and thus showing their integrability. In particular, we show that the so-called ABC system is completely integrable if it possesses one constant of the motion.   Pacs numbers: 45.20.Jj, 02.30.Ik, 05.45.Ac   Keywords: Lagrangian, integrable system, chaos, Lotka Volterra 
  In the present work the Calderbank-Pedersen description of four dimensional manifolds with self-dual Weyl tensor is used to obtain examples of quaternionic-kahler metrics with two commuting isometries. The eigenfunctions of the hyperbolic laplacian are found by use of Backglund transformations acting over solutions of the Ward monopole equation. The Bryant-Salamon construction of $G_2$ holonomy metrics arising as $R^3$ bundles over quaternionic-kahler base spaces is applied to this examples to find internal spaces of the M-theory that leads to an N=1 supersymmetry in four dimensions. Type IIA solutions will be obtained too by reduction along one of the isometries. The torus symmetry of the base spaces is extended to the total ones. 
  We investigate a system of 1,3, and 7-branes in type IIB string theory. We identify the 1-brane with the vortex of the Abelian Higgs model. We claim that in the S-dual, the theory on the 3-brane confines electric flux which we identify with the fundamental string and that the dual photon has a mass. We then investigate ways of breaking supersymmetry in the brane set up. 
  Quantum gravity phenomenology opens up the possibility of probing Planck scale physics. Thus, by exploiting the generic properties that a semiclassical state of the compound system fermions plus gravity should have, an effective dynamics of spin-1/2 particles is obtained within the framework of loop quantum gravity. Namely, at length scales much larger than Planck length and below the wave length of the fermion, the spin-1/2 dynamics in flat spacetime includes Planck scale corrections. In particular we obtain modified dispersion relations in vacuo for fermions. These corrections yield a time of arrival delay of the spin 1/2 particles with respect to a light signal and, in the case of neutrinos, a novel flavor oscillation. To detect these effects the corresponding particles must be highly energetic and should travel long distances. Hence Neutrino Bursts accompanying Gamma Ray Bursts or ultra high energy cosmic rays could be considered. Remarkably, future neutrino telescopes may be capable to test such effects. This paper provides a detailed account of the calculations and elaborates on results previously reported in a Letter. These are further amended by introducing a real parameter Upsilon aimed at encoding our lack of knowledge of scaling properties of the gravitational degrees of freedom. 
  There are two main sets of data for the observed spectrum of ultra high energy cosmic rays (those cosmic rays with energies greater than $\sim 4 \times 10^{18}$ eV), the High Resolution Fly's Eye (HiRes) collaboration group observations, which seem to be consistent with the predicted theoretical spectrum (and therefore with the theoretical limit known as the Greisen-Zatsepin-Kuzmin cutoff), and the observations from the Akeno Giant Air Shower Array (AGASA) collaboration group, which reveal an abundant flux of incoming particles with energies above $1 \times 10^{20}$ eV violating the Greisen-Zatsepin-Kuzmin cutoff. As an explanation of this anomaly it has been suggested that quantum-gravitational effects may be playing a decisive role in the propagation of ultra high energy cosmic rays. In this article we take the loop quantum gravity approach. We shall provide some techniques to establish and analyze new constraints on the loop quantum gravity parameters arising from both sets of data, HiRes and AGASA . We shall also study their effects on the predicted spectrum for ultra high energy cosmic rays. As a result we will state the possibility of reconciling the AGASA observations. 
  We discuss the application of the deformation quantization approach to perturbative quantum field theory. We show that the various forms of Wick's theorem are a direct consequence of the structure of the star products. We derive the scattering function for a free scalar field in interaction with a spacetime-dependent source. We show that the translation to operator formalism reproduces the known relations which lead to the derivation of the Feynman rules. 
  We discuss a concept of particle localization which is motivated from quantum field theory, and has been proposed by Brunetti, Guido and Longo and by Schroer. It endows the single particle Hilbert space with a family of real subspaces indexed by the space-time regions, with certain specific properties reflecting the principles of locality and covariance. We show by construction that such a localization structure exists also in the case of massive anyons in d=2+1, i.e. for particles with positive mass and with arbitrary spin s in the reals. The construction is completely intrinsic to the corresponding ray representation of the (proper orthochronous) Poincare group. Our result is of particular interest since there are no free fields for anyons, which would fix a localization structure in a straightforward way. We present explicit formulas for the real subspaces, expected to turn out useful for the construction of a quantum field theory for anyons. In accord with well-known results, only localization in string-like, instead of point-like or bounded, regions is achieved. We also prove a single-particle PCT theorem, exhibiting a PCT operator which acts geometrically correctly on the family of real subspaces. 
  We study the late time behavior of the boundary state representing the rolling tachyon constructed by Sen. It is found that the coupling of the rolling tachyon to massive modes of the closed string grows exponentially as the system evolves. We argue that the description of rolling tachyon by a boundary state is valid during the finite time determined by string coupling, and that energy could be dissipated to the bulk beyond this time. We also comment on the relation between the rolling tachyon boundary state and the spacelike D-brane boundary state. 
  We describe the asymptotic boundary of the general homogeneous plane wave spacetime, using a construction of the `points at infinity' from the causal structure of the spacetime as introduced by Geroch, Kronheimer and Penrose. We show that this construction agrees with the conformal boundary obtained by Berenstein and Nastase for the maximally supersymmetric ten-dimensional plane wave. We see in detail how the possibility to go beyond (or around) infinity arises from the structure of light cones. We also discuss the extension of the construction to time-dependent plane wave solutions, focusing on the examples obtained from the Penrose limit of Dp-branes. 
  A review is given of ideas in electromagnetic duality and connections to integrable field theories with soliton solutions. This leads on to a summary of recent work on Lorentzian algebras. 
  The nonrelativistic case of noncommutative scalar dipole field theory with quartic interaction on a two-dimensional spacetime is analyzed. As there are two parameters in the general quartic interaction we try a way to find their relation. To do this we first investigate the formulation of the quantum mechanics for a particle carrying the noncommutative dipole. We point out a problem therein and propose a possible method to solve it. We use this prescription to determined the quartic interaction term in the field theory once the noncommutative dipole is turned on. The two particle Schr\"odinger equation is then solved and the exact wavefunction of the bound state and associated spectrum are found. It is seen that the wavefunction has three center positions in the relative coordinates and the separation of the center is equal to the dipole length $L$ of the dipole field, exhibiting the nature of the noncommutative dipole field. 
  The properties of the Das-Popowicz Moyal momentum algebra that we introduce in hep-th/0207242 are reexamined in details and used to discuss some aspects of integrable models and 2d conformal field theories. Among the results presented, we setup some useful convention notations which lead to extract some non trivial properties of the Moyal momentum algebra. We use the particular sub-algebra sl(n)-{Sigma}_{n}^{(0,n)} to construct the sl(2)-Liouville conformal model and its sl(3)-Toda extension. We show also that the central charge, a la Feigin-Fuchs, associated to the spin-2 conformal current of the (theta)-Liouville model is given by c(theta)=1+24.theta^{2}. Moreover, the results obtained for the Das-Popowicz Mm algebra are applied to study systematically some properties of the Moyal KdV and Boussinesq hierarchies generalizing some known results. We discuss also the primarity condition of conformal $w_{\theta}$-currents and interpret this condition as being a dressing gauge symmetry in the Moyal momentum space. Some computations related to the dressing gauge group are explicitly presented. 
  Noncommutative version of D-dimensional relativistic particle is proposed. We consider the particle interacting with the configuration space variable $\theta^{\mu\nu}(\tau)$ instead of the numerical matrix. The corresponding Poincare invariant action has a local symmetry, which allows one to impose the gauge $\theta^{0i}=0, ~ \theta^{ij}=const$. The matrix $\theta^{ij}$ turns out to be the noncommutativity parameter of the gauge fixed formulation. Poincare transformations of the gauge fixed formulation are presented in the manifest form. Consistent coupling of NC relativistic particle to the electromagnetic field is constructed and discussed. 
  We continue our discussion of the q-state Potts models for q <= 4, in the scaling regimes close to their critical and tricritical points. In a previous paper, the spectrum and full S-matrix of the models on an infinite line were elucidated; here, we consider finite-size behaviour. TBA equations are proposed for all cases related to phi(21) and phi(12) perturbations of unitary minimal models. These are subjected to a variety of checks in the ultraviolet and infrared limits, and compared with results from a recently-proposed nonlinear integral equation. A nonlinear integral equation is also used to study the flows from tricritical to critical models, over the full range of q. Our results should also be of relevance to the study of the off-critical dilute A models in regimes 1 and 2. 
  In the limit of large, constant B-field (the ``Seiberg-Witten limit''), the derivative expansion for open-superstring effective actions is naturally expressed in terms of the symmetric products *n. Here, we investigate corrections around the large-B limit, for Chern-Simons couplings on the brane and to quadratic order in gauge fields. We perform a boundary-state computation in the commutative theory, and compare it with the corresponding computation on the noncommutative side. These results are then used to examine the possible role of Wilson lines beyond the Seiberg-Witten limit. To quadratic order in fields, the entire tree-level amplitude is described by a metric-dependent deformation of the *2 product, which can be interpreted in terms of a deformed (non-associative) version of the Moyal * product. 
  In this paper we calculate the divergent part of the one loop effective action for QED on noncommutative space using the background field method. The effective action is obtained up to the second order in the noncommutative parameter theta and in the classical fields. 
  In spacetimes of dimension greater than four it is natural to consider higher order (in R) corrections to the Einstein equations. In this letter generalized Israel junction conditions for a membrane in such a theory are derived. This is achieved by generalising the Gibbons-Hawking boundary term. The junction conditions are applied to simple brane world models, and are compared to the many contradictory results in the literature. 
  Recently we showed that in semiclassical 2D dilaton gravity the regularity of a black hole horizon may be compatible with divergencies of Polyakov-Liouville stresses on it, the temperature deviating from its Hawking value. This makes the question about thermal properties of such solutions non-trivial. We demonstrate that, adding to gravitation-dilaton part of the action certain counterterms, which diverge on the horizon but are finite outside it, one may achieve finiteness of the effective gravitation-dilaton couplings on the horizon. This gives for the entropy S the Bekenstein - Hawking value in the nonextreme case and S=0 in the extreme one similarly to what happens to ''standard '' black holes. 
  We extent our previous study on spherically symmetric braneworld solutions with induced gravity, including non-local bulk effects. We find the most general static four-dimensional black hole solutions with g_{tt}=-g_{rr}^{-1}. They satisfy a closed system of equations on the brane and represent the strong-gravity corrections to the Schwarzschild-(A)dS_4 spacetime. These new solutions have extra terms which give extra attraction relative to the Newtonian-(A)dS_4 force; however, the conventional limits are easily obtained. These terms, when defined asymptotically, behave like AdS_4 in this regime, while when defined at infinitely short distances predict either an additional attractive Newtonian potential or an attractive potential which scales approximately as sqrt(r). One of the solutions found gives extra deflection of light compared to Newtonian deflection. 
  We study the dependence of the free energy on the CP violating angle theta, in four-dimensional SU(N) gauge theories with N >= 3, and in the large-N limit.   Using the Wilson lattice formulation for numerical simulations, we compute the first few terms of the expansion of the ground-state energy F(theta) around theta = 0, F(theta) - F(0) = A_2 theta^2 (1 + b_2 theta^2 + ...). Our results support Witten's conjecture: F(theta) - F(0) = A theta^2 + O(1/N) for theta < pi.   We verify that the topological susceptibility has a nonzero large-N limit chi_infinity = 2A with corrections of O(1/N^2), in substantial agreement with the Witten-Veneziano formula which relates chi_infinity to the eta' mass. Furthermore, higher order terms in theta are suppressed; in particular, the O(theta^4) term b_2 (related to the eta' - eta' elastic scattering amplitude) turns out to be quite small: b_2 = -0.023(7) for N=3, and its absolute value decreases with increasing N, consistently with the expectation b_2 = O(1/N^2). 
  We reconsider light-cone superstring field theory on the maximally supersymmetric pp-wave background. We find that the results for the fermionic Neumann matrices given so far in the literature are incomplete and verify our expressions by relating them to the bosonic Neumann matrices and proving several non-trivial consistency conditions among them, as for example the generalization of a flat space factorization theorem for the bosonic Neumann matrices. We also study the bosonic and fermionic constituents of the prefactor and point out a subtlety in the relation between continuum and oscillator basis expressions. 
  Theory of pointlike magnetic monopole with an arbitrary magnetic charge is considered. It is shown that a proper description requires making use of nonunitary representations of the rotation group and the nonassociative generalization of the gauge group and fibre bundle theory. 
  In this article a self-contained exposition of proving perturbative renormalizability of a quantum field theory based on an adaption of Wilson's differential renormalization group equation to perturbation theory is given. The topics treated include the spontaneously broken SU(2) Yang-Mills theory.  Although mainly a coherent but selective review, the article contains also some simplifications and extensions with respect to the literature. 
  We discuss a D-dimensional Euclidean scalar field interacting with a scale invariant quantized metric. We assume that the metric depends on d-dimensional coordinates where d<D. We show that the interacting quantum fields have more regular short distance behaviour than the free fields. A model of a Gaussian metric is discussed in detail. In particular, in the \Phi^4 theory in four dimensions we obtain explicit lower and upper bounds for each term of the perturbation series. It turns out that there is no coupling constant renormalization in the \Phi^4 model in four dimensions. We show that in a particular range of the scale dimension there are models in D=4 without any divergencies. 
  The explicit constrtuction of states saturating uncertainty relations following from basic commutation rules of NCQM is given both in Fock space and coordinate representation 
  We argue that the higher space-time derivative terms that occur in closed bosonic string field theory are reminiscent of those found in the Landau theory of liquid-crystal transitions. We examine the lowest level approximation of the closed bosonic string and find evidence for the existence of a new vacuum that spontaneously breaks Lorentz and translation symmetries. This effect can be interpreted as a spontaneous compactification of the theory. 
  Skyrme theory on S^2 (Faddeev coset proposal), is analyzed with a generalization of 0-curvature integrability, based on gauge techniques. New expressions valid for models in the sphere are given. The relation of the minimum energy configurations to gauge vacua is clarified. Consequences of adding a potential term to break the SO(3) symmetry are discussed. 
  We study exact renormalisation group flows for background field dependent regularisations. It is shown that proper-time flows are approximations to exact background field flows for a specific class of regulators. We clarify the role of the implicit scale dependence introduced by the background field. Its impact on the flow is evaluated numerically for scalar theories at criticality for different approximations and regularisations. Implications for gauge theories are discussed. 
  Recently effective actions have been extensively used to describe tachyon condensation in string theory. While the various effective actions which have appeared in the literature have very similar properties for static configurations, they differ for time-dependent tachyons. In this paper we discuss general properties of non-linear effective Lagrangians which are first order in derivatives. In particular we show that some observed properties, such as asymptotically vanishing pressure, are rather generic features, although the quantative features differ. On the other hand we argue that certain features of marginal tachyon profiles are beyond the reach of any first order Lagrangian description. We also point out that an effective action, proposed earlier, captures the dynamics of tachyons well. 
  Due to the occurrence of large exceptional Lie groups in supergravity, calculations involving explicit Lie algebra and Lie group element manipulations easily become very complicated and hence also error-prone if done by hand. Research on the extremal structure of maximal gauged supergravity theories in various dimensions sparked the development of a library for efficient abstract multilinear algebra calculations involving sparse and non-sparse higher-rank tensors, which is presented here. 
  We present a toy model for five-dimensional heterotic M-theory where bulk three-branes, originating in 11 dimensions from M five-branes, are modelled as kink solutions of a bulk scalar field theory. It is shown that the vacua of this defect model correspond to a class of topologically distinct M-theory compactifications. Topology change can then be analysed by studying the time evolution of the defect model. In the context of a four-dimensional effective theory, we study in detail the simplest such process, that is the time evolution of a kink and its collision with a boundary. We find that the kink is generically absorbed by the boundary thereby changing the boundary charge. This opens up the possibility of exploring the relation between more complicated defect configurations and the topology of brane-world models. 
  The Casimir effect is investigated in light-cone quantization. It is shown that for spacelike separation of the walls enclosing the system the standard result for the pressure exerted on the walls is obtained. For walls separated in light-cone space direction no regularization of the quantum fluctuations exists which would yield a finite pressure. The origin of this failure and its implications for other vacuum properties are discussed by analyzing the Casimir effect as seen from a moving observer approaching the speed of light. The possibility for calculation of thermodynamic quantities in light-cone quantization via the Casimir effect is pointed out. 
  A proof of the Riemann's hypothesis (RH) about the non-trivial zeros of the Riemann zeta-function is presented. It is based on the construction of an infinite family of operators D^{(k,l)} in one dimension, and their respective eigenfunctions \psi_s (t), parameterized by continuous real indexes k and l. Orthogonality of the eigenfunctions is connected to the zeros of the Riemann zeta-function. Due to the fundamental Gauss-Jacobi relation and the Riemann fundamental relation Z (s') = Z (1-s'), one can show that there is a direct concatenation among the following symmetries, t goes to 1/t, s goes to \beta - s (\beta a real), and s' goes to 1 - s', which establishes a one-to-one correspondence between the label s of one orthogonal state to a unique vacuum state, and a zero s' of the \zeta. It is shown that the RH is a direct consequence of these symmetries, by arguing in particular that an exclusion of a continuum of the zeros of the Riemann zeta function results in the discrete set of the zeros located at the points s_n = 1/2 + i \lambda_n in the complex plane. 
  Three closely related issues will be discussed. Magnetic quarks having non-Abelian charges have been found recently to appear as the dominant infrared degrees of freedom in some vacua of softly broken N=2 supersymmetric QCD with SU(n_c) gauge group. Their condensation upon N=1 perturbation causes confinement and dynamical symmetry breaking. We argue that these magnetic quarks can be naturally related to the semiclassical non-Abelian monopoles of the type first discussed by Goddard, Nuyts, Olive and E. Weinberg. We discuss also general properties of non-Abelian vortices and discuss their relevance to the confinement in QCD. Finally, calculation by Douglas and Shenker of the tension ratios for vortices of different N-alities in the softly broken N=2 supersymmetric SU(N) Yang-Mills theory, is carried to the second order in the adjoint multiplet mass. A correction to the ratios violating the sine formula is found, showing that the latter is not a universal quantity. 
  The influence of hypersonic waves excited in a single crystal is investigated on the process of electron-positron pair creation by high-energy photons. The coherent part of the corresponding differential cross-section is derived as a function of the amplitude and wave number of the hypersound. The values of the parameters are specified for which the latter affects remarkably on the pair creation cross-section. It is shown that under certain conditions the presence of hypersonic waves can result in enhancement of the process cross-section. 
  We give central elements of the Yang-Baxter algebra for the $R$-matrix of the eight-vertex model, in the case when the crossing parameter is a rational multiple of one of the periods. 
  We present new solutions to Einstein-Maxwell-dilaton-axion (EMDA) gravity in four dimensions describing black holes which asymptote to the linear dilaton background. In the non-rotating case they can be obtained as the limiting geometry of dilaton black holes. The rotating solutions (possibly endowed with a NUT parameter) are constructed using a generating technique based on the Sp(4,R) duality of the EMDA system. In a certain limit (with no event horizon present) our rotating solutions coincide with supersymmetric Israel-Wilson-Perjes type dilaton-axion solutions. In presence of an event horizon supersymmetry is broken. The temperature of the static black holes is constant, and their mass does not depend on it, so the heat capacity is zero. We investigate geodesics and wave propagation in these spacetimes and find superradiance in the rotating case. Because of the non-asymptotically flat nature of the geometry, certain modes are reflected from infinity, in particular, all superradiant modes are confined. This leads to classical instability of the rotating solutions. The non-rotating linear dilaton black holes are shown to be stable under spherical perturbations. 
  By considering 5--dimensional cosmological models with a bulk filled with a pressureless scalar field; equivalently dust matter, and a negative cosmological constant, we have found a regular instantonic solution which is free from any singularity at the origin of the extra--coordinate. This instanton describes 5--dimensional asymptotically anti de Sitter wormhole, when the bulk has a topology R times S^4. Compactified brane-world instantons which are built up from such instantonic solution describe either a single brane or a string of branes. Their analytical continuation to the pseudo--Riemannian metric can give rise to either 4-dimensional inflating branes or solutions with the same dynamical behaviour for extra--dimension and branes, in addition to multitemporal solutions. Dust brane-world models with arbitrary dimensions (D >= 5) as well as other spatial topologies are also briefly discussed. 
  We complete the list of fluxbrane solutions in classical supergravities by introducing Melvin type space-times supported by antisymmetric forms of rank $D-1$ and their pseudoscalar duals. In IIB theory these solutions belong to the same family as the seven-brane and D-instanton. In current notation, a fluxbrane supported by the D-1 form is an F0 brane, its euclidean continuation is a cylindrical background which ``interacts'' with a pointlike instanton. The general F0 brane can have a transverse space S_k times R^{(D-k-2)} with 1 <= k <= D-2. For k=1 we find the complete solution containing four parameters, three of them associated with pointlike singularities on the Melvin-type background. The S-dual to the F0 brane in ten dimensions is the F8 fluxbrane of codimension one, or F-wall, similar fluxbranes exist in any dimensions if an appropriate form field is present. F-walls contain naked singularities unless one introduces source branes. In such a way one obtains new brane-world type solutions with two bulk scalar fields. Their relation to the supersymmetric brane-worlds is discussed. 
  In this paper the N=2 supersymmetric extension of the Schroedinger Hamiltonian with 1/r-potential in arbitrary space-dimensions is constructed. The supersymmetric hydrogen atom admits a conserved Laplace-Runge-Lenz vector which extends the rotational symmetry SO(d) to a hidden SO(d+1) symmetry. This symmetry of the system is used to determine the discrete eigenvalues with their degeneracies and the corresponding bound state wave functions. 
  We compute the boundary two point functions of operators corresponding to massive spin 1 and spin 2 de Sitter fields, by an extension of the ``S-Matrix'' approach developed for bulk scalars. In each case the two point functions are of the form required for conformal invariance of the dual boundary field theory. We emphasise that in the context of dS/CFT one should consider unitary representations of the Euclidean conformal group, without reference to analytic continuation of the boundary theory to Lorentzian signature. 
  The notion of Wigner particles is attached to irreducible unitary representations of the Poincare group, characterized by parameters m and s of mass and spin, respectively. However, the Lorentz symmetry is broken in theories with long-range interactions, rendering this approach inapplicable (infraparticle problem). A unified treatment of both particles and infraparticles via the concept of particle weights can be given within the framework of Local Quantum Physics. They arise as temporal limits of physical states in the vacuum sector and describe the asymptotic particle content. In this paper their definition and characteristic properties are worked out in detail. The existence of the temporal limits is established by use of suitably defined seminorms which are also essential in proving the characteristic features of particle weights. 
  Berenstein, Maldacena, and Nastase have proposed, as a limit of the strong form of the AdS/CFT correspondence, that string theory in a particular plane wave background is dual to a certain subset of operators in the N=4 super-Yang-Mills theory. Even though this is a priori a strong/weak coupling duality, the matrix elements of the string theory Hamiltonian, when expressed in gauge theory variables, are analytic in the 't Hooft coupling constant. This allows one to conjecture that, like the masses of excited string states, these can be recovered using perturbation theory in Yang-Mills theory.   In this paper we identify the difference between the generator of scale transformations and a particular U(1) R-symmetry generator as the operator dual to the string theory Hamiltonian for nonvanishing string coupling. We compute its matrix elements and find that they agree with the string theory prediction provided that the state-operator map is modified for nonvanishing string coupling. We construct this map explicitly and calculate the anomalous dimensions of the new operators. We identify the component arising from the modification of the state-operator map with the contribution of the string theory contact terms to the masses of string states. 
  We construct the string states $|O_{p}^J>_J$, $|O_{q}^{J_1}>_{{J_1}{J_2}}$ and $|O_{0}^{J_{1}J_{2}}>_{{J_1}{J_2}}$ in the Hilbert space of the quantum mechanical orbifold model so as to calculate the three point functions and the matrix elements of the light-cone Hamiltonian from the interacting string bit model. With these string states we show that the three point functions and the matrix elements of the Hamiltonian derived from the interacting string bit model up to $g^{2}_2$ order precisely match with those computed from the perturbative SYM theory in BMN limit. 
  In our previous papers it has been shown that quantum theory based on a Galois field (GFQT) possesses a new symmetry between particles and antiparticles, and for massless particles this symmetry (called the AB one) is compatible with all representation operators of the symmetry algebra. In the present paper, it is shown that the AB symmetry is compatible with all representation operators of the symmetry algebra in the general case. As a consequence of simple arithmetical considerations, this symmetry is compatible with the vacuum condition, only for particles with half-integer spin (in usual units). If the AB symmetry is combined with the spin-statistics theorem, one arrives at the following conclusions: in quantum theory based on a Galois field i) any neutral particle can be only composite but not elementary (this property has been proved for the massless case in hep-th/0207192); ii) any interaction can involve only an even number of creation/annihilation operators. 
  In this note we update the discussion of the BMN correspondence and string interactions in hep-th/0205089 to incorporate the effects of operator mixing. We diagonalize the matrix of two point functions of single and double trace operators, and compute the eigen-operators and their anomalous dimensions to order g_2^2 \lambda'. Surprisingly, operators in different R symmetry multiplets remain degenerate at this order. We also calculate the corresponding energy shifts in string theory, and find a discrepancy with field theory results, indicating possible new effects in light-cone string field theory. 
  The non-perturbative quantum geometry of the Universal Hypermultiplet (UH) is investigated in N=2 supergravity. The UH low-energy effective action is given by the four-dimensional quaternionic non-linear sigma-model having an U(1)xU(1) isometry. The UH metric is governed by the single real pre-potential that is an eigenfunction of the Laplacian in the hyperbolic plane. We calculate the classical pre-potential corresponding to the standard (Ferrara-Sabharwal) metric of the UH arising in the Calabi-Yau compactification of type-II superstrings. The non-perturbative quaternionic metric, describing the D-instanton contributions to the UH geometry, is found by requiring the SL(2,Z) modular invariance of the UH pre-potential. The pre-potential found is unique, while it coincides with the D-instanton function of Green and Gutperle, given by the order-3/2 Eisenstein series. As a by-product, we prove cluster decomposition of D-instantons in curved spacetime. The non-perturbative UH pre-potential interpolates between the perturbative (large CY volume) region and the superconformal (Landau-Ginzburg) region in the UH moduli space. We also calculate a non-perturbative scalar potential in the hyper-K"ahler limit, when an abelian isometry of the UH metric is gauged in the presence of D-instantons. 
  We investigate the Penrose limit of various brane solutions including Dp-branes, NS5-branes, fundamental strings, (p,q) fivebranes and (p,q) strings. We obtain special null geodesics with the fixed radial coordinate (critical radius), along which the Penrose limit gives string theories with constant mass. We also study string theories with time-dependent mass, which arise from the Penrose limit of the brane backgrounds. We examine equations of motion of the strings in the asymptotic flat region and around the critical radius. In particular, for (p,q) fivebranes, we find that the string equations of motion in the directions with the B field are explicitly solved by the spheroidal wave functions. 
  In complete analogy with Seiberg-Witten map defined in noncommutative geometry we introduce a new map between a q-deformed gauge theory and an ordinary gauge theory. The construction of this map is elaborated in order to fit the Hopf algebra structure. 
  A first principle derivation is given of the neutrino damping rate in real-time thermal field theory. Starting from the discontinuity of the neutrino self energy at the two loop level, the damping rate can be expressed as integrals over space phase of amplitudes squared, weighted with statistical factors that account for the possibility of particle absorption or emission from the medium. Specific results for a background composed of neutrinos, leptons, protons and neutrons are given. Additionally, for the real part of the dispersion relation we discuss the relation between the results obtained from the thermal field theory, and those obtained by the thermal average of the forward scattering amplitude. 
  There are several mathematical and physical reasons why Dirac's quantization must hold. How far one can go without it remains an open problem. The present work outlines a few steps in this direction. 
  We show that Pauli's spin-statistics relation remains valid in noncommutative quantum field theories (NC QFT), with the exception of some peculiar cases of noncommutativity between space and time. We also prove that, while the individual symmetries C and T, and in some cases also P, are broken, the CPT theorem still holds in general for noncommutative field theories, in spite of the inherent nonlocality and violation of Lorentz invariance. 
  There are fractionally supersymmetric pp-waves in 11 dimensional supergravity. We study the corresponding supersymmetric Yang-Mills matrix dynamics for M-theory and find its superalgebra and vacuum equations. We show that the ground state energy of the Hamiltonian with nontrivial dynamical superysmmetry can be zero, positive or negative depending on parameters. 
  We discuss the construction of multi-caloron solutions with non-trivial holonomy, both as approximate superpositions and exact self-dual solutions. The charge k SU(n) moduli space can be described by kn constituent monopoles. Exact solutions help us to understand how these constituents can be seen as independent objects, which seems not possible with the approximate superposition. An "impurity scattering" calculation provides relatively simple expressions. Like at zero temperature an explicit parametrization requires solving a quadratic ADHM constraint, achieved here for a class of axially symmetric solutions. We will discuss the properties of these exact solutions in detail, but also demonstrate that interesting results can be obtained without explicitly solving for the constraint. 
  We study the string modes in the pp-wave light-cone string field theory. First, we clarify the discrepancy between the Neumann coefficients for the supergravity vertex and the zero mode of the full string one. We also repeat our previous manipulation of the prefactor for the string modes and find that the prefactor reduces to the energy difference of the cos modes minus that of the sin modes. Finally, we discuss off-shell three-string processes. 
  We propose a new vector potential for the Abelian magnetic monopole. The potential is non-singular in the entire region around the monopole. We argue how the Dirac quantization condition can be derived for any choice of potential. 
  The stringy description for the instabilities in the $RR$ charged $D_{p}-\bar{D}_{p}$ pairs is now well understood in terms of the open string tachyon condensation. The quantum interpretation presumably via the stringy description for the instabilities in the $NSNS$-charged $F1-\bar{F1}$ and $NS5-\bar{NS5}$ pairs in IIA/IIB theories, however, has not been established yet. This would be partly because of the absence (for the $F1-\bar{F1}$ case) or our relatively poor understanding (for the $NS5-\bar{NS5}$ case) of their worldvolume (gauge theory) dynamics. In the present work, using the well-known quantum description for instabilities in the $RR$-charged $D_{p}-\bar{D}_{p}$ systems and in the M-theory brane-antibrane systems and invoking appropriate string dualities, the stringy nature of the instabilities in the $NSNS$-charged $F1-\bar{F1}$ and $NS5-\bar{NS5}$ systems has been uncovered. For the annihilations to string vacua, the quantum, stringy interpretations are simple extensions of Sen's conjecture for those in $RR$-charged brane-antibrane systems. 
  A string model with dynamical metric and torsion is proposed. The geometry of the string is described by an effective Lagrangian for the scalar and vector fields. The path integral quantization of the string is considered. 
  String Quantum Gravity is motivated and introduced. Advances in the study of the classical and quantum string dynamics in curved spacetime are reported: 1-New Classes of Exact Multistring solutions in curved spacetimes. 2-Mass spectrum of Strings in Curved Spacetimes. 3-The effect of a Cosmological Constant and of Spacial Curvature on Classical and Quantum Strings. 4-Classical splitting of Fundamental Strings. 5-The General String Evolution in constant Curvature Spacetimes. 6-The Conformal Invariance Effects. 
  New Developments in String Gravity and String Cosmology are reported: 1-String driven cosmology and its Predictions. 2-The primordial gravitational wave background in string cosmology. 3-Non-singular string cosmologies from Exact Conformal Field Theories. 4-Quantum Field Theory, String Temperature and the String Phase of de Sitter space-time, 5-Hawking Radiation in String Theory and the String Phase of Black Holes. 6-New Dual Relation between Quantum Field Theory regimes and String regimes in Curved Backgrounds, and the 'QFT/String Tango'. 7- New Coherent String States and Minimal Uncertainty Principle in WZWN models 
  We show that infinite variety of Poincar\'{e} bialgebras with nontrivial classical r-matrices generate nonsymmetric nonlinear composition laws for the fourmomenta. We also present the problem of lifting the Poincar\'{e} bialgebras to quantum Poincar\'{e} groups by using e.g. Drinfeld twist, what permits to provide the nonlinear composition law in any order of dimensionfull deformation parmeter $\lambda$ (from physical reasons we can put $\lambda = \lambda_{p}$ where $\lambda_{p}$ is the Planck lenght). The second infinite variety of composition laws for fourmomentum is obtained by nonlinear change of basis in Poincar\'{e} algebra, which can be performed for any choice of coalgebraic sector, with classical or quantum coproduct. In last Section we propose some modification of Hopf algebra scheme with Casimir-dependent deformation parameter, which can help to resolve the problem of consistent passage to macroscopic classical limit. 
  Here we present a class of solutions of M-theory flux branes, which are non-singular at origin. These class of solutions help us to determine the field strength at origin together with the behavior of it near origin. Further we show a way to find the attractor solutions of such flux branes. 
  We propose a new method to determine the unknown parameter associated to a self-consistent harmonic approximation. We check the validity of our technique in the context of the sine-Gordon model. As a non trivial application we consider the scaling regime of the 2D Ising model away from the critical point and in the presence of a magnetic field $h$. We derive an expression that relates the approximate correlation length $\xi$, $T-T_c$ and $h$. 
  We show how higher-genus su(N) fusion multiplicities may be computed as the discretized volumes of certain polytopes. The method is illustrated by explicit analyses of some su(3) and su(4) fusions, but applies to all higher-point and higher-genus su(N) fusions. It is based on an extension of the realm of Berenstein-Zelevinsky triangles by including so-called gluing and loop-gluing diagrams. The identification of the loop-gluing diagrams is our main new result, since they enable us to characterize higher-genus fusions in terms of polytopes. Also, the genus-2 0-point su(3) fusion multiplicity is found to be a simple binomial coefficient in the affine level. 
  Through defining irreducible loop integrals (ILIs), a set of consistency conditions for the regularized (quadratically and logarithmically) divergent ILIs are obtained to maintain the generalized Ward identities of gauge invariance in non-Abelian gauge theories. Overlapping UV divergences are explicitly shown to be factorizable in the ILIs and be harmless via suitable subtractions. A new regularization and renormalization method is presented in the initial space-time dimension of the theory.   The procedure respects unitarity and causality. Of interest, the method leads to an infinity free renormalization and meanwhile maintains the symmetry principles of the original theory except the intrinsic mass scale caused conformal scaling symmetry breaking and the anomaly induced symmetry breaking. Quantum field theories (QFTs) regularized through the new method are well defined and governed by a physically meaningful characteristic energy scale (CES) $M_c$ and a physically interesting sliding energy scale (SES) $\mu_s$ which can run from $\mu_s \sim M_c$ to a dynamically generated mass gap $\mu_s=\mu_c$ or to $\mu_s =0$ in the absence of mass gap and infrared (IR) problem.   It is strongly indicated that the conformal scaling symmetry and its breaking mechanism play an important role for understanding the mass gap and quark confinement. 
  We study a class of inhomogeneous and anisotropic $G_2$ string cosmological models. In the case of separable $G_2$ models we show that the governing equations reduce to a system of ordinary differential equations. We focus on a class of separable $G_2$ M-theory cosmological models, and study their qualitative behaviour (a class of models with time-reversed dynamics is also possible). We find that generically these inhomogeneous M-theory cosmologies evolve from a spatially inhomogeneous and negatively curved model with a non-trivial form field towards spatially flat and spatially homogeneous dilaton-moduli-vacuum solutions with trivial form--fields. The late time behaviour is the same as that of spatially homogeneous models previously studied. However, the inhomogeneities are not dynamically insignificant at early times in these models. 
  The one-loop renormalization of the abelian Higgs-Kibble model in a general 't Hooft gauge and with chiral fermions is fully worked out within dimensional renormalization scheme with a non-anticommuting $\gamma_5$. The anomalous terms introduced in the Slavnov-Taylor identities by the minimal subtraction algorithm are calculated and the asymmetric counterterms needed to restore the BRS symmetry, if the anomaly cancellation conditions are met, are computed. The computations draw heavily from regularized action principles and Algebraic Renormalization theory. 
  We show that the deformations of Virasoro and super Virasoro algebra, constructed earlier on an abstract mathematical background, emerge after Wick rotation, within an exact treatment of discrete-time free field models on a circle. The deformation parameter is $e^\lambda$, where $\lambda=\tau/\rho$ is the ratio of the discrete-time scale $\tau$ and the radius $\rho$ of the compact space. 
  Recent work claimed that the off-diagonal gluons (and ghosts) in pure Yang-Mills theories, with Maximal Abelian gauge fixing (MAG), attain a dynamical mass through an off-diagonal ghost condensate. This condensation takes place due to a quartic ghost interaction, unavoidably present in MAG for renormalizability purposes. The off-diagonal mass can be seen as evidence for Abelian dominance. We discuss why ghost condensation of the type discussed in those works cannot be the reason for the off-diagonal mass and Abelian dominance, since it results in a tachyonic mass. We also point out what the full mechanism behind the generation of a real mass might look like. 
  We develop an approach based on the Noether method to construct nilpotent BRST charges and BRST-invariant actions. We apply this approach first to the holomorphic part of the flat-space covariant superstring, and we find that the ghosts b, c_z which we introduced by hand in our earlier work, are needed to fix gauge symmetries of the ghost action. Then we apply this technique to the superparticle and determine its cohomology. Finally, we extend our results to the combined left- and right-moving sectors of the superstring. 
  We introduce a method to obtain deformed defects starting from a given scalar field theory which possesses defect solutions. The procedure allows the construction of infinitely many new theories that support defect solutions, analytically expressed in terms of the defects of the original theory. The method is general, valid for both topological and non-topological defects, and we show how it extends to quantum mechanics, and how it works when the scalar field couples to fermions. We illustrate the general procedure with several examples, which support kink-like or lump-like defects. 
  We obtain analytic, nonperturbative, approximate solutions of Yukawa theory in the one-fermion sector using light-front quantization. The theory is regulated in the ultraviolet by the introduction of heavy Pauli-Villars scalar and fermion fields, each with negative norm. In order to obtain a directly soluble problem, fermion-pair creation and annihilation are neglected, and the number of bosonic constituents is limited to one of either type. We discuss some of the features of the wave function of the eigensolution, including its endpoint behavior and spin and orbital angular momentum content. The limit of infinite Pauli-Villars mass receives special scrutiny. 
  Recently, investigations have begun into a holographic duality for two-dimensional de Sitter space. In this paper, we evaluate the associated central charge, using a modified version of the canonical Hamiltonian method that was first advocated by Catelani {\it et al}. Our computation agrees with that of a prior work (Cadoni {\it et al}), but we argue that the method used here is, perhaps, aesthetically preferable on holographic grounds. We also confirm an agreement between the Cardy and thermodynamic entropy, thus providing further support for the conjectured two-dimensional de Sitter/conformal field theory correspondence. 
  We study localization of gravity in flat space in superstring theory. We find that an induced Einstein-Hilbert term can be generated only in four dimensions, when the bulk is a non-compact Calabi-Yau threefold with non-vanishing Euler number. The origin of this term is traced to R^4 couplings in ten dimensions. Moreover, its size can be made much larger than the ten-dimensional gravitational Planck scale by tuning the string coupling to be very small or the Euler number to be very large. We also study the width of the localization and discuss the problems for constructing realistic string models with no compact extra dimensions. 
  By exploiting a correspondence between Random Regge triangulations (i.e., Regge triangulations with variable connectivity) and punctured Riemann surfaces, we propose a possible characterization of the SU(2) Wess-Zumino-Witten model on a triangulated surface of genus g. Techniques of boundary CFT are used for the analysis of the quantum amplitudes of the model at level k=1. These techniques provide a non-trivial algebra of boundary insertion operators governing a brane-like interaction between simplicial curvature and WZW fields. Through such a mechanism, we explicitly characterize the partition function of the model in terms of the metric geometry of the triangulation, and of the 6j symbols of the quantum group SU(2)_Q, at Q=e^{\sqrt{-1}\pi /3}. We briefly comment on the connection with bulk Chern-Simons theory. 
  We study the rest-frame instant form of a new formulation of relativistic perfect fluids in terms of new generalized Eulerian configuration coordinates. After the separation of the relativistic center of mass from the relative variables on the Wigner hyper-planes, we define orientational and shape variables for the fluid, viewed as a relativistic extended deformable body, by introducing dynamical body frames. Finally we define Dixon's multipoles for the fluid. 
  We describe in detail how a sliding scale is introduced in the renormalization of a QFT according to integer-dimensional implicit regularization scheme. We show that since no regulator needs to be specified at intermediate steps of the calculation, the introduction of a mass scale is a direct consequence of a set of renormalization conditions. As an illustration the one loop beta-function for QED and lambda*phi^4 theories are derived. They are given in terms of derivatives of appropriately systematized functions (related to definited parts of the amplitudes) with respect to a mass scale mu. Our formal scheme can be easily generalized to higher loop calculations. 
  We study the rolling tachyon condensate in the presence of a gauge field. The generic vacuum admits both a rolling tachyon, \dot{T}, and a uniform electric field, \vec{E}, which together affect the effective metric governing the fluctuations of open string modes. If one suppresses the gauge field altogether, the light-cone collapses completely. This is the Carrollian limit, with vanishing speed of light and no possible propagation of signals. In the presence of a gauge field, however, the lightcone is squeezed to the shape of a fan, allowing propagation of signals along the direction of \pm \vec{E} at speed |E|=<1. This shows that there are perturbative degrees of freedom propagating along electric flux lines. Such causal behavior appears to be a very general feature of tachyon effective Lagrangian with runway potentials. We speculate on how this may be connected to appearance of fundamental strings. 
  We present a geometrical description of N=8 supergravity, using central charge superspace. The essential properties of the multiplet, like self-duality properties of the vectors or the non-linear sigma model structure of the scalars, are found as consequences of constraints at 0 and 1/2 canonical dimension. We also present in detail how to derive from this geometrical formulation the supergravity transformations as well as the whole equations of motion for the component fields in order to compare them with the results already known and obtained in formulations on the component level. 
  We study the question of generalizing light-front field theories to finite temperature. We show that the naive generalization has serious problems and we identify the source of the difficulty. We provide a proper generalization of these theories to finite temperature based on a relativistic description of thermal field theories, both in the real and the imaginary time formalisms. Various issues associated with scalar and fermion theories, such as non-analyticity of self-energy, tensor decomposition are discussed in detail. 
  The $sl(2)$ affine Toda model coupled to matter (ATM) is shown to describe various features, such as the spectrum and string tension, of the low-energy effective Lagrangian of QCD$_{2}$ (one flavor and $N$ colors). The corresponding string tension is computed when the dynamical quarks are in the {\sl fundamental} representation of SU(N) and in the {\sl adjoint} representation of SU(2). 
  We summarize our findings on the quantum moduli constraints and superpotentials of an infinite family of moose extensions of $n_f = n_c$ SUSY QCD. For $n_c=2$, we perform concrete calculations using traditional integrating out techniques as well as Intriligator's ``integrating in'' technique. Checking the constraints and superpotentials in the limits of setting $\Lambda$'s to zero or integrating out mass terms, we find that the quantum moduli constraints are local in theory space and are equivalent to a consistent structure of ``splitting relations'' amongst the different theories. Extending the results to arbitrary $n_c$, we show that the splitting relations, along with a set of rules for flowing from a high energy theory to a low energy theory, incorporate much of the physics of the moose chain. The relations can be used both to simplify perturbative calculations of symmetry breaking and to incorporate nonperturbative effects. 
  Bekenstein proposed that the spectrum of horizon area of quantized black holes must be discrete and uniformly spaced. We examine this proposal in the context of spherically symmetric charged black holes in a general class of gravity theories. By imposing suitable boundary conditions on the reduced phase space of the theory to incorporate the thermodynamic properties of these black holes and then performing a simplifying canonical transformation, we are able to quantize the system exactly. The resulting spectra of horizon area, as well as that of charge are indeed discrete. Within this quantization scheme, near-extremal black holes (of any mass) turn out to be highly quantum objects, whereas extremal black holes do not appear in the spectrum, a result that is consistent with the postulated third law of black hole thermodynamics. 
  The little groups (i.e. the subgroups of Lorentz group, leaving invariant given configurations of tensorial charges) of unitary irreps of superstring/M-theory superalgebras are considered. It is noted, that in the case of $(n-1)/n$ (maximal supersymmetric) BPS configuration in any dimensions the non-zero supercharge is neutral w.r.t. the algebra of little group, which means that all members of supermultiplet are in the same representation of that algebra and hence of (generalized with tensorial charges) Poincare algebra. This situation is similar to two-dimensional case and shows that usual spin-statistics connection statement is insufficient in the presence of branes, because different little groups can appear. We discuss the rules for definition of statistics for representations of generalized Poincare, and note that a geometric quantization method seems to be most relevant for that purpose. 
  Recent progress in theoretical physics suggests that the dark energy in the universe might be resulted from the rolling tachyon field of string theory. Measurements to SNe Ia can be helpful to reconstruct the equation of state of the rolling tachyon which is a possible candidate of dark energy. We present a numerical analysis for the evolution of the equation of state of the rolling tachyon and derive the reconstruction equations for the equation of state as well as the potential. 
  The algebraic and geometric structures of deformations are analyzed concerning topological field theories of Schwarz type by means of the Batalin-Vilkovisky formalism. Deformations of the Chern-Simons-BF theory in three dimensions induces the Courant algebroid structure on the target space as a sigma model. Deformations of BF theories in $n$ dimensions are also analyzed. Two dimensional deformed BF theory induces the Poisson structure and three dimensional deformed BF theory induces the Courant algebroid structure on the target space as a sigma model. The deformations of BF theories in $n$ dimensions induce the structures of Batalin-Vilkovisky algebras on the target space. 
  We discuss the Green-Schwarz action for type IIB strings in general plane-wave backgrounds obtained as Penrose limits from any IIB supergravity solutions with vanishing background fermions. Using the normal-coordinate expansion in superspace, we prove that the light-cone action is necessarily quadratic in the fermionic coordinates. This proof is valid for more general pp-wave backgrounds under certain conditions. We also write down the complete quadratic action for general bosonic on-shell backgrounds in a form in which its geometrical meaning is manifest both in the Einstein and string frames. When the dilaton and 1-form field strength are vanishing, and the other field strengths are constant, our string-frame action reduces, up to conventions, to the one which has been written down using the supercovariant derivative. 
  According to Belinsky, Khalatnikov and Lifshitz, gravity near a space-like singularity reduces to a set of decoupled one-dimensional mechanical models at each point in space. We point out that these models fall into a class of conformal mechanical models first introduced by de Alfaro, Fubini and Furlan (DFF). The deformation used by DFF to render the spectrum discrete corresponds to a negative cosmological constant. The wave function of the universe is the zero-energy eigenmode of the Hamiltonian, also known as the spherical vector of the representation of the conformal group SO(1,2). A new class of conformal quantum mechanical models is constructed, based on the quantization of nilpotent coadjoint orbits, where the conformal group is enhanced to an ADE non-compact group for which the spherical vector is known. 
  We study O(N) symmetric supersymmetric models in three dimensions at finite temperature. These models are known to have an interesting phase structures. In particular, in the limit $N \to \infty$ one finds spontaneous breaking of scale invariance with no explicit breaking. Supersymmetry is softly broken at finite temperature and the peculiar phase structure and properties seen at T=0 are studied here at finite temperature. 
  We discuss the general form of the mass terms that can appear in the effective field theories of coordinate-dependent compactifications a la Scherk-Schwarz. As an illustrative example, we consider an interacting five-dimensional theory compactified on the orbifold S^1/Z_2, with a fermion subject to twisted periodicity conditions. We show how the same physics can be described by equivalent effective Lagrangians for periodic fields, related by field redefinitions and differing only in the form of the five-dimensional mass terms. In a suitable limit, these mass terms can be localized at the orbifold fixed points. We also show how to reconstruct the twist parameter from any given mass terms of the allowed form. Finally, after mentioning some possible generalizations of our results, we re-discuss the example of brane-induced supersymmetry breaking in five-dimensional Poincare' supergravity, and comment on its relation with gaugino condensation in M-theory. 
  We discuss two semiclassical string solutions on AdS_5\times S_5. In the first case, we consider a multiwrapped circular string pulsating in the radial direction of AdS_5, but fixed to a point on the S_5. We compute the energy of this motion as a function of a large quantum number $n$. We identify the string level with $mn$, where $m$ is the number of string wrappings.   Using the AdS/CFT correspondence, we argue that the bare dimension of the corresponding gauge invariant operator is $2n$ and that its anomalous dimension scales as \lambda^{1/4}\sqrt{mn}, for large $n$. Next we consider a multiwrapped circular string pulsating about two opposite poles of the $S_5$. We compute the energy of this motion as a function of a large quantum number, $n$ where again the string level is given as $mn$. We find that the dimension of the corresponding operator is 2n(1+f(m^2\lambda/(2n)^2)), where f(x) is computible as a series about x=0 and where it is analytic. We also compare this result to the BMN result for large J operators. 
  There has been recent interest in conformal twisted boundary conditions and their realisations in solvable lattice models. For the Ising and Potts quantum chains, these amount to boundary terms that are related to duality, which is a proper symmetry of the model at criticality. Thus, at criticality, the duality-twisted Ising model is translationally invariant, similar to the more familiar cases of periodic and antiperiodic boundary conditions. The complete finite-size spectrum of the Ising quantum chain with this peculiar boundary condition is obtained. 
  In recent work, it was shown that velocity-dependent forces between moving strings or branes lead to an accelerating expanding universe without assuming the existence of a cosmological constant. Here we show that the repulsive velocity-dependent force arises in more general contexts and can lead to cosmic structure formation. 
  We derive a renormalization group formalism for the Randall-Sundrum scenario, where the renormalization scale is set by a floating compactification radius. While inspired by the AdS/CFT conjecture, our results are derived concretely within higher-dimensional effective field theory. Matching theories with different radii leads to running hidden brane couplings. The hidden brane Lagrangian consists of four-dimensional local operators constructed from the induced value of the bulk fields on the brane. We find hidden Lagrangians which are non-trivial fixed points of the RG flow. Calculations in RS1 can be greatly simplified by ``running down'' the effective theory to a small radius. We demonstrate these simplifications by studying the Goldberger-Wise stabilization mechanism. In this paper, we focus on the classical and tree-level quantum field theory of bulk scalar fields, which demonstrates the essential features of the RG in the simplest context. 
  We describe new numerical methods to solve the static axisymmetric vacuum Einstein equations in more than four dimensions. As an illustration, we study the compactified non-uniform black string phase connected to the uniform strings at the Gregory-Laflamme critical point. We compute solutions with a ratio of maximum to minimum horizon radius up to nine. For a fixed compactification radius, the mass of these solutions is larger than the mass of the classically unstable uniform strings. Thus they cannot be the end state of the instability. 
  We investigate factorized scattering from a reflecting and transmitting impurity. Bulk scattering is non trivial, provided that the bulk scattering matrix depends separately on the spectral parameters of the colliding particles, and not only on their difference. We show that a specific extension of a boundary algebra encodes the underlying scattering theory. The total scattering operator is constructed in this framework and shown to be unitary. 
  We compute the O(1/N) correction to the stability critical exponent, omega, in the Landau-Ginzburg-Wilson model with O(N) x O(m) symmetry at the stable chiral fixed point and the stable direction at the unstable antichiral fixed point. Several constraints on the O(1/N) coefficients of the four loop perturbative beta-functions are computed. 
  We obtain the Penrose limit of NCYM theories in dimensions $3 \leq d \leq 6$ which originate from (D$(p-2)$, D$p$) supergravity bound state configurations for $2 \leq p \leq 5$ in the so-called NCYM limit. In most cases the Penrose limit does not lead to solvable string theories except for six-dimensional NCYM theory. We obtain the masses of various bosonic coordinates and observe that they are light-cone time dependent and their squares can be negative as has also been observed in other cases in the literature. When the non-commutative effect is turned off we recover the results of Penrose limit of ordinary D$p$-branes in the usual YM limit. We point out that for $d = 6$ NCYM theory, there exists another null geodesic in the neighborhood of which the Penrose limit leads to a solvable string theory. We briefly discuss the quantization of this theory and show that this pp-wave background is half supersymmetric. 
  Induced quantum gravity dynamics built over a Riemann surface is studied in arbitrary dimension. Local coordinates on the target space are given by means of the Laguerre-Forsyth construction. A simple model is proposed and pertubatively quantized. In doing so, the classical W-symmetry turns out to be preserved on-shell at any order of the $\hbar$ perturbative expansion. As a main result, due to quantum corrections, the target coordinates acquire a non-trivial character. 
  Possible short and semi-short representations for $\N=2$ and $\N=4$ superconformal symmetry in four dimensions are discussed. For $\N=4$ the well known short supermultiplets whose lowest dimension conformal primary operators correspond to $\half$-BPS or ${1\over 4}$-BPS states and are scalar fields belonging to the $SU(4)_r$ symmetry representations $[0,p,0]$ and $[q,p,q]$ and having scale dimension $\Delta =p$ and $\Delta = 2q+p$ respectively are recovered. The representation content of semi-short multiplets, which arise at the unitarity threshold for long multiplets, is discussed. It is shown how, at the unitarity threshold, a long multiplet can be decomposed into four semi-short multiplets. If the conformal primary state is spinless one of these becomes a short multiplet. For $\N=4$ a ${1\over 4}$-BPS multiplet need not have a protected dimension unless the primary state belongs to a $[1,p,1]$ representation. 
  We investigate here a supermatrix model with a mass term and a cubic interaction. It is based on the super Lie algebra osp(1|32,R), which could play a role in the construction of the eleven-dimensional M-theory. This model contains a massive version of the IIB matrix model, where some fields have a tachyonic mass term. Therefore, the trivial vacuum of this theory is unstable. However, this model possesses several classical solutions where these fields build noncommutative curved spaces and these solutions are shown to be energetically more favorable than the trivial vacuum. In particular, we describe in details two cases, the SO(3) \times SO(3) \times SO(3) (three fuzzy 2-spheres) and the SO(9) (fuzzy 8-sphere) classical backgrounds. 
  After some considerations and coincidences that appear when working in the light-cone gauge, in both the Mandelstam-Leibbrandt prescription and the covariantization method, we suspect that there must be some connection between them. This work shows that we were right and it is practically trivial to demonstrate that relationship. And since the covariantization method is not a prescription, this implies that the results of Mandelstam and Leibbrandt are not prescriptions too, but they are a identities from light-cone gauge coordinates. 
  These ICTP Trieste lecture notes review the pure spinor approach to quantizing the superstring with manifest D=10 super-Poincare invariance. The first section discusses covariant quantization of the superparticle and gives a new proof of equivalence with the Brink-Schwarz superparticle. The second section discusses the superstring in a flat background and shows how to construct vertex operators and compute tree amplitudes in a manifestly super-Poincare covariant manner. And the third section discusses quantization of the superstring in curved backgrounds which can include Ramond-Ramond flux. 
  We give a formulation of linearized minimal 5-dimensional supergravity in N = 1 superspace. Infinitesimal local 5D diffeomorphisms, local 5D Lorentz transformations, and local 5D supersymmetry are all realized as off-shell superfield transformations. Compactification on an S^1 / Z_2 orbifold and couplings to brane-localized supermultiplets are very simple in this formalism. We use this to show that 5-dimensional supergravity can naturally generate mu and B mu terms of the correct size in gaugino- or radion-mediated supersymmetry breaking. We also include a self-contained review of linearized minimal 4D supergravity in superspace. 
  The quantum gravity problem of N point particles interacting with the gravitational field in 2+1 dimensions is approached working out the phase-space functional integral. The maximally slicing gauge is adopted for a non compact open universe with the topology of the plane. The conjugate momenta to the gravitational field are related to a class of meromorphic quadratic differentials. The boundary term for the non compact space is worked out in detail. In the extraction of the physical degrees of freedom functional determinants related to the puncture formulation of string theory occur and cancel out in the final reduction. Finally the ordering problem in the definition of the functional integral is discussed. 
  A re-formulated, non-Hermitian version of the Witten's supersymmetric quantum mechanics is presented. Its use of pseudo-Hermitian (so called PT symmetric) Hamiltonians is reviewed and illustrated via several forms of an innovated supersymmetric partnership between strongly singular ("spiked") harmonic oscillators. 
  The Becker-Becker-Strominger formula, describing the string world-sheet instanton corrections to the four-fermion correlator in the Calabi-Yau compactified type-IIA superstrings, is calculated in the special case of the Calabi-Yau threefold realized in the intersection of two Del Pezzo surfaces. We also derive the selection rules in the supersymmetric GUT of the Pati-Salam type associated with our construction. 
  Type I - heterotic duality in D=10 predicts various relations and constraints on higher order F^n couplings at different string loop levels on both sides. We prove the vanishing of two-loop corrections to the heterotic F^4 terms, which is one of the basic predictions from this duality. Furthermore, we show that the heterotic F^5 and (CP even) F^6 couplings are not renormalized at one loop. These results strengthen the conjecture that in D=10 any Tr F^(2n) coupling appears only at the disc tree-level on type I side and at (n-1)-loop level on the heterotic side. Our non-renormalization theorems are valid in any heterotic string vacuum with sixteen supercharges. 
  We study almost-forward scattering in the context of usual and non-commutative QED. We study the semi-classical behaviour of particles undergoing this scattering process in the two theories, and show that the shock wave picture, effective in QED fails for NCQED. Further, we show that whereas in QED, there are no leading logarithmic contributions to the amplitude upto sixth order, uncancelled logarithms appear in NCQED. 
  Using graviton correlator on deSitter (dS) brane in 5d AdS or dS bulk we calculate the four-dimensional Newton potential. For flat brane in AdS bulk the Randall-Sundrum result is recovered. For flat brane in dS bulk the sign of subleading ($1/r^3$) term in Newton potential is negative if compare with AdS bulk. In accordance with dS/CFT correspondence this indicates that dual CFT should be non-unitary (for example, higher derivative conformal theory). 
  We review the formalism of holographic renormalization. We start by discussing mathematical results on asymptotically anti-de Sitter spacetimes. We then outline the general method of holographic renormalization. The method is illustrated by working all details in a simple example: a massive scalar field on anti-de Sitter spacetime. The discussion includes the derivation of the on-shell renormalized action, of holographic Ward identities, anomalies and RG equations, and the computation of renormalized one-, two- and four-point functions. We then discuss the application of the method to holographic RG flows. We also show that the results of the near-boundary analysis of asymptotically AdS spacetimes can be analytically continued to apply to asymptotically de Sitter spacetimes. In particular, it is shown that the Brown-York stress energy tensor of de Sitter spacetime is equal, up to a dimension dependent sign, to the Brown-York stress energy tensor of an associated AdS spacetime. 
  We evaluate various disk level four-point functions involving the massless scalar and tachyon vertex operators in the presence of background B-flux in superstring theory. By studying these amplitudes in specific limits, we find couplings of two scalars with two tachyons, and couplings of four tachyons on the world-volume of non-BPS D-branes of superstring theory. They are fully consistent with the non-commutative tachyonic Dirac-Born-Infeld effective action. They also fix the coefficient of $T^4$ term in the expansion of the tachyon potential around its maximum. 
  We show that the spacetime of the near-horizon limit of the extreme rotating d=5 black hole, which is maximally supersymmetric in N=2,d=5 supergravity for any value of the rotation parameter j in [-1,1], is locally isomorphic to a homogeneous non-symmetric spacetime corresponding to an element of the 1-parameter family of coset spaces SO(2,1)x SO(3)/SO(2)_j in which the subgroup SO(2)_j is a combination of the two SO(2) subgroups of SO(2,1) and SO(3). 
  The latest astrophysical data on the Supernova luminosity--distance -- redshift relations, primordial nucleosynthesis, value of Cosmic Microwave Background--temperature, and baryon asymmetry are considered as an evidence of relative measurement standard, field nature of time, and conformal symmetry of the physical world. We show how these principles of description of the universe help modern quantum field theory to explain the creation of the universe, time, and matter from the physical vacuum as a state with the lowest energy. 
  We show that Riemann surfaces, and separated variables immediately provide classical Poisson commuting Hamiltonians. We show that Baxter's equations for separated variables immediately provide quantum commuting Hamiltonians. The construction is simple, general, and does not rely on the Yang--Baxter equation. 
  A unified theory of four-dimensional gravity together with the standard model is presented, with supersymmetry breaking of M-theory at a TeV. Masses of the the known particles are derived. The cosmological constant is quantum generated to the observed value. Quantum corrections to the classical compactification are analyzed, and the scenario is stable. 
  In this work, we pursue further consequences of a general formalism for non-covariant gauges developed in an earlier work (hep-th/0205042). We carry out further analysis of the additional restrictions on renormalizations noted in that work. We use the example of the axial gauge A_3=0. We find that if multiplicative renormalization together with ghost-decoupling is to hold, the ``prescription-term'' (that defines a prescription) cannot be chosen arbitrarily but has to satisfy certain non-trivial conditions (over and above those implied by the validity of power counting) arising from the WT identities associated with the residual gauge invariance. We also give a restricted class of solutions to these conditions. 
  This work initiates the study of {\it orthogonal} symmetric polynomials in superspace. Here we present two approaches leading to a family of orthogonal polynomials in superspace that generalize the Jack polynomials. The first approach relies on previous work by the authors in which eigenfunctions of the supersymmetric extension of the trigonometric Calogero-Moser-Sutherland Hamiltonian were constructed. Orthogonal eigenfunctions are now obtained by diagonalizing the first nontrivial element of a bosonic tower of commuting conserved charges not containing this Hamiltonian. Quite remarkably, the expansion coefficients of these orthogonal eigenfunctions in the supermonomial basis are stable with respect to the number of variables. The second and more direct approach amounts to symmetrize products of non-symmetric Jack polynomials with monomials in the fermionic variables. This time, the orthogonality is inherited from the orthogonality of the non-symmetric Jack polynomials, and the value of the norm is given explicitly. 
  The quantum correlations of scalar fields are examined as a power series in derivatives. Recursive algebraic equations are derived and determine the amplitudes; all loop integrations are performed. This recursion contains the same information as the usual loop expansion. The approach is pragmatic and generalizable to most quantum field theories. 
  Assuming an Einstein-Gauss-Bonnet theory of gravitation in a ($D \geq 5$)-dimensional spacetime with boundary, we consider the problem of the boundary dynamics given the matter Lagrangian on it. The resulting equation is applied in particular on the derivation of the Friedmann eq. of a 3-brane, understood as the non-orientable boundary of a 5d spacetime. We briefly discuss the contradictory conclusions of the literature. 
  In this paper we describe the heterotic dual of the type IIB theory compactified to four dimensions on a toroidal orientifold in the presence of fluxes. The type IIB background is most easily described in terms of an M-theory compactification on a four-fold. The heterotic dual is obtained by performing a series of U-dualities. We argue that these dualities preserve supersymmetry and that the supergravity description is valid after performing them. The heterotic string is compactified on a manifold that is no longer Kahler and has torsion. These manifolds have to satisfy a number of constraints, for example, existence of an holomorphic three-form, size limits, torsional equations etc. We give an explicit form of the background and study the constraints associated to them. 
  We consider Dirac monopoles embedded into SU(N) gauge theory with theta-term for $\theta = 4\pi M $ (where $M$ is half-integer for $N = 2$ and is integer for $N>2$). Due to the theta - term those monopoles obtain the SU(N) charge and become the dyons. They belong to different irreducible representations of SU(N) (but not to all of them). The admitted representations are enumerated. Their minimal rank increases with increasing of $N$. The main result of the paper is the representation of the partition function of SU(N) model with theta-term (that contains singular gauge fields correspondent to the mentioned monopoles) as the vacuum average of the product of Wilson loops (considered along the monopole worldlines). This vacuum average should be calculated within the correspondent model without theta-term. 
  A quantum theory of gravity is described in the case of a positive cosmological constant in 3+1 dimensions. Both old and new results are described, which support the case that loop quantum gravity provides a satisfactory quantum theory of gravity. These include the existence of a ground state, discoverd by Kodama, which both is an exact solution to the constraints of quantum gravity and has a semiclassical limit which is deSitter spacetime. The long wavelength excitations of this state are studied and are shown to reproduce both gravitons and, when matter is included, quantum field theory on deSitter spacetime. Furthermore, one may derive directly from the Wheeler-deWitt equation, Planck scale, computable corrections to the energy-momentum relations for matter fields. This may lead in the next few years to experimental tests of the theory.   To study the excitations of the Kodama state exactly requires the use of the spin network representation, which is quantum deformed due to the cosmological constant. The theory may be developed within a single horizon, and the boundary states described exactly in terms of a boundary Chern-Simons theory. The Bekenstein bound is recovered and the N bound of Banks is given a background independent explanation.   The paper is written as an introduction to loop quantum gravity, requiring no prior knowledge of the subject. The deep relationship between quantum gravity and topological field theory is stressed throughout. 
  Recently Polchinski and Strassler reproduced the high energy QCD scaling at fixed angles from a gauge string duality inspired in AdS/CFT correspondence. In their approach a confining gauge theory is taken as approximately dual to an AdS space with an IR cut off. Considering such an approximation (AdS slice) we found a one to one holographic mapping between bulk and boundary scalar fields. Associating the bulk fields with dilatons and the boundary fields with glueballs of the confining gauge theory we also found the same high energy QCD scaling. Here, using this holographic mapping we give a simple estimate for the mass ratios of the glueballs assuming the AdS slice approximation to be valid at low energies. We also compare these results to those coming from supergravity and lattice QCD. 
  We consider the ground state of supermembrane on the maximally supersymmetric pp-wave background by using the quantum-mechanical procedure of de Wit-Hoppe-Nicolai. In the pp-wave case the ground state has non-trivial structure even in the zero-mode Hamiltonian, which is identical with that of superparticles on the pp-wave and resembles supersymmetric harmonic oscillators. The supergravity multiplet in the flat case is splitting with a certain energy difference. We explicitly construct the unique supersymmetric ground-state wave function of the zero-mode Hamiltonian, which is obviously normalizable. Moreover, we discuss the nonzero-mode Hamiltonian and construct an example for the ground-state wave function with a truncation of the variables. This special solution seems non-normalizable but its L^2-norm can be represented by an asymptotic series. 
  We study the fuzzy structure of the general Kaehler coset space G/S\otimes{U(1)}^k deformed by the Fedosov formalism. It is shown that the Killing potentials satisfy the fuzzy algebrae working in the Darboux coordinates. 
  The surface energy for a conformally flat spacetime which represents the Hawking wormhole in spherical (static) Rindler coordinates is computed using the Hawking - Hunter formalism for non asymptotically - flat spacetimes.  The physical gravitational Hamiltonian is proportional to the Rindler acceleration g of the hyperbolic observer and is finite on the event horizon ksi = b (b-the Planck length, ksi - the Minkowski interval). The corresponding temperature of the system of particles associated to the massless scalar field Psi, coupled conformally to Einstein's equations, is given by the Davies - Unruh temperature up to a constant factor of order unity. 
  We do not find any AdS_2 branes, neither in the H_3^+ WZNW model nor the SL(2,R) WZNW model. We then reexamine the case of the branes that possess a su(2) symmetry: we speculate that they would have to live on the boundary of AdS_3. This cannot be realized in an euclidean spacetime, but in the SL(2,R) WZNW model by analytical continuation. 
  We discuss the relation between matrix models and the Seiberg--Witten type (SW) theories, recently proposed by Dijkgraaf and Vafa. In particular, we prove that the partition function of the Hermitean one-matrix model in the planar (large $N$) limit coincides with the prepotential of the corresponding SW theory. This partition function is the logarithm of a Whitham $\tau$-function. The corresponding Whitham hierarchy is explicitly constructed. The double-point problem is solved. 
  Gauge theories with and without matter are formulated in the derivative expansion. Amplitudues are derived as a power series in the energy scales; there are simplifications as compared with the usual loop expansion. The incorporation and summation of Wilson loops is natural in this approach and allows for a derivation of the confining potential as well as the masses of the gauge invariant observables. 
  We derive the double scaling limit of eigenvalue correlations in the random matrix model at critical points and we relate the limiting correlation functions to a nonlinear hierarchy of ordinary differential equations. 
  Maximally supersymmetric gauge theories have experienced renewed interest due to the AdS/CFT correspondence and its conjectured S-duality. These gauge theories possess a large amount of symmetry and have quasi-integrable properties. We derive the amplitudes in the derivative expansion of the spontaneously broken examples and perform all loop integrations. The S-matrix is found via an algebraic recursion and at each order is SL(2,Z) invariant. 
  Dijkgraaf and Vafa (DV) have conjectured that the exact superpotential for a large class of N=1 SUSY gauge theories can be extracted from the planar limit of a certain holomorphic matrix integral. We test their proposal against existing knowledge for a family of deformations of N=4 SUSY Yang-Mills theory involving an arbitrary polynomial superpotential for one of the three adjoint chiral superfields. Specifically, we compare the DV prediction for these models with earlier results based on the connection between SUSY gauge theories and integrable systems. We find complete agreement between the two approaches. In particular we show how the DV proposal allows the extraction of the exact eigenvalues of the adjoint scalar in the confining vacuum and hence computes all related condensates of the finite-N gauge theory. We extend these results to include Leigh-Strassler deformations of the N=4 theory. 
  An sp-brane can be viewed as the creation and decay of an unstable D(p+1)-brane. It is argued that the decaying half of an sp-brane can be described by a variant of boundary Liouville theory. The pair creation of open strings by a decaying s-brane is studied in the minisuperspace approximation to the Liouville theory. In this approximation a Hagedorn-like divergence is found in the pair creation rate, suggesting the s-brane energy is rapidly transferred into closed string radiation. 
  The effective action of a free scalar field propagating in the generalized Godel spacetime is evaluated by the zeta-function regularization method. From the result we show that the renormalized stress energy tensor may be divergent at the chronology horizon. This gives a support to the chronology protection conjecture. 
  We use a perturbative method to evaluate the effective action of a free scalar field propagating in the Bianchi type I spacetime with large space anisotropy. The zeta- function regularization method is used to evaluate the action to the second order in the Schwinger perturbative formula. As the quantum corrections contain fourth derivative in the metric we apply the method of iterative reduction to reduce it to the second-order form to obtain the self-consistent solution of the semiclassical gravity theory, The reduced Einstein equation shows that the space anisotropy, which will be smoothed out during the evolution of universe, may play an important role in the dynamics of early universe. We quantize the corresponding minisuperspace model to investigate the behavior of the space anisotropy in the initial epoch. From the wavefunction of the Wheeler-DeWitt equation we see that the probability for the Bianchi type I spacetime with large anisotropy is less then that with a small anisotropy. 
  A quadratic semiclassical theory, regarding the interaction of gravity with a massive scalar quantum field, is considered in view of the renormalizable energy-momentum tensor in a multi-dimensional curved spacetime. According to it, a self-consistent coupling between the square curvature term R^{2} and the quantum field \Phi should be introduced in order to yield the "correct" renormalizable energy-momentum tensor in quadratic gravity theories. The subsequent interaction discards any higher-order derivative terms from the gravitational field equations, but, in the expence, it introduces a geometric source term in the wave equation for the quantum field. Unlike the conformal coupling case (R\Phi ^{2}), this term does not represent an additional "mass" and, therefore, the quantum field interacts with gravity not only through its mass (or energy) content (~\Phi ^{2}), but also, in a more generic way (R^{2}\Phi). Within this context, we propose a general method to obtain mode-solutions for the quantum field, by means of the associated Green's function in an anisotropic six-dimensional background. 
  Considering a Skyrme model with a peculiar gauging of the symmetry, monopole-like solutions exist through a topological lower bound. However, it was recently shown that these objects cannot form bound states in the limit of vanishing Skyrme coupling. Here we consider these monopoles in scalar-tensor gravity. A numerical study of the equations reveals that neither the coupling to gravity nor to the scalar dilaton nor to dilaton-gravity leads to bound multimonopole states. 
  A semiclassical string description is given for correlators of Wilson loops with local operators in N=4 SYM theory in the regime when operators carry parametrically large R-charge. The OPE coefficients of the circular Wilson loop in chiral primary operators are computed to all orders in the alpha' expansion in AdS_5xS^5 string theory. The results agree with field-theory predictions. 
  We calculate the effective potential for the WLPNGB in a world with a circular latticized extra dimension. The mass of the WLPNGB is calculated from the one-loop quantum effect of scalar fields at zero and finite temperature. We show that a series expansion by the modified Bessel functions is useful to calculate the one-loop effective potentials. 
  We define a class of orthosymplectic $osp(m;j|2n;\omega)$ and unitary $sl(m;j|n;\epsilon)$ superalgebras which may be obtained from $osp(m|2n)$ and $sl(m|n)$ by contractions and analytic continuations in a similar way as the special linear, orthogonal and the symplectic Cayley-Klein algebras are obtained from the corresponding classical ones. Casimir operators of Cayley-Klein superalgebras are obtained from the corresponding operators of the basic superalgebras. Contractions of $sl(2|1)$ and $osp(3|2)$ are regarded as an examples. 
  We study N=2 nonlinear two dimensional sigma models with boundaries and their massive generalizations (the Landau-Ginzburg models). These models are defined over either Kahler or bihermitian target space manifolds. We determine the most general local N=2 superconformal boundary conditions (D-branes) for these sigma models. In the Kahler case we reproduce the known results in a systematic fashion including interesting results concerning the coisotropic A-type branes. We further analyse the N=2 superconformal boundary conditions for sigma models defined over a bihermitian manifold with torsion. We interpret the boundary conditions in terms of different types of submanifolds of the target space. We point out how the open sigma models correspond to new types of target space geometry. For the massive Landau-Ginzburg models (both Kahler and bihermitian) we discuss an important class of supersymmetric boundary conditions which admits a nice geometrical interpretation. 
  In this note we show how Dijkgraaf and Vafa's hypothesis relating the exact superpotential of an N=1 theory to a matrix model can be used to describe all the massive vacua of the N=1*, or mass deformed N=4, theory including the Higgs vacuum. The matrix model computation of the superpotential for each massive vacuum independently yields a modular function of the associated effective coupling in that vacuum which agrees with previously derived results up to a vacuum-independent additive constant. The results in the different massive vacua can be related by the action of SL(2,Z) on the N=4 coupling, thus providing evidence for modular invariance of the underlying N=4 theory. 
  We consider gauge field theories in $D>4$ following the Wilson RG approach and show that they possess the ultraviolet fixed points where the gauge coupling is dimensionless in any space-time dimension. At the fixed point the anomalous dimensions of the field and vertex operators are known exactly. These fixed points are nonperturbative and correspond to conformal invariant theories. The same phenomenon also happens in supersymmetric theories with the Yukawa type interactions. 
  We consider a dynamical membrane world in a space-time with scalar bulk matter described by domain walls. Using the solutions to Einstein field equations and Israel conditions we investigate the possibility of having shortcuts for gravitons leaving the wall and returning subsequently. As it turns out, they usually appear under mild conditions.   In the comparison with photons following a geodesic inside the brane, we verify that shortcuts exist. For some Universes they are small, but there are cases where shortcuts are effective. In these cases we expect them to play a significant role in the solution of the horizon problem. 
  We extend the Bardakci-Thorn (BT) worldsheet formalism to supersymmetric non-abelian gauge theory. Our method covers the cases of N =1,2,4 extended supersymmetry. This task requires the introduction of spinor valued Grassmann variables on the worldsheet analogous to those of the supersymmetric formulation of superstring theory. As in the pure Yang-Mills case, the worldsheet formalism automatically generates the correct quartic vertices from the cubic vertices. We also discuss coupling renormalization to one loop order. 
  We discuss the perturbative behavior of the 1/2 BPS operators in N=2 SCFT on the example of two very similar quadrilinear composite operators made out of hypermultiplets. An explicit one-loop computation shows that one of them is protected while the other acquires an anomalous dimension. Although both operators are superconformal primaries in the free case, the quantum corrections make the latter become a 1/2 BPS descendant of the Konishi multiplet, while the former remains primary. The comparative study of these two operators at higher orders may be helpful in understanding the quantum properties of the Konishi multiplet. 
  Zhang and Hu have formulated an SU(2) quantum Hall system on the four-sphere, with interesting three-dimensional boundary dynamics including gapless states of nonzero helicity. In order to understand the local physics of their model we study the U(1) and SU(2) quantum Hall systems on flat R^4, with flat boundary R^3. In the U(1) case the boundary dynamics is essentially one dimensional. The SU(2) theory can be formulated on R^4 for any isospin I, but in order to obtain a flat boundary theory we must take I \to \infty as in Zhang and Hu. The theory simplifies in the limit, the boundary becoming a collection of one-dimensional systems. We also discuss general constraints on the emergence of gravity from nongravitational field theories. 
  I review our current understanding of the Worldformula, M theory, focusing on themes from the work of Heisenberg. 
  We present a topologically non-trivial generalization of gauged N=16 supergravity on the coset E_8 / SO(16) in three-dimensions. This formulation is based on a combination of BF-term and a Chern-Simons term for an SO(16) gauge field A_\m{}^{I J}. The fact that an additional vector field B_\m{}^{I J} is physical and propagating with couplings to \sigma-model fields makes our new gauging non-trivial and different from the conventional one. Even though the field strength of the A_\m{}^{I J}-field vanishes on-shell, the action is topologically non-trivial due to non-vanishing \pi_3-homotopy. We also present an additional modifications by an extra Chern-Simons term. As by-products, we give also an application to N=9 supergravity coupled to a \sigma-model on the coset F_4 / SO(9), and a new BF-Chern-Simons theory coupled to ^\forall N extended supergravity. 
  We study the classical dynamics of mechanical model obtained from the light-cone version of SU(2) Yang-Mills field theory under the supposition of gauge potential dependence only on ``time'' along the light-cone direction. The computer algebra system Maple was used strongly to compute and separate the complete set of constraints. In contrast to the instant form of Yang-Mills mechanics the constraints here represent a mixed form of first and second-class constraints and reduce the number of the physical degrees of freedom up to four canonical one. 
  We show that a kink and a topologically trivial soliton in the Gross-Neveu model form, in the large-N limit, a marginally stable static configuration, which is bound at threshold. The energy of the resulting composite system does not depend on the separation of its solitonic constituents, which serves as a modulus governing the profile of the compound soliton. Thus, in the large-N limit, a kink and a non-topological soliton exert no force on each other. 
  The irreducible Freedman-Townsend vertex is derived by means of the Hamiltonian deformation procedure based on local BRST cohomology. 
  Intertwining relations for $N$-particle Calogero-like models with internal degrees of freedom are investigated. Starting from the well known Dunkl-Polychronakos operators, we construct new kind of local (without exchange operation) differential operators. These operators intertwine the matrix Hamiltonians corresponding to irreducible representations of the permutation group $S_N$. In particular cases, this method allows to construct a new class of exactly solvable Dirac-like equations and a new class of matrix models with shape invariance. The connection with approach of multidimensional supersymmetric quantum mechanics is established. 
  We give an explicit construction of general classical solutions for the noncommutative CP^(N-1) model in two dimensions, showing that they correspond to integer values for the action and topological charge. We also give explicit solutions for the Dirac equation in the background of these general solutions and show that the index theorem is satisfied. 
  We study a family of interacting bosonic representations of the N=2 superconformal algebra. These models can be tensored with a conjugate theory to give the free theory. We explain how to use free fields to study interacting fields and their dimensions, and how we may identify different free fields as representing the same interacting field. We show how a lattice of identifying fields may be built up and how every free field may be reduced to a standard form, thus permitting the resolution of the spectrum. We explain how to build the extended algebra and show that there are a finite number of primary fields for this algebra for any of the models. We illustrate this by studying an example. 
  We analyze the renormalization properties of quantum field theories in de Sitter space and show that only two of the maximally invariant vacuum states of free fields lead to consistent perturbation expansions. One is the Euclidean vacuum, and the other can be viewed as an analytic continuation of Euclidean functional integrals on $RP^d$. The corresponding Lorentzian manifold is the future half of global de Sitter space with boundary conditions on fields at the origin of time. We argue that the perturbation series in this case has divergences at the origin, which render the future evolution of the system indeterminate without a better understanding of high energy physics. 
  All purely bosonic supersymmetric solutions of minimal supergravity in five dimensions are classified. The solutions preserve either one half or all of the supersymmetry. Explicit examples of new solutions are given, including a large family of plane-fronted waves and a maximally supersymmetric analogue of the G\"odel universe which lifts to a solution of eleven dimensional supergravity that preserves 20 supersymmetries. 
  The complete on-shell action of topological Einstein-Maxwell gravity in four-dimensions is presented. It is shown explicitly how this theory for SU(2) holonomy manifolds arises from four-dimensional Euclidean N=2 supergravity. The twisted local BRST symmetries and twisted local Lorentz symmetries are given and the action and stress tensor are shown to be BRST-exact. A set of BRST-invariant topological operators is given. The vector and antisymmetric tensor twisted supersymmetries and their algebra are also found. 
  We discuss semiclassical quantization of closed superstrings in AdS_5 x S^5. We consider two basic examples: point-like string boosted along large circle of S^5 and folded string rotating in AdS_5. In the first case we clarify the general structure of the sigma model perturbation theory for the energy of string states beyond the leading 1-loop order (related to the plane-wave limit). In the second case we argue that the large spin limit of the expression for the ground-state energy (i.e. for the dimension of the corresponding minimal twist gauge theory operator) has the form S + f(\lambda) \ln S to all orders in \alpha' ~ \lambda^{-1/2} expansion, in agreement with the AdS/CFT duality. We also suggest an extension of the semiclassical approach to near-conformal (near-AdS) cases on the example of the fractional D3-brane on conifold background. Expanded version of talks at the Third International Sakharov Conference on Physics, Moscow, June 24-29, 2002 and at Strings 2002, Cambridge, July 15-20,2002 
  Recently we have introduced a matrix model depending on two coupling constants $g^2$ and $\lambda$, which contains the fuzzy sphere as a background; to obtain the classical limit $g^2$ must depend on $N$ in a precise way. In this paper we show how to obtain the classical solitons of the $N \to \infty$ limit imposing the development $\lambda = \half + \frac{\lambda_0}{N}$; as a consequence at finite $N$ one obtains a noncommutative version of the solitons for the fuzzy sphere.  
  In the case of an invertible coordinate commutator matrix $\theta_{ij}$, we derive a general instanton solution of the noncommutative gauge theories on $d=2n$ planes given in terms of $n$ oscillators. 
  We study the phenomenological implications of the classical limit of the "stringy" commutation relations [x_i,p_j]=i hbar[(1+beta p^2) delta_{ij} + beta' p_i p_j]. In particular, we investigate the "deformation" of Kepler's third law and apply our result to the rotation curves of gas and stars in spiral galaxies. 
  We propose that for every event in de Sitter space, there is a CPT-conjugate event at its antipode. Such an ``elliptic'' $Z_2$-identification of de Sitter space provides a concrete realization of observer complementarity: every observer has complete information. It is possible to define the analog of an S-matrix for quantum gravity in elliptic de Sitter space that is measurable by all observers. In a holographic description, S-matrix elements may be represented by correlation functions of a dual (conformal field) theory that lives on the single boundary sphere. S-matrix elements are de Sitter-invariant, but have different interpretations for different observers. We argue that Hilbert states do not necessarily form representations of the full de Sitter group, but just of the subgroup of rotations. As a result, the Hilbert space can be finite-dimensional and still have positive norm. We also discuss the elliptic interpretation of de Sitter space in the context of type IIB* string theory. 
  By use of the AdS/CFT correspondence on orbifolds, models are derived which can contain the standard model of particle phenomenology. It will be assumed that the theory becomes conformally invariant at a renormalization-group fixed-point in the TeV region. A recent application to TeV unification is briefly mentioned. 
  Recent analysis suggests that the classical dynamics of a tachyon on an unstable D-brane is described by a scalar Born-Infeld type action with a runaway potential. The classical configurations in this theory at late time are in one to one correspondence with the configuration of a system of non-interacting (incoherent), non-rotating dust. We discuss some aspects of canonical quantization of this field theory coupled to gravity, and explore, following earlier work on this subject, the possibility of using the scalar field (tachyon) as the definition of time in quantum cosmology. At late `time' we can identify a subsector in which the scalar field decouples from gravity and we recover the usual Wheeler - de Witt equation of quantum gravity. 
  We elaborate an explicit example of dimensional reduction of the free massless Dirac operator with SU(3)-symmetry, defined on 12-dimensional manifold, which is the total space of principle SU(3)-bundle over 4-dimensional manifold. It turns out that after the dimensional reduction we obtain the "usual" Dirac operator, defined on 4-dimensional pseudo-Riemannian (non-flat) manifold but with mass term, acting on two spinor SU(3)-octets in the presence of gauge field with structure group SU(3) and source term depending on the stress tensor of the gauge field. The group of symmetry SU(3) is chosen because it is a classifying group for the Standard model and in the same time is complicated enough to demonstrate all the details in the general case. We pay attention and point out the crucial moments in the procedure of the dimensional reduction, where the new structures (gauge field with structure group SU(3), its stress tenzor, two spinor SU(3)-octets and mass term) arise. 
  We study the problem of vortex solutions in the background of an electrically charged black string. We show numerically that the Abelian Higgs field equations in the background of a four-dimensional black string have vortex solutions. These solutions which have axial symmetry, show that the black string can support the Abelian Higgs field as hair. This situation holds also in the case of the extremal black string. We also consider the self-gravity of the Abelian Higgs field and show that the effect of the vortex is to induce a deficit angle in the metric under consideration. 
  We consider quantum p-form fields interacting with a background dilaton. We calculate the variation with respect to the dilaton of a difference of the effective actions in the models related by a duality transformation. We show that this variation is defined essentially by the supertrace of the twisted de Rham complex. The supertrace is then evaluated on a manifold of an arbitrary dimension, with or without boundary. 
  Avoiding the problem of the existence of asymptotically constant spinors satysfying certain differential equations on a non-compact hypersrface we presented the proof of positivity of the ADM and Bondi energy in Einstein-Maxwell axion-dilaton gravity. In our attitude spinor fields defining the enwergy need only be defined near infinity and there satysfying propagation equations. 
  We briefly review the recent progress concerning the application of the hidden integrability to the derivation of the stringy/brane picture for the high energy QCD. 
  The scalar potentials of the non-semi-simple CSO(p,8-p)(p=7,6,5) gaugings of N=8 supergravity are studied for critical points. The CSO(7,1) gauging has no G_2-invariant critical points, the CSO(6,2) gauging has three new SU(3)-invariant AdS critical points and the CSO(5,3) gauging has no SO(5)-invariant critical points. The scalar potential of CSO(6,2) gauging in four dimensions we discovered provides the SU(3) invariant scalar potential of five dimensional SO(6) gauged supergravity. The nontrivial effective scalar potential can be written in terms of the superpotential which can be read off from A_1 tensor of the theory. We discuss first-order domain wall solutions by analyzing the supergravity scalar-gravity action and using some algebraic relations in a complex eigenvalue of A_1 tensor. We examine domain wall solutions of G_2 sectors of noncompact SO(7,1) and CSO(7,1) gaugings and SU(3) sectors of SO(6,2) and CSO(6,2) gaugings. They share common features with each sector of compact SO(8) gauged N=8 supergravity in four dimensions. We analyze the scalar potentials of the CSO(p,q,8-p-q) gauged supergravity we have found before. The CSO(p,6-p,2) gauge theory in four dimensions can be reduced from the SO(p,6-p) gauge theory in five dimensions. Moreover, the SO(p,5-p) gauge theory in seven dimensions reduces to CSO(p,5-p,3) gauge theory in four dimensions. Similarly, CSO(p,q-p,8-q) gauge theories in four dimensions are related to SO(p,q-p)(q=2,3,4,7) gauge theories in other dimensions. 
  We study the commutators of the kappa-deformed Poincare Algebra (kappaPA) in an arbitrary basis. It is known that the two recently studied doubly special relativity theories correspond to different choices of kappaPA bases. We present another such example. We consider the classical limit of kappaPA and calculate particle velocity in an arbitrary basis. It has standard properties and its expression takes a simple form in terms of the variables in the Snyder basis. We then study the particle trajectory explicitly for the case of a constant force. Assuming that the spacetime continuum, velocity, acceleration, etc. can be defined only at length scales greater than x_{min} ne 0, we show that the acceleration has a finite maximum. 
  A manifestly gauge invariant exact renormalization group for pure SU(N) Yang-Mills theory is proposed, along with the necessary gauge invariant regularisation which implements the effective cutoff. The latter is naturally incorporated by embedding the theory into a spontaneously broken SU(N|N) super-gauge theory, which guarantees finiteness to all orders in perturbation theory. The effective action, from which one extracts the physics, can be computed whilst manifestly preserving gauge invariance at each and every step. As an example, we give an elegant computation of the one-loop SU(N) Yang-Mills beta function, for the first time at finite N without any gauge fixing or ghosts. It is also completely independent of the details put in by hand, e.g. the choice of covariantisation and the cutoff profile, and, therefore, guides us to a procedure for streamlined calculations. 
  Using a new scaling limit as well as a new cut-off procedure, we show that $\phi^4$ theory on noncommutative ${\bf R}^4$ can be obtained from the corresponding theory on fuzzy ${\bf S}^2 \times {\bf S}^2$. The star-product on this noncommutative ${\bf R}^4$ is effectively local in the sense that the theory naturally has an ultra-violet cut-off $\Lambda$ which is inversely proportional to the noncommutativity $\theta$, i.e $ \Lambda= \frac{2}{\theta}$. We show that the UV-IR mixing in this case is absent to one loop in the $2-$point function and also comment on the $4-$point function. 
  We point out that the permanent confinement in a compact 2+1-dimensional U(1) Abelian Higgs model is destroyed by matter fields in the fundamental representation. The deconfinement transition is Kosterlitz-Thouless like. The dual theory is shown to describe a three-dimensional gas of point charges with logarithmic interactions which arises from an anomalous dimension of the gauge field caused by critical matter field fluctuations. The theory is equivalent to a sine-Gordon-like theory in 2+1 dimensions with an anomalous gradient energy proportional to $k^3$. The Callan-Symanzik equation is used to demonstrate that this theory has a massless and a massive phase. The renormalization group equations for the fugacity $y(l)$ and stiffness parameter $K(l)$ of the theory show that the renormalization of $K(l)$ induces an anomalous scaling dimension $\eta_y$ of $y(l)$. The stiffness parameter of the theory has a universal jump at the transition determined by the dimensionality and $\eta_y$. As a byproduct of our analysis, we relate the critical coupling of the sine-Gordon-like theory to an {\it a priori} arbitrary constant that enters into the computation of critical exponents in the Abelian Higgs model at the charged infrared-stable fixed point of the theory, enabling a determination of this parameter. This facilitates the computation of the critical exponent $\nu$ at the charged fixed point in excellent agreement with one-loop renormalization group calculations for the three-dimensional XY-model, thus confirming expectations based on duality transformations. 
  We investigate the generation of primordial gravitational waves from inflation in braneworld cosmologies with extra dimensions. Advantage of using primordial gravitational waves to probe extra dimensions is that their theory depends only on the geometry, not on the microscopic models of inflation and stabilization. D(D-3)/2 degrees of freedom of the free bulk gravitons are projected onto the 3d brane as tensor, vector and scalar modes. We found the following no-go results for a generic geometry of a five (or D) dimensional warped metric with four dimensional de Sitter (inflationary) slices and two (or one) edge of the world branes: Massive KK graviton modes are not generated from inflation (with the Hubble parameter H) due to the gap in the KK spectrum; the universal lower bound on the gap is sqrt{3/2} H. Massless scalar and vector projections of the bulk gravitons are absent, unlike in geometries with KK compactification.   A massless 4d tensor mode is generated from inflation with the amplitude H/M_P, where M_P is the effective Planck mass during inflation, derived from the D dimensional fundamental mass M_S and the volume of the inner dimensions. However, M_P for a curved dS braneworld may differ from that of the flat brane at low energies, either due to the H-dependence of the inner space volume or variations in the brane separation before stabilization. Thus the amplitude of gravitational waves from inflation in braneworld cosmology may be different from that predicted by inflation in 4d theory. 
  It is well known that the building blocks for state sum models of quantum gravity are given by 6j and 10j symbols. In this work we study the asymptotics of these symbols by using their expressions as group integrals. We carefully describe the measure involved in terms of invariant variables and develop new technics in order to study their asymptotics. Using these technics we compute the asymptotics of the various Euclidean and Lorentzian 6j-symbols. Finally we compute the asymptotic expansion of the 10j symbol which is shown to be non-oscillating in agreement with a recent result of Baez et al. We discuss the physical origin of this behavior and a way to modify the Barrett-Crane model in order to cure this disease. 
  We start from the well-known form of the interval of the special relativity, stare it, and build up an attempt to implement the causality from it. Some features appear to be new, they involve the mass of the particle and the structure of space-time. 
  It is shown that the four $(3 + 1)$-dimensional (4D) free Abelian 2-form gauge theory provides an example of (i) a class of field theoretical models for the Hodge theory, and (ii) a possible candidate for the quasi-topological field theory (q-TFT). Despite many striking similarities with some of the key topological features of the two $(1 + 1)$-dimensional (2D) free Abelian (and self-interacting non-Abelian) gauge theories, it turns out that the 4D free Abelian 2-form gauge theory is {\it not} an exact TFT. To corroborate this conclusion, some of the key issues are discussed. In particular, it is shown that the (anti-)BRST and (anti-)co-BRST invariant quantities of the 4D 2-form Abelian gauge theory obey the recursion relations that are reminiscent of the exact TFTs but the Lagrangian density of this theory is not found to be able to be expressed as the sum of (anti-)BRST and (anti-)co-BRST exact quantities as is the case with the {\it topological} 2D free Abelian (and self-interacting non-Abelian) gauge theories. 
  We investigate chiral anomaly for fermions in fundamental representation on noncommutative (fuzzy) 2-sphere. In spite that this system is realized by finite dimensional matrices and no regularization is necessary for either UV or IR, we can reproduce the correct chiral anomaly which is consistent with the calculations done in flat noncommutative space. Like the flat case, there are ambiguities to define chiral currents. We define chiral currents in a gauge-invariant way and a gauge-covariant way, and show that the corresponding anomalous chiral Ward-Takahashi identities take different forms. The Ward-Takahashi identity for the gauge-invariant current contains explicit nonlocality while that for the covariant one is given by a local expression. 
  We provide a purely perturbative (one loop) derivation of mirror symmetry for supersymmetric sigma models in two dimensions. 
  We show that the integral of the first Pontrjagin class is given by an integer and it is identified with instanton number of the U(n) gauge theory on noncommutative ${\bf R^4}$. Here the dimension of the vector space $V$ that appear in the ADHM construction is called Instanton number. The calculation is done in operator formalism and the first Pontrjagin class is defined by converge series. The origin of the instanton number is investigated closely, too. 
  A new, recursive method of calculating matrix elements of polynomial hamiltonians is proposed. It is particularly suitable for the recent algebraic studies of the supersymmetric Yang-Mills quantum mechanics in any dimensions. For the D=2 system with the SU(2) gauge group, considered here, the technique gives exact, closed expressions for arbitrary matrix elements of the hamiltonian and of the supersymmetric charge, in the occupation number representation. Subsequent numerical diagonalization provides the spectrum and restricted Witten index of the system with very high precision (taking into account up to $10^5$ quanta).   Independently, the exact value of the restricted Witten index is derived analytically for the first time. 
  We discuss the worldvolume description of intersecting D-branes, including the metric on the moduli space of deformations. We impose a choice of static gauge that treats all the branes on an equal footing and describes the intersection of D-branes as an embedded special Lagrangian three-surface. Some explicit solutions to these equations are given and their interpretation in terms of a superpotential on moduli space is discussed. These surfaces arise from flat direction of a non-Abelian superpotential and imply the existance of non-compact G_2 manifolds. 
  The low energy effective action including gauge field degrees of freedom on a non-BPS p=2 brane embedded in a N=1, D=4 target superspace is obtained through the method of nonlinear realizations of the associated super-Poincare symmetries. The invariant interactions of the gauge fields and the brane excitation modes corresponding to the Nambu-Goldstone degrees of freedom resulting from the broken space translational symmetry and the target space supersymmetries are determined. Brane localized matter field interactions with the gauge fields are obtained through the construction of the combined gauge and super-Poincare covariant derivatives for the matter fields. 
  In this paper we reinvestigate photon. We show that photon with a specific frequency can be identified with the Dirac magnetic monopole. We give a model of quantum electrodynamics from which we derive photon as a quantum loop of this model. This nonlinear loop model of photon is exactly solvable and thus may be regarded as a quantum soliton. From the winding numbers of this loop model of photon we derive the quantization property of energy of Planck's formula of radiation and the quantization property of electric charge. We show that the quantization property of electric charge is derived from the quantization property of energy of Planck's formula of radiation when photon with a specific frequency is identified with the magnetic monopole. From this nonlinear model of photon we also construct a model of electron which has a mass mechanism for generating mass to electron. 
  F-Lie algebras are natural generalisations of Lie algebras (F=1) and Lie superalgebras (F=2). We give finite dimensional examples of F-Lie algebras obtained by an inductive process from Lie algebras and Lie superalgebras. Matrix realizations of the $F-$Lie algebras constructed in this way from osp(2|m) are given. We obtain a non-trivial extension of the Poincar\'e algebra by an In\"on\"u-Wigner contraction of a certain $F-$Lie algebras with $F>2$. 
  We calculate the thermal partition function of DLCQ superstring on the maximally supersymmetric pp-wave background, which is realized as the Penrose limit of orbifolded $AdS_5\times S^5$ and known to be dual to the $\cN=2$ ``large'' quiver gauge theory as shown by S. Mukhi, M. Rangamani and E. Verlinde, hep-th/0204147. Making use of the path-integral technique, we derive the manifestly modular invariant expression and show the equivalence with the free energy of second quantized free superstring on this background. The ``virtual strings'' wound around the temporal circle play essential roles for realizing the modular invariance and for a simple analysis on the Hagedorn temperature. We also calculate the thermal one-loop amplitudes of open strings under the various backgrounds of the supersymmetric time-like and Euclidean D-branes, and confirm the existence of correct open-closed string duality. Furthermore, we extend these thermodynamical analysis to the 6-dimensional DLCQ pp-waves with general RR and NSNS flux. These superstring vacua are similarly derived from the supersymmetric (half SUSY) and non-supersymmetric orbifolds of $AdS_3 \times S^3 \times M^4$ ($M^4 =T^4 $ or $K3$) by the appropriate Penrose limits, giving rise to the SUSY enhancement as in the case of orbifolded $AdS_5 \times S^5$. 
  We study four-point correlation functions with logarithmic behaviour in Liouville field theory on a sphere, which consist of one kind of the local operators. We study them as non-integrated correlation functions of the gravitational sector of two-dimensional quantum gravity coupled to an ordinary conformal field theory in the conformal gauge. We also examine, in the (p,q) minimal conformal field theories, a condition of the appearance of logarithmic correlation functions of gravitationally dressed operators. 
  Peierls brackets are part of the space-time approach to quantum field theory, and provide a Poisson bracket which, being defined for pairs of observables which are group invariant, is group invariant by construction. It is therefore well suited for combining the use of Poisson brackets and the full diffeomorphism group in general relativity. The present paper provides an introduction to the topic, with applications to field theory and point Lagrangians. 
  We study topological open membranes of BF type in a manifest BV formalism. Our main interest is the effect of the bulk deformations on the algebra of boundary operators. This forms a homotopy Lie algebra, which can be understood in terms of a closed string field theory. The simplest models are associated to quasi-Lie bialgebras and are of Chern-Simons type. More generally, the induced structure is a Courant algebroid, or ``quasi-Lie bialgebroid'', with boundary conditions related to Dirac bundles. A canonical example is the topological open membrane coupling to a closed 3-form, modeling the deformation of strings by a C-field. The Courant algebroid for this model describes a modification of deformation quantization. We propose our models as a tool to find a formal solution to the quantization problem of Courant algebroids. 
  In three-dimensional Einstein-Maxwell gravity the electrostatic Banados-Teitelboim-Zanelli solution and the magnetostatic Hirschmann-Welch solution are connected by a duality mapping. Here we point out that a similar duality mapping exists among circularly symmetric electrostatic and magnetostatic spacetimes, and electric and magnetic stationary solutions, for a nonlinear electrodynamics coupled to three-dimensional Einstein gravity. 
  We construct the noncommutative deformation of the Maldacena-Nunez supergravity solution. The background describes a bound state of D5-D3 branes wrapping an S^2 inside a Calabi-Yau three-fold, and in the presence of a magnetic B-field. The dual field theory in the IR is a N=1 U(N) SYM theory with spatial noncommutativity. We show that, under certain conditions, the massive Kaluza-Klein states can be decoupled and that UV/IR mixing seems to be visible in our solution. By calculating the quark-antiquark potential via the Wilson loop we show confinement in the IR and strong repulsion at closer distances. We also compute the beta-function and show that it coincides with the recently calculated commutative one. 
  We examine two restructurings of the series relationship between the bare and renormalized coupling constant in dimensional regularization. In one of these restructurings, we are able to demonstrate via all-orders summation of leading and successive epsilon = 0 (dimensionality = 4) poles that the bare coupling vanishes in the dimension-4 limit. 
  We investigate coupled gravity and a bulk scalar field on a slice of AdS_5 bulk with special BPS branes at the two ends. With the special scalar potentials on the branes the scalar field does not stabilize the size of the orbifold. With a careful treatment of the general coordinate invariance the complete tensor-scalar spectrum is presented. There are two massless zero modes in the scalar sector, the radion and a dilatonic zero mode. The scalar KK modes acquire masses at the order of the warped mass scale. The four dimensional effective action is of tensor-scalar type. 
  The Yang-Mills (YM) and self-dual Yang-Mills (SDYM) equations on the noncommutative Euclidean four-dimensional space are considered. We introduce an ansatz for a gauge potential reducing the noncommutative SDYM equations to a difference form of the Nahm equations. By constructing solutions to the difference Nahm equations, we obtain solutions of the noncommutative SDYM equations. They are noncommutative generalizations of the known solutions to the SDYM equations such as the Minkowski solution, the one-instanton solution and others. Using the noncommutative deformation of the Corrigan-Fairlie-'t Hooft-Wilzek ansatz, we reduce the noncommutative YM equations to equations on a scalar field which have meron solutions in the commutative limit and show that they have no such solutions in the noncommutative case. To overcome this difficulty, another ansatz reducing the noncommutative YM equations to a system of difference equations on matrix-valued functions is used. For self-dual configurations this system is reduced to the difference Nahm equations. 
  The two-dimensional ghost systems with negative integral central charge received much attention in the last years for their role in a number of applications and in connection with logarithmic conformal field theory. We consider the free massive bosonic and fermionic ghost systems and concentrate on the non-trivial sectors containing the disorder operators. A unified analysis of the correlation functions of such operators can be performed for ghosts and ordinary complex bosons and fermions. It turns out that these correlators depend only on the statistics although the scaling dimensions of the disorder operators change when going from the ordinary to the ghost case. As known from the study of the ordinary case, the bosonic and fermionic correlation functions are the inverse of each other and are exactly expressible through the solution of a non-linear differential equation. 
  We develop a systematic procedure for deriving canonical string field theory from large N matrix models in the Berenstein-Maldacena-Nastase limit. The approach, based on collective field theory, provides a generalization of standard string field theory. 
  We discuss issues relating to the topology of Euclidean de Sitter space. We show that in (2+1) dimensions, the Euclidean continuation of the`causal diamond', i.e the region of spacetime accessible to a timelike observer, is a three-hemisphere. However, when de Sitter entropy is computed in a `stretched horizon' picture, then we argue that the correct Euclidean topology is a solid torus. The solid torus shrinks and degenerates into a three-hemisphere as one goes from the `stretched horizon' to the horizon, giving the Euclidean continuation of the causal diamond. We finally comment on the generalisation of these results to higher dimensions. 
  We construct the boundary WZNW functional for symmetry breaking D-branes on a group manifold which are localized along a product of a number of twisted conjugacy classes and which preserve an action of an arbitrary continuous subgroup. These branes provide a geometric interpretation for the algebraic formulation of constructing D-branes developed recently in hep-th/0203161. We apply our results to obtain new symmetry breaking and non-factorizing D-branes in the background SL(2,R) x SU(2). 
  We study cosmological consequences of non-constant brane world moduli in five dimensional brane world models with bulk scalars and two boundary branes. We focus on the case where the brane tension is an exponential function of the bulk scalar field, $U_b \propto \exp{(\alpha \phi)}$. In the limit $\alpha \to 0$, the model reduces to the two-brane model of Randall-Sundrum, whereas larger values of $\alpha$ allow for a less warped bulk geometry. Using the moduli space approximation, we derive the four-dimensional low-energy effective action from a supergravity-inspired five-dimensional theory. For arbitrary values of $\alpha$, the resulting theory has the form of a bi-scalar-tensor theory. We show that, in order to be consistent with local gravitational observations, $\alpha$ has to be small (less than $10^{-2}$) and the separation of the branes must be large. We study the cosmological evolution of the interbrane distance and the bulk scalar field for different matter contents on each branes. Our findings indicate that attractor solutions exist which drive the moduli fields towards values consistent with observations. The efficiency of the attractor mechanism crucially depends on the matter content on each branes. In the five-dimensional description, the attractors correspond to the motion of the negative tension brane towards a bulk singularity, which signals the eventual breakdown of the four-dimensional description and the necessity of a better understanding of the bulk singularity. 
  We discuss interacting quantum field theory in de Sitter space and argue that the Mottola-Allen vacuum ambiguity is an artifact of free field theory. The nature of the nonthermality of the MA-vacua is also clarified. We propose analyticity of correlation functions as a fundamental requirement of quantum field theory in curved spacetimes. In de Sitter space, this principle determines the vacuum unambiguously and facilitates the systematic development of perturbation theory. 
  We investigate the giant gravitons in the maximally supersymmetric IIB pp-wave from several viewpoints: (i) the dynamics of D3-branes, (ii) the world-sheet description and (iii) the correlation functions in the dual N=4 Yang-Mills theory. In particular, we derive the BPS equation of a D3-brane with magnetic flux, which is equivalent to multiple D-strings, and discuss the behavior of solutions in the presence of RR-flux. We find solutions which represent the excitations of the giant gravitons in that system. 
  We analyze the link between the occurrence of massless B-type D-branes for specific values of moduli and monodromy around such points in the moduli space. This allows us to propose a classification of all massless B-type D-branes at any point in the moduli space of Calabi-Yau's. This classification then justifies a previous conjecture due to Horja for the general form of monodromy. Our analysis is based on using monodromies around points in moduli space where a single D-brane becomes massless to generate monodromies around points where an infinite number become massless. We discuss the various possibilities within the classification. 
  We uncover a method of calculation that proceeds at every step without fixing the gauge or specifying details of the regularisation scheme. Results are obtained by iterated use of integration by parts and gauge invariance identities. The initial stages can even be computed diagrammatically. The method is formulated within the framework of an exact renormalization group for SU(N) Yang-Mills gauge theory, incorporating an effective cutoff through a manifest spontaneously broken SU(N|N) gauge invariance. We demonstrate the technique with a compact calculation of the one-loop beta function, achieving a manifestly universal result, and without gauge fixing, for the first time at finite N. 
  We study fluctuations around the warped conifold supergravity solution of Klebanov and Tseytlin [hep-th/0002159], known to be dual to a cascading N=1 gauge theory. Although this supergravity background is not asymptotically AdS, corresponding to a non-conformal field theory, it is possible to apply the usual methods of AdS/CFT duality to extract the high energy behavior of field theory correlators by solving linearized equations of motion for fluctuations around the background. We consider the Goldstone vector dual to the anomalous R-symmetry current and compute its mass, which exactly matches the general prediction of [hep-th/0009156]. We find the high energy 2-point functions for the R-current and two other vectors. As expected, the R-current 2-point function has a longitudinal part because R-symmetry is broken. We also calculate the high energy 2-point function of the energy-momentum tensor from fluctuations of modes in the graviton sector. This 2-point function has a trace part corresponding to broken conformal symmetry. 
  We study the physics of D-branes in the presence of constant Ramond-Ramond potentials. In the string field theory context, we first develop a general formalism to analyze open strings in gauge trivial closed string backgrounds, and then apply it both to the RNS string and within Berkovits' covariant formalism, where the results have the most natural interpretation. The most remarkable finding is that, in the presence of a Dp-brane, both a constant parallel NS-NS B-field and R-R C^(p-1)-field do not solve the open/closed equations of motion, and induce the same non-vanishing open string tadpole. After solving the open string equations in the presence of this tadpole, and after gauging away the closed string fields, one is left with a U(1) field strength on the brane given by F=(1/2)(B-*C^(p-1)), where * is Hodge duality along the brane world-volume. One observes that this result differs from the usually assumed result F=B. Technically, this is due to the fact that supersymmetric and bosonic string world-sheet theories are different. Note, however, that the usual F+B combination is still the combination which remains gauge invariant at the sigma-model level. Also, the standard result F=B is, in the D3-brane case, not compatible with S-duality. On the other hand our result, which is derived automatically given the general formalism, offers a non-trivial check of S-duality, to all orders in F, and this leads to a S-dual invariant Moyal deformation. In an appendix, we solve the source equation describing the open superstring in a generic NS-NS and R-R closed string background, within the super Poincare covariant formalism. 
  A new Einstein-Hilbert type (SGM) action describing gravitational interaction of Nambu-Goldstone(N-G) fermion of nonlinear supersymmetry(NL SUSY) is obtained by performing the Einstein gravity analogue geometrical arguments in high symmetric four dimensional (SGM) spacetime. All elementary particles except graviton are regarded as the composite eigenstates of SO(10) super-Poincar\'e algebra(SPA) composed of the fundamental N-G fermion ``superons'' of NL SUSY. Some phenomenological implications of the composite picture of SGM, the linearlization of SGM and N = 2 Volkov-Akulov model are discussed. 
  A slightly modified and regularized version of the non-relativistic limit of the relativistic anyon model considered by Jackiw and Nair yields particles associated with the twofold central extension of the Galilei group, with independent spin and exotic structure. 
  We identify natural degrees of freedom of polycrystalline materials -- affine transformations of grains -- with those of a three-dimensional lattice theory for $(T\otimes\Omega)(\mathbb{R}^3)$. We define a lattice Dirac operator on this space and identify its continuous limit with the free field limit of the whole fermionic sector of the standard model. Fermion doubling is used here as a tool to obtain the necessary number of steps of freedom. The correspondence extends to important structural properties (families, colors, flavor pairs, electromagnetic charge). We find a lattice version of chiral symmetry similar to the Ginsparg-Wilson approach.   This correspondence suggests to propose a ``polycrystalline ether''. Combined with GLET, a general Lorentz ether theory of gravity with GR limit, this becomes a concept for a theory of everything. The extension to gauge fields is the major open problem and requires new concepts. 
  We observe a correspondence between the zero modes of superconformal algebras $S'(2, 1)$ and W(4) and the Lie superalgebras formed by classical operators appearing in the K{\"a}hler and hyper-K{\"a}hler geometry. 
  Chapters : 1. Introduction to electric-magnetic duality 2. Classical duality in bosonic brane electrodynamics 3. Massless spin two gauge theory 4. Duality-symmetric actions and chiral forms 5. BRST quantization of duality-symmetric Maxwell's theory 6. Quantization conditions 7. Consistent deformations 
  It seems to me at this time that two recent developments may permit fast progress on our way to understand the symmetry structure of toroidally (compactified and) reduced M-theory. The first indication of a (possibly) thin spot in the wall that prevents us from deriving a priori the U-duality symmetries of these models is to be found in the analysis of the hyperbolic billiards that control the chaotic time evolution of (quasi)homogeneous anisotropic String, Supergravity or Einstein cosmologies near a spacelike singularity. What happens is that U-duality symmetry controls chaos via negative constant curvature. On the other hand it was noticed in 1982 that (symmetrizable) ''hyperbolic'' Kac-Moody algebras have maximal rank ten, exactly like superstring models and that two of these four rank ten algebras matched physical theories. My second reason for optimism actually predates also the previous breakthrough, it was the discovery in 1998 of surprising superalgebras extending U-dualities to all (p+1)-forms (associated to p-branes). They have a super-nonlinear sigma model structure similar to the symmetric space structure associated to 0-forms and they obey a universal self-duality field equation. As the set of forms is doubled to implement electric-magnetic duality, they obey a set of first order equations. More remains to be discovered but the beauty and subtlety of the structure cannot be a random emergence from chaos. In fact we shall explain how a third maximal rank hyperbolic algebra $BE_{10}$ controls heterotic cosmological chaos and how as predicted Einstein's General Relativity fits into the general picture. 
  We study bosonic strings in a Rindler background using a D-23 brane. This model is shown to be directly related to the orbifold approach to Rindler strings. We propose a duality between these pictures based on comparisons of their closed string spectra. 
  We discuss the generalization of the connection between the determinant of an operator entering a quadratic form and the associated Gaussian path-integral valid for grassmann variables to the paragrassmann case [$\theta^{p+1}=0$ with $p=1$ ($p>1$) for grassmann (paragrassamann) variables]. We show that the q-deformed commutation relations of the paragrassmann variables lead naturally to consider q-deformed quadratic forms related to multiparametric deformations of GL(n) and their corresponding $q$-determinants. We suggest a possible application to the study of disordered systems. 
  Simulations of SU(2) lattice gauge theory are used to establish a relation between the IR properties of Green functions and confinement. Using Landau gauge where the gauge configurations are restricted to the first Gribov regime, results on the ghost-, gluon form-factor and the running coupling constant are presented. Finally, we discuss the behavior of the ghost form-factor of the de-confined phase at high temperatures. 
  We introduce the notion of generalized Weyl system, and use it to define *-products which generalize the commutation relations of Lie algebras. In particular we study in a comparative way various *-products which generalize the k-Minkowski commutation relations. 
  The construction of field theories with space-time symmetries, including tensorial charges (i.e. of M-theory type), initiated in hep-th/9907011, is extended to include interaction. For SO(2,2) gravity in a tensorial space-time, with space-time symmetry consisting of Lorentz generators and "translations", represented by second-rank antisymmetric tensor, the cubic interaction terms are constructed by requirement of maintaining the gauge invariance property of theory. This interaction is essentially unique. 
  We investigate maximally symmetric backgrounds in nonsupersymmetric string vacua with D-branes and O-planes localized in the compact space. We find a class of solutions with a perturbative string coupling constant in all regions of spacetime. Depending on the particular model, we find either a time evolution with a big-bang type singularity or a space dependent background with generically orbifold singularities. We show that the result can be interpreted as a supersymmetric bulk with some symmetries broken by the boundaries. We also discuss an interesting connection to Lorentzian and Euclidian orbifolds. 
  A suitable counterterm for a Euclidean space lattice version of \phi^4_n theories, n\ge 4, is combined with several additional procedures so that in the continuum limit the resultant quantum field theory is nontrivial. Arguments to support this unconventional choice are presented. 
  We construct an explicit 5-dimensional supergravity model that realizes the "no scale" mechanism for supersymmetry breaking with no unstable moduli. Supersymmetry is broken by a constant superpotential localized on a brane, and the radion is stabilized by Casimir energy from supergravity and massive hypermultiplets. If the standard model gauge and matter fields are localized on a brane, then visible sector supersymmetry breaking is dominated by gravity loops and flavor-violating hypermultiplet loops, and gaugino masses are smaller than scalar masses. We present a realistic model in which the the standard model gauge fields are partly localized. In this model visible sector supersymmetry breaking is naturally gaugino mediated, while masses of the gravitino and gravitational moduli are larger than the weak scale. 
  Starting from the non-linear sigma model of the IIB string in the light-cone gauge, we analyze the role of RR fluxes in Holography. We find that the worldsheet theory of states with only left or right moving modes does not see the presence of RR fields threading a geometry. We use this significant simplification to compute part of the strong coupling spectrum of the two dimensional NCOS theory. We also reproduce the action of a closed string in a PP-wave background using this general formalism; and we argue for various strategies to find new systems where the closed string theory may be exactly solvable. 
  We show that a given set of first class constraints becomes abelian if one maps each constraint to the surface of other constraints. There is no assumption that first class constraints satisfy a closed algebra. The explicit form of the projection map is obtained at least for irreducible first class constraints. Using this map we give a method to obtain gauge fixing conditions such that the set of abelian first class constraints and gauge fixing conditions satisfy the symplectic algebra. 
  We have investigated the classical stability of magnetically charged black $p$-brane solutions for string theories that include the case studied by Gregory and Laflamme. It turns out that the stability behaves very differently depending on a coupling parameter between dilaton and gauge fields. In the case of Gregory and Laflamme, it has been known that the black brane instability decreases monotonically as the charge of black branes increases and finally disappears at the extremal point. For more general cases we found that, when the coupling parameter is small, black brane solutions become stable even before reaching to the extremal point. On the other hand, when the coupling parameter is large, black branes are always unstable and moreover the instability does not continue to decrease, but starts to increase again as they approach to the extremal point. However all extremal black branes are shown to be stable even in this case. It has also been shown that main features of the classical stability are in good agreement with the local thermodynamic behavior of the corresponding black hole system through the Gubser-Mitra conjecture. Some implications of our results are also discussed. 
  We propose two non-abelian Chern-Simons matrix models as the effective descriptions of 4-dimensional quantum Hall fluids. One of them describes a new type of 4-dimensional quantum Hall fluid on the space of quaternions, the other provides the description of non-commutative field theory for Zhang and Hu's 4-dimensional quantum Hall fluid of $S^4$. The complete sets of physical quantum states of these matrix models are determined, and the properties of quantum Hall fluids related with them are discussed. 
  We investigate one-flavor QCD with an additional chiral scalar field. For a large domain in the space of coupling constants, this model belongs to the same universality class as QCD, and the effects of the scalar become unobservable. This is connected to a ``bound-state fixed point'' of the renormalization flow for which all memory of the microscopic scalar interactions is lost. The QCD domain includes a microscopic scalar potential with minima at nonzero field. On the other hand, for a scalar mass term m^2 below a critical value m_c^2, the universality class is characterized by perturbative spontaneous chiral symmetry breaking which renders the quarks massive. Our renormalization group analysis shows how this universality class is continuously connected with the QCD universality class. 
  We prove inductively that every k-trace operator of SO(6)_R irrep with Young tableau partition {r_1,r_2,r_3}, constructed out of k chiral primaries in the twenty dimensional SO(6)_R irrep, leads to a quasi primary field with protected conformal dimension. Our argument is based on perturbative evaluations of certain four point functions up to order 1/N^2. 
  An extension of the Keski-Vakuri, Kraus and Wilczek (KKW) analysis to black hole spacetimes which are not Schwarzschild-type is presented. Preserving the regularity at the horizon and stationarity of the metric in order to deal with the across-horizon physics, a more general coordinate transformation is introduced. In this analysis the Hawking radiation is viewed as a tunnelling process which emanates from the non-Schwarzschild-type black hole. Expressions for the temperature and entropy of these non-Schwarzschild-type black holes are extracted. As a paradigm, in the context of this generalization, we consider the Garfinkle-Horowitz-Strominger (GHS) black hole as a dynamical background and we derive the modified temperature and entropy of GHS black hole. Deviations are eliminated and corresponding standard results are recovered to the lowest order in the emitted shell of energy. The extremal GHS black hole is found to be non-``frozen'' since it is characterized by a constant non-zero temperature. Furthermore, the modified extremality condition forbids naked singularities to form from the collapse of the GHS black hole. 
  Nonperturbative solutions to the nonlinear field equations in the NS sector of cubic as well as nonpolynomial superstring field theory can be obtained from a linear equation which includes a "spectral" parameter \lambda and a coboundary operator Q(\lambda). We borrow a simple ansatz from the dressing method (for generating solitons in integrable field theories) and show that classical superstring fields can be constructed from any string field T subject merely to Q(\lambda)T=0. Following the decay of the non-BPS D9 brane in IIA theory and shifting the background to the tachyon vacuum, we repeat the arguments in vacuum superstring field theory and outline how to compute classical solutions explicitly. 
  We consider dimensionally reduced versions of N=2 four- dimensional supersymmetric Yang-Mills theory and determine the one-loop effective Lagrangians associated with the motion over the corresponding moduli spaces. In the (0+1) case, the effective Lagrangian describes an N=4 supersymmetric quantum mechanics of the Diaconescu--Entin type. In (1+1) dimensions, the effective Lagrangian represents a twisted N=4 supesymmetric sigma model due to Gates, Hull, and Rocek. We discuss the genetic relationship between these two models and present the explicit results for all gauge groups. 
  We study holographic RG flows of N=2 matter couple AdS_3 supergravities which admit both compact and non-compact sigma manifolds. For the compact case the supersymmetric domain wall solution interpolates between a conformal IR region and flat spacetime and this corresponds to a deformation of the CFT by an irrelevant operator. When it is non-compact, the solution can be interpreted as a flow between an UV fixed point and a non-conformal(singular) IR region. This is an exact example of a deformation flow when the singularity is physical. We also find a non-supersymmetric deformation flow when the scalar potential has a second AdS vacua. The ratio of the central charges is rational for certain values of the size of the sigma model. Next, we analyze the spectrum of a massless scalar on our background by transforming the problem into Schroedinger form. The spectrum is continuous for the compact model, yet it can be both continuous (with or without mass gap) and discrete otherwise. Finally, 2-point functions are computed for two examples whose quantum mechanical potentials are of Calogero type. 
  We review in this lecture the relation between the Maldacena Conjecture, also known as AdS/CFT correspondence, and the so called Holographic principle that seems to be an essential ingredient for a quantum gravity theory. We also illustrate the idea of Holography by showing that the curvature of the anti-de Sitter space reduces the number of degrees of freedom making it possible to find a mapping between a quantum theory defined on the bulk and another defined on the corresponding boundary. 
  As part of a programme for the general study of the low-energy implications of supersymmetry breaking in brane-world scenarios, we study the nonlinear realization of supersymmetry which occurs when breaking N=2 to N=1 supergravity. We consider three explicit realizations of this supersymmetry breaking pattern, which correspond to breaking by one brane, by one antibrane or by two (or more) parallel branes. We derive the minimal field content, the effective action and supersymmetry transformation rules for the resulting N=1 theory perturbatively in powers of kappa = 1/M_{Planck}. We show that the way the massive gravitino and spin-1 fields assemble into N=1 multiplets implies the existence of direct brane-brane contact interactions at order O(kappa). This result is contrary to the O(kappa^2) predicted by the sequestering scenario but in agreement with recent work of Anisimov et al. Our low-energy approach is model independent and is a first step towards determining the low-energy implications of more realistic brane models which completely break all supersymmetries. 
  Using holographic renormalization, we study correlation functions throughout a renormalization group flow between two-dimensional superconformal field theories. The ultraviolet theory is an N=(4,4) CFT which can be thought of as a symmetric product of U(2) super WZW models. It is perturbed by a relevant operator which preserves one-quarter supersymmetry and drives the theory to an infrared fixed point. We compute correlators of the stress-energy tensor and of the relevant operators dual to supergravity scalars. Using the former, we put together Zamolodchikov's C function, and contrast it with proposals for a holographic C function. In passing, we address and resolve two puzzles also found in the case of five-dimensional bulk supergravity. 
  We construct quantum mechanical models which mimic many features of string theory. We use these models to gain improved descriptions of B fields and gerbes. We examine analogs of T duality, D branes, and mirror symmetry and derive quantum mechanical analogs of standard phenomena, such as the noncommutative geometry induced by a B field. 
  We study the linearly-realised worldsheet supersymmetries in the ``massive'' type II light-cone actions for pp-wave backgrounds. The pp-waves have have 16+N_sup Killing spinors, comprising 16 ``standard'' Killing spinors that occur in any wave background, plus N_sup ``supernumerary'' Killing spinors (0\le N_sup \le 16) that occur only for special backgrounds. We show that only the supernumerary Killing spinors give rise to linearly-realised worldsheet supersymmetries after light-cone gauge fixing, while the 16 standard Killing spinors describe only non-linearly realised inhomogeneous symmetries. We also study the type II actions in the physical gauge, and we show that although in this case the actions are not free, there are now linearly-realised supersymmetries coming both from the standard and the supernumerary Killing spinors. In the physical gauge, there are no mass terms for any worldsheet degrees of freedom, so the masses appearing in the light-cone gauge may be viewed as gauge artefacts. We obtain type IIA and IIB supergravity solutions describing solitonic strings in pp-wave backgrounds, and show how these are related to the physical-gauge fundamental string actions. We study the supersymmetries of these solutions, and find examples with various numbers of Killing spinors, including total numbers that are odd. 
  A spinning nonextremal D3-brane undergoes a phase transition to a naked singularity which, from the braneworld point of view, corresponds to the apparent graviton speed passing from subluminal to superluminal. We investigate this phase transition from the dual perspectives of braneworld scenarios and holography. We discuss the relevance of the thermodynamic stability domains of a spinning D3-brane to the physics of braneworld scenarios. We also describe various gravitational Lorentz violations which arise from static D3-branes. 
  We study aspects of the propagation of strings on BTZ black holes. After performing a careful analysis of the global spacetime structure of generic BTZ black holes, and its relation to the geometry of the SL(2,R) group manifold, we focus on the simplest case of the massless BTZ black hole. We study the SL(2,R) Wess-Zumino-Witten model in the worldsheet minisuperspace limit, taking into account special features associated to the Lorentzian signature of spacetime. We analyse the two- and three-point functions in the pointparticle limit. To lay bare the underlying group structure of the correlation functions, we derive new results on Clebsch-Gordan coefficients for SL(2,R) in a parabolic basis. We comment on the application of our results to string theory in singular time-dependent orbifolds, and to a Lorentzian version of the AdS/CFT correspondence. 
  The problem of confinement of fermions in 1+1 dimensions is approached with a linear potential in the Dirac equation by considering a mixing of Lorentz vector and scalar couplings. Analytical bound-states solutions are obtained when the scalar coupling is of sufficient intensity compared to the vector coupling. 
  We construct the stress tensors for the p-adic string model and for the pure tachyonic sector of open string field theory by naive metric covariantization of the action. Then we give the concrete energy density of a lump solution of the p-adic model. In the cubic open bosonic string field theory, we also give the energy density of a lump solution and pressure evolution of a rolling tachyon solution. 
  Noncommutative geometry governs the physics of quantum Hall (QH) effects. We introduce the Weyl ordering of the second quantized density operator to explore the dynamics of electrons in the lowest Landau level. We analyze QH systems made of $N$-component electrons at the integer filling factor $\nu=k\leq N$. The basic algebra is the SU(N)-extended W$_{\infty}$. A specific feature is that noncommutative geometry leads to a spontaneous development of SU(N) quantum coherence by generating the exchange Coulomb interaction. The effective Hamiltonian is the Grassmannian $G_{N,k}$ sigma model, and the dynamical field is the Grassmannian $G_{N,k}$ field, describing $k(N-k)$ complex Goldstone modes and one kind of topological solitons (Grassmannian solitons). 
  An attempt to get a non-trivial-mass structure of particles in a Randall-Sundrum type of 5-dimensional spacetime with q-deformed extra dimension is discussed. In this spacetime, the fifth dimensional space is boundary free, but there areises an elastic potential preventing free motion toward the fifth direction. The q-deformation is, then, introduced in such a way that the spacetime coordinates become non-commutative between 4-dimensional components and the fifth component. As a result of this q-deformation, there arises naturally an ultraviolet-cutoff effect for the propagators of particles embedded in this spacetime. 
  We find the parameters of the MSSM in terms of bulk supergravity fields for the D-brane model of Berenstein, Jejjala and Leigh (hep-ph/0105042). The model consists of a D3-brane at the singularity of a non-abelian orbifold \Delta_{27}, which gives the particles of the Supersymmetric Standard Model. We compute the action for the D-brane fields in the presence of both supersymmetric and supersymmetry breaking background fluxes. We get quark, lepton, gaugino, Higgsino, scalar partners and Higgs masses, as well as soft trilinear couplings, as functions of the background fields. This work develops a framework for connecting MSSM phenomenology to brane compactifications. 
  We find a new Penrose limit of AdS_5*S^5 that gives the maximally symmetric pp-wave background of type IIB string theory in a coordinate system that has a manifest space-like isometry. This induces a new pp-wave/gauge-theory duality which on the gauge theory side involves a novel scaling limit of N=4 SYM theory. The new Penrose limit, when applied to AdS_5*S^5/Z_M, yields a pp-wave with a space-like circle. The dual gauge theory description involves a triple scaling limit of an N=2 quiver gauge theory. We present in detail the map between gauge theory operators and string theory states including winding states, and verify agreement between the energy eigenvalues obtained from string theory and those computed in gauge theory, at least to one-loop order in the planar limit. We furthermore consider other related new Penrose limits and explain how these limits can be understood as part of a more general framework. 
  By employing D6-branes intersecting at angles in $D = 4$ type IIA strings, we construct {\em four stack string GUT models} (PS-I class), that contain at low energy {\em exactly the three generation Standard model} with no extra matter and/or extra gauge group factors. These classes of models are based on the Pati-Salam (PS) gauge group $SU(4)_C \times SU(2)_L \times SU(2)_R$. They represent deformations around the quark and lepton basic intersection number structure. The models possess the same phenomenological characteristics of some recently discussed examples (PS-A class) of four stack PS GUTS. Namely, there are no colour triplet couplings to mediate proton decay and proton is stable as baryon number is a gauged symmetry. Neutrinos get masses of the correct sizes. Also the mass relation  $m_e = m_d$ at the GUT scale is recovered.  Moreover, we clarify the novel role of {\em extra} branes, the latter having non-trivial intersection numbers with quarks and leptons and creating scalar singlets, needed for the satisfaction of RR tadpole cancellation conditions.  The presence of N=1 supersymmetry in sectors involving the {\em extra} branes is equivalent to the, model dependent, orthogonality conditions of the U(1)'s surviving massless the generalized Green-Schwarz mechanism.  The use of  {\em extra} branes creates mass couplings that predict the appearance of light fermion doublets up to the scale of electroweak scale symmetry breaking. 
  We compute the conserved quantities of the four-dimensional Kerr-Newman-dS (KNdS) black hole through the use of the counterterm renormalization method, and obtain a generalized Smarr formula for the mass as a function of the entropy, the angular momentum and the electric charge. The first law of thermodynamics associated to the cosmological horizon of KNdS is also investigated. Using the minimal number of intrinsic boundary counterterms, we consider the quasilocal thermodynamics of asymptotic de Sitter Reissner-Nordstrom black hole, and find that the temperature is equal to the product of the surface gravity (divided by $2\pi$) and the Tolman redshift factor. We also perform a quasilocal stability analysis by computing the determinant of Hessian matrix of the energy with respect to its thermodynamic variables in both the canonical and the grand-canonical ensembles and obtain a complete set of phase diagrams. We then turn to the quasilocal thermodynamics of four-dimensional Kerr-Newman-de Sitter black hole for virtually all possible values of the mass, the rotation and the charge parameters that leave the quasilocal boundary inside the cosmological event horizon, and perform a quasilocal stability analysis of KNdS black hole. 
  We analyze a potential that produces background charges which are automatically quantized. This introduces a new mechanism for charge quantization, although so far it has only been implemented for background charges. We show that this same mechanism can also lead to an alternative means of hiding extra dimensions that is analogous to the Kaluza-Klein approach. 
  We construct five different two-parameter massive deformations of the unique nine-dimensional N=2 supergravity. All of these deformations have a higher-dimensional origin via Scherk-Schwarz reduction and correspond to gauged supergravities. The gauge groups we encounter are SO(2), SO(1,1)^+, R, R^+ and the two-dimensional non-Abelian Lie group A(1), which consists of scalings and translations in one dimension.   We make a systematic search for half-supersymmetric domain walls and non-supersymmetric de Sitter space solutions. Furthermore, we discuss in which sense the supergravities we have constructed can be considered as low-energy limits of compactified superstring theory. 
  We analyze the transmission of supersymmetry breaking in brane-world models of pseudo-supersymmetry. In these models two branes preserve different halves of the bulk supersymmetry. Thus supersymmetry is broken although each sector of the model is supersymmetric when considered separately. The world-volume theory on one brane feels the breakdown of supersymmetry only through two-loop interactions involving a coupling to fields from the other brane. In a 5D toy model with bulk vectors, we compute the diagrams that contribute to scalar masses on one brane and find that the masses are proportional to the compactification scale up to logarithmic corrections, m^2 ~ (2 pi R)^{-2}(ln(2 pi R ms)-1.1), where ms is an ultraviolet cutoff. Thus, for large compactification radii, where this result is valid, the brane scalars acquire a positive mass squared. We also compute the three-loop diagrams relevant to the Casimir energy between the two branes and find E ~ (2 pi R)^{-4}((ln(2 pi R ms)-1.7)^2+0.2). For large radii, this yields a repulsive Casimir force. 
  I discuss effective field theories of brane-world models where different sectors break different halves of the extended bulk supersymmetry. It is shown how to consistently couple N=2 supersymmetric matter to N=1 superfields that lack N=2 partners but transform in a non-linear representation of the N=2 algebra. In particular, I explain how to couple an N=2 vector to N=1 chiral fields such that the second supersymmetry is non-linearly realized. This method is then used to study systems where different sectors break different halves of supersymmetry, which appear naturally in models of intersecting branes. 
  We resolve the controversy on the stability of the monopole condensation in the one-loop effective action of SU(2) QCD by calculating the imaginary part of the effective action with two different methods at one-loop order. Our result confirms that the effective action for the magnetic background has no imaginary part but the one for the electric background has a negative imaginary part. This assures that the monopole condensation is indeed stable, but the electric background becomes unstable due to the pair-annihilation of gluons. 
  Fermionic extensions of generic 2d gravity theories obtained from the graded Poisson-Sigma model (gPSM) approach show a large degree of ambiguity. In addition, obstructions may reduce the allowed range of fields as given by the bosonic theory, or even prohibit any extension in certain cases. In our present work we relate the finite W-algebras inherent in the gPSM algebra of constraints to algebras which can be interpreted as supergravities in the usual sense (Neuveu-Schwarz or Ramond algebras resp.), deformed by the presence of the dilaton field. With very straightforward and natural assumptions on them --like demanding rigid supersymmetry in a certain flat limit, or linking the anti-commutator of certain fermionic charges to the Hamiltonian constraint-- in the ``genuine'' supergravity obtained in this way the ambiguities disappear, as well as the obstructions referred to above. Thus all especially interesting bosonic models (spherically reduced gravity, the Jackiw-Teitelboim model etc.)\ under these conditions possess a unique fermionic extension and are free from new singularities. The superspace supergravity model of Howe is found as a special case of this supergravity action. For this class of models the relation between bosonic potential and prepotential does not introduce obstructions as well. 
  We present an overview of the ways in which D-brane charges are classified in terms of K-theory, emphasizing the natural physical interpretations of a homological classification within a topological setting. 
  We study deep inelastic scattering in gauge theories which have dual string descriptions. As a function of $gN$ we find a transition. For small $gN$, the dominant operators in the OPE are the usual ones, of approximate twist two, corresponding to scattering from weakly interacting partons. For large $gN$, double-trace operators dominate, corresponding to scattering from entire hadrons (either the original `valence' hadron or part of a hadron cloud.) At large $gN$ we calculate the structure functions. As a function of Bjorken $x$ there are three regimes: $x$ of order one, where the scattering produces only supergravity states; $x$ small, where excited strings are produced; and, $x$ exponentially small, where the excited strings are comparable in size to the AdS space. The last regime requires in principle a full string calculation in curved spacetime, but the effect of string growth can be simply obtained from the world-sheet renormalization group. 
  It has been conjectured that the (weakly coupled) Randall-Sundrum (RS) model with gauge fields in the bulk is dual to a (strongly coupled) 4D conformal field theory (CFT) with an UV cut-off and in which global symmetries of the CFT are gauged. We elucidate features of this dual CFT which are crucial for a complete understanding of the proposed duality. We argue that the limit of no (or small) brane-localized kinetic term for bulk gauge field on the RS side (often studied in the literature) is dual to no bare kinetic term for the gauge field which is coupled to the CFT global current. In this limit, the kinetic term for this gauge field in the dual CFT is ``induced'' by CFT loops. Then, this CFT loop contribution to the gauge field 1PI two-point function is dual (on the RS side) to the full gauge propagator (i.e., including the contribution of Kaluza-Klein and zero-modes) with both external points on the Planck brane. We also emphasize that loop corrections to the gauge coupling on the RS side are dual to sub-leading effects in a large-N expansion on the CFT side; these sub-leading corrections to the gauge coupling in the dual CFT are (in general) sensitive to the strong dynamics of the CFT. 
  We present a general procedure for determining possible (nonuniform) magnetic fields such that the Pauli equation becomes quasi-exactly solvable (QES) with an underlying $sl(2)$ symmetry. This procedure makes full use of the close connection between QES systems and supersymmetry. Of the ten classes of $sl(2)$-based one-dimensional QES systems, we have found that nine classes allow such construction. 
  We study bulk-boundary correlators in topological open membranes. The basic example is the open membrane with a WZ coupling to a 3-form. We view the bulk interaction as a deformation of the boundary string theory. This boundary string has the structure of a homotopy Lie algebra, which can be viewed as a closed string field theory. We calculate the leading order perturbative expansion of this structure. For the 3-form field we find that the C-field induces a trilinear bracket, deforming the Lie algebra structure. This paper is the first step towards a formal universal quantization of general quasi-Lie bialgebroids. 
  We present an improvement of the interacting string bit theory proposed in hep-th/0206059, designed to reproduce the non-planar perturbative amplitudes between BMN operators in N=4 gauge theory. Our formalism incorporates the effect of operator mixing and all non-planar corrections to the inner product. We use supersymmetry to construct the bosonic matrix elements of the light-cone Hamiltonian to all orders in g_2, and make a detailed comparison with the non-planar amplitudes obtained from gauge theory to order (g_2)^2. We find a precise match. 
  A new background field formulation of QCD is presented in which the background gluon field is not a classical field, but an operator made up of quantized quark fields. This background field allows colorless quark states to form exact quantum solutions of the QCD equations of motion for any value of the coupling constant. When matrix elements of the Hamiltonian are calculated in the context of these solution states, quark fields and gluon fluctuations completely decouple. Due to decoupling, gluon fluctuations around the background field can be ignored and only the quark part of the Hamiltonian need be considered when comparing colorless quark states. Despite nonlinear terms involving the background field, this pure quark Hamiltonian is completely diagonalizable and leads to energies for colorless quark configurations that are infinitely more negative than those for colored quark configurations. In this way, the method provides an explanation for confinement and can be used to calculate hadron masses directly from QCD. 
  We point out some intriguing analogies between field theoretic solitons (topological defects) and D-branes. Annihilating soliton-antisoliton pairs can produce stable solitons of lower dimensionality. Solitons that localize massless gauge fields in their world volume automatically imply the existence of open flux tubes ending on them and closed flux tubes propagating in the bulk. We discuss some aspects of this localization on explicit examples of unstable wall-anti-wall systems. The annihilation of these walls can be described in terms of tachyon condensation which renders the world-volume gauge field non-dynamical. During this condensation the world volume gauge fields (open string states) are resonantly excited. These can later decay into closed strings, or get squeezed into a network flux tubes similar to a network of cosmic strings formed at a cosmological phase transition. Although, as in the $D$-brane case, perturbatively one can find exact time-dependent solutions, when the energy of the system stays localized in the plane of the original soliton, such solutions are unstable with respect to decay into open and closed string states. Thus, when a pair of such walls annihilates, the energy is carried away (at least) by closed string excitations (``glueballs''), which are the lowest energy excitations about the bulk vacuum. Suggested analogies can be useful for the understanding of the complicated D-brane dynamics and of the production of topological defects and reheating during brane collision in the early universe. 
  We construct the Penrose limit backgrounds in closed forms along the generic null geodesics for the near-horizon geometries of D1, D3, D5, NS1 and NS5 branes. The Penrose limit metrics of D1, D5 and NS1 have non-trivial dependence of the light-cone time coordinate, while those of D3 and NS5 have no its dependence. We study the Penrose limits on the marginal 1/4 supersymmetric configurations of standard intersecting branes, such as the NS-NS intersection of NS1 and NS5, the R-R intersections of Dp and Dq over some spatial dimensions and the mix intersections of NS5 and Dp over (p -1)-dimensional spaces. They are classified into three types that correspond to the Penrose limits of D1, D3 and D5 backgrounds. 
  String modes in a pp-wave background are generically massive, and the worldvolume description of the branes is to be given by `massive' gauge theories. In this paper, we present a five dimensional super Yang-Mills action with the Kahler-Chern-Simons term plus the Myers term as a low energy worldvolume description of the longitudinal five branes in a maximally supersymmetric pp-wave background. We derive the action from the M-theory matrix model on the pp-wave. We utilize the previously found 4/32 BPS solution of rotating five branes with stacks of membranes, but, to obtain the static configuration, we reformulate the matrix model in a rotating coordinate system which provides the inertial frame for the branes. Expanding the matrix model around the solution, we first obtain a non-commutative field theory action naturally equipped with the full sixteen dynamical supersymmetries. In the commutative limit, we show only four supersymmetries survive, resulting in a novel five dimensional "N=1/2" theory. 
  I study the hypothetical thermodynamic system which saturates the so-called Hubble entropy bound and show that it is invariant under the S- and T-dualities of string theory as well as the interchanges of the eleventh dimension of M-theory. I also discuss how unique the entropy bound is under the dualities and some related issues. 
  We study the spectral representation and dispersion relations that follow from some basic assumptions and the reduced spacetime symmetries on noncommutative (NC) space. Kinematic variables involving the NC parameter appear naturally as parametric variables in this analysis. When subtractions are necessary to remove ultraviolet divergences, they are always made at the fixed values of these NC variables. This point is also illustrated by a perturbative analysis of self-energies. Our analysis of the reduced spacetime symmetries suggests a weaker microcausality requirement. Starting from it, we make a first attempt at dispersion relations for forward scattering. It turns out that the attempt is hampered by a new unphysical region specified by a given motion in the NC plane which does not seem to be surmountable using the usual tricks. Implications for a possible subtraction and renormalization scheme for NC field theory in which the ultraviolet-infrared (UV/IR) mixing is removed are also briefly commented on. 
  In this paper we compute the radiation of the massless closed string states due to the non-vanishing coupling to the rolling open string tachyon with co-dimensions larger than 2, and discuss the effect of back reaction to the motion of the rolling tachyon. We find that for small string coupling, the tachyonic matter remains in the late time, but it will completely evaporate away over a short time of few string scales if the string coupling is huge. Comment on the implication of our results to the g-theorem of boundary conformal field theory is given. 
  We show that the Ginsparg-Wilson (GW) relation can play an important role to define chiral structures in {\it finite} noncommutative geometries. Employing GW relation, we can prove the index theorem and construct topological invariants even if the system has only finite degrees of freedom. As an example, we consider a gauge theory on a fuzzy two-sphere and give an explicit construction of a noncommutative analog of the GW relation, chirality operator and the index theorem. The topological invariant is shown to coincide with the 1st Chern class in the commutative limit. 
  We study semiclassical rotating strings in AdS/CFT backgrounds that exhibit both confinement and finite-size effects. The energy versus spin dispersion relation for short strings is the expected Regge trajectory behaviour, with the same string tension as is measured by the Wilson loop. Long strings probe the interplay between confinement and finite-size effects. In particular, the dispersion relation for long strings shows a characteristic dependence on the string tension and the finite-size scale. 
  We formulate matrix string models on a class of exact string backgrounds with non constant RR-flux parameterized by a holomorphic prepotential function and with manifest (2,2) supersymmetry. This lifts these string theories to M-theory exposing the non perturbative string interaction which is studied by generalizing the instanton asymptotic expansion, well established in the flat background case, to this more general case. We obtain also a companion matrix model with four manifest supersymmetries in eleven dimensions. 
  We propose that perturbative quantum field theory and string theory can be consistently modified in the infrared to eliminate, in a radiatively stable manner, tadpole instabilities that arise after supersymmetry breaking. This is achieved by deforming the propagators of classically massless scalar fields and the graviton so as to cancel the contribution of their zero modes. In string theory, this modification of propagators is accomplished by perturbatively deforming the world-sheet action with bi-local operators similar to those that arise in double-trace deformations of AdS/CFT. This results in a perturbatively finite and unitary S-matrix (in the case of string theory, this claim depends on standard assumptions about unitarity in covariant string diagrammatics). The S-matrix is parameterized by arbitrary scalar VEVs, which exacerbates the vacuum degeneracy problem. However, for generic values of these parameters, quantum effects produce masses for the nonzero modes of the scalars, lifting the fluctuating components of the moduli. Warning: in the case of string theory, the simple prescription discussed in this paper fails to decouple BRST trivial modes from the physical S-matrix. A procedure aimed at correcting this is under investigation. 
  We propose a phenomenological approach to the cosmological constant problem based on generally covariant non-local and acausal modifications of four-dimensional gravity at enormous distances. The effective Newton constant becomes very small at large length scales, so that sources with immense wavelengths and periods -- such as the vacuum energy-- produce minuscule curvature. Conventional astrophysics, cosmology and standard inflationary scenaria are unaffected, as they involve shorter length scales. A new possibility emerges that inflation may ``self-terminate'' naturally by its own action of stretching wavelengths to enormous sizes. In a simple limit our proposal leads to a modification of Einstein's equation by a single additional term proportional to the average space-time curvature of the Universe. It may also have a qualitative connection with the dS/CFT conjecture. 
  We develop an unhiggsing procedure for finding the D-brane probe world volume gauge theory for blowups of geometries whose gauge theory data are known. As specific applications we unhiggs the well-studied theories for the cone over the third del Pezzo surface. We arrive at what we call pseudo del Pezzos and these will constitute a first step toward the understanding of higher, non toric del Pezzos. Moreover, our methods and results give further support for toric duality as well as obtaining superpotentials from global symmetry considerations. 
  As shown in our previous papers (hep-th/0209001 and reference therein), quantum theory based on a Galois field (GFQT) possesses a new symmetry between particles and antiparticles, which has no analog in the standard approach. In the present paper it is shown that this symmetry (called the AB one) is also compatible with supersymmetry. We believe this is a strong argument in favor of our assumption that the AB symmetry is a fundamental symmetry in the GFQT (and in nature if it is described by quantum theory over a Galois field). We also consider operatorial formulations of space inversion and X inversion in the GFQT. It is shown in particular that the well known fact, that the parity of bosons is real and the parity of fermions is imaginary, is a simple consequence of the AB symmetry. 
  In this writing we shall address certain beautiful inter-relations between the construction of 4-dimensional supersymmetric gauge theories and resolution of algebraic singularities, from the perspective of String Theory. We review in some detail the requisite background in both the mathematics, such as orbifolds, symplectic quotients and quiver representations, as well as the physics, such as gauged linear sigma models, geometrical engineering, Hanany-Witten setups and D-brane probes.   We investigate aspects of world-volume gauge dynamics using D-brane resolutions of various Calabi-Yau singularities, notably Gorenstein quotients and toric singularities. Attention will be paid to the general methodology of constructing gauge theories for these singular backgrounds, with and without the presence of the NS-NS B-field, as well as the T-duals to brane setups and branes wrapping cycles in the mirror geometry. Applications of such diverse and elegant mathematics as crepant resolution of algebraic singularities, representation of finite groups and finite graphs, modular invariants of affine Lie algebras, etc. will naturally arise. Various viewpoints and generalisations of McKay's Correspondence will also be considered.   The present work is a transcription of excerpts from the first three volumes of the author's PhD thesis which was written under the direction of Prof. A. Hanany - to whom he is much indebted - at the Centre for Theoretical Physics of MIT, and which, at the suggestion of friends, he posts to the ArXiv pro hac vice; it is his sincerest wish that the ensuing pages might be of some small use to the beginning student. 
  Free scalar fields in de Sitter space have a one-parameter family of states invariant under the de Sitter group, including the standard thermal vacuum. We show that, except for the thermal vacuum, these states are unphysical when gravitational interactions are included. We apply these observations to the quantum state of the inflaton, and find that, at best, dramatic fine tuning is required for states other than the thermal vacuum to lead to observable features in the CMBR anisotropy. 
  In a tachyon effective field theory of a non-BPS brane, we construct a classical solution representing a parallel brane-antibrane. The solution is made of a kink and an antikink placed at antipodal points of S^1. Estimation of the brane energy suggests an excitation of a string connecting the two branes, even though the theory is Abelian. We performed fluctuation analysis around the obtained solution, and find the structure of the supersymmetry breaking by the co-existence of the brane and the antibrane. We discuss possible processes of the pair-annihilation of the brane defect. 
  We derive expressions for three-body phase space that are explicitly symmetrical in the masses of the three particles, by three separate methods. 
  Lorentz-invariant non-commutative QED (NCQED) is constructed such that it should be a part of Lorentz-invariant non-commutative standard model (NCSM), a subject to be treated in later publications. Our NCSM is based on Connes' observation that the total fermion field in the standard model may be regarded as a bi-module over a flavor-color algebra. In this paper, it is shown that there exist two massless gauge fields in NCQED which are interchanged by $C'$ transformation. Since $C'$ is reduced to the conventional charge conjugation $C$ in the commutative limit, the two gauge fields become identical to the photon field in the same limit, which couples to only four spinors with charges $\pm 2,\pm 1.$ Following Carlson-Carone-Zobin, our NCQED respects Lorentz invariance employing Doplicher-Fredenhagen-Roberts' algebra instead of the usual algebra with constant $\theta^{\mu\nu}$. In the new version $\theta^{\mu\nu}$ becomes an integration variable. We show using a simple NC scalar model that the $\theta$ integration gives an {\it invariant} damping factor instead of the oscillating one to the nonplanar self-energy diagram in the one-loop approximation. Seiberg-Witten map shows that the $\theta$ expansion of NCQED generates exotic but well-motivated derivative interactions beyond QED with allowed charges being only $0, \pm 1, \pm 2$. 
  As a concrete application of the holographic correspondence to manifolds which are only asymptotically Anti-de Sitter, we take a closer look at the quaternionic Taub-NUT space. This is a four dimensional, non-compact, inhomogeneous, riemannian manifold with the interesting property of smoothly interpolating between two symmetric spaces, AdS_4 itself and the coset SU(2,1)/U(2). Even more interesting is the fact that the scalar curvature of the induced conformal structure at the boundary (corresponding to a squashed three-sphere) changes sign as we interpolate between these two limiting cases. Using twistor methods, we construct the bulk-to-bulk and bulk-to-boundary propagators for conformally coupled scalars on quaternionic Taub-NUT. This may eventually enable us to calculate correlation functions in the dual strongly coupled CFT on a squashed S^3 using the standard AdS/CFT prescription. 
  We examine the possibility that there may exist a logarithmic correction to the infrared asymptotic solution with power behavior which has recently been found for the gluon and Faddeev-Popov ghost propagators in the Landau gauge. We propose a new Ansatz to find a pair of solutions for the gluon and ghost form factors by solving the coupled Schwinger-Dyson equation under a simple truncation. This Ansatz enables us to derive the infrared and ultraviolet asymptotic solutions simultaneously and to understand why the power solution and the logarithmic solution is possible only in the infrared and ultraviolet limit respectively. Even in the presence of the logarithmic correction, the gluon propagator vanishes and the ghost propagator is enhanced in the infrared limit, and the gluon-ghost-antighost coupling constant has an infrared fixed point (but with a different $\beta$ function). This situation is consistent with Gribov-Zwanziger confinement scenario and color confinement criterion of Kugo and Ojima. 
  We show that the $1/p^2$ power corrections to the ultraviolet asymptotic solutions are allowed as consistent solutions of the coupled Schwinger-Dyson equation for the gluon and (Faddeev-Popov) ghost propagators in Yang-Mills theory. This result supports the existence of the vacuum condensate $<A_\mu^2>$ with mass dimension 2, as recently suggested by the operator product expansion and lattice simulations. We compare the solution with the result of operator product expansion. 
  We present a class of solutions to the Einstein-Maxwell equations in d-dimensions, all of which are asymptotically (anti)-de Sitter space-times. They describe electrically charged rotating solutions, which are generalizations of those found by Lemos (gr-qc/9404041). These solutions have toroidal, planar or cylindrical horizons and can be interpreted as black holes, or black strings/branes. We calculate the inverse temperature and entropy, and then we use the Brown-York stress-tensor to calculate mass and angular momenta of these solutions. 
  A tubular D3-brane with electromagnetic flux is considered. It is verified that the quantized electric and magnetic charges are the F-string and D-string charges of the brane, respectively. The D3-brane with parallel electric and magnetic fields collapses to the bound state of strings. A D3-brane with nonzero Poynting vector can be viewed as expanded FD strings. A fundamental string is viewed as a collapsed D3-brane and mapped to the (n,m) strings via electric-magnetic duality rotations. This mapping is the same as a weak-strong duality transformation and therefore reveals the equivalence of the electric-magnetic and weak-strong dualities. 
  We obtain a new explicit expression for the noncommutative (star) product on the fuzzy two-sphere which yields a unitary representation. This is done by constructing a star product, $\star_{\lambda}$, for an arbitrary representation of SU(2) which depends on a continuous parameter $\lambda$ and searching for the values of $\lambda$ which give unitary representations. We will find two series of values: $\lambda = \lambda^{(A)}_j=1/(2j)$ and $\lambda=\lambda^{(B)}_j =-1/(2j+2)$, where j is the spin of the representation of SU(2). At $\lambda = \lambda^{(A)}_j$ the new star product $\star_{\lambda}$ has poles. To avoid the singularity the functions on the sphere must be spherical harmonics of order $\ell \leq 2j$ and then $\star_{\lambda}$ reduces to the star product $\star$ obtained by Preusnajder. The star product at $\lambda=\lambda^{(B)}_j$, to be denoted by $\bullet$, is new. In this case the functions on the fuzzy sphere do not need to be spherical harmonics of order $\ell \leq 2j$. Because in this case there is no cutoff on the order of spherical harmonics, the degrees of freedom of the gauge fields on the fuzzy sphere coincide with those on the commutative sphere. Therefore, although the field theory on the fuzzy sphere is a system with finite degrees of freedom, we can expect the existence of the Seiberg-Witten map between the noncommutative and commutative descriptions of the gauge theory on the sphere. We will derive the first few terms of the Seiberg-Witten map for the U(1) gauge theory on the fuzzy sphere by using power expansion around the commutative point $\lambda=0$. 
  These lectures provide an introduction to the microscopic description of branes in curved backgrounds. After a brief reminder of the flat space theory, the basic principles and techniques of (rational) boundary conformal field theory are presented in the second lecture. The general formalism is then illustrated through a detailed discussion of branes on compact group manifolds. In the final lecture, many more recent developments are reviewed, including some results for non-compact target spaces. 
  We discuss the bulk Casimir effect (effective potential) for a conformal or massive scalar when the bulk represents five-dimensional AdS or dS space with two or one four-dimensional dS brane, which may correspond to our universe. Using zeta-regularization, the interesting conclusion is reached, that for both bulks in the one-brane limit the effective potential corresponding to the massive or to the conformal scalar is zero. The radion potential in the presence of quantum corrections is found. It is demonstrated that both the dS and the AdS braneworlds may be stabilized by using the Casimir force only. A brief study indicates that bulk quantum effects are relevant for brane cosmology, because they do deform the de Sitter brane. They may also provide a natural mechanism yielding a decrease of the four-dimensional cosmological constant on the physical brane of the two-brane configuration. 
  A number theoretical model of $1/f$ noise found in phase locked loops is developed. The dynamics of phases and frequencies involved in the nonlinear mixing of oscillators and the low-pass filtering is formulated thanks to the rules of the hyperbolic geometry of the half plane. A cornerstone of the analysis is the Ramanujan sums expansion of arithmetical functions found in prime number theory, and their link to Riemann hypothesis. 
  In the BMN approach to N=4 SYM a large class of correlators of interest are expressible in terms of expectation values of traces of words in a zero-dimensional Gaussian complex matrix model. We develop a loop-equation based, analytic strategy for evaluating such expectation values to any order in the genus expansion. We reproduce the expectation values which were needed for the calculation of the one-loop, genus one correction to the anomalous dimension of BMN-operators and which were earlier obtained by combinatorial means. Furthermore, we present the expectation values needed for the calculation of the one-loop, genus two correction. 
  There is good support for the embedding of the Standard Model fermions in the chiral 16 representation of SO(10). Such an embedding is provided by the realistic free fermionic heterotic-string models. In this talk we demonstrate the existence of solutions with 3 generations and SO(10) observable gauge group, in the case of compactification on a torus-fibred Calabi-Yau space over a Hirzebruch base surface. The SO(10)symmetry is broken to SU(5)xU(1) by a Wilson line. The overlap with the realistic free fermionic heterotic-string models is discussed. 
  We consider the one-dimensional XXX spin 1/2 Heisenberg antiferromagnet at zero temperature and zero magnetic field. We are interested in a probability of formation of a ferromagnetic string in the antiferromagnetic ground-state. We call it emptiness formation probability [EFP]. We suggest a new technique for computation of EFP in the inhomogeneous case. It is based on quantum Knizhnik-Zamolodchikov equation. We evalauted EFP for strings of the length six in the inhomogeneous case. The homogeneous limit confirms our hypothesis about the relation of quantum correlations to number theory. We also make a conjecture about a general structure of EFP for arbitrary lenght of the string \. 
  We study orientifolds of type IIB string theory in the plane-wave background supported by null RR 3-form flux F^{(3)}. We describe how to extract the RR tadpoles in the Green-Schwarz formalism in a general setting. Two models with orientifold groups {1, \Omega} and {1,\Omega I_4}, which are T-dual to each other, are considered. Consistency of these backgrounds requires 32 D9 branes for the first model and 32 D5 branes for the second one. We study the spectra and comment on the heterotic duals of our models. 
  We investigate solutions that are dynamically evolving between asymptotically de Sitter and asymptotically anti-de Sitter regions in the context of Einstein gravity coupled to general matter fields in d dimensions. We demonstrate the existence of a no go theorem whenever the matter content of the theory is 'reasonable', i.e. such that the weak energy condition is satisfied. We show that there exist solutions for which the energy conditions are violated in a finite region of the spacetime. We speculate on the holographic interpretation of these gravitational backgrounds by combining ideas from the AdS/CFT and the dS/CFT dualities. 
  New and rather controversial observations hint that the fine structure constant \alpha may have been smaller in the early universe, suggesting that some of the fundamental ``constants'' of physics may be dynamical. In a recent paper, Davies, Davis, and Lineweaver have argued that black hole thermodynamics favors theories in which the speed of light c decreases with time, and disfavors those in which the fundamental electric charge e increases. We show that when one considers the full thermal environment of a black hole, no such conclusion can be drawn: thermodynamics is consistent with an increase in \alpha whether it comes from a decrease in c, an increase in e, or a combination of the two. 
  We develop a method to derive the on-shell invariant quantum action of the supergravity in such a way that the quartic ghost interaction term is explicity determined. First, we reinvestigate the simple supergravity in terms of a principal superfibre bundle. This gives rise to the closed geometrical BRST algebra. Therefore we determine the open BRST algebra, which realizes the invariance of the classical action. Then, given a prescription to build the full quantum action, we obtain the quantum BRST algebra. Together with the constructed quantum action this allows us to recover the auxiliary fields and the invariant extension of the classical action. 
  The recent proposal on the correspondence between the ${\cal N}=4$ super Yang-Mills theory and string theory in the Penrose limit of the AdS$_5\times$S$^5$ geometry involves a few puzzles from the viewpoint of holographic principle, especially in connection with the interpretation of times. To resolve these puzzles, we propose to interpret the PP-wave strings on the basis of tunneling null geodesics connecting boundaries of the AdS geometry. Our approach predicts a direct and systematic identification of the S-matrix of Euclidean string theory in the bulk with the short-distance structure of correlation functions of super Yang-Mills theory on the AdS boundary, as an extension of the ordinary relation in supergravity-CFT correspondence. Holography requires an infinite number of contact terms for interaction vertices of string field theory and constrains their forms in a way consistent with supersymmetry. 
  A flow invariant in quantum field theory is a quantity that does not depend on the flow connecting the UV and IR conformal fixed points. We study the flow invariance of the most general sum rule with correlators of the trace Theta of the stress tensor. In even (four and six) dimensions we recover the results known from the gravitational embedding. We derive the sum rules for the trace anomalies a and a' in six dimensions. In three dimensions, where the gravitational embedding is more difficult to use, we find a non-trivial vanishing relation for the flow integrals of the three- and four-point functions of Theta. Within a class of sum rules containing finitely many terms, we do not find a non-vanishing flow invariant of type a in odd dimensions. We comment on the implications of our results. 
  As argued previously, amplitudes of quantum field theories on noncommutative space and time cannot be computed using naive path integral Feynman rules. One of the proposals is to use the Gell-Mann--Low formula with time-ordering applied before performing the integrations. We point out that the previously given prescription should rather be regarded as an interaction point time-ordering. Causality is explicitly violated inside the region of interaction. It is nevertheless a consistent procedure, which seems to be related to the interaction picture of quantum mechanics. In this framework we compute the one-loop self-energy for a space/time noncommutative \phi^4 theory. Although in all intermediate steps only three-momenta play a role, the final result is manifestly Lorentz covariant and agrees with the naive calculation. Deriving the Feynman rules for general graphs, we show, however, that such a picture holds for tadpole lines only. 
  We investigate some issues that are relevant for the derivation of experimental limits on the parameters of canonical noncommutative spacetimes. By analyzing a simple Wess-Zumino-type model in canonical noncommutative spacetime with soft supersymmetry breaking we explore the implications of ultraviolet supersymmetry on low-energy phenomenology. The fact that new physics in the ultraviolet can modify low-energy predictions affects significantly the derivation of limits on the noncommutativity parameters based on low-energy data. These are, in an appropriate sense here discussed, ``conditional limits''. We also find that some standard techniques for an effective low-energy description of theories with non-locality at short distance scales are only applicable in a regime where theories in canonical noncommutative spacetime lack any predictivity, because of the strong sensitivity to unknown UV physics. It appears useful to combine high-energy data, from astrophysics, with the more readily available low-energy data. 
  In this paper we continue our study f the exact solution in open bosonic string field theory. We present new solution in the string field theory defined on the background corresponding to the boundary conformal field theory describing D25-brane. Then we will study the fluctuation modes around this solution and we determine their basic properties from the linearised equation of motion of the string field theory defined above the classical solution. 
  We propose a possible scheme for getting the known QCD scaling laws within string theory. In particular, we consider amplitudes for exclusive scattering of hadrons at large momentum transfer, hadronic form factors and distribution functions. 
  It was recently shown how to break supersymmetry in certain $AdS_3$ spaces, without destabilizing the background, by using a ``double trace'' deformation which localizes on the boundary of space-time. By viewing spatial sections of $AdS_3$ as a compactification space, one can convert this into a SUSY breaking mechanism which exists uniformly throughout a large 3+1 dimensional space-time, without generating any dangerous tadpoles. This is a generalization of a Visser type infinite extra dimensions compactification. Although the model is not Lorentz invariant, the dispersion relation is relativistic at high enough momenta, and it can be arranged such that at the same kinematical regime the energy difference between between former members of a SUSY multiplet is large. 
  String theory has long ago been initiated by the quest for a theoretical explanation of the observed high-energy ``Regge behaviour'' of strong interaction amplitudes, but this 35-years-old puzzle is still unsoved. We discuss how modern tools like the AdS/CFT correspondence give a new insight on the problem. 
  We determine some particular values of the noncommutativity parameter \theta and show that the Murthy-Shankar approach is in fact a particular case of a more general one. Indeed, using the fractional quantum Hall effect (FQHE) experimental data, we give a measurement of \theta. This measurement can be obtained by considering some values of the filling factor \nu and other ingredients, magnetic field B and electron density \rho. Moreover, it is found that \theta can be quantized either fractionally or integrally in terms of the magnetic length l_0 and the quantization is exactly what Murthy and Shankar formulated recently for the FQHE. On the other hand, we show that the mapping of the FQHE in terms of the composite fermion basis has a noncommutative geometry nature and therefore there is a more general way than the Murthy-Shankar method to do this mapping. 
  We explore how various anomaly-cancelling terms in M-theory and string theory transform non-trivially into each other under duality. Specifically, we study the phenomenon in which bulk terms in M-theory get mapped to brane worldvolume terms in string theory. The key mathematical ingredient is G-index theory. 
  These notes give an introductory review on brane cosmology. This subject deals with the cosmological behaviour of a brane-universe, i.e. a three-dimensional space, where ordinary matter is confined, embedded in a higher dimensional spacetime. In the tractable case of a five-dimensional bulk spacetime, the brane (modified) Friedmann equation is discussed in detail, and various other aspects are presented, such as cosmological perturbations, bulk scalar fields and systems with several branes. 
  Witten's supersymmetric quantum mechanics may incorporate potentials with strong singularities after their appropriate regularization. This was proposed by Das and Pernice [Nucl. Phys. B 561 (1999) 357 and arXiv: hep-th/0207112]. We suggest that one of the most natural recipes of this type is generated by an infinitesimal complex shift of coordinates. The necessary regularization is then mediated by a non-Hermitian, PT symmetric intermediate Hamiltonian. 
  We extract from gauge theoretical calculations the matrix elements of the SYM dilatation operator. By the BMN correspondence this should coincide with the 3-string vertex of light cone string field theory in the pp-wave background. We find a mild but important discrepancy with the SFT results. If the modified $O(g_2)$ matrix elements are used, the $O(g_2^2)$ anomalous dimensions are exactly reproduced without the need for a contact interaction in the single string sector. 
  I give a brief review on motivations, basic postulates, and recent developments in Doubly Special Relativity Theory. 
  We compute the one loop partition function of type IIB string in plane wave R-R 5-form background $F^5$ using both path integral and operator formalisms and show that the two results agree perfectly. The result turns out to be equal to the partition function in the flat background. We also study the Tadpole cancellation for the unoriented closed and open string model in plane wave R-R 5-form background studied in hep-th/0203249 and find that the cancellation of the Tadpole requires the gauge group to be SO(8). 
  We study the quantum moduli spaces and dynamical superpotentials of four dimensional $SU(2)^r$ linear and ring moose theories with $\mathcal{N}=1$ supersymmetry and link chiral superfields in the fundamental representation. Nontrivial quantum moduli spaces and dynamical superpotentials are produced. When the moduli space is perturbed by generic tree level superpotentials, the vacuum space becomes discrete. The ring moose is in the Coulomb phase and we find two singular submanifolds with a nontrivial modulus that is a function of all the independent gauge invariants needed to parameterize the quantum moduli space. The massive theory near these singularities confines. The Seiberg-Witten elliptic curve that describes the quantum moduli space of the ring moose is produced. 
  Quantum moduli spaces of four dimensional $SU(2)^{r}$ linear and ring moose theories with $\mathcal{N}=1$ supersymmetry and link chiral superfields in the fundamental representation are produced starting from simple pure gauge theories of disconnected nodes. 
  We respond to, and comment upon, a number of points raised in a recent paper by Kofman, Linde, and Mukhanov. 
  We discuss the Freund-Rubin compactification with cosmological constant and the dilaton field, and examine the stability of the spacetimes at the low energy. The Minkowski or de Sitter spacetime can be obtained if the dilation field is turned off while we observe that the dilaton field does not permit such spacetimes but only the anti-de Sitter spacetime. The stability of the spacetime depends on the dimensions of the spacetime and the compactified space and the coupling constant of the dilaton field $a$. In the $a=1$ case, which corresponds to superstring theories, the anti-de Sitter vacuum is stable at least in the linear level. 
  We consider path integration of a fermionic oscillator with a one-parameter family of boundary conditions with respect to the time coordinate. The dependence of the fermion determinant on these boundary conditions is derived in a closed form with the help of the self-adjoint extension of differential operators. The result reveals its crucial dependence on them, contrary to the conventional understanding that this dependence becomes negligible over sufficiently long time evolution. An example in which such dependence plays a significant role is discussed in a model of supersymmetric quantum mechanics. 
  I discuss two types of non-supersymmetric string model constructions that give at low energy exactly the Standard model (SM) with no additional matter/and or gauge group factors. The construction is based on D6 branes intersecting at angles in a compactification of type IIA theory on a decomposable orientifolded $T^6$ torus. The first type is based on five and six stack SM-like constructions at the string scale while, the other construction is based on a four stack GUT left-right symmetric structure centered around the Pati-Salam $SU(4)_C \times SU(2)_L \times SU(2)_R$ gauge group. All classes of models exbibit important phenomenological properties including a stable proton and sizes of neutrino masses in consistency with neutrino oscillation experiments. The models are non-SUSY, but amazingly, they allow the existence of supersymmetric particles! 
  Braneworld effective action is constructed by two different methods based respectively on the Dirichlet and Neumann boundary value problems. The equivalence of these methods is shown due to nontrivial duality relations between special boundary operators of these two problems. Previously known braneworld action algorithms in two-brane Randall-Sundrum model are generalized to curved branes with deSitter and Anti-deSitter geometries. 
  An analysis of a spherically symmetric braneworld configuration is performed when the intrinsic curvature scalar is included in the bulk action. In the case when the electric part of the Weyl tensor is zero, all the exterior solutions are found; one of them is of the Schwarzschild-(A)dS(4) form, which is matched to a modified Oppenheimer-Volkoff interior solution. In the case when the electric part of the Weyl tensor is non zero, the exterior Schwarzschild-(A)dS(4) black hole solution is modified receiving corrections from the non-local bulk effects. A non-universal gravitational constant arises, depending on the density of the considered object and the Newton's law is modified for small and large distances; however, the conventional limits are easily obtained. 
  We determine the one-loop deformation of the conformal symmetry of a general N}=2 superconformally invariant Yang-Mills theory. The deformation is computed for several explicit examples which have a realization as world-volume theories on a stack of D3 branes. These include (i) N=4 SYM with gauge groups SU(N), USp(2N) and SO(N); (ii) USp(2N) gauge theory with one hypermultiplet in the traceless antisymmetric representation and four hypermultiplets in the fundamental; (iii) quiver gauge theory with gauge group SU(N)xSU(N) and two hypermultiplets in the bifundamental representations (N,\bar N) and (bar N,N). The existence of quantum corrections to the conformal transformations imposes restrictions on the effective action which we study on a subset of the Coulomb branch corresponding to the separation of one brane from the stack. In the N=4 case, the one-loop corrected transformations provide a realization of the conformal algebra; this deformation is shown to be one-loop exact. For the other two models, higher-loop corrections are necessary to close the algebra. Requiring closure, we infer the two-loop conformal deformation. 
  In the context of colliding brane worlds I discuss a toy cosmological model, developed in collaboration with E. Gravanis, which arguably produces inflation and a relaxing to zero cosmological ``constant'' hierarchically small as compared to the supersymmetry breaking (TeV) scale. Supersymmetry breaking is induced by compactification of the brane worlds on magnetized tori. The crucial ingredient is the non-criticality (non conformality) of string theory on the observable brane world induced at the collision, which is thus viewed as a cause for departure from equilibrium in this system. The hierarchical smallness of the present-era vacuum energy, as compared to the SUSY breaking scale, is thus attributed to relaxation. 
  Basic elements of the exact renormalization group method and recent results within this approach are reviewed. Topics covered are the derivation of equations for the effective action and relations between them, derivative expansion, solutions of fixed point equations and the calculation of the critical exponents, construction of the c-function and a description of the chiral phase transition. 
  The abelian Higgs model on the noncommutative plane admits both BPS vortices and non-BPS fluxons. After reviewing the properties of these solitons, we discuss several new aspects of the former. We solve the Bogomoln'yi equations perturbatively, to all orders in the inverse noncommutivity parameter, and show that the metric on the moduli space of k vortices reduces to the computation of the trace of a k-dimensional matrix. In the limit of large noncommutivity, we present an explicit expression for this metric. 
  We show how massive/gauged maximal supergravities in 11-n dimensions with SO(n-l,l) gauge groups (and other non-semisimple subgroups of Sl(n,R)) can be systematically obtained by dimensional reduction of ``massive 11-dimensional supergravity''. This series of massive/gauged supergravities includes, for instance, Romans' massive N=2A,d=10 supergravity for n=1, N=2,d=9 SO(2) and SO(1,1) gauged supergravities for n=2, and N=8,d=5 SO(6-l,l) gauged supergravity. In all cases, higher p-form fields get masses through the Stuckelberg mechanism which is an alternative to self-duality in odd dimensions. 
  In this talk, we show how the monodromy matrix, ${\hat{\cal M}}$, can be constructed for the two dimensional tree level string effective action. The pole structure of ${\hat{\cal M}}$ is derived using its factorizability property. It is shown that the monodromy matrix transforms non-trivially under the non-compact T-duality group, which leaves the effective action invariant and this can be used to construct the monodromy matrix for more complicated backgrounds starting from simpler ones. We construct, explicitly, ${\hat{\cal M}}$ for the exactly solvable Nappi-Witten model, both when B=0 and $B\neq 0$, where these ideas can be directly checked. 
  Integral transformations of the QCD invariant (running) coupling and of some related objects are discussed. Special attention is paid to the Fourier transformation, that is to transition from the space-time to the energy--momentum representation.   The conclusion is that the condition of possibility of such a transition provides us with one more argument against the real existence of unphysical singularities observed in the perturbative QCD.   The second conclusion relates to the way of "translation" of some singular long--range asymptotic behaviors to the infrared momentum region. Such a transition has to be performed with the due account of the Tauberian theorem. This comment relates to the recent ALPHA collaboration results on the asymptotic behavior of the QCD effective coupling obtained by lattice simulation. 
  The problem of finding boundary states in CFT, often rephrased in terms of "NIMreps" of the fusion algebra, has a natural extension to CFT on non-orientable surfaces. This provides extra information that turns out to be quite useful to give the proper interpretation to a NIMrep. We illustrate this with several examples. This includes a rather detailed discussion of the interesting case of the simple current extension of A_2 level 9, which is already known to have a rich structure. This structure can be disentangled completely using orientation information. In particular we find here and in other cases examples of diagonal modular invariants that do not admit a NIMrep, suggesting that there does not exist a corresponding CFT. We obtain the complete set of NIMreps (plus Moebius and Klein bottle coefficients) for many exceptional modular invariants of WZW models, and find an explanation for the occurrence of more than one NIMrep in certain cases. We also (re)consider the underlying formalism, emphasizing the distinction between oriented and unoriented string annulus amplitudes, and the origin of orientation-dependent degeneracy matrices in the latter. 
  Stochastic Loewner evolutions (SLE) are random growth processes of sets, called hulls, embedded in the two dimensional upper half plane. We elaborate and develop a relation between SLE evolutions and conformal field theories (CFT) which is based on a group theoretical formulation of SLE processes and on the identification of the proper hull boundary states. This allows us to define an infinite set of SLE zero modes, or martingales, whose existence is a consequence of the existence of a null vector in the appropriate Virasoro modules. This identification leads, for instance, to linear systems for generalized crossing probabilities whose coefficients are multipoint CFT correlation functions. It provides a direct link between conformal correlation functions and probabilities of stopping time events in SLE evolutions. We point out a relation between SLE processes and two dimensional gravity and conjecture a reconstruction procedure of conformal field theories from SLE data. 
  In this letter we study the process of Hawking radiation of a black hole assuming the existence of a limiting physical curvature scale. The particular model is constructed using the Limiting Curvature Hypothesis (LCH) and in the context of two-dimensional dilaton gravity. The black hole solution exhibits properties of the standard Schwarzschild solution at large values of the radial coordinate. However, near the center, the black hole is nonsingular and the metric becomes that of de Sitter spacetime. The Hawking temperature is calculated using the method of complex paths. We find that such black holes radiate eternally and never completely evaporate. The final state is an eternally radiating relic, near the fundamental scale, which should make a viable dark matter candidate. We briefly comment on the black hole information loss problem and the production of such black holes in collider experiments. 
  In this paper, we examine a generic theory of 1+1-dimensional gravity with coupling to a scalar field. Special attention is paid to a class of models that have a power-law form of dilaton potential and can capably admit black hole solutions. The study focuses on the formulation of a Lorentzian partition function. We incorporate the principles of Hamiltonian thermodynamics, as well as black hole spectroscopy, and find that the partition function can be expressed in a well-defined, calculable form. We then go on to extract the black hole entropy, including the leading-order quantum correction. As anticipated, this correction can be expressed as the logarithm of the classical entropy. Interestingly, the prefactor for this logarithmic correction disagrees, in both magnitude and sign, with the findings from a prior study (on the very same model). We comment on this discrepancy and provide a possible rationalization. 
  For many cases, the conditions to fully embed a classical solution of one field theory within a larger theory cannot be met. Instead, we find it useful to embed only the solution's asymptotic fields as this relaxes the embedding constraints. Such asymptotically embedded defects have a simple classification that can be used to construct classical solutions in general field theories. 
  Using its multiple integral representation, we compute the large distance asymptotic behavior of the emptiness formation probability of the XXZ spin-1/2 Heisenberg chain in the massless regime. 
  It has long been claimed that the antisymmetric tensor field of the second rank is pure longitudinal after quantization. In my opinion, such a situation is quite unacceptable. I repeat the well-known procedure of the derivation of the set of Proca equations. It is shown that it can be written in various forms. Furthermore, on the basis of the Lagrangian formalism I calculate dynamical invariants (including the Pauli-Lubanski vector of relativistic spin for this field). Even at the classical level the Pauli-Lubanski vector can be equal to zero after applications of well-known constraints. The importance of the normalization is pointed out for the problem of the description of quantized fields of maximal spin 1. The correct quantization procedure permits us to propose a solution of this puzzle in the modern field theory. Finally, the discussion of the connection of the Ogievetskii-Polubarinov-Kalb-Ramond field and the electrodynamic gauge is presented. 
  This work has been withdrawn. 
  We examine the mechanical matrix model that can be derived from the SU(2) Yang-Mills light-cone field theory by restricting the gauge fields to depend on the light-cone time alone. We use Dirac's generalized Hamiltonian approach. In contrast to its well-known instant-time counterpart the light-cone version of SU(2) Yang-Mills mechanics has in addition to the constraints, generating the SU(2) gauge transformations, the new first and second class constraints also. On account of all of these constraints a complete reduction in number of the degrees of freedom is performed. It is argued that the classical evolution of the unconstrained degrees of freedom is equivalent to a free one-dimensional particle dynamics. Considering the complex solutions to the second class constraints we show at this time that the unconstrained Hamiltonian system represents the well-known model of conformal mechanics with a ``strength'' of the inverse square interaction determined by the value of the gauge field spin. 
  By critical analyses of the order parameter of symmetry breaking, we have researched the phase transitions at high density in D=2 and D=3 Gross-Neveu (GN) model and shown that the gap equation obeyed by the dynamical fermion mass has the same effectivenesss as the effective potentials for such analyses of all the second order and some special first order phase transitions. In the meantime we also further ironed out a theoretical divergence and proven that in D=3 GN model a first order phase transition does occur in the case of zero temperature and finite chemical potential. 
  In this paper we propose a knot model of the $\pi$ mesons. In this knot model we give the formation of the $\pi^+$ meson by strong interaction and the decay of the $\pi^+$ meson to the moun $\mu^+$ and the moun neutrino $\nu_{\mu}$ by weak interaction. We give a mass machanism for the generation of the masses of the $\pi^0$ meson, $\pi^+$ meson, the moun $\mu^+$ and the moun neutrino $\nu_{\mu}$. 
  We systematically study VEVs of a gauge scalar field $\Sigma$ in a bulk U(1) vector multiplet and scalar fields in brane/bulk hypermultiplets charged under U(1) in the 5D $S^1/Z_2$ orbifold model with generic FI terms. A non-trivial VEV of $\Sigma$ generates bulk mass terms for U(1) charged fields, and their zero modes have non-trivial profiles. In particular, in the SUSY-breaking case, bosonic and fermionic zero modes have Gaussian profiles. Such non-trivial profiles are useful to explain hierarchical couplings. A toy model for SUSY breaking is studied, and it yields sizable $D$-term contributions to scalar masses. Because the overall magnitude of $D$-term contributions is the same everywhere in the bulk and also on both branes, we have to take into account these contributions and other SUSY-breaking terms to obtain a realistic description. We also give profiles and mass eigenvalues of higher modes. 
  We discuss low-energy heterotic M-theory with five-branes in four and five dimensions and its application to moving brane cosmology. 
  The application of the theory of scale relativity to microphysics aims at recovering quantum mechanics as a new non-classical mechanics on a non-derivable space-time. This program was already achieved as regards the Schr\"odinger and Klein Gordon equations, which have been derived in terms of geodesic equations in this framework: namely, they have been written according to a generalized equivalence/strong covariance principle in the form of free motion equations $D^2x/ds^2=0$, where $D/ds$ are covariant derivatives built from the description of the fractal/non-derivable geometry. Following the same line of thought and using the mathematical tool of Hamilton's bi-quaternions, we propose here a derivation of the Dirac equation also from a geodesic equation (while it is still merely postulated in standard quantum physics). The complex nature of the wave function in the Schr\"odinger and Klein-Gordon equations was deduced from the necessity to introduce, because of the non-derivability, a discrete symmetry breaking on the proper time differential element. By extension, the bi-quaternionic nature of the Dirac bi-spinors arises here from further discrete symmetry breakings on the space-time variables, which also proceed from non-derivability. 
  In a recent paper hep-ph/0208225 it has been claimed that to one-loop order in noncommutative phi^4 scalar field theory using dimensional regularization the UV and IR divergencies decouple. We point out that this statement is incorrect. 
  We investigate cosmological evolutions of the bulk scalar field $\phi(t)$ and the radion $d(t)$ in five-dimensional dilatonic two branes model. The bulk potential for the scalar field is taken as the exponential function $V_{bulk} \propto \exp(-2 \sqrt{2} b\phi)$, where $b$ is the parameter of the theory. This model includes Randall-Sundrum model (with $b=0$) and five-dimensional Ho\v{r}ava-Witten theory (with $b=1$). We consider matter on both branes and arbitrary potentials on the branes and in the bulk. These matter and potentials induce the cosmological expansion of the brane as well as the time evolution of the bulk scalar field and the radion. Starting with full five-dimensional equations, we derive four-dimensional effective equations which govern the low-energy dynamics of brane worlds. A correspondent five-dimensional geometry is also obtained. The effective four-dimensional theory on a positive tension brane is described by bi-scalar tensor theory. If the radion is stabilized, the effective theory becomes Brans-Dicke (BD) theory with BD parameter $1/2 b^2$. On the other hand, if the scalar field is stabilized, the effective theory becomes scalar-tensor theory with BD parameter $\frac{3}{2(3b^2+1)}\frac{\phi(t)}{1-\phi(t)}$ where $\phi$ is the BD field defined by radion $d(t)$. If we do not introduce the stabilization mechanism for these moduli fields, the acceptable late time cosmology can be realized only if the dilaton coupling $b$ is small ($b^2 < 1.6 \times 10^{-4}$) and the negative tension brane is sufficiently away from the positive tension brane. We also construct several models for inflationary brane worlds driven by potentials on the brane and in the bulk. 
  We compute a time-dependent noncommutativity parameter in a model with a time-dependent background, a space-time metric of the plane wave type supported by a Neveu-Schwarz two-form potential. This model is an open string version of the WZW model based on a non-semi-simple group previously studied by Nappi and Witten. The background we study is not conformally invariant. We consider a light-cone action for the sigma-model, compute the worldsheet propagator, and use it to derive a time-dependent noncommutativity parameter. 
  In the present work we investigate temperature effects on the spinor and scalar effetive QED in the context of Thermo Field Dynamics. Following Weisskopf's zero-point energy method, the problem of charge renormalization is reexamined and high temperature contributions are extracted from the thermal correction for the Lagrangian densities. 
  We propose a general formalism to compute exact correlation functions for Cardy's boundary states. Using the free-field construction of boundary states and applying the Coulomb-gas technique, it is shown that charge-neutrality conditions pick up particular linear combinations of conformal blocks. As an example we study the critical Ising model with free and fixed boundary conditions, and demonstrate that conventional results are reproduced. This formalism thus directly associates algebraically constructed boundary states with correlation functions which are in principle observable or numerically calculable. 
  We study branes residing in infinite volume space and of finite extent in the transverse directions. We calculate the graviton propagator in the harmonic gauge both inside and outside the brane and discuss its dependence on the thickness of the brane. Our treatment includes the full tensor structure of the propagator. We obtain two infinite towers of massive modes and tachyonic ghosts. In the thin-brane limit, we recover four-dimensional Einstein gravity. We compare our results to similar recent results by Dubovsky and Rubakov. 
  We propose a class of N=2 supersymmetric nonlinear sigma models on the noncompact Ricci-flat Kahler manifolds, interpreted as the complex line bundles over the hermitian symmetric spaces. Kahler potentials and Ricci-flat metrics for these manifolds with isometries are explicitly constructed by using the techniques of supersymmetric gauge theories. Each of the metrics contains a resolution parameter which controls the size of these base manifolds, and the conical singularity appears when the parameter vanishes. 
  We argue that the effective field theory on D3-branes in a plane-wave background with 3-form flux is a nonlocal deformation of Yang-Mills theory. In the case of NSNS flux, it is a dipole field theory with lightlike dipole vectors. For an RR 3-form flux the dipole theory is strongly coupled. We propose a weakly coupled S-dual description for it. The S-dual description is local at any finite order in string perturbation theory but becomes nonlocal when all perturbation theory orders are summed together. 
  Ghost condensates of dimension two are analysed in a class of nonlinear gauges in pure Yang-Mills theories. These condensates are related to the breaking of the SL(2,R) symmetry, present in these gauges. 
  We consider an eight-dimensional local octonionic theory with the seven-sphere playing the role of the gauge group. Duality conditions for two- and four-forms in eight dimensions are related. Dual fields--octonionic instantons--solve an 8D generalization of the Yang-Mills equation. Modifying the ADHM construction of 4D instantons, we find general $k$-instanton 8D solutions which depends on $16k-7$ effective parameters. 
  A new class of Weyl invariant backgrounds are presented in terms of the metric $G_{\mu\nu}$ and the anti-symmetric Kalb-Ramond fields $B_{\mu\nu}$. The ten-dimensional spacetime is a product of four-dimensional flat spacetime and curved six-dimensional spacetime having nonvanishing Ricci tensors. The non-vanishing Kalb-Ramond field strengths cannot be written globally as $H=dB$, being of the monopole type. Nevertheless they define homogeneous spacetime with no singularity. 
  We consider the Cauchy problem of the gravitational wave with the initial data distributed only around the brane in the one brane model of Randall and Sundrum, and examine its behavior as t\to\infty. Then we find its leading behavior is t^{-6} unlike an ordinary flat 5-dimensional space-time. Such a signal shows that the Huygens principle is violated on the 4-dimensional brane world even asymptotically and also shows the difference between compact and non-compact brane worlds. Some comments are also given related to AdS/CFT correspondence. 
  An essential aspect of noncommutative field theories is their bilocal nature. This feature, and its role in the IR/UV mixing, are discussed using a canonical quantization procedure developed recently. 
  We study counter-terms of one- and two-point Green functions of some special operators in ${\cal N}=1$ SYM from their SUGRA duals from the consideration of AdS/CFT or gauge/gravity correspondence. We consider both the Maldacena-Nunez solution and the Klebanov-Strassler-Tseytlin solution that are proposed to be SUGRA duals of ${\cal N}=1$ SYM. We obtain radial/energy-scale relation for each solution by comparing SUGRA calculations with the field theory results. Using these relations we evaluate the $\beta$-function of ${\cal N}=1$ SYM. We find that the leading order term can be accurately obtained for both solutions and the higher order terms exhibit some ambiguities. We discuss the origin of these ambiguities and conclude that more studies are needed to check whether these SUGRA solutions are exactly dual to ${\cal N}=1$ SYM. 
  We present a unified approach to representations of quantum mechanics on noncommutative spaces with general constant commutators of phase-space variables. We find two phases and duality relations among them in arbitrary dimensions. Conditions for physical equivalence of different representations of a given system are analysed. Symmetries and classification of phase spaces are discussed. Specially, the dynamical symmetry of a physical system is investigated. Finally, we apply our analyses to the two-dimesional harmonic oscillator and the Landau problem. 
  We consider a general N=(2,2) non-linear sigma model with a torsion. We show that the consistency of N=(2,2) supersymmetry implies that the target manifold is necessary equipped with two (in general, different) Poisson structures. Finally we argue that the Poisson geometry of the target space is a characteristic feature of the sigma models with extended supersymmetry. 
  We introduce simple and more advanced concepts that have played a key role in the development of supersymmetric systems. This is done by first describing various supersymmetric quantum mechanics models. Topics covered include the basic construction of supersymmetric field theories, the phase structure of supersymmetric systems with and without gauge particles, superconformal theories and infrared duality in both field theory and string theory. A discussion of the relation of conformal symmetry to a vanishing vacuum energy (cosmological constant) is included. 
  The canonical proper time formulation of relativistic dynamics provides a framework from which one can describe the dynamics of classical and quantum systems using the clock of those very systems. The framework utilizes a canonical transformation on the time variable that is used to describe the dynamics, and does not transform other dynamical variables such as momenta or positions. This means that the time scales of the dynamics are described in terms of the natural local time coordinates, which is the most meaningful parameterization of phenomena such as the approach to equilibrium, or the back reaction of interacting systems. We summarize the formalism of the canonical proper time framework, and provide example calculations of the eigenvalues of the hydrogen atom and near horizon description of a scalar field near a Schwarzschild black hole. 
  We compute the boundary stress-energies of time-dependent asymptotically AdS spacetimes in 5 and 7 dimensions, and find that their traces are equal to the respective 4 and 6 dimensional field-theoretic trace anomalies. This provides good supporting evidence in favour of the AdS/CFT correspondence in time-dependent backgrounds. 
  We present a method, based on loop equations, to compute recursively all the terms in the large $N$ topological expansion of the free energy for the 2-hermitian matrix model. We illustrate the method by computing the first subleading term, i.e. the free energy of a statistical physics model on a discretized torus. 
  We study extremal and non-extremal generalizations of the regular non-abelian solution found by Chamseddine and Volkov in 5d N=4 gauged supergravity, which has been shown by Maldacena and Nastase to describe a system of NS5-branes wrapping an three-sphere dual to three-dimensional U(N) N=1 supersymmetric Yang-Mills with Chern-Simons coupling k=N/2. All black hole solutions have a temperature larger than the Hagedorn temperature Tc of the little string theory and their entropy decreases as the temperature increases. This is a sign that the system is thermodynamically unstable above Tc. We have also found an analytical solution describing NS5-branes wrapped on a constant radius three-sphere and involving a linear dilaton. Its non-extremal generalization has a temperature equal to 2 Tc. 
  We present a quantum mechanical model of spherical supermembranes. Using superfields to represent the cartesian coordinates of the membrane, we are able to exactly determine its supersymmetric vacua. We find there are two classical vacua, one corresponding to an extended membrane and one corresponding to a point-like membrane. For the ${\mathcal N} = 2$ case, instanton effects then lift these vacua to massive states. For the ${\mathcal N} = 4$ case, there is no instanton tunneling, and the vacua remain massless. Similarities to spherical supermembranes as giant gravitons and in Matrix theory on pp-waves is discussed. 
  In this paper, we investigate the global monopole in asymptotically dS/Ads spacetime and find that the mass of the monopole in the asymptotically dS spacetime could be positive if the cosmological constant is greater than a critical value. This shows that the gravitational field of the global monopole could be attractive or repulsive depending on the value of the cosmological constant. 
  The spacetime dependent lagrangian formalism of references [1-2] is used to obtain a classical solution of Yang-Mills theory. This is then used to obtain an estimate of the vacuum expectation value of the Higgs field, viz. $\phi_{a}=A/e$, where $A$ is a constant and $e$ is the Yang-Mills coupling (related to the usual electric charge). The solution can also accommodate non-commuting coordinates on the boundary of the theory which may be used to construct D-brane actions.   PACS:11.15.-q ; 11.27+d ; 11.10.Ef 
  Measurements of radio emission from distant galaxies and quasars verify that the polarization vectors of these radiations are not randomly oriented as naturally expected. This peculiar phenomenon suggests that the spacetime intervening between the source and observer may be exhibiting some sort of optical activity, the origin of which is not known. In the present paper we provide a plausible explanation to this phenomenon by investigating the r\^ole played by a Chern-Simons-like term in the background of an ordinary or superconducting screwed cosmic string in a scalar-tensor gravity. We discuss the possibility that the excess in polarization of the light from radio-galaxies and quasars can be understood as if the electromagnetic waves emitted by these cosmic objects interact with a scalar-tensor screwed cosmic string through a Chern-Simons coupling. We use current astronomical data to constrain possible values for the coupling constant of this theory, and show that it turns out to be: $\lambda \sim 10^{-26}$ eV, which is two orders of magnitude larger than in string-inspired theories. 
  We demonstrate the relation between the Scherk-Schwarz mechanism and flipped gauged brane-bulk supergravities in five dimensions. We discuss the form of supersymmetry violating Scherk-Schwarz terms in pure supergravity and in supergravity coupled to matter. We point out that brane-induced supersymmetry breakdown in 5d Horava-Witten model is not of the Scherk-Schwarz type. We discuss in detail flipped super-bigravity, which is the locally supersymmetric extension of the (++) bigravity. 
  We consider bulk tachyon condensations in a non-linear sigma model whose low energy effective theory contains a nontrivial scalar potential. We argue that one would typically encounter a strong coupling background along a RG flow corresponding to a bulk tachyon condensation, beyond which the RG analysis would not be reliable. In a range of the flow in which the string coupling constant is small, we can show that the tachyon condensation actually decreases the central charge of the sigma model. 
  The relationship between the entropy of de Sitter (dS) Schwarzschild space and that of the CFT which lives on the brane by using Friedmann-Robertson-Walker (FRW) equations and Cardy-Verlinde formula is discussed. We show that when the time-like brane crosses the black hole horizon of dS Schwarzschild black hole, the entropy of the CFT exactly agrees with the black hole entropy of 5-dimensional dS background. 
  We investigate the black hole solutions in the $R^2$-gravity, where the action contains the square of the curvature. In case that the action does not contain the square of the Riemann tensor and in case that the $R^2$-terms are the Gauss-Bonnet (GB) combination, we find exact solutions. We investigate the thermodynamics of these theories and find the Hawking-Page like phase transition, which is the phase transition between the black hole (BH) spacetime and the pure anti-deSitter (AdS) spacetime. From the viewpoint of the AdS/CFT correspondence, such a phase transition may correspond to thermal transition of dual CFT.   An interesting feature of $R^2$-gravity is the possibility of the negative (or zero) dS (or AdS) BH entropy, which depends on the parameters of the $R^2$-terms. We speculate that the appearence of negative entropy may indicate a new type instability where a transition between dS (AdS) BH with negative entropy and AdS (dS) BH with positive entropy occurs.   We also apply the GB gravity to the brane cosmology, where the brane moves in the bulk AdS BH spacetime. By investigating the FRW-like equation, which describes the motion of the brane, we find the behavior of the matter on the brane. When the radius of the brane is large, the matter fields behave as CFT but when the radius is small, the brane universe behaves as the universe with dust or curvature dominant universe, depending on the parameters. 
  The digest of ideology interpreting D-branes on Calabi-Yau manifolds as objects of the derived category is given. 
  In this paper the consistency of the de Sitter invariant $\alpha $-vacua, which have been introduced as simple tools to study the effects of transplanckian physics, is investigated. In particular possible non renormalization problems are discussed, as well as non standard properties of Greens functions. We also discuss the non thermal properties of the $\alpha $-vacua and the necessity of $\alpha$ to change. The conclusion is that non of these problems necessarily exclude an application of the $\alpha $-vacua to inflation. 
  We investigate the condensate mechanism of the low-lying excitations in the matrix models of 4-dimensional quantum Hall fluids recently proposed by us. It is shown that there exist some hierarchies of 4-dimensional quantum Hall fluid states in the matrix models, and they are similar to the Haldane's hierarchy in the 2-dimensional quantum Hall fluids. However, these hierarchical fluid states appear consistently in our matrix models without any requirement of modifications of the matrix models. 
  We prove that no local diffeomorphism invariant two-dimensional theory of the metric and the dilaton without higher derivatives can describe the exact string black hole solution found a decade ago by Dijkgraaf, Verlinde and Verlinde. One of the key points in this proof is the concept of dilaton-shift invariance. We present and solve (classically) all dilaton-shift invariant theories of two-dimensional dilaton gravity. Two such models, resembling the exact string black hole and generalizing the CGHS model, are discussed explicitly. 
  We presented a new physical model that links the maximum speed of light with the minimal Planck scale into a maximal-acceleration Relativity principle in the spacetime tangent bundle and in phase spaces (cotangent bundle). The maximal proper-acceleration bound is a = c^2/ \Lambda in full agreement with the old predictions of Caianiello, the Finslerian geometry point of view of Brandt and more recent results in the literature. Inspired by the maximal-acceleration corrections to the Lamb shifts of one-electron atoms by Lambiase, Papini and Scarpetta, we derive the exact integral equation that governs the Renormalization-Group-like scaling dependence of the fractional change of the fine structure constant as a function of the cosmological redshift factor and a cutoff scale L_c, where the maximal acceleration relativistic effects are dominant. A particular physical model exists dominated entirely by the vacuum energy, when the cutoff scale is the Planck scale, with \Omega_\Lambda \sim 1 . The implications of this extreme case scenario are studied. 
  We discuss a new class of string and p-brane models where the string/brane tension appears as an additional dynamical degree of freedom instead of being introduced by hand as an ad hoc dimensionfull scale. The latter property turns out to have a significant impact on the string/brane dynamics. The dynamical tension obeys Maxwell (or Yang-Mills) equations of motion (in the string case) or their rank p gauge theory analogues (in the p-brane case), which in particular triggers a simple classical mechanism of (``color'') charge confinement. 
  We study the black hole shape using the C-metric solution for a matter trapped near the IR-brane in the ${AdS}_4$ space. In the AdS/CFT duality, the IR-brane is introduced by embedding a 2-brane at the ${AdS}_4$ radius $z=l$, while the UV-brane is defined by the ${AdS}_4$ boundary. We find the C-metric solution generates a negative tension IR-brane and a negative thermal energy gas of colliding particles. We analyze the momentum energy tensor at the ${AdS}_4$ boundary. We find that the negative energy black hole solution is entirely unstable even for a small perturbation in the Poincare coordinate space. However, such a black hole decays very rapidly due to the imaginary part emerges in the ADM mass. This imaginary part appears because of the orbifold constraints of the C-metric solution in the unusual coordinates. Moreover, this decay rate diverges at the UV-brane. This implies that the black hole evaporates instantaneously otherwise the boundary itself will collapse. 
  It is proven that the pure spinor superstring in an AdS_5 x S^5 background remains conformally invariant at one loop level in the sigma model perturbation theory. 
  The anti-de Sitter C-metric (AdS C-metric) is characterized by a quite interesting new feature when compared with the C-metric in flat or de Sitter backgrounds. Indeed, contrarily to what happens in these two last exact solutions, the AdS C-metric only describes a pair of accelerated black holes if the acceleration parameter satisfies A>1/L, where L is the cosmological length. The two black holes cannot interact gravitationally and their acceleration is totally provided by the pressure exerted by a strut that pushes the black holes apart. Our analysis is based on the study of the causal structure, on the description of the solution in the AdS 4-hyperboloid in a 5D Minkowski embedding spacetime, and on the physics of the strut. We also analyze the cases A=1/L and A<1/L that represent a single accelerated black hole in the AdS background. 
  We derive the low energy effective theory for two branes system solving the bulk geometry formally in the covariant curvature formalism developed by Shiromizu, Maeda and Sasaki. As expected, the effective theory looks like a Einstein-scalar system. Using this theory we can discuss the cosmology and non-linear gravity at low energy scales. 
  We study the finite theta correction to the metric of the moduli space of noncommutative multi-solitons in scalar field theory in (2+1) dimensions. By solving the equation of motion up to order O(theta^{-2}) explicitly, we show that the multi-soliton solution must have the same center for a generic potential term. We examine the condition that the multi-centered configurations are allowed. Under this condition, we calculate the finite theta correction to the metric of the moduli space of multi-solitons and argue the possibility of the non right-angle scattering of two solitons. We also obtain the potential between two solitons. 
  We study non-supersymmetric fermion mass and condensate deformations of the AdS/CFT Correspondence. The 5 dimensional supergravity flows are lifted to a complete and remarkably simple 10 dimensional background. A brane probe analysis shows that when all the fermions have an equal mass a positive mass is generated for all six scalar fields leaving non-supersymmetric Yang Mills theory in the deep infra-red. We numerically determine the potential, produced by the background, in the Schroedinger equation relevant to the study of O^++ glueballs. The potential is a bounded well, providing evidence of stability and for a discrete, confined spectrum. The geometry can also describe the supergravity background around an (unstable) fuzzy 5-brane. 
  In this paper, the generation of topological energy in models with large extra dimensions is investigated. The origin of this energy is attributed to a topological deformation of the standard Minkowski vacuum due to compactification of extra dimensions. This deformation is seen to give rise to an effective, finite energy density due to massive Kaluza-Klein modes of gravitation. It's renormalized value is seen to depend on the size of the extra dimensions instead of the UV cut-off of the theory. It is shown that if this energy density is to contribute to the observed cosmological constant, there will be extremely stringent bounds on the number of extra dimensions and their size. 
  We analyse the SU(2)_k WZNW models beyond the integrable representations and in particular the case of SU(2)_0. We find that these are good examples of logarithmic conformal field theories as indecomposable representations are naturally produced in the fusion of discrete irreducible representations. We also find extra, chiral and non-chiral, multiplet structure in the theory. The chiral fields, which we construct explicitly in SU(2)_0, generate extended algebras within the model. We also study the process of quantum hamiltonian reduction of SU(2)_0, giving the c=-2 triplet model, in both the free field approach and at the level of correlation functions. For rational level SU(2)_k this gives us a useful technique to study the h_{1,s} correlators of the c_{p,q} models and we find very similar structures to SU(2)_0. We also discuss LCFT as a limit of a sequence of ordinary CFTs and some of the subtleties that can occur. 
  In this work, we exploit the operator content of the $(D_{4}, A_{6})$ conformal algebra. By constructing a $Z_{2}$-invariants fusion rules of a chosen subalgebra and by resolving the bootstrap equations consistent with these rules, we determine the structure constants of the subalgebra. 
  We obtain the Penrose limit of six dimensional Non-Commutative Open String (NCOS$_6$) theory and show that in the neighborhood of a particular null geodesic it leads to an exactly solvable string theory (unlike their counterparts in four or in other dimensions). We describe the phase structure of this theory and discuss the Penrose limit in different phases including Open D-string (OD1) theory. We compute the string spectrum and discuss their relations with the states of various theories at different phases. We also consider the case of general null geodesic for which the Penrose limit leads to string theory in the time dependent pp-wave background and comment on the renormalization group flow in the dual theory. 
  Using the algebraic geometry method of Berenstein and Leigh (BL), hep-th/0009209 and hep-th/0105229), and considering singular toric varieties ${\cal V}_{d+1}$ with NC irrational torus fibration, we construct NC extensions ${\cal M}_{d}^{(nc)}$ of complex d dimension Calabi-Yau (CY) manifolds embedded in ${\cal V}_{d+1}^{(nc)}$. We give realizations of the NC $\mathbf{C}^{\ast r}$ toric group, derive the constraint eqs for NC Calabi-Yau (NCCY) manifolds ${\cal M}^{nc}_d$ embedded in ${\cal V}_{d+1}^{nc}$ and work out solutions for their generators. We study fractional $D$ branes at singularities and show that, due to the complete reducibility property of $\mathbf{C}^{\ast r}$ group representations, there is an infinite number of non compact fractional branes at fixed points of the NC toric group. 
  Using the canonical method developed for anomalous theories, we present the independent rederivation of the quantum relationship between the massive Thirring and the sine-Gordon models. The same method offers the possibility to obtain the Mandelstam soliton operators as a solution of Poisson brackets "equation" for the fermionic fields. We checked the anticommutation and basic Poisson brackets relations for these composite operators. The transition from the Hamiltonian to the corresponding Lagrangian variables produces the known Mandelstam's result. 
  We study the quantum stability of Type IIB orbifold and orientifold string models in various dimensions, including Melvin backgrounds, where supersymmetry (SUSY) is broken {\it \`a la} Scherk-Schwarz (SS) by twisting periodicity conditions along a circle of radius R. In particular, we compute the R-dependence of the one-loop induced vacuum energy density $\rho(R)$, or cosmological constant. For SS twists different from Z2 we always find, for both orbifolds and orientifolds, a monotonic $\rho(R)<0$, eventually driving the system to a tachyonic instability. For Z2 twists, orientifold models can have a different behavior, leading either to a runaway decompactification limit or to a negative minimum at a finite value R_0. The last possibility is obtained for a 4D chiral orientifold model where a more accurate but yet preliminary analysis seems to indicate that $R_0\to \infty$ or towards the tachyonic instability, as the dependence on the other geometric moduli is included. 
  We construct matter field theories in ``theory space'' that are fractal, and invariant under geometrical renormalization group (RG) transformations. We treat in detail complex scalars, and discuss issues related to fermions, chirality, and Yang-Mills gauge fields. In the continuum limit these models describe physics in a noninteger spatial dimension which appears above a RG invariant ``compactification scale,'' M. The energy distribution of KK modes above M is controlled by an exponent in a scaling relation of the vacuum energy (Coleman-Weinberg potential), and corresponds to the dimensionality. For truncated-s-simplex lattices with coordination number s the spacetime dimensionality is 1+(3+2ln(s)/ln(s+2)). The computations in theory space involve subtleties, owing to the 1+3 kinetic terms, yet the resulting dimensionalites are equivalent to thermal spin systems. Physical implications are discussed. 
  Ghost condensates of dimension two are analysed in pure SU(N) Yang-Mills theories by combining the local composite operators technique with the algebraic BRST renormalization. 
  Using the U(4) hybrid formalism, manifestly N=(2,2) worldsheet supersymmetric sigma models are constructed for the Type IIB superstring in Ramond-Ramond backgrounds. The Kahler potential in these N=2 sigma models depends on four chiral and antichiral bosonic superfields and two chiral and antichiral fermionic superfields. When the Kahler potential is quadratic, the model is a free conformal field theory which describes a flat ten-dimensional target space with Ramond-Ramond flux and non-constant dilaton. For more general Kahler potentials, the model describes curved target spaces with Ramond-Ramond flux that are not plane-wave backgrounds. Ricci-flatness of the Kahler metric implies the on-shell conditions for the background up to the usual four-loop conformal anomaly. 
  I review some aspects of non-critical strings in connection with Lorentz-Invariance violating approaches to quantum gravity. I also argue how non-critical strings may provide a unifying framework where string Cosmology and quantum gravity may be tackled together. 
  We analyze the possibility of constructing a supersymmetric invariant that contains the $R^4$ term among its components as a superpotential term in type IIB on-shell superspace. We consider a scalar superpotential, i.e. an arbitrary holomorphic function of a chiral scalar superfield. In general, IIB superspace does not allow for the existence of chiral superfields, but the obstruction vanishes for a specific superfield, the dilaton superfield. This superfield contains all fields of type IIB supergravity among its components, and its existence is implied by the solution of the Bianchi identities. The construction requires the existence of an appropriate chiral measure, and we find an obstruction to the existence of such a measure. The obstruction is closely related to the obstruction for the existence of chiral superfields and is non-linear in the fields. These results imply that the IIB superinvariant related to the $R^4$ term is not associated with a scalar chiral superpotential. 
  Quantum vacuum energy has been known to have observable consequences since 1948 when Casimir calculated the force of attraction between parallel uncharged plates, a phenomenon confirmed experimentally with ever increasing precision. Casimir himself suggested that a similar attractive self-stress existed for a conducting spherical shell, but Boyer obtained a repulsive stress. Other geometries and higher dimensions have been considered over the years. Local effects, and divergences associated with surfaces and edges have been studied by several authors. Quite recently, Graham et al. have re-examined such calculations, using conventional techniques of perturbative quantum field theory to remove divergences, and have suggested that previous self-stress results may be suspect. Here we show that the examples considered in their work are misleading; in particular, it is well-known that in two dimensions a circular boundary has a divergence in the Casimir energy for massless fields, while for general dimension $D$ not equal to an even integer the corresponding Casimir energy arising from massless fields interior and exterior to a hyperspherical shell is finite. It has also long been recognized that the Casimir energy for massive fields is divergent for $D\ne1$. These conclusions are reinforced by a calculation of the relevant leading Feynman diagram in $D$ and three dimensions. There is therefore no doubt of the validity of the conventional finite Casimir calculations. 
  We show an explicit connection between the solution to the equations of motion in the Gaussian functional approximation and the minimum of the (Gaussian) effective potential/action of the linear $\Sigma$ model, as well as with the N/D method in dispersion theory. The resulting equations contain analytic functions with branch cuts in the complex mass squared plane. Therefore the minimum of the effective action may lie in the complex mass squared plane. Many solutions to these equations can be found on the second, third, etc. Riemann sheets of the equation, though their physical interpretation is not clear. Our results and the established properties of the S-matrix in general, and of the N/D solutions in particular, guide us to the correct choice of the Riemann sheet. We count the number of states and find only one in each spin-parity and isospin channel with quantum numbers corresponding to the fields in the Lagrangian, i.e. to Castillejo-Dalitz-Dyson (CDD) poles. We examine the numerical solutions in both the strong and weak coupling regimes and calculate the Kallen-Lehmann spectral densities and then use them for physical interpretation. 
  We describe the general geometrical framework of brane world constructions in orientifolds of type IIA string theory with D6-branes wrapping 3-cycles in a Calabi-Yau 3-fold, and point out their immediate phenomenological relevance. These branes generically intersect in points, and the patterns of intersections govern the chiral fermion spectra and issues of gauge and supersymmetry breaking in the low energy effective gauge theory on their world volume. In particular, we provide an example of an intersecting brane world scenario on the quintic Calabi-Yau with the gauge group and the chiral spectrum of the Standard Model and discuss its properties in some detail. Additionally we explain related technical advancements in the construction of supersymmetric orientifold vacua with intersecting D-branes. Six-dimensional orientifolds of this type generalize the rather limited set of formerly known orbifolds of type I, and the presented techniques provide a short-cut to obtain their spectra. Finally, we comment on lifting configurations of intersecting D6-branes to M-theory on non-compact G_2 manifolds. 
  We propose a model action in 1+1 flat space-time (compact in spatial dimension) embedded in D flat space-time with a non dynamical space-time dependent two vector. For the above constrained system Dirac brackets of suitably defined co-ordinates turn out to be non zero and under specific choice of the two vector the model action reduces to a general action of open string in a noncommutative background where we find the natural embedding of the open string action of in presence of magnetic field on D-brane. 
  We study the constraint structure of the O(3) nonlinear sigma model in the framework of the Lagrangian, symplectic, Hamilton-Jacobi as well as the Batalin-Fradkin-Tyutin embedding procedure. 
  We report results on the construction of cosmological braneworld models in the context of the Einstein-Gauss-Bonnet gravity, which include the leading correction to the Einstein-Hilbert action suggested by superstring theory. We obtain and study the equations governing the dynamics of the standard cosmological models. We find that they can be written in the same form as in the case of the Randall-Sundrum model but with time-varying four-dimensional gravitational and cosmological constants. Finally, we discuss the cosmological evolution predicted by these models and their compatibility with observational data. 
  We construct integrable spin chains with inhomogeneous periodic disposition of the anisotropy parameter. The periodicity holds for both auxiliary (space) and quantum (time) directions. The integrability of the model is based on a set of coupled Yang-Baxter equations. This construction yields P-leg integrable ladder Hamiltonians. We analyse the corresponding quantum group symmetry. 
  We study four dimensional $Z_2 \times Z_2$ (shift)-orientifolds in presence of internal magnetic fields and NS-NS $B$-field backgrounds, describing in some detail one explicit example with N=1 supersymmetry. These models are related by $T$-duality to orientifolds with $D$-branes intersecting at angles and exhibit, due to the background fields, a rank reduction of the gauge group and multiple matter families. Moreover, the low-energy spectra are chiral and anomaly free if $D5$-branes are present along the magnetized directions. 
  We study Chern-Simons theory with a complex G_C or a real G x G gauge group on a manifold with boundary - this includes Lorentzian and Euclidean (anti-) de Sitter (E/A)dS gravity for G=SU(2) or G=SL(2,R). We show that there is a canonical choice of boundary conditions that leads to an unambiguous, fully covariant and gauge invariant, off-shell derivation of the boundary action - a G_C/G or G WZW model, coupled in a gauge invariant way to the boundary value of the gauge field. In particular, for (E/A)dS gravity, the boundary action is a WZW model with target space (E/A)dS_3, reminiscent of a worldsheet for worldsheet mechanism. We discuss in some detail the properties of the boundary theories that arise and we confront our results with various related constructions in the literature. 
  Diaconescu, Moore and Witten have shown that the topological part of the M-theory partition function is an invariant of an E8 gauge bundle over the 11-dimensional bulk. This presents a puzzle as an 11d gauge theory cannot exhibit linearly realized supersymmetry. One possibility is that the gauge theory is nonsupersymmetric and flows to 11d SUGRA only in the infrared, with SUSY arising as a low energy accidental degeneracy. Although no such gauge theory has been constructed, any such construction must satisfy a number of constraints in order to correctly reproduce the known 10-dimensional physics on each boundary component. We analyze these constraints and in particular use them to attempt an approximate construction of the 11d gravitino as a condensate of the gauge theory fields. 
  String solitons in AdS_5 contain information of N=4 SUSY Yang-Mills theories on the boundary. Recent proposals for rotating string solitons reproduce the spectrum for anomalous dimensions of Wilson operators for the boundary theory. There are possible extensions of this duality for lower supersymmetric and even for non-supesymmetric Yang-Mills theories. We explicitly demonstrate that the supersymmetric anomalous dimensions of Wilson operators in N=0,1 Yang-Mills theories behave, for large spin J, at the two-loop level in perturbation theory, like log J. We compile the analytic one- and two-loop results for the N=0 case which is known in the literature, as well as for the N=1 case which seems to be missing. 
  The aim of this work is to investigate the role played by the fermion mass and that of an external field on the fermionic Casimir energy density under S^1 X R^3 topology. Both twisted and untwisted spin connections are considered and the exact calculation is performed using a somewhat different approach based on the combination of the analytic regularization method through alpha-representation and Euler-Maclaurin summation formula. 
  We perform a one-loop calculation of the vacuum energy of a tachyon field in anti de-Sitter space with boundary conditions corresponding to the presence of a double-trace operator in the dual field theory. Such an operator can lead to a renormalization group flow between two different conformal field theories related to each other by a Legengre transformation in the large N limit. The calculation of the one-loop vacuum energy enables us to verify the holographic c-theorem one step beyond the classical supergravity approximation. 
  The occurence of 5d de Sitter space with 4d de Sitter brane is discussed on classical and quantum level. It is shown that quantum effects maybe produced by dual CFT living on the brane. Moreover, gravity trapping on the brane is proved via the presentation of 5d dS gravity as 4d gravity coupled with gauge theory. This supports the dS/CFT correspondence. Some open questions in 5d dS/4d CFT correspondence are briefly discussed. 
  We point out the existence of nonlinear $\sigma$-models on group manifolds which are left symmetric and right Poisson-Lie symmetric. We discuss the corresponding rich T-duality story with particular emphasis on two examples: the anisotropic principal chiral model and the $SL(2,C)/SU(2)$ WZW model. The latter has the de Sitter space as its (conformal) non-Abelian dual. 
  The phase structure of the finite SU(2)xSU(2) theory with N=2 supersymmetry, broken to N=1 by mass terms for the adjoint-valued chiral multiplets, is determined exactly by compactifying the theory on a circle of finite radius. The exact low-energy superpotential is constructed by identifying it as a linear combination of the Hamiltonians of a certain symplectic reduction of the spin generalized elliptic Calogero-Moser integrable system. It is shown that the theory has four confining, two Higgs and two massless Coulomb vacua which agrees with a simple analysis of the tree-level superpotential of the four-dimensional theory. In each vacuum, we calculate all the condensates of the adjoint-valued scalars. 
  The Hopf algebra of renormalisation in quantum field theory is described at a general level. The products of fields at a point are assumed to form a bialgebra B and renormalisation endows T(T(B)^+), the double tensor algebra of B, with the structure of a noncommutative bialgebra. When the bialgebra B is commutative, renormalisation turns S(S(B)^+), the double symmetric algebra of B, into a commutative bialgebra. The usual Hopf algebra of renormalisation is recovered when the elements of $T^1(B)$ are not renormalised, i.e. when Feynman diagrams containing one single vertex are not renormalised. When B is the Hopf algebra of a commutative group, a homomorphism is established between the bialgebra S(S(B)^+) and the Faa di Bruno bialgebra of composition of series. The relation with the Connes-Moscovici Hopf algebra of diffeomorphisms is given. Finally, the bialgebra S(S(B)^+) is shown to give the same results as the standard renormalisation procedure for the scalar field. 
  We introduce the 2PPI (2-point-particle-irreducible) expansion, which sums bubble graphs to all orders. We prove the renormalizibility of this summation. We use it on the Gross-Neveu model to calculate the mass gap and vacuum energy. After an optimization of the expansion, the final results are qualitatively good. 
  Charged scalar field is quantized in the background of a static d-2 - brane which is a core of the magnetic flux lines in flat d+1 - dimensional space-time. We find that vector potential of the magnetic core induces the energy-momentum tensor in the vacuum. The tensor components are periodic functions of the brane flux and holomorphic functions of space dimension. The dependence on the distance from the brane and on the coupling to the space-time curvature scalar is comprehensively analysed. 
  Subject of this work is a class of Chern-Simons field theories with non-semisimple gauge group, which may well be considered as the most straightforward generalization of an Abelian Chern-Simons field theory. As a matter of fact these theories, which are characterized by a non-semisimple group of gauge symmetry, have cubic interactions like those of non-abelian Chern-Simons field theories, but are free from radiative corrections. Moreover, at the tree level in the perturbative expansion,there are only two connected tree diagrams, corresponding to the propagator and to the three vertex originating from the cubic interaction terms. For such theories it is derived here a set of BRST invariant observables, which lead to metric independent amplitudes. The vacuum expectation values of these observables can be computed exactly. From their expressions it is possible to isolate the Gauss linking number and an invariant of the Milnor type, which describes the topological relations among three or more closed curves. 
  We study the role of rolling tachyons in the cosmological model with dilatonic gravity. In the string frame, flat space solutions of both initial-stage and late-time are obtained in closed form. In the Einstein frame, we show that every expanding solution is decelerating. 
  Employing the string bit formalism of hep-th/0209215, we identify the basis transformation that relates BMN operators in N=4 gauge theory to string states in the dual string field theory at finite g_2=J^2/N. In this basis, the supercharge truncates at linear order in g_2, and the mixing amplitude between 1 and 2-string states precisely matches with the (corrected) answer of hep-th/0206073 for the 3-string amplitude in light-cone string field theory. Supersymmetry then predicts the order g_2^2 contact term in the string bit Hamiltonian. The resulting leading order mass renormalization of string states agrees with the recently computed shift in conformal dimension of BMN operators in the gauge theory. 
  For m^2 < a^2 + q^2, with m, a, and q respectively the source mass, angular momentum per unit mass, and electric charge, the Kerr--Newman (KN) solution of Einstein's equation reduces to a naked singularity of circular shape, enclosing a disk across which the metric components fail to be smooth. By considering the Hawking and Ellis extended interpretation of the KN spacetime, it is shown first that, similarly to the electron-positron system, this solution presents four inequivalent classical states. Next, it is shown that due to the topological structure of the extended KN spacetime it does admit states with half-integral angular momentum. This last property is corroborated by the fact that, under a rotation of the space coordinates, those inequivalent states transform into themselves only after a 4pi rotation. As a consequence, it becomes possible to naturally represent them in a Lorentz spinor basis. The state vector representing the whole KN solution is then constructed, and its evolution is shown to be governed by the Dirac equation. The KN solution can thus be consistently interpreted as a model for the electron-positron system, in which the concepts of mass, charge and spin become connected with the spacetime geometry. Some phenomenological consequences of the model are explored. 
  A BCS-type wave function describes the ground state of the massless Thirring model in the chirally broken phase. The massless Thirring model with fermion fields quantized in the chirally broken phase bosonizes to the quantum field theory of the free massless (pseudo)scalar field (Eur. Phys. J. C20, 723 (2001)). The wave functions of the ground state of the free massless (pseudo)scalar field are obtained from the BCS-type wave function by averaging over quantum fluctuations of the Thirring fermion fields. We show that these wave functions are orthogonal, normalized and non-invariant under shifts of the massless (pseudo)scalar field. This testifies the spontaneous breaking of the field-shift symmetry in the quantum field theory of a free massless (pseudo)scalar field. We show that the vacuum-to-vacuum transition amplitude calculated for the bosonized BCS-type wave functions coincides with the generating functional of Green functions defined only by the contribution of vibrational modes (Eur. Phys. J. C 24, 653 (2002)) . This confirms the assumption that the bosonized BCS-type wave function is defined by the collective zero-mode (hep-th/0212226). We argue that the obtained result is not a counterexample to the Mermin-Wagner-Hohenberg and Coleman theorems. 
  We construct the supergravity solution for fully localized D2/D6 intersection. The near horizon limit of this solution is the supergravity dual of supersymmetric Yang-Mills theory in 2+1 dimensions with flavor. We use this solution to formulate mirror symmetry of 2+1 dimensional gauge theories in the language of AdS/CFT correspondence. We also construct the supergravity dual of a non-commutative gauge theory with fundamental matter. 
  In the present article we calculate the expectation values of of S$_{z}$ and S$^{2}$ operators for spin-1 and spin-3/2 particles by expanding a general wave function which includes all spin values. The results are same as in the stantard quantum mechanics. 
  We show that the duality between the self-dual and Maxwell-Chern-Simons theories in 2+1-dimensions survives when the space-time becomes noncommutative. Existence of the Seiberg-Witten map is crucial in the present analysis. It should be noted that the above models, being manifestly gauge variant and invariant respectively, transform differently under the Seiberg-Witten map. We also discuss this duality in the Stuckelberg formalism where the self-dual model is elevated to a gauge theory. The "`master"' lagrangian approach has been followed throughout. 
  In this brief note we show that the zero dimensional version of the field theory of tachyon matter proposed by Sen, provides an action integral formulation for the motion of a particle in the presence of Newtonian gravity and nonlinear damping (quadratic in velocity). 
  I suggest that classical General Relativity in four spacetime dimensions incorporates a Principal of Maximal Tension and give arguments to show that the value of the maximal tension is $c^4 \over 4 G$. The relation of this principle to other, possibly deeper, maximal principles is discussed, in particular the relation to the tension in string theory. In that case it leads to a purely classical relation between $G$ and the classical string coupling constant $\alpha ^\prime$ and the velocity of light $c$ which does not involve Planck's constant. 
  A model Hamiltonian that exhibits asymptotic freedom and a bound state, is used to show on example that similarity renormalization group procedure can be tuned to improve convergence of perturbative derivation of effective Hamiltonians, through adjustment of the generator of the similarity transformation. The improvement is measured by comparing the eigenvalues of perturbatively calculated renormalized Hamiltonians that couple only a relatively small number of effective basis states, with the exact bound state energy in the model. The improved perturbative calculus leads to a few-percent accuracy in a systematic expansion. 
  We discuss the nonlinear transformations of standard Poincar\'{e} symmetry in the context of recently introduced Doubly Special Relativity (DSR) theories. We introduce four classes of modified relativistic theories with three of them describing various DSR frameworks. We consider four examples of modified relativistic symmetries, which illustrate each of the considered class. 
  We describe the supersymmetrization of two formulations of free noncommutative planar particles -- in coordinate space with higher order Lagrangian [1] and in the framework of Faddeev and Jackiw [2,3], with first order action. In nonsupersymmetric case the first formulation after imposing subsidiary condition eliminating internal degrees of freedom provides the second formulation. In supersymmetric case one can also introduce the split into ``external'' and ``internal'' degrees of freedom both describing supersymmetric models. 
  Modification of nonrelativistic phase space structure based on fuzzy ordered sets (Fosets) structure investigated as a possible quantization framework. In this model particle's $m$ state corresponds to Foset element - fuzzy point. Due to fuzzy ordering its space coordinate $x$ acquires principal uncertainty $\sigma_x$. It's shown that proposed Mechanics on fuzzy phase space manifold reproduces the main quantum effects, in particular the interference of quantum states. 
  We suggest a general relation between theories of infinite number of higher-spin massless gauge fields in $AdS_{d+1}$ and large $N$ conformal theories in $d$ dimensions containing $N$-component vector fields. In particular, we propose that the singlet sector of the well-known critical 3-d O(N) model with the $(\phi^a \phi^a)^2$ interaction is dual, in the large $N$ limit, to the minimal bosonic theory in $AdS_4$ containing massless gauge fields of even spin. 
  Recently the anomalous dimension of twist two operators in N=4 SYM theory was computed by Gubser, Klebanov and Polyakov in the limit of large 't Hooft coupling using semi-classical rotating strings in AdS_5. Here we reproduce their results for large angular momentum by using the cusp anomaly of Wilson loops in Minkowski signature also computed within the AdS/CFT correspondence. In this case the anomalous dimension is related to an Euclidean worldsheet whose properties are completely determined by the symmetries of the problem. This gives support to the proposed identification of rotating strings and twist two operators. 
  A local supersymmetric extension with N=2 of the dimensional continuation of the Euler-Gauss-Bonnet density from eight to nine dimensions is constructed. The gravitational sector is invariant under local Poincare translations, and the full field content is given by the vielbein, the spin connection, a complex gravitino, and an Abelian one-form. The local symmetry group is shown to be super Poincare with N=2 and a U(1) central extension, and the full supersymmetric Lagrangian can be written as a Chern-Simons form. 
  The cosmological constant problem is how one chooses, without fine-tuning, one singular point $\Lambda_{eff}=0$ for the 4D cosmological constant. We argue that some recently discovered {\it weak self-tuning} solutions can be viewed as blowing-up this one point into a band of some parameter. These weak self-tuning solutions may have a virtue that only de Sitter space solutions are allowed outside this band, allowing an inflationary period. We adopt the hybrid inflation at the brane to exit from this inflationary phase and to enter into the standard Big Bang cosmology. 
  I generalize classical gravity/quantum gauge theory duality in AdS/CFT correspondence to (1+1)-dimensional non-relativistic quantum mechanical system. It is shown that (1+1)-dimensional non-relativistic quantum mechanical system can be reproduced from holographic projection of (2+1)-dimension classical gravity at semiclassical limit. In this explanation every quantum path in 2-dimension corresponds to a classical path of 3-dimension gravity under definite holographic projection. I consider free particle and harmonic oscillator as two examples and find their dual gravity description. 
  We show that the constraint structure in the chain by chain method can be investigated within the symplectic analysis of Faddeev-Jackiw formalism. 
  We complete the first stage of constructing a theory of fields not investigated before; these fields transform according to Lorentz group representations decomposable into an infinite direct sum of finite-dimensional irreducible representations. We consider only those theories that initially have a double symmetry: relativistic invariance and the invariance under the transformations of a secondary symmetry generated by the polar or the axial four-vector representation of the orthochronous Lorentz group. The high symmetry of the theory results in an infinite degeneracy of the particle mass spectrum with respect to spin. To eliminate this degeneracy, we postulate a spontaneous secondary-symmetry breaking and then solve the problems on the existence and the structure of nontrivial interaction Lagrangians. 
  The normalization of the states of the fermionic projector is analyzed. By considering the system in finite 4-volume and taking the infinite volume limit, it is made precise what "idempotence" of the fermionic projector means. It is shown that for each fermionic state, the probability integral has a well-defined infinite volume limit. 
  We study multiwrapped circular string pulsating in the radial direction of AdS black hole. We compute the energy of this string as a function of a large quantum number n. One then could associate it with energy and a quantum number of states in the dual finite temperature {\cal N}=4 SYM theory as well as three dimensional pure gauge theory. We observe that the n dependence of the energy has a universal form. We have also considered pulsating string in the background of the near-extremal D4-brane solution. Circular pulsating membrane in M-theory on AdS_7\times S^4 has also been studied. 
  I study some classes of RG flows in three dimensions that are classically conformal and have manifest g -> 1/g dualities. The RG flow interpolates between known (four-fermion, Wilson-Fischer, phi_3^6) and new interacting fixed points. These models have two remarkable properties: i) the RG flow can be integrated for arbitrarily large values of the couplings g at each order of the 1/N expansion; ii) the duality symmetries are exact at each order of the 1/N expansion. I integrate the RG flow explicitly to the order O(1/N), write correlators at the leading-log level and study the interpolation between the fixed points. I examine how duality is implemented in the regularized theory and verified in the results of this paper. 
  I discuss several issues about the irreversibility of the RG flow and the trace anomalies c, a and a'. First I argue that in quantum field theory: i) the scheme-invariant area Delta(a') of the graph of the effective beta function between the fixed points defines the length of the RG flow; ii) the minimum of Delta(a') in the space of flows connecting the same UV and IR fixed points defines the (oriented) distance between the fixed points; iii) in even dimensions, the distance between the fixed points is equal to Delta(a)=a_UV-a_IR. In even dimensions, these statements imply the inequalities 0 =< Delta(a)=< Delta(a') and therefore the irreversibility of the RG flow. Another consequence is the inequality a =< c for free scalars and fermions (but not vectors), which can be checked explicitly. Secondly, I elaborate a more general axiomatic set-up where irreversibility is defined as the statement that there exist no pairs of non-trivial flows connecting interchanged UV and IR fixed points. The axioms, based on the notions of length of the flow, oriented distance between the fixed points and certain "oriented-triangle inequalities", imply the irreversibility of the RG flow without a global a function. I conjecture that the RG flow is irreversible also in odd dimensions (without a global a function). In support of this, I check the axioms of irreversibility in a class of d=3 theories where the RG flow is integrable at each order of the large N expansion. 
  The thermal one- and two-graviton Green's function are computed using a temporal gauge. In order to handle the extra poles which are present in the propagator, we employ an ambiguity-free technique in the imaginary-time formalism. For temperatures T high compared with the external momentum, we obtain the leading T^4 as well as the subleading T^2 and log(T) contributions to the graviton self-energy. The gauge fixing independence of the leading T^4 terms as well as the Ward identity relating the self-energy with the one-point function are explicitly verified. We also verify the 't Hooft identities for the subleading T^2 terms and show that the logarithmic part has the same structure as the residue of the ultraviolet pole of the zero temperature graviton self-energy. We explicitly compute the extra terms generated by the prescription poles and verify that they do not change the behavior of the leading and sub-leading contributions from the hard thermal loop region. We discuss the modification of the solutions of the dispersion relations in the graviton plasma induced by the subleading T^2 contributions. 
  We consider the low energy limit of a stack of N M-branes at finite temperature. In this limit, the M-branes are well described, via the AdS/CFT correspondence, in terms of classical solutions to the eleven dimensional supergravity equations of motion. We calculate Minkowski space two-point functions on these M-branes in the long-distance, low-frequency limit, i.e. the hydrodynamic limit, using the prescription of Son and Starinets [hep-th/0205051]. From these Green's functions for the R-currents and for components of the stress-energy tensor, we extract two kinds of diffusion constant and a viscosity. The N dependence of these physical quantities may help lead to a better understanding of M-branes. 
  We present, in explicit matrix representation and a modernity befitting the community, the classification of the finite discrete subgroups of G_2 and compute the McKay quivers arising therefrom. Of physical interest are the classes of N=1 gauge theories descending from M-theory and of mathematical interest are possible steps toward a systematic study of crepant resolutions to smooth G_2 manifolds as well as generalised McKay Correspondences. This writing is a companion monograph to hep-th/9811183 and hep-th/9905212, wherein the analogues for Calabi-Yau three- and four-folds were considered. 
  We analyze the emergence of a minimal length for a large class of generalized commutation relations, preserving commutation of the position operators and translation invariance as well as rotation invariance (in dimension higher than one). We show that the construction of the maximally localized states based on squeezed states generally fails. Rather, one must resort to a constrained variational principle. 
  We show that the Bessel equation can be cast, by means of suitable transformations, into a system of two damped/amplified parametric oscillator equations. The relation with the group contraction mechanism is analyzed and the breakdown of loop-antiloop symmetry due to group contraction manifests itself as violation of time-reversal symmetry. A preliminary discussion of the relation between some infinite dimensional loop-algebras, such as the Virasoro-like algebra, and the Euclidean algebras e(2) and e(3) is also presented. 
  In this paper I developed a classical model of elementary particle that is associated with a membrane of finite size, surrounded by non-linear electromagnetic field. The form of local interaction which lead to bounded states of finite masses, charges and spins was constructed. To do this I added Kaluza-Klein Lagrangian on the surface, which is associated with extended particle, to quadratic in field potentials term (a la tachyon mass). This ensures that Lagrangian remains an even function of the field, but spontaneous symmetry breaking leads to nontrivial soliton-like solutions. I assumed that the particle has axial symmetry and that the surface has only one degree of freedom, i.e. is a disk with the radius determined from the equations of motion. The solution of system of two non-linear partial differential equations for the field potentials was obtained numerically by different methods. Several solutions with increasing orders of leading field multipoles and disk radius were obtained, and masses, electric charges and spins were calculated. In the framework of this model the ratio of a charge square to a double spin, i.e. the fine structure constant, which do not depend on parameters of the model, was calculated. 
  Non-vanishing fluxes in M-theory and string theory compactifications induce a superpotential in the lower dimensional theory. Gukov has conjectured the explicit form of this superpotential. We check this conjecture for the heterotic string compactified on a Calabi-Yau three-fold as well as for warped M-theory compactifications on Spin(7) holonomy manifolds, by performing a Kaluza-Klein reduction. 
  We establish N=(1/8,1) supersymmetric Yang-Mills vector multiplet with generalized self-duality in Euclidian eight-dimensions with the original full SO(8) Lorentz covariance reduced to SO(7). The key ingredient is the usage of octonion structure constants made compatible with SO(7) covariance and chirality in 8D. By a simple dimensional reduction together with extra constraints, we derive N=1/8+7/8 supersymmetric self-dual vector multiplet in 7D with the full SO(7) Lorentz covariance reduced to G_2. We find that extra constraints needed on fields and supersymmetry parameter are not obtained from a simple dimensional reduction from 8D. We conjecture that other self-dual supersymmetric theories in lower dimensions D =6 and 4 with respective reduced global Lorentz covariances such as SU(3) \subset SO(6) and SU(2) \subset SO(4) can be obtained in a similar fashion. 
  A new systematic method is developed to study to what extent the symmetry requirements alone, above all the invariance under 16 supersymmetries (SUSY), determine the completely off-shell effective action $\Gamma$ of a D-particle, i.e. without imposing any restrictions on its position $r^m(\tau)$ and spin $\theta_\alpha(\tau)$. Our method consists of (i) writing down the proper closure relations for general SUSY transformations $\delta_\epsilon$ (which necessarily involves $\Gamma$ itself) together with the invariance condition $\delta_\epsilon\Gamma=0$ (ii) and solving this coupled system of functional differential equations for $\delta_\epsilon$ and $\Gamma$ simultaneously, modulo field redefinitions, in a consistent derivative expansion scheme. Our analysis is facilitated by a novel classification scheme introduced for the terms in $\Gamma$. At order 2 and 4, although no assumption is made on the underlying theory, we reproduce the effective action previously obtained at the tree and the 1-loop level in Matrix theory respectively (modulo two constants), together with the quantum-corrected SUSY transformations which close properly. This constitutes a complete unambiguous proof of off-shell non-renormalization theorems. 
  We explain the motivation and main ideas underlying our proposal for a Lagrangian for Matrix Theory based on sixteen supercharges. Starting with the pedagogical example of a bosonic matrix theory we describe the appearance of a continuum spacetime geometry from a discrete, and noncommutative, spacetime with both Lorentz and Yang-Mills invariances. We explain the appearance of large N ground states with Dbranes and elucidate the principle of matrix Dbrane democracy at finite N. Based on the underlying symmetry algebras that hold at both finite and infinite N, we show why the supersymmetric matrix Lagrangian we propose does not belong to the class of supermatrix models which includes the BFSS and IKKT Matrix Models. We end with a preliminary discussion of a path integral prescription for the Hartle-Hawking wavefunction of the Universe derived from Matrix Theory. 
  We consider the N=1 super Yang-Mills theory with gauge group U(Nc) or SU(Nc) and one adjoint Higgs field with an arbitrary polynomial superpotential. We provide a purely field theoretic derivation of the exact effective superpotential W(S) for the glueball superfield S in the confining vacua. We show that the result matches with the Dijkgraaf-Vafa matrix model proposal. The proof brings to light a deep relationship between non-renormalization theorems first discussed by Intriligator, Leigh and Seiberg, and the fact that W(S) is given by a sum over planar diagrams. 
  The symmetries of the simplest non-abelian Toda equations are discussed. The set of characteristic integrals whose Hamiltonian counterparts form a W-algebra, is presented. 
  We consider the asymmetric orbifold that is obtained by acting with T-duality on a 4-torus, together with a shift along an extra circle. The chiral algebra of the resulting theory has non-trivial outer automorphisms that act as permutations on its simple factors. These automorphisms play a crucial role for constructing D-branes that couple to the twisted sector of the orbifold. 
  We compute the finite temperature correction to the induced fermion number for fermions coupled to a static SU(2) chiral background, using the derivative expansion technique. At zero temperature the induced fermion number is topological, being the winding number of the chiral background. At finite temperature however, higher order terms in the derivative expansion give nontopological corrections to the winding number. We use this result to show that the standard cancellation of the fractional parts of the fermion number inside and outside the bag in an SU(2) x SU(2) hybrid chiral bag model of the nucleon does not occur at nonzero temperature. 
  In the cubic string field theory, using the gauge invariant operators corresponding to the on-shell closed string vertex operators, we have explicitly evaluated the decay amplitudes of two open string tachyons or gauge fields to one closed string tachyon or graviton up to level two. We then evaluated the same amplitudes in the bosonic string theory, and shown that the amplitudes in both theories have exactly the same pole structure. We have also expanded the decay amplitudes in the bosonic string theory around the Mandelstam variable s=0, and shown that their leading contact terms are fully consistent with a tachyonic Dirac-Born-Infeld action which includes both open string and closed string tachyon. 
  A nonlocal quantum gravity theory is presented which is finite and unitary to all orders of perturbation theory. Vertex form factors in Feynman diagrams involving gravitons suppress graviton and matter vacuum fluctuation loops by introducing a low-energy gravitational scale, \Lambda_{Gvac} < 2.4\times 10^{-3} eV. Gravitons coupled to non-vacuum matter loops and matter tree graphs are controlled by a vertex form factor with the energy scale, \Lambda_{GM} < 1-10 TeV. 
  Lorentz symmetry violation (LSV) can be generated at the Planck scale, or at some other fundamental length scale, and naturally preserve Lorentz symmetry as a low-energy limit (deformed Lorentz symmetry, DLS). DLS can have important implications for ultra-high energy cosmic-ray physics (see papers physics/0003080 - hereafter referred to as I -, astro-ph/0011181 and astro-ph/0011182, and references quoted in these papers). A crucial question is how DLS can be extended to a deformed Poincar\'e symmetry (DPS), and what can be the dynamical origin of this phenomenon. In a recent paper (hep-th/0208064, hereafter referred to as II), we started a discussion of proposals to identify DPS with a symmetry incorporating the Planck scale (like doubly special relativity, DSR) and suggested new ways in similar directions. Implications for models of quadratically deformed relativistic kinematics (QDRK) and linearly deformed relativistic kinematics (LDRK) were also discussed. We pursue here our study of these basic problems, focusing on the possibility to relate deformed relativistic kinematics (DRK) to new space-time dimensions and compare our QDRK model, in the form proposed since 1997, which the Kirzhnits-Chechin (KCh) and Sato-Tati (ST) models. It is pointed out that, although the KCh model does not seem to work such as it was formulated, our more recent proposals can be related to suitable extensions of this model generalizing the Finsler algebras (even to situations where a preferred physical inertial frame exists) and using the Magueijo-Smolin transformation as a technical tool. 
  A family of spherically symmetric, static and self--dual Lorentzian wormholes is obtained in n--dimensional Einstein gravity. This class of solutions includes the n--dimensional versions of the Schwarzschild black hole and the spatial--Schwarzschild traversable wormhole. Using isotropic coordinates we study the geometrical structure of the solution, and delineate the domains of the free parameters for which wormhole, naked singular geometries and the Schwarzschild black hole are obtained. It is shown that, in the lower dimensional Einstein gravity without cosmological constant, we can not have self--dual Lorentzian wormholes. 
  Non-Abelian extensions of fluid dynamics, which can have applications to the quark-gluon plasma, are given. These theories are presented in a symplectic/Lagrangian formulation and involve a fluid generalization of the Kirillov-Kostant form well known in Lie group theory. In our simplest model the fluid flows with velocity v and in presence of non-Abelian chromoelectric/magnetic E^a / B^a fields, the fluid feels a Lorentz force of the form Q_a E^a + (v / c) \times Q_a B^a, where Q_a is a space-time local non-Abelian charge satisfying a fluid Wong equation [ (D_t + v \cdot D) Q ]_a = 0 with gauge covariant derivatives. 
  In standard Poincare and anti de Sitter SO(2,3) invariant theories, antiparticles are related to negative energy solutions of covariant equations while independent positive energy unitary irreducible representations (UIRs) of the symmetry group are used for describing both a particle and its antiparticle. Such an approach cannot be applied in de Sitter SO(1,4) invariant theory. We argue that it would be more natural to require that (*) one UIR should describe a particle and its antiparticle simultaneously. This would automatically explain the existence of antiparticles and show that a particle and its antiparticle are different states of the same object. If (*) is adopted then among the above groups only the SO(1,4) one can be a candidate for constructing elementary particle theory. It is shown that UIRs of the SO(1,4) group can be interpreted in the framework of (*) and cannot be interpreted in the standard way. By quantizing such UIRs and requiring that the energy should be positive in the Poincare approximation, we conclude that i) elementary particles can be only fermions. It is also shown that ii) C invariance is not exact even in the free massive theory and iii) elementary particles cannot be neutral. This gives a natural explanation of the fact that all observed neutral states are bosons. 
  We review the string/gauge theory duality relating Chern-Simons theory and topological strings on noncompact Calabi-Yau manifolds, as well as its mathematical implications for knot invariants and enumerative geometry. 
  The one-loop low-energy effective action for non-Abelian N=4 supersymmetric Yang-Mills theory is computed to order $F^6$ by use of heat kernel techniques in N=1 superspace. At the component level, the $F^5$ terms are found to be consistent with the form of the non-Abelian Born-Infeld action computed to this order by superstring methods. The $F^6$ terms will be of importance for comparison with superstring calculations. 
  In this paper we aim to determine the baryon numbers at which the minimal energy Skyrmion has icosahedral symmetry. By comparing polyhedra which arise as minimal energy Skyrmions with the dual of polyhedra that minimize the energy of Coulomb charges on a sphere, we are led to conjecture a sequence of magic baryon numbers, B=7,17,37,67,97,... at which the minimal energy Skyrmion has icosahedral symmetry and unusually low energy. We present evidence for this conjecture by applying a simulated annealing algorithm to compute energy minimizing rational maps for all degrees upto 40. Further evidence is provided by the explicit construction of icosahedrally symmetric rational maps of degrees 37, 47, 67 and 97. To calculate these maps we introduce two new methods for computing rational maps with Platonic symmetries. 
  Dijkgraaf and Vafa have conjectured that the effective superpotentials for N=1 four-dimensional supersymmetric gauge theories can be given by the planar diagrams of matrix models. We examine some special models with cubic and quartic tree level superpotentials for adjoint chiral superfield \Phi. We consider the effective superpotentials for the classical vacuum \Phi=0 for U(N) and SO(N)/Sp(N) gauge theories. We evaluate the effective superpotentials exactly in terms of the matrix model and in terms of closed string theory on Calabi-Yau geometry with fluxes. As a result we find their perfect agreements. 
  We review various attempts to localize the discrete spectra of semirelativistic Hamiltonians of the form H = \beta \sqrt{m^2 + p^2} + V(r) (w.l.o.g. in three spatial dimensions) as entering, for instance, in the spinless Salpeter equation. Every Hamiltonian in this class of operators consists of the relativistic kinetic energy \beta \sqrt{m^2 + p^2} (where \beta > 0 allows for the possibility of more than one particles of mass m) and a spherically symmetric attractive potential V(r), r = |x|. In general, accurate eigenvalues of a nonlocal Hamiltonian operator can only be found by the use of a numerical approximation procedure. Our main emphasis, however, is on the derivation of rigorous semi-analytical expressions for both upper and lower bounds to the energy levels of such operators. We compare the bounds obtained within different approaches and present relationships existing between the bounds. 
  We derive and generalize the RR twisted tadpole cancellation conditions necessary to obtain consistent D=4, Z_N orbifold compactifications of Type IIB string theory. At least two different types of branes (or antibranes with opposite RR charges) are introduced into the construction. The matter spectra and their contribution to the non-abelian gauge anomalies are computed. Their relation with the tadpole cancellation conditions is also reviewed. The presence of tachyons is a common feature for some of the non-supersymmetric systems of branes. 
  Abelian duality on the closed three-dimensional Riemannian manifold M is discussed. Partition functions for the ordinary U(1) gauge theory and a circle-valued scalar field theory on M are explicitly calculated and compared. It is shown that the both theories are mutually dual. 
  We examine the structure of winding toroidal and open cylindrical membranes, especially in cases where they are stretched between boundaries. Non-zero winding or stretching means that there are linear terms in the mode expansion of the coordinates obeying Dirichlet boundary conditions. A linear term acts as an outer derivation on the subalgebra of volume-preserving diffeomorphisms generated by single-valued functions, and obstructs the truncation to matrix theory obtained via non-commutativity with rational parameter. As long as only one of the two membrane directions is stretched, the possible consistent truncation is to coordinates taking values in representations of an affine algebra. We show that this consistent truncation of the supermembrane gives a precise microscopic derivation of matrix string theory with the representation content appropriate for the physical situation. The matrix superstring theory describing parallel M5-branes is derived. We comment on the possible applications of the construction to membrane quantisation in certain M-theory backgrounds. 
  We compute string field theory Hamiltonian matrix elements and compare them with matrix elements of the dilatation operator in gauge theory. We get precise agreement between the string field theory and gauge theory computations once the correct cubic Hamiltonian matrix elements in string field theory and a particular basis of states in gauge theory are used. We proceed to compute the matrix elements of the dilatation operator to order g_2^2 in this same basis. This calculation makes a prediction for string field theory Hamiltonian matrix elements to order g_2^2, which have not yet been computed. However, our gauge theory results precisely match the results of the recent computation by Pearson et al. of the order g_2^2 Hamiltonian matrix elements of the string bit model. 
  We use ideas on integrability in higher dimensions to define Lorentz invariant field theories with an infinite number of local conserved currents. The models considered have a two dimensional target space. Requiring the existence of Lagrangean and the stability of static solutions singles out a class of models which have an additional conformal symmetry. That is used to explain the existence of an ansatz leading to solutions with non trivial Hopf charges. 
  In this work we derive two important tools for working in the \kappa basis of string field theory. First we give an analytical expression for the finite part of the spectral density \rho_{fin}. This expression is relevant when both matter and ghost sectors are considered. Then we calculate the form of the matter part of the Virasoro generators L_n in the \kappa basis, which construct string field theory's derivation Q_{BRST}. We find that the Virasoro generators are given by one dimensional delta functions with complex arguments. 
  We connect a possible solution for the ``cosmological constant problem'' to the existence of a (postulated) conformal fixed point in a fundamental theory. The resulting cosmology leads to quintessence, where the present acceleration of the expansion of the universe is linked to a crossover in the flow of coupling constants. 
  We review various aspects of configurations of intersecting branes, including the conditions for preservation of supersymmetry. In particular, we discuss the projection conditions on the Killing spinors for given brane configurations and the relation to calibrations. This highlights the close connection between intersecting branes and branes wrapping supersymmetric cycles as well as special holonomy manifolds. We also explain how these conditions can be used to find supergravity solutions without directly solving the Einstein equations. The description of intersecting branes is considered both in terms of the brane worldvolume theories and as supergravity solutions. There are well-known simple procedures (harmonic function rules) for writing down the supergravity solutions for supersymmetric configurations of orthogonally intersecting branes. However, such solutions involve smeared or delocalised branes. We describe several methods of constructing solutions with less smearing, including some fully localised solutions. Some applications of these supergravity solutions are also considered - in particular the study of black holes and gauge theories. 
  We study various aspects of N=(4,4) type IIA GS superstring theory in the pp-wave background, which arises as the compactification of maximally supersymmetric eleven-dimensional pp-wave geometry along the spacelike isometry direction. We show the supersymmetry algebra of N=(4,4) worldsheet supersymmetry as well as non-linearly realized supersymmetry. We also give quantization of closed string and open string incorporating various boundary conditions. From the open string boundary conditions, we find configurations of D-branes which preserve half the supersymmetries. Among these we identify D4 brane configurations with longitudinal five brane configurations in matrix model on the eleven-dimensional pp-wave geometry. 
  We compute the large scale (macroscopic) correlations in ensembles of normal random matrices with an arbitrary measure and in ensembles of general non-Hermition matrices with a class of non-Gaussian measures. In both cases the eigenvalues are complex and in the large $N$ limit they occupy a domain in the complex plane. For the case when the support of eigenvalues is a connected compact domain, we compute two-, three- and four-point connected correlation functions in the first non-vanishing order in 1/N in a manner that the algorithm of computing higher correlations becomes clear. The correlation functions are expressed through the solution of the Dirichlet boundary problem in the domain complementary to the support of eigenvalues. The two-point correlation functions are shown to be universal in the sense that they depend only on the support of eigenvalues and are expressed through the Dirichlet Green function of its complement. 
  The idea that asymptotic de Sitter space can be described by a finite Hilbert Space implies that any quantum measurement has an irreducible innacuracy. We argue that this prevents any measurement from verifying the existence of the Poincare recurrences that occur in the mathematical formulation of quantum de Sitter (dS) space. It also implies that the mathematical quantum theory of dS space is not unique. There will be many different Hamiltonians, which give the same results, within the uncertainty in all possible measurements. 
  We introduce lattice models with explicit N=2 supersymmetry. In these interacting models, the supersymmetry generators Q^+ and Q^- yield the Hamiltonian H={Q^+,Q^-} on any graph. The degrees of freedom can be described as either fermions with hard cores, or as quantum dimers. The Hamiltonian of our simplest model contains a hopping term and a repulsive potential, as well as the hard-core repulsion. We discuss these models from a variety of perspectives: using a fundamental relation with conformal field theory, via the Bethe ansatz, and using cohomology methods. The simplest model provides a manifestly-supersymmetric lattice regulator for the supersymmetric point of the massless 1+1-dimensional Thirring (Luttinger) model. We discuss the ground-state structure of this same model on more complicated graphs, including a 2-leg ladder, and discuss some generalizations. 
  Throughout John Wheeler's career, he wrestled with big issues like the fundamental length, the black hole and the unification of quantum mechanics and relativity. In this essay, I argue that solid state physics -- historically the study of silicon, semiconductors and sand grains -- can give surprisingly deep insights into the big questions of the world. 
  We extend the method of matrix partition to obtain explicitly the gauge field for noncommutative ADHM construction in some general cases. As an application of this method we apply it to the U(2) 2-instanton and get explicit result for the gauge fields in the coincident instanton limit. We also easily apply it to the noncommutative 't Hooft instantons in the appendix. 
  The properties of a noncanonical 3D Wigner quantum oscillator, whose position and momentum operators generate the Lie superalgebra sl(1|3), are further investigated. Within each state space W(p), p=1,2,..., the energy E_q, q=0,1,2,3, takes no more than 4 different values. If the oscillator is in a stationary state \psi_q \in W(p) then measurements of the non-commuting Cartesian coordinates of the particle are such that their allowed values are consistent with it being found at a finite number of sites, called ``nests''. These lie on a sphere centered on the origin of fixed, finite radius \varrho_q. The nests themselves are at the vertices of a rectangular parallelepiped. In the typical cases (p>2) the number of nests is 8 for q=0 and 3, and varies from 8 to 24, depending on the state, for q=1 and 2. The number of nests is less in the atypical cases (p=1,2), but it is never less than two. In certain states in W(2) (resp. in W(1)) the oscillator is ``polarized'' so that all the nests lie on a plane (resp. on a line). The particle cannot be localized in any one of the available nests alone since the coordinates do not commute. The probabilities of measuring particular values of the coordinates are discussed. The mean trajectories and the standard deviations of the coordinates and momenta are computed, and conclusions are drawn about uncertainty relations. The rotational invariance of the system is also discussed. 
  The presence of cosmological fluctuations influences the background cosmology in which the perturbations evolve. This back-reaction arises as a second order effect in the cosmological perturbation expansion. The effect is cumulative in the sense that all fluctuation modes contribute to the change in the background geometry, and as a consequence the back-reaction effect can be large even if the amplitude of the fluctuation spectrum is small. We review two approaches used to quantify back-reaction. In the first approach, the effect of the fluctuations on the background is expressed in terms of an effective energy-momentum tensor. We show that in the context of an inflationary background cosmology, the long wavelength contributions to the effective energy-momentum tensor take the form of a negative cosmological constant, whose absolute value increases as a function of time since the phase space of infrared modes is increasing. This then leads to the speculation that gravitational back-reaction may lead to a dynamical cancellation mechanism for a bare cosmological constant, and yield a scaling fixed point in the asymptotic future in which the remnant cosmological constant satisfies $\Omega_{\Lambda} \sim 1$. We then discuss how infrared modes effect local observables (as opposed to mathematical background quantities) and find that the leading infrared back-reaction contributions cancel in single field inflationary models. However, we expect non-trivial back-reaction of infrared modes in models with more than one matter field. 
  We describe how to reduce the fuzzy four-sphere algebra to a set of four independent raising and lowering oscillator operators. In terms of them we derive the projector valued operators for the fuzzy four-sphere, which are the global definition of k-instanton connections over this noncommutative base manifold. 
  Using the algebraic geometric approach of Berenstein et {\it al} (hep-th/005087 and hep-th/009209) and methods of toric geometry, we study non commutative (NC) orbifolds of Calabi-Yau hypersurfaces in toric varieties with discrete torsion. We first develop a new way of getting complex $d$ mirror Calabi-Yau hypersurfaces $H_{\Delta}^{\ast d}$ in toric manifolds $M_{\Delta }^{\ast (d+1)}$ with a $C^{\ast r}$ action and analyze the general group of the discrete isometries of $H_{\Delta}^{\ast d}$. Then we build a general class of $d$ complex dimension NC mirror Calabi-Yau orbifolds where the non commutativity parameters $\theta_{\mu \nu}$ are solved in terms of discrete torsion and toric geometry data of $M_{\Delta}^{(d+1)}$ in which the original Calabi-Yau hypersurfaces is embedded. Next we work out a generalization of the NC algebra for generic $d$ dimensions NC Calabi-Yau manifolds and give various representations depending on different choices of the Calabi-Yau toric geometry data. We also study fractional D-branes at orbifold points. We refine and extend the result for NC $% (T^{2}\times T^{2}\times T^{2})/(\mathbf{{Z_{2}}\times {Z_{2})}}$ to higher dimensional torii orbifolds in terms of Clifford algebra. 
  We study strings on orbifolds of $AdS_{5}\times S^5$ by SU(2) discrete groups in the Penrose limit. We derive the degenerate metrics of pp wave of $AdS_{5}\times S^5/\Gamma$ using ordinary $ADE$ and affine $\wildtilde{ADE}$ singularities of complex surfaces and results on ${\cal N}=4$ CFT$_4$'s. We also give explicit metric moduli depencies for abelian and non abelian orbifolds. 
  In hep-th/0111281 the complete set of eigenvectors and eigenvalues of Neumann matrices was found. It was shown also that the spectral density contains a divergent constant piece that being regulated by truncation at level L equals (log L)/(2\pi). In this paper we find an exact analytic expression for the finite part of the spectral density. This function allows one to calculate finite parts of various determinants arising in string field theory computations. We put our result to some consistency checks. 
  Phase Space is the framework best suited for quantizing superintegrable systems--systems with more conserved quantities than degrees of freedom. In this quantization method, the symmetry algebras of the hamiltonian invariants are preserved most naturally. We illustrate the power and simplicity of the method through new applications to nonlinear sigma-models, specifically for Chiral Models and de Sitter N-spheres, where the symmetric quantum hamiltonians amount to compact and elegant expressions, in accord with the Groenewold-van Hove theorem. Additional power and elegance is provided by the use of Nambu Brackets (linked to Dirac Brackets) involving the extra invariants of superintegrable models. The quantization of Nambu Brackets is then successfully compared to that of Moyal, validating Nambu's original proposal, while invalidating other proposals. 
  In this paper we define a causal Lorentz covariant noncommutative (NC) classical Electrodynamics. We obtain an explicit realization of the NC theory by solving perturbatively the Seiberg-Witten map. The action is polynomial in the field strenght $F$, allowing to preserve both causality and Lorentz covariance. The general structure of the Lagrangian is studied, to all orders in the perturbative expansion in the NC parameter $\theta$. We show that monochromatic plane waves are solutions of the equations of motion to all orders. An iterative method has been developed to solve the equations of motion and has been applied to the study of the corrections to the superposition law and to the Coulomb law. 
  I give a pedagogical and historical introduction to axion physics, and briefly review the present status of axions in our understanding of particle physics and cosmology. This is a contribution to Continuous Advances in QCD 2002/Arkadyfest, held in honor of Arkady Vainshtein's 60th birthday. 
  After review the quantum Hall effect on the fuzzy two-sphere $S^2$ and Zhang and Hu's 4-sphere $S^4$, the incompressible quantum Hall fluid on $S^2$, $S^4$ and torus are discussed respectively. Next, the corresponding Laughlin wavefunctions on $S^2$ are also given out. The ADHM construction on $S^4$ is discussed. We also point out that on torus, the incompressible quantum Hall fluid is related to the integrable Gaudin model and the solution can be given out by the Yang Bethe ansatz. 
  The spacetime dependent lagrangian formalism of references [1-2] is used to obtain is used to obtain a classical solution of Yang-Mills theory. This is then used to obtain an estimate of the vacuum expectation value of the Higgs field,{\it viz.} $\phi_{a}=A/e$, where $A$ is a constant and $e$ is the Yang-Mills coupling (related to the usual electric charge).The solution can also accommodate non-commuting coordinates on the boundary of the theory which may be used to construct $D$-brane actions. The formalism is also used to obtain the Deser-Gomberoff-Henneaux-Teitelboim results [10] for dyon charge quantisation in abelian $p$-form theories in dimensions $D=2(p+1)$ for both even and odd $p$. PACS: 11.15.-q,11.27.+d,11.10.Ef 
  We examine the set of objects which can be built in type IIA string theory by matrix methods using an infinite number of D0-branes. In addition to stacks of ordinary Dp-branes and branes in background fields, we find exotic states which cannot be constructed by other means. These states exhibit strongly noncommutative geometry, (e.g., partial derivatives on them do not commute) and some are conjectured to have Z_N-valued charges similar to those of the type I D-instanton. Real-valued charges are forbidden by Dirac quantization, leading to a nontrivial relationship between noncommutative topological invariants. 
  Static, spherically symmetric solutions of the Einstein--Kalb--Ramond (KR) field equations are obtained. Besides an earlier known exact solution, we also find an approximate, asymptotically flat solution for which the metric coefficients are obtained as an infinte series in $\frac{1}{r}$. Subsequently, we study gravitational lensing and perihelion precession in these spacetimes and obtain explicit formulae which include corrections to these effects in the presence of the KR field. 
  We consider the 5D gauge unification of $SU(2)_L\times U(1)_Y$ into $SU(3)_W$ at a TeV scale. Compactification of the extra dimension on an orbifold $S^1/(Z_2\times Z'_2)$ allows fixed points where $SU(2)_L\times U(1)_Y$ representations can be assigned. We explain the long proton lifetime and the top-bottom mass hierarchy and etc. geometrically. We also show that local gauge anomalies on the orbifold can be exactly cancelled by a 5D Chern-Simons term with a jumping coefficient. 
  Dimensional reduction of (super-)gravity theories to 3 dimensions results in sigma models on coset spaces G/H, such as the E_8/SO(16) coset in the bosonic sector of 3 dimensional maximal supergravity. The reverse process, oxidation, is the reconstruction of a higher dimensional gravity theory from a coset sigma model. Using the group G as starting point, the higher dimensional models follow essentially from decomposition into subgroups. All equations of motion and Bianchi identities can be directly reconstructed from the group lattice, Kaluza-Klein modifications and Chern-Simons terms are encoded in the group structure. Manipulations of extended Dynkin diagrams encode matter content, and (string) dualities. The reflection symmetry of the ``magic triangle'' for E_n gravities, and approximate reflection symmetry of the older ``magic triangle'' of supergravities in 4 dimensions, are easily understood in this framework. 
  Quantum fluctuations of some systems vanish not only in the limit $\hbar\to 0$, but also as some other parameters (such as $1\over N$, the inverse of the number of `colors' of a Yang-Mills theory) vanish. These lead to new classical limits that are often much better approximations to the quantum theory. We describe two examples: the familiar Hartree--Fock-Thomas-Fermi methods of atomic physics as well as the limit of large spatial dimension. Then we present an approach of the Hecke operators on modular forms inspired by these ideas of quantum mechanics. It explains in a simple way why the spectra of these operators tend to the spectrum of random matrices for large weight for the modular forms. 
  Hopf structure of the prototype realizations of the W(2)-algebra and also N=1 superconformal algebra are obtained using the bosonic and also fermionic Feigin-Fuchs type of free massless scalar fields in the operator product expansion (OPE) language. 
  We have analyzed the interplay between Scherk-Schwarz supersymmetry breaking and general fermion (gaugino or gravitino) mass terms localized on the fixed point branes of the S^1/Z_2 orbifold. Analytic solutions for eigenfunctions and eigenvalues are found in all cases. All results are checked by numerical calculations that make use of regularized \delta-functions. Odd and generically also even fermions are discontinuous at the brane fixed points, but in all cases the combination that couples to the brane is continuous. For CP-even brane mass terms supersymmetry restoration can take place when their effects are cancelled by those of Scherk-Schwarz compactification. However such a cancellation can not occur for CP-odd brane mass terms. 
  We review some contributions on fusion rules that were inspired by the work of Sharp, in particular, the generating-function method for tensor-product coefficients that he developed with Patera. We also review the Kac-Walton formula, the concepts of threshold level, fusion elementary couplings, fusion generating functions and fusion bases. We try to keep the presentation elementary and exemplify each concept with the simple $\su(2)_k$ case. 
  In this paper we show that the matrix model techniques developed by Dijkgraaf and Vafa can be extended to compute quantum deformed moduli spaces of vacua in four dimensional supersymmetric gauge theories. The examples studied give the moduli space of a bulk D-brane probe in geometrically engineered theories, in the presence of fractional branes at singularities. 
  We introduce a technique for restoring general coordinate invariance into theories where it is explicitly broken. This is the analog for gravity of the Callan-Coleman-Wess-Zumino formalism for gauge theories. We use this to elucidate the properties of interacting massless and massive gravitons. For a single graviton with a Planck scale Mpl and a mass mg, we find that there is a sensible effective field theory which is valid up to a high-energy cutoff Lambda parametrically above mg. Our methods allow for a transparent understanding of the many peculiarities associated with massive gravitons, among them the need for the Fierz-Pauli form of the Lagrangian, the presence or absence of the van Dam-Veltman-Zakharov discontinuity in general backgrounds, and the onset of non-linear effects and the breakdown of the effective theory at large distances from heavy sources. The natural sizes of all non-linear corrections beyond the Fierz-Pauli term are easily determined. The cutoff scales as Lambda ~ (mg^4 Mpl)^(1/5) for the Fierz-Pauli theory, but can be raised to Lambda ~ (mg^2 Mpl)^(1/3) in certain non-linear extensions. Having established that these models make sense as effective theories, there are a number of new avenues for exploration, including model building with gravity in theory space and constructing gravitational dimensions. 
  We argue that there are no axially symmetric rotating monopole solutions for a Yang-Mills-Higgs theory in flat spacetime background. We construct axially symmetric Yang-Mills-Higgs solutions in the presence of a negative cosmological constant, carrying magnetic charge $n$ and a nonvanishing electric charge. However, these solution are also nonrotating. 
  Due to the quasi-exponential red-shifting which occurs during an inflationary period in the very early Universe, wavelengths which at the present time correspond to cosmological lengths are in general sub-Planckian during the early stages of inflation. This talk discusses two approaches to addressing this issue which both indicate that the standard predictions of inflationary cosmology - made using classical general relativity and weakly coupled scalar matter field theory - are not robust against changes in the physics on trans-Planckian scales. One approach makes use of modified dispersion relations for a usual free field scalar matter theory, the other uses some properties of space-time non-commutativity - a feature expected in string theory. Thus, it is possible that cosmological observations may be used as a window to explore trans-Planckian physics. We also speculate on a novel way of obtaining inflation based on modified dispersion relations for ordinary radiation. 
  In this paper we discuss the question of whether the entropy of cosmological horizon in Topological Reissner-Nordstr\"om- de Sitter spaces can be described by the Cardy-Verlinde formula, which is supposed to be an entropy formula of conformal field theory in any dimension. Furthermore, we find that the entropy of black hole horizon can also be rewritten in terms of the Cardy-Verlinde formula for these black holes in de Sitter spaces, if we use the definition due to Abbott and Deser for conserved charges in asymptotically de Sitter spaces. Our result is in favour of the dS/CFT correspondence. 
  We present a model of interacting quantum fields, formulated in a non-perturbative manner. One of the fields is treated semi-classically, the other is the photon field. The model has an interpretation of an electromagnetic field in a fluctuating spacetime.   The model is equivalent with the quantization of electromagnetism proposed recently by Czachor. Interesting features are that standard photon theory is recovered as a limiting case, and that localized field operators for the electromagnetic field exist as unbounded operators in Hilbert space. 
  We comment on the global structure of the spacetime that results from the collision of two bubbles filled with AdS space produced from the decay of a false vacuum through quantum tunnelling. 
  We first review the construction of the supersymmetric extension of the (quantum) Calogero-Moser-Sutherland (CMS) models. We stress the remarkable fact that this extension is completely captured by the insertion of a fermionic exchange operator in the Hamiltonian: sCMS models ({\it s} for supersymmetric) are nothing but special exchange-type CMS models. Under the appropriate projection, the conserved charges can thus be formulated in terms of the standard Dunkl operators. This is illustrated in the rational case, where the explicit form of the 4N (N being the number of bosonic variables) conserved charges is presented, together with their full algebra. The existence of 2N commuting bosonic charges settles the question of the integrability of the srCMS model. We then prove its superintegrability by displaying 2N-2 extra independent charges commuting with the Hamiltonian. In the second part, we consider the supersymmetric version of the trigonometric case (stCMS model) and review the construction of its eigenfunctions, the Jack superpolynomials. This leads to closed-form expressions, as determinants of determinants involving supermonomial symmetric functions. Here we focus on the main ideas and the generic aspects of the construction: those applicable to all models whether supersymmetric or not. Finally, the possible Lie superalgebraic structure underlying the stCMS model and its eigenfunctions is briefly considered. 
  We construct the parabosonic string formalism based on the paraquantization of both the center of mass variables and the excitation modes of the string. A critical study of the different commutators of the Poincar\'{e} algebra based on the redefinition of its generators and the direct treatment using trilinear relations is done. Space-time critical dimensions $D$ as functions of the paraquantization order $Q$ are obtained. 
  We construct a mathematical framework for twisted N=2 supersymmetric topological quantum field theory on a 4-manifold. Supersymmetry in flat space is defined and the twist homomorphism is constructed, giving us a supermanifold that is the total space of an odd vector bundle over the even 4-manifold. A special category of connections on this space is defined and a decomposition into so-called component fields is proved. The twisted supersymmetric action is computed, and the structure of the action, the decomposition, and the action of a special odd vector field are all shown to have a rich geometrical structure that was partially interpred by Atiyah and Jeffrey. In short, the action is an infinite-dimensional analogue of the Euler class of the vector bundle of self-dual 2-forms over the space of connections mod gauge. This geometrical insight serves two purposes: first, it motivates the study of anti-self-dual connections, intersection theory, and the action of the group of gauge transformations, all of which appear by themselves after the twist. Secondly, it sets the stage for an eventual proof of Witten's Conjecture, relating the Donaldson and Seiberg-Witten invariants. What we build here amounts to a mathematical treatment of a physical treatment of a mathematical construction of Donaldson. 
  Based on the non-abelian effective action for D1-branes, a new action for matrix string theory in non-trivial backgrounds is proposed. Once the background fields are included, new interactions bring the possibility of non-commutative solutions i.e. The Myers effect for ``string bits'' . 
  When coordinates are noncommutative, the Hall effect is reinvestigated. The Hall conductivity is expressed with noncommutative parameters, so that in the commutative limit it tends to the conventional result. 
  In this note we explicitly compute the graviton exchange graph for scalar fields with arbitrary conformal dimension \Delta in arbitrary spacetime dimension d. This results in an analytical function in \Delta as well as in d. 
  We find, at the Lagrangian off-shell level, the explicit equivalence transformation which relates the conformal mechanics of De Alfaro, Fubini and Furlan to the conformal mechanics describing the radial motion of the charged massive particle in the Bertotti-Robinson AdS$_2\times S^2$ background. Thus we demonstrate the classical equivalence of these two systems which are usually regarded as essentially different ``old'' and ``new'' conformal mechanics models. We also construct a similar transformation for N=2, $SU(1,1|1)$ superconformal mechanics in N=2 superfield formulation. Performing this transformation in the action of the N=2 superconformal mechanics, we find an off-shell superfield action of N=2 superextension of Bertotti-Robinson particle. Such an action has not been given before. We show its on-shell equivalence to the AdS$_2$ superparticle action derived from the spontaneous partial breaking of $SU(1,1|1)$ superconformal symmetry treated as the N=2 AdS$_2$ supersymmetry. 
  We consider stationary axially symmetric black holes in SU(2) Einstein-Yang-Mills-dilaton theory. We present a mass formula for these stationary non-Abelian black holes, which also holds for Abelian black holes. The presence of the dilaton field allows for rotating black holes, which possess non-trivial electric and magnetic gauge fields, but don't carry a non-Abelian charge. 
  In this paper we consider a scenario, consisting of a de Sitter phase followed by a phase described by a scale factor $a(t)\sim t^{q}$, where $1/3<q<1$, which can be viewed as an inflationary toy model. It is argued that this scenario naively could lead to an information paradox. We propose that the phenomenon of Poincar\'{e} recurrences plays a crucial role in the resolution of the paradox. We also comment on the relevance of these results to inflation and the CMBR. 
  In hep-th/0202087 it was argued that the operator L_0 is bad defined in kappa-basis as a kernel operator. Indeed, we show that L_0 is a difference operator. We also find a representation of L_1 and L_{-1} in a class of difference operators. 
  By employing D6-branes intersecting at angles in $D = 4$ type I strings, we construct {\em five stack} string GUT models (PS-II class), that contain at low energy {\em exactly the Standard model} with no extra matter and/or extra gauge group factors. These classes of models are based on the Pati-Salam (PS) gauge group $SU(4)_C \times SU(2)_L \times SU(2)_R$. They represent deformations around the quark and lepton basic intersection number structure.   The models possess the same phenomenological characteristics of some recently discussed examples (PS-A and PS-I class) of four stack PS GUTS. Namely, there are no colour triplet couplings to mediate proton decay and proton is stable as baryon number is a gauged symmetry. Neutrinos get masses of the correct sizes. Also the mass relation $m_e = m_d$ at the GUT scale is recovered.   The conditions for the non-anomalous U(1)'s to survive massless the Green-Schwarz mechanism are equivalent, to the conditions, coming from the presence of N=1 supersymmetry, in sectors involving the presence of {\em extra} branes and also required to guarantee the existence of the Majorana mass term for the right handed neutrinos. These conditions are independent from the number of {\em extra} U(1) branes. We also discuss the relative size of the leading worldsheet instanton correction to the trilinear Yukawa couplings in a general GUT model. 
  We derive and analyze the perturbation series for the classical effective action in quantum statistical mechanics, treated as a toy model for the dimensionally reduced effective action in quantum field theory at finite temperature. The first few terms of the series are computed for the harmonic oscillator and the quartic potential. 
  We consider the properties of an ensemble of universes as function of size, where size is defined in terms of the asymptotic value of the Hubble constant (or, equivalently, the value of the cosmological constant). We assume that standard model parameters depend upon size in a manner that we have previously suggested, and provide additional motivation for that choice. Given these assumptions, it follows that universes with different sizes will have different physical properties, and we estimate, very roughly, that only if a universe has a size within a factor of a square root of two or so of our own will it support life as we know it. We discuss implications of this picture for some of the basic problems of cosmology and particle physics, as well as the difficulties this point of view creates. 
  The possibility of noncommutative topological gravity arising in the same manner as Yang-Mills theory is explored. We use the Seiberg-Witten map to construct such a theory based on a SL(2,C) complex connection, from which the Euler characteristic and the signature invariant are obtained. This gives us a way towards the description of noncommutative gravitational instantons as well as noncommutative local gravitational anomalies. 
  It is shown how in principle for non-abelian gauge theories it is possible in the finite volume hamiltonian framework to make sense of calculating the expectation value of ||A||^2=\int d^3x(A^a_i(x))^2. Gauge invariance requires one to replace ||A||^2 by its minimum over the gauge orbit, which makes it a highly non-local quantity. We comment on the difficulty of finding a gauge invariant expression for ||A||^2_{min} analogous to that found for the abelian case, and the relation of this question to Gribov copies. We deal with these issues by implementing the hamiltonian on the so-called fundamental domain, with appropriate boundary conditions in field space, essential to correctly represent the physics of the problem. 
  The ${\cal N}=1$ SUSY nonlinear sigma models in four spacetime dimensions are studied to obtain BPS walls and junctions. A nonlinear sigma model with a single chiral scalar superfield is found which has a moduli space of the topology of $S^1$ and admits BPS walls and junctions connecting arbitrary points in moduli space. New BPS junction solutions connecting $N$ discrete vacua are also found for nonlinear sigma models with several chiral scalar superfields. More detailed exposition of our results can be found in Ref.\cite{NNS}. 
  We study noncommutative open string (NCOS) theories realized in string theory with time-dependent backgrounds. Starting from a noncommutative Yang-Mills theory (NCYM) with a constant space-space noncommutativity but in a time-dependent background and making an S-dual transformation, we show that the resulting theory is an NCOS also in a time-dependent background but now with a time-dependent time-space noncommutativity and a time-dependent string scale. The corresponding dual gravity description is also given. A general SL(2,Z) transformation on the NCYM results in an NCOS with a time-dependent time-space noncommutativity and a constant space-space noncommutativity, and also in a time-dependent background. 
  The rules to construct Lagrangian formulation for $\theta$-superfield theory of fields ($\theta$-STF) are introduced and considered on the whole in the framework of new superfield quantization method for general gauge theories. Algebraic, group-theoretic and analytic description aspects for supervariables over (Grassmann) algebras containing anticommuting generating element $\theta$ and interpreted further in particular as an "odd" time are examined. Superfunction $S_{L}(\theta)$, its global symmetries are defined on the extended space $T_{odd}{\cal M}_{cl} \times \{\theta\}$ parameterized by superfields ${\cal A}^{\imath}(\theta), \frac{d{{\cal A}}^{\imath}(\theta)}{d\theta \phantom{xxx}}, \theta$. Extremality properties of superfunctional $Z[{\cal A}]=\int d\theta S_{L}(\theta)$ and ones of corresponding Euler-Lagrange equations are analyzed. The direct and inverse problems of zero locus reduction for extended (anti)symplectic manifolds over ${\cal M}_{min}$ = $\{({\cal A}^{\imath}(\theta), C^{\alpha}(\theta))\}$, with (odd) even brackets, corresponding to initial $\theta$-superfield models are employed to construct iteratively the new interconnected models both embedded into manifolds above with reduced brackets and enlarging them with continued ones. Component (on $\theta$) formulation for $\theta$-STF variables and operations is produced providing the connection with standard gauge field theory. Realization of $\theta$-STF constructions is demonstrated on models of scalar, spinor, vector superfields which are used to formulate the $\theta$-superfield model with abelian two-parametric gauge supergroup. 
  In this paper we consider a strong-weak coupling duality of the N=2 super-Liouville field theory (SLFT). Without the self-duality found in other Liouville theories, the N=2 SLFT, we claim, is associated with a `dual' action by a transformation $b\to 1/b$ where $b$ is the coupling constant. To justify our conjecture, we compute the reflection amplitudes (or two-point functions) of the (NS) and the (R) operators of the N=2 SLFT based on the conjectured dual action and show that the results are consistent with known results. 
  We investigate compactifications with duality twists and their relation to orbifolds and compactifications with fluxes. Inequivalent compactifications are classified by conjugacy classes of the U-duality group and result in gauged supergravities in lower dimensions with nontrivial Scherk-Schwarz potentials on the moduli space. For certain twists, this mechanism is equivalent to introducing internal fluxes but is more general and can be used to stabilize some of the moduli. We show that the potential has stable minima with zero energy precisely at the fixed points of the twist group. In string theory, when the twist belongs to the T-duality group, the theory at the minimum has an exact CFT description as an orbifold. We also discuss more general twists by nonperturbative U-duality transformations. 
  We assume that QCD can be effectively described with string-like variables. The hadronic string is built over the chirally non-invariant QCD vacuum by means of the boundary interaction with background chiral fields associated with pions. By making this interaction compatible with the conformal symmetry of the string and with the unitarity constraint on chiral fields we reconstruct the equations of motion for the latter ones and furthermore recover the Lagrangian of non-linear sigma model of pion interactions. The estimated chiral structural constants of Gasser and Leutwyler fit well the phenomenological values. 
  We study the impact of non-Abelian T-duality transformations on a string based cosmological model. The implementation of the pre-big-bang scenario is investigated. We found a region of the dual phase where such a picture is possible. 
  After a brief review on Matrix String Theory on flat backgrounds, we formulate matrix string models on different pp-wave backgrounds. This will be done both in the case of constant and variable RR background flux for certain exact string geometries. We exhibit the non--perturbative representation of string interaction and show how the eigenvalue tunneling drives the WKB expansion to give the usual perturbative string interaction also in supersymmetric pp-wave background cases. 
  We investigate a model of two-dimensional gravity with arbitrary scalar potential obtained by gauging a deformation of de Sitter or more general algebras, which accounts for the existence of an invariant energy scale. We obtain explicit solutions of the field equations and discuss their properties. 
  For semisimple groups, possibly multiplied by U(1)'s, the number of Yang-Mills gauge fields is equal to the number of generators of the group. In this paper, it is shown that, for non-semisimple groups, the number of Yang-Mills fields can be larger. These additional Yang-Mills fields are not irrelevant because they appear in the gauge transformations of the original Yang-Mills fields. Such non-semisimple Yang-Mills theories may lead to physical consequences worth studying. The non-semisimple group with only two generators that do not commute is studied in detail. 
  We analyze and compare two families of topologies that have been proposed for representation spaces of chiral algebras by Huang and Gaberdiel & Goddard respectively. We show, in particular, that for suitable pairs the topology of Gaberdiel & Goddard is coarser. We also give a new proof that the chiral two-point blocks are continuous in the topology of Huang. 
  We construct representation of the Separated Variables (SoV) for the quantum SL(2,R) Heisenberg closed spin chain and obtain the integral representation for the eigenfunctions of the model. We calculate explicitly the Sklyanin measure defining the scalar product in the SoV representation and demonstrate that the language of Feynman diagrams is extremely useful in establishing various properties of the model. The kernel of the unitary transformation to the SoV representation is described by the same "pyramid diagram" as appeared before in the SoV representation for the SL(2,C) spin magnet. We argue that this kernel is given by the product of the Baxter Q-operators projected onto a special reference state. 
  This thesis is devoted to studying two important aspects of braneworld physics: their cosmology and their holography. We examine the Einstein equations induced on a general $(n-2)$-brane of arbitrary tension, embedded in some $n$-dimensional bulk. The brane energy-momentum tensor enters these equations both linearly and quadratically. From the point of view of a homogeneous and isotropic brane we see quadratic deviations from the FRW equations of the standard cosmology. There is also a contribution from a bulk Weyl tensor. We study this in detail when the bulk is AdS-Schwarzschild or Reissner-Nordstr\"om AdS. This contribution can be understood holographically. For the AdS-Schwarzschild case, we show that the geometry on a brane near the AdS boundary is just that of a radiation dominated FRW universe. The radiation comes from a field theory that is dual to the AdS bulk. We also develop a new approach which allows us to consider branes that are not near the AdS boundary. This time the dual field theory contributes quadratic energy density/pressure terms to the FRW equations. Remarkably, these take exactly the same form as for additional matter placed on the brane by hand, with no bulk Weyl tensor.   We also derive the general equations of motion for a braneworld containing a domain wall. For the critical brane, the induced geometry is identical to that of a vacuum domain wall in $(n-1)$-dimensional Einstein gravity. We develop the tools to construct a nested Randall-Sundrum scenario whereby we have a ``critical'' domain wall living on an anti-de Sitter brane. We also show how to construct instantons on the brane, and calculate the probability of false vacuum decay. 
  We show that the $E-S \sim \log S$ behaviour found for long strings rotating on $AdS_5\times S^5$ may be reproduced by membranes rotating on $AdS_4\times S^7$ and on a warped $AdS_5$ M-theory solution. We go on to obtain rotating membrane configurations with the same $E-K \sim \log K$ relation on $G_2$ holonomy backgrounds that are dual to ${\mathcal{N}}=1$ gauge theories in four dimensions. We study membrane configurations on $G_2$ holonomy backgrounds systematically, finding various other Energy-Charge relations. We end with some comments about strings rotating on warped backgrounds. 
  We construct intersecting D5-brane orbifold models that yield the (non-supersymmetric) standard model up to vector-like matter and charged-singlet scalars.  The models are constrained by the requirement that twisted tadpoles cancel, and that the gauge boson coupled to the weak hypercharge $U(1)_Y$ does not get a string-scale mass via a generalised Green-Schwarz mechanism. Gauge coupling constant ratios close to those measured are easily obtained for reasonable values of the parameters, consistently with having the string scale close to the electroweak scale, as required to avoid the hierarchy problem. 
  As a non-trivial check of the non-supersymmetric gauge/gravity duality, we use a near-extremal black brane background to compute the retarded Green's functions of the stress-energy tensor in N=4 super-Yang-Mills (SYM) theory at finite temperature. For the long-distance, low-frequency modes of the diagonal components of the stress-energy tensor, hydrodynamics predicts the existence of a pole in the correlators corresponding to propagation of sound waves in the N=4 SYM plasma. The retarded Green's functions obtained from gravity do indeed exhibit this pole, with the correct values for the sound speed and the rate of attenuation. 
  A precise form of the time evolution of rolling tachyons corresponding to a brane-antibrane pair is investigated by solving the Hamiltonian equations of motion under the assumption that in a region far away from branes the tachyon vacuum is almost already achieved, even at the beginning of the rolling. 
  This contribution discusses the recent progress in research of consistent supersymmetric interactions of AdS(5) higher spin gauge fields. 
  We give a non-technical outline of a program to study the (2,0) theories in six space-time dimensions. Away from the origin of their moduli space, these theories describe the interactions of tensor multiplets and self-dual spinning strings. We argue that if the ratio between the square of the energy of a process and the string tension is taken to be small, it should be possible to study the dynamics of such a system perturbatively in this parameter. As a first step in this direction, we perform a classical computation of the amplitude for scattering chiral tensor and scalar fields (i.e. the bosonic part of a tensor multiplet) against a self-dual spinnless string. 
  An overview of some of the developments in string theory over the past two years is given, focusing on four topics: realistic (standard model like) models from string theory, geometric engineering and theories with fluxes, the gauge theory-gravity correspondence, and time dependent backgrounds and string theory. Plenary talk at ICHEP'02, Amsterdam, July 24-31, 2002. 
  We propose nonlinear integral equations to describe the groundstate energy of the fractional supersymmetric sine-Gordon models. The equations encompass the N=1 supersymmetric sine-Gordon model as well as the phi_(id,id,adj) perturbation of the SU(2)_L x SU(2)_K /SU(2)_(L+K) models at rational level K. A second set of equations are proposed for the groundstate energy of the N=2 supersymmetric sine-Gordon model. 
  In a recent preprint, Brouder and Schmitt give a careful construction of a `renormalisation' Hopf algebra out of an arbitrary bialgebra. In this note, we point out that this is a special case of the construction of the cooperad of a bialgebra (Berger-Moerdijk) combined with the construction of a bialgebra from a cooperad (Frabetti-Van der Laan). 
  I describe the main features of new intersecting D4- and D5-brane orbifold models that yield the non-supersymmetric standard model, up to vector-like matter and, in some cases, extra U(1) factors in the gauge group. There are six-stack D4-brane models that have charged-singlet scalar tachyons and which either contain all of the Yukawa couplings to the tachyonic Higgs doublets that are needed to generate mass terms for the fermions at renormalisable level or possess an unwanted extra U(1) gauge symmetry after spontaneous symmetry breaking. In the D5-brane models a minimum of eight stacks is needed. 
  We show how the bosonic sector of the radion supermultiplet plus d=4, N=1 supergravity emerge from a consistent braneworld Kaluza-Klein reduction of D=5 M--theory. The radion and its associated pseudoscalar form an SL(2,R)/U(1) nonlinear sigma model. This braneworld system admits its own brane solution in the form of a 2-supercharge supersymmetric string. Requiring this to be free of singularities leads to an SL(2,Z) identification of the sigma model target space. The resulting radion mode has a minimum length; we suggest that this could be used to avoid the occurrence of singularities in brane-brane collisions. We discuss possible supersymmetric potentials for the radion supermultiplet and their relation to cosmological models such as the cyclic universe or hybrid inflation. 
  We study the spectrum of solvable string models on plane waves descending from non-conformal Dp-brane geometries. We mainly focus on S-dual F1/D1-waves in type IIB and type I/heterotic 10D superstrings. We derive the Kaluza-Klein spectrum of N=1,2 10D supergravities on D1/F1-waves. We compute helicity supertraces counting multiplicities and R-charges of string excitations in the plane wave geometry. The results are compared against the expectations coming from gauge/supergravity descriptions. In the type I case, the Klein, Annulus and Moebius one-loop amplitudes are computed for ten-dimensional D1-waves. We test the consistency of the open string descendant by showing that after modular transformations to the closed string channel, the three amplitudes combine themselves to reconstruct a complete square (|B>+|C>)^2. Tadpole conditions are also discussed. 
  We show the existence of classical solutions of a system of D3-branes oriented at an arbitrary angle with respect to each other, in a six dimensional pp-wave background obtained from AdS_3\times S^3\times R^4, with NS-NS and R-R 3-form field strength. These D-brane bound states are shown to preserve 1/16 of the supersymmetries. We also present more D-brane bound state solutions by applying T-duality symmetries. Finally, the probe analysis is discussed along with a brief outline of the open string construction. 
  The worldsheet renormalization group approach to tachyon condensation in string theory is reviewed. The open string case is summarized with examples, and closed string tachyon condensation on nonsupersymmetric orbifolds is examined in detail. The idea that the renormalization group dissipates the localized states associated to the defect is explored. 
  We construct lattices with alternating kinks and anti-kinks. The lattice is shown to be stable in certain models. We consider the forces between kinks and antikinks and find that the lattice dynamics is that of a Toda lattice. Such lattices are exotic metastable states in which the system can get trapped during a phase transition. 
  It has been proposed that the successful inflationary description of density perturbations on cosmological scales is sensitive to the details of physics at extremely high (trans-Planckian) energies. We test this proposal by examining how inflationary predictions depend on higher-energy scales within a simple model where the higher-energy physics is well understood. We find the best of all possible worlds: inflationary predictions are robust against the vast majority of high-energy effects, but can be sensitive to some effects in certain circumstances, in a way which does not violate ordinary notions of decoupling. This implies both that the comparison of inflationary predictions with CMB data is meaningful, and that it is also worth searching for small deviations from the standard results in the hopes of learning about very high energies. 
  We argue that pp-wave backgrounds can not admit event horizons. We also comment on pp-wave generalizations which would admit horizons and show that there exists a black string solution which asymptotes to a five dimensional plane wave background. 
  An exotic scenario of our universe is proposed in which our universe starts from zero space-time dimensions (0d), namely, a set of discrete points, it increases the (continuous) dimensionality during the cooling down of it, and finally arrives at the present four space-time dimensions (4d). This scenario may be called asymptotic disappearance (or discretization) of space-time (ADST) scenario. The final stage of the scenario is to generate dynamically 4d from the 3d, which is shown to be possible for both gauge theory and gravity, including the standard model. To examine the validity of the scenario, 4d QED generated from its 3d version at high energy is studied, in which one spatial dimension is discretized with a lattice constant $a$. From the LEP2 experiment on $e^+ e^- \to \gamma \gamma$, $a$ is constrained to satisfy $a \le 461$ GeV. Expected bound on $a$ at future $e^+ e^-$ linear collider is discussed. The finite size of $a$ modifies the dispersion relation, causing the violation of Lorentz symmetry. Then, the paradox of observing 20 TeV $\gamma$ rays from the active galaxy Markarian 501 may disappear, overcoming the GZK cut. 
  We study the direct interaction between global and local monopoles. While in two previous papers, the coupling between the two sectors was only indirect through the coupling to gravity, we here introduce a new term in the potential that couples the Goldstone field and the Higgs field directly. We investigate the influence of this term in curved space and compare it to the results obtained previously. 
  Discretized nonabelian gauge theories living on finite group spaces G are defined by means of a geometric action \int Tr F\wedge *F . This technique is extended to obtain a discrete version of the Born-Infeld action. 
  We analyze perturbative aspects of gauged matrix models, including those where classically the gauge symmetry is partially broken. Ghost fields play a crucial role in the Feynman rules for these vacua. We use this formalism to elucidate the fact that non-perturbative aspects of N=1 gauge theories can be computed systematically using perturbative techniques of matrix models, even if we do not possess an exact solution for the matrix model. As examples we show how the Seiberg-Witten solution for N=2 gauge theory, the Montonen-Olive modular invariance for N=1*, and the superpotential for the Leigh-Strassler deformation of N=4 can be systematically computed in perturbation theory of the matrix model/gauge theory (even though in some of these cases the exact answer can also be obtained by summing up planar diagrams of matrix models). 
  We investigate an exactly marginal N=1 supersymmetric deformation of SU(N) N=4 supersymmetric Yang-Mills theory discovered by Leigh and Strassler. We use a matrix model to compute the exact superpotential for a further massive deformation of the U(N) Leigh-Strassler theory. We then show how the exact superpotential and eigenvalue spectrum for the SU(N) theory follows by a process of integrating-in. We find that different vacua are related by an action of the SL(2,Z) modular group on the bare couplings of the theory extending the action of electric-magnetic duality away from the N=4 theory. We perform non-trivial tests of the matrix model results against semiclassical field theory analysis. We also show that there are interesting points in parameter space where condensates can diverge and vacua disappear. Based on the matrix model results, we propose an exact elliptic superpotential to describe the theory compactified on a circle of finite radius. 
  We analyze the relation between the short-distance behavior of quantum field theory and the strong-field limit of the background field formalism, for QED effective Lagrangians in self-dual backgrounds, at both one and two loop. The self-duality of the background leads to zero modes in the case of spinor QED, and these zero modes must be taken into account before comparing the perturbative beta function coefficients and the coefficients of the strong-field limit of the effective Lagrangian. At one-loop this is familiar from instanton physics, but we find that at two-loop the role of the zero modes, and the interplay between IR and UV effects in the renormalization, is quite different. Our analysis is motivated in part by the remarkable simplicity of the two-loop QED effective Lagrangians for a self-dual constant background, and we also present here a new independent derivation of these two-loop results. 
  Using the covariant N=2 harmonic supergraph techniques we calculate the one-loop low-energy effective action of N=4 super-Yang-Mills theory in Coulomb branch with gauge group SU(2) spontaneously broken down to U(1). The full dependence of the low-energy effective action on both the hypermultiplet and gauge fields is determined. The direct quantum calculation confirms the correctness of the exact N=4 SYM low-energy effective action derived in hep-th/0111062 on the purely algebraic ground by invoking a hidden N=2 supersymmetry which completes the manifest N=2 one to N=4. Our results provide an exhaustive solution to the problem of finding out the exact completely N=4 supersymmetric low-energy effective action for the theory under consideration. 
  In the 1920's, Madelung noticed that if the complex Schroedinger wavefunction is expressed in polar form, then its modulus squared and the gradient of its phase may be interpreted as the hydrodynamic density and velocity, respectively, of a compressible fluid. In this paper, we generalize Madelung's transformation to the quaternionic Schroedinger equation. The non-abelian nature of the full SU(2) gauge group of this equation leads to a richer, more intricate set of fluid equations than those arising from complex quantum mechanics. We begin by describing the quaternionic version of Madelung's transformation, and identifying its ``hydrodynamic'' variables. In order to find Hamiltonian equations of motion for these, we first develop the canonical Poisson bracket and Hamiltonian for the quaternionic Schroedinger equation, and then apply Madelung's transformation to derive non-canonical Poisson brackets yielding the desired equations of motion. These are a particularly natural set of equations for a non-abelian fluid, and differ from those obtained by Bistrovic et al. only by a global gauge transformation. Because we have obtained these equations by a transformation of the quaternionic Schroedinger equation, and because many techniques for simulating complex quantum mechanics generalize straightforwardly to the quaternionic case, our observation leads to simple algorithms for the computer simulation of non-abelian fluids. 
  The issue of justifying the matrix-theory proposal is revisited. We first discuss how the matrix-string theory is derived directly starting from the eleven dimensional supermembrane wrapped around a circle of radius $R=g_s\ell_s$, without invoking any stringy assumptions, such as S- and T-dualities. This derivation provides us a basis for studying both string ($R\to 0)$- and M ($R\to \infty$)-theory limits of quantum membrane theory in a single unified framework. In particular, we show that two different boosts of supermembrane, namely one of unwrapped membrane along the M-theory circle and the other of membrane wrapped about a transervse direction which is orthogonal to the M-theory circle, give the same matrix theory in the 11 dimensional limit, $R\to \infty$ (with $N\to \infty$). We also discuss briefly the nature of possible covariantized matrix (string) theories. 
  We consider a pure U(1) quantum gauge field theory on a general Riemannian compact four manifold. We compute the partition function with Abelian Wilson loop insertions. We find its duality covariance properties and derive topological selection rules. Finally, we show that, to have manifest duality, one must assume the existence of twisted topological sectors besides the standard untwisted one. 
  We study the decay of a very massive closed superstring (i.e. \alpha' M^2>> 1) in the unique state of maximum angular momentum. This is done in flat ten-dimensional spacetime and in the regime of weak string coupling, where the dominant decay channel is into two states of masses M_1, M_2. We find that the lifetime surprisingly grows with the first power of the mass M: T =c \alpha' M. We also compute the decay rate for each values of M_1, M_2. We find that, for large M, the dynamics selects only special channels of decay: modulo processes which are exponentially suppressed, for every decay into a state of given mass M_1, the mass M_2 of the other state is uniquely determined. 
  We construct the cubic interaction vertex and dynamically generated supercharges in light-cone superstring field theory in the pp-wave background. We show that these satisfy the pp-wave superalgebra at first order in string coupling. The cubic interaction vertex and dynamical supercharges presented here differ from the expressions previously given in the literature. Using this vertex we compute various string theory three-point functions and comment on their relation to gauge theory in the BMN limit. 
  The effects of higher order gravity terms on a dilatonic brane world model are discussed. For a single positive tension flat 3-brane, and one infinite extra dimension, we present a particular class of solutions with finite 4-dimensional Planck scale and no naked singularities. A `self-tuning' mechanism for relaxing the cosmological constant on the brane, without a drastic fine tuning of parameters, is discussed in this context. 
  We compute the density of open strings stretching between AdS2 branes in the Euclidean AdS3. This is done by solving the factorization constraint of a degenerate boundary field, and the result is checked by a Cardy-type computation. We mention applications to branes in the Minkowskian AdS3 and its cigar coset. 
  In this PhD thesis the low energy effective actions for D-branes and the M5-brane are used to analyze the dynamics of brane probes moving in the geometry created by diverse supergravity backgrounds. It begins with an introduction to string theory. Brane effective actions with local kappa symmetry are also reviewed. We then study the embedding of D(8-p)-branes in the background geometry of parallel Dp-branes. A BPS condition is found and the analytical solutions describe embeddings which represent branes joined by tubes, which are bundles of fundamental strings. We continue with the analysis of branes partially wrapped on spheres in the near-horizon geometry of branes and bound states. There is a finite number of static configurations as a consequence of a quantization rule for the worldvolume gauge field. These configurations can be interpretated as bound states and this is checked directly in a particular case by using the Myers polarization mechanism. Finally we find configurations of brane probes which behave as massless particles in the geometry of non-threshold bound states. They can be interpreted as gravitons blown up into a fuzzy sphere and sharing some directions with the background. 
  We propose a cosmological braneworld scenario in which two branes collide and emerge as reborn branes whose tensions have signs opposite to the original tensions of the respective branes. In this scenario, gravity on each of the branes is described by a scalar-tensor-type theory in which the radion plays the role of the gravitational scalar, and the branes are assumed to be inflating. However, the whole dynamics is different from those of the usual inflation, due to the non-trivial dynamics of the radion field. Transforming the conformal frame to the Einstein frame, this born-again scenario resembles the pre-big-bang scenario. Thus, our scenario has features of both inflation and pre-big-bang scenarios. In particular, gravitational waves produced from vacuum fluctuations may have a very blue spectrum, while the inflaton field gives rise to a standard scale-invariant spectrum. 
  We calculate the free energy, energy and entropy in the matrix quantum mechanical formulation of 2D string theory in a background strongly perturbed by tachyons with the imaginary Minkowskian momentum $\pm i/R$ (``Sine-Liouville'' theory). The system shows a thermodynamical behaviour corresponding to the temperature $T=1/(2\pi R)$. We show that the microscopically calculated energy of the system satisfies the usual thermodynamical relations and leads to a non-zero entropy. 
  We consider the brane universe in the bulk background of the topological Reissner-Nordstr\"om de Sitter black holes. We show that the thermodynamic quantities (including entropy) of the dual CFT take usual special forms expressed in terms of Hubble parameter and its time derivative at the moment, when the brane crosses the black hole horizon or the cosmological horizon. We obtain the generalized Cardy-Verlinde formula for the CFT with an charge and cosmological constant, for any values of the curvature parameter $k$ in the Friedmann equations. 
  We consider the noncommutativity parameter of the space-time as a bosonic worldsheet field. By finding a fermionic super-partner for it, we can find star products between the boson-boson, boson-fermion and fermion-fermion fields of superstring worldsheet and also between superfields of the worldsheet superspace. We find a two dimensional action for the noncommutativity parameter and its fermionic partner. We discuss the symmetries of this action. 
  We study the potential induced by imaginary self-dual 3-forms in compactifications of string theory and the cosmological evolution associated with it. The potential contains exponentials of the volume moduli of the compactification, and we demonstrate that the exponential form of the potential leads to a power law for the scale factor of the universe. This power law does not support accelerated expansion. We explain this result in terms of supersymmetry and comment on corrections to the potential that could lead to inflation or quintessence. 
  In this paper, we paraquantize the spinning string theory in the Neuveu-Shwarz model. Both the center of mass variables and the excitation modes of the string verify paracommutation relations. Except the $[p^{\mu},p^{\nu}]$ commutator, the two other commutators of Poincar\'e algebra are satisfied. With the sole use of trilinear relations we find existence possibilities of spinning strings at space-time dimensions other than D=10. 
  We investigate a \Pi-shape Wilson loop in N=4 super Yang--Mills theory, which lies partially at the light-cone, and consider an associated open superstring in AdS_5 x S^5. We discuss how this Wilson loop determines the anomalous dimensions of conformal operators with large Lorentz spin and present an explicit calculation in perturbation theory to order \lambda. We find the minimal surface in the supergravity approximation, that reproduces the Gubser, Klebanov and Polyakov prediction for the anomalous dimensions at large \lambda=g_YM^2 N, and discuss its quantum-mechanical interpretation. 
  We construct the holographic type nonlocal effective action in two-brane Randall-Sundrum model and show that it describes a phase transition between the local and nonlocal phases of the theory -- a cumulative effect of the tower of massive Kaluza-Klein modes. We show that the corresponding renormalization group flow interpolating between the limits of short and long interbrane separations can be dynamically mediated by a repulsive interbrane potential that gives rise to braneworld cosmological scenarios with diverging branes. 
  In this note we consider the homogeneous and isotropic cosmology in the finite-range gravity theory recently proposed by Babak and Grishchuk. In this scenario the universe undergoes late time accelerated expansion if both the massive gravitons present in the model are tachyons. We carry out the phase space analysis of the system and show that the late-time acceleration is an attractor of the model. 
  We show that noncommuting electric fields occur naturally in $\theta$-expanded noncommutative gauge theories. Using this noncommutativity, which is field dependent, and a hamiltonian generalisation of the Seiberg-Witten Map, the algebraic consistency in the lagrangian and hamiltonian formulations of these theories, is established. A comparison of results in different descriptions shows that this generalised map acts as canonical transformation in the physical subspace only. Finally, we apply the hamiltonian formulation to derive the gauge symmetries of the action. 
  The existence of bi-Hamiltonian structures for the rational Harmonic Oscillator (non-central harmonic oscillator with rational ratio of frequencies) is analyzed by making use of the geometric theory of symmetries. We prove that these additional structures are a consequence of the existence of dynamical symmetries of non-symplectic (non-canonical) type. The associated recursion operators are also obtained. 
  We study the behavior of matrix string theory in the strong coupling region, where it is expected to reduce to discrete light-cone type IIA superstring. In the large $N$ limit, the reduction corresponds to the double-dimensional reduction from wrapped supermembranes on $R^{10}\times S^1$ to type IIA superstrings on $R^{10}$ in the light-cone gauge, which is shown classically, however it is not obvious quantum mechanically. We analyze the problem in matrix string theory by using the strong coupling ($1/g$) expansion. We find that the quantum corrections do not cancel out at $\mathcal{O}(1/g^2)$. Detailed calculations can be seen in Ref.\cite{UY}. 
  We investigate the supersymmetric D-brane configurations in the pp-wave backgrounds proposed by Maldacena and Maoz. We study the surviving supersymmetry in a D-brane configuration from the worldvolume point of view. When we restrict ourselves to the background with N=(2,2) supersymmetry and no holomorphic Killing vector term, there are two types of supersymmetric D-branes: A-type and B-type. An A-type brane is wrapped on a special Lagrangian submanifold, and the imaginary part of the superpotential should be constant on its worldvolume. On the other hand, a B-type brane is wrapped on a complex submanifold, and the superpotential should be constant on its worldvolume. The results are almost consistent with the worldsheet theory in the lightcone gauge. The inclusion of gauge fields is also discussed and found BPS D-branes with the gauge field excitations. Furthermore, we consider the backgrounds with holomorphic Killing vector terms and N=(1,1) supersymmetric backgrounds. 
  We perform a systematic study of the Standard Model embedding in a D-brane configuration of type I string theory at the TeV scale. We end up with an attractive model and we study several phenomenological questions, such as gauge coupling unification, proton stability, fermion masses and neutrino oscillations. At the string scale, the gauge group is U(3)_color x U(2)_weak x U(1)_1 x U(1)_bulk. The corresponding gauge bosons are localized on three collections of branes; two of them describe the strong and weak interactions, while the last abelian factor lives on a brane which is extended in two large extra dimensions with a size of afew microns. The hypercharge is a linear combination of the first three U(1)s. All remaining U(1)s get masses at the TeV scale due to anomalies, leaving the baryon and lepton numbers as (perturbatively) unbroken global symmetries at low energies. The conservation of baryon number assures proton stability, while lepton number symmetry guarantees light neutrino masses that involve a right-handed neutrino in the bulk. The model predicts the value of the weak angle which is compatible with the experiment when the string scale is in the TeV region. It also contains two Higgs doublets that provide tree-level masses to all fermions of the heaviest generation, with calculable Yukawa couplings; one obtains a naturally heavy top and the correct ratio m_b/m_tau. We also study neutrino masses and mixings in relation to recent solar and atmospheric neutrino data. 
  We construct creation and annihilation operators for harmonic oscillators with minimal length uncertainty relations. We discuss a possible generalization to a large class of deformations of cannonical commutation relations. We also discuss dynamical symmetry of noncommutative harmonic oscillator. 
  We determine the Killing spinors for a class of magnetic brane solutions with Minkowski worldvolume of the theory of AdS Einstein Maxwell theories in d dimensions. We also obtain curved magnetic brane solutions with Ricci-flat worldvolumes. If we demand that the curved brane solution admits Killing spinors, then its worldvolume must admit parallel spinors. Classes of Ricci-flat worldvolumes admitting parallel spinors are discussed. 
  We study the scaling exponents for the expanding isotropic flat cosmological models. The dimension of space, the equation of state of the cosmic fluid and the scaling exponent for a physical variable are related by the Euler Beta function that controls the singular behavior of the global integrals. We encounter dual cosmological scenarios using the properties of the Beta function. For the entropy density integral we reproduce the Fischler-Susskind holographic bound. 
  We construct the classical W-algebras for some non-abelian Toda systems associated with the Lie groups GL(2n,R) and Sp(n,R). We start with the set of characteristic integrals and find the Poisson brackets for the corresponding Hamiltonian counterparts. The convenient block matrix representation for the Toda equations is used. The infinitesimal symmetry transformations generated by the elements of the W-algebras are presented. 
  We show that the partially topological twisted N=16, D=2 super Yang-Mill theory gives rise to a N_T=8 Hodge-type cohomological gauge theory with global SU(4) symmetry. 
  We study pairs of planar D-branes intersecting on null hypersurfaces, and other related configurations. These are supersymmetric and have finite energy density. They provide open-string analogues of the parabolic orbifold and null-fluxbrane backgrounds for closed superstrings. We derive the spectrum of open strings, showing in particular that if the D-branes are shifted in a spectator dimension so that they do not intersect, the open strings joining them have no asymptotic states. As a result, a single non-BPS excitation can in this case catalyze a condensation of massless modes, changing significantly the underlying supersymmetric vacuum state. We argue that a similar phenomenon can modify the null cosmological singularity of the time-dependent orbifolds. This is a stringy mechanism, distinct from black-hole formation and other strong gravitational instabilities, and one that should dominate at weak string coupling. A by-product of our analysis is a new understanding of the appearance of 1/4 BPS threshold bound states, at special points in the moduli space of toroidally-compactified type-II string theory. 
  We use the boundary state formalism for the bosonic string to calculate the emission amplitude for closed string states from particular D-branes. We show that the amplitudes obtained are exactly the same as those obtained from the world-sheet sigma model calculation, and that the construction enforces the requirement for integrated vertex operators, even in the off-shell case. Using the expressions obtained for the boundary state we propose higher order terms in the string loop expansion for the background considered. 
  We introduce a notion of $Q$-algebra that can be considered as a generalization of the notion of $Q$-manifold (a supermanifold equipped with an odd vector field obeying $\{Q,Q\} =0$). We develop the theory of connections on modules over $Q$-algebras and prove a general duality theorem for gauge theories on such modules. This theorem containing as a simplest case $SO(d,d,{\bf Z})$-duality of gauge theories on noncommutative tori can be applied also in more complicated situations. We show that $Q$-algebras appear naturally in Fedosov construction of formal deformation of commutative algebras of functions and that similar $Q$-algebras can be constructed also in the case when the deformation parameter is not formal. 
  We discuss the holographic principle in a radiation-dominated, closed Friedmann-Robertson-Walker (FRW) universe with a positive cosmological constant. By introducing a cosmological D-bound on the entropy of matter in the universe, we can write the Friedmann equation governing the evolution of the universe in the form of the Cardy formula. When the cosmological D-bound is saturated, the Friedmann equation coincides with the Cardy-Verlinde formula describing the entropy of radiation in the universe. As a concrete model, we consider a brane universe in the background of Schwarzschild-de Sitter black holes. It is found that the cosmological D-bound is saturated when the brane crosses the black hole horizon of the background. At that moment, the Friedmann equation coincides with the Cardy-Verlinde formula describing the entropy of radiation matter on the brane. 
  In this paper, I explain how gauge symmetry can be broken in a geometric way, \`{a} la Kaluza-Klein. In higher dimensional gravitational theories, one usually considers the extra dimensions to be ``frozen'' in time. However, the internal manifold is actually a dynamic entity. For example, its metric can change even if one expects its topological properties to be invariant. It is conceivable then, that at an earlier epoch the internal manifold made a geometric transition from say a maximally symmetric metric space to a less symmetric one. We know in a Kaluza-Klein reduction scheme, the massless gauge bosons are associated with the Killing vectors of the internal manifold. After the transition of the internal manifold, the gauge bosons associated with the broken Killing isometries will pick up a mass thereby breaking the gauge invariance partially. In this paper, I explore this idea, work out the mass of broken gauge bosons for some simple examples, and also point out how a mechanism similar to that of Higgs may be at work. 
  The static potential, corresponding to the interaction of two heavy sources is computed for $\mathcal{N}=4$ Super Yang Mills in the strong 't Hooft coupling regime by using the AdS/CFT conjecture and performing a computation of a rectangular Wilson loop at a finite distance of the boundary. 
  We construct open cosmic string solutions in Schwarzschild black hole and non-dilatonic black p-brane backgrounds. These strings can be thought to stretch between two D-branes or between a D-brane and the horizon in curved space-time. We study small fluctuations around these solutions and discuss their basic properties. 
  Properties of the rotating Kerr-Newman black hole solution allow to relate it with spinning particles. Singularity of black hole (BH) can be regularized by a metric deformation. In this case, as a consequence of the Einstein equations, a material source appears in the form of a relativistically rotating superconducting disk which replaces the former singular region. We show a relation of the BH regularization with confinement formation. By regularization, a phase transition occurs near the core of a charged black hole solution: from external electrovacuum to an internal superconducting state of matter. We discuss two models of such a kind, which demonstrate the appearance of a baglike structure and a mechanism of confinement based on dual Dirac's electrodynamics. First one is an approximate solution based on a supersymmetric charged domain wall, and second is an exact solution based on nonlinear electrodynamics. 
  We consider a relativistic superalgebra in the picture in which the time and spatial derivative cannot be presented in the operators of the particle. The supersymmetry generators as well as the Hamilton operators for the massive relativistic particles with spin zero and spin-1/2 are expressed in terms of the principal series of the unitary representations of the Lorentz group. We also consider the massless case. New Hamilton operators are conctructed for the massless particles with spin zero and spin 1/2. 
  The consistent, local, smooth deformations of the dual formulation of linearized gravity involving a tensor field in the exotic representation of the Lorentz group with Young symmetry type (D-3,1) (one column of length D-3 and one column of length 1) are systematically investigated. The rigidity of the Abelian gauge algebra is first established. We next prove a no-go theorem for interactions involving at most two derivatives of the fields. 
  We study five dimensional brane physics induced by an O(2) texture formed in one extra dimension. The model contains two 3-branes of nonzero tension, and the extra dimension is compact. The symmetry-breaking scale of the texture controls the particle hierarchy between the two branes. The TeV-scale particles are confined to the negative-tension brane where the observer sees gravity as essentially four dimensional. The effect of massive Kaluza-Klein gravitons is suppressed. 
  We consider the thermodynamic properties of $(d+1)$-dimensional spacetimes with NUT charges. Such spacetimes are asymptotically locally anti de Sitter (or flat), with non-trivial topology in their spatial sections, and can have fixed point sets of the Euclidean time symmetry that are either $(d-1)$-dimensional (called "bolts") or of lower dimensionality (pure "NUTs"). We compute the free energy, conserved mass, and entropy for 4, 6, 8 and 10 dimensions for each, using both Noether charge methods and the AdS/CFT-inspired counterterm approach. We then generalize these results to arbitrary dimensionality. We find in $4k+2$ dimensions that there are no regions in parameter space in the pure NUT case for which the entropy and specific heat are both positive, and so all such spacetimes are thermodynamically unstable. For the pure NUT case in $4k$ dimensions a region of stability exists in parameter space that decreases in size with increasing dimensionality. All bolt cases have some region of parameter space for which thermodynamic stability can be realized. 
  We discuss the restrictions imposed by the Konishi anomaly on the matrix model approach to the calculation of the effective superpotentials in N=1 SUSY gauge theories with different matter content. It is shown that they correspond to the anomaly deformed Virasoro $L_0$ constraints . 
  We consider the ground state energy of a spinor field in the background of a square well shaped magnetic flux tube. We use the zeta- function regularization and express the ground state energy as an integral involving the Jost function of a two dimensional scattering problem. We perform the renormalization by subtracting the contributions from first several heat kernel coefficients. The ground state energy is presented as a convergent expression suited for numerical evaluation. We discuss corresponding numerical calculations. Using the uniform asymptotic expansion of the special functions entering the Jost function we are able to calculate higher order heat kernel coefficients. 
  In the present note the expansion of the wave function of a massless particle (with the definite value of its helicity) over the untary irreducible representaions of the Lorentz group (defined on the light cone) is used as for the analog of the Fourier transformation for deriving of an equation in the relativistic configuration representation. 
  The structure and dynamics of an n-particle system are described with coupled nonlinear Heisenberg's commutator equations where the nonlinear terms are generated by the two-body interaction that excites the reference vacuum via particle-particle and particle-hole excitations. Nonperturbative solutions of the system are obtained with the use of dynamic linearization approximation and cluster transformation coefficients. The dynamic linearization approximation converts the commutator chain into an eigenvalue problem. The cluster coefficients factorize the matrix elements of the (n)-particles or particle-hole systems in terms of the matrix elements of the (n-1)-systems coupled to a particle-particle, particle-hole, and hole-hole boson. Group properties of the particle-particle, particle-hole, and hole-hole permutation groups simplify the calculation of these coefficients. The particle-particle vacuum-excitations generate superconductive diagrams in the dynamics of 3-quarks systems. Applications of the model to fermionic and bosonic systems are discussed. 
  We consider a model of brane world gravity in the context of non-conformal non-SUSY matter. In particular we modify the earlier strong coupling solution to the glueball spectrum in an $AdS^7$ Black Hole by introducing a Randall-Sundrum Planck brane as a UV cut-off. The consequence is a new normalizable zero mass tensor state, which gives rise to an effective Einstein-Hilbert theory of gravity, with exponentially small corrections set by the mass gap to the discrete glueball spectrum. However the simplest microscopic theory for the Planck brane is found to have a tachyonic instability in the radion mode. 
  Exact calculations are given for the Casimir energy for various fields in $R\times S^3$ geometry. The Green's function method naturally gives a result in a form convenient in the high-temperature limit, while the statistical mechanical approach gives a form appropriate for low temperatures. The equivalence of these two representations is demonstrated. Some discrepancies with previous work are noted. In no case, even for ${\cal N}=4$ SUSY, is the ratio of entropy to energy found to be bounded. 
  The dual superconductor picture of the QCD vacuum is thought to describe various aspects of the strong interaction including confinement. Ordinary superconductivity is described by the Ginzburg-Landau (GL) equation. In the present work we show that it is possible to arrive at a GL-like equation from pure SU(2) gauge theory. This is accomplished by using Abelian projection to split the SU(2) gauge fields into an Abelian subgroup and its coset. The two gauge field components of the coset part act as the effective, complex, scalar field of the GL equation. The Abelian part of the SU(2) gauge field is then analogous to the electromagnetic potential in the GL equation. An important aspect of the dual superconducting model is for the GL Lagrangian to have a spontaneous symmetry breaking potential, and the existence of Nielsen-Olesen flux tube solutions. Both of these require a tachyonic mass for the effective scalar field. Such a tachyonic mass term is obtained from the condensation of ghost fields. 
  We discuss the different possibilities of constructing the various energy-momentum tensors for noncommutative gauge field models. We use Jackiw's method in order to get symmetric and gauge invariant stress tensors--at least for commutative gauge field theories. The noncommutative counterparts are analyzed with the same methods. The issues for the noncommutative cases are worked out. 
  We study quark confinement in a system of two parallel domain walls interpolating different color dielectric media. We use the phenomenological approach in which the confinement of quarks appears considering the QCD vacuum as a color dielectric medium. We explore this phenomenon in QCD_2, where the confinement of the color flux between the domain walls manifests, in a scenario where two 0-branes (representing external quark and antiquark) are connected by a QCD string. We obtain solutions of the equations of motion via first-order differential equations. We find a new color confining potential that increases monotonically with the distance between the domain walls. 
  We apply the well-known Scherk-Schwarz supersymmetry breaking mechanism in an open string context. We construct a new Z_3\times Z_3^\prime model, containing only D9-branes, and rederive from a more geometric perspective the known Z_6^\prime\times Z_2^\prime model, containing D9, D5 and \bar D 5 branes. We show recent results about the study of quantum instability of these models. 
  We extend and test the method of Dijkgraaf and Vafa for computing the superpotential of N=1 theories to include flavors in the fundamental representation of the gauge group. This amounts to computing the contribution to the superpotential from surfaces with one boundary in the matrix integral. We compute exactly the effective superpotential for the case of gauge group U(N_c), N_f massive flavor chiral multiplets in the fundamental and one massive chiral multiplet in the adjoint, together with a Yukawa coupling. We compare up to sixth-order with the result obtained by standard field theory techniques in the already non trivial case of N_c=2 and N_f=1. The agreement is perfect. 
  An overview is presented of some cosmological aspects of string theory. Recent developments are emphasised, especially the attempts to derive inflation or alternatives to inflation from the dynamics of branes in string theory. Time dependent backgrounds with potential cosmological implications, such as those provided by negative tension branes and S-branes and the rolling string tachyon are also discussed. 
  (2+1)-dimensional Georgi-Glashow model, else called the Polyakov model, is explored at nonzero temperatures and in the regime when the Higgs boson is not infinitely heavy. The finiteness of the Higgs-boson mass leads to the appearance of the upper bound on the parameter of the weak-coupling approximation, necessary to maintain the stochasticity of the Higgs vacuum. The modification of the finite-temperature behavior of the model emerging due to the introduction of massless quarks is also discussed. 
  Within the context of traditional logarithmic grand unification at M_GUT = 10^(16) GeV, we show that it is nevertheless possible to observe certain GUT states such as X and Y gauge bosons at lower scales, perhaps even in the TeV range. We refer to such states as ``GUT precursors.'' Such states offer an interesting alternative possibility for new physics at the TeV scale, even when the scale of gauge coupling unification remains high, and suggest that it may be possible to probe GUT physics directly even within the context of high-scale gauge coupling unification. More generally, our results also suggest that it is possible to construct self-consistent ``hybrid'' models containing widely separated energy scales, and give rise to a Kaluza-Klein realization of non-trivial fixed points in higher-dimensional gauge theories. We also discuss how such theories may be deconstructed at high energies. 
  From the covariant bound on the entropy of partial light-sheets, we derive a version of Bekenstein's bound: S/M \leq pi x/hbar, where S, M, and x are the entropy, total mass, and width of any isolated, weakly gravitating system. Because x can be measured along any spatial direction, the bound becomes unexpectedly tight in thin systems. Our result completes the identification of older entropy bounds as special cases of the covariant bound. Thus, light-sheets exhibit a connection between information and geometry far more general, but in no respect weaker, than that initially revealed by black hole thermodynamics. 
  We study topological D-branes of type B in N=2 Landau-Ginzburg models, focusing on the case where all vacua have a mass gap. In general, tree-level topological string theory in the presence of topological D-branes is described mathematically in terms of a triangulated category. For example, it has been argued that B-branes for an N=2 sigma-model with a Calabi-Yau target space are described by the derived category of coherent sheaves on this space. M. Kontsevich previously proposed a candidate category for B-branes in N=2 Landau-Ginzburg models, and our computations confirm this proposal. We also give a heuristic physical derivation of the proposal. Assuming its validity, we can completely describe the category of B-branes in an arbitrary massive Landau-Ginzburg model in terms of modules over a Clifford algebra. Assuming in addition Homological Mirror Symmetry, our results enable one to compute the Fukaya category for a large class of Fano varieties. We also provide a (somewhat trivial) counter-example to the hypothesis that given a closed string background there is a unique set of D-branes consistent with it. 
  The Dirac operator for a manifold Q, and its chirality operator when Q is even dimensional, have a central role in noncommutative geometry. We systematically develop the theory of this operator when Q=G/H, where G and H are compact connected Lie groups and G is simple. An elementary discussion of the differential geometric and bundle theoretic aspects of G/H, including its projective modules and complex, Kaehler and Riemannian structures, is presented for this purpose. An attractive feature of our approach is that it transparently shows obstructions to spin- and spin_c-structures. When a manifold is spin_c and not spin, U(1) gauge fields have to be introduced in a particular way to define spinors. Likewise, for manifolds like SU(3)/SO(3), which are not even spin_c, we show that SU(2) and higher rank gauge fields have to be introduced to define spinors. This result has potential consequences for string theories if such manifolds occur as D-branes. The spectra and eigenstates of the Dirac operator on spheres S^n=SO(n+1)/SO(n), invariant under SO(n+1), are explicitly found. Aspects of our work overlap with the earlier research of Cahen et al.. 
  We present analytic monopole and dyon solutions whose energy is fixed by the electroweak scale. Our result shows that genuine electroweak monopole and dyon could exist whose mass scale is much smaller than the grand unification scale. 
  We present finite energy analytic monopole and dyon solutions whose size is fixed by the electroweak scale. We discuss two types of solutions. The first type is obtained by regularizing the recent solutions of Cho and Maison by modifying the coupling strength of the quartic self-interaction of $W$-boson in Weinberg-Salam model. The second is obtained by enlarging the gauge group $SU(2) \times U(1)$ to $SU(2) \times SU(2)$. Our result demonstrates that one could actually construct genuine electroweak monopole and dyon whose mass scale is much smaller than the grand unification scale, with a minor modification of the electroweak interaction without compromizing the underlying gauge invariance. 
  We discuss the holography and entropy bounds in Gauss-Bonnet gravity theory. By applying a Geroch process to an arbitrary spherically symmetric black hole, we show that the Bekenstein entropy bound always keeps its form as $S_{\rm B}=2\pi E R$, independent of gravity theories. As a result, the Bekenstein-Verlinde bound also remains unchanged. Along the Verlinde's approach, we obtain the Bekenstein-Hawking bound and Hubble bound, which are different from those in Einstein gravity. Furthermore, we note that when $HR=1$, the three cosmological entropy bounds become identical as in the case of Einstein gravity. But, the Friedmann equation in Gauss-Bonnet gravity can no longer be cast to the form of cosmological Cardy formula. 
  We study integrable realizations of conformal twisted boundary conditions for sl(2) unitary minimal models on a torus. These conformal field theories are realized as the continuum scaling limit of critical G = A,D,E lattice models with positive spectral parameter u > 0 and Coxeter number g. Integrable seams are constructed by fusing blocks of elementary local face weights. The usual A-type fusions are labelled by the Kac labels (r, s) and are associated with the Verlinde fusion algebra. We introduce a new type of fusion in the two braid limits u->\pm i\infty associated with the graph fusion algebra, and labelled by nodes $a,b\in G$ respectively. When combined with automorphisms, they lead to general integrable seams labelled by $x=(r,a,b,\kappa) \in (A_{g-2}, H, H, Z_2 )$ where H is the graph G itself for Type I theories and its parent for Type II theories. Identifying our construction labels with the conformal labels of Petkova and Zuber, we find that the integrable seams are in one-to-one correspondence with the conformal seams. The distinct seams are thus associated with the nodes of the Ocneanu quantum graph. The quantum symmetries and twisted partition functions are checked numerically for $|G| \leq 6$. We also show, in the case of $D_{2l}$, that the non-commutativity of the Ocneanu algebra of seams arises because the automorphisms do not commute with the fusions. 
  In this paper we compute the charge group for symmetry preserving D-branes on group manifolds for all simple, simply-connected, connected compact Lie groups G. 
  We explore the physical consequences of a recently discovered class of exact solutions to five dimensional Kaluza-Klein theory. We find a number of surprising features including: (1) In the presence of a Kaluza-Klein bubble, there are arbitrarily large black holes with topology S^3. (2) In the presence of a black hole or a black string, there are expanding bubbles (with de Sitter geometry) which never reach null infinity. (3) A bubble can hold two black holes of arbitrary size in static equilibrium. In particular, two large black holes can be close together without merging to form a single black hole. 
  The development of the notion of group contraction first introduced by E. In{\"o}n{\"u} and E.P. Wigner in 1953 is briefly reviewed. The fundamental role of the idea of degenerate transformations is stressed. The significance of contractions of algebraic structures for exactly solvable problems of mathematical physics is noticed. 
  Liouville field theory on an unoriented surface is investigated, in particular, the one point function on a RP^2 is calculated. The constraint of the one point function is obtained by using the crossing symmetry of the two point function. There are many solutions of the constraint and we can choose one of them by considering the modular bootstrap. 
  The spherical D2-brane solution is obtained without RR external background. The solution is shown to preserve $(1/4)$ supersymmetries. The configurations obtained depend on the integration constant $R_0$. For $R_0 \neq 0$ the shape of the solution is a deformed sphere. When, however, $R_0 = 0$, the D2-brane system seems to exhibit a brane-anti-brane configuration. 
  Since any fermionic operator \psi can be written as \psi=q+ip, where q and p are hermitian operators, we use the eigenvalues of q and p to construct a functional formalism for calculating matrix elements that involve fermionic fields. The formalism is similar to that for bosonic fields and does not involve Grassmann numbers. This makes possible to perform numerical fermionic lattice computations that are much faster than not only other algorithms for fermions, but also algorithms for bosons. 
  The presence of fields with negative mass-squared typically leads to some form of instability in standard field theories. The observation that, at least in the light-cone gauge, strings propagating in plane wave spacetimes can have worldsheet scalars with such tachyon-like masses suggests that the supergravity background may itself be unstable. To address this issue, we perform a perturbative analysis around the type IIB vacuum plane wave, the solution which most obviously generates worldsheet scalars with negative mass-squared. We argue that this background is perturbatively stable. 
  We describe various aspects of plane wave backgrounds. In particular, we make explicit a simple criterion for singularity by establishing a relation between Brinkmann metric entries and diffeomorphism-invariant curvature information. We also address the stability of plane wave backgrounds by analyzing the fluctuations of generic scalar modes. We focus our attention on cases where after fixing the light-cone gauge the resulting world sheet fields appear to have negative "mass terms". We nevertheless argue that these backgrounds may be stable. 
  The Skyrme model is a classical field theory which models the strong interaction between atomic nuclei. It has to be quantized in order to compare it to nuclear physics. When the Skyrme model is semi-classically quantized it is important to take the Finkelstein-Rubinstein constraints into account. The aim of this paper is to show how to calculate these FR constraints directly from the rational map ansatz using basic homotopy theory. We then apply this construction in order to quantize the Skyrme model in the simplest approximation, the zero mode quantization. This is carried out for up to 22 nucleons, and the results are compared to experiment. 
  We introduce a class of higher-order derivative models in (2,1) space-time dimensions. The models are described by a vector field, and contain a Proca-like mass term which prevents gauge invariance. We use the gauge embedding procedure to generate another class of higher-order derivative models, gauge-invariant and dual to the former class. We show that the results are valid in arbitrary (d,1) space-time dimensions when one discards the Chern-Simons and Chern-Simons-like terms. We also investigate duality at the quantum level, and we show that it is preserved in the quantum scenario. Other results include investigations concerning the gauge embedding approach when the vector field couples with fermionic matter, and when one adds nonlinearity. 
  We consider the Dirac equation in the magnetic-solenoid field (the field of a solenoid and a collinear uniform magnetic field). For the case of Aharonov-Bohm solenoid, we construct self-adjoint extensions of the Dirac Hamiltonian using von Neumann's theory of deficiency indices. We find self-adjoint extensions of the Dirac Hamiltonian in both above dimensions and boundary conditions at the AB solenoid. Besides, for the first time, solutions of the Dirac equation in the magnetic-solenoid field with a finite radius solenoid were found. We study the structure of these solutions and their dependence on the behavior of the magnetic field inside the solenoid. Then we exploit the latter solutions to specify boundary conditions for the magnetic-solenoid field with Aharonov-Bohm solenoid. 
  We discuss the cosmological implications of some recent advances in understanding the dynamics of tachyon condensation in string theory. 
  There is an approach due to Bazhanov and Reshetikhin for solving integrable RSOS models which consists of solving the functional relations which result from the truncation of the fusion hierarchy. We demonstrate that this is also an effective means of solving integrable vertex models. Indeed, we use this method to recover the known Bethe Ansatz solutions of both the closed and open XXZ quantum spin chains with U(1) symmetry. Moreover, since this method does not rely on the existence of a pseudovacuum state, we also use this method to solve a special case of the open XXZ chain with nondiagonal boundary terms. 
  We give a geometrical derivation of the Dirac equation by considering a spin-1/2 particle travelling with the speed of light in a cubic spacetime lattice. The mass of the particle acts to flip the multi-component wavefunction at the lattice sites. Starting with a difference equation for the case of one spatial and one time dimensions, we generalize the approach to higher dimensions. Interactions with external electromagnetic and gravitational fields are also considered. One logical interpretation is that only at the lattice sites is the spin-1/2 particle aware of its mass and the presence of external fields. 
  Following recent advances in the local theory of current-algebraic orbifolds we present the basic dynamics - including the {\it twisted KZ equations} - of each twisted sector of all outer-automorphic WZW orbifolds on so(2n). Physics- friendly Cartesian bases are used throughout, and we are able in particular to assemble two $Z_3$ triality orbifolds and three $S_3$ triality orbifolds on so(8). 
  We show that Liouville-von Neumann approach to quantum mechanical systems, which demands the existence of invariant operators, reproduces the time-dependent variational Gaussian approximation. We find the effective action of the time-dependent systems and show that many aspects of the dynamics are independent of the details of time evolution, e.g., the squeezing of the wave-function is determined by the effective potential of the final stage of time-evolution. 
  The survey summarizes briefly the results obtained recently in the Casimir effect studies considering the following subjects: i) account of the material characteristics of the media and their influence on the vacuum energy (for example, dilute dielectric ball); ii) application of the spectral geometry methods for investigating the vacuum energy of quantized fields with the goal to gain some insight, specifically, in the geometrical origin of the divergences that enter the vacuum energy and to develop the relevant renormalization procedure; iii) universal method for calculating the high temperature dependence of the Casimir energy in terms of heat kernel coefficients. 
  We investigate whether a recently proposed modulation of the power spectrum of primordial density fluctuations generated through transplankian (maybe stringy) effects during inflation can be observed. We briefly review the mechanism leading to the modulation and apply it to a generic slow-roll scenario of inflation. We then investigate how these primordial modulation effects leave an imprint in the cosmic microwave background radiation. Our conclusions are that for favourable parameter values already the presently flying MAP satellite will have a chance to detect such transplanckian oscillations in the pattern of temperature fluctuations on the sky, and that the upcoming Planck satellite will either detect them or put stringent limits related to the mass scale where the new effects appear. 
  The existence of a SL(2,R) symmetry is discussed in SU(N) Yang-Mills in the maximal Abelian Gauge. This symmetry, also present in the Landau and Curci-Ferrari gauge, ensures the absence of tachyons in the maximal Abelian gauge. In all these gauges, SL(2,R) turns out to be dynamically broken by ghost condensates. 
  The physics of angular momentum in even space dimensions can be surprisingly counter-intuitive. Three such suprises, all associated with the properties of supersymmetric rotating objects, are examined: (i) 5D black holes, (ii) Dyonic instantons and (iii) Supertubes. 
  We study matrix models related via the correspondence of Dijkgraaf and Vafa to supersymmetric gauge theories with matter in the fundamental. As in flavorless examples, measure factors of the matrix integral reproduce information about R-symmetry violation in the field theory. The models, studied previously as models of open strings, exhibit a large-M phase transition as the number of flavors is varied. This is the matrix model's manifestation of the end of asymptotic freedom. Using the relation to a quiver gauge theory, we extract the effective glueball superpotential and Seiberg-Witten curve from the matrix model. 
  We are investigating the properties of vacuum and boundary states in the CFT of free bosons under the conformal transformation. We show that transformed vacuum (boundary state) is given in terms of tau-functions of dispersionless KP (Toda) hierarchies. Applications of this approach to string field theory is considered. We recognize in Neumann coefficients the matrix of second derivatives of tau-function of dispersionless KP and identify surface states with the conformally transformed vacuum of free field theory. 
  We systematically investigate open strings in the plane wave background of type IIB string theory. We carefully analyze possible boundary conditions for open strings and find static as well as time-dependent branes. The branes fall into equivalence classes depending on whether they are related by the action of target space isometries. In particular static branes localized at the origin of transverse space and certain time-dependent branes fall into the same equivalence class. We analyze thoroughly the symmetries of all branes we discuss. Apart from symmetries descending from target space isometries, the worldsheet action being free admits a countably infinite number of other global worldsheet symmetries. We find that one can use such worldsheet symmetries to restore seemingly broken target space symmetries. In particular, we show that D-branes localized at arbitrary constant positions which were thought to be 1/4 supersymmetric in fact have sixteen supercharges whilst D-branes which were thought to be non-supersymmetric have eight supercharges. We discuss in detail the quantization in all cases. 
  We develop efficient algorithms for level-truncation computations in open bosonic string field theory. We determine the classical action in the universal subspace to level (18,54) and apply this knowledge to numerical evaluations of the tachyon condensate string field. We obtain two main sets of results. First, we directly compute the solutions up to level L=18 by extremizing the level-truncated action. Second, we obtain predictions for the solutions for L > 18 from an extrapolation to higher levels of the functional form of the tachyon effective action. We find that the energy of the stable vacuum overshoots -1 (in units of the brane tension) at L=14, reaches a minimum E_min = -1.00063 at L ~ 28 and approaches with spectacular accuracy the predicted answer of -1 as L -> infinity. Our data are entirely consistent with the recent perturbative analysis of Taylor and strongly support the idea that level-truncation is a convergent approximation scheme. We also check systematically that our numerical solution, which obeys the Siegel gauge condition, actually satisfies the full gauge-invariant equations of motion. Finally we investigate the presence of analytic patterns in the coefficients of the tachyon string field, which we are able to reliably estimate in the L -> infinity limit. 
  In Einstein gravity there is a simple procedure to build D-dimensional spacetimes starting from (D-1)-dimensional ones, by stacking any (D-1)-dimensional Ricci-flat metric into the extra-dimension. We analyze this procedure in the context of Einstein-Gauss-Bonnet gravity, and find that it can only be applied to metrics with a constant Krestschmann scalar. For instance, we show that solutions of the black-string type are not allowed in this framework. 
  First, we present a simple confining abelian pure gauge theory. Classically, its kinetic term is not positive definite, and it contains a simple UV regularized F^4 interaction. This provoques the formation of a condensate ~ F^2 such that, at the saddle point of the effective potential, the wave function normalization constant of the abelian gauge fields vanishes exactly. Then we study SU(2) pure Yang-Mills theory in an abelian gauge and introduce an additional auxiliary field for a BRST invariant condensate of dimension 2, which renders the charged sector massive. Under simple assumptions its effective low energy theory reduces to the confining abelian model discussed before, and the vev of rho is seen to scale correctly with the renormalization point. Under these assumptions, the confinement condition Z_eff = 0 also holds for the massive charged sector, which suppresses the couplings of the charged fields to the abelian gauge bosons in the infrared regime. 
  A first-quantized string (and membrane) theory is developed here by using a general wave function of the string (and membrane), analogously to the first-quantized quantum theory of a point particle. From the general wave function of the string (and membrane), the properties of the string (and membrane) such as its relation to Bosons, Fermions and spacetime are investigated. The string and membrane wave functions are found to be very useful and we can deduce Klein-Gordon equation, Dirac equation and the fundamental property of the spacetime from this new starting point. 
  We consider the N=2 gauge theory on N D7-branes wrapping K3, with D3-brane probes. In the large N limit, the D7-branes blow up to form an enhancon shell. We probe the region inside and outside the enhancon shell using the D3-branes, and compute the probe metric using the Seiberg-Witten formalism. Supergravity arguments suggest a flat interior up to 1/N corrections, and indeed our results for the D3-brane probes are consistent with that. By including the dynamics of the branes, these results, together with those of hep-th/0204050, demonstrate the robustness of the enhancon mechanism beyond patching together of supergravity solutions with D-brane source junction conditions. 
  Using N=1 superspace techniques in four dimensions we show how to perturbatively compute the superpotential generated for the glueball superfield upon integrating out massive charged fields. The technique applies to arbitrary gauge groups and representations. Moreover we show that for U(N) gauge theories admitting a large N expansion the computation dramatically simplifies and we prove the validity of the recently proposed recipe for computation of this quantity in terms of planar diagrams of matrix integrals. 
  We discuss analytic continuation from d-dimensional Lorentzian de Sitter (dS$_d$) to d-dimensional Lorentzian anti-de Sitter (AdS$_{d}$) spacetime. We show that AdS$_{d}$, with opposite signature of the metric, can be obtained as analytic continuation of a portion of dS$_d$. This implies that the dynamics of (positive square-mass) scalar particles in AdS$_{d}$ can be obtained from the dynamics of tachyons in dS$_d$. We discuss this correspondence both at the level of the solution of the field equations and of the Green functions. The AdS/CFT duality is obtained as analytic continuation of the dS/CFT duality. 
  We present an orbifold compactification of the minimal seven dimensional supergravity. The vacuum is a slice of AdS_7 where six-branes of opposite tension are located at the orbifold fixed points. The cancellation of gauge and gravitational anomalies restricts the gauge group and matter content on the boundaries. In addition anomaly cancellation fixes the boundary gauge couplings in terms of the gravitational constant, and the mass parameter of the Chern-Simons term. 
  We show how the worldvolume realization of the Hanany-Witten effect for a supersymmetric D5-brane in a D3 background also provides a classical realization of the `s-rule' exclusion principle. Despite the supersymmetry, the force on the D5-brane vanishes only in the D5 `ground state', which is shown to interpolate between 6-dimensional Minkowski space and an $OSp(4^*|4)$-invariant $adS_2\times S^4$ geometry. The M-theory analogue of these results is briefly discussed. 
  We compute the short distance expansion of fields or operators that live in the coadjoint representation of an infinite dimensional Lie algebra by using only properties of the adjoint representation and its dual. We explicitly compute the short distance expansion for the duals of the Virasoro algebra, affine Lie Algebras and the geometrically realized N-extended supersymmetric GR Virasoro algebra. 
  The realization that forthcoming experimental studies, such as the ones planned for the GLAST space telescope, will be sensitive to Planck-scale deviations from Lorentz symmetry has increased interest in noncommutative spacetimes in which this type of effects is expected. We focus here on $\kappa$-Minkowski spacetime, a much-studied example of Lie-algebra noncommutative spacetime, but our analysis appears to be applicable to a more general class of noncommutative spacetimes. A technical controversy which has significant implications for experimental testability is the one concerning the $\kappa$-Minkowski relation between group velocity and momentum. A large majority of studies adopted the relation $v = dE(p)/dp$, where $E(p)$ is the $\kappa$-Minkowski dispersion relation, but recently some authors advocated alternative formulas. While in these previous studies the relation between group velocity and momentum was introduced through ad hoc formulas, we rely on a direct analysis of wave propagation in $\kappa$-Minkowski. Our results lead conclusively to the relation $v = dE(p)/dp$. We also show that the previous proposals of alternative velocity/momentum relations implicitly relied on an inconsistent implementation of functional calculus on $\kappa$-Minkowski and/or on an inconsistent description of spacetime translations. 
  We calculate the holographic conformal anomaly and brane Newton potential when bulk is 5d AdS BH. It is shown that such anomaly is the same as in the case of pure AdS or (asymptotically) dS bulk spaces, i.e. it is (bulk) metric independent one. While Newton potential on the static brane in AdS BH is different from the one in pure AdS space, the gravity trapping still occurs for two branes system. This indicates to metric independence of gravity localization. 
  An exact expression for the quasinormal modes of scalar perturbations on a massless topological black hole in four and higher dimensions is presented. The massive scalar field is nonminimally coupled to the curvature, and the horizon geometry is assumed to have a negative constant curvature. 
  We consider the evolution of massive scalar fields in (asymptotically) de Sitter spacetimes of arbitrary dimension. Through the proposed dS/CFT correspondence, our analysis points to the existence of new nonlocal dualities for the Euclidean conformal field theory. A massless conformally coupled scalar field provides an example where the analysis is easily explicitly extended to 'tall' background spacetimes. 
  We construct a group field theory which realizes the sum of gravity amplitudes over all three dimensional topologies trough a perturbative expansion. We prove this theory to be uniquely Borel summable. This shows how to define a non-perturbative summation over triangulations including all topologies in the context of three dimensional discrete gravity. 
  We derive the scalar potential of the effective theory of type  IIB orientifold with 3-form fluxes turned on in presence of non abelian brane coordinates. N=4 supergravity predicts a positive semidefinite potential with vanishing cosmological constant in the vacuum of commuting coordinates, with a classical moduli space given by three radial moduli and three RR scalars which complete three copies of the coset (U(1,1+n)/U(1)\otimes U(1+n)), together with 6n D3-branes coordinates, n being the rank of the gauge group G. Implications for the super Higgs mechanism are also discussed. 
  Using recently developed numerical methods, we examine neutral compactified non-uniform black strings which connect to the Gregory-Laflamme critical point. By studying the geometry of the horizon we give evidence that this branch of solutions may connect to the black hole solutions, as conjectured by Kol. We find the geometry of the topology changing solution is likely to be nakedly singular at the point where the horizon radius is zero. We show that these solutions can all be expressed in the coordinate system discussed by Harmark and Obers. 
  We study brane embeddings in M-theory plane-waves and their supersymmetry. The relation with branes in AdS backgrounds via the Penrose limit is also explored. Longitudinal planar branes are originated from AdS branes while giant gravitons of AdS spaces become spherical branes which are realized as fuzzy spheres in the massive matrix theory. 
  The generalised calibration for a wrapped membrane is gauge equivalent to the supergravity three-form under which the membrane is electrically charged. Given the relevant calibration, one can go a long way towards constructing the supergravity solution for the wrapped brane. Applications of this method have been restricted since generalised calibrations have not yet been completely classified in spacetimes with non-vanishing flux. In this paper, we take a first step towards such a classification by studying membranes wrapping holomorphic curves. Supersymmetry preservation imposes a constraint on the Hermitean metric in the embedding space and it is found that this can be expressed as a restriction on possible generalised calibrations. Allowed calibrations in a particular spacetime are simply those which satisfy the constraint equation relevant to that background; in particular, we see that the previously considered Kahler calibrations are just a subclass of possible solutions. 
  A general class of solutions of string background equations is studied and its physical interpretations are presented. These solutions correspond to generalizations of the standard black p-brane solutions to surfaces with curvature k=-1,0. The relation with the recently introduced S-branes is provided. The mass, charge, entropy and Hawking temperature are computed, illustrating the interpretation in terms of negative tension branes. Their cosmological interpretation is discussed as well as their potential instability under small perturbations. 
  Implications of N=4 superconformal symmetry on Berenstein-Maldacena-Nastase (BMN) operators with two charge defects are studied both at finite charge J and in the BMN limit. We find that all of these belong to a single long supermultiplet explaining a recently discovered degeneracy of anomalous dimensions on the sphere and torus. The lowest dimensional component is an operator of naive dimension J+2 transforming in the [0,J,0] representation of SU(4). We thus find that the BMN operators are large J generalisations of the Konishi operator at J=0. We explicitly construct descendant operators by supersymmetry transformations and investigate their three-point functions using superconformal symmetry. 
  We argue that the  Born-Infeld solution on the D$9-$brane is unstable under inclusion of derivative corrections to Born-Infeld theory coming from string theory. More specifically, we find no electrostatic solutions to the first order corrected Born-Infeld theory on the D$9-$brane which give a finite value for the Lagrangian. 
  We give an expanded discussion of the proposal that spacetime supersymmetry representations may be viewed as having their origins in 1D theories that involve a special class of real Clifford algebras. These 1D theories reproduce the supersymmetric structures of spacetime supersymmetric theories after the latter are reduced on a 0-brane. 
  The cosmology of the Horava-Witten M-Theory reduced to five dimensions retaining the volume modulus is considered. Brane matter is considered as a perturbation on the vacuum solution, and the question of under what circumstances does the theory give rise to the standard RWF cosmology is examined. It is found that for static solutions, one obtains a consistent solution of the bulk field equations and the brane boundary conditions only for pure radiation on the branes. (A similar result holds if additional 5-branes are added in the bulk.) If one stabilizes the fifth dimension in an ad hoc manner, a similar inconsistency still occurs (at least for a Hubble constant that has no dependence on y, the fifth dimension.) Within this framework, the possibility of recovering the RWF cosmology still remains if the volume modulus and /or the distance between branes becomes time dependent, under which circumstances the Hubble constant must then depend on y (unless the fifth dimension and volume modulus expand at precisely the same rate). 
  We study the dual equivalence between the nonlinear generalization of the self-dual ($NSD_{B\wedge F}$) and the topologically massive $B\wedge F$ models with particular emphasis on the nonlinear electrodynamics proposed by Born and Infeld. This is done through a dynamical gauge embedding of the nonlinear self-dual model yielding to a gauge invariant and dynamically equivalent theory. We clearly show that nonpolinomial $NSD_{B\wedge F}$ models can be mapped, through a properly defined duality transformation, into $TM_{B\wedge F}$ actions. The general result obtained is then particularized for a number of examples, including the Born-Infeld-BF (BIBF) model that has experienced a revival in the recent literature. 
  We argue that $(0,1)$ heterotic string models with 4 non-heterotic spacetime dimensions may provide an instability of the vacuum to gravitational polarization in globally strong gravitational fields. This instability would be triggered during gravitational collapse by a non-local quantum switching process, and lead to the formation of regions of a high temperature broken symmetry phase which would be the equivalent in these string models of conventional black holes. 
  In this paper we study the non-maximally symmetric D-branes on the SU(2) group discussed in a previous article (hep-th/0205097). Using the two-form defined in hep-th/0205097 the DBI action on the branes is constructed. This action is checked for its agreement with CFT predictions. The geometry of the branes is analyzed in detail, and the singularities of branes covering the entire group are found. 
  The masses of the matter fields of N=2 Super-Yang-Mills theories can be defined as parameters of deformed supersymmetry transformations. The formulation used involves central charges for the matter fields. The explicit form of the deformed supersymmetry transformations and of the invariant Lagrangian in presence of the gauge supermultiplet are constructed. This works generalizes a former one, due to the same authors, which presented the free matter case. 
  The authors' recent works on off-shell boundary/crosscap states are reviewed. 
  Almost all known instanton solutions in noncommutative Yang-Mills theory have been obtained in the modified ADHM scheme. In this paper we employ two alternative methods for the construction of the self-dual U(2) BPST instanton on a noncommutative Euclidean four-dimensional space with self-dual noncommutativity tensor. Firstly, we use the method of dressing transformations, an iterative procedure for generating solutions from a given seed solution, and thereby generalize Belavin's and Zakharov's work to the noncommutative setup. Secondly, we relate the dressing approach with Ward's splitting method based on the twistor construction and rederive the solution in this context. It seems feasible to produce nonsingular noncommutative multi-instantons with these techniques. 
  We investigate various supersymmetric brane intersections. Motivated by the recent results on supertubes, we investigate general constraints in which parallel brane-antibrane configurations are supersymmetric. Dual descriptions of these configurations involve systems of branes in relative motion. In particular, we find new supersymmetric configurations which are not related to a static brane intersection by a boost. In these new configurations, the intersection point moves at the speed of light. These systems provide interesting time dependent backgrounds for open strings. 
  We demonstrate the relation between the Scherk-Schwarz mechanism and flipped gauged brane-bulk supergravities in five dimensions. We discuss the form of supersymmetry violating Scherk-Schwarz terms in pure supergravity and in supergravity coupled to matter. Although the Lagrangian mass terms that arise as the result of the Scherk-Schwarz redefinition of fields are naturally of the order of the inverse radius of the orbifold, the effective 4d physical mass terms are rather set by the scale \sqrt{|\bar{\Lambda}|}, where \bar{\Lambda} is the 4d cosmlogical constant. 
  We study a black hole domain wall system in dilaton gravity which is the low-energy limit of the superstring theory. We solve numerically equations of motion for real self-interacting scalar field and justify the existence of static axisymmetric field configuration representing the thick domain wall in the background of a charged dilaton black hole. It was also confirmed that the extreme dilaton black hole always expelled the domain wall. 
  The tachyonic regime of the quantum fluctuations of a self-interacting scalar field around its vacuum mean value is studied within a kinetic approach. We derive a quantum kinetic equation which determines the time evolution of the momentum distribution function of produced tachyonic modes and includes memory effects. The back-reaction of the quantum fluctuations on the vaccum mean field is taken into account, while their interaction is neglected. We show that the tachyonic modes do not correspond to real particles and contribute to the decay of the metastable vacuum state. 
  The first article in this series presented a thorough discussion of particle weights and their characteristic properties. In this part a disintegration theory for particle weights is developed which yields pure components linked to irreducible representations and exhibiting features of improper energy-momentum eigenstates. This spatial disintegration relies on the separability of the Hilbert space as well as of the C*-algebra. Neither is present in the GNS-representation of a generic particle weight so that we use a restricted version of this concept on the basis of separable constructs. This procedure does not entail any loss of essential information insofar as under physically reasonable assumptions on the structure of phase space the resulting representations of the separable algebra are locally normal and can thus be continuously extended to the original quasi-local C*-algebra. 
  We analyze the Hamiltonian of the compactified D=11 supermembrane with non-trivial central charge in terms of the matrix model constructed recently by some of the authors. Our main result provides a rigorous proof that the quantum Hamiltonian of the supersymmetric model has compact resolvent and thus its spectrum consists of a discrete set of eigenvalues with finite multiplicity. 
  Anthropic principle can help us to understand many properties of our world. However, for a long time this principle seemed too metaphysical and many scientists were ashamed to use it in their research. I describe here a justification of the weak anthropic principle in the context of inflationary cosmology and suggest a possible way to justify the strong anthropic principle using the concept of the multiverse. 
  Recently (hep-th/0104171) we considered N=2 super Yang-Mills with a N=2 mass breakingn term and showed the existence of BPS Z_{k}-string solutions for arbitrary simple gauge groups which are spontaneously broken to non-Abelian residual gauge groups. We also calculated their string tensions exactly. In doing so, we have considered in particular the hypermultiplet in the same representation as the one of a diquark condensate. In the present work we analyze some of the different phases of the theory and find that the magnetic fluxes of the monopoles are multiple of the fundamental Z_{k}-string flux, allowing for monopole confinement in one of the phase transitions of the theory. We also calculate the threshold length for a string breaking. Some of these confining theories can be obtained by adding a N=0 deformation term to N=2 or N=4 superconformal theories. 
  We discuss the Montonen-Olive electric-magnetic duality for the BPS massless monopole clouds in N=4 supersymmetric Yang-Mills theory with non-Abelian unbroken gauge symmetries. We argue that these low energy non-Abelian clouds can be identified as the duals of the infrared bremsstrahlung radiation of the non-Abelian massless particles. After we break the N=4 supersymmetry to N=1 by adding a superpotential, or to N=0 by further adding soft breaking terms, these non-Abelian clouds will generally condense and screen the non-Abelian charges of the massive monopole probes. The effective mass of these dual non-Abelian states is likely to persist as we lower the energy to the QCD scale, if all the non-Abelian Higgs particles are massive. This can be regarded as a manifestation of the non-Abelian dual Meissner effect above the QCD scale, and we expect it to continuously connect with the confinement as we lower the supersymmetry breaking scale to the QCD scale. 
  In the q-deformed theory the perturbation approach can be expressed in terms of two pairs of undeformed position and momentum operators. There are two configuration spaces. Correspondingly there are two q-perturbation Hamiltonians, one originates from the perturbation expansion of the potential in one configuration space, the other one originates from the perturbation expansion of the kinetic energy in another configuration space.   In order to establish a general foundation of the q-perturbation theory, two perturbation equivalence theorems are proved: (I) Equivalence theorem {\it I}: Perturbation expressions of the q-deformed uncertainty relations calculated by two pairs of undeformed operators are the same, and the two q-deformed uncertainty relations undercut Heisenberg's minimal one in the same style. (II) The general equivalence theorem {\it II}: for {\it any} potential (regular or singular) the expectation values of two q-perturbation Hamiltonians in the eigenstates of the undeformed Hamiltonian are equivalent to all orders of the perturbation expansion. As an example of singular potentials the perturbation energy spectra of the q-deformed Coulomb potential are studied. 
  We derive the superpotential of gauge theories having matter fields in the fundamental representation of gauge fields by using the method of Dijkgraaf and Vafa. We treat the theories with one flavour and reproduce a well-known non-perturbative superpotential for meson field. 
  Recently we have presented a new physical model that links the maximum speed of light with the minimal Planck scale into a maximal-acceleration Relativity principle in phase spaces . The maximal proper-acceleration bound is $a = c^2/ \Lambda$ where $ \Lambda$ is the Planck scale. The group transformation laws of this Maximal-acceleration Relativity theory under velocity and acceleration boosts are analyzed in full detail. For pure acceleration boosts it is shown that the minimal Planck-areas (maximal tension) are universal invariant quantities in any frame of reference. The implications of this minimal Planck-area (maximal tension) principle in future developments of string theory, $ W$-geometry and Quantum Gravity are briefly outlined. 
  It is known that YM_2 with gauge group SU(N) is equivalent to a string theory with coupling g_s=1/N, order by order in the 1/N expansion. We show how this results can be obtained from the bosonization of the fermionic formulation of YM_2, improving on results in the literature, and we examine a number of non-perturbative aspects of this string/YM correspondence. We find contributions to the YM_2 partition function of order exp{-kA/(\pi\alpha' g_s)} with k an integer and A the area of the target space, which would correspond, in the string interpretation, to D1-branes. Effects which could be interpreted as D0-branes are instead stricly absent, suggesting a non-perturbative structure typical of type 0B string theories. We discuss effects from the YM side that are interpreted in terms of the stringy exclusion principle of Maldacena and Strominger. We also find numerically an interesting phase structure, with a region where YM_2 is described by a perturbative string theory separated from a region where it is described by a topological string theory. 
  We study the optimization of nonperturbative renormalization group equations truncated both in fields and derivatives. On the example of the Ising model in three dimensions, we show that the Principle of Minimal Sensitivity can be unambiguously implemented at order $\partial^2$ of the derivative expansion. This approach allows us to select optimized cut-off functions and to improve the accuracy of the critical exponents $\nu$ and $\eta$. The convergence of the field expansion is also analyzed. We show in particular that its optimization does not coincide with optimization of the accuracy of the critical exponents. 
  The appearance of space/time non-commutativity in theories of open strings with a constant non-diagonal background metric is considered. We show that, even if the space-time coordinates commute, when there is a metric with a time-space component, no electric field and the boundary condition along the spatial direction is Dirichlet, a Moyal phase still arises in products of vertex operators. The theory is in fact dual to the non-commutatitive open string (NCOS) theory. The correct definition of the vertex operators for this theory is provided. We study the system also in the presence of a $B$ field. We consider the case in which the Dirichlet spatial direction is compactified and analyze the effect of these background on the closed string spectrum. We then heat up the system. We find that the Hagedorn temperature depends in a non-extensive way on the parameters of the background and it is the same for the closed and the open string sectors. 
  We discuss the definition of particle velocity in doubly relativity theories. The general formula relating velocity and four-momentum of particle is given. 
  The supersymmetric abelian Higgs model with N scalar fields admits multiple domain wall solutions. We perform a Callias-type index calculation to determine the number of zero modes of this soliton. We confirm that the most general domain wall has 2(N-1) zero modes, which can be interpreted as the positions and phases of (N-1) constituent domain walls. This implies the existence of moduli for a D-string interpolating between N D5-branes in IIB string theory. 
  We study supersymmetric intersecting D6-branes wrapping 3-cycles in the Type IIA T^6/Z_4 orientifold background. As a new feature, the 3-cycles in this orbifold space arise both from the untwisted and the Z_2 twisted sectors. We present an integral basis for the homology lattice, H_3(M,Z), in terms of fractional 3-cycles, for which the intersection form involves the Cartan matrix of E8. We show that these fractional D6-branes can be used to construct supersymmetric brane configurations realizing a three generation Pati-Salam model. Via brane recombination processes preserving supersymmetry, this GUT model can be broken down to a standard-like model. 
  We investigate the problem of mapping, through the Morita equivalence, odd dimensional noncommutative lattice gauge theories onto suitable matrix models. We specialize our analysis to noncommutative three dimensional QED (NCQED) and scalar QED (NCSQED), for which we explicitly build the corresponding Matrix Model. 
  The higher dimensional analogue of the Blau-Thompson model in D=5 is constructed by a N_T=1 topological twist of N=2, D=5 super Yang-Mills theory. Its dimenional reduction to D=4 and D=3 gives rise to the B-model and N_T=4 equivariant extension of the Blau-Thompson model, respectively. A further dimensional reduction to D=2 provides another example of a N_T=8 Hodge-type cohomological theory with global symmetry group SU(2) \otimes \bar SU(2). Moreover, it is shown that this theory possesses actually a larger global symmetry group SU(4) and and that it agrees with the N_T=8 topological twisting of N+16, D=2 super Yang-Mills theory. 
  We show that the recently obtained class of spacetimes for which all of the scalar curvature invariants vanish (which can be regarded as generalizations of pp-wave spacetimes) are exact solutions in string theory to all perturbative orders in the string tension scale. As a result the spectrum of the theory can be explicitly obtained, and these spacetimes are expected to provide some hints for the study of superstrings on more general backgrounds. Since these Lorentzian spacetimes suffer no quantum corrections to all loop orders they may also offer insights into quantum gravity. 
  We study renormalization group flows in unitary two dimensional sigma models with asymptotically flat target spaces. Applying an infrared cutoff to the target space, we use the Zamolodchikov c-theorem to demonstrate that the target space ADM energy of the UV fixed point is greater than that of the IR fixed point: spacetime energy decreases under world-sheet RG flow. This result mirrors the well understood decrease of spacetime Bondi energy in the time evolution process of tachyon condensation. 
  Through a nonperturbative analysis on a sextic triple-well potential, we reveal novel aspects of the dynamical property of the system in connection with N-fold supersymmetry and quasi-solvability. 
  We review recent progress in a fully dynamical Lagrangian description of the supergravity-superbrane interaction. It suggests that the interacting superfield action, when it exists, is gauge equivalent to the component action of dynamical supergravity interacting with the bosonic limit of the superbrane. 
  We construct spherically and axially symmetric monopoles in SU(5) Yang-Mills-Higgs theory both in flat and curved space as well as spherical and axial non-abelian, ''hairy'' black holes. We find that in analogy to the SU(2) case, the flat space monopoles are either non-interacting (in the BPS limit) or repelling. In curved space, however, gravity is able to overcome the repulsion for suitable choices of the Higgs coupling constants and the gravitational coupling. In addition, we confirm that indeed all qualitative features of (gravitating) SU(2) monopoles are found as well in the SU(5) case. For the non-abelian black holes, we compare the behaviour of the solutions in the BPS limit with that for non-vanishing Higgs self-coupling constants. 
  We review a recent progress in constructing the low-energy effective action of N=4 SYM theory. This theory is formulated in terms of N=2 harmonic superfields corresponding to N=2 vector multiplet and hypermultiplet. Such a formulation possesses the manifest N=2 supersymmetry and an extra hidden on-shell supersymmetry. Exploring the hidden N=2 supersymmetry we proved that the known non-holomorphic potentials of the form ln W ln \bar{W} can be explicitly completed by the appropriate hypermultiplet-dependent terms to the entire N=4 supersymmetric form. The non-logarithmic effective potentials do not admit an N=4 completion and, hence, such potentials cannot occur in N=4 supersymmetric theory. As a result we obtain the exact N=4 supersymmetric low-energy effective action in N=4 SYM theory. 
  We synthesize and extend the previous ideas about appearance of both noncommutative and Finsler geometry in string theory with nonvanishing B--field and/or anholonomic (super) frame structures \cite{vstring,vstr2,vnonc,vncf}. There are investigated the limits to the Einstein gravity and string generalizations containing locally anisotropic structures modeled by moving frames. The relation of anholonomic frames and nonlinear connection geometry to M--theory and possible noncommutative versions of locally anisotropic supergravity and D--brane physics is discussed. We construct and analyze new classes of exact solutions with noncommutative local anisotropy describing anholonomically deformed black holes (black ellipsoids) in string gravity, embedded Finsler--string two dimensional structures, solitonically moving black holes in extra dimensions and wormholes with noncommutativity and anisotropy induced from string theory. 
  We study the physics of N=1 super Yang-Mills theory with gauge group U(Nc) and one adjoint Higgs field, by using the recently derived exact effective superpotentials. Interesting phenomena occur for some special values of the Higgs potential couplings. We find critical points with massless glueballs and/or massless monopoles, confinement without a mass gap, and tensionless domain walls. We describe the transitions between regimes with different patterns of gauge symmetry breaking, or, in the matrix model language, between solutions with a different number of cuts. The standard large Nc expansion is singular near the critical points, with domain walls tensions scaling as a fractional power of Nc. We argue that the critical points are four dimensional analogues of the Kazakov critical points that are commonly found in low dimensional matrix integrals. We define a double scaling limit that yields the exact tension of BPS two-branes in the resulting N=1, four dimensional non-critical string theory. D-brane states can be deformed continuously into closed string solitonic states and vice-versa along paths that go over regions where the string coupling is strong. 
  We define the "maximally integrable" isotropic oscillator on CP(N) and discuss its various properties, in particular, the behaviour of the system with respect to a constant magnetic field. We show that the properties of the oscillator on CP(N) qualitatively differ in the N>1 and N=1 cases. In the former case we construct the ``axially symmetric'' system which is locally equivalent to the oscillator. We perform the Kustaanheimo-Stiefel transformation of the oscillator on CP(2) and construct some generalized MIC-Kepler problem. We also define a N=2 superextension of the oscillator on CP(N) and show that for N>1 the inclusion of a constant magnetic field preserves the supersymmetry of the system. 
  We examine the corrections to the lowest order gravitational interactions of massive particles arising from gravitational radiative corrections. We show how the masslessness of the graviton and the gravitational self interactions imply the presence of nonanalytic pieces sqrt{-q^2}, ln-q^2, etc. in the form factors of the energy-momentum tensor and that these correspond to long range modifications of the metric tensor g_{\mu\nu} of the form G^2m^2/r^2, G^2m\hbar/r^3, etc. The former coincide with well known solutions from classical general relativity, while the latter represent new quantum mechanical effects, whose strength and form is necessitated by the low energy quantum nature of the general relativity. We use these results to define a running gravitational charge. 
  We treat general relativity as an effective field theory, obtaining the full nonanalytic component of the scattering matrix potential to one-loop order. The lowest order vertex rules for the resulting effective field theory are presented and the one-loop diagrams which yield the leading nonrelativistic post-Newtonian and quantum corrections to the gravitational scattering amplitude to second order in G are calculated in detail. The Fourier transformed amplitudes yield a nonrelativistic potential and our result is discussed in relation to previous calculations. The definition of a potential is discussed as well and we show how the ambiguity of the potential under coordinate changes is resolved. 
  Field theories on the plane wave background are considered. We discuss that for such field theories one can only form 1+1 dimensional freely propagating wave packets. We analyze tree level four point functions of scalar field theory as well as scalars coupled to gauge fields in detail and show that these four point functions are well-behaved so that they can be interpreted as S-matrix elements for 2 particle $\to$ 2 particle scattering amplitudes. Therefore, at least classically, field theories on the plane wave background have S-matrix formulation. 
  We show that the Born-Infeld action with the Wess-Zumino terms for the Ramond-Ramond fields, which is the D3-brane effective action, is a solution to the Hamilton-Jacobi (H-J) equation of type IIB supergravity. Adopting the radial coordinate as time, we develop the ADM formalism for type IIB supergravity reduced on $S^5$ and derive the H-J equation, which is the classical limit of the Wheeler-De Witt equation and whose solutions are classical on-shell actions. The solution to the H-J equation reproduces the on-shell actions for the supergravity solution of a stack of D3-branes in a $B_2$ field and the near-horizon limit of this supergravity solution, which is conjectured to be dual to noncommutative Yang Mills and reduces to $AdS_5 \times S^5$ in the commutative limit. Our D3-brane effective action is that of a probe D3-brane, and the radial time corresponds to the vacuum expectation value of the Higgs field in the dual Yang Mills. Our findings can be applied to the study of the holographic renormalization group. 
  In this note we investigate the effective superpotential of an N=1 SU(N_c) gauge theory with one adjoint chiral multiplet and N_f fundamental chiral multiplets. We propose a matrix model prescription in which only matrix model diagrams with less than two boundaries contribute to the gauge theory effective superpotential. This prescription reproduces exactly the known gauge theory physics for all N_f and $N_c$. For N_f < N_c this is given by the Affleck-Dine-Seiberg superpotential. For N_f > N_c we present arguments leading to the conclusion that the dynamics of these theories is also reproduced by the matrix model. 
  We study the quantum mechanical consistency of noncommutative gauge theories by perturbatively analyzing the Wilsonian quantum effective action in the matrix formulation. In the process of integrating out UV states, we find new divergences having dual UV-IR interpretations and no analogues in ordinary quantum field theories. The appearance of these new UV-IR divergences has profound consequences for the renormalizability of the theory. In particular, renormalizability fails in any nonsupersymmetric noncommutative gauge theory. In fact, we argue that renormalizability generally fails in any noncommutative theory that allows quantum corrections beyond one-loop. Thus, it seems that noncommutative quantum theories are extremely sensitive to the UV, and only the softest UV behavior can be tolerated. 
  We further study the validity of the Monte Carlo Hamiltonian method. The advantage of the method, in comparison with the standard Monte Carlo Lagrangian approach, is its capability to study the excited states. We consider two quantum mechanical models: a symmetric one $V(x) = |x|/2$; and an asymmetric one $V(x)=\infty$, for $x < 0$ and $V(x)=x$, for $x \ge 0$. The results for the spectrum, wave functions and thermodynamical observables are in agreement with the analytical or Runge-Kutta calculations. 
  The complex non-local action functional is used in classical electrodynamics to describe the back-reaction effects for the charge moving in the constant homogeneous electromagnetic field. We apply the mass-shift method to obtain the higher order radiation effects in the non-relativistic cyclotron motion and generalize the method to the case of Bargmann-Michel-Telegdi particle. 
  The zeta-function regularization method is used to evaluate the renormalized effective action for massless conformally coupling scalar field propagating in a closed Friedman spacetime perturbed by a small rotation. To the second order of the rotational parameter in the model spacetime the analytic form of the effective action is obtained with the help of the Schwinger perturbation formula. After investigating the time evolution of the rotational parameter we find that the quantum field effect can produce an effect which damps the cosmological rotational in the early universe. 
  We calculate the one loop gluon vacuum energy in the background of a color magnetic vortex for SU(2) and SU(3). We use zeta functional regularization to obtain analytic expressions suitable for numerical treatment. The momentum integration is turned to the imaginary axis and fast converging sums/integrals are obtained. We investigate numerically a number of profiles of the background. In each case the vacuum energy turns out to be positive increasing in this way the complete energy and making the vortex configuration less stable. In this problem bound states (tachyonic modes) are present for all investigated profiles making them intrinsically unstable. 
  Braneworld effective action for two-brane model is constructed by two different methods based respectively on the Dirichlet and Neumann boundary value problems. The equivalence of these methods is shown due to nontrivial duality relations between special boundary operators of these two problems. 
  We show how within the Dijkgraaf-Vafa prescription one can derive superpotentials for matter fields. The ingredients forming the non-perturbative Affleck-Dine-Seiberg superpotentials arise from constrained matrix integrals, which are equivalent to classical complex Wishart random matrices. The mechanism is similar to the way the Veneziano-Yankielowicz superpotential arises from the matrix model measure. 
  A new representation of Lagrangians of 4D nonlinear electrodynamics is considered. In this new formulation, in parallel with the standard Maxwell field strength F, an auxiliary bispinor (tensor) field V is introduced. The gauge field strength appears only in bilinear terms of the full Lagrangian, while the interaction Lagrangian E depends on the auxiliary fields, E = E(V). Two types of self-duality inherent in the nonlinear electrodynamics models admit a simple characterization in terms of the function E. The continuous SO(2) duality symmetry between nonlinear equations of motion and Bianchi identities amounts to requiring E to be a function of the SO(2) invariant quartic combination |V|^4. The discrete self-duality (or self-duality under Legendre transformation) amounts to a weaker condition E(V)= E(iV). This approach can be generalized to a system of n Abelian gauge fields exhibiting U(n) duality. The corresponding interaction Lagrangian should be U(n) invariant function of n bispinor auxiliary fields. 
  We consider the Adler-Bardeen anomaly of the U(1) axial current in abelian and non-abelian gauge theories and present its algebraic characterization as well as an explicit evaluation proving regularization scheme independence of the anomaly. By extending the gauge coupling to an external space-time dependent field we get a unique definition for the quantum corrections of the topological term. It also implies a simple proof of the non-renormalization theorem of the Adler-Bardeen anomaly. We consider local gauge couplings in supersymmetric theories and find that there the renormalization of the gauge coupling is determined by the topological term in all loop orders except for one loop. It is shown that in one-loop order the quantum corrections to the topological term induce an anomalous breaking of supersymmetry, which is characterized by similar properties as the Adler-Bardeen anomaly. 
  The extension of coupling constants to space-time dependent fields, the local couplings, makes possible to derive the non-renormalization theorems of supersymmetry by an algebraic characterization of Lagrangian N=1 supermultiplets. For super-Yang-Mills theories the construction implies non-renormalization of the coupling beyond one-loop order. However, renormalization in presence of the local gauge coupling is peculiar due to a new anomaly in one-loop order, which appears as an anomaly of supersymmetry in the Wess-Zumino gauge. As an application we derive the closed all-order expression for the gauge $\beta$ function and prove the non-renormalization of general N=2 supersymmetric theories from a cancellation of the susy anomaly. 
  We present a qualitative model of the Coulomb branch of the moduli space of low-energy effective N=2 SQCD with gauge group SU(3) and up to five flavours of massive matter. Overall, away from double cores, we find a situation broadly similar to the case with no matter, but with additional complexity due to the proliferation of extra BPS states. We also include a revised version of the pure SU(3) model which can accommodate just the orthodox weak coupling spectrum. 
  We first discuss how the longstanding confusion in the literature concerning one-loop quantum corrections to 1+1 dimensional solitons has finally been resolved. Then we use 't Hooft and Veltman's dimensional regularization to compute the kink mass, and find that chiral domain wall fermions, induced by fermionic zero modes, lead to spontaneous parity violation and an anomalous contribution to the central charge such that the BPS bound becomes saturated. On the other hand, Siegel's dimensional reduction shifts this anomaly to the counter terms in the renormalized current multiplet. The superconformal anomaly is located in an evanescent counter term, and imposing supersymmetry, this counter term induces the same anomalous contribution to the central charge. Next we discuss a new regularization scheme: local mode regularization. The local energy density computed in this scheme satisfies the BPS equality (it is equal to the local central charge density). In an appendix we give a very detailed account of the DHN method to compute soliton masses applied to the supersymmetric kink. 
  Two families of sets, nonstationary and stationary, are obtained. Each nonstationary set $\psi_{p_v}$ consists of the solutions with the quantum number $p_v=p^0v-p_3.$ It can be obtained from the nonstationary set $\psi_{p_3}$ with quantum number $p_3$ by a boost along $x_3$-axis (along the direction of the electric field) with velocity $-v$. Similarly, any stationary set of solutions characterized by a quantum number $p_s=p^0-sp_3$ can be obtained from stationary solutions with quantum number $p^0$ by the same boost with velocity $-s$. All these sets are equivalent and the classification (i.e. ascribing the frequency sign and in-, out- indexes) in any set is determined by the classification in $\psi_{p_3}$-set, where it is beyond doubt. 
  We classify (up to local isometry) the maximally supersymmetric solutions of the eleven- and ten-dimensional supergravity theories. We find that the AdS solutions, the Hpp-waves and the flat space solutions exhaust them. 
  We derive effective actions for Spacelike branes (S-branes) and find a solution describing the formation of fundamental strings in the rolling tachyon background. The S-brane action is a Dirac-Born-Infeld action for Euclidean worldvolumes defined in the context of time-dependent tachyon condensation of non-BPS branes. It includes gauge fields and in particular a scalar field associated with translation along the time direction. We show that the BIon spike solutions constructed in this system correspond to the production of a confined electric flux tube (a fundamental string) at late time of the rolling tachyon. 
  We discuss the construction of four dimensional non-supersymmetric models obtained from configurations of D6-branes intersecting at angles. We present the first examples of string GUT models which break exactly to the Standard Model (SM) at low energy. Even though the models are non supersymmetric (SUSY), the demand that some open string sectors preserve N=1 SUSY creates gauge singlet scalars that break the extra anomaly free U(1)'s generically present in the models, predicting $s{\tilde \nu}_R$'s and necessarily creating Majorana mass terms for right handed neutrinos. 
  We evaluate the 4-point function of the auxiliary field in the critical O(N) sigma model at O(1/N) and show that it describes the exchange of tensor currents of arbitrary even rank l>0. These are dual to tensor gauge fields of the same rank in the AdS theory, which supports the recent hypothesis of Klebanov and Polyakov. Their couplings to two auxiliary fields are also derived. 
  We discuss the Penrose limit of the Chamseddine-Volkov BPS selfgravitating monopole in four dimensional N=4 supergravity theory with non-abelian gauge multiplets. We analyze the properties of the resulting supersymmetric pp-wave solutions when various Penrose limits are considered. Apart from the usual rescaling of coordinates and fields we find that a rescaling of the gauge coupling constant to zero is required, rendering the theory abelian. We also study the Killing spinor equations showing an enhancement of the supersymmetries preserved by the solutions and discuss the embedding of the pp-wave solution in $d=10$ dimensions. 
  The nonlinear integral equations describing the spectra of the left and right (continuous) quantum KdV equations on the cylinder are derived from integrable lattice field theories, which turn out to allow the Bethe Ansatz equations of a twisted ``spin -1/2'' chain. A very useful mapping to the more common nonlinear integral equation of the twisted continuous spin $+1/2$ chain is found. The diagonalization of the transfer matrix is performed. The vacua sector is analysed in detail detecting the primary states of the minimal conformal models and giving integral expressions for the eigenvalues of the transfer matrix. Contact with the seminal papers \cite{BLZ, BLZ2} by Bazhanov, Lukyanov and Zamolodchikov is realised. General expressions for the eigenvalues of the infinite-dimensional abelian algebra of local integrals of motion are given and explicitly calculated at the free fermion point. 
  We present a new approach to the quantization of the superstring. After a brief review of the classical Green-Schwarz formulation for the superstring and Berkovits' approach to its quantization based on pure spinors, we discuss our formulation without pure spinor constraints. In order to illustrate the ideas on which our work is based, we apply them to pure Yang-Mills theory. In the appendices, we include some background material for the Green-Schwarz and Berkovits formulations, in order that this presentation be self contained. 
  We show that the rigidly rotating quantum thermal distribution on flat space-time suffers from a global pathology which can be cured by introducing a cylindrical mirror if and only if it has a radius smaller than that of the speed-of-light cylinder. When this condition is met, we demonstrate numerically that the renormalized expectation value of the energy-momentum stress tensor corresponds to a rigidly rotating thermal bath up to a finite correction except on the mirror where there are the usual Casimir divergences. 
  We reconsider supersymmetric five dimensional rotating charged black holes, and their description in terms of D-branes. By wrapping some of the branes on K3, we are able to explore the role of the enhancon mechanism in this system. We verify that enhancon loci protect the black hole from violations of the Second Law of Thermodynamics which would have been achieved by the addition of certain D-brane charges. The same charges can potentially result in the formation of closed time-like curves by adding them to holes initially free of them, and so the enhancon mechanism forbids this as well. Although this latter observation is encouraging, it is noted that this mechanism alone does not eliminate closed time-like curves from these systems, but is in accord with earlier suggestions that they may not be manufactured, in this context, by physical processes. 
  Using mirror symmetry, we show that Chern-Simons theory on certain manifolds such as lens spaces reduces to a novel class of Hermitian matrix models, where the measure is that of unitary matrix models. We show that this agrees with the more conventional canonical quantization of Chern-Simons theory. Moreover, large N dualities in this context lead to computation of all genus A-model topological amplitudes on toric Calabi-Yau manifolds in terms of matrix integrals. In the context of type IIA superstring compactifications on these Calabi-Yau manifolds with wrapped D6 branes (which are dual to M-theory on G2 manifolds) this leads to engineering and solving F-terms for N=1 supersymmetric gauge theories with superpotentials involving certain multi-trace operators. 
  We explore the implication of applying the "minimally complex" representation to study intermediate states. We propose an interpretation of the "velocity variables" used in the "minimally complex" representation so as to make the construction consistent with the kinematics of the underlying scattering process under consideration. We also show that this choice of representation advocates the use of velocity basis over momentum basis in studying intermediate states. 
  We study the large $N$ limit of the class of U(N) ${\CN}=1$ SUSY gauge theories with an adjoint scalar and a superpotential $W(\P)$. In each of the vacua of the quantum theory, the expectation values $\la$Tr$\Phi^p$$\ra$ are determined by a master matrix $\Phi_0$ with eigenvalue distribution $\rho_{GT}(\l)$. $\rho_{GT}(\l)$ is quite distinct from the eigenvalue distribution $\rho_{MM}(\l)$ of the corresponding large $N$ matrix model proposed by Dijkgraaf and Vafa. Nevertheless, it has a simple form on the auxiliary Riemann surface of the matrix model. Thus the underlying geometry of the matrix model leads to a definite prescription for computing $\rho_{GT}(\l)$, knowing $\rho_{MM}(\l)$. 
  We compute the two closed string graviton - two open string scalar scattering amplitude on the disc to show that there is no second-derivative curvature - scalar coupling term R X^2 in the low-energy effective action of a D-brane in curved space in type II superstring theory. 
  We discuss mirror symmetry in generalized Calabi-Yau compactifications of type II string theories with background NS fluxes. Starting from type IIB compactified on Calabi-Yau threefolds with NS three-form flux we show that the mirror type IIA theory arises from a purely geometrical compactification on a different class of six-manifolds. These mirror manifolds have SU(3) structure and are termed half-flat; they are neither complex nor Ricci-flat and their holonomy group is no longer SU(3). We show that type IIA appropriately compactified on such manifolds gives the correct mirror-symmetric low-energy effective action. 
  BPS equations and wall solutions are studied keeping (part of) supersymmetry (SUSY) manifest. Using N=1 superfields, massive hyper-Kahler quotient is introduced to obtain massive N=2 (8 SUSY) nonlinear sigma models in four dimensions with T*CP(n) target manifold, which yield BPS wall solutions for the n=1 case. We also describe massive hyper-Kahler quotient of T*CP(n) by using the harmonic superspace formalism which preserves all SUSY manifestly, and BPS equations and wall solutions are obtained in the n=1 case. 
  A variational method is used to analyse compact U(1) gauge theory in 2+1-dimensions at finite temperature, T, weak coupling, g and where the fundamental magnetic monopoles have magnetic charge 2\pi n/g. The theory undergoes a critical transition from a confining phase at temperatures below T_c=2g^2/n^2\pi to a deconfined phase at temperatures above T_c. The free energy and all its derivatives are continuous at T_c, indicative of the BKT phase transition. The relevant gauge-invariant correlation functions decay exponentially at large distances. The spatial Wilson loop obeys the area law at all finite temperatures, even for the non-compact theory. The case n=2 corresponds to the compact U(1) theory considered as a low energy effective theory for the spontaneously broken Georgi--Glashow model. The results in this case agree with those derived previously for compact U(1) in this model using dimensional reduction of the Lagrangian. 
  The Dirac approach to constrained systems can be adapted to construct relativistic invariant theories on a noncommutative (NC) space. As an example, we propose and discuss relativistic invariant NC particle coupled to electromagnetic field by means of the standard term $A^\mu\dot x_\mu$. Poincare invariance implies deformation of the free particle NC algebra in the interaction theory. The corresponding corrections survive in the nonrelativistic limit. 
  The singular part of the \textit{operator product expansion} (OPE) of a pair of \textit{globally conformal invariant} (GCI) scalar fields $\phi$ of (integer) dimension $d$ can be written as a sum of the 2-point function of $\phi$ and $d-1$ bilocal conformal fields $V_{\nu}(x_1, x_2)$ of dimension $(\nu, \nu)$, $\nu = 1, ..., d-1$. As the correlation functions of $\phi(x)$ are proven to be rational [6], we argue that the correlation functions of $V_{\nu}$ can also be assumed rational. Each $V_{\nu}(x_1, x_2)$ is expanded into local symmetric tensor fields of \textit{twist} (dimension minus rank) $2\nu$. The case $d=2$, considered previously [5], is briefly reviewed and current work on the $d=4$ case (of a Lagrangean density in 4 space--time dimensions) is previewed. 
  We show that in strongly coupled N=4 SYM the binding energy of a heavy and a light quark is independent of the strength of the coupling constant. As a consequence we are able to show that in the presence of light quarks the analog of the QCD string can snap and color charges are screened. The resulting neutral mesons interact with each other only via pion exchange and we estimate the masses 
  We present a systematic derivation of multi-instanton amplitudes in terms of ADHM equivariant cohomology. The results rely on a supersymmetric formulation of the localization formula for equivariant forms. We examine the cases of N=4 and N=2 gauge theories with adjoint and fundamental matter. 
  I describe domain walls and domain wall lattices in non-Abelian models. 
  Drawing analogies with block spin techniques used to study continuum limits in critical phenomena, we attempt to block up D-branes by averaging over near neighbour elements of their (in general noncommuting) matrix coordinates, i.e.\ in a low energy description. We show that various D-brane (noncommutative) geometries arising in string theory appear to behave sensibly under blocking up, given certain key assumptions in particular involving gauge invariance. In particular, the (gauge-fixed) noncommutative plane, fuzzy sphere and torus exhibit a self-similar structure under blocking up, if some ``counterterm'' matrices are added to the resulting block-algebras. Applying these techniques to matrix representations of more general D-brane configurations, we find that blocking up averages over far-off-diagonal matrix elements and brings them in towards the diagonal, so that the matrices become ``less off-diagonal'' under this process. We describe heuristic scaling relations for the matrix elements under this process. Further, we show that blocking up does not appear to exhibit any ``chaotic'' behaviour, suggesting that there is sensible physics underlying such a matrix coarse-graining. We also discuss briefly interrelations of these ideas with B-fields and noncommutativity. 
  In these lectures, recent progress on multiloop superstring perturbation theory is reviewed. A construction from first principles is given for an unambiguous and slice-independent two-loop superstring measure on moduli space for even spin structure. A consistent choice of moduli, invariant under local worldsheet supersymmetry is made in terms of the super-period matrix. A variety of subtle new contributions arising from a careful gauge fixing procedure are taken into account.   The superstring measure is computed explicitly in terms of genus two theta-functions and reveals the importance of a new modular object of weight 6. For given even spin structure, the measure exhibits a behavior under degenerations of the worldsheet that is consistent with physical principles. The measure allows for a unique modular covariant GSO projection. Under this GSO projection, the cosmological constant, the 1-, 2- and 3- point functions of massless supergravitons vanish pointwise on moduli space. A certain disconnected part of the 4-point function is shown to be given by a convergent integral on moduli space. A general consistent formula is given for the two-loop cosmological constant in compactifications with central charge c=15 and with N=1 worldsheet supersymmetry. Finally, some comments are made on possible extensions of this work to higher loop order. 
  We address the localization of a scalar field, whose bulk-mass M is considered in a wide range including the tachyonic region,on a three-brane. The brane with non-zero cosmological constant $\lambda$ is embedded in five dimensional bulk space. We find in this case that the trapped scalar could have mass $m$ which has an upper bound and expressed as $m^2=m_0^2+\alpha M^2\leq \beta |\lambda|$ with the calculable numbers $m_0^2, \alpha, \beta$. We point out that this result would be important to study the stability of the brane and cosmological problems based on the brane-world. 
  Interacting AdS_4 higher spin gauge theories with N \geq 1 supersymmetry so far have been formulated as constrained systems of differential forms living in a twistor extension of 4D spacetime. Here we formulate the minimal N=1 theory in superspace, leaving the internal twistor space intact. Remarkably, the superspace constraints have the same form as those defining the theory in ordinary spacetime. This construction generalizes straightforwardly to higher spin gauge theories N>1 supersymmetry. 
  Exact massive S-matrices for two dimensional sigma models on symmetric spaces SU(2N)/Sp(N) and Sp(2P)/Sp(P)*Sp(P) are conjectured. They are checked by comparison of perturbative and non perturbative TBA calculations of free energy in a strong external field. We find the mass spectrum of the models and calculate their exact mass gap. 
  We derive the mass shell condition for N coincident D0 branes in codimension two i.e. in space-time dimension three. Using this we present the action for this system in first order formalism. Our analysis is restricted to flat space-time. 
  We consider no-scale extended supergravity models as they arise from string and M-theory compactifications in presence of fluxes. The special role of gauging axion symmetries for the Higgs and superHiggs mechanism is outlined. 
  I present a very simplistic toy model for the inflationary paradigm where the size of the universe undergoes a period of exponential growth. The basic assumption I make use of is that black holes might have a quantized area (mass) spectrum with a stable ground state and that the universe has started with a tightly packed collection of these objects alone. 
  We discuss the mathematical properties of six--dimensional non--K\"ahler manifolds which occur in the context of ${\cal N}=1$ supersymmetric heterotic and type IIA string compactifications with non--vanishing background H--field. The intrinsic torsion of the associated SU(3) structures falls into five different classes. For heterotic compactifications we present an explicit dictionary between the supersymmetry conditions and these five torsion classes. We show that the non--Ricci flat Iwasawa manifold solves the supersymmetry conditions with non--zero H--field, so that it is a consistent heterotic supersymmetric groundstate. 
  Some issues in relating the central extensions of the planar Galilei group to parameters in the corresponding relativistic theory are discussed. 
  We argue that the Einstein-Yang-Mills theory presents nontrivial solutions with a NUT charge. These solutions approach asymptotically the Taub-NUT spacetime. They are characterized by the NUT parameter, the mass and the node numbers of the magnetic potential and present both electric and magnetic potentials.=0D The existence of nontrivial Einstein-Yang-Mills solutions with NUT charge in the presence of a negative cosmological constant is also discussed. We use the counterterm subtraction method to calculate the boundary energy-momentum tensor and the mass of these configurations. Also, dyon black hole solutions with nonspherical event horizon topology are shown to exist for a negative cosmological constant. 
  We analyze the "geometric engineering" limit of a type II string on a suitable Calabi-Yau threefold to obtain an N=2 pure SU(2) gauge theory. The derived category picture together with Pi-stability of B-branes beautifully reproduces the known spectrum of BPS solitons in this case in a very explicit way. Much of the analysis is particularly easy since it can be reduced to questions about the derived category of CP1. 
  A class of D-branes for the type IIB plane-wave background is considered that preserve half the dynamical supersymmetries of the light-cone gauge. The D-branes of this type are the euclidean (or instantonic) (0,0), (0,4) and (4,0) branes (where (r,s) denotes a brane oriented with r axes in the first four directions transverse to the +,- light-cone and s axes in the second four directions). Corresponding lorentzian D-branes are (+,-;0,0), (+,-;0,4) and (+,-;4,0). These are constructed in two ways. The first uses a boundary state formalism which implements appropriate fermionic gluing conditions and the second is based on a direct quantisation of the open strings ending on the branes. In distinction to the D-branes considered earlier these have massless world-volume fermions but do not possess kinematical supersymmetries. Cylinder diagrams describing the overlap between a pair of boundary states displaced by some distance are evaluated. The open-string description of this system involves mode frequencies that are, in general, given by irrational solutions to transcendental equations. The closed-string and open-string descriptions are shown to be equivalent by a nontrivial implementation of the S modular transformation. A classical description of the D-instanton (the (0,0) case) in light-cone gauge is also given. 
  We perform a completely perturbative matrix model calculation of the physical low-energy quantities of the N=2 U(N) gauge theory. Within the matrix model framework we propose a perturbative definition of the periods a_i in terms of certain tadpole diagrams, and check our conjecture up to first order in the gauge theory instanton expansion. The prescription does not require knowledge of the Seiberg-Witten differential or curve. We also compute the N=2 prepotential F(a) perturbatively up to the first-instanton level finding agreement with the known result. 
  We investigate the late-time behavior of a universe containing a supergravity gas and wrapped 2-branes in the context of M-theory compactified on T^10. The supergravity gas tends to drive uniform expansion, while the branes impede the expansion of the directions about which they are wrapped. Assuming spatial homogeneity, we study the dynamics both numerically and analytically. At late times the radii obey power laws which are determined by the brane wrapping numbers, leading to interesting hierarchies of scale between the wrapped and unwrapped dimensions. The biggest hierarchy that could evolve from an initial thermal fluctuation produces three large unwrapped dimensions. We also study configurations corresponding to string winding, in which the M2-branes are all wrapped around the (small) 11th dimension, and show that this recovers the scenario discussed by Brandenberger and Vafa. 
  It is shown that non-commutative spaces, which are quotients of associative algebras by ideals generated by non-linear relations of a particular type, admit extremely simple formulae for deformed or star products. Explicit construction of these star products is carried out. Quantum gauge theories are formulated on these spaces, and the Seiberg-Witten map is worked out in detail. 
  We embed two 4D chiral multiplets of opposite representations in the 5D N=2 $SU(N+K)$ gauge theory compactified on an orbifold $S^1/(Z_2\times Z'_2)$. There are two types of orbifold boundary conditions in the extra dimension to obtain the 4D N=1 $SU(N)\times SU(K)\times U(1)$ gauge theory from the bulk: in Type I, one has the bulk gauge group at $y=0$ and the unbroken gauge group at $y=\pi R/2$ while in Type II, one has the unbroken gauge group at both fixed points. In both types of orbifold boundary conditions, we consider the zero mode(s) as coming from a bulk $(K+N)$-plet and brane fields at the fixed point(s) with the unbroken gauge group. We check the consistency of this embedding of fields by the localized anomalies and the localized FI terms. We show that the localized anomalies in Type I are cancelled exactly by the introduction of a bulk Chern-Simons term. On the other hand, in some class of Type II, the Chern-Simons term is not enough to cancel all localized anomalies even if they are globally vanishing. We also find that for the consistent embedding of brane fields, there appear only the localized log FI terms at the fixed point(s) with a U(1) factor. 
  The interaction between a parallel brane-antibrane and brane-antibrane is investigated by regarding the brane-antibrane pair as a kink or anti-kink type tachyon condensed state. As the kink-type tachyon condensed state is known as a non-BPS brane we expand the Lagrangian of tachyon effective field theory to the quadratic order in the off-diagonal fluctuation and then use the zeta-function regularization and Schwinger perturbative formula to evaluate the interaction within a kink-kink or a kink-antikink.  The results show that while the kink and kink has repulsive force the kink and anti-kink has attractive force and may annihilate by each others. We therefore evaluate the free energy at finite temperature and determine the critical temperature above which the stable state of kink-antikink system may be found. 
  We propose an extended BRST invariant Lagrangian quantization scheme of general gauge theories based on explicit realization of "modified triplectic algebra" in general coordinates. All the known Lagrangian quantization schemes based on the extended BRST symmetry are obtained by specifying the (free) parameters of that method. 
  Nonlinear sigma models (NLSM) in d=3 have many interesting and non-trivial features, which were explored poorly in contrast with NLSM in d=2 and d=4. We present a few results from our study of the perturbative and non-perturbative properties of three-dimensional (3D) NLSM. i) We have shown that cancellation of ultra-violet (UV) divergences takes place in 3D extended (N=2,4) supersymmetric NLSM in low orders of the 1/n expansion. ii) We consider noncommutative extension of the 3D CP(n) model, and study low-energy dynamics of BPS solitons in this model. We also discuss briefly dynamics of non-BPS solutions. 
  Compactifications of M-theory on manifolds with reduced holonomy arise as the local eleven-dimensional description of D6-branes wrapped on supersymmetric cycles in manifolds of lower dimension with a different holonomy group. Whenever the isometry group SU(2) is present, eight-dimensional gauged supergravity is a natural arena for such investigations. In this paper we use this approach and review the eleven dimensional description of D6-branes wrapped on coassociative 4-cycles, on deformed 3-cycles inside Calabi-Yau threefolds and on Kahler 4-cycles. 
  We illustrate a basic framework for analytic computations of Feynman graphs using the Moyal star formulation of string field theory. We present efficient methods of computation based on (a) the monoid algebra in noncommutative space and (b) the conventional Feynman rules in Fourier space. The methods apply equally well to perturbative string states or nonperturbative string states involving D-branes. The ghost sector is formulated using Moyal products with fermionic (b,c) ghosts. We also provide a short account on how the purely cubic theory and/or VSFT proposals may receive some clarification of their midpoint structures in our regularized framework. 
  We review our recent results on the on-shell description of sine-Gordon model with integrable boundary conditions. We determined the spectrum of boundary states together with their reflection factors by closing the boundary bootstrap and checked these results against WKB quantization and numerical finite volume spectra obtained from the truncated conformal space approach. The relation between a boundary resonance state and the semiclassical instability of a static classical solution is analyzed in detail. 
  Two dimensional SU(N) Yang-Mills theory is known to be equivalent to a string theory, as found by Gross in the large N limit, using the 1/N expansion. Later it was found that even a generalized YM theory leads to a string theory of the Gross type. In the standard YM theory case, Douglas and others found the string hamiltonian describing the propagation and the interactions of states made of strings winding on a cylindrical space-time. We address the problem of finding a similar hamiltonian for the generalized YM theory. As in the standard case we start by writing the theory as a theory of free fermions. Performing a bosonization, we express the hamiltonian in terms of the modes of a bosonic field, that are interpreted as in the standard case as creation and destruction operators for states of strings winding around the cylindrical space-time. The result is similar to the standard hamiltonian, but with new kinds of interaction vertices. 
  We generalize the worldline formalism to include spin 1/2 fields coupled to gravity. To this purpose we first extend dimensional regularization to supersymmetric nonlinear sigma models in one dimension. We consider a finite propagation time and find that dimensional regularization is a manifestly supersymmetric regularization scheme, since the classically supersymmetric action does not need any counterterm to preserve worldline supersymmetry. We apply this regularization scheme to the worldline description of Dirac fermions coupled to gravity. We first compute the trace anomaly of a Dirac fermion in 4 dimensions, providing an additional check on the regularization with finite propagation time. Then we come to the main topic and consider the one-loop effective action for a Dirac field in a gravitational background. We describe how to represent this effective action as a worldline path integral and compute explicitly the one- and two-point correlation functions, i.e. the spin 1/2 particle contribution to the graviton tadpole and graviton self-energy. These results are presented for the general case of a massive fermion. It is interesting to note that in the worldline formalism the coupling to gravity can be described entirely in terms of the metric, avoiding the introduction of a vielbein. Consequently, the fermion--graviton vertices are always linear in the graviton, just like the standard coupling of fermions to gauge fields. 
  With the general aim to classify BPS solutions in N=2, D=5 supergravities interacting with an arbitrary number of vector, tensor and hypermultiplets, here we begin considering the most general electrostatic, spherical-symmetric BPS solutions in the presence of hypermultiplet couplings. We discuss the properties of the BPS equations and the restrictions imposed by their integrability conditions. We exhibit explicit solutions for the case of static BPS black-holes coupled to one (the so called universal) hypermultiplet. 
  Computations in renormalizable perturbative quantum field theories reveal mathematical structures which go way beyond the formal structure which is usually taken as underlying quantum field theory. We review these new structures and the role they can play in future developments. 
  Two dimensional light cone world sheet massive models can be used to define good string backgrounds.In many cases these light cone world sheet lagrangians flow from a CFT in the UV to a theory of massive particles in the IR. The relevant symmetry in the IR, playing a similar role to Virasoro in the UV, are quantum affine Kac Moody algebras. Finite dimensional irreps of this algebra are associated with the spectrum of massive particles. The case of N=0 Sine Gordon at the N=2 point is associated with a Landau Ginzburg model that defines a good string background. For the world sheet symmetry $(N=2) \otimes U_{q}(\hat{Sl(2)})$ the N=2 piece is associated with the string conformal invariance and the $U_{q}(\hat{Sl(2)})$ piece with the world sheet RG. The two dimensional light cone world sheet massive model can be promoted to a CFT by adding extra light cone fields $X^{-}$ and $X^{+}$. From the point of view of the quantum affine symmetry these two fields are associated, respectively, with the center and the derivation of the affine Kac Moody algebra. 
  In the first part of this paper, we study the back-reaction of large-N light cone momentum on the maximally supersymmetric anti-pp-wave background. This gives the type IIA geometry of large-N D0-branes on curved space with fluxes. By taking an appropriate decoupling limit, we conjecture a new duality between string theory on that background and dual field theory on D0-branes which we derive by calculating linear coupling terms. Agreement of decoupling quantities, SO(3) \times SO(6) isometry and Higgs branch on both theories are shown. Also we find whenever dual field theory is weakly coupled, the curvature of the geometry is large. In the second part of this paper, we derive the supermembrane action on a general pp-wave background only through the properties of null Killing vector and through this, derive the Matrix model. 
  M-theory on the maximally supersymmetric plane wave background of eleven-dimensional supergravity admits spherical BPS transverse M5-branes with zero light-cone energy. We give direct evidence that the single M5-brane state corresponds to the trivial (X=0) classical vacuum in the large N limit of the plane wave matrix theory. In particular, we show that the linear fluctuation spectrum of the spherical fivebrane matches exactly with the set of exactly protected excited states about the X=0 vacuum in the matrix model. These states include geometrical fluctuations of the sphere, excitations of the worldvolume two-form field, and fermion excitations. In addition, we propose a description of multiple fivebrane states in terms of matrix model vacua.   Finally, we discuss how to obtain the continuum D2/M2 and NS5/M5 theories on spheres from the matrix model. The matrix model can be viewed as a regularization for these theories. 
  We consider a generalized connection in three dimensions and show that it emerges in Chern-Simons-Maxwell theories when one studies the disorder instanton operator. We generalize this construction to non-Abelian theories and find that the disorder operator (the 't Hooft operator) is equivalent to a generalized Wilson loop in a representation that depends on the Chern-Simons term. We speculate about the effective action of the disorder operator and its applications to the possible phases of the theory in the infra-red. We also show that fractional statistics can emerge in gauge theories without a Chern-Simons term if the generalized connection rather than the ordinary connection is used to couple charged particles. 
  We discuss Gubser-Klebanov-Polyakov proposal for the gauge/string theory correspondence for gauge theories in curved space. Specifically, we consider Klebanov-Tseytlin cascading gauge theory compactified on S^3. We explain regime when this gauge theory is a small deformation of the superconformal N=1 gauge theory on the world volume of regular D3-branes at the tip of the conifold. We study closed string states on the leading Regge trajectory in this background, and attempt to identify the dual gauge theory twist two operators. 
  We refine the dictionary of the gauge/gravity correspondence realizing N=1 super Yang-Mills by means of D5-branes wrapped on a resolved Calabi-Yau space. This is done by fixing an ambiguity on the correct interpretation of the holographic dual of the running gauge coupling and amounts to identify a specific 2-cycle in the dual ten-dimensional supergravity background. In doing so, we also discuss the role played in this context by gauge transformations in the relevant seven-dimensional gauged supergravity. While all nice properties of the duality are maintained, this modification of the dictionary has some interesting physical consequences and solves a puzzle recently raised in the literature. In this refined framework, it is also straightforward to see how the correspondence naturally realizes a geometric transition. 
  The O(3) nonlinear sigma model with boundary, in dimension two, is considered. An algorithm to determine all its soliton solutions that preserve a rotational symmetry in the boundary is exhibited. This nonlinear problem is reduced to that of clamped elastica in a hyperbolic plane. These solutions carry topological charges that can be holographically determined from the boundary conditions. As a limiting case, we give a wide family of new soliton solutions in the free O(3) nonlinear sigma model. 
  The massless Curci-Ferrari model with N_f flavours of quarks is renormalized to three loops in the MSbar scheme in an arbitrary covariant gauge with parameter alpha. The renormalization of the BRST invariant dimension two composite operator, 1/2 (A^a_\mu)^2 - alpha \bar{c}^a c^a, which corresponds to the mass operator in the massive Curci-Ferrari model, is determined by renormalizing the Green's function where the operator is inserted in a ghost two-point function. Consequently the anomalous dimension of the QCD Landau gauge operator, 1/2 (A^a_\mu)^2, and the (gauge independent) photon mass anomalous dimension in QED are both deduced at three loops. 
  We address the problems of fermions in light front QCD on a transverse lattice. We propose and numerically investigate different approaches of formulating fermions on the light front transverse lattice. In one approach we use forward and backward derivatives. There is no fermion doubling and the helicity flip term proportional to the fermion mass in the full light front QCD becomes an irrelevant term in the free field limit. In the second approach with symmetric derivative (which has been employed previously in the literature), doublers appear and their occurrence is due to the decoupling of even and odd lattice sites. We study their removal from the spectrum in two ways namely, light front staggered formulation and the Wilson fermion formulation. The numerical calculations in free field limit are carried out with both fixed and periodic boundary conditions on the transverse lattice and finite volume effects are studied. We find that an even-odd helicity flip symmetry on the light front transverse lattice is relevant for fermion doubling. 
  In this talk we review some generalizations of 't Hooft and Mandelstam ideas on confinement for theories with non-Abelian unbroken gauge groups. In order to do that, we consider N=2 super Yang-Mills with one flavor and a mass breaking term. One of the spontaneous symmetry breaking is accomplished by a scalar that can be in particular in the representation of the diquark condensate and therefore it can be thought as being itself the condensate. We analyze the phases of the theory. In the superconducting phase, we show the existence of BPS Z_k-strings and calculate exactly their string tension in a straightforward way. We also find that magnetic fluxes of the monopole and Z_k-strings are proportional to one another allowing for monopole confinement in a phase transition. We further show that some of the resulting confining theories can be obtained by adding a deformation term to N=2 or N=4 superconformal theories. 
  In a given 4d spacetime bakcground, one can often construct not one but a family of distinct N=2 string theories. This is due to the multiple ways N=2 superconformal algebra can be embedded in a given worldsheet theory. We formulate the principle of obtaining different physical theories by gauging different embeddings of the same symmetry algebra in the same ``pre-theory.'' We then apply it to N=2 strings and formulate the recipe for finding the associated parameter spaces of gauging. Flat and curved target spaces of both (4,0) and (2,2) signatures are considered. We broadly divide the gauging choices into two classes, denoted by alpha and beta, and show them to be related by T-duality. The distinction between them is formulated topologically and hinges on some unique properties of 4d manifolds. We determine what their parameter spaces of gauging are under certain simplicity ansatz for generic flat spaces (R^4 and its toroidal compactifications) as well as some curved spaces. We briefly discuss the spectra of D-branes for both alpha and beta families. 
  We present a powerful method to generate various equations which possess the Lax representations on noncommutative (1+1) and (1+2)-dimensional spaces. The generated equations contain noncommutative integrable equations obtained by using the bicomplex method and by reductions of the noncommutative (anti-)self-dual Yang-Mills equation. This suggests that the noncommutative Lax equations would be integrable and be derived from reductions of the noncommutative (anti-)self-dual Yang-Mills equation, which implies the noncommutative version of Richard Ward conjecture. The integrability and the relation to string theories are also discussed. 
  We refute the claim that previous works on the one-loop quantum mass of solitons had incorrectly dropped a surface term from a partial integration. Rather, the paper quoted in the title contains a fallacious derivation with two compensating errors. We also remark that the $\phi^2\cos^2\ln(\phi^2)$ model considered in that paper does not have solitons at the quantum level because at two-loop order the degeneracy of the vacua is lifted. This may be remedied, however, by a supersymmetric extension. 
  The SL(2, R) WZW model of strings on an ADS3 background is investigated in the spirit of J.Maldacena's and H.Ooguri's approach (hep-th/0001053) and (hep-th/0005183). Choosing a standard, but most general three-variable parametrization of the SL(2, R) group element g, the system of equations for the Operator Product Expansion (OPE) relations is analysed. In the investigated SL(2, R) case, this system is consistent if each three points on the complex plane lie on a certain hypersurface in CP3. A system of three nonlinear first-order differential equations has been obtained for the parametrization functions. It was demonstrated also how the mathematical apparatus of generalized functions and integral geometry can be implemented in order to modify the integral operators, entering the Kac-Moody and Virasoro algebras. 
  There exist local infinitesimal redefinitions of the fermionic fields, which may be used to modify the strength of the coupling for the interaction term in massless QED3. Under those (formally unitary) transformations, the functional integration measure changes by an anomalous Jacobian, which (after regularization) yields a term with the same structure as the quadratic parity-conserving term in the effective action. Besides, the Dirac operator is affected by the introduction of new terms, apart from the modification in the minimal coupling term. We show that the result coming from the Jacobian, plus the effect of those new terms, add up to reproduce the exact quadratic term in the effective action. Finally, we also write down the form a finite decoupling transformation would have, and comment on the unlikelihood of that transformation to yield a helpful answer to the non-perturbative evaluation of the fermionic determinant. 
  In the AdS/CFT correspondence, wrapped D3-branes (such as "giant gravitons") on the string theory side of the correspondence have been identified with Pfaffian, determinant and subdeterminant operators on the field theory side. We substantiate this identification by showing that the presence of pairs of such operators in a correlation function of a large N gauge theory naturally leads to a modified 't Hooft expansion including also worldsheets with boundaries. This happens independently of supersymmetry or conformal invariance. 
  Relations between two definitions of (total) angular momentum operator, as a generator of rotations and in the Lagrangian formalism, are explored in quantum field theory. Generally, these definitions result in different angular momentum operators, which are suitable for different purposes in the theory. From the spin and orbital angular momentum operators (in the Lagrangian formalism) are extracted additive terms which are conserved operators and whose sum is the total angular momentum operator. 
  We define the chiral zero modes' phase space of the G=SU(n) Wess-Zumino-Novikov-Witten model as an (n-1)(n+2)-dimensional manifold M_q equipped with a symplectic form involving a special 2-form - the Wess-Zumino (WZ) term - which depends on the monodromy M. This classical system exhibits a Poisson-Lie symmetry that evolves upon quantization into an U_q(sl_n) symmetry for q a primitive even root of 1. For each constant solution of the classical Yang-Baxter equation we write down explicitly a corresponding WZ term and invert the symplectic form thus computing the Poisson bivector of the system. The resulting Poisson brackets appear as the classical counterpart of the exchange relations of the quantum matrix algebra studied previously. We argue that it is advantageous to equate the determinant D of the zero modes' matrix to a pseudoinvariant under permutations q-polynomial in the SU(n) weights, rather than to adopt the familiar convention D=1. 
  Unified theory of gravitational interactions and electromagnetic interactions is discussed in this paper. Based on gauge principle, electromagnetic interactions and gravitational interactions are formulated in the same manner and are unified in a semi-direct product group of U(1) Abel gauge group and gravitational gauge group. 
  In the Berkooz-Douglas matrix model of M theory in the presence of longitudinal $M5$-brane, we investigate the effective dynamics of the system by considering the longitudinal $M5$-brane as the background and the spherical $M5$-brane related with the other space dimensions as the probe brane. Due to there exists the background field strength provided by the source of the longitudinal $M5$-brane, an open membrane should be ended on the spherical $M5$-brane based on the topological reason. The formation of the bound brane configuration for the open membrane ending on the 5-branes in the background of longitudinal 5-brane can be used to model the 4-dimensional quantum Hall system proposed recently by Zhang and Hu. The description of the excitations of the quantum Hall soliton brane configuration is established by investigating the fluctuations of $D0$-branes living on the bound brane around their classical solution derived by the transformations of area preserving diffeomorphisms of the open membrane. We find that this effective field theory for the fluctuations is an SO(4) non-commutative Chern-Simons field theory. The matrix regularized version of this effective field theory is given in order to allow the finite $D0$-branes to live on the bound brane. We also discuss some possible applications of our results to the related topics in M-theory and to the 4-dimensional quantum Hall system. 
  The classical formal equivalence upon a redefinition of the gauge connection between Chern-Simons theory and topological massive Yang-Mills theory in three-dimensional Euclidean space-time is analyzed at the quantum level within the BRST formulation of the Equivalence Theorem. The parameter controlling the change in the gauge connection is the inverse $\lambda$ of the topological mass. The BRST differential associated with the gauge connection redefinition is derived and the corresponding Slavnov-Taylor (ST) identities are proven to be anomaly-free. The Green functions of local operators constructed only from the ($\lambda$-dependent) transformed gauge connection, as well as those of BRST invariant operators, are shown to be independent of the parameter $\lambda$, as a consequence of the validity of the ST identities. The relevance of the antighost-ghost fields, needed to take into account at the quantum level the Jacobian of the change in the gauge connection, is analyzed. Their role in the identification of the physical states of the model within conventional perturbative gauge theory is discussed. 
  Consistent nontrivial interactions within a special class of covariant mixed-symmetry type tensor gauge fields of degree three are constructed from the deformation of the solution to the master equation combined with specific cohomological techniques. In spacetime dimensions strictly greater than four, the only consistent interaction terms are those gauge invariant under the original symmetry. Only in four spacetime dimensions the gauge symmetry is found deformed. 
  A classical phase space with a suitable symplectic structure is constructed together with functions which have Poisson brackets algebraically identical to the Lie algebra structure of the Lie group SU(n). In this phase space we show that the orbit of the generators corresponding to the simple roots of the Lie algebra give rise to fibres that are complex lines containing spheres. There are n-1 spheres on a fibre and they intersect in exactly the same way as the Cartan matrix of the Lie algebra. This classical phase space bundle,being compact,has a description as a variety.Our construction shows that the variety containing the intersecting spheres is exactly the one obtained by resolving the singularities of the variety {x_0}{x_1}-{{x_2}^n}=0 in {C^3}. A direct connection between this singular variety and the classical phase space corresponding to the Lie group SU(n) is thus established. 
  This paper reexamines the question of vacuum decay in theories of quantum gravity. In particular it suggests that decay into stable flat or AdS vacua, never occurs. Instead, vacuum decay occurs, if at all, into a cosmological spacetime. If the latter has negative cosmological constant, it undergoes a Big Crunch, which suggests that the whole picture is inconsistent. The question of decay of de Sitter space must be very carefully defined. 
  We study a new mechanism for the electromagnetic gauging of chiral bosons showing that new possibilities emerge for the interacting theory of chiral scalars. We introduce a chirally coupled gauge field necessary to mod out the degree of freedom that obstructs gauge invariance in a system of two opposite chiral bosons soldering them together. 
  We study the relation between the Type IIB (NSNS and RR) 2-form fields and the (complex) gauge coupling constant of the 4D N=2 SU(Nc) super Yang-Mills theory with Nf fundamental matters. We start from the analysis of the D2-brane world volume theory with heavy Nc quarks on the Nf D6 supergravity background. After a sequence of T- and S-dualities, we obtain the (generalized) 2-forms in the configuration with Nc D5-branes wrapping on the vanishing two-cycle under the influence of the background. These 2-forms shows the same behavior as the gauge coupling constant of the 4D N=2 super QCD. The background reduces to the Z_{Nf} orbifold in the twelve-dimensional space-time formally realized by introducing the two parameters as the additional space coordinates. The 10D gravity dual is suggested as the 2D flip in this twelve-dimensional space-time. In the case of Nf=2Nc, this gravity dual becomes AdS_5 \times S^5/Z_2 with D3-charge which depends on the constant generalized NSNS 2-form. This is the result expected from the M-theory QCD configuration. Based on the known exact result, we also discuss this configuration after including the nonperturbative effect. 
  We construct D-branes in the Nappi-Witten (NW) and Guadagnini-Martellini-Mintchev (GMM) gauged WZW models. For the $SL(2,R)\times SU(2)/U(1)\times U(1)$ NW and $SU(2)\times SU(2)/U(1)$ GMM models we present the explicit equations describing the D-brane hypersurfaces in their target spaces. In the latter case we show that the D-branes are classified according to the Cardy theorem. We also present the semiclassical mass computation and find its agreement with the CFT predictions. 
  The consistent form of the gauge anomaly is worked out at first order in $\theta$ for the noncommutative three-point function of the ordinary gauge field of certain noncommutative chiral gauge theories defined by means of the Seiberg-Witten map. We obtain that for any compact simple Lie group the anomaly cancellation condition of this three-point function reads $\traza \T^a \T^b \T^c = 0$, if one restricts the type of noncommutative counterterms that can be added to the classical action to restore the gauge symmetry to those which are renormalizable by power-counting. On the other hand, if the power-counting remormalizability paradigm is relinquished and one admits noncommutative counterterms (of the gauge fields, its derivatives and $\theta$) which are not power-counting renormalizable, then, the anomaly cancellation condition for the noncommutative three-point function of the ordinary gauge field becomes the ordinary one: $\traza \T^a \{\T^b,\T^c\} = 0$. 
  In this work we show that corrections to the Newton's second law appears if we assume that the phase space has a symplectic structure consistent with the rules of commutation of noncommutative quantum mechanis. In the central field case we find that the correction term breaks the rotational symmetry. In particular, for the Kepler problem, this term takes the form of a Coriolis force produced by the weak gravitational field far from a rotating massive object. 
  Gauge invariant generation of mass for supersymmetric U(1) vector field through use of a chiral Stueckelberg superfield is considered. When a Fayet-Iliopoulos D term is also present, no breaking of supersymmetry ever occurs so long as the Stueckelberg mass is not zero. A moduli space in which gauge symmetry is spontaneously broken arises in this case. 
  We propose that local Lorentz invariance is spontaneously violated at high energies, due to a nonvanishing vacuum expectation value of a vector field \phi^\mu, as a possible explanation of the observation of ultra-high energy cosmic rays with an energy above the GZK cutoff. Certain consequences of spontaneous breaking of Lorentz invariance in cosmology are discussed. 
  We formulate a new bootstrap principle which allows for the construction of particle spectra involving unstable as well as stable particles. We comment on the general Lie algebraic structure which underlies theories with unstable particles and propose several new scattering matrices. We find a new Lie algebraic decoupling rule, which predicts the renormalization group flow in dependence of the relative ordering of the resonance parameters. The proposals are exemplified for some concrete theories which involve unstable particles, such as the homogeneous sine-Gordon models and their generalizations. The new decoupling rule can be validated by means of our new bootstrap principle and also via the renormalization group flow, which we obtain from a thermodynamic Bethe ansatz analysis. 
  We discuss the renormalization properties of noncommutative non-gauge supersymmetric field theories. 
  Motivated by recent work of Dijkgraaf and Vafa, we study anomalies and the chiral ring structure in a supersymmetric U(N) gauge theory with an adjoint chiral superfield and an arbitrary superpotential. A certain generalization of the Konishi anomaly leads to an equation which is identical to the loop equation of a bosonic matrix model. This allows us to solve for the expectation values of the chiral operators as functions of a finite number of ``integration constants.'' From this, we can derive the Dijkgraaf-Vafa relation of the effective superpotential to a matrix model. Some of our results are applicable to more general theories. For example, we determine the classical relations and quantum deformations of the chiral ring of $\N=1$ super Yang-Mills theory with SU(N) gauge group, showing, as one consequence, that all supersymmetric vacua of this theory have a nonzero chiral condensate. 
  We review the derivation of a noncommutative version of the nonlinear sigma model on $\CPn$ and it's soliton solutions for finite $\theta$ emphasizing the similarities it bears to the GMS scalar field theory. It is also shown that unlike the scalar theory, some care needs to be taken in defining the topological charge of BPS solitons of the theory due to nonvanishing surface terms in the energy functional. Finally it is shown that, like its commutative analogue, the noncommutative $\CPn$-model also exhibits a non-BPS sector. Unlike the commutative case however, there are some surprises in the noncommutative case that merit further study. 
  Building upon an earlier proposal for the classification of fluxes, a sequence is proposed which generalizes the AHSS by computing type IIB string theory's group of conserved RR and also NS charges, which is conjectured to be a K-theory of dual pairs. As a test, the formalism of Maldacena, Moore and Seiberg (hep-th/0108100) is applied to classify D-branes, NS5-branes, F-strings and their dielectric counterparts in IIB compactified on a 3-sphere with both NS and RR background fluxes. The soliton spectra on the 3-sphere are then compared with the output of the sequence, as is the baryon spectrum in Witten's non-spin^c example, AdS^5xRP^5. The group of conserved charges is seen to change during Brown-Teitelboim-like phase transitions which change the effective cosmological constant. 
  I review the proposal of Berenstein-Douglas for a completely general definition of Seiberg duality. To give evidence for their conjecture I present the first example of a physical dual pair and explicitly check that it satisfies the requirements. Then I explicitly show that a pair of toric dual quivers is also dual according to their proposal. All these computations go beyond tilting modules, and really work in the derived category. I introduce all necessary mathematics where needed. 
  The physical mass scales that determine the behaviour of general (simply-laced) Homogeneous Sine-Gordon models are investigated by means of a study of their finite-size effects, using the thermodynamic Bethe ansatz. These models describe integrable multiparameter perturbations of the theory of level-$k$ $G$-parafermions, where $G$ is a Lie group. The parameters can be related to adjustable mass scales of stable and unstable particles. Our results confirm the presence of unstable particle states at generic values of $k$, as predicted at large $k$ by semiclassical arguments. 
  ``Dimension bubbles'' of the type previously studied by Blau and Guendelman [S.K. Blau and E.I. Guendelman, Phys. Rev. D40, 1909 (1989)], which effectively enclose a region of 5d spacetime and are surrounded by a region of 4d spacetime, can arise in a 5d theory with a compact extra dimension that is dimensionally reduced to give an effective 4d theory. These bubbles with thin domain walls can be stabilized against total collapse in a rather natural way by a scalar field which, as in the case with ``ordinary'' nontopological solitons, traps light scalar particles inside the bubble. 
  This paper has been withdrawn by the authors. It has been superseded by hep-th/0309154 
  We show that D-branes in the Euclidean $AdS_3$ can be naturally associated to the maximally isotropic subgroups of the Lu-Weinstein double of SU(2). This picture makes very transparent the residual loop group symmetry of the D-brane configurations and gives also immediately the D-branes shapes and the $\sigma$-model boundary conditions in the de Sitter T-dual of the $SL(2,C)/SU(2)$ WZW model. 
  The explicit (all-order in fermions) form of the kappa-symmetric $D3$ brane probe action was previously found in the two maximally supersymmetric type IIB vacua: flat space and $AdS_5 \times S^5$. Here we present the form of the action in the third maximally supersymmetric type IIB background: gravitational plane wave supported by constant null 5-form strength. We study $D3$ brane action in both covariant and light cone kappa symmetry gauges. Like the fundamental string action, the $D3$ brane action takes a simple form once written in the light cone kappa-symmetry gauge. We also consider the $\cN=4$ SYM theory in 4d plane wave background. Since some (super)symmetries of plane wave SYM action are friendly to (super)symmetries of the type IIB superstring in plane wave Ramond-Ramond background we suggest this SYM model may be useful in the context of AdS/CFT duality. We develop the Hamiltonian light cone gauge formulation for this theory. 
  The hedgehog Skyrme model on three-sphere admits very rich spectrum of solitonic solutions which can be encompassed by a strikingly simple scheme. The main result of this paper is the statement of the tripartite structure of solutions of the model and the discovery in what configurations these solutions appear. The model contains features of more complicated models in General Relativity and as such can give insight into them. 
  We present an improved effective action for the D-brane-anti-D-brane system obtained from boundary superstring field theory. Although the action looks highly non-trivial, it has simple explicit multi-vortex (i.e. codimension-2 multi-BPS D-brane) multi-anti-vortex solutions. The solutions have a curious degeneracy corresponding to different ``magnetic'' fluxes at the core of each vortex. We also generalize the brane anti-brane effective action that is suitable for the study of the inflationary scenario and the production of defects in the early universe. We show that when a brane and anti-brane are distantly separated, although the system is classically stable it can decay via quantum tunneling through the barrier. 
  We consider probe p-branes and Dp-branes dynamics in D-dimensional string theory backgrounds of general type. Unified description for the tensile and tensionless branes is used. We obtain exact solutions of their equations of motion and constraints in static gauge as well as in more general gauges. Their dynamics in the whole space-time is also analysed and exact solutions are found. 
  We describe a new class of supersymmetric string compactifications to 4d Minkowski space. These solutions involve type II strings propagating on (orientifolds of) non Calabi-Yau spaces in the presence of background NS and RR fluxes. The simplest examples have descriptions as cosets, generalizing the three-dimensional nilmanifold. They can also be thought of as twisted tori. We derive a formula for the (super)potential governing the light fields, which is generated by the fluxes and certain ``twists'' in the geometry. Detailed consideration of an example also gives strong evidence that in some cases, these exotic geometries are related by smooth transitions to standard Calabi-Yau or G2 compactifications of M-theory. 
  We show that the recently demonstrated absence of the van Dam-Veltman-Zakharov discontinuity for massive spin 3/2 with a Lambda term is an artifact of the tree level approximation, and that the discontinuity reappears at one loop. As a numerical check on the calculation, we rederive the vanishing of the one- loop beta function for D=11 supergravity on AdS_4 x S^7 level-by-level in the Kaluza-Klein tower. 
  We re-examine the projective lightcone limit of the gauge-invariant Green-Schwarz action on 5D anti-de Sitter x the five-sphere. It implies the usual holography for AdS5, but also (a complex) one for S5. The result is N=4 projective superspace, which unlike N=4 harmonic superspace can describe N=4 super Yang-Mills off shell. 
  We consider sl(2) minimal conformal field theories on a cylinder from a lattice perspective. To each allowed one-dimensional configuration path of the A_L Restricted Solid-on-Solid (RSOS) models we associate a physical state |h> and a monomial in a finite fermionic algebra. The orthonormal states produced by the action of these monomials on the primary states generate finite Virasoro modules with dimensions given by the finitized Virasoro characters $\chi^{(N)}_h(q)$. These finitized characters are the generating functions for the double row transfer matrix spectra of the critical RSOS models. We argue that a general energy-preserving bijection exists between the one-dimensional configuration paths and the eigenstates of these transfer matrices and exhibit this bijection for the critical and tricritical Ising models in the vacuum sector. Our results extend to Z_{L-1} parafermion models by duality. 
  We consider sl(2) minimal conformal field theories and the dual parafermion models. Guided by results for the critical A_L Restricted Solid-on-Solid (RSOS) models and its Virasoro modules expressed in terms of paths, we propose a general level-by-level algorithm to build matrix representations of the Virasoro generators and chiral vertex operators (CVOs). We implement our scheme for the critical Ising, tricritical Ising, 3-state Potts and Yang-Lee theories on a cylinder and confirm that it is consistent with the known two-point functions for the CVOs and energy-momentum tensor. Our algorithm employs a distinguished basis which we call the L_1-basis. We relate the states of this canonical basis level-by-level to orthonormalized Virasoro states. 
  Referring to the supermultiplet of N=1 supergravity (SUGRA) the linearization of N=1 SGM action describing the nonlinear supersymmetric (NL SUSY) gravitational interaction of superon (Nambu-Goldstone fermion) is attempted. The field contents of on-shell SUGRA supermultiplet are realized as the composites, though they have new SUSY transformations which closes on super-Poincar\'e algebra. Particular attentions are paid to the local Lorentz invariance. 
  Free string theory on the plane-wave background displays a discrete Z2 symmetry exchanging the two transverse SO(4) rotation groups. This symmetry should be respected also at the interacting level. We show that the zero mode structure proposed in hep-th/0208148 can be completed to a full kinematical vertex, contrary to claims appeared in the previous literature. We also comment on the relation with recent works on the string-bit formalism and on the comparison with the field theory side of the correspondence. 
  We explicitly write down the Feynman rules following the work of Dijkfraaf, Vafa and collaborators for N=1 super Yang-Mills having products of SU groups as the gauge group and matter chiral superfields in adjoint, fundamental, and bi-fundamental representations without baryonic perturbations. As an application of this, we show expectation values calculated by these methods satisfy the Konishi anomaly relation. 
  We investigate the effect of the noncommutative geometry on the classical orbits of particles in a central force potential. The relation is implemented through the modified commutation relations $[x_i, x_j]=i \theta_{ij} $. Comparison with observation places severe constraints on the value of the noncommutativity parameter. 
  We present a new class of charged rotating solutions in the Einstein-Gauss-Bonnet gravity with a negative cosmological constant. These solutions may be interpreted as black brane solutions with two inner and outer event horizons or an extreme black brane depending on the value of the mass parameter $m$. We also find that the Killing vectors are the null generators of the event horizon. The physical properties of the brane such as the temperature, the angular velocity, the entropy, the electric charge and potential are computed. We also compute the action and the Gibbs potential as a function of temperature and angular velocity for the uncharged solutions, and compute the angular momentum and the mass of the black brane through the use of Gibbs potential. We show that these thermodynamic quantities satisfy the first law of thermodynamics. We also perform a local stability analysis of the asymptotically AdS uncharged rotating black brane in various dimensions and show that they are locally stable for the whole phase space both in the canonical and grand-canonical ensemble. We found that the thermodynamic properties of Gauss-Bonnet rotating black branes are completely the same as those without the Gauss-Bonnet term, although the two solutions are quite different. 
  We find new supersymmetric solutions of the massive supergravity theory which can be constructed by generalized Scherk-Schwarz dimensional reduction of eleven dimensional supergravity, using the scaling symmetry of the equations of motion. Firstly, we construct field configurations which solve the ten dimensional equations of motion by reducing on the radial direction of Ricci-flat cones. Secondly, we will extend this result to the supersymmetric case by performing a dimensional reduction along the flow of a homothetic Killing vector which is the Euler vector of the cone plus a boost. 
  We study the noncommutative generalization of (euclidean) integrable models in two-dimensions, specifically the sine- and sinh-Gordon and the U(N) principal chiral models. By looking at tree-level amplitudes for the sinh-Gordon model we show that its na\"\i ve noncommutative generalization is {\em not} integrable. On the other hand, the addition of extra constraints, obtained through the generalization of the zero-curvature method, renders the model integrable. We construct explicit non-local non-trivial conserved charges for the U(N) principal chiral model using the Brezin-Itzykson-Zinn-Justin-Zuber method. 
  We give a recipe relating holomorphic quantities in supersymmetric field theory to their descendants in non-supersymmetric Z_2 orbifold field theories. This recipe, consistent with a recent proposal of Strassler, gives exact results for bifermion condensates, domain wall tensions and gauge coupling constants in the planar limit of the orbifold theories. 
  We discuss the causal structure of pp-wave spacetimes using the ideal point construction outlined by Geroch, Kronheimer, and Penrose. This generalizes the recent work of Marolf and Ross, who considered similar issues for plane wave spacetimes. We address the question regarding the dimension of the causal boundary for certain specific pp-wave backgrounds. In particular, we demonstrate that the pp-wave spacetime which gives rise to the N = 2 sine-Gordon string world-sheet theory is geodesically complete and has a one-dimensional causal boundary. 
  We present a linearized gravity investigation of the bent braneworld, where an AdS_4 brane is embedded in AdS_5. While we focus on static spherically symmetric mass distributions on the brane, much of the analysis continues to hold for more general configurations. In addition to the identification of the massive Karch-Randall graviton and a tower of Kaluza-Klein gravitons, we find a radion mode that couples to the trace of the energy-momentum tensor on the brane. The Karch-Randall radion arises as a property of the embedding of the brane in the bulk space, even in the context of a single brane model. 
  Some of the recent important developments in understanding string/ gauge dualities are based on the idea of highly symmetric motion of ``string solitons'' in $AdS_5\times S^5$ geometry originally suggested by Gubser, Klebanov and Polyakov. In this paper we study symmetric motion of short strings in the presence of antisymetric closed string B field. We compare the values of the energy and the spin in the case of non-vanishing B field with those obtained in the case of B=0. The presence of NS-NS antisymmetric field couples the fluctuation modes that indicates changes in the quantum corrections to the energy spectrum. 
  We obtain explicit formulas for the Neumann coefficients and associated quantities that appear in the three-string vertex for type IIB string theory in a plane-wave background, for any value of the mass parameter mu. The derivation involves constructing the inverse of a certain infinite-dimensional matrix, in terms of which the Neumann coefficients previously had been written only implicitly. We derive asymptotic expansions for large mu and find unexpectedly simple results, which are valid to all orders in 1/mu. Using BMN duality, these give predictions for certain gauge theory quantities to all orders in the modified 't Hooft coupling lambda'. A specific example is presented. 
  In this paper we show that in the presence of an anti-symmetric tensor $B$-background, Witten's star algebra for open string fields persists to possess the structure of a direct product of commuting Moyal pairs. The interplay between the noncommutativity due to three-string overlap and that due to the $B$-background is our main concern. In each pair of noncommutative directions parallel to the $B$-background, the Moyal pairs mix string modes in the two directions and are labeled, in addition to a continuous parameter, by {\it two} discrete values as well. However, the Moyal parameters are $B$-dependent only for discrete pairs. We have also demonstrated the large-$B$ contraction of the star algebra, with one of the discrete Moyal pairs dropping out while the other giving rise to the center-of-mass noncommutative function algebra. 
  We discuss the cosmological evolution of a 3-brane Universe in the presence of energy influx from the bulk. We show that this influx can lead to accelerated expansion on the brane, depending on the equations of state of the bulk and brane matter. The absorption of non-relativistic bulk matter by the brane at an increasing rate leads to a small positive acceleration parameter during the era of matter domination on the brane. On the other hand, the brane expansion remains decelerating during radiation domination. 
  It has been conjectured that string theory in a pp-wave background is dual to a sector of N=4 supersymmetric Yang-Mills theory.   We study the Hagedorn transition for free strings in this background. We find that the free energy at the transition point is finite suggesting a confinement/deconfinement transition in the gauge theory. In the limit of vanishing mass parameter the free energy matches that of free strings on an 8-torus with momentum/winding chemical potential. The entropy in the microcanonical ensemble with fixed energy and fixed momentum/winding is computed in each case. 
  In this paper, we use the matrix model of pure fundamental flavors (without the adjoint field) to check the Seiberg duality in the case of complete mass deformation. We show that, by explicit integration at both sides of electric and magnetic matrix models, the results agree with the prediction in the field theory. 
  We construct all complete metrics of cohomogeneity one G(2) holonomy with S^3 x S^3 principal orbits from gauged supergravity. Our approach rests on a generalization of the twisting procedure used in this framework. It corresponds to a non-trivial embedding of the special Lagrangian three-cycle wrapped by the D6-branes in the lower dimensional supergravity. There are constraints that neatly reduce the general ansatz to a six functions one. Within this approach, the Hitchin system and the flop transformation are nicely realized in eight dimensional gauged supergravity. 
  We review some aspects of the gravity duals of supersymmetric gauge theories, arising in the world-volume of D-branes wrapping supersymmetric cycles of special holonomy manifolds, within the framework of lower dimensional gauged supergravity. 
  This talk is about results obtained by Kirill Melnikov and myself pertaining to the canonical quantization of a massless scalar field in the presence of a Schwarzschild black hole. After a brief summary of what we did and how we reproduce the familiar Hawking temperature and energy flux, I focus attention on how our discussion differs from other treatments. In particular I show that we can define a system which fakes an equilibrium thermodynamic object whose entropy is given by the $A/4$ (where $A$ is the area of the black hole horizon), but for which the assignment of a classical entropy to the system is incorrect. Finally I briefly discuss a discretized version of the theory which seems to indicate that things work in a surprising way near $r=0$. 
  In an attempt to study asymptotically plane wave spacetimes which admit an event horizon, we find solutions to vacuum Einstein's equations in arbitrary dimension which have a globally null Killing field and rotational symmetry. We show that while such solutions can be deformed to include ones which are asymptotically plane wave, they do not posses a regular event horizon. If we allow for additional matter, such as in supergravity theories, we show that it is possible to have extremal solutions with globally null Killing field, a regular horizon, and which, in addition, are asymptotically plane wave. In particular, we deform the extremal M2-brane solution in 11-dimensional supergravity so that it behaves asymptotically as a 10-dimensional vacuum plane wave times a real line. 
  Previously known exactly solvable models of 2D semiclassical dilaton gravity admit, in the general case, only non-extreme black holes. It is shown that there exist exceptional degenerate cases, that can be obtained by some limiting transitions from the general exact solution, which include, in particular, extremal and ultraextremal black holes. We also analyze properties of extreme black holes without demanding exact solvability and show that for such solutions quantum backreaction forbids the existence of ultraextreme black holes. The conditions,under which divergencies of quantum stresses in a free falling frame can disappear, are found. We derive the closed equation with respect to the metric as a function of the dilaton field that enables one, choosing the form of the metric, to restore corresponding Lagrangian. It is demonstrated that exactly solvable models, found earlier, can be extended to include an electric charge only in two cases: either the dilaton-gravitation coupling is proportional to the potential term, or the latter vanishes. The second case leads to the effective potential with a negative amplitude and we analyze, how this fact affects the structure of spacetime. We also discuss the role of quantum backreaction in the relationship between extremal horizons and the branch of solutions with a constant dilaton. 
  The role of Wigner's little group, as an abelian gauge generator in different contexts, is studied. 
  It is shown that the local axial anomaly in $2-$dimensions emerges naturally if one postulates an underlying noncommutative fuzzy structure of spacetime . In particular the Dirac-Ginsparg-Wilson relation on ${\bf S}^2_F$ is shown to contain an edge effect which corresponds precisely to the ``fuzzy'' $U(1)_A$ axial anomaly on the fuzzy sphere . We also derive a novel gauge-covariant expansion of the quark propagator in the form $\frac{1}{{\cal D}_{AF}}=\frac{a\hat{\Gamma}^L}{2}+\frac{1}{{\cal D}_{Aa}}$ where $a=\frac{2}{2l+1}$ is the lattice spacing on ${\bf S}^2_F$, $\hat{\Gamma}^L$ is the covariant noncommutative chirality and ${\cal D}_{Aa}$ is an effective Dirac operator which has essentially the same IR spectrum as ${\cal D}_{AF}$ but differes from it on the UV modes. Most remarkably is the fact that both operators share the same limit and thus the above covariant expansion is not available in the continuum theory . The first bit in this expansion $\frac{a\hat{\Gamma}^L}{2}$ although it vanishes as it stands in the continuum limit, its contribution to the anomaly is exactly the canonical theta term. The contribution of the propagator $\frac{1}{{\cal D}_{Aa}}$ is on the other hand equal to the toplogical Chern-Simons action which in two dimensions vanishes identically . 
  We explore time-symmetric hypersurfaces containing apparent horizons of black objects in a 5d spacetime with one coordinate compactified on a circle. We find a phase transition within the family of such hypersurfaces: the horizon has different topology for different parameters. The topology varies from $S^3$ to $S^2 \times S^1$. This phase transition is discontinuous -- the topology of the horizon changes abruptly. We explore the behavior around the critical point and present a possible phase diagram. 
  We explicitly show that, in a system with T-duality symmetry, the configuration space volume degrees of freedom may hide on the surface boundary of the region of accessible states with energy lower than a fixed value. This means that, when taking the decompactification limit (big volume limit), a number of accessible states proportional to the volume is recovered even if no volume dependence appears when energy is high enough. All this behavior is contained in the exact way of computing sums by making integrals. We will also show how the decompactification limit for the gas of strings can be defined in a microcanonical description at finite volume. 
  Physical arguments stemming from the theory of black-hole thermodynamics are used to put constraints on the dynamics of closed-string tachyon condensation in Scherk--Schwarz compactifications. A geometrical interpretation of the tachyon condensation involves an effective capping of a noncontractible cycle, thus removing the very topology that supports the tachyons. A semiclassical regime is identified in which the matching between the tachyon condensation and the black-hole instability flow is possible. We formulate a generalized correspondence principle and illustrate it in several different circumstances: an Euclidean interpretation of the transition from strings to black holes across the Hagedorn temperature and instabilities in the brane-antibrane system. 
  We discuss various properties of the Seiberg-Witten curve for the E-string theory which we have obtained recently in hep-th/0203025. Seiberg-Witten curve for the E-string describes the low-energy dynamics of a six-dimensional (1,0) SUSY theory when compactified on R^4 x T^2. It has a manifest affine E_8 global symmetry with modulus \tau and E_8 Wilson line parameters {m_i},i=1,2,...,8 which are associated with the geometry of the rational elliptic surface. When the radii R_5,R_6 of the torus T^2 degenerate R_5,R_6 --> 0, E-string curve is reduced to the known Seiberg-Witten curves of four- and five-dimensional gauge theories. In this paper we first study the geometry of rational elliptic surface and identify the geometrical significance of the Wilson line parameters. By fine tuning these parameters we also study degenerations of our curve corresponding to various unbroken symmetry groups. We also find a new way of reduction to four-dimensional theories without taking a degenerate limit of T^2 so that the SL(2,Z) symmetry is left intact. By setting some of the Wilson line parameters to special values we obtain the four-dimensional SU(2) Seiberg-Witten theory with 4 flavors and also a curve by Donagi and Witten describing the dynamics of a perturbed N=4 theory. 
  Starting with the generators of the Poincar\'e group for arbitrary mass (m) and spin (s) a nonunitary transformation is implemented to obtain momenta with an absolute Planck scale limit. In the rest frame (for $m>0$) the transformed energy coincides with the standard one, both being $m$. As the latter tends to infinity under Lorentz transformations the former tends to a finite upper limit $m\coth(lm) = l^{-1}+ O(l)$ where $l$ is the Planck length and the mass-dependent nonleading terms vanish exactly for zero rest mass.The invariant $m^{2}$ is conserved for the transformed momenta. The speed of light continues to be the absolute scale for velocities. We study various aspects of the kinematics in which two absolute scales have been introduced in this specific fashion. Precession of polarization and transformed position operators are among them. A deformation of the Poincar\'e algebra to the SO(4,1) deSitter one permits the implementation of our transformation in the latter case. A supersymmetric extension of the Poincar\'e algebra is also studied in this context. 
  A new model for monopole catalysis of nucleon decay is proposed. Unlike in the earlier one, the only light fields in this model are the photon and Skyrme (pion) field. The model admits the 't Hooft- Polyakov monopole and Skyrmion as classical solutions, while baryon number non-conservation occurs through an anomaly involving an intermediate mass axial vector field resembling W- and Z-bosons. By considering spherically symmetric monopole-Skyrmion configurations, we find that the Skyrmion looses essentially all its mass when interacting with the monopole, which is phenomenologically identical to the monopole catalysis of nucleon decay. Short-distance monopole-Skyrmion physics in this model is interesting too, as there exist almost degenerate metastable monopole-Skyrmion bound states separated by substantial energy barriers. Yet the heights of the latter are much smaller than the physical nucleon mass, so the complete disappearance of the normal undeformed Skyrmion remains perfectly possible in Skyrmion-monopole scattering. 
  Gravitational corrections in N=1 and N=2 supersymmetric gauge theories are obtained from topological string amplitudes. We show how they are recovered in matrix model computations. This provides a test of the proposal by Dijkgraaf and Vafa beyond the planar limit. Both, matrix model and topological string theory, are used to check a conjecture of Nekrasov concerning these gravitational couplings in Seiberg-Witten theory. Our analysis is performed for those gauge theories which are related to the cubic matrix model, i.e. pure SU(2) Seiberg-Witten theory and N=2 U(N) SYM broken to N=1 via a cubic superpotential. We outline the computation of the topological amplitudes for the local Calabi-Yau manifolds which are relevant for these two cases. 
  We explicitly construct cubic interaction light-cone Hamiltonian for the chiral primary system involving the metric fields and the self-dual four-form fields in the IIB pp-wave supergravity. The background fields representing pp-waves exhibit SO(4)*SO(4)*Z_2 invariance. It turns out that the interaction Hamiltonian is precisely the same as that for the dilaton-axion system, except for the fact that the chiral primary system fields have the opposite parity to that of the dilaton-axion fields under the Z_2 transformation that exchanges two SO(4)'s. 
  We study D-branes in the null-brane background. Using the covariant formalism of the worldsheet theory, we construct the boundary states describing D-branes on the null-brane. From the cylinder amplitudes, we find that the D-branes with codimension zero or two have time-dependent effective tensions. 
  We provide a derivation for the particle number densities on phase space for scalar and fermionic fields in terms of Wigner functions. Our expressions satisfy the desired properties: for bosons the particle number is positive, for fermions it lies in the interval between zero and one, and both are consistent with thermal field theory. As applications we consider the Bunch-Davies vacuum and fermionic preheating after inflation. 
  Testing the BMN correspondence at non-zero string coupling g_s requires a one-loop string field theory calculation. At order g_s^2, matrix elements of the light-cone string field theory Hamiltonian between single-string states receive two contributions: the iterated cubic interaction, and a contact term {Q, Q} whose presence is dictated by supersymmetry. In this paper we calculate the leading large mu p^+ alpha' contribution from both terms for the set of intermediate states with two string excitations. We find precise agreement with the basis-independent order g_2^2 results from gauge theory. 
  Holographic considerations may provide a glimpse of quantum gravity beyond what is currently accessible by other means. Here we apply holography to inflationary cosmology. We argue that the appropriate holographic bound on the total entropy in the inflationary perturbations is given by the {\it difference} of the area of the apparent horizon in the slow-roll approximation and its exact value near the end of inflation. This implies that the effective field theory of the inflaton weakly coupled to gravity requires modifications below the Planck scale. For a conventional model of inflation this scale is of order of the GUT scale, $\Lambda_{UV} \la 10^{16} GeV$, but could be considerably lower. Signatures of such new physics could show up in the CMB. 
  We study a defect conformal field theory describing D3-branes intersecting over two space-time dimensions. This theory admits an exact Lagrangian description which includes both two- and four-dimensional degrees of freedom, has (4,4) supersymmetry and is invariant under global conformal transformations. Both two- and four-dimensional contributions to the action are conveniently obtained in a two-dimensional (2,2) superspace. In a suitable limit, the theory has a dual description in terms of a probe D3-brane wrapping an AdS_3 x S^1 slice of AdS_5 x S^5. We consider the AdS/CFT dictionary for this set-up. In particular we find classical probe fluctuations corresponding to the holomorphic curve wy=c\alpha^{\prime}. These fluctuations are dual to defect fields containing massless two-dimensional scalars which parameterize the classical Higgs branch, but do not correspond to states in the Hilbert space of the CFT. We also identify probe fluctuations which are dual to BPS superconformal primary operators and to their descendants. A non-renormalization theorem is conjectured for the correlators of these operators, and verified to order g^2. 
  A bosonic string in twenty six dimensions is effectively reduced to four dimensions by eleven Majorana fermions which are vectors in the bosonic represetation SO(d-1,1). By dividing the fermions in two groups, actions can be written down which are world sheet supersymmetric, 2-d local and local 4-d supersymmetric. The novel string is anomally free, free of ghosts and the partition function is modular invariant. 
  Considering our (3+1)-dimensional space-time as, in some way, discrete or l attice with a parameter $a=\lambda_P$, where $\lambda_P$ is the Planck length, we have investigated the additional contributions of lattice artifact monopoles to beta-functions of the renormalisation group equations for the running fine structure constants $\alpha_i(\mu)$ (i=1,2,3 correspond to the U(1), SU(2) and SU(3) gauge groups of the Standard Model) in the Family Replicated Gauge Group Model (FRGGM) which is an extension of the Standard Model at high energies. It was shown that monopoles have $N_{fam}$ times smaller magnetic charge in FRGGM than in SM ($N_{fam}$ is the number of families in FRGGM). We have estimated al so the enlargement of a number of fermions in FRGGM leading to the suppression of the asymptotic freedom in the non-Abelian theory. We have shown that, in contrast to the case of AntiGUT when the FRGGM undergoes the breakdown at $\mu=\mu_G\sim 10^{18}$ GeV, we have the possibility of unification if the FRGGM-breakdown occurs at $\mu_G\sim 10^{14}$ GeV. By numerical calculations we obtained an example of the unification of all gauge interactions (including gravity) at the scale $\mu_{GUT}\approx 10^{18.4}$ GeV. We discussed the possibility of $[SU(5)]^3$ or $[SO(10)]^3$ (SUSY or not SUSY) unifications. 
  We analytically calculate the topological charge of the noncommutative ADHM U(N) k-instanton using the Corrigan's identity and find that the result is exactly the instanton number k. 
  We consider reduced Super Yang-Mills Theory in $d+1$ dimensions, where $d=2,3,5,9$. We present commutators to prove that for $d=3,5$ and 9 a possible ground state must be a $Spin(d)$ singlet. We also discuss the case $d=2$, where we give an upper bound on the total angular momentum and show that for odd dimensional gauge group no $Spin(d)$ invariant state exists in the Hilbert space. 
  The quantum Yang-Mills theory describing dual ($\tilde g$) and non-dual ($g$) charges and revealing the generalized duality symmetry was developed by analogy with the Zwanziger formalism in QED. 
  We present important elements of a gauge and diffeomorphism invariant formulation of the moduli space approximation to soliton dynamics. We argue that explicit velocity-dependent modifications are determined entirely from gauge and diffeomorphism invariance. We illustrate the formalism for the case of a Yang-Mills theory on a curved spacetime background. 
  Wilson loops in ${\cal N}=4$ supersymmetric Yang-Mills theory correspond at strong coupling to extremal surfaces in $AdS_5$. We study a class of extremal surfaces known as special Legendrian submanifolds. The "hemisphere" corresponding to the circular Wilson loop is an example of a special Legendrian submanifold, and we give another example. We formulate the necessary conditions for the contour on the boundary of $AdS_5$ to be the boundary of the special Legendrian submanifold and conjecture that these conditions are in fact sufficient. We call the solutions of these conditions "special contact Wilson loops". The first order equations for the special Legendrian submanifold impose a constraint on the functional derivatives of the Wilson loop at the special contact contour which should be satisfied in the Yang-Mills theory at strong coupling. 
  We find a concise relation between the moduli $\tau, \rho$ of a rational Narain lattice $\Gamma(\tau,\rho)$ and the corresponding momentum lattices of left and right chiral algebras via the Gauss product. As a byproduct, we find an identity which counts the cardinality of a certain double coset space defined for isometries between the discriminant forms of rank two lattices. 
  In three-dimensional quantum electrodynamics (QED$_{3}$) with massive gauge boson, we investigate the Dyson-Schwinger equation for the fermion self-energy in the Landau gauge and find that chiral symmetry breaking (CSB) occurs when the gauge boson mass $\xi$ is smaller than a finite critical value $\xi_{cv}$ but is suppressed when $\xi > \xi_{cv}$. We further show that the critical value $\xi_{cv}$ does not qualitatively change after considering higher order corrections from the wave function renormalization and vertex function. Based on the relation between CSB and the gauge boson mass $\xi$, we give a field theoretical description of the competing antiferromagnetic and superconducting orders and, in particular, the coexistence of these two orders in high temperature superconductors. When the gauge boson mass $\xi$ is generated via instanton effect in a compact QED$_{3}$ of massless fermions, our result shows that CSB coexists with instanton effect in a wide region of $\xi$, which can be used to study the confinement-deconfinement phase transition. 
  System of a D-brane in bosonic string theory on a constant $B$ field background is studied in order to obtain further insight into the bulk-boundary duality. Boundary states which describe arbitrary numbers of open-string tachyons and gluons are given. UV behaviors of field theories on the non-commutative world-volume are investigated by using these states. We take zero-slope limits of generating functions of one-loop amplitudes of gluons (and open-string tachyons) in which the region of the small open-string proper time is magnified. Existence of $B$ field allows the limits to be slightly different from the standard field theory limits of closed-string. They enable us to capture world-volume theories at a trans-string scale. In this limit the generating functions are shown to be factorized by two curved open Wilson lines (and their analogues) and become integrals on the space of paths with a Gaussian distribution around straight lines. These indicate a possibility that field theories on the non-commutative world-volume are topological at such a trans-string scale. We also give a proof of the Dhar-Kitazawa conjecture by making an explicit correspondence between the closed-string states and the paths. Momentum eigenstates of closed-string or momentum loops also play an important role in these analyses. 
  In this paper we give explicit gauge invariant Lagrangian formulation for massive theories based on mixed symmetry tensors \Phi_{[\mu\nu],\alpha}, T_{[\mu\nu\alpha],\beta} and R_{[\mu\nu],[\alpha\beta]} both in Minkowski as well as in (Anti) de Sitter spaces. In particular, we study all possible massless and partially massless limits for such theories in (A)dS. 
  In this paper we continue the investigation, within the context of the Dijkgraaf-Vafa Programme, of Seiberg duality in matrix models as initiated in hep-th/0211202, by allowing degenerate mass deformations. In this case, there are some massless fields which remain and the theory has a moduli space. With this illustrative example, we propose a general methodology for performing the relevant matrix model integrations and addressing the corresponding field theories which have non-trivial IR behaviour, and which may or may not have tree-level superpotentials. 
  We study the conformal field theories corresponding to current superalgebras $osp(2|2)^{(1)}_k$ and $osp(2|2)^{(2)}_k$. We construct the free field realizations, screen currents and primary fields of these current superalgebras at general level $k$. All the results for $osp(2|2)^{(2)}_k$ are new, and the results for the primary fields of $osp(2|2)^{(1)}_k$ also seem to be new. Our results are expected to be useful in the supersymmetric approach to Gaussian disordered systems such as random bond Ising model and Dirac model. 
  We study the hermitean and normal two matrix models in planar approximation for an arbitrary number of eigenvalue supports. Its planar graph interpretation is given. The study reveals a general structure of the underlying analytic complex curve, different from the hyperelliptic curve of the one matrix model. The matrix model quantities are expressed through the periods of meromorphic generating differential on this curve and the partition function of the multiple support solution, as a function of filling numbers and coefficients of the matrix potential, is shown to be the quasiclassical tau-function. The relation to softly broken N=1 supersymmetric Yang-Mills theories is discussed. A general class of solvable multimatrix models with tree-like interactions is considered. 
  In this paper, we present a general modified dispersion relation derived from q-deformed noncommutative theory and apply it to the ultrahigh energy cosmic ray and the TeV-photon paradoxes--threshold anomalies. Our purpose is not only trying to solve these puzzles by noncommutative theory but also to support noncommutative theory through the coincidence of the region in the parameter space for resolving the threshold anomalies with the one from the q-deformed noncommutative theory. 
  The Moyal star formulation of string field theory is reviewed. The various versions of the star product are compared and related to one another in a regulated theory that resolves associativity anomalies. A summary of computations and challenges is given. 
  The transformation properties of a Kalb-Ramond field are those of a gauge field; however, it is not clear which is the group structure these transformations are associated with. The purpose of this letter is to establish a basic framework in order to clarify the group structure underneath the 2-form gauge potential. 
  We investigate the behaviour of a model with two oppositely charged scalar fields. In the Bogomol'nyi limit this may be seen as the scalar sector of N=1 supersymmetric QED, and it has been shown that cosmic strings form. We examine numerically the model out of the Bogomol'nyi limit, and show that this remains the case. We then add supersymmetry-breaking mass terms to the supersymmetric model, and show that strings still survive.  Finally we consider the extension to N=2 supersymmetry with supersymmetry-breaking mass terms, and show that this leads to the formation of stable cosmic strings, unlike in the unbroken case. 
  We provide evidence of the relation between supersymmetric gauge theories and matrix models beyond the planar limit. We compute gravitational R^2 couplings in gauge theories perturbatively, by summing genus one matrix model diagrams. These diagrams give the leading 1/N^2 corrections in the large N limit of the matrix model and can be related to twist field correlators in a collective conformal field theory. In the case of softly broken SU(N) N=2 super Yang-Mills theories, we find that these exact solutions of the matrix models agree with results obtained by topological field theory methods. 
  In the framework of perturbative quantum field theory a new, universal renormalization condition (called Master Ward Identity) was recently proposed by one of us (M.D.) in a joint paper with F.-M. Boas. The main aim of the present paper is to get a better understanding of the Master Ward Identity by analyzing its meaning in classical field theory. It turns out that it is the most general identity for classical local fields which follows from the field equations. It is equivalent to a generalization of the Schwinger-Dyson Equation and is closely related to the Quantum Action Principle of Lowenstein and Lam.    As a byproduct we give a self-contained treatment of Peierls' manifestly covariant definition of the Poisson bracket. 
  By using symplectic Majorana spinors as Grassmann coordinates in a superspace associated with the supersymmetric extension of the isometry group on the spherical surface S_2, it proves possible to formulate supersymmetric models on S_2 using superspace techniques. 
  We present a general procedure for calculating one-loop ``Casimir'' energy densities for a scalar field coupled to a fixed potential in renormalized quantum field theory. We implement direct subtraction of counterterms computed precisely in dimensional regularization with a definite renormalization scheme. Our procedure allows us to test quantum field theory energy conditions in the presence of background potentials spherically symmetric in some dimensions and independent of others. We explicitly calculate the energy density for several examples. For a square barrier, we find that the energy is negative and divergent outside the barrier, but there is a compensating divergent positive contribution near the barrier on the inside. We also carry out calculations with exactly solvable $\sech^2$ potentials, which arise in the study of solitons and domain walls. 
  We consider the prepotential of Dijkgraaf and Vafa (DV) as one more (and in fact, singular) example of the Seiberg-Witten (SW) prepotentials and discuss its properties from this perspective. Most attention is devoted to the issue of complete system of moduli, which should include not only the sizes of the cuts (in matrix model interpretation), but also their positions, i.e. the number of moduli should be almost doubled, as compared to the DV consideration. We introduce the notion of regularized DV system (not necessarilly related to matrix model) and discuss the WDVV equations. These definitely hold before regularization is lifted, but an adequate limiting procedure, preserving all ingredients of the SW theory, remains to be found. 
  We demonstrate the effectivity of the covariant background field method by means of an explicit calculation of the 3-loop beta-function for a pure Yang-Mills theory. To maintain manifest background invariance throughout our calculation, we stay in coordinate space and treat the background field non-perturbatively. In this way the presence of a background field does not increase the number of vertices and leads to a relatively small number of vacuum graphs in the effective action. Restricting to a covariantly constant background field in Fock-Schwinger gauge permits explicit expansion of all quantum field propagators in powers of the field strength only. Hence, Feynman graphs are at most logarithmically divergent. At 2-loop order only a single Feynman graph without subdivergences needs to be calculated. At 3-loop order 24 graphs remain. Insisting on manifest background gauge invariance at all stages of a calculation is thus shown to be a major labor saving device. All calculations were performed with Mathematica in view of its superior pattern matching capabilities. Finally, we describe briefly the extension of such covariant methods to the case of supergravity theories. 
  A family of spherically symmetric solutions with horizon in the model with multi-component anisotropic fluid (MCAF) is obtained. The metric of any solution contains n-1 Ricci-flat ``internal space'' metrics and for certain equations of state (p_i = \pm \rho) coincides with the metric of intersecting black brane solution in the model with antisymmetric forms. Examples of simulation of intersecting M2 and M5 black branes are considered. The post-Newtonian parameters beta and gamma corresponding to the 4-dimensional section of the metric are calculated. 
  This review describes the role of magnetic symmetry in 2+1 dimensional gauge theories. In confining theories without matter fields in fundamental representation the magnetic symmetry is spontaneously broken. Under some mild assumptions, the low-energy dynamics is determined universally by this spontaneous breaking phenomenon. The degrees of freedom in the effective theory are magnetic vortices. Their role in confining dynamics is similar to that played by pions and sigma in the chiral symmetry breaking dynamics.  I give an explicit derivation of the effective theory in (2+1)-dimensional weakly coupled confining models and argue that it remains qualitatively the same in strongly coupled (2+1)-dimensional gluodynamics. Confinement in this effective theory is a very simple classical statement about the long range interaction between topological solitons, which follows (as a result of a simple direct classical calculation) from the structure of the effective Lagrangian. I show that if fundamentally charged dynamical fields are present the magnetic symmetry becomes local rather than global. The modifications to the effective low energy description in the case of heavy dynamical fundamental matter are discussed. This effective lagrangian naturally yields a bag like description of baryonic excitations. I also discuss the fate of the magnetic symmetry in gauge theories with the Chern-Simons term. 
  We study the corrections induced by a baryon vertex to the superpotential of SQCD with gauge group SU(N) and N quark flavors. We first compute the corrections order by order using a standard field theory technique and derive the corresponding glueball superpotential by "integrating in" the glueball field. The structure of the corrections matches with the expectations from the recently introduced perturbative techniques. We then compute the first non-trivial contribution using this new technique and find exact quantitative agreement. This involves cancellations between diagrams that go beyond the planar approximation. 
  We study the D3/D(-1) brane system and show how to compute instanton corrections to correlation functions of gauge theories in four dimensions using open string techniques. In particular we show that the disks with mixed boundary conditions that are typical of the D3/D(-1) system are the sources for the classical instanton solution. This can then be recovered from simple calculations of open string scattering amplitudes in the presence of D-instantons. Exploiting this fact we also relate this stringy description to the standard instanton calculus of field theory. 
  This paper is concerned with a lattice model which is suited to square-rectangle transformations characterized by two strain components. The microscopic model involves nonlinear and competing interactions, which play a key role in the stability of soliton solutions and emerge from interactions as a function of particle pairs and noncentral type or bending forces. Special attention is devoted to the continuum approximation of the two-dimensional discrete system with the view of including the leading discreteness effects at the continuum description. The long time evolution of the localized structures is governed by an asymptotic integrable equation of the Kadomtsev-Petviashvili I type which allows the explicit construction of moving multi-solitons on the lattice. Numerical simulation performed at the discrete system investigate the stability and dynamics of multi-soliton in the lattice space. 
  A new derivation is given of four-point functions of charge $Q$ chiral primary multiplets in N=4 supersymmetric Yang-Mills theory. A compact formula, valid for arbitrary $Q$, is given which is manifestly superconformal and analytic in the internal bosonic coordinates of analytic superspace. This formula allows one to determine the spacetime four-point function of any four component fields in the multiplets in terms of the four-point function of the leading chiral primary fields. The leading term is expressed in terms of $1/2 Q(Q-1)$ functions of two conformal invariants and a number of single variable functions. Crossing symmetry reduces the number of independent functions, while the OPE implies that the single-variable functions arise from protected operators and should therefore take their free form. This is the partial non-renormalisation property of such four-point functions which can be viewed as a consequence of the OPE and the non-renormalisation of three-point functions of protected operators. 
  We explore the consequences of assuming that the bounded space-time subsets contain a finite number of degrees of freedom. A physically natural hypothesis is that this number is additive for spatially separated subsets. We show that this assumption conflicts with the Lorentz symmetry of Minkowski space since it implies that a conserved current determines the number of degrees of freedom. However, the entanglement across boundaries can lead to a subadditive property for the degrees of freedom of spatially separated sets. We show that this condition and the Poincare symmetry lead to the Bousso covariant entropy bound for Minkowski space. 
  We use matrix model technology to study the N=2 U(N) gauge theory with N_f massive hypermultiplets in the fundamental representation. We perform a completely perturbative calculation of the periods a_i and the prepotential F(a) up to the first instanton level, finding agreement with previous results in the literature. We also derive the Seiberg-Witten curve and differential from the large-M solution of the matrix model. We show that the two cases N_f<N and N \le N_f < 2N can be treated simultaneously. 
  This paper has been withdrawn by the author. 
  We derive an exact equation for simple self non-intersecting Wilson loops in non-abelian gauge theories with gauge fields interacting with fermions in 2-dimensional Euclidean space. 
  We construct solutions of type IIB supergravity with non-trivial Ramond-Ramond 5-form in ten dimensions by replacing the transverse flat space of pp-wave backgrounds with exact $N=(4,4)$ $c=4$ superconformal field theory blocks. These solutions, which also include a dilaton and (in some cases) an anti-symmetric tensor field, lead to integrable models on the world-sheet in the light-cone gauge of string theory. In one instance we demonstrate explicitly the emergence of the complex sine-Gordon model, which coincides with integrable perturbations of the corresponding superconformal building blocks in the transverse space. In other cases we arrive at the supersymmetric Liouville theory or at the complex sine-Liouville model. For axionic instantons in the transverse space, as for the (semi)-wormhole geometry, we obtain an entire class of supersymmetric pp-wave backgrounds by solving the Killing spinor equations as in flat space, supplemented by the appropriate chiral projections; as such, they generalize the usual Neveu-Schwarz five-brane solution of type IIB supergravity in the presence of a Ramond-Ramond 5-form. We also present some further examples of interacting light-cone models and we briefly discuss the role of dualities in the resulting string theory backgrounds. 
  We formulate the $O(3) \s-$ model on fuzzy sphere and construct the Hopf term. We show that the field can be expanded in terms of the ladder operators of Holstein-Primakoff realisation of SU(2) algebra and the corresponding basis set can be classified into different topological sectors by the magnetic quantum numbers. We obtain topological charge $Q$ and show that $-2j\le Q \le2j$. We also construct BPS solitons. Using the covariantly conserved current, we construct the Hopf term and show that its value is $Q^2$ as in the commutative case. We also point out the interesting relation of physical space to deformed SU(2) algebra. 
  We demonstrate by explicit calculation that the first two terms in the CIV-DV prepotential for the two-cut case satisfy the generalized WDVV equations, just as in all other known examples of hyperelliptic Seiberg-Witten models. The WDVV equations are non-trivial in this situation, provided the set of moduli is extended as compared to the Dijkgraaf-Vafa suggestion and includes also moduli, associated with the positions of the cuts (not only with their lengths). Expression for the extra modulus dictated by WDVV equation, however, appears different from a naive expectation implied by the Whitham theory. Moreover, for every value of the "quantum-deformation parameter" 1/g_3, we actually find an entire one-parameter family of solutions to the WDVV equations, of which the conventional prepotential is just a single point. 
  The temperature phase transition in the $N$-component scalar field theory with spontaneous symmetry breaking is investigated using the method combining the second Legendre transform and with the consideration of gap equations in the extrema of the free energy. After resummation of all super daisy graphs an effective expansion parameter, $(1/2N)^{1/3}$, appears near $T_c$ for large $N$. The perturbation theory in this parameter accounting consistently for the graphs beyond the super daisies is developed. A certain class of such diagrams dominant in 1/N is calculated perturbatively. Corrections to the characteristics of the phase transition due to these contributions are obtained and turn out to be next-to-leading order as compared to the values derived on the super daisy level and do not alter the type of the phase transition which is weakly first-order. In the limit $N$ goes to infinity the phase transition becomes second order. A comparison with other approaches is done. 
  We use the superspace method of hep-th/0211017 to prove the matrix model conjecture for N=1 USp(N) and SO(N) gauge theories in four dimensions. We derive the prescription to relate the matrix model to the field theory computations. We perform an explicit calculation of glueball superpotentials. The result is consistent with field theory expectations. 
  We analyze the rather unusual properties of some exact solutions in 2D dilaton gravity for which infinite quantum stresses on the Killing horizon can be compatible with regularity of the geometry. In particular, the Boulware state can support a regular horizon. We show that such solutions are contained in some well-known exactly solvable models (for example, RST). Formally, they appear to account for an additional coefficient $B$ in the solutions (for the same Lagrangian which contains also ''traditional'' solutions) that gives rise to the deviation of temperature $T$ from its Hawking value $T_{H}$. The Lorentzian geometry, which is a self-consistent solution of the semiclassical field equations, in such models, is smooth even at $B\neq 0$ and there is no need to put B=0 ($T=T_{H}$) to smooth it out$.$ We show how the presence of $B\neq 0$ affects the structure of spacetime. In contrast to ''usual'' black holes, full fledged thermodynamic interpretation, including definite value of entropy, can be ascribed (for a rather wide class of models) to extremal horizons, not to nonextreme ones. We find also new exact solutions for ''usual'' black holes (with $T=T_{H}$). The properties under discussion arise in the \QTR{it}{weak}-coupling regime of the effective constant of dilaton-gravity interaction. Extension of features, traced in 2D models, to 4D dilaton gravity leads, for some special models, to exceptional nonextreme black holes having no own thermal properties. 
  The Seiberg-Witten limit of fermionic N=2 string theory with nonvanishing B-field is governed by noncommutative self-dual Yang-Mills theory (ncSDYM) in 2+2 dimensions. Conversely, the self-duality equations are contained in the equation of motion of N=2 string field theory in a B-field background. Therefore finding solutions to noncommutative self-dual Yang-Mills theory on R^{2,2} might help to improve our understanding of nonperturbative properties of string (field) theory. In this paper, we construct nonlinear soliton-like and multi-plane wave solutions of the ncSDYM equations corresponding to certain D-brane configurations by employing a solution generating technique, an extension of the so-called dressing approach. The underlying Lax pair is discussed in two different gauges, the unitary and the hermitean gauge. Several examples and applications for both situations are considered, including abelian solutions constructed from GMS-like projectors, noncommutative U(2) soliton-like configurations and interacting plane waves. We display a correspondence to earlier work on string field theory and argue that the solutions found here can serve as a guideline in the search for nonperturbative solutions of nonpolynomial string field theory. 
  Dynamical symmetries of Born-Infeld theory can be absorbed into the spacetime geometry, giving rise to relativistic kinematics with an additional invariant acceleration scale. The standard Poincare group P is thereby enhanced to its pseudo-complexified version, which is isomorphic to P x P. We construct the irreducible representations of this group, which yields the particle spectrum of a relativistic quantum theory that respects a maximal acceleration. It is found that each standard relativistic particle is associated with a `pseudo'-partner of equal spin but generically different mass. These pseudo-partners act as Pauli-Villars regulators for the other member of the doublet, as is found from the explicit construction of quantum field theory on pseudo-complex spacetime. Conversely, a Pauli-Villars regularised quantum field theory on real spacetime possesses a field phase space with integrable pseudo-complex structure, which gives rise to a quantum field theory on pseudo-complex spacetime.   This equivalence between maximal acceleration kinematics, pseudo-complex quantum field theory, and Pauli-Villars regularisation rigorously establishes a conjecture on the regularising property of the maximal acceleration principle in quantum field theory, by Nesterenko, Feoli, Lambiase and Scarpetta. 
  In this paper we show that in field theories with topologically stable kinks and flat directions in their potential, a so-called dynamical vacuum selection (DVS) takes place in the non-trivial, soliton sector of the theory. We explore this DVS mechanism using a specific model. For this model we show that there is only a static kink solution when very specific boundary conditions are met, very similar to the case of vortices in two dimensions. In the case of other boundary conditions a scalar cloud is expelled to infinity, leaving a static kink behind. Other circumstances under which DVS may or may not take place are discussed as well. 
  We study boundary conditions for the bosonic, spinning (NSR) and Green-Schwarz open string, as well as for 1+1 dimensional supergravity. We consider boundary conditions that arise from (1) extremizing the action, (2) BRST, rigid or local supersymmetry, or kappa(Siegel)-symmetry of the action, (3) closure of the set of boundary conditions under the symmetry transformations, and (4) the boundary limits of bulk Euler-Lagrange equations that are ``conjugate'' to other boundary conditions. We find corrections to Neumann boundary conditions in the presence of a bulk tachyon field. We discuss a boundary superspace formalism. We also find that path integral quantization of the open string requires an infinite tower of boundary conditions that can be interpreted as a smoothness condition on the doubled interval; we interpret this to mean that for a path-integral formulation of open strings with only Neumann boundary conditions, the description in terms of orientifolds is not just natural, but is actually fundamental. 
  We consider geometrical characteristics of monopole clusters of the lattice SU(2) gluodynamics. We argue that the polymer approach to the field theory is an adequate means to describe the monopole clusters. Both finite-size and the infinite, or percolating clusters are considered. We find out that the percolation theory allows to reproduce the observed distribution of the finite-size clusters in their length and radius. Geometrical characteristics of the percolating cluster reflect, in turn, the basic properties of the ground state of a system with a gap. 
  We calculate small correction terms to gravitational potential on Randall-Sundrum brane with an induced Einstein term. The behaviors of the correction terms depend on the magnitudes of $AdS$ radius $k^{-1}$ and a characteristic length scale $\l$ of model. We represent the gravitational potential for arbitrary $k$ and $\l$ at all distances. 
  Models of particle physics based on manifolds of $G_2$ holonomy are in most respects much more complicated than other string-derived models, but as we show here they do have one simplification: threshold corrections to grand unification are particularly simple. We compute these corrections, getting completely explicit results in some simple cases. We estimate the relation between Newton's constant, the GUT scale, and the value of $\alpha_{GUT}$, and explore the implications for proton decay. In the case of proton decay, there is an interesting mechanism which (relative to four-dimensional SUSY GUT's) enhances the gauge boson contribution to $p\to\pi^0e^+_L$ compared to other modes such as $p\to \pi^0e^+_R$ or $p\to \pi^+\bar\nu_R$. Because of numerical uncertainties, we do not know whether to intepret this as an enhancement of the $p\to \pi^0e^+_L$ mode or a suppression of the others. 
  We present two families of exterior differential systems (EDS) for causal embeddings of orthonormal frame bundles over Riemannian spaces of dimension q = 2,3,4,5.. into orthonormal frame bundles over flat spaces of higher dimension. We calculate Cartan characters showing that these EDS are dynamical field theories. The first family includes a new non-isometric embedding EDS for classical Einstein vacuum relativity (q = 4). The second, generated only by 2-forms, is a family of classical "stringy" or Kaluza-type (q = 5) integrable systems. Cartan forms are found for all these dynamical systems. 
  A natural extension of the Dijkgraaf-Vafa proposal is to include fields in the fundamental representation of the gauge group. In this paper we use field theory techniques to analyze gauge theories whose tree level superpotential is a generic polynomial in multi-trace operators constructed out of such fields. We show that the effective superpotential is generated by planar diagrams with at most one (generalized) boundary. This justifies the proposal put forward in hep-th/0211075.   We then proceed to extend the gauge theory-matrix model duality to include baryonic operators. We obtain the full moduli space of vacua for an U(N) theory with N flavors. We also outline a program leading to a string theory justification of the gauge theory-matrix model correspondence with fundamental matter. 
  We have analyzed IIB matrix model based on the improved mean field approximation (IMFA) and have obtained a clue that the four-dimensional space-time appears as its most stable vacuum. This method is a systematic way to give an improved perturbation series and was first applied to IIB matrix model by Nishimura and Sugino. In our previous paper we reformed this method and proposed a criterion for convergence of the improved series, that is, the appearance of the ``plateau''. In this paper, we perform higher order calculations, and find that our improved free energy tends to have a plateau, which shows that IMFA works well in IIB matrix model. 
  The $n$ integrals in involution for the motion on the $n$-dimensional ellipsoid under the influence of a harmonic force are explicitly found. The classical separation of variables is given by the inverse momentum map. In the quantum case the Schr\"odinger equation separates into one-dimensional equations that coincide with those obtained from the classical separation of variables. We show that there is a more general orthogonal parametrisation of Jacobi type that depends on two arbitrary real parameters. Also if there is a certain relation between the spring constants and the ellipsoid semiaxes the motion under the influence of such a harmonic potential is equivalent to the free motion on the ellipsoid. 
  Utilizing the techniques recently developed for N=1 super Yang-Mills theories by Dijkgraaf, Vafa and collaborators, we derive the linearity principle of Intriligator, Leigh and Seiberg, for the confinement phase of the theories with semi-simple gauge groups and matters in a non-chiral representation which satisfies a further technical assumption. 
  Earlier we have shown that interacting electron-positron and electromagnetic fields can be considered as a certain microscopic distortion of pseudo-Euclidean properties of the Minkovsky 4-space-time. The known Dirac and Maxwell equations prove to be group-theoretical relations describing this distortion (nonmetrized closed 4-manifold). Here we apply the above geometrical approach to obtain equations for a neutrino interacting with its weak field. These equations contain some new terms and demonstrate geometrical mechanisms of gauge-invariance and P-T violation. Equations are also proposed for gravitational field and its microscopic quantum sources. 
  Recent results on zero modes of the Abelian Dirac operator in three dimensions support to some degree the conjecture that the Chern-Simons action admits only certain quantized values for gauge fields that lead to zero modes of the corresponding Dirac operator. Here we show that this conjecture is wrong by constructing an explicit counter-example. 
  Skyrme theories on S^3 and S^2, are analyzed using the generalized zero curvature in any dimensions. In the first case, new symmetries and integrable sectors, including the B =1 skyrmions, are unraveled. In S^2 the relation to QCD suggested by Faddeev is discussed 
  We review recent work on the holographic duals of type II and heterotic matrix string theories described by warped AdS_3 supergravities. In particular, we compute the spectra of Kaluza-Klein primaries for type I, II supergravities on warped AdS_3xS^7 and match them with the primary operators in the dual two-dimensional gauge theories. The presence of non-trivial warp factors and dilaton profiles requires a modification of the familiar dictionary between masses and ``scaling'' dimensions of fields and operators. We present these modifications for the general case of domain wall/QFT correspondences between supergravities on warped AdS_{d+1}xS^q geometries and super Yang-Mills theories with 16 supercharges. 
  Superspace power-counting rules give estimates for the loop order at which divergences can first appear in non-renormalisable supersymmetric field theories. In some cases these estimates can be improved if harmonic superspace, rather than ordinary superspace, is used. The new estimates are in agreement with recent results derived from unitarity calculations for maximally supersymmetric Yang-Mills theories in five and six dimensions. For N=8 supergravity in four dimensions, we speculate that the onset of divergences may correspondingly occur at the six loop order. 
  It is shown that two definitions for an exterior differential in superspace, giving the same exterior calculus, yet lead to different results when applied to the Poisson bracket. A prescription for the transition with the help of these exterior differentials from the given Poisson bracket of definite Grassmann parity to another bracket is introduced. It is also indicated that the resulting bracket leads to generalization of the Schouten-Nijenhuis bracket for the cases of superspace and brackets of diverse Grassmann parities. It is shown that in the case of the Grassmann-odd exterior differential the resulting bracket is the bracket given on exterior forms. The above-mentioned transition with the use of the odd exterior differential applied to the linear even/odd Poisson brackets, that correspond to semi-simple Lie groups, results, respectively, in also linear odd/even brackets which are naturally connected with the Lie superalgebra. The latter contains the BRST and anti-BRST charges and can be used for calculation of the BRST operator cohomology. 
  We study generic one-loop (string) amplitudes where an integration over the fundamental region F of the modular group is needed. We show how the known lattice-reduction technique used to unfold F to a more suitable region S can be modified to rearrange generic modular invariant amplitudes. The main aim is to unfold F to the strip and, at the same time, to simplify the form of the integrand when it is a sum over a finite number of terms, like in one-loop amplitudes for closed strings compactified on orbifolds. We give a general formula and a recipe to compute modular invariant amplitudes. As an application of the technique we compute the one-loop vacuum energy \rho_n for a generic \Z_n freely acting orbifold, generalizing the result that this energy is less than zero and drives the system to a tachyonic divergence, and that \rho_n<\rho_m if n>m. 
  Dynamics of confining vacua which appear as deformed superconformal theory with a non-Abelian gauge symmetry, is studied by taking a concrete example of the sextet vacua of ${\cal N}=2$, SU(3) gauge theory with $n_f=4$, with equal quark masses. We show that the low-energy "matter" degrees of freedom of this theory consist of four magnetic monopole doublets of the low-energy effective SU(2) gauge group, one dyon doublet, and one electric doublet. We find a mechanism of cancellation of the beta function, which naturally but nontrivially generalizes that of Argyres-Douglas. Study of our SCFT theory as a limit of six colliding ${\cal N}=1$ vacua, suggests that the confinement in the present theory occurs in an essentially different manner from those vacua with dynamical Abelianization, and involves strongly interacting non-Abelian magnetic monopoles. 
  We give a simple proof that the Neumann coefficients of surface states in Witten's SFT satisfy the Hirota equations for dispersionless KP hierarchy. In a similar way we show that the Neumann coefficients for the three string vertex in the same theory obey the Hirota equations of the dispersionless Toda Lattice hierarchy. We conjecture that the full (dispersive) Toda Lattice hierachy and, even more attractively a two--matrix model, may underlie open SFT. 
  The Shimura-Taniyama conjecture states that the Mellin transform of the Hasse-Weil L-function of any elliptic curve defined over the rational numbers is a modular form. Recent work of Wiles, Taylor-Wiles and Breuil-Conrad-Diamond-Taylor has provided a proof of this longstanding conjecture. Elliptic curves provide the simplest framework for a class of Calabi-Yau manifolds which have been conjectured to be exactly solvable. It is shown that the Hasse-Weil modular form determined by the arithmetic structure of the Fermat type elliptic curve is related in a natural way to a modular form arising from the character of a conformal field theory derived from an affine Kac-Moody algebra. 
  In this paper we study the Penrose limit of AdS_5 orbifolds. The orbifold can be either in the pure spatial directions or space and time directions. For the AdS_5/\Gamma\times S^5 spatial orbifold we observe that after the Penrose limit we obtain the same result as the Penrose limit of AdS_5\times S^5/\Gamma. We identify the corresponding BMN operators in terms of operators of the gauge theory on R\times S^3/\Gamma. The semi-classical description of rotating strings in these backgrounds have also been studied. For the spatial AdS orbifold we show that in the quadratic order the obtained action for the fluctuations is the same as that in S^5 orbifold, however, the higher loop correction can distinguish between two cases. 
  We use geometrical conformal field theory methods to investigate tachyon fluctuations about the butterfly projector state in Vacuum String Field Theory. We find that the on-shell condition for the tachyon field is equivalent to the requirement that the quadratic term in the string-field action vanish on shell. This further motivates the interpretation of the butterfly state as a D-brane. We begin a calculation of the tension of the butterfly, and conjecture that this will match the case of the sliver and further strengthen this interpretation. 
  Cachazo and Vafa studied $\mathcal{N}=1$ dynamics of U(N) gauge theory from a viewpoint of type IIB superstring compactified on a Calabi-Yau manifold with fluxes. They proved the equivalence between the dynamics and that of $\mathcal{N}=2$ supersymmetric U(N) gauge theory deformed by certain superpotential terms. We generalize their results to gauge theories with massive flavors in fundamental representation for classical gauge groups. When the additional tree level superpotential takes the form of the square of an adjoint chiral superfield we derive Affleck-Dine-Seiberg potentials. By turning off the flux, we obtain the Seiberg-Witten curves of $\mathcal{N}=2$ gauge theories. 
  The boundary states for AdS_2 D-branes in Lorentzian AdS_3 space-time are presented. AdS_2 D-branes are algebraically defined by twisted Dirichlet boundary conditions and are located on twisted conjugacy classes of SL(2,R). Using free field representation of symmetry currents in the SL(2,R) WZNW model, the twisted Dirichlet gluing conditions among currents are translated to matching conditions among free fields and then to boundary conditions among the modes of free fields. The Ishibashi states are written as coherent states on AdS_3 in the free field formalism and it is shown that twisted Dirichlet boundary conditions are satisfied on them. The tree-level amplitude of propagation of closed strings between two AdS_2 D-branes is evaluated and by comparing which to the characters of sl(2,R) Kac-Moody algebra it is shown that only states in the principal continuous series representation of sl(2,R) Kac-Moody algebra contributes to the amplitude and thus they are the only ones that couple to AdS_2 D-branes. The form of the character of sl(2,R) principal continuous series and the boundary condition among the zero modes are used to determine the physical boundary states for AdS_2 D-branes. 
  We investigate a string model defined by a special plane-wave metric ds^2 = 2dudv - l(u) x^2 du^2 + dx^2 with l(u) = k/u^2 and k=const > 0. This metric is a Penrose limit of some cosmological, Dp-brane and fundamental string backgrounds. Remarkably, in Rosen coordinates the metric has a ``null cosmology'' interpretation with flat spatial sections and scale factor which is a power of the light-cone time u. We show that: (i) This spacetime is a Lorentzian homogeneous space. In particular, like Minkowski space, it admits a boost isometry in u,v. (ii) It is an exact solution of string theory when supplemented by a u-dependent dilaton such that its exponent (i.e. effective string coupling) goes to zero at u=infinity and at the singularity u=0, reducing back-reaction effects. (iii) The classical string equations in this background become linear in the light-cone gauge and can be solved explicitly in terms of Bessel's functions; thus the string model can be directly quantized. This allows one to address the issue of singularity at the string-theory level. We examine the propagation of first-quantized point-particle and string modes in this time-dependent background. Using certain analytic continuation prescription we argue that string propagation through the singularity can be smooth. 
  It has recently been shown that the uniqueness theorem for stationary black holes cannot be extended to five dimensions. However, uniqueness is an important assumption of the string theory black hole entropy calculations. This paper partially justifies this assumption by proving two uniqueness theorems for supersymmetric black holes in five dimensions. Some remarks concerning general properties of non-supersymmetric higher dimensional black holes are made. It is conjectured that there exist new families of stationary higher dimensional black hole solutions with fewer symmetries than any known solution. 
  We apply the proposal of Dijkgraaf and Vafa to analyze N=1 gauge theory with SO(N) and Sp(N) gauge groups with arbitrary tree-level superpotentials using matrix model techniques. We derive the planar and leading non-planar contributions to the large M SO(M) and Sp(M) matrix model free energy by applying the technology of higher-genus loop equations and by straightforward diagrammatics. The loop equations suggest that the RP^2 free energy is given as a derivative of the sphere contribution, a relation which we verify diagrammatically. With a refinement of the proposal of Dijkgraaf and Vafa for the effective superpotential, we find agreement with field theory expectations. 
  In the AdS/CFT correspondence a chiral primary is described by a supergravity solution with mass equaling angular momentum. For AdS_3 X S^3 we are led to consider three special families of metrics with this property: metrics with conical defects, Aichelburg-Sexl type metrics generated by rotating particles, and metrics generated by giant gravitons. We find that the first two of these are special cases of the complete family of chiral primary metrics which can be written down using the general solution in hep-th/0109154, but they correspond to two extreme limits - the conical defect metrics map to CFT states generated by twist operators that are all identical, while the Aichelburg-Sexl metrics yield a wide dispersion in the orders of these twists. The giant graviton solutions in contrast do not represent configurations of the D1-D5 bound state; they correspond to fragmenting this system into two or more pieces. We look at the large distance behavior of the supergravity fields and observe that the excitation of these fields is linked to the existence of dispersion in the orders and spins of the twist operators creating the chiral primary in the CFT. 
  Particles colliding at impact parameter much larger than the effective gravitational radius can not classically form a black hole and just scatter off the radial potential barrier separating the particles. We show that the process of the black hole production can still go quantum-mechanically via familiar mechanism of the under-barrier tunneling. The mechanism is illustrated for collision of a trans-Planckian particle and a lighter particle. Our analysis reveals instability of trans-Planckian particles against transition to the phase of black hole. 
  The SO(4) isometry of the extreme Reissner-Nordstrom black hole of N=1, D=5 supergravity can be partly broken, without breaking any supersymmetry, in two different ways. The ``right'' solution is a rotating black hole (BMPV); the ``left'' is interpreted as a black hole in a Godel universe. In ten dimensions, both spacetimes are described by deformations of the D1-D5-pp-wave system with the property that the non-trivial Closed Timelike Curves of the five dimensional manifold are absent in the universal covering space of the ten dimensional manifold. In the decoupling limit, the BMPV deformation is normalizable. It corresponds to the vev of an IR relevant operator of dimension \Delta=1. The Godel deformation is sub-leading in \alpha' unless we take an infinite vorticity limit; in such case it is a non-normalizable perturbation. It corresponds to the insertion of a vector operator of dimension \Delta=5. Thus we conclude that from the dual (1+1)-CFT viewpoint the SO(4) R-symmetry is broken `spontaneously' in the BMPV case and explicitly in the Godel case. 
  We calculate the scalar potential in the gauged N=2 supergravity with a single hypermultiplet, whose generic quaternionic moduli space metric has an abelian isometry. This isometry is gauged by the use of a graviphoton gauge field. The hypermultiplet metric and the scalar potential are both governed by the single real potential that is a solution to the 3d (integrable) continuous Toda equation. An explicit solution, controlled by the Eisenstein series E_{3/2}, is found in the case of the D-instanton-corrected universal hypermultilet moduli space metric having an U(1)xU(1) isometry, with one of the isometries being gauged. 
  A new type of superstring in four dimension is proposed which has the central charge 26. The Neveu Schwarz and the Ramond vacua are both tachyonic. The self energy of the scalar tachyon cancels from the contribution of the fermionic loop of the Ramond sector. The NS tachyonic vacuum is used to construct a massless graviton. Coulombic vector excitations of zero mass, referred as a `Newtonian' graviton are also shown to exist. The propagators are explicitly evaluated. Following the method of Weinberg, we deduce the Einstein's field equation of general relativity. 
  We consider topological quantum mechanics as an example of topological field theory and show that its special properties lead to numerous interesting relations for topological corellators in this theory. We prove that the generating function $\mathcal{F}$ for thus corellators satisfies the anticommutativity equation $(\mathcal{D}- \mathcal{F})^2=0$. We show that the commutativity equation $[dB,dB]=0$ could be considered as a special case of the anticommutativity equation. 
  We briefly review the possible Poisson structures on the chiral WZNW phase space and discuss the associated Poisson-Lie groupoids. Many interesting dynamical r-matrices appear naturally in this framework. Particular attention is paid to the special cases in which these r-matrices satisfy the classical dynamical Yang-Baxter equation or its Poisson-Lie variant. 
  We consider quantum electrodynamics with additional coupling of spinor fields to the space-time independent axial vector violating both Lorentz and CPT symmetries. The Fock-Schwinger proper time method is used to calculate the one-loop effective action up to the second order in the axial vector and to all orders in the space-time independent electromagnetic field strength. We find that the Chern-Simons term is not radiatively induced and that the effective action is CPT invariant in the given approximation. 
  We propose a way to classify all supersymmetric configurations of D=11 supergravity using the G-structures defined by the Killing spinors. We show that the most general bosonic geometries admitting a Killing spinor have at least a local SU(5) or an (Spin(7)\ltimes R^8)x R structure, depending on whether the Killing vector constructed from the Killing spinor is timelike or null, respectively. In the former case we determine what kind of local SU(5) structure is present and show that almost all of the form of the geometry is determined by the structure. We also deduce what further conditions must be imposed in order that the equations of motion are satisfied. We illustrate the formalism with some known solutions and also present some new solutions including a rotating generalisation of the resolved membrane solutions and generalisations of the recently constructed D=11 Godel solution. 
  We study the effect of the Born-Infeld electric field on the supersymmetric configuration of various composite D-branes. We show that the generic values of the electric field do not affect the supersymmetry but, as it approaches $1/2\pi\alpha'$ keeping the magnetic field finite, various combinations of the magnetic fields allow up to 8 supersymmetries. We also explore the unbroken supersymmetries for two intersecting D-strings which are in uniform or relative motion. For a finite uniform Lorentz boost, 16 supersymmetries are guaranteed only when they are parallel. For an infinite one, 8 supersymmetries are preserved only when both the D-strings are oriented to the forward or backward direction of the boost. Under a finite relative boost, 8 supersymmetries are preserved only when the intersecting angle is less than $\pi/2$ and the intersecting point moves at the speed of light. As for an infinite relative boost, 8 supersymmetries are preserved regardless of the values of the intersecting angle. 
  In this paper, we give a proof of the equivalence of ${\cal N}=1$ $SO/Sp$ gauge theories deformed from ${\cal N}=2$ by the superpotential of adjoint field $\Phi$, the dual type IIB superstring theory on CY threefold geometries with fluxes and orientifold action after geometric transition. Furthermore, by relating the geometric picture to the matrix model, we show the equivalence between the field theory and the corresponding matrix model. 
  The multi-centre metrics are a family of euclidean solutions of the empty space Einstein equations with self-dual curvature. For this full class, we determine which metrics do exhibit an extra conserved quantity quadraic in the momenta, induced by a Killing-Stackel tensor. Our results bring to light several metrics which correspond to classically integrable dynamical systems. They include, as particular cases, the Eguchi-Hanson and Taub-NUT metrics. 
  We construct the supersymmetric effective action at order alpha'^4 of the abelian open superstring. It includes the alpha'^4 terms in the abelian Born-Infeld action, and in particular the leading derivative correction of the form d^4F^4. Besides linear supersymmetry this sector of the open string effective action also has a nonlinear supersymmetry. The terms d^4F^4 and their fermionic partners have an arbitrary coefficient, and we discuss the possible fate of such coefficients when higher orders in alpha' are included. 
  I study some aspects of the renormalization of quantum field theories with infinitely many couplings in arbitrary space-time dimensions. I prove that when the space-time manifold admits a metric of constant curvature the propagator is not affected by terms with higher derivatives. More generally, certain lagrangian terms are not turned on by renormalization, if they are absent at the tree level. This restricts the form of the action of a non-renormalizable theory, and has applications to quantum gravity. The new action contains infinitely many couplings, but not all of the ones that might have been expected. In quantum gravity, the metric of constant curvature is an extremal, but not a minimum, of the complete action. Nonetheless, it appears to be the right perturbative vacuum, at least when the curvature is negative, suggesting that the quantum vacuum has a negative asymptotically constant curvature. The results of this paper give also a set of rules for a more economical use of effective quantum field theories and suggest that it might be possible to give mathematical sense to theories with infinitely many couplings at high energies, to search for physical predictions. 
  In discussions of Bousso's Covariant Entropy Bound, the Null Energy Condition is always assumed, as a sufficient {\em but not necessary} condition which helps to ensure that the entropy on any lightsheet shall necessarily be finite.   The spectacular failure of the Strong Energy Condition in cosmology has, however, led many astrophysicists and cosmologists to consider models of dark energy which violate {\em all} of the energy conditions, and indeed the current data do not completely rule out such models. The NEC also has a questionable status in brane cosmology: it is probably necessary to violate the NEC in the bulk in order to obtain a "self-tuning" theory of the cosmological constant. In order to investigate these proposals, we modify the Karch-Randall model by introducing NEC-violating matter into $AdS_5$ in such a way that the brane cosmological constant relaxes to zero. The entropy on lightsheets remains finite. However, we still find that the spacetime is fundamentally incompatible with the Covariant Entropy Bound machinery, in the sense that it fails the Bousso-Randall consistency condition. We argue that holography probably forbids all {\em cosmological} violations of the NEC, and that holography is in fact the fundamental physical principle underlying the cosmological version of the NEC. 
  We study the braneworld effective action in the two-brane Randall-Sundrum model. In the framework of this essentially-nonlocal action we reveal the origin of an infinite sequence of gravitational wave modes -- the usual massless one as well as the tower of Kaluza-Klein massive ones. Mixing of the modes, which parametrically depends on the background value of the modulus of the extra dimension, can be interpreted as radion-induced gravitational-wave oscillations, a classical analogue to meson and neutrino oscillations. We show that these oscillations arising in M-theory-inspired braneworld setups could lead to effects detectable by gravitational-wave interferometers. 
  We study the possibility of extending ghost systems with higher spin to a logarithmic conformal field theory. In particular we are interested in c=-26 which turns out to behave very differently to the already known c=-2 case. The energy momentum tensor cannot be built anymore by a combination of derivatives of generalized symplectic fermion fields.   Moreover, the logarithmically extended theory is only consistent when considered on nontrivial Riemann surfaces. This results in a LCFT with some unexpected properties. For instance the Virasoro mode L_0 is diagonal and for certain values of the deformation parameters even the whole global conformal group is non-logarithmic. 
  We continue the study of supersymmetric type IIB pp wave solutions by Maldacena and Maoz (hep-th/0207284), who showed Ramond-Ramond five-forms can induce potential terms in the light cone string actions which are nonlinear sigma models with special holonomy target spaces. We show nonvanishing Ramond-Ramond three-forms provide extra potential terms involving Killing vectors in the string action and identify the supersymmetry requirements. In particular, in solutions with (1,1) worldsheet supersymmetry, the Killing vectors are required to be self-dual in Spin(7). 
  We argue that in the pure N=1 Super Yang-Mills theory gauge symmetry is spontaneously broken to the maximal Abelian subgroup. In particular, colored gluino condensate is nonzero. It invalidates, in a subtle way, the so-called strong-coupling instanton calculation of the (normal) gluino condensate and resolves the long-standing paradox why its value does not agree with that obtained by other methods. 
  We study a non local version of the sine-Gordon model connected to a many-body system with backward and umklapp scattering processes. Using renormalization group methods we derive the flow equations for the couplings and show how non locality affects the gap in the spectrum of charge-density excitations. We compare our results with previous predictions obtained through the self-consistent harmonic approximation. 
  We analyze the geometric engineering of the N=2 SU(2) gauge theories with $1\leq N_f\leq 3$ massive hypermultiplets in the vector representation. The set of partial differential equations satisfied by the periods of the Seiberg-Witten differential is obtained from the Picard-Fuchs equations of the local B-model. The differential equations and its solutions are consistent with the massless case. We show that the Yukawa coupling of the local A-model gives rise to the correct instanton expansion in the gauge theory, and propose the pattern of the distribution of the world-sheet instanton number from it. As a side result, we obtain the asymptotic form of the instanton number in the gauge theories with massless hypermultiplets. 
  In this paper we consider a somewhat unconventional approach for deriving worldvolume theories for D3 branes probing Calabi-Yau singularities. The strategy consists of extrapolating the calculation of F-terms to the large volume limit. This method circumvents the inherent limitations of more traditional approaches used for orbifold and toric singularities. We illustrate its usefulness by deriving quiver theories for D3 branes probing singularities where a Del Pezzo surface containing four, five or six exceptional curves collapses to zero size. In the latter two cases the superpotential depends explicitly on complex structure parameters. These are examples of probe theories for singularities which can currently not be computed by other means. 
  A review of the construction of a Weyl-invariant spinning-membrane action that is $polynomial$ in the fields, without a cosmological constant term, comprised of quadratic and quartic-derivative terms, and where supersymmetry is linearly realized, is presented. The action is invariant under a $modified$ supersymmetry transformation law which is derived from a new $ Q + K + S $ sum-rule based on the 3D-superconformal algebra . 
  Using a fixed-energy amplitude in Randall-Sundrum single brane scenario, we compute the Newton potential on the brane. It is shown that the correction terms to the Newton potential involve a logarithmic factor. Especially, when the distance between two point masses are very small compared to $AdS$ radius, the contribution of KK spectrum becomes dominant compared to the usual inversely square law. This fact may be used to prove the existence of an extra dimension experimentally. 
  The oxidation program of hep-th/0210178 is extended to cover oxidation of 3-d sigma model theories on a coset G/H, with G non-compact (but not necessarily split), and H the maximal compact subgroup. We recover the matter content, the equations of motion and Bianchi identities from group lattice and Cartan involution. Satake diagrams provide an elegant tool for the computations, the maximal oxidation dimension, and group disintegration chains can be directly read off. We give a complete list of theories that can be recovered from oxidation of a 3 dimensional coset sigma model on G/H, where G is a simple non-compact group. 
  We investigate in detail the structure of mesonic vacua of N=1 U(Nc) supersymmetric gauge theory with Nf flavors from the matrix model. We show that the Witten index from the matrix model calculation agrees with a result from field theoretical analysis. We also discuss the relationship between a diagrammatic summation and direct matrix integration with insertion of a variable changing delta function. Using this formalism, we obtain the quantum moduli space and evidence of the Seiberg duality from the matrix models. 
  We consider the supersymmetric vector multiplet in a purely quantum framework. We obtain some discrepancies with respect to the literature in the expression of the super-propagator and we prove that the model is consistent only for positive mass. The gauge structure is constructed purely deductive and leads to the necessity of introducing scalar ghost superfields, in analogy to the usual gauge theories. The construction of a consistent supersymmetric gauge theory based on the vector model depends crucially one the definition of gauge invariance. We find some significant difficulties to impose a supersymmetric gauge invariance condition for the usual expressions from the literature. 
  It has recently been proposed by Mersini et al. 01, Bastero-Gil and Mersini 02 that the dark energy could be attributed to the cosmological properties of a scalar field with a non-standard dispersion relation that decreases exponentially at wave-numbers larger than Planck scale (k_phys > M_Planck). In this scenario, the energy density stored in the modes of trans-Planckian wave-numbers but sub-Hubble frequencies produced by amplification of the vacuum quantum fluctuations would account naturally for the dark energy. The present article examines this model in detail and shows step by step that it does not work. In particular, we show that this model cannot make definite predictions since there is no well-defined vacuum state in the region of wave-numbers considered, hence the initial data cannot be specified unambiguously. We also show that for most choices of initial data this scenario implies the production of a large amount of energy density (of order M_Planck^4) for modes with momenta of order M_Planck, far in excess of the background energy density. We evaluate the amount of fine-tuning in the initial data necessary to avoid this back-reaction problem and find it is of order H/M_Planck. We also argue that the equation of state of the trans-Planckian modes is not vacuum-like. Therefore this model does not provide a suitable explanation for the dark energy. 
  We consider the process of photon decay in quantum electrodynamics with a CPT-violating Chern-Simons-like term added to the action. For a simplified model with only the quadratic Maxwell and Chern-Simons-like terms and the quartic Euler-Heisenberg term, we obtain a nonvanishing probability for the decay of a particular photon state into three others. 
  We treat D-dimensional black holes with Killing horizon for extended Gauss-Bonnet gravity. We use Carlip method and impose boundary conditions on horizon what enables us to identify Virasoro algebra and evaluate its central charge and Hamiltonian eigenvalue. The Cardy formula allows then to calculate the number of states and thus provides for microscopic interpretation of entropy. 
  The classification of the octonionic realizations of the one-dimensional extended supersymmetries is here furnished. These are non-associative realizations which, albeit inequivalent, are put in correspondence with a subclass of the already classified associative representations for 1D extended supersymmetries. Examples of dynamical systems invariant under octonionic realizations of the extended supersymmetries are given. We cite among the others the octonionic spinning particles, the N=8 KdV, etc. Possible applications to supersymmetric systems arising from dimensional reduction of the octonionic superstring and M-theory are mentioned. 
  By introducing an appropriate parent action and considering a perturbative approach, we establish, up to fourth order terms in the field and for the full range of the coupling constant, the equivalence between the noncommutative Yang-Mills-Chern-Simons theory and the noncommutative, non-Abelian Self-Dual model. In doing this, we consider two different approaches by using both the Moyal star-product and the Seiberg-Witten map. 
  We discuss the derivation of the CIV-DV prepotential for arbitrary power n+1 of the original superpotential in the N=1 SUSY YM theory (for arbitrary number n of cuts in the solution of the planar matrix model in the Dijkgraaf-Vafa interpretation). The goal is to hunt for structures, not so much for exact formulas, which are necessarily complicated, before the right language is found to represent them. Some entities, reminiscent of representation theory, clearly emerge, but a lot of work remains to be done to identify the relevant ones. As a practical application, we obtain a cubic (first non-perturbative) contribution to the prepotential for any n. 
  The main obstacle in attempts to construct a consistent quantum gravity is the absence of independent flat time. This can in principle be cured by going out to higher dimensions. The modern paradigm assumes that the fundamental theory of everything is some form of string theory living in space of more than four dimensions. We advocate another possibility that the fundamental theory is a form of D=4 higher-derivative gravity. This class of theories has a nice feature of renormalizability so that perturbative calculations are feasible. There are also finite N=4 supersymmetric conformal supergravity theories. This possibility is particularly attractive. Einstein's gravity is obtained in a natural way as an effective low-energy theory. The N=1 supersymmetric version of the theory has a natural higher-dimensional interpretation due to Ogievetsky and Sokatchev, which involves embedding of our curved Minkowsky space-time manifold into flat 8-dimensional space. Assuming that a variant of the finite N=4 theory also admit a similar interpretation, this may eventually allow one to construct consistent quantum theory of gravity. We argue, however, that even though future gravity theory will probably use higher dimensions as construction scaffolds, its physical content and meaning should refer to 4 dimensions where observer lives. 
  Using a particular structure for the Lagrangian action in a one-dimensional Thirring model and performing the Dirac's procedure, we are able to obtain the algebra for chiral currents which is entirely defied on the constraint surface in the corresponding hamiltonian description of the theory. 
  A new method to compute the symplectic structure of a quantum field theory with non trivial boundary conditions is proposed. Following the suggestion in \cite{ho:gnus, ardalan}, we regard that the boundary conditions are second class constraints in the sense of the Dirac's method. However, we show that this proposal is more useful if we consider an inverse of the Holographic map between a theory defined in the boundary to another with constraints but without boundary. 
  We calculate the linearized metric perturbations in the five dimensional two-brane model of Randall and Sundrum. In a carefully chosen gauge, we write down and decouple Einstein equations for the perturbations and get the final and simple perturbative metric ansatz. This ansatz turns out to be equal to the linear expansion of the metric solution of Charmousis et al. \cite{rubakov}. We show that this ansatz, the metric ansatz of Boos et al. \cite{boos} and the one of Das and Mitov \cite{das} are not incompatible, as it appears on the surface, but completely equivalent by an allowed gauge transformation that we give. 
  We construct consistent brane-world Kaluza-Klein reductions involving the radion mode that measures the separation of the domain-wall branes. In these new examples, we can obtain matter supermultiplets coupled to supergravity on the brane, starting from pure gauged supergravity in the higher dimension. This contrasts with previously-known examples of consistent brane-world reductions involving the radion, where either pure supergravity reduced to pure supergravity, or else supergravity plus matter reduced to supergravity plus matter. As well as considering supersymmetric reductions, we also show that there exist broader classes of consistent reductions of bosonic systems. These include examples where the lower-dimensional theory has non-abelian Yang-Mills fields and yet the scalar sector has a potential that admits Minkowski spacetime as a solution. Combined with a sphere reduction to obtain the starting point for the brane-world reduction, this provides a Kaluza-Klein mechanism for obtaining non-abelian gauge symmetries from the geometry of the reduction, whilst still permitting a Minkowski vacuum in the lower dimension. 
  We propose the extension of some structural aspects that have successfully been applied in the development of the theory of quantum fields propagating on a general spacetime manifold so as to include superfield models on a supermanifold. We only deal with the limited class of supermanifolds which admit the existence of a smooth body manifold structure. Our considerations are based on the Catenacci-Reina-Teofillatto-Bryant approach to supermanifolds. In particular, we show that the class of supermanifolds constructed by Bonora-Pasti-Tonin satisfies the criterions which guarantee that a supermanifold admits a Hausdorff body manifold. This construction is the closest to the physicist's intuitive view of superspace as a manifold with some anticommuting coordinates, where the odd sector is topologically trivial. The paper also contains a new construction of superdistributions and useful results on the wavefront set of such objects. Moreover, a generalization of the spectral condition is formulated using the notion of the wavefront set of superdistributions, which is equivalent to the requirement that all of the component fields satisfy, on the body manifold, a microlocal spectral condition proposed by Brunetti-Fredenhagen-K\"ohler. 
  The concept of self-dual supersymmetric nonlinear electrodynamics is generalized to a curved superspace of N = 1 supergravity, for both the old minimal and the new minimal versions of N = 1 supergravity. We derive the self-duality equation, which has to be satisfied by the action functional of any U(1) duality invariant model of a massless vector multiplet, and construct a family of self-dual nonlinear models. This family includes a curved superspace extension of the N = 1 super Born-Infeld action. The supercurrent and supertrace in such models are proved to be duality invariant. The most interesting and unexpected result is that the requirement of nonlinear self-duality yields nontrivial couplings of the vector multiplet to Kahler sigma models. We explicitly derive the couplings to general Kahler sigma models in the case when the matter chiral multiplets are inert under the duality rotations, and more specifically to the dilaton-axion chiral multiplet when the group of duality rotations is enhanced to SL(2,R). 
  A formula for the power spectrum of curvature perturbations having any initial conditions in inflation is obtained. Based on the physical conditions before inflation, the possibility exists that the initial state of scalar perturbations is not only the Bunch-Davies state, but also a more general state (a squeezed state). For example, the derived formula for the power spectrum is calculated using simple toy cosmological models. When there exists a radiation-dominated period before inflation, the behavior of the scalar perturbation is revealed not to vary greatly; however, from large scales to small scales the power spectrum of the curvature perturbations oscillates around the normal value. In addition, when inflation has a large break and the breaking time is a radiation- dominated period, a large enhancement is revealed to occur which depends on the length of the breaking time. 
  We quantize in light cone gauge the bosonic sector of string theory on Anti-de Sitter space in the zero curvature radius limit. We find that the worldsheet falls apart into a theory of free partons and map the Hilbert space of the string theory to the Hilbert space of a free scalar in light-front description. We outline how the string worldsheet reproduces the field theory at weak coupling. 
  We briefly review the construction of N=2 WZW models in terms of Manin triples. We analyse the restrictions which should be imposed on the gluing conditions of the affine currents in order to preserve half of the bulk supersymmetry. In analogy with the Kahler case there are two types of D-branes, A- and B-types which have a nice algebraic interpretation in terms of the Manin triple. 
  The BRST quantization of matrix Chern-Simons theory is carried out, the symmetries of the theory are analysed and used to constrain the form of the effective action. 
  We apply stochastic quantization method to the bosonic part of IIB matrix model, i.e., a naive zero volume limit of large N Yang-Mills theory, to construct a collective field theory of Wilson loops. The Langevin equation for Wilson loops can be interpreted as the time evolution of closed string fields. The corresponding Fokker-Planck hamiltonian deduces a closed string field theory which describes interacting Wilson loops with manifest Lorentz invariance. 
  It is argued from geometrical, group-theoretical and physical points of view that in the framework of QCD it is not only necessary but also possible to modify the Dirac equation so that correspondence principle holds valid. The Dirac wave equation for a quark is proposed and some consequences are considered. In particular, it is shown that interquark potential expresses the Coulomb law for the quarks and, in fact, coincides with the known Cornell potential. 
  A general and systematic construction of Non Abelian affine Toda models and its symmetries is proposed in terms of its underlying Lie algebraic structure. It is also shown that such class of two dimensional integrable models naturally leads to the construction of a pair of actions related by T-duality transformations 
  Thermodynamics of 5d SdS black hole is considered. Thermal fluctuations define the (sub-dominant) logarithmic corrections to black hole entropy and then to Cardy-Verlinde formula and to FRW brane cosmology. We demonstrate that logarithmic terms (which play the role of effective cosmological constant) change the behavior of 4d spherical brane in dS, SdS or Nariai bulk. In particularly, bounce Universe occurs or 4d dS brane expands to its maximum and then shrinks. The entropy bounds are also modified by next-to-leading terms. Out of braneworld context the logarithmic terms may suggest slight modification of standard FRW cosmology. 
  In the light-cone closed string and toroidal membrane theories, we associate the global constraints with gauge symmetries. In the closed string case, we show that the physical states defined by the BRS charge satisfy the level-matching condition. In the toroidal membrane case, we show that the Faddeev-Popov ghost and anti-ghost corresponding to the global constraints are essentially free even if we adopt any gauge fixing condition for the local constraint. We discuss the quantum double-dimensional reduction of the wrapped supermembrane with the global constraints. 
  An elementary introduction to perturbative renormalization and renormalization group is presented. No prior knowledge of field theory is necessary because we do not refer to a particular physical theory. We are thus able to disentangle what is specific to field theory and what is intrinsic to renormalization. We link the general arguments and results to real phenomena encountered in particle physics and statistical mechanics. 
  Dimensional reduction of theories involving (super-)gravity gives rise to sigma models on coset spaces of the form G/H, with G a non-compact group, and H its maximal compact subgroup. The reverse process, called oxidation, is the reconstruction of the possible higher dimensional theories, given the lower dimensional theory. In 3 dimensions, all degrees of freedom can be dualized to scalars. Given the group G for a 3 dimensional sigma model on the coset G/H, we demonstrate an efficient method for recovering the higher dimensional theories, essentially by decomposition into subgroups. The equations of motion, Bianchi identities, Kaluza-Klein modifications and Chern-Simons terms are easily extracted from the root lattice of the group G. We briefly discuss some aspects of oxidation from the E_{8(8)}/SO(16) coset, and demonstrate that our formalism reproduces the Chern-Simons term of 11-d supergravity, knows about the T-duality of IIA and IIB theory, and easily deals with self-dual tensors, like the 5-tensor of IIB supergravity. 
  We calculate the expectation value of an arbitrary product of characteristic polynomials of complex random matrices and their hermitian conjugates. Using the technique of orthogonal polynomials in the complex plane our result can be written in terms of a determinant containing these polynomials and their kernel. It generalizes the known expression for hermitian matrices and it also provides a generalization of the Christoffel formula to the complex plane. The derivation we present holds for complex matrix models with a general weight function at finite-N, where N is the size of the matrix. We give some explicit examples at finite-N for specific weight functions. The characteristic polynomials in the large-N limit at weak and strong non-hermiticity follow easily and they are universal in the weak limit. We also comment on the issue of the BMN large-N limit. 
  The D-branes of the maximally supersymmetric plane-wave background are described. 
  We consider the coupling of a single Dirac fermion to the three component unit vector field which appears as an order parameter in the Faddeev model. Classically, the coupling is determined by requiring that it preserves a certain local frame independence. But quantum mechanically the separate left and right chiral fermion number currents suffer from a frame anomaly. We employ this anomaly to compute the fermion number of a knotted soliton. The result coincides with the self-linking number of the soliton. In particular, the anomaly structure of the fermions relates directly to the inherent chiral properties of the soliton. Our result suggests that interactions between fermions and knotted solitons can lead to phenomena akin the Callan-Rubakov effect. 
  On promoting the type IIA side of the N=1 Heterotic/type IIA dual pairs of [1] to M-theory on a `barely G_2 Manifold' of [2], by spectrum-matching we show a possible triality between Heterotic on a self-mirror Calabi-Yau, M-theory on the above `barely G_2-Manifold' constructed from the Calabi-Yau on the type IIA side and $F$-theory on an elliptically fibered Calabi-Yau 4-fold fibered over a trivially rationally ruled CP^1 x E base, E being the Enriques surface. We raise an apparent puzzle on the F-theory side, namely, the Hodge data of the 4-fold derived can not be obtained by a naive freely acting orbifold of CY_3(3,243) x T^2 as one might have guessed on the basis of arguments related to dualities involving string, M and (definition of) F theories. There are some interesting properties of the antiholomorphic involution used in \cite{VW} for constructing the type IIA orientifold and by us in constructing the 'barely G_2 manifold', that we also study. 
  In this report we review a structure of cubic Vacuum Superstring Field Theory and known solutions to its equation of motion. 
  We study families of PP-wave solutions of type-IIB supergravity that have (light-cone) time dependent metrics and RR five-form fluxes. They arise as Penrose limits of supergravity solutions that correspond to rotating or continuous distributions of D3-branes. In general, the solutions preserve sixteen supersymmetries. On the dual field theory side these backgrounds describe the BMN limit of N=4 SYM when some scalars in the field theory have non-vanishing expectation values. We study the perturbative string spectrum and in several cases we are able to determine it exactly for the bosons as well as for the fermions. We find that there are special states for particular values of the light-cone constant P_+. 
  Witten has presented an argument for the vanishing of the cosmological constant in 2+1 dimensions. This argument is crucially tied to the specific properties of (2+1)-dimensional gravity. We argue that this reasoning can be deconstructed to 3+1 dimensions under certain conditions. Our observation is also tied to a possibility that there exists a well-defined UV completion of (3+1)-dimensional gravity. 
  Dirac's quantization of magnetic monopole strength is derived without reference to a (singular, patched) vector potential. 
  Closed string tachyon condensation resolves the singularities of nonsupersymmetric orbifolds, however the resolved space typically has fewer D-brane charges than that of the orbifold. The description of the tachyon condensation process via a gauged linear sigma model enables one to track the topology as one passes from the sigma model's ``orbifold phase'' to its resolved, ``geometric phase,'' and thus to follow how the D-brane charges disappear from the effective spacetime dynamics. As a mathematical consequence, our results point the way to a formulation of a ``quantum McKay correspondence'' for the resolution of toric orbifold singularities. 
  We construct a seven-dimensional brane world in a slice of AdS_7, where the boundary matter content is fixed by the cancellation of anomalies. The seven-dimensional minimal N=2 gauged supergravity is compactified on the orbifold S^1/Z_2, and the supersymmetric bulk-boundary Lagrangian is consistently derived for boundary vector and hypermultipets up to fermionic bilinear terms. Anomaly cancellation then fixes the boundary gauge coupling in terms of the seven-dimensional Planck mass, and a topological mass parameter of the Chern-Simons term. In addition for gauge groups containing the standard model, anomaly cancellation restricts the gauge groups on the six-dimensional boundaries to be only one of the exceptional groups. There are also special values of the separation of the two boundaries, where the boundary couplings become singular, and lead to a possible phase transition in the boundary theory. Furthermore, by the AdS/CFT correspondence our brane world is dual to a six-dimensional conformal field theory, suggesting that our bulk theory describes the strong coupling dynamics of six-dimensional theories. 
  We consider Penrose limits of the Klebanov-Strassler and Maldacena-Nunez holographic duals to N =1 supersymmetric Yang-Mills. By focusing in on the IR region we obtain exactly solvable string theory models. These represent the nonrelativistic motion and low-lying excitations of heavy hadrons with mass proportional to a large global charge. We argue that these hadrons, both physically and mathematically, take the form of heavy nonrelativistic strings; we term them "annulons." A simple toy model of a string boosted along a compact circle allows us considerable insight into their properties. We also calculate the Wilson loop carrying large global charge and show the effect of confinement is quadratic, not linear, in the string tension. 
  The fundamental concepts of Riemannian geometry, such as differential forms, vielbein, metric, connection, torsion and curvature, are generalized in the context of non-commutative geometry. This allows us to construct the Einstein-Hilbert-Cartan terms, in addition to the bosonic and fermionic ones in the Lagrangian of an action functional on non-commutative spaces. As an example, and also as a prelude to the Standard Model that includes gravitational interactions, we present a model of chiral spinor fields on a curved two-sheeted space-time with two distinct abelian gauge fields. In this model, the full spectrum of the generalized metric consists of pairs of tensor, vector and scalar fields. They are coupled to the chiral fermions and the gauge fields leading to possible parity violation effects triggered by gravity. 
  In order to study the thermodynamic properties of brane-antibrane systems, we compute the finite temperature effective potential of tachyon T in this system on the basis of boundary string field theory. At low temperature, the minimum of the potential shifts towards T=0 as the temperature increases. In the D9-antiD9 case, the sign of the coefficient of |T|^2 term of the potential changes slightly below the Hagedorn temperature. This means that a phase transition occurs near the Hagedorn temperature. On the other hand, the coefficient is kept negative in the Dp-antiDp case with p <= 8, and thus a phase transition does not occur. This leads us to the conclusion that only a D9-antiD9 pair and no other (lower dimensional) brane-antibrane pairs are created near the Hagedorn temperature. We also discuss a phase transition in NS9B-antiNS9B case as a model of the Hagedorn transition of closed strings. 
  Supersymmetric pure Yang-Mills theory is formulated with a local, i.e. space-time dependent, complex coupling in superspace. Super-Yang-Mills theories with local coupling have an anomaly, which has been first investigated in the Wess-Zumino gauge and there identified as an anomaly of supersymmetry. In a manifest supersymmetric formulation the anomaly appears in two other identities: The first one describes the non-renormalization of the topological term, the second relates the renormalization of the gauge coupling to the renormalization of the complex supercoupling. Only one of the two identities can be maintained in perturbation theory. We discuss the two versions and derive the respective beta function of the local supercoupling, which is non-holomorphic in the first version, but directly related to the coupling renormalization, and holomorphic in the second version, but has a non-trivial, i.e.anomalous, relation to the beta function of the gauge coupling. 
  Using the result of a matrix model computation of the exact glueball superpotential, we investigate the relevant mass perturbations of the Leigh-Strassler marginal ``q'' deformation of N=4 supersymmetric gauge theory. We recall a conjecture for the elliptic superpotential that describes the theory compactified on a circle and identify this superpotential as one of the Hamiltonians of the elliptic Ruijsenaars-Schneider integrable system. In the limit that the Leigh-Strassler deformation is turned off, the integrable system reduces to the elliptic Calogero-Moser system which describes the N=1^* theory. Based on these results, we identify the Coulomb branch of the partially mass-deformed Leigh-Strassler theory as the spectral curve of the Ruijsenaars-Schneider system. We also show how the Leigh-Strassler deformation may be obtained by suitably modifying Witten's M theory brane construction of N=2 theories. 
  We review orientifold constructions in the presence of magnetic backgrounds both in the open and closed sectors. Generically, the resulting orientifold models have a nice geometric description in terms of rotated D-branes and/or O-planes. In the case of multiple magnetic backgrounds, some amount of supersymmetry is recovered if the magnetic fields are suitably chosen and part of the original D-branes and/or O-planes are transmuted into new ones. 
  We consider a cosmological brane moving in a static five-dimensional bulk spacetime endowed with a scalar field whose potential is exponential. After studying various cosmological behaviours for the homogeneous background, we investigate the fluctuations of the brane that leave spacetime unaffected. A single mode embodies these fluctuations and obeys a wave equation which we study for bouncing and ever-expanding branes. 
  In this work we review the derivation of Dirac and Weinberg equations based on a ``principle of indistinguishability'' for the (j,0) and (0,j) irreducible representations (irreps) of the Homogeneous Lorentz Group (HLG). We generalize this principle and explore its consequences for other irreps containing spin >= 1. We rederive Ahluwalia-Kirchbach equation using this principle and conclude that it yields O(p^{2j}) equations of motion for any representation containing spin j and lower spins. We also use the obtained generators of the HLG for a given representation to explore the possibility of the existence of first order equations for that representation. We show that, except for j=1/2, there exists no Dirac-like equation for the (j,0)\oplus (0,j) representation nor for the (1/2,1/2) representation. We rederive Kemmer-Duffin-Petieau (KDP) equation for the (1,0)\oplus (1/2,1/2)\oplus(0,1) representation by this method and show that the (1,1/2)\oplus (1/2,1) representation satisfies a Dirac-like equation which describes a multiplet of j=3/2 and j=1/2 with masses m and m/2} respectively. 
  We consider the exact superpotential of N=1 super Yang-Mills theory with gauge group SO(N) and arbitrary tree-level polynomial superpotential of one adjoint Higgs field. A field-theoretic derivation of the glueball superpotential is given, based on factorization of the N=2 Seiberg-Witten curve. Following the conjecture of Dijkgraaf and Vafa, the result is matched with the corresponding SO(N) matrix model prediction. The verification involves an explicit solution of the first non-trivial loop equation, relating the spherical free energy to that of the non-orientable surfaces with topology $RP^2$. 
  The local cohomology of an extended BRST differential which includes global N=1 supersymmetry and Poincare transformations is completely and explicitly computed in four-dimensional supersymmetric gauge theories with super-Yang-Mills multiplets, chiral matter multiplets and linear multiplets containing 2-form gauge potentials. In particular we determine to first order all N=1 supersymmetric and Poincare invariant consistent deformations of these theories that preserve the N=1 supersymmetry algebra on-shell modulo gauge transformations, and all Poincare invariant candidate gauge and supersymmetry anomalies. When the Yang-Mills gauge group is semisimple and no linear multiplets are present, we find that all such deformations can be constructed from standard superspace integrals and preserve the supersymmetry transformations in a formulation with auxiliary fields, and the candidate anomalies are exhausted by supersymmetric generalizations of the well-known chiral anomalies. In the general case there are additional deformations and candidate anomalies which are relevant especially to the deformation of free theories and the general classification of interaction terms in supersymmetric field theories. 
  We construct a model for noncommutative gravity in four dimensions, which reduces to the Einstein-Hilbert action in the commutative limit. Our proposal is based on a gauge formulation of gravity with constraints. While the action is metric independent, the constraints insure that it is not topological. We find that the choice of the gauge group and of the constraints are crucial to recover a correct deformation of standard gravity. Using the Seiberg-Witten map the whole theory is described in terms of the vierbeins and of the Lorentz transformations of its commutative counterpart. We solve explicitly the constraints and exhibit the first order noncommutative corrections to the Einstein-Hilbert action. 
  We demonstrate how to compute real-time Green's functions for a class of finite temperature field theories from their AdS gravity duals. In particular, we reproduce the two-by-two Schwinger-Keldysh matrix propagator from a gravity calculation. Our methods should work also for computing higher point Lorentzian signature correlators. We elucidate the boundary condition subtleties which hampered previous efforts to build a Lorentzian-signature AdS/CFT correspondence. For two-point correlators, our construction is automatically equivalent to the previously formulated prescription for the retarded propagator. 
  I agree with the authors of hep-th/0211149 that the claim made in Phys.Lett. B542, 282 (2002) is incorrect and that the derivation of its main formula, although correct, contains two compensating errors. In this reply the main formula of Phys.Lett. B542, 282 (2002) is rederived. This new derivation shows that not only the energy momentum cut off regularization method still works in the calculation of the soliton quantum mass corrections, but also that the so called mode number regularization emerges naturally from it. 
  A 1/2-BPS family of time dependent plane wave spacetimes which give rise to exactly solvable string backgrounds is presented. In particular a solution which interpolates between Minkowski spacetime and the maximally supersymmetric homogeneous plane wave along a timelike direction is analyzed. We work in d=4, N=2 supergravity, but the results can be easily extended to d=10,11. The conformal boundary of a particular class of solutions is studied. 
  We prove that the asymptotic field of a Skyrme soliton of any degree has a non-trivial multipole expansion. It follows that every Skyrme soliton has a well-defined leading multipole moment. We derive an expression for the linear interaction energy of well-separated Skyrme solitons in terms of their leading multipole moments. This expression can always be made negative by suitable rotations of one of the Skyrme solitons in space and iso-space.We show that the linear interaction energy dominates for large separation if the orders of the Skyrme solitons' multipole moments differ by at most two. In that case there are therefore always attractive forces between the Skyrme solitons. 
  Well-separated Skyrme solitons of arbitrary degree attract after a suitable relative rotation in space and iso-space, provided the orders of the solitons' leading multipoles do not differ by more than two. I summarise the derivation of this result, obtained jointly with Manton and Singer, and discuss to what extent its combination with earlier results of Esteban allows one to deduce the existence of minima of the Skyrme energy functional. 
  Motivated by application of current superalgebras in the study of disordered systems such as the random XY and Dirac models, we investigate $gl(2|2)$ current superalgebra at general level $k$. We construct its free field representation and corresponding Sugawara energy-momentum tensor in the non-standard basis. Three screen currents of the first kind are also presented. 
  Bethe-Salpeter equation for the massive particles with spin 1 is considered. The scattering amplitude decomposition of the particles with spin 1 by relativistic tensors is derived. The transformation coefficients from helicity amplitudes to invariant functions is found. The integral equations system for invariant functions is obtained and partial decomposition of this system is performed. Equivalent system of the integral equation for the partial helicity amplitudes is presented. 
  Dijkgraaf and Vafa have conjectured the correspondences between topological string theories, ${\cal N}=1$ gauge theories and matrix models. By the use of this conjecture, we calculate the quantum deformations of Calabi-Yau threefolds with ADE singularities from ADE multi-matrix models. We obtain the effective superpotentials of the dual quiver gauge theories in terms of the geometric engineering for the deformed geometries. We find the Veneziano-Yankielowicz terms in the effective superpotentials. 
  We considered real, p-adic and adelic noncommutative scalar solitons and obtained some new results. 
  We cast M-brane interactions including intersecting membranes and five-branes in manifestly gauge invariant form using an arrangement of higher dimensional Dirac surfaces. We show that the noncommutative gauge symmetry present in the doubled M-theory formalism involving dual 3-form and 6-form gauge fields is preserved in a form quantised over the integers. The proper context for discussing large noncommutative gauge transformations is relative cohomology, in which the 3-form transformation parameters become exact when restricted to the five-brane worldvolume. We show how this structure yields the lattice of M-theory charges and gives rise to the conjectured 7D Hopf-Wess-Zumino term. 
  We consider N = 1 supersymmetric U(N) field theories in four dimensions with adjoint chiral matter and a multi-trace tree-level superpotential. We show that the computation of the effective action as a function of the glueball superfield localizes to computing matrix integrals. Unlike the single-trace case, holomorphy and symmetries do not forbid non-planar contributions. Nevertheless, only a special subset of the planar diagrams contributes to the exact result. Some of the data of this subset can be computed from the large-N limit of an associated multi-trace Matrix model. However, the prescription differs in important respects from that of Dijkgraaf and Vafa for single-trace superpotentials in that the field theory effective action is not the derivative of a multi-trace matrix model free energy. The basic subtlety involves the correct identification of the field theory glueball as a variable in the Matrix model, as we show via an auxiliary construction involving a single-trace matrix model with additional singlet fields which are integrated out to compute the multi-trace results. Along the way we also describe a general technique for computing the large-N limits of multi-trace Matrix models and raise the challenge of finding the field theories whose effective actions they may compute. Since our models can be treated as N = 1 deformations of pure N =2 gauge theory, we show that the effective superpotential that we compute also follows from the N = 2 Seiberg-Witten solution. Finally, we observe an interesting connection between multi-trace local theories and non-local field theory. 
  In this note we investigate U(N) gauge theories with matter in the fundamental and adjoint representations of the gauge group, interacting via generalized Yukawa terms of the form Tr[Q \Phi^n {\tilde Q}]. We find agreement between the matrix model and the gauge theory descriptions of these theories. The analysis leads to a partial description of the Higgs branch of the gauge theory. We argue that the transition between phases with different unbroken flavor symmetry groups is related to the appearance of cuts in the matrix model computation. 
  In this paper we examine fermionic type characters (Universal Chiral Partition Functions) for general 2D conformal field theories with a bilinear form given by a matrix of the form K \oplus K^{-1}. We provide various techniques for determining these K-matrices, and apply these to a variety of examples including (higher level) WZW and coset conformal field theories. Applications of our results to fractional quantum Hall systems and (level restricted) Kostka polynomials are discussed. 
  A universal symmetric truncation of the bosonic string Hilbert space yields all known closed fermionic string theories in ten dimensions, their D-branes and their open descendants. We highlight the crucial role played by group theory and two-dimensional conformal field theory in the construction and emphasize the predictive power of the truncation. Such circumstantial evidence points towards the existence of a mechanism which generates space-time fermions out of bosons dynamically within the framework of bosonic string theory. 
  Effective hadron models commonly require the computation of functional determinants. In the static case these are one--loop vacuum polarization energies, known as Casimir energies. In this talk I will present general methods to efficiently compute renormalized one--loop vacuum polarization energies and energy densities and apply these methods to construct soliton solutions within a variational approach. This calculational method is particularly useful to study singular limits that emerge in the discussion of the {\it classical} Casimir problem which is usually posed as the response of a fluctuating quantum field to externally imposed boundary conditions. 
  We analyze the structure of supersymmetric Godel-like cosmological solutions of string theory. Just as the original four-dimensional Godel universe, these solutions represent rotating, topologically trivial cosmologies with a homogeneous metric and closed timelike curves. First we focus on "phenomenological" aspects of holography, and identify the preferred holographic screens associated with inertial comoving observers in Godel universes. We find that holography can serve as a chronology protection agency: The closed timelike curves are either hidden behind the holographic screen, or broken by it into causal pieces. In fact, holography in Godel universes has many features in common with de Sitter space, suggesting that Godel universes could represent a supersymmetric laboratory for addressing the conceptual puzzles of de Sitter holography. Then we initiate the investigation of "microscopic" aspects of holography of Godel universes in string theory. We show that Godel universes are T-dual to pp-waves, and use this fact to generate new Godel-like solutions of string and M-theory by T-dualizing known supersymmetric pp-wave solutions. 
  In this paper we study the effects of noncommutativity on a closed superstring propagating in the spacetime that is compactified on tori. The effects of compactification and noncommutativity appear in the momentum, quantization, supercurrent, super-conformal generators and in the boundary state of the closed superstring emitted from a D$_p$-brane with the NS$\otimes$NS background $B$-field. 
  We clarify aspects of the holographic AdS/CFT correspondence that are typical of Lorentzian signature, to lay the foundation for a treatment of time-dependent gravity and conformal field theory phenomena. We provide a derivation of bulk-to-boundary propagators associated to advanced, retarded and Feynman bulk propagators, and provide a better understanding of the boundary conditions satisfied by the bulk fields at the horizon. We interpret the subleading behavior of the wavefunctions in terms of specific vacuum expectation values, and compute two-point functions in our framework. We connect our bulk methods to the closed time path formalism in the boundary field theory. 
  In this paper, we systematically study the effective action for non-commutative QED in the static limit at high temperature. When $\theta p^{2}\ll 1$, where $\theta$ represents the magnitude of the parameter for non-commutativity and $p$ denotes a typical external three momentum, we show that this leads naturally to a derivative expansion in this model. The study of the self-energy, in this limit, leads to nontrivial $\theta$ dependent corrections to the electric and magnetic masses, which exist only above a certain critical temperature. The three point and the four point amplitudes are also studied as well as their relations to the Ward identities in this limit. We determine the closed form expression for the current involving only the spatial components of the gauge field and present the corresponding static effective action, which is gauge invariant. 
  We construct the 4D N=1 supergravity which describes the low-energy limit of 6D supergravity compactified on a sphere with a monopole background a la Salam and Sezgin. This provides a simple setting sharing the main properties of realistic string compactifications such as flat 4D spacetime, chiral fermions and N=1 supersymmetry as well as Fayet-Iliopoulos terms induced by the Green-Schwarz mechanism. The matter content of the resulting theory is a supersymmetric SO(3)xU(1) gauge model with two chiral multiplets, S and T. The expectation value of T is fixed by the classical potential, and S describes a flat direction to all orders in perturbation theory. We consider possible perturbative corrections to the Kahler potential in inverse powers of $Re S$ and $Re T$, and find that under certain circumstances, and when taken together with low-energy gaugino condensation, these can lift the degeneracy of the flat direction for $Re S$. The resulting vacuum breaks supersymmetry at moderately low energies in comparison with the compactification scale, with positive cosmological constant. It is argued that the 6D model might itself be obtained from string compactifications, giving rise to realistic string compactifications on non Ricci flat manifolds. Possible phenomenological and cosmological applications are briefly discussed. 
  The Euclidean action and entropy are computed in string-generated gravity models with quadratic curvatures, and used to argue that a negative mass extremal metric is the background for hyperbolic (k=-1) black hole spacetimes, k being the curvature constant of the event horizon. The entropy associated with a black hole is always positive for k=(0,1) family. The positivity of energy condition also ensures that the k=-1 (extremal) entropy is non-negative. 
  Unlike the Schwarzschild black string in the Randall-Sundrum scenario which is known to have the geodesics reaching the AdS-horizon terminating there, the D=7 extremal BPS selfdual string of N=2 gauged supergravity potentially differs from this result. I give a complete proof that timelike radial trajectories of the selfdual string solution that escapes to r=\infty do not see a curvature singularity on the horizon at z=\infty. 
  We consider a noncommutative theory developed in a curved background. We show that the Moyal product has to be conveniently modified and, consequently, some of its old properties are lost compared with the flat case. We also address the question of diffeomorphism symmetry. 
  We consider the exact effective superpotential of N=1 U(N_c) super Yang-Mills theory with N_f massive flavors an additional adjoint Higgs field. We use the proposal of Dijkgraaf and Vafa to calculate the superpotential in terms of a matrix model with a large number of flavors. We do this by gauging the flavor symmetry and forcing this sector in a classical vacuum. This gives rise to a 2-matrix model of ADE type A_2, and large flavors. This approach allows us to add an arbitrary polynomial tree level superpotential for the Higgs field, and use strict large N methods in the matrix model. 
  We investigate the asymptotic dynamics of topological anti-de Sitter supergravity in two dimensions. Starting from the formulation as a BF theory, it is shown that the AdS_2 boundary conditions imply that the asymptotic symmetries form a super-Virasoro algebra. Using the central charge of this algebra in Cardy's formula, we exactly reproduce the thermodynamical entropy of AdS_2 black holes. Furthermore, we show that the dynamics of the dilaton and its superpartner reduces to that of superconformal transformations that leave invariant one chiral component of the stress tensor supercurrent of a two-dimensional conformal field theory. This dynamics is governed by a supersymmetric extension of the de Alfaro-Fubini-Furlan model of conformal quantum mechanics. Finally, two-dimensional de Sitter gravity is also considered, and the dS_2 entropy is computed by counting CFT states. 
  Quenched reduction is revisited from the modern viewpoint of field-orbifolding. Fermions are included and it is shown how the old problem of preserving anomalies and field topology after reduction is solved with the help of the overlap construction. 
  We deconstruct the fifth dimension of the 5D SYM theory with SU(M) gauge symmetry and Chern-Simons level k=M and show how the 5D moduli space follows from the non-perturbative analysis of the 4D quiver theory. The 5D coupling h=1/(g_5)^2 of the un-broken SU(M) is allowed to take any non-negative values, but it cannot be continued to h<0 and there are no transitions to other phases of the theory. The alternative UV completions of the same 5D SYM -- via M theory on the C^3/Z_2M orbifold or via the dual five-brane web in type IIB string theory -- have identical moduli spaces: h >= 0 only, and no flop transitions. We claim these are intrinsic properties of the SU(M) SYM theory with k=M. 
  We investigate the quantum consistency of p-form Maxwell-Chern-Simons electrodynamics in 3p+2 spacetime dimensions (for p odd). These are the dimensions where the Chern--Simons term is cubic, i.e., of the form FFA. For the theory to be consistent at the quantum level in the presence of magnetic and electric sources, we find that the Chern--Simons coupling constant must be quantized. We compare our results with the bosonic sector of eleven dimensional supergravity and find that the Chern--Simons coupling constant in that case takes its corresponding minimal allowed value. 
  We make a number of observations about matter-ghost string phase, which may eventually lead to a formal connection between matroid theory and string theory. In particular, in order to take advantage of the already established connection between matroid theory and Chern-Simons theory, we propose a generalization of string theory in terms of some kind of Kahler metric. We show that this generalization is closely related to the Kahler-Chern-Simons action due to Nair and Schiff. In addition, we discuss matroid/string connection via matroid bundles and a Schild type action, and we add new information about the relationship between matroid theory, D=11 supergravity and Chern-Simons formalism. 
  We discuss the cosmological constant problem in the context of higher codimension brane world scenarios with infinite-volume extra dimensions. In particular, by adding higher curvature terms in the bulk action we are able to find smooth solutions with the property that the 4-dimensional part of the brane world-volume is flat for a range of positive values of the brane tension. 
  We have examined the localization of gauge bosons on the three-brane embedded in 5D bulk space-time, and we find two kinds of branes on which both graviton and gauge bosons can be trapped. One is the dS brane with a positive cosmological constant, and the other is the one with zero cosmological constant or 4D Minkowski brane. Then, which brane is realized would depend on the observation of the cosmological constant of our world. 
  In this paper we study half-supersymmetric (D6,D8) bound state brane configuration of massive type IIA supergravity. We show that this bound state can also be generated by using massive T-duality rules of type-II supergravities in D=9, starting from D7-branes. We write down corresponding Killing spinors and find that these backgrounds indeed preserve 16 supersymmetries like any other D$p$-brane bound state with $B$-field. We also make a point on the massive nature of the $B$-field in this background. The Seiberg-Witten limits to obtain 9-dimensional NCYMs are also discussed, but the full understanding of such gauge theories remains unanswered. 
  On example of the model field system we demonstrate that quantum fluctuations of non-abelian gauge fields leading to radiative corrections to Higgs potential and spontaneous symmetry breaking can generate order region in phase space of inherently chaotic classical field system. We demonstrate on the example of another model field system that quantum fluctuations do not influence on the chaotic dynamics of non-abelian Yang--Mills fields if the ratio of bare coupling constants of Yang--Mills and Higgs fields is larger then some critical value. This critical value is estimated. 
  Recently it has been suggested that an increase in the fine structure constant alpha with time would decrease the entropy of a Reissner-Nordstrom black hole, thereby violating the second law of thermodynamics. In this note we point out that, at least for a certain class of charged dilaton black holes related to string theory, the entropy does not change under adiabatic variations of alpha and one might expect it to increase for non-adiabatic changes. 
  The dynamics of n slowly moving fundamental monopoles in the SU(n+1) BPS Yang-Mills-Higgs theory can be approximated by geodesic motion on the 4n-dimensional hyperkahler Lee-Weinberg-Yi manifold. In this paper we apply a variational method to construct some scaling geodesics on this manifold. These geodesics describe the scattering of n monopoles which lie on the vertices of a bouncing polyhedron; the polyhedron contracts from infinity to a point, representing the spherically symmetric n-monopole, and then expands back out to infinity. For different monopole masses the solutions generalize to form bouncing nested polyhedra. The relevance of these results to the dynamics of well separated SU(2) monopoles is also discussed. 
  In this paper we generalize the O(p+1,p+1) solution generating technique (this is a method used to deform Dp-branes by turning on a NS-NS B-field) to M-theory, in order to be able to deform M5-brane supergravity solutions directly in eleven dimensions, by turning on a non zero three form A. We find that deforming the M5-brane, in some cases, corresponds to performing certain SL(2,R) transformations of the Kahler structure parameter for the three-torus, on which the M5-brane has been compactified. We show that this new M-theory solution generating technique can be reduced to the O(p+1,p+1) solution generating technique with p=4. Further, we find that it implies that the open membrane metric and generalized noncommutativity parameter are manifestly deformation independent for electric and light-like deformations. We also generalize the O(p+1,p+1) method to the type IIA/B NS5-brane in order to be able to deform NS5-branes with RR three and two forms, respectively. In the type IIA case we use the newly obtained solution generating technique and deformation independence to derive a covariant expression for an open D2-brane coupling, relevant for OD2-theory. 
  We study theoretical aspects of the rotating black hole production and evaporation in the extra dimension scenarios with TeV scale gravity, within the mass range in which the higher dimensional Kerr solution provides good description. We evaluate the production cross section of black holes taking their angular momenta into account. We find that it becomes larger than the Schwarzschild radius squared, which is conventionally utilized in literature, and our result nicely agrees with the recent numerical study by Yoshino and Nambu within a few percent error for higher dimensional case. In the same approximation to obtain the above result, we find that the production cross section becomes larger for the black hole with larger angular momentum. Second, we derive the generalized Teukolsky equation for spin 0, 1/2 and 1 brane fields in the higher dimensional Kerr geometry and explicitly show that it is separable in any dimensions. For five-dimensional (Randall-Sundrum) black hole, we obtain analytic formulae for the greybody factors in low frequency expansion and we present the power spectra of the Hawking radiation as well as their angular dependence. Phenomenological implications of our result are briefly sketched. 
  I review the basic features of four dimensional Z_2 x Z_2 (shift) orientifolds with internal magnetic fields, describing two examples with N=1 supersymmetry. As in the corresponding six-dimensional examples, D9-branes magnetized along four internal directions can mimic D5-branes, even in presence of multiplets of image branes localized on different fixed tori. Chiral low-energy spectra can be obtained if the model also contains D5-branes parallel to the magnetized directions. 
  In this paper we propose a superfield description for all Bianchi-type cosmological models. The action is invariant under world-line local $n=4$ supersymmetry with $SU(2)_{local}XSU(2)_{global}$ internal symmetry. Due to the invariance of the action we obtain the constraints, which form a closed superalgebra of the $n=4$ supersymmetric quantum mechanics. This procedure provides the inclusion of supermatter in a sistematic way. 
  We study a class of near-BPS operators for a complex 2-parameter family of N=1 superconformal Yang-Mills theories that can be obtained by a Leigh-Strassler deformation of N=4 SYM theory. We identify these operators in the large N and large R-charge limit and compute their exact scaling dimensions using N=1 superspace methods. From these scaling dimensions we attempt to reverse-engineer the light-cone worldsheet theory that describes string propagation on the Penrose limit of the dual geometry. 
  We construct eight-stack intersecting D5-brane models, with an orbifold transverse space, that yield the (non-supersymmetric) standard model up to vector-like leptons. The matter includes right-chiral neutrinos and the models have the renormalisable Yukawa couplings to tachyonic Higgs doublets needed to generate mass terms for {\it all} matter, including the vector-like leptons. The models are constrained by the requirement that twisted tadpoles cancel, that the gauge boson coupled to the weak hypercharge $U(1)_Y$ does not get a string-scale mass via a generalised Green-Schwarz mechanism, and that there are no surviving, unwanted gauged U(1) symmetries coupled to matter.  Gauge coupling constant ratios close to those measured are easily obtained for reasonable values of the parameters, consistently with having the string scale close to the electroweak scale, as required to avoid the hierarchy problem. Unwanted (colour-triplet, charged-singlet, and neutral-singlet) scalar tachyons can be removed by a suitable choice of the parameters. 
  We study the growth of fluctuations in collapsing cosmologies, extending old work of Lifshitz and Khalatnikov. As examples of systems where the fluctuations have a different composition than the background we study scalar fields with general improvement terms. Fluctuations always grow, and often dominate the homogeneous background. We argue that even for very dilute fluctuations, scattering processes inevitably lead to a dense gas of black holes. This leads us to hypothesize that the generic final state of a Big Crunch is described by a collapsing $p=\rho$ FRW cosmology. We conjecture that the black hole fluid is invariant under the conformal Killing symmetry of this metric, so that the final state is in fact stationary. 
  The status of string theory is reviewed, and major recent developments - especially those in going beyond perturbation theory in the string theory and quantum field theory frameworks - are discussed. This analysis helps better understand the role and place of string theory in the modern picture of the physical world. Even though quantum field theory describes a wide range of experimental phenomena, it is emphasized that there are some insurmountable problems inherent in it - notably the impossibility to formulate the quantum theory of gravity on its basis - which prevent it from being a fundamental physical theory of the world of microscopic distances. It is this task, the creation of such a theory, which string theory, currently far from completion, is expected to solve. In spite of its somewhat vague current form, string theory has already led to a number of serious results and greatly contributed to progress in the understanding of quantum field theory. It is these developments which are our concern in this review. 
  We study nonconformal quantum scalar fields and averages of their local observables (such as <phi^2>^{ren} and T_{ab}^{ren}) in a spacetime of a 2-dimensional black hole. In order to get an analytical approximation for these expressions the WKB approximation is often used. We demonstrate that at the horizon WKB approximation is violated for a nonconformal field, that is when the field mass or/and the parameter of non-minimal coupling do not vanish. We propose a new "uniform approximation" which solves this problem. We use this approximation to obtain an improved analytical approximation for <phi^2>^{ren} in the 2-dimensional black hole geometry. We compare the obtained results with numerical calculations. 
  We study four-point correlation functions of 1/2-BPS operators in N=4 SYM which are dual to massive KK modes in AdS_5 supergravity. On the field theory side, the procedure of inserting the SYM action yields partial non-renormalisation of the four-point amplitude for such operators. In particular, if the BPS operators have dimensions equal to three or four, the corresponding four-point amplitude is determined by one or two independent functions of the two conformal cross-ratios, respectively. This restriction on the amplitude does not merely follow from the superconformal Ward identities, it also encodes dynamical information related to the structure of the gauge theory Lagrangian. The dimension 3 BPS operator is the AdS/CFT dual of the first non-trivial massive Kaluza-Klein mode of the compactified type IIB supergravity, whose interactions go beyond the level of the five-dimensional gauged N=8 supergravity. We show that the corresponding effective Lagrangian has a surprisingly simple sigma-model-type form with at most two derivatives. We then compute the supergravity-induced four-point amplitude for the dimension 3 operators. Remarkably, this amplitude splits into a "free" and an "interacting" parts in exact agreement with the structure predicted by the insertion procedure. The underlying OPE fulfills the requirements of superconformal symmetry and unitarity. 
  We investigate a U(1) gauge invariant quantum mechanical system on a 2D noncommutative space with coordinates generating a generalized deformed oscillator algebra. The Hamiltonian is taken as a quadratic form in gauge covariant derivatives obeying the nonlinear Dolan-Grady relations. This restricts the structure function of the deformed oscillator algebra to a quadratic polynomial. The cases when the coordinates form the su(2) and sl(2,R) algebras are investigated in detail. Reducing the Hamiltonian to 1D finite-difference quasi-exactly solvable operators, we demonstrate partial algebraization of the spectrum of the corresponding systems on the fuzzy sphere and noncommutative hyperbolic plane. A completely covariant method based on the notion of intrinsic algebra is proposed to deal with the spectral problem of such systems. 
  The role of general two-impurity multi-trace operators in the BMN correspondence is explored. Surprisingly, the anomalous dimensions of all two-impurity multi-trace BMN operators to order g_2^2\lambda' are completely determined in terms of single-trace anomalous dimensions. This is due to suppression of connected field theory diagrams in the BMN limit and this fact has important implications for some string theory processes on the PP-wave background. We also make gauge theory predictions for the matrix elements of the light-cone string field theory Hamiltonian in the two string-two string and one string-three string sectors. 
  The aim of this work is to present a general theory of coset models G/H in which different left and right actions of H on G are gauged. Our main results include a formula for their modular invariant partition function, the construction of a large set of boundary states and a general description of the corresponding brane geometries. The paper concludes with some explicit applications to the base of the conifold and to the time-dependent Nappi-Witten background. 
  In another application of the methods of Henneaux, Teitelboim, and Vergara developed for diffeomorphisms invariant models, the CGHS theory of 2D black holes is focused in order to obtain the true degrees of freedom, the simplectic structure and the {\it effective} Hamiltonian that rules the dynamics in reduced phase-space. 
  We discuss how to obtain the superpotential of the baryons and mesons for SU(N) gauge theories with N flavour matter fields from matrix integral. We apply the mean-field approximation for the matrix integral. Assuming the planar limit of the self-consistency equation, we show that the result almost agrees with the field theoretical result. 
  We study the properties of the energy-momentum tensor in non-commutative gauge theories by coupling them to a weak external gravitational field. In particular, we show that the stress tensor of such a theory coincides exactly with that derived from a theory where a Seiberg-Witten map has been implemented (namely, the procedure is commutative). Various other interesting features are also discussed. 
  A novel geometrical model of a two-brane has been constructed, which is  inherently noncommutative, without external interactions. We show that the noncommutative spacetime is to be interpreted as having an internal angular momentum throughout. Subsequently, the elementary excitations - i.e., point particles - living on the brane are endowed with a spin. The study of them reveals in a natural way various signatures of a noncommutative quantum theory, such as dipolar nature of the basic excitations \cite{jab} and momentum dependent shifts in the interaction point \cite{big}. The observation \cite{sw} that noncommutative and ordinary field theories are alternative descriptions of the same underlying theory, is corroborated here by showing that they are gauge equivalent. Also, treating the present model as an explicit example, we show that, even classically, in the presence of constraints, the equivalence between Nambu-Goto and Polyakov formulations is subtle. 
  We discuss properties of a 3-brane in an asymptotic 5-dimensional de-Sitter spacetime. It is found that a Minkowski solution can be obtained without fine-tuning. In the model, the tiny observed positive cosmological constant is interpreted as a curvature of 5-dimensional manifold, but the Minkowski spacetime, where we live, is a natural 3-brane perpendicular to the fifth coordinate axis. 
  We study the fermionic zero modes of BPS semilocal magnetic vortices in N=2 supersymmetric QED with a Fayet-Iliopoulos term and two matter hypermultiplets of opposite charge. There is a one-parameter family of vortices with arbitrarily wide magnetic cores. Contrary to the situation in pure Nielsen-Olesen vortices, new zero modes are found which get their masses from Yukawa couplings to scalar fields that do not wind and are non-zero at the core. We clarify the relation between fermion mass and zero modes. The new zero modes have opposite chiralities and therefore do not affect the net counting (left minus right) of zero modes coming from index theorems but manage to evade other index theorems in the literature that count the total number (left plus right) of zero modes in simpler systems. 
  We compute the Euclidean actions of a $d$-dimensional charged rotating black brane both in the canonical and the grand-canonical ensemble through the use of the counterterms renormalization method, and show that the logarithmic divergencies associated with the Weyl anomalies and matter field vanish. We obtain a Smarr-type formula for the mass as a function of the entropy, the angular momenta, and the electric charge, and show that these quantities satisfy the first law of thermodynamics. Using the conserved quantities and the Euclidean actions, we calculate the thermodynamics potentials of the system in terms of the temperature, angular velocities, and electric potential both in the canonical and grand-canonical ensembles. We also perform a stability analysis in these two ensembles, and show that the system is thermally stable. This is commensurate with the fact that there is no Hawking-Page phase transition for a black object with zero curvature horizon. Finally, we obtain the logarithmic correction of the entropy due to the thermal fluctuation around the equilibrium. 
  We define a 3-generator algebra obtained by replacing the commutators by anticommutators in the defining relations of the angular momentum algebra. We show that integer spin representations are in one to one correspondence with those of the angular momentum algebra. The half-integer spin representations, on the other hand, split into two representations of dimension j + 1/2. The anticommutator spin algebra is invariant under the action of the quantum group SO_q(3) with q=-1. 
  We study the properties of matrix models with soft confining potentials. Their precise mathematical characterization is that their weight function is not determined by its moments. We mainly rely on simple considerations based on orthogonal polynomials and the moment problem. In addition, some of these models are equivalent, by a simple mapping, to matrix models that appear in Chern-Simons theory. The models can be solved with q deformed orthogonal polynomials (Stieltjes-Wigert polynomials), and the deformation parameter turns out to be the usual $q$ parameter in Chern-Simons theory. In this way, we give a matrix model computation of the Chern-Simons partition function on $S^{3}$ and show that there are infinitely many matrix models with this partition function. 
  The fermionic oscillator defined by the algebraic relations cc^*+c^*c=1 and c^{2}=0 admits the homogeneous group O(2) as its invariance group. We show that, the structure of the inhomogeneous invariance group of this oscillator is a quantum group. 
  We study the effects produced by nonlinear electrodynamics in spacetimes conformal to Bianchi metrics. In the presence of Born-Infeld type fields these models accelerate, expand and isotropize. This effect is compared with the corresponding to a linear electromagnetic field; it turns out that for the same geometry, Maxwell fields does not favour inflation as much as Born-Infeld field. The behavior of the nonlinear radiation is analyzed in terms of the equation of state. The energy conditions are analyzed as well, showing that the Born-Infeld field violates the strong energy condition. 
  We present a review of the homological algebra tools involved in the standard de Rham theory and their subsequent generalizations relevant for the understanding of free massless higher spin gauge structure. M-theory arguments suggest the existence of an extension of (Abelian) S-duality symmetry for non-Abelian gauge theories, like the four dimensional Yang-Mills or Einstein theories. Some no-go theorems prove that this extension, if it exists, should fall outside the scope of local perturbative field theory. 
  We consider astrophysics of large black holes localized on the brane in the infinite Randall-Sundrum model. Using their description in terms of a conformal field theory (CFT) coupled to gravity, deduced in Ref. [1], we show that they undergo a period of rapid decay via Hawking radiation of CFT modes. For example, a black hole of mass ${\rm few} \times M_\odot$ would shed most of its mass in $\sim 10^4 - 10^5$ years if the AdS radius is $L \sim 10^{-1}$ mm, currently the upper bound from table-top experiments. Since this is within the mass range of X-ray binary systems containing a black hole, the evaporation enhanced by the hidden sector CFT modes could cause the disappearance of X-ray sources on the sky. This would be a striking signature of RS2 with a large AdS radius. Alternatively, for shorter AdS radii, the evaporation would be slower. In such cases, the persistence of X-ray binaries with black holes already implies an upper bound on the AdS radius of $L \la 10^{-2}$ mm, an order of magnitude better than the bounds from table-top experiments. The observation of primordial black holes with a mass in the MACHO range $M \sim 0.1 - 0.5 M_\odot$ and an age comparable to the age of the universe would further strengthen the bound on the AdS radius to $L \la {\rm few} \times 10^{-6} $ mm. 
  In spacetime dimensions larger than 2, whenever a global symmetry G is spontaneously broken to a subgroup H, and G and H are Lie groups, there are Nambu-Goldstone modes described by fields with values in G/H. In two-dimensional spacetimes as well, models where fields take values in G/H are of considerable interest even though in that case there is no spontaneous breaking of continuous symmetries. We consider such models when the world sheet is a two-sphere and describe their fuzzy analogues for G=SU(N+1), H=S(U(N-1)xU(1)) ~ U(N) and G/H=CP^N. More generally our methods give fuzzy versions of continuum models on S^2 when the target spaces are Grassmannians and flag manifolds described by (N+1)x(N+1) projectors of rank =< (N+1)/2. These fuzzy models are finite-dimensional matrix models which nevertheless retain all the essential continuum topological features like solitonic sectors. They seem well-suited for numerical work. 
  We solve for spectrum, obtain explicitly and study group properties of eigenfunctions of Dirac operator on the Riemann sphere $S^2$. The eigenvalues $\lambda$ are nonzero integers. The eigenfunctions are two-component spinors that belong to representations of SU(2)-group with half-integer angular momenta $l = |\lambda| - \half$. They form on the sphere a complete orthonormal functional set alternative to conventional spherical spinors. The difference and relationship between the spherical spinors in question and the standard ones are explained. 
  Motivated by the search for potentially exactly solvable time-dependent string backgrounds, we determine all homogeneous plane wave (HPW) metrics in any dimension and find one family of HPWs with geodesically complete metrics and another with metrics containing null singularities. The former generalises both the Cahen-Wallach (constant $A_{ij}$) metrics to time-dependent HPWs, $A_{ij}(t)$, and the Ozsvath-Sch\"ucking anti-Mach metric to arbitrary dimensions. The latter is a generalisation of the known homogeneous metrics with $A_{ij}\sim 1/t^2$ to a more complicated time-dependence. We display these metrics in various coordinate systems, show how to embed them into string theory, and determine the isometry algebra of a general HPW and the associated conserved charges. We review the Lewis-Riesenfeld theory of invariants of time-dependent harmonic oscillators and show how it can be deduced from the geometry of plane waves. We advocate the use of the invariant associated with the extra (timelike) isometry of HPWs for lightcone quantisation, and illustrate the procedure in some examples. 
  We describe intersecting M5-branes, as well as M5-branes wrapping the holomorphic curve xy=c, in terms of a limit of a defect conformal field theory with two-dimensional (4,0) supersymmetry. This dCFT describes the low-energy theory of intersecting D3-branes at a C^2/Z_k orbifold. In an appropriate k -> infinity limit, two compact spatial directions are generated. By identifying moduli of the M5-M5 intersection in terms of those of the dCFT, we argue that the SU(2)_L R-symmetry of the (4,0) defect CFT matches the SU(2) R-symmetry of the N =2, d=4 theory of the M5-M5 intersection. We find a 't Hooft anomaly in the SU(2)_L R-symmetry, suggesting that tensionless strings give rise to an anomaly in the SU(2) R-symmetry of intersecting M5-branes. 
  Brane-like vertex operators, defining backgrounds with the ghost-matter mixing in NSR superstring theory, play an important role in a world-sheet formulation of D-branes and M theory, being creation operators for extended objects in the second quantized formalism. In this paper we show that dilaton's beta function in ghost-matter mixing backgrounds becomes stochastic. The renormalization group (RG) equations in ghost-matter mixing backgrounds lead to non-Markovian Fokker-Planck equations which solutions describe superstrings in curved space-times with brane-like metrics.We show that Feigenbaum universality constant $\delta=4,669...$ describing transitions from order to chaos in a huge variety of dynamical systems, appears analytically in these RG equations. We find that the appearance of this constant is related to the scaling of relative space-time curvatures at fixed points of the RG flow. In this picture the fixed points correspond to the period doubling of Feigenbaum iterational schemes. 
  We study large N conformal field theories perturbed by relevant double-trace deformations. Using the auxiliary field trick, or Hubbard-Stratonovich transformation, we show that in the infrared the theory flows to another CFT. The generating functionals of planar correlators in the ultraviolet and infrared CFT's are shown to be related by a Legendre transform. Our main result is a universal expression for the difference of the scale anomalies between the ultraviolet and infrared fixed points, which is of order 1 in the large N expansion. Our computations are entirely field theoretic, and the results are shown to agree with predictions from AdS/CFT. We also remark that a certain two-point function can be computed for all energy scales on both sides of the duality, with full agreement between the two and no scheme dependence. 
  We obtain a unified picture for the conifold singularity resolution. We propose that gauged supergravity, through a novel prescription for the twisting, provides an appropriate framework to smooth out singularities in the context of gravity duals of supersymmetric gauge theories. 
  We analyze large gauge invariance in combined nonabelian and thermal QFT and their physical consequences for D=3 effective actions. After briefly reviewing the structure of bundles and large gauge transformations that arise in non-simply connected 3-manifolds and gauge groups, we discuss their connections to Chern-Simons terms and Wilson-Polyakov loops. We then provide an invariant characterization of the ``abelian'' fluxes encountered in explicit computations of finite temperature effective actions. In particular we relate, and provide explicit realizations of, these fluxes to a topological index that measures the obstruction to global diagonalization of the loops around compactified time. We also explore the fate of, and exhibit some everywhere smooth, large transformations for non-vanishing index in various topologies. 
  We consider the interplay of duality symmetries and gauged isometries of supergravity models giving N-extended, spontaneously broken supergravity with a no-scale structure. Some examples, motivated by superstring and M-theory compactifications are described. 
  We explicitly construct N=1 worldvolume supersymmetric minimal off-shell Goldstone superfield actions for two options of 1/2 partial spontaneous breaking of AdS5 supersymmetry SU(2,2|1) corresponding to its nonlinear realizations in the supercosets with the AdS5 and AdS5 X S1 bosonic parts. The relevant Goldstone supermultiplets are comprised, respectively, by improved tensor and chiral N=1 superfields. The second action is obtained from the first one by duality transformation. In the bosonic sectors they yield static-gauge Nambu-Goto actions for L3-brane on AdS5 and scalar 3-brane on AdS5 X S1. 
  We discuss the dynamics of a spherically symmetric dark radiation vaccum in the Randall-Sundrum brane world scenario. Under certain natural assumptions we show that the Einstein equations on the brane form a closed system. For a de Sitter brane we determine exact dynamical and inhomogeneous solutions which depend on the brane cosmological constant, on the dark radiation tidal charge and on its initial configuration. We define the conditions leading to singular or globally regular solutions. We also analyse the localization of gravity near the brane and show that a phase transition to a regime where gravity propagates away from the brane may occur at short distances during the collapse of positive dark energy density. 
  Compactifying a higher-dimensional theory defined in R^{1,3+n} on an n-dimensional manifold {\cal M} results in a spectrum of four-dimensional (bosonic) fields with masses m^2_i = \lambda_i, where - \lambda_i are the eigenvalues of the Laplacian on the compact manifold. The question we address in this paper is the inverse: given the masses of the Kaluza-Klein fields in four dimensions, what can we say about the size and shape (i.e. the topology and the metric) of the compact manifold? We present some examples of isospectral manifolds (i.e., different manifolds which give rise to the same Kaluza-Klein mass spectrum). Some of these examples are Ricci-flat, complex and K\"{a}hler and so they are isospectral backgrounds for string theory. Utilizing results from finite spectral geometry, we also discuss the accuracy of reconstructing the properties of the compact manifold (e.g., its dimension, volume, and curvature etc) from measuring the masses of only a finite number of Kaluza-Klein modes. 
  We find explicit supergravity solutions which describe branes in the AdS_3 x S^3 background. These solutions preserve 8 of the 16 supersymmetries of this background, and are consistent with kappa-symmetry. These represent new 1/2-BPS states of string theory. 
  The original ideas about noncommuting coordinates are recalled. The connection between U(1) gauge fields defined on noncommuting coordinates and fluid mechanics is explained. 
  We consider strings with large spin in AdS_3xS^3xM with NS-NS background. We construct the string configurations as solutions of SL(2,R) WZW theory. We compute the relation between the space-time energy and spin, and show that the anomalous correction is constant, and not logarithmic in the spin. This is in contrast to the S-dual background with R-R charge where the anomalous correction is logarithmic. 
  We point out a problem with the stability of composite (global-magnetic) monopoles recently proposed by J. Spinelly, U. de Freitas and E.R. Bezerra de Mello [Phys. Rev. D66, 024018 (2002)]. 
  We analyze the decomposition of tensor products between infinite dimensional (unitary) and finite-dimensional (non-unitary) representations of SL(2,R). Using classical results on indefinite inner product spaces, we derive explicit decomposition formulae, true modulo a natural cohomological reduction, for the tensor products. 
  We review the concept of S-branes introduced by Gutperle and Strominger hep-th/0202210. Using the effective spacetime description of the rolling tachyon worldsheets discussed by Sen, we analyze the possibility that the gravitational backreaction of tachyon matter is important in the time-dependent process. We show that this is indeed the case in the example of the S0-brane in 4-dimensional Einstein-Maxwell theory. This talk is based on hep-th/0207235. 
  We generalize the Brane Gas Cosmological Scenario to M-theory degrees of freedom, namely $M5$ and $M2$ branes. Without brane intersections, the Brandenberger Vafa(BV) arguments applied to M-theory degrees of freedom generically predict a large 6 dimensional spacetime. We show that intersections of $M5$ and $M2$ branes can instead lead to a large 4 dimensional spacetime. One dimensional intersections in 11D is related to (2,0) little strings (LST) on NS5 branes in type IIA. The gas regime of membranes in M-theory corresponds to the thermodynamics of LST obtained from holography. We propose a mechanism whereby LST living on the worldvolume of NS5 (M5)-branes wrapping a five dimensional torus, annihilate most efficiently in 3+1 dimensions leading to a large 3+1 dimensional spacetime. We also show that this picture is consistent with the gas approximation in M-theory. 
  We present a variational method which uses a quartic exponential function as a trial wave-function to describe time-dependent quantum mechanical systems. We introduce a new physical variable $y$ which is appropriate to describe the shape of wave-packet, and calculate the effective action as a function of both the dispersion $\sqrt{< \hat{q}^2>}$ and $y$. The effective potential successfully describes the transition of the system from the false vacuum to the true vacuum. The present method well describes the long time evolution of the wave-function of the system after the symmetry breaking, which is shown in comparison with the direct numerical computations of wave-function. 
  The problem of dark energy arises due to its self-gravitating properties. Therefore explaining vacuum energy may become a question for the realm of quantum gravity, that can be addressed within string theory context. In this talk I concentrate on a recent, string-inspired model, that relies on nonlinear physics of short-distance perturbation modes, for explaining dark energy without any fine-tuning. Dark energy can be observationally probed by its equation of state, w. Different models predict different types of equations of state and string-inspired ones have a time dependent w(z) as their unique signature. Exploring the link between dark energy and string theory may provide indirect evidence for the latter, by means of precision cosmology data. 
  We investigate the algebraic structure of supersymmetric Ward- and Slavnov-Taylor identities (STI) in field theories with global supersymmetry. We develop diagrammatical methods for STI in supersymmetric gauge theories within a generalized BRST formalism. These identities are used as consistency checks for the implementation of phenomenological supersymmetric models of elementary particle physics in Monte Carlo simulations. We discuss the infrastructure necessary for generalizing the matrix element generator O'Mega to supersymmetric field theories. 
  We explore spherically symmetric stationary solutions, generated by ``stars'' with regular interiors, in purely massive gravity. We reexamine the claim that the resummation of non-linear effects can cure, in a domain near the source, the discontinuity exhibited by the linearized theory as the mass m of the graviton tends to zero. First, we find analytical difficulties with this claim, which appears not to be robust under slight changes in the form of the mass term. Second, by numerically exploring the inward continuation of the class of asymptotically flat solutions, we find that, when m is ``small'', they all end up in a singularity at a finite radius, well outside the source, instead of joining some conjectured ``continuous'' solution near the source. We reopen, however, the possibility of reconciling massive gravity with phenomenology by exhibiting a special class of solutions, with ``spontaneous symmetry breaking'' features, which are close, near the source, to general relativistic solutions and asymptote, for large radii, a de Sitter solution of curvature ~m^2. 
  The Epstein--Glaser type T-subtraction introduced by one of the authors in a previous paper is extended to the Lorentz invariant framework. The advantage of using our subtraction instead of Epstein and Glaser's standard W-subtraction method is especially important when working in Minkowski space, as then the counterterms necessary to keep Lorentz invariance are simplified. We show how T-renormalization of primitive diagrams in the Lorentz invariant framework directly relates to causal Riesz distributions. A covariant subtraction rule in momentum space is found, sharply improving upon the BPHZL method for massless theories. 
  The method which allows for asymptotic expansion of the one-loop effective action W=ln det A is formulated. The positively defined elliptic operator A= U + M^2 depends on the external classical fields taking values in the Lie algebra of the internal symmetry group G. Unlike the standard method of Schwinger - DeWitt, the more general case with the nondegenerate mass matrix M=diag(m1,m2,...) is considered. The first coefficients of the new asymptotic series are calculated and their relationship with the Seeley-DeWitt coefficients is clarified. 
  A parafermionic conformal theory with the symmetry Z_5 is constructed, based on the second solution of Fateev-Zamolodchikov for the corresponding parafermionic chiral algebra.   The primary operators of the theory, which are the singlet, doublet 1, doublet 2, and disorder operators, are found to be accommodated by the weight lattice of the classical Lie algebra B_2. The finite Kac tables for unitary theories are defined and the formula for the conformal dimensions of primary operators is given. 
  We obtain a $\frac 14$-supersymmetric 6-brane solution of IIA Supergravity by T-dualizing the supertube recently found. The resulting $C_{\it 1}$ electric charge is related to the original $D0$-brane charge. The uplifted solution to eleven dimensions results to be a purely geometrical configuration, which can be interpreted as a bound state of a Taub-NUT space and a pp-wave. Being the non trivial part of the metric pseudo-Riemannian, the resulting reduced holonomy group is non-compact and locally isomorphic to a semidirect product of an Abelian four dimensional group and SU(2). 
  We calculate one-loop scattering amplitudes for gravitons and two-forms in dimensions greater than four. The string based Kawai-Lewellen-Tye relationships allow gravitons and two-forms to be treated in a unified manner. We use the results to determine the ultra-violet infinities present in these amplitudes and show how these determine the renormalised one-loop action in six and eight dimensions. 
  We discuss several examples of three-dimensional critical phenomena that can be described by Landau-Ginzburg-Wilson $\phi^4$ theories. We present an overview of field-theoretical results obtained from the analysis of high-order perturbative series in the frameworks of the $\epsilon$ and of the fixed-dimension d=3 expansions. In particular, we discuss the stability of the O(N)-symmetric fixed point in a generic N-component theory, the critical behaviors of randomly dilute Ising-like systems and frustrated spin systems with noncollinear order, the multicritical behavior arising from the competition of two distinct types of ordering with symmetry O($n_1$) and O($n_2$) respectively. 
  We use fractional and wrapped branes to describe perturbative and nonperturbative properties of the gauge theories living on their worldvolume. (Talk given at the 35th International Symposium Ahrenshoop on the Theory of Elementary Particles, Wernsdorf, August 26-30, 2002.) 
  We study Schwinger pair creation of charged Kaluza-Klein particles from a static KK electric field. We find that the gravitational backreaction of the electric field on the geometry - which is incorporated via the electric KK Melvin solution - prevents the electrostatic potential from overcoming the rest mass of the KK particles, thus impeding the tunneling mechanism which is often thought of as responsible for the pair creation. However, we find that pair creation still occurs with a finite rate formally similar to the classic Schwinger result, but via an apparently different mechanism, involving a combination of the Unruh effect and vacuum polarization due to the E-field. 
  A non-trivial q-deformation of the fermionic oscillator proposed by us in 1991 is recalled in view of the result of the paper cited in the title. 
  In the Penrose limit, AdS*S space turns into a Cahen-Wallach (CW) space whose Killing vectors satisfy a Heisenberg algebra. This algebra is mapped onto the holographic screen on the boundary of AdS. I show that the Heisenberg algebra on the boundary of AdS may be obtained directly from the CW space by appropriately constraining the states defined on it. The transformations generated by the constraint are similar to gauge transformations. The ``holographic screen'' on the CW space is thus obtained as a ``gauge-fixing'' condition. 
  We show that the BMN operators in D=4 N=4 super Yang Mills theory proposed as duals of stringy oscillators in a plane wave background have a natural quantum group construction in terms of the quantum deformation of the SO(6) $R$ symmetry. We describe in detail how a q-deformed U(2) subalgebra generates BMN operators, with $ q \sim e^{2 i \pi \over J}$. The standard quantum co-product as well as generalized traces which use $q$-cyclic operators acting on tensor products of Higgs fields are the ingredients in this construction. They generate the oscillators with the correct (undeformed) permutation symmetries of Fock space oscillators. The quantum group can be viewed as a spectrum generating algebra, and suggests that correlators of BMN operators should have a geometrical meaning in terms of spaces with quantum group symmetry. 
  We show how one can get solitonic strings in a six-dimensional (2,0) supersymmetric theory by incorporating a nonlinear interaction term. We derive a zero force condition between parallel strings, and compute a metric on a moduli space which is $R^4$ when the strings are far apart. When compactifying the strings on a two-torus we show that, in the limit of vanishing two-torus, one regains the moduli space of two widely separated dyons of equal magnetic charges in four dimensions. 
  Gravitational wave solutions to Einstein's equations and their generation are examined in D-dimensional flat spacetimes. First the plane wave solutions are analyzed; then the wave generation is studied with the solution for the metric tensor being obtained with the help of retarded D-dimensional Green's function. Due to the difficulties in handling the wave tails in odd dimensions we concentrate our study in even dimensions. We compute the metric quantities in the wave zone in terms of the energy momentum tensor at retarded time. Some special cases of interest are studied: first the slow motion approximation, where the $D$-dimensional quadrupole formula is deduced. Within the quadrupole approximation, we consider two cases of interest, a particle in circular orbit and a particle falling radially into a higher dimensional Schwarzschild black hole. Then we turn our attention to the gravitational radiation emitted during collisions lasting zero seconds, i.e., hard collisions. We compute the gravitational energy radiated during the collision of two point particles, in terms of a cutoff frequency. In the case in which at least one of the particles is a black hole, we argue this cutoff frequency should be close to the lowest gravitational quasinormal frequency. In this context, we compute the scalar quasinormal frequencies of higher dimensional Schwarzschild black holes. Finally, as an interesting new application of this formalism, we compute the gravitational energy release during the quantum process of black hole pair creation. These results might be important in light of the recent proposal that there may exist extra dimensions in the Universe, one consequence of which may be black hole creation at the Large Hadron Collider at CERN. 
  The exact solution of the noncompact SL(2,C) Heisenberg spin magnet reveals a hidden symmetry of the energy spectrum. To understand its origin, we solve the spectral problem for the model within quasiclassical approach. In this approach, the integrals of motion satisfy the Bohr-Sommerfeld quantization conditions imposed on the orbits of classical motion. In the representation of the separated coordinates, the latter wrap around a Riemann surface defined by the spectral curve of the model. A novel feature of the obtained quantization conditions is that they involve both the alpha- and beta-periods of the action differential on the Riemann surface, thus allowing us to find their solutions by exploring the full modular group of the spectral curve. We demonstrate that the quasiclassical energy spectrum is in a good agreement with the exact results. 
  Scalar fields are studied on fuzzy $S^4$ and a solution is found for the elimination of the unwanted degrees of freedom that occur in the model. The resulting theory can be interpreted as a Kaluza-Klein reduction of CP^3 to S^4 in the fuzzy context. 
  The lowest (``vector'') and next-lowest (``scalar'') bound-state masses of the massive Schwinger model have been determined recently to a very high accuracy numerically on the lattice. Therefore, improved results for these bound-state masses from analytical calculations are of some interest. Here, we provide such improved results by employing both standard and renormal-ordered (fermion) mass perturbation theory, as well as a consistency condition between the two perturbative calculations. The resulting bound-state masses are in excellent agreement with the lattice results for small and intermediate fermion mass, and remain within 10% of the exact results even in the limit of very large fermion mass. 
  We discuss the action of SL(2,Z) on local operators in D=4, N=4 SYM theory in the superconformal phase. The modular property of the operator's scaling dimension determines whether the operator transforms as a singlet, or covariantly, as part of a finite or infinite dimensional multiplet under the SL(2,Z) action. As an example, we argue that operators in the Konishi multiplet transform as part of a (p,q) PSL(2,Z) multiplet. We also comment on the non-perturbative local operators dual to the Konishi multiplet. 
  The representation of the superalgebra SO(2,1) which is given by Eq. (3.1) and which resulted in the relativistic wave equation (4.1) is fully reducible. In fact, its even part is the direct sum of two spin 1/2 representations of the Lorentz group and does not represent spin 3/2 particle as we claimed. Consequently, we retract our claim and withdraw the manuscript. 
  The problem of decomposition of unitary irreps of (super) tensorial (i.e. extended with tensorial charges) Poincar{\'e} algebra w.r.t. its different subalgebras is considered. This requires calculation of little groups for different configurations of tensor charges. Particularly, for preon states (i.e. states with maximal supersymmetry) in different dimensions the particle content is calculated, i.e. the spectrum of usual Poincar{\'e} representations in the preon representation of tensorial Poincar{\'e}. At d=4 results coincide with (and may provide another point of view on) the Vasiliev's results in field theories in generalized space-time. The translational subgroup of little groups of massless particles and branes is shown to be (and coincide with, at d=4) a subgroup of little groups of "pure branes" algebras, i.e. tensorial Poincar{\'e} algebras without vector generators. At 11d it is shown that, contrary to lower dimensions, spinors are not homogeneous space of Lorentz group, and one have to distinguish at least 7 different kinds of preons. 
  In five-dimensional supergravity, an exact solution of BPS wall is found for a gravitational deformation of the massive Eguchi-Hanson nonlinear sigma model. The warp factor decreases for both infinities of the extra dimension. Thin wall limit gives the Randall-Sundrum model without fine-tuning of input parameters. We also obtain wall solutions with warp factors which are flat or increasing in one side, by varying a deformation parameter of the potential. 
  It is argued that the familiar algebra of the non-commutative space-time with $c$-number $\theta^{\mu\nu}$ is inconsistent from a theoretical point of view. Consistent algebras are obtained by promoting $\theta^{\mu\nu}$ to an anti-symmetric tensor operator ${\hat\theta}^{\mu\nu}$. The simplest among them is Doplicher-Fredenhagen-Roberts (DFR) algebra in which the triple commutator among the coordinate operators is assumed to vanish. This allows us to define the Lorentz-covariant operator fields on the DFR algebra as operators diagonal in the 6-dimensional $\theta$-space of the hermitian operators, ${\hat\theta}^{\mu\nu}$. It is shown that we then recover Carlson-Carone-Zobin (CCZ) formulation of the Lorentz-invariant non-commutative gauge theory with no need of compactification of the extra 6 dimensions. It is also pointed out that a general argument concerning the normalizability of the weight function in the Lorentz metric leads to a division of the $\theta$-space into two disjoint spaces not connected by any Lorentz transformation so that the CCZ covariant moment formula holds true in each space, separately. A non-commutative generalization of Connes' two-sheeted Minkowski space-time is also proposed. Two simple models of quantum field theory are reformulated on $M_4\times Z_2$ obtained in the commutative limit. 
  We construct some N=1 supersymmetric three-family SU(5) Grand Unified Models from type IIA orientifolds on $\IT^6/(\IZ_2\times \IZ_2)$ with D6-branes intersecting at general angles. These constructions are supersymmetric only for special choices of untwisted moduli. We show that within the above class of constructions there are no supersymmetric three-family models with 3 copies of {\bf 10}-plets unless there are simultaneously some {\bf 15}-plets. We systematically analyze the construction of such models and their spectra. The M-theory lifts of these brane constructions become purely geometrical backgrounds: they are singular $G_2$ manifolds where the Grand Unified gauge symmetries and three families of chiral fermions are localized at codimension 4 and codimension 7 singularities respectively. We also study somepreliminary phenomenological features of the models. 
  As argued in our previous papers, it would be more natural to modify the standard approach to quantum theory by requiring that i) one unitary irreducible representation (UIR) of the symmetry algebra should describe a particle and its antiparticle simultaneously. This would automatically explain the existence of antiparticles and show that a particle and its antiparticle are different states of the same object. If i) is adopted then among the Poincare, so(2,3) and so(1,4) algebras only the latter is a candidate for constructing elementary particle theory. We extend our analysis in hep-th/0210144 and prove that: 1) UIRs of the so(1,4) algebra can indeed be interpreted in the framework of i) and cannot be interpreted in the framework of the standard approach; 2) as a consequence of a new symmetry (called AB one) between particles and antiparticles for UIRs satisfying i), elementary particles described by UIRs of the so(1,4) algebra can be only fermions; 3) as a consequence of the AB symmetry, the vacuum condition can be consistent only for particles with the half-integer spin (in conventional units) and therefore only such particles can be elementary. In our approach the well known fact that fermions have imaginary parity is a consequence of the AB symmetry. 
  We give the supergravity duals of commutative and noncommutative non-abelian gauge theories with N=2 in 2+1 dimensions. The moduli space on the Coulomb branch of these theories is studied using supergravity. 
  We obtain a (Abelian) two form field as a connection on a flat space-time and its corresponding field strength is canonically constructed. 
  We study a duality, recently conjectured by Klebanov and Polyakov, between higher-spin theories on AdS_4 and O(N) vector models in 3-d. These theories are free in the UV and interacting in the IR. At the UV fixed point, the O(N) model has an infinite number of higher-spin conserved currents. In the IR, these currents are no longer conserved for spin s>2. In this paper, we show that the dual interpretation of this fact is that all fields of spin s>2 in AdS_4 become massive by a Higgs mechanism, that leaves the spin-2 field massless. We identify the Higgs field and show how it relates to the RG flow connecting the two CFTs, which is induced by a double trace deformation. 
  The local composite operator A^2 is analysed in pure Yang-Mills theory in the Landau gauge within the algebraic renormalization. It is proven that the anomalous dimension of A^2 is not an independent parameter, being expressed as a linear combination of the gauge beta function and of the anomalous dimension of the gauge fields. 
  In this letter we present a derivation, from the D0-brane picture, of the background monopole field and in general of the full dynamics of the Yang-Mills theory on the dielectric D2-brane of Myers. To do this we study the large N limit of the fuzzy sphere relevant to the dielectric solution. In contrast to the usual interpretation, where the commutative D2-brane picture arises directly from the large N limit of the D0-brane picture, we find that a residual non-commutativity must be preserved, in order to make the connection by means of the Seiberg-Witten map. 
  In hep-th/0211011 we started a systematic investigation of open strings in the plane wave background. In this paper we continue the analysis by discussing the superalgebras of conserved charges, the spectra of open strings, and the spectra of DBI fluctuations around D-brane embeddings. We also derive the gluing conditions for corresponding boundary states and analyze their symmetries. All results are consistent with each other, and confirm the existence of additional supersymmetries as previously discussed. We further show that for every symmetry current one can construct a (countably) infinite number of related currents that contain more worldsheet derivatives, and discuss non-local symmetries. 
  We review a recent construction of the free field equations for totally symmetric tensors and tensor-spinors that exhibits the corresponding linearized geometry. These equations are not local for all spins >2, involve unconstrained fields and gauge parameters, rest on the curvatures introduced long ago by de Wit and Freedman, and reduce to the local (Fang-)Fronsdal form upon partial gauge fixing. We also describe how the higher-spin geometry is realized in free String Field Theory, and how the gauge fixing to the light cone can be effected. Finally, we review the essential features of local compensator forms for the higher-spin bosonic and fermionic equations with the same unconstrained gauge symmetry. 
  The original {\it membrane at the end of the universe} corresponds to a probe $M2$-brane of signature $(2,1)$ occupying the $S^{2} \times S^{1}$ boundary of the $(10,1)$ spacetime $AdS_{4} \times S^{7}$, and is described by an $OSp(4/8)$ SCFT. However, it was subsequently generalized to other worldvolume signatures $(s,t)$ and other spacetime signatures $(S,T)$. An interesting special case is provided by the $(3,0)$ brane at the end of the de Sitter universe $dS_{4}$ which has recently featured in the $dS/CFT$ correspondence. The resulting CFT contains the one recently proposed as the holographic dual of a four-dimensional de Sitter cosmology.   Supersymmetry restricts $S,T,s,t$ by requiring that the corresponding bosonic symmetry $O(s+1,t+1) \times O(S-s,T-t)$ be a subgroup of a superconformal group. The case of $dS_{4} \times AdS_{7}$ is `doubly holographic' and may be regarded as the near horizon geometry of $N_{2}$ $M2$-branes or equivalently, under interchange of conformal and R symmetry, of $N_{5}$ $M5$-branes, provided $N_{2}=2N_{5}{}^{2}$. The same correspondence holds in the pp-wave limit of conventional $M$-theory. 
  Quantum Yang-Mills theory and the Wilson loop can be rewritten identically in terms of local gauge-invariant variables being directly related to the metric of the dual space. In this formulation, one reveals a hidden high local symmetry of the Yang-Mills theory, which mixes up fields with spins up to J=N for the SU(N) gauge group. In the simplest case of the SU(2) group the dual space seems to tend to the de Sitter space in the infrared region. This observation suggests a new mechanism of gauge-invariant mass generation in the Yang-Mills theory. 
  We examine the fluctuations around a Dp-brane solution in an unstable D-brane system using boundary states and also boundary string field theory. We show that the fluctuations correctly reproduce the fields on the Dp-brane. Plugging these into the action of the unstable D-brane system, we recover not only the tension and RR charge, but also full effective action of the Dp-brane exactly. Our method works for general unstable D-brane systems and provides a simple proof of D-brane descent/ascent relations under the tachyon condensation. In the lowest dimensional unstable D-brane system, called K-matrix theory, D-branes are described in terms of operator algebra. We show the equivalence of the geometric and algebraic descriptions of a D-brane world-volume manifold using the equivalence between path integral and operator formulation of the boundary quantum mechanics. As a corollary, the Atiyah-Singer index theorem is naturally obtained by looking at the coupling to RR-fields. We also generalize the argument to type I string theory. 
  The history of discovery of bosonic string theory is well documented. This theory evolved as an attempt to find a multidimensional analogue of Euler's beta function. Such an analogue had in fact been known in mathematics literature at least in 1922 and was studied subsequently by mathematicians such as Selberg, Weil and Deligne among others. The mathematical interpretation of this multidimensional beta function is markedly different from that described in physics literature. This paper aims to bridge the gap between the existing treatments. Preserving all results of conformal field theories intact, developed formalism employing topological, algebro-geometric, number-theoretic and combinatorial metods is aimed to provide better understanding of the Veneziano amplitudes and, thus, of string theories. 
  We consider holographic RG flow solutions with eight supersymmetries and study the geometry transverse to the brane. For both M2-branes and for D3-branes in F-theory this leads to an eight-manifold with only a four-form flux. In both settings there is a natural four-dimensional hyper-Kahler slice that appears on the Coulomb branch. In the IIB theory this hyper-Kahler manifold encodes the Seiberg-Witten coupling over the Coulomb branch of a U(1) probe theory. We focus primarily upon a new flow solution in M-theory. This solution is first obtained using gauged supergravity and then lifted to eleven dimensions. In this new solution, the brane probes have an Eguchi-Hanson moduli space with the M2-branes spread over the non-trivial 2-sphere. It is also shown that the new solution is valid for a class of orbifold theories. We discuss how the hyper-Kahler structure on the slice extends to some form of G-structure in the eight-manifold, and describe how this can be computed. 
  Following the new gauging fixing method of D'Hoker and Phong, we study two-loop superstrings in hyperelliptic language. By using hyperelliptic representation of genus 2 Riemann surface we derive a set of identities involving the Szeg\"o kernel. These identities are used to prove the vanishing of the cosmological constant and the non-renormalization theorem point-wise in moduli space by doing the summation over all the 10 even spin structures. Modular invariance is maintained at every stage of the computation explicitly. The 4-particle amplitude is also computed and an explicit expression for the chiral integrand is obtained. We use this result to show that the perturbative correction to the $R^4$ term in type II superstring theories is vanishing at two loops.   In this paper, a summary of the main results is presented with detailed derivations to be provided in two subsequent publications. 
  We investigate supersymmetry in one-dimensional quantum mechanics with point interactions. We clarify a class of point interactions compatible with supersymmetry and present N=2 supersymmetric models on a circle with two point interactions as well as a superpotential. A hidden su(2) structure inherent in the system plays a crucial role to construct the N=2 supercharges. Spontaneous breaking of supersymmetry due to point interactions and an extension to higher N-extended supersymmetry are also discussed. 
  In these lecture notes from the 2002 Cargese summer school we review the progress that has been made towards finding a string theory for QCD (or for pure (super)Yang-Mills theory) following the discovery of the AdS/CFT correspondence. We start with a brief review of the AdS/CFT correspondence and a general discussion of its application to the construction of a string theory for QCD. We then discuss in detail two possible paths towards a QCD string theory, one which uses a mass deformation of the N=4 super Yang-Mills theory (the Polchinski-Strassler background) and the other using a compactification of "little string theory" on a 2-sphere (the Maldacena-Nunez solution). A third approach (the Klebanov-Strassler solution) is described in other lectures of this school. We briefly assess the advantages and disadvantages of all three approaches. 
  We calculate a class of two-point boundary correlators in 2D quantum gravity using its microscopic realization as loop gas on a random surface. We find a perfect agreement with the two-point boundary correlation function in Liouville theory, obtained by V. Fateev, A. Zamolodchikov and Al. Zamolodchikov. We also give a geometrical meaning of the functional equation satisfied by this two-point function. 
  This paper is devoted to studying some properties of the Courant algebroids: we explain the so-called "conducting bundle construction" and use it to attach the Courant algebroid to Dixmier-Douady gerbe (following ideas of P. Severa). We remark that WZNW-Poisson condition of Klimcik and Strobl (math.SG/0104189) is the same as Dirac structure in some particular Courant algebroid. We propose the construction of the Lie algebroid on the loop space starting from the Lie algebroid on the manifold and conjecture that this construction applied to the Dirac structure above should give the Lie algebroid of symmetries in the WZNW-Poisson $\sigma$-model, we show that it is indeed true in the particular case of Poisson $\sigma$-model. 
  We show that the degenerate positive-norm physical propagating fields of the open bosonic string can be gauged to the higher rank fields at the same mass level. As a result, their scattering amplitudes can be determined from those of the higher spin fields. This phenomenon arises from the existence of two types of zero-norm states with the same Young representations as those of the degenerate positive-norm states in the old covariant first quantized (OCFQ) spectrum. This is demonstrated by using the lowest order gauge transformation of Witten's string field theory (WSFT) up to the fourth massive level (spin-five), and is found to be consistent with conformal field theory calculation based on the first quantized generalized sigma-model approach. In particular, on-shell conditions of zero-norm states in OCFQ stringy gauge transformation are found to correspond, in a one-to-one manner, to the background ghost fields in off-shell gauge transformation of WSFT. The implication of decoupling of scalar modes on Sen's conjectures was also briefly discussed. 
  This is a very brief review of some results from hep-th/0112154 and hep-th/0209191. In holographic renormalization, we studied the RG flow of a 2d N=(4,4) CFT perturbed by a relevant operator, flowing to a conformal fixed point in the IR. Here, the supergravity dual is displayed, and the computation of correlators is discussed. The sample stress-energy correlator given here provides an opportunity to explicitly compare Zamolodchikov's C-function to the proposal for a "holographic C-function". 
  The vanishing of the cosmological constant and the non-renormali-zation theorem are verified at two loops by explicit computation using the hyperelliptic language and the newly obtained chiral measure of D'Hoker and Phong. A set of identities is found which is used in the verification of the non-renormalization theorem and leads to a great simplification of the calculation of the four-particle amplitude at two loops. 
  In this talk we discuss symmetry preserving D-branes on a line of a marginally deformed SU(2) WZW model. A semiclassical and a quantum theoretical approach are presented. 
  We study the impact of topological phase transitions of the internal Calabi-Yau threefold on the space-time geometry of five-dimensional extremal black holes and black strings. For flop transitions and SU(2) gauge symmetry enhancement we show that solutions can always be continued and that the behaviour of metric, gauge fields and scalars can be characterized in a model independent way. Then we look at supersymmetric solutions which describe naked singularities rather than geometries with a horizon. For black strings we show that the solution cannot become singular as long as the scalar fields take values inside the Kahler cone. For black holes we establish the same result for the elliptic fibrations over the Hirzebruch surfaces F_0, F_1, F_2. These three models exhibit a behaviour similar to the enhancon, since one runs into SU(2) enhancement before reaching the apparent singularity. Using the proper continuation inside the enhancon radius one finds that the solution is regular. 
  Following [1] we further apply the octonionic structure to supersymmetric D=11 $M$-theory. We consider the octonionic $2^{n+1} \times 2^{n+1}$ Dirac matrices describing the sequence of Clifford algebras with signatures ($9+n,n$) ($n=0,1,2, ...$) and derive the identities following from the octonionic multiplication table. The case $n=1$ ($4\times 4$ octonion-valued matrices) is used for the description of the D=11 octonionic $M$ superalgebra with 52 real bosonic charges; the $n=2$ case ($8 \times 8$ octonion-valued matrices) for the D=11 conformal $M$ algebra with 232 real bosonic charges. The octonionic structure is described explicitly for $n=1$ by the relations between the 528 Abelian O(10,1) tensorial charges $Z_\mu $Z_{\mu\nu}$, $Z_{\mu >... \mu_5}$ of the $M$-superalgebra. For $n=2$ we obtain 2080 real non-Abelian bosonic tensorial charges $Z_{\mu\nu}, Z_{\mu_1 \mu_2 \mu_3}, Z_{\mu_1 ... \mu_6}$ which, suitably constrained describe the generalized D=11 octonionic conformal algebra. Further, we consider the supersymmetric extension of this octonionic conformal algebra which can be described as D=11 octonionic superconformal algebra with a total number of 64 real fermionic and 239 real bosonic generators. 
  It is shown how to calculate simple vacuum diagrams in light-cone quantum field theory. As an application, I consider the one-loop effective potential of phi^4 theory. The standard result is recovered both with and without the inclusion of zero modes having longitudinal momentum k^+ = 0. 
  I consider supergravity solutions of D5 branes wrapped on supersymmetric 2-cycles and use them to discuss relevant features of four-dimensional N=1 super Yang-Mills theories with gauge group SU(N). In particular, using a gravitational dual of the gaugino condensate, it is shown that is possible to obtain the complete NSVZ $\beta$-function. It is also described how different aspects of the gauge theory are nicely encoded in this supergravity solution. 
  We propose a simple conformal mechanics model which is classically equivalent to a charged massive particle propagating near the AdS_2\times S^2 horizon of an extreme Reissner-Nordstr\"om black hole. The equivalence holds for any finite value of the black hole mass and with both the radial and angular degrees of freedom of the particle taken into account. It is ensured by the existence of a canonical transformation in the Hamiltonian formalism. Using this transformation, we construct the Hamiltonian of a N=4 superparticle on AdS_2\times S^2 background. 
  We considered the solutions of the Friedmann equation in several setups, arguing that the Weierstra$\ss$ form of the solutions leads to connections with some Conformal Field Theory on a torus. Thus a link with the Cardy entropy formula is obtained in a quite natural way. The argument is shown to be valid in a four dimensional radiation dominated universe with a cosmological constant as well as in four further different Universes. 
  We investigate deconstruction of five dimensional supersymmetric abelian gauge theories compactified on $S_1/Z_2$, with various sets of bulk and matter multiplets. The problem of anomalies, chirality and stability in the deconstructed theories is discussed. We find that for most of the 5d brane/bulk matter assignments there exists the deconstructed version. There are, however, some exceptions. 
  It has been shown by Polchinski and Strassler that the scaling of high energy QCD scattering amplitudes can be obtained from string theory. They considered an AdS slice as an approximation for the dual space of a confining gauge theory. Here we use this approximation to estimate in a very simple way the ratios of scalar glueball masses imposing Dirichlet boundary conditions on the string dilaton field. These ratios are in good agreement with the results in the literature. We also find that they do not depend on the size of the slice. 
  We derive the one loop mixing matrix for anomalous dimensions in N=4 Super Yang-Mills. We show that this matrix can be identified with the Hamiltonian of an integrable SO(6) spin chain with vector sites. We then use the Bethe ansatz to find a recipe for computing anomalous dimensions for a wide range of operators. We give exact results for BMN operators with two impurities and results up to and including first order 1/J corrections for BMN operators with many impurities. We then use a result of Reshetikhin's to find the exact one-loop anomalous dimension for an SO(6) singlet in the limit of large bare dimension. We also show that this last anomalous dimension is proportional to the square root of the string level in the weak coupling limit. 
  In this paper we assume the de Sitter Space version of Black Hole Complementarity which states that a single causal patch of de Sitter space is described as an isolated finite temperature cavity bounded by a horizon which allows no loss of information. We discuss the how the symmetries of de Sitter space should be implemented. Then we prove a no go theorem for implementing the symmetries if the entropy is finite. Thus we must either give up the finiteness of the de Sitter entropy or the exact symmetry of the classical space. Each has interesting implications for the very long time behavior. We argue that the lifetime of a de Sitter phase can not exceed the Poincare recurrence time. This is supported by recent results of Kachru, Kallosh, Linde and Trivedi. 
  We construct a large family of supergravity solutions that describe BPS excitations on AdS_3 x S^3 with angular momentum on S^3. These solutions take into account the full backreaction on the metric. We find that as we increase the energy of the excitation, the energy gap to the next non-BPS excitation decreases. These solutions can be viewed as Kaluza-Klein monopole ``supertubes'' which are completely non-singular geometries. We also make some remarks on supertubes in general. 
  For $M$-theory on the $G_2$ holonomy manifold given by the cone on ${\bf S^3}\x {\bf S^3}$ we consider the superpotential generated by membrane instantons and study its transformations properties, especially under monodromy transformations and triality symmetry. We find that the latter symmetry is, essentially, even a symmetry of the superpotential. As in Seiberg/Witten theory, where a flat bundle given by the periods of an universal elliptic curve over the $u$-plane occurs, here a flat bundle related to the Heisenberg group appears and the relevant universal object over the moduli space is related to hyperbolic geometry. 
  Within the Dijkgraaf-Vafa correspondence, we study the complete factorization of the Seiberg-Witten curve for U(N_c) gauge theory with N_f<N_c massive flavors. We obtain explicit expressions, from random matrix theory, for the moduli, parametrizing the curve. These moduli characterize the submanifold of the Coulomb branch where all monopoles become massless. We find that the matrix model reveals some non-trivial structures of the gauge theory. In particular the moduli are additive with respect to adding extra matter and increasing the number of colors. 
  The present theory is based on the assumption that at the very small (Planck scale) distances our space-time is discrete, and this discreteness influences on the Planck scale physics. Considering our (3+1)-dimensional space-time as a regular hypercubic lattice with a parameter $a=\lambda_\text{P}$, where $\lambda_\text{P}$ is the Planck length, we have investigated a role of lattice artifact monopoles which is essential near the Planck scale if the Family replicated gauge group model (FRGGM) is an extension of the Standard Model at high energies. It was shown that monopoles have $N$ times smaller magnetic charge in FRGGM than in SM ($N$ is the number of families in FRGGM). These monopoles can give an additional contribution to beta-functions of the renormalisation group equations for the running fine structure constants $\alpha_\text{i}(\mu)$ (i=1,2,3 correspond to the U(1), SU(2), and SU(3) gauge groups of the Standard Model). We have used the Dirac relation for renormalised electric and magnetic charges. Also we have estimated the enlargement of a number of fermions in FRGGM leading to the suppression of the asymptotic freedom in the non-Abelian theory. Different role of monopoles in the vicinity of the Planck scale gives rise or to AntiGUT, or to the new possibility of unification of gauge interactions (including gravity) at the scale $\mu_\text{GUT}\approx 10^{18.4}$ GeV. We discussed the possibility of the [SU(5)]$^3$ SUSY or [SO(10)]$^3$ SUSY unifications. 
  The fuzzy algebra of S^4 is discussed by quantum deformation. To this end we embed the classical S^4 in the Kaehler coset space SO(5)/U(2). The harmonic functions of S^4 are constructed in terms of the complex coordinates of SO(5)/U(2). Being endowed with the symplectic structure they can be deformed by the Fedosov formalism. We show that they generate the fuzzy algebra \hat A_\infty (S^4) under the * product defined therein, by using the Darboux coordinate system. The fuzzy spheres of higher even dimensions can be discussed similarly. We give basic arguments for the generalization as well. 
  We compute string scattering amplitudes in an orbifold of Minkowski space by a boost, and show how certain divergences in the four point function are associated with graviton exchange near the singularity. These divergences reflect large tree-level backreaction of the gravitational field. Near the singularity, all excitations behave like massless fields on a 1+1 dimensional cylinder. For excitations that are chiral near the singularity, we show that divergences are avoided and that the backreaction is milder. We discuss the implications of this for some cosmological spacetimes. Finally, in order to gain some intuition about what happens when backreaction is taken into account, we study an open string rolling tachyon background as a toy model that shares some features with R^{1,1}/Z. 
  We investigate the scalar potential of gauged N=4 supergravity with matter. The extremum in the SU(1,1)/U(1) scalars is obtained for an arbitrary number of matter multiplets. The constraints on the matter scalars are solved in terms of an explicit parametrisation of an  SO(6,6+n) element. For the case of six matter multiplets we discuss both compact and noncompact gauge groups.  In an example involving noncompact groups and four scalars we find a potential with an absolute minimum and a positive cosmological constant. 
  I study a relativistic open string coupling through its endpoints to a plane wave with arbitrary temporal profile. The string's transverse oscillations respond linearly to the external field. This makes it possible to solve the classical equations, and to calculate the quantum-mechanical S-matrix in closed form. I analyze the dynamics of the string as the characteristic frequency and duration of the pulse are continuously varied. I derive, in particular, the multipole expansion in the adiabatic limit of very long wavelengths, and discuss also more violent phenomena such as shock waves, cusps and null brane intersections. Apart from their relevance to the study of time-dependence in superstring theory, these results could have other applications, such as the teleportation of gravitational wave bursts by cosmic strings. 
  In this note we extend previous work on massless Ramond spectra of open strings connecting D-branes wrapped on complex manifolds, to consider D-branes wrapped on smooth complex orbifolds. Using standard methods, we calculate the massless boundary Ramond sector spectra directly in BCFT, and find that the states in the spectrum are counted by Ext groups on quotient stacks (which provide a notion of homological algebra relevant for orbifolds). Subtleties that cropped up in our previous work also appear here. We also use the McKay correspondence to relate Ext groups on quotient stacks to Ext groups on (large radius) resolutions of the quotients. As stacks are not commonly used in the physics community, we include pedagogical discussions of some basic relevant properties of stacks. 
  We compute explicitly the four-particle amplitude in superstring theories by using the hyperelliptic language and the newly obtained chiral measure of D'Hoker and Phong. Although the algebra of the intermediate steps is a little bit involved, we obtain a quite simple expression for the four-particle amplitude. As expected, the integrand is independent of all the insertion points. As an application of the obtained result, we show that the perturbative correction to the $R^4$ term in type II superstring theories is vanishing point-wise in (even) moduli space at two loops. 
  We give a pedagogical introduction to Affleck and Ludwig's g-theorem, distinguishing its applications in field theory vs string theory. We clarify the recent proposal that the vacuum degeneracy $g$ of a noncompact worldsheet sigma model with a continuous spectrum of scaling dimensions is lowered under renormalization group flow while preserving the central charge. As an illustration we argue that the IR stable endpoint of the relevant flow of the worldsheet RG induced by a thermal tachyon in the type II string is the noncompact supersymmetric vacuum with lower $g$. Note Added (Sep 2005). 
  We construct smooth Calabi-Yau threefolds Z, torus-fibered over a dP_9 base, with fundamental group Z_2 X Z_2. To do this, the structure of rational elliptic surfaces is studied and it is shown that a restricted subset of such surfaces admit at least a Z_2 X Z_2 group of automorphisms. One then constructs Calabi-Yau threefolds X as the fiber product of two such dP_9 surfaces, demonstrating that the involutions on the surfaces lift to a freely acting Z_2 X Z_2 group of automorphisms on X. The threefolds Z are then obtained as the quotient Z=X/(Z_2 X Z_2). These Calabi-Yau spaces Z admit stable, holomorphic SU(4) vector bundles which, in conjunction with Z_2 X Z_2 Wilson lines, lead to standard-like models of particle physics with naturally suppressed nucleon decay. 
  We consider brane-induced gravity model in more than one extra dimensions, regularized by assuming that the bulk gravity is soft in ultraviolet. We study linear theory about flat multi-dimensional space-time and flat brane. We first find that this model allows for violation of equivalence between gravitational and inertial masses of brane matter. We then observe that the model has a scalar ghost field localized near the brane, as well as quasi-localized massive graviton. Pure tensor structure of four-dimensional gravity on the brane at intermediate distances is due to the cancellation between the extra polarization of the massive graviton, and the ghost. This is completely analogous to the situation in the GRS model. 
  We study string theory in supersymmetric time-dependent backgrounds. In the framework of general relativity, supersymmetry for spacetimes without flux implies the existence of a covariantly constant null vector, and a relatively simple form of the metric. As a result, the local nature of any such spacetime can be easily understood. We show that we can view any such geometry as a sequence of solutions to lower-dimensional Euclidean gravity. If we choose the lower-dimensional solutions to degenerate at some light-cone time, we obtain null singularities, which may be thought of as generalizations of the parabolic orbifold singularity. We find that in string theory, many such null singularities get repaired by $\alpha'$-corrections - in particular, by worldsheet instantons. As a consequence, the resulting string theory solutions do not suffer from any instability. Even though the CFT description of these solutions is not always valid, they can still be well understood after taking the effects of light D-branes into account; the breakdown of the worldsheet conformal field theory is purely gauge-theoretic, not involving strong gravitational effects. 
  We study the cancellations among Feynman diagrams that implement the Ward and Slavnov-Taylor identities corresponding to the conserved supersymmetry current in supersymmetric quantum field theories. In particular, we show that the Faddeev-Popov ghosts of gauge- and supersymmetries never decouple from the physical fields, even for abelian gauge groups. The supersymmetric Slavnov-Taylor identities provide efficient consistency checks for automatized calculations and can verify the supersymmetry of Feynman rules and the numerical stability of phenomenological predictions simultaneously. 
  We consider a supersymmetric U(N) gauge theory with matter fields in the adjoint, fundamental and anti-fundamental representations. As in the framework which was put forward by Dijkgraaf and Vafa, this theory can be described by a matrix model. We analyze this theory along the lines of [F. Cachazo, M. Douglas, N.S. and E. Witten, ``Chiral Rings and Anomalies in Supersymmetric Gauge Theory'' hep-th/0211170] and show the equivalence of the gauge theory and the matrix model. In particular, the anomaly equations in the gauge theory is identified with the loop equations in the matrix model. 
  We investigate the ground state of a free massless (pseudo)scalar field in 1+1-dimensional space-time. We argue that in the quantum field theory of a free massless (pseudo)scalar field without infrared divergences (Eur. Phys. J. C24, 653 (2002)) the ground state can be represented by a tensor product of wave functions of the fiducial vacuum and of the collective zero-mode, describing the motion of the ``center of mass'' of a free massless (pseudo)scalar field. We show that the bosonized version of the BCS wave function of the ground state of the massless Thirring model obtained in (Phys.Lett. B563, 231 (2003)) describes the ground state of the free massless (pseudo)scalar field. 
  We show that in the Randall-Sundrum model the Casimir energy of bulk gauge fields (or any of their supersymmetric relatives) has contributions that depend logarithmically on the radion. These contributions satisfactorily stabilize the radion, generating a large hierarchy of scales without fine-tuning. The logarithmic behaviour can be understood, in a 4D holographic description, as the running of gauge couplings with the infrared cut-off scale. 
  We study excited spherical branes (``giant gravitons'') in $AdS \times S$ spacetimes with background flux. For large excitation, these branes may be treated semiclassically. We compute their spectra using Bohr-Sommerfeld quantization and use the AdS/CFT correspondence to relate them to anomalous dimensions in the dual field theory at strong coupling, expressed as a series expansion in powers of 1/N. These effects resemble those due to $k$-body forces between quarks in Hartree-Fock models of baryons at large $N$. For branes expanded in AdS, we argue that the anomalous dimensions are due to loop corrections to the effective action. 
  Based on the decomposition of SU(2) gauge field, we derive a generalization of the decomposition theory for the SU(N) gauge field. We thus obtain the invariant electro-magnetic tensors of SU(N) groups and the extended Wu-Yang potentials. The sourceless solutions are also discussed. 
  We find the temperature of the phase transition in the (2+1)d Georgi-Glashow model. The critical temperature is shown to depend on the gauge coupling and on the ratio of Higgs and gauge boson masses. In the BPS limit of light Higgs the previous result by Dunne, Kogan, Kovner, and Tekin is reproduced. 
  By applying the method of Dijkgraaf-Vafa, we study matrix model related to supersymmetric SO(N_c) gauge theory with N_f flavors of quarks in the vector representation found by Intriligator-Seiberg. By performing the matrix integral over tree level superpotential characterized by light meson fields (mass deformation) in electric theory, we reproduce the exact effective superpotential in the gauge theory side. Moreover, we do similar analysis in magnetic theory. It turns out the matrix descriptions of both electric and magnetic theories are the same: Seiberg duality in the gauge theory side. 
  Let us imagine that there is an overall quantum theory (not necessarily recognized yet) of matter and energy ({\it i.e.}, of elementary fermions and bosons) interacting with the physical spacetime (treated on a quantum level). Since states of quantum spacetime are so far not observed directly, they ought to be projected out from the overall Hilbert space (much like states of a quantum medium in the optical model often constructed in nuclear physics). Then, in the reduced Hilbert space only states of quantum matter and energy are left, but now endowed with the energy width that enters through an antiHermitian interaction-like operator, a remainder of their coupling to the quantum spacetime. We postulate that such an energy width involves an averaged coupling of quantum matter and energy to a classical field of time deviations from the uniform time run (in the classical spacetime of special relativity). The well known time-temperature analogy helps us to fix other postulates leading altogether to a quantum theory we call chronodynamics (a loose analogue of thermodynamics of small deviations from thermal equilibrium). 
  The membrane instanton superpotential for $M$-theory on the $G_2$ holonomy manifold given by the cone on ${\bf S^3}\x {\bf S^3}$ is given by the dilogarithm and has Heisenberg monodromy group in the quantum moduli space. We compare this to a Heisenberg group action on the type IIA hypermultiplet moduli space for the universal hypermultiplet, to metric corrections from membrane instantons related to a twisted dilogarithm for the deformed conifold and to a flat bundle related to a conifold period, the Heisenberg group and the dilogarithm appearing in five-dimensional Seiberg/Witten theory. 
  It is shown that the SO(3) gauge field configurations can be completely characterised by certain gauge invariant vector fields. The singularities of these vector fields describe the topological aspects of the gauge field configurations. The topological (or monopole) charge is expressed in terms of an Abelian vector potential. 
  Non-supersymmetric orbifolds of N=1 super Yang-Mills theories are conjectured to inherit properties from their supersymmetric parent. We examine this conjecture by compactifying the Z_2 orbifold theories on a spatial circle of radius R. We point out that when the orbifold theory lies in the weakly coupled vacuum of its parent, fractional instantons do give rise to the conjectured condensate of bi-fundamental fermions. Unfortunately, we show that quantum effects render this vacuum unstable through the generation of twisted operators. In the true vacuum state, no fermion condensate forms. Thus, in contrast to super Yang-Mills, the compactified orbifold theory undergoes a chiral phase transition as R is varied. 
  In the context of N=2 supergravity we explain the occurrence of partial super-Higgs with vanishing vacuum energy and moduli stabilization in a model suggested by superstring compactifications on type IIB orientifolds with 3-form fluxes.   The gauging of axion symmetries of the quaternionic manifold, together with the use of degenerate symplectic sections for special geometry, are the essential ingredients of the construction. 
  The lower order terms of the heat kernel expansion at coincident points are computed in the context of finite temperature quantum field theory for flat space-time and in the presence of general gauge and scalar fields which may be non Abelian and non stationary. The computation is carried out in the imaginary time formalism and the result is fully consistent with invariance under topologically large and small gauge transformations. The Polyakov loop is shown to play a fundamental role. 
  We study the departures from the classical synchrotron radiation due to noncommutativity of coordinates. We find that these departures are significant, but do not give tight bounds on the magnitude of the noncommutative parameter. On the other hand, these results could be used in future investigations in this direction. We also find an acausal behavior for the electromagnetic field due to the presence in the theory of two different speeds of light. This effect naturally arises even if only \theta^{12} is different from zero. 
  A consistent gauging of maximal supergravity requires that the T-tensor transforms according to a specific representation of the duality group. The analysis of viable gaugings is thus amenable to group-theoretical analysis, which we explain and exploit for a large variety of gaugings. We discuss the subtleties in four spacetime dimensions, where the ungauged Lagrangians are not unique and encoded in an E_7(7)\Sp(56,R)/GL(28) matrix. Here we define the T-tensor and derive all relevant identities in full generality. We present a large number of examples in d=4,5 spacetime dimensions which include non-semisimple gaugings of the type arising in (multiple) Scherk-Schwarz reductions. We also present some general background material on the latter as well as some group-theoretical results which are necessary for using computer algebra. 
  We discuss the (dual-)gauge transformations and BRST cohomology for the two (1 + 1)-dimensional (2D) free Abelian one-form and four (3 + 1)-dimensional (4D) free Abelian 2-form gauge theories by exploiting the (co-)BRST symmetries (and their corresponding generators) for the Lagrangian densities of these theories. For the 4D free 2-form gauge theory, we show that the changes on the antisymmetric polarization tensor e^{\mu\nu} (k) due to (i) the (dual-)gauge transformations corresponding to the internal symmetry group, and (ii) the translation subgroup T(2) of the Wigner's little group, are connected with each-other for the specific relationships among the parameters of these transformation groups. In the language of BRST cohomology defined w.r.t. the conserved and nilpotent (co-)BRST charges, the (dual-)gauge transformed states turn out to be the sum of the original state and the (co-)BRST exact states. We comment on (i) the quasi-topological nature of the 4D free 2-form gauge theory from the degrees of freedom count on e^{\mu\nu} (k), and (ii) the Wigner's little group and the BRST cohomology for the 2D one-form gauge theory {\it vis-{\`a}-vis} our analysis for the 4D 2-form gauge theory. 
  It is well known that BRST symmetry plays a fundamental role in constructing quantum gauge theories. Yet, at the classical level, it constitutes the modern language to study constrained systems. First, this letter reviews the Sp(2) covariant quantisation of gauge theories, so that the geometrical interpretation of gauge theories in terms of quasi-principal fibre bundles Q(Ms, Gs) is the main scenario. It is then described the Sp(2) algebra for ordinary Yang-Mills theory. Some basic set theory and topology terms are reviewed to proceed then to use theorems on left inverse and right inverse of functions together with groups (Ms, Gs) to have triangular compositions of functions on manifolds. Henceforth, it is so the purpose of this letter to present topological structures to gauge theories, in particular to construct commutative diagrams for cocycle conditions of the Sp(2) symmetry. 
  Big bang/crunch curvature singularities in exact CFT string backgrounds can be removed by turning on gauge fields. This is described within a family of {SL(2)xSU(2)xU(1)_x}/{U(1)xU(1)} quotient CFTs. Uncharged incoming wavefunctions from the ``whiskers'' of the extended universe can be fully reflected if and only if a big bang/crunch curvature singularity, from which they are scattered, exists. Extended BTZ-like singularities remain as long as U(1)_x is compact. 
  This note announces the proof of a conjecture of H. Verlinde, according to which the spaces of Liouville conformal blocks and the Hilbert spaces from the quantization of the Teichm\"uller spaces of Riemann surfaces carry equivalent representations of the mapping class group. This provides a basis for the geometrical interpretation of quantum Liouville theory in its relation to quantized spaces of Riemann surfaces. 
  The simple current construction of orientifolds based on rational conformal field theories is reviewed. When applied to SO(16) level 1, one can describe all ten-dimensional orientifolds in a unified framework. 
  These notes are based on lectures presented at the 2001 Les Houches Summerschool ``Unity from Duality: Gravity, Gauge Theory and Strings'' 
  We propose an effective description of 0-brane black holes, in which the black hole is modeled as a gas of non-interacting quasi-particles in the dual quantum mechanics. This simple model is shown to account for many of the static thermodynamic properties of the black hole. It also accounts for dynamical properties, such as the rate at which energy gets thermalized by the black hole. We use the model to show that the entropy of the quantum mechanics is proportional to the black hole horizon area in Planck units. 
  String theory avoids the ultraviolet infinities that arise in trying to quantize gravity. It is also more predictive than conventional quantum field theory, one aspect of this being the way that it contributed to the emergence of the concept of ``supersymmetry'' of particle interactions. There are hints from the successes of supersymmetric unified theories of particle interactions that supersymmetry is relevant to elementary particles at energies close to current accelerator energies; if this is so, it will be confirmed experimentally and supersymmetry is then also likely to be important in cosmology, in connection with dark matter, baryogenesis, and/or inflation. Magnetic monopoles play an important role in the structure of string theory, and thus should certainly exist, if string theory is correct, though they may have been diluted by inflation to an unobservable level. The monopole mass in many attractive models is near the Planck mass, but, if unification of elementary particle forces with gravity occurs near TeV energies through large or warped extra dimensions, as in some recent models, then monopoles should be below 100 TeV and in an astrophysical context would be ultrarelativistic. In such models, supersymmetry would definitely be expected at TeV energies. 
  We present new rolling tachyon solutions describing the classical decay of D-branes. Our methods are simpler than those appearing in recent works, yet our results are exact in classical string theory. The role of pressure in the decay is studied using tachyon profiles with spatial variation. In this case the final state involves an array of codimension one D-branes rather than static, pressureless tachyon matter. 
  We present a spacetime diffeomorphism invariant formulation of the geodesic approximation to soliton dynamics. 
  An indication for the existence of a collective Myers solution in the non-abelian D0-brane Born-Infeld action is the presence of a tachyonic mode in fluctuations around the standard diagonal background. We show that this computation for non-abelian D0-branes in curved space has the geometric interpretation of computing the eigenvalues of the geodesic deviation operator for U(N)-valued coordinates. On general grounds one therefore expects a geometric Myers effect in regions of sufficiently negative curvature. We confirm this by explicit computations for non-abelian D0-branes on a sphere and a hyperboloid. For the former the diagonal solution is stable, but not so for the latter. We conclude by showing that near the horizon of a Schwarzschild black hole one also finds a tachyonic mode in the fluctuation spectrum, signaling the possibility of a near-horizon gravitationally induced Myers effect. 
  A physical applicability of normed split-algebras, such as hyperbolic numbers, split-quaternions and split-octonions is considered. We argue that the observable geometry can be described by the algebra of split-octonions. In such a picture physical phenomena are described by the ordinary elements of chosen algebra, while zero divisors (the elements of split-algebras corresponding to zero norms) give raise the coordinatization of space- time. It turns to be possible that two fundamental constants (velocity of light and Planck constant) and uncertainty principle have geometrical meaning and appears from the condition of positive definiteness of norms. The property of non-associativity of octonions could correspond to the appearance of fundamental probabilities in four dimensions. Grassmann elements and a non-commutativity of space coordinates, which are widely used in various physical theories, appear naturally in our approach. 
  We construct several classes of exact supersymmetric supergravity solutions describing D4 branes polarized into NS5 branes and F-strings polarized into D2 branes. These setups belong to the same universality class as the perturbative solutions used by Polchinski and Strassler to describe the string dual of N=1* theories. The D4-NS5 setup can be interpreted as a string dual to a confining 4+1 dimensional theory with 8 supercharges, whose properties we discuss. By T-duality, our solutions give Type IIB supersymmetric backgrounds with polarized branes. 
  We show that a suitable rescaling of the matrix model coupling constant makes manifest the duality group of the N=2 SYM theory with gauge group SU(2). This is done by first identifying the possible modifications of the SYM moduli preserving the monodromy group. Then we show that in matrix models there is a simple rescaling of the pair $(S_D,S)$ which makes them dual variables with $\Gamma(2)$ monodromy. We then show that, thanks to a crucial scaling property of the free energy derived perturbatively by Dijkgraaf, Gukov, Kazakov and Vafa, this redefinition corresponds to a rescaling of the free energy which in turn fixes the rescaling of the coupling constant. Next, we show that in terms of the rescaled free energy one obtains a nonperturbative relation which is the matrix model counterpart of the relation between the $u$--modulus and the prepotential of N=2 SYM. This suggests considering a dual formulation of the matrix model in which the expansion of the prepotential in the strong coupling region, whose QFT derivation is still unknown, should follow from perturbation theory. The investigation concerns the SU(2) gauge group and can be generalized to higher rank groups. 
  We present an improved version of our earlier work on summing the planar graphs in phi^3 field theory. The present treatment is also based on our world sheet formalism and the mean field approximation, but it makes use of no further approximations. We derive a set of equations between the expectation values of the world sheet fields, and we investigate them in certain limits. We show that the equations can give rise to (metastable) string forming solutions. 
  In Calabi-Yau fourfold compactifications of M-theory with flux, we investigate the possibility of partial supersymmetry breaking in the three-dimensional effective theory. To this end, we place the effective theory in the framework of general N=2 gauged supergravities, in the special case where only translational symmetries are gauged. This allows us to extract supersymmetry-breaking conditions, and interpret them as conditions on the 4-form flux and Calabi-Yau geometry. For N=2 unbroken supersymmetry in three dimensions we recover previously known results, and we find a new condition for breaking supersymmetry from N=2 to N=1, i.e. from four to two supercharges. An example of a Calabi-Yau hypersurface in a toric variety that satisfies this condition is provided. 
  It is shown in detail that the dynamics of the Einstein-dilaton-p-form system in the vicinity of a spacelike singularity can be asymptotically described, at a generic spatial point, as a billiard motion in a region of Lobachevskii space (realized as an hyperboloid in the space of logarithmic scale factors). This is done within the Hamiltonian formalism, and for an arbitrary number of spacetime dimensions $D \geq 4$. A key role in the derivation is played by the Iwasawa decomposition of the spatial metric, and by the fact that the off-diagonal degrees of freedom, as well as the p-form degrees of freedom, get ``asymptotically frozen'' in this description. For those models admitting a Kac-Moody theoretic interpretation of the billiard dynamics we outline how to set up an asymptotically equivalent description in terms of a one-dimensional non-linear sigma-model formally invariant under the corresponding Kac-Moody group. 
  We argue that the giant graviton configurations known from the literature have a complementary, microscopical description in terms of multiple gravitational waves undergoing a dielectric (or magnetic moment) effect. We present a non-Abelian effective action for these gravitational waves with dielectric couplings and show that stable dielectric solutions exist. These solutions agree in the large $N$ limit with the giant graviton configurations in the literature. 
  We investigate two models in non-commutative (NC) field theory by means of Monte Carlo simulations. Even if we start from the Euclidean lattice formulation, such simulations are only feasible after mapping the systems onto dimensionally reduced matrix models. Using this technique, we measure Wilson loops in 2d NC gauge theory of rank 1. It turns out that they are non-perturbatively renormalizable, and the phase follows an Aharonov-Bohm effect if we identify \theta = 1/B. Next we study the 3d \lambda \phi^{4} model with two NC coordinates, where we present new results for the correlators and the dispersion relation. We further reveal the explicit phase diagram. The ordered regime splits into a uniform and a striped phase, as it was qualitatively conjectured before. We also confirm the recent observation by Ambjorn and Catterall that such stripes occur even in d=2, although they imply the spontaneous breaking of translation symmetry. However, in d=3 and d=2 we observe only patterns of two stripes to be stable in the range of parameters investigated. 
  There has been disagreement in the literature on whether the hydrogen atom spectrum receives any tree-level correction due to noncommutativity. Here we shall clarify the issue and show that indeed a general argument on the structure of proton as a nonelementary particle leads to the appearance of such corrections. As a showcase, we evaluate the corrections in a simple nonrelativistic quark model with a result in agreement with the previous one we had obtained by considering the electron moving in the external electric field of proton. Thus the previously obtained bound on the noncommutativity parameter, $\theta < (10^4 GeV)^{-2}$, using the Lamb shift data, remains valid. 
  Commutativity of the diagram of the maps connecting three one--particle state, implied by the Equivalence Postulate (EP), gives a cocycle condition which unequivocally leads to the quantum Hamilton--Jacobi equation. Energy quantization is a direct consequences of the local homeomorphicity of the trivializing map. We review the EP and show that the quantum potential for two free particles, which depends on constants which may have a geometrical interpretation, plays the role of interaction term that admits solutions which do not vanish in the classical limit. 
  We present a modification of the Berkovits superparticle. This is firstly in order to covariantly quantize the pure spinor ghosts, and secondly to covariantly calculate matrix elements of a generic operator between two states. We proceed by lifting the pure spinor ghost constraints and regaining them through a BRST cohomology. We are then able to perform a BRST quantization of the system in the usual way, except for some interesting subtleties. Since the pure spinor constraints are reducible, ghosts for ghosts terms are needed, which have so far been calculated up to level 4. Even without a completion of these terms, we are still able to calculate arbitrary matrix elements of a physical operator between two physical states. 
  We show that after the Seiberg-Witten map is performed the action for noncommutative field theories can be regarded as a coupling to a field dependent gravitational background. This gravitational background depends only on the gauge field. Charged and uncharged fields couple to different backgrounds and we find that uncharged fields couple more strongly than the charged ones. We also show that the background is that of a gravitational plane wave. A massless particle in this background has a velocity which differs from the velocity of light and we find that the deviation is larger in the uncharged case. This shows that noncommutative field theories can be seen as ordinary theories in a gravitational background produced by the gauge field with a charge dependent gravitational coupling. 
  We study flop-transitions for M-theory on Calabi-Yau three-folds and their applications to cosmology in the context of the effective five-dimensional supergravity theory. In particular, the additional hypermultiplet which becomes massless at the transition is included in the effective action. We find the potential for this hypermultiplet which includes quadratic and quartic terms as well as additional dependence on the Kahler moduli. By constructing explicit cosmological solutions, it is demonstrated that a flop-transition can indeed by achieved dynamically, as long as the hypermultiplet is set to zero. Once excitations of the hypermultiplet are taken into account we find that the transition is generically not completed but the system is stabilised close to the transition region. Regions of moduli space close to flop-transitions can, therefore, be viewed as preferred by the cosmological evolution. 
  An explicit realization of 't Hooft's loop operator in continuum Yang-Mills theory is given. 
  The defect conformal field theory describing intersecting D3-branes at a C^2/Z_k orbifold is used to (de)construct the theory of intersecting M5-branes, as well as M5-branes wrapping the holomorphic curve xy=c. The possibility of a 't Hooft anomaly due to tensionless strings at the intersection is discussed. This note is based on a talk given by Zachary Guralnik at the 35th International Symposium Ahrenshoop on the Theory of Elementary Particles. 
  This is the written version of a talk I gave at the 35th Symposium Ahrenshoop in Berlin, Germany, August 2002. It is an exposition of joint work with S. Doplicher, K. Fredenhagen, and Gh. Piacitelli [1]. The violation of unitarity found in quantum field theory on noncommutative spacetimes in the context of the so-called modified Feynman rules is linked to the notion of time ordering implicitely used in the assumption that perturbation theory may be done in terms of Feynman propagators. Two alternative approaches which do not entail a violation of unitarity are sketched. An outlook upon our more recent work is given. 
  The classical and quantum features of Nambu mechanics are analyzed and fundamental issues are resolved. The classical theory is reviewed and developed utilizing varied examples. The quantum theory is discussed in a parallel presentation, and illustrated with detailed specific cases. Quantization is carried out with standard Hilbert space methods. With the proper physical interpretation, obtained by allowing for different time scales on different invariant sectors of a theory, the resulting non-Abelian approach to quantum Nambu mechanics is shown to be fully consistent. 
  These talks present an overview of a tentative theory of large distance physics. For each large distance L (in dimensionless units), the theory gives two complementary descriptions of spacetime physics: quantum field theory at distances larger than L, string scattering amplitudes at distances smaller than L. The mechanism of the theory is a certain 2d nonlinear model, the lambda-model, whose target manifold is the manifold of general nonlinear models of the worldsurface, the background spacetimes for string scattering. So far, the theory has only been formulated and its basic working described, in general terms. The theory's only claims to interest at present are matters of general principle. It is a self-contained nonperturbative theory of large distance physics, operating entirely at large distance. The lambda-model constructs an actual QFT at large distance, a functional integral over spacetime fields. It constructs an effective background spacetime for string scattering at relatively small distances. It is background independent, dynamically. Nothing is adjustable in its formulation. It is a mechanical theory, not an S-matrix theory. String scattering takes place at small distances within a mechanical large distance environment. The lambda-model constructs QFT in a way that offers possibilities of novel physical phenomena at large distances. The task now is to perform concrete calculations in the lambda-model, to find out if it produces a physically useful QFT. 
  We rigorously derive an effective quantum mechanical Hamiltonian from N=4 gauge theory in the BMN limit. Its eigenvalues yield the exact one-loop anomalous dimensions of scalar two-impurity BMN operators for all genera. It is demonstrated that this reformulation vastly simplifies computations. E.g. the known anomalous dimension formula for genus one is reproduced through a one-line calculation. We also efficiently evaluate the genus two correction, finding a non-vanishing result. We comment on multi-trace two-impurity operators and we conjecture that our quantum-mechanical reformulation could be extended to higher quantum loops and more impurities. 
  Gravity on noncommutative analogues of compact spaces can give a finite mode truncation of ordinary commutative gravity. We obtain the actions for gravity on the noncommutative two-sphere and on the noncommutative ${\bf CP}^2$ in terms of finite dimensional $(N\times N)$-matrices. The commutative large $N$ limit is also discussed. 
  We write in superspace the lagrangian containing the fourth power of the Weyl tensor in the "old minimal" d=4, N=2 supergravity, without local SO(2) symmetry. Using gauge completion, we analyze the lagrangian in components. We find out that the auxiliary fields which belong to the Weyl and compensating vector multiplets have derivative terms and therefore cannot be eliminated on-shell. Only the auxiliary fields which belong to the compensating nonlinear multiplet do not get derivatives and could still be eliminated; we check that this is possible in the leading terms of the lagrangian. We compare this result to the similar one of "old minimal" N=1 supergravity and we comment on possible generalizations to other versions of N=1,2 supergravity. 
  We apply a universal normal Calabi-Yau algebra to the construction and classification of compact complex $n$-dimensional spaces with SU(n) holonomy and their fibrations. This algebraic approach includes natural extensions of reflexive weight vectors to higher dimensions and a `dual' construction based on the Diophantine decomposition of invariant monomials. The latter provides recurrence formulae for the numbers of fibrations of Calabi-Yau spaces in arbitrary dimensions, which we exhibit explicitly for some Weierstrass and K3 examples. 
  Within the class of integrable Calogero models associated with (semi-)simple Lie algebras and with symmetric pairs of Lie algebras identified in a previous paper, we analyze whether and to what extent it is possible to find a gauge transformation that takes the traditional Lax pair with its dynamical R-matrix to a new Lax pair with a numerical R-matrix. 
  In this note, following the work of Seiberg in hep-th/0211234 for the conjecture between the field theory and matrix model in the case with massive fundamental flavors, we generalize it to the case with massless fundamental flavors. We show that with a little modifications, the analysis given by Seiberg can be used directly to the case of massless flavors. Furthermore, this new method explains the insertion of delta functions in the matrix model given by Demasure and Janik in hep-th/0211082. 
  Following recent advances in the local theory of current-algebraic orbifolds, we study various geometric properties of the general WZW orbifold, the general coset orbifold and a large class of (non-linear) sigma model orbifolds. Phase-space geometry is emphasized for the WZW orbifolds - while for the sigma model orbifolds we construct the corresponding {\it sigma model orbifold action}, which includes the previously-known general WZW orbifold action and general coset orbifold action as special cases. We focus throughout on the {\it twisted Einstein tensors} with diagonal monodromy, including the twisted Einstein metric, the twisted B field and the twisted torsion field of each orbifold sector. Finally, we present strong evidence for a conjectured set of {\it twisted Einstein equations} which should describe those sigma model orbifolds in this class which are also 1-loop conformal. 
  We derive the necessary and sufficient condition for Type A N-fold supersymmetry by direct calculation of the intertwining relation and show the complete equivalence between this analytic construction and the sl(2) construction based on quasi-solvability. An intimate relation between the pair of algebraic Hamiltonians is found. The classification problem on Type A N-fold supersymmetric models is investigated by considering the invariance of both the Hamiltonians and N-fold supercharge under the GL(2,K) transformation. We generalize the Bender-Dunne polynomials to all the Type A N-fold supersymmetric models without requiring the normalizability of the solvable sector. Although there is a case where weak orthogonality of them is not guaranteed, this fact does not cause any difficulty on the generalization. It is shown that the anti-commutator of the Type A N-fold supercharges is expressed as the critical polynomial of them in the original Hamiltonian, from which we establish the complete Type A N-fold superalgebra. A novel interpretation of the critical polynomials in view of polynomial invariants is given. 
  Using the eternal BTZ black hole as a concrete example, we show how spacelike singularities and horizons can be described in terms of AdS/CFT amplitudes. Our approach is based on analytically continuing amplitudes defined in Euclidean signature. This procedure yields finite Lorentzian amplitudes. The naive divergences associated with the Milne type singularity of BTZ are regulated by an $i\epsilon$ prescription inherent in the analytic continuation and a cancellation between future and past singularities.   The boundary description corresponds to a tensor product of two CFTs in an entangled state, as in previous work. We give two bulk descriptions corresponding to two different analytic continuations. In the first, only regions outside the horizon appear explicitly, and so amplitudes are manifestly finite. In the second, regions behind the horizon and on both sides of the singularity appear, thus yielding finite amplitudes for virtual particles propagating through the black hole singularity. This equivalence between descriptions only outside and both inside and outside the horizon is reminiscent of the ideas of black hole complementarity. 
  In this note we derive the low-energy effective action of type IIB theory compactified on half-flat manifolds and we show that this precisely coincides with the low-energy effective action of type IIA theory compactified on a Calabi-Yau manifold in the presence of NS three-form fluxes. We provide in this way a further check of the recently formulated conjecture that half-flat manifolds appear as mirror partners of Calabi-Yau manifolds when NS fluxes are turned on. 
  The instanton partition function of N=2, D=4 SU(2) gauge theory is obtained by taking the field theory limit of the topological open string partition function, given by a Chern-Simons theory, of a CY3-fold. The CY3-fold on the open string side is obtained by geometric transition from local F_0 which is used in the geometric engineering of the SU(2) theory. The partition function obtained from the Chern-Simons theory agrees with the closed topological string partition function of local F_0 proposed recently by Nekrasov. We also obtain the partition functions for local F_1 and F_2 CY3-folds and show that the topological string amplitudes of all local Hirzebruch surfaces give rise to the same field theory limit. It is shown that a generalization of the topological closed string partition function whose field theory limit is the generalization of the instanton partition function, proposed by Nekrasov, can be determined easily from the Chern-Simons theory. 
  Relativistic resonances and decaying states are described by representations of Poincar\'e transformations, similar to Wigner's definition of stable particles. To associate decaying state vectors to resonance poles of the $S$-matrix, the conventional Hilbert space assumption (or asymptotic completeness) is replaced by a new hypothesis that associates different dense Hardy subspaces to the in- and out-scattering states. Then one can separate the scattering amplitude into a background amplitude and one or several ``relativistic Breit-Wigner'' amplitudes, which represent the resonances per se. These Breit-Wigner amplitudes have a precisely defined lineshape and are associated to exponentially decaying Gamow vectors which furnish the irreducible representation spaces of causal Poincar\'e transformations into the forward light cone. 
  We analyze the construction of non-supersymmetric three generation six-stack Pati-Salam (PS) $SU(4)_C \times SU(2)_L \times SU(2)_R$ GUT classes of models, by localizing D6-branes intersecting at angles in four dimensional orientifolded toroidal compactifications of type IIA. Special role in the models is played by the presence of extra branes needed to satisfy the RR tadpole cancellation conditions. The models contain at low energy {\em exactly the Standard model} with no extra matter and/or extra gauge group factors. They are build such that they represent deformations around the quark and lepton basic intersection number structure. The models possess the same phenomenological characteristics of some recently discussed examples (PS-A, PS-I; PS-II GUT classes; hep-th/0203187, hep-th/0209202; hep-th/0210004) of four and five stack PS GUTS respectively. Namely, there are no colour triplet couplings to mediate proton decay and proton is stable as baryon number is a gauged symmetry.   The mass relation $m_e = m_d$ at the GUT scale is recovered. Even though more complicated, than in lower stack GUTS, the conditions of the non-anomalous U(1)'s to survive massless the generalized Green-Schwarz mechanism are solved consistently by the angle conditions coming from the presence of N=1 supersymmetric sectors involving the presence of {\em extra} branes and also required for the existence of a Majorana mass term for the right handed neutrinos. 
  A relativistic resonance which was defined by a pole of the $S$-matrix, or by a relativistic Breit-Wigner line shape, is represented by a generalized state vector (ket) which can be obtained by analytic extension of the relativistic Lippmann-Schwinger kets. These Gamow kets span an irreducible representation space for Poincar\'e transformations which, similar to the Wigner representations for stable particles, are characterized by spin (angular momentum of the partial wave amplitude) and complex mass (position of the resonance pole). The Poincar\'e transformations of the Gamow kets, as well as of the Lippmann-Schwinger plane wave scattering states, form only a semigroup of Poincar\'e transformations into the forward light cone. Their transformation properties are derived. From these one obtains an unambiguous definition of resonance mass and width for relativistic resonances. The physical interpretation of these transformations for the Born probabilities and the problem of causality in relativistic quantum physics is discussed. 
  We consider the uniqueness problem of a negative eigenvalue in the spectrum of small fluctuations about a bounce solution in a multidimensional case. Our approach is based on the concept of conjugate points from Morse theory and is a natural generalization of the nodal theorem approach usually used in one dimensional case. We show that bounce solution has exactly one conjugate point at $\tau=0$ with multiplicity one. 
  We evaluate chiral anomaly on the noncommutative torus with the overlap Dirac operator satisfying the Ginsparg-Wilson relation in arbitrary even dimensions. Utilizing a topological argument we show that the chiral anomaly is combined into a form of the Chern character with star products. 
  We start from a parity-breaking MCS QED$_{3}$ model with spontaneous breaking of the gauge symmetry as a framework for evaluation of the electron-electron interaction potential and for attainment of numerical values for the e-e bound state. Three expressions are obtained for the potential according to the polarization state of the scattered electrons. In an energy scale compatible with Condensed Matter electronic excitations, these three potentials become degenerated. The resulting potential is implemented in the Schrodinger equation and the variational method is applied to carry out the electronic binding energy. The resulting binding energies in the scale of 10-100 meV and a correlation length in the scale of 10-30 Angs. are possible indications that the MCS-QED$_{3}$ model adopted may be suitable to address an eventual case of e-e pairing in the presence of parity-symmetry breakdown. The data analyzed here suggest an energy scale of 10-100 meV to fix the breaking of the U(1)-symmetry.   PACS numbers: 11.10.Kk 11.15.Ex 74.20.-z 74.72.-h ICEN-PS-01/17 
  We derive effective equations of motion for a massless charged particle coupled to the dynamical electromagnetic field having regard to the radiation back reaction. It is shown that unlike the massive case not all the divergences resulting from the self-action of the particle are Lagrangian, i.e. can be canceled out by adding appropriate counterterms to the original action. Besides, the order of renormalized differential equations governing the effective dynamics turns out to be greater than the order of the corresponding Lorentz-Dirac equation for a massive particle. For the case of homogeneous external field the first radiative correction to the Lorentz equation is explicitly derived via the reduction of order procedure. 
  The new first order, rheonomic, kappa-supersymmetric formalism recently introduced by us for the world-volume action of the D3-brane is extended to the case of D5-branes. This extension requires the dual formulation of the Free Differential Algebra of type IIB supergravity in terms of 6-form gauge potentials which was so far missing and is given here. Furthermore relying on our new approach we are able to write the D5-world volume action in a manifestly SL(2,R) covariant form. This is important in order to solve the outstanding problem of finding the appropriate boundary actions of D3-branes on smooth ALE manifolds with twisted fields. The application of our results to this problem is however postponed to a subsequent publication. 
  We construct globally regular as well as non-abelian black hole solutions of a higher order curvature Einstein-Yang-Mills (EYM) model in $d=5$ dimensions. This model consists of the superposition of the first two members of the gravitational hierarchy (Einstein plus first Gauss-Bonnet(GB)) interacting with the superposition of the first two members of the $SO(d)$ Yang--Mills hierarchy. 
  The structure of the Euler-Lagrange equations for a general Lagrangian theory is studied. For these equations we present a reduction procedure to the so-called canonical form. In the canonical form the equations are solved with respect to highest-order derivatives of nongauge coordinates, whereas gauge coordinates and their derivatives enter in the right hand sides of the equations as arbitrary functions of time. The reduction procedure reveals constraints in the Lagrangian formulation of singular systems and, in that respect, is similar to the Dirac procedure in the Hamiltonian formulation. Moreover, the reduction procedure allows one to reveal the gauge identities between the Euler-Lagrange equations. Thus, a constructive way of finding all the gauge generators within the Lagrangian formulation is presented. At the same time, it is proven that for local theories all the gauge generators are local in time operators. 
  Recent cosmological observations suggest the existence of a positive cosmological constant $\Lambda$ with the magnitude $\Lambda(G\hbar/c^3) \approx 10^{-123}$. This review discusses several aspects of the cosmological constant both from the cosmological (sections 1-6) and field theoretical (sections 7-11) perspectives. The first section introduces the key issues related to cosmological constant and provides a brief historical overview. This is followed by a summary of the kinematics and dynamics of the standard Friedmann model of the universe paying special attention to features involving the cosmological constant. Section 3 reviews the observational evidence for cosmological constant, especially the supernova results, constraints from the age of the universe and a few others. Theoretical models (quintessence, tachyonic scalar field, ...) with evolving cosmological `constant' are described from different perspectives in the next section. Constraints on dark energy from structure formation and from CMBR anisotropies are discussed in the next two sections. The latter part of the review (sections 7-11) concentrates on more conceptual and fundamental aspects of the cosmological constant. Section 7 provides some alternative interpretations of the cosmological constant which could have a bearing on the possible solution to the problem. Several relaxation mechanisms have been suggested in the literature to reduce the cosmological constant to the currently observed value and some of these attempts are described in section 8. Next section gives a brief description of the geometrical structure of the de Sitter spacetime and the thermodynamics of the de Sitter universe is taken up in section 10. The last section deals with the role of string theory in the cosmological constant problem. 
  We define a level for a large class of Lorentzian Kac-Moody algebras. Using this we find the representation content of very extended $A_{D-3}$ and $E_8$ (i.e. $E_{11}$) at low levels in terms of $A_{D-1}$ and $A_{10}$ representations respectively. The results are consistent with the conjectured very extended $A_8$ and $E_{11}$ symmetries of gravity and maximal supergravity theories given respectively in hep-th/0104081 and hep-th/0107209. We explain how these results provided further evidence for these conjectures. 
  We define and compute the energy of higher curvature gravity theories in arbitrary dimensions. Generically, these theories admit constant curvature vacua (even in the absence of an explicit cosmological constant), and asymptotically constant curvature solutions with non-trivial energy properties. For concreteness, we study quadratic curvature models in detail. Among them, the one whose action is the square of the traceless Ricci tensor always has zero energy, unlike conformal (Weyl) gravity. We also study the string-inspired Einstein-Gauss-Bonnet model and show that both its flat and Anti-de-Sitter vacua are stable. 
  This paper could have been entitled "D branes and strings from flesh and blood." We study field theoretic prototypes of D branes/strings. To this end we consider (2+1)-dimensional domain walls in (3+1)-dimensional N=2 SQCD with SU(2) gauge group and two quark flavors in the fundamental representation. This theory is perturbed by a small mass term of the adjoint matter which, in the leading order in the mass parameter, does not break N=2 supersymmetry, and reduces to a (generalized) Fayet-Iliopoulos term in the effective low-energy N=2 SQED. We find 1/2 BPS-saturated domain wall solution interpolating between two quark vacua at weak coupling, and show that this domain wall localizes a U(1) gauge field. To make contact with the brane/string picture we consider the Abrikosov-Nielsen-Olesen magnetic flux tube in one of two quark vacua and demonstrate that it can end on the domain wall. We find an explicit 1/4 BPS-saturated solution for the wall/flux tube junction. We verify that the end point of the flux tube on the wall plays the role of an electric charge in the dual (2+1)-dimensional SQED living on the wall. Flow to N=1 theory is discussed. Our results lead us to a conjecture regarding the notorious "missing wall" in the solution of Kaplunovsky et al. 
  We study M theory compactifications on manifolds of $G_2$-holonomy with gauge and matter fields supported at singularities. We show that, under certain topological conditions, the combination of background $G$-flux and background fields at the singularities induces a potential for the moduli with an isolated minimum. The theory in the minimum is supersymmetric and has a negative cosmological constant in the simplest case. In a more realistic scenario, we find that the fundamental scale is around 10 Tev and the heirarchy between the four dimensional Planck and electroweak scales may be explained by the value of a topological invariant. Hyperbolic three-manifolds enter the discussion in an interesting way. 
  Nonlinear realizations superfield techniques, pertinent to the description of partial breaking of global N=2 supersymmetry in a flat d=4 super Minkowski background, are generalized to the case of partially broken N=1 AdS5 supersymmetry SU(2,2|1). We present, in an explicit form, off-shell manifestly N=1, d=4 supersymmetric minimal Goldstone superfield actions for two patterns of partial breaking of SU(2,2|1) supersymmetry. They correspond to two different nonlinear realizations of the latter, in the supercosets with the AdS5 and AdS5\times S1 bosonic parts. The relevant worldvolume Goldstone supermultiplets are accommodated, respectively, by improved tensor and chiral N=1, d=4 superfields. The second action is obtained from the first one by dualizing the improved tensor Goldstone multiplet into a chiral Goldstone one. In the bosonic sectors, the first and second actions yield static-gauge Nambu-Goto actions for a L3-brane on AdS5 and a scalar 3-brane on AdS5\times S1. 
  The correlator of a Wilson loop with a local operator in N=4 SYM theory can be represented by a string amplitude in AdS(5)xS(5). This amplitude describes an overlap of the boundary state, which is associated with the loop, with the string mode, which is dual to the local operator. For chiral primary operators with a large R charge, the amplitude can be calculated by semiclassical techniques. We compare the semiclassical string amplitude to the SYM perturbation theory and find an exact agrement to the first two non-vanishing orders. 
  In this note we establish the explicit relation between correlation functions in ${\cal N} = 4$ SUSY Yang--Mills theory on $S^3\times R$ in a double scaling limit and scattering amplitudes of the String Theory on the pp-wave background. The relation is found for two-- and three--point correlation functions in these theories. As a by product we formulate a dictionary for the correspondence between these theories: in particular, we find some unknown relations between the parameters in these theories. Furthermore, we argue that the String Theory on the pp--wave background is related to four--dimensional SUSY Yang--Mills theory rather than to the reduced matrix quantum mechanics. We identify string theory excitations which are related to the Kaluza--Klein excitation of the Yang--Mills theory on $S^3$. 
  We give the full list of types of static (homogeneous)solutions within a wide family of exactly solvable 2D dilaton gravities with backreaction of conformal fields. It includes previously known solutions as particular cases. Several concrete examples are considered for illustration. They contain a black hole and cosmological horizon in thermal equilibrium, extremal and ultraextremal horizons, etc. In particular, we demonstrate that adS and dS geometries can be \QTR{it}{exact} solutions of semiclassical field equations for a \QTR{it}{nonconstant} dilaton field. 
  Toric Duality arises as an ambiguity in computing the quiver gauge theory living on a D3-brane which probes a toric singularity. It is reviewed how, in simple cases Toric Duality is Seiberg Duality. The set of all Seiberg Dualities on a single node in the quiver forms a group which is contained in a larger group given by a set of Picard-Lefschetz transformations. This leads to elements in the group (sometimes called fractional Seiberg Duals) which are not Seiberg Duality on a single node, thus providing a new set of gauge theories which flow to the same universality class in the Infra Red. 
  This contribution to the Proceedings of the Workshop on Integrable Theories, Solitons and Duality in Sao Paulo in July 2002 summarizes results from the papers hep-th/0112023 and math.QA/0208043. We derive the non-local conserved charges in the sine-Gordon model and affine Toda field theories on the half-line. They generate new kinds of symmetry algebras that are coideals of the usual quantum groups. We show how intertwiners of tensor product representations of these algebras lead to solutions of the reflection equation. We describe how this method for finding solutions to the reflection equation parallels the previously known method of using intertwiners of quantum groups to find solutions to the Yang-Baxter equation. 
  The representation of the superalgebra so(2,1) that resulted in the relativistic wave equation (2.1) and Eq. (2.4) is fully reducible. In fact, its even part that lead to Eq. (3.1) is the direct sum of two spin 1/2 representations of the Lorentz group and does not represent spin 3/2 particle as we claimed. Consequently, we retract our claim and withdraw the manuscript. 
  We study the exact renormalization group of the four dimensional phi4 theory perturbatively. We reformulate the differential renormalization group equations as integral equations that define the continuum limit of the theory directly with no need for a bare theory. We show how the self-consistency of the integral equations leads to the determination of the interaction vertices in the continuum limit. The inductive proof of the existence of a solution to the integral equations amounts to a proof of perturbative renormalizability, and it consists of nothing more than counting the scale dimensions of the interaction vertices. Universality is discussed within a context of the exact renormalization group. 
  Proceeding from a nonlinear realization of the most general N=4, d=1 superconformal symmetry, associated with the supergroup D(2,1;alpha), we construct a new model of nonrelativistic N=4 superconformal mechanics. In the bosonic sector it combines the worldline dilaton with the fields parametrizing the R-symmetry coset S^2 ~ SU(2)/U(1). We present invariant off-shell N=4 and N=2 superfield actions for this system and show the existence of an independent N=4 superconformal invariant which extends the dilaton potential. The extended supersymmetry requires this potential to be accompanied by a d=1 WZW term on S^2. We study the classical dynamics of the bosonic action and the geometry of its sigma-model part. It turns out that the relevant target space is a cone over S^2 for any non-zero alpha \neq \pm 1/2. The constructed model is expected to be related to the `relativistic' N=4 mechanics of the AdS_2 times S^2 superparticle via a nonlinear transformation of the fields and the time variable. 
  We calculate the partition function $Z(t)$ and the asymptotic integrated level density $N(E)$ for Yang-Mills-Higgs Quantum Mechanics for two and three dimensions ($n = 2, 3$). Due to the infinite volume of the phase space $\Gamma$ on energy shell for $n= 2$, it is not possible to disentangle completely the coupled oscillators ($x^2 y^2$-model) from the Higgs sector. The situation is different for $n = 3$ for which $\Gamma$ is finite. The transition from order to chaos in these systems is expressed by the corresponding transitions in $Z(t)$ and $N(E)$, analogous to the transitions in adjacent level spacing distribution from Poisson distribution to Wigner-Dyson distribution. We also discuss a related system with quartic coupled oscillators and two dimensional quartic free oscillators for which, contrary to YMHQM, both coupling constants are dimensionless. 
  We study vortex-like configuration in Maxwell-Chern-Simons Electrodynamics. Attention is paid to the similarity it shares with the Nielsen-Olesen solutions at large distances. A magnetic symmetry between a point-like and an azimuthal-like current in this framework is also pointed out. Furthermore, we address the issue of a neutral and spinless particle interacting with a charged vortex, and obtain that the Aharonov-Casher-type phase depends upon mass and distance parameters. 
  The bulk gauge fields on 5d AdS black hole are discussed. We construct the bulk (and the corresponding brane) gauge propagator when black hole has large radius. The properties of gauge and ghost propagators are studied in both, minkowski or euclidean signature. In euclidean formulation the propagator structure corresponds to the one of theory at finite temperature (which depends on coordinates). The decoupling of KK modes and localization of gauge fields on flat brane is demonstrated. We show that with such a bulk there is no natural solution of hierarchy problem. 
  For a given complex n-fold M we present an explicit construction of all complex (n+1)-folds which are principal holomorphic T2-fibrations over M. For physical applications we consider the case of M being a Calabi-Yau 2-fold. We show that for such M, there is a subclass of the 3-folds that we construct, which has natural families of non-Kaehler SU(3)-structures satisfying the conditions for N = 1 supersymmetry in the heterotic string theory compactified on the 3-folds. We present examples in the aforementioned subclass with M being a K3-surface and a 4-torus. 
  We study the process of relaxation back to thermal equilibrium in $(1+1)$-dimensional conformal field theory at finite temperature. When the size of the system is much larger than the inverse temperature, perturbations decay exponentially with time. On the other hand, when the inverse temperature is large, the relaxation is oscillatory with characteristic period set by the size of the system. We then analyse the intermediate regime in two specific models, namely free fermions, and a strongly coupled large $\tt k$ conformal field theory which is dual to string theory on $(2+1)$-dimensional anti-de Sitter spacetime. In the latter case, there is a sharp transition between the two regimes in the ${\tt k}=\infty$ limit, which is a manifestation of the gravitational Hawking-Page phase transition. In particular, we establish a direct connection between quasinormal and normal modes of the gravity system, and the decaying and oscillating behaviour of the conformal field theory. 
  This paper presents a first example of parasupersymmetric relativistic quantum-mechanical model with non-oscillator-like interaction: the Coulomb problem for the modified Stueckelberg equation, describing a relativistic massive spin-1 particle in the electromagnetic field of a point charge. 
  We discuss the vacuum energy density and the cosmological constant of dS$_5$ brane world with a dilaton field. It is shown that a stable AdS$_4$ brane can be constructed and gravity localization can be realized. An explicit relation between the dS bulk cosmological constant and the brane cosmological constant is obtained. The discrete mass spectrum of the massive scalar field in the AdS$_4$ brane is used to acquire the relationship between the brane cosmological constant and the vacuum energy density. The vacuum energy density in the brane gotten by this method is in agreement with astronomical observations. 
  It is shown that the Lagrangian density of the supersymmetric 3-brane can be regarded as a component of an infinite-dimensional supermultiplet of N=2, D=4 supersymmetry spontaneously broken down to N=1. The latter is described by N=1 Hermitian bosonic matrix superfield V_{mn} = V^\dagger_{nm}, [V_{mn}] = m+n, m,n=0,1,... in which the component V_{01} is identified with a chiral Goldstone N=1 multiplet associated with central charge of the N=2, D=4 superalgebra, and V_{11} obeys a specific nonlinear recursive equation providing the possibility to express V_{11} (as well as the other components V_{mn}) covariantly in terms of V_{01}. We demonstrate that the solution of V_{11} gives the right \emph{PBGS} action for the super-3-brane. 
  The possibility of obtaining singularity free cosmological solutions in four dimensional effective actions motivated by string theory is investigated. In these effective actions, in addition to the Einstein-Hilbert term, the dilatonic and the axionic fields are also considered as well as terms coming from the Ramond-Ramond sector. A radiation fluid is coupled to the field equations, which appears as a consequence of the Maxwellian terms in the Ramond-Ramond sector. Singularity free bouncing solutions in which the dilaton is finite and strictly positive are obtained for models with flat or negative curvature spatial sections when the dilatonic coupling constant is such that $\omega < - 3/2$, which may appear in the so called $F$ theory in 12 dimensions. These bouncing phases are smoothly connected to the radiation dominated expansion phase of the standard cosmological model, and the asymptotic pasts correspond to very large flat spacetimes. 
  This is an expanded version of lectures given in Hangzhou and Beijing, on the symplectic forms common to Seiberg-Witten theory and the theory of solitons. Methods for evaluating the prepotential are discussed. The construction of new integrable models arising from supersymmetric gauge theories are reviewed, including twisted Calogero-Moser systems and spin chain models with twisted monodromy conditions. A practical framework is presented for evaluating the universal symplectic form in terms of Lax pairs. A subtle distinction between a Lie algebra and a Lie group version of this symplectic form is clarified, which is necessary in chain models. 
  The holographic renormalization group (RG) is reviewed in a self-contained manner. The holographic RG is based on the idea that the radial coordinate of a space-time with asymptotically AdS geometry can be identified with the RG flow parameter of the boundary field theory. After briefly discussing basic aspects of the AdS/CFT correspondence, we explain how the notion of the holographic RG comes out in the AdS/CFT correspondence. We formulate the holographic RG based on the Hamilton-Jacobi equations for bulk systems of gravity and scalar fields, as was introduced by de Boer, Verlinde and Verlinde. We then show that the equations can be solved with a derivative expansion by carefully extracting local counterterms from the generating functional of the boundary field theory. The calculational methods to obtain the Weyl anomaly and scaling dimensions are presented and applied to the RG flow from the N=4 SYM to an N=1 superconformal fixed point discovered by Leigh and Strassler. We further discuss a relation between the holographic RG and the noncritical string theory, and show that the structure of the holographic RG should persist beyond the supergravity approximation as a consequence of the renormalizability of the nonlinear sigma model action of noncritical strings. As a check, we investigate the holographic RG structure of higher-derivative gravity systems, and show that such systems can also be analyzed based on the Hamilton-Jacobi equations, and that the behaviour of bulk fields are determined solely by their boundary values. We also point out that higher-derivative gravity systems give rise to new multicritical points in the parameter space of the boundary field theories. 
  We describe the infrared R-operation for subtraction of infrared divergencies in Feynman diagrams 
  Based on a novel first class algebra, we develop an extension of the pure spinor (PS) formalism of Berkovits, in which the PS constraints are removed. By using the homological perturbation theory in an essential way, the BRST-like charge $Q$ of the conventional PS formalism is promoted to a bona fide nilpotent charge $\hat{Q}$, the cohomology of which is equivalent to the constrained cohomology of $Q$. This construction requires only a minimum number (five) of additional fermionic ghost-antighost pairs and the vertex operators for the massless modes of open string are obtained in a systematic way. Furthermore, we present a simple composite "$b$-ghost" field $B(z)$ which realizes the important relation $T(z) = \{\hat{Q}, B(z)\} $, with $T(z)$ the Virasoro operator, and apply it to facilitate the construction of the integrated vertex. The present formalism utilizes U(5) parametrization and the manifest Lorentz covariance is yet to be achieved. 
  We study the cosmological evolution of a scalar field that couples to the trace $T=T^{a}_a$ of energy momentum tensor of all the fields (including itself). In the case of a shallow exponential potential, the presence of coupling to the trace $T$ in the field equation makes the energy density of the scalar field decrease faster thereby hastening the commencement of radiation domination. This effect gradually diminishes at later epochs allowing the scalar field to dominate the energy density again. We interpret this phase as the current epoch of cosmic acceleration with $\Omega_{\phi}=0.7$. A variant of this model can lead to accelerated expansion at the present epoch followed by a $a(t)\propto t^{2/3}$ behaviour as $t\to \infty$, making the model free from future event horizon. The main features of the model are independent of initial conditions. However, fine tuning of parameters is necessary for viable evolution. 
  We identify the impurity interactions of the recently proposed CFT description of a bilayer Quantum Hall system at filling nu =m/(pm+2) in Mod. Phys. Lett. A 15 (2000) 1679. Such a CFT is obtained by m-reduction on the one layer system, with a resulting pairing symmetry and presence of quasi-holes. For the m=2 case boundary terms are shown to describe an impurity interaction which allows for a localized tunnel of the Kondo problem type. The presence of an anomalous fixed point is evidenced at finite coupling which is unstable with respect to unbalance and flows to a vacuum state with no quasi-holes. 
  We propose a new formulation for the AEther of Dirac based on a lagrangian approach. We analyse how the presence of a particular self-interaction term in the lagrangian lead us to a description of the aether as being a medium with conductivity which is governed by macroscopic Maxwell equations with a polarization tensor M_{\alpha\beta} depending on the vector potential. These results are then applied to the analysis of the amplification of the primordial magnetic induction in a curved background of Friedmann's geometry. 
  When phase space coordinates are noncommutative, especially including arbitrarily noncommutative momenta, the Hall effect is reinvestigated. A minimally gauge-invariant coupling of electromagnetic field is introduced by making use of Faddeev-Jackiw formulation for unconstrained and constrained systems. We find that the parameter of noncommutative momenta makes an important contribution to the Hall conductivity. 
  We discuss examples of systems which can be quantized consistently, although they do not admit a Lagrangian description. 
  We discuss the relation between the dual formulations of Type IIB supergravity emphasizing the differences between Lorentz and Euclidean signature. We demonstrate how the SL(2,R) symmetry of the usual action is manifested in the solution of the equations of motion with Euclidean signature for the dual theory. 
  We obtain a gauged supergravity theory in three dimensions with eight real supersymmetries by means of a Scherk-Schwarz reduction of pure N=(1,0) supergravity in six dimension on the SU(2) group manifold. The SU(2) Yang-Mills fields in the model propagate, since they have an ordinary kinetic term in addition to Chern-Simons couplings. The other propagating degrees of freedom consist of a dilaton, five scalars which parameterise the coset SL(3,R)/SO(3), three vector fields in the adjoint of SU(2), and twelve spin 1/2 fermions. The model admits an AdS_3 vacuum solution. We also show how a charged black hole solution can be obtained, by performing a dimensional reduction of the rotating self-dual string of six-dimensional (1,0) supergravity. 
  We suggest a new model of string theory with world-sheet supersymmetry. It possesses an additional global fermionic symmetry which is similar in many ways to BRST symmetry. The spectrum consists of massless states of Rarita-Schwinger fields describing infinite tower of half-integer spins. 
  In a recent paper [1] a new generalization of the Killing motion, the {\it gauged motion}, has been introduced for stationary spacetimes where it was shown that the physical symmetries of such spacetimes are well described through this new symmetry. In this article after a more detailed study in the stationary case we present the definition of gauged motion for general spacetimes. The definition is based on the gauged Lie derivative induced by a threading family of observers and the relevant reparametrization invariance. We also extend the gauged motion to the case of Kaluza-Klein theories. 
  Classical geometry of de Sitter spacetime is reviewed in arbitrary dimensions. Topics include coordinate systems, geodesic motions, and Penrose diagrams with detailed calculations. 
  We consider the quasi-de Sitter geometry of the inflationary universe. We calculate the energy flux of the slowly rolling background scalar field through the quasi-de Sitter apparent horizon and set it equal to the change of the entropy (1/4 of the area) multiplied by the temperature, dE=TdS. Remarkably, this thermodynamic law reproduces the Friedmann equation for the rolling scalar field. The flux of the slowly rolling field through the horizon of the quasi-de Sitter geometry is similar to the accretion of a rolling scalar field onto a black hole, which we also analyze. Next we add inflaton fluctuations which generate scalar metric perturbations. Metric perturbations result in a variation of the area entropy. Again, the equation dE=TdS with fluctuations reproduces the linearized Einstein equations. In this picture as long as the Einstein equations hold, holography does not put limits on the quantum field theory during inflation. Due to the accumulating metric perturbations, the horizon area during inflation randomly wiggles with dispersion increasing with time. We discuss this in connection with the stochastic decsription of inflation. We also address the issue of the instability of inflaton fluctuations in the ``hot tin can'' picture of de Sitter horizon. 
  Boundary conditions play an important role in the ADHMN construction of BPS monopole solutions. In this paper we show how different types of boundary conditions can be related to each other by removing monopoles to spatial infinity. In particular, we use this method to show how the jumping data naturally emerge. The results can be interpreted in the D-brane picture and provide a better understanding of the derivation of the ADHMN construction from D-branes. We comment briefly on the cases with non-Abelian unbroken symmetry and massless monopoles. 
  We present noncommutative nonlinear supersymmetric theories. The first example is a non-polynomial Akulov-Volkov-type lagrangian with noncommutative nonlinear global supersymmetry in arbitrary space-time dimensions. The second example is the generalization of this lagrangian to Dirac-Born-Infeld lagrangian with nonlinear supersymmetry realized in dimensions D=2,3,4, 6 and 10. 
  Taking as starting point a Lorentz and CPT non-invariant Chern-Simons-like model defined in 1+3 dimensions, we proceed realizing its dimensional reduction to D=1+2. One then obtains a new planar model, composed by the Maxwell-Chern-Simons (MCS) sector, a Klein-Gordon massless scalar field, and a coupling term that mixes the gauge field to the external vector, $v^{\mu}$. In spite of breaking Lorentz invariance in the particle frame, this model may preserve the CPT symmetry for a single particular choice of $v^{\mu}$. Analyzing the dispersion relations, one verifies that the reduced model exhibits stability, but the causality can be jeopardized by some modes. The unitarity of the gauge sector is assured without any restriction, while the scalar sector is unitary only in the space-like case.   PACS numbers: 11.10.Kk; 11.30.Cp; 11.30.Er; 12.60.-i 
  We discuss the effective theory for the close limit of two branes in a covariant way. To do so we solve the five dimensional Einstein equation along the direction of the extra dimension. Using the Taylor expansion we solve the bulk spacetimes and derive the effective theory describing the close limit. We also discuss the radion dynamics and braneworld black holes for the close limit in our formulation. 
  Correlation functions of chiral primary operators and their superconformal descendants in the N=4 supersymmetric SU(N) Yang-Mills theory are studied in detail in a one-instanton background and at large N. Whereas earlier calculations were restricted to correlation functions that are saturated by the 16 exact superconformal fermionic moduli, here the effect of the set of additional fermionic moduli associated with the embeddings of the SU(2) instanton in SU(N) is considered. The presence of the extra fermionic modes is essential for matching Yang-Mills instanton effects in various correlation functions with D-instanton effects in type IIB string theory via the AdS/CFT conjecture. The leading terms of this kind on the string side contribute at order 1/(alpha'), which is the same order as the R^4 interaction. For example, the instanton contributions to correlation functions of higher dimensional chiral primary operators are seen to match amplitudes involving Kaluza-Klein excitations of the supergravity fields, as expected. Another example is the matching of certain multi-fermion correlation functions which correspond to certain multi-fermion interactions required by supersymmetry of the IIB string effective action. Careful analysis of a variety of competing effects makes it possible to decipher contributions corresponding to higher derivative interactions in the IIB effective action. In this manner it is possible to check for the presence of terms of order alpha'. Comments are also made on the structure of instanton contributions to near-extremal correlation functions of chiral primary operators. 
  Stellar structure in braneworlds is markedly different from that in ordinary general relativity. As an indispensable first step towards a more general analysis, we completely solve the ``on brane'' 4-dimensional Gauss and Codazzi equations for an arbitrary static spherically symmetric star in a Randall--Sundrum type II braneworld. We then indicate how this on-brane boundary data should be propagated into the bulk in order to determine the full 5-dimensional spacetime geometry. Finally, we demonstrate how this procedure can be generalized to solid objects such as planets. 
  Recent astrophysical observations suggest that the value of fine structure constant $\alpha=e^2/\hbar c$ may be slowly increasing with time. This may be due to an increase of $e$ or a decrease of $c$, or both. In this article, we argue from model independent considerations that this variation should be considered adiabatic. Then, we examine in detail the consequences of such an adiabatic variation in the context of a specific model of quantized charged black holes. We find that the second law of black hole thermodynamics is obeyed, regardless of the origin of the variation, and that interesting constraints arise on the charge and mass of black holes. Finally, we estimate the work done on a black hole of mass $M$ due to the proposed $\alpha$ variation. 
  It is argued that the (NS-sector) superstring field equations are integrable, i.e. their solutions are obtainable from linear equations. We adapt the 25-year-old solution-generating "dressing" method and reduce the construction of nonperturbative superstring configurations to a specific cohomology problem. The application to vacuum superstring field theory is outlined. 
  By the appropriate use of the Fock-Schwinger gauge properties, we derive the closed integral form of the `point-split' non-local background gauge connection originally expressed as a finite sum. This is achieved in the limit when the finite sum becomes infinite. With this closed integral form of the connection, we obtain the same exact results in the calculation of one-loop effective Lagrangian accommodating arbitrary orders of covariant field derivatives in quantum field theory of arbitrary spacetime dimensions and of arbitrary gauge group. Particularly, we display the one-loop effective Lagrangian for real boson fields up to 8 mass dimensions-the same result obtained when the connection was yet in the finite sum form. 
  In the context of the AdS/CFT correspondence the general form of a three-form flux perturbation to the AdS_5 x S^5 solution in the type IIB supergravity which preserves N=2 supersymmetry is obtained. The arbitrary holomorphic function appearing in the N=1 case studied by Grana and Polchinski is restricted to a quadratic function of the coordinates transverse to the D3-branes. 
  N=1 and 2 superconformal boundary conditions are shown to be the consequence of a boundary on the worldsheet superspace with positive codimension in the anticommuting subspace. In addition to the well-known boundary conditions, I also find two new infinite series of N=2 boundary states. Their free field realizations are given. A self-contained development of 2d superspace leads to new perspectives on this subject. 
  We review a recent proposal towards a microscopic understanding of the entropy of non-supersymmetric spacetimes -- with emphasis on the Schwarzschild black hole. The approach is based at an intermediate step on the description of the non-supersymmetric spacetime in terms of dual Euclidean brane pairs of type-II string-theory or M-theory. By counting specific chain structures on the brane-complex, it is shown that one can reproduce the exact Schwarzschild black hole entropy plus its logarithmic correction. 
  In these lecture notes, I describe the motivation behind a recent formulation of a non-perturbative gravitational path integral for Lorentzian (instead of the usual Euclidean) space-times, and give a pedagogical introduction to its main features. At the regularized, discrete level this approach solves the problems of (i) having a well-defined Wick rotation, (ii) possessing a coordinate-invariant cutoff, and (iii) leading to_convergent_ sums over geometries. Although little is known as yet about the existence and nature of an underlying continuum theory of quantum gravity in four dimensions, there are already a number of beautiful results in d=2 and d=3 where continuum limits have been found. They include an explicit example of the inequivalence of the Euclidean and Lorentzian path integrals, a non-perturbative mechanism for the cancellation of the conformal factor, and the discovery that causality can act as an effective regulator of quantum geometry. 
  We show that after mapping each element of a set of second class constraints to the surface of the other ones, half of them form a subset of abelian first class constraints. The explicit form of the map is obtained considering the most general Poisson structure. We also introduce a proper redefinition of second class constraints that makes their algebra symplectic. 
  Exact vacuum expectation values of the third level descendent fields $<(\partial\phi)^3({\bar\partial}\phi)^3e^{a\phi}>$ in the Bullough-Dodd model are proposed. By performing quantum group restrictions, we obtain $<L_{-3}{\bar L}_{-3}{\Phi}_{lk}>$ in perturbed minimal conformal field theories. 
  We work out boundary conditions for the covariant open string in the type IIA plane wave background, which corresponds to the D-branes in the type IIA theory. We use the kappa symmetric string action and see what kind of boundary conditions should be imposed to retain kappa symmetry. We find half BPS as well as quarter BPS branes and the analysis agrees with the previous work in the light cone gauge if the result is available. Finally we find that D0-brane is non-supersymmetric. 
  We conduct a systematic search for a viable string/M-theory cosmology, focusing on cosmologies that include an era of slow-roll inflation, after which the moduli are stabilized and the Universe is in a state with an acceptably small cosmological constant. We observe that the duality relations between different cosmological backgrounds of string/M-theory moduli space are greatly simplified, and that this simplification leads to a truncated moduli space within which possible cosmological solutions lie. We review some known challenges to four dimensional models in the "outer", perturbative, region of moduli space, and use duality relations to extend them to models of all of the (compactified) perturbative string theories and 11D supergravity, including brane world models. We conclude that cosmologies restricted to the outer region are not viable, and that the most likely region of moduli space in which to find realistic cosmology is the "central", non-perturbative region, with coupling and compact volume both of order unity, in string units. 
  We discuss the sizes of a black hole in the M theory pp-wave background, and how the transverse size can be reproduced in the matrix model. 
  The correspondence between del Pezzo surfaces and field theory models over the complex numbers or for split real forms is extended to other real forms, in particular to those compatible with supersymmetry. Specifically, all theories of the Magic triangle that reduce to the pure supergravities in four dimensions correspond to singular real del Pezzo surfaces and the same is true for the Magic square of N=2 SUGRAS. A real del Pezzo surface is the invariant set under an antilinear involution of a complex one. This conjugation induces an involution of the Picard group that preserves the anticanonical class and the intersection form. The known non-split U-duality algebras are embedded into superBorcherds algebras defined by their Cartan matrix (minus the intersection form) and fixed by the anti-involution. These data may be described by Tits-Satake bicoloured diagrams. As in the split case, oxidation results from blowing down disjoint real P^1's of self-intersection -1. The singular del Pezzo surfaces of interest are obtained by degenerating regular surfaces upon contraction of real curves of self-intersection -2. We use the finite classification of real simple singularities to exhibit the relevant normal surfaces. We also give a general construction of more magic triangles like a type I split magic triangle and prove their (approximate) symmetry with respect to their diagonal, this symmetry argument was announced in our previous paper for the split case. 
  We study how to generate new Lie algebras $\mathcal{G}(N_0,..., N_p,...,N_n)$ from a given one $\mathcal{G}$. The (order by order) method consists in expanding its Maurer-Cartan one-forms in powers of a real parameter $\lambda$ which rescales the coordinates of the Lie (super)group $G$, $g^{i_p} \to \lambda^p g^{i_p}$, in a way subordinated to the splitting of $\mathcal{G}$ as a sum $V_0 \oplus ... \oplus V_p \oplus ... \oplus V_n$ of vector subspaces. We also show that, under certain conditions, one of the obtained algebras may correspond to a generalized \.In\"on\"u-Wigner contraction in the sense of Weimar-Woods, but not in general. The method is used to derive the M-theory superalgebra, including its Lorentz part, from $osp(1|32)$. It is also extended to include gauge free differential (super)algebras and Chern-Simons theories, and then applied to D=3 CS supergravity. 
  This is a brief introduction to the subject of Conformal Field Theory on surfaces with boundaries and crosscaps, which describes the perturbative expansion of open string theory. 
  String theory is a quantum theory that reproduces the results of General Relativity at long distances but is completely different at short distances. Mathematically, string theory is based on a very new -- and little understood -- framework for geometry that reduces to ordinary differential geometry when the curvature is asymptotically small. In the 1990's, many interesting results were obtained about the behavior of string theory in spacetimes that develop singularities. In many cases, the physics at the singularity is governed by an effective Lagrangian constructed using an interesting bit of classical geometry such as the association of A-D-E groups with certain hypersurface singularities or the ADHM construction of instantons. In other examples, the physics at the singularity cannot be described in classical terms but involves a non-Gaussian conformal field theory. 
  Precanonical quantization of pure Yang-Mills fields, which is based on the covariant De Donder-Weyl (DW) Hamiltonian formalism, and its connection with the functional Schrodinger representation in the temporal gauge are discussed. The YM mass gap problem is related to a finite dimensional spectral problem for a generalized Clifford-valued magnetic Schr\"odinger operator in the space of gauge potentials which represents the DW Hamiltonian operator. 
  We introduce a notion of universality classes for the Gregory-Laflamme instability and determine, in the supergravity approximation, the stability of a variety of solutions, including the non-extremal D3-brane, M2-brane, and M5-brane. These three non-dilatonic branes cross over from instability to stability at a certain non-extremal mass. Numerical analysis suggests that the wavelength of the shortest unstable mode diverges as one approaches the cross-over point from above, with a simple critical exponent which is the same in all three cases. 
  In this paper we study strings with quantized masses in the pp-wave background. We obtain these strings from the membrane theory. For achieving to this, one of the membrane and one of the spacetime directions will be identified and wrapped. From the action of strings in the pp-wave background, we obtain its mass dual action. Some properties of the closed and open strings in this background will be studied. 
  We derive recurrence relations between phase space expressions in different dimensions by confining some of the coordinates to tori or spheres of radius $R$ and taking the limit as $R \to \infty$. These relations take the form of mass integrals, associated with extraneous momenta (relative to the lower dimension), and produce the result in the higher dimension. 
  We give an elementary introduction to the structure of supergravity theories. This leads to a table with an overview of supergravity and supersymmetry theories in dimensions 4 to 11. The basic steps in constructing supergravity theories are considered: determination of the underlying algebra, the multiplets, the actions, and solutions. Finally, an overview is given of the geometries that result from the scalars of supergravity theories. 
  N=1 supersymmetric U(N) gauge theory with adjoint matter $\Phi$ and a polynomial superpotential $\Tr W(\Phi)$ has been much studied recently. The classical theory has several vacua labeled by integers $(N_1,N_2,...,N_k)$, with the classical unbroken gauge group $\prod_i U(N_i)$. Quantum mechanically, each classical vacuum leads to $\prod_i N_i$ different vacua. As the parameters of $W(\Phi)$ are varied, these vacua change in a continuous (and holomorphic) fashion. We find that vacua associated with $(N_1,N_2,...,N_k)$ can be continuously transformed to vacua with $(\tilde N_1,\tilde N_2,...,\tilde N_k)$, thus leading to a new kind of duality. Traditional order parameters, like the Wilson loop and 't Hooft loop, sometimes distinguish different phases. We also find phases that are not distinguished by conventional order parameters. The whole picture of the phase diagram is reminiscent of the phase diagram of $M$-theory. 
  The non-perturbation and perturbation structures of the q-deformed probability currents are studied. According to two ways of realizing the q-deformed Heisenberg algebra by the undeformed operators, the perturbation structures of two q-deformed probability currents are explored in detail. Locally the structures of two perturbation q-deformed probability currents are different, one is explicitly potential dependent; the other is not. But their total contributions to the whole space are the same. 
  In the framework of chiral perturbation theory we demonstrate the equivalence of the supersymmetric and the replica methods in the symmetry breaking classes of Dyson indices \beta=1 and \beta=4. Schwinger-Dyson equations are used to derive a universal differential equation for the finite volume partition function in sectors of fixed topological charge, \nu. All dependence on the symmetry breaking class enters through the Dyson index \beta. We utilize this differential equation to obtain Virasoro constraints in the small mass expansion for all \beta and in the large mass expansion for \beta=2 with arbitrary \nu. Using quenched chiral perturbation theory we calculate the first finite volume correction to the chiral condensate demonstrating how, for all \betathere exists a region in which the two expansion schemes of quenched finite volume chiral perturbation theory overlap. 
  The superfield light cone gauge formulation of $\cN =4$ SYM in plane wave background is developed. We find a realization of superconformal symmetries in terms of a light cone superfield. An oscillator realization of superconformal symmetries is also obtained. 
  Some recent studies have considered a Randall-Sundrum-like brane world evolving in the background of an anti-de Sitter Reissner-Nordstrom black hole. For this scenario, it has been shown that, when the bulk charge is non-vanishing, a singularity-free ``bounce'' universe will always be obtained. However, for the physically relevant case of a de Sitter brane world, we have recently argued that, from a holographic (c-theorem) perspective, such brane worlds may not be physically viable. In the current paper, we reconsider the validity of such models by appealing to the so-called ``causal entropy bound''. In this framework, a paradoxical outcome is obtained: these brane worlds are indeed holographically viable, provided that the bulk charge is not too small. We go on to argue that this new finding is likely the more reliable one. 
  By performing the matrix integral over the tree level superpotential of N=1 supersymmetric SO(N)/Sp(N) gauge theories obtained from N=2 SQCD by adding the mass term for the adjoint scalar field, the exact effective superpotential in terms of meson field contains the nonperturbative ADS superpotential as well as the classical tree level superpotential. By completing the meson matrix integral with the help of saddle point equation, we find the free energy contributions from matter part in terms of glueball field, the adjoint field mass and quark mass. By extremizing the effective superpotential with respect to the glueball field, we analyze the vacuum structure and describe the behavior of two limiting cases:zero limit of quark mass and infinity limit of adjoint field mass. We also study the magnetic theory. 
  We consider a $\sigma$-model formulation of open string theory in the presence of D-branes. We perform two-loop computations and discuss gravitational corrections to Born-Infeld action when branes are non-trivially embedded in a curved ambient space. In particular for the case of a stack of $N$ coincident D-branes we analyze couplings of the form $R_{ijkl}[\Phi^i,\Phi^j][\Phi^k,\Phi^l]$. 
  We review the quantum instability of the Savvidy-Nielsen-Olesen (SNO) vacuum of the one-loop effective action of SU(2) QCD, and point out a critical defect in the calculation of the functional determinant of the gluon loop in the SNO effective action. We prove that the gauge invariance, in particular the color reflection invariance, exclude the unstable tachyonic modes from the gluon loop integral. This guarantees the stability of the magnetic condensation in QCD. 
  An action for a massless graviton interacting with a massive tensor field is proposed. The model is based on coupling the metric tensor to an SP(4) gauge theory spontaneously broken to $SL(2,C)$. The symmetry breaking is achieved by using a Higgs multiplet containing a scalar field and a vector field related by a constraint. We show that in the non-unitary gauge and for the Fierz-Pauli form of the mass term, the six degrees of freedom of the massive tensor are identified with two tensor helicities, two vector helicities of the Goldstone vector, and two scalars present in the Goldstone multiplet. The propagators of this system are well behaved, in contrast to the system consisting of two tensors. 
  We show how, modulo the distinction between the antipode and the "twisted" or "renormalized" antipode, Connes and Kreimer's algebraic paradigm trivializes the proofs of equivalence of the (corrected) Dyson-Salam, Bogoliubov-Parasiuk-Hepp and Zimmermann procedures for renormalizing Feynman amplitudes. We discuss the outlook for a parallel simplification of computations in quantum field theory, stemming from the same algebraic approach. 
  The symmetry algebra of massless fields living on the 3-dimensional conformal boundary of AdS(4) is shown to be isomorphic to 3d conformal higher spin algebra (AdS(4) higher spin algebra). A simple realization of this algebra on the free flat 3d massless matter fields is given in terms of an auxiliary Fock module. 
  We work out the decomposition of the indefinite Kac Moody algebras ${E_{10}}$ and ${E_{11}}$ w.r.t. their respective subalgebras $A_9$ and $A_{10}$ at low levels. Tables of the irreducible representations with their outer multiplicities are presented for ${E_{10}}$ up to level $\ell = 18$ and for ${E_{11}}$ up to level $\ell =10$. On the way we confirm and extend existing results for ${E_{10}}$ root multiplicities, and for the first time compute non-trivial root multiplicities of ${E_{11}}$. 
  In this paper I review some old and new works on the computation of two-loop 4-particle amplitude in superstring theory. I also present the proof by Iengo, showing the vanishing of the term related to the two-loop correction to the $R^4$ term. Finally I will present some recent works on two-loop computation in hyperelliptic language following the new gauging fixing method of D'Hoker and Phong. 
  It is shown that a simple model of 2N-Grassmann variables with a four-body coupling involves caustics when the integral has been converted to a bosonic form with the aid of the auxiliary field. Approximation is then performed to assure validity of the auxiliary field method(AFM). It turns out that even in N=2, the smallest case in which a four-body interaction exists, AFM does work more excellently if higher order effects, given by a series in terms of $1/N^{1/3}$ around a caustic and of 1/N around a saddle point, would be taken into account. 
  We review the recently discovered ansatz that describes non-extremal charged dilatonic branes of string/M-theory with a transverse circle. The ansatz involves a new coordinate system that interpolates between the spherical and cylindrical case, and reduces the equations of motion to a set of equations on one unknown function of two variables. The function is independent of the charge, so that the ansatz can also be used to construct neutral black holes on cylinders and near-extremal charged dilatonic branes with a transverse circle. The construction enables us to argue that, for sufficiently large mass, there exists a neutral solution that breaks translational invariance in the circle direction, and has larger entropy than that of the neutral black string of the same mass. 
  The low-energy effective action of type-I superstring theory in ten dimensions is obtained performing a truncation of type-IIB supergravity in a background where D9-branes are present. The open sector corresponds to the first order in the low-energy expansion of the D9-brane action in a type-I background. In hep-th/9901055 it was shown that there are two ways of performing a type-I truncation of the D9-brane action, and the resulting truncated action was obtained in a flat background. We extend this result to a generic type-I background, and argue that the two different truncations are in correspondence with the open sector of the low-energy effective action of the two different consistent ten-dimensional type-I string theories, namely the SO(32) superstring and the $USp(32)$ non-supersymmetric string. 
  We derive an explicit formula for the low energy limits of the one-loop, on-shell, massive N-photon amplitudes, for arbitrary N and all helicity assignments, in scalar and spinor QED. The two-loop corrections to the same amplitudes are obtained for up to the ten point case. All photon amplitudes with an odd number of `+' helicities are shown to vanish in this limit to all loop orders. 
  We show that there is only one physically acceptable vacuum state for quantum fields in de Sitter space-time which is left invariant under the action of the de Sitter-Lorentz group $SO(1,d)$ and supply its physical interpretation in terms of the Poincare invariant quantum field theory (QFT) on one dimension higher Minkowski spacetime. We compute correlation functions of the generalized vertex operator $:e^{i\hat{S}(x)}:$, where $\hat{S}(x)$ is a massless scalar field, on the $d$-dimensional de Sitter space and demonstrate that their limiting values at timelike infinities on de Sitter space reproduce correlation functions in $(d-1)$-dimensional Euclidean conformal field theory (CFT) on $S^{d-1}$ for scalar operators with arbitrary real conformal dimensions. We also compute correlation functions for a vertex operator $e^{i\hat{S}(u)}$ on the \L obaczewski space and find that they also reproduce correlation functions of the same CFT. The massless field $\hat{S}(u)$ is the nonlocal transform of the massless field $\hat{S}(x)$ on de Sitter space introduced by one of us. 
  In the present talk I shall review the construction of N=2 supergravity models exhibiting stable de Sitter vacua. These solutions represent the first instance of stable backgrounds with positive cosmological constant in the framework of extended supergravities (N >=2). After briefly reviewing the role of de Sitter space--times in inflationary cosmology, I shall describe the main ingredients which were necessary for the construction of gauged N=2 supergravity models admitting stable solutions of this kind. 
  The quantization of a scalar field in AdS leads to two kinds of normalizable modes, usually called regular and irregular modes. The regular one is easily taken into account in the standard prescription for the AdS/CFT correspondence. The irregular mode requires a modified prescription which we argue is not completely satisfactory. We discuss an alternative quantization in AdS which incorporates boundary terms in a natural way. Within this quantization scheme we present an improved prescription for the AdS/CFT correspondence which can be applied to both, regular and irregular modes. Boundary conditions other than Dirichlet are naturally treated in this new improved setting. 
  The Dirichlet boundary-value problem and isoperimetric inequalities for positive definite regular solutions of the vacuum Einstein equations are studied in arbitrary dimensions for the class of metrics with boundaries admitting a U(1) action. We show that in the case of non-trivial bundles Taub-Bolt infillings are double-valued whereas Taub-Nut and Eguchi-Hanson infillings are unique. In the case of trivial bundles, there are two Schwarzschild infillings in arbitrary dimensions. The condition of whether a particular type of filling in is possible can be expressed as a limitation on squashing through a functional dependence on dimension in each case. The case of the Eguchi-Hanson metric is solved in arbitrary dimension. The Taub-Nut and the Taub-Bolt are solved in four dimensions and methods for arbitrary dimension are delineated. For the case of Schwarzschild, analytic formulae for the two infilling black hole masses in arbitrary dimension have been obtained. This should facilitate the study of black hole dynamics/thermodynamics in higher dimensions. We found that all infilling solutions are convex. Thus convexity of the boundary does not guarantee uniqueness of the infilling. Isoperimetric inequalities involving the volume of the boundary and the volume of the infilling solutions are then investigated. In particular, the analogues of Minkowski's celebrated inequality in flat space are found and discussed providing insight into the geometric nature of these Ricci-flat spaces. 
  We construct analogs of the flat space Kaluza-Klein (KK) monopoles in locally Anti-de Sitter (AdS) spaces for $D \ge 5+1$. We show that, unlike the flat space KK monopole, there is no five dimensional static KK monopole in AdS that smoothly reduces to the flat space one as the cosmological constant goes to zero. Thus, one needs at least two extra dimensions, one of which is compact, to get a static KK monopole in cosmological backgrounds. 
  As was shown by Aharony, Hanany and Kol and independently by Sen, type IIB string theory admits configurations where strings of different charges (p_i, q_i) form so-called string networks. We argue that these networks blow up into "supersheets": supersymmetric spinning cylindrical D3-branes carrying electric and magnetic fluxes. These supersheets are three-dimensional generalizations of the supertubes that were constructed by Mateos and Townsend. We calculate the mass of both systems for arbitrary values of the parameters and find exact agreement. 
  Curvature-squared corrections for D-brane actions in type II string theory were derived by Bachas, Bain and Green. Here we write down a generalisation of these corrections to all orders in $F$, the field strength of the U(1) gauge field on the brane. Some of these terms are needed to restore consistency with T-duality. 
  The one-loop contribution to vacuum polarization is calculated for the adjoint fermions in three dimensional noncommutative spaces, both at zero and finite temperature. At zero temperature, we confirm a previously found result for the parity odd part and subsequently analyze the even parity part, which exhibits UV/IR mixing. We discuss in detail, two regimes of the high temperature behavior of the parity odd part. When the thermal wavelength is much smaller, as compared to the noncommutativity scale, we find an interesting Fermi-Bose transmutation in the nonplanar part. 
  We present a cosmological model in 1+m+p dimensions, where in m-dimensional space there are uniformly distributed p-branes wrapping over the extra p-dimensions. We find that during cosmological evolution m-dimensional space expands with the exact power-law corresponding to pressureless matter while the extra p-dimensions contract. Adding matter, we also obtain solutions having the same property. We show that this might explain in a natural way why the extra dimensions are small compared to the observed three spatial directions. 
  We review the construction of chiral four-dimensional compactifications of type IIA string theory with intersecting D6-branes. Such models lead to four-dimensional theories with non-abelian gauge interactions and charged chiral fermions. We discuss the application of these techniques to building of models with spectrum as close as possible to the Standard Model, and review their main phenomenological properties. We also emphasize the advantages/disadvantages of carrying out this idea using supersymmetric of non-supersymmetric models. 
  The twisting function describing a nonstandard (super-Jordanian) quantum deformation of $osp(1|2)$ is given in explicite closed form. The quantum coproducts and universal R-matrix are presented. The non-uniqueness of the twisting function as well as two real forms of the deformed $osp(1|2)$ superalgebras are considered. One real quantum $osp(1|2)$ superalgebra is interpreted as describing the $\kappa$-deformation of D=1, N=1 superconformal algebra, which can be applied as a symmetry algebra of N=1 superconformal mechanics. 
  Starting from N=1 scalar and vector supermultiplets in 2+1 dimensions, we construct superfields which constitute Lagrangians invariant under N=2 supersymmetries. We first recover the N=2 supersymmetric Abelian-Higgs model and then the N=2 pure super Yang-Mills model. The conditions for this elevation are consistent with previous results found by other authors. 
  We study the thermal partition function of superstring on the pp-wave background with the circle compactification along a transverse direction. We calculate it in the two ways: the operator formalism and the path-integral calculation. The former gives the finite result with no subtlety of the Wick rotation, which only contains the contributions of physical states. On the other hand, the latter yields the manifestly modular invariant expression, even though we only have the winding modes along the transverse circle (no Kaluza-Klein excitations). We also check the equivalence of these two analyses. The DLCQ approach makes the path-integration quite easy. Remarkably, we find that the contributions from the transverse winding sectors disappear in the non-DLCQ limit, while they indeed contribute in the DLCQ model, depending non-trivially on the longitudinal quantum numbers. 
  The PP-wave/SYM proposal in its original form emphasizes a duality relation between the masses of the string states and the anomalous dimensions of the corresponding BMN operators in gauge theory, the mass--dimension type duality. In this paper, we give evidence in favour of another duality relation of the vertex--correlator type, which relates the coefficients of 3-point correlators of BMN operators in gauge theory to 3-string vertices in lightcone string field theory in the pp-wave background. We verify that all the available field theory results in the literature, as well as the newly obtained ones, for the 3-point functions are successfully reproduced from our proposal. 
  We explain how structures related to octonions are ubiquitous in M-theory. All the exceptional Lie groups, and the projective Cayley line and plane appear in M-theory. Exceptional G_2-holonomy manifolds show up as compactifying spaces, and are related to the M2 Brane and 3-form. We review this evidence, which comes from the initial 11-dim structures. Relations between these objects are stressed, when extant and understood. We argue for the necessity of a better understanding of the role of the octonions themselves (in particular non-associativity) in M-theory. 
  The timelike boundary Liouville (TBL) conformal field theory consisting of a negative norm boson with an exponential boundary interaction is considered. TBL and its close cousin, a positive norm boson with a non-hermitian boundary interaction, arise in the description of the $c=1$ accumulation point of $c<1$ minimal models, as the worldsheet description of open string tachyon condensation in string theory and in scaling limits of superconductors with line defects. Bulk correlators are shown to be exactly soluble. In contrast, due to OPE singularities near the boundary interaction, the computation of boundary correlators is a challenging problem which we address but do not fully solve. Analytic continuation from the known correlators of spatial boundary Liouville to TBL encounters an infinite accumulation of poles and zeros. A particular contour prescription is proposed which cancels the poles against the zeros in the boundary correlator $d(\o) $ of two operators of weight $\o^2$ and yields a finite result. A general relation is proposed between two-point CFT correlators and stringy Bogolubov coefficients, according to which the magnitude of $d(\o)$ determines the rate of open string pair creation during tachyon condensation. The rate so obtained agrees at large $\o$ with a minisuperspace analysis of previous work. It is suggested that the mathematical ambiguity arising in the prescription for analytic continuation of the correlators corresponds to the physical ambiguity in the choice of open string modes and vacua in a time dependent background. 
  Boundary charges in gauge theories (like the ADM mass in general relativity) can be understood as integrals of linear conserved n-2 forms of the free theory obtained by linearization around the background. These forms are associated one-to-one to reducibility parameters of this background (like the time-like Killing vector of Minkowski space-time). In this paper, closed n-2 forms in the full interacting theory are constructed in terms of a one parameter family of solutions to the full equations of motion that admits a reducibility parameter. These forms thus allow one to apply Stokes theorem without bulk contributions and, provided appropriate fall-off conditions are satisfied, they reduce asymptotically near the boundary to the conserved n-2 forms of the linearized theory. As an application, the first law of black hole mechanics in asymptotically anti-de Sitter space-times is derived. 
  We study the mutual consistency of twisted boundary conditions in the coset conformal field theory G/H. We calculate the overlap of the twisted boundary states of G/H with the untwisted ones, and show that the twisted boundary states are consistently defined in the diagonal modular invariant. The overlap of the twisted boundary states is expressed by the branching functions of a twisted affine Lie algebra. As a check of our argument, we study the diagonal coset theory so(2n)_1 \oplus so(2n)_1/so(2n)_2, which is equivalent with the orbifold S^1/\Z_2. We construct the boundary states twisted by the automorphisms of the unextended Dynkin diagram of so(2n), and show their mutual consistency by identifying their counterpart in the orbifold. For the triality of so(8), the twisted states of the coset theory correspond to neither the Neumann nor the Dirichlet boundary states of the orbifold and yield the conformal boundary states that preserve only the Virasoro algebra. 
  N=1, D=4 Superstring possessing $SO(6)\otimes SO(5)$ symmetry action and with the same gauge symmetry obtained from zero mass spectrum of vector meson as well, is constructed from the bosonic string in twenty six dimensions. Without breaking supersymmetry, the gauge symmetry of the model descends to the supersymmetric standard model of the electroweak scale in four dimension.It is proved that there can be three generations in the model 
  We investigate the previously proposed cyclic regime of the Kosterlitz-Thouless renormalization group (RG) flows. The period of one cycle is computed in terms of the RG invariant. Using bosonization, we show that the theory has $U_q (\hat{sl(2)})$ quantum affine symmetry, with $q$ {\it real}. Based on this symmetry, we study two possible S-matrices for the theory, differing only by overall scalar factors. We argue that one S-matrix corresponds to a continuum limit of the XXZ spin chain in the anti-ferromagnetic domain $\Delta < -1$. The latter S-matrix has a periodicity in energy consistent with the cyclicity of the RG. We conjecture that this S-matrix describes the cyclic regime of the Kosterlitz-Thouless flows.   The other S-matrix we investigate is an analytic continuation of the usual sine-Gordon one. It has an infinite number of resonances with masses that have a Russian doll scaling behavior that is also consistent with the period of the RG cycles computed from the beta-function. Closure of the bootstrap for this S-matrix leads to an infinite number of particles of higher spin with a mass formula suggestive of a string theory. 
  General structure of BRST-invariant constraint algebra is established, in its commutator and antibracket forms, by means of formulation of algebra-generating equations in yet more extended phase space. New ghost-type variables behave as fields and antifields with respect to quantum antibrackets. Explicit form of BRST-invariant gauge algebra is given in detail for rank-one theories with Weyl- and Wick- ordered ghost sector. A gauge-fixed unitarizing Hamiltonian is constructed, and the formalism is shown to be physically equivalent to the standard BRST-BFV approach. 
  We discuss isotropic and homogeneous D-brane-world cosmology with non-Abelian Born-Infeld (NBI) matter on the brane. In the usual Friedmann-Robertson-Walker (FRW) model the scale non-invariant NBI matter gives rise to an equation of state which asymptotes to the string gas equation $p=-\epsilon/3$ and ensures a start-up of the cosmological expansion with zero acceleration. We show that the same state equation in the brane-world setup leads to the Tolman type evolution as if the conformal symmetry was effectively restored. This is not precisely so in the NBI model with symmetrized trace, but the leading term in the expansion law is still the same. A cosmological sphaleron solution on the D-brane is presented. 
  It is demonstrated that there are smooth Yang-Mills potentials which correspond to monopoles and vortices of one-half winding number. They are the generic configurations, in contrast to the integral winding number configurations like the 't Hooft-Polyakov monopole. 
  Following the work of Kinnersley and Walker for flat spacetimes, we have analyzed the anti-de Sitter C-metric in a previous paper. In the de Sitter case, Podolsky and Griffiths have established that the de Sitter C-metric (dS C-metric) found by Plebanski and Demianski describes a pair of accelerated black holes in the dS background with the acceleration being provided (in addition to the cosmological constant) by a strut that pushes away the two black holes or, alternatively, by a string that pulls them. We extend their analysis mainly in four directions. First, we draw the Carter-Penrose diagrams of the massless uncharged dS C-metric, of the massive uncharged dS C-metric and of the massive charged dS C-metric. These diagrams allow us to clearly identify the presence of two dS black holes and to conclude that they cannot interact gravitationally. Second, we revisit the embedding of the dS C-metric in the 5D Minkowski spacetime and we represent the motion of the dS C-metric origin in the dS 4-hyperboloid as well as the localization of the strut. Third, we comment on the physical properties of the strut that connects the two black holes. Finally, we find the range of parameters that correspond to non-extreme black holes, extreme black holes, and naked particles. 
  We generalize to noncommutative cylinder the solution generation technique, originally suggested for gauge theories on noncommutative plane. For this purpose we construct partial isometry operators and complete set of orthogonal projectors in the algebra of the cylinder, and an isomorphism between the free module and its direct sum with the Fock module on the cylinder. We construct explicitly the gauge theory soliton and evaluate the spectrum of perturbations about this soliton. 
  We study the realization of anomalous Ward identities in deconstructed (latticized) supersymmetric theories. In a deconstructed four-dimensional theory with N=2 supersymmetry, we show that the chiral symmetries only appear in the infrared and that the anomaly is reproduced in the usual framework of lattice perturbation theory with Wilson fermions. We then realize the theory on the world-volume of fractional D-branes on an orbifold. In this brane realization, we show how deconstructed theory anomalies can be computed via classical supergravity. Our methods and observations are more generally applicable to deconstructed/latticized supersymmetric theories in various dimensions. 
  We study the decay of unstable D$p$-branes when the world-volume gauge field is turned on. We obtain the relevant Dp-brane boundary state with electric and magnetic fields by boosting and rotating the rolling tachyon boundary state of a D(p-1)-brane and then T-dualizing along one of the transverse directions. A simple recipe to turn on the gauge fields in the boundary state is given. We find that the effect of the electric field is to parametrically enhance coupling of closed string oscillation modes along the electric field direction and provide an intuitive understanding of the result in the T-dualized picture. We also analyze the system by using the effective field theory and compare the result with the boundary state approach. 
  Quarks and leptons charges and interactions are derived from gauge theories associated with symmetries. Their space-time labels come from representations of the non-compact algebra of Special Relativity. Common to these descriptions are the Lie groups stemming from their invariances. Does Nature use Exceptional Groups, the most distinctive among them? We examine the case for and against their use. They do indeed appear in charge space, as the Standard Model fits naturally inside the exceptional group $E_6$. Further, the advent of the $E_8\times E_8$ Heterotic Superstring theory adds credibility to this venue. On the other hand, their use as space-time labels has not been as evident as they link spinors and tensors under space rotations, which flies in the face of the spin-statistics connection. We discuss a way to circumvent this difficulty in trying to generalize eleven-dimensional supergravity. 
  An algorithm is devised to generate characteristic identities between Maxwell fieldstrength invariants (traced over Lorentz indices and disregarding ordering) that suffer linear dependence in certain dimensionalities as they have been originally obtained using a Maple routine. These relations between invariants are then applied to simplify the Abelian Born-Infeld (ABI) effective action in arbitrary degree of fieldstrength invariants. I have explicitly displayed the simplified ABI action in 4, 6, 8, 10, and 12 space-time dimensions relevant in Dp-branes. 
  We review and compare different variational formulations for the Schr\"{o}dinger field. Some of them rely on the addition of a conveniently chosen total time derivative to the hermitic Lagrangian. Alternatively, the Dirac-Bergmann algorithm yields the Schr\"{o}dinger equation first as a consistency condition in the full phase space, second as canonical equation in the reduced phase space. The two methods lead to the same (reduced) Hamiltonian. As a third possibility, the Faddeev-Jackiw method is shown to be a shortcut of the Dirac method. By implementing the quantization scheme for systems with second class constraints, inconsistencies of previous treatments are eliminated. 
  The problem of confinement of neutral fermions in two-dimensional space-time is approached with a pseudoscalar double-step potential in the Dirac equation. Bound-state solutions are obtained when the coupling is of sufficient intensity. The confinement is made plausible by arguments based on effective mass and anomalous magnetic interaction. 
  Pure gauge representation for general vacuum background fields (Cartan forms) in the generalized $AdS$ superspace identified with $OSp(L,M)$ is found. This allows us to formulate dynamics of free massless fields in the generalized $AdS$ space-time and to find their (generalized) conformal and higher spin field transformation laws. Generic solution of the field equations is also constructed explicitly. The results are obtained with the aid of the star product realization of ortosymplectic superalgebras. 
  Matrix descriptions of higher dimensional spherical branes are investigated. It is known that a fuzzy 2k-sphere is described by the coset space SO(2k+1)/U(k) and has some extra dimensions. It is shown that a fuzzy 2k-sphere is comprised of n^{\frac{k(k-1)}{2}} spherical D(2k-1)-branes and has a fuzzy 2(k-1)-sphere at each point. We can understand the relationship between these two viewpoints by the dielectric effect. Contraction of the algebra is also discussed. 
  We describe the classical $o(3,2)$ $r$-matrices as generating the quantum deformations of either D=3 conformal algebra with mass-like deformation parameters or D=4 $AdS$ algebra with dimensionless deformation parameters. We describe the quantization of classical $o(3,2)$ $r$-matrices via Drinfeld twist method which locates the deformation in the coalgebra sector.   Further we obtain the quantum $o(3,2)$ algebra in a convenient Hopf algebra form by considering suitable deformation maps from classical to deformed $o(3,2)$ algebra basis. It appears that if we pass from $\kappa$-deformed D =3 conformal algebra basis to the deformed D=4 $AdS$ generators basis the role of dimensionfull parameter is taken over by the $AdS$ radius $R$. We provide also the bilinear $o(3,2)$ Casimir which we express using the deformed D=3 conformal basis. 
  Fermionic-type character formulae are presented for charged irreduciblemodules of the graded parafermionic conformal field theory associated to the coset $osp(1,2)_k/u(1)$. This is obtained by counting the weakly ordered `partitions' subject to the graded $Z_k$ exclusion principle. The bosonic form of the characters is also presented. 
  We study the four-point correlator of 1/2-BPS operators of weight 4 in N=4 SYM, which are dual to massive KK modes in AdS_5 supergravity. General field-theoretic arguments lead to a partially non-renormalized form of the amplitude that depends on two a priori independent functions of the conformal cross-ratios. We explicitly compute the amplitude in the large N limit at one loop (order g^2) and in AdS_5 supergravity. Surprisingly, the one-loop result shows that the two functions determining the amplitude coincide while in the supergravity regime they are distinctly different. We discuss the possible implications of this perturbative degeneracy for the AdS/CFT correspondence. 
  We calculate, using holographic duality, the thermal two-point function in finite temperature little string theory. The analysis of those correlators reveals possible instabilities of the thermal ensemble, as in previous discussions of the thermodynamics of little string theory. We comment on the dependence of the instability on the spatial volume of the system. 
  We consider the two-dimensional model of W3-gravity within Lagrangian quantization methods for general gauge theories. We use the Batalin-Vilkovisky formalism to study the arbitrariness in the realization of the gauge algebra. We obtain a one-parametric non-analytic extension of the gauge algebra, and a corresponding solution of the classical master equation, related via an anticanonical transformation to a solution corresponding to an analytic realization. We investigate the possibility of closed solutions of the classical master equation in the Sp(2)-covariant formalism and show that such solutions do not exist in the approximation up to the third order in ghost and auxiliary fields. 
  Some simple observations concerning certain kappa-deformations in the framework of 2D dilaton gravity formulated as Poisson-sigma models are discussed, and the question of (non-)triviality is addressed. 
  We study the Nonlinear (Polynomial, N-fold,...) Supersymmetry algebra in one-dimensional QM. Its structure is determined by the type of conjugation operation (Hermitian conjugation or transposition) and described with the help of the Super-Hamiltonian projection on the zero-mode subspace of a supercharge. We show that the SUSY algebra with transposition symmetry is always polynomial in the Hamiltonian if supercharges represent differential operators of finite order. The appearance of the extended SUSY with several (complex or real) supercharges is analyzed in details and it is established that no more than two independent supercharges may generate a Nonlinear superalgebra which can be appropriately specified as {\cal N} = 2 SUSY. In this case we find a non-trivial hidden symmetry operator and rephrase it as a non-linear function of the Super-Hamiltonian on the physical state space. The full {\cal N} = 2 Non-linear SUSY algebra includes "central charges" both polynomial and non-polynomial (due to a symmetry operator) in the Super-Hamiltonian. 
  We discuss how in the presence of a nontrivial RR two-form field strength and nontrivial dilaton the conditions of preserving supersymmetry on six-dimensional manifolds lead to generalized monopole and Killing spinor equations. We show that the manifold is K\"ahler in the ten-dimensional string frame if F_0^{(1,1)}=0. We then determine explicitly the intrinsic torsion of the SU(3)-structure on six-manifolds that result via Kaluza-Klein reduction from seven-manifolds with G_2-structure of generic intrinsic torsion. Lastly we give explicitly the intrinsic torsion of the SU(3)-structure for an N=1 supersymmetric background in the presence of nontrivial RR two-form field strength and nontrivial dilaton. 
  We present an explicit relation between representations of the Virasoro algebra and polynomial martingales in stochastic Loewner evolutions (SLE). We show that the Virasoro algebra is the spectrum generating algebra of SLE martingales. This is based on a new representation of the Virasoro algebra, inspired by the Borel-Weil construction, acting on functions depending on coordinates parametrizing conformal maps. 
  We propose a generic supersymmetric and kappa-invariant action for describing coincident D0-branes with non-abelian matter fields on their worldline. The action is shown to be in agreement with the Matrix Theory limit of the ND0-brane effective action. 
  A generalized harmonic oscillator on noncommutative spaces is considered. Dynamical symmetries and physical equivalence of noncommutative systems with the same energy spectrum are investigated and discussed. General solutions of three-dimensional noncommutative harmonic oscillator are found and classified according to dynamical symmetries. We have found conditions under which three-dimensional noncommutative harmonic oscillator can be represented by ordinary, isotropic harmonic oscillator in effective magnetic field. 
  It is shown that the quantization of a superparticle propagating in an N=1, D=4 superspace extended with tensorial coordinates results in an infinite tower of massless spin states satisfying the Vasiliev unfolded equations for free higher spin fields in flat and AdS_4 N=1 superspace. The tensorial extension of the AdS_4 superspace is proved to be a supergroup manifold OSp(1|4). The model is manifestly invariant under an OSp(N|8) (N=1,2) superconformal symmetry. As a byproduct, we find that the Cartan forms of arbitrary Sp(2n) and OSp(1|2n) groups are GL(2n) flat, i.e. they are equivalent to flat Cartan forms up to a GL(2n) rotation. This property is crucial for carrying out the quantization of the particle model on OSp(1|4) and getting the higher spin field dynamics in super AdS_4, which can be performed in a way analogous to the flat case. 
  We present a simple group representation analysis of massive, and particularly ``partially massless'', fields of arbitrary spin in de Sitter spaces of any dimension. The method uses bulk to boundary propagators to relate these fields to Euclidean conformal ones at one dimension lower. These results are then used to revisit an old question: can a consistent de Sitter supergravity be constructed, at least within its intrinsic horizon? 
  We investigate dynamics of the homogeneous time-dependent SU(2) Yang-Mills fields governed by the non-Abelian Born-Infeld lagrangian which arises in superstring theory as a result of summation of all orders in the string slope parameter $\alpha'$. It is shown that generically the Born-Infeld dynamics is less chaotic than that in the ordinary Yang-Mills theory, and at high enough field strength the Yang-Mills chaos is stabilized. More generally, a smothering effect of the string non-locality on behavior of classical fields is conjectured. 
  The construction of D-branes in N=2 superconformal minimal models, based on free field realization of N=2 super-Virasoro algebra unitary modules is represented. 
  We prove that the quasiclassical tau-function of the multi-support solutions to matrix models, proposed recently by Dijkgraaf and Vafa to be related to the Cachazo-Intrilligator-Vafa superpotentials of the N=1 supersymmetric Yang-Mills theories, satisfies the Witten-Dijkgraaf-Verlinde-Verlinde equations. 
  We compute the cosmological perturbations generated in the brane world inflation driven by the bulk inflaton. Different from the model that the inflation is a brane effect, we exhibit the modification of the power spectrum of scalar perturbations due to the existence of the fifth dimension. With the change of the initial vacuum, we investigate the dependence of the correction of the power spectrum on the choice of the vacuum. 
  We introduce the relation between the holographic entropy bounds and the inflationary universe. First the holographic entropy bounds for radiation-dominated universe, radiation-dominated universe with a positive cosmological constant are introduced. For an exact de Sitter phase, we use the maximal entropy bound. We classify the inflation based on the quasi-de Sitter spacetime into three steps: slow-roll period of inflation, epoch of reheating, and radiation-dominated era. Then we study how to apply three entropy bounds to the three steps of the inflation.  Finally we discuss our results. 
  The non-perturbative (D-instanton corrected) low-energy effective action of the universal hypermultiplet in the type-IIA string theory compactified on a Calabi-Yau threefold is calculated in 4d, N=2 supergravity. The action is given by the quaternionic non-linear sigma-model whose metric is governed by the Eisenstein series E_{3/2}. The U(1)xU(1) isometry and SL(2,Z) modular invariance play the key role in our construction. 
  (R-channel) TBA is elaborated to find the effective central charge dependence on the boundary parameters for the massless boundary sine-Gordon model with the coupling constant $(8\pi) /\beta^2 = 1+ \lambda $ with $\lambda$ a positive integer. Numerical analysis of the massless boundary TBA demonstrates that at an appropriate boundary parameter range (cusp point) there exists a singularity crossing phenomena and this effect should be included in TBA to have the right behavior of the effective central charge. 
  We study Born-Infeld type effective action for unstable D3-brane system including a tachyon and an Abelian gauge field, and find the rolling tachyon with constant electric and magnetic fields as the most general homogeneous solution. Tachyonic vacua are characterized by magnitudes of the electric and magnetic fields and the angle between them. Analysis of small fluctuations in this background shows that the obtained configuration may be interpreted as a fluid consisting of string-like objects carrying electric and magnetic fields. They are stretched along one direction and the rolling tachyon move in a perpendicular direction to the strings. Direction of the propagating waves coincides with that of strings with velocity equal to electric field. 
  The renormalization-group functions of the two-dimensional n-vector \lambda \phi^4 model are calculated in the five-loop approximation. Perturbative series for the \beta-function and critical exponents are resummed by the Pade-Borel-Leroy techniques. An account for the five-loop term shifts the Wilson fixed point location only briefly, leaving it outside the segment formed by the results of the lattice calculations. This is argued to reflect the influence of the non-analytical contribution to the \beta-function. The evaluation of the critical exponents for n = 1, n = 0 and n = -1 in the five-loop approximation and comparison of the results with known exact values confirm the conclusion that non-analytical contributions are visible in two dimensions. For the 2D Ising model, the estimate \omega = 1.31 for the correction-to-scaling exponent is found. 
  Starting with a Lie algebroid ${\cal A}$ over a space $M$ we lift its action to the canonical transformations on the affine bundle ${\cal R}$ over the cotangent bundle $T^*M$. Such lifts are classified by the first cohomology $H^1({\cal A})$. The resulting object is a Hamiltonian algebroid ${\cal A}^H$ over ${\cal R}$ with the anchor map from $\G({\cal A}^H)$ to Hamiltonians of canonical transformations. Hamiltonian algebroids generalize Lie algebras of canonical transformations. We prove that the BRST operator for ${\cal A}^H$ is cubic in the ghost fields as in the Lie algebra case. The Poisson sigma model is a natural example of this construction. Canonical transformations of its phase space define a Hamiltonian algebroid with the Lie brackets related to the Poisson structure on the target space. We apply this scheme to analyze the symmetries of generalized deformations of complex structures on Riemann curves $\Si_{g,n}$ of genus $g $with $n$ marked points .We endow the space of local $\GL$-opers with the Adler-Gelfand-Dikii (AGD) Poisson brackets. It allows us to define a Hamiltonian algebroid over the phase space of $W_N$-gravity on $\Si_{g,n}$. The sections of the algebroid are Volterra operators on $\Si_{g,n}$ with the Lie brackets coming from the AGD bivector. The symplectic reduction defines the finite-dimensional moduli space of $W_N$-gravity and in particular the moduli space of the complex structures $\bp$ on $\Si_{g,n}$ deformed by the Volterra operators. 
  We study a family of kinetic operators in string field theory describing the theory around the closed string vacuum. Those operators are based on the analytical classical solutions of Takahashi and Tanimoto and are analogous to the pure ghost action usually referred to as "vacuum string field theory," but are much more general, and less singular than the pure ghost operator. The closed string vacuum is related to the D-brane vacuum by large, singular, gauge transformations or field redefinition, and all those different representations are related to each other by small gauge transformations. We try to clarify the nature of this singular gauge transformation. We also show that by choosing the Siegel gauge one recovers the propagator proposed in hep-th/0207266 that generates closed string surfaces. 
  We study partially supersymmetric plane-wave like deformations of string theories and M-theory on brane backgrounds. These deformations are dual to nonlocal field theories. We calculate various expectation values of configurations of closed as well as open Wilson loops and Wilson surfaces in those theories. We also discuss the manifestation of the nonlocality structure in the supergravity backgrounds. A plane-wave like deformation of little string theory has also been studied. 
  We develop a perturbative approach to the solution of the scalar wave equation for a large AdS black hole. In three dimensions, our method coincides with the known exact solution. We discuss the five-dimensional case in detail and apply our procedure to the Heun equation. We calculate the quasi-normal modes analytically and obtain good agreement with numerical results for the low-lying frequencies. 
  After a short presentation of KMS states and modular theory as the unifying description of thermalizing systems we propose the absence of transverse vacuum fluctuations in the holographic projections as the mechanism for an area behavior (the transverse area) of localization entropy as opposed to the volume dependence of ordinary heat bath entropy. Thermalization through causal localization is not a property of QM, but results from the omnipresent vacuum polarization in QFT and does not require a Gibbs type ensemble avaraging (coupling to a heat bath). 
  The super-AdS_5 X S^5 and the four-dimensional N=4 superconformal algebras play important roles in superstring theories. It is often discussed the roles of the osp(1|32) algebra as a maximal extension of the superalgebras in flat background. In this paper we show that the su(2,2|4), the super-AdS_5 X S^5 algebra or the superconformal algebra, is not a restriction of the osp(1|32) though the bosonic part of the former is a subgroup of the latter. There exist only two types of u(1) extension of the super-AdS_5 X S^5 algebra if the bosonic AdS_5 X S^5 covariance is imposed. Possible significance of the results is also discussed briefly. 
  Returning to the old problems in ordinary QED, by an appropriate extension of the dimensional regularization method in noncommutative space we try to provide a quite coherent look into NCQED.The renormalisation of theories, the $\beta$ function,the vacuum polarisation of photon, the general structure of vertex fermion-photon, the anomalous magnetic moment (AMM) of fermions and the validity of Ward identity at the one-loop level are reinvestigated. 
  A general method of the BRST--anti-BRST symmetric conversion of second-class constraints is presented. It yields a pair of commuting and nilpotent BRST-type charges that can be naturally regarded as BRST and anti-BRST ones. Interchanging the BRST and anti-BRST generators corresponds to a symmetry between the original second-class constraints and the conversion variables, which enter the formalism on equal footing. 
  Simulations in compact U(1) lattice gauge theory in 4D show now beyond any reasonable doubts that the phase transition separating the Coulomb from the confined phase is of first order, albeit a very weak one. This settles the issue from the numerical side. On the analytical side, it was suggested some time ago, based on the qualitative analogy between the phase diagram of such a model and the one of scalar QED obtained by soft breaking the N=2 Seiberg-Witten model down to N=0, that the phase transition should be of second order. In this work we take a fresh look at this issue and show that a proper implementation of the Seiberg-Witten model below the supersymmetry breaking scale requires considering some new radiative corrections. Through the Coleman-Weinberg mechanism this turns the second order transition into a weakly first order one, in agreement with the numerical results. We comment on several other aspects of this continuum model. 
  We analyze the nonrelativistic quantum scattering problem of a charged particle by an Abelian magnetic monopole in the background of a global monopole. In addition to the magnetic and geometric effects, we consider the influence of the electrostatic self-interaction on the charged particle. Moreover, for the specific case where the electrostatic self-interaction becomes attractive, charged particle-monopole bound system can be formed and the respective energy spectrum is hydrogen-like one. 
  The purpose of the ``bootstrap program'' for integrable quantum field theories in 1+1 dimensions is to construct a model in terms of its Wightman functions explicitly. In this article, this program is mainly illustrated in terms of the sine-Gordon and the sinh-Gordon model and (as an exercise) the scaling Ising model. We review some previous results on sine-Gordon breather form factors and quantum operator equations. The problem to sum over intermediate states is attacked in the short distance limit of the two point Wightman function for the sinh-Gordon and the scaling Ising model. 
  An explicitly exact superconformal description is provided to some classes of Type II string theories in non constant RR backgrounds. This is done by applying the manifest (2,2) approach of Berkovits and Maldacena to Type II strings and by studying the condition of exact conformal invariance of certain supersymmetric backgrounds. We find a new set of exact type IIA strings with non constant RR 2-form and 4-form curvatures and for type IIB with non constant 3-form curvature. 
  We present a three dimensional novel massive N=2 super Yang-Mills action as a low energy effective worldvolume description of the D2-branes on a pp-wave. The action contains the Myers term, mass terms for three Higgs, and terms mixing the electric fields with other two Higgs.   We derive the action in three different ways, from the M-theory matrix model, from the supermembrane action, and from the Dirac-Born-Infeld action. We verify the consistent mutual agreement and comment how each approach is complementary to another. In particular, we give the eleven dimensional geometric interpretation of the vacua in the worldvolume theory as the membranes tilted to the eleventh direction with the giant gravitons around. 
  The results of our research on noncommutative perturbative quantum field theory and its relation to string theory are exposed with details. 1) We give an introduction to noncommutative quantum field theory and its derivation from open string theory in an antisymmetric background. 2) We perform a perturbative Wilson loop calculation for 2D NCYM. We compare the LCG results for the WML and the PV prescription. With WML the loop is well-defined and regular in the commutative limit. With PV the result is singular. This is intriguing: in the commutative theory their difference is related to topological excitations, moreover PV provides a point-like potential. 3) Commutative 2D YM exhibits an interplay between geometrical and U(N) gauge properties: in the exact expression of a Wilson loop with n windings a scaling intertwines n and N. In the NC case the interplay becomes tighter due to the merging of space-time and ``internal'' symmetries. Surprisingly, in our up to O(g^6) (and beyond) crossed graphs calculations the scaling we mentioned occurs for large n, N and theta. 4) We discuss the breakdown of perturbative unitarity of noncommutative electric-type QFT in the light of strings. We consider the analytic structure of string loop two-point functions suitably continuing them off-shell, and then study the Seiberg-Witten limit. In this way we pick up how the unphysical tachyonic branch cut appears in the NC field theory. 
  We study D3-branes in an NS5-branes background defined by an arbitrary 4d harmonic function. Using a gauge-invariant formulation of Born-Infeld dynamics as well as the supersymmetry condition, we find the general solution for the $\omega$-field. We propose an interpretation in terms of the Myers effect. 
  The complete energy spectrum for the Dirac oscillator via R-deformed Heisenberg algebra is investigated. 
  These lectures provide an introduction to the subject of tachyon condensation in the open bosonic string. The problem of tachyon condensation is first described in the context of the low-energy Yang-Mills description of a system of multiple D-branes, and then using the language of string field theory. An introduction is given to Witten's cubic open bosonic string field theory. The Sen conjectures on tachyon condensation in open bosonic string field theory are introduced, and evidence confirming these conjectures is reviewed. 
  We give a model-independent derivation of general intersecting rules for spacelike branes (S-branes) in arbitrary dimensions $d$. This is achieved by directly solving bosonic field equations for supergravity coupled to a dilaton and antisymmetric tensor fields with minimal ans\"{a}tze. We compare the results with those in eleven-dimensional supergravity and other solutions. 
  We study the constraint structure of the topologically massive theory with one- and two-form fields in the framework of Batalin-Fradkin-Tyutin embedding procedure. Through this analysis we obtain a new type of Wess-Jumino action with novel symmetry, which is originated from the topological coupling term, as well as the St\"uckelberg action related to the explicit gauge breaking mass terms from the original theory. 
  An exact representation of the Euclidean fermion determinant in two dimensions for centrally symmetric, finite-ranged Abelian background fields is derived. Input data are the wave function inside the field's range and the scattering phase shift with their momenta rotated to the positive imaginary axis and fixed at the fermion mass for each partial-wave. The determinant's asymptotic limit for strong coupling and small fermion mass for square-integrable, unidirecitonal magnetic fields is shown to depend only on the chiral anomaly. The concept of duality is extended from one to two-variable fields, thereby relating the two-dimensional Euclidean determinant for a class of background magnetic fields to the pair production probability in four dimensions for a related class of electric pulses. Additionally, the ``diamagnetic'' bound on the two-dimensional Euclidean determinant is related to the negative sign of dImS_eff/dm^2 in four dimensions in the strong coupling, small mass limit, where S_eff is the one-loop effective action. 
  In this talk I discuss intersecting brane configurations coming from explicit metrics with G_2 holonomy. An example of a 7-manifold which representing a R^3 bundle over a self-dual Einstein space is described and the potential appearing after compactification over the 6-d twistor space is derived. 
  We observe a relation between closed strings tachyons and one-loop instabilities in non-supersymmetric non-commutative gauge theories. In particular we analyze the spectra of type IIB string theory on C^3/Z_N orbifold singularities and the non-commutative field theory that lives on D3 branes located at the singularity. We find a surprising correspondence between the existence or not of one-loop low-momentum instabilities in the non-commutative field theory and the existence or not of tachyons in the closed string twisted sectors. Moreover, the relevant piece of the non-commutative field theory effective action is suggestive of an exchange of closed string modes. This suggests that non-commutative field theories retain some information about the dynamics of the underlying string configuration. Finally, we also comment on a possible relation between closed string tachyon condensation and field theory tachyon condensation. 
  We discuss a formulation of quantum field theory on quantum space time where the perturbation expansion of the S-matrix is term by term ultraviolet finite.   The characteristic feature of our approach is a quantum version of the Wick product at coinciding points: the differences of coordinates q_j - q_k are not set equal to zero, which would violate the commutation relation between their components. We show that the optimal degree of approximate coincidence can be defined by the evaluation of a conditional expectation which replaces each function of q_j - q_k by its expectation value in optimally localized states, while leaving the mean coordinates (q_1 + ... + q_n)/n invariant.   The resulting procedure is to a large extent unique, and is invariant under translations and rotations, but violates Lorentz invariance. Indeed, optimal localization refers to a specific Lorentz frame, where the electric and magnetic parts of the commutator of the coordinates have to coincide *).   Employing an adiabatic switching, we show that the S-matrix is term by term finite. The matrix elements of the transfer matrix are determined, at each order in the perturbative expansion, by kernels with Gaussian decay in the Planck scale. The adiabatic limit and the large scale limit of this theory will be studied elsewhere.   --   *) S. Doplicher, K. Fredenhagen, and J.E.Roberts, Commun. Math. Phys. 172, 187 (1995) [arXiv:hep-th/0303037] 
  Unstable D-branes or brane-antibrane systems can decay to lower-dimensional branes. In the effective field theory description, the final state branes are defects in the tachyon field which describes the initial instability. We study the dynamical formation of codimension one defects (kinks) using Sen's ansatz for the tachyon Lagrangian. It is shown that the slope of the kink diverges within a finite amount of time after the tachyon starts to roll. We discuss the relevance for reheating after brane-antibrane inflation. 
  We present two applications of quantum integrable systems. First, we predict that it is possible to generate high harmonics from solid state devices by demostrating that the emission spectrum for a minimally coupled laser field of frequency $\omega$ to an impurity system of a quantum wire, contains multiples of the incoming frequency. Second, evaluating expressions for the conductance in the high temperature regime we show that the caracteristic filling fractions of the Jain sequence, which occur in the fractional quantum Hall effect, can be obtained from quantum wires which are described by minimal affine Toda field theories. 
  Decoherence associated with super-Hubble modes in de Sitter space may have a dual description, in which it is attributed to interaction of sub-Hubble modes with an ``environment'' residing just inside the observer's horizon. We present a version of such description, together with some consistency checks, which it is shown to pass. 
  A systematic construction is presented of 1/4 BPS operators in N=4 superconformal Yang-Mills theory, using either analytic superspace methods or components. In the construction, the operators of the classical theory annihilated by 4 out of 16 supercharges are arranged into two types. The first type consists of those operators that contain 1/4 BPS operators in the full quantum theory. The second type consists of descendants of operators in long unprotected multiplets which develop anomalous dimensions in the quantum theory. The 1/4 BPS operators of the quantum theory are defined to be orthogonal to all the descendant operators with the same classical quantum numbers. It is shown, to order $g^2$, that these 1/4 BPS operators have protected dimensions. 
  In this paper we analyse the non-hyperelliptic Seiberg-Witten curves derived from M-theory that encode the low energy solution of N=2 supersymmetric theories with product gauge groups. We consider the case of a SU(N_1)xSU(N_2) gauge theory with a hypermultiplet in the bifundamental representation together with matter in the fundamental representations of SU(N_1) and SU(N_2). By means of the Riemann bilinear relations that hold on the Riemann surface defined by the Seiberg--Witten curve, we compute the logarithmic derivative of the prepotential with respect to the quantum scales of both gauge groups. As an application we develop a method to compute recursively the instanton corrections to the prepotential in a straightforward way. We present explicit formulas for up to third order on both quantum scales. Furthermore, we extend those results to SU(N) gauge theories with a matter hypermultiplet in the symmetric and antisymmetric representation. We also present some non-trivial checks of our results. 
  We propose an alternative interpretation for the meaning of noncommutativity of the string-inspired field theories and quantum mechanics. Arguments are presented to show that the noncommutativity generated in the stringy context should be assumed to be only between the particle coordinate observables, and not of the spacetime coordinates. Some implications of this fact for noncomutative field theories and quantum mechanics are discussed. In particular, a consistent interpretation is given for the wavefunction in quantum mechanics. An analysis of the noncommutative theories in the Schr\"odinger formulation is performed employing a generalized quantum Hamilton-Jacobi formalism. A formal structure for noncommutative quantum mechanics, richer than the one of noncommutative quantum field theory, comes out. Conditions for the classical and commutative limits of these theories have also been determined and applied in some examples. 
  Modular theory of operator algebras and the associated KMS property are used to obtain a unified description for the thermal aspects of the standard heat bath situation and those caused by quantum vacuum fluctuations from localization. An algebraic variant of lightfront holography reveals that the vacuum polarization on wedge horizons is compressed into the lightray direction. Their absence in the transverse direction is the prerequisite to an area (generalized Bekenstein-) behavior of entropy-like measures which reveal the loss of purity of the vacuum due to restrictions to wedges and their horizons. Besides the well-known fact that localization-induced (generalized Hawking-) temperature is fixed by the geometric aspects, this area behavior (versus the standard volume dependence) constitutes the main difference between localization-caused and standard thermal behavior. 
  The continuous and discrete symmetries of the Dirac-type operators produced by particular Killing-Yano tensors are studied in manifolds of arbitrary dimensions. The Killing-Yano tensors considered are covariantly constant and realize certain square roots of the metric tensor. Such a Killing-Yano tensor produces simultaneously a Dirac-type operator and the generator of a one-parameter Lie group connecting this operator with the standard Dirac one. The Dirac operators are related among themselves through continuous or discrete transformations. It is shown that the groups of the continuous symmetry can be only U(1) and SU(2), specific to (hyper-)Kahler spaces, but arising even in cases when the requirements for these special geometries are not fulfilled. The discrete symmetries are also studied obtaining the discrete groups Z_4 and Q. The briefly presented examples are the Euclidean Taub-NUT space and the Minkowski spacetime. 
  We study the classical breaking of a highly excited (closed or open) string state on the leading Regge trajectory, represented by a rotating soliton solution, and we find the resulting solutions for the outgoing two pieces, describing two specific excited string states. This classical picture reproduces very accurately the precise analytical relation of the masses $M_1$ and $M_2$ of the decay products found in a previous quantum computation. The decay rate is naturally described in terms of a semiclassical formula. We also point out some interesting features of the evolution after the splitting process. 
  The unitarity of the NS supersymmetric coset SL(2,R)/U(1) is studied for the discrete representations. The results are applied to the proof of the no-ghost theorem for fermionic strings in AdS(3) x N in the NS sector. A no-ghost theorem is proved for states in flowed discrete representations. 
  The convergence of integrals over charge densities is discussed in relation with the problem of electric charge and (non-local) charged states in Quantum Electrodynamics (QED). Delicate, but physically relevant, mathematical points like the domain dependence of local charges as quadratic forms and the time smearing needed for strong convergence of integrals of charge densities are analyzed. The results are applied to QED and the choice of time smearing is shown to be crucial for the removal of vacuum polarization effects responible for the time dependence of the charge (Swieca phenomenon). The possibility of constructing physical charged states in the Feynman-Gupta-Bleuler gauge as limits of local states vectors is discussed, compatibly with the vanishing of the Gauss charge on local states. A modification by a gauge term of the Dirac exponential factor which yields the physical Coulomb fields from the Feynman-Gupta-Bleuler fields is shown to remove the infrared divergence of scalar products of local and physical charged states, allowing for a construction of physical charged fields with well defined correlation functions with local fields. 
  Various approaches by the author and collaborators to define gravitational fluctuations associated with a noncommutative space are reviewed. 
  Peierls brackets are part of the space-time approach to quantum field theory, and provide a Poisson bracket which, being defined for pairs of observables which are group invariant, is group invariant by construction. It is therefore well suited for combining the use of Poisson brackets and the full diffeomorphism group in general relativity. The present paper provides an introduction to the topic, with applications to gauge field theory. 
  The q-deformed kink of the $\lambda\phi^4-$model is obtained via the normalisable ground state eigenfunction of a fluctuation operator associated with the q-deformed hyperbolic functions. From such a bosonic zero-mode the q-deformed potential in 1+1 dimensions is found, and we show that the q-deformed kink solution is a kink displaced away from the origin. 
  A recently proposed method for the characterization and analysis of local equilibrium states in relativistic quantum field theory is applied to a simple model. Within this model states are identified which are locally (but not globally) in thermal equilibrium and it is shown that their local thermal properties evolve according to macroscopic equations. The largest space-time regions in which local equilibrium states can exist are timelike cones. Thus, although the model does not describe dissipative effects, such states fix in a natural manner a time direction. Moreover, generically they determine a distinguished space-time point where a singularity in the temperature (a hot bang) must have occurred if local equilibrium prevailed thereafter. The results illustrate how the breaking of the time reflection symmetry at macroscopic scales manifests itself in a microscopic setting. 
  I consider gravity induced over a smooth (finite thickness) soliton. Graviton kinetic term is coupled to bulk scalar that develops solitonic vacuum expectation value. Couplings of Kaluza-Klein modes to soliton-localized matter are suppressed, giving rise to crossover distance $r_c=M_{P}^2/M_{*}^3$ between 4D and 5D behavior. This system can be viewed as a finite thickness brane regularization of the model of Dvali, Gabadadze and Porrati. 
  After a pedagogical review of elementary cosmology, I go on to discuss some obstacles to obtaining inflationary or accelerating universes in M/String Theory. In particular, I give an account of an old No-Go Theorem to this effect. I then describe some recent ideas about the possible r\^ole of the tachyon in cosmology. I stress that there are many objections to a naive inflationary model based on the tachyon, but there remains the possiblity that the tachyon was important in a possible pre-inflationary Open-String Era preceding our present Closed String Era. 
  We present two classes of regular supergravity backgrounds dual to supersymmetric and non-supersymmetric gauge theories living on the world-volume of wrapped branes. In particular we consider the Maldacena Nunez and the Klebanov Strassler models, describing N=1 and N=2 theories, and find their non supersymmetric generalization by explicitly solving the second order equation of motion. We also study various aspects of these solutions, including the supersymmetry breaking issue and the vacuum energy. 
  We study the classical decay of unstable scalar solitons in noncommutative field theory in 2+1 dimensions. This can, but does not have to, be viewed as a toy model for the decay of D-branes in string theory. In the limit that the noncommutativity parameter \theta is infinite, the gradient term is absent, there are no propagating modes and the soliton does not decay at all. If \theta is large, but finite, the rotationally symmetric decay channel can be described as a highly excited nonlinear oscillator weakly coupled to a continuum of linear modes. This system is closely akin to those studied in the context of discrete breathers. We here diagonalize the linear problem and compute the decay rate to first order using a version of Fermi's Golden Rule, leaving a more rigorous treatment for future work. 
  M(atrix) theory in PP-wave background possesses a discrete set of classical vacua, all of which preserves 16 supersymmetry and interpretable as collection of giant gravitons. We find Euclidean instanton solutions that interpolate between them, and analyze their properties. Supersymmetry prevents direct mixing between different vacua but still allows effect of instanton to show up in higher order effective interactions, such as analog of v^4 interaction of flat space effective theory. An explicit construction of zero modes is performed, and Goldstone zero modes, bosonic and fermionic, are identified. We further generalize this to massive M(atrix) theory that includes fundamental hypermultiplets, corresponding to insertion of longitudinal fivebranes in the background. After a brief comparison to their counterpart in AdS\times S, we close with a summary. 
  We present a formula for a classical $r$-matrix of an integrable system obtained by Hamiltonian reduction of some free field theories using pure gauge symmetries. The framework of the reduction is restricted only by the assumption that the respective gauge transformations are Lie group ones. Our formula is in terms of Dirac brackets, and some new observations on these brackets are made. We apply our method to derive a classical $r$-matrix for the elliptic Calogero-Moser system with spin starting from the Higgs bundle over an elliptic curve with marked points. In the paper we also derive a classical Feigin-Odesskii algebra by a Poisson reduction of some modification of the Higgs bundle over an elliptic curve. This allows us to include integrable lattice models in a Hitchin type construction. 
  We study the classical stability of a class of S-brane geometries having cosmological horizons. By considering the perturbations of the metric in these geometries we establish that their horizons are unstable in the sense that an observer trying to cross the horizon experiences an infinite flux of radiation at the instant of crossing. The backreaction of this radiation is likely to convert the horizons into curvature singularities, similar to the instability of the internal Cauchy horizon of the Reissner-Nordstrom black hole. We also compute the particle production by the time-dependent fields in the future regions of these geometries, and find that the spectrum of produced particles is thermal, with temperature coinciding with the Hawking temperature computed by euclideanizing the metric in the static region. Possible implications of these results are discussed. 
  A holographic interpretation for some specific Ricci flat string backgrounds of the form $A_6\times C_4$ is proposed. The conjecture is that there is a Four-dimensional Euclidean Conformal Field Theory (ECFT) defined on a codimension two {\em submanifold} of the manifold $A_6$ (where one of the two remaining {\em holographic} coordinates of $A_6$ is timelike, and the other one spacelike), with central charge proportional to the radius of curvature of the six-dimensional manifold, $c\sim l^4$. 
  By analyzing the odd parity part of the gauge field two and three point vertex functions, the one-loop radiative correction to the Chern-Simons coefficient is computed in noncommutative Chern-Simons-Higgs model at zero and at high temperature. At high temperature, we show that the static limit of this correction is proportional to $T$ but the first noncommutative correction increases as $T\log T$. Our results are analytic functions of the noncommutative parameter. 
  We study the heterotic string compactified on K3 x T^2 near the line T=U, where the effective action becomes singular due to an SU(2) gauge symmetry enhancement. By `integrating in' the light W^\pm vector multiplets we derive a quantum corrected effective action which is manifestly SU(2) invariant and non-singular. This effective action is found to be consistent with a residual SL(2,Z) quantum symmetry on the line T=U. In an appropriate decompactification limit, we recover the known SU(2) invariant action in five dimensions. 
  We derive the low energy effective action of the STU-model in four and five dimensions near the line T=U, where SU(2) gauge symmetry enhancement occurs. By `integrating in' the light W^{\pm} bosons together with their superpartners, the quantum corrected effective action becomes non-singular at T=U and manifestly SU(2) invariant. The four-dimensional theory is found to be consistent with modular invariance and the five-dimensional decompactification limit. This talk concisely summarizes results that have been obtained in collaboration with J. Louis and T. Mohaupt. 
  Supersymmetry is considered in spaces of constant curvature (spherical, de Sitter and Anti-de Sitter spaces) of two, three and four dimensions. 
  The paradigmatic Unruh radiation is an ideal and simple case of stationary vacuum radiation patterns related to worldlines defined as Frenet-Serret curves. We review the corresponding body of literature as well as the experimental proposals that have been suggested to detect these types of quantum field radiation patterns. Finally, we comment on a few other topics related to the Unruh effect 
  We consider toy cosmological models in which a classical, homogeneous, spinor field provides a dominant or sub-dominant contribution to the energy-momentum tensor of a flat Friedmann-Robertson-Walker universe. We find that, if such a field were to exist, appropriate choices of the spinor self-interaction would generate a rich variety of behaviors, quite different from their widely studied scalar field counterparts. We first discuss solutions that incorporate a stage of cosmic inflation and estimate the primordial spectrum of density perturbations seeded during such a stage. Inflation driven by a spinor field turns out to be unappealing as it leads to a blue spectrum of perturbations and requires considerable fine-tuning of parameters. We next find that, for simple, quartic spinor self-interactions, non-singular cyclic cosmologies exist with reasonable parameter choices. These solutions might eventually be incorporated into a successful past- and future-eternal cosmological model free of singularities. In an Appendix, we discuss the classical treatment of spinors and argue that certain quantum systems might be approximated in terms of such fields. 
  We consider the entropy bounds recently conjectured by Fischler, Susskind and Bousso, and proven in certain cases by Flanagan, Marolf and Wald (FMW). One of the FMW derivations supposes a covariant form of the Bekenstein entropy bound, the consequences of which we explore. The derivation also suggests that the entropy contained in a vacuum spacetime, e.g. Schwarzschild, is related to the shear on congruences of null rays. We find evidence for this intuition, but in a surprising way. We compare the covariant entropy bound to certain earlier discussions of black hole entropy, and comment on the separate roles of quantum mechanics and gravity in the entropy bound. 
  We argue that the quantized non-Abelian gauge theory can be obtained as the infrared limit of the corresponding classical gauge theory in a higher dimension. We show how the transformation from classical to quantum field theory emerges, and calculate Planck's constant from quantities defined in the underlying classical gauge theory. 
  Charged particles in static electric and magnetic fields have Landau levels and tunneling states from the vacuum. Using the instanton method of Phys. Rev. D 65, 105002 (2002), we obtain the formulae for the pair-production rate in spinor and scalar QED, which sum over all Landau levels and recover exactly the well-known results. The pair-production rates are calculated for an electric field of finite extent, with a constant magnetic field also present, and are shown to have boundary effects. 
  Noncommutative Chern-Simons theory can be classically mapped to commutative Chern-Simons theory by the Seiberg-Witten map. We provide evidence that the equivalence persists at the quantum level by computing two and three-point functions of field strengths on the commutative side and their Seiberg-Witten transforms on the noncommutative side to the first nontrivial order in perturbation theory. 
  Conserved operator quantities in quantum field theory can be defined via the Noether theorem in the Lagrangian formalism and as generators of some transformations. These definitions lead to generally different conserved operators which are suitable for different purposes. Some relations involving conserved operators are analyzed. 
  We consider CIV-DV prepotential F for N=1 SU(n) SYM theory at the extremum of the effective superpotential and prove the relation $2F-S dF/dS = - 2 u_2 Lambda^2n /(n^2-1)$ 
  We describe an expedient way to derive the CIV-DV prepotential in power series expansion in S_i. This is based on integrations of equations for its derivatives with respect to additional (Whitham) moduli T_m. For illustrative purposes, we calculate explicitly the leading terms of the expansion and explicitly check some components of the WDVV equations to the leading order. Extension to any higher order is simple and algorithmic. 
  We derive equations of motion for the tachyon field living on an unstable non-BPS D-brane in the level truncated open cubic superstring field theory in the first non-trivial approximation. We construct a special time dependent solution to this equation which describes the rolling tachyon. It starts from the perturbative vacuum and approaches one of stable vacua in infinite time. We investigate conserved energy functional and show that its different parts dominate in different stages of the evolution. We show that the pressure for this solution has its minimum at zero time and goes to minus energy at infinite time. 
  We study inflation and reheating in a brane world model derived from Type IIA string theory. This particular setup is based on a model of string mediated supersymmetry breaking. The inflaton is one of the transverse scalars of a D4-brane which has one of its spatial dimensions stretched between two NS5-branes, so that it is effectively three-dimensional. This D4-brane is attracted to a D6-brane that is separated from the 5-branes by a fixed amount. The potential of the transverse scalar due to the D4/D6 interaction makes a good inflaton potential. As the D4-brane slides along the two 5-branes towards the 6-brane it begins to oscillate near the minimum of the potential. The inflaton field couples to the massless Standard Model fields through Yukawa couplings. In the brane picture these couplings are introduced by having another D6'-brane intersect the D4-brane such that the 4-6' strings, whose lowest lying modes are the Standard Model matter, couple to the scalar mode of the 4-4 strings, the inflaton. The inflaton can decay into scalar and spinor particles on the 4-6' strings, reheating the universe. Observational data is used to place constraints on the parameters of the model. 
  We study compactifications of Type IIB string theory on a K3 \times T^2/Z_2 orientifold in the presence of RR and NS flux. We find the most general supersymmetry preserving, Poincare invariant, vacua in this model. All the complex structure moduli and some of the Kahler moduli are stabilised in these vacua. We obtain in an explicit fashion the restrictions imposed by supersymmetry on the flux, and the values of the fixed moduli. Some T-duals and Heterotic duals are also discussed, these are non-Calabi-Yau spaces. A superpotential is constructed describing these duals. 
  We examine the possibility of rolling tachyon to play the dual roll of inflaton at early epochs and dark matter at late times. We argue that enough inflation can be generated with the rolling tachyon either by invoking the large number of branes or brane world assisted inflation. However, reheating is problematic in this model. 
  We investigate the weak-interaction emission of spin-1/2 fermions from decaying (and non-decaying) particles endowed with uniform circular motion. The decay of swirling protons and the neutrino- antineutrino emission from circularly moving electrons are analyzed in some detail. The relevance of our results to astrophysics is commented. 
  We study dynamics of rolling tachyon and Abelian gauge field on unstable D-branes, of which effective action is given by Born-Infeld type nonlocal action. Possible cosmological evolutions are also discussed. In the Einstein frame of string cosmology, every expanding flat universe is proven to be decelerating. 
  Summations and relations involving the Hurwitz and Riemann zeta-functions are extended first to Barnes zeta-functions and then to zeta-functions of general type. The analysis is motivated by the evaluation of determinants on spheres which are treated both by a direct expansion method and by regularised sums. Comments on existing calculations are made. A Kaluza--Klein technique is introduced providing a determinant interpretation of the Glaisher-Kinkelin-Bendersky constants which are then generalised to arbitrary zeta-functions. This technique allows an improved treatment of sphere determinants. 
  We analyse possible cosmological scenarios on a brane where the brane acts as a dynamical boundary of various black holes with anti-de Sitter or de Sitter asymptotics. In many cases, the brane is found to describe completely non-singular universe. In some cases, quantum gravity era of the brane-universe can also be avoided by properly tuning bulk parameters. We further discuss the creation of a brane-universe by studying its wave function. This is done by employing Wheeler-De Witt equation in the mini superspace formalism. 
  The method of scaling algebras, which has been introduced earlier as a means for analyzing the short-distance behaviour of quantum field theories in the setting of the model-independent, operator-algebraic approach, is extended to the case of fields carrying superselection charges. A criterion for the preservance of superselection charges in the short-distance scaling limit is proposed. Consequences of this preservance of superselection charges are studied. The conjugate charge of a preserved charge is also preserved, and the preservance of all charges of a quantum field theory in the scaling limit leads to equivalence of local and global intertwiners between superselection sectors. 
  Chern--Simons type Lagrangians in $d=3$ dimensions are analyzed from the point of view of their covariance and globality. We use the transgression formula to find out a new fully covariant and global Lagrangian for Chern--Simons gravity: the price for establishing globality is hidden in a bimetric (or biconnection) structure. Such a formulation allows to calculate from a global and simpler viewpoint the energy-momentum complex and the superpotential both for Yang--Mills and gravitational examples. 
  We demonstrate the existence of a class of N=1 supersymmetric nonperturbative vacua of Horava-Witten M-theory compactified on a torus fibered Calabi-Yau 3-fold Z with first homotopy group \pi_{1}(Z)= Z2, having the following properties: 1) SO(10) grand unification group, 2) net number of three generations of chiral fermions in the observable sector, and 3) potentially viable matter Yukawa couplings. These vacua correspond to semistable holomorphic vector bundles V_{Z} over Z having structure group SU(4)_C, and generically contain M5-branes in the bulk space. The nontrivial first homotopy group allows Wilson line breaking of the SO(10) symmetry. Additionally, we propose how the 11-dimensional Horava-Witten M-theory framework may be used to extend the perturbative calculation of the top quark Yukawa coupling in the realistic free-fermionic models to the nonperturbative regime. The basic argument being that the relevant coupling couples twisted-twisted-untwisted states and can be calculated at the level of the Z2 X Z2 orbifold without resorting to the full three generation models. 
  The most fruitful approach to studying low energy soliton dynamics in field theories of Bogomol'nyi type is the geodesic approximation of Manton. In the case of vortices and monopoles, Stuart has obtained rigorous estimates of the errors in this approximation, and hence proved that it is valid in the low speed regime. His method employs energy estimates which rely on a key coercivity property of the Hessian of the energy functional of the theory under consideration. In this paper we prove an analogous coercivity property for the Hessian of the energy functional of a general sigma model with compact K\"ahler domain and target. We go on to prove a continuity property for our result, and show that, for the CP^1 model on S^2, the Hessian fails to be globally coercive in the degree 1 sector. We present numerical evidence which suggests that the Hessian is globally coercive in a certain equivariance class of the degree n sector for n>1. We also prove that, within the geodesic approximation, a single CP^1 lump moving on S^2 does not generically travel on a great circle. 
  We treat the two-dimensional Achucarro-Ortiz black hole (also known as (1+1) dilatonic black hole) as a Casimir-type system. The stress tensor of a massless scalar field satisfying Dirichlet boundary conditions on two one-dimensional "walls" ("Dirichlet walls") is explicitly calculated in three different vacua. Without employing known regularization techniques, the expression in each vacuum for the stress tensor is reached by using the Wald's axioms. Finally, within this asymptotically non-flat gravitational background, it is shown that the equilibrium of the configurations, obtained by setting Casimir force to zero, is controlled by the cosmological constant. 
  We compute the anomalous dimensions of BMN operators with two covariant derivative impurities at the planar level up to first order in the effective coupling lambda'. The result equals those for two scalar impurities as well as for mixed scalar and vector impurities given in the literature. Though the results are the same, the computation is very different from the scalar case. This is basically due to the existence of a non-vanishing overlap between the derivative impurity and the ``background'' field Z. We present details of these differences and their consequences. 
  Stability of AdS space allows scalar fields to have negative mass squared as long as the Breitenlohner-Freedman bound is satisfied. In a compactification of AdS instead, to avoid instabilities, a tachyonic bulk mass must be supplemented by appropriate brane actions. In this paper we clarify the meaning of the lower bound in the Randall-Sundrum scenario with two branes and explain how the instability disappears in the infinite space limit. A CFT interpretation is also given as radiative symmetry breaking. 
  We present an approach to the construction of action principles for differential equations, and apply it to field theory in order to construct systematically, for integrable equations which are based on a Nijenhuis (or hereditary) operator, a ladder of action principles which is complementary to the well-known multi-Hamiltonian formulation. We work out results for the Korteweg-de Vries (KdV) equation, which is a member of the positive hierarchy related to a hereditary operator. Three negative hierarchies of (negative) evolution equations are defined naturally from the hereditary operator as well, in the context of field theory. The Euler-Lagrange equations arising from the action principles are equivalent to the original evolution equation + deformations, which are obtained in terms of the positive and negative evolution vectors. We recognize the Liouville, Sinh-Gordon, Hunter-Zheng and Camassa-Holm equations as negative equations. The ladder for KdV is directly mappable to a ladder for any of these negative equations and other positive equations (e.g., the Harry-Dym and a special case of the Krichever-Novikov equations): a new nonlocal action principle for the deformed system Sinh-Gordon + spatial translation vector is presented. Several nonequivalent, nonlocal time-reparametrization invariant action principles for KdV are constructed. Hamiltonian and Symplectic operators are obtained in factorized form. Alternative Lax pairs for all negative flows are constructed, using the flows and the hereditary operator as only input. From this result we prove that all positive and negative equations in the hierarchies share the same sets of local and nonlocal constants of the motion for KdV, which are explicitly obtained using the local and nonlocal action principles for KdV. 
  Motivated by ultra-high-energy cosmic ray physics, we discuss all the possible alternatives to the familiar Lorentz transformations of the momentum and the energy of a particle. Starting from natural physical requirements, we exclude all the possibilities, apart from the ones which arise from the usual four-vector transformations by means of a change of coordinates in the mass-shell. This result confirms the remark, given in a preceding paper, that, in a theory without preferred inertial frames, one can always define a linearly transforming energy parameter to which the GZK cutoff argument can be applied. We also discuss the connections between the conservation and the transformation properties of energy-momentum and the relation between energy-momentum and velocity. 
  Recently it has been demonstrated by Dienes and Mafi, that the physics of toroidal compactified models of extra dimensions can depend on the shape angle of the torus. Toroidal compactification has also recently been used as a regulator for numerical solutions of supersymmetric fields theories in 2+1 dimensions. The question is; does the shape angle of the torus also affect the physics in this situation? Clearly a numerical solution should be independent of the shape of the space we compactify on. We show that within the context of standard DLCQ, that toroidal compactification is only allowed for a specific set of shape angles and for that set of shape angles the numerical solutions are unchanged. 
  We discuss the instability of the Veneziano-Yankielowicz effective action (or its supersymmetric ground-state) with respect to higher order derivative terms. As such terms must be present in an effective action, the V-Y action alone cannot describe the dynamics of SYM consistently. We introduce an extension of this action, where all instabilities are removed by means of a much richer structure of the Kaehler potential. We demonstrate that the dominant contributions to the effective potential are determined by the non-holomorphic part of the action and we prove that the non-perturbative ground-state can be equipped with stable dynamics. Making an expansion near the resulting ground-state to second order in the derivatives never leads back to the result by Veneziano and Yankielowicz. As a consequence new dynamical effects arise, which are interpreted as the formation of massive states in the boson sector (glueballs) and are accompanied by dynamical supersymmetry breaking. As this regime of the dynamics is not captured by standard semi-classical analysis (instantons etc.), our results do not contradict these calculations but investigate the physics of the system beyond these approximations. 
  In this first of a series of four articles, it is shown how a hamiltonian quantum dynamics can be formulated based on a generalization of classical probability theory using the notion of quasi-invariant measures on the classical phase space for our description of dynamics. This is based on certain distributions rather than invariant Gibbs measures. The first quantization is by functorial analytic continuation of real probability amplitudes, mathematically effecting the introduction of correlation between otherwise independent subsystems, and whose physical consequence is the incorporation of Breit-Wigner resonances associated to Gamow vectors into our description of dynamics. The resulting quantum dynamics admits a natural field theory interpretation. This basic formalism will be employed in subsequent installments of the series. 
  In [1] (hep-th/0211069), the author has discussed the quantum parameter space of the N=1 super Yang-Mills theory with one adjoint Higgs field Phi, tree-level superpotential W_tree = m (Phi^2)/2 + g (Phi^3)/3$, and gauge group U(Nc). In particular, full details were worked out for U(2) and U(3). By discussing higher rank gauge groups like U(4), for which the classical parameter space has a large number of disconnected components, we show that the phenomena discussed in [1] are generic. It turns out that the quantum space is connected. The classical components are related in the quantum theory either through standard singularities with massless monopoles or by branch cuts without going through any singularity. The branching points associated with the branch cuts correspond to new strong coupling singularities, which are not associated with vanishing cycles in the geometry, and at which glueballs can become massless. The transitions discussed recently by Cachazo, Seiberg and Witten are special instances of those phenomena. 
  In this second of a series of four articles, a pair of quantized free oscillators is transformed into a resonant system of coupled oscillators by analytic continuation which is performed algebraically by the group of complex symplectic transformations, creating dynamical representations of numerous semi-groups from the Hamiltonian free system. The free oscillators are the quantum analogue of action angle variable solutions for the coupled oscillators and quantum resonances, including Breit-Wigner resonances. Among the exponentially decaying Breit-Wigner resonances are hamiltonian systems in which energy transfers from one oscillator to the other. There are significant mathematical constraints in order that complex spectra be accommodate in a well defined formalism, which may be met by using the commutative real algebra $\C (1,i)$ as the ring of scalars in place of the field of complex numbers. By including distributional solutions to the Schr{\"o}dinger equation, placing us in a rigged Hilbert space, and by using the Hamiltonian as a generator of canonical transformation of the space of states, the Schr{\"o}dinger equation is the equation for parallel transport of generalized energy eigenvectors, explicitly establishing the Hamiltonian as the generator of dynamical time translations in this formalism. 
  In this third of a series of four articles, we continue the study of the representations of the dynamical transformations of systems of correlated quantized oscillators. By including generalized wave function solutions to Schr{\"o}dinger's equation (belonging to a rigged Hilbert space), and by considering the algebra of observables as a whole, the presence of Devaney chaos, hyperbolic quasi-invariant measures, torus actions, ergodicity and entropy generation associated to the non-invertible decay of Gamow vectors and their associated to Breit-Wigner resonances is shown. A weak (local) form of the second law of thermodynamics is demonstrated through the decay of resonances. Both coherence formation and decoherence (decay) are associated with irreversibility and may be associated with entropy growth, which is due to the dynamical time evolution of resonances. Hilbert space is the manifold of stationary states. There is a fractal structure associated with dynamical time evolution of resonances in the space of generalized states, and the statistical nature of the exponential decay may be identified with quasi-trapping. Equilibrium states may be regarded as strange attractors with respect to dynamical time evolution. 
  The hamiltonian quantum dynamical structures in the Gel'fand triplets of spaces used in preceding installments to describe correlated hamiltonian dynamics on phase space by quasi-invariant measures are shown to possess a covering structure, which is constructed explicitly using the properties of Clifford algebras. The unitary Clifford algebras are constructed from the intersection of the orthogonal and common symplectic (Weyl) Clifford algebras of the complexification of canonical phase space. A well defined spin geometry exists for the unitary Clifford algebras. Unitary Clifford algebras provide bosonic and fermionic representations through alternative topological completions of the same structure, and physically represent the stable states of the system. Unitary Clifford algebras are used to define dynamical gauge bundles for two, three and four correlated (unified) fields. The generic dynamical gauge group for four pairs of canonical variables is shown to be equivalent to $U(4)\times U(4)$, with the spectrum effectively determined by $S[U(4)\times U(3)]$ due to the constraint of geodesic transport of the generators of the dynamical group. It is conjectured this is an unified version of U(4) gauge gravity. An isomorphism is shown explicitly demonstrating the ability to associate these structures over four pairs of canonical variables with covariant structures in a non-trivial spacetime with $(+,-,-,-)$ local signature. The covariance of the identity of elementary particles follows, and is of dynamical origin. An area of fundamental conflict exists between notions of noncompact hamiltonian dynamics and general covariance, and a resolution is proposed for the present constructions. 
  We study new compactifications of the SO(32) heterotic string theory on compact complex non-Kahler manifolds. These manifolds have many interesting features like fewer moduli, torsional constraints, vanishing Euler character and vanishing first Chern class, which make the four-dimensional theory phenomenologically attractive. We take a particular compact example studied earlier and determine various geometrical properties of it. In particular we calculate the warp factor and study the sigma model description of strings propagating on these backgrounds. The anomaly cancellation condition and enhanced gauge symmetry are shown to arise naturally in this framework, if one considers the effect of singularities carefully.   We then give a detailed mathematical analysis of these manifolds and construct a large class of them. The existence of a holomorphic (3,0) form is important for the construction. We clarify some of the topological properties of these manifolds and evaluate the Betti numbers. We also determine the superpotential and argue that the radial modulus of these manifolds can actually be stabilized. 
  We study closed string tachyon condensation on general non-supersymmetric orbifolds of C^2. Extending previous analyses on Abelian cases, we present the classification of quotients by discrete finite subgroups of GL(2; C) as well as the generalised Hirzebruch-Jung continued fractions associated with the resolution data. Furthermore, we discuss the intimate connexions with certain generalised versions of the McKay Correspondence. 
  The generalized supersymmetries admitting abelian bosonic tensorial central charges are classified in accordance with their division algebra structure (over ${\bf R}$, ${\bf C}$, ${\bf H}$ or ${\bf O}$). It is shown in particular that in D=11 dimensions, the $M$-superalgebra admits a consistent octonionic formulation, involving 52 real bosonic generators (in place of the 528 of the standard $M$-superalgebra). The octonionic $M5$ (super-5-brane) sector coincides with the octonionic $M1$ and $M2$ sectors, while in the standard formulation these sectors are all independent. The octonionic conformal and superconformal $M$-algebras are explicitly constructed. They are respectively given by the $Sp(8|{\bf O})$ ($OSp(1,8|{\bf O})$) (super)algebra of octonionic-valued (super)matrices, whose bosonic subalgebra consists of 232 (and respectively 239) generators. 
  We study aspects of superstring vacua of non-compact special holonomy manifolds with conical singularities constructed systematically using soluble N = 1 superconformal field theories (SCFT's). It is known that Einstein homogeneous spaces G/H generate Ricci flat manifolds with special holonomies on their cones R_+ x G/H, when they are endowed with appropriate geometrical structures, namely, the Sasaki-Einstein, tri-Sasakian, nearly Kahler, and weak G_2 structures for SU(n), Sp(n), G_2, and Spin(7) holonomies, respectively. Motivated by this fact, we consider the string vacua of the type: R^{d-1,1} x (N = 1 Liouville) x (N=1 supercoset CFT on G/H) where we use the affine Lie algebras of G and H in order to capture the geometry associated to an Einstein homogeneous space G/H. Remarkably, we find the same number of spacetime and worldsheet SUSY's in our ``CFT cone'' construction as expected from the analysis of geometrical cones over G/H in many examples. We also present an analysis on the possible Liouville potential terms (cosmological constant type operators) which provide the marginal deformations resolving the conical singularities. 
  An inflationary brane model driven by a bulk inflaton with exponential potential is proposed. We find a family of exact solutions that describe power-law inflation on the brane. These solutions enable us to derive exact solutions for metric perturbations analytically. By calculating scalar and tensor perturbations, we obtain a spectrum of primordial fluctuations at the end of the inflation. The amplitudes of scalar and tensor perturbations are enhanced in the same way if the energy scale of the inflation is sufficiently higher than the tension of the brane. Then the relative amplitude of scalar and tensor perturbations is not suppressed even for high-energy inflation. This is a distinguishable feature from the inflation model driven by inflaton on the brane where tensor perturbations are suppressed for high-energy inflation. We also point out that massive Kaluza-Klein modes are not negligible at high-frequencies on 3-space of our brane. 
  A graphical representation of supersymmetry is presented. It clearly expresses the chiral flow appearing in SUSY quantities, by representing spinors by {\it directed lines} (arrows). The chiral suffixes are expressed by the directions (up, down, left, right) of the arrows. The SL(2,C) invariants are represented by {\it wedges}. Both the Weyl spinor and the Majorana spinor are treated. We are free from the messy symbols of spinor suffixes. The method is applied to the 5D supersymmetry. Many applications are expected. The result is suitable for coding a computer program and is highly expected to be applicable to various SUSY theories (including Supergravity) in various dimensions. 
  Calculations of the two-loop $\beta$-function for N=1 supersymmetric electrodynamics are compared for regularizations by higher derivatives and by the dimensional reduction. The renormalized effective action are found to be the same for both regularizations. However, unlike the dimensional reduction, the higher derivative regularization does not lead to anomaly puzzle, because it allows to perform correct calculation of diagrams with insertions of counterterms. In particular, using this method a contribution of diagrams with insertions of counterterms is calculated exactly to all orders. This contribution appears to be 0 if the theory is regularized by the dimensional reduction. We argue, that this result follows from mathematical inconsistency of the dimensional reduction and is responsible for the anomaly puzzle. 
  We study cosmological aspects of braneworld models with a warped dimension and an arbitrary number of compact dimensions. With a stabilized radion, a number of different cosmological bulk solutions are found in a general case. Both one and two brane models are considered. The Friedmann equation is calculated in each case. Particular attention is paid to six dimensional models where we find that the usual Friedmann equation can typically be recovered without fine-tuning. 
  In this paper we consider a charged massless scalar quantum field operator in the spacetime of an idealized cosmic string, i.e., an infinitely long, straight and static cosmic string, which presents a magnetic field confined in a cylindrical tube of finite radius. Three distinct situations are taking into account in this analysis: {\it{i)}} a homogeneous field inside the tube, {\it{ii)}} a magnetic field proportional to $1/r$ and {\it{iii)}} a cylindrical shell with $\delta$-function. In these three cases the axis of the infinitely long tube of radius $R$ coincides with the cosmic string. In order to study the vacuum polarization effects outside the tube, we explicitly calculate the Euclidean Green function associated with this system for the three above situations, considering points in the region outside the tube. 
  It is shown that three series of diagrams entering the calculation of some hot $QCD$ process, are mass (or collinear) singularity free, indeed. This generalizes a result which was recently established up to the third non trivial order of (thermal) Perturbation Theory. 
  The Maldacena-Nunez solution is generalized to include a number of integration constants, one of which controls the resolution of the singularity of the wrapped D5-brane background. Some features of the dual pure N=1 super Yang-Mills (SYM) theory are calculated, amongst which the gluino condensate, the beta function of the gauge coupling and a brane probe potential, which is related to the Veneziano-Yankielowicz effective potential. Each integration constant has a precise meaning in the dual SYM theory, e.g., the amount of non-perturbative SYM physics captured by the gravity configuration is described by the singularity resolution parameter. 
  We study a six-dimensional braneworld model with infinite warped extra dimensions in the case where the four-dimensional brane is described by a topological vortex of a U(1) symmetry-breaking Abelian Higgs model in presence of a negative cosmological constant. A detailed analysis of the microscopic parameters leading to a finite volume space-time in the extra dimensions is numerically performed. As previously shown, we find that a fine-tuning is required to avoid any kind of singularity on the brane. We then discuss the stability of the vortex by investigating the scalar part of the gauge-invariant perturbations around this fine-tuned configuration. It is found that the hyperstring forming Higgs and gauge fields, as well as the background metric warp factors, cannot be perturbed at all, whereas transverse modes can be considered stable. The warped space-time structure that is imposed around the vortex thus appears severely constrained and cannot generically support nonempty universe models. The genericness of our conclusions is discussed; this will shed some light on the possibility of describing our space-time as a general six-dimensional warped braneworld. 
  We give a new derivation of the quasinormal frequencies of Schwarzschild black holes in d>=4 and Reissner-Nordstrom black holes in d=4, in the limit of infinite damping. For Schwarzschild in d>=4 we find that the asymptotic real part is T_Hawking.log(3) for scalar perturbations and for some gravitational perturbations; this confirms a result previously obtained by other means in the case d=4. For Reissner-Nordstrom in d=4 we find a specific generally aperiodic behavior for the quasinormal frequencies, both for scalar perturbations and for electromagnetic-gravitational perturbations. The formulae are obtained by studying the monodromy of the perturbation analytically continued to the complex plane; the analysis depends essentially on the behavior of the potential in the "unphysical" region near the black hole singularity. 
  We propose an exact expression for the unintegrated form of the star gauge invariant axial anomaly in an arbitrary even dimensional gauge theory. The proposal is based on the inverse Seiberg-Witten map and identities related to it, obtained earlier by comparing Ramond-Ramond couplings in different decsriptions. The integrated anomalies are expressed in terms of a simplified version of the Elliott formula involving the noncommutative Chern character. These anomalies, under the Seiberg-Witten transformation, reduce to the ordinary axial anomalies. Compatibility with existing results of anomalies in noncommutative theories is established. 
  For a quantum mechanical system with broken supersymmetry, we present a simple method of determining the ground state when the corresponding energy eigenvalue is sufficiently small. A concise formula is derived for the approximate ground state energy in an associated, well-separated, asymmetric double-well-type potential. Our discussion is also relevant for the analysis of the fermion bound state in the kink-antikink scalar background. 
  We study the noncommutative version of the extended ADHM construction in the eight dimensional U(1) Yang-Mills theory. This construction gives rise to the solutions of the BPS equations in the Yang-Mills theory, and these solutions preserve at least 3/16 of supersymmetries. In a wide subspace of the extended ADHM data, we show that the integer $k$ which appears in the extended ADHM construction should be interpreted as the $D4$-brane charge rather than the $D0$-brane charge by explicitly calculating the topological charges in the case that the noncommutativity parameter is anti-self-dual. We also find the relationship with the solution generating technique and show that the integer $k$ can be interpreted as the charge of the $D0$-brane bound to the $D8$-brane with the $B$-field in the case that the noncommutativity parameter is self-dual. 
  It is argued that a phenomenologically viable grand unification model from superstring is $SU(3)^3$, the simplest gauge group among the grand unifications of the electroweak hypercharge embedded in semi-simple groups. We construct a realistic 4D $SU(3)^3$ model with the GUT scale $\sin^2\theta_W^0= \frac38$ in a $Z_3$ orbifold with Wilson line(s). By two GUT scale vacuum expectation values, we obtain a rank 4 supersymmetric standard model below the GUT scale, and predict three more strange families. 
  We study semi-classically the dynamics of string solitons in the Maldacena-Nunez background, dual in the infra-red to N=1, d=4 SYM. For closed string configurations rotating in the S^2 x R space wrapped by the stack of N D-branes we find a behavior that indicates the decoupling of the stringy Kaluza-Klein modes with sufficiently large SO(3) quantum numbers. We show that the spectrum of a pulsating string configuration in S^2 coincides with that of a N=2 super Sine-Gordon model. Closed string configurations spinning in the transversal S^3 give a relation of the energy and the conserved angular momentum identical to that obtained for configurations spinning in the S^5 of the AdS_5 x S^5, dual to N =4 SYM. In order to obtain non-trivial relations between the energy and the spin, we also consider conical-like configurations stretching along a radial variable in the unwrapped directions of the system of D-branes and simultaneously along the transversal direction. We find that in this precise case, these configurations are unstable --contrary to other backgrounds, where we show that they are stable. We point out that in the Poincare-like coordinates used for the Maldacena-Nunez background it seems that it is not possible to reproduce the well-known field theory relation between the energy and the angular momentum. We reach a similar conclusion for the Klebanov-Strassler background, by showing that the conical-like configurations are also unstable. 
  We investigate the spatially inhomogeneous decay of an unstable D-brane and construct an asymptotic solution which describes a codimension one D-brane and the tachyon matter in boundary string field theory. In this solution, the tachyon matter exists around the lower-dimensional D-brane. 
  We study string-gas cosmology in dilaton gravity, inspired by the fact that it naturally arises in a string theory context. Our main interest is the thermodynamical treatment of the string-gas and the resulting implications for the cosmology. Within an adiabatic approximation, thermodynamical equilibrium and a small, toroidal universe as initial conditions, we numerically solve the corresponding equations of motions in two different regimes describing the string-gas thermodynamics: (i) the Hagedorn regime, with a single scale factor, and (ii) an almost-radiation dominated regime, which includes the leading corrections due to the lightest Kaluza Klein and winding modes, with two scale factors. The scale factor in the Hagedorn regime exhibits very slow time evolution with nearly constant energy and negligible pressure. By contrast, in case (ii) we find interesting cosmological solutions where the large dimensions continue to expand and the small ones are kept undetectably small. 
  Affine Kac-Moody algebras give rise to interesting systems of differential equations, so-called Knizhnik-Zamolodchikov equations. The monodromy properties of their solutions can be encoded in the structure of a modular tensor category on (a subcategory of) the representation category of the affine Lie algebra. We discuss the relation between these solutions and physical correlation functions in two-dimensional conformal field theory. In particular we report on a proof for the existence of the latter on world sheets of arbitrary topology. 
  In this paper the relevance of holographic entropy bounds in the context of inflation is investigated. We distinguish between entropy on large and small scales and confront the entropy of quantum fluctuations in an inflating cosmology with the appropriate entropy bounds. In conclusion we do not find any constraints on inflation from holography, but some suggestions for future studies are given. 
  We discuss the application of the method of characteristics to the renormalization-group equation for the perturbative QCD series within the electron-positron annihilation cross-section. We demonstrate how one such renormalization-group improvement of this series is equivalent to a closed-form summation of the first four towers of renormalization-group accessible logarithms to all orders of perturbation theory. 
  In this letter we consider what happens to an M2-brane probe when Minkowski space is dimensionally reduced to a fluxbrane solution of IIA supergravity. Given that fluxbrane reductions generally break supersymmetry, we look at how supersymmetry is realised on the D2-brane probe after dualisation. We also show how to extend this to more general non-linear sigma models. 
  We construct a classical mechanics Hamiltonian which exhibits spontaneous symmetry breaking akin the Coleman-Weinberg mechanism, dimensional transmutation, and asymptotically free self-similarity congruent with the beta-function of four dimensional Yang-Mills theory. Its classical equations of motion support stable periodic orbits and in a three dimensional projection these orbits are self-linked into topologically nontrivial, toroidal knots. 
  The thermodynamics of type IIB superstring theory in the maximally supersymmetric plane wave background is studied. We compute the thermodynamic partition function for non-interacting strings exactly and the result differs slightly from previous computations. We clarify some of the issues related to the Hagedorn temperature in the limits of small and large constant RR 5-form. We study the thermodynamic behavior of strings in the case of $AdS_3 \times S^3 \times T^4$ geometries in the presence of NS-NS and RR 3-form backgrounds. We also comment on the relationship of string thermodynamics and the thermodynamic behavior of the sector of Yang-Mills theory which is the holographic dual of the string theory. 
  We investigate the possibility of having an event horizon within several classes of metrics that asymptote to the maximally supersymmetric IIB plane wave. We show that the presence of a null Killing vector (not necessarily covariantly constant) implies an effective separation of the Einstein equations into a standard and a wave component. This feature may be used to generate new supergravity solutions asymptotic to the maximally supersymmetric IIB plane wave, starting from standard seed solutions such as branes or intersecting branes in flat space. We find that in many cases it is possible to preserve the extremal horizon of the seed solution. On the other hand, non-extremal deformations of the plane wave solution result in naked singularities. More generally, we prove a no-go theorem against the existence of horizons for backgrounds with a null Killing vector and which contain at most null matter fields. Further attempts at turning on a nonzero Hawking temperature by introducing additional matter have proven unsuccessful. This suggests that one must remove the null Killing vector in order to obtain a horizon. We provide a perturbative argument indicating that this is in fact possible. 
  The interaction energy and force between widely separated strings is analyzed in a field theory having applications to superconducting cosmic strings, the SO(5) model of high-temperature superconductivity, and solitons in nonlinear optics. The field theory has two order parameters, one of which is broken in the vacuum (giving rise to strings), the other of which is unbroken in the vacuum but which could nonetheless be broken in the core of the string. If this does occur, there is an effect on the energetics of widely separated strings. This effect is important if the length scale of this second order parameter is longer than that of the other fields in the problem. 
  We consider warped compactifications that solve the 10 dimensional supergravity equations of motion at a point, stabilize the position of a D3-brane world, and admit a warp factor that violates Lorentz invariance along the brane. This gives a string embedding of ``asymmetrically warped'' models which we use to calculate stringy (\alpha') corrections to standard model dispersion relations, paying attention to the maximum speeds for different particles. We find, from the dispersion relations, limits on gravitational Lorentz violation in these models, improving on current limits on the speed of graviton propagation, including those derived from field theoretic loops. We comment on the viability of models that use asymmetric warping for self-tuning of the brane cosmological constant. 
  Three-dimensional gravity with a minimally coupled self-interacting scalar is considered. The fall-off of the fields at infinity is assumed to be slower than that of a localized distribution of matter, so that the asymptotic symmetry group is the conformal group. The counterterm Lagrangian needed to render the action finite is found by demanding that the action attain an extremum for the boundary conditions implied by the above fall-off of the fields at infinity. These counterterms explicitly depend on the scalar field. As a consequence, the Brown-York stress-energy tensor acquires a non trivial contribution from the matter sector. Static circularly symmetric solutions with a regular scalar field are explored for a one-parameter family of potentials. Their masses are computed via the Brown-York quasilocal stress-energy tensor, and they coincide with the values obtained from the Hamiltonian approach. The thermal behavior, including the transition between different configurations, is analyzed, and it is found that the scalar black hole can decay into the BTZ solution irrespective of the horizon radius. It is also shown that the AdS/CFT correspondence yields the same central charge as for pure gravity. 
  We study small fluctuations of the stringy wormhole solutions of graviton-dilaton-axion system in arbitrary dimensions. We show under O($d$)-symmetric harmonic perturbation that the Euclidean wormhole solutions are unstable in flat space irrespective of dimensions and in anti de Sitter space of $d=3$. 
  In Ref. 3, we presented an asymptotic formula for the fermion-induced effective energy in 3+1 dimensions in the presence of a cylindrically symmetric inhomogeneous strong magnetic field. However, there are some points which were not clearly explained. In fact, the arguments, which led us to the asymptotic formula, are based on a numerical study of the integral of Eq. (10), as we will see in the main part of this paper. The aim of this work is to present this study in detail. 
  An explicit construction for the monodromy of the Liouville conformal blocks in terms of Racah-Wigner coefficients of the quantum group U_q(sl(2,R)) is proposed. As a consequence, crossing-symmetry for four point functions is analytically proven, and the expression for the correlator of three boundary operators is obtained. 
  The renormalized Feynman propagator for a scalar field in the background of a cosmic dispiration (a disclination plus a screw dislocation) is derived, opening a window to investigate vacuum polarization effects around a cosmic string with dislocation, as well as in the bulk of an elastic solid carrying a dispiration. The use of the propagator is illustrated by computing vacuum fluctuations. In particular it is shown that the dispiration polarizes the vacuum giving rise to an energy momentum tensor which, as seen from a local inertial frame, presents non vanishing off-diagonal components. Such a new effect resembles that where an induced vacuum current arises around a needle solenoid carrying a magnetic flux (the Aharonov-Bohm effect), and may have physical consequences. Connections with a closely related background, namely the spacetime of a spinning cosmic string, are briefly addressed. 
  We review some facts about AdS2xS2 branes in AdS3xS3 with a Neveu-Schwarz background, and consider the case of Ramond-Ramond backgrounds. We compute the spectrum of quadratic fluctuations in the low-energy approximation and discuss the open-string geometry. 
  This brief review deals with prepotentials inspired by N=1 SUSY considerations due to Cachazo, Intrilligator, Vafa. These prepotentials associated with matrix models following Dijkgraaf and Vafa should be considered as given on an enlarged moduli space that includes Whitham times (couplings of the superpotential). This moduli space is nothing but the whole moduli space of (decorated) hyperelliptic curves. Corresponding prepotentials are logarithms of (quasiclassical) tau-functions and satisfy the WDVV equations. 
  We introduce a gauge invariant and string independent two-point fermion correlator which is analyzed in the context of the Schwinger model (QED_2). We also derive an effective infrared worldline action for this correlator, thus enabling the computation of its infrared behavior. Finally, we briefly discuss possible perspectives for the string independent correlator in the QED_3 effective models for the normal state of HTc superconductors. 
  We investigate both analytically and numerically the evolution of scalar perturbations generated in models which exhibit a smooth transition from a contracting to an expanding Friedmann universe. We find that the resulting spectral index in the late radiation dominated universe depends on which of the $\Psi$ or \$zeta$ variables passes regularly through the transition. The results can be parameterized through the exponent $q$ defining the rate of contraction of the universe. For $q \geq -1/2$ we find that there are no stable cases where both variables are regular during the transition. In particular, for $0<q\ll 1$, we find that the resulting spectral index is close to scale invariant if $\Psi$ is regular, whereas it has a steep blue behavior if $\zeta$ is regular. We also show that as long as $q\leqslant 1$, perturbations arising from the Bardeen potential remain small during contraction in the sense that there exists a gauge in which all the metric and matter perturbation variables are small. 
  The Morse problem is investigated in relativistic quantum mechanics. 
  We study a new class of infinite-dimensional Lie algebras W_\infty(p,q) generalizing the standard W_\infty algebra, viewed as a tensor operator algebra of SU(1,1) in a group-theoretic framework. Here we interpret W_\infty(p,q) either as an infinite continuation of the pseudo-unitary symmetry U(p,q), or as a "higher-U(p,q)-spin extension" of the diffeomorphism algebra diff(p,q) of the N=p+q torus U(1)^N. We highlight this higher-spin structure of W_\infty(p,q) by developing the representation theory of U(p,q) (discrete series), calculating higher-spin representations, coherent states and deriving K\"ahler structures on flag manifolds. They are essential ingredients to define operator symbols and to infer a geometric pathway between these generalized W_\infty symmetries and algebras of symbols of U(p,q)-tensor operators. Classical limits (Poisson brackets on flag manifolds) and quantum (Moyal) deformations are also discussed. As potential applications, we comment on the formulation of diffeomorphism-invariant gauge field theories, like gauge theories of higher-extended objects, and non-linear sigma models on flag manifolds. 
  Given a Quantum Field Theory, with a particular content of fields and a symmetry associated with them, if one wants to study the evolution of the couplings via a Wilsonian renormalisation group, there is still a freedom on the construction of a flow equation, allowed by scheme independence. In the present thesis, making use of this choice, we first build up a generalisation of the Polchinski flow equation for the massless scalar field, and, applying it to the calculation of the beta function at one loop for the characteristic self-interaction, we test its universality beyond the already known cutoff independence. Doing so we also develop a method to perform the calculation with this generalised flow equation for more complex cases. In the second part of the thesis, the method is extended to SU(N) Yang-Mills gauge theory, regulated by incorporating it in a spontaneously broken SU(N|N) supergauge group. Making use of the freedom allowed by scheme independence, we develop a flow equation for a SU(N|N) gauge theory, which preserves the invariance step by step throughout the flow and demonstrate the technique with a compact calculation of the one-loop beta function for the SU(N) Yang-Mills physical sector of SU(N|N), achieving a manifestly universal result, and without gauge fixing, for the first time at finite N. 
  The Casimir forces on two parallel plates in conformally flat de Sitter background due to conformally coupled massless scalar field satisfying mixed boundary conditions on the plates is investigated. In the general case of mixed boundary conditions formulae are derived for the vacuum expectation values of the energy-momentum tensor and vacuum forces acting on boundaries.  Different cosmological constants are assumed for the space between and outside of the plates to have general results applicable to the case of domain wall formations in the early universe. 
  We use the matrix model to describe the N=2 SO(N)/Sp(N) supersymmetric gauge theories with massive hypermultiplets in the fundamental representation. By taking the tree level superpotential perturbation made of a polynomial of a scalar chiral multiplet, the effective action for the eigenvalues of chiral multiplet can be obtained. By varying this action with respect to an eigenvalue, a loop equation is obtained. By analyzing this equation, we derive the Seiberg-Witten curve within the context of matrix model. 
  We perform a quantization of 4-dimensional Nambu-Goto theory of open string in light cone gauge, related in Lorentz-invariant way with the world sheet. This allows to obtain a quantum theory without anomalies in Lorentz group. We consider a special type of gauge fixing conditions, imposed in oscillator sector of the theory, which lead to a relatively simple Hamiltonian mechanics, convenient for canonical quantization. We discuss the algebraic and geometric properties of this mechanics and determine its mass spectrum for the states of spin singlet S=0. 
  Within the framework of Relativistic Schroedinger Theory (an alternative form of quantum mechanics for relativistic many-particle systems) it is shown that a general N-particle system must occur in one of two forms: either as a ``positive'' or as a ``negative'' mixture, in analogy to the fermion-boson dichotomy of matter in the conventional theory. The pure states represent a limiting case between the two types of mixtures which themselves are considered as the RST counterparts of the entangled (fermionic or bosonic) states of the conventional quantum theory. Both kinds of mixtures are kept separated from dynamical as well as from topological reasons. The 2-particle configurations (N=2) are studied in great detail with respect to their geometric and topological properties which are described in terms of the Euler class of an appropriate bundle connection. If the underlying space-time manifold (as the base space of the fibre bundles applied) is parallelisable, the 2-particle configurations can be thought to be generated geometrically by an appropriate (2+2) splitting of the local tangent space. 
  Supersymmetric backgrounds in string and M-theory of the Godel Universe type are studied. We find several new Godel Universes that preserve up to 20 supersymmetries. In particular, we obtain an interesting Godel Universe in M-theory with 18 supersymmetries which does not seem to be dual to a pp-wave. We show that not only T-duality but also the type-IIA/M-theory S-duality can give supersymmetric Godel Universes from pp-waves. We find solutions that can interpolate between Godel Universes and pp-waves. We also compute the string spectrum on two type IIA Godel Universes. Furthermore, we obtain the spectrum of D-branes on a Godel Universe and find the supergravity solution for a D4-brane on a Godel Universe. 
  We have proposed a method in the context of BFFT approach that leads to truncation of the infinite series regarded to constraints in the extended phase space, as well as other physical quantities (such as Hamiltonian). This has been done for cases where the matrix of Poisson brackets among the constraints is symplectic or constant. The method is applied to Proca model, single self dual chiral bosons and chiral Schwinger models as examples. 
  We study a gravitational model whose vacuum sector is invariant under conformal transformations. In this model we investigate the anomalous gravitational coupling of the large-scale matter. In this kind of coupling the large-sale matter is taken to couple to a metric which is different but conformally related to the metric appearing explicitly in the vacuum sector. The effect of the conformal symmetry breaking of the large-scale matter would lead in general to a variable strength of the anomalous gravitational coupling. This feature is used to derive a relativistic particle concept which shares the essential dynamical characteristics of the particle concept used in the causal interpretation of quantum mechanics with respect to the form of the Hamilton-Jacobi equation. The basic aspect of this result is that it relates the variable strength of the anomalous gravitational coupling of the large-scale matter to the appearance of a term similar to a quantum potential term. Some of the general characteristics of the corresponding pilot wave are discussed. 
  We introduce a class of finite dimensional nonlinear superalgebras $L = L_{\bar{0}} + L_{\bar{1}}$ providing gradings of $L_{\bar{0}} = gl(n) \simeq sl(n) + gl(1)$. Odd generators close by anticommutation on polynomials (of degree $>1$) in the $gl(n)$ generators. Specifically, we investigate `type I' super-$gl(n)$ algebras, having odd generators transforming in a single irreducible representation of $gl(n)$ together with its contragredient. Admissible structure constants are discussed in terms of available $gl(n)$ couplings, and various special cases and candidate superalgebras are identified and exemplified via concrete oscillator constructions. For the case of the $n$-dimensional defining representation, with odd generators $Q_{a}, \bar{Q}{}^{b}$, and even generators ${E^{a}}_{b}$, $a,b = 1,...,n$, a three parameter family of quadratic super-$gl(n)$ algebras (deformations of $sl(n/1)$) is defined. In general, additional covariant Serre-type conditions are imposed, in order that the Jacobi identities be fulfilled. For these quadratic super-$gl(n)$ algebras, the construction of Kac modules, and conditions for atypicality, are briefly considered. Applications in quantum field theory, including Hamiltonian lattice QCD and space-time supersymmetry, are discussed. 
  A non-minimal photon-torsion axial coupling in the quantum electrodynamics (QED) framework is considered. The geometrical optics in Riemann-Cartan spacetime is considering and a plane wave expansion of the electromagnetic vector potential is considered leading to a set of the equations for the ray congruence. Since we are interested mainly on the torsion effects in this first report we just consider the Riemann-flat case composed of the Minkowskian spacetime with torsion. It is also shown that in torsionic de Sitter background the vacuum polarisation does alter the propagation of individual photons, an effect which is absent in Riemannian spaces. It is shown that the cosmological torsion background inhomogeneities induce Lorentz violation and massive photon modes in this QED. 
  It is shown that the problem of the recursive restoration of the Slavnov-Taylor (ST) identities at the quantum level for anomaly-free gauge theories is equivalent to the problem of parameterizing the local approximation to the quantum effective action in terms of ST functionals, associated with the cohomology classes of the classical linearized ST operator. The ST functionals of dimension <=4 correspond to the invariant counterterms, those of dimension >4 generate the non-symmetric counterterms upon projection on the action-like sector. At orders higher than one in the loop expansion there are additional contributions to the non-invariant counterterms, arising from known lower order terms. They can also be parameterized by using the ST functionals. We apply the method to Yang-Mills theory in the Landau gauge with an explicit mass term introduced in a BRST-invariant way via a BRST doublet. Despite being non-unitary, this model provides a good example where the method devised in the paper can be applied to derive the most general solution for the action-like part of the quantum effective action, compatible with the fulfillment of the ST identities and the other relevant symmetries of the model, to all orders in the loop expansion. The full dependence of the solution on the normalization conditions is given. 
  Gravity coupled to an arbitrary number of antisymmetric tensors and scalar fields in arbitrary space-time dimensions is studied in a context of general, static, spherically symmetric solutions with many orthogonally intersecting branes. Neither supersymmetry nor harmonic gauge is assumed. It is shown that the system reduces to a Toda-like system after an adequate redefinition of transverse radial coordinate $r$. Duality $r \to 1/r$ in the set of solutions is observed. 
  We present a noncommutative version of the Burgers equation which possesses the Lax representation and discuss the integrability in detail. We find a noncommutative version of the Cole-Hopf transformation and succeed in the linearization of it. The linearized equation is the (noncommutative) diffusion equation and exactly solved. We also discuss the properties of some exact solutions. The result shows that the noncommutative Burgers equation is completely integrable even though it contains infinite number of time derivatives. Furthermore, we derive the noncommutative Burgers equation from the noncommutative (anti-)self-dual Yang-Mills equation by reduction, which is an evidence for the noncommutative Ward conjecture. Finally, we present a noncommutative version of the Burgers hierarchy by both the Lax-pair generating technique and the Sato's approach. 
  The applicability of the space-time formulation of the gluonic sector of QCD in terms of the Polyakov worldline path integral, via the use of the background field gauge fixing method, is extended to multi-gluon loop configurations. Relevant master formulas are derived for the computation of effective action terms. 
  Using the Ocneanu quantum geometry of ADE diagrams (and of other diagrams belonging to higher Coxeter-Dynkin systems), we discuss the classification of twisted partition functions for affine and minimal models in conformal field theory and study several examples associated with the WZW, Virasoro and W_{3} cases. 
  A Fermi ball is a kind of non-topological soliton, which is thought to arise from the spontaneous breaking of an approximate $Z_2$ symmetry and to contribute to cold dark matter. We consider a simple model in which fermion fields with multi-flavors are coupled to a scalar field through Yukawa coupling, and examine how the number of the fermion flavors affects the stability of the Fermi ball against the fragmentation. (1)We find that the Fermi ball is stable against the fragmentation in most cases even in the lowest order thin-wall approximation. (2)We then find that in the other specific cases, the stability is marginal in the lowest order thin-wall approximation, and the next-to-leading order correction determines the stable region of the coupling constants; We examine the simplest case where the total fermion number $N_i$ and the Yukawa coupling constant $G_i$ of each flavor $i$ are common to the flavor, and find that the Fermi ball is stable in the limited region of the parameters and has the broader region for the larger number of the flavors. 
  The proper inclusion of flavor in the Dijkgraaf-Vafa proposal for the solution of N=1 gauge theories through matrix models has been subject of debate in the recent literature. We here reexamine this issue by geometrically engineering fundamental matter with type IIB branes wrapped on non-compact cycles in the resolved geometry, and following them through the geometric transition. Our approach treats massive and massless flavor fields on equal footing, including the mesons. We also study the geometric transitions and superpotentials for finite mass of the adjoint field. All superpotentials we compute reproduce the field theory results. Crucial insights come from T-dual brane constructions in type IIA. 
  We present a new model of inflation in which the inflaton is the extra component of a gauge field in a 5d theory compactified on a circle. The chief merit of this model is that the potential comes only from non-local effects so that its flatness is not spoiled by higher dimensional operators or quantum gravity corrections. The model predicts a red spectrum (n ~ 0.96) and a significant production of gravitational waves (r ~ 0.11). We also comment on the relevance of this idea to quintessence. 
  We show that a dispiration (a disclination plus a screw dislocation) polarizes the vacuum of a scalar field giving rise to an energy momentum tensor which, as seen from a local inertial frame, presents non vanishing off-diagonal components. The results may have applications in cosmology (chiral cosmic strings) and condensed matter physics (materials with linear defects). 
  We examine the interrelation between the (2,0) supersymmetric six dimensional effective action for the A_1 theory, and the corresponding low-energy theory for the collective coordinates associated to supersymmetric selfdual BPS strings. We argue that this low-energy theory is a two-dimensional N = 4 supersymmetric sigma model. 
  We give an overview of D-branes in the maximally supersymmetric plane wave background of IIB supergravity. We start by reviewing the results of the probe analysis. We then present the open string analysis and show how certain spacetime symmetries are restored using worldsheet symmetries. We discuss the construction of these branes as boundary states and summarize what is known about the dual gauge theory description. 
  By treating the bulk-quantized Yang-Mills theory as a constrained system we obtain a consistent gauge-fixed BRST hamiltonian in the minimal sector. This provides an independent derivation of the 5-d lagrangian bulk action. The ground state is independent of the (anti)ghosts and is interpreted as the solution of the Fokker-Planck equation, thus establishing a direct connection to the Fokker-Planck hamiltonian. The vacuum state correlators are shown to be in agreement with correlators in lagrangian 5-d formulation. It is verified that the complete propagators remain parabolic in one-loop dimensional regularization. 
  We examine the extent to which the action for the membrane of M-theory (the eleven-dimensional construct which underlies and unifies all of the known string theories) simplifies in the so-called Open Membrane (OM) limit, a limit which lies at the root of the various manifestations of noncommutativity in the string context. In order for the discussion to be relatively self-contained, we start out by reviewing why the strings of ten-dimensional string theory are in fact membranes (M2-branes) living in eleven dimensions. After that, we recall the definition of OM theory, as well as the arguments showing that it is part of a larger, eleven-dimensional structure known as Galilean or Wrapped M2-brane (WM2) theory. WM2 theory is a rich theoretical construct which is interesting for several reasons, in particular because it is essentially a toy model of M-theory. We then proceed to deduce a membrane action for OM/WM2 theory, and spell out its implications for the four different types of M2-branes one can consider in this setting. For two of these types, the action in question can be simplified by gauge-fixing to a form which implies a discrete membrane spectrum. The boundary conditions for the remaining two cases do not allow this same gauge choice, and so their dynamics remain to be unraveled. 
  We take as starting point the planar model arising from the dimensional reduction of the Maxwell Electrodynamics with the (Lorentz-violating) Carroll-Field-Jackiw term. We then write and study the extended Maxwell equations and the corresponding wave equations for the potentials. The solution to these equations show some interesting deviations from the usual MCS Electrodynamics, with background-dependent correction terms. In the case of a time-like background, the correction terms dominate over the MCS sector in the region far from the origin, and establish the behaviour of a massless Electrodynamics (in the electric sector). In the space-like case, the solutions indicate the clear manifestation of spatial anisotropy, which is consistent with the existence of a privileged direction is space. 
  We consider a supersymmetric extension of the algebra associated with three and four dimensional Anti de Sitter space. A representation of the supersymmetry operators in superspace is given. Supersymmetry invariant models are constructed for the superspace associated with AdS3. 
  We study matrix models related to $SO/Sp$ gauge theories with flavors. We give the effective superpotentials for gauge theories with arbitrary tree level superpotential up to first instanton level. For quartic tree level superpotential we obtained exact one-cut solution. We also derive Seiberg-Witten curve for these gauge theories from matrix model argument. 
  We consider a two-dimensional non-commutative inverted oscillator in the presence of a constant magnetic field, coupled to the system in a ``symplectic'' and ``Poisson'' way. We show that it has a discrete energy spectrum for some value of the magnetic field. 
  We consider a spherical thick 3-brane immersed in a five-dimensional bulk spacetime. We demonstrate how the thick brane equation of motion expanded in powers of the thickness of the brane can be obtained from the expected junction conditions on the boundaries of thick brane with the two embedding spacetimes. It is shown that the finite thickness leads to a faster collapse of the spherical shell. 
  We show how a large class of boundary RG flows in two-dimensional conformal field theories can be summarized in a single rule. This rule is a generalization of the 'absorption of the boundary spin'-principle of Affleck and Ludwig and applies to all theories which have a description as a coset model. We give a formulation for coset models with arbitrary modular invariant partition function and present evidence for the conjectured rule. The second half of the article contains an illustrated section of examples where the rule is applied to unitary minimal models of the A- and D-series, in particular the 3-state Potts model, and to parafermion theories. We demonstrate how the rule can be used to compute brane charge groups in the example of N=2 minimal models. 
  We construct low energy effective Lagrangians for 3d N=4 supersymmetric Yang-Mills theory with any gauge group. They represent supersymmetric sigma models at hyper-Kahlerian manifolds of dimension 4r (r is the rang of the group). In the asymptotic region, perturbatively exact explicit expression for the metric are written. We establish the relationship of this metric with the TAUB-NUT metric describing the perturbatively exact effective Lagrangians for unitary groups and monopole moduli spaces: the former is obtained out of the latter by a proper hyper-Kahlerian reduction. We describe in details the reduction procedure for SO/Sp/G_2 gauge groups, where it can also be given a natural interpretation in D-brane language. We conjecture that the exact nonperturbative metrics can be obtained by a similar hyper-Kahlerian reduction from the corresponding multidimensional Atiyah-Hitchin metrics. 
  Following the RG flow of an N=1 quiver gauge theory and applying Seiberg duality whenever necessary defines a duality cascade, that in simple cases has been understood holographically. It has been argued that in certain cases, the dualities will pile up at a certain energy scale called the duality wall, accompanied by a dramatic rise in the number of degrees of freedom. In string theory, this phenomenon is expected to occur for branes at a generic threefold singularity, for which the associated quiver has Lorentzian signature. We here study sequences of Seiberg dualities on branes at the C_3/Z_3 orbifold singularity. We use the naive beta functions to define an (unphysical) scale along the cascade. We determine, as a function of initial conditions, the scale of the wall as well as the critical exponent governing the approach to it. The position of the wall is piecewise linear, while the exponent appears to be constant. We comment on the possible implications of these results for physical walls. 
  We consider compactification of the SO(32) heterotic string on a 6-dimensional Z_3 orbifold with one discrete Wilson line. A complete set of all possible embeddings is given, 159 in all. The unbroken subgroups of SO(32) are tabulated. The extended gauge symmetry SU(3)^3, recently discussed by J. E. Kim [hep-th/0301177] for semi-realistic E_8 x E_8 heterotic string models, occurs for several embeddings, as well as other groups that may be of interest in unified string models. The extent to which extra gauge group factors can be hidden is discussed and compared to the E_8 x E_8 case. Along flat directions where an effective hidden sector exists, the embeddings described here provide for hidden gauge groups that are not possible in the E_8 x E_8 heterotic string. 
  By using the systematic approach of parent action method, we derive one Weyl-noninvariant and two Weyl-invariant actions of bosonic $p$-branes ($p\geq 2$) starting from the Nambu-Goto action, and establish the duality symmetries in this set of four actions. Moreover, we discover a new bosonic $p$-brane action (including the string theory) and deduce two corresponding Weyl-invariant formulations by proposing a new special parent action. We find that the same duality symmetries as those mentioned above exist in this new set of actions. The new $p$-brane actions are also briefly analyzed. 
  Withdrawn by arXiv administration because the text and equations were plagiarized almost entirely verbatim from hep-th/9610131 . 
  Some methods of the ``unfolded dynamics'' machinery particularly useful for the analysis of higher spin gauge theories are summarized. A formulation of 4d conformal higher spin theories in Sp(8) invariant space-time with matrix coordinates and its extension to Sp(2M) invariant space-times are discussed. A new result on the global characterizaton of causality of physical events in the Sp(2M) invariant space-time is announced. 
  The dualised formulation of the symmetric space sigma model is peformed for a general scalar coset G/K where G is a maximally non-compact group and K is it's maximal compact subgroup.By using the twisted self-duality condition the general form of the first-order equations are obtained.The results are applied to the example of SL(2,R)/SO(2) scalar manifold of the IIB supergravity. 
  We start by reviewing the formulation of noncommutative quantum mechanics as a constrained system. Then, we address to the problem of field theories defined on a noncommutative space-time manifold. The Moyal product is introduced and the appearance of the UV/IR mechanism is exemplified. The emphasis is on finding and analyzing noncommutative quantum field theories which are renormalizable and free of nonintegrable infrared singularities. In this last connection we give a detailed discussion of the quantization of the noncommutative Wess-Zumino model as well as of its low energy behavior. 
  We compute the $O(\lambda)$ correction to the Casimir energy for the massive $\lambda\phi^4$ model confined between a pair of parallel plates. The calculations are made with Dirichlet and Neumann boundary conditions. The correction is shown to be sensitive to the boundary conditions, except in the zero mass limit, in which case our results agree with those found in the literature. 
  We discuss the asymptotic properties of quantum states density for fundamental $p-$branes which can yield a microscopic interpretation of the thermodynamic quantities in M-theory. The matching of BPS part of spectrum for superstring and supermembrane gives the possibility of getting membrane's results via string calculations. In the weak coupling limit of M-theory the critical behavior coincides with the first order phase transition in standard string theory at temperature less than the Hagedorn's temperature $T_H$. The critical temperature at large coupling constant is computed by considering M-theory on manifold with topology ${\mathbb R}^9\otimes{mathbb T}^2$. Alternatively we argue that any finite temperature can be introduced in the framework of membrane thermodynamics. 
  We outline the construction of metastable de Sitter vacua of type IIB string theory. Our starting point is highly warped IIB compactifications with nontrivial NS and RR three-form fluxes. By incorporating known corrections to the superpotential from Euclidean D-brane instantons or gaugino condensation, one can make models with all moduli fixed, yielding a supersymmetric AdS vacuum. Inclusion of a small number of anti-D3 branes in the resulting warped geometry allows one to uplift the AdS minimum and make it a metastable de Sitter ground state. The lifetime of our metastable de Sitter vacua is much greater than the cosmological timescale of 10^10 years. We also prove, under certain conditions, that the lifetime of dS space in string theory will always be shorter than the recurrence time. 
  The standard quantum states of $n$ complex Grassmann variables with a free-particle Lagrangian transform as a spinor of SO(2n). However, the same `free-fermion' model has a non-linearly realized $SU(n|1)$ symmetry; it can be viewed as the mechanics of a `particle' on the Grassmann-odd coset space $SU(n|1)/U(n)$. We implement a quantization of this model for which the states with non-zero norm transform as a representation of $SU(n|1)$, the representation depending on the U(1) charge of the wave-function. For $n=2$ the wave-function can be interpreted as a BRST superfield. 
  The Ginsparg-Wilson algebra is the algebra underlying the Ginsparg-Wilson solution of the fermion doubling problem in lattice gauge theory. The Dirac operator of the fuzzy sphere is not afflicted with this problem. Previously we have indicated that there is a Ginsparg-Wilson operator underlying it as well in the absence of gauge fields and instantons. Here we develop this observation systematically and establish a Dirac operator theory for the fuzzy sphere with or without gauge fields, and always with the Ginsparg-Wilson algebra. There is no fermion doubling in this theory. The association of the Ginsparg-Wilson algebra with the fuzzy sphere is surprising as the latter is not designed with this algebra in mind. The theory reproduces the integrated U(1)_A anomaly and index theory correctly. 
  We provide a general scheme for dualizing higher-spin gauge fields in arbitrary irreducible representations of GL(D,R). We also give a recipe for constructing Fronsdal-like field equations and Lagrangians for such exotic fields. 
  The nonlinear realization of conformal so(2,d) symmetry for relativistic systems and the dynamical conformal so(2,1) symmetry of nonrelativistic systems are investigated in the context of AdS/CFT correspondence. We show that the massless particle in d-dimensional Minkowski space can be treated as the system confined to the border of the AdS_{d+1} of infinite radius, while various nonrelativistic systems may be canonically related to a relativistic (massless, massive, or tachyon) particle on the AdS_2 X S^{d-1}. The list of nonrelativistic systems "unified" by such a correspondence comprises the conformal mechanics model, the planar charge-vortex and 3-dimensional charge-monopole systems, the particle in a planar gravitational field of a point massive source, and the conformal model associated with the charged particle propagating near the horizon of the extreme Reissner-Nordstrom black hole. 
  The paper proposes an algorithm for regularization of the self-energy expressions for a Dirac particle that meets the relativistic and gauge invariance requirements. 
  We compute the two-body one-loop effective action for the matrix theory in the pp-wave background, and compare it to the effective action on the supergravity side in the same background. Agreement is found for the effective actions on both sides. This points to the existence of a supersymmetric nonrenormalization theorem in the pp-wave background. 
  The baryon vertex of IIB superstring theory on $AdS_{5}\times \mathbb{R}P^{5}$, for the case of orthogonal groups, is studied. The energy of the three brane decayed from an original five brane is calculated explicitly. The radius of this decayed three brane, for a BPS configuration, is also given and interpreted. 
  Generalizing the noncommutative harmonic oscillator construction, we propose a new extension of quantum field theory based on the concept of "noncommutative fields". Our description permits to break the usual particle-antiparticle degeneracy at the dispersion relation level and introduces naturally an ultraviolet and an infrared cutoff. Phenomenological bounds for these new energy scales are given. 
  In this work we explain the construction of the thermal vacuum for the bosonic string, as well that of the thermal boundary state interpreted as a $D_{p}$-brane at finite temperature. In both case we calculate the respective entropy using the entropy operator of the Thermo Field Dynamics Theory. We show that the contribution of the thermal string entropy is explicitly present in the $D_{p}$-brane entropy. Furthermore, we show that the Thermo Field approach is suitable to introduce temperature in boundary states. 
  We study string interactions among string states with arbitrary impurities in the Type IIB plane wave background using string field theory. We reproduce all string amplitudes from gauge theory by computing matrix elements of the dilatation operator in a previously proposed basis of states. A direct correspondence is found between the string field theory and gauge theory Feynman diagrams. 
  In the framework of nonlinear realizations we rederive the action of the N=2 SuperConformal Quantum Mechanics (SCQM). We propose also the WZNW -- like construction of the interaction term in the lagrangian with the help of Cartan's Omega forms. 
  The one-loop energy density of an infinitely thin static magnetic vortex in SU(2) Yang-Mills theory is evaluated using the Schroedinger picture. Both the gluonic fluctuations as well as the quarks in the vortex background are included. The energy density of the magnetic vortex is discussed as a function of the magnetic flux. The center vortices correspond to local minima in the effective potential. These minima are degenerated with the perturbative vacuum if the fermions are ignored. Inclusion of fermions lifts this degeneracy, raising the vortex energy above the energy of the perturbative vacuum. 
  In this note we provide an explicit example of type IIB supersymmetric D3-branes solution on a pp-wave like background, consisting in the product of an eight-dimensional pp-wave times a two-dimensional flat space. An interesting property of our solution is the fully localization of the D3-branes (i.e. the solution depends on all the transverse coordinates). Then we show the generalization to other Dp-branes and to the D1/D5 system. 
  In this work we probe the Born-Infeld (BI) black hole in the isolated horizon framework. It turns out that the BI black hole is consistent with the heuristic model for colored black holes proposed by Ashtekar et al [(2001) Class.Quant.Grav. v. 18, 919-940]. The model points to the unstability of the BI black hole. 
  We point out that the worldvolume coordinate functions $\hat{x}^\mu(\xi)$ of a $p$-brane, treated as an independent object interacting with dynamical gravity, are Goldstone fields for spacetime diffeomorphisms gauge symmetry. The presence of this gauge invariance is exhibited by its associated Noether identity, which expresses that the source equations follow from the gravitational equations. We discuss the spacetime counterpart of the Higgs effect and show that a $p$-brane does not carry any local degrees of freedom, extending early known general relativity features. Our considerations are also relevant for brane world scenarios. 
  Hamiltonian systems with linearly dependent constraints (irregular systems), are classified according to their behavior in the vicinity of the constraint surface. For these systems, the standard Dirac procedure is not directly applicable. However, Dirac's treatment can be slightly modified to obtain, in some cases, a Hamiltonian description completely equivalent to the Lagrangian one. A recipe to deal with the different cases is provided, along with a few pedagogical examples. 
  We calculate the D-instanton corrections (with all D-instanton numbers) to the quantum moduli space metric of a single matter hypermultiplet with toric isometry, in the effective N=2 supergravity arising in type-IIA superstrings compactified on a Calabi-Yau (CY) threefold of Hodge number h_{2,1}=1. The non-perturbative quaternionic hypermultiplet metric is derived by resolution of a complex orbifold singularity, thus generalizing the known (Ooguri-Vafa) solution in flat spacetime to N=2 supergravity. 
  The Heisenberg, interaction, and Schr\"odinger pictures of motion are considered in Lagrangian (canonical) quantum field theory. The equations of motion (for state vectors and field operators) are derived for arbitrary Lagrangians which are polynomial or convergent power series in field operators and their first derivatives. The general links between different time-dependent pictures of motion are derived. It is pointed that all of them admit covariant formulation, similar to the one of interaction picture. A new picture, called the momentum picture, is proposed. It is a 4-dimensional analogue of the Schr\"odinger picture of quantum mechanics as in it the state vectors are spacetime-dependent, while the field operators are constant relative to the spacetime. The equations of motion in momentum picture are derived and partially discussed. In particular, the ones for the field operators turn to be of algebraic type. The general idea of covariant pictures of motion is presented. The equations of motion in these pictures are derived. 
  We study SUSY-intertwining for non-Hermitian Hamiltonians with special emphasis to the two-dimensional generalized Morse potential, which does not allow for separation of variables. The complexified methods of SUSY-separation of variables and two-dimensional shape invariance are used to construct particular solutions - both for complex conjugated energy pairs and for non-paired complex energies. 
  We consider different variants of factorization of a 2x2 matrix Schroedinger/Pauli operator in two spatial dimensions. They allow to relate its spectrum to the sum of spectra of two scalar Schroedinger operators, in a manner similar to one-dimensional Darboux transformations. We consider both the case when such factorization is reduced to the ordinary 2-dimensional SUSY QM quasifactorization and a more general case which involves covariant derivatives. The admissible classes of electromagnetic fields are described and some illustrative examples are given. 
  We study different Penrose limits of supergravity solution of NS5-brane in the presence of RR field. Although in the case of NS5-brane we get a 4-dimensional plane wave, in the case with RR field we will get two different plane waves; a 4-dimensional and a 3-dimensional one. These plane wave solutions are the backgrounds that a particular string solution feels at one loop approximation. Using the one loop correction one can identify a particular subsector of LST/deformed LST which is dual to type II string theories on these plane waves. 
  The NAHE-set, that underlies the realistic free fermionic models, corresponds to Z2XZ2 orbifold at an enhanced symmetry point, with (h_{11},h_{21})=(27,3). Alternatively, a manifold with the same data is obtained by starting with a Z2XZ2 orbifold at a generic point on the lattice and adding a freely acting Z2 involution. In this paper we study type I orientifolds on the manifolds that underly the NAHE-based models by incorporating such freely acting shifts. We present new models in the Type I vacuum which are modulated by Z2^n for n=2,3. In the case of n=2, the Z2XZ2 structure is a composite orbifold Kaluza-Klein shift arrangement. The partition function provides a simpler spectrum with chiral matter. For n=3, the case discussed is a Z2 modulation of the T6/(Z2 X Z2) spectrum. The additional projection shows an enhanced closed and open sector with chiral matter. The brane stacks are correspondingly altered from those which are present in the Z2 X Z2 orbifold. In addition, we discuss the models arising from the open sector with and without discrete torsion. 
  Precision cosmological data hint that a dark energy with equation of state $w = P/\rho < -1$ and hence dubious stability is viable. Here we discuss for any $w$ nucleation from $\Lambda > 0$ to $\Lambda = 0$ in a first-order phase transition. The critical radius is argued to be at least of galactic size and the corresponding nucleation rate is glacial, thus underwriting the dark energy's stability and rendering remote any microscopic effect. 
  In this paper we will study some aspects of dS/CFT correspondence. We will focus on the relation between Witten's non-standard de Sitter inner product and correlators in the holographic dual conformal field theory. We will argue that from the definition of Witten's inner product and conjecture that the Hilbert space of initial states of massive scalar field on $\mathcal{I}^-$ in de Sitter space corresponds to the Hilbert space of states of Euclidean CFT on $\mathcal{I}^-$, we can obtain CFT correlators in any vacuum state. 
  In this paper, a general theory on unification of non-Abelian SU(N) gauge interactions and gravitational gauge interactions is discussed. SU(N) gauge interactions and gravitational gauge interactions are formulated on the similar basis and are unified in a semi-direct product group GSU(N). Based on this model, we can discuss unification of fundamental interactions of Nature. 
  We use the low energy effective theory of string theory to investigate condensations of closed string tachyons propagating in the bulk. The c-function is related to the total energy of the system via the effective action. A possible modification of the c-theorem is discussed. We also deduce endpoints of the decays by investigating scalar potential of gauged supergravities. A string theory in the flat spacetime would be a possible endpoint. 
  We show how the full holomorphic geometry of local Calabi-Yau threefold compactifications with N=1 supersymmetry can be obtained from matrix models. In particular for the conifold geometry we relate F-terms to the general amplitudes of c=1 non-critical bosonic string theory, and express them in a quiver or, equivalently, super matrix model. Moreover we relate, by deconstruction, the uncompactified c=1 theory to the six-dimensional conformal (2,0) theory. Furthermore, we show how we can use the idea of deconstruction to connect 4+k dimensional supersymmetric gauge theories to a k-dimensional internal bosonic gauge theory, generalizing the relation between 4d theories and matrix models. Examples of such bosonic systems include unitary matrix models and gauged matrix quantum mechanics, which deconstruct 5-dimensional supersymmetric gauge theories, and Chern-Simons gauge theories, which deconstruct gauge theories living on branes wrapped over cycles in Calabi-Yau threefolds. 
  We clarify the relationship between black hole entropy and the number of degrees of freedom in the dual QFT with a cut-off. We show that simple gravity arguments predict the correct cut-off procedure. 
  We show that many numerically established properties of Q-balls can be understood in terms of analytic approximations for a certain type of potential. In particular, we derive an explicit formula between the energy and the charge of the Q-ball valid for a wide range of the charge Q. 
  The magnetic field redefinition in Jain's composite fermion model for the fractional quantum Hall effect is shown to be effectively described by a mean-field approximation of a model containing a Maxwell-Chern-Simons gauge field non-minimally coupled to matter. Also an explicit non-relativistic limit of the non-minimal (2+1)D Dirac equation is derived. 
  The status of the `BRST-invariant' condensate of mass dimension two in QCD is explained. The condensate is only invariant under an `on-shell' BRST symmetry which includes a partial gauge-fixing. The on-shell BRST symmetry represents the residual gauge symmetry under gauge transformations which preserve the partial gauge fixing. The gauge-invariant operators which correspond to the BRST-invariant condensate are identified in the Lorentz and maximal Abelian gauges and are shown to be invariant under the residual gauge transformations. 
  A light-front Hamiltonian reproducing the results of two-dimensional quantum electrodynamics in the Lorentz coordinates is constructed using the bosonization procedure and an analysis of the bosonic perturbation theory in all orders in the fermion mass. The resulting Hamiltonian involves a supplementary counterterm in addition to the usual terms appearing in the naive light-front quantization. This term is proportional to a linear combination of zeroth fermion modes (which are multiplied by a factor compensating the charge and fermion number). The coefficient of the counterterm has no ultraviolet divergence, depends on the value of the fermion condensate in the \theta-vacuum, and is linear in this value for a small fermion mass. 
  We consider a class of warped higher dimensional brane models with topology $M \times \Sigma \times S^1/Z_2$, where $\Sigma$ is a $D_2$ dimensional manifold. Two branes of codimension one are embedded in such a bulk space-time and sit at the orbifold fixed points. We concentrate on the case where an exponential warp factor (depending on the distance along the orbifold) accompanies the Minkowski $M$ and the internal space $\Sigma$ line elements. We evaluate the moduli effective potential induced by bulk scalar fields in these models, and we show that generically this can stabilize the size of the extra dimensions. As an application, we consider a scenario where supersymmetry is broken not far below the cutoff scale, and the hierarchy between the electroweak and the effective Planck scales is generated by a combination of redshift and large volume effects. The latter is efficient due to the shrinking of $\Sigma$ at the negative tension brane, where matter is placed. In this case, we find that the effective potential can stabilize the size of the extra dimensions (and the hierarchy) without fine tuning provided that the internal space $\Sigma$ is flat. 
  In this paper a class of multi-Chern-Simons field theories which is relevant to the statistical mechanics of polymer systems is investigated. Motivated by the problems which one encounters in the treatment of these theories, a general procedure is presented to eliminate the Chern-Simons fields from their action. In this way it has been possible to derive an expression of the partition function of topologically linked polymers which depends explicitly on the topological numbers and does not have intractable nonlocal terms as it happened in previous approaches. The new formulation of multi-Chern-Simons field theories is then used to remove and clarify some inconsistencies and ambiguities which apparently affect field theoretical models of topologically linked polymers. Finally, the limit of disentangled polymers is discussed. 
  Within the program of holographic renormalization, we discuss the computation of three-point correlation functions along RG flows. We illustrate the procedure in two simple cases. In an RG flow to the Coulomb branch of N=4 SYM theory we derive a compact and finite expression for the three-point function of lowest CPO's dual to inert scalars. In the GPPZ flow, that captures some features of N=1 SYM theory, we compute the three-point function with insertion of two inert scalars and one active scalar that mixes with the stress tensor. By amputating the external legs at the mass poles we extract the trilinear coupling of the corresponding superglueballs. Finally we outline the procedure for computing three-point functions with insertions of the stress tensor as well as of (broken) R-symmetry currents. 
  We show, by direct computations of bosonic string spectra, the O(d,d;Z) (d=1,2) T-duality in the maximally supersymmetric IIB plane-wave background compactified on S^1 and T^2. Only half of the ordinary set of zero modes appear in the Hamiltonian. This "half" Narain lattice is proved to be stabilized by the T-duality group. 
  We construct 7-dimensional compact Einstein spaces with conical singularities that preserve 1/8 of the supersymmetries of M-theory. Mathematically they have weak G_2-holonomy. We show that for every non-compact G_2-holonomy manifold which is asymptotic to a cone on a 6-manifold Y, there is a corresponding weak G_2-manifold with two conical singularities which, close to the singularities, looks like a cone on Y. Our construction provides explicit metrics on these weak G_2-manifolds. We completely determine the cohomology of these manifolds in terms of the cohomology of Y. 
  In this paper we present a covariant quantization of the ``massive'' spin-2 field on de Sitter (dS) space. By ``massive'' we mean a field which carries a specific principal series representation of the dS group. The work is in the direct continuation of previous ones concerning the scalar, the spinor and the vector cases. The quantization procedure, independent of the choice of the coordinate system, is based on the Wightman-Garding axiomatic and on analyticity requirements for the two-point function in the complexified pseudo-Riemanian manifold. Such a construction is necessary in view of preparing and comparing with the dS conformal spin-2 massless case (dS linear quantum gravity) which will be considered in a forthcoming paper and for which specific quantization methods are needed. 
  We analyse the effect of general brane kinetic terms for bulk scalars, fermions and gauge bosons in theories with extra dimensions, with and without supersymmetry. We find in particular a singular behaviour when these terms contain derivatives orthogonal to the brane. This is brought about by $\delta(0)$ divergences arising at second and higher order in perturbation theory. We argue that this behaviour can be smoothed down by classical renormalization. 
  The problems of attempting inflationary model-building in a theory containing a dilaton are explained. In particular, I study the shape of the dilaton potential today and during inflation, based on a weakly-coupled heterotic string model where corrections to the Kahler potential are assumed to be responsible for dilaton stabilization. Although no specific model-building is attempted, if the inflationary energy density is related to the scale of gaugino condensation, then the dilaton may be stabilized close enough to today's value that there is no significant change in the GUT scale coupling. This can occur in a very wide range of models, and helps to provide some justification for the standard predictions of the spectral index. I explain how this result can ultimately be traced to the supersymmetry structure of the theory. 
  We derive a local, gauge invariant action for the SU(N) non-linear sigma-model in 2+1 dimensions. In this setting, the model is defined in terms of a self-interacting pseudo vector-field \theta_\mu, with values in the Lie algebra of the group SU(N). Thanks to a non-trivially realized gauge invariance, the model has the correct number of degrees of freedom: only one polarization of \theta_\mu, like in the case of the familiar Yang-Mills theory in 2+1 dimensions. Moreover, since \theta_\mu is a pseudo-vector, the physical content corresponds to one massless pseudo-scalar field in the Lie algebra of SU(N), as in the standard representation of the model. We show that the dynamics of the physical polarization corresponds to that of the SU(N) non-linear sigma model in the standard representation, and also construct the corresponding BRST invariant gauge-fixed action. 
  We use the Lorentzian AdS/CFT prescription to find the poles of the retarded thermal Green's functions of ${\cal N=4}$ SU(N) SYM theory in the limit of large N and large 't Hooft coupling. In the process, we propose a natural definition for quasinormal modes in an asymptotically AdS spacetime, with boundary conditions dictated by the AdS/CFT correspondence. The corresponding frequencies determine the dispersion laws for the quasiparticle excitations in the dual finite-temperature gauge theory. Correlation functions of operators dual to massive scalar, vector and gravitational perturbations in a five-dimensional AdS-Schwarzschild background are considered. We find asymptotic formulas for quasinormal frequencies in the massive scalar and tensor cases, and an exact expression for vector perturbations. In the long-distance, low-frequency limit we recover results of the hydrodynamic approximation to thermal Yang-Mills theory. 
  We derive anticommutators of supercharges with a brane charge for a D-particle in AdS(2) x S(2) and pp-wave backgrounds. A coset GL(2|2)/(GL(1))^4 and its Penrose limit are used with the supermatrix-valued coordinates for the AdS and the pp-wave spaces respectively. The brane charges have position dependence, and can be absorbed into bosonic generators by shift of momenta which results in closure of the superalgebras. 
  BPS wall solutions are obtained for N=2 supersymmetric nonlinear sigma model with Eguchi-Hanson target manifold in a manifestly supersymmetric manner. The model is constructed by a massive hyper-Kahler quotient method both in the N=1 superfield and in the N=2 superfield (harmonic superspace). We describe the model in simple terms and give relations between various parameterizations which are useful to describe the model and the solution. Some more details can be found in our previous paper [hep-th/0211103]. This article is dedicated to Professor Hiroshi Ezawa on the occasion of his seventieth birthday. 
  A localized configuration is found in the 5D bulk-boundary theory on an $S_1/Z_2$ orbifold model of Mirabelli-Peskin. A bulk scalar and the extra (fifth) component of the bulk vector constitute the configuration. $\Ncal=1$ SUSY is preserved. The effective potential of the SUSY theory is obtained using the background field method. The vacuum is treated in a general way by allowing its dependence on the extra coordinate. Taking into account the {\it supersymmetric boundary condition}, the 1-loop full potential is obtained. The scalar-loop contribution to the Casimir energy is also obtained. Especially we find a {\it new} type which depends on the brane configuration parameters besides the $S_1$ periodicity parameter. 
  In the Randall-Sundrum scenario we investigate the dynamics of a spherically symmetric 3-brane world when matter fields are present in the bulk. To analyze the 5-dimensional Einstein equations we employ a global conformal transformation whose factor characterizes the $Z_2$ symmetric warp. We find a new set of exact dynamical collapse solutions which localize gravity in the vicinity of the brane for a stress-energy tensor of conformal weight -4 and a warp factor that depends only on the coordinate of the fifth dimension. Geometries which describe the dynamics of inhomogeneous dust and generalized dark radiation on the brane are shown to belong to this set. The conditions for singular or globally regular behavior and the static marginally bound limits are discussed for these examples. Also explicitly demonstrated is complete consistency with the effective point of view of a 4-dimensional observer who is confined to the brane and makes the same assumptions about the bulk degrees of freedom. 
  We use analytic and numerical methods to obtain the solution of the Q-ball equation of motion. In particular, we show that the profile function of the three-dimensional Q-ball can be accurately approximated by the symmetrized Woods-Saxon distribution. 
  We study the existence of classical soliton solutions with intrinsic angular momentum in Yang-Mills-Higgs theory with a compact gauge group $\mathcal{G}$ in (3+1)-dimensional Minkowski space. We show that for \textit{symmetric} gauge fields the Noether charges corresponding to \textit{rigid} spatial symmetries, as the angular momentum, can be expressed in terms of \textit{surface} integrals. Using this result, we demonstrate in the case of $\mathcal{G}=SU(2)$ the nonexistence of stationary and axially symmetric spinning excitations for all known topological solitons in the one-soliton sector, that is, for 't Hooft--Polyakov monopoles, Julia-Zee dyons, sphalerons, and also vortices. 
  Hamiltonian systems with functionally dependent constraints (irregular systems), for which the standard Dirac procedure is not directly applicable, are discussed. They are classified according to their behavior in the vicinity of the constraint surface into two fundamental types. If the irregular constraints are multilinear (type I), then it is possible to regularize the system so that the Hamiltonian and Lagrangian descriptions are equivalent. When the constraints are power of a linear function (type II), regularization is not always possible and the Hamiltonian and Lagrangian descriptions may be dynamically inequivalent. It is shown that the inequivalence between the two formalisms can occur if the kinetic energy is an indefinite quadratic form in the velocities. It is also shown that a system of type I can evolve in time from a regular configuration into an irregular one, without any catastrophic changes. Irregularities have important consequences in the linearized approximation to nonlinear theories, as well as for the quantization of such systems. The relevance of these problems to Chern-Simons theories in higher dimensions is discussed. 
  We study how to obtain a sufficiently flat inflaton potential that is natural from the particle physics point of view. Supersymmetry, which is broken during inflation, cannot protect the potential against non-renormalizable operators violating slow-roll. We are therefore led to consider models based on non-linearly realized symmetries. The basic scenario with a single four-dimensional pseudo Nambu Goldstone boson requires the spontaneous breaking scale to be above the Planck scale, which is beyond the range of validity of the field theory description, so that quantum gravity corrections are not under control. A nice way to obtain consistent models with large field values is to consider simple extensions in extra-dimensional setups. We also consider the minimal structures necessary to obtain purely four-dimensional models with spontaneous breaking scale below M_P; we show that they require an approximate symmetry that is supplemented by either the little-Higgs mechanism or supersymmetry to give trustworthy scenarios. 
  The absorption cross section for scattering of fermions off an extreme BTZ black hole is calculated. It is shown that, as in the case of scalar particles, an extreme BTZ black hole exhibits a vanishing absorption cross section, which is consistent with the vanishing entropy of such object. Additionally, we give a general argument to prove that the particle flux near the horizon is zero. Finally we show that the {\it reciprocal space} introduced previously in \cite{gm} gives rise to the same result and, therefore, it could be considered as the space where the scattering process takes place in an AdS spacetime. 
  We briefly discuss the twisting procedure applied to the $\kappa$-deformed space-time. It appears that one can consider only two kinds of such twistings: in space and time directions. For both types of twisitngs we introduce related phase spaces and consider briefly their properties. We discuss in detail the changes of duality relations under the action of twist. The Jordanian twisted space-time and phase space in D=2 are also commented. 
  Consistent Hamiltonian interactions that can be added to an abelian free BF-type class of theories in any n greater or equal to 4 spacetime dimensions are constructed in the framework of the Hamiltonian BRST deformation based on cohomological techniques. The resulting model is an interacting field theory in higher dimensions with an open algebra of on-shell reducible first-class constraints. We argue that the Hamiltonian couplings are related to a natural structure of Poisson manifold on the target space. 
  In this letter we will dicuss the possibility of a resummation procedure in order to cure the UV/IR-mixing problem of noncommutative field theories. The method is presented for a scalar phi^4 theory on Euclidean space. Finally, we sketch the idea of resummation for U(1)-gauge theories. 
  Analytical approximations for ${< \phi^2 >}$ and ${< T^{\mu}_{\nu} >}$ of a quantized scalar field in static spherically symmetric spacetimes are obtained. The field is assumed to be both massive and massless, with an arbitrary coupling $\xi$ to the scalar curvature, and in a zero temperature vacuum state. The expressions for ${< \phi^2 >}$ and ${< T^{\mu}_{\nu} >}$ are divided into low- and high-frequency parts. The contributions of the high-frequency modes to these quantities are calculated for an arbitrary quantum state. As an example, the low-frequency contributions to ${< \phi^2 >}$ and ${< T^{\mu}_{\nu} >}$ are calculated in asymptotically flat spacetimes in a quantum state corresponding to the Minkowski vacuum (Boulware quantum state). The limits of the applicability of these approximations are discussed. 
  We note that the existence of physical states which are coherent superpositions of states with even and odd numbers of fermions means the existence, together with x,y,z,t, of additional spinor dimensions of space-time. A system with variable number of electrons is described in which such superpositions are realized. Experiments with mesoscopic condensed matter systems are suggested which generalize the experiment of Nakamura et al. and may provide direct observation of such superpositions and, thereby, justify the reality of a superspace with additional spinor dimensions introduced in quantum field theory to account for supersymmetry. The nature of additional dimensions of space-time is elucidated for nonrelativistic systems. 
  We analyse the physical boundary conditions at infinity for metric fluctuations and gauge functions in the RS2 model with matter on the brane. We argue that due to these boundary conditions the radion field cannot be gauged out in this case. Thus, it represents a physical degree of freedom of the model. 
  The problem of scattering of neutral fermions in two-dimensional space-time is approached with a pseudoscalar potential step in the Dirac equation. Some unexpected aspects of the solutions beyond the absence of Klein\'{}s paradox are presented. An apparent paradox concerning the uncertainty principle is solved by introducing the concept of effective Compton wavelength. Added plausibility for the existence of bound-state solutions in a pseudoscalar double-step potential found in a recent Letter is given. 
  We point out the existence of a class of non-Gaussian yet free "quantum field theories" in 0+0 dimensions, based on a cubic action classified by simple Lie groups. A "three-pronged" version of the Wick theorem applies. 
  We incorporate running parameters and anomalous dimensions into the framework of the exact renormalization group. We modify the exact renormalization group differential equations for a real scalar field theory, using the anomalous dimensions of the squared mass and the scalar field. Following a previous paper in which an integral equation approach to the exact renormalization group was introduced, we reformulate the modified differential equations as integral equations that define the continuum limit directly in terms of a running squared mass and self-coupling constant. Universality of the continuum limit under an arbitrary change of the momentum cutoff function is discussed using the modified exact renormalization group equations. 
  Static topologically-nontrivial configurations in sigma-models, for spatial dimension D \geq 2, are unstable. The question addressed here is whether such sigma-model solitons can be stabilized by steady rotation in internal space; that is, rotation in a global SO(2) symmetry. This is the mechanism which stabilizes Q-balls (non-topological solitons). The conclusion is that the Q-mechanism can stabilize topological solitons in D=2 spatial dimensions, but not for D=3. 
  This is a brief review of the Schrodinger's factorization method and its relations to supersymmetric quantum mechanics and its nonlinear (parastatistical, etc) modifications, self-similar infinite soliton potentials, quantum algebras, coherent states, Ising chains, discretized random matrices and 2D lattice Coulomb gases. 
  We consider warp compactifications of M-theory on 7-manifolds in the presence of 4-form fluxes and investigate the constraints imposed by supersymmetry. As long as the 7-manifold supports only one Killing spinor we infer from the Killing spinor equations that non-trivial 4-form fluxes will necessarily curve the external 4-dimensional space. On the other hand, if the 7-manifold has at least two Killing spinors, there is a non-trivial Killing vector yielding a reduction of the 7-manifold to a 6-manifold and we confirm that 4-form fluxes can be incorporated if one includes non-trivial SU(3) structures. 
  The structure of the electroweak theory is suggested by classical geometrical ideas. A nonlinear map is constructed, from a 12-dimensional linear space of three Weyl spinors onto the 12-dimensional tangent bundle of the Stiefel manifold of orthonormal tetrads associated with the Lorentz group -- except, inevitably, for a set of measure zero. In the approach of this paper, the electroweak field is more natural than the Dirac field. This may be just a curiosity since it may not survive quantization, but it suggests a path to bosonization of the electroweak field in (3+1) dimensions. 
  We describe the supergravity solutions dual to D6-branes with both time-dependent and time-independent B-fields. These backgrounds generalize the Taub-NUT metric in two key ways: they have asymmetric warp factors and background fluxes. In the time-dependent case, the warping takes a novel form. Kaluza-Klein reduction in these backgrounds is unusual, and we explore some of the new features. In particular, we describe how a localized gauge-field emerges with an analogue of the open string metric and coupling. We also describe a gravitational analogue of the Seiberg-Witten map. This provides a framework in supergravity both for studying non-commutative gauge theories, and for constructing novel warped backgrounds. 
  We set up a consistent renormalizable perturbation theory of a scalar field in a nontrivial alpha vacuum in de Sitter space. Although one representation of the effective action involves non-local interactions between anti-podal points, we show the theory leads to causal physics, and we prove a spectral theorem for the interacting two-point function. We construct the renormalized stress energy tensor and show this develops no imaginary part at leading order in the interactions, consistent with stability. 
  By combining a renormalization group argument relating the charge e and mass m of the proton by e^2 ln m ~ 0.1 pi (in Planck units) with the Carter-Carr-Rees anthropic argument that gives an independent approximate relation m ~ e^20 between these two constants, both can be crudely estimated. These equations have the factor of 0.1 pi and the exponent of 20 which depend upon known discrete parameters (e.g., the number of generations of quarks and leptons, and the number of spatial dimensions), but they contain NO continuous observed parameters. Their solution gives the charge of the proton correct to within about 8%, though the mass estimate is off by a factor of about 1000 (16% error on a logarithmic scale). When one adds a fudge factor of 10 previously given by Carr and Rees, the agreement for the charge is within about 2%, and the mass is off by a factor of about 3 (2.4% error on a logarithmic scale). If this 10 were replaced by 15, the charge agrees within 1.1% and the mass itself agrees within 0.7%. 
  Many solutions of General Relativity appear to allow the possibility of time travel. This was initially a fascinating discovery, but geometries of this type violate causality, a basic physical law which is believed to be fundamental. Although string theory is a proposed fundamental theory of quantum gravity, geometries with closed timelike curves have resurfaced as solutions to its low energy equations of motion. In this paper, we will study the class of solutions to low energy effective supergravity theories related to the BMPV black hole and the rotating Wave--D1--D5--brane system. Time travel appears to be possible in these geometries. We will attempt to build the causality violating regions and propose that stringy effects prohibit their construction. We will show how the geometry is corrected and that, once corrected, causality is preserved. We will track our chronology protection proposal in the dual conformal field theory. The absence of closed timelike curves in the geometry coincides with the preservation of unitarity in the conformal field theory. The agent of chronology protection for the geometries studied here mirrors the enhancon mechanism, a mechanism string theory employs to resolve a class of naked singularities. 
  We investigate the vacuum expectation values for the energy-momentum tensor of a massive scalar field with general curvature coupling and obeying the Robin boundary condition on a spherical shell in the $D+1$-dimensional global monopole background. The expressions are derived for the Wightman function, the vacuum expectation values of the field square, the vacuum energy density, radial and azimuthal stress components in both regions inside and outside the shell. A regularization procedure is carried out by making use of the generalized Abel-Plana formula for the series over zeros of cylinder functions. This formula allows us to extract from the vacuum expectation values the parts due to the global monopole gravitational field in the situation without a boundary, and to present the boundary induced parts in terms of exponentially convergent integrals, useful, in particular, for numerical calculations. The asymptotic behavior of the vacuum densities is investigated near the sphere surface and at large distances. We show that for small values of the parameter describing the solid angle deficit in global monopole geometry the boundary induced vacuum stresses are strongly anisotropic. 
  We calculate the difference of one-loop vacuum energies for massive scalar field in five-dimensional AdS black hole. (The same is done in five-dimensional deSitter space). In each case this difference is specified by the boundary conditions corresponding to the double-trace operator (massive term) and it describes RG flow in the manner discussed by Gubser-Mitra for pure AdS space. For AdS black hole there occurs instability which is the manifestation of the Hawking-Page phase transition. For stable phase of AdS black hole as well as for deSitter bulk, c-function found beyond the leading order approximation shows the monotonic behaviour consistent with c-theorem. 
  The Moyal *-deformed noncommutative version of Burgers' equation is considered. Using the *-analog of the Cole-Hopf transformation, the linearization of the model in terms of the linear heat equation is found. Noncommutative q-deformations of shock soliton solutions and their interaction are described 
  We review some recent results concerning the quantitative analysis of the universality classes of two-dimensional statistical models near their critical point. We also discuss the exact calculation of the two--point correlation functions of disorder operators in a free theory of complex bosonic and fermionic field, correlators ruled by a Painleve differential equation. 
  The use of the AdS/CFT correspondence to arrive at quiver gauge field theories is dicussed, focusing on the orbifolded case without supersymmetry. An abelian orbifold with the finite group $Z_{p}$ can give rise to a $G = SU(N)^p$ gauge group with chiral fermions and complex scalars in different bi-fundamental representations of $G$. The precision measurements at the $Z$ resonance suggest the values $p = 12$ and $N = 3$, and a unifications scale $M_U \sim 4$ TeV.The robustness and predictivity of such grand unification is discussed. 
  Some recent studies have implied that quantum fluctuations will prevent a near-extremal black hole from ever attaining a state of precise extremality. In this paper, we consider the analogous situation for the scenario of a nearly degenerate Schwarzschild-de Sitter black hole (of arbitrary dimensionality). For this purpose, we utilize a holographic type of duality between the solutions of interest and the near-massless sector of (two-dimensional) Jackiw-Teitelboim theory. After explicitly demonstrating this duality, we go on to argue that, on the basis of one-loop considerations, a similar censorship applies in this de Sitter context. That is, quantum back-reaction effects will conspire to prohibit a non-degenerate Schwarzschild-de Sitter spacetime from continuously evolving into a degenerate (i.e., Nariai) black hole solution. 
  We construct chiral N=(1,0) self-dual supergravity in Euclidean eight-dimensions with reduced holonomy Spin(7), including all the higher-order interactions in a closed form. We first establish the non-chiral N=(1,1) superspace supergravity in eight-dimensions with SO(8) holonomy without self-duality, as the foundation of the formulation. In order to make the whole computation simple, and the generalized self-duality compatible with supersymmetry, we adopt a particular set of superspace constraints similar to the one originally developed in ten-dimensional superspace. The intrinsic properties of octonionic structure constants make local supersymmetry, generalized self-duality condition, and reduced holonomy Spin(7) all consistent with each other. 
  We carefully analyze the supersymmetry algebra of closed strings and open strings in a type IIB plane wave background. We use eight component chiral spinors, SO(8) Majorana-Weyl spinors, in light-cone gauge to provide a useful basis of string field theory calculation in the plane wave. We consider the two classes of D-branes, $D_\pm$-branes, and give a worldsheet derivation of conserved supercurrents for all half BPS D-branes preserving 16 supersymmetries in the type IIB plane wave background. We exhaustively provide the supersymmetry algebra of the half BPS branes as well. We also point out that the supersymmetry algebra distinguishes the two SO(4) directions with relative sign which is consistent with the Z_2 symmetry of the string action. 
  By putting a confined inter source, we construct a model which can give us convergent solution from free field equation. On the other hand, the solution of new field equation can be separated into two parts, one part is just same as the one in Quantum Field Theory and make it survived in this model, and the other part, which we will see doesn't take energy and momentum, just gives us a negative propagator which can soften quadratic divergence. 
  Motivated by the work of Mersini, the particle production related to the tunneling in false vacuum decay is carefully investigated in the thin-wall approximation. It is shown that in this case the particle production is exponentially suppressed even when the momentum is comparable to the curvature scale of the bubble. The number of created particles is ultraviolet finite. 
  We argue that the AdS dual of the three dimensional critical O(N) vector model can be evaluated using the Legendre transform that relates the generating functionals of the free UV and the interacting IR fixed points of the boundary theory. As an example, we use our proposal to evaluate the minimal bulk action of the scalar field that it is dual to the spin-zero ``current'' of the O(N) vector model. We find that the cubic bulk self interaction coupling vanishes. We briefly discuss the implications of our results for higher spin theories and comment on the bulk-boundary duality for subleading N. 
  We calculate 3-point correlation functions of Delta-BMN operators with 3 scalar impurites in N=4 supersymmetric gauge theory. We use these results to test the pp-wave/SYM duality correspondence of the vertex--correlator type. This correspondence relates the coefficients of 3-point correlators of Delta-BMN operators in gauge theory to the 3-string vertex in lightcone string field theory in the pp-wave background. We verify the vertex--correlator duality equation of hep-th/0301036 at the 3 scalar impurites level for supergravity and for string modes. 
  We argue that a consistent definition of the velocity of a particle in generalizations of special relativity with two observer-independent scales should be independent from the mass of the particle. This request rules out the definition $v_i=\partial p_0/\partial p_i$, but allows for other definitions proposed in the literature. 
  This note is the written version of a talk which was presented at the Cargese 2002 summer school. It gives a brief introduction to the paper hep-th/0205281, written in collaboration with R. Dijkgraaf and E. Verlinde. In this paper, we calculate the euclidean partition function of the type IIA NS five-brane wrapped on an arbitrary Calabi-Yau space in a double-scaling decoupling limit and in the presence of a flat RR 3-form background field. The result is the product of a theta function, coming from the classical fluxes of the self-dual tensor field, and a factor representing the quantum contributions. The quantum factor turns out to be related to topological B-model string amplitudes, and both factors satisfy a holomorphic anomaly equation. The result can teach us more about little string theories and about instanton corrections to four-dimensional effective quantities. 
  We present an exact solution for a factorizable brane-world spacetime with two extra dimensions and explicit brane sources. The compactification manifold has the topology of a two-sphere, and is stabilized by a bulk cosmological constant and magnetic flux. The geometry of the sphere is locally round except for conical singularities at the locations of two antipodal branes, deforming the sphere into an American-style football. The bulk magnetic flux needs to be fine-tuned to obtain flat geometry on the branes. Once this is done, the brane geometry is insensitive to the brane vacuum energy, which only affects the conical deficit angle of the extra dimensions. Solutions of this form provide a new arena in which to explore brane-world phenomenology and the effects of extra dimensions on the cosmological constant problem. 
  A class of metrics $g_{ab}(x^i)$ describing spacetimes with horizons (and associated thermodynamics) can be thought of as a limiting case of a family of metrics $g_{ab}(x^i;\lambda)$ {\it without horizons} when $\lambda\to 0$. I construct specific examples in which the curvature corresponding $g_{ab}(x^i;\lambda)$ becomes a Dirac delta function and gets concentrated on the horizon when the limit $\lambda\to 0$ is taken, but the action remains finite. When the horizon is interpreted in this manner, one needs to remove the corresponding surface from the Euclidean sector, leading to winding numbers and thermal behaviour. In particular, the Rindler spacetime can be thought of as the limiting case of (horizon-free) metrics of the form [$g_{00}=\epsilon^2+a^2x^2; g_{\mu\nu}=-\delta_{\mu\nu}$] or [$g_{00} = - g^{xx} = (\epsilon^2 +4 a^2 x^2)^{1/2}, g_{yy}=g_{zz}=-1]$ when $\epsilon\to 0$. In the Euclidean sector, the curvature gets concentrated on the origin of $t_E-x$ plane in a manner analogous to Aharanov-Bohm effect (in which the the vector potential is a pure gauge everywhere except at the origin) and the curvature at the origin leads to nontrivial topological features and winding number. 
  We consider a brane scenario with a massive scalar field in the five-dimensional bulk. We study the scalar states that are localized on the brane, which is assumed to be de Sitter. These localized scalar modes are massive in general, their effective four-dimensional mass depending on the mass of the five-dimensional scalar field, on the Hubble parameter in the brane and on the coupling between the brane tension and the bulk scalar field. We then introduce a purely four-dimensional approach based on an effective potential for the projection of the scalar field in the brane, and discuss its regime of validity. Finally, we explore the quasi-localized scalar states, which have a non-zero width that quantifies their probability of tunneling from the brane into the bulk. 
  This article provides a basic introduction to some concepts of non-commutative geometry. The importance of quantum groups and quantum spaces is stressed. Canonical non-commutativity is understood as an approximation to the quantum group case. Non-commutative gauge theory and the non-commutative Standard Model are formulated on a space-time satisfying canonical non-commutativity relations. We use *-formalism and Seiberg-Witten maps. 
  Calibration technology provides us with a fast and elegant way to find the supergravity solutions for BPS wrapped M-branes. Its true potential had however remained untapped due to the absence of a classification of calibrations in spacetimes with non-trivial flux. The applications of this method were thus limited in practise to M-branes wrapping Kahler calibrated cycles. In this paper, we catagorize a type of generalised calibrations which exist in supergravity backgrounds and contain Kahler calibrations as a sub-class. This broadens the arena of brane configurations whose supergravity solutions are accessible through the calibration 'short-cut' method. 
  We study the process of compactification as a topology change. It is shown how the mediating spacetime topology, or cobordism, may be simplified through surgery. Within the causal Lorentzian approach to quantum gravity, it is shown that any topology change in dimensions $\geq 5$ may be achieved via a causally continuous cobordism. This extends the known result for 4 dimensions. Therefore, there is no selection rule for compactification at the level of causal continuity. Theorems from surgery theory and handle theory are seen to be very relevant for understanding topology change in higher dimensions. Compactification via parallelisable cobordisms is particularly amenable to study with these tools. 
  Kaigorodov spaces arise, after spherical compactification, as near horizon limits of M2, M5, and D3-branes with a particular pp-wave propagating in a world volume direction. We show that the uncompactified near horizon configurations K\times S are solutions of D=11 or D=10 IIB supergravity which correspond to perturbed versions of their AdS \times S analogues. We derive the Penrose-Gueven limits of the Kaigorodov space and the total spaces and analyse their symmetries. An Inonu-Wigner contraction of the Lie algebra is shown to occur, although there is a symmetry enhancement. We compare the results to the maximally supersymmetric CW spaces found as limits of AdS\times S spacetimes: the initial gravitational perturbation on the brane and its near horizon geometry remains after taking non-trivial Penrose limits, but seems to decouple. One particuliar limit yields a time-dependent homogeneous plane-wave background whose string theory is solvable, while in the other cases we find inhomogeneous backgrounds. 
  Noncommutative Maxwell-Chern-Simons theory in 3-dimensions is defined in terms of star product and noncommutative fields. Seiberg-Witten map is employed to write it in terms of ordinary fields. A parent action is introduced and the dual action is derived. For spatial noncommutativity it is studied up to second order in the noncommutativity parameter \theta. A new noncommutative Chern-Simons action is defined in terms of ordinary fields, inspired by the dual action. Moreover, a transformation between noncommuting and ordinary fields is proposed. 
  We study a set of chiral symmetries contained in degenerate operators beyond the `minimal' sector of the c(p,q) models. For the operators h_{(2j+2)q-1,1}=h_{1,(2j+2)p-1} at conformal weight [ (j+1)p-1 ][ (j+1)q -1 ], for every 2j \in N, we find 2j+1 chiral operators which have quantum numbers of a spin j representation of SU(2). We give a free-field construction of these operators which makes this structure explicit and allows their OPEs to be calculated directly without any use of screening charges. The first non-trivial chiral field in this series, at j=1/2, is a fermionic or para-fermionic doublet. The three chiral bosonic fields, at j=1, generate a closed W-algebra and we calculate the vacuum character of these triplet models. 
  We study geometric transitions for topological strings on compact Calabi-Yau hypersurfaces in toric varieties. Large N duality predicts an equivalence between topological open and closed string theories connected by an extremal transition. We develop new open string enumerative techniques and perform a high precision genus zero test of this conjecture for a certain class of toric extremal transitions. Our approach is based on a) an open string version of Gromov-Witten theory with convex obstruction bundle and b) an extension of Chern-Simons theory treating the framing as a formal variable. 
  In here the matrix model approach, by Dijkgraaf and Vafa, is used in order to obtain the effective superpotential for a certain deformation of N=4 SYM discovered by Leigh and Strassler. An exact solution to the matrix model Lagrangian is found and is expressed in terms of elliptic functions. 
  Turning on background fields in string theory sometimes has an alternative interpretation as a deformation of the target space geometry. A particularly well-known case is the NS-NS two form B, which gives rise to space-time non-commutativity. In this note we point out that this phenomenon extends to ten-dimensional superspace when employing a covariant quantization of the superstring, generalizing an observation by Ooguri and Vafa in four dimensions. In particular, we will find that RR field strengths give rise to a non-zero $\{\theta,\theta\}$ anti-commutator, just as in four dimensions, whereas the gravitino yields a non-zero value for $[x,\theta]$. 
  A new mneumonic device is shown to emerge in connection with O(7) numerical tensors exhibiting duality and reflecting the natural 7=(4+3) splitting of 7-dimensional space. Then Desargues' and Pappus' theorems are shown to be connected through a geometry that makes use of octonionic numbers exhibiting this duality. Construction of exceptional Hilbert space based on Jordan algebras and exceptional projective geometries is illustrated. A brief discussion of the Moufang plane and non-Desarguesian geometries is presented. 
  Traditional derivations of the Planck mass ignore the role of charge and spin in general relativity. From the Kerr-Newman null surface and horizon radii, quantized charge and spin dependence are introduced in an extended Planck scale of mass. Spectra emerge with selection rules dependent upon the choice of Kerr-Newman radius to link with the Compton wavelength. The appearance of the fine structure constant suggests the possibility of a variation in time of the extended Planck mass, which may be much larger than the variation in the traditional one. There is a suggestion of a connection with the $\alpha$ value governing high-energy radiation in Z-boson production and decay. 
  RR fluxes representing different cohomology classes may correspond to the same twisted K-theory class. We argue that such fluxes are related by monodromies, generalizing and sometimes T-dual to the familiar monodromies of a D7-brane. A generalized theta angle is also transformed, but changes by a multiple of 2pi. As an application, NS5-brane monodromies modify the twisted K-theory classification of fluxes. Furthermore, in the noncompact case K-theory does not distinguish flux configurations in which dG is nontrivial in compactly supported cohomology. Such fluxes are realized as the decay products of unstable D-branes that wrapped nontrivial cycles. This is interpreted using the E8 bundle formalism. 
  A detailed investigation is presented of the energy-momentum tensor approach to the evaluation of the force acting on a rigid Casimir cavity in a weak gravitational field. Such a force turns out to have opposite direction with respect to the gravitational acceleration. The order of magnitude for a multi-layer cavity configuration is derived and experimental feasibility is discussed, taking into account current technological resources. 
  We study {\cal N}=2 SO(2N+1) SYM theory in the context of matrix model. By adding a superpotential of the scalar multiplet, W(\Phi), of degree 2N+2, we reduce the theory to {\cal N}=1. The 2N+1 distinct critical points of W(\Phi) allow us to choose a vacuum in such a way to break the gauge group to its maximal abelian subgroup. We compute the free energy of the corresponding matrix model in the planar limit and up to two vertices. This result is then used to work out the effective superpotential of {\cal N}=1 theory up to one-instanton correction. At the final step, by scaling the superpotential to zero, the effective U(1) couplings and the prepotential of the {\cal N}=2 theory are calculated which agree with the previous results. 
  This article treats the generalisation to brane dynamics of the covariant canonical variational procedure leading to the construction of a conserved bilinear symplectic current in the manner originally developped by Witten, Zuckerman and others in the context of field theory. After a general presentation, including a review of the relationships between the various (Lagrangian, Eulerian and other) relevant kinds of variation, the procedure is illustrated by application to the particularly simple case of branes of the Dirac-Goto-Nambu type, in which internal fields are absent. 
  We discuss properties of a 4-dimensional Schwarzschild black hole in a spacetime where one of the spatial dimensions is compactified. As a result of the compactification the event horizon of the black hole is distorted. We use Weyl coordinates to obtain the solution describing such a distorted black hole. This solution is a special case of the Israel-Khan metric. We study the properties of the compactified Schwarzschild black hole, and develop an approximation which allows one to find the size, shape, surface gravity and other characteristics of the distorted horizon with a very high accuracy in a simple analytical form. We also discuss possible instabilities of a black hole in the compactified space. 
  We consider sound propagation on M5- and M2-branes in the hydrodynamic limit. In particular, we look at the low energy description of a stack of N M-branes at finite temperature. At low energy, the M-branes are well described, via the AdS/CFT correspondence, in terms of classical solutions to the eleven dimensional supergravity equations of motion. From this gravitational description, we calculate Lorentzian signature two-point functions of the stress-energy tensor on these M-branes in the long-distance, low-frequency limit, i.e. the hydrodynamic limit. The poles in these Green's functions show evidence for sound propagation in the field theory living on the M-branes. 
  We extend the method of calculation of propagators in maximally symmetric spaces (Minkowski, dS, AdS and their Euclidean versions) in terms of intrinsic geometric objects to the case of massive spin 3/2 field. We obtain the propagator for arbitrary space-time dimension and mass in terms of Heun's function, which is a generalization of the hypergeometric function appearing in the case of other spins. As an application of this result we calculate the conformal dimension of the dual operator in the recently proposed dS/CFT correspondence both for spin 3/2 and for spin 1/2. We find that, in agreement with the expectation from analytic continuation from AdS, the conformal dimension of the dual operator is {\it always} complex (i.e. it is complex for every space-time dimension and value of the mass parameter). We comment on the implications of this result for fermions in dS/CFT. 
  We show that with every classical system possessing first class constraints that form a natural Lie algebra, we can associate a superalgebra that admits the constraint Lie algebra as a subalgebra. An odd generator of this superalgebra that commutes with the constraints is shown to be the BRST operator whose form follows from a non linear coset representation of the superalgebra. We further show the existence of the superalgebra for all Yang-Mills theories and for 26-dimensional bosonic strings. 
  We provide a systematic study on the possibility of supersymmetry (SUSY) for one dimensional quantum mechanical systems consisting of a pair of lines $\R$ or intervals [-l, l] each having a point singularity. We consider the most general singularities and walls (boundaries) at $x = \pm l$ admitted quantum mechanically, using a U(2) family of parameters to specify one singularity and similarly a U(1) family of parameters to specify one wall. With these parameter freedoms, we find that for a certain subfamily the line systems acquire an N = 1 SUSY which can be enhanced to N = 4 if the parameters are further tuned, and that these SUSY are generically broken except for a special case. The interval systems, on the other hand, can accommodate N = 2 or N = 4 SUSY, broken or unbroken, and exhibit a rich variety of (degenerate) spectra. Our SUSY systems include the familiar SUSY systems with the Dirac $\delta(x)$-potential, and hence are extensions of the known SUSY quantum mechanics to those with general point singularities and walls. The self-adjointness of the supercharge in relation to the self-adjointness of the Hamiltonian is also discussed. 
  The Wilsonian renormalization group (WRG) equation is used to derive a new class of scale invariant field theories with nonvanishing anomalous dimensions in 2-dimensional ${\cal N}=2$ supersymmetric nonlinear sigma models. When the coordinates of the target manifolds have nontrivial anomalous dimensions, vanishing of the $\beta$ function suggest the existence of novel conformal field theories whose target space is not Ricci flat. We construct such conformal field theories with ${\bf U}(N)$ symmetry. The theory has one free parameter a corresponding to the anomalous dimension of the scalar fields. The new conformal field theories are well behaved for positive a and have the central charge 3N, while they have curvature singularities at the boundary for a<0. When the target space is of complex 1-dimension, we obtain the explicit form of the Lagrangian, which reduces to two different kinds of free field theories in weak and in strong coupling limit. As a consistency test, the anomalous dimensions are reproduced in these two limits. The target space in this case looks like a semi-infinite cigar with one-dimension compactified to a circle. 
  We prove the uniqueness theorem for static higher dimensional charged black holes spacetime containing an asymptotically flat spacelike hypersurface with compact interior and with both degenerate and non-degenerate components of the event horizon. 
  We confirm earlier hints that the conventional phase diagram of the discrete chiral Gross-Neveu model in the large N limit is deficient at non-zero chemical potential. We present the corrected phase diagram constructed in mean field theory. It has three different phases, including a kink-antikink crystal phase. All transitions are second order. The driving mechanism for the new structure of baryonic matter in the Gross-Neveu model is an Overhauser type instability with gap formation at the Fermi surface. 
  We consider the massless tricritical Ising model M(4,5) perturbed by the thermal operator in a cylindrical geometry and apply integrable boundary conditions, labelled by the Kac labels (r,s), that are natural off-critical perturbations of known conformal boundary conditions. We derive massless thermodynamic Bethe ansatz (TBA) equations for all excitations by solving, in the continuum scaling limit, the TBA functional equation satisfied by the double-row transfer matrices of the A_4 lattice model of Andrews, Baxter and Forrester (ABF) in Regime IV. The resulting TBA equations describe the massless renormalization group flow from the tricritical to critical Ising model. As in the massive case of Part I, the excitations are completely classified in terms of (m,n) systems but the string content changes by one of three mechanisms along the flow. Using generalized q-Vandemonde identities, we show that this leads to a flow from tricritical to critical Ising characters. The excited TBA equations are solved numerically to follow the continuous flows from the UV to the IR conformal fixed points. 
  A set of Markov processes corresponding to systems of hard-core particles interacting along the line are shown to be solvable via a dynamic matrix product ansatz (DMPA). We show that quantum spin Hamiltonians can be treated by the DMPA as well, and demonstrate how the DMPA, originally formulated for systems with open ends, works for periodic systems. 
  It is shown that, to the lowest order in $\hbar,$ the particle production related to the tunneling that leads to the false vacuum decay is described by the orthogonal part of fluctuation field with respect to the bounce solution. As a simple example the spatially homogeneous tunneling is considered in order to illustrate the consequences coming from such a restriction of the fluctuation field. 
  We study the dyon electric charge of D6-branes as eleven dimensional KK-monopoles. We observe that the dyon charge is intimately related with the existence of gauge connections and antisymmetric fields on the brane world volume. 
  We study free massive fermionic ghosts, in the presence of an extended line of impurities, relying on the Lagrangian formalism. We propose two distinct defect interactions, respectively, of relevant and marginal nature. The corresponding scattering theories reveal the occurrence of resonances and instabilities in the former case and the presence of poles with imaginary residues in the latter. Correlation functions of the thermal and disorder operators are computed exactly, exploiting the bulk form factors and the matrix elements relative to the defect operator. In the marginal situation, the one-point function of the disorder operator displays a critical exponent continuously varying with the interaction strength. 
  We describe a ten dimensional supergravity geometry which is dual to a gauge theory that is non-supersymmetric Yang Mills in the infra-red but reverts to $N$=4 super Yang Mills in the ultra-violet. A brane probe of the geometry shows that the scalar potential of the gauge theory is stable. We discuss the infra-red behaviour of the solution. The geometry describes a Schroedinger equation potential that determines the glueball spectrum of the theory; there is a mass gap and a discrete spectrum. The glueball mass predictions match previous AdS/CFT Correspondence computations in the non-supersymmetric Yang Mills theory, and lattice data, at the 10% level. (Based on a talk presented at SCGT02 in Nagoya, Japan) 
  In this paper we extend previous work on calculating massless boundary Ramond sector spectra of open strings to include cases with nonzero flat B fields. In such cases, D-branes are no longer well-modelled precisely by sheaves, but rather they are replaced by `twisted' sheaves, reflecting the fact that gauge transformations of the B field act as affine translations of the Chan-Paton factors. As in previous work, we find that the massless boundary Ramond sector states are counted by Ext groups -- this time, Ext groups of twisted sheaves. As before, the computation of BRST cohomology relies on physically realizing some spectral sequences. Subtleties that cropped up in previous work also appear here. 
  The spectrum of the Dirac Hamiltonian in the background of crossing vortices is studied. To exploit the index theorem, and in analogy to the lattice the space-time manifold is chosen to be the four-torus $\T^4$. For sake of simplicity we consider two idealized cases: infinitely fat and thin transversally intersecting vortices. The time-dependent spectrum of the Dirac Hamiltonian is calculated and in particular the influence of the vortex crossing on the quark spectrum is investigated. For the infinitely fat intersecting vortices it is found that zero modes of the four-dimensional Dirac operator can be expressed in terms of those eigenspinors of the Euclidean time-dependent Dirac Hamiltonian, which cross zero energy. For thin intersecting vortices the time gradient of the spectral flow of the Dirac Hamiltonian is steepest at the time at which the vortices cross each other. 
  An approximate hadronic symmetry based on spin and flavor independence and broken by spin and mass dependent terms is shown to follow from QCD. This symmetry justifies the SU(6) classification scheme, but is more general in allowing its supersymmetric extension based on a diquark-antiquark symmetry. It will be shown that the same supersymmetry is also implied in the skyrmion type effective Lagrangian which could be extracted from QCD. Predictions of the Skyrme model is improved by using different realizations of the chiral group. 
  Vertex symmetry for interacting fermions will be shown to lead to a Lagrangian exhibiting $SU(2N)_W$ invariance associated with the subgroup $SU(2N)_q \times SU(2N)_{\bar{q}}$ generated by $C$-odd and $C$-even spin operators. Approximate $SU(6)_W$ vertex symmetry as well as chiral invariance will then be shown to follow from a principle of maximum smoothness (M\"oller-Rosenfeld) of the bound state quark wave function. 
  In a recent paper by Johan Hansson [hep-ph/0208137] it is claimed that the non-appearance of quarks and gluons as physical particles is an automatic result of the nonabelian nature of the color interaction in quantum chromodynamics. It is shown that the arguments given by Hansson are insufficient to support his claim by giving simple counter arguments. 
  We show that the string bit model suffers from doubling in the fermionic sector. The doubling leads to strong violation of supersymmetry in the limit $N\to\infty$. Since there is an exact correspondence between string bits and the algebra of BMN operators even at finite $N$, doubling is expected also on the side of super-Yang--Mills theory. We discuss the origin of the doubling in the BMN sector. 
  We compute the Yukawa couplings among chiral fields in toroidal Type II compactifications with wrapping D6-branes intersecting at angles. Those models can yield realistic standard model spectrum living at the intersections. The Yukawa couplings depend both on the Kahler and open string moduli but not on the complex structure. They arise from worldsheet instanton corrections and are found to be given by products of complex Jacobi theta functions with characteristics. The Yukawa couplings for a particular intersecting brane configuration yielding the chiral spectrum of the MSSM are computed as an example. We also show how our methods can be extended to compute Yukawa couplings on certain classes of elliptically fibered CY manifolds which are mirror to complex cones over del Pezzo surfaces. We find that the Yukawa couplings in intersecting D6-brane models have a mathematical interpretation in the context of homological mirror symmetry. In particular, the computation of such Yukawa couplings is related to the construction of Fukaya's category in a generic symplectic manifold. 
  We show that the $c=1$ bosonic string theory at finite temperature has two matrix-model realizations related by a kind of duality transformation. The first realization is the standard one given by the compactified matrix quantum mechanics in the inverted oscillator potential. The second realization, which we derive here, is given by the normal matrix model. Both matrix models exhibit the Toda integrable structure and are associated with two dual cycles (a compact and a non-compact one) of a complex curve with the topology of a sphere with two punctures. The equivalence of the two matrix models holds for an arbitrary tachyon perturbation and in all orders in the string coupling constant. 
  In hep-th/0004063 Pilch and Warner (PW) constructed N=2 supersymmetric RG flow corresponding to the mass deformation of the N=4 SU(N) Yang-Mills theory. In this paper we present exact deformations of PW flow when the gauge theory 3-space is compactified on S^3. We consider also the case with the gauge theory world-volume being dS_4 instead of R^{3,1}. The solution is constructed in five-dimensional gauged supergravity and is further uplifted to 10d. 
  This paper deals with magnetic equations of the type dH=J where the current J is a delta-function on a brane worldvolume and H a p-form field strength. In many situations in M-theory this equation needs to be solved for H in terms of a potential. A standard universality class of solutions, involving Dirac-branes, gives rise to strong intermediate singularities in H which in many physically relevant cases lead to inconsistencies. In this paper we present an alternative universality class of solutions for magnetic equations in terms of Chern-kernels, and provide relevant applications, among which the anomaly-free effective action for open M2-branes ending on M5-branes. The unobservability of the Dirac-brane requires a Dirac quantization condition; we show that the requirement of ``unobservability'' of the Chern-kernel leads in M-theory to classical gravitational anomalies which cancel precisely their quantum counterparts. 
  We consider a deformation of N=1 supersymmetric gauge theories in four dimensions, which we call the C-deformation, where the gluino field satisfies a Clifford-like algebra dictated by a self-dual two-form, instead of the standard Grassmannian algebra. The superpotential of the deformed gauge theory is computed by the full partition function of an associated matrix model (or more generally a bosonic gauge theory), including non-planar diagrams. In this identification, the strength of the two-form controls the genus expansion of the matrix model partition function. For the case of pure N=1 Yang-Mills this deformation leads to the identification of the all genus partition function of c=1 non-critical bosonic string at self-dual radius as the glueball superpotential. Though the C-deformation violates Lorentz invariance, the deformed F-terms are Lorentz invariant and the Lorentz violation is screened in the IR. 
  We study the physics of a single discrete gravitational extra dimension using the effective field theory for massive gravitons. We first consider a minimal discretization with 4D gravitons on the sites and nearest neighbor hopping terms. At the linear level, 5D continuum physics is recovered correctly, but at the non-linear level the theory becomes highly non-local in the discrete dimension. There is a peculiar UV/IR connection, where the scale of strong interactions at high energies is related to the radius of the dimension. These new effects formally vanish in the limit of zero lattice spacing, but do not do so quickly enough to reproduce the continuum physics consistently in an effective field theory up to the 5D Planck scale. Nevertheless, this model does make sense as an effective theory up to energies parametrically higher than the compactification scale. In order to have a discrete theory that appears local in the continuum limit, the lattice action must have interactions between distant sites. We speculate on the relevance of these observations to the construction of finite discrete theories of gravity in four dimensions. 
  We provide a precise definition and analysis of quantum causal histories (QCH). A QCH consists of a discrete, locally finite, causal pre-spacetime with matrix algebras encoding the quantum structure at each event. The evolution of quantum states and observables is described by completely positive maps between the algebras at causally related events. We show that this local description of evolution is sufficient and that unitary evolution can be recovered wherever it should actually be expected. This formalism may describe a quantum cosmology without an assumption of global hyperbolicity; it is thus more general than the Wheeler-DeWitt approach. The structure of a QCH is also closely related to quantum information theory and algebraic quantum field theory on a causal set. 
  We first study a free particle on an $(n-1)$-sphere in an extended phase space, where the originally second-class Hamiltonian and constraints are now in strong involution. This allows for a Schr\"odinger representation and a Hamilton-Jacobi formulation of the model. We thereby obtain the free particle energy spectrum corresponding to that of a rigid rotator. We extend these considerations to a modified version of the field theoretical O(3) nonlinear sigma model, and obtain the corresponding energy spectrum as well as BRST Lagrangian. 
  Quaternionic and octonionic realizations of Clifford algebras and spinors are classified and explicitly constructed in terms of recursive formulas. The most general free dynamics in arbitrary signature space-times for both quaternionic and octonionic spinors is presented. In the octonionic case we further provide a systematic list of results and tables expressing, e.g., the relations of the octonionic Clifford algebras with the $G_2$ cosets over the Lorentz algebras, the identities satisfied by the higher-rank antisymmetric octonionic tensors and so on. Applications of these results range from the classification of octonionic generalized supersymmetries, the construction of octonionic superstrings, as well as the investigations concerning the recently discovered octonionic $M$-superalgebra and its superconformal extension. 
  We discuss the use of the language of operads to try to understand holography and through that quantum gravity. As we find out, the Deligne Conjecture and the action of the Grothendieck-Teichmuller group play an important role. 
  We study the unitarity bounds of the scattering amplitudes in the extra dimensional gauge theory where the gauge symmetry is broken by the boundary condition. The estimation of the amplitude of the diagram including four massive gauge bosons in the external lines shows that the asymptotic power behavior of the amplitude is canceled. The calculation will be done in the 5 dimensional standard model and the SU(5) grand unified theory, whose 5th dimensional coordinate is compactified on $S^1/Z_2$. The broken gauge theories through the orbifolding preserve the unitarity at high energies similarly to the broken gauge theories where the gauge bosons obtain their masses through the Higgs mechanism. We show that the contributions of the Kaluza-Klein states play a crucial role in conserving the unitarity. 
  We show that in applications of variational theory to quantum field theory it is essential to account for the correct Wegner exponent omega governing the approach to the strong-coupling, or scaling limit. Otherwise the procedure either does not converge at all or to the wrong limit. This invalidates all papers applying the so-called delta-expansion to quantum field theory. 
  Using the Nahm transform we investigate doubly periodic charge one SU(2) instantons with radial symmetry. Two special points where the Nahm zero modes have softer singularities are identified as the locations of instanton core constituents. For a square torus this constituent picture is closely reflected in the action density. In rectangular tori with large aspect ratios the cores merge to form monopole-like objects. For particular values of the parameters the torus can be cut in half yielding two copies of a twisted charge 1/2 instanton. These findings are illustrated with plots of the action density within a two-dimensional slice containing the constituents. 
  We consider evolution of compact extra dimensions in cosmology and try to see whether wrapped branes can prevent the expansion of the internal space. Some difficulties of Brandenberger and Vafa mechanism for decompactification are pointed out. In both pure Einstein and dilaton gravities, we study cosmology of winding brane gases in a continuum approximation. The energy momentum tensor is obtained by coupling the brane action to the gravity action and we present several exact solutions for various brane configurations. T-duality invariance of the solutions are established in dilaton gravity. Our results indicate that phenomenologically the most viable scenario can be realized when there is only one brane wrapping over all extra dimensions. 
  The response of the one loop effective action for a gauge theory with local couplings $g(x),\theta(x)$ under a local Weyl rescaling of the background metric is calculated. Apart from terms which may be removed by local contributions to the effective action the result is compatible with $Sl(2,R)$ symmetry acting on $g,\theta$. Two loop effects are also discussed. 
  We survey noncommutative spacetimes with coordinates being enveloping algebras of Lie algebras. We also explain how to do differential geometry on noncommutative spaces that are obtained from commutative ones via a Moyal-product type cocycle twist, such as the noncommutative torus, $\theta$-spaces and Clifford algebras. The latter are noncommutative deformations of the finite lattice $(\Z_2)^n$ and we compute their noncommutative de Rham cohomology and moduli of solutions of Maxwell's equations. We exactly quantize noncommutative U(1)-Yang-Mills theory on $\Z_2\times\Z_2$ in a path integral approach. 
  The light-front quantization of gauge theories in light-cone gauge provides a frame-independent wavefunction representation of relativistic bound states, simple forms for current matrix elements, explicit unitary, and a trivial vacuum. The light-front Hamiltonian form of QCD provides an alternative to lattice gauge theory for the computation of nonperturbative quantities such as the hadronic spectrum and the corresponding eigenfunctions. In the case of the electroweak theory, spontaneous symmetry breaking is represented by the appearance of zero modes of the Higgs field. Light-front quantization then leads to an elegant ghost-free theory of massive gauge particles, automatically incorporating the Lorentz and 't Hooft conditions, as well as the Goldstone boson equivalence theorem. 
  In these talks we review some of the recent results on open strings and noncommutative gauge theories, starting from the early calculations of open strings in a constant electromagnetic background. We discuss both the neutral string and the charged string. In the latter case, the scaling limit that leads to noncommutative abelian gauge theory can be generalized to a scaling limit in which multiple noncommutativity parameters enter. Our approach corresponds to expanding a theory with U(N) Chan-Paton factors around a background U(1)^N gauge field with different magnetic fields in each U(1). This scaling limit can be interpreted in terms of a matrix model. We also describe an open string model with a time-dependent noncommutativity parameter. This model is the open string version of a WZW model based on a non-semi-simple group. It has a time-dependent background, and a spacetime metric of the plane wave type supported by a Neveu-Schwarz two-form potential. 
  We give a simplified method to generate two types of zero-norm states in the old covariant first quantized (OCFQ) spectrum of open bosonic string. Zero-norm states up to the fourth massive level and general formulas of some zero-norm tensor states at an arbitrary mass level are calculated. On-shell Ward identities generated by zero-norm states and the factorization property of stringy vertex operators can then be used to argue that the string-tree scattering amplitudes of the degenerate lower spin propagating states are fixed by those of higher spin propagating states at each fixed mass level. This decoupling phenomenon is, in contrast to Gross's high-energy symmetries, valid to all energy. As examples, we explicitly demonstrate this stringy phenomenon up to the fourth massive level (spin-five), which justifies the calculation of two other previous approaches based on the massive worldsheet sigma-model and Witten's string field theory (WSFT). 
  We present a method for extracting tunnelling amplitudes from perturbation expansions which are always divergent and not Borel-summable. We show that they can be evaluated by an analytic continuation of variational perturbation theory. The power of the method is illustrated by calculating the imaginary parts of the partition function of the anharmonic oscillator in zero spacetime dimensions and of the ground state energy of the anharmonic oscillator for all negative values of the coupling constant $g$ and show that they are in excellent agreement with the exactly known values. As a highlight of the theory we recover from the divergent perturbation expansion of the tunnelling amplitude the action of the instanton and the effects of higher loop fluctuations around it. 
  In this thesis we review the fundamental framework of boundary string field theory (BSFT) and apply it to the tachyon condensation on non-BPS systems in the superstring theory. The boundary string field theory can be regarded as a natural extension of the nonlinear sigma model. By using this theory we can describe the tachyon condensation exactly and also obtain the effective actions on non-BPS systems consisting of the Dirac-Born-Infeld type action and the Wess-Zumino type action. Especially the Wess-Zumino action is written by superconnection and coincides with the mathematical argument by K-theory. Moreover we also discuss the tachyon condensation keeping the conformal invariance (on-shell). The exact argument by using the boundary state formalism gives a good support to the conjecture of the tachyon condensation and it is also consistent with boundary string field theory. 
  A velocity of a point particle in the kappa-Minkowski spacetime is investigated. Characteristic points of the spacetime are that the Poincare group becomes a quantum group with kappa, which is a mass dimension parameter, and is a kind of non-commutative geometry. We consider a particle in a coordinate space instead of it in a momentum space which is discussed in many articles. We see that the particle's velocity has an uncertainty which depends on a length of particle's propagation. 
  We study superstring theories on the Penrose limit of the enhancon geometry realized by the D(p+4)-branes wrapped on a K3 surface. We first examine the null geodesics with fixed radius in general brane backgrounds, which give solvable superstring theories with constant masses. In most cases, the superstring theories contain negative mass-squared. We clarify a condition that the world-sheet free fields have positive mass-squared. We then apply this condition to the enhancon geometry and find that the null geodesics with fixed radius exist only for p=0 case. They define the superstring theories with positive mass-squared. For p>0 case, we show that there is no null geodesic with fixed radius. We also discuss the decoupling limit which gives the dual geometry of super Yang-Mills theory with 8 supercharges. We discuss the K3-volume dependence of the superstring spectrum. 
  We discuss the 2-point-particle-irreducible (2PPI) expansion, which sums bubble graphs to all orders, in the context of SU(N) Yang-Mills theory in the Landau gauge. Using the method we investigate the possible existence of a gluon condensate of mass dimension two, <A^2>, and the corresponding non-zero vacuum energy. This condensate gives rise to a dynamically generated mass for the gluon. 
  We consider solutions of six dimensional Einstein equations with two compact dimensions. It is shown that one can introduce 3-branes in this background in such a way that the effective four dimensional cosmological constant is completely independent of the brane tensions. These tensions are completely arbitrary, without requiring any fine tuning. We must, however, fine tune bulk parameters in order to obtain a sufficiently small value for the observable cosmological constant. We comment in the effective four dimensional description of this effect at energies below the compactification scale. 
  We examine the gauge generating nature of the translational subgroup of Wigner's little group for the case of massless tensor gauge theories and show that the gauge transformations generated by the translational group is only a subset of the complete set of gauge transformations. We also show that, just like the case of topologically massive gauge theories, translational groups act as generators of gauge transformations in gauge theories obtained by extending massive gauge noninvariant theories by a Stuckelberg mechanism. The representations of the translational groups that generate gauge transformations in such Stuckelberg extended theories can be obtained by the method of dimensional descent. We illustrate these with the examples of Stuckelberg extended first class versions of Proca, Einstein-Pauli-Fierz and massive Kalb-Ramond theories in 3+1 dimensions. A detailed analysis of the partial gauge generation in massive and massless 2nd rank symmetric gauge theories is provided. The gauge transformations generated by translational group in 2-form gauge theories are shown to explicitly manifest the reducibility of gauge transformations in these theories. 
  The relationship between on-shell tree level scattering amplitudes of open and closed strings, discovered some time ago by Kawai, Lewellen and Tye, is used at field theory level (at $O(\alpha'^3)$) to establish a link between the general relativity and the non-abelian Yang-Mills effective actions. Insisting at the effective Lagrangian level that any tree $N$ point gravity on-shell scattering amplitude is directly factorisable into a sum of $N$ point left-right products of non-abelian Yang-Mills tree on-shell scattering amplitudes, non-trivial mappings of the effective general relativity operators into the effective non-abelian Yang-Mills operators are derived. Implications of such mapping relations of the field operators are discussed. 
  We study the properties of anti--de Sitter black holes with a Gauss-Bonnet term for various horizon topologies (k=0, \pm 1) and for various dimensions, with emphasis on the less well understood k=-1 solution. We find that the zero temperature (and zero energy density) extremal states are the local minima of the energy for AdS black holes with hyperbolic event horizons. The hyperbolic AdS black hole may be stable thermodynamically if the background is defined by an extremal solution and the extremal entropy is non-negative. We also investigate the gravitational stability of AdS spacetimes of dimensions D>4 against linear perturbations and find that the extremal states are still the local minima of the energy. For a spherically symmetric AdS black hole solution, the gravitational potential is positive and bounded, with or without the Gauss-Bonnet type corrections, while, when k=-1, a small Gauss-Bonnet coupling, namely, \alpha << {l}^2 (where l is the curvature radius of AdS space), is found useful to keep the potential bounded from below, as required for stability of the extremal background. 
  A superstring action is quantised with Neveu Schwarz(NS) and Ramond(R) boundary conditions. The zero mass states of the NS sector are classified as the vector gluons, W-mesons, $B_{\mu}$-mesons and scalars containing Higgs. The fifteen zero mass fermions are obtained from the Ramond sector. A space time supersymmetric Hamiltonian of the Standard Model is presented without any conventional SUSY particles. 
  We discuss the isometry group structure of three-dimensional black holes and Chern-Simons invariants. Aspects of the holographic principle relevant to black hole geometry are analyzed. 
  Supersymmetric non-linear sigma-models are described by a field dependent Kaehler metric determining the kinetic terms. In general it is not guaranteed that this metric is always invertible. Our aim is to investigate the symmetry structure of supersymmetric models in four dimensional space-time in which metric singularities occur. For this purpose we study a simple anomaly-free extension of the supersymmetric CP^1 model from a classical point of view. We show that the metric singularities can be regularized by the addition of a soft supersymmetry-breaking mass parameter. 
  We discuss the computation of the leading corrections to D-brane solutions due to higher derivative terms in the corresponding low energy effective action. We develop several alternative methods for analyzing the problem. In particular, we derive an effective one-dimensional action from which the field equations for spherically symmetric two-block brane solutions can be derived, show how to obtain first order equations, and discuss a few other approaches. We integrate the equations for extremal branes and obtain the corrections in terms of integrals of the evaluation of the higher derivative terms on the lowest order solution. To obtain completely explicit results one would need to know all leading higher derivative corrections which at present are not available. One of the known higher derivative terms is the R^4 term, and we obtain the corrections to the D3 brane solution due to this term alone. We note, however, that (unknown at present) higher terms depending on F_5 are expected to modify our result. We analyze the thermodynamics of brane solutions when such quantum corrections are present. We find that the R^4 term induces a correction to the tension and the electric potential of the D3 brane but not to its charge, and the tension is still proportional to the electric potential times the charge. In the near-horizon limit the corrected solution becomes AdS_5 \times S^5 with the same cosmological constant as the lowest order solution but a different value of the (constant) dilaton. 
  We analyze the classical stability of string cosmologies driven by the dynamics of orientifold planes. These models are related to time-dependent orbifolds, and resolve the orbifold singularities which are otherwise problematic by introducing orientifold planes. In particular, we show that the instability discussed by Horowitz and Polchinski for pure orbifold models is resolved by the presence of the orientifolds. Moreover, we discuss the issue of stability of the cosmological Cauchy horizon, and we show that it is stable to small perturbations due to in-falling matter. 
  The stochastic quantization method is applied to N = 1 supersymmetric Yang-Mills theory, in particular in 4 and 10 dimensions. In the 4 dimensional case, based on Ito calculus, the Langevin equation is formulated in terms of the superfield formalism. The stochastic process manifestly preserves both the global N = 1 supersymmetry and the local gauge symmetry. The expectation values of the local gauge invariant observables in SYM_4 are reproduced in the equilibrium limit. In the superfield formalism, it is impossible in SQM to choose the so-called Wess-Zumino gauge in such a way to gauge away the auxiliary component fields in the vector multiplet, while it is shown that the time development of the auxiliary component fields is determined by the Langevin equations for the physical component fields of the vector multiplet in an '' almost Wess-Zumino gauge ''. The physical component expressions of the superfield Langevin equation are naturally extended to the 10 dimensional case, where the spinor field is Majorana-Weyl. By taking a naive zero volume limit of the SYM_10, the IIB matirx model is studied in this context. 
  Quaternionic formulation of D=4 conformal group and of its associated twistors and their relation to harmonic analyticity is presented. Generalization of $SL(2,\cal{C})$ to the D=4 conformal group SO(5,1) and its covering group $SL(2,\cal{Q})$ that generalizes the euclidean Lorentz group in $R^4$ [namely $SO(3,1)\approx SL(2,\cal{C})$ which allow us to obtain the projective twistor space $CP^3$] is shown. Quasi-conformal fields are introduced in D=4 and Fueter mappings are shown to map self-dual sector onto itself (and similarly for the anti-self-dual part). Differentiation of Fueter series and various forms of differential operators are shown, establishing the equivalence of Fueter analyticity with twistor and harmonic analyticity. A brief discussion of possible octonion analyticity is provided. 
  We find a new class of time-dependent brane solutions in supergravities in arbitrary dimensions $D$. These are general intersecting light-like branes (null-branes), and their superposition and intersection rules are obtained. This is achieved by directly solving bosonic field equations for supergravity coupled to a dilaton and antisymmetric tensor fields. We discuss their possible significance. 
  We derive an effective Abelian gauge theory (EAGT) of a modified SU(2) Yang-Mills theory. The modification is made by explicitly introducing mass terms of the off-diagonal gluon fields into pure SU(2) Yang-Mills theory, in order that Abelian dominance at a long-distance scale is realized in the modified theory. In deriving the EAGT, the off-diagonal gluon fields involving longitudinal modes are treated as fields that produce quantum effects on the diagonal gluon field and other fields relevant at a long-distance scale. Unlike earlier papers, a necessary gauge fixing is carried out without spoiling the global SU(2) gauge symmetry. We show that the EAGT allows a composite of the Yukawa and the linear potentials which also occurs in an extended dual Abelian Higgs model. This composite potential is understood to be a static potential between color-electric charges. In addition, we point out that the EAGT involves the Skyrme-Faddeev model. 
  The heat kernel method is extended to the case of finite temperature. Special emphasis is given to the study of gauge theories. Due to the compactness of space in the Euclidean time direction (inverse temperature) the field strength cannot completely characterize the gauge fields. This is just a manifestation of the Aharonov-Bohm effect. The field strength has to be supplemented by the Polyakov loop. Only if the latter is taken into account one obtains gauge covariant results for the generalized Seeley-DeWitt coefficients of the heat kernel expansion. 
  We investigate some issues regarding quantum corrections for de Sitter branes in a bulk AdS(5) spacetime. The one-loop effective action for a Majorana spinor field is evaluated and compared with the scalar field result. We also evaluate the cocycle function for various boundary conditions, finding that the quantum corrections naturally induce higher order curvature terms in the original action and, in general, it is not possible to eliminate the cocycle function by renormalisation. In the one brane limit care must be taken on how one extracts physical results. The effective potential is found to be zero on the conformally related cylinder. However, using the actual metric, the contribution from the cocycle function is non-zero and must be included. Subtleties with any zero modes are also discussed. 
  As a step to understand general patterns of integrability in 1+1 quantum field theories with supergroup symmetry, we study in details the case of $OSP(1/2)$. Our results include the solutions of natural generalizations of models with ordinary group symmetry: the $UOSP(1/2)_{k}$ WZW model with a current current perturbation, the $UOSP(1/2)$ principal chiral model, and the $UOSP(1/2)\otimes UOSP(1/2)/UOSP(1/2)$ coset models perturbed by the adjoint. Graded parafermions are also discussed. A pattern peculiar to supergroups is the emergence of another class of models, whose simplest representative is the $OSP(1/2)/OSP(0/2)$ sigma model, where the (non unitary) orthosymplectic symmetry is realized non linearly (and can be spontaneously broken). For most models, we provide an integrable lattice realization. We show in particular that integrable $osp(1/2)$ spin chains with integer spin flow to $UOSP(1/2)$ WZW models in the continuum limit, hence providing what is to our knowledge the first physical realization of a super WZW model. 
  The supertube and BIon spike solutions are examined in a general curved target space. The criteria for the existence of these solutions are explicitly derived. Also the equation which the general BIon solution should satisfy is derived. 
  The description of string-theoretic s-branes at g_s=0 as exact worldsheet CFTs with a (lambda cosh X^0) or (lambda e^(X^0)) boundary interaction is considered. Due to the imaginary-time periodicity of the interaction under X^0 -> X^0 + 2 pi i, these configurations have intriguing similarities to black hole or de Sitter geometries. For example, the open string pair production as seen by an Unruh detector is thermal at temperature T = 1/4 pi. It is shown that, despite the rapid time dependence of the s-brane, there exists an exactly thermal mixed state of open strings. The corresponding boundary state is constructed for both the bosonic and superstring cases. This state defines a long-distance Euclidean effective field theory whose light modes are confined to the s-brane. At the critical value of the coupling lambda=1/2, the boundary interaction simply generates an SU(2) rotation by pi from Neumman to Dirichlet boundary conditions. The lambda=1/2 s-brane reduces to an array of sD-branes (D-branes with a transverse time dimension) on the imaginary time axis. The long range force between a (bosonic) sD-brane and an ordinary D-brane is shown from the annulus diagram to be 11/12 times the force between two D-branes. The linearized time-dependent RR field F=dC produced by an sD-brane in superstring theory is explicitly computed and found to carry a half unit of s-charge Q_s=\int_S *F=1/2, where S is any transverse spacelike slice. 
  We give an introduction to a new approach to the covariant quantization of superstrings. After a brief review of the classical Green--Schwarz superstring and Berkovits' approach to its quantization based on pure spinors, we discuss our covariant formulation without pure spinor constraints. We discuss the relation between the concept of grading, which we introduced to define vertex operators, and homological perturbation theory, and we compare our work with recent work by others. In the appendices, we include some background material for the Green-Schwarz and Berkovits formulations, in order that this presentation be self contained. 
  We study the wave equation for a minimally coupled massive scalar in D-dimensional de Sitter space. We compute the absorption cross section to investigate its cosmological horizon in the southern diamond. By analogy of the quantum mechanics, it is found that there is no absorption in de Sitter space. This means that de Sitter space is usually in thermal equilibrium, like the black hole in anti de Sitter space. It confirms that the cosmological horizon not only emits radiation but also absorbs that previously emitted by itself at the same rate, keeping the curvature radius of de Sitter space fixed. 
  It is shown that despite the possibility of a breakdown of hyperbolicity, the SU(2) Skyrme model can never exhibit bulk violations of Einstein causality because its energy momentum tensor satisfies the dominant energy condition. It also satisfies the strong energy condition. The Born-Infeld-Skyrme model also satisfies both the dominant and strong energy conditions. 
  We extend the results of Cachazo, Seiberg and Witten to N=1 supersymmetric gauge theories with gauge groups SO(2N), SO(2N+1) and Sp(2N). By taking the superpotential which is an arbitrary polynomial of adjoint matter \Phi as a small perturbation of N=2 gauge theories, we examine the singular points preserving N=1 supersymmetry in the moduli space where mutually local monopoles become massless. We derive the matrix model complex curve for the whole range of the degree of perturbed superpotential. Then we determine a generalized Konishi anomaly equation implying the orientifold contribution. We turn to the multiplication map and the confinement index K and describe both Coulomb branch and confining branch. In particular, we construct a multiplication map from SO(2N+1) to SO(2KN-K+2) where K is an even integer as well as a multiplication map from SO(2N) to SO(2KN-2K+2) (K is a positive integer), a map from SO(2N+1) to SO(2KN-K+2) (K is an odd integer) and a map from Sp(2N) to Sp(2KN+2K-2). Finally we analyze some examples which show some duality: the same moduli space has two different semiclassical limits corresponding to distinct gauge groups. 
  We propose an analytic framework to study the nonperturbative solutions of Witten's open string field theory. The method is based on the Moyal star formulation where the kinetic term can be split into two parts. The first one describes the spectrum of two identical half strings which are independent from each other. The second one, which we call midpoint correction, shifts the half string spectrum to that of the standard open string. We show that the nonlinear equation of motion of string field theory is exactly solvable at zeroth order in the midpoint correction. An infinite number of solutions are classified in terms of projection operators. Among them, there exists only one stable solution which is identical to the standard butterfly state. We include the effect of the midpoint correction around each exact zeroth order solution as a perturbation expansion which can be formally summed to the complete exact solution. 
  In this paper we make two observations related to discrete torsion. First, we observe that an old obscure degree of freedom (momentum/translation shifts) in (symmetric) string orbifolds is related to discrete torsion. We point out how our previous derivation of discrete torsion from orbifold group actions on B fields includes these momentum lattice shift phases, and discuss how they are realized in terms of orbifold group actions on D-branes. Second, we describe the M theory dual of IIA discrete torsion, a duality relation to our knowledge not previously understood. We show that IIA discrete torsion is encoded in analogues of the shift orbifolds above for the M theory C field. 
  Projecting on a suitable subset of coordinates, a picture is constructed in which the conformal boundary of $AdS_5\times S^5$ and that of the plane wave resulting in the Penrose limit are located at the same line. In a second line of arguments all $AdS_5\times S^5$ and plane wave geodesics are constructed in their integrated form. Performing the Penrose limit, the approach of null geodesics reaching the conformal boundary of $AdS_5\times S^5$ to that of the plane wave is studied in detail. At each point these null geodesics of $AdS_5\times S^5$ form a cone which degenerates in the limit. 
  In this article we construct the chirality and Dirac operators on noncommutative AdS_2. We also derive the discrete spectrum of the Dirac operator which is important in the study of the spectral triple associated with AdS_2. It is shown that the degeneracy of the spectrum present in the commutative AdS_2 is lifted in the noncommutative case. The way we construct the chirality operator is suggestive of how to introduce the projector operators of the corresponding projective modules on this space. 
  We consider a massive scalar field with arbitrary coupling in $\mathbf{S}^{1}\times \mathbf{S}^{3}$ space, which mimics the thermal expanding universe, and calculate explicitly all relevant thermodynamical functions in the low- and high-temperature regimes, extending previous analysis of entropy bounds and entropy/energy ratios performed in the conformal case. For high temperatures, new mass-dependent entropy ratios are established which, differently to the conformal limit, fulfil Bekenstein's and Verlinde's bounds in the physical region. 
  Recently it was conjectured by Gibbons and Townsend that the large n limit of an N=4 superconformal extension of the n-particle Calogero model might provide a microscopic description of the extreme Reissner-Nordstrom black hole near the horizon. In this paper a possibility to construct an SU(1,1|2) invariant extension of the Calogero model is considered. We treat in detail the two-particle case and comment on some peculiarities intrinsic to n>2 generalizations. 
  We propose a new Doubly Special Relativity theory based on the generalization of the $\kappa$-deformation of the Poincar\'e algebra acting along one of the null directions. We recall the quantum Hopf structure of such deformed Poincar\'e algebra and use it to derive the phase space commutation relations. As in the DSR based on the standard quantum $\kappa$-Poincar\'e algebra we find that the space time is non-commutative. We investigate the fate of the properties of Special Relativity in the null basis: the split of the algebra of Lorentz and momentum generators into kinematical and dynamical parts, the action of the kinematical boost $M^{+-}$, and the emergence of the two dimensional Galilean symmetry. 
  We systematically analyse the necessary and sufficient conditions for the preservation of supersymmetry for bosonic geometries of the form R^{1,9-d} \times M_d, in the common NS-NS sector of type II string theory and also type I/heterotic string theory. The results are phrased in terms of the intrinsic torsion of G-structures and provide a comprehensive classification of static supersymmetric backgrounds in these theories. Generalised calibrations naturally appear since the geometries always admit NS or type I/heterotic fivebranes wrapping calibrated cycles. Some new solutions are presented. In particular we find d=6 examples with a fibred structure which preserve N=1,2,3 supersymmetry in type II and include compact type I/heterotic geometries. 
  Electromagnetic plane waves provide examples of time-dependent open string backgrounds free of $\alpha'$ corrections. The solvable case of open strings in a quadrupolar wave front, analogous to pp-waves for closed strings, is discussed. In light-cone gauge, it leads to non-conformal boundary conditions similar to those induced by tachyon condensates. A maximum electric gradient is found, at which macroscopic strings with vanishing tension are pair-produced -- a non-relativistic analogue of the Born-Infeld critical electric field. Kinetic instabilities of quadrupolar electric fields are cured by standard atomic physics techniques, and do not interfere with the former dynamic instability. A new example of non-conformal open-closed duality is found. Propagation of open strings in time-dependent wave fronts is discussed. 
  We discuss a new mechanism of obtaining a period of cosmological inflation in the context of string theory. This mechanism is based on embedded defects which form dynamically on higher dimensional D-branes. Such defects generate topological inflation, but unlike topological inflation from stable defects, here there is a natural graceful exit from inflation: the decay of the embedded defect. We demonstrate the idea in the context of a brane-antibrane annihilation process. The graceful exit mechanism suggested here applies generically to all realizations of inflation on D-branes. 
  We cautiously reanalyze some easily confused notions on particle horizon problem in this paper and then we give a new answer to the particle horizon problem. This answer is independent of physics plunging into Planck time. 
  The correlation functions of open Wilson line operators in two-dimensional Yang-Mills theory on the noncommutative torus are computed exactly. The correlators are expressed in two equivalent forms. An instanton expansion involves only topological numbers of Heisenberg modules and enables extraction of the weak-coupling limit of the gauge theory. A dual algebraic expansion involves only group theoretic quantities, winding numbers and translational zero modes, and enables analysis of the strong-coupling limit of the gauge theory and the high-momentum behaviour of open Wilson lines. The dual expressions can be interpreted physically as exact sums over contributions from virtual electric dipole quanta. 
  We consider non-supersymmetric large N orientifold field theories. Specifically, we discuss a gauge theory with a Dirac fermion in the anti-symmetric tensor representation. We argue that, at large N and in a large part of its bosonic sector, this theory is non-perturbatively equivalent to N=1 SYM, so that exact results established in the latter (parent) theory also hold in the daughter orientifold theory. In particular, the non-supersymmetric theory has an exactly calculable bifermion condensate, exactly degenerate parity doublets, and a vanishing cosmological constant (all this to leading order in 1/N). 
  We show how to cast an interacting system of M--branes into manifestly gauge-invariant form using an arrangement of higher-dimensional Dirac surfaces. Classical M--theory has a cohomologically nontrivial and noncommutative set of gauge symmetries when written using a ``doubled'' formalism containing 3-form and 6-form gauge fields. We show how the arrangement of Dirac surfaces allows an integral subgroup of these symmetries to be preserved at the quantum level. The proper context for discussing these large gauge transformations is relative cohomology, in which the 3-form transformation parameters become exact when restricted to the five-brane worldvolume. This structure yields the correct lattice of M-theory brane charges. 
  We show how the Dijkgraaf-Vafa matrix model proposal can be extended to describe five-dimensional gauge theories compactified on a circle to four dimensions. This involves solving a certain quantum mechanical matrix model. We do this for the lift of the N=1* theory to five dimensions. We show that the resulting expression for the superpotential in the confining vacuum is identical with the elliptic superpotential approach based on Nekrasov's five-dimensional generalization of Seiberg-Witten theory involving the relativistic elliptic Calogero-Moser, or Ruijsenaars-Schneider, integrable system. 
  We study examples where conformal invariance implies triviality of the underlying quantum field theory. 
  In order to evaluate the Feynman path integral in noncommutative quantum mechanics, we consider properties of a Lagrangian related to a quadratic Hamiltonian with noncommutative spatial coordinates. A quantum-mechanical system with noncommutative coordinates is equivalent to another one with commutative coordinates. We found connection between quadratic classical Lagrangians of these two systems. We also shown that there is a subclass of quadratic Lagrangians, which includes harmonic oscillator and particle in a constant field, whose connection between ordinary and noncommutative regimes can be expressed as a linear change of position in terms of a new position and velocity. 
  The effective potential for an on-shell BRST invariant gluon-ghost condensate of mass dimension 2 in the Curci-Ferrari gauge in SU(N) Yang-Mills is analysed by combining the local composite operator technique with the algebraic renormalization. We pay attention to the gauge parameter independence of the vacuum energy obtained in the considered framework and discuss the Landau gauge as an interesting special case. 
  A partial wave analysis using the basis of the total angular momentum operator J_3 is carried out for the first order Born amplitude of a Dirac particle in an Aharonov-Bohm (AB)potential. It is demonstrated that the s-partial wave contributes to the scattering amplitude in contrast to the case with scalar non-relativistic particles.We suggest that this explains the fact that the first order Born amplitude of a Dirac particle coincides with the exact amplitude expanded to the same order, where it does not for a scalar particle. An interesting algebra involving the Dirac velocity operator and the angular observables is discovered and its consequences are exploited. 
  Suggested correspondence between a black hole and a highly excited elementary string is explored. Black hole entropy is calculated by computing the density of states for an open excited string. We identify the square root of oscillator number of the excited string with Rindler energy of black hole to obtain an entropy formula which, not only agrees at the leading order with the Bekenstein-Hawking entropy, but also reproduces the logarithmic correction obtained for black hole entropy in the quantum geometry framework. This provides an additional supporting evidence for correspondence between black holes and strings. 
  The holographic principle relates (classical) gravitational waves in the bulk to quantum fluctuations and the Weyl anomaly of a conformal field theory on the boundary (the brane). One can thus argue that linear perturbations in the bulk of static black holes located on the brane be related to the Hawking flux and that (brane-world) black holes are therefore unstable. We try to gain some information on such instability from established knowledge of the Hawking radiation on the brane. In this context, the well-known trace anomaly is used as a measure of both the validity of the holographic picture and of the instability for several proposed static brane metrics. In light of the above analysis, we finally consider a time-dependent metric as the (approximate) representation of the late stage of evaporating black holes which is characterized by decreasing Hawking temperature, in qualitative agreement with what is required by energy conservation. 
  We study the rotating tubular D2-brane as a time dependent supersymmetric solution of type-IIA string theory. We show that the Poynting angular momentum of the supertube can be replaced by the mechanical angular momentum without disturbing the 8 supersymmetries. Unlike the non-rotating supertube, whose cross section can take an arbitrary shape, the rotating supertube admits only the circular cross section. When there is no electric field on the world volume, the supersymmetry dictates the angular velocity of the tubular D2-brane to be inversely proportional to the magnetic field. This rotating supertube can be considered as the `blown-up' configuration of an array of spinning D0-particles and is T-dual to the spiraling D-helix whose pitch moves at the speed of light. 
  The analysis of the combinatorics resulting from the perturbative expansion of the transition amplitude in quantum field theories, and the relation of this expansion to the Hausdorff series leads naturally to consider an infinite dimensional Lie subalgebra and the corresponding enveloping Hopf algebra, to which the elements of this series are associated. We show that in the context of these structures the power sum symmetric functionals of the perturbative expansion are Hopf primitives and that they are given by linear combinations of Hall polynomials, or diagrammatically by Hall trees. We show that each Hall tree corresponds to sums of Feynman diagrams each with the same number of vertices, external legs and loops. In addition, since the Lie subalgebra admits a derivation endomorphism, we also show that with respect to it these primitives are cyclic vectors generated by the free propagator, and thus provide a recursion relation by means of which the (n+1)-vertex connected Green functions can be derived systematically from the n-vertex ones. 
  We analyze $N=2,1,0$ vacua of type IIB string theory on $K3\times T^2/Z_2$ in presence of three-form fluxes from a four dimensional supergravity viewpoint. The quaternionic geometry of the $K3$ moduli space together with the special geometry of the NS and R-R dilatons and of the $T^2$-complex structure moduli play a crucial role in the analysis. The introduction of fluxes corresponds to a particular gauging of N=2, D=4 supergravity. Our results agree with a recent work of Tripathy and Trivedi. The present formulation shows the power of supergravity in the study of effective theories with broken supersymmetry. 
  In this paper we obtain a bound $\Lambda_{\rm NC} < 150$ TeV on the scale of space-time noncommutativity considering photon-neutrino interactions. We compute "star-dipole moments" and "star-charge radii" originating from space-time noncommutativity and compare them with the dipole moments calculated in the neutrino-mass extended standard model (SM). The computation depends on the nature of the neutrinos, Dirac versus Majorana, their mass and the energy scale. We focus on Majorana neutrinos. The "star-charge radius" is found to be $r^* = \sqrt{|< r^2_{\nu}>_{\rm NC}|} =|3\sum_{i=1}^3 ({\theta}^{0i})^2|^{1/4} < 1.6 \times 10^{-19}$ cm at $\Lambda_{\rm NC} = 150$ TeV. 
  We consider a gauge-invariant Hamiltonian analysis for Yang-Mills theories in three spatial dimensions. The gauge potentials are parametrized in terms of a matrix variable which facilitates the elimination of the gauge degrees of freedom. We develop an approximate calculation of the volume element on the gauge-invariant configuration space. We also make a rough estimate of the ratio of $0^{++}$ glueball mass and the square root of string tension by comparison with $(2+1)$-dimensional Yang-Mills theory. 
  In string field theory there is a fundamental object, the algebra of string field states \A, that must be understood better from a mathematical point of view. In particular we are interested in finding, if possible, a C^* structure over it, or possibly over a subalgebra U \subset \A. In this paper we define a * operation on \A, and then using a particular description of Witten's star product, the Moyal's star product, we find an appropriate pre C^* algebra S(R^{2n}) on a finite dimensional manifold, where the finite dimensionality is obtained with a cutoff procedure on the string oscillators number. Then we show that using an inductive limit we obtain a pre C^* algebra S \subset \A that can be completed to a C^* algebra. 
  For an effective treatment of the evaporation process of a large black hole the problem concerning the role played by the fluctuations of the (vacuum) stress tensor close to the horizon is addressed. We present arguments which establish a principal relationship between the outward fluctuations of the stress tensor close to the horizon and quantities describing the onset of the evaporation process. This suggest that the evaporation process may be described by a fluctuation-dissipation theorem relating the noise of the horizon to the black hole evaporation rate. 
  In this paper, the BRST symmetry transformation is presented for the noncommutative U(N) gauge theory. The nilpotency of the charge associated to this symmetry is then proved. As a consequence for the space-like non-commutativity parameter, the Hilbert space of physical states is determined by the cohomology space of the BRST operator as in the commutative case. Further, the unitarity of the S-matrix elements projected onto the subspace of physical states is deduced. 
  Starting from a self-dual formulation of gravity, we obtain a noncommutative theory of pure Einstein theory in four dimensions. In order to do that, we use Seiberg-Witten map. It is shown that the noncommutative torsion constraint is solved by the vanishing of commutative torsion. Finally, the noncommutative corrections to the action are computed up to second order. 
  We study nonsingular cosmological scenarios in a general $D$-dimensional string effective action of the dilaton-modulus-axion system in the presence of the matter source. In the standard dilatonic Brans-Dicke parameter ($\omega=-1$) with radiation, we analytically obtain singularity-free bouncing solutions where the universe starts out in a state with a finite curvature and evolves toward the weakly coupled regime. We apply this analytic method to the string-gas cosmology including the massive state in addition to the leading massless state (radiation), with and without the axion. We numerically find bouncing solutions which asymptotically approach an almost radiation-dominant universe with a decreasing curvature irrespective of the presence of the axion, implying that inclusion of the matter source is crucial for the existence of such solutions for $\omega=-1$. In the theories with $\omega \ne -1$, it is possible to obtain complete regular bouncing solutions with a finite dilaton and curvature in both past and future asymptotics for the general dimension, $D$. We also discuss the case where dilatonic higher-order corrections are involved to the tree-level effective action and demonstrate that the presence of axion/modulus fields and the matter source does not significantly affect the dynamics of the dilaton-driven inflation and the subsequent graceful exit. 
  We investigate a non-perturbative vacuum in open string field theory expanded around the analytic classical solution which has been found in the universal Fock space generated by matter Virasoro generators and ghost oscillators. We carry out level-truncation analyses up to level (6,18) in the theory around one-parameter families of the solution. We observe that the value of the vacuum energy cancels the D-brane tension as the approximation level is increased, but this non-perturbative vacuum disappears at the boundary of the parameter space. These results provide strong evidence for the conjecture that, although the universal solutions are pure gauge in almost all the parameter space, they are regarded as the tachyon vacuum solution at the boundary. 
  Quantum field theory in the near-horizon region of a black hole predicts the existence of an infinite number of degenerate modes. Such a degeneracy is regulated in the brick wall model by the introduction of a short distance cutoff. In this Letter we show that states of the brick wall model with non zero energy admit a further degeneracy for any given finite value of the cutoff. The black hole entropy is calculated within the brick wall model taking this degeneracy into account. Modes with complex frequencies however do not exhibit such a degeneracy. 
  Within a broad class of inflationary models we critically analyze the way initial quantum fluctuations on a new-physics hypersurface (NPH) affect standard predictions for large-scale cosmological perturbations. We find that these so-called transplanckian effects depend crucially on the definition of the "vacuum state" in particular on which Hamiltonian is minimized on the NPH in order to select such a state. Transplanckian effects can be made much smaller than previously suggested if sufficiently "adiabatic" Hamiltonians are minimized. 
  We develop the technique of Thermodynamic Bethe Ansatz to investigate the ground state and the spectrum in the thermodynamic limit of the staggered $XXZ$ models proposed recently as an example of integrable ladder model. This model appeared due to staggered inhomogeneity of the anisotropy parameter $\Delta$ and the staggered shift of the spectral parameter. We give the structure of ground states and lowest lying excitations in two different phases which occur at zero temperature. 
  Using a quantum field theoretic setting, we present evidence for dimensional reduction of any sub-volume of Minkowksi space. First, we show that correlation functions of a class of operators restricted to a sub-volume of D-dimensional Minkowski space scale as its surface area. A simple example of such area scaling is provided by the energy fluctuations of a free massless quantum field in its vacuum state. This is reminiscent of area scaling of entanglement entropy but applies to quantum expectation values in a pure state, rather than to statistical averages over a mixed state. We then show, in a specific case, that fluctuations in the bulk have a lower-dimensional representation in terms of a boundary theory at high temperature. 
  The free Schr\"odinger equation with mass M can be turned into a non-massive Klein-Gordon equation via Fourier transformation with respect to M. The kinematic symmetry algebra sch_d of the free d-dimensional Schr\"odinger equation with M fixed appears therefore naturally as a parabolic subalgebra of the complexified conformal algebra conf_d+2 in d+2 dimensions. The explicit classification of the parabolic subalgebras of conf_3 yields physically interesting dynamic symmetry algebras. This allows us to propose a new dynamic symmetry group relevant for the description of ageing far from thermal equilibrium, with a dynamical exponent z=2. The Ward identities resulting from the invariance under conf_d+2 and its parabolic subalgebras are derived and the corresponding free-field energy-momentum tensor is constructed. We also derive the scaling form and the causality conditions for the two- and three-point functions and their relationship with response functions in the context of Martin-Siggia-Rose theory. 
  In this paper it is exactly proved that the standard transformations of the three-dimensional (3D) vectors of the electric and magnetic fields E and B are not relativistically correct transformations. Thence the 3D vectors E and B are not well-defined quantities in the 4D spacetime and, contrary to the general belief, the usual Maxwell equations with the 3D E and B are not in agreement with the special relativity. The 4-vectors E^a and B^a, as well-defined 4D quantities, are introduced instead of ill-defined 3D E and B. The proof is given in the tensor and the Clifford algebra formalisms. 
  The existence of ring-like structures in exact hopfion solutions is shown. 
  We investigate the correspondence between the D2-brane, which is described by the abelian Born-Infeld action, and multiple D0-branes, which are done by the nonabelian Born-Infeld action. We construct effective actions for the fuzzy cylinder, sphere and plane formed via D0-branes and compare these actions with those for the cylindrical, spherical and planar D2-brane. We show that in the continuous limit, the effective actions for the fuzzy D0-branes precisely coincide with those for the single D2-brane if both the fuzziness of the D0-brane and the area occupied per a unit of magnetic flux on the D2-brane are equal to $(2\pi\ell_s)^2$. 
  We present new evidence for the conjecture that BPS correlation functions in the N=2 supersymmetric gauge theories are described by an auxiliary two dimensional conformal field theory. We study deformations of the N=2 supersymmetric gauge theory by all gauge-invariant chiral operators. We calculate the partition function of the N=2 theory on R^4 with appropriately twisted boundary conditions. For the U(1) theory with instantons (either noncommutative, or D-instantons, depending on the construction) the partition function has a representation in terms of the theory of free fermions on a sphere, and coincides with the tau-function of the Toda lattice hierarchy. Using this result we prove to all orders in string loop expansion that the effective prepotential (for U(1) with all chiral couplings included) is given by the free energy of the topological string on CP^1. Gravitational descendants play an important role in the gauge fields/string correspondence. The dual string is identified with the little string bound to the fivebrane wrapped on the two-sphere. We also discuss the theory with fundamental matter hypermultiplets. 
  Explicit supersymmetry breaking is studied in higher dimensional theories by having boundaries respect only a subgroup of the bulk symmetry. If the boundary symmetry is the maximal subgroup allowed by the boundary conditions imposed on the fields, then the symmetry can be consistently gauged; otherwise gauging leads to an inconsistent theory. In a warped fifth dimension, an explicit breaking of all bulk supersymmetries by the boundaries is found to be inconsistent with gauging; unlike the case of flat 5D, complete supersymmetry breaking by boundary conditions is not consistent with supergravity. Despite this result, the low energy effective theory resulting from boundary supersymmetry breaking becomes consistent in the limit where gravity decouples, and such models are explored in the hope that some way of successfully incorporating gravity can be found. A warped constrained standard model leads to a theory with one Higgs boson with mass expected close to the experimental limit. A unified theory in a warped fifth dimension is studied with boundary breaking of both SU(5) gauge symmetry and supersymmetry. The usual supersymmetric prediction for gauge coupling unification holds even though the TeV spectrum is quite unlike the MSSM. Such a theory may unify matter and Higgs in the same SU(5) hypermultiplet. 
  By combining the concepts of graviton and matroid, we outline a new gravitational theory which we call gravitoid theory. The idea of this theory emerged as an attempt to link the mathematical structure of matroid theory with M-theory. Our observations are essentially based on the formulation of matroid bundle due to MacPherson and Anderson-Davis. Also, by considering the oriented matroid theory, we add new observations about the link between the Fano matroid and D=11 supergravity which was discussed in some of our recent papers. In particular we find a connection between the affine matroid AG(3,2) and the $G_ {2}-$symmetry of D=11 supergravity. 
  The chiral ring of classical supersymmetric Yang-Mills theory with gauge group $Sp(N)$ or SO(N) is computed, extending previous work (of Cachazo, Douglas, Seiberg, and the author) for SU(N). The result is that, as has been conjectured, the ring is generated by the usual glueball superfield $S\sim \Tr W_\alpha W^\alpha$, with the relation $S^h=0$, $h$ being the dual Coxeter number. Though this proposition has important implications for the behavior of the quantum theory, the statement and (for the most part) the proofs amount to assertions about Lie groups with no direct reference to gauge theory. 
  These notes comprise the first of two articles devoted to the construction of exact solutions of noncommutative gauge theory in two spacetime dimensions. This first part deals solely with the classical theory on a noncommutative torus. Topics covered include a mathematical introduction to the geometry of the noncommutative torus, the definition, properties and symmetries of noncommutative Yang-Mills theory, and the complete solution of the classical field equations. 
  We propose a modified mode-expansion of the bulk fields in a BPS domain wall background to obtain the effective theory on the wall. The broken SUSY is nonlinearly realized on each mode defined by our mode-expansion. Our work clarifies a relation between two different approaches to derive the effective theory on a BPS wall, {\it i.e.} the nonlinear realization approach and the mode-expansion approach. We also discuss a further modification that respects the Lorentz and $U(1)_R$ symmetries broken by the wall. 
  The existence of anomalous symmetry-breaking solutions of the SO(2,1) commutator algebra is explicitly extended beyond the case of scale-invariant contact interactions. In particular, the failure of the conservation laws of the dilation and special conformal charges is displayed for the two-dimensional inverse square potential. As a consequence, this anomaly appears to be a generic feature of conformal quantum mechanics and not merely an artifact of contact interactions. Moreover, a renormalization procedure traces the emergence of this conformal anomaly to the ultraviolet sector of the theory, within which lies the apparent singularity. 
  Motivated by the study of branes in curved backgrounds, we investigate the construction of non-perturbative extensions of the super-isometry algebra osp*(8|4) of the AdS_7xS^4 background of M-theory. This algebra is not a subalgebra of osp(1|32) and its non-perturbative extension can therefore not be obtained by embedding in this simple superalgebra. We show how, instead, it is possible to construct an extension directly by solving the Jacobi identities. This requires, in addition to the expected non-perturbative charges, the introduction of new charges which appear in the {Q,Q} bracket only via a linear combination with the bosonic generators of the isometry algebra. The resulting extended algebra has the correct flat-space limit, but it is not simple and the non-perturbative charges do not commute with the super-isometry generators. We comment on the consequences of this structure for the representation theory and on possible alternatives to our construction. 
  Motivated by some recent speculative attempts to model the dark energy, scalar fields with negative kinetic energy coupled to gravity without a cosmological constant are considered. It is shown that in the presence of an ordinary fluid, any solution of the vacuum Einstein equations with cosmological constant is a solution provided $\rho-P={\Lambda \over 4 \pi G}$. The solutions can be interpreted as a steady state in which matter or entropy is being continuously created (or destroyed). The motion of the matter is not determined by the background Einstein spacetime, many different matter flows can be found giving rise to the same metric. Solutions without ordinary matter are also considered. Anti-gravitating multi-solutions and repulsive solutions which can chase ordinary matter or black holes are exhibited. These results may also have applications to gravity theories with higher derivatives. 
  The interpretation of D-branes in terms of open strings has lead to much interest in boundary conditions of two-dimensional conformal field theories (CFTs). These studies have deepened our understanding of CFT and allowed us to develop new computational tools. The construction of CFT correlators based on combining tools from topological field theory and non-commutative algebra in tensor categories, which we summarize in this contribution, allows e.g. to discuss, apart from boundary conditions, also defect lines and disorder fields. 
  Recently, it was pointed out that quantum orders and the associated projective symmetry groups can produce and protect massless gauge bosons and massless fermions in local bosonic models. In this paper, we demonstrate that a state with such kind of quantum orders can be viewed as a string-net condensed state. The emerging gauge bosons and fermions in local bosonic models can be regarded as a direct consequence of string-net condensation. The gauge bosons are fluctuations of nets of large closed strings which are condensed in the ground state. The ends of condensed open strings are the charged particles of the corresponding gauge field. For certain types of strings, the ends of strings can even be fermions. According to the string-net picture, fermions always carry gauge charges. This suggests the existence of a new discrete gauge field that couples to neutrinos and neutrons. We also discuss how chiral symmetry that protects massless Dirac fermions can emerge from the projective symmetry of quantum order. 
  The bosonization and duality rules in three-dimensions are applied to analyze some features of superfluids and superconductors. The energy of an ensemble of vortices in a superfluid is recovered by means of a kind of bound which, to some extent, shares similarity with the Bogomol'nyi bound. In the case of superconductors, after recasting the partition function in the form of a pure effective gauge theory, the existence of finite energy vortex solutions is discussed 
  On the basis of the Berkovits pure spinor formalism of covariant quantization of supermembrane, we attempt to construct a M(atrix) theory which is covariant under $SO(1,10)$ Lorentz group. We first construct a bosonic M(atrix) theory by starting with the first-order formalism of bosonic membrane, which precisely gives us a bosonic sector of M(atrix) theory by BFSS. Next we generalize this method to the construction of M(atrix) theory of supermembranes. However, it seems to be difficult to obtain a covariant and supersymmetric M(atrix) theory from the Berkovits pure spinor formalism of supermembrane because of the matrix character of the BRST symmetry. Instead, in this paper, we construct a supersymmetric and covariant matrix model of 11D superparticle, which corresponds to a particle limit of covariant M(atrix) theory. By an explicit calculation, we show that the one-loop effective potential is trivial, thereby implying that this matrix model is a free theory at least at the one-loop level. 
  The action and the thermodynamics of a rotating black hole in the presence of a positive cosmological constant are analyzed. Since there is no spatial infinity, one must bring in, instead, a platform where the parameters characterizing the thermodynamic ensemble are specified. In the present treatment the platform in question is taken to be one of the two horizons, which is considered as a boundary. If the boundary is taken to be the cosmological horizon one deals with the action and thermodynamics of the black hole horizon. Conversely, if one takes the black hole horizon as the boundary, one deals with the action and thermodynamics of the cosmological horizon. The two systems are different. Their energy and angular momenta are equal in magnitude but have opposite sign. In either case, the energy and the angular momentum are obtained as surface terms on the boundary, according to the standard Hamiltonian procedure. The temperature and the rotational chemical potential are also expressed in terms of magnitudes on the boundary. If, in the resulting expressions, one continues the cosmological constant to negative values, the black hole thermodynamic parameters defined on the cosmological horizon coincide with those calculated at spatial infinity in the asymptotically anti-de Sitter case. 
  There are currently many string inspired conjectures about the structure of the low-energy effective action for super Yang-Mills theories which require explicit multi-loop calculations. In this paper, we develop a manifestly covariant derivative expansion of superspace heat kernels and present a scheme to evaluate multi-loop contributions to the effective action in the framework of the background field method. The crucial ingredient of the construction is a detailed analysis of the properties of the parallel displacement propagators associated with Yang-Mills supermultiples in N-extended superspace. 
  Just as the single fluxbrane is quantum mechanically unstable to the nucleation of a locally charged spherical brane, so intersecting fluxbranes are unstable to various decay modes. Each individual element of the intersection can decay via the nucleation of a spherical brane, but uncharged spheres can also be nucleated in the region of intersection. For special values of the fluxes, however, intersecting fluxbranes are supersymmetric, and so are expected to be stable. We explicitly consider the instanton describing the decay modes of the two--element intersection (an F5-brane in the string theory context), and show that in dimensions greater than four the action for the decay mode of the supersymmetric intersection diverges. This observation allows us to show that stable intersecting fluxbranes should also exist in type 0A string theory. 
  We present a derivation of the entropy of black holes in induced gravity models based on conformal properties of induced gravity constituents near the horizon. The four-dimensional (4D) theory is first reduced to a tower of two-dimensional (2D) gravities such that each 2D theory is induced by fields with certain momentum $p$ along the horizon. We demonstrate that in the vicinity of the horizon constituents of the 2D induced gravities are described by conformal field theories (CFT) with specific central charges depending on spin and non-minimal couplings and with specific correlation lengths depending on the masses of fields and on the momentum $p$. This enables one to use CFT methods to count partial entropies $s(p)$ in each 2D sector. The sum of partial entropies correctly reproduces the Bekenstein-Hawking entropy of the 4D induced gravity theory. Our results indicate that earlier attempts of the derivation of the entropy of black holes based on a near-horizon CFT may have a microscopic realization. 
  It is shown that a sufficient condition for color confinement is given by Z_3^{-1}=0, where Z_3 denotes the renormalization constant of the color gauge field. 
  A family of superpotentials is constructed which may be relevant to supersymmetry breaking in 4 dimensional (0,1) heterotic string models. The scale of supersymmetry breaking, as well as the coupling constant, would be stable and could not run away to zero. 
  The gauge symmetries of a general dynamical system can be systematically obtained following either a Hamiltonean or a Lagrangean approach. In the former case, these symmetries are generated, according to Dirac's conjecture, by the first class constraints. In the latter approach such local symmetries are reflected in the existence of so called gauge identities. The connection between the two becomes apparent, if one works with a first order Lagrangean formulation. We thereby confirm Dirac's conjecture. Our analysis applies to arbitrary constrained systems with first and second class constraints, and thus extends a previous analysis by one of the authors to such general systems. We illustrate our general results in terms of several examples. 
  The applications of the existing Liouville theories for the description of the longitudinal dynamics of non-critical Nambu-Goto string are analyzed. We show that the recently developed DOZZ solution to the Liouville theory leads to the cut singularities in tree string amplitudes. We propose a new version of the Polyakov geometric approach to Liouville theory and formulate its basic consistency condition - the geometric bootstrap equation. Also in this approach the tree amplitudes develop cut singularieties. 
  We study both global as well as local (Nielsen-Olesen) strings in de Sitter space. While these type of topological defects have been studied in the background of a de Sitter metric previously, we study here the full set of coupled equations. We find only ``closed'' solutions. The behaviour of the metric tensor of these solutions resembles that of ``supermassive'' strings with a curvature singularity at the cosmological horizon. For global strings (and the composite defect) we are able to construct solutions which are regular on the interval from the origin to the cosmological horizon if the global string core lies completely inside the horizon. 
  Here we address the problem of bosonizing massive fermions without making expansions in the fermion masses in both massive $QED_2$ and $QED_3$ with $ N $ fermion flavors including also a Thirring coupling. We start from two point correlators involving the U(1) fermionic current and the gauge field.   From the tensor structure of those correlators we prove that the U(1) current must be identically conserved (topological) in the corresponding bosonized theory both in D=2 and D=3 dimensions. We find an effective generating functional in terms of bosonic fields which reproduces those two point correlators and from that we obtain a map of the Lagrangian density $\bar{\psi}^{r} (i \partial / - m){\psi}^{r}$ into a bosonic one in both dimensions. This map is nonlocal but it is independent of the eletromagnetic and Thirring couplings, at least in the quadratic approximation for the fermionic determinant. 
  Key issues of classical and quantum strings in gravitational plane waves, shock waves and spacetime singularities are synthetically understood. This includes the string mass and mode number excitations, energy-momentum tensor, scattering amplitudes, vaccum polarization and wave-string polarization effect. The role of the real pole singularities characteristic of the tree level string spectrum (real mass resonances) and that of spacetime singularities is clearly exhibited. This throws light on the issue of singularities in string theory which can be thus classified and fully physically characterized in two different sets: strong singularities (poles of order equal or larger than 2, and black holes), where the string motion is collective and non oscillating in time, outgoing and scattering states do not appear, the string does not cross the singularities, and weak singularities (poles of order smaller than 2, Dirac delta, and conic/orbifold singularities) where the whole string motion is oscillatory in time, outgoing and scattering states exist, and the string crosses the singularities. Commom features of strings in singular plane backgrounds and in inflationary backgrounds are explicitly exhibited. The string dynamics and the scattering/excitation through the singularities (whatever their kind: strong or weak) is fully physically consistent and meaningful. 
  It is argued that the phenomenon of a flux tube in quantum chromodynamics is closely connected with a spontaneously symmetry breakdown of gauge theory. It is shown that in the presence of a mass term in the SU(2) gauge theory the Nielsen-Olesen equations describe the flux tube surrounded by an external field. 
  Quantum particles confined to surfaces in higher dimensional spaces are acted upon by forces that exist only as a result of the surface geometry and the quantum mechanical nature of the system. The dynamics are particularly rich when confinement is implemented by forces that act normal to the surface. We review this confining potential formalism applied to the confinement of a particle to an arbitrary manifold embedded in a higher dimensional Euclidean space. We devote special attention to the geometrically induced gauge potential that appears in the effective Hamiltonian for motion on the surface. We emphasize that the gauge potential is only present when the space of states describing the degrees of freedom normal to the surface is degenerate. We also distinguish between the effects of the intrinsic and extrinsic geometry on the effective Hamiltonian and provide simple expressions for the induced scalar potential. We discuss examples including the case of a 3-dimensional manifold embedded in a 5-dimensional Euclidean space. 
  We study superstring theories on AdS(3) x N backgrounds yielding N=2,3,4 extended superconformal symmetries in the dual boundary CFT. In each case the necessary constraints on the internal worldsheet theory N are found. 
  We present new supersymmetric domain wall and string solutions of five-dimensional N = 2 gauged supergravity coupled to an arbitrary number of vector multiplets. Using the techniques of very special geometry allows to obtain the most general domain wall preserving half of the supersymmetries. This solution, which describes a renormalization group flow in the dual field theory, is given in terms of Weierstrass elliptic functions. The magnetically charged, one quarter supersymmetric string solutions are shown to be closely related to Liouville theory. We furthermore investigate general product space compactifications, and show that topological transitions from AdS_3 x S^2 to AdS_3 x H^2 can occur when one moves in moduli space. 
  In this lecture I make some educated guesses, about the landscape of string theory vacua. Based on the recent work of a number of authors, it seems plausible that the lanscape is unimaginably large and diverse. Whether we like it or not, this is the kind of behavior that gives credence to the Anthropic Principle. I discuss the theoretical and conceptual issues that arise in developing a cosmology based on the diversity of environments implicit in string theory. 
  With the aim of exploring a massive model corresponding to the perturbation of the conformal model [hep-th/0211094] the nonlinear integral equation for a quantum system consisting of left and right KdV equations coupled on the cylinder is derived from an integrable lattice field theory. The eigenvalues of the energy and of the transfer matrix (and of all the other local integrals of motion) are expressed in terms of the corresponding solutions of the nonlinear integral equation. The analytic and asymptotic behaviours of the transfer matrix are studied and given. 
  We calculate the one-loop corrections to gauge couplings in N=1 supersymmetric brane world models, which are realized in an type IIA orbifold/orientifold background with several stacks of D6 branes wrapped on 3-cycles with non-vanishing intersections. Contributions arise from both N=1 and N=2 open string subsectors. In contrast to what is known from ordinary orbifold theories, N=1 subsectors give rise to moduli-dependent one-loop corrections. 
  String theory gives rise to various mechanisms to generate primordial inflation, of which ``brane inflation'' is one of the most widely considered. In this scenario, inflation takes place while two branes are approaching each other, and the modulus field representing the separation between the branes plays the role of the inflaton field. We study the phase space of initial conditions which can lead to a sufficiently long period of cosmological inflation, and find that taking into account the possibility of nonvanishing initial momentum can significantly change the degree of fine tuning of the required initial conditions. 
  We discuss D-braneworld cosmology, that is, the brane is described by the Born-Infeld action. Compared with the usual Randall-Sundrum braneworld cosmology where the brane action is the Nambu-Goto one, we can see some drastic changes at the very early universe: (i)universe may experience the rapid accelerating phase (ii)the closed universe may avoid the initial singularity. We also briefly address the dynamics of the cosmology in the open string metric, which might be favorer than the induced metric from the view point of the D-brane. 
  We discuss the dynamics of a particular two-dimensional (2D) physical system in the four dimensional (4D) (non-)commutative phase space by exploiting the consistent Hamiltonian and Lagrangian formalisms based on the symplectic structures defined on the 4D (non-)commutative cotangent manifolds. The noncommutativity exists {\it equivalently} in the coordinate or the momentum planes embedded in the 4D cotangent manifolds. The signature of this noncommutativity is reflected in the derivation of the first-order Lagrangians where we exploit the most general form of the Legendre transformation defined on the (non-)commutative (co-)tangent manifolds. The second-order Lagrangian, defined on the 4D {\it tangent manifold}, turns out to be the {\it same} irrespective of the noncommutativity present in the 4D cotangent manifolds for the discussion of the Hamiltonian formulation. A connection with the noncommutativity of the dynamics, associated with the quantum groups on the q-deformed 4D cotangent manifolds, is also pointed out. 
  We show that, contrary to common belief, the propagation of a spin-2 field in an electromagnetic background is {\em causal}. The proof will be given in the Fierz formalism which, as we shall see, is free of the ambiguity present in the more usual Einstein representation. We shall also review the proof in this latter representation. 
  We analyse the moduli spaces of superconformal field theories (SCFTs). For N=2 we find an enhanced moduli space which in geometrical terms corresponds to tori with two independent complex structures. To explain the precise relation with the moduli space of SCFTs on K3 surfaces as described by Aspinwall and Morrison, we discuss some subtleties with the precise interpretation of the N=2 and N=4 moduli spaces. We also explain why in some cases the SYZ-description of mirror symmetry as fibrewise T-duality seems to break down.   Using gluing matrices we give an algebraic description of D-branes and construct the corresponding boundary states. We study how isomorphisms of the SCFTs act on D-branes. Finally we give a geometrical interpretation of our algebraic constructions and make contact with the geometrical D-brane categories and Kontsevich's homological mirror symmetry conjecture. 
  On the example of the three-dimensional Ising model, we show that nonperturbative renormalization group equations allow one to obtain very accurate critical exponents. Implementing the order $\partial^4$ of the derivative expansion leads to $\nu=0.632$ and to an anomalous dimension $\eta=0.033$ which is significantly improved compared with lower orders calculations. 
  A synthetic report of the advances in the study of classical and quantum string dynamics in curved backgrounds is provided, namely: the new feature of multistring solutions; the effect of a cosmological constant and of spacial curvature on classical and quantum strings; classical splitting of fundamental strings;the general string evolution in constant curvature spacetimes;the conformal invariant effects;strings on plane waves, shock waves and spacetime singularities and its spectrum. New developments in string gravity and string cosmology are reported: string driven cosmology and its predictions;the primordial gravitation wave background; non-singular string cosmologies from exact conformal field theories;QFT, string temperature and the string phase of de Sitter space; the string phase of black holes;new dual relation between QFT regimes and string regimes and the 'QFT/String Tango'; new coherent string states and minimal uncertainty principle in string theory 
  The exact fermion propagator in a classical time-dependent gauge field is derived by solving the equation of motion for the Dirac Green's functions. From the retarded propagator obtained in this way the momentum spectrum for the produced fermion pairs is calculated. Different approximations and the exact solution for the propagator and the momentum spectrum are presented. 
  It was recently pointed out that the physics of a single discrete gravitational extra dimension exhibits a peculiar UV/IR connection relating the UV scale to the radius of the effective extra dimension. Here we note that this non-locality is a manifestation of holography, encoding the correct scaling of the number of fundamental degrees of freedom of the UV theory. This in turn relates the Wilsonian RG flow in the UV theory to the effective gravitational dynamics in the extra dimension. The relevant holographic c-function is determined by the expression for the holographic bound. Holography in this context is a result of the requirements of unitarity and diffeomorphism invariance. We comment on the relevance of this observation for the cosmological constant problem. 
  We use the Konishi anomaly equations to construct the exact effective superpotential of the glueball superfields in various N=1 supersymmetric gauge theories. We use the superpotentials to study in detail the structure of the spaces of vacua of these theories. We consider chiral and non-chiral SU(N) models, the exceptional gauge group G(2) and models that break supersymmetry dynamically. 
  By considering the vacuum polarization, we study the effects of geometry on electrostatic self--energy of a test charge near the black hole horizon and also in regions with strong and weak curvature in static two dimensional curved backgrounds. We also discuss the relation of ultraviolet behavior of the gauge field propagator and charge confinement. 
  Based on some recent work of the authors, we focus on the relationship between the Casimir energy of a Majorana spinor field for a Euclidean Einstein universe $S^4\times R$ and for a Euclidean de Sitter brane ($S^4$) embedded in AdS(5). This is for a conformally coupled massless field. Interestingly, the one brane effective potential is zero and the results are equivalent, as for the scalar case, when evaluated on the conformally related cylinder. However, using the actual metric this equivalence no longer holds because a non-trivial contribution from the path integral measure (known as the cocycle function) is non-zero. 
  A covariant nature of the Langevin equation in Ito calculus is clarified in applying stochastic quantization method to U(N) and SU(N) lattice gauge theories. The stochastic process is expressed in a manifestly general coordinate covariant form as a collective field theory on the group manifold. A geometric interpretation is given for the Langevin equation and the corresponding Fokker-Planck equation in the sense of Ito. 
  A consistent physical theory of quantum mechanics can be built on a complex Hamiltonian that is not Hermitian but instead satisfies the physical condition of space-time reflection symmetry (PT symmetry). Thus, there are infinitely many new Hamiltonians that one can construct that might explain experimental data. One would think that a quantum theory based on a non-Hermitian Hamiltonian violates unitarity. However, if PT symmetry is not broken, it is possible to use a previously unnoticed physical symmetry of the Hamiltonian to construct an inner product whose associated norm is positive definite. This construction is general and works for any PT-symmetric Hamiltonian. The dynamics is governed by unitary time evolution. This formulation does not conflict with the requirements of conventional quantum mechanics. There are many possible observable and experimental consequences of extending quantum mechanics into the complex domain, both in particle physics and in solid state physics. 
  Minkowski space can be sliced, outside the lightcone, in terms of Euclidean Anti-de Sitter and Lorentzian de Sitter slices. In this paper we investigate what happens when we apply holography to each slice separately. This yields a dual description living on two spheres, which can be interpreted as the boundary of the light cone. The infinite number of slices gives rise to a continuum family of operators on the two spheres for each separate bulk field. For a free field we explain how the Green's function and (trivial) S-matrix in Minkowski space can be reconstructed in terms of two-point functions of some putative conformal field theory on the two spheres. Based on this we propose a Minkowski/CFT correspondence which can also be applied to interacting fields. We comment on the interpretation of the conformal symmetry of the CFT, and on generalizations to curved space. 
  We compute the graviton two scalar off-shell interaction vertex at tree level in Type IIB superstring theory on the pp-wave background using the light-cone string field theory formalism. We then show that the tree level vertex vanishes when all particles are on-shell and conservation of p_{+} and p_{-} are imposed. We reinforce our claim by calculating the same vertex starting from the corresponding SUGRA action expanded around the pp-wave background in the light-cone gauge. 
  This is a study of holomorphic matrix models, the matrix models which underlie the conjecture of Dijkgraaf and Vafa. I first give a systematic description of the holomorphic one-matrix model. After discussing its convergence sectors, I show that certain puzzles related to its perturbative expansion admit a simple resolution in the holomorphic set-up. Constructing a `complex' microcanonical ensemble, I check that the basic requirements of the conjecture (in particular, the special geometry relations involving chemical potentials) hold in the absence of the hermicity constraint. I also show that planar solutions of the holomorphic model probe the entire moduli space of the associated algebraic curve. Finally, I give a brief discussion of holomorphic $ADE$ models, focusing on the example of the $A_2$ quiver, for which I extract explicitly the relevant Riemann surface. In this case, use of the holomorphic model is crucial, since the Hermitian approach and its attending regularization would lead to a singular algebraic curve, thus contradicting the requirements of the conjecture. In particular, I show how an appropriate regularization of the holomorphic $A_2$ model produces the desired smooth Riemann surface in the limit when the regulator is removed, and that this limit can be described as a statistical ensemble of `reduced' holomorphic models. 
  Realistic heterotic string models require the presence of the Wilson lines. We study the effect of continuous Wilson lines on the heterotic Yukawa couplings and find that the presence of continuous Wilson lines affects the magnitude of the twisted Yukawa couplings resulting in their stronger hierarchy. 
  Hydrodynamic fluctuations at non-zero temperature can cause slow relaxation toward equilibrium even in observables which are not locally conserved. A classic example is the stress-stress correlator in a normal fluid, which, at zero wavenumber, behaves at large times as t^{-3/2}. A novel feature of the effective theory of hydrodynamic fluctuations in supersymmetric theories is the presence of Grassmann-valued classical fields describing macroscopic supercharge density fluctuations. We show that hydrodynamic fluctuations in supersymmetric theories generate essentially the same long-time power-law tails in real-time correlation functions that are known in simple fluids. In particular, a t^{-3/2} long-time tail must exist in the stress-stress correlator of N=4 supersymmetric Yang-Mills theory at non-zero temperature, regardless of the value of the coupling. Consequently, this feature of finite-temperature dynamics can provide an interesting test of the AdS/CFT correspondence. However, the coefficient of this long-time tail is suppressed by a factor of 1/N_c^2. On the gravitational side, this implies that these long-time tails are not present in the classical supergravity limit; they must instead be produced by one-loop gravitational fluctuations. 
  The role of quantum effects in brane-world cosmology is investigated. It is shown in time-independent formulation that quantum creation of deSitter branes in five-dimensional (A)dS bulk occurs with also account of brane quantum CFT contribution. The surface action is chosen to include cosmological constant and curvature term. (The time-dependent formulation of quantum-corrected brane FRW equations is shown to be convenient for comparison with Supernovae data). The particles creation on deSitter brane is estimated and is shown to be increased due to KK modes. The deSitter brane effective potential due to bulk quantum matter on 5d AdS space is found. It may be used to get the observable cosmological constant in the minimum of the potential (stabilization). The appearence of the entropy bounds from bulk field equation is also mentioned. 
  The high-energy limit of stringy symmetries, derived from the decoupling of two types of zero-norm states in the old covariant first quantized (OCFQ) spectrum of open bosonic string, are used to reproduce Gross's linear relations among high-energy scattering amplitudes of different string states with the same momenta. Moreover, the proportionality constants between scattering amplitudes of different string states are calculated for the first few low-lying levels. These proportionality constants are, as suggested by Gross from the saddle point calculation of high-energy string-loop amplitudes, independent of the scattering angle $\phi_{CM}$ and the order $\chi $ of string perturbation theory. The decoupling of degenerate positive-norm states, which is valid to \QTR{it}{all} energy, can also be derived from these stringy symmetries. The high-energy limit of this decoupling is found to be consistent with the work of Gross and Manes. 
  We study a class of time dependent solutions of the vacuum Einstein equations which are plane waves with weak null singularities. This singularity is weak in the sense that though the tidal forces diverge at the singularity, the rate of divergence is such that the distortion suffered by a freely falling observer remains finite. Among such weak singular plane waves there is a sub-class which do not exhibit large back reaction in the presence of test scalar probes. String propagation in these backgrounds is smooth and there is a natural way to continue the metric beyond the singularity. This continued metric admits string propagation without the string becoming infinitely excited. We construct a one parameter family of smooth metrics which are at a finite distance in the space of metrics from the extended metric and a well defined operator in the string sigma model which resolves the singularity. 
  We study the necessary and sufficient conditions on Abelianizable first class constraints. The necessary condition is derived from topological considerations on the structure of gauge group. The sufficient condition is obtained by applying the theorem of implicit function in calculus and studying the local structure of gauge orbits. Since the sufficient condition is necessary for existence of proper gauge fixing conditions, we conclude that in the case of a finite set of non-Abelianizable first class constraints, the Faddeev-Popov determinant is vanishing for any choice of subsidiary constraints. This result is explicitly examined for SO(3) gauge invariant model. 
  We investigate intersecting D6-branes on an orientifold of type IIA theory in the orbifold background T^6/(Z_4 x Z_2) with the emphasis on finding chiral spectra. RR tadpole cancellation conditions and the scalar potential at disc level are computed. The general chiral spectrum is displayed, and two supersymmetric models with a Pati-Salam group are shown,one with four generations and the other one with three generations and exactly the chiral matter content of the SM plus right handed neutrinos. 
  We consider Type IIB orientifold models on Calabi-Yau spaces with three-form G-flux turned on. These fluxes freeze some of the complex structure moduli and the complex dilaton via an F-term scalar potential. By introducing pairs of D9-anti-D9 branes with abelian magnetic fluxes it is possible to freeze also some of the Kaehler moduli via a D-term potential. Moreover, such magnetic fluxes in general lead to chiral fermions, which make them interesting for string model-building. These issues are demonstrated in a simple toy model based on a Z_2 x Z_2' orbifold. 
  We calculate the symmetry currents for the type IIB superstring on a maximally supersymmetric plane wave background using the N=(2,2) superconformally covariant U(4) formulation developed by Berkovits, Maldacena and Maoz. An explicit realization of the U(4) generators together with 16 fermionic generators is obtained in terms of the N=(2,2) worldsheet fields. Because the action is no longer quadratic, we use a light-cone version to display the currents in terms of the covariant worldsheet variables. 
  In this note we show that a simple modification of supersymmetric quantum mechanics involving a mass term for half the fermions naturally leads to a derivation of the integral formula for the chi_y genus, a quantity that interpolates between the Euler characteristic and arithmetic genus. We note that this modification naturally arises in the moduli space dynamics of monopoles or instantons in theories with 16 supercharges partially broken to 8 supercharges by mass terms. 
  A simple model for the reheating of the universe after inflation is studied in which an essentially inhomogeneous scalar field representing matter is coupled to an essentially homogeneous scalar inflaton field. Through this coupling, the potential determining the evolution of the inflaton field is made time-dependent. Due to this the frequency of parametric resonance becomes time dependent, making the reheating process especially effective. All fields including the gravitational field are initially simplified by expanding each in terms of the respective homogeneity or inhomogeneity. Employing only the lowest order of this expansion, we space-average, and introduce all independent averages as new variables. This leads to a hierarchy of equations for the spatial moments of the fields and their derivatives. A small expansion parameter permits a truncation to lowest order, yielding a closed system of 5 coupled nonlinear first order differential equations. For a parabolic potential, the energy densities of the matter and inflaton fields oscillate chaotically around each other from the end of inflation until they reach extremely small values. The average period of preponderance of one of the two continuously increases in this process. We discuss that this may provide one clue to a solution of the coincidence problem. For a Mexican hat potential we can easily obtain and understand dynamical symmetry breaking. 
  Torus-fibered Calabi-Yau threefolds Z, with base dP_9 and fundamental group pi_1(Z)=Z_2 X Z_2, are reviewed. It is shown that Z=X/(Z_2 X Z_2), where X=B X_{P_1} B' are elliptically fibered Calabi-Yau threefolds that admit a freely acting Z_2 X Z_2 automorphism group. B and B' are rational elliptic surfaces, each with a Z_2 X Z_2 group of automorphisms. It is shown that the Z_2 X Z_2 invariant classes of curves of each surface have four generators which produce, via the fiber product, seven Z_2 X Z_2 invariant generators in H_4(X,Z). All invariant homology classes are computed explicitly. These descend to produce a rank seven homology group H_4(Z,Z) on Z. The existence of these homology classes on Z is essential to the construction of anomaly free, three family standard-like models with suppressed nucleon decay in both weakly and strongly coupled heterotic superstring theory. 
  We present a new Lorentz gauge invariant U(1) topological field theory in Riemann-Cartan spacetime manifold $U_{4}$. By virtue of the decomposition theory of U(1) gauge potential and the $ \phi $--mapping topological current theory, it is proved that the U(1) complex scalar field $\phi (x)$ can be looked upon as the order parameter field in our Universe, and the set of zero points of $\phi (x)$ create the cosmic strings as the spacetime defects in the early Universe. In the standard cosmology this complex scalar order parameter field possesses negative pressure, provides an accelerating expansion of Universe and be able to explain the inflation in the early Universe. Therefore this complex scalar field is not only the order parameter field created the cosmic strings, but also reasonably behaves as the quintessence, the dark energy. 
  In this Note, we study bosonization of the noncommutative massive Thirring model in 2+1- dimensions. We show that, contrary to the duality between massive Thirring model and Maxwell-Chern-Simons model in ordinary spacetime, in the low energy (or large fermion mass) limit, their noncommutative versions are not equivalent, in the same approximation. 
  We consider multidimensional gravitational models with a nonlinear scalar curvature term and form fields in the action functional. In our scenario it is assumed that the higher dimensional spacetime undergoes a spontaneous compactification to a warped product manifold. Particular attention is paid to models with quadratic scalar curvature terms and a Freund-Rubin-like ansatz for solitonic form fields. It is shown that for certain parameter ranges the extra dimensions are stabilized. In particular, stabilization is possible for any sign of the internal space curvature, the bulk cosmological constant and of the effective four-dimensional cosmological constant. Moreover, the effective cosmological constant can satisfy the observable limit on the dark energy density. Finally, we discuss the restrictions on the parameters of the considered nonlinear models and how they follow from the connection between the D-dimensional and the four-dimensional fundamental mass scales. 
  We discuss type IIB orientifolds with D-branes, and NSNS and RR field strength fluxes. The D-brane sectors lead to open string spectra with non-abelian gauge symmetry and charged chiral fermions. The closed string field strengths generate a scalar potential stabilizing most moduli. We describe the construction of N=1 supersymmetric models in the context of orientifolds of IIB theory on T^6/Z_2 x Z_2, containing D9-branes with world-volume magnetic fluxes, and illustrate model building possibilities with several explicit examples. We comment on a T-dual picture with D8-branes on non-Calabi-Yau half-flat geometries, and discuss some of the topological properties of such configurations. We also explore the construction of models with fluxes and with D3-branes at singularities and present a non-supersymmetric 3-family SU(5) model. 
  We introduce a new map between a q-deformed gauge theory on a general GL_{q}(N)-covariant quantum hyperplane and an ordinary gauge theory in a full analogy with Seiberg-Witten map. Perturbative analysis of the q-deformed QED at the classical level is presented and gauge fixing a la BRST is discussed 
  We identify a canonical transformation which maps the chiral Gross-Neveu model onto a recently proposed Cooper pair model. Baryon number and axial charge are interchanged. The same physics can be described either as chiral symmetry breaking (quark-antiquark pairing) or as superconductivity (quark-quark pairing). 
  The notion of Galois currents in Rational Conformal Field Theory is introduced and illustrated on simple examples. This leads to a natural partition of all theories into two classes, depending on the existence of a non-trivial Galois current. As an application, the projective kernel of a RCFT, i.e. the set of all modular transformations represented by scalar multiples of the identity, is described in terms of a small set of easily computable invariants. 
  The propagator of a spin zero particle in coordinate space is derived supposing that the particle propagates rectilinearly always at the speed of light and changes its direction in some random points due to a scattering process.The average path between two scatterings is of the order of the Compton length. 
  According to a recent proposal, the so-called Barbero-Immirzi parameter of Loop Quantum Gravity can be fixed, using Bohr's correspondence principle, from a knowledge of highly-damped black hole oscillation frequencies. Such frequencies are rather difficult to compute, even for Schwarzschild black holes. However, it is now quite likely that they may provide a fundamental link between classical general relativity and quantum theories of gravity. Here we carry out the first numerical computation of very highly damped quasinormal modes (QNM's) for charged and rotating black holes. In the Reissner-Nordstr\"om case QNM frequencies and damping times show an oscillatory behaviour as a function of charge. The oscillations become faster as the mode order increases. At fixed mode order, QNM's describe spirals in the complex plane as the charge is increased, tending towards a well defined limit as the hole becomes extremal. Kerr QNM's have a similar oscillatory behaviour when the angular index $m=0$. For $l=m=2$ the real part of Kerr QNM frequencies tends to $2\Omega$, $\Omega$ being the angular velocity of the black hole horizon, while the asymptotic spacing of the imaginary parts is given by $2\pi T_H$. 
  The spontaneous symmetry breaking in noncommutative $\lambda\Phi^4$ theory has been analyzed by using the formalism of the effective action for composite operators in the Hartree-Fock approximation. It turns out that there is no phase transition to a constant vacuum expectation of the field and the broken phase corresponds to a nonuniform background. By considering $<\phi(x)>=A \cos(\vec Q \cdot \vec x)$ the generated mass gap depends on the angles among the momenta $\vec k$ and $\vec Q$ and the noncommutativity parameter $\vec\theta$. The order of the transition is not easily determinable in our approximation. 
  In gravitational theories with extra dimensions, it is argued that the existence of a positive vacuum energy generically implies catastrophic instability of our four-dimensional world. The most generic instability is a decompactification transition to growth of the extra dimensions, although other equally bad transitions may take place. This follows from simple considerations based on the form of the potential for the size modulus of the extra dimensions, and apparently offers a resolution of the conundrums presented by eternal de Sitter space. This argument is illustrated in the context of string theory with a general discussion of potentials generated by fluxes, wrapped branes, and stringy corrections. Moreover, it is unlikely that the present acceleration of the Universe represents an ongoing transition in a quintessence scenario rolling towards decompactification, unless the higher dimensional theory has a cosmological constant. 
  We use the matrix model -- gauge theory correspondence of Dijkgraaf and Vafa in order to construct the geometry encoding the exact gaugino condensate superpotential for the N=1 U(N) gauge theory with adjoint and symmetric or anti-symmetric matter, broken by a tree level superpotential to a product subgroup involving U(N_i) and SO(N_i) or Sp(N_i/2) factors. The relevant geometry is encoded by a non-hyperelliptic Riemann surface, which we extract from the exact loop equations. We also show that O(1/N) corrections can be extracted from a logarithmic deformation of this surface. The loop equations contain explicitly subleading terms of order 1/N, which encode information of string theory on an orientifolded local quiver geometry. 
  We investigate the relationship between supersymmetric gauge theories with moduli spaces and matrix models. Particular attention is given to situations where the moduli space gets quantum corrected. These corrections are controlled by holomorphy. It is argued that these quantum deformations give rise to non-trivial relations for generalized resolvents that must hold in the associated matrix model. These relations allow to solve a sector of the associated matrix model in a similar way to a one-matrix model, by studying a curve that encodes the generalized resolvents. At the level of loop equations for the matrix model, the situations with a moduli space can sometimes be considered as a degeneration of an infinite set of linear equations, and the quantum moduli space encodes the consistency conditions for these equations to have a solution. 
  We study the dynamical behavior of the dilaton in the background of three-dimensional Kerr-de Sitter space which is inspired from the low-energy string effective action. The Kerr-de Sitter space describes the gravitational background of a point particle whose mass and spin are given by $1-M$ and $J$ and its curvature radius is given by $\ell$. In order to study the propagation of metric, dilaton, and Kalb-Ramond two-form, we perform the perturbation analysis in the southern diamond of Kerr-de Sitter space including a conical singularity. It reveals a mixing between the dilaton and other unphysical fields. Introducing a gauge-fixing, we can disentangle this mixing completely and obtain one decoupled dilaton equation. However it turns out to be a tachyon with $m^2=-8/\ell^2$. We compute the absorption cross section for the dilatonic tachyon to extract information on the cosmological horizon of Kerr-de Sitter space. It is found that there is no absorption of the dilatonic tachyon in the presence of the cosmological horizon of Kerr-de Sitter space. 
  S(pacelike)D-branes are objects arising naturally in string theory when Dirichlet boundary conditions are imposed on the time direction. SD-brane physics is inherently time-dependent. Previous investigations of gravity fields of SD-branes have yielded undesirable naked spacelike singularities. We set up the problem of coupling the most relevant open-string tachyonic mode to massless closed-string modes in the bulk, with backreaction and Ramond-Ramond fields included. We find solutions numerically in a self-consistent approximation; our solutions are naturally asymptotically flat and time-reversal asymmetric. We find completely nonsingular evolution; in particular, the dilaton and curvature are well-behaved for all time. The essential mechanism for spacetime singularity resolution is the inclusion of full backreaction between the bulk fields and the rolling tachyon. Our analysis is not the final word on the story, because we have to make some significant approximations, most notably homogeneity of the tachyon on the unstable branes. Nonetheless, we provide significant progress in plugging a gaping hole in prior understanding of the gravity fields of SD-branes. 
  Higher spin tensor gauge fields have natural gauge-invariant field equations written in terms of generalised curvatures, but these are typically of higher than second order in derivatives. We construct geometric second order field equations and actions for general higher spin boson fields, and first order ones for fermions, which are non-local but which become local on gauge-fixing, or on introducing auxiliary fields. This generalises the results of Francia and Sagnotti to all representations of the Lorentz group. 
  We propose uncertainty relations for the different coordinates of spacetime events, motivated by Heisenberg's principle and by Einstein's theory of classical gravity. A model of Quantum Spacetime is then discussed where the commutation relations exactly implement our uncertainty relations.   We outline the definition of free fields and interactions over QST and take the first steps to adapting the usual perturbation theory. The quantum nature of the underlying spacetime replaces a local interaction by a specific nonlocal effective interaction in the ordinary Minkowski space. A detailed study of interacting QFT and of the smoothing of ultraviolet divergences is deferred to a subsequent paper.   In the classical limit where the Planck length goes to zero, our Quantum Spacetime reduces to the ordinary Minkowski space times a two component space whose components are homeomorphic to the tangent bundle TS^2 of the 2-sphere. The relations with Connes' theory of the standard model will be studied elsewhere. 
  The problem of constructing gauge invariant currents in terms of light-cone bound-state wave functions is solved by utilising the gauging of equations method. In particular, it is shown how to construct perturbative expansions of the electromagnetic current in the light-cone formalism, such that current conservation is satisfied at each order of the perturbation theory. 
  By using the hybrid formalism, superstrings in four-dimensional NS-NS plane waves are studied in a manifest supersymmetric manner. This description of the superstring is obtained by a field redefinition of the RNS worldsheet fields and defined as a topological N=4 string theory. The Hilbert space consists of two types of representations describing short and long strings. We study the physical spectrum to find boson-fermion asymmetry in the massless spectrum of the short string. Some massive spectrum of the short string and the massless spectrum of the long string are also studied. 
  Cho and Pak reply to Lamm et al. [hep-th/0007108] comment on "A Convergent Series for the Effective Action of QED" [hep-th/0006057]. 
  The Leuven workshop on the `Quantum Structure of Space-time and the Geometrical Nature of the Fundamental Interactions' had a special session dedicated to the memory of Sonia Stanciu. This is the summary of a talk delivered by the author on this occasion. 
  We discuss a Penrose limit of an elliptic brane configuration with $N_1$ NS5 and $N_2$ D4 branes. This background is T-dual to $N_1$ D3 branes at a fixed point of a $\mathbf{C}^3/\mathbf{Z}_{N_2}$ singularity and the T-duality survives the Penrose limit. The triple scaling limit of $N_1$ and $N_2$ gives rise to IIA pp-wave solution with a space-like compact direction. We identify the quiver gauge theory operators and argue that upon exchange of the momentum along the compact direction and the winding number these operators coincide with the operators derived in the dual type IIB description. We also find a new Penrose limit of the type IIB background and the corresponding limit in the type IIA picture. In the coordinate system we use there are two manifest space-like isometries. The quiver gauge theory operator duals of the string states are built of three bosonic fields. 
  The modular invariance of the one-loop partition function of the closed bosonic string in four dimensions in the presence of certain homogeneous exact pp-wave backgrounds is studied. In the absence of an axion field the partition function is found to be modular invariant. In the presence of an axion field modular invariace is broken. This can be attributed to the light-cone gauge which breaks the symmetry in the $\sigma$-, $t$-directions. Recovery of this broken modular invariance suggests the introduction of twists in the world-sheet directions. However, one needs to go beyond the light-cone gauge to introduce such twists. 
  In an earlier paper, we showed that the causal boundary of any homogeneous plane wave satisfying the null convergence condition consists of a single null curve. In Einstein-Hilbert gravity, this would include any homogeneous plane wave satisfying the weak null energy condition. For conformally flat plane waves such as the Penrose limit of $AdS_5 \times S^5$, all spacelike curves that reach infinity also end on this boundary and the completion is Hausdorff. However, the more generic case (including, e.g., the Penrose limits of $AdS_4 \times S^7$ and $AdS_7 \times S^4$) is more complicated. In one natural topology, not all spacelike curves have limit points in the causal completion, indicating the need to introduce additional points at `spacelike infinity'--the endpoints of spacelike curves. We classify the distinct ways in which spacelike curves can approach infinity, finding a {\it two}-dimensional set of distinct limits. The dimensionality of the set of points at spacelike infinity is not, however, fixed from this argument. In an alternative topology, the causal completion is already compact, but the completion is non-Hausdorff. 
  We have covariantized the Lagrangians of the U(1)_V * U(1)_A models, which have U(1)_V * U(1)_A gauge symmetry in two dimensions, and studied their symmetric structures. The special property of the U(1)_V * U(1)_A models is the fact that all these models have an extra time coordinate in the target space-time. The U(1)_V * U(1)_A models coupled to two-dimensional gravity are string models in 26+2 dimensional target space-time for bosonic string and in 10+2 dimensional target space-time for superstring. Both string models have two time coordinates. In order to construct the covariant Lagrangians of the U(1)_V * U(1)_A models the generalized Chern-Simons term plays an important role. The supersymmetric generalized Chern-Simons action is also proposed. The Green-Schwarz type of U(1)_V * U(1)_A superstring model has another fermionic local symmetry as well as \kappa-symmetry. The supersymmetry of target space-time is different from the standard one. 
  We demonstrate an inflationary solution to the cosmological horizon problem during the Hagedorn regime in the early universe. Here the observable universe is confined to three spatial dimensions (a three-brane) embedded in higher dimensions. The only ingredients required are open strings on D-branes at temperatures close to the string scale. No potential is required. Winding modes of the strings provide a negative pressure that can drive inflation of our observable universe. Hence the mere existence of open strings on branes in the early hot phase of the universe drives Hagedorn inflation, which can be either power law or exponential. We note the amusing fact that, in the case of stationary extra dimensions, inflationary expansion takes place only for branes of three or less dimensions. 
  We study flux tubes on Higgs branches with curved geometry in supersymmetric gauge theories. As a first example we consider N=1 QED with one flavor of charges and with Higgs branch curved by adding a Fayet-Iliopoulos (FI) term. We show that in a generic vacuum on the Higgs branch flux tubes exist but become ``thick''. Their internal structure in the plane orthogonal to the string is determined by ``BPS core'' formed by heavy fields and long range ``tail '' associated with light fields living on the Higgs branch. The string tension is given by the tension of ``BPS core'' plus contribution coming from the ``tail''. Next we consider N=2 QCD with gauge group SU(2)and two flavors of fundamental matter (quarks) with the same mass. We perturb this theory by the mass term for the adjoint field which to the leading order in perturbation parameter do not break N=2 supersymmetry and reduces to FI term. The Higgs branch has Eguchi-Hanson geometry. We work out string solution in the generic vacuum on the Higgs branch and calculate its string tension. We also discuss if these strings can turn into semilocal strings, the possibility related to the confinement/deconfinement phase transition. 
  We determine the general coupling of a system of scalars and antisymmetric tensors, with at most two derivatives and undeformed gauge transformations, for both rigid and local N=2 supersymmetry in four-dimensional spacetime. Our results cover interactions of hyper, tensor and double-tensor multiplets and apply among others to Calabi-Yau threefold compactifications of Type II supergravities. As an example, we give the complete Lagrangian and supersymmetry transformation rules of the double-tensor multiplet dual to the universal hypermultiplet. 
  We derive the Lagrangian and the transformation laws of N=4 gauged supergravity coupled to matter multiplets whose sigma-model of the scalars is SU(1,1)/U(1)x SO(6,6+n)/SO(6)xSO(6+n) and which corresponds to the effective Lagrangian of the Type IIB string compactified on the T^6/Z_2 orientifold with fluxes turned on and in presence of n D3-branes. The gauge group is T^12 x G where G is the gauge group on the brane and T^12 is the gauge group on the bulk corresponding to the gauged translations of the R-R scalars coming from the R-R four--form. The N=4 bulk sector of this theory can be obtained as a truncation of the Scherk-Schwarz spontaneously broken N=8 supergravity. Consequently the full bulk spectrum satisfies quadratic and quartic mass sum rules, identical to those encountered in Scherk-Schwarz reduction gauging a flat group. This theory gives rise to a no scale supergravity extended with partial super-Higgs mechanism. 
  A multidimensional gravitational model with several scalar fields and form fields is considered. A wide class of generalized pp-wave solutions defined on a product of n+1 Ricci-flat spaces is obtained. Certain examples of solutions (e.g. in supergravitational theories) are singled out. For special cone-type internal factor spaces the solutions are written in Brinkmann form. An example of pp-wave solution is obtained using Penrose limit of a solution defined on a product of two Einstein spaces. 
  It is well-known that the standard no-ghost theorem is valid as long as the background has the light-cone directions. We prove the no-ghost theorem for the NSR string when only the timelike direction is flat. This is done by the BRST quantization, using the technique of Frenkel, Garland and Zuckerman and our previous results for the bosonic string. The theorem actually applies as long as the timelike direction is written as a u(1) SCFT. 
  For the problem of calculating bound states in quantum field theory, the light-cone representation offers advantages over the more common equal-time representation. It also has subtleties and disadvantages compared to the equal-time representation. If current efforts to use the light-cone representation to solve for the properties of hadrons in QCD are to succeed, at least two problems have to be solved: we must find the induced operators; we must develop an effective procedure of regularization and renormalization. In this paper I will try to explain what an induced operator is and say what we know about them and will report on recent attempts to develop an effective procedure of regularization and renormalization. 
  Using topological Yang-Mills theory as example, we discuss the definition and determination of observables in topological field theories (of Witten-type) within the superspace formulation proposed by Horne. This approach to the equivariant cohomology leads to a set of bi-descent equations involving the BRST and supersymmetry operators as well as the exterior derivative. This allows us to determine superspace expressions for all observables, and thereby to recover the Donaldson-Witten polynomials when choosing a Wess-Zumino-type gauge. 
  We consider a geometric regularization for the class of conifold transitions relating D-brane systems on noncompact Calabi-Yau spaces to certain flux backgrounds. This regularization respects the SL(2,Z) invariance of the flux superpotential, and allows for computation of the relevant periods through the method of Picard-Fuchs equations. The regularized geometry is a noncompact Calabi-Yau which can be viewed as a monodromic fibration, with the nontrivial monodromy being induced by the regulator. It reduces to the original, non-monodromic background when the regulator is removed. Using this regularization, we discuss the simple case of the local conifold, and show how the relevant field-theoretic information can be extracted in this approach. 
  We argue that the Penrose limit of a general string background is a generalization of the Seiberg-Sen limit describing M(atrix) theory as the DLCQ of M theory in flat space. The BMN theory of type IIB strings on the maximally supersymmetric pp-wave background is understood as the exact analogue of the BFSS M(atrix) theory, namely, a DLCQ of IIB string theory on $AdS_5 \times S^5$ in the limit of infinite longitudinal momentum. This point of view is used to explain some features of the BMN duality. 
  New algebraic approach to analytical calculations of D-dimensional integrals for multi-loop Feynman diagrams is proposed. We show that the known analytical methods of evaluation of multi-loop Feynman integrals, such as integration by parts and star-triangle relation methods, can be drastically simplified by using this algebraic approach. To demonstrate the advantages of the algebraic method of analytical evaluation of multi-loop Feynman diagrams, we calculate ladder diagrams for the massless $\phi^3$ theory. Using our algebraic approach we show that the problem of evaluation of special classes of Feynman diagrams reduces to the calculation of the Green functions for specific quantum mechanical problems. In particular, the integrals for ladder massless diagrams in the $\phi^3$ scalar field theory are given by the Green function for the conformal quantum mechanics. 
  The tachyon effective field theory describing the dynamics of a non-BPS D-brane in superstring theory has an infinitely thin but finite tension kink solution describing a codimension one BPS D-brane. We study the world-volume theory of massless modes on the kink, and show that the world volume action has precisely the Dirac-Born-Infeld (DBI) form without any higher derivative corrections. We generalize this to a vortex solution in the effective field theory on a brane-antibrane pair. As in the case of the kink, the vortex is infinitely thin, has finite energy density, and the world-volume action on the vortex is again given exactly by the DBI action on a BPS D-brane. We also discuss the coupling of fermions and restoration of supersymmetry and $\kappa$-symmetry on the world-volume of the kink. Absence of higher derivative corrections to the DBI action on the soliton implies that all such corrections are related to higher derivative corrections to the original effective action on the world-volume of a non-BPS D-brane or brane-antibrane pair. 
  The radiation from oscillating kink in (1+1) dimensional relativistic $\phi^4$ model is considered. Both analytical and numerical approaches are presented and the comparison between these methods is discussed. Acceleration of the kink in external radiation is calculated and numerical results are also presented. 
  Being gauge non-invariant, a Chern-Simons (2k-1)-form seen as a Lagrangian of gauge theory on a (2k-1)-dimensional manifold leads to the gauge conservation law of a modified Noether current. 
  We argue that existing methods for the perturbative computation of anomalous dimensions and the disentanglement of mixing in N = 4 gauge theory can be considerably simplified, systematized and extended by focusing on the theory's dilatation operator. The efficiency of the method is first illustrated at the one-loop level for general non-derivative scalar states. We then go on to derive, for pure scalar states, the two-loop structure of the dilatation operator. This allows us to obtain a host of new results. Among these are an infinite number of previously unknown two-loop anomalous dimensions, new subtleties concerning 't Hooft's large N expansion due to mixing effects of degenerate single and multiple trace states, two-loop tests of various protected operators, as well as two-loop non-planar results for two-impurity operators in BMN gauge theory. We also put to use the recently discovered integrable spin chain description of the planar one-loop dilatation operator and show that the associated Yang-Baxter equation explains the existence of a hitherto unknown planar ``axial'' symmetry between infinitely many gauge theory states. We present evidence that this integrability can be extended to all loops, with intriguing consequences for gauge theory, and that it leads to a novel integrable deformation of the XXX Heisenberg spin chain. Assuming that the integrability structure extends to more than two loops, we determine the planar three-loop contribution to the dilatation operator. 
  A small comment on the paper with the mentioned title by Carl M. Bender, Dorje C. Brody and Hugh F. Jones. 
  A new approach to the inverse scattering problem proposed by Schroer, is applied to two-dimensional integrable quantum field theories. For any two-particle S-matrix S_2 which is analytic in the physical sheet, quantum fields are constructed which are localizable in wedge-shaped regions of Minkowski space and whose two-particle scattering is described by the given S_2. These fields are polarization-free in the sense that they create one-particle states from the vacuum without polarization clouds. Thus they provide examples of temperate polarization-free generators in the presence of non-trivial interaction. 
  We study F-terms describing coupling of the supergravity to N=1 supersymmetric gauge theories which admit large N expansions. We show that these F-terms are given by summing over genus one non-planar diagrams of the large N expansion of the associated matrix model (or more generally bosonic gauge theory). The key ingredient in this derivation is the observation that the chiral ring of the gluino fields is deformed by the supergravity fields, generalizing the C-deformation which was recently introduced. The gravity induced part of the C-deformation can be derived from the Bianchi identities of the supergravity, but understanding gravitational corrections to the F-terms requires a non-traditional interpretation of these identities. 
  We investigate the Kepler problem using a symplectic structure consistent with the commutation rules of the noncommutative quantum mechanics. We show that a noncommutative parameter of the order of $10^{-58} \text m^2$ gives observable corrections to the movement of the solar system. In this way, modifications in the physics of smaller scales implies modifications at large scales, something similar to the UV/IR mixing. 
  We explore the phases of N = 1 supersymmetric U(N) gauge theories with fundamental matter that arise as deformations of N = 2 SQCD by the addition of a superpotential for the adjoint chiral multiplet. As the parameters in the superpotential are varied, the vacua of this theory sweep out various branches, which in some cases have multiple semiclassical limits. In such limits, we recover the vacua of various product gauge group theories, with flavors charged under some group factors. We describe in detail the structure of the vacua in both classical and quantum regimes, and develop general techniques such as an addition and a multiplication map which relate vacua of different gauge theories. We also consider possible indices characterizing different branches and potential relationships with matrix models. 
  We argue that two-dimensional (0,2) gauged linear sigma models are not destabilized by instanton generated world-sheet superpotentials. We construct several examples where we show this to be true. The general proof is based on the Konishi anomaly for (0,2) theories. 
  The Bousso bound requires that one quarter the area of a closed codimension two spacelike surface exceeds the entropy flux across a certain lightsheet terminating on the surface. The bound can be violated by quantum effects such as Hawking radiation. It is proposed that at the quantum level the bound be modified by adding to the area the quantum entanglement entropy across the surface. The validity of this quantum Bousso bound is proven in a two-dimensional large N dilaton gravity theory. 
  Newton potential has been evaluated for the case of dS brane embedded in Minkowski, dS$_5$ and AdS$_5$ bulks. We point out that only the AdS$_5$ bulk might be consistent with the Newton's law from the brane-world viewpoint when we respect a small cosmological constant observed at present universe. 
  In this paper we study Hitchin system on singular curves. Some examples of such system were first considered by N. Nekrasov (hep-th/9503157), but our methods are different. We consider the curves which can be obtained from the projective line by gluing several points together or by taking cusp singularities. (More general cases of gluing subschemas will be considered in the next paper). It appears that on such curves all ingredients of Hitchin integrable system (moduli space of vector bundles, dualizing sheaf, Higgs field etc.) can be explicitly described, which may deserve independent interest. As a main result we find explicit formulas for the Hitchin hamiltonians. We also show how to obtain the Hitchin integrable system on such a curve as a hamiltonian reduction from a more simple system on some finite-dimensional space. In this paper we also work out the case of a degenerate curve of genus two and find the analogue of the Narasimhan-Ramanan parameterization of SL(2)-bundles. We describe the Hitchin system in such coordinates. As a demonstration of the efficiency of our approach we also rederive the rational and trigonometric Calogero systems from the Hitchin system on cusp and node with a marked point. 
  A derivative expansion of the effective average action beyond first order yields renormalization group functional flow equations which are used for the computation of critical exponents of the Ising universality class. The critical exponent nu in D=3 is consistent with high-precision methods. 
  In the framework of Lagrangian formulation, some q-deformed physical systems are considered. The q-deformed Legendre transformation is obtained for the free motion of a non-relativistic particle on a quantum line. This is subsequently exploited to obtain the Lagrangians for the q-deformed harmonic oscillator and q-deformed relativistic free particle. The Euler-Lagrange equations of motion are derived in a consistent manner with the corresponding Hamilton's equations. The Lagrangian for the q-deformed relativistic particle is endowed with the q-deformed gauge symmetry and reparametrization invariance which are shown to be equivalent only for $ q = \pm 1$. 
  A remarkable feature of D-branes is the appearance of a nonabelian gauge theory in the description of several (nearly) coincident branes. This nonabelian structure plays an important role in realizing various geometric effects with D-branes. In particular, the branes' transverse displacements are described by matrix-valued scalar fields and so noncommutative geometry naturally appears in this framework. I review the action governing this nonabelian theory, as well as various related physical phenomena such as the dielectric effect, giant gravitons and fuzzy funnels. 
  A Brane evolving in the background of a charged AdS black-hole displays in general a bouncing behaviour with a smooth transition from a contracting to an expanding phase. We examine in detail the conditions and consequences of this behaviour in various cases. For a cosmological-constant-dominated Brane, we obtain a singularity-free, inflationary era which is shown to be compatible only with an intermediate-scale fundamental Planck mass. For a radiation-dominated Brane, the bouncing behaviour can occur only for background-charge values exceeding those allowed for non-extremal black holes. For a matter-dominated Brane, the black-hole mass affects the proper volume or the expansion rate of the Brane. We also consider the Brane evolving in an asymmetric background of two distinct charged AdS black hole spacetimes being bounded by the Brane and find that, in the case of an empty critical Brane, bouncing behaviour occurs only if the black-hole mass difference is smaller than a certain value. The effects of a Brane curvature term on the bounce at early and late times are also investigated. 
  We study N=1 supersymmetric U(N) gauge theory coupled to an adjoint scalar superfiled with a cubic superpotential containing a multi trace term. We show that the field theory results can be reproduced from a matrix model which its potential is given in terms of a linearized potential obtained from the gauge theory superpotential by adding some auxiliary nondynamical field. Once we get the effective action from this matrix model we could integrate out the auxiliary field getting the correct field theory results. 
  A geometric treatment of T-duality as an operation which acts on differential forms in superspace allows us to derive the complete set of T-duality transformation rules which relate the superfield potentials of D=10 type IIA supergravity with those of type IIB supergravity including Ramond-Ramond superfield potentials and fermionic supervielbeins. We show that these rules are consistent with the superspace supergravity constraints. 
  Closed string tachyons have long been somewhat mysterious. We note that there is often a regime in the classical moduli space in which one can systematically compute the effective action for such fields. In this regime, the tachyon is light, and cannot be integrated out. Instead, one must consider the combined dynamics of gravitons, moduli, tachyons and other light fields. We compute the action and find that the quartic term for the tachyon is positive in the field definition where the tachyon has no derivative coupling to the radion. We study the evolution of isotropic, homogeneous configurations and find that typically the system is driven to regions where the calculation is no longer under control. 
  We study the tensor product of the {\it higher spin representations} (see the definition in Sect. 2.2) of the elliptic quantum group $E_{\tau,\eta}(sl_n)$. The transfer matrices associated with the $E_{\tau,\eta}(sl_n)$-module are exactly diagonalized by the nested Bethe ansatz method. Some special cases of the construction give the exact solution for the $Z_n$ Belavin model and for the elliptic $A_{n-1}$ Ruijsenaars-Schneider model. 
  The motion of point charged particles is considered in an even dimensional Minkowski space-time. The potential functions corresponding to the massless scalar and the Maxwell fields are derived algorithmically. It is shown that in all even dimensions particles lose energy due to acceleration. 
  A special class of mixed-symmetry type tensor gauge fields of degrees two and three in four dimensions is investigated from the perspective of the Lagrangian deformation procedure based on cohomological BRST techniques. It is shown that the deformed solution to the master equation can be taken to be nonvanishing only at the first order in the coupling constant. As a consequence, we deduce an interacting model with deformed gauge transformations, an open gauge algebra and undeformed reducibility functions. The resulting coupled Lagrangian action contains a quartic vertex and some ``mass'' terms involving only the tensor of degree two. We discuss in what sense the results of the deformation procedure derived here are complementary to recent others. 
  In these lecture notes I give an introduction to deformation quantization. The quantization problem is discussed in some detail thereby motivating the notion of star products. Starting from a deformed observable algebra, i.e. the star product algebra, physical applications require to study representations of this algebra. I review the recent development of a representation theory including techniques like Rieffel induction and Morita equivalence. Applications beyond quantization theory are found in noncommutative field theories. 
  We consider a class of non-supersymmetric gauge theories obtained by orbifolding the N=4 super-Yang-Mills theories. We focus on the resulting quiver theories in their deconstructed phase, both at small and large coupling, where a fifth dimension opens up. In particular we investigate the r\^ole played by this extra dimension when evaluating the rectangular Wilson loops encoding the interaction potential between quarks located at different points in the orbifold. The large coupling potential of the deconstructed quiver theory is determined using the AdS/CFT correspondence and analysing the corresponding minimal surface solution for the dual gravitational metric. At small coupling, the potential between quarks decreases with their angular distance while at strong coupling we find a linear dependence at large distance along the (deconstructed) fifth dimension. 
  We obtain the exact non-perturbative solution of a scalar field theory defined on a space with noncommuting position and momentum coordinates. The model describes non-locally interacting charged particles in a background magnetic field. It is an exactly solvable quantum field theory which has non-trivial interactions only when it is defined with a finite ultraviolet cutoff. We propose that small perturbations of this theory can produce solvable models with renormalizable interactions. 
  We present a conformal field theory calculation of four-point and three-point correlation functions for the bosonic twist fields arising at the intersections of D-branes wrapping (supersymmetric) homology cycles of Type II orientifold compactifications. Both the quantum contribution from local excitations at the intersections and the world-sheet disk instanton corrections are computed. As a consequence we obtain the complete expression for the Yukawa couplings of chiral fermions with the Higgs fields. The four-point correlation functions in turn lead to the determination of the four-point couplings in the effective theory, and may be of phenomenological interest. 
  Witten's observables of topological Yang-Mills theory, defined as classes of an equivariant cohomology, are reobtained as the BRST cohomology classes of a superspace version of the theory. 
  In this note we study supersymmetric solutions of 11 dimensional SUGRA. Amongst others, we find a new solution, which preserves 3/8 of the original supersymmetry. 
  Inspired by the intimate relationship between Voiculescu's noncommutative probability theory (of type A) and large-N matrix models in physics, we look for physical models related to noncommutative probability theory of type B. These turn out to be fermionic matrix-vector models at the double large-N limit. In the context of string theory, they describe different orbifolded string worldsheets with boundaries. Their critical exponents coincide with that of ordinary string worldsheets, but their renormalised tree-level one-boundary amplitudes differ. 
  Closed string tachyon condensation in spacetime generates potentials on the worldsheet that model two-dimensional inflationary cosmology. These models illustrate and elucidate a variety of aspects of inflation, in particular the generation of quantum fluctuations and their back-reaction on geometry. We exhibit a class of Liouville gravity models coupled to matter that can exhibit, for example: (a) pure de Sitter gravity; (b) slow-roll inflation; (c) topological inflation; and (d) graceful exit into an FRW phase. The models also provide a quantitative testing ground for ideas about the origin of inflation, such as the various `no-boundary/tunnelling' proposals, and the `eternal/chaotic' inflationary scenario. We suggest an alternative mechanism for quantum creation of cosmological spacetimes which, in the context of the model, provides a natural explanation for why the typical FRW cosmology at large scales underwent a period of inflation at small scale. 
  We relate classical and quantum Dirac and Nambu brackets. At the classical level, we use the relations between the two brackets to gain some insight into the Jacobi identity for Dirac brackets, among other things. At the quantum level, we suggest that the Nambu bracket is the preferred method for introducing constraints, although at the expense of some unorthodox behavior, which we describe in detail. 
  We consider asymptotically anti-de Sitter black holes in $d$-spacetime dimensions in the thermodynamically stable regime. We show that the Bekenstein-Hawking entropy and its leading order corrections due to thermal fluctuations can be reproduced by a weakly interacting fluid of bosons and fermions (`dual gas') in $\Delta=\alpha(d-2)+1$ spacetime dimensions, where the energy-momentum dispersion relation for the constituents of the fluid is assumed to be $\epsilon = \kappa p^\alpha$. We examine implications of this result for entropy bounds and the holographic hypothesis. 
  Investigating the solitons in the non-commutative $CP^{1}$ model, we have found a new set of BPS solitons which does not have counterparts in the commutative model. 
  We construct the so-called theta vectors on noncommutative T^4, which correspond to the theta functions on commutative tori with complex structures. Following the method of Dieng and Schwarz, we first construct holomorphic connections and then find the functions satisfying the holomorphic conditions, the theta vectors. The holomorphic structure in the noncommutative T^4 case is given by a 2x2 complex matrix, and the consistency requires its off-diagonal elements to be the same. We also construct the tensor product of these functions satisfying the consistency requirement. 
  A class of exact solutions of the Skyrme model are obtained. They are described by the Weierstrass $\wp$-function or the Jacobi elliptic function. They are not solitonic but of wave character. They supply us with examples of the superposition of three plane waves in the Skyrme model. 
  In this paper, we study the phase space of phantom model with O(\emph{N}) symmetry in exponential potential. Different from the model without O(\emph{N}) symmetry, the introduction of the symmetry leads to a lower bound $w>-3$ on the equation of state for the existence of stable phantom dominated attractor phase. The reconstruction relation between the potential of O(\textit{N}) phantom system and red shift has been derived. 
  We review various generalizations of supersymmetry and discuss their relationship. In particular, we show how supersymmetry, parasupersymmetry, fractional supersymmetry, orthosupersymmetry, and the Z_n-graded topological symmetries are related. 
  Cosmological consequences of the brane world scenario are reviewed in a pedagogical manner. According to the brane world idea, the standard model particles are confined on a hyper--surface (a so--called brane), which is embedded in a higher--dimensional spacetime (the so--called bulk). We begin our review with the simplest consistent brane world model: a single brane embedded in a five--dimensional Anti-de Sitter space--time. Then we include a scalar field in the bulk and discuss in detail the difference with the Anti-de Sitter case. The geometry of the bulk space--time is also analysed in some depth. Finally, we investigate the cosmology of a system with two branes and a bulk scalar field. We comment on brane collisions and summarize some open problems of brane world cosmology. 
  We study solutions of Einstein's equations corresponding codimension n>2 global topological defects with de Sitter slices. We analyze a class of solutions that are cylindrically symmetric and admit positive, negative or zero bulk cosmological constant. We derive the relevant graviton equations. For an extended brane, the properties of the solution depend on apropriate boundary conditions that the exterior solutions have to satisfy near the core. As an alternative we consider matching copies of the exterior solution related by symmetry. We show that we can get localization only when the bulk cosmological constant is negative. We obtain a condition on the global defect symmetry breaking scale which ultimately controls the size of the n-1 internal dimensions at the position of the brane. The induced metric on the brane, in the case of mirror spacetimes, is a direct product of a de Sitter space and an (n-1)-sphere, while the metric of the embedding spacetime is a warped product and the actual size of the (n-1)-sphere changes as we move along the radial direction. The solutions possess naked singularities, which nevertheless satisfy no-flow conditions. 
  A solution of the (4+n)-dimensional vacuum Einstein equations is found for which spacetime is compactified on a compact hyperbolic manifold of time-varying volume to a flat four-dimensional FLRW cosmology undergoing accelerated expansion in Einstein conformal frame. This shows that the `no-go' theorem forbidding acceleration in `standard' (time-independent) compactifications of string/M-theory does not apply to `cosmological' (time-dependent) hyperbolic compactifications. 
  We obtain an elliptic system of monopole equations on 8-manifolds with Spin(7) holonomy by minimizing an action involving negative spinors coupled to an Abelian gauge fields. 
  The acceleration-dependent system with noncommuting coordinates, proposed by Lukierski, Stichel and Zakrzewski [Ann. Phys. 260, 224 (1997)] is derived as the non-relativistic limit of Mathisson's classical electron [Acta Physica Polonica 6, 218 (1937)], further discussed by Weyssenhoff and Raabe [Acta Physica Polonica 9, 7 (1947)]. The two-parameter centrally extended Galilean symmetry of the model is recoved using elementary methods. The relation to Schr\"odinger's Zitternde Elektron is indicated. 
  The `Landau--Ginzburg' theory of Girvin and MacDonald, modified by adding the natural magnetic term, is shown to admit stable topological as well as non-topological vortex solutions. The system is the commun $\lambda\to0$ limit of two slightly different non-relativistic Maxwell--Chern--Simons models of the type introduced recently by Manton. The equivalence with the model of Zhang, Hansson and Kivelson is demonstrated. 
  We investigate the properties of localized anomalous U(1)'s in heterotic string theory on the orbifold T^6/Z_3. We argue that the local four dimensional and original ten dimensional Green-Schwarz mechanisms can be implemented simultaneously, making the theory manifestly gauge invariant everywhere, in the bulk and at the fixed points. We compute the shape of the Fayet-Iliopoulos tadpoles, and cross check this derivation for the four dimensional auxiliary fields by a direct calculation of the tadpoles of the internal gauge fields. Finally we study some resulting consequences for spontaneous symmetry breaking, and derive the profile of the internal gauge field background over the orbifold. 
  We derive the actions for type II Green-Schwarz strings up to second order in the fermions, for general bosonic backgrounds. We base our work on the so-called normal coordinate expansion. The resulting actions are $\kappa$-symmetric and, for the case of surviving background supersymmetries, supersymmetric. We first obtain the type IIa superstring action from the 11D supermembrane by double dimensional reduction. Then, by means of a generalization of T-duality we derive the type IIb superstring action. The resulting actions are surprisingly simple and elegant. 
  In the presence of a short-distance cutoff, the choice of a vacuum state in an inflating, non-de Sitter universe is unavoidably ambiguous. The ambiguity is related to the time at which initial conditions for the mode functions are specified and to the way the expansion of the universe affects those initial conditions. In this paper we study the imprint of these uncertainties on the predictions of inflation. We parametrize the most general set of possible vacuum initial conditions by two phenomenological variables. We find that the generated power spectrum receives oscillatory corrections whose amplitude is proportional to the Hubble parameter over the cutoff scale. In order to further constrain the phenomenological parameters that characterize the vacuum definition, we study gravitational particle production during different cosmological epochs. 
  With the aim of extending the gauge theory -- matrix model connection to more general matter representations, we prove that for various two-index tensors of the classical gauge groups, the perturbative contributions to the glueball superpotential reduce to matrix integrals. Contributing diagrams consist of certain combinations of spheres, disks, and projective planes, which we evaluate to four and five loop order. In the case of $Sp(N)$ with antisymmetric matter, independent results are obtained by computing the nonperturbative superpotential for $N=4,6$ and 8. Comparison with the Dijkgraaf-Vafa approach reveals agreement up to $N/2$ loops in matrix model perturbation theory, with disagreement setting in at $h=N/2+1$ loops, $h$ being the dual Coxeter number. At this order, the glueball superfield $S$ begins to obey nontrivial relations due to its underlying structure as a product of fermionic superfields. We therefore find a relatively simple example of an ${\cal N}=1$ gauge theory admitting a large $N$ expansion, whose dynamically generated superpotential differs from the one obtained in the matrix model approach. 
  The present-day significance of Yang's quantized space-time algebra (YST) is pointed out from the holographic viewpoint. One finds that the D-dimensional YST and its modified version (MYST) have the background symmetry SO(D+1,1) and SO(D,2), which are well known to underlie the dS/CFT and AdS/CFT correspondences, respectively. This fact suggests a new possibility of dS/YST and AdS/MYST in parallel with dS/CFT and AdS/CFT, respectively. The spatial components of quantized space-time and momentum operators of YST have discrete eigenvalues and their respective minimums, $a$ and 1/R, without contradiction to Lorentz-covariance. With respect to MYST, the spatial components of space-time operators and the time component (energy) of momentum operators have discrete eigenvalues and their respective minimums, $a$ and 1/R, in contrast to YST. This discrete structure of YST and MYST, which CFT lacks entirely, may provide a theoretical ground for the unified regularization or cutoff of ultraviolet and infrared divergences familiar in the UV/IR connection in the holographic hypothesis. 
  Superpartner correspondence of states of colored particle in external chromomagnetic field given by non-commuting constant vector potentials is determined. Squared Dirac equation for this particle is solved exactly and explicit expressions of superpartner states are found. The wave functions of states with definite energy are found. Supersymmetry algebra and superpartner states in three dimensional motion case are considered. 
  We calculate the coefficients of three-point functions of BMN operators with two vector impurities. We find that these coefficients can be obtained from those of the three-point functions of scalar BMN operators by interchanging the coefficient for the symmetric-traceless representation with the coefficient for the singlet. We conclude that the Z_2 symmetry of the pp-wave string theory is not manifest at the level of field theory three-point correlators. 
  A QED-based mechanism, breaking translational invariance of the vacuum at sufficiently small distance scales, is suggested as an explanation for the vacuum energy pressure that accelerates the universe. Very-small-scale virtual vacuum currents are assumed to generate small-scale electromagnetic fields corresponding to the appearance of a 4-potential $A_{\mu}^{ext} (x)$, which is itself equal to the vev of the operator $ A_{\mu}(x)$ in the presence of that $A_{\mu}^{ext}(x)$. The latter condition generates a bootstrap-like equation for $A_{\mu}^{ext}(x)$ which has an approximate, tachyonic-like solution corresponding to propagation outside the light cone, and damping inside; this solution is given in terms of a mass parameter M that turns out to be on the order of the Planck mass if only the simplest, electron vacuum-bubble is included; if the muon and tau bubbles are included, M decreases to $\sim 10^6 - 10^7$ GeV. A multiplicative 4-vector $v_{\mu}$, whose magnitude is determined by a comparison with the average mass density needed to produce the observed acceleration is introduced, and characterizes the distance d over which the fields so produced may be expected to be coherent; the present analysis suggests that d can lie anywhere in the range from $10^{-5} cm$ (corresponding to a "spontaneous vacuum phase change") to $10^{-13}cm$ (representing a "polarization of the QED vacuum" by quark-antiquark pairs of the QCD vacuum). Near the light-cone, such electric fields become large, introducing the possibility of copious charged-particle pair production, whose back-reaction-fields tend to diminish the vacuum electric field. The possibility of an experimental test of the resulting plasma at large momentum transfers is discussed. 
  We discuss domain walls and vacuum energy density (cosmological constant) in N=1 gluodynamics and in non-supersymmetric large N orientifold field theories which have been recently shown to be planar equivalent (in the boson sector) to N=1 gluodynamics. A relation between the vanishing force between two parallel walls and vanishing cosmological constant is pointed out. This relation may explain why the cosmological constant vanishes in the orientifold field theory at leading order although the hadronic spectrum of this theory does not contain fermions in the limit N-->infinity. The cancellation is among even and odd parity bosonic contributions, due to NS-NS and R-R cancellations in the annulus amplitude of the underlying string theory. We use the open-closed string channel duality to describe interaction between the domain walls which is interpreted as the exchange of composite ``dilatons'' and ``axions'' coupled to the walls. Finally, we study some planar equivalent pairs in which both theories in the parent-daughter pair are non-supersymmetric. 
  In this paper, we analyze the anisotropy of the scale factor in the Kantowski-Sachs spacetime. We show that the anisotropy will not increase when the expansion rate is greater than certain values while it will increase when the expansion rate is less than that value or the Universe is contracting. It is manifested that the matter dominated and radiation dominated era favor the flat spacetime if the anisotropy does not develop significantly. The relation between the cosmological anisotropy and the red-shift of the supernovae, which could be used to verify the anisotropy through the observation, has been derived. 
  We show how Moore's observation, in the context of toroidal compactifications in type IIB string theory, concerning the complex multiplication structure of black hole attractor varieties, can be generalized to Calabi-Yau compactifications with finite fundamental groups. This generalization leads to an alternative general framework in terms of motives associated to a Calabi-Yau variety in which it is possible to address the arithmetic nature of the attractor varieties in a universal way via Deligne's period conjecture. 
  In this work, we present the gravitational field generated by a cosmic string carrying a timelike current in the scalar-tensor gravities. The mechanism of formation and evolution of wakes is fully investigated in this framework. We show explicitly that the inclusion of electromagnetic properties for the string induces logarithmic divergences in the accretion problem. 
  Performing a Scherk-Schwarz dimensional reduction of D=11 supergravity on a three-dimensional group manifold we construct five D=8 gauged maximal supergravities whose gauge groups are the three-dimensional (non-)compact subgroups of SL(3,R). These cases include the Salam-Sezgin SO(3) gauged supergravity. We construct the most general half-supersymmetric domain wall solutions to these five gauged supergravities. The generic form is a triple domain wall solution whose truncations lead to double and single domain wall solutions. We find that one of the single domain wall solutions has zero potential but nonzero superpotential.   Upon uplifting to 11 dimensions each domain wall becomes a purely gravitational 1/2 BPS solution. The corresponding metric has a 7+4 split with a Minkowski 7-metric and a 4-metric that corresponds to a gravitational instanton. These instantons generalize the SO(3) metric of Belinsky, Gibbons, Page and Pope (which includes the Eguchi-Hanson metric) to the other Bianchi types of class A. 
  It would be extremely useful to know whether a particular low energy effective theory might have come from a compactification of a higher dimensional space. Here, this problem is approached from the ground up by considering theories with multiple interacting massive gravitons. It is actually very difficult to construct discrete gravitational dimensions which have a local continuum limit. In fact, any model with only nearest neighbor interactions is doomed. If we could find a non-linear extension for the Fierz-Pauli Lagrangian for a graviton of mass mg which does not break down until the scale Lambda_2=(mg Mpl)^(1/2), this could be used to construct a large class of models whose continuum limit is local in the extra dimension. But this is shown to be impossible: a theory with a single graviton must break down by Lambda_3 = (mg^2 Mpl)^(1/3). Next, we look at how the discretization prescribed by the truncation of the KK tower of an honest extra diemsinon rasies the scale of strong coupling. It dictates an intricate set of interactions among various fields which conspire to soften the strongest scattering amplitudes and allow for a local continuum limit. A number of canditate symmetries associated with locality in the discretized dimension are also discussed. 
  We investigate supersymmetric QCD with N_c+1 flavors using an extension of the recently proposed relation between gauge theories and matrix models. The impressive agreement between the two sides provides a beautiful confirmation of the extension of the gauge theory-matrix model relation to this case. 
  The model of Dvali, Gabadadze, and Porrati (DGP) gives a simple geometrical setup in which gravity becomes 5-dimensional at distances larger than a length scale \lambda_{DGP}. We show that this theory has strong interactions at a length scale \lambda_3 ~ (\lambda_{DGP}^2 / M_P)^{1/3}. If \lambda_{DGP} is of order the Hubble length, then the theory loses predictivity at distances shorter than \lambda_3 ~ 1000 km. The strong interaction can be viewed as arising from a longitudinal `eaten Goldstone' mode that gets a small kinetic term only from mixing with transverse graviton polarizations, analogous to the case of massive gravity. We also present a negative-energy classical solution, which can be avoided by cutting off the theory at the same scale scale \lambda_3. Finally, we examine the dynamics of the longitudinal Goldstone mode when the background geometry is curved. 
  We consider deSitter universe and Nariai universe induced by quantum CFT with classical phantom matter and perfect fluid. The model represents the combination of trace-anomaly driven inflation and phantom driven deSitter universe. The similarity of phantom matter with quantum CFT indicates that phantom scalar may be the effective description for some quantum field theory. It is demonstrated that it is easier to achieve the acceleration of the scale factor preserving the energy conditions in such unified model. Some properties of unified theory (anti-gravitating solutions, negative ADM mass Nariai solution, relation with steady state) are briefly mentioned. 
  Thirty years ago, Coleman proved that no continuous symmetry is broken spontaneously in a two-dimensional relativistic quantum field theory. In his argument, however, it is difficult to understand the physical meaning of the assumption of no infrared divergence. I derive the same result directly from the cluster property of a local field regarded as a physically acceptable assumption. 
  We introduce a matrix model for noncommutative gravity, based on the gauge group $U(2) \otimes U(2)$. The vierbein is encoded in a matrix $Y_{\mu}$, having values in the coset space $U(4)/ (U(2) \otimes U(2))$, while the spin connection is encoded in a matrix $X_\mu$, having values in $U(2) \otimes U(2)$. We show how to recover the Einstein equations from the $\theta \to 0$ limit of the matrix model equations of motion. We stress the necessity of a metric tensor, which is a covariant representation of the gauge group in order to set up a consistent second order formalism. We finally define noncommutative gravitational instantons as generated by $U(2) \otimes U(2)$ valued quasi-unitary operators acting on the background of the Matrix model. Some of these solutions have naturally self-dual or anti-self-dual spin connections. 
  We investigate quantum corrections in non-commutative gauge theory on fuzzy sphere. We study translation invariant models which classically favor a single fuzzy sphere with U(1) gauge group. We evaluate the effective actions up to the two loop level. We find non-vanishing quantum corrections at each order even in supersymmetric models. In particular the two loop contribution favors U(n) gauge group over U(1) contrary to the tree action in a deformed IIB matrix model with a Myers term. We further observe close correspondences to 2 dimensional quantum gravity. 
  We determine the partition functions of ${\cal N}=4$ super Yang-Mills gauge theory for some $ADE$ gauge groups on $K3$, under the assumption that they are holomorphic. Our partition functions satisfy the gap condition and Montonen-Olive duality at the same time, like the SU(N) partition functions of Vafa and Witten. As a result we find a close relation between Hecke operator and $S$-duality of ${\cal N}=4$ super Yang-Mills for $ADE$ gauge group on $K3$. 
  We investigate the fermionic sector of a given theory, in which massive and charged Dirac fermions interact with an Abelian gauge field, including a non standard contribution that violates both Lorentz and CPT symmetries. We offer an explicit calculation in which the radiative corrections due to the fermions seem to generate a Chern-Simons-like effective action. Our results are obtained under the general guidance of dimensional regularization, and they show that there is no room for Lorentz and CPT violation in both commutative and noncommutative spacetime. 
  By using the relativistic top theory, we derive a relativistic top deviation equation. This equation turns out to be a generalization of the geodesic deviation equation for a pair of nearby point particles. In fact, we show that when the spin angular momentum tensor associated to the top vanishes, such a relativistic top deviation equation reduces to the geodesic deviation equation for spinless point particles. Just as the geodesic deviation equation for spinless particles can be used to investigate the detection of gravitational waves, our generalized formula for a relativistic top can be used to study the gravitational wave background. Our formulation may be of special interest to detect the inflationary gravitational waves via the polarization of the cosmic background radiation. 
  We discuss tree level three and four point scattering amplitudes in type II string models with matter fields localized at the intersections of D-brane wrapping cycles. Using conformal field theory techniques we calculate the four fermion amplitudes. These give "contact" interactions that can lead to flavour changing effects. We show how in the field theory limit the amplitudes can be interpreted as the exchange of Kaluza-Klein excitations, string oscillator states and stretched heavy string modes. 
  Brane-induced gravity in five dimensions (Dvali--Gabadadze--Porrati model) exhibits modification of gravity at ultra-large distances, $r\gg r_c = M_{Pl}^2/M^3$ where $M$ is the five-dimensional gravity scale. This makes the model potentially interesting for explaining the observed acceleration of the Universe. We argue, however, that it has an intrinsic intermediate energy scale $(M^9/M_{Pl}^4)^{1/5}$. At higher energies, the model is strongly coupled. For $r_c$ of order of the present Hubble size, the strong coupling regime occurs at distanced below tens of metres. 
  We construct a parafermionic conformal theory with the symmetry Z_N, for N odd, based on the second solution of Fateev-Zamolodchikov for the corresponding parafermionic chiral algebra. Primary operators are classified according to their transformation properties under the dihedral group D_N, as singlet, doublet 1,2,...,(N-1)/2, and disorder operators. In an assumed Coulomb gas scenario, the corresponding vertex operators are accommodated by the weight lattice of the Lie algebra B_(N-1)/2. The unitary theories are representations of the coset SO_n(N) x SO_2(N) / SO_{n+2}(N), with n=1,2,... . Physically, they realise the series of multicritical points in statistical theories having a D_N symmetry. 
  We consider Minkowski compactifications of M-theory on generic seven-dimensional manifolds. After analyzing the conditions on the four-form flux, we establish a set of relations between the components of the intrinsic torsion of the internal manifold and the components of the four-form flux needed for preserving supersymmetry. The existence of two nowhere vanishing vectors on any seven-manifold with G_2 structure plays a crucial role in our analysis, leading to the possibility of four-dimensional compactifications with N=1 and N=2 supersymmetry. 
  In this paper, starting from the common foundation of Connes' noncommutative geometry (NCG) [1,2,3,4], various possible alternatives in the formulation of a theory of gravity in noncommutative spacetime are discussed in detail. The diversity in the final physical content of the theory is shown to the the consequence of the arbitrariness in each construction steps. As an alternative in the last step, when the staructure equations are to be solved, a minimal set of constraints on the torsion and connection is found to determine all the geometric notions in terms of metric. In the Connes-Lott model of noncommutative spacetime, in order to keep the full spectrum of the discretized Kaluza-Klein theory [5], it is necessary to include the torsion in the generalized Einstein-Hilbert-Cartan action. 
  The study of renormalization of Yang-Mills fields in the light-front gauge has always been a delicate subject in that divergent {\em non-local} terms arise from the calculations of Feynman diagrams. In this short paper we show that this happened because of a deficiency in the gauge fixing procedure that results in an incorrect propagator and propose a cure for it by considering the {\em correct} propagator for the gauge potential. We explicitly show that the use of our correct propagator in the light-front leads to a vacuum polarization tensor at the one-loop level that is free of non-local terms. 
  Liouville, SL(2,R)/U(1) and SL(2,R)/R_+ coset structures are completely described by gauge invariant Hamiltonian reduction of the SL(2,R) WZNW theory. 
  We consider a dynamical brane world in a six-dimensional spacetime containing a singularity. Using the Israel conditions we study the motion of a 4-brane embedded in this setup. We analyse the brane behaviour when its position is perturbed about a fixed point and solve the full nonlinear dynamics in the several possible scenarios. We also investigate the possible gravitational shortcuts and calculate the delay between graviton and photon signals and the ratio of the corresponding subtended horizons. 
  Canonical quantisation of the free-field zero modes q, p on a half-line p>0 provides for WZNW coset theories self-adjoint vertex operatorson account of hidden symmetries generated by an S-matrix. 
  We investigate real-time tachyon dynamics of unstable D-brane carrying fundamental string charge. We construct the boundary state relevant for rolling of modulated tachyon with gauge fields excited on the world-volume, and study spatial distribution of the fundamental string charge and current as the D-brane decays. We find that, in contrast to homogeneous tachyon rolling, spatial modulation of the tachyon field triggers density wave of strings when electric field is turned on, and of string anti-string pairs when magnetic field is turned on. We show that the energy density and the fundamental string charge density are locked together, and evolve into a localized delta-function array (instead of evolving into a string fluid) until a critical time set by initial condition of rolling tachyon. When the gauge fields approach the critical limit, the fundamental strings produced become BPS-like. We also study the dynamics via effective field theory, and find agreement. 
  Using the nonlinear realizations of the Virasoro group we construct the action of the Conformal Quantum Mechanics (CQM) with additional harmonic potential. We show that $SL(2,R)$ invariance group of this action is nontrivially embedded in the reparametrization group of the time which is isomorphic to the centerless Virasoro group. We generalize the consideration to the Ermakov systems and construct the action for the time dependent oscillator. Its symmetry group is also the $SL(2,R) \sim SU(1,1)$ group embedded in the Virasoro group in a more complicated way. 
  We study parity symmetries and crosscap states in classes of N=2 supersymmetric quantum field theories in 1+1 dimensions, including non-linear sigma models, gauged WZW models, Landau-Ginzburg models, and linear sigma models. The parity anomaly and its cancellation play important roles in many of them. The case of the N=2 minimal model are studied in complete detail, from all three realizations -- gauged WZW model, abstract RCFT, and LG models. We also identify mirror pairs of orientifolds, extending the correspondence between symplectic geometry and algebraic geometry by including unorientable worldsheets. Through the analysis in various models and comparison in the overlapping regimes, we obtain a global picture of orientifolds and D-branes. 
  We consider topology changing processes in SU(2)--Higgs theory. In the Standard Model of particle physics they are accompanied by baryon--and lepton--number non--conservation. At fixed energy and multiplicity of initial state, these processes are described by classical Theta--instanton solutions. We describe these solutions and calculate the suppression exponents for the probabilities of the topology changing transitions at relatively low energies. 
  The comparison of symmetries in the interior and the exterior of a domain wall is relevant when discussing the correspondence between domain walls and branes, and also when studying the interaction of walls and magnetic monopoles. I discuss the symmetries in the context of an SU(N) times Z_2 model (for odd N) with a single adjoint scalar field. Situations in which the wall interior has less symmetry than the vacuum are easy to construct while the reverse situation requires significant engineering of the scalar potential. 
  The paper is withdrawn by the author. Parts of the contents are expanded into separate papers; hep-th/0308015 LOCALIZED TACHYON MASS AND A G-THEOREM ANALOGUE, hep-th/0308028 COMMENTS ON THE FATE OF UNSTABLE ORBIFOLDS, hep-th/0308029 CHIRAL RINGS AND GSO PROJECTION IN ORBIFOLDS. 
  We compute the emission of closed string radiation from homogeneous rolling tachyons. For an unstable decaying D$p$-brane the radiated energy is infinite to leading order for $p\leq 2$ and finite for $p>2$. The closed string state produced by a decaying brane is closely related to the state produced by D-instantons at a critical Euclidean distance from $t=0$. In the case of a D0 brane one can cutoff this divergence so that we get a finite energy final state which would be the state that the brane decays into. 
  In M-theory vacua with vanishing 4-form F, one can invoke the ordinary Riemannian holonomy H \subset SO(1,10) to account for unbroken supersymmetries n=1, 2, 3, 4, 6, 8, 16, 32. However, the generalized holonomy conjecture, valid for non-zero F, can account for more exotic fractions of supersymmetry, in particular 16<n<32. The conjectured holonomies are given by H \subset G where G are the generalized structure groups G=SO(d-1,1) x G(spacelike), G=ISO(d-1) x G(null) and G=SO(d) x G(timelike) with 1<=d<11. For example, G(spacelike)=SO(16), G(null)=[SU(8) x U(1)] \ltimes R^{56} and G(timelike)=SO*(16) when d=3. Although extending spacetime symmetries, there is no conflict with the Coleman-Mandula theorem. The holonomy conjecture rules out certain vacua which are otherwise permitted by the supersymmetry algebra. 
  We calculate circular Wilson loop expectation value of pure ${\cal N}=1$ super Yang-Mills from the Klebanov-Strassker-Tseytlin solution of supergravity and the proposed gauge/gravity duality. The calculation is performed numerically via searching world-sheet minimal surface. It is shown that Wilson loop exhibits area law for large radius which implies that ${\cal N}=1$ super Yang-Mills is confined at large distance or low energy scale. Meanwhile, Wilson loop exhibits logarithmic behavior for small radius and it indicates asymptotical freedom of ${\cal N}=1$ super Yang-Mills at short distance or high energy scale. 
  We analyze the one-dimensional Dirac oscillator in a thermal bath. We found that the heat capacity is two times greater than the heat capacity of the one-dimensional harmonic oscillator for higher temperatures. 
  We have developed a matrix model for FQH states at filling factor \nu_{k_1k_2} going beyond the Laughlin theory. To illustrate our idea, we have considered an FQH system of a finite number N=(N_{1}+N_{2}) of electrons with filling factor \nu_{k_{1}k_{2}} = \nu_{p_{1}p_{2}}=\frac{p_{2}}{p_{1}p_{2}-1}; p_{1} is an odd integer and p_{2} is an even integer. The \nu_{p_{1}p_{2}} series corresponds just to the level two of the Haldane hierarchy; it recovers the Laughlin series \nu_{p_{1}} =\frac{1}{p_{1}} by going to the limit p_{2} large and contains several observable FQH states such as \nu = 2/3, 2/5, >.... 
  In this note, we give a method to derive the Seiberg duality by the matrix model. The key fact we used is that the effective actions given by matrix model method should be identical for both electric and magnetic theories. We demonstrate our method for SQCD with U(N), SO(N) and Sp(N) gauge groups. 
  The field equations of the auxiliary fields are nonlinear and free of derivatives. Hence, it is argued, a Legendre transform to generate the 1PI Generating Functionals is not correct for the auxiliary fields. A corrected formulation of the BRS symmetry must be constructed by integrating the auxiliary fields. This necessarily destroys manifest supersymmetry. It also generates a new "Physical BRS-ZJ" cohomology problem. The resulting "Physical BRS-ZJ cohomology" is then described. The cohomology contains a rich spectrum of potential supersymmetry anomalies in certain composite operators that have spinor indices. Some examples of these composite operators are set out explicitly. The examples depend on certain Constraint Equations that arise from simple Generating Functions. The Constraint Equations constrain the mass and Yukawa coefficients of the theory. The potential supersymmetry anomalies have calculable one-loop coefficients that depend on the solutions to the Constraint Equations. If these coefficients are non-zero, the anomalies will then necessarily break supersymmetry with a calculable pattern, probably with a zero cosmological constant. 
  This work presents a classification of all smooth 't Hooft-Jackiw-Nohl-Rebbi instantons over Gibbons-Hawking spaces. That is, we find all smooth SU(2) Yang-Mills instantons over these spaces which arise by conformal rescalings of the metric with suitable functions.    Since the Gibbons-Hawking spaces are hyper-Kahler gravitational instantons, the rescaling functions must be positive harmonic. By using twistor methods we present integral formulae for the kernel of the Laplacian associated to these spaces. These integrals are generalizations of the classical Whittaker-Watson formula. By the aid of these we prove that all 't Hooft instantons have already been found in a recent paper.    This result also shows that actually all such smooth 't Hooft-Jackiw-Nohl-Rebbi instantons describe singular magnetic monopoles over the flat three-space with zero magnetic charge moreover the reducible ones generate the the full L^2 cohomology of the Gibbons-Hawking spaces. 
  Regularization of quantum field theories introduces a mass scale which breaks axial rotational and scaling invariances. We demonstrate from first principles that axial torsion and torsion trace modes have non-transverse vacuum polarization tensors, and become massive as a result. The underlying reasons are similar to those responsible for the Adler-Bell-Jackiw (ABJ) and scaling anomalies. Since these are the only torsion components that can couple minimally to spin 1/2 particles, the anomalous generation of masses for these modes, naturally of the order of the regulator scale, may help to explain why torsion and its associated effects, including CPT violation in chiral gravity, have so far escaped detection. As a simpler manifestation of the reasons underpinning the ABJ anomaly than triangle diagrams, the vacuum polarization demonstration is also pedagogically useful. In addition it is shown that the teleparallel limit of a Weyl fermion theory coupled only to the left-handed spin connection leads to a counter term which is the Samuel-Jacobson-Smolin action of chiral gravity in four dimensions. 
  Though a Chern-Simons (2k-1)-form is not gauge-invariant and it depends on a background connection, this form seen as a Lagrangian of gauge theory on a (2k-1)-dimensional manifold leads to the energy-momentum conservation law. 
  We review both the construction of conformal blocks in quantum Liouville theory and the quantization of Teichm\"uller spaces as developed by Kashaev, Checkov and Fock. In both cases one assigns to a Riemann surface a Hilbert space acted on by a representation of the mapping class group. According to a conjecture of H. Verlinde, the two are equivalent. We describe some key steps in the verification of this conjecture. 
  We reconsider the construction of exponential fields in the quantized Liouville theory. It is based on a free-field construction of a continuous family or chiral vertex operators. We derive the fusion and braid relations of the chiral vertex operators. This allows us to simplify the verification of locality and crossing symmetry of the exponential fields considerably. The calculation of the matrix elements of the exponential fields leads to a constructive derivation of the formula proposed by Dorn/Otto and the brothers Zamolodchikov. 
  We present a novel derivation of the duality between the two-dimensional Euclidean black hole and supersymmetric Liouville theory. We realise these (1+1)-dimensional conformal field theories on the worldvolume of domain walls in a (2+1)-dimensional gauge theory. We show that there exist two complementary descriptions of the domain wall dynamics, resulting in the two mirror conformal field theories. In particular, effects which are usually attributed to worldsheet instantons are captured by the classical scattering of domain walls. 
  It is shown that the four-particle amplitude of superstring theory at two loops obtained in [1,2] is equivalent to the previously obtained results in [3,4,5]. Here the ${\bf Z}_2$ symmetry in hyperelliptic Riemann surface plays an important role in the proof. 
  We study the low-lying excitations of Type IIA superstring theory in a plane wave background with 24 supersymmetries. In the light-cone gauge, the superstring action has ${\cal N}=(4,4)$ supersymmetry and is exactly solvable, since it is quadratic in superstring coordinates. We obtain explicitly the spectrum of the Type IIA supergravity fluctuation modes in the plane wave background and give its correspondence with the spectrum of string states from the zero-mode sector of the light-cone superstring Hamiltonian. 
  We present a class of scalar field cosmologies with a dynamically evolving Newton parameter $G$ and cosmological term $\Lambda$. In particular, we discuss a class of solutions which are consistent with a renormalization group scaling for $G$ and $\Lambda$ near a fixed point. Moreover, we propose a modified action for gravity which includes the effective running of $G$ and $\Lambda$ near the fixed point. A proper understanding of the associated variational problem is obtained upon considering the four-dimensional gradient of the Newton parameter. 
  We discuss supersymmetry breaking induced by simultaneous presence of a Wilson-line type superpotential and boundary-localized Fayet-Iliopoulos terms in a four dimensional theory based on deconstruction of five-dimensional abelian gauge theories on orbifolds. Large hierarchy between the scale of supersymmetry breaking and the fundamental scale can be generated dynamically. The model has several potentially interesting phenomenological applications. We also discuss the conditions that are necessary for interpreting our $U(1)^N$ model as an ultra-violet completion of some 5d theory. In particular, the corresponding 5d theory contains Chern-Simons couplings. 
  In this paper we construct the supergravity solutions for the orthogonally intersecting null scissors and the fluxed D-strings. We name the latter as the super-crosses according to their shape. It turns out that the smeared solutions are U-dual related to the intersecting $(p,q)$-strings. Their open string properties are also studied. As a by-product, we clarify the supersymmetry conditions of D2-D2 pairs with most generic fluxes. 
  We give an explicit formulation of the time dependent AdS/CFT correspondence when there are multiple vacua present in Lorentzian signature. By computing sample two point functions we show how different amplitudes are related by cosmological particle production. We illustrate our methods in two example spacetimes: (a) a ``bubble of nothing'' in AdS space, and (b) an asymptotically locally AdS spacetime with a bubble of nothing on the boundary. In both cases the alpha vacua of de Sitter space make an interesting appearance. 
  In many aspects the most complicated noncommutative spaces correspond to foliated manifolds with nonvanishing Godbillon-Vey class. We argue that gauge invariance probably prevents a foliated manifold from creating resilient leaves and thus resulting in having nonvanishing GV-class. So these spaces which are of the highest noncommutativity are not likely to appear in gauge theories. 
  We explore the effects of a heavy fermion doublet in a simplified version of the standard electroweak theory. We integrate out the doublet and compute the exact effective energy functional of spatially varying gauge and Higgs fields. We perform a variational search for a local minimum of the effective energy and do not find evidence for a soliton carrying the quantum numbers of the decoupled fermion doublet. The fermion vacuum polarization energy offsets the gain in binding energy previously argued to be sufficient to stabilize a fermionic soliton. The existence of such a soliton would have been a natural way to maintain anomaly cancellation at the level of the states. We also see that the sphaleron energy is significantly increased due to the quantum corrections of the heavy doublet. We find that when the doublet is slightly heavier than the quantum--corrected sphaleron, its decay is exponentially suppressed owing to a new barrier. This barrier exists only for an intermediate range of fermion masses, and a heavy enough doublet is indeed unstable. 
  We review in a pedagogical manner some of the efforts aiming to extend the gauge/gravity correspondence to non-conformal supersymmetric gauge theories in four dimensions. After giving a general overview, we discuss in detail two specific examples: fractional D-branes on orbifolds and D-branes wrapped on supersymmetric cycles of Calabi-Yau spaces. We explore in particular which gauge theory information can be extracted from the corresponding supergravity solutions, and what the remaining open problems are. We also briefly explain the connection between these and other approaches, such as fractional branes on conifolds, branes suspended between branes, M5-branes on Riemann surfaces and M-theory on G2-holonomy manifolds, and discuss the role played by geometric transitions in all that. 
  We compute the correlation functions mixing the powers of two non-commuting random matrices within the same trace. The angular part of the integration was partially known in the literature: we pursue the calculation and carry out the eigenvalue integration reducing the problem to the construction of the associated biorthogonal polynomials. The generating function of these correlations becomes then a determinant involving the recursion coefficients of the biorthogonal polynomials. 
  When calculating higher terms of the epsilon-expansion of massive Feynman diagrams, one needs to evaluate particular cases of multiple inverse binomial sums. These sums are related to the derivatives of certain hypergeometric functions with respect to their parameters. Exploring this connection and using it together with an approach based on generating functions, we analytically calculate a number of such infinite sums, for an arbitrary value of the argument which corresponds to an arbitrary value of the off-shell external momentum. In such a way, we find a number of new results for physically important Feynman diagrams. Considered examples include two-loop two- and three-point diagrams, as well as three-loop vacuum diagrams with two different masses. The results are presented in terms of generalized polylogarithmic functions. As a physical example, higher-order terms of the epsilon-expansion of the polarization function of the neutral gauge bosons are constructed. 
  We present a method for evaluating divergent non-Borel-summable series by an analytic continuation of variational perturbation theory. We demonstrate the power of the method by an application to the exactly known partition function of the anharmonic oscillator in zero spacetime dimensions. In one spacetime dimension we derive the imaginary part of the ground state energy of the anharmonic oscillator for {\em all negative values of the coupling constant $g$, including the nonanalytic tunneling regime at small-$g$. As a highlight of the theory we retrieve from the divergent perturbation expansion the action of the critical bubble and the contribution of the higher loop fluctuations around the bubble. 
  The obstruction for a perturbative reconstruction of the five-dimensional bulk metric starting from the four-dimensional metric at the boundary,that is, the Dirichlet problem, is computed in dimensions $6\leq d\leq 10$ and some comments are made on its general structure and, in particular, on its relationship with the conformal anomaly, which we compute in dimension $d=8$. 
  Topological euclidean gravity is built in eight dimensions for manifolds with $Spin(7) \subset SO(8)$ holonomy. In a previous work, we considered the construction of an eight-dimensional topological theory describing the graviton and one graviphoton. Here we solve the question of determining a topological model for the combined system of a metric and a Kalb--Ramond two-form gauge field. We then recover the complete $N=1, D=8$ supergravity theory in a twisted form. We observe that the generalized self-duality conditions of our model correspond to the octonionic string equations. 
  A number of physical systems exhibit a particular form of asymptotic conformal invariance: within a particular range of distances, they are characterized by a long-range conformal interaction (inverse square potential), the absence of dimensional scales, and an SO(2,1) symmetry algebra. Examples from molecular physics to black holes are provided and discussed within a unified treatment. When such systems are physically realized in the appropriate strong-coupling regime,the occurrence of quantum symmetry breaking is possible. This anomaly is revealed by the failure of the symmetry generators to close the algebra in a manner shown to be independent of the renormalization procedure. 
  We consider cosmological solutions in 5d locally supersymmetric theories including boundary actions, with either opposite tension branes for identical brane chiralities or equal tension branes for flipped brane chiralities. We analyse the occurrence of supersymmetry breakdown in both situations. We find that supersymmetry as seen by a brane observer is broken due to the motion of the brane in the bulk. When the brane energy-momentum tensor is dominated by the brane tension, the 4d vacuum energy cosmological constant on the observable brane is positive and proportional to the inverse square of the brane local time. We find that the mass splitting within supersymmetric multiplets living on the brane is of the order of the inverse of the brane local time, examplifying the tight relation between the vacuum energy scale and the supersymmetry breaking scale. 
  We study the noncommutative extensions of certain integrable field theories, namely the sine- and sinh-Gordon (sG and shG) models, and the U(N) principal chiral model (pcm). We argue that the Moyal deformations of the sG and shG models are not integrable, by looking at tree-level amplitudes where there is particle production. By considering the noncommutative generalization of the zero-curvature method, it is possible to define integrable versions of the noncommutative sG and shG models, which introduce extra constraints. The noncommutative pcm is shown to be integrable and we discuss the existence of non-trivial non-local conserved charges, and the associated noncommutative zero-curvature condition. 
  The parquet approximation in the matrix Higgs model is considered. We demonstrate analytically that in the large $N$ limit the parquet approximation gives an satisfying agreement with the exact results. It is shown that the parquet planar series can be derived by means of the generating functional. 
  We study the bound states of brane/antibrane systems by examining the motion of a probe antibrane moving in the background fields of N source branes. The classical system resembles the point-particle central force problem, and the orbits can be solved by quadrature. Generically the antibrane has orbits which are not closed on themselves. An important special case occurs for some Dp-branes moving in three transverse dimensions, in which case the orbits may be obtained in closed form, giving the standard conic sections but with a nonstandard time evolution along the orbit. Somewhat surprisingly, in this case the resulting elliptical orbits are exact solutions, and do not simply apply in the limit of asymptotically-large separation or non-relativistic velocities. The orbits eventually decay through the radiation of massless modes into the bulk and onto the branes, and we estimate this decay time. Applications of these orbits to cosmology are discussed in a companion paper. 
  In certain Lorentz-covariant higher-derivative field theories of spins < or =1, would-be ultraviolet divergences generate color-singlet poles as infrared divergences. Absence of higher-order poles implies ten-dimensional supersymmetric Yang-Mills with bound-state supergravity, in close analogy with open string theory. 
  Using S(pacelike)-branes defined through rolling tachyon solutions, we show how the dynamical formation of D(irichlet)-branes and strings in tachyon condensation can be understood. Specifically we present solutions of S-brane actions illustrating the classical confinement of electric and magnetic flux into fundamental strings and D-branes. The role of S-branes in string theory is further clarified and their RR charges are discussed. In addition, by examining ``boosted'' S-branes, we find what appears to be a surprising dual S-brane description of strings and D-branes, which also indicates that the critical electric field can be considered as a self-dual point in string theory. We also introduce new tachyonic S-branes as Euclidean counterparts to non-BPS branes. 
  We propose a Matrix Theory approach to Romans' massive Type IIA supergravity. It is obtained by applying the procedure of Matrix Theory compactifications to Hull's proposal of the Massive Type IIA String Theory as M-Theory on a twisted torus. The resulting Matrix Theory is a super-Yang Mills theory on large N three-branes with a space dependent non-commutativity parameter, which is also independently derived by a T-duality approach. We give evidence showing that the energies of a class of physical excitations of the super-Yang Mills theory show the correct symmetry expected from Massive Type IIA string theory in a lightcone quantization. 
  The search for regular black hole solutions in classical gravity leads us to consider a core of Euclidean signature in the interior of a black hole. Solutions of Lorentzian and Euclidean general relativity match in such a way that energy densities and pressures of an isotropic perfect fluid form are everywhere finite and continuous. Although the weak energy condition cannot be satisfied for these solutions in general relativity, it can be when higher derivative terms are added. A numerical study shows how the transition becomes smoother in theories with more derivatives. As an alternative to the Euclidean core, we also discuss a closely related time dependent orbifold construction with a smooth space-like boundary inside the horizon. 
  The Dirac equation is solved for a pseudoscalar Coulomb potential in a two-dimensional world. An infinite sequence of bounded solutions are obtained. These results are in sharp contrast with those ones obtained in 3+1 dimensions where no bound-state solutions are found. Next the general two-dimensional problem for pseudoscalar power-law potentials is addressed consenting us to conclude that a nonsingular potential leads to bounded solutions. The behaviour of the upper and lower components of the Dirac spinor for a confining linear potential nonconserving- as well as conserving-parity, even if the potential is unbounded from below, is discussed in some detail. 
  We analyse some quantum multiplets associated with extended supersymmetries. We study in detail the general form of the causal (anti)commutation relations. The condition of positivity of the scalar product imposes severe restrictions on the (quantum) model. It is problematic if one can find out quantum extensions of the standard model with extended supersymmetries. 
  Using functional derivatives with respect to the free correlation function we derive a closed set of Schwinger-Dyson equations in phi^4-theory. Its conversion to graphical recursion relations allows us to systematically generate all connected and one-particle irreducible Feynman diagrams for the two- and four-point function together with their weights. 
  We will review the algebras which have been conjectured as symmetries in M-theory. The Borcherds algebras, which are the most general Lie algebras under control, seem natural candidates. 
  The present paper contains two interrelated developments. First, are proposed new generalized Verma modules. They are called k-Verma modules, k\in N, and coincide with the usual Verma modules for k=1. As a vector space a k-Verma module is isomorphic to the symmetric tensor product of k copies of the universal enveloping algebra U(g^-), where g^- is the subalgebra of lowering generators in the standard triangular decomposition of a simple Lie algebra g = g^+ \oplus h \oplus g^- . The second development is the proposal of a procedure for the construction of multilinear intertwining differential operators for semisimple Lie groups G . This procedure uses k-Verma modules and coincides for k=1 with a procedure for the construction of linear intertwining differential operators. For all k central role is played by the singular vectors of the k-Verma modules. Explicit formulae for series of such singular vectors are given. Using these are given explicitly many new examples of multilinear intertwining differential operators. In particular, for G = SL(2,R) are given explicitly all bilinear intertwining differential operators. Using the latter, as an application are constructed (n/2)-differentials for all n\in 2N, the ordinary Schwarzian being the case n=4. As another application, in a Note Added we propose a new hierarchy of nonlinear equations, the lowest member being the KdV equation. 
  Gauge invariant regularization of quantum field theory in the framework of Light-Front (LF) Hamiltonian formalism via introducing a lattice in transverse coordinates and imposing boundary conditions in LF coordinate $x^-$ for gauge fields on the interval $|x^-|\le L$ is considered. The remaining ultraviolet divergences in the longitudinal momentum $p_-$ are removed by gauge invariant finite mode regularization. We find that LF canonical formalism for the introduced regularization does not contain usual most complicated second class constraints connecting zero and nonzero modes of gauge fields. The described scheme can be used either for the regularization of conventional gauge theory or for gauge invariant formulation of effective low-energy models on the LF. The lack of explicit Lorentz invariance in our approach leads to difficulty with defining the vacuum state. We discuss this difficulty, particulary, in the connection with the problem of taking the limit of continuous space. 
  We try to determine the partition function of ${\cal N}=4$ super Yang-Mills theoy for ADE gauge group on K3 by self-dualizing our previous ADE partition function. The resulting partition function satisfies gap condition. 
  We find classical solutions of D-branes in pp-wave spacetime with nonconstant NS-NS flux. We also present Dp-Dp' bound state solutions in this background. We further analyze the supersymmetric properties of these brane solutions by solving the type IIB killing spinor equations explicitly. 
  In this article we continue with the microscopical investigation of giant graviton configurations in AdS_m x S^n spacetimes, initiated in hep-th/0207199. Using dualities and a Matrix theory derivation we propose an action that describes multiple Type IIB gravitons. This action contains multipole moment couplings to the Type IIB background potentials. Using these couplings, we study, from the microscopical point of view, the giant graviton and dual giant graviton configurations in the AdS_5 x S^5 background. In both cases the gravitons expand into a non-commutative 3-sphere, that is defined as an S^1-bundle over a fuzzy 2-sphere. When the number of gravitons is large we find perfect agreement with the Abelian, macroscopical description of giant gravitons in this spacetime, given in the literature. 
  We study the finite-temperature properties of the supersymmetric version of (2+1)D Georgi-Glashow model. As opposed to its nonsupersymmetric counterpart, the parity symmetry in this theory at zero temperature is spontaneously broken by the bilinear photino condensate. We find that as the temperature is raised, the deconfinement and the parity restoration occur in this model at the same point $T_c=g^2/8\pi$. The transition is continuous, but is not of the Ising type as in nonsupersymmetric Georgi-Glashow model, but rather of the Berezinsky-Kosterlitz-Thouless type as in $Z_4$-invariant spin model. 
  An assessment is offered of the progress that the major approaches to quantum gravity have made towards the goal of constructing a complete and satisfactory theory. The emphasis is on loop quantum gravity and string theory, although other approaches are discussed, including dynamical triangulation models (euclidean and lorentzian) regge calculus models, causal sets, twistor theory, non-commutative geometry and models based on analogies to condensed matter systems. We proceed by listing the questions the theories are expected to be able to answer. We then compile two lists: the first details the actual results so far achieved in each theory, while the second lists conjectures which remain open. By comparing them we can evaluate how far each theory has progressed, and what must still be done before each theory can be considered a satisfactory quantum theory of gravity. We find there has been impressive recent progress on several fronts. At the same time, important issues about loop quantum gravity are so far unresolved, as are key conjectures of string theory. However, there is a reasonable expectation that experimental tests of lorentz invariance at Planck scales may in the near future make it possible to rule out one or more candidate quantum theories of gravity. 
  We consider the one-loop renormalization of dimension four composite operators and the energy-momentum tensor in noncommutative \phi^4 scalar field theory. Proper operator bases are constructed and it is proved that the bare composite operators are expressed via renormalized ones, with the help of a mixing matrix, whose explicit form is calculated. The corresponding matrix elements turn out to differ from the commutative theory. The canonically defined energy-momentum tensor is not finite and must be replaced by the "improved" one, in order to provide finiteness. The suitable "improving" terms are found. 
  Inspired by factorized scattering from delta-type impurities in (1+1)-dimensional space-time, we propose and analyse a generalization of the Zamolodchikov-Faddeev algebra. Distinguished elements of the new algebra, called reflection and transmission generators, encode the particle-impurity interactions. We describe in detail the underlying algebraic structure. The relative Fock representations are explicitly constructed and a general factorized scattering theory is developed in this framework. 
  The dynamics and radiation of positronium is investigated in intense laser fields. 
  We consider a classical spinning particle in the frame of the relativistic physics by means of a covariant Hamiltonian and of a generalization of Poisson brackets which take into account the gauge fields. We obtain different equations of motion and evolution in this context and we compare our results with those of Bargmann-Michel-Telegdi. An extension to the case of a curved space and a link towards quantum theory are given at the end of the paper. 
  In the Matrix Quantum Mechanical formulation of 2D string theory it is possible to introduce arbitrary tachyonic perturbations. In the case when the tachyonic momenta form a lattice, the theory is known to be integrable and, therefore, it can be used to describe the corresponding string theory. We study the backgrounds of string theory obtained from these matrix model solutions. They are found to be flat but the perturbations can change the global structure of the target space. They can lead either to a compactification, or to the presence of boundaries depending on the choice of boundary conditions. Thus, we argue that the tachyon perturbations have a dual description in terms of the unperturbed theory in spacetime with a non-trivial global structure. 
  This article contains an overview of some recent attempts at understanding supergravity and string duals of four dimensional gauge theories using the AdS/CFT correspondence. We discuss the general philosophy underlying the various ways to realize Super Yang-Mills theories in terms of systems of branes. We then review some of the existing duals for N=2 and N=1 theories. We also discuss differences and similarities with realistic theories. 
  We elaborate on the duality-symmetric nonlinear electrodynamics in a new formulation with auxiliary tensor fields. The Maxwell field strength appears only in bilinear terms of the corresponding generic Lagrangian, while the self-interaction is presented by a function E depending on the auxiliary fields. Two types of dualities inherent in the nonlinear electrodynamics models admit a simple off-shell characterization in terms of this function. In the standard formulation, the continuous U(1) duality symmetry is nonlinearly realized on the Maxwell field strength. In the new setting, the same symmetry acts as linear U(1) transformations of the auxiliary field variables. The nonlinear U(1) duality condition proves to be equivalent to the linear U(1) invariance of the self-interaction E. The discrete self-duality (or self-duality by Legendre transformation) amounts to a weaker reflection symmetry of E. For a class of duality- symmetric Lagrangians we introduce an alternative representation with the auxiliary scalar field and find new explicit examples of such systems. 
  We study supersymmetric intersections of M2 and M5 branes with different pp-waves of M-theory. We consider first M-brane probes in the background of pp-waves and determine under which conditions the embedding is supersymmetric. We particularize our formalism to the case of pp-waves with 32, 24 and 20 supersymmetries. We also construct supergravity solutions for the brane-wave system. Generically these solutions are delocalised along some directions transverse to the brane and preserve the same number of supersymmetries as in the brane probe approach. 
  We discuss systematic approaches to the classification of string/M theory vacua, and physical questions this might help us resolve. To this end, we initiate the study of ensembles of effective Lagrangians, which can be used to precisely study the predictive power of string theory, and in simple examples can lead to universality results. Using these ideas, we outline an approach to estimating the number of vacua of string/M theory which can realize the Standard Model. 
  We show how the existence of non-trivial finite-energy time-dependent classical lumps is restricted by a generalized virial theorem. For simple model Lagrangians, bounds on energies follow. 
  The matrix model with mass term has a nontrivially classical solution which is known to represent a noncommutative fuzzy sphere. The fuzzy sphere has a lower energy then that of the trivial solution. In this letter we investigate the quantum correction of the energy of the fuzzy sphere by using the Gaussian variational technique, in contrast to the other studying in which only the small fluctuation was considered. Our result, which only considering the boson part, shows that the quantum correction does not change the stability of the fuzzy sphere. 
  We present new classes of supersymmetric Standard-like models from type IIA $\IT^6/(\IZ_2\times \IZ_2)$ orientifold with intersecting D6-branes. D6-branes can wrap general supersymmetric three-cycles of $\IT^6=\IT^2\times \IT^2\times \IT^2$, and any $\IT^2$ is allowed to be tilted. The models still suffer from additional exotics, however we obtained solutions with fewer Higgs doublets, as well as models with all three families of left-handed quarks and leptons arising from the same intersecting sector, and examples of a genuine left-right symmetric model with three copies of left-handed and right-handed families of quarks and leptons. 
  Using a reformulation of the method of (p,q) webs, we study the four-dimensional N=1 quiver theories from M-theory on seven-dimensional manifolds with G_2 holonomy. We first construct such manifolds as U(1) quotients of eight-dimensional toric hyper-K\"ahler manifolds, using N=4 supersymmetric sigma models. We show that these geometries, in general, are given by real cones on \bf S^2 bundles over complex two-dimensional toric varieties, \cal \bf V^2= {{\bf C}^{r+2}/ {{\bf C}^*}^r}. Then we discuss the connection between the physics content of M-theory on such G_2 manifolds and the method of (p,q) webs. Motivated by a result of Acharya and Witten [hep-th/0109152], we reformulate the method of $(p,q)$ webs and reconsider the derivation of the gauge theories using toric geometry Mori vectors of \cal \bf V^2 and brane charge constraints. For {\bf WP^2}_{w_1,w_2, w_3}, we find that the gauge group is given by G=U(w_1n)\times U(w_2n)\times U(w_3n). This is required by the anomaly cancellation condition. 
  In this note we will study solution of open bosonic string field theory based on action of operators from chiral algebra of boundary conformal field theory on identity element of string field theory star algebra. We will demonstrate that the string field theory action for fluctuation fields around this classical solution can be mapped to the string field theory action defined through the new boundary conformal field theory that arises from the original one through the marginal deformation inserted on the world-sheet boundary. 
  In N=2 SYM theory with matter non-conservation of the S-charge of a BPS state under monodromies leads to so-called BPS state "decay". A mechanism for such a behavior on the semiclassical level could be established through consideration of soliton-fermion bound state. Solutions to classical equations of motion allow to observe the BPS state "decay" in vivo. 
  We give a direct Lie algebraic characterisation of conformal inclusions of chiral current algebras associated with compact, reductive Lie algebras. We use straightforward quantum field theoretic arguments and prove a long standing conjecture of Schellekens and Warner on grounds of unitarity and positivity of energy. We explore the structures found to characterise ``conformal covariance subalgebras'' and ``coset current algebras''. 
  We compute the complete contribution to the stress-energy tensor in the minimal bosonic higher spin theory in D=4 that is quadratic in the scalar field. We find arbitrarily high derivative terms, and that the total sign of the stress-energy tensor depends on the parity of the scalar field. 
  We derive the low energy effective action for the dilatonic braneworld. In the case of the single-brane model, we find the effective theory is described by the Einstein-scalar theory coupled to the dark radiation. Remarkably, the dark radiation is not conserved in general due to a coupling to the bulk scalar field. The effective action incorporating Kaluza-Klein (KK) corrections is obtained and the role of the AdS/CFT correspondence in the dilatonic braneworld is revealed. In particular, it is shown that CFT matter would not be confined to the braneworld in the presence of the bulk scalar field. The relation between our analysis and the geometrical projection method is also clarified. In the case of the two-brane model, the effective theory reduces to a scalar-tensor theory with a non-trivial coupling between the radion and the bulk scalar field. 
  We provide a simple low energy description of recombination of intersecting D-branes using super Yang-Mills theory. The recombination is realized by condensation of an off-diagonal tachyonic fluctuation localized at the intersecting point. The recombination process is equivalent to brane-antibrane annihilation, thus our result confirms Sen's conjecture on tachyon condensation, although we work in the super Yang-Mills theory whose energy scale is much lower than alpha'. We also discuss the decay width of non-parallelly separated D-branes. 
  The alternating integrable spin chain and the $RSOS(q_{1},q_{2};p)$ model in the presence of a quantum impurity are investigated. The boundary free energy due to the impurity is derived, the ratios of the corresponding $g$ functions at low and high temperature are specified and their relevance to boundary flows in unitary minimal and generalized coset models is discussed. Finally, the alternating spin chain with diagonal and non--diagonal integrable boundaries is studied, and the corresponding boundary free energy and $g$ functions are derived. 
  A set of brackets for classical dissipative systems, subject to external random forces, are derived. The method is inspired to the old procedure found by Peierls, for deriving the canonical brackets of conservative systems, starting from an action principle. It is found that an adaptation of Peierls' method is applicable also to dissipative systems, when the friction term can be described by a linear functional of the coordinates, as is the case in the classical Langevin equation, with an arbitrary memory function. The general expression for the brackets satisfied by the coordinates, as well as by the external random forces, at different times, is determined, and it turns out that they all satisfy the Jacobi identity. Upon quantization, these classical brackets are found to coincide with the commutation rules for the quantum Langevin equation, that have been obtained in the past, by appealing to microscopic conservative quantum models for the friction mechanism. 
  We solve for the expectation values of chiral operators in supersymmetric U(N) gauge theories with matter in the adjoint, fundamental and anti-fundamental representations. A simple geometric picture emerges involving a description by a meromorphic one-form on a Riemann surface. The equations of motion are equivalent to a condition on the integrality of periods of this form. The solution indicates that all semiclassical phases with the same number of U(1) factors are continuously connected. 
  We address dynamical supersymmetry breaking within a N=1 supersymmetric Standard-like Model based on a Z_2 x Z_2 Type IIA orientifold with intersecting D6-branes. The model possesses an additional, confining gauge sector with the USp(2)_A x USp(2)_B x USp(4) gauge group, where the gaugino condensation mechanism allows for the breaking of supersymmetry and stabilizes moduli. We derive the leading contribution to the non-perturbative effective superpotential and determine numerically the minima of the supergravity potential. These minima break supersymmetry and fix two undetermined moduli, which in turn completely specify the gauge couplings at the string scale. For this specific construction the minima have a negative cosmological constant. We expect that for other supersymmetric Standard-like models with intersecting D6-branes, which also possess confining gauge sectors, the supersymmetry breaking mechanism would have qualitatively similar features. 
  We find the effective action for any D-brane in a general bosonic background of supergravity. The results are explicit in component fields up to second order in the fermions and are obtained in a covariant manner. No interaction terms between fermions and the field $f=b+F$, characteristic of the bosonic actions, are considered. These are reserved for future work. In order to obtain the actions, we reduce directly from the M2-brane world-volume action to the D2-brane world-volume action. Then, by means of T-duality, we obtain the other Dp-brane actions. The resulting Dp-brane actions can be written in a single compact and elegant expression. 
  We construct a large-N twisted reduced model of the four-dimensional super Yang-Mills theory coupled to one adjoint matter. We first consider a non-commutative version of the four-dimensional superspace, and then give the mapping rule between matrices and functions on this space explicitly. The supersymmetry is realized as a part of the internal $U(\infty)$ gauge symmetry in this reduced model. Our reduced model can be compared with the Dijkgraaf-Vafa theory that claims the low-energy glueball superpotential of the original gauge theory is governed by a simple one-matrix model. We show that their claim can be regarded as the large-N reduction in the sense that the one-matrix model they proposed can be identified with our reduced model. The map between matrices and functions enables us to make direct identities between the free energies and correlators of the gauge theory and the matrix model. As a by-product, we can give a natural explanation for the unconventional treatment of the one-matrix model in the Dijkgraaf-Vafa theory where eigenvalues lie around the top of the potential. 
  We obtain smooth M-theory solutions whose geometry is a warped product of AdS_5 and a compact internal space that can be viewed as an S^4 bundle over S^2. The bundle can be trivial or twisted, depending on the even or odd values of the two diagonal monopole charges. The solution preserves N=2 supersymmetry and is dual to an N=1 D=4 superconformal field theory, providing a concrete framework to study the AdS_5/CFT_4 correspondence in M-theory. We construct analogous embeddings of AdS_4, AdS_3 and AdS_2 in massive type IIA, type IIB and M-theory, respectively. The internal spaces have generalized holonomy and can be viewed as S^n bundles over S^2 for n=4, 5 and 7. Surprisingly, the dimensions of spaces with generalized holonomy includes D=9. We also obtain a large class of solutions of AdS\times H^2. 
  Departing from the observation that the Penrose limit of AdS_3 x S^3 is a group contraction in the sense of Inonu and Wigner, we explore the relation between the symmetric D-branes of AdS_3 x S^3 and those of its Penrose limit, a six-dimensional symmetric plane wave analogous to the four-dimensional Nappi--Witten spacetime. Both backgrounds are Lie groups admitting bi-invariant lorentzian metrics and symmetric D-branes wrap their (twisted) conjugacy classes. We determine the (twisted and untwisted) symmetric D-branes in the plane wave background and we prove the existence of a space-filling D5-brane and, separately, of a foliation by D3-branes with the geometry of the Nappi--Witten spacetime which can be understood as the Penrose limit of the AdS_2 x S^2 D3-brane in AdS_3 x S^3. Parenthetically we also derive a simple criterion for a symmetric plane wave to be isometric to a lorentzian Lie group. In particular we observe that the maximally supersymmetric plane wave in IIB string theory is isometric to a lorentzian Lie group, whereas the one in M-theory is not. 
  Recently, gauged supergravities in three dimensions with Yang-Mills and Chern-Simons type interactions have been constructed. In this article, we demonstrate that any gauging of Yang-Mills type with semisimple gauge group G_0, possibly including extra couplings to massive Chern-Simons vectors, is equivalent on-shell to a pure Chern-Simons type gauging with non-semisimple gauge group $G_0 \ltimes T \subset G$, where T is a certain translation group, and where G is the maximal global symmetry group of the ungauged theory. We discuss several examples. 
  We construct boundary states for D-branes which carry traveling waves in the covariant formalism. We compute their vacuum amplitudes to investigate their interactions. In non-compact space, the vacuum amplitudes become trivial as is common in plane wave geometries. However, we found that if they are compactified in the traveling direction, then the amplitudes are affected by non-trivial time dependent effects. The interaction between D-branes with waves traveling in the opposite directions (`pulse-antipulse scattering') are also computed. Furthermore, we apply these ideas to open string tachyon condensation with traveling waves. 
  In this paper, we study the fractional decomposition of the quantum enveloping affine algebras $U_Q(\hat A(n))$ and $U_Q(\hat{C}(n))$ with vanishing central charge in the limit $Q\to q=e^{\frac{2i\pi}k}$ . This decomposition is based on the bosonic representation and can be related to the fractional supersymmetry and $k$-fermionic spin. The equivalence between the quantum affine algebras and the classical ones in the fermionic realization is also established. 
  We discuss the algebra of $N\times N$ matrices as a reduced quantum plane. A $3-$nilpotent deformed differential calculus involving a complex parameter $q$ is constructed. The two cases, $q$ $3^{rd}$ and $N^{th}$ root of unity are completely treated. As application, a gauge field theory for the particular cases $n=2$ and $n=3$ is established. 
  We study an RNS string with one end fixed on a $D0$-brane and the other end free as a qualitative guide to the spectrum of hadrons containing one very heavy quark. The mixed boundary conditions break half of the world-sheet supersymmetry. Boson-fermion masses can still be matched if space-time is 9 dimensional; thus SO(8) triality still plays a role in the spectrum, although full space-time supersymmetry does not survive. We quantize the system in a temporal-like gauge where $X^0 \sim \tau$. Only odd $\alpha$ and even $d$ R modes remain, while the NS oscillators $b$ become odd-integer moded. Although the gauge choice eliminates negative-norm states at the outset, there are still even-moded Virasoro and even(odd) moded super-Virasoro constraints to be imposed in the NS(R) sectors. The Casimir energy is now positive in both sectors; there are no tachyons. States for $\alpha' M^2 \leq 3$ are explicitly constructed and found to be organized into SO(8) irreps by (super)constraints, which include a novel "$\sqrt{L_0}$" operator in the NS and $\Gamma^0 \pm I$ in the R sectors. GSO projections are not allowed. The pre-constraint states above the ground state have matching multiplicities, indicating spacetime supersymmetry is broken by the (super)constraints. A distinctive physical signature of the system is a slope twice that of the open RNS string. When both ends are fixed, all leading and subleading trajectories are eliminated, resulting in a spectrum qualitatively similar to the $J/\psi$ and $\Upsilon$ particles. 
  Candelas and Font introduced the notion of a `top' as half of a three dimensional reflexive polytope and noticed that Dynkin diagrams of enhanced gauge groups in string theory can be read off from them. We classify all tops satisfying a generalized definition as a lattice polytope with one facet containing the origin and the other facets at distance one from the origin. These objects torically encode the local geometry of a degeneration of an elliptic fibration. We give a prescription for assigning an affine, possibly twisted Kac-Moody algebra to any such top (and more generally to any elliptic fibration structure) in a precise way that involves the lengths of simple roots and the coefficients of null roots. Tops related to twisted Kac-Moody algebras can be used to construct string compactifications with reduced rank of the gauge group. 
  In this thesis I present a new type of brane - H-brane - where the role of time in String Theory is considered as a primary concept in the search for still unknown black hole physics. Using the basic tools of boundary conformal field theory, we describe the H-branes both in the open and closed string channels and examine how they fit naturally in the known D-brane moduli space. In particular we compute their Ishibashi states and quantize the system using the first order formalism. We find that the geometry associated to the target null coordinates is non-commutative. 
  We study the quantum mechanics of BMN operators with two scalar impurities and arbitrarily many traces, at one loop and all genus. We prove an operator identity which partially elucidates the structure of this quantum mechanics, provides some support for a conjectured formula for the free all genus two-point functions, and demonstrates that a single O(g_2^2) contact term arises in the Hamiltonian as a result of transforming from the natural gauge theory basis to the string basis. We propose to identify the S-matrix of this quantum mechanics with the S-matrix of string theory in the plane-wave background. 
  Liouville theory with a negative norm boson and no screening charge corresponds to an exact classical solution of closed bosonic string theory describing time-dependent bulk tachyon condensation. A simple expression for the two point function is proposed based on renormalization/analytic continuation of the known results for the ordinary (positive-norm) Liouville theory. The expression agrees exactly with the minisuperspace result for the closed string pair-production rate (which diverges at finite time). Puzzles concerning the three-point function are presented and discussed. 
  We investigate the correspondence between superstring theory on pp-wave background with NSNS-flux and superconformal field theory on a symmetric orbifold. This correspondence can be regarded as the ``Penrose limit'' of AdS_3/CFT_2 correspondence. Superstring theory on the Penrose limit of AdS_3 x S^3 (x M^4) (M^4 = T^4 or K3) with NSNS-flux can be described by a generalization of Nappi-Witten model. We quantize this system in the covariant gauge and obtain the spectrum of superstring theory. In the dual CFT point of view, the Penrose limit means concentrating on the subsector of almost BPS states with large R-charges. We show that stringy states can be embedded in the single-particle Hilbert space of symmetric orbifold theory. 
  The cubic interaction vertex and the dynamical supercharges are constructed for open strings ending on D7-branes, in light-cone superstring field theory in PP-wave background. In this context, we write down the symmetry generators in terms of the relevant group structure: SU(2) x SU(2) x SO(2) x SO(2), originating from the eight transverse directions in the PP-wave background and use the expressions to explicitly construct the vertex at the level of stringy zero modes. The results are further generalized to include all the stringy excitations as well. 
  Using a technique \cite{holgernorma2002} to construct a basis for spinors and ``families'' of spinors in terms of Clifford algebra objects, we define other Clifford algebra objects, which transform the state of one ''family'' of spinors into the state of another ''family'' of spinors, changing nothing but the ''family'' number. The proposed transformation works - as does the technique - for all dimensions and any signature and might open a path to understanding families of quarks and leptons\cite{norma92,norma93,normaixtapa2001,pikanorma2002}. 
  We examine a tensionless limit of a SL(2,Z) set of background solutions to IIB supergravity theory, obtained by performing an infinite boost. This yields a solution that corresponds to taking the original string tension to zero. The limit reproduces ordinary Minkowski space except for a delta-like singularity along the string. We study the field content and the energy momentum tensor. 
  We evaluate the fermion propagator in parity-conserving QED_3 with N flavours, in the context of an IR domain approximation. This provides results which are non-perturbative in the loopwise expansion sense. We include fermion-loop effects, and show that they are relevant to the chiral symmetry breaking phenomenon, that can be understood in this context. 
  We have extended previous analysis of the bulk/brane supersymmetrizations involving non-zero brane mass terms of bulk fermions (gravitini) and twisting of boundary conditions. We have constructed new brane/bulk models that may be relevant for realistic model building. In particular, we have built a model with the Randall-Sundrum bosonic sector, orthogonal projection operators on the branes in the fermionic sector, and an unbroken N=1 supersymmetry. We have also constructed 5d super-bigravity with static vacuum and unbroken N=1 supersymmetry, which may be viewed as a deconstruction of 5d supergravity. 
  The classical theory for a massive free particle moving on the group manifold $AdS_3 \cong SL(2, \mathbb{R})$ is analysed in detail. In particular a symplectic structure and two different sets of canonical coordinates are explicitly found, corresponding to the Cartan and Iwasawa decomposition of the group. Canonical quantization is performed in two different ways; by imposing the future-directed constraint before and after quantization. It is found that this leads to different quantum theories. The Hilbert space of either theory decomposes into the sum of certain irreducible representations of $sl(2,\mathbb{R}) \oplus sl(2,\mathbb{R})$; however, depending on how the constraint is imposed we get different representations. Quantization of the mass occurs, although a continuum exists in the unconstrained theory corresponding to particles that can reverse their direction in time. A quantization in terms of the ``chiral'' variables of the theory is also carried out giving the same results. Comparisons are made between QFT in $AdS_3$ and the quantum mechanics derived, and it is found that one of the quantum theories is consistent with the Breitenlohner-Freedman bound. 
  We investigate the charges and fluxes that can occur in higher-order Abelian gauge theories defined on compact space-time manifolds with boundary. The boundary is necessary to supply a destination to the electric lines of force emanating from brane sources, thus allowing non-zero net electric charges, but it also introduces new types of electric and magnetic flux. The resulting structure of currents, charges, and fluxes is studied and expressed in the language of relative homology and de Rham cohomology and the corresponding abelian groups. These can be organised in terms of a pair of exact sequences related by the Poincar\'e-Lefschetz isomorphism and by a weaker flip symmetry exchanging the ends of the sequences. It is shown how all this structure is brought into play by the imposition of the appropriately generalised Maxwell's equations. The requirement that these equations be integrable restricts the world-volume of a permitted brane (assumed closed) to be homologous to a cycle on the boundary of space-time. All electric charges and magnetic fluxes are quantised and satisfy the Dirac quantisation condition. But through some boundary cycles there may be unquantised electric fluxes associated with quantised magnetic fluxes and so dyonic in nature. 
  In this paper we find examples of moduli stabilization and runaway behavior which can be treated exactly. This is shown for supersymmetric field theories which can be realized on the world volume of D-branes. From a geometric point of view, these field theories lift moduli spaces of vacua by deforming lines of singularities where supersymmetric fractional branes can be located in the geometry without D-branes. 
  We realize the CFT with target a lens space SU(2)/Z_l as a simple current construction. This allows us to compute the boundary states and the annuli coefficients, and in particular to study the B-type branes, in purely algebraic terms. Several issues, like the appearance of fractional branes and symmetry breaking boundary conditions, can be addressed more directly in this approach than in a more geometric treatment. 
  We construct new intersecting S-brane solutions in 11-dimensional supergravity which do not have supersymmetric analogs. They are obtained by letting brane charges to be proportional to each other. Solutions fall into two categories with respect to whether there is a non-diagonal term to be cancelled in the field equations or not. In each case we show that they can be constructed by using a simple set of rules which is similar to the harmonic function rule of the usual static p-branes. Furthermore, we study an intersection where the Chern-Simons term makes a non-zero contribution to the field equations. We show that this configuration has a singularity like other S-branes. 
  Noncommutative U(N) gauge theories at different N may be often thought of as different sectors of a single theory: the U(1) theory possesses a sequence of vacua labeled by an integer parameter N, and the theory in the vicinity of the N-th vacuum coincides with the U(N) noncommutative gauge theory. We construct noncommutative domain walls on fuzzy cylinder, separating vacua with different gauge theories. These domain walls are solutions of BPS equations in gauge theory with an extra term stabilizing the radius of the cylinder. We study properties of the domain walls using adjoint scalar and fundamental fermion fields as probes. We show that the regions on different sides of the wall are not disjoint even in the low energy regime -- there are modes penetrating from one region to the other. We find that the wall supports a chiral fermion zero mode. Also, we study non-BPS solution representing a wall and an antiwall, and show that this solution is unstable. We suggest that the domain walls emerge as solutions of matrix model in large class of pp-wave backgrounds with inhomogeneous field strength. In the M-theory language, the domain walls have an interpretation of a stack of branes of fingerstall shape inserted into a stack of cylindrical branes. 
  Flux compactifications of $M$ theory on certain $G_2$-manifolds have vacua in which all moduli are determined.These vacua have discrete parameters - the flux quanta. Small changes in these quanta produce small changes in the moduli and these can lead to large changes in the Yukawa couplings between Higgses, quarks and leptons. We argue that a large number of vacua exist with a wide variety of Yukawa hierarchies. 
  We study exclusive scattering of `hadrons' at high energy and fixed angle in (nonconformal) noncommutative gauge theories. Via gauge-string duality, we show that the noncommutativity renders the scattering soft, leading to exponential suppression. The result fits with the picture that, in noncommutative gauge theory, fundamental partons consist of extremely soft constituents and `hadrons' are made out of open Wilson lines. 
  In order to study the thermodynamic properties of brane-antibrane systems in the toroidal background, we compute the finite temperature effective potential of tachyon T in this system on the basis of boundary string field theory. We first consider the case that all the radii of the target space torus are about the string scale. If the Dp-antiDp pair is extended in all the non-compact directions, the sign of the coefficient of |T|^2 term of the potential changes slightly below the Hagedorn temperature. This means that a phase transition occurs near the Hagedorn temperature. On the other hand, if the Dp-antiDp pair is not extended in all the non-compact directions, the coefficient is kept negative, and thus a phase transition does not occur. Secondly, we consider the case that some of the radii of the target space torus are much larger than the string scale and investigate the behavior of the potential for each value of the radii and the total energy. If the Dp-antiDp pair is extended in all the non-compact directions, a phase transition occurs for large enough total energy. 
  Using the Schwarzschild-type coordinates in stead of the global ones we reconstruct the classical rotating closed string solutions in the AdS*5 x S*5 backgrounds. They are explicitly described by the Jacobi elliptic and trigonometrical functions of worldsheet coordinates. We study the orbiting closed string configurations in the near-horizon geometries of Dp, NS1 and NS5 branes, and derive the energy and spin of them, whose relation takes a simple form for short strings. Specially in the D5 and NS5 backgrounds we have a linear relation that the energy of the point-like string is proportional to the spin, which is associated with the spectrum of strings in the pp-wave geometries obtained by taking a special Penrose limit on the D5 and NS5 backgrounds. 
  We point out that the recently proposed model of a flat 4-dimensional universe with accelerated expansion in string/M-theory is a special case of time-dependent solutions that the author found under the name of ``S-branes.'' We also show that similar accelerating models can be obtained from S-branes if the internal space is chosen to be hyperbolic or flat spaces. 
  We show that the on-shell S-matrix elements of four open string massless scalars, two scalars and two tachyons, and four open string tachyons in the super string theory can be written in a unique form. We then propose an off-shell extension for the S-matrix element of four scalars which is consistent, in the low energy limit, with the Dirac-Born-Infeld effective action. Using a similar off-shell extension for the S-matrix element of two scalars and two tachyons and for the S-matrix element of four tachyons, we show that they are fully consistent with the tachyonic DBI action. 
  Using the idea of Itzykson-Zuber integral, unitary-matrix integration of 2D Yang-Mills action is presented. The uniqueness of the solution of heat equation enables us to integrate out the unitary-matrix parts of hermite matrices and to obtain the expression of integration over vectors, also in this case. 
  After recalling episodes from Pascual Jordan's biography including his pivotal role in the shaping of quantum field theory and his much criticized conduct during the NS regime, I draw attention to his presentation of the first phase of development of quantum field theory in a talk presented at the 1929 Kharkov conference. He starts by giving a comprehensive account of the beginnings of quantum theory, emphasising that particle-like properties arise as a consequence of treating wave-motions quantum-mechanically. He then goes on to his recent discovery of quantization of ``wave fields'' and problems of gauge invariance. The most surprising aspect of Jordan's presentation is however his strong belief that his field quantization is a transitory not yet optimal formulation of the principles underlying causal, local quantum physics. The expectation of a future more radical change coming from the main architect of field quantization already shortly after his discovery is certainly quite startling. I try to answer the question to what extent Jordan's 1929 expectations have been vindicated. The larger part of the present essay consists in arguing that Jordan's plea for a formulation without ``classical correspondence crutches'', i.e. for an intrinsic approach (which avoids classical fields altogether), is successfully addressed in past and recent publications on local quantum physics. 
  We consider the general properties of effective field theories. We note that the freedom to fix the renormalization conditions in the effective field theory is not as great as it seems. The consideration of minimal requirements of correctness of the perturbative scheme based on Dyson's formula for S-matrix leads to severe restriction on the essential parameters of a theory and, hence, on the allowed set of renormalization conditions. In the first part we make a short review of the structure of localizable effective field theories. We discuss the conditions needed to ensure the correctness of the very first step of perturbative calculation of the S-matrix -- the construction of the tree-level amplitudes. In the second part we consider the examples demonstrating the main stages of derivation and analysis of the system of bootstrap equations. 
  When M-theory is compactified on G_2-holonomy manifolds with conical singularities, charged chiral fermions are present and the low-energy four-dimensional theory is potentially anomalous. We reconsider the issue of anomaly cancellation, first studied by Witten. We propose a mechanism that provides local cancellation of all gauge and mixed gauge-gravitational anomalies, i.e. separately for each conical singularity. It is similar in spirit to the one used to cancel the normal bundle anomaly in the presence of five-branes. It involves smoothly cutting off all fields close to the conical singularities, resulting in an anomalous variation of the 3-form C and of the non-abelian gauge fields present if there are also ADE singularities. 
  We study the qualitative features of the QCD phase diagram in the context of the linear quark-meson model with two flavours, using the exact renormalization group. We identify the universality classes of the second-order phase transitions and calculate critical exponents. In the absence of explicit chiral-symmetry breaking through the current quark masses, we discuss in detail the tricritical point and demonstrate that it is linked to the Gaussian fixed point. In its vicinity we study the universal crossover between the Gaussian and O(4) fixed points, and the weak first-order phase transitions. In the presence of explicit chiral-symmetry breaking, we study in detail the critical endpoint of the line of first-order phase transitions. We demonstrate the decoupling of pion fluctuations and identify the Ising universality class as the relevant one for the second-order phase transition. 
  We provide new expressions for the scattering amplitudes in the soliton-antisoliton sector of the elliptic sine-Gordon model in terms of cosets of the affine Weyl group corresponding to infinite products of q-deformed gamma functions. When relaxing the usual restriction on the coupling constants, the model contains additional bound states which admit an interpretation as breathers. These breather bound states are unavoidably accompanied by Tachyons. We compute the complete S-matrix describing the scattering of the breathers amonst themselves and with the soliton-antisoliton sector. We carry out various reductions of the model, one of them leading to a new type of theory, namely an elliptic version of the minimal D(n+1)-affine Toda field theory. 
  We consider the $GSO(-)$ sector of the open superstring using the formalism with four-dimensional hybrid variables. This sector is defined by the world sheet hybrid variables $(\theta^{\alpha},\bar{\theta}^{\dot\alpha})$ with antiperiodic boundary conditions. The corresponding spectrum of states and conditions for physical vertices are described. In particular we construct explicitly the lower level $GSO(-)$ vertex operators corresponding to the tachyon and the massless fermions. Using these new vertices, the tachyon and massless sector contribution to the superstring field theory action of Berkovits is evaluated. In this way we have included the Ramond sector and we end by discussing some features of the action 
  Starting from an anomaly-free Abelian Higgs model coupled to gravity in a 6-dimensional space-time we construct an effective four-dimensional theory of charged fermions interacting with U(1) Abelian gauge field and gravity, both localised near the core of a Nielsen-Olesen vortex configuration. We show that an anomaly free theory in 6-dimensions can give rise to an anomalous theory in D=4, which suggests a possibility of consistent regularisation of abelian anomalous chiral gauge theories in four dimensions. We also show that the spectrum of charged bulk fermions has a mass gap. 
  A solution to the BI action describing an expanded D3-brane is considered. The configuration is not supersymmetric but is supported against collapse by the Poynting vector. In the uniform magnetic field limit, the brane magnetic charge grows without limit while the energy and the radius of the brane remain finite. In this limit the brane consists of a large number of small D-strings forming a hypertube and kept in equilibrium by the electric flux along the hypertube. 
  Gravity may be "locally localized" over a wide range of length scales on a d-dimensional anti-de Sitter (AdS) brane living inside AdS_{d+1}. In this paper we examine this phenomenon from the point of view of the holographic dual "defect conformal field theory". The mode expansion of bulk fields on the gravity side is shown to be precisely dual to the "boundary operator product expansion" of operators as they approach the defect. From the field theory point of view, the condition for localization is that a "reduced operator" appearing in this expansion acquires negative anomalous dimension. In particular, a very light localized graviton exists when a mode arising from the reduction of the ambient stress-energy tensor to the defect has conformal dimension Delta ~ d-1. The part of the stress tensor containing the defect dynamics has dimension Delta = d-1 in the free theory, but we argue that it acquires an anomalous dimension in the interacting theory, and hence does not participate in localization in the regime of small backreaction of the brane. We demonstrate that such an anomalous dimension is consistent with the conservation of the full stress-energy tensor. Finally, we analyze how to compute the anomalous dimensions of reduced operators from gravity at leading order in the interactions with the brane. 
  We present a variational method for deriving relativistic two-fermion wave equations in a Hamiltonian formulation of QED. A reformulation of QED is performed, in which covariant Green functions are used to solve for the electromagnetic field in terms of the fermion fields. The resulting modified Hamiltonian contains the photon propagator directly. The reformulation permits one to use a simple Fock-space variational trial state to derive relativistic fermion-antifermion wave equations from the corresponding quantum field theory. We verify that the energy eigenvalues obtained from the wave equation agree with known results for positronium. 
  Analyticity of gluon and Faddeev--Popov ghost propagators and their form factors on the complex momentum-squared plane is exploited to continue analytically the ultraviolet asymptotic form calculable by perturbation theory into the infrared non-perturbative solution. We require the non-perturbative multiplicative renormalizability to write down the renormalization group equation. These requirements enable one to settle the value of the exponent characterizing the infrared asymptotic solution with power behavior which was originally predicted by Gribov and has recently been found as approximate solutions of the coupled truncated Schwinger--Dyson equations. For this purpose, we have obtained all the possible superconvergence relations for the propagators and form factors in both the generalized Lorentz gauge and the modified Maximal Abelian gauge. We show that the transverse gluon propagators are suppressed in the infrared region to be of the massive type irrespective of the gauge parameter, in agreement with the recent result of numerical simulations on a lattice. However, this method alone is not sufficient to specify some of the ghost propagators which play the crucial role in color confinement. Combining the above result with the renormalization group equation again, we find an infrared enhanced asymptotic solution for the ghost propagator. The coupled solutions fulfill the color confinement criterion due to Kugo and Ojima and also Nishijima, at least, in the Lorentz--Landau gauge. We also point out that the solution in compatible with color confinement leads to the existence of the infrared fixed point in pure Yang--Mills theory without dynamical quarks. Finally, the Maximal Abelian gauge is also examined in connection with quark confinement. 
  The decay constant of the QCD axion is required by observation to be small compared to the Planck scale. In theories of "natural inflation," and certain proposed anthropic solutions of the cosmological constant problem, it would be interesting to obtain a large decay constant for axion-like fields from microscopic physics. String theory is the only context in which one can sensibly address this question. Here we survey a number of periodic fields in string theory in a variety of string vacua. In some examples, the decay constant can be parameterically larger than the Planck scale but the effective action then contains appreciable harmonics of order $f_A/M_p$. As a result, these fields are no better inflaton candidates than Planck scale axions. 
  The phenomenon of creation of strings, occurring when particles pass through a domain wall and related to the Hanany-Witten effect via dualities, is discussed in ten and nine dimensions. We consider both the particle actions in massive backgrounds as well as the 1/4-supersymmetric particle-string-domain wall supergravity solutions and discuss their physical interpretation. In 10D we discuss the D0-F1-D8 system in massive IIA theory while in 9D the SL(2,R)-generalisation is constructed. It consists of (p,q)-particles, (r,s)-strings and the double domain wall solution of the three different 9D gauged supergravities where a subgroup of SL(2,R) is gauged. 
  I discuss several aspects of CP non-invariance in the strongly interacting theory of quarks and gluons. I use a simple effective Lagrangian technique to map out the region of quark masses where CP symmetry is spontaneously broken. I then turn to the possible explicit CP violation arising from a complex quark mass. After summarizing the definition of the renormalized theory as a limit, I argue that attempts to remove the CP violation by making the lightest quark mass vanish are not well defined. I close with some warnings for lattice simulations. 
  We study aspects of the interaction between a D-brane and an anti-D-brane in the maximally supersymmetric plane wave background of type IIB superstring theory, which is equipped with a mass parameter mu. An early such study in flat spacetime (mu=0) served to sharpen intuition about D-brane interactions, showing in particular the key role of the ``stringy halo'' that surrounds a D-brane. The halo marks the edge of the region within which tachyon condensation occurs, opening a gateway to new non-trivial vacua of the theory. It seems pertinent to study the fate of the halo for non--zero mu. We focus on the simplest cases of a Lorentzian brane with p=1 and an Euclidean brane with p=-1, the D--instanton. For the Lorentzian brane, we observe that the halo is unaffected by the presence of non--zero mu. This most likely extends to other (Lorentzian) p. For the Euclidean brane, we find that the halo is affected by non-zero mu. As this is related to subtleties in defining the exchange amplitude between Euclidean branes in the open string sector, we expect this to extend to all Euclidean branes in this background. 
  This thesis is designed for a comprehensive review of noncommutative (BPS) solitons with applications to D-brane dynamics including our works. We focus on noncommutative instantons and monopoles and study various aspects of the exact solutions by using Atiyah-Drinfeld-Hitchin-Manin (ADHM) and Nahm constructions. Finally we propose noncommutative extensions of integrable systems and soliton theories in lower dimensions in collaboration with Kouichi Toda, which would pioneer a new study area of integrable systems. Appendix is devoted to a brief and systematic review of formal aspects of ADHM/Nahm construction and Nahm transformation on commutative spaces. This article is also a step to a comprehensive review of ADHM/Nahm construction on both commutative and noncommutative spaces. Comments are welcome. 
  The paradigmatic Unruh radiation is an ideal and simple case of stationary scalar vacuum radiation patterns related to worldlines defined as Frenet-Serret curves. We briefly review the corresponding body of theoretical literature as well as the proposals that have been suggested to detect these types of quantum field radiation patterns 
  It was generally believed that, in general relativity, the fundamental laws of nature should be invariant or covariant under a general coordinate transformation. In general relativity, the equivalence principle tells us the existence of a local inertial coordinate system and the fundamental laws in the local inertial coordinate system which are the same as those in inertial reference system. Then, after a general coordinate transformation, the fundamental laws of nature in arbitrary coordinate system or in arbitrary curved space-time can be obtained. However, through a simple example, we find that, under a general coordinate transformation, basic physical equations in general relativity do not transform covariantly, especially they do not preserve their forms under the transformation from a local inertial coordinate system to a curved space-time. The origination of the violation of the general covariance is then studied, and a general theory on general coordinate transformations is developed. Because of the the existence of the non-homogeneous term, the fundamental laws of nature in arbitrary curved space-time can not be expressed by space-time metric, physical observable and their derivatives. In other words, basic physical equations obtained from the equivalence principle and the principle of general covariance are different from those in general relativity. Both the equivalence principle and the principle of general covariance can not be treated as foundations of general relativity. So, what are the foundations of General Relativity? Such kind of essential problems on General Relativity can be avoided in the physical picture of gravity. Quantum gauge theory of gravity, which is founded in the physics picture of gravity, does not have such kind of fundamental problems. 
  The analysis of the gauge principle as a mere passive symmetry requirement leads to the conclusion that the connection term in the covariant derivative is flat and that local phase transformations are without any empirical significance in analogy to coordinate transformations. Nevertheless, the Aharonov-Bohm effect shows the physical significance of the non-trivial holonomy of a flat connection. On this basis the proposal of a new kind of charge, the phase charge, is made, understood as the coupling strength of the particle to the holonomy. The equivalence of phase and usual field charge must be tested experimentally in terms of an Aharonov-Bohm effect with muons or tauons, for instance. 
  We investigate the breakdown of supersymmetry at finite temperature. While it has been proven that temperature always breaks supersymmetry, the nature of this breaking is less clear. On the one hand, a study of the Ward-Takahashi identities suggests a spontaneous breakdown of supersymmetry without the existence of a Goldstino, while on the other hand it has been shown that in any supersymmetric plasma there should exist a massless fermionic collective excitation, the phonino. Aim of this work is to unify these two approaches. For the Wess-Zumino model, it is shown that the phonino exists and contributes to the supersymmetric Ward-Takahashi identities in the right way displaying that supersymmetry is broken spontaneously with the phonino as the Goldstone fermion. 
  The critical effective potential is the nonperturbative part of the effective action at a phase transition. It equals the scale invariant effective average potential and can be calculated from the renormalization group flow of the effective average action. In some cases this requires only the solution of an ordinary differential equation without actually simulating the renormalization group flow. Here the Ising model is examined beyond leading order and with full field dependent effective potential. 
  In the framework of started in Ref.[1] construction procedure of the general superfield quantization method for gauge theories in Lagrangian formalism the rules for Hamiltonian formulation of general superfield theory of fields (GSTF) are introduced and are on the whole considered.   Mathematical means developed in [1] for Lagrangian formulation of GSTF are extended to use in Hamiltonian one. Hamiltonization for Lagrangian formulation of GSTF via Legendre transform of superfunction $S_{L}\bigl({\cal A}(\theta),{\stackrel{\circ}{\cal A}}(\theta),\theta\bigr)$ with respect to ${\stackrel{\circ}{{\cal A}^{\imath}}}(\theta)$ is considered. As result on the space $T^{\ast}_{odd}{\cal M}_{cl}\times \{\theta\}$ parametrized by classical superfields ${\cal A}^{\imath}(\theta)$, superantifields ${\cal A}^{\ast}_{\imath}(\theta)$ and odd Grassmann variable $\theta$ the superfunction $S_{H}({\cal A}(\theta),{\cal A}^{\ast}(\theta),\theta)$ is defined. Being equivalent to different types of Euler-Lagrange equations the distinct Hamiltonian systems are investigated. Translations along $\theta$ for superfunctions on $T^{\ast}_{odd}{\cal M}_{cl}\times \{\theta\}$ being associated with these systems are studied. Various types of antibrackets and differential operators acting on $C^{k}\bigl(T^{\ast}_{odd}{\cal M}_{cl} \times \{\theta\} \bigr)$ are considered. Component (on $\theta$)formulation for GSTF quantities and operations is produced. Analogy between ordinary Hamiltonian classical mechanics and GSTF in Hamiltonian formulation is proposed. Realization of the GSTF general scheme is demonstrated on 6 models. 
  The aim of this review is to present the list of by now a significant collection of quantum integrable models, ultralocal as well as nonultralocal, in a systematic way stressing on their underlying unifying algebraic structures. We restrict to quantum and statistical models belonging to trigonometric and rational classes with (2 x 2)- Lax operators. The ultralocal models can be classified successfully through their associated quantum algebra and are governed by the Yang-Baxter equation, while the nonultralocal models, the theory of which is still in the developmental stage, allow systematization through the braided Yang-Baxter equation. Along with the known integrable models some possible directions for investigation in this field and generation of such new models are suggested. 
  We develop a method to compute the Casimir effect for arbitrary geometries. The method is based on the string-inspired worldline approach to quantum field theory and its numerical realization with Monte-Carlo techniques. Concentrating on Casimir forces between rigid bodies induced by a fluctuating scalar field, we test our method with the parallel-plate configuration. For the experimentally relevant sphere-plate configuration, we study curvature effects quantitatively and perform a comparison with the ``proximity force approximation'', which is the standard approximation technique. Sizable curvature effects are found for a distance-to-curvature-radius ratio of a/R >~ 0.02. Our method is embedded in renormalizable quantum field theory with a controlled treatment of the UV divergencies. As a technical by-product, we develop various efficient algorithms for generating closed-loop ensembles with Gaussian distribution. 
  We propose a mechanism by which electric charges deconfine in an Abelian Higgs model with matter fields belonging to the fundamental representation of the gauge group. Kosterlitz-Thouless like recursion relations for a scale-dependent stiffness parameter and fugacity are given, showing that for a logarithmic potential between point charges in any dimension, there exists a stable fixed point at zero fugacity, with a dimensionality dependent universal jump in the stiffness parameter at the phase transition. 
  We obtain a general class of time-dependent, asymptotically de Sitter backgrounds which solve the first order bosonic equations that extremize the action for supergravity with gauged non-compact $R$-symmetry. These backgrounds correspond only to neutral fields with the correct sign of kinetic energy. Within N=2 five-dimensional supergravity with vector-superfields we provide examples of multi-centered charged black holes in asymptotic de Sitter space, whose spatial part is given by a time-dependent hyper-K\"ahler space. Reducing these backgrounds to four dimensions yields asymptotically de Sitter multi-centered charged black hole backgrounds and we show that they are related to an instanton configuration by a massive T-duality over time. Within N=2 gauged supergravity in four (and five)-dimensions with hyper-multiplets there could also be neutral cosmological backgrounds that are regular and correspond to the different de Sitter spaces at early and late times. 
  By combining two distinct renormalization group transformations, opposing scale transformations, we obtain a composite transformation which does not rescale the system, and drives it to a "geometrical" fixed point, controlling the effective geometry and locality. The latticized (deconstructed) action for an extra-dimensional field theory becomes a ``perfect action,'' with a linear ladder spectrum for N KK-modes. 
  We study the matrix model/gauge theory connection for three different N=1 models: U(N) x U(N) with matter in bifundamental representations, U(N) with matter in the symmetric representation, and U(N) with matter in the antisymmetric representation. Using Ward identities, we explicitly show that the loop equations of the matrix models lead to cubic algebraic curves. We then establish the equivalence of the matrix model and gauge theory descriptions in two ways. First, we derive generalized Konishi anomaly equations in the gauge theories, showing that they are identical to the matrix-model equations. Second, we use a perturbative superspace analysis to establish the relation between the gauge theories and the matrix models. We find that the gauge coupling matrix for U(N) with matter in the symmetric or antisymmetric representations is_not_ given by the second derivative of the matrix-model free energy. However, the matrix-model prescription can be modified to give the gauge coupling matrix. 
  Brane inflation in superstring theory predicts that cosmic strings (but not domain walls or monopoles) are produced towards the end of the inflationary epoch. Here, we discuss the production, the spectrum and the evolution of such cosmic strings, properties that differentiate them from those coming from an abelian Higgs model. As D-branes in extra dimensions, some type of cosmic strings will dissolve rapidly in spacetime, while the stable ones appear with a spectrum of cosmic string tensions. Moreover, the presence of the extra dimensions reduces the interaction rate of the cosmic strings in some scenarios, resulting in an order of magnitude enhancement of the number/energy density of the cosmic string network when compared to the field theory case. 
  We propose an M-theory lift picture of the exchange among type IIA orientifold two-planes. This consists in wrapping a M5-brane on a three-cycle in the transverse space of the M-theory orientifold plane OM2. A flux quantization condition for the three-form self-dual field strength, on the worldvolume of the M5-brane is computed. This condition establishes the value which explains the relative charge between two different OM2-planes. Also, we find that the exchange of the four types of orientifold two-planes in string theory, has a common picture in M-theory. Moreover, we find that the assignment of the extra charge is fixed by cohomology and by the flux quantization of the field strength G in M-theory. We conclude that cohomology is sufficient to describe some orientifold properties in M-theory, that at string theory level, only K-theory is able to explain. 
  We study the superpotential for the heterotic string compactified on non-Kahler complex manifolds. We show that many of the geometrical properties of these manifolds can be understood from the proposed superpotential. In particular we give an estimate of the radial modulus of these manifolds. We also show, how the torsional constraints can be obtained from this superpotential. 
  The gauge-fixed action of a `spacetime-filling' D3-brane with dilaton-axion coupling is formulated in N=1 superspace. We investigate its symmetries by paying special attention to a possible non-linearly realized extra (hidden) supersymmetry, and emphasize the need of a linear superfield coupled to an abelian Chern-Simons superfield to represent a dilaton-axion supermultiplet in the off-shell manifestly supersymmetric approach. 
  We consider supersymmetric sigma models on the Kahler target spaces, with twisted mass. The Kahler spaces are assumed to have holomorphic Killing vectors. Introduction of a superpotential of a special type is known to be consistent with N=2 superalgebra (Alvarez-Gaume and Freedman). We show that the algebra acquires central charges in the anticommutators {Q_L, Q_L} and {Q_R, Q_R}. These central charges have no parallels, and they can exist only in two dimensions. The central extension of the N=2 superalgebra we found paves the way to a novel phenomenon -- spontaneous breaking of a part of supersymmetry. In the general case 1/2 of supersymmetry is spontaneously broken (the vacuum energy density is positive), while the remaining 1/2 is realized linearly. In the model at hand the standard fermion number is not defined, so that the Witten index as well as the Cecotti-Fendley-Intriligator-Vafa index are useless. We show how to construct an index for counting short multiplets in internal algebraic terms which is well-defined in spite of the absence of the standard fermion number. Finally, we outline derivation of the quantum anomaly in {\bar Q_L, Q_R}. 
  A regular bouncing universe is obtained in the context of a dilaton-gravity brane world scenario. The scale factor starts in a contracting inflationary phase both in the Einstein and in the string frame, it then undergoes a bounce (due to interaction with the bulk Weyl tensor), and subsequently enters into a decelerated expanding era. This graceful exit is obtained at low curvature and low coupling, and without violating the Null Energy Condition. 
  We complete the construction of the Moyal star formulation of bosonic open string field theory (MSFT) by providing a detailed study of the fermionic ghost sector. In particular, as in the case of the matter sector, (1) we construct a map from Witten's star product to the Moyal product, (2) we propose a regularization scheme which is consistent with the matter sector and (3) as a check of the formalism, we derive the ghost Neumann coefficients algebraically directly from the Moyal product. The latter satisfy the Gross-Jevicki nonlinear relations even in the presence of the regulator, and when the regulator is removed they coincide numerically with the expression derived from conformal field theory. After this basic construction, we derive a regularized action of string field theory in the Siegel gauge and define the Feynman rules. We give explicitly the analytic expression of the off-shell four point function for tachyons, including the ghost contribution. Some of the results in this paper have already been used in our previous publications. This paper provides the technical details of the computations which were omitted there. 
  Weak-strong coupling duality relations are shown to be present in the quantum-mechanical many-body system with the interacting potential proportional to the pair-wise inverse-squared distance in addition to the harmonic potential. Using duality relations we have solved the problem of families interacting by the inverse-squared interaction. Owing to duality, the coupling constants of the families are mutually inverse. The spectrum and eigenfunctions are determined mainly algebraically owing to O(2,1) dynamical symmetry. The constructed Hamiltonian for families and appropriate solutions are of hierarchical nature. 
  We construct supergravity solutions dual to microstates of the D1-D3-D5 system with nonzero B field moduli. Just like the D1-D5 solutions in hep-th/0109154 these solutions are generically nonsingular everywhere, with the `throat' closing smoothly near r=0. We write expressions relating the asymptotic supergravity fields to the integral brane charges. We study the infall of a D1 brane down the throat of the geometries. This test brane `bounces' off the smooth end for generic initial conditions. The details of the bounce depend on both the choice of D1-D3-D5 microstate and the direction of approach of the infalling D1 brane. In the dual field theory description we see that the tachyon mode starts to condense, but the tachyon bounces back up the potential hill without reaching the deepest point of the potential. 
  The quantum vacuum effects are investigated for a massive scalar field with general curvature coupling and obeying the Robin boundary conditions given on two concentric spherical shells with radii $a $ and $b$ in the $D+1$-dimensional global monopole background. The expressions are derived for the Wightman function, the vacuum expectation values of the field square, the vacuum energy density, radial and azimuthal stress components in the region between the shells. A regularization procedure is carried out by making use of the generalized Abel-Plana formula for the series over zeros of combinations of the cylinder functions. This formula allows us to extract from the vacuum expectation values the parts due to a single sphere on background of the global monopole gravitational field, and to present the "interference" parts in terms of exponentially convergent integrals, useful, in particular, for numerical evaluations. The vacuum forces acting on the boundaries are presented as a sum of the self--action and interaction terms. The first one contains well known surface divergences and needs a further regularization. The interaction forces between the spheres are finite for all values $a<b$ and are attractive for a Dirichlet scalar. The asymptotic behavior of the vacuum densities is investigated (i) in the limits $a\to 0$ and $b\to \infty $, (ii) in the limit $a,b\to \infty $ for fixed value $b-a$, and (iii) for small values of the parameter associated with the solid angle deficit in global monopole geometry. We show that in the case (ii) the results for two parallel Robin plates on the Minkowski bulk are rederived to the leading order. 
  We discuss realizations of the SL(2,R) current algebra in the hyperbolic basis using free scalar fields. It has been previously shown by Satoh how such a realization can be used to describe the principal continuous representations of SL(2,R). We extend this work by introducing another realization that corresponds to the principal discrete representations of SL(2,R). We show that in these realizations spectral flow can be interpreted as twisting of a free scalar field. Finally, we discuss how these realizations can be obtained from the BTZ Lagrangian. 
  With a special choice of gauge the operator of the Klein-Fock-Gordon equation in homogeneous electric field respects boost symmetry. Using this symmetry we obtain solutions for the scalar massive field equation in such a background (boost modes in the electric field). We calculate the spectrum of particles created by the electric field, as seen by an accelerated observer at spatial infinity of the right wedge of Minkowski spacetime. It is shown that the spectrum and the total number of created pairs measured by a remote uniformly accelerated observer in Minkowski spacetime are precisely the same as for inertial observers. 
  We present an extension of previous results (hep-th/0105215)on the quantization of general gauge theories within the BRST-antBRST invatiant Lagrangian scheme in general coordinates, namely, we consider the case when the base manifold of fields and antifields is a supermanifold desribed in terms of both bosonic and fermionic variables. 
  Bousso's entropy bound for two-dimensional gravity is investigated in the lightcone gauge. It is shown that due to the Weyl anomaly, the null component of the energy-momentum tensor takes a nonvanishing value, and thus, combined with the conditions that were recently proposed by Bousso, Flanagan and Marolf, a holographic entropy bound similar to Bousso's is expected to hold in two dimensions. A connection of our result to that of Strominger and Thompson is also discussed. 
  We review the most general, local, superconformal boundary conditions for the two-dimensional N=1 and N=2 non-linear sigma models, and analyse them for the N=1 and N=2 supersymmetric WZW models. We find that the gluing map between the left and right affine currents is generalised in a very specific way as compared to the constant Lie algebra automorphisms that are known. 
  We extend the formalism introduced in the paper hep-th/0209050 to compute correlation functions in the AdS/CFT correspondence. We show how the on-shell action of a scalar field in a compactification of AdS space can be obtained by flowing in the space of (non-local) theories with different sizes of the extra-dimension. Our method is relevant in particular for holographic computations in the Randall-Sundrum scenario with one or two branes and it allows the inclusion of brane-localized actions in a systematic way. The method can also be generalized to other backgrounds and does not rely on explicit knowledge of the solutions of the wave-equations. 
  In this note we examine some semiclassical features of D-branes in the SL(2)/U(1) gauged WZW model and determine the small fluctuation spectra for one class of branes. We compare our results with expectations from the CFT side. 
  In this work we construct two classes of exact solutions for the most general time-dependent Dirac Hamiltonian in 1+1 dimensions. Some problems regarding to some formal solutions in the literature are discussed. Finally the existence of a generalized Lewis-Riesenfeld invariant connected with such solutions is discussed. 
  The ``little group'' for massless particles (namely, the Lorentz transformations $\Lambda$ that leave a null vector invariant) is isomorphic to the Euclidean group E2: translations and rotations in a plane. We show how to obtain explicitly the rotation angle of E2 as a function of $\Lambda$ and we relate that angle to Berry's topological phase. Some particles admit both signs of helicity, and it is then possible to define a reduced density matrix for their polarization. However, that density matrix is physically meaningless, because it has no transformation law under the Lorentz group, even under ordinary rotations. 
  This is an expository paper which aims at explaining a physical point of view on the K-theoretic classification of D-branes. We combine ideas of renormalization group flows between boundary conformal field theories, together with spacetime notions such as anomaly cancellation and D-brane instanton effects. We illustrate this point of view by describing the twisted K-theory of the special unitary groups SU(N). 
  We study geometric engineering of four-dimensional N=1 gauge models from M-theory on a seven-dimensional manifold with G_2 holonomy. The manifold is constructed as a K3 fibration over a three-dimensional base space with ADE geometry. The resulting gauge theory is discussed in the realm of (p,q) webs. We discuss how the anomaly cancellation condition translates into a condition on the associated affine ADE Lie algebras. 
  We study various $Sp(2M)$ invariant field equations corresponding to rank $r$ tensor products of the Fock (singleton) representation of $Sp(2M)$. These equations are shown to describe localization on ``branes'' of different dimensions embedded into the generalized space-time $\M_M$ with matrix (i.e., ``central charge'') coordinates. The case of bilinear tensor product is considered in detail. The conserved currents built from bilinears of rank 1 fields in $\M_M$ are shown to satisfy the field equations of the rank 2 fields in $\M_M$. Also, the rank 2 fields in $\M_M$ are shown to be equivalent to the rank 1 fields in $\M_{2M}$. 
  We evaluate the propagator of scalar and spinor in three dimensional quantum electrodynamics with the use of Ward-Identity for soft-photon emission vertex.We work well in position space to treat infrared divergences in our model. Exponentiation of one-photon matrix element yields a full propagator in position space.It has a simple form as free propagator multiplied by quantum correction.And it shows a new type of mass singularity.But this is not an integrable function so that analysis in momentum space is not easy.Term by term integral converges and they have a logarithmic singularity associated with renormalized mass in perturbation theory.Renormalization constant vanishes for weak coupling,which suggests confinement of charged particle.There exsists a critical coupling constant above which the vacuum expectation value of pair condensation is finite. 
  We investigate a large class of supersymmetric magnetic brane solutions supported by U(1) gauge fields in AdS gauged supergravities. We obtain first-order equations in terms of a superpotential. In particular, we find systems which interpolate between AdS_{D-2}\times \Omega^2 (where \Omega^2=S^2 or H^2) in the horizon and AdS_D-type geometry in the asymptotic region, for 4\le D\le 7. The boundary geometry of the AdS_D-type metric is Minkowski_{D-3}\times \Omega^2. This provides smooth supergravity solutions for which the boundary of the AdS spacetime compactifies spontaneously. These solutions indicate the existence of a large class of superconformal field theories in diverse dimensions whose renormalization group flow runs from the UV to the IR fixed point. We show that the same set of first-order equations also admits solutions which are asymptotically AdS_{D-2}\times \Omega^2 but singular at small distance. This implies that the stationary AdS_{D-2}\times \Omega^2 solutions typically lie on the inflection points of the modulus space. 
  We consider a recently proposed two-dimensional Abelian model for a Hodge theory, which is neither a Witten type nor a Schwarz type topological theory. It is argued that this model is not a good candidate for a Hodge theory since, on-shell, the BRST Laplacian vanishes. We show, that this model allows for a natural extension such that the resulting topological theory is of Witten type and can be identified with the twisted N=16, D=2 super Maxwell theory. Furthermore, the underlying basic cohomology preserves the Hodge-type structure and, on-shell, the BRST Laplacian does not vanish. 
  It is shown how to solve the Euclidean equations of motion of a point particle in a general potential and in the presence of a four-Fermi term. The classical action in this theory depends explicitly on a set of four fermionic collective coordinates. The corrections to the classical action due to the presence of fermions are of topological nature in the sense that they depend only on the values of the fields at the boundary points $\tau \to \pm \infty$. As an application, the Sine-Gordon model with a four-Fermi term is solved explicitly and the corrections to the classical action are computed. 
  We pursue the study of string interactions in the PP-wave background and show that the proposal of hep-th/0211188 can be extended to a full supersymmetric vertex. Then we compute some string amplitudes in both the bosonic and fermionic sector, finding agreement with the field theory results at leading order in lambda'. 
  We consider global monopoles in asymptotic de Sitter/ Anti- de Sitter space-time. We present the by our numerical analysis confirmed asymptotic behaviour of the metric and Goldstone field functions. We find that the appearance of horizons in this model depends strongly on the sign and value of the cosmological constant as well as on the value of the gravitational coupling. In Anti-de Sitter (AdS) space, we find that for a fixed value of the cosmological constant, global monopoles without horizons exist only up to a critical value of the gravitational coupling. Moreover, we observe (in contrast to another recent study) that the introduction of a cosmological constant can NOT render a positive mass of the global monopole. 
  Doubly Special Relativity (DSR) is a class of theories of relativistic motion with two observer-independent scales. We investigate the velocity of particles in DSR, defining velocity as the Poisson bracket of position with the appropriate hamiltonian, taking care of the non-trivial structure of the DSR phase space. We find the general expression for four-velocity, and we show further that the three-velocity of massless particles equals 1 for all DSR theories. The relation between the boost parameter and velocity is also clarified. 
  Renormalization group evolution of QCD composite light-cone operators, built from two and more quark and gluon fields, is responsible for the logarithmic scaling violations in diverse physical observables. We analyze spectra of anomalous dimensions of these operators at large conformal spins at weak and strong coupling with the emphasis on the emergence of a dual string picture. The multi-particle spectrum at weak coupling has a hidden symmetry due to integrability of the underlying dilatation operator which drives the evolution. In perturbative regime, we demonstrate the equivalence of the one-loop cusp anomaly to the disk partition function in two-dimensional Yang-Mills theory which admits a string representation. The strong coupling regime for anomalous dimensions is discussed within the two pictures addressed recently, -- minimal surfaces of open strings and rotating long closed strings in AdS background. In the latter case we find that the integrability implies the presence of extra degrees of freedom -- the string junctions. We demonstrate how the analysis of their equations of motion naturally agrees with the spectrum found at weak coupling. 
  In this Reply I present some arguments in favor of the stability of the topological defect composed by global and magnetic monopoles. 
  In this paper the Seiberg-Witten map for a time-dependent background related to a null-brane orbifold is studied. The commutation relations of the coordinates are linear, i.e. it is an example of the Lie algebra type. The equivalence map between the Kontsevich star product for this background and the Weyl-Moyal star product for a background with constant noncommutativity parameter is also studied. 
  Segal proposed ultraquantum commutation relations with two ultraquantum constants hbar' and hbar'' besides Planck's quantum constant hbar with a variable i. The Heisenberg quantum algebra is a contraction - in a more general sense than that of Inonu and Wigner - of the Segal ultraquantum algebra. The usual constant i arises as a vacuum order-parameter in the quantum limit where hbar' and hbar'' approach zero. One physical consequence is a discrete spectrum for canonical variable and space-time coordinates. Another is an interconvention of time and energy accompnying space-time meltdown (disorder), with a fundamental conversion factor of some kilograms of energy per second. 
  We compute the meson spectrum of an N=2 super Yang-Mills theory with fundamental matter from its dual string theory on AdS_5 x S_5 with a D7-brane probe. For scalar and vector mesons with arbitrary R-charge the spectrum is computed in closed form by solving the equations for D7-brane fluctuations; for matter with non-zero mass m_q it is discrete, exhibits a mass gap of order m_q / sqrt(g_s N) and furnishes representations of SO(5) even though the manifest global symmetry of the theory is only SO(4). The spectrum of mesons with large spin J is obtained from semiclassical, rotating open strings attached to the D7-brane. It displays Regge-like behaviour for J << sqrt(g_s N), whereas for J >> sqrt(g_s N) it corresponds to that of two non-relativistic quarks bound by a Coulomb potential. Meson interactions, baryons and `giant gauge bosons' are briefly discussed. 
  We introduce the brane-bulk interaction to discuss a limitation of the cosmological Cardy-Verlinde formula which is useful for the holographic description of brane cosmology. In the presence of the brane-bulk interaction, we cannot find the entropy representation of the first Friedmann equation (the cosmological Cardy-Verlinde formula). In the absence of the interaction, the cosmological Cardy-Verlinde formula is established even for the time-dependent charged AdS background. Hence, if there exists a dynamic exchange of energy between the brane and the bulk (that is, if $\tilde T^t~_y \not=0$), we cannot achieve the cosmological holographic principle on the brane. 
  Bound and scattering state Schr\"odinger functions of nonrelativistic quantum mechanics as representation matrix elements of space and time are embedded into residual representations of spacetime as generalizations of Feynman propagators. The representation invariants arise as singularities of rational representation functions in the complex energy and complex momentum plane. The homogeneous space $GL(2,C)/U(2)$ with rank 2, the orientation manifold of the unitary hypercharge-isospin group, is taken as model of nonlinear spacetime. Its representations are characterized by two continuous invariants whose ratio will be related to gauge field coupling constants as residues of the related representation functions. Invariants of product representations define unitary Poincar\'e group representations with masses for free particles in tangent Minkowski spacetime. 
  Some of the recent important developments in understanding string/ gauge dualities are based on the idea of highly symmetric motion of ``string solitons'' in $AdS_5\times S^5$ geometry originally suggested by Gubser, Klebanov and Polyakov. In this paper we study symmetric motion of certain string configurations in so called Pilch-Warner geometry. The two-form field $A_2$ breaks down the supersymmetry to $\mathcal{N}=1$ but for the string configurations considered in this paper the classical values of the energy and the spin are the same as for string in $AdS\times S^5$. Although trivial at classical level, the presence of NS-NS antisymmetric field couples the fluctuation modes that indicates changes in the quantum corrections to the energy spectrum. We compare our results with those obtained in the case of pp-wave limit in hep-th/0206045. 
  In the SO(N_c) gauge theory with N_f quarks for N_f=N_c-2, its instanton effects indicate the signal of dynamical flavor symmetry breakdown of SU(N_f) to SO(N_f), which is not described by the conventional "magnetic" degree's of freedom. It is argued that this breaking is well described by our effective superpotential consisting of "electric" quarks and gluons instead of monopoles of SO(N_c). The low-energy particles include the Nambu-Goldstone superfields associated with this breakdown. The proposed superpotential is found to exhibit the holomorphic decoupling property and the anomaly-matching property on a residual chiral U(1) symmetry. 
  The spectrum and degeneracies of the Dirac operator are analysed on compact coset spaces when there is a non-zero homogeneous background gauge field which is compatible with the symmetries of the space, in particular when the gauge field is derived from the spin-connection. It is shown how the degeneracy of the lowest Landau level in the recently proposed higher dimensional quantum Hall effect is related to the Atiyah-Singer index theorem for the Dirac operator on a compact coset space. 
  Assuming spherical symmetry we analyse the dynamics of an inhomogeneous dark radiation vaccum on a Randall and Sundrum 3-brane world. Under certain natural conditions we show that the effective Einstein equations on the brane form a closed system. On a de Sitter brane and for negative dark energy density we determine exact dynamical and inhomogeneous solutions which depend on the brane cosmological constant, on the dark radiation tidal charge and on its initial configuration. We also identify the conditions leading to the formation of a singularity or of regular bounces inside the dark radiation vaccum. 
  We review here the quantum mechanics of some noncommutative theories in which no state saturates simultaneously all the non trivial Heisenberg uncertainty relations. We show how the difference of structure between the Poisson brackets and the commutators in these theories generically leads to a harmonic oscillator whose positions and momenta mean values are not strictly equal to the ones predicted by classical mechanics.   This raises the question of the nature of quasi classical states in these models. We propose an extension based on a variational principle. The action considered is the sum of the absolute values of the expressions associated to the non trivial Heisenberg uncertainty relations. We first verify that our proposal works in the usual theory i.e we recover the known Gaussian functions. Besides them, we find other states which can be expressed as products of Gaussians with specific hyper geometrics.   We illustrate our construction in two models defined on a four dimensional phase space: a model endowed with a minimal length uncertainty and the non commutative plane. Our proposal leads to second order partial differential equations. We find analytical solutions in specific cases. We briefly discuss how our proposal may be applied to the fuzzy sphere and analyze its shortcomings. 
  Continuing on our previous work, we consider a class of higher dimensional brane models with the topology of $AdS_{D_1+1} \times \Sigma$, where $\Sigma$ is a one-parameter compact manifold and two branes of codimension 1 are located at the orbifold fixed points. We consider a set-up where such a solution arises from Einstein-Yang-Mills theory and evaluate the one-loop effective potential induced by gauge fields and by a generic bulk scalar field. We show that this type of brane models resolves the gauge hierarchy between the Planck and electroweak scales through redshift effects due to the warp factor $a=e^{-\pi kr}$. The value of $a$ is then fixed by minimizing the effective potential. We find that, as in the Randall Sundrum case, the gauge field contribution to the effective potential stabilises the hierarchy without fine-tuning as long as the laplacian $\Delta_\Sigma$ on $\Sigma$ has a zero eigenvalue. Scalar fields can stabilise the hierarchy depending on the mass and the non-minimal coupling. We also address the quantum self-consistency of the solution, showing that the classical brane solution is not spoiled by quantum effects. 
  The effects of quantum corrections to a conformally invariant scalar field theory on a curved manifold of positive constant curvature with boundary are considered in the context of a renormalisation procedure. The renormalisation of the theory to second order in the scalar self-coupling pursued herein involves explicit calculations of up to third loop-order and reveals that, in addition to the renormalisation of the scalar self-coupling and scalar field, the removal of all divergences necessitates the introduction of conformally non-invariant counterterms proportional to $ R\Phi^2$ and $ K\Phi^2$ in the bare scalar action as well as counterterms proportional to $ RK^2$, $ R^2$ and $ RK$ in the gravitational action. The substantial backreaction effects and their relevance to the renormalisation procedure are analysed. 
  Dirac showed that the existence of magnetic monopoles would imply quantization of electric charge. I discuss the converse, and propose two `principles of completeness' which I illustrate with various examples. 
  We suggest a new cosmological scenario which naturally guarantees the smallness of scalar masses and VEVs, without invoking supersymmetry or any other (non-gravitationaly coupled) new physics at low energies. In our framework, the scalar masses undergo discrete jumps due to nucleation of closed branes during (eternal) inflation. The crucial point is that the step size of variation decreases in the direction of decreasing scalar mass. This scenario yields exponentially large domains with a distribution of scalar masses, which is sharply peaked around a hierarchically small value of the mass. This value is the "attractor point" of the cosmological evolution. 
  In this paper we study the midpoint structure of the algebra of open strings from the standpoint of the operator/Moyal formalism. We construct a split string description for the continuous Moyal product of hep-th/0202087, study the breakdown of associativity in the star algebra, and identify in infinite sequence of new (anti)commutative coordinates for the star product in in the complex plane. We also explain how poles in the open string non(anti)commutativity parameter correspond to certain ``null'' operators which annihilate the vertex, implying that states proportional to such operators tend to have vanishing star product with other string fields. The existence of such poles, we argue, presents an obstruction to realizing a well-defined formulation of the theory in terms of a Moyal product. We also comment on the interesting, but singular, representation $L_0$ which has appeared prominently in the recent studies of Bars {\it et al}. 
  We argue that the Dirac-Born-Infeld (DBI) action coupled to a tachyon, that is known to reproduce some aspects of open string dynamics, can be obtained from open string theory in a certain limit, which generalizes the limit leading to the usual DBI action. This helps clarify which aspects of the full open string theory are captured by this action. 
  We develop the derivative expansion of the one-loop ${\cal N}=4$ SYM effective action depending both on ${\cal N}=2$ vector multiplet and on hypermultiplet background fields. We get a new derivation of the complete ${\cal N}=4$ supersymmetric low-energy effective action obtained in hep-th/0111062 and find subleading corrections to it. A problem of ${\cal N}=4$ supersymmetry of the results is discussed. Using the formalism of ${\cal N}=2$ harmonic superspace and exploring on-shell hidden ${\cal N}=2$ supersymmetry of ${\cal N}=4$ SYM theory we construct the appropriate hypermultiplet-depending contributions. The hidden ${\cal N}=2$ supersymmetry requirements allow to get a leading, in hypermultiplet derivatives, part of the correct ${\cal N}=4$ supersymmetric functional containing $F^{8}$ among the component fields. 
  The wave function renormalization constant $Z$, the probability to find the bare particle in the physical particle, usually satisfies the unitarity bound $0 \leq Z \leq 1$ in field theories without negative metric states. This unitarity bound implies the positivity of the anomalous dimension of the field in the one-loop approximation. In nonlinear sigma models, however, this bound is apparently broken because of the field dependence of the canonical momentum. The contribution of the bubble diagrams to the anomalous dimension can be negative, while the contributions from more than two particle states satisfies the positivity of the anomalous dimension as expected.  We derive the genuine unitarity bound of the wave function renormalization constant. 
  We present a phase-space analysis of cosmology containing multiple scalar fields with positive and negative exponential potentials. We show that there exist power-law multi-kinetic-potential scaling solutions for sufficiently flat positive potentials or steep negative potentials. The former is the unique late-time attractor and the well-known assisted inflationary solution, but the later is never unstable in an expanding universe. Moreover, for steep negative potentials there exist a kinetic-dominated regime in which each solution is a late-time attractor. We briefly discuss the physical consequences of these results. 
  Nonlinear field equations for totally symmetric bosonic massless fields of all spins in any dimension are presented. 
  The recent claim in hep-th/0302225 that, contrary to all previous work, massive charged s=2 fields propagate causally is false. 
  We address solutions of brane-world with cosmological constant $\lambda$ by introducing the dilaton in 5d bulk, and we examine the localization of graviton, gauge bosons and dilaton. For those solutions, we find that both graviton and gauge bosons can be trapped for either sign, positive or negative, and wide range of $\lambda$ due to the non-trivial dilaton. While the dilaton can not be trapped on the brane. 
  We reconstruct the canonical operators $p_i,q_i$ of the quantum closed Toda chain in terms of Sklyanin's separated variables. 
  We perform a systematic study of the one-loop renormalizability of all Poisson-Lie T-dualizable $\si$-models with two-dimensional targets. We show that whatever Drinfeld double and whatever matrix of coupling constants we consider the corresponding $\si$-model is always one-loop renormalizable in the strict field theoretical sense. Moreover, in all cases, the RG flow in the space of the coupling constants is compatible with the Poisson-Lie T-duality. 
  A new formulation of simple D=4 supergravity in terms of the geometry of superspace is presented. The formulation is derived from the gauge theory of the inhomogeneous orthosymplectic group IOSp(3,1|4) on a (4,4)-dimensional base supermanifold by imposing constraints and taking a limit. Both the constraints and the limiting procedure have a clear {\it a priori} physical motivation, arising from the relationship between IOSp(3,1|4) and the super Poincar\'{e} group. The construction has similarities with the space-time formulation of Newtonian gravity. 
  We study the quantization problem of relativistic scalar and spinning particles interacting with a radiation electromagnetic field by using the path integral and the external source method. The spin degrees of freedom are described in terms of Grassmann variables and the Feynman kernel is obtained through functional integration on both Bose and Fermi variables. We provide rigourous proof that the Feynman amplitudes are only determined by the classical contribution and we explicitly evaluate the propagators. 
  From the viewpoint of the singular quantum mechanics the effect of the energy-dependent coupling constant for $\delta$-function potential is examined. The energy-dependence of the coupling constant naturally generates the time-derivative in the boundary condition of the Euclidean propagator. This is explicitly confirmed by making use of the simple 1d model. The result is applied to the linearized gravity fluctuation equation for the brane-world scenario with 4d induced gravity. Our approach generates $5d$ Newton potential at a certain intermediate range of distance between two test massive sources. For other range of distance 4d Newton potential is recovered. 
  We consider a blown-up 3-brane, with the resulting geometry R^(3,1) \times S^(N-1), in an infinite-volume bulk with N > 2 extra dimensions. The action on the brane includes both an Einstein term and a cosmological constant. Similar setups have been proposed both to reproduce 4-d gravity on the brane, and to solve the cosmological constant problem. Here we obtain a singularity-free solution to Einstein's equations everywhere in the bulk and on the brane, which allows us to address these question explicitely. One finds, however, that the proper volume of S^(N-1) and the cosmological constant on the brane have to be fine-tuned relatively to each other, thus the cosmological constant problem is not solved. Moreover the scalar propagator on the brane behaves 4-dimensionally over a phenomenologically acceptable range only if the warp factor on the brane is huge, which aggravates the Weak Scale - Planck Scale hierarchy problem. 
  We study the direct interaction of an antisymmetric Kalb-Ramond field with a scalar particle derived from a gauge principle. The method outlined in this paper to define a covariant derivative is applied to a simple model leading to a linear coupling between the fields. Although no conserved Noether charge exists, a conserved topological current comes out from the antisymmetry properties of the Kalb-Ramond field. Some interesting features of this current are pointed out. 
  Derivative corrections to the Wess--Zumino couplings of open-string effective actions are computed at all orders in derivatives, taking the open-string metric into account. This leads to a set of deformed star-products beyond the Seiberg--Witten limit, and allows to reinterpret the couplings in terms of a deformed integration prescription along a Wilson line in the non-commutative set-up. Moreover, the recursive definition of the star-products induces deformations of U(1) non-commutative Yang--Mills theory. 
  In this paper it is shown that the entropy of the black hole horizon in the Achucarro-Ortiz spacetime, which is the most general two-dimensional black hole derived from the three-dimensional rotating BTZ black hole, can be described by the Cardy-Verlinde formula. The latter is supposed to be an entropy formula of conformal field theory in any dimension. 
  We consider four-dimensional N=2 supersymmetric gauge theories with gauge group U(N) on R^3 x S^1, in the presence of a classical superpotential. The low-energy quantum superpotential is obtained by simply replacing the adjoint scalar superfield in the classical superpotential by the Lax matrix of the integrable system that underlies the 4d field theory. We verify in a number of examples that the vacuum structure obtained in this way matches precisely that in 4d, although the degrees of freedom that appear are quite distinct. Several features of 4d field theories, such as the possibility of lifting vacua from U(N) to U(tN), become particularly simple in this framework. It turns out that supersymmetric vacua give rise to a reduction of the integrable system which contains information about the field theory but also about the Dijkgraaf-Vafa matrix model. The relation between the matrix model and the quantum superpotential on R^3 x S^1 appears to involve a novel kind of mirror symmetry. 
  We study light-cone gauge quantization of IIB strings in AdS_5 \times S^5 for small radius in Poincare coordinates. A picture of strings made up of noninteracting bits emerges in the zero radius limit. In this limit, each bit behaves like a superparticle moving in the AdS_5 \times S^5 background, carrying appropriate representations of the super conformal group PSU(2,2|4). The standard Hamiltonian operator which causes evolution in the light-cone time has continuous eigenvalues and provides a basis of states which is not suitable for comparing with the dual super Yang-Mills theory. However, there exist operators in the light-cone gauge which have discrete spectra and can be used to label the states. We obtain the spectrum of single bit states and construct multi-bit states in this basis. There are difficulties in the construction of string states from the multi-bit states, which we discuss. A non-zero value of the radius introduces interactions between the bits and the spectrum of multi-bit states gets modified. We compute the leading perturbative corrections at small radius for a few simple cases. Potential divergences in the perturbative corrections, arising from strings near the boundary, cancel. This encourages us to believe that our perturbative treatment could provide a framework for a rigorous and detailed testing of the AdS/CFT conjecture, once the difficulties in the construction of string states are resolved. 
  We study differential and integral relations for the quantum Jost solutions associated with an integrable derivative nonlinear Schrodinger (DNLS) model. By using commutation relations between such Jost solutions and the basic field operators of DNLS model, we explicitly construct first few quantum conserved quantities of this system including its Hamiltonian. It turns out that this quantum Hamiltonian has a new kind of coupling constant which is quite different from the classical one. This modified coupling constant plays a crucial role in our comparison between the results of algebraic and coordinate Bethe ansatz for the case of DNLS model. We also find out the range of modified coupling constant for which the quantum $N$-soliton state of DNLS model has a positive binding energy. 
  All purely bosonic supersymmetric solutions of minimal gauged supergravity in five dimensions are classified. The solutions fall into two classes depending on whether the Killing vector constructed from the Killing spinor is time-like or null. When it is timelike, the solutions are determined by a four-dimensional Kahler base-manifold, up to an anti-holomorphic function, and generically preserve 1/4 of the supersymmetry. When it is null we provide a precise prescription for constructing the solutions and we show that they generically preserve 1/4 of the supersymmetry. We show that $AdS_5$ is the unique maximally supersymmetric configuration. The formalism is used to construct some new solutions, including a non-singular deformation of $AdS_5$, which can be uplifted to obtain new solutions of type IIB supergravity 
  Gauge fields are special in the sense that they are invariant under gauge transformations and \emph{``ipso facto''} they lead to problems when we try quantizing them straightforwardly. To circumvent this problem we need to specify a gauge condition to fix the gauge so that the fields that are connected by gauge invariance are not overcounted in the process of quantization. The usual way we do this in the light-front is through the introduction of a Lagrange multiplier, $(n\cdot A)^{2}$, where $n_{\mu}$ is the external light-like vector, i.e., $n^{2}=0$, and $A_{\mu}$ is the vector potential. This leads to the usual light-front propagator with all the ensuing characteristics such as the prominent $(k\cdot n)^{-1}$ pole which has been the subject of much research. However, it has been for long recognized that this procedure is incomplete in that there remains a residual gauge freedom still to be fixed by some ``ad hoc'' prescription, and this is normally worked out to remedy some unwieldy aspect that emerges along the way. In this work we propose \emph{two} Lagrange multipliers with distinct coefficients for the light-front gauge that leads to the correctly defined propagator with no residual gauge freedom left. This is accomplished via $(n\cdot A)^2+(\partial \cdot A)^2$ terms in the Lagrangian density. These lead to a well-defined and exact though Lorentz non invariant propagator. 
  We perform the matching required to compute the leading effective boundary contribution to the QED lagrangian in the presence of a conducting surface, once the electron is integrated out. Our result resolves a confusion in the literature concerning the interpretation of the leading such correction to the Casimir energy. It also provides a useful theoretical laboratory for brane-world calculations in which kinetic terms are generated on the brane, since a lot is known about QED near boundaries. 
  In this paper we give explicit first order Lagrangian formulation for mixed symmetry tensor fields \Phi_{[\mu\nu],\alpha}, T_{[\mu\nu\alpha],\beta} and R_{[\mu\nu],[\alpha\beta]}. We show that such Lagrangians could be written in a very suggestive form similar to the well known tetrad formalism in gravity. Such description could simplify the investigations of possible interactions for these fields. Some examples of interactions are given. 
  The large-group behavior of the non-local two dimensional generalized Yang-Mills theories (nlgYM$_2$'s) on arbitrary closed non-orientable surfaces is investigated. It is shown that all order of $\phi^{2k}$ model of these theories have thired order phase transition only on projective plane (RP$^2$). Also the phase structure of $\phi^2 + \frac{\gamma}{4}\phi^4$ model of nlgYM$_2$ is studied and it is found that for $\gamma >0$, this model has third order phase transition only on RP$^2$ and for $\gamma<0$ it has third order phase transition on any closed non-orientable surfaces except RP$^2$ and Klein bottel. 
  We look at some dynamic geometries produced by scalar fields with both the "right" and the "wrong" sign of the kinetic energy. We start with anisotropic homogeneous universes with closed, open and flat spatial sections. A non-singular solution to the Einstein field equations representing an open anisotropic universe with the ghost field is found. This universe starts collapsing from $t \to -\infty$ and then expands to $t \to \infty$ without encountering singularities on its way. We further generalize these solutions to those describing inhomogeneous evolution of the ghost fields. Some interesting solutions with the plane symmetry are discussed. These have a property that the same line element solves the Einstein field equations in two mirror regions $|t|\geq z$ and $|t|\leq z$, but in one region the solution has the \emph{right} and in the other, the \emph{wrong} signs of the kinetic energy. We argue, however, that a physical observer can not reach the mirror region in a finite proper time. Self-similar collapse/expansion of these fields are also briefly discussed. 
  Numerical methods have allowed the construction of vacuum non-uniform strings. For sufficient non-uniformity, the local geometry about the minimal horizon sphere (the "waist") was conjectured to be a cone metric. We are able to test this conjecture explicitly giving strong evidence in favour of it. We also show how to extend the conjecture to weakly charged strings. 
  We find self-adjoint extensions of the rational Calogero model in presence of the harmonic interaction. The corresponding eigenfunctions may describe the near-horizon quantum states of certain types of black holes. 
  In this paper we show that the entropy of cosmological horizon in 4-dimensional Topological Kerr-Newman-de Sitter spaces can be described by the Cardy-Verlinde formula, which is supposed to be an entropy formula of conformal field theory in any dimension. Furthermore, we find that the entropy of black hole horizon can also be rewritten in terms of the Cardy-Verlinde formula for these black holes in de Sitter spaces, if we use the definition due to Abbott and Deser for conserved charges in asymptotically de Sitter spaces. Such result presume a well-defined dS/CFT correspondence, which has not yet attained the credibility of its AdS analogue. 
  We show that some higher derivative theories have a BRST symmetry. This symmetry is due to the higher derivative structure and is not associated to any gauge invariance. If physical states are defined as those in the BRST cohomology then the only physical state is the vacuum. All negative norm states, characteristic of higher derivative theories, are removed from the physical sector. As a consequence, unitarity is recovered but the S-matrix is trivial. We show that a class of higher derivative quantum gravity theories have this BRST symmetry so that they are consistent as quantum field theories. Furthermore, this BRST symmetry may be present in both relativistic and non-relativistic systems. 
  In conventional gauge theory, a charged point particle is described by a representation of the gauge group. If we propagate the particle along some path, the parallel transport of the gauge connection acts on this representation. The Lagrangian density of the gauge field depends on the curvature of the connection which can be calculated from the holonomy around (infinitesimal) loops. For Abelian symmetry groups, say G=U(1), there exists a generalization, known as p-form electrodynamics, in which (p-1)-dimensional charged objects can be propagated along p-surfaces and in which the Lagrangian depends on a generalized curvature associated with (infinitesimal) closed p-surfaces. In this article, we use Lie 2-groups and ideas from higher category theory in order to formulate a discrete gauge theory which generalizes these models at the level p=2 to possibly non-Abelian symmetry groups. An important feature of our model is that it involves both parallel transports along paths and generalized transports along surfaces with a non-trivial interplay of these two types of variables. Our main result is the geometric picture, namely the assignment of non-Abelian quantities to geometrical objects in a coordinate free way. We construct the precise assignment of variables to the curves and surfaces, the generalized local symmetries and gauge invariant actions and we clarify which structures can be non-Abelian and which others are always Abelian. A discrete version of connections on non-Abelian gerbes is a special case of our construction. Even though the motivation sketched so far suggests applications mainly in string theory, the model presented here is also related to spin foam models of quantum gravity and may in addition provide some insight into the role of centre monopoles and vortices in lattice QCD. 
  We briefly introduce the conception on Euler-Lagrange cohomology groups on a symplectic manifold $(\mathcal{M}^{2n}, \omega)$ and systematically present the general form of volume-preserving equations on the manifold from the cohomological point of view. It is shown that for every volume-preserving flow generated by these equations there is an important 2-form that plays the analog role with the Hamiltonian in the Hamilton mechanics. In addition, the ordinary canonical equations with Hamiltonian $H$ are included as a special case with the 2-form $\frac{1}{n-1} H \omega$. It is studied the other volume preserving systems on $({\cal M}^{2n}, \omega)$. It is also explored the relations between our approach and Feng-Shang's volume-preserving systems as well as the Nambu mechanics. 
  Schwarzschild black hole being thermodynamically unstable, corrections to its entropy due to small thermal fluctuations cannot be computed. However, a thermodynamically stable Schwarzschild solution can be obtained within a cavity of any finite radius by immersing it in an isothermal bath. For these boundary conditions, classically there are either two black hole solutions or no solution. In the former case, the larger mass solution has a positive specific heat and hence is locally thermodynamically stable. We find that the entropy of this black hole, including first order fluctuation corrections is given by: ${\cal S} = S_{BH} - \ln[\f{3}{R} (S_{BH}/4\p)^{1/2} -2]^{-1} + (1/2) \ln(4\p)$, where $S_{BH}=A/4$ is its Bekenstein-Hawking entropy and $R$ is the radius of the cavity. We extend our results to four dimensional Reissner-Nordstr\"om black holes, for which the corresponding expression is: ${\cal S} = S_{BH} - \f{1}{2} \ln [ {(S_{BH}/\p R^2) ({3S_{BH}}/{\p R^2} - 2\sqrt{{S_{BH}}/{\p R^2 -\a^2}}) \le(\sqrt{{S_{BH}}/{\p R^2}} - \a^2 \ri)}/ {\le({S_{BH}}/{\p R^2} -\a^2 \ri)^2} ]^{-1} +(1/2)\ln(4\p).$ Finally, we generalise the stability analysis to Reissner-Nordstr\"om black holes in arbitrary spacetime dimensions, and compute their leading order entropy corrections. In contrast to previously studied examples, we find that the entropy corrections in these cases have a different character. 
  We present a new form of solution to the quantum Knizhnik-Zamolodchikov equation on level -4 in a special case corresponding to the Heisenberg XXX spin chain. Our form is equivalent to the integral representation obtained by Jimbo and Miwa in 1996 . An advantage of our form is that it is reduced to the product of single integrals. This fact is deeply related to a cohomological nature of our formulae. Our approach is also based on the deformation of hyper-elliptic integrals and their main property -- deformed Riemann bilinear relation. Jimbo and Miwa also suggested a nice conjecture which relates solution of the qKZ on level -4 to any correlation function of the XXX model. This conjecture together with our form of solution to the qKZ makes it possible to prove a conjecture that any correlation function of the XXX model can be expressed in terms of the Riemann zeta-function at odd arguments and rational coefficients suggested in our previous papers. This issue will be discussed in our next publication. 
  We implement in systems of fermions the formalism of pseudoclassical paths that we recently developed for systems of bosons and show that quantum states of fermionic fields can be described, in the Heisenberg picture, as linear combinations of randomly distributed paths that do not interfere between themselves and obey classical Dirac equations. Every physical observable is assigned a time-dependent value on each path in a way that respects the anticommutative algebra between quantum operators and we observe that these values on paths do not necessarily satisfy the usual algebraic relations between classical observables. We use these pseudoclassical paths to define the dynamics of quantum fluctuations in systems of fermions and show that, as we found for systems of bosons, the dynamics of fluctuations of a wide class of observables that we call "collective" observables can be approximately described in terms of classical stochastic concepts. Finally, we apply this formalism to describe the dynamics of local fluctuations of globally conserved fermion numbers. 
  We analyze proton decay via dimension six operators in certain GUT-like models derived from Type IIA orientifolds with $D6$-branes. The amplitude is parametrically enhanced by a factor of $\alpha_{GUT}^{-1/3}$ relative to the coresponding result in four-dimensional GUT's. Nonetheless, even assuming a plausible enhancement from the threshold corrections, we find little overall enhancement of the proton decay rate from dimension six operators, so that the predicted lifetime from this mechanism remains close to $10^{36}$ years. 
  Extending a computation which appeared recently in hep-th/0301173, we compute the transmission and reflection coefficients for massless uncharged scalars and gravitational waves scattered by d>=4 Schwarzschild or d=4 Reissner-Nordstrom black holes, in the limit of large imaginary frequencies. The transmission coefficient has an interpretation as the "greybody factor" which determines the spectrum of Hawking radiation. The result has an interesting structure and we speculate that it may admit a simple dual description; curiously, for Reissner-Nordstrom the result suggests that this dual description should involve both the inner and outer horizons. We also discuss some numerical evidence in favor of the formulas of hep-th/0301173. 
  Particle physics has for some time made extensive use of extended field configuations such as solitons, instantons, and sphalerons. However, no direct use has yet been made of the quite extensive literature on ``localized wave'' configurations developed by the engineering, optics, and mathematics communities. In this article I will exhibit a particularly simple ``physical wavelet'' -- it is a Lorentz covariant classical field configuration that lives in physical Minkowski space. The field is everwhere finite and nonsingular, and has quadratic falloff in both space and time. The total energy is finite, the total action is zero, and the field configuration solves the wave equation. These physical wavelets can be constructed for both complex and real scalar fields, and can be extended to the Maxwell and Yang-Mills fields in a straightforward manner. Since these wavelets are finite energy, they are guaranteed to be classically present at finite temperature; since they are zero action, they can contribute to the quantum mechanical path integral at zero ``cost''. 
  We study quantization via star products. We investigate a quantization scheme in which a quantum theory is described entirely in terms of the function space without reference to a Hilbert space, unlike the formulation employing the Wigner functions. The associative law plays an essential role in excluding the unwanted solutions to the stargen-value equation. This is demonstrated explicitly with the $D$-dimensional harmonic oscillator. 
  Three-loop quantum corrections to the effective action are calculated for N=1 supersymmetric electrodynamics, regularized by higher derivatives. Using the obtained results we investigate the anomaly puzzle in the considered model. 
  We point out that the solution of $(4+n)$-dimensional gravity coupled to the dilaton and an $n$-form field strength can give rise to a flat 4-dimensional universe (with a scale factor) of the type proposed recently under time dependent compactifications. The compact internal spaces could be hyperbolic, flat or spherical and the solution is identical to the space-like two brane or S2-brane. As has been shown previously for SM2 solution with a fixed field strength we show that for $n=7$ (where the dilaton is vanishing and with a general field strength), 6 the corresponding SM2 and SD2 solutions can give accelerating cosmologies in Einstein frame for both hyperbolic and flat internal spaces, thereby meeting the challenge of obtaining such a solution from M/String theory compactifications. 
  We investigate classical dynamics of the bosonic string in the background metric, antisymmetric and dilaton fields. We use canonical methods to find Hamiltonian in terms of energy-momentum tensor components. The later are secondary constraints of the theory. Due to the presence of the dilaton field the Virasoro generators have nonlinear realization. We find that, in the curve space-time, opposite chirality currents do not commute. As a consequence of the two-dimensional general covariance, the energy-momentum tensor components satisfy two Virasoro algebras, even in the curve space-time. We obtain new gauge symmetry which acts on both world-sheet and space-time variables, and includes world-sheet Weyl transformation. We emphasize that background antisymmetric and dilaton fields are the origin of space-time torsion and space-time nonmetricity, respectively. 
  In the present article, we study the space-time geometry felt by probe bosonic string moving in antisymmetric and dilaton background fields. This space-time geometry we shall call the stringy geometry. In particular, the presence of the antisymmetric field leads to the space-time torsion, and the presence of the dilaton field leads to the space-time nonmetricity. We generalize the geometry of surfaces embedded in space-time to the case when torsion and nonmetricity are present. We define the mean extrinsic curvature for Minkowski signature and introduce the concept of mean torsion. Its orthogonal projection defines the dual mean extrinsic curvature. In this language, one field equation is just the equality of mean extrinsic curvature and dual mean extrinsic curvature, which we call self-duality relation. In the torsion and nonmetricity free case, the world-sheet is a minimal surface, specified by the requirement that mean extrinsic curvature vanishes. Generally, it is stringy self-dual (anti self-dual) surface. In the presence of the dilaton field, which breaks conformal invariance, the conformal factor which connects intrinsic and induced metrics, is determined as a function of the dilaton field itself. We also derive the integration measure for the space-time with stringy nonmetricity. 
  We propose a new reduction mechanism which allows one to construct n-particle (super)conformal theories with pairwise interaction starting from a composite system involving n(n-1)/2+1 copies of the ordinary (super)conformal mechanics. Applications of the scheme include an N=4 superconformal extension for a complexification of the Calogero model and a D(2,1|\alpha)-invariant n-particle system. 
  A prescription for calculating low-energy one-loop higher-mass dimensional effective Lagrangians for non-Abelian field theories is constructed in the spirit of quasilocal background field method. Basis of Lorentz and gauge-invariant monomials of similar mass-dimensions acting as building blocks are matrix-multiplied in a specified order (usually dictated by a permutation of tensorial indices) generating the much needed invariants. The same set of building blocks is used to generate higher-order corrections for a specific mass-dimension. Though the gauge group, the spacetime dimensions, the order of corrections that can be included, and the mass-dimensions that can be formed are all kept arbitrary in the prescription, we constructed basis invariants from 3 up to 12 mass-dimensions to accommodate higher-order corrections up to fourth-order. With these basis, we pursued solving the zeroth-order corrections leading to invariants from 2 up to 16 mass-dimensions, for first-order from 4 up to 8 mass-dimensions, second and third order corrections from 6 up to 8 mass-dimensions. As a result, we have reproduced the zeroth-order corrections showing dependence on the covariant derivative of the background matrix potential. Previous calculation was done up to 12 mass-dimensions but this dependence was not shown in closed form. For higher-order corrections, the case for 4 up to 6 mass-dimensions are also reproduced. Finally, we calculated the case for 8 mass-dimensions which is reduced only by exploiting the antisymmetry of the fieldstrength tensor and the freedom to throw away total derivatives. 
  M-theory compactifies on a seven-dimensional time-dependent hyperbolic or flat space to a four-dimensional FLRW cosmology undergoing a period of accelerated expansion in Einstein conformal frame. The strong energy condition is violated by the scalar fields produced in the compactification, as is necessary to evade the no-go theorem for time-independent compactifications. The four-form field strength of eleven-dimensional supergravity smoothly switches on during the period of accelerated expansion in hyperbolic compactifications, whereas in flat compactifications, the three-form potential smoothly changes its sign. For small acceleration times, this behaviour is like a phase transition of the three-form potential, during which the cosmological scale factor approximately doubles. 
  We apply the Cornwall-Jackiw-Tomboulis (CJT) formalism to the scalar $\lambda \phi^{4}$ theory in canonical-noncommutative spacetime. We construct the CJT effective potential and the gap equation for general values of the noncommutative parameter $\theta_{\mu\nu}$. We observe that under the hypothesis of translational invariance, which is assumed in the effective potential construction, differently from the commutative case ($\theta_{\mu\nu}= 0$), the renormalizability of the gap equation is incompatible with the renormalizability of the effective potential. We argue that our result, is consistent with previous studies suggesting that a uniform ordered phase would be inconsistent with the infrared structure of canonical noncommutative theories. 
  I suggest wave equations for the scalar, pseudoscalar, vector, and pseudovector fields with different masses for spin zero and one states. Tensor, matrix, and quaternion formulations of fields with two mass and spin states are considered. This is the generalization of the Dirac-K\"ahler equation on the case of different masses of fields with spin one and zero. The equation matrices obtained are simple linear combinations of matrix elements in the 16-dimensional space. Spin projection operators and solutions of equations (for spin one) in the form of matrix-dyads are obtained. The canonical quantization of fields under consideration is studied. The anomalous interaction of the scalar, pseudoscalar, vector, and pseudovector fields with the external electromagnetic field is considered. Three constants which characterize the anomalous magnetic moment and quadrupole electric moment of a particle are introduced. 
  We propose a set of conventional Bethe Ansatz equations and a corresponding expression for the eigenvalues of the transfer matrix for the open spin-1/2 XXZ quantum spin chain with nondiagonal boundary terms, provided that the boundary parameters obey a certain linear relation. 
  We propose that the euclidean bilocal collective field theory of critical large-N vector models provides a complete definition of the proposed dual theory of higher spin fields in anti de-Sitter spaces. We show how this bilocal field can be decomposed into an infinite number of even spin fields in one more dimension. The collective field has a nontrivial classical solution which leads to a O(N) thermodynamic entropy characteristic of the lower dimensional theory, as required by general considerations of holography. A subtle cancellation of the entropy coming from the bulk fields in one higher dimension with O(1) contributions from the classical solution ensures that the subleading terms in thermodynamic quantities are of the expected form. While the spin components of the collective field transform properly under dilatational, translational and rotational isometries of $AdS$, special conformal transformations mix fields of different spins indicating a need for a nonlocal map between the two sets of fields. We discuss the nature of the propagating degrees of freedom through a hamiltonian form of collective field theory and argue that nonsinglet states which are present in an euclidean version are related to nontrivial backgrounds. 
  Four dimensional N=2 supergravity has regular, stationary, asymptotically flat BPS solutions with intrinsic angular momentum, describing bound states of separate extremal black holes with mutually nonlocal charges. Though the existence and some properties of these solutions were established some time ago, fully explicit analytic solutions were lacking thus far. In this note, we fill this gap. We show in general that explicit solutions can be constructed whenever an explicit formula is known in the theory at hand for the Bekenstein-Hawking entropy of a single black hole as a function of its charges, and illustrate this with some simple examples. We also give an example of moduli-dependent black hole entropy. 
  It has been shown recently [10] that Cauchy transforms of orthogonal polynomials appear naturally in general correlation functions containing ratios of characteristic polynomials of random NxN Hermitian matrices. Our main goal is to investigate the issue of universality of large N asymptotics for those Cauchy transforms for a wide class of weight functions. Our analysis covers three different scaling regimes: the "hard edge", the "bulk" and the "soft edge" of the spectrum, thus extending the earlier results known for the bulk. The principal tool is to show that for finite matrix size N the auxiliary "wave functions" associated with the Cauchy transforms obey the same second order differential equation as those associated with the orthogonal polynomials themselves. 
  The construction of a $N_T=3$ cohomological gauge theory on an hyper-K"ahler eight-fold, whose group theoretical descrition was given by Blau and Thompson, is performed explicitly. 
  We apply a recent proposal for defining conserved mass in asymptotically de Sitter spacetimes to the class of Taub-Bolt-dS spacetimes. We compute the action, entropy and conserved mass of these spacetimes, and find that in certain instances the mass and entropy can exceed that of pure de Sitter spacetime, in violation of recent suggestive conjectures to the contrary. 
  The shifted quantization condition of the M-theory 4-form G_4 is well-known. The most naive generalization to type IIA string theory fails, an orientifold counterexample was found by Hori in hep-th/9805141. In this note we use D2-brane anomaly cancellation to find the corresponding shifted quantization condition in IIA. Our analysis is consistent with the known O4-plane tensions if we include a torsion correction to the usual construction of G_4 from C_3, B and G_2. The resulting Bianchi identities enforce that RR fluxes lift to K-theory classes. 
  In this paper we reinterpret the Poisson structure of the Hitchin-type system in cohomological terms. The principal ingredient of a new interpretation in the case of the Beauville system is the meromorphic cohomology of the spectral curve, and the main result is the identification of the Riemann bilinear form and the symplectic structure of the model. Eventual perspectives of this approach lie in the quantization domain. 
  String representations of the Wilson loop are constructed in the SU(N)-version of compact QED in three and four dimensions. This is done exactly in the case of the fundamental Wilson loop and in the large-N limit in the case of the adjoint Wilson loop. Using for concreteness the three-dimensional fundamental case, it is demonstrated how the resulting SU(N)-generalization of the so-called theory of confining strings can be obtained in various ways. In its weak-field limit (corresponding to the limit of low monopole densities), this theory enables one to fix the value of the string tension, which cannot be fixed when deduced from the mean magnetic field inside a flat contour (the derivation of this field is also presented). Moreover, the obtained theory enables one to find also the coupling constants of terms in the expansion of the nonlocal string effective action, which are higher in the derivatives than the Nambu-Goto term (some of these terms vanish at the flat surface). In the four-dimensional case with the theta-term, the critical values of the theta-parameter, at which the problem of crumpling of large world sheets might be solved, are found in both the fundamental case and in the large-N limit of the adjoint case. These values are only accessible provided the electric coupling constant is larger than a certain value, in accordance with the known fact that confinement in the four-dimensional compact QED holds in the strong-coupling regime. 
  In this paper we recall the construction of Doubly Special Relativity (DSR) as a theory with energy-momentum space being the four dimensional de Sitter space. Then the bases of the DSR theory can be understood as different coordinate systems on this space. We investigate the emerging geometrical picture of Doubly Special Relativity by presenting the basis independent features of DSR that include the non-commutative structure of space-time and the phase space algebra. Next we investigate the relation between our geometric formulation and the one based on quantum $\kappa$-deformations of the Poincar\'e algebra. Finally we re-derive the five-dimensional differential calculus using the geometric method, and use it to write down the deformed Klein-Gordon equation and to analyze its plane wave solutions. 
  It is argued that free QFT can be defined on the event horizon of a Schwarzschild-like spacetime and that this theory is unitarily and algebraically equivalent to QFT in the bulk (near the horizon). Under that unitary equivalence the bulk hidden SL(2,R) symmetry found in a previous work becomes manifest on the event horizon, it being induced by a group of horizon diffeomorphisms. The class of generators of that group can be enlarged to include a full Virasoro algebra of fields which are defined on the event horizon. These generators have a quantum representation in QFT on the event horizon and thus in the bulk. 
  N=2 closed strings have been recently divided in hep-th/0211147 to two T-dual families denoted by \alpha and \beta. In (2,2) signature both families have one scalar in the spectrum. The scalar in the \beta-string is known to be a deformation of the target space K\"ahler potential and the dynamics is that of self-dual gravity. In this paper we compute the effective action of the scalar in the \alpha-string. The scalar is a deformation of a potential that determines the metric, torsion and dilaton. The scalar is free and the dynamics is that of a self-dual curvature with torsion. 
  Modifications of the electromagnetic Maxwell Lagrangian in four dimensions have been considered by some authors. One may include an explicit massive term (Proca) and a topological but not Lorentz-invariant term within certain observational limits.   We find the dual-corresponding gauge invariant version of this theory by using the recently suggested gauge embedding method. We enforce this dualisation procedure by showing that, in many cases, this is actually a constructive method to find a sort of parent action, which manifestly establishes duality. We also use the gauge invariant version of this theory to formulate a Batalin-Vilkovisky quantization and present a detailed discussion on the excitation spectrum. 
  We carry out an investigation into the possibility of developing a Bohmian interpretation based on the continuous motion of point particles for noncommutative quantum mechanics. The conditions for such an interpretation to be consistent are determined, and the implications of its adoption for noncommutativity are discussed. A Bohmian analysis of the noncommutative harmonic oscillator is carried out in detail. By studying the particle motion in the oscillator orbits, we show that small-scale physics can have influence at large scales, something similar to the IR-UV mixing. 
  Light-front wavefunctions provide a frame-independent representation of hadrons in terms of their physical quark and gluon degrees of freedom. The light-front Hamiltonian formalism provides new nonperturbative methods for obtaining the QCD spectrum and eigensolutions, including resolvant methods, variational techniques, and discretized light-front quantization. A new method for quantizing gauge theories in light-cone gauge using Dirac brackets to implement constraints is presented. In the case of the electroweak theory, this method of light-front quantization leads to a unitary and renormalizable theory of massive gauge particles, automatically incorporating the Lorentz and 't Hooft conditions as well as the Goldstone boson equivalence theorem. Spontaneous symmetry breaking is represented by the appearance of zero modes of the Higgs field leaving the light-front vacuum equal to the perturbative vacuum. I also discuss an "event amplitude generator" for automatically computing renormalized amplitudes in perturbation theory. The importance of final-state interactions for the interpretation of diffraction, shadowing, and single-spin asymmetries in inclusive reactions such as deep inelastic lepton-hadron scattering is emphasized 
  Free scalar field theory in the sector with a large number of particles can be interpreted as bosonic string theory on anti-de Sitter space of vanishing radius. Different ways of writing the field theory Hamiltonian translate to different ways of reparametrizing the world-sheet sigma coordinate. Adding a mass term in the field theory corresponds to cutting off the warped AdS direction, with cut-off inversely proportional to the mass. The string theory has neither tachyon, nor critical dimension. 
  We show that the source of RR field computed from the boundary state describing the decay of a non-BPS brane is reproduced by a particular form of the Wess-Zumino term in the tachyon effective action. We also obtain a simple expression of the S-charge associated with rolling tachyons. 
  Unified N=2 Maxwell-Einstein supergravity theories (MESGTs) are supergravity theories in which all the vector fields, including the graviphoton, transform in an irreducible representation of a simple global symmetry group of the Lagrangian. As was established long time ago, in five dimensions there exist only four unified Maxwell-Einstein supergravity theories whose target manifolds are symmetric spaces. These theories are defined by the four simple Euclidean Jordan algebras of degree three. In this paper, we show that, in addition to these four unified MESGTs with symmetric target spaces, there exist three infinite families of unified MESGTs as well as another exceptional one. These novel unified MESGTs are defined by non-compact (Minkowskian) Jordan algebras, and their target spaces are in general neither symmetric nor homogeneous. The members of one of these three infinite families can be gauged in such a way as to obtain an infinite family of unified N=2 Yang-Mills-Einstein supergravity theories, in which all vector fields transform in the adjoint representation of a simple gauge group of the type SU(N,1). The corresponding gaugings in the other two infinite families lead to Yang-Mills-Einstein supergravity theories coupled to tensor multiplets. 
  The fermion propagators in the fivebrane background of type II superstring theories are calculated. The propagator can be obtained by explicitly evaluating the transition amplitude between two specific NS-R boundary states by the propagator operator in the non-trivial world-sheet conformal field theory for the fivebrane background. The propagator in the field theory limit can be obtained by using point boundary states. We can explicitly investigate the lowest lying fermion states propagating in the non-trivial ten-dimensional space-time of the fivebrane background: M^6 x W_k^(4), where W_k^(4) is the group manifold of SU(2)_k x U(1). The half of the original supersymmetry is spontaneously broken, and the space-time Lorentz symmetry SO(9,1) reduces to SO(5,1) in SO(5,1) x SO(4) \subset SO(9,1) by the fivebrane background. We find that there are no propagations of SO(4) (local Lorentz) spinor fields, which is consistent with the arguments on the fermion zero-modes in the fivebrane background of low-energy type II supergravity theories. 
  It is shown that it is possible to define quantum field theory of a massless scalar free field on the Killing horizon of a 2D-Rindler spacetime. Free quantum field theory on the horizon enjoys diffeomorphism invariance and turns out to be unitarily and algebraically equivalent to the analogous theory of a scalar field propagating inside Rindler spacetime, nomatter the value of the mass of the field in the bulk. More precisely, there exists a unitary transformation that realizes the bulk-boundary correspondence under an appropriate choice for Fock representation spaces. Secondly, the found correspondence is a subcase of an analogous algebraic correspondence described by injective *-homomorphisms of the abstract algebras of observables generated by abstract quantum free-field operators. These field operators are smeared with suitable test functions in the bulk and exact 1-forms on the horizon. In this sense the correspondence is independent from the chosen vacua. It is proven that, under that correspondence the ``hidden'' $SL(2,\bR)$ quantum symmetry found in a previous work gets a clear geometric meaning, it being associated with a group of diffeomorphisms of the horizon itself. 
  We discuss an O(N) exension of the Sine-Gordon (S-G)equation which allows us to perform an expansion around the leading order in large-N result using Path-Integral methods. In leading order we show our methods agree with the results of a variational calculation at large-N. We discuss the striking differences for a non-polynomial interaction between the form for the effective potential in the Gaussian approximation that one obtains at large-N when compared to the N=1 case. This is in contrast to the case when the classical potential is a polynomial in the field and no such drastic differences occur. We find for our large-N extension of the Sine-Gordon model that the unbroken ground state is unstable as one increases the coupling constant (as it is for the original S-G equation) and we determine the stability criteria. 
  We review the properties of characters of the N=4 SCA in the context of a non-linear sigma model on $K3$, how they are used to span the orbits, and how the orbits produce topological invariants like the elliptic genus. We derive the same expression for the $K3$ elliptic genus using three different Gepner models ($1^6$, $2^4$ and $4^3$ theories), detailing the orbits and verifying that their coefficients $F_i$ are given by elementary modular functions. We also reveal the orbits for the $1^3 2^2$, $1^4 4$ and $1^2 4^2$ theories. We derive relations for cubes of theta functions and study the function $ {1\over\eta} \sum_{n\in \Z} (-1)^n (6n+1)^k q^{(6n+1)^2 /24} $ for $k=1,2,3,4$. 
  We construct the cubic interaction vertex and dynamically generated supercharges in light-cone superstring field theory for a large class half-supersymmetric D-branes in the plane-wave background. We show that these satisfy the plane-wave superalgebra at first order in string coupling. The cubic interaction vertex and dynamical supercharges presented here are given explicitly in terms of oscilators and can be used to compute three-point functions of open strings with endpoints on half-supersymmetric D-branes. 
  We reconsider the question of which Calabi-Yau compactifications of the heterotic string are stable under world-sheet instanton corrections to the effective space-time superpotential. For instance, compactifications described by (0,2) linear sigma models are believed to be stable, suggesting a remarkable cancellation among the instanton effects in these theories. Here, we show that this cancellation follows directly from a residue theorem, whose proof relies only upon the right-moving world-sheet supersymmetries and suitable compactness properties of the (0,2) linear sigma model. Our residue theorem also extends to a new class of "half-linear" sigma models. Using these half-linear models, we show that heterotic compactifications on the quintic hypersurface in CP^4 for which the gauge bundle pulls back from a bundle on CP^4 are stable. Finally, we apply similar ideas to compute the superpotential contributions from families of membrane instantons in M-theory compactifications on manifolds of G_2 holonomy. 
  The quantum theory involving noncommutative tensionless p-branes is studied following path integral methods. Our procedure allow a simple treatment for generally covariant noncommutative extended systems and it contains, as a particular case, the thermodynamics and the quantum tensionless string theory. The effect induced by noncommutativity in the field space is to produce a confinement among pairing of null p-branes. 
  4D Einstein gravity coupled to scalars and abelian gauge fields in its 2-Killing vector reduction is shown to be quasi-renormalizable to all loop orders at the expense of introducing infinitely many essential couplings. The latter can be combined into one or two functions of the `area radius' associated with the two Killing vectors. The renormalization flow of these couplings is governed by beta functionals expressible in closed form in terms of the (one coupling) beta function of a symmetric space sigma-model. Generically the matter coupled systems are asymptotically safe, that is the flow possesses a non-trivial UV stable fixed point at which the trace anomaly vanishes. The main exception is a minimal coupling of 4D Einstein gravity to massless free scalars, in which case the scalars decouple from gravity at the fixed point. 
  A path integral formula for the associative star-product of two superfields is proposed. It is a generalization of the Kontsevich-Cattaneo-Felder's formula for the star-product of functions of bosonic coordinates. The associativity of the star-product imposes certain conditions on the background of our sigma model. For generic background the action is not supersymmetric. The supersymmetry invariance of the action constrains the background and leads to a simple formula for the star-product. 
  We use Ward identities derived from the generalized Konishi anomaly in order to compute effective superpotentials for SU(N), SO(N) and $Sp(N)$ supersymmetric gauge theories coupled to matter in various representations. In particular we focus on cubic and quartic tree level superpotentials. With this technique higher order corrections to the perturbative part of the effective superpotential can be easily evaluated. 
  Boundary integrable models with N=2 supersymmetry are considered. For the simplest boundary N=2 superconformal minimal model with a Chebyshev bulk perturbation we show explicitly how fermionic boundary degrees of freedom arise naturally in the boundary perturbation in order to maintain integrability and N=2 supersymmetry. A new boundary reflection matrix is obtained for this model and N=2 boundary superalgebra is studied. A factorized scattering theory is proposed for a N=2 supersymmetric extension of the boundary sine-Gordon model with either (i) fermionic or (ii) bosonic and fermionic boundary degrees of freedom. Exact results are obtained for some quantum impurity problems: the boundary scaling Lee-Yang model, a massive deformation of the anisotropic Kondo model at the filling values g=2/(2n+3) and the boundary Ashkin-Teller model. 
  All consistent interactions in a three-dimensional theory with tensor gauge fields of degrees two and three are obtained by means of the deformation of the solution to the master equation combined with cohomological techniques. The local BRST cohomology of this model allows the deformation of the Lagrangian action, accompanying gauge symmetries and gauge algebra. The relationship with the Chern--Simons theory is discussed. 
  We illustrate the importance of mass scales and their relation in the specific case of the linear sigma model within the context of its one loop Ward identities. In the calculation it becomes apparent the delicate and essential connection between divergent and finite parts of amplitudes. The examples show how to use mass scales identities which are absolutely necessary to manipulate graphs involving several masses. Furthermore, in the context of the Implicitly Regularization, finite(physical) and divergent (counterterms) parts of the amplitude can and must be written in terms of a single scale which is the renormalization group scale. This facilitates, e.g., obtaining symmetric counterterms and immediately lead to the proper definition of Renormalization Group Constants. 
  We study quivers in the context of matrix models. We introduce chains of generalized Konishi anomalies to write the quadratic and cubic equations that constrain the resolvents of general affine and non-affine quiver gauge theories, and give a procedure to calculate all higher-order relations. For these theories we also evaluate, as functions of the resolvents, VEV's of chiral operators with two and four bifundamental insertions. As an example of the general procedure we explicitly consider the two simplest quivers A2 and A1(affine), obtaining in the first case a cubic algebraic curve, and for the affine theory the same equation as that of U(N) theories with adjoint matter, successfully reproducing the RG cascade result. 
  We give a simple interpretation of the recent solutions for cosmologies with a transient accelerating phase obtained from compactification in hyperbolic manifolds, or from S-brane solutions of string/M-theory. In the four-dimensional picture, these solutions correspond to bouncing the radion field off its exponential potential. Acceleration occurs at the turning point, when the radion stops and the potential energy momentarily dominates. The virtues and limitations of these approaches become quite transparent in this interpretation. 
  One-loop corrections to kink masses in a family of (1+1)-dimensional field theoretical models with two real scalar fields are computed. A generalized DHN formula applicable to potentials with and without reflection is obtained. It is shown how half-bound states arising in the spectrum of the second order fluctuation operator for one-component topological kinks and the vacuum play a central r$\hat{{\rm o}}$le in the computation of the kink Casimir energy. The issue of whether or not the kink degeneracy exhibited by this family of models at the classical level survives one-loop quantum fluctuations is addressed. 
  We compute the corrections to heterotic-string backgrounds with (2,0) world-sheet supersymmetry, up to two loops in sigma-model perturbation theory. We investigate the conditions for these backgrounds to preserve spacetime supersymmetry and we find that a sufficient requirement for consistency is the applicability of the $\partial\bar\partial$-lemma. In particular, we investigate the $\alpha'$ corrections to (2,0) heterotic-string compactifications and we find that the Calabi-Yau geometry of the internal space is deformed to a Hermitian one. We show that at first order in $\alpha'$, the heterotic anomaly-cancellation mechanism does not induce any lifting of moduli. We explicitly compute the corrections to the conifold and to the U(n)-invariant Calabi-Yau metric at first order in $\alpha'$. We also find a generalization of the gauge-field equations, compatible with the Donaldson equations on conformally-balanced Hermitian manifolds. 
  We discuss the k dependence of the k-string tension sigma_k in SU(N) supersymmetric gluodynamics. As well known, at large N the k-string consists, to leading order, of k noninteracting fundamental strings, so that sigma_k=k sigma_1. We argue, both from field-theory and string-theory side, that subleading corrections to this formula run in powers of 1/N^2 rather than 1/N, thus excluding the Casimir scaling. We suggest a heuristic model allowing one to relate the k-string tension in four-dimensional gluodynamics with the tension of the BPS domain walls (k-walls). In this model the domain walls are made of a net of strings connected to each other by baryon vertices. The relation emerging in this way leads to the sine formula sigma_ k ~ Lambda^2 N sin pi k/N. We discuss possible corrections to the sine law, and present arguments that they are suppressed by 1/k factors. We explain why the sine law does not hold in two dimensions. Finally, we discuss the applicability of the sine formula for non-supersymmetric orientifold field theories. 
  An exact and general solution is presented for a previously open problem. We show that the superconformal R-symmetry of any 4d SCFT is exactly and uniquely determined by a maximization principle: it is the R-symmetry, among all possibilities, which (locally) maximizes the combination of 't Hooft anomalies a_{trial}(R) \equiv (9 Tr R^3-3 Tr R)/32. The maximal value of a_{trial} is then, by a result of Anselmi et. al., the central charge \it{a} of the SCFT. Our a_{trial} maximization principle almost immediately ensures that the central charge \it{a} decreases upon any RG flow, since relevant deformations force a_{trial} to be maximized over a subset of the previously possible R-symmetries. Using a_{trial} maximization, we find the exact superconformal R-symmetry (and thus the exact anomalous dimensions of all chiral operators) in a variety of previously mysterious 4d N=1 SCFTs. As a check, we verify that our exact results reproduce the perturbative anomalous dimensions in all perturbatively accessible RG fixed points. Our result implies that N =1 SCFTs are algebraic: the exact scaling dimensions of all chiral primary operators, and the central charges \it{a} and \it{c}, are always algebraic numbers. 
  We find a nonsupersymmetric dilatonic deformation of $AdS_5$ geometry as an exact nonsingular solution of the type IIB supergravity. The dual gauge theory has a different Yang-Mills coupling in each of the two halves of the boundary spacetime divided by a codimension one defect. We discuss the geometry of our solution in detail, emphasizing the structure of the boundary, and also study the string configurations corresponding to Wilson loops. We also show that that the background is stable under small scalar perturbations. 
  We study a variety of problems in the spectral theory of automorphic forms using entirely analytic techniques such as Selberg trace formula, asymptotics of Whittaker functions and behavior of heat kernels. Error terms for Weyl's law and an analog of Selberg's eigenvalue conjecture for $SL_3({\bf Z})$ is given. We prove the following: Let $\cal H$ be the homogeneous space associated to the group $PGL_3(\bf R)$. Let $X = \Gamma{\backslash SL_3({\bf Z}})$ and consider the first non-trivial eigenvalue $\lambda_1$ of the Laplacian on $L^2(X)$. Using geometric considerations, we prove the inequality $\lambda_1 > 3pi^2/10> 2.96088.$ Since the continuous spectrum is represented by the band $[1,\infty)$, our bound on $\lambda_{1}$ can be viewed as an analogue of Selberg's eigenvalue conjecture for quotients of the hyperbolic half space. Brief comment on relevance of automorphic forms to applications in high energy physics is given. 
  Five-dimensional braneworld cosmology with deSitter (inflationary) brane universe induced by classical and quantum matter is discussed. It is shown that negative energy phantom field with quantum CFT supports the creation of deSitter universe. On the same time, pure phantom or dust with quantum effects, or Chaplygin gas with quantum effects may naturally lead to the occurence of Anti-deSitter brane universe but not deSitter one. It is also interesting that unlike to four-dimensional gravity, for phantom with (or without) quantum contribution the standard cosmological energy conditions may be effectively satisfied. 
  We obtain a solution to eleven-dimensional supergravity that consists of M2-branes embedded in a dielectric distribution of M5-branes. Contrary to normal expectations, this solution has maximal supersymmetry for a brane solution (i.e. sixteen supercharges). While the solution is constructed using gauged supergravity in four dimensions, the complete eleven-dimensional solution is given. In particular, we obtain the Killing spinors explicitly, and we find that they are characterised by a duality rotation of the standard Dirichlet projection matrix for M2-branes. 
  I shall use a few personal reminiscences of my time as a student and colleague of Dirac in Cambridge to introduce some reflections on the nature of research in theoretical physics. I shall discuss and illustrate the approach of Dirac to his own research and the pervasiveness of the influence his example has provided.   I shall discuss how ideas produced at all stages of his career have proved to be extraordinarily visionary, still motivating and exerting an influence on research many years after his death. Examples include his celebrated equation and prediction of the existence of antiparticles, the magnetic monopole and its quantisation condition, the quantum theory of dynamical systems with constraints and the membrane theory of the electron. 
  To investigate the possibility of a ghost-antighost condensate the coupled Dyson--Schwinger equations for the gluon and ghost propagators in Yang--Mills theories are derived in general covariant gauges, including ghost-antighost symmetric gauges. The infrared behaviour of these two-point functions is studied in a bare-vertex truncation scheme which has proven to be successful in Landau gauge. In all linear covariant gauges the same infrared behaviour as in Landau gauge is found: The gluon propagator is infrared suppressed whereas the ghost propagator is infrared enhanced. This infrared singular behaviour provides indication against a ghost-antighost condensate. In the ghost-antighost symmetric gauges we find that the infrared behaviour of the gluon and ghost propagators cannot be determined when replacing all dressed vertices by bare ones. The question of a BRST invariant dimension two condensate remains to be further studied. 
  We demonstrate that an equidistant area spectrum for the link variables in loop quantum gravity can reproduce both the thermodynamics and the quasinormal mode properties of black holes. 
  The properties of a non-canonical 3D Wigner quantum oscillator, whose position and momentum operators generate the Lie superalgebra sl(1|3), are further investigated. Within each state space W(p), p=1,2,..., the energy E_q, q=0,1,2,3, takes no more than 4 different values. If the oscillator is in a stationary state \psi_q\in W(p) then measurements of the non-commuting Cartesian coordinates of the particle are such that their allowed values are consistent with it being found at a finite number of sites, called ``nests''. These lie on a sphere centered on the origin of fixed, finite radius \varrho_q. The nests themselves are at the vertices of a rectangular parallelepiped. In the typical cases (p>2) the number of nests is 8 for q=0 and 3, and varies from 8 to 24, depending on the state, for q=1 and 2. The number of nests is less in the atypical cases (p=1,2), but it is never less than two. In certain states in W(2) (resp. in W(1)) the oscillator is ``polarized'' so that all the nests lie on a plane (resp. on a line). The particle cannot be localized in any one of the available nests alone since the coordinates do not commute. The probabilities of measuring particular values of the coordinates are discussed. The mean trajectories and the standard deviations of the coordinates and momenta are computed, and conclusions are drawn about uncertainty relations. 
  The contraction of a spin-1/2 representation of the de Sitter group SO(3,2) yields a translation operator that consists of the usual momentum operator plus a second order term, the "momentum spin" as described by F. Guersey. The contribution of momentum spin to the kinematics of a multiparticle system in a tangential space of anti de Sitter space is analyzed. It is shown that it can be described by a perturbation term with the structure of the interaction term of quantum electrodynamics. An evaluation of the corresponding coupling constant reproduces Wyler's heuristic formula for the electromagnetic coupling constant. 
  We derive the Konishi anomaly equations for N=1 supersymmetric gauge theories based on the classical gauge groups with matter in two-index tensor and fundamental representations, thus extending the existing results for U(N). A general formula is obtained which expresses solutions to the Konishi anomaly equation in terms of solutions to the loop equations of the corresponding matrix model. This provides an alternative to the diagrammatic proof that the perturbative part of the glueball superpotential $W_{\rm eff}$ for these matter representations can be computed from matrix model integrals, and further shows that the two approaches always give the same result. The anomaly approach is found to be computationally more efficient in the cases we studied. Also, we show in the anomaly approach how theories with a traceless two-index tensor can be solved using an associated theory with a traceful tensor and appropriately chosen coupling constants. 
  Following earlier work by Polyakov and Gubser, Klebanov and Polyakov, we attempt to clarify the structure of vertex operators representing string states which have large (``semiclassical'') values of AdS energy (equal to 4-d dimension \Delta) and angular momentum J in S^5 or spin S in AdS_5. We comment on the meaning of semiclassical limit in the context of \alpha' perturbative expansion for the 2-d anomalous dimensions of the corresponding vertex operators. We consider in detail the leading-order 1-loop renormalization of these operators in AdS_5 x S^5 sigma model (ignoring fermionic contributions). We find new examples of operators for which, as in the case considered in hep-th/0110196, the 1-loop anomalous dimension can be made small by tuning quantum numbers. We also comment on a possibility of deriving the semiclassical relation between \Delta and J or S from the marginality condition for the vertex operators, using a stationary phase approximation in the path integral expression for their 2-point correlator on a complex plane. 
  A method for systematically including topological degrees of freedom in perturbation theory is developed. This is not bound by the restrictions of semi-classical techniques. The Yang-Mills theory in three Euclidean dimensions is considered here. A well-defined separation of the topological and the ``spin wave'' degrees of freedom is obtained, motivated by a singular gauge. This has ``photons'' distorting the spherically symmetric magnetic fields of Dirac monopoles, and massless charged vector bosons ``W'' scattering off the latter. It is explicitly shown that the Dirac string does not contribute. The mode of the charged vector bosons with total angular momentum J=0 provides precisely the core to give a finite energy to the monopole. The radial equation for W is remarkably simplified and only two polarization states survive exactly for the anomalous magnetic moment required by the Yang-Mills interaction. 
  A new noncommutative model invariant with respect to U(1) gauge group is proposed. The model is free of nonintegrable infrared singularities. Its commutative classical limit describes a free scalar field. Generalization to U(N) models is also considered. 
  Extension procedure for supermanifold ${\cal M}_{cl}$ of superfields ${\cal A}^{\imath}(\theta)$, ghost number construction are considered. Classical and $\hbar$-deformed generating (master) equations, existence theorems for their solutions are formulated in $T^{\ast}_{odd}{\cal M}_{min}$, $T^{\ast}_{odd}{\cal M}_{ext}$. Analogous scheme is realized for BV similar generating equations. Master equations versions for GSQM and BV similar scheme are deformed in powers of superfields ${\stackrel{\circ}{\Gamma}}{}^p(\theta)$ = $\bigl({\stackrel{\circ}{\Phi}}{}^B(\theta)$, ${\stackrel{\circ}{\Phi}}{}^{\ast}_B(\theta)\bigr)$ into supermanifold $T_{odd}(T^{\ast}_{odd}{\cal M}_{ext})$. Arbitrariness in a choice of solutions for these equations is described. Investigation of formal Hamiltonian systems for II class theories [2] defined via corresponding master equations solutions is conducted. Gauge fixing for those theories is described by two ways. Functional integral of superfunctions on $T_{odd}(T^{\ast}_{odd}{\cal M}_{ext})$ is defined. Properties for generating functionals of Green's superfunctions are studied. $\theta$-component quantization formulation, connection with BV method and superfield quantization [3] are established. Quantization scheme realization is demonstrated on 6 models. 
  This thesis starts with a review of BPS M-branes and their supergravity solutions. These solutions can be obtained in various ways. We describe the harmonic function rule and the Fayyazuddin-Smith metric ansatz in detail, illustrating both methods by examples of M-branes wrapping holomorphic curves. Another, simpler way, is to use calibrations. In order for a wrapped brane to be BPS, the cycle it wraps must be calibrated; given the relevant calibration, there exists a procedure which yields the supergravity solution almost immediately. In the past, this method was applied only to Kahler calibrations, as these were the only ones known to exist in backgrounds with non-trivial flux. We extend this method to a wider domain, using a constraint to classify possible calibrations. A rule is given which can generate the required constraint for any given M-brane wrapped on a holomorphic cycle. Ways in which this constraint can be satisfied are also discussed. 
  Reducible representations of CAR and CCR are applied to second quantization of Dirac and Maxwell fields. The resulting field operators are indeed operators and not operator-valued distributions. Examples show that the formalism may lead to a finite quantum field theory. 
  We show that the scattering amplitude of four open string scalars or tachyons on the world-volume of a D$_p$-brane in the bosonic string theory can be written in a universal form. The difference between this amplitude and the corresponding amplitude in the superstring theory is in an extra tachyonic pole. We show that in an $\alpha'$ expansion and for slowly varying fields, the amplitude is consistent with the tachyonic DBI action in which the even part of the tachyon potential is $V(T)=e^{-(\sqrt{\pi} T/\alpha)^2}$ with $\alpha=1$ for bosonic theory and $\alpha=\sqrt{2}$ for superstring theory. 
  In this paper, the knotlike cosmic strings in the Riemann-Cartan space-time of the early universe are discussed. It has been revealed that the cosmic strings can just originate from the zero points of the complex scalar quintessence field. In these strings we mainly study the knotlike configurations. Based on the integral of Chern-Simons 3-form a topological invariant for knotlike cosmic strings is constructed, and it is shown that this invariant is just the total sum of all the self-linking and linking numbers of the knots family. Furthermore, it is also pointed out that this invariant is preserved in the branch processes during the evolution of cosmic strings. 
  We present a comprehensive study of the cosmological solutions of 6D braneworld models with azimuthal symmetry in the extra dimensions, moduli stabilization by flux or a bulk scalar field, and which contain at least one 3-brane that could be identified with our world. We emphasize an unusual property of these models: their expansion rate depends on the 3-brane tension either not at all, or in a nonstandard way, at odds with the naive expected dimensional reduction of these systems to 4D general relativity at low energies. Unlike other braneworld attempts to find a self-tuning solution to the cosmological constant problem, the apparent failure of decoupling in these models is not associated with the presence of unstabilized moduli; rather it is due to automatic cancellation of the brane tension by the curvature induced by the brane. This provides some corroboration for the hope that these models provide a distinctive step toward understanding the smallness of the observed cosmological constant. However, we point out some challenges for obtaining realistic cosmology within this framework. 
  We study the regularization of theories of ``brane induced'' gravity in codimension $N>1$. The brane can be interpreted as a thin dielectric with a large dielectric constant, embedded in a higher dimensional space. The kinetic term for the higher dimensional graviton is enhanced over the brane. A four dimensional gravitation is found on the brane at distances smaller than a critical distance $r<r_c$, and is due to the exchange of a massive resonant graviton. The crossover scale $r_c$ is determined by the mass of the resonance. The suppression of the couplings of light Kaluza-Klein modes to brane matter results in a higher dimensional force law at large distances. We show that the resulting theory is free of ghosts or tachyons. 
  We propose a generalization of spin algebra using three-index objects. There is a possibility that a triple commutation relation among three-index objects implies a kind of uncertainty relation among their expectation values. 
  We investigate the description of the region behind the event horizon in rotating black holes in the AdS/CFT correspondence, using the rotating BTZ black hole as a concrete example. We extend a technique introduced by Kraus, Ooguri and Shenker [hep-th/0212277], based on analytically continuing amplitudes defined in a Euclidean space, to include rotation. In the rotating case, boundary amplitudes again have two different bulk descriptions, involving either integration only over the regions outside the black holes' event horizon, or integration over this region and the region between the event horizon and the Cauchy horizon (inner horizon). We argue that generally, the holographic map will relate the field theory to the region bounded by the Cauchy horizons in spacetime. We also argue that these results suggest that the holographic description of black holes will satisfy strong cosmic censorship. 
  It is shown that the extra supersymmetry of tensionless superstring and super $p$-brane is accompanied by the presence of new bosonic gauge symmetries.   It permits to use composed coordinates encoding all physical degrees of freedom of the model and invariant under these gauge symmetries and the enhanced $\kappa$-symmetry.   It is proved that the composed gauge invariant coordinates coincide with the componets of symplectic supertwistor realizing a linear representation of the hidden OSp(1,2M) symmetry of the super $p$-brane Lagrangian. A connection of the presented gauge symmetries with massless higher spin gauge theories and a symmetric phase of $M$/string-theory is discussed. 
  We obtain the asymptotic density of open p-brane states with zero-modes included. The resulting logarithmic correction to the p-brane entropy has a coefficient - \frac{p + 2}{2 p}, and is independent of the dimension of the embedding spacetime. Such logarithmic corrections to the entropy, with precisely this coefficient, appear in two other contexts also: a gas of massless particles in p-dimensional space, and a Schwarzschild black hole in (p + 2)-dimensional anti de Sitter spacetime. 
  The time-varying density of D-branes and anti-D-branes in an expanding universe is calculated. The D-brane anti-brane annihilation rate is shown to be too small to compete with the expansion rate of a FRW type universe and the branes over-close the universe. This brane problem is analogous to the old monopole problem. Interestingly however, it is shown that small dimension D-branes annihilate more slowly than high dimension branes. Hence, an initially brany universe may be filled with only low dimension branes at late times. When combined with an appropriate late inflationary theory this leads to an attractive dynamical way to create a realistic braneworld scenario. 
  By constructing a nilpotent extended BRST operator $\bs$ that involves the N=2 global supersymmetry transformations of one chirality, we show that the standard N=2 off-shell Super Yang Mills Action can be represented as an exact BRST term $\bs \Psi$, if the gauge fermion $\Psi$ is allowed to depend on the inverse powers of supersymmetry ghosts. By using this nonanalytical structure of the gauge fermion (via inverse powers of supersymmetry ghosts), we give field redefinitions in terms of composite fields of supersymmetry ghosts and N=2 fields and we show that Witten's topological Yang Mills theory can be obtained from the ordinary Euclidean N=2 Super Yang Mills theory directly by using such field redefinitions. In other words, TYM theory is obtained as a change of variables (without twisting). As a consequence it is found that physical and topological interpretations of N=2 SYM are intertwined together due to the requirement of analyticity of global SUSY ghosts. Moreover, when after an instanton inspired truncation of the model is used, we show that the given field redefinitions yield the Baulieu-Singer formulation of Topological Yang Mills. 
  We show that the least energy conditions in the gauged nonlinear sigma model with Chern-Simons term lead to exact soliton-like solutions which have the same features as domain walls. We will derive and discuss the corresponding solutions, and compute the total energy, charge, and spin of the resulting system. 
  We show how to get the one-loop beta function and the chiral anomaly of N=1 Super QCD from a stack of fractional N D3-branes localized inside the world-volume of 2M fractional D7-branes on the orbifold C^3/(Z_2 x Z_2). They are obtained by analyzing the classical supergravity background generated by such a brane configuration, in the spirit of the gauge/gravity correspondence. 
  The role non-Abelian magnetic monopoles play in the dynamics of confinement is discussed by examining carefully a class of supersymmetric gauge theories as theoretical laboratories. In particular, in the so-called $r$-vacua of softly broken ${N}=2$ supersymmmetric $SU(n_c)$ QCD, the Goddard-Olive-Nuyts-Weinberg monopoles appear as the dominant low-energy effective degrees of freedom. Even more interesting is the physics of confining vacua which are deformations of nontrivial superconformal theories. We argue that in such cases, occurring in the $r= {n_f \over 2}$ vacua of $SU(n_c)$ theories or in all of confining vacua of $USp(2n_c)$ or $SO(n_f)$ theories with massless flavors, a new mechanism of confinement involving strongly interacting non-Abelian magnetic monopoles is at work. 
  The infinite matrices in Witten's vertex are easy to diagonalize. It just requires some SL(2,R) lore plus a Watson-Sommerfeld transformation. We calculate the eigenvalues of all Neumann matrices for all scale dimensions s, both for matter and ghosts, including fractional s which we use to regulate the difficult s=0 limit. We find that s=1 eigenfunctions just acquire a p term, and x gets replaced by the midpoint position. 
  We show that renormalized non-commutative scalar field theories do not reduce to their planar sector in the limit of large non-commutativity. This follows from the fact that the RG equation of the Wilson-Polchinski type which describes the genus zero sector of non-commutative field theories couples generic planar amplitudes with non-planar amplitudes at exceptional values of the external momenta. We prove that the renormalization problem can be consistently restricted to this set of amplitudes. In the resulting renormalized theory non-planar divergences are treated as UV divergences requiring appropriate non-local counterterms. In 4 dimensions the model turns out to have one more relevant (non-planar) coupling than its commutative counterpart. This non-planar coupling is ``evanescent'': although in the massive (but not in the massless) case its contribution to planar amplitudes vanishes when the floating cut-off equals the renormalization scale, this coupling is needed to make the Wilsonian effective action UV finite at all values of the floating cut-off. 
  The duality map between gauge theories and strings suggests that when the gauge theory is in the weak coupling regime the dual string tension effectively tends to zero, $\alpha' \to \infty$. This observation of Sundborg and Witten initiates a fresh interest to the old problem of tensionless limit of standard string theory and to the description of its genuine symmetries. We approach this problem formulating tensionless string theory by means of geometrical concept of surface perimeter. The perimeter action uniquely leads to a tensionless string theory. 
  Symmetry breaking can produce ``Alice'' strings, which alter scattered charges and carry monopole number and charge when twisted into loops. Alice behavior arises algebraically, when a string's untraced Wilson loop obstructs unbroken symmetries -- a fragile criterion. We give a topological criterion, compelling Alice behavior or deforming it away. Our criterion, that \pi_o(H) acts nontrivially on \pi_1(H), links topologically Alice strings to topological monopoles. We twist Alice loops to form monopoles, and find nematic and He-3A Alice strings are topologically Alice, carrying fundamental monopole charge when twisted into loops. 
  Symmetry breaking can produce ``Alice'' strings, which alter scattered charges and carry monopole number and charge when twisted into loops. We apply recent topological results, fixing Alice strings' stability and prescribing their twisting into loops with monopole charge, to several models. We show that Alice strings of condensed matter systems (nematic liquid crystals, He-3A, and related systems of non-chiral Bose condensates and amorphous chiral superconductors) are topologically Alice, and carry fundamental monopole charge when twisted into loops. They might thus be observed indirectly, not as strings, but as loop-like point defects. Other models yield Alice loops that carry only deposited, and not fundamental, charge. 
  We study time dependent solutions in cubic open string field theory which are expected to describe the configuration of the rolling tachyon. We consider the truncated system consisting of component fields of level zero and two, which are expanded in terms of cosh n x^0 modes. For studying the large time behavior of the solution we need to know the coefficients of all and, in particular, large n modes. We examine numerically the coefficients of the n-th mode, and find that it has the leading n-dependence of the form (-\beta)^n \lambda^{-n^2} multiplied by a peculiar subleading part with peaks at n=2^m=4,8,16,32,64,128,.... This behavior is also reproduced analytically by solving simplified equations of motion of the tachyon system. 
  The vacuum amplitude of the closed membrane theory is investigated using the fact that any three-dimensional manifold has the corresponding Heegaard diagram (splitting) although it is not unique. We concentrate on the topological aspect with the geometry considered only perturbatively. In the simplest case where the action describes the free fields we find that the genus one amplitudes (lens space) are obtained from the S3 amplitude by merely renormalizing the membrane tension. The amplitudes corresponding to the Heegaard diagram of genus two or higher can be calculated as the Coulomb amplitudes with arbitrary charge distributed on a knot or a link which corresponds to the set of branch points of a given regular or an irregular covering space. We also discuss the case of membrane instanton. 
  In this paper, we show that the propagator of the dual of a general Proca-like theory, derived from the gauging iterative Noether Dualization Method, can be written by means of a simple relation between known propagators. This result is also a demonstration that the Lagrangian obtained by dualization describes the same physical particles as the ones present in the original theory at the expense of introducing new non-physical (ghosts) excitations. 
  Focusing on gauge degrees of freedom specified by a 1+3 dimensions model hosting a Maxwell term plus a Lorentz and CPT non-invariant Chern-Simons-like contribution, we obtain a minimal extension of such a system to a supersymmetric environment. We comment on resulting peculiar self-couplings for the gauge sector, as well as on background contribution for gaugino masses. Furthermore, a non-polynomial generalization is presented. 
  The topology of configuration space may be responsible in part for the existence of sphalerons. Here, sphalerons are defined to be static but unstable finite-energy solutions of the classical field equations. Another manifestation of the nontrivial topology of configuration space is the phenomenon of spectral flow for the eigenvalues of the Dirac Hamiltonian. The spectral flow, in turn, is related to the possible existence of anomalies. In this review, the interconnection of these topics is illustrated for three particular sphalerons of SU(2) Yang-Mills-Higgs theory. 
  We study cosmological solutions to the low-energy effective action of heterotic string theory including possible leading order $\alpha'$ corrections and a potential for the dilaton. We consider the possibility that including such stringy corrections can resolve the initial cosmological singularity. Since the exact form of these corrections is not known the higher-derivative terms are constructed so that they vanish when the metric is de Sitter spacetime. The constructed terms are compatible with known restrictions from scattering amplitude and string worldsheet beta-function calculations. Analytic and numerical techniques are used to construct a singularity-free cosmological solution. At late times and low-curvatures the metric is asymptotically Minkowski and the dilaton is frozen. In the high-curvature regime the universe enters a de Sitter phase. 
  The most general parallelizable pp-wave backgrounds which are non-dilatonic solutions in the NS-NS sector of type IIA and IIB string theories are considered. We demonstrate that parallelizable pp-wave backgrounds are necessarily homogeneous plane-waves, and that a large class of homogeneous plane-waves are parallelizable, stating the necessary conditions. Such plane-waves can be classified according to the number of preserved supersymmetries. In type IIA, these include backgrounds preserving 16, 18, 20, 22 and 24 supercharges, while in the IIB case they preserve 16, 20, 24 or 28 supercharges. An intriguing property of parallelizable pp-wave backgrounds is that the bosonic part of these solutions are invariant under T-duality, while the number of supercharges might change under T-duality. Due to their \alpha' exactness, they provide interesting backgrounds for studying string theory. Quantization of string modes, their compactification and behaviour under T-duality are studied. In addition, we consider BPS $Dp$-branes, and show that these $Dp$-branes can be classified in terms of the locations of their world volumes with respect to the background $H$-field. 
  We consider the Skyrme model using the explicit parameterization of the rotation group SO(3) through elements of its algebra. Topologically nontrivial solutions already arise even in the one-dimensional case because the fundamental group of SO(3) is Z_2. We explicitly find and analyze one-dimensional static solutions. Among them, there are topologically nontrivial solutions with finite energy among them. We propose a new class of projective models whose target spaces are arbitrary real projective spaces RP^d. 
  The mechanism of recombination of intersecting D-branes is investigated within the framework of effective tachyon field theory. We regard the branes as the kink-type tachyon condensed states and use the effective two-tachyon Lagrangian to study the off-diagonal fluctuations therein. It is seen that there is a tachyonic mode in the off-diagonal fluctuations, which signs the instability of the intersecting D-branes. We diagonalize the two-by-two tachyonic matrix field and see that the corresponding eigenfunctions could be used to describe the new recombined branes. Our prescription can be extended to discuss the general behavior of the recombination of intersection D-branes with arbitrary shapes. We present in detail the physical reasons behind the mathematical process of diagonalizing the tachyonic matrix field. 
  We study aspects of the accelerating cosmologies obtained from the compactification of vacuum solution and S2-branes of superstring/M theories. Parameter dependence of the resulting expansion of our universe and internal space is examined for all cases. We find that accelerating expansions are obtained also from spherical internal spaces, albeit the solution enters into contracting phase eventually. The relation between the models of SM2- and SD2-branes are also discussed, and a potential problem with SD2-brane model is noted. 
  Boundary states that preserve supersymmetry are constructed for fractional D-strings with travelling waves on a ${\bf C}^3/ {{\bf Z}_2\times {\bf Z}_2}$ orbifold. The gravitational radiation emitted between two D-strings with antiparallel travelling waves is calculated. 
  Scattering and electron-positron pair production by a one-dimensional potential is considered in the framework of the $S-$matrix formalism. The solutions of the Dirac equation are classified according to frequency sign. The Bogoliubov transformation relating the in- and out-states are given. We show that the norm of a solution of the wave equation is determined by the largest amplitude of its asymptotic form when $x_3\to \pm\infty$. For a number of potentials we give the explicit expressions for the complete in- and out-sets of orthonormalized wave functions. We note that in principle virtual vacuum processes in external field influence the phase of wave function of scattered particle.. 
  We study the quantisation of complex, finite-dimensional, compact, classical phase spaces C, by explicitly constructing Hilbert-space vector bundles over C. We find that these vector bundles split as the direct sum of two holomorphic vector bundles: the holomorphic tangent bundle T(C), plus a complex line bundle N(C). Quantum states (except the vacuum) appear as tangent vectors to C. The vacuum state appears as the fibrewise generator of N(C). Holomorphic line bundles N(C) are classified by the elements of Pic(C), the Picard group of C. In this way Pic(C) appears as the parameter space for nonequivalent vacua. Our analysis is modelled on, but not limited to, the case when C is complex projective space. 
  Smooth-throat wormholes are treated on as possessing quantum fluctuation energy with scalar massive field as its source. Heat kernel coefficients of the Laplace operator are calculated in background of the arbitrary-profile throat wormhole with the help of the zeta-function approach. Two specific profile are considered. Some arguments are given that the wormholes may exist. It serves as a solution of semiclassical Einstein equations in the range of specific values of length and certain radius of wormhole's throat and constant of non-minimal connection. 
  We study compactifications of Einstein gravity on product spaces in vacuum and their acceleration phases. Scalar potentials for the dimensionally reduced effective theory are found to be of exponential form and exact solutions are obtained for a class of product spaces. The inflation in our solutions is not sufficient for the early universe. We comment on the possibility of obtaining sufficient inflation by compactification in general. 
  The field theoretical renormalization group equations have many common features with the equations of dynamical systems. In particular, the manner how Callan-Symanzik equation ensures the independence of a theory from its subtraction point is reminiscent of self-similarity in autonomous flows towards attractors. Motivated by such analogies we propose that besides isolated fixed points, the couplings in a renormalizable field theory may also flow towards more general, even fractal attractors. This could lead to Big Mess scenarios in applications to multiphase systems, from spin-glasses and neural networks to fundamental string (M?) theory. We consider various general aspects of such chaotic flows. We argue that they pose no obvious contradictions with the known properties of effective actions, the existence of dissipative Lyapunov functions, and even the strong version of the c-theorem. We also explain the difficulties encountered when constructing effective actions with chaotic renormalization group flows and observe that they have many common virtues with realistic field theory effective actions. We conclude that if chaotic renormalization group flows are to be excluded, conceptually novel no-go theorems must be developed. 
  We prove that, under suitable assumptions, operationally motivated data completely determine a space-time in which the quantum systems can be interpreted as evolving. At the same time, the dynamics of the quantum system is also determined. To minimize technical complications, this is done in the example of three-dimensional Minkowski space. 
  We study Born-Infeld type tachyonic effective action of unstable D2-brane with a runaway potential and find rich structure of static regular solitonic solutions. There exists only periodic array of tachyon kink-antikinks in pure tachyonic theory, however, in the presence of electromagnetic fields, solutions include periodic arrays, topological tachyon kinks, half kink, and bounces. Computed tension of each kink or single unit of the periodic array has $T_{1}=\sqrt{2}\pi T_{2}$ or that with a multiplicative factor depending on electric field. When both electric and magnetic fields are turned on, fundamental string charge density has a confined component in addition to a constant piece. These evidences imply that the obtained codimension-1 objects are likely to be interpreted as D1-brane (and D1F1) or array of D1$\bar{{\rm D}}1$ (and D1F1-$\bar{{\rm D}}1$F1) as was the case without the electromagnetic fields. Generalization to unstable D$p$-branes is straightforward. 
  We find exact rotating and non-rotating neutral black hole solutions in the Godel universe of the five dimensional minimal supergravity theory. We also describe the embedding of this solution in M-theory. After dimensional reduction and T-duality, we obtain a supergravity solution corresponding to placing a black string in a pp-wave background. 
  We offer a physicists' proof that center-vortex theory requires the area in the Wilson-loop area law to involve an extremal area. Area-law dynamics is determined by integrating over Wilson loops only, not over surface fluctuations for a fixed loop. Fluctuations leading to to perimeter-law corrections come from loop fluctuations as well as integration over finite -thickness center-vortex collective coordinates. In d=3 (or d=2+1) we exploit a contour form of the extremal area in isothermal which is similar to d=2 (or d=1+1) QCD in many respects, except that there are both quartic and quadratic terms in the action. One major result is that at large angular momentum \ell in d=3+1 the center-vortex extremal-area picture yields a linear Regge trajectory with Regge slope--string tension product \alpha'(0)K_F of 1/(2\pi), which is the canonical Veneziano/string value. In a curious effect traceable to retardation, the quark kinetic terms in the action vanish relative to area-law terms in the large-\ell limit, in which light-quark masses \sim K_F^{1/2} are negligible. This corresponds to string-theoretic expectations, even though we emphasize that the extremal-area law is not a string theory quantum-mechanically. We show how some quantum trajectory fluctuations as well as non-leading classical terms for finite mass yield corrections scaling with \ell^{-1/2}. We compare to old semiclassical calculations of relativistic q\bar{q} bound states at large \ell, which also yield asymptotically-linear Regge trajectories, finding agreement with a naive string picture (classically, not quantum-mechanically) and disagreement with an effective-propagator model. We show that contour forms of the area law can be expressed in terms of Abelian gauge potentials, and relate this to old work of Comtet. 
  The issue of holographic principle in the PP-wave limit of the AdS/CFT correspondence is discussed, in the hope of clarifying some confusions in the literature. We show that, in the plane-wave limit, the relation between the partition function in the bulk and the gauge-invariant correlation functions on the boundary should be interpreted on the basis of a tunneling picture in the semi-classical approximation which is appropriate for the plane-wave limit. This leads to a natural relation between Euclidean S-matrix in the bulk and the short-distance operator-product expansion of the so-called BMN operators on the boundary. 
  We present a variant of the supereigenvalue model proposed before by Alvarez-Gaume, Itoyama, Manes, and Zadra. This model derives a set of three planar loop equations which takes the same form as the set of three anomalous Ward-Takahashi identities on the gaugino condensates recently derived by Cachazo, Douglas, Seiberg and Witten. Another model which implements N=2 superVirasoro constraints is constructed for comparison. 
  The particle algebras generated by the creation/annihilation operators for bosons and for fermions are shown to possess quantum invariance groups. These structures and their sub(quantum)groups are investigated. 
  Alice electrodynamics (AED) is a theory of electrodynamics in which charge conjugation is a local gauge symmetry. In this paper we investigate a charge instability in alice electrodynamics in (2+1)-dimensions due to this local charge conjugation. The instability manifests itself through the creation of a pair of alice fluxes. The final state is one in which the charge is completely delocalized, i.e., it is carried as cheshire charge by the flux pair that gets infinitely separated. We determine the decay rate in terms of the parameters of the model. The relation of this phenomenon with other salient features of 2-dimensional compact QED, such as linear confinement due to instantons/monopoles, is discussed. 
  We discuss a model in which high energy brane corrections allow a single scalar field to describe inflation at early epochs and quintessence at late times. The reheating mechanism in the model originates from Born-Infeld matter whose energy density mimics cosmological constant at very early times and manifests itself as radiation subsequently. For most of the inflationary evolution the Born-Infeld matter remains subdominant to the the scalar field. Shortly before the end of inflation driven by the scalar field, the energy density of Born-Infeld matter starts scaling as radiation and drops by several orders of magnitudes at the epoch inflation ends. The problem of over production of gravity wave background in scenarios based upon reheating through gravitational particle production is successfully resolved by suitably fixing the initial value of radiation energy density at the end of inflation. No additional fine tuning of the parameters is required for a viable evolution. 
  Overall, brane inflation is compatible with the recent analysis of the WMAP data. Here we explore the constraints of WMAP and 2dFGRS data on the various brane inflationary scenarios. Brane inflation naturally ends with the production of cosmic strings, which may provide a way to distinguish these models observationally. We argue that currently available data cannot exclude a non-negligible contribution from cosmic strings definitively. We perform a partial statistical analysis of mixed models that include a sub-dominant contribution from cosmic strings. Although the data favor models without cosmic strings, we conclude that they cannot definitively rule out a cosmic-string-induced contribution of $\sim 10 %$ to the observed temperature, polarization and galaxy density fluctuations. These results imply that $G\mu \lesssim 3.5\times 10^{-7}(\lambda/0.25)\sqrt{B/0.1}$, where $\lambda$ is a dimensionless parameter related to the interstring distance, and $B$ measures the importance of perturbations induced by cosmic strings. We argue that, conservatively, the data available currently still permit $B\lesssim 0.1$. Precision measurements sensitive to the B-mode polarization produced by vector density perturbation modes driven by the string network could provide evidence for these models. Accurate determinations of $n_s(k)$, the scalar fluctuation index, could also distinguish among various brane inflation models. 
  In this paper we present new results on the core instability of the 't Hooft Polyakov monopoles we reported on before. This instability, where the spherical core decays in a toroidal one, typically occurs in models in which charge conjugation is gauged. In this paper we also discuss a third conceivable configuration denoted as ``split core'', which brings us to some details of the numerical methods we employed. We argue that a core instability of 't Hooft Polyakov type monopoles is quite a generic feature of models with charged Higgs particles. 
  A result from Palmer, Beatty and Tracy suggests that the two-point function of certain spinless scaling fields in a free Dirac theory on the Poincare disk can be described in terms of Painleve VI transcendents. We complete and verify this description by fixing the integration constants in the Painleve VI transcendent describing the two-point function, and by calculating directly in a Dirac theory on the Poincare disk the long distance expansion of this two-point function and the relative normalization of its long and short distance asymptotics. The long distance expansion is obtained by developing the curved-space analogue of a form factor expansion, and the relative normalization is obtained by calculating the one-point function of the scaling fields in question. The long distance expansion in fact provides part of the solution to the connection problem associated with the Painleve VI equation involved. Calculations are done using the formalism of angular quantization. 
  In this letter, we determine the particle and the string light cone in the pp-wave background. The result is a deformed version of the flat one. We point out the light cone exhibits an intriguing periodicity in the light cone time direction x^+ with a period \sim 1/ \mu. Our results also suggest that a quantum theory in the pp-wave background can be formulated consistently only if the background is periodic in the light cone time x^+. 
  Sen has recently drawn attention to an exact time-dependent Boundary Conformal Field Theory with the space-time interpretation of brane creation and annihilation. An interesting limit of this BCFT is formally equivalent to an array of D-branes located in imaginary time. This raises the question: what is the meaning of D-branes in imaginary time? The answer we propose is that D-branes in imaginary time define purely closed string backgrounds. In particular we prove that the disk scattering amplitude of m closed strings off an arbitrary configuration of imaginary branes is equivalent to a sphere amplitude with m+1 closed string insertions. The extra puncture is a specific closed string state, generically normalizable, that depends on the details of the brane configuration. We study in some detail the special case of the array of imaginary D-branes related to Sen's BCFT and comment on its space-time interpretation. We point out that a certain limit of our set-up allows to study classical black hole creation and suggests a relation between Choptuik's critical behavior and a phase-transition a` la Gregory-Laflamme. We speculate that open string field theory on imaginary D-branes is dual to string theory on the corresponding closed string background. 
  Based on local gauge invariance, four different kinds of fundamental interactions in Nature are unified in a theory which has $SU(3)_c \otimes SU(2)_L \otimes U(1) \otimes_s Gravitational Gauge Group$ gauge symmetry. In this approach, gravitational field, like electromagnetic field, intermediate gauge field and gluon field, is represented by gauge potential. Four kinds of fundamental interactions are formulated in the similar manner, and therefore can be unified in a direct or semi-direct product group. The model discussed in this paper can be regarded as extension of the standard model to gravitational interactions. The model discussed in this paper is a renormalizable quantum model, so it can be used to study quantum effects of gravitational interactions. 
  The three dimensional nonlinear sigma model is unrenormalizable in perturbative method. By using the $\beta$ function in the nonperturbative Wilsonian renormalization group method, we argue that ${\cal N}=2$ supersymmetric nonlinear $\sigma$ models are renormalizable in three dimensions. When the target space is an Einstein-K\"{a}hler manifold with positive scalar curvature, such as C$P^N$ or $Q^N$, there are nontrivial ultraviolet (UV) fixed point, which can be used to define the nontrivial continuum theory. If the target space has a negative scalar curvature, however, the theory has only the infrared Gaussian fixed point, and the sensible continuum theory cannot be defined. We also construct a model which interpolates between the C$P^N$ and $Q^N$ models with two coupling constants. This model has two non-trivial UV fixed points which can be used to define the continuum theory. Finally, we construct a class of conformal field theories with ${\bf SU}(N)$ symmetry, defined at the fixed point of the nonperturbative $\beta$ function. These conformal field theories have a free parameter corresponding to the anomalous dimension of the scalar fields. If we choose a specific value of the parameter, we recover the conformal field theory defined at the UV fixed point of C$P^N$ model and the symmetry is enhanced to ${\bf SU}(N+1)$. 
  We propose a useful method for deriving the effective theory for a system where BPS and anti-BPS domain walls coexist. Our method respects an approximately preserved SUSY near each wall. Due to the finite width of the walls, SUSY breaking terms arise at tree-level, which are exponentially suppressed. A practical approximation using the BPS wall solutions is also discussed. We show that a tachyonic mode appears in the matter sector if the corresponding mode function has a broader profile than the wall width. 
  For models with several time-dependent components generalized entropies can be defined. This is shown for the Bianchi type IX model. We first derive the Cardy-Verlinde formula under the assumption that the first law of thermodynamics is valid. This leads to an explicit expression of the total entropy associated with this type of universes. Assuming the validity of the Cardy entropy formula, we obtain expressions for the corresponding Bekenstein, Bekenstein-Hawking and Hubble entropies. We discuss the validity of the Cardy-Verlinde formula and possible extensions of the outlined procedure to other time-dependent models. 
  We consider solitonic solutions of the DBI tachyon effective action for a non-BPS brane. When wrapped on a circle, these solutions are regular and have a finite energy. We show that in the decompactified limit, these solitons give Sen's infinitely thin finite energy kink -- interpreted as a BPS brane -- provided that some conditions on the potential hold. In particular, if for large $T$ the potential is exponential, $V = e^{-T^a}$, then Sen's solution is only found for $a<1$. For power-law potentials $V = 1/T^b$, one must have $b>1$. If these conditions are not satisfied, we show that the lowest energy configuration is the unstable tachyon vacuum with no kinks. We examine the stability of the solitons and the spectrum of small perturbations. 
  We solve closed string theory in all regular homogeneous plane-wave backgrounds with homogeneous NS three-form field strength and a dilaton. The parameters of the model are constant symmetric and anti-symmetric matrices k_{ij} and f_{ij} associated with the metric, and a constant anti-symmetric matrix h_{ij} associated with the NS field strength. In the light-cone gauge the rotation parameters f_{ij} have a natural interpretation as a constant magnetic field. This is a generalisation of the standard Landau problem with oscillator energies now being non-trivial functions of the parameters f_{ij} and k_{ij}. We develop a general procedure for solving linear but non-diagonal equations for string coordinates, and determine the corresponding oscillator frequencies, the light-cone Hamiltonian and level matching condition. We investigate the resulting string spectrum in detail in the four-dimensional case and compare the results with previously studied examples. Throughout we will find that the presence of the rotation parameter f_{ij} can lead to certain unusual and unexpected features of the string spectrum like new massless states at non-zero string levels, stabilisation of otherwise unstable (tachyonic) modes, and discrete but not positive definite string oscillator spectra. 
  We show that a large class of supersymmetric compactifications, including all simply connected Calabi-Yau and G_2 manifolds, have classical configurations with negative energy density as seen from four dimensions. In fact, the energy density can be arbitrarily negative -- it is unbounded from below. Nevertheless, positive energy theorems show that the total ADM energy remains positive. Physical consequences of the negative energy density include new thermal instabilities, and possible violations of cosmic censorship. 
  We consider the model with the dilaton and twisted moduli fields, which is inspired by type I string models. Stabilization of their vacuum expectation values is studied. We find the stabilization of the twisted moduli field has different aspects from the dilaton stabilization. 
  Quantum mechanical fluctuations in an interval give rise to the Casimir effect, which destabilizes the size of the interval. This can be problematic in constructing Kaluza-Klein theories. We consider the possibility that a breakdown of the Poincar\'e symmetry in an extra dimension can solve this instability problem. As a specific example, we consider the space-time with a $\kappa$-deformed Poincar\'e algebra, calculate the Casimir force between two plates, and find that we can have an interval with a stable size. 
  The anomalous Ward identity is derived for $N = 2$ SUSY Yang-Mills theories, which is resulted out of Wrapping of $D_5$ branes on Supersymmetric two cycles.   From the Ward identity One obtains the Witten-Dijkgraaf-Verlinde-Verlinde equation and hence can solve for the pre-potential. This way one avoids the problem of enhancon which maligns the non-perturbative behaviour of the Yang-Mills theory resulted out of Wrapped branes. 
  Very recent CMB data of WMAP offers an opportunity to test inflation models, in particular, the running of spectral index is quite new and can be used to rule out some models. We show that an noncommutative spacetime inflation model gives a good explanation of these new results. In fitting the data, we also obtain a relationship between the noncommutative parameter (string scale) and the ending time of inflation. 
  The large-group behavior of the nonlocal YM$_2$'s and gYM$_2$'s on a cylinder or a disk is investigated. It is shown that this behavior is similar to that of the corresponding local theory, but with the area of the cylinder replaced by an effective area depending on the dominant representation. The critical areas for nonlocal YM$_2$'s on a cylinder with some special bounary conditions are also obtained. 
  In the framework of the quantum inverse scattering method, we consider a problem of constructing local operators for two-dimensional quantum integrable models, especially for the lattice versions of the nonlinear Schrodinger and sine-Gordon models. We show that a certain class of local operators can be constructed from the matrix elements of the monodromy matrix in a simple way. They are closely related to the quantum projectors and have nice commutation relations with the half of the matrix elements of the elementary monodromy matrix. The form factors of these operators can be calculated by using the standard algebraic Bethe ansatz techniques. 
  We consider the Cartan subalgebra of any very extended algebra G+++ where G is a simple Lie algebra and let the parameters be space-time fields. These are identified with diagonal metrics and dilatons. Using the properties of the algebra, we find that for all very extensions G+++ of simple Lie algebras there are theories of gravity and matter, which admit classical solutions carrying representations of the Weyl group of G+++. We also identify the T and S-dualities of superstrings and of the bosonic string with Weyl reflections and outer automorphisms of well-chosen very extended algebras and we exhibit specific features of the very extensions. We take these results as indication that very extended algebras underlie symmetries of any consistent theory of gravity and matter, and might encode basic information for the construction of such theory. 
  We develop a matrix model to describe bilayered quantum Hall fluids for a series of filling factors. Considering two coupling layers, and starting from a corresponding action, we construct its vacuum configuration at \nu=q_iK_{ij}^{-1}q_j, where K_{ij} is a 2\times 2 matrix and q_i is a vector. Our model allows us to reproduce several well-known wave functions. We show that the wave function \Psi_{(m,m,n)} constructed years ago by Yoshioka, MacDonald and Girvin for the fractional quantum Hall effect at filling factor {2\over m+n} and in particular \Psi_{(3,3,1)} at filling {1\over 2} can be obtained from our vacuum configuration. The unpolarized Halperin wave function and especially that for the fractional quantum Hall state at filling factor {2\over 5} can also be recovered from our approach. Generalization to more than 2 layers is straightforward. 
  We describe a supersymmetric example of the holographic duality between 3 dimensional vector models and higher spin gauge theories in AdS_4, first proposed in a bosonic context by Klebanov and Polyakov. We argue that a particular Fradkin-Vasiliev type supersymmetric higher spin gauge theory in AdS_4 with 16 supersymmetries is dual to the singlet sector of bilinear operators of a free N = 4, SU(N) vector model defined on the boundary. Starting from the duality between type IIB on AdS_5 \times S^5 with a D5-brane as an AdS_4 \times S^2 subspace and the 4 dimensional SU(N) SYM with a defect, we recover the duality between our vector model and the higher spin gauge theory. In this case, we propose that the higher spin gauge theory is a truncation of the open string theory on the world volume of the D5-brane in its tensionless string limit. We also comment on a possible Higgs mechanism in our model. 
  We study the gauged sigma model and its mirror Landau-Ginsburg model corresponding to type IIA on the Fermat degree-24 hypersurface in WCP^4[1,1,2,8,12] (whose blow-up gives the smooth CY_3(3,243)) away from the orbifold singularities, and its orientifold by a freely-acting antiholomorphic involution. We derive the Picard-Fuchs equation obeyed by the period integral as defined in the work of Cecotti and Hori-Vafa, of the parent N=2 type IIA theory of Kachru and Vafa. We obtain the Meijer's basis of solutions to the equation in the large {\it and} small complex structure limits (on the mirror Landau-Ginsburg side) of the abovementioned Calabi-Yau, and make some remarks about the monodromy properties associated based on the work of Morrison, at the same and another MATHEMATICAlly interesting point. Based on a recently shown N=1 four-dimensional triality (hep-th/0212054) between Heterotic on the self-mirror Calabi-Yau CY_3(11,11), M theory on (CY_3(3,243) x S^1)/Z_2 and F-theory on an elliptically fibered CY_4 with the base given by CP^1 x Enriques surface, we first give a heuristic argument that there can be no superpotential generated in the orientifold of of CY_3(3,243), and then explicitly verify the same using mirror symmetry formulation of Hori-Vafa for the abovementioned hypersurface away from its orbifold singularities. We then discuss briefly the sigma model and the mirror Landau-Ginsburg model corresponding to the resolved Calabi-Yau as well. 
  We show that the supertube configurations exist in all supersymmetric type IIA backgrounds which are purely geometrical and which have, at least, one flat direction. In other words, they exist in any spacetime of the form R^{1,1} x M_8, with M_8 any of the usual reduced holonomy manifolds. These generalised supertubes preserve 1/4 of the supersymmetries preserved by the choice of the manifold M_8. We also support this picture with the construction of their corresponding family of IIA supergravity backgrounds preserving from 1/4 to 1/32 of the total supercharges. 
  We discuss in detail the properties of gravity with a negative cosmological constant as viewed in Cherns-Simons theory on a line times a disc. We reanalyze the problem of computing the BTZ entropy, and show how the demand of unitarity and modular invariance of the boundary conformal field theory severely constrain proposals in this framework. 
  In the framework of $Z_N$ orbifolds, we discuss effects of heterotic string backgrounds including discrete Wilson lines on the Yukawa matrices and their connection to CP violation. 
  We study a simple model of p-adic closed and open strings. It sheds some light on the dynamics of tachyon condensation for both types of strings. We calculate the effect of static and decaying D-brane configurations on the closed string background. For closed string tachyons we find lumps analogous to D-branes. By studying their fluctuation spectrum and the D-branes they admit, we argue that closed string lumps should be interpreted as spacetimes of lower dimensionality described by some noncritical p-adic string theory. 
  The 7--particle form factors of the fundamental spin field of the O(3) nonlinear $\sigma$--model are constructed. We calculate the corresponding contribution to the spin--spin correlation function, and compare with predictions from the spectral density scaling hypothesis. The resulting approximation to the spin--spin correlation function agrees well with that computed in renormalized (asymptotically free) perturbation theory in the expected energy range. Further we observe simple lower and upper bounds for the sum of the absolute square of the form factors which may be of use for analytic estimates. 
  When the interaction potential is suitably reordered, the Moyal field theory admits two types of Galilean symmetries, namely the conventional mass-parameter-centrally-extended one with commuting boosts, but also the two-fold centrally extended ``exotic'' Galilean symmetry, where the commutator of the boosts yields the noncommutative parameter. In the free case, one gets an ``exotic'' two-parameter central extension of the Schroedinger group. The conformal symmetry is, however, broken by the interaction. 
  Brane gas cosmology is a scenario inspired by string theory which proposes a simple resolution to the initial singularity problem and gives a dynamical explanation for the number of spatial dimensions of our universe. In this work we have studied analytically and numerically the late-time behaviour of these type of cosmologies taking a proper care of the annihilation of winding modes. This has help us to clarify and extend several aspects of their dynamics. We have found that the decay of winding states into non-winding states behaving like a gas of ordinary non-relativistic particles precludes the existence of a late expansion phase of the universe and obstructs the growth of three large spatial dimensions as we observe today. We propose a generic solution to this problem by considering the dynamics of a gas of non-static branes. We have also obtained a simple criterion on the initial conditions to ensure the small string coupling approximation along the whole dynamical evolution, and consequently, the consistency of an effective low-energy description. Finally, we have reexamined the general conditions for a loitering period in the evolution of the universe which could serve as a mechanism to resolve the {\sl brane problem} - a problem equivalent to the {\sl domain wall problem} in standard cosmology - and discussed the scaling properties of a self-interacting network of winding modes taking into account the effects of the dilaton dynamics. 
  We argue that the N=1 higher-spin theory on AdS4 is holographically dual to the N=1 supersymmetric critical O(N) vector model in three dimensions. This appears to be a special form of the AdS/CFT correspondence in which both regular and irregular bulk modes have similar roles and their interplay leads simultaneously to both the free and the interacting phases of the boundary theory. We study various boundary conditions that correspond to boundary deformations connecting, for large-N, the free and interacting boundary theories. We point out the importance of parity in this holography and elucidate the Higgs mechanism responsible for the breaking of higher-spin symmetry for subleading N. 
  Field theoretic models possessing a global internal fermionic shift symmetry are considered. When such a symmetry is realized locally, spin 3/2 fields appear naturally as gauge fields. Implementation of the gauging procedure requires not only the usual replacement of ordinary derivatives by covariant derivatives containing the spin 3/2 fields, but also the inclusion of additional monomials. The Higgs mechanism and the high energy Nambu-Goldstone fermion equivalence theorem are explicitly demonstrated. 
  Numerical solutions of Einstein's and scalar-field equations are found for a global defect in a higher-dimensional spacetime. The defect has a $(3+1)$-dimensional core and a ``hedgehog'' scalar-field configuration in $n=3$ extra dimensions. For sufficiently low symmetry-breaking scales $\eta$, the solutions are characterized by a flat worldsheet geometry and a constant solid deficit angle in the extra dimensions, in agreement with previous work. For $\eta$ above the higher-dimensional Planck scale, we find that static-defect solutions are singular. The singularity can be removed if the requirement of staticity is relaxed and defect cores are allowed to inflate. We obtain an analytic solution for the metric of such inflating defects at large distances from the core. The three extra dimensions of the nonsingular solutions have a ``cigar'' geometry. Although our numerical solutions were obtained for defects of codimension $n=3$, we argue that the conclusions are likely to apply to all $n\geq 3$. 
  We study the entropy of concrete de Sitter flux compactifications and deformations of them containing D-brane domain walls. We determine the relevant causal and thermodynamic properties of these "D-Sitter" deformations of de Sitter spacetimes. We find a string scale correspondence point at which the entropy localized on the D-branes (and measured by probes sent from an observer in the middle of the bubble) scales the same with large flux quantum numbers as the entropy of the original de Sitter space, and at which Bousso's bound is saturated by the D-brane degrees of freedom (up to order one coefficients) for an infinite range of times. From the geometry of a static patch of D-Sitter space and from basic relations in flux compactifications, we find support for the possibility of a low energy open string description of the static patch of de Sitter space. 
  In a previous work, it was shown that the 8-dimensional topological quantum field theory for a metric and a Kalb-Ramond 2-form gauge field determines N = 1 D = 8 supergravity. It is shown here that, the combination of this TQFT with that of a 3-form determines N = 2 D = 8 supergravity, that is, an untruncated dimensional reduction of N = 1 D = 11 supergravity. Our construction holds for 8-dimensional manifolds with Spin(7) \subset SO(8) holonomy. We suggest that the origin of local Poincare supersymmetry is the gravitational topological symmetry. We indicate a mechanism for the lift of the TQFT in higher dimensions, which generates Chern-Simons couplings. 
  Nonperturbative treatments of the UV limit of pure gravity suggest that it admits a stable fixed point with positive Newton's constant and cosmological constant. We prove that this result is stable under the addition of a scalar field with a generic potential and nonminimal couplings to the scalar curvature. There is a fixed point where the mass and all nonminimal scalar interactions vanish while the gravitational couplings have values which are almost identical to the pure gravity case. We discuss the linearized flow around this fixed point and find that the critical surface is four-dimensional. In the presence of other, arbitrary, massless minimally coupled matter fields, the existence of the fixed point, the sign of the cosmological constant and the dimension of the critical surface depend on the type and number of fields. In particular, for some matter content, there exist polynomial asymptotically free scalar potentials, thus providing a solution to the well-known problem of triviality. 
  We provide a method and the results for the calculation of the holonomy of a Yang-Mills connection in an arbitrary triangular path, in an expansion (developed here to fifth order) in powers of the corresponding segments. The results might have applications in generalizing to Yang-Mills fields previous calculations of the corrections to particle dynamics induced by loop quantum gravity, as well as in the field of random lattices. 
  We propose a new interpretation of the c=1 matrix model as the world-line theory of N unstable D-particles, in which the hermitian matrix is provided by the non- abelian open string tachyon. For D-particles in 1+1-d string theory, we find a direct quantitative match between the closed string emission due to a rolling tachyon and that due to a rolling eigenvalue in the matrix model. We explain the origin of the double-scaling limit, and interpret it as an extreme representative of a large equivalence class of dual theories. Finally, we define a concrete decoupling limit of unstable D-particles in IIB string theory that reduces to the c=1 matrix model, suggesting that 1+1-d string theory represents the near-horizon limit of an ultra-dense gas of IIB D-particles. 
  Recently it was observed that the hyperbolic compactification of M/string theory related to S-branes may lead to a transient period of acceleration of the universe. We study time evolution of the corresponding effective 4d cosmological model supplemented by cold dark matter and show that it is marginally possible to describe observational data for the late-time cosmic acceleration in this model. However, investigation of the compactification 11d -> 4d suggests that the Compton wavelengths of the KK modes in this model are of the same order as the size of the observable part of the universe. Assuming that this problem, as well as several other problems of this scenario, can be resolved, we propose a possible solution of the cosmological coincidence problem due to relation between the dark energy density and the effective dimensionality of the universe. 
  Using only general properties of the tachyon potential we show that inflation may be generic when many branes and anti-branes become coincident. Inflation may occur because of: (1) the assistance of the many diagonal tachyon fields; (2) when the tachyons condense in a staggered fashion; or (3) when some of them condense very late. We point out that such inflation is in some sense a stringy implementation of chaotic inflation and may have important applications for ``regularizing'' a lopsided or singular cosmological compact surface. 
  We study gravitational corrections to the effective superpotential in theories with a single adjoint chiral multiplet, using the generalized Konishi anomaly and the gravitationally deformed chiral ring. We show that the genus one correction to the loop equation in the corresponding matrix model agrees with the gravitational corrected anomaly equations in the gauge theory. An important ingrediant in the proof is the lack of factorization of chiral gauge invariant operators in presence of a supergravity background. We also find a genus zero gravitational correction to the superpotential, which can be removed by a field redefinition. 
  The construction of dual theories for linearized gravity in four dimensions is considered. Our approach is based on the parent Lagrangian method previously developed for the massive spin-two case, but now considered for the zero mass case. This leads to a dual theory described in terms of a rank two symmetric tensor, analogous to the usual gravitational field, and an auxiliary antisymmetric field. This theory has an enlarged gauge symmetry, but with an adequate partial gauge fixing it can be reduced to a gauge symmetry similar to the standard one of linearized gravitation. We present examples illustrating the general procedure and the physical interpretation of the dual fields. The zero mass case of the massive theory dual to the massive spin-two theory is also examined, but we show that it only contains a spin-zero excitation. 
  We construct the covariant nonlocal action for recently suggested long-distance modifications of gravity theory motivated by the cosmological constant and cosmological acceleration problems. This construction is based on the special nonlocal form of the Einstein-Hilbert action explicitly revealing the fact that this action within the covariant curvature expansion begins with curvature-squared terms. Nonlocal form factors in the action of both quantum and brane-induced nature are briefly discussed. In particular, it is emphasized that for certain class of quantum initial value problems nonlocal nature of the Euclidean action does not contradict the causality of effective equations of motion. 
  We study the BRST quantization of bosonic and NSR strings propagating in AdS(3) x N backgrounds. The no-ghost theorem is proved using the Frenkel-Garland-Zuckerman method. Regular and spectrally-flowed representations of affine SL(2,R) appear on an equal footing. Possible generalizations to related curved backgrounds are discussed. 
  We study the wave equation for a massive scalar field in three-dimensional AdS-black hole and dS (de Sitter) spaces to find what is the difference and similarity between two spaces. Here the AdS-black hole is provided by the J=0 BTZ black hole. To investigate its event (cosmological) horizons, we compute the absorption cross section, quasinormal modes, and study the AdS(dS)/CFT correspondences. Although there remains an unclear point in defining the ingoing flux near infinity of the BTZ black hole, quasinormal modes are obtained and the AdS/CFT correspondence is confirmed. However, we do not find quasinormal modes and thus do not confirm the assumed dS/CFT correspondence. This difference between AdS-black hole and dS spaces is very interesting, because their global structures are similar to each other. 
  We construct a manifestly SO(4) x SO(4) invariant, supersymmetric extension of the closed string cubic interaction vertex and dynamical supercharges in light-cone string field theory on the plane wave space-time. We find that the effective vertex for states built out of bosonic creation oscillators coincides with the one previously constructed in the SO(8) formalism and conjecture that in general the two formulations are physically equivalent. Further evidence for this claim is obtained from the discrete Z_2-symmetry of the plane wave and by computing the mass-shift of the simplest stringy state using perturbation theory. We verify that the leading non-planar correction to the anomalous dimension of the dual gauge theory operators is correctly recovered. 
  We compute the billiards that emerge in the Belinskii-Khalatnikov-Lifshitz (BKL) limit for all pure supergravities in D=4 spacetime dimensions, as well as for D=4, N=4 supergravities coupled to k (N=4) Maxwell supermultiplets. We find that just as for the cases N=0 and N=8 investigated previously, these billiards can be identified with the fundamental Weyl chambers of hyperbolic Kac-Moody algebras. Hence, the dynamics is chaotic in the BKL limit. A new feature arises, however, which is that the relevant Kac-Moody algebra can be the Lorentzian extension of a twisted affine Kac-Moody algebra, while the N=0 and N=8 cases are untwisted. This occurs for N=5, N=3 and N=2. An understanding of this property is provided by showing that the data relevant for determining the billiards are the restricted root system and the maximal split subalgebra of the finite-dimensional real symmetry algebra characterizing the toroidal reduction to D=3 spacetime dimensions. To summarize: split symmetry controls chaos. 
  Moduli spaces of conformal field theories corresponding to current-current deformations are discussed. For WZW models, CFT and sigma model considerations are compared. It is shown that current-current deformed WZW models have WZW-like sigma model descriptions with non-bi-invariant metrics, additional B-fields and a non-trivial dilaton. 
  We quantise complex, infinite-dimensional projective space CP(H). We apply the result to quantise a complex, finite-dimensional, classical phase space C whose symplectic volume is infinite, by holomorphically embedding it into CP(H). The embedding is univocally determined by requiring it to be an isometry between the Bergman metric on C and the Fubini-Study metric on CP(H). Then the Hilbert-space bundle over C is the pullback, by the embedding, of the Hilbert-space bundle over CP(H). 
  In this paper we obtain strings that propagate in the quantized pp-wave backgrounds. We can obtain these strings from the solutions of membrane. The other way is the propagation of a massless string in a spacetime with two time dimensions. This string sweeps a worldvolume, which enables us to obtain other strings in the quantized pp-wave backgrounds in the spacetime with one time direction. The associated algebras and Hamiltonians of these massive strings will be studied. 
  A quantum-mechanical technique is used within the framework of U(2) super-Yang-Mills theory to investigate what happens in the process after recombination of two D-p-branes at one angle. Two types of initial conditions are considered, one of which with $p=4$ is a candidate of inflation mechanism. It is observed that the branes' shapes come to have three extremes due to localization of tachyon condensation. Furthermore, open string pairs connecting the decaying D-branes are shown to be created; most part of the released energy is used to create them. It also strongly suggests that creation of closed strings happens afterward. Closed strings as gravitational radiation from the D-branes are also shown to be created. A few speculations are also given on implications of the above phenomena for an inflation model. 
  The \gamma-trace anomaly of supersymmetry current in a supersymmetric gauge theory shares a superconformal anomaly multiplet with the chiral R-symmetry anomaly and the Weyl anomaly, and its holographic reproduction is a valuable test to the AdS/CFT correspondence conjecture. We investigate how the \gamma-trace anomaly of the supersymmetry current of {\cal N}=1 four-dimensional supersymmetric gauge theory in an {\cal N}=1 conformal supergravity background can be extracted out from the ${\cal N}=2$ gauged supergravity in five dimensions. It is shown that the reproduction of this super-Weyl anomaly originates from the following two facts: First the {\cal N}=2 bulk supersymmetry transformation converts into {\cal N}=1 superconformal transformation on the boundary, which consists of {\cal N}=1 supersymmetry transformation and special conformal supersymmetry (or super-Weyl) transformation; second the supersymmetry variation of the bulk action of five-dimensional gauged supergravity is a total derivative. The non-compatibility of supersymmetry and the super-Weyl transformation invariance yields the holographic supersymmetry current anomaly. Furthermore, we speculate on that the contribution from the external gauge and gravitational background fields to the superconformal anomaly may have different holographic origin. 
  We obtain the thermal one loop free energy and the Hagedorn temperature of IIA superstring theory on the pp-wave geometry which comes from the circle compactification of the maximally supersymmetric eleven dimensional one. We use both operator and path integral methods and find the complete agreement between them in the free energy expression. In particular, the free energy in the $\mu \to \infty$ limit is shown to be identical with that of IIB string theory on maximally supersymmetric pp-wave, which indicates the universal thermal behavior of strings in the large class of pp-wave backgrounds. We show that the zero point energy and the modular properties of the free energy are naturally incorporated into the path integral formalism. 
  The N=1 Super Sinh-Gordon model with spontaneously broken supersymmetry is considered. Explicit expressions for form-factors of the trace of the stress energy tensor Theta, the energy operator epsilon, as well as the order and disorder operators sigma and mu are proposed. 
  We consider a $R^{1,d}/Z_2$ orbifold, where $Z_2$ acts by time and space reversal, also known as the embedding space of the elliptic de Sitter space. The background has two potentially dangerous problems: time-nonorientability and the existence of closed time-like curves. We first show that closed causal curves disappear after a proper definition of the time function. We then consider the one-loop vacuum expectation value of the stress tensor. A naive QFT analysis yields a divergent result. We then analyze the stress tensor in bosonic string theory, and find the same result as if the target space would be just the Minkowski space $R^{1,d}$, suggesting a zero result for the superstring. This leads us to propose a proper reformulation of QFT, and recalculate the stress tensor. We find almost the same result as in Minkowski space, except for a potential divergence at the initial time slice of the orbifold, analogous to a spacelike Big Bang singularity. Finally, we argue that it is possible to define local S-matrices, even if the spacetime is globally time-nonorientable. 
  In this letter, we emphasize the effect that the inclusion of a electromagnetic properties for a string brings logarithmic divergences to the accretion problem and the formation and evolution of wakes can break down. 
  A simple method for breaking gauge groups by orbifolding is presented. We extend the method of Kac and Peterson to include Wilson lines. The complete classification of the gauge group breaking, e.g. from heterotic string, is now possible. From this Dynkin diagram technique, one can easily visualize the origin and the symmetry pattern of the surviving gauge group. 
  The theory of self-dual bosonic lumps immersed in the Cayley-calibrated space of octonions has a large class of exact finite quaternionic power series solutions. 
  In 1938, Stueckelberg introduced a scalar field which makes an Abelian gauge theory massive but preserves gauge invariance. The Stueckelberg mechanism is the introduction of new fields to reveal a symmetry of a gauge--fixed theory. We first review the Stueckelberg mechanism in the massive Abelian gauge theory.   We then extend this idea to the standard model, stueckelberging the hypercharge U(1) and thus giving a mass to the physical photon. This introduces an infrared regulator for the photon in the standard electroweak theory, along with a modification of the weak mixing angle accompanied by a plethora of new effects. Notably, neutrinos couple to the photon and charged leptons have also a pseudo-vector coupling. Finally, we review the historical influence of Stueckelberg's 1938 idea, which led to applications in many areas not anticipated by the author, such as strings. We describe the numerous proposals to generalize the Stueckelberg trick to the non-Abelian case with the aim to find alternatives to the standard model. Nevertheless, the Higgs mechanism in spontaneous symmetry breaking remains the only presently known way to give masses to non-Abelian vector fields in a renormalizable and unitary theory. 
  The Lorentzian Kac-Moody algebra E11, obtained by doubly overextending the compact E8, is decomposed into representations of its canonical hyperbolic E10 subalgebra. Whereas the appearing representations at levels 0 and 1 are known on general grounds, higher level representations can currently only be obtained by recursive methods. We present the results of such an analysis up to height 120 in E11 which comprises representations on the first five levels. The algorithms used are a combination of Weyl orbit methods and standard methods based on the Peterson and Freudenthal formulae. In the appendices we give all multiplicities of E10 occuring up to height 340 and for E11 up to height 240. 
  We study four dimensional N=2 SO/SP supersymmetric gauge theory on R^3\times S^1 deformed by a tree level superpotential. We will show that the exact superpotential can be obtained by making use of the Lax matrix of the corresponding integrable model which is the periodic Toda lattice. The connection between vacua of SO(2N) and SO(2kN-2k+2) can also be seen in this framework. Similar analysis can also be applied for SO(2N+1) and SP(2N). 
  We consider a toy metric in four dimensional space-time defined in terms of a recursive hierarchical prescription. The matter distribution turns out to be extremely inhomogeneous. Surprisingly, for very large samples the average mass density tends (very slowly) to a constant. There is no trace of fractal dimension left. 
  Gluon and ghost condensates of dimension two and their relevance for Yang-Mills theories are briefly reviewed 
  We investigate holography on an (n-1)-dimensional brane embedded in a background of AdS black holes, in n-dimensional Gauss-Bonnet gravity. We demonstrate that for a critical brane near the AdS boundary, the Friedmann equation corresponds to that of the standard cosmology driven by a CFT dual to the AdS bulk. We show that there is no holographic description for non-critical branes, or when the brane is further away from the AdS boundary. We then derive a Cardy-Verlinde formula for the dual CFT on the critical brane near the boundary. This gives us insight into the remarkable correspondence between Cardy-Verlinde formulae and Friedmann equations in Einstein gravity. 
  We investigate the geometry of the matrix model associated with an N=1 super Yang-Mills theory with three adjoint fields, which is a massive deformation of N=4. We study in particular the Riemann surface underlying solutions with arbitrary number of cuts. We show that an interesting geometrical structure emerges where the Riemann surface is related on-shell to the Donagi-Witten spectral curve. We explicitly identify the quantum field theory resolvents in terms of geometrical data on the surface. 
  We determine the moduli space of classical solutions to the field equations of Poisson Sigma Models on arbitrary Riemann surfaces for Poisson structures with vanishing Poisson form class. This condition ensures the existence of a presymplectic form on the target Poisson manifold which agrees with the induced symplectic forms of the Poisson tensor upon pullback to the leaves. The dimension of the classical moduli space as a function of the genus of the worldsheet Sigma and the corank k of the Poisson tensor is determined as k(rank(H^1(Sigma))+1). Representatives of the classical solutions are provided using the above mentioned presymplectic 2-forms, and possible generalizations to cases where such a form does not exist are discussed. The results are compared to the known moduli space of classical solutions for two-dimensional BF and Yang-Mills theories. 
  We construct consistent non-linear Kaluza Klein reduction ansatze for a subset of fields arising from the reduction of IIB* and M* theory on dS_5 x H^5 and dS_4 x AdS_7, respectively. These reductions yield four and five-dimensional de Sitter supergravities, albeit with wrong sign kinetic terms. We also demonstrate that the ansatze may be used to lift multi-centered de Sitter black hole solutions to ten and eleven dimensions. The lifted dS_5 black holes correspond to rotating E4-branes of IIB* theory. 
  In our previous papers, we prove the no-ghost theorem without light-cone directions (hep-th/0005002, hep-th/0303051). We point out that our results are valid for more general backgrounds. In particular, we prove the no-ghost theorem for AdS_3 in the context of the BRST quantization (with the standard restriction on the spin). We compare our BRST proof with the OCQ proof and establish the BRST-OCQ equivalence for AdS_3. The key in both approaches lies in the certain structure of the matter Hilbert space as a product of two Verma modules. We also present the no-ghost theorem in the most general form. 
  Motivated by attempts to extend AdS/CFT duality to non-BPS states we consider classical closed string solutions with several angular momenta in different directions of AdS_5 and S^5. We find a novel solution describing a circular closed string located at a fixed value of AdS_5 radius while rotating simultaneously in two planes in AdS_5 with equal spins S. This solution is a direct generalization of a two-spin flat-space solution where the string rotates in two orthogonal planes while always lying on a 3-sphere. Similar solution exists for a string rotating in S^5: it is parametrized by the angular momentum J of the center of mass and the two equal SO(6) angular momenta J_2=J_3=J' in the two rotation planes. The remarkably simple case is of J=0 where the energy depends on J' as E=[(2J')^2 + {\l}]^{1/2} with {\l}being the string tension or `t Hooft coupling. We discuss interpolation of the E(J') relation to weak coupling by identifying the N=4 SYM theory operator that should be dual to the corresponding semiclassical string state and utilizing existing results for its perturbative anomalous dimension. This opens up a possibility of studying AdS/CFT duality in this new non-BPS sector. We also investigate stability of these classical solutions under small perturbations and comment on several generalizations. 
  We investigate the possibility of self-tuning of the effective 4D cosmological constant in 6D supergravity, to see whether it could naturally be of order 1/r^4 when compactified on two dimensions having Kaluza-Klein masses of order 1/r. In the models we examine supersymmetry is broken by the presence of non-supersymmetric 3-branes (on one of which we live). If r were sub-millimeter in size, such a cosmological constant could describe the recently-discovered dark energy. A successful self-tuning mechanism would therefore predict a connection between the observed size of the cosmological constant, and potentially observable effects in sub-millimeter tests of gravity and at the Large Hadron Collider. We do find self tuning inasmuch as 3-branes can quite generically remain classically flat regardless of the size of their tensions, due to an automatic cancellation with the curvature and dilaton of the transverse two dimensions. We argue that in some circumstances six-dimensional supersymmetry might help suppress quantum corrections to this cancellation down to the bulk supersymmetry-breaking scale, which is of order 1/r. We finally examine an explicit realization of the mechanism, in which 3-branes are inserted into an anomaly-free version of Salam-Sezgin gauged 6D supergravity compactified on a 2-sphere with nonzero magnetic flux. This realization is only partially successful due to a topological constraint which relates bulk couplings to the brane tension, although we give arguments why these relations may be stable against quantum corrections. 
  We show that a simple change of the classical boson-fermion coupling constant, $2\alpha \to 2\alpha n $, $n\in \N$, in the superconformal mechanics model gives rise to a radical change of a symmetry: the modified classical and quantum systems are characterized by the nonlinear superconformal symmetry. It is generated by the four bosonic integrals which form the so(1,2) x u(1) subalgebra, and by the 2(n+1) fermionic integrals constituting the two spin-n/2 so(1,2)-representations and anticommuting for the order n polynomials of the even generators. We find that the modified quantum system with an integer value of the parameter $\alpha$ is described simultaneously by the two nonlinear superconformal symmetries of the orders relatively shifted in odd number. For the original quantum model with $|\alpha|=p$, $p\in \N$, this means the presence of the order 2p nonlinear superconformal symmetry in addition to the osp(2|2) supersymmetry. 
  We construct three quasi-supersymmetric $G^3$ GUT models with $S_3$ symmetry and gauge coupling unification from intersecting D6-branes on Type IIA orientifolds. The Standard Model fermions and Higgs doublets can be embedded into the bifundamental representations in these models, and there is no any other unnecessary massless representation. Especially in Model I with gauge group $U(4)^3$, we just have three-family SM fermions and three pairs of Higgs particles. The $G^3$ gauge symmetry in these models can be broken down to the Standard Model gauge symmetry by introducing light open string states. And 1 TeV scale supersymmetry breaking soft masses imply the reasonable intermediate string scale. 
  We investigate the quantum structure of Witten's cubic open bosonic string field theory by computing the one-loop contribution to the open string tadpole using both oscillator and conformal field theory methods. We find divergences and a breakdown of BRST invariance in the tadpole diagram arising from tachyonic and massless closed string states, and we discuss ways of treating these problems. For a Dp-brane with sufficiently many transverse dimensions, the tadpole can be rendered finite by analytically continuing the closed string tachyon by hand; this diagram then naturally incorporates the (linearized) shift of the closed string background due to the presence of the brane. We observe that divergences at higher loops will doom any straightforward attempt at analyzing general quantum effects in bosonic open string field theory on a Dp-brane of any dimension, but our analysis does not uncover any potential obstacles to the existence of a sensible quantum open string field theory in the supersymmetric case. 
  We propose TBA integral equations for multiparticle soliton (fermion) states in the Sine-Gordon (massive Thirring) model. This is based on T-system and Y-system equations, which follow from the Bethe Ansatz solution in the light-cone lattice formulation of the model. Even and odd charge sectors are treated on an equal footing, corresponding to periodic and twisted boundary conditions, respectively. The analytic properties of the Y-system functions are conjectured on the basis of the large volume solution of the system, which we find explicitly. A simple relation between the TBA Y-functions and the counting function variable of the alternative non-linear integral equation (Destri-deVega equation) description of the model is given. 
  We study a gauge fixed action of open string field theory expanded around the universal solution which has been found as an analytic classical solution with one parameter a. For a>-1/2, we are able to reproduce open string scattering amplitudes in the theory fixed in the Siegel gauge. At a=-1/2, all scattering amplitudes vanish and there is no open string excitation in the gauge fixed theory. These results support the conjecture that the universal solution can be regarded as pure gauge or the tachyon vacuum solution. 
  We look for solutions in Einstein gravity corresponding to inflating braneworlds of arbitrary dimension and co-dimension. These solutions correspond to isolated sources (no long range fields). Using dynamical systems techniques, we show that there exists a unique solution corresponding to a black $p$-brane with a regular horizon at the location of the brane. The solution is {\it not} however asymptotically flat, but has global deficit angles. 
  It is well known that, due to vanishing theorems, there are no nontrivial finite action solutions to the Abelian Seiberg-Witten (SW) monopole equations on Euclidean four-dimensional space R^4. We show that this is no longer true for the noncommutative version of these equations, i.e., on a noncommutative deformation R^4_\theta of R^4 there exist smooth solutions to the SW equations having nonzero topological charge. We introduce action functionals for the noncommutative SW equations and construct explicit regular solutions. All our solutions have finite energy. We also suggest a possible interpretation of the obtained solutions as codimension four vortex-like solitons representing D(p-4)- and anti-D(p-4)-branes in a Dp-anti-Dp brane system in type II superstring theory. 
  This paper discusses Skyrmions on the 3-sphere coupled to fermions. The resulting Dirac equation commutes with a generalized angular momentum G. For G = 0 the Dirac equation can be solved explicitly for a constant Skyrme configuration and also for a SO(4) symmetric hedgehog configuration. We discuss how the spectrum changes due to the presence of a non-trivial winding number, and also consider more general Skyrme configurations numerically. 
  One hundred years after its creator's birth, the Dirac equation stands as the cornerstone of XXth Century physics. But it is much more, as it carries the seeds of supersymmetry. Dirac also invented the light-cone, or "front form" dynamics, which plays a crucial role in string theory and in elucidating the finiteness of N=4 Yang-Mills theory. The light-cone structure of eleven-dimensional supergravity (N=8 supergravity in four dimensions) suggests a group-theoretical interpretation of its divergences. We speculate they could be compensated by an infinite number of triplets of massless higher spin fields, each obeying a Dirac-like equation associated with the coset $F_4/SO(9)$. The divergences are proportional to the trace over a non-compact structure containing the compact form of $F_4$. Its nature is still unknown, but it could show the way to $M$-theory. 
  We consider in the Randall-Sundrum scenario the dynamics of a spherically symmetric 3-brane world when matter fields exist in the bulk. We determine exact 5-dimensional solutions which localize gravity near the brane and describe the dynamics of homogeneous polytropic matter on the brane. We show that these geometries are associated with a well defined conformal class of bulk matter fields. We analyze the effective polytropic dynamics on the brane identifying conditions which define it as singular or as globally regular. 
  We calculate quantum corrections to the mass of the vortex in N=2 supersymmetric abelian Higgs model in (2+1) dimensions. We put the system in a box and apply the zeta function regularization. The boundary conditions inevitably violate a part of the supersymmetries. Remaining supersymmetry is however enough to ensure isospectrality of relevant operators in bosonic and fermionic sectors. A non-zero correction to the mass of the vortex comes from finite renormalization of couplings. 
  These notes comprise the second part of two articles devoted to the construction of exact solutions of noncommutative gauge theory in two spacetime dimensions. Here we shall deal with the quantum field theory. Topics covered include an investigation of the symmetries of quantum gauge theory on the noncommutative torus within the path integral formalism, the derivation of the exact expression for the vacuum amplitude, and the classification of instanton contributions. A section dealing with a new, exact combinatorial solution of gauge theory on a two-dimensional fuzzy torus is also included. 
  We discuss a new class of brane models (extending both p-brane and Dp-brane cases) where the brane tension appears as an additional dynamical degree of freedom instead of being put in by hand as an ad hoc dimensionfull scale. Consistency of dynamics naturally involves the appearence of additional higher-rank antisymmetric tensor gauge fields on the world-volume which can couple to charged lower-dimensional branes living on the original Dp-brane world-volume. The dynamical tension has the physical meaning of electric-type field strength of the additional higher-rank world-volume gauge fields. It obeys Maxwell (or Yang-Mills) equations of motion (in the string case p=1) or their higher-rank gauge theory analogues (in the Dp-brane case). This in particular triggers a simple classical mechanism of ("color") charge confinement. 
  This review of bosonic string field theory is concentrated on two main subjects. In the first part we revisit the construction of the three string vertex and rederive the relevant Neumann coefficients both for the matter and the ghost part following a conformal field theory approach. We use this formulation to solve the VSFT equation of motion for the ghost sector. This part of the paper is based on a new method which allows us to derive known results in a simpler way. In the second part we concentrate on the solution of the VSFT equation of motion for the matter part. We describe the construction of the three strings vertex in the presence of a background B field. We determine a large family of lump solutions, illustrate their interpretation as D-branes and study the low energy limit. We show that in this limit the lump solutions flow toward the so-called GMS solitons. 
  We compute glueball superpotentials for four-dimensional, N=1 supersymmetric gauge theories, with arbitrary gauge groups and massive matter representations. This is done by perturbatively integrating out massive, charged fields. The Feynman diagram computations simplify, and are related to the corresponding matrix model. This leads to a natural notion of ``projection to planar diagrams'' for arbitrary gauge groups and representations. We discuss a general ambiguity in the glueball superpotential W(S) for terms, S^n, whose order, n, is greater than the dual Coxeter number. This ambiguity can be resolved for all classical gauge groups, (A,B,C,D), via a natural embedding in an infinite rank supergroup. We use this to address some recently raised puzzles. For exceptional groups, we compute the superpotential terms for low powers of the glueball field and propose an all-order completion for some examples including N=1^* for all simply-laced groups. We also comment on compactification of these theories to lower dimensions. 
  We study black holes in AdS-like spacetimes, with the horizon given by an arbitrary positive curvature Einstein metric. A criterion for classical instability of such black holes is found in the large and small black hole limits. Examples of large unstable black holes have a B\"ohm metric as the horizon. These, classically unstable, large black holes are locally thermodynamically stable. The gravitational instability has a dual description, for example by using the $AdS_7 \times S^4$ version of the AdS/CFT correspondence. The instability corresponds to a critical temperature of the dual thermal field theory defined on a curved background. 
  We calculate the vacuum energy of a spinor field in the background of a Nielsen-Olesen vortex. We use the method of representing the vacuum energy in terms of the Jost function on the imaginary momentum axis. Renormalization is carried out using the heat kernel expansion and zeta functional regularization. With this method well convergent sums and integrals emerge which allow for an efficient numerical calculation of the vacuum energy in the given case where the background is not known analytically but only numerically. The vacuum energy is calculated for several choices of the parameters and it turns out to give small corrections to the classical energy. 
  We evaluate the noncommutative Chern-Simons action induced by fermions interacting with an Abelian gauge field in a noncommutative massive Thirring model in (2+1)-dimensional spacetime. This calculation is performed in the Dirac and Majorana representations. We observe that in Majorana representation when $\theta$ goes to zero we do not have induced Chern-Simons term in the dimensional regularization scheme. 
  It is shown that the supersymmetric extension of the Stelle-West formalism permits the construction of an action for $(3+1)$-dimensional N=1 supergravity with cosmological constant genuinely invariant under the $OSp(4/1).$ Since the action is invariant under the supersymmetric extension of the $AdS$ group, the supersymmetry algebra closes off shell without the need for auxiliary fields. The limit case $m\to 0$, i.e.$(3+1)$ -dimensional N=1 supergravity invariant under the Poincar\'{e} supergroup is also discussed. 
  We prove the dispersionless Hirota equations for the dispersionless Toda, dispersionless coupled modified KP and dispersionless KP hierarchies using an idea from classical complex analysis. We also prove that the Hirota equations characterize the tau functions for each of these hierarchies. As a result, we establish the links between the hierarchies. 
  We summarize how to describe D-branes in a matrix theory based on unstable D-instantons, which we call K-matrix theory, and explicity show that D-branes can be constructed as bound states of infinitly many unstable D-instantons. We examine the fluctuations around Dp-brane solutions in the matrix theory and show that they correctly reproduce fields on the Dp-brane world-volume. Plugging them into the action of the matrix theory, we precisely obtain the Dp-brane action as the effective action of the fluctuations. 
  We investigate an AdS_4 x L_2 D5-brane in AdS_5 x X_5 space-time, in the context of AdS/dCFT correspondence. Here, X_5 is a Sasaki-Einstein manifold and L_2 is a submanifold of X_5. This brane has the same supersymmetry as the 3-dimensional N=1 superconformal symmetry if L_2 is a special Legendrian submanifold in X_5. In this case, this brane is supposed to correspond to a superconformal wall defect in 4-dimensional N=4 super Yang-Mills theory. We construct these new string backgrounds and show they have the correct supersymmetry, also in the case with non-trivial gauge flux on L_2. The simplest new example is AdS_4 x T^2 brane in AdS_5 x S^5. We construct the brane solution expressing the RG flow between two different defects. We also perform similar analysis for an AdS_3 x L_3 M5-brane in AdS_4 x X_7, for a weak G_2 manifold X_7 and its submanifold L_3. This system has the same supersymmetry as 2-dimensional N=(1,0) global superconformal symmetry, if L_3 is an associative submanifold. 
  We re-examine the issue of local counter terms in the analysis of quantum anomalies. We analyze two-dimensional theories and show that the notion of local counter terms need to be carefully defined depending on the physics contents such as whether one is analyzing gauge theory or bosonization. It is shown that a part of the Jacobian, which is apparently spurious and eliminated by a local counter term corresponding to the mass term of the gauge field in gauge theory, cannot be removed by a {\it local} counter term and plays a central role by giving the kinetic term of the bosonized field in the context of path integral bosonization. 
  Antisymmetric tensor gauge fields phi_{ab}(eta) are formulated on the surface of a sphere S_4t(eta^2 = a^2) embedded in five dimensions. The free field model is equivalent to a scalar model on this sphere. Interactions with gauge fields are discussed. It is feasable to formulate models for interactions with U(1) gauge fields A_a(eta) that are akin to those of Freedman and Townsend in flat space. In addition, it proves possible to have a novel interaction of phi_{ab}with A_a and a spinor field Psi(eta) on S_4 with both Abelian and non-Abelian gauge invariance. In these models, A_a plays the role of a Stueckelberg field. 
  We study the gauge structure of vacuum string field theory expanded around the D-brane solution, namely, the gauge transformation and the transversality condition of the massless vector fluctuation mode. We find that the gauge transformation on massless vector field is induced as an anomaly; an infinity multiplied by an infinitesimal factor. The infinity comes from the singularity at the edge of the eigenvalue distribution of the Neumann matrix, while the infinitesimal factor from the violation of the equation of motion of the fluctuation modes due to the regularization for the infinity. However, the transversality condition cannot be obtained even if we take into account the anomaly contribution. 
  The tachyon effective field theory describing the dynamics of a non-BPS D-$p$-brane has electric flux tube solutions where the electric field is at its critical value and the tachyon is at its vacuum. It has been suggested that these solutions have the interpretation of fundamental strings. We show that in order that an electric flux tube can `end' on a kink solution representing a BPS D-$(p-1)$-brane, the electric flux must be embedded in a tubular region inside which the tachyon is finite rather than at its vacuum where it is infinite. Energetic consideration then forces the transverse `area' of this tube to vanish. We suggest a possible interpretation of the original electric flux tube solutions around the tachyon vacuum as well as of tachyon matter as system of closed strings at density far above the Hagedorn density. 
  We consider the superposition of infinitely many instantons on a circle in R^4. The construction yields a self-dual solution of the Yang-Mills equations with action density concentrated on the ring. We show that this configuration is reducible in which case magnetic charge can be defined in a gauge invariant way. Indeed, we find a unit charge monopole (worldline) on the ring. This is an analytic example of the correlation between monopoles and action/topological density, however with infinite action. We show that both the Maximal Abelian Gauge and the Laplacian Abelian Gauge detect the monopole, while the Polyakov gauge does not. We discuss the implications of this configuration. 
  We develop a renormalization group formalism for the compactified Randall-Sundrum scenario wherein the extra-dimensional radius serves as the scaling parameter. Couplings on the hidden brane scale as we move within local effective field theories with varying size of the warped extra dimension. We consider this RG approach applied to U(1) gauge theories and gravity. We use this method to derive a low energy effective theory. 
  We consider compactifications of six dimensional gravity in four dimensional Minkowski or de Sitter space times a two dimensional sphere, S^2. As has been recently pointed out, it is possible to introduce 3-branes in these backgrounds with arbitrary tension without affecting the effective four dimensional cosmological constant, since its only effect is to induce a deficit angle in the sphere. We show that if a monopole like configuration of a 6D U(1) gauge field is used to produce the spontaneous compactification of the two extra dimensions in a sphere a fine tuning between brane and bulk parameters is reintroduced once the quantization condition for the gauge field is taken into account, so the 4D cosmological constant depends on the brane tension. This problem is absent if instead of the monopole we consider a four form field strength in the bulk to obtain the required energy-momentum tensor. Also, making use of the four form field, we generalize the solution to an arbitrary number of dimensions (\ge 6), keeping always four noncompact dimensions and compactifying the rest in a n-dimensional sphere. We show that a (n+1)-brane with arbitrary tension can be introduced in this background without affecting the effective 4D cosmological constant. 
  Using Supergravity on $AdS_7\times S^4$ we calculate the bulk one-loop contribution to the conformal anomaly of the (2,0) theory describing $N$ coincident M5 branes. When this is added to the tree-level result, and an additional subleading order contribution calculated by Tseytlin, it gives an expression for the anomaly that interpolates correctly between the large $N$ theory and the free (2,0) tensor theory corresponding to N=1. Thus we can argue that we have identified the exact $N$-dependence of the anomaly, which may have a simple protected form valid away from the large $N$ limit. 
  Non-degenerate perturbation theory, which was used to calculate the scale dimension of operators on the gauge theory side of the correspondence, breaks down when effects of triple trace operators are included. We interpret this as an instability of excited single-string states in the dual string theory for decay into the continuum of degenerate 3-string states. We apply time-dependent perturbation theory to calculate the decay widths from gauge theory. These widths are new gauge theory data which can be compared with future calculations in light cone string field theory. 
  We derive both Lagrangian and Hamiltonian mechanics as gauge theories of Newtonian mechanics. Systematic development of the distinct symmetries of dynamics and measurement suggest that gauge theory may be motivated as a reconciliation of dynamics with measurement. Applying this principle to Newton's law with the simplest measurement theory leads to Lagrangian mechanics, while use of conformal measurement theory leads to Hamilton's equations. 
  Kachru, Kallosh, Linde, & Trivedi recently constructed a four-dimensional de Sitter compactification of IIB string theory, which they showed to be metastable in agreement with general arguments about de Sitter spacetimes in quantum gravity. In this paper, we describe how discrete flux choices lead to a closely-spaced set of vacua and explore various decay channels. We find that in many situations NS5-brane meditated decays which exchange NSNS 3-form flux for D3-branes are comparatively very fast. 
  We construct a solution of the BFSS matrix theory, which is a counterpart of the BIon solution representing a fundamental string ending on a bound state of a D2-brane and D0-branes. We call this solution the `fuzzy BIon' and show that this configuration preserves 1/4 supersymmetry of type IIA superstring theory. We also construct an effective action for the fuzzy BIon by analyzing the nonabelian Born-Infeld action for D0-branes. When we take the continuous limit, with some conditions, this action coincides with the effective action for the BIon configuration. 
  We analyze the ghost condensates <f^{abc}c^{b}c^{c}>, <f^{abc}\oc^{b}\oc^{c}> and <f^{abc}\oc^{b}c^{c}> in Yang-Mills theory in the Curci-Ferrari gauge. By combining the local composite operator formalism with the algebraic renormalization technique, we are able to give a simultaneous discussion of <f^{abc}c^{b}c^{c}>, <f^{abc}\oc^{b}\oc^{c}> and <f^{abc}\oc^{b}c^{c}>, which can be seen as playing the role of the BCS, respectively Overhauser effect in ordinary superconductivity. The Curci-Ferrari gauge exhibits a global continuous symmetry generated by the Nakanishi-Ojima (NO) algebra. This algebra includes, next to the (anti-)BRST transformation, a SL(2,R) subalgebra. We discuss the dynamical symmetry breaking of the NO algebra through these ghost condensates. Particular attention is paid to the Landau gauge, a special case of the Curci-Ferrari gauge. 
  We consider world-sheet instanton effects in N=1 string orientifolds of noncompact toric Calabi-Yau threefolds. We show that unoriented closed string topological amplitudes can be exactly computed using localization techniques for holomorphic maps with involution. Our results are in precise agreement with mirror symmetry and large N duality predictions. 
  Some ideas and remarks are presented concerning a possible Lagrangian approach to the study of internal boundary conditions relating integrable fields at the junction of two domains. The main example given in the article concerns single real scalar fields in each domain and it is found that these may be free, of Liouville type, or of sinh-Gordon type. 
  We compare the matrix model and integrable system approaches to calculating the exact vacuum structure of general N=1 deformations of either the basic N=2 theory or its generalization with a massive adjoint hypermultiplet, the N=2* theory. We show that there is a one-to-one correspondence between arbitrary critical points of the Dijkgraaf-Vafa glueball superpotential and equilibrium configurations of the associated integrable system. The latter being either the periodic Toda chain, for N=2, or the elliptic Calogero-Moser system, for N=2*. We show in both cases that the glueball superpotential at the crtical point equals the associated Hamiltonian. Our discussion includes an analysis of the vacuum structure of the N=1* theory with an arbitrary tree-level superpotential for one of the adjoint chiral fields. 
  We apply the path-integral formalism to compute the anomalies in general orbifold gauge theories (including possible non-trivial Scherk-Schwarz boundary conditions) where a gauge group G is broken down to subgroups H_f at the fixed points y=y_f. Bulk and localized anomalies, proportional to \delta(y-y_f), do generically appear from matter propagating in the bulk. The anomaly zero-mode that survives in the four-dimensional effective theory should be canceled by localized fermions (except possibly for mixed U(1) anomalies). We examine in detail the possibility of canceling localized anomalies by the Green-Schwarz mechanism involving two- and four-forms in the bulk. The four-form can only cancel anomalies which do not survive in the 4D effective theory: they are called globally vanishing anomalies. The two-form may cancel a specific class of mixed U(1) anomalies. Only if these anomalies are present in the 4D theory this mechanism spontaneously breaks the U(1) symmetry. The examples of five and six-dimensional Z_N orbifolds are considered in great detail. In five dimensions the Green-Schwarz four-form has no physical degrees of freedom and is equivalent to canceling anomalies by a Chern-Simons term. In all other cases, the Green-Schwarz forms have some physical degrees of freedom and leave some non-renormalizable interactions in the low energy effective theory. In general, localized anomaly cancellation imposes strong constraints on model building. 
  For a recently proposed pure gauge theory in three dimensions, without a Chern-Simons term, we calculate the static interaction potential within the structure of the gauge-invariant variables formalism. The result coincides with that of the Maxwell-Chern-Simons theory in the short distance regime, which shows the confining nature of the potential. 
  Within the approach of Supersymmetric Quantum Mechanics associated with the variational method a recipe to construct the superpotential of three dimensional confined potentials in general is proposed. To illustrate the construction, the energies of the Harmonic Oscillator and the Hulth\'en potential, both confined in three dimensions are evaluated. Comparison with the corresponding results of other approximative and exact numerical results is presented. 
  The original ideas about noncommuting coordinates are recalled. The connection between U(1) gauge fields defined on noncommuting coordinates and fluid mechanics is explained. Non-Abelian fluid mechanics is described. 
  We study NSR strings in the Nappi-Witten background, which is the Penrose limit of a certain NS5-brane supergravity solution. We solve the theory in the light-cone gauge, obtaining the spectrum, which is space-time supersymmetric. In light of the LST/NS5-brane duality, this spectrum should be in correspondence with the states of little string theory in the appropriate limit. A semiclassical analysis verifies that the relationship between energy and angular momentum, after a field redefinition, matches the known result for a flat background. 
  In this work we compute the vacuum expectation values of the energy-momentum tensor and the average value of a massive, charged scalar field in the presence of a magnetic flux cosmic string for both zero- and finite-temperature cases. 
  It is discussed that the Ernst--Schwarzschild metric describing a nonrotating black hole in the external magnetic field admits the solutions of the Dirac monopole types for the corresponding Maxwell equations. The given solutions are obtained in explicit form and a possible influence of the conforming Dirac monopoles on Hawking radiation is also outlined. 
  The global multiplicative properties of Laplace type operators acting on irreducible rank one symmetric spaces are considered. The explicit form of the multiplicative anomaly is derived and its corresponding value is calculated exactly, for important classes of locally symmetric spaces and different dimensions. 
  The $n+1$-dimensional Milne Universe with extra free directions is used to construct simple FRW cosmological string models in four dimensions, describing expansion in the presence of matter with $p=k \rho $, $k=(4-n)/3n$. We then consider the n=2 case and make SL(2,Z) orbifold identifications. The model is surprisingly related to the null orbifold with an extra reflection generator. The study of the string spectrum involves the theory of harmonic functions in the fundamental domain of SL(2,Z). In particular, from this theory one can deduce a bound for the energy gap and the fact that there are an infinite number of excitations with a finite degeneracy. We discuss the structure of wave functions and give examples of physical winding states becoming light near the singularity. 
  A mass-like quantum Weyl-Poincare algebra is proposed to describe, after the identification of the deformation parameter with the Planck length, a new relativistic theory with two observer-independent scales (or DSR theory). Deformed momentum representation, finite boost transformations, range of rapidity, energy and momentum, as well as position and velocity operators are explicitly studied and compared with those of previous DSR theories based on kappa-Poincare algebra. The main novelties of the DSR theory here presented are the new features of momentum saturation and a new type of deformed position operators. 
  The equations of motion and boundary conditions for the fluctuations around a classical open string, in a curved space-time with torsion, are considered in compact and world-sheet covariant form. The rigidly rotating open strings in Anti de Sitter space with and without torsion are investigated in detail. By carefully analyzing the tangential fluctuations at the boundary, we show explicitly that the physical fluctuations (which at the boundary are combinations of normal and tangential fluctuations) are finite, even though the world-sheet is singular there. The divergent 2-curvature thus seems less dangerous than expected, in these cases. The general formalism can be straightforwardly used also to study the (bosonic part of the) fluctuations around the closed strings, recently considered in connection with the AdS/CFT duality, on AdS_5 \times S^5 and AdS_3 \times S^3 \times T^4. 
  The singular behavior of conformal interactions is examined within a comparative analysis of renormalization frameworks. The effective approach--inspired by the effective-field theory program--and its connection with the core framework are highlighted. Applications include black-hole thermodynamics, molecular dipole-bound anions, the Efimov effect, and various regimes of QED. 
  Path integral quantization of dilaton gravity in two dimensions is applied to the CGHS model to the first nontrivial order in matter loops. Our approach is background independent as geometry is integrated out exactly. The result is an effective shift of the Killing norm: the apparent horizon becomes smaller. The Hawking temperature which is constant to leading order receives a quantum correction. As a consequence, the specific heat becomes positive and proportional to the square of the black hole mass. 
  We study gauge theories based on abelian $p-$ forms on real compact hyperbolic manifolds. The tensor kernel trace formula and the spectral functions associated with free generalized gauge fields are analyzed. 
  We present an algebraic construction of the orthogonal eigenfunctions of the supersymmetric extension of the rational Calogero-Moser-Sutherland model with harmonic confinement. These eigenfunctions are the superspace extension of the generalized Hermite (or Hi-Jack) polynomials. The conserved quantities of the rational supersymmetric model are related to their trigonometric relatives through a similarity transformation. This leads to a simple expression between the corresponding eigenfunctions: the generalized Hermite superpolynomials are written as a differential operator acting on the corresponding Jack superpolynomials. As an aside, the maximal superintegrability of the supersymmetric rational Calogero-Moser-Sutherland model is demonstrated. 
  Supersymmetric solutions of 11-dimensional supergravity can be classified according to the holonomy of the supercovariant derivative arising in the Killing spinor condition. It is shown that the holonomy must be contained in $\SL(32,\R)$. The holonomies of solutions with flux are discussed and examples are analysed. In extending to M-theory, account has to be taken of the phenomenon of ` supersymmetry without supersymmetry'. It is argued that including the fermionic degrees of freedom in M-theory requires a formulation with a local $\SL(32,\R)$ symmetry, analogous to the need for local Lorentz symmetry in coupling spinors to gravity. 
  The correspondences proposed previously between higher spin gauge theories and free singleton field theories were recently extended into a more complete picture by Klebanov and Polyakov in the case of the minimal bosonic theory in D=4 to include the strongly coupled fixed point of the 3d O(N) vector model. Here we propose an N=1 supersymmetric version of this picture. We also elaborate on the role of parity in constraining the bulk interactions, and in distinguishing two minimal bosonic models obtained as two different consistent truncations of the minimal N=1 model that retain the scalar or the pseudo-scalar field. We refer to these models as the Type A and Type B models, respectively, and conjecture that the latter is holographically dual to the 3d Gross-Neveu model. In the case of the Type A model, we show the vanishing of the three-scalar amplitude with regular boundary conditions. This agrees with the O(N) vector model computation of Petkou, thereby providing a non-trivial test of the Klebanov-Polyakov conjecture. 
  We solve the two-component Dirac equation in the presence of a spatially one dimensional symmetric cusp potential. We compute the scattering and bound states solutions and we derive the conditions for transmission resonances as well as for supercriticality. 
  We continue the development of a systematic procedure for deriving closed string pp wave string field theory from the large N Berenstein-Maldacena-Nastase limit. In the present paper the effects of the Yang-Mills interaction are considered in detail for general BMN states. The SFT interaction with the appropriate operator insertion at the interaction point is demonstrated. 
  We study the vacuum stability of a model of massless scalar and fermionic fields minimally coupled to a Chern-Simons field. The classical Lagrangian only involves dimensionless parameters, and the model can be thought as a (2+1) dimensional analog of the Coleman-Weinberg model. By calculating the effective potential, we show that dynamical symmetry breakdown occurs in the two-loop approximation. The vacuum becomes asymmetric and mass generation, for the boson and fermion fields takes place. Renormalization group arguments are used to clarify some aspects of the solution. 
  We study the perturbative approach to the Wilsonian integration of noncommutative gauge theories in the matrix representation. We begin by motivating the study of noncommutative gauge theories and reviewing the matrix formulation. We then systematically develop the perturbative treatment of UV states and calculate both the leading and next to leading order one- and two-loop corrections to the quantum effective action. Throughout, we discuss how our formalism clarifies problems associated with UV-IR mixing, a particular emphasis being placed on the dipole structure imposed by noncommutative gauge invariance. Ultimately, using the structural understanding developed in this work, we are able to determine the exact form of perturbative corrections in the UV regime defined by $\theta\Lambda^2\gg 1$. Finally, we apply our results to the analysis of the divergence structure and show that 3+1 and higher dimensional noncommutative theories that allow renormalization beyond one-loop are not self-consistent. 
  We analyze the relation between the concept of auxiliary variables and the Inverse problem of the calculus of variations to construct a Lagrangian from a given set of equations of motion. The problem of the construction of a consistent second order dynamics from a given first order dynamics is investigated. At the level of equations of motion we find that this reduction process is consistent provided that the mapping of the boundary data be taken properly into account. At the level of the variational principle we analyze the obstructions to construct a second order Lagrangian from a first order one and give an explicit formal non-local Lagrangian that reproduce the second order projected dynamics. Finally we apply our ideas to the so called ``Noncommutative classical dynamics''. 
  We consider general aspects of the realization of R and non-R flavor symmetries in the AdS_5 x H_5 dual of 4d N=1 superconformal field theories. We find a general prescription for computing the charges under these symmetries for baryonic operators, which uses only topological information (intersection numbers) on H_5. We find and discuss a new correspondence between the nodes of the SCFT quiver diagrams and certain divisors in the associated geometry. We also discuss connections between the non-R flavor symmetries and the enhanced gauge symmetries in non-conformal theories obtained by adding wrapped branes. 
  We introduce a transformation for converting a series in a parameter, \lambda, to a series in the inverse of the parameter \lambda^{-1}. By applying the transform on simple examples, it becomes apparent that there exist relations between convergent and divergent series, and also between large- and small-coupling expansions. The method is also applied to the divergent series expansion of Euler-Heisenberg-Schwinger result for the one-loop effective action for constant background magnetic (or electric) field. The transform may help us gain some insight about the nature of both divergent (Borel or non-Borel summable series) and convergent series and their relationship, and how both could be used for analytical and numerical calculations. 
  The AdS/CFT correspondence relates dibaryons in superconformal gauge theories to holomorphic curves in Kaehler-Einstein surfaces. The degree of the holomorphic curves is proportional to the gauge theory conformal dimension of the dibaryons. Moreover, the number of holomorphic curves should match, in an appropriately defined sense, the number of dibaryons. Using AdS/CFT backgrounds built from the generalized conifolds of Gubser, Shatashvili, and Nekrasov (1999), we show that the gauge theory prediction for the dimension of dibaryonic operators does indeed match the degree of the corresponding holomorphic curves. For AdS/CFT backgrounds built from cones over del Pezzo surfaces, we are able to match the degree of the curves to the conformal dimension of dibaryons for the n'th del Pezzo surface, n=1,2,...,6. Also, for the del Pezzos and the A_k type generalized conifolds, for the dibaryons of smallest conformal dimension, we are able to match the number of holomorphic curves with the number of possible dibaryon operators from gauge theory. 
  We incorporate massive flavored fundamental quarks in the supergravity dual of N=1 SYM by introducing D7 brane probes to the Klebanov Strassler background. We find probe configurations that solve the D7 equations of motion. We compute the quadratic fluctuations of the D7 brane and extract the spectrum of vector and pseudo scalar flavored mesons. The spectra found are discrete and exhibit a mass gap of the order of the glueball mass. 
  We study the N=1 version of Argyres-Douglas (AD) points by making use of the recent developments in understanding the dynamics of the chiral sector of N=1 gauge theories. We shall consider N=1 U(N) gauge theories with an adjoint matter and look for the tree-level superpotential W(x) which reproduces the N=2 AD points via the factorization equation relating the N=1 and N=2 curves. We find that the following superpotentials generate the N=2 AD points: (1) W'(x)=x^N \pm 2\Lambda^N, (2) W'(x)=x^n, N-1\ge n \ge N/2+1.   In case (1) the physics is essentially the same as the N=2 theory even in the presence of the superpotential. There seems to be an underlying structure of N-reduced KP hierarchy in the system.   Case (2) occurs at the intersection of a number of N=1 vacua with massless monopoles. This branch of vacua is characterized by having s_+=0 or s_-=0 where s_{\pm} denotes the number of double roots in P_N(x)\pm 2\Lambda^N. It is possible to show that the mass gap in fact vanishes at this AD point. We conjecture that it represents a new class of N=1 superconformal field theory. 
  Lagrangian and Hamiltonian dynamics of de Azcarraga-Lukierski N=2 massive superparticle is considered in the framework of twistor-like Lorentz-harmonic approach. The emphasis is on the study of the interaction with external Abelian gauge superfield. The requirement of preservation of all gauge symmetries of the free model including kappa-symmetry yields correct expressions for the superfield strength constraints and determines the form of nonminimal interaction. We also show that for de Azcarraga-Lukierski N=2 massive superparticle the pullback of field strength 2-superform to the superworld line is not integrable in contrast to the massless superparticle. 
  We derive the spectrum of Kaluza-Klein descendants of string excitations on AdS_5 x S^5. String states are organized in long multiplets of the AdS supergroup SU(2,2|4,R) with a rich pattern of shortenings at the higher spin enhancement point \lambda=0. The string states holographically dual to the higher spin currents of SYM theory in the strict zero coupling limit are identified together with the corresponding Goldstone particles responsible for the Higgsing of the higher spin symmetry at \lambda\neq 0. Exploiting higher spin symmetry we propose a very simple yet effective mass formula and establish a one-to-one correspondence between the complete spectrum of \Delta_0 <= 4 string states and relevant/marginal single-trace deformations in N=4 SYM theory at large N. To this end, we describe how to efficiently enumerate scaling operators in `free' YM theory, with the inclusion of fermionic `letters', by resorting to Polya theory. Comparison between the spectra of 1/4-BPS states is also presented. Finally, we discuss how to organize the spectrum of N=4 SYM theory in SU(2,2|4,R) supermultiplets by means of some kind of `Eratostenes's sieve'. 
  Building a consistent Quantum Theory of Gravity is one of the most challenging aspects of modern theoretical physics. In the past couple of years, new attempts have been made along the path of ``asymptotic safety'' through the use of Exact Renormalisation Group Equations, which hinge on the existence of a non-trivial fixed point of the flow equations. We will first summarize the major results that have been obtained along these lines, then we will consider the effect of introducing matter fields into the theory. Our analyses show that in order to preserve the existence of the fixed point one must satisfy some constraints on the matter content of the theory. 
  We consider closed-string tachyon condensation in the twisted sectors on the C/Z_{2n+1} \times R^{7,1} orbifold. We calculate the localized energy density in the fixed plane on the orbifold at the one-loop level, and we obtain the decay rate per unit volume of the fixed plane to leading order. We show that the decay rate increases monotonically as a function of n. 
  This is a note on the coupled supergravity-tachyon matter system, which has been earlier proposed as a candidate for the effective space-time description of S-branes. In particular, we study an ansatz with the maximal ISO(p+1)xSO(8-p,1) symmetry, for general brane dimensionality p and homogeneous brane distribution in transverse space \rho_\perp. A simple application of singularity theorems shows that (for p \le 7) the most general solution with these symmetries is always singular. (This invalidates a recent claim in the literature.) We include a few general comments about the possibility of describing the decay of unstable D-branes in purely gravitational terms. 
  We discuss the treatment of squeezed states as excitations in the Euclidean vacuum of de Sitter space. A comparison with the treatment of these states as candidate no-particle states, or alpha-vacua, shows important differences already in the free theory. At the interacting level alpha-vacua are inconsistent, but squeezed state excitations seem perfectly acceptable. Indeed, matrix elements can be renormalized in the excited states using precisely the standard local counterterms of the Euclidean vacuum. Implications for inflationary scenarios in cosmology are discussed. 
  We prove the integrality of the open instanton numbers in two examples of counting holomorphic disks on local Calabi-Yau threefolds: the resolved conifold and the degenerate $ \P \times \P $. Given the B-model superpotential, we extract by hand all Gromow-Witten invariants in the expansion of the A-model superpotential. The proof of their integrality relies on enticing congruences of binomial coefficients modulo powers of a prime. We also derive an expression for the factorial $(p^k-1)!$ modulo powers of the prime $p$. We generalise two theorems of elementary number theory, by Wolstenholme and by Wilson. 
  We construct an N=1 theory with gauge group U(nN) and degree n+1 tree level superpotential whose matrix model spectral curve develops an A_{n+1} Argyres-Douglas singularity. We evaluate the coupling constants of the low-energy U(1)^n theory and show that the large N expansion is singular at the Argyres-Douglas points. Nevertheless, it is possible to define appropriate double scaling limits which are conjectured to yield four dimensional non-critical string theories as proposed by Ferrari. In the Argyres-Douglas limit the n-cut spectral curve degenerates into a solution with n/2 cuts for even n and (n+1)/2 cuts for odd n. 
  Recently, a singularity theorem for full SD-brane spacetimes was given in hep-th/0305055. We comment on the relation between this and previous work as well as provide a more geometric formulation interpreted as a no-go theorem. We then point out that some setups of physical interest escape the theorem: cosmological applications, half-SDp-branes and decaying unstable Dp-branes for general p. We also provide indications that the space-filling full SD8-brane (in d=10) escapes as well, because of the important role of Ramond-Ramond fields. In any case, tachyon cosmology is not ruled out by the no-go theorem. Lastly, we remark upon interesting directions for potential generalizations of the theorem, and quantum corrections. 
  We study the large N degeneracy in the structure of the four-point amplitudes of 1/2-BPS operators of arbitrary weight k in perturbative N=4 SYM theory. At one loop (order g^2) this degeneracy manifests itself in a smaller number of independent conformal invariant functions describing the amplitude, compared to AdS_5 supergravity results. To study this phenomenon at the two-loop level (order g^4) we consider a particular N=2 hypermultiplet projection of the general N=4 amplitude. Using the formalism of N=2 harmonic superspace we then explicitly compute this four-point correlator at two loops and identify the corresponding conformal invariant functions. In the cases of 1/2-BPS operators of weight k=3 and k=4 the one-loop large N degeneracy is lifted by the two-loop corrections. However, for weight k > 4 the degeneracy is still there at the two-loop level. This behavior suggests that for a given weight k the degeneracy will be removed if perturbative corrections of sufficiently high order are taken into account. These results are in accord with the AdS/CFT duality conjecture. 
  We compute the annulus diagram corresponding to the interaction of a fractional D3 brane with a gauge field on its world-volume and a stack of N fractional D3 branes on the orbifolds C^2 /Z_2 and C^3/Z_2 x Z_2. We show that its logarithmic divergence can be equivalently understood as due either to massless open string states circulating in the loop or to massless closed string states exchanged between two boundary states. This follows from the fact that, under open/closed string duality, massless states in the open and closed string channels are matched into each other without mixing with massive states. This explains why the perturbative properties of many gauge theories living on the worldvolume of less supersymmetric and nonconformal branes have been recently obtained from their corresponding supergravity solution. 
  A previous study of the Kawai, Lewellen and Tye (KLT) relations between gravity and gauge theories, imposed by the relationship of closed and open strings, are here extended in the light of general relativity and Yang-Mills theory as effective field theories. We discuss the possibility of generalizing the traditional KLT mapping in this effective setting. A generalized mapping between the effective Lagrangians of gravity and Yang-Mills theory is presented, and the corresponding operator relations between gauge and gravity theories at the tree level are further explored. From this generalized mapping remarkable diagrammatic relations are found, -- linking diagrams in gravity and Yang-Mills theory, -- as well as diagrams in pure effective Yang-Mills theory. Also the possibility of a gravitational coupling to an antisymmetric field in the gravity scattering amplitude is considered, and shown to allow for mixed open-closed string solutions, i.e., closed heterotic strings. 
  We use the fermion zero-modes in the background of multi-caloron solutions with non-trivial holonomy as a probe for constituent monopoles. We find in general indication for an extended structure. However, for well separated constituents these become point-like. We analyse this in detail for the SU(2) charge 2 case, where one is able to solve the relevant Nahm equation exactly, beyond the piecewize constant solutions studied previously. Remarkably the zero-mode density can be expressed in the high temperature limit as a function of the conserved quantities that classify the solutions of the Nahm equation. 
  We discuss the thermodynamics of the N=2*, SU(N) gauge theory at large 't Hooft coupling. The tool we use is the non-extremal deformation of the supergravity solution of Pilch and Warner (PW) [hep-th/0004063], dual to N=4, SU(N) gauge theory softly broken to N=2. We construct the exact non-extremal solution in five-dimensional gauged supergravity and further uplift it to ten dimensions. Turning to the thermodynamics, we analytically compute the leading correction in m/T to the free energy of the non-extremal D3 branes due to the PW mass deformation, and find that it is positive. We also demonstrate that the mass deformation of the non-extremal D3 brane geometry induces a temperature dependent gaugino condensate. We find that the standard procedure of extracting the N=2* gauge theory thermodynamic quantities from the dual supergravity leads to a violation of the first law of thermodynamics. We speculate on a possible resolution of this paradox. 
  Assuming that a quantum field theory with a $\theta$-vacuum term in the action shows non-trivial $\theta$-dependence and provided that some reasonable properties of the probability distribution function of the order parameter hold, we argue that the theory either breaks spontaneously CP at $\theta = \pi$ or shows a singular behavior at some critical $\theta_c$ between 0 and $\pi$. This result, which applies to any model with a pure imaginary contribution to the euclidean action consisting in a quantized charge coupled to a phase, as QCD, is illustrated with two simple examples; one of them intimately related to Witten's result on SU(N) in the large $N$ limit. 
  Solving the exact renormalisation group equation a la Wilson-Polchinski perturbatively, we derive a power-counting theorem for general matrix models with arbitrarily non-local propagators. The power-counting degree is determined by two scaling dimensions of the cut-off propagator and various topological data of ribbon graphs. As a necessary condition for the renormalisability of a model, the two scaling dimensions have to be large enough relative to the dimension of the underlying space. In order to have a renormalisable model one needs additional locality properties--typically arising from orthogonal polynomials--which relate the relevant and marginal interaction coefficients to a finite number of base couplings. The main application of our power-counting theorem is the renormalisation of field theories on noncommutative R^D in matrix formulation. 
  We investigate classical formation of a trap surface in $D$-dimensional Einstein gravity in the process of a head-on collision of two high-energy particles, which are treated as Aichelburg-Sexl shock waves. From the condition of the trap surface volume local maximality we deduce an explicit form of the inner trap surface. Imposing the continuity condition on the fronts we obtain a time-dependent solution for the trap surface. We discuss trap surface appearance and evolution. 
  We revisit the {\it origin of structures problem} of standard Friedmann-Robertson-Walker cosmology to point out an unjustified approximation in the prevalent analysis. We follow common procedures in statistical mechanics to revise the issue without the disputed approximation. Our conclusions contradict the current wisdom and reveal and unexpected scenario for the origin of primordial cosmological structures. We show that standard physics operating in the cosmic plasma during the radiation dominated expansion of the universe produce at the time of decoupling scale invariant density anisotropies over cosmologically large comoving volumes. Scale invariance is shown to be a direct consequence of the causality constrains imposed by the short FRW comoving horizon at decoupling, which strongly suppress the power spectrum of density fluctuations with cosmologically large comoving wavelength. The global amplitude of these cosmological density anisotropies is fixed by the power spectrum in comoving modes whose wavelength is shorter than the causal horizon at the time and can be comparable to the amplitude of the primordial cosmological inhomogeneities imprinted in the cosmic microwave background radiation. 
  D4-D8 and D3-D7 systems are studied and a possible holographic dual of large N QCD (SU(N) gauge fields and fundamental quarks) is sought. A candidate system is found, for which however no explicit solution is available. Susy is broken by having a $D7-\bar{D7}$ condensing to a D5. The mechanism for supersymmetry breaking is then used to try to construct a Standard Model embedding. One can either obtain too few low energy fields or too many. The construction requires TeV scale string theory. 
  As is well-known, nonunitary RCFTs are distinguished from unitary ones in a number of ways, two of which are that the vacuum 0 doesn't have minimal conformal weight, and that the vacuum column of the modular S matrix isn't positive. However there is another primary field, call it o, which has minimal weight and has positive S column. We find that often there is a precise and useful relationship, which we call the Galois shuffle, between primary o and the vacuum; among other things this can explain why (like the vacuum) its multiplicity in the full RCFT should be 1. As examples we consider the minimal WSU(N) models. We conclude with some comments on fractional level admissible representations of affine algebras. As an immediate consequence of our analysis, we get the classification of an infinite family of nonunitary WSU(3) minimal models in the bulk. 
  We introduce a new prescription for renormalizing Feynman diagrams. The prescription is similar to BPHZ, but it is mass independent, and works in the massless limit as the MS scheme with dimensional regularization. The prescription gives a diagrammatic solution to Wilson's exact renormalization group differential equation. 
  One has believed that low energy effective theories of the Higgs branch of gauged linear sigma models correspond to supersymmetric nonlinear sigma models, which have been already investigated by many works. In this paper we discuss a explicit derivation of supersymmetric nonlinear sigma models from gauged linear sigma models. In this process we construct Kahler potentials of some two-dimensional toric varieties explicitly. Thus we will be able to study some algebraic varieties in the language of differential geometry. 
  Dilaton gravity in two dimensions is briefly reviewed from the perspective of three dilaton potentials: One determines classical physics ("the good", denoted by w), the second is relevant for semi-classical (and quantum) effects ("the muggy", denoted by I) and the third could be responsible for nonperturbative quantum effects ("the bad", denoted by Z).   This paper is based upon lectures given in Cernowitz in October/November 2002 at The XIV International Hutsulian Workshop Mathematical Theories and their Physical and Technical Applications. 
  These notes are based on lectures given at the Clay School on Geometry and String Theory, Isaac Newton Institute, Cambridge, 25 March - 19 April 2002. They attempt to provide an elementary and somewhat self contained discussion of the construction of supergravity solutions describing branes wrapping calibrated cycles, emphasising the geometrical aspects and focusing on D=11 supergravity. Following a discussion of the role of special holonomy backgrounds in D=11 supergravity, the basic membrane and fivebrane solutions are reviewed and the connection with the AdS/CFT correspondence is made. The world-volume description of branes is introduced and used to argue that branes wrapping calibrated cycles in special holonomy manifolds preserve supersymmetry. The corresponding supergravity solutions are constructed first in an auxiliary gauged supergravity theory which is obtained via Kaluza-Klein reduction. 
  We discuss the existence of glueball states for N=1 SYM within the Maldacena-Nunez model. We find that for this model the existence of an area law in the Wilson loop operator does not imply the existence of a discrete glueball spectrum. We suggest that implementing the model with an upper hard cut-off can amend the lack of spectrum. As a result the model can be only interpreted in the infra-red region. A direct comparison with the lattice data allows us to fix the scale up to where the model is sensible to describe low-energy observables. Nevertheless, taking for granted the lattice results, the resulting spectrum does not follow the general trends found in other supergravity backgrounds. We further discuss the decoupling of the non-singlet Kaluza-Klein states by analysing the associated supergravity equation of motion. The inclusion of non-commutative effects is also analysed and we find that leads to an enhancement on the value of the masses. 
  We give a simplified proof for the perturbative renormalizability of theories with massive vector particles. For renormalizability it is sufficient that the vector particle is treated as an gauge field, corresponding to an Abelian gauge group. Contrary to the non-Abelian case one does not need the Higgs mechanism to create the appropriate mass terms. The proof uses ``Stueckelberg's trick'' and the Ward-Takahashi identities from local Abelian gauge invariance. The simplification is due to the fact that, again contrary to the non-Abelian case, no BRST analysis is needed. 
  It is a well known result that the scalar field is composed of two chiral particles (Floreanini-Jackiw particles) of opposite chiralities. Also, that a Siegel particle spectrum is formed by a nonmover field (a Hull's noton) and a FJ particle. In this work we show that in a scalar field spectrum, in a curved expanding universe scenario, we can find two dynamical chiral fields. 
  We compute the moduli Kahler potential for M-theory on a compact manifold of G_2 holonomy in a large radius approximation. Our method relies on an explicit G_2 structure with small torsion, its periods and the calculation of the approximate volume of the manifold. As a verification of our result, some of the components of the Kahler metric are computed directly by integration over harmonic forms. We also discuss the modification of our result in the presence of co-dimension four singularities and derive the gauge-kinetic functions for the massless gauge fields that arise in this case. 
  We classify non-dilatonic NS-NS type II supergravity backgrounds admitting a consistent absolute parallelism. They are all given by parallelised Lie groups admitting scalar flat bi-invariant lorentzian metrics. There are seven different classes, some of them containing moduli. For each class we determine the amount of supersymmetry which is preserved: there are examples with 16, 18, 20, 22, 24, 28 and 32 supersymmetries. 
  In this note we show that in a two-dimensional non-commutative space the area operator is quantized, this outcome is compared with the result obtained by Loop Quantum Gravity methods. 
  The non-compact CFT of a class of NS-supported pp-wave backgrounds is solved exactly. The associated tree-level covariant string scattering amplitudes are calculated. The S-matrix elements are well-defined, dual but not analytic as a function of $p^+$. They have poles corresponding to physical intermediate states with $p^+\not =0$ and logarithmic branch cuts due to on-shell exchange of spectral-flow images of $p^+=0$ states. When $\mu\to 0$ a smooth flat space limit is obtained. The $\mu\to\infty$ limit, unlike the case of RR-supported pp-waves, gives again a flat space theory. 
  The spectral principle of Connes and Chamseddine is used as a starting point to define a discrete model for Euclidean quantum gravity. Instead of summing over ordinary geometries, we consider the sum over generalized geometries where topology, metric and dimension can fluctuate. The model describes the geometry of spaces with a countable number $n$ of points, and is related to the Gaussian unitary ensemble of Hermitian matrices. We show that this simple model has two phases. The expectation value $<n>$, the average number of points in the universe, is finite in one phase and diverges in the other. We compute the critical point as well as the critical exponent of $<n>$. Moreover, the space-time dimension $\delta$ is a dynamical observable in our model, and plays the role of an order parameter. The computation of $<\delta>$ is discussed and an upper bound is found, $<\delta> < 2$. 
  The known problem of fermion parity is considered on the base of investigating possible linear single-valued representations of spinor coverings of the extended Lorentz group. It is shown that in the frame of this theory does not exist, as separate concepts, P-parity and T-parity for a fermion, instead only some unified concept of (PT)-parity can be determined in a group-theoretical language. 
  Requiring covariance of Maxwell's equations without {\it a priori} imposing charge invariance allows for both spin-1 and spin-1/2 transformations of the complete Maxwell field and current. The spin-1/2 case yields new transformation rules, with new invariants, for all traditional Maxwell field and source quantities. The accompanying spin-1/2 representations of the Lorentz group employ the Minkowski metric, and consequently the primary spin-1/2 Maxwell invariants are also spin-1 invariants; for example, $\Phi^2 - {\bf A}^2$, ${\bf E}^2 - {\bf B}^2 + 2i {\bf E} \bm{\cdot} {\bf B} - ({\partial}_{0}{\Phi} + {\bm{\nabla \cdot}}{\bf A})^2$. The associated Maxwell Lagrangian density is also the same for both spin-1 and spin-1/2 fields. However, in the spin-1/2 case, standard field and source quantities are complex and both charge and gauge invariance are lost. Requiring the potentials to satisfy the Klein-Gordon equation equates the Maxwell and field-potential equations with two Dirac equations of the Klein-Gordon mass, and thus one complex Klein-Gordon Maxwell field describes either two real vector fields or two Dirac fields, all of the same mass. 
  Using an effective field theory approach, we address the effects on the gauge couplings of one and two additional compact dimensions in the presence of a constant background (gauge) field. Such background fields are a generic presence in models with extra dimensions and can be employed for gauge symmetry breaking mechanisms in the context of 4D N=1 supersymmetric models. The structure of the ultraviolet (UV) and infrared (IR) divergences that the gauge couplings develop in the presence of Wilson line vev's is investigated. One-loop radiative corrections to the gauge couplings due to overlapping effects of the compact dimensions and Wilson line vev's are computed for generic 4D N=1 models. Values of Wilson lines vev's corresponding to points (in the ``moduli'' space) of enhanced gauge symmetry cannot be smoothly reached {\it perturbatively} from those corresponding to the broken phase. The one-loop corrections are compared to their (heterotic) string counterpart in the ``field theory'' limit alpha'->0 to show remarkably similar results when no massless states are present in a Kaluza-Klein tower. An additional correction to the gauge coupling exists in the effective field theory approach when for specific Wilson lines vev's massless Kaluza-Klein states are present. This correction is not recoverable by the limit alpha'->0 of the (infrared regularised) string because the infrared regularisation limit and the limit alpha'->0 of the string result do not commute. 
  The Casimir forces on two parallel plates in conformally flat domain wall background due to conformally coupled massless scalar field satisfying mixed boundary conditions on the plates is investigated. In the general case of mixed boundary conditions formulae are derived for the vacuum expectation values of the energy-momentum tensor and vacuum forces acting on boundaries. 
  The basic theorem of the Lagrangian formulation for general superfield theory of fields (GSTF) is proved. The gauge transformations of general type (GTGT) and gauge algebra of generators of GTGT (GGTGT) as the consequences of the above theorem are studied.   It is established the gauge algebra of GGTGT contains the one of generators of gauge transformations of special type (GGTST) as one's subalgebra. In the framework of Lagrangian formulation for GSTF the nontrivial superfield model generalizing the model of Quantum Electrodynamics and belonging to the class of gauge theory of general type (GThGT) with Abelian gauge algebra of GGTGT is constructed. 
  For (planar) closed self-avoiding loops we construct a "holographic" map from the loop equations of large-N QCD_4 to an effective action defined over infinite rank Hitchin bundles. The effective action is constructed densely embedding Hitchin systems into the functional integral of a partially quenched or twisted Eguchi-Kawai model, by means of the resolution of identity into the gauge orbits of the microcanonical ensemble and by changing variables from the moduli fields of Hitchin systems to the moduli of the corresponding holomorphic de Rham local systems. The key point is that the contour integral that occurs in the loop equations for the de Rham local systems can be reduced to the computation of a residue in a certain regularization. The outcome is that, for self-avoiding loops, the original loop equations are implied by the critical equation of an effective action computed in terms of the localisation determinant and of the Jacobian of the change of variables to the de Rham local systems. We check, at lowest order in powers of the moduli fields, that the localisation determinant reproduces exactly the first coefficient of the beta function. 
  The gauge-invariant Chern-Simons-type Lorentz- and CPT-breaking term is here re-assessed and issues like causality, unitarity, spontaneous gauge-symmetry breaking are investigated. Moreover, we obtain a minimal extension of such a system to a supersymmetric environment. We comment on resulting peculiar self-couplings for the gauge sector, as well as on background contribution for gaugino masses. 
  We show that the effective actions of D-brane and M-brane are solutions to the Hamilton-Jacobi (H-J) equations in supergravities. This fact means that these effective actions are on-shell actions in supergravities. These solutions to the H-J equations reproduce the supergravity solutions that represent D-branes in a $B_2$ field, M2 branes and the M2-M5 bound states. The effective actions in these solutions are those of a probe D-brane and a probe M-brane. Our findings can be applied to the study of the gauge/gravity correspondence, especially the holographic renormalization group, and a search for new solutions of supergravity. 
  Recently, a set of thermodynamic Bethe ansatz equations is proposed by Dorey, Pocklington and Tateo for unitary minimal models perturbed by \phi_{1,2} or \phi_{2,1} operator. We examine their results in view of the lattice analogues, dilute A_L models at regime 1 and 2. Taking M_{5,6}+\phi_{1,2} and M_{3,4}+\phi_{2,1} as the simplest examples, we will explicitly show that the conjectured TBA equations can be recovered from the lattice model in a scaling limit. 
  In the context of tachyon effective theory coupled to Born-Infeld electromagnetic fields, we obtain all possible singularity-free static flat configurations of codimension one on unstable Dp-branes. Computed tension and string charge density suggest that the obtained kinks are D(p-1) or D(p-1)F1-branes. 
  In this paper we study general properties of noncommutative field theories obtained from the Seiberg-Witten limit of string theories in the presence of an external B-field. We analyze the extension of the Wightman axioms to this context and explore their consequences, in particular we present a proof of the CPT theorem for theories with space-space noncommutativity. We analyze as well questions associated to the spin-statistics connections, and show that noncommutative N=4, U(1) gauge theory can be softly broken to N=0 satisfying the axioms and providing an example where the Wilsonian low energy effective action can be constructed without UV/IR problems, after a judicious choice of soft breaking parameters is made. We also assess the phenomenological prospects of such a theory, which are in fact rather negative. 
  We formulate Dirac-Kaehler fermion action by introducing a new Clifford product with noncommutative differential form on a lattice. Hermiticity of the Dirac-Kaehler action requires to choose the lattice structure having both orientabilities on a link. The Kogut-Susskind fermion and the staggered fermion actions are derived directly from the Dirac-Kaehler fermion formulated by the Clifford product. The lattice QCD action with Dirac-Kaehler matter fermion is also derived via an inner product defined by the Clifford product. 
  This is an exposition of recent progress in the categorical approach to D-brane physics. I discuss the physical underpinnings of the appearance of homotopy categories and triangulated categories of D-branes from a string field theoretic perspective, and with a focus on applications to homological mirror symmetry. 
  We consider an N=1 SU(N_c) SUSY gauge theory with N_f \geq N_c matter multiplets transforming in the fundamental and antifundamental representations of the gauge group. Using the Konishi anomaly and a non-anomalous conservation law, we derive a system of partial differential equations that determine the low energy effective superpotential as a function of the mesonic and baryonic vacuum expectation values. We apply the formalism to the cases of N_f = N_c and N_f = N_c +1 where the equations are easily integrated and recover the known results. We further apply the formalism to derive a system of partial differential equations to determine the low energy effective superpotential for the Seiberg dual theories. Finally we briefly discuss the associated matrix models via the Dijkgraaf-Vafa conjecture. 
  We study the light-front Schwinger model at finite temperature following the recent proposal in \cite{alves}. We show that the calculations are carried out efficiently by working with the full propagator for the fermion, which also avoids subtleties that arise with light-front regularizations. We demonstrate this with the calculation of the zero temperature anomaly. We show that temperature dependent corrections to the anomaly vanish, consistent with the results from the calculations in the conventional quantization. The gauge self-energy is seen to have the expected non-analytic behavior at finite temperature, but does not quite coincide with the conventional results. However, the two structures are exactly the same on-shell. We show that temperature does not modify the bound state equations and that the fermion condensate has the same behavior at finite temperature as that obtained in the conventional quantization. 
  We discuss the notion of tensionless limit in quantum bosonic string theory, especially in flat Minkowski space, noncompact group manifolds (e.g., SL(2,R)) and coset manifolds (e.g., AdS). We show that in curved space typically there exists a critical value of the tension which is related to the critical value of the level of the corresponding affine algebra. We argue that at the critical level the sring theory becomes tensionless and that there exists a huge new symmetry of the theory. We dicuss the appearence of the higher spin massless states at the critical level. 
  Pair creation of strings in time-dependent backgrounds is studied from an effective field theory viewpoint, and some possible cosmological applications are discussed. Simple estimates suggest that excited strings may have played a significant role in preheating, if the string tension as measured in four-dimensional Einstein frame falls a couple of orders of magnitude below the four-dimensional Planck scale. 
  We point out that the presence of energetic cosmic rays above the GZK cutoff may be explained by fundamental non-linearities in quantum mechanics at the Plank level. 
  We correct an unfortunate error in an earlier work of the author, and show that in center-vortex QCD (gauge group SU(3)) the baryonic area law is the so-called $Y$ law, described by a minimal area with three surfaces spanning the three quark world lines and meeting at a central Steiner line joining the two common meeting points of the world lines. (The earlier claim was that this area law was a so-called $\Delta$ law, involving three extremal areas spanning the three pairs of quark world lines.) We give a preliminary discussion of the extension of these results to $SU(N), N>3$. These results are based on the (correct) baryonic Stokes' theorem given in the earlier work claiming a $\Delta$ law. The $Y$-form area law for SU(3) is in agreement with the most recent lattice calculations. 
  We study the soldering formalism in the context of abelian p-form theories. We develop further the fusion process of massless antisymmetric tensors of different ranks into a massive p-form and establish its duality properties. To illustrate the formalism we consider two situations. First the soldering mass generation mechanism is compared with the Higgs and Julia-Toulouse mechanisms for mass generation due to condensation of electric and magnetic topological defects. We show that the soldering mechanism interpolates between them for even dimensional spacetimes, in this way confirming the Higgs/Julia-Toulouse duality proposed by Quevedo and Trugenberger \cite{QT} a few years ago. Next, soldering is applied to the study of duality group classification of the massive forms. We show a dichotomy controlled by the parity of the operator defining the symplectic structure of the theory and find their explicit actions. 
  We study the level-expansion structure of the NS string field theory actions, mainly focusing on the modified (i.e. 0-picture in the NS sector) cubic superstring field theory. This theory has a non-trivial structure already at the quadratic level due to presence of the picture-changing operator. It is explicitly shown how the usual Maxwell and tachyon actions can be obtained after integrating out the auxiliary fields. We then discuss the reality of the action in the CFT language for all of modified cubic, Witten's cubic and Berkovits' non-polynomial theories. The tachyon condensation problems in modified cubic theory are re-examined. We also carry out level truncation analysis in vacuum superstring field theory proposed in our previous paper, and find some difficulties in both of cubic and non-polynomial formulations. 
  We study a string theory inspired model for hybrid inflation in the context of a brane-antibrane system partially compactified on a compact submanifold of (a caricature of) a Calabi-Yau manifold. The interbrane distance acts as the inflaton, whereas the end of the inflationary epoch is brought about by the rapid rolling of the tachyon. The number of e-foldings is sufficiently large and is controlled by the initial conditions. The slow roll parameters, however, are essentially determined by the geometry and have little parametric dependence. Primordial density fluctuations can be made consistent with current data at the cost of reducing the string scale. 
  The PCT theorem is shown to be valid in quantum field theory formulated on noncommutative spacetime by exploiting the properties of the Wightman functions defined in such a set up. 
  We find that the IIA Matrix models defined on the non-compact $C^3/Z_6$,  $C^2/Z_2$ and $C^2/Z_4$ orbifolds preserve supersymmetry where the fermions are on-mass-shell Majorana-Weyl fermions. In these examples supersymmetry is preserved both in the orbifolded space and in the non-orbifolded space at the same time. The Matrix model on $C^3/Z_6$ orbifold has the same ${\cal N}=2$ supersymmetry as the case of  $C^3/Z_3$ orbifold which was pointed out previously.  On the other hand the Matrix models on $C^2/Z_2$ and $C^2/Z_4$ orbifold have a half of the ${\cal N}=2$ supersymmetry. We further find that the Matrix model on $C^2/Z_2$ orbifold with a parity-like identification preserves ${\cal N}=2$ supersymmetry. 
  Recently it has been argued that, because tachyonic matter satisfies the Strong Energy Condition [SEC], there is little hope of avoiding the singularities which plague S-Brane spacetimes. Meanwhile, however, Townsend and Wohlfarth have suggested an ingenious way of circumventing the SEC in such situations, and other suggestions for actually violating it in the S-Brane context have recently been proposed. Of course, the natural context for discussions of [effective or actual] violations of the SEC is the theory of asymptotically deSitter spacetimes, which tend to be less singular than ordinary FRW spacetimes. However, while violating or circumventing the SEC is necessary if singularities are to be avoided, it is not at all clear that it is sufficient. That is, we can ask: would an asymptotically deSitter S-brane spacetime be non-singular? We show that this is difficult to achieve; this result is in the spirit of the recently proved "S-brane singularity theorem". Essentially our results suggest that circumventing or violating the SEC may not suffice to solve the S-Brane singularity problem, though we do propose two ways of avoiding this conclusion. 
  We identify type IIA orientifolds that are dual to M-theory compactifications on manifolds with G_2-holonomy. We then discuss the construction of crosscap states in Gepner models. (Based on a talk presented by S.G. at PASCOS 2003 held at the Tata Institute of Fundamental Research, Mumbai during Jan. 3-8, 2003.) 
  We construct plane wave backgrounds with time-dependent profiles corresponding to Penrose limits of NS5-branes with transverse space symmetry group broken from SO(4) to SO(2)\times Z_N. We identify the corresponding exact theory as the five-dimensional Logarithmic Conformal Field Theory (CFT) arising from the contraction of the SU(2)_{N'}/U(1)\times SL(2,R)_{-N} exact CFT, times R^5. We study several general aspects and construct the free field representation in this theory. String propagation and spectra are also considered and explicitly solved in the light-cone gauge. 
  We study the general features of the dynamics of the phantom field in the cosmological context. In the case of inverse coshyperbolic potential, we demonstrate that the phantom field can successfully drive the observed current accelerated expansion of the universe with the equation of state parameter $w_{\phi} < -1$. The de-Sitter universe turns out to be the late time attractor of the model. The main features of the dynamics are independent of the initial conditions and the parameters of the model. The model fits the supernova data very well, allowing for $-2.4 < w_{\phi} < -1$ at 95 % confidence level. 
  We construct the set of theories which share the property that the tree-level amplitudes nullify even if both initial and final states contain the same type of particles. The origin of this phenomenon lies in the fact that the reduced classical dynamics describes isochronic systems. 
  We study the duality symmetry in p-form models containing a generalized $B_q\wedge F_{p+1}$ term in spacetime manifolds of arbitrary dimensions. The equivalence between the $B_q\wedge F_{p+1}$ self-dual ($SD_{B\wedge F}$) and the $B_q\wedge F_{p+1}$ topologically massive ($TM_{B\wedge F}$) models is established using a gauge embedding procedure, including the minimal coupling to conserved charged matter current. The minimal coupling adopted for both tensor fields in the self-dual representation is transformed into a non minimal magnetic like coupling in the topologically massive representation but with the currents swapped. It is known that to establish this equivalence a current-current interaction term is needed to render the matter sector unchanged. We show that both terms arise naturally from the embedding adopted. Comparison with Higgs/Julia-Toulouse duality is established. 
  We explore the relationship between classical quasinormal mode frequencies and black hole quantum mechanics in 2+1 dimensions. Following a suggestion of Hod, we identify the real part of the quasinormal frequencies with the fundamental quanta of black hole mass and angular momentum. We find that this identification leads to the correct quantum behavior of the asymptotic symmetry algebra, and thus of the dual conformal field theory. Finally, we suggest a further connection between quasinormal mode frequencies and the spectrum of a set of nearly degenerate ground states whose multiplicity may be responsible for the Bekenstein-Hawking entropy. 
  Discrete interaction models for the classical harmonic oscillator are used for introducing new mathematical generalizations in the usual continuous formalism. The inverted harmonic potential and generalized discrete hyperbolic and trigonometric functions are defined. 
  It is shown that light front thermal field theory is equivalent to conventional thermal field theory. The proof is based on the use of spectral representations, and applies to all Lagrangians for which such equivalence has been proven at zero temperature. It is also pointed out that conventional spectral functions can be used to express light-front finite temperature free propagators. As an application of our approach, we derive the light-front finite temperature spin 1/2 fermion propagator in full Dirac space. 
  Attempts to solve Yang-Mills theory must eventually face the problem of analyzing the theory at intermediate values of the coupling constant. In this regime neither perturbation theory nor the gravity dual are adequate, and one must consider the full string theory in the appropriate background. We suggest that in some nontrivial cases the world sheet theory may be exactly solvable. For the Green-Schwarz superstring on AdS_5 x S^5 we find an infinite set of nonlocal classically conserved charges, of the type that exist in integrable field theories. 
  When the gravitational Chern-Simons term is reduced from 3 to 2 dimensions, the lower dimensional theory supports a symmetry breaking solution and an associated kink. Kinks in general relativity bear a close relation to flat space kinks, governed by identical potentials. 
  We discuss the (dual-)gauge transformations for the gauge-fixed Lagrangian density and establish their intimate connection with the translation subgroup T(2) of the Wigner's little group for the free one-form Abelian gauge theory in four $(3 + 1)$-dimensions (4D) of spacetime. Though the relationship between the usual gauge transformation for the Abelian massless gauge field and T(2) subgroup of the little group is quite well-known, such a connection between the dual-gauge transformation and the little group is a new observation. The above connections are further elaborated and demonstrated in the framework of Becchi-Rouet-Stora-Tyutin (BRST) cohomology defined in the quantum Hilbert space of states where the Hodge decomposition theorem (HDT) plays a very decisive role. 
  Using a boundary prescription motivated by the Ads-Cft conjecture, I study the thermodynamical properties of the class of Kerr-Bolt-Ads spacetime.The stability conditions and the complete phase diagrams are investigated. 
  The relationship (resemblance and/or contrast) between quantum and classical integrability in Ruijsenaars-Schneider systems, which are one parameter deformation of Calogero-Moser systems, is addressed. Many remarkable properties of classical Calogero and Sutherland systems (based on any root system) at equilibrium are reported in a previous paper (Corrigan-Sasaki). For example, the minimum energies, frequencies of small oscillations and the eigenvalues of Lax pair matrices at equilibrium are all "integer valued". In this paper we report that similar features and results hold for the Ruijsenaars-Schneider type of integrable systems based on the classical root systems. 
  We treat the events determined by a quantum physical state in a noncommutative space-time, generalizing the analogous treatment in the usual Minkowski space-time based on positive-operator-valued measures (POVMs). We consider in detail the model proposed by Snyder in 1947 and calculate the POVMs defined on the real line that describe the measurement of a single coordinate. The approximate joint measurement of all the four space-time coordinates is described in terms of a generalized Wigner function (GWF). We derive lower bounds for the dispersion of the coordinate observables and discuss the covariance of the model under the Poincare' group. The unusual transformation law of the coordinates under space-time translations is interpreted as a failure of the absolute character of the concept of space-time coincidence. The model shows that a minimal length is compatible with Lorents covariance. 
  We discuss the renormalization properties of noncommutative supersymmetric theories. We also discuss how the gauge field plays a role similar to gravity in noncommutative theories. 
  We analyze three aspects of N=1 heterotic string compactifications on elliptically fibered Calabi-Yau threefolds: stability of vector bundles, five-brane instanton transitions and chiral matter. First we show that relative Fourier-Mukai transformation preserves absolute stability. This is relevant for vector bundles whose spectral cover is reducible. Then we derive an explicit formula for the number of moduli which occur in (vertical) five-brane instanton transitions provided a certain vanishing argument applies. Such transitions increase the holonomy of the heterotic vector bundle and cause gauge changing phase transitions. In an M-theory description the transitions are associated with collisions of bulk five-branes with one of the boundary fixed planes. In F-theory they correspond to three-brane instanton transitions. Our derivation relies on an index computation with data localized along the curve which is related to the existence of chiral matter in this class of heterotic vacua. Finally, we show how to compute the number of chiral matter multiplets for this class of vacua allowing to discuss the associated Yukawa couplings. 
  We embed second class constrained systems by a formalism that combines concepts of the BFFT method and the unfixing gauge formalism. As a result, we obtain a gauge-invariant system where the introduction of the Wess-Zumino (WZ) field is essential. The initial phase-space variables are gauging with the introduction of the WZ field, a procedure that resembles the St\"uckelberg field- shifting formalism. In some cases, it is possible to eliminate the WZ field and, therefore, obtain an invariant system written only as a function of the original phase-space variables. We apply this formalism to important physical models: the reduced-SU(2) Skyrme model and the two dimensional chiral bosons field theory. In these systems, the gauge-invariant Hamiltonians are derived in a very simple way when compared with other usual formalisms. 
  We propose a discretisation scheme based on the Dirac-Kahler formalism (DK) in which the algebraic relations between continuum operators ${\wedge, d, \star}$ are captured by their discrete analogues, allowing the construction of the relevant projection operators necessary to prevent species doubling. We thus avoid the traditional form of species doubling as well as spectral doubling, which does not occur in the DK setting. Chirality is also captured, since we have $\star$ from geometric discretisation. 
  Physical consistency of quantum fields in anti-de Sitter space time requires that the space must be compactified by the inclusion of a boundary where appropriate conditions are imposed. An interpretation for the presence of this boundary is found taking AdS as a limiting case of the space generated by a large number of coincident branes. The compactification of AdS leads to a discretization of the spectrum of bulk fields. As a consequence, we find a one to one mapping between the quantum states of scalar fields in AdS bulk and boundary. Using this mapping as an approximation for the dual relation between string dilaton field and scalar QCD glueballs the high energy QCD scaling is reproduced. We also use this map to estimate the ratio of scalar glueball masses. 
  We shall discuss issues of duality and topological mass generation in diverse dimensions. Particular emphasis will be given to the mass generation mechanism from interference between self and anti self-dual components, as disclosed by the soldering formalism. This is a gauge embedding procedure derived from an old algorithm of second-class constraint conversion used by the author to approach anomalous gauge theories. The problem of classification of the electromagnetic duality groups, both massless and massive, that is closely related will be discussed. Particular attention will be paid to a new approach to duality based on the soldering embedding to tackle the problem of mass generation by topological mechanisms in arbitrary dimensions including the couplings to dynamical matter, nonlinear cases and nonabelian symmetries. 
  A manifestly invariant renormalization scheme of N=1 nonabelian supersymmetric gauge theories is proposed. 
  A theoretical analysis of higher-order corrections to D=11 supergravity is given in a superspace framework. It is shown that any deformation of D=11 supergravity for which the lowest-dimensional component of the four-form $G_4$ vanishes is trivial. This implies that the equations of motion of D=11 supergravity are specified by an element of a certain spinorial cohomology group and generalises previous results obtained using spinorial or pure spinor cohomology to the fully non-linear theory. The first deformation of the theory is given by an element of a different spinorial cohomology group with coefficients which are local tensorial functions of the massless supergravity fields. The four-form Bianchi Identities are solved, to first order and at dimension $-{1/2}$, in the case that the lowest-dimensional component of $G_4$ is non-zero. Moreover, it is shown how one can calculate the first-order correction to the dimension-zero torsion and thus to the supergravity equations of motion given an explicit expression for this object in terms of the supergravity fields. The version of the theory with both a four-form and a seven-form is discussed in the presence of the five-brane anomaly-cancelling term. It is shown that the supersymmetric completion of this term exists and it is argued that it is the unique anomaly-cancelling invariant at this dimension which is at least quartic in the fields. This implies that the first deformation of the theory is completely determined by the anomaly term from which one can, in principle, read off the corrections to all of the superspace field strength tensors. 
  Motivated by recent accelerating cosmological model, we derive the solutions to vacuum Einstein equation in $(d+1)$-dimensional Minkowski space with $n$-dimensional hyperbolic manifold. The conditions of accelerating expansion are given in such a set up. 
  We determine coherent states peaked at classical space-time of the Schwarzschild black hole in the frame-work of canonical quantisation of general relativity. The information about the horizon is naturally encoded in the phase space variables, and the perturbative quantum fluctuations around the classical geometry depend on the distance from the horizon. For small black holes, space near the vicinity of the singularity appears discrete with the singular point excluded from the spectrum. 
  We construct a cubic field theory which provides all genus amplitudes of the topological A-model for all non-compact Calabi-Yau toric threefolds. The topology of a given Feynman diagram encodes the topology of a fixed Calabi-Yau, with Schwinger parameters playing the role of Kahler classes of Calabi-Yau. We interpret this result as an operatorial computation of the amplitudes in the B-model mirror which is the Kodaira-Spencer quantum theory. The only degree of freedom of this theory is an unconventional chiral scalar on a Riemann surface. In this setup we identify the B-branes on the mirror Riemann surface as fermions related to the chiral boson by bosonization. 
  We investigate B-type topological Landau-Ginzburg theory with one variable, with D2-brane boundary conditions. We find that the allowed brane configurations are determined in terms of the possible factorizations of the superpotential, and compute the corresponding open string chiral rings. These are characterized by bosonic and fermionic generators that satisfy certain relations. Moreover we show that the disk correlators, being continuous functions of deformation parameters, satisfy the topological sewing constraints, thereby proving consistency of the theory. In addition we show that the open string LG model is, in its content, equivalent to a certain triangulated category introduced by Kontsevich, and thus may be viewed as a concrete physical realization of it. 
  For matrix models with measure on the Lie algebra of SO/Sp, the sub-leading free energy is given by F_{1}(S)=\pm{1/4}\frac{\del F_{0}(S)}{\del S}. Motivated by the fact that this relationship does not hold for Chern-Simons theory on S^{3}, we calculate the sub-leading free energy in the matrix model for this theory, which is a Gaussian matrix model with Haar measure on the group SO/Sp. We derive a quantum loop equation for this matrix model and then find that F_{1} is an integral of the leading order resolvent over the spectral curve. We explicitly calculate this integral for quadratic potential and find agreement with previous studies of SO/Sp Chern-Simons theory. 
  This paper is continuation of our previous papers hep-th/0209246 and hep-th/0304077 .   We discuss in more detail a new form of solution to the quantum Knizhnik-Zamolodchikov equation [qKZ] on level -4 obtained in the paper hep-th/0304077 for the Heisenberg XXX spin chain. The main advantage of this form is it's explicit reducibility to one-dimensional integrals. We argue that the deep mathematical reason for this is some special cohomologies of deformed Jacobi varieties. We apply this new form of solution to the correlation functions using the Jimbo-Miwa conjecture. A formula (46) for the correlation functions obtained in this way is in a good agreement with the ansatz for the emptiness formation probability from the paper hep-th/0209246. Our previous conjecture on a structure of correlation functions of the XXX model in the homogeneous limit through the Riemann zeta functions at odd arguments is a corollary of the formula (46). 
  We compute topological correlators in Landau-Ginzburg models on a Riemann surface with arbitrary number of handles and boundaries. The boundaries may correspond to arbitrary topological D-branes of type B. We also allow arbitrary operator insertions on the boundary and in the bulk. The answer is given by an explicit formula which can be regarded as an open-string generalization of C. Vafa's formula for closed-string topological correlators. We discuss how to extend our results to the case of Landau-Ginzburg orbifolds. 
  We review the construction of time-dependent backgrounds with space-like singularities. We mainly consider exact CFT backgrounds. The algebraic and geometric aspects of these backgrounds are discussed. Physical issues, results and difficulties associated with such systems are reviewed. Finally, we present some new results: a two dimensional cosmology in the presence of an Abelian gauge field described within a family of (SL(2)xU(1))/(U(1)xZ) quotient CFTs. 
  The multiplet of superconformal anomalous currents in the case $(1,0)$, $d=6$ is derived. The supersymmetric multiplet of anomalies contains the trace of the energy momentum tensor, the gamma trace of the supercurrent and some topological vector current with divergence equal to the $R$-current anomaly. The extension of this consideration to the $(2,0)$ case and some application and motivation coming from $AdS_{7}/CFT_{6}$ correspondence is discussed. 
  In the recent literature one can find calculations of various one--loop amplitudes, like anomalies, tadpoles and vacuum energies, on specific types of orbifolds, like S^1/Z_2. This work aims to give a general description of such one--loop computations for a large class of orbifold models. In order to achieve a high degree of generality, we formulate these calculations as evaluations of traces of operators over orbifold Hilbert spaces. We find that in general the result is expressed as a sum of traces over hyper surfaces with local projections, and the derivatives perpendicular to these hyper surfaces are rescaled. These local projectors naturally takes into account possible non--periodic boundary conditions. As the examples T^6/Z_4 and T^4/D_4 illustrate, the methods can be applied to non--prime as well as non--Abelian orbifolds. 
  Two dimensional charged black holes in string theory can be obtained as exact (SL(2,R)xU(1))/U(1) quotient CFTs. The geometry of the quotient is induced from that of the group, and in particular includes regions beyond the black hole singularities. Moreover, wavefunctions in such black holes are obtained from gauge invariant vertex operators in the SL(2,R) CFT, hence their behavior beyond the singularity is determined. When the black hole is charged we find that the wavefunctions are smooth at the singularities. Unlike the uncharged case, scattering waves prepared beyond the singularity are not fully reflected; part of the wave is transmitted through the singularity. Hence, the physics outside the horizon of a charged black hole is sensitive to conditions set behind the past singularity. 
  We have found a (classical) competition between duality and gauge symmetries when trying to obtain an explicit dual to the non-nonabelian version of the self-dual model proposed by Townsend, Pilch and van Nieuwenhuizen\cite{TPvN} (NASD) either through gauge embedding procedures or gauge invariant master approach. We found that the theory dual to NASD {\it is not} the Yang-Mills-Chern-Simons model. We then proved that a model gauge invariant and dual equivalent to NASD cannot be achieved because of this conflict. Other nonabelian self-dual formulation proposed in \cite{KLRvN} was studied and its dual obtained. We discuss the consequences of the dual equivalence found here in the context of 3D non-abelian bosonization. 
  In this paper we analyse the vacuum polarization effects due to a magnetic flux on massless fermionic fields in a cosmic string background. Three distinct configurations of magnetic fields are considered. In all of them the magnetic fluxes are confined in a long cylindrical tube of finite radius. 
  We present supersymmetric positive definite scalar products together with natural Krein structures of supersymmetries. 
  A fully quantum version of the Witten-Olive analysis of the central charge in the N=1 Wess-Zumino model in $d=2$ with a kink solution is presented by using path integrals in superspace. We regulate the Jacobians with heat kernels in superspace, and obtain all superconformal anomalies as one Jacobian factor. The conserved quantum currents differ from the Noether currents by terms proportional to field equations, and these terms contribute to the anomalies. We identify the particular variation of the superfield which produces the central charge current and its anomaly; it is the variation of the auxiliary field. The quantum supersymmetry algebra which includes the contributions of superconformal anomalies is derived by using the Bjorken-Johnson-Low method instead of semi-classical Dirac brackets. We confirm earlier results that the BPS bound remains saturated at the quantum level due to equal anomalies in the energy and central charge. 
  In systems of intersecting branes, we consider sets of directions in which one type of brane is pointlike, with transverse fluctuations described by matrix coordinates X, and the other set of branes is space-filling, with a local symmetry associated to its worldvolume gauge field. Under this symmetry, massless fields associated with p-p' strings should transform in the fundamental representation, \Phi \to U(X) \Phi, but this transformation rule is ill-defined when X is a general matrix. In this paper, we make sense of this transformation rule for \Phi and show that imposing gauge invariance using the resulting rule places strong constraints on the effective actions, determining infinite series of terms in the \alpha' expansion. We describe the most general invariant effective actions and note that these are written most simply in terms of covariant objects built from \Phi which transform like fundamental fields living on the whole space-filling brane. Our description leads us to introduce several interesting structures, including Wilson lines from ordinary points to matrix locations, pull-backs of fields from matrix geometries to ordinary space, and delta functions which localize to matrix configurations. 
  We show that contrary to first expectations realistic three generation supersymmetric intersecting brane world models give rise to phenomenologically interesting predictions about gauge coupling unification. Assuming the most economical way of realizing the matter content of the MSSM via intersecting branes we obtain a model independent relation among the three gauge coupling constants at the string scale. In order to correctly reproduce the experimentally known values of sin^2[theta_W(M_z)] and alpha_s(M_z) this relation leads to natural gauge coupling unification at a string scale close to the standard GUT scale 2 x 10^16 GeV. Additional vector-like matter can push the unification scale up to the Planck scale. 
  We show that in four or more spacetime dimensions, the Einstein equations for gravitational perturbations of maximally symmetric vacuum black holes can be reduced to a single 2nd-order wave equation in a two-dimensional static spacetime for a gauge-invariant master variable, irrespective of the mode of perturbations. Our formulation applies to the case of vanishing as well as non-vanishing cosmological constant Lambda. The sign of the sectional curvature K of each spatial section of equipotential surfaces is also kept general. In the four-dimensional Schwarzschild background, this master equation for a scalar perturbation is identical to the Zerilli equation for the polar mode and the master equation for a vector perturbation is identical to the Regge-Wheeler equation for the axial mode. Furthermore, in the four-dimensional Schwarzschild-anti-de Sitter background with K=0,1, our equation coincides with those derived by Cardoso and Lemos recently. As a simple application, we prove the perturbative stability and uniqueness of four-dimensional non-extremal spherically symmetric black holes for any Lambda. We also point out that there exists no simple relation between scalar-type and vector-type perturbations in higher dimensions, unlike in four dimensions. Although we only treat maximally symmetric black holes in the present paper, the final master equations are valid even when the hirozon geometry is described by a generic Einstein manifold. 
  Recent results on the annulus partition function in Liouville field theory are applied to non-critical string theory, both below and above the critical dimension. Liouville gravity coupled to $c\le 1$ matter has a dual formulation as a matrix model. Two well-known matrix model results are reproduced precisely using the worldsheet formulation: (1) the correlation function of two macroscopic loops, and (2) the leading non-perturbative effects. The latter identifies the eigenvalue instanton amplitudes of the matrix approach with disk instantons of the worldsheet approach, thus demonstrating that the matrix model is the effective dynamics of a D-brane realization of $d\le 1$ non-critical string theory. In the context of string theory above the critical dimension, i.e. $d\ge 25$, Liouville field theory realizes two-dimensional de Sitter gravity on the worldsheet. In this case, appropriate D-brane boundary conditions on the annulus realize the S-matrix for two-dimensional de Sitter gravity. 
  The generalized covariant entropy bound is the conjecture that the entropy of the matter present on any non-expanding null hypersurface L will not exceed the difference between the areas, in Planck units, of the initial and final spatial 2-surfaces bounding L. The generalized Bekenstein bound is a special case which states that the entropy of a weakly gravitating isolated matter system will not exceed the product of its mass and its width. Here we show that both bounds can be derived directly from the following phenomenological assumptions: that entropy can be computed by integrating an entropy current which vanishes on the initial boundary and whose gradient is bounded by the energy density. Though we note that any local description of entropy has intrinsic limitations, we argue that our assumptions apply in a wide regime. We closely follow the framework of an earlier derivation, but our assumptions take a simpler form, making their validity more transparent in some examples. 
  We examined 5d thick brane worlds constructed by a real scalar field. The solutions are obtained in terms of a simple form of smooth warp factor. For the case of dS brane, we found a disease of the solutions. For example, it is impossible to construct thick-brane worlds with our simple smoothing. This result is independent of supersymmetry or other symmetries. 
  We study the Penrose limit of a supersymmetric IIB background, with non-trivial NS 3-form field strength, obtaining a solution with the smallest number of supercharges (i.e. 16) allowed; we write down explicitly the superalgebra of the theory, build the supersymmetric associated IIB string sigma model and make conjectures on the dual gauge theory. 
  BRST cohomology methods are used to explain the origin of the SL(2,R) symmetry in Yang-Mills theories. Clear evidence is provided for the unphysical nature of this symmetry. This is obtained from the analysis of a local functional of mass dimension two and constitutes a no-go statement for giving a physical meaning to condensates associated with the symmetry breaking of SL(2,R). 
  In spacetimes admitting Yano tensors the classical theory of the spinning particle possesses enhanced worldline supersymmetry. Quantum mechanically generators of extra supersymmetries correspond to operators that in the classical limit commute with the Dirac operator and generate conserved quantities. We show that the result is preserved in the full quantum theory, that is, Yano symmetries are not anomalous. This was known for Yano tensors of rank two, but our main result is to show that it extends to Yano tensors of arbitrary rank. We also describe the conformal Yano equation and show that is invariant under Hodge duality. There is a natural relationship between Yano tensors and supergravity theories. As the simplest possible example, we show that when the spacetime admits a Killing spinor then this generates Yano and conformal Yano tensors. As an application, we construct Yano tensors on maximally symmetric spaces: they are spanned by tensor products of Killing vectors. 
  We consider the quantization of a scalar kappa-deformed field up to the point of obtaining an expression for its vacuum energy. The expression is given by the half sum of the field frequencies, as in the non-deformed case, but with the frequencies obeying the kappa-deformed dispersion relation. We consider a set of kappa-deformed Maxwell equations and show that for the purpose of calculating the Casimir energy the Maxwell field, as in the non-deformed case, behaves as a pair of scalar fields. Those results provide a foundation for computing the Casimir energy starting from the the half sum of field frequencies. A method of calculation starting from this expression is briefly described. 
  In the tensionless limit of string theory on flat background all the massive tower of states gets squeezed to a common zero mass level and the free theory is described by an infinite amount of massless free fields with arbitrary integer high spin. We notice that in this situation the very notion of critical dimension gets lost, the apparency of infinite global symmetries takes place, and the closed tensionless string can be realized as a constrained subsystem of the open one in a natural way. Moreover, we study the tensionless limit of the Witten's cubic sting field theory and find that the theory in such a limit can be represented as an infinite set of free arbitrary higher spin excitations plus an interacting sector involving their zero-modes only. 
  Newton law arising due to the gravity localized on the general singular brane embedded in $AdS_5$ bulk is examined in the absence or presence of the 4d induced Einstein term. For the RS brane, apart from the subleading correction, Newton potential obeys 4d-type and $5d$-type gravitational law at long- and short-ranges if it were not for the induced Einstein term. The 4d induced Einstein term generates an intermediate range at short distance, in which the $5d$ Newton potential $1/r^2$ emerges. For Neumann brane the long-range behavior of Newton potential is exponentially suppressed regardless of the existence of the induced Einstein term. For Dirichlet brane the expression of Newton potential is dependent on the renormalized coupling constant $v^{ren}$. At particular value of $v^{ren}$ Newton potential on Dirichlet brane exhibits a similar behavior to that on RS brane. For other values the long-range behavior of Newton potential is exponentially suppressed as that in Neumann brane. 
  In a quantum field with spacetime invariance governed by the Poincare algebra the one-loop effective action is equal to the sum of zero modes frequencies, which is the vacuum energy of the field. The first Casimir invariant of the Poincare algebra provides the proper time Hamiltonian in Schwinger's proper time representation of the effective action. We consider here a massive neutral scalar field with spacetime invariance governed by the so called kappa-deformed Poincare algebra. We show here that if in the kappa-deformed theory the first Casimir invariant of the algebra is also used as the proper-time hamiltonian the effective action appears with a real and an imaginary part. The real part is equal to half the sum of kappa-deformed zero mode frequencies, which gives the vacuum energy of the kappa-deformed field. In the limit in which the deformation disappears this real part reduces to half of the sum of zero mode frequencies of the usual scalar field. The imaginary part is proportional to the sum of the squares of the kappa-deformed zero mode frequencies. This part is a creation rate of field excitations in the situations in which it gives rise to a finite physically meaningful quantity. This is the case when the field is submitted to boundary conditions and properly renormalized, as we show in a related paper. 
  In a related paper we have obtained that the effective action for a kappa-deformed quantum field theory has a real and an imaginary part. The real part is half the sum of the kappa-deformed zero mode frequencies, while the imaginary part is proportional to the sum of the squares of the zero mode frequencies, being proportional to the inverse of kappa. Here we calculate this imaginary part for the kappa-deformed electromagnetic field confined between two perfectly conducting parallel plates. After renormalization this imaginary part gives a creation rate of kappa-deformed electromagnetic radiation. This creation rate goes to zero at the appropriate limits, namely: when the deformation disappears or at infinite separation of the plates. The result agrees with previously obtained results and shed light on them by exhibiting the creation rate as originated in a sum of zero modes. Let us note that due to the rather complicated kappa-deformed electromagnetic dispersion relation we were led to the theorem of the argument in order to sum the squares of the kappa-deformed frequencies. 
  We consider unstable D0-branes of two dimensional string theory, described by the boundary state of Zamolodchikov and Zamolodchikov [hep-th/0101152] multiplied by the Neumann boundary state for the time coordinate $t$. In the dual description in terms of the $c=1$ matrix model, this D0-brane is described by a matrix eigenvalue on top of the upside down harmonic oscillator potential. As suggested by McGreevy and Verlinde [hep-th/0304224], an eigenvalue rolling down the potential describes D-brane decay. As the eigenvalue moves down the potential to the asymptotic region it can be described as a free relativistic fermion. Bosonizing this fermion we get a description of the state in terms of a coherent state of the tachyon field in the asymptotic region, up to a non-local linear field redefinition by an energy-dependent phase. This coherent state agrees with the exponential of the closed string one-point function on a disk with Sen's marginal boundary interaction for $t$ which describes D0-brane decay. 
  The most straightforward use of AdS/CFT correspondence gives versions of QCD where quarks are in adjoint representations. Using an asymmetric orbifold approach we obtain nonsupersymmetric QCD with four quark flavors in fundamental representations of color. 
  We calculate the power spectrum of metric fluctuations in inflationary cosmology starting with initial conditions which are imposed mode by mode when the wavelength equals some critical length $\ell_{_{\rm C}}$ corresponding to a new energy scale $M_{_{\rm C}}$ at which trans-Planckian physics becomes important. In this case, the power spectrum can differ from what is calculated in the usual framework (which amounts to choosing the adiabatic vacuum state). The fractional difference in the results depends on the ratio $\sigma_0$ between the Hubble expansion rate $H_{\rm inf}$ during inflation and the new energy scale $M_{_{\rm C}}$. We show how and why different choices of the initial vacuum state (stemming from different assumptions about trans-Planckian physics) lead to fractional differences which depend on different powers of $\sigma_0$. As we emphasize, the power in general also depends on whether one is calculating the power spectrum of density fluctuations or of gravitational waves. 
  The Renormalization Group flow equations obtained by means of a proper time regulator are used to analyze the restoration of the discrete chiral symmetry at non-zero density and temperature in the Gross-Neveu model in d=2+1 dimensions. The effects of the wave function renormalization of the auxiliary scalar field on the transition have been studied. The analysis is performed for a number of fermion flavors N_f=12 and the limit of large N_f is also considered. The results are compared with those coming from lattice simulations. 
  The Cartan-Penrose (CP) equation is interpreted as a connection between a spinor at a point in spacetime, and a pair of holographic screens on which the information at that point may be projected. Local SUSY is thus given a physical interpretation in terms of the ambiguity of the choice of holographic screen implicit in the work of Bousso. The classical CP equation is conformally invariant, but quantization introduces metrical information via the B(ekenstein)-H(awking)-F(ischler)-S(usskind)-B(ousso) connection between area and entropy. A piece of the classical projective invariance survives as the $(-1)^F$ operation of Fermi statistics. I expand on a previously discussed formulation of quantum cosmology, using the connection between SUSY and screens. 
  We discuss the cosmological constant problem in the context of higher codimension brane world scenarios with infinite-volume extra dimensions. 
  The principle of equivalence provides a description of gravity in terms of the metric tensor and determines how gravity affects the light cone structure of the space-time. This, in turn, leads to the existence of observers (in any space-time) who do not have access to regions of space-time bounded by horizons. To take into account this generic possibility, it is necessary to demand that \emph{physical theories in a given coordinate system must be formulated entirely in terms of variables that an observer using that coordinate system can access}. This principle is powerful enough to obtain the following results: (a) The action principle of gravity must be of such a structure that, in the semiclassical limit, the action of the unobserved degrees of freedom reduces to a boundary contribution $A_{\rm boundary}$ obtained by integrating a four divergence. (b) When the boundary is a horizon, $A_{\rm boundary}$ essentially reduces to a single, well-defined, term. (c) This boundary term must have a quantized spectrum with uniform spacing, $\Delta A_{boundary}=2\pi\hbar$, in the semiclassical limit. Using this principle in conjunction with the usual action principle in gravity, we show that: (i) The area of any one-way membrane is quantized. (ii) The information hidden by a one-way membrane leads to an entropy which is always one-fourth of the area of the membrane, in the leading order. (iii) In static space-times, the action for gravity can be given a purely thermodynamic interpretation and the Einstein equations have a formal similarity to laws of thermodynamics. 
  Characteristic length scale of the post-Newtonian corrections to the gravitational field of a body is given by its gravitational radius r_g. The role of this scale in quantum domain is discussed in the context of the low-energy effective theory. The question of whether quantum gravity effects appear already at r_g leads to the question of correspondence between classical and quantum theories, which in turn can be unambiguously resolved considering the issue of general covariance. The O(\hbar^0) loop contributions turn out to violate the principle of general covariance, thus revealing their essentially quantum nature. The violation is O(1/N), where N is the number of particles in the body. This leads naturally to a macroscopic formulation of the correspondence principle. 
  It is shown that Euclidean field theory with polynomial interaction, can be regularized using the wavelet representation of the fields. The connections between wavelet based regularization and stochastic quantization are considered. 
  We argue that two dimensional classical SU(2) Yang-Mills theory describes the embedding of Riemann surfaces in three dimensional curved manifolds. Specifically, the Yang-Mills field strength tensor computes the Riemannian curvature tensor of the ambient space in a thin neighborhood of the surface. In this sense the two dimensional gauge theory then serves as a source of three dimensional gravity. In particular, if the three dimensional manifold is flat it corresponds to the vacuum of the Yang-Mills theory. This implies that all solutions to the original Gauss-Codazzi surface equations determine two dimensional integrable models with a SU(2) Lax pair. Furthermore, the three dimensional SU(2) Chern-Simons theory describes the Hamiltonian dynamics of two dimensional Riemann surfaces in a four dimensional flat space-time. 
  We compute the supergravity loop contributions to the visible sector scalar masses in the simplest 5D `brane-world' model. Supersymmetry is assumed to be broken away from the visible brane and the contributions are UV finite due to 5D locality. We perform the calculation with N = 1 supergraphs, using a formulation of 5D supergravity in terms of N = 1 superfields. We compute contributions to the 4D effective action that determine the visible scalar masses, and we find that the mass-squared terms are negative. 
  We present explicit analytic form of general warped solutions of the string inspired dilaton gravity system with bulk cosmological constant in 5 dimensions. The general solution allows for either nonvanishing effective 4-dimensional cosmological constant or the nontrivial 4-dimensional dilaton but not both. 
  In this paper, a inflationary model from the rotation of D4-brane is constructed. We show that for a very wide rage of parameter, this model satisfies the observation and find that regarded as inflaton, the rotation of branes may be more nature than the distance between branes. Our model offers a new avenue for brane inflation. 
  We consider a four dimensional space-time symmetry which is a non trivial extension of the Poincar\'e algebra, different from supersymmetry and not contradicting {\sl a priori} the well-known no-go theorems. We investigate some field theoretical aspects of this new symmetry and construct invariant actions for non-interacting fermion and non-interacting boson multiplets. In the case of the bosonic multiplet, where two-form fields appear naturally, we find that this symmetry is compatible with a local U(1) gauge symmetry, only when the latter is gauge fixed by a `t Hooft-Feynman term. 
  In the chaotic quantization approach one replaces the Gaussian white noise of the Parisi-Wu approach of stochastic quantization by a deterministic chaotic process on a very small scale. We consider suitable coupled chaotic noise processes as generated by Tchebyscheff maps, and show that the vacuum energy of these models is minimized for coupling constants that coincide with running standard model couplings at energy scales given by the known fermion and boson masses. Chaotic quantization thus allows to predict fundamental constants of nature from first principles. At the same time, it provides a natural framework to understand the dynamical origin of vacuum energy in our universe. 
  We discuss the Ward identity and the low-energy theorem for the divergence of the axial-vector current in the massless Thirring model with fermion fields quantized in the chirally broken phase (Eur. Phys. J. C20, 723 (2001)). The Ward identity and the low-energy theorem are analysed in connection with Witten's criterion for Goldstone bosons (Nucl. Phys. B145, 110 (1978)). We show that the free massless (pseudo)scalar field bosonizing the massless Thirring model in the chirally broken phase satisfies Witten's criterion to interpret quanta of this field as Goldstone bosons. As has been shown in hep-th/0210104 and hep-th/0212226, Goldstone's criterion, the non-invariance of the wave function of the ground state, is also fulfilled. 
  We study both the direct and the double dimensional reduction of space-like branes of M-theory and point out some peculiarities in the process unlike their time-like counterpart. In particular, we show how starting from SM2 and SM5-brane solutions we can obtain SD2 and SNS5-brane as well as SNS1 and SD4-brane solutions of string theory by direct and double dimensional reductions respectively. In the former case we need to use delocalized SM-brane solutions, whereas in the latter case we need to use anisotropic SM-brane solutions in the directions which are compactified. 
  The analysis of the extremal structure of the scalar potentials of gauged maximally extended supergravity models in five, four, and three dimensions, and hence the determination of possible vacuum states of these models is a computationally challenging task due to the occurrence of the exceptional Lie groups $E_6$, $E_7$, $E_8$ in the definition of these potentials. At present, the most promising approach to gain information about nontrivial vacua of these models is to perform a truncation of the potential to submanifolds of the $G/H$ coset manifold of scalars which are invariant under a subgroup of the gauge group and of sufficiently low dimension to make an analytic treatment possible.   New tools are presented which allow a systematic and highly effective study of these potentials up to a previously unreached level of complexity. Explicit forms of new truncations of the potentials of four- and three-dimensional models are given, and for N=16, D=3 supergravities, which are much more rich in structure than their higher-dimensional cousins, a series of new nontrivial vacua is identified and analysed. 
  We express all correlation functions in timelike boundary Liouville theory as unitary matrix integrals and develop efficient techniques to evaluate these integrals. We compute large classes of correlation functions explicitly, including an infinite number of terms in the boundary state of the rolling tachyon. The matrix integrals arising here also determine the correlation functions of gauge invariant operators in two dimensional Yang-Mills theory, suggesting an equivalence between the rolling tachyon and QCD_2. 
  I determine the twisted K-theory of all compact simply connected simple Lie groups. The computation reduces via the Freed-Hopkins-Teleman theorem to the CFT prescription, and thus explains why it gives the correct result. Finally I analyze the exceptions noted by Bouwknegt et al. 
  We study the supergravity dual to the confinement/deconfinement phase transition for the N=4 SU(N) SYM on R x S^3 with a chemical potential conjugate to a U(1)\subset SO(6)_R charge. The appropriate supergravity system is a single charge black hole in D=5 N=8 gauged supergravity. Application of the gauge/string theory holographic renormalization approach leads to new expressions for the black hole ADM mass and its generalized free energy. We comment on the relation of this phase transition to the Hagedorn transition for strings in the maximally supersymmetric plane wave background with null RR five form field strength. 
  A higher dimensional gravity invariant both under local Lorentz rotations and under local Anti de Sitter boosts is constructed. It is shown that such a construction is possible both when odd dimensions and when even dimensions are considered. It is also proved that such actions have the same coefficients as those obtained by Troncoso and Zanelli. 
  In this letter we calculate the emission of gravity waves by the binary pulsar in the framework of five dimensional braneworlds. We consider only spacetimes with one compact extra-dimension. We show that the presence of additional degrees of freedom, especially the 'gravi-scalar' leads to a modification of Einstein's quadrupole formula. We compute the induced change for the binary pulsar PSR 1913+16 where it amounts to about 20% which is by far excluded by present experimental data. 
  Using Berkovits' Superstring Filed Theory action including the Ramond sector, we calculate the contribution to this action coming from the tachyon and massless fermions from both $GSO$ sectors. Some features of the action are discussed in the end, settling the ground for a more systematic treatment of spacetime fermions in superstring field theory. 
  String compactifications with non-abelian gauge fields localized on D-branes, with background NSNS and RR 3-form fluxes, and with non-trivial warp factors, can naturally exist within T-dual versions of type I string theory. We develop a systematic procedure to construct the effective bosonic Lagrangian of type I T-dualized along a six-torus, including the coupling to gauge multiplets on D3-branes and the modifications due to 3-form fluxes. Looking for solutions to the ten-dimensional equations of motion, we find warped products of Minkowski space and Ricci-flat internal manifolds. Once the warp factor is neglected, the resulting no-scale scalar potential of the effective four-dimensional theory combines those known for 3-form fluxes and for internal Yang-Mills fields and stabilizes many of the moduli. We perform an explicit comparison of our expressions to those obtained from N=4 gauged supergravity and find agreement. We also comment on the possibility to include D9-branes with world-volume gauge fluxes in the background with 3-form fluxes. 
  We extend the results of Mirabelli and Peskin to supergravity. We study the compactification on S_1/Z_2 of Zucker's off-shell formulation of 5D supergravity and its coupling to matter at the fixed points. We clarify some issues related to the off-shell description of supersymmetry breaking a la Scherk-Schwarz (here employed only as a technical tool) discussing how to deal with singular gravitino wave functions.   We then consider `visible' and `hidden' chiral superfields localized at the two different fixed points and communicating only through 5D supergravity. We compute the one-loop corrections that mix the two sectors and the radion superfield. Locality in 5D ensures the calculability of these effects, which transmit supersymmetry breaking from the hidden to the visible sector. In the minimal set-up visible-sector scalars get a universal squared mass m_0^2 < 0. In general (e.g. in presence of a sizable gravitational kinetic term localized on the hidden brane) the radion-mediated contribution to m_0^2 can be positive and dominant. Although we did not build a complete satisfactory model, brane-to-brane effects can cure the tachyonic sleptons predicted by anomaly mediation by adding a positive m_0^2 which is universal up to subleading flavour-breaking corrections. 
  We investigate the classical stability of the higher-dimensional Schwarzschild black holes against linear perturbations, in the framework of a gauge-invariant formalism for gravitational perturbations of maximally symmetric black holes, recently developed by the authors. The perturbations are classified into the tensor, vector, and scalar-type modes according to their tensorial behaviour on the spherical section of the background metric, where the last two modes correspond respectively to the axial- and the polar-mode in the four-dimensional situation. We show that, for each mode of the perturbations, the spatial derivative part of the master equation is a positive, self-adjoint operator in the $L^2$-Hilbert space, hence that the master equation for each tensorial type of perturbations does not admit normalisable negative-modes which would describe unstable solutions.   On the same Schwarzschild background, we also analyse the static perturbation of the scalar mode, and show that there exists no static perturbation which is regular everywhere outside the event horizon and well-behaved at spatial infinity. This checks the uniqueness of the higher-dimensional spherically symmetric, static, vacuum black hole, within the perturbation framework.   Our strategy for the stability problem is also applicable to the other higher-dimensional maximally symmetric black holes with non-vanishing cosmological constant. We show that all possible types of maximally symmetric black holes (thus, including the higher-dimensional Schwarzschild-de Sitter and Schwarzschild-anti-de Sitter black holes) are stable against the tensor and the vector perturbations. 
  We discuss a model in which the boundary condition decides the four dimensional cosmological constant. It is reviewed in a primitive way that boundary conditions are required by the action principle. 
  We propose a "Copenhagen interpretation" for spacetime noncommutativity. The goal is to be able to predict results of simple experiments involving signal propagation directly from commutation relations. A model predicting an energy dependence of the speed of photons of the order E/E_Planck is discussed in detail. Such effects can be detectable by the GLAST telescope, to be launched in 2006. 
  This thesis first gives a background and review of the article hep-th/0102038, where we substantiated a conjectured duality between two a priori unrelated gauge theories, N=2 quiver theory and N=2 Seiberg-Witten theory. These gauge theories have different realisations as the worldvolume theories on different D-brane configurations. We showed that there is an identity between the spaces of vacua (moduli spaces) arising in the two theories, which suggests that the corresponding string theory pictures are dual. Second, the thesis gives a background and description of the analysis performed in the three articles hep-th/0111161, hep-th/0202069 and hep-th/0304013, where we derived the most general, local, superconformal boundary conditions of the two-dimensional nonlinear sigma model. This model describes the dynamics of open strings, and the boundary conditions dictate the geometry of D-branes. In the last article we studied these boundary conditions for the special case of WZW models. 
  We continue our study of nonperturbative superpotentials of four-dimensional N=2 supersymmetric gauge theories with gauge group U(N) on R^3 x S^1, broken to N=1 due to a classical superpotential. In a previous paper, hep-th/0304061, we discussed how the low-energy quantum superpotential can be obtained by substituting the Lax matrix of the underlying integrable system directly into the classical superpotential. In this paper we prove algebraically that this recipe yields the correct factorization of the Seiberg-Witten curves, which is an important check of the conjecture. We will also give an independent proof using the algebraic-geometrical interpretation of the underlying integrable system. 
  Many quantum field theoretical models possess non-trivial solutions which are stable for topological reasons. We construct a self-consistent example for a self-interacting scalar field--the quantum (or dressed) kink--using a two particle irreducible effective action in the Hartree approximation. This new solution includes quantum fluctuations determined self-consistently and nonperturbatively at the 1-loop resummed level and allowed to backreact on the classical mean-field profile. This dressed kink is static under the familiar Hartree equations for the time evolution of quantum fields. Because the quantum fluctuation spectrum is lower lying in the presence of the defect, the quantum kink has a lower rest energy than its classical counterpart. However its energy is higher than well-known strict 1-loop results, where backreaction and fluctuation self-interactions are omitted. We also show that the quantum kink exists at finite temperature and that its profile broadens as temperature is increased until it eventually disappears. 
  We compute the decay of an unstable D9 brane in type IIA string theory including backreaction effects using an effective field theory approach. The open string tachyon on the brane is coupled consistently to the space-time metric, the dilaton and the RR 9-form. The purpose of this note is to address the fate of the open string energy density, which remains constant if no interaction with the closed string modes is included. Our computations show that taking only into account the coupling to the massless closed strings the total energy stored in the open string sector vanishes asymptotically, independently how small one chooses g_s. We find also the large time behaviour of the fields in the Einstein and string frames. 
  We point out that gravity on dS_n gives rise to a localized graviton on dS_{n-1}. This way one can derive a recursion relation for the entropy of dS spaces, which might have interesting implications for dS holography. In the same spirit we study domain walls interpolating between dS spaces with different cosmological constant. Our observation gives an easy way to calculate what fraction of the total entropy can be accessed by an observer stuck on the bubble wall. 
  We argue that the demand of background independence in a quantum theory of gravity calls for an extension of standard geometric quantum mechanics. We discuss a possible kinematical and dynamical generalization of the latter by way of a quantum covariance of the state space. Specifically, we apply our scheme to the problem of a background independent formulation of Matrix Theory. 
  We investigate two classes of D-branes in 2-d string theory, corresponding to extended and localized branes, respectively. We compute the string emission during tachyon condensation and reinterpret the results within the $c=1$ matrix model. As in hep-th/0304224, we find that the extended branes describe classical eigenvalue trajectories, while, as found in hep-th/0305159, the localized branes correspond to the quantum field that creates and destroys eigenvalues. The matrix model relation between the classical probe and the local collective field precisely matches with the descent relation between the boundary states of D-strings and D-particles. 
  Generalizing self-duality on R^2 x S^2 to higher dimensions, we consider the Donaldson-Uhlenbeck-Yau equations on R^{2n} x S^2 and their noncommutative deformation for the gauge group U(2). Imposing SO(3) invariance (up to gauge transformations) reduces these equations to vortex-type equations for an abelian gauge field and a complex scalar on R^{2n}_\theta. For a special S^2-radius R depending on the noncommutativity \theta we find explicit solutions in terms of shift operators. These vortex-like configurations on R^{2n}_\theta determine SO(3)-invariant multi-instantons on R^{2n}_\theta x S^2_R for R=R(\theta). The latter may be interpreted as sub-branes of codimension 2n inside a coincident pair of noncommutative Dp-branes with an S^2 factor of suitable size. 
  We calculate in the strong coupling and large N limit the energy emitted by an accelerated external charge in ${\cal N}=4$ SU(N) Yang-Mills theory, using the AdS/CFT correspondence. We find that the energy is a local functional of the trajectory of the charge. It coincides up to an overall factor with the Lienard formula of the classical electrodynamics. In the AdS description the radiated energy is carried by a nonlinear wave on the string worldsheet for which we find an exact solution. 
  We study non-critical superstrings propagating in $d \le 6$ dimensional Minkowski space or equivalently, superstrings propagating on the two-dimensional Euclidean black hole tensored with d-dimensional Minkowski space. We point out a subtlety in the construction of supersymmetric theories in these backgrounds, and explain how this does not allow a consistent geometric interpretation in terms of fields propagating on a cigar-like spacetime. We explain the global symmetries of the various theories by using their description as the near horizon geometry of wrapped NS5-brane configurations. In the six-dimensional theory, we present a CFT description of the four-dimensional moduli space and the global O(3) symmetry. The worldsheet action invariant under this symmetry contains both the N=2 sine-Liouville interaction and the cigar metric, thereby providing an example where the two interactions are naturally present in the same worldsheet lagrangian already at the non-dynamical level. 
  Domain wall networks on the surface of a soliton are studied in a simple theory. It consists of two complex scalar fields, in (3+1)-dimensions, with a global U(1) x Z_n symmetry, where n>2. Solutions are computed numerically in which one of the fields forms a Q-ball and the other field forms a network of domain walls localized on the surface of the Q-ball. Examples are presented in which the domain walls lie along the edges of a spherical polyhedron, forming junctions at its vertices. It is explained why only a small restricted class of polyhedra can arise as domain wall networks. 
  We consider general approach to exactly solvable 2D dilaton cosmology with one-loop backreaction from conformal fields taken into account. It includes as particular cases previous models discussed in literature. We list different types of solutions and investigate their properties for simple models, typical for string theory. We find a rather rich class of everywhere regular solutions which exist practically in every type of analyzed solutions. They exhibit different kinds of asymptotic behavior in past and future, including inflation, superinflation, deflation, power expansion or contraction. In particular, for some models the dS spacetime with a time-dependent dilaton field is the exact solution of field equations. For some kinds of solutions the weak energy condition is violated independent of a specific model. We find also the solutions with a singularity which is situated in an infinite past (or future), so at any finite moment of a comoving time the universe is singularity-free. It is pointed out that for some models the spacetime may be everywhere regular even in spite of infinitely large quantum backreaction in an infinite past. 
  Operator product expansions (OPE) for the product of a scalar field with its conjugate are presented as infinite sums of bilocal fields V_k (x_1, x_2) of dimension (k,k). For a {\it globally conformal invariant} (GCI) theory we write down the OPE of V_k into a series of {\it twist} (dimension minus rank) 2k symmetric traceless tensor fields with coefficients computed from the (rational) 4-point function of the scalar field.   We argue that the theory of a GCI hermitian scalar field L(x) of dimension 4 in D = 4 Minkowski space such that the 3-point functions of a pair of L's and a scalar field of dimension 2 or 4 vanish can be interpreted as the theory of local observables of a conformally invariant fixed point in a gauge theory with Lagrangian density L(x). 
  Scalar field theories with quartic interaction are quantized on fuzzy $S^2$ and fuzzy $S^2\times S^2$ to obtain the 2- and 4-point correlation functions at one-loop. Different continuum limits of these noncommutative matrix spheres are then taken to recover the quantum noncommutative field theories on the noncommutative planes ${\mathbb R}^2$ and ${\mathbb R}^4$ respectively. The canonical limit of large stereographic projection leads to the usual theory on the noncommutative plane with the well-known singular UV-IR mixing. A new planar limit of the fuzzy sphere is defined in which the noncommutativity parameter ${\theta}$, beside acting as a short distance cut-off, acts also as a conventional cut-off ${\Lambda}=\frac{2}{\theta}$ in the momentum space. This noncommutative theory is characterized by absence of UV-IR mixing. The new scaling is implemented through the use of an intermediate scale that demarcates the boundary between commutative and noncommutative regimes of the scalar theory. We also comment on the continuum limit of the $4-$point function. 
  The construction of supersymmetric invariant integrals is discussed in a superspace setting. The formalism is applied to D=4, N=4 SYM and used to construct the F^2, F^4 and (F^5 + \del^2 F^4) terms in the effective action of coincident D-branes. The results are in agreement with those obtained by other methods. A simple derivation of the abelian \del^4 F^4 invariant is given and generalised to the non-abelian case. We also find some double-trace invariants. The invariants are interpreted in terms of superconformal multiplets: the F^2 and F^4 terms are given by one-half BPS multiplets, the (F^5+\del^2F^4) arises as a full superspace integral of the Konishi multiplet K and the abelian \del^4 F^4 term comes from integrating the fourth power of the field strength superfield. Counterparts of the abelian invariants are exhibited for the D=6,(2,0) tensor multiplet and the D=3, N=8 scalar multiplet. The method is also applied to D=4, N=8 supergravity. All invariants in the linearised theory (with SU(8) symmetry) which arise from partial superspace integrals are constructed. 
  We discuss conformal invariance of the massless Thirring model. We show that conformal symmetry of the massless Thirring model is dynamically broken due to the constant of motion caused by the equations of motion. This confirms the existence of the chirally broken phase in the massless Thirring model (Eur. Phys. J. C20, 723 (2001), which is accompanied by the appearance of massless (pseudo)scalar Goldstone bosons (Eur. Phys. J. C24, 653 (2002), hep-th/0210104 and hep-th/0305174). 
  Quantization of gravity suggests that a finite region of space has a finite number of degrees of freedom or `bits'. What happens to these bits when spacetime expands, as in cosmological evolution? Using gravity/field theory duality we argue that bits `fuse together' when space expands. 
  We define a class of topological A-models on a collection of Riemann surfaces, whose boundaries are sewn together along the seams. The target spaces for the Riemann surfaces are the Grassmanians Gr_{m_i,n} with the common value of n, and the boundary conditions at the seams demand that the spaces C^{m_i}\subset C^n present the orthogonal decomposition of C^n. The whole construction is a QFT interpretation of a part of Khovanov's categorification of the sl(3) HOMFLY polynomial. 
  There are many theories of quantum gravity, depending on asymptotic boundary conditions, and the amount of supersymmetry. The cosmological constant is one of the fundamental parameters that characterize different theories. If it is positive, supersymmetry must be broken. A heuristic calculation shows that a cosmological constant of the observed size predicts superpartners in the TeV range. This mechanism for SUSY breaking also puts important constraints on low energy particle physics models. This essay was submitted to the Gravity Research Foundation Competition and is based on a longer article, which will be submitted in the near future. 
  Recent analysis of the observation data indicates that the equation of state of the dark energy might be smaller than -1, which leads to the introduction of phantom models featured by its negative kinetic energy to account for the regime of equation of state $w<-1$. In this paper, we generalize the idea to the Born-Infield type Lagrangian with negative kinetic energy term and give the condition for the potential, under which the late time attractor solution exists and also analyze a viable cosmological model in such a scheme. 
  We analyze the possibility of nonperturbative renormalizability of gauge theories in D > 4 dimensions. We develop a scenario, based on Weinberg's idea of asymptotic safety, that allows for renormalizability in extra dimensions owing to a non-Gaussian ultraviolet stable fixed point. Our scenario predicts a critical dimension D_cr beyond which the UV fixed point vanishes, such that renormalizability is possible for D <= D_cr. Within the framework of exact RG equations, the critical dimension for various SU(N) gauge theories can be computed to lie near five dimensions: 5 ~< D_cr < 6. Therefore, our results exclude nonperturbative renormalizability of gauge theories in D=6 and higher dimensions. 
  We show the connection between the extended center of the quantum group in roots of unity and the restriction of the $XXZ$ model. We also give explicit expressions for operators that respect the restriction and act on the state space of the restricted models. The formulas for these operators are verified by explicit calculation for third-degree roots; they are conjectured to hold in the general case. 
  We consider the behavior of the photon number integral under inversion, concentrating on euclidean space. The discussion may be framed in terms of an additive differential $I$ which arises under inversions. The quantity $\int \int I$ is an interesting integral invariant whose value characterizes different configurations under inversion. 
  We find broad classes of solutions to the field equations for d-dimensional gravity coupled to an antisymmetric tensor of arbitrary rank and a scalar field with non-vanishing potential. Our construction generates these configurations from the solution of a single nonlinear ordinary differential equation, whose form depends on the scalar potential. For an exponential potential we find solutions corresponding to brane geometries, generalizing the black p-branes and S-branes known for the case of vanishing potential. These geometries are singular at the origin with up to two (regular) horizons. Their asymptotic behaviour depends on the parameters of the model. When the singularity has negative tension or the cosmological constant is positive we find time-dependent configurations describing accelerating universes. Special cases give explicit brane geometries for (compact and non-compact) gauged supergravities in various dimensions, as well as for massive 10D supergravity, and we discuss their interrelation. Some examples lift to give new solutions to 10D supergravity. Limiting cases with a domain wall structure preserve part of the supersymmetries of the vacuum. We also consider more general potentials, including sums of exponentials. Exact solutions are found for these with up to three horizons, having potentially interesting cosmological interpretation. We give several additional examples which illustrate the power of our techniques. 
  In this paper, we explicitly construct a series of projectors on integral noncommutative orbifold $T^2/Z_4$ by extended $GHS$ constrution. They include integration of two arbitary functions with $Z_4$ symmetry. Our expressions possess manifest $Z_{4}$ symmetry. It is proved that the expression include all projectors with minimal trace and in their standard expansions, the eigen value functions of coefficient operators are continuous with respect to the arguments $k$ and $q$. Based on the integral expression, we alternately show the derivative expression in terms of the similar kernal to the integral one.Since projectors correspond to soliton solutions of the field theory on the noncommutative orbifold, we thus present a series of corresponding solitons. 
  We study an SU(N) gauge-Higgs model with N_F massless fundamental fermions on M^3 \otimes S^1. The model has two kinds of order parameters for gauge symmetry breaking: the component gauge field for the S^1 direction (Hosotani mechanism) and the Higgs field (Higgs mechanism). We find that the model possesses three phases called Hosotani, Higgs and coexisting phases for N=odd, while for N=even, the model has only two phases, the Hosotani and coexisting phases. The phase structure depends on a parameter of the model and the size of the extra dimension. The critical radius and the order of the phase transition are determined. We also consider the case that the representation of matter fields under the gauge group is changed. We find some models, in which there is only one phase, independent of parameters of the models as well as the size of the extra dimension. 
  Quantum mechanical WKB-method is elaborated for the known quantum Kepler problem in curved 3-space models Euclide, Riemann and Lobachevsky in the framework of the complex variable function theory. Generalized Schr\"{o}dinger, Klein-Fock hydrogen atoms are considered. Exact energy levels are found and their exactness is proved on the base of exploration into $n$-degree terms of the WKB-series. Dirac equation is solved too, but only approximate energy spectrum is established. 
  We examine Killing spinor equations of the general eleven-dimensional pp-wave backgrounds, which contain a scalar H(x^m,x^-) in the metric and a three-form \xi(x^m,x^-) in the flux. Considering non-harmonic extra Killing spinors, we show that if the backgrounds admit at least one extra Killing spinor in addition to the standard 16 Killing spinors, they can be reduced to the form with H=A_{mn}(x^-)x^mx^n and \xi(x^-) modulo coordinate transformations. We further examine the cases in which the extra Killing spinor is characterized by a set of Cartan matrices. The super-isometry algebras of the resulting backgrounds are also derived. 
  We rederive the thermodynamical properties of a non interacting gas in the presence of a minimal uncertainty in length. Apart from the phase space measure which is modified due to a change of the Heisenberg uncertainty relations, the presence of an ultraviolet cut-off plays a tremendous role.  The theory admits an intrinsic temperature above which the fermion contribution to energy density, pressure and entropy is negligible. 
  We study effective gravitational F-terms, obtained by integrating an $U(N)$ adjoint chiral superfield $\Phi$ coupled to the ${\cal N}=1$ gauge chiral superfield $W_\alpha$ and supergravity, to arbitrary orders in the gravitational background. The latter includes in addition to the ${\cal N}=1$ Weyl superfield $G_{\alpha\beta\gamma}$, the self-dual graviphoton field strength $F_{\alpha\beta}$ of the parent, broken ${\cal N}=2$ theory. We first study the chiral ring relations resulting from the above non-standard gravitational background and find agreement, for gauge invariant operators, with those obtained from the dual closed string side via Bianchi identities for ${\cal N}=2$ supergravity coupled to vector multiplets. We then derive generalized anomaly equations for connected correlators on the gauge theory side, which allow us to solve for the basic one-point function $\langle {\rm Tr} W^2/(z-\Phi)\rangle$ to all orders in $F^2$. By generalizing the matrix model loop equation to the generating functional of connected correlators of resolvents, we prove that the gauge theory result coincides with the genus expansion of the associated matrix model, after identifying the expansion parameters on the two sides. 
  We give a rigorous calculation of the large N limit of the partition function of SU(N) gauge theory on a 2D cylinder in the case where one boundary holomony is a so-called special element of type $\rho$. By MacDonald's identity, the partition function factors in this case as a product over positive roots and it is straightforward to calculate the large N asymptotics of the free energy. We obtain the unexpected result that the free energy in these cases is asymptotic to N times a functional of the limit densities of eigenvalues of the boundary holonomies. This appears to contradict the predictions of Gross-Matysin and Kazakov-Wynter that the free energy should have a limit governed by the complex Burgers equation. 
  The geometrical description of BPS 3-string junction in the F-theory background is given by lifting a string junction in IIB into F-theory and constructing a holomorphic curve in K3 with respect to a special complex structure of K3. The holomorphic curve is fibration of 1-cycles of the elliptic fiber over the geodesic string junction. The F-theory picture in this paper provides an unifying description of both string and string junction, and is advantageous over the M-theory picture of them. 
  The implementation of the Background Field Method (BFM) for quantum field theories is analysed within the Batalin-Vilkovisky (BV) formalism. We provide a systematic way of constructing general splittings of the fields into classical and quantum parts, such that the background transformations of the quantum fields are linear in the quantum variables. This leads to linear Ward-Takahashi identities for the background invariance and to great simplifications in multiloop computations. In addition, the gauge fixing is obtained by means of (anti)canonical transformations generated by the gauge-fixing fermion. Within this framework we derive the BFM for the N=2 Super-Yang-Mills theory in the Wess-Zumino gauge viewed as the twisted version of Donaldson-Witten topological gauge theory. We obtain the background transformations for the full BRST differential of N=2 Super-Yang-Mills (including gauge transformations, SUSY transformations and translations). The BFM permits all observables of the supersymmetric theory to be identified easily by computing the equivariant cohomology of the topological theory. These results should be regarded as a step towards the construction of a super BFM for the Minimal Supersymmetric Standard Model. 
  An explicit operator mapping in the form of a similarity transformation is constructed between the RNS formalism and an extension of the pure spinor formalism (to be called EPS formalism) recently proposed by the present authors. Due to the enlarged field space of the EPS formalism, where the pure spinor constraints are removed, the mapping is completely well-defined in contrast to the one given previously by Berkovits in the original pure spinor (PS) formalism. This map provides a direct demonstration of the equivalence of the cohomologies of the RNS and the EPS formalisms and is expected to be useful for better understanding of various properties of the PS and EPS formalisms. Furthermore, the method of construction, which makes systematic use of the nilpotency of the BRST charges, should find a variety of applications. 
  The localization of vector multiplets is examined using the {\cal N}=1 supersymmetric U(1) gauge theory with the Fayet-Iliopoulos term coupled to charged chiral multiplets in four dimensions. The vector field becomes localized on a BPS wall connecting two different vacua that break the gauge symmetry. The vacuum expectation values of charged fields vanish (approximately) around the center of the wall, causing the Higgs mechanism to be ineffective. The mass of the localized vector multiplet is found to be the inverse width of the wall. The model gives an explicit example of this general phenomenon. A five-dimensional version of the model can also be constructed if we abandon supersymmetry. 
  In this paper, three methods for describing the conformal transformations of the S-matrix in quantum field theory are proposed. They are illustrated by applying the algebraic renormalization procedure to the quantum scalar field theory, defined by the LSZ reduction mechanism in the BPHZ renormalization scheme. Central results are shown to be independent of scheme choices and derived to all orders in loop expansions. Firstly, the local Callan-Symanzik equation is constructed, in which the insertion of the trace of the energy-momentum tensor is related to the beta function and the anomalous dimension. With this result, the Ward identities for the conformal transformations of the Green functions are derived. Then the conformal transformations of the S-matrix defined by the LSZ reduction procedure are calculated. Secondly, the conformal transformations of the S-matrix in the functional formalism are related to charge constructions. The commutators between the charges and the S-matrix operator are written in a compact way to represent the conformal transformations of the S-matrix. Lastly, the massive scalar field theory with local coupling is introduced in order to control breaking of the conformal invariance further. The conformal transformations of the S-matrix with local coupling are calculated. 
  We study deformations of $Z_2 \times Z_2$ (shift-)orientifolds in four dimensions in the presence of both uniform Abelian internal magnetic fields and quantized NS-NS $B_{ab}$ backgrounds, that are shown to be equivalent to asymmetric shift-orbifold projections. These models are related by $T$-duality to orientifolds with $D$-branes intersecting at angles. As in corresponding six-dimensional examples, $D9$-branes magnetized along two internal directions acquire a charge with respect to the R-R six form, contributing to the tadpole of the orthogonal $D5$-branes (``brane transmutation''). The resulting models exhibit rank reduction of the gauge group and multiple matter families, due both to the quantized $B_{ab}$ and to the background magnetic fields. Moreover, the low-energy spectra are chiral and anomaly free if additional $D5$-branes longitudinal to the magnetized directions are present, and if there are no Ramond-Ramond tadpoles in the corresponding twisted sectors of the undeformed models. 
  In the framework of the matrix model/gauge theory correspondence, we consider supersymmetric U(N) gauge theory with $U(1)^N$ symmetry breaking pattern. Due to the presence of the Veneziano--Yankielowicz effective superpotential, in order to satisfy the $F$--term condition $\sum_iS_i=0$, we are forced to introduce additional terms in the free energy of the corresponding matrix model with respect to the usual formulation. This leads to a matrix model formulation with a cubic potential which is free of parameters and displays a branched structure. In this way we naturally solve the usual problem of the identification between dimensionful and dimensionless quantities. Furthermore, we need not introduce the $\N=1$ scale by hand in the matrix model. These facts are related to remarkable coincidences which arise at the critical point and lead to a branched bare coupling constant. The latter plays the role of the $\N=1$ and $\N=2$ scale tuning parameter. We then show that a suitable rescaling leads to the correct identification of the $\N=2$ variables. Finally, by means of the the mentioned coincidences, we provide a direct expression for the $\N=2$ prepotential, including the gravitational corrections, in terms of the free energy. This suggests that the matrix model provides a triangulation of the istanton moduli space. 
  We analyze the dynamics of D2-branes on SU(2) within a recently proposed matrix model, which works for finite radius of SU(2). The spectrum of single-brane excitations turns out to be free of tachyonic modes. It is similar to the spectrum found using DBI and CFT calculations, however the triplet of rotational zero modes is missing. This is attributed to a naive treatment of the quantum symmetries of the model. The mass of the lightest states connecting two different branes is also calculated, and found to be proportional to the arc length for small angles. 
  The relativistic two-particle quantum mixtures are studied from the topological point of view. The mixture field variables can be transformed in such a way that a kinematical decoupling of both particle degrees of freedom takes place with a residual coupling of purely algebraic nature ("exchange coupling"). Both separated sets of particle variables induce a certain map of space-time onto the corresponding "exchange groups", i.e. SU(2) and SU(1,1), so that for the compact case (SU(2)) there arises a pair of winding numbers, either odd or even, which are a topological characteristic of the two-particle Hamiltonian. 
  In the semi-light cone gauge $g_{ab}=e^{2\phi}\delta_{ab}$, $\bar{\gamma}^+\theta=0$, we evaluate the $\phi$-dependent effective action for the pp-wave Green-Schwarz (GS) superstring in both harmonic and group coordinates. When we compute the fermionic $\phi$-dependent effective action in harmonic coordinates, we find a new triangular one-loop Feynman diagram. We show that the bosonic $\phi$-dependent effective action cancels with the fermionic one, indicating that the pp-wave GS superstring is a conformal field theory. We introduce the group coordinates preserving $SO(4)\times SO(4)$ and conformal symmetry. Group coordinates are interesting because vertex operators take simple forms in them. The new feature in group coordinates is that there are logarithmic divergences from n-gons, so that the divergent structure is more complicated than in harmonic coordinates. After summing over all contributions from n-gons, we show that in group coordinates, the GS superstring on pp-wave RR background is still a conformal field theory. 
  We study classical dynamics of an open string tachyon $T$ of unstable D$p$-brane coupled to the gauge field $A_\mu$. In the vacuum with vanishing potential, V=0, two fluid-like degrees of freedom, string fluid and tachyon matter, survive the tachyon condensation. We offer general analysis of the associated Hamiltonian dynamics in arbitrary background. The canonical field equations are organized into two sets, fluid equations of motion augmented by an integrability condition. We show that a large class of motionless and degenerate family of classical solutions exist and represent arbitrary transverse distribution of tachyon matter and flux lines. We further test their stability by perturbing the fluid equation up to the second order.   Second half of this note considers possibility of $V \neq 0$ in the dynamics. We incorporate $V$ in the Hamiltonian equation of motion and consider interaction between domain walls and string fluid. During initial phase of tachyon condensation, topological defect at T=0 is shown to attract nearby and parallel flux lines. The final state is fundamental strings absorbed and spread in some singular D$(p-1)$ brane soliton. When string fluid is transverse to the domain wall, the latter is known to turn into a smooth solution. We point out that a minimal solution of this sort exists and saturates a BPS energy bound of fundamental string ending on a D$(p-1)$ brane. 
  We study the holographic description of string theory in a plane wave spacetime by taking the Penrose limit of the usual AdS/CFT correspondence. We consider three-point functions with two BMN operators and one non-BMN operator; the latter should go over to a perturbation of the dual CFT. On the string side we take the Penrose limit of the metric perturbation produced by the non-BMN operator, and the BMN state propagates in this perturbed background. The work of Lee, Minwalla, Rangamani, and Seiberg shows that for chiral operators the AdS three-point functions agree with those calculated in the free gauge theory. However, when this is reduced to an effective plane wave amplitude by truncating the amplitude to propagate from the AdS boundary to the Penrose geodesic, we find a puzzling mismatch. We discuss possible resolutions, and future directions. 
  We investigate the Einstein equation with a positive cosmological constant for $4n+4$-dimensional metrics on bundles over Quaternionic K\"ahler base manifolds whose fibers are 4-dimensional Bianchi IX manifolds. The Einstein equations are reduced to a set of non-linear ordinary differential equations. We numerically find inhomogeneous compact Einstein spaces with orbifold singularity. 
  We study kink (domain wall) solutions in a model consisting of two complex scalar fields coupled to two independent Abelian gauge fields in a Lagrangian that has $U(1)\times U(1)$ gauge plus $\mathbb{Z}_2$ discrete symmetry. We find consistent solutions such that while the U(1) symmetries of the fields are preserved while in their respective vacua, they are broken on the domain wall. The gauge field solutions show that the domain wall is sandwiched between domains with constant magnetic fields. 
  Recently, Hirayama and Yamashita have presented an ansatz that allows us to construct a class of solutions for the SU(2) Skyrme model. Though these solutions are not solitonic, they provide us with an example on how the plane wave solutions arise in nonlinear field theories. In this paper, we investigate the applicability of the ansatz for the SU(3) Skyrme model. We explicitly construct a class of solutions for the SU(3) model, which in the simplest circumstance is reduced to a combination of the plane waves and Weierstrass elliptic functions. We also discuss some properties of these solutions. For example, the intrinsic structure of these solutions is found to describe an asymmetrical top rotating in the complex three-dimensional space. 
  We investigate the presence of defects in systems described by real scalar field in (D,1) spacetime dimensions. We show that when the potential assumes specific form, there are models which support stable global defects for D arbitrary. We also show how to find first-order differential equations that solve the equations of motion, and how to solve models in D dimensions via soluble problems in D=1. We illustrate the procedure examining specific models and finding explicit solutions. 
  We show that AdS two-point functions can be obtained by connecting two points in the interior of AdS space with one point on its boundary by a dual pair of Dobrev's boundary-to-bulk intertwiners and integrating over the boundary point. 
  In this paper we calculate the divergent part of the one loop effective action for QED on noncommutative space using the background field method. The effective action is obtained up to the second order in the noncommutativity parameter $\theta$ and in the classical fields. 
  In the context of (4+d)-dimensional general relativity, we propose an inflationary scenario wherein 3 spatial dimensions grow large, while d extra dimensions remain small. Our model requires that a self-interacting d-form acquire a vacuum expectation value along the extra dimensions. This causes 3 spatial dimensions to inflate, whilst keeping the size of the extra dimensions nearly constant. We do not require an additional stabilization mechanism for the radion, as stable solutions exist for flat, and for negatively curved compact extra dimensions. From a four-dimensional perspective, the radion does not couple to the inflaton; and, the small amplitude of the CMB temperature anisotropies arises from an exponential suppression of fluctuations, due to the higher-dimensional origin of the inflaton. The mechanism triggering the end of inflation is responsible, both, for heating the universe, and for avoiding violations of the equivalence principle due to coupling between the radion and matter. 
  The behavior of solitons in integrable theories is strongly constrained by the integrability of the theory, that is by the existence of an infinite number of conserved quantities that these theories are known to possess. As a result the soliton scattering of such theories are expected to be trivial (with no change of direction, velocity or shape). In this paper we present an extended review on soliton scattering of two spatial dimensional integrable systems which have been derived as dimensional reductions of the self-dual Yang-Mills-Higgs equations and whose scattering properties are highly non-trivial. 
  We show that a self-tuning mechanism of the cosmological constant could work in 5D non-compact space-time with a $Z_2$ symmetry in the presence of a massless scalar field. The standard model matter fields live only on the 4D brane. The change of vacuum energy on the brane (brane cosmological constant) by, for instance, electroweak and QCD phase transitions, just gives rise to dynamical shifts of the profiles of the background metric and the scalar field in the extra dimension, keeping 4D space-time flat without any fine-tuning. To avoid naked singularities in the bulk, the brane cosmological constant should be negative. We introduce an additional brane-localized 4D Einstein-Hilbert term so as to provide the observed 4D gravity with the non-compact extra dimension. With a general form of brane-localized gravity term allowed by the symmetries, the low energy Einstein gravity is successfully reproduced on the brane at long distances. We show this phenomenon explicitly for the case of vanishing bulk cosmological constant. 
  We review in detail the construction of {\em all} stable static fermion bags in the 1+1 dimensional Gross-Neveu model with $N$ flavors of Dirac fermions, in the large $N$ limit. In addition to the well known kink and topologically trivial solitons (which correspond, respectively, to the spinor and antisymmetric tensor representations of O(2N)), there are also threshold bound states of a kink and a topologically trivial soliton: the heavier topological solitons (HTS). The mass of any of these newly discovered HTS's is the sum of masses of its solitonic constituents, and it corresponds to the tensor product of their O(2N) representations. Thus, it is marginally stable (at least in the large $N$ limit). Furthermore, its mass is independent of the distance between the centers of its constituents, which serves as a flat collective coordinate, or a modulus. There are no additional stable static solitons in the Gross-Neveu model. We provide detailed derivation of the profiles, masses and fermion number contents of these static solitons. For pedagogical clarity, and in order for this paper to be self-contained, we also included detailed appendices on supersymmetric quantum mechanics and on reflectionless potentials in one spatial dimension, which are intimately related with the theory of static fermion bags. In particular, we present a novel simple explicit formula for the diagonal resolvent of a reflectionless Schr\"odinger operator with an arbitrary number of bound states. In additional appendices we summarize the relevant group representation theoretic facts, and also provide a simple calculation of the mass of the kinks. 
  A new solution to Einstein equations in (1+5)-spacetime with an embedded (1+3) brane is given. This solution localizes the zero modes of all kinds of matter fields and 4-gravity on the (1+3) brane by an increasing, transverse gravitational potential. This localization occurs despite the fact that the gravitational potential is not a decreasing exponential, and asymptotically approaches a finite value rather than zero. 
  N=(1,0) supergravity in six dimensions admits AdS_3\times S^3 as a vacuum solution. We extend our recent results presented in hep-th/0212323, by obtaining the complete N=4 Yang-Mills-Chern-Simons supergravity in D=3, up to quartic fermion terms, by S^3 group manifold reduction of the six dimensional theory. The SU(2) gauge fields have Yang-Mills kinetic terms as well as topological Chern-Simons mass terms. There is in addition a triplet of matter vectors. After diagonalisation, these fields describe two triplets of topologically-massive vector fields of opposite helicities. The model also contains six scalars, described by a GL(3,R)/SO(3) sigma model. It provides the first example of a three-dimensional gauged supergravity that can obtained by a consistent reduction of string-theory or M-theory and that admits AdS_3 as a vacuum solution. There are unusual features in the reduction from six-dimensional supergravity, owing to the self-duality condition on the 3-form field. The structure of the full equations of motion in N=(1,0) supergravity in D=6 is also elucidated, and the role of the self-dual field strength as torsion is exhibited. 
  The validity of the tree-unitarity criterion for scattering amplitudes on the noncommutative space-time is considered, as a condition that can be used to shed light on the problem of unitarity violation in noncommutative quantum field theories when time is noncommutative. The unitarity constraints on the partial wave amplitudes in the noncommutative space-time are also derived. 
  We consider a quantum deformation of the wave equation on a cosmological background as a toy-model for possible trans-Planckian effects. We compute the power spectrum of scalar and tensor fluctuations for power-law inflation, and find a noticeable deviation from the standard result. We consider de Sitter inflation as a special case, and find that the resulting power spectrum is scale invariant. For both inflationary scenarios the departure from the standard spectrum is sensitive to the size of the deformation parameter. A modulation in the power spectrum appears to be a generic feature of the model. 
  We study the Hamiltonian structure of the gauge symmetry enhancement in the enlarged CP(N) model coupled with U(2) Chern-Simons term, which contains a free parameter governing explicit symmetry breaking and symmetry enhancement. After giving a general discussion of the geometry of constrained phase space suitable for the symmetry enhancement, we explicitly perform the Dirac analysis of our model and compute the Dirac brackets for the symmetry enhanced and broken cases. We also discuss some related issues. 
  We study the theory of the Lorentz group (1/2,0)+(0,1/2) representation in the helicity basis of the corresponding 4-spinors. As Berestetski, Lifshitz and Pitaevskii mentioned, the helicity eigenstates are not the parity eigenstates. Relations with the Gelfand-Tsetlin-Sokolik-type quantum field theory are discussed. Finally, a new form of the parity operator (which commutes with the Hamiltonian) is proposed in the Fock space. 
  We construct a supergravity solution describing a charged rotating black ring with S^2xS^1 horizon in a five dimensional asymptotically flat spacetime. In the neutral limit the solution is the rotating black ring recently found by Emparan and Reall. We determine the exact value of the lower bound on J^2/M^3, where J is the angular momentum and M the mass; the black ring saturating this bound has maximum entropy for the given mass. The charged black ring is characterized by mass M, angular momentum J, and electric charge Q, and it also carries local fundamental string charge. The electric charge distributed uniformly along the ring helps support the ring against its gravitational self-attraction, so that J^2/M^3 can be made arbitrarily small while Q/M remains finite. The charged black ring has an extremal limit in which the horizon coincides with the singularity. 
  We deform the standard four dimensional $\N=1$ superspace by making the odd coordinates $\theta$ not anticommuting, but satisfying a Clifford algebra. Consistency determines the other commutation relations of the coordinates. In particular, the ordinary spacetime coordinates $x$ cannot commute. We study chiral superfields and vector superfields and their interactions. As in ordinary noncommutative field theory, a change of variables allows us to express the gauge interactions in terms of component fields which are subject to standard gauge transformation laws. Unlike ordinary noncommutative field theories, the change of the Lagrangian is a polynomial in the deformation parameter. Despite the deformation, the noncommutative theories still have an antichiral ring with all its usual properties. We show how these theories with precisely this deformation arise in string theory in a graviphoton background. 
  It is shown that the tachyons which originate from unstable D-branes carry energy and momentum at the velocity $\beta = c^2/v,$ where $v$ is the phase velocity which is greater than $c.$ For an observer who moves with the velocity $\beta,$ tachyon is observed to be moving from one of the ground states of the tachyon potential to the potential hill. It is found that tachyon either passes over the hill or bounces back to the original ground state. Another possible solution is the case which is margial to these, that is tachyon reaches to the top of the potential hill and stays there forever. 
  We obtain D=6, N=(1,1) de Sitter supergravity from a hyperbolic reduction of the massive type IIA* theory. We construct a smooth cosmological solution in which the co-moving time runs from an infinite past, which is dS_4\times S^2, to an infinite future, which is a dS_6-type spacetime with the boundary R^3\times S^2. This provides an effective four-dimensional cosmological model with two compact extra dimensions forming an S^2. Interestingly enough, although the solution is time-dependent, it arises from a first-order system via a superpotential construction. We lift the solutions back to D=10, and in particular obtain two smooth embeddings of dS_4 in massive type IIA*, with the internal space being either H^4\times S^2 or an H^4 bundle over S^2. We also obtain the analogous D=5 and D=4 solutions. We show that there exist cosmological solutions that describe an expanding universe with the expansion rate significantly larger in the past than in the future. 
  It is pointed out that the existence of bare mass terms for matter fields changes gauge symmetry patterns through the Hosotani mechanism. As a demonstration, we study an SU(2) gauge model with massive adjoint fermions defined on $M^4\otimes S^1$. It turns out that the vacuum structure changes at certain critical values of $mL$, where $m~(L)$ stands for the bare mass (the circumference of $S^1$). The gauge symmetry breaking patterns are different from models with massless adjoint fermions. We also consider a supersymmmetric SU(2) gauge model with adjoint hypermultiplets, in which the supersymmetry is broken by bare mass terms for the gaugino and squark fields instead of the Scherk-Schwarz mechanism. 
  We find that localized quantum N-body soliton states exist for a derivative nonlinear Schrodinger (DNLS) model within an extended range of coupling constant (\xi_q) given by 0 < | \xi_q | < 1/\hbar \tan [\pi/(N-1)]. We also observe that soliton states with both positive and negative momentum can appear for a fixed value of \xi_q. Thus the chirality property of classical DNLS solitons is not preserved at the quantum level. Furthermore, it is found that the solitons with positive (negative) chirality have positive (negative) binding energy. 
  We consider a D-brane coupled with gravity in type IIB supergravity on S^5 and derive the effective theory on the D-brane in two different ways, that is, holographic and geometrical projection methods. We find that the effective equations on the brane obtained by these methods coincide. The theory on the D-brane described by the Born-Infeld action is not like Einstein-Maxwell theory in the lower order of the gradient expansion, i.e., the Maxwell field does not appear in the theory. Thus the careful analysis and statement for cosmology on self-gravitating D-brane should be demanded in realistic models. 
  We investigate the stability of a spatially homogeneous and isotropic non-singular cosmological model. We show that the complete set of independent perturbations (the electric part of the perturbed Weyl tensor and the perturbed shear) are regular and well behaved functions which have no divergences, contrary to previous claims in the literature. 
  The AdS/CFT transformation relates two nonlinear realizations of (super)conformal groups: their realization in the appropriate field theories in Minkowski space with a Goldstone dilaton field and their realization as (super)isometry groups of AdS (super)spaces. It exists already at the classical level and maps the field variables and space-time coordinates of the given (super)conformal field theory in $d$-dimensional Minkowski space ${\cal M}_d$ on the variables of a scalar codimension one (super)brane in AdS$_{d+1}$ in a static gauge, the dilaton being mapped on the transverse AdS brane coordinate. We explain the origin of this coordinate map and describe some its implications, in particular, in $d=1$ models of conformal and superconformal mechanics. We also give a suggestive geometric interpretation of this AdS/CFT transform in the pure bosonic case in the framework of an extended $2d+1$-dimensional conformal space involving extra coordinates associated with the generators of dilatations and conformal boosts. 
  We discuss the theory and phenomenology of the interplay between the massless graviton and its massive Kaluza-Klein modes in the Randall-Sundrum two-brane model. The equations of motion of the transverse traceless degrees of freedom are derived by means of a Green function approach as well as from an effective nonlocal action. The second procedure clarifies the extraction of the particle content from the nonlocal action and the issue of its diagonalization. The situation discussed is generic for the treatment of two-brane models if the on-brane fields are used as the dynamical degrees of freedom. The mixing of the effective graviton modes of the localized action can be interpreted as radion-induced gravitational-wave oscillations, a classical analogy to meson and neutrino oscillations. We show that these oscillations arising in M-theory-motivated braneworld setups could lead to effects detectable by gravitational-wave interferometers. The implications of this effect for models with ultra-light gravitons are discussed. 
  We examine the extension of the Klebanov-Witten gauge/gravity correspondence away from the low-energy conformal limit, to a duality involving the full, asymptotically Ricci-flat background describing three-branes on the conifold. After a discussion of the nature of this duality at the string theory level (prior to taking any limits), we concentrate on the intermediate-energy regime where excited string modes are negligible but the branes are still coupled to the bulk. Building upon previous work, we are able to characterize the effective D3-brane worldvolume action in this regime as an IR deformation of the Klebanov-Witten N=1 superconformal gauge theory by a specific dimension-eight operator. In addition, we compute the two-point functions of the operators dual to all partial waves of the dilaton on the conifold-three-brane background, and subject them to various checks. 
  In this talk I briefly review recent developments in quantum field theories on a noncommutative Euclidean space, with Heisenberg-like commutation relations between coordinates. I will be concentrated on new physics learned from this simplest class of non-local field theories, which has applications to both string theory and condensed matter systems, and possibly to particle phenomenology. 
  We review the integrable structure of the Dirichlet boundary problem in two dimensions. The solution to the Dirichlet boundary problem for simply-connected case is given through a quasiclassical tau-function, which satisfies the Hirota equations of the dispersionless Toda hierarchy, following from properties of the Dirichlet Green function. We also outline a possible generalization to the case of multiply-connected domains related to the multi-support solutions of matrix models. 
  I discuss algorithms for the evaluation of Feynman integrals. These algorithms are based on Hopf algebras and evaluate the Feynman integral to (multiple) polylogarithms. 
  We study a class of D=11 BPS spacetimes that describe M-branes wrapping supersymmetric 2 and 4-cycles of Calabi-Yau 3-folds. We analyze the geometrical significance of the supersymmetry constraints and gauge field equations of motion for these spacetimes. We show that the dimensional reduction to D=5 yields known BPS black hole and black string solutions of D=5, N=2 supergravity. The usual ansatz for the dimensional reduction is valid only in the linearized regime of slowly varying moduli and small gauge field strengths. Our identification of the massless D=5 modes with D=11 quantities extends beyond this regime and should prove useful in constructing non-linear ansatze for Calabi-Yau dimensional reductions of supergravity theories. 
  String theory suggests the existence of a minimum length scale. An exciting quantum mechanical implication of this feature is a modification of the uncertainty principle. In contrast to the conventional approach, this generalised uncertainty principle does not allow to resolve space time distances below the Planck length. In models with extra dimensions, which are also motivated by string theory, the Planck scale can be lowered to values accessible by ultra high energetic cosmic rays (UHECRs) and by future colliders, i.e. $M_f\approx$ 1 TeV. It is demonstrated that in this novel scenario, short distance physics below $1/M_f$ is completely cloaked by the uncertainty principle. Therefore, Planckian effects could be the final physics discovery at future colliders and in UHECRs. As an application, we predict the modifications to the $e^+e^- \to f^+f^-$ cross-sections. 
  Using matrix-model methods we study three different N=2 models: U(N) x U(N) with matter in the bifundamental representation, U(N) with matter in the symmetric representation, and U(N) with matter in the antisymmetric representation. We find that the (singular) cubic Seiberg-Witten curves (and associated Seiberg-Witten differentials) implied by the matrix models, although of a different form from the ones previously proposed using M-theory, can be transformed into the latter and are thus physically equivalent. We also calculate the one-instanton corrections to the gauge-coupling matrix using the perturbative expansion of the matrix model. For the U(N) theories with symmetric or antisymmetric matter we use the modified matrix-model prescription for the gauge-coupling matrix discussed in ref. [hep-th/0303268]. Moreover, in the matrix model for the U(N) theory with antisymmetric matter, one is required to expand around a different vacuum than one would naively have anticipated. With these modifications of the matrix-model prescription, the results of this paper are in complete agreement with those of Seiberg-Witten theory obtained using M-theory methods. 
  We study open and closed string interactions in the Type IIB plane wave background using open+closed string field theory. We reproduce all string amplitudes from the dual N=2 Sp(N) gauge theory by computing matrix elements of the dilatation operator. A direct diagrammatic correspondence is found between string theory and gauge theory Feynman diagrams. The prefactor and Neumann matrices of open+closed string field theory are separately realized in terms of gauge theory quantities. 
  We study D-branes in topologically twisted N=2 minimal models using the Landau-Ginzburg realization. In the cases of A and D-type minimal models we provide what we believe is an exhaustive list of topological branes and compute the corresponding boundary OPE algebras as well as all disk correlators. We also construct examples of topological branes in E-type minimal models. We compare our results with the boundary state formalism, where possible, and find agreement. 
  In this paper we will study quantum field theory of fluctuation modes around the rolling tachyon solution on non-BPS D-brane effective action. The goal of this paper is to study particle production during the decay of non-BPS D-brane and explore possible relation with minisuperspace calculation. We find that the number of particles produced on half S-brane exponentially grows for large time which suggests that linearised approximation breaks down and also that backreaction of fluctuation field on classical solution should be taken into account. 
  For each one of the Lie algebras $\mathfrak{gl}_{n}$ and $\widetilde {\mathfrak{gl}}_{n}$, we constructed a family of integrable generalizations of the Toda chains characterized by two integers $m_{+}$ and $m_{-}$. The Lax matrices and the equations of motion are given explicitly, and the integrals of motion can be calculated in terms of the trace of powers of the Lax matrix $L$. For the case of $m_{+}=m_{-}$, we find a symmetric reduction for each generalized Toda chain we found, and the solution to the initial value problems of the reduced systems is outlined. We also studied the spectral curves of the periodic $(m_{+},m_{-})$-Toda chains, which turns out to be very different for different pairs of $m_{+}$ and $m_{-}$. Finally we also obtained the nonabelian generalizations of the $(m_{+},m_{-})$-Toda chains in explicit form. 
  We determine the quasinormal frequencies for all gravitational perturbations of the d-dimensional Schwarzschild black hole, in the infinite damping limit. Using the potentials for gravitational perturbations derived recently by Ishibashi and Kodama, we show that in all cases the asymptotic real part of the frequency is proportional to the Hawking temperature with a coefficient of log 3. Via the correspondence principle, this leads directly to an equally spaced entropy spectrum. We comment on the possible implications for the spacing of eigenvalues of the Virasoro generator in the associated near-horizon conformal algebra. 
  By studying some bouncing universe models dominated by a specific class of hydrodynamical fluids, we show that the primordial cosmological perturbations may propagate smoothly through a general relativistic bounce. We also find that the purely adiabatic modes, although almost always fruitfully investigated in all other contexts in cosmology, are meaningless in the bounce or null energy condition (NEC) violation cases since the entropy modes can never be neglected in these situations: the adiabatic modes exhibit a fake divergence that is compensated in the total Bardeen gravitational potential by inclusion of the entropy perturbations. 
  We solved the Schr{\"o}dinger equation for a particle in a uniform magnetic field in the n-dimensional torus. We obtained a complete set of solutions for a broad class of problems; the torus T^n = R^n / {\Lambda} is defined as a quotient of the Euclidean space R^n by an arbitrary n-dimensional lattice {\Lambda}. The lattice is not necessary either cubic or rectangular. The magnetic field is also arbitrary. However, we restrict ourselves within potential-free problems; the Schr{\"o}dinger operator is assumed to be the Laplace operator defined with the covariant derivative. We defined an algebra that characterizes the symmetry of the Laplacian and named it the magnetic algebra. We proved that the space of functions on which the Laplacian acts is an irreducible representation space of the magnetic algebra. In this sense the magnetic algebra completely characterizes the quantum mechanics in the magnetic torus. We developed a new method for Fourier analysis for the magnetic torus and used it to solve the eigenvalue problem of the Laplacian. All the eigenfunctions are given in explicit forms. 
  We propose an efficient way to obtain a correct Veneziano-Yankielowicz type integration constant of the effective glueball superpotential $W_{eff}(S,g,\Lambda)$, even for massless theories. Applying our method, we show some $\mathcal{N} = 1$ theories do not have such an effective glueball superpotential, even though they have isolated vacua. In these cases, $S = 0$ typically. 
  In order to check the validity of auxiliary field method in the Nambu--Jona-Lasinio model, the one-loop (=quantum) effects of auxiliary fields to the gap equation are considered with N-component fermion models in 4 and 3 dimensions. N is not assumed so large but regarded as a loop expansion parameter. To overcome infrared divergences caused by the Nambu-Goldstone bosons, an intrinsic fermion mass is assumed. It is shown that the loop expansion can be justified by this intrinsic mass whose lower limit is also given. It is found that due to quantum effects, chiral symmetry breaking ($\chi$SB) is restored in D=4 and D=3 when the four-Fermi coupling is large. However, $\chi$SB is enhanced in a small coupling region in D=3. 
  We examine Killing spinor equations of the general IIB pp-wave backgrounds, which contain a scalar H(x^m,x^-) in the metric and a self-dual four-form \xi(x^m,x^-) in the self-dual five-form flux. Considering non-harmonic extra Killing spinors, we find that if the backgrounds admit at least one extra Killing spinor in addition to 16 standard Killing spinors, backgrounds can be reduced to the form with H=A_{mn}(x^-)x^mx^n and \xi(x^-), modulo coordinate transformations. We examine further the cases in which the extra Killing spinors are characterized by a set of Cartan matrices. Solving Killing spinor equations, we find IIB pp-wave backgrounds which admit 18, 20, 24 and 32 Killing spinors. 
  This paper completes the work, initiated in [hep-th/9906003,hep-th/0301204], further referred as Parts I and II, concerning to Dirac's quantization of Nambu-Goto theory of open string, formulated in the space-time of dimension d=4. Here we perform more detailed study of Gribov's copies in the classical mechanics and determine the quantum spectrum of masses for the arbitrary spin case. 
  We compute the first radiative correction to the Casimir energy of a massive scalar field with a quartic self-interaction in the presence of two parallel plates. Three kinds of boundary conditions are considered: Dirichlet-Dirichlet, Neumann-Neumann and Dirichlet-Neumann. We use dimensional and analytical regularizations to obtain our physical results. 
  We investigate the mass spectrum of a scalar field in a world with latticized and circular continuum space where background fields takes a topological configuration. We find that the mass spectrum is related to the characteristic values of Mathieu functions. The gauge symmetry breaking in a similar spacetime is also discussed. 
  In the study of certain noncommutative versions of Minkowski spacetime there is still a large ambiguity concerning the characterization of their symmetries. Adopting as our case study the kappaMinkowski noncommutative space-time, on which a large literature is already available, we propose a line of analysis of noncommutative-spacetime symmetries that relies on the introduction of a Weyl map (connecting a given function in the noncommutative Minkowski with a corresponding function in commutative Minkowski) and of a compatible notion of integration in the noncommutative spacetime. We confirm (and we establish more robustly) previous suggestions that the commutative-spacetime notion of Lie-algebra symmetries must be replaced, in the noncommutative-spacetime context, by the one of Hopf-algebra symmetries. We prove that in kappaMinkowski it is possible to construct an action which is invariant under a Poincare-like Hopf algebra of symmetries with 10 generators, in which the noncommutativity length scale has the role of relativistic invariant. The approach here adopted does leave one residual ambiguity, which pertains to the description of the translation generators, but our results, independently of this ambiguity, are sufficient to clarify that some recent studies (gr-qc/0212128 and hep-th/0301061), which argued for an operational indistiguishability between theories with and without a length-scale relativistic invariant, implicitly assumed that the underlying spacetime would be classical. 
  Including {\it world-sheet orientation-reversing automorphisms} $\hat{h}_{\sigma} \in H_-$ in the orbifold program, we construct the operator algebras and twisted KZ systems of the general WZW {\it orientation orbifold} $A_g (H_-) /H_-$. We find that the orientation-orbifold sectors corresponding to each $\hat{h}_{\sigma} \in H_-$ are {\it twisted open} WZW strings, whose properties are quite distinct from conventional open-string orientifold sectors. As simple illustrations, we also discuss the classical (high-level) limit of our construction and free-boson examples on abelian $g$. 
  We consider a scalar field interacting with a quantized metric varying on a submanifold (e.g. a scalar field interacting with a quantized gravitational wave). We explicitly sum up the perturbation series for the time-ordered vacuum expectation values of the scalar field. As a result we obtain a modified non-canonical short distance behavior. 
  The procedures to overcome nonrenormalizability of \phi^4_n, n\ge5, quantum field theory models that were presented in a recent paper are extended to address nonrenormalizability of \phi^p_3, p=8,10,12,..., models. The principles involved in these procedures are based on the hard-core picture of nonrenormalizability. 
  A perturbative calculation of the correlator of three parallel open Wilson lines is performed for the U(N) theory in two non-commutative space-time dimensions. In the large-N planar limit, the perturbative series is fully resummed and asymptotically leads to an exponential increase of the correlator with the lengths of the lines, in spite of an interference effect between lines with the same orientation. This result generalizes a similar increase occurring in the two-line correlator and is likely to persist when more lines are considered provided they share the same direction. 
  We study gravity duals of large N non-supersymmetric gauge theories with matter in the fundamental representation by introducing a D7-brane probe into deformed AdS backgrounds. In particular, we consider a D7-brane probe in both the AdS Schwarzschild black hole solution and in the background found by Constable and Myers, which involves a non-constant dilaton and S^5 radius. Both these backgrounds exhibit confinement of fundamental matter and a discrete glueball and meson spectrum. We numerically compute the quark condensate and meson spectrum associated with these backgrounds. In the AdS-black hole background, a quark-bilinear condensate develops only at non-zero quark mass. We speculate on the existence of a third order phase transition at a critical quark mass where the D7 embedding undergoes a geometric transition. In the Constable-Myers background, we find a chiral symmetry breaking condensate as well as the associated Goldstone boson in the limit of small quark mass. The existence of the condensate ensures that the D7-brane never reaches the naked singularity at the origin of the deformed AdS space. 
  We find the general form of supersymmetric invariant two point functions. By imposing supersymmetric positivity we obtain the general supersymmetric K\"allen-Lehmann representation. 
  I discuss the role of Hochschild cohomology in Quantum Field Theory with particular emphasis on Dyson--Schwinger equations. 
  We impose a certain class of boundary conditions on Killing horizon and show for Lagrangians with arbitrary curvature dependence that one can identify a Virasoro algebra with nontrivial central charge and calculable Hamiltonian eigenvalue. Entropy can then be calculated from Cardy formula. 
  We consider the partition function for a matrix model with a global unitary invariant energy function. We show that the averages over the partition function of global unitary invariant trace polynomials of the matrix variables are the same when calculated with any choice of a global unitary fixing, while averages of such polynomials without a trace define matrix-valued correlation functions, that depend on the choice of unitary fixing. The unitary fixing is formulated within the standard Faddeev-Popov framework, in which the squared Vandermonde determinant emerges as a factor of the complete Faddeev-Popov determinant. We give the ghost representation for the FP determinant, and the corresponding BRST invariance of the unitary-fixed partition function. The formalism is relevant for deriving Ward identities obeyed by matrix-valued correlation functions. 
  Duality is investigated for higher spin ($s \geq 2$), free, massless, bosonic gauge fields. We show how the dual formulations can be derived from a common "parent", first-order action. This goes beyond most of the previous treatments where higher-spin duality was investigated at the level of the equations of motion only. In D=4 spacetime dimensions, the dual theories turn out to be described by the same Pauli-Fierz (s=2) or Fronsdal ($s \geq 3$) action (as it is the case for spin 1). In the particular s=2 D=5 case, the Pauli-Fierz action and the Curtright action are shown to be related through duality. A crucial ingredient of the analysis is given by the first-order, gauge-like, reformulation of higher spin theories due to Vasiliev. 
  An approach to the Holographic Renormalization Group in the context of Rehren duality - a structural form of the AdS-CFT correspondence, in the context of Local Quantum Physics (Algebraic QFT) - is proposed. Special attention to the issue of UV/IR connection is paid. 
  I propose to formalize quantum theories as topological quantum field theories in a generalized sense, associating state spaces with boundaries of arbitrary (and possibly finite) regions of space-time. I further propose to obtain such ``general boundary'' quantum theories through a generalized path integral quantization. I show how both, non-relativistic quantum mechanics and quantum field theory can be given a ``general boundary'' formulation. Surprisingly, even in the non-relativistic case, features normally associated with quantum field theory emerge from consistency conditions. This includes states with arbitrary particle number and pair creation. I also note how three dimensional quantum gravity is an example for a realization of both proposals and suggest to apply them to four dimensional quantum gravity. 
  In this work we propose an exact solution of the c=1 Liouville model, i.e. of the world-sheet theory that describes the homogeneous decay of a closed string tachyon. Our expressions are obtained through careful extrapolation from the correlators of Liouville theory with c > 25. In the c=1 limit, we find two different theories which differ by the signature of Liouville field. The Euclidean limit coincides with the interacting c=1 theory that was constructed by Runkel and Watts as a limit of unitary minimal models. The couplings for the Lorentzian limit are new. In contrast to the behavior at c > 1, amplitudes in both c=1 models are non-analytic in the momenta and consequently they are not related by Wick rotation. 
  We consider the $N$-components 3-dimensional massive Gross-Neveu model compactified in one spatial direction, the system being constrained to a slab of thickness $L$. We derive a closed formula for the effective renormalized $L$-dependent coupling constant in the large-N limit, using bag-model boundary conditions. For values of the fixed coupling constant in absence of boundaries $\lambda \geq \lambda_c \simeq 19.16$, we obtain ultra-violet asymptotic freedom (for $L \to 0$) and confinement for a length $L^{(c)}$ such that $2.07 m^{-1} < L^{(c)} \lesssim 2.82 m^{-1}$, $m$ being the fermionic mass. Taking for $m$ an average of the masses of the quarks composing the proton, we obtain a confining legth $L^{(c)}_p$ which is comparable with an estimated proton diameter. 
  de Sitter space-time has a one complex parameter family of invariant vacua for the theory of a free, massive scalar field. For most of these vacua, in an interacting scalar theory the one loop corrections diverge linearly for large values of the loop momentum. These divergences are not of a form that can be removed by a de Sitter invariant counterterm, except in the case of the Euclidean, or Bunch-Davies, vacuum. 
  Use of the AdS/CFT correspondence to arrive at phenomenological gauge field theories is dicussed, focusing on the orbifolded case without supersymmetry. An abelian orbifold with the finite group $Z_{p}$ can give rise to a $G = SU(N)^p$ gauge group with chiral fermions and complex scalars in different bi-fundamental representations of $G$. The precision measurements at the $Z$ resonance suggest the values $p = 12$ and $N = 3$, and a unifications scale $M_U \sim 4$ TeV. Robustness and predictivity of such grand unification is discussed. 
  Novel theories appear on the world-volume of branes by orienting B fields along various directions of the branes. We review some of the earlier developments and explore many new examples of these theories. In particular, among other things, we study the pinning effect of branes near conifold like singularities and brane-antibrane theories with different fluxes on their world-volumes. We show that all these theories arise from different limits of an M-theory configuration with appropriately chosen G-fluxes. This gives us a way to study them from a unified framework in M-theory. 
  We study classes of five-dimensional cosmological solutions with negative curvature, which are obtained from static solutions by an exchange of a spatial and temporal coordinate, and in some cases by an analytic continuation. Such solutions provide a suitable laboratory to address the time-dependent AdS/CFT correspondence. For a specific example we address in detail the calculation of the boundary stress-energy and the Wilson line and find disagreement with the standard AdS/CFT correspondence. We trace these discrepancies to the time-dependent effects, such as particle creation, which we further study for specific backgrounds. We also identify specific time-dependent backgrounds that reproduce the correct conformal anomaly. For such backgrounds the calculation of the Wilson line in the adiabatic approximation indicates only a Coulomb repulsion. 
  It has been conjectured recently that the field theory limit of the topological string partition functions, including all higher genus contributions, for the family of CY3-folds giving rise to N=2 4D SU(N) gauge theory via geometric engineering can be obtained from gauge instanton calculus. We verify this surprising conjecture by calculating the partition functions for such local CYs using diagrammatic techniques inspired by geometric transitions. Determining the Gopakumar-Vafa invariants for these geometries to all orders in the fiber wrappings allows us to take the field theory limit. 
  In this paper, we describe a way to construct a class of dark energy models that admit late time de Sitter attractor solution. In the canonical scalar and Born-Infeld scalar dark energy models, we show mathematically that a simple sufficient condition for the existence of a late time de Sitter like attractor solution is that the potentials of the scalar field have non-vanishing minimum while this condition becomes that the potentials have non-vanishing maximum for the phantom models. These attractor solutions correspond to an equation of state $w=-1$ and a cosmic density parameter $\Omega_{\phi}=1$, which are important features for a dark energy model that can meet the current observations. 
  Underlying a general noncommutative algebra with both noncommutative coordinates and noncommutative momenta in a (1+1)-dimensional spacetime, a chiral boson Lagrangian with manifest Lorentz covariance is proposed by linearly imposing a generalized self-duality condition on a noncommutative generalization of massless real scalar fields. A significant property uncovered for noncommutative chiral bosons is that the left- and right-moving chiral scalars cannot be distinguished from each other, which originates from the noncommutativity of coordinates and momenta. An interesting result is that Dirac's method can be consistently applied to the constrained system whose Lagrangian explicitly contains space and time. The self-duality of the noncommutative chiral boson action does not exist. 
  We examine the equivalence between the Konishi anomaly equations and the matrix model loop equations in N=1* gauge theories, the mass deformation of N=4 supersymmetric Yang-Mills. We perform the superfunctional integral of two adjoint chiral superfields to obtain an effective N=1 theory of the third adjoint chiral superfield. By choosing an appropriate holomorphic variation, the Konishi anomaly equations correctly reproduce the loop equations in the corresponding three-matrix model. We write down the field theory loop equations explicitly by using a noncommutative product of resolvents peculiar to N=1* theories. The field theory resolvents are identified with those in the matrix model in the same manner as for the generic N=1 gauge theories. We cover all the classical gauge groups. In SO/Sp cases, both the one-loop holomorphic potential and the Konishi anomaly term involve twisting of index loops to change a one-loop oriented diagram to an unoriented diagram. The field theory loop equations for these cases show certain inhomogeneous terms suggesting the matrix model loop equations for the RP2 resolvent. 
  The Kaluza-Klein reduction of the 3d gravitational Chern-Simons term to a 2d theory is equivalent to a Poisson-sigma model with fourdimensional target space and degenerate Poisson tensor of rank 2. Thus two constants of motion (Casimir functions) exist, namely charge and energy. The application of well-known methods developed in the framework of first order gravity allows to construct all classical solutions straightforwardly and to discuss their global structure. For a certain fine tuning of the values of the constants of motion the solutions of hep-th/0305117 are reproduced. Possible generalizations are pointed out. 
  Chiral QED with a generalized Fadeevian regularization is considered. Imposing a chiral constraint a gauged version of Floranini-Jackiw lagrangian is constructed. The imposition of the chiral constarint has spoiled t he manifestly Lorentz covariance of the theory. The phase space structure for this theory has been det ermined. It is found that spectrum changes drastically but it is Lorentz invariant. Chiral fermion di sappears from the spectra and the photon anquire mass as well. Poincare algebra has been calculated to show physicial Lorentz invariance explicitely. 
  We complete the classification of parallelizable NS-NS backgrounds in type II supergravity by adding the dilatonic case to the result of Figueroa-O'Farrill on the non-dilatonic case. We also study the supersymmetry of these parallelizable backgrounds. It is shown that all the dilatonic parallelizable backgrounds have sixteen supersymmetries. 
  A functional method is discussed, where the quantum fluctuations of a theory are controlled by a mass parameter and the evolution of the theory with this parameter is connected to its renormalization. It is found, in the framework of the gradient expansion, that the coupling constant of a N=1 Wess-Zumino theory in 2+1 dimensions does not get quantum corrections. 
  Using the method developed by Cherkis and Hashimoto we construct partially localized D3/D5(2), D4/D4(2) and M5/M5(3) supergravity solutions where one of the harmonic functions is given in an integral form. This is a generalization of the already known near-horizon solutions. The method fails for certain intersections such as D1/D5(1) which is consistent with the previous no-go theorems. We point out some possible ways of bypassing these results. 
  The leading terms in the tree-level effective action for the massless fields of the bosonic open string are calculated by integrating out all massive fields in Witten's cubic string field theory. In both the abelian and nonabelian theories, field redefinitions make it possible to express the effective action in terms of the conventional field strength. The resulting actions reproduce the leading terms in the abelian and nonabelian Born-Infeld theories, and include (covariant) derivative corrections. 
  In some four-dimensional orientifolds, U(1) gauge fields that are free of four-dimensional anomalies can still be massive. It is shown that this is due to mass-generating six-dimensional anomalies. Six-dimensional anomalies affect four-dimensional masses via decompactifications. 
  Dimensional reductions of pure Einstein gravity on cosets other than tori are inconsistent. The inclusion of specific additional scalar and p-form matter can change the situation. For example, a D-dimensional Einstein-Maxwell-dilaton system, with a specific dilaton coupling, is known to admit a consistent reduction on S^2= SU(2)/U(1), of a sort first envisaged by Pauli. We provide a new understanding, by showing how an S^3=SU(2) group-manifold reduction of (D+1)-dimensional Einstein gravity, of a type first indicated by DeWitt, can be broken into in two steps; a Kaluza-type reduction on U(1) followed by a Pauli-type coset reduction on S^2. More generally, we show that any D-dimensional theory that itself arises as a Kaluza U(1) reduction from (D+1) dimensions admits a consistent Pauli reduction on any coset of the form G/U(1). Extensions to the case G/H are given. Pauli coset reductions of the bosonic string on G= (G\times G)/G are believed to be consistent, and a consistency proof exists for S^3=SO(4)/SO(3). We examine these reductions, and arguments for consistency, in detail. The structures of the theories obtained instead by DeWitt-type group-manifold reductions of the bosonic string are also studied, allowing us to make contact with previous such work in which only singlet scalars are retained. Consistent truncations with two singlet scalars are possible. Intriguingly, despite the fact that these are not supersymmetric models, if the group manifold has dimension 3 or 25 they admit a superpotential formulation, and hence first-order equations yielding domain-wall solutions. 
  Using a recently developed generalized Weyl formalism, we construct an asymptotically flat, static vacuum Einstein solution that describes a superposition of multiple five-dimensional Schwarzschild black holes. The spacetime exhibits a U(1)\times U(1) rotational symmetry. It is argued that for certain choices of parameters, the black holes are collinear and so may be regarded as a five-dimensional generalization of the Israel-Khan solution. The black holes are kept in equilibrium by membrane-like conical singularities along the two rotational axes; however, they still distort one another by their mutual gravitational attraction. We also generalize this solution to one describing multiple charged black holes, with fixed mass-to-charge ratio, in Einstein-Maxwell-dilaton theory. 
  The Non-Commutative (NC) CP(1) model is studied from field theory perspective. Our formalism and definition of the NC CP(1) model differs crucially from the existing one \cite{nccp}. Due to the U(1) gauge invariance, the Seiberg-Witten map is used to convert the NC action to an action in terms of ordinary spacetime degrees of freedom and the subsequent theory is studied. The NC effects appear as (NC parameter) $\theta $-dependent interaction terms. The expressions for static energy, obtained from both the symmetric and canonical forms of the energy momentum tensor, are {\it {identical}}, when only spatial noncommutativity is present. Bogomolny analysis reveals a lower bound in the energy in an unambiguous way, suggesting the presence of a new soliton. However, the BPS equations saturating the bound are not compatible to the full variational equation of motion. This indicates that the definitions of the energy momentum tensor for this particular NC theory, (the NC theory is otherwise consistent and well defined), are inadequate, thus leading to the "energy crisis". A collective coordinate analysis corroborates the above observations. It also shows that the above mentioned mismatch between the BPS equations and the variational equation of motion is small. 
  We study the worldvolume supersymmetries of M2 branes in the maximally supersymmetric plane wave background of M theory. For certain embeddings the standard probe analysis indicates that the worldvolume theory has less than 16 supersymmetries. We show that at the quadratic level the worldvolume theory admits additional linearly realized supersymmetries, and that the spectra of the branes are organized into multiplets of these symmetries. We find however that these supersymmetries are not respected by worldvolume interactions. Our analysis was motivated by recent work showing that D-branes in the maximally supersymmetric plane wave background of IIB string theory admit supersymmetries beyond those of the probe analysis. The construction of the additional supercharges in this case was specific to a string worldsheet that is a strip and the present results suggest that string interactions do not preserve these symmetries. 
  Yukawa's space-time approach is briefly summarized, which starts just before his meson theory and finally comes to the theory of elementary domain. Recent rapid development in superstring theory is reviewed, focussing our attention on several important topics, which seem deeply related to Yukawa's concern. The importance of the idea of noncommutative space-time is emphasized. 
  It is shown that the gravitational redshift as predicted by Einstein's theory, is modified in presence of second rank antisymmetric tensor (Kalb-Ramond) field in a string inspired background spacetime.In presence of extra dimensions, the Randall-Sundrum brane world scenario is found to play a crucial role in suppressing this additional shift. The bound on the value of the warp factor is determined from the redshift data and is found to be in good agreement with that determined from the requirements of Standard model. 
  We re-evaluate the zero point Casimir energy for the case of a massive scalar field in $\mathbf{R}^{1}\times\mathbf{S}^{3}$ space, allowing also for deviations from the standard conformal value $\xi =1/6$, by means of zero temperature zeta function techniques. We show that for the problem at hand this approach is equivalent to the high temperature regularization of the vacuum energy, as conjectured in a previous publication. Two different, albeit equally valid, ways of doing the analytic continuation are described. 
  We carefully analyse the use of the effective action in dynamical problems, in particular the conditions under which the equation $\frac{\delta \Ga} {\delta \phi}=0$ can be used as a quantum equation of motion, and the relation between the asymptotic states involved in the definition of $\Ga$ and the initial state of the system. By considering the quantum mechanical example of a double-well potential, where we can get exact results for the time evolution of the system, we show that an approximation to the effective potential in the quantum equation of motion that correctly describes the dynamical evolution of the system is obtained with the help of the wilsonian RG equation (already at the lowest order of the derivative expansion), while the commonly used one-loop effective potential fails to reproduce the exact results. 
  In this paper we define and study a matrix model describing the M-theory plane wave background with a single Horava-Witten domain wall. In the limit of infinite mu, the matrix model action becomes quadratic and we can identify the matrix Hamiltonian with a regularized Hamiltonian for hemispherical membranes that carry fermionic degrees of freedom on their boundaries. The number of fermionic degrees of freedom must be sixteen; this condition arises naturally in the framework of the matrix model. We can also prove the exact E_8 symmetry of the spectrum around the membrane vacua at infinite mu, which arises as a current algebra at level one just as in the heterotic string. We also find the full E_8 gauge multiplet as well as the multiple-gluon states, carried by collections of hemispherical membranes. Finally we discuss the dual description of the hemispherical membranes in terms of spherical fivebranes immersed in the domain wall; we identify the correct vacuum of the matrix model and make some preliminary comparisons with the (1,0) superconformal field theory. 
  Recently proposed nonlocal and nonperturbative late time behavior of the heat kernel is generalized to curved spacetimes. Heat kernel trace asymptotics is dominated by two terms one of which represents a trivial covariantization of the flat-space result and another one is given by the Gibbons-Hawking integral over asymptotically-flat infinity. Nonlocal terms of the effective action generated by this asymptotics might underly long- distance modifications of the Einstein theory motivated by the cosmological constant problem. New mechanisms of the cosmological constant induced by infrared effects of matter and graviton loops are briefly discussed. 
  We study a non-standard decoupling limit of the D1/D5-brane system, which interpolates between the near-horizon geometry of the D1/D5 background and the near-horizon limit of the pure D5-brane geometry. The S-dual description of this background is actually an exactly solvable two-dimensional (worldsheet) conformal field theory: {null-deformed SL(2,R)} x SU(2) x T^4 or K3. This model is free of strong-coupling singularities. By a careful treatment of the SL(2,R), based on the better-understood SL(2,R) / U(1) coset, we obtain the full partition function for superstrings on SL(2,R) x SU(2) x K3. This allows us to compute the partition functions for the J^3 and J^2 current-current deformations, as well as the full line of supersymmetric null deformations, which links the SL(2,R) conformal field theory with linear dilaton theory. The holographic interpretation of this setup is a renormalization-group flow between the decoupled NS5-brane world-volume theory in the ultraviolet (Little String Theory), and the low-energy dynamics of super Yang--Mills string-like instantons in six dimensions. 
  Recently a mass deformation of the maximally supersymmetric Yang-Mills quantum mechanics has been constructed from the supermembrane action in eleven dimensional plane-wave backgrounds. However, the origin of this plane-wave matrix theory in terms of a compactification of a higher dimensional Super Yang-Mills model has remained obscure. In this paper we study the Kaluza-Klein reduction of D=4, N=4 Super Yang-Mills theory on a round three-sphere, and demonstrate that the plane-wave matrix theory arises through a consistent truncation to the lowest lying modes. We further explore the relation between the dilatation operator of the conformal field theory and the hamiltonian of the quantum mechanics through perturbative calculations up to two-loop order. In particular we find that the one-loop anomalous dimensions of pure scalar operators are completely captured by the plane-wave matrix theory. At two-loop level this property ceases to exist. 
  The entanglement entropy of the event horizon is known to be plagued by the UV divergence due to the infinitely blue-shifted near horizon modes. In this paper we calculate the entanglement entropy using the transplanckian dispersion relation, which has been proposed to model the quantum gravity effects. We show that, very generally, the entropy is rendered UV finite due to the suppression of high energy modes effected by the transplanckian dispersion relation. 
  Oblique Dp-branes in the maximally supersymmetric type IIB plane-wave background are constructed in terms of boundary states, as well as from the open string point of view. These Dp-branes, whose existence was anticipated by Hikida and Yamaguchi from general supersymmetry arguments, have an isometry that is a subgroup of the diagonal SO(4) symmetry of the background. The oblique D3-brane is found to preserve four dynamical and four kinematical supersymmetries while the oblique D5-brane preserves one half of both the dynamical and kinematical supersymmetries. We also discuss the open-string boundary conditions for curved D7- and D5-branes, and analyze their supersymmetry. 
  We demonstrate that certain supersymmetric Goedel-like universe solutions of supergravity are not solutions of string theory. This is achieved by realizing that supertubes are BPS states in these spaces, and under certain conditions, when wrapping closed timelike curves, some world-volume modes develop negative kinetic terms. Since these universes are homogeneous, this instability takes place everywhere in space-time. We also construct a family of supergravity solutions which locally look like the Goedel universe inside a domain wall made out of supertubes, but have very different asymptotic structure. One can adjust the volume inside the domain wall so there will be no closed timelike curves, and then those spaces seem like perfectly good string backgrounds. 
  P-term inflation is a version of hybrid inflation which naturally appears in some brane inflation models. It was introduced in the framework of N=2 supersymmetric gauge theory where superconformal SU(2,2|2) symmetry is broken down to N=2 supersymmetry by the vev of the auxiliary triplet field P. Depending on the direction of this vev, one can get either D-term inflation or F-term inflation with a particular relation between Yukawa and gauge coupling, or a mix of these models. We show that F and D models, before coupling to gravity is included, are related by a change of variables. Coupling of this model to N=1 supergravity breaks this symmetry and introduces a class of P-term models interpolating between D-term and F-term inflation. The difference between these models is determined by the direction of the vector P, which depends on the fluxes in the underlying D3/D7 model of brane inflation. We discuss cosmological consequences of various versions of P-term inflation. 
  Using the non-symmetric-connection approach proposed by Osborn, we demonstrate that, for a bosonic string in a specially chosen plane-fronted gravitational wave and an axion background, the conformal anomaly vanishes at the two-loop level. Under some conditions, the anomaly vanishes at all orders. 
  Procedure of finding of the Bogomolny-Prasad-Sommerfield monopole solutions in the Georgi-Glashow model is investigated in detail on the backgrounds of three space models of constant curvature: Euclid, Riemann, Lobachevski's. Classification of possible solutions is given. It is shown that among all solutions there exist just three ones which reasonably and in a one-to-one correspondence can be associated with respective geometries. It is pointed out that the known non-singular BPS-solution in the flat Minkowski space can be understood as a result of somewhat artificial combining the Minkowski space background with a possibility naturally linked up with the Lobachewski geometry. The standpoint is brought forth that of primary interest should be regarded only three specifically distinctive solutions -- one for every curved space background. In the framework of those arguments the generally accepted status of the known monopole BPS-solution should be critically reconsidered and even might be given away. 
  Motivated by formal similarities between the continuum limit of the Ising model and the Unruh effect, this paper connects the notion of an Ishibashi state in boundary conformal field theory with the Tomita--Takesaki theory for operator algebras. A geometrical approach to the definition of Ishibashi states is presented, and it is shownthat, when normalisable the Ishibashi states are cyclic separating states, justifying the operator state correspondence. When the states are not normalisable Tomita--Takesaki theory offers an alternative approach based on left Hilbert algebras, opening the way to extensions of our construction and the state-operator correspondence. 
  T-duality acts on circle bundles by exchanging the first Chern class with the fiberwise integral of the H-flux, as we motivate using E_8 and also using S-duality. We present known and new examples including NS5-branes, nilmanifolds, Lens spaces, both circle bundles over RP^n, and the AdS^5 x S^5 to AdS^5 x CP^2 x S^1 with background H-flux of Duff, Lu and Pope. When T-duality leads to M-theory on a non-spin manifold the gravitino partition function continues to exist due to the background flux, however the known quantization condition for G_4 fails. In a more general context, we use correspondence spaces to implement isomorphisms on the twisted K-theories and twisted cohomology theories and to study the corresponding Grothendieck-Riemann-Roch theorem. Interestingly, in the case of decomposable twists, both twisted theories admit fusion products and so are naturally rings. 
  We study gauged five-dimensional supergravity on the interval [0,\pi R]. We find a set of boundary conditions with respect to which the theory is locally supersymmetric. For theories with detuned brane tensions (\Lambda_4<0), we show that these boundary conditions can be used to spontaneously break global supersymmetry. For the original, tuned Randall-Sundrum scenario (\Lambda_4=0), we prove that the locally supersymmetric boundary conditions are also globally supersymmetric. We lift the theory from [0,\pi R] to S^1 and R, with arbitrary twists for the fermions, and cast these results in the language of the Scherk-Schwarz mechanism. 
  We study the conditions for spontaneous symmetry breaking of the (2+1)-dimensional noncommutative phi^6 model in the small-theta limit. In this regime, considering the model as a cutoff theory, it is reasonable to assume translational invariance as a property of the vacuum state and study the conditions for spontaneous symmetry breaking by an effective potential analysis. An investigation of up to the two loop level reveals that noncommutative effects can modify drastically the shape of the effective potential. Under reasonable conditions, the nonplanar sector of the theory can become dominant and induce symmetry breaking for values of the mass and coupling constants not reached by the commutative counterpart. 
  We propose an action for gravity on a fuzzy sphere, based on a matrix model. We find striking similarities with an analogous model of two dimensional gravity on a noncommutative plane, i.e. the solution space of both models is spanned by pure U(2) gauge transformations acting on the background solution of the matrix model, and there exist deformations of the classical diffeomorphisms which preserve the two-dimensional noncommutative gravity actions. 
  In this article we derive the full interacting effective actions for supersymmetric D-branes in arbitrary bosonic type II supergravity backgrounds. The actions are presented in terms of component fields up to second order in fermions. As one expects, the actions are built from the supercovariant derivative operator and the $\kappa$-symmetry projector. The results take a compact and elegant form exhibiting $\kappa$-symmetry, as well as supersymmetry in a background with Killing spinors. We give the explicit transformation rules for these symmetries in all cases, including the M2-brane. As an example, we analyze the N=2 super-worldvolume field theory defined by a test D4-brane in the supergravity background produced by a large number of D0-branes. This example displays rigid supersymmetry in a curved spacetime. 
  In a previous paper, we developed a numerical method to obtain a static black hole localized on a 3-brane in the Randall-Sundrum infinite braneworld, and presented examples of numerical solutions that describe small localized black holes. In this paper we quantitatively analyze the behavior of the numerically obtained black hole solutions, focusing on thermodynamic quantities. The thermodynamic relations show that the localized black hole deviates smoothly from a five-dimensional Schwarzschild black hole, which is a solution in the limit of a small horizon radius. We compare the thermodynamic behavior of these solutions with that of the exact solution on the 2-brane in the 4D braneworld. We find similarities between them. 
  We studied the phase structures of ${\cal N}=1$ supersymmetric $SO(N_c)$ gauge theory with $N_f$ flavors in the vector representation as we deformed the ${\cal N}=2$ supersymmetric QCD by adding the superpotential of arbitrary polynomial for the adjoint chiral scalar field. Using weak and strong coupling analyses, we determined the most general factorization forms for various breaking patterns. We observed all kinds of smooth transitions for quartic superpotential. 
  A two-dimensional (2D) scalar-tensor gravity theory is used to describe the near-horizon near-extremal behavior of black 3-branes solutions of type IIB string theory. The asymptotic symmetry group of the 2D, asymptotically anti de Sitter (AdS) metric, is generated by a Virasoro algebra. The (non-constant) configuration of the 2D scalar field, which parametrizes the volume of the 3-brane, breaks the conformal symmetry and produces a divergent central charge in the Virasoro algebra. Using a renormalization procedure we find a finite value for the central charge and by means of the Cardy formula, the entropy of the black 3-brane in terms of microstates of the conformal field theory living on the boundary of the 2D spacetime. We find for the entropy as a function of the temperature the power-law behavior $S\propto T^3$. Unfortunately, owing to a scale symmetry of the 2D model the proportionality constant is undetermined. The black 3-brane case is a nice example of how finite temperature effects in higher dimensional AdS/CFT dualities can be described by a AdS$_2$/CFT$_1$ duality endowed with a scalar field that breaks the conformal symmetry and produces a non-vanishing central charge. 
  We discuss how holographic bounds can be applied to the quantum fluctuations of the inflaton. In general the holographic principle will lead to a bound on the UV cutoff scale of the effective theory of inflation, but it will depend on the coarse-graining prescription involved in calculating the entropy. We propose that the entanglement entropy is a natural measure of the entropy of the quantum perturbations, and show which kind of bound on the cutoff it leads to. Such bounds are related to whether the effects of new physics will show up in the CMB. 
  The theory of cosmological perturbations has become a cornerstone of modern quantitative cosmology since it is the framework which provides the link between the models of the very early Universe such as the inflationary Universe scenario (which yield causal mechanisms for the generation of fluctuations) and the wealth of recent high-precision observational data. In these lectures, I provide an overview of the classical and quantum theory of cosmological fluctuations. Crucial points in both the current inflationary paradigm of the early Universe and in some proposed alternatives are that, first, the perturbations are generated on microscopic scales as quantum vacuum fluctuations, and, second, that via an accelerated expansion of the background geometry (or by a contraction of the background), the wavelengths of the fluctuations become much larger than the Hubble radius for a long period of cosmic evolution. Hence, both Quantum Mechanics and General Relativity are required in order to understand the generation and evolution of fluctuations.   After a review of the Newtonian theory of perturbations, I discuss first the classical relativistic theory of fluctuations, and then their quantization. Briefly summarized are two new applications of the theory of cosmological fluctuations: the trans-Planckian ``problem'' of inflationary cosmology and the study of the back-reaction of cosmological fluctuations on the background space-time geometry. 
  Central role played by certain non-Abelian monopoles (of Goddard-Nuyts-Olive-Weinberg type) in the infrared dynamics in many confining vacua of softly broken ${\cal N}=2$ supersymmetric gauge theories, has recently been clarified. We discuss here the main lessons to be learned from these studies for the confinement nechanism in QCD. 
  The embedding diagrams of representations of the N=2 superconformal algebra with central charge c=3 are given. Some non-unitary representations possess subsingular vectors that are systematically described. The structure of the embedding diagrams is largely defined by the spectral flow symmetry. As an additional consistency check the action of the spectral flow on the characters is calculated. 
  I present a point of view about what M Theory is and how it is related to the real world, which departs in certain crucial respects from conventional wisdom. I argue against the possibility of a background independent formulation of the theory, or of a Poincare invariant, Supersymmetry violating vacuum state. A fundamental assumption is black hole dominance of high energy physics. Much of this paper is a compilation of things I have said elsewhere. I review a crude argument for the critical exponent connecting the gravitino mass and the cosmological constant, and propose a framework for finding a quantum theory of de Sitter space. 
  We present self-dual N=2 supergravity in superspace for Euclidean seven dimensions with the reduced holonomy G_2 \subset SO(7), including all higher-order terms. As its foundation, we first establish N=2 supergravity without self-duality in Euclidean seven dimensions. We next show how the generalized self-duality in terms of octonion structure constants can be consistently imposed on the superspace constraints. We found two self-dual N=2 supergravity theories possible in 7D, depending on the representations of the two spinor charges of N=2. The first formulation has both of the two spinor charges in the {\bf 7} of G_2 with 24 + 24 on-shell degrees of freedom. The second formulation has both charges in the {\bf 1} of G_2 with 16 + 16 on-shell degrees of freedom. 
  High precision measurements of the Casimir effect and recent applications to micro electromechanical systems raise the question of how large the Casimir force can be made in an arbitrarily small device. Using a simple model for the metal boundary in which the metal is perfectly conducting at frequencies below plasma frequency omega_p and perfectly transparent above such frequency, I find that the Casimir force for plate separations a<lambda_p/2, where lambda_p is the plasma wavelength is given by -(h omega_p^4)/(24 pi^2 c^3) which is independent of a. This result is considered the maximum value of the Casimir force for non-ideal metallic boundaries as calculated by quantum field theory. It differs from predictions of non retarded Van der Waals theory. Implications of this result for geometries different from the planar one and in particular for the hollow metallic sphere are discussed. 
  An exact solution of domain wall junction is obtained in N=2 supersymmetric (SUSY) QED with three massive hypermultiplets. The junction preserves two out of eight SUSY. Both a (magnetic) Fayet-Iliopoulos (FI) term and complex masses for hypermultiplets are needed to obtain the junction solution. There are zero modes corresponding to spontaneously broken translation, SUSY, and U(1). All broken and unbroken SUSY charges are explicitly worked out in the Wess-Zumino gauge in N=1 superfields as well as in components. The relation to models in five dimensions is also clarified. 
  We apply the supersymmetry approach to one-dimensional quantum systems with spatially-dependent mass, by including their ordering ambiguities dependence. In this way we extend the results recently reported in the literature. Furthermore, we point out a connection between these systems and others with constant masses. This is done through convenient transformations in the coordinates and wavefunctions. 
  We investigate the possible influence of very-high-energy physics on inflationary predictions focussing on whether effective field theories can allow effects which are parametrically larger than order H^2/M^2, where M is the scale of heavy physics and H is the Hubble scale at horizon exit. By investigating supersymmetric hybrid inflation models, we show that decoupling does not preclude heavy-physics having effects for the CMB with observable size even if H^2/M^2 << O(1%), although their presence can only be inferred from observations given some a priori assumptions about the inflationary mechanism. Our analysis differs from the results of hep-th/0210233, in which other kinds of heavy-physics effects were found which could alter inflationary predictions for CMB fluctuations, inasmuch as the heavy-physics can be integrated out here to produce an effective field theory description of low-energy physics. We argue, as in hep-th/0210233, that the potential presence of heavy-physics effects in the CMB does not alter the predictions of inflation for generic models, but does make the search for deviations from standard predictions worthwhile. 
  We attempt the linearization of N=1 SGM action describing the nonlinear supersymmetric gravitational interaction of superon(Nambu-Goldstone fermion). We find that 80+80 field contents may give the off-shell supermultiplet of the supergravity-like linearized theory and they are realized explicitly up to $O(\psi^2)$ as the composites, though they have modified SUSY transformations which closes on super-Poincar\'e algebra. Particular attentions are paid to the local Lorentz invariance in the minimal interaction. 
  We apply the quartic exponential variational approximation to the symmetry breaking phenomena of scalar field in three and four dimensions. We calculate effective potential and effective action for the time-dependent system by separating the zero mode from other non-zero modes of the scalar field and treating the zero mode quantum mechanically. It is shown that the quantum mechanical properties of the zero mode play a non-trivial role in the symmetry breaking of the scalar $\lambda \phi^4$ theory. 
  We develop a Coulomb gas formalism for boundary conformal field theory having a $W$ symmetry and illustrate its operation using the three state Potts model. We find that there are free-field representations for six $W$ conserving boundary states, which yield the fixed and mixed physical boundary conditions, and two $W$ violating boundary states which yield the free and new boundary conditions. Other $W$ violating boundary states can be constructed but they decouple from the rest of the theory. Thus we have a complete free-field realization of the known boundary states of the three state Potts model. We then use the formalism to calculate boundary correlation functions in various cases. We find that the conformal blocks arising when the two point function of $\phi_{2,3}$ is calculated in the presence of free and new boundary conditions are indeed the last two solutions of the sixth order differential equation generated by the singular vector. 
  We show that the chiral symmetry breaking occurs in the vacuum of the massless Nambu-Jona-Lasinio (NJL) and Thirring models without a Goldstone boson. The basic reason of non-existence of the massless boson is due to the fact that the new vacuum after the symmetry breaking acquires nonzero fermion mass which inevitably leads to massive bosons. The new vacuum has a finite condensate of $<\bar \psi \psi>$ with the chiral current conservation. Thus, it contradicts the Goldstone theorem, and we show that the proof of the Goldstone theorem cannot be justified any more for fermion field theory models with regularizations. 
  For any classical Lie algebra $g$, we construct a family of integrable generalizations of Toda mechanics labeled a pair of ordered integers $(m,n)$. The universal form of the Lax pair, equations of motion, Hamiltonian as well as Poisson brackets are provided, and explicit examples for $\mathfrak{g}=B_{r},C_{r},D_{r}$ with $m,n\leq3$ are also given. For all $m,n$, it is shown that the dynamics of the $(m,n-1)$- and the $(m-1,n)$-Toda chains are natural reductions of that of the $(m,n)$-chain, and for $m=n$, there is also a family of symmetrically reduced Toda systems, the $(m,m)_{\mathrm{Sym}}$-Toda systems, which are also integrable. In the quantum case, all $(m,n)$-Toda systems with $m>1$ or $n>1$ describe the dynamics of standard Toda variables coupled to noncommutative variables. Except for the symmetrically reduced cases, the integrability for all $(m,n)$-Toda systems survive after quantization. 
  The boson mass and the condensate in two dimensional QCD with $SU(N_c)$ colors are numerically evaluated with the Bogoliubov vacuum. It is found that the boson mass is finite at the massless limit, and it is well described by the phenomenological expression $ {\cal M}_{N_c}={2\over 3}\sqrt{{N_cg^2\over{3\pi}}} $ for large $N_c$. Also, the condensate values agree very well with the prediction by the $1/N_c$ expansion. The validity of the naive light cone method is examined, and it turns out that the light cone prescription of the boson mass with the trivial vacuum is accidentally good for QED$_2$. But it is not valid for QCD$_2$. Further, at the massless fermion limit, the chiral symmetry is spontaneouly broken without anomaly, and the Goldstone theorem does not hold for QCD$_2$. 
  We propose a new class of universal amplitude ratios which involve the first terms of the short distance expansion of the correlators of a statistical model in the vicinity of a critical point. We will describe the critical system with a conformal field theory (UV fixed point) perturbed by an appropriate relevant operator. In two dimensions the exact knowledge of the UV fixed point allows for accurate predictions of the ratios and in many nontrivial integrable perturbations they can even be evaluated exactly. In three dimensional O(N) scalar systems feasible extensions of some existing results should allow to obtain perturbative expansions for the ratios. By construction these universal ratios are a perfect tool to explore the short distance properties of the underlying quantum field theory even in regimes where the correlation length and one point functions are not accessible in experiments or simulations. 
  We discuss partition functions of N=(4,4) type IIA string theory on the pp-wave background. This theory is shown to be modular invariant. The boundary states are constructed and possible D-brane instantons are classified. Then we calculate cylinder amplitudes in both closed and open string descriptions and check the open/closed string duality. Furthermore we consider general properties of modular invariant partition functions in the case of pp-waves. 
  We consider N =1 compactifications to four dimensions of heterotic string theory in the presence of fluxes. We show that up to order O(\alpha'^2) the associated action can be written as a sum of squares of BPS-like quantities. In this way we prove that the equations of motion are solved by backgrounds which fulfill the supersymmetry conditions and the Bianchi identities. We also argue for the expression of the related superpotential and discuss the radial modulus stabilization for a class of examples. 
  A quantum deformation of the conformal algebra of the Minkowskian spacetime in $(3+1)$ dimensions is identified with a deformation of the $(4+1)$-dimensional AdS algebra. Both Minkowskian and AdS first-order non-commutative spaces are explicitly obtained, and the former coincides with the well known $\kappa$-Minkowski space. Next, by working in the conformal basis, a new non-commutative Minkowskian spacetime is constructed through the full (all orders) dual quantum group spanned by deformed Poincar\'e and dilation symmetries. Although Lorentz invariance is lost, the resulting non-commutative spacetime is quantum group covariant, preserves space isotropy and, furthermore, can be interpreted as a generalization of the $\kappa$-Minkowski space in which a variable fundamental scale (Planck length) appears. 
  In this paper we study the conjectured dual operators to a near maximal giant graviton and their open string fluctuations in the large $N$ limit. Using matrix model estimates we show that the spectrum of states near the D-brane operator is consistent with a Fock space of open plus closed string states. We also give an argument that these operators, in spite of having large R charge of order N, are amenable to being studied with standard perturbative techniques, which organize themselves in a 1/N expansion. Also the spectrum of operators dual to massless fluctuations on the D-brane is shown to be protected from weak to strong coupling at leading order, so it is possible to read the shape of the dual operator by understanding how the spherical harmonics of the D-brane fluctuations appear. 
  We use the results of hep-th/0007174 on the simple current classification of open unoriented CFTs to construct half supersymmetry preserving crosscap states for rational Calabi-Yau compactifications. We show that the corresponding orientifold fixed planes obey the BPS-like relation M=exp(i*phi)Q. To prove this relation, it is essential that the worldsheet CFT properly includes the degrees of freedom from the uncompactified space-time component. The BPS-phase phi can be identified with the automorphism type of the crosscap states. To illustrate the method we compute crosscap states in Gepner models with each k_i odd. 
  We compute the NSVZ beta functions for N = 1 four-dimensional quiver theories arising from D-brane probes on singularities, complete with anomalous dimensions, for a large set of phases in the corresponding duality tree. While these beta functions are zero for D-brane probes, they are non-zero in the presence of fractional branes. As a result there is a non-trivial RG behavior. We apply this running of gauge couplings to some toric singularities such as the cones over Hirzebruch and del Pezzo surfaces. We observe the emergence in string theory, of ``Duality Walls,'' a finite energy scale at which the number of degrees of freedom becomes infinite, and beyond which Seiberg duality does not proceed. We also identify certain quiver symmetries as T-duality-like actions in the dual holographic theory. 
  The current algebra for gauge theories like QCD at finite temperature and density is studied. We start considering, the massless Thirring model at finite temperature and density, finding an explicit expression for the current algebra. The central charge only depends on the coupling constant and there are not new effects due to temperature and density. From this calculation, we argue how to compute the central charge for $QCD_4$ and we argue why the central charge in four dimensions could be modified by finite temperature and density. 
  We introduce a Nambu-Poisson bracket in the geometrical description of the D=11 M5-brane. This procedure allows us, under some assumptions, to eliminate the local degrees of freedom of the antisymmetric field in the M5-brane Hamiltonian and to express it as a D=11 p-brane theory invariant under symplectomorphisms. A regularization of the M5-brane in terms of a multi D1-brane theory invariant under the $SU(N)\times SU(N)$ group in the limit when $N\to\infty$ is proposed. Also, a regularization for the D=10 D4-brane in terms of a multi D0-brane theory is suggested. 
  We show that a solution of the type IIB supergravity representing D3-branes in the presence of a 2-form background has 16 supersymmetries by explicitly constructing the transformation parameter of unbroken supersymmetry. This solution is dual to a noncommutative Yang-Mills theory in a certain limit. 
  We consider how the time-dependent decay process of an unstable D-brane should be described in the full (quantum) open-closed string theory. It is argued that the system, starting from the unstable D-brane configuration, will evolve in time into the time-independent open string tachyon vacuum configuration which we assume to be finite, with the total energy conserved. As a concrete realization of this idea, we construct a toy model describing the open and closed string tachyons which admits such a time-dependent solution. The structure of our model has some resemblance to that of open-closed string field theory. 
  We give a general framework for constructing supersymmetric solutions in the presence of non-trivial fluxes of tensor gauge fields. This technique involves making a general Ansatz for the metric and then defining the Killing spinors in terms of very simple projectors on the spinor fields. These projectors and, through them, the spinors, are determined algebraically in terms of the metric Ansatz. The Killing spinor equations then fix the tensor gauge fields algebraically, and, with the Bianchi identities, provide a system of equations for all the metric functions. We illustrate this by constructing an infinite family of massive flows that preserve eight supersymmetries in M-theory. This family constitutes all the radially symmetric Coulomb branch flows of the softly broken, large N scalar-fermion theory on M2-branes. We reduce the problem to the solution of a single, non-linear partial differential equation in two variables. This equation governs the flow of the fermion mass, and the function that solves it then generates the entire M-theory solution algebraically in terms of the function and its first derivatives. While the governing equation is non-linear, it has a very simple perturbation theory from which one can see how the Coulomb branch is encoded. 
  We explicitly construct the supersymmetry transformations for the N=2 supersymmetric RG flow solution of chiral IIB supergravity. We show that the metric, dilaton/axion, five-index tensor and half of the three index tensor are determined algebraically in terms of the Killing spinor of the unbroken supersymmetry. The algebraic nature of the solution allows us to generalize this construction to a new class of N=2 supersymmetric solutions of IIB supergravity. Each solution in this class is algebraically determined by supersymmetry and is parametrized by a single function of two variables that satisfies a non-linear equation akin to the Laplace equation on the space transverse to the brane. 
  In this article we calculate several divergent amplitudes in phi^4-theory on non-commutative space-time in the framework of Interaction Point Time Ordered Perturbation Theory (IPTOPT), continuing work done in hep-th/0209253. On the ground of these results we find corresponding Feynman rules which allow for a much easier diagrammatic calculation of amplitudes. The most important feature of the present theory is the lack of the UV/IR mixing problem in all amplitudes calculated so far. Although we are not yet able to give a rigorous proof, we provide a strong argument for this result to hold in general. Together with the found Feynman rules this opens promising vistas towards the systematic renormalization of non-commutative field theories. 
  In this letter we will extend the analysis given by Al. Zamolodchikov for the scaling Yang-Lee model on the sphere to the Ising model in a magnetic field. A numerical study of the partition function and of the vacuum expectation values (VEV) is done by using the truncated conformal space (TCS) approach. Our results strongly suggest that the partition function is an entire function of the coupling constant. 
  In the past decades, time ordered perturbation theory was very successful in describing relativistic scattering processes. It was developed for local quantum field theories. However, there are field theories which are governed by non-local interactions, for example non-commutative quantum field theory (NCQFT). Filk (Phys. Lett. B 376 (1996) 53) first studied NCQFT perturbatively obtaining the usual Feynman propagator and additional phase factors as the basic elements of perturbation theory. However, this treatment is only applicable for cases, where the deformation of space-time does not involve time. Thus, we generalize Filk's approach in two ways: First, we study non-local interactions of a very general type able to embed NCQFT. And second, we also include the case, where non-locality involves time. A few applications of the obtained formalism will also be discussed. 
  Type IIB strings on many pp-wave backgrounds, supported either by 5-form or 3-form fluxes, have negative light-cone zero-point energy. This raises the question of their stability and poses possible problems in the definition of their thermodynamic properties. After having pointed out the correct way of calculating the zero-point energy, an issue not fully discussed in literature, we show that these Type IIB strings are classically stable and have well defined thermal properties, exhibiting a Hagedorn behavior. 
  We calculate the vacuum averages of the energy-momentum tensor associated with a massless left-handed spinor fields due to magnetic fluxes on idealized cosmic string spacetime. In this analysis three distinct configurations of magnetic fields are considered: {\it{i)}} a homogeneous field inside the tube, {\it{ii)}} a magnetic field proportional to $1/r$ and {\it{iii)}} a cylindrical shell with $\delta$-function. In these three cases the axis of the infinitely long tubes of radius $R$ coincides with the cosmic string. In order to proceed with these calculations we explicitly obtain the Euclidean Feynman propagators associated with these physical systems. As we shall see, these propagators possess two distinct parts. The first are the standard ones, i.e., corresponding to the spinor Green functions associated with the massless fermionic fields on the idealized cosmic string spacetime with a magnetic flux running through the line singularity. The second parts are new, they are due to the finite thickness of the radius of the tubes. As we shall see these extra parts provide relevant contributions to the vacuum averages of the energy-momentum tensor. 
  We study both analytically and numerically a coupled system of spherically symmetric SU(2) Yang-Mills-dilaton equation in 3+1 Minkowski space-time. It has been found that the system admits a hidden scale invariance which becomes transparent if a special ansatz for the dilaton field is used. This choice corresponds to transition to a frame rotated in the $\ln r-t$ plane at a definite angle. We find an infinite countable family of self-similar solutions which can be parametrized by the $N$ - the number of zeros of the relevant Yang-Mills function. According to the performed linear perturbation analysis, the lowest solution with N=0 only occurred to be stable. The Cauchy problem has been solved numerically for a wide range of smooth finite energy initial data. It has been found that if the initial data exceed some threshold, the resulting solutions in a compact region shrinking to the origin, attain the lowest N=0 stable self-similar profile, which can pretend to be a global stable attractor in the Cauchy problem. The solutions live a finite time in a self-similar regime and then the unbounded growth of the second derivative of the YM function at the origin indicates a singularity formation, which is in agreement with the general expectations for the supercritical systems. 
  It is shown that an effective theory with meron degrees of freedom produces confinement in SU(2) Yang Mills theory. This effective theory is compatible with center symmetry. When the scale is set by the string tension, the action density and topological susceptibility are similar to those arising in lattice QCD. 
  We revisit the proposal that the resolution of the Cosmological Constant Problem involves a sub-millimeter breakdown of the point-particle approximation for gravitons. No fundamental description of such a breakdown, which simultaneously preserves the point-particle nature of matter particles, is yet known. However, basic aspects of the self-consistency of the idea, such as preservation of the macroscopic Equivalence Principle while satisfying quantum naturalness of the cosmological constant, are addressed in this paper within a Soft Graviton Effective Theory. It builds on Weinberg's analysis of soft graviton couplings and standard heavy particle effective theory, and minimally encompasses the experimental regime of soft gravity coupled to hard matter. A qualitatively distinct signature for short-distance tests of gravity is discussed, bounded by naturalness to appear above approximately 20 microns. 
  Exactly soluble string theories describing a particular hadronic sector of certain confining gauge theories have been obtained recently as Penrose-Gueven limits of the dual supergravity backgrounds. The effect of taking the Penrose-Gueven limit on the gravity side translates, in the gauge theory side, into an effective truncation to hadrons of large U(1) charge (annulons). We present an exact calculation of the finite temperature partition function for the hadronic states corresponding to a Penrose-Gueven limit of the Maldacena-Nunez embedding of N=1 SYM into string theory. It is established that the theory exhibits a Hagedorn density of states.   Motivated by this exact calculation we propose a semiclassical string approximation to the finite temperature partition function for confining gauge theories admitting a supergravity dual, by performing an expansion around classical solutions characterized by temporal windings. This semiclassical approximation reveals a hadronic energy density of states of Hagedorn type, with the coefficient determined by the gauge theory string tension as expected for confining theories. We argue that our proposal captures primarily information about states of pure N=1 SYM, given that this semiclassical approximation does not entail a projection onto states of large U(1) charge. 
  We consider a dynamical approach to the cosmological constant.  There is a scalar field with a potential whose minimum occurs at a generic, but negative, value for the vacuum energy, and it has a non-standard kinetic term whose coefficient diverges at zero curvature as well as the standard kinetic term. Because of the divergent coefficient of the kinetic term, the lowest energy state is never achieved. Instead, the cosmological constant automatically stalls at or near zero. The merit of this model is that it is stable under radiative corrections and leads to stable dynamics, despite the singular kinetic term. The model is not complete, however, in that some reheating is required. Nonetheless, our approach can at the very least reduce fine-tuning by 60 orders of magnitude or provide a new mechanism for sampling possible cosmological constants and implementing the anthropic principle. 
  A prescription is developed for matching general relativistic perturbations across singularities of the type encountered in the ekpyrotic and cyclic scenarios i.e. a collision between orbifold planes. We show that there exists a gauge in which the evolution of perturbations is locally identical to that in a model space-time (compactified Milne mod Z_2) where the matching of modes across the singularity can be treated using a prescription previously introduced by two of us. Using this approach, we show that long wavelength, scale-invariant, growing-mode perturbations in the incoming state pass through the collision and become scale-invariant growing-mode perturbations in the expanding hot big bang phase. 
  We present an intuitive picture of the chiral symmetry breaking in the NJL and Thirring models. For the current current interaction model with the massless fermion, the vacuum is realized with the chiral symmetry broken phase, and the fermion acquires the induced mass. With this finite fermion mass, we calculate the boson mass, and find that the boson is always massive, and there exists no Goldstone boson in the Nambu-Jona-Lasinio (NJL) model. 
  Dynamic perturbation equations are derived for a generic stationary state of an elastic string model -- of the kind appropriate for representing a superconducting cosmic string -- in a flat background. In the case of a circular equilibrium (i.e. vorton) state of a closed string loop it is shown that the fundamental axisymmetric ($n=0$) and lowest order ($n=1$) nonaxisymmetric perturbation modes can never be unstable. However, stability for modes of higher order ($n\geq 2$) is found to be non-trivially dependent on the values of the characteristic propagation velocity, $c$ say, of longitudinal perturbations and of the corresponding extrinsic perturbation velocity, $v$ say. For each mode number the criterion for instability is the existence of nonreal roots for a certain cubic eigenvalue equation for the corresponding mode frequency. A very simple sufficient but not necessary condition for reality of the roots and therefore absence of instability is that the characteristic velocity ratio, $c/v$ be greater than or equal to unity. Closer examination of the low velocity (experimentally accessible) nonrelativistic regime shows that in that limit the criterion for instability is just that the dimensionless characteristic ratio $c/v$ be less than a critical value $\chi_c$ whose numerical value is approximately $1\over 2$. In the relativistic regime that is relevant to superconducting cosmic strings the situation is rather delicate, calling for more detailed investigation that is postponed for future work. 
  We study various aspects of D-branes in the two families of closed N=2 strings denoted by \alpha and \beta in hep-th/0211147. We consider two types of N=2 boundary conditions, A-type and B-type. We analyse the D-branes geometry. We compute open and closed string scattering amplitudes in the presence of the D-branes and discuss the results. We find that, except the space filling D-branes, the B-type D-branes decouple from the bulk. The A-type D-branes exhibit inconsistency. We construct the D-branes effective worldvolume theories. They are given by a dimensional reduction of self-dual Yang-Mills theory in four dimensions. We construct the D-branes gravity backgrounds. Finally, we discuss possible N=2 open/closed string dualities. 
  We complement the low-energy gravi-dilaton effective action of string theory with a non-local, general-covariant dilaton potential, and obtain homogeneous solutions describing a non-singular (bouncing-curvature) cosmology. We then compute, both analytically and numerically, the spectrum of amplified scalar and tensor perturbations, and draw some general lessons on how to extract observable consequences from pre-big bang and ekpyrotic scenarios. 
  This is a very brief but selfcontained review of the concept of quantum group symmetries and their anomalies.   Remarkably, general constructions can be very simply illustrated on the standard harmonic oscillator which is shown to possess a non-commutative and non-cocommutative anomalous quantum group symmetry. 
  Using a recently developed approach for solving the three dimensional Dirac equation with spherical symmetry, we obtain the two-point Green's function of the relativistic Dirac-Morse problem. This is accomplished by setting up the relativistic problem in such a way that makes comparison with the nonrelativistic problem highly transparent and results in a mapping of the latter into the former. The relativistic bound states energy spectrum is easily obtained by locating the energy poles of the Green's function components in a simple and straightforward manner. 
  The local composite gluon-ghost operator $({1/2}A^{a\mu}A_{\mu}^{a}+\alpha \bar{c}^{a}c^{a})$ is analysed in the framework of the algebraic renormalization in SU(N) Yang-Mills theories in the Landau, Curci-Ferrari and maximal abelian gauges. We show, to all orders of perturbation theory, that this operator is multiplicatively renormalizable. Furthermore, its anomalous dimension is not an independent parameter of the theory, being given by a general expression valid in all these gauges. We also verify the relations we obtain for the operator anomalous dimensions by explicit 3-loop calculations in the MSbar scheme for the Curci-Ferrari gauge. 
  We show that the holonomy of the supercovariant connection for M-theory backgrounds with $N$ Killing spinors reduces to a subgroup of $SL(32-N,\bR)\st (\oplus^N \bR^{32-N})$. We use this to give the necessary and sufficient conditions for a background to admit $N$ Killing spinors. We show that there is no topological obstruction for the existence of up to 22 Killing spinors in eleven-dimensional spacetime. We investigate the symmetry superalgebras of supersymmetric backgrounds and find that their structure constants are determined by an antisymmetric matrix. The Lie subalgebra of bosonic generators is related to a real form of a symplectic group. We show that there is a one-one correspondence between certain bases of the Cartan subalgebra of $sl(32, \bR)$ and supersymmetric planar probe M-brane configurations. A supersymmetric probe configuration can involve up to 31 linearly independent planar branes and preserves one supersymmetry. The space of supersymmetric planar probe M-brane configurations is preserved by an $SO(32,\bR)$ subgroup of $SL(32, \bR)$. 
  We show how to systematically construct higher-derivative terms in effective actions in harmonic superspace despite the infinite redundancy in their description due to the infinite number of auxiliary fields. Making an assumption about the absence of certain superspace Chern-Simons-like terms involving vector multiplets, we write all 3- and 4-derivative terms on Higgs, Coulomb, and mixed branches. Among these terms are several with only holomorphic dependence on fields, and at least one satisfies a non-renormalization theorem. These holomorphic terms include a novel 3-derivative term on mixed branches given as an integral over 3/4 of superspace. As an illustration of our method, we search for Wess-Zumino terms in the low energy effective action of N=2 supersymmetric QCD. We show that such terms occur only on mixed branches. We also present an argument showing that the combination of space-time locality with supersymmetry implies locality in the anticommuting superspace coordinates of for unconstrained superfields. 
  These lectures contain a pedagogical review of non-perturbative results from holomorphy and Seiberg duality with applications to dynamical SUSY breaking. Background material on anomalies, instantons, unitarity bounds from superconformal symmetry, and gauge mediation are also included. 
  We construct black hole solutions to Einstein-Born-Infeld gravity with a cosmological constant. Since an elliptic function appears in the solutions for the metric, we construct horizons numerically. The causal structure of these solutions differ drastically from their counterparts in Einstein-Maxwell gravity with a cosmological constant. The charged de-Sitter black holes can have up to three horizons and the charged anti-de Sitter black hole can have one or two depending on the parameters chosen. 
  A procedure for computing the dimensions of the moduli spaces of reducible, holomorphic vector bundles on elliptically fibered Calabi-Yau threefolds X is presented. This procedure is applied to poly-stable rank n+m bundles of the form V + pi* M, where V is a stable vector bundle with structure group SU(n) on X and M is a stable vector bundle with structure group SU(m) on the base surface B of X. Such bundles arise from small instanton transitions involving five-branes wrapped on fibers of the elliptic fibration. The structure and physical meaning of these transitions are discussed. 
  We analyse the dynamics of an open membrane, both for the free case and when it is coupled to a background three-form, whose boundary is attached to $p$-branes. The role of boundary conditions and constraints in the Nambu-Goto and Polyakov formulations is studied. The low-energy approximation that effectively reduces the membrane to an open string is examined in detail. Noncommutative features of the boundary string coordinates, where the cylindrical membrane is attached to the D$p$-branes, are revealed by algebraic consistency arguments and not by treating boundary conditions as primary constraints, as is usually done. The exact form of the noncommutative algebra is obtained in the low-energy limit. 
  Some configurations of general dielectric D2-brane with both electric and magnetic fields are investigated. We find two classical stable solutions describing a BIon configuration with $S^2$ structure and a cylindrical tube respectively. Both of them can be regarded as the blown-up objects of the Born-Infeld string. The dependence of the geometry of these configurations on the Born-Infeld fields is analyzed. Also we find that similar BIon configuration exists as a fuzzy object in a system with N D0-branes. Furthermore this configuration is demonstrated to be more stable than the usual dielectric spherical D2-brane. 
  In this work the construction of supergravity duals to the noncommutative ${\cal N}=4$ SYM theory in the infinite momentum frame but with constant momentum density is attempted. In the absence of the content of noncommutativity, it has been known for some time that the previous $AdS_{5}/CFT_{4}$ correspondence should be replaced by the $K_{5}/CFT_{4}$ (with $K_{(p+2)}$ denoting the generalized Kaigorodov spacetime) correspondence with the pp-wave propagating on the BPS brane worldvolume. Interestingly enough, putting together the two contents, i.e., the introduction of noncommutativity and at the same time that of the pp-wave along the brane worldvolume, leads to quite nontrivial consequences such as the emergence of ``time-space'' noncommutativity in addition to the ``space-space'' noncommutativity in the manifold on which the dual gauge theory is defined. Taking the gravity decoupling limit, it has been realized that for small $u$, the solutions all reduce to $K_{5}\times S^5$ geometry confirming our expectation that the IR dynamics of the dual gauge theory should be unaffected by the noncommutativity while as $u\to \infty$, the solutions start to deviate significantly from $K_{5}\times S^5$ limit indicating that the UV dynamics of the dual gauge theory would be heavily distorted by the effect of noncommutativity. 
  We study chiral SU(N) supersymmetric gauge theories with matter in the antifundamental and antisymmetric representations. For SU(5) with two families, we show how to reproduce the non-perturbatively generated superpotential, and we discuss dynamical supersymmetry breaking purely in terms of the Konishi anomaly. We apply the same technique in general to SU(N) with one family. We also briefly comment on the chiral ring for these theories. 
  A detailed numerical stability analysis of the static, spherically symmetric globally regular solutions of the Einstein-Yang-Mills equations with a positive cosmological constant, Lambda, is carried out. It is found that the number of unstable modes in the even parity sector is n for solutions with n=1,2 nodes as Lambda varies. The solution with n=3 nodes exhibits a rather surprising behaviour in that the number of its unstable modes jumps from 3 to 1 as Lambda crosses (from below) a critical value. In particular the topologically 3-sphere type solution with n=3 nodes has only a single unstable mode. 
  We discuss in detail the relation between the gauge fixed and gauge invariant BRST cohomology. We showed previously that in certain gauges some cohomology classes of the gauge-fixed BRST differential do not correspond to gauge invariant observables. We now show that in addition ``accidental'' conserved currents may appear. These correspond one-to-one to observables that become trivial in this gauge. We explicitly show how the gauge-fixed BRST cohomology appears in the context of the Quantum Noether Method. 
  We determine the gluino condensate $< \Tr \lambda^2 >$ in the pure ${\cal N}=1$ super Yang-Mills theory (SYM) for the classical gauge groups $SU(r+1)$, $SO(2r+1)$, $USp(2r)$ and $SO(2r)$, by deforming the pure ${\cal N}=2$ SYM theory with the adjoint scalar multiplet mass, following the work by Finnell and Pouliot, and Ritz and Vainshtein. The value of the gluino condensate agrees in all cases with what was found in the weak coupling istanton calculation. 
  We study the Dirac and the Laplacian operators on orientable Riemann surfaces of arbitrary genus g. In particular we compute their determinants with twisted boundary conditions along the b-cycles. All the ingredients of the final results (including the normalizations) are explicitly written in terms of the Schottky parametrization of the Riemann surface. By using the bosonization equivalence, we derive a multi-loop generalization of the well-known g=1 product formulae for the Theta-functions. We finally comment on the applications of these results to the perturbative theory of open charged strings. 
  As was recently found in hep-th/0304255, there exists a simple non-supersymmetric classical solution describing a closed string rotating in S^5 and located at the center of AdS_5. It is parametrized by the angular momentum J of the center of mass and two equal SO(6) angular momenta J' in the two other orthogonal rotation planes. The dual N=4 SYM operators should be scalar operators in SU(4) representations [0,J-J',2J'] or [J'-J,0,J'+J]. This solution is stable if J' > 3/2 J and for large J + 2 J' its classical energy admits an expansion in positive powers of g_eff = \lambda/(J + 2 J')^2: E= J + 2 J' + g_eff J' + ... . This suggests a possibility of a direct comparison with perturbative SYM results for the corresponding anomalous dimensions in the sector with g_eff << 1, by analogy with the BMN case. We conjecture that all quantum sigma model string corrections are then subleading at large J', so that the classical formula for the energy is effectively exact to all orders in \lambda. It could then be interpolated to weak coupling, representing a prediction for the anomalous dimensions on the SYM side. We test this conjecture by computing the 1-loop superstring sigma model correction to the classical energy. 
  We present a class of black string spacetimes which asymptote to maximally symmetric plane wave geometries. Our construction will rely on a solution generating technique, the null Melvin twist, which deforms an asymptotically flat black string spacetime to an asymptotically plane wave black string spacetime while preserving the event horizon. 
  The decay of an unstable D-brane via closed string emission and open string pair production is considered in subcritical string theory with a spacelike linear dilaton. The decay rate is given by the imaginary part of the annulus, which has ambiguities corresponding to the choices of incoming closed and open string vacua. An exact expression for the full annulus diagram is computed with a natural choice of incoming vacua. It is found that the ultraviolet divergences present in critical string theory in both of these processes are absent for any nonzero spacelike dilaton. Implications for the vexing issue of the tachyon dust are discussed. 
  We review the solutions of O(N) and U(N) quantum field theories in the large $N$ limit and as 1/N expansions, in the case of vector representations. Since invariant composite fields have small fluctuations for large $N$, the method relies on constructing effective field theories for composite fields after integration over the original degrees of freedom. We first solve a general scalar $U(\phib^2)$ field theory for $N$ large and discuss various non-perturbative physical issues such as critical behaviour. We show how large $N$ results can also be obtained from variational calculations.We illustrate these ideas by showing that the large $N$ expansion allows to relate the $(\phib^2)^2$ theory and the non-linear $\sigma$-model, models which are renormalizable in different dimensions. Similarly, a relation between $CP(N-1)$ and abelian Higgs models is exhibited. Large $N$ techniques also allow solving self-interacting fermion models. A relation between the Gross--Neveu, a theory with a four-fermi self-interaction, and a Yukawa-type theory renormalizable in four dimensions then follows. We discuss dissipative dynamics, which is relevant to the approach to equilibrium, and which in some formulation exhibits quantum mechanics supersymmetry. This also serves as an introduction to the study of the 3D supersymmetric quantum field theory. Large $N$ methods are useful in problems that involve a crossover between different dimensions. We thus briefly discuss finite size effects, finite temperature scalar and supersymmetric field theories. We also use large $N$ methods to investigate the weakly interacting Bose gas. The solution of the general scalar $U(\phib^2)$ field theory is then applied to other issues like tricritical behaviour and double scaling limit. 
  We present a simple algebraic argument for the conclusion that the low energy limit of a quantum theory of gravity must be a theory invariant, not under the Poincare group, but under a deformation of it parameterized by a dimensional parameter proportional to the Planck mass. Such deformations, called kappa-Poincare algebras, imply modified energy-momentum relations of a type that may be observable in near future experiments. Our argument applies in both 2+1 and 3+1 dimensions and assumes only 1) that the low energy limit of a quantum theory of gravity must involve also a limit in which the cosmological constant is taken very small with respect to the Planck scale and 2) that in 3+1 dimensions the physical energy and momenta of physical elementary particles is related to symmetries of the full quantum gravity theory by appropriate renormalization depending on Lambda l^2_{Planck}. The argument makes use of the fact that the cosmological constant results in the symmetry algebra of quantum gravity being quantum deformed, as a consequence when the limit \Lambda l^2_{Planck} -> 0 is taken one finds a deformed Poincare invariance. We are also able to isolate what information must be provided by the quantum theory in order to determine which presentation of the kappa-Poincare algebra is relevant for the physical symmetry generators and, hence, the exact form of the modified energy-momentum relations. These arguments imply that Lorentz invariance is modified as in proposals for doubly special relativity, rather than broken, in theories of quantum gravity, so long as those theories behave smoothly in the limit the cosmological constant is taken to be small. 
  This is the first paper in a series in which an attempt is made to formulate a perturbation theory around the the Chern-Simons state of quantum gravity discovered by Kodama. It is based on an extension of the theory of 't Hooft Deser and Jackew describing point particles in 3D gravity to four spacetime dimensions. General covariance now requires the basic excitations to be extended in one spatial dimension rather than pointlike. As a consequence the symmetry of the Kodama state, which is the (anti)deSitter symmetry, is realized 'holographically' on a timelike boundary, the generators of this symmetry being related to quasilocal energy-momenta. As GR induces a Chern-Simons theory on the boundary, the deSitter symmetry of the vacuum must be q-deformed with the deformation parameter related to the cosmological constant. It is proposed to introduce excitations around this vacuum by putting punctures on the boundary to each of which is associate a vector in some representation of deSitter group projected to the boundary. By equations of motion those punctures must be connected by continuous lines of fluxes of an SO(3,1) gauge field. It is also argued that quasilocal masses and spins of these excitations must satisfy a relation of Regge type, which may point on a possible relation between non-perturbative quantum gravity and string theory. 
  A possible way to resolve the singularities of general relativity is proposed based on the assumption that the description of space-time using commuting coordinates is not valid above a certain fundamental scale. Beyond that scale it is assumed that the space-time has noncommutative structure leading in turn to a resolution of the singularity. As a first attempt towards realizing the above programme a modification of the Kasner metric is constructed which is commutative only at large time scales. At small time scales, near the singularity, the commutation relations among the space coordinates diverge. We interpret this result as meaning that the singularity has been completely delocalized. 
  We study decay of unstable D-branes in string theory in the presence of electric field, and show that the classical open string theory results for various properties of the final state agree with the properties of closed string states into which the system is expected to decay. This suggests a duality between tree level open string theory on unstable D-branes and closed strings at high density. 
  The heat kernel expansion is a very convenient tool for studying one-loop divergences, anomalies and various asymptotics of the effective action. The aim of this report is to collect useful information on the heat kernel coefficients scattered in mathematical and physical literature. We present explicit expressions for these coefficients on manifolds with and without boundaries, subject to local and non-local boundary conditions, in the presence of various types of singularities (e.g., domain walls). In each case the heat kernel coefficients are given in terms of several geometric invariants. These invariants are derived for scalar and spinor theories with various interactions, Yang-Mills fields, gravity, and open bosonic strings. We discuss the relations between the heat kernel coefficients and quantum anomalies, corresponding anomalous actions, and covariant perturbation expansions of the effective action (both "low-" and "high-energy" ones). 
  We apply recently developed integrable spin chain and dilatation operator techniques in order to compute the planar one-loop anomalous dimensions for certain operators containing a large number of scalar fields in N =4 Super Yang-Mills. The first set of operators, belonging to the SO(6) representations [J,L-2J,J], interpolate smoothly between the BMN case of two impurities (J=2) and the extreme case where the number of impurities equals half the total number of fields (J=L/2). The result for this particular [J,0,J] operator is smaller than the anomalous dimension derived by Frolov and Tseytlin [hep-th/0304255] for a semiclassical string configuration which is the dual of a gauge invariant operator in the same representation. We then identify a second set of operators which also belong to [J,L-2J,J] representations, but which do not have a BMN limit. In this case the anomalous dimension of the [J,0,J] operator does match the Frolov-Tseytlin prediction. We also show that the fluctuation spectra for this [J,0,J] operator is consistent with the string prediction. 
  The Kaigorodov space is a homogeneous Einstein space and it describes a pp-wave propagating in anti-de Sitter space. It is conjectured in the literature that M-theory or string theory on the Kaigorodov space times a compact manifold is dual to a conformal field theory in an infinitely-boosted frame with constant momentum density. In this note we present a charged generalization of the Kaigorodov space by boosting a non-extremal charged domain wall to the ultrarelativity limit where the boost velocity approaches the speed of light. The finite boost of the domain wall solution gives the charged generalization of the Carter-Novotn\'y-Horsk\'y metric. We study the thermodynamics associated with the charged Carter-Novotn\'y-Horsk\'y space and discuss its relation to that of the static black domain walls and its implications in the domain wall/QFT (quantum field theory) correspondence. 
  We investigate the coherent bremsstrahlung by relativistic electrons in a single crystal excited by hypersonic vibrations. The formula for the corresponding differential cross-section is derived in the case of a sinusoidal wave. The conditions are specified under which the influence of the hypersound is essential. The case is considered in detail when the electron enters into the crystal at small angles with respect to a crystallographic axis. It is shown that in dependence of the parameters, the presence of hypersonic waves can either enhance or reduce the bremsstrahlung cross-section. 
  We explore the holographic principle in the context of asymptotically flat spacetimes. In analogy with the AdS/CFT scenario we analyse the asympotically symmetry group of this class of spacetimes, the so called Bondi-Metzner-Sachs (BMS) group. We apply the covariant entropy bound to relate bulk entropy to boundary symmetries and find a quite different picture with respect to the asymptotically AdS case. We then derive the covariant wave equations for fields carrying BMS representations to investigate the nature of the boundary degrees of freedom. We find some similarities with 't Hooft S-matrix proposal and suggest a possible mechanism to encode bulk data. 
  We find the minimal energy solution describing a folded closed string located at the center of AdS_5 and rotating simultaneously in two planes in S^5 with two arbitrary SO(6) angular momenta J_1 and J_2. In the case when J_1=J_2=J' we observe the precise agreement between the leading coefficient in the large J' expansion of the energy of this solution and the minimal eigen-value of the 1-loop anomalous dimension matrix of the corresponding [J',0,J'] SYM operators obtained by Beisert, Minahan, Staudacher and Zarembo in hep-th/0306139. We find also perfect agreement between string and SYM results in cases of states with unequal spins J_1 and J_2 dual to [J_2,J_1-J_2,J_2] operators. This represents a remarkable quantitative test of the AdS/CFT duality in a non-supersymmetric (and non-near-BPS) sector. 
  A new kind of q-deformed charged coherent states is constructed in Fock space of two-mode q-boson system with su_{q}(2) covariance and a resolution of unity for these states is derived. We also present a simple way to obtain these coherent states using state projection method. 
  We first review the representations of the six-dimensional (2,0) superalgebra on a free tensor multiplet and on a free string. We then construct a supersymmetric Lagrangian describing a free tensor multiplet. (It also includes a decoupled anti self-dual part of the three-form field strength.) This field theory is then rewritten in variables appropriate for analyzing a situation where the R-symmetry is spontaneously broken by the vacuum expectation values of the scalar moduli fields. Finally, we construct a supersymmetric and kappa-symmetric action for a free string. 
  We study time dependent solutions in dilaton gravity which correspond to the decay of conical spacetimes. In string theory this can be interpreted as a strong coupling limit of the decay of a non-supersymmetric orbifold spacetime with localized tachyons. 
  We construct a string bit model in the pp-wave background in which fermion doubling produces the correct spectrum of string states. 
  Godel universe in M-theory is a supersymmetric and homogeneous background with rotation and four-form magnetic flux. It is known that, as seen in inertial frame of co-moving observer, all geodesics with zero orbital angular momentum orbit inside `surface of light velocity' (CTC horizon). To learn if other probes can travel beyond the CTC horizon, we study dynamics of M-graviton and, in particular, M2-brane, whose motion is affected by Lorentz force exerted by the four-form magnetic flux and by nonzero orbital angular momentum. Classically, we find that both probes gyrate closed orbits, but diameter and center of gyration depends on sign and magnitude of probe's energy, charge and orbital angular momentum. For M2-brane, orbits in general travel outside the CTC horizon. Quantum-mechanically, we find that wave function and excitation energy levels are all self-similar. We draw analogy of probe's dynamics with Landau problem for charged particle in magnetic field. 
  The holographic bound states that the entropy in a region cannot exceed one quarter of the area (in Planck units) of the bounding surface. A version of the holographic principle that can be applied to cosmological spacetimes has recently been given by Fischler and Susskind. This version can be shown to fail in closed spacetimes and they concluded that the holographic bound may rule out such universes. In this paper I give a modified definition of the holographic bound that holds in a large class of closed universes. Fischler and Susskind also showed that the dominant energy condition follows from the holographic principle applied to cosmological spacetimes with $a(t)=t^p$. Here I show that the dominant energy condition can be violated by cosmologies satisfying the holographic principle with more general scale factors. 
  The purpose of this paper is to describe a relationship between the moduli space of vortices and the moduli space of instantons. We study charge k vortices in U(N) Yang-Mills-Higgs theories and show that the moduli space is isomorphic to a special Lagrangian submanifold of the moduli space of k instantons in non-commutative U(N) Yang-Mills theories. This submanifold is the fixed point set of a U(1) action on the instanton moduli space which rotates the instantons in a plane. To derive this relationship, we present a D-brane construction in which the dynamics of vortices is described by the Higgs branch of a U(k) gauge theory with 4 supercharges which is a truncation of the familiar ADHM gauge theory. We further describe a moduli space construction for semi-local vortices, lumps in the CP(N) and Grassmannian sigma-models, and vortices on the non-commutative plane. We argue that this relationship between vortices and instantons underlies many of the quantitative similarities shared by quantum field theories in two and four dimensions. 
  It is intriguing to consider the possibility that the Big Bang of the standard (3+1) dimensional cosmology originated from the collision of two branes within a higher dimensional spacetime, leading to the production of a large amount of entropy. In this paper we study, subject to certain well-defined assumptions, under what conditions such a collision leads to an expanding universe. We assume the absence of novel physics, so that ordinary (4+1) -dimensional Einstein gravity remains a valid approximation. It is necessary that the fifth dimension not become degenerate at the moment of collision. First the case of a symmetric collision of infinitely thin branes having a hyperbolic or flat spatial geometry is considered. We find that a symmetric collision results in a collapsing universe on the final brane unless the pre-existing expansion rate in the bulk just prior to the collision is sufficiently large in comparison to the momentum transfer in the fifth dimension. Such prior expansion may either result from negative spatial curvature or from a positive five-dimensional cosmological constant. The relevance of these findings to the Colliding Bubble Braneworld Universe scenario is discussed. Finally, results from a numerical study of colliding thick-wall branes is presented, which confirm the results of the thin-wall approximation. 
  The statistical computation of the (2+1)-dimensional Kerr-de Sitter space in the context of the {\it classical} Virasoro algebra for an asymptotic isometry group has been a mystery since first, the degeneracy of the states has the right value only at the infinite boundary which is casually disconnected from our universe, second, the analyses were based on the unproven Cardy's formula for complex central charge and conformal weight. In this paper, I consider the entropy in Carlip's "would-be gauge" degrees of freedom approach instead. I find that it agree with the Bekenstein-Hawking entropy but there are no the above problems. Implications to the dS/CFT are noted. 
  We present a superfield construction of Hamiltonian quantization with N=2 supersymmetry generated by two fermionic charges Q^a. As a byproduct of the analysis we also derive a classically localized path integral from two fermionic objects \Sigma^a that can be viewed as ``square roots'' of the classical bosonic action under the product of a functional Poisson bracket. 
  Non-trivial, consistent interactions of a free, massless tensor field t_{\mu \nu |\alpha \beta} with the mixed symmetry of the Riemann tensor are studied in the following cases: self-couplings, cross-interactions with a Pauli-Fierz field and cross-couplings with purely matter theories. The main results, obtained from BRST cohomological techniques under the assumptions on smoothness, locality, Lorentz covariance and Poincar\'{e} invariance of the deformations, combined with the requirement that the interacting Lagrangian is at most second-order derivative, can be synthesized into: no consistent self-couplings exist, but a cosmological-like term; no cross-interactions with the Pauli-Fierz field can be added; no non-trivial consistent cross-couplings with the matter theories such that the matter fields gain gauge transformations are allowed. 
  We derive the Hamilton equations of motion for a constrained system in the form given by Dirac, by a limiting procedure, starting from the Lagrangean for an unconstrained system. We thereby ellucidate the role played by the primary constraints and their persistance in time. 
  We prove strong ellipticity of chiral bag boundary conditions on even dimensional manifolds. From a knowledge of the heat kernel in an infinite cylinder, some basic properties of the zeta function are analyzed on cylindrical product manifolds of arbitrary even dimension. 
  We consider in the paper some consequences of the nonsymmetric Kaluza-Klein (Jordan-Thiry) theory with spontaneous symmetry breaking connecting to the existence of the warp factor and communnication in extra-dimensions. We consider a toy model of a time machine. 
  We derive the analytical properties of the elastic forward scattering amplitude of two scalar particles from the axioms of the noncommutative quantum field theory. For the case of only space-space noncommutativity, i.e. $\theta_{0i}=0$, we prove the dispersion relation which is similar to the one in commutative quantum field theory. The proof in this case is based on the existence of the analog of the usual microcausality condition and uses the Lehmann-Symanzik-Zimmermann (LSZ) or equivalently the Bogoliubov-Medvedev-Polivanov (BMP) reduction formalisms. The existence of the latter formalisms is also shown. We remark on the general noncommutative case, $\theta_{0i}\neq0$, as well as on the nonforward scattering amplitude and mention their peculiarities. 
  To solve the relativistic bound-state problem one needs to systematically and simultaneously decouple the high-energy from the low-energy modes and the many-body from the few-particle states using a consistent renormalization scheme. In a recent paper we have shown that one such approach can be a combination of the coupled cluster method as used in many-body theory and the Wilsonian exact renormalization group. Even though the method is intrinsically non-perturbative, one can easily implement a loop expansion within it. In this letter we provide further support for this aspect of our formalism by obtaining results for the two-loop renormalized $\phi^{4}$ theory. We show that the non-unitary representation inherent in our method leads to an economic computation and does not produce any non-hermiticity in the relevant terms. 
  Axially symmetric monopole anti-monopole dipole solutions to the second order equations of a simple SU(2) Yang-Mills-Higgs model featuring a quartic Skyrme-like term are constructed numerically. The effect of varying the Skyrme coupling constant on these solutions is studied in some detail. 
  We prove that M-theory plane waves with extra supersymmetries are necessarily homogeneous (but possibly time-dependent), and we show by explicit construction that such time-dependent plane waves can admit extra supersymmetries. To that end we study the Penrose limits of Goedel-like metrics, show that the Penrose limit of the M-theory Goedel metric (with 20 supercharges) is generically a time-dependent homogeneous plane wave of the anti-Mach type, and display the four extra Killings spinors in that case. We conclude with some general remarks on the Killing spinor equations for homogeneous plane waves. 
  Scale invariant theories which contain maximal rank gauge field strengths (of $D$ indices in $D$ dimensions) are studied. The integration of the equations of motion of these gauge fields leads to the s.s.b. of scale invariance. The cases in study are: i) the spontaneous generation of $r^{-1}$ potentials in particle mechanics in a theory that contains only $r^{-2}$ potentials in the scale invariant phase, ii) mass generation in scalar field theories iii) generation of non trivial dilaton potentials in generally covariant theories, iv) spontaneous generation of confining behavior in gauge theories. The possible origin of these models is discussed. 
  We build up the anticommutator algebra for the fermionic coordinates of open superstrings attached to branes with antisymmetric tensor fields. We use both Dirac quantization and the symplectic Faddeev Jackiw approach. In the symplectic case we find a way of generating the boundary conditions as zero modes of the symplectic matrix by taking a discretized form of the action and adding terms that vanish in the continuous limit. This way boundary conditions can be handled as constraints. 
  A full rational CFT, consistent on all orientable world sheets, can be constructed from the underlying chiral CFT, i.e. a vertex algebra, its representation category C, and the system of chiral blocks, once we select a symmetric special Frobenius algebra A in the category C [I]. Here we show that the construction of [I] can be extended to unoriented world sheets by specifying one additional datum: a reversion on A - an isomorphism from the opposed algebra of A to A that squares to the twist. A given full CFT on oriented surfaces can admit inequivalent reversions, which give rise to different amplitudes on unoriented surfaces, in particular to different Klein bottle amplitudes.   We study the classification of reversions, work out the construction of the annulus, Moebius strip and Klein bottle partition functions, and discuss properties of defect lines on non-orientable world sheets. As an illustration, the Ising model is treated in detail. 
  We study three-dimensional Chern-Simons theory with complex gauge group SL(2,C), which has many interesting connections with three-dimensional quantum gravity and geometry of hyperbolic 3-manifolds. We show that, in the presence of a single knotted Wilson loop in an infinite-dimensional representation of the gauge group, the classical and quantum properties of such theory are described by an algebraic curve called the A-polynomial of a knot. Using this approach, we find some new and rather surprising relations between the A-polynomial, the colored Jones polynomial, and other invariants of hyperbolic 3-manifolds. These relations generalize the volume conjecture and the Melvin-Morton-Rozansky conjecture, and suggest an intriguing connection between the SL(2,C) partition function and the colored Jones polynomial. 
  We derive the torsion constraints and show the consistency of equations of motion of four-dimensional Type II supergravity in superspace, with Type II sigma model. This is achieved by coupling the four-dimensional compactified Type II Berkovits' superstring to an N=2 curved background and requiring that the sigma-model has superconformal invariance at tree-level. We compute this in a manifestly 4D N=2 supersymmetric way. The constraints break the target conformal and SU(2) invariances and the dilaton will be a conformal, $SU(2)\times U(1)$ compensator. For Type II superstring in four dimensions, worldsheet supersymmetry requires two different compensators. One type is described by chiral and anti-chiral superfields. This compensator can be identified with a vector multiplet. The other Type II compensator is described by twist-chiral and twist-anti-chiral superfields and can be identified with a tensor hypermultiplet. Also, the superconformal invariance at tree-level selects a particular gauge, where the matter is fixed, but not the compensators. After imposing the reality conditions, we show that the Type II sigma model at tree-level is consistent with the equations of motion for Type II supergravity in the string gauge. 
  Using the properties of defect lines, we study boundary renormalisation group flows. We find that when there exists a flow between maximally symmetric boundary conditions "a" and "b" then there also exists a boundary flow between "c x a" and "c x b" where "x" denotes the fusion product. We also discuss applications of this simple observation. 
  We report on our results of D3-brane probing a large class of generalised type IIB supergravity solutions presented very recently in the literature. The structure of the solutions is controlled by a single non-linear differential equation. These solutions correspond to renormalisation group flows from pure N=4 supersymmetric gauge theory to an N=2 gauge theory with a massive adjoint scalar. The gauge group is SU(n) with n large. After presenting the general result, we focus on one of the new solutions, solving for the specific coordinates needed to display the explicit metric on the moduli space. We obtain an appropriately holomorphic result for the coupling. We look for the singular locus, and interestingly, the final result again manifests itself in terms of a square root branch cut on the complex plane, as previously found for a set of solutions for which the details are very different. This, together with the existence of the single simple non-linear differential equation, is further evidence in support of an earlier suggestion that there is a very simple model --perhaps a matrix model with relation to the Calogero-Moser integrable system-- underlying this gauge theory physics. 
  We reassess the problem of renormalization in finite temperature field theory (FTFT). A new point of view elucidates the relation between the ultraviolet divergences for T=0 and $T \not= 0$ theories and makes clear the reason why the ultraviolet behavior keeps unaffected when we consider the FTFT version associated to a given quantum field theory (QFT). The strength of the derivation one lies on the H\"ormander's criterion for the existence of products of distributions in terms of the wavefront sets of the respective distributions. The approach allows us to regard the FTFT both imaginary and real time formalism at once in a unified way in the contour ordered formalism. 
  We explore physics behind the horizon in eternal AdS Schwarzschild black holes. In dimension d >3, where the curvature grows large near the singularity, we find distinct but subtle signals of this singularity in the boundary CFT correlators. Building on previous work, we study correlation functions of operators on the two disjoint asymptotic boundaries of the spacetime by investigating the spacelike geodesics that join the boundaries. These dominate the correlators for large mass bulk fields. We show that the Penrose diagram for d>3 is not square. As a result, the real geodesic connecting the two boundary points becomes almost null and bounces off the singularity at a finite boundary time t_c \neq 0. If this geodesic were to dominate the correlator there would be a "light cone" singularity at t_c. However, general properties of the boundary theory rule this out. In fact, we argue that the correlator is actually dominated by a complexified geodesic, whose properties yield the large mass quasinormal mode frequencies previously found for this black hole. We find a branch cut in the correlator at small time (in the limit of large mass), arising from coincidence of three geodesics. The t_c singularity, a signal of the black hole singularity, occurs on a secondary sheet of the analytically continued correlator. Its properties are computationally accessible. The t_c singularity persists to all orders in the 1/m expansion, for finite \alpha', and to all orders in g_s. Certain leading nonperturbative effects can also be studied. The behavior of these boundary theory quantities near t_c gives, in principle, significant information about stringy and quantum behavior in the vicinity of the black hole singularity. 
  Some properties of the higher grading integrable generalizations of the conformal affine Toda systems are studied. The fields associated to the non-zero grade generators are Dirac spinors. The effective action is written in terms of the Wess-Zumino-Novikov-Witten (WZNW) action associated to an affine Lie algebra, and an off-critical theory is obtained as the result of the spontaneous breakdown of the conformal symmetry. Moreover, the off-critical theory presents a remarkable equivalence between the Noether and topological currents of the model. Related to the off-critical model we define a real and local Lagrangian provided some reality conditions are imposed on the fields of the model. This real action model is expected to describe the soliton sector of the original model, and turns out to be the master action from which we uncover the weak-strong phases described by (generalized) massive Thirring and sine-Gordon type models, respectively. The case of any (untwisted) affine Lie algebra furnished with the principal gradation is studied in some detail.   The example of $\hat{sl}(n) (n=2,3)$ is presented explicitly. 
  The general solutions in the models of closed and open superstring and super p-branes with exotic fractions of the N=1 supersymmetry are considered and the spontaneously broken character of the $OSp(1,2M)$ symmetry of the models is established. It is shown that extending these models by Wess-Zumino terms generates the Dirichlet boundary conditions for superstring and super p-branes. Using the generalized Wess-Zumino terms new $OSp(1,2M)$ invariant super p-brane and Dp-brane-like actions preserving $\frac{M-1}{M}$ fraction of supersymmetry are proposed. For M=32 these models suggest new superbrane vacua of M-theory preserving 31 from 32 global supersymmetries. 
  It is shown that the continuum limit of the spin 1/2 Heisenberg XYZ model is far from sufficient for the site number of 16. Therefore, the energy spectrum of the XYZ model obtained by Kolanovic et al. has nothing to do with the massive Thirring model, but it shows only the spectrum of the finite size effects. 
  We propose a systematic method to construct quasi-solvable quantum many-body systems having permutation symmetry. By the introduction of elementary symmetric polynomials and suitable choice of a solvable sector, the algebraic structure of sl(M+1) naturally emerges. The procedure to solve the canonical-form condition for the two-body problem is presented in detail. It is shown that the resulting two-body quasi-solvable model can be uniquely generalized to the M-body system for arbitrary M under the consideration of the GL(2,K) symmetry. An intimate relation between quantum solvability and supersymmetry is found. With the aid of the GL(2,K) symmetry, we classify the obtained quasi-solvable quantum many-body systems. It turns out that there are essentially five inequivalent models of Inozemtsev type. Furthermore, we discuss the possibility of including M-body (M>=3) interaction terms without destroying the quasi-solvability. 
  We propose an algorithm for the construction of higher order gauge field theories from a superfield formulation within the Batalin-Vilkovisky formalism. This is a generalization of the superfield algorithm recently considered by Batalin and Marnelius. This generalization seems to allow for non-topological gauge field theories as well as alternative representations of topological ones. A five dimensional non-abelian Chern-Simons theory and a topological Yang-Mills theory are treated as examples. 
  Noncommutative field theories with commutator of the coordinates of the form $[x^{\mu},x^{\nu}]=i \Lambda_{\quad \omega}^{\mu \nu}x^{\omega}$ are studied. Explicit Lorentz invariance is mantained considering $\Lambda $ a Lorentz tensor. It is shown that a free quantum field theory is not affected. Since invariance under translations is broken, the conservation of energy-momentum is violated, obeying a new law which is expressed by a Poincar\'e-invariant equation. The resulting new kinematics is studied and applied to simple examples and to astrophysical puzzles, such as the observed violation of the GZK cutoff.   The $\lambda $$\Phi ^{4}$ quantum field theory is also considered in this context. In particular, self interaction terms violate the usual conservation of energy-momentum and, hence, the radiative correction to the propagator is altered. The correction to first order in $\lambda $ is calculated. The usual UV divergent terms are still present, but a new type of term also emerges, which is IR divergent, violates momentum conservation and implies a correction to the dispersion relation. 
  The large order growth of string perturbation theory in $c\le 1$ conformal field theory coupled to world sheet gravity implies the presence of $O(e^{-{1\over g_s}})$ non-perturbative effects, whose leading behavior can be calculated in the matrix model approach. Recently it was proposed that the same effects should be reproduced by studying certain localized D-branes in Liouville Field Theory, which were constructed by A. and Al. Zamolodchikov. We discuss this correspondence in a number of different cases: unitary minimal models coupled to Liouville, where we compare the continuum analysis to the matrix model results of Eynard and Zinn-Justin, and compact c=1 CFT coupled to Liouville in the presence of a condensate of winding modes, where we derive the matrix model prediction and compare it to Liouville theory. In both cases we find agreement between the two approaches. The c=1 analysis also leads to predictions about properties of D-branes localized in the vicinity of the tip of the cigar in SL(2)/U(1) CFT with c=26. 
  In cosmological backgrounds, there can be 'partially massless' higher spin fields which have fewer degrees of freedom than their massive partners. The equations for the partially massless spin-2 fields are usually taken to be the linearized Einstein equations augmented with a 'tuned' Pauli-Fierz mass. Here, we add more powers of curvatures and show that for the string-generated Einstein-Gauss-Bonnet model, partially massless spin-2 fields have real mass in AdS, in contrast to the Einstein level result. We discuss the implication of this for the AdS/CFT applications and briefly study the C^4-corrected AdS_5 x S^5$ solution in type IIB SUGRA. 
  We perform the generalised dimensional reduction of D=11 supergravity over three-dimensional group manifolds as classified by Bianchi. Thus, we construct eleven different maximal D=8 gauged supergravities, two of which have an additional parameter. One class of group manifolds (class B) leads to supergravities that are defined by a set of equations of motion that cannot be integrated to an action.   All 1/2 BPS domain wall solutions are given. We also find a non-supersymmetric domain wall solution where the single transverse direction is time. This solution describes an expanding universe and upon reduction gives the Einstein-de Sitter universe in D=4. The uplifting of the different solutions to M-theory and the isometries of the corresponding group manifold are discussed. 
  We investigate the role of the cosmological constant in the holographic description of a radiation-dominated universe $C_2/R^4$ with a positive cosmological constant $\Lambda$. In order to understand the nature of cosmological term, we first study the newtonian cosmology. Here we find two aspects of the cosmological term: entropy ($\Lambda \to S_{\rm \Lambda}$) and energy ($\Lambda \to E_{\rm \Lambda}$). Also we solve the Friedmann equation parametrically to obtain another role. In the presence of the cosmological constant, the solutions are described by the Weierstrass elliptic functions on torus and have modular properties. In this case one may expect to have a two-dimensional Cardy entropy formula but the cosmological constant plays a role of the modular parameter $\tau(C_2,\Lambda)$ of torus. Consequently the entropy concept of the cosmological constant is very suitable for establishing the holographic entropy bounds in the early universe. This contrasts to the role of the cosmological constant as a dark energy in the present universe. 
  In this article, an alternative interpretation of the Seiberg-Witten map in non-commutative field theory is provided. We show that the Seiberg-Witten map can be induced in a geometric way, by a field dependent co-ordinate transformation that connects noncommutative and ordinary space-times. Furthermore, in continuation of our earlier works, it has been demonstrated here that the above (field dependent co-ordinate) transformations are present in a gauge fixed version of the relativistic spinning particle model, embedded in the Batalin-Tyutin extended space. We emphasize that the space-time non-commutativity emerges naturally from the particle {\it {spin}} degrees of freedom. Contrary to similarly motivated works, the non-commutativity is not imposed here in an {\it{ad-hoc}} manner. 
  We exploit the geometrical superfield formalism to derive the local, covariant and continuous Becchi-Rouet-Stora-Tyutin (BRST) symmetry transformations and the non-local, non-covariant and continuous dual-BRST symmetry transformations for the free Abelian one-form gauge theory in four $(3 + 1)$-dimensions (4D) of spacetime. Our discussion is carried out in the framework of BRST invariant Lagrangian density for the above 4D theory in the Feynman gauge. The geometrical origin and interpretation for the (dual-)BRST charges (and the transformations they generate) are provided in the language of translations of some superfields along the Grassmannian directions of the six ($ 4 + 2)$-dimensional supermanifold parametrized by the four spacetime and two Grassmannian variables. 
  We construct a combined non-perturbative path integral over geometries and topologies for two-dimensional Lorentzian quantum gravity. The Lorentzian structure is used in an essential way to exclude geometries with unacceptably large causality violations. The remaining sum can be performed analytically and possesses a unique and well-defined double-scaling limit, a property which has eluded similar models of Euclidean quantum gravity in the past. 
  We provide an integral formula for the free energy of the two-matrix model with polynomial potentials of arbitrary degree (or formal power series). This is known to coincide with the tau-function of the dispersionless two--dimensional Toda hierarchy. The formula generalizes the case studied by Kostov, Krichever, Mineev-Weinstein, Wiegmann, Zabrodin and separately Takhtajan in the case of conformal maps of Jordan curves. Finally we generalize the formula found in genus zero to the case of spectral curves of arbitrary genus with certain fixed data. 
  We consider classes of T_6 orientifolds, where the orientifold projection contains an inversion I_{9-p} on 9-p coordinates, transverse to a Dp-brane. In absence of fluxes, the massless sector of these models corresponds to diverse forms of N=4 supergravity, with six bulk vector multiplets coupled to N=4 Yang--Mills theory on the branes. They all differ in the choice of the duality symmetry corresponding to different embeddings of SU(1,1)\times SO(6,6+n) in Sp(24+2n,R), the latter being the full group of duality rotations. Hence, these Lagrangians are not related by local field redefinitions. When fluxes are turned on one can construct new gaugings of N=4 supergravity, where the twelve bulk vectors gauge some nilpotent algebra which, in turn, depends on the choice of fluxes. 
  We derive intersecting brane solutions in pp-wave spacetime by solving the supergravity field equations explicitly. The general intersection rules are presented. We also generalize the construction to the non-extremal solutions. Both the extremal and non-extremal solutions presented here are asymptotic to BFHP plane waves. We find that these solutions do not have regular horizons. 
  In the presence of a minimal uncertainty in length, there exists a critical temperature above which the thermodynamics of a gas of radiation changes drastically.   We find that the equilibrium temperature of a system composed of a Schwarzschild black hole surrounded by radiation is unaffected by these modifications. This is in agreement with works related to the robustness of the Hawking evaporation. The only change the deformation introduces concerns the critical volume at which the system ceases to be stable.   On the contrary, the evolution of the very early universe is sensitive to the new behavior. We readdress the shortcomings of the standard big bang model(flatness, entropy and horizon problems) in this context, assuming a minimal coupling to general relativity. Although they are not solved, some qualitative differences set in. 
  We study a model of 2D QFT with boundary interaction, in which two-component massless Bose field is constrained to a circle at the boundary. We argue that this model is integrable at two values of the topological angle, $\theta =0$ and $\theta=\pi$. For $\theta=0$ we propose exact partition function in terms of solutions of ordinary linear differential equation. The circular brane model is equivalent to the model of quantum Brownian dynamics commonly used in describing the Coulomb charging in quantum dots, in the limit of small dimensionless resistance $g_0$ of the tunneling contact. Our proposal translates to partition function of this model at integer charge. 
  We show that the boundary states are idempotent B*B=B with respect to the star product of HIKKO type closed string field theory. Variations around the boundary state correctly reproduce the open string spectrum with the gauge symmetry. We explicitly demonstrate it for the tachyonic and massless vector modes. The idempotency relation may be regarded as the equation of motion of closed string field theory at a possible vacuum. 
  We discuss plane wave backgrounds of string theory and their relation to Goedel-like universes. This involves a twisted compactification along the direction of propagation of the wave, which induces closed timelike curves. We show, however, that no such curves are geodesic. The particle geodesics and the preferred holographic screens we find are qualitatively different from those in the Goedel-like universes. Of the two types of preferred screen, only one is suited to dimensional reduction and/or T-duality, and this provides a ``holographic protection'' of chronology. The other type of screen, relevant to an observer localized in all directions, is constructed both for the compact and non-compact plane waves, a result of possible independent interest. We comment on the consistency of field theory in such spaces, in which there are closed timelike (and null) curves but no closed timelike (or null) geodesics. 
  We have shown that the Unruh quantization scheme can be realized in Minkowski spacetime in the presence of Bose-Einstein condensate containing infinite average number of particles in the zero boost mode and located basically inside the light cone. Unlike the case of an empty Minkowski spacetime the condensate provides the boundary conditions necessary for the Fulling quantization of the part of the field restricted only to the Rindler wedge of Minkowski spacetime. 
  We investigate a conformal invariant gravitational model which is taken to hold at pre-inflationary era. The conformal invariance allows to make a dynamical distinction between the two unit systems (or conformal frames) usually used in cosmology and elementary particle physics. In this model we argue that when the universe suffers phase transitions, the resulting mass scales introduced by particle physics should have variable contributions to vacuum energy density. These variations are controlled by the conformal factor that appears as a dynamical field. We then deal with the cosmological consequences of this model. In particular, we shall show that there is an inflationary phase at early times. At late times, on the other hand, it provides a mechanism which makes a large effective cosmological constant relax to a sufficiently small value consistent with observations. Moreover, we shall show that the conformal factor acts as a quintessence field that leads the universe to accelerate at late times. 
  We consider scalar field theory with space and space-time-dependent non-commutativity. In perturbation theory, we find that the structure of the UV/IR mixing is quite different from cases with constant non-commutativity. In particular, UV/IR mixing becomes intertwined in an interesting way with violations of momentum conservation. 
  In two previous papers we have analyzed the C-metric in a background with a cosmological constant, namely the de Sitter (dS) C-metric, and the anti-de Sitter (AdS) C-metric, following the work of Kinnersley and Walker for the flat C-metric. These exact solutions describe a pair of accelerated black holes in the flat or cosmological constant background, with the acceleration A being provided by a strut in-between that pushes away the two black holes. In this paper we analyze the extremal limits of the C-metric in a background with generic cosmological constant. We follow a procedure first introduced by Ginsparg and Perry in which the Nariai solution, a spacetime which is the direct topological product of the 2-dimensional dS and a 2-sphere, is generated from the four-dimensional dS-Schwarzschild solution by taking an appropriate limit, where the black hole event horizon approaches the cosmological horizon. Similarly, one can generate the Bertotti-Robinson metric from the Reissner-Nordstrom metric by taking the limit of the Cauchy horizon going into the event horizon of the black hole, as well as the anti-Nariai by taking an appropriate solution and limit. Using these methods we generate the C-metric counterparts of the Nariai, Bertotti-Robinson and anti-Nariai solutions, among others. One expects that the solutions found in this paper are unstable and decay into a slightly non-extreme black hole pair accelerated by a strut or by strings. Moreover, the Euclidean version of these solutions mediate the quantum process of black hole pair creation, that accompanies the decay of the dS and AdS spaces. 
  We demonstrate that a clear physical content and relevance can be attributed to the on-shell BRST-invariant mixed gluon--ghost condensate of mass dimension two which was recently proposed by the author. We argue that a gauge invariant observable is associated with the mixed condensate. 
  By considering the continuum scaling limit of the $A_{4}$ RSOS lattice model of Andrews-Baxter-Forrester with integrable boundaries, we derive excited state TBA equations describing the boundary flows of the tricritical Ising model. Fixing the bulk weights to their critical values, the integrable boundary weights admit a parameter $\xi $ which plays the role of the perturbing boundary field $\phi_{1,3}$ and induces the renormalization group flow between boundary fixed points. The boundary TBA equations determining the RG flows are derived in the $\mathcal{B}_{(1,2)}\to \mathcal{B}_{(2,1)}$ example. The induced map between distinct Virasoro characters of the theory are specified in terms of distribution of zeros of the double row transfer matrix. 
  We elaborate in more details why lattice calculation in [Kolanovic et al, Phys. Rev. D 62, 025021 (2000)] was done correctly and argue that incresing the number of sites is not expected to change our conclusions on the mass spectrum. 
  Exact BPS solutions of multi-walls are obtained in five-dimensional supergravity. The solutions contain 2n parameters similarly to the moduli space of the corresponding global SUSY models and have a smooth limit of vanishing gravitational coupling. The models are constructed as gravitational deformations of massive T^*CP^n nonlinear sigma models by using the off-shell formulation of supergravity and massive quaternionic quotient method with U(1)*U(1) gauging. We show that the warp factor can have at most single stationary point in this case. We also obtain BPS multi-wall solutions even for models which reduce to generalizations of massive T^*CP^n models with only U(1)^n isometry in the limit of vanishing gravitational coupling. At particular values of parameters, isometry of the quaternionic manifolds is enhanced. 
  We present a novel calculation of color triplet excitations in two dimensional QCD with SU(2) colors. It is found that the lowest energy of the color triplet excitations is proportional to the box length $L$, and can be written as ${\cal M}_C={L\over{2\pi}}{g^2\over{\pi}} $. Therefore, the color triplet excited states go to infinity when the system size becomes infinity. The properties of the color triplet states such as the wave functions are studied for the finite box length. 
  We construct the effective potential for the dimension two composite operator 1/2 A^{a 2}_\mu in QCD with massless quarks in the Landau gauge for an arbitrary colour group at two loops. For SU(3) we show that an estimate for the effective gluon mass decreases as N_f increases. 
  We find a new family of supersymmetric vacuum solutions in the six-dimensional chiral gauged N=(1,0) supergravity theory. They are generically of the form AdS_3 x S^3, where the 3-sphere is squashed homogeneously along its Hopf fibres. The squashing is freely adjustable, corresponding to changing the 3-form charge, and the solution is supersymmetric for all squashings. In a limit where the length of the Hopf fibres goes to zero, one recovers, after a compensating rescaling of the fibre coordinate, a solution that is locally the same as the well-known (Minkowski)_4 x S^2 vacuum of this theory. It can now be viewed as a fine tuning of the new more general family. The traditional "Cosmological Constant Problem" is replaced in this theory by the problem of why the four-dimensional (Minkowski)_4 x S^2 vacuum should be selected over other members of the equally supersymmetric AdS_3 x S^3 family. We also obtain a family of dyonic string solutions in the gauged N=(1,0) theory, whose near-horizon limits approach the AdS_3 times squashed S^3 solutions. 
  We construct a Chern-Simons type gauged N=8 supergravity in three spacetime dimensions with gauge group SO(4) x T_\infty over the infinite dimensional coset space SO(8,\infty)/(SO(8) x SO(\infty)), where T_\infty is an infinite dimensional translation subgroup of SO(8,\infty). This theory describes the effective interactions of the (infinitely many) supermultiplets contained in the two spin-1 Kaluza-Klein towers arising in the compactification of N=(2,0) supergravity in six dimensions on AdS_3 x S^3 with the massless supergravity multiplet. After the elimination of the gauge fields associated with T_\infty, one is left with a Yang Mills type gauged supergravity with gauge group SO(4), and in the vacuum the symmetry is broken to the (super-)isometry group of AdS_3 x S^3, with infinitely many fields acquiring masses by a variant of the Brout-Englert-Higgs effect. 
  A set of consistent Poisson brackets for an open string in generic spacetime background and NS-NS $B$-field is constructed. Upon quantization, this set of Poisson brackets lead to spacial \emph{commutative} $D$-branes at the string ends, showing that noncommutativity between spacial coordinates on the $D$-branes can be avoided. 
  We describe a procedure for finding Kaluza-Klein monopole solutions in deconstructed four and five dimensional supersymmetric gauge theories. In the deconstruction of a four dimensional theory, the KK monopoles are finite-action solutions of the Euclidean equations of motion of the finite lattice spacing theory. The "lattice" KK monopoles can be viewed as constituents of continuum-limit four dimensional instantons. In the five dimensional case, the KK monopoles are static finite-energy stringlike configurations, wrapped and twisted around the compact direction, and can similarly be interpreted as constituents of five dimensional finite-energy gauge solitons. We discuss the quantum numbers and zero modes of the towers of deconstructed KK monopoles and their significance for understanding anomalies and nonperturbative effects in deconstruction. 
  In this paper, using the Weyl-Wigner-Moyal formalism for quantum mechanics, we develop a {\it quantum-deformed} exterior calculus on the phase-space of an arbitrary hamiltonian system. Introducing additional bosonic and fermionic coordinates we construct a super-manifold which is closely related to the tangent and cotangent bundle over phase-space. Scalar functions on the super-manifold become equivalent to differential forms on the standard phase-space. The algebra of these functions is equipped with a Moyal super-star product which deforms the pointwise product of the classical tensor calculus. We use the Moyal bracket algebra in order to derive a set of quantum-deformed rules for the exterior derivative, Lie derivative, contraction, and similar operations of the Cartan calculus. 
  We introduce a master action in noncommutative space, out of which we obtain the action of the noncommutative Maxwell-Chern-Simons theory. Then, we look for the corresponding dual theory at both first and second orders in the noncommutative parameter. At the first order, the dual theory happens to be, precisely, the action obtained from the usual commutative Self-Dual model by generalizing the Chern-Simons term to its noncommutative version, including a cubic term. Since this resulting theory is also equivalent to the noncommutative massive Thirring model in the large fermion mass limit, we remove, as a byproduct, the obstacles arising in the generalization to noncommutative space, and to the first nontrivial order in the noncommutative parameter, of the bosonization in three dimensions. Then, performing calculations at the second order in the noncommutative parameter, we explicitly compute a new dual theory which differs from the noncommutative Self-Dual model, and further, differs also from other previous results, and involves a very simple expression in terms of ordinary fields. In addition, a remarkable feature of our results is that the dual theory is local, unlike what happens in the non-Abelian, but commutative case. We also conclude that the generalization to noncommutative space of bosonization in three dimensions is possible only when considering the first non-trivial corrections over ordinary space. 
  The group embeddings used in orbifolding the AdS/CFT correspondence to arrive at quiver gauge field theories are studied for both supersymmetric and non-supersymmetric cases. For an orbifold $AdS_5 \times S^5/\Gamma$ the conditions for embeddings of the finite group $\Gamma$ in the $SU(4) \sim O(6)$ isotropy of $S^5$ are stated in the form of consistency rules, both for Abelian and Non-Abelian $\Gamma$. 
  One of the most disturbing difficulties in thinking about the cosmological constant is that it is not stable under radiative corrections. The feedback mechanism proposed in [hep-th/0306108] is a dynamical way to protect a zero or small cosmological constant against radiative corrections. Hence, while this by itself does not solve the cosmological constant problem, it can help solving the problem. In the present paper we investigate stability and gravity in this approach and show that the feedback mechanism is both classically and quantum mechanically stable and has self-consistent, stable dynamics. 
  We propose that stretched horizons can be described in terms of a gas of non-interacting quasiparticles. The quasiparticles are unstable, with a lifetime set by the imaginary part of the lowest quasinormal mode frequency. If the horizon arises from an AdS/CFT style duality the quasiparticles are also the effective low-energy degrees of freedom of the finite-temperature CFT. We analyze a large class of models including Schwarzschild black holes, non-extremal Dp-branes, the rotating BTZ black hole and de Sitter space, and we comment on degenerate horizons. The quasiparticle description makes manifest the relationship between entropy and area. 
  We show that the reduction of a planar free spin-1/2 particle system by the constraint fixing its total angular momentum produces the one-dimensional Akulov-Pashnev-Fubini-Rabinovici superconformal mechanics model with the nontrivially coupled boson and fermion degrees of freedom. The modification of the constraint by including the particle's spin with the relative weight $n\in \N$, $n>1$, and subsequent application of the Dirac reduction procedure (`first quantize and then reduce') give rise to the anomaly free quantum system with the order $n$ nonlinear superconformal symmetry constructed recently in hep-th/0304257. We establish the origin of the quantum corrections to the integrals of motion generating the nonlinear superconformal algebra, and fix completely its form. 
  In my lecture I consider integrals over moduli spaces of supersymmetric gauge field configurations (instantons, Higgs bundles, torsion free sheaves).   The applications are twofold: physical and mathematical; they involve supersymmetric quantum mechanics of D-particles in various dimensions, direct computation of the celebrated Seiberg-Witten prepotential, sum rules for the solutions of the Bethe ansatz equations and their relation to the Laumon's nilpotent cone. As a by-product we derive some combinatoric identities involving the sums over Young tableaux. 
  We discuss the model consisting of tachyon which may have the negative kinetic energy plus scalar phantom and plus conformal quantum matter. It is demonstrated that such a model naturally admits two deSitter phases where the early universe inflation is produced by quantum effects and the late time accelerating universe is caused by phantom/tachyon. The energy conditions bounds for such cosmology are derived. It is interesting that effective equation of state may change its sign which depends from the proper choice of the combination of phantom/tachyon and quantum effects. 
  We discuss an open supermembrane theory on the maximally supersymmetric pp-wave background in eleven dimensions. The boundary surfaces of an open supermembrane are studied by using the covariant supermembrane theory. In particular, we find the configurations of M5-branes and 9-branes preserving a half of supersymmetries at the origin. 
  We have studied the scalar perturbation of static charged dilaton black holes in 2+1 dimensions. The black hole considered here is a solution to the low-energy string theory in 2+1 dimensions. It is asymptotic to the anti-de Sitter space. The exact values of quasinormal modes for the scalar perturbations are calculated. For both the charged and uncharged cases, the quasinormal frequencies are pure-imaginary leading to purely damped modes for the perturbations. 
  We consider a deformed superspace in which the coordinates \theta do not anticommute, but satisfy a Clifford algebra. We present results on the properties of N=1/2 supersymmetric theories of chiral superfields in deformed superspace, taking the Wess-Zumino model as the prototype. We prove new (non)renormalization theorems: the F-term is radiatively corrected and becomes indistinguishable from the D-term, while the Fbar-term is not renormalized. Supersymmetric vacua are critical points of the antiholomorphic superpotential. The vacuum energy is zero to all orders in perturbation theory. We illustrate these results with several examples. 
  In this paper we propose a quiver model of matrix quantum mechanics with 8 supercharges which, on a Higgs branch, deconstructs the worldsheet of Matrix String Theory. This discrete model evades the fermion doubling problem and, in the continuum limit, enhances the number of supersymmetries to sixteen. Our model is motivated by orbifolding the Matrix Model, and the deconstruction {\it ansatz} exhibits a duality between target space compactification and worldsheet deconstruction. 
  Recently an alternative description of 2d supergravities in terms of graded Poisson-Sigma models (gPSM) has been given. As pointed out previously by the present authors a certain subset of gPSMs can be interpreted as "genuine" supergravity, fulfilling the well-known limits of supergravity, albeit deformed by the dilaton field. In our present paper we show that precisely that class of gPSMs corresponds one-to-one to the known dilaton supergravity superfield theories presented a long time ago by Park and Strominger. Therefore, the unique advantages of the gPSM approach can be exploited for the latter: We are able to provide the first complete classical solution for any such theory. On the other hand, the straightforward superfield formulation of the point particle in a supergravity background can be translated back into the gPSM frame, where "supergeodesics" can be discussed in terms of a minimal set of supergravity field degrees of freedom. Further possible applications like the (almost) trivial quantization are mentioned. 
  We study basic properties of supermanifolds endowed with an even (odd) symplectic structure and a connection respecting this symplectic structure. Such supermanifolds can be considered as generalization of Fedosov manifolds to the supersymmetric case. Choosing an apporpriate definition of inverse (second-rank) tensor fields on supermanifolds we consider the symmetry behavior of tensor fields as well as the properties of the symplectic curvature and of the Ricci tensor on even (odd) Fedosov supermanifolds. We show that for odd Fedosov supermanifolds the scalar curvature, in general, is non-trivial while for even Fedosov supermanifolds it necessarily vanishes. 
  We show how the recently proposed CFT for a bilayer Quantum Hall system at filling nu=m/pm+2, the Twisted Model (TM), is equivalent to the system of two massless scalar bosons with a magnetic boundary interaction as introduced in Nucl. Phys. B443 (1995) 444, at the so called magic points. We are then able to describe, within such a framework, the dissipative quantum mechanics of a particle confined to a plane and subject to an external magnetic field normal to it. Such an analogy is further developed in terms of the TM boundary states, by describing the interaction between an impurity with a Hall system. 
  We consider intersecting hypersurfaces in curved spacetime with gravity governed by a class of actions which are topological invariants in lower dimensionality. Along with the Chern-Simons boundary terms there is a sequence of intersection terms that should be added in the action functional for a well defined variational principle. We construct them in the case of Characteristic Classes, obtaining relations which have a general topological meaning. Applying them on a manifold with a discontinuous connection 1-form we obtain the gravity action functional of the system and show that the junction conditions can be found in a simple algebraic way. At the sequence of intersections there are localised independent energy tensors, constrained only by energy conservation. We work out explicitly the simplest non trivial case. 
  We investigate the algebraic properties of the quantum counterpart of the classical canonical transformations using the symbol-calculus approach to quantum mechanics. In this framework we construct a set of pseudo-differential operators which act on the symbols of operators, i.e., on functions defined over phase-space. They act as operatorial left- and right- multiplication and form a $W_{\infty}\times W_{\infty}$- algebra which contracts to its diagonal subalgebra in the classical limit. We also describe the Gel'fand-Naimark-Segal (GNS) construction in this language and show that the GNS representation-space (a doubled Hilbert space) is closely related to the algebra of functions over phase-space equipped with the star-product of the symbol-calculus. 
  The dynamics of even topological open membranes relies on Nambu Brackets. Consequently, such 2p-branes can be quantized through the consistent quantization of the underlying Nambu dynamical structures. This is a summary construction relying on the methods detailed in refs hep-th/0205063 and hep-th/0212267. 
  Nonunitary versions of Newtonian gravity leading to wavefunction localization admit natural special-relativistic generalizations. They include the first consistent relativistic localization models. At variance with the unified model of localization and gravity, the purely localizing version requires negative energy fields, which however are less harmful than usual and can be used to build ultraviolet-finite theories. 
  If the vacuum manifold of a field theory has the appropriate topological structure, the theory admits topological structures analogous to the D-branes of string theory, in which defects of one dimension terminate on other defects of higher dimension. The shapes of such defects are analyzed numerically, with special attention paid to the intersection regions. Walls (co-dimension 1 branes) terminating on other walls, global strings (co-dimension 2 branes) and local strings (including gauge fields) terminating on walls are all considered. Connections to supersymmetric field theories, string theory and condensed matter systems are pointed out. 
  We study the most general supersymmetric warped M-theory backgrounds with non-trivial G-flux of the type R^{1,2} x M_8 and AdS_3 x M_8. We give a set of necessary and sufficient conditions for preservation of supersymmetry which are phrased in terms of G-structures and their intrinsic torsion. These equations may be interpreted as calibration conditions for a static ``dyonic'' M-brane, that is, an M5-brane with self-dual three-form turned on. When the electric flux is turned off we obtain the supersymmetry conditions and non-linear PDEs describing M5-branes wrapped on associative and special Lagrangian three-cycles in manifolds with G_2 and SU(3) structures, respectively. As an illustration of our formalism, we recover the 1/2-BPS dyonic M-brane, and also construct some new examples. 
  Motivated by Ooguri and Vafa, we study superstrings in flat R^4 in a constant self-dual graviphoton background. The supergravity equations of motion are satisfied in this background which deforms the N=2 d=4 flat space super-Poincare algebra to another algebra with eight supercharges. A D-brane in this space preserves a quarter of the supercharges; i.e. N=1/2 supersymmetry is realized linearly, and the remaining N=3/2 supersymmetry is realized nonlinearly. The theory on the brane can be described as a theory in noncommutative superspace in which the chiral fermionic coordinates $\theta^\alpha$ of N=1 d=4 superspace are not Grassman variables but satisfy a Clifford algebra. 
  We construct Cardy states in the Kazama-Suzuki model G/H x U(1), which satisfy the boundary condition twisted by the automorphisms of the coset theory. We classify all the automorphisms of G/H x U(1) induced from those of the G theory. The automorphism group contains at least a Z_2 as a subgroup corresponding to the charge conjugation. We show that in several models there exist extra elements other than the charge conjugation and that the automorphism group can be larger than Z_2. We give the explicit form of the twisted Cardy states which are associated with the non-trivial automorphisms. It is shown that the resulting states preserve the N=2 superconformal algebra. As an illustration of our construction, we give a detailed study for two hermitian symmetric space models SU(4)/SU(2) x SU(2) x U(1) and SO(8)/SO(6) x U(1) both at level one. We also study the action of the level-rank duality on the Cardy states and find the relation with the exceptional Cardy states originated from a conformal embedding. 
  Noncommutative Chern-Simons gauge theory coupled to nonrelativistic scalars or spinors is shown to admit the ``exotic'' two-parameter-centrally extended Galilean symmetry, realized in a unique way consistent with the Seiberg-Witten map. Nontopological spinor vortices and topological external-field vortices are constructed by reducing the problem to previously solved self-dual equations. 
  We calculate the most general effective potential for the massless Thirring model in dependence of the local fields of all fermion-antifermion collective excitations. We analyse the minima of this potential describing different vacua of the quantum system. We confirm the existence of the absolute minimum found in EPJC 20, 723 (2001) corresponding to the chirally broken phase of the massless Thirring model. As has been shown in EPJC 20, 723 (2001) this minimum is stable under quantum fluctuations. 
  In the context of the formalism proposed by Stelle-West and Grignani-Nardelli, it is shown that Chern-Simons supergravity can be consistently obtained as a dimensional reduction of (3+1)-dimensional supergravity, when written as a gauge theory of the Poincare group. The dimensional reductions are consistent with the gauge symmetries, mapping (3+1)-dimensional Poincare supergroup gauge transformations onto (2+1)-dimensional Poincare supergroup ones. 
  A fuzzy circle and a fuzzy 3-sphere are constructed as subspaces of fuzzy complex projective spaces, of complex dimension one and three, by modifying the Laplacians on the latter so as to give unwanted states large eigenvalues. This leaves only states corresponding to fuzzy spheres in the low energy spectrum (this allows the commutative algebra of functions on the continuous sphere to be approximated to any required degree of accuracy). The construction of a fuzzy circle opens the way to fuzzy tori of any dimension, thus circumventing the problem of power law corrections in possible numerical simulations on these spaces. 
  We consider the brane-localized supersymmetry breaking in 5D compactified on $S^1/Z_2$. In case of a bulk gaugino with arbitrary brane masses for its even and odd modes, we find the mass spectrum and the wave functions of gaugino. We show that the gaugino masses at the distant brane are soft in the usual sense in the effective field theory with zero modes of bulk gauge fields and they are also extremely soft in view of the one-loop finite mass of a brane scalar in the KK regularization. 
  The role of the measurement process in resolving the gauge ambiguity of the effective gravitational potential is reexamined. The motion of a classical point-like particle in the field of an arbitrary linear source, and in the field of another point-like particle is investigated. It is shown that in both cases the value of the gravitational field read off from the one-loop effective action of the testing particle depends on the Feynman weighting parameter. The found dependence is essential in that it persists in the expression for the gravitational potential. This result disproves the general conjecture about gauge independence of the effective equations of motion of classical point-like particles. 
  The pp-wave/SYM correspondence is an equivalence relation, H_{string}= Delta -J, between the Hamiltonian H_{string} of string field theory in the pp-wave background and the dilatation operator Delta in N=4 Super Yang-Mills in the double scaling limit. We calculate matrix elements of these operators in string field theory and in gauge theory.In the string theory Hilbert space we use the natural string basis,and in the gauge theory we use the basis which is isomorphic to it. States in this basis are specific linear combinations of the original BMN operators, and were constructed previously for the case of two scalar impurities. We extend this construction to incorporate BMN operators with vector and mixed impurities. This enables us to verify from the gauge theory perspective two key properties of the three-string interaction vertex of Spradlin and Volovich: (1) the vanishing of the three-string amplitude for string states with one vector and one scalar impurity; and (2) the relative minus sign in the string amplitude involving states with two vector impurities compared to that with two scalar impurities. This implies a spontaneous breaking of the Z_2 symmetry of the string field theory in the pp-wave background. Furthermore, we calculate the gauge theory matrix elements of Delta -J for states with an arbitrary number of scalar impurities. In all cases we find perfect agreement with the corresponding string amplitudes derived from the three-string vertex. 
  A general form for all supersymmetric solutions of minimal supergravity in six dimensions is obtained. Examples of new supersymmetric solutions are presented. It is proven that the only maximally supersymmetric solutions are flat space, AdS_3 x S^3 and a plane wave. As an application of the general solution, it is shown that any supersymmetric solution with a compact horizon must have near-horizon geometry R^{1,1} x T^4, R^{1,1} x K3 or identified AdS_3 x S^3. 
  We study the IIB engineering of N=1 gauge theories with unitary gauge group and matter in the adjoint and (anti)symmetric representations. We show that such theories can be obtained as Z2 orientifolds of Calabi-Yau A2 fibrations, and discuss the explicit T-duality transformation to an orientifolded Hanany-Witten construction. The low energy dynamics is described by a geometric transition of the orientifolded background. Unlike previously studied cases, we show that the orientifold 5-`plane' survives the transition, thus bringing a nontrivial contribution to the effective superpotential. We extract this contribution by using matrix model results and compare with geometric data. A Higgs branch of our models recovers the engineering of SO/Sp theories with adjoint matter through an O5-`plane' T-dual to an O6-plane. We show that the superpotential agrees with that produced by engineering through an O5-`plane' dual to an O4-plane, even though the orientifold of this second construction is replaced by fluxes after the transition. 
  We study the N=1/2 supersymmetric theory on noncommutative superspace which is a deformation of usual superspace. We consider deformed Wess-Zumino model as an example and show vanishing of vacuum energy, renormalization of superpotential and nonvanishing of tadpole. We find that the perturbative effective action has terms which are not written in the star deformation. Also we consider gauge theory on noncommutative superspace and observe that gauge group is restricted. We generalize the star deformation to include noncommutativity between bosonic coordinates and fermionic coordinates. 
  We study N=2 supersymmetric four dimensional gauge theories, in a certain N=2 supergravity background, called Omega-background. The partition function of the theory in the Omega-background can be calculated explicitly. We investigate various representations for this partition function: a statistical sum over random partitions, a partition function of the ensemble of random curves, a free fermion correlator. These representations allow to derive rigorously the Seiberg-Witten geometry, the curves, the differentials, and the prepotential. We study pure N=2 theory, as well as the theory with matter hypermultiplets in the fundamental or adjoint representations, and the five dimensional theory compactified on a circle. 
  We study a torus-like compactification of type IIB maximally supersymmetric PP-wave background. As the most general case, we discuss a T^8 compactification of all the transverse directions. A nontrivial structure of the isometry group requires an additional light-like compactification. This additional S^1 fiber is twisted on the T^8. We determine the spectrum of closed strings in this twisted torus background and compute the thermal partition function. 
  We attempt the linearization of the nonlinear supersymmetry encoded in the Einstein-Hilbert-type action {\it superon-graviton model(SGM)} describing the nonlinear supersymmetric gravitational interaction of Nambu-Goldstone fermion {\it superon}. We discuss the linearizaton procedure in detail by the heuristic arguments referring to supergravity, in which particular attentions are paid to the local Lorentz invariance in the minimal interaction. We show explicitly up to the leading order that 80(bosons)+80(fermions) may be the minimal off-shell supermultiplet of the linearized theory. 
  We consider a dynamics of Higgs' Fields in the framework of cosmological models involving a scalar field from Nonsymmetric Kaluza Klein (Jordan-Thiry) Theory. The scalar field plays here a role of a quintessece field. We consider phase transitions in cosmological models of the second and of the first order due to an evolution of Higgs' field. We develop inflationary models including calculation of an amount of an inflation. 
  We show how it is possible to integrate out chiral matter fields in N=1 supersymmetric theories and in this way derive in a simple diagrammatic way the $N_f S \log S - S \log \det X$ part of the Veneziano-Yankielowicz-Taylor superpotential. 
  Using the world-line method we resum the scalar one-loop effective action. This is based on an exact expression for the one-loop action obtained for a background potential and a Taylor expansion of the potential up to quadratic order in x-space. We thus reproduce results of Masso and Rota very economically. An alternative resummation scheme is suggested using ``center of mass'' based loops which is equivalent under the assumption of vanishing third and higher derivatives in the Taylor expansion but leads to simplified expressions. In an appendix some general issues concerning the relation between world-line integrals with fixed end points versus integrals with fixed center are clarified. We finally note that this method is also very valuable for gauge field effective actions where it is based on the Euler--Heisenberg type resummation. 
  Supersymmetry of the Wess-Zumino (N=1, D=4) multiplet allows field equations that determine a larger class of geometries than the familiar Kahler manifolds, in which covariantly holomorphic vectors rather than a scalar superpotential determine the forces. Indeed, relaxing the requirement that the field equations be derivable from an action leads to complex flat geometry. The Batalin-Vilkovisky formalism is used to show that if one requires that the field equations be derivable from an action, we once again recover the restriction to Kahler geometry, with forces derived from a scalar superpotential. 
  We present a phase-space analysis of cosmology containing multiple scalar fields with a positive or negative cross-coupling exponential potential. We show that there exist power-law kinetic-potential-scaling solutions for a sufficiently flat positive potential or for a steep negative potential. The former is the unique late-time attractor, but it is difficult to yield assisted inflation. The later is never stable in an expanding universe. Moreover, for a steep negative potential there exists a kinetic-dominated regime in which each solution is a late-time attractor. In the presence of ordinary matter these scaling solutions with a negative cross-coupling potential are found unstable. We briefly discuss the physical consequences of these results. 
  The Ward identities for amplitudes at the tree level are derived from symmetries of the corresponding classical dynamical systems. The results are applied to some 2 into n amplitudes. 
  We introduce a finite dimensional matrix model approximation to the algebra of functions on a disc based on noncommutative geometry. The algebra is a subalgebra of the one characterizing the noncommutative plane with a * product and depends on two parameters N and theta. It is composed of functions which decay exponentially outside a disc. In the limit in which the size of the matrices goes to infinity and the noncommutativity parameter goes to zero the disc becomes sharper. We introduce a Laplacian defined on the whole algebra and calculate its eigenvalues. We also calculate the two--points correlation function for a free massless theory (Green's function). In both cases the agreement with the exact result on the disc is very good already for relatively small matrices. This opens up the possibility for the study of field theories on the disc with nonperturbative methods. The model contains edge states, a fact studied in a similar matrix model independently introduced by Balachandran, Gupta and Kurkcuoglu. 
  In this note we study the puzzle posed by two M5-branes intersecting on a string (or equivalently, a single M5-brane wrapping a holomorphic four-cycle in C^4). It has been known for a while that this system is different from all other configurations built using self-intersecting M-branes; in particular the corresponding supergravity solution exhibits various curious features which have remained unexplained. We propose that the resolution to these puzzles lies in the existence of a non-zero two-form on the M5-brane world-volume. 
  We propose an heuristic rule for the area transformation on the non-commutative plane. The non-commutative area preserving transformations are quantum deformation of the classical symplectic diffeomorphisms. Area preservation condition is formulated as a field equation in the non-commutative Chern-Simons gauge theory. The higher dimensional generalization is suggested and the corresponding algebraic structure - the infinite dimensional $\sin$-Lie algebra is extracted. As an illustrative example the second-quantized formulation for electrons in the lowest Landau level is considered. 
  We study "fuzzy funnel" solutions to the non-Abelian equations of motion of the D-string. Our funnel describes n^6/360 coincident D-strings ending on n^3/6 D7-branes, in terms of a fuzzy six-sphere which expands along the string. We also provide a dual description of this configuration in terms of the world volume theory of the D7-branes. Our work makes use of an interesting non-linear higher dimensional generalization of the instanton equations. 
  By using graded (super) Lie algebras, we can construct noncommutative superspace on curved homogeneous manifolds. In this paper, we take a flat limit to obtain flat noncommutative superspace. We particularly consider $d=2$ and $d=4$ superspaces based on the graded Lie algebras $osp(1|2)$, $su(2|1)$ and $psu(2|2)$. Jacobi identities of supersymmetry algebras and associativities of star products are automatically satisfied. Covariant derivatives which commute with supersymmetry generators are obtained and chiral constraints can be imposed. We also discuss that these noncommutative superspaces can be understood as constrained systems analogous to the lowest Landau level system. 
  We study three types of star products in SFT: the ghosts, the twisted ghosts and the matter. We find that their Neumann coefficients are related to each other in a compact way which includes the Gross-Jevicki relation between matter and ghost sector: we explicitly show that the same relation, with a minus sign, holds for the twisted and nontwisted ghost (which are different but define the same solution). In agreement with this, we prove that matter and twisted ghost coefficients just differ by a minus sign. As a consistency check, we also compute the spectrum of the twisted ghost vertices from conformal field theory and, using equality of twisted and reduced slivers, we derive the spectrum of the non twisted ghost star. 
  In this review, we summarize recent findings that show how standard 4-d Einstein gravity coupled to a conformal field theory can become massive in Anti de Sitter Space. Key ingredients in this phenomenon are non-standard ``transparent'' boundary condition given to the CFT fields and the fact that AdS space is not globally hyperbolic, due to the presence of a time-like boundary. 
  We construct Witten-type string field theory vertices for a fermionic first order system with conformal weights (0,1) in the operator formulation using delta-function overlap conditions as well as the Neumann function method. The identity, the reflector and the interaction vertex are treated in detail paying attention to the zero mode conditions and the U(1) charge anomaly. The Neumann coefficients for the interaction vertex are shown to be intimately connected with the coefficients for bosons allowing a simple proof that the reparametrization anomaly of the fermionic first order system cancels the contribution of two real bosons. This agrees with their contribution c=-2 to the central charge. The overlap equations for the interaction vertex are shown to hold. Our results have applications in N=2 string field theory, Berkovits' hybrid formalism for superstring field theory, the \eta\xi-system and the twisted bc-system used in bosonic vacuum string field theory. 
  This paper discusses the formulation of the non-commutative Chern-Simons (CS) theory where the spatial slice, an infinite strip, is a manifold with boundaries. As standard star products are not correct for such manifolds, the standard non-commutative CS theory is not also appropriate here. Instead we formulate a new finite-dimensional matrix CS model as an approximation to the CS theory on the strip. A work which has points of contact with ours is due to Lizzi, Vitale and Zampini where the authors obtain a description for the fuzzy disc. The gauge fields in our approach are operators supported on a subspace of finite dimension N+\eta of the Hilbert space of eigenstates of a simple harmonic oscillator with N, \eta \in Z^+ and N \neq 0. This oscillator is associated with the underlying Moyal plane. The resultant matrix CS theory has a fuzzy edge. It becomes the required sharp edge when N and \eta goes to infinity in a suitable sense. The non-commutative CS theory on the strip is defined by this limiting procedure. After performing the canonical constraint analysis of the matrix theory, we find that there are edge observables in the theory generating a Lie algebra with properties similar to that of a non-abelian Kac-Moody algebra. Our study shows that there are (\eta+1)^2 abelian charges (observables) given by the matrix elements (\cal A_i)_{N-1 N-1} and (\cal A_i)_{nm} (where n or m \geq N) of the gauge fields, that obey certain standard canonical commutation relations. In addition, the theory contains three unique non-abelian charges, localized near the N^th level. We show that all non-abelian edge observables except these three can be constructed from the abelian charges above. Using the results of this analysis we discuss the large N and \eta limit. 
  The Bose--Fermi recoupling of particles arising from the $\bZ_{2}$--grading of the irreducible representations of SU(2) is responsible for the Pauli exclusion principle. We demonstrate from fundamental physical assumptions how to extend this to gradings, other than the $\bZ_{2}$ grading, arising from other groups. This requires non--associative recouplings where phase factors arise due to {\it rebracketing} of states. In particular, we consider recouplings for the $\bZ_{3}$--grading of SU(3) colour and demonstrate that all the recouplings graded by triality leading to the Pauli exclusion principle demand quark state confinement. Note that quark state confinement asserts that only ensembles of triality zero are possible, as distinct from spatial confinement where particles are confined to a small region of space by a {\it confining force} such as given by the dynamics of QCD. 
  As a first step to a detailed study of orientifolds of Gepner models associated with Calabi-Yau manifolds, we construct crosscap states associated with anti-holomorphic involutions (with fixed points) of Calabi-Yau manifolds. We argue that these orientifolds are dual to M-theory compactifications on (singular) seven-manifolds with $G_2$ holonomy. Using the spacetime picture as well as the M-theory dual, we discuss aspects of the orientifold that should be obtained in the Gepner model. This is illustrated for the case of the quintic. 
  Here we consider a gravitational action having local Poincare invariance which is given by the dimensional continuation of the Euler density in ten dimensions. It is shown that the local supersymmetric extension of this action requires the algebra to be the maximal extension of the N=1 super-Poincare algebra. The resulting action is shown to describe a gauge theory for the M-algebra, and is not the eleven-dimensional supergravity theory of Cremmer-Julia-Scherk. The theory admits a class of vacuum solutions of the form S^{10-d} x Y_{d+1}, where Y_{d+1} is a warped product of R with a d-dimensional spacetime. It is shown that a nontrivial propagator for the graviton exists only for d=4 and positive cosmological constant. Perturbations of the metric around this solution reproduce linearized General Relativity around four-dimensional de Sitter spacetime. 
  For certain class of triangles (with angles proportional to $\fr{\pi}{N}$, $N\geq 3$) we formulate image method by making use of the group $G_N$ generated by reflections with respect to the three lines which form the triangle under consideration. We formulate the renormalization procedure by classification of subgroups of $G_N$ and corresponding fixed points in the triangle. We also calculate Casimir energy for such geometries, for scalar massless fields. More detailed calculation is given for odd $N$. 
  Recently a proposal for the non-abelian effective D-brane action was given through fourth order in alpha'. As the resulting expressions turned out to be quite involved, some checks of this result are called for. In the present paper we calculate the spectrum in the presence of constant magnetic background fields and compare it to the string theoretical result. Apart from a small typo in the original expression (the overall sign of thefourth order term), we obtain perfect agreement. We discuss potential applications. 
  This paper explores the use of a deformation by a root of unity as a tool to build models with a finite number of states for applications to quantum gravity. The initial motivation for this work was cosmological breaking of supersymmetry. We explain why the project was unsuccessful. What is left are some observations on supersymmetry for q-bosons, an analogy between black holes in de Sitter and properties of quantum groups, and an observation on a noncommutative quantum mechanics model with two degrees of freedom, depending on one parameter. When this parameter is positive, the spectrum has a finite number of states; when it is negative or zero, the spectrum has an infinite number of states. This exhibits a desirable feature of quantum physics in de Sitter space, albeit in a very simple, non-gravitational context. 
  We conjecture a time dependent Lambda(t), in terms of the Gaussian curvature of the causal horizon, which is nonvanishing even in Minkowski space due to the lack of informations beyond the light cone. Using the Heisenberg Principle, the corresponding energy of the quantum fluctuations, localized on the past or future null horizon, is proportional to sqrt(Lambda(t)). We compute Lambda(t) for the (conformally flat) Lorentzian Hawking wormhole geometry, written in static spherical Rindler coordinates. 
  The Bogomolny equations for Yang-Mills-Higgs monopoles follow from a system of linear equations which may be solved through a parametric Riemann-Hilbert problem. We extend this approach to noncommutative R^3 and use it to (re)construct noncommutative Dirac, Wu-Yang, and BPS monopole configurations in a unified manner. In all cases we write down the underlying matrix-valued functions for multi-monopoles and solve the corresponding Riemann-Hilbert problems for charge one. 
  We prove that all SYM theories that have a quantum modified moduli space $\m$ defined by a single constraint equation have trivial homotopy groups $\pi_j(\m)$ for $j=0,1,2,3$ and 4. This implies that none of these theories admit skyrmions or vortexes, a fact that had only been proved for supersymmetric QCD with $N_f=N_c$ and $Sp(2n)$ with $2n+2$ fundamentals, whereas those of them with a nontrivial $H^5 (\m)$ admit Wess-Zumino-Witten terms in their effective actions. Contrary to expectations, examples of quantum modified moduli spaces with a trivial $H^5 (\m)$ are found in the literature. 
  We study issues of duality and dual equivalence in non-commutative manifolds. In particular the question of dual equivalence for the actions of the non-commutative extensions of the self-dual model (NC-SD) in 3D space-time and the Maxwell-Chern-Simons model (MCS-SD) is investigate. We show that former model {\it is not} dual equivalent the non-commutative extension of the Maxwell-Chern-Simons model, as widely believed, but a to deformed version of it that is disclosed here. Our results are not restrict to any finite order in the Seiberg-Witten expansion involving the non-commutative parameter $\theta$. 
  In this note we demonstrate that vortices in a non-relativistic Chern-Simons theory form a quantum Hall fluid. We show that the vortex dynamics is controlled by the matrix mechanics previously proposed by Polychronakos as a description of the quantum Hall droplet. As the number of vortices becomes large, they fill the plane and a hydrodynamic treatment becomes possible, resulting in the non-commutative theory of Susskind. Key to the story is the recent D-brane realisation of vortices and their moduli spaces. 
  We propose a simple form for the superalgebra of M2 and M5-brane probes in arbitrary supersymmetric backgrounds of 11D supergravity, extending previous results in the literature. In particular, we identify the topological charges in the algebras and find BPS bounds for the energies. The charges are given by the integral over a brane's spatial worldvolume of a certain closed form built out of the Killing spinors and background fields. The existence of such closed forms for arbitrary supersymmetric backgrounds generalises the existence of calibration forms for special holonomy manifolds. 
  We consider the Hamiltonian BRST quantization of a noncommutative non abelian gauge theory. The Seiberg-Witten map of all phase-space variables, including multipliers, ghosts and their momenta, is given in first order in the noncommutative parameter $\theta$. We show that there exists a complete consistence between the gauge structures of the original and of the mapped theories, derived in a canonical way, once we appropriately choose the map solutions. 
  A possible way of defining M theory as the CS theory for the supergroup $OSp(1|32)\times OSp(1|32)$ is investigated, based on the approach by Horava in hep-th/9712130. In the high energy limit (expansion in M), where only the highest ($R^5$) terms survive in the action, the supergroup contracts to the D'Auria-Fre M theory supergroup. Then the contracted equations of motion are solved by the usual 11d supergravity equations of motion, linearized in everything but the vielbein. These two facts suggest that the whole nonlinear 11d sugra should be obtainable somehow in the contraction limit. Type IIB also arises as a contraction of the $OSp(1|32)\times OSp(1|32)$ theory. The presence of a cosmological constant in 11d constraints the parameter M experimentally to be of the order of the inverse horizon size, $1/L_0$. Then the 11d Planck mass $M_{P,11}\sim 10GeV$ (hopefully higher: $>TeV$ due to uncertainties). Unfortunately, the most naive attempt at cosmological implications for the theory is excluded experimentally. Interestingly, the low energy expansion (high M) of the CS theory, truncated to the gravitational sector, gives much better phenomenology. 
  The behaviour of matrix string theory in the background of a type IIA pp wave at small string coupling, g_s << 1, is determined by the combination M g_s where M is a dimensionless parameter proportional to the strength of the Ramond-Ramond background. For M g_s << 1, the matrix string theory is conventional; only the degrees of freedom in the Cartan subalgebra contribute, and the theory reduces to copies of the perturbative string. For M g_s >> 1, the theory admits degenerate vacua representing fundamental strings blown up into fuzzy spheres with nonzero lightcone momenta. We determine the spectrum of small fluctuations around these vacua. Around such a vacuum all N-squared degrees of freedom are excited with comparable energies. The spectrum of masses has a spacing which is independent of the radius of the fuzzy sphere, in agreement with expected behaviour of continuum giant gravitons. Furthermore, for fuzzy spheres characterized by reducible representations of SU(2) and vanishing Wilson lines, the boundary conditions on the field are characterized by a set of continuous angles which shows that generically the blown up strings do not ``close''. 
  We rewrite the Born-Infeld Lagrangian, which is originally given by the determinant of a $4 \times 4$ matrix composed of the metric tensor $g$ and the field strength tensor $F$, using the determinant of a $(4 \cdot 2^n) \times (4 \cdot 2^n)$ matrix $H_{4 \cdot 2^{n}}$. If the elements of $H_{4 \cdot 2^{n}}$ are given by the linear combination of $g$ and $F$, it is found, based on the representation matrix for the multiplication operator of the Cayley-Dickson algebras, that $H_{4 \cdot 2^{n}}$ is distinguished by a single parameter, where distinguished matrices are not similar matrices. We also give a reasonable condition to fix the paramete 
  We generalise the electric-magnetic duality in standard Maxwell theory to its non-commutative version. Both space-space and space-time non-commutativity are necessary. The duality symmetry is then extended to a general class of non-commutative gauge theories that goes beyond non-commutative electrodynamics. As an application of this symmetry, plane wave solutions are analysed. Dispersion relations following from these solutions show that general non-commutative gauge theories other than electrodynamics admits two waves with distinct polarisations propagating at different velocities in the same direction. 
  In this paper, I develop the Soldering formalism in a new domain - the noncommutative planar field theories. The Soldering mechanism fuses two distinct theories showing opposite or complimentary properties of some symmetry, taking into account the interference effects. The above mentioned symmetry is hidden in the composite (or soldered) theory. In the present work it is shown that a pair of noncommutative Maxwell-Chern-Simons theories, having opposite signs in their respective topological terms, can be consistently soldered to yield the Proca model (Maxwell theory with a mass term) with corrections that are at least quadratic in the noncommutativity parameter. We further argue that this model can be thought of as the noncommutative generalization of the Proca theory of ordinary spacetime. It is well-known that abelian noncommutative gauge theory bears a close structural similarity with non-abelian gauge theory. This fact is manifested in a non-trivial way if the present work is compared with existing literature, where soldering of non-abelian models are discussed. Thus the present work further establishes the robustness of the soldering programme. The subtle role played by gauge invariance, (or the lack of it), in the above soldering process, is revealed in an interesting way. 
  We find a nonsemisimple fusion algebra F_p associated with each (1,p) Virasoro model. We present a nonsemisimple generalization of the Verlinde formula which allows us to derive F_p from modular transformations of characters. 
  We analyze the group-theoretical ramifications of the Nambu-Goldstone [NG] theorem in the self-consistent relativistic variational Gaussian wave functional approximation to spinless field theories. In an illustrative example we show how the Nambu-Goldstone theorem would work in the O(N) symmetric $\phi^4$ scalar field theory, if the residual symmetry of the vacuum were lesser than O(N-1), e.g. if the vacuum were O(N-2), or O(N-3),... symmetric. [This does not imply that any of the "lesser" vacua is actually the absolute energy minimum: stability analysis has not been done.] The requisite number of NG bosons would be (2N - 3), or (3N - 6), ... respectively, which may exceed N, the number of elementary fields in the Lagrangian. We show how the requisite new NG bosons would appear even in channels that do not carry the same quantum numbers as one of N "elementary particles" (scalar field quanta, or Castillejo-Dalitz-Dyson [CDD] poles) in the Lagrangian, i.e. in those "flavour" channels that have no CDD poles. The corresponding Nambu-Goldstone bosons are composites (bound states) of pairs of massive elementary (CDD) scalar fields excitations. As a nontrivial example of this method we apply it to the physically more interesting 't Hooft $\sigma$ model (an extended $N_{f} = 2$ bosonic linear $\sigma$ model with four scalar and four pseudoscalar fields), with spontaneously and explicitly broken chiral $O(4) \times O(2) \simeq SU_{\rm R} (2) \times SU_{\rm L}(2) \times U_{\rm A}(1)$ symmetry. 
  Maximal and non-maximal supergravities in three dimensions allow for a large variety of semisimple (Chern-Simons) gauge groups. In this paper, we analyze non-semisimple and complex gauge groups that satisfy the pertinent consistency relations for a maximal (N=16) gauged supergravity to exist. We give a general procedure how to generate non-semisimple gauge groups from known admissible semisimple gauge groups by a singular boost within E_{8(8)}. Examples include the theories with gauge group SO(8) x T_{28} that describe the reduction of IIA/IIB supergravity on the seven-sphere. In addition, we exhibit two `strange embeddings' of the complex gauge group SO(8,C) into (real) E_{8(8)} and prove that both can be consistently gauged. We discuss the structure of the associated scalar potentials as well as their relation to those of D>3 gauged supergravities. 
  In order to establish an explicit connection between four-dimensional Hall effect on $S^4$ and six-dimensional Hall effect on $\DC P^3$, we perform the Hamiltonian reduction of a particle moving on $\DC P^3$ in a constant magnetic field to the four-dimensional Hall mechanics (i.e. a particle on $S^4$ in a SU(2) instanton field). This reduction corresponds to fixing the isospin of the latter system. 
  We classify maximally supersymmetric backgrounds (vacua) of chiral (1,0) and (2,0) supergravities in six dimensions and, by reduction, also those of the minimal N=2 supergravity in five dimensions. Up to R-symmetry, the (2,0) vacua are in one-to-one correspondence with (1,0) vacua, and these in turn are locally isometric to Lie groups admitting a bi-invariant lorentzian metric with anti-selfdual parallelising torsion, which we classify. We then show that the five-dimensional vacua are homogeneous spaces arising canonically as the spaces of right cosets of spacelike one-parameter subgroups. 
  We present a string inspired 3D Euclidean field theory as the starting point for a modified Ricci flow analysis of the Thurston conjecture. In addition to the metric, the theory contains a dilaton, an antisymmetric tensor field and a Maxwell-Chern Simons field. For constant dilaton, the theory appears to obey a Birkhoff theorem which allows only nine possible classes of solutions, depending on the signs of the parameters in the action. Eight of these correspond to the eight Thurston geometries, while the ninth describes the metric of a squashed three sphere. It therefore appears that one can construct modified Ricci flow equations in which the topology of the geometry is encoded in the parameters of an underlying field theory. 
  In this paper, we analyse the Einstein and Einstein-Maxwell billiards for all spatially homogeneous cosmological models corresponding to 3 and 4 dimensional real unimodular Lie algebras and provide the list of those models which are chaotic in the Belinskii, Khalatnikov and Lifschitz (BKL) limit. Through the billiard picture, we confirm that, in D=5 spacetime dimensions, chaos is present if off-diagonal metric elements are kept: the finite volume billiards can be identified with the fundamental Weyl chambers of hyperbolic Kac-Moody algebras. The most generic cases bring in the same algebras as in the inhomogeneous case, but other algebras appear through special initial conditions. 
  We investigate the emission of gravitons by a cosmological brane into an Anti de Sitter five-dimensional bulk spacetime. We focus on the distribution of gravitons in the bulk and the associated production of `dark radiation' in this process. In order to evaluate precisely the amount of dark radiation in the late low-energy regime, corresponding to standard cosmology, we study numerically the emission, propagation and bouncing off the brane of bulk gravitons. 
  We work out boundary states for Type IIA string theory on a plane wave background. By directly utilizing the channel duality, the induced conditions from the open string boundary conditions are imposed on the boundary states. The resulting boundary states correctly reproduce the partition functions of the open string theory for $DpDp$ and $DpD\bar{p}$ cases where $Dp$ branes are half BPS brane if located at the origin of the plane wave background. 
  Three-manifolds can be obtained through surgery of framed links in $S^3$. We study the meaning of surgery procedures in the context of topological strings. We obtain U(N) three-manifold invariants from U(N) framed link invariants in Chern-Simons theory on $S^3$. These three-manifold invariants are proportional to the trivial connection contribution to the Chern-Simons partition function on the respective three-manifolds. Using the topological string duality conjecture, we show that the large $N$ expansion of U(N) Chern-Simons free energies on three-manifolds, obtained from some class of framed links, have a closed string expansion. These expansions resemble the closed string $A$-model partition functions on Calabi-Yau manifolds with one Kahler parameter. We also determine Gopakumar-Vafa integer coefficients and Gromov-Witten rational coefficients corresponding to Chern-Simons free energies on some three-manifolds. 
  We study massless and massive graviton modes that bind on thick branes which are supergravity domain walls solutions in $D$-dimensional supergravity theories where only the supergravity multiplet and the scalar supermultiplet are turned on. The domain walls are bulk solutions provided by tachyon potentials. Such domain walls are regarded as BPS branes of one lower dimension that are formed due to tachyon potentials on a non-BPS D-brane. 
  We review our recent works on the supersymmetrization of the leading string correction (the R^4 term) to N=1,2 supergravity theories in four dimensions. We show that, in the "old minimal" formulations of these theories, when going on-shell in the presence of this correction, the auxiliary fields which come from multiplets with physical fields cannot be eliminated, but those ones that come from compensating multiplets without any physical fields can be eliminated. We conjecture similar results for other versions of these theories. 
  30-component, of the first order, equation for a spin 2 particle, equivalent to the second order Pauli-Fierz one, is generalized to presence of an external electromagnetic field as well as a curved background space-time geometry. The essential property of the generally covariant wave equation obtained is that here from the very beginning, in accordance with requirement of the Pauli-Fierz approach, a set of additional relations on 30-component wave function for eliminating complementary spin 0 and spin 1 fields is present at the starting equation. 
  We show that the global aspects of Abelian and center projection of a SU(2) gauge theory on an arbitrary manifold are naturally described in terms of smooth Deligne cohomology. This is achieved through the introduction of a novel type of differential topological structure, called Cho structure. Half integral monopole charges appear naturally in this framework. 
  Using superspace projection operators we provide a classification of (3/2,2) off-shell supermultiplets which are realized in terms of a real axial vector superfield, with or without compensating superfields. Any linearized supergravity action is shown to be a superposition of those corresponding to (i) old minimal supergravity, (ii) new minimal supergravity and (iii) the novel (3/2,2) off-shell supermultiplet with 12+12 degrees of freedom obtained in hep-th/0201096. 
  In this paper we obtain the space-time generated by a time-like current-carrying superconducting screwed cosmic string(TCSCS). This gravitational field is obtained in a modified scalar-tensor theory in the sense that torsion is taken into account. We show that this solution is comptible with a torsion field generated by the scalar field $\phi $. The analysis of gravitational effects of a TCSCS shows up that the torsion effects that appear in the physical frame of Jordan-Fierz can be described in a geometric form given by contorsion term plus a symmetric part which contains the scalar gradient. As an important application of this solution, we consider the linear perturbation method developed by Zel'dovich, investigate the accretion of cold dark matter due to the formation of wakes when a TCSCS moves with speed $v$ and discuss the role played by torsion. Our results are compared with those obtained for cosmic strings in the framework of scalar-tensor theories without taking torsion into account. 
  We formulate noncommutative self-dual N=4 supersymmetric Yang-Mills theory in D=2+2 dimensions. As in the corresponding commutative case, this theory can serve as the possible master theory of all the noncommutative supersymmetric integrable models in lower dimensions. As a by-product, noncommutative self-dual N=2 supersymmetric Yang-Mills theory is obtained in D=2+2. We also perform a dimensional reduction of the N=2 theory further into N=(2,2) in D=1+1, as a basis for more general future applications. As a typical example, we show how noncommutative integrable matrix N=(1,0) supersymmetric KdV equations in D=1+1 arise from this theory, via the Yang-Mills gauge groups GL(n, R) or SL(2n, R). 
  We present a systematic study of accelerating cosmologies obtained from M/string theory compactifications of hyperbolic spaces with time-varying volume. A set of vacuum solutions where the internal space is a product of hyperbolic manifolds is found to give qualitatively the same accelerating four-dimensional FLRW universe behavior as a single hyperbolic space. We also examine the possibility that our universe is a hyperbolic space and provide exact Milne type solutions, as well as intersecting S-brane solutions. When both the usual 4D spacetime and the m-dimensional internal space are hyperbolic, we find eternally accelerating cosmologies for $m\geq 7$, with and without form field backgrounds. In particular, the effective potential for a magnetic field background in the large 3 dimensions is positive definite with a local minimum and thus enhances the eternally accelerating expansion. 
  In this paper we extend our recent results (hep-th/0304067) on the first order formulation for the massless mixed symmetry tensor fields to the case of massive fields both in Minkowski as well as in (Anti) de Sitter spaces (including all possible massless and partially massless limits). Main physical results are essentially the same as in hep-th/0211233. 
  The generic rotating BTZ black hole, obtained by identifications in AdS3 space through a discrete subgroup of its isometry group, is investigated within a Lie theoretical context. This space is found to admit a foliation by two-dimensional leaves, orbits of a two-parameter subgroup of SL(2,R) and invariant under the BTZ identification subgroup. A global expression for the metric is derived, allowing a better understanding of the causal structure of the black hole. 
  We demonstrate how Sen's singular kink solution of the Born-Infeld tachyon action can be constructed by taking the appropriate limit of initially regular profiles. It is shown that the order in which different limits are taken plays an important role in determining whether or not such a solution is obtained for a wide class of potentials. Indeed, by introducing a small parameter into the action, we are able circumvent the results of a recent paper which derived two conditions on the asymptotic tachyon potential such that the singular kink could be recovered in the large amplitude limit of periodic solutions. We show that this is explained by the non-commuting nature of two limits, and that Sen's solution is recovered if the order of the limits is chosen appropriately. 
  In this letter we construct the kink D1-brane super D-helix solution and its T-dual the D2-brane supertube using the effective action of non-BPS tachyonic D-branes . In the limit of zero angular momentum, both types of solutions collapse to zero radius, giving rise respectively to a degenerate string configuration corresponding to a particle travelling with the speed of light and to a static straight string configuration. These solutions share all the properties of fundamental strings and do not have the pathological behavior of other solutions previously found in this context. A short discussion on the ``generalized gauge transformations'' suggested by Sen is used to justify the validity of our approach. 
  We study the lowest-order modifications of the static potential for Born-Infeld electrodynamics and for the $\theta$-expanded version of the noncommutative U(1) gauge theory, within the framework of the gauge-invariant but path-dependent variables formalism. The calculation shows a long-range correction ($1/r^5$-type) to the Coulomb potential in Born-Infeld electrodynamics. However, the Coulomb nature of the potential (to order $e^2$) is preserved in noncommutative electrodynamics. 
  We propose a novel infinite-volume brane world scenario where we live on a non-inflating spherical 3-brane, whose radius is somewhat larger than the present Hubble size, embedded in higher dimensional bulk. Once we include higher curvature terms in the bulk, we find completely smooth solutions with the property that the 3-brane world-volume is non-inflating for a continuous range of positive values of the brane tension, that is, without fine-tuning. In particular, our solution, which is a near-BPS background with supersymmetry broken on the brane around TeV, is controlled by a single integration constant. 
  We discuss aspects of the dictionary between brane configurations in del Pezzo geometries and dibaryons in the dual superconformal quiver gauge theories. The basis of fractional branes defining the quiver theory at the singularity has a K-theoretic dual exceptional collection of bundles which can be used to read off the spectrum of dibaryons in the weakly curved dual geometry. Our prescription identifies the R-charge R and all baryonic U(1) charges Q_I with divisors in the del Pezzo surface without any Weyl group ambiguity. As one application of the correspondence, we identify the cubic anomaly tr R Q_I Q_J as an intersection product for dibaryon charges in large-N superconformal gauge theories. Examples can be given for all del Pezzo surfaces using three- and four-block exceptional collections. Markov-type equations enforce consistency among anomaly equations for three-block collections. 
  We study the first non trivial gravitational corrections to the F-terms of ${\cal N}=1$ SYM SO/Sp gauge theories, with matter in some representations, by using the generalized Konishi anomaly method. We derive equations at genus one for the operators in the chiral ring and compare them with the loop equations of the corresponding matrix models, finding agreement. We find that for adjoint representation the genus 0 contributions to such corrections can be adsorbed by a field redefinition; remarkably, this is not the case for matter in the (anti-)symmetric representation of ($Sp$) $SO$. 
  We show that AdS black hole solutions admit an analytical continuation to become magnetic flux-branes. Although a BPS AdS black hole generally has a naked singularity, the BPS flux-brane can be regular everywhere with an appropriate choice of U(1)-charges. This flux-brane interpolates from AdS_{D-2} \times H^2 at small distance to an asymptotic AdS_D-type metric with an AdS_{D-2}\times S^1 boundary. We also obtain a smooth cosmological solution of de Sitter Einstein-Maxwell gravity which flows from dS_2\times S^{D-2} in the infinite past to a dS_D-type metric, with an S^{D-2}\times S^1 boundary, in the infinite future. 
  The problem of consistent Hamiltonian structure for O(N) nonlinear sigma model in the presence of five different types of boundary conditions is considered in detail. For the case of Neumann, Dirichlet and the mixture of these two types of boundaries, the consistent Poisson brackets are constructed explicitly, which may be used, e.g. for the construction of current algebras in the presence of boundary. While for the mixed boundary conditions and the mixture of mixed and Dirichlet boundary conditions, we prove that there is no consistent Poisson brackets, showing that the mixed boundary conditions are incompatible with all nontrivial subgroups of $O((N)$. 
  Gauge theory of gravity is formulated based on principle of local gauge invariance. Because the model has strict local gravitational gauge symmetry, gauge theory of gravity is a perturbatively renormalizable quantum model. However, in the original model, all gauge gravitons are massless. We want to ask that whether there exists massive gravitons in Nature? In this paper, we will propose a gauge model with massive gravitons. The mass term of gravitational gauge field is introduced into the theory without violating the strict local gravitational gauge symmetry. Massive gravitons can be considered to be possible origin of dark energy and dark matter in the Universe. 
  The dark energy equation of state for theories with either a discretuum or continuum distribution of vacua is investigated. In the discretuum case the equation of state is constant $w=p/\rho=-1$. The continuum case may be realized by an action with large wave function factor $Z$ for the dark energy modulus and generic potential. This form of the action is quantum mechanically stable and does not lead to measurable long range forces or violations of the equivalence principle. In addition, it has a special property which may be referred to as super-technical naturalness which results in a one-parameter family of predictions for the cosmological evolution of the dark energy equation of state as a function of redshift $w=w(z)$. The discretuum and continuum predictions will be tested by future high precision measurements of the expansion history of the universe. Application of large $Z$-moduli to a predictive theory of $Z$-inflation is also considered. 
  We examine an axion string coupled to a Majorana fermion. It is found that there exist a Majorana-Weyl zero mode on the string. Due to the zero mode, the axion strings obey non-abelian statistics. 
  We construct supersymmetric deformations of general, locally supersymmetric, nonlinear sigma models in three spacetime dimensions, by extending the pure supergravity theory with a Chern-Simons term and gauging a subgroup of the sigma model isometries, possibly augmented with R-symmetry transformations. This class of models is shown to include theories with standard Yang-Mills Lagrangians, with optional moment interactions and topological mass terms. The results constitute a general classification of three-dimensional gauged supergravities. 
  We evaluate the effective actions of supersymmetric matrix models on fuzzy S^2\times S^2 up to the two loop level. Remarkably it turns out to be a consistent solution of IIB matrix model. Based on the power counting and SUSY cancellation arguments, we can identify the 't Hooft coupling and large N scaling behavior of the effective actions to all orders. In the large N limit, the quantum corrections survive except in 2 dimensional limits. They are O(N) and O(N^{4\over 3}) for 4 and 6 dimensional spaces respectively. We argue that quantum effects single out 4 dimensionality among fuzzy homogeneous spaces. 
  A world-volume model of non-critical 3-brane is quantized in a strong coupling phase where fluctuations of the conformal mode become dominant. This phase, called the conformal-mode dominant phase, is realized at the very high energy far beyond the Planck mass scale. We separately treat the conformal mode and the traceless mode and quantize the conformal mode "non-perturbatively", while the traceless mode is treated in the perturbation which is renormalizable and asymptotically free. In the conformal-mode dominant phase, the coupling of the traceless mode vanishes and the world-volume dynamics is described as a four dimensional conformal field theory (CFT_4). We canonically quantize this model on R*S^3 where the dynamical metric fields are expanded using spherical tensor harmonics on S^3. Conformal charges and conformal algebra are constructed. They give strong constraints on physical states. We find that all negative-metric modes are related to positive-metric modes through the charges so that they are not independent physical modes themselves. An infinite number of physical states satisfying the conformal invariance conditions are constructed. In appendix, we construct spherical vector, tensor harmonics on S^3 in the useable form using Wigner D-functions and Clebsch-Gordan coefficients and calculate integrals of their products. 
  We derive thermodynamic Bethe ansatz equations describing the vacuum energy of the SU(2N)/Sp(N) nonlinear sigma model on a cylinder geometry. The starting points are the recently-proposed amplitudes for the scattering among the physical, massive excitations of the theory. The analysis fully confirms the correctness of the S-matrix. We also derive closed sets of functional relations for the pseudoenergies (Y-systems). These relations are shown to be the k-->infinity limit of the Sp(k+1)-related systems studied some years ago by Kuniba and Nakanishi in the framework of lattice models. 
  We derive an exact quantum equation of motion for the photon Wigner operator in non-commutative QED, which is gauge covariant. In the classical approximation, this reduces to a simple transport equation which describes the hard thermal effects in this theory. As an example of the effectiveness of this method we show that, to leading order, this equation generates in a direct way the Green amplitudes calculated perturbatively in quantum field theory at high temperature. 
  We study the propagating gravitational waves as a tool to probe the extra dimensions. In the set-up with one compact extra dimension and non-gravitational physics resigning on the 4-dimensional subspace (brane) of 5-dimensional spacetime we find the Green's function describing the propagation of 5-dimensional signal along the brane. The Green's function has a form of the sum of contributions from large number of images due to the compactness of the fifth dimension. Additionally, a peculiar feature of the causal wave propagation in five dimensions (making a five-dimensional spacetime very much different from the familiar four-dimensional case) is that the entire region inside the past light-cone contributes to the signal at the observation point. The 4-dimensional propagation law is nevertheless reproduced at large (compared to the size of extra dimension) intervals from the source as a superposition of signals from large number of images. The fifth dimension however shows up in the form of corrections to the purely 4-dimensional picture. We find three interesting effects: a tail effect for a signal of finite duration, screening at the forefront of this signal and a frequency-dependent amplification for a periodic signal. We discuss implications of these effects in the gravitational wave astronomy and estimate the sensitivity of gravitational antenna needed for detecting the extra dimension. 
  Some of the peculiar electrodynamical effects associated with gauged ``dimension bubbles'' are presented. Such bubbles, which effectively enclose a region of 5d spacetime, can arise from a 5d theory with a compact extra dimension. Bubbles with thin domain walls can be stabilized against total collapse by the entrapment of light charged scalar bosons inside the bubble, extending the idea of a neutral dimension bubble to accommodate the case of a gauged U(1) symmetry. Using a dielectric approach to the 4d dilaton-Maxwell theory, it is seen that the bubble wall is almost totally opaque to photons, leading to a new stabilization mechanism due to trapped photons. Photon dominated bubbles very slowly shrink, resulting in a temperature increase inside the bubble. At some critical temperature, however, these bubbles explode, with a release of radiation. 
  Motivated by recent suggestions that highly damped black hole quasinormal modes (QNM's) may provide a link between classical general relativity and quantum gravity, we present an extensive computation of highly damped QNM's of Kerr black holes. We do not limit our attention to gravitational modes, thus filling some gaps in the existing literature. The frequency of gravitational modes with l=m=2 tends to \omega_R=2 \Omega, \Omega being the angular velocity of the black hole horizon. If Hod's conjecture is valid, this asymptotic behaviour is related to reversible black hole transformations. Other highly damped modes with m>0 that we computed do not show a similar behaviour. The real part of modes with l=2 and m<0 seems to asymptotically approach a constant value \omega_R\simeq -m\varpi, \varpi\simeq 0.12 being (almost) independent of a. For any perturbing field, trajectories in the complex plane of QNM's with m=0 show a spiralling behaviour, similar to the one observed for Reissner-Nordstrom (RN) black holes. Finally, for any perturbing field, the asymptotic separation in the imaginary part of consecutive modes with m>0 is given by 2\pi T_H (T_H being the black hole temperature). We conjecture that for all values of l and m>0 there is an infinity of modes tending to the critical frequency for superradiance (\omega_R=m) in the extremal limit. Finally, we study in some detail modes branching off the so--called ``algebraically special frequency'' of Schwarzschild black holes. For the first time we find numerically that QNM multiplets emerge from the algebraically special Schwarzschild modes, confirming a recent speculation. 
  I examine the effect of trying to impose a Dirichlet boundary condition on a scalar field by coupling it to a static background. The zero point -- or Casimir -- energy of the field diverges in the limit that the background forces the field to vanish. This divergence cannot be absorbed into a renormalization of the parameters of the theory. As a result, the Casimir energy of a surface on which a Dirichlet boundary condition is imposed, and other quantities like the surface tension, which are obtained by deforming the surface, depend on the physical cutoffs that characterize the coupling between the field and the matter on the surface. In contrast, the energy density away from the surface and forces between rigid surfaces are finite and independent of these complications 
  We continue the analysis of hep-th/0303060 in the one-loop sector and present the complete psu(2,2|4) dilatation operator of N=4 Super Yang-Mills theory. This operator generates the matrix of one-loop anomalous dimensions for all local operators in the theory. Using an oscillator representation we show how to apply the dilatation generator to a generic state. By way of example, we determine the planar anomalous dimensions of all operators up to and including dimension 5.5, where we also find some evidence for integrability. Finally, we investigate a number of subsectors of N=4 SYM in which the dilatation operator simplifies. Among these we find the previously considered so(6) and su(2) subsectors, a su(2|4) subsector isomorphic to the BMN matrix model at one-loop, a u(2|3) supersymmetric subsector of nearly eighth-BPS states and, last but not least, a non-compact sl(2) subsector whose dilatation operator lifts uniquely to the full theory. 
  We show that deviations of the quantum state of the inflaton from the thermal vacuum of inflation may leave an imprint in the CMB anisotropies. The quantum dynamics of the inflaton in such a state produces corrections to the inflationary fluctuations, which may be observable. Because these effects originate from IR physics below the Planck scale, they will dominate over any trans-Planckian imprints in any theory which obeys decoupling. Inflation sweeps away these initial deviations and forces its quantum state closer to the thermal vacuum. We view this as the quantum version of the cosmic no-hair theorem. Such imprints in the CMB may be a useful, independent test of the duration of inflation, or of significant features in the inflaton potential about 60 e-folds before inflation ended, instead of an unlikely discovery of the signatures of quantum gravity. The absence of any such substructure would suggest that inflation lasted uninterrupted much longer than ${\cal O}(100)$ e-folds. 
  As a first application of our renormalisation group approach to non-local matrix models [hep-th/0305066], we prove (super-)renormalisability of Euclidean two-dimensional noncommutative \phi^4-theory. It is widely believed that this model is renormalisable in momentum space arguing that there would be logarithmic UV/IR-divergences only. Although momentum space Feynman graphs can indeed be computed to any loop order, the logarithmic UV/IR-divergence appears in the renormalised two-point function -- a hint that the renormalisation is not completed. In particular, it is impossible to define the squared mass as the value of the two-point function at vanishing momentum. In contrast, in our matrix approach the renormalised N-point functions are bounded everywhere and nevertheless rely on adjusting the mass only. We achieve this by introducing into the cut-off model a translation-invariance breaking regulator which is scaled to zero with the removal of the cut-off. The naive treatment without regulator would not lead to a renormalised theory. 
  It is proven that the classical pure spinor superstring in an AdS_5 x S^5 background has a flat current depending on a continuous parameter. This generalizes the recent result of Bena, et al. for the classical Green-Schwarz superstring. 
  We calculate the effective action for nonabelian gauge bosons up to quartic order using WZW-like open superstring field theory. After including level zero and level one contributions, we obtain with 75% accuracy the Yang-Mills quartic term. We then prove that the complete effective action reproduces the exact Yang-Mills quartic term by analytically performing a summation over the intermediate massive states. 
  We discuss k-strings in the large-N Yang-Mills theory and its supersymmetric extension. Whereas the tension of the bona fide (stable) QCD string is expected to depend only on the N-ality of the representation, tensions that depend on specific representation R are often reported in the lattice literature. In particular, adjoint strings are discussed and found in certain simulations. We clarify this issue by systematically exploiting the notion of the quasi-stable strings which becomes well-defined at large N. The quasi-stable strings with representation-dependent tensions decay, but the decay rate (per unit length per unit time) is suppressed as Lambda^2 F(N) where F(N) falls off as a function of N. It can be determined on the case-by-case basis. The quasi-stable strings eventually decay into stable strings whose tension indeed depends only on the N-ality.   We also briefly review large-N arguments showing why the Casimir formula for the string tension cannot be correct, and present additional arguments in favor of the sine formula. Finally, we comment on the relevance of our estimates to Euclidean lattice measurements. 
  Using the holographic machinery built up in a previous work, we show that the hidden SL(2,R) symmetry of a scalar quantum field propagating in a Rindler spacetime admits an enlargement in terms of a unitary positive-energy representation of Virasoro algebra, with central charge c=1, defined in the Fock representation. The Virasoro algebra of operators gets a manifest geometrical meaning if referring to the holographically associated QFT on the horizon: It is nothing but a representation of the algebra of vector fields defined on the horizon equipped with a point at infinity. All that happens provided the Virasoro ground energy h vanishes and, in that case, the Rindler Hamiltonian is associated with a certain Virasoro generator. If a suitable regularization procedure is employed, for h=1/2, the ground state of that generator corresponds to thermal states when examined in the Rindler wedge, taking the expectation value with respect to Rindler time. This state has inverse temperature 1/(2beta), where beta is the parameter used to define the initial SL(2,R) unitary representation. (As a consequence the restriction of Minkowski vacuum to Rindler wedge is obtained by fixing h=1/2 and 2beta=beta_U, the latter being Unruh's inverse temperature). Finally, under Wick rotation in Rindler time, the pair of QF theories which are built up on the future and past horizon defines a proper two-dimensional conformal quantum field theory on a cylinder. 
  We classify all supersymmetric solutions of minimal gauged supergravity in four dimensions. There are two classes of solutions that are distinguished by the norm of the Killing vector constructed from the Killing spinor. If the Killing vector is timelike, the solutions are determined by the geometry of a two-dimensional base-manifold. When it is lightlike, the most general BPS solution is given by an electrovac AdS travelling wave. This supersymmetric configuration was previously unknown. Generically the solutions preserve one quarter of the supersymmetry. Also in the timelike case we show that there exist new BPS solutions, which are of Petrov type I, and are thus more general than the previously known type D configurations. These geometries can be uplifted to obtain new solutions of eleven-dimensional supergravity. 
  Basic idea of Randall-Sundrum brane world model I and II is reviewed with detailed calculation. After introducing the brane world metric with exponential warp factor, metrics of Randall-Sundrum models are constructed. We explain how Randall-Sundrum model I with two branes makes the gauge hierarchy problem much milder, and then derive Newtonian gravity in Randall-Sundrum model II with a single brane by considering small fluctuations. 
  We consider theories containing gravity, at most one dilaton and form field strengths. We show that the existence of particular BPS solutions of intersecting extremal closed branes select the theories, which upon dimensional reduction to three dimensions possess a simple simply laced Lie group symmetry G. Furthermore these theories can be fully reconstructed from the dynamics of such branes and of their openings. Amongst such theories are the effective actions of the bosonic sector of M-theory and of the bosonic string. The BPS intersecting brane solutions form representations of a subgroup of the group of Weyl reflections and outer automorphisms of the triple Kac-Moody extension G+++ of the G algebra, which cannot be embedded in the overextended Kac-Moody subalgebra G++ characterising the cosmological Kasner solutions. 
  In (2+1) dimensions, the Maxwell term $-(1/4) F_{\alpha\beta}F^{\alpha\beta}$ can be replaced by the Chern-Simons three-form $(\kappa/4)\epsilon^{\alpha\beta\gamma}A_\alpha F_{\beta\gamma}$, yielding a novel type of `electromagnetism'. This has been proposed for studying the Quantum Hall Effect as well as High-Temperature Superconductivity. The gauge field can be coupled to a scalar field either relativistically or non-relativistically. In both cases, one admits finite-energy, vortex solutions. 
  We develop the relation between de Sitter holography and inflation in detail, with particular attention to cosmic density perturbations. We set up the Hamilton-Jacobi formalism to present a systematic treatment of the logarithmic corrections to a scale invariant spectrum. Our computations can be interpreted without reference to holography, as strong infra-red effects in gravity. This point of view may be relevant for the fine-tuning problems inherent to inflation. 
  I review aspects of string theory on plane wave backgrounds emphasising the connection to gauge theory given by the BMN correspondence. Topics covered include the Penrose limit and its role in deriving the BMN duality from AdS/CFT, light-cone string field theory in the maximally supersymmetric plane wave and extensions of the correspondence to less supersymmetric backgrounds. 
  We discuss a way of producing anisotropies in the spectrum of superheavy Dark matter, which are due to the distortion of the inflationary space time induced by the recoil of D-particles upon their scattering with ordinary string matter in the Early Universe. We calculate such distortions by world-sheet Liouville string theory (perturbative) methods. The resulting anisotropies are found to be proportional to the average recoil velocity and density of the D-particles. In our analysis we employ a regulated version of de Sitter space, allowing for graceful exit from inflation. This guarantees the asymptotic flatness of the space time, as required for a consistent interpretation, within an effective field theory context, of the associated Bogolubov coefficients as particle number densities. The latter are computed by standard WKB methods. 
  We investigate an inflationary model of the universe based on the assumption that space-time is noncommutative in the very early universe. We analyze the effects of space-time noncommutativity on the quantum fluctuations of an inflaton field and investigate their contributions to the cosmic microwave background (CMB). We show that the angular power spectrum l(l+1)C_l generically has a sharp damping for lower l if we assume that the last scattering surface is traced back to fuzzy spheres at the times when large-scale modes cross the Hubble horizon. 
  The conformal gravity is one of the most important models of quantum gravity with higher derivatives. We investigate the role of the Gauss-Bonnet term in this theory. The coincidence limit of the second coefficient of the Schwinger-DeWitt expansion is evaluated in an arbitrary dimension $n$. In the limit $n=4$ the Gauss-Bonnet term is topological and its contribution cancels. This cancellation provides an efficient test for the correctness of calculation and, simultaneously, clarifies the long-standing general problem concerning the role of the topological term in quantum gravity. For $n\neq 4$ the Gauss-Bonnet term becomes dynamical in the classical theory and relevant at the quantum level. In particular, the renormalization group equations in dimension $n=4-\epsilon$ manifest new fixed points due to quantum effects of this term. 
  If the present acceleration of the universe is due to a cosmological constant, \lambda, then the entropy of the microwave background is bounded. It cannot exceed \lambda^{-3/4} \sim 10^{91}, which is much less than the entropy of empty de Sitter space \lambda^{-1} \sim 10^{122}. This is due to the limited efficiency of storing entropy by local field theoretical degrees of freedom. The observed entropy of the microwave background is of O(10^{85}). 
  In a certain kinematic limit, where the effects of spacetime curvature (and other background fields) greatly simplify, the light-cone gauge world-sheet action for a type IIB superstring on AdS_5 x S^5 reduces to that of a free field theory. It has been conjectured by Berenstein, Maldacena, and Nastase that the energy spectrum of this string theory matches the dimensions of operators in the appropriately defined large R-charge large-N_c sector of N=4 supersymmetric Yang--Mills theory in four dimensions. This holographic equivalence is thought to be exact, independent of any simplifying kinematic limits. As a step toward verifying this larger conjecture, we have computed the complete set of first curvature corrections to the spectrum of light-cone gauge string theory that arises in the expansion of AdS_5 x S^5 about the plane-wave limit. The resulting spectrum has the complete dependence on lambda = g_YM^2 N_c; corresponding results in the gauge theory are known only to second order in lambda. We find precise agreement to this order, including the N=4 extended supermultiplet structure. In the process, we demonstrate that the complicated schemes put forward in recent years for defining the Green--Schwarz superstring action in background Ramond-Ramond fields can be reduced to a practical (and correct) method for quantizing the string. 
  The cubic interaction vertex of light-cone string field theory in the plane-wave background has a simple effective form when considering states with only bosonic excitations. This simple effective interaction vertex is used in this paper to calculate the three string interaction matrix elements for states of arbitrary bosonic excitation and these results are used to examine certain decay modes on the mass-shell. It is shown that the matrix elements of one string to two string decays involving only bosonic excitations will vanish to all orders in 1/mu on the mass-shell when the number of excitations on the initial string is less than or equal to two, but in general will not vanish when the number of excitations is greater than two. Also, a truncated calculation of the mass-shell matrix elements for one string to three string decays of two excitation states is performed and suggests that these matrix elements do not vanish on the mass-shell. There is, however, a quantitative discrepancy between this last result and its (also non-vanishing) gauge theory prediction from the BMN correspondence. 
  We calculate the thermal partition functions of open strings on the S-brane backgrounds (the bouncing or rolling tachyon backgrounds) both in the bosonic and superstring cases. According to hep-th/0302146, we consider the discretized temperatures compatible with the pure imaginary periodicity of tachyon profiles. The ``effective Hagedorn divergence'' is shown to appear no matter how low temperature is chosen (including zero-temperature). This feature is likely to be consistent with the large rate of open string pair production discussed in hep-th/0209090 and also emission of closed string massive modes hep-th/0303139. We also discuss the possibility to remove the divergence by considering the space-like linear dilaton backgrounds as in hep-th/0306132. 
  The many-fermion Lagrangian which includes the 't Hooft six-quark flavor mixing interaction (N_f=3) and the U_L(3)\times U_R(3) chiral symmetric four-quark Nambu -- Jona-Lasinio (NJL) type interactions is bosonized by the path integral method. The method of the steepest descents is used to derive the effective quark-mesonic Lagrangian with linearized many-fermion vertices. We obtain, additionally to the known lowest order stationary phase result of Reinhardt and Alkofer, the next to leading order (NLO) contribution arising from quantum fluctuations of auxiliary bosonic fields around their stationary phase trajectories (the Gaussian integral contribution). Using the gap equation we construct the effective potential, from which the structure of the vacuum can be settled. For some set of parameters the effective potential has several extrema, that in the case of SU(2)_I\times U(1)_Y flavor symmetry can be understood on topological grounds. With increasing strength of the fluctuations the spontaneously broken phase gets unstable and the trivial vacuum is restored. The effective potential reveals furthermore the existence of logarithmic singularities at certain field expectation values, signalizing caustic regions. 
  The O(3) Skyrme system in two space dimensions admits topological soliton solutions. This paper studies the transition between the high-density crystalline phase of such solitons and the low-density phase where there are multi-Skyrmions localized in space. The details depend crucially on the choice of the potential function. Two such choices are investigated: in the first system, multi-Skyrmions at low density form a ring; while in the second, there is an explicit crystal solution, and the preferred low-density configurations are chunks of this crystal. The second system is a particularly good analogue of three-dimensional Skyrmions. 
  The Standard Model data suggests that the quantum gravity vacuum should accommodate two pivotal ingredients. The existence of three chiral generations and their embedding in chiral 16 SO(10) representations. The Z2XZ2 orbifolds are examples of perturbative heterotic string vacua that yield these properties. The exploration of these models in the nonperturbative framework of M-theory is discussed. A common prediction of these contructions is the existence of super-heavy meta-stable states due to the Wilson-line breaking of the GUT symmetries. Cosmic ray experiments in the forthcoming years offer an exciting experimental window to the phenomenology of such states. 
  We describe the generalized kappa-deformations of D=4 relativistic symmetries with finite masslike deformation parameter kappa and an arbitrary direction in kappa-deformed Minkowski space being noncommutative. The corresponding bicovariant differential calculi on kappa-deformed Minkowski spaces are considered. Two distinguished cases are discussed: 5D noncommutative differential calculus (kappa-deformation in time-like or space-like direction), and 4D noncommutative differential calculus having the classical dimension (noncommutative kappa-deformation in light-like direction). We introduce also left and right vector fields acting on functions of noncommutative Minkowski coordinates, and describe the noncommutative differential realizations of kappa-deformed Poincare algebra. The kappa-deformed Klein-Gordon field on noncommutative Minkowski space with noncommutative time (standard kappa-deformation) as well as noncommutative null line (light-like kappa-deformation) are discussed. Following our earlier proposal (see {1,2]) we introduce an equivalent framework replacing the local noncommutative field theory by the nonlocal commutative description with suitable nonlocal star product multiplication rules. The modification of Pauli--Jordan commutator function is described and the kappa-dependence of its light-cone behaviour in coordinate space is explicitly given. The problem with the kappa-deformed energy- momentum conservation law is recalled. 
  Non commutative superspaces can be introduced as the Moyal-Weyl quantization of a Poisson bracket for classical superfields. Different deformations are studied corresponding to constant background fields in string theory. Supersymmetric and non supersymmetric deformations can be defined, depending on the differential operators used to define the Poisson bracket. Some examples of deformed, 4 dimensional lagrangians are given. For extended superspace (N>1), some new deformations can be defined, with no analogue in the N=1 case. 
  This paper studies various aspects of the world volume dynamics of the M-theory five-brane, including: non-BPS solutions and solution generating symmetries; the scattering properties of world volume solutions; and the equivalence with probe brane dynamics. 
  On the space of three-dimensional conformal field theories with U(1) symmetry and a chosen coupling to a background gauge field, there is a natural action of the group $SL(2,{\bf Z})$. The generator $S$ of $SL(2,{\bf Z})$ acts by letting the background gauge field become dynamical, an operation considered recently by Kapustin and Strassler in explaining three-dimensional mirror symmetry. The other generator $T$ acts by shifting the Chern-Simons coupling of the background field. This $SL(2,{\bf Z})$ action in three dimensions is related by the AdS/CFT correspondence to $SL(2,{\bf Z})$ duality of low energy U(1) gauge fields in four dimensions. 
  Recently it was established that the one-loop planar dilatation generator of N=4 Super Yang-Mills theory may be identified, in some restricted cases, with the Hamiltonians of various integrable quantum spin chains. In particular Minahan and Zarembo established that the restriction to scalar operators leads to an integrable vector so(6) chain, while recent work in QCD suggested restricting to twist operators, containing mostly covariant derivatives, yields certain integrable Heisenberg XXX chains with non-compact spin symmetry sl(2). Here we unify and generalize these insights and argue that the complete one-loop planar dilatation generator of N=4 is described by an integrable su(2,2|4) super spin chain. We also write down various forms of the associated Bethe ansatz equations, whose solutions are in one-to-one correspondence with the set of all one-loop planar anomalous dimensions in the N=4 gauge theory. We finally speculate on the non-perturbative extension of these integrable structures, which appears to involve non-local deformations of the conserved charges. 
  We investigate a new way of realizing a period of cosmological inflation in the context of brane gas cosmology. It is argued that a gas of co-dimension one branes, out of thermal equilibrium with the rest of the matter, has an equation of state which can - after stabilization of the dilaton - lead to power-law inflation of the bulk. The most promising implementation of this mechanism might be in Type IIB superstring theory, with inflation of the three large spatial dimensions triggered by ``stabilized embedded 2-branes''. Possible applications and problems with this proposal are discussed. 
  By considering the effects of string winding and momentum modes on a time dependent background, we find a method by which six compact dimensions become stabilized naturally at the self-dual radius while three dimensions grow large. 
  We investigate the relationship between the holographic temperature bound and the slow-roll inflation. For this purpose we introduce the holographic temperature bound for a radiation matter :$T \ge T_{\rm H}$. Here $T_{\rm H}$ is the Hubble temperature which arises from the cosmological holographic description of for a radiation-dominated universe. For the quasi-de Sitter phase of slow-roll inflation, we find that the holographic temperature bound of $T_{\rm GH} \ge T_{\rm H}$ is guaranteed with the Gibbons-Hawking temperature $T_{\rm GH}$. When $T_{\rm GH}= T_{\rm H}$, inflation ends. 
  Recently, a principle for state confinement has been proposed in a category theoretic framework and to accomodate this result the notion of a pre-monoidal category was developed. Here we describe an algebraic approach for the construction of such categories. We introduce a procedure called twining which breaks the quasi-bialgebra structure of the universal enveloping algebras of semi-simple Lie algebras and renders the category of finite-dimensional modules pre-monoidal. The category is also symmetric, meaning that each object of the category provides representations of the symmetric groups, which allows for a generalised boson-fermion statistic to be defined. Exclusion and confinement principles for systems of indistinguishable particles are formulated as an invariance with respect to the symmetric group actions. We apply the construction to several examples and in particular show that the symmetries which can be associated to colour, spin and flavour degrees of freedom of quarks do lead naturally to confinement of states. 
  We show that a non-associative structure applied to the algebra of Fermi operators with su(3) colour degrees of freedom leads to a consistent Fermi statistic for the tensor operators of the colour algebra. A consequence of this construction is that leads to quark confinement, without the need to resort to a confining force. Confinement arises as a symmetry constraint in much the same manner as the Pauli exclusion principle. 
  We study dyonic instantons in (4+1) dimensional Yang-Mills theory. Especially we consider the most general two instanton solution given by the Jackiw-Nohl-Rebbi ansatz and find its dyonic version. By exploring the zeros of the Higgs field, we rederive the porism structure of triangles in this solution and also find the magnetic monopole string loop. This leads to the identification of dyonic instanton with the supertube inserted between D4 branes. 
  We develop a technique for computing expected numbers of vacua in Gaussian ensembles of supergravity theories, and apply it to derive an asymptotic formula for the index counting all flux supersymmetric vacua with signs in Calabi-Yau compactification of type IIb string theory, which becomes exact in the limit of a large number of fluxes. This should give a reasonable estimate for actual numbers of vacua in string theory, for CY's with small b_3. 
  We study solutions of the Knizhnik-Zamolodchikov equation for discrete representations of SU(2)_k at rational level k+2=p/q using a regular basis in which the braid matrices are well defined for all spins. We show that at spin J=(j+1)p-1 for half integer j there are always a subset of 2j+1 solutions closed under the action of the braid matrices. For integer j these fields have integer conformal dimension and all the 2j+1 solutions are monodromy free. The action of the braid matrices on these can be consistently accounted for by the existence of a multiplet of chiral fields with extra SU(2) quantum numbers (m=-j,...,j). In the quantum group SU_q(2), with q=e^{\f{-i \pi}{k+2}}, there is an analogous structure and the related representations are trivial with respect to the standard generators but transform in a spin j representation of SU(2) under the extended center. 
  In this paper we examine Casimir effect in the case of tachyonic field, which is connected with particles with negative four-momentum square. We consider here only the case of one dimensional, scalar field. In order to describe tachyonic field, we use the absolute synchronization scheme preserving Lorentz invariance. The renormalized vacuum energy is calculated by means of Abel-Plana formula. Finaly, the Casimir energy and Casimir force as the functions of distance are obtained. In order to compare the resulting formula with the standard one, we calculate the Casimir energy and Casimir force for massive, scalar field. 
  Six-dimensional N=(1,0) Einstein-Maxwell gauged supergravity is known to admit a (Minkowski)_4\times S^2 vacuum solution with four-dimensional N=1 supersymmetry. The massless sector comprises a supergravity multiplet, an SU(2) Yang-Mills vector multiplet, and a scalar multiplet. In this paper it is shown that, remarkably, the six-dimensional theory admits a fully consistent dimensional reduction on the 2-sphere, implying that all solutions of the four-dimensional N=1 supergravity can be lifted back to solutions in six dimensions. This provides a striking realisation of the idea, first proposed by Pauli, of obtaining a theory that includes Yang-Mills fields by dimensional reduction on a coset space. We address the cosmological constant problem within this model, and find that if the Kaluza-Klein mass scale is taken to be 10^{-3} eV (as has recently been suggested) then four-dimensional gauge-coupling constants for bulk fields must be of the order of 10^{-31}. We also suggest a link between a modification of the model with 3-branes, and a five-dimensional model based on an S^1/Z_2 orbifold. 
  In the unitary gauge the unphysical degrees of freedom of spontaneously broken gauge theories are eliminated. The Feynman rules are simpler than in other gauges, but it is non-renormalizable by the rules of power counting. On the other hand, it is formally equal to the limit $\xi \to 0$ of the renormalizable R$_{\xi}$-gauge. We consider perturbation theory to one-loop order in the R$_{\xi}$-gauge and in the unitary gauge for the case of the two-dimensional abelian Higgs model. An apparent conflict between the unitary gauge and the limit $\xi \to 0$ of the R$_{\xi}$-gauge is resolved, and it is demonstrated that results for physical quantities can be obtained in the unitary gauge. 
  We study the phases and geometry of the N=1 A_2 quiver gauge theory using matrix models and a generalized Konishi anomaly. We consider the theory both in the Coulomb and Higgs phases. Solving the anomaly equations, we find that a meromorphic one-form sigma(z)dz is naturally defined on the curve Sigma associated to the theory. Using the Dijkgraaf-Vafa conjecture, we evaluate the effective low-energy superpotential and demonstrate that its equations of motion can be translated into a geometric property of Sigma: sigma(z)dz has integer periods around all compact cycles. This ensures that there exists on Sigma a meromorphic function whose logarithm sigma(z)dz is the differential. We argue that the surface determined by this function is the N=2 Seiberg-Witten curve of the theory. 
  We study the Penrose limit of Type IIB duals of softly broken N=1 SU(N) gauge theories in four dimensions, obtained as deformations of the Maldacena-Nunez and Klebanov-Strassler backgrounds. We extract the string spectrum on the resulting pp-wave backgrounds and discuss some properties of the conjectured dual gauge theory hadrons, the so called "Annulons". The string zero-point energy on the light-cone is nontrivial, due to the loss of linearly realized worldsheet supersymmetry, and negative, even in the unbroken supersymmetric case. This causes the appearance of non-perturbative corrections to the hadronic mass spectrum. We briefly discuss the thermodynamic behavior of these string models, calculating the corresponding Hagedorn temperatures. 
  We present a new development in our approach to the covariant quantization of superstrings in 10 dimensions which is based on a gauged WZNW model. To incorporate worldsheet diffeomorphisms we need the quartet of ghosts $(b_{zz},c^{z}, \b_{zz}, \g^{z})$ for topological gravity. The currents of this combined system form an N=2 superconformal algebra. The model has vanishing central charge and contains two anticommuting BRST charges, $Q_{S}=Q_{W} + \oint \g^{z} b_{zz} + \oint \eta_{z}$ and $Q_{V} = \oint c^{z} \Big(T^{W}_{zz} + {1\over 2} T^{top}_{zz}\Big) + \g^{z} (B^{W}_{zz} + {1\over 2} B^{top}_{zz} \Big)$, where $\eta_{z}$ is obtained by the usual fermionization of $\b_{zz}, \g^{z}$. Physical states form the cohomology of $Q_{S}+Q_{V}$, have nonnegative grading, and are annihilated by $b_{0}$ and $\beta_{0}$. We no longer introduce any ghosts by hand, and the formalism is completely Lorentz covariant. 
  We study Neumann coefficients of the various vertices in the Witten's open string field theory (SFT). We show that they are not independent, but satisfy an infinite set of algebraic relations. These relations are identified as so-called Hirota identities. Therefore, Neumann coefficients are equal to the second derivatives of tau-function of dispersionless Toda Lattice hierarchy (this tau-function is just the partition sum of normal matrix model). As a result, certain two-vertices of SFT are identified with the boundary states, corresponding to boundary conditions on an arbitrary curve. Such two-vertices can be obtained by the contraction of special surface states with Witten's three vertex. We analyze a class of SFT surface states,which give rise to boundary states under this procedure. We conjecture that these special states can be considered as describing D-branes and other non-perturbative objects as "solitons" in SFT. We consider some explicit examples, one of them is a surface states corresponding to orientifold. 
  The problem of membrane topology in the matrix model of M-theory is considered. The matrix regularization procedure, which makes a correspondence between finite-sized matrices and functions defined on a two-dimensional base space, is reexamined. It is found that the information of topology of the base space manifests itself in the eigenvalue distribution of a single matrix. The precise manner of the manifestation is described. The set of all eigenvalues can be decomposed into subsets whose members increase smoothly, provided that the fundamental approximations in matrix regularization hold well. Those subsets are termed as eigenvalue sequences. The eigenvalue sequences exhibit a branching phenomenon which reflects Morse-theoretic information of topology.   Furthermore, exploiting the notion of eigenvalue sequences, a new correspondence rule between matrices and functions is constructed. The new rule identifies the matrix elements directly with Fourier components of the corresponding function, evaluated along certain orbits. The rule has semi-locality in the base space, so that it can be used for all membrane topologies in a unified way. A few numerical examples are studied, and consistency with previously known correspondence rules is discussed. 
  We show that the class of four-dimensional Taub-Bolt(NUT) spacetimes with positive cosmological constant for some values of NUT charges are stable and have entropies that are greater than that of de Sitter spacetime, in violation of the entropic N-bound conjecture. We also show that the maximal mass conjecture, which states "any asymptotically dS spacetime with mass greater than dS has a cosmological singularity", can be violated as well. Our calculation of conserved mass and entropy is based on an extension of the path integral formulation to asymptotically de Sitter spacetimes. 
  We study the noncommutative superspace of arbitrary dimensions in a systematic way. Superfield theories on a noncommutative superspace can be formulated in two folds, through the star product formalism and in terms of the supermatrices. We elaborate the duality between them by constructing the isomorphism explicitly and relating the superspace integrations of the star product Lagrangian or the superpotential to the traces of the supermatrices. We show there exists an interesting fine tuned commutative limit where the duality can be still maintained. Namely on the commutative superspace too, there exists a supermatrix model description for the superfield theory. We interpret the result in the context of the wave particle duality. The dual particles for the superfields in even and odd spacetime dimensions are D-instantons and D0-branes respectively to be consistent with the T-duality. 
  We propose a new type of cosmological model in which it is postulated that not only the temperature but also the curvature is limited by the mass scale of the Hagedorn temperature. We find that the big bang of this universe is smoothly connected to the big crunch of the previous universe through a Hagedorn universe, in which the temperature and curvature remain very close to their limiting values. In this way, we obtain the picture of a cyclic universe. By estimating the entropy gained in each big crunch and big bang, we reach the conclusion that our universe has repeated this process about forty times after it was created at the Planck scale. We also show that the model gives a scale-invariant spectrum of curvature perturbations. 
  L-infinity morphisms are studied from the point of view of perturbative quantum field theory, as generalizations of Feynman expansions. The connection with the Hopf algebra approach to renormalization is exploited. Using the coalgebra structure (Forest Formula), the weights of the corresponding expansions are proved to be cycles of the DG-coalgebra of Feynman graphs.   The properties of integrals over configuration spaces (Feynman integrals) are investigated. The aim is to develop a cohomological approach in order to construct the coefficients of formality morphisms using an algebraic machinery, as an alternative to the analytical approach using integrals over configuration spaces.   The connection with a related TQFT is mentioned, supplementing the Feynman path integral interpretation of Kontsevich formula. 
  We study supersymmetric Sp(N) gauge theories with an antisymmetric tensor and degree n+1 tree level superpotential. The generalized Konishi anomaly equations derived in hep-th/0304119 and hep-th/0304138 are used to compute the low energy superpotential of the theory. This is done by imposing a certain integrality condition on the periods of a meromorphic one form. Explicit computations for Sp(2), Sp(4), Sp(6) and Sp(8) with cubic superpotential are done and full agreement with the results of the dynamically generated superpotential approach is found. As a byproduct, we find a very precise map from Sp(N) to a U(N+2n) theory with one adjoint and a degree n+1 tree level superpotential. 
  A renormalization group procedure for effective particles is applied to quantum chromodynamics of one flavor of quarks with large mass m in order to calculate light-front Hamiltonians for heavy quarkonia, H_lambda, using perturbative expansion in the coupling constant alpha_lambda. lambda is the renormalization group parameter with the interpretation of an inverse of the spatial size of the color charge distribution in the effective quarks and gluons. The eigenvalue equation for H_lambda couples quark-anti-quark states with sectors of a larger number of constituents. The coupling to states with more than one effective gluon, and interactions in the quark-anti-quark-gluon sector, are removed at the price of introducing an ansatz for the gluon mass, mu^2. The simplified equation is used to evaluate a new Hamiltonian of order alpha_lambda that acts only in the effective quark-anti-quark sector and in the non-relativistic limit turns out to contain the Coulomb term with Breit-Fermi orrections and spin-independent harmonic oscillator term with frequency omega = [(4/3)(alpha_lambda/pi)]^1/2 lambda (lambda/m)^2 (pi/1152)^1/4. The latter originates from the hole excavated in the overlapping quark self-interaction gluon clouds by the exchange of effective gluons between the quarks. The new term is largely independent of the details of mu^2 and in principle can fit into the ball park of phenomenology. The first approximation can be improved by including more terms in H_lambda and solving the eigenvalue equations numerically. 
  We consider an Einstein-Yang-Mills Lagrangian in a five dimensional space-time including a cosmological constant. Assuming all fields to be independent of the extra coordinate, a dimensional reduction leads to an effective (3+1)-dimensional Einstein-Yang-Mills-Higgs-dilaton model where the cosmological constant induces a Liouville potential in the dilaton field. We construct spherically symmetric solutions analytically in specific limits and study the generic solutions for vanishing dilaton coupling numerically. We find that in this latter case the solutions bifurcate with the branch of (Anti-) de Sitter-Reissner-Nordstrom ((A)dSRN) solutions. 
  We study N=(4,4) superconformal field theories with left and right central charge c=6 which allow geometric interpretations on specific quartic hypersurfaces in CP^3. Namely, we recall the proof that the Gepner model (2)^4 admits a geometric interpretation on the Fermat quartic and give an independent cross-check of this result, providing a link to the "mirror moonshine phenomenon" on K3. We clarify the role of Shioda-Inose structures in our proof and thereby generalize it: We introduce "very attractive quartics" and show how on each of them a superconformal field theory can be constructed explicitly. 
  We study the Casimir energy of a minimally coupled, real, massless scalar field outside a spherically symmetric background potential. We obtain a general expression for the null energy condition in d dimensions and explicit expressions for a perfectly reflecting spherical boundary in 3+1 and 2+1 dimensions. In these cases, the null energy condition is always violated for radial motion and obeyed for azimuthal motion. Nevertheless, the averaged null energy condition is always obeyed. 
  We study a novel class of nonsingular time-symmetric cosmological bounces. In this class of four dimensional models the bounce is induced by a perfect fluid with a negative energy density. Metric perturbations are solved in an analytic way all through the bounce. The conditions for generating a scale invariant spectrum of tensor and scalar metric perturbations are discussed. 
  A new static partially twisted solution of N=4, SO(4) gauged supergravity in D=11 is obtained in this work using Cveti\^c et al embedding of four dimensional into eleven dimensional supergravities. In four dimensions we get two solutions: an asymptotic one corresponding to $AdS_4$ and a near horizon fixed point solution of the form $AdS_2\times H_2$. Hence, while the former solution has 32 supercharges the latter turns out to have only 4 conserved. Moreover, we managed to find an exact interpolating solution, thus connecting the above two. Aiming at a future study of $AdS/CFT$ duality for the theory at hand we derived the Penrose limit of the four dimensional solutions. Interestingly the pp-wave limit of the near horizon solution suggests itself as being of the supernumerary supersymmetric type. In D=11 we exhibit the uplift of the four dimensional solutions: one associated to $AdS_4\times S^7$ and the other to a foliation of $AdS_2\times H_2 \times S^7$, as well as their pp-wave limits. 
  It is shown that the $M$-algebra related with the $M$ theory comes in two variants. Besides the standard $M$ algebra based on the real structure, an alternative octonionic formulation can be consistently introduced. This second variant has striking features. It involves only 52 real bosonic generators instead of 528 of the standard $M$ algebra and moreover presents a novel and surprising feature, its octonionic $M5$ (super-5-brane) sector is no longer independent, but coincides with the octonionic $M1$ and $M2$ sectors. This is in consequence of the non-associativity of the octonions. An octonionic version of the superconformal $M$-algebra also exists. It is given by $OSp(1,8|{\bf O})$ and admits 239 bosonic and 64 fermionic generators. It is speculated that the octonionic $M$-algebra can be related to the exceptional Lie and Jordan algebras that apparently play a special role in the Theory Of Everything. 
  It is suggested \cite{CDTT} that current cosmic acceleration arises due to modification of General Relativity by the terms with negative powers of curvature. We show that time-dependent (hyperbolic) compactifications of string/M-theory lead to the effective 4d gravity which naturally contains such terms. The same may be achieved in braneworld by the proper choice of the boundary action. Hence, such a model which seems to eliminate the need for dark energy may have the origin in M-theory. 
  The superembedding formalism is used to study correction terms to the dynamics of the M2 brane in a flat background. This is done by deforming the standard embedding constraint. It is shown rigorously that the first such correction occurs at dimension four. Cohomological techniques are used to determine this correction explicitly. The action is derived to quadratic order in fermions, and the modified $\k$-symmetry transformations are given. 
  We construct a model where cosmological perturbations are analytically solved based on dilatonic brane worlds. A bulk scalar field has an exponential potential in the bulk and an exponential coupling to the brane tension. The bulk scalar field yields a power-law inflation on the brane. The exact background metric can be found including the back-reaction of the scalar field. Then exact solutions for cosmological perturbations which properly satisfy the junction conditions on the brane are derived. These solutions provide us an interesting model to understand the connection between the behavior of cosmological perturbations on the brane and the geometry of the bulk. Using these solutions, the behavior of an anisotropic stress induced on the inflationary brane by bulk gravitational fields is investigated. 
  We show that in a multi D\={D} branes system with high temperature, there may exist a thermal cosmological phase before usual tachyon inflation. Though this thermal phase can be very transitory, it may has some interesting applications for early tachyon/brane cosmology. 
  U(n) Yang-Mills theory on the fuzzy sphere S^2_N is quantized using random matrix methods. The gauge theory is formulated as a matrix model for a single Hermitian matrix subject to a constraint, and a potential with two degenerate minima. This allows to reduce the path integral over the gauge fields to an integral over eigenvalues, which can be evaluated for large N. The partition function of U(n) Yang-Mills theory on the classical sphere is recovered in the large N limit, as a sum over instanton contributions. The monopole solutions are found explicitly. 
  We study four dimensional supersymmetric gauge theory on the noncommutative superspace, recently proposed by Seiberg. We construct the gauge-invariant action of N=1 super Yang-Mills theory with chiral and antichiral superfields, which has N=1/2 supersymmetry on the noncommutative superspace. We also construct the action of N=2 super Yang-Mills theory. It is shown that this theory has only N=1/2 supersymmetry. 
  We clarify the properties of the behavior of classical cosmological perturbations when the Universe experiences a bounce. This is done in the simplest possible case for which gravity is described by general relativity and the matter content has a single component, namely a scalar field in a closed geometry. We show in particular that the spectrum of scalar perturbations can be affected by the bounce in a way that may depend on the wave number, even in the large scale limit. This may have important implications for string motivated models of the early Universe. 
  We calculate the potential for $Dp-\Dbar p$ pair and show that the coincident $Dp-\Dbar p$ system has $(11-p)$ tachyonic modes, with $(9-p)$ of them due to radiative corrections. We propose that the decay width of an unstable non-BPS-$Dp$-brane to closed strings is given by the imaginary part of the one-loop contribution to the effective potential of the open string tachyon mode. 
  In this paper we will consider the minisuperspace approach to S-branes dynamics in the Schr\"odinger picture description. Time-evolution of vacuum wave functional for quantum field theory on S-brane is studied. Open string pair production is calculated. The analysis of density matrix for mixed states is also performed. 
  We study monopole and dyon solutions to the equations of motion of the bosonic sector of N = 4 gauged supergravity in four dimensional space-time. A static, spherically symmetric ansatz for the metric, gauge fields, dilaton and axion leads to soliton solutions which, in the electrically charged case, have compact spatial sections. Both analytical and numerical results for the solutions are presented. 
  We show how to determine the lowest order mixing of all scalar with two-fermion two impurity BMN operators in the antisymmetric representation of SO(4). Differentiation on harmonic superspace allows one to derive two-loop anomalous dimensions of gauge invariant operators from this knowledge: the value for the second anomalous correction to the dimension is essentially the square of the two-fermion admixture. The method effectively increases the loop order by one. For low J we find agreement to all orders in N with results obtained upon diagonalisation of the N=4 dilation operator.   We give a formula for the generalised Konishi anomaly and display its role in the mixing. For J=2 we resolve the mixing up to order $g^2$ in the singlet representation. The sum of the anomaly and the naive variation of the leading two-fermion admixtures to the singlets is exactly equal to the two-fermion terms in the antisymmetric descendants. 
  Perturbing the Seiberg-Witten curves for N=2 U(N_c) and SU(N_c) super Yang-Mills theory with N_f<N_c flavours with a mass term for the adjoint field completely lifts the quantum vacuum degeneracy. The generated N=1 effective superpotential can be obtained from the factorization formulae of Seiberg-Witten curves with matter. We show that the Affleck-Dine-Seiberg superpotential emerges. Moreover it appears additive with respect to the classical superpotential for the meson superfields, as expected from the Intriligator-Leigh-Seiberg linearity principle. 
  We propose that type 0B string theory in two dimensions admits a dual description in terms of a one dimensional bosonic matrix model of a hermitian matrix. The potential in the matrix model is symmetric with respect to the parity-like Z_2 transformation of the matrix. The two sectors in the theory, namely the NSNS and RR scalar sectors correspond to two classes of operators in the matrix model, even and odd under the Z_2 symmetry respectively. We provide evidence that the matrix model successfully reconstructs the perturbative S-matrix of the string theory, and reproduces the closed string emission amplitude from unstable D-branes. Following recent work in two dimensional bosonic string, we argue that the matrix model can be identified with the theory describing N unstable D0-branes in type 0B theory. We also argue that type 0A theory is described in terms of the quantum mechanics of brane-antibrane systems. 
  We discuss the compactification of type IIB supergravity with fluxes to generate a potential for the moduli. In particular we resolve an apparent conflict with the no-go theorem for de Sitter space. It is shown that a positive potential for certain moduli is possible in situations where the volume modulus has no critical point. We also point out that the derivation of the potential is strictly valid only for a trivial warp factor. To go beyond that seems to require the inclusion of all the Kaluza-Klein excitations. We end with a discussion of the stabilization of the volume modulus. 
  It is shown that gravity in 2+1 dimensions coupled to point particles provides a nontrivial example of Doubly Special Relativity (DSR). This result is obtained by interpretation of previous results in the field and by exhibiting an explicit transformation between the phase space algebra for one particle in 2+1 gravity found by Matschull and Welling and the corresponding DSR algebra. The identification of 2+1 gravity as a $DSR$ system answers a number of questions concerning the latter, and resolves the ambiguity of the basis of the algebra of observables.   Based on this observation a heuristic argument is made that the algebra of symmetries of ultra high energy particle kinematics in 3+1 dimensions is described by some DSR theory. 
  We attempt to generalize the effective action for the D-brane-anti-D-brane system obtained from boundary superstring field theory (BSFT) by adding an extra D-brane to it to obtain a covariantized action for 2 D-branes and 1 anti-D-brane. We discuss the approximations made to obtain the effective action in closed form. Among other properties, this effective action admits solitonic solutions of codimension 2 (vortices) when one of the D-brane is far separated from the brane-anti-brane pair. 
  We consider systems of two free particles in de Sitter invariant quantum theory and calculate the mean value of the mass operator for such systems. It is shown that, in addition to the well known relativistic contribution (and de Sitter antigravity which is small when the de Sitter radius is large), there also exists a contribution caused by the fact that certain decomposition coefficients have different phases. Such a contribution is negative and proportional to the particle masses in the nonrelativistic approximation. In particular, for a class of two-body wave functions the mean value is described by standard Newtonian gravity and post Newtonian corrections in General Relativity. This poses the problem whether gravity can be explained without using the notion of interaction at all. We discuss a hypothesis that gravity is a manifestation of Galois fields in quantum physics. 
  In this paper we construct generalised Penrose limits for the solutions of massive type IIA supergravity. We consider a Freund-Rubin type solution and apply these {\it massive} Penrose limits and obtain supersymmetric pp-wave which is a standard type IIA background. We point out that results in this paper are easily generalised for the cases of gauged supergravities. 
  We improve the study of the lack of perturbative unitarity of noncommutative space-time quantum field theories derived from open string theory in electric backgrounds, enforcing the universality of the mechanism by which a tachyonic branch cut appears when the Seiberg-Witten limit freezes the string in an unstable vacuum. The main example is realized in the context of the on-shell four-tachyon amplitude of the bosonic string, and the dependence of the phenomenon on the brane-worldvolume dimension is analysed. We discuss the possibility of a proof in superstring theory, and finally mention the NCOS limit in this framework. 
  A review is given of the Peierls bracket formalism in field theory, and of a new, recent application of this concept to the analysis of dissipative systems. 
  We study quantum aspects of field theories defined on N=1/2 superspace, where both bosonic and fermionic coordinates are made non(anti)commutative. We compute the one-loop effective superpotential, and we find that planar and nonplanar contributions exhibit markedly different behavior. Planar diagrams yield an effective superpotential proportional to N_c \Phi \log \Phi. For nonplanar diagrams, we show that ultraviolet-infrared mixing takes place and explain why some nonplanar diagrams are ultraviolet-divergent when bosonic noncommutativity is turned off. Each nonplanar diagram is not expressible as a star product of background fields, but, once resummed appropriately, they are expressed as a star product involving open Wilson lines in superspace. The open Wilson lines are responsible for ultraviolet-infrared mixing. We comment on an intriguing relation of our result to the Dijkgraaf-Vafa correspondence between gauge theories and matrix models. 
  D-branes intersecting at an arbitrary fixed angle generically constitute a configuration unstable toward recombination. The reconnection of the branes nucleates at the intersection point and involves a generalization of the process of brane decay of interest to non-perturbative string dynamics as well as cosmology. After reviewing the string spectrum of systems of angled branes, we show that worldsheet twist superfields may be used in the context of Boundary Superstring Field Theory to describe the dynamics. Changing the angle between the branes is seen from the worldsheet as spectral flow with boundary insertions flowing from bosonic to fermionic operators. We calculate the complete tachyon potential and the low energy effective action as a function of angle and find an expression that interpolates between the brane-antibrane and the Dirac-Born-Infeld actions. The potential captures the mechanism of D-brane recombination and provides for interesting new physics for tachyon decay. 
  The aim of the present article is to give physical meaning to the ingredients of standard gauge field theory in the framework of the scale relativity theory. Owing to the principle of the relativity of scales, the scale-space is not absolute. Therefore, the scale variables are functions of the space-time coordinates, so that we expect a coupling between the displacement in space-time and the dilation/contraction of the scale variables, which are identified with gauge transformations. The gauge fields naturally appear as a new geometric contribution to the total variation of the scale variables. The gauge charges emerge as the generators of the scale transformation group applied to a generalized action (now identified with the scale relativistic invariant) and are therefore the conservative quantities which find their origin in the symmetries of the scale-space. We recover the expression for the covariant derivative of non-Abelian gauge theory. Under the gauge transformations, the fermion multiplets and the boson field transform in such a way that the Lagrangian, which is here derived instead of being set as a founding axiom, remains invariant. We have therefore obtained gauge theories as a consequence of scale symmetries issued from a geometric fractal space-time description, which we apply to peculiar examples of the electroweak and grand unified theories. 
  We construct a new family of N-fold supersymmetric systems which is referred to as ``type B''. A higher derivative representation of the N-fold supercharge for this new family is given by a deformation of the type A N-fold supercharge. By utilizing the same method as in the sl(2) construction of type A N-fold supersymmetry, we show that this family includes two of the quasi-solvable models of Post-Turbiner type. 
  A Bethe Ansatz solution of the open spin-1/2 XXZ quantum spin chain with nondiagonal boundary terms has recently been proposed. Using a numerical procedure developed by McCoy et al., we find significant evidence that this solution can yield the complete set of eigenvalues for generic values of the bulk and boundary parameters satisfying one linear relation. Moreover, our results suggest that this solution is practical for investigating the ground state of this model in the thermodynamic limit. 
  We derive the anomalous transformation law of the quantum stress tensor for a 2D massless scalar field coupled to an external dilaton. This provides a generalization of the Virasoro anomaly which turns out to be consistent with the trace anomaly. We apply these results to compute vacuum polarization of a spherical star based on the equivalence principle. 
  We suggest a new large-N_c limit for multi flavor QCD. Since fundamental and two-index antisymmetric representations are equivalent in SU(3), we have the option to define SU(N_c) QCD keeping quarks in the latter. We can then define a new 1/N_c expansion (at fixed number of flavors N_f) that shares appealing properties with the topological (fixed N_f/N_c) expansion while being more suitable for theoretical analysis. In particular, for N_f=1, our large-N_c limit gives a theory that we recently proved to be equivalent, in the bosonic sector, to N=1 supersymmetric gluodynamics. Using known properties of the latter, we derive several qualitative and semi-quantitative predictions for N_f=1 massless QCD which can be easily tested in lattice simulations. Finally, we comment on possible applications for pure SU(3) Yang-Mills theory and real QCD. 
  We show that the $E_{11}$ representation that contains the space-time translation generators also contains the rank two and five totally anti-symmetric representations of $A_{10}$. By studying the behaviour of these latter $A_{10}$ representations under SL(32), which we argue is contained in the Cartan involution invariant sub-algebra of $E_{11}$, we find that the rank two and five totally anti-symmetric representations must be identified with the central charges of the eleven dimensional supersymmetry algebra. 
  We study systematically, through two loops, the divergence structure of the supersymmetric WZ model defined on the N=1/2 nonanticommutative superspace. By introducing a spurion field to represent the supersymmetry breaking term F^3 we are able to perform our calculations using conventional supergraph techniques. Divergent terms proportional to F, F^2 and F^3 are produced (the first two are to be expected on general grounds) but no higher-point divergences are found. By adding ab initio F and F^2 terms to the original lagrangian we render the model renormalizable. We determine the renormalization constants and beta functions through two loops, thus making it possible to study the renormalization group flow of the nonanticommutation parameter. 
  The vector perturbations induced on the brane by gravitational waves propagating in the bulk are studied in a cosmological framework. Cosmic expansion arises from the brane motion in a non-compact five-dimensional anti-de Sitter spacetime. By solving the vector perturbation equations in the bulk, for generic initial conditions, we find that they give rise to growing modes on the brane in the Friedmann-Lema\^{\i}tre era. Among these modes, we exhibit a class of normalizable perturbations, which are exponentially growing with respect to conformal time on the brane. The presence of these modes is, at least, strongly constrained by the current observations of the cosmic microwave background (CMB). We estimate the anisotropies they induce in the CMB, and derive quantitative constraints on the allowed amplitude of their primordial spectrum. Our results provide stringent constraints for all braneworld models with bulk inflation. 
  These lectures give an introduction to the novel duality relating type IIB string theory in a maximally supersymmetric plane-wave background to N=4, d=4, U(N) Super Yang-Mills theory in a particular large N and large R-charge limit due to Berenstein, Maldacena and Nastase. In the first part of these lectures the duality is derived from the AdS/CFT correspondence by taking a Penrose limit of the AdS_5 x S^5 geometry and studying the corresponding double-scaling limit on the gauge theory side. The resulting free plane-wave superstring is then quantized in light-cone gauge. On the gauge theory side of the correspondence the composite Super Yang-Mills operators dual to string excitations are identified, and it is shown how the string spectrum can be mapped to the planar scaling dimensions of these operators. In the second part of these lectures we study the correspondence at the interacting respectively non-planar level. On the gauge theory side it is demonstrated that the large N large R-charge limit in question preserves contributions from Feynman graphs of all genera through the emergence of a new genus counting parameter - in agreement with the string genus expansion for non-zero g_s. Effective quantum mechanical tools to compute higher genus contributions to the scaling dimensions of composite operators are developed and explicitly applied in a genus one computation. We then turn to the interacting string theory side and give an elementary introduction into light-cone superstring field theory in a plane-wave background and point out how the genus one prediction from gauge theory can be reproduced. Finally, we summarize the present status of the plane-wave string/gauge theory duality. 
  Energy densities of the quantum states that are superposition of two multi-electron-positron states are examined. It is shown that the energy densities can be negative only when two multi-particle states have the same number of electrons and positrons or when one state has one more electron-positron pair than the other. In the cases in which negative energy could arise, we find that the energy is that of a positive constant plus a propagating part which oscillates between positive and negative, and the energy can dip to negative at some places at for a certain period of time if the quantum states are properly manipulated. It is demonstrated that the negative energy densities satisfy the quantum inequality. Our results also reveal that for a given particle content, the detection of negative energy is an operation that depends on the frame where any measurement is to be performed. This suggests that the sign of energy density for a quantum state may be a coordinate-dependent quantity in quantum theory. 
  Starting from the $\mathcal{N}=2$ SYM$_{4}$ quiver theory living on wrapped $% N_{i}D5$ branes around $S_{i}^{2}$ spheres of deformed ADE fibered Calabi-Yau threefolds (CY3) and considering deformations using \textit{% massive} vector multiplets, we explicitly build a new class of $\mathcal{N}% =1 $ quiver gauge theories. In these models, the quiver gauge group $% \prod_{i}U(N_{i}) $ is spontaneously broken down to $% \prod_{i}SU(N_{i}) $ and Kahler deformations are shown to be given by the real part of the integral $(2,1) $ form of CY3. We also give the superfield correspondence between the $\mathcal{N}=1$ quiver gauge models derived here and those constructed in hep-th/0108120 using complex deformations. Others aspects of these two dual $\mathcal{N}=1$ supersymmetric field theories are discussed. 
  In these lectures we start with a pedagogical introduction of the properties of open and closed superstrings and then, using the open/closed string duality, we construct the boundary state that provides the description of the maximally supersymmetric Dp branes in terms of the perturbative string formalism. We then use it for deriving the corresponding supergravity solution and the Born-Infeld action and for studying the properties of the maximally supersymmetric gauge theories living on their worldvolume. In the last section of these lectures we extend these results to less supersymmetric and non-conformal gauge theories by considering fractional branes of orbifolds and wrapped branes.   Lectures given at the School "Frontiers in Number Theory, Physics and Geometry", Les Houches, March 2003. 
  We present a new non-abelian generalization of the Born-Infeld Lagrangian. It is based on the observation that the basic quantity defining it is the generalized volume element, computed as the determinant of a linear combination of metric and Maxwell tensors. We propose to extend the notion of determinant to the tensor product of space-time and a matrix representation of the gauge group. We compute such a Lagrangian explicitly in the case of the SU(2) gauge group and then explore the properties of static, spherically symmetric solutions in this model. We have found a one-parameter family of finite energy solutions. In the last section, the main properties of these solutions are displayed and discussed. 
  A supersymmetric string model in the D=11 superspace maximally extended by antisymmetric tensor bosonic coordinates, $\Sigma^{(528|32)}$, is proposed. It possesses 30 $\kappa$-symmetries and 32 target space supersymmetries. The usual preserved supersymmetry-$\kappa$-symmetry correspondence suggests that it describes the excitations of a BPS state preserving all but two supersymmetries. The model can also be formulated in any $\Sigma^{({n(n+1)\over 2}|n)}$ superspace, n=32 corresponding to D=11. It may also be treated as a `higher--spin generalization' of the usual Green--Schwarz superstring. Although the global symmetry of the model is a generalization of the super--Poincar\'e group, ${\Sigma}^{({n(n+1)\over 2}|n)}\times\supset Sp(n)$, it may be formulated in terms of constrained OSp(2n|1) orthosymplectic supertwistors. We work out this supertwistor realization and its Hamiltonian dynamics.   We also give the supersymmetric p-brane generalization of the model. In particular, the $\Sigma^{(528|32)}$ supersymmetric membrane model describes excitations of a 30/32 BPS state, as the $\Sigma^{(528|32)}$ supersymmetric string does, while the supersymmetric 3-brane and 5-brane correspond, respectively, to 28/32 and 24/32 BPS states. 
  We present the solution for effective couplings on a running de Sitter brane. A renormalization group formalism is developed for linearized gravity in a two de Sitter brane model. It is shown that the effective tension remains a constant similarly to the zero cosmological constant case. We also discuss the suppression of higher derivative terms. 
  Relation between one-dimensional Schroedinger equation and the vacuum eigenvalues of the Q-operators is extended to their higher-level eigenvalues. 
  A unified description of all interactions could be based on a higher-dimensional theory involving only spinor fields. The metric arises as a composite object and the gravitational field equations contain torsion-corrections as compared to Einstein gravity. Lorentz symmetry in spinor space is only global, implying new goldstone-boson-like gravitational particles beyond the graviton. However, the Schwarzschild and Friedman solutions are unaffected at one loop order. Our generalized gravity seems compatible with all present observations. 
  We present new static axially symmetric solutions of SU(2) Yang-Mills-Higgs theory, representing chains of monopoles and antimonopoles in static equilibrium. They correspond to saddlepoints of the energy functional and exist both in the topologically trivial sector and in the sector with topological charge one. 
  We define N=4, d=1 harmonic superspace HR^{1+2|4} with an SU(2)/U(1) harmonic part, SU(2) being one of two factors of the R-symmetry group SU(2)xSU(2) of N=4, d=1 Poincar\'e supersymmetry. We reformulate, in this new setting, the models of N=4 supersymmetric quantum mechanics associated with the off-shell multiplets (3, 4, 1) and (4, 4, 0). The latter admit a natural description as constrained superfields living in an analytic subspace of HR^{1+2|4}. We construct the relevant superfield actions consisting of a sigma-model as well as a superpotential parts and demonstrate that the superpotentials can be written off shell in a manifestly N=4 supersymmetric form only in the analytic superspace. The constraints implied by N=4 supersymmetry for the component bosonic target-space metrics, scalar potentials and background one-forms automatically follow from the harmonic superspace description. The analytic superspace is shown to be closed under the most general N=4, d=1 superconformal group D(2,1;\alpha). We give its action on the analytic superfields comprising the (3, 4, 1) and (4, 4, 0) multiplets, reveal a surprising relation between the latter and present the corresponding superconformally invariant actions. The harmonic superspace approach suggests a natural generalization of these multiplets, with a [2(n+1), 4n, 2(n-1)] off-shell content for n>2. 
  We investigate the relation between Connes-Kreimer Hopf algebra approach to renomalization and deformation quantization. Both approaches rely on factorization, the correspondence being established at the level of Wiener-Hopf algebras, and double Lie algebras/Lie bialgebras, via r-matrices.   It is suggested that the QFTs obtained via deformation quantization and renormalization correspond to each other in the sense of Kontsevich/Cattaneo-Felder. 
  We classify almost all classical string configurations, considered in the framework of the semi-classical limit of the string/gauge theory duality. Then, we describe a procedure for obtaining the conserved quantities and the exact classical string solutions in general string theory backgrounds, when the string embedding coordinates depend non-linearly on the worldsheet time parameter. 
  We investigate possible extensions of the (2+1) dimensional $CP^{N-1}$ model to the noncommutative space. Up to the leading nontrivial order of 1/N, we prove that the model restricted to the left fundamental representation of the gauge group is renormalizable and does not have dangerous infrared divergences. In contrast, if the basic field $\phi$ transforms in accord with the adjoint representation, infrared singularities are present in the two point function of the auxiliary gauge field and also in the leading correction to the self-energy of the $\phi$ field. These infrared divergences may produce nonintegrable singularities leading at higher orders to a breakdown of the 1/N expansion. Gauge invariance of the renormalization procedure is also discussed. 
  I consider N=1 U(N) gauge theory with matter in the adjoint, fundamental and anti-fundamental representations. Focusing on the equations defining the Riemann surface that describes the quantum theory, the gaugino condensates (and related superpotentials) are calculated in the limit of SU(N) gauge group, both in the pure theory and in the presence of matter. In the case without fundamental matter it is investigated the structure of the space of vacua. In particular it is discussed how different vacua can be related, in a way which finally helps to count them. 
  Two different matrix models for QCD with a non-vanishing quark chemical potential are shown to be equivalent by mapping the corresponding partition functions. The equivalence holds in the phase with broken chiral symmetry. It is exact in the limit of weak non-Hermiticity, where the chemical potential squared is rescaled with the volume. At strong non-Hermiticity it holds only for small chemical potential. The first model proposed by Stephanov is directly related to QCD and allows to analyze the QCD phase diagram. In the second model suggested by the author all microscopic spectral correlation functions of complex Dirac operators can be calculated in the broken phase. We briefly compare those predictions to complex Dirac eigenvalues from quenched QCD lattice simulations. 
  A cosmology inspired structure for phase space is introduced, which leads to finitization and lattice-like discretization of position and momentum eigenvalues in a preferred, cosmic frame. Lorentz invariance is broken at very high energies, at present inaccessible. The divergent perturbation terms in quantum electrodynamics become finite and small; this could become a requirement leading to model restrictions in other perturbative theories. So the very success of the usual renormalization procedures is simply explained by their finitization, and is viewed as indicating the reality of the lattice. 
  The arising of central extensions is discussed in two contexts. At first classical counterparts of quantum anomalies (deserving being named as "classical anomalies") are associated with a peculiar subclass of the non-equivariant maps. Further, the notion of "residual symmetry" for theories formulated in given non-vanishing EM backgrounds is introduced. It is pointed out that this is a Lie-algebraic, model-independent, concept. 
  We present a new $(p - 1)$-brane solution to Einstein's equations in a general space-time dimension. This solution is a natural generalization of the stringlike defect solution with codimension 2 in 6 space-time dimensions, which has been recently discovered by Gogberashvili and Singleton, to a general $(p - 1)$-brane solution with codimension $n$ in general $D = p + n$ space-time dimensions. It is shown that all the local fields are localized on the brane only through the gravitational interaction although this solution does not have a warp factor and takes a finite value in the radial infinity. Thus, this solution is a solution in an arbitrary space-time dimension realizing the idea of "gravitational trapping" of the whole bulk fields on the brane within the framework of a local field theory. Some problems associated with this solution and localization are pointed out. 
  In order to investigate whether space coordinates are intrinsically noncommutative, we make use of the Hall effect on the two-dimensional plane.   We calculate the Hall conductivity in such a way that the noncommutative U(1) gauge invariance is manifest. We find that the noncommutativity parameter theta does not appear in the Hall conductivity itself, but the particle number density of electron depends on theta.   We point out that the peak of particle number density differs from that of the charge density. 
  Evolution of extended data is considered in various flow problems, using Nambu brackets as a tool applicable to all cases. Extra dimensions, N-brackets, and extended structures are first employed to linearize the Euler equations. N-bracket induced evolutions of strings, membranes, and other d-branes are then discussed in detail. 
  We study the late-time tails appearing in the propagation of massless fields (scalar, electromagnetic and gravitational) in the vicinities of a D-dimensional Schwarzschild black hole. We find that at late times the fields always exhibit a power-law falloff, but the power-law is highly sensitive to the dimensionality of the spacetime. Accordingly, for odd D>3 we find that the field behaves as t^[-(2l+D-2)] at late times, where l is the angular index determining the angular dependence of the field. This behavior is entirely due to D being odd, it does not depend on the presence of a black hole in the spacetime. Indeed this tails is already present in the flat space Green's function. On the other hand, for even D>4 the field decays as t^[-(2l+3D-8)], and this time there is no contribution from the flat background. This power-law is entirely due to the presence of the black hole. The D=4 case is special and exhibits, as is well known, the t^[-(2l+3)] behavior. In the extra dimensional scenario for our Universe, our results are strictly correct if the extra dimensions are infinite, but also give a good description of the late time behaviour of any field if the large extra dimensions are large enough. 
  Supersymmetric extensions of Hamilton-Jacobi separable Liouville mechanical systems with two degrees of freedom are defined. It is shown that supersymmetry can be implemented in this type of systems in two independent ways. The structure of the constants of motion is unveiled and the entanglement between integrability and supersymmetry is explored. 
  The standard model fermion spectrum, including a right handed neutrino, can be obtained as a zero-mode of the Dirac operator on a space which is the product of complex projective spaces of complex dimension two and three. The construction requires the introduction of topologically non-trivial background gauge fields. By borrowing from ideas in Connes' non-commutative geometry and making the complex spaces `fuzzy' a matrix approximation to the fuzzy space allows for three generations to emerge. The generations are associated with three copies of space-time. Higgs' fields and Yukawa couplings can be accommodated in the usual way. 
  A semiclassical approach is used to obtain Lorentz covariant expressions for the form factors between the kink states of a quantum field theory with degenerate vacua. Implemented on a cylinder geometry it provides an estimate of the spectral representation of correlation functions in a finite volume. Illustrative examples of the applicability of the method are provided by the Sine-Gordon and the broken \phi^4 theories in 1+1 dimensions. 
  We study a multispecies one-dimensional Calogero model with two- and three-body interactions. Using an algebraic approach (Fock space analysis), we construct ladder operators and find infinitely many, but not all, exact eigenstates of the model Hamiltonian. Besides the ground state energy, we deduce energies of the excited states. It turns out that the spectrum is linear in quantum numbers and that the higher-energy levels are degenerate. The dynamical symmetry responsible for degeneracy is SU(2). We also find the universal critical point at which the model exhibits singular behaviour. Finally, we make contact with some special cases mentioned in the literature. 
  We show that the holonomy of the supercovariant connection of IIB supergravity is contained in $SL(32, \bR)$. We also find that the holonomy reduces to a subgroup of $SL(32-N)\st (\oplus^N \bR^{32-N})$ for IIB supergravity backgrounds with $N$ Killing spinors. We give the necessary and sufficient conditions for a IIB background to admit $N$ Killing spinors. A IIB supersymmetric probe configuration can involve up to 31 linearly independent planar branes and preserves one supersymmetry. 
  We discuss a class of phantom ($p < - \varrho$) cosmological models. Except for phantom we admit various forms of standard types of matter and discuss the problem of singularities for these cosmologies. The singularities are different from those of standard matter cosmology since they appear for infinite values of the scale factor. We also find an interesting relation between the phantom models and standard matter models which is like the duality symmetry of string cosmology. 
  A relativistic quantum model of particle scattering near the horizon of a microscopic black hole unifies gravity and the harmonic-oscillator force. The model is obtained by modifying a harmonic-oscillator nonstandard Lagrangian for a closed system of relativistic quarks to eliminate any cosmic background frame. Formulated in terms of the Planck units, the model has only one parameter, the cosmic number N2. Other results include (1) quark confinement through cluster decomposition rather than a binding potential, (2) lepton substructure without violation of known experimental results, (3) a spontaneously broken symmetry between leptons and hadrons with leptons behaving as free particles and hadrons which scatter with Veneziano-type amplitudes, (5) realistic particle masses and sizes, and (6) a solution of a cosmic-number problem in cosmology. 
  We consider the chiral ring of the pure N=1 supersymmetric gauge theory with SU(N) gauge group and show that the classical relation S^{N^2}=0 is modified to the exact quantum relation (S^N-\Lambda^{3N})^N=0. 
  We present a linearized treatment of the Karch-Randall braneworld where an AdS_4 or dS_4 brane is embedded in AdS_5. We examine the quasi-zero graviton mode in detail and reproduce the graviton mass by elementary means for the AdS_5 case. We also determine the axially symmetric, static excitations of the vacuum and demonstrate that they reproduce the 4D AdS_4 and dS_4 Schwarzschild metrics on the brane. 
  Recent advances in understanding the propagation of perturbations through the transition from big crunch to big bang (esp. Tolley et al. hep-th/0306109) make it possible for the first time to consider the full set of phenomenological constraints on the scalar field potential in cyclic models of the universe. We show that cyclic models require a comparable degree of tuning to that needed for inflationary models. The constraints are reduced to a set of simple design rules including "fast-roll" parameters analogous to the "slow-roll" parameters in inflation. 
  The Bethe ansatz equations for the spin 1/2 Heisenberg XXZ spin chain are numerically solved, and the energy eigenvalues are determined for the anti-ferromagnetic case. We examine the relation between the XXZ spin chain and the Thirring model, and show that the spectrum of the XXZ spin chain is different from that of the regularized Thirring model. 
  We show that localized N-body soliton states exist for a quantum integrable derivative nonlinear Schrodinger model for several non-overlapping ranges (called bands) of the coupling constant \eta. The number of such distinct bands is given by Euler's \phi-function which appears in the context of number theory. The ranges of \eta within each band can also be determined completely using concepts from number theory such as Farey sequences and continued fractions. We observe that N-body soliton states appearing within each band can have both positive and negative momentum. Moreover, for all bands lying in the region \eta > 0, soliton states with positive momentum have positive binding energy (called bound states), while the states with negative momentum have negative binding energy (anti-bound states). 
  The paper is devoted to the hierarchy problem of a symmetry breaking in the Nonsymmetric Kaluza-Klein (Jordan-Thiry) Theory. The basic idea consists in a deformation of a vacuum states manifold to the cartesian product of vacuum states manifolds of every stage of a symmetry breaking. 
  In Article I, a harmonic-oscillator model of a universe of n quarks is infinitesimally modified to eliminate the background reference frame. As a result, quark trajectories exhibit the unification of gravity and the harmonic oscillator near the horizon of a quantum black hole, a region that is approximately flat in space-time. Constituent quarks are confined to composite particles by cluster decomposition rather than a binding force. Here, the composite-particles are input for a perturbation model of particle-exchange interactions. As in Article I, the Hamiltonian cannot be expressed as H=H0+HI where H0 generates the unperturbed equations of motion. In the present article, H0 annihilates the initial state. Quark substructures yield exchange particles of various masses and angular momenta and thus a natural unification of forces. The background-frame elimination in the fundamental model implies causality and unitarity in the effective model. Feynman propagators are derived and first-order scattering amplitudes are calculated for elastic and inelastic scattering. 
  We review some facts about various T-dualities and sigma models on group manifolds, with particular emphasis on supersymmetry. We point out some of the problems in reconciling Poisson-Lie duality with the bi-hermitean geometry of N=2 supersymmetric sigma models. A couple of examples of supersymmetric models admitting Poisson-Lie duality are included. 
  We study the cosmology of the Randall-Sundrum brane-world where the Einstein-Hilbert action is modified by curvature correction terms: a four-dimensional scalar curvature from induced gravity on the brane, and a five-dimensional Gauss-Bonnet curvature term. The combined effect of these curvature corrections to the action removes the infinite-density big bang singularity, although the curvature can still diverge for some parameter values. A radiation brane undergoes accelerated expansion near the minimal scale factor, for a range of parameters. This acceleration is driven by the geometric effects, without an inflaton field or negative pressures. At late times, conventional cosmology is recovered. 
  We report on the gauged supergravity interpretation of certain compactifications of superstring theories with p-form fluxes turned on. We discuss in particular the interplay of duality symmetries in type IIB orientifolds and gauged isometries in the corresponding supergravity models. Turning on fluxes is generally described by the gauging of some nilpotent Lie group whose generators correspond to axion symmetries of R-R and N-S scalars. 
  We develop a novel approach to gravity that we call `matrix general relativity' (MGR) or `gravitational chromodynamics' (GCD or GQCD for quantum version). Gravity is described in this approach not by one Riemannian metric (i.e. a symmetric two-tensor field) but by a multiplet of such fields, or by a matrix-valued symmetric two-tensor field that satisfies certain conditions. We define the matrix extensions of standard constructions of differential geometry including connections and curvatures, and finally, an invariant functional of the new field that reduces to the standard Einstein action functional in the commutative (diagonal) case. Our main idea is the analogy with Yang-Mills theory (QCD and Standard Model). We call the new degrees of freedom of gravity associated with the matrix structure `gravitational color' or simply `gravicolor' and introduce a new gauge symmetry associated with this degree of freedom. As in the Standard Model there are two possibilities. First of all, it is possible that at high energies (say at Planckian scale) this symmetry is exact (symmetric phase), but at low energies it is badly broken, so that one tensor field remains massless (and gives general relativity) and the other ones become massive with the masses of Planckian scale. Second possibilty is that the additional degrees of freedom of gravitational field are confined within the Planckian scale. What one sees at large distances are singlets (invariants) of the new gauge symmetry. 
  We propose an approach to formulating string theory in a curved spacetime, which is based on the connection between the states of the WZW model for the isometry group of a background spacetime metric and the representations of the corresponding quantum group. In this approach the string states scattering amplitudes are defined by certain evaluations of the theta spin networks for the associated quantum group. We examine the evaluations given by the spin network invariants defined by the spin foam state sum model associated to the two-dimensional BF theory for the background isometry group. We show that the corresponding string amplitudes are well defined if the spacetime manifold is compact and admits a group metric. We compute the simplest scattering amplitudes in the case of the SU(2) background isometry group, and we provide arguments that these are the amplitudes of a topological string theory. 
  We study type IIB supergravity solutions with four supersymmetries that interpolate between two types widely considered in the literature: the dual of Becker and Becker's compactifications of M-theory to 3 dimensions and the dual of Strominger's torsion compactifications of heterotic theory to 4 dimensions. We find that for all intermediate solutions the internal manifold is not Calabi-Yau, but has SU(3) holonomy in a connection with a torsion given by the 3-form flux. All 3-form and 5-form fluxes, as well as the dilaton, depend on one function appearing in the supersymmetry spinor, which satisfies a nonlinear differential equation. We check that the fields corresponding to a flat bound state of D3/D5-branes lie in our class of solutions. The relations among supergravity fields that we derive should be useful in studying new gravity duals of gauge theories, as well as possibly compactifications. 
  Form Factor Perturbation Theory is applied to study the spectrum of the O(3) non--linear sigma model with the topological term in the vicinity of $\theta = \pi$. Its effective action near this value is given by the non--integrable double Sine--Gordon model. Using previous results by Affleck and the explicit expressions of the Form Factors of the exponential operators $e^{\pm i\sqrt{8\pi} \phi(x)}$, we show that the spectrum consists of a stable triplet of massive particles for all values of $\theta$ and a singlet state of higher mass. The singlet is a stable particle only in an interval of values of $\theta$ close to $\pi$ whereas it becomes a resonance below a critical value $\theta_c$. 
  Using the AdS/CFT correspondence we study the holographic principle and the CFT/FRW relations in the near-horizon AdS(5)xS(5) geometry with a probe D3-brane playing the role of the boundary to this space. The motion of the probe D3-brane in the bulk, induces a cosmological evolution on the brane. As the brane crosses the horizon of the bulk black hole, it probes the holography of the dual CFT. We test the holographic principle and we find corrections to CFT/FRW relations in various physical cases: for radially moving, spinning and electrically charged D3-brane and for a NS/NS B-field in the bulk. 
  We investigate a possible unified theory of all interactions which is based only on fundamental spinor fields. The vielbein and metric arise as composite objects. The effective quantum gravitational theory can lead to a modification of Einstein's equations due to the lack of local Lorentz-symmetry. We explore the generalized gravity with global instead of local Lorentz symmetry in first order of a systematic derivative expansion. At this level diffeomorphisms and global Lorentz symmetry allow for two new invariants in the gravitational effective action. The one which arises in the one loop approximation to spinor gravity is consistent with all present tests of general relativity and cosmology. This shows that local Lorentz symmetry is tested only very partially by present observations. In contrast, the second possible new coupling is severely restricted by present solar system observations. 
  We study intersecting D-branes in a type IIB plane wave background using Green-Schwarz worldsheet formulation. We consider all possible $D_\pm$-branes intersecting at angles in the plane wave background and identify their residual supersymmetries. We find, in particular, that $D_\mp - D_\pm$ brane intersections preserve no supersymmetry. We also present the explicit worldsheet expressions of conserved supercharges and their supersymmetry algebras. 
  The inverse time ordered Green's function is defined in a covariant three-dimensional formalism using the diagram approach.The equation of motion of a two-quark composite system in an external electromagnetic field up to the second order in the field strength is obtained. The polarizability of the $\pi$-meson is calculated using this formalism. The mean square radius of the two-quark system is calculated from the wave functions of the system. In addition, radiative corrections are calculated for both the polarizability and the mean square radius. 
  We give the explicit form of the half-string representation in the continuous kappa basis. We show the comma structure of the three-vertex, when expanded around an arbitrary projector, and that the zero-mode must be replaced by the mid-point degree of freedom in the half-string representation. The treatment of the ghost sector enables us to calculate the normalization of the vertices. The simplicity of this formalism is demonstrated with some applications, such as gauge transformations and identification of subalgebras. 
  A general formalism is developed that allows the construction of a field theory on quantum spaces which are deformations of ordinary spacetime. The symmetry group of spacetime (Poincar\' e group) is replaced by a quantum group. This formalism is demonstrated for the kappa-deformed Poincar\'e algebra and its quantum space. The algebraic setting is mapped to the algebra of functions of commuting variables with a suitable star-product. Fields are elements of this function algebra. The Dirac and Klein-Gordon equation are defined and an action is found from which they can be derived. 
  The casimir pressure on a non ideal conducting slab is calculated. Using a simple model for the conductivity according to which the slab is perfectly conducting at frequencies below plasma frequency omega_p and perfectly transparent above such frequency, it is found that the vacuum pressure on each surface of the slab is (h omega_p^4)/(24 pi^2 c^3) which is finite without removal of any divergence 
  In a toy model we derive the gravitational equation on a self-gravitating curved D-brane. The effective theory on the brane is drastically changed from the ordinal Einstein equation. The net cosmological constant on the brane depends on a tuning between the brane tension and the brane charges. Moreover, non-zero matter stress tensor exists if the net cosmological constant is not zero. This fact indicates a direct connection between matters on the brane and the dark energy. 
  We carefully review the basic examples of anomaly cancellation in M-theory: the 5-brane anomalies and the anomalies on S^1/Z_2. This involves cancellation between quantum anomalies and classical inflow from topological terms. To correctly fix all coefficients and signs, proper attention is paid to issues of orientation, chirality and the Euclidean continuation. Independent of the conventions chosen, the Chern-Simons and Green-Schwarz terms must always have the same sign. The reanalysis of the reduction to the heterotic string on S^1/Z_2 yields a surprise: a previously neglected factor forces us to slightly modify the Chern-Simons term, similar to what is needed for cancelling the normal bundle anomaly of the 5-brane. This modification leads to a local cancellation of the anomaly, while maintaining the periodicity on S^1. 
  We derive a duality-symmetric action for type IIA D=10 supergravity by the Kaluza-Klein dimensional reduction of the duality-symmetric action for D=11 supergravity with the 3-form and 6-form gauge field. We then double the bosonic fields arising as a result of the Kaluza-Klein dimensional reduction and add mass terms to embrace the Romans's version, so that in its final form the bosonic part of the action contains the dilaton, NS-NS and RR potentials of the standard type IIA supergravity as well as their duals, the corresponding duality relations are deduced directly from the action. We discuss the relation of our approach to the doubled field formalism by Cremmer, Julia, Lu and Pope, complete the extension of this construction to the supersymmetric case and lift it onto the level of the proper duality-symmetric action. We also find a new dual formulation of type IIA D=10 supergravity in which the NS-NS two-form potential is replaced with its six-form counterpart. A truncation of this dual model produces the Chamseddine's version of N=1, D=10 supergravity. 
  We study the renormalization group flows of two-dimensional metrics in sigma models and demonstrate that they provide a continual analogue of the Toda field equations based on the infinite dimensional algebra G(d/dt;1). The resulting Toda field equation is a non-linear generalization of the heat equation, which is integrable in target space and shares the same dissipative properties in time. We provide the general solution of the renormalization group flows in terms of free fields, via Backlund transformations, and present some simple examples that illustrate the validity of their formal power series expansion in terms of algebraic data. We study in detail the sausage model that arises as geometric deformation of the O(3) sigma model, and give a new interpretation to its ultra-violet limit by gluing together two copies of Witten's two-dimensional black hole in the asymptotic region. We also provide some new solutions that describe the renormalization group flow of negatively curved spaces in different patches, which look like a cane in the infra-red region. Finally, we revisit the transition of a flat cone C/Z_n to the plane, as another special solution, and note that tachyon condensation in closed string theory exhibits a hidden relation to the infinite dimensional algebra G(d/dt;1) in the regime of gravity. Its exponential growth holds the key for the construction of conserved currents and their systematic interpretation in string theory, but they still remain unknown. 
  The Konishi anomalies for noncommutative N=1 supersymmetric U(1) gauge theory arising from planar and nonplanar diagrams are calculated. Whereas planar Konishi anomaly is the expected \star-deformation of the commutative anomaly, nonplanar anomaly reflects the important features of nonplanar diagrams of noncommutative gauge theories, such as UV/IR mixing and the appearance of nonlocal open Wilson lines. We use the planar and nonplanar Konishi anomalies to calculate the effective superpotential of the theory. In the limit of vanishing |\Theta p|, with \Theta the noncommutativity parameter, the noncommutative effective superpotential depends on a gauge invariant superfield, which includes supersymmetric Wilson lines, and has nontrivial dependence on the gauge field supermultiplet. 
  We describe a duality relation between configurations of M5-branes in M-theory and type IIB theory on Taub-NUT geometries with NSNS and RR 3-form field strength fluxes. The flux parameters are controlled by the angles between the M5-brane and the (T)duality directions. For one M5-brane, the duality leads to a family of supersymmetric flux configurations which interpolates between imaginary self-dual fluxes and fluxes similar to the Polchinski-Strassler kind. For multiple M5-branes, the IIB configurations are related to fluxes for twisted sector fields in orbifolds. The dual M5-brane picture also provides a geometric interpretation for several properties of flux configurations (like the supersymmetry conditions, their contribution to tadpoles, etc), and for many non-trivial effects in the IIB side. Among the latter, the dielectric effect for probe D3-branes is dual to the recombination of probe M5-branes with background ones; also, a picture of a decay channel for non-supersymmetric fluxes is suggested. 
  A family of modified $(0,1)$ heterotic string models in D=4 is constructed in which the strings incorporate $R$ flux tubes which may in special cases support a local spacetime superconformal symmetry consistent with quantum mechanics. There is an intrinsic Goldstino multiplet, so that supersymmetry breaking can be driven by any process that generates a non-zero value for the superpotential. The superconformal anomaly freedom of these models may be related to a naturally vanishing $\Lambda$. The $R$ charge of such models may also play a role in producing a generation-ordered fermion spectrum. \ 
  We study black holes in three-dimensional Chern-Simons gravity with a negative cosmological constant. In particular, we identify how the Chern-Simons interactions between a scattering particle and a black hole project the particle wavefunction onto a wavefunction in the black hole background. We also analyze the set of space-times that should be allowed in the theory and the way in which boundary conditions affect the spectrum of space-times. 
  Soliton equations are derived which characterize the boundary CFT a la Callan et al. Soliton fields of classical soliton equations are shown to appear as a neutral bound state of a pair of soliton fields of BCFT. One soliton amplitude under the influence of the boundary is calculated explicitly and is shown that it is frozen at the Dirichlet limit. 
  We extend a recent scenario of Kachru, Kallosh, Linde and Trivedi to fix the string moduli fields by using a combination of fluxes and non-perturbative superpotentials, leading to de Sitter vacua. In our scenario the non-perturbative superpotential is taken to be the N=1^* superpotential for an SU(N) theory, originally computed by Dorey and recently rederived using the techniques of Dijkgraaf-Vafa. The fact that this superpotential includes the full instanton contribution gives rise to the existence of a large number of minima, increasing with N. In the absence of supersymmetry breaking these correspond to supersymmetric anti de Sitter vacua. The introduction of antibranes lifts the minima to a chain of (non-supersymmetric) de Sitter minima with the value of the cosmological constant decreasing with increasing compactification scale. Simpler cases are also discussed, including a finite instanton sum, as in the racetrack scenario. The relative semiclassical stability of these vacua is studied. Possible cosmological implications of these potentials are also discussed. 
  This article is based on the covariant canonical formalism and corresponding symplectic structure on phase space developed by Witten, Zuckerman and others in the context of field theory. After recalling the basic principles of this procedure, we construct the conserved bilinear symplectic current for generic elastic string models. These models describe current carrying cosmic strings evolving in an arbitrary curved background spacetime. Particular attention is paid to the special case of the chiral string for which the worldsheet current is null. Different formulations of the chiral string action are discussed in detail, and as a result the integrability property of the chiral string is clarified. 
  Using Einstein, Landau-Lifshitz, Papapetrou and Weinberg energy-momentum complexes we explicitly evaluate the energy and momentum distributions associated with a non-static and circularly symmetric three-dimensional spacetime. The gravitational background under study is an exact solution of the Einstein's equations in the presence of a cosmological constant and a null fluid. It can be regarded as the three-dimensional analogue of the Vaidya metric and represents a non-static spinless (2+1)-dimensional black hole with an outflux of null radiation. All four above-mentioned prescriptions give exactly the same energy and momentum distributions for the specific black hole background. Therefore, the results obtained here provide evidence in support of the claim that for a given gravitational background, different energy-momentum complexes can give identical results in three dimensions. Furthermore, in the limit of zero cosmological constant the results presented here reproduce the results obtained by Virbhadra who utilized the Landau-Lifshitz energy-momentum complex for the same (2+1)-dimensional black hole background in the absence of a cosmological constant. 
  The Casimir energy is evaluated for massless scalar fields under Dirichlet or Neumann boundary conditions, and for the electromagnetic field with perfect conductor boundary conditions on one and two infinite parallel plates moving by uniform proper acceleration through the Fulling--Rindler vacuum in an arbitrary number of spacetime dimension. For the geometry of a single plate the both regions of the right Rindler wedge, (i) on the right (RR region) and (ii) on the left (RL region) of the plate are considered. The zeta function technique is used, in combination with contour integral representations. The Casimir energies for separate RR and RL regions contain pole and finite contributions. For an infinitely thin plate taking RR and RL regions together, in odd spatial dimensions the pole parts cancel and the Casimir energy for the whole Rindler wedge is finite. In $d=3$ spatial dimensions the total Casimir energy for a single plate is negative for Dirichlet scalar and positive for Neumann scalar and the electromagnetic field. The total Casimir energy for two plates geometry is presented in the form of a sum of the Casimir energies for separate plates plus an additional interference term. The latter is negative for all values of the plates separation for both Dirichlet and Neumann scalars, and for the electromagnetic field. 
  We respond to `Comments on the U(2) ADHM two-instanton' [Y. Tian, Phys. Lett. B 566 (2003) 183]. 
  The Wess-Zumino model on N=1/2 nonanticommutative superspace, which contains the dimension-6 term F^3, is shown to be renormalizable to all orders in perturbation theory, upon adding F and F^2 terms to the original Lagrangian. The renormalizability is possible, even with this higher-dimension operator, because the Lagrangian is not hermitian. Such deformed field theories arise naturally in string theory with a graviphoton background. 
  Geometric discretisation draws analogies between discrete objects and operations on a complex with continuum ones on a manifold. We generalise the theory to the cubic case and incorporate metric, by adding volume factors to our discrete Hodge star and then by modifying our inner product which leads to the same result. 
  A method for computing the open string mirror map and superpotential for noncompact Calabi-Yaus, following the physical computations of Lerche and Mayr, is presented. It is also shown that the obvious extension of these techniques to the compact case is not consistent. As an example, the local CP^2 case is worked out in 2 ways. 
  We realize the physical N-anyon Hilbert spaces, introduced previously via unitary representations of the group of diffeomorphisms of the plane, as N-fold braided-symmetric tensor products of the 1-particle Hilbert space. This perspective provides a convenient Fock space construction for nonrelativistic anyon quantum fields along the more usual lines of boson and fermion fields, but in a braided category. We see how essential physical information is thus encoded. In particular we show how the algebraic structure of our anyonic Fock space leads to a natural anyonic exclusion principle related to intermediate occupation number statistics, and obtain the partition function for an idealised gas of fixed anyonic vortices. 
  In this dissertation we explore various aspects of the AdS/CFT correspondence. As string quantization on general backgrounds with fluxes is very difficult, one often uses the duality at the level of canonical fields of supergravity and the corresponding half-BPS operators in SYM, since they both belong to the shortest multiplets of the superconformal group SU(2,2|4). In addition to half-BPS operators, there are others with non-renormalization properties. One such class of operators is the quarter-BPS operators, which are dual to threshold bound states of elementary supergravity excitations. Their scaling dimension is also determined by their internal quantum numbers. Extended superspace methods make it simple to identify and remove descendant pieces from quarter-BPS candidates. We also compute three-point functions involving quarter-BPS operators, and explain how their non-renormalization translates into statements about the dual supergravity quantities. But we can go beyond discussing protected operators. The GS superstring on AdS(5) x S(5) can be quantized exactly in the limit where the AdS(5) radius R goes to infinity and the R-charge J scales like R^2. BMN states are then dual to single trace operators with certain phases inserted (BMN operators). BMN operators are another natural generalization of half-BPS operators. The perturbative expansion of scaling dimensions of BMN operators is in powers of gN/J^2. Moreover, one can do perturbation theory around the infinite R, J ~ R^2 limit. Both expansions have the same regime of validity in string theory and in SYM. We calculate the first 1/R^2 and 1/J corrections, and find complete agreement between the dual quantities. 
  Inflation predicts a primordial gravitational wave spectrum that is slightly ``red,'' i.e., nearly scale-invariant with slowly increasing power at longer wavelengths. In this paper, we compute both the amplitude and spectral form of the primordial tensor spectrum predicted by cyclic/ekpyrotic models. The spectrum is blue and exponentially suppressed compared to inflation on long wavelengths. The strongest observational constraint emerges from the requirement that the energy density in gravitational waves should not exceed around 10 per cent of the energy density at the time of nucleosynthesis. 
  We apply an improved renormalization group analysis for pure Yang-Mills theory at one loop order and obtained the result that a non-perturbatively generated pole mass of gluon emerges as $M_P^2/\Lambda^2 \simeq 0.66$, where $\Lambda$ is the $MS$-bar scale. 
  A string in four dimensions is constructed by supplementing it with forty four Majorana fermions. The later are represented by eleven vectors in the bosonic representation $SO(D-1,1)$. The central charge is 26. The fermions are grouped in such a way that the resulting action is world sheet supersymmetric. The energy momentum and current generators satisfy the super-Virasoro algebra. GSO projections are necessary for proving modular invariance. Space-time supersymmetry algebra is deduced and is substantiated for specific modes of zero mass. The symmetry group of the model can descend to the low energy standard model group $SU (3) \times SU_L (2) \times U_Y (1)$ through the Pati-Salam group. 
  We investigate gauge invariant operators corresponding to on-shell closed string states in open string field theory. Using both oscillator representation and conformal mapping techniques, we calculate a two closed string tachyon amplitude that connects two gauge invariant operators by an open string propagator.We find that this amplitude is in a complete agreement with the usual disc amplitude. 
  We consider Uncertainty Principles which take into account the role of gravity and the possible existence of extra spatial dimensions. Explicit expressions for such Generalized Uncertainty Principles in 4+n dimensions are given and their holographic properties investigated. In particular, we show that the predicted number of degrees of freedom enclosed in a given spatial volume matches the holographic counting only for one of the available generalizations and without extra dimensions. 
  Laughlin's Ansatz to explain the fractional Quantum Hall effect is derived by coupling a particle associated with ``exotic'' the two-fold central extension of the planar Galilei group. The reduced system is identical to the one used to describe the dynamics of vortices in an incompressible planar fluid. 
  The Veneziano-Yankielowicz glueball superpotential for an arbitrary N=1 SUSY pure gauge theory with classical gauge group is derived using an approach following recent work of Dijkgraaf, Vafa and others. These non-perturbative terms, which had hitherto been included by hand in the above approach, are thus seen to arise naturally, and the approach is rendered self-contained. By minimising the glueball superpotential for theories with fundamental matter added, the expected vacuum structure with gaugino condensation and chiral symmetry breaking is obtained. Various possible extensions are also discussed. 
  Some cosmological implications of ultraviolet quantum effects leading to a condensation of Born-Infeld matter are considered. It is shown that under very general conditions the quantum condensate can not act as phantom matter if its energy density is positive. On the other hand, it behaves as an effective cosmological constant in the limit where quantum induced contributions to the energy-momentum tensor dominate over the classical effects. 
  We solve the topological Poisson Sigma model for a Poisson-Lie group $G$ and its dual $G^*$. We show that the gauge symmetry for each model is given by its dual group that acts by dressing transformations on the target. The resolution of both models in the open geometry reveals that there exists a map from the reduced phase of each model ($P$ and $P^*$) to the main symplectic leaf of the Heisenberg double ($D_{0}$) such that the symplectic forms on $P$, $P^{*}$ are obtained as the pull-back by those maps of the symplectic structure on $D_{0}$. This uncovers a duality between $P$ and $P^{*}$ under the exchange of bulk degrees of freedom of one model with boundary degrees of freedom of the other one. We finally solve the Poisson Sigma model for the Poisson structure on $G$ given by a pair of $r$-matrices that generalizes the Poisson-Lie case. The Hamiltonian analysis of the theory requires the introduction of a deformation of the Heisenberg double. 
  We consider the compactification of (d+n)-dimensional pure gravity and of superstring/M-theory on an n-dimensional internal space to a d-dimensional FLRW cosmology, with spatial curvature k=-1,0,+1, in Einstein conformal frame. The internal space is taken to be a product of Einstein spaces, each of which is allowed to have arbitrary curvature and a time-dependent volume. By investigating the effective d-dimensional scalar potential, which is a sum of exponentials, it is shown that such compactifications, in the k=0,+1 cases, do not lead to large amounts of accelerating expansion of the scale factor of the resulting FLRW universe, and, in particular, not to inflation. The case k=-1 admits solutions with eternal accelerating expansion for which the acceleration, however, tends to zero at late times. 
  We argue that there is an equivalence of M-theory on T^3 \times A_{N-1} with a four-dimensional non-supersymmetric quiver gauge theory on the Higgs branch. The quiver theory in question has gauge group SU(N)^{N_4N_6N_8} and is considered in a strong coupling and large N_{4,6,8} limit. We provide field- and string-theoretical evidence for the equivalence making use of the deconstruction technique. In particular, we find wrapped M2-branes in the mass spectrum of the quiver theory at low energies. 
  Slavnov-Taylor 1.0 is a Mathematica package which allows us to perform automatic symbolic computation in BRST formalism. This article serves as a self-contained guide to prospective users, and indicates the conventions and approximations used. 
  We study a chiral N=1, U(N) field theory in the context of the Dijkgraaf-Vafa correspondence. Our model contains one adjoint, one conjugate symmetric and one antisymmetric chiral multiplet, as well as eight fundamentals. We compute the generalized Konishi anomalies and compare the chiral ring relations they induce with the loop equations of the (intrinsically holomorphic) matrix model defined by the tree-level superpotential of the field theory. Surprisingly, we find that the matrix model is well-defined only if the number of flavors equals two! Despite this mismatch, we show that the 1/N expansion of the loop equations agrees with the generalized Konishi constraints. This indicates that the matrix model - gauge theory correspondence should generally be modified when applied to theories with net chirality. We also show that this chiral theory produces the same gaugino superpotential as a nonchiral SO(N) model with a single symmetric multiplet and a polynomial superpotential. 
  We study various aspects of orientifold projections of Type IIB closed string theory on Gepner points in different dimensions. The open string sector is introduced, in the usual constructive way, in order to cancel RR charges carried by orientifold planes. Moddings by cyclic permutations of the internal N=2 superconformal blocks as well as by discrete phase symmetries are implemented. Reduction in the number of generations, breaking or enhancements of gauge symmetries and topology changes are shown to be induced by such moddings. Antibranes sector is also considered; in particular we show how non supersymmetric models with antibranes and free of closed and open tachyons do appear in this context. A systematic study of consistent models in D=8 dimensions and some illustrative examples in D=6 and D=4 dimensions are presented. 
  We search for tubular solutions in unstable D3-brane. With critical electric field E=1, solutions representing supertubes, which are supersymmetric bound states of fundamental strings, D0-branes, and a cylindrical D2-brane, are found and shown to exhibit BPS-like property. We also point out that boosting such a {\it tachyon tube} solution generates string flux winding around the tube, resulting in helical electric fluxes on the D2-brane. We also discuss issues related to fundamental string, absence of magnetic monopole, and finally more tachyon tubes with noncritical electric field. 
  The Schouten-Nijenhuis bracket is generalized for the superspace case and for the Poisson brackets of opposite Grassmann parities. Quite a number of generalizations for the differential analog of the special Yang-Baxter equations is also proposed. 
  We utilize a diagrammatic notation for invariant tensors to construct the Young projection operators for the irreducible representations of the unitary group U(n), prove their uniqueness, idempotency, and orthogonality, and rederive the formula for their dimensions. We show that all U(n) invariant scalars (3n-j coefficients) can be constructed and evaluated diagrammatically from these U(n) Young projection operators. We prove that the values of all U(n) 3n-j coefficients are proportional to the dimension of the maximal representation in the coefficient, with the proportionality factor fully determined by its S[k] symmetric group value. We also derive a family of new sum rules for the 3-j and 6-j coefficients, and discuss relations that follow from the negative dimensionality theorem. 
  The one-loop structure of the trace anomaly is investigated using different regularizations and renormalization schemes: dimensional, proper time and Pauli-Villars. The universality of this anomaly is analyzed from a very general perspective. The Euler and Weyl terms of the anomalous trace of the stress tensor are absolutely universal. The pure derivative $ \square R$-term is shown to be universal only if the regularization breaks conformal symmetry softly. If the breaking of conformal symmetry by the regularization method is hard the coefficient of this term might become arbitrary which points out the presence of an ambiguous $ \int\sqrt{-g} R^2$-term in the effective quantum action. These ambiguities arise in some prescriptions of dimensional and Pauli-Villars regularizations. We discuss the implications of these results for anomaly-induced inflationary scenarios and AdS/CFT correspondence. 
  Using a generalized Weyl formalism, we show how stationary, axisymmetric solutions of the four-dimensional vacuum Einstein equation can be turned into static, axisymmetric solutions of five-dimensional dilaton gravity coupled to a two-form gauge field. This procedure is then used to obtain new solutions of the latter theory describing pairs of extremal magnetic black holes with opposite charges, known as black diholes. These diholes are kept in static equilibrium by membrane-like conical singularities stretching along two different directions. We also present solutions describing diholes suspended in a background magnetic field, and with unbalanced charges. 
  We study the boundary correlation functions in Liouville theory and in solvable statistical models of 2D quantum gravity. In Liouville theory we derive functional identities for all fundamental boundary structure constants, similar to the one obtained for the boundary two-point function by Fateev, Zamolodchikov and Zamolodchikov. All these functional identities can be written as difference equations with respect to one of the boundary parameters. Then we switch to the microscopic realization of 2D quantum gravity as a height model on a dynamically triangulated disc and consider the boundary correlation functions of electric, magnetic and twist operators. By cutting open the sum over surfaces along a domain wall, we derive difference equations identical to those obtained in Liouville theory. We conclude that there is a complete agreement between the predictions of Liouville theory and the discrete approach. 
  We studied the phase structures of N=1 supersymmetric USp(2N_c) gauge theory with N_f flavors in the fundamental representation as we deformed the N=2 supersymmetric QCD by adding the superpotential for adjoint chiral scalar field. We determined the most general factorization curves for various breaking patterns, for example, the two different breaking patterns of quartic superpotential. We observed all kinds of smooth transitions for quartic superpotential. Finally we discuss the intriguing role of USp(0) in the phase structure and the possible connection with observations made recently in hep-th/0304271 (Aganagic, Intriligator, Vafa and Warner) and in hep-th/0307063 (Cachazo). 
  We show that solitonic solutions of the classical string action on the AdS_5 x S^5 background that carry charges (spins) of the Cartan subalgebra of the global symmetry group can be classified in terms of periodic solutions of the Neumann integrable system. We derive equations which determine the energy of these solitons as a function of spins. In the limit of large spins J, the first subleading 1/J coefficient in the expansion of the string energy is expected to be non-renormalised to all orders in the inverse string tension expansion and thus can be directly compared to the 1-loop anomalous dimensions of the corresponding composite operators in N=4 super YM theory. We obtain a closed system of equations that determines this subleading coefficient and, therefore, the 1-loop anomalous dimensions of the dual SYM operators. We expect that an equivalent system of equations should follow from the thermodynamic limit of the algebraic Bethe ansatz for the SO(6) spin chain derived from SYM theory. We also identify a particular string solution whose classical energy exactly reproduces the one-loop anomalous dimension of a certain set of SYM operators with two independent R charges J_1, J_2. 
  Motivated by the recent interest on models with varying constants and whether black hole physics can constrain such theories, two-dimensional charged stringy black holes are considered. We exploit the role of two-dimensional stringy black holes as toy models for exploring paradoxes which may lead to constrains on a theory. A two-dimensional charged stringy black hole is investigated in two different settings. Firstly, the two-dimensional black hole is treated as an isolated object and secondly, it is contained in a thermal environment. In both cases, it is shown that the temperature and the entropy of the two-dimensional charged stringy black hole are decreased when its electric charge is increased in time. By piecing together our results and previous ones, we conclude that in the context of black hole thermodynamics one cannot derive any model independent constraints for the varying constants. Therefore, it seems that there aren't any varying constant theories that are out of favor with black hole thermodynamics. 
  We calculate the spectrum of the linearized supergravity around the maximally supersymmetric pp-wave background in eleven dimensions. The resulting spectrum agrees with that of zero-mode Hamiltonian of a supermembrane theory on the pp-wave background. We also discuss the connection with the Kaluza-Klein zero modes of AdS_4 x S^7 background. 
  By applying a set of Hassan-Sen transformations and string dualities to the Kerr-Godel solution of minimal D=5 supergravity we derive a four parameter family of five dimensional solutions in type II string theory. They describe rotating, charged black holes in a rotating background. For zero background rotation, the solution is D=5 Kerr-Newman; for zero charge it is Kerr-Godel. In a particular extremal limit the solution describes an asymptotically Godel BMPV black hole. 
  We consider two dimensional supergravity coupled to $\hat c=1$ matter. This system can also be interpreted as noncritical type 0 string theory in a two dimensional target space. After reviewing and extending the traditional descriptions of this class of theories, we provide a matrix model description. The 0B theory is similar to the realization of two dimensional bosonic string theory via matrix quantum mechanics in an inverted harmonic oscillator potential; the difference is that we expand around a non-perturbatively stable vacuum, where the matrix eigenvalues are equally distributed on both sides of the potential. The 0A theory is described by a quiver matrix model. 
  It was shown in hep-th/0301099 that the leading UV/IR mixing effects in noncommutative gauge theories on D-branes are able to capture information about the closed string spectrum of the parent string theory. The analysis was carried out for D-branes on nonsupersymmetric C^3/Z_N orbifolds of Type IIB. In this paper we consider D-branes on twisted circles compactifications of Type II string theory. We find that the signs of the leading UV/IR mixing effects know about the mass gap between the lowest modes in NSNS and RR closed string towers. Moreover, the relevant piece of the field theory effective action can be reproduced purely in the language of closed strings. Remarkably this approach unifies in a single structure, that of a closed string exchange between D-branes, both the leading planar and nonplanar effects associated to the absence of supersymmetry. 
  We propose that there is a unique expansion for the string theory S-matrix elements of tachyons that corresponds to non-abelian tachyon action. For those S-matrix elements which, in their expansion, there are the Feynman amplitudes resulting from the non-abelian kinetic term, we give a prescription on how to find the expansion. The gauge invariant action is an $\alpha'$ expanded action, and the tachyon mass $m$ which appears as coefficient of many different couplings, is arbitrary. We then analyze in details the S-matrix element of four tachyons and the S-matrix element of two tachyons and two gauge fields, in both bosonic and superstring theories, in favor of this proposal. In the superstring theory, the leading terms of the non-abelian gauge invariant couplings are in agreement with the symmetrised trace of the direct non-abelian generalization of the tachyonic Born-Infeld action in which the tachyon potential is consistent with $V(T)=e^{\pi\alpha' m^2T^2}$. In the bosonic theory, on the other hand, the leading terms are those appear in superstring case as well as some other gauge invariant couplings which spoils the symmetrised trace prescription. These latter terms are zero in the abelian case. 
  We compare the glueball mass spectrum of an effective N=1 pure super Yang-Mills theory formulated in terms of a three-form supermultiplet with the available lattice data. These confirm the presence of four scalars and two Majorana fermions but the detailed mass spectrum is difficult to reconcile with the effective supersymmetric theory. By imposing supersymmetry and using two of four bosonic masses we get a prediction for the remaining masses as well as the mixing angles. We find that the mass of the three-form dominates over the contribution of the Veneziano-Yankielowicz-Dijkgraaf-Vafa term. As a byproduct we introduce a Fayet-Iliopoulos term for the three-form multiplet and show that it generates a glueball condensate. 
  We give a comprehensive review of various methods to define currents and the energy-momentum tensor in classical field theory, with emphasis on a geometric point of view. The necessity of ``improving'' the expressions provided by the canonical Noether procedure is addressed and given an adequate geometric framework. The main new ingredient is the explicit formulation of a principle of ``ultralocality'' with respect to the symmetry generators, which is shown to fix the ambiguity inherent in the procedure of improvement and guide it towards a unique answer: when combined with the appropriate splitting of the fields into sectors, it leads to the well-known expressions for the current as the variational derivative of the matter field Lagrangian with respect to the gauge field and for the energy-momentum tensor as the variational derivative of the matter field Lagrangian with respect to the metric tensor. In the second case, the procedure is shown to work even when the matter field Lagrangian depends explicitly on the curvature, thus establishing the correct relation between scale invariance, in the form of local Weyl invariance ``on shell'', and tracelessness of the energy-momentum tensor, required for a consistent definition of the concept of a conformal field theory. 
  The oscillator representation is used for the non-perturbative description of vacuum particle creation in a strong time-dependent electric field in the framework of scalar QED. It is shown that the method can be more effective for the derivation of the quantum kinetic equation (KE) in comparison with the Bogoliubov method of time-dependent canonical transformations. This KE is used for the investigation of vacuum creation in periodical linear and circular polarized electric fields and also in the case of the presence of a constant magnetic field, including the back reaction problem. In particular, these examples are applied for a model illustration of some features of vacuum creation of electron-positron plasma within the planned experiments on the X-ray free electron lasers. 
  We analyze further the possibility of obtaining localized black hole solutions in the framework of Randall-Sundrum-type brane-world models. We consider black hole line-elements analytic at the horizon, namely, generalizations of the Painleve and Vaidya metrics, which are taken to have a decaying dependence of the horizon on the extra dimension. These backgrounds have no other singularities apart from the standard black hole singularity which is localized in the direction of the fifth dimension. Both line-elements can be sustained by a regular, shell-like distribution of bulk matter of a non-standard form. Of the two, the Vaidya line-element is shown to provide the most attractive, natural choice: despite the scaling of the horizon, the 5D spacetime has the same topological structure as the one of a RS-Schwarzschild spacetime and demands a minimal bulk energy-momentum tensor. 
  We evaluate the one-loop prefactor in the false vacuum decay rate in a theory of a self interacting scalar field in 3+1 dimensions. We use a numerical method, established some time ago, which is based on a well-known theorem on functional determinants. The proper handling of zero modes and of renormalization is discussed. The numerical results in particular show that quantum corrections become smaller away from the thin-wall case. In the thin-wall limit the numerical results are found to join into those obtained by a gradient expansion. 
  We identify the strong coupling fishnet diagram with a certain Ising spin configuration in the lightcone worldsheet description of planar Tr\Phi^3 field theory. Then, using a mean field formalism, we take the remaining planar diagrams into account in an average way. The fishnet spin configuration requires two mean fields \phi,\phi' where the fishnet diagram is the case \phi=1,\phi'=0. For general values of these fields, the system is then approximated as a light-cone quantized string with a field dependent effective string tension T_{eff}(\phi,\phi'). We also calculate the worldsheet energy density E(\phi,\phi'), and find the field values that minimize it in the presence of a transverse space infra-red cutoff \epsilon>0. The criterion for string formation is that the tension in this minimum energy state remains non-zero as \epsilon goes to 0. In the most simple-minded implementation of the mean field method, we find that the tension vanishes for weak and moderate coupling, but for very large coupling stays non-zero. A more elaborate treatment, taking temporal correlations into account, removes this ``phase transition'' and the string tension of the minimum energy state vanishes for all values of the coupling when \epsilon goes to 0. Our mean field analysis thus suggests that the ``fishnet phase'' of Tr\Phi^3 theory is unstable, and there is no string formation for any value of the coupling. This is probably a reasonable outcome given the instability of the underlying theory. 
  We review how the identification of gauge theory operators representing string states in the pp-wave/BMN correspondence and their associated anomalous dimension reduces to the determination of the eigenvectors and the eigenvalues of a simple quantum mechanical Hamiltonian and analyze the properties of this Hamiltonian. Furthermore, we discuss the role of random matrices as a tool for performing explicit evaluation of correlation functions. 
  We discuss entropy bounds for a class of two-dimensional gravity models. While the Bekenstein bound can be proved to hold in general for weakly gravitating matter, the analogous of the holographic bound is not universal, but depends on the specific model considered 
  The recently found non-BPS multi-wall configurations in the N=1 supergravity in four dimensions is shown to have no tachyonic scalar fluctuations without additional stabilization mechanisms. Mass of radion (lightest massive fluctuation) is found to be proportional to $Lambda {\rm e}^{-\pi\Lambda R/2}$, where $\Lambda $ is the inverse width of the wall and $ R$ is the radius of compactified dimension. We obtain localized massless graviton and gravitino forming a supermultiplet with respect to the Killing spinor. The relation between the bulk energy density and the boundary energy density (cosmological constants) is an automatic consequence of the field equation and Einstein equation. In the limit of vanishing gravitational coupling, the Nambu-Goldstone modes are reproduced. 
  Accuracy of a relativistic weak-coupling expansion procedure for solving the Hamiltonian bound-state eigenvalue problem in theories with asymptotic freedom is measured using a well-known matrix model. The model is exactly soluble and simple enough to study the method up to sixth order in the expansion. The procedure is found in this case to match the precision of the best available benchmark method of the altered Wegner flow equation, reaching the accuracy of a few percent. 
  We study the causality violation in the non-local phi ^{4}-theory (as formulated by Kleppe and Woodard) containing a finite mass scale Lambda . Starting from the Bogoliubov-Shirkov criterion for causality, we construct and study combinations of S-matrix elements that signal the violation of causality in the one loop approximation. We find that the causality violation in the exclusive process \phi +\phi --> \phi +\phi grows with energy, but the growth with energy, (for low to moderate energies) is suppressed to all orders compared to what one would expect purely from dimensional considerations. We however find that the causality violation in other processes such as \phi +\phi--> \phi +\phi +\phi +\phi grows with energy as expected from dimensional considerations at low to moderate energies. For high enough energies comparable to the mass scale Lambda, however, we find a rapid (exponential-like) growth in the degree of causality violation. We suggest a scenario, based on an earlier work, that will enable one to evade a large theoretical causality violation at high energies, should it be unobserved experimentally. 
  In this letter we use the spurion field approach adopted in hep-th/0307099 in order to show that by adding F and F^2 terms to the original lagrangian, the N=1/2 Wess-Zumino model is renormalizable to all orders in perturbation theory. We reformulate in superspace language the proof given in the recent work hep-th/0307165 in terms of component fields. 
  The properties of gauge-invariant composite operators and their correlation functions in N=4 SYM are discussed in the analytic superspace formalism. A complete classification of the different types of operators in the theory is given. Operators can be either protected or unprotected according to whether they do not or do have anomalous dimensions, and the analytic superspace formalism allows one to identify which type a given operator is in a straightforward manner. A simple discussion is given of the behaviour of reducible multiplets at threshold. It is pointed out that there is a class of ``semi-protected'' operators which do not have anomalous dimensions but which do not necessarily have non-renormalised three-point functions when the other two operators in the correlator are protected, although two-point functions of such operators are non-renormalised. A complete discussion of superconformal invariants in analytic superspace is given. The paper includes a modified discussion of the transformation rules of analytic superfields which clarifies the $U(1)_Y$ properties of operators and correlation functions and, in particular, explicit examples are given of three-point correlation functions which violate this symmetry. A tensor, $\cE$, invariant under $SL(n|m)$ but not under $GL(n|m)$, is introduced and used in the discussion of $U(1)_Y$ and in the construction of invariants. 
  We construct the derivative corrections to the four-point vertices in the abelian open string effective action to all orders in alpha'. The result is based on the structure of the string four-point function. Supersymmnetry of these vertices is guaranteed by the supersymmetry of the F^4 term in the effective action. By this construction we establish the existence of an infinite number of supersymmetry invariants, the number of invariants at order alpha'^n grows linearly with n. 
  To set up a self-consistent quantum field theory of degenerate systems, the unperturbed state should be described by a density matrix instead of a pure state. This increases the combinatorial complexity of the many-body equations. Hopf algebraic techniques are used to deal with this complexity and show that the Schwinger-Dyson equations are modified in a non-trivial way. The hierarchy of Green functions is derived for degenerate systems, and the case of a single electron in a two-fold degenerate orbital is calculated in detail. 
  A general solution of the Batalin-Vilkovisky master equation was formulated in terms of generalized fields. Recently, a superfields approach of obtaining solutions of the Batalin-Vilkovisky master equation is also established. Superfields formalism is usually applied to topological quantum field theories. However, generalized fields method is suitable to find solutions of the Batalin-Vilkovisky master equation either for topological quantum field theories or the usual gauge theories like Yang-Mills theory. We show that by truncating some components of superfields with appropriate actions, generalized fields formalism of the usual gauge theories result. We demonstrate that for some topological quantum field theories and the relativistic particle both of the methods possess the same field contents and yield similar results. Inspired by the observed relations we give the solution of the BV-master equation for on-shell N=1 supersymmetric Yang-Mills theory utilizing superfields. 
  It is pointed, that effects of refraction of electromagnetic radiation in the medium, formed by the magnetized vacuum, become essential already for relatively soft photons, not hard enough to create an electron-positron pair, including those belonging to soft gamma-, X-ray, optic and radio- range, if the magnetic field B exceeds the critical value of Bcr=m^2/e=4.4 10^13 Gauss. Three leading terms in the asymptotic expansion of the one-loop polarization operator in a constant magnetic field are found for B>>Bcr, and the corresponding refraction index is shown to depend only on the propagation direction of the photon relative to the external field. It is established, that the refraction index for one of polarization modes unlimitedly grows with the field, while the other is saturated at a moderate level. The photon capture effect is extended to soft photons. The results may be essential in studying reflection, refraction and splitting of X-rays, light and radio waves by magnetic fields of magnetars, as well as in considering emission of such waves by charged particles . 
  We present a new class of static axially symmetric solutions of SU(2) Yang-Mills-Higgs theory, where the Higgs field vanishes on rings centered around the symmetry axis. Associating a magnetic dipole moment with each Higgs vortex ring, the dipole moments add for solutions in the trivial topological sector, whereas they cancel for magnetically charged solutions. 
  We study the supergravity solutions describing non-extremal enhancons. There are two branches of solutions: a `shell branch' connected to the extremal solution, and a `horizon branch' which connects to the Schwarzschild black hole at large mass. We show that the shell branch solutions violate the weak energy condition, and are hence unphysical. We investigate linearized perturbations of the horizon branch and the extremal solution numerically, completing an investigation initiated in a previous paper. We show that these solutions are stable against the perturbations we consider. This provides further evidence that these latter supergravity solutions are capturing some of the true physics of the enhancon. 
  We formulate Feynman path integral on a non commutative plane using coherent states. The propagator for a free particle exhibits UV cut-off induced by the parameter of non commutativity. 
  We study adding flavors into the Maldacena-N\u{u}nez background. It is achieved by introducing spacetime filling D9-branes or intersecting D5$'$-branes into the background with a wrapping D5-brane. Both D9-branes and D5$'$-branes can be spacetime filling from the 5D bulk point of view. At the probe limit it corresponds to introducing non-chiral fundamental flavors into the dual N=1 SYM. We propose a method to twist the fundamental flavor which has to involve open string charge. It reflects the fact that coupling fundamental matter to SYM in the dual string theory corresponds to adding an open string sector 
  The manner in which continuum center vortices generate topological charge density is elucidated using an explicit example. The example vortex world-surface contains one lone self-intersection point, which contributes a quantum 1/2 to the topological charge. On the other hand, the surface in question is orientable and thus must carry global topological charge zero due to general arguments. Therefore, there must be another contribution, coming from vortex writhe. The latter is known for the lattice analogue of the example vortex considered, where it is quite intuitive. For the vortex in the continuum, including the limit of an infinitely thin vortex, a careful analysis is performed and it is shown how the contribution to the topological charge induced by writhe is distributed over the vortex surface. 
  Modulo some natural generalizations to noncompact spaces, we show in this letter that Moyal planes are nonunital spectral triples in the sense of Connes. The action functional of these triples is computed, and we obtain the expected result, ie the noncommutative Yang-Mills action associated with the Moyal product. In particular, we show that Moyal gauge theory naturally fit into the rigorous framework of noncommutative geometry. 
  We propose a Boundary Conformal Field Theory description of hole states in the c=1 matrix model. 
  We obtain a new gauged D=6, N=(1,1) pure supergravity by a generalised consistent Kaluza-Klein reduction of M-theory on K3 \times R. The reduction requires a conspiratory gauging of both the Cremmer-Julia type global (rigid) symmetry and the homogeneous rescaling symmetry of the supergravity equations of motion. The gauged supergravity is different from the Romans D=6 gauged supergravity in that the four vector fields in our new theory are all abelian. We show that it admits a supersymmetric (Minkowski)_4\times S^2 vacuum, which can be lifted to D=11 where it becomes the near-horizon geometry of two intersecting M5-branes wrapping on a supersymmetric two-cycle of K3. 
  We obtain models of chaotic, slow--roll, hybrid and D--term inflation from the Hanany--Witten brane configuration and its deformations. The deformations are given by the different orientations of the branes and control the parameters of the scalar potential such as the inflaton mass, Yukawa couplings and the anomalous D--term. The different inflationary models are continuously connected and arise in different limits of the parameter space. We describe a compactified version of the brane construction that also leads to models of inflation. 
  We discuss duality between the linear and chiral dilaton formulations, in the presence of super-Yang-Mills instanton corrections to the effective action. In contrast to previous work on the subject, our approach appeals directly to explicit instanton calculations and does not rely on the introduction of an auxiliary Veneziano-Yankielowicz superfield. We discuss duality in the case of an axion that has a periodic scalar potential, and find that the bosonic fields of the dual linear multiplet have a modified interpretation. We note that symmetries of the axion potential manifest themselves as symmetries of the equations of motion for the linear multiplet. We also make some brief remarks regarding dilaton stabilization. We point out that corrections recently studied by Dijkgraaf and Vafa can be used to stabilize the axion in the case of a single super-Yang-Mills condensate. 
  In 1992, E.E.Podkletnov and R.Nieminen find that, under certain conditions, ceramic superconductor with composite structure has revealed weak shielding properties against gravitational force. In classical Newton's theory of gravity and even in Einstein's general theory of gravity, there are no grounds of gravitational shielding effects. But in quantum gauge theory of gravity, the gravitational shielding effects can be explained in a simple and natural way. In quantum gauge theory of gravity, gravitational gauge interactions of complex scalar field can be formulated based on gauge principle. After spontaneous symmetry breaking, if the vacuum of the complex scalar field is not stable and uniform, there will be a mass term of gravitational gauge field. When gravitational gauge field propagates in this unstable vacuum of the complex scalar field, it will decays exponentially, which is the nature of gravitational shielding effects. The mechanism of gravitational shielding effects is studied in this paper, and some main properties of gravitational shielding effects are discussed. 
  A proper understanding of boundary-value problems is essential in the attempt of developing a quantum theory of gravity and of the birth of the universe. The present paper reviews these topics in light of recent developments in spectral geometry, i.e. heat-kernel asymptotics for the Laplacian in the presence of Dirichlet, or Robin, or mixed boundary conditions; completely gauge-invariant boundary conditions in Euclidean quantum gravity; local vs. non-local boundary-value problems in one-loop Euclidean quantum theory via path integrals. 
  A novel geometric model of a noncommutative plane has been constructed. We demonstrate that it can be construed as a toy model for describing and explaining the basic features of physics in a noncommutative spacetime from a field theory perspective. The noncommutativity is induced internally through constraints and does not require external interactions. We show that the noncommutative space-time is to be interpreted as having an {\it internal} angular momentum throughout. Subsequently, the elementary excitations - {\it i.e.} point particles - living on this plane are endowed with a {\it spin}. This is explicitly demonstrated for the zero-momentum Fourier mode. The study of these excitations reveals in a natural way various {\it stringy} signatures of a noncommutative quantum theory, such as dipolar nature of the basic excitations \cite{jab} and momentum dependent shifts in the interaction point \cite{big}. The observation \cite{sw} that noncommutative and ordinary field theories are alternative descriptions of the same underlying theory, is corroborated here by showing that they are gauge equivalent.   Also, treating the present model as an explicit example, we show that, even classically, in the presence of additional constraints, (besides the usual ones due to reparameterization invariances), the equivalence between Nambu-Goto and Polyakov formulations is subtle. 
  For quasiexactly solvable (QES) potentials a certain number of wave functions and energy levels can be analytically calculated. The complexity of an explicit calculation of the energy levels grows with the dimension of the QES sector. For a class of such systems the generating function of the secular polynomials is also an initial condition solution of the Schr\"odinger equation. This generating function is used to obtain approximate energy levels in the limit of a large QES sector. This new method combines the WKB approximation with the saddle point approximation. 
  We discuss the scalar propagator on generic AdS x S backgrounds. For the conformally flat situations and masses corresponding to Weyl invariant actions, the propagator is powerlike in the sum of the chordal distances with respect to AdS(d+1) and S(d'+1). In these cases we analyze its source structure. In all other cases the propagator depends on both chordal distances separately. There an explicit formula is found for certain special mass values. For pure AdS we show how the well known propagators in the Weyl invariant case can be expressed as linear combinations of simple powers of the chordal distance. For AdS(5) x S(5) we relate our propagator to the expression in the plane wave limit and find a geometric interpretation of the variables occurring in the known explicit construction on the plane wave. As a byproduct of comparing different techniques, including the KK mode summation, a theorem for summing certain products of Legendre and Gegenbauer functions is derived. 
  The field strength is defined for the orthosymplectic non-degenerate graded Lie algebra on three even and two odd generators. We show that a pair of Grassman-odd scalar fields find their place as a constituent part of the graded gauge potential on the equal footing with an ordinary, i.e. Grassman-even, one-form taking values in the proper Lie subalgebra, su(2), of the graded Lie algebra. Some possibilities of constructing a meaningful variational principle are discussed. 
  We extend the formalism of Hamiltonian string bit models of quantum gravity type in two spacetime dimensions to include couplings to particles. We find that the single-particle closed and open universe models respectively behave like empty open and closed universes, and that a system of two distinguishable particles in a closed universe behaves like an empty closed universe. We then construct a metamodel that contains all such models, and find that its transition amplitude is exactly the same as the sl(2) gravity model. 
  {We carry out a fluctuation analysis of the non-abelian Born-Infeld action up to order F^6 in the presence of a background intersecting D-branes system. We compare the mass spectrum which is obtained in our analysis to the spectrum derived from the worldsheet analysis. The mass spectra completely agree with each other. This result provides a strong support for the claim that the action we analyze here, which was proposed in hep-th/0208044 by Koerber and Sevrin, is the correct low energy effective action of string theory. 
  We consider the thermodynamic and cosmological properties of brane gases in the early universe. Working in the low energy limit of M-theory we assume the universe is a homogeneous but anisotropic 10-torus containing wrapped 2-branes and a supergravity gas. We describe the thermodynamics of this system and estimate a Hagedorn temperature associated with excitations on the branes. We investigate the cross-section for production of branes from the thermal bath and derive Boltzmann equations governing the number of wrapped branes. A brane gas may lead to decompactification of three spatial dimensions. To investigate this possibility we adopt initial conditions in which we fix the volume of the torus but otherwise assume all states are equally likely. We solve the Einstein-Boltzmann equations numerically, to determine the number of dimensions with no wrapped branes at late times; these unwrapped dimensions are expected to decompactify. Finally we consider holographic bounds on the initial volume, and find that for allowed initial volumes all branes typically annihilate before freeze-out can occur. 
  We illustrate how boundary states are recovered when going from a noncommutative manifold to a commutative one with a boundary. Our example is the noncommutative plane with a defect, whose commutative limit was found to be a punctured plane - so here the boundary is one point. Defects were introduced by removing states from the standard harmonic oscillator Hilbert space. For Chern-Simons theory, the defect acts as a source, which was found to be associated with a nonlinear deformation of the $w_\infty$ algebra. The undeformed $w_\infty$ algebra is recovered in the commutative limit, and here we show that its spatial support is in a tiny region near the puncture. 
  We propose an extension of the definition of vertex algebras in arbitrary space-time dimensions together with their basic structure theory. An one-to-one correspondence between these vertex algebras and axiomatic quantum field theory (QFT) with global conformal invariance (GCI) is constructed. 
  We present a new fermionic solution of the supersymmetric matrix model. The solution satisfies the commutation and anticommutation relations for noncommutative superspace. Therefore the solution can be considered as an implementation of noncommutative superspace on the matrix model. 
  The effect of the initial conditions in inflation on scalar and tensor perturbations is investigated. Formulae for the power spectra of gravitational waves and curvature perturbations for any initial conditions in inflation are derived, and the ratio of scalar to tensor perturbations and spectral index are calculated. The formulation is applied to some simplified pre-inflationary cosmological models, and the differences of the ratio and spectral index are investigated with respect to two matching conditions. In addition, the present power spectrum of gravitational waves is derived. The proposed formulation is preliminarily shown to be a possible test of the appropriateness of a given pre-inflationary model. 
  We prove the uniqueness of the supersymmetric Salam-Sezgin (Minkowski)_4\times S^2 ground state among all nonsingular solutions with a four-dimensional Poincare, de Sitter or anti-de Sitter symmetry. We construct the most general solutions with an axial symmetry in the two-dimensional internal space, and show that included amongst these is a family that is non-singular away from a conical defect at one pole of a distorted 2-sphere. These solutions admit the interpretation of 3-branes with negative tension. 
  We study the evolution of Born-Infeld-type phantom in the second Randall-Sundrum brane scenario, and find that there exists attractor solution for the potential with a maximum, which implies a cosmological constant at the late time. Especially, we discuss the BI model of constant potential without and with dust matter. In the weak tension limit of the brane, we obtain an exact solution for the BI phantom and scale factor and show that there is no big rip during the evolution of the brane. 
  We develop a quantum theoretical formalism to derive the brane gravity from its origin -- brane-generating dynamics in higher dimensions. Based on it, we discuss characteristic properties of the brane induced gravity, as well as brane induced field theory. 
  Axioms for nonunital spectral triples, extending those introduced in the unital case by Connes, are proposed. As a guide, and for the sake of their importance in noncommutative quantum field theory, the spaces $\R^{2N}$ endowed with Moyal products are intensively investigated. Some physical applications, such as the construction of noncommutative Wick monomials and the computation of the Connes--Lott functional action, are given for these noncommutative hyperplanes. 
  In this paper we construct the Nonsymmetric Jordan-Thiry Theory unifying N.G.T., the Yang-Mills' field, the Higgs' fields and scalar forces in a geometric manner. In this way we get masses from higher dimensions. We discuss spontaneous symmetry breaking, the Higgs' mechanism and a mass generation in the theory. The scalar field (as in the classical Jordan-Thiry Theory) is connected to the effective gravitational constant. This field is massive and has Yukawa-type behaviour. We derive the equation of motion for a test particle from conservation laws in the hydrodynamic limit. We consider a truncation procedure for a tower of massive scalar fields using Friedrichs' theory and an approximation procedure for the lagrangian involving Higgs' field. The geodetic equations on the Jordan-Thiry manifold are considered with an emphasis to terms involving Higgs' field. We consider also field equations in linear approximation.   We consider a dynamics of Higgs' field in the framework of cosmological models involving the scalar field. The scalar field plays here a role of a quintessence field. We consider phase transition in cosmological models of the second and the first order. We consider a warp factor known from some modern approaches. We consider a toy model of a time-machine. We consider a mass of a quintessence particle, various properties of a quintessence field. We calculate a speed of sound in a quintessence and fluctuations of a quintessence caused by primordial metric fluctuations. 
  We consider a deformation of N=1 four dimensional Minkowski superspace where odd coordinates $\theta^{\alpha}$ do not anticommute. We define supersymmetric and associative star product and show how the remaining (anti)commutation relations among the superspace coordinates are modified. In particular, the even coordinates do not commute as well. We also study chiral and vector superfields and their interactions. Suprisingly we find that ordinary undeformed N=1 supersymmetric field theories are compatible with the deformed supersymmetry considered. 
  Using Katz, Klemm and Vafa geometric engineering method of $\mathcal{N}=2$ supersymmetric QFT$_{4}$s and results on the classification of generalized Cartan matrices of Kac-Moody (KM) algebras, we study the un-explored class of $\mathcal{N}=2$ CFT$_{4}$s based on \textit{indefinite} singularities. We show that the vanishing condition for the general expression of holomorphic beta function of $\mathcal{N}=2$ quiver gauge QFT$_{4}$s coincides exactly with the fundamental classification theorem of KM algebras. Explicit solutions are derived for mirror geometries of CY threefolds with \textit{% hyperbolic} singularities. 
  These notes are a writeup of lectures given at the twelfth Oporto meeting on ``Geometry, Topology, and Physics,'' and at the Adelaide workshop ``Strings and Mathematics 2003,'' primarily geared towards a physics audience. We review current work relating boundary states in the open string B model on Calabi-Yau manifolds to sheaves. Such relationships provide us with a mechanism for counting open string states in situations where the physical spectrum calculation is nearly intractable -- after translating to mathematics, such calculations become easy. We describe several different approaches to these models, and also describe how these models are changed by varying physical circumstances -- flat B field backgrounds, orbifolds, and nonzero Higgs vevs. We also discuss mathematical interpretations of operator products, and how such mathematical interpretations can be checked physically. One of the motivations for this work is to understand the precise physical relationship between boundary states in the open string B model and derived categories in mathematics, and we outline what is currently known of the relationship. 
  Solutions to the four-dimensional Euclidean Weyl equation in the background of a general JNR N-instanton are known to be normalisable and regular throughout four-space. We show that these solutions are asymptotically given by a linear combination of simple singular solutions to the free Weyl equation, which can be interpreted as localised spinors. The `spinorial' data parameterising the asymptotics of the delocalised solutions to the Weyl equation in the presence of the instanton almost determines the background instanton, yet not completely. However, it captures the geometry and symmetry of the underlying instanton configuration. 
  We show that Dirac-Born-Infeld theory possesses a hidden invariance that enhances the local O(1,p) Lorentz symmetry on a Dirichlet p-brane to an O(1,p) x O(1,p) gauge group, encoding both an invariant velocity and acceleration (or length) scale. This enlarged gauge group predicts consequences for the kinematics of observers on Dirichlet branes, with admissible accelerations being bounded from above. An important lesson beyond string theory is that a fundamental length scale can be implemented into the kinematics of general relativity, whilst preserving both space-time as a smooth manifold and local Lorentz symmetry, contrary to common belief. We point out consequences for string phenomenology, classical gravity and atomic physics. 
  We study the evolution of stable D-branes of C/Z_n and twisted circle theories in the process of closed string tachyon condensation. We interpret the fractional branes in these backgrounds as type II branes wrapping (`blown up') cycles, and trace the evolution of the corresponding cycles under tachyon condensation. We also study RG flows of the corresponding N=2 boundary conformal field theories. We find flows along which fractional D-branes either disappear or evolve into other fractional D-branes, and other flows along which bulk branes either disappear or evolve into stable branes. 
  We argue that the quartic fermionic potential of five-dimensional Chern-Simons supergravity induces spontaneous symmetry breaking, in a phenomenon bearing a close connection with the Nambu and Jona-Lasinio model. 
  A calculation by E.D. Jones of the cosmological mass scale for the phase transition from pre-geometric to physical description as about 5 Tev could be interpreted as a prediction of an effective threshold for novel physical effects in particle-particle collisions. 
  We show how super BF theory in any dimension can be quantized as a spin foam model, generalizing the situation for ordinary BF theory. This is a first step toward quantizing supergravity theories. We investigate in particular 3-dimensional (p=1,q=1) supergravity which we quantize exactly. We obtain a super-Ponzano-Regge model with gauge group OSp(1|2). A main motivation for our approach is the implementation of fermionic degrees of freedom in spin foam models. Indeed, we propose a description of the fermionic degrees of freedom in our model. Complementing the path integral approach we also discuss aspects of a canonical quantization in the spirit of loop quantum gravity. Finally, we comment on 2+1-dimensional quantum supergravity and the inclusion of a cosmological constant. 
  This thesis is devoted to the study of a class of constructions based on Superstring Theory, baptized in the literature as Intersecting Brane Worlds. In particular we explore several issues regarding the proposal of Intersecting Brane Worlds as string-based models yielding semi-realistic low-energy physics. We find that they provide an interesting framework where, for instance, just the Standard Model chiral content and gauge group can be obtained.   Although many of the results presented in this work are valid for more general constructions, we center on configurations of D-branes intersecting at angles. We construct several classes of such compactifications which may yield realistic D=4 physics. We build several explicit examples giving the Standard Model chiral spectrum, and then proceed to analyze some of the related phenomenology. We pay special attention to features such as the structure of U(1) global symmetries, the absence of open-string tachyons, the appearance of light extra matter and the possibility of lowering the string scale in such scenarios. Finally, we investigate the relationship between low-energy field theory quantities, such as FI-parameters and Yukawa couplings, with the geometrical objects underlying the string construction. We find that their description is closely related to calibration theory and to the construction of Fukaya's category. 
  We search for the possibility to have supersymmetric string bits at finite discretization $J$. From a general setup we find that the string bits can be made supersymmetric modulo a single defect mode which is not expected to have any sensible effect in the continuum limit. 
  We construct non-supersymmetric type I string models which correspond to consistent flat-space solutions of all classical equations of motion. Moreover, the one-loop vacuum energy is naturally fixed by the size of compact extra dimensions which, in the two-dimensional case, can be lowered to a fraction of a millimetre. This class of models has interesting non-abelian gauge groups and can accommodate chiral fermions. In the large radius limit, supersymmetry is recovered in the bulk, while D-brane excitations, although non-supersymmetric, exhibit Fermi-Bose degeneracy at all mass levels. We also give some evidence for a suppression of higher-loop corrections to the vacuum energy. 
  We discuss the first string theory examples of three generation non-supersymmetric SU(5) and {\em flipped} SU(5) GUTS, which break to the Standard model at low energy, without extra matter and/or gauge group factors. Our GUT examples are based on IIA $Z_3$ orientifolds with D6-branes intersecting at non-trivial angles. These theories necessarily satisfy RR tadpoles and are free of NSNS tadpoles as the complex structure moduli are frozen (even though a dilaton tadpole remains) to discrete values. We identify appropriately the bifundamental Higgses responsible for electroweak symmetry breaking. In this way, the neutrino see-saw mechanism get nicely realized in these constructions. Moreover, as baryon number is not a gauged symmetry gauge mediated dimension six operators do contribute to proton decay; however proton lifetime may be safely enhanced by appropriately choosing a high GUT scale. An accompanying natural doublet-triplet splitting guarantees the suppression of scalar mediated proton decay modes and the stability of triplet scalar masses against higher dimensional non-renormalizable operators. 
  This is the 4-th paper in the series devoted to a systematic study of the problem of mathematically correct formulation of the rules needed to manage an effective field theory. Here we consider the problem of constructing the full set of essential parameters in the case of the most general effective scattering theory containing no massless particles with spin J > 1/2. We perform the detailed classification of combinations of the Hamiltonian coupling constants and select those which appear in the expressions for renormalized S-matrix elements at a given loop order. 
  We investigate the appearance of closed timelike curves in quotients of plane waves along spacelike isometries. First we formulate a necessary and sufficient condition for a quotient of a general spacetime to preserve stable causality. We explicitly show that the plane waves are stably causal; in passing, we observe that some pp-waves are not even distinguishing. We then consider the classification of all quotients of the maximally supersymmetric ten-dimensional plane wave under a spacelike isometry, and show that the quotient will lead to closed timelike curves iff the isometry involves a translation along the u direction. The appearance of these closed timelike curves is thus connected to the special properties of the light cones in plane wave spacetimes. We show that all other quotients preserve stable causality. 
  Holographic considerations are used in the scrutiny of a special class of brane-world cosmologies. Inherently to this class, the brane typically bounces, at a finite size, as a consequence of a charged black hole in the bulk. Whereas a prior treatment [hep-th/0301010] emphasized a brane that is void of standard-model matter, the analysis is now extended to include an intrinsic (radiation-dominated) matter source. An interesting feature of this generalized model is that a bounce is no longer guaranteed but, rather, depends on the initial conditions. Ultimately, we demonstrate that compliance with an appropriate holographic bound is a sufficient prerequisite for a bounce to occur. 
  A Hamiltonian formalism is used to describe ensembles of fields in terms of two canonically conjugate functionals (one being the field probability density). The postulate that a classical ensemble is subject to nonclassical fluctuations of the field momentum density, of a strength determined solely by the field uncertainty, is shown to lead to a unique modification of the ensemble Hamiltonian. The modified equations of motion are equivalent to the quantum equations for a bosonic field, and thus this exact uncertainty principle provides a new approach to deriving and interpreting the properties of quantum ensembles. The examples of electromagnetic and gravitational fields are discussed. In the latter case the exact uncertainty approach specifies a unique operator ordering for the Wheeler-DeWitt and Ashtekar-Wheeler-DeWitt equations. 
  Recently a new formalism has been developed for the covariant quantization of superstrings. We study properties of Dp-branes and p-branes in this new framework, focusing on two different topics: effective actions and boundary states for Dp-branes. We present a derivation of the Wess-Zumino terms for super (D)p-branes using BRST symmetry. To achieve this we derive the BRST symmetry for superbranes, starting from the approach with/without pure spinors, and completely characterize the WZ terms as elements of the BRST cohomology. We also develope the boundary state description of Dp-branes by analyzing the boundary conditions for open strings in the completely covariant (i.e., without pure spinors) BRST formulation. 
  We investigate the vacuum expectation values of the energy-momentum tensor and the fermionic condensate associated with a massive spinor field obeying the MIT bag boundary condition on a spherical shell in the global monopole spacetime. In order to do that it was used the generalized Abel-Plana summation formula. As we shall see, this procedure allows to extract from the vacuum expectation values the contribution coming from to the unbounded spacetime and explicitly to present the boundary induced parts. As to the boundary induced contribution, two distinct situations are examined: the vacuum average effect inside and outside the spherical shell. The asymptotic behavior of the vacuum densities is investigated near the sphere center and surface, and at large distances from the sphere. In the limit of strong gravitational field corresponding to small values of the parameter describing the solid angle deficit in global monopole geometry, the sphere-induced expectation values are exponentially suppressed. As a special case we discuss the fermionic vacuum densities for the spherical shell on background of the Minkowski spacetime. Previous approaches to this problem within the framework of the QCD bag models have been global and our calculation is a local extension of these contributions. 
  The generally adopted approach in theory of relativistic strings and membranes, is similar to use of Lagrange coordinates in continious media mechanics. One can use an alternative approach, which is similar to use of Euler coordinates. Under such approach the consideration of thick (delocalized) membranes is natural. Membrane kinematics, which coorespond to Euler coordinates is constructed. Variables, similar to Hamiltonian variables, are introduced by means of Legander transformation. The case of free membranes appears to be degenerate. The examples of exact solutions of Einstein equations with delocalized membranes are presented. 
  Lorentzian simplicial quantum gravity is a non-perturbatively defined theory of quantum gravity which predicts a positive cosmological constant. Since the approach is based on a sum over space-time histories, it is perturbatively non-renormalizable even in three dimensions. By mapping the three-dimensional theory to a two-matrix model with ABAB interaction we show that both the cosmological and the (perturbatively) non-renormalizable gravitational coupling constant undergo additive renormalizations consistent with canonical quantization. 
  The Drinfeld-Manin construction of U(N) instanton is reformulated in the ADHM formulism, which gives explicit general solutions of the ADHM constraints for U(N) (N>=2k-1) k-instantons. For the N<2k-1 case, implicit results are given systematically as further constraints, which can be used to the collective coordinate integral. We find that this formulism can be easily generalized to the noncommutative case, where the explicit solutions are as well obtained. 
  We consider Neveu-Schwarz pp-waves with spacetime supersymmetry. Upon compactification of a spacelike direction, these backgrounds develop Closed Null Curves (CNCs) and Closed Timelike Curves (CTCs), and are U-dual to supersymmetric Godel type universes. We study classical and quantum strings in this background, with emphasis on the strings winding around the compact direction. We consider two types of strings: long strings stabilized by NS flux and rotating strings which are stabilized against collapse by angular momentum. Some of the latter strings wrap around CNCs and CTCs, and are thus a potential source of pathology. We analyze the partition function, and in particular discuss the effects of these string states. Although our results are not conclusive, the partition function seems to be dramatically altered due to the presence of CNCs and CTCs. We discuss some interpretations of our results, including a possible sign of unitary violation. 
  A new class of integrable theories of 0+1 and 1+1 dimensional dilaton gravity coupled to any number of scalar fields is introduced. These models are reducible to systems of independent Liouville equations whose solutions must satisfy the energy and momentum constraints. The constraints are solved thus giving the explicit analytic solution of the theory in terms of arbitrary chiral fields. In particular, these integrable theories describe spherically symmetric black holes and branes of higher dimensional supergravity theories as well as superstring motivated cosmological models. 
  We are studying quantum corrections in the earlier proposed string theory based on world-sheet action which measures the linear sizes of the surfaces. At classical level the string tension is equal to zero and as it was demonstrated in the previous studies one loop correction to the classical world-sheet action generates Nambu-Goto area term, that is nonzero string tension. We extend this analysis computing the world-sheet effective action in the second order of the loop expansion. 
  We study loop corrections to the universal dilaton supermultiplet for type IIA strings compactified on Calabi-Yau threefolds. We show that the corresponding quaternionic kinetic terms receive non-trivial one-loop contributions proportional to the Euler number of the Calabi-Yau manifold, while the higher-loop corrections can be absorbed by field redefinitions. The corrected metric is no longer Kahler. Our analysis implies in particular that the Calabi-Yau volume is renormalized by loop effects which are present even in higher orders, while there are also one-loop corrections to the Bianchi identities for the NS and RR field strengths. 
  The reduction of higher dimensional supergravities to low dimensional dilaton gravity theories is outlined. Then a recently proposed new class of integrable theories of 0+1 and 1+1 dimensional dilaton gravity coupled to any number of scalar fields is described in more detail. These models are reducible to systems of independent Liouville equations whose solutions are not independent because they must satisfy the energy and momentum constraints. The constraints are solved, thus giving the explicit analytic solution of the theory in terms of arbitrary chiral fields. In particular, these integrable theories describe spherically symmetric black holes and branes of higher dimensional supergravity theories as well as superstring motivated cosmological models. 
  We propose a new reformulation of Yang-Mills theory in which three- and four-gluon self-interactions are eliminated at the price of introducing a sufficient number of auxiliary fields. We discuss the validity of this reformulation in the possible applications such as dynamical gluon mass generation, color confinement and glueball mass calculation. Moreover, we set up a new $1/N_c$ color expansion in the $SU(N_c)$ Yang-Mills theory based on this reformulation. In fact, we give the Feynman rules of the $1/N_c$ expansion in the manifestly Lorentz covariant gauge. The Yang-Mills theory is defined on a non-trivial vacuum where color--singlet transverse gluon pair condensations take place by the attractive gluonic self-interactions. This vacuum condensation provides a common non-vanishing mass for all the gluons with color symmetry being preserved. It is shown that the auxiliary fields become dynamical by acquiring the kinetic term due to quantum corrections. Then the static potential between a pair of color charges is derived as a combination of the Yukawa-type potential and the linear potential with non-vanishing string tension. The mass of the lightest scalar glueball is calculated as the ratio to the gluon mass. The explicit calculations are performed as a partial resummation of the leading order diagrams for the small 't Hooft coupling. 
  We present a new method to calculate the spectrum of (slow-roll) inflationary perturbations, inspired by the conjectured dS/CFT correspondence. We show how the standard result for the spectrum of inflationary perturbations can be obtained from deformed CFT correlators, whose behavior is determined by the Callan-Symanzik equation. We discuss the possible advantages of this approach and end with some comments on the role of holography in dS/CFT and its relation to the universal nature of the spectrum of inflationary perturbations. 
  The Weyl-gauge ($A_0^a=0)$ QCD Hamiltonian is unitarily transformed to a representation in which it is expressed entirely in terms of gauge-invariant quark and gluon fields. In a subspace of gauge-invariant states we have constructed that implement the non-Abelian Gauss's law, this unitarily transformed Weyl-gauge Hamiltonian can be further transformed and, under appropriate circumstances, can be identified with the QCD Hamiltonian in the Coulomb gauge. To circumvent the problem that this Hamiltonian, which is expressed entirely in terms of gauge-invariant variables, must be used with nonnormalizable complicated states that implement the non-Abelian Gauss's law, we demonstrate an isomorphism that materially facilitates the application of this Hamiltonian to a variety of physical processes, including the evaluation of $S$-matrix elements. This isomorphism relates the gauge-invariant representation of the Hamiltonian and the required set of gauge-invariant states to a Hamiltonian of the same functional form but dependent on ordinary unconstrained Weyl-gauge fields operating within a space of `standard' perturbative states. The fact that the gauge-invariant chromoelectric field is not hermitian has important implications for the functional form of the Hamiltonian finally obtained. When this nonhermiticity is taken into account, the `extra' perturbative terms in Christ and Lee's Coulomb-gauge Hamiltonian are natural outgrowths of the formalism. When this nonhermiticity is neglected, the Hamiltonian used in the earlier work of Gribov and others results. 
  Structure group SU(4) gauge vacua of both weakly and strongly coupled heterotic superstring theory compactified on torus-fibered Calabi-Yau threefolds Z with Z_2 x Z_2 fundamental group are presented. This is accomplished by constructing invariant, stable, holomorphic rank four vector bundles on the simply connected cover of Z. Such bundles can descend either to Hermite-Yang-Mills instantons on Z or to twisted gauge fields satisfying the Hermite-Yang-Mills equation corrected by a non-trivial flat B-field. It is shown that large families of such instantons satisfy the constraints imposed by particle physics phenomenology. The discrete parameter spaces of those families are presented, as well as a lower bound on the dimension of the continuous moduli of any such vacuum. In conjunction with Z_2 x Z_2 Wilson lines, these SU(4) gauge vacua can lead to standard-like models at low energy with an additional U(1)_{B-L} symmetry. This U(1)_{B-L} symmetry is very helpful in naturally suppressing nucleon decay. 
  The Higgs branch of N=2 supersymmetric gauge theories with non-Abelian gauge groups are described by hyper-Kahler (HK) nonlinear sigma models with potential terms. With the non-Abelian HK quotient by U(M) and SU(M) gauge groups, we give the massive HK sigma models that are not toric in the N=1 superfield formalism and the harmonic superspace formalism. The U(M) quotient gives N!/[M! (N-M)!] discrete vacua that may allow various types of domain walls, whereas the SU(M) quotient gives no discrete vacua. 
  Non(anti)commutative gauge theories are supersymmetric Yang-Mills and matter system defined on a deformed superspace whose coordinates obey non(anti)commutative algebra. We prove that these theories in four dimensions with N=1/2 supersymmetry are renormalizable to all orders in perturbation theory. Our proof is based on operator analysis and symmetry arguments. In a case when the Grassman-even coordinates are commutative, deformation induced by non(anti)commutativity of the Grassman-odd coordinates contains operators of dimension-four or higher. Nevertheless, they do not lead to power divergences in a loop diagram because of absence of operators Hermitian-conjugate to them. In a case when the Grassman-even coordinates are noncommutative, the ultraviolet-infrared mixing makes the theory renormalizable by the planar diagrams, and the deformed operators are not renormalized at all. We also elucidate relation at quantum level between non(anti)commutative deformation and N=1/2 supersymmetry. We point out that the star product structure dictates a specific relation for renormalization among the deformed operators. 
  A class of exactly solvable models of domain walls are worked out in D=4 ${\cal N}=1$ supergravity. We develop a method to embed globally supersymmetric theories with exact BPS domain wall solutions into supergravity, by introducing a gravitationally deformed superpotential. The gravitational deformation is natural in the spirit of maintaining the K\"ahler invariance. The solutions of the warp factor and the Killing spinor are also obtained. We find that three distinct behaviors of warp factors arise depending on the value of a constant term in the superpotential : exponentially decreasing in both sides of the wall, flat in one side and decreasing in the other, and increasing in one side and decreasing in the other. Only the first possibility gives the localized massless graviton zero mode. Models with multi-walls and models with runaway vacua are also discussed. 
  The problem of construction of fiber bundle over the moduli space of the Skyrme model is considered. We analyse an extension of the original Skyrme model which includes the minimal interaction with fermions. An analogy with modili space of the fermion-monopole system is used to construct a fiber bundle structure over the skyrmion moduli space. The possibility of the non-trivial holonomy appearance is considered. It is shown that the effect of the fermion interaction turns the $n$-skyrmion moduli space into a real vector bundle with natural $SO(2n+1)$ connection. 
  Non-uniform black strings in the two-brane system are investigated using the effective action approach. It is shown that the radion acts as a non-trivial hair of the black strings. From the brane point of view, the black string appears as the deformed dilatonic black hole which becomes dilatonic black hole in the single brane limit and reduces to the Reissner-Nordstr\"om black hole in the close limit of two-branes. The stability of solutions is demonstrated using the catastrophe theory. From the bulk point of view, the black strings are proved to be non-uniform. Nevertheless, the zeroth law of black hole thermodynamics still holds. 
  The extended BRST cohomology of N=2 super Yang-Mills theory is discussed in the framework of Algebraic Renormalization. In particular, N=2 supersymmetric descent equations are derived from the cohomological analysis of linearized Slavnov-Taylor operator $\B$. It is then shown that both off- and on-shell N=2 super Yang-Mills actions are related to a lower-dimensional gauge invariant field polynomial $Tr\f^2$ by solving these descent equations. Moreover, it is found that these off- and on-shell solutions differ only by a $\B-$exact term, which can be interprated as a consequence of the fact that the cohomology of both cases are the same. 
  Arguably the simplest model of a cosmological singularity in string theory, the Lorentzian orbifold $\Real^{1,1}/boost$ is known to lead to severe divergences in scattering amplitudes of untwisted states, indicating a large backreaction toward the singularity. In this work we take a first step in investigating whether condensation of twisted states may remedy this problem and resolve the spacelike singularity. By using the formal analogy with charged open strings in an electric field, we argue that, contrary to earlier claims, twisted sectors do contain physical scattering states, which can be viewed as charged particles in an electric field. Correlated pairs of twisted states will therefore be produced, by the ordinary Schwinger mechanism. For open strings in an electric field, on-shell wave functions for the zero-modes are determined, and shown to analytically continue to non-normalizable modes of the usual Landau harmonic oscillator in Euclidean space. Closed strings scattering states of the Milne orbifold continue to non-normalizable modes in an unusual Euclidean orbifold of $\Real^2$ by a rotation by an irrational angle. Irrespective of the formal analogy with the Milne Universe, open strings in a constant electric field, or colliding D-branes, may also serve as a useful laboratory to study time-dependence in string theory. 
  We show that the effective action for the edge excitations of a quantum Hall droplet of fermions in higher dimensions is generically given by a chiral bosonic action. We explicitly analyze the quantum Hall effect on complex projective spaces ${\bf CP}^k$ with a U(1) background magnetic field. The edge excitations are described by abelian bosonic fields on $S^{2k-1}$ with only one spatial direction along the boundary of the droplet relevant for the dynamics. Our analysis also leads to an action for edge excitations for the case of the Zhang-Hu four dimensional quantum Hall effect defined on $S^4$ with an SU(2) background magnetic field, using the fact that ${\bf CP}^3$ is an $S^2$ bundle over $S^4$. 
  The one-loop quantum corrections to the mass and central charge of the N=2 vortex in 2+1 dimensions are determined using supersymmetry-preserving dimensional regularization by dimensional reduction of the corresponding N=1 model with Fayet-Iliopoulos term in 3+1 dimensions. Both the mass and the central charge turn out to have nonvanishing one-loop corrections which however are equal and thus saturate the Bogomolnyi bound. We explain BPS saturation by standard multiplet shortening arguments, correcting a previous claim in the literature postulating the presence of a second degenerate short multiplet at the quantum level. 
  We propose an alternative for the Clebsch decomposition of currents in fluid mechanics, in terms of complex potentials taking values in a Kahler manifold. We reformulate classical relativistic fluid mechanics in terms of these complex potentials and rederive the existence of an infinite set of conserved currents. We perform a canonical analysis to find the explicit form of the algebra of conserved charges. The Kahler-space formulation of the theory has a natural supersymmetric extension in 4-D space-time. It contains a conserved current, but also a number of additional fields complicating the interpretation. Nevertheless, we show that an infinite set of conserved currents emerges in the vacuum sector of the additional fields. This sector can therefore be identified with a regime of supersymmetric fluid mechanics. Explicit expressions for the current and the density are obtained. 
  We work out some properties of a recently proposed globally N = 1 supersymmetric extension of relativistic fluid mechanics in four-dimensional Minkowski space. We construct the lagrangean, discuss its symmetries and the corresponding conserved Noether charges. We reformulate the theory in hamiltonian formulation, and rederive the (supersymmetry and internal) transformations generated by these charges. Super-Poincare algebra is also realized in this formulation. 
  We study the covariance properties of the equations satisfied by the generating functions of the chiral operators R and T of supersymmetric SO(N)/Sp(N) theories with symmetric/antisymmetric tensors. It turns out that T is an affine connection. As such it cannot be integrated along cycles on Riemann surfaces. This explains the discrepancies observed by Kraus and Shigemori. Furthermore, by means of the polynomial defining the Riemann surface, seen as quadratic-differential, one can construct an affine connection that added to T leads to a new generating function which can be consistently integrated. Remarkably, thanks to an identity, the original equations are equivalent to equations involving only one-differentials. This provides a geometrical explanation of the map recently derived by Cachazo in the case of Sp(N) with antisymmetric tensor. Finally, we suggest a relation between the Riemann surfaces with rational periods which arise in studying the Laplacian on special Riemann surfaces and the integrality condition for the periods of T. 
  A complete quantum field theoretic study of charged and neutral particle creation in a rapidly/adiabatically expanding Friedman-Robertson-Walker metric for an O(4) scalar field theory with quartic interactions (admitting a phase transition) is given. Quantization is carried out by inclusion of quantum fluctuations. We show that the quantized Hamiltonian admits an su(1,1) invariance. The squeezing transformation diagonalizes the Hamiltonian and shows that the dynamical states are squeezed states. Allowing for different forms of the expansion parameter, we show how the neutral and charged particle production rates change as the expansion is rapid or adiabatic. The effects of the expansion rate versus the symmetry restoration rate on the squeezing parameter is shown. 
  We study nonabelian vortices (flux tubes) in SU(N) gauge theories, which are responsible for the confinement of (nonabelian) magnetic monopoles. In particular a detailed analysis is given of ${\cal N}=2$ SQCD with gauge group SU(3) deformed by a small adjoint chiral multiplet mass. Tuning the bare quark masses (which we take to be large) to a common value $m$, we consider a particular vacuum of this theory in which an SU(2) subgroup of the gauge group remains unbroken. We consider $5 \ge N_f \ge 4$ flavors so that the SU(2) sub-sector remains non asymptotically free: the vortices carrying nonabelian fluxes may be reliably studied in a semi-classical regime. We show that the vortices indeed acquire exact zero modes which generate global rotations of the flux in an $SU(2)_{C+F}$ group. We study an effective world-sheet theory of these orientational zero modes which reduces to an ${\cal N}=2$ O(3) sigma model in (1+1) dimensions. Mirror symmetry then teaches us that the dual SU(2) group is not dynamically broken. 
  The modified gravity, which eliminates the need for dark energy and which seems to be stable, is considered. The terms with positive powers of the curvature support the inflationary epoch while the terms with negative powers of the curvature serve as effective dark energy, supporting current cosmic acceleration. The equivalent scalar-tensor gravity may be compatible with the simplest solar system experiments. 
  In the paper we consider some consequences of the Nonsymmetric Kaluza-Klein (Jordan-Thiry) Theory. We calculate: primordial fluctuation spectrum functions, spectral indices, and first derivatives of spectral indices. 
  The physics of spontaneous chiral symmetry breaking in the case of the simultaneous presence of a magnetic field and a fermionic quartic interaction is discussed for both local and nonlocal kernels in 2+1 and 3+1 dimensions. The approach is based on the use of Valatin-Bogoliubov canonical transformations, which allow, in the absence of fermionic quartic terms, to completely diagonalize the Hamiltonian and construct the vacuum state. 
  This paper has been withdrawn because of serious errors. 
  We show that noncommutative gauge theories with arbitrary compact gauge group defined by means of the Seiberg-Witten map have the same one-loop anomalies as their commutative counterparts. This is done in two steps. By explicitly calculating the $\epsilon^{\m_1\m_2\m_3\m_4}$ part of the renormalized effective action, we first find the would-be one-loop anomaly of the theory to all orders in the noncommutativity parameter $\theta^{\m\n}$. And secondly we isolate in the would-be anomaly radiative corrections which are not BRS trivial. This gives as the only true anomaly occurring in the theory the standard Bardeen anomaly of commutative spacetime, which is set to zero by the usual anomaly cancellation condition. 
  Inspired by a formal resemblance of certain q-expansions of modular forms and the master field formalism of matrix models in terms of Cuntz operators, we construct a Hermitian one-matrix model, which we dub the ``modular matrix model.'' Together with an N=1 gauge theory and a special Calabi-Yau geometry, we find a modular matrix model that naturally encodes the Klein elliptic j-invariant, and hence, by Moonshine, the irreducible representations of the Fischer-Griess Monster group. 
  We investigate the Hawking radiation in the gauge-Higgs-Yukawa theory. The ballistic model is proposed as an effective description of the system. We find that a spherical domain wall around the black hole is formed by field dynamics rather than thermal phase-transition. The formation is a general property of the black hole whose Hawking temperature is equal to or greater than the energy scale of the theory. The formation of the electroweak wall and that of the GUT wall are shown. We also find a phenomenon of the spontaneous charging-up of the black hole by the wall. The Hawking radiation drives a mechanism of the charge-transportation into the black hole when C- and CP-violation are assumed. The mechanism can strongly transport the hyper-charge into a black hole of the electroweak scale. 
  The Hawking radiation in the vacuum of the spontaneous symmetry breaking in the gauge-Higgs-Yukawa theory is investigated by a general relativistic formulation of the the ballistic model. The restoration of the symmetry on the horizon and the formation of the spherical domain wall around the black hole are shown even if the Hawking temperature is lower than the critical temperature of the phase transition in the gauge-Higgs-Yukawa theory. When the Hawking temperature is much lower than the critical temperature, the domain wall closely near the horizon is formed. The wall is formed by the field dynamics rather than the thermal phase transition. 
  We provide a general method for studying manifestly $O(n+1)$ covariant formulation of $p$-form gauge theories by stereographically projecting these theories, defined in flat Euclidean space, onto the surface of a hypersphere. The gauge fields in the two descriptions are mapped by conformal Killing vectors while conformal Killing spinors are necessary for the matter fields, allowing for a very transparent analysis and compact presentation of results. General expressions for these Killing vectors and spinors are given. The familiar results for a vector gauge theory are reproduced. 
  D-brane configurations containing fundamental strings are constructed as classical solutions of Yang-Mills theory. The fundamental strings in these systems stretch between D-branes. In the case of D1-branes, this construction gives smooth (classical) resolutions of string junctions and string networks. Using a non-abelian Yang-Mills analysis of the string current, the string charge density is computed and is shown to have support in the region between the D-brane world-volumes. The 't Hooft-Polyakov monopole is analyzed using similar methods, and is shown to contain D-strings whose flux has support off the D-brane world-volume defined by the Higgs scalar field, when this field is interpreted in terms of a transverse dimension. The constructions presented here are used to give a qualitative picture of tachyon condensation in the Yang-Mills limit, where fundamental strings and lower-dimensional D-branes arise in a volume of space-time where brane-antibrane annihilation has occurred. 
  Present knowledge of higher-derivative terms in string effective actions is, with a few exceptions, restricted to the NS-NS sector, a situation which prevents the development of a variety of interesting applications for which the RR terms are relevant. We here provide the formalism as well as efficient techniques to determine the latter directly from string-amplitude calculations. As an illustration of these methods, we compute the dependence of the type-IIB action on the three- and five-form RR field strengths at four-point, genus-one, order-(alpha')^3 level. We explicitly verify that our results are in accord with the SL(2,Z) S-duality invariance of type-IIB string theory. Extensions of our method to other bosonic terms in the type-II effective actions are discussed as well. 
  We calculate a total amount of an inflation during two de Sitter phases in our cosmological modells and masses of quintessence particles in both de Sitter phases.. 
  We construct new backgrounds of d-dimensional gravity with a negative cosmological constant coupled to a m-form field strength. We find a class of magnetically charged anti-de Sitter black holes with m-dimensional Einstein horizon of positive, zero or negative curvature. We also construct a new magnetic co-dimension four brane for the case of m=3. This brane obeys a charge quantization condition of the form q \sim L^2 where q is the magnetic 3-form charge and L is the AdS radius. In addition, we work out some time-dependent solutions. 
  We find the leading RG logs in $\phi^4$ theory for any Feynman diagram with 4 external edges. We obtain the result in two ways. The first way is to calculate the relevant terms in Feynman integrals. The second way is to use the RG invariance based on the Lie algebra of graphs introduced by Connes and Kreimer. The non-RG logs, such as $(\ln s/t)^n$, are discussed. 
  We describe new solutions of Yang-Mills-Higgs theories consisting of magnetic monopoles in a phase with fully broken gauge symmetry. Rather than spreading out radially, the magnetic field lines form flux tubes. The solution is topologically stable and, when embedded in N=2 SQCD, preserves 1/4 of the supercharges. From the perspective of the flux-tube the monopole appears as a kink. Many monopoles may be threaded onto a single flux tube and placed at arbitrary separation to create a stable, BPS necklace of solitons. 
  In N=1 supersymmetric U(N) gauge theory with adjoint matter $\Phi$ and polynomial tree-level superpotential $W(\Phi)$, the massless fluctuations about each quantum vacuum are generically described by $U(1)^n$ gauge theory for some n. However, by tuning the parameters of $W(\Phi)$ to non-generic values, we can reach singular vacua where additional fields become massless. Using both the matrix model prescription and the strong-coupling approach, we study in detail three examples of such singularities: the singularities of the n=1 branch, intersections of n=1 and n=2 branches, and a class of N=1 Argyres-Douglas points. In all three examples, we find that the matrix model description of the low-energy physics breaks down in some way at the singularity. 
  We present an exactly soluble charged wormhole model in two dimensions by adding infalling chiral fermions on the static wormhole. The infalling energy due to the infalling charged matter requires the classical back reaction of the geometry, which is solved by taking into account of the nontrivial nonchiral exotic energy. Finally, we obtain the exact expression for the size of the throat depending on the total amount of the infalling net energy and discuss the interesting transition from the AdS spacetime to the wormhole geometry. 
  A new solution to Einstein's equations with a negative bulk cosmological constant in infinite (1+5)-spacetime is found. It is shown that the zero modes of all kinds of matter fields and 4-gravity are localized in (1+3) subspace by the increasing and limited from above warp factor. 
  For all affine Toda field theories we propose a new type of generic boundary bootstrap equations, which can be viewed as a very specific combination of elementary boundary bootstrap equations. These equations allow to construct generic solutions for the boundary reflection amplitudes, which are valid for theories related to all simple Lie algebras, that is simply laced and non-simply laced. We provide a detailed study of these solutions for concrete Lie algebras in various representations. 
  Using geometric engineering method of 4D $\mathcal{N}=2$ quiver gauge theories and results on the classification of Kac-Moody (KM) algebras, we show on explicit examples that there exist three sectors of $\mathcal{N}=2$ infrared CFT$_{4}$s. Since the geometric engineering of these CFT$_{4}$s involve type II strings on K3 fibered CY3 singularities, we conjecture the existence of three kinds of singular complex surfaces containing, in addition to the two standard classes, a third indefinite set. To illustrate this hypothesis, we give explicit examples of K3 surfaces with H$_{3}^{4}$ and E$_{10}$ hyperbolic singularities. We also derive a hierarchy of indefinite complex algebraic geometries based on affine $A_{r}$ and T$%_{(p,q,r)}$ algebras going beyond the hyperbolic subset. Such hierarchical surfaces have a remarkable signature that is manifested by the presence of poles. 
  We study aspects of confinement in two deformed versions of the AdS/CFT correspondence - the GPPZ dual of N=1* Yang Mills, and the Yang Mills* N=0 dual. Both geometries describe discrete glueball spectra which we calculate numerically. The results agree at the 10% level with previous AdS/CFT computations in the Klebanov Strassler background and AdS Schwarzchild respectively. We also calculate the spectra of bound states of the massive fermions in these geometries and show that they are light, so not decoupled from the dynamics. We then study the behaviour of Wilson loops in the 10d lifts of these geometries. We find a transition from AdS-like strings in the UV to strings that interact with the unknown physics of the central singularity of the space in the IR. 
  Standard quantum mechanics is viewed as a limit of a cut system with artificially restricted dimension of a Hilbert space. Exact spectrum of cut momentum and coordinate operators is derived and the limiting transition to the infinite dimensional Hilbert space is studied in detail. The difference between systems with discrete and continuous energy spectra is emphasized. In particular a new scaling law, characteristic for nonlocalized, states is found. Some applications for supersymmetric quantum mechanics are briefly outlined. 
  We study IR/UV mixing effects in noncommutative supersymmetric Yang-Mills theories with gauge group U(N) using background field perturbation theory. We compute three- and four-point functions of background fields, and show that the IR/UV mixed contributions to these correlators can be reproduced from an explicitly gauge-invariant effective action, which is expressed in terms of open Wilson lines. 
  We construct explicitly the grade star Hermitian adjoint representation of osp(2/1; C) graded Lie algebra. Its proper Lie subalgebra, the even part of the graded Lie algebra osp(2/1; C), is given by su(2) compact Lie algebra. The Baker-Campbell-Hausdorff formula is considered and reality conditions for the Grassman-odd transformation parameters, which multiply the pair of odd generators of the graded Lie algebra, are clarified. 
  We study the Casimir problem for a fermion coupled to a static background field in one space dimension. We examine the relationship between interactions and boundary conditions for the Dirac field. In the limit that the background becomes concentrated at a point (a ``Dirac spike'') and couples strongly, it implements a confining boundary condition. We compute the Casimir energy for a masslike background and show that it is finite for a stepwise continuous background field. However the total Casimir energy diverges for the Dirac spike. The divergence cannot be removed by standard renormalization methods. We compute the Casimir energy density of configurations where the background field consists of one or two sharp spikes and show that the energy density is finite except at the spikes. Finally we define and compute an interaction energy density and the force between two Dirac spikes as a function of the strength and separation of the spikes. 
  While 2-dimensional quantum systems are known to exhibit non-permutation, braid group statistics, it is widely expected that quantum statistics in 3-dimensions is solely determined by representations of the permutation group. This expectation is false for certain 3-dimensional systems, as was shown by the authors of ref. [1,2,3]. In this work we demonstrate the existence of ``cyclic'', or $Z_n$, {\it non-permutation group} statistics for a system of n > 2 identical, unknotted rings embedded in $R^3$. We make crucial use of a theorem due to Goldsmith in conjunction with the so called Fuchs-Rabinovitch relations for the automorphisms of the free product group on n elements. 
  We investigate deformations of four-dimensional N=(1,1) euclidean superspace induced by nonanticommuting fermionic coordinates. We essentially use the harmonic superspace approach and consider nilpotent bi-differential Poisson operators only. One variant of such deformations (termed chiral nilpotent) directly generalizes the recently studied chiral deformation of N=(1/2,1/2) superspace. It preserves chirality and harmonic analyticity but generically breaks N=(1,1) to N=(1,0) supersymmetry. Yet, for degenerate choices of the constant deformation matrix N=(1,1/2) supersymmetry can be retained, i.e. a fraction of 3/4. An alternative version (termed analytic nilpotent) imposes minimal nonanticommutativity on the analytic coordinates of harmonic superspace. It does not affect the analytic subspace and respects all supersymmetries, at the expense of chirality however. For a chiral nilpotent deformation, we present non(anti)commutative euclidean analogs of N=2 Maxwell and hypermultiplet off-shell actions. 
  One of the powerful techniques to analyze the 5 dimensional Super Yang Mills theory with a massive hypermultiplet (N=1*) is provided by the AdS/CFT correspondence. It predicts that, for certain special values of the hypermultiplet mass, this theory develops nonperturbative branches of the moduli space as well as new light degrees of freedom.   We use the higher dimensional generalization of the matrix model/gauge theory correspondence and recover all the prediction of the supergravity analysis. We construct the map between the four dimensional holomorphic superpotential and the five dimensional action and explicitly show that the superpotential is flat along the nonperturbative branches. This is the first instance in which the Dijkgraaf-Vafa method is used to analyze intrinsically higher dimensional phenomena. 
  We study a new type of warped compactifications of M-theory on eight manifolds for which nowhere vanishing covariantly constant spinors of indefinite chirality on the internal manifold can be found. We derive the constraints on the fluxes and the warp factor following from supersymmetry and the equations of motion. We show, that the lift of Type IIB PP-waves to M-theory is a special type of solution of this general class of models. We comment on the relation between the Type IIB version of such compactifications as a dual description of the Polchinski-Strassler solution describing a four-dimensional confining gauge theory. 
  We study the localized tachyon condensation (LTC) of non-supersymmetric orbifold backgrounds in their mirror Landau-Ginzburg picture. Using he existence of four copies of (2,2) worldsheet supersymmetry, we show at the CFT level, that the minimal tachyon mass in twisted sectors shows somewhat analogous properties of c- or g-function. Namely, $m := |\alpha' M^2_{min}| $ satisfies $m(M) \geq m(D_1\oplus D_2)={\rm max} \{m(D_1),m(D_2)\}$.   $c$- $g$- $m$- functions share the common property $ f(M)\geq f(D_1\oplus D_2)$ for $f=c,g,m$, although they have different behavior in detail. 
  Symmetry restoring phase transitions in three dimension Gross-Neveu model are shown to be second order at finite temperature $T$ and first order at T=0 and finite chemical potential $\mu$ by critical analysis of the dynamical fermion mass based on the gap equation. The latter is further verified by effective potential analysis. The resulting tricritical point is $(T,\mu)=(0,m(0))$, where $m(0)$ is the dynamical fermion mass at $T=\mu=0$. Physical difference between the above second and first order phase transition is illustrated by means of variations of thermodynamical particle density. 
  In this article we consider an open string attached to a D$_p$-brane, in the presence of the pp-wave and background gauge fields. The effects of the string mass on the open string propagator, open string metric and the noncommutativity parameter are studied. Some free matrices appear in the propagator and in the open string variables. The symmetries of the string propagator and consistency with the zero mass case put some restrictions on these matrices. 
  The Casimir stress on a cylinderical shell in background of conformally flat space-time for massless scalar field is investigated. In the general case of Robin (mixed) boundary condition formulae are derived for the vacuum expectation values of the energy-momentum tensor and vacuum forces acting on boundaries. The special case of the dS bulk is considered then different cosmological constants are assumed for the space inside and outside of the shell to have general results applicable to the case of cylindrical domain wall formations in the early universe. 
  We present a coherent state quantization of the dynamics of a relativistic test particle on a one-sheet hyperboloid embedded in a three-dimensional Minkowski space. The group SO_0(1,2) is considered to be the symmetry group of the system. Our procedure relies on the choice of coherent states of the motion on a circle. The coherent state realization of the principal series representation of SO_0(1,2) seems to be a new result. 
  We describe Stochastic Loewner Evolution on arbitrary Riemann surfaces with boundary using Conformal Field Theory methods. We propose in particular a CFT construction for a probability measure on (clouded) paths, and check it against known restriction properties. The probability measure can be thought of as a section of the determinant bundle over moduli spaces of Riemann surfaces. Loewner evolutions have a natural description in terms of random walk in the moduli space, and the stochastic diffusion equation translates to the Virasoro action of a certain weight-two operator on a uniformised version of the determinant bundle. 
  We consider a non-anticommutative N=2 superspace with an SU(2) singlet and Lorentz scalar deformation parameter, $\{\theta^{\alpha i},\theta^{\beta j}\}_\star = -2iP \e^{\alpha\beta}\e^{ij}$. We exploit this unique feature of the N=2 case to construct a deformation of the non-Abelian super-Yang-Mills theory which preserves the full N=2 supersymmetry together with the SU(2) R symmetry and Lorentz invariance. The resulting action describes a kind of "heterotic special geometry" with antiholomorphic prepotential $\bar f(\bar\phi) = Tr (\bar\phi^2 (1+P\bar\phi)^{-2})$. 
  When de Sitter first introduced his celebrated spacetime, he claimed, following Schwarzschild, that its spatial sections have the topology of the real projective space RP^3 (that is, the topology of the group manifold SO(3)) rather than, as is almost universally assumed today, that of the sphere S^3. (In modern language, Schwarzschild was disturbed by the non-local correlations enforced by S^3 geometry.) Thus, what we today call "de Sitter space" would not have been accepted as such by de Sitter. There is no real basis within classical cosmology for preferring S^3 to RP^3, but the general feeling appears to be that the distinction is in any case of little importance. We wish to argue that, in the light of current concerns about the nature of de Sitter space, this is a mistake. In particular, we argue that the difference between "dS(S^3)" and "dS(RP^3)" may be very important in attacking the problem of understanding horizon entropies. In the approach to de Sitter entropy via Schwarzschild-de Sitter spacetime, we find that the apparently trivial difference between RP^3 and S^3 actually leads to very different perspectives on this major question of quantum cosmology. 
  In this short note we apply Schrodinger picture description of the minisuperspace approach to the closed string tachyon condensation. We will calculate the rate of produced closed string and we will show that the density of high massive closed string modes reaches the string density in time of order one in string units. 
  As an extension of the so-called BMN conjecture, we investigate the plane-wave limit for possible holographic connection between bulk string theories in non-conformal backgrounds of D$p$-branes and the corresponding supersymmetric gauge theories for $p<5$. Our work is based on the tunneling picture for dominant null trajectories of strings in the limit of large angular momentum. The tunneling null trajectories start from the near-horizon boundary and return to the boundary, and the resulting backgrounds are time-dependent for general D$p$-branes except for $p=3$. We develop a general method for extracting diagonalized two-point functions for boundary theories as Euclidean (bulk) S-matrix in the time-dependent backgrounds. For the case of D0-brane, two-point functions of supergravity modes are shown to agree with the results derived previously by the perturbative analysis of supergravity. We then discuss the implications of the holography for general cases of D$p$-branes including the stringy excitations. All the cases ($p\ne 3, p<5$) exhibit interesting infra-red behaviors, which are different from free-field theories, suggesting the existence of quite nontrivial fixed-points in dual gauge theories. 
  We show that one can obtain naturally the Cornell confining potential from the spontaneous symmetry breaking of scale invariance in gauge theory. At the classical level a confining force is obtained and at the quantum level, using a gauge invariant but path-dependent variables formalism, the Cornell confining potential is explicitly obtained. 
  An M/string-theory origin for the six-dimensional Salam-Sezgin chiral gauged supergravity is obtained, by embedding it as a consistent Pauli-type reduction of type I or heterotic supergravity on the non-compact hyperboloid ${\cal H}^{2,2}$ times $S^1$. We can also obtain embeddings of larger, non-chiral, gauged supergravities in six dimensions, whose consistent truncation yields the Salam-Sezgin theory. The lift of the Salam-Sezgin (Minkowski)$_4\times S^2$ ground state to ten dimensions is asymptotic at large distances to the near-horizon geometry of the NS5-brane. 
  We propose a new version of holographic principle. This proposal extends the holographic principle based on the lightsheet to the one constraining the entropy passing through bulk hypersurface of timelike geodesics by the boundary area divided by 4G. We give a proof of the proposal in the classical regime based on a simple local entropy condition. 
  We study the localized tachyon condensation in their mirror Landau-Ginzburg picture. We completely determine the decay mode of an unstable orbifold $C^r/Z_n$, $r=1,2,3$ under the condensation of a tachyon with definite R-charge and mass by extending the Vafa's work hep-th/0111105. Here, we give a simple method that works uniformly for all $C^r/Z_n$. For $C^2/Z_n$, where method of toric geometry works, we give a proof of equivalence of our method with toric one. For $C^r/Z_n$ cases, the orbifolds decay into sum of $r$ far separated orbifolds. 
  The GSO projection in the twisted sector of orbifold background is sometimes subtle and incompatible descriptions are found in literatures. Here, from the equivalence of partition functions in NSR and GS formalisms, we give a simple rule of GSO projection for the chiral rings of string theory in $\C^r/\Z_n$, $r=1,2,3$. Necessary constructions of chiral rings are given by explicit mode analysis. 
  We investigate a decay of a bulk tachyon with a Kaluza-Klein momentum in bosonic and Type 0 string theories compactified on S^1. Potential for the tachyon has a (local) minimum. A decay of the tachyon would lead the original theory to a strongly coupled theory. An endpoint of the decay would exist if the strong coupling limit exists and it is a stable theory. 
  The aim of this note is to propose an interpretation for the full (non-chiral) correlation functions of the Liouville conformal field theory within the context of the quantization of spaces of Riemann surfaces. 
  Elasticity property (i.e. no-particle creation) is used in the tree level scattering of scalar particles in 1+1 dimensions to construct the affine Toda field theory(ATFT) associated with root systems of groups $a_2^{(2)}$ and $c_2^{(1)}$. A general prescription is given for constructing ATFT (associated with rank two root systems) with two self conjugate scalar fields. It is conjectured that the same method could be used to obtain the other ATFT associated with higher rank root systems. 
  In the context of the PT-symmetric version of quantum electrodynamics, it is argued that the C operator introduced in order to define a unitary inner product has nothing to do with charge conjugation. 
  We deal with scalar field coupled to gravity in five dimensions in warped geometry. We investigate models described by potentials that drive the system to support thick brane solutions that engender internal structure. We find analytical expressions for the brane solutions, and we show that they are all linearly stable. 
  It is shown that a version of PT-symmetric electrodynamics based on an axial-vector current coupling massless fermions to the photon possesses anomalies and so is rendered nonrenormalizable. An alternative theory is proposed based on the conventional vector current constructed from massive Dirac fields, but in which the PT transformation properties of electromagnetic fields are reversed. Such a theory seems to possess many attractive features. 
  Superfield realizations of Lorentz-violating extensions of the Wess-Zumino model are presented. These models retain supersymmetry but include terms that explicitly break the Lorentz symmetry. The models can be understood as arising from superspace transformations that are modifications of the familiar one in the Lorentz-symmetric case. 
  We find the complete chiral ring relations of the supersymmetric U(N) gauge theories with matter in adjoint representation. We demonstrate exact correspondence between the solutions of the chiral ring and the supersymmetric vacua of the gauge theory. The chiral ring determines the expectation values of chiral operators and the low energy gauge group. All the vacua have nonzero gaugino condensation. We study the chiral ring relations obeyed by the gaugino condensate. These relations are generalizations of the formula $S^N=\Lambda^{3N}$ of the pure ${\cal N} =1$ gauge theory. 
  We give a mathematically rigorous construction of the moduli space and vacuum geometry of a class of quantum field theories which are N=2 supersymmetric Wess-Zumino models on a cylinder. These theories have been proven to exist in the sense of constructive quantum field theory, and they also satisfy the assumptions used by Vafa and Cecotti in their study of the geometry of ground states. Since its inception, the Vafa-Cecotti theory of topological-antitopological fusion, or tt* geometry, has proven to be a powerful tool for calculations of exact quantum string amplitudes. However, tt* geometry postulates the existence of certain vector bundles and holomorphic sections built from the ground states. Our purpose in the present article is to give a mathematical proof that this postulate is valid within the context of the two-dimensional N=2 supersymmetric Wess-Zumino models. We also give a simpler proof in the case of holomorphic quantum mechanics. 
  We present a new 3-brane solution to Einstein's equations in (1+5)-spacetime with a negative bulk cosmological constant. This solution is a stringlike defect solution with decreasing scale function approaching a finite non-zero value in the radial infinity. It is shown that all local fields are localized on the brane only through the gravitational interaction. 
  Based on Haldane's spherical geometrical formalism of two-dimensional quantum Hall fluids, the relation between the noncommutative geometry of $S^2$ and the two-dimensional quantum Hall fluids is exhibited. If the number of particles $N$ is infinitely large, two-dimensional quantum Hall physics can be precisely described in terms of the noncommutative U(1) Chern-Simons theory proposed by Susskind, like in the case of plane. However, for the finite number of particles on two-sphere, the matrix-regularized version of noncommutative U(1) Chern-Simons theory involves in spinor oscillators. We establish explicitly such a finite matrix model on two-sphere as an effective description of fractional quantum Hall fluids of finite extent. The complete sets of physical quantum states of this matrix model are determined, and the properties of quantum Hall fluids related to them are discussed. We also describe how the low-lying excitations are constructed in terms of quasiparticle and quasihole excitations in the matrix model. It is shown that there consistently exists a Haldane's hierarchical structure of two-dimensional quantum Hall fluid states in the matrix model. These hierarchical fluid states are generated by the parent fluid state for particles by condensing the quasiparticle and quasihole excitations level by level, without any requirement of modifications of the matrix model. 
  Taking resort to Haldane's spherical geometry we can visualize fractional quantum Hall effect on the noncommutative manifold $M_4 \times Z_N$ with $N>2$ and odd. The discrete space leads to the deformation of symplectic structure of the continuous manifold such that the symplectic area is given by $\triangle p.\triangle q=2\pi m \hbar$ with $m$ an odd integer which is related to the Berry phase and the filling factor is given by $\frac{1}{m}$. We here argue that this is equivalent to the noncommutative field theory as prescribed by Susskind and Polychronakos which is characterized by area preserving diffeomorphism. The filling factor $\frac{1}{m}$ is determined from the change in chiral anomaly and hence the Berry phase as envisaged by the star product. 
  We consider threshold production of arbitrary number of Higgs particles by fermion-antifermion pair. The nullification of amplitudes for the special case Higgs mass = 2 * fermion mass is explained in terms of hidden symmetry of the reduced effective dynamics. 
  We present the exact solution of a scalar field theory defined with noncommuting position and momentum variables. The model describes charged particles in a uniform magnetic field and with an interaction defined by the Groenewold-Moyal star-product. Explicit results are presented for all Green's functions in arbitrary even spacetime dimensionality. Various scaling limits of the field theory are analysed non-perturbatively and the renormalizability of each limit examined. A supersymmetric extension of the field theory is also constructed in which the supersymmetry transformations are parametrized by differential operators in an infinite-dimensional noncommutative algebra. 
  By carefully analyzing the radial part of the wave-equation for a scalar field in AdS, we show that for a particular range of boundary conditions on the scalar field, the radial spectrum contains a bound state. Using the AdS/CFT correspondence, we interpret this peculiar phenomenon as being dual to an unstable double trace deformation of the boundary conformal field theory. We thus show how the bulk theory holographically detects whether a boundary perturbation is stable. 
  We study N=1 supersymmetric four-dimensional solutions of massive Type IIA supergravity with intersecting D6-branes in the presence NS-NS three-form fluxes. We derive N=1 supersymmetry conditions for the D6-brane and flux configurations in an internal manifold $X_6$ and derive the intrinsic torsion (or SU(3)-structure) related to the fluxes. In the absence of fluxes, N=1 supersymmetry implies that D6-branes wrap supersymmetric three-cycles of $X_6$ that intersect at angles of SU(3) rotations and the geometry is deformed by SU(3)-structures. The presence of fluxes breaks the SU(3) structures to SU(2) and the D6-branes intersect at angles of SU(2) rotations; non-zero mass parameter corresponds to D8-branes which are orthogonal to the common cycle of all D6-branes. The anomaly inflow indicates that the gauge theory on intersecting (massive) D6-branes is not chiral. 
  We utilise the duality between M theory and Type IIA string theory to show the existence of Freund-Rubin compactifications of M theory on 7-manifolds with singularities supporting chiral fermions. This leads to a concrete way to study phenomenologically interesting quantum gravity vacua using a holographically dual three dimensional field theory. 
  Recent work has shown that unstable D-branes in two dimensional string theory are represented by eigenvalues in a dual matrix model. We elaborate on this proposal by showing how to systematically include higher order effects in string perturbation theory. The full closed string state produced by a rolling open string tachyon corresponds to a sum of string amplitudes with any number of boundaries and closed string vertex operators. These contributions are easily extracted from the matrix model. As in the AdS/CFT correspondence, the sum of planar diagrams in the open string theory is directly related to the classical theory in the bulk, i.e. sphere diagrams. We also comment on the description of static D-branes in the matrix model, in terms of a solution representing a deformed Fermi sea. 
  The scalar field theory and the scalar electrodynamics quantized in the flat gap are considered. The dynamical effects arising due to the boundary presence with two types of boundary conditions (BC) satisfied by scalar fields are studied. It is shown that while the Neumann BC lead to the usual scalar field mass generation, the Dirichlet BC give rise to the dynamical mechanism of spontaneous symmetry breaking. Due to the later, there arises the possibility of the new type phase transition from the normal to spontaneously broken phase. The decreasing in the characteristic size of the quantization region (the gap size here) and increasing in the temperature compete with each other, tending to transport the system in the spontaneously broken and in the normal phase, respectively. The system evolves with a combined parameter, simultaneously reflecting the change in temperature and in the size. As a result, at the critical value of this parameter there occurs the phase transition from the normal phase to the spontaneously broken one. In particular, the usual massless scalar electrodynamics transforms to the Higgs model. 
  We provide Wilsonian proof for renormalizability of four-dimensional quantum field theories with ${\cal N}=1/2$ supersymmetry. We argue that the non-hermiticity inherent to these theories permits assigning noncanonical scaling dimension both for the Grassman coordinates and superfields. This reassignment can be done in such a way that the non(anti)commutativity parameter is dimensionless, and then the rest of the proof ammounts to power counting. The renormalizability is also stable against adding standard four-dimensional soft-breaking terms to the theory. However, with the new scaling dimension assignments, some of these terms are not just relevant deformations of the theory but become marginal. 
  We study solitonic solutions of a deformed Wess-Zumino model in 2 dimensions, corresponding to a deformation of the usual ${\cal N}=1,D=2$ superspace to the one with non-anticommuting odd supercoordinates. The deformation turns out to add a kinetic term for the auxiliary field besides the known $F^{3}$ term coming from the deformation of the cubic superpotential. Both these modifications are proportional to the effective deformation parameter $\lambda \equiv \det C$, where $C$ denotes the non-anticommutativity matrix. We find a modified ``orbit'' equation which on the EOM relates the auxiliary and the scalar components of the scalar superfield as a first order correction to the usual relation in terms of the small parameter $\lambda $. Subsequently, we obtain the modified form of the first order BPS equation for the scalar field and find its solution to first order in $\lambda $. Issues such as modification of the BPS mass formula and a non-linear realization of the ${\cal N}=1$ supersymmetry are discussed. 
  We present two new solutions to Einstein's equations in (1+5)-spacetime with a positive bulk cosmological constant. One solution has increasing and another solution decreasing bounded scale function without singularities in the range from the origin r=0 to the radial infinity. For the both solutions it is shown that all local fields are localized near the origin r=0 in the extra space through the gravitational interaction. 
  An interesting observation was reported by Corrigan-Sasaki that all the frequencies of small oscillations around equilibrium are " quantised" for Calogero and Sutherland (C-S) systems, typical integrable multi-particle dynamics. We present an analytic proof by applying recent results of Loris-Sasaki. Explicit forms of `classical' and quantum eigenfunctions are presented for C-S systems based on any root systems. 
  We study the P"oschl-Teller equation in complex domain and deduce infinite families of TQ and Bethe ansatz equations, classified by four integers. In all these models the form of T is very simple, while Q can be explicitly written in terms of the Heun function. At particular values there is a interesting interpretation in terms of finite lattice spin (L-2)/2 XXZ quantum chain with Delta= cos(pi/L) (for free-free boundary conditions), or Delta=-cos(pi/L) (for periodic boundary conditions). This result generalises the findings of Fridkin, Stroganov and Zagier. We also discuss the continuous (field theory) limit of these systems in view of the so-called ODE/IM correspondence. 
  By using a variant of quantum inverse scattering method (QISM) which is directly applicable to field theoretical systems, we derive all possible commutation relations among the operator valued elements of the monodromy matrix associated with an integrable derivative nonlinear Schrodinger (DNLS) model. From these commutation relations we obtain the exact Bethe eigenstates for the quantum conserved quantities of DNLS model. We also explicitly construct the first few quantum conserved quantities including the Hamiltonian in terms of the basic field operators of this model. It turns out that this quantum Hamiltonian has a new kind of coupling constant which is quite different from the classical one. This fact allows us to apply QISM to generate the spectrum of quantum DNLS Hamiltonian for the full range of its coupling constant. 
  We investigate the embedding of brane inflation into stable compactifications of string theory. At first sight a warped compactification geometry seems to produce a naturally flat inflaton potential, evading one well-known difficulty of brane-antibrane scenarios. Careful consideration of the closed string moduli reveals a further obstacle: superpotential stabilization of the compactification volume typically modifies the inflaton potential and renders it too steep for inflation. We discuss the non-generic conditions under which this problem does not arise. We conclude that brane inflation models can only work if restrictive assumptions about the method of volume stabilization, the warping of the internal space, and the source of inflationary energy are satisfied. We argue that this may not be a real problem, given the large range of available fluxes and background geometries in string theory. 
  It has long been known that, in higher-dimensional general relativity, there are black hole solutions with an arbitrarily large angular momentum for a fixed mass. We examine the geometry of the event horizon of such ultra-spinning black holes and argue that these solutions become unstable at large enough rotation. Hence we find that higher-dimensional general relativity imposes an effective `Kerr-bound' on spinning black holes through a dynamical decay mechanism. Our results also give indications of the existence of new stationary black holes with `rippled' horizons of spherical topology. We consider various scenarios for the possible decay of ultra-spinning black holes, and finally discuss the implications of our results for black holes in braneworld scenarios. 
  We study Big Crunch/Big Bang cosmologies that correspond to exact world-sheet superconformal field theories of type II strings. The string theory spacetime contains a Big Crunch and a Big Bang cosmology, as well as additional ``whisker'' asymptotic and intermediate regions. Within the context of free string theory, we compute, unambiguously, the scalar fluctuation spectrum in all regions of spacetime. Generically, the Big Crunch fluctuation spectrum is altered while passing through the bounce singularity. The change in the spectrum is characterized by a function $\Delta$, which is momentum and time-dependent. We compute $\Delta$ explicitly and demonstrate that it arises from the whisker regions. The whiskers are also shown to lead to ``entanglement'' entropy in the Big Bang region. Finally, in the Milne orbifold limit of our superconformal vacua, we show that $\Delta\to 1$ and, hence, the fluctuation spectrum is unaltered by the Big Crunch/Big Bang singularity. We comment on, but do not attempt to resolve, subtleties related to gravitational backreaction and light winding modes when interactions are taken into account. 
  The charges of D-branes in Kazama-Suzuki coset models are analyzed. We provide the calculation of the corresponding twisted equivariant K-theory, and in the case of Grassmannian cosets, su(n+1)/u(n), compare this to the charge lattices that are derived from boundary conformal field theory. 
  We discuss the convergence of level truncation in bosonic open string field theory. As a test case we consider the calculation of the quartic tachyon coupling $\gamma_4$. We determine the exact contribution from states up to level L=28 and discuss the $L\to\infty$ extrapolation by means of the BST algorithm. We determine in a self-consistent way both the coupling and the exponent $\omega$ of the leading correction to $\gamma_4$ at finite $L$ that we assume to be $\sim 1/L^\omega$. The results are $\gamma_4 = -1.7422006(9)$ and $|\omega-1|\lesssim 10^{-4}$.} 
  We investigate the $h$-deformed quantum (super)group of $2\times 2$ matrices and use a kind of contraction procedure to prove that the $n$-th power of this deformed quantum (super)matrix is quantum (super)matrix with the deformation parameter $nh$. 
  The relative coefficients of higher derivative interactions of the IIB effective action of the form C^4, (D F_5)^4, F_5^8, ... (where C is the Weyl tensor and F_5 is the five-form field strength) are motivated by supersymmetry arguments. It is shown that the classical supergravity solution for N parallel D3-branes is unaltered by this combination of terms. The non-vanishing of \zeroC^2 in this background (where \zero C is the background value of the Weyl tensor) leads to effective O(1/alpha') interactions, such as C^2 and Lambda^8 (where Lambda is the dilatino). These contain D-instanton contributions in addition to tree and one-loop terms. The near horizon limit of the N D3-brane system describes a multi-AdS_5xS^5 geometry that is dual to \calN=4 SU(N) Yang-Mills theory spontaneously broken to S(U(M_1)x...xU(M_r)). Here, the N D3-branes are grouped into r coincident bunches with M_r in each group, with M_r/N = m_r fixed as N goes to infinity. The boundary correlation function of eight Lambda's is constructed explicitly. The second part of the paper considers effects of a constrained instanton in this large-N Yang-Mills theory by an extension of the analysis of Dorey, Hollowood and Khoze of the one-instanton measure at finite N. This makes precise the correspondence with the supergravity D-instanton measure at leading order in the 1/N expansion. However, the duality between instanton-induced correlation functions in Yang-Mills theory and the dual supergravity is somewhat obscured by complications relating to the structure of constrained instantons. 
  We prove that the two interaction Hamiltonians of light-cone closed superstring field theory in the plane-wave background present in the literature are identical. 
  We estimate the very long time behaviour of correlation functions in the presence of eternal black holes. It was pointed out by Maldacena (hep-th 0106112) that their vanishing would lead to a violation of a unitarity-based bound. The value of the bound is obtained from the holographic dual field theory. The correlators indeed vanish in a semiclassical bulk approximation. We trace the origin of their vanishing to the continuum energy spectrum in the presence of event horizons. We elaborate on the two very long time scales involved: one associated with the black hole and the other with a thermal gas in the vacuum background. We find that assigning a role to the thermal gas background, as suggested in the above work, does restore the compliance with a time-averaged unitarity bound. We also find that additional configurations are needed to explain the expected time dependence of the Poincar\'e recurrences and their magnitude. It is suggested that, while a semiclassical black hole does reproduce faithfully ``coarse grained'' properties of the system, additional dynamical features of the horizon may be necessary to resolve a finer grained information-loss problem. In particular, an effectively formed stretched horizon could yield the desired results. 
  We present warped compactification solutions of six-dimensional supergravity, which are generalizations of the Randall-Sundrum warped brane world to codimension two and to a supersymmetric context. In these solutions the dilaton varies over the extra dimensions, and this makes the electroweak hierarchy only power-law sensitive to the proper radius of the extra dimensions (as opposed to being exponentially sensitive as in the RS model). Warping changes the phenomenology of these models because the Kaluza-Klein gap can be much larger than the internal space's inverse proper radius. We provide examples both for Romans' nonchiral supergravity and Salam-Sezgin chiral supergravity, and in both cases the solutions break all of the supersymmetries of the models. We interpret the solution as describing the fields sourced by a 3-brane and a boundary 4-brane (Romans' supergravity) or by one or two 3-branes (Salam-Sezgin supergravity), and we identify the topological constraints which are required by this interpretation. For both types of solutions the 3-branes are flat for all topologically-allowed values of the brane tensions. We identify the general mechanism for and limitations of the self-tuning of the effective 4D cosmological constant in higher-dimensional supergravity which these models illustrate. 
  Little groups for preon branes (i.e. configurations of branes with maximal (n-1)/n fraction of survived supersymmetry) for dimensions d=2,3,...,11 are calculated for all massless, and partially for massive orbits. For massless orbits little groups are semidirect product of d-2 translational group $T_{d-2}$ on a subgroup of (SO(d-2) $\times$ R-invariance) group. E.g. at d=9 the subgroup is exceptional $G_2$ group. It is also argued, that 11d Majorana spinor invariants, which distinguish orbits, are actually invariant under d=2+10 Lorentz group. Possible applications of these results include construction of field theories in generalized space-times with brane charges coordinates, different problems of group's representations decompositions, spin-statistics issues. 
  Quantum-mechanical initial conditions for the fluctuations of the geometry can be assigned in excess of a given physical wavelength. The two-point functions of the scalar and tensor modes of the geometry will then inherit corrections depending on which Hamiltonian is minimized at the initial stage of the evolution. The energy density of the background geometry is compared with the energy-momentum pseudo-tensor of the fluctuations averaged over the initial states, minimizing each different Hamiltonian. The minimization of adiabatic Hamiltonians leads to initial states whose back-reaction on the geometry is negligible. The minimization of non-adiabatic Hamiltonians, ultimately responsible for large corrections in the two-point functions, is associated with initial states whose energetic content is of the same order as the energy density of the background. 
  We study the classical and quantum cosmology of a $(4+d)$-dimensional spacetime minimally coupled to a scalar field and present exact solutions for the resulting field equations for the case where the universe is spatially flat. These solutions exhibit signature transition from a Euclidean to a Lorentzian domain and lead to stabilization of the internal space, in contrast to the solutions which do not undergo signature transition. The corresponding quantum cosmology is described by the Wheeler-DeWitt equation which has exact solutions in the mini-superspace, resulting in wavefunctions peaking around the classical paths. Such solutions admit parametrizations corresponding to metric solutions of the field equations that admit signature transition. 
  Recent investigations involving the decay of unstable D-branes in string theory suggest that the tree level open string theory which describes the dynamics of the D-brane already knows about the closed string states produced in the decay of the brane. We propose a specific conjecture involving quantum open string field theory to explain this classical result, and show that the recent results in two dimensional string theory are in exact accordance with this conjecture. 
  Starting from the non-BPS D$(p+1)$-brane action we derive an effective action in $(p+1)$ space dimensions by studying the fluctuations of various bosonic fields around the time-like tachyonic kink solution (obtained by Wick rotation of the space-like tachyonic kink solution) of the non-BPS brane. In real time this describes the dynamics of a space-like or Euclidean brane in $(p+1)$-dimensions containing a Dirac-Born-Infeld part and a Wess-Zumino part. The WZ part is purely imaginary and so the action is complex if it represents the source of the time-dependent background of type II string theory i.e. the S-brane. On the other hand, the WZ part as well as the action is real if it represents the source in type II$^\ast$ string theory i.e. the E-brane. The DBI part is the same as obtained before using different method. This is then further illustrated by considering brane probe in space-like brane background. 
  Energy-momentum is an important conserved quantity whose definition has been a focus of many investigations in general relativity. Unfortunately, there is still no generally accepted definition of energy and momentum in general relativity. Attempts aimed at finding a quantity for describing distribution of energy-momentum due to matter, non-gravitational and gravitational fields resulted in various energy-momentum complexes whose physical meaning have been questioned. The problems associated with energy-momentum complexes resulted in some researchers even abandoning the concept of energy-momentum localization in favour of the alternative concept of quasi-localization. However, quasi-local masses have their inadequacies, while the remarkable work of Virbhadra and some others, and recent results of Cooperstock and Chang {\it et al.} have revived an interest in various energy-momentum complexes. Hence in this work we use energy-momentum complexes to obtain the energy distributions in various space-times.   We elaborate on the problem of energy localization in general relativity and use energy-momentum prescriptions of Einstein, Landau and Lifshitz, Papapetrou, Weinberg, and M{\o}ller to investigate energy distributions in various space-times. It is shown that several of these energy-momentum complexes give the same and acceptable results for a given space-time. This shows the importance of these energy-momentum complexes. Our results agree with Virbhadra's conclusion that the Einstein's energy-momentum complex is still the best tool for obtaining energy distribution in a given space-time. The Cooperstock hypothesis for energy localization in GR is also supported. 
  We use recent results of Intriligator and Wecht [hep-th/0304128] to study the phase structure of $\NN=1$ super Yang-Mills theory with gauge group $SU(N_c)$, a chiral superfield in the adjoint, and $N_f$ chiral superfields in the fundamental representation of the gauge group. Our discussion sheds new light on [hep-th/0304128] and supports the conjecture that the central charge $a$ decreases under RG flows and is non-negative in unitary four dimensional conformal field theories. 
  We study the evolution of gravitational waves(GWs) after inflation in a brane-world cosmology embedded in five-dimensional anti-de Sitter spacetime. Contrary to the standard four-dimensional results, the GWs at the high-energy regime in brane-world model suffer from the effects of the non-standard cosmological expansion and the excitation of the Kaluza-Klein modes(KK-modes), which can affect the amplitude of stochastic gravitational wave background significantly. To investigate these two high-energy effects quantitatively, we numerically solve the wave equation of the GWs in the radiation dominated epoch at relatively low-energy scales. We show that the resultant GWs is suppressed by the excitation of the KK modes. The created KK modes are rather soft and escape away from the brane to the bulk gravitational field. The results are also compared to the semi-analytic prediction from the low-energy approximation and the evolved amplitude of GWs on the brane reasonably matches the numerical simulations. 
  Recently, a number of intriguing results have been obtained for strongly coupled ${\cal N}=4$ Supersymmetric Yang-Mills theory in vacuum and matter, using the AdS/CFT correspondence. In this work, we provide a physical picture supporting and explaining most of these results within the gauge theory. The modified Coulomb's law at strong coupling forces static charges to communicate via the high frequency modes of the gauge/scalar fields. Therefore, the interaction between even relativistically moving charges can be approximated by a potential. At strong coupling, WKB arguments yield a series of deeply bound states, whereby the large Coulomb attraction is balanced by centrifugation. The result is a constant density of light bound states at {\bf any} value of the strong coupling, explaining why the thermodynamics and kinetics are coupling constant independent. In essence, at strong coupling the matter is not made of the original quasiparticles but of much lighter (binary) composites. A transition from weak to strong coupling is reminiscent to a transition from high to low $T$ in QCD. We establish novel results for screening in vacuum and matter through a dominant set of diagrams some of which are in qualitative agreement with known strong coupling results. 
  In this note we consider higher-loop contributions to the planar dilatation operator of N=4 SYM in the su(2) subsector of two complex scalar fields. We investigate the constraints on the form of this object due to interactions of two excitations in the BMN limit. We then consider two scenarios to uniquely fix some higher-loop contributions: (i) Higher-loop integrability fixes the dilatation generator up to at least four-loops. Among other results, this allows to conjecture an all-loop expression for the energy in the near BMN limit. (ii) The near plane-wave limit of string theory and the BMN correspondence fix the dilatation generator up to three-loops. We comment on the difference between both scenarios. 
  We consider the tricritical Ising model on a strip or cylinder under the integrable perturbation by the thermal $\phi_{1,3}$ boundary field. This perturbation induces five distinct renormalization group (RG) flows between Cardy type boundary conditions labelled by the Kac labels $(r,s)$. We study these boundary RG flows in detail for all excitations. Exact Thermodynamic Bethe Ansatz (TBA) equations are derived using the lattice approach by considering the continuum scaling limit of the $A_4$ lattice model with integrable boundary conditions. Fixing the bulk weights to their critical values, the integrable boundary weights admit a thermodynamic boundary field $\xi$ which induces the flow and, in the continuum scaling limit, plays the role of the perturbing boundary field $\phi_{1,3}$. The excitations are completely classified, in terms of string content, by $(m,n)$ systems and quantum numbers but the string content changes by either two or three well-defined mechanisms along the flow. We identify these mechanisms and obtain the induced maps between the relevant finitized Virasoro characters. We also solve the TBA equations numerically to determine the boundary flows for the leading excitations. 
  We investigate the structure of local anomalies of heterotic E_8 x E_8' theory on T^6/Z_4. We show that the untwisted states lead to anomalies in ten, six and four dimensions. At each of the six dimensional fixed spaces of this orbifold the twisted states ensure, that the anomalies factorize separately. As some of these twisted states live on T^2/Z_2, they give rise to four dimensional anomalies as well. At all four dimensional fixed points at worst a single Abelian anomaly can arise. Since the anomalies in all these dimensions factorize in a universal way, they can be canceled simultaneously. In addition, we show that for all U(1) factors at the four dimensional fixed points at least logarithmically divergent Fayet--Ilopoulos tadpoles are generated. 
  We obtain the energy distribution in the Kerr-Newman metric with the help of Bergmann-Thomson energy-momentum complex. We find that the energy-momentum definitions prescribed by Einstein, Landau-Lifshitz, Papapetrou, Weinberg, and Bergmann-Thomson give the same and acceptable result and also support the {\em Cooperstock Hypothesis} for energy localization in general relativity. The repulsive effect due to the electric charge and rotation parameters of the metric is also reflected from the energy distribution expression. 
  We investigate the evolution of gravitational wave perturbations about a brane cosmology embedded in a five-dimensional anti-de Sitter bulk. During slow-roll inflation in a Randall-Sundrum brane-world, the zero mode of the 5-dimensional graviton is generated, while the massive modes remain in their vacuum state. When the zero mode re-enters the Hubble radius during radiation domination, massive modes are generated. We show that modes decouple in the low-energy/near-brane limit and develop perturbative techniques to calculate the mode-mixing at finite energy. 
  The main focus of the present work is to study the Feynman's proof of the Maxwell equations using the NC geometry framework. To accomplish this task, we consider two kinds of noncommutativity formulations going along the same lines as Feynman's approach. This allows us to go beyond the standard case and discover non-trivial results. In fact, while the first formulation gives rise to the static Maxwell equations, the second formulation is based on the following assumption $m[x_{j},\dot{x_{k}}]=i\hbar \delta_{jk}+im\theta_{jk}f.$ The results extracted from the second formulation are more significant since they are associated to a non trivial $\theta $-extension of the Bianchi-set of Maxwell equations. We find $div_{\theta}B=\eta_{\theta}$ and $\frac{\partial B_{s}}{\partial t}+\epsilon_{kjs}\frac{\partial E_{j}}{\partial x_{k}}=A_{1}\frac{d^{2}f}{dt^{2}}+A_{2}\frac{df}{dt}+A_{3},$ where $\eta_{\theta}$, $A_{1}$, $A_{2}$ and $A_{3}$ are local functions depending on the NC $\theta $-parameter. The novelty of this proof in the NC space is revealed notably at the level of the corrections brought to the previous Maxwell equations. These corrections correspond essentially to the possibility of existence of magnetic charges sources that we can associate to the magnetic monopole since $div_{\theta}B=\eta_{\theta}$ is not vanishing in general. 
  In this paper we construct an expanding phase with phantom matter, in which the scale factor expands very slowly but the Hubble parameter increases gradually, and assume that this expanding phase could be matched to our late observational cosmology by the proper mechanism. We obtain the nearly scale-invariant spectrum of adiabatic fluctuations in this scenario, different from the simplest inflation and usual ekpyrotic/cyclic scenario, the tilt of nearly scale-invariant spectrum in this scenario is blue. Although there exists an uncertainty surrounding the way in which the perturbations propagate through the transition in our scenario, which is dependent on the detail of possible "bounce" physics, compared with inflation and ekpyrotic/cyclic scenario, our work may provide another feasible cosmological scenario generating the nearly scale-invariant perturbation spectrum. 
  The gauge fixing procedure for N=1 supersymmetric Yang-Mills theory (SYM) is proposed in the context of the stochastic quantization method (SQM). The stochastic gauge fixing, which was formulated by Zwanziger for Yang-Mills theory, is extended to SYM_4 in the superfield formalism by introducing a chiral and an anti-chiral superfield as the gauge fixing functions. It is shown that SQM with the stochastic gauge fixing reproduces the probability distribution of SYM_4, defined by the Faddeev-Popov prescription, in the equilibrium limit with an appropriate choice of the stochastic gauge fixing functions. We also show that the BRST symmetry of the corresponding stochastic action and the power counting argument in the superfield formalism ensure the renormalizability of SYM_4 in this context. 
  We investigate the $g_2$-invariant bulk (1+1D, factorized) $S$-matrix constructed by Ogievetsky, using the bootstrap on the three-point coupling of the vector multiplet to constrain its CDD ambiguity. We then construct the corresponding boundary $S$-matrix, demonstrating it to be consistent with $Y(g_2,a_1\times a_1)$ symmetry. 
  We discuss the holographic c-function and describe an algorithm for the practical computation of the changing central charge in arbitrary RG-flows. One example is worked out in detail. The renormalisation procedure of [hep-th/0112150]necessary to obtain the central charge is reviewed. 
  We extend a deformation prescription recently introduced and present some new soluble nonlinear problems for kinks and lumps. In particular, we show how to generate models which present the basic ingredients needed to give rise to "dimension bubbles," having different macroscopic space dimensions on the interior and the exterior of the bubble surface. Also, we show how to deform a model possessing lumplike solutions, relevant to the discussion of tachyonic excitations, to get a new one having topological solutions. 
  This diploma thesis has three major objectives. Firstly, we give an elementary introduction to M-theory compactifications, which are obtained from an analysis of its low-energy effective theory, eleven-dimensional supergravity. In particular, we show how the requirement of N=1 supersymmetry in four dimensions leads to compactifications on G_2-manifolds. We also examine the Freund-Rubin solution as well as the M2- and M5-brane. Secondly, we review the construction of realistic theories in four dimensions from compactifications on G_2-manifolds. It turns out that this can only be achieved if the manifolds are allowed to carry singularities of various kinds. Thirdly, we are interested in the concept of anomalies in the framework of M-theory. We present some basic material on anomalies and examine three cases where anomalies play a prominent role in M-theory. We review M-theory on R^10 x S^1/Z_2 where anomalies are a major ingredient leading to the duality between M-theory and the E_8 x E_8 heterotic string. A detailed calculation of the tangent and normal bundle anomaly in the case of the M5-brane is also included. It is known that in this case the normal bundle anomaly can only be cancelled if the topological term of eleven-dimensional supergravity is modified in a suitable way. Finally, we present a new mechanism to cancel anomalies which are present if M-theory is compactified on G_2-manifolds carrying singularities of codimension seven. In order to establish local anomaly cancellation we once again have to modify the topological term of eleven-dimensional supergravity as well as the Green-Schwarz term. 
  We study the Z_N flux tubes and monopole confinement in deformed N=2* super Yang-Mills theories. In order to do that we consider an N=4 super Yang-Mills theory with an arbitrary gauge group G and add some N=2, N=1 and N=0 deformation terms. We analyze some possible vacuum solutions and phases of the theory, depending on the deformation terms which are added. In the Coulomb phase for the N=2* theory, G is broken to U(1)^r and the theory has monopole solutions. Then, by adding some deformation terms, the theory passes to the Higgs or color superconducting phase, in which G is broken to its center C_G. In this phase we construct the Z_N flux tubes ansatz and obtain the BPS string tension. We show that the monopole magnetic fluxes are linear integer combinations of the string fluxes and therefore the monopoles can become confined. Then, we obtain a bound for the threshold length of the string-breaking. We also show the possible formation of a confining system with 3 different monopoles for the SU(3) gauge group. Finally we show that the BPS string tensions of the theory satisfy the Casimir scaling law. 
  Based on the recently considered classical string configurations, in the framework of the semi-classical limit of the string/gauge theory correspondence, we describe a procedure for obtaining exact classical string solutions in general string theory backgrounds, when the string embedding coordinates depend non-linearly on the worldsheet spatial parameter. The tensionless limit, corresponding to small t'Hooft coupling on the field theory side, is also considered. Applying the developed approach, we first reproduce some known results. Then, we find new string solutions - with two spins in AdS_5 black hole background and in AdS_5 x S^5 with two spins and up to nine independent conserved R-charges. 
  Recently, interesting braneworld cosmologies in the Randall-Sundrum scenario have been constructed using a bulk spacetime which corresponds to a charged AdS black hole. In particular, these solutions appear to `bounce', making a smooth transition from a contracting to an expanding phase. By considering the spacetime geometry more carefully, we demonstrate that generically in these solutions the brane will encounter a singularity in the transition region. 
  We make contact between the infinite-dimensional non-local symmetry of the typeIIB superstring on AdS5xS5 worldsheet theory and a non-abelian infinite-dimensional symmetry algebra for the weakly coupled superconformal gauge theory. We explain why the planar limit of the one-loop dilatation operator is the Hamiltonian of a spin chain, and show that it commutes with the g*2 N = 0 limit of the non-abelian charges. 
  By means of an identity that equates elliptic genus partition function of a supersymmetric sigma model on the $N$-fold symmetric product $S^N X$ of $X$ ($S^N X=X^N/S_N$, $S_N$ is the symmetric group of $N$ elements) to the partition function of a second quantized string theory, we derive the asymptotic expansion of the partition function as well as the asymptotic for the degeneracy of spectrum in string theory. The asymptotic expansion for the state counting reproduces the logarithmic correction to the black hole entropy. 
  The exact Li$\acute{e}$nard-Wiechert solutions for the point charge in arbitrary motion are shown to be null fields everywhere. These are used as a basis to introduce extended electromagnetic field equations that have null field solutions with fractional charges that combine with absolute confining potentials. 
  We analyze in detail the recursive construction of the Seiberg-Witten map and give an exhaustive description of its ambiguities. The local BRST cohomology for noncommutative Yang-Mills theory is investigated in the framework of the effective commutative Yang-Mills type theory. In particular, we show how some of the conformal symmetries get obstructed by the noncommutative deformation. 
  We consider the Dirac equation with a magnetic-solenoid field (the superposition of the Aharonov--Bohm solenoid field and a collinear uniform magnetic field). Using von Neumann's theory of the self-adjoint extensions of symmetric operators, we construct a one-parameter family and a two-parameter family of self-adjoint Dirac Hamiltonians in the respective 2+1 and 3+1 dimensions. Each Hamiltonian is specified by certain asymptotic boundary conditions at the solenoid. We find the spectrum and eigenfunctions for all values of the extension parameters. We also consider the case of a regularized magnetic-solenoid field (with a finite-radius solenoid field component) and study the dependence of the eigenfunctions on the behavior of the magnetic field inside the solenoid. The zero-radius limit yields a concrete self-adjoint Hamiltonian for the case of the magnetic-solenoid field. In addition, we consider the spinless particle in the regularized magnetic-solenoid field. By the example of the radial Dirac Hamiltonian with the magnetic-solenoid field, we present an alternative, more simple and efficient, method for constructing self-adjoint extensions applicable to a wide class of singular differential operators. 
  It is proved that, even if the gauge symmetry has been broken spontaneously at tree level, supersymmetry would never break through any finite orders of perturbation if it is not broken classically. 
  It is shown that, by allowing a transmutation between a boson and a fermion, the system with both bosons and fermions will have the statistical distribution function of an anyon. 
  The model of spinning particle, based on the Kerr-Newman solution with |a}>>m, is discussed. It is shown that the Kerr singular ring can be considered as a string with an orientifold world-sheet. Orientifold adds to the Kerr ring an extra peculiar point, the fixed point of the world-sheet parity operator. It is shown that the Kerr string represents a new type of the string solutions and turns out to be an open D-string with joined ends which are in the circular light-like motion along the Kerr ring. 
  The study of BTZ blackhole physics and the cosmological horizon of 3D de Sitter spaces are carried out in unified way using the connections to the Chern Simons theory on three manifolds with boundary. The relations to CFT on the boundary is exploited to construct exact partition functions and obtain logarithmic corrections to Bekenstein formula in the asymptotic regime. Comments are made on the dS/CFT correspondence frising from these studies. 
  We consider radial oscillations of supertube probes in the Godel-type background which is U-dual to the compactified pp-wave obtained from the Penrose limit of the NS five-brane near horizon geometry. The supertube probe computation can be carried over directly to a string probe calculation on the U-dual background. The classical equations of motion are solved explicitly. In general, the probe is not restricted to travel unidirectionally through any global time coordinate. In particular, we find geodesics that close. 
  Using concepts developed in string theory, Cohen, Moore, Nelson and Polchinski calculated the propagator for a relativistic point particle. Following these authors we extend the technique to include the case of closed world lines. The partition function found corresponds to the Feynmann and Schwinger proper time formalism. We also explicitly verify that the partition function is equivalent to the usual path length action partition function. As an example of a sum over closed world lines, we compute the Euler-Heisenberg effective Lagrangian in a novel way. 
  We propose an approach of lattice gauge theory based on a homotopic interpretation of its degrees of freedom. The basic idea is to dress the plaquettes of the lattice to view them as elementary homotopies between nearby paths. Instead of using a unique $G$-valued field to discretize the connection 1-form, $A$, we use an $\AG$-valued field $U$ on the edges, which plays the role of the 1-form $\ad_A$, and a $G$-valued field $V$ on the plaquettes, which corresponds to the Faraday tensor, $F$. The 1-connection, $U$, and the 2-connection, $V$, are then supposed to have a 2-curvature which vanishes. This constraint determines $V$ as a function of $U$ up to a phase in $Z(G)$, the center of $G$. The 3-curvature around a cube is then Abelian and is interpreted as the magnetic charge contained inside this cube. Promoting the plaquettes to elementary homotopies induces a chiral splitting of their usual Boltzmann weight, $w=v\bar{v}$, defined with the Wilson action. We compute the Fourier transform, $\hat{v}$, of this chiral Boltzmann weight on $G=SU_3$ and we obtain a finite sum of generalized hypergeometric functions. The dual model describes the dynamics of three spin fields : $\lambda_P\in{\hat{G}}$ and $m_P\in{\hat{Z(G)}}\simeq\Z_3$, on each oriented plaquette $P$, and $\epsilon_{ab}\in{\hat{\OG}}\simeq\Z_2$, on each oriented edge $(ab)$. Finally, we sketch a geometric interpretation of this spin system in a fibered category modeled on the category of representations of $G$. 
  It is shown how two-dimensional states corresponding to D-branes arise in orbifolds of topologically massive gauge and gravity theories. Brane vertex operators naturally appear in induced worldsheet actions when the three-dimensional gauge theory is minimally coupled to external charged matter and the orbifold relations are carefully taken into account. Boundary states corresponding to D-branes are given by vacuum wavefunctionals of the gauge theory in the presence of the matter, and their various constraints, such as the Cardy condition, are shown to arise through compatibility relations between the orbifold charges and bulk gauge invariance. We show that with a further conformally-invariant coupling to a dynamical massless scalar field theory in three dimensions, the brane tension is naturally set by the bulk mass scales and arises through dynamical mechanisms. As an auxilliary result, we show how to describe string descendent states in the three-dimensional theory through the construction of gauge-invariant excited wavefunctionals. 
  We elaborate a full superfield description of the interacting system of dynamical D=4, N=1 supergravity and dynamical superstring. As far as minimal formulation of the simple supergravity is used, such a system should contain as well the tensor (real linear) multiplet which describes the dilaton and the two-superform gauge field whose pull-back provides the Wess-Zumino term for the superstring. The superfield action is given by the sum of the Wess-Zumino action for D=4, N=1 superfield supergravity, the superfield action for the tensor multiplet in curved superspace and the Green-Schwarz superstring action. The latter includes the coupling to the tensor multiplet both in the Nambu-Goto and in the Wess-Zumino terms. We derive superfield equations of motion including, besides the superfield supergravity equations with the source, the source-full superfield equations for the linear multiplet. The superstring equations keep the same form as for the superstring in supergravity and 2-superform background. The analysis of gauge symmetries shows that the superfield description of the interacting system is gauge equivalent to the dynamical system described by the sum of the spacetime, component action for supergravity interacting with tensor multiplet and of the purely bosonic string action. 
  We consider the possibility that the UV completeness of a fundamental theory is achieved by a modification of propagators at large momenta. We assume that general covariance is preserved at all energies, and focus on the coupling of a scalar field to the background geometry as an example. Naively, one expects that the gravitational interaction, like Yukawa interactions, will be regularized by a propagator which decays to zero sufficiently fast above some cutoff scale, but we show that in order to avoid the ultra-violet divergence, the propagator should approach to a nonzero constant. This incompatibility between the regularizations of gravitational and Yukawa interactions suggests that a symmetry of the particle spectrum is needed for a UV complete fundamental theory. 
  We obtain D=4 de Sitter gravity coupled to SU(2) Yang-Mills gauge fields from an explicit and consistent truncation of D=11 supergravity via Kaluza-Klein dimensional reduction on a non-compact space. The ``internal'' space is a smooth hyperbolic 7-space (H^7) written as a foliation of two 3-spheres, on which the SU(2) Yang-Mills fields reside. The positive cosmological constant is completely fixed by the SU(2) gauge coupling constant. The explicit reduction ansatz enables us to lift any of the D=4 solutions to D=11. In particular, we obtain dS_2 in M-theory, where the nine-dimensional transverse space is an H^7 bundle over S^2. We also obtain a new smooth embedding of dS_3 in D=6 supergravity. 
  We present evidence that the supersymmetric matrix model of Marinari and Parisi represents the world-line theory of N unstable D-particles in type II superstring theory in two dimensions. This identification suggests that the matrix model gives a holographic description of superstrings in a two-dimensional black hole geometry. 
  In this paper we compute leading order correction due to small statistical fluctuations around equilibrium, to the Cardy-Verlinde entropy formula (which is supposed to be an entropy formula of conformal field theory in any dimension) of a Topological Reissner-Nordstrom black hole in de Sitter space. 
  A covariant quantization method is developed for the off-shell superparticle in 10 dimensions. On-shell it is consistent with lightcone quantization, while off-shell it gives a noncommutative superspace that realizes non-linearly a hidden 11-dimensional super Poincare symmetry. The non-linear commutation rules are then used to construct the supersymmetric generalization of the covariant Moyal star product in noncommutative superspace. As one of the possible applications, we propose this new product as the star product in supersymmetric string field theory. Furthermore, the formalism introduces new techniques and concepts in noncommutative (super)geometry. 
  We argue that already at classical level the energy-momentum tensor for a scalar field on manifolds with boundaries in addition to the bulk part contains a contribution located on the boundary. Using the standard variational procedure for the action with the boundary term, the expression for the surface energy-momentum tensor is derived for arbitrary bulk and boundary geometries. Integral conservation laws are investigated. The corresponding conserved charges are constructed and their relation to the proper densities is discussed. Further we study the vacuum expectation value of the energy-momentum tensor in the corresponding quantum field theory. It is shown that the surface term in the energy-momentum tensor is essential to obtain the equality between the vacuum energy, evaluated as the sum of the zero-point energies for each normal mode of frequency, and the energy derived by the integration of the corresponding vacuum energy density. As an application, by using the zeta function technique, we evaluate the surface energy for a quantum scalar field confined inside a spherical shell. 
  In this paper, we calculate the stress-energy tensor for a quantized massless conformally coupled scalar field in the background of a conformally flat brane-world geometries, where the scalar field satisfying Robin boundary conditions on two parallel plates. In the general case of Robin boundary conditions formula are derived for the vacuum expectation values of the energy-momentum tensor. Further the surface energy per unit area are obtained . As an application of the general formula we have considered the important special case of the AdS$_{4+1}$ bulk, moreover application to the Randall-Sundrum scenario is discused. In this specific example for a certain choice of Robin coefficients, one could make the effective cosmological constant vanish. 
  The cosmological curvature perturbation may be generated when some `curvaton' field, different from the inflaton, oscillates in a background of unperturbed radiation. In its simplest form the curvaton paradigm requires the Hubble parameter during inflation to be bigger than $10^7\GeV$, but this bound may be evaded if the curvaton field (or an associated tachyon) is strongly coupled to a field which acquires a large value at the end of inflation. As a result the curvaton paradigm might be useful in improving the viability of low-scale inflation models, in which the supersymmetry-breaking mechanism is the same as the one which operates in the vacuum. 
  For the minimal O(N) sigma model, which is defined to be generated by the O(N) scalar auxiliary field alone, all n-point functions, till order 1/N included, can be expressed by elementary functions without logarithms. Consequently, the conformal composite fields of m auxiliary fields possess at the same order such dimensions, which are m times the dimension of the auxiliary field plus the order of differentiation. 
  We reformulate the conditions of Liouville integrability in the language of Gozzi et al.'s quantum BRST anti-BRST description of classical mechanics. The Das-Okubo geometrical Lax equation is particularly suited to this approach. We find that the Lax pair and inverse scattering wavefunction appear naturally in certain sectors of the quantum theory. 
  A family of cosmological solutions with $(n+1)$ Ricci-flat spaces in the theory with several scalar fields and multiple exponential potential is obtained when coupling vectors in exponents obey certain relations. Two subclasses of solutions with power-law and exponential behaviour of scale factors are singled out. It is proved that power-law solutions may take place only when coupling vectors are linearly independent and exponential dependence occurs for linearly dependent set of coupling vectors. A subfamily of solutions with accelerated expansion is singled out. A generalized isotropization behaviours of certain classes of general solutions are found. In quantum case exact solutions to Wheeler-DeWitt equation are obtained and special "ground state" wave functions are considered. 
  We review the construction of the $D$-branes at finite temperature as boundary states in the Fock space of thermal perturbative closed string. This is a talk presented by I. V. V. at Common Trends in Cosmology and Particle Physics June 2003, Balatonfured, Hungary. 
  In a field-theoretical context, we consider the Euclidean $(\phi^4+\phi^6)_D$ model compactified in one of the spatial dimensions. We are able to determine the dependence of the transition temperature ($T_{c}$)for a system described by this model, confined between two parallel planes, as a function of the distance($L$) separating them. We show that $T_{c}$ is a concave function of $L^{-1}$. We determine a minimal separation below which the transition is suppressed. 
  We study the evolution of bubble spacetimes in vacuum and electrovac scenarios by numerical means. We find strong evidence against the formation of naked singularities in (i) scenarios with negative masses displaying initially collapsing conditions and (ii) scenarios with negative masses displaying initially expanding conditions, previously reported to give rise to such singularities. Additionally, we show that the presence of strong gauge fields implies that an initially collapsing bubble bounces back and expands. By fine-tuning the strength of the gauge field we find that the solution approaches a static bubble solution. 
  We extend recent remarkable progress in the comparison of the dynamical energy spectrum of rotating closed strings in AdS_5xS^5 and the scaling weights of the corresponding non-near-BPS operators in planar N=4 supersymmetric gauge theory. On the string side the computations are feasible, using semiclassical methods, if angular momentum quantum numbers are large. This results in a prediction of gauge theory anomalous dimensions to all orders in the `t Hooft coupling lambda. On the gauge side the direct computation of these dimensions is feasible, using a recently discovered relation to integrable (super) spin chains, provided one considers the lowest order in lambda. This one-loop computation then predicts the small-tension limit of the string spectrum for all (i.e. small or large) quantum numbers. In the overlapping window of large quantum numbers and small effective string tension, the string theory and gauge theory results are found to match in a mathematically highly non-trivial fashion. In particular, we compare energies of states with (i) two large angular momenta in S^5, and (ii) one large angular momentum in AdS_5 and S^5 each, and show that the solutions are related by an analytic continuation. Finally, numerical evidence is presented on the gauge side that the agreement persists also at higher (two) loop order. 
  We obtain a diagonal solution of the dual reflection equation for elliptic $A^{(1)}_{n-1}$ SOS model. The isomorphism between the solutions of the reflection equation and its dual is studied. 
  We use the recently found matrix description of noncritical superstring theory of Type 0A to compute tachyon scattering amplitudes in a background with a RR flux. We find that after the string coupling is multiplicatively renormalized, the amplitudes in any genus become polynomial in the RR flux. We propose that in the limit where both the string coupling and the RR flux go to infinity, the theory has a weakly-coupled description in terms of another superstring theory with a vanishingly small RR flux. This duality exchanges the inverse string coupling and the 0-brane charge. The dual superstring theory must have a peculiar property that its only field-theoretic degree of freedom is a massless RR scalar. 
  We study four dimensional N=2 G_2 supersymmetric gauge theory on R^3\times S^1 deformed by a tree level superpotential. We will show that the exact superpotential can be obtained by making use of the Lax matrix of the corresponding integrable model which is the periodic Toda lattice based on the dual of the affine G_2 Lie algebra. At extrema of the superpotential the Seiberg-Witten curve typically factorizes, and we study the algebraic equations underlying this factorization. For U(N) theories the factorization was closely related to the geometrical engineering of such gauge theories and to matrix model descriptions, but here we will find that the geometrical interpretation is more mysterious. Along the way we give a method to compute the gauge theory resolvent and a suitable set of one-forms on the Seiberg-Witten curve. We will also find evidence that the low-energy dynamics of G_2 gauge theories can effectively be described in terms of an auxiliary hyperelliptic curve. 
  We study analytical aspects of a generic q-deformation with q real, by relating it with discrete scale invariance. We show how models of conformal quantum mechanics, in the strong coupling regime and after regularization, are also discrete scale invariant. We discuss the consequences of their distinctive spectra, characterized by functional behavior. The role of log-periodic behavior and q-periodic functions is examined, and we show how q-deformed zeta functions, characterized by complex poles, appear. As an application, we discuss one-loop effects in discretely self-similar space-times. 
  It has been often conjectured that the correct theory of quantum gravity will act as a UV regulator in the low energy limit of quantum field theory. Earlier work has shown that if the path integral defining the quantum field theory propagator is modified, so that the amplitude is invariant under the duality transformation l--> 1/l where l is the length of the path, then the propagator is UV-finite and exhibits a ``zero-point length'' of the spacetime. Since string theory uses extended structures and has a T-duality, these results should also emerge directly from string theory. We show, by explicit path integral computation, that this is indeed the case. The lowest order string theory correction to the propagator is the same as that obtained by the hypothesis of path integral duality. 
  We consider the tachyon potentials in closed and open-closed string theories. In doing so, we apply technique which proved to be useful in studying the open string tachyon potentials to the problem of interest. 
  The time evolution of self-interacting spherically symmetric scalar fields in Minkowski spacetime is investigated based on the use of Green's theorem. It is shown that a massive Klein-Gordon field can be characterized by the formation of certain expanding shell structures where all the shells are built up by very high frequency oscillations. This oscillation is found to be modulated by the product of a simple time decaying factor of the form $t^{-{3}/{2}}$ and of an essentially self-similar expansion. Apart from this self-similar expansion the developed shell structure is preserved by the evolution. In particular, the energy transported by each shell appears to be time independent. 
  Five dimensional gravity coupled, both in the bulk and on a brane, to a scalar Liouville field yields a geometry confined to a strip around the brane and with time dependent scale factors for the four geometry. In various limits known models can be recovered as well as a temporally expanding four geometry with a warp factor falling exponentially away from the brane. The effective theory on the brane has a time dependent Planck mass and ``cosmological constant''. Although the scale factor expands, the expansion is not an acceleration. 
  We study a generalization of the Callan-Harvey mechanism to the case of a non local mass. Using a 2+1 model as a concrete example, we show that both the existence and properties of localized zero modes can also be consistently studied when the mass is non local. After dealing with some general properties of the resulting integral equations, we show how non local masses naturally arise when radiative corrections are included. We do that for a 2+1 dimensional example, and also evaluate the zero mode of the resulting non local Dirac operator. 
  $Z_n$ Belavin model with open boundary condition is studied. The double-row transfer matrices of the model are diagonalized by algebraic Bethe ansatz method in terms of the intertwiner and the face-vertex correspondence relation. The eigenvalues and the corresponding Bethe ansatz equations are obtained. 
  We extend the formulation for perturbations of maximally symmetric black holes in higher dimensions developed by the present authors in a previous paper (hep-th/0305147) to a charged black hole background whose horizon is described by an Einstein manifold. For charged black holes, perturbations of electromagnetic fields are coupled to the vector and scalar modes of metric perturbations non-trivially. We show that by taking appropriate combinations of gauge-invariant variables for these perturbations, the perturbation equations for the Einstein-Maxwell system are reduced to two decoupled second-order wave equations describing the behaviour of the electromagnetic mode and the gravitational mode, for any value of the cosmological constant. These wave equations are transformed into Schr\"odinger-type ODEs through a Fourier transformation with respect to time. Using these equations, we investigate the stability of generalised black holes with charge. We also give explicit expressions for the source terms of these master equations with application to the emission problem of gravitational waves in mind. 
  An ideal gas of twodimensional Dirac fermions in the background of a pointlike magnetic vortex with arbitrary flux is considered. We find that this system acquires fractional electric charge at finite temperatures and determine the functional dependence of the thermal average and quadratic fluctuation of the charge on the temperature, the vortex flux, and the continuous parameter of the boundary condition at the location of the vortex. 
  Exploiting the path integral approach al la Batalin and Vilkovisky, we show that any anomaly-free Quantum Field Theory (QFT) comes with a family parametrized by certain moduli space M, which tangent space at the point corresponding to the initial QFT is given by the space of all observables. Furthermore the tangent bundle over M is equipped with flat quantum connection, which can be used to determine all correlation functions of the family of QFTs. We also argue that considering family of QFTs is an inevitable step, due to the fact that the products of quantum observables are not quantum observables in general, which leads to a new "global" perspective on quantum world. We also uncover structure of $d$-algebra in the large class of d-dimensional QFT. This leads to an universal quantization machine for d-algebras decorated by algebro-differential-topology of (d+1)-manifolds as well as a new perspective on differential-topology of low dimensions. This paper is a summary of a forthcoming paper of this author. 
  We propose the form of the Liouville action satisfying Polyakov conjecture on the accessory parameters for the hyperbolic singularities on the Riemann sphere. 
  In this letter, it is pointed out that the two matrix model defined by the action S=(1/2)(tr A^2+tr B^2)-(alpha_A/4) tr A^4-(alpha_B/4) tr B^4-(beta/2) tr(AB)^2 can be solved in the large N limit using a generalization of the solution of Kazakov and Zinn-Justin (who considered the symmetric case alpha_A=alpha_B). This model could have useful applications to 3D Lorentzian gravity. 
  We consider generalised Scherk Schwarz reductions of supergravity and superstring theories with twists by electromagnetic dualities that are symmetries of the equations of motion but not of the action, such as the S-duality of D=4, N=4 super-Yang-Mills coupled to supergravity. The reduction cannot be done on the action itself, but must be done either on the field equations or on a duality invariant form of the action, such as one in the doubled formalism in which potentials are introduced for both electric and magnetic fields. The resulting theory in odd-dimensions has massive form fields satisfying a self-duality condition $dA \sim m*A$. We construct such theories in D=3,5,7. 
  We discuss the evolution of cosmological tensor perturbations in the RSII model. In Gaussian normal coordinates the wave equation is non-separable, so we use the near-brane limit to perform the separation and study the evolution of perturbations. Massive excitations, which may also mix, decay outside the horizon which could lead to some novel cosmological signatures. 
  We consider scalar field theory in de Sitter space with a general vacuum invariant under the continuously connected symmetries of the de Sitter group. We begin by reviewing approaches to define this as a perturbative quantum field theory. One approach leads to Feynman diagrams with pinch singularities in the general case, which renders the theory perturbatively ill-defined. Another approach leads to well-defined perturbative correlation functions on the imaginary time continuation of de Sitter space. When continued to real-time, a path integral with a non-local action generates the time-ordered correlators. Curiously, observables built out of local products of the fields show no sign of this non-locality. However once one couples to gravity, we show acausal effects are unavoidable and presumably make the theory ill-defined. The Bunch-Davies vacuum state is the unique de Sitter invariant state that avoids these problems. 
  The two-loop (Euler-Heisenberg-type) effective action for N = 2 supersymmetric QED is computed using the N = 1 superspace formulation. The effective action is expressed as a series in supersymmetric extensions of F^{2n}, where n=2,3,..., with F the field strength. The corresponding coefficients are given by triple proper-time integrals which are evaluated exactly. As a by-product, we demonstrate the appearance of a non-vanishing F^4 quantum correction at the two-loop order. The latter result is in conflict with the conclusion of hep-th/9710142 that no such quantum corrections are generated at two loops in generic N = 2 SYM theories on the Coulomb branch. We explain a subtle loophole in the relevant consideration of hep-th/9710142 and re-derive the F^4 term from harmonic supergraphs. 
  We investigate a stabilization of extra dimensions in a ten-dimensional Kaluza-Klein theory and IIB supergravity. We assume (A)dS_5 x S^5 compactifcation, and calculate quantum effects to find an effective potential for the radius of internal space. The effective potential has a minimum, and if the universe is created on the top of the potential hill, the universe evolves from dS_5 to AdS_5 after exponential expansion. The internal space S^5 stays to be small and its radius becomes constant. Our model in IIB supergravity contains the 4-form gauge field with classically vacuum expectation value, which is role of ten-dimensional cosmological constant. If the universe evolves into AdS, the five-dimensional Randall & Sundrum setup with stabilized dilaton is obtained from the type IIB supergravity model. 
  High density phase transitions in a 4 dimensional Nambu-Jona-Lasinio model containing a single symmetry breaking order parameter coming from the fermion-antifermion condensates are researched and expounded by means of both the gap equation and the effective potential approach. The phase transitions are proven to be second order at a high temperature $T$; however at T=0, they are first- or second- order, depending on whether $\Lambda/m(0)$, the ratio of the momentum cutoff $\Lambda$ in the fermion loop integrals to the dynamical fermion mass $m(0)$ at zero temperature, is less than 3.387 or not. The former condition can not be satisfied in some models. The discussions further show complete effectiveness of the critical analysis based on the gap equation for second order phase transitions including determination of the condition of their occurrence. 
  By means of critical behaviors of the dynamical fermion mass in four-fermion interaction models, we have shown by explicit calculations that when T=0 the particle density will have a discontinuous jumping across the critical chemical potential $\mu_c$ in 2D and 3D Gross-Neveu (GN) model and these physically explain the first order feature of corresponding symmetry restoring phase transitions. For second order phase transitions in 3D GN model when $T\to 0$ and in 4D Nambu-Jona-Lasinio (NJL) model when T=0, it has been proven that the particle density itself will be continuous across $\mu_c$ but its derivative over the chemical potential $\mu$ will have a discontinuous jumping. The results give a physical explanation of implications of the tricritical point $(T,\mu)=(0,\mu_c)$ in 3D GN model. The discussions also show effectiveness of the critical analysis approach of phase transitions. 
  We study certain relevant boundary perturbations of Liouville theory and discuss implications of our results for the brane dynamics in noncritical string theories. Our results include   (i) There exist monodromies in the parameter $\mu_{\rm B}$ of the Neumann-type boundary condition that create an admixture represented by the Dirichlet type boundary condition.   (ii) Certain renormalization group flows can be studied perturbatively, which allows one to determine the results of the corresponding brane decays.   (iii) There exists a simple renormalization group flow that can be calculated exactly. In all the cases that we have studied, the RG flow acts like a covering transformation for the mondromies mentioned under (i). 
  We classify the simply-connected supersymmetric parallelisable backgrounds of heterotic supergravity. They are all given by parallelised Lie groups admitting a bi-invariant lorentzian metric. We find examples preserving 4, 8, 10, 12, 14 and 16 of the 16 supersymmetries. 
  We discuss a general procedure for arriving at the Hamilton-Jacobi equation of second-class constrained systems, and illustrate it in terms of a number of examples by explicitely obtaining the respective Hamilton principal function, and verifying that it leads to the correct solution to the Euler-Lagrange equations. 
  In the present paper, degeneration phenomena in conformal field theories are studied. For this purpose, a notion of convergent sequences of CFTs is introduced. Properties of the resulting limit structure are used to associate geometric degenerations to degenerating sequences of CFTs, which, as familiar from large volume limits of non-linear sigma models, can be regarded as commutative degenerations of the corresponding ``quantum geometries''.   As an application, the large level limit of the A-series of unitary Virasoro minimal models is investigated in detail. In particular, its geometric interpretation is determined. 
  We study the spectrum of BPS domain walls within the parameter space of N=1 U(N) gauge theories with adjoint matter and a cubic superpotential. Using a low energy description obtained by compactifying the theory on R^3 x S^1, we examine the wall spectrum by combining direct calculations at special points in the parameter space with insight drawn from the leading order potential between minimal walls, i.e those interpolating between adjacent vacua. We show that the multiplicity of composite BPS walls -- as characterised by the CFIV index -- exhibits discontinuities on marginal stability curves within the parameter space of the maximally confining branch. The structure of these marginal stability curves for large N appears tied to certain singularities within the matrix model description of the confining vacua. 
  Based solely on the arguments relating Friedmann equation and the Cardy formula we derive a bound for the number of e-folds during inflation for a standard Friedmann-Robertson-Walker as well as non-standard four dimensional cosmology induced by a Randall-Sundrum-type model. 
  We investigate the boundary bootstrap programme for finding exact reflection matrices of integrable boundary quantum field theories with N=1 boundary supersymmetry. The bulk S-matrix and the reflection matrix are assumed to take the form S=S_1S_0, R=R_1R_0, where S_0 and R_0 are the S-matrix and reflection matrix of some integrable non-supersymmetric boundary theory that is assumed to be known, and S_1 and R_1 describe the mixing of supersymmetric indices. Under the assumption that the bulk particles transform in the kink and boson/fermion representations and the ground state is a singlet we present rules by which the supersymmetry representations and reflection factors for excited boundary bound states can be determined. We apply these rules to the boundary sine-Gordon model, to the boundary a_2^(1) and a_4^(1) affine Toda field theories, to the boundary sinh-Gordon model and to the free particle. 
  First, we diagonalize the bc-ghost 3-string Neumann matrices using the technique described in hep-th/0304158. Their eigenvalues are in complete agreement with the previous authors. Second, we diagonalize the N-string gluing vertices for the bosonized ghost system. And third, we verify the descent and associativity relations for the combined bosonic matter+ghost gluing vertices. We find that in order for these relations to be true, the vertices must be normalized by the factor Z_N. Here Z_N is the partition function of the bosonic matter+ghost CFT on the gluing surface, which is the unit disc with the Neumann boundary conditions and the midpoint cone like singularity specifying by the angle excess \pi(N-2). 
  In this paper we consider the stability of some inflating brane-world models in quantum cosmology. It is shown that whereas the singular model based on the construction of inflating branes from Euclidean five-dimensional anti-de Sitter space is unstable to tensorial cosmological perturbations in the bulk, the nonsingular model which uses a five-dimensional asymptotically anti-de Sitter wormhole to construct the inflating branes is stable to these perturbations. 
  The status of accelerating four-dimensional universes obtained by time-dependent compactifications of 10 or 11 dimensional supergravity is reviewed, as is the `no-go' theorem that they evade. All flat cosmologies for a simple exponential potential are found explicitly. It is noted that transient acceleration is generic, and unavoidable for `flux' compactifications. Included is an eternally accelerating flat cosmology without a future event horizon. 
  We consider an arbitrary U(1) charged matter non-minimally coupled to the self-dual field in $d=2+1$. The coupling includes a linear and a rather general quadratic term in the self-dual field. By using both a Lagragian gauge embedding and a master action approaches we derive the dual Maxwell Chern-Simons type model and show the classical equivalence between the two theories. At quantum level the master action approach in general requires the addition of an awkward extra term to the Maxwell Chern-Simons type theory. Only in the case of a linear coupling in the self-dual field the extra term can be dropped and we are able to establish the quantum equivalence of gauge invariant correlation functions in both theories. 
  In this paper, we consider the D-brane, especially ${\rm D}_-$-brane, in the pp-wave background, which has the eight dynamical and the eight kinematical supercharges. Since the pp-wave background has not a SO(8) but a ${\rm SO}(4) \times {\rm SO}(4)$ symmetric group, the D-brane world-volume theory has a non-trivial symmetric group which depends on the configuration of D-brane. Here, we will analyze the open string spectrum consistent with the non-trivial symmetric group and, at the low energy limit, classify the field contents of the D-brane world-volume theory which come from the fermionic zero modes of the open string. 
  We reveal an intimate connection between the quantum knot invariant for torus knot T(s,t) and the character of the minimal model M(s,t), where s and t are relatively prime integers. We show that Kashaev's invariant, i.e., the N-colored Jones polynomial at the N-th root of unity, coincides with the Eichler integral of the character. 
  Vacuum condensates of dimension two and their relevance for the dynamical mass generation for gluons in Yang-Mills theories are discussed 
  Twistor formulation of massive arbitrary spin particle has been constructed. Twistor space of such particle is formed two twistors and two complex scalars which form together 'bosonic supertwistor'. The formulation is deduced from space-time one for spinning particle by means of introducing auxiliary harmonic variables and consequent partial fixing of gauges. It is carried out the canonical quantization of twistor massive particle with nonzero spin. It is found the eigenvalues of Casimir operators on particle states and harmonic expansion of wave function in spectrum. 
  We study soliton solutions of a modified non-linear Schroedinger (MNLS) equation. Using an Ansatz for the time and azimuthal angle dependence previously considered in the studies of the spinning Q-balls, we construct multi-node solutions of MNLS as well as spinning generalisations. 
  I review type IIB string compactifications in which the three-form field strengths satisfy a self-duality condition on the internal manifold. I begin with an overview of the models, giving preliminary formulae and several points of view from which they can be understood. Then I describe windows into the small radius behavior of the compactifications, which is more complicated than compactifications without fluxes. I discuss details of the flux-generated potential and nonperturbative corrections to it. These nonperturbative corrections allow a discussion of the cosmological constant and possible mechanisms for the universe to decay from one energy state to another. I conclude with comments on related topics and interesting directions for future study.   As this review is a PhD dissertation, I will indicate my own contributions to the subject. However, it is my hope that this document will be a useful and relatively comprehensive review, especially to graduate students. In particular, the early part of the document is almost entirely a literature review. 
  We derive the radiation pressure force on a non-relativistic moving plate in 1+1 dimensions. We assume that a massless scalar field satisfies either Dirichlet or Neumann boundary conditions (BC) at the instantaneous position of the plate. We show that when the state of the field is invariant under time translations, the results derived for Dirichlet and Neumann BC are equal. We discuss the force for a thermal field state as an example for this case. On the other hand, a coherent state introduces a phase reference, and the two types of BC lead to different results. 
  For a cosmological Randall-Sundrum braneworld with anisotropy, i.e., of Bianchi type, the modified Einstein equations on the brane include components of the five-dimensional Weyl tensor for which there are no evolution equations on the brane. If the bulk field equations are not solved, this Weyl term remains unknown, and many previous studies have simply prescribed it ad hoc. We construct a family of Bianchi braneworlds with anisotropy by solving the five-dimensional field equations in the bulk. We analyze the cosmological dynamics on the brane, including the Weyl term, and shed light on the relation between anisotropy on the brane and Weyl curvature in the bulk. In these models, it is not possible to achieve geometric anisotropy for a perfect fluid or scalar field -- the junction conditions require anisotropic stress on the brane. But the solutions can isotropize and approach a Friedmann brane in an anti-de Sitter bulk. 
  A new action for eleven dimensional supergravity on a manifold with boundary is presented. The action is a possible low energy limit of $M$-theory. Previous problems with infinite constants in the action are overcome and a new set of boundary conditions relating the behaviour of the supergravity fields to matter fields are obtained. One effect of these boundary conditions is that matter fields generate gravitational torsion. 
  Due to the fact that only matter fields have phase, frequently is believed that the gauge principle can induce gauge fields only in quantum systems. But this is not necessary. This paper, of pedagogical scope, presents a classical system constituted by a particle in a classical potential, which is used as a model to illustrate the gauge principle and the spontaneous symmetry breaking. Those concepts appear in the study of second order phase transitions. Ferroelectricity, ferromagnetism, superconductivity, plasmons in a free electron gas, and the mass of vector bosons in the gauge field Yang-Mills theories, are some of the phenomena in which these transitions occur. 
  With the help of numerical simulations we study N-soliton scattering (N=3,4) in the (2+1)-dimensional CP^1 model with periodic boundary conditions. When the solitons are scattered from symmetrical configurations the scattering angles observed agree with the earlier \pi/N predictions based on the model on R_2 with standard boundary conditions. When the boundary conditions are not symmetric the angles are different from \pi/N. We present an explanation of our observed patterns based on a properly formulated geodesic approximation. 
  An intriguing connection, based on duality symmetry, between ordinary (commutative) Born-Infeld type theory and non-commutative Maxwell type theory, is pointed out. Both discrete as well as continuous duality transformations are considered and their implications for self duality condition and Legendre transformations are analysed. 
  A new geometrical interpretation of chiral perturbation theory based on topological QCD is presented in picture format. This work is a written summary of a talk given at NAPP 2003 in Dubrovnik, Croatia. 
  Recent numerical calculations have shown that the ground state of the Gross-Neveu model at finite density is a crystal. Guided by these results, we can now present the analytical solution to this problem in terms of elliptic functions. The scalar potential is the superpotential of the non-relativistic Lame Hamiltonian. This model can also serve as analytically solvable toy model for a relativistic superconductor in the Larkin-Ovchinnikov-Fulde-Ferrell phase. 
  The effects of the generalized uncertainty principle({\bf GUP}) on the cosmological constant problem are discussed in the Schwarzchild-deSitter spacetime, through studying the corrections to its thermodynamics. We derive the correction to the Hawking temperature of the cosmological horizon, by a heuristic method enlighten by gr-qc/0106080 . The logarithmic correction to the Bekenstein-Hawking entropy is also obtained. For an ordinary star (not a black hole), the probable ratio of the vacuum energy to the total energy within the cosmological horizon is 2/3, which roughly coincides with the evidences from the astronomical observations. For a black hole, the ratio tends to decrease. An inequality associating the energy density with the length of system is put forward for understanding the smallness of the cosmological constant, and the relation between the Bekenstein entropy bound and the Bekenstein-Hawking entropy is also briefly discussed. 
  We show that the entropy of Schwarzschild black holes in any dimension can be described by a gas of free string bits at the stretched horizon. The number of string bits is equal to the black hole entropy and energy dependent. For an asymptotic observer the bit gas is at the Hawking temperature. We show that the same description is also valid for de Sitter space--times in any dimension. 
  Four-dimensional colliding plane wave (CPW) solutions have played an important role in understanding the classical non-linearities of Einstein's equations. In this note, we investigate CPW solutions in $2n+2$--dimensional Einstein gravity with a $n+1$-form flux. By using an isomorphism with the four-dimensional problem, we construct exact solutions analogous to the Szekeres vacuum solution in four dimensions. The higher-dimensional versions of the Khan-Penrose and Bell-Szekeres CPW solutions are studied perturbatively in the vicinity of the light-cone. We find that under small perturbations, a curvature singularity is generically produced, leading to both space-like and time-like singularities. For $n=4$, our results pertain to the collision of two ten-dimensional type IIB Blau - Figueroa o'Farrill - Hull - Papadopoulos plane waves. 
  We discuss a systematic method of analytically calculating the asymptotic form of quasi-normal frequencies of a four-dimensional Schwarzschild black hole by expanding around the zeroth-order approximation to the wave equation proposed by Motl and Neitzke. We obtain an explicit expression for the first-order correction and arbitrary spin. Our results are in agreement with the results from WKB and numerical analyses in the case of gravitational waves. 
  The symplectic projector method is applied to derive the local physical degrees of freedom of a particle moving freely on an arbitrary surface. The dependence of the projector on the coordinates and momenta of the particle is discussed. 
  We investigate dilaton stabilization in a higher-dimensional theory. The background geometry is based on an eleven-dimensional Kaluza-Klein/supergravity model, which is assumed to be a product of four-dimensional de Sitter (dS_4) spacetime and a seven sphere. The dilaton potential has a local minimum resulting from contributions of the cosmological constant, the curvature of the internal spacetime and quantum effects of the background scalar, vector, spinor, and tensor fields. The dilaton settles down to the local minimum, and the scale of the extra dimensions eventually become time independent. Our four-dimensional universe evolves from dS_4 into AdS_4 after stabilization of the extra dimension. 
  We study zero modes of N=1/2 supersymmetric Yang-Mills action in the background of instantons. In this background, because of a quartic antichiral fermionic term in the action, the fermionic solutions of the equations of motion are not in general zero modes of the action. Hence, when there are fermionic solutions, the action is no longer minimized by instantons. By deforming the instanton equation in the presence of fermions, we write down the zero mode equations. The solutions satisfy the equations of motion, and saturate the BPS bound. The deformed instanton equations imply that the finite action solutions have U(1) connections which are not flat anymore. 
  Following hep-th/0305177, we write the boundary state of half S-brane in bosonic string theory as a grand canonical partition function of a unitary matrix model. From this representation, it follows that the annulus amplitude can be written as a grand canonical partition function of a unitary two-matrix model. We also show that the contribution of the exponentially growing couplings to the timelike oscillators can be resummed in a certain annulus amplitude. 
  We study transverse asymptotically flat spacetimes without horizons that arise from brane matter sources. We assume that asymptotically there is a spatial translation Killing vector that is tangent to the brane. Such spacetimes are characterized by a tension, analogous to the ADM mass, which is a gravitational charge associated with the asymptotic spatial translation Killing vector. Using spinor techniques, we prove that the purely gravitational contribution to the spacetime tension is positive definite. 
  We study classes of D-branes embedded in various AdS^m x S^n x S^p x T^q backgrounds, which nontrivially mix the target-space submanifolds. Mixing is achieved either via diagonal geometric embedding or through a mixed worldvolume flux which has one index in the sphere and one index in the AdS part. Branes of the former type wrap calibrated cycles in the target space, while those of the latter type wrap non-supersymmetric target space cycles which are stabilised only after the mixed worldvolume flux is turned on. In the second part of the paper we study two qualitatively different Penrose limits of these diagonal branes. In the first case we look at geodesics which do not belong to the worldvolume of brane. In order to get a nontrivial result, one needs to bring the brane closer and closer to the geodesic while taking the limit. The result is a D-brane with a worldvolume relativistic pulse. In the second case the Penrose geodesic belongs to the worldvolume and the resulting brane is of the ``oblique'' type: it is diagonally embedded between different SO groups of the target space pp-wave. 
  Flux compactifications of string theory exhibiting the possibility of discretely tuning the cosmological constant to small values have been constructed. The highly tuned vacua in this discretuum have curvature radii which scale as large powers of the flux quantum numbers, exponential in the number of cycles in the compactification. By the arguments of Susskind/Witten (in the AdS case) and Gibbons/Hawking (in the dS case), we expect correspondingly large entropies associated with these vacua. If they are to provide a dual description of these vacua on their Coulomb branch, branes traded for the flux need to account for this entropy at the appropriate energy scale. In this note, we argue that simple string junctions and webs ending on the branes can account for this large entropy, obtaining a rough estimate for junction entropy that agrees with the existing rough estimates for the spacing of the discretuum. In particular, the brane entropy can account for the (A)dS entropy far away from string scale correspondence limits. 
  The modified gravity with $\ln R$ or $R^{-n} (\ln R)^m$ terms which grow at small curvature is discussed. It is shown that such a model which has well-defined newtonian limit may eliminate the need for dark energy and may provide the current cosmic acceleration. It is demonstrated that $R^2$ terms are important not only for early time inflation but also to avoid the instabilities and the linear growth of the gravitational force. It is very interesting that the condition of no linear growth for gravitational force coincides with the one for scalar mass in the equivalent scalar-tensor theory to be very large. Thus, modified gravity with $R^2$ term seems to be viable classical theory. 
  A few years ago H. Morales and the author introduced a type of generalized derivative that contained both vector and scalar boson fields. Here it is shown how to construct a full-fledged generalized Yang-Mills theory through the introduction of "extended field" multiplets. These are mixed fields that include both a vector and a scalar part. It is shown how the standard model of high energy physics appears naturally in a Yang-Mills theory that uses extended field multiplets through two spontaneous symmetry breakings, one due to the VEV of a scalar field and another to the VEV of a vector field. 
  It is shown that flat zero-energy solutions (vacua) of the 5d Kaluza-Klein theory admit a non-trivial homotopy structure generated by certain Kaluza-Klein excitations. These vacua consist of an infinite set of homotopically different spacetimes denoted by $\mathcal{M}^{(n)}_5$, among which $\mathcal{M}^{(0)}_5$ and $\mathcal{M}^{(1)}_5$ are especially identified as $M_{4} \times S^{1}$ and $M_5$, the ground states of the 5d Kaluza-Klein theory and the 5d general relativity, respectively (where $M_k$ represents the $k$-dimensional Minkowski space). 
  In the Light-Front Dynamics, the wave function equations and their numerical solutions, for two fermion bound systems, are presented. Analytical expressions for the ladder one-boson exchange interaction kernels corresponding to scalar, pseudoscalar, pseudovector and vector exchanges are given. Different couplings are analyzed separately and each of them is found to exhibit special features. The results are compared with the non relativistic solutions. 
  The microlocal space-time of extended hadrons, considered to be anisotropic is specified here as a special Finsler space. For this space the classical field equation is obtained from a property of the field on the neighbouring points of the autoparallel curve. The quantum field equation has also been derived for the bispinor field of a free lepton in this Finslerian microspace through its quantum generalization below a fundamental length-scale. The bispinor can be decomposed as a direct product of two spinsors, one depending on the position coordinates and the other on the directional arguments of the Finsler space. The former one represents the spinor of the macrospace, an associated Riemannian space-time of the Finsler space, and satisfies the Dirac equation. The directional variable-dependent spinor satisfies a different equation which is solved here. This spinor-part of the bispinor field for a constituent of the hadron can give rise to an additional quantum number for generating the internal symmetry of hadrons. Also, it is seen that in the process of separating the bispinor field and its equation an epoch-dependent mass term arises. Although, this part of the particle-mass has no appreciable contribution in the present era it was very significant for the very early period of the universe after its creation. Finally, the field equations for a particle in an external electromagnetic field for this Finslerian microlocal space-time and its associated Riemannian macrospaces have been found. 
  The local composite operator $A_{\mu}^{2}$ is analysed within the algebraic renormalization in Yang-Mills theories in linear covariant gauges. We establish that it is multiplicatively renormalizable to all orders of perturbation theory. Its anomalous dimension is computed to two-loops in the MSbar scheme. 
  The stringy picture behind the integrable spin chains governing the evolution equations in Yang-Mills theory is discussed. It is shown that one-loop dilatation operator in N=4 theory can be expressed in terms of two-point functions on 2d worldsheet. Using the relation between Neumann integrable system and the spin chains it is argued that the transition to the finite gauge theory coupling implies the discretization of the worldsheet. We conjecture that string bit model for the discretized worldsheet corresponds to the representation of the integrable spin chains in terms of the separated variables. 
  A canonical formalism for Lagrangians of maximal nonlocality is established. The method is based on the familiar Legendre transformation to a new function which can be derived from the maximally nonlocal Lagrangian. The corresponding canonical equations are derived through the standard procedure in local theory and appear much like those local ones, though the implication of the equations is largely expanded. 
  Free ${\cal N}=4$ Super Yang-Mills theory (in the large $N$ limit) is dual to an, as yet, intractable closed string theory on $AdS_5\times S^5$. We aim to implement open-closed string duality in this system and thereby recast the free field correlation functions as amplitudes in $AdS$. The basic strategy is to implement this duality directly on planar field theory correlation functions in the worldline (or first quantised) formulation. The worldline loops (remnants of the worldsheet holes) close to form tree diagrams. These tree diagrams are then to be manifested as tree amplitudes in $AdS$ by a change of variables on the worldline moduli space (i.e. Schwinger parameter space). Restricting to twist two operators, we are able to carry through this program for two and three point functions. However, it appears that this strategy can be implemented for four and higher point functions as well. An analogy to electrical networks is very useful in this regard. 
  Weak coherent states share many properties of the usual coherent states, but do not admit a resolution of unity expressed in terms of a local integral. They arise e.g. in the case that a group acts on an inadmissible fiducial vector. Motivated by the recent Affine Quantum Gravity Program, the present work studies the path integral representation of the affine weak coherent state matrix elements of the unitary time-evolution operator. Since weak coherent states do not admit a resolution of unity, it is clear that the standard way of constructing a path integral, by time slicing, is predestined to fail. Instead, a well-defined path integral with Wiener measure, based on a continuous-time regularization, is used to approach this problem. The dynamics is rigorously established for linear Hamiltonians, and the difficulties presented by more general Hamiltonians are addressed. 
  In this Diploma-thesis models of gauge field theory on noncommutative spaces are studied. On the canonically deformed plane we discuss the dependence of the established gauge theory on the choice of the star product. Furthermore, gauge field theory on the Eq(2)-symmetric plane is studied using two different approaches, a star product approach and an algebraic approach. 
  The scope of this work concerns the adaptation of the parallelizability pp-wave (Ppp-wave) process to D=10 type IIB string backgrounds in the presence of the non-trivial anti-self dual R-R 5-form $\QTR{cal}{F}$. This is important in the sense that it gives rise to some unsuspected properties. In fact, exact solutions of type IIB string backgrounds on Ppp-waves are discussed. For the $u$-dependence of the dilaton field $\Phi $, we establish explicitly a correspondence between type IIB supergravity equations of motion and 2d-conformal Liouville field theory. We show also that the corresponding conserved conformal current $T(\Phi)$ coincides exactly with the trace of the symmetric matrix $\mu_{ij}$ appearing in the quadratic front factor $F=\mu _{ij}x^{i}x^{j}$ of the Ppp-wave. Furthermore, we consider the transverse space dependence of the dilaton $\Phi $ and show that the supergravity equations are easily solved for the linear realization of the dilaton field. Other remarkable properties related to this case are also discussed. 
  Using a super-realization of the Wigner-Heisenberg algebra a new realization of the q-deformed Wigner oscillator is implemented. 
  The N=2 supersymmetry in quantum mechanics involving two-component eigenfunction is investigated. 
  We extend to quantum mechanics the technique of stochastic subordination, by means of which one can express any semi-martingale as a time-changed Brownian motion. As examples, we considered two versions of the q-deformed Harmonic oscillator in both ordinary and imaginary time and show how these various cases can be understood as different patterns of time quantization rules. 
  We calculate corrections to the Bekenstein-Hawking entropy formula for the five-dimensional topological AdS (TAdS)-black holes and topological de Sitter (TdS) spaces due to thermal fluctuations. We can derive all thermal properties of the TdS spaces from those of the TAdS black holes by replacing $k$ by $-k$. Also we obtain the same correction to the Cardy-Verlinde formula for TAdS and TdS cases including the cosmological horizon of the Schwarzschild-de Sitter (SdS) black hole. Finally we discuss the AdS/CFT and dS/CFT correspondences and their dynamic correspondences. 
  Hamiltonians with inverse square interaction potential occur in the study of a variety of physical systems and exhibit a rich mathematical structure. In this talk we briefly mention some of the applications of such Hamiltonians and then analyze the case of the N-body rational Calogero model as an example. This model has recently been shown to admit novel solutions, whose properties are discussed. 
  We formulate Noncommutative Qauntum Field Theory in terms of fields defined as mean value over coherent states of the noncommutative plane. No *-product is needed in this formulation and noncommutativity is carried by a modified Fourier transform of fields. As a result the theory is UV finite and the cutoff is provided by the noncommutative parameter theta. 
  We derive analytically one-loop corrections to the effective Polyakov-line operator in the branched-polymer approximation of the reduced four-dimensional supersymmetric Yang-Mills integrals. 
  We study the time evolution of freely rolling moduli in the context of M-theory on a G_2 manifold. This free evolution approximates the correct dynamics of the system at sufficiently large values of the moduli when effects from non-perturbative potentials and flux are negligible. Moduli fall into two classes, namely bulk moduli and blow-up moduli. We obtain a number of non-trivial solutions for the time-evolution of these moduli. As a generic feature, we find the blow-up moduli always expand asymptotically at early and late time. 
  We discuss a method of calculating analytically the asymptotic form of quasi-normal frequencies for large AdS black holes in five dimensions. In this case, the wave equation reduces to a Heun equation. We show that the Heun equation may be approximated by a Hypergeometric equation at large frequencies. Thus we obtain the asymptotic form of quasi-normal frequencies in agreement with numerical results. We also present a simple monodromy argument that leads to the same results. We include a comparison with the three-dimensional case in which exact expressions are derived. 
  We continue and extend earlier work on the summation of planar graphs in phi^3 field theory, based on a local action on the world sheet. The present work employs a somewhat different version of the self consistent field (meanfield) approximation compared to the previous work on the same subject. Using this new approach, we are able to determine in general the asymptotic forms of the solutions, and in the case of one solution, even its exact form. This solution leads to formation of an unstable string, in agreement with the previous work. We also investigate and clarify questions related to Lorentz invariance and the renormalization of the solution. 
  We consider a system consisting of a particle in the harmonic approximation, having frequency $\bar{\omega}$, coupled to a scalar field inside a spherical reflecting cavity of diameter $L$. By introducing {\it dressed} coordinates we define {\it dressed} states which allow a non-perturbative unified description of the radiation process, for weak and strong coupling regimes. We perform a study of the energy distribution in a small cavity, with the initial condition that the particle is in the first excited state. In the {\it weak} coupling regime, we conclude for the quasi-stability of the excited particle. For instance, for a frequency $\bar{\omega}$ of the order $\bar{\omega}\sim 4.00\times 10^{14}/s$ (in the visible red), starting from the initial condition that the particle is in the first excited level, we find that for a cavity with diameter $L\sim 1.0\times 10^{-6}m$, the probability that the particle be at any time still in the first excited level, will be of the order of 97%. For appropriate cavity dimensions, which are of the same order of those ensuring stability for weak coupling, we ensure for strong coupling the complete decay of the particle to the ground state in a small ellapsed time. Also we consider briefly the effects of a quartic interaction up to first order in the interaction parameter $\lambda$. We obtain for a large cavity an explicit $\lambda$-dependent expression for the particle radiation process. This formula is obtained in terms of the corresponding exact expression for the linear case and we conclude for the enhancement of the particle decay induced by the quartic interaction. 
  Exact solutions of the relativistic many-body problem are presented 
  We show explicitly how the Newton-Hooke groups act as symmetries of the equations of motion of non-relativistic cosmological models with a cosmological constant. We give the action on the associated non-relativistic spacetimes and show how these may be obtained from a null reduction of 5-dimensional homogeneous pp-wave Lorentzian spacetimes. This allows us to realize the Newton-Hooke groups and their Bargmann type central extensions as subgroups of the isometry groups of the pp-wave spacetimes. The extended Schrodinger type conformal group is identified and its action on the equations of motion given. The non-relativistic conformal symmetries also have applications to time-dependent harmonic oscillators. Finally we comment on a possible application to Gao's generalization of the matrix model. 
  A systematic construction is given for N=1 open string boundary coupling to Abelian and non-Abelian Dp-brane worldvolume fields, in general curved backgrounds. The basic ingredient is a set of four ``boundary vectors'' that provide a unified description of boundary conditions and boundary couplings. We then turn to the problem of apparent inconsistency of non-Abelian worldvolume scalar couplings (obtained by T-duality), with general covariance. It means that the couplings cannot be obtained from a covariant action by gauge fixing ordinary general coordinate transformations (GCT). It is shown that the corresponding worldsheet theory has the same problem, but is also invariant under certain matrix-valued coordinate transformations (MCT) that can be used to restore its covariance. The same transformations act on the worldvolume, leading to a covariant action. Then the non-Abelian Dp-brane action obtained by T-duality corresponds to gauge fixing the MCT and not GCT, hence the apparent incompatibility with general covariance. 
  Heterotic M-Theory is a promising candidate for that corner of M-theory which makes contact with the real world. However, while the theory requires one of its expansion parameters, $\epsilon$, to be perturbatively small, a successful phenomenology requires $\epsilon = {\cal O}(1)$. We show that the constraint to have small $\epsilon$ is actually unnecessary: instead of the original flux compactification background valid to linear order in $\epsilon$ one has to use its appropriate non-linear extension, the exact background solution. The exact background is determined by supersymmetry and consequently one expects the tree-level cosmological constant to vanish which we demonstrate in detail, thereby verifying once more the consistency of this background. Furthermore we show that the exact background represents precisely the 11d origin of the 5d domain wall solution which is an exact solution of the effective 5d heterotic M-theory. We also comment on singularities and the issue of chirality changing transitions in the exact background. The exact background is then applied to determine Newton's Constant for vacua with an M5 brane on the basis of a recent stabilization mechanism for the orbifold length. For vacua without M5 brane we obtain a correction to the lower bound on Newton's Constant which brings it in perfect agreement with the measured value. 
  A quantum-mechanical technique is used within the framework of U(2) super-Yang-Mills theory to investigate processes after recombination of two D-p-branes at one angle. Two types of initial conditions are considered, one of which with $p=4$ is a candidate of inflation mechanism. It is shown that the branes' shapes come to have three extremes due to localization of tachyon condensation. ``Pair-creations'' of open stings connecting the recombined branes is also observed. The appearance of closed strings is also discussed; the decaying branes are shown to radiate non-vanishing gravitational wave, which may be interpreted as evidence of closed string appearance. A few speculations are also given on implications of the above phenomena for an inflation model. 
  It is shown that the N=3 harmonic-superfield equations of motion are invariant with respect to the 4-th supersymmetry. The SU(3) harmonics are also used to analyze a more flexible form of superfield constraints for the Abelian N=4 vector multiplet and its N=3 decomposition. An alternative unusual representation of the N=4 supersymmetry is realized on infinite multiplets of analytic superfields in the N=3 harmonic superspace. U(1) charges of superfields in these multiplets are parametrized by an integer- valued parameter which plays the role of the discrete coordinate. Each superfield term of the N=3 Yang-Mills action has the infinite-dimensional N=4 generalization. The gauge group of this model contains an infinite number of superfield parameters. 
  In this note we reconsider the minisuperspace toy models for rolling and bouncing tachyons. We show that the theories require to choose boundary conditions at infinity since particles in an exponentially unbounded potential fall to infinity in finite world-sheet time. Using standard techniques from operator theory, we determine the possible boundary conditions and we compute the corresponding energy spectra and minisuperspace 3-point functions. Based on this analysis we argue in particular that world-sheet models of S-branes possess a discrete spectrum of conformal weights containing both positive and negative values. Finally, some suggestions are made for possible relations with previous studies of the minisuperspace theory. 
  We investigate the quantization of the bosonic string model which has a local U(1)_V * U(1)_A gauge invariance as well as the general coordinate and Weyl invariance on the world-sheet. The model is quantized by Lagrangian and Hamiltonian BRST formulations {\'a} la Batalin, Fradkin and Vilkovisky and noncovariant light-cone gauge formulation. Upon the quantization the model turns out to be formulated consistently in 26+2-dimensional background spacetime involving two time-like coordinates. 
  We give a full nonlinear numerical treatment of time-dependent 5d braneworld geometry, which is determined self-consistently by potentials for the scalar field in the bulk and at two orbifold branes, supplemented by boundary conditions at the branes. We describe the BraneCode, an algorithm which we designed to solve the dynamical equations numerically. We applied the BraneCode to braneworld models and found several novel phenomena of the brane dynamics. Starting with static warped geometry with de Sitter branes, we found numerically that this configuration is often unstable due to a tachyonic mass of the radion during inflation. If the model admits other static configurations with lower values of de Sitter curvature, this effect causes a violent re-structuring towards them, flattening the branes, which appears as a lowering of the 4d effective cosmological constant. Braneworld dynamics can often lead to brane collisions. We found that in the presence of the bulk scalar field, the 5d geometry between colliding branes approaches a universal, homogeneous, anisotropic strong gravity Kasner-like asymptotic, irrespective of the bulk/brane potentials. The Kasner indices of the brane directions are equal to each other but different from that of the extra dimension. 
  We investigate scalar perturbations from inflation in braneworld cosmologies with extra dimensions. For this we calculate scalar metric fluctuations around five dimensional warped geometry with four dimensional de Sitter slices. The background metric is determined self-consistently by the (arbitrary) bulk scalar field potential, supplemented by the boundary conditions at both orbifold branes. Assuming that the inflating branes are stabilized (by the brane scalar field potentials), we estimate the lowest eigenvalue of the scalar fluctuations - the radion mass. In the limit of flat branes, we reproduce well known estimates of the positive radion mass for stabilized branes. Surprisingly, however, we found that for de Sitter (inflating) branes the square of the radion mass is typically negative, which leads to a strong tachyonic instability. Thus, parameters of stabilized inflating braneworlds must be constrained to avoid this tachyonic instability. Instability of "stabilized" de Sitter branes is confirmed by the BraneCode numerical calculations in the accompanying paper hep-th/0309001. If the model's parameters are such that the radion mass is smaller than the Hubble parameter, we encounter a new mechanism of generation of primordial scalar fluctuations, which have a scale free spectrum and acceptable amplitude. 
  The modern quantum theory is based on the assumption that quantum states are represented by elements of a complex Hilbert space. It is expected that in future quantum theory the number field will be not postulated but derived from more general principles. We consider the choice of the number field in quantum theory based on a Galois field (GFQT) discussed in our previous publications. Since any Galois field is not algebraically closed, in the general case there is no guarantee that even a Hermitian operator necessarily has eigenvalues. We assume that the symmetry algebra is the Galois field analog of the de Sitter algebra so(1,4) and consider spinless irreducible representations of this algebra. It is shown that the Galois field analog of complex numbers is the minimal extension of the residue field modulo $p$ for which the representations are fully decomposable. 
  We give a method to construct Calabi-Yau metrics on G-invariant vector bundles over Kahler coset spaces G/H using supersymmetric nonlinear realizations with matter coupling. As a concrete example we discuss the CP^N model coupled with matter. The canonical line bundle is reproduced by the singlet matter and the cotangent bundle with a new non-compact Calabi-Yau metric which is not hyper-Kahler is obtained by the anti-fundamental matter. 
  We provide a complete solution of closed strings propagating in Nappi-Witten space. Based on the analysis of geodesics we construct the coherent wavefunctions which approximate as closely as possible the classical trajectories. We then present a new free field realization of the current algebra using the gamma, beta ghost system. Finally we construct the quantum vertex operators, for the tachyon, by representing the wavefunctions in terms of the free fields. This allows us to compute the three- and four-point amplitudes, and propose the general result for N-point tachyon scattering amplitude. 
  In this work, we develop a general framework in which Noncommutative Quantum Mechanics (NCQM) is showed to be equivalent to Quantum Mechanics (QM) on a suitable transformed Quantum Phase Space (QPS). Imposing some constraints on this particular transformation, we firstly find that the product of the two noncommutativity parameters possesses a lower bound in direct relation with Heisenberg incertitude relations, and secondly that the two parameters are equivalent but with opposite sign, up to a dimension factor depending on the physical system under study. This means that "noncommutativity" is represented by a unique parameter which may play the role of a "fundamental constant" characterizing the whole NCQPS. Within our framework, we treat some physical systems on NCQPS : free particle, harmonic oscillator, system of two-charged particles, Hydrogen atom. Among the obtained results, we discover a new phenomenon which consists to see a free particle on NCQPS as equivalent to a harmonic oscillator with Larmor frequency depending on one noncommutativity parameter, representing the same particle in the presence of a magnetic field. For the other examples, additional correction terms appear in the expression of the energy spectrum. Finally, in the two-particle system case, we emphasize the fact that for two opposite charges noncommutativity is effectively perceived with opposite sign. 
  The thermodynamical partition function of the Duffin-Kemmer-Petiau theory is evaluated using the imaginary-time formalism of quantum field theory at finite temperature and path integral methods. The DKP partition function displays two features: (i) full equivalence with the partition function for charged scalar particles and charged massive spin 1 particles; and (ii) the zero mode sector which is essential to reproduce the well-known relativistic Bose-Einstein condensation for both theories. 
  We review how instanton solutions at finite temperature can be seen as boundstates of constituent monopoles, discuss some speculations concerning their physical relevance and the lattice evidence for their presence in a dynamical context. 
  TBA integral equations are proposed for 1-particle states in the sausage- and SS-models and their $\sigma$-model limits. Combined with the ground state TBA equations the exact mass gap is computed in the O(3) and O(4) nonlinear $\sigma$-model and the results are compared to 3-loop perturbation theory and Monte Carlo data. 
  We study the integrable structure of the Dirichlet boundary problem in two dimensions and extend the approach to the case of planar multiply-connected domains. The solution to the Dirichlet boundary problem in multiply-connected case is given through a quasiclassical tau-function, which generalizes the tau-function of the dispersionless Toda hierarchy. It is shown to obey an infinite hierarchy of Hirota-like equations which directly follow from properties of the Dirichlet Green function and from the Fay identities. The relation to multi-support solutions of matrix models is briefly discussed. 
  We find a regular analytic 1st order deformation of the Klebanov-Strassler background. From the dual gauge theory point of view the deformation describes supersymmetry soft breaking gaugino mass terms. We calculate the difference in vacuum energies between the supersymmetric and the non-supersymmetric solutions and find that it matches the field theory prediction. We also discuss the breaking of the $U(1)_R$ symmetry and the space-time dependence of the gaugino bilinears two point function. Finally, we determine the Penrose limit of the non-supersymmetric background and write down the corresponding plane wave string theory. This string describes ``annulons''-heavy hadrons with mass proportional to large global charge. Surprisingly the string spectrum has two fermionic zero modes. This implies that the sector in the non-supersymmetric gauge theory which is the dual of the annulons is supersymmetric. 
  It is well-known that the sum over topologies in quantum gravity is ill-defined, due to a super-exponential growth of the number of geometries as a function of the space-time volume, leading to a badly divergent gravitational path integral. Not even in dimension 2, where a non-perturbative quantum gravity theory can be constructed explicitly from a (regularized) path integral, has this problem found a satisfactory solution. -- In the present work, we extend a previous 2d Lorentzian path integral, regulated in terms of Lorentzian random triangulations, to include space-times with an arbitrary number of handles. We show that after the imposition of physically motivated causality constraints, the combined sum over geometries and topologies is well-defined and possesses a continuum limit which yields a concrete model of space-time foam in two dimensions. 
  We estimate the quark condensate in one-flavor massless QCD from the known value of the gluino condensate in SUSY Yang-Mills theory using our newly proposed "orientifold" large-N expansion. The numerical result for the quark condensate renormalized at the scale 2 GeV is then given as a function of alpha_s(2 GeV) and of possible corrections from sub-leading terms. Our value can be compared with the quark condensate in (quenched) lattice QCD or with the one extracted from the Gell-Mann--Oakes--Renner relation by virtue of non-lattice determinations of the quark masses. In both cases we find quite a remarkable agreement. 
  We have conclusively established the duality between noncommutative  Maxwell-Chern-Simons theory and Self-Dual model, the latter in ordinary spacetime, to the first non-trivial order in the noncommutativity parameter $\theta^{\mu\nu}$, with $\theta^{0i}=0$. This shows that the former theory is free for marginally noncommutative spacetimes. A $\theta$-generalized covariant mapping between the variables of the two models in question has been derived explicitly, that converts one model to the other, including the symplectic structure and action. 
  We study one- and two-soliton solutions of noncommutative Chern-Simons theory coupled to a nonrelativistic or a relativistic scalar field. In the nonrelativistic case, we find a tower of new stationary time-dependent solutions, all with the same charge density, but with increasing energies. The dynamics of these solitons cannot be studied using traditional moduli space techniques, but we do find a nontrivial symplectic form on the phase space indicating that the moduli space is not flat. In the relativistic case we find the metric on the two soliton moduli space. 
  I give a brief summary of the results reported in hep-th 0306013 in collaboration with G. Amelino-Camelia and F. D'Andrea. I focus on the analysis of the symmetries of $\kappa$-Minkowski noncommutative space-time, described in terms of a Weyl map. The commutative space-time notion of Lie-algebra symmetries must be replaced by the one of Hopf-algebra symmetries. However, in the Hopf algebra sense, it is possible to construct an action in $\kappa$-Minkowski which is invariant under a 10-generators Poincar\'e-like symmetry algebra. 
  We study the closed string emission from an unstable D$p$-brane with constant background electric field in bosonic string theory. The average total number density and the average total energy density of emitted closed strings are explicitly calculated in the presence of electric field. It is explicitly shown that the energy density in the UV region becomes finite whenever the background electric field is switched on. The energy density converted into closed strings in the presence of electric field is negligibly small compared with the D-brane tension in the weak string coupling limit. 
  Previous results on form factors for the scaling Ising and the sinh-Gordon models are extended to general $Z_{N}$-Ising and affine $A_{N-1}$-Toda quantum field theories. In particular result for order, disorder parameters and para-fermi fields $\sigma_{Q}(x), \mu_{\tilde{Q}}(x)$ and $\psi_{Q}(x)$ are presented for the $Z_{N}$-model. For the $A_{N-1}$-Toda model all form factors for exponentials of the Toda fields are proposed. The quantum field equation of motion is proved and the mass and wave function renormalization are calculated exactly. 
  It is known that in the zeta function regularization and in the Fujikawa method chiral anomaly is defined through a coefficient in the heat kernel expansion for the Dirac operator. In this paper we apply the heat kernel methods to calculate boundary contributions to the chiral anomaly for local (bag) boundary conditions. As a by-product some new results on the heat trace asymptotics are also obtained. 
  We show how to write a set of brackets for the Langevin equation, describing the dissipative motion of a classical particle, subject to external random forces. The method does not rely on an action principle, and is based solely on the phenomenological description of the dissipative dynamics as given by the Langevin equation. The general expression for the brackets satisfied by the coordinates, as well as by the external random forces, at different times, is determined, and it turns out that they all satisfy the Jacobi identity. Upon quantization, these classical brackets are found to coincide with the commutation rules for the quantum Langevin equation, that have been obtained in the past, by appealing to microscopic conservative quantum models for the friction mechanism. 
  We study Ward identities for simple processes with external gauge bosons in the time-ordered perturbation theory approach to time-like noncommutative gauge theories. We demonstrate that these Ward identities cannot be satisfied when all orders in the noncommutativity parameters theta_i0 are taken into account. We conclude that in time-ordered perturbation theory one cannot solve the unitarity problem of time-like noncommutative quantum field theories. 
  We take advantage of the fact that in lambda phi ^4 problems a large field cutoff phi_max makes perturbative series converge toward values exponentially close to the exact values, to make optimal choices of phi_max. For perturbative series terminated at even order, it is in principle possible to adjust phi_max in order to obtain the exact result. For perturbative series terminated at odd order, the error can only be minimized. It is however possible to introduce a mass shift in order to obtain the exact result. We discuss weak and strong coupling methods to determine the unknown parameters. The numerical calculations in this article have been performed with a simple integral with one variable. We give arguments indicating that the qualitative features observed should extend to quantum mechanics and quantum field theory. We found that optimization at even order is more efficient that at odd order. We compare our methods with the linear delta-expansion (LDE) (combined with the principle of minimal sensitivity) which provides an upper envelope of for the accuracy curves of various Pade and Pade-Borel approximants. Our optimization method performs better than the LDE at strong and intermediate coupling, but not at weak coupling where it appears less robust and subject to further improvements. We also show that it is possible to fix the arbitrary parameter appearing in the LDE using the strong coupling expansion, in order to get accuracies comparable to ours. 
  To achieve a maximal locality in a trivial field theory, we maximize the ultraviolet cutoff of the theory by fine tuning the infrared values of the parameters. This optimization procedure is applied to the scalar theory in $D+1$ dimensions ($D \geq 4$) with one extra dimension compactified on a circle with radius $R$. The optimized, infrared values of the parameters are then compared with the corresponding ones of the uncompactified theory in $D$ dimensions, which is assumed to be the low-energy effective theory. We find that these values approximately agree with each other, as long as $R^{-1} \gsim s M$ is satisfied, where $s\simeq 10,50,50, 100$ for $D=4,5,6,7$, and $M$ is a typical scale of the $D$-dimensional theory. This result supports the previously made claim that the maximization of the ultraviolet cutoff in an nonrenormalizable field theory can give the theory more predictive power. 
  The perturbative vacuum of type 0 string theory is unstable due to the existence of the closed string tachyon. This instability can be removed by S^1 compactification with twisted boundary condition for the tachyon field. We show that even in this situation unwrapped NS5-branes are unstable and decay to bubbles of nothing smoothly without tunneling any potential barrier. We discuss a relation between the closed string tachyon condensation and the instability of NS5-branes. 
  We propose a matrix model on a homogeneous plane-wave background with 20 supersymmetries. This background is anti-Mach type and is equivalent to the time-dependent background. We study supersymmetries in this theory and calculate the superalgebra. The vacuum energy of the abelian part is also calculated. In addition we find classical solutions such as graviton solution, fuzzy sphere and hyperboloid. 
  We study the Polyakov line in Yang-Mills matrix models, which include the IKKT model of IIB string theory. For the gauge group SU(2) we give the exact formulae in the form of integral representations which are convenient for finding the asymptotic behaviour. For the SU(N) bosonic models we prove upper bounds which decay as a power law at large momentum p. We argue that these capture the full asymptotic behaviour. We also indicate how to extend the results to some correlation functions of Polyakov lines. 
  It is shown that non-composite intersecting S-brane solutions received recently by N. Ohta [Phys. Lett. B 558, 213 (2003); hep-th/0301095] are special case of cosmological-type solutions with composite p-branes obtained in our earlier publications. 
  We evaluate the sphere level S-matrix element of two tachyons and two massless NS states, the S-matrix element of four tachyons, and the S-matrix element of two tachyons and two Ramon-Ramond vertex operators, in type 0 theory. We then find an expansion for theses amplitudes that their leading order terms correspond to a covariant tachyon action. To the order considered, there are no $T^4$, $T^2(\prt T)^2$, $T^2H^2$, nor $T^2R$ tachyon couplings, whereas, the tachyon couplings $F\bF T$ and $T^2F^2$ are non-zero. 
  We analyse the condensation of closed string tachyons on the $C/Z_N$ orbifold. We construct the potential for the tachyons upto the quartic interaction term in the large $N$ limit. In this limit there are near marginal tachyons. The quartic coupling for these tachyons is calculated by subtracting from the string theory amplitude for the tachyons, the contributions from the massless exchanges, computed from the effective field theory. We argue that higher point interaction terms are are also of the same order in 1/N as the quartic term and are necessary for existence of the minimum of the tachyon potential that is consistent with earlier analysis. 
  We find the gravitational resonance (quasinormal) modes of the higher dimensional Schwarzschild and Reissner-Nordstrem black holes. The effect on the quasinormal behavior due to the presence of the $\lambda$ term is investigated. The QN spectrum is totally different for different signs of $\lambda$. In more than four dimensions there excited three types of gravitational modes: scalar, vector, and tensor. They produce three different quasinormal spectra, thus the isospectrality between scalar and vector perturbations, which takes place for D=4 Schwarzschild and Schwarzschild-de-Sitter black holes, is broken in higher dimensions. That is the scalar-type gravitational perturbations, connected with deformations of the black hole horizon, which damp most slowly and therefore dominate during late time of the black hole ringing. 
  We describe a new regularization of quantum field theory on the noncommutative torus by means of one-dimensional matrix models. The construction is based on the Elliott-Evans inductive limit decomposition of the noncommutative torus algebra. The matrix trajectories are obtained via the expansion of fields in a basis of new noncommutative solitons described by projections and partial isometries. The matrix quantum mechanics are compared with the usual zero-dimensional matrix model regularizations and some applications are sketched. 
  Does the existence of dark energy suggest that there is more to the graviton than we think we know? 
  We consider the Horava-Witten based model with 5-branes situated near the distant orbifold plane and with vanishing instanton numbers on the physical plane. This model has a toric fibered Calabi-Yau with del Pezzo base dP_7 which allows three generations with Standard Model gauge group at the GUT scale. Previous analysis showed that the quark hierarchy at the electroweak scale could be achieved qualitatively without undue fine tuning due to the effects of the 5-branes on the Kahler potential. We extend here this analysis to include the leptons. A new mechanism is introduced to obtain neutrino masses by assuming massless right handed neutrinos exist in the particle spectrum, which allows a cubic holomorphic term to exist in the Kahler metric, l_L*H_2*nu_R, scaled by the 11D Planck mass. After transferring this term to the superpotential, this term gives rise to neutrino masses of the correct size at the electroweak scale. With natural choices of the Yukawa and Kahler matrix entries, it is possible to fit all mass, CKM and MNS experimental data. The model predicts mu -> e + gamma decay at a rate that should be detectable for much of the SUSY parameter space in the next round of experiments. 
  In this work, I investigate the noncommutative Poisson algebra of classical observables corresponding to a proposed general Noncommutative Quantum Mechanics, \cite{1}. I treat some classical systems with various potentials and some Physical interpretations are given concerning the presence of noncommutativity at large scales (Celeste Mechanics) directly tied to the one present at small scales (Quantum Mechanics) and its possible relation with UV/IR mixing. 
  Motivated by describing time-evolutions of noncommutative worlds, I discuss symmetry-preserving evolutions of noncommutative worlds of finite dimensional representation spaces. An interesting issue in such evolutions is that there can be transitions of representation spaces of symmetry, and therefore the evolutions are generally non-unitary or symmetry-violating. The central idea of this paper is that a main world evolves by emitting baby worlds which compensate the violations of the symmetry of the main world. Tracing out the states of the baby worlds, the symmetry-preserving evolutions of the density matrices of the main world are obtained. I give a simple example with SU(2) symmetry, which can be regarded as an evolving quantum two-sphere. This simple model has some attractive features resembling our universe: it gets born from a vacuum and its entropy of geometric origin grows. I also discuss the evolutions in Heisenberg picture. 
  The loop equations for a chain of hermitian random matrices are computed explicitely, including the 1/N^2 corrections. To leading order, the master loop equation reduces to an algebraic equation, whose solution can be written in terms of geometric properties of the underlying algebraic curve. In particular we compute the free energy, the resolvents, the 2-loop functions and some mixed one loop functions. We also initiate the calculation of the 1/N^2 expansion. 
  Perturbative corrections to N=1/2 supersymmetric U(N) gauge theory at one-loop order are studied. It is shown that whereas the quantum corrections to N=1 sector of the theory are not affected by the C-deformation, the non(anti)commutativity parameter C receives one-loop perturbative corrections. These perturbative corrections are computed by performing an explicit one-loop calculation of the three and four-point functions of the theory. The running of the non(anti)commutativity parameter C is also studied using an appropriate Callan-Symanzik equation. 
  We consider the Lagrangian particle model introduced in [hep-th/9612017] for zero mass but nonvanishing second central charge of the planar Galilei group. Extended by a magnetic vortex or a Coulomb potential the model exibits conformal symmetry. In the former case we observe an additional SO(2,1) hidden symmetry. By either a canonical transformation with constraints or by freezing scale and special conformal transformations at $t=0$ we reduce the six-dimensional phase-space to the physically required four dimensions. Then we discuss bound states (bounded solutions) in quantum dynamics (classical mechanics). We show that the Schr\"odinger equation for the pure vortex case may be transformed into the Morse potential problem thus providing us with an explanation of the hidden SO(2,1) symmetry. 
  We calculate the interaction potential between widely separated D-branes in PP-wave backgrounds in string theory as well as in low-energy supergravity. Timelike and spacelike orientations are qualitatively different but in both cases the effective brane tensions and RR charges take the same values as in Minkowski space in accordance with the expectations from the sigma model perturbation theory. 
  The collinear factorization properties of two-loop scattering amplitudes in dimensionally-regulated N=4 super-Yang-Mills theory suggest that, in the planar ('t Hooft) limit, higher-loop contributions can be expressed entirely in terms of one-loop amplitudes. We demonstrate this relation explicitly for the two-loop four-point amplitude and, based on the collinear limits, conjecture an analogous relation for n-point amplitudes. The simplicity of the relation is consistent with intuition based on the AdS/CFT correspondence that the form of the large N_c L-loop amplitudes should be simple enough to allow a resummation to all orders. 
  We use the generalized Konishi anomaly equations and R-symmetry anomaly to compute the exact perturbative and non-perturbative gravitational F-terms of four-dimensional N=1 supersymmetric gauge theories. We formulate the general procedure for computation and consider chiral and non-chiral SU(N) gauge theories. 
  We re-consider the self tuning idea in brane world models of finite volume. We notice that in a large class of self tuning models, the four dimensional physics is sensitive to the vacuum energy on the brane. In particular the compactification volume changes each time the tension of the brane is modified: consequently, observable constants, as the effective Planck mass or masses of matter fields, change as well. We notice that the self tuning mechanism and the stabilization mechanism of the size of the extra dimensions are generically in apparent conflict. We focus on a self tuning model in six spacetime dimensions to analyze how the above considerations are explicitely realized. 
  We study brane recombination for supersymmetric configurations of intersecting branes in terms of the world-volume field theory. This field theory contains an impurity, corresponding to the degrees of freedom localized at the intersection. The Higgs branch, on which the impurity fields condense, consists of vacua for which the intersection is deformed into a smooth calibrated manifold. We show this explicitly using a superspace formalism for which the calibration equations arise naturally from F- and D-flatness. 
  We derive the effective superpotential for an N=1 SU(N_c) gauge theory with one massless adjoint field and N_f massless fundamental flavors and cubic tree-level superpotential for the adjoint field. This is a generalization of the Affleck-Dine-Seiberg superpotential to gauge theories with one massless adjoint matter field. Using Kutasov's generalization of Seiberg duality, we then find the effective superpotential for a related theory with massive fundamental flavors. 
  Using the Ashtekar-Sen variables of loop quantum gravity, a new class of exact solutions to the equations of quantum cosmology is found for gravity coupled to a scalar field, that corresponds to inflating universes. The scalar field, which has an arbitrary potential, is treated as a time variable, reducing the hamiltonian constraint to a time-dependent Schroedinger equation. When reduced to the homogeneous and isotropic case, this is solved exactly by a set of solutions that extend the Kodama state, taking into account the time dependence of the vacuum energy. Each quantum state corresponds to a classical solution of the Hamiltonian-Jacobi equation. The study of the latter shows evidence for an attractor, suggesting a universality in the phenomena of inflation. Finally, wavepackets can be constructed by superposing solutions with different ratios of kinetic to potential scalar field energy, resolving, at least in this case, the issue of normalizability of the Kodama state. 
  We propose a paradigm for the inflation and the vanishing cosmological constant in a unified way with the self-tuning solutions of the cosmological constant problem. Here, we consider a time-varying cosmological constant in self-tuning models of the cosmological constant. As a specific example, we demonstrate it with a 3-form field in 5D. 
  A self-consistent treatment of two and three point functions in models with trilinear interactions forces them to have opposite anomalous dimensions. We indicate how the anomalous dimension can be extracted nonperturbatively by solving and suitably truncating the topologies of the full set of Dyson-Schwinger equations. The first step requires a sensible ansatz for the full vertex part which conforms to first order perturbation theory at least. We model this vertex to obtain typical transcendental equations between anomalous dimension and coupling constant $g$ which coincide with know results to order $g^4$. 
  We review the approach to calculation of one-loop effective action in ${\cal N}=2,4$ SYM theories. We compute the non-holomorphic corrections to low-energy effective action (higher derivative terms) in ${\cal N}=2$, SU(2) SYM theory coupled to hypermultiplets on a non-abelian background for $R_{\xi}$-gauge fixing conditions. A general procedure for calculating the gauge parameters depending contributions to one-loop superfield effective action is developed. The one-loop non-holomorphic effective potential is exactly found in terms of the Euler dilogarithm function for a specific choice of gauge parameters.We also discuss the calculations of hypermultiplet dependence of ${\cal N}=4$ SYM effective action. 
  A hermitian one-matrix model with an even quartic potential exhibits a third-order phase transition when the cuts of the matrix model curve coalesce. We use the known solutions of this matrix model to compute effective superpotentials of an N=1, SU(N) supersymmetric Yang-Mills theory coupled to an adjoint superfield, following the techniques developed by Dijkgraaf and Vafa. These solutions automatically satisfy the quantum tracelessness condition and describe a breaking to SU(N/2) x SU(N/2) x U(1). We show that the value of the effective superpotential is smooth at the transition point, and that the two-cut (broken) phase is more favored than the one-cut (unbroken) phase below the critical scale. The U(1) coupling constant diverges due to the massless monopole, thereby demonstrating Ferrari's general formula. We also briefly discuss the implication of the Painleve II equation arising in the double scaling limit. 
  In the 6D brane world model with a 4-form flux on a sphere $S^2$ for self-tuning the cosmological constant, we comment on the fine-tuning problem in view of the quantization of the dual 2-form flux and the orbifolding case $S^2/Z_2$. 
  The 2 into n scattering with final particles at rest is discussed. The comparison with purely soft processes allows to identify symmetries responsible for vanishing of certain 2 into n amplitudes. Some examples are given. 
  Using dispersive techniques, it is possible to avoid ultraviolet divergences in the calculation of Feynman diagrams, making subsequent regularization of divergent diagrams unnecessary. We give a simple introduction to the most important features of such dispersive techniques in the framework of the so-called finite causal perturbation theory. The method is also applied to the 'divergent' general massive two-loop sunrise selfenergy diagram, where it leads directly to an analytic expression for the imaginary part of the diagram in accordance with the literature, whereas the real part can be obtained by a single integral dispersion relation. It is pointed out that dispersive methods have been known for decades and have been applied to several nontrivial Feynman diagram calculations. 
  Gauge theory on the q-deformed two-dimensional Euclidean plane R^2_q is studied using two different approaches. We first formulate the theory using the natural algebraic structures on R^2_q, such as a covariant differential calculus, a frame of one-forms and invariant integration. We then consider a suitable star product, and introduce a natural way to implement the Seiberg-Witten map. In both approaches, gauge invariance requires a suitable ``measure'' in the action, breaking the E_q(2)-invariance. Some possibilities to avoid this conclusion using additional terms in the action are proposed. 
  We review some surprising links which have been discovered in the last few years between the theory of certain ordinary differential equations, and particular integrable lattice models and quantum field theories in two dimensions. An application of this correspondence to a problem in non-Hermitian (PT-symmetric) quantum mechanics is also discussed. 
  This thesis examines the interaction of both bosonic and superstrings with various backgrounds with a view to understanding the interplay between tachyon condensation and world-sheet conformal invariance, and to understanding the overlap of d-branes and closed string modes. We develop the boundary state and show that in a background of interest to tachyon condensation the conformal invariance of the string world-sheet is broken, which suggests a generalized boundary state obtained by integrating over the conformal group of the disk. We find that this prescription reproduces particle emission amplitudes calculated from the string sigma model for both on- and off-shell boundary interactions. The boundary state appears as a coherent superposition of closed string states, and using this observation a proposal for calculating amplitudes beyond tree level is developed. The application of this technique to more general, time dependent backgrounds is also discussed. 
  We study some aspects of type 0 strings propagating in the two dimensional black hole geometry, corresponding to the exact SL(2)/U(1) SCFT background. 
  We propose, within a perturbative string theory example, a cosmological way to generate a large hierarchy between the observed Planck mass and the fundamental string scale. Time evolution results in three large space dimensions, one additional dimension transverse to our world and five small internal dimensions with a very slow time evolution. The evolution of the string coupling and internal space generate a large Planck mass. However due to an exact compensation between the time evolution of the internal space and that of the string coupling, the gauge and Yukawa couplings on our Universe are time independent. 
  We construct a five--dimensional, asymptotically Goedel, three--charge black hole via dimensional reduction of an asymptotically plane wave, rotating D1-D5-brane solution of type IIB supergravity. This latter is itself constructed via the solution generating procedure of Garfinkle and Vachaspati, applied to the standard rotating D1-D5-brane solution. Taking all charges to be equal gives a "BMPV Goedel black hole", which is closely related to that recently found by Herdeiro. We emphasise, however, the importance of our ten--dimensional microscopic description in terms of branes. We discuss various properties of the asymptotically Goedel black hole, including the physical bound on the rotation of the hole, the existence of closed timelike curves, and possible holographic protection of chronology. 
  In this paper we continue our studies of Hitchin systems on singular curves (started in hep-th/0303069). We consider a rather general class of curves which can be obtained from the projective line by gluing two subschemes together (i.e. their affine part is: Spec $\{f \in \CC[z]: f(A(\ep))=f((B(\ep)); \ep^N=0 \}$, where $A(\ep), B(\ep)$ are arbitrary polynomials) . The most simple examples are the generalized cusp curves which are projectivizations of Spec $\{f \in \CC[z]: f'(0)=f''(0)=...f^{N-1}(0)=0 \}$). We describe the geometry of such curves; in particular we calculate their genus (for some curves the calculation appears to be related with the iteration of polynomials $A(\ep), B(\ep)$ defining the subschemes). We obtain the explicit description of moduli space of vector bundles, the dualizing sheaf, Higgs field and other ingredients of the Hitchin integrable systems; these results may deserve the independent interest. We prove the integrability of Hitchin systems on such curves. To do this we develop $r$-matrix formalism for the functions on the truncated loop group $GL_n(\CC[z]), z^N=0$. We also show how to obtain the Hitchin integrable systems on such curves as hamiltonian reduction from the more simple system on some finite-dimensional space. 
  Lorentz covariance is the fundamental principle of every relativistic field theory which insures consistent physical descriptions. Even if the space-time is noncommutative, field theories on it should keep Lorentz covariance. In this paper, the nonabelian gauge theory on noncommutative spacetime is defined and its Lorentz invariance is maintained based on the idea of Carlson, Carone and Zobin. The deviation from the standard model in particle physics has not yet observed, and so any model beyond standard model must reduce to it in some approximation. Noncommutative gauge theory must also reproduce standard model in the limit of noncommutative parameter $\theta^{\mu\nu}\to0$. Referring to Jur$\check{\text{c}}$o {\it et. al.}, we will construct the nonabelian gauge theory that deserves to formulate standard model. BRST symmetry is very important to quantize nonabelian gauge theory and construct the covariant canonical formulation. It is discussed about the fields in noncommutative gauge theory without considering those components. Scale symmetry of ghost fields is also discussed. 
  I review the analysis of (2+1)-dimensional Yang-Mills ($YM_{2+1})$ theory via the use of gauge-invariant matrix variables. The vacuum wavefunction, string tension, the propagator mass for gluons, its relation to the magnetic mass for $YM_{3+1}$ at nonzero temperature and the extension of our analysis to the Yang-Mills-Chern-Simons theory are discussed. A possible extension to 3+1 dimensions is also briefly considered. 
  Recently, corrections to the standard Einstein-Hilbert action are proposed to explain the current cosmic acceleration in stead of introducing dark energy. We discuss the Palatini formulation of the modified gravity with a $\ln R$ term suggested by Nojiri and Odintsov. We show that in the Palatini formulation, the $\ln R$ gravity can drive a current exponential accelerated expansion and it reduces to the standard Friedmann evolution for high redshift region. We also discuss the equivalent scalar-tensor formulation of the theory. We indicate that the $\ln R$ gravity may still have a conflict with electron-electron scattering experiment which stimulates us to pursue a more fundamental theory which can give the $\ln R$ gravity as an effective theory. Finally, we discuss a problem faced with the extension of the $\ln R$ gravity by adding $R^m$ terms. 
  The tadpole cancellation in the unoriented Liouville theory is discussed. Using two different methods -- the free field method and the boundary-crosscap state method, we derive one-loop divergences. Both methods require two D1-branes with the symplectic gauge group to cancel the orientifold tadpole divergence. However, the finite part left is different in each method and this difference is studied. We also discuss the validity of the free field method and the possible applications of our result. 
  We study the mechanism by which gravitational actions reproduce the trace anomalies of the holographically related conformal field theories. Two universal features emerge: a) the ratios of type B trace anomalies in any even dimension are independent of the gravitational action, being uniquely determined by the underlying algebraic structure b) the normalization of the type A and the overall normalization of the type B anomalies are given by action dependent expressions with the dimension dependence completely fixed. 
  We discuss the properties of codimension-two branes and compare them to codimension-one branes. In particular, we show that for deficit angle branes the brane energy momentum tensor is uniquely related to integration constants in the bulk solution. We investigate chiral fermions whose wave functions are concentrated on the brane, while all their properties in the effective four-dimensional world can be inferred from the tail of the wave function in the bulk, thereby realizing a holographic principle. We propose holographic branes for which the knowledge of the bulk geometry is sufficient for the computation of all relevant properties of the observable particles, independently of the often unknown detailed physics of the branes. 
  The Casimir force in a system consisting of two parallel conducting plates in the presence of compactified universal extra dimensions (UXD) is analyzed. The Casimir force with UXDs differs from the force obtained without extra dimensions. A new power law for the Casimir force is derived. By comparison to experimental data the size R of the universal extra dimensions can be restricted to R < 10 nm for one extra dimension. 
  We provide explicit formulas for the orthogonal eigenfunctions of the supersymmetric extension of the rational Calogero-Moser-Sutherland model with harmonic confinement, i.e., the generalized Hermite (or Hi-Jack) polynomials in superspace. The construction relies on the triangular action of the Hamiltonian on the supermonomial basis. This translates into determinantal expressions for the Hamiltonian's eigenfunctions. 
  A new class of $A^{(1)}_n$ integrable lattice models is presented. These are interaction-round-a-face models based on fundamental nimrep graphs associated with the $A^{(1)}_n$ conjugate modular invariants, there being a model for each value of the rank and level. The Boltzmann weights are parameterized by elliptic theta functions and satisfy the Yang-Baxter equation for any fixed value of the elliptic nome q. At q=0, the models provide representations of the Hecke algebra and are expected to lead in the continuum limit to coset conformal field theories related to the $A^{(1)}_n$ conjugate modular invariants. 
  In higher dimensional field theories with compactified dimensions there are three standard ways to do perturbative calculations: i) by the summation over Kaluza-Klein towers; ii) by the summation over winding numbers making use of the Poisson-resummation formula and iii) by using mixed propagators, where the coordinates of the four infinite dimensions are Fourier-transformed to momentum space while those of the compactified dimensions are kept in configuration space. The third method is broadly used in finite temperature field theory calculations. One of its advantage is that one can easily separate the ultraviolet divergent terms of the uncompactified theory from the non-local finite corrections arising from windings around the compact dimensions. In this note we demonstrate the use of this formalism by calculating one-loop self-energy corrections in a 5D theory formulated on the manifold M_4 \otimes S_1 and on the orbifold M_4 \otimes S_1/Z_2. 
  We construct four dimensional intersecting D6-brane models that have locally the spectrum of the N=1 Supersymmetric Standard Model. All open {\em visible} string sectors share the same N=1 supersymmetry. As expected in these supersymmetric classes of models, where the D6-branes wrap a toroidal orientifold of type IIA, the hierarchy may be stabilized if the string scale is low, e.g. below 30 TeV. We analyze the breaking of supersymmetry in the vicinity of the supersymmetric point by turning on complex structure deformations as Fayet-Iliopoulos terms. Positive masses for all squarks and sleptons, to avoid charge/colour breaking minima, may be reached when also two loop contributions may be included. In the ultimate version of the present models N=1 supersymmetry may be broken by gauge mediation. The constructions with four, five and six stacks of D6-branes at $M_s$ are build directly. Next by the use of brane recombination we are able to show that there is a continuous, RR homology flow, between six, five and four stack models. Moreover, we examine the gauge coupling constants of the Standard Model $SU(3)_C$, $SU(2)_L$, $U(1)_Y$ at the string scale in the presence of a non-zero antisymmetric NS B-field. 
  Newton potential for DGP brane-world scenario is examined when the extra dimension is semi-infinite. The final form of the potential involves a self-adjoint extension parameter $\alpha$, which plays a role of an additional mass (or distance) scale. The striking feature of Newton potential in this setup is that the potential behaves as seven-dimensional in long range when $\alpha$ is nonzero. For small $\alpha$ there is an intermediate range where the potential is five-dimensional. Five-dimensional Newton constant decreases with increase of $\alpha$ from zero. In the short range the four-dimensional behavior is recovered. The physical implication of this result is discussed in the context of the accelerating behavior of universe. 
  In this paper the nonperturbative analysis of the spectrum for one-particle excitations of the electron-positron field (EPF) is considered in the paper. A standard form of the quantum electrodynamics (QED) is used but the charge of the "bare" electron $e_0$ is supposed to be of a large value. It is shown that in this case the quasi-particle can be formed with a non-zero averaged value of the scalar component of the electromagnetic field (EMF). Self-consistent equations for the distribution of charge density in the "physical" electron (positron) are derived. A variational solution of these equations is obtained and it defines the finite renormalization of the charge and mass of the electron (positron). It is found that the coupling constant between EPF and EMF and mass of the "bare" electron can be connected with the observed values of the fine structure constant and the mass of the "physical" electron. It is also shown that although the non-renormalized QED corresponds to the strong coupling between EPF and EMF, the interaction between "physical" electron (positron) with EMF is defined by the observed value of the coupling constant. It is proved that the translational motion of the "physical" particle is separated from its internal degrees of freedom. As a result the dependence of the one-particle excitation energy on its total momentum corresponds to the relativistic spectrum of a free particle with the observed value of mass. Regularization of the terms of a series of the perturbation theory is due to the form-factor of the "physical" electron (positron). 
  Parent actions for component fields are utilized to derive the dual of supersymmetric U(1) gauge theory in 4 dimensions. Generalization of the Seiberg-Witten map to the component fields of noncommutative supersymmetric U(1) gauge theory is analyzed. Through this transformation we proposed parent actions for noncommutative supersymmetric U(1) gauge theory as generalization of the ordinary case.Duals of noncommutative supersymmetric U(1) gauge theory are obtained. Duality symmetry under the interchange of fields with duals accompanied by the replacement of the noncommutativity parameter \Theta_{\mu\nu} with \tilde{\Theta}_{\mu \nu} = \epsilon_{\mu\nu\rho\sigma}\Theta^{\rho\sigma} of the non--supersymmetric case is broken at the level of actions. We proposed a noncommutative parent action for the component fields which generates actions possessing this duality symmetry. 
  We consider a closed string field theory with an arbitrary matter current as a source of the closed string field. We find that the source must satisfy a constraint equation as a consequence of the BRST invariance of the theory. We see that it corresponds to the covariant conservation law for the matter current, and the equation of motion together with this constraint equation determines the classical behavior of both the closed string field and the matter. We then consider the boundary state (D-brane) as an example of a source. We see that the ordinary boundary state cannot be a source of the closed string field when the string coupling g turns on. By perturbative expansion, we derive a recursion relation which represents the bulk backreaction and the D-brane recoil. We also make a comment on the rolling tachyon boundary state. 
  An interesting example of the deep interrelation between Physics and Mathematics is obtained when trying to impose mathematical boundary conditions on physical quantum fields. This procedure has recently been re-examined with care. Comments on that and previous analysis are here provided, together with considerations on the results of the purely mathematical zeta-function method, in an attempt at clarifying the issue. Hadamard regularization is invoked in order to fill the gap between the infinities appearing in the QFT renormalized results and the finite values obtained in the literature with other procedures. 
  There are two types of non(anti-)commutative deformation of D=4, N=1 supersymmetric field theories and D=2, N=2 theories. One is based on the non-supersymmetric star product and the other is based on the supersymmetric star product . These deformations cause partial breaking of supersymmetry in general. In case of supersymmetric star product, the chirality is broken by the effect of the supersymmetric star product, then it is not clear that lagrangian or observables including F-terms preserve part of supersymmetry. In this article, we investigate the ring structure whose product is defined by the supersymmetric star product. We find the ring whose elements correspond to 1/2 SUSY F-terms. Using this, the 1/2 SUSY invariance of the Wess-Zumino model is shown easily and directly. 
  A cross between two well-known integrable multi-particle dynamics, an affine Toda molecule and a Sutherland system, is introduced for any affine root system. Though it is not completely integrable but partially integrable, or quasi exactly solvable, it inherits many remarkable properties from the parents. The equilibrium position is algebraic, i.e. proportional to the Weyl vector. The frequencies of small oscillations near equilibrium are proportional to the affine Toda masses, which are essential ingredients of the exact factorisable S-matrices of affine Toda field theories. Some lower lying frequencies are integer times a coupling constant for which the corresponding exact quantum eigenvalues and eigenfunctions are obtained. An affine Toda-Calogero system, with a corresponding rational potential, is also discussed. 
  We prove the equivalence of many-photon Green functions in statistical quantum field Duffin-Kemmer-Petiau (DKP) and Klein-Gordon-Fock (KGF) theories using functional path integral formalism for partition functional in statistical quantum (finite temperature) field theory. We also calculate the polarization operators in these theories in one-loop approximation, and demonstrate their coincidence. 
  We clarify the existence of two different types of truncations of the field content in a theory, the consistency of each type being achieved by different means. A proof is given of the conditions to have a consistent truncation in the case of dimensional reductions induced by independent Killing vectors. We explain in what sense the tracelessness condition found by Scherk and Scharwz is not only a necessary condition but also a {\it sufficient} one for a consistent truncation. The reduction of the gauge group is fully performed showing the existence of a sector of rigid symmetries. We show that truncations originated by the introduction of constraints will in general be inconsistent, but this fact does not prevent the possibility of correct upliftings of solutions in some cases. The presence of constraints has dynamical consequences that turn out to play a fundamental role in the correctness of the uplifting procedure. 
  We review the recently developed relation between the traditional algebraic approach to conformal field theories and the more recent probabilistic approach based on stochastic Loewner evolutions. It is based on implementing random conformal maps in conformal field theories. 
  We review the circumstances under which test particles can be localized around a spacetime section \Sigma_0 smoothly contained within a codimension-1 embedding space M. If such a confinement is possible, \Sigma_0 is said to be totally geodesic. Using three different methods, we derive a stability condition for trapped test particles in terms of intrinsic geometrical quantities on \Sigma_0 and M; namely, confined paths are stable against perturbations if the gravitational stress-energy density on M is larger than that on \Sigma_0, as measured by an observed travelling along the unperturbed trajectory. We confirm our general result explicitly in two different cases: the warped-product metric ansatz for (n+1)-dimensional Einstein spaces, and a known solution of the 5-dimensional vacuum field equation embedding certain 4-dimensional cosmologies. We conclude by defining a confinement energy condition that can be used to classify geometries incorporating totally geodesic submanifolds, such as those found in thick braneworld and other 5-dimensional scenarios. 
  D0-branes are unstable in the presence of an R-R field strength background. A fuzzy two-sphere is classically stable under such a background, this phenomenon being called the Myers effect. We analyze this effect from the viewpoint of tachyon condensation. It is explicitly shown that a fuzzy two-sphere is realized by the condensation of tachyons which appear from strings connecting different D0-branes. The formation of a fuzzy $CP^{2}$ is also investigated by considering the SU(3) invariant R-R field strength background. We find that the dynamics of the D-branes depends on the properties of the associated algebra. 
  We investigate maximally symmetric brane world solutions with a scalar field. Five-dimensional bulk gravity is described by a general lagrangian which yields field equations containing no higher than second order derivatives. This includes the Gauss-Bonnet combination for the graviton. Stability and gravitational properties of such solutions are considered, and we particularily emphasise the modifications induced by the higher order terms. In particular it is shown that higher curvature corrections to Einstein theory can give rise to instabilities in brane world solutions. A method for analytically obtaining the general solution for such actions is outlined. Genericaly, the requirement of a finite volume element together with the absence of a naked singularity in the bulk imposes fine-tuning of the brane tension. A model with a moduli scalar field is analysed in detail and we address questions of instability and non-singular self-tuning solutions. In particular, we discuss a case with a normalisable zero mode but infinite volume element. 
  In this talk I describe some applications of random matrix models to the study of N=1 supersymmetric Yang-Mills theories with matter fields in the fundamental representation. I review the derivation of the Veneziano-Yankielowicz-Taylor/Affleck-Dine-Seiberg superpotentials from constrained random matrix models (hep-th/0211082), a field theoretical justification of the logarithmic matter contribution to the Veneziano-Yankielowicz-Taylor superpotential (hep-th/0306242) and the random matrix based solution of the complete factorization problem of Seiberg-Witten curves for N=2 theories with fundamental matter (hep-th/0212212). 
  We consider the anisotropic effect in the quantum Hall systems by applying a confining potential that is not of parabolic type. This can be done by extending Susskind--Polychronakos's approach to involve the matrices of two coupled harmonic oscillators. Starting from its action, we employ a unitary transformation to diagonalize the model. The operators for building up the anisotropic ground state and creating the collective excitations can be constructed explicitly. Evaluating the area of the quantum Hall droplet, we obtain the corresponding filling factor which is found to depend on the anisotropy parameter and to vary with the magnetic field strength. This can be used to obtain the observed anisotropic filling factors, i.e. {9\over 2}, {11\over 2} and others. 
  We show how to reconstruct a field theory from the spectrum of bound states on a topological defect. We apply our recipe to the case of kinks in 1+1 dimensions with one or two bound states. Our recipe successfully yields the sine-Gordon and $\lambda \phi^4$ field theories when suitable bound state spectra are assumed. The recipe can also be used to globally reconstruct the inflaton potential of inflationary cosmology if the inflaton produces a topological defect. We discuss how defects can provide ``smoking gun'' evidence for a class of inflationary models. 
  We analyze the canonical structure of a continuum model of ferromagnets and clarify known difficulties in defining a momentum density. The moments of the momentum density corresponding to volume-preserving coordinate transformations can be defined, but a nonsingular definition of the other moments requires an enlargement of the phase space which illuminates a close relation to fluid mechanics. We also discuss the nontrivial connectivity of the phase space for two and three dimensions and show how this feature can be incorporated in the quantum theory, working out the two-dimensional case in some detail. 
  Constrained systems are fundamental to understanding of several physical realities. Even so the Hall effect is one of more revisited issue we can still find new approaches to obtain old and new important relations. In this paper a semi classical formulation is considered where an Chern-Simons gauge invariant theory is constructed for a Schroedinger field. The main idea is to describe both classical and integer Hall effect. We build up the constraints this model by means of the Fadeev-Jackiw quantization algorithm. In a second step we consider a noncommutative extension to the action. In this extended approach noncommutative constraints relations are obtained and guide us to an interesting adjustment factor on the conductivity expression. 
  We investigate the static solutions of Callan and Maldecena and Gibbons to lowest order Dirac-Born-Infeld theory. Among them are charged wormhole solutions connecting branes to anti-branes. It is seen that there are no such solutions when the separation between the brane and anti-brane is smaller than some minimum value. The minimum distance coincides with the energy minimum, and depends monotonically on the charge. Making the charge sufficiently large, such that the minimum separation is much bigger than $ \sqrt{\alpha '}$, may suppress known quantum processes leading to decay of the brane-anti-brane system. For this to be possible the zeroth order wormhole solutions should be reasonable approximations of solutions in the full $D-$brane theory. With this in mind we address the question of whether the zeroth order solutions are stable under inclusion of higher order corrections to the Dirac-Born-Infeld action. 
  The stability of Yang-Mills bundles over the usual $S^4$ space-time manifold is investigated according to the topological methods. The necessary gauge- and topological invaraint criterion for the exsitence of the related critical points is defined. It is shown that according to this criterion there exists no critical point even for the action functional of the standard U(1) gauge theory of electrodynamics on a $S^4$ manifold in view of its topological structure and therefore such a theory can not be stable. We will discuss also a general consequence of this result according to which for a stable U(1) Yang-Mills theory over a compact 4-manifold, this manifold should possess some self consistent compact 2-manifold substructure. These results are also in agreement with the known very general result for the {\it structural stability} of dynamical systems. 
  We formulate quantum mechanics in the two-dimensional torus without using position operators. We define an algebra with only momentum operators and shift operators and construct irreducible representation of the algebra. We show that it realizes quantum mechanics of a charged particle in a uniform magnetic field. We prove that any irreducible representation of the algebra is unitary equivalent to each other. This work provides a firm foundation for the noncommutative torus theory. 
  Recently, it was shown that the entropy of the black hole horizon in the Achucarro-Ortiz spacetime can be described by the Cardy-Verlinde formula. In this paper, we compute the self-gravitational corrections to the Cardy-Verlinde formula of the two-dimensional Achucarro-Ortiz black hole. These corrections stem from the effect of self-gravitation and they are derived in the context of Keski-Vakkuri, Kraus and Wilczek (KKW) analysis. The black hole under study is therefore treated as a dynamical background. The self-gravitational corrections to the entropy as given by the Cardy-Verlinde formula of Achucarro-Ortiz black hole, are found to be positive. This result provides evidence in support of the claim that the holographic bound is not universal in the framework of two-dimensional gravity models. 
  Analysis of string interactions indicates a weakening of gravity at the string length scale, thus avoiding black holes and their singularities. 
  Supersymmetric field theories of scalars and fermions in 4-D space-time can be cast in the formalism of Kaehler geometry. In these lectures I review Kaehler geometry and its application to the construction and analysis of supersymmetric models on Kaehler coset manifolds. It is shown that anomalies can be eliminated by the introduction of line-bundle representations of the coset symmetry groups. Such anomaly-free models can be gauged consistently and used to construct alternatives to the usual MSSM and supersymmetric GUTs. 
  We propose a new class of inflationary solutions to the standard cosmological problems (horizon, flatness, monopole,...), based on a modification of old inflation. These models do not require a potential which satisfies the normal inflationary slow-roll conditions. Our universe arises from a single tunneling event as the inflaton leaves the false vacuum. Subsequent dynamics (arising from either the oscillations of the inflaton field or thermal effects) keep a second field trapped in a false minimum, resulting in an evanescent period of inflation (with roughly 50 e-foldings) inside the bubble. This easily allows the bubble to grow sufficiently large to contain our present horizon volume. Reheating is accomplished when the inflaton driving the last stage of inflation rolls down to the true vacuum, and adiabatic density perturbations arise from moduli-dependent Yukawa couplings of the inflaton to matter fields. Our scenario has several robust predictions, including virtual absence of gravity waves, a possible absence of tilt in scalar perturbations, and a higher degree of non-Gaussianity than other models. It also naturally incorporates a solution to the cosmological moduli problem. 
  We present a Kahler potential for four dimensional heterotic M-theory which includes moduli describing a gauge five brane living on one of the orbifold fixed planes. This result can also be thought of as describing compactifications of either of the weakly coupled heterotic strings in the presence of a gauge five brane. This is the first example of a Kahler potential in these theories which includes moduli describing background gauge field configurations. Our results are valid when the solitons width is much smaller than the size scale of the Calabi-Yau threefold and can be used to provide a more complete description of some moving brane scenarios. We point out that, in general, it is not consistent to truncate away the gauge five brane moduli in a simple manner. 
  In this paper, by using the factorization equation of the ${\cal N}=2$ theory we study ${\cal N}=1$ supersymmetric SO(N) gauge theory in Argyres-Douglas points . We suppose that maximal confinement occurs in the system and give a tree level superpotential. Then, we obtain general Picard-fuchs equations for glueball superfield which are hypergeometric equations having regular singular points corresponding to Argyres-Douglas points. Furthermore, we study the solution of these differential equations and calculate the effective superpotential. Finally, we study scaling behavior of the chiral operators and coupling constant around the AD points. 
  In this work we establish the correspondence between solutions to the Friedmann--Robertson--Walker cosmologies for perfect fluid and scalar field sources, where both ones fulfill state equations of the form $p+\rho=\gamma f(\rho)$, not necessarily linear ones. Such state equations are of common use in the case of matter--fluids, nevertheless, for a scalar field, they introduce relationships on the potential and kinetic scalar field energies which restrict the set of solutions. A theorem on this respect is demonstrated: From any given (3+1)--cosmological solution, obeying the quoted state equations, one can derive its (2+1)--cosmological counterpart or vice-versa. Some applications are given. 
  Supersymmetry is formulated for integrable models based on the $sl(2|1)$ loop algebra endowed with a principal gradation. The symmetry transformations which have half-integer grades generate supersymmetry. The $sl(2|1)$ loop algebra leads to N=2 supersymmetric mKdV and sinh-Gordon equations. The corresponding N=1 mKdV and sinh-Gordon equations are obtained via reduction induced by twisted automorphism. Our method allows for a description of a non-local symmetry structure of supersymmetric integrable models. 
  We present non-Abelian gaugings of supermembrane for general isometries for compactifications from eleven-dimensions, starting with Abelian case as a guide. We introduce a super Killing vector in eleven-dimensional superspace for a non-Abelian group G associated with the compact space B for a general compactification, and couple it to a non-Abelian gauge field on the world-volume. As a technical tool, we use teleparallel superspace with no manifest local Lorentz covariance. Interestingly, the coupling constant is quantized for the non-Abelian group G, due to its generally non-trivial mapping \pi_3(G). 
  Including world-sheet orientation-reversing automorphisms in the orbifold program, we recently reported the twisted operator algebra and twisted KZ equations in each open-string sector of the general WZW orientation orbifold. In this paper we work out the corresponding classical description of these sectors, including the {\it WZW orientation-orbifold action} -- which is naturally defined on the solid half cylinder -- and its associated WZW orientation-orbifold branes. As a generalization, we also obtain the {\it sigma-model orientation-orbifold action}, which describes a much larger class of open-string orientation-orbifold sectors. As special cases, this class includes twisted open-string {\it free boson} examples, the open-string WZW sectors above and the open-string sectors of the {\it general coset orientation orbifold}. Finally, we derive the {\it orientation- orbifold Einstein equations}, in terms of twisted Einstein tensors -- which hold when the twisted open-string sigma-model sectors are 1-loop conformal. 
  We construct boundary states in a particular c=1 conformal field theory, the SU(2)_1/G orbifold with G a binary finite subgroup of SU(2). These states preserve the conformal symmetry, at least, but break rational symmetries of the SU(2)_1/G orbifold in general. 
  We investigate the possibility that supersymmetry is not a fundamental symmetry of nature, but emerges as an accidental approximate global symmetry at low energies. This can occur if the visible sector is non-supersymmetric at high scales, but flows toward a strongly-coupled superconformal fixed point at low energies; or, alternatively, if the visible sector is localized near the infrared brane of a warped higher-dimensional spacetime with supersymmetry broken only on the UV brane. These two scenarios are related by the AdS/CFT correspondence. In order for supersymmetry to solve the hierarchy problem, the conformal symmetry must be broken below 10^{11} GeV. Accelerated unification can naturally explain the observed gauge coupling unification by physics below the conformal breaking scale. In this framework, there is no gravitino and no reason for the existence of gravitational moduli, thus eliminating the cosmological problems associated with these particles. No special dynamics is required to break supersymmetry; rather, supersymmetry is broken at observable energies because the fixed point is never reached. In 4D language, this can be due to irrelevant supersymmetry breaking operators with approximately equal dimensions. In 5D language, the size of the extra dimension is stabilized by massive bulk fields. No small input parameters are required to generate a large hierarchy. Supersymmetry can be broken in the visible sector either through direct mediation or by the F term of the modulus associated with the breaking of conformal invariance. 
  In the present paper we discuss the relevance for de Sitter fields of the mass and spin interpretation of the parameters appearing in the theory. We show that these apparently conceptual interrogations have important consequences concerning the field theories. Among these, it appeared that several authors were using masses which they thought to be different, but which corresponded to a common unitary irreducible representation (UIR), hence to identical physicals systems. This could actually happen because of the arbitrariness of their mass definition in the de Sitter (dS) space. The profound cause of confusion however is to be found in the lack of connexion between the group theoretical approach on the one hand, and the usual field equation (in local coordinates) approach on the other hand. This connexion will be established in the present paper and by doing so we will get rid of any ambiguity by giving a consistent and univocal definition of a "mass" term uniquely defined with respect to a specific UIR of the de Sitter group. 
  We investigate a system of two coupled harmonic oscillators on the non-commutative plane \RR^2_{\theta} by requiring that the spatial coordinates do not commute. We show that the system can be diagonalized by a suitable transformation, i.e. a rotation with a mixing angle \alpha. The obtained eigenstates as well as the eigenvalues depend on the non-commutativity parameter \theta. Focusing on the ground state wave function before the transformation, we calculate the density matrix \rho_0(\theta) and find that its traces {\rm Tr}(\rho_{0}(\theta)) and {\rm Tr}(\rho_0^2(\theta)) are not affected by the non-commutativity. Evaluating the Wigner function on \RR^2_{\theta} confirms this. The uncertainty relation is explicitly determined and found to depend on \theta. For small values of \theta, the relation is shifted by a \theta^2 term, which can be interpreted as a quantum correction. The calculated entropy does not change with respect to the normal case. We consider the limits \alpha=1 and \alpha={\pi\over 2}. In first case, by identifying \theta to the squared magnetic length, one can recover basic features of the Hall system. 
  We discuss the topology of the symmetry groups appearing in compactified (super-)gravity, and discuss two applications. First, we demonstrate that for 3 dimensional sigma models on a symmetric space G/H with G non-compact and H the maximal compact subgroup of G, the possibility of oxidation to a higher dimensional theory can immediately be deduced from the topology of H. Second, by comparing the actual symmetry groups appearing in maximal supergravities with the subgroups of SL(32,R) and Spin(32), we argue that these groups cannot serve as a local symmetry group for M-theory in a formulation of de Wit-Nicolai type. 
  Intersecting branes have been the subject of string model building for several years. This work introduces in detail the toroidal and Z_N-orientifolds, where the main discussion employs the picture of intersecting D6-branes. The derivation of the R-R and NS-NS tadpole cancellation conditions in CFT is shown in detail. Various aspects of the massless spectrum are discussed, involving spacetime anomalies, the generalized Green-Schwarz mechanism and possible gauge breaking mechanisms. Both N=1 SUSY and non-SUSY approaches to low energy model building are treated. Firstly, the problem of complex structure instabilities in toroidal OmegaR-orientifolds is approached by a Z_3-orbifolded model, including a stable non-SUSY 3-generation standard-like model. It descends naturally from a flipped SU(5) GUT. Secondly, supersymmetric models on the Z_4-orbifold are discussed, involving exceptional 3-cycles and the explicit construction of fractional D-branes. A three generation Pati-Salam model that even can be broken down to a MSSM-like model is constructed as an example, involving non-flat and non-factorizable branes. Finally, the possibility that unstable closed and open string moduli could play the role of the inflaton is being explored. In the closed string sector, the slow-rolling requirement can only be fulfilled for very specific cases, where some moduli are frozen and a special choice of coordinates is taken. In the open string sector, inflation is not possible at all. 
  The limits of linear electrodynamics are reviewed, and possible directions of nonlinear extension are explored. The central theme is that the qualitative character of the empirical successes of quantum electrodynamics must be used as a guide for understanding the nature of the nonlinearity of electrodynamics at the subatomic level. Some established theories of nonlinear electrodynamics, namely, those of Mie, Born, and Infeld are presented in the language of the modern geometrical and topological methods of mathematical physics. The manner by which spacetime curvature and topology can affect electromagnetism is also reviewed. Finally, the phenomena of nonlinear optics are discussed as a possible guide to building one's intuition regarding the process of extending electrodynamics into nonlinearity in a manner that is consistent with the qualitative and empirical results of quantum electrodynamics. 
  We show that the existence of algebraic forms of quantum, exactly-solvable, completely-integrable $A-B-C-D$ and $G_2, F_4, E_{6,7,8}$ Olshanetsky-Perelomov Hamiltonians allow to develop the {\it algebraic} perturbation theory, where corrections are computed by pure linear algebra means. A Lie-algebraic classification of such perturbations is given. In particular, this scheme admits an explicit study of anharmonic many-body problems. The approach also allows to calculate the ratios of a certain generalized Dyson-Mehta integrals algebraically, which are interested by themselves. 
  We study an extended QCD model in (1+1) dimensions obtained from QCD in 4D by compactifying two spatial dimensions and projecting onto the zero-mode subspace. We work out this model in the large $N_c$ limit and using light cone gauge but keeping the equal-time quantization. This system is found to induce a dynamical mass for transverse gluons -- adjoint scalars in QCD(2), and to undergo a chiral symmetry breaking with the full quark propagators yielding non-tachyonic, dynamical quark masses, even in the chiral limit. We study quark-antiquark bound states which can be classified in this model by their properties under Lorentz transformations inherited from 4D. The scalar and pseudoscalar sectors of the theory are examined and in the chiral limit a massless ground state for pseudoscalars is revealed with a wave function generalizing the so called 't Hooft pion solution. 
  Biconformal supergravity models provide a new gauging of the superconformal group relevant to the Maldacena conjecture. Using the group quotient method to biconformally gauge SU(2,2|N), we generate a 16-dim superspace. We write the most general even- and odd-parity actions linear in the curvatures, the bosonic sector of which is known to descend to general relativity on a 4-dim manifold. 
  We introduce the free N=1 supersymmetric derivation ring and prove the existence of an exact sequence of supersymmetric rings and linear transformations. We apply necessary and sufficient conditions arising from this exact supersymmetric sequence to obtain the essential relations between conserved quantities, gradients and the N=1 super KdV hierarchy. We combine this algebraic approach with an analytic analysis of the super heat operator.We obtain the explicit expression for the Green's function of the super heat operator in terms of a series expansion and discuss its properties. The expansion is convergent under the assumption of bounded bosonic and fermionic potentials. We show that the asymptotic expansion when $t\to0^+$ of the Green's function for the super heat operator evaluated over its diagonal generates all the members of the N=1 super KdV hierarchy. 
  In the previous works, we proposed the stochastic quantization method (SQM) approach to N=1 supersymmetric Yang-Mills theory (SSYM). In four dimensions, in particular, we obtained the superfield Langevin equation and the corresponding Fokker-Planck equation which describe the underlying stochastic process manifestly preserving the global supersymmetry as well as the local gauge symmetry. The stochastic gauge fixing procedure was also applied to SSYM_4 in the superfield formalism. In this note, we apply the background field methd to SSYM_4 in terms of the stochastic action principle in SQM approach. The one-loop $\beta$-function for the gauge coupling agrees with that given by the path-integral approach, thereby confirming that the stochastic gauge fixing procedure with the background local gauge invariant Zwanziger's gauge fixing functions simulates the contributions from the Nielsen-Kallosh ghost as well as the Faddeev-Popov ghost at the one-loop level. We also show the equivalence of the stochastic effective action in the background field method to the standard one in SQM. 
  It is shown that a class of rotating strings in AdS_5 x S^5 with SO(6) angular momenta (J,J',J') preserve 1/8-supersymmetry for large J,J', in which limit they are effectively tensionless; when J=0, supersymmetry is enhanced to 1/4. These results imply that recent checks of the AdS/CFT correspondence actually test a nearly-BPS sector. 
  In the Randall-Sundrum scenario we analyze the dynamics of a spherically symmetric 3-brane when the bulk is filled with matter fields. Considering a global conformal transformation whose factor is the $Z_2$ symmetric warp we find a new set of exact dynamical solutions for which gravity is bound to the brane. The set corresponds to a certain class of conformal bulk fields. We discuss the geometries which describe the dynamics on the brane of polytropic dark energy. 
  We introduce a novel type of phase diagram for black holes and black strings on cylinders. The phase diagram involves a new asymptotic quantity called the relative binding energy. We plot the uniform string and the non-uniform string solutions in this new phase diagram using data of Wiseman. Intersection rules for branches of solutions in the phase diagram are deduced from a new Smarr formula that we derive. 
  We consider the spontaneous breaking of supersymmetry in five dimensional supergravity with boundaries where the supersymmetry breaking mechanism is provided by the even part of the bulk prepotential. The supersymmetric action comprises boundary brane terms with detuned tensions. The two branes have opposite tensions. We analyse the possible vacua with spontaneously broken supersymmetry. A class of solutions corresponds to rotated branes in an $AdS_5$ bulk. In particular parallel branes which are rotated with respect to the bulk preserve 1/4 of supersymmetry. We analyse more general vacua using the low energy effective action for gravity coupled to the radion field. Supersymmetry breaking implies that the radion field acquires a potential which is negative and unbounded from below. This potential is modified when coupling the boundary branes to a bulk four--form field. For a brane charge larger than the deficit of brane tension, the radion potential is bounded from below while remaining flat when the charge equals the deficit of brane tension. 
  We study a class of Wilson Loops in N =4, D=4 Yang-Mills theory belonging to the chiral ring of a N=2, d=1 subalgebra. We show that the expectation value of these loops is independent of their shape. Using properties of the chiral ring, we also show that the expectation value is identically 1. We find the same result for chiral loops in maximally supersymmetric Yang-Mills theory in three, five and six dimensions. In seven dimensions, a generalized Konishi anomaly gives an equation for chiral loops which closely resembles the loop equations of the three dimensional Chern-Simons theory. 
  Boundary quantum field theory is investigated in the Lagrangian framework. Models are defined perturbatively around the Neumann boundary condition. The analyticity properties of the Green functions are analyzed: Landau equations, Cutkosky rules together with the Coleman-Norton interpretation are derived. Illustrative examples as well as argument for the equivalence with other perturbative expansions are presented. 
  We consider SUSY sine-Gordon theory in the framework of perturbed conformal field theory. Using an argument from Zamolodchikov, we obtain the vacuum structure and the kink adjacency diagram of the theory, which is cross-checked against the exact S matrix prediction, first-order perturbed conformal field theory (PCFT), the NLIE method and truncated conformal space approach. We provide evidence for consistency between the usual Lagrangian description and PCFT on the one hand, and between PCFT, NLIE and a massgap formula conjectured by Baseilhac and Fateev, on the other. In addition, we extend the NLIE description to all the vacua of the theory. 
  We construct exact vortex solutions to the equations of motion of the Abelian Higgs model defined in non commutative space, analyzing in detail the properties of these solutions beyond the BPS point. We show that our solutions behave as smooth deformations of vortices in ordinary space time except for parity symmetry breaking effects induced by the non commutative parameter $\theta$. 
  Various nonsupersymmetric theories at large but finite $N$ are argued to permit light scalars and large hierarchies without fine-tuning. In a dual string description, the hierarchy results from competition between classical and quantum effects. In some cases the flow may end when a string mode becomes tachyonic and condenses, thereby realizing a quantum-mechanically stable Randall-Sundrum hierarchy scenario. Among possible applications, it is suggested that lattice simulation of \nfour Yang-Mills at large 't Hooft coupling may be easier than expected, and that supersymmetry may naturally be an approximate symmetry of our world. (This letter is a writeup of work presented at Aspen in summer 2002.) 
  A version of non-Abelian monopole equations is explored through dimensional reductions, with often the addition of algebraic conditions. On zero curvature spaces, spinor related extensions of integrable systems have been generated, and certain reduced one-dimensional systems have been discussed with respect to integrability, as well as solutions found. 
  The nonlocal electrodynamics of accelerated systems is discussed in connection with the development of Lorentz-invariant nonlocal field equations. Nonlocal Maxwell's equations are presented explicitly for certain linearly accelerated systems. In general, the field equations remain nonlocal even after accelerated motion has ceased. 
  Twisted supersymmetric theories on a product of two Riemann surfaces possess non-local holomorphic currents in a BRST cohomology. The holomorphic currents act as vector fields on the chiral ring. The OPE's of these currents are invariant under the renormalization group flow up to BRST-exact terms. In the context of electric-magnetic duality, the algebra generated by the holomorphic currents in the electric theory is isomorphic to the one on the magnetic side. For the currents corresponding to global symmetries this isomorphism follows from 't Hooft anomaly matching conditions. The isomorphism between OPE's of the currents corresponding to non-linear transformations of fields of matter imposes non-trivial conditions on the duality map of chiral ring. We consider in detail the $SU(N_c)$ SQCD with matter in fundamental and adjoint representations, and find agreement with the duality map proposed by Kutasov, Schwimmer and Seiberg. 
  The gaugino propagator is calculated by explicitly considering the propagation of a heterotic string between two different points in space-time using the non-trivial world-sheet conformal field theory for the fivebrane background. We find that there are no propagations of gaugino which is in the spinor representation of the non-trivial four-dimensional space of the fivebrane background. This result is consistent with the arguments on the fermion zero-modes of the fivebrane background in the low-energy heterotic supergravity theory. Furthermore, assuming the continuous limit to the flat space-time background at the place far away from the fivebrane, we suggest an effective propagator which is effective only at the place far away from the fivebrane in the flat space-time limit. From the effective propagator we evaluate a possible gaugino pair condensation. The result is consistent with the suggested scenario of the gaugino condensation in the fivebrane background in the low-energy heterotic supergravity theory. 
  We obtain an interesting realization of the de Rham cohomological operators of differential geometry in terms of the noncommutative q-superoscillators for the supersymmetric quantum group GL_{qp} (1|1). In particular, we show that a unique superalgebra, obeyed by the bilinears of fermionic and bosonic noncommutative q-(super)oscillators of GL_{qp} (1|1), is exactly identical to that obeyed by the de Rham cohomological operators. A set of discrete symmetry transformation for a set of GL_{qp} (1|1) covariant superalgebras turns out to be the analogue of the Hodge duality * operation of differential geometry. A connection with an extended BRST algebra obeyed by the nilpotent (anti-)BRST and (anti-)co-BRST charges, the ghost charge and a bosonic charge (which is equal to the anticommutator of (anti-)BRST and (anti-)co-BRST charges) is also established. 
  We present a brief review of the fuzzy disc, the finite algebra approximating functions on a disc, which we have introduced earlier. We also present a comparison with recent papers of Balachandran, Gupta and K\"urk\c{c}\"{u}o\v{g}lu, and of Pinzul and Stern, aimed at the discussion of edge states of a Chern-Simons theory. 
  The two-dimensional Ashkin-Teller model provides the simplest example of a statistical system exhibiting a line of critical points along which the critical exponents vary continously. The scaling limit of both the paramagnetic and ferromagnetic phases separated by the critical line are described by the sine-Gordon quantum field theory in a given range of its dimensionless coupling. After computing the relevant matrix elements of the order and disorder operators in this integrable field theory, we determine the universal amplitude ratios along the critical line within the two-particle approximation in the form factor approach. 
  Casimir forces are conventionally computed by analyzing the effects of boundary conditions on a fluctuating quantum field. Although this analysis provides a clean and calculationally tractable idealization, it does not always accurately capture the characteristics of real materials, which cannot constrain the modes of the fluctuating field at all energies. We study the vacuum polarization energy of renormalizable, continuum quantum field theory in the presence of a background field, designed to impose a Dirichlet boundary condition in a particular limit. We show that in two and three space dimensions, as a background field becomes concentrated on the surface on which the Dirichlet boundary condition would eventually hold, the Casimir energy diverges. This result implies that the energy depends in detail on the properties of the material, which are not captured by the idealized boundary conditions. This divergence does not affect the force between rigid bodies, but it does invalidate calculations of Casimir stresses based on idealized boundary conditions. 
  We study whether modification of gravity at large distances is possible in warped backgrounds with two branes and a brane-induced term localized on one of the branes. We find that there are three large regions in the parameter space where the theory is weakly coupled up to high energies. In one of these regions gravity on the brane is four-dimensional at arbitrarily large distances, and the induced Einstein term results merely in the renormalization of the 4d Planck mass. In the other two regions the behavior of gravity changes at ultra-large distances; however, radion becomes a ghost. In parts of these regions, both branes have positive tensions, so the only reason for the appearance of the ghost field is the brane-induced term. In between these three regions, there are domains in the parameter space where gravity is strongly coupled at phenomenologically unacceptable low energy scale. 
  A generalized vector particle theory with the use of an extended set of Lorentz group irredicible representations, including scalar, two 4-vectors, and antisymmetric 2-rang tensor, is investigated. Initial equations depend upon four complex parameters $\lambda_{i}$, obeying two supplementary conditions, so restriction of the model to the case of electrically neutral vector particle is not a trivial task. A special basis in the space of 15-component wave functions is found where instead of four $\lambda_{i}$ only one real-valued quantity $\sigma$, a bilinear combination of $\lambda_{i}$, is presented. This $\lambda$-parameter is interpreted as an additional electromagnetic characteristic of a charged vector particle, polarizability. It is shown that in this basis $C$-operation is reduced to the complex conjugation only, without any accompanying linear transformation. Restriction to a massless vector particle is determined.   Extension of the whole theory to the case of Riemannian space-time is accomplished. Two methods of obtaining corresponding generally covariant wave equations are elaborated: of tensor- and of tetrad-based ones. Their equivalence is proved. It is shown that in case of pure curved space-time models without Cartan torsion no specific additional interaction terms because of non-flat geometry arise. The conformal symmetry of a massless generally covariant equation is demonstrated explicitly. A canonical tensor of energy-momentum $T_{\beta \alpha}$ is constructed, its conservation law happens to involves the Riemann curvature tensor. Within the framework of known ambiguity of any energy-momentum tensor, a new tensor $\bar{T}_{\beta \alpha}$ is suggested to be used, which obeys a common conservation law. 
  We analyze the restoration of the Slavnov-Taylor (ST) identities for pure massless Yang-Mills theory in the Landau gauge within the BPHZL renormalization scheme with IR regulator. We obtain the most general form of the action-like part of the symmetric regularized action, obeying the relevant ST identities and all other relevant symmetries of the model, to all orders in the loop expansion. We also give a cohomological characterization of the fulfillment of BPHZL IR power-counting criterion, guaranteeing the existence of the limit where the IR regulator goes to zero. The technique analyzed in this paper is needed in the study of the restoration of the ST identities for those models, like the MSSM, where massless particles are present and no invariant regularization scheme is known to preserve the full set of ST identities of the theory. 
  In this paper we calculate leading order correction due to small statistical fluctuations around equilibrium, to the Bekenstein-Hawking entropy formula for the Achucarro-Oritz black hole, which is the most general two-dimensional black hole derived from the three-dimensional rotating Banados-Teitelboim-Zanelli black hole. Then we obtain the same correction to the Cardy-Verlinde entropy formula (which is supposed to be an entropy formula of conformal field theory in any dimension). 
  We construct boundary states for supertubes in the flat spacetime. The T-dual objects of supertubes are moving spiral D1-branes (D-helices). Since we can obtain these D-helices from the usual D1-branes via null deformation, we can construct the boundary states for these moving D-helices in the covariant formalism. Using these boundary states, we calculate the vacuum amplitude between two supertubes in the closed string channel and read the open string spectrum via the open closed duality. We find there are critical values of the energy for on-shell open strings on the supertubes due to the non-trivial stringy correction. We also consider supertubes in the type IIA Godel universe in order to use them as probes of closed timelike curves. This universe is the T-dual of the maximally supersymmetric type IIB PP-wave background. Since the null deformations of D-branes are also allowed in this PP-wave, we can construct the boundary states for supertubes in the type IIA Godel universe in the same way. We obtain the open string spectrum on the supertube from the vacuum amplitude between supertubes. As a consequence, we find that the tachyonic instability of open strings on the supertube, which is the signal of closed time like curves, disappears due to the stringy correction. 
  Here we discuss two many-particle quantum systems, which are obtained by adding some nonhermitian but PT (i.e. combined parity and time reversal) invariant interaction to the Calogero model with and without confining potential. It is shown that the energy eigenvalues are real for both of these quantum systems. For the case of extended Calogero model with confining potential, we obtain discrete bound states satisfying generalised exclusion statistics. On the other hand, the extended Calogero model without confining term gives rise to scattering states with continuous spectrum. The scattering phase shift for this case is determined through the exchange statistics parameter. We find that, unlike the case of usual Calogero model, the exclusion and exchange statistics parameter differ from each other in the presence of PT invariant interaction. 
  Two-point correlation functions of spin operators in the minimal models ${{\cal M}}_{p,p'}$ perturbed by the field $\Phi_{13}$ are studied in the framework of conformal perturbation theory. The first-order corrections for the structure functions are derived analytically in terms of gamma functions. Together with the exact vacuum expectation values of local operators, this gives the short-distance expansion of the correlation functions. The long-distance behaviors of these correlation functions in the case ${{\cal M}}_{2,2n+1}$ have been worked out using a form-factor bootstrap approach.   The results of numerical calculations demonstrate that the short- and long-distance expansions match at the intermediate distances. Including the descendent operators in the OPE drastically improves the convergency region. The combination of the two methods thus describes the correlation functions at all length scales with good precision. 
  Exact time-dependent solutions of c=1 string theory are described using the free fermion formulation. One such class of solutions describes draining of the Fermi sea and has a spacetime interpretation as closed string tachyon condensation. A second class of solutions, corresponding to droplets of Fermi liquid orbiting in phase space, describes closed cosmologies which bounce through singularities. 
  We present some exact scalar potentials for the dimensionally reduced theory and examine the possibility of obtaining accelerating 4d cosmology from String/M-theory, more generally, hyperbolic and flux compactification. In the hyperbolic case, even in the zero-flux limit, the scalar potential is positive for the 4d effective theory as required to get an accelerating universe, and thereby evading the ``no-go theorem'' given for static internal space. When we turn on the gauge fields as source terms at the cosmological background with potential V\propto exp(-2c\phi), we find eternally accelerating cosmologies when the 4d space-time is flat and c\geq 1, or hyperbolic and 1<c<\sqrt{2}. 
  Supersymmetric gauge theories have had a significant impact on our understanding of QCD and of field theory in general. The phases of N=1 supersymmetric QCD (SQCD) are discussed, and the possibility of similar phases in non-supersymmetric QCD is emphasized. It is described how duality in SQCD links many previously known duality transformations that were thought to be distinct, including Olive-Montonen duality of N=4 supersymmetric gauge theory and quark-hadron duality in (S)QCD. A connection between Olive-Montonen duality and the confining strings of Yang-Mills theory is explained, in which a picture of confinement via non-abelian monopole condensation -- a generalized dual Meissner effect -- emerges explicitly. Similarities between supersymmetric and ordinary QCD are discussed, as is a non-supersymmetric QCD-like ``orbifold'' of N=1 Yang-Mills theory. I briefly discuss the recent discovery that gauge theories and string theories are more deeply connected than ever previously realized. Specific questions for lattice gauge theorists to consider are raised in the context of the first two topics. (Published in ``At the Frontier of Particle Physics: Handbook of QCD'', M. Shifman, editor) 
  Moller Scattering and Bhabha Scattering on noncommutative space-time is restudied. It is shown that the noncommutative correction of scattering cross sections is not monotonous enhancement with the total energy of colliding electrons, there is an optimum collision energy to get the greatest noncommutative correction. Most surprisingly, there is a linear relation between the noncommutative QED threshold energy and the optimum collision energy. 
  We study a modified non-linear Schroedinger equation on a 2 dimensional sphere with radius R aiming to describe electron-phonon interactions on fullerenes and fullerides. These electron-phonon interactions are known to be important for the explanation of the high transition temperature of superconducting fullerides. Like in the $R\to \infty$ limit, we are able to construct non-spinning as well as spinning solutions which are characterised by the number of nodes of the wave function. These solutions are closely related to the spherical harmonic functions. For small R, we discover specific branches of the solutions. Some of the branches survive in the $R\to\infty$ limit and the solutions obtained on the plane ($R=\infty$) are recovered. 
  We develop a gluing algorithm for Gromov-Witten invariants of toric Calabi-Yau threefolds based on localization and gluing graphs. The main building block of this algorithm is a generating function of cubic Hodge integrals of special form. We conjecture a precise relation between this generating function and the topological vertex at fractional framing. 
  We construct the Baxter Q-operator and the representation of the Separated Variables (SoV) for the homogeneous open SL(2,R) spin chain. Applying the diagrammatical approach, we calculate Sklyanin's integration measure in the separated variables and obtain the solution to the spectral problem for the model in terms of the eigenvalues of the Q-operator. We show that the transition kernel to the SoV representation is factorized into the product of certain operators each depending on a single separated variable. As a consequence, it has a universal pyramid-like form that has been already observed for various quantum integrable models such as periodic Toda chain, closed SL(2,R) and SL(2,C) spin chains. 
  We present a method where derivations of star-product algebras are used to build covariant derivatives for noncommutative gauge theory. We write down a noncommutative action by linking these derivations to a frame field induced by a nonconstant metric. An example is given where the action reduces in the classical limit to scalar electrodynamics on a curved background. We further use the Seiberg-Witten map to extend the formalism to arbitrary gauge groups. A proof of the existence of the Seiberg-Witten-map for an abelian gauge potential is given for the formality star-product. We also give explicit formulas for the Weyl ordered star-product and its Seiberg-Witten-maps up to second order. 
  After reviewing the general structure of supersymmetric intersecting brane world models, we discuss the issue of stringy gauge coupling unification for a natural class of MSSM-like models. 
  We explicitly construct a solution of eight-dimensional gauged supergravity representing D6-branes wrapped on six-cycles inside Calabi-Yau fourfolds. The solution preserves two supercharges and asymptotically is a cone with the coset space SU(2)^4/U(1)^3 as its base. It is shown to correspond to an M-theory compactification on a Calabi-Yau manifold with SU(5) holonomy and we discuss in detail its geometrical and topological features. We also construct a family of related higher dimensional metrics having SU(n+1) holonomy, which of course have no brane interpretation. 
  The recent discovery of non-perturbatively stable two-dimensional string backgrounds and their dual matrix models allows the study of complete scattering matrices in string theory. In this note we adapt work of Moore, Plesser, and Ramgoolam on the bosonic string to compute the exact S-matrices of 0A and 0B string theory in two dimensions. Unitarity of the 0B theory requires the inclusion of massless soliton sectors carrying RR scalar charge as asymptotic states. We propose a regularization of IR divergences and find transition probabilities that distinguish the otherwise energetically degenerate soliton sectors. Unstable D-branes can decay into distinct soliton sectors. 
  Numerous topics in three and four dimensional supersymmetric gauge theories are covered. The organizing principle in this presentation is scaling (Wilsonian renormalization group flow.) A brief introduction to scaling and to supersymmetric field theory, with examples, is followed by discussions of nonrenormalization theorems and beta functions. Abelian gauge theories are discussed in some detail, with special focus on three-dimensional versions of supersymmetric QED, which exhibit solitons, dimensional antitransmutation, duality, and other interesting phenomena. Many of the same features are seen in four-dimensional non-abelian gauge theories, which are discussed in the final sections. These notes are based on lectures given at TASI 2001. 
  Recently, we proposed a non-local relativistic formulation of MOND (Modified Newtonian Dynamics). The equations of motion were not derived, rather they were inferred from the result one would obtain by using the Schwinger-Keldysh formalism. The formalism simultaneously ensures causality of the field equations and the reality of in-out operator amplitudes. This point was avoided in our previous paper as its discussion was too far afield. Here we first demonstrate the features non-local actions generally possess: namely acausal equations of motion and non-real in-out operator amplitudes; and secondly how the Schwinger-Keldysh formalism works to provide the characteristics we usually desire from effective theories. 
  Large N geometric transitions and the Dijkgraaf-Vafa conjecture suggest a deep relationship between the sum over planar diagrams and Calabi-Yau threefolds. We explore this correspondence in details, explaining how to construct the Calabi-Yau for a large class of M-matrix models, and how the geometry encodes the correlators. We engineer in particular two-matrix theories with potentials W(X,Y) that reduce to arbitrary functions in the commutative limit. We apply the method to calculate all correlators <tr X^{p}> and <tr Y^{p}> in models of the form W(X,Y)=V(X)+U(Y)-XY and W(X,Y)=V(X)+YU(Y^{2})+XY^{2}. The solution of the latter example was not known, but when U is a constant we are able to solve the loop equations, finding a precise match with the geometric approach. We also discuss special geometry in multi-matrix models, and we derive an important property, the entanglement of eigenvalues, governing the expansion around classical vacua for which the matrices do not commute. 
  We present the covariant gravitational equations to describe a four-dimensional brane world in the case with the Gauss-Bonnet term in a bulk spacetime, assuming that gravity is confined on the $Z_2$ symmetric brane. It contains some components of five-dimensional Weyl curvature ($E_{\mu\nu}$) which describes all effects from the bulk spacetime just as in the case of the Randall-Sundrum second model. Applying this formalism to cosmology, we derive the generalized Friedmann equation and calculate the Weyl curvature term, which is directly obtained from a black hole solution. 
  We discuss that the Ginsparg-Wilson relation, which has the key role in the recent development of constructing lattice chiral gauge theory, can play an important role to define chiral structures in finite matrix models and noncommutative geometries. 
  Commutative supersymmetric Yang-Mills is known to be renormalizable for ${\cal N} = 1, 2$, while finite for ${\cal N} = 4$. However, in the noncommutative version of the model (NCSQED$_4$) the UV/IR mechanism gives rise to infrared divergences which may spoil the perturbative expansion. In this work we pursue the study of the consistency of NCSQED$_4$ by working systematically within the covariant superfield formulation. In the Landau gauge, it has already been shown for ${\cal N} = 1$ that the gauge field two-point function is free of harmful UV/IR infrared singularities, in the one-loop approximation. Here we show that this result holds without restrictions on the number of allowed supersymmetries and for any arbitrary covariant gauge. We also investigate the divergence structure of the gauge field three-point function in the one-loop approximation. It is first proved that the cancellation of the leading UV/IR infrared divergences is a gauge invariant statement. Surprisingly, we have also found that there exist subleading harmful UV/IR infrared singularities whose cancellation only takes place in a particular covariant gauge. Thus, we conclude that these last mentioned singularities are in the gauge sector and, therefore, do not jeopardize the perturbative expansion and/or the renormalization of the theory. 
  It is shown that the one-loop two-point amplitude in {\it Lorentz-invariant} non-commutative (NC) $\phi^3$ theory is finite after subtraction in the commutative limit and satisfies the usual cutting rule, thereby eliminating the unitarity problem in Lorentz-non-invariant NC field theory in the approximation considered. 
  In N=1 supersymmetric SO(N)/USp(2N) gauge theories with the tree-level superpotential W(\Phi) that is an arbitrary polynomial of the adjoint matter \Phi, the massless fluctuations about each quantum vacuum are described by U(1)^n gauge theory. By turning on the parameters of W(\Phi) to the special values, the singular vacua where the additional fields become massless can be reached. Using the matrix model prescription, we study the intersections of n=0 and n=1 branches. The general formula for the matrix model curve at the singularity which is valid for arbitrary N is obtained and this generalizes the previous results for small values of N from strong-coupling approach. Applying the analysis to the degenerated case, we also obtain a general matrix model curve which is not only valid at a special point but also on the whole branch. 
  The study of Skyrmions predicts that there is an icosahedrally symmetric charge seventeen SU(2) Yang-Mills instanton in which the topological charge density, for fixed Euclidean time, is localized on the edges of the truncated icosahedron of the buckyball. In this paper the existence of such an instanton is proved by explicit construction of the associated ADHM data. A topological charge density isosurface is displayed which verifies the buckyball structure of the instanton. 
  Intersecting D-brane worlds provide phenomenologically appealing constructions of four dimensional low energy string vacua. In this talk, a Z_4 x Z_2 orbifold background is taken into account. It is possible to obtain supersymmetric and stable chiral models. In particular, a three generation model with Pati-Salam gauge group and no exotic chiral matter is presented. 
  It is pointed out that the usual $\theta$-algebra assumed for non-commuting coordinates is not $P$- and $T$-invariant, unless one {\it formally} transforms the non-commutativity parameter $\theta^{\mu\nu}$ in an appropriate way. On the other hand, the Lorentz-covariant DFR algebra, which `relativitizes' the $\theta$-algebra by replacing $\theta^{\mu\nu}$ with a second-rank antisymmetric tensor operator $\htheta^{\mu\nu}$, is $C$-, $ P$- and $T$-invariant. It is then proved that $C, P$ and $T$ are separately conserved in Lorentz-invariant Non-Commutative QED. 
  We derive the anomalous transformation law of the quantum stress tensor for a 2D massless scalar field coupled to an external dilaton. This provides a generalization of the Virasoro anomaly which turns out to be consistent with the trace anomaly. We apply it together with the equivalence principle to compute the expectation values of the covariant quantum stress tensor on a curved background. Finally we briefly illustrate how to evaluate vacuum polarization and Hawking radiation effects from these results. 
  We obtain spinning and rotating closed string solutions in AdS_5 \times T^{1,1} background, and show how these solutions can be mapped onto rotating closed strings embedded in configurations of intersecting branes in type IIA string theory. Then, we discuss spinning closed string solutions in the UV limit of the Klebanov-Tseytlin background, and also properties of classical solutions in the related intersecting brane constructions in the UV limit. We comment on extensions of this analysis to the deformed conifold background, and in the corresponding intersecting brane construction, as well as its relation to the deep IR limit of the Klebanov-Strassler solution. We briefly discuss on the relation between type IIA brane constructions and their related M-theory descriptions, and how solitonic solutions are related in both descriptions. 
  We study the dimensional reduction of fermions, both in the symmetric and in the broken phase of the 3-d Gross-Neveu model at large N. In particular, in the broken phase we construct an exact solution for a stable brane world consisting of a domain wall and an anti-wall. A left-handed 2-d fermion localized on the domain wall and a right-handed fermion localized on the anti-wall communicate with each other through the 3-d bulk. In this way they are bound together to form a Dirac fermion of mass m. As a consequence of asymptotic freedom of the 2-d Gross-Neveu model, the 2-d correlation length \xi = 1/m increases exponentially with the brane separation. Hence, from the low-energy point of view of a 2-d observer, the separation of the branes appears very small and the world becomes indistinguishable from a 2-d space-time. Our toy model provides a mechanism for brane stabilization: branes made of fermions may be stable due to their baryon asymmetry. Ironically, our brane world is stable only if it has an extreme baryon asymmetry with all states in this ``world'' being completely filled. 
  We discuss thermalization in de Sitter space and argue, from two different points of view, that the typical time needed for thermalization is of order $R^{3}/l_{pl}^{2}$, where $R$ is the radius of the de Sitter space in question. This time scale gives plenty of room for non-thermal deviations to survive during long periods of inflation. We also speculate in more general terms on the meaning of the time scale for finite quantum systems inside isolated boxes, and comment on the relation to the Poincar\'{e} recurrence time. 
  We illustrate a method for computing the number of physical states of open string theory at the stable tachyonic vacuum in level truncation approximation. The method is based on the analysis of the gauge-fixed open string field theory quadratic action that includes Fadeev-Popov ghost string fields. Computations up to level 9 in the scalar sector are consistent with Sen's conjecture about the absence of physical open string states at the tachyonic vacuum. We also derive a long exact cohomology sequence that relates relative and absolute cohomologies of the BRS operator at the non-perturbative vacuum. We use this exact result in conjunction with our numerical findings to conclude that the higher ghost number non-perturbative BRS cohomologies are non-empty. 
  We quantize non-commutative Maxwell theory canonically in the background field gauge for weak and slowly varying background fields. We determine the complete basis for expansion under such an approximation. As an application, we derive the Wigner function which determines the leading order high temperature behavior of the perturbative amplitudes of non-commutative Maxwell theory. To leading order, we also give a closed form expression for the distribution function for the non-commutative $U (1)$ gauge theory at high temperature. 
  We show that it is possible to formulate gravity with a complex vierbein based on SL(2,C) gauge invariance. The proposed action is a four-form where the metric is not introduced but results as a function of the complex vierbein. This formulation is based on the first order formalism. The novel feature here is that integration of the spin-connection gauge field gives rise to kinetic terms for a massless graviton, a massive graviton with the Fierz-Pauli mass term, and a scalar field. The resulting theory is equivalent to bigravity. We then show that by extending the gauge group to GL(2,C} the formalism can be easily generalized to apply to a noncommutative space with the star product. We give the deformed action and derive the Seiberg-Witten map for the complex vierbein and gauge fields. 
  We consider a generic scenario of spontaneous breaking of supersymmetry in the hidden sector within N=1 supersymmetric orientifold compactifications of type II string theories with D-branes that support semi-realistic chiral gauge theories. The soft breaking terms in the visible sector of the models are computed in a standard way without specifying the breaking mechanism, which leads to expressions that generalize those formerly known for heterotic or type I string models. The elements of the effective tree level supergravity action relevant for this, such as the Kahler metric for the matter fields, the superpotential of the visible sector and the gauge kinetic functions, are specified by dimensional reduction and duality arguments. As phenomenological applications we argue that gauge coupling unification can only occur in special regions of the moduli space; we show that flavor changing neutral currents can be suppressed sufficiently for a wide range of parameters, and we briefly address the issues of CP violation, electric dipole moments and dark matter, as well. 
  We propose that the double scaling behavior of the unitary matrix models, and that of the complex matrix models, is related to type 0B and 0A fermionic string theories. The particular backgrounds involved correspond to $\hat c<1 $ matter coupled to super-Liouville theory. We examine in detail the $\hat c=0$ or pure supergravity case, which is related to the double scaling limit around the Gross-Witten transition, and find that reversing the sign of the Liouville superpotential interchanges the 0A and 0B theories. We also find smooth transitions between weakly coupled string backgrounds with D-branes, and backgrounds with Ramond-Ramond fluxes only. Finally, we discuss matrix models with multicritical potentials that are conjectured to correspond to 0A/0B string theories based on $(2, 4k)$ super-minimal models. 
  We present a non--perturbative study of the (1+1)--dimensional massless Thirring model by using path integral methods. The regularization ambiguities -coming from the computation of the fermionic determinant- allow to find new solution types for the model. At quantum level the Ward identity for the 1PI 2-point function for the fermionic current separates such solutions in two phases or sectors, the first one has a local gauge symmetry that is implemented at quantum level and the other one without this symmetry. The symmetric phase is a new solution which is unrelated to the previous studies of the model and, in the non--symmetric phase there are solutions that for some values of the ambiguity parameter are related to well-known solutions of the model. We construct the Schwinger--Dyson equations and the Ward identities. We make a detailed analysis of their UV divergence structure and, after, we perform a non--perturbative regularization and renormalization of the model. 
  We examine recent claims of a large set of flux compactification solutions of string theory. We conclude that the arguments for AdS solutions are plausible. The analysis of meta-stable dS solutions inevitably leads to situations where long distance effective field theory breaks down. We then examine whether these solutions are likely to lead to a description of the real world. We conclude that one must invoke a strong version of the anthropic principle. We explain why it is likely that this leads to a prediction of low energy supersymmetry breaking, but that many features of anthropically selected flux compactifications are likely to disagree with experiment. 
  We describe a block-spin-like transformation on a simplified subset of the space of supersymmetric quiver gauge theories that arise on the worldvolumes of D-brane probes of orbifold geometries, by sequentially Higgsing the gauge symmetry in these theories. This process flows to lower worldvolume energies in the regions of the orbifold moduli space where the closed string blowup modes, and therefore the expectation values of the bifundamental scalars, exhibit a hierarchy of scales. Lifting to the ``upstairs'' matrices of the image branes makes contact with the matrix coarse-graining defined in a previous paper. We describe the structure of flows we observe under this process. The quiver lattice for $\BC^2/\BZ_N$ in this region of moduli space deconstructs an inhomogeneous, fractal-like extra dimension, in terms of which our construction describes a coarse-graining of the deconstruction lattice. 
  A two-parameter deformed superoscillator system with SUq1/q2(n|m)-covariance is presented and used to construct a two-parameter deformed N=2 SUSY algebra. The Fock space representation of the algebra is discussed and the deformed Hamiltonian for such generalized superoscillators is obtained. 
  The term higher gauge theory refers to the generalization of gauge theory to a theory of connections at two levels, essentially given by 1- and 2-forms. So far, there have been two approaches to this subject. The differential picture uses non-Abelian 1- and 2-forms in order to generalize the connection 1-form of a conventional gauge theory to the next level. The integral picture makes use of curves and surfaces labeled with elements of non-Abelian groups and generalizes the formulation of gauge theory in terms of parallel transports. We recall how to circumvent the classic no-go theorems in order to define non-Abelian surface ordered products in the integral picture. We then derive the differential picture from the integral formulation under the assumption that the curve and surface labels depend smoothly on the position of the curves and surfaces. We show that some aspects of the no-go theorems are still present in the differential (but not in the integral) picture. This implies a substantial structural difference between non-perturbative and perturbative approaches to higher gauge theory. We finally demonstrate that higher gauge theory provides a geometrical explanation for the extended topological symmetry of BF-theory in both pictures. 
  We review the last year progress in understanding supersymmetric SU(2) Yang-Mills quantum mechanics in four and ten space-time dimensions. The four dimensional system is now well under control and the precise spectrum is obtained in all channels. In D=10 some new results are also available. 
  In light of WMAP results, we examine the observational constraint on the P-term inflation. With the tunable parameter $f$, P-term inflation contains richer physics than D-term and F-term inflationary models. We find the logarithmic derivative spectral index with $n>1$ on large scales and $n<1$ on small scales in agreement to observation. We obtained a reasonable range for the choice of the gauge coupling constant $g$ in order to meet the requirements of WMAP observation and the expected number of the e-foldings. Although tuning $f$ and $g$ we can have larger values for the logarithmic derivative of the spectral index, it is not possible to satisfy all observational requirements for both, the spectral index and its logarithmic derivative at the same time. 
  We calculate the vacuum expectation value of the stress-energy tensor for a quantised bulk scalar field in the Randall-Sundrum model, and discuss the consequences of its local behaviour for the self-consistency of the model. We find that, in general, the stress-energy tensor diverges in the vicinity of the branes. Our main conclusion is that the stress-energy tensor is sufficiently complicated that it has implications for the effective potential, or radion stabilisation, methods that have so far been used. 
  We show that there is a natural action of SL(2,Z) on the two-point functions of the energy momentum tensor and of higher-spin conserved currents in three-dimensional CFTs. The dynamics behind the S-operation of SL(2,Z) is that of an irrelevant current-current deformation and we point out its similarity to the dynamics of a wide class of three-dimensional CFTs. The holographic interpretation of our results raises the possibility that many three-dimensional CFTs have duals on AdS4 with SL(2,Z) duality properties at the linearized level. 
  In the $n-$dimensional Freund-Rubin model with an antisymmetric tensor field of rank $s-1$, the dimension of the external spacetime we live in must be $min(s, n-s)$. This result is a generalization of the previous result in the $d=11$ supergravity case, where $s = 4$. 
  We study a close relationship between the topological anti-de Sitter (TAdS)-black holes and topological de Sitter (TdS) spaces including the Schwarzschild-de Sitter (SdS) black hole in five-dimensions. We show that all thermal properties of the TdS spaces can be found from those of the TAdS black holes by replacing $k$ by $-k$. Also we find that all thermal information for the cosmological horizon of the SdS black hole is obtained from either the hyperbolic-AdS black hole or the Schwarzschild-TdS space by substituting $m$ with $-m$. For this purpose we calculate thermal quantities of bulk, (Euclidean) conformal field theory (ECFT) and moving domain wall by using the A(dS)/(E)CFT correspondences. Further we compute logarithmic corrections to the Bekenstein-Hawking entropy, Cardy-Verlinde formula and Friedmann equation due to thermal fluctuations. It implies that the cosmological horizon of the TdS spaces is nothing but the event horizon of the TAdS black holes and the dS/ECFT correspondence is valid for the TdS spaces in conjunction with the AdS/CFT correspondence for the TAdS black holes. 
  By solving the two variable differential equations which arise from finding the eigenfunctions for the Casimir operator for $O(d,2)$ succinct expressions are found for the functions, conformal partial waves, representing the contribution of an operator of arbitrary scale dimension $\Delta$ and spin $\ell$ together with its descendants to conformal four point functions for $d=4$, recovering old results, and also for $d=6$. The results are expressed in terms of ordinary hypergeometric functions of variables $x,z$ which are simply related to the usual conformal invariants. An expression for the conformal partial wave amplitude valid for any dimension is also found in terms of a sum over two variable symmetric Jack polynomials which is used to derive relations for the conformal partial waves. 
  The symmetry between the creation of pairs of massless bosons or fermions by accelerated mirror in 1+1 space and the emission of single photons or scalar quanta by electric or scalar charge in 3+1 space is deepened in this paper. The relation of Bogoliubov coefficients with Fourier's components of current or charge density leads to the coicidence of the spin of any disturbances bilinear in scalar or spinor field with the spin of quanta emitted by the electric or scalar charge. The mass and invariant momentum transfer of these disturbances are essential for the relation of Bogoliubov coefficients with Green's functions of wave equations both for 1+1 and 3+1 spaces. Namely the relation (20) leads to the coincidence of the self-action changes and vacuum-vacuum amplitudes for the accelerated mirror in 1+1 space and charge in 3+1 space. Thus, both invariants of the Lorentz group, spin and mass, perform intrinsic role in established symmetry. The symmetry embraces not only the processes of real quanta radiation. It extends also to the processes of the mirror and the charge interactions with the fields carring spacelike momenta. These fields accompany their sources and define the Bogoliubov matrix coefficients \alpha^{B,F}. It is shown that the traces of \alpha^{B,F} describe the vector and scalar interactions of accelerated mirror with uniformly moving detector. This interpretation rests essentially on the relation (100) between the propagators of the waves with spacelike momenta in 2- and 4-dimentional spaces. The traces of \alpha^{B,F} coincide actually with the products of the mass shift \Delta m_{1,0} of accelerated electric or scalar charge and the proper time of the shift formation. The symmetry fixes the value of the bare fine structure constant \alpha_0=1/4\pi. 
  The UV/IR mixing in the \lambda \phi^4 model on a non-commutative (NC) space leads to new predictions in perturbation theory, including Hartree-Fock type approximations. Among them there is a changed phase diagram and an unusual behavior of the correlation functions. In particular this mixing leads to a deformation of the dispersion relation. We present numerical results for these effects in d=3 with two NC coordinates. 
  We show that the construction of super-Calogero model with OSp(2|2) supersymmetry is not unique. In particular, we find a new co-ordinate representation of the generators of the OSp(2|2) superalgebra that appears as the dynamical supersymmetry of the rational super-Calogero model. Both the quadratic and the cubic Casimir operators of the OSp(2|2) are necessarily zero in this new representation, while they are, in general, nonzero for the super-Calogero model that is currently studied in the literature. The Scasimir operator that exists in the new co-ordinate representation is not present in the case of the existing super-Calogero model. We also discuss the case of N free superoscillators and superconformal quantum mechanics for which the same conclusions are valid. 
  Weak coherent states share many properties of the usual coherent states, but do not admit a resolution of unity expressed in terms of a local integral. They arise e.g. in the case that a group acts on an inadmissible fiducial vector. Motivated by the recent Affine Quantum Gravity Program, the present article studies the path integral representation of the affine weak coherent state matrix elements of the unitary time-evolution operator. Since weak coherent states do not admit a resolution of unity, it is clear that the standard way of constructing a path integral, by time slicing, is predestined to fail. Instead a well-defined path integral with Wiener measure, based on a continuous-time regularization, is used to approach this problem. The dynamics is rigorously established for linear Hamiltonians, and the difficulties presented by more general Hamiltonians are addressed. 
  The electrostatic configurations of the Born-Infeld field in the 2-dimensional Euclidean plane are obtained by means of a non-analytical complex mapping which captures the structure of equipotential and field lines. The electrostatic field reaches the Born-Infeld limit value when the field lines become tangent to an epicycloid around the origin. The total energy by unit of length remains finite. 
  The q-deformed traces and orbits for the two parametric quantum groups $GL_{qp}(2)$ and $GL_{qp}(1|1)$ are defined. They are subsequently used in the construction of $q$-orbit invariants for these groups. General $qp$-(super)oscillator commutation relations are obtained which remain invariant under the coactions of groups $GL_{qp}(2)$ and $GL_{qp}(1|1)$. The $GL_{qp}(2)$ covariant deformed algebra is deduced in terms of the bilinears of bosonic $qp$-oscillators which turns out to be a central extension of the Witten-type deformation of $sl(2)$ algebra. In the case of the supergroup $GL_{qp}(1|1)$, the corresponding covariant algebras contain $N = 2$ supersymmetric quantum mechanical subalgebras. 
  We propose a new mechanism for obtaining de Sitter vacua in type IIB string theory compactified on (orientifolded) Calabi-Yau manifolds similar to those recently studied by Kachru, Kallosh, Linde and Trivedi (KKLT). dS vacuum appears in KKLT model after uplifting an AdS vacuum by adding an anti-D3-brane, which explicitly breaks supersymmetry. We accomplish the same goal by adding fluxes of gauge fields within the D7-branes, which induce a D-term potential in the effective 4D action. In this way we obtain dS space as a spontaneously broken vacuum from a purely supersymmetric 4D action. We argue that our approach can be directly extended to heterotic string vacua, with the dilaton potential obtained from a combination of gaugino condensation and the D-terms generated by anomalous U(1) gauge groups. 
  We compute the three loop $\beta$ function of the Wess-Zumino model to motivate implicit regularization (IR) as a consistent and practical momentum-space framework to study supersymmetric quantum field theories. In this framework which works essentially in the physical dimension of the theory we show that ultraviolet are clearly disantangled from infrared divergences. We obtain consistent results which motivate the method as a good choice to study supersymmetry anomalies in quantum field theories. 
  In this paper we present a solution for Kaluza-Klein magnetic monopole in a five-dimensional global monopole spacetime. This new solution is a generalization of previous ones obtained by D. Gross and M. Perry (Nucl. Phys. B {\bf 226}, 29 (1983)) containing a magnetic monopole in a Ricci-flat formalism, and by A. Banerjee, S. Charttejee and A. See (Class. Quantum Grav. {\bf 13}, 3141 (1996)) for a global monopole in a five-dimensional spacetime, setting zero specific integration constant. Also we analyse the classical motion of a massive charged test particle on this manifold and present the equation for classical trajectory obeyed by this particle. 
  In backgrounds with compact dimensions there may exist several phases of black objects including the black-hole and the black-string. The phase transition between them raises puzzles and touches fundamental issues such as topology change, uniqueness and Cosmic Censorship. No analytic solution is known for the black hole, and moreover, one can expect approximate solutions only for very small black holes, while the phase transition physics happens when the black hole is large. Hence we turn to numerical solutions. Here some theoretical background to the numerical analysis is given, while the results will appear in a forthcoming paper. Goals for a numerical analysis are set. The scalar charge and tension along the compact dimension are defined and used as improved order parameters which put both the black hole and the black string at finite values on the phase diagram. Predictions for small black holes are presented. The differential and the integrated forms of the first law are derived, and the latter (Smarr's formula) can be used to estimate the ``overall numerical error''. Field asymptotics and expressions for physical quantities in terms of the numerical ones are supplied. Techniques include ``method of equivalent charges'', free energy, dimensional reduction, and analytic perturbation for small black holes. 
  We study Seiberg duality of quiver gauge theories associated to the complex cone over the second del Pezzo surface. Homomorphisms in the path algebra of the quivers in each of these cases satisfy relations which follow from a superpotential of the corresponding gauge theory as F-flatness conditions. We verify that Seiberg duality between each pair of these theories can be understood as a derived equivalence between the categories of modules of representation of the path algebras of the quivers. Starting from the projective modules of one quiver we construct tilting complexes whose endomorphism algebra yields the path algebra of the dual quiver. Finally, we present a general scheme for obtaining Seiberg dual quiver theories by constructing quivers whose path algebras are derived equivalent. We also discuss some combinatorial relations between this approach and some of the other approaches which has been used to study such dualities. 
  In this paper we complement our recent result on the explicit formula for the planar limit of the free energy of the two-matrix model by computing the second and third order observables of the model in terms of canonical structures of the underlying genus g spectral curve. In particular we provide explicit formulas for any three-loop correlator of the model. Some explicit examples are worked out. 
  The three-dimensional noncommutative supersymmetric QED is investigated within the superfield approach. We prove the absence of UV/IR mixing in the theory at any loop order and demonstrate its one-loop finiteness. 
  Ruijsenaars-Schneider models associated with $A_{n-1}$ root system with a discrete coupling constant are studied. The eigenvalues of the Hamiltonian are givein in terms of the Bethe ansatz formulas. Taking the "non-relativistic" limit, we obtain the spectrum of the corresponding Calogero-Moser systems in the third formulas of Felder et al [20]. 
  We study N=2 supersymmetric gauge theories with $d=6$ bulk and $d=4$ brane fields charged under a U(1) gauge symmetry. Radiatively induced Fayet-Iliopoulos terms lead to an instability of the bulk fields. We compute the profile of the bulk zero modes and observe the phenomenon of spontaneous localization towards the position of the branes. While this mechanism is quite similar to the $d=5$ case, the mass spectrum of the excited Kaluza-Klein modes shows a crucial difference. 
  We define the oscillator and Coulomb systems on four-dimensional spaces with U(2)-invariant Kahler metric and perform their Hamiltonian reduction to the three-dimensional oscillator and Coulomb systems specified by the presence of Dirac monopoles. We find the Kahler spaces with conic singularity, where the oscillator and Coulomb systems on three-dimensional sphere and two-sheet hyperboloid are originated. Then we construct the superintegrable oscillator system on three-dimensional sphere and hyperboloid, coupled to monopole, and find their four-dimensional origins. In the latter case the metric of configuration space is non-Kahler one. Finally, we extend these results to the family of Kahler spaces with conic singularities. 
  Trajectories in the space of the unitarily inequivalent representations of the canonical commutation relations are shown to be classical trajectories. Under convenient conditions, they may exhibit properties typical of chaotic behavior in classical nonlinear dynamics. Quantum noise in fluctuating random force in the system--environment coupling and system--environment entanglement is also discussed. 
  We analyse the very-extended Kac-Moody algebras as representations in terms of certain A_d subalgebras and find the generators at low levels. Our results for low levels agree precisely with the bosonic field content of the theories containing gravity, forms and scalars which upon reduction to three dimensions can be described by a non-linear realisation. We explain how the Dynkin diagrams of the very-extended algebras encode information about the field content and generalised T-duality transformations. 
  We point out a close relation between a family of Goedel-type solutions of 3+1 General Relativity and the Landau problem in S^2, R^2 and H_2; in particular, the classical geodesics correspond to Larmor orbits in the Landau problem. We discuss the extent of this relation, by analyzing the solutions of the Klein-Gordon equation in these backgrounds. For the R^2 case, this relation was independently noticed in hep-th/0306148. Guided by the analogy with the Landau problem, we speculate on the possible holographic description of a single chronologically safe region. 
  We demonstrate that by employing the correspondence between gauge theories in geometric and in deconstructed extra dimensions, it is possible to transfer the methods for calculating finite Casimir energy densities in higher dimensions to the four-dimensional deconstruction setup. By this means, one obtains an unambiguous and well-defined prescription to determine finite vacuum energy contributions of four-dimensional quantum fields which have a higher-dimensional correspondence. Thereby, large kink masses lead to an exponentially suppressed Casimir effect. For a specific model we hence arrive at a small and positive contribution to the cosmological constant in agreement with observations. 
  We map out and explore the zoo of possible 4d N=1 superconformal theories which are obtained as RG fixed points of N=1 SQCD with N_f fundamental and N_a adjoint matter representations. Using "a-maximization," we obtain exact operator dimensions at all RG fixed points and classify all relevant, Landau-Ginzburg type, adjoint superpotential deformations. Such deformations can be used to RG flow to new SCFTs, which are then similarly analyzed. Remarkably, the resulting 4d SCFT classification coincides with Arnold's ADE singularity classification. The exact superconformal R-charge and the central charge a are computed for all of these theories. RG flows between the different fixed points are analyzed, and all flows are verified to be compatible with the conjectured a-theorem. 
  By directly solving the equations of motion we obtain the time dependent solutions of supergravities with dilaton and a $q$-form field-strength in arbitrary dimensions. The metrics are assumed to have the symmetries ISO($p+1$) $\times$ SO($d-p-2,1$) and can be regarded as those of the magnetically charged Euclidean or space-like branes. When we impose the extremality condition, we find that the magnetic charges of the branes become imaginary and the corresponding real solutions then represent the E$p$-branes of type II$^\ast$ theories (for the field-strengths belonging to the RR sector). On the other hand, when the extremality condition is relaxed we find real solutions in type II theories which resemble the solutions found by Kruczenski-Myers-Peet. In $d=10$ they match exactly. We point out the relations between the solutions found in this paper and those of Chen-Gal'tsov-Gutperle in arbitrary dimensions. Although there is no extremal limit for these solutions, we find another class of solutions, which resemble the solutions in the extremal case with imaginary magnetic charges and the corresponding real solutions can be regarded as the non-BPS E$p$-brane solutions of type II$^\ast$ theories (for the field-strengths in RR sector). 
  We investigate the gravitational energy emission of an ultrarelativistic particle radially falling into a D-dimensional black hole. We numerically integrate the equations describing black hole gravitational perturbations and obtain energy spectra, total energy and angular distribution of the emitted gravitational radiation. The black hole quasinormal modes for scalar, vector, and tensor perturbations are computed in the WKB approximation. We discuss our results in the context of black hole production at the TeV scale. 
  Extension of Feynman's path integral to quantum mechanics of noncommuting spatial coordinates is considered. The corresponding formalism for noncommutative classical dynamics related to quadratic Lagrangians (Hamiltonians) is formulated. Our approach is based on the fact that a quantum-mechanical system with a noncommutative configuration space may be regarded as another effective system with commuting spatial coordinates. Since path integral for quadratic Lagrangians is exactly solvable and a general formula for probability amplitude exists, we restricted our research to this class of Lagrangians. We found general relation between quadratic Lagrangians in their commutative and noncommutative regimes. The corresponding noncommutative path integral is presented. This method is illustrated by two quantum-mechanical systems in the noncommutative plane: a particle in a constant field and a harmonic oscillator. 
  The chaotic inflationary model of the early universe, proposed by Linde is explored in the brane world considering matter described by a minimally coupled self interacting scalar field. We obtain cosmological solutions which admit evolution of a universe either from a singularity or without a singularity. It is found that a very weakly coupled self-interacting scalar field is necessary for a quartic type potential in the brane world model compared to that necessary in general relativity. In the brane world sufficient inflation may be obtained even with an initial scalar field having value less than the Planck scale. It is found that if the universe is kinetic energy dominated to begin with, it transits to an inflationary stage subsequently. 
  Non-supersymmetric Yang-Mill gauge theory in 4-dimension is shown to be dual to 4-dimensional non-supersymmetric string theory in a twisted AdS2(n)xT2 spacetime background. The partition function of a generic hadron is calculated to illustrate the mathematical structure of the twisted QCD topology. The physical implications of the twisted QCD topology are discussed. 
  The multi-centre metrics are a family of euclidean solutions of the empty space Einstein equations with self-dual curvature. For this full class, we determine which metrics do exhibit an extra conserved quantity quadratic in the momenta, induced by a Killing-St\" ackel tensor. Our systematic approach brings to light a subclass of metrics which correspond to new classically integrable dynamical systems. Within this subclass we analyze on the one hand the separation of coordinates in the Hamilton-Jacobi equation and on the other hand the construction of some new Killing-Yano tensors. 
  We propose a new duality involving topological strings in the limit of large string coupling constant.   The dual is described in terms of a classical statistical mechanical model of crystal melting, where the temperature is inverse of the string coupling constant. The crystal is a discretization of the toric base of the Calabi-Yau with lattice length $g_s$. As a strong evidence for this duality we recover the topological vertex in terms of the statistical mechanical probability distribution for crystal melting.  We also propose a more general duality involving the dimer problem on periodic lattices and topological A-model string on arbitrary local toric threefolds. The $(p,q)$ 5-brane web, dual to Calabi-Yau, gets identified with the transition regions of rigid dimer configurations. 
  We review the proof of a conjecture concerning the reality of the spectra of certain PT-symmetric quantum mechanical systems, obtained via a connection between the theories of ordinary differential equations and integrable models. Spectral equivalences inspired by the correspondence are also discussed. 
  This paper is concerned with magnetic monopole solutions of SU(3) Yang-Mills-Higgs system beyond the Bogomol'nyi-Prasad-Sommerfield limit. The different SU(2) embeddings, which correspond to the fundamental monopoles, as well the embedding along composite root are studied. The interaction of two different fundamental monopoles is considered. Dissolution of a single fundamental non-BPS SU(3) monopole in the limit of the minimal symmetry breaking is analysed. 
  Liouville field theory on the pseudosphere is considered (Dirichlet conditions). We compute explicitely the bulk-boundary structure constant with two different methods: first we use a suggestion made by Hosomichi in JHEP 0111 (2001) that relates this quantity directly to the bulk-boundary structure constant with Neumann conditions, then we do a direct computation. Agreement is found. 
  We analyze the Landau problem and quantum Hall effect on $S^3$ taking a constant background field proportional to the spin connection on $S^3$. The effective strength of the field can be tuned by changing the dimension of the representation to which the fermions belong. The effective action for the edge excitations of a quantum Hall droplet in the limit of a large number of fermions is obtained. We find that the appropriate space for many of these considerations is $S^2 \times S^2$, which plays a role similar to that of ${\bf CP}^3$ vis-a-vis $S^4$. We also give a method of representing the algebra of functions on fuzzy $S^3/{\bf Z}_2$ in terms of finite dimensional matrices. 
  We show that long-time, long-distance fluctuations of plane-symmetric horizons exhibit universal hydrodynamic behavior. By considering classical fluctuations around black-brane backgrounds, we find both diffusive and shear modes. The diffusion constant and the shear viscosity are given by simple formulas, in terms of metric components. For a given metric, the answers can be interpreted as corresponding kinetic coefficients in the holographically dual theory. For the near-extremal Dp, M2 and M5 branes, the computed kinetic coefficients coincide with the results of independent AdS/CFT calculations. In all the examples, the ratio of shear viscosity to entropy density is equal to \hbar/(4\pi k_B), suggesting a special meaning of this value. 
  The relation between coordinates and the solutions of the stationary Schrodinger equation in the noncommutative algebra of functions on $R^{2N}$ is discussed. We derive this relation for a certain class of wave functions for which the quantum prepotentials depend linearly on the coordinates similarly to the commutative case. Also, the differential equation satisfied by the prepotentials is given. 
  We consider different M2-brane configurations in the M-theory AdS_7xS^4 background, with field theory dual A_{N-1}(2,0) SCFT. New membrane solutions are found and compared with the recently obtained ones. 
  In the recent years, field theory on non-commutative (NC) spaces has attracted a lot of attention. Most literature on this subject deals with perturbation theory, although the latter runs into grave problems beyond one loop. Here we present results from a fully non-perturbative approach. In particular, we performed numerical simulations of the \lambda \phi^{4} model with two NC spatial coordinates, and a commutative Euclidean time. This theory is lattice discretized and then mapped onto a matrix model. The simulation results reveal a phase diagram with various types of ordered phases. We discuss the suitable order parameters, as well as the spatial and temporal correlators. The dispersion relation clearly shows a trend towards the expected IR singularity. Its parameterization provides the tool to extract the continuum limit. 
  We study the 3-form flux $H_{\m\n\l}$ associated with the semi-classical geometry of $G/H$ gauged WZW models. We derive a simple, general expression for the flux in an orthonormal frame and use it to explicitly verify conformal invariance to the leading order in $\a'$. For supersymmetric models, we briefly revisit the conditions for enhanced supersymmetry. We also discuss some examples of non-abelian cosets with flux. 
  It has been proposed that the entropy of any object must satisfy fundamental (holographic or Bekenstein) bounds set by the object's size and perhaps its energy. However, most discussions of these bounds have ignored the possibility that objects violating the putative bounds could themselves become important components of Hawking radiation. We show that this possibility cannot a priori be neglected in existing derivations of the bounds. Thus this effect could potentially invalidate these derivations; but it might also lead to observational evidence for the bounds themselves. 
  We study the space-time invariances of the bosonic relativistic particle and bosonic relativistic string using general formulations obtained by incorporating the Hamiltonian constraints into the formalism. We point out that massless particles and tensionless strings have a larger set of space-time invariances than their massive and tensionful partners, respectively. We also show that it is possible to use the reparametrization invariance of the string formulation we present to reach the classical conformal equations of motion without the use of two-dimensional Weyl scalings of the string world sheet. Finally, we show that it is possible to fix a gauge with an enlarged number of space-time invariances in which every point of the free tensionless string moves as if it were an almost-free massless particle. The existence of such a string motion agrees with what is expected from gauge theory-string duality. 
  We expand our previous analysis on fivebrane and membrane instanton solutions in the universal hypermultiplet, including near-extremal multi-centered solutions and mixed fivebrane-membrane charged instantons. The results are most conveniently described in terms of a double-tensor multiplet. 
  Motivated by the study of quantum fields in a Friedman-Robertson-Walker (FRW) spacetime, the one-loop effective action for a scalar field defined in the ultrastatic manifold $R\times H^3/\Gamma$, $H^3/\Gamma$ being the finite volume, non-compact, hyperbolic spatial section, is investigated by a generalisation of zeta-function regularisation. It is shown that additional divergences may appear at one-loop level. The one-loop renormalisability of the model is discussed and making use of a generalisation of zeta-function regularisation, the one-loop renormalisation group equations are derived. 
  The covariant quantization of the tensionless free bosonic (open and closed) strings in AdS spaces is obtained. This is done by representing the AdS space as an hyperboloid in a flat auxiliary space and by studying the resulting string constrained hamiltonian system in the tensionless limit. It turns out that the constraint algebra simplifies in the tensionless case in such a way that the closed BRST quantization can be formulated and the theory admits then an explicit covariant quantization scheme. This holds for any value of the dimension of the AdS space. 
  We clarify the discussion of N=2 supersymmetric boundary conditions for the classical d=2, N=(2,2) Non-Linear Sigma Model on an infinite strip. Our conclusions about the supersymmetric cycles match the results found in the literature. However, we find a constraint on the boundary action that is not satisfied by many boundary actions used in the literature. 
  Planarity was introduced by 't Hooft in his topological classification of diagrams in the large-N limit of U(N) gauge theories. Planarity also occurs in noncommutative field theories where amplitudes possess invariance only under cyclic permutations, a feature inherited from the parent string theory. In noncommutative gauge theories both kinds of planarity merge in a context which turns out to be particularly intriguing in the two-dimensional case where gauge invariant correlators can be explicitly computed. 
  The time dependent formation of an electric flux tube (fundamental string) is reviewed. The main tool used for analysis is the Spacelike brane, which is a kink in time of the rolling tachyon. Both the S-brane and rolling tachyon are attempts to extend the D-brane concept to time dependent backgrounds. While S-branes are similar to Euclidean counterparts of the more familiar timelike D-branes, S-branes can smoothly change their worldvolume signature from spacelike to timelike which we interpret as the formation of a topological defect. 
  We construct dual descriptions of (0,2) gauged linear sigma models. In some cases, the dual is a (0,2) Landau-Ginzburg theory, while in other cases, it is a non-linear sigma model. The duality map defines an analogue of mirror symmetry for (0,2) theories. Using the dual description, we determine the instanton corrected chiral ring for some illustrative examples. This ring defines a (0,2) generalization of the quantum cohomology ring of (2,2) theories. 
  We calculate analytically asymptotic values of quasi-normal frequencies of four-dimensional Kerr black holes by solving the Teukolsky wave equation. We obtain an expression for arbitrary spin of the wave in agreement with Hod's proposal which is based on Bohr's correspondence principle. However, the range of frequencies is bounded from above by $1/a$, where $a$ is the angular momentum per unit mass of the black hole. Our argument is only valid in the small-$a$ limit which includes the Schwarzschild case. 
  We show that the celebrated Painleve equations for the Ising correlation functions follow in a simple way from the Ward Identities associated with local Integrals of Motion of the doubled Ising field theory. We use these Ward Identities to derive the equations determining the matrix elements of the product $\sigma(x)\sigma(x')$ between any particle states. The result is then applied in evaluating the leading mass corrections in the Ising field theory perturbed by an external magnetic field. 
  We study two-dimensional supersymmetric non-linear sigma-models with boundaries. We derive the most general family of boundary conditions in the non-supersymmetric case. Next we show that no further conditions arise when passing to the N=1 model. We present a manifest N=1 off-shell formulation. The analysis is greatly simplified compared to previous studies and there is no need to introduce non-local superspaces nor to go (partially) on-shell. Whether or not torsion is present does not modify the discussion. Subsequently, we determine under which conditions a second supersymmetry exists. As for the case without boundaries, two covariantly constant complex structures are needed. However, because of the presence of the boundary, one gets expressed in terms of the other one and the remainder of the geometric data. Finally we recast some of our results in N=2 superspace and discuss applications. 
  We use the (M,n) phase diagram recently introduced in hep-th/0309116 to investigate the phase structure of black holes and strings on cylinders. We first prove that any static neutral black object on a cylinder can be put into an ansatz for the metric originally proposed in hep-th/0204047, generalizing a result of Wiseman. Using the ansatz, we then show that all branches of solutions obey the first law of thermodynamics and that any solution has an infinite number of copies. The consequences of these two results are analyzed. Based on the new insights and the known branches of solutions, we finally present an extensive discussion of the possible scenarios for the Gregory-Laflamme instability and the black hole/string transition. 
  The Born--Infeld theory of a D3-brane intersecting with(p,q) strings is reconsidered. From the assumption that the electromagnetic fields are those of a dyon, and using the kappa invariance of the action, the explicit scalar field and its charge are derived. Considering perturbations orthogonal to both branes, the SL(2,Z)-invariant S-matrix is obtained. Owing to the selfduality of the brane the latter can be evaluated explicitly in both high and low energy regions. 
  We discuss the Whitham deformation of the effective superpotential in the Dijkgraaf-Vafa (DV) theory. It amounts to discussing the Whitham deformation of an underlying (hyper)elliptic curve. Taking the elliptic case for simplicity we derive the Whitham equation for the period, which governs flowings of branch points on the Riemann surface. By studying the hodograph solution to the Whitham equation it is shown that the effective superpotential in the DV theory is realized by many different meromorphic differentials. Depending on which meromorphic differential to take, the effective superpotential undergoes different deformations. This aspect of the DV theory is discussed in detail by taking the N=1^* theory. We give a physical interpretation of the deformation parameters. 
  By explicit calculation of the effect of a ghost-dependent canonical transformation of BRST-charge, we derive the corresponding transformation law for structure coefficients of hamiltonian gauge algebra under rotation of constraints.We show the transformation law to deviate from the behaviour (expected naively) characteristic to a genuine connection. 
  We combine old ideas about exact renormalization-group-flow (RGF) equations with the Vilkovisky-De Witt (VDW) approach to reparametrization invariant effective actions and arrive at a new, exact, gauge-invariant RGF equation. The price to be paid for such a result is that both the action and the RGF equation depend explicitly upon the base point (in field space) needed for the VDW construction. We briefly discuss the complications originating from this fact and possible ways to overcome them. 
  We propose a massless nonminimally coupled scalar field as a mechanism for stabilizing the size of the extradimension in the Randall-Sundrum I scenario. Without needing to introduce self interactions terms we obtain a potential for the modulus field that sets the size of the fifth dimension. The minumum of this potential yields appropriate values of the compactification scale for small values of the coupling $\xi$. 
  Numerical solutions of Einstein, scalar, and gauge field equations are found for static and inflating defects in a higher-dimensional spacetime. The defects have $(3+1)$-dimensional core and magnetic monopole configuration in $n=3$ extra dimensions. For symmetry-breaking scale $\eta$ below the critical value $\eta_c$, the defects are characterized by a flat worldsheet geometry and asymptotically flat extra dimensions. The critical scale $\eta_c$ is comparable to the higher-dimensional Planck scale and has some dependence on the gauge and scalar couplings. For $\eta=\eta_c$, the extra dimensions degenerate into a `cigar', and for $\eta>\eta_c$ all static solutions are singular. The singularity can be removed if the requirement of staticity is relaxed and defect cores are allowed to inflate. The inflating solutions have de Sitter worldsheets and cigar geometry in the extra dimensions. Exact analytic solutions describing the asymptotic behavior of these inflating monopoles are found and the parameter space of these solutions is analyzed. 
  We develop a systematic algorithm to construct, classify and study exact solutions of type II A/B supergravity which are time--dependent and homogeneous and hence represent candidate cosmological backgrounds. Using the formalism of solvable Lie algebras to represent the geometry of non--compact coset manifolds U/H we are able to reduce the supergravity field equations to the geodesic equations in U/H and rephrase these latter in a completely algebraic setup by means of the so called Nomizu operator representation of covariant derivatives in solvable group manifolds. In this way a systematic method of integration of supergravity equations is provided. We show how the possible D=3 solutions are classified by non--compact subalgebras G of E8(8) and their ten--dimensional physical interpretation (oxidation) depends on the classification of the different embeddings of G inside E8(8). We give some preliminary examples of explicit solutions based on the simplest choice G=A2. We also show how, upon oxidation, these solutions provide a smooth and exact realization of the bouncing phenomenon on Weyl chamber walls envisaged by the cosmological billiards of Damour et al. We also show how this physical phenomenon is triggered by the presence of euclidean $D$--branes possibly interpretable at the microscopic level as S--branes. We outline how our analysis could be extended to a wider setup where, by further reducing to $D=2,1$, more general backgrounds could be constructed applying our method to the infinite algebras E9 and E10. 
  In this note, we first explain the equivalence between the interaction Hamiltonian of Green-Schwarz light-cone gauge superstring field theory and the twist field formalism known from matrix string theory. We analyze the role of the large N limit in matrix string theory, in particular in relation with conformal perturbation theory around the orbifold SCFT that reproduces light-cone string perturbation theory. We show how the scaling with N is directly related to measures on the moduli space of Riemann surfaces. The scaling dimension 3 of the Mandelstam vertex as reproduced by the twist field interaction is in this way related to the dimension 3(h-1) of the moduli space. We analyze the structure and scaling of the higher order twist fields that represent the contact terms. We find one relevant twist field at each order. More generally, the structure of string field theory seems more transparent in the twist field formalism. Finally we also investigate the modifications necessary to describe the pp-wave backgrounds in the light-cone gauge and we interpret a diagram from the BMN limit as a stringy diagram involving the contact term. 
  The geometric form of standard quantum mechanics is compatible with the two postulates: 1) The laws of physics are invariant under the choice of experimental setup and 2) Every quantum observation or event is intrinsically statistical. These postulates remain compatible within a background independent extension of quantum theory with a local intrinsic time implying the relativity of the concept of a quantum event. In this extension the space of quantum events becomes dynamical and only individual quantum events make sense observationally. At the core of such a general theory of quantum relativity is the three-way interplay between the symplectic form, the dynamical metric and non-integrable almost complex structure of the space of quantum events. Such a formulation provides a missing conceptual ingredient in the search for a background independent quantum theory of gravity and matter. The crucial new technical element in our scheme derives from a set of recent mathematical results on certain infinite dimensional almost Kahler manifolds which replace the complex projective spaces of standard quantum mechanics. 
  We propose a way to recover Lorentz invariance of the perturbative S matrix in the Discrete Light-Cone Quantization (DLCQ) in the continuum limit without spoiling the trivial vacuum. 
  We discuss the 2PPI expansion, a summation of the bubble graphs up to all orders, by means of the 2D Gross-Neveu toy model, whose exact mass gap and vacuum energy are known. Then we use the expansion to give analytical evidence that a dimension two gluon condensate exists for pure Yang-Mills in the Landau gauge. This <A^2> condensate consequently gives rise to a dynamical gluon mass. 
  We further develop an algorithmic and diagrammatic computational framework for very general exact renormalization groups, where the embedded regularisation scheme, parametrised by a general cutoff function and infinitely many higher point vertices, is left unspecified. Calculations proceed iteratively,by integrating by parts with respect to the effective cutoff, thus introducing effective propagators, and differentials of vertices that can be expanded using the flow equations; many cancellations occur on using the fact that the effective propagator is the inverse of the classical Wilsonian two-point vertex. We demonstrate the power of these methods by computing the beta function up to two loops in massless four dimensional scalar field theory, obtaining the expected universal coefficients, independent of the details of the regularisation scheme. 
  Primary superfields for a two dimensional Euclidean superconformal field theory are constructed as sections of a sheaf over a graded Riemann sphere. The construction is then applied to the N=3 Neveu-Schwarz case. Various quantities in the N=3 theory are calculated and discussed, such as formal elements of the super-Mobius group, and the two-point function. 
  We construct an interaction between a (2,0) tensor multiplet in six dimensions and a self-dual string. The interaction is a sum of a Nambu-Goto term, with the tension of the string given by the modulus of the scalar fields of the tensor multiplet, and a non-local Wess-Zumino term, that encodes the electromagnetic coupling of the string to the two-form gauge field of the tensor multiplet. The interaction is invariant under global (2,0) supersymmetry, modulo the equations of motion of a free tensor multiplet. It is also invariant under a local fermionic kappa-symmetry, as required by the BPS-property of the string. 
  In the framework of an extended BRST formalism, it is shown that the four $(3 + 1)$-dimensional (4D) free Abelian 2-form (notoph) gauge theory presents an example of a tractable field theoretical model for the Hodge theory. 
  We give a short introduction to AdS/CFT and its plane wave limit. 
  The generalized Verlinde formulae expressing traces of mapping classes corresponding to automorphisms of certain Riemann surfaces, and the congruence relations on allowed modular representations following from them are presented. The surfaces considered are families of algebraic curves given by suitably chosen equations, the modular curve $\mathcal{X}(11)$, and a factor curve of $\mathcal{X}(8)$. The examples of modular curves illustrate how the study of arithmetic properties of suitable modular representations can be used to gain information on automorphic properties of Riemann surfaces. 
  It is well-known that the charge of fermion is 0 or $\pm1$ in the U(1) gauge theory on noncommutative spacetime. Since the deviation from the standard model in particle physics has not yet observed, and so there may be no room to incorporate the noncommutative U(1) gauge theory into the standard model because the quarks have fractional charges. However, it is shown in this article that there is the noncommutative gauge theory with arbitrary charges which symmetry is for example SU(3+1)$\ast$. This enveloping gauge group consists of elements $\exp i {\sum_{a=0}^8 T^a \alpha^a(x,\theta)\ast+ Q \beta(x,\theta)\ast} $ with $Q=\text{diag}(e,e,e,e') $ and the restriction $\lim_{\theta\to0}\alpha^0(x,\theta)=0.$ This type of gauge theory is emergent from the spontaneous breakdown of the noncommutative SU(N)$\ast$ or SO(N)$\ast$ gauge theory in which the gauge field contains the 0 component $A_\mu^0(x,\theta)$. $A_\mu^0(x,\theta)$ can be eliminated by gauge transformation. Thus, the noncommutative gauge theory with arbitrary U(1) charges can not exist alone, but it must coexist with noncommutative nonabelian gauge theory. This suggests that the spacetime noncommutativity requires the grand unified theory which spontaneously breaks down to the noncommutative standard model with fractionally charged quarks. 
  In three spacetime dimensions, where no graviton propagates, pure gravity is known to be finite. It is natural to inquire whether finiteness survives the coupling with matter. Standard arguments ensure that there exists a subtraction scheme where no Lorentz-Chern-Simons term is generated by radiative corrections, but are not sufficiently powerful to ensure finiteness. Therefore, it is necessary to perform an explicit (two-loop) computation in a specific model. I consider quantum gravity coupled with Chern-Simons U(1) gauge theory and massless fermions and show that renormalization originates four-fermion divergent vertices at the second loop order. I conclude that quantum gravity coupled with matter, as it stands, is not finite in three spacetime dimensions. 
  As it stands, quantum gravity coupled with matter in three spacetime dimensions is not finite. In this paper I show that an algorithmic procedure that makes it finite exists, under certain conditions. To achieve this result, gravity is coupled with an interacting conformal field theory C. The Newton constant and the marginal parameters of C are taken as independent couplings. The values of the other irrelevant couplings are determined iteratively in the loop- and energy-expansions, imposing that their beta functions vanish. The finiteness equations are solvable thanks to the following properties: the beta functions of the irrelevant couplings have a simple structure; the irrelevant terms made with the Riemann tensor can be reabsorbed by means of field redefinitions; the other irrelevant terms have, generically, non-vanishing anomalous dimensions. The perturbative expansion is governed by an effective Planck mass that takes care of the interactions in the matter sector. As an example, I study gravity coupled with Chern-Simons U(1) gauge theory with massless fermions, solve the finiteness equations and determine the four-fermion couplings to two-loop order. The construction of this paper does not immediately apply to four-dimensional quantum gravity. 
  I show that under certain conditions it is possible to define consistent irrelevant deformations of interacting conformal field theories. The deformations are finite or have a unique running scale ("quasi-finite"). They are made of an infinite number of lagrangian terms and a finite number of independent parameters that renormalize coherently. The coefficients of the irrelevant terms are determined imposing that the beta functions of the dimensionless combinations of couplings vanish ("quasi-finiteness equations"). The expansion in powers of the energy is meaningful for energies much smaller than an effective Planck mass. Multiple deformations can be considered also. I study the general conditions to have non-trivial solutions. As an example, I construct the Pauli deformation of the IR fixed point of massless non-Abelian Yang-Mills theory with N_c colors and N_f <~ 11N_c/2 flavors and compute the couplings of the term F^3 and the four-fermion vertices. Another interesting application is the construction of finite chiral irrelevant deformations of N=2 and N=4 superconformal field theories. The results of this paper suggest that power-counting non-renormalizable theories might play a role in the description of fundamental physics. 
  We construct effective Lagrangians of the Veneziano-Yankielowicz (VY) type for two non-supersymmetric theories which are orientifold daughters of supersymmetric gluodynamics (containing one Dirac fermion in the two-index antisymmetric or symmetric representation of the gauge group). Since the parent and daughter theories are planar equivalent, at N\to\infty the effective Lagrangians in the orientifold theories basically coincide with the bosonic part of the VY Lagrangian.   We depart from the supersymmetric limit in two ways. First, we consider finite (albeit large) values of N. Then 1/N effects break supersymmetry. We suggest seemingly the simplest modification of the VY Lagrangian which incorporates these 1/N effects, leading to a non-vanishing vacuum energy density. We analyze the spectrum of the finite-N non-supersymmetric daughters. For N=3 the two-index antisymmetric representation (one flavor) is equivalent to one-flavor QCD. We show that in this case the scalar quark-antiquark state is heavier than the corresponding pseudoscalar state, `` eta' ''. Second, we add a small fermion mass term. The fermion mass term breaks supersymmetry explicitly. The vacuum degeneracy is lifted. The parity doublets split. We evaluate the splitting. Finally, we include the theta-angle and study its implications. 
  We present the first two leading terms of the 1/N (genus) expansion of the free energy for ensembles of normal and complex random matrices. The results are expressed through the support of eigenvalues (assumed to be a connected domain in the complex plane). In particular, the subleading (genus-1) term is given by the regularized determinant of the Laplace operator in the complementary domain with the Dirichlet boundary conditions. An explicit expression of the genus expansion through harmonic moments of the domain gives some new representations of the mathematical objects related to the Dirichlet boundary problem, conformal analysis and spectral geometry. 
  In this work we discuss a generalization for the thermal Bogoliubov transformation in the context of a hermitian general SU(1,1) transformation generator. The TFD tilde conjugation rules are redefined using an appropriated Tomita-Takesaki modular operator. 
  The discrete Lax operators with the spectral parameter on an algebraic curve are defined. A hierarchy of commuting flows on the space of such operators is constructed. It is shown that these flows are linearized by the spectral transform and can be explicitly solved in terms of the theta-functions of the spectral curves. The Hamiltonian theory of the corresponding systems is analyzed. The new type of completely integrable Hamiltonian systems associated with the space of rank $r=2$ discrete Lax operators on a {\it variable} base curve is found. 
  We discuss the possibility to absorb all anomalies in the supersymmetry algebra of the N=(1,1) Wess-Zumino model in d=1+1 by a local counter term. This counter term corresponds to the change of the vacuum parameter $v_0^2$ in the model and the transition to an unconventional but admissible renormalization scheme. It does not modify the physical consequences such as BPS saturation, and thus the situation is rather different from gauge theory where local counter terms are required to absorb spurious gauge anomalies. 
  This text collects useful results concerning the quasi-Hopf algebra $\D $. We give a review of issues related to its use in conformal theories and physical mathematics. Existence of such algebras based on 3-cocycles with values in $ {R} / {Z} $ which mimic for finite groups Chern-Simons terms of gauge theories, open wide perspectives in the so called "classification program". The modularisation theorem proved for quasi-Hopf algebras by two authors some years ago makes the computation of topological invariants possible. An updated, although partial, bibliography of recent developments is provided. 
  In the plane-wave matrix model, the background configuration of two membrane fuzzy spheres, one of which rotates around the other one in the SO(6) symmetric space, is allowed as a classical solution. We study the one-loop quantum corrections to this background in the path integral formulation. Firstly, we show that each fuzzy sphere is stable under the quantum correction. Secondly, the effective potential describing the interaction between fuzzy spheres is obtained as a function of r, which is the distance between two fuzzy spheres. It is shown that the effective potential is flat and hence the fuzzy spheres do not feel any force. The possibility on the existence of flat directions is discussed. 
  We study the localization of gravity on a string-like topological defect within a 6-dimensional space-time. Assuming zero cosmological constant we find complete numerical solutions to a set of first-order, Bogomol'nyi-Prasad-Sommeferld (BPS)-like, equations for the metric and the scalar field, where the dynamics of the latter are dictated by a supergravity-type potential. Our axially symmetric solutions have no deficit angle and factorize as $AdS_5 \times S_1 $ far from the core. They are regular everywhere, providing complete smooth cigar-like geometries. The total energy of these configurations depends only on the boundary conditions for the warp factor and it is shown to vanish. 
  Employing Maxwell's equations as the field theory of the photon, quantum mechanical operators for spin, chirality, helicity, velocity, momentum, energy and position are derived. The photon ``Zitterbewegung'' along helical paths is explored. The resulting non-commutative geometry of photon position and the quantum version of the Pythagorean theorem is discussed. The distance between two photons in a polarized beam of given helicity is shown to have a discrete spectrum. Such a spectrum should become manifest in measurements of two photon coincidence counts. The proposed experiment is briefly described. 
  We compute the boundary energy and the Casimir energy for both the spin-1/2 XXZ quantum spin chain and (by means of the light-cone lattice construction) the massive sine-Gordon model with both left and right boundaries. We also derive a nonlinear integral equation for the ground state of the sine-Gordon model on a finite interval. These results, which are based on a recently-proposed Bethe Ansatz solution, are for general values of the bulk coupling constant, and for both diagonal and nondiagonal boundary interactions. However, the boundary parameters are restricted to obey one complex (two real) constraints. 
  The Gaussian expansion has been developed since early 80s as a powerful analytical method, which enables nonperturbative studies of various systems using `perturbative' calculations. Recently the method has been used to suggest that 4d space-time is generated dynamically in a matrix model formulation of superstring theory. Here we clarify the nature of the method by applying it to exactly solvable one-matrix models with various kinds of potential including the ones unbounded from below and of the double-well type. We also formulate a prescription to include a linear term in the Gaussian action in a way consistent with the loop expansion, and test it in some concrete examples. We discuss a case where we obtain two distinct plateaus in the parameter space of the Gaussian action, corresponding to different large-N solutions. This clarifies the situation encountered in the dynamical determination of the space-time dimensionality in the previous works. 
  We investigate non-trivial topological structures in Discrete Light Cone Quantization (DLCQ) through the example of the broken symmetry phase of the two dimensional $\phi^4$ theory using anti periodic boundary condition (APBC). We present evidence for degenerate ground states which is both a signature of spontaneous symmetry breaking and mandatory for the existence of kinks. Guided by a constrained variational calculation with a coherent state ansatz, we then extract the vacuum energy and kink mass and compare with classical and semi - classical results. We compare the DLCQ results for the number density of bosons in the kink state and the Fourier transform of the form factor of the kink with corresponding observables in the coherent variational kink state. 
  An exact matrix integral is evaluated for a $2\times 2$ 3-dimensional matrix model with Myers term. We derive weak and strong coupling expansions of the effective action. We also calculate the expectation values of the quadratic and cubic operators. Implications for non-commutative gauge theory on fuzzy sphere are discussed. 
  We study the extension of integrable equations which possess the Lax representations to noncommutative spaces. We construct various noncommutative Lax equations by the Lax-pair generating technique and the Sato theory. The Sato theory has revealed essential aspects of the integrability of commutative soliton equations and the noncommutative extension is worth studying. We succeed in deriving various noncommutative hierarchy equations in the framework of the Sato theory, which is brand-new. The existence of the hierarchy would suggest a hidden infinite-dimensional symmetry in the noncommutative Lax equations. We finally show that a noncommutative version of Burgers equation is completely integrable because it is linearizable via noncommutative Cole-Hopf transformation. These results are expected to lead to the completion of the noncommutative Sato theory. 
  In the models of brane construction, the isometry of a compactified space might be broken by branes. In four-dimensional effective Lagrangian, the breaking of the isometry is seen as the spontaneous breaking of the corresponding effective symmetry. Then it seems natural to expect that there are various kinds of defects that will be implemented by the spontaneous symmetry breaking. These defects are parametrized by the brane positions. In this paper we consider two kinds of such ``brane defects'', which are formed by the local fluctuations of the locations of branes along their transversal directions. The fluctuation of a brane position might leads to winding (or wraping) around a non-contractible circle of the compactified space. These ``primary'' brane defects are already discussed by several authors. On the other hand, if there are multiple branes in the compactified space and their configuration in a compactified space is determined by the potential that depends only on their relative positions, one might find incidental symmetry in the effective potential, which is spontaneously broken by branes. We examined the latter ``incidental'' symmetry breakings and stable defect configurations. We paid special attention to the difference between ``primary'' brane defects. 
  The classical solution to the Liouville equation in the case of three hyperbolic singularities of its energy-momentum tensor is derived and analyzed. The recently proposed classical Liouville action is explicitly calculated in this case. The result agrees with the classical limit of the three point function in the DOZZ solution of the quantum Liouville theory. 
  We introduce a new map between a (q,h)-deformed gauge theory and ordinary gauge theory in a full analogy with Seiberg-Witten map. 
  The coset construction is the most important tool to construct rational conformal field theories with known chiral data. For some cosets at small level, so-called maverick cosets, the familiar analysis using selection and identification rules breaks down. Intriguingly, this phenomenon is linked to the existence of exceptional modular invariants. Recent progress in CFT, based on studying algebras in tensor categories, allows for a universal construction of the chiral data of coset theories which in particular also applies to maverick cosets. 
  In this paper we continue previous work on counting open string states between D-branes by considering open strings between D-branes with nonzero Higgs vevs, and in particular, nilpotent Higgs vevs, as arise, for example, when studying D-branes in orbifolds. Ordinarily Higgs vevs can be interpreted as moving the D-brane, but nilpotent Higgs vevs have zero eigenvalues, and so their interpretation is more interesting -- for example, they often correspond to nonreduced schemes, which furnishes an important link in understanding old results relating classical D-brane moduli spaces in orbifolds to Hilbert schemes, resolutions of quotient spaces, and the McKay correspondence. We give a sheaf-theoretic description of D-branes with Higgs vevs, including nilpotent Higgs vevs, and check that description by noting that Ext groups between the sheaves modelling the D-branes, do in fact correctly count open string states. In particular, our analysis expands the types of sheaves which admit on-shell physical interpretations, which is an important step for making derived categories useful for physics. 
  In this work, a general definition of convolution between two arbitrary Tempered Ultradistributions is given. When one of the Tempered Ultradistributions is rapidly decreasing this definition coincides with the definition of J. Sebastiao e Silva. In the four-dimensional case, when the Tempered Ultradistributions are even in the variables $k^0$ and $\rho$ (see Section 5) we obtain an expression for the convolution, which is more suitable for practical applications. The product of two arbitrary even (in the variables $x^0$ and $r$) four dimensional distributions of exponential type is defined via the convolution of its corresponding Fourier Transforms. With this definition of convolution, we treat the problem of singular products of Green Functions in Quantum Field Theory. (For Renormalizable as well as for Nonrenormalizable Theories). Several examples of convolution of two Tempered Ultradistributions are given. In particular we calculate the convolution of two massless Wheeeler's propagators and the convolution of two complex mass Wheeler's propagators. 
  This thesis is devoted to the construction and study of D-branes in some curved space-times in string theory. On the one hand, those D-branes are described geometrically as submanifolds subject to Born-Infeld effective dynamics. On the other hand, they can be built microscopically using boundary conformal field theory. We use and compare those two approaches. We also improve them technically : we rewrite Born-Infeld dynamics in a gauge-invariant way, and formulate precise analyticity requirements for the density of open strings on certain D-branes. Our results include the effective description of symmetric D-branes in compact groups, the determination of the complete spectrum of open strings on AdS2 D-branes in AdS3, the exact construction of some D-branes in the cigar SL(2)/U(1), and a geometric description of all D3-branes in NS5-brane backgrounds. 
  We consider the cosmology of the reduced 5D Horava-Witten M-Theory (HW) with volume modulus and treating matter on the orbifold planes to first order. It is seen that one can recover the FRW cosmology in the Hubble expansion era with relativistic matter, but if a solution exist with non-relativistic (massive) matter it must be non-static with a Hubble constant that depends on the fifth dimension. (The same result holds when 5-branes are present.) This difficulty is traced to the fact that in HW, the volume modulus couples to the bulk and brane cosmological constants (so that the net 4D constant vanishes naturally). This situation is contrasted with the Randall-Sundrum 1 model (which is here treated without making the stiff potential approximation) where the radion field does not couple to the cosmological constants (and so one must instead fine tune the net constant to zero). One finds that non-relativistic matter is accommodated there by changing the distance between the end branes. 
  D-branes are by now an integral part of our toolbox towards understanding nature. In this review we will describe recent progress in their use to realize fundamental interactions. The realization of the Standard Model and relevant physics and problems will be detailed. New ideas on realizing 4-dimensional gravity use the brane idea in an important way. Such approaches will be reviewed and compared to the standard paradigm of compactification. Branes can play a pivotal role both in early- and late-universe cosmology mainly via the brane-universe paradigm. Brane realizations of various cosmological ideas (early inflation, sources for dark matter and dark energy, massive gravity etc) will be also reviewed. 
  We carry out a numerical study of fluctuations in the spectrum of regular graphs. Our experiments indicate that the level spacing distribution of a generic k-regular graph approaches that of the Gaussian Orthogonal Ensemble of random matrix theory as we increase the number of vertices. A review of the basic facts on graphs and their spectra is included. 
  We consider solitonic solutions of coupled scalar systems, whose Lagrangian has a potential term (quasi-supersymmetric potential) consisting of the square of derivative of a superpotential. The most important feature of such a theory is that among soliton masses there holds a Ritz-like combination rule (e.g. $M_{12}+M_{23}=M_{13}$), instead of the inequality ($M_{12}+M_{23}<M_{13}$) which is a stability relation generally seen in N=2 supersymmetric theory. The promotion from N=1 to N=2 theory is considered. 
  The self-duality equations for gauge fields in pseudoeuclidean spaces of eight and seven dimensions are considered. Some new classes of solutions of the equations are found. 
  We present the first example of a Kahler potential for heterotic M-theory which includes gauge bundle moduli. These moduli describe the background gauge field configurations living on the orbifold fixed planes. We concentrate on the bundle moduli describing the size and SU(2) orientation of a gauge five brane - a soliton which is primarily composed of these gauge fields. Our results are valid when the width of this object is small compared to the overall size of the Calabi-Yau threefold. We find that, in general, it is not consistent to truncate away these moduli in a simple manner. 
  We present a systematic discussion of supersymmetric solutions of 2D dilaton supergravity. In particular those solutions which retain at least half of the supersymmetries are ground states with respect to the bosonic Casimir function (essentially the ADM mass). Nevertheless, by tuning the prepotential appropriately, black hole solutions may emerge with an arbitrary number of Killing horizons. The absence of dilatino and gravitino hair is proven. Moreover, the impossibility of supersymmetric dS ground states and of nonextremal black holes is confirmed, even in the presence of a dilaton. In these derivations the knowledge of the general analytic solution of 2D dilaton supergravity plays an important role. The latter result is addressed in the more general context of gPSMs which have no supergravity interpretation.   Finally it is demonstrated that the inclusion of non-minimally coupled matter, a step which is already nontrivial by itself, does not change these features in an essential way. 
  Braneworld scenarios with compact extra-dimensions need the volume of the extra space to be stabilized. Goldberger and Wise have introduced a simple mechanism, based on the presence of a bulk scalar field, able to stabilize the radius of the Randall-Sundrum model. Here, we transpose the same mechanism to generic single-brane and two-brane models, with one extra dimension and arbitrary scalar potentials in the bulk and on the branes. The single-brane construction turns out to be always unstable, independently of the bulk and brane potentials. In the case of two branes, we derive some generic criteria ensuring the stabilization or destabilization of the system. 
  In order to address the issues raised by the recent discovery of non-uniqueness of black holes in five dimensions, we construct a solution of string theory at low energies describing a five-dimensional spinning black ring with three charges that can be interpreted as D1-brane, D5-brane, and momentum charges. The solution possesses closed timelike curves (CTCs) and other pathologies, whose origin we clarify. These pathologies can be avoided by setting any one of the charges, e.g. the momentum, to zero. We argue that the D1-D5-charged black ring, lifted to six dimensions, describes the thermal excitation of a supersymmetric D1-D5 supertube, which is in the same U-duality class as the D0-F1 supertube. We explain how the stringy microscopic description of the D1-D5 system distinguishes between a spherical black hole and a black ring with the same asymptotic charges, and therefore provides a (partial) resolution of the non-uniqueness of black holes in five dimensions. 
  We develop a novel approach to gravity in which gravity is described by a matrix-valued symmetric two-tensor field and construct an invariant functional that reduces to the standard Einstein-Hilbert action in the commutative limit. We also introduce a gauge symmetry associated with the new degrees of freedom. 
  We study several different kinds of bound states built from D-branes and orientifolds. These states are to atoms what branonium - the bound state of a brane and its anti-brane - is to positronium, inasmuch as they typically involve a light brane bound to a much heavier object with conserved charges which forbid the system's decay. We find the fully relativistic motion of a probe Dp'-brane in the presence of source Dp-branes is integrable by quadratures. Keplerian conic sections are obtained for special choices for p and p' and the systems are shown to be equivalent to nonrelativistic systems. Their quantum behaviour is also equivalent to the corresponding non-relativistic limit. In particular the p=6, p'=0 case is equivalent to a non-relativistic dyon in a magnetic monopole background, with the trajectories in the surface of a cone. We also show that the motion of the probe branes about D6-branes in IIA theory is equivalent to the motion of the corresponding probes in the uplift to M-theory in 11 dimensions, for which there are no D6-branes but their fields are replaced by a particular Taub-NUT geometry. We further discuss the interactions of D-branes and orientifold planes having the same dimension. this system behaves at large distances as a brane-brane system but at shorter distances it does not have the tachyon instability. 
  We continue our investigation of the quantum equivalence between commutative and noncommutative Chern-Simons theories by computing the complete set of two-loop quantum corrections to the correlation function of a pure open Wilson line and an open Wilson line with a field strength insertion, on the noncommutative side in a covariant gauge. The conjectured perturbative equivalence between the free commutative theory and the apparently interacting noncommutative one requires that the sum of these corrections vanish, and herein we exhibit the remarkable cancellations that enforce this. From this computation we speculate on the form of a possible all-order result for this simplest nonvanishing correlator of gauge invariant observables. 
  We demonstrate that the near-horizon physics, the Hawking radiation and the reflection off the radial potential barrier, can be understood entirely within a conformal field theory picture in terms of one- and two-point functions in the boundary Liouville theory. An important element in this demonstration is the notion of {\it horizon state}, the Hawking radiation being interpreted as a result of the transition of horizon state to the ordinary states propagating outside black hole horizon. 
  We show that the algebraic aspects of Lie symmetries and generalized symmetries in nonrelativistic and relativistic quantum mechanics can be preserved in linear lattice theories. The mathematical tool for symmetry preserving discretizations on regular lattices is the umbral calculus. 
  We study a previously introduced bi-local gauge invariant reformulation of two dimensional QCD, called 2d HadronDynamics. The baryon arises as a topological soliton in HadronDynamics. We derive an interacting parton model from the soliton model, thus reconciling these two seemingly different points of view. The valence quark model is obtained as a variational approximation to HadronDynamics. A succession of better approximations to the soliton picture are obtained. The next simplest case corresponds to a system of interacting valence, `sea' and anti-quarks. We also obtain this `embellished' parton model directly from the valence quark system through a unitary transformation. Using the solitonic point of view, we estimate the quark and anti-quark distributions of 2d QCD. Possible applications to Deep Inelastic Structure Functions are pointed out. 
  Recently the effective action for the 4-point functions in abelian open superstring theory has been derived, giving an explicit construction of the bosonic and fermionic terms of this infinite $\alpha'$ series. In the present work we generalize this result to the nonabelian case. We test our result, at ${\alpha'}^3$ and ${\alpha'}^4$ order, with several existing versions for these terms, finding agreement in most of the cases. We also apply these ideas to derive the effective action for the 4-point functions of the NS-NS sector of closed superstring theory, to all order in $\alpha'$. 
  An n-particle 3-dimensional Wigner quantum oscillator model is constructed explicitly. It is non-canonical in that the usual coordinate and linear momentum commutation relations are abandoned in favour of Wigner's suggestion that Hamilton's equations and the Heisenberg equations are identical as operator equations. The construction is based on the use of Fock states corresponding to a family of irreducible representations of the Lie superalgebra sl(1|3n) indexed by an A-superstatistics parameter p. These representations are typical for p\geq 3n but atypical for p<3n. The branching rules for the restriction from sl(1|3n) to gl(1) \oplus so(3) \oplus sl(n) are used to enumerate energy and angular momentum eigenstates. These are constructed explicitly and tabulated for n\leq 2. It is shown that measurements of the coordinates of the individual particles gives rise to a set of discrete values defining nests in the 3-dimensional configuration space. The fact that the underlying geometry is non-commutative is shown to have a significant impact on measurements of particle separation. In the atypical case, exclusion phenomena are identified that are entirely due to the effect of A-superstatistics. The energy spectrum and associated degeneracies are calculated for an infinite-dimensional realisation of the Wigner quantum oscillator model obtained by summing over all p. The results are compared with those applying to the analogous canonical quantum oscillator. 
  We study D-branes with plane waves of arbitrary profiles as examples of time-dependent backgrounds in string theory. We show how to reproduce the quantum mechanical (one-to-one) open-string S-matrix starting from the closed-string boundary state for the D-branes, thereby establishing the channel duality of this calculation. The required Wick rotation to a Lorentzian worldsheet singles out as 'prefered' time coordinate the open-string light-cone time. 
  A gauge invariant flow equation is derived by applying a Wilsonian momentum cut-off to gauge invariant field variables. The construction makes use of the geometrical effective action for gauge theories in the Vilkovisky-DeWitt framework. The approach leads to modified Nielsen identities that pose non-trivial constraints on consistent truncations. We also evaluate the relation of the present approach to gauge fixed formulations as well as discussing possible applications. 
  We point out the existence of some simple string solitons in AdS_5 x S^5, which at the same time are spinning in AdS_5 and pulsating in S^5, or vice-versa. This introduces an additional arbitrary constant into the scaling relations between energy and spin or R-charge. The arbitrary constant is not an angular momentum, but can be related to the amplitude of the pulsation. We discuss the solutions in detail and consider the scaling relations. Pulsating multi spin or multi R-charge solutions can also be constructed. 
  We represent B fields and higher p-form potentials on a manifold M as connections on affine bundles over M. We realize D branes on M as special submanifolds of these affine bundles. We check the physical relevance of this representation by showing that the properties of the resulting branes closely correspond to those of their string theory counterparts. As a simple application of this geometric understanding of the B field, we show how to obtain supersymmetry algebra deformations induced by 2 and 3 form potentials. 
  We discuss compactifications of heterotic string theory to four dimensions in the presence of H-fluxes, which deform the geometry of the internal manifold, and a gaugino condensate which breaks supersymmetry. We focus on the compensation of the two effects in order to obtain vacua with zero cosmological constant and we comment on the effective superpotential describing these vacua. 
  Due to the Unruh effect, accelerated and inertial observers differ in their description of a given quantum state. The implications of this effect are explored for the entropy assigned by such observers to localized objects that may cross the associated Rindler horizon. It is shown that the assigned entropies differ radically in the limit where the number of internal states $n$ becomes large. In particular, the entropy assigned by the accelerated observer is a bounded function of $n$. General arguments are given along with explicit calculations for free fields. The implications for discussions of the generalized second law and proposed entropy bounds are also discussed. 
  We describe a monopole-like order parameter for the confinement-deconfinement transition in gauge theories where dynamical charges and monopoles coexist. It has been recently proposed in a collaboration with J. Froehlich. It avoids an inconsistency in the treatment of small scales present in earlier definitions of monopole fields by respecting Dirac's quantization condition for electromagnetic fluxes. An application to SU(2) lattice Yang-Mills theory is outlined, naturally fitting in the 't Hooft scenario for confinement. 
  We present a comprehensive analysis of branes in the Euclidean 2D black hole (cigar). In particular, exact boundary states and annulus amplitudes are provided for D0-branes which are localized at the tip of the cigar as well as for two families of extended D1 and D2-branes. Our results are based on closely related studies for the Euclidean AdS3 model and, as predicted by the conjectured duality between the 2D black hole and the sine-Liouville model, they share many features with branes in Liouville theory. New features arise here due to the presence of closed string modes which are localized near the tip of the cigar. The paper concludes with some remarks on possible applications. 
  In \cN = 2, 4 superconformal field theories in four space-time dimensions, the quantum corrections with four derivatives are believed to be severely constrained by non-renormalization theorems. The strongest of these is the conjecture formulated by Dine and Seiberg in hep-th/9705057 that such terms are generated only at one loop. In this note, using the background field formulation in \cN = 1 superspace, we test the Dine-Seiberg proposal by comparing the two-loop F^4 quantum corrections in two different superconformal theories with the same gauge group SU(N): (i) \cN = 4 SYM (i.e. \cN = 2 SYM with a single adjoint hypermultiplet); (ii) \cN = 2 SYM with 2N hypermultiplets in the fundamental. According to the Dine-Seiberg conjecture, these theories should yield identical two-loop F^4 contributions from all the supergraphs involving quantum hypermultiplets, since the pure \cN = 2 SYM and ghost sectors are identical provided the same gauge conditions are chosen. We explicitly evaluate the relevant two-loop supergraphs and observe that the F^4 corrections generated have different large N behaviour in the two theories under consideration. Our results are in conflict with the Dine-Seiberg conjecture. 
  Fuzzy spaces are obtained by quantizing adjoint orbits of compact semi-simple Lie groups. Fuzzy spheres emerge from quantizing S^2 and are associated with the group SU(2) in this manner. They are useful for regularizing quantum field theories and modeling spacetimes by non-commutative manifolds. We show that fuzzy spaces are Hopf algebras and in fact have more structure than the latter. They are thus candidates for quantum symmetries. Using their generalized Hopf algebraic structures, we can also model processes where one fuzzy space splits into several fuzzy spaces. For example we can discuss the quantum transition where the fuzzy sphere for angular momentum J splits into fuzzy spheres for angular momenta K and L. 
  We derive boundary states which describe configurations of multiple parallel branes with arbitrary open string states interactions in bosonic string theory. This is obtained by a careful discussion of the factorization of open/closed string states amplitudes taking care of cycles needed by ensuring vertexes commutativity: in particular the discussion reveals that already at the tree level open string knows of the existence of closed string. We also give a formal expression for computing pure closed string amplitudes using the open string formalism. 
  We analyze the validity of the generalized covariant entropy bound near the apparent horizon of isotropic expanding cosmological models. We encounter violations of the bound for cosmic times smaller than a threshold. By introducing an infrared cutoff we are able to mantain the bound for a radiation dominated universe. We study different physical mechanisms to restore the bound, as a non-additivity of the entropy at a fundamental level and/or a cosmological uncertainty relation. 
  We propose a geometrical interpretation for the discrete torsion appearing in the algebraic formulation of quotients of WZW models by discrete abelian subgroups. Part of the discrete torsion corresponds to the choice of action of the subgroup, yielding different quotient spaces. Another part corresponds to the set of different choices of connection for the H field in each of these spaces. The former is for instance used to describe generalized lens spaces L(n,p). 
  We use light-like Wilson loops and the AdS/CFT correspondence to compute the anomalous dimensions of twist two operators in the cascading (Klebanov-Strassler) theory. The computation amounts to find a minimal surface in the UV region of the KS background which is described by the Klebanov-Tseytlin solution. The result is similar to the one for SU(N), N=4 SYM but with N replaced by an effective, scale dependent N. We perform also a calculation using a rotating string and find agreement. In fact we use a double Wick rotated version of the rotating string solution which is an Euclidean world-sheet ending on a light-like line in the boundary. It gives the same result as the rotating string for the N=4 case but is more appropiate than the rotating string in the N=1 case. 
  We evaluate the spin-orbit and spin-spin interaction between two fermions in strongly coupled gauge theories in their Coulomb phase. We use the quasi-instantaneous character of Coulomb's law at strong coupling to resum a class of ladder diagrams. For ${\cal N}=4$ SYM we derive both weak and strong coupling limits of the the spin-orbit and spin-spin interactions, and find that in the latter case these interactions are subleading corrections and do not seriously affect the deeply bound Coulomb states with large angular momentum, pointed out in our previous paper. The results are important for understanding of the regime of intermediate coupling, which is the case for QCD somewhat above the chiral transition temperature. 
  Within an adiabatic approximation, thermodynamical equilibrium and a small, nine dimensional, toroidal universe as initial conditions, we analyze the evolution of the dimensions in two different regimes: (i) the Hagedorn regime, with a single scale factor with a nearly constant time evolution (ii) an almost-radiation dominated regime, including the leading corrections due to the lightest Kaluza Klein and winding modes, in which for some initial conditions the large dimensions continue to expand and the small ones remain small. 
  These lecture notes present an elementary introduction to light-cone string field theory, with an emphasis on its application to the study of string interactions in the plane wave limit of AdS/CFT. We summarize recent results and conclude with a list of open questions. 
  We explore the possibility that the dark energy is due to a potential of a scalar field and that the magnitude and the slope of this potential in our part of the universe are largely determined by anthropic selection effects. We find that, in some models, the most probable values of the slope are very small, implying that the dark energy density stays constant to very high accuracy throughout cosmological evolution. In other models, however, the most probable values of the slope are such that the slow roll condition is only marginally satisfied, leading to a re-collapse of the local universe on a time-scale comparable to the lifetime of the sun. In the latter case, the effective equation of state varies appreciably with the redshift, leading to a number of testable predictions. 
  We discuss an open supermembrane theory in the AdS_4 x S^7 and AdS_7 x S^4 backgrounds. The possible Dirichlet branes of an open supermembrane are classified by analyzing the covariant Wess-Zumino term. All of the allowed configurations are related to those on the pp-wave background via the Penrose limit. 
  Complete set of Maxwell equations is represented by single equation using the Riemann-Silberstein vector ${\bf F}={\bf E}+i{\bf B}$. We demonstrate that the Fourier form of invariants $E^2-B^2$ and ${\bf EB}$ is proportional to dissipated power and equal to zero respectively. 
  We compute the Weyl anomaly for an abelian Wilson surface by using a regularization that respects the gauge invariance. We then study the loop space on which lives a one-form connection. We restrict ourselves to the subsector consisting of only circular loops, and derive a Maxwell type action on this restricted loop space. 
  Recently, corrections to the standard Einstein-Hilbert action are proposed to explain the current cosmic acceleration in stead of introducing dark energy. In the Palatini formulation of those modified gravity models, there is an important observation due to Arkani-Hamed: matter loops will give rise to a correction to the modified gravity action proportional to the Ricci scalar of the metric. In the presence of such term, we show that the current forms of modified gravity models in Palatini formulation, specifically, the 1/R gravity and $\ln R$ gravity, will have phantoms. Then we study the possible instabilities due to the presence of phantom fields. We show that the strong instability in the metric formulation of 1/R gravity indicated by Dolgov and Kawasaki will not appear and the decay timescales for the phantom fields may be long enough for the theories to make sense as effective field theory . On the other hand, if we change the sign of the modification terms to eliminate the phantoms, some other inconsistencies will arise for the various versions of the modified gravity models. Finally, we comment on the universal property of the Palatini formulation of the matter loops corrected modified gravity models and its implications. 
  The standard forms of supersymmetry and supergravity are inextricably wedded to Lorentz invariance. Here a Lorentz-violating form of supergravity is proposed. The superpartners have exotic properties that are not possible in a theory with exact Lorentz symmetry and microcausality. For example, the bosonic sfermions have spin 1/2 and the fermionic gauginos have spin 1. The theory is based on a phenomenological action that is shown to follow from a simple microscopic and statistical picture. 
  We study bosonic string theory in the light-cone gauge on AdS_3 spacetime with zero radius of curvature (in string units) R/\sqrt{\alpha^\prime}=0. We find that the worldsheet theory admits an infinite number of conserved quantities which are naturally interpreted as spacetime charges and which form a representation of (two commuting copies of) a Virasoro algebra. Near the boundary of AdS_3 these charges are found to be isomorphic to the infinite set of asymptotic Killing vectors of AdS_3 found originally by Brown and Henneaux. In addition to the spacetime Virasoro algebra, there is a worldsheet Virasoro algebra that generates diffeomorphisms of the spatial coordinate of the string worldsheet. We find that if the worldsheet Virasoro algebra has a central extension then the spacetime Virasoro algebra acquires a central extension via a mechanism similar to that encountered in the context of the SL(2,R) WZW model.Our observations are consistent with a recently proposed duality between bosonic strings on zero radius AdS_d+1 and free field theory in d dimensions. 
  The survey summarizes briefly the results obtained recently in the Casimir effect studies considering the following subjects: i) account of the material characteristics of the media and their influence on the vacuum energy (for example, dilute dielectric ball); ii) application of the spectral geometry methods for investigating the vacuum energy of quantized fields with the goal to gain some insight, specifically, in the geometrical origin of the divergences that enter the vacuum energy and to develop the relevant renormalization procedure; iii) universal method for calculating the high temperature dependence of the Casimir energy in terms of heat kernel coefficients. 
  A class of exact solutions of the Faddeev model, that is, the modified SO(3) nonlinear sigma model with the Skyrme term, is obtained in the four dimensional Minkowskian spacetime. The solutions are interpreted as the isothermal coordinates of a Riemannian surface. One special solution of the static vortex type is investigated numerically. It is also shown that the Faddeev model is equivalent to the mesonic sector of the SU(2) Skyrme model where the baryon number current vanishes. 
  With a q-deformed quantum mechanical framework, features of the uncertainty relation and a novel formulation of the Schr\"odinger equation are considered. 
  For a special time-dependent homogeneous plane wave background that includes a null singularity we construct the closed string propagators. We carry out the summation over the oscillator modes and extract the worldsheet spacetime structures of string propagators specially near the singularity. We construct the closed string propagators in a time-independent smooth homogeneous plane wave background characterized by the constant dilaton, the constant null NS-NS field strength and the constant magnetic field. By expressing them in terms of the hypergeometric function we reveal the background field dependences and the worldsheet spacetime structures of string propagators. The conformal invariance condition for the constant dilaton plays a role to simplify the expressions of string propagators. 
  We discuss the modified gravity which may produce the current cosmic aceleration of the universe and eliminates the need for dark energy. It is shown that such models where the action quickly grows with the decrease of the curvature define the FRW universe with the minimal curvature. It is required the infinite time to reach the minimal curvature during the universe evolution. It is demonstrated that quantum effects of conformal fields may strongly suppress the instabilities discovered in modified gravity. We also briefly speculate on the modification of gravity combined with the presence of the cosmological constant dark energy. 
  We derive a boundary action of N=2 super-Liouville theory which preserves both N=2 supersymmetry and conformal symmetry by imposing explicitly $T={\bar T}$ and $G={\bar G}$. The resulting boundary action shows a new duality symmetry. 
  In the spherically symmetric case the requirements of regularity of density and pressures and finiteness of the ADM mass $m$, together with the weak energy condition, define the family of asymptotically flat globally regular solutions to the Einstein minimally coupled equations which includes the class of metrics asymptotically de Sitter as $r\to 0$. A source term connects smoothly de Sitter vacuum in the origin with the Minkowski vacuum at infinity and corresponds to anisotropic vacuum defined macroscopically by the algebraic structure of its stress-energy tensor invariant under boosts in the radial direction. Dependently on parameters, geometry describes vacuum nonsingular black holes, and self-gravitating particle-like structures whose ADM mass is related to both de Sitter vacuum trapped in the origin and smooth breaking of space-time symmetry. The geometry with the regular de Sitter center has been applied to estimate geometrical limits on sizes of fundamental particles, and to evaluate the gravito-electroweak unification scale from the measured mass-squared differences for neutrino. 
  We present a new (1+3)-brane solution to Einstein equations in (1+5)-space. As distinct from previous models this solution is free of singularities in the full 6-dimensional space-time. The gravitational potential transverse to the brane is an increasing (but not exponentially) function and asymptotically approaches a finite value. The solution localizes the zero modes of all kinds of matter fields and Newtonian gravity on the brane. An essential feature of the model is that different kind of matter fields have different localization radii. 
  Quantization of electrodynamics in curved space-time in the Lorenz gauge and with arbitrary gauge parameter makes it necessary to study Green functions of non-minimal operators with variable coefficients. Starting from the integral representation of photon Green functions, we link them to the evaluation of integrals involving Gamma functions. Eventually, the full asymptotic expansion of the Feynman photon Green function at small values of the world function, as well as its explicit dependence on the gauge parameter, are obtained without adding by hand a mass term to the Faddeev--Popov Lagrangian. Coincidence limits of second covariant derivatives of the associated Hadamard function are also evaluated, as a first step towards the energy-momentum tensor in the non-minimal case. 
  The extension of the Veneziano-Yankielowicz effective Lagrangian with terms including covariant derivatives is discussed. This extension is important to understand glue-ball dynamics of the theory. Though the superpotential remains unchanged, the physical spectrum exhibits completely new properties. 
  We derive a finite set of nonlinear integral equations for describing the finite size dependence of the ground state energy of the O(4) nonlinear sigma model. By modifying the kernel functions of these equations we propose nonlinear integral equations for the finite size effects in the SS-model. The equations are formulated in terms of two complex valued unknown functions and they are valid for arbitrary real values of the couplings. 
  We discuss the geometric engineering and large n transition for an N=1 U(n) chiral gauge theory with one adjoint, one conjugate symmetric, one antisymmetric and eight fundamental chiral multiplets. Our IIB realization involves an orientifold of a non-compact Calabi-Yau A_2 fibration, together with D5-branes wrapping the exceptional curves of its resolution as well as the orientifold fixed locus. We give a detailed discussion of this background and of its relation to the Hanany-Witten realization of the same theory. In particular, we argue that the T-duality relating the two constructions maps the Z_2 orientifold of the Hanany-Witten realization into a Z_4 orientifold in type IIB. We also discuss the related engineering of theories with SO/Sp gauge groups and symmetric or antisymmetric matter. 
  We consider the perturbation theory in the fermion mass (chiral perturbation theory) for the two-dimensional quantum electrodynamics. With this aim, we rewrite the theory in the equivalent bosonic form in which the interaction is exponential and the fermion mass becomes the coupling constant. We reformulate the bosonic perturbation theory in the superpropagator language and analyze its ultraviolet behavior. We show that the boson Green's functions without vacuum loops remain finite in all orders of the perturbation theory in the fermion mass. 
  We find asymptotically anti de Sitter solutions in N=8 supergravity which have negative total energy. This is possible since the boundary conditions required for the positive energy theorem are stronger than those required for finite mass (and allowed by string theory). But stability of the anti de Sitter vacuum is still ensured by the positivity of a modified energy, which includes an extra surface term. Some of the negative energy solutions describe classical evolution of nonsingular initial data to naked singularities. Since there is an open set of such solutions, cosmic censorship is violated generically in supergravity. Using the dual field theory description, we argue that these naked singularities will be resolved in the full string theory. 
  For a (2+1)-dimensional topologically massive Born-Infeld theory, we compute the interaction potential within the structure of the gauge-invariant but path-dependent variables formalism. The result is equivalent to that of $QED_3$ with a Thirring interaction term among fermions, in the short distance regime. 
  We find a large class of supersymmetric solutions in the $AdS_3 \times S^3$ background with NS fluxes. This two-parameter family of solutions preserves 8 of the 16 supersymmetries of the background. 
  We identify a deformation of the N=2 supersymmetric sigma model on a Calabi-Yau manifold X which has the same effect on B-branes as a noncommutative deformation of X. We show that for hyperkahler X such deformations allow one to interpolate continuously between the A-model and the B-model. For generic values of the noncommutativity and the B-field, properties of the topologically twisted sigma-models can be described in terms of generalized complex structures introduced by N. Hitchin. For example, we show that the path integral for the deformed sigma-model is localized on generalized holomorphic maps, whereas for the A-model and the B-model it is localized on holomorphic and constant maps, respectively. The geometry of topological D-branes is also best described using generalized complex structures. We also derive a constraint on the Chern character of topological D-branes, which includes A-branes and B-branes as special cases. 
  We continue our study of heterotic compactifications on non-Kahler complex manifolds with torsion. We give further evidence of the consistency of the six-dimensional manifold presented earlier and discuss the anomaly cancellation and possible supergravity description for a generic non-Kahler complex manifold using the newly proposed superpotential. The manifolds studied in our earlier papers had zero Euler characteristics. We construct new examples of non-Kahler complex manifolds with torsion in lower dimensions, that have non-zero Euler characteristics. Some of these examples are constructed from consistent backgrounds in F-theory and therefore are solutions to the string equations of motion. We discuss consistency conditions for compactifications of the heterotic string on smooth non-Kahler manifolds and illustrate how some results well known for Calabi-Yau compactifications, including counting the number of generations, apply to the non-Kahler case. We briefly address various issues regarding possible phenomenological applications. 
  We investigate braneworld cosmology based on the D-brane initiated in our previous paper. The brane is described by a Born-Infeld action and the gauge field is contained. The higher order corrections of an inverse string tension will be addressed. The results obtained by the truncated argument are altered by the higher order corrections. The equation of state of the gauge field on the brane is radiation-like in low energy scales and almost dust-like fluid in high energy scales. Our model is, however, limited below a critical finite value of the energy density. For the description of full history of our universe the presence of a S-brane might be essential. 
  We study the braneworld solutions based on a solvable model of 5d gauged supergravity with two scalars of conformal dimension three, which correspond to bilinear operators of fermions in the dual $\mathcal{N}=4$ super Yang-Mills theory on the boundary. An accelerating braneworld solution is obtained when both scalars are taken as the form of deformations of the super Yang-Mills theory and the bulk supersymmetry is broken. This solution is smoothly connected to the Poincare invariant brane in the limit of vanishing cosmological constant. The stability of this brane-solution and the correspondence to the gauge theory are addressed. 
  In this paper we discuss the contribution of planar diagrams to gravitational F-terms for N=1 supersymmetric gauge theories admitting a large N description. We show how the planar diagrams lead to a universal contribution at the extremum of the glueball superpotential, leaving only the genus one contributions, as was previously conjectured. We also discuss the physical meaning of gravitational F-terms. 
  We examine giant gravitons with a worldvolume magnetic flux $q$ in type IIA pp-wave background and find that they can move away from the origin along $x^4$ direction in target space satisfying $Rx^4=-q$. This nontrivial relation can be regarded as a complementary relation of the giant graviton on IIA pp-wave and is shown to be connected to the spacetime uncertainty principle. The giant graviton is also investigated in a system of N D0-branes as a fuzzy sphere solution. It is observed that $x^4$ enters into the fuzzy algebra as a deformation parameter. Such a background dependent Myers effect guarantees that we again get the crucial relation of our giant graviton. In the paper, we also find a BIon configuration on the giant graviton in this background. 
  In this letter we show how the method of [4] for the calculation of two-point functions in d+1-dimensional AdS space can be simplified. This results in an algorithm for the evaluation of the two-point functions as linear combinations of Legendre functions of the second kind. This algorithm can be easily implemented on a computer. For the sake of illustration, we displayed the results for the case of symmetric traceless tensor fields with rank up to l=4. 
  This work propose an alternative and systematic way to obtain a canonical Lagrangian formulation for rotational systems. This will be done in the symplectic framework and with the introduction of extra variables which enlarge the phase space. In fact, this formalism provides a remarkable and new result to compute the canonical Lagrangian formulation for rotational systems, {\it i.e.}, the obstruction to the construction of a canonical formalism can be solved in an arbitrary way and, consequently, a set of dynamically equivalent Lagrangian descriptions can be computed. 
  In this paper we construct Seiberg-Witten maps for superfields on canonically deformed N=1, d=4 Minkowski and Euclidean superspace. On Minkowski superspace we show that the Seiberg-Witten map is not compatible with locality, (anti)chirality and supersymmetry at the same time. On Euclidean superspace we show that there exists a local, chiral and supersymmetric Seiberg-Witten map for chiral superfields if we take the noncommutativity parameter to be selfdual, and a local, antichiral and supersymmetric Seiberg-Witten map for antichiral superfields if we take the noncommutativity parameter to be antiselfdual, respectively. 
  In this short note we will study effective action for unstable D-brane in linear dilaton background. We will solve the equation of motion for large T and we will calculate the stress energy tensor. Then we compare our results with the calculations performed using exact conformal field theory description of the open string worldsheet theory. 
  A simple technique for the construction of gravity theories in Born-Infeld style is presented, and the properties of some of these novel theories are investigated. They regularize the positive energy Schwarzschild singularity, and a large class of models allows for the cancellation of ghosts. The possible correspondence to low energy string theory is discussed. By including curvature corrections to all orders in alpha', the new theories nicely illustrate a mechanism that string theory might use to regularize gravitational singularities. 
  We analyze the quantum process in which a cosmic string breaks in a de Sitter (dS) background, and a pair of neutral or charged black holes is produced at the ends of the string. The energy to materialize and accelerate the pair comes from the positive cosmological constant and, in addition, from the string tension. The compact saddle point solutions without conical singularities (instantons) or with conical singularities (sub-maximal instantons) that describe this process are constructed through the analytical continuation of the dS C-metric. Then, we explicitly compute the pair creation rate of the process. In particular, we find the nucleation rate of a cosmic string in a dS background, and the probability that it breaks and a pair of black holes is produced. Finally we verify that, as occurs with pair production processes in other background fields, the pair creation rate of black holes is proportional to exp(S), where the gravitational entropy of the black hole, S, is given by one quarter of the area of the horizons present in the saddle point solution that mediates the process. 
  Using a discrete spectrum proposed for expectation values of canonical variables in black hole coherent states, the semiclassical entropy associated with the Schwarzschild space-time is derived to be the area of the apparent horizon. 
  We derive Yang-Mills vertex operators for (super)string theory whose BRST invariance requires only the free gauge-covariant field equation and no gauge condition. Standard conformal field theory methods yield the three-point vertices directly in gauge-invariant form. 
  A link between matroid theory and $p$-branes is discussed. The Schild type action for $p$-branes and matroid bundle notion provide the two central structures for such a link. We use such a connection to bring the duality concept in matroid theory to $p$-branes physics. Our analysis may be of particular interest in M-theory and in matroid bundle theory. 
  We examine gauge theories on Minkowski space-time times fuzzy coset spaces. This means that the extra space dimensions instead of being a continuous coset space S/R are a corresponding finite matrix approximation. The gauge theory defined on this non-commutative setup is reduced to four dimensions and the rules of the corresponding dimensional reduction are established. We investigate in particular the case of the fuzzy sphere including the dimensional reduction of fermion fields. 
  The description of quantum field systems with meta-stable vacuum is motivated by studies of many physical problems (the decay of disoriented chiral condensate, the resonant decay of CP-odd meta-stable states, self-consistent model of QGP pre-equilibrium evolution, the phase transition problem in the systems with broken symmetry etc). A non-perturbative approach based on the kinetic description within the framework of the quasi-particle representation was proposed here. We restrict ourselves to scalar field theory with potentials of polynomial type. The back reaction mechanism, i.e. the particle production influence on background field is also discussed. Using the oscillator representation, we derive the generalized kinetic equation with non-pertrubative source term for description of particle-antiparticle creation under action of background field and equation of motion for it. As an illustrative example we consider one-component scalar theory with double-well potential. On this example, we study some features of proposed approach, in particular, the selection problem of stable vacuum state, what allows to avoid appearance of tachyonic regimes. The similar analysis is possible for some other models of such kind: the Friedberg-Lee model, the non-linear of $\eta$ -- meson model of Witten--Di Vecchia--Veneziano, end so on. 
  We consider superstring theories on pp-wave backgrounds which result in an integrable ${\cal N}=(2,2)$ supersymmetric Landau-Ginzburg theory on the worldsheet. We obtain exact eigenvalues of the light-cone gauge superstring hamiltonian in the massive and interacting world-sheet theory with superpotential $Z^3-Z$. We find the modes of the supergravity part of the string spectrum, and their space-time interpretation. Because the system is effectively at strong coupling on the worldsheet, these modes are not in one-to-one correspondence with the usual type IIB supergravity modes in the $p_{-} \to 0$ limit. However, the above correspondence holds in the $\alpha'\to 0$ limit. 
  In this Letter we show that the vacuum polarization of quantum fields in an anti-de Sitter space naturally gives rise to a small but nonzero cosmological constant in a brane world living in it. To explain the extremely small ratio of mass density in the cosmological constant to the Planck mass density in our universe (\approx 10^{-123}) as suggested by cosmological observations, all we need is a four-dimensional brane world (our universe) living in a five-dimensional anti-de Sitter space with a curvature radius r_0 \sim 10^{-3}cm and a fundamental Planck energy M_P \sim 10^9 GeV, and a scalar field with a mass m \sim r_0^{-1}\sim 10^{-2}eV. Probing gravity down to a scale \sim 10^{-3}cm, which is attainable in the near future, will provide a test of the model. 
  Spherically symmetric solutions of the SU(N) Einstein-Yang-Mills-Higgs system are constructed using the harmonic map ansatz. The problem reduces to solving a set of ordinary differential equations for the appropriate profile functions. In the SU(2) case, we recover the equations studied in great detail previously, while for the SU(N) (N > 2) case we find new solutions. In the SU(3) case we see that our expressions are the gravitating analogues of the solutions obtained through the SO(3) embedding into SU(2). 
  The debate between loop quantum gravity and string theory is sometimes lively, and it is hard to present an impartial view on the issue. Leaving any attempt to impartiality aside, I report here, instead, a conversation on this issue, overheard in the cafeteria of a Major American University. 
  We study the duality of $\cN=1$ gauge theories in the presence of a massless adjoint field and massive quarks by calculating the superpotential using the Dighkgraaf-Vafa matrix model and by comparing with the previous result coming from Kutasov duality. The Kutasov duality method gives a result in which one instanton term is absent. The matrix model method confirms it and also show that the absence of the one instanton term is related to the masslessness of the adjoint field. 
  We consider solitonic solutions of the DBI tachyon effective action for a non-BPS brane in the presence of an electric field. We find that for a constant electric field $\tilde E\le 1$, regular solitons compactified on a circle admit a singular and decompactified limit corresponding to Sen's proposal provided the tachyon potential satisfies some restrictions. On the other hand for the critical electric field $\tilde E=1$, regular and finite energy solitons are constructed without any restriction on the potential. 
  We present a path integral representation for massless spin one-half particles. It is shown that this gives us a super-symmetric, P-and T-non-invariant pseudoclassical model for relativistic massless spinning particles. Dirac quantization of this model is considered. 
  We generalize the classification of all supersymmetric solutions of pure N=2, D=4 gauged supergravity to the case when external sources are included. It is shown that the source must be an electrically charged dust. We give a particular solution to the resulting equations, that describes a Goedel-type universe preserving one quarter of the supersymmetries. 
  We show that the massive noncommutative U(1) theory is embedded in a gauge theory using an alternative systematic way, which is based on the symplectic framework. The embedded Hamiltonian density is obtained after a finite number of steps in the iterative symplectic process, oppositely to the result proposed using the BFFT formalism. This alternative formalism of embedding shows how to get a set of dynamically equivalent embedded Hamiltonian densities. 
  Gauge theories that have been first quantized using the Hamiltonian BRST operator formalism are described as classical Hamiltonian BRST systems with a BRST charge of the form <\Psi,\Omega\Psi>_{even} and with natural ghost and parity degrees for all fields. The associated proper solution of the classical Batalin-Vilkovisky master equation is constructed from first principles. Both of these formulations can be used as starting points for second quantization. In the case of time reparametrization invariant systems, the relation to the standard <\Psi,\Omega\Psi>_{odd} master action is established. 
  We use two different methods to obtain Asymptotically Locally Flat hyperkahler metrics of type D_k. 
  I study the physical Fock space of the tensionless string theory with perimeter action, exploring its new gauge symmetry algebra. The cancellation of conformal anomaly requires the space-time to be 13-dimensional. All particles are massless and there are no tachyon states in the spectrum. The zero mode conformal operator defines the levels of the physical Fock space. All levels can be classified by the highest Casimir operator W of the little group E(11) for massless particles in 11-dimensions. The ground state is infinitely degenerated and contains massless gauge fields of arbitrary large integer spin, realizing the irreducible representations of E(11) of fixed helicity. The excitation levels realize CSR representations of little group E(11) with an infinite number of helicities. After inspection of the first excitation level, which, as I prove, is a physical null state, I conjecture that all excitation levels are physical null states. In this theory the tensor field of the second rank does not play any distinctive role and therefore one can suggest that in this model there is no gravity. 
  In the Randall-Sundrum compactification of AdS$_5$ with detuned brane tensions, supersymmetry can be spontaneously broken by a non-trivial Wilson line for the graviphoton. The supersymmetry breaking vanishes in the tuned limit. This effect is equivalent to supersymmetry breaking by Scherk-Schwarz boundary conditions. 
  We propose a modification of the Sp(2) covariant superfield quantization to realize a superalgebra of generating operators isomorphic to the massless limit of the corresponding superalgebra of the osp(1,2) covariant formalism. The modified scheme ensures the compatibility of the superalgebra of generating operators with extended BRST symmetry without imposing restrictions eliminating superfield components from the quantum action. The formalism coincides with the Sp(2) covariant superfield scheme and with the massless limit of the osp(1,2) covariant quantization in particular cases of gauge-fixing and solutions of the quantum master equations. 
  We investigate the spectrum of the gauge theory with Chern-Simons term on the noncommutative plane, a modification of the description of the Quantum Hall fluid recently proposed by Susskind. We find a series of the noncommutative massive ``plane wave'' solutions with polarization dependent on the magnitude of the wave-vector. The mass of each branch is fixed by the quantization condition imposed on the coefficient of the noncommutative Chern-Simons term. For the radially symmetric ansatz a vortex-like solution is found and investigated. We derive a nonlinear difference equation describing these solutions and we find their asymptotic form. These excitations should be relevant in describing the Quantum Hall transitions between plateaus and the end transition to the Hall Insulator. 
  The finite-size scaling properties of the quantum Ising chain with different types of generalized defects are studied. These not only mean an alteration of the coupling constant as previously examined, but an additional arbitrary transformation in the algebra of observables at one site of the chain. One can distinguish between two classes of generalized defects: those which do not affect the finite-size integrability of the Ising chain, and on the other hand those that destroy this property. In this context, finite-size integrability is always understood as a synonym for the possibility to write the Hamiltonian of the finite chain as a bilinear expression in fermionic operators by means of a Jordan-Wigner transformation. Concerning the first type of defect, an exact solution for the scaling spectrum is obtained for the most universal defect that preserves the global Z_2 symmetry of the chain. It is shown that in the continuum limit this yields the same result as for one properly chosen `ordinary' defect, that is changing the coupling constant only, and thus the finite-size scaling spectra can be described by irreps of a shifted u(1) Kac-Moody algebra. The other type of defect is examined by means of numerical finite-size calculations. In contrast to the first case, these suggest a non-continuous dependence of the scaling dimensions on the defect parameters. A conjecture for the operator content involving only one primary field of a Virasoro algebra with central charge c=1/2 is given. 
  We show that the Goedel solution of five-dimensional gauged supergravity contains either closed time-like curves through every space-time point or none at all, dependent on the rotational parameter. In addition, we present a deformation of that solution with a parameter kappa which characterizes the symmetry of four-dimensional base space: for kappa = 1, 0, -1 it has spherical, flat and hyperbolic symmetry, respectively. Also investigated are the causal properties of the lifted solution in 10 dimensions. 
  Quantum fluctuations of a certain class of bulk operators defined in spatial sub-volumes of Minkowski space-time, have an unexpected area scaling property. We wish to present evidence that such area scaling may be ascribed to a boundary theory. We first highlight the implications of area scaling with two examples in which the boundary area of the spatial regions is not monotonous with their volume. Next, we prove that the covariance of two operators that are restricted to two different regions in Minkowski space scales linearly with their mutual boundary area. Finally, we present an example which demonstrates why this implies an underlying boundary theory. 
  A question is addressed pertinent to models of fundamental fermions in a world of high dimensions. Tex extra compactified dimensions are needed to accommodate quarks and leptons of each generation in a single spinor space carrying a representation of the spin group Spin(10). We present arguments to support a special choice of the geometry. 
  Some quantum properties of QED3 are studied with the help of an exact evolution equation of the effective action with the bare fermion mass. The resulting effective theory and the occurrence of a dynamical mass are discussed in the framework of the gradient expansion. 
  The gist of using the light cone gauge lies in the well known property of ghosts decoupling. But from the BRST point of view this is a stringency since for the construction of a nilpotent operator (from a Lie algebra) the presence of ghosts are mandatory. We will show that this is a foible which has its origins in the very fact of using just one light cone vector ($n_\mu$) instead of working with both light cone vectors ($n_\mu$ and $m_\mu$) to fulfill the light cone base vectors. This will break out ghost decoupling from theory but allowing now a consistent BRST theory for the light cone gauge. 
  A QED-based bootstrap mechanism, appearing at sufficiently small space-time scales, is suggested as an explanation for the "dark" vacuum energy that may be able to accelerate the universe. Very small-scale vacuum currents are allowed to generate small-scale electromagnetic fields, which become significant near the light cone. The resulting lepton-pair production, some of which may be tachyonic, has ramifications for the convergence of all QED perturbative processes, as well as providing possible mechanisms for "dark" matter, galactic gamma-ray bursts, and ultra-high-energy cosmic rays. 
  We describe the first convergent numerical method to determine static black hole solutions (with S^3 horizon) in 5d compactified spacetime. We obtain a family of solutions parametrized by the ratio of the black hole size and the size of the compact extra dimension. The solutions satisfy the demanding integrated first law. For small black holes our solutions approach the 5d Schwarzschild solution and agree very well with new theoretical predictions for the small corrections to thermodynamics and geometry. The existence of such black holes is thus established. We report on thermodynamical (temperature, entropy, mass and tension along the compact dimension) and geometrical measurements. Most interestingly, for large masses (close to the Gregory-Laflamme critical mass) the scheme destabilizes. We interpret this as evidence for an approach to a physical tachyonic instability. Using extrapolation we speculate that the system undergoes a first order phase transition. 
  We investigate for N = 1 supersymmetry (SUSY) the relation between a scalar supermultiplet of linear SUSY and a nonlinear (NL) SUSY model including apparently pathological higher derivaive terms of a Nambu-Goldstone (N-G) fermion besides the Volkov-Akulov (V-A) action. SUSY invariant relations with higher derivative terms of the N-G fermion, which connect the linear and NL SUSY models, are constructed at leading orders by heuristic arguments. We discuss a higher derivative action of the N-G fermion in the NL SUSY model, which apparently includes a (Weyl) ghost field. By using this relation, we also explicitly prove an equivalence between the standard NL SUSY V-A model and our NL SUSY model with the pathological higher derivatives as an example with respect to the universality of NL SUSY actions with the N-G fermion. 
  We study the SUSY breaking of the covariant gauge-fixing term in SUSY QED and observe that this corresponds to a breaking of the Lorentz gauge condition by SUSY. Reasoning by analogy with SUSY's violation of the Wess-Zumino gauge, we argue that the SUSY transformation, already modified to preserve Wess-Zumino gauge, should be further modified by another gauge transformation which restores the Lorentz gauge condition. We derive this modification and use the resulting transformation to derive a Ward identitiy relating the photon and photino propagators without using ghost fields. Our transformation also fulfills the SUSY algebra, modulo terms that vanish in Lorentz gauge. 
  In these lectures, we review the physics of time-dependent orbifolds of string theory, with particular attention to orbifolds of three-dimensional Minkowski space. We discuss the propagation of free particles in the orbifold geometries, together with their interactions. We address the issue of stability of these string vacua and the difficulties in defining a consistent perturbation theory, pointing to possible solutions. In particular, it is shown that resumming part of the perturbative expansion gives finite amplitudes. Finally we discuss the duality of some orbifold models with the physics of orientifold planes, and we describe cosmological models based on the dynamics of these orientifolds. 
  Compact abelian gauge theories in $d=2+1$ dimensions arise often as an effective field-theoretic description of models of quantum insulators. In this paper we review some recent results about the compact abelian Higgs model in $d=2+1$ in that context. 
  We study exact renormalisation group equations for the 3d Ising universality class. At the Wilson-Fisher fixed point, symmetric and antisymmetric correction-to-scaling exponents are computed with high accuracy for an optimised cutoff to leading order in the derivative expansion. Further results are derived for other cutoffs including smooth, sharp and background field cutoffs. An estimate for higher order corrections is given as well. We establish that the leading antisymmetric corrections to scaling are strongly subleading compared to the leading symmetric ones. 
  We construct two-dimensional conformal field theories with a Z_N symmetry, based on the second solution of Fateev-Zamolodchikov for the parafermionic chiral algebra. Primary operators are classified according to their transformation properties under the dihedral group (Z_N x Z_2, where Z_2 stands for the Z_N charge conjugation), as singlets, [(N-1)/2] different doublets, and a disorder operator. In an assumed Coulomb gas scenario, the corresponding vertex operators are accommodated by the Kac table based on the weight lattice of the Lie algebra B_{(N-1)/2} when N is odd, and D_{N/2} when N is even. The unitary theories are representations of the coset SO_n(N) x SO_2(N) / SO_{n+2}(N), with n=1,2,.... We suggest that physically they realize the series of multicritical points in statistical systems having a Z_N symmetry. 
  In a two-dimensional toy model, motivated from five-dimensional heterotic M-theory, we study the collision of scalar field kinks with boundaries. By numerical simulation of the full two-dimensional theory, we find that the kink is always inelastically reflected with a model-independent fraction of its kinetic energy converted into radiation. We show that the reflection can be analytically understood as a fluctuation around the scalar field vacuum. This picture suggests the possibility of spontaneous emission of kinks from the boundary due to small perturbations in the bulk. We verify this picture numerically by showing that the radiation emitted from the collision of an initial single kink eventually leads to a bulk populated by many kinks. Consequently, processes changing the boundary charges are practically unavoidable in this system. We speculate that the system has a universal final state consisting of a stack of kinks, their number being determined by the initial energy. 
  We detail numerical methods to compute the geometry of static vacuum black holes in 6 dimensional gravity compactified on a circle. We calculate properties of these Kaluza-Klein black holes for varying mass, while keeping the asymptotic compactification radius fixed. For increasing mass the horizon deforms to a prolate ellipsoid, and the geometry near the horizon and axis decompactifies. We are able to find solutions with horizon radii approximately equal to the asymptotic compactification radius. Having chosen 6-dimensions, we may compare these solutions to the non-uniform strings compactified on the same radius of circle found in previous numerical work. We find the black holes achieve larger masses and horizon volumes than the most non-uniform strings. This sheds doubt on whether these solution branches can merge via a topology changing solution. Further work is required to resolve whether there is a maximum mass for the black holes, or whether the mass can become arbitrarily large. 
  This Ph.D. thesis was submitted to the Catholic University of Leuven. We discuss two main topics: various formulations of duality-symmetric theories and D=5 supergravity (both construction and supersymmetric solutions). The thesis is structured as follows: Ch. 1 Introduction; Ch. 2 String theory toolkit; Ch. 3 Free duality-symmetric theories; Ch. 4 Interacting duality-symmetric theories; Ch. 5 Conformal supergravity in D=5: Weyl multiplets; Ch. 6 Conformal supergravity in D=5: matter and gauge-fixing; Ch. 7 Supersymmetric solutions in D=5 gauged supergravity. 
  Motivated by the renormalization group (RG) approach to $c=0$ matrix model of Bre\'zin and Zinn-Justin, we develop a RG scheme for $c=1$ matrix model on a circle and analyze how the two coupling constants in double scaling limit with critical exponent flow with the change in length scale. The RG flow equations produce a non-trivial fixed point with the correct string susceptibility exponent and the expected logarithmic scaling violation of the $c=1$ theory. The change of world-sheet free energy with length scale indicates a sign change as we increase the temperature, indicating a phase transition due to liberation of the non-singlet states. At low temperature, the RG analysis also lead to T-duality of the singlet sector free energy. The RG flow to the $c=1$ fixed point can be understood as the decay of unstable $D0$-branes with open string rolling tachyon to the 2D closed string theory described by the end point of the flow. The amplitude of the decay is extracted from the change of the world-sheet free energy described by the RG process and is in accordance with the prediction from boundary Liouville theory. 
  We address the renormalized effective action for a Randall-Sundrum brane running in 5d bulk space. The running behavior of the brane action is obtained by shifting the brane-position without changing the background and the fluctuations. After an appropriate renormalization, we obtain an effective, low energy braneworld action, in which the effective 4d Planck mass is independent of the running-position. We address some implications of this effective action. 
  We construct bosonic and fermionic matrix-vector models which describe orbifolded string worldsheets at a limit in which the dimension of the vector space and the matrix order are taken to infinity. We evaluate tree-level one-loop or multiloop amplitudes of these string worldsheets by means of Schwinger-Dyson equations and derive their expressions at the multicritical points. Some of these amplitudes resemble or are closely related to those of ordinary multicritical Hermitian matrix models by a constant factor, whereas some differ significantly. 
  We discuss a quantum extension of the holographic RG flow equation obtained previously from the classical Hamiltonian constraint in the bulk AdS supergravity. The Wheeler-DeWitt equation is proposed to generate the extended RG flow and to produce 1/N subleading corrections systematically. Our formulation in five dimensions is applied to the derivation of the Weyl anomaly of boundary N=4 SU(N) super-Yang-Mills theory beyond the large N limit. It is shown that subleading 1/N^2 corrections arising from fields in AdS_5 supergravity agree with those obtained recently by Mansfield et al. using their Schroedinger equation, thereby guaranteeing to reproduce the exact form of the boundary Weyl anomaly after summing up all of the KK modes. 
  In recent work, we have developed a variational principle for large N multi-matrix models based on the extremization of non-commutative entropy. Here, we test the simplest variational ansatz for our entropic variational principle with Monte-Carlo measurements. In particular, we study the two matrix model with action Tr[{m^2 \over 2} (A_1^2 + A_2^2) - {1 \over 4} [A_1,A_2]^2] which has not been exactly solved. We estimate the expectation values of traces of products of matrices and also those of traces of products of exponentials of matrices (Wilson loop operators). These are compared with a Monte-Carlo simulation. We find that the simplest wignerian variational ansatz provides a remarkably good estimate for observables when $m^2$ is of order unity or more. For small values of m^2 the wignerian ansatz is not a good approximation: the measured correlations grow without bound, reflecting the non-convergence of matrix integrals defining the pure commutator squared action. Comparison of this ansatz with the exact solution of a two matrix model studied by Mehta is also summarized. Here the wignerian ansatz is a good approximation both for strong and weak coupling. 
  We discuss the geometric engineering of SO/Sp gauge theories with symmetric or antisymmetric tensor matter and show that the `mysterious' rank zero gauge group factors observed by a few authors can be traced back to the effects of an orientifold which survives the geometric transition. By mapping the Konishi constraints of such models to those of the U(N) theory with adjoint matter, we show that the required shifts in the ranks of the unbroken gauge group components is due to the flux contribution of the orientifold after the transition. 
  The problem of the stabilization of moduli is discussed within the context of compactified strongly coupled heterotic string theory. It is shown that all geometric, vector bundle and five-brane moduli are completely fixed, within a phenomenologically acceptable range, by non-perturbative physics. This result requires, in addition to the full space of moduli, non-vanishing Neveu-Schwarz flux, gaugino condensation with threshold corrections and the explicit form of the Pfaffians in string instanton superpotentials. The stable vacuum presented here has a negative cosmological constant. The possibility of ``lifting'' this to a metastable vacuum with positive cosmological constant is briefly discussed. 
  Even though matrix model partition functions do not exhaust the entire set of tau-functions relevant for string theory, they seem to be elementary building blocks for many others and they seem to properly capture the fundamental symplicial nature of quantum gravity and string theory. We propose to consider matrix model partition functions as new special functions. This means they should be investigated and put into some standard form, with no reference to particular applications. At the same time, the tables and lists of properties should be full enough to avoid discoveries of unexpected peculiarities in new applications. This is a big job, and the present paper is just a step in this direction. Here we restrict our consideration to the finite-size Hermitean 1-matrix model and concentrate mostly on its phase/branch structure arising when the partition function is considered as a D-module. We discuss the role of the CIV-DV prepotential (as generating a possible basis in the linear space of solutions to the Virasoro constraints, but with a lack of understanding of why and how this basis is distinguished) and evaluate first few multiloop correlators, which generalize semicircular distribution to the case of multitrace and non-planar correlators. 
  Renormalization-group (RG) flow equations have been derived for the generalized sine-Gordon model (GSGM) and the Coulomb gas (CG) in d >= 3 of dimensions by means of Wegner's and Houghton's, and by way of the real-space RG approaches. The UV scaling laws determined by the leading-order terms of the flow equations are in qualitative agreement for all dimensions d >= 3, independent of the dimensionality, and in sharp contrast to the special case d = 2. For the 4-dimensional GSGM it is demonstrated explicitly (by numerical calculations), that the blocked potential tends to a constant effective potential in the infrared (IR) limit, satisfying the requirements of periodicity and convexity. The comparison of the RG flows for the three-dimensional GSGM, the CG, and the vortex-loop gas reveals a significant dependence on the renormalization schemes and the approximations used. 
  We calculate tree level three and four point scattering amplitudes in type II string models with matter fields localized at the intersections of D-brane wrapping cycles. The analysis of the three point amplitude is performed in the context of Yukawa couplings and it is seen that a natural mechanism for the generation of a mass hierarchy arises. The four point amplitude for fermions at the intersection of four independent stacks of D-branes is then determined. 
  This study of gauge field theories on kappa-deformed Minkowski spacetime extends previous work on field theories on this example of a noncommutative spacetime. We construct deformed gauge theories for arbitrary compact Lie groups using the concept of enveloping algebra-valued gauge transformations and the Seiberg-Witten formalism. Derivative-valued gauge fields lead to field strength tensors as the sum of curvature- and torsion-like terms. We construct the Lagrangians explicitly to first order in the deformation parameter. This is the first example of a gauge theory that possesses a deformed Lorentz covariance. 
  We study an exactly marginal deformation of N=4 SUSY Yang-Mills with gauge group U(N) using field theory and string theory methods. The classical theory has a Higgs branch for rational values of the deformation parameter. We argue that the quantum theory also has an S-dual confining branch which cannot be seen classically. The low-energy effective theory on these branches is a six-dimensional non-commutative gauge theory with sixteen supercharges. Confinement of magnetic and electric charges, on the Higgs and confining branches respectively, occurs due to the formation of BPS-saturated strings in the low energy theory. The results also suggest a new way of deconstructing Little String Theory as a large-N limit of a confining gauge theory in four dimensions. 
  Large N topological string dualities have led to a class of proposed open/closed dualities for superstrings. In the topological string context, the worldsheet derivation of these dualities has already been given. In this paper we take the first step in deriving the full ten-dimensional superstring dualities by showing how the dualities arise on the superstring worldsheet at the level of F terms. As part of this derivation, we show for F-term computations that the hybrid formalism for the superstring is equivalent to a $\hat c=5$ topological string in ten-dimensional spacetime. Using the $\hat c=5$ description, we then show that the D brane boundary state for the ten-dimensional open superstring naturally emerges on the worldsheet of the closed superstring dual. 
  We present a self-contained review of the Plane-wave/super-Yang-Mills duality, which states that strings on a plane-wave background are dual to a particular large R-charge sector of N=4, D=4 superconformal U(N) gauge theory. This duality is a specification of the usual AdS/CFT correspondence in the "Penrose limit''. The Penrose limit of AdS_5 S^5 leads to the maximally supersymmetric ten dimensional plane-wave (henceforth "the'' plane-wave) and corresponds to restricting to the large R-charge sector, the BMN sector, of the dual superconformal field theory. After assembling the necessary background knowledge, we state the duality and review some of its supporting evidence. We review the suggestion by 't Hooft that Yang-Mills theories with gauge groups of large rank might be dual to string theories and the realization of this conjecture in the form of the AdS/CFT duality. We discuss plane-waves as exact solutions of supergravity and their appearance as Penrose limits of other backgrounds, then present an overview of string theory on the plane-wave background, discussing the symmetries and spectrum. We then make precise the statement of the proposed duality, classify the BMN operators, and mention some extensions of the proposal. We move on to study the gauge theory side of the duality, studying both quantum and non-planar corrections to correlation functions of BMN operators, and their operator product expansion. The important issue of operator mixing and the resultant need for re-diagonalization is stressed. Finally, we study strings on the plane-wave via light-cone string field theory, and demonstrate agreement on the one-loop correction to the string mass spectrum and the corresponding quantity in the gauge theory. A new presentation of the relevant superalgebra is given. 
  We argue that stable, maximally symmetric compactifications of string theory to 1+1 dimensions are in conflict with holography. In particular, the finite horizon entropies of the Rindler wedge in 1+1 dimensional Minkowski and anti de Sitter space, and of the de Sitter horizon in any dimension, are inconsistent with the symmetries of these spaces. The argument parallels one made recently by the same authors, in which we demonstrated the incompatibility of the finiteness of the entropy and the symmetries of de Sitter space in any dimension. If the horizon entropy is either infinite or zero the conflict is resolved. 
  The worldsheet representation of the sum of the planar diagrams of scalar $\Phi^3$ field theory and ${\cal N}=0,1,2,4$ supersymmetric Yang-Mills theory is explained. This was a talk given to the Light Cone Workshop: Hadrons and Beyond, 5-9 August 2003, University of Durham. 
  We present several higher-dimensional spacetimes for which observers living on 3-branes experience an induced metric which bounces. The classes of examples include boundary branes on generalised S-brane backgrounds and probe branes in D-brane/anti D-brane systems. The bounces we consider normally would be expected to require an energy density which violates the weak energy condition, and for our co-dimension one examples this is attributable to bulk curvature terms in the effective Friedmann equation. We examine the features of the acceleration which provides the bounce, including in some cases the existence of positive acceleration without event horizons, and we give a geometrical interpretation for it. We discuss the stability of the solutions from the point of view of both the brane and the bulk. Some of our examples appear to be stable from the bulk point of view, suggesting the possible existence of stable bouncing cosmologies within the brane-world framework. 
  We find Lorentzian solutions of spacetime noncommutative gauge theories that are localized exponentially in space and time. Together with time translational invariance of the theories, we argue that perurbative S matrix formulation of such theories is problematic in the sense that the S matrix based on free in and out states misses the spacetime localized degrees. We show that, in 3+1 dimensions, the problem disappears for the cases where the noncommutativity becomes purely spatial by an appropriate Lorentz transformation or the noncommutativity is lightlike with its electric and magnetic parts orthogonal to each other. 
  In the recent years, Hopf algebras have been introduced to describe certain combinatorial properties of quantum field theories. I will give a basic introduction to these algebras and review some occurrences in particle physics. 
  We first review the canonical formalism with general space-like hypersurfaces developed by Dirac by rederiving the Hamilton-Jacobi equations which are satisfied by on-shell actions defined on such hypersurfaces. We compare the case of gravitational systems with that of the flat space. Next, we remark as a supplement to our previous results that the effective actions of D-brane and M-brane given by arbitrary embedding functions are on-shell actions of supergravities. 
  We show numerically that the Abelian Higgs field equations in the background of a four-dimensional rotating charged black string have vortex solutions. These solutions which have axial symmetry show that the rotating black string can support the Abelian Higgs field as hair. We find that one encounters with an electric field coupled to the Higgs scalar field for the case of rotating black string. This electric field is due to an electric charge per unit length, which increases as the rotation parameter becomes larger. We also find that the vortex thickness decreases as the rotation parameter grows up. Finally we consider the self-gravity of the Abelian Higgs field and show that the effect of the vortex is to induce a deficit angle in the metric under consideration which decreases as the rotation parameter increases. 
  Matrix model is used as a regularization of field theory on non-commutative torus. However, there exists an example that the product of the large-N limit of matrices does not coincide with that of the corresponding fields. We propose a new procedure for regularizing fields on a non-commutative torus by matrices with the help of the projection in the representation space, so that the products of the matrices coincide with those of the corresponding fields in the large-N limit. 
  The Loop Variable method that has been developed for the U(1) bosonic open string is generalized to include non-Abelian gauge invariance by incorporating "Chan-Paton" gauge group indices. The scale transformation symmetry $k(s) \to k(s) \lambda (s)$ that was responsible for gauge invariance in the U(1) case continues to be a symmetry. In addition there is a "rotation" symmetry. Both symmetries crucially involve the massive modes. However it is plausible that only a linear combination, which is the usual Yang-Mills transformation on massless fields, has a smooth (world sheet) continuum limit. We also illustrate how an infinite number of terms in the equation of motion in the cutoff theory add up to give a term that has a smooth continuum limit, and thus contributes to the low energy Yang-Mills equation of motion. 
  We exploit a gauge invariant approach for the analysis of the equations governing the dynamics of active scalar fluctuations coupled to the fluctuations of the metric along holographic RG flows. In the present approach, a second order ODE for the active scalar emerges rather simply and makes it possible to use the Green's function method to deal with (quadratic) interaction terms. We thus fill a gap for active scalar operators, whose three-point functions have been inaccessible so far, and derive a general, explicitly Bose symmetric formula thereof. As an application we compute the relevant three-point function along the GPPZ flow and extract the irreducible trilinear couplings of the corresponding superglueballs by amputating the external legs on-shell. 
  Massless localized vector field is obtained in five-dimensional supersymmetric (SUSY) QED coupled to tensor multiplets as a half BPS solution. The four-dimensional gauge coupling is obtained as a topological charge. We also find all the (bosonic) massive modes exactly for a particular value of a parameter, demonstrating explicitly the existence of a mass gap. The four-dimensional Coulomb law is shown to hold for sources placed on the wall. 
  Following our previous papers (hep-th/0212158 and hep-th/0303126) we complete the construction of the parafermionic theory with the symmetry Z_N based on the second solution of Fateev-Zamolodchikov for the corresponding parafermionic chiral algebra. In the present paper we construct the Z_N parafermionic theory for N even. Primary operators are classified according to their transformation properties under the dihedral group (Z_N x Z_2, where Z_2 stands for the Z_N charge conjugation), as two singlets, doublet 1,2,...,N/2-1, and a disorder operator. In an assumed Coulomb gas scenario, the corresponding vertex operators are accommodated by the Kac table based on the weight lattice of the Lie algebra D_{N/2}. The unitary theories are representations of the coset SO_n(N) x SO_2(N) / SO_{n+2}(N), with n=1,2,.... We suggest that physically they realise the series of multicritical points in statistical systems having a Z_N symmetry. 
  We develop the method of anholonomic frames with associated nonlinear connection (in brief, N--connection) structure and show explicitly how geometries with local anisotropy (various type of Finsler--Lagrange--Cartan--Hamilton geometry) can be modeled in the metric--affine spaces. There are formulated the criteria when such generalized Finsler metrics are effectively induced in the Einstein, teleparallel, Riemann--Cartan and metric--affine gravity. We argue that every generic off--diagonal metric (which can not be diagonalized by coordinate transforms) is related to specific N--connection configurations. We elaborate the concept of generalized Finsler--affine geometry for spaces provided with arbitrary N--connection, metric and linear connection structures and characterized by gravitational field strengths, i. e. by nontrivial N--connection curvature, Riemannian curvature, torsion and nonmetricity. We apply a irreducible decomposition techniques (in our case with additional N--connection splitting) and study the dynamics of metric--affine gravity fields generating Finsler like configurations. The classification of basic eleven classes of metric--affine spaces with generic local anisotropy is presented. 
  The anholonomic frame method is generalized for non--Riemannian gravity models defined by string corrections to the general relativity and metric-affine gravity (MAG) theories. Such spacetime configurations are modeled as metric-affine spaces provided with generic off-diagonal metrics (which can not be diagonalized by coordinate transforms) and anholonomic frames with associated nonlinear connection (N-connection) structure. We investigate the field equations of MAG and string gravity with mixed holonomic and anholonomic variables. There are proved the main theorems on irreducible reduction to effective Einstein-Proca equations with respect to anholonomic frames adapted to N-connections. String corrections induced by the antisymmetric H-fields are considered. There are also proved the theorems and criteria stating a new method of constructing exact solutions with generic off-diagonal metric ansatz depending on 3-5 variables and describing various type of locally anisotropic gravitational configurations with torsion, nonmetricity and/or generalized Finsler-affine effective geometry. We analyze solutions, generated in string gravity, when generalized Finsler-affine metrics, torsion and nonmetricity interact with three dimensional solitons. 
  We construct new classes of exact solutions in metric--affine gravity (MAG) with string corrections by the antisymmetric $H$--field. The solutions are parametrized by generic off--diagonal metrics possessing noncommutative symmetry associated to anholonomy framerelations and related nonlinear connection (N--connection) structure. We analyze the horizon and geodesic properties of a class of off--diagonal metrics with deformed spherical symmetries. The maximal analytic extension of ellipsoid type metrics are constructed and the Penrose diagrams are analyzed with respect to adapted frames. We prove that for small deformations (small eccentricities) there are such metrics that the geodesic behaviour is similar to the Schwarzcshild one. The new class of spacetimes do not possess Killing symmetries even in the limits to the general relativity and, in consequence, they are not prohibited by black hole uniqueness theorems. Such static ellipsoid (rotoid) configurations are compatible with the cosmic cenzorship criteria. We study the perturbations of two classes of static black ellipsoid solutions of four dimensional gravitational field equations. We conclude that such anisotropic black hole objects may be stable with respect to the perturbations parametrized by the Schrodinger equations in the framework of the one--dimensional inverse scattering theory. 
  We argue that the origin of non-perturbative corrections exp(-2\pi R n\mu) in the c=1 matrix model is (1,n) D-branes of Zamolodchikovs. We confirm this identification comparing the flow of these corrections under the Sine--Liouville perturbation in the two approaches. 
  We analyse different N=4 supergravities coupled to six vector multiplets corresponding to low-energy descriptions of the bulk sector of T6/Z2 orientifolds with p-brane in IIB (p odd) and in IIA (p even) superstrings. When fluxes are turned on, a gauging emerges corresponding to some non-semisimple Lie algebra related to nilpotent algebras N_p inside so(6,6), with dimension 15 + (p-3)(9-p). The non-metric axions have Stueckelberg couplings that induce a spontaneous breaking of gauge symmetries. In four cases the gauge algebra is non-abelian with a non-commutative structure of the compactification torus, due to fluxes of NS-NS and R-R forms. 
  The classical action of a two dimensional N=2 supersymmetric theory, characterized by a general K\"{a}hler potential, is written down on a non(anti)commutative superspace. The action has a power series expansion in terms of the determinant of the non(anti)commutativity parameter $C^{\alpha\beta}$. The theory is explicitly shown to preserve half of the N=2 supersymmetry, to all orders in (det C)^n. The results are further generalized to include arbitrary superpotentials as well. 
  We extend the analysis of hep-th/0304045 to the bosonic case and find the one-derivative effective action valid in the vicinity of rolling tachyons with an energy not larger than that of the original D-brane. For on-shell tachyons rolling down the well-behaved side of the potential in this theory, the energy is conserved and the pressure eventually decreases exponentially. For tachyons rolling down the ''wrong'' side, the pressure instead blows up in a finite time. 
  We compare the standard single scalar field inflationary predictions with those of an inflationary phase driven by a tachyon field. A slow-roll formalism is defined for tachyon inflation, and we derive the spectra of scalar and tensor perturbations as well as the consistency relations. At lowest order the predictions of standard and tachyon inflation are shown to be the same. Higher order deviations are present and their observational relevance is discussed. We then study some typical inflationary tachyon potentials, discuss their observational consequences and compare them with recent data. All the models predict a negative and very small running of the scalar spectral index, and they consistently lie within the 1\sigma contour of the data set. However, the regime of blue scalar spectral index and large gravity waves cannot be explored by these models. Finally, a new exact solution of the unperturbed and perturbed coupled gravity and tachyon equations is also presented. 
  We propose a new, self-consistent and dynamical scenario which gives rise to well-defined initial conditions for five-dimensional brane-world cosmologies with radion stabilization. At high energies, the five-dimensional effective theory is assumed to have a scale invariance so that it admits an expanding scaling solution as a future attractor. The system automatically approaches the scaling solution and, hence, the initial condition for the subsequent low-energy brane cosmology is set by the scaling solution. At low energies, the scale invariance is broken and a radion stabilization mechanism drives the dynamics of the brane-world system. We present an exact, analytic scaling solution for a class of scale-invariant effective theories of five-dimensional brane-world models which includes the five-dimensional reduction of the Horava-Witten theory, and provide convincing evidence that the scaling solution is a future attractor. 
  We present a general approach to construct a class of generalized topological field theories with constraints by means of generalized differential calculus and its application to connection theory. It turns out that not only the ordinary BF formulations of general relativity and Yang-Mills theories, but also the N=1,2 chiral supergravities can be reformulated as these constrained generalized topological field theories once the free parameters in the Lagrangian are specially chosen. We also show that the Chern-Simons action on the boundary may naturally be induced from the generalized topological action in the bulk, rather than introduced by hand. 
  In this paper, we investigate the dynamics and the evolution of the scale factor of a probe Dp-brane which move in the background of source Dp-branes. Action of the probe brane is described by the Born-Infeld action and the interaction with the background R-R field. When the probe brane moves away from the source branes, it expands by power law, whose index depends on the dimension of the brane. If the energy density of the gauge field on the brane is subdominant, the expansion is decelerating irrespective of the dimension of the brane. On the other hand, when the probe brane is a Nambu-Goto brane, the energy density of the gauge field can be dominant, in which case accelerating expansion occurs for $p \leq 4$. The accelerating expansion stops when the brane has expanded sufficiently so that the energy density of the gauge field become subdominant. 
  We propose a superfield formalism of Lagrangian BRST-antiBRST quantization of arbitrary gauge theories in general coordinates with the base manifold of fields and antifields desribed in terms of both bosonic and fermionic variables. 
  We consider a natural generalisation of the Laplace type operators for the case of non-commutative (Moyal star) product. We demonstrate existence of a power law asymptotic expansion for the heat kernel of such operators on T^n. First four coefficients of this expansion are calculated explicitly. We also find an analog of the UV/IR mixing phenomenon when analysing the localised heat kernel. 
  The aim of this contribution is to explain how Connes derives the standard model of electromagnetic, weak and strong forces from noncommutative geometry. The reader is supposed to be aware of two other derivations in fundamental physics: the derivation of the Balmer-Rydberg formula for the spectrum of the hydrogen atom from quantum mechanics and Einstein's derivation of gravity from Riemannian geometry. 
  We show the existence of an infinite set of non-local classically conserved charges on the Green-Schwarz closed superstring in a pp-wave background. We find that these charges agree with the Penrose limit of non-local classically conserved charges recently found for the $AdS_5 \times S^5$ Green-Schwarz superstring. The charges constructed in this paper could help to understand the role played by these on the full $AdS_5 \times S^5$ background. 
  We present an M-theory proof of the anomaly of Freed and Witten which in general shifts the quantisation law for the U(1) gauge field on a D6-brane. The derivation requires an understanding of how fields on the D6-brane lift to M-theory, together with a localisation formula which we prove using a U(1)-index theorem. We also show how the anomaly is related to the K-theory classification of Ramond-Ramond fields. In addition we discuss the M-theory origin of the D6-brane effective action, and illustrate the general arguments with a concrete example. 
  We explore the validity of the generalized Bekenstein bound, S <= pi M a. We define the entropy S as the logarithm of the number of states which have energy eigenvalue below M and are localized to a flat space region of width a. If boundary conditions that localize field modes are imposed by fiat, then the bound encounters well-known difficulties with negative Casimir energy and large species number, as well as novel problems arising only in the generalized form. In realistic systems, however, finite-size effects contribute additional energy. We study two different models for estimating such contributions. Our analysis suggests that the bound is both valid and nontrivial if interactions are properly included, so that the entropy S counts the bound states of interacting fields. 
  We give a counterexample to the large $N$ asymptotics of character values $\chi_R(U)$ of irreducible characters of SU(N) conjectured in papers of Gross-Matytsin and Kazakov-Wynter in 1995. Our counterexample is based on Kostant's calculation of values of SU(N) characters on Coxeter elements. 
  We perform three tests on our proposal to implement diffeomorphism invariance in the non-abelian D0-brane DBI action as a basepoint independence constraint between matrix Riemann normal coordinate systems. First we show that T-duality along an isometry correctly interchanges the potential and kinetic terms in the action. Second, we show that the method to impose basepoint independence using an auxiliary dN^2-dimensional non-linear sigma model also works for metrics which are curved along the brane, provided a physical gauge choice is made at the end. Third, we show that without alteration this method is applicable to higher order in velocities. Testing specifically to order four, we elucidate the range of validity of the symmetrized trace approximation to the non-abelian DBI action. 
  We use Einstein, Landau-Lifshitz, Papapetrou and Weinberg energy-momentum complexes to evaluate energy distribution of a regular black hole. It is shown that for a regular black hole, these energy-momentum complexes give the same energy distribution. This supports Cooperstock hypothesis and also Aguirregabbiria et al. conclusions. Further, we evaluate energy distribution using M$\ddot{o}$ller's prescription. This does not exactly coincide with ELLPW energy expression but, at large distances, they become same. 
  We find the Goldstino action descending from the N=1 Goldstone-Maxwell superfield action associated with the spontaneous partial supersymmetry breaking, N=2 to N=1, in superspace. The new Goldstino action has higher (second-order) spacetime derivatives, while it can be most compactly described as a solution to the simple recursive relation. Our action seems to be related to the standard (having only the first-order derivatives) Akulov-Volkov action for Goldstino via a field redefinition. 
  We discuss gauge theories for commutative but non-associative algebras related to the $ SO(2k+1)$ covariant finite dimensional fuzzy $2k$-sphere algebras. A consequence of non-associativity is that gauge fields and gauge parameters have to be generalized to be functions of coordinates as well as derivatives. The usual gauge fields depending on coordinates only are recovered after a partial gauge fixing.The deformation parameter for these commutative but non-associative algebras is a scalar of the rotation group. This suggests interesting string-inspired algebraic deformations of spacetime which preserve Lorentz-invariance. 
  We discuss the interplay between freely acting orbifold actions, discrete deformations and internal uniform magnetic fields in four-dimensional orientifold models. 
  We study space-time symmetries in Non-Commutative (NC) gauge theory in the (constrained) Hamiltonian framework. The specific example of NC CP(1) model, posited in \cite{sg}, has been considered. Subtle features of Lorentz invariance violation in NC field theory were pointed out in \cite{har}. Out of the two - Observer and Particle - distinct types of Lorentz transformations, symmetry under the former, (due to the translation invariance), is reflected in the conservation of energy and momentum in NC theory. The constant tensor $\theta_{\mu\nu}$ (the noncommutativity parameter) destroys invariance under the latter.   In this paper we have constructed the Hamiltonian and momentum operators which are the generators of time and space translations respectively. This is related to the Observer Lorentz invariance. We have also shown that the Schwinger condition and subsequently the Poincare algebra is not obeyed and that one can not derive a Lorentz covariant dynamical field equation. These features signal a loss of the Particle Lorentz symmetry. The basic observations in the present work will be relevant in the Hamiltonian study of a generic noncommutative field theory. 
  We propose a symmetry law for a doublet of different form fields, which resembles gauge transformations for matter fields. This may be done for general Lie groups, resulting in an extension of Lie algebras and group manifolds. It is also shown that non-associative algebras naturally appear in this formalism, which are briefly discussed.    Afterwards, a general connection which includes a two-form field is settled-down, solving the problem of setting a gauge theory for the Kalb-Ramond field for generical groups.    Topological Chern-Simons theories can also be defined in four dimensions, and this approach clarifies their relation to the so-called $B \wedge F$-theories. We also revise some standard aspects of Kalb-Ramond theories in view of these new perspectives.    Since this gauge connection is built upon a pair of fields consisting of a one-form and a two-form, one may define Yang-Mills theories as usually and, remarkably, also {\it minimal coupling} with bosonic matter, where the Kalb-Ramond field appears naturally as mediator; so, a new associated conserved charge can be defined. For the Abelian case, we explicitly construct the minimal interaction between $B$-field and matter following a "gauge principle" and find a novel conserved tensor current. This is our most significative result from the physical viewpoint.    This framework is also generalized in such a way that any $p$-rank tensor may be formulated as a gauge field. 
  We study U(N) SQCD with N_f <= N flavors of quarks and antiquarks by engineering it with a configuration of fractional D3-branes on a C^3 / Z_2 x Z_2 orbifold. In particular we show how the moduli space of the gauge theory naturally emerges from the classical geometry produced by the D3-branes, and how the non-perturbatively generated superpotential is recovered from geometrical data. 
  We show that a Goedel-like deformation of AdS3 in heterotic string theory can be realized as an exact string background. Indeed this class of solutions is obtained as an exactly marginal deformation of the conformal field theory describing the NS5/F1 heterotic background. It can also be embedded in type II superstrings as a Kaluza-Klein reduction. We compute the spectrum of this model as well as the genus one modular invariant partition function. We discuss the issue of closed timelike curves and the propagation of long strings. They destabilize completely the background, although we construct another exact string background that may describe the result of the condensation of these long strings. Closed timelike curves are avoided in that case. 
  We show that fractional flux from Wilson lines can stabilize the moduli of heterotic string compactifications on Calabi-Yau threefolds. We observe that the Wilson lines used in GUT symmetry breaking naturally induce a fractional flux. When combined with a hidden-sector gaugino condensate, this generates a potential for the complex structure moduli, Kahler moduli, and dilaton. This potential has a supersymmetric AdS minimum at moderately weak coupling and large volume. Notably, the necessary ingredients for this construction are often present in realistic models. We explore the type IIA dual phenomenon, which involves Wilson lines in D6-branes wrapping a three-cycle in a Calabi-Yau, and comment on the nature of the fractional instantons which change the Chern-Simons invariant. 
  Charged Black holes in Gauss-Bonnet extended gravity are studied. The electromagnetic field is coupled non-minimally, as in U(2,2) Chern-Simons theory. We find that the geometrical properties of the solution exhibit ``phase transitions" as one varies the mass and charge. The full phase diagram for all values of the ADM mass and charge is displayed. 
  Warped supergravities with lower-dimensional branes provide consistent field-theoretic framework for discussion of important aspects of modern string theory physics. In particular, it is possible to study reliably supersymmetry breakdown and its transmission between branes. In this talk we present two versions of locally supersymmetrized bigravity - one with unbroken N=1,d=4 supersymmetry, and the second one with all supersymmetries broken by boundary conditions. 
  In this paper we illustrate an interesting example of low scale inflation with an extremely large number of e-foldings. This realization can be implemented easily in hybrid inflation model where usually inflation ends via phase transition. However this phase transition can be so prolong that there is a subsequent epoch of slow roll inflation governed by the dynamics of two fields. This second bout of inflation can even resolve the $\eta$ problem which plagues certain kind of inflationary models. However we also notice that for extremely low scale inflation it is hard to obtain the right amplitude for the scalar density perturbations. In this regard we invoke alternative mechanisms for generating fluctuations. We also describe how to ameliorate the cosmological moduli problem in this context. 
  In the BRST-BFV scheme for noncommutative D-branes with constant NS $B$-field, introducing ghost degrees of freedom we construct the gauge fixed Hamiltonian and corresponding effective Lagrangian invariant under nilpotent BRST charge. It is also shown that the presence of auxiliary variables introduced via the improved Dirac formalism plays a crucial role in the construction of the BRST invariant Lagrangian. 
  Witten has argued that charges of Type I D-branes in the presence of an H-flux, take values in twisted KO-theory. We begin with the study of real bundle gerbes and their holonomy. We then introduce the notion of real bundle gerbe KO-theory which we establish is a geometric realization of twisted KO-theory. We examine the relation with twisted K-theory, the Chern character and provide some examples. We conclude with some open problems. 
  The vacuum problem of light-cone quantum field theory is reanalysed from a functional-integral point of view. 
  We derive a discrete path integral for massless fermions on a hypercubic spacetime lattice with null faces. The amplitude for a path with N steps and B bends is +/- (1/2)^N (i/sqrt{3})^B. 
  To build genuine generators of the rotations group in noncommutative quantum mechanics, we show that it is necessary to extend the noncommutative parameter $\theta $ to a field operator, which one proves to be only momentum dependent. We find consequently that this field must be obligatorily a dual Dirac monopole in momentum space. Recent experiments in the context of the anomalous Hall effect provide for a monopole in the crystal momentum space. We suggest a connection between the noncommutative field and the Berry curvature in momentum space which is at the origine of the anomalous Hall effect. 
  Inspired by the Poisson Sigma Model and its relation to 2d gravity, we consider models governing morphisms from TSigma to any Lie algebroid E, where Sigma is regarded as d-dimensional spacetime manifold. We address the question of minimal conditions to be placed on a bilinear expression in the 1-form fields, S^ij(X) A_i A_j, so as to permit an interpretation as a metric on Sigma. This becomes a simple compatibility condition of the E-tensor S with the chosen Lie algebroid structure on E. For the standard Lie algebroid E=TM the additional structure is identified with a Riemannian foliation of M, in the Poisson case E=T^*M with a sub-Riemannian structure which is Poisson invariant with respect to its annihilator bundle. (For integrable image of S, this means that the induced Riemannian leaves should be invariant with respect to all Hamiltonian vector fields of functions which are locally constant on this foliation). This provides a huge class of new gravity models in d dimensions, embedding known 2d and 3d models as particular examples. 
  We discuss possible choices for boundary conditions in the AdS/CFT correspondence, and calculate the renormalisation group flow induced by a double-trace perturbation. In running from the UV to the IR there is a unit shift in the central charge. The discrepancy between our result and results obtained by other authors is accounted for by the discovery that there is a non-trivial flow for perturbations induced by bulk fields with masses saturating the Breitenlohner-Freedman bound. 
  We show that Yang-Mills matrix integrals remain convergent when a Myers term is added, and stay in the same topological class as the original model. It is possible to add a supersymmetric Myers term and this leaves the partition function invariant. 
  I will give a brief summary of an approach to string phenomenology which is inspired by AdS/CFT correspondence and which has been pursued for the last five years. Finite-N non-SUSY theories as discussed here are not obtainable from AdS/CFT although a speculation, currently under study, is that key UV properties of infinite-N theories which may be so obtained can survive, at least in some(one?) cases for the finite-N case. Future work will study the (non-)occurrence of quadratic divergences in the resultant finite-N gauge theories. 
  We show that even in the simple framework of pure Kaluza-Klein gravity the shape moduli can generate potentials supporting inflation and/or quintessence. Using the shape moduli as the inflaton or quintessence-field has the additional benefit of being able to explain symmetry breaking in a natural geometric way. A numerical analysis suggests that in these models it may be possible to obtain sufficient e-foldings during inflation as well as a small cosmological constant at the current epoch (without fine tuning), while preserving the constraint coming from the fine structure constant. 
  We construct a family of five-dimensional gauged supergravity actions which describe flop transitions of M-theory compactified on Calabi-Yau threefolds. While the vector multiplet sector can be treated exactly, we use the Wolf spaces X(1+N) = U(1+N,2)/(U(1+N) x U(2)) to model the universal hypermultiplet together with N charged hypermultiplets corresponding to winding states of the M2-brane. The metric, the Killing vectors and the moment maps of these spaces are obtained explicitly by using the superconformal quotient construction of quaternion-Kahler manifolds. The inclusion of the extra hypermultiplets gives rise to a non-trivial scalar potential which is uniquely fixed by M-theory physics. 
  We study five-dimensional Kasner cosmologies in a time-dependent Calabi-Yau compactification of M-theory undergoing a flop transition. The dynamics of the additional states, which become massless at the transition point and give rise to a scalar potential, are taken into account using a recently constructed gauged supergravity action. Due to the dynamics of these states the moduli do not show the usual run-away behavior but oscillate around the transition region. Moreover, the solutions typically exhibit short periods of accelerated expansion. We also analyze the interplay between the geometries of moduli space and space-time. 
  We use the framework of noncommutative geometry to define a discrete model for fluctuating geometry. Instead of considering ordinary geometry and its metric fluctuations, we consider generalized geometries where topology and dimension can also fluctuate. The model describes the geometry of spaces with a countable number $n$ of points. The spectral principle of Connes and Chamseddine is used to define dynamics.We show that this simple model has two phases. The expectation value $<n>$, the average number of points in the universe, is finite in one phase and diverges in the other. Moreover, the dimension $\delta$ is a dynamical observable in our model, and plays the role of an order parameter. The computation of $<\delta>$ is discussed and an upper bound is found, $<\delta> < 2$. We also address another discrete model defined on a fixed $d=1$ dimension, where topology fluctuates. We comment on a possible spontaneous localization of topology. 
  We calculate logarithmic corrections to the Bekenstein-Hawking entropy for three-dimensional BTZ black hole with J=0 and Kerr-de Sitter (KdS) space with J=0 including the Schwarzschild-de Sitter (SdS) solution due to thermal fluctuations. It is found that there is no distinction between the event horizon of the BTZ black hole and the cosmological horizon of KdS space. We obtain the same correction to the Cardy formula for BTZ, KdS, and SdS cases. We discuss AdS/CFT and dS/ECFT correspondences in connection with logarithmic corrections. 
  We systematically analyze supersymmetric D-branes in the type IIB plane wave background using Green-Schwarz superstring theory. We find several new supersymmetric oblique and curved D-branes. The supersymmetries preserved by various configurations of these D-branes including their intersections are also identified. In particular, we show that $D_+$-branes of type (+,-,n,n) for n=1,2,3,4 preserve 4 dynamical supersymmetries by introducing gauge field excitations and newly discovered oblique D5- and D7-branes also preserve four or two dynamical supersymmetries with gauge field condensates. Furthermore we find new curved D-branes preserving four dynamical supersymmetries. 
  An analysis of the couplings of the 210 dimensional SO(10) vector multiplet to matter is given. Specifically we give an $SU(5)\times U(1)$ decomposition of the vector couplings $\bar{16}_{\pm}-16_{\pm}-210$, where $16_{\pm}$ is the semispinor of SO(10) chirality ${\pm}$, using a recently derived basic theorem. The analysis is carried out using the Wess-Zumino gauge. However, we also consider the more general situation where all components of the vector multiplet enter in the couplings with the chiral fields. Here elimination of the auxiliary fields leads to a sigma model type nonlinear Lagrangian. Interactions of the type analysed here may find applications in effective theories with the 210 vector arising as a condensate. The analysis presented here completes the explicit computation of all lowest order couplings involving the $16_{\pm}$ of spinors with Higgs and vectors multiplets using the basic theorem. 
  We construct covariant coordinate transformations on the fuzzy sphere and utilize these to construct a covariant map from a gauge theory on the fuzzy sphere to a gauge theory on the ordinary sphere. We show that this construction coincides with the Seiberg-Witten map on the Moyal plane in the appropriate limit. The analysis takes place in the algebra and is independent of any star-product representation. 
  We introduce two classes of rotating solutions of Einstein-Maxwell gravity in $n+1$ dimensions which are asymptotically anti-de Sitter type. They have no curvature singularity and no horizons. The first class of solutions, which has a conic singularity yields a spacetime with a longitudinal magnetic field and $k$ rotation parameters. We show that when one or more of the rotation parameters are non zero, the spinning brane has a net electric charge that is proportional to the magnitude of the rotation parameters. The second class of solutions yields a spacetime with an angular magnetic field and $% \kappa$ boost parameters. We find that the net electric charge of these traveling branes with one or more nonzero boost parameters is proportional to the magnitude of the velocity of the brane. We also use the counterterm method inspired by AdS/CFT correspondence and calculate the conserved quantities of the solutions. We show that the logarithmic divergencies associated to the Weyl anomalies and matter field are zero, and the $r$ divergence of the action can be removed by the counterterm method. 
  We construct non-Abelian N=2 on-shell vector multiplets in five and in four dimensions. Closing of the supersymmetry algebra imposes dynamical constraints on the fields, and these constraints should be interpreted as equations of motion. If these field equations should not be derivable from an action, we find that supersymmetry allows a broader class of target-space geometries than the familiar rigid (very) special manifolds. These theories moreover have more general potentials due to the possibility of including Fayet-Iliopoulos terms in the non-Abelian case. We show that by introducing an action, we recover the standard results. Finally, we relate the five- and the four-dimensional theories through dimensional reduction and discuss the corresponding generalised r-map. 
  We demonstrate that the recently found agreement between one-loop scaling dimensions of large dimension operators in N=4 gauge theory and energies of spinning strings on AdS_5 x S^5 extends to the eigenvalues of an infinite number of hidden higher commuting charges. This dynamical agreement is of a mathematically highly intricate and non-trivial nature. In particular, on the gauge side the generating function for the commuting charges is obtained by integrable quantum spin chain techniques from the thermodynamic density distribution function of Bethe roots. On the string side the generating function, containing information to arbitrary loop order, is constructed by solving exactly the Backlund equations of the integrable classical string sigma model. Our finding should be an important step towards matching the integrable structures on the string and gauge side of the AdS/CFT correspondence. 
  We investigate the renormalization group flow of the Yukawa model with a fixed momentum cutoff at the leading order in 1/N, where N is the number of the fermion species. We demonstrate the scale invariance of coupling constants of the Nambu-Jona-Lasinio model. 
  We revisit the construction of topological Yang-Mills theories of the Witten type with arbitrary space-time dimension and number of ``shift supersymmetry'' generators, using a superspace formalism. The super-BF structure of these theories is exploited in orderto determine their actions uniquely, up to the ambiguities due to the fixing of the Yang-Mills and BF gauge invariance. UV finiteness to all orders of perturbation theory is proved in a gauge of the Landau type. 
  A structure of the radiation-ball which is identified as a Schwarzschild black hole is found out by investigating the backreaction of Hawking radiation into space-time. The structure consists of the radiation which is gravitationally bounded in the ball and of a singularity. The entropy of the radiation in the ball is proportional to the surface-area of the ball and nearly equals to the Bekenstein entropy. The Hawking radiation is regarded as a leak-out from the ball. There arises no information paradox because there exists no horizon in the structure. 
  We study classical supertube probes on supergravity backgrounds which are sourced by over-rotating supertubes, and which therefore contain closed timelike curves. We show that the BPS probes are stable despite the appearance of negative kinetic terms in the probe action. By studying the radial oscillations of these probes, we show that closed geodesics exist on these backgrounds. 
  We discuss the potential and mass-matrix of gauged N=4 matter coupled supergravity for the case of six matter multiplets, extending previous work by considering the dependence on all scalars. We consider all semi-simple gauge groups and analyse the potential and its first and second derivatives in the origin of the scalar manifold. Although we find in a number of cases an extremum with a positive cosmological constant, these are not stable under fluctuations of all scalar fields. 
  We consider a semiclassical multiwrapped circular string pulsating on S_5, whose center of mass has angular momentum J on an S_3 subspace. Using the AdS/CFT correspondence we argue that the one-loop anomalous dimension of the dual operator is a simple rational function of J/L, where J is the R-charge and L is the bare dimension of the operator. We then reproduce this result directly from a super Yang-Mills computation, where we make use of the integrability of the one-loop system to set up an integral equation that we solve. We then verify the results of Frolov and Tseytlin for circular rotating strings with R-charge assignment (J',J',J). In this case we solve for an integral equation found in the O(-1) matrix model when J'< J and the O(+1) matrix model if J'> J. The latter region starts at J'=L/2 and continues down, but an apparent critical point is reached at J'=4J. We argue that the critical point is just an artifact of the Bethe ansatz and that the conserved charges of the underlying integrable model are analytic for all J' and that the results from the O(-1) model continue onto the results of the O(+1) model. 
  A series of exact BPS solutions are found for single and double domain walls in N=2 supersymmetric (SUSY) QED for finite gauge coupling constants. Vector fields are found to be massive, although it is localized on the wall. Massless modes can be assembled into a chiral scalar multiplet of the preserved N=1 SUSY, after an appropriate gauge choice. The low-energy effective Lagrangian for the massless fields is obtained for the finite gauge coupling. The inter-wall force is found to be much stronger than the known infinite coupling case. The previously proposed expansion in inverse powers of the gauge coupling has pathological oscillations, and does not converge to the correct finite coupling result. 
  We study the issue of radion stabilization within five-dimensional supersymmetric theories compactified on the orbifold S^1/Z_2. We break supersymmetry by the Scherk-Schwarz mechanism and explain its implementation in the off-shell formulation of five dimensional supergravity in terms of the tensor and linear compensator multiplets. We show that radion stabilization may be achieved by radiative corrections in the presence of five-dimensional fields which are quasi-localized on the boundaries through the presence of Z_2 odd mass terms. For the mechanism to work the number of quasi-localized fields should be greater than 2+N_V-N_h where N_V and N_h are the number of massless gauge- and hypermultiplets in the bulk. The radion is stabilized in a metastable Minkowski vacuum with a lifetime much larger than cosmological time-scales. The radion mass is in the meV range making it interesting for present and future measurements of deviations from the gravitational inverse-square law in the submillimeter range. 
  We derive some non-perturbative results in 1+1 and 2+1 dimensions within the context of the particle path-integral representation for a Dirac field propagator in the presence of an external field, in a formulation introduced by Migdal in 1986. We consider the specific properties of the path-integral expressions corresponding to the 1+1 and 2+1 dimensional cases, presenting a derivation of the chiral anomaly in the former and of the Chern-Simons current in the latter. We also discuss particle propagation in constant electromagnetic field backgrounds. 
  We determine the position space fermion propagator in three dimensional QED based on Ward-identity and spectral representation.There is a new type of mass singularity which governs the long distance behaviour.It leads the propagator vanish at large distance more stongly than the mass term does.This term corresponds to Dynamical mass.Momentum space proagator is compared with the analysis of Schwinger-Dyson equation and our solution contains a non-perubative effects beyond the quenched approximation with bare vertex. 
  Instanton contributions to the anomalous dimensions of gauge-invariant composite operators in the N=4 supersymmetric SU(N) Yang-Mills theory are studied in the one-instanton sector. Independent sets of scalar operators of bare dimension 2, 3, 4 and 5 are constructed in all the allowed representations of the SU(4) R-symmetry group and their two-point functions are computed in the semiclassical approximation. Analysing the moduli space integrals the sectors in which the scaling dimensions receive non-perturbative contributions are identified. The requirement that the integrations over the fermionic collective coordinates which arise in the instanton background are saturated leads to non-renormalisation properties for a large class of operators. Instanton-induced corrections to the scaling dimensions are found only for dimension 4 SU(4) singlets and for dimension 5 operators in the representation [0,1,0] of SU(4). In many cases the non-renormalisation results are argued to be specific to operators of small dimension, but for some special sectors it is shown that they are valid for arbitrary dimension. Comments are also made on the implications of the results on the form of the instanton contributions to the dilation operator of the theory and on the possibility of realising its action on the instanton moduli space. 
  We provide a general overview of the current state of the art in three generation model building proposals - using intersecting D-brane toroidal compactifications of IIA string theories - which have, only, the SM at low energy. In this context, we focus on these model building directions, where natural non-supersymmetric constructions based on  $SU(4)_C \times SU(2)_L \times SU(2)_R$, SU(5) and flipped SU(5) GUT groups, have at low energy only the Standard Model. In the flipped SU(5) GUTS, the special build up structure of the models accommodates naturally a see-saw mechanism and a new solution to the doublet-triplet splitting problem. 
  We investigate unoriented strings and superstrings in two dimensions and their dual matrix quantum mechanics. Most of the models we study have a tachyon tadpole coming from the RP^2 worldsheet which needs to be cancelled by a renormalization of the worldsheet theory. We find evidence that the dual matrix models describe the renormalized theory. The singlet sector of the matrix models is integrable and can be formulated in terms of fermions moving in an external potential and interacting via the Calogero-Moser potential. We show that in the double-scaling limit the latter system exhibits particle-hole duality and interpret it in terms of the dual string theory. We also show that oriented string theories in two dimensions can be continuously deformed into unoriented ones by turning on non-local interactions on the worldsheet. We find two unoriented superstring models for which only oriented worldsheets contribute to the S-matrix. A simple explanation for this is found in the dual matrix model. 
  Within the formulation of a q-deformed Quantum Mechanics a qualitative undercut of the q-deformed uncertainty relation from the Heisenberg uncertainty relation is revealed. When $q$ is some fixed value not equal to one, recovering of ordinary quantum mechanics and the corresponding recovering condition are discussed. 
  Little String Theory (LST) is a still somewhat mysterious theory that describes the dynamics near a certain class of time-like singularities in string theory. In this paper we discuss the topological version of LST, which describes topological strings near these singularities. For 5+1 dimensional LSTs with sixteen supercharges, the topological version may be described holographically in terms of the N=4 topological string (or the N=2 string) on the transverse part of the near-horizon geometry of NS5-branes. We show that this topological string can be used to efficiently compute the half-BPS F^4 terms in the low-energy effective action of the LST. Using the strong-weak coupling string duality relating type IIA strings on K3 and heterotic strings on T^4, the same terms may also be computed in the heterotic string near a point of enhanced gauge symmetry. We study the F^4 terms in the heterotic string and in the LST, and show that they have the same structure, and that they agree in the cases for which we compute both of them. We also clarify some additional issues, such as the definition and role of normalizable modes in holographic linear dilaton backgrounds, the precise identifications of vertex operators in these backgrounds with states and operators in the supersymmetric Yang-Mills theory that arises in the low energy limit of LST, and the normalization of two-point functions. 
  A superfield formulation is presented of the central charge anomaly in quantum corrections to solitons in two-dimensional theories with N=1 supersymmetry. Extensive use is made of the superfield supercurrent, that places the supercurrent J^{mu}_{alpha}, energy-momentum tensor Theta^{mu nu} and topological current zeta^{mu} in a supermultiplet, to study the structure of supersymmetry and related superconformal symmetry in the presence of solitons. It is shown that the supermultiplet structure of (J^{mu}_{alpha}, Theta^{mu nu}, zeta^{mu}) is kept exact while the topological current zeta^{\mu} acquires a quantum modification through the superconformal anomaly. In addition, the one-loop superfield effective action is explicitly constructed to verify the BPS saturation of the soliton spectrum as well as the effect of the anomaly. 
  Models, describing relativistic particles, where Lagrangian densities depend linearly on both the curvature and the torsion of the trajectories, are revisited in D=3 space forms. The moduli spaces of trajectories are completely and explicitly determined using the Lancret program. The moduli subspaces of closed solitons in the three sphere are also determined. 
  Chaotic inflation on the brane is considered in the context of stochastic inflation. It is found that there is a regime in which eternal inflation on the brane takes place. The corresponding probability distributions are found in certain cases. The stationary probability distribution over a comoving volume and the creation probability of a de Sitter braneworld yield the same exponential behaviour. Finally, nonperturbative effects are briefly discussed. 
  Using Schr\"odinger functional methods, we show that in the ${\cal N}=4$ SYM/Type IIB Supergravity correspondence the renormalisation of the boundary Newton and gravitational constants arising from bulk fields cancels when we sum over all the Kaluza-Klein modes of Supergravity. This accords with the expected finiteness of ${\cal N}=4$ SYM, and it is expected that other renormalisations cancel in a similar way. 
  I review in this talk different approaches to the construction of vortex and instanton solutions in noncommutative field theories. 
  At very early times, the universe was not in a vacuum state. Under the assumtion that the deviation from equillibrium was large, in particular that it is higher than the scale of inflation, we analyse the conditions for local transitions between states that are related to different vacua. All pathways lead to an attractor solution of a description of the universe by eternal inflation with domains that have different low energy parameters. The generic case favors transitions between states that have significantly different parameters rather than jumps between nearby states in parameter space. I argue that the strong CP problem presents a potential difficulty for this picture, more difficult than the hierarchy problem or the cosmological constant problem. Finally, I describe how the spectrum of quark masses may be a probe of the early dynamics of vacuum states. As an example, by specializing to the case of intersecting braneworld models, I show that the observed mass spectrum, which is approximately scale invariant, corresponds to a flat distribution in the intersection area of the branes, with a maximum area A_max ~ 100 alpha'. 
  (Anti)causal boundary conditions being imposed on the (seemingly) Hermitian Quantum Theory (HQT) as described in standard textbooks lead to an (Anti)Causal Quantum Theory ((A)CQT) with an indefinite metric. Therefore, an (anti)causal neutral scalar field is not Hermitian, as one (anti)causal neutral scalar field consists of a non-Hermitian linear combination of two (Hermitian) acausal fields. Fundamental symmetries in (A)CQT are addressed. The quantum theoretical (transition) probability and antiparticle concepts are revised. Imaginary parts of cross sections and refraction indices are related. 
  The coupled cluster method (CCM) is one of the most successful and universally applicable techniques in quantum many-body theory. The intrinsic nonlinear and non-perturbative nature of the method is considered to be one of its advantages. We present here a combination of CCM with the Wilsonian renormalization group which leads to a powerful framework for construction of effective Hamiltonian field theories. As a toy example we obtain the two-loop renormalized $\phi^{4}$ theory. 
  Recent released WMAP data show a low value of quadrupole in the CMB temperature fluctuations, which confirms the early observations by COBE. In this paper, a scenario, in which a contracting phase is followed by an inflationary phase, is constructed. We calculate the perturbation spectrum and show that this scenario can provide a reasonable explanation for lower CMB anisotropies on large angular scales. 
  The time-time component of the gluon propagator in the Coulomb gauge is believed to provide a long-range confining force. We give the result, including finite parts, for the $ D_{00} $ propagator to order $ g^2 $ in the Coulomb gauge. 
  We explicitly construct the creation operators for the quantum field configurations associated to quantum membranes (2-branes) in BF and generalized Chern-Simons theories in a spacetime of dimension D=5. The creation operators for quantum excitations carrying topological charge are also obtained in the same theories. For the case of D=5 generalized Chern-Simons theory, we show that this operator actually creates an open string with a topological charge at its tip. It is shown that a duality structure exists in general, relating the membrane and topological excitation operators and the corresponding dual algebra is derived. Composite topologically charged membranes are shown to possess generalized statistics that may, in particular, be fermionic. This is the first step for the bosonization procedure in these theories. Potential applications in the full quantization of 2-branes is also briefly discussed. 
  We present a list of all inequivalent bosonic covariant free particle wave equations in a flat spacetime of arbitrary dimension. The wave functions are assumed to have a finite number of components. We relate these wave equations to equivalent versions obtained following other approaches. 
  Motivated by the work of Klebanov and Polyakov [hep-th/0210114] on the relationship of the large N O(N) vector model in three-dimensions to AdS_4 and higher spin representations, we attempt to find analogous connections for AdS_5. Since the usual O(N) vector model in four-dimensions is inconsistent, we consider the (consistent) large N gauged vector model and a N=1 supersymmetric analogue in four-dimensions. Both these theories have UV and IR fixed points, and are candidates for a (\alpha')^-1 expansion in AdS_5, a conjectured AdS_5/CFT correspondence and higher-spin representations in the bulk theory. 
  Nucleation of branes by a four-form field has recently been considered in string motivated scenarios for the neutralization of the cosmological constant. An interesting question in this context is whether the nucleation of stacks of coincident branes is possible, and if so, at what rate does it proceed. Feng et al. have suggested that, at high ambient de Sitter temperature, the rate may be strongly enhanced, due to large degeneracy factors associated with the number of light species living on the worldsheet. This might facilitate the quick relaxation from a large effective cosmological constant down to the observed value. Here, we analyse this possibility in some detail. In four dimensions, and after the moduli are stabilized, branes interact via repulsive long range forces. Because of that, the Coleman-de Luccia (CdL) instanton for coincident brane nucleation may not exist, unless there is some short range interaction which keeps the branes together. If the CdL instanton exists, we find that the degeneracy factor depends only mildly on the ambient de Sitter temperature, and does not switch off even in the case of tunneling from flat space. This would result in catastrophic decay of the present vacuum. If, on the contrary, the CdL instanton does not exist, coindident brane nucleation may still proceed through a "static" instanton, representing pair creation of critical bubbles -- a process somewhat analogous to thermal activation in flat space. In that case, the branes may stick together due to thermal symmetry restoration, and the pair creation rate depends exponentially on the ambient de Sitter temperature, switching off sharply as the temperature approaches zero. Such static instanton may be well suited for the "saltatory" relaxation scenario proposed by Feng et al. 
  The thermal instability of the giant graviton is investigated within the BMN matrix model. We calculate the one-loop thermal correction of the quantum fluctuation around the trivial vacuum and giant graviton respectively. From the exact formula of the free energy we see that at low temperature the giant graviton is unstable and will dissolve into vacuum fluctuation. However, at sufficient high temperature the trivial vacuum fluctuation will condense to form the giant graviton configuration. The transition temperature of the giant graviton is determined in our calculation. 
  We present a calculation of critical phenomena directly in continuous dimension d employing an exact renormalization group equation for the effective average action. For an Ising-type scalar field theory we calculate the critical exponents nu(d) and eta(d) both from a lowest--order and a complete first--order derivative expansion of the effective average action. In particular, this can be used to study critical behavior as a function of dimensionality at fixed temperature. 
  In this paper we continue the program, initiated in Ref. hep-th/0112246, to investigate an integrable noncommutative version of the sine-Gordon model. We discuss the origin of the extra constraint which the field function has to satisfy in order to guarantee classical integrability. We show that the system of constraint plus dynamical equation of motion can be obtained by a suitable reduction of a noncommutative version of 4d self-dual Yang-Mills theory. The field equations can be derived from an action which is the sum of two WZNW actions with cosine potentials corresponding to a complexified noncommutative U(1) gauge group. A brief discussion of the relation with the bosonized noncommutative Thirring model is given. In spite of integrability we show that the S-matrix is acasual and particle production takes place. 
  We analyze gauge theories based on abelian $p-$forms in real compact hyperbolic manifolds. The explicit thermodynamic functions associated with skew--symmetric tensor fields are obtained via zeta--function regularization and the trace tensor kernel formula. Thermodynamic quantities in the high--temperature expansions are calculated and the entropy/energy ratios are established. 
  We reproduce Chang's duality condition in a regularized $\phi^4_{1+1}$ theory quantized on a light front. The regularization involves higher derivatives in the Lagrangian, renders the model finite in the ultraviolet, and does not require introduction of a finite size of the system. It is demonstrated that the light-front quantization is a natural way to treat systems with higher derivatives. The phase transition is related to the presence of tachyons in the regularized theory. Prospects for computing the critical coupling in this formulation are briefly discussed. 
  In the previous submission of this paper we claimed to have solved the problem of constructing a consistent non-abelian Born Infeld action which is $\kappa$--supersymmetric to all orders. Our method was based on an extension of target superspace to $\mathrm{N} \times \mathrm{N}$ matrix--valued objects for all the items appearing in the geometric construction. Then we applied the \textit{double first order formalism} introduced by us in the previous construction of abelian brane actions to the non--abelian case and we claimed that the Bosonic action given by the prescription of the \textit{symmetrized trace} could be promoted to a fully kappa supersymmetric one. Unfortunately there is an internal inconsistency, relative to U(N) gauge invariance in the matricisation of target superspace geometry and there is a subtle inconsistency in the variation of the symmetric trace action due to the loss of associativity of the underlying symmetrized product. Because of that the entire construction previously submitted does not stand on its feet. Consequently the present resubmission is done in order to disclaim our previously claimed result and also to express our sincere gratitude to our friends Paul Howe and Ulf Lindstrom who, with their constructive criticism and private correspondence, have helped us to understand where the bugs in our construction were. 
  We quantise a Poisson structure on H^{n+2g}, where H is a semidirect product group of the form $G\ltimes\mathfrak{g}^*$. This Poisson structure arises in the combinatorial description of the phase space of Chern-Simons theory with gauge group $G\ltimes\mathfrak{g}^*$ on $R \times S_{g,n}$, where S_{g,n} is a surface of genus g with n punctures. The quantisation of this Poisson structure is a key step in the quantisation of Chern-Simons theory with gauge group $G\ltimes\mathfrak{g}^*$. We construct the quantum algebra and its irreducible representations and show that the quantum double D(G) of the group G arises naturally as a symmetry of the quantum algebra. 
  We present classical solutions of Dp-branes $(p \ge 5)$ in plane wave spacetime with nonconstant R-R 3-form flux. We also show the existence of a system of D3-branes in this background. We further analyze the supersymmetric properties of these branes by solving type II Killing spinor equations explicitly. 
  We consider a model with a charged vector field along with a Cremmer-Scherk-Kalb-Ramond (CSKR) matter field coupled to a U(1) gauge potential. We obtain a natural Lorentz symmetry violation due to the local U(1) spontaneous symmetry breaking mechanism triggered by the imaginary part of the vector matter. The choice of the unitary gauge leads to the decoupling of the gauge-KR sector from the Higgs-KR sector. The excitation spectrum is carefully analyzed and the physical modes are identified. We propose an identification of the neutral massive spin-1 Higgs-like field with the massive Z' boson of the so-called mirror matter models. 
  Causality on the gravity side of the AdS/CFT correspondence restricts motion on the moduli space of the N=4 super Yang Mills theory by imposing a speed limit on how fast the scalar field may roll. This effect can be traced to higher derivative operators arising from integrating out light degrees of freedom near the origin. In the strong coupling limit of the theory, the dynamics is well approximated by the Dirac-Born-Infeld Lagrangian for a probe D3-brane moving toward the horizon of the AdS Poincare patch, combined with an estimate of the (ultimately suppressed) rate of particle and string production in the system. We analyze the motion of a rolling scalar field explicitly in the strong coupling regime of the field theory, and extend the analysis to cosmological systems obtained by coupling this type of field theory to four dimensional gravity. This leads to a mechanism for slow roll inflation for a massive scalar at subPlanckian VEV without need for a flat potential (realizing a version of k-inflation in a microphysical framework). It also leads to a variety of novel FRW cosmologies, some of which are related to those obtained with tachyon matter. 
  We write an explicit local action for a large extra dimensions stabilization scenario due to Arkani-Hamed, Hall, Smith and Weiner (AHSW). Our action allows the AHSW proposal to be generalized to non-Poincare invariant configurations, supersymmetric extensions, quantum effective field theory, and cosmological scenarios. The central step in constructing the action is working in terms of the four form gauge field which is the "electric-magnetic" dual to the "magnetic" scalar of AHSW. The action is manifestly invariant under higher general coordinate invariance. 
  The Bekenstein bound takes the holographic principle into the realm of flat space, promising new insights on the relation of non-gravitational physics to quantum gravity. This makes it important to obtain a precise formulation of the bound. Conventionally, one specifies two macroscopic quantities, mass and spatial width, which cannot be simultaneously diagonalized. Thus, the counting of compatible states is not sharply defined. The resolution of this and other formal difficulties leads naturally to a definition in terms of discretized light-cone quantization. In this form, the area difference specified in the covariant bound converts to a single quantum number, the harmonic resolution K. The Bekenstein bound then states that the Fock space sector with K units of longitudinal momentum contains no more than exp(2 pi^2 K) independent discrete states. This conjecture can be tested unambiguously for a given Lagrangian, and it appears to hold true for realistic field theories, including models arising from string compactifications. For large K, it makes contact with more conventional but less well-defined formulations. 
  We study the question of whether the linearization of the Kodama state around classical deSitter spacetime is normalizable in the inner product of the theory of linearized gravitons on deSitter spacetime. We find the answer is no in the Lorentzian theory. However, in the Euclidean theory the corresponding linearized Kodama state is delta-functional normalizable. We discuss whether this result invalidates the conjecture that the full Kodama state is a good physical state for quantum gravity with positive cosmological constant. 
  We consider supergravity solutions corresponding to D7 branes wrapped on Kahler manifolds with a U(1)_R twist such that some supersymmetry is preserved. We find a class of 1/4-BPS backgrounds where a D7-brane is wrapped on a T^2 torus with a metric of non-constant curvature. Similarly to the flat D7-brane case, the solution has a singularity at finite radius. We also discuss the case where the D7-brane is wrapped on a 4-dimensional non-compact manifold.  The field theories on the D7 brane have N=1 supersymmetry in 6 and 4 dimensions respectively. 
  This paper was withdrawn by the author. 
  A remarkable consequence of the AdS/CFT correspondence is the nonperturbative derivation of dimensional counting rules for hard scattering processes. Using string/gauge duality we derive the QCD power behavior of light-front Fock-state hadronic wavefunctions for hard scattering in the large-$r$ region of the AdS space from the conformal isometries which determine the scaling of string states as we approach the boundary from the interior of AdS space. The nonperturbative scaling results are obtained for spin-zero and spin-$\half$ hadrons and are extended to include the orbital angular momentum dependence of the constituents in the light-front Fock-expansion. The correspondence with string states is considered for hadronic states of arbitrary orbital angular momentum for a given hadron of spin $\leq 2$. We examine the implications of the color configuration of hadronic Fock-states for the QCD structure of scattering amplitudes at large $N_C$. Quark interchange amplitudes emerge as the dominant large $N_C$ scattering mechanisms for conformal QCD. 
  We consider D-branes of open superstrings in the AdS_5 x S^5 background. The possible configurations of D-branes preserving half of supersymmetries are classified by analyzing the kappa-invariance of an open superstring in a covariant manner. We also revisit the classification of D-branes in the pp-wave background. It is shown that Penrose limits of the possible D-branes in the AdS_5 x S^5 give all of the D-branes in the pp-wave. In addition, 1/4 supersymmetric D-string, which is related to the D-string preserving 8 dynamical supersymmetries in the pp-wave, is presented. We also discuss the relation between our branes and the AdS branes in a brane probe analysis. 
  We generalized the 't Hooft-Veltman method of unitary regulators to put forward a path-integral framework for finite, alternative theories to a given quantum field theory. And we demonstrated that the proposed framework is feasible by providing a finite alternative to the quantum field theory of a single, self-interacting real scalar field. Here we give two properties of self-energy that make the corresponding scattering matrix unitary. We show that the perturbative self-energy has these two properties at least up to the second order in the coupling constant. 
  In this talk, we discuss four-dimensional N=1 affine $ADE$ quiver gauge models using the geometric engineering method in M-theory on $G_2$ manifolds with K3 fibrations. 
  We investigate $4+d$ dimensional fermionic models in which the system in codimension-$d$ supports a topologically stable solution, and in which the fermion may be localised to the brane, with power law in 'instanton' backgrounds and exponentially in 'soliton' backgrounds. When the fermions are isoscalars, the mechanism fails, while for isospinor fermions it is successful. As backgrounds we consider instantons of Yang--Mills and sigma models in even codimensions, solitons of sigma models in odd codimensions, as well as solitons of Higgs and Goldstone models in all codimensions. 
  We show the three-loop integrability of large N plane-wave matrix theory in a subsector of states comprised of two complex light scalar fields. This is done by diagonalizing the theory's Hamiltonian in perturbation theory and taking the large N limit. At one-loop level the result is known to be equal to the Heisenberg spin-1/2 chain, which is a well-known integrable system. Here, integrability implies the existence of hidden conserved charges and results in a degeneracy of parity pairs in the spectrum. In order to confirm integrability at higher loops, we show that this degeneracy is not lifted and that (corrected) conserved charges exist. Plane-wave matrix theory is intricately connected to N=4 Super Yang-Mills, as it arises as a consistent reduction of the gauge theory on a three-sphere. We find that after appropriately renormalizing the mass parameter of the plane-wave matrix theory the effective Hamiltonian is identical to the dilatation operator of N=4 Super Yang-Mills theory in the considered subsector. Our results therefore represent a strong support for the conjectured three-loop integrability of planar N=4 SYM and are in disagreement with a recent dual string theory finding. Finally, we study the stability of the large N integrability against nonsupersymmetric deformations of the model. 
  We extend the argument of Gomis and Mehen for violation of unitarity in field theories with space-time noncommutativity to dipole field theories. In dipole field theories with a timelike dipole vector, we present 1-loop amplitudes that violate the optical theorem. A quantum mechanical system with nonlocal potential of finite extent in time also shows violation of unitarity. 
  In this paper I give an explicit conformal field theory description of (2+1)-dimensional BTZ black hole entropy. In the boundary Liouville field theory I investigate the reducible Verma modules in the elliptic sector, which correspond to certain irreducible representations of the quantum algebra U_q(sl_2) \odot U_{\hat{q}}(sl_2). I show that there are states that decouple from these reducible Verma modules in a similar fashion to the decoupling of null states in minimal models. Because ofthe nonstandard form of the Ward identity for the two-point correlation functions in quantum Liouville field theory, these decoupling states have positive-definite norms. The explicit counting from these states gives the desired Bekenstein-Hawking entropy in the semi-classical limit when q is a root of unity of odd order. 
  We apply the method of geometric transition and compute all genus topological closed string amplitudes compactified on local {\bf F}_0 by making use of the Chern-Simons gauge theory. We find an exact agreement of the results of our computation with the formula proposed recently by Nekrasov for {\cal N}=2 SU(2) gauge theory with two parameters \beta and \hbar. \beta is related to the size of the fiber of {\bf F}_0 and \hbar corresponds to the string coupling constant.   Thus Nekrasov's formula encodes all the information of topological string amplitudes on local {\bf F}_0 including the number of holomorphic curves at arbitrary genus. By taking suitable limits \beta and/or \hbar \to 0 one recovers the four-dimensional Seiberg-Witten theory and also its coupling to external graviphoton fields.   We also compute topological string amplitude for the local 2nd del Pezzo surface and check the consistency with Nekrasov's formula of SU(2) gauge theory with a matter field in the vector representation. 
  We derive and solve the compositeness condition for the SU(N_c) gauge boson coupling with N_s scalar fields at the next-to-leading order in 1/N_s and the leading order in ln(\Lambda^2) (\Lambda is the compositeness scale). It turns out that the argument of gauge-boson compositeness (with a large \Lambda) is successful only when N_s/N_c>22, in which the asymptotic freedom fails, as is in the previously investigated case with fermionic matter. 
  A formulation of the N_T=1, D=8 Euclidean super Yang-Mills theory with generalized self-duality and reduced Spin(7)-invariance is given which avoids the peculiar extra constraints of Nishino and Rajpoot, hep-th/0210132. Its reduction to 7 dimensions leads to the G_2-invariant N_T=2, D=7 super Yang-Mills theory which may be regarded as a higher-dimensional analogue of the N=2, D=3 super-BF theory. When reducing further that G_2-invariant theory to 3 dimensions one gets the N_T=2 super-BF theory coupled to a spinorial hypermultiplet. 
  The free energies of the conformal field theories dual to charged adS and rotating adS black holes show Hawking-Page phase transition. We study the transition by constructing boundary free energies in terms of order parameters. This is done by employing Landau's phenomenological theory of first order phase transition. The Cardy-Verlinde formula is then showed to follow quite naturally. We further make some general observations on the Cardy-Verlinde formula and the first order phase transition. 
  Recent work on exact renormalization group flow equations has pointed out the possibility to study critical phenomena in continuous dimension D of space. In an investigation of the O(N) model the dimension N of the fields may be seen as a continuous parameter as well. One may conjecture that a variation of D or N in the vicinity of a second order phase transition yields the same critical exponents as a variation of the temperature. A numerical computation confirms this. 
  We emphasize that the group-theoretical considerations leading to SO(10) unification of electro-weak and strong matter field components naturally extend to space-time components, providing a truly unified description of all generation degrees of freedoms in terms of a single chiral spin representation of one of the groups SO(13,1), SO(9,5), SO(7,7) or SO(3,11). The realization of these groups as higher dimensional space-time symmetries produces unification of all fundamental fermions is a single space-time spinor. 
  Based on the geometric interpretation of the Dirac equation as an evolution equation on the three-dimensional exterior bundle /(R^3), we propose the bundle (T x / x /)(R^3) as a geometric interpretation of all standard model fermions. The generalization to curved background requires an ADM decomposition M^4=M^3 x R and gives the bundle (T x / x /)(M^3). As a consequence of the geometric character of the bundle there is no necessity to introduce a tetrad or triad formalism. Our space-geometric interpretation associates colors as well as fermion generations with directions in space, electromagnetic charge with the degree of a differential form, and weak interactions with the Hodge star operator.   The space-geometric interpretation leads to different physical predictions about the connection of the SM with gravity, but gives no such differences on Minkowski background. 
  We propose a twisted D=N=2 superspace formalism. The relation between the twisted super charges including the BRST charge, vector and pseudo scalar super charges and the N=2 spinor super charges is established. We claim that this relation is essentially related with the Dirac-K\"ahler fermion mechanism. We show that a fermionic bilinear form of twisted N=2 chiral and anti-chiral superfields is equivalent to the quantized version of BF theory with the Landau type gauge fixing while a bosonic bilinear form leads to the N=2 Wess-Zumino action. We then construct a Yang-Mills action described by the twisted N=2 chiral and vector superfields, and show that the action is equivalent to the twisted version of the D=N=2 super Yang-Mills action, previously obtained from the quantized generalized topological Yang-Mills action with instanton gauge fixing. 
  The local and manifestly covariant Lagrangian interactions in four spacetime dimensions that can be added to a ``free'' model that describes a generic matter theory and an abelian BF theory are constructed by means of deforming the solution to the master equation on behalf of specific cohomological techniques. 
  Supersymmetric orientifolds of four dimensional Gepner Models are constructed in a systematic way. For all levels of the Gepner model being odd the generic expression for both the A-type and the B-type Klein bottle amplitude is derived. The appearing massless tadpoles are canceled by introducing appropriate boundary states of Recknagel/Schomerus(RS). After determining the Moebius strip amplitude we extract general expressions for the tadpole cancellation conditions. We discuss the issue of chirality for such supersymmetric orientifold models and finally present a couple of examples in detail. 
  A precise correspondence between freely-acting orbifolds (Scherk-Schwarz compactifications) and string vacua with NSNS flux turned on is established using T-duality.   We focus our attention to a certain non-compact Z_2 heterotic freely-acting orbifold with N=2 supersymmetry (SUSY). The geometric properties of the T-dual background are studied. As expected, the space is non-Kahler with the most generic torsion compatible with SUSY. All equations of motion are satisfied, except the Bianchi identity for the NSNS field, that is satisfied only at leading order in derivatives, i.e. without the curvature term. We point out that this is due to unknown corrections to the standard heterotic T-duality rules. 
  We investigate the idea of a "general boundary" formulation of quantum field theory in the context of the Euclidean free scalar field. We propose a precise definition for an evolution kernel that propagates the field through arbitrary spacetime regions. We show that this kernel satisfies an evolution equation which governs its dependence on deformations of the boundary surface and generalizes the ordinary (Euclidean) Schroedinger equation. We also derive the classical counterpart of this equation, which is a Hamilton-Jacobi equation for general boundary surfaces. 
  In this talk, we describe our present understanding of thermal field theories on the light-front with an application to Schwinger model. 
  The similarity renormalization group procedure formulated in terms of effective particles is briefly reviewed in a series of selected examples that range from the model matrix estimates of its numerical accuracy to issues of the Poincare symmetry in constituent theories to derivation of the Schroedinger equation for quarkonia in QCD. 
  We explore the space of solutions of the classical equations of motion in the Euclidean electroweak theory. We sketch a topological prescription that finds known solutions and indicates the existence of novel ones. All spatially-varying, time-independent solutions are unstable. However, if we consider quantum fluctuations around static classical configurations, it may be possible to find stable solutions called quantum solitons. Such objects carry a conserved quantum number, in analogy with a topological soliton carrying a topological charge. We explain the mechanism and motivation for the existence of a quantum soliton and describe our search for one within a spherical ansatz. We also comment on promising candidates outside the ansatz. 
  The equivalence of the chain method and Hamilton-Jacobi formalism is demonstrated. The stabilization algorithm of Hamilton-Jacobi formalism is clariffied and two examples are presented in details. 
  The cosmological constant problem is usually considered an inevitable feature of any effective theory capturing well-tested gravitational and matter physics, without regard to the details of short-distance gravitational couplings. In this paper, a subtle effective description avoiding the problem is presented in a first quantized language, consistent with experiments and the Equivalence Principle. First quantization allows a minimal domain of validity to be carved out by cutting on the proper length of particle worldlines. This is facilitated by working in (locally) Euclidean spacetime, although considerations of unitarity are still addressed by analytic continuation from Lorentzian spacetime. The new effective description demonstrates that the cosmological constant problem {\it is} sensitive to short-distance details of gravity, which can be probed experimentally. ``Fat Gravity'' toy models are presented, illustrating how gravity might shut off at short but testable distances, in a generally covariant manner that suppresses the cosmological constant. This paper improves on previous work by allowing generalizations to massless matter, non-trivial spins, non-perturbative phenomena, and multiple (metastable) vacua. 
  The complete one-loop, planar dilatation operator of the N=4 superconformal gauge theory was recently derived and shown to be integrable. Here, we present further compelling evidence for a generalisation of this integrable structure to higher orders of the coupling constant. For that we consider the su(2|3) subsector and investigate the restrictions imposed on the spin chain Hamiltonian by the symmetry algebra. This allows us to uniquely fix the energy shifts up to the three-loop level and thus prove the correctness of a conjecture in hep-th/0303060. A novel aspect of this spin chain model is that the higher-loop Hamiltonian, as for N=4 SYM in general, does not preserve the number of spin sites. Yet this dynamic spin chain appears to be integrable. 
  We consider open superstring partition function Z on the disc in time-dependent tachyon background T= f(x_i) e^{m x_0} where the profile function f depends on spatial coordinates. We compute Z to second order in derivatives of f and compare the result with some previously suggested effective actions depending only on the first derivatives of the tachyon field. We also compute the target-space stress-energy tensor in this background and demonstrate its conservation in the ``on-shell'' case of the linear profile f= f_0 + q_i x_i corresponding to a marginal perturbation. We comment on the role of the rolling tachyon with linear spatial profile in the decay of an unstable D-brane. 
  We study non-abelian monopole operators in the infrared limit of three-dimensional SU(N_c) and N=4 SU(2) gauge theories. Using large N_f expansion and operator-state isomorphism of the resulting superconformal field theories, we construct monopole operators which are (anti-)chiral primaries and compute their charges under the global symmetries. Predictions of three-dimensional mirror symmetry for the quantum numbers of these monopole operators are verified. 
  We analyse the existence of closed timelike curves in spacetimes which possess an isometry. In particular we check which discrete quotients of such spaces lead to closed timelike curves. As a by-product of our analysis, we prove that the notion of existence or non-existence of closed timelike curves is a T-duality invariant notion, whenever the direction along which we apply such transformations is everywhere spacelike. Our formalism is straightforwardly applied to supersymmetric theories. We provide some new examples in the context of D-branes and generalized pp-waves. 
  We study the hidden symmetries arising in the dimensional reduction of d=5, N=2 supergravity to three dimensions. Extending previous partial results for the bosonic part, we give a derivation that includes fermionic terms, shedding light on the appearance of the local hidden symmetry SO(4) in the reduction. 
  We derive general and complete expressions for N-point tree-level amplitudes in Type II string models with matter fields localised at D-brane intersections. 
  We present some explicit computations checking a particular form of gradient formula for a boundary beta function in two-dimensional quantum field theory on a disc. The form of the potential function and metric that we consider were introduced in hep-th/9210065, hep-th/9311177 in the context of background independent open string field theory. We check the gradient formula to the third order in perturbation theory around a fixed point. Special consideration is given to situations when resonant terms are present exhibiting logarithmic divergences and universal nonlinearities in beta functions. The gradient formula is found to work to the given order. 
  We find the metric of small black holes on cylinders, i.e. neutral and static black holes with a small mass in d-dimensional Minkowski-space times a circle. The metric is found using an ansatz for black holes on cylinders proposed in hep-th/0204047. We use the new metric to compute corrections to the thermodynamics which is seen to deviate from that of the (d+1)-dimensional Schwarzschild black hole. Moreover, we compute the leading correction to the relative binding energy which is found to be non-zero. We discuss the consequences of these results for the general understanding of black holes and we connect the results to the phase structure of black holes and strings on cylinders. 
  We introduce $3N\times 3N$ Lax pair with spectral parameter for Calogero-Inozemtsev model. The one degree of freedom case appears to have $2\times 2$ Lax representation. We derive it from the elliptic Gaudin model via some reduction procedure and prove algebraic integrability. This Lax pair provides elliptic linear problem for the Painlev{\'e} VI equation in elliptic form. 
  We study the supersymmetric extension of the Faddeev model in four dimensions. The Faddeev model contains three dimensional soliton solutions and we are interested in how these solitons are affected by supersymmetry. We consider both the N=1 and N=2 extensions and find that in neither case it is possible to supersymmetrize the model without adding additional bosonic terms. There are essentially two ways of constructing the supersymmetric theory, one that will lead to a model which allows for solitons and another that gives a model where solitons are excluded.   The N=2 model is studied since extending supersymmetry is the natural way of including topological charges in the algebra. A lower bound to the mass is obtained by computing the central charge. The result is that it is possible to have a non-trivial lower bound on the mass, this in principle allows for massive solitons. 
  Stacks of D3-branes placed at the tip of a cone over a del Pezzo surface provide a way of geometrically engineering a small but rich class of gauge/gravity dualities. We develop tools for understanding the resulting quiver gauge theories using exceptional collections. We prove two important results for a general quiver gauge theory: 1) we show the ordering of the nodes can be determined up to cyclic permutation and 2) we derive a simple formula for the ranks of the gauge groups (at the conformal point) in terms of the numbers of bifundamentals. We also provide a detailed analysis of four node quivers, examining when precisely mutations of the exceptional collection are related to Seiberg duality. 
  We consider the large-$D$ limit of Einstein gravity. It is observed that a consistent leading large-$D$ graph limit exists, and that it is built up by a subclass of planar diagrams. The graphs in the effective field theory extension of Einstein gravity are investigated in the same context, and it is seen that an effective field theory extension of the basic Einstein-Hilbert theory will not upset the latter leading large-$D$ graph limit, {\it i.e.}, the same subclass of planar diagrams will dominate at large-$D$ in the effective field theory. The effective field theory description of large-$D$ quantum gravity limit will be renormalizable, and the resulting theory will thus be completely well defined up to the Planck scale at $\sim 10^{19}$ GeV. The $(\frac1D)$ expansion in gravity is compared to the successful $(\frac1N)$ expansion in gauge theory (the planar diagram limit), and dissimilarities and parallels of the two expansions are discussed. We consider the expansion of the effective field theory terms and we make some remarks on explicit calculations of $n$-point functions. 
  The twisted butterfly state solves the equation of motion of vacuum string field theory in the singular limit. The finiteness of the energy density of the solution is an important issue, but possible conformal anomaly resulting from the twisting has prevented us from addressing this problem. We present a description of the twisted regulated butterfly state in terms of a conformal field theory with a vanishing central charge which consists of the ordinary bc ghosts and a matter system with c=26. Various quantities relevant to vacuum string field theory are computed exactly using this description. We find that the energy density of the solution can be finite in the limit, but the finiteness depends on the subleading structure of vacuum string field theory. We further argue, contrary to our previous expectation, that contributions from subleading terms in the kinetic term to the energy density can be of the same order as the contribution from the leading term which consists of the midpoint ghost insertion. 
  We calculate the back-reaction of long wavelength cosmological perturbations on a general relativistic measure of the local expansion rate of the Universe. Specifically, we consider a cosmological model in which matter is described by two scalar matter fields, one being the inflaton and the other one representing a matter field which is used as a clock. We analyze back-reaction in a phase of inflaton-driven slow-roll inflation, and find that the leading infrared back-reaction terms contributing to the evolution of the expansion rate do not vanish when measured at a fixed value of the clock field. We also analyze the back-reaction of entropy modes in a specific cosmological model with negative square mass for the entropy field and find that back-reaction can become significant. Our work provides evidence that, in general, the back-reaction of infrared fluctuations could be locally observable. 
  We study a Newtonian cosmological model in the context of a noncommutative space. It is shown that the trajectories of a test particle undergo modifications such that it no longer satisfies the cosmological principle. For the case of a positive cosmological constant, spiral trajectories are obtained and corrections to the Hubble constant appear. It is also shown that, in the limit of a strong noncommutative parameter, the model is closely related to a particle in a G\"odel-type metric. 
  We construct explicit BPS and non-BPS solutions of the U(2k) Yang-Mills equations on the noncommutative space R^{2n}_\theta x S^2 with finite energy and topological charge. By twisting with a Dirac multi-monopole bundle over S^2, we reduce the Donaldson-Uhlenbeck-Yau equations on R^{2n}_\theta x S^2 to vortex-type equations for a pair of U(k) gauge fields and a bi-fundamental scalar field on R^{2n}_\theta. In the SO(3)-invariant case the vortices on R^{2n}_\theta determine multi-instantons on R^{2n}_\theta x S^2. We show that these solutions give natural physical realizations of Bott periodicity and vector bundle modification in topological K-homology, and can be interpreted as a blowing-up of D0-branes on R^{2n}_\theta into spherical D2-branes on R^{2n}_\theta x S^2. In the generic case with broken rotational symmetry, we argue that the D0-brane charges on R^{2n}_\theta x S^2 provide a physical interpretation of the Adams operations in K-theory. 
  We consider the gravitation-dilaton theory (not necessarily exactly solvable), whose potentials represent a generic linear combination of an exponential and linear functions of the dilaton. A black hole, arising in such theories, is supposed to be enclosed in a cavity, where it attains thermal equilibrium, whereas outside the cavity the field is in the Boulware state. We calculate quantum corrections to the Hawking temperature $T_{H}$, with the contribution from the boundary taken into account. Vacuum polarization outside the shell tend to cool the system. We find that, for the shell to be in the thermal equilibrium, it cannot be placed too close to the horizon. The quantum corrections to the mass due to vacuum polarization vanish in spite of non-zero quantum stresses. We discuss also the canonical boundary conditions and show that accounting for the finiteness of the system plays a crucial role in some theories (e.g., CGHS), where it enables to define the stable canonical ensemble, whereas consideration in an infinite space would predict instability. 
  We have analyzed the breakdown of global supersymmetry by a non-vanishing expectation value of the fifth component of the graviphoton on warped S^1/Z_2 orbifolds. It has been demonstrated that the setups where such a breakdown is possible correspond to the models where the true gauge symmetry on the orbifold, respecting the Z_2-parities and periodicity, is broken by boundary terms. In the tuned models, giving Randall-Sundrum warp factor, gauge symmetry stays intact, and any <A_5> can be gauged away without violating supersymmetry. 
  We consider the structure of the cosmological singularity in Veneziano's inflationary model. The problem of choosing initial data in the model is shown to be unsolved -- the spacetime in the asymptotically flat limit can be filled with an arbitrary number of gravitational and scalar field quanta. As a result, the universe acquires a domain structure near the singularity, with an anisotropic expansion of its own being realized in each domain. 
  Exact microscopic spectral correlation functions are derived by means of the replica limit of the Toda lattice equation. We consider both Hermitian and non-Hermitian theories in the Wigner-Dyson universality class (class A) and in the chiral universality class (class AIII). In the Hermitian case we rederive two-point correlation functions for class A and class AIII as well as several one-point correlation functions in class AIII. In the non-Hermitian case the spectral density of non-Hermitian complex random matrices in the weak non-Hermiticity limit is obtained directly from the replica limit of the Toda lattice equation. In the case of class A, this result describes the spectral density of a disordered system in a constant imaginary vector potential (the Hatano-Nelson model) which is known from earlier work. New results are obtained for the spectral density in the weak non-Hermiticity limit of a quenched chiral random matrix model at nonzero chemical potential. These results apply to the ergodic or $\epsilon$ domain of quenched QCD at nonzero chemical potential. The spectral density obtained is different from the result derived by Akemann for a closely related model, which is given by the leading order asymptotic expansion of our result. In all cases, the replica limit of the Toda lattice equation explains the factorization of spectral one- and two-point functions into a product of a bosonic (noncompact integral) and a fermionic (compact integral) partition function. We conclude that the fermionic, the bosonic and the supersymmetric partition functions are all part of a single integrable hierarchy. This is the reason that it is possible to obtain the supersymmetric partition function, and its derivatives, from the replica limit of the Toda lattice equation. 
  We compute the prepotential of N=2 supersymmetric gauge theories in four dimensions obtained by toroidal compactifications of gauge theories from 6 dimensions, as a function of Kahler and complex moduli of T^2. We use three different methods to obtain this: matrix models, geometric engineering and instanton calculus. Matrix model approach involves summing up planar diagrams of an associated gauge theory on T^2. Geometric engineering involves considering F-theory on elliptic threefolds, and using topological vertex to sum up worldsheet instantons. Instanton calculus involves computation of elliptic genera of instanton moduli spaces on R^4. We study the compactifications of N=2* theory in detail and establish equivalence of all these three approaches in this case. As a byproduct we geometrically engineer theories with massive adjoint fields. As one application, we show that the moduli space of mass deformed M5-branes wrapped on T^2 combines the Kahler and complex moduli of T^2 and the mass parameter into the period matrix of a genus 2 curve. 
  We discuss the Reissner-Nordstrom-de Sitter black holes in the context of dS/CFT correspondence by using static and planar coordinates. The boundary stress tensor and the mass of the solutions are computed. Also, we investigate how the RG flow is changed for different foliations. The Kastor-Traschen multi-black hole solution is considered as well as AdS counterparts of these configurations. In particular, we find that in planar coordinates the black holes appear like punctures in the dual boundary theory. 
  We construct higher dimensional quantum Hall systems based on fuzzy spheres. It is shown that fuzzy spheres are realized as spheres in colored monopole backgrounds. The space noncommutativity is related to higher spins which is originated from the internal structure of fuzzy spheres. In $2k$-dimensional quantum Hall systems, Laughlin-like wave function supports fractionally charged excitations, $q=m^{-{1/2}k(k+1)}$ (m is odd). Topological objects are ($2k-2$)-branes whose statistics are determined by the linking number related to the general Hopf map. Higher dimensional quantum Hall systems exhibit a dimensional hierarchy, where lower dimensional branes condense to make higher dimensional incompressible liquid. 
  It is shown that the Spin(7)-invariant super Yang-Mills theory in euight dimensions, which relies on the existence of the Caley invariant, permits the construction of a cohomological extension, which relies on the existence of the eight-dimensional analogue of the Pontryagin invariant arising from a quartic chiral primary operator. 
  We study the eigenvalue problem of the rational Calogero model with the coupling of the inverse-square interaction as a complex number. We show that although this model is manifestly non-invariant under the combined parity and time-reversal symmetry ${\cal{PT}}$, the eigenstates corresponding to the zero value of the generalized angular momentum have real energies. 
  We analyze non-supersymmetric four dimensional open string models of type IIB string theory compactified on $T^2\times K3$ with Scherk-Schwarz deformation acting on an $S^1$ of the $T^2$ torus. We find that there are always two solutions to the tadpole conditions that are shown to be connected via Wilson lines in an non-trivial way. These models although non-supersymmetric, are free of R-R and NS-NS tadpoles. 
  In a recent paper by A. Das and X. Zhou [Phys. Rev. D 68, 065017 (2003)] it is claimed that explicit evaluation of the thermal photon self-energy in the Schwinger model gives off-shell thermal Green functions that are different in light-front and conventional quantizations. We show that the claimed difference originates from an erroneous simplification of the fermion propagator used in the light-front calculation. 
  We study the four-dimensional Z_2 random-plaquette lattice gauge theory as a model of topological quantum memory, the toric code in particular. In this model, the procedure of quantum error correction works properly in the ordered (Higgs) phase, and phase boundary between the ordered (Higgs) and disordered (confinement) phases gives the accuracy threshold of error correction. Using self-duality of the model in conjunction with the replica method, we show that this model has exactly the same mathematical structure as that of the two dimensional random-bond Ising model, which has been studied very extensively. This observation enables us to derive a conjecture on the exact location of the multicritical point (accuracy threshold) of the model, p_c=0.889972..., and leads to several nontrivial results including bounds on the accuracy threshold in three dimensions. 
  Within the framework of the covariant formulation of Light-Front Dynamics, we develop a nonperturbative renormalization scheme in the fermion model supposing that the composite fermion is a superposition of the "bare" fermion and a fermion+boson state. We first assume the constituent boson to be spinless. Then we address the case of gauge bosons in the Feynman and in the Light-Cone gauges. For all these cases the fermion state vector and the necessary renormalization counterterms are calculated analytically. It turns out that in Light-Front Dynamics, to restore the rotational invariance, an extra counterterm is needed, having no any analogue in Feynman approach. For gauge bosons the results obtained in the two gauges are compared with each other. In general, the number of spin components of the two-body (fermion+boson) wave function depends on the gauge. But due to the two-body Fock space truncation, only one non-zero component survives for each gauge. And moreover, the whole solutions for the state vector, found for the Feynman and Light-Cone gauges, are the same (except for the normalization factor). The counterterms are however different 
  We propose that in quantum gravity one needs to impose a final state boundary condition at black hole singularities. This resolves the apparent contradiction between string theory and semiclassical arguments over whether black hole evaporation is unitary. 
  The causal structure of the flat brane universe of RSII type is re-investigated to clarify the boundary conditions for stochastic gravitational waves. In terms of the Gaussian normal coordinate of the brane, a singularity of the equation for gravitational waves appears in the bulk. We show that this singularity corresponds to the ``seam singularity'' which is a singular subspace on the brane universe. Based upon the causal structure, we discuss the boundary conditions for gravitational waves in the bulk. Introducing a null coordinate, we propose a numerical procedure to solve gravitational waves with appropriate boundary conditions and show some examples of our numerical results. This procedure could be also applied in scalar type perturbations. The problem in the choice of the initial condition for gravitational waves is briefly discussed. 
  We find a one-parameter family of long-lived physical string states in type II superstring theory. We compute the decay rate by an exact numerical evaluation of the imaginary part of the one-loop propagator. Remarkably, the lifetime rapidly increases with the mass. We find a power-law dependence of the form $T = const. g^{-2} Mass^\alpha$, where the value of $\alpha $ depends on the parameter characterizing the state. For the most stable state in this family, one has $\alpha ~= 5$. The dominant decay channel of these massive string states is by emission of soft massless particles. The quantum states can be viewed semiclassically as closed strings which cannot break during the classical evolution. 
  The Hamiltonian of the simplest super $p$-brane model preserving 3/4 of the D=4 N=1 supersymmetry in the centrally extended symplectic superspace is derived and its symmetries are described. The constraints of the model are covariantly separated into the first- and the second-class sets and the Dirac brackets (D.B.) are constructed. We show the D.B. noncommutativity of the super $p$-brane coordinates and find the D.B. realization of the $OSp(1|8)$ superalgebra. Established is the coincidence of the D.B. and Poisson bracket realizations of the $OSp(1|8)$ superalgebra on the constraint surface and the absence there of anomaly terms in the commutation relations for the quantized generators of the superalgebra. 
  We demonstrate that weakly coupled, large N, d-dimensional SU(N) gauge theories on a class of compact spatial manifolds (including S^{d-1} \times time) undergo deconfinement phase transitions at temperatures proportional to the inverse length scale of the manifold in question. The low temperature phase has a free energy of order one, and is characterized by a stringy (Hagedorn) growth in its density of states. The high temperature phase has a free energy of order N^2. These phases are separated either by a single first order transition that generically occurs below the Hagedorn temperature or by two continuous phase transitions, the first of which occurs at the Hagedorn temperature. These phase transitions could perhaps be continuously connected to the usual flat space deconfinement transition in the case of confining gauge theories, and to the Hawking-Page nucleation of AdS_5 black holes in the case of the N=4 supersymmetric Yang-Mills theory. We suggest that deconfinement transitions may generally be interpreted in terms of black hole formation in a dual string theory. Our analysis proceeds by first reducing the Yang-Mills partition function to a (0+0)-dimensional integral over a unitary matrix U, which is the holonomy (Wilson loop) of the gauge field around the thermal time circle in Euclidean space; deconfinement transitions are large N transitions in this matrix integral. 
  A certain sector of the matrix model for M-theory on a plane wave background has recently been shown to produce the transverse five-brane. We consider this theory at finite temperature. We find that, at a critical temperature it has a Gross-Witten phase transition which corresponds to deconfinement of the matrix model gauge theory. We interpret the phase transition as the Hagedorn transition of M-theory and of type II string theory in the five-brane background. We also show that there is no Hagedorn behaviour in the transverse membrane background case. 
  This is a review of some beautiful matrix models related to the moduli space of Riemann surfaces as well as to noncritical c=1 string theory at self-dual radius. These include the Penner model and the W-infinity model, which have different origins but are equivalent to each other. In the final section, which is new material, it is shown that these models are also equivalent to a Liouville matrix model. We speculate that this might be interpreted in terms of N D-instantons of the c=1 string. 
  We present a new version of holographic cosmology, which is compatible with present observations. A primordial $p=\rho$ phase of the universe is followed by a brief matter dominated era and a brief period of inflation, whose termination heats the universe. The flatness and horizon problems are solved by the $p=\rho$ dynamics. The model is characterized by two parameters, which should be calculable in a more fundamental approach to the theory. For a large range in the phenomenologically allowed parameter space, the observed fluctuations in the cosmic microwave background were generated during the $p=\rho$ era, and are exactly scale invariant. The scale invariant spectrum cuts off sharply at both upper and lower ends, and this may have observational consequences. We argue that the amplitude of fluctuations is small but cannot yet calculate it precisely. 
  An algebraic formulation of the stringy geometry on simple current orbifolds of the WZW models of type A_N is developed within the framework of Reflection Equation Algebras, REA_q(A_N). It is demonstrated that REA_q(A_N) has the same set of outer automorphisms as the corresponding current algebra A^{(1)}_N which is crucial for the orbifold construction. The CFT monodromy charge is naturally identified within the algebraic framework. The ensuing orbifold matrix models are shown to yield results on brane tensions and the algebra of functions in agreement with the exact BCFT data. 
  We present nonperturbative light-front energy eigenstates in the broken phase of a two dimensional $\frac{\lambda}{4!}\phi^4$ quantum field theory using Discrete Light Cone Quantization and extrapolate the results to the continuum limit. We establish degeneracy in the even and odd particle sectors and extract the masses of the lowest two states and the vacuum energy density for $\lambda=0.5$ and 1.0. We present two novel results: the Fourier transform of the form factor of the lowest excitation as well as the number density of elementary constituents of that state. A coherent state with kink - antikink structure is revealed. 
  We show that, in any conformal field theory, the weights of all bulk primary fields that couple to N phi_{2,1} fields on the boundary are given by the spectrum of an N-particle Calogero-Sutherland model. The corresponding correlation function is simply related to the N-particle wave function. Examples are discussed for the minimal models and for the non-unitary O(n) model. 
  We test the spectrum of string theory on AdS_5 x S^5 derived in hep-th/0305052 against that of single-trace gauge invariant operators in free N=4 super Yang-Mills theory. Masses of string excitations at critical tension are derived by extrapolating plane-wave frequencies at g_{YM}=0 down to finite J. On the SYM side, we present a systematic description of the spectrum of single-trace operators and its reduction to PSU(2,2|4) superconformal primaries via a refined Eratostenes' supersieve. We perform the comparison of the resulting SYM/string spectra of charges and multiplicities order by order in the conformal dimension \Delta up to \Delta=10 and find perfect agreement. Interestingly, the SYM/string massive spectrum exhibits a hidden symmetry structure larger than expected, with bosonic subgroup SO(10,2) and thirty-two supercharges. 
  We present a model of dark energy based on the string effective action, and on the assumption that the dilaton is strongly coupled to dark matter. We discuss the main differences between this class of models and more conventional models of quintessence,uncoupled to dark matter. This paper is based on talks presented at the "VII Congresso Nazionale di Cosmologia" (Osservatorio Astronomico di Roma, Monte Porzio Catone, November 2002), and at the Meeting "Dark Energy Day" (University of Milano-Bicocca, November 2002). To appear in the Proc. of the International Conference on ``Thinking, Observing and Mining the Universe" (Sorrento, September 2003), eds. G. Longo and G. Miele (World Scientific, Singapore). 
  A power-counting renormalizable model into which massive Yang-Mills theory is embedded is analyzed. The model is invariant under a nilpotent BRST differential s. The physical observables of the embedding theory, defined by the cohomology classes of s in the Faddeev-Popov neutral sector, are given by local gauge-invariant quantities constructed only from the field strength and its covariant derivatives. 
  The aim of this paper is to study orientifolds of c=1 conformal field theories. A systematic analysis of the allowed orientifold projections for c=1 orbifold conformal field theories is given. We compare the Klein bottle amplitudes obtained at rational points with the orientifold projections that we claim to be consistent for any value of the orbifold radius. We show that the recently obtained Klein bottle amplitudes corresponding to exceptional modular invariants, describing bosonic string theories at fractional square radius, are also in agreement with those orientifold projections. 
  One possibility to explain the current accelerated expansion of the universe may be related with the presence of cosmologically evolving scalar whose mass depends on the local matter density (chameleon cosmology). We point out that matter quantum effects in such scalar-tensor theory produce the chameleon scalar field dependent conformal anomaly. Such conformal anomaly adds higher derivative terms to chameleon field equation of motion. As a result, the principal possibility for instabilities appears. These instabilities seem to be irrelevant at small curvature but may become dangerous in the regions where gravitational field is strong. 
  A main purpose of this paper is to explain how the theory of higher spin fields in flat D=4 space and in AdS(4) emerges as a result of the quantization of a superparticle propagating in so called tensorial superspaces which have the property of a `generalized conformal' or simply General Linear (GL) flatness. 
  It is a well known result that the scalar field spectrum is composed of two chiral particles (Floreanini-Jackiw particles) of opposite chiralities. Also, that a Siegel particle spectrum is formed by a nonmover field (a Hull's noton) and a FJ particle. We show here that, in fact, the spectrum of the chiral boson on a circle has a particle not present in its currently well known spectrum. 
  Proceeding from nonlinear realizations of the most general N=4, d=1 superconformal symmetry associated with the supergroup D(2,1;\alpha), we construct all known and two new off-shell N=4, d=1 supermultiplets as properly constrained N=4 superfields. We find plenty of nonlinear interrelations between the multiplets constructed and present a few examples of invariant superfield actions for them. The superconformal transformation properties of these multiplets are explicit within our method. 
  We construct sphaleron solutions in Weinberg-Salam theory, which possess only discrete symmetries. Related to rational maps of degree N, these sphalerons carry baryon number Q_B=N/2. The energy density of these sphalerons reflects their discrete symmetries. We present an N=3 sphaleron with tetrahedral energy density, an N=4 sphaleron with cubic energy density, and an N=5 sphaleron with octahedral energy density. 
  We study the vacuum polarization (Casimir) energy in renormalizable, continuum quantum field theory in the presence of a background field, designed to impose Dirichlet boundary conditions on the fluctuating quantum field. In two and three spatial dimensions the Casimir energy diverges as a background field becomes concentrated on the surface on which the Dirichlet boundary condition would eventually hold. This divergence does not affect the force between rigid bodies, but it does invalidate calculations of Casimir stresses based on idealized boundary conditions. 
  In the framework of superfield approach, we derive the local, covariant, continuous and nilpotent (anti-)BRST and (anti-)co-BRST symmetry transformations on the U(1) gauge field $(A_\mu)$ and the (anti-)ghost fields $((\bar C)C)$ of the Lagrangian density of the two $(1 + 1)$-dimensional QED by exploiting the (dual-)horizontality conditions defined on the four $(2 + 2)$-dimensional supermanifold. The long-standing problem of the derivation of the above symmetry transformations for the matter (Dirac) fields $(\bar\psi, \psi)$ in the framework of superfield formulation is resolved by a new set of restrictions on the $(2 + 2)$-dimensional supermanifold. These new physically interesting restrictions on the supermanifold owe their origin to the invariance of conserved currents of the theory. The geometrical interpretation for all the above transformations is provided in the framework of superfield formalism. 
  We generalize the construction of Snyder to a Lorentz covariant noncommutative superspace. 
  The basic aspects of the momentum picture of motion in Lagrangian quantum field theory are given. Under some assumptions, this picture is a 4-dimensional analogue of the Schr\"odinger picture: in it the field operators are constant, spacetime-independent, while the state vectors have a simple, exponential, spacetime-dependence. The role of these assumptions is analyzed. The Euler-Lagrange equations in momentum picture are derived and attention is paid on the conserved operators in it. 
  A general class of rotating closed string solutions in AdS_5 x S^5 is shown to be described by a Neumann-Rosochatius one-dimensional integrable system. The latter represents an oscillator on a sphere or a hyperboloid with an additional ``centrifugal'' potential. We expect that the reduction of the AdS_5 x S^5 sigma model to the Neumann-Rosochatius system should have further generalizations and should be useful for uncovering new relations between integrable structures on the two sides of the AdS/CFT duality. We find, in particular, new circular rotating string solutions with two AdS_5 and three S^5 spins. As in other recently discussed examples, the leading large-spin correction to the classical energy turns out to be proportional to the square of the string tension or the 't Hooft coupling \lambda, suggesting that it can be matched onto the one-loop anomalous dimensions of the corresponding ``long'' operators on the SYM side of the AdS/CFT duality. 
  We construct a supermatrix model which has a classical solution representing the noncommutative (fuzzy) two-supersphere. Expanding supermatrices around the classical background, we obtain a gauge theory on a noncommutative superspace on sphere. This theory has $osp(1|2)$ supersymmetry and $u(2L+1|2L)$ gauge symmetry. We also discuss a commutative limit of the model keeping radius of the supersphere fixed. 
  We derive the scalar potential in zero mode effective action arising from a de Sitter brane embedded in five dimensions with bulk cosmological constant $\Lambda$. The scalar potential for a scalar field canonically normalized is given by the sum of exponential potentials. In the case of $\Lambda=0$ and $\Lambda>0$, we point out that the scalar potential has an unstable local maximum at the origin and exponentially vanishes for large positive scalar field. In the case of $\Lambda<0$, the scalar potential has an unstable local maximum at the origin and a stable local minimum, it is shown that the positive cosmological constant in brane is reduced by negative potential energy of scalar at minimum. 
  The quantum Euclidean space R_{q}^{N} is a kind of noncommutative space which is obtained from ordinary Euclidean space R^{N} by deformation with parameter q. When N is odd, the structure of this space is similar to R_{q}^{3}. Motivated by realization of R_{q}^{3} by differential operators in R^{3}, we give such realization for R_{q}^{5} and R_{q}^{7} cases and generalize our results to R_{q}^{N} (N odd) in this paper, that is, we show that the algebra of R_{q}^{N} can be realized by differential operators acting on C^{infinite} functions on undeformed space R^{N}. 
  The main purpose of this paper is to rethink the relativity issue within the framework of the fundamental postulates of quantum mechanics. The aspect of so-called ``double special relativity'' (DSR) is a starting point in our discussion. The three elementary ideas were involved to show that special relativity may be treated as an integral part of quantum mechanics. These ideas (or observations) are: (1) the necessity of distinguishing the two time meanings, namely: (i) the vital one referring to description of system evolution, and (ii) the frozen one, referring to energy measure by means of inverse time units; (2) the existence of the energy-momentum (and time-distance) comparison scale in relativistic description; and (3) a possibility of introduction of mass by means of a light-cone frame description. The resulting quantum-mechanical analysis allows us to find diagonal form of the Lorentz-boost transformation matrix and thus to relate the interval invariant relativity principle with the principles of quantum mechanics. The manner, in which the diagonal form of the transformation matrix was found, shows that covariant description itself is a preferred frame description and that the time that undergoes relativistic transformation rules is the frozen time, whereas the vital time is the Lorentz invariant. A generalized form of the Heisenberg uncertainty principle proposed by Witten (Phys. Today, Apriel 1996), is derived. It turns out, that this form is equivalent to the one know from the analysis of the covariant harmonic oscillator given by Kim and Noz (the book, 1986). As a by-product of this analysis one finds that special relativity itself preserves the Planck length, however, a particle cannot be seen any longer as a material point, but rather as an extended quantum object. 
  We survey and investigate some computational aspects of the Fourier-Mukai transform. 
  Extending the results of a previous paper, we consider boundary conditions for spinor fields and other fields of non-zero spin in the AdS/CFT correspondence. We calculate the RG-flow induced by double trace perturbations dual to bulk spinor fields. For spinors there is a half-unit shift in the central charge in running from the UV to the IR, in accordance with the c-theorem. 
  We show that the cosmological constant may be reduced by thermal production of membranes by the cosmological horizon, analogous to a particle ``going over the top of the potential barrier", rather than tunneling through it. The membranes are endowed with charge associated with the gauge invariance of an antisymmetric gauge potential. In this new process, the membrane collapses into a black hole, thus the net effect is to produce black holes out of the vacuum energy associated with the cosmological constant. We study here the corresponding Euclidean configurations ("thermalons"), and calculate the probability for the process in the leading semiclassical approximation. 
  We obtain the Killing equations and the corresponding infinitesimal isometries for the ten dimensional space generated by a large number of coincident D3-branes. In a convenient limit this space becomes an $AdS_5\times S^5$ which is relevant for the AdS/CFT correspondence. In this case, using Poincare coordinates, we also write down the Killing equations and infinitesimal isometries. Then we obtain a simple realization of the isomorphism between $AdS$ isometries and the boundary conformal group. 
  Closed string theories on orbifolds contain both untwisted and twisted states. The latter are normally assumed to live exactly at the orbifold fixed points. We perform a calculation of a gauge field tadpole amplitude and show that off-shell both the twisted and untwisted states give rise to non-trivial momentum profiles over the orbifold C^3/Z_3. These profiles take the form of Gaussian distributions integrated over the fundamental domain of the modular parameter of the torus. The propagators of the internal coordinate fields on the torus world sheet determine the width of the Gaussian profiles. These propagators are determined up to a single normal ordering constant which must be bounded below to allow the existence of the coordinate space representation of these Gaussians. Apart from the expected massless states, massive and even tachyonic string excitations contribute to the profiles in some anomalous U(1) models. However, when a tadpole is integrated over the internal dimensions, these tachyonic contributions cancel in a non-trivial manner. 
  The possibility of interaction among multiverses is studied assuming that in the first instants of the big-bang, many disjoint regions were created producing many independent universes (multiverses). Many of these mini-universes were unstable and they decayed, but other remained as topological remnant (like domain walls or baby universes) or possibly as mini-black-holes. In this paper, we study the quantum statistical mechanics of multiverses assuming that in the first instants of the big-bang, the relativistic symmetry was only an approximate symmetry and the interaction among multiverses was produced by non-local communication. The breaking of the relativistic symmetry induces on each multiverse a tiny harmonic interaction. The oscillation frequency for each multiverse is proportional to 1/B, where B is the noncommutativity parameter. We argue that B can identified as the primordial magnetic field, {\it i.e.} $\sim 10^{-16} {GeV}^2$. This tiny frequency could suggest that the relativistics invariance --from the cosmological point of view-- is almost exact and the multiverses could be not detected using the presently astronomical observations. 
  Recent work has provided a direct string calculation of the internal coordinate dependence of gauge field tadpoles on the orbifold C^3/Z_3. We investigate the structure of these profiles in momentum and coordinate space representations using both analytical and numerical methods. The twisted states are determined to be localized within a few string lengths of the singularity. This provides an example of how delta-like singularities, which are typical for field theories on orbifolds, are avoided in string theory. A systematic expansion of the full string result allows us to speculate on how the leading stringy effects can be incorporated in a field theoretical description. In one heterotic model, we find that even though tachyons are not present in the physical spectrum, they totally dominate the tadpole profile in the vicinity of the fixed point. 
  We summarize our recent results of studying five-dimensional Kasner cosmologies in a time-dependent Calabi-Yau compactification of M-theory undergoing a topological flop transition. The dynamics of the additional states, which become massless at the transition point and give rise to a scalar potential, helps to stabilize the moduli and triggers short periods of accelerated cosmological expansion. 
  In these notes we provide a pedagogical introduction to the subject of tachyon condensation in Witten's cubic bosonic open string field theory. We use both the low-energy Yang-Mills description and the language of string field theory to explain the problem of tachyon condensation on unstable D-branes. We give a self-contained introduction to open string field theory using both conformal field theory and overlap integrals. Our main subjects are the Sen conjectures on tachyon condensation in open string field theory and the evidence that supports these conjectures. We conclude with a discussion of vacuum string field theory and projectors of the star-algebra of open string fields. We comment on the possible role of string field theory in the construction of a nonperturbative formulation of string theory that captures all possible string backgrounds. 
  At the leading order, the low-energy effective field equations in string theory admit solutions of the form of products of Minkowski spacetime and a Ricci-flat Calabi-Yau space. The equations of motion receive corrections at higher orders in \alpha', which imply that the Ricci-flat Calabi-Yau space is modified. In an appropriate choice of scheme, the Calabi-Yau space remains Kahler, but is no longer Ricci-flat. We discuss the nature of these corrections at order {\alpha'}^3, and consider the deformations of all the known cohomogeneity one non-compact Kahler metrics in six and eight dimensions. We do this by deriving the first-order equations associated with the modified Killing-spinor conditions, and we thereby obtain the modified supersymmetric solutions. We also give a detailed discussion of the boundary terms for the Euler complex in six and eight dimensions, and apply the results to all the cohomogeneity one examples. 
  There is a class of single trace operators in ${\cal N}=4$ Yang-Mills theory which are related by the AdS/CFT correspondence to classical string solutions. Interesting examples of such solutions corresponding to periodic trajectories of the Neumann system were studied recently. In our paper we study a generalization of these solutions. We consider strings moving with large velocities. We show that the worldsheet of the fast moving string can be considered as a perturbation of the degenerate worldsheet, with the small parameter being the relativistic factor $\sqrt{1-v^2}$. The series expansion in this relativistic factor should correspond to the perturbative expansion in the dual Yang-Mills theory. The operators minimizing the anomalous dimension in the sector with given charges correspond to periodic trajectories in the mechanical system which is closely related to the product of two Neumann systems. 
  We study the thermodynamic properties associated with black hole horizon and cosmological horizon for the Gauss-Bonnet solution in de Sitter space. When the Gauss-Bonnet coefficient is positive, a locally stable small black hole appears in the case of spacetime dimension $d=5$, the stable small black hole disappears and the Gauss-Bonnet black hole is always unstable quantum mechanically when $d \ge 6$. On the other hand, the cosmological horizon is found always locally stable independent of the spacetime dimension. But the solution is not globally preferred, instead the pure de Sitter space is globally preferred. When the Gauss-Bonnet coefficient is negative, there is a constraint on the value of the coefficient, beyond which the gravity theory is not well defined. As a result, there is not only an upper bound on the size of black hole horizon radius at which the black hole horizon and cosmological horizon coincide with each other, but also a lower bound depending on the Gauss-Bonnet coefficient and spacetime dimension. Within the physical phase space, the black hole horizon is always thermodynamically unstable and the cosmological horizon is always stable, further, as the case of the positive coefficient, the pure de Sitter space is still globally preferred. This result is consistent with the argument that the pure de Sitter space corresponds to an UV fixed point of dual field theory. 
  We give a complete account of the Schr\"odinger representation approach to the calculation of the Weyl anomaly of ${\cal N}=4$ SYM from the AdS/CFT correspondence. On the AdS side, the $1/N^2$ correction to the leading order result receives contributions from all the fields of Type IIB Supergravity, the contribution of each field being given by a universal formula. The correct matching with the CFT result is thus a highly non-trivial test of the correspondence. 
  In hep-th/0310278, Blankleider and Kvinikhidze propose an alternate thermal propagator for the fermions in the light-front Schwinger model. We show that such a propagator does not describe correctly the thermal behavior of fermions in this theory and, as a consequence, the claims made in their paper are not correct. 
  We explore ``weak'' supersymmetric systems whose algebra involves, besides Poincare generators, extra bosonic generators not commuting with supercharges. This allows one to have inequal number of bosonic and fermionic 1--particle states in the spectrum. Coleman--Mandula and Haag--Lopuszanski--Sohnius theorems forbid the presence of such extra bosonic charges in {\it interacting} theory for $d \geq 3$. However, these theorems do not apply in one or two dimensions. For $d=1$, we construct a nontrivial interacting system characterized by weak supersymmetric algebra. It is related to ``n--fold'' supersymmetric systems and to quasi-exactly solvable systems studied earlier. 
  We analyze the Seiberg-Witten curve of the six-dimensional N=(1,1) gauge theory compactified on a torus to four dimensions. The effective theory in four dimensions is a deformation of the N=2* theory. The curve is naturally holomorphically embedding in a slanted four-torus--actually an abelian surface--a set-up that is natural in Witten's M-theory construction of N=2 theories. We then show that the curve can be interpreted as the spectral curve of an integrable system which generalizes the N-body elliptic Calogero-Moser and Ruijsenaars-Schneider systems in that both the positions and momenta take values in compact spaces. It turns out that the resulting system is not simply doubly elliptic, rather the positions and momenta, as two-vectors, take values in the ambient abelian surface. We analyze the two-body system in some detail. The system we uncover provides a concrete realization of a Beauville-Mukai system based on an abelian surface rather than a K3 surface. 
  Effective superpotentials obtained by integrating out matter in super Yang-Mills and conformal supergravity backgrounds in N=1 SUSY theories are considered. The pure gauge and supergravity contributions (generalizing Veneziano-Yankielowicz terms) are derived by considering the case with matter fields in the fundamental representation of the gauge group. These contributions represent quantum corrections to the tree-level Yang-Mills and conformal supergravity actions. The classical equations of motion following from the conformal supergravity action require the background to be (super)conformally flat. This condition is unchanged by quantum corrections to the effective superpotential, irrespective of the matter content of the theory. 
  This paper describes my talk given to the 27th Johns Hopkins Workshop: Symmetries and Mysteries of M Theory, G\"oteborg, Sweden, 24-26 August, 2003. After a brief introduction to the lightcone worldsheet formalism for summing the planar diagrams of field theory, I explain how the {\it uv} divergences of quantum field theory translate to the new language of string. It is shown through one loop that, at least for scalar cubic vertices, the counter-terms necessary for Poincar\'e invariance in space-time dimensions $D\leq 6$ are indeed local on the worldsheet. The extension to cover the case of gauge field vertices will be more complicated due to the extra divergences at $p^+=0$ in lightcone gauge. 
  Using the procedure of the marked point fusion, there are obtained integrable systems with poles in the matrix of the Lax operator order higher than one, considered Hamiltonians, symplectic structure and symmetries of these systems. Also, taking the Inozemtsev Limit procedure it was found the Toda-like system having nontrivial commutative relations between the phase space variables. 
  We study the equivalence among orientifold three-planes in the context of the Atiyah-Hirzebruch Spectral Sequence. This equivalence refers to the fact that two different cohomology classes in the same cohomology group, which classify the orientifolds, are lifted to the same trivial class in K-theory. The physical interpretation of this mathematical algorithm is given by the role of D-brane instantons. By following some recent ideas, we extend the sequence to include a classification of NS-NS fluxes. We find that such equivalences, in the low energy limit of the dynamics on the worldvolume of type IIB D3-branes on top of the orientifolds, are interpreted as the SL(2,Z) duality in four dimensional N=4 SYM theories. 
  Tracing over the degrees of freedom inside (or outside) a sub-volume V of Minkowski space in a given quantum state |psi>, results in a statistical ensemble described by a density matrix rho. This enables one to relate quantum fluctuations in V when in the state |psi>, to statistical fluctuations in the ensemble described by rho. These fluctuations scale linearly with the surface area of V. If V is half of space, then rho is the density matrix of a canonical ensemble in Rindler space. This enables us to `derive' area scaling of thermodynamic quantities in Rindler space from area scaling of quantum fluctuations in half of Minkowski space. When considering shapes other than half of Minkowski space, even though area scaling persists, rho does not have an interpretation as a density matrix of a canonical ensemble in a curved, or geometrically non-trivial, background. 
  In this paper we consider spacetimes in vacuum general relativity --possibly coupled to a scalar field-- with a positive cosmological constant $\Lambda$. We employ the Isolated Horizons (IH) formalism where the boundary conditions imposed are that of two horizons, one of black hole type and the other, serving as outer boundary, a cosmological horizon. As particular cases, we consider the Schwarzschild-de Sitter spacetime, in both 2+1 and 3+1 dimensions. Within the IH formalism, it is useful to define two different notions of energy for the cosmological horizon, namely, the "mass" and the "energy". Empty de Sitter space provides an striking example of such distinction: its horizon energy is zero but the horizon mass takes a finite value given by $\pi/(2\sqrt\Lambda)$. For both horizons we study their thermodynamic properties, compare our results with those of Euclidean Hamiltonian methods and construct some generalized Bekenstein entropy bounds. We discuss these new entropy bounds and compare them with some recently proposed entropy bounds in the cosmological setting. 
  In this note we develop fuzzy versions of the supersymmetric non-linear sigma model on the supersphere S^(2,2). In hep-th/0212133 Bott projectors have been used to obtain the fuzzy CP^1 model. Our approach utilizes the use of supersymmetric extensions of these projectors. Here we obtain these (super) -projectors and quantize them in a fashion similar to the one given in hep-th/0212133. We discuss the interpretation of the resulting model as a finite dimensional matrix model. 
  We analyze the presence of an ambiguity in the pion electromagnetic form factor within a renormalizable version of the Nambu-Jona-Lasinio model. We found out that the ambiguity present on the evaluation of the form factor decouples from its transversal part, confirming previous results obtained ignoring the ambiguity. This result helps us to understand the role played by finite but undetermined quantities in Quantum Field Theories. 
  We propose a new reformulation of Yang-Mills theory in which three- and four-gluon self-interactions are eliminated at the price of introducing a sufficient number of auxiliary fields. We discuss the validity of this reformulation in the possible applications such as dynamical gluon mass generation, color confinement and glueball mass calculation. We emphasize the transverse-gluon pair condensation as the basic mechanism for dynamical mass generation. The confinement is realized as a consequence of a fact that the auxiliary fields become dynamical in the sense that they acquire the kinetic term due to quantum corrections. 
  A generalization of the thermodynamic uncertainty relations is proposed. It is done by introducing of an additional term proportional to the interior energy into the standard thermodynamic uncertainty relation that leads to existence of the lower limit of inverse temperature. The authors are of the opinion that the approach proposed may lead to proof of these relations. To this end, the statistical mechanics deformation at Planck scale. The statistical mechanics deformation is constructed by analogy to the earlier quantum mechanical results. As previously, the primary object is a density matrix, but now the statistical one. The obtained deformed object is referred to as a statistical density pro-matrix. This object is explicitly described, and it is demonstrated that there is a complete analogy in the construction and properties of quantum mechanics and statistical density matrices at Plank scale (i.e. density pro-matrices). It is shown that an ordinary statistical density matrix occurs in the low-temperature limit at temperatures much lower than the Plank's. The associated deformation of a canonical Gibbs distribution is given explicitly. 
  Lorentz covariance is the fundamental principle of every relativistic field theory which insures consistent physical descriptions. Even if the space-time is noncommutative, field theories on it should keep Lorentz covariance. In this letter, it is shown that the field theory on noncommutative spacetime is Lorentz covariant if the noncommutativity emerges from the algebra of spacetime operators described by Doplicher, Fredenhagen and Roberts. 
  We examine the absorption cross section of the massive scalar field for the higher-dimensional extended object. Adopting the usual quantum mechanical matching conditions between the asymptotic and near-horizon solutions in radial equation, we check whether or not the universal property of the absorption cross section, which is that the low-energy cross section is proportional to the surface area of horizon, is maintained when the mass effect is involved. It is found that the mass effect in general does not break the universal property of the cross section if particular conditions are required to the spacetime geometry. However, the mass-dependence of the cross section is very sensitive to the spacetime property in the near-horizon regime. 
  The genuine Kaluza-Klein-like theories (with no fields in addition to gravity with torsion) have difficulties with the existence of massless spinors after the compactification of some of dimensions of space\cite{witten}. We demonstrate in this letter on an example of a flat torus - as a compactified part of an (1+5)-dimensional space - that for constant fields and appropriate boundary conditions there exists in the 1+3 dimensional space a massless solution, which is mass protected and chirally coupled with a Kaluza-Klein charge to the corresponding gauge field. 
  We study a general relativistic particle action obtained by incorporating the Hamiltonian constraints into the formalism as a toy model for general relativity and string theory. We show how a non-vanishing cosmological constant and a weakening of gravity at short distances may be interpreted as evidences for the existence of gravitational dipoles. 
  This paper carries on the investigation of the non-unitary su(2)_{-1/2} WZW model. An essential tool in our first work on this topic was a free-field representation, based on a c=-2 \eta\xi ghost system, and a Lorentzian boson. It turns out that there are several ``versions'' of the \eta\xi system, allowing different su(2)_{-1/2} theories. This is explored here in details. In more technical terms, we consider extensions (in the c=-2 language) from the small to the large algebra representation and, in a further step, to the full symplectic fermion theory. In each case, the results are expressed in terms of su(2)_{-1/2} representations. At the first new layer (large algebra), continuous representations appear which are interpreted in terms of relaxed modules. At the second step (symplectic formulation), we recover a logarithmic theory with its characteristic signature, the occurrence of indecomposable representations. To determine whether any of these three versions of the su(2)_{-1/2} WZW is well-defined, one conventionally requires the construction of a modular invariant. This issue, however, is plagued with various difficulties, as we discuss. 
  Reasonable-looking models of inflation are compared, taking into account the possibility that the curvature perturbation might originate from some "curvaton" field different from the inflaton. 
  The twisted Eguchi-Kawai (TEK) model provides a non-perturbative definition of noncommutative Yang-Mills theory: the continuum limit is approached at large $N$ by performing suitable double scaling limits, in which non-planar contributions are no longer suppressed. We consider here the two-dimensional case, trying to recover within this framework the exact results recently obtained by means of Morita equivalence. We present a rather explicit construction of classical gauge theories on noncommutative toroidal lattice for general topological charges. After discussing the limiting procedures to recover the theory on the noncommutative torus and on the noncommutative plane, we focus our attention on the classical solutions of the related TEK models. We solve the equations of motion and we find the configurations having finite action in the relevant double scaling limits. They can be explicitly described in terms of twist-eaters and they exactly correspond to the instanton solutions that are seen to dominate the partition function on the noncommutative torus. Fluxons on the noncommutative plane are recovered as well. We also discuss how the highly non-trivial structure of the exact partition function can emerge from a direct matrix model computation. The quantum consistency of the TEK formulation is eventually checked by computing Wilson loops in a particular limit. 
  We use the Dijkgraaf-Vafa technique to study massive vacua of 6D SU(N) SYM theories on tori with R-symmetry twists. One finds a matrix model living on the compactification torus with a genus 2 spectral curve. The Jacobian of this curve is closely related to a twisted four torus T in which the Seiberg-Witten curves of the theory are embedded. We also analyze R-symmetry twists in a bundle with nontrivial first Chern class which yields intrinsically 6D SUSY breaking and a novel matrix integral whose eigenvalues float in a sea of background charge. Next we analyze the underlying integrable system of the theory, whose phase space we show to be a system of N-1 points on $T$. We write down an explicit set of Poisson commuting Hamiltonians for this system for arbitrary N and use them to prove that equilibrium configurations with respect to all Hamiltonians correspond to points in moduli space where the Seiberg-Witten curve maximally degenerates to genus 2, thereby recovering the matrix model spectral curve. We also write down a conjecture for a dual set of Poisson commuting variables which could shed light on a particle-like interpretation of the system. 
  We investigate N-extended supersymmetry in one-dimensional quantum mechanics on a circle with point singularities. For any integer n, N=2n supercharges are explicitly constructed and a class of point singularities compatible with supersymmetry is clarified. Key ingredients in our construction are n sets of discrete transformations, each of which forms an su(2) algebra of spin 1/2. The degeneracy of the spectrum and spontaneous supersymmetry breaking are briefly discussed. 
  This Resource Letter provides a guide to some of the the introductory and review literature in string theory. It is in no way complete, though it is intended to be of use to students at several levels. Owing to the nature of the subject, even much of the introductory literature is quite technical by the standards of many Resource Letters, requiring prior knowledge of quantum field theory and general relativity. This Resource Letter is thus somewhat different from others. The first part describes a few more popular accounts of string theory, which are primarily addressed to the general public, but those with an understanding of basic physics will be able to read them more deeply and so obtain a useful rough orientation to the field. The second part describes resources that are available at the advanced undergraduate level, and the balance describes string resources for more advanced students. The latter range from general introductions to recent review articles on branes and black holes, gauge/gravity duality, string field theory, non-commutative geometry, non-BPS branes, tachyon condensation, phenomenology, brane worlds, orbifolds, Calabi-Yau manifolds, and holography. 
  The E. Cartan's equations defining "simple" spinors (renamed "pure" by C. Chevalley) are interpreted as equations of motions for fermion multiplets in momentum spaces which, in a constructive approach based bilinearly on those spinors, result compact and lorentzian, naturally ending up with a ten dimension space.   The equations found are most of those traditionally adopted ad hoc by theoretical physics in order to represent the observed phenomenology of elementary particles. In particular it is shown how, the known internal symmetry groups, in particular those of the standard model, might derive from the 3 complex division algebras correlated with the associated Clifford algebras. They also explain the origin of charges, the tendency of fermions to appear in charged-neutral doublets, as well as the origin of families.   The adoption of the Cartan's conjecture on the non elementary nature of euclidean geometry (bilinearly generated by simple or pure spinors) might throw light on several problematic aspects of particle physics. 
  The classic argument by Polyakov showing that monopoles produce confinement in the Higgs phase of the Georgi-Glashow model is generalized to study the spectrum of k-strings. We find that the leading-order low-density approximation yields Casimir scaling in the weakly-coupled 3-d SU(N) Georgi-Glashow model. Corrections to the Casimir formula are considered. When k is of the order of N, the non-diluteness effect is of the same order as the leading term, indicating that non-diluteness can significantly change the Casimir-scaling behavior. The correction produced by the propagating Higgs field is also studied and found to increase, together with the non-diluteness effect, the Casimir-scaling ratio. Furthermore, a correction due to closed k-strings is also computed and is shown to yield the same k-dependence as the one due to non-diluteness, but with the opposite sign and a nontrivial N-dependence. Finally, we consider the possible implications of our analysis for the SU(N) analogue of compact QED in four dimensions. 
  We calculate the radiated energy to $O(\hbar)$ from a charged wave-packet in the uniform magnetic field. In the high-speed and weak-field limit, while the non-commutativity of the system reduces the classical radiation, the additional corrections originated from the velocity uncertainty of the wave-packet leads to an enhancement of the radiation. 
  It is shown that Euclidean field theory with polynomial interaction, can be regularized using the wavelet representation of the fields. The connections between wavelet based regularization and stochastic quantization are considered with $\phi^3$ field theory taken as an example. 
  We investigate the Hamiltonian formulation of quantum scalar fields in a static quantum metric. We derive a functional integral formula for the propagator. We show that the quantum metric substantially changes the behaviour of the scalar propagator and the effective Yukawa potential. 
  The effective gravitational mass as well as the energy and momentum distributions of a radiating charged particle in Einstein's universe are evaluated. The Moller's energy-momentum complex is employed for this computation. The spacetime under study is a generalization of Bonnor and Vaidya spacetime in the sense that the metric is described in the cosmological background of Einstein's universe in lieu of the flat background. Several spacetimes are limiting cases of the one considered here. Particularly for the Reissner-Nordstrom black hole background, our results are exactly the same with those derived by Cohen and Gautreau using Whittaker's theorem and by Cohen and de Felice using Komar's mass. Furthermore, the power output for the spacetime under consideration is obtained. 
  It is known that hidden U(1) gauge factors can couple to visible U(1)'s through Kinetic Mixing. This phenomenon is shown generically to occur in nonsupersymmetric string set-ups, between branes and antibranes. Kinetic Mixing acts either to give millicharges (of e.g. hypercharge) to would-be hidden sector fermions, or to generate an enhanced communication of supersymmetry breaking that dominates over the usual gravitational suppression. In either case, the conclusion is that the string scale in nonsupersymmetric brane configurations has a generic upper bound of M_s <~ 10^8 GeV. 
  In this work we show how to define the action of a scalar field in a such a way that Robin boundary condition is implemented dynamically, i.e., as a consequence of the stationary action principle. We discuss the quantization of that system via functional integration. Using this formalism, we derive an expression for the Casimir energy of a massless scalar field under Robin boundary conditions on a pair of parallel plates, characterized by constants $c_1$ and $c_2$. Some special cases are discussed; in particular, we show that for some values of $c_1$ and $c_2$ the Casimir energy as a function of the distance between the plates presents a minimum. We also discuss the renormalization at one-loop order of the two-point Green function in the $\lambda\phi^4$ theory submitted to Robin boundary condition on a plate. 
  We review the attempts to construct black hole/string solutions in asymptotically plane wave spacetimes. First, we demonstrate that geometries admitting a covariantly constant null Killing vector cannot admit event horizons, which implies that pp-waves can't describe black holes. However, relaxing the symmetry requirements allows us to generate solutions which do possess regular event horizons while retaining the requisite asymptotic properties. In particular, we present two solution generating techniques and use them to construct asymptotically plane wave black string/brane geometries. 
  A Hamiltonian treatment of the gravitational collapse of thin shells is presented. The direct integration of the canonical constraints reproduces the standard shell dynamics for a number of known cases. The formalism is applied in detail to three dimensional spacetime and the properties of the (2+1)-dimensional charged black hole collapse are further elucidated. The procedure is also extended to deal with rotating solutions in three dimensions. The general form of the equations providing the shell dynamics implies the stability of black holes, as they cannot be converted into naked singularities by any shell collapse process. 
  It is generally believed the way to resolve the black hole information paradox in string theory is to embed the black hole in anti-deSitter spacetime -- without of course claiming that Schwarzschild-AdS is a realistic spacetime. Here we propose that, similarly, the best way to study topologically non-trivial versions of de Sitter spacetime from a stringy point of view is to embed them in an anti-de Sitter orbifold bulk, again without claiming that this is literally how de Sitter arises in string theory. Our results indicate that string theory may rule out the more complex spacetime topologies which are compatible with local de Sitter geometry, while still allowing the simplest versions. 
  Motivated by the conjecture that the cosmological constant problem is solved by strong quantum effects in the infrared we use the exact flow equation of Quantum Einstein Gravity to determine the renormalization group behavior of a class of nonlocal effective actions. They consist of the Einstein-Hilbert term and a general nonlinear function $F_k(V)$ of the Euclidean spacetime volume $V$. For the $V + V \ln V$-invariant the renormalization group running enormously suppresses the value of the renormalized curvature which results from Planck-size parameters specified at the Planck scale. One obtains very large, i.e., almost flat universes without finetuning the cosmological constant. A critical infrared fixed point is found where gravity is scale invariant. 
  We discuss classical and quantum aspects of the dynamics of a family of domain walls arising in a generalized Wess-Zumino model. These domain walls can be embedded in ${\cal N}=1$ supergravity as exact solutions and are composed of two basic lumps. 
  We develop a formalism that allows a complete classification of four-dimensional Z2XZ2 heterotic string models. Three generation models in a sub-class of these compactifications are related to the existence of three twisted sectors in Z2XZ2 orbifolds. In the work discussed here we classify the sub-class of these models that produce spinorial representations from all three twisted planes, and including symmetric shifts on the internal lattice. We show that perturbative three generation models are not obtained solely with symmetric shifts on complex tori, but necessitate the action of an asymmetric shift. In a subclass of these models we show that their chiral content is predetermined by the choice of the N=4 lattice. The implication of the results and possible geometrical interpretation are briefly discussed. 
  Using the techniques of out-of-equilibrium field theory, we study the influence on the properties of cosmological perturbations generated during inflation on observable scales coming from fluctuations corresponding today to scales much bigger than the present Hubble radius. We write the effective action for the coarse-grained inflaton perturbations integrating out the sub-horizon modes, which manifest themselves as a colored noise and lead to memory effects. Using the simple model of a scalar field with cubic self-interactions evolving in a fixed de Sitter background, we evaluate the two- and three-point correlation function on observable scales. Our basic procedure shows that perturbations do preserve some memory of the super-horizon-scale dynamics, in the form of scale-dependent imprints in the statistical moments. In particular, we find a blue tilt of the power-spectrum on large scales, in agreement with the recent results of the WMAP collaboration which show a suppression of the lower multipoles in the Cosmic Microwave Background anisotropies, and a substantial enhancement of the intrinsic non-Gaussianity on large scales 
  We consider branes as "points" in an infinite dimensional brane space ${\cal M}$ with a prescribed metric. Branes move along the geodesics of ${\cal M}$. For a particular choice of metric the equations of motion are equivalent to the well known equations of the Dirac-Nambu-Goto branes (including strings). Such theory describes "free fall" in ${\cal M}$-space. In the next step the metric of ${\cal M}$-space is given the dynamical role and a corresponding kinetic term is added to the action. So we obtain a background independent brane theory: a space in which branes live is ${\cal M}$-space and it is not given in advance, but comes out as a solution to the equations of motion. The embedding space ("target space") is not separately postulated. It is identified with the brane configuration. 
  The time evolution of strongly exited SU(2) Bogomolny-Prasad-Sommerfield (BPS) magnetic monopoles in Minkowski spacetime is investigated by means of numerical simulations based on the technique of conformal compactification and on the use of hyperboloidal initial value problem. It is found that an initially static monopole does not radiate the entire energy of the exciting pulse toward future null infinity. Rather, a long-lasting quasi-stable `breathing state' develops in the central region and certain expanding shell structures -- built up by very high frequency oscillations -- are formed in the far away region. 
  The spherically symmetric magnetic monopole in an SU(2) gauge theory coupled to a massless Higgs field is shown to possess an infinite number of resonances or quasinormal modes. These modes are eigenfunctions of the isospin 1 perturbation equations with complex eigenvalues, $E_n=\omega_n-i\gamma_n$, satisfying the outgoing radiation condition. For $n\to\infty$, their frequencies $\omega_n$ approach the mass of the vector boson, $M_W$, while their lifetimes $1/\gamma_n$ tend to infinity. The response of the monopole to an arbitrary initial perturbation is largely determined by these resonant modes, whose collective effect leads to the formation of a long living breather-like excitation characterized by pulsations with a frequency approaching $M_W$ and with an amplitude decaying at late times as $t^{-5/6}$. 
  We compute radiative corrections in five and six dimensional field theories, using winding modes in mixed momentum-coordinate space. This method provides a simple way of finding UV divergencies, finite corrections and localized terms when the space is compactified on orbifolds. As an application we compute the finite piece of scalar masses, the logarithmic contributions to the couplings and the effect of localized parallel and perpendicular kinetic terms. We apply it to get a two loop effective potential that can stabilize large extra dimensions. 
  We develop a general formalism for covariant Hamiltonian evolution of supersymmetric (field) theories by making use of the fact that these can be represented on the exterior bundle over their bosonic configuration space as generalized Dirac-Kaehler systems of the form $(d \pm d^\dag)\ket{\psi} = 0$. By using suitable deformations of the supersymmetry generators we find covariant Hamiltonians for target spaces with general gravitational and Kalb-Ramond field backgrounds and discuss their perturbation theory.   As an example, these results are applied to the study of curvature corrections of superstring spectra for $AdS_3 x S^3$ close to its pp-wave limit. 
  We use the `branes within branes' approach to study the appearance of stable $ (p-2)-branes and unstable (p-1)-branes in type II string theory from p-brane--p-antibrane pairs.Our goal is to describe the emergence of these lower dimensional branes from brane-antibrane pairs in string theory using a tractable gauge theory language. This is achieved by suspending the original p-brane--p-antibrane pair between two (p+2)-branes, and describing its dynamics in terms of the worldvolume gauge theory on the spectator (p+2)-branes. Instantons, monopoles, sphalerons and their higher-dimensional generalizations in this worldvolume gauge theory correspond to stable (BPS) and unstable (non-BPS) branes in string theory. Collisions of stable branes with corresponding antibranes and production of lower-dimensional branes in string theory are described in a straightforward way in gauge theory. Tachyonic modes on the p-brane--p-antibrane worldvolume do not appear in our analysis since we work on the worldvolume of the spectator (p+2)-branes. Our results on brane descent relations are in agreement with Sen's tachyon condensation approach. 
  We give an introduction to the recently established connection between supersymmetric gauge theories and matrix models. We begin by reviewing previous material that is required in order to follow the latest developments. This includes the superfield formulation of gauge theories, holomorphy, the chiral ring, the Konishi anomaly and the large N limit. We then present both the diagrammatic proof of the connection and the one based on the anomaly. Our discussion is entirely field theoretical and self contained. 
  In order for Dirac theory to be gauge invariant it can be shown that the Schwinger term must be zero. However, it can also be shown that for the vacuum state to be the lowest energy state the Schwinger term must be nonzero. Therefore there is an inconsistency in Dirac theory involving the evaluation of the Schwinger term. This inconsistency is discussed along with a possible way to resolve it. 
  We classify the possible bosonic and Type 0 unoriented string theories in two dimensions, and find their dual matrix(-vector) models. There are no RP^2 R-R tadpoles in any of the models, but many of them possess a massless tachyon tadpole. Thus all the models we find are consistent two-dimensional string vacua, but some get quantum corrections to their classical tachyon background. Where possible, we solve the tadpole cancellation condition, and find all the tachyon tadpole-free theories. 
  We show that all two-dimensional conformal field theories possess a hidden sl(2,R) affine symmetry. More precisely, we add appropriate ghost fields to an arbitrary CFT, and we use them to construct the currents of sl(2,R). We then define a BRST operator whose cohomology defines a physical subspace where the extended theory coincides with the original CFT. We use the sl(2,R) algebra to construct candidate wave functions for 3-d quantum gravity coupled to matter, and we discuss their viability. 
  We discuss the Schwarzschild solution in the Dvali-Gabadadze-Porrati (DGP) model. We obtain a perturbative expansion and find the explicit form of the lowest-order contribution. By keeping off-diagonal terms in the metric, we arrive at a perturbative expansion which is valid both far from and near the Schwarzschild radius. We calculate the lowest-order contribution explicitly and obtain the form of the metric both on the brane and in the bulk. As we approach the Schwarzschild radius, the perturbative expansion yields the standard four-dimensional Schwarzschild solution on the brane which is non-singular in the decoupling limit. This non-singular behavior is similar to the Vainshtein solution in massive gravity demonstrating the absence of the van Dam-Veltman-Zakharov (vDVZ) discontinuity in the DGP model. 
  An exponential potential of the form $V\sim \exp(-2c \phi/M_p)$ arising from the hyperbolic or flux compactification of higher-dimensional theories is of interest for getting short periods of accelerated cosmological expansions. Using a similar potential but derived for the combined case of hyperbolic-flux compactification, we study the four-dimensional flat (and open) FLRW cosmologies and give analytic (and numerical) solutions with exponential behavior of scale factors. We show that, for the M-theory motivated potentials, the cosmic acceleration of the universe can be eternal if the spatial curvature of the 4d spacetime is negative, while the acceleration is only transient for a spatially flat universe. We also comment on the size of the internal space and its associated geometric bounds on massive Kaluza-Klein excitations. 
  The asymmetric ABAB-matrix model describes the transfer matrix of three-dimensional Lorentzian quantum gravity. We study perturbatively the scaling of the ABAB-matrix model in the neighbourhood of its symmetric solution and deduce the associated renormalization of three-dimensional Lorentzian quantum gravity. 
  We study operator mixing, due to planar one-loop corrections, for composite operators in D=4 supersymmetric theories. We present some N=1,2 Yang-Mills and Wess-Zumino models, in which the planar one-loop anomalous dimension matrix in the sector of holomorphic scalars is identified with the Hamiltonian of an integrable quantum spin chain of SU(3) or SU(2) symmetry, even if the theory is away from the conformal points. This points to a more universal origin of the integrable structure beyond superconformal symmetry. We also emphasize the role of the superpotential in the appearance of the integrable structure. The computations of operator mixing in our examples by solving Bethe Ansatz equations show some new features absent in N=4 SYM. 
  We look at general braneworlds in six-dimensional Einstein-Gauss-Bonnet gravity. We find the general matching conditions for the Einstein-Gauss-Bonnet braneworld, which remarkably turn out to give precisely the four-dimensional Einstein equations for the induced metric and matter on the brane, even when the extra dimensions are non-compact and have infinite volume. We also show that relaxing regularity of the curvature in the vicinity of the brane, or alternatively having a finite width brane, gives rise to an additional possible correction to the Einstein equations, which contains information on the brane's embedding in the bulk and cannot be determined from knowledge of the braneworld alone. We comment on the advantages and disadvantages of each possibility, and the relevance of these results regarding a possible solution of the cosmological constant problem. 
  We derive expressions for three-body phase space that are explicitly symmetrical in the masses of the three particles. We study geometrical properties of the variables involved in elliptic integrals and demonstrate that it is convenient to use the Jacobian zeta function to express the results in four and six dimensions. 
  In this paper we first derive solutions which can be interpreted as branes wrapping nontrivial curved manifolds, and then study their cosmological implications. We find that at early times the branes tend to shrink the internal manifold, while allowing the ``unwrapped'' dimensions to expand in congruence with what has already been observed in the case when the internal manifold is flat (tori). However, at late times the internal curvature terms become important leading to potentially interesting differences. 
  The distance between BPS branes in string theory corresponds to a flat direction in the effective potential. Small deviations from supersymmetry may lead to a small uplifting of this flat direction and to brane inflation. However, this scenario can work only if the BPS properties of the branes and the corresponding flatness of the inflaton potential are preserved in the theories with the stable volume compactification. We present an ``inflaton trench'' mechanism that keeps the inflaton potential flat due to shift symmetry, which is related to near BPS symmetry in our model. 
  Much about the confinement and dynamical symmetry breaking in QCD might be learned from models with supersymmetry. In particular, models based on N=2 supersymmetric theories with gauge groups SU(N), SO(N) and $USp(2 N)$ and with various number of flavors, give deep dynamical hints about these phenomena. For instance, the BPS non-abelian monopoles can become the dominant degrees of freedom in the infrared due to quantum effects. Upon condensation (which can be triggered in these class of models by perturbing them with an adjoint scalar mass) they induce confinement with calculable pattern of dynamical symmetry breaking. This may occur either in a weakly interacting regime or in a strongly coupled regime (in the latter, often the low-energy degrees of freedom contain relatively non-local monopoles and dyons simultaneously and the system is near a nontrivial fixed-point).   Also, the existence of sytems with BPS {\it non-abelian vortices} has been shown recently. These results point toward the idea that the ground state of QCD is a sort of dual superconductor of non-abelian variety. 
  A generic property of curved manifolds is the existence of focal points. We show that branes located at focal points of the geometry satisfy special properties. Examples of backgrounds to which our discussion applies are AdS_m x S^n and plane wave backgrounds. As an example, we show that a pair of AdS_2 branes located at the north and south pole of the S^5 in AdS_5 x S^5 are half supersymmetric and that they are dual to a two-monopole solution of N=4 SU(N) SYM theory. Our second example involves spacelike branes in the (Lorentzian) plane wave. We develop a modified lightcone gauge for the open string channel, analyze in detail the cylinder diagram and establish open-closed duality. When the branes are located at focal points of the geometry the amplitude acquires most of the characteristics of flat space amplitudes. In the open string channel the special properties are due to stringy modes that become massless. 
  A class of non-linear eigenvalue problems defined in the form of operator polynomials is investigated. The problems are related to wave equations which appear in a relativistic quantum field theory. Spectral asymptotics for this class are found explicitly. The properties of operator polynomials are analyzed for scalar, spinor and gauge fields. It is also shown how to use these results in finite temperature theories. 
  The original Calogero and Sutherland models describe N quantum particles on the line interacting pairwise through an inverse square and an inverse sinus-square potential. They are well known to be integrable and solvable. Here we extend the Calogero and Sutherland Hamiltonians by means of new interactions which are PT-symmetric but not self adjoint. Some of these new interactions lead to integrable PT-symmetric Hamiltonians; the algebraic properties further reveal that they are solvable as well. We also consider PT-symmetric interactions which lead to a new quasi-exactly solvable deformation of the Calogero and Sutherland Hamiltonians. 
  A new symmetry-preserving loop regularization method proposed in \cite{ylw} is further investigated. It is found that its prescription can be understood by introducing a regulating distribution function to the proper-time formalism of irreducible loop integrals. The method simulates in many interesting features to the momentum cutoff, Pauli-Villars and dimensional regularization. The loop regularization method is also simple and general for the practical calculations to higher loop graphs and can be applied to both underlying and effective quantum field theories including gauge, chiral, supersymmetric and gravitational ones as the new method does not modify either the lagrangian formalism or the space-time dimension of original theory. The appearance of characteristic energy scale $M_c$ and sliding energy scale $\mu_s$ offers a systematic way for studying the renormalization-group evolution of gauge theories in the spirit of Wilson-Kadanoff and for exploring important effects of higher dimensional interaction terms in the infrared regime. 
  We discuss linearized gravity from the point of view of a gauge theory. In (3+1)-dimensions our analysis allows to consider linearized gravity in the context of the MacDowell-Mansouri formalism. Our observations may be of particular interest in the strong-weak coupling duality for linearized gravity, in Randall-Sundrum brane world scenario and in Ashtekar formalism. 
  We consider D7-branes in the gauge theory/string theory correspondence, using a probe approximation. The D7-branes have four directions embedded holomorphically in a non-compact Calabi-Yau 3-fold (which for specificity we take to be the conifold) and their remaining four directions are parallel to a stack of D3-branes transverse to the Calabi-Yau space. The dual gauge theory, which has $\mathcal{N}=1$ supersymmetry, contains quarks which transform in the fundamental representation of the gauge group, and we identify the interactions of these quarks in terms of a superpotential. By activating three-form fluxes in the gravity background, we obtain a dual gauge theory with a cascade of Seiberg dualities. We find a supersymmetric supergravity solution for the leading backreaction effects of the D7-branes, valid for asymptotically large radius. The cascading theory with flavors exhibits the interesting phenomenon that the rate of the cascade slows and can stop as the theory flows to the infrared. 
  The spin-1/2 XXZ Heisenberg chain with two types of boundary terms is considered. For the first type, the Hamiltonian is hermitian but not for the second type which includes the U_{q}[SU(2)] symmetric case. It is shown that for a certain `tuning' between the anisotropy angle and the boundary terms the spectra present unexpected degeneracies. These degeneracies are related to the structure of the irreducible representations of the Virasoro algebras for c<1. 
  We present a derivation of the chiral ring relations, arising in ${\cal N}=1$ gauge theories in the presence of (anti-)self-dual background gravitational field $G_{\alpha\beta\gamma}$ and graviphoton field strength $F_{\alpha\beta}$. These were previously considered in the literature in order to prove the relation between gravitational F-terms in the gauge theory and coefficients of the topological expansion of the related matrix integral. We consider the spontaneous breaking of ${\cal N} =2$ to ${\cal N} =1$ supergravity coupled to vector- and hyper-multiplets, and take a rigid limit which keeps a non-trivial $G_{\alpha\beta\gamma}$ and $F_{\alpha\beta}$ with a finite supersymmetry breaking scale. We derive the resulting effective, global, ${\cal N}=1$ theory and show that the chiral ring relations are just a consequence of the standard ${\cal N}=2$ supergravity Bianchi identities . We can also obtain models with matter in different representations and in particular quiver theories. We also show that, in the presence of non-trivial $F_{\alpha\beta}$, consistency of the Konishi-anomaly loop equations with the chiral ring relations, demands that the gauge kinetic function and the superpotential, a priori unrelated for an ${\cal N}=1$ theory, should be derived from a prepotential, indicating an underlying ${\cal N}=2$ structure. 
  The Dirac equation is exactly solved for a pseudoscalar linear plus Coulomb-like potential in a two-dimensional world. This sort of potential gives rise to an effective quadratic plus inversely quadratic potential in a Sturm-Liouville problem, regardless the sign of the parameter of the linear potential, in sharp contrast with the Schroedinger case. The generalized Dirac oscillator already analyzed in a previous work is obtained as a particular case. 
  We argue that the chiral anomaly of $\Ncal = 1$ super Yang-Mills theory admits a dual description as spontaneous symmetry breaking in M theory on $G_2$ holonomy manifolds. We identify an angle of the $G_2$ background dual to the anomalous $U(1)_R$ current in field theory. This angle is not an isometry of the metric and we therefore develop a theory of ``massive isometry'' to describe fluctuations about such angles. Another example of a massive isometry occurs in the Atiyah-Hitchin metric. 
  An interesting interplay between self-duality, the Kodama (Chern-Simons) state and knot invariants is shown to emerge in the quantum theory of an Abelian gauge theory. More precisely, when a self-dual representation of the CCR is chosen, the corresponding vacuum in the Schroedinger representation is precisely given by the Kodama state. Several consequences of this construction are explored. 
  The near-horizon geometry of a large class of extremal and near-extremal black holes in string and M theory contains three-dimensional asymptotically anti-de Sitter space. Motivated by this structure, we are led naturally to a discrete set of complex frequencies defined in terms of the monodromy at the inner and outer horizons of the black hole. We show that the correspondence principle, whereby the real part of these ``non-quasinormal frequencies'' is identified with certain fundamental quanta, leads directly to the correct quantum behavior of the near-horizon Virasoro algebra, and thus the black hole entropy. Remarkably, for the rotating black hole in five dimensions we also reproduce the fractionization of conformal weights predicted in string theory. 
  The supersymmetry in quantum mechanics and shape invariance condition are applied as an algebraic method to solve the Dirac-Coulomb problem. The ground state and the excited states are investigated using new generalized ladder operators. 
  It has been found that the states of the 2-charge extremal D1-D5 system are given by smooth geometries that have no singularity and no horizon individually, but a `horizon' does arise after `coarse-graining'. To see how this concept extends to the 3-charge extremal system, we construct a perturbation on the D1-D5 geometry that carries one unit of momentum charge $P$. The perturbation is found to be regular everywhere and normalizable, so we conclude that at least this state of the 3-charge system behaves like the 2-charge states. The solution is constructed by matching (to several orders) solutions in the inner and outer regions of the geometry. We conjecture the general form of `hair' expected for the 3-charge system, and the nature of the interior of black holes in general. 
  N=2 gauge theories broken down to N=1 by a tree level superpotential are necessarily at the points in the moduli space where the Seiberg-Witten curve factorizes. We find exact solution to the factorization problem of Seiberg-Witten curves associated with the breaking of the U(N_c) gauge group down to two factors U(N_1)xU(N_2). The result is a function of three discrete parameters and two continuous ones. We find discrete identifications between various sets of parameters and comment on their relation to the global structure of N=1 vacua and their various possible dual descriptions. In an appendix we show directly that integrality of periods leads to factorization. 
  A new approach to generalised Casimir type of problems is derived within the context of renormalisable quantum field theory (QFT). We study the simplest case of a massive fluctuating boson field coupled to a time-independent background potential. We use analytic properties of scattering data to compute the relevant Green's functions at imaginary momenta, which in turn yields a simple and efficient method to compute (one-loop) vacuum energy densities in QFT. Renormalisation is easily performed in the perturbative sector by identifying low order Feynman diagrams with the first few Born approximation to the Green's function. Numerical examples illustrate the efficiency of our approach. 
  We identify a class of point-particle models that exhibit a target-space duality. This duality arises from a construction based on supersymmetric quantum mechanics with a non-vanishing central charge. Motivated by analogies to string theory, we are led to speculate regarding mechanisms for restricting the background geometry. 
  Following a recently proposed confinement generating scenario \cite{Di}, we provide a new string inspired model with a massive dilaton and a general dilaton-gluon coupling. By solving analytically the equations of motion, we derive a new class of confining interquark potentials, which includes most of the QCD motivated potential forms given in the literature. 
  It is well known that velocities does not commute in presence of an electromagnetic field. This property implies that angular algebra symmetries, such as the sO(3) and Lorentz algebra symmetries, are broken. To restore these angular symmetries we show the necessity of adding the Poincare momentum M to the simple angular momentum L. These restorations performed succesively in a flat space and in a curved space lead in each cases to the generation of a Dirac magnetic monopole. In the particular case of the Lorentz algebra we consider an application of our theory to the gravitoelectromagnetism. In this last case we establish a qualitative relation giving the mass spectrum for dyons. 
  We prove an equivalence, in the large N limit, between certain U(N) gauge theories containing adjoint representation matter fields and their orbifold projections. Lattice regularization is used to provide a non-perturbative definition of these theories; our proof applies in the strong coupling, large mass phase of the theories. Equivalence is demonstrated by constructing and comparing the loop equations for a parent theory and its orbifold projections. Loop equations for both expectation values of single-trace observables, and for connected correlators of such observables, are considered; hence the demonstrated non-perturbative equivalence applies to the large N limits of both string tensions and particle spectra. 
  We derive the BPS equations satisfied by lump solitons in $(2+1)$-dimensional sigma models with toric 8-dimensional hyper-K\"ahler (${HK}_8$) target spaces and check they preserve 1/2 of the supersymmetry. We show how these solitons are realised in M theory as M2-branes wrapping holomorphic 2-cycles in the $\bE^{1,2}\times {HK}_8$ background. Using the $\kappa$-symmetry of a probe M2-brane in this background we determine the supersymmetry they preserve, and note that there is a discrepancy in the fraction of supersymmetry preserved by these solitons as viewed from the low energy effective sigma model description of the M2-brane dynamics or the full M theory. Toric ${HK}_8$ manifolds are dual to a Hanany-Witten setup of D3-branes suspended between 5-branes. In this picture the lumps correspond to vortices of the three dimensional ${\mathcal N}=3$ or ${\mathcal N}=4$ theory. 
  The trace anomaly for a conformally invariant scalar field theory on a curved manifold of positive constant curvature with boundary is considered. In the context of a perturbative evaluation of the theory's effective action explicit calculations are given for those contributions to the conformal anomaly which emerge as a result of free scalar propagation as well as from scalar self-interactions up to second order in the scalar self-coupling. The renormalisation-group behaviour of the theory is, subsequently, exploited in order to advance the evaluation of the conformal anomaly to third order in the scalar self-coupling. In effect, complete contributions to the theory's conformal anomaly are evaluated up to fourth-loop order. 
  We study conditions on the topological D-branes of types A and B obtained by requiring a proper matching of the spectral flow operators on the boundary. These conditions ensure space-time supersymmetry and stability of D-branes. In most cases, we reproduce the results of Marino-Minasian-Moore-Strominger, who studied the same problem using the supersymmetric Born-Infeld action. In some other cases, corresponding to coisotropic A-branes, our stability condition is new. Our results enable us to define an analogue of the Maslov class and grading for coisotropic A-branes. We expect that they play a role in a conjectural generalization of the Floer homology. 
  It is shown that the gauge theory of relativistic 3-Branes can be formulated in a conformally invariant way if the embedding space is six-dimensional. The implementation of conformal invariance requires the use of a modified measure, independent of the metric in the action. Brane-world scenarios without the need of a cosmological constant in 6D are constructed. Thus, no ``old'' cosmological constant problem appears at this level. 
  We summarize the field-theory/matrix model correspondence for a chiral N=1 model with matter in the adjoint, antisymmetric and conjugate symmetric representations as well as eight fundamentals to cancel the chiral anomaly. The associated holomorphic matrix model is consistent only for two fundamental fields, which requires a modification of the original Dijkgraaf-Vafa conjecture. The modified correspondence holds in spite of this mismatch. 
  We construct, in the framework of the N=4 SYM theory, a supermultiplet of twist-two conformal operators and study their renormalization properties. The components of the supermultiplet have the same anomalous dimension and enter as building blocks into multi-particle quasipartonic operators. The latter are determined by the condition that their twist equals the number of elementary constituent fields from which they are built. A unique feature of the N=4 SYM is that all quasipartonic operators with different SU(4) quantum numbers fall into a single supermultiplet. Among them there is a subsector of the operators of maximal helicity, which has been known to be integrable in the multi-color limit in QCD, independent of the presence of supersymmetry. In the N=4 SYM theory, this symmetry is extended to the whole supermultiplet of quasipartonic operators and the one-loop dilatation operator coincides with a Hamiltonian of integrable SL(2|4) Heisenberg spin chain. 
  We study the stretched membrane of a black hole as consisting of a perfect fluid. We find that the pressure of this fluid is negative and the specific heat is negative too. A surprising result is that if we are to assume the fluid be composed of some quanta, then the dispersion relation of the fundamental quantum is $E=m^2/k$, with $m$ at the scale of the Planck mass. There are two possible interpretation of this dispersion relation, one is the noncommutative spacetime on the stretched membrane, another is that the fundamental quantum is microscopic black holes. 
  In this paper we study braneworld cosmology when the bulk space is a charged black hole in de Sitter space (Topological Reissner-Nordstr\"om de Sitter Space) in general dimension, then we compute leading order correction to the Friedmann equation that arise from logarithmic corrections to the entropy in the holographic-branworld cosmological framwork. Finally we consider the holographic entropy bounds in this senario, we show the entropy bounds are also modified by logarithmic term. 
  A class of Z_2-graded Lie algebra and Lie superalgebra extensions of the pseudo-orthogonal algebra of a spacetime of arbitrary dimension and signature is investigated. They have the form g = g_0 + g_1, with g_0 = so(V) + W_0 and g_1 = W_1, where the algebra of generalized translations W = W_0 + W_1 is the maximal solvable ideal of g, W_0 is generated by W_1 and commutes with W. Choosing W_1 to be a spinorial so(V)-module (a sum of an arbitrary number of spinors and semispinors), we prove that W_0 consists of polyvectors, i.e. all the irreducible so(V)-submodules of W_0 are submodules of \Lambda V. We provide a classification of such Lie (super)algebras for all dimensions and signatures. The problem reduces to the classification of so(V)-invariant \Lambda^k V-valued bilinear forms on the spinor module S. 
  In this paper we construct Einstein spaces with negative Ricci curvature in various dimensions. These spaces -- which can be thought of as generalised AdS spacetimes -- can be classified in terms of the geometry of the horospheres in Poincare-like coordinates, and can be both homogeneous and static. By using simple building blocks, which in general are homogeneous Einstein solvmanifolds, we give a general algorithm for constructing Einstein metrics where the horospheres are any product of generalised Heisenberg geometries, nilgeometries, solvegeometries, or Ricci-flat manifolds. Furthermore, we show that all of these spaces can give rise to black holes with the horizon geometry corresponding to the geometry of the horospheres, by explicitly deriving their metrics. 
  We compare the classical scattering of kinks in (1+1) Higgs model with its analogous noncommutative counterpart. While at a classical level we are able to solve the scattering at all orders finding a smooth solution, at a noncommutative level we present only perturbative results, suggesting the existence of a smooth solution also in this case. 
  We argue that a modification of the super-AdS algebras which accounts for the presence of D-branes requires not only the inclusion of bosonic brane charges, but also the inclusion of new fermionic ones. We show that such fermionic brane charges are indeed present in the matrix model and the supermembrane in the pp-wave limit of the corresponding backgrounds. We briefly comment on an AdS version of Sezgin's M-algebra inspired by this observation. 
  We study the role that tachyon fields may play in cosmology as compared to the well-established use of minimally coupled scalar fields. We first elaborate on a kind of correspondence existing between tachyons and minimally coupled scalar fields; corresponding theories give rise to the same cosmological evolution for a particular choice of the initial conditions but not for any other. This leads us to study a specific one-parameter family of tachyonic models based on a perfect fluid mixed with a positive cosmological constant. For positive values of the parameter one needs to modify Sen's action and use the sigma process of resolution of singularities. The physics described by this model is dramatically different and much richer than that of the corresponding scalar field. For particular choices of the initial conditions the universe, that does mimick for a long time a de Sitter-like expansion, ends up in a finite time in a special type of singularity that we call a "big brake". This singularity is characterized by an infinite deceleration. 
  We determine the necessary and sufficient conditions on the metric and the four-form for the most general bosonic supersymmetric configurations of D=11 supergravity which admit a null Killing spinor i.e. a Killing spinor which can be used to construct a null Killing vector. This class covers all supersymmetric time-dependent configurations and completes the classification of the most general supersymmetric configurations initiated in hep-th/0212008. 
  A simple, but effcient way of calculating regularized Casimir energies suitable for non-trivial frequency spectra is briefly described and applied to the case of a kappa-deformed scalar field theory. The results are consistent with the ones obtained by other means. 
  We re-evaluate the zero point Casimir energy for the case of a massive scalar field in $\mathbf{R}^{1}\times\mathbf{S}^{3}$ space, allowing also for deviations from the standard conformal value $\xi =1/6$, by means of zero temperature zeta function techniques. We show that for the problem at hand this approach is equivalent to the high temperature regularisation of the vacuum energy. 
  We solve the equation of motion of Witten's cubic open string field theory in a series expansion using the regulated butterfly state. The expansion parameter is given by the regularization parameter of the butterfly state, which can be taken to be arbitrarily small. Unlike the case of level truncation, the equation of motion can be solved for an arbitrary component of the Fock space up to a positive power of the expansion parameter. The energy density of the solution is well-defined and remains finite even in the singular butterfly limit, and it gives approximately 68% of the D25-brane tension for the solution at the leading order. Moreover, it simultaneously solves the equation of motion of vacuum string field theory, providing support for the conjecture at this order. We further improve our ansatz by taking into account next-to-leading terms, and find two numerical solutions which give approximately 88% and 109%, respectively, of the D25-brane tension for the energy density. These values are interestingly close to those by level truncation at level 2 without gauge fixing studied by Rastelli and Zwiebach and by Ellwood and Taylor. 
  Kovtun, Son and Starinets proposed a bound on the viscosity of any fluid in terms of its entropy density. The bound is saturated by maximally supersymmetric theories at strong coupling, but can also easily be challenged experimentally to within a factor of 10 already today. We argue that this bound follows directly from the generalized covariant entropy bound, bringing holography within the reach of experimental investigation. 
  Following hep-th/0211098 we study the matrix model which describes the topological A-model on T^{*}(S^{3}/\ZZ_p). We show that the resolvent has square root branch cuts and it follows that this is a p cut single matrix model. We solve for the resolvent and find the spectral curve. We comment on how this is related to large N transitions and mirror symmetry. 
  We give the results for all the one-loop propagators, including finite parts, in the Coulomb gauge. In finite parts we find new non-rational functions in addition to the single logarithms of the Feynman gauge. Of course, the two gauges must agree for any gauge invariant function. 
  We consider compactifications of M-theory on 7-manifolds in the presence of 4-form fluxes, which leave at least four supercharges unbroken. We focus especially on the case, where the 7-manifold supports two spinors which are SU(3) singlets and the fluxes appear as specific SU(3) structures. We derive the constraints on the fluxes imposed by supersymmetry and calculate the resulting 4-dimensional superpotential. 
  Perturbative spectra and related factorization properties of one-loop open string amplitudes in the presence of a constant external background B are analysed in detail. While the pattern of the closed string spectrum, obtained after a careful study of the properly symmetrized amplitudes, turns out to be unaffected by the presence of B, a series of double open-string poles, which would be absent when B is turned off, can couple owing to a partial symmetry loss. These features are studied first in a bosonic setting and then generalized in the more satisfactory superstring context. When the background is of an ``electric'' type, a classical perturbative instability is produced beyond a critical value of the electric field. In the Seiberg-Witten limit this instability is the origin of the unphysical tachyonic cut occurring in the non-planar amplitudes of the corresponding noncommutative field theories. 
  We study Born-Infeld strings in a six dimensional brane world scenario recently suggested by Giovannini, Meyer and Shaposhnikov (GMS). In the limit of the Einstein-Abelian-Higgs model, we classify the solutions found by GMS. Especially, we point out that the warped solutions, which lead to localisation of gravity, are the - by the presence of the cosmological constant - deformed inverted string solutions. Further, we construct the Born-Infeld analogues of the anti-warped solutions and determine the domain of existence of these solutions, while a analytic argument leads us to a "no-go" hypothesis: solutions which localise gravity do NOT exist in a 6 dimensional Einstein-Born-Infeld-Abelian-Higgs (EBIAH) brane world scenario. This latter hypothesis is confirmed by our numerical results. 
  When string theory is compactified on a six-dimensional manifold with a nontrivial NS flux turned on, mirror symmetry exchanges the flux with a purely geometrical composite NS form associated with lack of integrability of the complex structure on the mirror side. Considering a general class of T^3-fibered geometries admitting SU(3) structure, we find an exchange of pure spinors (e^{iJ} and \Omega) in dual geometries under fiberwise T-duality, and study the transformations of the NS flux and the components of intrinsic torsion. A complementary study of action of twisted covariant derivatives on invariant spinors allows to extend our results to generic geometries and formulate a proposal for mirror symmetry in compactifications with NS flux. 
  In this talk, we review the instability due to radiatively induced FI tadpoles in ${\cal N}=2$ supersymmetric gauge theories on orbifolds in six dimensions. Even with the localized FI tadpoles, we have the unbroken supersymmetry at the expense of having the spontaneous localization of bulk zero mode. We find the non-decoupling of massive modes unlike the 5D case. We also comment on the local anomaly cancellation. 
  Multigravity theories are constructed from the discretization of the extra dimension of five dimensional gravity. Using an ADM decomposition, the discretization is performed while maintaining the four dimensional diffeomorphism invariance on each site. We relate the Goldstone bosons used to realize nonlinearly general covariance in discretized gravity to the shift fields of the higher dimensional metric. We investigate the scalar excitations of the resulting theory and show the absence of ghosts and massive modes; this is due to a local symmetry inherited from the reparametrization invariance along the fifth dimension. 
  We determine solutions to 5D Einstein gravity with a discrete fifth dimension. The properties of the solutions depend on the discretization scheme we use and some of them have no continuum counterpart. In particular, we find that the neglect of the lapse field (along the discretized direction) gives rise to Randall-Sundrum type metric with a negative tension brane. However, no brane source is required. We show that this result is robust under changes in the discretization scheme. The inclusion of the lapse field gives rise to solutions whose continuum limit is gauge fixed by the discretization scheme. We find however one particular scheme which leads to an undetermined lapse reflecting the reparametrization invariance of the continuum theory. We also find other solutions, with no continuum counterpart with changes in the metric signature or avoidance of singularity. We show that the models allow a continuous mass spectrum for the gravitons with an effective 4D interaction at small scales. We also discuss some cosmological solutions. 
  This thesis considers one and two dimensional supersymmetric nonlinear sigma models. First there is a discussion of the geometries of one and two dimensional sigma models, with rigid supersymmetry.   For the one-dimensional case, the supersymmetry is promoted to a local one and the required gauge fields are introduced. The most general Lagrangian, including these gauge fields, is found. The constraints of the system are analysed, and its Dirac quantisation is investigated.   In the next chapter we introduce equivariant cohomology which is used later in the thesis.   Then actions are constructed for (p,0)- and (p,1)- supersymmetric, $1 \leq p \leq 4$, two-dimensional gauge theories coupled to non-linear sigma model matter with a Wess-Zumino term.   The scalar potential for a large class of these models is derived. It is then shown that the Euclidean actions of the (2,0) and (4,0)-supersymmetric models without Wess-Zumino terms are bounded by topological charges which involve the equivariant extensions of the Kahler forms of the sigma model target spaces evaluated on the two-dimensional spacetime.   Similar bounds for Euclidean actions of appropriate gauge theories coupled to non-linear sigma model matter in higher spacetime dimensions are given which now involve the equivariant extensions of the Kahler forms of the sigma model target spaces and the second Chern character of gauge fields. It is found that the BPS configurations are generalisations of abelian and non-abelian vortices. 
  We describe how a stable effective theory in which particles of the same fermion number attract may spontaneously break Lorentz invariance by giving non-zero fermion number density to the vacuum (and therefore dynamically generating a chemical potential term). This mecanism yields a finite vacuum expectation value $<\bar\psi\gamma^\mu\psi>$ which we consider in the context of proposed models that require such a breaking of Lorentz invariance in order to yield composite degrees of freedom that act approximately like gauge bosons. We also make general remarks about how the background source provided by $<\bar\psi\gamma^\mu\psi>$ could relate to work on signals of Lorentz violation in electrodynamics. 
  We consider a fermionic determinant associated to a non covariant Quantum Field Theory used to describe a non relativistic system in (1+1) dimensions. By exploiting the freedom that arises when Lorentz invariance is not mandatory, we determine the heat-kernel regulating operator so as to reproduce the correct dispersion relations of the bosonic excitations. We also derive the Hamiltonian of the functionally bosonized model and the corresponding currents. In this way we were able to establish the precise heat-kernel regularization that yields complete agreement between the path-integral and operational approaches to the bosonization of the Tomonaga-Luttinger model. 
  Well-defined non-perturbative formulations of the physics of string theories, sometimes with D-branes present, were identified over a decade ago, from a careful study of double scaled matrix models. Following recent work which recasts some of those old results in the context of type 0 string theory, a study is made of a much larger family of models, which are proposed as type 0A models of the entire superconformal minimal series coupled to gravity. This gives many further examples of important physical phenomena, including non-perturbative descriptions of transitions between D-branes and fluxes, tachyon condensation, and holography. In particular, features of a large family of non-perturbatively stable string equations are studied, and results are extracted which pertain to type 0A string theory, with D-branes and fluxes, in this large class of backgrounds. For the entire construction to work, large parts of the spectrum of the supergravitationally dressed superconformal minimal models and that of the gravitationally dressed bosonic conformal minimal models must coincide, and it is shown how this happens. The example of the super-dressed tricritical Ising model is studied in some detail. 
  We calculate the anomalous magnetic moment of the electron in the Chern-Simons theory in 2+1 dimensions with and without a Maxwell term, both at zero temperature as well as at finite temperature. In the case of the Maxwell-Chern-Simons (MCS) theory, we find that there is an infrared divergence, both at zero as well as at finite temperature, when the tree level Chern-Simons term vanishes, which suggests that a Chern-Simons term is essential in such theories. At high temperature, the thermal correction in the MCS theory behaves as $\frac{1}{\beta} \ln \beta M$, where $\beta$ denotes the inverse temperature and $M$, the Chern-Simons coefficient. On the other hand, we find no thermal correction to the anomalous magnetic moment in the pure Chern-Simons (CS) theory. 
  We have constructed a modified BFT method that preserves the chain structure of constraints. This method has two advantages: first, it leads to less number of primary constraints such that the remaining constraints emerge automatically; and second, it gives less number of independent gauge parameters. We have applied the method for bosonized chiral Schwiger model. We have constructed a gauge invariant embedded Lagrangian for this model. 
  We consider ten-dimensional supersymmetric Yang-Mills theory (10D SUSY YM theory) and its dimensional reductions, in particular, BFSS and IKKT models. We formulate these theories using algebraic techniques based on application of differential graded Lie algebras and associative algebras as well as of more general objects, L_{\infty}- and A_{\infty}- algebras.   We show that using pure spinor formulation of 10D SUSY YM theory equations of motion and isotwistor formalism one can interpret these equations as Maurer-Cartan equations for some differential Lie algebra. This statement can be used to write BV action functional of 10D SUSY YM theory in Chern-Simons form. The differential Lie algebra we constructed is closely related to differential associative algebra Omega of (0, k)-forms on some supermanifold; the Lie algebra is tensor product of Omega and matrix algebra .   We construct several other algebras that are quasiisomorphic to Omega and, therefore, also can be used to give BV formulation of 10D SUSY YM theory and its reductions. In particular, Omega is quasiisomorphic to the algebra B constructed by Berkovits. The algebras Omega_0 and B_0 obtained from Omega and B by means of reduction to a point can be used to give a BV-formulation of IKKT model.   We introduce associative algebra SYM as algebra where relations are defined as equations of motion of IKKT model and show that Koszul dual to the algebra B_0 is quasiisomorphic to SYM. 
  A radiation-ball solution which is identified as a Reissner-Nordstrom black hole is found out. The radiation-ball, which is derived by analyzing the backreaction of the Hawking radiation into space-time, consists of radiation trapped in a ball by a deep gravitational potential and of a singularity. The Hawking radiation is regarded as a leak-out of the radiation from the ball. The gravitational potential becomes deep as the charge becomes large, however, the basic structure of the ball is independent of the charge. The extremal-charged black hole corresponds with the fully frozen ball by the infinite red-shift. The total entropy of the radiation in the ball, which is independent of the charge, obeys the area-law and is near the Bekenstein entropy. 
  We propose a simple low-energy classical experiment in which the effects of noncommutativity can be clearly separated from commutative physics. The ensuing bound on the noncommutative scale is remarkable, especially in view of its elementary derivation. 
  We construct colliding plane wave solutions in higher dimensional gravity theory with dilaton and higher form flux, which appears naturally in the low energy theory of string theory. Especially, the role of the junction condition in constructing the solutions is emphasized. Our results not only include the previously known CPW solutions, but also provide a wide class of new solutions that is not known in the literature before. We find that late time curvature singularity is always developed for the solutions we obtained in this paper. This supports the generalized version of Tipler's theorem in higher dimensional supergravity. 
  Let $n$ be any natural number. Let $K$ be any $n$-dimensional knot in $S^{n+2}$. We define a supersymmetric quantum system for $K$ with the following properties. We firstly construct a set of functional spaces (spaces of fermionic \{resp. bosonic\} states) and a set of operators (supersymmetric infinitesimal transformations) in an explicit way. Thus we obtain a set of the Witten indexes for $K$. Our Witten indexes are topological invariants for $n$-dimensional knots. Our Witten indexes are not zero in general. If $K$ is equivalent to the trivial knot, all of our Witten indexes are zero. Our Witten indexes restrict the Alexander polynomials of $n$-knots. If one of our Witten indexes for an $n$-knot $K$ is nonzero, then one of the Alexander polynomials of $K$ is nontrivial. Our Witten indexes are connected with homology with twisted coefficients. Roughly speaking, our Witten indexes have path integral representation by using a usual manner of supersymmetric theory. 
  Using Ward identities of N=1/2 supersymmetric Yang-Mills theory, we show that while the partition function and antichiral gluino condensates remain invariant under the $C$ deformation, chiral gluino correlators can get contributions from all gauge fields with instanton numbers $k\leq 1$. In particular, a Ward identity of the $U(1)_R$ symmetry allows us to determine the explicit dependence of chiral gluino correlators on the deformation parameter. 
  We discuss a regular non-supersymmetric deformation of the Klebanov-Strassler solution related to gaugino mass terms in the dual gauge theory. We also describe the Penrose limit of the new background and the corresponding plane wave string theory. 
  We review a special class of semiclassical string states in AdS_5 x S^5 which have a regular expansion of their energy in integer powers of the ratio of the square of string tension (`t Hooft coupling) and the square of large angular momentum in S^5. They allow one to quantitatively check the AdS/CFT duality in non-supersymmetric sector of states and also help to uncover the role of integrable structures on the two sides of the string theory -- gauge theory duality. 
  Quantum real numbers are proposed by performing a quantum deformation of the standard real numbers $\R$. We start with the q-deformed Heisenberg algebra $\cLLq$ which is obtained by the Moyal $\ast$-deformation of the Heisenberg algebra generated by $a$ and $\ad$. By representing $\cLLq$ as the algebras of $q$-differentiable functions, we derive quantum real lines from the base spaces of these functional algebras. We find that these quantum lines are discrete spaces. In particular, for the case with $q = e^{2\pi i \frac{1}{N}} $, the quantum real line is composed of fuzzy, i.e., fluctuating points and nontrivial infinitesimal structure appears around every standard real number. 
  We study the boundary N=2 Liouville theory based on the ``modular bootstrap'' approach. As fundamental conformal blocks we introduce the ``extended characters'' that are defined as the proper sums over spectral flows of irreducible characters of the N=2 superconformal algebra (SCA) and clarify their modular transformation properties in models with rational central charges. We then try to classify the Cardy states describing consistent D-branes based on the modular data. We construct the analogues of ZZ-branes (hep-th/0101152), localized at the strong coupling region, and the FZZT-branes (hep-th/0001012, hep-th/0009138), which extend along the Liouville direction. The former is shown to play important roles to describe the BPS D-branes wrapped around vanishing cycles in deformed Calabi-Yau singularities, reproducing the correct intersection numbers of vanishing cycles. We also discuss the non-BPS D-branes in 2d type 0 (and type II) string vacua composed of the N=2 Liouville with $\hat{c}(\equiv c/3)=5$. Unstable D0-branes are found as the ZZ-brane analogues mentioned above, and the FZZT-brane analogues are stable due to the existence of mass gap despite the lack of GSO projection. 
  Optical activity of electromagnetic waves in a string inspired Kalb-Ramond cosmological background is studied in presence of extra spacetime dimension. The Kalb-Ramond-electromagnetic coupling which originates from the gauge anomaly cancelling Chern-Simons term in a string inspired model, is explicitly calculated following Randall-Sundrum braneworld conjecture. It is shown that the Randall-Sundrum scenario leads to an enormous enhancement of the optical rotation of a plane polarized electromagnetic wave propagating on the visible brane.Absence of any experimental support in favour of such a large rotation in astrophysical experiments on distant galactic radio waves indicates an apparent conflict between Randall-Sundrum brane world scenario and the presence of Kalb-Ramond antisymmetric tensor field in the background spacetime. 
  Recently, Witten showed that there is a natural action of the group SL(2,Z) on the space of 3 dimensional conformal field theories with U(1) global symmetry and a chosen coupling of the symmetry current to a background gauge field on a 3-fold N. He further argued that, for a class of conformal field theories, in the nearly Gaussian limit, this SL(2,Z) action may be viewed as a holographic image of the well-known SL(2,Z) Abelian duality of a pure U(1) gauge theory on AdS-like 4-folds M bounded by N, as dictated by the AdS/CFT correspondence. However, he showed that explicitly only for the generator T; for the generator S, instead, his analysis remained conjectural. In this paper, we propose a solution of this problem. We derive a general holographic formula for the nearly Gaussian generating functional of the correlators of the symmetry current and, using this, we show that Witten's conjecture is indeed correct when N=S^3. We further identify a class of homology 3-spheres N for which Witten's conjecture takes a particular simple form. 
  We consider the effects of a two-form field on the late-time dynamics of brane gas cosmologies. Assuming thermal equilibrium of winding states, we find that the presence of a form field allows a late stage of expansion of the Universe even when the winding degrees of freedom decay into a pressureless gas of string loops. Finally, we suggest to understand the dimensionality of the Universe not as a result of the thermal equilibrium condition but rather as a consequence of the symmetries of the geometry. 
  It is recommended that lattice QCD representations of the fermion determinant, including the discretization of the Dirac operator, be checked in the continuum limit against known QED determinant results. Recent work on the massive QED fermion determinant in two dimensions is reviewed. A feasible approach to the four-dimensional QED determinant with O(2) x O(3) symmetric background fields is briefly discussed. 
  We derive a set of necessary and sufficient conditions for obtaining N=1 backgrounds of M-theory and type IIA strings in the presence of fluxes. Our metrics are warped products of four-dimensional Minkowski space-time with a curved internal manifold. We classify the different solutions for irreducible internal manifolds as well as for manifolds with $S^1$ isometries by employing the formalism of group structures and intrinsic torsion. We provide examples within these various classes along with general techniques for their construction. In particular, we generalize the Hitchin flow equations so that one can explicitly build irreducible 7-manifolds with 4-form flux. We also show how several of the examples found in the literature fit in our framework and suggest possible generalizations. 
  We consider brane-world models with a Schwarzschild-AdS black hole bulk. In the particular case of a flat black hole horizon geometry, we study the behaviour of the brane cosmological equations when T-duality transformations act on the bulk. We find that the scale factor is inverted and that either the Friedmann equation or the energy conservation equation are unchanged. However, these become both invariant if we include a tension in the brane action. In this case, the T-duality in the bulk is completely equivalent to the scale factor duality on the brane. 
  We review some recent results concerning integrable quantum field theories in 1+1 space-time dimensions which contain unstable particles in their spectrum. Recalling first the main features of analytic scattering theories associated to integrable models, we subsequently propose a new bootstrap principle which allows for the construction of particle spectra involving unstable as well as stable particles. We describe the general Lie algebraic structure which underlies theories with unstable particles and formulate a decoupling rule, which predicts the renormalization group flow in dependence of the relative ordering of the resonance parameters. We extend these ideas to theories with an infinite spectrum of unstable particles. We provide new expressions for the scattering amplitudes in the soliton-antisoliton sector of the elliptic sine-Gordon model in terms of infinite products of q-deformed gamma functions. When relaxing the usual restriction on the coupling constants, the model contains additional bound states which admit an interpretation as breathers. For that situation we compute the complete S-matrix of all sectors. We carry out various reductions of the model, one of them leading to a new type of theory, namely an elliptic version of the minimal SO(n)-affine Toda field theory. 
  We study a relation between topological quantum field theory and the Kodama (Chern-Simons) state. It is shown that the Kodama (Chern-Simons) state describes a topological state with unbroken diffeomorphism invariance in Yang-Mills theory and Einstein's general relativity in four dimensions. We give a clear explanation of "why" such a topological state exists. 
  We study a longstanding problem of identification of the fermion-monopole symmetries. We show that the integrals of motion of the system generate a nonlinear classical Z_2-graded Poisson, or quantum super- algebra, which may be treated as a nonlinear generalization of the $osp(2|2)\oplus su(2)$. In the nonlinear superalgebra, the shifted square of the full angular momentum plays the role of the central charge. Its square root is the even osp(2|2) spin generating the u(1) rotations of the supercharges. Classically, the central charge's square root has an odd counterpart whose quantum analog is, in fact, the same osp(2|2) spin operator. As an odd integral, the osp(2|2) spin generates a nonlinear supersymmetry of De Jonghe, Macfarlane, Peeters and van Holten, and may be identified as a grading operator of the nonlinear superconformal algebra. 
  Description of the smallest quantum groupoid associated with the A2 diagram.   From the talk: ``Quantum groupoids and Ocneanu bialgebras for Coxeter-Dynkin systems'' given at the XV Colloquio Latinoamericano de Algebra, Cocoyoc, Mexico, July 20th-26th, 2003. 
  We argue that in the context of string theory a large number N of connected degenerate vacua that mix will lead to a ground state with much lower energy, essentially because of the standard level repulsion of quantum theory for the wavefunction of the Universe. We imagine a history where initial quantum fluctuations give an energy density $\sim m_{susy}^2m_{Pl}^2$, but the universe quickly cascades to an energy density $\sim m_{susy}^2m_{Pl}^2/N$. Then at various phase transitions there are large contributions to the energy density and rearrangement of levels, followed again by a rapid cascade to the ground state or near it. If this mechanism is correct, the ground state of the theory describing our world would be a superposition of a large number of connected string vacua,with shared superselection sets of properties such as three families etc. The observed value of the cosmological constant in terms of the Planck mass, the scale of supersymmetry breaking and the number of connected string vacua. 
  Recently a description of linearized 5D supergravity in 4D, N =1 superspace was presented. By analyzing the on-shell component Lagrangian, this description was proven to be Lorentz invariant in five dimensions. This paper describes a geometric formulation of the 5D supergravity in 4D, N =1 superspace. Using this new formalism, the on-shell structure of the previous description is verified from purely superspace methods. This geometric description provides a connection to dimensionally reduced manifestly supersymmetric 5D supergravity. 
  We argue that brane anti-brane inflation in string theory de-Sitter vacua of Kachru-Kallosh-Linde-Trivedi (KKLT) is captured by the dynamics of a D3-brane probe in the local KKLT model constructed in hep-th/0203041. This provides a framework to study in a controllable way corrections to the inflationary slow roll parameter \eta due to conformal symmetry breaking in a warped geometry throat. We compute the leading correction to \eta for the inflation in the Klebanov-Tseytlin throat geometry. We find that in certain regime this correction tends to decrease \eta. Computations in a different regime suggest however that it is unlikely that \eta << 1 can be achieved with the D3-brane throat inflation. 
  We study (anti-) instantons in super Yang-Mills theories defined on a non anticommutative superspace. The instanton solution that we consider is the same as in ordinary SU(2) N=1 super Yang-Mills, but the anti-instanton receives corrections to the U(1) part of the connection which depend quadratically on fermionic coordinates, and linearly on the deformation parameter C. By substituting the exact solution into the classical Lagrangian the topological charge density receives a new contribution which is quadratic in C and quartic in the fermionic zero-modes. The topological charge turns out to be zero. We perform an expansion around the exact classical solution in presence of a fermionic background and calculate the full superdeterminant contributing to the one-loop partition function. We find that the one-loop partition function is not modified with respect to the usual N=1 super Yang-Mills. 
  Based on some important properties of $dS$ space, we present a Beltrami model ${\cal B}_\Lambda$ that may shed light on the observable puzzle of $dS$ space and the paradox between the special relativity principle and cosmological principle. In ${\cal B}_\Lambda$, there are inertial-type coordinates and inertial-type observers. Thus, the classical observables can be defined for test particles and light signals. In addition, by choosing the definition of simultaneity the Beltrami metric is transformed to the Robertson-Walker-like metric. It is of positive spatial curvature of order $\Lambda$. This is more or less indicated already by the CMB power spectrum from WMAP and should be further confirmed by its data in large scale. 
  Exact time-dependent solutions of nonrelativistic noncommutative Chern - Simons gauge theory are presented in closed analytic form. They are different from (indeed orthogonal to) those discussed recently by Hadasz, Lindstrom, Rocek and von Unge. Unlike theirs, our solutions can move with an arbitrary constant velocity, and can be obtained from the previously known static solutions by the recently found ``exotic'' boost symmetry. 
  Vacuum expectation values of the energy-momentum tensor for a conformally coupled scalar field is investigated in de Sitter (dS) spacetime in presence of a curved brane on which the field obeys the Robin boundary condition with coordinate dependent coefficients. To generate the corresponding vacuum densities we use the conformal relation between dS and Rindler spacetimes and the results previously obtained by one of the authors for the Rindler counterpart. The resulting energy-momentum tensor is non-diagonal and induces anisotropic vacuum stresses. The asymptotic behaviour of this tensor is investigated near the dS horizon and the boundary. 
  The `Chern-Simons Quantum Mechanics' of a particle on CP(n|m) is shown to yield the fuzzy descriptions of these superspaces, for which we construct the non-(anti)commuting position operators. For a particle on the supersphere CP(1|1) = SU(2|1)/U(1|1), the particle's wave-function at fuzziness level 2s is shown to be a degenerate irrep of SU(2|1) describing a supermultiplet of SU(2) spins (s-1\2, s). 
  We investigate confinement from new global defect structures in three spatial dimensions. The global defects arise in models described by a single real scalar field, governed by special scalar potentials. They appear as electrically, magnetically or dyonically charged structures. We show that they induce confinement, when they are solutions of effective QCD-like field theories in which the vacua are regarded as color dielectric media with an anti-screening property. As expected, in three spatial dimensions the monopole-like global defects generate the Coulomb potential as part of several confining potentials. 
  In this paper the SU(N) Einstein-Skyrme system is considered. We express the chiral field (which is not a simple embedding of the SU(2) one) in terms of harmonic maps. In this way, SU(N) spherical symmetric equations can be obtained easily for any $N$ and the gravitating skyrmion solutions of these equations can be studied. In particular, the SU(3) case is considered in detail and three different types of gravitating skyrmions with topological charge 4, 2 and 0, respectively, are constructed numerically. Note that the configurations with topological charge 0 correspond to mixtures of skyrmions and antiskyrmions. 
  According to one of many equivalent definitions of twistors a (null) twistor is a null geodesic in Minkowski spacetime. Null geodesics can intersect at points (events). The idea of Penrose was to think of a spacetime point as a derived concept: points are obtained by considering the incidence of twistors. One needs two twistors to obtain a point. Twistor is thus a ``square root'' of a point. In the present paper we entertain the idea of quantizing the space of twistors. Twistors, and thus also spacetime points become operators acting in a certain Hilbert space. The algebra of functions on spacetime becomes an operator algebra. We are therefore led to the realm of non-commutative geometry. This non-commutative geometry turns out to be related to conformal field theory and holography. Our construction sheds an interesting new light on bulk/boundary dualities. 
  An action principle is described which unifies general relativity and topological field theory. An additional degree of freedom is introduced and depending on the value it takes the theory has solutions that reduce it to 1) general relativity in Palatini form, 2) general relativity in the Ashtekar form, 3) $F\wedge F$ theory for SO(5) and 4) $BF$ theory for SO(5). This theory then makes it possible to describe explicitly the dynamics of phase transition between a topological phase and a gravitational phase where the theory has local degrees of freedom. We also find that a boundary between adymnamical and topological phase resembles an horizon. 
  The frame-like covariant Lagrangian formulation of bosonic and fermionic mixed-symmetry type higher spin massless fields propagating on the AdS(d) background is proposed. Higher spin fields are described in terms of gauge p-forms which carry tangent indices representing certain traceless tensor or gamma transversal spinor-tensor representations of the AdS(d) algebra o(d-1,2) (or o(d,1) for bosonic fields in dS(d)). Manifestly gauge invariant Abelian higher spin field strengths are introduced for the general case. We describe the general framework and demonstrate how it works for the mixed-symmetry type fields associated with the three-cell "hook" and arbitrary two-row rectangular tableaux. The manifestly gauge invariant actions for these fields are presented in a simple form. The flat limit is also analyzed. 
  We consider the effects of homogeneous Dirichlet's boundary conditions in the scalar electrodynamics with self-interaction. We have found for a critical scale of the compactification length that symmetry is restored and scalar field develops mass and vector field does not. 
  We propose new formulas for singular vectors in Verma modules over the affine Lie superalgebra $\hat{sl}(2|1)$. We analyze the coexistence of singular vectors of different types and identify the twisted modules $N_{h,k;\theta}$ arising as submodules and quotient modules of $\hat{sl}(2|1)$ Verma modules. We show that with the twists (spectral flow transformations) properly taken into account, a resolution of irreducible representations can be constructed consisting of only the $N_{h,k;\theta}$ modules. 
  Diagrammatic rules are developed for simplifying two-loop QED diagrams with propagators in a constant self-dual background field. This diagrammatic analysis, using dimensional regularization, is used to explain how the fully renormalized two-loop Euler-Heisenberg effective Lagrangian for QED in a self-dual background field is naturally expressed in terms of one-loop diagrams. The connection between the two-loop and one-loop vacuum diagrams in a background field parallels a corresponding connection for free vacuum diagrams, without a background field, which can be derived by simple algebraic manipulations. It also mirrors similar behavior recently found for two-loop amplitudes in N=4 SUSY Yang-Mills theory. 
  We present a new method to compute quantum energies in presence of a background field. The method is based on the string-inspired worldline approach to quantum field theory and its numerical realization with Monte-Carlo techniques. Our procedure is applied to the study of the Casimir force between rigid bodies induced by a fluctuating real scalar field. We test our method with the classic parallel-plate configuration and study curvature effects quantitatively for the sphere-plate configuration. The numerical estimates are compared with the ``proximity force approximation''. Sizable curvature effects are found for a distance-to-curvature-radius ratio of a/R>0.02. 
  We derive one-point functions of the N=2 super-Liouville theory on a half line using the modular transformations of the characters in terms of the bulk and boundary cosmological constants. We also show that these results are consistent with conformal bootstrap equations which are based on the bulk and boundary actions. We provide various independent checks for our results. 
  The Poisson brackets of the SU(2)_k WZNW zero modes are derived directly, using Euler angles parametrization. 
  The distribution of individual Dirac eigenvalues is derived by relating them to the density and higher eigenvalue correlation functions. The relations are general and hold for any gauge theory coupled to fermions under certain conditions which are stated. As a special case, we give examples of the lowest-lying eigenvalue distributions for QCD-like gauge theories without making use of earlier results based on the relation to Random Matrix Theory. 
  A direct very simple proof that there can be no closed trapped surfaces (ergo no black hole regions) in spacetimes with all curvature scalar invariants vanishing is given. Explicit examples of the recently introduced ``dynamical horizons'' which nevertheless do not enclose any trapped region are presented too. 
  We consider the behaviour of the cosmological acceleration for time-dependent hyperbolic and flux compactifications of M-theory, with an exponential potential. For flat and closed cosmologies it is seen that a positive acceleration is always transient for both compactifications. For open cosmologies, both compactifications can give at late times periods of positive acceleration. As a function of proper time this acceleration has a power law decay and can be either positive, negative or oscillatory. 
  A brief sketch of computer methods of involutivity analysis of differential equations is presented in context of its application to study degenerate Lagrangian systems. We exemplify the approach by a detailed consideration of a finite-dimensional model, the so-called light-cone SU(2) Yang-Mills mechanics. All algorithms are realized in computer algebra system Maple. 
  Kovtun, Son and Starinets proposed a bound on the shear viscosity of any fluid in terms of its entropy density. We argue that this bound is always saturated for gauge theories at large 't Hooft coupling, which admit holographically dual supergravity description. 
  The highly non-trivial structure of the $\theta$--vacuum encodes many of the fundamental properties of gauge theories. In particular, the response of the vacuum to the $\theta$--term perturbation is sensitive to the existence of confinement, chiral symmetry breaking, etc. We analyze the dependence of the vacuum energy density on theta around two special values, $\theta=0$ and $\theta=\pi$. The existence or not of singular behaviors associated to spontaneous breaking of CP symmetry in these vacua has been a controversial matter for years. We clarify this important problem by means of continuum non-perturbative techniques. The results show the absence of first order cusp singularities on the vacuum energy density at $\theta=0$ and $\theta=\pi$ for some gauge theories. This smooth dependence of the energy on $\theta$ might have implications for long standing cosmological problems like the baryonic asymmetry and the cosmological constant problem. 
  Extending our recent work (\arXiv{\tt hep-th/0310106}) we study the nonsinglet sector of $c=1$ matrix model by renormalization group analysis for a gauged matrix quantum mechanics on circle with an appropriate gauge breaking term to incorporate the effect of world-sheet vortices. The flow equations indicate BKT phase transition around the self-dual radius and the nontrivial fixed points of the flow exhibit black hole like phases for a range of temperatures beyond the self-dual point. One class of fixed point interpolate between $c=1$ for $R > 1$ and $c=0$ as $R \to 0$ via black hole phase that emerges after the phase transition. The other two classes of nontrivial fixed points also develop black hole like behavior beyond R=1. From a thermodynamic study of the free energy obtained from the Callan-Symanzik equations we show that all these unstable phases do have negative specific heat. The thermodynamic quantities indicate that the system does undergo a first order phase transition near the Hagedorn temperature, around which the new phase is formed, and exhibits one loop finite energy correction to the Hagedorn density of states. The flow equations also suggest a deformation of the target space geometry through a running of the compactification radius where the scale is given by the dilaton. Remarkably there is a regime where cyclic flow is observed. 
  We give a geometrical definition of the asymptotic flatness at null infinity in spacetimes of even dimension $d$ greater than 4 within the framework of conformal infinity. Our definition is shown to be stable against perturbations to linear order. We also show that our definition is stringent enough to allow one to define the total energy of the system viewed from null infinity as the generator conjugate to an asymptotic time translation. We derive an expression for the generator conjugate within the Hamiltonian framework, and propose to take this notion of energy as the natural generalisation of the Bondi energy to higher dimensions. Our definitions of asymptotic flatness and the Bondi energy formula differ qualitatively from the corresponding definitions in $d=4$; although the asymptotic structure of null infinity in higher dimensions parallels that in 4-dimensions in some ways, the latter seems to be a rather special case on the whole compared to general $d>4$. Our definitions and constructions do not work in odd spacetime dimensions, essentially because the unphysical metric seems to have insufficient regularity properties at null infinity in that case. 
  We study the time evolution of the one-loop diagram in Sen's rolling tachyon background. We find that at least in the long cylinder case they grow rapidly at late time, due to the exponential growth of the timelike oscillator terms in the boundary state. This can also be interpreted as the virtual open string pair creation in the decaying brane. This behavior indicates a breakdown of this rolling tachyon solution at some point during the evolution. We also discuss the closed string emission from this one-loop diagram, and the evolution of a one-loop diagram connecting a decaying brane to a stable brane, which is responsible for the physical open string creation on the stable brane. 
  We show that the amount of particle production in an arbitrary cosmological background can be determined using only the late-time positive-frequency modes. We don't refer to modes at early times, so there is no need for a Bogolubov transformation. We also show that particle production can be extracted from the Feynman propagator in an auxiliary spacetime. This provides a first-quantized formalism for computing particle production which, unlike conventional Bogolubov transformations, may be amenable to a string-theoretic generalization. 
  We consider N=1 supersymmetric U(N), SO(N), and Sp(N) gauge theories, with two-index tensor matter and added tree-level superpotential, for general breaking patterns of the gauge group. By considering the string theory realization and geometric transitions, we clarify when glueball superfields should be included and extremized, or rather set to zero; this issue arises for unbroken group factors of low rank. The string theory results, which are equivalent to those of the matrix model, refer to a particular UV completion of the gauge theory, which could differ from conventional gauge theory results by residual instanton effects. Often, however, these effects exhibit miraculous cancellations, and the string theory or matrix model results end up agreeing with standard gauge theory. In particular, these string theory considerations explain and remove some apparent discrepancies between gauge theories and matrix models in the literature. 
  We review the worldsheet analysis for intersecting branes with focus on small and large angles. For small angles, we review the Yang-Mills fluctuation analysis in hep-th/0303204 and find an additional family of massless modes. They are the components of a Goldstone scalar corresponding to the spontaneously broken U(2)-gauge symmetry. For branes at large angles, we derive an effective tachyon field theory from BSFT results. We show how the gauge symmetry of this system implies a mass spectrum which is consistent with the worldsheet analysis. 
  The transformation properties of a Kalb-Ramond field are those of a gauge potential. However, it is not clear what is the group structure to which these transformations are associated. In this paper, we complete a program started in previous articles in order to clarify this question. Using the spectral theorem, we improve and generalize previous approaches and find the possible group structures underneath the 2-form gauge potential as extensions of Lie groups, when its representations are assumed to act into any tensor (or spinor) space with inner product.   We also obtain a fundamental representation where a two-form field turns out to be a connection on a flat Euclidean basis manifold, with a corresponding canonical curvature. However, we show that these objects are not associated to space-time tensors and, in particular, that a standard Yang-Mills action is not relativistically invariant, except (as expected) in the Abelian case. This is our main result, from the physical point of view. 
  We analyze the physical (reduced) space of non-local theories, around the fixed points of these systems, by analyzing: i) the Hamiltonian constraints appearing in the 1+1 formulation of those theories, ii) the symplectic two form in the surface on constraints.   P-adic string theory for spatially homogeneous configurations has two fixed points. The physical phase space around $q=0$ is trivial, instead around $q=\frac 1g$ is infinite dimensional. For the special case of the rolling tachyon solutions it is an infinite dimensional lagrangian submanifold. In the case of string field theory, at lowest truncation level, the physical phase space of spatially homogeneous configurations is two dimensional around $q=0$, which is the relevant case for the rolling tachyon solutions, and infinite dimensional around $q=\frac {M^2}g$. 
  The Lanczos potential $L_{abc}$ acts as a tensor potential for the spin-2 field strength $W_{abcd}$ in an role similar to that of the vector potential $A_a$ for the Maxwell tensor $F_{ab}$. After some general considerations inspired by the example of electromagnetism, we consider the linear spin-2 theory and a Born-Infeld type action in terms of $L_{abc}$. 
  Using the prescription of [1] for defining period integrals in the Landau-Ginsburg theory for compact Calabi-Yau's, we obtain the Picard-Fuchs equation and the Meijer basis of solutions for the compact Calabi-Yau CY_3(3,243) expressed as a degree-24 Fermat hypersurface AFTER resolution of the orbifold singularities. This is similar in spirit to the method of obtaining Meijer basis of solutions in [2] for the case wherein one is away from the orbifold singularities, and one is considering the large-base limit of the Calabi-Yau. The importance of the method lies in the ease with which one can consider the large AND small complex structure limits, as well as the ability to get the "ln"-terms in the periods without having to parametrically differentiate infinite series. We consider in detail the evaluation of the monodromy matrix in the large and small complex structure limits. We also consider the action of the freely acting antiholomorphic involution of [2],[3] on D=11 supergravity compactified on CY_3(3,243) x S^1 [4] and obtain the Kaehler potential for the same in the limit of large volume of the Calabi-Yau. As a by-product, we also give a conjecture for the action of the orientation-reversing antiholomorphic involution on the periods, given its action on the cohomology, using a canonical (co)homology basis. Finally, we also consider showing a null superpotential on the orientifold of type IIA on CY_3(3,243), having taken care of the orbifold singularities, thereby completing the argument initiated in [2]. 
  Within an effective field theory framework we compute the most general structure of the one-loop corrections to the 4D gauge couplings in one- and two-dimensional orbifold compactifications with non-vanishing constant gauge background (Wilson lines). Although such models are non-renormalisable, we keep the analysis general by considering the one-loop corrections in three regularisation schemes: dimensional regularisation (DR), Zeta-function regularisation (ZR) and proper-time cut-off regularisation (PT). The relations among the results obtained in these schemes are carefully addressed. With minimal re-definitions of the parameters involved, the results obtained for the radiative corrections can be applied to most orbifold compactifications with one or two compact dimensions. The link with string theory is discussed. We mention a possible implication for the gauge couplings unification in such models. 
  The effect of NS 5 branes on an orientifold is studied. The orientifold is allowed to pass through a pile of k NS branes forming a regularized CHS geometry. Its effect on open strings in its vicinity is used to study the change in the orientifold charge induced by the NS branes. 
  This paper discusses the problem of inflation in the context of Friedmann-Robertson-Walker Cosmology. We show how, after a simple change of variables, one can quantize the problem in a way which parallels the classical discussion. The result is that two of the Einstein equations arise as exact equations of motion and one of the usual Einstein equations (suitably quantized) survives as a constraint equation to be imposed on the space of physical states. However, the Friedmann equation, which is also a constraint equation and which is the basis of the Wheeler-DeWitt equation, acquires a welcome quantum correction that becomes significant for small scale factors. We discuss the extension of this result to a full quantum mechanical derivation of the anisotropy ($\delta \rho /\rho$) in the cosmic microwave background radiation, and the possibility that the extra term in the Friedmann equation could have observable consequences. Finally, we suggest interesting ways in which these techniques can be generalized to cast light on the question of chaotic or eternal inflation. In particular, we suggest one can put an experimental lower bound on the distance to a universe with a scale factor very different from our own, by looking at its effects on our CMB radiation. 
  We attempt to obtain realistic glueball Regge trajectories from the gauge/string correspondence. To this end we study closed spinning string configurations in two supergravity backgrounds: Klebanov-Strassler (KS) and Maldacena-Nunez (MN) which are dual to confining gauge theories. These backgrounds represent two embeddings of N=1 SYM, in the large rank limit, in string theory. The classical configuration we consider is that of a folded closed string spinning in a supergravity region with vanishing transverse radius (\tau = 0) which is dual to the IR of the gauge theory. Classically, a spinning string yields a linear Regge trajectory with zero intercept. By performing a semi-classical analysis we find that quantum effects alter both the linearity of the trajectory and the vanishing classical intercept: J:=\alpha(t)= \alpha_0 + \alpha' t +\beta\sqrt{t}. Two features of our Regge trajectories are compatible with the experimental Pomeron trajectory: positive intercept and positive curvature. The fact that both KS and MN string backgrounds give the same functional expression of the Regge trajectories suggests that in fact we are observing string states dual to N=1 SYM. 
  The slow-roll inflation is a beautiful paradigm, yet the inflaton potential can hardly be sufficiently flat when unknown gravitational effects are taken into account. However, the hybrid inflation models constructed in D = 4 N = 1 supergravity can be consistent with N = 2 supersymmetry, and can be naturally embedded into string theory. This article discusses the gravitational effects carefully in the string model, using D = 4 supergravity description. We adopt the D3--D7 system of Type IIB string theory compactified on K3 x T^2/Z_2 orientifold for definiteness. It turns out that the slow-roll parameter can be sufficiently small despite the non-minimal Kahler potential of the model. The conditions for this to happen are clarified in terms of string vacua. We also find that the geometry obtained by blowing up singularity, which is necessary for the positive vacuum energy, is stabilized by introducing certain 3-form fluxes. 
  The q-deformation of the Lie algebras underlying the standard field theories leads to a pair of dual algebras. We describe a simple choice of possible field theories based on these derived algebras. One of these approximates the standard Lie theory of point particles, while the other is proposed as a field theory of knotted solitons. 
  The derivation of the nilpotent (anti-)BRST symmetries for the matter fields, present in any arbitrary interacting gauge theory, has been an long-standing problem in the framework of superfield approach to BRST formalism. These nilpotent (anti-)BRST symmetries for the Dirac fields are derived in the superfield formulation for the interacting Abelian gauge theory in four $(3 + 1)$-dimensions (4D) of spacetime. The same type of symmetries are deduced for the 4D complex scalar fields having a gauge invariant interaction with the U(1) gauge field. The above interacting theories are considered on the six $(4 + 2)$-dimensional supermanifold parametrized by four {\it even} spacetime coordinates and a couple of {\it odd} elements of the Grassmann algebra. The invariance of the conserved matter (super)currents and the horizontality condition on the (super)manifolds play very important roles in the above derivations. The geometrical origin and interpretation for all the above off-shell nilpotent symmetries are provided in the framework of superfield formalism. 
  We construct the multiplicatively renormalizable effective potential for the mass dimension two local composite operator A^2 in linear covariant gauges. We show that the formation of <A^2> is energetically favoured and that the gluons acquire a dynamical mass due to this gluon condensate. We also discuss the gauge parameter independence of the resultant vacuum energy. 
  A precise zeta-function calculation shows that the contribution of the vacuum energy to the observed value of the cosmological constant can possibly have the desired order of magnitude albeit the sign strongly depends on the topology of the universe. The non-renormalizable, infinite contributions which have been recently shown to occur when one physically imposes boundary conditions on quantum fields (Casimir calculations) are considered. It is shown that using a Hadamard regularization in addition to the zeta method, the ordinary, finite results in the literature are exactly recovered. 
  A new framework for exploiting information about the renormalization group (RG) behavior of gravity in a dynamical context is discussed. The Einstein-Hilbert action is RG-improved by replacing Newton's constant and the cosmological constant by scalar functions in the corresponding Lagrangian density. The position dependence of $G$ and $\Lambda$ is governed by a RG equation together with an appropriate identification of RG scales with points in spacetime. The dynamics of the fields $G$ and $\Lambda$ does not admit a Lagrangian description in general. Within the Lagrangian formalism for the gravitational field they have the status of externally prescribed ``background'' fields. The metric satisfies an effective Einstein equation similar to that of Brans-Dicke theory. Its consistency imposes severe constraints on allowed backgrounds. In the new RG-framework, $G$ and $\Lambda$ carry energy and momentum. It is tested in the setting of homogeneous-isotropic cosmology and is compared to alternative approaches where the fields $G$ and $\Lambda$ do not carry gravitating 4-momentum. The fixed point regime of the underlying RG flow is studied in detail. 
  We introduce a $(5+m)$-dimensional vacuum description of five-dimensional bulk inflaton models with exponential potentials that makes analysis of cosmological perturbations simple and transparent. We show that various solutions, including the power-law inflation model recently discovered by Koyama and Takahashi, are generated from known $(5+m)$-dimensional vacuum solutions of pure gravity. We derive master equations for all types of perturbations, and each of them becomes a second order differential equation for one master variable supplemented by simple boundary conditions on the brane. One exception is the case for massive modes of scalar perturbations. In this case, there are two independent degrees of freedom, and in general it is difficult to disentangle them into two separate sectors. 
  We consider a new class of solutions (dressed slivers) in Vacuum String Field Theory, which represent D25-branes. For each dressed sliver we introduce a deformation parameter and define a family of states which are characterized by new abelian star-subalgebras. We show that this deformation parameter can be used as a regulator: it allows us to define for each such solution a finite norm and energy density. Finally we show how to generalize these results to parallel coincident and to lower dimensional branes. 
  The ground state energy of a quantum field in the background of classical field configurations is considered.   The subject of the ground state energy in framework of the quantum field theory is explained.   The short review of calculation methods (generalized zeta function and heat kernel expansion) and their mathematical foundations is given. We use the zeta-functional regularization and express the ground state energy as an integral involving the Jost function of a two dimensional scattering problem.We perform the renormalization by subtracting the contributions from first several heat kernel coefficients. The ground state energy is presented as a convergent expression suited for numerical evaluation.   The investigation for three models has been carried out: scalar quantum field on the background of scalar string with rectangular shape, spinor vacuum polarized by magnetic string of the similar shape and spinor vacuum interacting with the Nielsen-Olesen vortex. Using the uniform asymptotic expansion of the special functions entering the Jost function we are also able to calculate higher order heat kernel coefficients.   Several features of vacuum energy have been investigated numerically. We discuss corresponding numerical results. 
  We consider a Lorentz-violating modification to the fermionic Lagrangian of QED that is known to produce a finite Chern-Simons term at leading order. We compute the second order correction to the one-loop photon self-energy in the massless case using an exact propagator and a nonperturbative formulation of the theory. This nonperturbative theory assigns a definite value to the coefficient of the induced Chern-Simons term; however, we find that the theory fails to preserve gauge invariance at higher orders. We conclude that the specific nonperturbative value of the Chern-Simons coefficient has no special significance. 
  We study two related problems in the context of a supergravity dual to N=1 SYM. One of the problems is finding kappa symmetric D5-brane probes in this particular background. The other is the use of these probes to add flavors to the gauge theory. We find a rich and mathematically appealing structure of the supersymmetric embeddings of a D5-brane probe in this background. Besides, we compute the mass spectrum of the low energy excitations of N=1 SQCD (mesons) and match our results with some field theory aspects known from the study of supersymmetric gauge theories with a small number of flavors. 
  We calculate correlation functions for vertex operators with negative integer exponentials of a periodic Liouville field, and derive the general case by continuing them as distributions. The path-integral based conjectures of Dorn and Otto prove to be conditionally valid only. We formulate integral representations for the generic vertex operators and indicate structures which are related to the Liouville S-matrix. 
  Recently, an impressive agreement was found between anomalous dimensions of certain operators in N=4 SYM and rotating strings with two angular momenta in the bulk of AdS5xS5. A one-loop field theory computation, which involves solving a Heisenberg chain by means of the Bethe ansatz agrees with the large angular momentum limit of a rotating string. We point out that the Heisenberg chain can be equally well solved by using a sigma model approach. Moreover we also show that a certain limit, akin to the BMN limit, leads exactly to the same sigma model for a string rotating with large angular momentum. The agreement is then at the level of the action. As an upshot we propose that the rotating string should be identified with a coherent, semi-classical state built out of the eigenstates of the spin chain. The agreement is then complete. For example we show that the mean value of the spin <S> gives, precisely, the position of the string in the bulk. This suggests a more precise formulation of the AdS/CFT correspondence in the large-N limit and also indicates a way to obtain string theory duals of other gauge theories. 
  We derive an oscillator form for the Butterflies in terms of Sliver matrix S and its twisted version T as was already done for the Wedges in term of T. We write a General Squeezed state depending on a matrix U and we show in a compact way the interpolation between Identity state and the Sliver and between the Nothing state and the Sliver, growing in powers of T and S matrices, respectively, in the choice of such matrix U. Furthermore, we define a class of states which we call Laguerre states and we give a formal derivation of such interpolating state in terms of them. 
  We examine the AdS Higgs phenomenon for spin-1 fields, and demonstrate that graviphotons pick up a dynamically generated mass in AdS_4, once matter boundary conditions are relaxed. We perform an explicit one-loop calculation of the graviphoton mass, and compare this result with the mass generated for the graviton in AdS. In this manner, we obtain a condition for unbroken supersymmetry. With this condition, we examine both N=2 and N=4 gauged supergravities coupled to matter multiplets, and find that for both cases the ratio between dynamically generated graviton and graviphoton masses is consistent with unbroken supersymmetry. 
  We discuss commuting flows and conservation laws for Lax hierarchies on noncommutative spaces in the framework of the Sato theory. On commutative spaces, the Sato theory has revealed essential aspects of the integrability for wide class of soliton equations which are derived from the Lax hierarchies in terms of pseudo-differential operators. Noncommutative extension of the Sato theory has been already studied by the author and Kouichi Toda, and the existence of various noncommutative Lax hierarchies are guaranteed. In the present paper, we present conservation laws for the noncommutative Lax hierarchies with both space-space and space-time noncommutativities and prove the existence of infinite number of conserved densities. We also give the explicit representations of them in terms of Lax operators. Our results include noncommutative versions of KP, KdV, Boussinesq, coupled KdV, Sawada-Kotera, modified KdV equations and so on. 
  It was pointed out that brane-anti-brane inflation without warped geometry is not viable due to compactification effects (in the simplified scenario where the inflaton is decoupled from the compactification moduli). We show that the inflationary scenario with branes at a small angle in this simplified scenario remains viable. We also point out that brane-anti-brane inflation may still be viable under some special conditions. We also discuss a way to treat potentials in compact spaces that should be useful in the analysis of more realistic brane inflationary scenarios. 
  This work is a natural continuation of our recent study in quantizing relativistic particles. There it was demonstrated that, by applying a consistent quantization scheme to a classical model of a spinless relativistic particle as well as to the Berezin-Marinov model of 3+1 Dirac particle, it is possible to obtain a consistent relativistic quantum mechanics of such particles. In the present article we apply a similar approach to the problem of quantizing the massive 2+1 Dirac particle. However, we stress that such a problem differs in a nontrivial way from the one in 3+1 dimensions. The point is that in 2+1 dimensions each spin polarization describes different fermion species. Technically this fact manifests itself through the presence of a bifermionic constant and of a bifermionic first-class constraint. In particular, this constraint does not admit a conjugate gauge condition at the classical level. The quantization problem in 2+1 dimensions is also interesting from the physical viewpoint (e.g. anyons). In order to quantize the model, we first derive a classical formulation in an effective phase space, restricted by constraints and gauges. Then the condition of preservation of the classical symmetries allows us to realize the operator algebra in an unambiguous way and construct an appropriate Hilbert space. The physical sector of the constructed quantum mechanics contains spin-1/2 particles and antiparticles without an infinite number of negative-energy levels, and exactly reproduces the one-particle sector of the 2+1 quantum theory of a spinor field. 
  This paper is devoted to study gauge embedding of either commutative and noncommutative theories in the framework of the symplectic formalism. We illustrate our ideas in the Proca model, the irrotational fluid model and the noncommutative self-dual model. In the process of this new path of embedding, the infinitesimal gauge generators of the gauge embedded theory are easily and directly chosen. Among other advantages, this enables a greater control over the final Lagrangian and puts some light on the so called ''arbitrariness problem". 
  In five-dimensional supergravity, an exact solution of BPS wall is found for a gravitational deformation of the massive Eguchi-Hanson nonlinear sigma model. The warp factor decreases for both infinities of the extra dimension. Thin wall limit gives the Randall-Sundrum model without fine-tuning of input parameters. We also obtain wall solutions with warp factors which are flat or increasing in one side, by varying a deformation parameter of the potential. 
  We analyze the possibility to construct a self-consistent gauge field theory in $D>4$. We first look for the cancellation of the UV divergences in SUSY theories. Then, following the Wilson RG approach, we study the RG equation for the gauge coupling in perturbative and nonperturbative regimes. In the first case the power low running is discussed. In the second case it is shown that there exist the ultraviolet fixed point where the gauge coupling is dimensionless in any space-time dimension. This fixed point is nonperturbative and corresponds to scale invariant theory. The same phenomenon also happens in supersymmetric theory in D=6. 
  Particular class of AdS(d) mixed-symmetry bosonic massless fields corresponding to arbitrary two-column Young tableaux is considered. Unique gauge invariant free actions are found and equations of motion are analyzed. 
  We study the Poisson sigma model which can be viewed as a topological string theory. Mainly we concentrate our attention on the Poisson sigma model over a group manifold G with a Poisson-Lie structure. In this case the flat connection conditions arise naturally. The boundary conditions (D-branes) are studied in this model. It turns out that the D-branes are labelled by the coisotropic subgroups of G. We give a description of the moduli space of classical solutions over Riemann surfaces both without and with boundaries. Finally we comment briefly on the duality properties of the model. 
  We study the spectrum of fluctuations about static solutions in 1+1 dimensional non-commutative scalar field models. In the case of soliton solutions non-commutativity leads to creation of new bound states. In the case of static singular solutions an infinite tower of bound states is produced whose spectrum has a striking similarity to the spectrum of confined quark states. 
  Calorons (periodic instantons) are anti-self-dual (ASD) connections on S^1 \times R^3 and form an intermediate case between instantons and monopoles. The ADHM and Nahm constructions of instantons and monopoles can be regarded as generalizations of a correspondence between ASD connections on the 4-torus, often referred to as the Nahm transform. This thesis describes how the Nahm transform can be extended to the case of calorons. It is shown how calorons can be constructed from Nahm data similar to that for monopoles, but defined over the circle. The inverse transformation, from the caloron to the Nahm data, is also described. 
  In a braneworld context, the radion is a massless mode coupling to the trace of the matter stress tensor. Since the radion also governs the separation between branes, it is expected to decouple from the physical spectrum in single brane scenarios, such as the one-brane Randall-Sundrum model. However, contrary to expectations, we demonstrate that the Karch-Randall radion always remains as a physical excitation, even in the single brane case. Here, the radion measures the distance not between branes, but rather between the brane and the anti-de Sitter boundary on the other side of the bulk. 
  We calculate gaugino and meson condensates in N=1 SQCD theory with SU(N_c) gauge group and N_f < 2N_c matter flavours, by deforming the pure N=2 Super-Yang-Mills plus fundamental matter action with a mass term for the adjoint scalar superfield. This follows similar recent work by Konishi and Ricco hep-th/0306128 for the case without fundamental matter. 
  We study the action of time dependent canonical and coordinate transformations in phase space quantum mechanics. We extend the covariant formulation of the theory by providing a formalism that is fully invariant under both standard and time dependent coordinate transformations. This result considerably enlarges the set of possible phase space representations of quantum mechanics and makes it possible to construct a causal representation for the distributional sector of Wigner quantum mechanics. 
  Based on the effective field theory philosophy, a universal form of the scaling laws could be easily derived with the scaling anomalies naturally clarified as the decoupling effects of underlying physics. In the novel framework, the conventional renormalization group equations and Callan-Symanzik equations could be reproduced as special cases and a number of important and difficult issues around them could be clarified. The underlying theory point of view could envisage a harmonic scaling law that help to fix the form of the loop amplitudes through anomalies, and the heavy field decoupling can be incorporated in this underlying theory approach in a more unified manner. 
  We study the asymptotic form of the Gopakumar-Vafa invariants at all genera for Calabi-Yau toric threefolds which have the structure of fibration of the A_n singularity over P^1. We claim that the asymptotic form is the inverse Laplace transform of the corresponding instanton amplitude in the prepotential of N=2 SU(n+1) gauge theory coupled to external graviphoton fields, which is given by the logarithm of the Nekrasov's partition function. 
  In this paper, by using the quasi-normal frequencies for non-rotating BTZ black hole derived by Cardos and Lemos, also via Bohr-Sommerfeld quantization for an adiabatic invariant, $I=\int {dE\over \omega(E)}$, which $E$ is the energy of system and $\omega(E)$ is vibrational frequency, we leads to an equally spaced mass spectrum. The result for the area of event horizon is $A_{n}=2\pi \sqrt{\frac{nm \hbar}{\Lambda}}$ which is not equally spaced, in contrast with area spectrum of black hole in higher dimension. 
  In this work a new method is developed to investigate the Aharonov-Casher effect in a noncommutative space. It is shown that the holonomy receives non-trivial kinematical corrections. 
  We discuss how to extract renormalized from bare Polyakov loops in SU(N) lattice gauge theories at nonzero temperature in four spacetime dimensions. Single loops in an irreducible representation are multiplicatively renormalized without mixing, through a renormalization constant which depends upon both representation and temperature. The values of renormalized loops in the four lowest representations of SU(3) were measured numerically on small, coarse lattices. We find that in magnitude, condensates for the sextet and octet loops are approximately the square of the triplet loop. This agrees with a large $N$ expansion, where factorization implies that the expectation values of loops in adjoint and higher representations are just powers of fundamental and anti-fundamental loops. For three colors, numerically the corrections to the large $N$ relations are greatest for the sextet loop, $\leq 25%$; these represent corrections of $\sim 1/N$ for N=3. The values of the renormalized triplet loop can be described by an SU(3) matrix model, with an effective action dominated by the triplet loop. In several ways, the deconfining phase transition for N=3 appears to be like that in the $N=\infty$ matrix model of Gross and Witten. 
  Using a recently proposed group-theoretical approach, we explore novel gaugings of maximal supergravity in four dimensions with gauge group embeddings that can be generated by fluxes of IIB string theory. The corresponding potentials are positive without stationary points. Some allow domain wall solutions which can be elevated to ten dimensions. Appropriate truncations describe type-IIB flux compactifications on T^6 orientifolds leading to non-maximal, four-dimensional, supergravities. 
  We review recent progress in gauging maximal supergravity theories 
  We identify the goldstino fields that give mass to the Kaluza Klein modes of five dimensional supergravity, when supersymmetry breaking is induced by brane effects. We then proof the four dimensional Equivalence Theorem that, in renormalizable gauges, allows for the replacement of Kaluza Klein modes of helicity $\pm1/2$ gravitinos in terms of goldstinos. Finally we identify the five dimensional renormalizable gauge fixing that leads to the Equivalence Theorem. 
  This publication is an exercise which extends to two variables the Christoffel's construction of orthogonal polynomials for potentials of one variable with external sources. We generalize the construction to biorthogonal polynomials. We also introduce generalized Schur polynomials as a set of orthogonal, symmetric, non homogeneous polynomials of several variables, attached to Young tableaux. 
  In this talk, we describe our recent results on the supersymmetrization of the Harry Dym hierarchy as well as a newly constructed deformed Harry Dym hierarchy which is integrable with two arbitrary parameters. In various limits of these parameters, the deformed hierarchy reduces to various known integrable systems. 
  The effective action for the interacting massive scalar field in curved space-time is derived using the heat-kernel method. Starting from this effective action, we establish a smooth quadratic form of the low-energy decoupling for the four-scalar coupling constant and for the nonminimal interaction parameter. The evolution of this parameter from the conformal value 1/6 at high energies down to the IR regime is investigated within the two toy models with negative and positive four-scalar coupling constants. 
  In this thesis I review cosmological and astrophysical exact models for Randall-Sundrum-type braneworlds and their physical implications. I present new insights and show their analogies with quantum theories via the holographic idea. In astrophysics I study the two fundamental models of a spherically symmetric static star and spherically symmetric collapsing objects. I show how matching for the pressure of a static star encodes braneworld effects. In addition I study the problem of the vacuum exterior conjecturing a uniqueness theorem. Furthermore I show that a collapsing dust cloud in the braneworld has a non-static exterior, in contrast to the General Relativistic case. This non-static behaviour is linked to the presence of a "surplus potential energy" that must be released, producing a non-zero flux of energy. Via holography this can be connected with the Hawking process, giving an indirect measure of the brane tension. In cosmology I investigate the generalization of the Randall-Sundrum-type model obtained by introducing the Gauss-Bonnet combination into the action. I elucidate the junction conditions necessary to study the brane model and obtain the cosmological dynamics, showing that, even in the thin shell limit for the brane, the Gauss-Bonnet term implies a non-trivial internal structure for the matter and geometry distributions. Independently of the gravitational theory used, I show how to derive the modified Friedman equation and how it is related to the black hole solution of the theory. Via holography I also show how to interpret quantum mechanically the mass of this black hole from a four-dimensional perspective in the simplest Randall-Sundrum-type scenario. 
  The open-closed vertex in the maximally supersymmetric type IIB plane-wave light-cone string field theory is considered and an explicit solution for the bosonic part of the vertex is derived, valid for all values of the mass parameter, \mu. This vertex is of relevance to IIB plane-wave orientifolds, as well as IIB plane-wave strings in the presence of D-branes and their gauge theory duals. Methods of complex analysis are used to develop a systematic procedure for obtaining the solution. This procedure is first applied to the vertex in flat space and then extended to the plane-wave case. The plane-wave solution for the vertex requires introducing certain ``\mu-deformed Gamma functions'', which are generalizations of the ordinary Gamma function. The behaviour of the Neumann matrices is graphically illustrated and their large-\mu asymptotics are analysed. 
  The spectrum of created particles during the tunneling process, leading to the decay of a false vacuum state, is studied numerically in the thick-wall approximation. It is shown that in this case the particle production is very intensive for small momenta. The number of created particles is nearly constant $n(p)\approx 1$ for $4\leq p\leq500$. 
  The anomalies in five-dimensional orbifold theories are examined in a generic type of non-factorizable geometries. In spite of complicated fermion wavefunctions, the shape of anomaly is found to be identical to that of flat theories. In particular it is split evenly on the orbifold fixed points. This result also follows from the arguments on the AdS/CFT correspondence and an anomaly cancellation mechanism. The cancellation with Chern-Simons term works if the four-dimensional effective theory is free from chiral anomalies. We also discuss the Fayet-Iliopoulos (FI) term in warped supersymmetric theories. Unlike the gauge anomaly, FI divergences reside not only on the orbifold fixed points but also in the whole five-dimensional bulk. The effect of the FI term is to generate supersymmetric masses for charged hypermultiplets, which are no longer constant but have metric factor dependence. We calculate the spectrum and wavefunctions of Kaluza-Klein modes in the presence of the FI term and discuss phenomenological implications to quark-lepton masses and large scale hierarchies. 
  We compute analytically the tetrahedron graph in Liouville theory on the pseudosphere. The result allows to extend the check of the bootstrap formula of Zamolodchikov and Zamolodchikov to third order perturbation theory of the coefficient G3. We obtain complete agreement. 
  We argue that T-duality and F-theory appear automatically in the E_8 gauge bundle perspective of M-theory. The 11-dimensional supergravity four-form determines an E_8 bundle. If we compactify on a two-torus, this data specifies an LLE_8 bundle where LG is a centrally-extended loopgroup of G. If one of the circles of the torus is smaller than sqrt(alpha') then it is also smaller than a nontrivial circle S in the LLE_8 fiber and so a dimensional reduction on the total space of the bundle is not valid. We conjecture that S is the circle on which the T-dual type IIB theory is compactified, with the aforementioned torus playing the role of the F-theory torus. As tests we reproduce the T-dualities between NS5-branes and KK-monopoles, as well as D6 and D7-branes where we find the desired F-theory monodromy. Using Hull's proposal for massive IIA, this realization of T-duality allows us to confirm that the Romans mass is the central extension of our LE_8. In addition this construction immediately reproduces the conjectured formula for global topology change from T-duality with H-flux. 
  We study quantum electrodynamics coupled to the matter field on singular background, which we call defect. For defect on the infinite plane we calculated the fermion propagator and mean electromagnetic field. We show that at large distances from the defect plane, the electromagnetic field is constant what is in agreement with the classical results. The quantum corrections determining the field near the plane are calculated in the leading order of perturbation theory. 
  We examine a stationary but non-static asymptotically AdS_3 spacetime with two causally connected conformal boundaries, each of which is a ``null cylinder'', namely a cylinder with a null direction identified. This spacetime arises from three different perspectives: (i) as a non-singular, causally regular orbifold of global AdS_3 by boosts, (ii) as a Penrose-like limit focusing on the horizon of extremal BTZ black holes, and (iii) as an S^1 fibration over AdS_2. Each of these perspectives sheds an interesting light on holography. Examination of the conformal boundary of the spacetime shows that the dual to the space should involve DLCQ limits of the D1-D5 conformal field theory. The Penrose-like limit approach leads to a similar conclusion, by isolating a sector of the complete D1-D5 CFT that describes the physics in the vicinity of the horizon of an extremal black hole. As such this is a holographic description of the universal horizon dynamics of the extremal black holes in AdS_3 and also of the four and five dimensional stringy black holes whose states were counted in string theory. The AdS_2 perspective draws a connection to a 0+1d quantum mechanical theory. Various dualities lead to a Matrix model description of the spacetime. Many interesting issues that are related to both de Sitter physics and attempts to ``see behind a horizon'' using AdS/CFT arise from (a) the presence of two disconnected components to the boundary, and (b) the analytic structure of bulk physics in the complex coordinate plane. 
  In this paper we study the nonperturbative corrections to the generalized Konishi anomaly that come from the strong coupling dynamics of the gauge theory. We consider U(N) gauge theory with adjoint and Sp(N) or SO(N) gauge theory with symmetric or antisymmetric tensor. We study the algebra of chiral rotations of the matter field and show that it does not receive nonperturbative corrections. The algebra implies Wess-Zumino consistency conditions for the generalized Konishi anomaly which are used to show that the anomaly does not receive nonperturbative corrections for superpotentials of degree less than 2l+1 where 2l=3c(Adj)-c(R) is the one-loop beta function coefficient. The superpotentials of higher degree can be nonperturbatively renormalized because of the ambiguities in the UV completion of the gauge theory. We discuss the implications for the Dijkgraaf-Vafa conjecture. 
  Quantum Mechanics of the Early Universe is considered as deformation of a well-known Quantum Mechanics. Similar to previous works of the author, the principal approach is based on deformation of the density matrix with concurrent development of the wave function deformation in the respective Schr{\"o}dinger picture, the associated deformation parameter being interpreted as a new small parameter. It is demonstrated that the existence of black holes in the suggested approach in the end twice causes nonunitary transitions resulting in the unitarity. In parallel this problem is considered in other terms: entropy density, Heisenberg algebra deformation terms, respective deformations of Statistical Mechanics, - all showing the identity of the basic results. From this an explicit solution for Hawking's informaion paradox has been derived. 
  The Lovelock gravity consists of the dimensionally extended Euler densities. The geometry and horizon structure of black hole solutions could be quite complicated in this gravity, however, we find that some thermodynamic quantities of the black holes like the mass, Hawking temperature and entropy, have simple forms expressed in terms of horizon radius. The case with black hole horizon being a Ricci flat hypersurface is particularly simple. In that case the black holes are always thermodynamically stable with a positive heat capacity and their entropy still obeys the area formula, which is no longer valid for black holes with positive or negative constant curvature horizon hypersurface. In addition, for black holes in the gravity theory of Ricci scalar plus a $2n$-dimensional Euler density with a positive coefficient, thermodynamically stable small black holes always exist in $D=2n+1$ dimensions, which are absent in the case without the Euler density term, while the thermodynamic properties of the black hole solutions with the Euler density term are qualitatively similar to those of black holes without the Euler density term as $D>2n+1$. 
  We describe the computation of SUSY-breaking terms on a D3-brane in a quite general type IIB supergravity background. We apply it to study the SUSY-breaking induced on the D3-brane world-volume by the presence of NSNS and RR 3-form fluxes. We provide explicit general formulae for the SUSY-breaking soft terms valid for the different types of fluxes, leading to different patterns of soft terms. Imaginary anti-selfdual fluxes with G_3 a pure (3,0)-form lead to soft terms corresponding to dilaton-dominated SUSY-breaking. More general SUSY-breaking patterns are discussed, arising from more general fluxes, or from distant anti-D3-branes. The known finiteness properties of dilaton-dominated soft terms are understood in terms of holography. The above results are interpreted in the context of the 4d effective supergravity theory, where flux components correspond to auxiliary fields of e.g. the 4d dilaton and overall volume modulus. We present semirealistic Type IIB orientifold examples with (meta)stable vacua leading to non-vanishing soft terms of the dilaton-domination type. Such models have many of the ingredients of the recent construction of deSitter vacua in string theory. We finally explore possible phenomenological applications of this form of SUSY-breaking, where we show that soft terms are of order M_s^2/M_p. Thus a string scale of order M_s=10^{10} GeV, and compactification scale three orders of magnitude smaller could explain the smallness of the weak scale versus the Planck mass. 
  The charges of the twisted D-branes of certain WZW models are determined. The twisted D-branes are labelled by twisted representations of the affine algebra, and their charge is simply the ground state multiplicity of the twisted representation. It is shown that the resulting charge group is isomorphic to the charge group of the untwisted branes, as had been anticipated from a K-theory calculation. Our arguments rely on a number of non-trivial Lie theoretic identities. 
  We analyze the question of $U_{\star} (1)$ gauge invariance in a flat non-commutative space where the parameter of non-commutativity, $\theta^{\mu\nu} (x)$, is a local function satisfying Jacobi identity (and thereby leading to an associative Kontsevich product). We show that in this case, both gauge transformations as well as the definitions of covariant derivatives have to modify so as to have a gauge invariant action. We work out the gauge invariant actions for the matter fields in the fundamental and the adjoint representations up to order $\theta^{2}$ while we discuss the gauge invariant Maxwell theory up to order $\theta$. We show that despite the modifications in the gauge transformations, the covariant derivative and the field strength, Seiberg-Witten map continues to hold for this theory. In this theory, translations do not form a subgroup of the gauge transformations (unlike in the case when $\theta^{\mu\nu}$ is a constant) which is reflected in the stress tensor not being conserved. 
  We discuss some aspects of noncommutative quantum field theories obtained from the Seiberg-Witten limit of string theories in the presence of an external B-field. General properties of these theories are studied as well as the phenomenological potential of noncommutative QED. 
  Considering complex $n$-dimension Calabi-Yau homogeneous hyper-surfaces $% \mathcal{H}_{n}$ with discrete torsion and using Berenstein and Leigh algebraic geometry method, we study Fractional D-branes that result from stringy resolution of singularities. We first develop the method introduced in hep-th/0105229 and then build the non commutative (NC) geometries for orbifolds $\mathcal{O}=\mathcal{H}_{n}/\mathbf{Z}_{n+2}^{n}$ with a discrete torsion matrix $t_{ab}=exp[{\frac{i2\pi}{n+2}}{(\eta_{ab}-\eta_{ba})}]$, $\eta_{ab} \in SL(n,\mathbf{Z})$. We show that the NC manifolds $% \mathcal{O}^{(nc)}$ are given by the algebra of functions on the real $% (2n+4) $ Fuzzy torus $\mathcal{T}_{\beta_{ij}}^{2(n+2)}$ with deformation parameters $\beta_{ij}=exp{\frac{i2\pi}{n+2}}{[(\eta_{ab}^{-1}-\eta_{ba}^{-1})} q_{i}^{a} q_{j}^{b}]$ with $q_{i}^{a}$'s being charges of $% \mathbf{Z}_{n+2}^{n}$. We also give graphic rules to represent $\mathcal{O}% ^{(nc)}$ by quiver diagrams which become completely reducible at orbifold singularities. It is also shown that regular points in these NC geometries are represented by polygons with $(n+2)$ vertices linked by $(n+2)$ edges while singular ones are given by $(n+2)$ non connected loops. We study the various singular spaces of quintic orbifolds and analyze the varieties of fractional $D$ branes at singularities as well as the spectrum of massless fields. Explicit solutions for the NC quintic $\mathcal{Q}^{(nc)}$ are derived with details and general results for complex $n$ dimension orbifolds with discrete torsion are presented. 
  In this work a generalization of the Bogoliubov transformation is developed to describe a space compactified fermionic field. The method is the fermion counterpart of the formalism introduced earlier for bosons (J. C. da Silva, A. Matos Neto, F.C. Khanna and A.E. Santana, Phys. Rev. A 66 (2002) 052101), and is based on the thermofield dynamics approach. We analyze the energy-momentum tensor for the Casimir effect of a free massless fermion system in a N-dimensional box in contact with a heat bath. As a particular situation we calculate the Casimir energy and pressure for the field in a 3-dimensional box. One interesting result is that the attractive or repulsive nature of the Casimir pressure can change depending on the rate among the sizes. This fact is analyzed with specific examples. 
  We realize the exceptional superconformal algebra $CK_6$, spanned by 32 fields, inside the Lie superalgebra of pseudodifferential symbols on the supercircle $S^{1|3}$. We obtain a one-parameter family of irreducible representations of $CK_6$ in a superspace spanned by 8 fields. 
  We briefly review a recent progress in constructing the low-energy effective action in ${\cal N}=2,4$ super Yang-Mills theories. Using superfield methods we study the one- and two-loop contributions to the effective action in the Coulomb and non-Abelian phases. General structure of low-energy corrections to the effective action is discussed. 
  Non-uniform black strings in the two-brane system are investigated using the effective action approach. It is shown that the radion acts as a non-trivial hair of black strings. The stability of solutions is demonstrated using the catastrophe theory. The black strings are shown to be non-uniform. 
  We discuss type IIB orientifolds with D-branes, and NSNS and RR field strength fluxes, with D-brane sectors leading to open string spectra with non-abelian gauge symmetry and charged chiral fermions. The closed string field strengths generate a scalar potential stabilizing most moduli. Hence the models combine the advantages of leading to phenomenologically interesting (and even semirealistic) chiral open string spectra, and of stabilizing the dilaton and most geometric moduli. We describe the explicit construction of two classes of non-supersymmetric models on $\IT^6$ and orbifolds/orientifolds thereof, with chiral gauge sector arising from configurations of D3-branes at singularities, and from D9-branes with non-trivial world-volume magnetic fields. The latter examples yield the chiral spectrum of just the Standard Model. 
  We consider non-compact Z_N orientifold models of type IIB superstring theory with four-dimensional gravity induced on a set of coincident D3-branes. For the models with odd N the contribution to the one-loop renormalization of the Planck mass is shown to come only from the torus and to be O(N) as the contributions from annulus, Moebius strip and Klein bottle cancel. One can therefore realize the Dvali-Gabadadze-Porrati idea that four-dimensional gravity is induced by quantum effects at the one-loop level by considering large N. 
  We describe an integrable model, related to the Gaudin magnet, and its relation to the matrix model of Brezin, Itzykson, Parisi and Zuber. Relation is based on Bethe ansatz for the integrable model and its interpretation using orthogonal polynomials and saddle point approximation. Lagre $N$ limit of the matrix model corresponds to the thermodynamic limit of the integrable system. In this limit (functional) Bethe ansatz is the same as the generating function for correlators of the matrix models. 
  We exhibit a Hopf superalgebra structure of the algebra of field operators of quantum field theory (QFT) with the normal product. Based on this we construct the operator product and the time-ordered product as a twist deformation in the sense of Drinfeld. Our approach yields formulas for (perturbative) products and expectation values that allow for a significant enhancement in computational efficiency as compared to traditional methods. Employing Hopf algebra cohomology sheds new light on the structure of QFT and allows the extension to interacting (not necessarily perturbative) QFT. We give a reconstruction theorem for time-ordered products in the spirit of Streater and Wightman and recover the distinction between free and interacting theory from a property of the underlying cocycle. We also demonstrate how non-trivial vacua are described in our approach solving a problem in quantum chemistry. 
  Gauge fields of mixed symmetry, corresponding to arbitrary representations of the local Lorentz group of the background spacetime, arise as massive modes in compactifications of superstring theories. We describe bosonic gauge field theories on constant curvature spaces whose fields are in irreducible representations of the general linear group corresponding to Young tableaux with two columns. The gauge-invariant actions for such fields are given and generally require the use of auxiliary fields and additional mass-like terms. We examine these theories in various (partially) massless regimes in which each of the mass-like parameters vanishes. We also make some comments about how the structure extends for gauge fields corresponding to arbitrary Young tableaux. 
  We present a tentative formulation of theories of gravity with suitable matter content, including in particular pure gravity in D dimensions, the bosonic effective actions of M-theory and of the bosonic string, in terms of actions invariant under very-extended Kac-Moody algebras G+++. We conjecture that they host additional degrees of freedom not contained in the conventional theories. The actions are constructed in a recursive way from a level expansion for all very-extended algebras G+++. They constitute non-linear realisations on cosets, a priori unrelated to space-time, obtained from a modified Chevalley involution. Exact solutions are found for all G+++. They describe the algebraic properties of BPS extremal branes, Kaluza-Klein waves and Kaluza-Klein monopoles. They illustrate the generalisation to all G+++ invariant theories of the well-known duality properties of string theories by expressing duality as Weyl invariance in G+++. Space-time is expected to be generated dynamically. In the level decomposition of E8+++ = E11, one may indeed select an A10 representation of generators Pa which appears to engender space-time translations by inducing infinite towers of fields interpretable as field derivatives in space and time. 
  The axioms of topological electromagnetism are refined by the introduction of the de Rham homology of k-vector fields on orientable manifolds and the use of Poincare duality in place of Hodge duality. The central problem of defining the electromagnetic constitutive law is elaborated upon in the linear and nonlinear cases. The manner by which the spacetime metric might follow from the constitutive law is examined in the linear case. The possibility that the intersection form of the spacetime manifold might play a role in defining a topological basis for the constitutive law is explored. The manner by which wave motion might follow from the electromagnetic structure is also discussed. 
  We discuss string spectra in the low-tension limit using the BRST formalism, with emphasis on the role of triplets of totally symmetric tensors and spinor-tensors and their generalizations to cases with mixed symmetry and to (A)dS backgrounds. We also present simple compensator forms of the field equations for individual higher-spin gauge fields that display the {unconstrained} gauge symmetry of a previous non-local construction and reduce upon partial gauge fixing to the (Fang-)Fronsdal equations. For Bose fields we also show how a local Lagrangian formulation with {unconstrained} gauge symmetry is determined by a previous BRST construction. 
  We use free boson techniques to investigate A-D-E-quiver matrix models. Certain higher spin fields in the free boson formulation give rise to higher order loop equations valid at finite N. These fields form a special kind of W-algebra, called Casimir algebra. We compute explicitly the loop equations for A_r and D_r quiver models and check that at large N they are related to a deformation of the corresponding singular Calabi-Yau geometry. 
  The Hamiltonian treatment for the collapse of thin shells for a family of Lanczos-Lovelock theories is studied. This formalism allows us to carry out a concise analysis of these theories. It is found that the black holes solution can be created by collapsing a thin shell. Naked singularities cannot be formed by this mechanism. Among the different Lanczos-Lovelock's theories, the Chern-Simons' theory corresponds to an exceptional case, because naked singularities can emerge from the collapse of a thin shell. This kind of theory does not possess a gravitational self-interaction analogous to the Newtonian case. 
  We calculate the quasinormal modes (QNMs) for gravitational perturbations of the Schwarzschild black hole in the five dimensional (5D) spacetime with a continued fraction method. For all the types of perturbations (scalar-gravitational, vector-gravitational, and tensor-gravitational perturbations), the QNMs associated with l=2, l=3, and l=4 are calculated. Our numerical results are summarized as follows: (i) The three types of gravitational perturbations associated with the same angular quantum number l have a different set of the quasinormal (QN) frequencies; (ii) There is no purely imaginary frequency mode; (iii) The three types of gravitational perturbations have the same asymptotic behavior of the QNMs in the limit of the large imaginary frequencies. In Hawking temperature units these frequencies are given by log3 + i2pi(n+1/2) as n goes to infinity, where n is the mode number. 
  The aim of the present talk is to show that monopoles cannot play any role in the Standard Model (SM) and in its usual extensions up to the Planck scale: $M_{Pl}=1.22\cdot 10^{19}$ GeV, because they have a huge charge and are completely confined or screened. The possibility of the extension of the SM with Family Replicated Gauge Group (FRGG) symmetry of the type $(SMG)^N=[SU(3)_c]^N\times [SU(2)_L]^N\times [U(1)_Y]^N$ is briefly discussed. It was shown that the Abelian monopoles (existing also in non-Abelian theories) in FRGG model have $N^*$ times smaller magnetic charge than in the SM, where $N^*=N(N+1)/2$. These monopoles can appear at the high energies in the FRGG-model and give additional contributions to the beta-functions of the renormalisation group equations for the running constants $\alpha_i(\mu)$, where i=1,2,3 correspond to the U(1), SU(2) and SU(3) gauge groups of the SM. 
  Recently the study of braneworld on the self-gravitating D-brane has been initiated and derived the gravitational equation on the brane by holographic and geometrical projection methods. Surprisingly, in common with these two methods, the matter on the brane cannot be the source of the gravity on the brane at leading order. In this paper we will propose the low energy effective action on the D-brane coupled with gravity which derives the same results. 
  Neglecting the effect of particle production at the moment of bubble nucleation, the spectrum of created particles during the bubble expansion is evaluated in the thin-wall approximation. It is shown that the expanding thin-walled bubble makes the dominant contribution to the particle production. 
  We report on new results in Witten's cubic string field theory for the off-shell factor in the 4-tachyon amplitude that was not fully obtained explicitly before. This is achieved by completing the derivation of the Veneziano formula in the Moyal star formulation of Witten's string field theory (MSFT). We also demonstrate detailed agreement of MSFT with a number of on-shell and off-shell computations in other approaches to Witten's string field theory. We extend the techniques of computation in MSFT, and show that the j=0 representation of SL(2,R) generated by the Virasoro operators $L_{0},L_{\pm1}$ is a key structure in practical computations for generating numbers. We provide more insight into the Moyal structure that simplifies string field theory, and develop techniques that could be applied more generally, including nonperturbative processes. 
  We present a method to obtain soliton solutions to relativistic system of coupled scalar fields. This is done by examining the energy associated to static field configurations. In this case we derive a set of first-order differential equations that solve the equations of motion when the energy saturates its lower bound. To illustrate the general results, we investigate some systems described by polynomial interactions in the coupled fields. 
  Using a non-perturbative functional method, where the quantum fluctuations are gradually set up,it is shown that the interaction of a N=1 Wess-Zumino model in 2+1 dimensions does not get renormalized. This result is valid in the framework of the gradient expansion and aims at compensating the lack of non-renormalization theorems. 
  We show that the Gauss-Bonnet correction to Einstein gravity induces a gravitational tachyon mode, namely an unstable spin 2 fluctuation, in the Randall-Sundrum I model. We demonstrate that this instability is generically related to the presence of a negative tension brane in the set-up, with or without $Z_2$-symmetry across it. Indeed it is shown that the tachyon mode is a bound state localised on any negative tension brane of co-dimension one, embedded in anti-de Sitter background. We discuss the possible resolution of this instability by the inclusion of induced gravity terms on the branes or by an effective four-dimensional cosmological constant. 
  We work with $N-$dimensional compact real hyperbolic space $X_{\Gamma}$ with universal covering $M$ and fundamental group $\Gamma$. Therefore, $M$ is the symmetric space $G/K$, where $G=SO_1(N,1)$ and $K=SO(N)$ is a maximal compact subgroup of $G$. We regard $\Gamma$ as a discrete subgroup of $G$ acting isometrically on $M$, and we take $X_{\Gamma}$ to be the quotient space by that action: $X_{\Gamma}=\Gamma\backslash M = \Gamma\backslash G/K$. The natural Riemannian structure on $M$ (therefore on $X$) induced by the Killing form of $G$ gives rise to a connection $p-$form Laplacian ${\frak L}_p$ on the quotient vector bundle (associated with an irreducible representation of K). We study gauge theories based on abelian $p-$forms on the real compact hyperbolic manifold $X_{\Gamma}$. The spectral zeta function related to the operator ${\frak L}_p$, considering only the co-exact part of the $p-$forms and corresponding to the physical degrees of freedom, can be represented by the inverse Mellin transform of the heat kernel. The explicit thermodynamic fuctions related to skew-symmetric tensor fields are obtained by using the zeta-function regularization and the trace tensor kernel formula (which includes the identity and hyperbolic orbital integrals). Thermodynamic quantities in the high and low temperature expansions are calculated and new entropy/energy ratios established. 
  Horowitz and Maldacena have suggested that the unitarity of the black hole S-matrix can be reconciled with Hawking's semiclassical arguments if a final-state boundary condition is imposed at the spacelike singularity inside the black hole. We point out that, in this scenario, departures from unitarity can arise due to interactions between the collapsing body and the infalling Hawking radiation inside the event horizon. The amount of information lost when a black hole evaporates depends on the extent to which these interactions are entangling. 
  We study N_f D6-brane probes in the supergravity background dual to N_c D4-branes compactified on a circle with supersymmetry-breaking boundary conditions. In the limit in which the resulting Kaluza--Klein modes decouple, the gauge theory reduces to non-supersymmetric, four-dimensional QCD with N_c colours and N_f << N_c flavours. As expected, this decoupling is not fully realised within the supergravity/Born--Infeld approximation. For N_f = 1 and massless quarks, m_q = 0, we exhibit spontaneous chiral symmetry breaking by a quark condensate, <\bar{psi} \psi> \neq 0, and find the associated massless `pion' in the spectrum. The latter becomes massive for m_q > 0, obeying the Gell-Mann--Oakes--Renner relation: M_pi^2= - m_q <\bar{psi} \psi> / \f_pi^2. In the case N_f > 1 we provide a holographic version of the Vafa--Witten theorem, which states that the U(N_f) flavour symmetry cannot be spontaneously broken. Further we find N_f^2 - 1 unexpectedly light pseudo-scalar mesons in the spectrum. We argue that these are not (pseudo) Goldstone bosons and speculate on the string mechanism responsible for their lightness. We then study the theory at finite temperature and exhibit a phase transition associated with a discontinuity in the chiral condensate. D6/anti-D6 pairs are also briefly discussed. 
  We extend the investigation of nonextremal enhancons, finding the most general solutions with the correct symmetry and charges. There are two families of solutions. One of these contains a solution with a regular horizon found previously; this previous example is shown to be the unique solution with a regular horizon. The other family generalises a previous nonextreme extension of the enhancon, producing solutions with shells which satisfy the weak energy condition. We argue that identifying a unique solution with a shell requires input beyond supergravity. 
  Recent observations confirm that our universe is flat and consists of a dark energy component $\Omega_{DE}\simeq 0.7$. This dark energy is responsible for the cosmic acceleration as well as determines the feature of future evolution of the universe. In this paper, we discuss the dark energy of universe in the framework of scalar-tensor cosmology. It is shown that the dark energy is the main part of the energy density of the gravitational scalar field and the future universe will expand as $a(t)\sim t^{1.3}$. 
  String theory is the most promising candidate for the theory unifying all interactions including gravity. It has an extremely difficult dynamics. Therefore, it is useful to study some its simplifications. One of them is non-critical string theory which can be defined in low dimensions. A particular interesting case is 2D string theory. On the one hand, it has a very rich structure and, on the other hand, it is solvable. A complete solution of 2D string theory in the simplest linear dilaton background was obtained using its representation as Matrix Quantum Mechanics. This matrix model provides a very powerful technique and reveals the integrability hidden in the usual CFT formulation.   This thesis extends the matrix model description of 2D string theory to non-trivial backgrounds. We show how perturbations changing the background are incorporated into Matrix Quantum Mechanics. The perturbations are integrable and governed by Toda Lattice hierarchy. This integrability is used to extract various information about the perturbed system: correlation functions, thermodynamical behaviour, structure of the target space. The results concerning these and some other issues, like non-perturbative effects in non-critical string theory, are presented in the thesis. 
  We study "multitrace" deformations of large N master fields in models with a mass gap. In particular, we determine the conditions for the multitrace couplings to drive tachyonic instabilities. These tachyons represent new local instabilities of the associated nonlocal string theories. In the particular case of Dp-branes at finite temperature, we consider topology-changing phase transitions and the effect of multitrace perturbations on the corresponding phase diagrams. 
  We construct the exact solution of one (anti)instanton in N=1/2 super Yang-Mills theory defined on non(anti)commutative superspace. We first identify N = 1/2 superconformal invariance as maximal spacetime symmetry. For gauge group U(2), SU(2) part of the solution is given by the standard (anti)instanton, but U(1) field strength also turns out nonzero. The solution is SO(4) rotationally symmetric. For gauge group U(N), in contrast to the U(2) case, we show that the entire U(N) part of the solution is deformed by non(anti)commutativity and fermion zero-modes. The solution is no longer rotationally symmetric; it is polarized into an axially symmetric configuration because of the underlying non(anti)commutativity. We compute the `information metric' of one (anti) instanton. We find that moduli space geometry is deformed from hyperbolic space (Euclidean anti-de Sitter space) in a way anticipated from reduced spacetime symmetry. Remarkably, the volume measure of the moduli space turns out to be independent of the non(anti)commutativity. Implications to D-branes in Ramond- Ramond flux background and Maldacena's gauge-gravity correspondence are discussed. 
  We construct the general action for Abelian vector multiplets in rigid 4-dimensional Euclidean (instead of Minkowskian) N=2 supersymmetry, i.e., over space-times with a positive definite instead of a Lorentzian metric. The target manifolds for the scalar fields turn out to be para-complex manifolds endowed with a particular kind of special geometry, which we call affine special para-Kahler geometry. We give a precise definition and develop the mathematical theory of such manifolds. The relation to the affine special Kahler manifolds appearing in Minkowskian N=2 supersymmetry is discussed. Starting from the general 5-dimensional vector multiplet action we consider dimensional reduction over time and space in parallel, providing a dictionary between the resulting Euclidean and Minkowskian theories. Then we reanalyze supersymmetry in four dimensions and find that any (para-)holomorphic prepotential defines a supersymmetric Lagrangian, provided that we add a specific four-fermion term, which cannot be obtained by dimensional reduction. We show that the Euclidean action and supersymmetry transformations, when written in terms of para-holomorphic coordinates, take exactly the same form as their Minkowskian counterparts. The appearance of a para-complex and complex structure in the Euclidean and Minkowskian theory, respectively, is traced back to properties of the underlying R-symmetry groups. Finally, we indicate how our work will be extended to other types of multiplets and to supergravity in the future and explain the relevance of this project for the study of instantons, solitons and cosmological solutions in supergravity and M-theory. 
  The equations of 10 or 11 dimensional supergravity admit supersymmetric compactifications on 7-manifolds of $G_2$ holonomy, but these supergravity vacua are deformed away from special holonomy by the higher-order corrections of string or M-theory. We find simple expressions for the first-order corrections to the Einstein and Killing spinor equations in terms of the calibrating 3-form of the leading-order G_2-holonomy background. We thus obtain, and solve explicitly, systems of first-order equations describing the corrected metrics for most of the known classes of cohomogeneity-one 7-metrics with G_2 structures 
  We study the moduli space of the boundary conformal field theories describing an unstable D-brane of type II string theory compactified on a circle of critical radius. This moduli space has two branches, -- a three dimensional branch S^3/Z_2 and a two dimensional branch described by a square torus T^2. These two branches are joined along a circle. We compare this with the moduli space of classical solutions of tachyon effective field theory compactified on a circle of critical radius. This moduli space has a very similar structure to that of the boundary conformal field theory with the only difference that the S^3 of the S^3/Z_2 component becomes a deformed S^3. This provides one more indication that the tachyon effective field theory captures qualitatively the dynamics of the tachyon on an unstable D-brane. 
  In this work, we discuss the interaction between anti-symmetric rank-two tensor matter and topological Yang-Mills fields. The matter field considered here is the rank-2 Avdeev-Chizhov tensor matter field in a suitably extended $N_{T}=2$ SUSY. We start off from the $N_{T}=2$, D=4 superspace formulation and we go over to Riemannian manifolds. The matter field is coupled to the topological Yang-Mills field. We show that both actions are obtained as $Q-$exact forms, which allows us to write the energy-momentum tensor as $Q-$exact observables. 
  We study the embedding of cosmic strings, related to the Abrikosov-Nielsen-Olesen vortex solution, into d=4, N=1 supergravity. We find that the local cosmic string solution which saturates the BPS bound of supergravity with $D$-term potential for the Higgs field and with constant Fayet--Iliopoulos term, has 1/2 of supersymmetry unbroken. We observe an interesting relation between the gravitino supersymmetry transformation, positive energy condition and the deficit angle of the cosmic string. We argue that the string solutions with magnetic flux with F-term potential cannot be supersymmetric, which leads us to a conjecture that D1-strings (wrapped D(1+q)-branes) of string theory in the effective 4d supergravity are described by the D-term strings that we study in this paper. We give various consistency checks of this conjecture, and show that it highlights some generic properties of non-BPS string theory backgrounds, such as brane-anti-brane systems. Supersymmetry breaking by such systems can be viewed as FI D-term breaking, which implies, under certain conditions, the presence of gauged R-symmetry on such backgrounds. The D-term nature of the brane-anti-brane energy can also provide information on the superpotential for the tachyon, which Higgses the R-symmetry. In this picture, the inter-brane force can be viewed as a result of the world-volume gauge coupling renormalization by the open string loops. 
  We discuss the gauge coupling renormalization in orbifold field theories in which the 4-dimensional graviton and/or matter fields are quasi-localized in extra dimension to generate hierarchically different mass scales and/or Yukawa couplings. In such theories, there can be large calculable Kaluza-Klein threshold corrections to low energy gauge couplings, enhanced by the logarithms of small warp factor and/or of small Yukawa couplings. We present the results on those Kaluza-Klein threshold corrections in generic 5-dimensional theory on $S^1/Z_2\times Z_2$ containing arbitrary 5-dimensional gauge, spinor and scalar fields. 
  We study the formation of D and F-cosmic strings in D-brane annihilation after brane inflation. We show that D-string formation by quantum de Sitter fluctuations is severely suppressed, due to suppression of RR field fluctuations in compact dimensions. We discuss the resonant mechanism of production of D and F-strings, which are formed as magnetic and electric flux tubes of the two orthogonal gauge fields living on the world-volume of the unstable brane. We outline the subsequent cosmological evolution of the D-F string network. We also compare the nature of these strings with the ordinary cosmic strings and point out some differences and similarities. 
  M-theory effects prevent five-dimensional domain-wall and black-hole solutions from developing curvature singularities. While so far this analysis was performed for particular models, we now present a model-independent proof that these solutions do not have naked singularities as long as the Kahler moduli take values inside the extended Kahler cone. As a by-product we obtain information on the regularity of the Kahler-cone metric at boundaries of the Kahler cone and derive relations between the geometry of moduli space and space-time. 
  We consider cosmological models with a scalar field with equation of state $w\ge 1$ that contract towards a big crunch singularity, as in recent cyclic and ekpyrotic scenarios. We show that chaotic mixmaster oscillations due to anisotropy and curvature are suppressed, and the contraction is described by a homogeneous and isotropic Friedmann equation if $w>1$. We generalize the results to theories where the scalar field couples to p-forms and show that there exists a finite value of $w$, depending on the p-forms, such that chaotic oscillations are suppressed. We show that $Z_2$ orbifold compactification also contributes to suppressing chaotic behavior. In particular, chaos is avoided in contracting heterotic M-theory models if $w>1$ at the crunch. 
  We study recombinations of D-brane systems intersecting at more than one angle using super Yang-Mills theory. We find the condensation of an off-diagonal tachyon mode relates to the recombination, as was clarified for branes at one angle in hep-th/0303204. For branes at two angles, after the tachyon mode between two D2-branes condensed, D2-brane charge is distributed in the bulk near the intersection point. We also find that, when two intersection angles are equal, the off-diagonal lowest mode is massless, and a new stable non-abelian configuration, which is supersymmetric up to a quadratic order in the fluctuations, is obtained by the deformation by this mode. 
  We study the quantum effects in a brane-world model in which a positive constant curvature brane universe is embedded in a higher-dimensional bulk AdS black hole, instead of the usual portion of the AdS$_5$. By using zeta regularisation, in the large mass regime, we explicitly calculate the one-loop effective potential due to the bulk quantum fields and show that it leads to a non-vanishing cosmological constant, which can definitely acquire a positive value. 
  The quantum periodic XXZ chain with alternating spins is studied. The properties of the related R-matrix and Hamiltonians are discussed. A compact expression for the ground state energy is obtained. The corresponding conformal anomaly is found via the finite-size computations and also by means of the Bethe ansatz method. In the presence of an external magnetic field, the magnetic susceptibility is derived. The results are also generalized to the case of a chain containing several different spins. 
  We consider a more general initial condition satisfying the minimal uncertainty relationship. We calculate the power spectrum of a simple model in inflationary cosmology. The results depend on perturbations generated below a fundamental scale, e.g. the Planck scale. 
  If the source of the current accelerating expansion of the universe is a positive cosmological constant, Banks and Fischler argued that there exists an upper limit of the total number of e-foldings of inflation. We further elaborate on the upper limit in the senses of viewing the cosmological horizon as the boundary of a cavity and of the holographic D-bound in a de Sitter space. Assuming a simple evolution model of inflation, we obtain an expression of the upper limit in terms of the cosmological constant, the energy density of inflaton when the inflation starts, the energy density as the inflation ends, and reheating temperature. We discuss how the upper limit is modified in the different evolution models of the universe. The holographic D-bound gives more high upper limit than the entropy threshold in the cavity. For the most extremal case where the initial energy density of inflation is as high as the Planck energy, and the reheating temperature is as low as the energy scale of nucleosynthesis, the D-bound gives the upper limit as 146 and the entropy threshold as 122. For reasonable assumption in the simplest cosmology, the holographic D-bound leads to a value about 85, while the cavity model gives a value around 65 for the upper limit, which is close to the value in order to solve the flatness problem and horizon problem in the hot big bang cosmology. 
  In this talk I give an overview of the work done during the last 15 years in collaboration with the late Adrian Patrascioiu. In this work we accumulated evidence against the commonly accepted view that theories with nonabelian symmetry -- either two dimensional nonlinear $\sigma$ models or four dimensional Yang-Mills theories -- have the property of asymptotic freedom (AF) usually ascribed to them. 
  We discuss and compare several geometric structures which imply an upper bound to the acceleration of a particle measured in its rest system. While all of them have the same implications on the motion of a point particle, they differ in other important respects. In particular, they have different symmetry groups, which influence in a different way the search for an underlying dynamical theory. 
  We consider the RNS superstrings in $AdS_3 \times S^3 \times {\cal M}$, where $\cal M$ may be $K3$ or $T^4$, based on $SL(2,R)$ and SU(2) WZW models. We construct the physical states and calculate the spectrum. A subsector of this theory describes strings propagating in the six dimensional plane wave obtained by the Penrose limit of $AdS_3 \times S^3 \times {\cal M}$. We reproduce the plane wave spectrum by taking $J$ and the radius to infinity. We show that the plane wave spectrum actually coincides with the large $J$ spectrum at fixed radius, i.e. in $AdS_3 \times S^3$. Relation to some recent topics of interest such as the Frolov-Tseytlin string and strings with critical tension or in zero radius $AdS$ are discussed. 
  We provide a conceptual unified description of the quantum properties of black holes (BH), elementary particles, de Sitter (dS) and Anti de Sitter (AdS) string states.The conducting line of argument is the classical-quantum (de Broglie, Compton) duality here extended to the quantum gravity (string) regime (wave-particle-string duality). The semiclassical (QFT) and quantum (string) gravity regimes are respectively characterized and related: sizes, masses, accelerations and temperatures. The Hawking temperature, elementary particle and string temperatures are shown to be the same concept in different energy regimes and turn out the precise classical-quantum duals of each other; similarly, this result holds for the BH decay rate, heavy particle and string decay rates; BH evaporation ends as quantum string decay into pure (non mixed) radiation. Microscopic density of states and entropies in the two (semiclassical and quantum) gravity regimes are derived and related, an unifying formula for BH, dS and AdS states is provided in the two regimes. A string phase transition towards the dS string temperature (which is shown to be the precise quantum dual of the semiclassical (Hawking-Gibbons) dS temperature) is found and characterized; such phase transition does not occurs in AdS alone. High string masses (temperatures) show a further (square root temperature behaviour) sector in AdS. From the string mass spectrum and string density of states in curved backgrounds, quantum properties of the backgrounds themselves are extracted and the quantum mass spectrum of BH, dS and AdS radii obtained. 
  We extend the four-dimensional gauged supergravity analysis of type IIB vacua on $K3\times T^2/Z_2$ to the case where also D3 and D7 moduli, belonging to N=2 vector multiplets, are turned on. In this case, the overall special geometry does not correspond to a symmetric space, unless D3 or D7 moduli are switched off. In the presence of non--vanishing fluxes, we discuss supersymmetric critical points which correspond to Minkowski vacua, finding agreement with previous analysis. Finally, we point out that care is needed in the choice of the symplectic holomorphic sections of special geometry which enter the computation of the scalar potential. 
  In brane inflation, the relative brane position in the bulk of a brane world is the inflaton. For branes moving in a compact manifold, the approximate translational (or shift) symmetry is necessary to suppress the inflaton mass, which then allows a slow-roll phase for enough inflation. Following recent works, we discuss how inflation may be achieved in superstring theory. Imposing the shift symmetry, we obtain the condition on the superpotential needed for inflation and suggest how this condition may be naturally satisfied. 
  We use the \hat c=1 matrix model to compute the potential energy V(C) for (the zero mode of) the RR scalar in two-dimensional type 0B string theory. The potential is induced by turning on a background RR flux, which in the matrix model corresponds to unequal Fermi levels for the two types of fermions. Perturbatively, this leads to a linear runaway potential, but non-perturbative effects stabilize the potential, and we find the exact expression V(C)=\frac{1}{2\pi}\int da\arccos [\cos(C)/\sqrt{1+e^{-2\pi a}}]. We also compute the finite-temperature partition function of the 0B theory in the presence of flux. The perturbative expansion is T-dual to the analogous result in type 0A theory, but non-perturbative effects (which depend on C) do not respect naive R\to 1/R duality. The model can also be used to study scattering amplitudes in background RR fluxes. 
  We find an interpretation of the recent connection found between topological strings on Calabi-Yau threefolds and crystal melting: Summing over statistical mechanical configuration of melting crystal is equivalent to a quantum gravitational path integral involving fluctuations of Kahler geometry and topology. We show how the limit shape of the melting crystal emerges as the average geometry and topology of the quantum foam at the string scale. The geometry is classical at large length scales, modified to a smooth limit shape dictated by mirror geometry at string scale and is a quantum foam at area scales g_s \alpha'. 
  There is a growing interest in the logical possibility that exceptional mathematical structures (exceptional Lie and superLie algebras, the exceptional Jordan algebra, etc.) could be linked to an ultimate "exceptional" formulation for a Theory Of Everything (TOE). The maximal division algebra of the octonions can be held as the mathematical responsible for the existence of the exceptional structures mentioned above. In this context it is quite motivating to systematically investigate the properties of octonionic spinors and the octonionic realizations of supersymmetry. In particular the $M$-algebra can be consistently defined for two structures only, a real structure, leading to the standard $M$-algebra, and an octonionic structure. The octonionic version of the $M$-algebra admits striking properties induced by octonionic $p$-forms identities. 
  Representations of the superalgebra $osp(2|2)$ and current superalgebra $osp(2|2)^{(1)}_k$ in the standard basis are investigated. All finite-dimensional typical and atypical representations of $osp(2|2)$ are constructed by the vector coherent state method. Primary fields of the non-unitary conformal field theory associated with $osp(2|2)^{(1)}_k$ in the standard basis are constructed for arbitrary level $k$. 
  Auxiliary field methods in D=2 (or 3), N=2 supersymmetric (SUSY) nonlinear sigma models (NLSMs) are studied. For these models auxiliary fields as Lagrange multipliers belong to a vector or a chiral superfield, which gives a Kahler quotient of complexified gauge group or a holomorphic constraint on it, respectively. Using these, NLSMs on all Hermitian symmetric spaces were formulated previously. In this paper, we formulate new SUSY NLSMs on some rank-two Kahler coset spaces as SUSY gauge theories with two Fayet-Iliopoulos parameters. 
  We showed in hep-th/0303210 that the Dijkgraaf-Vafa theory can be regarded as large-N reduction in the case of $\mathcal{N}=1$ supersymmetric U(N) gauge theories, with single adjoint matter. We generalize this to gauge theories with gauge groups being the products of some unitary groups coupled to bifundamental or fundamental matter. We show that some large-N reduced models of these theories are supermatrix models, whose free energy is equivalent to the prepotentials of the original gauge theories. The supermatrix model in our approach should be taken in the Veneziano limit $N_c,N_f \to \infty $ with $N_f/N_c$ fixed. 
  The title "Time Asymmetric Quantum Theory: the Theory of Resonances" (without questionmark) of the CFIF workshop (23.-26.7.2003, Lisbon, Portugal) implies that the theoretical description of resonances is uniquely described by the formalism of A. Bohm et al. reflecting the title of the workshop. Our presentation in this workshop tries to introduce an apparently inequivalent, alternative feasible relativistic formalism provided by the author under the name "(Anti)Causal Quantum Theory" which is compared to the former. 
  In this paper we present a discrete, non-perturbative formulation for type IIB string theory. Being a supersymmetric quiver matrix mechanics model in the framework of M(atrix) theory, it is a generalization of our previous proposal of compactification via orbifolding for deconstructed IIA strings. In the continuum limit, our matrix mechanics becomes a $(2+1)$-dimensional Yang-Mills theory with 16 supercharges. At the discrete level, we are able to construct explicitly the solitonic states that correspond to membranes wrapping on the compactified torus in target space. These states have a manifestly $SL(2,\integer)$-invariant spectrum with correct membrane tension, and give rise to an emergent flat dimension when the compactified torus shrinks to vanishing size. 
  We give a covariant characterisation of the Penrose plane wave limit: the plane wave profile matrix $A(u)$ is the restriction of the null geodesic deviation matrix (curvature tensor) of the original spacetime metric to the null geodesic, evaluated in a comoving frame. We also consider the Penrose limits of spacetime singularities and show that for a large class of black hole, cosmological and null singularities (of Szekeres-Iyer ``power-law type''), including those of the FRW and Schwarzschild metrics, the result is a singular homogeneous plane wave with profile $A(u)\sim u^{-2}$, the scale invariance of the latter reflecting the power-law behaviour of the singularities. 
  We present two new classes of magnetic brane solutions in Einstein-Maxwell-Gauss-Bonnet gravity with a negative cosmological constant. The first class of solutions yields an $(n+1)$-dimensional spacetime with a longitudinal magnetic field generated by a static magnetic brane. We also generalize this solution to the case of spinning magnetic branes with one or more rotation parameters. We find that these solutions have no curvature singularity and no horizons, but have a conic geometry. In these spacetimes, when all the rotation parameters are zero, the electric field vanishes, and therefore the brane has no net electric charge. For the spinning brane, when one or more rotation parameters are non zero, the brane has a net electric charge which is proportional to the magnitude of the rotation parameter. The second class of solutions yields a spacetime with an angular magnetic field. These solutions have no curvature singularity, no horizon, and no conical singularity. Again we find that the net electric charge of the branes in these spacetimes is proportional to the magnitude of the velocity of the brane. Finally, we use the counterterm method in the Gauss-Bonnet gravity and compute the conserved quantities of these spacetimes. 
  Constrained Hamiltonian systems are investigated by using the Hamilton-Jacobi method. Integration of a set of equations of motion and the action function is discussed. It is shown that we have two types of integrable systems: a) ${\it Partially integrable systems}$, where the set of equations of motion is only integrable. b) {\it Completely integrable systems}, where the set of equations of motion and the action function is integrable. Two examples are studied. 
  The CPT anomaly of certain chiral gauge theories has been established previously for flat multiply connected spacetime manifolds M of the type R^3 x S^1, where the noncontractible loops have a minimal length. In this article, we show that the CPT anomaly also occurs for manifolds where the noncontractible loops can be arbitrarily small. Our basic calculation is performed for a flat noncompact manifold with a single "puncture," namely M = R^2 x (R^2 {0}). A hypothetical spacetime foam might have many such punctures (or other structures with similar effects). Assuming the multiply connected structure of the foam to be time independent, we present a simple model for photon propagation, which generalizes the single-puncture result. This model leads to a modified dispersion law of the photon. Observations of high-energy photons (gamma-rays) from explosive extragalactic events can then be used to place an upper bound on the typical length scale of these punctures. 
  Recently, Diaconescu, Moore and Witten provided a nontrivial link between K-theory and M-theory, by deriving the partition function of the Ramond-Ramond fields of Type IIA string theory from an E8 gauge theory in eleven dimensions. We give some relations between twisted K-theory and M-theory by adapting the method of Diaconescu-Moore-Witten and Moore-Saulina. In particular, we construct the twisted K-theory torus which defines the partition function, and also discuss the problem from the E8 loop group picture, in which the Dixmier-Douady class is the Neveu-Schwarz field. In the process of doing this, we encounter some mathematics that is new to the physics literature. In particular, the eta differential form, which is the generalization of the eta invariant, arises naturally in this context. We conclude with several open problems in mathematics and string theory. 
  Induced supersymmetry representations on composite operators are studied. In superspace the ensuing transformation rules (trivially) lead to an effective superfield. On the other hand, an induced representation must exist for non-linear (``on-shell'') supersymmetry as well. As this choice of the representation is physically irrelevant, any formulation of an effective action starting from the superspace representation must equally well be possible in a non-linear representation. We show that this leads to very relevant constraints on the formulation of effective actions in terms of composite operators. These ideas are applied to the simplest case of such a theory, N=1 SYM. It is shown that soft supersymmetry breaking within that theory forces one to include besides the Lagrangian multiplet S all currents of the super-conformal structure, embedded in a supergravity background, as relevant fields. 
  We compute the cross section for scattering of light string probes by randomly excited closed strings. For high energy probes, the cross section factorizes and can be used to define effective form factors for the excited targets. These form factors are well defined without the need for infinite subtractions and contain information about the shape and size of typical strings. For highly excited strings the elastic form factor can be written in terms of the `plasma dispersion function', which describes charge screening in high temperature plasmas. 
  Recently we have described a mechanical system which exhibits spontaneous breaking of Z_2 symmetry and related topological kinks called compactons. The corresponding field potential is not differentiable at its global minima. Therefore, standard derivation of dispersion relation $\omega(k)$ for small perturbations around the ground state can not be applied. In the present paper we obtain the dispersion relation. It turns out that evolution equation remains nonlinear even for arbitrarily small perturbations. The shape of the resulting running wave is piecewise combined from cosh functions. We also analyse dynamics of the symmetry breaking transition. It turns out that the number of produced compacton-anticompacton pairs strongly depends on the form of initial perturbation of the unstable former ground state. 
  Four-dimensional massive N=2 nonlinear sigma models and BPS wall solutions are studied in the off-shell harmonic superspace approach in which N=2 supersymmetry is manifest. The general nonlinear sigma model can be described by an analytic harmonic potential which is the hyper-Kahler analog of the Kahler potential in N=1 theory. We examine the massive nonlinear sigma model with multi-center four-dimensional target hyper-Kahler metrics and derive the corresponding BPS equation. We study in some detail two particular cases with the Taub-NUT and double Taub-NUT metrics. The latter embodies, as its two separate limits, both Taub-NUT and Eguchi-Hanson metrics. We find that domain wall solutions exist only in the double Taub-NUT case including its Eguchi-Hanson limit. 
  We discuss the matching of the BPS part of the spectrum for (super)membrane, which gives the possibility of getting membrane's results via string calculations. In the small coupling limit of M--theory the entropy of the system coincides with the standard entropy of type IIB string theory (including the logarithmic correction term). The thermodynamic behavior at large coupling constant is computed by considering M--theory on a manifold with topology ${\mathbb T}^2\times{\mathbb R}^9$. We argue that the finite temperature partition functions (brane Laurent series for $p \neq 1$) associated with BPS $p-$brane spectrum can be analytically continued to well--defined functionals. It means that a finite temperature can be introduced in brane theory, which behaves like finite temperature field theory. In the limit $p \to 0$ (point particle limit) it gives rise to the standard behavior of thermodynamic quantities. 
  We review boundary rigidity theorems assessing that, under appropriate conditions, Riemannian manifolds with the same spectrum of boundary geodesics are isometric. We show how to apply these theorems to the problem of reconstructing a $d+1$ dimensional, negative curvature space-time from boundary data associated to two-point functions of high-dimension local operators in a conformal field theory. We also show simple, physically relevant examples of negative-curvature spaces that fail to satisfy in a subtle way some of the assumptions of rigidity theorems. In those examples, we explicitly show that the spectrum of boundary geodesics is not sufficient to reconstruct the metric in the bulk. We also survey other reconstruction procedures and comment on their possible implementation in the context of the holographic AdS/CFT duality. 
  We calculate the most general causal N=1 three-dimensional, gauge invariant action coupled to matter in superspace and derive its component form using Ectoplasmic integration theory. One example of such an action can be obtained by compactifying M-theory on a Spin(7) holonomy manifold taking non-vanishing fluxes into account. We show that the resulting three-dimensional theory is in agreement with the more general construction. The scalar potential resulting from Kaluza-Klein compactification stabilizes all the moduli fields describing deformations of the metric except for the radial modulus. This potential can be written in terms of the superpotential previously discussed in the literature. 
  We have studied the scalar perturbation of static charged dilaton black holes in 3+1 dimensions. The black hole considered here is a solution to the low-energy string theory in 3+1 dimensions. The quasinormal modes for the scalar perturbations are calculated using the WKB method. The dilaton coupling constant has a considerable effect on the values of quasi normal modes. It is also observed that there is a linear relation between the quasi normal modes and the temperature for large black holes. 
  We describe a D--brane inflation model which consists of two fractional D3 branes separated on a transverse $T^2 \times K3$. Inflation arises due to the resolved orbifold singularity of $K3$ which corresponds to an anomalous D--term on the brane. We show that D--brane inflation in the bulk corresponds to D--term inflation on the brane. The inflaton and the trigger field parametrize the interbrane distances on $T^2$ an $K3$ respectively. After inflation the branes reach a supersymmetric configuration in which they are at the origin of $T^2$ but separated along the $K3$ directions. 
  The non-commutative geometry of deformation quantization appears in string theory through the effect of a B-field background on the dynamics of D-branes in the topological limit. For arbitrary backgrounds, associativity of the star product is lost, but only cyclicity is necessary for a description of the effective action in terms of a generalized product. In previous work we showed that this property indeed emerges for a non-associative product that we extracted from open string amplitudes in curved background fields. In the present note we extend our investigation through second order in a complete derivative expansion. We establish cyclicity with respect to the Born--Infeld measure and find a logarithmic correction that modifies the Kontsevich formula in an arbitrary background satisfying the generalized Maxwell equation. This equation is the physical equivalent of a divergence-free non-commutative parameter, which is required for cyclicity already in the associative case. 
  String (membrane) theory could be considered as degenerate case of relativistic continuous media theory. The paper presents models of media, which are continuous distributions of interacting membranes, strings or particles. 
  We show that depending on the direction of deformation of $\kappa$-Poincar\'e algebra (time-like, space-like, or light-like) the associated phase spaces of single particle in Doubly Special Relativity theories have the energy-momentum spaces of the form of de Sitter, anti-de Sitter, and flat space, respectively. 
  p-Adic mathematical physics emerged as a result of efforts to find a non-Archimedean approach to the spacetime and string dynamics at the Planck scale. One of its main achievements is a successful formulation and development of p-adic and adelic quantum mechanics, which have complex-valued wave functions of p-adic and adelic arguments, respectively. Various aspects of these quantum mechanics are reviewed here. In particular, the corresponding Feynman's path integrals, some minisuperspace cosmological models, and relevant approache to string theory, are presented. As a result of adelic approach, p-adic effects exhibit a spacetime and some other discreteness, which depend on the adelic quantum state of the physical system under consideration. Besides review, this article contains also some new results. 
  The classical analysis of Kazakov and Kostov of the Makeenko-Migdal loop equation in two-dimensional gauge theory leads to usual partial differential equations with respect to the areas of windows formed by the loop. We extend this treatment to the case of U(N) Yang-Mills defined on the noncommutative plane. We deal with all the subtleties which arise in their two-dimensional geometric procedure, using where needed results from the perturbative computations of the noncommutative Wilson loop available in the literature. The open Wilson line contribution present in the non-commutative version of the loop equation drops out in the resulting usual differential equations. These equations for all N have the same form as in the commutative case for N to infinity. However, the additional supplementary input from factorization properties allowing to solve the equations in the commutative case is no longer valid. 
  The dynamics of topological open branes is controlled by Nambu Brackets. Thus, they might be quantized through the consistent quantization of the underlying Nambu brackets, including odd ones: these are reachable systematically from even brackets, whose more tractable properties have been detailed before. 
  We study the action of the mapping class group of an oriented genus g surface with n punctures and a disc removed on a Poisson algebra which arises in the combinatorial description of Chern-Simons gauge theory when the gauge group is a semidirect product $G\ltimes\mathfrak{g}^*$. We prove that the mapping class group acts on this algebra via Poisson isomorphisms and express the action of Dehn twists in terms of an infinitesimally generated G-action. We construct a mapping class group representation on the representation spaces of the associated quantum algebra and show that Dehn twists can be implemented via the ribbon element of the quantum double D(G) and the exchange of punctures via its universal R-matrix. 
  For a classical superconformal gauge theory in a conformal supergravity background, its chiral R-symmetry anomaly, Weyl anomaly and super-Weyl anomaly constitute a supermultiplet. We review how these anomalies arise from the five-dimensional gauged supergravity in terms of the AdS/CFT correspondence at the gravity level. The holographic production of this full superconformal anomaly multiplet provides a support and test to the celebrated AdS/CFT conjecture. 
  We describe the construction of string theory models with semirealistic spectrum in a sector of (anti) D3-branes located at an orbifold singularity at the bottom of a highly warped throat geometry, which is a generalisation of the Klebanov-Strassler deformed conifold. These models realise the Randall-Sundrum proposal to naturally generate the Planck/electroweak hierarchy in a concrete string theory embedding, and yielding interesting chiral open string spectra. We describe examples with Standard Model gauge group (or left-right symmetric extensions) and three families of SM fermions, with correct quantum numbers including hypercharge. The dilaton and complex structure moduli of the geometry are stabilised by the 3-form fluxes required to build the throat. We describe diverse issues concerning the stabilisation of geometric Kahler moduli, like blow-up modes of the orbifold singularities, via D term potentials and gauge theory non-perturbative effects, like gaugino condensation. This local geometry, once embedded in a full compactification, could give rise to models with all moduli stabilised, and with the potential to lead to de Sitter vacua. Issues of gauge unification, proton stability, supersymmetry breaking and Yukawa couplings are also discussed. 
  We present a general formula for the topology and H-flux of the T-dual of a type two compactification. Our results apply to T-dualities with respect to any free circle action. In particular we find that the manifolds on each side of the duality are circle bundles whose curvatures are given by the integral of the dual H-flux over the dual circle. As a corollary we conjecture an obstruction to multiple T-dualities, generalizing an obstruction known to exist on the twisted torus. Examples include SU(2) WZW models, Lens spaces and the supersymmetric string theory on the non-spin AdS^5xCP^2xS^1 compactification. 
  We study the four dimensional effective action of a system of D6-branes wrapped on the K3 manifold times a torus, allowing the volume of the internal manifolds to remain dynamical. An unwrapped brane is at best a Dirac monopole of the dual R-R sector field to which it couples. After wrapping, a brane is expected to behave as a BPS monopole, where the Higgs vacuum expectation value is set by the size of the K3. We determine the moduli space of an arbitrary number of these wrapped branes by introducing a time dependent perturbation of the static solution, and expanding the supergravity equations of motion to determine the dynamics of this perturbation, in the low velocity limit. The result is the hyper-Kahler generalisation of the Euclidean Taub-NUT metric presented by Gibbons and Manton. We note that our results also pertain to the behavior of bound states of Kaluza-Klein monopoles and wrapped NS5-branes in the T^4 compactified heterotic string. 
  Motivated by the Dijkgraaf-Vafa correspondence, we consider the matrix model duals of N=1 supersymmetric SU(Nc) gauge theories with Nf flavors. We demonstrate via the matrix model solutions a relation between vacua of theories with different numbers of colors and flavors. This relation is due to an N=2 nonrenormalization theorem which is inherited by these N=1 theories. Specializing to the case Nf=Nc, the simplest theory containing baryons, we demonstrate that the explicit matrix model predictions for the locations on the Coulomb branch at which monopoles condense are consistent with the quantum modified constraints on the moduli in the theory. The matrix model solutions include the case that baryons obtain vacuum expectation values. In specific cases we check explicitly that these results are also consistent with the factorization of corresponding Seiberg-Witten curves. Certain results are easily understood in terms of M5-brane constructions of these gauge theories. 
  We review the generalized Witten-Nester spinor stability argument for flat domain wall solutions of gravitational theories. Neither the field theory nor the solution need be supersymmetric. Nor is the space-time dimension restricted. We develop the non-trivial extension required for AdS-sliced domain walls and apply this to show that the recently proposed "Janus" solution of Type IIB supergravity is stable non-perturbatively for a broad class of deformations. Generalizations of this solution to arbitrary dimension and a simple curious linear dilaton solution of Type IIB supergravity are byproducts of this work. 
  We construct the mirror of the Beauville manifold. The Beauville manifold is a Calabi-Yau manifold with non-abelian fundamental group. We use the conjecture of Batyrev and Borisov to find the previously misidentified mirror of its universal covering space, $\mathbb{P}^7[2,2,2,2]$. The monomial-divisor mirror map is essential in identifying how the fundamental group of the Beauville manifold acts on the mirror of $\mathbb{P}^7[2,2,2,2]$. Once we find the mirror of the Beauville manifold, we confirm the existence of the threshold bound state around the conifold point, which was originally conjectured in hep-th/0106262. We also consider how the quantum symmetry group acts on the D-branes that become massless at the conifold point and show the action proposed in hep-th/0102018 is compatible with mirror symmetry. 
  For the meron configuration of the SU(2) gauge field in the four dimensional Minkowskii spacetime, the decomposition into an isovector field $\bn$, isoscalar fields $\rho$ and $\sigma$, and a U(1) gauge field $C_{\mu}$ is attained by solving the consistency condition for $\bn$. The resulting $\bn$ turns out to possess two singular points, behave like a monopole-antimonopole pair and reduce to the conventional hedgehog in a special case. The $C_{\mu}$ field also possesses singular points, while $\rho$ and $\sigma$ are regular everywhere. 
  An analysis is performed of instanton configurations in pure Euclidean Yang-Mills theory containing small Lorentz-violating perturbations that maintain gauge invariance. Conventional topological arguments are used to show that the general classification of instanton solutions involving the topological charge is the same as in the standard case. Explicit solutions are constructed for general gauge invariant corrections to the action that are quadratic in the curvature. The value of the action is found to be unperturbed to lowest order in the Lorentz-violating parameters. 
  The many low energy modes near a black hole horizon give the thermal atmosphere a divergent entropy which becomes of order $A/4G$ with a Planck scale cut-off. However, Sorkin has given a Newtonian argument for 3+1 Schwarzschild black holes to the effect that fluctuations of such modes provide the horizon with a non-zero quantum mechanical width. This width then effectively enforces a cut-off at much larger distances so that the entropy of the thermal atmosphere is negligible in comparison with $A/4G$ for large black holes. We generalize and improve this result by giving a relativistic argument valid for any spherical black hole in any dimension. The result is again a cut-off $L_c$ at a geometric mean of the Planck scale and the black hole radius; in particular, $L_c^d \sim \frac{R}{T_H} \ell_p^{d-2}$. With this cut-off, the entropy of the thermal atmosphere is again parametrically small in comparison with the Bekenstein-Hawking entropy of the black hole. The effect of a large number $N$ of fundamental fields and the discrepancies from naive predictions of a stretched horizon model are also discussed. 
  We find out that baryon numbers of the matter fallen into black holes are rapidly washed-out by investigating the radiation-ball description of the black holes. The radiation-ball solution, which was derived by analyzing the backreaction of the Hawking radiation into space-time and is identified as a black hole, consists of the radiation gravitationally-trapped into the ball and of a singularity. The baryon number of the black hole is defined as that of the radiation in the ball. The sphaleron processes of the Standard Model work in the ball because the proper temperature of the radiation is Planck scale and the Higgs vev becomes zero. The decay-rate of the baryon number becomes \dot{B}/B = -alpha_W^4 / r_BH for the Schwarzschild black hole of radius r_BH. When we assume the baryon number violating processes of the GUT, we find more rapid decay-rate. We can regard the black holes as the baryon-reactors which convert the baryonic matter into energy of radiation. 
  Using the quasi-normal modes frequency of extremal Reissner-Nordstr\"om black holes, we obtain area spectrum for these type of black holes. We show that the area and entropy black hole horizon are equally spaced. Our results for the spacing of the area spectrum differ from that of schwarzschild black holes. 
  New solutions to the classical equations of motion of a bosonic matrix-membrane are given. Their continuum limit defines 3-manifolds (in Minkowski space) whose mean curvature vanishes. Part of the construction are minimal surfaces in S^7, and their discrete analogues. 
  Typical de Sitter (dS) vacua of gauged supergravity correspond to saddle points of the potential and often the unstable mode runs into a singularity. We explore the possibility to obtain dS points where the unstable mode goes on both sides into a supersymmetric smooth vacuum. Within N=2 gauged supergravity coupled to the universal hypermultiplet, we have found a potential which has two supersymmetric minima (one of them can be flat) and these are connected by a de Sitter saddle point. In order to obtain this potential by an Abelian gauging, it was important to include the recently proposed quantum corrections to the universal hypermultiplet sector. Our results apply to four as well as five dimensional gauged supergravity theories. 
  One-loop quantities in QFT can be computed in an efficient way using the worldline formalism. The latter rests on the ability of calculating 1D path integrals on the circle. In this paper we give a systematic discussion for treating zero modes on the circle of 1D path integrals for both bosonic and supersymmetric nonlinear sigma models, following an approach originally introduced by Friedan. We use BRST techniques and place a special emphasis on the issue of reparametrization invariance. Various examples are extensively analyzed to verify and test the general set-up. In particular, we explicitly check that the chiral anomaly, which can be obtained by the semiclassical approximation of a supersymmetric 1D path integral, does not receive higher order worldline contributions, as implied by supersymmetry. 
  In this paper we first show that within the Hamiltonian description of general relativity, the central charge of a near horizon asymptotic symmetry group is zero, and therefore that the entropy of the system cannot be estimated using Cardy's formula. This is done by mapping a static black hole to a two dimensional space. We explain how such a charge can only appear to a static observer who chooses to stay permanently outside the black hole. Then an alternative argument is given for the presence of a universal central charge. Finally we suggest an effective quantum theory on the horizon that is compatible with the thermodynamics behaviour of the black hole. 
  A four dimensional non-trivial extension of the Poincar\'e algebra different from supersymmetry is explicitly studied. Representation theory is investigated and an invariant Lagrangian is exhibited. Some discussion on the Noether theorem is also given. 
  Macroscopic fundamental and Dirichlet strings have several potential instabilities: breakage, tachyon decays, and confinement by axion domain walls. We investigate the conditions under which metastable strings can exist, and we find that such strings are present in many models. There are various possibilities, the most notable being a network of (p,q) strings. Cosmic strings give a potentially large window into string physics. 
  Motivated by issues in string theory and M-theory, we provide a pedestrian introduction to automorphic forms and theta series, emphasizing examples rather than generality. 
  We give a precise formulation of the M-theory 3-form potential C in a fashion applicable to topologically nontrivial situations. In our model the 3-form is related to the Chern-Simons form of an E8 gauge field. This leads to a precise version of the Chern-Simons interaction of 11-dimensional supergravity on manifolds with and without boundary. As an application of the formalism we give a formula for the electric C-field charge, as an integral cohomology class, induced by self-interactions of the 3-form and by gravity. As further applications, we identify the M-theory Chern-Simons term as a cubic refinement of a trilinear form, we clarify the physical nature of Witten's global anomaly for 5-brane partition functions, we clarify the relation of M-theory flux quantization to K-theoretic quantization of RR charge, and we indicate how the formalism can be applied to heterotic M-theory. 
  This report presents some studies of the gauge/string theory correspondence, a deep relation that is possible to establish between quantum field theories with local gauge symmetry and superstring theories including gravity. In its original version, known as AdS/CFT duality, the correspondence involves N=4 Super Yang-Mills theory in four space-time dimensions, which is a superconformal theory with a high degree of supersymmetry, thus very far from describing the physical world.   We explore extensions of the correspondence towards less supersymmetric and non-conformal gauge theories. Specifically, we study gauge theories in three and four dimensions, with eight or four preserved supersymmetries and exhibiting a scale anomaly, by means of supergravity solutions describing D-brane configurations of type II string theory. We show how relevant information on these gauge theories can be extracted from the dual classical solutions, both at the perturbative (e.g. running coupling constant, chiral anomaly) and non-perturbative level (e.g. effective superpotential). 
  Adding fundamental matter of mass m_Q to N=4 Yang Mills theory, we study quarkonium, and "generalized quarkonium" containing light adjoint particles. At large 't Hooft coupling the states of spin<=1 are anomalously light (Kruczenski et al., hep-th/0304032). We examine their form factors, and show these hadrons are unlike any known in QCD. By a traditional yardstick they appear infinite in size (as with strings in flat space) but we show that this is a failure of the yardstick. All of the hadrons are actually of finite size ~ \sqrt{g^2N}/m_Q, regardless of their radial excitation level and of how many valence adjoint particles they contain. Certain form factors for spin-1 quarkonia vanish in the large-g^2N limit; thus these hadrons resemble neither the observed J/Psi quarkonium states nor rho mesons. 
  Bulk supergravity on a manifold with boundary must be supplemented by boundary conditions that preserve local supersymmetry. This "downstairs" picture has certain advantages over the equivalent "upstairs" picture, expressed in terms of orbifolds. In particular, Scherk-Schwarz supersymmetry breaking can be described much more simply in the downstairs picture. Nevertheless, physics on the fundamental domain can always be lifted upstairs, so long as fields are allowed to be discontinuous across the boundary. In this talk we apply these considerations to five-dimensional supergravity in a warped Randall-Sundrum background. 
  We study a free particle system residing on a torus to investigate its Becci-Rouet-Stora-Tyutin symmetries associated with its St\"uckelberg coordinates, ghosts and anti-ghosts. By exploiting zeibein frame on the toric geometry, we evaluate energy spectrum of the system to describe the particle dynamics. We also investigate symplectic structures involved in the second-class system on the torus. 
  To describe a massive graviton in 4D Minkowski space-time one introduces a quadratic term in the Lagrangian. This term, however, can lead to a readjustment or instability of the background instead of describing a massive graviton on flat space. We show that for all local Lorentz-invariant mass terms Minkowski space is unstable. We start with the Pauli-Fierz (PF) term that is the only local mass term with no ghosts in the linearized approximation. We show that nonlinear completions of the PF Lagrangian give rise to instability of Minkowski space. We continue with the mass terms that are not of a PF type. Although these models are known to have ghosts in the linearized approximations, nonlinear interactions can lead to background change due to which the ghosts are eliminated. In the latter case, however, the graviton perturbations on the new background are not massive. We argue that a consistent theory of a massive graviton on flat space can be formulated in theories with extra dimensions. They require an infinite number of fields or non-local description from a 4D point of view. 
  We consider a class of \textit{factorizable} Poisson brackets which includes almost all reasonable Poisson structures. A particular case of the factorizable brackets are those associated with symplectic Lie algebroids. The BRST theory is applied to describe the geometry underlying these brackets as well as to develop a deformation quantization procedure in this particular case. This can be viewed as an extension of the Fedosov deformation quantization to a wide class of \textit{irregular} Poisson structures. In a more general case, the factorizable Poisson brackets are shown to be closely connected with the notion of $n$-algebroid. A simple description is suggested for the geometry underlying the factorizable Poisson brackets basing on construction of an odd Poisson algebra bundle equipped with an abelian connection. It is shown that the zero-curvature condition for this connection generates all the structure relations for the $n$-algebroid as well as a generalization of the Yang-Baxter equation for the symplectic structure. 
  We discuss four dimensional effective actions of string theory flux compactifications. These effective actions describe four dimensional gravity coupled to overall Kahler modulus of the compactification manifold. We demonstrate the agreement between ten dimensional equations of motion of supergravity with localized branes, and equations of motion derived from the effective action. The agreement is lost however if one evaluates the full effective action on the equations of motion for a subset of the supergravity modes, provided these modes depend on-shell on the Kahler modulus. 
  We consider the effective superpotentials of N=1 SU(N_c) and U(N_c) supersymmetric gauge theories that are obtained from the N=2 theory by adding a tree-level superpotential. We show that several of the techniques for computing the effective superpotential are implicitly regularized by 2N_c massive fundamental quarks, i.e. the theory is embedded in the finite theory with nontrivial UV fixed point. In order to study N=1 and N=2 theories with fundamentals, we explicitly factorize the Seiberg-Witten curve for N_f != 0 and compare to the known form of the N=1 superpotential. N=2 gauge theories have an underlying integrable structure, and we obtain results on a new Lax matrix for N_f = N_c. 
  We have studied the most general N=2 supergravity in five dimensions in context with the orbifold theory based on $M_4 \times S^1/Z_2$. Various ways to treat the supersymmetry with singular sources placed in orbifold fixed points were proposed in past. Supersymmetric branes were consistently introduced in a bulk where a gauged supergravity was present. In this paper we find that in the $N=2,D=5$ supergravity with general gauging, the possibility to obtain a supersymmetric brane world is constrained. Imposing the compatibility of supersymmetry transformation rules with the orbifold condition, we find the necessary and sufficient condition to obtain supersymmetric branes and bulk independently. We comment that the same condition guarantees naturally the presence of singular BPS solutions. 
  This paper has been withdrawn since the material in these notes has been updated, and extended, in hep-th/0409031 and hep-th/0409033, and in work to follow. 
  The IR/UV mixing in the non-commutative (NC) field theory is investigated in Carlson-Carone-Zobin (CCZ) formalism of Lorentz-invariant NC field theory provided that the fields are `independent' of the `internal' coordinates $\theta^{\mu\nu}$. A new regularization scheme called NC regularizatioon is then proposed, which removes the Lorentz-invariant IR singularity from the theory. It requires the usual UV limit $\Lambda\to \infty$ to be accompanied with the commutative limit $a\to 0$ with $\Lambda^2a^2$ fixed, where $a$ is the length parameter in the theory. The new UV limit gives the usual renormalized amplitude of the one-loop self-energy diagram of $\phi^3$ model. It is shown that the new regularization is gauge-invariant, that is, the non-transverse part of the vacuum polarization in QED is automatically transverse in Lorentz-invariant NCQED but the two transverse pieces, one of which is already transverse in QED, possesses Lorentz-invariant IR singularity which should be `subtracted off' at zero external momentum squared. The subtraction leads to the same result as the renormalized one by Pauli-Villars or dimensional regularizations. Other diagrams with three-point vertices which contribute to the photon self-energy in Lorentz-non-invariant NCQED all vanish due to Lorentz invariance under the assumption adopted, while the tadpole diagram gives a finite contribution to the charge renormalization which vanishes if $ Lambda^2a^2\to 0$. Lorentz-invariant NC $\phi^4$ and scalar Yukawa models are also discussed in the one-loop approximation. A comment is made that Lorentz-invariance might lead to a decoupling of U(1) part from SU(N) in NC U(N) gauge theory. 
  In this paper we study the tachyonic inflation in brane world cosmology with Gauss-Bonnet term in the bulk. We obtain the exact solution of slow roll equations in case of exponential potential. We attempt to implement the proposal of Lidsey and Nunes, astro-ph/0303168, for the tachyon condensate rolling on the Gauss-Bonnet brane and discuss the difficulties associated with the proposal. 
  Recent many physicists suggest that the dark energy in the universe might result from the Born-Infeld(B-I) type scalar field of string theory. The universe of B-I type scalar field with potential can undergo a phase of accelerating expansion. The corresponding equation of state parameter lies in the range of $\displaystyle -1<\omega<-{1/3}$. The equation of state parameter of B-I type scalar field without potential lies in the range of $0\leq\omega\leq1$. We find that weak energy condition and strong energy condition are violated for phantom B-I type scalar field. The equation of state parameter lies in the range of $\omega<-1$. 
  We study the Euclidean supersymmetric D=11 M-algebras. We consider two such D=11 superalgebras: the first one is N=(1,1) self-conjugate complex-Hermitean, with 32 complex supercharges and 1024 real bosonic charges, the second is N=(1,0) complex-holomorphic, with 32 complex supercharges and 528 bosonic charges, which can be obtained by analytic continuation of known Minkowski M-algebra. Due to the Bott's periodicity, we study at first the generic D=3 Euclidean supersymmetry case. The role of complex and quaternionic structures for D=3 and D=11 Euclidean supersymmetry is elucidated. We show that the additional 1024-528=496 Euclidean tensorial central charges are related with the quaternionic structure of Euclidean D=11 supercharges, which in complex notation satisfy SU(2) pseudo-Majorana condition. We consider also the corresponding Osterwalder-Schrader conjugations as implying for N=(1,0) case the reality of Euclidean bosonic charges. Finally, we outline some consequences of our results, in particular for D=11 Euclidean supergravity. 
  In analogy to the harmonic analysis for the Poincar\'e group with its irreducible representations characterizing free particles, the harmonic analysis for a nonlinear spacetime model as homogenous space of the extended Lorentz group GL(C^2) is given. What the Dirac energy-momentum measures are for free particles, are multipole measures in the analysis of nonlinear spacetime - they are related to spacetime interactions. The representations induced from the nonlinear spacetime fixgroup U(2) connect representations of external (spacetimelike) degrees of freedom with those of internal (hypercharge-isospinlike) ones as seen in the standard model of electroweak and strong interactions. The methods used are introduced and exemplified with the nonrelativistic Kepler dynamics in an interpretation as harmonic analysis of time and position functions. 
  We consider the topological B-model on local Calabi-Yau geometries. We show how one can solve for the amplitudes by using W-algebra symmetries which encodes the symmetries of holomorphic diffeomorphisms of the Calabi-Yau. In the highly effective fermionic/brane formulation this leads to a free fermion description of the amplitudes. Furthermore we argue that topological strings on Calabi-Yau geometries provide a unifying picture connecting non-critical (super)strings, integrable hierarchies, and various matrix models. In particular we show how the ordinary matrix model, the double scaling limit of matrix models, and Kontsevich-like matrix model are all related and arise from studying branes in specific local Calabi-Yau three-folds. We also show how A-model topological string on P^1 and local toric threefolds (and in particular the topological vertex) can be realized and solved as B-model topological string amplitudes on a Calabi-Yau manifold. 
  In this paper we will study the quantum field theory of fluctuation modes around the classical solution that describes tachyon condensation on unstable D-brane.We will calculate the number of particle produced near the beginning of the rolling tachyon process. We will perform this calculation for different tachyon effective actions and we will find that the rate of the particle production strongly depends on the form of the effective action used for the description of the early stage of the tachyon condensation. 
  We investigate scalar perturbations from inflation in a bulk inflaton braneworld model. Using the generalized longitudinal gauge, we derive and solve the full set of scalar perturbation equations. Our exact results support the recent argument that for the de Sitter brane the square of the radion mass is not positive, showing that unlike the flat brane case, the de Sitter brane is not stable. 
  To obtain the one loop effective action for a given superfield theory, one encounters the notion such as the `supertrace' of a differential operator on superspace. We develop, in a systematic way for the superspace of arbitrary dimension, a method to determine the supertrace precisely. We present a formula to express the supertrace explicitly as the superspace integral, which enables us to write the one loop effective action within the superfield formalism and still maintain the manifest supersymmetry. In the second part of the paper, we apply the result to a three dimensional N=1 supersymmetric CP(N) model in the auxiliary superfield formalism. The model contains a novel topological interaction term. We show in the large N limit the one loop effective action is given by the supersymmetric Maxwell-Chern-Simons theory. 
  Massive vector fields can be described in a gauge invariant way with the introduction of compensating fields. In the unitary gauge one recovers the original formulation. Although this gauging mechanism can be extended to noncommutative spaces in a straightforward way, non trivial aspects show up when we consider the Seiberg-Witten map. As we show here, only a particular class of its solutions leads to an action that admits the unitary gauge fixing. 
  We calculate the topological string amplitudes of Calabi-Yau toric threefolds corresponding to 4D, N=2, SU(2) gauge theory with N_f=0,1,2,3,4 fundamental hypermultiplets by using the method of the geometric transition and show that they reproduce Nekrasov's formulas for instanton counting. We also determine the asymptotic forms of the Gopakumar-Vafa invariants of the Calabi-Yau threefolds including those at higher genera from instanton amplitudes of the gauge theory. 
  We find classical open string solutions in the $AdS_5\times S^5/\Zop_2$ orientifold with angular momenta along the five-sphere. The energy of these solutions has an expansion in integral powers of $\lambda$ with sigma-model corrections suppressed by inverse powers of $J$ - the total angular momentum. This gives a prediction for the exact anomalous dimensions of operators in the large $N$ limit of an ${\cal N}=2$ $Sp(N)$ Super-Yang-Mills theory with matter. We also find a simple map between open and closed string solutions. This gives a prediction for an all-loop planar relationship between the anomalous dimensions of single-trace and two-quark operators in the dual gauge theory. 
  Positive frequency Wightman function and vacuum expectation value of the energy-momentum tensor are computed for a massive scalar field with general curvature coupling parameter subject to Robin boundary conditions on two parallel plates located on $D+1$ - dimensional AdS background. The general case of different Robin coefficients on separate plates is considered. The mode summation method is used with a combination of a variant of the generalized Abel-Plana formula for the series over zeros of combinations of cylinder functions. This allows us to extract manifestly the parts due to the AdS spacetime without boundaries and boundary induced parts. The asymptotic behavior of the vacuum densities near the plates and at large distances is investigated. The vacuum forces acting on the boundaries are presented as a sum of the self-action and interaction forces. The first one contains well-known surface divergences and needs further regularization. The interaction forces between the plates are attractive for Dirichlet scalar. We show that threre is a region in the space of parameters defining the boundary conditions in which the interaction forces are repulsive for small distances and attractive for large distances. An application to the Randall-Sundrum braneworld with arbitrary mass terms on the branes is discussed. 
  We obtain background independent solutions for an open string ending on D-brane, in variable external fields. Explicit solution of the boundary conditions is given for background metric and NS-NS two-form gauge field, depending on the coordinates of the transverse to the Dp-brane directions. Extension of the constraint algebra is proposed and discussed from both Hamiltonian and Lagrangian approach viewpoint. 
  We study the superstrings suspended between a D2- and an anti-D2-brane. We quantize the string in the presence of some general configuration of gauge fields over the (anti-)D-brane world volumes. The interstring can move only in a specific direction that is normal to the difference of the electric fields of each (anti-)D-branes. Especially when the electric fields are the same, the interstring cannot move. We obtain the condition for the tachyons to disappear from the spectrum. 
  In this contribution we will review briefly the supersymmetric Lagrangian approach to the supergravity-superbrane interaction which was developed in collaboration with J. A. de Azcarraga, J.M. Izquierdo, J. Lukierski and J. M. Isidro. The main accent will be made on the pure gauge nature of the (super)brane coordinate functions in the presence of dynamical (super)gravity described by an action rather than as a fixed background. This pure gauge nature just reflects the fact that the coordinate functions are Goldstone fields corresponding to the spontaneously broken diffeomorphism gauge symmetry of the interacting system. Moreover, a brane does not carry any local degrees of freedom in such an interacting system. This fact related with fundamental properties of General Relativity (discussed already at 1916) can be treated as a peculiarity of the spacetime Higgs effect which occurs in General Relativity in the presence of material particles, strings and branes. 
  In view of the newly conjectured Kac-Moody symmetries of supergravity theories placed in eleven and ten dimensions, the relation between these symmetry groups and possible compactifications are examined. In particular, we identify the relevant group cosets that parametrise the vacuum solutions of AdS x S type. 
  We consider the effect of string inhomogeneities on the time dependent background of Brane Gas Cosmology. We derive the equations governing the linear perturbations of the dilaton-gravity background in the presence of string matter sources. We focus on long wavelength fluctuations and find that there are no instabilities. Thus, the predictions of Brane Gas Cosmology are robust against the introduction of linear perturbations. In particular, we find that the stabilization of the extra dimensions (moduli) remains valid in the presence of dilaton and string perturbations. 
  In four dimensional N=1 supersymmetric field theory it is often the case that the $U(1)_R$ current that becomes part of the superconformal algebra at the infrared fixed point is conserved throughout the renormalization group (RG) flow. We show that when that happens, the central charge $a$ decreases under RG flow. The main tool we employ is an extension of recent ideas on ``$a$-maximization'' away from fixed points of the RG. This extension is useful more generally in studying RG flows in supersymmetric theories. 
  We propose a theoretically consistent modification of gravity in the infrared, which is compatible with all current experimental observations. This is an analog of Higgs mechanism in general relativity, and can be thought of as arising from ghost condensation--a background where a scalar field \phi has a constant velocity, <\dot\phi> = M^2. The ghost condensate is a new kind of fluid that can fill the universe, which has the same equation of state, \rho = -p, as a cosmological constant, and can hence drive de Sitter expansion of the universe. However, unlike a cosmological constant, it is a physical fluid with a physical scalar excitation, which can be described by a systematic effective field theory at low energies. The excitation has an unusual low-energy dispersion relation \omega^2 \sim k^4 / M^2. If coupled to matter directly, it gives rise to small Lorentz-violating effects and a new long-range 1/r^2 spin dependent force. In the ghost condensate, the energy that gravitates is not the same as the particle physics energy, leading to the possibility of both sources that can gravitate and antigravitate. The Newtonian potential is modified with an oscillatory behavior starting at the distance scale M_{Pl}/M^2 and the time scale M_{Pl}^2/M^3. This theory opens up a number of new avenues for attacking cosmological problems, including inflation, dark matter and dark energy. 
  We propose a new scenario for early cosmology, where an inflationary de Sitter phase is obtained with a ghost condensate. The transition to radiation dominance is triggered by the ghost itself, without any slow-roll potential. Density perturbations are generated by fluctuations around the ghost condensate and can be reliably computed in the effective field theory. The fluctuations are scale invariant as a consequence of the de Sitter symmetries, however, the size of the perturbations are parametrically different from conventional slow-roll inflation, and the inflation happens at far lower energy scales. The model makes definite predictions that distinguish it from standard inflation, and can be sharply excluded or confirmed by experiments in the near future. The tilt in the scalar spectrum is predicted to vanish (n_s=1), and the gravity wave signal is negligible. The non-Gaussianities in the spectrum are predicted to be observable: the 3-point function is determined up to an overall order 1 constant, and its magnitude is much bigger than in conventional inflation, with an equivalent f_NL ~ 100, not far from the present WMAP bounds. 
  The class of effectively closed infinite-genus surfaces, defining the completion of the domain of string perturbation theory, can be included in the category $O_G$, which is characterized by the vanishing capacity of the ideal boundary. The cardinality of the maximal set of endpoints is shown to be $2^{\mit N}$. The product of the coefficient of the genus-g superstring amplitude in four dimensions by $2^g$ in the $g\to \infty$ limit is an exponential function of the genus with a base comparable in magnitude to the unified gauge coupling. The value of the string coupling is consistent with the characteristics of configurations which provide a dominant contribution to a finite vacuum amplitude. 
  We investigate the accelerating phases of cosmologies supported by a metric, scalars and a single exponential scalar potential. The different solutions can be represented by trajectories on a sphere and we find that quintessence happens within the "arctic circle" of the sphere.   Furthermore, we obtain multi-exponential potentials from 3D group manifold reductions of gravity, implying that such potentials can be embedded in gauged supergravities with an M-theory origin. We relate the double exponential case to flux compactifications on maximally symmetric spaces and S-branes. In the triple exponential case our analysis suggests the existence of two exotic S(D-3)-branes in D dimensions. 
  We derive maps relating the currents and energy-momentum tensors in noncommutative (NC) gauge theories with their commutative equivalents. Some uses of these maps are discussed. Especially, in NC electrodynamics, we obtain a generalization of the Lorentz force law. Also, the same map for anomalous currents relates the Adler-Bell-Jackiw type NC covariant anomaly with the standard commutative-theory anomaly. For the particular case of two dimensions, we discuss the implications of these maps for the Sugawara-type energy-momentum tensor. 
  The presence of RR and NS three-form fluxes in type IIB string compactification on a Calabi-Yau orientifold gives rise to a nontrivial superpotential W for the dilaton and complex structure moduli. This superpotential is computable in terms of the period integrals of the Calabi-Yau manifold. In this paper, we present explicit examples of both supersymmetric and nonsupersymmetric solutions to the resulting 4d N=1 supersymmetric no-scale supergravity, including some nonsupersymmetric solutions with relatively small values of W. Our examples arise on orientifolds of the hypersurfaces in $WP^{4}_{1,1,1,1,4}$ and $WP^{4}_{1,1,2,2,6}$. They serve as explicit illustrations of several of the ingredients which have played a role in the recent proposals for constructing de Sitter vacua of string theory. 
  We examined analytically a cosmological black hole domain wall system. Using the C-metric construction we derived the metric for the spacetime describing an infinitely thin domain wall intersecting a cosmological black hole. We studied the behaviour of the scalar field describing a self-interacting cosmological domain wall and find the approximated solution valid for large distances. The thin wall approximation and the back raection problem were elaborated finding that the topological kink solution smoothed out singular behaviour of the zero thickness wall using a core topological and hence thick domain wall. We also analyze the nucleation of cosmological black holes on and in the presence of a domain walls and conclude that the domain wall will nucleate small black holes on it rather than large ones inside. 
  Evaluating Kaluza-Klein (KK) corrections is indispensable to test the braneworld scenario. In this paper, we propose a novel symmetry approach to a 4-dimensional effective action with KK corrections for the Randall-Sundrum two-brane system. The result can be used to assess the validity of the low energy approximation. Also, our result provides the basis for predicting CMB spectrum with KK corrections and the study of the transition from black strings to black holes. 
  It is shown that both the electric and magnetic dipole moment vectors of hydrogen atom in the excited states with wave function $$ u_n^{(\pm)} = {1\over\sqrt 2} [R_{n,n-1}(r) Y_{n-1,\pm (n-2)}(\theta\phi) \pm R_{n,n-2}(r) Y_{n-2,\pm (n-2)}(\theta\phi)]$$ align themselves in the direction of an external uniform electric field which is characteristic of magneto-electric effect. These states are found to have magnetic charge $g={3n\over (n-2)e}$ on account of this effect. This result is confirmed by an independent method. An experiment is suggested to fabricate these states and detect the magnetic charge. It ma be worth noting that inspite of many experimental searchs, magnetic charge, whose existence has been theorized both in electrodynamics and non-abelian gauge theories, none have been found so far, nor there exist any suggstion as to where these are to be found. 
  After the short survey of the Klein Paradox in 3-dimensional relativistic equations, we present a detailed consideration of Dirac modified equation, which follows by one particle infinite overweighting in Salpeter Equation. It is shown, that the separation of angular variables and reduction to radial equation is possible by using standard methods in momentum space. The kernel of the obtained radial equation differs from that of spinless Salpeter equation in bounded regular factor. That is why the equation has solutions of confined type for infinitely increasing potential. 
  Cohomology spaces of the Poisson superalgebra realized on smooth Grassmann-valued functions with compact support on $R^{2n}$ ($C^{2n}) are investigated under suitable continuity restrictions on cochains. The first and second cohomology spaces in the trivial representation and the zeroth and first cohomology spaces in the adjoint representation of the Poisson superalgebra are found for the case of a constant nondegenerate Poisson superbracket for arbitrary n>0. The third cohomology space in the trivial representation and the second cohomology space in the adjoint representation of this superalgebra are found for arbitrary n>1. 
  We analyse the orbital motion of a light anti D6-brane in the presence of a stack of heavy, distant D6-branes in ten dimensions, taking account of possible time-variations in the background moduli fields. The Coulomb-like central potential arising through brane-antibrane interactions is then modified to include time-dependent prefactors, which generally preclude the existence of stable elliptical orbits. 
  We present the first examples of cosmological solutions to four-dimensional heterotic models which include an evolving bundle modulus. The particular bundle modulus we consider corresponds to the width of a gauge five brane. As such our solutions can be used to describe the evolution in one of these models after a small instanton transition. We find that certain properties are generic to these solutions, regardless of initial conditions. This enables us to make some definite statements about the dynamics subsequent to a small instanton transition despite the fact that we cannot microscopically describe the process itself. We also show that an effective description of the small instanton transition by a continuous matching of fields and their first derivatives is precluded by the form of the respective low-energy theories before and after the transition. 
  I propose a way of unambiguously parallel transporting fields on non-Abelian flux tubes, or strings, by means of two gauge fields. One gauge field transports along the tube, while the other transports normal to the tube. Ambiguity is removed by imposing an integrability condition on the pair of fields. The construction leads to a gauge theory of mathematical objects known as Lie 2-groups, which are known to result also from the parallel transport of the flux tubes themselves. The integrability condition is also shown to be equivalent to the assumption that parallel transport along nearby string configurations are equal up to arbitrary gauge transformations. Attempts to implement this condition in a field theory leads to effective actions for two-form fields. 
  We study rotating strings with multiple spins in the background of $AdS_5\times T^{1,1}$, which is dual to a $\cN=1$ superconformal field theory with global symmetry $SU(2)\times SU(2)\times U(1)$ via the AdS/CFT correspondence. We analyse the limiting behaviour of macroscopic strings and discuss the identification of the dual operators and how their anomalous dimensions should behave as the global charges vary. A class of string solutions we find are dual to operators in SU(2) subsector, and our result implies that the one-loop planar dilatation operator restricted to the SU(2) subsector should be equivalent to the hamiltonian of the integrable Heisenberg spin chain. 
  Euclidean quantum gravity is studied with renormalisation group methods. Analytical results for a non-trivial ultraviolet fixed point are found for arbitrary dimensions and gauge fixing parameter in the Einstein-Hilbert truncation. Implications for quantum gravity in four dimensions are discussed. 
  In hep-th/0310120, Goheer, Kleban and Susskind argued that the holographic principle is inconsistent with the existence of stable, Lorentz invariant, 1+1 dimensional compactifications. We note some difficulties with their analysis and present two novel backgrounds of string theory with 1+1 noncompact dimensions that satisfy their conditions yet possess 16 or 24 supersymmetries. It is difficult to believe that such backgrounds could be unstable. 
  The Euler-Heisenberg effective action in a self-dual background is remarkably simple at two-loop. This simplicity is due to the inter-relationship between self-duality, helicity and supersymmetry. Applications include two-loop helicity amplitudes, beta-functions and nonperturbative effects. The two-loop Euler-Heisenberg effective Lagrangian for QED in a self-dual background field is naturally expressed in terms of one-loop quantities. This mirrors similar behavior recently found in two-loop amplitudes in N=4 SUSY Yang-Mills theory. 
  At present no theory of a massive graviton is known that is consistent with experiments at both long and short distances. The problem is that consistency with long distance experiments requires the graviton mass to be very small. Such a small graviton mass however implies an ultraviolet cutoff for the theory at length scales far larger than the millimeter scale at which gravity has already been measured. In this paper we attempt to construct a model which avoids this problem. We consider a brane world setup in warped AdS spacetime and we investigate the consequences of writing a mass term for the graviton on a the infrared brane where the local cutoff is of order a large (galactic) distance scale. The advantage of this setup is that the low cutoff for physics on the infrared brane does not significantly affect the predictivity of the theory for observers localized on the ultraviolet brane. For such observers the predictions of this theory agree with general relativity at distances smaller than the infrared scale but go over to those of a theory of massive gravity at longer distances. A careful analysis of the graviton two-point function, however, reveals the presence of a ghost in the low energy spectrum. A mode decomposition of the higher dimensional theory reveals that the ghost corresponds to the radion field. We also investigate the theory with a brane localized mass for the graviton on the ultraviolet brane, and show that the physics of this case is similar to that of a conventional four dimensional theory with a massive graviton, but with one important difference: when the infrared brane decouples and the would-be massive graviton gets heavier than the regular Kaluza--Klein modes, it becomes unstable and it has a finite width to decay off the brane into the continuum of Kaluza-Klein states. 
  The rules for superfield Lagrangian quantization method for general gauge theories on a basis of their generalization to special superfield models within a so-called $\theta$-superfield theory of fields ($\theta$-STF) are formulated. The $\theta$-superfield generating functionals of Green's functions together with effective action are constructed. Their properties including new interpretation and superfield realization of BRST transformations, Ward identities are studied. 
  The two-dimensional Ising model is the simplest model of statistical mechanics exhibiting a second order phase transition. While in absence of magnetic field it is known to be solvable on the lattice since Onsager's work of the forties, exact results for the magnetic case have been missing until the late eighties, when A.Zamolodchikov solved the model in a field at the critical temperature, directly in the scaling limit, within the framework of integrable quantum field theory. In this article we review this field theoretical approach to the Ising universality class, with particular attention to the results obtained starting from Zamolodchikov's scattering solution and to their comparison with the numerical estimates on the lattice. The topics discussed include scattering theory, form factors, correlation functions, universal amplitude ratios and perturbations around integrable directions. Although we restrict our discussion to the Ising model, the emphasis is on the general methods of integrable quantum field theory which can be used in the study of all universality classes of critical behaviour in two dimensions. 
  We study topological aspects of matrix models and noncommutative cohomological field theories (N.C.CohFT). N.C.CohFT have symmetry under the arbitrary infinitesimal noncommutative parameter $\theta$ deformation. This fact implies that N.C.CohFT possess a less sensitive topological property than K-theory, but the classification of manifolds by N.C.CohFT has a possibility to give a new view point of global characterization of noncommutative manifolds. To investigate properties of N.C.CohFT, we construct some models whose fixed point loci are given by sets of projection operators. Particularly, the partition function on the Moyal plane is calculated by using a matrix model. The moduli space of the matrix model is a union of Grassman manifolds. The partition function of the matrix model is calculated using the Euler number of the Grassman manifold. Identifying the N.C.CohFT with the matrix model, we get the partition function of the N.C.CohFT. To check the independence of the noncommutative parameters, we also study the moduli space in the large $\theta$ limit and the finite $\theta$, for the Moyal plane case. If the partition function of N.C.CohFT is topological in the sense of the noncommutative geometry, then it should have some relation with K-theory. Therefore we investigate certain models of CohFT and N.C.CohFT from the point of view of K-theory. These observations give us an analogy between CohFT and N.C.CohFT in connection with K-theory. Furthermore, we verify it for the Moyal plane and noncommutative torus cases that our partition functions are invariant under the those deformations which do not change the K-theory. Finally, we discuss the noncommutative cohomological Yang-Mills theory. 
  Coloring planar Feynman diagrams in spinor quantum electrodynamics, is a non trivial model soluble without computer. Four colors are necessary and sufficient. 
  We show that the boundary states satisfy a nonlinear relation (the idempotency equation) with respect to the star product of closed string field theory. This relation is universal in the sense that various D-branes, including the infinitesimally deformed ones, satisfy the same equation, including the coefficient. This paper generalizes our analysis (hep-th/0306189) in the following senses. (1) We present a background-independent formulation based on conformal field theory. It illuminates the geometric nature of the relation and allows us to more systematically analyze the variations around the D-brane background. (2) We show that the Witten-type star product satisfies a similar relation but with a more divergent coefficient. (3) We determine the coefficient of the relation analytically. The result shows that the alpha parameter can be formally factored out, and the relation becomes universal. We present a conjecture on vacuum theory based on this computation. 
  A brief review is given of three recent results concerning classical solutions of gravitational theories: (1) With asymptotically anti de Sitter boundary conditions, there are matter theories satisfying the positive energy theorem which violate cosmic censorship. (2) Despite supersymmetry, there are solutions to N=8 supergravity in which the total gravitational energy is arbitrarily negative. This theory can also violate cosmic censorship. (3) A large class of supersymmetric compactifications (including all simply connected Calabi-Yau manifolds) have solutions with negative four dimensional effective energy density. 
  We present a class of exact vacuum solutions corresponding to de Sitter and warm inflation models in the framework of scalar-tensor cosmologies. We show that in both cases the field equations reduce to planar dynamical systems with constraints. Then, we carry out a qualitative analysis of the models by examining the phase diagrams of the solutions near the equilibrium points. 
  Starting from the previously constructed effective supergravity theory below the scale of U(1) breaking in orbifold compactifications of the weakly coupled heterotic string, we study the effective theory below the scale of supersymmetry breaking by gaugino and matter condensation in a hidden sector. Issues we address include vacuum stability, soft supersymmetry-breaking masses in the observable sector, and the masses of the various moduli fields, including those associated with flat directions at the U(1)-breaking scale, and of their fermionic superpartners. The consistent treatment of U(1) breaking together with condensation yields qualitatively new results. 
  We derive the effective potentials for composite operators in a Nambu-Jona-Lasinio (NJL) model at zero and finite temperature and show that in each case they are equivalent to the corresponding effective potentials based on an auxiliary scalar field. The both effective potentials could lead to the same possible spontaneous breaking and restoration of symmetries including chiral symmetry if the momentum cutoff in the loop integrals is large enough, and can be transformed to each other when the Schwinger-Dyson (SD) equation of the dynamical fermion mass from the fermion-antifermion vacuum (or thermal) condensates is used. The results also generally indicate that two effective potentials with the same single order parameter but rather different mathematical expressions can still be considered physically equivalent if the SD equation corresponding to the extreme value conditions of the two potentials have the same form. 
  Nonlinear supersymmetric(NLSUSY) general relativity(GR) is considered and a new fundamental action of the vacuum Einstein-Hilbert(EH)-type is obtained by the Einstein gravity analogue geomtrical arguments on new spacetime inspired by NLSUSY. The new action of NLSUSY GR is unstable and breaks spontaneously to EH action with matter(or New spacetime is unstable and induces spontaneously the phase transition to Riemann spacetime with matter), which may be the real shape of the big bang of the universe and give naturally the unified description of spacetime and matter. Some implications for the low energy particle physics and the cosmology are discussed. 
  We work out the phase-space functional integral of the gravitational field in 2+1 dimensions interacting with N point particles in an open universe. 
  It is shown that the SO(3) isometries of the Euclidean Taub-NUT space combine a linear three-dimensional representation with one induced by a SO(2) subgroup, giving the transformation law of the fourth coordinate under rotations. This explains the special form of the angular momentum operator on this manifold which leads to a new type of spherical harmonics and spinors. 
  We demonstrate that neutral Dirac particles in external electric fields, which are equivalent to generalized Dirac oscillators, are physical examples of quasi-exactly solvable systems. Electric field configurations permitting quasi-exact solvability of the system based on the $sl(2)$ symmetry are discussed separately in spherical, cylindrical, and Cartesian coordinates. Some exactly solvable field configurations are also exhibited. 
  We begin with a brief introduction on N=1 gauge theories, focusing on the importance of the effective superpotential in light of the new techniques to compute it systematically. We then proceed to consider theories for which the Konishi anomaly proves to be enough to solve exactly for the effective superpotential. As an example we study a chiral SO(10) gauge theory, where we also discuss the occurrence of dynamical supersymmetry breaking. 
  The analytical bounce solution is derived in terms of the polygamma function in the Caldeira-Leggett's dissipative quantum tunneling model. The classical action for the bounce solution lies between the upper and lower bounds in the full range of $\alpha$, where $\alpha$ is a dissipation coefficient. The bounce peak point increases from 1 to 4/3 with increase of $\alpha$. In spite of various nice features we have shown that the solution we have derived is not exact one by making use of the zero mode argument in the linearized fluctuation equation. However, our solution can be a starting point for approximate computation of the prefactor in this model. 
  Glueballs have a natural interpretation as closed strings in Yang-Mills theory. Their stability requires that the string carries a nontrivial twist, or then it is knotted. Since a twist can be either left-handed or right-handed, this implies that the glueball spectrum must be degenerate. This degeneracy becomes consistent with experimental observations, when we identify the $\eta_L(1410)$ component of the $\eta(1440)$ pseudoscalar as a $0^{-+}$ glueball, degenerate in mass with the widely accepted $0^{++}$ glueball $f_0(1500)$. In addition of qualitative similarities, we find that these two states also share quantitative similarity in terms of equal production ratios, which we view as further evidence that their structures must be very similar. We explain how our string picture of glueballs can be obtained from Yang-Mills theory, by employing a decomposed gauge field. We also consider various experimental consequences of our proposal, including the interactions between glueballs and quarks and the possibility to employ glueballs as probes for extra dimensions: The coupling of strong interactions to higher dimensions seems to imply that absolute color confinement becomes lost. 
  We discuss some consequences of the fact that symmetry groups appearing in compactified (super-)gravity may be non-simply connected. The possibility to add fermions to a theory results in a simple criterion to decide whether a 3-dimensional coset sigma model can be interpreted as a dimensional reduction of a higher dimensional theory. Similar criteria exist for higher dimensional sigma models, though less decisive. Careful examination of the topology of symmetry groups rules out certain proposals for M-theory symmetries, which are not ruled out at the level of the algebra's. We conclude with an observation on the relation between the ``generalized holonomy'' proposal, and the actual symmetry groups resulting from E_10 and E_11 conjectures. 
  We study string interactions in the fermionic formulation of the c=1 matrix model. We give a precise nonperturbative description of the rolling tachyon state in the matrix model, and discuss S-matrix elements of the c=1 string. As a first step to study string interactions, we compute the interaction of two decaying D0-branes in terms of free fermions. This computation is compared with the string theory cylinder diagram using the rolling tachyon ZZ boundary states. 
  The quantum correction to the entanglement entropy of the event horizon is plagued by the UV divergence due to the infinitely blue-shifted near horizon modes. The resolution of this UV divergence provides an excellent window to a better understanding and control of the quantum gravity effects. We claim that the key to resolve this UV puzzle is the transplanckian dispersion relation. We calculate the entanglement entropy using a very general type of transplanckian dispersion relation such that high energy modes above a certain scale are cutoff, and show that the entropy is rendered UV finite. We argue that modified dispersion relation is a generic feature of string theory, and this boundedness nature of the dispersion relation is a general consequence of the existence of a minimal distance in string theory. 
  We review the supergravity derivation of some non-perturbatively generated effective superpotentials for N=1 gauge theories. Specifically, we derive the Veneziano-Yankielowicz superpotential for pure N=1 Super Yang-Mills theory from the warped deformed conifold solution, and the Affleck-Dine-Seiberg superpotential for N=1 SQCD from a solution describing fractional D3-branes on a C^3 / Z_2 x Z_2 orbifold. 
  Open Wilson lines are known to be the observables of noncommutative gauge theory with Moyal-Weyl star product. We generalize these objects to more general star products. As an application we derive a formula for the inverse Seiberg-Witten map for star products with invertible Poisson structures. 
  We construct a string-inspired nonsingular cosmological scenario in which an inflaton field is driven up the potential before the graceful exit by employing a low-energy string effective action with an orbifold compactification. This sets up an initial condition for the inflaton to lead to a sufficient amount of $e$-foldings and to generate a nearly scale-invariant primordial density perturbation during slow-roll inflation. Our scenario provides an interesting possibility to explain a suppressed power spectrum at low multipoles due to the presence of the modulus-driven phase prior to slow-roll inflation and thus can leave a strong imprint of extra dimensions in observed CMB anisotropies. 
  I review the conceptual, algebraical, and geometrical structure of Doubly Special Relativity. I also speculate about the possible relevance of DSR for quantum gravity phenomenology. 
  We compute the finite size effects in the ground state energy, equivalently the effective central charge c_{eff}, based on S-matrix theories recently conjectured to describe a cyclic regime of the Kosterlitz-Thouless renormalization group flows. The effective central charge has periodic properties consistent with renormalization group predictions. Whereas c_{eff} for the massive case has a singularity in the very deep ultra-violet, we argue that the massless version is non-singular and periodic on all length scales. 
  We extend a path-integral approach to bosonization previously developed in the framework of equilibrium Quantum Field Theories, to the case in which time-dependent interactions are taken into account. In particular we consider a non covariant version of the Thirring model in the presence of a dynamic barrier at zero temperature. By using the Closed Time Path (Schwinger-Keldysh) formalism, we compute the Green's function and the Total Energy Density of the system. Since our model contains the Tomonaga Luttinger model as a particular case, we make contact with recent results on non-equilibrium electronic systems. 
  An interacting scalar field theory in de Sitter space is non-renormalizable for a generic alpha-vacuum state. This pathology arises since the usual propagator used allows for a constructive interference among propagators in loop corrections, which produces divergences that are not proportional to standard counterterms. This interference can be avoided by defining a new propagator for the alpha-vacuum based on a generalized time-ordering prescription. The generating functional associated with this propagator contains a source that couples to the field both at a point and at its antipode. To one loop order, we show that a set of theories with very general antipodal interactions is causal and renormalizable. 
  A free scalar field propagating in de Sitter space has a one parameter family of invariant states called the alpha-vacua. In an interacting theory, all except a unique state, the Bunch-Davies vacuum, produce non-renormalizable divergences. This talk provides a brief introduction to the origin and the form of these divergences. 
  We demonsrate that the spectral curve of the matrix model for Chern-Simons theory on the Lens space S^{3}/\ZZ_p is precisely the Riemann surface which appears in the mirror to the blownup, orbifolded conifold. This provides the first check of the $A$-model large $N$ duality for T^{*}(S^{3}/\ZZ_p), p>2. 
  We model the QCD Dirac operator as a power-law random banded matrix (RBM) with the appropriate chiral symmetry. Our motivation is the form of the Dirac operator in a basis of instantonic zero modes with a corresponding gauge background of instantons. We compare the spectral correlations of this model to those of an instanton liquid model (ILM) and find agreement well beyond the Thouless energy. In the bulk of the spectrum the (dimensionless) Thouless energy of the RBM scales with the square root of system size in agreement with the ILM and chiral perturbation theory. Near the origin the scaling of the (dimensionless) Thouless energy in the RBM remains the same as in the bulk which agrees with chiral perturbation theory but not with the ILM. Finally we discuss how this RBM should be modified in order to describe the spectral correlations of the QCD Dirac operator at the finite temperature chiral restoration transition. 
  First, we review a result in our previous paper, of how a ten-dimensional superparticle, taken off-shell, has a hidden eleven-dimensional superPoincare symmetry. Then, we show that the physical sector is defined by three first-class constraints which preserve the full eleven-dimensional symmetry. Applying the same concepts to the eleven dimensional superparticle, taken off-shell, we discover a hidden twelve dimensional superPoincare symmetry that governs the theory. 
  We propose a simple model of extra-dimensional radius stabilization in a supersymmetric Randall-Sundrum model. In our model, we introduce only a bulk hypermultiplet and source terms (tadpole terms) on each boundary branes. With appropriate choice of model parameters, we find that the radius can be stabilized by supersymmetric vacuum conditions. Since the radion mass can be much larger than the gravitino mass and even the original supersymmetry breaking scale, radius stability is ensured even in the presence of supersymmetry breaking. We find a parameter region in which unwanted scalar masses induced by quantum corrections through the bulk hypermultiplet and a bulk gravity multiplet are suppressed and the anomaly mediation contribution dominates. 
  Rolling tachyon in linear dilaton background is examined by using an effective field theory with gauge field on an unstable D-brane in bosonic string theory. Several solutions are identified with tachyon matter equipped with constant electromagnetic field in linear dilaton background. The time evolution of effective coupling and the nature of propagating fluctuation modes around the tachyon matter are also studied. 
  We consider the problem of the characteristics of mass spectra in the doubly symmetric theory of fields transforming under the proper Lorentz group representations decomposable into an infinite direct sum of finite-dimensional irreducible representations. We show that there exists a range of free parameters of the theory where the mass spectra of fermions are quite satisfactory from the physical standpoint and correspond to the picture expected in the parton model of hadrons. 
  The scale invariance of the coupling constant in the induced gauge theory due to its compositeness condition is demonstrated in the renormalization group flow of the finite-cutoff gauge theory at the leading order in 1/N, where N is the number of the matter fermion species. 
  We show that the renormalization group beta functions in the Nambu-Jona-Lasinio model identically vanish in all order due to the compositeness condition. Accordingly the effective coupling constants are entirely fixed and do not run with the renormalization scale. 
  We begin by reviewing the results on the decay of unstable D-branes in type II string theory, and the open-closed string duality proposal that arises from these studies. We then apply this proposal to the study of tachyon driven cosmology, namely cosmological solutions describing the decay of unstable space filling D-branes. This naturally gives rise to a time reversal invariant bounce solution with positive spatial curvature. In the absence of a bulk cosmological constant the universe always begins with a big bang and ends in a big crunch. In the presence of a bulk cosmological constant one may get non-singular cosmological solutions for some special range of initial conditions on the tachyon. 
  Bundle gerbes are a higher version of line bundles, we present nonabelian bundle gerbes as a higher version of principal bundles. Connection, curving, curvature and gauge transformations are studied both in a global coordinate independent formalism and in local coordinates. These are the gauge fields needed for the construction of Yang-Mills theories with 2-form gauge potential. 
  We consider giant gravitons on the maximally supersymmetric plane-wave background of type IIB string theory. Fixing the light-cone gauge, we work out the low energy effective light-cone Hamiltonian of the three-sphere giant graviton. At first order, this is a U(1) gauge theory on R x S^3. We place sources in this effective gauge theory. Although non-vanishing net electric charge configurations are disallowed by Gauss' law, electric dipoles can be formed. From the string theory point of view these dipoles can be understood as open strings piercing the three-sphere, generalizing the usual BIons to the giant gravitons, BIGGons. Our results can be used to give a two dimensional (worldsheet) description of giant gravitons, similar to Polchinski's description for the usual D-branes, in agreement with the discussions of hep-th/0204196. 
  We present a two-parameter family of AdS solutions to the two-dimensional type 0A effective action. 
  In this paper which is based on the results obtained in collaboration with Igor Bandos and Dmitri Sorokin I present the extention of the "doubled fields" formalism by Cremmer, Julia, Lu and Pope to the supersymmetric case and lifting this construction onto the level of the proper duality-symmetric action for D=11 supergravity. Further extension of the doubled field formulation to include dynamical sources is also discussed. 
  Using the group-theoretical approach to the inverse scattering method the supersymmetric Korteweg-de Vries equation is obtained by application of the Drinfeld-Sokolov reduction to osp(1|2) loop superalgebra. The direct and inverse scattering problems are discussed for the corresponding Lax pair. 
  The integrable structure of the two dimensional superconformal field theory is considered. The classical counterpart of our constructions is based on the $\hat{osp}(1|2)$ super-KdV hierarchy. The quantum version of the monodromy matrix associated with the linear problem for the corresponding L-operator is introduced. Using the explicit form of the irreducible representations of $\hat{osp}_q(1|2)$, the so-called "fusion relations" for the transfer matrices considered in different representations of $\hat{osp}_q(1|2)$ are obtained. The possible integrable perturbations of the model (primary operators, commuting with integrals of motion) are classified and the relation with the supersymmetric $\hat{osp}(1|2)$ Toda field theory is discussed. 
  Our main aim in this thesis is to address the results and prospects of boundary logarithmic conformal field theories: theories with boundaries that contain the above Jordan cell structure. We have investigated c_{p,q} boundary theory in search of logarithmic theories and have found logarithmic solutions of two-point functions in the context of the Coulomb gas picture. Other two-point functions have also been studied in the free boson construction of BCFT with SU(2)_k symmetry. In addition, we have analyzed and obtained the boundary Ishibashi state for a rank-2 Jordan cell structure [hep-th/0103064]. We have also examined the (generalised) Ishibashi state construction and the symplectic fermion construction at c=-2 for boundary states in the context of the c=-2 triplet model. The differences between two constructions are interpreted, resolved and extended beyond each case. 
  We revisit the 5D gravity model by Dvali, Gabadadze, and Porrati (DGP). Within their framework it was shown that even in 5D non-compact Minkowski space $(x^\mu,z)$, the Newtonian gravity can emerge on a brane at short distances by introducing a brane-localized 4D Einstein-Hilbert term $\delta(z)M_4^2\sqrt{|\bar{g}_4|}\bar{R}_4$ in the action. Based on this idea, we construct simple setups in which graviton standing waves can arise, and we introduce brane-localized $z$ derivative terms as a correction to $\delta(z)M_4^2\sqrt{|\bar{g}_4|}\bar{R}_4$. We show that the gravity potential of brane matter becomes $-\frac{1}{r}$ at {\it long} distances, because the brane-localized $z$ derivative terms allow only a smooth graviton wave function near the brane. Since the bulk gravity coupling may be arbitrarily small, strongly interacting modes from the 5D graviton do not appear. We note that the brane metric utilized to construct $\delta(z)M_4^2\sqrt{|\bar{g}_4|}\bar{R}_4$ can be relatively different from the bulk metric by a conformal factor, and show that the graviton tensor structure that the 4D Einstein gravity predicts are reproduced in DGP type models. 
  Using both the matrix model prescription and the strong-coupling approach, we describe the intersections of n=0 and n=1 non-degenerated branches for quartic (polynomial of adjoint matter) tree-level superpotential in N=1 supersymmetric SO(N)/USp(2N) gauge theories with massless flavors. We also apply the method to the degenerated branch. The general matrix model curve on the two cases we obtain is valid for arbitrary N and extends the previous work from strong-coupling approach. For SO(N) gauge theory with equal massive flavors, we also obtain the matrix model curve on the degenerated branch for arbitrary N. Finally we discuss on the intersections of n=0 and n=1 non-degenerated branches for equal massive flavors. 
  We study the {\em quantum} decay of D0-branes in two-dimensional 0B string theory. The quantum nature of the branes provides a natural cut-off for the closed string emission rate. We find exact quantum mechanical wavefunctions for the decaying branes and show how one can include the effects of the Fermi sea for any string coupling (Fermi energy). 
  We discuss a previous attempt at a microscopic counting of the entropy of asymptotically flat non-extremal black-holes. This method used string dualities to relate 4 and 5 dimensional black holes to the BTZ black hole. We show how the dualities can be justified in a certain limit, equivalent to a near horizon limit, but the resulting spacetime is no longer asymptotically flat. 
  The number of M-theory vacuum supersymmetries, 0 <= n <= 32, is given by the number of singlets appearing in the decomposition of the 32 of SL(32,R) under H \subset SL(32,R) where H is the holonomy group of the generalized connection which incorporates non-vanishing 4-form. Here we compute this generalized holonomy for the n=16 examples of the M2-brane, M5-brane, M-wave, M-monopole, for a variety of their n=8 intersections and also for the n>16 pp waves. 
  Several reductions of the bosonic BMN matrix model equations to ordinary point particle Hamiltonian dynamics in the plane (or R^3) are given - as well as a few explicit solutions (some of which, as N->infinity, correspond to membranes rotating with constant angular velocity, others to higher dimensional objects). 
  Recently the multitude of vacua in string theory have led some authors to advocate the anthropic principle as a possible resolution for the contrived set of parameters that seem to govern our universe. I suggest that string theories should be viewed as effective theories, and hence of limited utility rather than as ``theories of everything''. I propose that quantum gravity should admit a form of determinism and that the self-dual points under phase-space duality should play a prominent role in the vacuum selection principle. 
  We consider a model of 2D quantum field theory on a disk, whose bulk dynamics is that of a two-component free massless Bose field (X,Y), and interaction occurs at the boundary, where the boundary values (X_B, Y_B) are constrained to special curve - the ``paperclip brane''. The interaction breaks conformal invariance, but we argue that it preserves integrability. We propose exact expression for the disk partition function (and more general overlap amplitudes < P | B > of the boundary state with all primary states) in terms of solutions of certain ordinary linear differential equations. 
  We compute the first radiative correction to the Casimir energy in the $(d+1)$-dimensional $\lambda|\phi|^{4}$ model submitted to quasi-periodic boundary conditions in one spatial direction. Our results agree with the ones found in the literature for periodic and anti-periodic boundary conditions, special cases of the quasi-periodic boundary conditions. 
  We study both bosonic and supersymmetric (p,q) minimal models coupled to Liouville theory using the ground ring and the various branes of the theory.   From the FZZT brane partition function, there emerges a unified, geometric description of all these theories in terms of an auxiliary Riemann surface M_{p,q} and the corresponding matrix model. In terms of this geometric description, both the FZZT and ZZ branes correspond to line integrals of a certain one-form on M_{p,q}. Moreover, we argue that there are a finite number of distinct (m,n) ZZ branes, and we show that these ZZ branes are located at the singularities of M_{p,q}. Finally, we discuss the possibility that the bosonic and supersymmetric theories with (p,q) odd and relatively prime are identical, as is suggested by the unified treatment of these models. 
  Perturbative scattering amplitudes in Yang-Mills theory have many unexpected properties, such as holomorphy of the maximally helicity violating amplitudes. To interpret these results, we Fourier transform the scattering amplitudes from momentum space to twistor space, and argue that the transformed amplitudes are supported on certain holomorphic curves. This in turn is apparently a consequence of an equivalence between the perturbative expansion of ${\cal N}=4$ super Yang-Mills theory and the $D$-instanton expansion of a certain string theory, namely the topological $B$ model whose target space is the Calabi-Yau supermanifold $\Bbb{CP}^{3|4}$. 
  Combining the benefits of D-branes and background fluxes in string compactifications opens up the possibility to explore phenomenologically interesting brane world models with stabilized moduli. However, it is difficult to determine interaction effects among open strings and fluxes in the effective action. We derive the full bosonic Lagrangian of a (spontaneously broken) N=4 supersymmetric model with D3-branes and NSNS and RR 3-form fluxes in an orientifold of type IIB, that, without fluxes, would be T-dual to type I theory. In the limit where backreaction in form of a warp factor is neglected, the effective action can be obtained through a procedure that combines dimensional reduction and T-duality, and it is found to be in agreement with results from gauged supergravity. This provides evidence for the consistency of this commonly used approximation scheme. 
  One of the most intriguing features of string thermodynamics is thermal duality, which relates the physics at temperature T to the physics at inverse temperature 1/T. Unfortunately, the traditional definitions of thermodynamic quantities such as entropy and specific heat are not invariant under thermal duality transformations. In this paper, we propose several novel approaches towards dealing with this issue. One approach yields a "bootstrap" method of extracting possible exact, closed-form solutions for finite-temperature effective potentials in string theory. Another approach involves extending the usual definition of entropy by introducing additional terms which are suppressed by powers of the string scale. At high temperatures, however, these string corrections become significant and lead to a variety of surprising new phenomena as the temperature is increased. 
  It has recently been suggested that an isotropic singularity may be a generic feature of brane cosmologies, even in the inhomogeneous case. Using the covariant and gauge-invariant approach we present a detailed analysis of linear perturbations of the isotropic model ${\cal F}_b$ which is a past attractor in the phase space of homogeneous Bianchi models on the brane. We find that for matter with an equation of state parameter $\gamma > 1$, the dimensionless variables representing generic anisotropic and inhomogeneous perturbations decay as $t\to 0$, showing that the model ${\cal F}_b$ is asymptotically stable in the past. We conclude that brane universes are born with isotropy naturally built-in, contrary to standard cosmology. The observed large-scale homogeneity and isotropy of the universe can therefore be explained as a consequence of the initial conditions if the brane-world paradigm represents a description of the very early universe. 
  We study the condensation of localized tachyon in non-supersymmetric orbifold ${\C^2/\Z_n}$. We first show that the G-parities of chiral primaries are preserved under the condensation of localized tachyon(CLT) given by the chiral primaries. Using this, we finalize the proof of the conjecture that the lowest-tachyon-mass-squared increases under CLT at the level of type II string with full consideration of GSO projection. We also show the equivalence between the $G$-parity given by $G=[jk_1/n]+ [jk_2/n]$ coming from partition function and that given by $G=\{jk_1/n\}k_2 -\{jk_2/n\}k_1$ coming from the monomial construction for the chiral primaires in the dual mirror picture. 
  We investigate various excited states of Sine-Gordon model on a strip with Dirichlet boundary conditions on both boundaries using a Non Linear Integral Equation (NLIE) approach. 
  Widely spread cruel misconceptions and mistakes in the calculation of multi-loop superstring amplitudes are exposed. Correct calculations are given. It is shown that the cardinal mistake in the gauge fixing procedure presents ab ovo in the Verlinde papers. The mistake was reproduced in following proposals including the recent papers. The modular symmetry of the multi-loop superstring amplitudes is clarified, an incorrectness of previous conjectures being shown. It is shown that the Berezin-type integral versus boson and fermion moduli is doubt under non-split transformations mixing fermion integration variables to the boson integration ones. In particular, due to singularities in moduli of the given spin structure, the integral can be finite or divergent dependently on the integration variables employed. Hence, unlike naive expectations, the multi-loop superstring amplitude is ambiguous. Nevertheless, the ambiguity is totally resolved by the requirement to preserve local symmetries of the superstring amplitude. In the Verlinde world-sheet description it includes, among other thing, the requirement that the amplitude is independent of the gravitino field locations. In action the resolution of the ambiguity in the Verlinde scheme is achieved by going to the supercovariant gauge. As it has been argued earlier, the resulted arbitrary-loop amplitudes are finite. 
  In this paper we discuss how to generalize the concept of nucleation to the p-branes with form fields. And we try to get ready for calculating the decay width of the dielectric brane. 
  We re-consider the procedure of ``taking a square root of the Dirac equation'' on the superspace and show that it leads to the well known superfield W_\alpha and to the proper equations of motion for the components, i.e. the Maxwell equations and the massless Dirac equation. 
  Calorons (periodic instantons) interpolate between monopoles and instantons, and their holonomy gives approximate Skyrmion configurations. We show that, for each caloron charge N \leq 4, there exists a one-parameter family of calorons which are symmetric under subgroups of the three-dimensional rotation group. In each family, the corresponding symmetric monopoles and symmetric instantons occur as limiting cases. Symmetric calorons therefore provide a connection between symmetric monopoles, symmetric instantons and Skyrmions. 
  The two-loop chiral measure for superstring theories compactified on $\bZ_2$ reflection orbifolds is constructed from first principles for even spin structures. This is achieved by a careful implementation of the chiral splitting procedure in the twisted sectors and the identification of a subtle worldsheet supersymmetric and supermoduli dependent shift in the Prym period. The construction is generalized to compactifications which involve more general NS backgrounds preserving worldsheet supersymmetry. The measures are unambiguous and independent of the gauge slice.   Two applications are presented, both to superstring compactifications where 4 dimensions are $\bZ_2$-twisted and where the GSO projection involves a chiral summation over spin structures. The first is an orbifold by a single $\bZ_2$-twist; here, orbifolding reproduces a supersymmetric theory and it is shown that its cosmological constant indeed vanishes. The second model is of the type proposed by Kachru-Kumar-Silverstein and additionally imposes a $\bZ_2$-twist by the parity of worldsheet fermion number; it is shown here that the corresponding cosmological constant does not vanish pointwise on moduli space. 
  I motivate a proposal for modeling, at weak string coupling, the ``Big Bounce" transition from a growing-curvature phase to standard (FRW) cosmology in terms of a pressure-less dense gas of "string-holes" (SH), string states lying on the correspondence curve between strings and black holes. During this phase SH evolve in such a way that temperature and (string-frame) curvature remain $O(M_s)$ and (a cosmological version of) the holographic entropy bound remains saturated. This reasoning also appears to imply a new interpretation of the Hagedorn phase transition in string theory. 
  We report on the construction of four dimensional gauged supergravity models that can be interpreted as type IIB orientifold compactification in presence of 3-form fluxes and D3--branes. We mainly address our attention to the symplectic embedding of the U-duality group of the theory and the consequent choice of the gauge group, whose four dimensional killing vectors are the remnant of the ten dimensional fluxes. We briefly discuss the structure of the scalar potential arising from the gauging and the properties of the killing vectors in order to preserve some amount of supersymmetry. 
  We find new explicit solutions describing closed strings spinning with equal angular momentum in two independent planes in AdS(5). These are 2N-folded strings in the radial direction and also winding M times around an angular direction. Thus in spacetime they consist of 2N segments. Solutions fulfilling the closed string periodicity conditions exist provided N/M>2, i.e. the strings must be folded at least six times in the radial coordinate. The strings are spinning, or actually orbiting, similarly to solutions found previously in black hole spacetimes, but unlike the one-spin solutions in AdS which spin around their center. For long strings we recover the logaritmic scaling relation between energy and spin known from the one-spin case, but different from other known two-spin cases. 
  The generic structure of 1-, 2- and 3-point functions of fields residing in indecomposable representations of arbitrary rank are given. These in turn determine the structure of the operator product expansion in logarithmic conformal field theory. The crucial role of zero modes is discussed in some detail. 
  In a previous paper (hep-th/0306142) we have started to explore the holographic principle in the case of asymptotically flat space-times and analyzed in particular different aspects of the Bondi-Metzner-Sachs (BMS) group, namely the asymptotic symmetry group of any asymptotically flat space-time. We continue this investigation in this paper. Having in mind a S-matrix approach with future and past null infinity playing the role of holographic screens on which the BMS group acts, we connect the IR sectors of the gravitational field with the representation theory of the BMS group. We analyze the (complicated) mapping between bulk and boundary symmetries pointing out differences with respect to the AdS/CFT set up. Finally we construct a BMS phase space and a free hamiltonian for fields transforming w.r.t BMS representations. The last step is supposed to be an explorative investigation of the boundary data living on the degenerate null manifold at infinity. 
  The results on the heat kernel expansion for the electromagnetic field in the background of dielectric media are briefly reviewed. The common approaches to the calculation of the heat kernel coefficients are discussed from the viewpoint of their applicability to the electromagnetic field interacting with dielectric body of arbitrary form. Using the toy-model of scalar photons we develop multiple reflection expansion method which seems the most promising one when the field obeys dielectric-like matching conditions on an arbitrary interface and show that the heat kernel coefficients are expressible through geometric invariants of the latter. 
  We discuss how a topology (the Zariski topology) on a space can appear to break down at small distances due to D-brane decay. The mechanism proposed coincides perfectly with the phase picture of Calabi-Yau moduli spaces. The topology breaks down as one approaches non-geometric phases. This picture is not without its limitations, which are also discussed. 
  In this paper we pursue further a programme initiated in a previous work and aimed at the construction, classification and property investigation of time dependent solutions of supergravity (superstring backgrounds) through a systematic exploitation of U-duality hidden symmetries. This is done by first reducing to D=3 where the bosonic part of the theory becomes a sigma model on E_{8(8)}/SO(16), solving the equations through an algorithm that produces general integrals for any chosen regular subalgebra G_r of E_{8(8)} and then oxiding back to D=10. Different oxidations and hence different physical interpretations of the same solutions are associated with different embeddings of G_r. We show how such embeddings constitute orbits under the Weyl group and we study the orbit space. This is relevant to associate candidate superstring cosmological backgrounds to space Dp-brane configurations that admit microscopic descriptions. In particular in this paper we show that there is just one Weyl orbit of A_r subalgebras for r < 6$. The orbit of the previously found A_2 solutions, together with space--brane representatives contains a pure metric representative that corresponds to homogeneous Bianchi type 2A cosmologies in D=4 based on the Heisenberg algebra. As a byproduct of our methods we obtain new exact solutions for such cosmologies with and without matter. We present a thorough investigation of their properties. 
  A matrix algebra is constructed which consists of the necessary degrees of freedom for a finite approximation to the algebra of functions on the family of orthogonal Grassmannians of real dimension 2N, known as complex quadrics. These matrix algebras contain the relevant degrees of freedom for describing truncations of harmonic expansions of functions on N-spheres. An Inonu-Wigner contraction of the quadric gives the co-tangent bundle to the commutative sphere in the continuum limit. It is shown how the degrees of freedom for the sphere can be projected out of a finite dimensional functional integral, using second-order Casimirs, giving a well-defined procedure for construction functional integrals over fuzzy spheres of any dimension. 
  We examine spherical p-branes in AdS_m x S^n, that wrap an S^p in either AdS_m (p=m-2) or S^n (p=n-2). We first construct a two-spin giant solution expanding in S^n and has spins both in AdS_m and S^n. For (m,n)={(5,5),(4,7),(7,4)}, it is 1/2 supersymmetric, and it reduces to the single-spin giant graviton when the AdS spin vanishes. We study some of its basic properties such as instantons, non-commutativity, zero-modes, and the perturbative spectrum. All vibration modes have real and positive frequencies determined uniquely by the spacetime curvature, and evenly spaced. We next consider the (0+1)-dimensional sigma-models obtained by keeping generally time-dependent transverse coordinates, describing warped product of a breathing-mode and a point-particle on S^n or AdS_m x S^1. The BPS bounds show that the only spherical supersymmetric solutions are the single and the two-spin giants. Moreover, we integrate the sigma-model and separate the canonical variables. We quantize exactly the point-particle part of the motion, which in local coordinates gives Poschl-Teller type potentials, and calculate its contribution to the anomalous dimension. 
  We consider a classical (string) field theory of $c=1$ matrix model which was developed earlier in hep-th/9207011 and subsequent papers. This is a noncommutative field theory where the noncommutativity parameter is the string coupling $g_s$. We construct a classical solution of this field theory and show that it describes the complete time history of the recently found rolling tachyon on an unstable D0 brane. 
  We treat the N-extended supergravity in 2+1 space-time dimensions as a Yang-Mills gauge field with Chern-Simons action associated to the N-extended Poincar\'{e} supergroup. We fix the gauge of this theory within the Batalin-Vilkovisky scheme. 
  A matrix quantum mechanics with potential $V={q^2 \over r^2}$ and an SL(2,R) conformal symmetry is conjectured to be dual to two-dimensional type 0A string theory on AdS$_2$ with $q$ units of RR flux. 
  We review recent results on the interplay between the five-dimensional space-time and the internal manifold in Calabi-Yau compactifications of M-theory. Black string, black hole and domain wall solutions as well as Kasner type cosmologies cannot develop a naked singularity as long as the moduli take values inside the Kahler cone. 
  We argue that topological matrix models (matrix models of the Kontsevich type) are examples of exact open/closed duality. The duality works at finite N and for generic `t Hooft couplings. We consider in detail the paradigm of the Kontsevich model for two-dimensional topological gravity. We demonstrate that the Kontsevich model arises by topological localization of cubic open string field theory on N stable branes. Our analysis is based on standard worldsheet methods in the context of non-critical bosonic string theory. The stable branes have Neumann (FZZT) boundary conditions in the Liouville direction. Several generalizations are possible. 
  The boundary beta-function generates the renormalization group acting on the universality classes of one-dimensional quantum systems with boundary which are critical in the bulk but not critical at the boundary. We prove a gradient formula for the boundary beta-function, expressing it as the gradient of the boundary entropy s at fixed non-zero temperature. The gradient formula implies that s decreases under renormalization except at critical points (where it stays constant). At a critical point, the number exp(s) is the ``ground-state degeneracy,'' g, of Affleck and Ludwig, so we have proved their long-standing conjecture that g decreases under renormalization, from critical point to critical point. The gradient formula also implies that s decreases with temperature except at critical points, where it is independent of temperature. The boundary thermodynamic energy u then also decreases with temperature. It remains open whether the boundary entropy of a 1-d quantum system is always bounded below. If s is bounded below, then u is also bounded below. 
  We discuss the linearized, gravitational self-interaction of a brane of arbitrary codimension in a spacetime of arbitrary dimension. We find that in the codimension two case the gravitational self-force is exactly zero for a Nambu-Goto equation of state, generalizing a previous result for a string in four dimensions. For the case of a 3-brane, this picks out the case of a six-dimensional brane-world model as having special properties which we discuss. In particular, we see that bare tension on the brane has no effect locally, suppressing the cosmological constant problem. 
  We investigate several properties of Ginsparg-Wilson fermion on fuzzy 2-sphere. We first examine chiral anomaly up to the second order of the gauge field and show that it is indeed reduced to the correct form of the Chern character in the commutative limit. Next we study topologically non-trivial gauge configurations and their topological charges. We investigate 't Hooft-Polyakov monopole type configuration on fuzzy 2-sphere and show that it has the correct commutative limit. We also consider more general configurations in our formulation. 
  Extending the usual $\mathbf{C}^{\ast r}$ actions of toric manifolds by allowing asymmetries between the various $\mathbf{C}^{\ast}$ factors, we build a class of non commutative (NC) toric varieties $\mathcal{V}%_{d+1}^{(nc)}$. We construct NC complex $d$ dimension Calabi-Yau manifolds embedded in $\mathcal{V}_{d+1}^{(nc)}$ by using the algebraic geometry method. Realizations of NC $\mathbf{C}^{\ast r}$ toric group are given in presence and absence of quantum symmetries and for both cases of discrete or continuous spectrums. We also derive the constraint eqs for NC Calabi-Yau backgrounds $\mathcal{M}_{d}^{nc}$ embedded in $\mathcal{V}_{d+1}^{nc}$ and work out their solutions. The latters depend on the Calabi-Yau condition $% \sum_{i}q_{i}^{a}=0$, $q_{i}^{a}$ being the charges of $\mathbf{C}^{\ast r}$% ; but also on the toric data ${q_{i}^{a},\nu_{i}^{A};p_{I}^{\alpha},\nu _{iA}^{\ast}} $ of the polygons associated to $\mathcal{V}%_{d+1}$. Moreover, we study fractional $D$ branes at singularities and show that, due to the complete reducibility property of $\mathbf{C}^{\ast r}$ group representations, there is an infinite number of fractional $D$ branes. We also give the generalized Berenstein and Leigh quiver diagrams for discrete and continuous $\mathbf{C}^{\ast r}$ representation spectrums. An illustrating example is presented. 
  The integrable perturbation of the degenerate boundary condition (d) by the $\phi_{1,3}$ boundary field generates a renormalization group flow down to the superposition of Cardy boundary states (+)&(-). Exact Thermodynamic Bethe Ansatz (TBA) equations for all the excited states are derived here extending the results of a previous paper to this case. As an intermediate step, the non-Cardy boundary conformal sector (+)&(-) is also described as the scaling limit of an A_4 lattice model with appropriate integrable boundary conditions and produces the first example of superposition of finitized Virasoro characters. 
  Current-current deformations for WZW models of semisimple compact groups are discussed in a sigma model approach. We start with the abelian rank one group U(1). Afterwards, we keep the rank one but allow for non abelian structures by considering SU(2). Finally, we present the general case of rank larger than one. 
  In this paper we discuss the connection between the deformed matrix model and two dimensional black holes in the light of the new developements involving fermionic type 0A-string theory. We argue that many of the old results can be carried over to this new setting and that the original claims about the deformed matrix model are essentially correct. We show the agreement between correlation functions calculated using continuum and matrix model techniques. We also explain how detailed properties of the space time metric of the extremal black hole of type 0A are reflected in the deformed matrix model. 
  It is shown that the interactions between the fermion and the gravitational fields are due to the torsion field. The torsion field is considered to be a potential one, like the electromagnetic and gravitational fields. The field equations are obtained, which describe the interactions of the torsion field with the conventional physical fields. The general covariant Lagrangian of the gravitational field, based on the torsion field, is derived. Experiment is proposed to test the theory. 
  The bosonization equivalence between the 2-dimensional Dirac and Laplacian operators can be used to derive new interesting identities involving Theta functions. We use these formulae to compute the multiloop partition function of the bosonic open string in presence of a constant electromagnetic field. 
  A brief review is given of the recent solution of a non-compact CFT describing a NS-supported pp-wave background. We will first explain how to compute the three and four-point correlators using current algebra techniques, thereby showing that some generic features of the non-compact WZW models become very clear in this simple context. We will then present the Penrose limit as a contraction of an U(1)xSU(2)_k WZW model, an approach that could prove useful in order to understand holography for pp-wave space-times. We will finally comment on the string amplitudes and on the existence of two flat space limits. 
  The paper has been withdrawn by authors. The issues studied in this paper were changed so much that we have published a new paper considering these issues. See hep-th/0406074 
  We study RR flux backgrounds in two dimensional type 0 string theories. In particular, we study the relation between the 0A matrix model and the extremal black hole in two dimensions. Using T-duality we find a dual flux background in type 0B theory and propose its matrix model description. When the Fermi level is set to zero this system remains weakly coupled and exhibits a larger symmetry related to the structure of flux vacua. Finally, we construct a two dimensional type IIB background as an orbifold of the 0B background. 
  We study the structure of closed timelike curves (CTCs) for the near horizon limit of the five dimensional BMPV black hole, in its overrotating regime. We argue that Bousso's holographic screens are inside the chronologically safe region, extending a similar observation of Boyda et al. hep-th/0212087 for Goedel type solutions. We then extend this result to quite generic axisymmetric spacetimes with CTCs, showing that causal geodesics can't escape the chronologically safe region. As a spin-off of our results, we fill a gap in the identification of all maximally supersymmetric solutions of minimal five dimensional supergravity, bringing this problem to a full conclusion. 
  Using superspace techniques we construct the general theory describing D=4, N=2 supergravity coupled to an arbitrary number of vector and scalar--tensor multiplets. The scalar manifold of the theory is the direct product of a special Kaehler and a reduction of a Quaternionic-Kaehler manifold. We perform the electric gauging of a subgroup of the isometries of such manifold as well as ``magnetic'' deformations of the theory discussing the consistency conditions arising in this process. The resulting scalar potential is the sum of a symplectic invariant part (which in some instances can be recast into the standard form of the gauged N =2 theory) and of a non--invariant part, both giving new deformations. We also show the relation of such theories to flux ompactifications of type II string theories. 
  We review the current knowledge of higher-derivative terms in string effective actions, the various approaches that have been used to obtain them and their applications. 
  The ``dilaton'', the Goldstone boson of spontaneously broken conformal field theories (in flat spacetime), is argued to provide a surprisingly provocative scalar analog of gravity. Many precise parallels and contrasts are drawn. In particular, the Equivalence Principle, the Cosmological Constant Problem, and the tension between them is shown to be closely replicated. Also, there is a striking transition when mass is compressed within the (analog) Schwarzchild radius. The scalar analogy may provide a simpler context in which to think about some of the puzzles posed by real gravity. 
  We present a simple method to derive the general exact solution describing monopole scalar radiation coupled to gravity in 2+1 dimensions. The solution confirms the conjecture of Adams, Polchinski, and Silverstein regarding the late time behavior of the decay of the C/Z_n orbifold in type II string theory. 
  In this work, a general definition of convolution between two arbitrary four dimensional Lorentz invariant (fdLi) Tempered Ultradistributions is given, in both: Minkowskian and Euclidean Space (Spherically symmetric tempered ultradistributions). The product of two arbitrary fdLi distributions of exponential type is defined via the convolution of its corresponding Fourier Transforms. Several examples of convolution of two fdLi Tempered Ultradistributions are given. In particular we calculate exactly the convolution of two Feynman's massless propagators. An expression for the Fourier Transform of a Lorentz invariant Tempered Ultradistribution in terms of modified Bessel distributions is obtained in this work (Generalization of Bochner's formula to Minkowskian space). At the same time, and in a previous step used for the deduction of the convolution formula, we obtain the generalization to the Minkowskian space, of the dimensional regularization of the perturbation theory of Green Functions in the Euclidean configuration space given in ref.[12]. As an example we evaluate the convolution of two n-dimensional complex-mass Wheeler's propagators. 
  We use the permutation symmetry between the product of several group manifolds in combination with orbifolds and T-duality to construct new classes of symmetry breaking branes on products of group manifolds. The resulting branes mix the submanifolds and break part of the diagonal chiral algebra of the theory. We perform a Langrangian analysis as well as a boundary CFT construction of these branes and find agreement between the two methods. 
  Thermal duality, which relates the physics of closed strings at temperature T to the physics at the inverse temperature 1/T, is one of the most intriguing features of string thermodynamics. Unfortunately, the classical definitions of thermodynamic quantities such as entropy and specific heat are not invariant under the thermal duality symmetry. In this paper, we investigate whether there might nevertheless exist special solutions for the string effective potential such that the duality symmetry will be preserved for all thermodynamic quantities. Imposing this as a constraint, we derive a series of unique functional forms for the complete temperature-dependence of the required string effective potentials. Moreover, we demonstrate that these solutions successfully capture the leading behavior of a variety of actual one-loop effective potentials for duality-covariant finite-temperature string ground states. This leads us to conjecture that our solutions might actually represent the exact effective potentials when contributions from all orders of perturbation theory are included. 
  In a recent companion paper, we observed that the rules of ordinary thermodynamics generally fail to respect thermal duality, a symmetry of string theory under which the physics at temperature T is related to the physics at the inverse temperature 1/T. Even when the free energy and internal energy exhibit the thermal duality symmetry, the entropy and specific heat are defined in such a way that this symmetry is destroyed. In this paper, we propose a modification of the traditional definitions of these quantities, yielding a manifestly duality-covariant thermodynamics. At low temperatures, these modifications produce "corrections" to the standard definitions of entropy and specific heat which are suppressed by powers of the string scale. These corrections may nevertheless be important for the full development of a consistent string thermodynamics. We find, for example, that the string-corrected entropy can be smaller than the usual entropy at high temperatures, suggesting a possible connection with the holographic principle. We also discuss some outstanding theoretical issues prompted by our approach. 
  We point out that the existence of global symmetries in a field theory is not an essential ingredient in its relation with an integrable model. We describe an obvious construction which, given an integrable spin chain, yields a field theory whose 1-loop scale transformations are generated by the spin chain Hamiltonian. We also identify a necessary condition for a given field theory to be related to an integrable spin chain.   As an example, we describe an anisotropic and parity-breaking generalization of the XXZ Heisenberg spin chain and its associated field theory. The system has no nonabelian global symmetries and generally does not admit a supersymmetric extension without the introduction of more propagating bosonic fields. For the case of a 2-state chain we find the spectrum and the eigenstates. For certain values of its coupling constants the field theory associated to this general type of chain is the bosonic sector of the Leigh-Strassler deformation of N=4 SYM theory. 
  We study a closed model of a universe filled with viscous fluid and quintessence matter components. The dynamical equations imply that the universe might look like an accelerated flat Friedmann-Robertson-Walker (FRW) universe at low redshift. We consider here dissipative processes which obey a causal thermodynamics. Here, we account for the entropy production via causal dissipative inflation. 
  We compute the two-derivative low-energy effective action for the radion in the (supersymmetric) Randall-Sundrum scenario with detuned brane tensions. At the classical level, a potential automatically stabilizes the distance between the branes. In the supersymmetric case, supersymmetry can be broken spontaneously by a vacuum expectation value for the fifth component of the graviphoton. 
  We review our present understanding of heterotic compactifications on non-Kahler complex manifolds with torsion. Most of these manifolds can be obtained by duality chasing a consistent F-theory compactification in the presence of fluxes. We show that the duality map generically leads to non-Kahler spaces on the heterotic side, although under some special conditions we recover Kahler compactifications. The dynamics of the heterotic theory is governed by a new superpotential and minimizing this superpotential reproduces all the torsional constraints. This superpotential also fixes most of the moduli, including the radial modulus. We discuss some new connections between Kahler and non-Kahler compactifications, including some phenomenological aspects of the latter compactifications. 
  We study the RG flow of N=1 world-volume gauge theories of D3-brane probes on certain singular Calabi-Yau threefolds. Taking the gauge theories out of conformality by introducing fractional branes, we compute the NSVZ beta-function and follow the subsequent RG flow in the cascading manner of Klebanov-Strassler. We study the duality trees that blossom from various Seiberg dualities and encode possible cascades. We observe the appearance of duality walls, a finite limit energy scale in the UV beyond which the dualization cascade cannot proceed. Diophantine equations of the Markov type characterize the dual phases of these theories. We discuss how the classification of Markov equations for different geometries into families relates the RG flows of the corresponding gauge theories. 
  Non-uniform black strings in the two-brane system are investigated using the effective action approach. It is shown that the radion acts as a non-trivial hair of black strings. The stability of solutions is demonstrated using the catastrophe theory. The black strings are shown to be non-uniform. 
  A class of Dp and Dp-D(p+4)-brane solutions are constructed in the Penrose limit of the linear-dilaton geometry. The classical solutions are shown to break all space-time supersymmetries. In the worldsheet description, the branes preserve as many supersymmetries as D-branes in flat space. It is shown that these supersymmetries do not have zero modes on the worldsheet and hence do not admit local space-time realizations. This indicates that, unlike the perturbative spectrum, the D-brane spectrum of strings in the linear-dilaton pp-wave is not similar to the flat space case. 
  Brans type I black holes is a peculiar spherically symmetric solution found in geometrized gravity theories since the azimuthal factor of its horizon is divergent or vanishing under the classical approach of $r=2M$. However, if we regard that the spherically symmetric solution is available only when all physical quantities of black holes are meaningful, then our investigation would be restricted to a special range of parameters and hence indicate a definite holographic relation to type-I black holes in Brans-Dicke theory. After that, we are able to investigate this holographic relation by making use of the brick wall method. Drawn a comparison between the arising result and a simulated entropy formula derived from the thermodynamical evolution, a variable cut-off factor $\alpha$ of Brans type I black holes is ultimately carried out. 
  We derive stringy symmetries with conserved charges of arbitrarily high spins from the decoupling of two types of zero-norm states in the old covariant first quantized (OCFQ) spectrum of open bosonic string. These symmetries are valid to all energy and all loop orders in string perturbation theory. The high-energy limit of these stringy symmetries can then be used to fix the proportionality constants between scattering amplitudes of different string states algebraically without referring to Gross and Mende's saddle point calculation of high-energy string-loop amplitudes. These proportionality constants are, as conjectured by Gross, independent of the scattering angle and the order of string perturbation theory. However, we also discover some new nonzero components of high-energy amplitudes not found previously by Gross and Manes. These components are essential to preserve massive gauge invariances or decouple massive zero-norm states of string theory. A set of massive scattering amplitudes and their high energy limit are calculated explicitly to justify our results. 
  This paper is presented on the occasion of 60-th birthday of Jose Adolfo de Azcarraga who in his very rich scientific curriculum vitae has also a chapter devoted to studies of quantum-deformed symmetries, in particular deformations of relativistic and Galilean space-time symmetries [1-4].   In this paper we provide new steps toward describing the $\kappa$-deformed D=4 conformal group transformations. We consider the quantization of D=4 conformal group with dimensionful deformation parameter $\kappa$. Firstly we discuss the noncommutativity following from the Lie-Poisson structure described by the light-cone $\kappa$-Poincar\'{e} $r$-matrix. We present complete set of D=4 conformal Lie-Poisson brackets and discuss their quantization. Further we define the light-cone $\kappa$-Poincar\'{e} quantum $R$-matrix in O(4,2) vector representation and discuss the inclusion of noncommutative conformal translations into the framework of $\kappa$-deformed conformal quantum group. The problem with real structure of $\kappa$-deformed conformal group is pointed out. 
  We reexamine the problem of operator mixing in N = 4 SYM. Particular attention is paid to the correct definition of composite gauge invariant local operators, which is necessary for the computation of their anomalous dimensions beyond lowest order. As an application we reconsider the case of operators with naive dimension Delta_0=4, already studied in the literature. Stringent constraints from the resummation of logarithms in power behaviours are exploited and the role of the generalized N = 4 Konishi anomaly in the mixing with operators involving fermions is discussed. A general method for the explicit (numerical) resolution of the operator mixing and the computation of anomalous dimensions is proposed. We then resolve the order g^2 mixing for the 15 (purely scalar) singlet operators of naive dimension \Delta_0=6. Rather surprisingly we find one isolated operator which has a vanishing anomalous dimension up to order g^4, belonging to an apparently long multiplet. We also solve the order g^2 mixing for the 26 operators belonging to the representation 20' of SU(4). We find an operator with the same one-loop anomalous dimension as the Konishi multiplet. 
  Using the stationary formulation of the toroidally compactified heterotic string theory in terms of a pair of matrix Ernst potentials we consider the four-dimensional truncation of this theory with no U(1) vector fields excited. Imposing one time-like Killing vector permits us to express the stationary effective action as a model in which gravity is coupled to a matrix Ernst potential which, under certain parametrization, allows us to interpret the matter sector of this theory as a double Ernst system. We generate a web of string vacua which are related to each other via a set of discrete symmetries of the effective action (some of them involve S-duality transformations and possess non-perturbative character). Some physical implications of these discrete symmetries are analyzed and we find that, in some particular cases, they relate rotating black holes coupled to a dilaton with no Kalb--Ramond field, static black holes with non-trivial dilaton and antisymmetric tensor fields, and rotating and static naked singularities. Further, by applying a nonlinear symmetry, namely, the so-called normalized Harrison transformation, on the seed field configurations corresponding to these neutral backgrounds, we recover the U(1)^n Abelian vector sector of the four-dimensional action of the heterotic string, charging in this way the double Ernst system which corresponds to each one of the neutral string vacua, i.e., the stationary and the static black holes and the naked singularities. 
  We estimate the instanton-induced vacuum energy in non-commutative U(1) Yang-Mills theory in four dimensions. In the dilute gas approximation, it is found to be plagued by infrared divergences, as a result of UV/IR mixing. 
  The AdS/CFT correspondence is an exact duality between string theory in anti-de Sitter space and conformal field theories on its boundary. Inspired in this correspondence some relations between strings and non conformal field theories have been found. Exact dualities in the non conformal case are intricate but approximations can reproduce important physical results. A simple approximation consists in taking just a slice of the AdS space with a size related to an energy scale. Here we will discuss how this approach can be used to reproduce the scaling of high energy QCD scattering amplitudes. Also we show that very simple estimates for glueball mass ratios can emerge from such an approximation. 
  In this paper we compute the N=1 effective low energy action for a stack of N space-time filling D3-branes in generic type IIB Calabi-Yau orientifolds with non-trivial background fluxes by reducing the Dirac-Born-Infeld and Chern-Simons actions. Specifically, we determine the Kahler potential for the excitations of the D-brane including their couplings to all bulk moduli fields. In the effective theory, N=1 supergravity is spontaneously broken by the presence of fluxes and we compute the induced soft supersymmetry breaking terms. We find an interesting structure in the resulting soft terms with generically universal soft scalar masses. 
  Nonabelian magnetic monopoles of Goddard-Nuyts-Olive-Weinberg type have recently been shown to appear as the dominant infrared degrees of freedom in a class of softly broken ${\cal N}=2$ supersymmetric gauge theories in which the gauge group $G$ is broken to various nonabelian subgroups $H $ by an adjoint Higgs VEV. When the low-energy gauge group $H$ is further broken completely by e.g. squark VEVs, the monopoles representing $\pi_2(G/H)$ are confined by the nonabelian vortices arising from the breaking of $H$, discussed recently (hep-th/0307278). By considering the system with $G=SU(N+1)$, $ H = {SU(N) \times U(1) {\o}{\mathbb Z}_N}$, as an example, we show that the total magnetic flux of the minimal monopole agrees precisely with the total magnetic flux flowing along the single minimal vortex.   The possibility for such an analysis reflects the presence of free parameters in the theory - the bare quark mass $m$ and the adjoint mass $\mu$ - such that for $m \gg \mu$ the topologically nontrivial solutions of vortices and of unconfined monopoles exist at distinct energy scales. 
  We consider the geometric transition and compute the all-genus topological string amplitudes expressed in terms of Hopf link invariants and topological vertices of Chern-Simons gauge theory. We introduce an operator technique of 2-dimensional CFT which greatly simplifies the computations. We in particular show that in the case of local Calabi-Yau manifolds described by toric geometry basic amplitudes are written as vacuum expectation values of a product vertex operators and thus appear quite similar to the Veneziano amplitudes of the old dual resonance models. Topological string amplitudes can be easily evaluated using vertex operator algebra. 
  We diagonalise the Hamiltonian of the Temperley-Lieb loop model with open boundaries using a coordinate Bethe Ansatz calculation. We find that in the groundstate sector of the loop Hamiltonian, but not in other sectors, a certain constraint on the parameters has to be satisfied. This constraint has a natural interpretation in the Temperley-Lieb algebra with boundary generators. The spectrum of the loop model contains that of the quantum spin-1/2 XXZ chain with non-diagonal boundary conditions. We hence derive a recently conjectured solution of the complete spectrum of this XXZ chain. We furthermore point out a connection with recent results for the two-boundary sine-Gordon model. 
  Instanton matrix models (IMM) for two dimensional string theories are obtained from the matrix quantum mechanics (MQM) of the T-dual theory. In this paper we study the connection between the IMM and MQM, which amounts to understand T-duality from the viewpoint of matrix models. We show that type 0A and type 0B matrix models perturbed by purely closed string momentum modes (or purely winding modes) have the integrable structure of Toda hierarchies, extending the well known results for c=1 string. In particular, we show that type 0A(0B) MQM perturbed by momentum modes has the same integrable structure as type 0B(0A) MQM perturbed by winding modes, which is a nontrivial check of the T-duality between the matrix models. The MQM deformed by NS-NS winding modes are used to study type 0 string in 2D black holes. We also find an intriguing connection between the IMM and the MQM via tachyon condensation. The array of alternating D-instantons and anti-D-instantons separated at the critical distance plays a key role in this picture. We discuss its implications on sD-branes in two dimensional string theories. 
  We consider dimensional reduction of the lightlike holography of the covariant entropy bound from D+1 dimensional geometry of M X S to the D dimensional geometry M. With a warping factor, the local Bekenstein bound in D+1 dimensions leads to a more refined form of the bound from the D dimensional view point. With this new local Bekenstein bound, it is quite possible to saturate the lightlike holography even with nonvanishing expansion rate. With a Kaluza-Klein gauge field, the dimensional reduction implies a stronger bound where the energy momentum tensor contribution is replaced by the energy momentum tensor with the electromagnetic contribution subtracted. 
  The trace over the degrees of freedom located in a subset of the space transforms the vacuum state into a density matrix with non zero entropy. This geometric entropy is believed to be deeply related to the entropy of black holes. Indeed, previous calculations in the context of quantum field theory, where the result is actually ultraviolet divergent, have shown that the geometric entropy is proportional to the area for a very special type of subsets. In this work we show that the area law follows in general from simple considerations based on quantum mechanics and relativity. An essential ingredient of our approach is the strong subadditive property of the quantum mechanical entropy. 
  We construct an associative differential algebra on a two-parameter quantum plane associated with a nilpotent endomorphism $d$ in the two cases $d^{2}=0$ and $d^3=0$ $(d^2\neq 0).$ The correspondent curvature is derived and the related non commutative gauge field theory is introduced. 
  In the Dirac bracket approach to dynamical systems with second class constraints observables are represented by elements of a quotient Dirac bracket algebra. We describe families of new realizations of this algebra through quotients of the original Poisson algebra. Explicite expressions for generators and brackets of the algebras under consideration are found. 
  We continue our study of the IIB matrix model on fuzzy $S^2 \times S^2$. Especially in this paper we focus on the case where the size of one of $S^2\times S^2$ is different from the other. By the power counting and SUSY cancellation arguments, we can identify the 't Hooft coupling and large $N$ scaling behavior of the effective action to all orders. We conclude that the most symmetric $S^2 \times S^2$ configuration where the both $S^2$s are of the same size is favored at the two loop level. In addition we develop a new approach to evaluate the amplitudes on fuzzy $S^2 \times S^2$. 
  Interpretation of D1 and D0-branes in 1+1 string theory as classical and quantum eigen-values in dual c=1 Matrix Quantum Mechanics (MQM) was recently suggested. MQM is known to be equivalent to a system of N free fermions (eigen-values). By considering quantum mechanics of fermions in the presence of classical eigen-value we are able to calculate explicitly the perturbation of the shape of Fermi-sea due to the interaction with the brane. We see that the shape of the Fermi-sea depending on the position of the classical eigen-value can exhibit critical behavior, such as development of cusp. On quantum level we build explicitly the operator describing quantum eigen-value. This is a vertex operator in bosonic CFT. Its expectation value between vacuum and Dirichlet boundary state is equal to the correct wave-function of the fermion. This supports the conjecture that quantum eigen-value corresponds to D0-brane. We also show that c=1 MQM can be obtained as analytical continuation of the system of 2d electrons in magnetic field which is studied in Quantum Hall Effect. 
  We discuss how the geometry of $D2$-$D0$ branes may be related to Gromov-Witten theory of Calabi-Yau threefolds. 
  Explicit form of two-point and three-point Sp(2M) invariant Green functions is found. 
  We argue that a semi-infinite D6-brane ending on an NS5-brane can be obtained from the condensation of the tachyon on the unstable D9-brane of type IIA theory. The construction uses a combination of the descriptions of these branes as solitons of the worldvolume theory of the D9-brane. The NS5-brane, in particular, involves a gauge bundle which is operator valued, and hence is better thought of as a gerbe. 
  We consider four-dimensional massive gravity with the Fierz-Pauli mass term. The analysis of the scalar sector has revealed recently that this theory becomes strongly coupled above the energy scale \Lambda = (M_{Pl}^2 m^4)^{1/5} where m is the mass of the graviton. We confirm this scale by explicit calculations of the four-graviton scattering amplitude and of the loop correction to the interaction between conserved sources. 
  We consider the decomposition of the adjoint and fundamental representations of very extended Kac-Moody algebras G+++ with respect to their regular A type subalgebra which, in the corresponding non-linear realisation, is associated with gravity. We find that for many very extended algebras almost all the A type representations that occur in the decomposition of the fundamental representations also occur in the adjoint representation of G+++. In particular, for E8+++, this applies to all its fundamental representations. However, there are some important examples, such as An+++, where this is not true and indeed the adjoint representation contains no generator that can be identified with a space-time translation. We comment on the significance of these results for how space-time can occur in the non-linear realisation based on G+++. Finally we show that there is a correspondence between the A representations that occur in the fundamental representation associated with the very extended node and the adjoint representation of G+++ which is consistent with the interpretation of the former as charges associated with brane solutions. 
  We discuss the role played by the divergences appearing in the interaction between a fractional D3 brane dressed with an SU(N) gauge field and a stack of N fractional D3 branes on the orbifolds C^2/Z_2 and C^3/(Z_2 x Z_2). In particular we show that the logarithmic divergences in the closed string channel, interpreted as due to twisted massless tadpoles, are mapped, under open/closed string duality, in the logarithmic ones in the open string channel, due to the massless states circulating in the annulus diagram and corresponding to the one-loop divergences that one finds in the gauge theory living in the world volume of the brane. This result provides a quantitative evidence of why the chiral and scale anomalies of the supersymmetric and non conformal gauge theories supported by the world volume of the branes can be inferred from supergravity calculations. 
  This work applies the formalism developed in our earlier paper to de Sitter space. After exactly solving the relevant Heisenberg equations of motion we give a detailed discussion of the subtleties associated with defining physical states and the emergence of the classical theory. This computation provides the striking result that quantum corrections to this long wavelength limit of gravity eliminate the problem of the big crunch. We also show that the same corrections lead to possibly measureable effects on the CMB radiation. Finally, for the sake of completeness we discuss the special case, $\Lambda=0$, and its relation to Minkowski space. 
  The instantaneous formulations for the relativistic Bethe-Salpeter (BS) and the radiative transitions between the bound-states are achieved if the BS kernel is instantaneous. It is shown that the original Salpeter instantaneous equation set up on the BS equation with an instantaneous kernel should be extended to involve the `small (negative energy) component' of the BS wave functions. As a precise example of the extension for the bound states with one kind of quantum number, the way to reduce the novel extended instantaneous equation is presented. How to guarantee the gauge invariance for the radiative transitions which is formulated in terms of BS wave functions, especially, which is formulated in the instantaneous formulation, is shown. It is also shown that to `guarantee' the gauge invariance for the radiative transitions in instantaneous formulation, the novel instantaneous equation for the bound states plays a very important role. Prospects on the applications and consequences of the obtained instantaneous formulations are outlined. 
  We show how the properties of the cosmological billiards provide useful information (spacetime dimension and $p$-form spectrum) on the oxidation endpoint of the oxidation sequence of gravitational theories. We compare this approach to the other available methods: $GL(n,R)$ subgroups and the superalgebras of dualities. 
  We briefly review some modern developments in higher spin field theory and their links with superstring theory. The analysis is based on various BRST constructions allowing to derive the Lagrangians for massive and massless higher spin fields on flat or constant curvature backgrounds of arbitrary dimensions. 
  We study some cosmological consequences of the five dimensional, two brane Randall-Sundrum scenario. We integrate over the extra dimensions and in four dimensions the action reduces to that of scalar tensor gravity. The radius of the compact extra dimension is taken to be time dependent.It is shown that the radius of the extra dimension rapidly approaches a constant nonzero separation of branes. A radion dominated universe cannot undergo accelerated expansion in the absence of a potential.It is shown that a simple quadratic potential with minimum at zero leads to constant nonzero separation of branes in a similar level but now accelerated expansion is possible. After stabilization the quadratic potential contributes an effective cosmological constant term.We show that with a suitable tuning of parameters the requirements for solving the hierarchy problem and getting an effective dark energy can be satisfied simultaneously. 
  We show that B-type $\Pi$-stable D-branes do not in general reduce to the (Gieseker-) stable holomorphic vector bundles used in mathematics to construct moduli spaces. We show that solutions of the almost Hermitian Yang--Mills equations for the non-linear deformations of Yang--Mills instantons that appear in the low-energy geometric limit of strings exist iff they are $\pi$-stable, a geometric large volume version of $\Pi$-stability. This shows that $\pi$-stability is the correct physical stability concept. We speculate that this string-canonical choice of stable objects, which is encoded in and derived from the central charge of the string-\emph{algebra}, should find applications to algebraic geometry where there is no canonical choice of stable \emph{geometrical} objects. 
  A qualitative discussion of possible features of a composite graviton arising out of open string field theory is provided along with expected modifications to the graviton propagator due to such a composite graviton model. Possible implications for the cosmological constant problem are also discussed. 
  The non-Abelian plane waves, first found in flat spacetime by Coleman and subsequently generalized to give pp-waves in Einstein-Yang-Mills theory, are shown to be 1/2 supersymmetric solutions of a wide variety of N=1 supergravity theories coupled to scalar and vector multiplets, including the theory of SU(2) Yang-Mills coupled to an axion \sigma and dilaton \phi recently obtained as the reduction to four-dimensions of the six-dimensional Salam-Sezgin model. In this latter case they provide the most general supersymmetric solution. Passing to the Riemannian formulation of this theory we show that the most general supersymmetric solution may be constructed starting from a self-dual Yang-Mills connection on a self-dual metric and solving a Poisson equation for e^\phi. We also present the generalization of these solutions to non-Abelian AdS pp-waves which allow a negative cosmological constant and preserve 1/4 of supersymmetry. 
  Building on our previous results, we study D-brane/string prototypes in weakly coupled (3+1)-dimensional supersymmetric field theory engineered to support (2+1)-dimensional domain walls, "non-Abelian" strings and various junctions. Our main but not exclusive task is the study of localization of non-Abelian gauge fields on the walls. The model we work with is N=2 QCD, with the gauge group SU(2)x$U(1) and N_f=4 flavors of fundamental hypermultiplets (referred to as quarks), perturbed by the Fayet-Iliopoulos term of the U(1) factor. In the limit of large but almost equal quark mass terms a set of vacua exists in which this theory is at weak coupling. We focus on these vacua (called the quark vacua). We study elementary BPS domain walls interpolating between selected quark vacua, as well as their bound state, a composite wall. The composite wall is demonstrated to localize a non-Abelian gauge field on its world sheet. Next, we turn to the analysis of recently proposed "non-Abelian" strings (flux tubes) which carry orientational moduli corresponding to rotations of the "color-magnetic" flux direction inside a global O(3). We find a 1/4-BPS solution for the string ending on the composite domain wall. The end point of this string is shown to play the role of a non-Abelian (dual) charge in the effective world volume theory of non-Abelian (2+1)-dimensional vector fields confined to the wall. 
  In the framework of the 5D low-energy effective field theory of the heterotic string with no vector fields excited, we combine two non-linear methods in order to construct a solitonic field configuration. We first apply the inverse scattering method on a trivial vacuum solution and obtain an stationary axisymmetric two-soliton configuration consisting of a massless gravitational field coupled to a non-trivial chargeless dilaton and to an axion field endowed with charge. The implementation of this method was done following a scheme previously proposed by Yurova. We also show that within this scheme, is not possible to get massive gravitational solitons at all. We then apply a non-linear Lie-Backlund matrix transformation of Ehlers type on this massless solution and get a massive rotating axisymmetric gravitational soliton coupled to axion and dilaton fields endowed with charges. We study as well some physical properties of the constructed massless and massive solitons and discuss on the effect of the generalized solution generating technique on the seed solution and its further generalizations. 
  The algebraic definition of charges for symmetry-preserving D-branes in Wess-Zumino-Witten models is shown to coincide with the geometric definition, for all simple Lie groups. The charge group for such branes is computed from the ambiguities inherent in the geometric definition. 
  Boundary interactions of closed-string with open-strings are examined intended to provide a constructive formulation of boundary string field theory. As an illustration, we consider the BPS $D$-brane of the type II superstring in a constant NS-NS two-form background, and study the boundary interaction with arbitrary configurations of gauge field on the brane. The boundary interaction is presented, within the world-sheet cut-off theories, as an off-shell boundary state in the closed-string Hilbert space. It is regarded as a closed-string theoretical counterpart of the Wilson loop in the world-volume gauge theory. We show that the action of the closed-string BRST operator on the boundary state is translated into the non-linear BRST transformation of the open-string fields on the world-volume. In particular, the BRST invariance condition at the $\alpha'$-order becomes the non-linear equation of motion for the non-commutative gauge theory. The action of the closed-string BRST operator on the boundary state is also shown to be identified with the beta functions of the world-sheet renormalization group flow. 
  We study gravitational perturbations in the Randall-Sundrum two-brane background with scalar-curvature terms in the action for the branes, allowing for positive as well as negative bulk gravitational constant. In the zero-mode approximation, we derive the linearized gravitational equations, which have the same form as in the original Randall-Sundrum model but with different expressions for the effective physical constants. We develop a generic method for finding tachyonic modes in the theory, which, in the model under consideration, may exist only if the bulk gravitational constant is negative. In this case, if both brane gravitational constants are nonzero, the theory contains one or two tachyonic mass eigenvalues in the gravitational sector. If one of the brane gravitational constants is set to zero, then either a single tachyonic mass eigenvalue is present or tachyonic modes are totally absent depending on the relation between the nonzero brane gravitational constant and brane separation. In the case of negative bulk gravitational constant, the massive gravitational modes have ghost-like character, while the massless gravitational mode is not a ghost in the case where tachyons are absent. 
  Values for the bulk Witten indices for D=10 Yang-Mills integrals for some regular simple groups of rank 4 and 5 are calculated by employing the BRST deformation technique by Moore, Nekrasov and Shatashvili. The results cannot be reconciled with the double assumption that the number of normalizable ground states is given by certain simple partition functions given by Kac and Smilga as well as that the corresponding boundary term is always negative. 
  We establish a holographic description of chiral symmetry breaking for a particular confining large N non-supersymmetric gauge theory with matter in the fundamental representation. The gravity dual of this theory is obtained by introducing a D7-brane probe into a deformed Anti-de Sitter background found by Constable and Myers. We numerically compute the \bar\psi\psi condensate and meson spectrum by solving the equation of motion of the D7-brane probe in this background. We find a symmetry breaking condensate as well as the associated, pion-like, Goldstone boson in the limit of small quark mass. The existence of the condensate ensures that the D7-brane never reaches the naked singularity at the origin of the deformed AdS space. 
  We present an elegant method to prove the invariance of the Chern-Simons part of the non-Abelian action for N coinciding D-branes under the R-R and NS-NS gauge transformations, by carefully defining what is meant by a background gauge transformation in the non-Abelian world volume action. We study as well the invariance under massive gauge transformations of the massive Type IIA supergravity and show that no massive dielectric couplings are necessary to achieve this invariance. We show that this result is consistent with (massive) T-duality from the non-Abelian action for N D9-branes. 
  We describe the minimal configurations of the compact D=11 Supermembrane and D-branes when the spatial part of the world-volume is a K\"ahler manifold. The minima of the corresponding hamiltonians arise at immersions into the target space minimizing the K\"ahler volume. Minimal immersions of particular K\"ahler manifolds into a given target space are known to exist. They have associated to them a symplectic matrix of central charges. We reexpress the Hamiltonian of the D=11 Supermembrane with a symplectic matrix of central charges, in the light cone gauge, using the minimal immersions as backgrounds and the $Sp\parent{2g,\mathbb{Z}}$ symmetry of the resulting theory, $g$ being the genus of the K\"ahler manifold. The resulting theory is a symplectic noncommutative Yang-Mills theory coupled with the scalar fields transverse to the Supermembrane. We prove that both theories are exactly equivalent. A similar construction may be performed for the Born-Infeld action. Finally, the noncommutative formulation is used to show that the spectrum of the reguralized Hamiltonian of the above mentioned D=11 Supermembrane is a discrete set of eigenvalues with finite multiplicity. 
  We develop the BPS preon conjecture to analyze the supersymmetric solutions of D=11 supergravity. By relating the notions of Killing spinors and BPS preons, we develop a moving G-frame method (G=GL(32,R), SL(32,R) or Sp(32,R)) to analyze their associated generalized holonomies. As a first application we derive here the equations determining the generalized holonomies of k/32 supersymmetric solutions and, in particular, those solving the necessary conditions for the existence of BPS preonic (31/32) solutions of the standard D=11 supergravity. We also show that there exist elementary preonic solutions, i.e. solutions preserving 31 out of 32 supersymmetries in a Chern--Simons type supergravity. We present as well a family of worldvolume actions describing the motion of pointlike and extended BPS preons in the background of a D'Auria-Fre type OSp(1|32)-related supergravity model. We discuss the possible implications for M-theory. 
  Six-dimensional orbifold models where the Higgs field is identified with some internal component of a gauge field are considered. We classify all possible T^2/Z_N orbifold constructions based on a SU(3) electroweak gauge symmetry. Depending on the orbifold twist, models with two, one or zero Higgs doublets can be obtained. Models with one Higgs doublet are particularly interesting because they lead to a prediction for the Higgs mass, which is twice the W boson mass at leading order: m_H=2 m_W. The electroweak scale is quadratically sensitive to the cut-off, but only through very specific localized operators. We study in detail the structure of these operators at one loop, and identify a class of models where they do not destabilize the electroweak scale at the leading order. This provides a very promising framework to construct realistic and predictive models of electroweak symmetry breaking. 
  We describe cosmic D--term strings as D3 branes wrapped on a resolved conifold. The matter content that gives rise to D--term strings is shown to describe the world--volume theory of a space--filling D3 brane transverse to the conifold which itself is a wrapped D5 brane. We show that, in this brane theory, the tension of the wrapped D3 brane mathces that of the D--term string. We argue that there is a new type of cosmic string which arises from fractional D1 branes on the world--volume of a fractional D3 brane. 
  We propose a multi-graviton theory with non-nearest-neighbor couplings in the theory space. The resulting four-dimensional discrete mass spectrum reflects the structure of a latticized extra dimension. For a plausible mass spectrum motivated by the discretized Randall-Sundrum brane-world, the induced cosmological constant turns out to be positive and may serve as a quite simple model for the dark energy of our accelerating universe. 
  We use the variational approximation with double Gaussian type trial wave-functional approximation, in which we use the square root of the dispersion of the zero-mode wave-function as an order parameter, to study the out of equilibrium quantum dynamics of time-dependent second order phase transitions in (3+1) dimensions. We study the time evolution of symmetric states of scalar $\lambda \phi^4$ theory in several situations by properly treating the effect of the $\lambda\phi^4$ interaction. We also calculate the effective action and the effective potential of the theory with the precarious renormalization. We show that the presence of a quenching of the mass-squared leads to second order phase transition nontrivially since the vacuum structure changes by absorbing the energy required for quenching, even though there is no symmetry breaking in the effective potential of the theory without quenching process. We also calculate the equal time correlation function, and then evaluate the correlation length as a function of the mass-squared. The time dependence of the correlation length varies depending on how the mass-squared changes in time. For constant mass-squared it gives the classical Cahn-Allen relation, and it leads to different relations for other time-dependence of the mass-squared. We also show that there exists a propagating spatial correlation after termination of the phase transition process in addition to the correlation corresponding to the formation and growth of domains. 
  We discuss the possibility of the spontaneous symmetry breaking characterized by order parameters with higher dimensionful composite fields. By analyzing general Ginzburg-Landau potential for a complex scalar field \phi=\phi_1 + i \phi_2 with O(2) symmetry, we demonstrate that a phase characterized by < \phi_1^2 - \phi_2^2 > \neq 0 with < \phi_1 >=< \phi_2 >=0 is realized in a certain parameter region. To clarify the driving force to favor this phase, we study the O(2) \phi^6 theory in three dimensions. 
  It is well known that a vector potential cannot be defined over the whole surface of a sphere around a magnetic monopole. A recent claim to the contrary is shown to have problems. It is explained however that a potential of the proposed type works if two patches are used instead of one. A general derivation of the Dirac quantization condition attempted with a single patch is corrected by introducing two patches. Further, the case of more than two patches using the original Wu-Yang type of potential is discussed in brief. 
  Qualitative characteristics and the rigorous definition of a concept of the double symmetry is given. We use some double symmetry for constructing a theory of fields not investigated before which transform as the proper Lorentz group representations decomposable into infinite direct sums of finite-dimentional irreducible representations. All variants of the double-symmetric free Lagrangian of such fields are in brief described. The solution of the problem of changing Lagrangian mass terms due to a spontaneous breaking of the secondary symmetry is stated. The general properties of the mass spectra of fermions in the considered theory are given. It is pointed out a region of free parameters where the theoretical mass spectra qualitatively correspond to a picture typical for the parton model of hadrons. 
  The renormalization group equations of two-dimensional sigma models describe geometric deformations of their target space when the world-sheet length changes scale from the ultra-violet to the infra-red. These equations, which are also known in the mathematics literature as Ricci flows, are analyzed for the particular case of two-dimensional target spaces, where they are found to admit a systematic description as Toda system. Their zero curvature formulation is made possible with the aid of a novel infinite dimensional Lie algebra, which has anti-symmetric Cartan kernel and exhibits exponential growth. The general solution is obtained in closed form using Backlund transformations, and special examples include the sausage model and the decay process of conical singularities to the plane. Thus, Ricci flows provide a non-linear generalization of the heat equation in two dimensions with the same dissipative properties. Various applications to dynamical problems of string theory are also briefly discussed. Finally, we outline generalizations to higher dimensional target spaces that exhibit sufficient number of Killing symmetries. 
  We build the Z$_{3}$ invariants fusion rules associated to the (D$_{4}$,A$_{6}$) conformal algebra. This algebra is known to describe the tri-critical Potts model. The 4-pt correlation functions of critical fields are developed in the bootstrap approach, and in the other hand, they are written in term of integral representation of the conformal blocks. By comparing both the expressions, one can determine the structure constantes of the operator algebra. 
  We classify all irreducible, almost commutative geometries whose spectral action is dynamically non-degenerate. Heavy use is made of Krajewski's diagrammatic language. The motivation for our definition of dynamical non-degeneracy stems from particle physics where the fermion masses are non-degenerate. 
  The goal of the present paper is to calculate the determinant of the Dirac operator with a mass in the cylindrical geometry. The domain of this operator consists of functions that realize a unitary one-dimensional representation of the fundamental group of the cylinder with $n$ marked points. The determinant represents a version of the isomonodromic $\tau$-function, itroduced by M. Sato, T. Miwa and M. Jimbo. It is calculated by comparison of two sections of the $\mathrm{det}^*$-bundle over an infinite-dimensional grassmannian. The latter is composed of the spaces of boundary values of some local solutions to Dirac equation. The principal ingredients of the computation are the formulae for the Green function of the singular Dirac operator and for the so-called canonical basis of global solutions on the 1-punctured cylinder. We also derive a set of deformation equations satisfied by the expansion coefficients of the canonical basis in the general case and find a more explicit expression for the $\tau$-function in the simplest case $n=2$. 
  In the field theories with twistor structure particles can be identified with (spacially bounded) caustics of null geodesic congruences defined by the twistor field. As a realization, we consider the ``algebrodynamical'' approach based on the field equations which originate from noncommutative analysis (over the algebra of biquaternions) and lead to the complex eikonal field and to the set of gauge fields associated with solutions of the eikonal equation. Particle-like formations represented by singularities of these fields possess ``elementary'' electric charge and other realistic ``quantum numbers'' and manifest self-consistent time evolution including transmutations. Related concepts of generating ``World Function'' and of multivalued physical fields are discussed. The picture of Lorentz invariant light-formed aether and of matter born from light arises then quite naturally. The notion of the Time Flow identified with the flow of primodial light (``pre-Light'') is introduced in the context. Popularization and development of the paper math-ph/0311006. 
  An infinite set of operator-valued relations in Liouville field theory is established. These relations are enumerated by a pair of positive integers $(m,n)$, the first $(1,1)$ representative being the usual Liouville equation of motion. The relations are proven in the framework of conformal field theory on the basis of exact structure constants in the Liouville operator product expansions. Possible applications in 2D gravity are discussed. 
  In this paper the hamiltonian analysis of the pure Chern-Simons theory on the noncommutative plane is performed. We use the techniques of geometric quantization to show that the classical reduced phase space of the theory has nontrivial topology and that quantization of the symplectic structure on this space is possible only if the Chern-Simons coefficient is quantized. Also we show that the physical Hilbert space of the theory is one dimensional and give an explicit expression for the vacuum wavefunction. This vacuum state is found to be related to the noncommutative Wess-Zumino-Witten action. 
  We consider d=10 N=1 supersymmetry algebra with maximal number of tensor charges Z and introduce a class of orbits of Z, invariant w.r.t. the $T_8$ subgroup of massless particles' little group $T_8\ltimes SO(8)$. For that class of orbits we classify all possible orbits and little groups, which appear to be semidirect products of $T_8\ltimes SO(k_1)\times ... SO(k_n)$ form, with $k_1+...+k_n=8$, where compact factor is embedded into SO(8) by triality map. We define actions of little groups on supercharge Q and construct corresponding supermultiplets. In some particular cases we show the existence of supermultiplets with both Fermi and Bose sectors consisting of the same representations of tensorial Poincare. In addition, complete classification of supermultiplets (not restricted to $T_8$-invariant orbits) with rank-2 matrix of supersymmetry charges anticommutator, is given. 
  We stress that the dS/CFT correspondence should be formulated using unitary principal series representations of the de Sitter isometry group/conformal group, rather than highest-weight representations as originally proposed. These representations, however, are infinite-dimensional, and so do not account for the finite gravitational entropy of de Sitter space in a natural way. We then propose to replace the classical isometry group by a q-deformed version. This is carried out in detail for two-dimensional de Sitter and we find that the unitary principal series representations deform to finite-dimensional unitary representations of the quantum group. We believe this provides a promising microscopic framework to account for the Bekenstein-Hawking entropy of de Sitter space. 
  The three string vertex for Type IIB superstrings in a maximally supersymmetric plane-wave background can be constructed in a light-cone gauge string field theory formalism. The detailed formula contains certain Neumann coefficients, which are functions of a momentum fraction y and a mass parameter \mu. This paper reviews the derivation of useful explicit expressions for these Neumann coefficients generalizing flat-space (\mu = 0) results obtained long ago. These expressions are then used to explore the large \mu asymptotic behavior, which is required for comparison with dual perturbative gauge theory results. The asymptotic formulas, exact up to exponentially small corrections, turn out to be surprisingly simple. 
  In this paper we study T-duality for principal torus bundles with H-flux. We identify a subset of fluxes which are T-dualizable, and compute both the dual torus bundle as well as the dual H-flux. We briefly discuss the generalized Gysin sequence behind this construction and provide examples both of non T-dualizable and of T-dualizable H-fluxes. 
  We construct new solutions of the vacuum Einstein field equations with cosmological constant. These solutions describe spacetimes with non-trivial topology that are asymptotically dS, AdS or flat. For a negative cosmological constant these solutions are NUT charged generalizations of the topological black hole solutions in higher dimensions. We also point out the existence of such NUT charged spacetimes in odd dimensions and we explicitly construct such spaces in 5 and 7 dimensions. The existence of such spacetimes with non-trivial topology is closely related to the existence of the cosmological constant. Finally, we discuss the global structure of such solutions and possible applications in string theory. 
  I propose a general form for the boundary coupling of B-type topological Landau-Ginzburg models. In particular, I show that the relevant background in the open string sector is a (generally non-Abelian) superconnection of type (0,1) living in a complex superbundle defined on the target space, which I allow to be a non-compact Calabi-Yau manifold. This extends and clarifies previous proposals. Generalizing an argument due to Witten, I show that BRST invariance of the partition function on the worldsheet amounts to the condition that the (0,<= 2) part of the superconnection's curvature equals a constant endomorphism plus the Landau-Ginzburg potential times the identity section of the underlying superbundle. This provides the target space equations of motion for the open topological model. 
  We review different applications of quasinormal modes to black hole physics, string theory and thermalisation in quantum field theory. In particular, we describe the the relation between quasi normal modes of AdS black holes and the time scale for thermalisation in strongly coupled, large $N$ conformal field theory. We also discuss the problem of unitarity within this approach. 
  We motivate inflationary scenarios with many scalar fields, and give a complete formulation of adiabatic and entropy perturbations. We find that if the potential is very flat, or if the theory has a SO(N) symmetry, the calculation of the fluctuation spectrum can be carried out in terms of merely two variables without any further assumption. We do not have to assume slow roll or SO(N) invariance for the background fields. We give some examples to show that, even if the slow roll assumption holds, the spectrum of fluctuations can be quite different from the case when there is a single inflaton. 
  Large-distance modification of gravity may be the mechanism for solving the cosmological constant problem. A simple model of the large-distance modification -- four-dimensional (4D) gravity with the hard mass term-- is problematic from the theoretical standpoint. Here we discuss a different model, the brane-induced gravity, that effectively introduces a soft graviton mass. We study the issues of unitarity, analyticity and causality in this model in more than five dimensions. We show that a consistent prescription for the poles of the Green's function can be specified so that 4D unitarity is preserved. However, in certain instances 4D analyticity cannot be maintained when theory becomes higher dimensional. As a result, one has to sacrifice 4D causality at distances of the order of the present-day Hubble scale. This is a welcome feature for solving the cosmological constant problem, as was recently argued in the literature. We also show that, unlike the 4D massive gravity, the model has no strong-coupling problem at intermediate scales. 
  Godel-type metrics are introduced and used in producing charged dust solutions in various dimensions. The key ingredient is a (D-1)-dimensional Riemannian geometry which is then employed in constructing solutions to the Einstein-Maxwell field equations with a dust distribution in D dimensions. The only essential field equation in the procedure turns out to be the source-free Maxwell's equation in the relevant background. Similarly the geodesics of this type of metric are described by the Lorentz force equation for a charged particle in the lower dimensional geometry. It is explicitly shown with several examples that Godel-type metrics can be used in obtaining exact solutions to various supergravity theories and in constructing spacetimes that contain both closed timelike and closed null curves and that contain neither of these. Among the solutions that can be established using non-flat backgrounds, such as the Tangherlini metrics in (D-1)-dimensions, there exists a class which can be interpreted as describing black-hole-type objects in a Godel-like universe. 
  In the quantum adelic field (string) theory models, vacuum energy -- cosmological constant vanish. The other (alternative ?) mechanism is given by supersymmetric theories. Some observations on prime numbers, zeta -- function and fine structure constant are also considered. 
  An effective U(1) gauge invariant theory is constructed for a non-commutative Schrodinger field coupled to a background U(1)_{\star} gauge field in 2+1-dimensions using first order Seiberg-Witten map. We show that this effective theory can be cast in the form of usual Schrodinger action with interaction terms of noncommutative origin provided the gauge field is of ``background'' type with constant magnetic field. The Galilean symmetry is investigated and a violation is found in the boost sector. We also consider the problem of Hall conductivity in this framework. 
  The hyperbolic spin chain is used to elucidate the notion of spontaneous symmetry breaking for a non-amenable internal symmetry group, here SO(1,2). The noncompact symmetry is shown to be spontaneously broken -- something which would be forbidden for a compact group by the Mermin-Wagner theorem. Expectation functionals are defined through the L \to \infty limit of a chain of length L; the functional measure is found to have its weight mostly on configurations boosted by an amount increasing at least powerlike with L. This entails that despite the non-amenability a certain subclass of noninvariant functions is averaged to an SO(1,2) invariant result. Outside this class symmetry breaking is generic. Performing an Osterwalder-Schrader reconstruction based on the infinite volume averages one finds that the reconstructed quantum theory is different from the original one. The reconstructed Hilbert space is nonseparable and contains a separable subspace of ground states of the reconstructed transfer operator on which SO(1,2) acts in a continuous, unitary and irreducible way. 
  The concepts of apartments and buildings were suggested by Tits for description of the Weyl-Coxeter reflection groups. We use these and many additional facts from the theory of reflection and pseudo-reflection groups along with results from the algebraic and symplectic geometry of toric varieties in order to obtain the tachyon-free Veneziano-like multiparticle scattering amplitudes and the partition function generating these amplitudes.Although the obtained amplitudes reproduce the tachyon-free spectra of both open and closed boisonic string, the generating (partition) function is not that of the traditional bosonic string. It is argued that it is directly related to the N=2 sypersymmetric quantum mechanical model proposed by Witten in 1982 in connection with his development of the Morse theory.Such partition function can be independently obtained with help of the results by Solomon (published in 1963) on invariants of finite (pseudo) reflection groups. Although the formalism developed in this work is also applicable to conformal field theories (CFT), it leaves all CFT results unchanged 
  We give a brief review of the application of some topological solutions in field theory. 
  The Virasoro minimal models with boundary are described in the Landau-Ginzburg theory by introducing a boundary potential, function of the boundary field value. The ground state field configurations become non-trivial and are found to obey the soliton equations. The conformal invariant boundary conditions are characterized by the reparametrization-invariant data of the boundary potential, that are the number and degeneracies of the stationary points. The boundary renormalization group flows are obtained by varying the boundary potential while keeping the bulk critical: they satisfy new selection rules and correspond to real deformations of the Arnold simple singularities of A_k type. The description of conformal boundary conditions in terms of boundary potential and associated ground state solitons is extended to the N=2 supersymmetric case, finding agreement with the analysis of A-type boundaries by Hori, Iqbal and Vafa. 
  Free totally symmetric arbitrary spin massive bosonic and fermionic fields propagating in AdS(d) are investigated. Using the light cone formulation of relativistic dynamics we study bosonic and fermionic fields on an equal footing. Light-cone gauge actions for such fields are constructed. Interrelation between the lowest eigenvalue of the energy operator and standard mass parameter for arbitrary type of symmetry massive field is derived. 
  We present a mechanism through which a certain class of short-distance cutoff affects the CMB anisotropies at large angular scales. Our analysis is performed in two steps. The first is given in an intuitive way, using the property of the inflationary universe that quantum fluctuations of an inflaton field become classical after crossing the Hubble horizon. We give a condition for a cutoff to yield a damping on large scales, and show that the holographic cutoff introduced in the preceding paper (hep-th/0307029) does satisfy the condition. The second analysis is carried out by setting an initial condition such that each mode of inflaton starts as the vacuum fluctuation of the Hamiltonian when being released from the constraint of cutoff. The first intuitive discussion is then shown to be correct qualitatively. 
  In this contribution we present the superfield T-duality rules relating type IIA and type IIB supergravity potentials for the case when both type IIA and type IIB superspaces have (at least) one isometry direction. We also give a brief review of T-duality and discuss the main steps of our approach to the derivation of the superfield T-duality rules, including the treatment of T-duality as an operation acting on differential forms rather than on the superspace coordinates. 
  Decoupling limits of physical interest occur in regions of space--time where the string coupling diverges. This is illustrated in the celebrated example of five-branes. There are several ways to overcome this strong-coupling problem. We review those which are somehow related to two-dimensional conformal field theories. One method consists of distributing the branes over transverse space, either on a circle or over a sphere. Those distributions are connected to conformal field theories by T-dualities or lead to a new kind of sigma model where the target space is a patchwork of pieces of exact conformal-field-theory target spaces. An alternative method we discuss is the introduction of diluted F-strings, which trigger a marginal deformation of an AdS$_3\times S^3\times T^4$ background with a finite string coupling. Our discussion raises the question of finding brane configurations, their spectrum, their geometry, and their interpretation in terms of two-dimensional conformal models. 
  The 2D quantum gravity on a disc, or the non-critical theory of open strings, is known to exhibit an integrable structure, the boundary ground ring, which determines completely the boundary correlation functions. Inspired by the recent progress in boundary Liouville theory, we extend the ground ring relations to the case of non-vanishing boundary Liouville interaction known also as FZZT brane in the context of the 2D string theory. The ring relations yield an over-determined set of functional recurrence equations for the boundary correlation functions. The ring action closes on an infinite array of equally spaced FZZT branes for which we propose a matrix model realization. In this matrix model the boundary ground ring is generated by a pair of complex matrix fields. 
  We consider the long standing problem in field theories of bosons that the boson vacuum does not consist of a `sea', unlike the fermion vacuum. We show with the help of supersymmetry considerations that the boson vacuum indeed does also consist of a sea in which the negative energy states are all "filled", analogous to the Dirac sea of the fermion vacuum, and that a hole produced by the annihilation of one negative energy boson is an anti-particle. Here, we must admit that it is only possible if we allow --as occurs in the usual formalism anyway-- that the "Hilbert space" for the single particle bosons is not positive definite. This might be formally coped with by introducing the notion of a double harmonic oscillator, which is obtained by extending the condition imposed on the wave function. This double harmonic oscillator includes not only positive energy states but also negative energy states. We utilize this method to construct a general formalism for a boson sea analogous to the Dirac sea, irrespective of the existence of supersymmetry. The physical result is consistent with that of the ordinary second quantization formalism. We finally suggest applications of our method to the string theories. 
  We consider supermembranes ending on M5-branes, with the aim of deriving the appropriate matrix theories describing different situations. Special attention is given to the case of non-vanishing (selfdual) C-field. We identify the relevant deformation of the six-dimensional super-Yang-Mills theory whose dimensional reduction is the matrix theory for membranes in the presence of M5-branes. Possible applications and limitations of the models are discussed. 
  The quantum electrodynamics in presence of background external fields is developed. Modern methods of local quantum physics allow to formulate the theory on arbitrarily strong possibly time-dependent external fields. Non-linear observables which depend only locally on the external field are constructed. The tools necessary for this formulation, the parametrices of the Dirac operator, are investigated. 
  We have investigated some issues relevant for the possibility to construct physical theories on the $\kappa$-Minkowski noncommutative spacetime. The notion of field in $\kappa$-Minkowski has been introduced by generalizing the Weyl system/map formalism and a comparative study of the star products arising from this generalization has been done. A line of analysis of the symmetries of $\kappa$-Minkowski has been proposed that relies on the possibility to find a "maximally"-symmetric action which is invariant under a 10-generator Poincar\'e-like symmetry algebra. The equation of motion for scalar particles has been obtained by a generalized variational principle. An extension of the Dirac equation for spin-1/2 particles has been proposed by using a five-dimensional differential calculus on $\kappa$-Minkowski. 
  N=2 supersymmetric U(N) Yang-Mills theory softly broken to N=1 by the superpotential of the adjoint scalar fields is discussed from the viewpoint of the Whitham deformation theory for prepotential. With proper identification of the superpotential we derive the matrix model curve from the condition that the mixed second derivatives of the Whitham prepotential have a nontrivial kernel. 
  We construct a supermatrix model whose classical background gives two-dimensional noncommutative supersphere. Quantum fluctuations around it give the supersymmetric gauge theories on the fuzzy supersphere constructed by Klimcik. This model has a parameter $\beta$ which can tune masses of the particles in the model and interpolate various supersymmetric gauge theories on sphere. 
  The perturbative framework of the space-time non-commutative real scalar field theory is formulated, based on the unitary S-matrix. Unitarity of the S-matrix is explicitly checked order by order using the Heisenberg picture of Lagrangian formalism of the second quantized operators, with the emphasis of the so-called minimal realization of the time-ordering step function and of the importance of the $\star$-time ordering. The Feynman rule is established and is presented using $\phi^4$ scalar field theory. It is shown that the divergence structure of space-time non-commutative theory is the same as the one of space-space non-commutative theory, while there is no UV-IR mixing problem in this space-time non-commutative theory. 
  We extend earlier work on the origin of the Bekenstein-Hawking entropy to higher-dimensional spacetimes. The mechanism of counting states is shown to work for all spacetimes associated with a Euclidean doublet $(E_1,M_1)+(E_2,M_2)$ of electric-magnetic dual brane pairs of type II string-theory or M-theory wrapping the spacetime's event horizon plus the complete internal compactification space. Non-Commutativity on the brane worldvolume enters the derivation of the Bekenstein-Hawking entropy in a natural way. Moreover, a logarithmic entropy correction with prefactor 1/2 is derived. 
  The role of the gravitational sector in the Lorentz- and CPT-violating Standard-Model Extension (SME) is studied. A framework is developed for addressing this topic in the context of Riemann-Cartan spacetimes, which include as limiting cases the usual Riemann and Minkowski geometries. The methodology is first illustrated in the context of the QED extension in a Riemann-Cartan background. The full SME in this background is then considered, and the leading-order terms in the SME action involving operators of mass dimension three and four are constructed. The incorporation of arbitrary Lorentz and CPT violation into general relativity and other theories of gravity based on Riemann-Cartan geometries is discussed. The dominant terms in the effective low-energy action for the gravitational sector are provided, thereby completing the formulation of the leading-order terms in the SME with gravity. Explicit Lorentz symmetry breaking is found to be incompatible with generic Riemann-Cartan geometries, but spontaneous Lorentz breaking evades this difficulty. 
  Many approaches to a semiclassical description of gravity lead to an integer black hole entropy. In four dimensions this implies that the Schwarzschild radius obeys a formula which describes the distance covered by a Brownian random walk. For the higher-dimensional Schwarzschild-Tangherlini black hole, its radius relates similarly to a fractional Brownian walk. We propose a possible microscopic explanation for these random walk structures based on microscopic chains which fill the interior of the black hole. 
  In this paper a new small parameter associated with the density matrix deformation (density pro-matrix)studied in previous works of the author is introduced into the Generalized Quantum Mechanics (GQM), i.e. quantum mechanics involving description of the Early Universe. It is noted that this parameter has its counterpart in the generalized statistical mechanics. Both parameters offer a number of merits: they are dimensionless, varying over the interval from 0 to 1/4 and assuming in this interval a discreet series of values. Besides, their definitions contain all the fundamental constants. These parameters are very small for the conventional scales and temperatures, e.g. the value of the first parameter is on the order of $\approx 10^{-66+2n}$, where $10^{-n}$ is the measuring scale and the Planck scale $\sim 10^{-33}cm$ is assumed. The second one is also too small. It is demonstrated that relative to the first of these parameters the Universe may be considered as a nonuniform lattice in the four-dimensional hypercube with dimensionless finite-length (1/4)edges. And the time variable is also described by one of the above-mentioned dimensions due to the second parameter and generalized uncertainty relations in thermodynamics. In this context the lattice is understood as a deformation rather than approximation. 
  Some recent studies of the properties of D-particles suggest that in string theory a rather conventional description of spacetime might be available up to scales that are significantly smaller than the Planck length. We test this expectation by analyzing the localization of a space-time event marked by the collision of two D-particles. We find that a spatial coordinate of the event can indeed be determined with better-than-Planckian accuracy, at the price of a rather large uncertainty in the time coordinate. We then explore the implications of these results for the popular quantum-gravity intuition which assigns to the Planck length the role of absolute limit on localization. 
  The interaction vertex for a fermionic first order system of weights (1,0) such as the twisted bc-system, the fermionic part of N=2 string field theory and the auxiliary \eta\xi system of N=1 strings is formulated in the Moyal basis. In this basis, the Neumann matrices are diagonal; as usual, the eigenvectors are labeled by \kappa\in\R. Oscillators constructed from these eigenvectors make up two Clifford algebras for each nonzero value of \kappa. Using a generalization of the Moyal-Weyl map to the fermionic case, we classify all projectors of the star-algebra which factorize into projectors for each \kappa-subspace. At least for the case of squeezed states we recover the full set of bosonic projectors with this property. Among the subclass of ghost number-homogeneous squeezed state projectors, we find a single class of BPZ-real states parametrized by one (nearly) arbitrary function of \kappa. This class is shown to contain the generalized butterfly states. Furthermore, we elaborate on sufficient and necessary conditions which have to be fulfilled by our projectors in order to constitute surface states. As a byproduct we find that the full star product of N=2 string field theory translates into a canonically normalized continuous tensor product of Moyal-Weyl products up to an overall normalization. The divergent factors arising from the translation to the continuous basis cancel between bosons and fermions in any even dimension. 
  We perform the stochastic quantization of scalar QED based on a generalization of the stochastic gauge fixing scheme and its geometric interpretation. It is shown that the stochastic quantization scheme exactly agrees with the usual path integral formulation. 
  We briefly review the AdS3/CFT2 correspondence and the holographic issues that arise in the Penrose limit. Exploiting current algebra techniques, developped by D'Appollonio and Kiritsis for the closely related Nappi-Witten model, we obtain preliminary results for bosonic string amplitudes in the resulting Hpp-wave background and comment on how to extend them to the superstring. 
  Multidimensional cosmological-type model with n Einstein factor spaces in the theory with l scalar fields and multiple exponential potential is considered. The dynamics of the model near the singularity is reduced to a billiard on the (N-1)-dimensional Lobachevsky space H^{N-1}, N = n+l. It is shown that for n > 1 the oscillating behaviour near the singularity is absent and solutions have an asymptotical Kasner-like behavior. For the case of one scale factor (n =1) billiards with finite volumes (e.g. coinciding with that of the Bianchi-IX model) are described and oscillating behaviour of scalar fields near the singularity is obtained. 
  In this paper we work in perturbative quantum gravity and we introduce a new effective model for gravity. Expanding the Einstein-Hilbert Lagrangian in graviton field powers we have an infinite number of terms. In this paper we study the possibility of an interpretation of more than three graviton interacting vertices as effective vertices of a most fundamental theory that contain tensor fields. Here we introduce a Lagrangian model named I.T.B (Intermediate Tensor Boson) where four gravitational "pseudo-currents" that contain two gravitons couple to three massive tensorial fields of rank 1, 3, 5 respectively. We show that the exchange of those massive particles reproduces, at low energy, the interacting vertices for four or more gravitons. In a particular version, the model contains a dimensionless coupling constant "g" and the mass M of the intermediate bosons as free parameters. The universal gravitational constant G is shown to be proportional to the inverse of mass squared of mediator fields. A foresighting choice of the dimensionless coupling constant could lower the energy scale where quantum gravity aspects show up. 
  We propose a conceptual framework that leads to an abstract characterization for the exact solvability of Calabi-Yau varieties in terms of abelian varieties with complex multiplication. The abelian manifolds are derived from the cohomology of the Calabi-Yau manifold, and the conformal field theoretic quantities of the underlying string emerge from the number theoretic structure induced on the varieties by the complex multiplication symmetry. The geometric structure that provides a conceptual interpretation of the relation between geometry and the conformal field theory is discrete, and turns out to be given by the torsion points on the abelian varieties. 
  Exploiting the SU(2) Skyrmion Lagrangian with second-class constraints associated with Lagrange multiplier and collective coordinates, we convert the second-class system into the first-class one in the Batalin-Fradkin-Tyutin embedding through introduction of the St\"uckelberg coordinates. In this extended phase space we construct the "canonical" quantum operator commutators of the collective coordinates and their conjugate momenta to describe the Schr\"odinger representation of the SU(2) Skyrmion, so that we can define isospin operators and their Casimir quantum operator and the corresponding eigenvalue equation possessing integer quantum numbers, and we can also assign via the homotopy class $\pi_{4}(SU(2))=Z_{2}$ half integers to the isospin quantum number for the solitons in baryon phenomenology. Different from the semiclassical quantization previously performed, we exploit the "canonical" quantization scheme in the enlarged phase space by introducing the St\"uckelberg coordinates, to evaluate the baryon mass spectrum having global mass shift originated from geometrical corrections due to the $S^{3}$ compact manifold involved in the topological Skyrmion. Including ghosts and anti-ghosts, we also construct Becci-Rouet-Stora-Tyutin invariant effective Lagrangian. 
  I argue that string creation may have played a role in reheating the universe after inflation. For strings in four dimensions that arise from branes wrapping cycles in the extra dimensions, estimates from effective field theory show that the string tension need only fall a couple of orders of magnitude below the Planck scale in order for string creation to extract a significant fraction of the energy in coherent motion of the inflaton field. I also comment on a special four-dimensional background which involves only Neveu-Schwarz fields and offers the possibility of studying closed string creation on the worldsheet. 
  We construct new models of N=8 superconformal mechanics associated with the off-shell N=8, d=1 supermultiplets (3,8,5) and (5,8,3). These two multiplets are derived as N=8 Goldstone superfields and correspond to nonlinear realizations of the N=8, d=1 superconformal group OSp(4^*|4) in its supercosets OSp(4^*|4)/U(1)_R x SO(5) and OSp(4^*|4)/SU(2)_R x SO(4), respectively. The irreducibility constraints for these superfields automatically follow from appropriate superconformal covariant conditions on the Cartan superforms. The N=8 superconformal transformations of the superspace coordinates and the Goldstone superfields are explicitly given. Interestingly, each N=8 supermultiplet admits two different off-shell N=4 decompositions, with different N=4 superconformal subgroups SU(1,1|2) and OSp(4^*|2) of OSp(4^*|4) being manifest as superconformal symmetries of the corresponding N=4, d=1 superspaces. We present the actions for all such N=4 splittings of the N=8 multiplets considered. 
  We propose an exactly-solvable model of the quantum oscillator on the class of K\"ahler spaces (with conic singularities), connected with two-dimensional complex projective spaces. Its energy spectrum is nondegenerate in the orbital quantum number, when the space has non-constant curvature. We reduce the model to a three-dimensional system interacting with the Dirac monopole. Owing to noncommutativity of the reduction and quantization procedures, the Hamiltonian of the reduced system gets non-trivial quantum corrections. We transform the reduced system into a MIC-Kepler-like one and find that quantum corrections arise only in its energy and coupling constant. We present the exact spectrum of the generalized MIC-Kepler system. The one-(complex) dimensional analog of the suggested model is formulated on the Riemann surface over the complex projective plane and could be interpreted as a system with fractional spin. 
  We investigate the infrared behaviour of gluon and ghost propagators in Landau gauge QCD by means of an exact renormalisation group equation. We explain how, in general, the infrared momentum structure of Green functions can be extracted within this approach. An optimisation procedure is devised to remove residual regulator dependences. In Landau gauge QCD this framework is used to determine the infrared leading terms of the propagators. The results support the Kugo-Ojima confinement scenario. Possible extensions are discussed. 
  The electron would decay into a photon and neutrino if the law of electric charge conservation is not respected. Such a decay would cause vacancy in closed shells of atoms giving rise to emission of x-rays and Auger electrons. Experimental searches for such very rare decay have given an estimate for the life time to be greater than $2.7 \times 10^{23}$ years. The simplest theoretical model which would give rise to such a decay is one where the electron is regarded as the first excited state and neutrino as the ground state of a fundamental spin 1/2 particle bound to a scalar particle by a super strong force and the photon is considered as a bound state of a fundamental spin 1/2 fermion-antifermion pair. The fine structure constant of the super strong coupling is found to be unity from the masslessness of the neutrino and the lower bound of the mass of the fundamental particles is estimated by using quantum mechanical formula for photon emission by atoms and found to be $10^{22}$ GeV from the bound for electron decay time indicating thereby that the composite nature of electron, neutrino and the photon would be revealed in the Planckian energy regime. A model based on extension of $SU(2)\otimes SU(2)$ symmetry of Dirac equation to $SU(3)\otimes SU(3)$ gives a lower bound for the mass of the gauge boson mediating the decay to be $10^9 GeV$ which is the geometric mean of the masses of the electron and the fundamental particles. 
  We review an approach for computing non-perturbative, exact superpotentials for Type II strings compactified on Calabi-Yau manifolds, with extra fluxes and D-branes on top. The method is based on an open string generalization of mirror symmetry, and takes care of the relevant sphere and disk instanton contributions. We formulate a framework based on relative (co)homology that uniformly treats the flux and brane sectors on a similar footing. However, one important difference is that the brane induced potentials are of much larger functional diversity than the flux induced ones, which have a hidden N=2 structure and depend only on the bulk geometry. This introductory lecture is meant for an audience unfamiliar with mirror symmetry. 
  It is well known that string theory can be formulated as two dimensional gravity coupled to matter. In the 2d gravity formulation the central charge of the matter together with a hidden dimension from the conformal factor or Liouville mode determines the Target space dimension. Also the vacuum amplitude of the 2d gravity formulation implies important constraints on the Target space theory associated with modular invariance. In this paper we study a three dimensional gravity approach to M-theory. We find that there are three hidden Liouville type fields coming from the 3d gravity sector and that these together with the number of zero modes of the matter fields determine an eleven dimensional Target space of M theory. We investigate the perturbative vacuum amplitude for the 3d gravity approach to M theory and constraints imposed from SL(3,Z) modular invariance using a method of Dolan and Nappi together with a sum over spin structures which generalizes the SL(2,Z) invariance found in string theory. To introduce gauge fields in M-theory we study the vacuum amplitude on a three annulus and introduce interactions with two dimensional matter on a boundary in analogy with the introduction of gauge fields for open string theory. We study a three dimensional version of M-theory from the 3d gravity perspective and show how it relates to two dimensional type 0A string theory described by a 2d superLiouville theory with c=1 matter and, on manifolds with boundary, to a E8xSO(8) 2d heterotic string. We discuss a nonperturbative 3d gravity approach to M-theory and the expansion about e=0 in the Chern-Simons gauge formulation of the theory. Finally we study the interaction of fermionic matter with 3d gravity to investigate the origins of conformal dimension and Liouville effective action from a 3d gravity approach. 
  In this thesis I study various aspects of theories in the two most studied examples of noncommutative spacetimes: canonical spacetime ($[x_{\mu},x_{\nu}]=\theta_{\mu\nu}$) and $\kappa$-Minkowski spacetime ($[x_{i},t]=\kappa^{-1} x_{i}$). In the first part of the thesis I consider the description of the propagation of "classical" waves in these spacetimes. In the case of $\kappa$-Minkowski this description is rather nontrivial, and its phenomenological implications are rather striking.   In the second part of the thesis I examine the structure of quantum field theory in noncommutative spacetime, with emphasis on the simple case of the canonical spacetime. I find that the so-called IR/UV mixing can affect significantly the phase structure of a quantum field theory and also forces us upon a certain revision of the strategies used in particle-physics phenomenology to constrain the parameters of a model. 
  We discuss two different nonlinear generalizations of the osp(2|2) supersymmetry which arise in superconformal mechanics and fermion-monopole models. 
  We construct the fermionic sector and supersymmetry transformation rules of a variant N=(1,1) supergravity theory obtained by generalized Kaluza-Klein reduction from seven dimensions. We show that this model admits both (Minkowski)_4 x S^2 and (Minkowski)_3 x S^3 vacua. We perform a consistent Kaluza-Klein reduction on S^2 and obtain D=4, N=2 supergravity coupled to a vector multiplet, which can be consistently truncated to give rise to D=4, N=1 supergravity with a chiral multiplet. 
  The theory of relativistic strings is considered in frames of Hamiltonian formalism and Dirac's quantization procedure. A special gauge fixing condition is formulated, related with the world sheet of the string in Lorentz-invariant way. As a result, a new set of Lorentz-invariant canonical variables is constructed, in which a consistent quantization of bosonic open string theory could be done in Minkowski space-time of dimension $d=4$. The obtained quantum theory possesses spin-mass spectrum with Regge-like behavior. 
  Correlation functions of discrete primary fields in the c=1 boundary conformal field theory of a scalar field in a critical periodic boundary potential are computed using the underlying SU(2) symmetry of the model. Bulk amplitudes are unambigously determined and we give a prescription for amplitudes involving discrete boundary fields. 
  We propose a stringy version of the old inflation scenario which does not require any slow-roll inflaton potential and is based on a specific example of string compatification with warped metric. Our set-up admits the presence of anti-D3-branes in the deep infrared region of the metric and a false vacuum state with positive vacuum energy density. The latter is responsible for the accelerated period of inflation. The false vacuum exists only if the number of anti-D3-branes is smaller than a critical number and the graceful exit from inflation is attained if a number of anti-D3-branes travels from the ultraviolet towards the infrared region. The cosmological curvature perturbation is generated through the curvaton mechanism. 
  We study the relations between all the vacua of Lorentzian and Euclidean d=4,5,6 SUGRAs with 8 supercharges, finding a new limiting procedure that takes us from the over-rotating near-horizon BMPV black hole to the Godel spacetime. The timelike compactification of the maximally supersymmetric Godel solution of N=1,d=5 SUGRA gives a maximally supersymmetric solution of pure Euclidean N=2,d=4 with flat space but non-trivial anti-selfdual vector field flux (``flacuum'') that, on the one hand, can be interpreted as an U(1) instanton on the 4-torus and that, on the other hand, coincides with the graviphoton background shown by Berkovits and Seiberg to produce the C-deformation introduced recently by Ooguri and Vafa. We construct flacuum solutions in other theories such as Euclidean type IIA supergravity. 
  A problem of the hidden ${\cal N}=2$ supersymmetry deformation for next-to-leading terms in the effective action for ${\cal N}=4$ SYM theory is discussed. Using formulation of the theory in ${\cal N}=2$ harmonic superspace and exploring the on-shell hidden ${\cal N}=2$ supersymmetry of ${\cal N}=4$ SYM theory, we construct the appropriate hypermultiplet-depending contributions for $F^6$ term in the Schwinger-De Witt expansion of the effective action. The procedure involves deformed hidden ${\cal N}=2$ supersymmetry and allows one to obtain self-consistently the correct ${\cal N}=4$ supersymmetric functional containing $F^{6}$ among the component fields. 
  Using formulation of ${\cal N}=4$ SYM theory in terms of ${\cal N}=1$ superfields superfields we construct the derivative expansion of the one-loop ${\cal N}=4$ SYM effective action in background fields corresponding to constant Abelian strength $F_{mn}$ and constant hypermultiplet. Any term of the effective action derivative expansion can be rewritten in terms of ${\cal N}=2$ superfields. The action is manifestly ${\cal N}=2$ supersymmetric but on-shell hidden ${\cal N}=2$ supersymmetry is violated. We propose a procedure which allows to restore the hidden ${\cal N}=2$ invariance. 
  The unitary S-matrix for the space-time non-commutative QED is constructed using the $\star$-time ordering which is needed in the presence of derivative interactions. Based on this S-matrix, perturbation theory is formulated and Feynman rule is presented. The gauge invariance is explicitly checked to the lowest order, using the Compton scattering process. The gauge fixing condition dependency of the classical solution of the vacuum is also discussed. 
  We explore the geometry of the superconformal moduli of the NSR superstring theory in order to construct the consistent sigma-model for the NSR strings, free of picture-changing ambiguities. The sigma-model generating functional is constructed by the integration over the bosonic and anticommuting moduli, corresponding to insertions of the vertex operators in scattering amplitudes. In particular, the integration over the bosonic moduli results in the appearance of picture-changing operators for the b-c system. Important example of the b-c pictures involves the unintegrated and integrated forms of the vertex operators. We derive the BRST-invariant expressions for the b-c picture-changing operators for open and closed strings and study some of their properties. We also show that the superconformal moduli spaces of the NSR superstring theory contain the global singularities, leading to the appearance of non-perturbative solitonic D-brane creation operators. 
  The question of how perturbations evolve through a bounce in the Cyclic and Ekpyrotic models of the Universe is still a matter of ongoing debate. In this report we show that the collision between boundary branes is in most cases singular even in the full 5-D formalism, and that first order perturbation theory breaks down for at least one perturbation variable. Only in the case that the boundary branes approach each other with constant velocity shortly before the bounce, can a consistent, non singular solution be found. It is then possible to follow the perturbations explicitly until the actual collision. In this case, we find that if a scale invariant spectrum developed on the hidden brane, it will get transferred to the visible brane during the bounce. 
  The quantum dynamics of a bulk-boundary theory is closely examined by the use of the background field method. As an example we take the Mirabelli-Peskin model, which is composed of 5D super Yang-Mills (bulk) and 4D Wess-Zumino (boundary). Singular interaction terms play an important role of canceling the divergences coming from the KK-mode sum. Some new regularization of the momentum integral is proposed. An interesting background configuration of scalar fields is found. It is a localized solution of the field equation. In this process of the vacuum search, we present a new treatment of the vacuum with respect to the extra coordinate. The "supersymmetric" effective potential is obtained at the 1-loop full (w.r.t. the coupling) level. This is the bulk-boundary generalization of the Coleman-Weinberg's case. Renormalization group analysis is done where the correct 4d result is reproduced. The Casimir energy is calculated and is compared with the case of the Kaluza-Klein model. 
  We study N=2 supersymmetric U(1) gauge theory in the noncommutative harmonic superspace with nonanticommutative fermionic coordinates. We examine the gauge transformation which preserves the Wess-Zumino gauge by harmonic expansions of component fields. The gauge transformation is shown to depend on the deformation parameters and the anti-holomorphic scalar field. We compute the action explicitly up to the third order in component fields and discuss the field redefinitions so that the component fields transform canonically. 
  In this paper we work in perturbative Quantum Gravity coupled to Scalar Matter at tree level and we introduce a new effective model in analogy with the Fermi theory of weak interaction and in relation with a previous work where we have studied only the gravity and its self-interaction. This is an extension of the I.T.B model (Intermediate-Tensor-Boson) for gravity also to gravitational interacting scalar matter. We show that in a particular gauge the infinite series of interactions containing "n" gravitons and two scalars could be rewritten in terms of only two Lagrangians containing a massive field, the graviton and, obviously, the scalar field. Using the S-matrix we obtain that the low energy limit of the amplitude reproduce the local Lagrangian for the scalar matter coupled to gravity. 
  We discuss thermodynamics of fuzzy spheres in a matrix model on a pp-wave background. The exact free energy in the fuzzy sphere vacuum is computed in the \mu -> \infty limit for an arbitrary matrix size N. The trivial vacuum dominates the fuzzy sphere vacuum at low temperature while the fuzzy sphere vacuum is more stable than the trivial vacuum at sufficiently high temperature. Our result supports that the fluctuations around the trivial vacuum would condense to form an irreducible fuzzy sphere above a certain temperature. 
  Taking the Mirabelli-Peskin model, we examine the Casimir effect in the brane world. It is compared with that of the ordinary Kaluza-Klein theory. 
  We investigate the integrable structures in an N=2 superconfomal Sp(N) Yang-Mills theory with matter, which is dual to an open+closed string system. We restrict ourselves to the BMN operators that correspond to free string states. In the closed string sector, an integrable structure is inherited from its parent theory, N=4 SYM. For the open string sector, the planar one-loop mixing matrix for gauge invariant holomorphic operators is identified with the Hamiltonian of an integrable SU(3) open spin chain. Using the K-matrix formalism we identify the integrable open-chain boundary conditions that correspond to string boundary conditions. The solutions to the algebraic Bethe ansatz equations (ABAE) with a few impurities are shown to recover the anomalous dimensions that exactly match the spectrum of free open string in the plane-wave background. We also discuss the properties of the solutions of ABAE beyond the BMN regime. 
  We investigate a four-dimensional world, embedded into a five-dimensional spacetime, and find the five-dimensional Riemann tensor via generalisation of the Gauss (--Codacci) equations. We then derive the generalised equations of the four-dimensional world and also show that the square of the dilaton field is equal to the Newton's constant. We find plausable constant and non-constant solutions for the dilaton. 
  Scale invariant theories which contain (in $4-D$) a four index field strength are considered. The integration of the equations of motion of these $4-index$ field strength gives rise to scale symmetry breaking. The phenomena of mass generation and confinement are possible consequences of this. 
  In flux compactifications of M-theory a superpotential is generated whose explicit form depends on the structure group of the 7-dimensional internal manifold. In this note, we discuss superpotentials for the structure groups: G_2, SU(3) or SU(2). For the G_2 case all internal fluxes have to vanish. For SU(3) structures, the non-zero flux components entering the superpotential describe an effective 1-dimensional model and a Chern-Simons model if there are SU(2) structures. 
  A Lagrangian approach is proposed and developed to study defects within affine Toda field theories. In particular, a suitable Lax pair is constructed together with examples of conserved charges. It is found that only those models based on $a_r^{(1)}$ data appear to allow defects preserving integrability. Surprisingly, despite the explicit breaking of Lorentz and translation invariance, modified forms of both energy and momentum are conserved. Some, but apparently not all, of the higher spin conserved charges are also preserved after the addition of contributions from the defect. This fact is illustrated by noting how defects may preserve a modified form of just one of the spin 2 or spin -2 charges but not both of them. 
  The interaction of a magnetic flux vortex with weak external fields is considered in the framework of the Abelian Higgs model. The approach is based on the calculation of the zero-mode excitation probability in the external field. The excitation of the field configuration is found perturbatively. As an example we consider the effect of interaction with an external current. The linear in the scalar field perturbation is also considered. 
  A simple geometric procedure is proposed for constructing Wick symbols on cotangent bundles to Riemannian manifolds. The main ingredient of the construction is a method of endowing the cotangent bundle with a formal K\"ahler structure. The formality means that the metric is lifted from the Riemannian manifold $Q$ to its phase space $T^\ast Q$ in the form of formal power series in momenta with the coefficients being tensor fields on the base. The corresponding K\"ahler two-form on the total space of $T^\ast Q$ coincides with the canonical symplectic form, while the canonical projection of the K\"ahler metric on the base manifold reproduces the original metric. Some examples are considered, including constant curvature space and nonlinear sigma models, illustrating the general construction. 
  Continuous formal deformations of the Poisson superbracket defined on compactly supported smooth functions on R^n taking values in a Grassmann algebra are described up to an equivalence transformation. It is shown that there are additional deformations which are different from the standard Moyal bracket. 
  We construct a few Euclidean supergravity solutions with multiple boundaries. We consider examples where the corresponding boundary field theory is well defined on each boundary. We point out that these configurations are puzzling from the AdS/CFT point of view. A proper understanding of the AdS/CFT dictionary for these cases might yield some information about the physics of closed universes. 
  The matrix model computations of effective superpotential terms in N=1 supersymmetric gauge theories in four dimensions have been proposed to apply more generally to gauge theories in higher dimensions. We discuss aspects of five-dimensional gauge theory compactified on a circle, which leads to a unitary matrix model. 
  In this paper we study the gravitational dielectric phenomena of a D2-brane in the background of Kaluza-Klein monopoles and D6-branes. In both cases the spherical D2-brane with nonzero radius becomes classical solution of the D2-brane action. We also investigate the gravitational Myers effect in the background of D6-branes. This phenomenon occurs since the tension of the D2-brane balances with the repulsive force between D0-branes and D6-branes. 
  An overview of string theory in the maximally supersymmetric plane-wave background is given, and some supersymmetric D-branes are discussed. 
  We present a line by line derivation of canonical quantum mechanics stemming from the compatibility of the statistical geometry of distinguishable observations with the canonical Poisson structure of Hamiltonian dynamics. This viewpoint can be naturally extended to provide a conceptually novel, non-perturbative formulation of quantum gravity. Possible observational implications of this new approach are briefly mentioned. 
  We propose a realization of inverted hybrid inflation scenario in the context of n=2 supersymmetric quantum cosmology. The spectrum of density fluctuations is calculated in the de Sitter regimen as a function of the gravitino and the Planck mass, and explicit forms for the wave function of the universe are found in the WKB regimen for a FRW closed and flat universes. 
  The first results, both positive and negative, recently obtained in the area of constructing stationary spinning solitons in flat Minkowski space in 3+1 dimensions are discussed. 
  Exploiting the gauge/gravity correspondence we find the spectrum of hadronic-like bound states of adjoint particles with a large global charge in several confining theories. In particular, we consider an embedding of four-dimensional N=1 supersymmetric Yang-Mills into IIA string theory, two classes of three-dimensional gauge theories and the softly broken version of one of them. In all cases we describe the low energy excitations of a heavy hadron with mass proportional to its global charge. These excitations include: the hadron's nonrelativistic motion, its stringy excitations and excitations corresponding to the addition of massive constituents. Our analysis provides ample evidence for the universality of such hadronic states in confining theories admitting supergravity duals. Besides, we find numerically a new smooth solution that can be thought of as a non-supersymmetric deformation of G_2 holonomy manifolds. 
  We study quantum field theory in six dimensions with two of them compactified on a square. A simple boundary condition is the identification of two pairs of adjacent sides of the square such that the values of a field at two identified points differ by an arbitrary phase. This allows a chiral fermion content for the four-dimensional theory obtained after integrating over the square. We find that nontrivial solutions for the field equations exist only when the phase is a multiple of \pi/2, so that this compactification turns out to be equivalent to a T^2/Z_4 orbifold associated with toroidal boundary conditions that are either periodic or anti-periodic. The equality of the Lagrangian densities at the identified points in conjunction with six-dimensional Lorentz invariance leads to an exact Z_8\times Z_2 symmetry, where the Z_2 parity ensures the stability of the lightest Kaluza-Klein particle. 
  After briefly reviewing the methods that allow us to derive consistently new Lie (super)algebras from given ones, we consider enlarged superspaces and superalgebras, their relevance and some possible applications. 
  We study soft supersymmetry breaking in local models of type II string theory compactifications with branes and fluxes. In such models, magnetic fluxes can be treated as auxiliary fields in N=2 SUSY multiplets. These multiplets appear as ``spurion superfields'' in the low-energy effective action for the local model. We discuss the pattern of SUSY breaking from N=2 to N=1 to N=0 in these models, and then identify the fields leading to soft SUSY breaking terms in various examples. In the final section, we reconsider arguments for the Dijkgraaf-Vafa conjecture in gauge theories with softly broken supersymmetry. 
  As is well known, classical General Relativity does not constrain the topology of the spatial sections of our Universe. However, the Brane-World approach to cosmology might be expected to do so, since in general any modification of the topology of the brane must be reflected in some modification of that of the bulk. Assuming the truth of the Adams-Polchinski-Silverstein conjecture on the instability of non-supersymmetric AdS orbifolds, evidence for which has recently been accumulating, we argue that indeed many possible topologies for accelerating universes can be ruled out because they lead to non-perturbative instabilities. 
  We consider the evolution of scalar perturbations in a class of non-singular bouncing universes obtained with higher-order corrections to the low-energy bosonic string action. We show that previous studies have relied on a singular evolution equation for the perturbations. From a simple criterium we show that scalar perturbations cannot be described at all times by an homogeneous second-order perturbation equation in pre-big bang type universes if we are to regularise the background evolution with higher-order curvature and string coupling corrections, and we propose a new system of first-order coupled differential equations. Given a bouncing cosmological background with inflation driven by the kinetic energy of the dilaton field, we obtain numerically the final power spectra generated from the vacuum quantum fluctuations of the metric and the dilaton field during inflation. Our result shows that both Bardeen's potential, $\Phi(\eta,k)$, and the curvature perturbation in the uniform curvature gauge, ${\mathcal R}(\eta,k)$, lead to a blue spectral distribution long after the transition. 
  We consider the effects of a gas of closed strings (treated quantum mechanically) on a background where one dimension is compactified on a circle. After we address the effects of a time dependent background on aspects of the string spectrum that concern us, we derive the energy-momentum tensor for a string gas and investigate the resulting space-time dynamics. We show that a variety of trajectories are possible for the radius of the compactified dimension, depending on the nature of the string gas, including a demonstration within the context of General Relativity (i.e. without a dilaton) of a solution where the radius of the extra dimension oscillates about the self-dual radius, without invoking matter that violates the various energy conditions. In particular, we find that in the case where the string gas is in thermal equilibrium, the radius of the compactified dimension dynamically stabilizes at the self-dual radius, after which a period of usual Friedmann-Robertson-Walker cosmology of the three uncompactified dimensions can set in. We show that our radion stabilization mechanism requires a stringy realization of inflation as scalar field driven inflation invalidates our mechanism. We also show that our stabilization mechanism is consistent with observational bounds. 
  Fuzzy spheres appear as classical solutions in a matrix model obtained via dimensional reduction of 3-dimensional Yang-Mills theory with the Chern-Simons term. Well-defined perturbative expansion around these solutions can be formulated even for finite matrix size, and in the case of $k$ coincident fuzzy spheres it gives rise to a regularized U($k$) gauge theory on a noncommutative geometry. Here we study the matrix model nonperturbatively by Monte Carlo simulation. The system undergoes a first order phase transition as we change the coefficient ($\alpha$) of the Chern-Simons term. In the small $\alpha$ phase, the large $N$ properties of the system are qualitatively the same as in the pure Yang-Mills model ($\alpha =0$), whereas in the large $\alpha$ phase a single fuzzy sphere emerges dynamically. Various `multi fuzzy spheres' are observed as meta-stable states, and we argue in particular that the $k$ coincident fuzzy spheres cannot be realized as the true vacuum in this model even in the large $N$ limit. We also perform one-loop calculations of various observables for arbitrary $k$ including $k=1$. Comparison with our Monte Carlo data suggests that higher order corrections are suppressed in the large $N$ limit. 
  The advent of phenomenological quantum gravity has ushered us in the search for experimental tests of the deviations from general relativity predicted by quantum gravity or by string theories, and as a by--product of this quest the possible modifications that some field equations, for instance, the motion equation of spin--1/2--particles, have already been considered. In the present work a modified Dirac equation, whose extra term embraces a second--order time derivative, is taken as mainstay, and three different experimental proposals to detect it are put forward. The novelty in these ideas is that two of them do not fall within the extant approaches in this context, to wit, red--shift, atomic interferometry, or Hughes--Drever type--like experiments. 
  This thesis contains an introductory chapter on orbifolds. The following chapter explains the foundations of orientifolds. Chapters 4-7 present own research. In chapter 4 we quantize open strings with linear boundary conditions, as they show up in electro-magnetic fields. We quantize the zero-modes for toroidal compactifications, too. As an application we calculate the commutator of the coordinate fields in the case of general constant Neveu-Schwarz U(1)-field strengths. Thereby we confirm previous results on non-commutativity of open string theories in Neveu-Schwarz backgrounds. Chapter 5 reviews the results of a former publication [1] on asymmetric orientifolds, supplemented by some recent insights in connection with chapter 4. Chapter 6 summarizes publication [2] where we investigated to what extend one can build phenomenologically interesting models from toroidal orientifolds. By turning on magnetic fluxes on D9-branes we induce chiral fermions. Most calculations are performed in an (equivalent) T-dual picture. Here the number of chiral fermions is given by the topological intersection number of D-branes. In orientifolds of toroidal compactifications one obtains either non-chiral or non-supersymmetric orientifold solutions. However both properties can be reconciled in orientifolds that are obtained from specific supersymmetric orbifold compactifications. In chapter 7 we present the ``sigma Omega'' -Orientifold on a T^6/Z(4) orbifold. As a very attractive example we investigate a supersymmetric U(4) x U(2)^3_L x U(2)^3_R model that is broken to an MSSM-like model by switching on suitable background fields in the LEEA. This chapter is based on our publication [3]. An appendix is supplemented with formulas applied in the text, as well as proofs to two theorems. 
  We demonstrate that the one-loop dilatation generator for the scalar sector of a certain perturbation of N=4 Super Yang-Mills with fundamentals is the Hamiltonian of an integrable spin chain with open boundary conditions. The theory is a supersymmetric defect conformal field theory (dCFT) with the fundamentals in hypermultiplets confined to a codimension one defect. We obtain a K-matrix satisfying a suitably generalized form of the boundary Yang-Baxter equation, study the Bethe ansatz equations and demonstrate how Dirichlet and Neumann boundary conditions arise in field theory, and match to existing results in the plane wave limit. 
  The first examples of supersymmetric, asymptotically AdS5, black hole solutions are presented. They form a 1-parameter family of solutions of minimal five-dimensional gauged supergravity. Their angular momentum can never vanish. The solutions are obtained by a systematic analysis of supersymmetric solutions with Killing horizons. Other new examples of such solutions are obtained. These include solutions for which the horizon is a homogeneous Nil or SL(2,R) manifold. 
  We evaluate bulk distribution of energies, pressures and various D-brane/F-string charges generated by nontrivial matrix configurations in nonabelian D-brane effective field theories, using supergravity source density formulas derived originally in Matrix theory. Off-diagonal elements of worldvolume nonabelian fields, especially transverse scalar fields, induce various interesting bulk structures exhibiting the shape of branes. First, we study the energy distribution of string-brane networks generated in the bulk by the Yang-Mills monopoles and the 1/4 BPS dyons, and confirm force balance of them. An application to the Yang-Mills description of recombination of intersecting D-branes gives results indicating presence of the tachyon matter. Second, we analyse the shape of fuzzy D-branes given by nonabelian scalar fields which are mutually noncommutative. We employ fuzzy S^2, fuzzy S^4 and fuzzy cylinder/supertube as matrix configurations of N D0-branes representing higher dimensional noncommutative D-branes. We find that in the continuum (large N) limit the D-brane charge distributions become in the expected shape of a sphere or a cylinder with an infinitesimal thickness. However, the distributions found for finite N are difficult to interpret, which leaves a puzzle for a possible dual description in terms of higher dimensional D-branes. A resolution is provided with use of an ordering ambiguity in the charge density formulas. 
  A brief review of a superanalysis over real and $p$-adic superspaces is presented. Adelic superspace is introduced and an adelic superanalysis, which contains real and $p$-adic superanalysis, is initiated. 
  We review the Penrose limit of the Type IIB dual of softly broken N=1 SYM in four dimensions obtained as a deformation of the Maldacena-Nunez background. We extract the string spectrum on the resulting pp-wave background and discuss some properties of the conjectured dual gauge theory hadrons, the so called ``Annulons''. 
  Noncommutative U(N) gauge theories at different N may be often thought of as different sectors of a single theory. For instance, U(1) theory possesses a sequence of vacua labeled by an integer parameter N, and the theory in the vicinity of the N-th vacuum coincides with the U(N) noncommutative gauge theory. We construct domain walls on noncommutative plane, which separate vacua with different gauge groups in gauge theory with adjoint scalar field. The scalar field has nonminimal coupling to the gauge field, such that the scale of noncommutativity is determined by the vacuum value of the scalar field. The domain walls are solutions of the BPS equations in the theory. It is natural to interprete the domain wall as a stack of D-branes plus a stack of folded D-branes. We support this interpretation by the analysis of small fluctuations around domain walls, and suggest that such configurations of branes emerge as solutions of the Matrix model in large class of pp-wave backgrounds with inhomogeneous field strength. We point out that the folded D-brane per se provides an explicit realization of the "mirror world" idea, and speculate on some phenomenological consequences of this scenario. 
  We show that the frame of the Kerr spinning particle consists of two topologically coupled strings. One of them is the Kerr singular ring representing a string with an orientifold world-sheet. It has electromagnetic excitations (traveling waves), which corresponds to the old model of the Kerr's microgeon. The excitations induce the appearance of the extra axial string which is topologically coupled to the Kerr circular string and is a carrier of pp-waves and fermionic zero modes. This string can be described by the Witten field model for superconducting strings. 
  Noncommuting spatial coordinates are studied in the context of a charged particle moving in a strong non-uniform magnetic field. We derive a relation involving the commutators of the coordinates, which generalizes the one realized in a strong constant magnetic field. As an application, we discuss the noncommutativity in the magnetic field present in a magnetic mirror. 
  These are lecture notes for two lectures delivered at the Les Houches workshop on Number Theory, Physics, and Geometry, March 2003. They review two examples of interesting interactions between number theory and string compactification, and raise some new questions and issues in the context of those examples. The first example concerns the role of the Rademacher expansion of coefficients of modular forms in the AdS/CFT correspondence. The second example concerns the role of the ``attractor mechanism'' of supergravity in selecting certain arithmetic Calabi-Yau's as distinguished compactifications. 
  The Matrix Theory that has been proposed for various pp wave backgrounds is discussed. Particular emphasis is on the existence of novel nontrivial supersymmetric solutions of the Matrix Theory. These correspond to branes of various shapes (ellipsoidal, paraboloidal, and possibly hyperboloidal) that are unexpected from previous studies of branes in pp wave geometries. 
  Nontrivial translation matrices occur for spin (A,B)+(C,D) with |A-C| = |B-D| = 1/2, necessarily associating a (C,D) field with a spin (A,B) field. Including translation matrices in covariant non-unitary Poincare representations also introduces new gauge terms in the construction of massless particle fields from canonical unitary fields. In the usual procedure without spacetime translation matrices, gauge terms arise from `translations' of the massless little group; the little group combines spacetime rotations and boosts making a group isomorphic with the Euclidean group E2, including E2 translations. The usual remedy is to invoke gauge invariance. But here, the spacetime translation gauge terms can cancel the little group gauge terms, trading the need for gauge invariance with the need to specify displacements and to freeze two little group degrees of freedom that are not wanted anyway. The cancelation process restricts the helicity to A-B-1 for A-C = -(B-D) = 1/2 and A-B+1 for A-C = -(B-D) = -1/2. However, the cancelation only works for the little group standard momentum and specific transformations and, in general, gauge invariance is still needed to obtain massless particle fields. Expressions for massless particle fields for each spin type are found. 
  It is sufficient to prove that there is an excess of prime factors in the product of repunits with odd prime bases defined by the sum of divisors of the integer $N=(4k+1)^{4m+1}\prod_{i=1}^\ell ~ q_i^{2\alpha_i}$ to establish that there do not exist any odd integers with equality between $\sigma(N)$ and 2N. The existence of distinct prime divisors in the repunits in $\sigma(N)$ follows from a theorem on the primitive divisors of the Lucas sequences $U_{2\alpha_i+1}(q_i+1,q_i)$ and $U_{2\alpha_j+1}(q_j+1,q_j)$ with $q_i,q_j,2\alpha_i+1,2\alpha_j+1$ being odd primes. The occurrence of new prime divisors in each quotient ${{(4k+1)^{4m+2}-1}\over {4k}}$, ${{q_i^{2\alpha_i+1}-1}\over {q_i-1}}, i=1,...,\ell$ also implies that the square root of the product of $2(4k+1)$ and the sequence of repunits will not be rational unless the primes are matched. Although twelve solutions to the rationality condition for the existence of odd perfect numbers are obtained, it is verified that they all satisfy ${{\sigma(N)}\over N}\ne 2$ because the repunits in the product representing $\sigma(N)$ introduce new prime divisors. Minimization of the number of prime divisors in $\sigma(N)$ leads to an infinite set of repunits of increasing mangitude or prime equations with no integer solutions. It is then proven that there exist no odd perfect numbers. 
  Eleven dimensional supergravity compactified on $T^{10}$ admits classical solutions describing what is known as billiard cosmology - a dynamics expressible as an abstract (billiard) ball moving in the 10-dimensional root space of the infinite dimensional Lie algebra E10, occasionally bouncing off walls in that space. Unlike finite dimensional Lie algebras, E10 has negative and zero norm roots, in addition to the positive norm roots. The walls above are related to physical fluxes that, in turn, are related to positive norm roots (called real roots) of E10. We propose that zero and negative norm roots, called imaginary roots, are related to physical branes. Adding `matter' to the billiard cosmology corresponds to adding potential terms associated to imaginary roots. The, as yet, mysterious relation between E10 and M-theory on $T^{10}$ can now be expanded as follows: real roots correspond to fluxes or instantons, and imaginary roots correspond to particles and branes (in the cases we checked). Interactions between fluxes and branes and between branes and branes are classified according to the inner product of the corresponding roots (again in the cases we checked). We conclude with a discussion of an effective Hamiltonian description that captures some features of M-theory on $T^{10}.$ 
  Superpotentials in ${\cal N}=2$ supersymmetric classical mechanics are no more than the Hamilton characteristic function of the Hamilton-Jacobi theory for the associated purely bosonic dynamical system. Modulo a global sign, there are several superpotentials ruling Hamilton-Jacobi separable supersymmetric systems, with a number of degrees of freedom greater than one. Here, we explore how supersymmetry and separability are entangled in the quantum version of this kind of system. We also show that the planar anisotropic harmonic oscillator and the two-Newtonian centers of force problem admit two non-equivalent supersymmetric extensions with different ground states and Yukawa couplings. 
  We show that gauge invariant extensions of the local functional $\cO = \frac12\int d^4x A^2$ have long range non localities which can only be ``renormalised'' with reference to a specific gauge. Consequently, there is no gauge independent way of claiming the perturbative renormalisability of these extensions. In particular, they are not renormalisable in the modern sense of Weinberg and Gomis. Critically, our study does not support the view that ghost fields play an indispensable role in the extension of a local operator into a non-local one as claimed recently in the literature. 
  The fibre bundle formalism inherent to the construction of non-abelian Kaluza-Klein theories is presented and its associated dimensional reduction process analysed: is performed the dimensional reduction of G-invariant matter and gauge fields over a multidimensional universe; the harmonic decomposition of non-symmetric fields over its internal space established and their dimensional reduction done. The spontaneous compactification process is presented. It is shown that during the dimensional reduction process two types of symmetry breaking can occur: a geometric followed by a spontaneous symmetry breaking. This last is connected to a scalar field resulting from the dimensional reduction process itself. We determine explicitly the scalar potential form leading to that symmetry breaking for the case in which the internal space is symmetric and an analysis for the general case is performed. The principal problems found in the construction of realistic Kaluza-Klein models are discussed and some solutions reviewed. The hierarchy problem is examined and its solution within Kaluza-Klein models with a large volume internal space considered. As an alternative solution, the Randall-Sundrum model is presented and its principal properties analysed. In special, the stability of the hierarchy is studied. 
  We investigate whether the (planar, two complex scalar) dilatation operator of N=4 gauge theory can be, perturbatively and, perhaps, non-perturbatively, described by an integrable long range spin chain with elliptic exchange interaction. Such a chain was introduced some time ago by Inozemtsev. In the limit of sufficiently ``long'' operators a Bethe ansatz exists, which we apply at the perturbative two- and three-loop level. Spectacular agreement is found with spinning string predictions of Frolov and Tseytlin for the two-loop energies of certain large charge operators. However, we then go on to show that the agreement between perturbative gauge theory and semi-classical string theory begins to break down, in a subtle fashion, at the three-loop level. This corroborates a recently found disagreement between three-loop gauge theory and near plane-wave string theory results, and quantitatively explains a previously obtained puzzling deviation between the string proposal and a numerical extrapolation of finite size three-loop anomalous dimensions. At four loops and beyond, we find that the Inozemtsev chain exhibits a generic breakdown of perturbative BMN scaling. However, our proposal is not necessarily limited to perturbation theory, and one would hope that the string theory results can be recovered from the Inozemtsev chain at strong 't Hooft coupling. 
  Rotating IIB strings in AdS_5 x S^5 become ultra-relativistic, and hence effectively tensionless, in the limit of large angular momentum on S^5. We have shown previously that such tensionless strings may preserve supersymmetry. Here we extend this result to include a class of supersymmetric tensionless strings with arbitrary SO(6) angular momentum. We close with some general comments on tensionless strings in AdS_5 x S^5. 
  Classical and quantum symmetries of super $p$-branes preserving exotic 3/4 fraction of $N=1 D=4$ global supersymmetry are studied. Classical realization of the algebra of global and world-volume symmetries is constructed and its quantum generalizations are analyzed. Established is that the status of the conformal supersymmetry $OSp(1|8)$ as a proper quantum symmetry of brane depends both on the choice of its vacuum state and the associated ordering of $\hat{\cal Q}$ and $\hat{\cal P}$ operators. 
  Holography principle imposes a stringent constraint on the scale of quantum gravity $M_*$ in brane-world scenarios, where all matter is confined on the brane. The thermodynamic entropy of astrophysical black holes and sub-horizon volumes during big bang nucleosynthesis exceed the relevant bounds unless $M_* > 10^{(4-6)}$ TeV, so a hierarchy relative to the weak scale is unavoidable. We discuss the implications for extra dimensions as well as holography. 
  In this paper, we give a general axiomatization of anomalies in closed and open conformal field theories. In particular, we generalize Segal's notion of modular functor to a setting where the ``set of labels'' is a 2-vector space. In the case of open conformal field theory, the ``set of $D$-branes'' is a 3-vector space. We also define a ``topological group completion'' of the symmetric bimonoidal category of finite-dimensional vector spaces, and propose it as a candidate for labelling conformal field theories whose modular functors are super-vector spaces. 
  In this contribution we describe how to obtain instanton effects in four dimensional gauge theories by computing string scattering amplitudes in D3/D(-1) brane systems. In particular we show that the disks with mixed boundary conditions, which are typical of the D3/D(-1) system, are the sources for the classical instanton solution. 
  Using the quasi-normal modes frequency of near extremal Schwarzschild-de Sitter black holes, we obtain area and entropy spectrum for black hole horizon. By using Boher-Sommerfeld quantization for an adiabatic invariant $I=\int {dE\over \omega(E)}$, which $E$ is the energy of system and $\omega(E)$ is vibrational frequency, we leads to an equally spaced mass spectrum. In the other term we extend directly the Kunstatter's approach [7] to determine mass and entropy spectrum of near extremal Schwarzschild-de Sitter black holes which is asymptotically de Sitter rather than asymptotically flat. 
  A twistorial formulation of a particle of arbitrary spin has been constructed. Equations of the twistor formulation are obtained for massive and massless spinning particles. The twistor space of the massive particle is formed by two twistors and two complex scalars. In the massive case, integral transformations relating twistor fields with usual space-time fields have been constructed. 
  We give an introduction to the heat kernel technique and zeta function. Two applications are considered. First we derive the high temperature asymptotics of the free energy for boson fields in terms of the heat kernel expansion and zeta function. Another application is chiral anomaly for local (MIT bag) boundary conditions. 
  In a previous paper (hep-th/0304045) it has been argued that tachyonic Dirac-Born-Infeld (DBI) actions can be obtained from open string theory in a limit, which generalizes the usual massless DBI limit. In the present note we review this construction focusing on a key property of the proposed tachyon effective actions: how they reproduce appropriate Veneziano amplitudes in a suitably defined kinematical region. Possible extensions and interesting open problems are briefly discussed. 
  In the matrix model formulation of two dimensional noncritical string theory, a D0 brane is identified with a single eigenvalue excitation. In terms of open string quantities (i.e fermionic eigenvalues) the classical limit of a macroscopically large number of D0 branes has a smooth classical limit : they are described by a filled region of phase space whose size is O(1) and disconnected from the Fermi sea. We show that while this has a proper description in terms of a {\em single} bosonic field at the quantum level, the classical limit is rather nontrivial. The quantum dispersions of bosonic quantities {\em survive in the classical limit} and appear as additional fields in a semiclassical description. This reinforces the fact that while the open string field theory description of these D-branes (i.e. in terms of fermions) has a smooth classical limit, a closed string field theory description (in terms of a single boson) does not. 
  The recent investigations of pure Landau gauge SU(3) Yang-Mills theories which are based on the truncated Schwinger-Dyson equations (SDE) indicate an infrared power law behavior of the gluon and the ghost propagators. It has been shown that the gluon propagator vanishes (or finite) in the infrared limit, while the ghost propagator is more singular than a massless pole, and also that there exists an infrared fixed point of the running gauge coupling. In this paper we reexamine this picture by means of the exact (non-perturbative) renormalization group (ERG) equations under some approximation scheme, in which we treat not only two point functions but also four point vertices in the effective average action with retaining their momentum dependence. Then it is shown that the gluon and the ghost propagators with the infrared power law behavior are obtained as an attractive solution starting from rather arbitrary ultraviolet bare actions. Here it is found to be crucial to include the momentum dependent four point vertices in the ERG framework, since otherwise the RG flows diverge at finite scales. The features of the ERG analyses in comparison with the SDE are also discussed. 
  We analyze the quantum process in which a cosmic string breaks in an anti-de Sitter (AdS) background, and a pair of charged or neutral black holes is produced at the ends of the strings. The energy to materialize and accelerate the pair comes from the strings tension. In an AdS background this is the only study done in the process of production of a pair of correlated black holes with spherical topology. The acceleration $A$ of the produced black holes is necessarily greater than (|L|/3)^(1/2), where L<0 is the cosmological constant. Only in this case the virtual pair of black holes can overcome the attractive background AdS potential well and become real. The instantons that describe this process are constructed through the analytical continuation of the AdS C-metric. Then, we explicitly compute the pair creation rate of the process, and we verify that (as occurs with pair creation in other backgrounds) the pair production of nonextreme black holes is enhanced relative to the pair creation of extreme black holes by a factor of exp(Area/4), where Area is the black hole horizon area. We also conclude that the general behavior of the pair creation rate with the mass and acceleration of the black holes is similar in the AdS, flat and de Sitter cases, and our AdS results reduce to the ones of the flat case when L=0. 
  For the closed relativistic string carrying a pointlike mass the exact solutions of the dynamical equations are obtained and studied. These solutions describe states of the mentioned system moving in Minkowski space and also in the space that is the direct product of Minkowski space and a compact manifold (torus). 
  We clarify a point concerning the ultraviolet behaviour of the Quantum Field Theory of gravity, under the assumption of the existence of an ultraviolet Fixed Point. We explain why Newton's constant should to scale like the inverse of the square of the cutoff, even though it is technically inessential. As a consequence of this behaviour, the existence of an UV Fixed Point would seem to imply that gravity has a built-in UV cutoff when described in Planck units, but not necessarily in other units. 
  We construct an approximation to field theories on the noncommutative torus based on soliton projections and partial isometries which together form a matrix algebra of functions on the sum of two circles. The matrix quantum mechanics is applied to the perturbative dynamics of scalar field theory, to tachyon dynamics in string field theory, and to the Hamiltonian dynamics of noncommutative gauge theory in two dimensions. We also describe the adiabatic dynamics of solitons on the noncommutative torus and compare various classes of noncommutative solitons on the torus and the plane. 
  We study the entropy of a FRW universe filled with dark energy (cosmological constant, quintessence or phantom). For general or time-dependent equation of state $p=w\rho$ the entropy is expressed in terms of energy, Casimir energy, and $w$. The correspondent expression reminds one about 2d CFT entropy only for conformal matter. At the same time, the cosmological Cardy-Verlinde formula relating three typical FRW universe entropies remains to be universal for any type of matter. The same conclusions hold in modified gravity which represents gravitational alternative for dark energy and which contains terms growing at low curvature. It is interesting that BHs in modified gravity are more entropic than in Einstein gravity. Finally, some hydrodynamical examples testing new shear viscosity bound, which is expected to be the consequence of the holographic entropy bound, are presented for the early universe in the plasma era and for the Kasner metric. It seems that the Kasner metric provides a counterexample to the new shear viscosity bound. 
  We study the universal properties of the phase diagram of QCD near the critical point using the exact renormalization group. For two-flavour QCD and zero quark masses we derive the universal equation of state in the vicinity of the tricritical point. For non-zero quark masses we explain how the universal equation of state of the Ising universality class can be used in order to describe the physical behaviour near the line of critical points. The effective exponents that parametrize the growth of physical quantities, such as the correlation length, are given by combinations of the critical exponents of the Ising class that depend on the path along which the critical point is approached. In general the critical region, in which such quantities become large, is smaller than naively expected. 
  We compute for various perturbed conformal field theories the vacuum energies by means of the thermodynamic Bethe ansatz. Depending on the infrared and ultraviolet divergencies of the models, governed by the scaling dimensions of the underlying perturbed conformal field theory in the ultraviolet, the vacuum energies exhibit different types of characteristics. In particular, for the homogeneous sine-Gordon models we observe that once the conformal dimension of the perturbing scalar field is smaller or greater than 1/2, the vacuum energies are positive or negative, respectively. This behaviour indicates the need for additional ultraviolet counterterms in the latter case. At the transition points we obtain an infinite vacuum energy, which is partly explainable with the presence of several free Fermions in the models studied. 
  We show how to properly gauge fix all the symmetries of the Ponzano-Regge model for 3D quantum gravity. This amounts to do explicit finite computations for transition amplitudes. We give the construction of the transition amplitudes in the presence of interacting quantum spinning particles. We introduce a notion of operators whose expectation value gives rise to either gauge fixing, introduction of time, or insertion of particles, according to the choice. We give the link between the spin foam quantization and the hamiltonian quantization. We finally show the link between Ponzano-Regge model and the quantization of Chern-Simons theory based on the double quantum group of SU(2) 
  The generalized Einstein - Maxwell field equations which arise from a truncated bosonic part of the low - energy string gravity effective action in four dimensions (the so called Einstein-Maxwell - axion - dilaton theory) are considered. The integrable structure of these field equations for D=4 space-times with two commuting isometries is elucidated. We express the dynamical part of the reduced equations as integrability conditions of some overdetermined $4\times 4$-matrix linear system with a spectral parameter. The remaining part of the field equations are expressed as the conditions of existence for this linear system of two $4\times 4$-matrix integrals of special structures. This provides a convenient base for a generalization to these equations of various solution generating methods developed earlier in General Relativity. 
  We investigate a deformed matrix model proposed by Kazakov et.al. in relation to Witten's two-dimensional black hole. The existing conjectures assert the equivalence of the two by mapping each to a deformed c=1 theory called the sine-Liouville theory. We point out that the matrix theory in question may be naturally interpreted as a gauged quantum mechanics deformed by insertion of an exponentiated Wilson loop operator, which gives us more direct and holographic map between the two sides. The matrix model in the usual scaling limit must correspond to the bosonic SL(2,R)/U(1) theory in genus expansion but exact in \alpha'. We successfully test this by computing the Wilson loop expectation value and comparing it against the bulk computation. For the latter, we employ the \alpha'-exact geometry proposed by Dijkgraaf, Verlinde, and Verlinde, which was further advocated by Tseytlin. We close with comments on open problems. 
  I construct field theory on an evolving fuzzy two-sphere, which is based on the idea of evolving non-commutative worlds of the previous paper. The equations of motion are similar to the one that can be obtained by dropping the time-derivative term of the equation derived some time ago by Banks, Peskin and Susskind for pure-into-mixed-state evolutions. The equations do not contain an explicit time, and therefore follow the spirit of the Wheeler-de Witt equation. The basic properties of field theory such as action, gauge invariance and charge and momentum conservation are studied. The continuum limit of the scalar field theory shows that the background geometry of the corresponding continuum theory is given by ds^2 = -dt^2+ t d Omega^2, which saturates locally the cosmic holographic principle. 
  In this scientific preface to the first issue of International Journal of Geometric Methods in Modern Physics, we briefly survey some peculiarities of geometric techniques in quantum models. 
  We show that fluctuations of bulk operators that are restricted to some region of space scale as the surface area of the region, independently of its geometry. Specifically, we consider two point functions of operators that are integrals over local operator densities whose two point functions falls off rapidly at large distances, and does not diverge too strongly at short distances. We show that the two point function of such bulk operators is proportional to the area of the common boundary of the two spatial regions. Consequences of this, relevant to the holographic principle and to area-scaling of Unruh radiation are briefly discussed. 
  The existence of phantom energy in a universe which evolves to eventually show a big rip doomsday is a possibility which is not excluded by present observational constraints. In this letter it is argued that the field theory associated with a simple quintessence model is compatible with a field definition which is interpretable in terms of a rank-three axionic tensor field, whenever we consider a perfect-fluid equation of state that corresponds to the phantom energy regime. Explicit expressions for the axionic field and its potential, both in terms of an imaginary scalar field, are derived which show that these quantities both diverge at the big rip, and that the onset of phantom-energy dominance must take place just at present. 
  First we briefly outline the general construction of Dp-brane models with dynamical tensions. We then proceed to a more detailed discussion of a modified string model where the string tension is related to the potential of (an external) world-sheet electric current. We show that cancellation of the pertinent conformal anomaly on the quantum level requires the dynamical string tension to be a square of a free massless world-sheet scalar field. 
  We discuss hypermultiplets admitting degenerate discrete vacua and BPS domain walls interpolating them. This talk is based on the original papers, hep-th/0307274, hep-th/0211103 and hep-th/0302028. 
  We study O($N+1$) nonlinear $\sigma$-model in $(p+1+N)$-dimensional curved spacetime with negative cosmological constant, and find a new $\sigma$-lump solution with half-integer winding and divergent energy. When the spatial structure of $N$ extra-dimensions is determined by this global defect, a black $\sigma p$-brane surrounded by the degenerated horizon is formed and its near-horizon geometry is identified as a warp geometry of cigar type. 
  A constructive procedure is proposed for formulation of linear differential equations invariant under global symmetry transformations forming a semi-simple Lie algebra f. Under certain conditions f-invariant systems of differential equations are shown to be associated with f-modules that are integrable with respect to some parabolic subalgebra of f. The suggested construction is motivated by the unfolded formulation of dynamical equations developed in the higher spin gauge theory and provides a starting point for generalization to the nonlinear case. It is applied to the conformal algebra o(M,2) to classify all linear conformally invariant differential equations in Minkowski space. Numerous examples of conformal equations are discussed from this perspective. 
  We consider a prescription for introducing deformed dispersion relations in the bosonic string action. We find that in a subset of such theories it remains true that the embedding coordinates propagate linearly on the worldsheet. While both the string modes and the center of mass propagate with deformed dispersion relations, the speed of light remains energy independent. We consider the canonical quantization of these strings, and find that it is possible to choose theories so that ghost modes still decouple, as usual. However the Virasoro algebra now exhibits an energy dependent central charge. We also find that there are examples where the tachyon is eliminated from the spectrum of the free bosonic string. 
  The Schouten-Nijenhuis bracket is generalized for the superspace case and for the Poisson brackets of opposite Grassmann parities. 
  We calculate genus one corrections to Hermitian one-matrix model solution with arbitrary number of cuts directly from the loop equation confirming the answer previously obtained from algebro-geometrical considerations and generalizing it to the case of arbitrary potentials. 
  The Casimir energies and pressures for a massless scalar field associated with $\delta$-function potentials in 1+1 and 3+1 dimensions are calculated. For parallel plane surfaces, the results are finite, coincide with the pressures associated with Dirichlet planes in the limit of strong coupling, and for weak coupling do not possess a power-series expansion in 1+1 dimension. The relation between Casimir energies and Casimir pressures is clarified,and the former are shown to involve surface terms. The Casimir energy for a $\delta$-function spherical shell in 3+1 dimensions has an expression that reduces to the familiar result for a Dirichlet shell in the strong-coupling limit. However, the Casimir energy for finite coupling possesses a logarithmic divergence first appearing in third order in the weak-coupling expansion, which seems unremovable. The corresponding energies and pressures for a derivative of a $\delta$-function potential for the same spherical geometry generalizes the TM contributions of electrodynamics. Cancellation of divergences can occur between the TE ($\delta$-function) and TM (derivative of $\delta$-function) Casimir energies. These results clarify recent discussions in the literature. 
  We study the space-time invariances of the relativistic particle action for both the massive and massless cases. While the massive action has only the invariances associated to the Poincare algebra, we find that the invariances of the massless action give rise to the conformal algebra in four dimensions. For the free massless particle, a new invariance of the action permits the construction of an extension of the conformal algebra. The conclusion is that two distinct symmetry breaking mechanisms are necessary to arrive at the Poincare algebra. 
  Our understanding of the four basic concepts of Physics -- space, time, matter and force -- has undergone radical change in the course of work on unification, starting with Maxwell's unification of electricity with magnetism, all the way to present day string theory. What started as four independent concepts, with space and time postulated and the possible forms of matter and force arbitrarily chosen, now appear as different aspects of a rich and novel dynamically determined structure. 
  We generalize classical Yang-Mills theory by extending nonlinear constitutive equations for Maxwell fields to non-Abelian gauge groups. Such theories may or may not be Lagrangian. We obtain conditions on the constitutive equations specifying the Lagrangian case, of which recently-discussed non-Abelian Born-Infeld theories are particular examples. Some models in our class possess nontrivial Galilean (c goes to infinity) limits; we determine when such limits exist, and obtain them explicitly. 
  We extend Bousso's notion of a lightsheet - a surface where entropy can be defined in a way so that the entropy bound is satisfied - to more general surfaces. Intuitively these surfaces may be regarded as deformations of the Bousso choice; in general, these deformations will be timelike and so we refer to them as `timesheets'. We show that a timesheet corresponds to a section of a certain twistor bundle over a given spacelike two-surface B. We further argue that increasing the entropy flux through a given region corresponds to increasing the volume of certain regions in twistor space. We further argue that in twistor space, it might be possible to give a purely topological characterization of a lightsheet, at least for suitably simple spacetimes. 
  In four dimensional general relativity, the fact that a Killing vector in a vacuum spacetime serves as a vector potential for a test Maxwell field provides one with an elegant way of describing the behaviour of electromagnetic fields near a rotating Kerr black hole immersed in a uniform magnetic field. We use a similar approach to examine the case of a five dimensional rotating black hole placed in a uniform magnetic field of configuration with bi-azimuthal symmetry, that is aligned with the angular momenta of the Myers-Perry spacetime. Assuming that the black hole may also possess a small electric charge we construct the 5-vector potential of the electromagnetic field in the Myers-Perry metric using its three commuting Killing vector fields. We show that, like its four dimensional counterparts, the five dimensional Myers-Perry black hole rotating in a uniform magnetic field produces an inductive potential difference between the event horizon and an infinitely distant surface. This potential difference is determined by a superposition of two independent Coulomb fields consistent with the two angular momenta of the black hole and two nonvanishing components of the magnetic field. We also show that a weakly charged rotating black hole in five dimensions possesses two independent magnetic dipole moments specified in terms of its electric charge, mass, and angular momentum parameters. We prove that a five dimensional weakly charged Myers-Perry black hole must have the value of the gyromagnetic ratio g=3. 
  We construct supersymmetric composite models of quarks and leptons from type IIA T^6/(Z_2 x Z_2) orientifolds with intersecting D6-branes. In case of T^6 = T^2 x T^2 x T^2 with no tilted T^2, a composite model of the supersymmetric SU(5) grand unified theory with three generations is constructed. In case of that one T^2 is tilted, a composite model with SU(3)_c x SU(2)_L x U(1)_Y gauge symmetry with three generations is constructed. These models are not realistic, but contain fewer additional exotic particles and U(1) gauge symmetries due to the introduction of the compositeness of quarks and leptons. The masses of some exotic particles are naturally generated through the Yukawa interactions among "preons". 
  The evolution of the extra dimension is investigated in the context of brane world cosmology. New cosmological solutions are found. In particular, solutions in the form of waves travelling along the extra dimension are identified. 
  In the present work some examples of toric hyperkahler metrics in eight dimensions are constructed. First it is described how the Calderbank-Pedersen metrics arise as a consequence of the Joyce description of selfdual structures in four dimensions, the Jones-Tod correspondence and a result due to Tod and Przanowski. It is also shown that any quaternionic Kahler metric with $T^2$ isometry is locally isometric to a Calderbank-Pedersen one. The Swann construction of hyperkahler metrics in eight dimensions is applied to them to find hyperkahler examples with $U(1)\times U(1)$ isometry. The connection with the Pedersen-Poon toric hyperkahler metrics is explained and it is shown that there is a class of solutions of the generalized monopole equation in $\mathbb{R}^2 \otimes Im\mathbb{H}$ related to eigenfunctions of certain linear equation. This hyperkahler examples are lifted to solutions of the D=11 supergravity and type IIA and IIB backgrounds are found by use of dualities. As before, all the description is achieved in terms of a single eigenfunction F. Some explicit F are found, together with the Toda structure corresponding to the trajectories of the Killing vectors of the Calderbank-Pedersen bases. 
  In this talk we discuss the geometric realization of B fields and higher p-form potentials on a manifold M as connections on affine bundles over M. We realize D branes on M as special submanifolds of these affine bundles. As an application of this geometric understanding of the B field, we give a simple geometric explanation for the Chern-Simons modification of the field strength of the B field. 
  We rewrite the N=(2,2) non-linear sigma model using auxiliary spinorial superfields defining the model on ${\cal T}\oplus^ *{\cal T}$, where ${\cal T}$ is the tangent bundle of the target space. This is motivated by possible connections to Hitchin's generalized complex structures. We find the general form of the second supersymmetry compatible with that of the original model. 
  We discuss exact results for the full nonperturbative effective superpotentials of four dimensional $\mathcal{N}=1$ supersymmetric U(N) gauge theories with additional chiral superfield in the adjoint representation and the free energies of the related zero dimensional bosonic matrix models with polynomial potentials in the planar limit using the Dijkgraaf-Vafa matrix model prescription and integrating in and out. The exact effective superpotentials are produced including the leading Veneziano-Yankielowicz term directly from the matrix models. We also discuss how to use integrating in and out as a tool to do random matrix integrals in the large $N$ limit. 
  With the non-Abelian Hyper-Kahler quotient by U(M) and SU(M) gauge groups, we give the massive Hyper-Kahler sigma models that are not toric in the N=1 superfield formalism. The U(M) quotient gives N!/[M! (N-M)!] (N is a number of flavors) discrete vacua that may allow various types of domain walls, whereas the SU(M) quotient gives no discrete vacua. We derive BPS domain wall solution in the case of N=2 and M=1 in the U(M) quotient model. 
  We analyze the divergent part of the one-loop effective action for the noncommutative SU(2) gauge theory coupled to the fermions in the fundamental representation. We show that the divergencies in the 2-point and the 3-point functions in the $\theta$-linear order can be renormalized, while the divergence in the 4-point fermionic function cannot. 
  It is shown that the non-abelian vectorial model, proposed by C.R.Hagen is obtained using the self-interaction mechanism. The equivalence between this model and the non-abelian topologically massive one is studied showing that the existing equivalence between the abelian models is not sustained. 
  We point out that a field \phi charged under a global U(1) symmetry generally allows for a starred localized extension with the transformation rule, \phi\to U_L\star\phi\star U_R^{-1}. This results in a double gauging of the global U(1) symmetry on noncommutative space. We interpret the gauge theory so obtained in terms of the gauge fields that in the commutative limit appear naturally and are respectively the gauge field responsible for the charge and a decoupled vector field. The interactions are shown to be very different from those obtained by assigning a transformation rule of \phi\to U\star\phi or \phi\star U^{-1}. 
  In certain backgrounds string theory exhibits quantum Hall-like behavior. These backgrounds provide an explicit realization of the effective non-commutative gauge theory description of the fractional quantum Hall effect (FQHE), and of the corresponding large N matrix model. I review results on the string theory realization of the two-dimensional fractional quantum Hall fluid (FQHF), and describe new results on the stringy description of higher-dimensional analogs. 
  Lagrangian constraints of the spin 2 selfdual theory in a 2+1 flat space-time are studied and the one degree of freedom reduced action is obtained. From this formulation, the quantum operator algebra is computed and the spin contribution on transformation generators is explored. 
  We describe how the ingredients and results of the Seiberg-Witten solution to N=2 supersymmetric U(N) gauge theory may be obtained from a matrix model. 
  The thin shell collapse leading to the formation of charged rotating black holes in three dimensions is analyzed in the light of a recently developed Hamiltonian formalism for these systems. It is proposed to demand, as a way to reconcile the properties of an infinitely extended solenoid in flat space with a magnetic black hole in three dimensions, that the magnetic field should vanish just outside the shell. The adoption of this boundary condition results in an exterior solution with a magnetic field different from zero at a finite distance from the shell. The interior solution is also found and assigns another interpretation, in a different context, to the magnetic solution previously obtained by Cl\'{e}ment and by Hirschmann and Welch. 
  As is well known, the universally accepted theory as quantum gravity (QG) doesn't exist. One of the main reasons for that is that quantized general relativity is perturbatively nonrenormalizable. But there are several theories whose low-energy effective action is general relativity such as super string theory, supergravity and so on. On the other hand there is the prospect that quantized general relativity will become consistent nonperturbatively, for example by way of the $\epsilon$ - expanded analysis in 2 + $\epsilon$ gravity and the implication of the exact renormalization group equation (ERGE) and so on. Thus, the reason that we can't check which is right, is because of no constraint from experiments. By the way, the early Universe can be considered a good laboratory for QG, because of the high energy scale. In this period it is thought that the effect of QG governed the Universe. Therefore to construct the Universe model out of consideration of a certain type of QG can be thought of the test of the theory. Here the result of quantized general relativity, which is treated nonperturbatively by ERGE, is checked. The application of QG improved Einstein equation to the very early Universe shows several characteristic properties. At first the Universe become free from the initial singularity. The second is the origin of the cosmological time is uniquely decided. Furthermore this result doesn't destroy the success that the current cosmology attained. 
  The splitting of a $Q$-deformed boson, in the $Q\to q=e^{\frac{\QTR{rm}{2\pi i}}{\QTR{rm}{k}}}$ limit, is discussed. The equivalence between a $Q$-fermion and an ordinary one is established. The properties of the quantum (super)Virasoro algebras when their deformation parameter $Q$ goes to a root of unity, are investigated. These properties are shown to be related to fractional supersymmetry and $k$-fermionic spin. 
  String-inspired cosmologies, whereby a non-singular curvature bounce is induced by a general-covariant, $T$-duality-invariant, non-local dilaton potential, are used to study numerically how inhomogeneities evolve and to compare the outcome with analytic expressions obtained through different matching conditions across the bounce. Good agreement is found if continuity across the bounce is assumed to hold for $\cal{R}$, the curvature perturbation on comoving hypersurfaces, rather than for the Bardeen potential. 
  We present explicit formulae for q-exponentials on quantum spaces which could be of particular importance in physics, i.e. the q-deformed Minkowski-space and the q-deformed Euclidean space with two, three or four dimensions. Furthermore, these formulae can be viewed as 2-, 3- or 4-dimensional analogues of the well-known q-exponential function. 
  This talk adapts the available formalism to study a class of heterotic M-theory vacua with SO(10) grand unification group. Compactification to four dimensions with N = 1 supersymmetry is achieved on a torus fibered Calabi-Yau 3-fold Z = X / tau_{X} with first homotopy group pi_{1}(Z) = Z_{2}. Here X is an elliptically fibered Calabi-Yau 3-fold which admits two global sections and \tau_{X} is a freely acting involution on X. The vacua in this class have net number of three generations of chiral fermions in the observable sector and may contain M5-branes in the bulk space which wrap holomorphic curves in Z. Vacua with nonvanishing and vanishing instanton charges in the observable sector are considered. The latter case corresponds to potentially viable matter Yukawa couplings. Since pi_{1}(Z) = Z_{2}, the grand unification group can be broken with Z_{2} Wilson lines.   The motivation is to use the above formalism to extend realistic free-fermionic models to the nonperturbative regime. The correspondence between these models and Z_{2} x Z_{2} orbifold compactification of the weakly coupled 10-dimensional heterotic string identifies associated Calabi-Yau 3-folds which possess the structure of the above Z and X. A nonperturbative extension of the top quark Yukawa coupling is discussed. 
  We argue that bound states of branes have a size that is of the same order as the horizon radius of the corresponding black hole. Thus the interior of a black hole is not `empty space with a central singularity', and Hawking radiation can pick up information from the degrees of freedom of the hole. 
  We calculate the one-loop corrections to the mass and central charge of the BPS monopole in N=2 super-Yang-Mills theory in 3+1 dimensions using a supersymmetry-preserving version of dimensional regularization adapted to solitons. In the renormalization scheme where previous studies have indicated vanishing quantum corrections, we find nontrivial corrections that we identify as the 3+1 dimensional analogue of the anomaly in the conformal central charge of the N=1 supersymmetric kink in 1+1 dimensions. As in the latter case, the associated contribution to the ordinary central charge has exactly the required magnitude to preserve BPS saturation at the one-loop level. It also restores consistency of calculations involving sums over zero-point energies with the low-energy effective action of Seiberg and Witten. 
  Quantum vacuum energy has been known to have observable consequences since 1948 when Casimir calculated the force of attraction between parallel uncharged plates, a phenomenon confirmed experimentally with ever increasing precision. Casimir himself suggested that a similar attractive self-stress existed for a conducting spherical shell, but Boyer obtained a repulsive stress. Other geometries and higher dimensions have been considered over the years. Local effects, and divergences associated with surfaces and edges have been investigated by several authors. Quite recently, Graham et al. have re-examined such calculations, using conventional techniques of perturbative quantum field theory to remove divergences, and have suggested that previous self-stress results may be suspect. Here we show that most of the examples considered in their work are misleading; in particular, it is well-known that in two dimensions a circular boundary has a divergence in the Casimir energy for massless fields, while for general dimension $D$ not equal to an even integer the corresponding Casimir energy arising from massless fields interior and exterior to a hyperspherical shell is finite. It has also long been recognized that the Casimir energy for massive fields is divergent for curved boundaries. These conclusions are reinforced by a calculation of the relevant leading Feynman diagram in $D$ dimensions. Divergences do occur in third order, as has been recognized for many years, but this logarithmic divergence is of questionable relevance to real shells. 
  A possible link between Holography and Information Theory is presented. Using the relation between the Shannon and Boltzmann formulas Holography can be seen as the best encoding scheme. 
  Including translation matrices in covariant non-unitary Poincare representations alters the construction of massive particle fields from canonical unitary fields. The conventional procedure without spacetime translation matrices determines covariant fields that transform by matrix representations of the homogeneous Lorentz group combined with a differential operator representation of the Poincare group that acts on spacetime coordinates. The differential operator part generates translations nontrivially with the momentum operator proportional to the gradient, but the finite dimensional matrix part represents translations trivially, i.e. the momentum matrices vanish. This paper generalizes the construction of a massive particle field to produce a field that also transforms according to a finite dimensional non-unitary representation of translations generated by nonzero momentum matrices. The more general field evaluated at spacetime coordinates x results when the translation matrix for a displacement x is applied to the conventional field evaluated at x. 
  The large-N reduced models have been proposed as the nonperturbative formulation of the superstring theory. One of the most promising candidates is the IIB matrix model. While there have been a lot of interesting discoveries of the IIB matrix model in relation to the gravity, we have a lot of problems to surmount, if a large-N reduced model is to be an eligible framework to unify the gravitational interaction. Firstly, it is still an enigma how we can realize the local Lorentz invariant matrix model. In addition, we need to understand how we can describe the curved spacetime more manifestly, in terms of a large-N reduced model.   This thesis discusses several attempts to address these issues concerning the gravitational interaction. This thesis is based on the following author's works hep-th/0102168, hep-th/0204078, hep-th/0209057 and hep-th/0401038. 
  We establish a connection between the trace anomaly and a thermal radiation in the context of the standard cosmology. This is done by solving the covariant conservation equation of the stress tensor associated with a conformally invariant quantum scalar field. The solution corresponds to a thermal radiation with a temperature which is given in terms of a cut-off time excluding the spacetime regions very close to the initial singularity. We discuss the interrelation between this result and the result obtained in a two-dimensional schwarzschild spacetime. 
  Clifford number representation for linear electrodynamics with dyon sources is considered. Source function for the appropriate system of the first order equations for electromagnetic field is obtained. The field of an arbitrary moving point dyon is derived. 
  The one-loop effective action for a scalar field defined in the ultrastatic space-time where non standard logarithmic terms in the asymptotic heat-kernel expansion are present, is investigated by a generalisation of zeta-function regularisation. It is shown that additional divergences may appear at one-loop level. The one-loop renormalisability of the model is discussed and the one-loop renormalisation group equations are derived. 
  Kaluza-Klein reduction of the 3-dimensional gravitational Chern-Simons term leads to a 2-dimensional theory that supports a symmetry breaking solution and an associated kink interpolating between AdS and dS geometries. 
  A generalisation of discrete torsion is introduced in which different discrete torsion phases are considered for the different fixed points or twist fields of a twisted sector. The constraints that arise from modular invariance are analysed carefully. As an application we show how all the different resolutions of the T^7/Z_2^3 orbifold of Joyce have an interpretation in terms of such generalised discrete torsion orbifolds. Furthermore, we show that these manifolds are pairwise identified under G_2 mirror symmetry. From a conformal field theory point of view, this mirror symmetry arises from an automorphism of the extended chiral algebra of the G_2 compactification. 
  It is well known that Yukawa potentials permit bound states in the Schrodinger equation only if the ratio of the exchanged mass to bound mass is below a critical multiple of the coupling constant. However, arguments suggested by the Darwin term imply a more complex situation. By numerically studying the Dirac equation with a Yukawa potential we investigate this amplification effect. 
  We discuss new insights into the quantum physics of solitons developed since 1997: why quantum corrections to the mass M and the central charge Z of solitons in supersymmetric (susy) field theories in 1+1 and 2+1 dimensions are nonvanishing, despite the fact that the zero-point energies of bosons and fermions seem to cancel each other, and the central charge is an integral of a total space derivative which naively seems to get contributions only from regions far removed from the soliton. Crucial are: (1) the requirement that the regularization scheme not only makes calculations finite, but it also should preserve (ordinary) supersymmetry, (2) the renormalization condition that tadpoles vanish in the trivial vacuum, (3) an anomaly in the central charge which is actually needed to saturate the Bogomolnyi bound, (4) the influence of the winding of classical fields on the quantum fields far away from the soliton. 
  We prove that the real four-dimensional Euclidean noncommutative \phi^4-model is renormalisable to all orders in perturbation theory. Compared with the commutative case, the bare action of relevant and marginal couplings contains necessarily an additional term: an harmonic oscillator potential for the free scalar field action. This entails a modified dispersion relation for the free theory, which becomes important at large distances (UV/IR-entanglement). The renormalisation proof relies on flow equations for the expansion coefficients of the effective action with respect to scalar fields written in the matrix base of the noncommutative R^4. The renormalisation flow depends on the topology of ribbon graphs and on the asymptotic and local behaviour of the propagator governed by orthogonal Meixner polynomials. 
  Supersymmetric, asymptotically AdS5, black hole solutions of five dimensional gauged supergravity coupled to arbitrarily many abelian vector multiplets are presented. The general nature of supersymmetric solutions of this theory is discussed. All maximally supersymmetric solutions of this theory (with or without gauging) are obtained. 
  By embedding Einstein's original formulation of GR into a broader context we show that a dynamic covariant description of gravitational stress-energy emerges naturally from a variational principle. A tensor $T^G$ is constructed from a contraction of the Bel tensor with a symmetric covariant second degree tensor field $\Phi$ and has a form analogous to the stress-energy tensor of the Maxwell field in an arbitrary space-time. For plane-fronted gravitational waves helicity-2 polarised (graviton) states can be identified carrying non-zero energy and momentum. 
  The four (electro-magnetic, weak, strong and gravitational) interactions are described by singular Lagrangians and by Dirac-Bergmann theory of Hamiltonian constraints. As a consequence a subset of the original configuration variables are {\it gauge variables}, not determined by the equations of motion. Only at the Hamiltonian level it is possible to separate the gauge variables from the deterministic physical degrees of freedom, the {\it Dirac observables}, and to formulate a well posed Cauchy problem for them both in special and general relativity. Then the requirement of {\it causality} dictates the choice of {\it retarded} solutions at the classical level. However both the problems of the classical theory of the electron, leading to the choice of ${1\over 2}  (retarded + advanced)$ solutions, and the regularization of quantum field teory, leading to the Feynman propagator, introduce {\it anticipatory} aspects. The determination of the relativistic Darwin potential as a semi-classical approximation to the Lienard-Wiechert solution for particles with Grassmann-valued electric charges, regularizing the Coulomb self-energies, shows that these anticipatory effects live beyond the semi-classical approximation (tree level) under the form of radiative corrections, at least for the electro-magnetic interaction. 
  A dimensionally reduced expression for the QCD fermion determinant at finite temperature and chemical potential is derived which sheds light on the determinant's dependence on these quantities. This is done via a partial zeta regularisation, formally applying a general formula for the zeta-determinant of a differential operator in one variable with operator-valued coefficients. The resulting expression generalises the known one for the free fermion determinant, obtained via Matsubara frequency summation, to the case of general background gauge field; moreover there is no undetermined overall factor. Rigorous versions of this result are obtained in a continuous time--lattice space setting. The determinant expression reduces to a remarkably simple form in the low temperature limit. A program for how to use this to obtain insight into the QCD phase transition at zero temperature and nonzero density is outlined. 
  We derive stringy Ward identities from the decoupling of two types of zero-norm states in the old covariant first quantized (OCFQ) spectrum of open bosonic string. These Ward identities are valid to all energy and all loop orders in string perturbation theory. The high-energy limit of these stringy Ward identities can then be used to fix the proportionality constants between scattering amplitudes of different string states algebraically without referring to Gross and Mende's saddle point calculation of high-energy string-loop amplitudes. As examples, all Ward identities for the mass level 4 and 6 are derived, their high-energy limits are calculated and the proportionality constants between scattering amplitudes of different string states are determined. In addition to those identified before, we discover some new nonzero components of high-energy amplitudes not found previously by Gross and Manes. These components are essential to preserve massive gauge invariances or decouple massive zero-norm states of string theory. A set of massive scattering amplitudes and their high energy limits are calculated explicitly for each mass level to justify our results. 
  We consider supersymmetric quantum mechanical systems in arbitrary dimensions on curved spaces with nontrivial gauge fields. The square of the Dirac operator serves as Hamiltonian. We derive a relation between the number of supercharges that exist and restrictions on the geometry of the underlying spaces as well as the admissible gauge field configurations. From the superalgebra with two or more real supercharges we infer the existence of integrability conditions and obtain a corresponding superpotential. This potential can be used to deform the supercharges and to determine zero modes of the Dirac operator. The general results are applied to the Kahler spaces CP^n. 
  We construct fractional branes in Landau-Ginzburg orbifold categories and study their behavior under marginal closed string perturbations. This approach is shown to be more general than the rational boundary state construction. In particular we find new D-branes on the quintic -- such as a single D0-brane -- which are not restrictions of bundles on the ambient projective space. We also exhibit a family of deformations of the D0-brane in the Landau-Ginzburg category parameterized by points on the Fermat quintic. 
  I describe some phenomenological contexts in which it is possible to investigate effects induced by (string-motivated) canonical noncommutative spacetime. Due to the peculiar structure of the theory the usual criteria adopted for the choice of experimental contexts in which to test a theory may not be applicable here; care is required in taking into account the effects of IR/UV mixing. This invites one to consider contexts involving particles of relatively high energies, like high-energy cosmic rays and certain high-energy gamma rays observed from distant astrophysical sources. 
  We systematically construct and study Type II Orientifolds based on Gepner models which have N=1 supersymmetry in 3+1 dimensions. We classify the parity symmetries and construct the crosscap states. We write down the conditions that a configuration of rational branes must satisfy for consistency (tadpole cancellation and rank constraints) and spacetime supersymmetry. For certain cases, including Type IIB orientifolds of the quintic and a two parameter model, one can find all solutions in this class. Depending on the parity, the number of vacua can be large, of the order of 10^{10}-10^{13}. For other models, it is hard to find all solutions but special solutions can be found -- some of them are chiral. We also make comparison with the large volume regime and obtain a perfect match. Through this study, we find a number of new features of Type II orientifolds, including the structure of moduli space and the change in the type of O-planes under navigation through non-geometric phases. 
  We present a short review of hep-th/0306170. In the context of AdS/CFT correspondence, we explore what information from behind the horizon of the bulk black hole geometry can be found in boundary CFT correlators. In particular, we argue that the CFT correlators contain distinct, albeit subtle, signals of the black hole singularity. 
  We study boundary renormalization group flows of N=2 minimal models using Landau-Ginzburg description of B-type. A simple algebraic relation of matrices is relevant. We determine the pattern of the flows and identify the operators that generate them. As an application, we show that the charge lattice of B-branes in the level k minimal model is Z_{k+2}. We also reproduce the fact that the charge lattice for the A-branes is Z^{k+1}, applying the B-brane analysis on the mirror LG orbifold. 
  The double triangle algebra(DTA) associated to an ADE graph is considered. A description of its bialgebra structure based on a reconstruction approach is given. This approach takes as initial data the representation theory of the DTA as given by Ocneanu's cell calculus. It is also proved that the resulting DTA has the structure of a weak *-Hopf algebra. As an illustrative example, the case of the graph A3 is described in detail. 
  Deformation of N=2 quiver gauge theories by adjoint masses leads to fixed manifolds of N=1 superconformal field theories. We elaborate on the role of the complex three-form flux in the IIB duals to these fixed point theories, primarily using field theory techniques. We study the moduli space at a fixed point and find that it is either the two (complex) dimensional ALE space or three-dimensional generalized conifold, depending on the type of three-form flux that is present. We describe the exactly marginal operators that parameterize the fixed manifolds and find the operators which preserve the dimension of the moduli space. We also study deformations by arbitrary superpotentials W(\Phi_i) for the adjoints. We invoke the a-theorem to show that there are no dangerously irrelevant operators like Tr\Phi_i^{k+1}, k>2 in the N=2 quiver gauge theories. The moduli space of the IR fixed point theory generally contains orbifold singularities if W(\Phi_i) does not give a mass to the adjoints. Finally we examine some nonconformal N=1 quiver theories. We find evidence that the moduli space at the endpoint of a Seiberg duality cascade is always a three-dimensional generalized conifold. In general, the low-energy theory receives quantum corrections. In several non-cascading theories we find that the moduli space is a generalized conifold realized as a monodromic fibration. 
  A linear system, which generates a Moyal-deformed two-dimensional soliton equation as integrability condition, can be extended to a three-dimensional linear system, treating the deformation parameter as an additional coordinate. The supplementary integrability conditions result in a first order differential equation with respect to the deformation parameter, the flow of which commutes with the flow of the deformed soliton equation. In this way, a deformed soliton hierarchy can be extended to a bigger hierarchy by including the corresponding deformation equations. We prove the extended hierarchy properties for the deformed AKNS hierarchy, and specialize to the cases of deformed NLS, KdV and mKdV hierarchies. Corresponding results are also obtained for the deformed KP hierarchy. A deformation equation determines a kind of Seiberg-Witten map from classical solutions to solutions of the respective `noncommutative' deformed equation. 
  The problems which arise for a relativistic quantum mechanics are reviewed and critically examined in connection with the foundations of quantum field theory. The conflict between the quantum mechanical Hilbert space structure, the locality property and the gauge invariance encoded in the Gauss' law is discussed in connection with the various quantization choices for gauge fields 
  We study numerically the continuum limit corresponding to the non-trivial fixed point of Dyson's hierarchical model. We discuss the possibility of using the critical amplitudes as input parameters. We determine numerically the leading and subleading critical amplitudes of the zero-momentum $2l$-point functions in the symmetric phase up to l=10 for randomly chosen local measures. In the infinite cutoff limit, the dimensionless renormalized coupling constants are in very good approximation universal (independent of the choice of the local measure). In addition, ratios of subleading amplitudes also appear to be universal. If we neglect very small log-periodic corrections, the non-universal features of the 2l-point functions appear to depend only the non-universal features of the 2-point function. We infer that when 2l becomes large, the dimensionless renormalized couplings grow as (2l)! despite the non-perturbative nature of our calculation, while the universal ratios of subleading amplitudes grow linearly. 
  It is shown that, with the only exception of $n=2$, the Einstein-Hilbert action in $n+D+d$ dimensions, with $n$ times, is invariant under the duality transformation $a\to \frac{1}{a}$ and $b\to \frac{1}{b}$, where $a$ is a Friedmann-Robertson-Walker scale factor in $D$ dimensions and $b$ a Brans-Dicke scalar field in $d$ dimensions respectively. We investigate the $2+D+d$ dimensional cosmological model in some detail. 
  I show that the expectation value of the composite field $T{\bar T}$, built from the components of the energy-momentum tensor, is expressed exactly through the expectation value of the energy-momentum tensor itself. The relation is derived in two-dimensional quantum field theory under broad assumptions, and does not require integrability. 
  It has been known for quite some time that the N=4 super Yang-Mills equations defined on four-dimensional Euclidean space are equivalent to certain constraint equations on the Euclidean superspace R^(4|16). In this paper we consider the constraint equations on a deformed superspace R^(4|16)_\hbar a la Seiberg and derive the deformed super Yang-Mills equations. In showing this, we propose a super Seiberg-Witten map. 
  We explicitly construct A-type orientifolds of supersymmetric Gepner models. In order to reduce the tadpole cancellation conditions to a treatable number we explicitly work out the generic form of the one-loop Klein bottle, annulus and Moebius strip amplitudes for simple current extensions of Gepner models. Equipped with these formulas, we discuss two examples in detail to provide evidence that in this setting certain features of the MSSM like unitary gauge groups with large enough rank, chirality and family replication can be achieved. 
  We consider a scalar field theory on AdS, and show that the usual AdS/CFT prescription is unable to map to the boundary a part of the information arising from the quantization in the bulk. We propose a solution to this problem by defining the energy of the theory in the bulk through the Noether current corresponding to time displacements, and, in addition, by introducing a proper generalized AdS/CFT prescription. We also show how this extended formulation could be used to consistently describe double-trace interactions in the boundary. The formalism is illustrated by focusing on the non-minimally coupled case using Dirichlet boundary conditions. 
  We review two ways in which smooth cosmological evolution between two de Sitter phases can be obtained from M/string-theory. Firstly, we perform a hyperbolic reduction of massive IIA* theory to D=6 N=(1,1) SU(2)xU(1) gauged de Sitter supergravity, which supports smooth cosmological evolution between dS_4 x S^2 and a dS_6-type geometry. Secondly, we obtain four-dimensional de Sitter gravity with SU(2) Yang-Mills gauge fields from a hyperbolic reduction of standard eleven-dimensional supergravity. The four-dimensional theory supports smooth cosmological evolution between dS_2 x S^2 and a dS_4-type geometry. Although time-dependent, these solutions arise from a first-order system via a superpotential construction. For appropriate choices of charges, these solutions describe an expanding universe whose expansion rate is significantly larger in the past than in the future, as required for an inflationary model. 
  We study properties of supergravity theories with non-compact gaugings, and their higher-dimensional interpretations via consistent reductions on the inhomogeneous non-compact hyperboloidal spaces {\cal H}^{p,q}. The gauged supergravities are free of ghosts, despite the non-compactness of the gauge groups. We give a general discussion of the existence of stationary points in the scalar potentials of such supergravities. These are of interest since they can be associated with de Sitter vacuum configurations. We give explicit results for consistent reductions on {\cal H}^{p,q} in various examples, derived from analytic continuation of previously-known consistent sphere reductions. In addition we also consider black hole and cosmological solutions, for specific examples of non-compact gaugings in D=4,5,7. 
  We review some recent developments in the subject of quantum corrections to soliton mass and central charge. We consider in particular approaches which use local densities for these corrections, as first discussed by Hidenaga Yamagishi. We then consider dimensional regularization of the supersymmetric kink in 1+1 dimensions and an extension of this method to a 2+1-dimensional gauge theory with supersymmetric abelian Higgs vortices as the solitons. 
  We show that a certain class of short-distance cutoff can give rise to large suppression on the CMB anisotropies at large angular scales. 
  Theory of massless scalar field $\phi$ with interaction $g \phi^3$ in six-dimensional space is considered. A possibility of initial scale invariance breaking, which results in a spontaneous arising of effective interaction $G \phi^4$, is studied by application of Bogolubov quasi-averages approach. It is shown, that compensation equation for form-factor of this interaction in approximation up to the third order in $G$ has a non-trivial solution. The conditions imposed on form-factor value at zero and scalar field mass $m$ fix the unique solution, which gives relations between parameters of interaction $g \phi^3$ and parameters $G $ and $m$. Arguments are laid down in favour of a stability of the non-trivial solution. 
  The aim of this report is to give an overview of the duality between type IIB string theory on the maximally supersymmetric PP-wave and the BMN sector of N=4 Super Yang-Mills theory. The general features of the string and the field theory descriptions are reviewed, but the main focus of this report is on the comparison between the two sides of the duality. In particular, it is first explained how free IIB strings emerge on the gauge theory side and then the generalizations of this relation to the full interacting theory are considered. An ``historical'' approach is taken and the various proposals presented in the literature are described. 
  In this review we describe the general geometrical framework of brane world constructions in orientifolds of type IIA string theory with D6-branes wrapping 3-cycles in a Calabi-Yau 3-fold. These branes generically intersect in points on the internal space, and the patterns of intersections govern the chiral fermion spectra. We discuss how the open string spectra in intersecting brane models are constructed, how the Standard Model can be embedded, and also how supersymmetry can be realized in this class of string vacua. After the general considerations we specialize the discussion to the case of orbifold backgrounds with intersecting D6-branes and to the quintic Calabi-Yau manifold. Then, we discuss parts of the effective action of intersecting brane world models. Specifically we compute from the Born-Infeld action of the wrapped D-branes the tree-level, D-term scalar potential, which is important for the stability of the considered backgrounds as well as for questions related to supersymmetry breaking. Second, we review the recent computation concerning of gauge coupling unification and also of one-loop gauge threshold corrections in intersecting brane world models. Finally we also discuss some aspects of proton decay in intersecting brane world models. 
  This paper aims at presenting the first steps towards a formulation of the Exact Renormalization Group Equation in the Hopf algebra setting of Connes and Kreimer. It mostly deals with some algebraic preliminaries allowing to formulate perturbative renormalization within the theory of differential equations. The relation between renormalization, formulated as a change of boundary condition for a differential equation, and an algebraic Birkhoff decomposition for rooted trees is explicited. 
  I explain how to construct noncommutative BPS configurations in four and lower dimensions by solving linear matrix equations. Examples are instantons in D=4 Yang-Mills, monopoles in D=3 Yang-Mills-Higgs, and (moving) solitons in D=2+1 Yang-Mills-Higgs. Some emphasis is on the latter as a showcase for the dressing method. 
  We solve the tensionless string in a constant plane wave background and obtain a hugely degenerate spectrum. This is the case for a large class of plane wave backgrounds. We show that the solution can also be derived as a consistent limit of the quantized tensile theory of IIB strings in a pp-wave. This is in contrast to the situation for several other backgrounds. 
  After recalling the definition of black holes, and reviewing their energetics and their classical thermodynamics, one expounds the conjecture of Bekenstein, attributing an entropy to black holes, and the calculation by Hawking of the semi-classical radiation spectrum of a black hole, involving a thermal (Planckian) factor. One then discusses the attempts to interpret the black-hole entropy as the logarithm of the number of quantum micro-states of a macroscopic black hole, with particular emphasis on results obtained within string theory. After mentioning the (technically cleaner, but conceptually more intricate) case of supersymmetric (BPS) black holes and the corresponding counting of the degeneracy of Dirichlet-brane systems, one discusses in some detail the ``correspondence'' between massive string states and non-supersymmetric Schwarzschild black holes. 
  We show that for all very special quaternionic manifolds a different N=1 reduction exists, defining a Kaehler Geometry which is ``dual'' to the original very special Kaehler geometry with metric G_{a\bar{b}}= - \partial_a \partial_b \ln V (V={1/6}d_{abc}\lambda^a \lambda^b \lambda^c). The dual metric g^{ab}=V^{-2} (G^{-1})^{ab} is Kaehler and it also defines a flat potential as the original metric. Such geometries and some of their extensions find applications in Type IIB compactifications on Calabi--Yau orientifolds. 
  We discuss the mechanism of truncations driven by the imposition of constraints. We show how the consistency of such truncations is controlled, and give general theorems that establish conditions for the correct uplifting of solutions. We show in some particular examples how one can get correct upliftings from 7d supergravities to 10d type IIB supergravity, even in cases when the truncation is not initially consistent by its own. 
  The Green-Schwarz action for a non-BPS p=2 brane embedded in a N=1, D=4 target superspace is shown to be equivalent to the Nambu-Goto-Akulov-Volkov action obtained via the nonlinear realization of the associated target space super-Poincare symmetries. Introducing a U(1) gauge field strength as a Lagrange multiplier, this p=2 brane action is re-cast into its equivalent dual form non-BPS D2-brane Born-Infeld action. Following the procedure given by Sen, the Green-Schwarz action for a non-BPS D2-brane is determined directly. From it, conversely, the dual form non-BPS p=2 brane action is derived. The p=2 brane and the D2-brane actions obtained by these two approaches are different in form. Through explicitly determined field redefinitions, these actions are shown to be equivalent. 
  We study decoupling in FRW spacetimes, emphasizing a Lagrangian description throughout. To account for the vacuum choice ambiguity in cosmological settings, we introduce an arbitrary boundary action representing the initial conditions. RG flow in these spacetimes naturally affects the boundary interactions. As a consequence the boundary conditions are sensitive to high-energy physics through irrelevant terms in the boundary action. Using scalar field theory as an example, we derive the leading dimension four irrelevant boundary operators. We discuss how the known vacuum choices, e.g. the Bunch-Davies vacuum, appear in the Lagrangian description and square with decoupling. For all choices of boundary conditions encoded by relevant boundary operators, of which the known ones are a subset, backreaction is under control. All, moreover, will generically feel the influence of high-energy physics through irrelevant (dimension four) boundary corrections. Having established a coherent effective field theory framework including the vacuum choice ambiguity, we derive an explicit expression for the power spectrum of inflationary density perturbations including the leading high energy corrections. In accordance with the dimensionality of the leading irrelevant operators, the effect of high energy physics is linearly proportional to the Hubble radius H and the scale of new physics L= 1/M. 
  In general or normal random matrix ensembles, the support of eigenvalues of large size matrices is a planar domain (or several domains) with a sharp boundary. This domain evolves under a change of parameters of the potential and of the size of matrices. The boundary of the support of eigenvalues is a real section of a complex curve. Algebro-geometrical properties of this curve encode physical properties of random matrix ensembles. This curve can be treated as a limit of a spectral curve which is canonically defined for models of finite matrices. We interpret the evolution of the eigenvalue distribution as a growth problem, and describe the growth in terms of evolution of the spectral curve. We discuss algebro-geometrical properties of the spectral curve and describe the wave functions (normalized characteristic polynomials) in terms of differentials on the curve. General formulae and emergence of the spectral curve are illustrated by three meaningful examples. 
  Using the loop equations we find an explicit expression for genus 1 correction in hermitian two-matrix model in terms of holomorphic objects associated to spectral curve arising in large N limit. Our result generalises known expression for $F^1$ in hermitian one-matrix model. We discuss the relationship between $F^1$, Bergmann tau-function on Hurwitz spaces, G-function of Frobenius manifolds and determinant of Laplacian over spectral curve. 
  Implications of the SL(2,R) symmetry of the c = 1 matrix models are explored. Based on the work of de Alfaro, Fubini and Furlan, we note that when the Fermi sea is drained, the matrix model for 2 dimensional string theory in the linear dilaton background is equivalent to the matrix model of AdS_2 recently proposed by Strominger, for which SL(2,R) is an isometry. Utilizing its Lie algebra, we find that a topological property of AdS_2 is responsible for quantizing D0-brane charges in type 0A theory. We also show that the matrix model faithfully reflects the relation between the Poincare patch and global coordinates of AdS_2. 
  It is known that the T-dual of a circle bundle with H-flux (given by a Neveu-Schwarz 3-form) is the T-dual circle bundle with dual H-flux. However, it is also known that torus bundles with H-flux do not necessarily have a T-dual which is a torus bundle. A big puzzle has been to explain these mysterious "missing T-duals.'' Here we show that this problem is resolved using noncommutative topology. It turns out that every principal 2-torus-bundle with H-flux does indeed have a T-dual, but in the missing cases (which we characterize), the T-dual is non-classical and is a bundle of noncommutative tori. The duality comes with an isomorphism of twisted K-theories, just as in the classical case. The isomorphism of twisted cohomology which one gets in the classical case is replaced by an isomorphism of twisted cyclic homology. 
  An action principle that applies uniformly to any number N of supercharges is proposed. We perform the reduction to the N=0 partition function by integrating out superpartner fields. As a new feature for theories of extended supersymmetry, the canonical Pfaffian measure factor is a result of a Gaussian integration over a superpartner. This is mediated through an explicit choice of direction n^a in the \theta-space, which the physical sector does not depend on. Also, we re-interpret the metric g^{ab} in the Susy algebra [D^a,D^b] = g^{ab}\partial_t as a symplectic structure on the fermionic \theta-space. This leads to a superfield formulation with a general covariant \theta-space sector. 
  In the symplectic Lagrangian framework we newly embed an irreducible massive vector-tensor theory into a gauge invariant system, which has become reducible, by extending the configuration space to include an additional pair of scalar and vector fields, which give the desired Wess-Zumino action. A comparision with the BFT Hamiltonian embedding approach is also done. 
  We employ the effective field theory method to systematically study the short-range interaction in two-body sector in 2, 3 and 4 spacetime dimensions, respectively. The phi**4 theory is taken as a specific example and matched onto the nonrelativistic effective theory to one loop level. An exact, Lorentz-invariant expression for the S-wave amplitude is presented, from which the nonperturbative information can be easily extracted. We pay particular attention to the renormalization group analysis in the 3 dimensions, and show that relativistic effects qualitatively change the renormalization group flow of higher-dimensional operators. There is one ancient claim that triviality of the 4-dimensional phi**4 theory can be substantiated in the nonrelativistic limit. We illustrate that this assertion arises from treating the interaction between two nonrelativistic particles as literally zero-range, which is incompatible with the Uncertainty Principle. The S-wave effective range in this theory is identified to be approximately 16/3pi times the Compton wavelength. 
  We combine I. background independent Loop Quantum Gravity (LQG) quantization techniques, II. the mathematically rigorous framework of Algebraic Quantum Field Theory (AQFT) and III. the theory of integrable systems resulting in the invariant Pohlmeyer Charges in order to set up the general representation theory (superselection theory) for the closed bosonic quantum string on flat target space. While we do not solve the, expectedly, rich representation theory completely, we present a, to the best of our knowledge new, non -- trivial solution to the representation problem. This solution exists 1. for any target space dimension, 2. for Minkowski signature of the target space, 3. without tachyons, 4. manifestly ghost -- free (no negative norm states), 5. without fixing a worldsheet or target space gauge, 6. without (Virasoro) anomalies (zero central charge), 7. while preserving manifest target space Poincar\'e invariance and 8. without picking up UV divergences. The existence of this stable solution is exciting because it raises the hope that among all the solutions to the representation problem (including fermionic degrees of freedom) we find stable, phenomenologically acceptable ones in lower dimensional target spaces, possibly without supersymmetry, that are much simpler than the solutions that arise via compactification of the standard Fock representation of the string. Moreover, these new representations could solve some of the major puzzles of string theory such as the cosmological constant problem. The solution presented in this paper exploits the flatness of the target space in several important ways. In a companion paper we treat the more complicated case of curved target spaces. 
  In this note we present several ideas toward the solution to the giant graviton puzzle - the apparent multiplicity of supergravity states dual to field theory chiral primary operators. We use the fact that, for certain ranges of the angular momentum, giant gravitons can be mapped into vacua of a dual theory to argue that the sphere and $AdS$ giant gravitons have very different boundary descriptions, and that an unpolarized KK graviton is unphysical in the regime where giant gravitons exist. We also show that a generic boundary state can correspond to different giant graviton configurations, which have non-overlapping ranges of validity. 
  In these proceedings we review the main results concerning superspace geometries with nonanticommutative spinorial variables and field theories formulated on them. In particular, we report on the quantum properties of the WZ model formulated in the N=1/2 nonanticommutative superspace. 
  Motivated by the representation of the super Virasoro constraints as generalized Dirac-K{\"a}hler constraints $(d \pm d^\dagger)|\psi> = 0$ on loop space, examples of the most general continuous deformations $d \to e^{-W} d e^W$ are considered which preserve the superconformal algebra at the level of Poisson brackets. The deformations which induce the massless NS and NS-NS backgrounds are exhibited. Hints for a manifest realization of S-duality in terms of an algebra isomorphism are discussed. It is shown how the first order theory of 'canonical deformations' is reproduced and how the deformation operator $W$ encodes vertex operators and gauge transformations. 
  We show that, in gauge theory of principal connections, any gauge non-invariant Lagrangian can be completed to the BRST-invariant one. The BRST extension of the global Chern-Simons Lagrangian is present. 
  Properties of nonlinear higher spin gauge theories of totally symmetric massless higher spin fields in anti-de Sitter space of any dimension are discussed with the emphasize on the general aspects of the approach. 
  A condensed introduction to quantum gauge theories is given in the perturbative S-matrix framework; path integral methods are used nowhere. This approach emphasizes the fact that it is not necessary to start from classical gauge theories which are then subject to quantization, but it is also possible to recover the classical group structure and coupling properties from purely quantum mechanical principles. As a main tool we use a free field version of the Becchi-Rouet-Stora-Tyutin gauge transformation, which contains no interaction terms related to a coupling constant. This free gauge transformation can be formulated in an analogous way for quantum electrodynamics, Yang-Mills theories with massless or massive gauge bosons and quantum gravity. 
  We apply two different numerical methods to solve for the boundstate of two 0-branes in three dimensions. One method is developed by us in this work and we compare it to a method existing in the literature. In spite of considering only three dimensional Minkowski space we obtain interesting results which should give some basic understanding of the behaviour of 0-branes. 
  In order to overcome ambiguity problem on identification of mathematical objects in noncommutative theory with physical observables, quantum mechanical system coupled to the NC U(1) gauge field in the noncommutative space is reformulated by making use of the unitarized Seiberg-Witten map, and applied to the Aharonov-Bohm and Hall effects of the NC U(1) gauge field. Retaining terms only up to linear order in the NC parameter \theta, we find that the AB topological phase and the Hall conductivity have both the same formulas as those of the ordinary commutative space with no \theta-dependence. 
  We have calculated the one loop effective potential of the vector multiplets arising from the compactification to five dimensions of heterotic M-theory on a Calabi-Yau manifold with h^{1,1}>1. We find that extensive cancellations between the fermionic and bosonic sectors of the theory cause the effective potential to vanish, with the exception of a higher order curvature term of the type which might arise from string corrections. 
  The Poincar\'e sector of a recently deformed conformal algebra is proposed to describe, after the identification of the deformation parameter with the Planck length, the symmetries of a new relativistic theory with two observer-independent scales (or DSR theory). Also a new non-commutative space-time is proposed. It is found that momentum space exhibits the same features of the DSR proposals preserving Lorentz invariance in a deformed way. The space-time sector is a generalization of the well known non-commutative $\kappa$-Minkowski space-time which however does not preserve Lorentz invariance, not even in the deformed sense. It is shown that this behavior could be expected in some attempts to construct DSR theories starting from the Poincar\'e sector of a deformed symmetry larger than Poincar\'e symmetry, unless one takes a variable Planck length. It is also shown that the formalism can be useful in analyzing the role of quantum deformations in the ``AdS-CFT correspondence". 
  The Foldy-Wouthuysen transformation for relativistic spin-1 particles interacting with nonuniform electric and uniform magnetic fields is performed. The Hamilton operator in the Foldy-Wouthuysen representation is determined. It agrees with the Lagrangian obtained by Pomeransky, Khriplovich, and Sen'kov. The validity of the Corben-Schwinger equations is confirmed. However, an attempt to generalize these equations in order to take into account the own quadrupole moment of particles was not successful. The known second-order wave equations are incorrect because they contain non-Hermitian terms. The correct second-order wave equation is derived. 
  We extend the graviphoton-corrected prepotential of five-dimensional pure U(N) super Yang-Mills, which was originally proposed by Nekrasov, by incorporating the effect of the five-dimensional Chern-Simons term. This extension allows us to reproduce by a gauge theory calculation the partition functions of corresponding topological A-model on local toric Calabi-Yau manifolds X^m_N for all m=0,1,...,N. The original proposal corresponds to the case m=0. 
  This thesis is devoted to the study of three problems on the Wess-Zumino-Witten (WZW) and Chern-Simons (CS) supergravity theories in the Hamiltonian framework: 1) The two-dimensional super WZW model coupled to supergravity is constructed. The canonical representation of Kac-Moody algebra is extended to the super Kac-Moody and Virasoro algebras. Then, the canonical action is constructed, invariant under local supersymmetry transformations. The metric tensor and Rarita-Schwinger fields emerge as Lagrange multipliers of the components of the super energy-momentum tensor. 2) In higher dimensions, CS theories are irregular systems, that is, they have constraints which are functionally dependent in some sectors of phase space. In these cases, the standard Dirac procedure must be redefined, as it is shown in the simplified case of finite number of degrees of freedom. Irregular systems fall into two classes depending on their behavior in the vicinity of the constraint surface. In one case, it is possible to regularize the system without ambiguities, while in the other, regularization is not always possible and the Hamiltonian and Lagrangian descriptions may be dynamically inequivalent. Irregularities have important consequences in the linearized approximation of nonlinear theories. 3) The dynamics of CS supergravity theory in D=5, based on the supersymmetric extension of the AdS algebra, su(2,2|4), is analyzed. A class of backgrounds is found, providing a regular and generic effective theory. Some of these backgrounds are shown to be BPS states. The charges for the simplest choice of asymptotic conditions are obtained, and they satisfy a supersymmetric extension of the classical WZW(4) algebra, associated to su(2,2|4). 
  The imaginary part of the two point functions of the superconformal anomalous currents are extracted from the cross-sections of semiclassical absorption of dilaton, RR-2 form and gravitino by the wrapped D5 branes. From the central terms of the two point functions anomalous Ward identity is established which relates the exact pre-potential of the ${\cal N}=2$ SUSY Yang-Mills theory with the vacuum expectation value of the anomaly multiplet. From the Ward identity, WDVV (Witten-Dijkgraaf-Verlinde-Verlinde) equation can be derived which is solved for the exact pre-potential. 
  Motivated by the recent interest in quantization of black hole area spectrum, we consider the area spectrum of Kerr and extremal Kerr black holes. Based on the proposal by Bekenstein and others that the black hole area spectrum is discrete and equally spaced, we implement Kunstatter's method to derive the area spectrum for the Kerr and extremal Kerr black holes. The real part of the quasinormal frequencies of Kerr black hole used for this computation is of the form $m\Omega$ where $\Omega$ is the angular velocity of the black hole horizon. The resulting spectrum is discrete but not as expected uniformly spaced. Thus, we infer that the function describing the real part of quasinormal frequencies of Kerr black hole is not the correct one. This conclusion is in agreement with the numerical results for the highly damped quasinormal modes of Kerr black hole recently presented by Berti, Cardoso and Yoshida. On the contrary, extremal Kerr black hole is shown to have a discrete area spectrum which in addition is evenly spaced. The area spacing derived in our analysis for the extremal Kerr black hole area spectrum is not proportional to $\ln 3$. Therefore, it does not give support to Hod's statement that the area spectrum $A_{n}=(4l^{2}_{p}ln 3)n$ should be valid for a generic Kerr-Newman black hole. 
  The Schouten-Nijenhuis bracket is generalized for the superspace case and for the Poisson brackets of opposite Grassmann parities. 
  The three--dimensional magnetic solution to the Einstein--Maxwell field equations have been considered by some authors. Several interpretations have been formulated for this magnetic spacetime. Up to now this solution has been considered as a two--parameter self--consistent field. We point out that the parameter related to the mass of this solution is just a pure gauge and can be rescaled to minus one. This implies that the magnetic metric has really a simple form and it is effectively one-parameter solution, which describes a distribution of a radial magnetic field in a 2+1 anti--de Sitter background space--time. We consider an alternative interpretation to the Dias--Lemos one for the magnetic field source. 
  A nonlinear generalisation of Schrodinger's equation is obtained using information-theoretic arguments. The nonlinearities are controlled by an intrinsic length scale and involve derivatives to all orders thus making the equation mildly nonlocal. The nonlinear equation is homogeneous, separable, conserves probability, but is not invariant under spacetime symmetries. Spacetime symmetries are recovered when a dimensionless parameter is tuned to vanish, whereby linearity is simultaneously established and the length scale becomes hidden. It is thus suggested that if, in the search for a more basic foundation for Nature's Laws, an inference principle is given precedence over symmetry requirements, then the symmetries of spacetime and the linearity of quantum theory might both be emergent properties that are intrinsically linked. Supporting arguments are provided for this point of view and some testable phenomenological consequences highlighted. The generalised Klien-Gordon and Dirac equations are also studied, leading to the suggestion that nonlinear quantum dynamics with intrinsically broken spacetime symmetries might be relevant to understanding the problem of neutrino mass(lessness) and oscillations: Among other observations, this approach hints at the existence of a hidden discrete family symmetry in the Standard Model of particle physics. 
  We study a class of holomorphic matrix models. The integrals are taken over middle dimensional cycles in the space of complex square matrices. As the size of the matrices tends to infinity, the distribution of eigenvalues is given by a measure with support on a collection of arcs in the complex planes. We show that the arcs are level sets of the imaginary part of a hyperelliptic integral connecting branch points. 
  We find all the higher dimensional solutions of the Einstein-Maxwell theory that are the topological product of two manifolds of constant curvature. These solutions include the higher dimensional Nariai, Bertotti-Robinson and anti-Nariai solutions, and the anti-de Sitter Bertotti-Robinson solutions with toroidal and hyperbolic topology (Plebanski-Hacyan solutions). We give explicit results for any dimension D>3. These solutions are generated from the appropriate extremal limits of the higher dimensional near-extreme black holes in a de Sitter, and anti-de Sitter backgrounds. Thus, we also find the mass and the charge parameters of the higher dimensional extreme black holes as a function of the radius of the degenerate horizon. 
  The field equations of the original Kaluza's theory are analyzed and it is shown that they lead to modification of Einstein's equations. The appearing extra energy-momentum tensor is studied and an example is given where this extra energy-momentum tensor is shown to allow four-dimensional Schwarzschild geometry to accommodate electrostatics. Such deviation from Reissner-Nordstrom geometry can account for the interpretation of Schwarzschild geometry as resulting not from mass only, but from the combined effects of mass and electric charge, even electric charge alone. 
  We find instanton/cosmological solutions with biaxial Bianchi-IX symmetry, involving non-trivial spatial dependence of the $\bbbc P^{1}$- and $\bbbc P^{2}$-sigma-models coupled to gravity. Such manifolds arise in N=1, $d=4$ supergravity with supermatter actions and hence the solutions can be embedded in supergravity. There is a natural way in which the standard coordinates of these manifolds can be mapped into the four-dimensional physical space. Due to its special symmetry, we start with $\bbbc P^{2}$ with its corresponding scalar Ansatz; this further requires the spacetime to be $SU(2) \times U(1)$-invariant. The problem then reduces to a set of ordinary differential equations whose analytical properties and solutions are discussed. Among the solutions there is a surprising, special-family of exact solutions which owe their existence to the non-trivial topology of $\bbbc P^{2}$ and are in 1-1 correspondence with matter-free Bianchi-IX metrics. These solutions can also be found by coupling $\bbbc P^{1}$ to gravity. The regularity of these Euclidean solutions is discussed -- the only possibility is bolt-type regularity. The Lorentzian solutions with similar scalar Ansatz are all obtainable from the Euclidean solutions by Wick rotation. 
  We search for regular tachyon kinks in an extended model, which includes the tachyon action recently proposed to describe the tachyon field. The extended model that we propose adds a new contribution to the tachyon action, and seems to enrich the present scenario for the tachyon field. We have found stable tachyon kinks of regular profile, which may appropriately lead to the singular kink found by Sen sometime ago. Also, under specific conditions we may find periodic array of kink-antikink configurations. 
  A description of the bosonic sector of ten-dimensional N=1 supergravity as a non-linear realisation is given. We show that if a suitable extension of this theory were invariant under a Kac-Moody algebra, then this algebra would have to contain a rank eleven Kac-Moody algebra, that can be identified to be a particular real form of very-extended D_8. We also describe the extension of N=1 supergravity coupled to an abelian vector gauge field as a non-linear realisation, and find the Kac-Moody algebra governing the symmetries of this theory to be very-extended B_8. Finally, we discuss the related points for the N=1 supergravity coupled to an arbitrary number of abelian vector gauge fields. 
  We compare the entropy as a function of energy of excited strings and black strings in an asymptotically plane wave background at the level of the correspondence principle. For the plane wave supported by the NSNS 3-form flux, neither the entropy formula nor the cross-over scale is affected by the presence of the flux and the correspondence is found to hold. For the plane wave supported by the RR 3-form flux, both the entropy and the cross-over point are modified, but the correspondence is still found to hold. 
  We consider Feynman's path integral approach to quantum mechanics with a noncommutativity in position and momentum sectors of the phase space. We show that a quantum-mechanical system with this kind of noncommutativity is equivalent to the another one with usual commutative coordinates and momenta. We found connection between quadratic classical Hamiltonians, as well as Lagrangians, in their commutative and noncommutative regimes. The general procedure to compute Feynman's path integral on this noncommutative phase space with quadratic Lagrangians (Hamiltonians) is presented. Using this approach, a particle in a constant field, ordinary and inverted harmonic oscillators are elaborated in detail. 
  The role of integrable systems in string theory is discussed. We remind old examples of the correspondence between stringy partition functions or effective actions and integrable equations, based on effective application of the matrix model technique. Then we turn to a new example, coming from the Nekrasov deformation of the Seiberg-Witten prepotential. In the last case the deformed theory is described by a different statistical model, which becomes equivalent to a partition function of a topological string. The full partition function of string theory arises therefore always as a certain "quantization" of its quasiclassical geometry. 
  We combine and exploit ideas from Coset Space Dimensional Reduction (CSDR) methods and Non-commutative Geometry. We consider the dimensional reduction of gauge theories defined in high dimensions where the compact directions are a fuzzy space (matrix manifold). In the CSDR one assumes that the form of space-time is M^D=M^4 x S/R with S/R a homogeneous space. Then a gauge theory with gauge group G defined on M^D can be dimensionally reduced to M^4 in an elegant way using the symmetries of S/R, in particular the resulting four dimensional gauge is a subgroup of G. In the present work we show that one can apply the CSDR ideas in the case where the compact part of the space-time is a finite approximation of the homogeneous space S/R, i.e. a fuzzy coset. In particular we study the fuzzy sphere case. 
  It is well-known that Scherk-Schwarz compactifications in string theory have a tachyon in the closed string spectrum appearing for a critical value of a compact radius. The tachyon can be removed by an appropriate orientifold projection in type II strings, giving rise to tachyon-free compactifications. We present explicit examples of this type in various dimensions, including six and four-dimensional chiral examples, with softly broken supersymmetry in the closed sector and non-BPS configurations in the open sector. These vacua are interesting frameworks for studying various cosmological issues. We discuss four-dimensional cosmological solutions and moduli stabilization triggered by nonperturbative effects like gaugino condensation on D-branes and fluxes. 
  We show how to formulate Yang-Mills Theory in \m{2+1} dimensions as a hamitonian system within a simplicial regularization and construct its quantization, with special attention to the mass gap. An approximate conformal invariance of the hamiltonian of this theory is useful to construct a continuum limit. 
  There exists a freedom in a class of four-dimensional electroweak theories proposed by Arkani-Hamed et al. relying on deconstruction and Coleman-Weinberg mechanism. The freedom comes from the winding modes of the link variable (Wilson operator) connecting non-nearest neighbours in the discrete fifth dimension. Using this freedom, dynamical breaking of SU(2) gauge symmetry, mass hierarchy patterns of fermions and Cabbibo-Kobayashi-Maskawa matrix may be obtained. 
  A scalar field in (2+1) dimensional Minkowski space-time is considered. Postulating noncommutative spatial coordinates, one is able to determine the (UV finite) vacuum expectation value of the quantum field energy momentum tensor. Calculation for the (3+1) case has been performed considering only two noncommutative coordinates. The results lead to a vacuum energy with a lowered degree of divergence, with respect to that of ordinary commutative theory. 
  We study the quotients of n+1-dimensional anti-de Sitter space by one-parameter subgroups of its isometry group SO(2,n) for general n. We classify the different quotients up to conjugation by O(2,n). We find that the majority of the classes exist for all n \geq 2. There are two special classes which appear in higher dimensions: one for n \geq 3 and one for n \geq 4. The description of the quotient in the majority of cases is thus a simple generalisation of the AdS_3 quotients. 
  This paper contains a classification of smooth Kaluza--Klein reductions (by one-parameter subgroups) of the maximally supersymmetric anti de Sitter backgrounds of supergravity theories. We present a classification of one-parameter subgroups of isometries of anti de Sitter spaces, discuss the causal properties of their orbits on these manifolds, and discuss their action on the space of Killing spinors. We analyse the problem of which quotients admit a spin structure. We then apply these results to write down the list of smooth everywhere spacelike supersymmetric quotients of AdS_3 x S^3 (x R^4), AdS_4 x S^7, AdS_5 x S^5 and AdS_7 x S^4, and the fraction of supersymmetry preserved by each quotient. The results are summarised in tables which should be useful on their own. The paper also includes a discussion of supersymmetry of singular quotients. 
  We consider a complex scalar field in (p+3)-dimensional bulk with a negative cosmological constant and study global vortices in two extra-dimensions. We reexamine carefully the coupled scalar and Einstein equations, and show that the boundary value of scalar amplitude at infinity of the extra-dimensions should be smaller than vacuum expectation value. The brane world has a cigar-like geometry with an exponentially decaying warp factor and a flat thick p-brane is embedded. Since a coordinate transformation identifies the obtained brane world as a black p-brane world bounded by a horizon, this strange boundary condition of the scalar amplitude is understood as existence of a short scalar hair. 
  Dirac's hole theory and quantum field theory are generally thought to be equivalent. In fact field theory can be derived from hole theory through the process of second quantization. However, it can be shown that problems worked in both theories yield different results. The reason for the difference between the two theories will be examined and the effect the this difference has on the way calculations are done in quantum theory will be examined. 
  Certain time-like singularities are shown to be resolved already in classical General Relativity once one passes from particle probes to scalar waves. The time evolution can be defined uniquely and some general conditions for that are formulated. The Reissner-Nordstrom singularity allows for communication through the singularity and can be termed "beam splitter" since the transmission probability of a suitably prepared high energy wave packet is 25%. The high frequency dependence of the cross section is w^{-4/3}. However, smooth geometries arbitrarily close to the singular one require a finite amount of negative energy matter. The negative-mass Schwarzschild has a qualitatively different resolution interpreted to be fully reflecting. These 4d results are similar to the 2d black hole and are generalized to an arbitrary dimension d>4. 
  It has been noticed that confinement effects can be described by the addition of a $ \sqrt {- F_{\mu \nu}^a F^{a\mu \nu}} $ term in the Lagrangian density. We now study the combined effect of such "confinement term" and that of a mass term. The surprising result is that the interplay between these two terms gives rise to a Coulomb interaction. Our picture has a certain correspondence with the quasiconfinement picture described by Giles, Jaffe and de Rujula for QCD with symmetry breaking. 
  We re-examine the problem of strong coupling in a regularized version of DGP (or ``brane-induced'') gravity. We find that the regularization of ref. hep-th/0304148 differs from DGP in that it does not exhibit strong coupling or ghosts up to cubic order in the interactions. We suggest that the nonlocal nature of the theory, when written in terms of the 4-D metric, is a plausible reason for this phenomenon. Finally, we briefly discuss the possible behavior of the model at higher-order in perturbation theory. 
  We describe some recent progress in our understanding of Yang-Mills theories formulated on noncommutative spaces and in particular how to formulate the standard model on such spaces. 
  In this article, we describe giant gravitons in AdS_5 x S^5 moving along generic trajectories in AdS_5. The giant graviton dynamics is solved by proving that the D3-brane effective action reduces to that of a massive point particle in AdS_5 and therefore the solutions are in one to one correspondence with timelike geodesics of AdS_5. All these configurations are related via isometries of the background, which induce target space symmetries in the world volume theory of the D-brane. Hence, all these configurations preserve the same amount of supersymmetry as the original giant graviton, i.e. half of the maximal supersymmetry. Multiparticle configurations of two or more giant gravitons are also considered. In particular, a binary system preserving one quarter of the supersymmetries is found, providing a non trivial time-dependent supersymmetric solution. A short study on the dual CFT description of all the above states is given, including a derivation of the exact induced isometry map in the CFT side of the correspondence. 
  Affine quantum gravity involves (i) affine commutation relations to ensure metric positivity, (ii) a regularized projection operator procedure to accomodate first- and second-class quantum constraints, and (iii) a hard-core interpretation of nonlinear interactions to understand and potentially overcome nonrenormalizability. In this program, some of the less traditional mathematical methods employed are (i) coherent state representations, (ii) reproducing kernel Hilbert spaces, and (iii) functional integral representations involving a continuous-time regularization. Of special importance is the profoundly different integration measure used for the Lagrange multiplier (shift and lapse) functions. These various concepts are first introduced on elementary systems to help motivate their application to affine quantum gravity. 
  We will describe some mathematical ideas of K. T. Chen on calculus on loop spaces. They seem useful to understand non-abelian Yang--Mills theories. 
  A rigorous treatment is given of the Green's function of the N=1 supersymmetric heat equation in one spatial dimension with a distribution initial value. The asymptotic expansion of the supersymmetric Green's function as t tends to 0+ is also derived. The coefficients of the expansion generate all the members of the supersymmetric N=1 KdV hierarchy. 
  We state some remarks on `$n$-dimensional Feynman diagrams' ($n\in\N$). There are different features between in the case of $n$-dimensional Feynman diagrams($n\geqq3$) and in the 1-, 2-dimensional case. 
  Following up on a recent paper by two of us (DM and SS), demonstrating the large enhancement in observable optical activity in radiation from high redshift sources arising from the string-based coupling of bulk Kalb-Ramond field to the Maxwell Chern Simons three-form on the brane in a Randall-Sundrum braneworld, we exhibit here a similar enhancement in parity-violating temperature-polarization correlations, yet unseen, in the CMB anisotropy due to a generalized parity-violating Kalb-Ramond axion-photon interaction proposed earlier by one of us (PM). The non-observation of such correlations in CMB anisotropies would necessitate unnatural fine tuning of the Kalb-Ramond axion parameters. As a stringy realization of Randall-Sundrum braneworld scenario is yet to be understood properly, our work indicates the need of a careful investigation to establish the connection between string-based phenomenological models and the Randall-Sundrum braneworld scenario 
  This is an exposition of joint work with S. Doplicher, K. Fredenhagen, and G. Piacitelli on field theory on the noncommutative Minkowski space. The limit of coinciding points is modified compared to ordinary field theory in a suitable way which allows for the definition of so-called regularized field monomials as interaction terms. Employing these in the Hamiltonian formalism results in an ultraviolet finite S-matrix. 
  The whole idea of holography as put forward by Gerard 't Hooft assumes that data on a boundary determine physics in the volume. This corresponds to a Dirichlet problem for euclidean signature, or to a Goursat (characteristic) problem in the lorentzian setting. Is this last aspect of the problem that is explored here for Ricci flat spaces with vanishing cosmological constant. 
  We study the cosmological properties of a codimension two brane world that sits at the intersection between two four branes, in the framework of six dimensional Einstein-Gauss-Bonnet gravity. Due to contributions of the Gauss-Bonnet terms, the junction conditions require the presence of localized energy density on the codimension two defect. The induced metric on this surface assumes a FRW form, with a scale factor associated to the position of the brane in the background; we can embed on the codimension two defect the preferred form of energy density. We present the cosmological evolution equations for the three brane, showing that, for the case of pure AdS$_6$ backgrounds, they acquire the same form of the ones for the Randall-Sundrum II model. When the background is different from pure AdS$_6$, the cosmological behavior is potentially modified in respect to the typical one of codimension one brane worlds. We discuss, in a particular model embedded in an AdS$_6$ black hole, the conditions one should satisfy in order to obtain standard cosmology at late epochs. 
  For a very large class of potentials, $V(\vec{x})$, $\vec{x}\in R^2$, we prove the universality of the low energy scattering amplitude, $f(\vec{k}', \vec{k})$. The result is $f=\sqrt{\frac{\pi}{2}}\{1/log k)+O(1/(log k)^2)$. The only exceptions occur if $V$ happens to have a zero energy bound state. Our new result includes as a special subclass the case of rotationally symmetric potentials, $V(|\vec{x}|)$. 
  The sigma model approach to the closed bosonic string on the affine-metric manifold is considered. The two-loop metric counterterms for the nonlinear two-dimensional sigma model with affine-metric target manifold are calculated. The correlation of the metric and affine connection is considered as the result of the ultraviolet finiteness (or beta-function vanishing) condition for the nonlinear sigma model. The examples of the nonflat nonRiemannian manifolds resulting in the trivial metric beta-function are suggested. 
  A perturbative analysis of a massive Chern-Simons is presented. The mass term is introduced in a BRST invariant way. With these method we prove that the number of independent renormalisation of massive Chern-Simons is equal to topologically massive Yang-Mills and different from pure Chern-Simons. 
  We have investigated the classical stability of charged black $D3$-branes in type IIB supergravity under small perturbations. For s-wave perturbations it turns out that black $D3$-branes are unstable when they have small charge density. As the charge density increases for given mass density, however, the instability decreases down to zero at a certain finite value of the charge density, and then black $D3$-branes become stable all the way down to the extremal point. It has also been shown that such critical value at which its stability behavior changes agrees very well with the predicted one by the thermodynamic stability behavior of the corresponding black hole system through the Gubser-Mitra conjecture. Unstable mode solutions we found involve non-vanishing fluctuations of the self-dual five-form field strength. Some implications of our results are also discussed. 
  It is shown that the pure spinor formulation of the heterotic superstring in a generic gravitational and super Yang-Mills background has vanishing one-loop beta functions. 
  Recently Sekino and Yoneya proposed a way to regularize the world volume theory of membranes wrapped around $S^1$ by matrices and showed that one obtains matrix string theory as a regularization of such a theory. We show that this correspondence between matrix string theory and wrapped membranes can be obtained by using the usual M(atrix) theory techniques. Using this correspondence, we construct the super-Poincare generators of matrix string theory at the leading order in the perturbation theory. It is shown that these generators satisfy 10 dimensional super-Poincar\'e algebra without any anomaly. 
  The thermodynamics of Schwarzschild black holes within an isothermal cavity and the associated Euclidean Dirichlet boundary-value problem are studied for four and higher dimensions in anti-de Sitter (AdS) space. For such boundary conditions classically there always exists a unique hot AdS solution and two or no Schwarzschild-AdS black-hole solutions depending on whether or not the temperature of the cavity-wall is above a minimum value, the latter being a function of the radius of the cavity. Assuming the standard area-law of black-hole entropy, it was known that larger and smaller holes have positive and negative specific heats and hence are locally thermodynamically stable and unstable respectively. In this paper we present the first derivation of this by showing that the standard area law of black-hole entropy holds in the semi-classical approximation of the Euclidean path integral for such boundary conditions. We further show that for wall-temperatures above a critical value a phase transition takes hot AdS to the larger Schwarzschild-AdS within the cavity. The larger hole thus can be globally thermodynamically stable above this temperature. The phase transition can occur for a cavity of arbitrary radius above a (corresponding) critical temperature. In the infinite cavity limit this picture reduces to that considered by Hawking and Page. The case of five dimensions is found to be rather special since exact analytic expressions can be obtained for the masses of the two holes as functions of cavity radius and temperature thus solving exactly the Euclidean Dirichlet problem. This makes it possible to compute the on-shell Euclidean action as functions of them from which other quantities of interest can be evaluated exactly. 
  To test the strong-gravity regime in Randall-Sundrum braneworlds, we consider black holes bound to a brane. In a previous paper, we studied numerical solutions of localized black holes whose horizon radii are smaller than the AdS curvature radius. In this paper, we improve the numerical method and discuss properties of the six dimensional (6D) localized black holes whose horizon radii are larger than the AdS curvature radius. At a horizon temperature $\mathcal{T} \approx 1/2\pi \ell$, the thermodynamics of the localized black hole undergo a transition with its character changing from a 6D Schwarzschild black hole type to a 6D black string type. The specific heat of the localized black holes is negative, and the entropy is greater than or nearly equal to that of the 6D black strings with the same thermodynamic mass. The large localized black holes show flattened horizon geometries, and the intrinsic curvature of the horizon four-geometry becomes negative near the brane. Our results indicate that the recovery mechanism of lower-dimensional Einstein gravity on the brane works even in the presence of the black holes. 
  We show that the BFT embedding method is problematic for mixed systems (systems possessing both first and second class constraints). The Chern-Simons theory as an example is worked out in detail. We give two methods to solve the problem leading to two different types of finite order BFT embedding for Chern-Simons theory. 
  Berezin integration over fermionic degrees of freedom as a standard tool of quantum field theory is analysed from the viewpoint of noncommutative geometry. It is shown that among the variety of contradictory integration prescriptions existing in the current literature, there is only one unique minimal set of consistent rules, which is compatible with Connes' normalized cyclic cohomology of the Gra{\ss}mann algebra. 
  We propose the notion of the oscillator on K\"ahler space and consider its supersymmetrization in the presence of a constant magnetic field. 
  We review recent work identifying soft SUSY-breaking terms in local type II string models with branes and magnetic fluxes. We then make a new observation about the configuration space of D-branes in Calabi-Yau backgrounds, and identify vevs for nonperturbative charged hypermultiplets in Calabi-Yau backgrounds with N=2 Fayet-Iliopoulos terms. 
  We review some recent developments on BPS string solutions and monopole confinement in the Higgs or (color) superconducting phase of deformed N=2 and N=4 super Yang-Mills theories. In particular, the monopole magnetic fluxes are shown to be always integer linear combinations of string fluxes. Moreover, a bound for the threshold length of the string breaking is obtained. When the gauge group SU(N) is broken to Z_N, the BPS string tension satisfies the Casimir scaling law. Furthermore in the SU(3) case the string solutions are such that they allow the formation of a confining system with three monopoles. 
  Taking as a starting point a Lorentz non-invariant Abelian-Higgs model defined in 1+3 dimensions, we carry out its dimensional reduction to D=1+2, obtaining a new planar model composed by a Maxwell-Chern-Simons-Proca gauge sector, a massive scalar sector, and a mixing term (involving the fixed background (v^{\mu}) that imposes the Lorentz violation to the reduced model. The propagators of the scalar and massive gauge field are evaluated and the corresponding dispersion relations determined. Based on the poles of the propagators, a causality and unitarity analysis is carried out at tree-level. One then shows that the model is totally causal and unitary. 
  In this paper we will study tachyon effective action for Dp-brane in bosonic string theory in the linear dilaton background. We obtain the tachyon effective Lagrangian from boundary state coeficient of Dp-brane in the linear dilaton background and compare it with tachyon effective Lagrangians that were proposed in previous papers. 
  A non-local toy model whose interaction consists of smeared, non-local field operators is presented. We work out the Feynman rules and propose a power counting formula for arbitrary graphs. Explicit calculations for one loop graphs show that their contribution is finite for sufficient smearing and agree with the power counting formula. UV/IR mixing does not occur. 
  We find a closed-form for the distribution function (defined in terms of a Wigner operator) for hot coloured particles in a background gluon field, in the hard thermal loop approximation. We verify that the current is the same as that derived from the known effective action. 
  We discuss supersymmetric black holes embedded in a Goedel-type universe with cosmological constant in five dimensions. The spacetime is a fibration over a four-dimensional Kaehler base manifold, and generically has closed timelike curves. Asymptotically the space approaches a deformation of AdS_5, which suggests that the appearance of closed timelike curves should have an interpretation in some deformation of D=4, N=4 super-Yang-Mills theory. Finally, a Goedel-de Sitter universe is also presented and its causal structure is discussed. 
  The three-point functions of two scalar fields $\sigma$ and the higher spin field $h^{(\ell)}$ of HS(4) on the one side and of their proposed holographic images $\alpha$ and $\mathcal{J^{(\ell)}}$ of the minimal conformal O(N) sigma model of dimension three on the other side are evaluated at leading perturbative order and compared in order to fix the coupling constant of HS(4). This necessitates a careful analysis of the local current $\Psi^{(\ell)}$ to which $h^{(\ell)}$ couples in HS(4) and which is bilinear in $\sigma$. 
  The possibility of a Chern-Simons like term generation in an extended model of QED, in which a Lorentz and CPT non-covariant interaction term for fermions is present, has been investigated at finite temperature and in the presence of a background color magnetic field. To this end, the photon polarization operator in an external constant axial-vector field has been considered. One-loop contributions to its antisymmetric component due to fermions in the linear order of the axial-vector field have been obtained. Moreover, the first nontrivial correction to the induced CS term due to the presence of a weak constant homogeneous color magnetic field has been derived. 
  We consider a dynamical brane world in a six dimensional spacetime containing a singularity. Using the Israel conditions we study the motion of a 4-brane embedded in this setup. We analize the brane behavior when its position is perturbed about a fixed point and solve the full non-linear dynamics in the several possible scenarios. We also investigate the possible gravitational shortcuts and calculate the delay between graviton and photon signals and the ratio of the corresponding subtended horizons. 
  We will discuss an integrable structure for weakly coupled superconformal Yang-Mills theories, describe certain equivalences for the Yangian algebra, and fill a technical gap in our previous study of this subject. 
  The kappa-deformation of the 2+1 anti-de Sitter, Poincare and de Sitter groups is studied within a unified approach by considering explicitly the curvature of the spacetime (cosmological constant) as a parameter. The Drinfel'd-double construction as well as the Poisson-Lie structure underlying the kappa-deformation are explicitly given, and the three quantum groups are obtained as Weyl quantizations of such Poisson-Lie algebras. As a consequence, the non-commutative 2+1 spacetimes and 4D spaces of worldlines are derived. The former generalize the kappa-Minkowski space to the (anti)de Sitter ones, and the latter can be interpreted as a new possibility to introduce non-commutative velocity spaces. Furthermore, provided that the deformation parameter is identified with the Planck length, quantum (anti)de Sitter algebras are presented both in the known basis arising in 2+1 quantum gravity and in a new one which generalizes the bicrossproduct kappa-Poincare basis. Finally, the existence of a kind of "duality" between the cosmological constant and the Planck length is also commented. 
  A model of chiral bosons on a non-commutative field space is constructed and new generalized bosonization (fermionization) rules for these fields are given. The conformal structure of the theory is characterized by a level of the Kac-Moody algebra equal to $(1+ \theta^2)$ where $\theta$ is the non-commutativity parameter and chiral bosons living in a non-commutative fields space are described by a rational conformal field theory with the central charge of the Virasoro algebra equal to 1. The non-commutative chiral bosons are shown to correspond to a free fermion moving with a speed equal to $ c^{\prime} = c \sqrt{1+\theta^2} $ where $c$ is the speed of light. Lorentz invariance remains intact if $c$ is rescaled by $c \to c^{\prime}$. The dispersion relation for bosons and fermions, in this case, is given by $\omega = c^{\prime} | k|$. 
  We find a closed form for Seiberg-Witten (SW) map between ordinary and noncommutative (NC) Dirac-Born-Infeld actions. We show that NC Maxwell action after the exact SW map can be regarded as ordinary Maxwell action coupling to a metric deformed by gauge fields. We also show that reversed procedure by inverse SW map leads to a similar interpretation in terms of induced NC geometry. This implies that noncommutativity in field theory can be interpreted as field dependent fluctuations of spacetime geometry, which genuinely realizes an interesting idea recently observed by Rivelles. 
  A gauge-invariant field is found which describes physical configurations, i.e. gauge orbits, of non-Abelian gauge theories. This is accomplished with non-Abelian generalizations of the Poincare'-Hodge formula for one-forms. In a particular sense, the new field is dual to the gauge field. Using this field as a coordinate, the metric and intrinsic curvature are discussed for Yang-Mills orbit space for the (2+1)- and (3+1)-dimensional cases. The sectional, Ricci and scalar curvatures are all formally non-negative. An expression for the new field in terms of the Yang-Mills connection is found in 2+1 dimensions. The measure on Schroedinger wave functionals is found in both 2+1 and 3+1 dimensions; in the former case, it resembles Karabali, Kim and Nair's measure. We briefly discuss the form of the Hamiltonian in terms of the dual field and comment on how this is relevant to the mass gap for both the (2+1)- and (3+1)-dimensional cases. 
  We compute string amplitudes on pp-waves supported by NS-NS 3-form fluxes and arising in the Penrose limit of AdS3xS3xM. We clarify the role of the non-chiral accidental SU(2) symmetry of the background. We comment on the extension of our results to the superstring and propose a holographic formula in the BMN limit of the AdS3/CFT2 correspondence valid for any correlator. 
  In the framework of superfield approach to Becchi-Rouet-Stora-Tyutin (BRST) formalism, the derivation of the (anti-)BRST nilpotent symmetries for the matter fields, present in any arbitrary interacting gauge theory, has been a long-standing problem. In our present investigation, the local, covariant, continuous and off-shell nilpotent (anti-)BRST symmetry transformations for the Dirac fields $(\psi, \bar\psi)$ are derived in the framework of the augmented superfield formulation where the four $(3 + 1)$-dimensional (4D) interacting non-Abelian gauge theory is considered on the six $(4 + 2)$-dimensional supermanifold parametrized by the four even spacetime coordinates $x^\mu$ and a couple of odd elements ($\theta$ and $\bar\theta$) of the Grassmann algebra. The requirement of the invariance of the matter (super)currents and the horizontality condition on the (super)manifolds leads to the derivation of the nilpotent symmetries for the matter fields as well as the gauge- and the (anti-)ghost fields of the theory in the general scheme of the augmented superfield formalism. 
  The work contains a detailed investigation of free neutral (Hermitian) or charged (non-Hermitian) scalar fields and the describing them (system of) Klein-Gordon equation(s) in momentum picture of motion. A form of the field equation(s) in terms of creation and annihilation operators is derived. An analysis of the (anti-)commutation relations on its base is presented. The concept of the vacuum and the evolution of state vectors are discussed. 
  We study semi-classical multi-spin strings in the non-supersymmetric backgrounds of $AdS$ Black Hole and Witten's confining model. We consider constant radius strings with rotations along the isometries of the backgrounds. In the $AdS$ Black Hole, solutions exist only if there is a non-zero spin in the Black Hole part. In contrast with the $AdS$ background, we find solutions which although have no rotation in the $S^5$ part, have a regular $\lambda$ expansion in the expression for large energy. In the near-extremal $D_3$ and $D_4$ backgrounds, we find that strings have to be located on the confining wall. We also discuss the stability of solutions by considering the fluctuation Lagrangians. 
  We address the question of which phase space functionals might represent a quantum state. We derive necessary and sufficient conditions for both pure and mixed phase space quantum states. From the pure state quantum condition we obtain a formula for the momentum correlations of arbitrary order and derive explicit expressions for the wavefunctions in terms of time dependent and independent Wigner functions. We show that the pure state quantum condition is preserved by the Moyal (but not by the classical Liouville) time evolution and is consistent with a generic stargenvalue equation. As a by-product Baker's converse construction is generalized both to an arbitrary stargenvalue equation, associated to a generic phase space symbol, as well as to the time dependent case. These results are properly extended to the mixed state quantum condition, which is proved to imply the Heisenberg uncertainty relations. Globally, this formalism yields the complete characterization of the kinematical structure of Wigner quantum mechanics. The previous results are then succinctly generalized for various quasi-distributions. Finally, the formalism is illustrated through the simple examples of the harmonic oscillator and the free Gaussian wave packet. As a by-product, we obtain in the former example an integral representation of the Hermite polynomials. 
  We review recent developments (up to January 2004) of the Liouville field theory and its matrix model dual. This review consists of three parts. In part I, we review the bosonic Liouville theory. After briefly reviewing the necessary background, we discuss the bulk structure constants (the DOZZ formula) and the boundary states (the FZZT brane and the ZZ brane). Various applications are also presented. In part II, we review the supersymmetric extension of the Liouville theory. We first discuss the bulk structure constants and the branes as in the bosonic Liouville theory, and then we present the matrix dual descriptions with some applications. In part III, the Liouville theory on unoriented surfaces is reviewed. After introducing the crosscap state, we discuss the matrix model dual description and the tadpole cancellation condition. This review also includes some original material such as the derivation of the conjectured dual action for the N = 2 Liouville theory from other known dualities and the comparison of the Liouville crosscap state with the c = 0 unoriented matrix model. This is based on my master's thesis submitted to Department of Physics, Faculty of Science, University of Tokyo on January 2004. 
  We present a Matrix theory action and Matrix configurations for spherical 4-branes. The dimension of the representations is given by N=2(2j+1) (j=1/2,1,3/2,...). The algebra which defines these configurations is not invariant under SO(5) rotations but under SO(3) \otimes SO(2). We also construct a non-commutative product for field theories on S^4 in terms of that on S^2. An explicit formula of the non-commutative product which corresponds to the N=4 dim representation of the non-commutative S^4 algebra is worked out. Because we use S^2 \otimes S^2 parametrization of S^4, our S^4 is doubled and the non-commutative product and functions on S^4 are indeterminate on a great circle (S^1) on S^4. We will however, show that despite this mild singularity it is possible to write down a finite action integral of the non-commutative field thoery on S^4. NS-NS B field background on S^4 which is associated with our Matrix S^4 configurations is also constructed. 
  In this paper it is shown that an i phi^3 field theory is a physically acceptable field theory model (the spectrum is positive and the theory is unitary). The demonstration rests on the perturbative construction of a linear operator C, which is needed to define the Hilbert space inner product. The C operator is a new, time-independent observable in PT-symmetric quantum field theory. 
  We examine the algebraic structure of the matrix regularization for the wrapped membrane on $R^{10}\times S^1$ in the light-cone gauge. We give a concrete representation for the algebra and obtain the matrix string theory having the boundary conditions for the matrix variables corresponding to the wrapped membrane, which is referred to neither Seiberg and Sen's arguments nor string dualities. We also embed the configuration of the multi-wrapped membrane in matrix string theory. 
  We consider the coupling of fermions to the three-dimensional noncommutative $CP^{N-1}$ model. In the case of minimal coupling, although the infrared behavior of the gauge sector is improved, there are dangerous (quadratic) infrared divergences in the corrections to the two point vertex function of the scalar field. However, using superfield techniques we prove that the supersymmetric version of this model with ``antisymmetrized'' coupling of the Lagrange multiplier field is renormalizable up to the first order in $\frac{1}{N}$. The auxiliary spinor gauge field acquires a nontrivial (nonlocal) dynamics with a generation of Maxwell and Chern-Simons noncommutative terms in the effective action. Up to the 1/N order all divergences are only logarithimic so that the model is free from nonintegrable infrared singularities. 
  Recently the revised phase diagram of the (large N) Gross-Neveu model in 1+1 dimensions with discrete chiral symmetry has been determined numerically. It features three phases, a massless and a massive Fermi gas and a kink-antikink crystal. Here we investigate the phase diagram by analytical means, mapping the Dirac-Hartree-Fock equation onto the non-relativistic Schroedinger equation with the (single gap) Lame potential. It is pointed out that mathematically identical phase diagrams appeared in the condensed matter literature some time ago in the context of the Peierls-Froehlich model and ferromagnetic superconductors. 
  We consider the effect of the ultraviolet (UV) or short wavelength modes on the background of Brane Gas Cosmology. We find that the string matter sources are negligible in the UV and that the evolution is given primarily by the dilaton perturbation. We also find that the linear perturbations are well behaved and the predictions of Brane Gas Cosmology are robust against the introduction of linear perturbations. In particular, we find that the stabilization of the extra dimensions (moduli) remains valid in the presence of dilaton and string perturbations. 
  Recently it has been proposed that gluon scattering amplitudes in gauge theory can be computed from the D-instanton expansion of the topological B-model on P^{3|4}, although only maximally helicity violating (MHV) amplitudes have so far been obtained from a direct B-model calculation. In this note we compute the simplest non-MHV gluon amplitudes (++--- and +-+--) from the B-model as an integral over the moduli space of degree 2 curves in P^{3|4} and find perfect agreement with Yang-Mills theory. 
  The high-energy behavior of string scattering in warped spacetimes is studied to all orders in perturbation theory. If one assumes that the theory is finite, the amplitudes exactly fall as powers of momentum. 
  We study how the noncommutative spacetime affects on inflation. First we obtain the noncommutative power spectrum of the curvature perturbations produced during inflation in the slow-roll approximation. This is the explicit $k$-dependent power spectrum up to first order in slow-roll parameters $\epsilon_1 \delta_1$ including the noncommutative parameter $\mu$. In order to test the role of $\mu$ further, we calculate the noncommutative power spectrum using the slow-roll expansion. We find corrections which arise from the change of pivot scale and a noncommutative parameter with $\mu\not=$ constant. It turns out that the noncommutative parameter $\mu$ could be considered as a zeroth order slow-roll parameter and the noncommutative spacetime effect suppresses the power spectrum. 
  We consider a 4+N-dimensional brane world with 2 co-dimension 1 branes in an empty bulk. The two branes have N-1 of their extra dimensions compactified on a sphere S^(N-1), whereas the ordinary 4 spacetime directions are Poincare invariant. An essential input are induced stress-energy tensors on the branes providing different tensions for the spherical and flat part of the branes. The junction conditions - notably through their extra dimensional components - fix both the distance between the branes as well as the size of the sphere. As a result, we demonstrate, that there are no scalar Kaluza-Klein states at all (massless or massive), that would correspond to a radion or a modulus field of S^(N-1). We also discuss the effect of induced Einstein terms on the branes and show that their coefficients are bounded from above, otherwise they lead to a graviton ghost. 
  We discuss the scalar propagator on generic AdS_{d+1} x S^{d'+1} backgrounds. For the conformally flat situations and masses corresponding to Weyl invariant actions the propagator is powerlike in the sum of the chordal distances with respect to AdS_{d+1} and S^{d'+1}. In all other cases the propagator depends on both chordal distances separately. We discuss the KK mode summation to construct the propagator in brief. For AdS_5 x S^5 we relate our propagator to the expression in the BMN plane wave limit and find a geometric interpretation of the variables occurring in the known explicit construction on the plane wave. 
  This is a short overview of spatially flat (or open) four-dimensional accelerating cosmologies for some simple exponential potentials obtained by string or M theory compactification on some non-trivial curved spaces, which may lead to some striking results, e.g., the observed cosmic acceleration and the scale of the dark energy from first principles. 
  We review the mechanism of gaugino condensation in the framework of the $d=10$ heterotic string and its $d=11$ extension of Horava and Witten. In particular we emphasize the relation between the gaugino condensate and the flux of the antisymmetric tensor fields of higher dimensional supergravity. Its potential role for supersymmetry breakdown and moduli stabilization is investigated. 
  In these notes we review recent work on describing D-branes with nonzero Higgs vevs in terms of sheaves, which gives a physical on-shell D-brane interpretation to more sheaves than previously understood as describing D-branes. The mathematical ansatz for this encoding is checked by comparing open string spectra between D-branes with nonzero Higgs vevs to Ext groups between the corresponding sheaves. We illustrate the general methods with a few examples. 
  It has been believed that topology and signature change of the universe can only happen accompanied by singularities, in classical, or instantons, in quantum, gravity. In this note, we point out however that in the braneworld context, such an event can be understood as a classical, smooth event. We supply some explicit examples of such cases, starting from the Dirac-Born-Infeld action. Topology change of the brane universe can be realised by allowing self-intersecting branes. Signature change in a braneworld is made possible in an everywhere Lorentzian bulk spacetime. In our examples, the boundary of the signature change is a curvature singularity from the brane point of view, but nevertheless that event can be described in a completely smooth manner from the bulk point of view. 
  Quantization of the nonlinear supersymmetry faces a problem of a quantum anomaly. For some classes of superpotentials, the integrals of motion admit the corrections guaranteeing the preservation of the nonlinear supersymmetry at the quantum level. With an example of the system realizing the nonlinear superconformal symmetry, we discuss the nature of such corrections and speculate on their possible general origin. 
  Fermion-antifermion pair-production in the presence of classical fields is described based on the retarded and advanced fermion propagators. They are obtained by solving the equation of motion for the Dirac Green's functions with the respective boundary conditions to all orders in the field. Subsequently, various approximation schemes fit for different field configurations are explained. This includes longitudinally boost-invariant forms. Those occur frequently in the description of ultrarelativistic heavy-ion collisions in the semiclassical limit. As a next step, the gauge invariance of the expression for the expectation value of the number of produced fermion-antifermion pairs as a functional of said propagators is investigated in detail. Finally, the calculations are carried out for a longitudinally boost-invariant model-field, taking care of the last issue, especially. 
  At the final stage of unstable D-brane decay in the effective field theory approach, all energy and momentum of the initial state are taken up by two types of fluids, known as string fluid and tachyon matter. In this note, we compare motion of this fluid system to that of macroscopic collection of stretched closed strings and find a precise match at classical level. The string fluid reflects low frequency undulation of the stretched strings while the tachyon matter encodes the average effect of high frequency oscillations turned on those strings. In particular, the combined fluid system has been known to have a reduced speed of light, depending on the composition, and we show that this property is exactly reproduced in classical motion on the closed string side. Finally we illustrate how the tachyon matter may be viewed as an effective degrees of freedom carrying high frequency energy-momentum of Nambu-Goto strings by coarse-graining the dynamics of the latter. 
  We calculate effective potentials in scalar field theories on the maximally supersymmetric pp-wave background in ten dimensions. For this purpose we have to work in the light-cone formulation, and hence we introduce two methods to compute them in the light-cone frame. One is to use the Yan's formula for evaluating one-loop correction terms. The other is to introduce a cut-off for the light-cone momentum. These methods are also confirmed in the case of Minkowski spacetime. 
  We have studied the noncommutative extension of the relativistic Chern-Simons-Higgs model, in the first non-trivial order in $\theta$, with only spatial noncommutativity. Both Lagrangian and Hamiltonian formulations of the problem have been discussed, with the focus being on the canonical and symmetric forms of the energy-momentum tensor. In the Hamiltonian scheme, constraint analysis and the induced Dirac brackets have been provided. The spacetime translation generators and their actions on the fields are discussed in detail.   The effects of noncommutativity on the soliton solutions have been analysed thoroughly and we have come up with some interesting observations. Considering the {\it{relative}} strength of the noncommutative effects, we have shown that there is a universal character in the noncommutative correction to the magnetic field - it depends {\it{only}} on $\theta$. On the other hand, in the cases of all other observables of physical interest, such as the potential profile, soliton mass or the electric field, $\theta$ as well as $\tau$, (comprising solely of commutative Chern-Simons-Higgs model parameters), appear on similar footings. This phenomenon is a new finding which has come up in the present analysis.   Lastly, we have pointed out a generic problem in the NC extension of the models, in the form of a mismatch between the BPS dynamical equation and the full variational equations of motion, to $O(\theta)$. This mismatch indicates that the analysis is not complete as it brings in to fore the ambiguities in the definition of the energy-momentum tensor in a noncommutative theory. 
  The periodic bounce of Born-Infeld theory of $D3$-branes is derived, and the BPS limit of infinite period is discussed as an example of tachyon condensation. The explicit bounce solution to the Born--Infeld action is interpreted as an unstable fundamental string stretched between the brane and its antibrane. 
  In the literature there are two different ways of describing an invariant star product on $S^2$. We show that the products are actually the same. We also calculate the canonical trace and use the Fedosov-Nest-Tsygan index theorem to obtain the characteristic class of this product. 
  We present a superstring-inspired version of D-term inflation which does not lead to cosmic string formation and appears to satisfy the current CMB constraints. It differs from minimal D-term inflation by a second pair of charged superfields which makes the strings non-topological (semilocal). The strings are also BPS, so the scenario is expected to survive supergravity corrections. The second pair of charged superfields arises naturally in several brane and conifold scenarios, but its effect on cosmic string formation had not been noticed so far. 
  Using the integrable spin chain picture we study the one-loop anomalous dimension of certain single trace scalar operators of N=4 SYM expected to correspond to semi-classical string states on AdS_5 x S^5 with three large angular momenta (J_1,J_2,J_3) on S^5. In particular, we investigate the analyticity structure encoded in the Bethe equations for various distributions of Bethe roots. In a certain region of the parameter space our operators reduce to the gauge theory duals of the folded string with two large angular momenta and in another region to the duals of the circular string with angular momentum assignment (J,J',J'), J>J'. In between we locate a critical line. We propose that the operators above the critical line are the gauge theory duals of the circular elliptic string with three different spins and support this by a perturbative calculation. 
  We show the natural relation between the Wigner Hamiltonian and the conformal Hamiltonian. It is presented a model in (super)conformal quantum mechanics with (super)conformal symmetry in the Wigner-Heisenberg algebra picture $ [x,p_{x}]= i(1+c{\bf P})$ (${\bf P}$ being the parity operator). In this context, the energy spectrum, the Casimir operator, creation and annihilation operators are defined. This superconformal Hamiltonian is similar to the super-Hamiltonian of the Calogero model and it is also an extension of the super-Hamiltonian for the Dirac Oscillator. 
  A method for computing the low-energy non-perturbative properties of SUSY GFT, starting from the microscopic lagrangian model, is presented. The method relies on covariant SUSY Feynman graph techniques, adapted to low energy, and Renormalization-Group-improved perturbation theory. We apply the method to calculate the glueball superpotential in N=1 SU(2) SYM and obtain a potential of the Veneziano-Yankielowicz type. 
  We examine a new application of the Holstein-Primakoff realization of the simple harmonic oscillator Hamiltonian. This involves the use of infinite-dimensional representations of the Lie algebra $su(2)$. The representations contain nonstandard raising and lowering operators, which are nonlinearly related to the standard $a^{\dag}$ and $a$. The new operators also give rise to a natural family of two-oscillator couplings. These nonlinear couplings are not generally self-adjoint, but their low-energy limits are self-adjoint, exactly solvable, and stable. We discuss the structure of a theory involving these couplings. Such a theory might have as its ultra-low-energy limit a Lorentz-violating Abelian gauge theory, and we discuss the extremely strong astrophysical constraints on such a model. 
  We write the IIB Green-Schwarz action in certain general classes of curved backgrounds threaded with Ramond-Ramond fluxes. The fixing of the kappa symmetry in the light-cone gauge and the use of supergravity Bianchi identities simplify the task. We find an expression that truncates to quartic order in the spacetime spinors and relays interesting information about the vacuum structure of the worldsheet theory. The results are particularly useful in exploring integrable string dynamics in the context of the holographic duality. 
  Modifications to the primordial power spectrum of inflationary density perturbations have been studied recently using a boundary effective field theory approach. In the approximation of a fluctuating quantum field on a fixed background, the generic effect of new physics is encoded in parameters of order H/M. Here, we point out that the back-reaction on the metric can be neglected only when these parameters obey certain bounds that may put them beyond the reach of observation. 
  We give the classification of the positive energy (lowest weight) unitary irreducible representations of the superalgebras osp(1|2n,R). 
  We find an exact solution of BPS wall in five-dimensional supergravity using a gravitational deformation of the massive Eguchi-Hanson nonlinear sigma model. The warp factor decreases for both infinities of the extra dimension. Our solution requires no fine-tuning between boundary and bulk cosmological constants, in contrast to the Randall-Sundrum model. Wall solutions are also obtained with warp factors which are flat or increasing in one side by varying a deformation parameter. 
  We discuss a special ``symplectic'' class of N = 4 supersymmetric sigma models in (0+1) dimension with 5r bosonic and 4r complex fermionic degrees of freedom. These models can be described off shell by N = 2 superfields (so that only half of supersymmetries are manifest) and also by N = 4 superfields in the framework of the harmonic superspace approach. Using the latter, we derive the general form of the relevant bosonic target metric. 
  It was proposed that the Calabi-Yau geometry can be intrinsically connected with some new symmetries, some new algebras. In order to do this it has been analyzed the graphs constructed from K3-fibre CY_d (d \geq 3) reflexive polyhedra. The graphs can be naturally get in the frames of Universal Calabi-Yau algebra (UCYA) and may be decode by universal way with changing of some restrictions on the generalized Cartan matrices associated with the Dynkin diagrams that characterize affine Kac-Moody algebras. We propose that these new Berger graphs can be directly connected with the generalizations of Lie and Kac-Moody algebras. 
  We present an overview of recent nonperturbative results in the theory of heat kernel and its late time asymptotics responsible for the infrared behavior of quantum effective action for massless theories. In particular, we derive the generalization of the Coleman-Weinberg potential to physical situations when the field is not homogeneous throughout the whole spacetime. This generalization represents a new nonlocal and nonperturbative action accounting for the effects of a transition domain between the spacetime interior and its infinity. In four dimensions these effects delocalize the logarithmic Coleman-Weinberg potential, while in $d>4$ they are dominated by new powerlike and renormalization-independent nonlocal structure. Nonperturbative behavior of the heat kernel is also constructed in curved spacetime with asymptotically-flat geometry, and its conformal properties are analyzed for conformally invariant scalar field. The problem of disentangling the local cosmological term from nonlocal effective action is discussed. 
  In matrix models, higher dimensional D-branes are obtained by imposing a noncommutative relation to coordinates of lower dimensional D-branes. On the other hand, a dual description of this noncommutative space is provided by higher dimensional D-branes with gauge fields. Fuzzy spheres can appear as a configuration of lower dimensional D-branes in a constant R-R field strength background. In this paper, we consider a dual description of higher dimensional fuzzy spheres by introducing nonabelian gauge fields on higher dimensional spherical D-branes. By using the Born-Infeld action, we show that a fuzzy $2k$-sphere and spherical D$2k$-branes with a nonabelian gauge field whose Chern character is nontrivial are the same objects when $n$ is large. We discuss a relationship between the noncommutative geometry and nonabelian gauge fields. Nonabelian gauge fields are represented by noncommutative matrices including the coordinate dependence. A similarity to the quantum Hall system is also studied. 
  In this letter, an alternative string theory in twistor space is proposed for describing perturbative N=4 super-Yang-Mills theory. Like the recent proposal of Witten, this string theory uses twistor worldsheet variables and has manifest spacetime superconformal invariance. However, in this proposal, tree-level super-Yang-Mills amplitudes come from open string tree amplitudes as opposed to coming from D-instanton contributions. 
  We clarify the structure of N=1 supergravity in 1+3 dimensions with constant FI terms. The FI terms induce non-vanishing R-charges for the fermions and the superpotential. Therefore the D-term inflation model in supergravity with constant FI terms has to be revisited.   We also investigate the case of the so-called anomalous U(1) when a chiral superfield is shifted under U(1). In such a case, in the context of string theory, the FI terms originate from the derivative of the Kaehler potential and they are inevitably field-dependent. This raises an issue of stabilization of the relevant field in applications to cosmology.   The recently suggested equivalence between the D-term strings and D-branes of type II theory shows that brane-anti-brane systems produce FI terms in the effective 4d theory, with the Ramond-Ramond axion shifting under the U(1) symmetry. This connection gives the possibility to interpret many unknown properties of D-\bar{D} systems in the more familiar language of 4d supergravity D-terms, and vice versa. For instance, the shift of the axion field in both cases restricts the possible forms of the moduli-stabilizing superpotential. We provide some additional consistency checks of the correspondence of D-term-strings to D-branes and show that instabilities of the two are closely related. Surviving cosmic D-strings of type II theory may be potentially observed in the form of D-term strings of 4d supergravity. 
  The main problem of inflation in string theory is finding the models with a flat potential, consistent with stabilization of the volume of the compactified space. This can be achieved in the theories where the potential has (an approximate) shift symmetry in the inflaton direction. We will identify a class of models where the shift symmetry uniquely follows from the underlying mathematical structure of the theory. It is related to the symmetry properties of the corresponding coset space and the period matrix of special geometry, which shows how the gauge coupling depends on the volume and the position of the branes. In particular, for type IIB string theory on K3xT^2/Z with D3 or D7 moduli belonging to vector multiplets, the shift symmetry is a part of SO(2,2+n) symmetry of the coset space [SU(1,1)/ U(1)]x[SO(2,2+n)/(SO(2)x SO(2+n)]. The absence of a prepotential, specific for the stringy version of supergravity, plays a prominent role in this construction, which may provide a viable mechanism for the accelerated expansion and inflation in the early universe. 
  Motivated by several pieces of evidence, in order to show that extreme black holes cannot be obtained as limits of non-extremal black holes, in this article we calculate explicitly quasinormal modes for Ba\~{n}ados, Teitelboim and Zanelli (BTZ) extremal black hole and we showed that the imaginary part of the frequency is zero. We obtain exact result for the scalar an fermionic perturbations. We also showed that the frequency is bounded from below for the existence of the normal modes (non-dissipative modes). 
  The gauge connections corresponding to electromagnetism, Yang-Mills theory and Einstein gravity can be derived by assuming specific commutation relations between the phase-space variables of a first quantized theory. Extending the procedure to noncommuting coordinates leads to new types of dynamics, which are explored. In particular, the conditions for the coexistence of an electromagnetic background and a noncommutative two-from are found, as well as a generalized mechanism of dimensional reduction. The noncommutative deformation of a gravitational background is also constructed. The present formulation suggests some simple experimental tests of noncommutativity. 
  We analyze diffeomorphism invariance in inflationary spacetimes regulated by a boundary at late time. We present the action for quadratic fluctuations in the presence of a boundary, and verify that it is gauge invariant precisely when the correct local counterterms are included. The scaling behavior of bulk correlation functions at the boundary is determined by Callan-Symanzik equations which predict scaling violations in agreement with the standard inflationary predictions for spectral indices of the CMB. 
  I will discuss the development of inflationary theory and its present status, including recent progress in describing de Sitter space and inflationary universe in string theory. 
  We study the pp limit of AdS(3) x S(3) at the interaction level. We find the interacting Hamiltonian for the bosonic fields of D=6 SUGRA in the pp-wave background, and compare it to the cubic couplings of the full AdS(3) x S(3). We show how the pp-wave theory vertex arises in the large J limit. Our analysis also provides some insight into the origin of specific prefactors which appear in the pp-wave interaction. 
  The anti-de Sitter/conformal field theory duality conjecture raises the question of how the perturbative expansion in the conformal field theory can resum to a simple function. We exhibit a relation between the one-loop and two-loop amplitudes whose generalization to higher-point and higher-loop amplitudes would answer this question. We also provide evidence for the first of these generalizations. 
  In this doctoral thesis we study zero-mode spectra of Matrix theory and eleven-dimensional supergravity on the plane-wave background. This background is obtained via the Penrose limit of AdS_4 x S^7 and AdS_7 x S^4. The plane-wave background is a maximally supersymmetric spacetime supported by non-vanishing constant four-form flux in eleven-dimensional spacetime. First, we discuss the Matrix theory on the plane-wave background suggested by Berenstein, Maldacena and Nastase. We construct the Hamiltonian, 32 supercharges and their commutation relations. We discuss a spectrum of one specific supermultiplet which represents the center of mass degrees of freedom of N D0-branes. This supermultiplet would also represent a superparticle of the eleven-dimensional supergravity in the large-N limit. Second, we study the linearized supergravity on the plane-wave background in eleven dimensions. Fixing the bosonic and fermionic fields in the light-cone gauge, we obtain the spectrum of physical modes. We obtain the fact that the energies of the states in Matrix theory completely correspond to those of fields in supergravity. Thus, we find that the Matrix theory on the plane-wave background contains the zero-mode spectrum of the eleven-dimensional supergravity completely. Through this result, we can argue the Matrix theory on the plane-wave as a candidate of quantum extension of eleven-dimensional supergravity, or as a candidate which describes M-theory. 
  We obtain necessary and sufficient conditions for a supersymmetric field configuration in the N=(1,0) U(1) or SU(2) gauged supergravities in six dimensions, and impose the field equations on this general ansatz. It is found that any supersymmetric solution is associated to an $SU(2)\ltimes \mathbb{R}^4$ structure. The structure is characterized by a null Killing vector which induces a natural 2+4 split of the six dimensional spacetime. A suitable combination of the field equations implies that the scalar curvature of the four dimensional Riemannian part, referred to as the base, obeys a second order differential equation. Bosonic fluxes introduce torsion terms that deform the $SU(2)\ltimes\mathbb{R}^4$ structure away from a covariantly constant one. The most general structure can be classified in terms of its intrinsic torsion. For a large class of solutions the gauge field strengths admit a simple geometrical interpretation: in the U(1) theory the base is K\"{a}hler, and the gauge field strength is the Ricci form; in the SU(2) theory, the gauge field strengths are identified with the curvatures of the left hand spin bundle of the base. We employ our general ansatz to construct new supersymmetric solutions; we show that the U(1) theory admits a symmetric Cahen-Wallach$_4\times S^2$ solution together with a compactifying pp-wave. The SU(2) theory admits a black string, whose near horizon limit is $AdS_3\times S_3$. We also obtain the Yang-Mills analogue of the Salam-Sezgin solution of the U(1) theory, namely $R^{1,2}\times S^3$, where the $S^3$ is supported by a sphaleron. Finally we obtain the additional constraints implied by enhanced supersymmetry, and discuss Penrose limits in the theories. 
  We consider the unitarity of D=9,10,11 conformal supersymmetry using the recently established classification of the UIRs of the superalgebras osp(1|2n,R). 
  The Hamiltonian counterpart of classical Lagrangian field theory is covariant Hamiltonian field theory where momenta correspond to derivatives of fields with respect to all world coordinates. In particular, classical Lagrangian and covariant Hamiltonian field theories are equivalent in the case of a hyperregular Lagrangian, and they are quasi-equivalent if a Lagrangian is almost-regular. In order to quantize covariant Hamiltonian field theory, one usually attempts to construct and quantize a multisymplectic generalization of the Poisson bracket. In the present work, the path integral quantization of covariant Hamiltonian field theory is suggested. We use the fact that a covariant Hamiltonian field system is equivalent to a certain Lagrangian system on a phase space which is quantized in the framework of perturbative field theory. We show that, in the case of almost-regular quadratic Lagrangians, path integral quantizations of associated Lagrangian and Hamiltonian field theories are equivalent. 
  We point out that aspects of quantum mechanics can be derived from the holographic principle, using only a perturbative limit of classical general relativity. In flat space, the covariant entropy bound reduces to the Bekenstein bound. The latter does not contain Newton's constant and cannot operate via gravitational backreaction. Instead, it is protected by - and in this sense, predicts - the Heisenberg uncertainty principle. 
  By a direct CFT computation, the spectrum of the topological B-model is compared to Ext groups of sheaves. A match can only be made if abstract vector bundles on holomorphic submanifolds are twisted by the canonical $\mathrm{Spin}^c$ structure of its support in describing physical branes. 
  In an attempt to restore the unitarity of the evaporation process, Horowitz and Maldacena recently proposed a boundary-condition constraint for the final quantum state of an evaporating black hole at its singularity. Gottesman and Preskill have argued that the proposed constraint must lead to nonlinear evolution of the initial (collapsing) quantum state. Here we show that in fact this evolution allows signaling, making it detectable outside the event horizon with entangled-probe experiments of the kind we proposed recently. As a result the Horowitz-Maldacena proposal may be subject to terrestrial tests. 
  We calculate the mass of the lowest lying spin two glueball in N=1 super Yang-Mills from the dual Klebanov-Strassler background. We show that the Regge trajectory obtained is linear; the 0++, 1-- and 2++ states lie on a line of slope 0.23 -measured in units of the conifold deformation. We also compare mass ratios with lattice data and find agreement within one standard deviation. 
  We study deformations of four-dimensional N=(1,1)Euclidean superspace induced by non-anticommuting fermionic coordinates. We essentially use the harmonic superspace approach and consider nilpotent bi-differential Poisson operators only, which generalizes the recently studied chiral deformation of N=(1/2,1/2) superspace. We present non-anticommutative Euclidean analogs of N=2 Maxwell and hypermultiplet off-shell actions. The talk is based on the paper hep-th/0308012. 
  We continue with the program of hep-th/0308184 to implement open-closed string duality on free gauge field theory (in the large $N$ limit). In this paper we consider correlators such as $\la \prod_{i=1}^n \Tr\Phi^{J_i}(x_i)\ra$. The Schwinger parametrisation of this $n$-point function exhibits a partial gluing up into a set of basic skeleton graphs. We argue that the moduli space of the planar skeleton graphs is exactly the same as the moduli space of genus zero Riemann surfaces with $n$ holes. In other words, we can explicitly rewrite the $n$-point (planar) free field correlator as an integral over the moduli space of a sphere with $n$ holes. A preliminary study of the integrand also indicates compatibility with a string theory on $AdS$. The details of our argument are quite insensitive to the specific form of the operators and generalise to diagrams of higher genus as well. We take this as evidence of the field theory's ability to reorganise itself into a string theory. 
  We study the Pauli equation on non-commutative plane. It is shown that the Supersymmetry algebra holds to all orders in the non-commutative parameter $\theta$ in case the gyro-magnetic ratio $g$ is 2. Using Seiberg-Witten map, the first order in $\theta$ correction to the spectrum is obtained in the case of uniform magnetic field. We find that the eigenstates in the non-commutative case are identical to the commutative case provided the magnetic field $B$ is everywhere replaced by $B(1+B\theta)$. 
  Massive hypermultiplets admit degenerate discrete vacua only if they form nonlinear sigma models or have gauge interaction. We discuss BPS domain walls in these theories. This talk is based on the original papers hep-th/0307274 and hep-th/0211103. 
  Recent work has shown that boundary correlators in AdS-Schwarzschild can probe the geometry near the singularity. In this paper we aim to analyze the specific signatures of the singularity, show how significant they can be, and uncover the origins of these large effects in explicitly outside the horizon descriptions.   We add perturbations to the metric localized near the singularity and explore their effects on the boundary correlators. Then we use analyticity arguments to show how this information arises from the Euclidean path integral of a free scalar field in the bulk. 
  It was conjectured recently that the string worldsheet theory for the fast moving string in AdS times a sphere becomes effectively first order in the time derivative and describes the continuous limit of an integrable spin chain. In this paper we will try to make this statement more precise. We interpret the first order theory as describing the long term evolution of the tensionless string perturbed by a small tension. The long term evolution is a Hamiltonian flow on the moduli space of periodic trajectories. It should correspond to the renormgroup flow on the field theory side. 
  Using techniques developed in a previous paper three-point functions in field theories described by holographic renormalization group flows are computed. We consider a system of one active scalar and one inert scalar coupled to gravity. For the GPPZ flow, their dual operators create states that are interpreted as glueballs of the N=1 SYM theory, which lies at the infrared end of the renormalization group flow. The scattering amplitudes for three-glueball processes are calculated providing precise predictions for glueball decays in N=1 SYM theory. Numerical results for low-lying glueballs are included. 
  The paper is withdrawn by the author due to an oversimplified and misleading approach which was taken initially as a starting point. 
  Vacuum effects in (2+1)-dimensional quantum electrodynamics (QED) with the topological Chern-Simons term are considered. The photon polarization operator is studied and the decay rate for the electron-positron photoproduction is presented as a function of the photon energy and external field strength. The radiatively induced electron mass shift in an external magnetic field is investigated. The electron self-energy in topologically massive (2+1)-QED at finite temperature and density is studied. The parity breaking part of the Yang-Mills action in the framework of the $SU(2)\times U(1)$ gauge field model at finite temperature is considered. 
  We develop a Born-Infeld type theory for gravity in any dimension. We show that in four dimensions our formalism allows a self-dual (or anti-self dual) Born-Infeld gravity description. Moreover, we show that such a self-dual action is reduced to both the Deser-Gibbons and the Jacobson-Smolin-Samuel action of Ashtekar formulation. A supersymmetric generalization of our approach is outlined. 
  Within the tachyon condensation approach, we find that a D(p-2)-brane is stable inside Dp-branes when the bulk is compactified. It is a codimension-2 soliton of the Dp-brane action with coupling to the bulk (p-1)-form RR field. We discuss the properties of such solitons. They may appear as detectable cosmic strings in our universe. 
  Hawking's argument for information loss in black hole evaporation rests on the assumption of independent Hilbert spaces for the interior and exterior of a black hole. We argue that such independence cannot be established without incorporating strong gravitational effects that undermine locality and invalidate the use of quantum field theory in a semiclassical background geometry. These considerations should also play a role in a deeper understanding of horizon complementarity. 
  We compute complete leading logarithms in $\Phi^4$ theory with the help of Connes and Kreimer RG equations. These equations are defined in the Lie algebra dual to the Hopf algebra of graphs. The results are compared with calculations in parquet approximation. An interpretation of the new RG equations is discussed. 
  We investigate a string-inspired scenario associated with a rolling massive scalar field on D-branes and discuss its cosmological implications. In particular, we discuss cosmological evolution of the massive scalar field on the ant-D3 brane of KKLT vacua. Unlike the case of tachyon field, because of the warp factor of the anti-D3 brane, it is possible to obtain the required level of amplitude of density perturbations. We study the spectra of scalar and tensor perturbations generated during the rolling scalar inflation and show that our scenario satisfies the observational constraint coming from the Cosmic Microwave Background anisotropies and other observational data. We also implement the negative cosmological constant arising from the stabilization of the modulus fields in the KKLT vacua and find that this leads to a successful reheating in which the energy density of the scalar field effectively scales as a pressureless dust. The present dark energy can be also explained in our scenario provided that the potential energy of the massive rolling scalar does not exactly cancel with the amplitude of the negative cosmological constant at the potential minimum. 
  Theories of gravity coupled to forms and dilatons may admit as solutions zero binding energy configurations of intersecting closed extremal branes. In such configurations, some branes may open on host closed branes. Properties of extremal branes reveal symmetries of the underlying theory which are well known in M-theory but transcend supersymmetry. From these properties it is possible to reconstruct all actions, comprising in particular pure gravity in D dimensions, the bosonic effective actions of M-theory and of the bosonic string, which upon dimensional reduction to three dimensions are invariant under the maximally non-compact simple simply laced Lie groups G. Moreover the features of extremal branes suggest the existence of a much larger symmetry, namely the `very-extended' Kac-Moody algebras G+++. This motivates the construction of explicit non-linear realisations of all simple G+++, which hopefully contain new degrees of freedom such as those encountered in string theories. They are defined without a priori reference to space-time and are proposed as substitutes for original field theoretic models of gravity, forms and dilatons. From the G+++ invariant theories, all algebraic properties of extremal branes are recovered from exact solutions, and there are indications that space-time is hidden in the infinite symmetry structure. The transformation properties of the exact solutions, which possibly induce new solutions foreign to conventional theories, put into evidence the general group-theoretical origin of `dualities' for all G+++. These dualities apparently do not require an underlying string theory. 
  We study the physical Fock space of the tensionless string theory with perimeter action which has pure massless spectrum. The states are classified by the Wigner's little group for massless particles. The ground state contains infinite many massless fields of fixed helicity, the excitation levels realize CSR representations. We demonstrate that the first and the second excitation levels are physical null states. 
  If an M2-brane intersects an M5-brane the canonical Wess-Zumino action is plagued by a Dirac-anomaly, i.e. a non-integer change of the action under a change of Dirac-brane. We show that this anomaly can be eliminated at the expense of a gravitational anomaly supported on the intersection manifold. Eventually we check that the last one is cancelled by the anomaly produced by the fermions present. This provides a quantum consistency check of these intersecting configurations. 
  We consider a generalisation of the DGP model, by adding a second brane with localised curvature, and allowing for a bulk cosmological constant and brane tensions. We study radion and graviton fluctuations in detail, enabling us to check for ghosts and tachyons. By tuning our parameters accordingly, we find bigravity models that are free from ghosts and tachyons. These models will lead to large distance modifications of gravity that could be observable in the near future. 
  We find the high overtones of gravitational and electromagnetic quasinormal spectrum of the Schwarzschild-de Sitter black hole. The calculations show that the real parts of the electromagnetic modes asymptotically approach zero. The gravitational modes show more peculiar behavior at large $n$: the real part oscillates as a function of imaginary even for very high overtones and these oscillations settles to some "profile" which just repeats itself with further increasing of the overtone number $n$. This lets us judge that $Re \omega$ is not a constant as $n \to \infty$ but rather some oscillating function. The spacing for imaginary part $ Im \omega_{n+1}-Im \omega_{n}$ for both gravitational and electromagnetic perturbations at high $n$ oscillates as a function of $n$ around the value $k_{e}$, where $k_{e}$ is the surface gravity. In addition we find the lower QN modes for which the values obtained with numerical methods are in a very good agreement with those obtained through the 6th order WKB technique. 
  Using the simplest model for a bouncing universe, namely that for which gravity is described by pure general relativity, the spatial sections are positively curved and the matter content is a single scalar field, we obtain the transition matrix relating cosmological perturbation modes between the contracting and expanding phases. We show that this case provides a specific example in which this relation explicitely depends on the perturbation scale whenever the null energy condition (NEC) is close to be violated. 
  A two-parametric family of integrable models (the SS model) that contains as particular cases several well known integrable quantum field theories is considered. After the quantum group restriction it describes a wide class of integrable perturbed conformal field theories. Exponential fields in the SS model are closely related to the primary fields in these perturbed theories. We use the bosonization approach to derive an integral representation for the form factors of the exponential fields in the SS model. The same representations for the sausage model and the cosine-cosine model are obtained as limiting cases. The results are tested at the special points, where the theory contains free particles. 
  We calculate analytically the asymptotic form of quasi-normal frequencies for massive scalar perturbations of large black holes in AdS_5. We solve the wave equation, which reduces to a Heun equation, perturbatively and calculate the wave function as an expansion in 1/m, where m is the mass of the mode. The zeroth-order results agree with expressions derived by numerical analysis. We also calculate the first-order corrections to frequencies explicitly and show that they are of order 1/m. 
  We discuss a new constraint for determining the superconformal U(1)_R symmetry of 4d N=1 SCFTs: It is the unique one which locally maximizes a(R) = 3Tr R^3-Tr R. This constraint comes close to proving the conjectured "a-theorem" for N=1 SCFTs. Using this "a-maximization", exact results can now be obtained for previously inaccessible 4d N=1 SCFTs. We apply this method to a rich class of examples: 4d N=1 SQCD with added matter chiral superfields in the adjoint representation. We classify a zoo of SCFTs, finding that Arnold's ADE singularity classification arises in classifying these theories via all possible relevant Landau-Ginzburg superpotentials. We verify that all RG flows are indeed compatible with the "a-theorem" conjecture, a_{IR}<a_{UV}, in every case 
  We define extended SL(2,R)/U(1) characters which include a sum over winding sectors. By embedding these characters into similarly extended characters of N=2 algebras, we show that they have nice modular transformation properties. We calculate the modular matrices of this simple but non-trivial non-rational conformal field theory explicitly . As a result, we show that discrete SL(2,R) representations mix with continuous SL(2,R) representations under modular transformations in the coset conformal field theory. We comment upon the significance of our results for a general theory of non-rational conformal field theories. 
  Quintessential inflation describes a scenario in which both inflation and dark energy (quintessence) are described by the same scalar field. In conventional braneworld models of quintessential inflation gravitational particle production is used to reheat the universe. This reheating mechanism is very inefficient and results in an excessive production of gravity waves which violate nucleosynthesis constraints and invalidate the model. We describe a new method of realizing quintessential inflation on the brane in which inflation is followed by `instant preheating' (Felder, Kofman & Linde 1999). The larger reheating temperature in this model results in a smaller amplitude of relic gravity waves which is consistent with nucleosynthesis bounds. The relic gravity wave background has a `blue' spectrum at high frequencies and is a generic byproduct of successful quintessential inflation on the brane. 
  We discuss the appearance of non-supersymmetric D6-brane GUT model constructions. We focus on the construction of the first examples of flipped SU(5) and SU(5) GUTS which have only the SM at low energy. These constructions are based on 4D compactifications in Z3 toroidal orientifolds of type IIA with D6-branes intersecting at angles. 
  We find de Sitter and flat space solutions with all moduli stabilized in four dimensional supergravity theories derived from the heterotic and type II string theories, and explain how all the previously known obstacles to finding such solutions can be removed. Further, we argue that if the compact manifold allows a large enough space of discrete topological choices then it is possible to tune the parameters of the four dimensional supergravity such that a hierarchy is created and the solutions lie in the outer region of moduli space in which the compact volume is large in string units, the string coupling is weak, and string perturbation theory is valid. We show that at least two light chiral superfields are required for this scenario to work, however, one field is sufficient to obtain a minimum with an acceptably small and negative cosmological constant. We discuss cosmological issues of the scenario and the possible role of anthropic considerations in choosing the vacuum of the theory. We conclude that the most likely stable vacuua are in or near the central region of moduli space where string perturbation theory is not strictly valid, and that anthropic considerations cannot help much in choosing a vacuum. 
  We review the basic results concerning the structure of effective action in N=4 supersymmetric Yang-Mills theory in Coulomb phase. Various classical formulations of this theory are considered. We show that the low-energy effective action depending on all fileds of N=4 vector multiplet can be exactly found. This result is discussed on the base of algebraic analysis exploring the general harmonic superspace techniques and on the base of straightforward quantum field theory calculations using the N=2 supersymmetric background field method. We study the one-loop effective action beyond leading low-energy approximation and construct supersymmetric generalization of Heisenberg-Euler-Schwinger effective action depending on all fields of N=4 vector multiplet. We also consider the derivation of leading low-enrgy effective action at two loops. 
  We discuss the signature of space-time in the context of the E_11 -conjecture. In this setting, the space-time signature depends on the choice of basis for the ``gravitational sub-algebra'' A_10, and Weyl transformations connect interpretations with different signatures of space-time. Also the sign of the 4-form gauge field term in the Lagrangian enters as an adjustable sign in a generalized signature. Within E_11, the combination of space-time signature (1,10) with conventional sign for the 4-form term, appropriate to M-theory, can be transformed to the signatures (2,9) and (5,6) of Hull's M*- and M'-theories (as well as (6,5), (9,2) and (10,1)). Theories with other signatures organize in orbits disconnected from these theories. We argue that when taking E_11 seriously as a symmetry algebra, one cannot discard theories with multiple time-directions as unphysical. We also briefly explore links with the SL(32,R) conjecture. 
  The non-conformal analog of abelian T-duality transformations relating pairs of axial and vector integrable models from the non abelian affine Toda family is constructed and studied in detail. 
  We demonstrate the existence of an infinite number of local commuting charges for classical solutions of the string sigma model on AdS_5 x S^5 associated with a certain circular three-spin solution spinning with large angular momenta in three orthogonal directions on the five-sphere. Using the AdS/CFT correspondence we find agreement to one-loop with the tower of conserved higher charges in planar N=4 super Yang-Mills theory associated with the dual composite single-trace operator in the highest weight representation (J_1,J_2,J_2) of SO(6). The agreement can be explained by the presence of integrability on both sides of the duality. 
  We compute the one-loop \beta-functions describing the renormalisation of the coupling constant \lambda and the frequency parameter \Omega for the real four-dimensional duality-covariant noncommutative \phi^4-model, which is renormalisable to all orders. The contribution from the one-loop four-point function is reduced by the one-loop wavefunction renormalisation, but the \beta_\lambda-function remains non-negative. Both \beta_\lambda and \beta_\Omega vanish at the one-loop level for the duality-invariant model characterised by \Omega=1. Moreover, \beta_\Omega also vanishes in the limit \Omega \to 0, which defines the standard noncommutative \phi^4-quantum field theory. Thus, the limit \Omega \to 0 exists at least at the one-loop level. 
  Starting from the recent classification of quotients of Freund--Rubin backgrounds in string theory of the type AdS_{p+1} x S^q by one-parameter subgroups of isometries, we investigate the physical interpretation of the associated quotients by discrete cyclic subgroups. We establish which quotients have well-behaved causal structures, and of those containing closed timelike curves, which have interpretations as black holes. We explain the relation to previous investigations of quotients of asymptotically flat spacetimes and plane waves, of black holes in AdS and of Godel-type universes. 
  In recent years a Hopf algebraic structure underlying the process of renormalization in quantum field theory was found. It led to a Birkhoff factorization for (regularized) Hopf algebra characters, i.e. for Feynman rules. In this work we would like to show that this Birkhoff factorization finds its natural formulation in terms of a classical r-matrix, coming from a Rota-Baxter structure underlying the target space of the regularized Hopf algebra characters. Working in the rooted tree Hopf algebra, the simple case of the Hopf subalgebra of ladder trees is treated in detail. The extension to the general case, i.e. the full Hopf algebra of rooted trees or Feynman graphs is briefly outlined. 
  It is argued here that the quantum computation of the vacuum pressure must take into account the contribution of zero-point oscillations of a rank-three gauge field. The field A_{\mu\nu\rho} possesses no radiative degrees of freedom, its sole function being that of polarizing the vacuum through the formation of \textit{finite} domains characterized by a non-vanishing, constant, but otherwise arbitrary pressure. This extraordinary feature, rather unique among quantum fields, is exploited to associate the A_{\mu\nu\rho} field with the ``bag constant'' of the hadronic vacuum, or with the cosmological term in the cosmic case. We find that the quantum fluctuations of A_{\mu\nu\rho} are inversely proportional to the confinement volume and interpret the result as a Casimir effect for the hadronic vacuum. With these results in hands and by analogy with the electromagnetic and string case, we proceed to calculate the Wilson loop of the three-index potential coupled to a ``test'' relativistic bubble. From this calculation we extract the static potential between two opposite points on the surface of a spherical bag and find it to be proportional to the enclosed volume. 
  This is a write-up of lectures intended for (under)graduate students. Contents: Scalar Ansatz (KP hierarchy). Fermionic Fock space. Fermi-Bose correspondence. KP hierarchy via free fermions. Formal distributions and locality. Operator product expansion. Vertex algebras. Free fermions. Virasoro algebra in KdV. 
  We construct the boundary ground ring in c < 1 open string theories with non-zero boundary cosmological constant (FZZT brane), using the Coulomb gas representation. The ring relations yield an over-determined set of functional recurrence equations for the boundary correlation functions, which involve shifts of the target space momenta of the boundary fields as well as the boundary parameters on the different segments of the boundary. 
  The basic BRST cohomological properties of a free, massless tensor field with the mixed symmetry of the Riemann tensor are studied in detail. It is shown that any non-trivial co-cycle from the local BRST cohomology group can be taken to stop at antighost number three, its last component belonging to the cohomology of the exterior longitudinal derivative and containing non-trivial elements from the (invariant) characteristic cohomology. 
  The anomalous Ward-Takahashi identity for the superconformal symmetry in the four-dimensional N=1 supersymmetric Yang-Mills theory is studied in terms of the stochastic quantization method (SQM). By applying the background field method to the SQM approach, we derive the superconformal anomaly in the one-loop approximation and show that the supersymmetric stochastic gauge fixing term does not contribute to the anomaly. 
  There is evidence that string theory possesses a large discretuum of stable and/or metastable ground states, with zero or four supersymmetries in four dimensions. I discuss critically the nature of this evidence. Assuming this "landscape" exists, anthropic explanations of some quantities are almost inevitable. I explain that this landscape is likely to lead to a prediction of low energy supersymmetry. But we argue that many features of low energy physics are not anthropic and, as currently understood, the landscape picture will get them wrong. This indicates that this viewpoint is potentially falsifiable. Moreover, if it is correct, many questions must be answered through more conventional scientific explanations. This is based on talks presented at the conference QTS3 at the University of Cincinnati and at the ITP in 2003. 
  We study the stability of the RS brane embedded in the AdS$_5$ vacuum of 5d gauged supergravity, where many tachyonic scalars exist. We consider a model in which these scalars couple to the brane such that the BPS conditions are satisfied to preserve the bulk supersymmetric configuration. In this case, we find that these tachyons are not trapped on the brane and only the massless dilaton is localized. As a result, the braneworld is stable. Further, the effective action of trapped fields is studied by using the brane running method developed recently, and we find that the action is independent of the brane position. 
  The issue concerning rigorous methods recently developed in deriving the asymptotics of quasi-normal modes is revisited and applied to a generic non rotating multi-horizon black holes solution. Some examples are illustrated and the single horizon cases are also considered. As a result, the asymptotics for large angular momentum parameter is shown to depend on the difference between the maximal or Nariai black hole mass and the ordinary black hole mass. The extremal limit is also discussed and the exact evaluation of the quasi-normal frequencies related to the Nariai space-time is presented, as a consistent check of the general asymptotic formula. 
  We consider the coupling of A_{\mu\nu\rho} to the generic current of matter field, later identified with the spin density current of a Dirac field. In fact, one of the objectives of this paper is to investigate the impact of the quantum fluctuations of A_{\mu\nu\rho} on the effective dynamics of the spinor field. The consistency of the field equations, even at the classical level, requires the introduction of a mass term for A_{\mu\nu\rho}. In this case, the Casimir vacuum pressure includes a contribution that is explicitly dependent on the mass of A_{\mu\nu\rho} and leads us to conclude that the mass term plays the same role as the infrared cutoff needed to regularize the finite volume partition functional previously calculated in the massless case. Remarkably, even in the presence of a mass term, A_{\mu\nu\rho} contains a mixture of massless and massive spin-0 fields so that the resulting equation is still gauge invariant. This is yet another peculiar, but physically relevant property of A_{\mu\nu\rho} since it is reflected in the effective dynamics of the spinor fields and confirms the confining property of A_{\mu\nu\rho} already expected from the earlier calculation of the Wilson loop. 
  Anomalous U(1)s are omnipresent in realizations of the Standard Model using D-branes. Such models are typically non-supersymmetric, and the anomalous U(1) masses are potentially relevant for experiment. In this paper, the string calculation of anomalous U(1) masses (hep-th/0204153) is extended to non-supersymmetric orientifolds. 
  The Yang-Mills Schr\"odinger equation is variationally solved in Coulomb gauge for the vacuum sector using a trial wave functional, which is strongly peaked at the Gribov horizon. We find the absence of gluons in the infrared and also a confining quark potential. 
  With some assumptions, the algebra between Ishibashi states in string field theory can be reduced to a commutative ring. From this viewpoint, Cardy states can be identified with its idempotents. The algebra can be identified with a fusion ring for the rational conformal field theory and a group ring for the orbifold. This observation supports our previous observation that boundary states satisfy a universal idempotency relation under closed string star product. 
  Recently, Giusto and Halpern reported the open-string description of a certain basic class of untwisted open WZW strings, including their associated non-commutative geometry and open-string KZ equations. In this paper, we combine this development with results from the theory of current-algebraic orbifolds to find the open-string description of a corresponding basic class of {\it twisted} open WZW strings, which begin and end on different WZW branes. The basic class of twisted open WZW strings is in 1-to-1 correspondence with the twisted sectors of all closed-string WZW orbifolds, and moreover, the basic class can be decomposed into a large collection of open-string WZW orbifolds. At the classical level, these open-string orbifolds exhibit new {\it twisted non-commutative geometries}, and we also find the relevant {\it twisted open-string KZ equations} which describe these orbifolds at the quantum level. In a related development, we also formulate the closed-string description (in terms of twisted boundary states) of the {\it general} twisted open WZW string. 
  Recently Hertog, Horowitz, and Maeda have argued that cosmic censorship can be generically violated in string theory in anti-de Sitter spacetime by considering a collapsing bubble of a scalar field whose mass saturates the Breitenlohner-Freedman bound. We study this system numerically and find that for various choices of initial data black holes form rather than naked singularities, implying that in these cases cosmic censorship is upheld. 
  We give a systematic derivation of the consistency conditions which constrain open-closed disk amplitudes of topological strings. They include the A-infinity relations (which generalize associativity of the boundary product of topological field theory), as well as certain homotopy versions of bulk-boundary crossing symmetry and Cardy constraint. We discuss integrability of amplitudes with respect to bulk and boundary deformations, and write down the analogs of WDVV equations for the space-time superpotential. We also study the structure of these equations from a string field theory point of view. As an application, we determine the effective superpotential for certain families of D-branes in B-twisted topological minimal models, as a function of both closed and open string moduli. This provides an exact description of tachyon condensation in such models, which allows one to determine the truncation of the open string spectrum in a simple manner. 
  We present a model of supersymmetric non-Abelian Chern-Simons theories in three-dimensions with arbitrarily many supersymmetries, called alephnull-extended supersymmetry. The number of supersymmetry N equals the dimensionality of any non-Abelian gauge group G as N = dim G. Due to the supersymmetry parameter in the adjoint representation of a local gauge group G, supersymmetry has to be local. The minimal coupling constant is to be quantized, when the homotopy mapping is nontrivial: \pi_3(G) = Z. Our results indicate that there is still a lot of freedom to be explored for Chern-Simons type theories in three dimensions, possibly related to M-theory. 
  The cyclic SOS model is considered on the basis of Smirnov's form factor bootstrap approach. Integral solutions to the quantum Knizhnik-Zamolodchikov equations of level 0 are presented. 
  Recently, Carlip proposed a derivation of the entropy of the two-dimensional dilatonic black hole by investigating the Virasoro algebra associated with a newly introduced near-horizon conformal symmetry. We point out not only that the algebra of these conformal transformations is not well defined on the horizon, but also that the correct use of the eigenvalue of the operator $L_0$ yields vanishing entropy. It has been shown that these problems can be resolved by choosing a different basis of the conformal transformations which is regular even at the horizon. We also show the generalization of Carlip's derivation to any higher dimensional case in pure Einstein gravity. The entropy obtained is proportional to the area of the event horizon, but it also depends linearly on the product of the surface gravity and the parameter length of a horizon segment in consideration. We finally point out that this derivation of black hole entropy is quite different from the ones proposed so far, and several features of this method and some open issues are also discussed. 
  A classical spinning particle based on the Kerr-Newman black hole (BH) solution is considered. For parameters of spinning particles $|a|>>m$, the BH horizons disappear and BH image is drastically changed. We show that it turns into a skeleton formed by two coupled stringy systems. One of them is the Kerr singular ring which can be considered as a circular D-string with an orientifold world-sheet.   Analyzing the aligned to the Kerr congruence electromagnetic excitations of this string, we obtain the second stringy system which consists of two axial half-infinite chiral D-strings. These axial strings are similar to the Dirac monopole strings but carry the induced chiral traveling pp-waves. Their field structure can be described by the field model suggested by Witten for the cosmic superconducting strings.   We discuss a relation of this stringy system to the Dirac equation and argue that this stringy system can play a role of a classical carrier of the wave function. 
  In this letter, we elaborate on the SL(2,Z) action on three dimensional conformal field theories with U(1) symmetry introduced by Witten, by trying to give an explicit verification of the claim regarding holographic dual of the S operation in AdS/CFT correspondence. A consistency check with the recently proposed prescription on boundary condition of bulk fields when we deform the boundary CFT in a non-standard manner is also discussed. 
  A q-analogue of four dimensional conformally invariant field theory based on the quantum algebra U_{q}(so(4,2)) is proposed. The two- and three-point correlation functions are calculated. The construction is elaborated in order to fit the Hopf algebra structure. 
  We describe firstly the basic features of quantum $\kappa$-Poincar\'{e} symmetries with their Hopf algebra structure. The quantum $\kappa$-Poincar\'{e} framework in any basis relates rigidly the quantum $\kappa$-Poincar\'{e} algebra with quantum $\kappa$-Poincar\'{e} group, noncommutative space-time and $\kappa$-deformed phase space. Further we present the approach of Doubly Special Relativity (DSR) theories, which introduce(in the version DSR1) kinematically the frame - independent fundamental mass parameter as described by maximal three-momentum $|\overrightarrow{p}|=\kappa c$. We argue why the DSR theories in one-particle sector can be treated as the part of quantum $\kappa$-Poincar\'{e} framework. The DSR formulation has been extended to multiparticle states either in a way leading to nonlinear description of classical relativistic symmetries, or providing the identification of DSR approach with full quantum $\kappa$-Poincar\'{e} framework. 
  We investigate the gauging of a three-dimensional deformation of the anti-de Sitter algebra, which accounts for the existence of an invariant energy scale. By means of the Poisson sigma model formalism, we obtain explicit solutions of the field equations, which reduce to the BTZ black hole in the undeformed limit. 
  We consider type II superstring compactifications on the singular Spin(7) manifold constructed as a cone on SU(3)/U(1). Based on a toric realization of the projective space CP^2, we discuss how the manifold can be viewed as three intersecting Calabi-Yau conifolds. The geometric transition of the manifold is then addressed in this setting. The construction is readily extended to higher dimensions where we speculate on possible higher-dimensional geometric transitions. Armed with the toric description of the Spin(7) manifold, we discuss a brane/flux duality in both type II superstring theories compactified on this manifold. 
  We investigate the general features of renormalization group flows near superconformal fixed points of four dimensional N=1 supersymmetric gauge theories with gravity duals. The gauge theories we study arise as the world-volume theory on a set of D-branes at a Calabi-Yau singularity where a del Pezzo surface shrinks to zero size. Based mainly on field theory analysis, we find evidence that such flows are often chaotic and contain exotic features such as duality walls. For a gauge theory where the del Pezzo is the Hirzebruch zero surface, the dependence of the duality wall height on the couplings at some point in the cascade has a self-similar fractal structure. For a gauge theory dual to CP^2 blown up at a point, we find periodic and quasi-periodic behavior for the gauge theory couplings that does not violate the a-conjecture. Finally, we construct supergravity duals for these del Pezzos that match our field theory beta functions. 
  It has recently been proposed that the D-instanton expansion of the open topological B-model on P^{3|4} is equivalent to the perturbative expansion of N=4 super Yang-Mills theory in four dimensions. In this note we extend the results of hep-th/0402016 and recover the gauge theory results for all n-point googly amplitudes by computing the integral over the moduli space of curves of degree n-3 in P^{3|4}. 
  We give a brief review of our approach to the quantization of superstrings. New is a covariant derivation of the measure at tree level and a path integral formula for this measure. 
  In the framework of the augmented superfield formalism, the local, covariant, continuous and off-shell (as well as on-shell) nilpotent (anti-)BRST symmetry transformations are derived for a $(0 + 1)$-dimensional free scalar relativistic particle that provides a prototype physical example for the more general reparametrization invariant string- and gravitational theories. The trajectory (i.e. the world-line) of the free particle, parametrized by a monotonically increasing evolution parameter $\tau$, is embedded in a $D$-dimensional flat Minkowski target manifold. This one-dimensional system is considered on a $(1 + 2)$-dimensional supermanifold parametrized by an even element $\tau$ and a couple of odd elements ($\theta$ and $\bar\theta$) of a Grassmannian algebra. The horizontality condition and the invariance of the conserved (super)charges on the (super)manifolds play very crucial roles in the above derivations of the nilpotent symmetries. The geometrical interpretations for the nilpotent (anti-)BRST charges are provided in the framework of augmented superfield approach. 
  Extending the work of Gaiotto, Itzhaki and Rastelli in hep-th/0304192, we derive a general prescription for computing amplitudes involving a periodic array of D-branes in imaginary time to arbitrary order. We use this prescription to show that closed string amplitudes with b boundaries are identical to closed string amplitudes with b additional insertions of a particular physical closed string state. We perform an explicit computation for the annulus, and argue on the basis of open and closed string field theory for higher order amplitudes. We also discuss possible subtleties in the prescription related to collisions of boundaries and insertions, and argue that they are harmless. This verifies the proposal that a periodic array of D-branes in imaginary time corresponds to a pure closed string background. 
  We investigate the stability of inflating branes embedded in an O(2) texture formed in one extra dimension. The model contains two 3-branes of nonzero tension, and the extra dimension is compact. When the gravitational perturbation is applied, the vacuum energy which is responsible for inflation on the branes stabilizes the branes if the symmetry-breaking scale of the texture is smaller than some critical value. This critical value is determined by the particle-hierarchy scale between the two branes, and is smaller than the 5D Planck-mass scale. The scale of the vacuum energy can be considerably low in providing the stability. This stability story is very different from the flat-brane case which always suffers from the instability due to the gravitational perturbation. 
  In the light of the proposal of hep-th/0207195, we discuss in detail the issue of the cosmological constant, explaining how can string theory naturally predict the value which is experimentally observed, without low-energy supersymmetry. 
  A Lorentz-noninvariant modification of the kinematic dispersion law was proposed in [hep-th/0211237], claimed to be derivable from from q-deformed noncommutative theory, and argued to evade ultrahigh energy threshold anomalies (trans-GKZ-cutoff cosmic rays and TeV-photons) by raising the respective thresholds. It is pointed out that such dispersion laws do not follow from deformed oscillator systems, and the proposed dispersion law is invalidated by tachyonic propagation, as well as photon instability, in addition to the process considered. 
  We interpret the A and B model topological strings on CP^{3|4} as equivalent to open N=2 string theory on spacetime with signature (2,2), when covariantized with respect to SO(2,2) and supersymmetrized a la Siegel. We propose that instantons ending on Lagrangian branes wrapping RP^{3|4} deform the self-dual N=4 Yang-Mills sector to ordinary Yang-Mills by generating a `t Hooft like expansion. We conjecture that the A and B versions are S-dual to each other. We also conjecture that mirror symmetry may explain the recent observations of Witten that twistor transformed N=4 Yang-Mills amplitudes lie on holomorphic curves. 
  We construct the monodromy matrix for a class of gauged WZWN models in the plane wave limit and discuss various properties of such systems. 
  In this lecture I address the issue of possible large distance modification of gravity and its observational consequences. Although, for the illustrative purposes we focus on a particular simple generally-covariant example, our conclusions are rather general and apply to large class of theories in which, already at the Newtonian level, gravity changes the regime at a certain very large crossover distance $r_c$. In such theories the cosmological evolution gets dramatically modified at the crossover scale, usually exhibiting a "self-accelerated" expansion, which can be differentiated from more conventional "dark energy" scenarios by precision cosmology. However, unlike the latter scenarios, theories of modified-gravity are extremely constrained (and potentially testable) by the precision gravitational measurements at much shorter scales. Despite the presence of extra polarizations of graviton, the theory is compatible with observations, since the naive perturbative expansion in Newton's constant breaks down at a certain intermediate scale. This happens because the extra polarizations have couplings singular in $1/r_c$. However, the correctly resummed non-linear solutions are regular and exhibit continuous Einsteinian limit. Contrary to the naive expectation, explicit examples indicate that the resummed solutions remain valid after the ultraviolet completion of the theory, with the loop corrections taken into account. 
  We obtain an explicit expression relating the writhing number, $W[C]$, of the quantum path, $C$, with any value of spin, $s$, of the particle which sweeps out that closed curve. We consider a fractal approach to the fractional spin particles and, in this way, we make clear a deeper connection between the Gauss-Bonnet theorem with the spin-statistics relation via the concept of Hausdorff dimension, $h$, associated to the fractal quantum curves of the particles:   \frac{h}{2+2s}=W[C]=\frac{1}{4\pi}\oint_{C}d x_{\alpha}\oint_{C}d x_{\beta} \epsilon^{\alpha\beta\gamma} \frac{(x-y)_{\gamma}}{|x-y|^3}. 
  We investigate quantum corrections to the effective action of the universal hypermultiplet in the language of projective superspace. We rederive the recently found one-loop correction to the universal hypermultiplet moduli space geometry. The deformed metric is described as a superspace action in terms of a single function, homogeneous of first degree. Our framework leads us to a natural proposal for the nonperturbative moduli space metric induced by five-brane instantons. 
  We study zeta-functions for a one parameter family of quintic threefolds defined over finite fields and for their mirror manifolds and comment on their structure. The zeta-function for the quintic family involves factors that correspond to a certain pair of genus 4 Riemann curves. The appearance of these factors is intriguing since we have been unable to `see' these curves in the geometry of the quintic. Having these zeta-functions to hand we are led to comment on their form in the light of mirror symmetry. That some residue of mirror symmetry survives into the zeta-functions is suggested by an application of the Weil conjectures to Calabi-Yau threefolds: the zeta-functions are rational functions and the degrees of the numerators and denominators are exchanged between the zeta-functions for the manifold and its mirror. It is clear nevertheless that the zeta-function, as classically defined, makes an essential distinction between Kahler parameters and the coefficients of the defining polynomial. It is an interesting question whether there is a `quantum modification' of the zeta-function that restores the symmetry between the Kahler and complex structure parameters. We note that the zeta-function seems to manifest an arithmetic analogue of the large complex structure limit which involves 5-adic expansion. 
  Various asymmetric orbifold models based on chiral shifts and chiral reflections are investigated. Special attention is devoted to the consistency of the models with two fundamental principles for asymmetric orbifolds : modular invariance and the existence of a proper Hilbert space formulation for states and operators. The interplay between these two principles is non-trivial. It is shown, for example, that their simultaneous requirement forces the order of a chiral reflection to be 4, instead of the naive 2. A careful explicit construction is given of the associated one-loop partition functions. At higher loops, the partition functions of asymmetric orbifolds are built from the chiral blocks of associated symmetric orbifolds, whose pairings are determined by degenerations to one-loop. 
  We propose a variant of the KKLT (A)dS flux vacuum construction which does not require an antibrane to source the volume modulus. The strategy is to find nonzero local minima of the no-scale potential in the complex structure and dilaton directions in moduli space. The corresponding no-scale potential expanded about this point sources the volume modulus in the same way as does the antibrane of the KKLT construction. We exhibit explicit examples of such nonzero local minima of the no-scale potential in a simple toroidal orientifold model. 
  The physical states in a world-volume model of a non-critical 3-brane are systematically constructed using techniques of four-dimensional conformal field theories on R*S^3 developed recently. Invariant combinations of creation modes under a special conformal transformation provide building blocks of physical states. Any state can be created by acting with such building blocks on a conformally invariant vacuum in an invariant way under the other conformal charges: the Hamiltonian and rotation generators on S^3. We explicitly construct building blocks for scalar, vector and gravitational fields, and classify them as finite types. 
  We construct a closed form of the action of the supersymmetric $CP^N$ sigma model on noncommutative superspace in four dimensions. We show that this model has $\mathcal{N}={1/2}$ supersymmetry and that the transformation law is not modified. The supersymmetric $CP^N$ sigma model on noncommutative superspace in two dimensions is obtained by dimensional reducing the model in four dimensions. 
  The formalism of graded Poisson-sigma models allows the construction of N=(2,2) dilaton supergravity in terms of a minimal number of fields. For the gauged chiral U(1) symmetry the full action, involving all fermionic contributions, is derived. The twisted chiral case follows by simple redefinition of fields. The equivalence of our approach to the standard second order one in terms of superfields is presented, although for the latter so far only the bosonic part of the action seems to have been available in the literature. It is shown how ungauged models can be obtained in a systematic way and some relations to relevant literature in superstring theory are discussed. 
  In this paper, a complex daor field which can be regarded as the square root of space-time metric is proposed to represent gravity. The locally complexified geometry is set up, and the complex spin connection constructs a bridge between gravity and SU(1,3) gauge field. Daor field equations in empty space are acquired, which are one-order differential equations and not conflict with Einstein's gravity theory. 
  We show that the commonly considered half BPS solutions of eleven dimensional supergravity and the ten dimensional type II theories, when expressed in terms of $E_{11}$ group elements, take the universal form $\exp(-{1\over 2}ln N \beta\cdot H)\exp((1-N)E_\beta)$. Using this formula we find new potential solutions to the $E_{11}$ non-linearly realisations corresponding to active fields which are beyond those in the supergravity approximations. These include the space filling nine brane of the IIB theory. We use $E_{11}$ to give a correspondence between the fields of the eleven dimensional and the IIA and IIB non-linear realisations without assuming any dimensional reduction. As one consequence, we find the eleven dimensional origin of the eight brane solution of the massive IIA theory. 
  We show that the generalized holonomy groups of ungauged supergravity theories with 8 real supercharges must be contained in SL(2-\nu,H)\ltimes{\nu H^{2-\nu}}\subseteq SL(2,H), where SL(2,H) is the generalized structure group. Here n=4\nu is the number of preserved supersymmetries, so the allowed values are limited to n=0,4,8. In particular, solutions of ungauged supergravities in four, five and six dimensions are examined and found to explicitly follow this pattern. We also argue that the G-structure has to be a subgroup of this generalized holonomy group, which may provide a possible classification for supergravity vacua with respect to the number of supercharges. 
  We compute the form of the Lagrangian of N=1 supersymmetric theories with gauged axion symmetries. It turns out that there appear generalized Chern-Simons terms that were not considered in previous superspace formulations of general N=1 theories. Such gaugings appear in supergravities arising from flux compactifications of superstrings, as well as from Scherk-Schwarz generalized dimensional reduction in M-theory. We also present the dual superspace formulation where axion chiral multiplets are dualized into linear multiplets. 
  Gauge symmetries generally appear as a constraint algebra, under which one expects all physical states to be singlets. However, quantum anomalies and boundary conditions introduce central charges and change this picture, thus causing gauge/diffeomorphism modes to become physical. We expose a cohomological (Higgs-less) generation of mass in U(N)-gauge invariant Yang-Mills theories through non-trivial representations of the gauge group. This situation is also present in black hole evaporation, where the Virasoro algebra turns out to be the relevant subalgebra of surface deformations of the horizon of an arbitrary black hole. 
  We construct finite size, supersymmetric, tubular D-brane configurations with three charges, two angular momenta and several brane dipole moments. In type IIA string theory these are tubular configurations with D0, D4 and F1 charge, as well as D2, D6 and NS5 dipole moments. These multi-charge generalizations of supertubes might have interesting consequences for the physics of the D1-D5-P black hole. We study the relation of the tubes to the spinning BMPV black hole, and find that they have properties consistent with describing some of the hair of this black hole. 
  During the black hole radiation, the interior contains all the matter of the initial black hole, together with the negative energy quanta entangled with the exterior Hawking radiation. Neither the initial matter nor the negative energy quanta evaporate from the black hole interior. Therefore, the information is not lost during the radiation. The black hole mass eventually drops to zero in semiclassical gravity, but this semiclassical state has an infinite temperature and still contains all the initial matter together with the negative energy entangled with the exterior radiation. 
  We investigate the high-energy behavior of the scattering amplitudes in the extra dimensional gauge theory where the gauge symmetry is broken by the boundary condition. We study, in particular, the 5-dimensional SU(5) grand unified theory whose 5th-dimensional coordinate is compactified on S^1/Z_2. We pay attention to the gauge symmetry compatible with the boundary condition on the orbifold and give the BRST formalism of the 4D theory which is obtained through integration of the 5D theory along the extra dimension. We derive the 4D equivalence theorem on the basis of the Slavnov-Taylor identities. We calculate the amplitudes of the process including four massive gauge bosons in the external lines and compare them with the ones for the connected reactions where the gauge fields are replaced by the corresponding would-be NG-like fields. We explicitly confirm the equivalence theorem to hold. 
  Extending a result of D.V. Vassilevich, we obtain the asymptotic expansion for the trace of a "spatially" regularized heat operator associated with a generalized Laplacian defined with integral Moyal products. The Moyal hyperplanes corresponding to any skewsymmetric matrix $\Theta$ being spectral triples, the spectral action introduced in noncommutative geometry by A. Chamseddine and A. Connes is computed. This result generalizes the Connes-Lott action previously computed by Gayral for symplectic $\Theta$. 
  In this work we propose two Lagrange multipliers with distinct coefficients for the light-front gauge that leads to the complete (non-reduced) propagator. This is accomplished via $(n\cdot A)^{2}+(\partial \cdot A)^{2}$ terms in the Lagrangian density. These lead to a well-defined and exact though Lorentz non invariant light front propagator. 
  We present new self-gravitating solutions in five dimensions that describe circular strings, i.e., rings, electrically coupled to a two-form potential (as e.g., fundamental strings do), or to a dual magnetic one-form. The rings are prevented from collapsing by rotation, and they create a field analogous to a dipole, with no net charge measured at infinity. They can have a regular horizon, and we show that this implies the existence of an infinite number of black rings, labeled by a continuous parameter, with the same mass and angular momentum as neutral black rings and black holes. We also discuss the solution for a rotating loop of fundamental string. We show how more general rings arise from intersections of branes with a regular horizon (even at extremality), closely related to the configurations that yield the four-dimensional black hole with four charges. We reproduce the Bekenstein-Hawking entropy of a large extremal ring through a microscopic calculation. Finally, we discuss some qualitative ideas for a microscopic understanding of neutral and dipole black rings. 
  An approach to special relativistic dynamics using the language of spinors and twistors is presented. Exploiting the natural conformally invariant symplectic structure of the twistor space, a model is constructed which describes a relativistic massive, spinning and charged particle, minimally coupled to an external electro-magnetic field. On the two-twistor phase space the relativistic Hamiltonian dynamics is generated by a Poincare scalar function obtained from the classical limit (appropriately defined by us) of the second order, to an external electro-magnetic field minimally coupled, Dirac operator. In the so defined relativistic classical limit there are no Grassman variables. Besides, the arising equation that describes dynamics of the relativistic spin differs significantly from the so called Thomas Bergman Michel Telegdi equation. 
  The effective four-dimensional, linearised gravity for a brane world model with higher order curvature terms and a bulk scalar field is analysed. Large and small distance gravitational laws are derived. The model has a single brane embedded in a five-dimensional bulk spacetime, and the scalar field represents the dilaton or a moduli field. The quadratic, Gauss-Bonnet curvature term (and corresponding higher kinetic terms for the scalar) is also included in the bulk action. It is particularly natural to include such terms in a brane world model. Boundary terms and junction conditions for the higher order terms are given.   The extra terms allow additional solutions of the field equations, which give better agreement with observational constraints. Brans-Dicke gravity is obtained on the brane. The scalar and tensor perturbations are affected differently by the higher gravity terms, and this provides a way for the scalar modes to be suppressed relative to the tensor ones. Another new (but less useful) feature is the appearance of instabilities for some parameter ranges. 
  We present a detailed analysis of the thermodynamics of two dimensional black hole solutions to type 0A with q units of electric and magnetic flux. We compute the free energy and derived quantities such as entropy and mass for an arbitrary non-extremal black hole. The free energy is non-vanishing, in contrast to the case of dilatonic 2-d black holes without electric and magnetic fluxes. The entropy of the extremal black holes is obtained, and we find it to be proportional to q^2, the square of the RR flux. We compare these thermodynamics quantities with those from candidate matrix model duals. 
  We analyse the most general supersymmetric solutions of D=11 supergravity consisting of a warped product of five-dimensional anti-de-Sitter space with a six-dimensional Riemannian space M_6, with four-form flux on M_6. We show that M_6 is partly specified by a one-parameter family of four-dimensional Kahler metrics. We find a large family of new explicit regular solutions where M_6 is a compact, complex manifold which is topologically a two-sphere bundle over a four-dimensional base, where the latter is either (i) Kahler-Einstein with positive curvature, or (ii) a product of two constant-curvature Riemann surfaces. After dimensional reduction and T-duality, some solutions in the second class are related to a new family of Sasaki-Einstein spaces which includes T^{1,1}/Z_2. Our general analysis also covers warped products of five-dimensional Minkowski space with a six-dimensional Riemannian space. 
  Astrophysical, terrestrial, and space-based searches for Lorentz violation are very briefly reviewed. Such searches are motivated by the fact that all superunified theories (and other theories that attempt to include quantum gravity) have some potential for observable violations of Lorentz invariance. Another motivation is the exquisite sensitivity of certain well-designed experiments and observations to particular forms of Lorentz violation. We also review some new predictions of a specific Lorentz-violating theory: If a fundamental energy \bar{m} in this theory lies below the usual GZK cutoff E_{GZK}, the cutoff is shifted to infinite energy; i.e., it no longer exists. On the other hand, if \bar{m} lies above E_{GZK}, there is a high-energy branch of the fermion dispersion relation which provides an alternative mechanism for super-GZK cosmic-ray protons. 
  We describe a model of D--brane inflation on fractional D3 branes transverse to a resolved and deformed conifold. The resolution and the deformation are both necessary for inflation. The fractional branes slowly approach each other along the $S^3$ and separate along the $S^2$ in the base of the conifold. We show that on the brane this corresponds to hybrid inflation. We describe the model also in terms of intersecting branes. 
  FRW solutions of the string theory low-energy effective actions are described, yielding a dilaton which first decreases and then increases. We study string creation in these backgrounds and find an exponential divergence due to an initial space-like singularity. We conjecture that this singularity may be removed by the effects of back-reaction, leading to a solution which at early times is de Sitter space. 
  The boundary state associated with the rolling tachyon solution on an unstable D-brane contains a part that decays exponentially in the asymptotic past and the asymptotic future, but it also contains other parts which either remain constant or grow exponentially in the past or future. We argue that the time dependence of the latter parts is completely determined by the requirement of BRST invariance of the boundary state, and hence they contain information about certain conserved charges in the system. We also examine this in the context of the unstable D0-brane in two dimensional string theory where these conserved charges produce closed string background associated with the discrete states, and show that these charges are in one to one correspondence with the symmetry generators in the matrix model description of this theory. 
  We show that a model based on a D3-brane--anti-D3-brane system at finite temperature, proposed previously as a microscopic description of the non-rotating black threebrane of type IIB supergravity arbitrarily far from extremality, can also successfully reproduce the entropy of the rotating threebrane with arbitrary charge (including the neutral case, which corresponds to the Kerr black hole in seven dimensions). Our results appear to confirm in particular the need for a peculiar condition on the energy of the two gases involved in the model, whose physical interpretation remains to be elucidated. 
  We construct the Z_k parafermions as diagonal affine cosets and apply them to the quantum Hall effect. This realization is particularly convenient for the analysis of the Z_k pairing rules, the modular S-matrices, the W_k symmetry and quantum group symmetry. The results are used for the computation of the mesoscopic chiral persistent currents in presence of Aharonov-Bohm flux. 
  We derive the four dimensional N=1/2 super Yang-Mills theory from tree-level computations in RNS open string theory with insertions of closed string Ramond-Ramond vertices. We also study instanton configurations in this gauge theory and their ADHM moduli space, using systems of D3 and D(-1) branes in a R-R background. 
  We consider a massive scalar field theory in anti-de Sitter space, in both minimally and non-minimally coupled cases. We introduce a relevant double-trace perturbation at the boundary, by carefully identifying the correct source and generating functional for the corresponding conformal operator. We show that such relevant double-trace perturbation introduces changes in the coefficients in the boundary terms of the action, which in turn govern the existence of a bound state in the bulk. For instance, we show that the usual action, containing no additional boundary terms, gives rise to a bound state, which can be avoided only through the addition of a proper boundary term. Another notorious example is that of a conformally coupled scalar field, supplemented by a Gibbons-Hawking term, for which there is no associated bound state. In general, in both minimally and non-minimally coupled cases, we explicitly compute the boundary terms which give rise to a bound state, and which ones do not. In the non-minimally coupled case, and when the action is supplemented by a Gibbons-Hawking term, this also fixes allowed values of the coupling coefficient to the metric. We interpret our results as the fact that the requirement to satisfy the Breitenlohner-Freedman bound does not suffice to prevent tachyonic behavior from existing in the bulk, as it must be supplemented by additional conditions on the coefficients in the boundary terms of the action. 
  We show that eternal inflation is compatible with holography. In particular, we emphasize that if a region is asymptotically de Sitter in the future, holographic arguments by themselves place no bound on the number of past e-foldings. We also comment briefly on holographic restrictions on the production of baby universes. 
  We describe the scattering of D-strings stretched between D3-branes, working from the D-string perspective. From the D3-brane perspective the ends of the D-strings are magnetic monopoles, and so the scattering we describe is equivalent to monopole scattering. Our aim is to test the prediction made by Manton for the energy radiated during monopole scattering. 
  We derive the TBA system of equations from the S-matrix describing integrable massive perturbation of the coset $G_l \times G_m / G_{l+m}$ by the field $(1,1,adj)$ for all the infinite series of the simple Lie algebras $G=A,B,C,D$. In the cases A,C, where the full S-matrices are known, the derivation is exact, while B,D cases dictate some natural assumption about the form of the crossing- -unitarizing prefactor for any two fundam. reps of the algebras. In all the cases the derived systems are transformed to the corresponding functional Y-system and shown to have the correct high temperature (UV) asymptotic in the ground state, reproducing the correct central charge of the coset. Some specific particular cases of the considered S-matrices are discussed. 
  If the Lie group of a non-Abelian theory is replaced by the corresponding q-group, one is led to replace the Lie algebra by two dual algebras. The first of these lies close to the Lie algebra that it is replacing while the second introduces new degrees of freedom. We interpret the theory based on the first algebra as a modification of standard field theory while we propose that the new degrees of freedom introduced by the second algebra describe solitonic rather than point particle sources. We have earlier found that the modified q-electroweak theory differs very little from the standard theory. Here we find a similar result for q-gravity. Both of the modified theories are incomplete, however, and must be completed by the solitonic sector. We propsoe that the solitonic sector of both q-electroweak and q-gravity have the symmetry of knots associated with SU_q(2). Since the Lorentz group is here deformed, there is no longer the standard classification of particles described by mass and spin. There is instead a classification of irreducible structures determined by SU_q(2). 
  The conservation of energy implies that an isolated radiating black hole cannot have an emission spectrum that is precisely thermal. Moreover, the no-hair theorem is only approximately applicable. We consider the implications for the black hole information puzzle. 
  We compute, on the disk, the non-linear tachyon $\beta$-function, $\beta^T$, of the open bosonic string theory. $\beta^T$ is determined both in an expansion to the third power of the field and to all orders in derivatives and in an expansion to any power of the tachyon field in the leading order in derivatives. We construct the Witten-Shatashvili (WS) space-time effective action $S$ and prove that it has a very simple universal form in terms of the renormalized tachyon field and $\beta^T$. The expression for $S$ is well suited to studying both processes that are far off-shell, such as tachyon condensation, and close to the mass-shell, such as perturbative on-shell amplitudes. We evaluate $S$ in a small derivative expansion, providing the exact tachyon potential. The normalization of $S$ is fixed by requiring that the field redefinition that maps $S$ into the tachyon effective action derived from the cubic string field theory is regular on-shell. The normalization factor is in precise agreement with the one required for verifying all the conjectures on tachyon condensation. The coordinates in the space of couplings in which the tachyon $\beta$-function is non linear are the most appropriate to study RG fixed points that can be interpreted as solitons of $S$, $i.e.$ D-branes. 
  We formulate a noncommutative description of topological half-flat gravity in four dimensions. BRST symmetry of this topological gravity is deformed through a twisting of the usual BRST quantization of noncommutative gauge theories. Finally it is argued that resulting moduli space of instantons is characterized by the solutions of a noncommutative version of the Plebanski's heavenly equation. 
  Cohomological Yang-Mills theory is formulated on a noncommutative differentiable four manifold through the $\theta$-deformation of its corresponding BRST algebra. The resulting noncommutative field theory is a natural setting to define the $\theta$-deformation of Donaldson invariants and they are interpreted as a mapping between the Chevalley-Eilenberg homology of noncommutative spacetime and the Chevalley-Eilenberg cohomology of noncommutative moduli of instantons. In the process we find that in the weak coupling limit the quantum theory is localized at the moduli space of noncommutative instantons. 
  We consider, in five dimensions, the effective action from heterotic string which includes quantum gravity corrections up to (a')^2. The expansion, in the string frame, is in terms of |a'R|, where R is the scalar curvature and uses the third order Euler density, next to the Gauss-Bonnet term. For a positive tension brane and infinite extra dimension, the logarithmic class of solutions is less dependent from fine-tuning problems than in previous formulations. More importantly, the model suggests that in the full non-perturbative formulation, the string scale can be much lower than the effective Planck mass, without the string coupling to be vanishingly small. Also a less severe fine-tuning of the brane tension in needed. 
  In this letter, I indicate that complex daor field should also have spinor suffixes. The gravitation and gauge fields are unified under the framework of daor field. I acquire the elegant coupling equation of gravitation and gauge fields, from which Einstein's gravitational equation can be deduced. 
  The work is devoted to the generalization of the Dirac equation for a flat locally anisotropic, i.e., Finslerian space-time. At first we reproduce the corresponding metric and a group of the generalized Lorentz transformations, which has the meaning of the relativistic symmetry group of such event space. Next, proceeding from the requirement of the generalized Lorentz invariance we find a generalized Dirac equation in its explicit form. An exact solution of the nonlinear generalized Dirac equation is also presented. 
  The holographic principle is tested by examining the logarithmic and higher order corrections to the Bekenstein-Hawking entropy of black holes. For the BTZ black hole, I find some disagreement in the principle for a holography screen at spatial infinity beyond the leading order, but a holography with the screen at the horizon does not, with an appropriate choice of a period parameter, which has been undetermined at the leading order, in Carlip's horizon-CFT approach for black hole entropy in any dimension. Its higher dimensional generalization is considered to see a universality of the parameter choice. The horizon holography from Carlip's is compared with several other realizations of a horizon holography, including induced Wess-Zumino-Witten model approaches and quantum geometry approach, but none of the these agrees with Carlip's, after clarifications of some confusions. Some challenging open questions are listed finally. 
  We study supergravity duals of strongly coupled four dimensional gauge theories formulated on compact quotients of hyperbolic spaces. The resulting background geometries are represented by Euclidean wormholes, which complicates establishing the precise gauge theory/string theory correspondence dictionary. These backgrounds suffer from the non-perturbative instabilities arising from the D3 - anti-D3 pair production in the background four-form potential. We discuss conditions for suppressing this Schwinger-like instability. We find that Euclidean wormholes arising in this construction develop a naked singularity, before they can be stabilized. 
  Considering the conformal scaling gauge symmetry as a fundamental symmetry of nature in the presence of gravity, a scalar field is required and used to describe the scale behavior of universe. In order for the scalar field to be a physical field, a gauge field is necessary to be introduced. A gauge invariant potential action is constructed by adopting the scalar field and a real Wilson-like line element of the gauge field. Of particular, the conformal scaling gauge symmetry can be broken down explicitly via fixing gauge to match the Einstein-Hilbert action of gravity. As a nontrivial background field solution of pure gauge has a minimal energy in gauge interactions, the evolution of universe is then dominated at earlier time by the potential energy of background field characterized by a scalar field. Since the background field of pure gauge leads to an exponential potential model of a scalar field, the universe is driven by a power-law inflation with the scale factor $a(t) \sim t^p$. The power-law index $p$ is determined by a basic gauge fixing parameter $g_F$ via $p = 16\pi g_F^2[1 + 3/(4\pi g_F^2) ]$. For the gauge fixing scale being the Planck mass, we are led to a predictive model with $g_F=1$ and $p\simeq 62$. 
  We study the effective potential of a scalar field based on the 5D gauged supergravity for the RS1 brane model in terms of the brane running method. The scalar couples to the brane such that the BPS conditions are satisfied for the bulk configuration. The resulting effective potential implies that the interbrane distance is undetermined in this case, and we need a small BPS breaking term on the brane to stabilize the interbrane distance at a finite length. We also discuss the relationship to the Goldberger-Wise model. 
  The average of the ratio of powers of the spectral determinants of the Dirac operator in the $\epsilon$-regime of QCD is shown to satisfy a Toda lattice equation. The quenched limit of this Toda lattice equation is obtained using the supersymmetric method. This super symmetric approach is then shown to be equivalent to taking the replica limit of the Toda lattice equation. Among other, the factorization of the microscopic spectral correlation functions of the QCD Dirac operator into fermionic and bosonic partition functions follows naturally from both approaches. While the replica approach relies on an analytic continuation in the number of flavors no such assumptions are made in the present approach where the numbers of flavors in the Toda lattice equation are strictly integer. 
  Long-range properties of the two-point correlation function of the electromagnetic field produced by an elementary particle are investigated. Using the Schwinger-Keldysh formalism it is shown that this function is finite in the coincidence limit outside the region of particle localization. In this limit, the leading term in the long-range expansion of the correlation function is calculated explicitly, and its gauge independence is proved. The leading contribution turns out to be of zero order in the Planck constant, and the relative value of the root mean square fluctuation of the Coulomb potential is found to be 1/\sqrt{2}, confirming the result obtained previously within the S-matrix approach. It is shown also that in the case of a macroscopic body, the \hbar^0 part of the correlation function is suppressed by a factor 1/N, where N is the number of particles in the body. Relation of the obtained results to the problem of measurability of the electromagnetic field is mentioned. 
  We present new theoretical results on the spectrum of the quantum field theory of the Double Sine Gordon model. This non-integrable model displays different varieties of kink excitations and bound states thereof. Their mass can be obtained by using a semiclassical expression of the matrix elements of the local fields. In certain regions of the coupling-constants space the semiclassical method provides a picture which is complementary to the one of the Form Factor Perturbation Theory, since the two techniques give information about the mass of different types of excitations. In other regions the two methods are comparable, since they describe the same kind of particles. Furthermore, the semiclassical picture is particularly suited to describe the phenomenon of false vacuum decay, and it also accounts in a natural way the presence of resonance states and the occurrence of a phase transition. 
  Using BRST-cohomological techniques, we analyze the consistent deformations of theories describing free tensor gauge fields whose symmetries are represented by Young tableaux made of two columns of equal length p, p>1. Under the assumptions of locality and Poincare invariance, we find that there is no consistent deformation of these theories that non-trivially modifies the gauge algebra and/or the gauge transformations. Adding the requirement that the deformation contains no more than two derivatives, the only possible deformation is a cosmological-constant-like term. 
  Dixon's multipoles for a system of N relativistic positive-energy scalar particles are evaluated in the rest-frame instant form of dynamics. The Wigner hyper-planes (intrinsic rest frame of the isolated system) turn out to be the natural framework for describing multipole kinematics. Classical concepts like the {\it barycentric tensor of inertia} turn out to be extensible to special relativity only by means of the quadrupole moments of the isolated system. Two new applications of the multipole technique are worked out for systems of interacting particles and fields. In the rest-frame of the isolated system of either free or interacting positive energy particles it is possible to define a unique world-line which embodies the properties of the most relevant centroids introduced in the literature as candidates for the collective motion of the system. This is no longer true, however, in the case of open subsystems of the isolated system. While effective mass, 3-momentum and angular momentum in the rest frame can be calculated from the definition of the {\it subsystem energy-momentum tensor}, the definitions of effective center of motion and effective intrinsic spin of the subsystem are not unique. Actually, each of the previously considered centroids corresponds to a different world-line in the case of open systems. The pole-dipole description of open subsystems is compared to their description as effective extended objects. Hopefully, the technique developed here could be instrumental for the relativistic treatment of binary star systems in metric gravity. 
  We investigate boundary conditions for the nonlinear sigma model on the compact symmetric space $G/H$, where $H \subset G$ is the subgroup fixed by an involution $\sigma$ of $G$. The Poisson brackets and the classical local conserved charges necessary for integrability are preserved by boundary conditions in correspondence with involutions which commute with $\sigma$. Applied to $SO(3)/SO(2)$, the nonlinear sigma model on $S^2$, these yield the great circles as boundary submanifolds. Applied to $G \times G/G$, they reproduce known results for the principal chiral model. 
  It has recently been shown that a non-Hermitian Hamiltonian H possessing an unbroken PT symmetry (i) has a real spectrum that is bounded below, and (ii) defines a unitary theory of quantum mechanics with positive norm. The proof of unitarity requires a linear operator C, which was originally defined as a sum over the eigenfunctions of H. However, using this definition to calculate C is cumbersome in quantum mechanics and impossible in quantum field theory. An alternative method is devised here for calculating C directly in terms of the operator dynamical variables of the quantum theory. This new method is general and applies to a variety of quantum mechanical systems having several degrees of freedom. More importantly, this method is used to calculate the C operator in quantum field theory. The C operator is a new time-independent observable in PT-symmetric quantum field theory. 
  In recent years, there has been considerable interest in theories formulated in anti-de Sitter (AdS) spacetime. However, AdS spacetime fails to be globally hyperbolic, so a classical field satisfying a hyperbolic wave equation on AdS spacetime need not have a well defined dynamics. Nevertheless, AdS spacetime is static, so the possible rules of dynamics for a field satisfying a linear wave equation are constrained by our previous general analysis--given in paper II--where it was shown that the possible choices of dynamics correspond to choices of positive, self-adjoint extensions of a certain differential operator, $A$. In the present paper, we reduce the analysis of electromagnetic, and gravitational perturbations in AdS spacetime to scalar wave equations. We then apply our general results to analyse the possible dynamics of scalar, electromagnetic, and gravitational perturbations in AdS spacetime. In AdS spacetime, the freedom (if any) in choosing self-adjoint extensions of $A$ corresponds to the freedom (if any) in choosing suitable boundary conditions at infinity, so our analysis determines all of the possible boundary conditions that can be imposed at infinity. In particular, we show that other boundary conditions besides the Dirichlet and Neumann conditions may be possible, depending on the value of the effective mass for scalar field perturbations, and depending on the number of spacetime dimensions and type of mode for electromagnetic and gravitational perturbations. 
  The exact bosonic Neumann matrices of the cubic vertex in plane-wave light-cone string field theory are derived using the contour integration techniques developed in our earlier paper. This simplifies the original derivation of the vertex. In particular, the Neumann matrices are written in terms of \mu-deformed Gamma-functions, thus casting them into a form that elegantly generalizes the well-known flat-space solution. The asymptotics of the \mu-deformed Gamma-functions allow one to determine the large-\mu behaviour of the Neumann matrices including exponential corrections. We provide an explicit expression for the first exponential correction and make a conjecture for the subsequent exponential correction terms. 
  Some aspects of the multiplicative anomaly of zeta determinants are investigated. A rather simple approach is adopted and, in particular, the question of zeta function factorization, together with its possible relation with the multiplicative anomaly issue is discussed. We look primordially into the zeta functions instead of the determinants themselves, as was done in previous work. That provides a supplementary view, regarding the appearance of the multiplicative anomaly. Finally, we briefly discuss determinants of zeta functions that are not in the pseudodifferential operator framework. 
  We construct a unique (2,0) supersymmetric action in six dimensions, describing a tensor multiplet interacting with a self-dual string. It is a sum of four terms: A free kinetic term for the tensor multiplet fields integrated over Minkowski space, a Nambu-Goto type kinetic term for the string integrated over the string world-sheet, a Wess-Zumino type electromagnetic coupling integrated over the world-volume of a Dirac membrane attached to the string, and a direct interaction of such Dirac membranes. In addition to supersymmetry, the action is also invariant under a local symmetry, which allows us to choose the Dirac membrane world-volume freely and eliminate half of the fermionic degrees of freedom on the string world-sheet. 
  The generalization of the Gross-Neveu model for noncommutative 3+1 space-time has been analyzed. We find indications that the chiral symmetry breaking occurs for an inhomogeneous background as in the LOFF phase in condensed matter. 
  We derive harmonic superspaces for N=2,3,4 SYM theory in four dimensions from superstring theory. The pure spinors in ten dimensions are dimensionally reduced and yield the harmonic coordinates. Two anticommuting BRST charges implement Grassmann analyticity and harmonic analyticity. The string field theory action produces the action and field equations for N=3 SYM theory in harmonic superspace. 
  We perform a thorough phase-plane analysis of the flow defined by the equations of motion of a FRW universe filled with a tachyonic fluid plus a barotropic one. The tachyon potential is assumed to be of inverse square form, thus allowing for a two-dimensional autonomous system of equations. The Friedmann constraint, combined with a convenient choice of coordinates, renders the physical state compact. We find the fixed-point solutions, and discuss whether they represent attractors or not. The way the two fluids contribute at late-times to the fractional energy density depends on how fast the barotropic fluid redshifts. If it does it fast enough, the tachyonic fluid takes over at late times, but if the opposite happens, the situation will not be completely dominated by the barotropic fluid; instead there will be a residual non-negligible contribution from the tachyon subject to restrictions coming from nucleosynthesis. 
  Letting the mass depend on the spin-field coupling as $M^2=m^2-(eg/2c^2)F_{\alpha\beta}S^{\alpha\beta}$, we propose a new set of relativistic planar equations of motion for spinning anyons. Our model can accommodate any gyromagnetic ratio $g$ and provides us with a novel version of the Bargmann-Michel-Telegdi equations in 2+1 dimensions. The system becomes singular when the field takes a critical value, and, for $g\neq2$, the only allowed motions are those which satisfy the Hall law. For each $g\neq2,0$ a secondary Hall effect arises also for another critical value of the field. The non-relativistic limit of our equations yields new models which generalize our previous ``exotic'' model, associated with the two-fold central extension of the planar Galilei group. 
  Recently, it has been proposed that the deformed matrix model describes a two-dimensional type 0A extremal black hole. In this paper, the thermodynamics of 0A charged non-extremal black holes is investigated. We observe that the free energy of the deformed matrix model to leading order in 1/q can be seen to agree to that of the extremal black hole. We also speculate on how the deformed matrix model is able to describe the thermodynamics of non-extremal black holes. 
  Adelic quantum mechanics is formulated. The corresponding model of the harmonic oscillator is considered. The adelic harmonic oscillator exhibits many interesting features. One of them is a softening of the uncertainty relation. 
  I shall present nonperturbative calculations of the electron's magnetic moment using the light-cone representation. 
  Canonical methods are not sufficient to properly quantize space-like axial gauges. In this paper, we obtain guiding principles which allow the construction of an extended Hamiltonian formalism for pure space-like axial gauge fields. To do so, we clarify the general role residual gauge fields play in the space-like axial gauge Schwinger model. In all the calculations we fix the gauge using a rule, $n{\cdot}A=0$, where $n$ is a space-like constant vector and we refer to its direction as $x_-$. Then, to begin with, we construct a formulation in which the quantization surface is space-like but not parallel to the direction of $n$. The quantization surface has a parameter which allows us to rotate it, but when we do so we keep the direction of the gauge field fixed. In that formulation we can use canonical methods. We bosonize the model to simplify the investigation. We find that the antiderivative, $({\partial}_-)^{-1}$, is ill-defined whatever quantization coordinates we use as long as the direction of $n$ is space-like. We find that the physical part of the dipole ghost field includes infrared divergences. However, we also find that if we introduce residual gauge fields in such a way that the dipole ghost field satisfies the canonical commutation relations, then the residual gauge fields are determined so as to regularize the infrared divergences contained in the physical part. The propagators then take the form prescribed by Mandelstam and Leibbrandt. We make use of these properties to develop guiding principles which allow us to construct consistent operator solutions in the pure space-like case where the quantization surface is parallel to the direction of $n$ and canonical methods do not suffice. 
  In this paper we construct complete macroscopic operators in two dimensional type 0 string theory. They represent D-branes localized in the time direction. We give another equivalent description of them as deformed Fermi surfaces. We also discuss a continuous array of such D-branes and show that it can be described by a matrix model with a deformed potential. For appropriate values of parameters, we find that it has an additional new sector hidden inside its strongly coupled region. 
  SU(2) gauge theory coupled to massless fermions in the adjoint representation is quantized in light-cone gauge by imposing the equal-time canonical algebra. The theory is defined on a space-time cylinder with "twisted" boundary conditions, periodic for one colour component (the diagonal 3- component) and antiperiodic for the other two. The focus of the study is on the non-trivial vacuum structure and the fermion condensate. It is shown that the indefinite-metric quantization of free gauge bosons is not compatible with the residual gauge symmetry of the interacting theory. A suitable quantization of the unphysical modes of the gauge field is necessary in order to guarantee the consistency of the subsidiary condition and allow the quantum representation of the residual gauge symmetry of the classical Lagrangian: the 3-colour component of the gauge field must be quantized in a space with an indefinite metric while the other two components require a positive-definite metric. The contribution of the latter to the free Hamiltonian becomes highly pathological in this representation, but a larger portion of the interacting Hamiltonian can be diagonalized, thus allowing perturbative calculations to be performed. The vacuum is evaluated through second order in perturbation theory and this result is used for an approximate determination of the fermion condensate. 
  We investigate how the uncertainty of noncommutative spacetime affects on inflation. For this purpose, the noncommutative parameter $\mu_0$ is taken to be a zeroth order slow-roll parameter. We calculate the noncommutative power spectrum up to second order using the slow-roll expansion. We find corrections arisen from a change of the pivot scale and the presence of a variable noncommutative parameter, when comparing with the commutative power spectrum. The power-law inflation is chosen to obtain explicit forms for the power spectrum, spectral index, and running spectral index. In cases of the power spectrum and spectral index, the noncommutative effect of higher-order corrections compensates for a loss of higher-order corrections in the commutative case. However, for the running spectral index, all higher-order corrections to the commutative case always provide negative spectral indexes, which could explain the recent WMAP data. 
  A new infinite series of Einstein metrics is constructed explicitly on S^2 x S^3, and the non-trivial S^3-bundle over S^2, containing infinite numbers of inhomogeneous ones. They appear as a certain limit of a nearly extreme 5-dimensional AdS Kerr black hole. In the special case, the metrics reduce to the homogeneous Einstein metrics studied by Wang and Ziller. We also construct an inhomogeneous Einstein metric on the non-trivial S^{d-2}-bundle over S^2 from a d-dimensional AdS Kerr black hole. Our construction is a higher dimensional version of the method of Page, which gave an inhomogeneous Einstein metric on CP^2\sharp\bar{CP^2}. 
  This article critically reviews the proposal for addressing the cosmological constant problem within the framework of supersymmetric large extra dimensions (SLED), as recently proposed in hep-th/0304256. After a brief restatement of the cosmological constant problem, a short summary of the proposed mechanism is given. The emphasis is on the perspective of the low-energy effective theory in order to see how it addresses the problem of why low-energy particles like the electron do not contribute too large a vacuum energy. This is followed by a discussion of the main objections, which are grouped into the following five topics:  (1) Weinberg's No-Go Theorem.  (2) Are hidden tunings of the theory required, and a problem?  (3) Why should the mechanism not rule out earlier epochs of inflation?  (4) How big are quantum effects, and which are the most dangerous?  (5) Can the mechanism be consistent with cosmological constraints?   It is argued that there are plausible reasons why the mechanism can thread the potential objections, but that a definitive proof that it does depends on addressing well-defined technical points. These points include identifying what fixes the size of the extra dimensions, checking how topological obstructions renormalize and performing specific calculations of quantum corrections. More detailed studies of these issues, which are well reach within our present understanding of extra-dimensional theories, are currently underway. As such, the jury remains out concerning the proposal, although the prospects for acquittal still seem good. 
  With the deconstruction technique, the geometric information of a torus can be encoded in a sequence of orbifolds. By studying the Matrix Theory on these orbifolds as quiver mechanics, we present a formulation that (de)constructs the torus of {\em generic shape} on which Matrix Theory is ``compactified''. The continuum limit of the quiver mechanics gives rise to a $(1+2)$-dimensional SYM. A hidden (fourth) dimension, that was introduced before in the Matrix Theory literature to argue for the electric-magnetic duality, can be easily identified in our formalism. We construct membrane wrapping states rigorously in terms of Dunford calculus in the context of matrix regularization. Unwanted degeneracy in the spectrum of the wrapping states is eliminated by using $SL(2,\integer)$ symmetry and the relations to the FD-string bound states. The dual IIB circle emerges in the continuum limit, constituting a critical evidence for IIB/M duality. 
  Massive type IIA supergravity admits a warped AdS_6 x S^4 vacuum solution, which is expected to be dual to an N=2, D=5 super-conformal Yang-Mills theory. We study solutions for strings rotating or spinning in this background. The warp factor plays no essential role when the string spins in the AdS_6, implying a commonality in the leading Regge trajectories between the D=4 and D=5 super-conformal field theories. The warp factor does, however, become important when the string rotates in the S^4, in particular for long strings, which have the the relation E- 3J/2=c_1 + c_2/J^5+..., where the angular momentum J is large. This relation is qualitatively different from that for long strings in the AdS_5 x S^5 background. We also study Penrose limits of the AdS_6 x S^4 solution, one of which gives rise to a free massive string theory with time-dependent masses. 
  Harmonic superspace can be used to construct higher derivative terms in N=2 supersymmetric effective actions despite the infinite redundancy in their description due to the infinite number of auxiliary fields. We are able to write down all of the 3- and 4-derivative terms on the Higgs, Coulomb, and mixed branches, modulo the possible existence of superspace Chern-Simons-like terms, which we discuss. Many of the terms we find are holomorphic, and at least one is shown to not receive quantum corrections. 
  We study the matching conditions of intersecting brane worlds in Lovelock gravity in arbitrary dimension. We show that intersecting various codimension 1 and/or codimension 2 branes one can find solutions that represent energy-momentum densities localized in the intersection, providing thus the first examples of infinitesimally thin higher codimension braneworlds that are free of singularities and where the backreaction of the brane in the background is fully taken into account. 
  In this Letter we discuss the issues of the graceful exit from inflation and of matter creation in the context of a recent scenario \cite{RHBrev} in which the back-reaction of long wavelength cosmological perturbations induces a negative contribution to the cosmological constant and leads to a dynamical relaxation of the bare cosmological constant. The initially large cosmological constant gives rise to primordial inflation, during which cosmological perturbations are stretched beyond the Hubble radius. The cumulative effect of the long wavelength fluctuations back-reacts on the background geometry in a form which corresponds to the addition of a negative effective cosmological constant to the energy-momentum tensor. In the absence of an effective scalar field driving inflation, whose decay can reheat the Universe, the challenge is to find a mechanism which produces matter at the end of the relaxation process. In this Letter, we point out that the decay of a condensate representing the order parameter for a ``flat'' direction in the field theory moduli space can naturally provide a matter generation mechanism. The order parameter is displaced from its vacuum value by thermal or quantum fluctuations, it is frozen until the Hubble constant drops to a sufficiently low value, and then begins to oscillate about its ground state. During the period of oscillation it can decay into Standard Model particles similar to how the inflaton decays in scalar-field-driven models of inflation. 
  We study a class of solutions to the SL(2,R)_k Knizhnik-Zamolodchikov equation. First, logarithmic solutions which represent four-point correlation functions describing string scattering processes on three-dimensional Anti-de Sitter space are discussed. These solutions satisfy the factorization ansatz and include logarithmic dependence on the SL(2,R)-isospin variables. Different types of logarithmic singularities arising are classified and the interpretation of these is discussed. The logarithms found here fit into the usual pattern of the structure of four-point function of other examples of AdS/CFT correspondence. Composite states arising in the intermediate channels can be identified as the phenomena responsible for the appearance of such singularities in the four-point correlation functions. In addition, logarithmic solutions which are related to non perturbative (finite k) effects are found. By means of the relation existing between four-point functions in Wess-Zumino-Novikov-Witten model formulated on SL(2,R) and certain five-point functions in Liouville quantum conformal field theory, we show how the reflection symmetry of Liouville theory induces particular Z_2 symmetry transformations on the WZNW correlators. This observation allows to find relations between different logarithmic solutions. This Liouville description also provides a natural explanation for the appearance of the logarithmic singularities in terms of the operator product expansion between degenerate and puncture fields. 
  We discuss the AdS/CFT duality from the perspective of integrable systems and establish a direct relationship between the dimension of single trace local operators composed of two types of scalar fields in N=4 super Yang-Mills and the energy of their dual semiclassical string states in AdS(5) X S(5). The anomalous dimensions can be computed using a set of Bethe equations, which for ``long'' operators reduces to a Riemann-Hilbert problem. We develop a unified approach to the long wavelength Bethe equations, the classical ferromagnet and the classical string solutions in the SU(2) sector and present a general solution, governed by complex curves endowed with meromorphic differentials with integer periods. Using this solution we compute the anomalous dimensions of these long operators up to two loops and demonstrate that they agree with string-theory predictions. 
  In the context of a gauge theoretical formulation, higher dimensional gravity invariant under the AdS group is dimensionally reduced to Euler-Chern-Simons gravity. The dimensional reduction procedure of Grignani-Nardelli [Phys. Lett. B 300, 38 (1993)] is generalized so as to permit reducing D-dimensional Lanczos Lovelock gravity to d=D-1 dimensions. 
  A new technique named Generalized Borel Transform (GBT) is applied to the generating functional of the $\Phi^{4}$ theory in zero dimensions with degenerate minima. The analytical solution of this function, obtained in the non perturbative regime, is compared with those estimations predicted by large order perturbation theory. It was established that the GBT is a very efficient technique to capture these contributions. On the other hand, renormalons associated to the resummation of those perturbative series were not found to be the genuine source of the non perturbative contributions of this model. 
  We study Skyrmion dynamics in a (2+1)-dimensional Skyrme model. The system contains a dimensionless parameter alpha, with alpha=0 corresponding to the O(3) sigma-model. If two Skyrmions collide head-on, then they can either coalesce or scatter -- this depends on alpha and on the incident speed v, and is affected by transfer of energy to and from the internal vibrational modes of the Skyrmions. We classify these internal modes and compute their spectrum, for a range of values of alpha. In particular, we find that there is a fractal-like structure of scattering windows, analogous to those seen for kink-antikink scattering in 1+1 dimensions. 
  We extend the study of the quantum mechanics of BMN gauge theory to the sector of three scalar impurities at one loop and all genus. The relevant matrix elements of the non-planar one loop dilatation operator are computed in the gauge theory basis. After a similarity transform the BMN gauge theory prediction for the corresponding piece of the plane wave string Hamiltonian is derived and shown to agree with light-cone string field theory. In the three-impurity sector single string states are unstable for the decay into two-string states at leading order in g_2. The corresponding decay widths are computed. 
  We propose new Wightman functions as vacuum expectation values of products of field operators in the \nc space-time. These Wightman functions involve the $\star$-product. In the case of only space-space \ncy ($\theta_{0i}=0$), we prove the CPT theorem using the \nc form of the Wightman functions. As a byproduct, one arrives at the general conclusion of the following theorem that the violation of CPT invariance implies the violation of not only Lorentz invariance, but also its subgroup of symmetry $SO(1,1)\times SO(2)$. We also show that the spin-statistics theorem, for the simplest case of a scalar field, holds within this formalism. 
  We study the dynamics of a confining vacuum in N=2 USp(4) gauge theory with n_f=4. The vacuum appears to be a deformed conformal theory with nonabelian gauge symmetry. The low-energy degrees of freedom consist of four nonabelian magnetic monopole doublets of the effective SU(2) colour group, two dyon doublets and one electric doublet. In this description the flavour quantum number is carried only by the monopoles. We argue that confinement is caused by the condensation of these monopoles, and involves strongly interacting nonabelian degrees of freedom. 
  We find a strong coupling expansion around the non-trivial extremum of the Yang-Mills action. It is shown that the developed formalism is the Gribov ambiguity free since each order of the developed perturbation theory is transparently gauge invariant. The success is a consequence of the restriction: calculations are not going beyond the norm of the $S$-matrix element. 
  We study the \N=4 SYM theory with SU(N) gauge group in the large N limit, deformed by giving equal mass to the four adjoint fermions. With this modification, a potential is dynamically generated for the six scalars in the theory, \phi^i. We show that the resulting theory is stable (perturbatively in the 't Hooft coupling), and that there are some indications that <\phi>=0 is the vacuum of the theory. Using the AdS/CFT correspondence, we compare the results to the corresponding supergravity computation, i.e. brane probing a deformed AdS_5 x S^5 background, and we find qualitative agreement. 
  In spacetimes with compact dimensions there exist several black object solutions including the black-hole and the black-string. These solutions may become unstable depending on their relative size and the relevant length scale set by the compact dimensions. The transition between these solutions raises puzzles and addresses fundamental questions such as topology change, uniquenesses and cosmic censorship. Here, we consider black strings wrapped over the compact circle of a $d$-dimensional cylindrical spacetime. We construct static perturbative non-uniform string solutions around the instability point of a uniform string. First we compute the instability mass for a large range of dimensions, $d$, and find that it follows essentially an exponential law $\gamma^d$, where $\gamma$ is a constant. Then we determine that there is a critical dimension, $d_*=13$, such that for $d\leq d_*$ the phase transition between the uniform and the non-uniform strings is of first order, while for $d>d_*$, it is, surprisingly, of higher order. 
  In this paper we first give a simple parametrization of the scalar coset manifold of the only known anomaly free chiral gauged supergravity in six dimensions in the absence of linear multiplets, namely gauged minimal supergravity coupled to a tensor multiplet, E_6 x E_7 x U(1)_R Yang-Mills multiplets and suitable number of hypermultiplets. We then construct the potential for the scalars and show that it has a unique minimum at the origin. We also construct a new BPS dyonic string solution in which U(1)_R x U(1) gauge fields, in addition to the metric, dilaton and the 2-form potential, assume nontrivial configurations in any U(1)_R gauged 6D minimal supergravity coupled to a tensor multiplet with gauge symmetry G\supseteq U(1). The solution preserves 1/4 of the 6D supersymmetries and can be trivially embedded in the anomaly free model, in which case the U(1) activated in our solution resides in E_7. 
  We analyze the cosmological consequences of locked inflation, a model recently proposed by Dvali and Kachru that can produce significant amounts of inflation without requiring slow-roll. We pay particular attention to the end of inflation in this model, showing that a secondary phase of saddle inflation can follow the locked inflationary era. However, this subsequent period of inflation results in a strongly scale dependent spectrum that can lead to massive black hole formation in the primordial universe. Avoiding this disastrous outcome puts strong constraints on the parameter space open to models of locked inflation. 
  We consider large N zero-coupling d-dimensional U(N) gauge theories, with N_f matter fields in the fundamental representation on a compact spatial manifold S^{d-1} x time, with N_f/N finite. The Gauss' law constraint induces interactions among the fields, in spite of the zero-coupling. This class of theories undergo a 3rd order deconfinement phase transition at a temperature T_c proportional to the inverse length scale of the compact manifold.   The low-temperature phase has a free-energy of {\cal O} (N^2_f), interpreted as that of a gas of (color singlet) mesons and glueballs. The high-temperature (deconfinement) phase has a free energy of order N^2 f (N_f/N, T), which is interpreted as that of a gas of gluons and of fundamental and anti-fundamental matter states. This suggests the existence of a dual string theory, and a transition to a black hole at high temperature. 
  We exploit the reparametrization symmetry of a relativistic free particle to impose a gauge condition which upon quantization implies space-time noncommutativity. We show that there is an algebraic map from this gauge back to the standard `commuting' gauge. Therefore the Poisson algebra, and the resulting quantum theory, are identical in the two gauges. The only difference is in the interpretation of space-time coordinates. The procedure is repeated for the case of a coupling with a constant electromagnetic field, where the reparametrization symmetry is preserved. For more arbitrary interactions, we show that standard dynamical system can be rendered noncommutative in space and time by a simple change of variables. 
  We give an explicit form of the PST-type SL(2;R)-covariant super D3-brane action for the flat Minkowski background. To this end, we follow the prescription developed by Hatsuda and Kamimura. As an application of the action, we obtain the supercharge of the action by using the standard Noether's method and calculate the Poisson bracket algebra of the supercharge. The central charge of the supersymmetry algebra is given in a manifestly SL(2;R)-covariant way. 
  Massive gravity models in 2+1 dimensions, such as those obtained by adding to Einstein's gravity the usual Fierz-Pauli, or the more complicated Ricci scalar squared ($R^2$), terms, are tree level unitary. Interesting enough these seemingly harmless systems have their unitarity spoiled when they are augmented by a Chern-Simons term. Furthermore, if the massive topological term is added to $R + R_{\mu\nu}^2$ gravity, or to $R + R_{\mu\nu}^2 + R^2$ gravity (higher-derivative gravity), which are nonunitary at the tree level, the resulting models remain nonunitary. Therefore, unlike the common belief, as well as the claims in the literature, the coexistence between three-dimensional massive gravity models and massive topological terms is conflicting. 
  Recently the authors have introduced a new gauged supergravity theory with a positive definite potential in D=6, obtained through a generalised Kaluza-Klein reduction from D=7. Of particular interest is the fact that this theory admits certain Minkowski x Sphere vacua. In this paper we extend the previous results by constructing gauged supergravities with positive definitive potentials in diverse dimensions, together with their vacuum solutions. In addition, we prove the supersymmetry of the generalised reduction ansatz. We obtain a supersymmetric solution with no form-field fluxes in the new gauged theory in D=9. This solution may be lifted to D=10, where it acquires an interpretation as a time-dependent supersymmetric cosmological solution supported purely by the dilaton. A further uplift to D=11 yields a solution describing a pp-wave. 
  Non-relativistic charged particles and strings coupled with abelian gauge fields are quantized in a geometric representation that generalizes the Loop Representation. We consider three models: the string in self-interaction through a Kalb-Ramond field in four dimensions, the topological interaction of two particles due to a BF term in 2+1 dimensions, and the string-particle interaction mediated by a BF term in 3+1 dimensions. In the first case one finds that a consistent "surface-representation" can be built provided that the coupling constant is quantized. The geometrical setting that arises corresponds to a generalized version of the Faraday's lines picture: quantum states are labeled by the shape of the string, from which emanate "Faraday`s surfaces". In the other models, the topological interaction can also be described by geometrical means. It is shown that the open-path (or open-surface) dependence carried by the wave functional in these models can be eliminated through an unitary transformation, except by a remaining dependence on the boundary of the path (or surface). These feature is closely related to the presence of anomalous statistics in the 2+1 model, and to a generalized "anyonic behavior" of the string in the other case. 
  Motivated in part by string theory, we consider the idea that the standard LambdaCDM cosmological model might be modified by the effect of a long-range scalar dark matter interaction. The variant of this widely-discussed notion considered here is suggested by the Brandenberger-Vafa picture for why we perceive three spatial dimensions. In this picture there may be at least two species of dark matter particles, with scalar ``charges'' such that the scalar interaction attracts particles with like sign and repels unlike signs. The net charge vanishes. Under this condition the evolution of the mass distribution in linear perturbation theory is the same as in the LambdaCDM cosmology, and both models therefore can equally well pass the available cosmological tests. The physics can be very different on small scales, however: if the scalar interaction has the strength suggested by simple versions of the string scenario, nonlinear mass concentrations are unstable against separation into charged halos with properties unlike the standard model prediction and possibly of observational interest. 
  Recently (Phys. Lett. A302 (2002) 253, hep-th/0208210; hep-th/0403146) employing bounded infinite-dimensional representations of the rotation group we have argued that one can obtain the consistent monopole theory with generalized Dirac quantization condition, $2\kappa\mu \in \mathbb Z$, where $\kappa$ is the weight of the Dirac string. Here we extend this proof to the unbounded infinite-dimensional representations. 
  Just as point objects are parallel transported along curves, giving holonomies, string-like objects are parallel transported along surfaces, giving surface holonomies. Composition of these surfaces correspond to products in a category theoretic generalization of the gauge group, called a 2-group. I consider two different ways of constructing surface holonomies, one by using a pair of one and two form connections, and another by using a pair of one-form connections. Both procedures result in the structure of a 2-group. 
  In the on-shell formulation of $D=5 $, N=2 supergravity, compactified on $S^1/Z_2$, we extend the results of Mirabelli and Peskin describing the interaction of the bulk fields with matter which is assumed to be confined on the brane. The novel characteristics of this approach are : Propagation of both gravity and gauge fields in the bulk, which offers an alternative for a unified description of models in extra dimensions and use of the on-shell formulation avoiding the complexity of off-shell schemes which involve numerous auxiliary fields. We also allow for nontrivial superpotential interactions of the chiral matter fields. The method we employ uses the N{\" o}ther procedure and our findings are useful for building models advocating propagation of the gauge degrees of freedom in the bulk, in addition to gravity. 
  We examine the effect of non-local deformations on the applicability of interaction point time ordered perturbation theory (IPTOPT) based on the free Hamiltonian of local theories. The usual argument for the case of quantum field theory (QFT) on a noncommutative (NC) space (based on the fact that the introduction of star products in bilinear terms does not alter the action) is not applicable to IPTOPT due to several discrepancies compared to the naive path integral approach when noncommutativity involves time. These discrepancies are explained in detail. Besides scalar models, gauge fields are also studied. For both cases, we discuss the free Hamiltonian with respect to non-local deformations. 
  The critical properties of the real phi^4 scalar field theory are studied numerically on the fuzzy sphere. The fuzzy sphere is a matrix (non commutative) discretisation of the algebra of functions on the usual two dimensional sphere. It is also one of the simplest examples of a non commutative space to study field theory on. Aside from the usual disordered and uniform phases present in the commutative scalar field theory, we find and discuss in detail a new phase with spontaneously broken rotational invariance, called matrix phase because the geometry of the fuzzy sphere, as expressed by the kinetic term, becomes negligible there. This gives some further insight on the effect of UV-IR mixing, the unusual behaviour which arises naturally when taking the commutative limit of a non commutative field theory. 
  We study quantum effects on moduli dynamics arising from the production of particles which are light at special points in moduli space. The resulting forces trap the moduli at these points, which often exhibit enhanced symmetry. Moduli trapping occurs in time-dependent quantum field theory, as well as in systems of moving D-branes, where it leads the branes to combine into stacks. Trapping also occurs in an expanding universe, though the range over which the moduli can roll is limited by Hubble friction. We observe that a scalar field trapped on a steep potential can induce a stage of acceleration of the universe, which we call trapped inflation. Moduli trapping ameliorates the cosmological moduli problem and may affect vacuum selection. In particular, rolling moduli are most powerfully attracted to the points with the largest number of light particles, which are often the points of greatest symmetry. Given suitable assumptions about the dynamics of the very early universe, this effect might help to explain why among the plethora of possible vacuum states of string theory, we appear to live in one with a large number of light particles and (spontaneously broken) symmetries. In other words, some of the surprising properties of our world might arise not through pure chance or miraculous cancellations, but through a natural selection mechanism during dynamical evolution. 
  We present a countably infinite number of new explicit co-homogeneity one Sasaki-Einstein metrics on S^2 x S^3, in both the quasi-regular and irregular classes. These give rise to new solutions of type IIB supergravity which are expected to be dual to N=1 superconformal field theories in four-dimensions with compact or non-compact R-symmetry and rational or irrational central charges, respectively. 
  The SO(10) embedding of the Standard Model spectrum is supported by evidence for neutrino masses. This thesis adapts the available formalism to study a class of heterotic M-theory vacua with SO(10) grand unification group. Compactification to four dimensions with N = 1 supersymmetry is achieved on a torus fibered Calabi-Yau 3-fold Z = X / tau_{X} with first homotopy group pi_{1}(Z) = Z_{2}. Here X is an elliptically fibered Calabi-Yau 3-fold which admits two global sections and tau_{X} is a freely acting involution on X. The vacua in this class have net number of three generations of chiral fermions in the observable sector and may contain M5-branes in the bulk space which wrap holomorphic curves in Z. Vacua with nonvanishing and vanishing instanton charges in the observable sector are considered. The latter case corresponds to potentially viable matter Yukawa couplings. Since pi_{1}(Z) = Z_{2}, the grand unification group can be broken with Z_{2} Wilson lines.   Realistic free-fermionic models preserve the SO(10) embedding of the Standard Model spectrum. These models have a stage in their construction which corresponds to Z_{2} x Z_{2} orbifold compactification of the weakly coupled 10-dimensional heterotic string. This correspondence identifies associated Calabi-Yau 3-folds which possess the structure of the above Z and X. This, in turn, allows the above formalism to be used to study heterotic M-theory vacua associated with realistic free-fermionic models. It is argued how the top quark Yukawa coupling in these models can be reproduced in the heterotic M-theory limit. 
  In this letter, we study the open spinning strings and their SYM duals. A new class of folded open spinning strings is found. At planar one-loop level in SYM, by solving the thermodynamic limit of the Bethe ansatz equations for an integrable open spin chain, we find good agreement with string theory predictions for energies of both circular and folded two-spin solutions. A universal relation between the open and closed spinning strings is verified in the spin chain approach. 
  We present a new class of "dielectric" N=1 supersymmetric solutions of IIB supergravity. This class contains not just the ten-dimensional lift of the Leigh-Strassler renormalization group flow, but also the Coulomb branch deformation of this flow in which the branes are allowed to spread in a radially symmetric manner, preserving the SU(2) global symmetry. We use the "algebraic Killing spinor" technique, illustrating how it can be adapted to N=1 supersymmetric flows. 
  We use the techniques of "algebraic Killing spinors" to obtain a family of holographic flow solutions with four supersymmetries in M-theory. The family of supersymmetric backgrounds constructed here includes the non-trivial flow to the (2+1)-dimensional analog of the Leigh-Strassler fixed point as well as generalizations that involve the M2-branes spreading in a radially symmetric fashion on the Coulomb branch of this non-trivial fixed point theory. In spreading out, these M2-branes also appear to undergo dielectric polarization into M5-branes. Our results naturally extend the earlier applications of the "algebraic Killing spinor" method and also generalize the harmonic Ansatz in that our entire family of new supersymmetric backgrounds is characterized by the solutions of a single, second-order, non-linear PDE. We also show that our solution is a natural hybrid of special holonomy and the "dielectric deformation" of the canonical supersymmetry projector on the M2-branes. 
  The principle of local covariance which was recently introduced admits a generally covariant formulation of quantum field theory. It allows a discussion of structural properties of quantum field theory as well as the perturbative construction of renormalized interacting models on generic curved backgrounds and opens in principle the way towards a background independent perturbative quantization of gravity. 
  The problem of interpreting quantum theory on a large (e.g. cosmological) scale has been commonly conceived as a search for objective reality in a framework that is fundamentally probabilistic. The Everett programme attempts to evade the issue by the reintroduction of determinism at the global level of a ``state vector of the universe''. The present approach is based on the recognition that, like determinism, objective reality is an unrealistic objective. It is shown how an objective theory of an essentially subjective reality can be set up using an appropriately weighted probability measure on the relevant set of Hilbert subspaces. It is suggested that an entropy principle (superseding the weak anthropic principle) should be used to provide the weighting that is needed. 
  A supersymmetric (SUSY) model of radius stabilization is constructed for the S^1/Z_2 warped compactifications with a hypermultiplet in five dimensions. Requiring the continuity of scalar field across the boundaries, we obtain radius stabilization preserving SUSY, realizing the SUSY extension of the Goldberger-Wise mechanism. Even if we allow discontinuities of the Z_2 odd field across the boundary, we always obtain SUSY preservation but obtain the radius stabilization only when the discontinuity is fixed by other mechanism. 
  We show that the general solution of scalar field cosmology in $d$ dimensions with exponential potentials for flat Robertson-Walker metric can be found in a straightforward way by introducing new variables which completely decouple the system. The explicit solution shows the region of parameters where the expansion has eternal acceleration, transient periods of acceleration, or no period of acceleration at all. In the cases of transient acceleration, the energy density exhibits a plateau during the accelerated expansion, where $p\cong -\rho$, due to dominance of potential energy. We determine the interval of accelerated expansion in terms of a simple formula. In particular, it shows that the period of accelerated expansion decreases in higher dimensions. 
  The maximally symmetric D-branes of string theory on the non-simply connected Lie group SU(n)/Z_d are analysed using conformal field theory methods, and their charges are determined. Unlike the well understood case for simply connected groups, the charge equations do not determine the charges uniquely, and the charge group associated to these D-branes is therefore in general not cyclic. The precise structure of the charge group depends on some number theoretic properties of n, d, and the level of the underlying affine algebra k. The examples of SO(3)=SU(2)/Z_2 and SU(3)/Z_3 are worked out in detail, and the charge groups for SU(n)/Z_d at most levels k are determined explicitly. 
  We begin with a discussion on two apparently disconnected topics - one related to nonperturbative superpotential generated from wrapping an M2-brane around a supersymmetric three cycle embedded in a G_2-manifold evaluated by the path-integral inside a path-integral approach of [1], and the other centered around the compact Calabi-Yau CY_3(3,243) expressed as a blow-up of a degree-24 Fermat hypersurface in WCP^4[1,1,2,8,12]. For the former, we compare the results with the ones of Witten on heterotic world-sheet instantons [2]. The subtopics covered in the latter include an N=1 triality between Heterotic, M- and F-theories, evaluation of RP^2-instanton superpotential, Picard-Fuchs equation for the mirror Landau-Ginsburg model corresponding to CY_3(3,243), D=11 supergravity corresponding to M-theory compactified on a `barely' G_2 manifold involving CY_3(3,243) and a conjecture related to the action of antiholomorphic involution on period integrals. We then show an indirect connection between the two topics by showing a connection between each one of the two and Witten's MQCD [3]. As an aside, we show that in the limit of vanishing "\zeta", a complex constant that appears in the Riemann surfaces relevant to definining the boundary conditions for the domain wall in MQCD, the infinite series of [4] used to represent a suitable embedding of a supersymmetric 3-cycle in a G_2-mannifold, can be summed. 
  Based on the Beltrami-de Sitter spacetime, we present the Newton-Hooke model under the Newton-Hooke contraction of the $BdS$ spacetime with respect to the transformation group, algebra and geometry. It is shown that in Newton-Hooke space-time, there are inertial-type coordinate systems and inertial-type observers, which move along straight lines with uniform velocity. And they are invariant under the Newton-Hooke group. In order to determine uniquely the Newton-Hooke limit, we propose the Galilei-Hooke's relativity principle as well as the postulate on Newton-Hooke universal time. All results are readily extended to the Newton-Hooke model as a contraction of Beltrami-anti-de Sitter spacetime with negative cosmological constant. 
  Maximal and non-maximal supergravities in three spacetime dimensions allow for a large variety of semisimple and non-semisimple gauge groups, as well as complex gauge groups that have no analog in higher dimensions. In this contribution we review the recent progress in constructing these theories and discuss some of their possible applications. 
  Classical stability behaviors of various static black brane backgrounds under small perturbations have been summarized briefly. They include cases of black strings in AdS$_5$ space, charged black $p$-brane solutions in the type II supergravity, and the BTZ black string in four-dimensions. The relationship between dynamical stability and local thermodynamic stability - the so-called Gubser-Mitra conjecture - has also been checked for those cases. 
  The condensation of closed string tachyons localized at the fixed point of a C^d/\Gamma orbifold can be studied in the framework of renormalization group flow in a gauged linear sigma model. The evolution of the Higgs branch along the flow describes a resolution of singularities via the process of tachyon condensation. The study of the fate of D-branes in this process has lead to a notion of a ``quantum McKay correspondence.'' This is a hypothetical correspondence between fractional branes in an orbifold singularity in the ultraviolet with the Coulomb and Higgs branch branes in the infrared. In this paper we present some nontrivial evidence for this correspondence in the case C^2/Z_n by relating the intersection form of fractional branes to that of ``Higgs branch branes,'' the latter being branes which wrap nontrivial cycles in the resolved space. 
  The main BRST cohomological properties of a free, massless tensor field that transforms in an irreducible representation of GL(D,R), corresponding to a rectangular, two-column Young diagram with k>2 rows are studied in detail. In particular, it is shown that any non-trivial co-cycle from the local BRST cohomology group H(s|d) can be taken to stop either at antighost number (k+1) or k, its last component belonging to the cohomology of the exterior longitudinal derivative H(gamma) and containing non-trivial elements from the (invariant) characteristic cohomology H^{inv}(delta|d). 
  It is known that string theory compactifications leading to low energy effective theories with different chiral matter content ({\it e.g.} different numbers of standard model generations) are connected through phase transitions, described by non-trivial quantum fixed point theories.   We point out that such compactifications are also connected on a purely classical level, through transitions that can be described using standard effective field theory. We illustrate this with examples, including some in which the transition proceeds entirely through supersymmetric configurations. 
  We study q-stars with various symmetries in anti de Sitter spacetime in 3+1 dimensions. Comparing with the case of flat spacetime, we find that the value of the field at the center of the soliton is larger when the other parameters show a more complicated behavior. We also investigate their phase space when the symmetry is local and the effect of the charge to its stability. 
  Inflation has been the leading early universe scenario for two decades, and has become an accepted element of the successful `cosmic concordance' model. However, there are many puzzling features of the resulting theory. It requires both high energy and low energy inflation, with energy densities differing by a hundred orders of magnitude. The questions of why the universe started out undergoing high energy inflation, and why it will end up in low energy inflation, are unanswered. Rather than resort to anthropic arguments, we have developed an alternative cosmology, the Cyclic universe, in which the universe exists in a very long-lived attractor state determined by the laws of physics. The model shares inflation's phenomenological successes without requiring an epoch of high energy inflation. Instead, the universe is made homogeneous and flat, and scale-invariant adiabatic perturbations are generated during an epoch of low energy acceleration like that seen today, but preceding the last big bang. Unlike inflation, the model requires low energy acceleration in order for a periodic attractor state to exist. The key challenge facing the scenario is that of passing through the cosmic singularity at t=0. Substantial progress has been made at the level of linearised gravity, which is reviewed here. The challenge of extending this to nonlinear gravity and string theory remains. 
  We analyze the creation of particles in two dimensions under the action of conformal transformations. We focus our attention on Mobius transformations and compare the usual approach, based on the Bogolubov coefficients, with an alternative but equivalent viewpoint based on correlation functions. In the latter approach the absence of particle production under full Mobius transformations is manifest. Moreover, we give examples, using the moving-mirror analogy, to illustrate the close relation between the production of quanta and energy. 
  In this paper we analyse the effect of the anomalous magnetic moment on the non-relativistic quantum motion of a neutral particle in magnetic and electric fields produced by linear sources of constant current and charge density, respectively. 
  We analyze the Carroll-Field-Jackiw (CFJ) modification of electrodynamics reformulated as the ordinary Maxwell theory with an additional special axion field. In this form, the CFJ model appears as a special case of the pre-metric approach recently developed by Hehl and Obukhov. This embedding turns out to be non-trivial. Particularly, the pre-metric energy-momentum tensor does not depend on the axion. This is in contrast to the CFJ energy-momentum tensor which involves the axion addition explicitly. We show that the relation between these two quantities is similar to the correspondence between the Noether conserved tensor and the Hilbert symmetric tensor. As a result the CFJ energy-momentum tensor appears as the unique conserved closure of the pre-metric one. Another problem is in the description of the birefringence effect, which in the pre-metric framework does not depend on the axion. The comparison with the CFJ model shows that the corresponding wave propagation (Fresnel) equation has to be extended by a derivative term, which is non zero for the axion field. In this way, the CFJ birefringence effect is derived in the metric-free approach. Consequently the Lorentz and CPT violating models can be embedded without contradictions in the pre-metric approach to electrodynamics. This correspondence can be useful for both constructions. 
  We construct a kappa-symmetric Green-Schwarz action for type IIA string theory on AdS_2. As a candidate holographic dual, we consider superconformal matrix quantum mechanics, given by the Marinari-Parisi model with vanishing or logarithmic superpotential. We derive that the super-eigenvalues form a consistent subsector, and that their dynamics reduces to that of the supersymmetric Calogero-Moser model. The classical string action and the matrix model both have an infinite set of conserved charges, that include an N=2 target space super-Virasoro symmetry. As a microscopic test of the duality, we reproduce the exact form of the Calogero interaction via a string worldsheet calculation. 
  The quantum states of the supertube are counted by directly quantizing the linearized Born-Infeld action near the round tube. The result is an entropy $S = 2\pi \sqrt{2 (Q_{D0}Q_{F1}-J)}$, in accord with conjectures in the literature. As a result, supertubes may be the generic D0-F1 bound state. Our approach also shows directly that supertubes are marginal bound states with a discrete spectrum. We also discuss the relation to recent suggestions of Mathur et al involving three-charge black holes. 
  For a 4-dimensional spatially-flat Friedmann-Robertson-Walker universe with a scalar field $\phi(x)$, potential $V(\phi)$ and constant equation of state $w=p/\rho$, we show that an expanding solution characterized by $\epsilon=3(1+w)/2$ produces the same scalar perturbations as a contracting solution with $\hat{\epsilon}=1/\epsilon$. The same symmetry applies to both the dominant and subdominant scalar perturbation modes. This result admits a simple physical interpretation and generalizes to $d$ spacetime dimensions if we define $\epsilon \equiv [(2d-5)+(d-1)w]/(d-2)$. 
  It is shown how metastable de Sitter vacua might arise from heterotic M-theory. The balancing of its two non-perturbative effects, open membrane instantons against gaugino condensation on the hidden boundary, which act with opposing forces on the interval length, is used to stabilize the orbifold modulus (dilaton) and other moduli. The non-perturbative effects break supersymmetry spontaneously through F-terms which leads to a positive vacuum energy density. In contrast to the situation for the weakly coupled heterotic string, the charged scalar matter fields receive non-vanishing vacuum expectation values and therefore masses in a phenomenologically relevant regime. It is important that in order to obtain these de Sitter vacua we are not relying on exotic effects or fine-tuning of parameters. Vacua with more realistic supersymmetry breaking scales and gravitino masses are obtained by breaking the hidden $E_8$ gauge group down to groups of smaller rank. Also small values for the open membrane instanton Pfaffian are favored in this respect. Finally we outline how the incorporation of additional flux superpotentials can be used to stabilize the remaining moduli. 
  We propose an alternative axiomatic description for non-commutative field theories (NCFT) based on some ideas by Soloviev to nonlocal quantum fields. The local commutativity axiom is replaced by the weaker condition that the fields commute at sufficiently large spatial separations, called asymptotic commutativity, formulated in terms of the theory of analytic functionals. The question of a possible violation of the CPT and Spin-Statistics theorems caused by nonlocality of the commutation relations $[\hat{x}_\mu,\hat{x}_\nu]=i\theta_{\mu\nu}$ is investigated. In spite of this inherent nonlocality, we show that the modification aforementioned is sufficient to ensure the validity of these theorems for NCFT. We restrict ourselves to the simplest model of a scalar field in the case of only space-space non-commutativity. 
  We discuss all possible compactifications on flat three-dimensional smooth spaces. In particular, various fields are studied on a box with opposite sides identified, after two of them are rotated by $\pi$, and their spectra are obtained. The compactification of a general 7D supersymmetric theory in such a box is considered and the corresponding four-dimensional theory is studied, in relation to the boundary conditions chosen. The resulting spectrum, according to the allowed field boundary conditions, corresponds to partially or completely broken supersymmmetry. We briefly discuss also the breaking of gauge symmetries under the proposed box compactification. 
  The present status of the uniqueness and stability issue of black holes in four and higher dimensions is overviewed with focus on the perturbative analysis of this issue for static black holes in higher dimensions as well as for those in four dimensions with cosmological constant. 
  We construct a time-dependent solution in vacuum string field theory and investigate whether the solution can be regarded as a rolling tachyon solution. First, compactifying one space direction on a circle of radius R, we construct a space-dependent solution given as an infinite number of *-products of a string field with center-of-mass momentum dependence of the form e^{-b p^2/4}. Our time-dependent solution is obtained by an inverse Wick rotation of the compactified space direction. We focus on one particular component field of the solution, which takes the form of the partition function of a Coulomb system on a circle with temperature R^2. Analyzing this component field both analytically and numerically using Monte Carlo simulation, we find that the parameter b in the solution must be set equal to zero for the solution to approach a finite value in the large time limit x^0\to\infty. We also explore the possibility that the self-dual radius R=\sqrt{\alpha'} is a phase transition point of our Coulomb system. 
  In the framework of quantum field theory (QFT) on noncommutative (NC) space-time with $SO(1,1)\times SO(2)$ symmetry, which is the feature arising when one has only space-space noncommutativity ($\theta_{0i}=0$), we prove that the Jost-Lehmann-Dyson representation, based on the causality condition usually taken in connection with this symmetry, leads to the mere impossibility of drawing any conclusion on the analyticity of the $2\to 2$-scattering amplitude in $\cos\Theta$, $\Theta$ being the scattering angle. A physical choice of the causality condition rescues the situation and as a result an analog of Lehmann's ellipse as domain of analyticity in $\cos\Theta$ is obtained. However, the enlargement of this analyticity domain to Martin's ellipse and the derivation of the Froissart bound for the total cross-section in NC QFT is possible {\it only} in the special case when the incoming momentum is orthogonal to the NC plane. This is the first example of a nonlocal theory in which the cross-sections are subject to a high-energy bound. For the general configuration of the direction of the incoming particle, although the scattering amplitude is still analytic in the Lehmann ellipse, no bound on the total cross-section has been derived. This is due to the lack of a simple unitarity constraint on the partial-wave amplitudes, which could be used in this case. High-energy upper bounds on the total cross-section, among others, are also obtained for an arbitrary flat (noncompact) dimension of NC space-time. 
  The existence of Jost-Lehmann-Dyson representation analogue has been proved in framework of space-space noncommutative quantum field theory. On the basis of this representation it has been found that some class of elastic amplitudes admits an analytical continuation into complex \cos\vartheta plane and corresponding domain of analyticity is Martin ellipse. This analyticity combined with unitarity leads to Froissart-Martin upper bound on total cross section. 
  A model with brane-localized curvature terms in a warped background is proposed. The model is based on the Randall-Sundrum solution for the background metric and possesses some interesting features, in particular, the radion field is absent in it. Although there is no modification of gravity at long distances, the model predicts deviations from Newton's law at short distances. This effect can be observed in the experiments for testing gravity at sub-millimeter scales. 
  M5-branes wrapping a holomorphic curve in a Calabi-Yau manifold can be used to construct four-dimensional N=1 gauge theories. In this paper we will consider M5-brane configurations corresponding to N=2 theories broken to N=1 by a superpotential for the adjoint scalar field. These M5-brane configurations can be obtained by lifting suitable intersecting brane configurations in type IIA, or equivalently by T-dualizing IIB configurations with branes and/or fluxes. We will show that turning on non-trivial expectation values for the glueball superfields corresponds to non-holomorphic deformations of the M5-brane. We compute the superpotential and show it agrees with that computed by Dijkgraaf and Vafa. Several aspects of the gauge theory, such as the appearance of non-holomorphic one-forms with integer periods on the Seiberg-Witten curve, have a natural interpretation from the M5-brane point of view. We also explain the interpretation of the superpotential in terms of the twisted (2,0) theory living on the fivebrane. 
  The "square root" of the Dirac operator derived on the superspace is used to construct supersymmetric field equations. In addition to the recently found solution - a vector supermultiplet I demonstrate how a chiral supermultiplet follows as the solution. Both vector and chiral supermultiplets are shown to obey appropriate (massless) equations of motion. This procedure yields thus a complete set of fields and their equations necessary to construct renormalizable supersymmetric theories. The problem of masses and interaction is also discussed. 
  In a fermionic quantum vacuum, the parameters k_\mu of a CPT-violating Chern-Simons-like action term induced by CPT-violating parameters of the fermionic sector depend on the universality class of the system. As a concrete example, we consider the Dirac Hamiltonian of a massive fermionic quasiparticle and add a particular term with purely-spacelike CPT-violating parameters b_\mu=(0,{\bf b}). A quantum phase transition separates two phases, one with a fully-gapped fermion spectrum and the other with topologically-protected Fermi points (gap nodes). The emergent Chern-Simons ``vector'' k_\mu=(0,{\bf k}) now consists of two parts. The regular part, {\bf k}^{reg}, is an analytic function of |{\bf b}| across the quantum phase transition and may be nonzero due to explicit CPT violation at the fundamental level. The anomalous (nonanalytic) part, {\bf k}^{anom}, comes solely from the Fermi points and is proportional to their splitting. In the context of condensed-matter physics, the quantum phase transition may occur in the region of the BEC-BCS crossover for Cooper pairing in the p-wave channel. For elementary particle physics, the splitting of Fermi points may lead to neutrino oscillations, even if the total electromagnetic Chern-Simons-like term cancels out. 
  We show that for every positive curvature Kahler-Einstein manifold in dimension 2n there is a countably infinite class of associated Sasaki-Einstein manifolds X_{2n+3} in dimension 2n+3. When n=1 we recover a recently discovered family of supersymmetric AdS_5 x X_5 solutions of type IIB string theory, while when n=2 we obtain new supersymmetric AdS_4 x X_7 solutions of D=11 supergravity. Both are expected to provide new supergravity duals of superconformal field theories. 
  We introduce Weyl's scale invariance as an additional local symmetry in the standard model of electroweak interactions. An inevitable consequence is the introduction of general relativity coupled to scalar fields a la Dirac and an additional vector particle we call the Weylon. We show that once Weyl's scale invariance is broken, the phenomenon (a) generates Newton's gravitational constant G_N and (b) triggers spontaneous symmetry breaking in the normal manner resulting in masses for the conventional fermions and bosons. The scale at which Weyl's scale symmetry breaks is of order Planck mass. If right-handed neutrinos are also introduced, their absence at present energy scales is attributed to their mass which is tied to the scale where scale invariance breaks. 
  This article presents a precise description of the interplay between the symmetries of a quantum or classical theory with spacetime interpretation, and some of its physical properties relating to causality, horizons and positive energy. Our major result is that the existence of static metrics on spacetimes and that of positive energy representations of symmetry groups, are equivalent to the existence of particular Adjoint-invariant convex cones in the symmetry algebras. This can be used to study backgrounds of supergravity and string theories through their symmetry groups. Our formalism is based on Segal's approach to infinitesimal causal structures on manifolds. The Adjoint action in the symmetry group is shown to correspond to changes of inertial frames in the spacetime, whereas Adjoint-invariance encodes invariance under changes of observers. This allows us to give a group theoretical description of the horizon structure of spacetimes, and also to lift causal structures to the Hilbert spaces of quantum theories. Among other results, by setting up the Dirac procedure for the complexified universal algebra, we classify the physically inequivalent observables of quantum theories. We illustrate this by finding the different Hamiltonians for stationary observers in AdS_2. 
  We consider the Einstein-Yang-Mills Lagrangian in a (4+n)-dimensional space-time. Assuming the matter and metric fields to be independent of the n extra coordinates, a spherical symmetric Ansatz for the fields leads to a set of coupled ordinary differential equations. We find that for n > 1 only solutions with either one non-zero Higgs field or with all Higgs fields constant exist. We construct the analytic solutions which fulfill this conditions for arbitrary n, namely the Einstein-Maxwell-dilaton solutions. We also present generic solutions of the effective 4-dimensional Einstein-Yang-Mills-Higgs-dilaton model, which possesses n Higgs triplets coupled in a specific way to n independent dilaton fields. These solutions are the abelian Einstein-Maxwell- dilaton solutions and analytic non-abelian solutions, which have diverging Higgs fields. In addition, we construct numerically asymptotically flat and finite energy solutions for n=2. 
  The symmetry structure of non-abelian affine Toda model based on the coset $SL(3)/SL(2)\otimes U(1)$ is studied. It is shown that the model possess non-abelian Noether symmetry closing into a q-deformed $SL(2)\otimes U(1)$ algebra. Specific two vertex soliton solutions are constructed. 
  We consider N=1 superpotentials corresponding to gaugings of an underlying extended supergravity for a chiral multiplet in the SU(1,1)/U(1) manifold of curvature 2/3. We analyze the resulting D=4 scalar potentials, and show that they can describe different N=1 phases of higher-dimensional supergravities, with broken or unbroken supersymmetry, flat or curved backgrounds, sliding or stabilized radius. As an application, we discuss the D=4 effective theory of the detuned supersymmetric Randall-Sundrum model in two different approximation schemes. 
  We argue that a brane world with a warped, infinite extra dimension allows for the inflaton to decay into the bulk so that after inflation, the effective dark energy disappears from our brane. This is achieved by the redshifting of the decay products into infinity of the 5th dimension. As a consequence, all matter and CMB density perturbations could have their origin in the decay of a MSSM flat direction rather than the inflaton. We also discuss a string theoretical model where reheating after inflation may not affect the observable brane. 
  We construct matter-coupled N=2 supergravity in five dimensions, using the superconformal approach. For the matter sector we take an arbitrary number of vector-, tensor- and hyper-multiplets. By allowing off-diagonal vector-tensor couplings we find more general results than currently known in the literature. Our results provide the appropriate starting point for a systematic search for BPS solutions, and for applications of M-theory compactifications on Calabi-Yau manifolds with fluxes. 
  We start by pointing out that certain Riemann surfaces appear rather naturally in the context of wave equations in the black hole background. For a given black hole there are two closely related surfaces. One is the Riemann surface of complexified ``tortoise'' coordinate. The other Riemann surface appears when the radial wave equation is interpreted as the Fuchsian differential equation. We study these surfaces in detail for the BTZ and Schwarzschild black holes in four and higher dimensions. Topologically, in all cases both surfaces are a sphere with a set of marked points; for BTZ and 4D Schwarzschild black holes there is 3 marked points. In certain limits the surfaces can be characterized very explicitly. We then show how properties of the wave equation (quasi-normal modes) in such limits are encoded in the geometry of the corresponding surfaces. In particular, for the Schwarzschild black hole in the high damping limit we describe the Riemann surface in question and use this to derive the quasi-normal mode frequencies with the log(3) as the real part. We then argue that the surfaces one finds this way signal an appearance of an effective string. We propose that a description of this effective string propagating in the black hole background can be given in terms of the Liouville theory living on the corresponding Riemann surface. We give such a stringy description for the Schwarzschild black hole in the limit of high damping and show that the quasi-normal modes emerge naturally as the poles in 3-point correlation function in the effective conformal theory. 
  As an alternative to the usual Feynman graphs, tree amplitudes in Yang-Mills theory can be constructed from tree graphs in which the vertices are tree level MHV scattering amplitudes, continued off shell in a particular fashion. The formalism leads to new and relatively simple formulas for many amplitudes, and can be heuristically derived from twistor space. 
  We analyze the global theory of boundary conditions for a constrained quantum system with classical configuration space a compact Riemannian manifold $M$ with regular boundary $\Gamma=\partial M$. The space $\CM$ of self-adjoint extensions of the covariant Laplacian on $M$ is shown to have interesting geometrical and topological properties which are related to the different topological closures of $M$. In this sense, the change of topology of $M$ is connected with the non-trivial structure of $\CM$. The space $\CM$ itself can be identified with the unitary group $\CU(L^2(\Gamma,\C^N))$ of the Hilbert space of boundary data $L^2(\Gamma,\C^N)$. A particularly interesting family of boundary conditions, identified as the set of unitary operators which are singular under the Cayley transform, $\CC_-\cap \CC_+$ (the Cayley manifold), turns out to play a relevant role in topology change phenomena. The singularity of the Cayley transform implies that some energy levels, usually associated with edge states, acquire an infinity energy when by an adiabatic change the boundary condition reaches the Cayley submanifold $\CC_-$. In this sense topological transitions require an infinite amount of quantum energy to occur, although the description of the topological transition in the space $\CM$ is smooth. This fact has relevant implications in string theory for possible scenarios with joint descriptions of open and closed strings. In the particular case of elliptic self--adjoint boundary conditions, the space $\CC_-$ can be identified with a Lagrangian submanifold of the infinite dimensional Grassmannian. The corresponding Cayley manifold $\CC_-$ is dual of the Maslov class of $\CM$. 
  We derive conditions for the existence of four-dimensional \N=1 supersymmetric flux vacua of massive type IIA string theory with general supergravity fluxes turned on. For an SU(3) singlet Killing spinor, we show that such flux vacua exist only when the internal geometry is nearly-K\"ahler. The geometry is not warped, all the allowed fluxes are proportional to the mass parameter and the dilaton is fixed by a ratio of (quantized) fluxes. The four-dimensional cosmological constant, while negative, becomes small in the vacuum with the weak string coupling. 
  Exact, non-singular, time-dependent solutions of Maxwell-Einstein gravity with and without dilatons are constructed by double Wick rotating a variety of static, axisymmetric solutions. This procedure transforms arrays of charged or neutral black holes into s-brane (spacelike brane) solutions, i.e. extended, short-lived spacelike defects. Along the way, new static solutions corresponding to arrays of alternating-charge Reissner-Nordstrom black holes, as well as their dilatonic generalizations, are found. Their double Wick rotation yields s-brane solutions which are periodic in imaginary time and potential large-N duals for the creation/decay of unstable D-branes in string theory. 
  We consider twisted tachyons on C/Z_N orbifolds of bosonic closed string theory. It has been conjectured that these tachyonic instabilities correspond to decays of the orbifolds into flat space or into orbifolds with smaller deficit angles. We examine this conjecture using closed string field theory, with the string field truncated to low-level tachyons. We compute the tachyon potentials for C/Z_2 and C/Z_3 orbifolds and find critical points at depths that generate about 70% of the expected change in the deficit angle. We find that both twisted fields and untwisted modes localized near the apex of the cone acquire vacuum expectation values and contribute to the potential. 
  Entropy bounds render quantum corrections to the cosmological constant $